Estudio desde primeros principios de mecanismos de apantallado

Transcription

Universidad de Cantabria
Facultad de Ciencias
Dpto. Ciencias de la Tierra y
Física de la Materia Condensada
Estudio desde primeros principios de
mecanismos de apantallado del campo de
depolarización en condensadores nanométricos.
TESIS DOCTORAL
Pablo Aguado Puente
Dpto. de Ciencias de la Tierra y Fı́sica de la Materia Condensada
First-principles study
of screening mechanisms of the
depolarizing field in nanosized capacitors
Memoria presentada por
Pablo Aguado Puente
para optar al tı́tulo de
Doctor por la Universidad de Cantabria
Memoria dirigida por
Dr. Javier Junquera Quintana
Junio 2011
Avda. Los Castros, s/n.
39005 Santander
D. Javier Junquera Quintana, Profesor Titular del departamento de Ciencias de la Tierra y
Física de la Materia Condensada de la Universidad de Cantabria
INFORMA:
Que el trabajo presentado en esta memoria, titulado “Estudio desde primeros principios de
mecanismos de apantallado del campo de depolarización en condensadores
nanométricos.” ha sido realizado bajo su dirección por D. Pablo Aguado Puente en el
Departamento de Ciencias de la Tierra y Física de la Materia Condensada de la Universidad de
Cantabria y EMITE su conformidad para que dicha memoria sea presentada y tenga lugar,
posteriormente, su correspondiente defensa para optar al título de Doctor por la Universidad de
Cantabria.
En Santander, a 21 de junio de 2011
Fdo.: Javier Junquera Quintana
Mi agradecimiento
Quiero darle las gracias a Javier por lo mucho que me ha enseñado en estos años, de
ciencia y de lo que no es ciencia, por ponerme las cosas fáciles, por la confianza que
siempre ha tenido en mı́ y por lo contagioso de su entusiasmo por lo que hace. En
estos años he tenido la suerte de conocer y trabajar con mucha gente generosa y siempre
dispuesta a colaborar. A Max Stengel, a cuyo continuo bombardeo de ideas debo una
parte importante de esta tesis, más de tres años y veintisiete páginas de artı́culo después.
To Philippe Ghosez, Eric Bousquet, Pavlo Zubko and Pablo Garcı́a for the enriching
discussions and all the things I have learned from them. My sincere acknowledgement
to Patrycja Paruch for her hospitality during my stay in Geneva and for showing me
how a ferroelectric looks like in the real world. To the rest of the people at Geneva,
I really enjoyed the time I spent there. To Karin Rabe for her kindness and valuable
scientific advise during my stay at Rutgers. To Morrel Cohen for the, unfortunately few,
but insightful discussions we had during my visit.
A Fernando y los de altas presiones por acogerme como mascota. A Lucie que no
sabe jugar a los dardos. A los del despacho, a la Jefa, a Trueba que nunca se queja, a
Susana, a Rosa, a Marcos, a Diego, a Carlos, a Cristina y a Pincho. A Echeandı́a porque
si no se le echarı́a de menos.
A Elisa porque ella lo dejaba ası́, a la Matahierbas, a Kus y al Chopo. A Spirit que
está de fiesta. A Trufa y a Coco. A los blogs. A la noche de hoy porque ya casi es de
dı́a. A Calvin y a Hobbes. Al chocolate. A Alba. A mis padres por los legos y bizcochos
y por su cariño.
i
Este trabajo de investigación ha sido realizado gracias a una beca FPU del Ministerio de
Educación (Ref. AP2006-02958), que también ha cubierto las estancias en las Universidades
de Ginebra (Suiza) y Rutgers (EE.UU.), ası́ como gracias a la financiación del Ministerio de
Educación y Ciencia (Proyecto Ref. FIS2006-02261), del Ministerio de Ciencia e Innovación
(Proyecto Ref. FIS2009-12721-C04-02) y el Séptimo programa Marco de la Unión Europea
(Proyecto OxIDes: Oxides Interface Design). Los recursos computacionales han sido proporcionados por el grupo ATC de la Universidad de Cantabria y la Red Española de Supercomputación.
Glossary
AFD
B1-WC
BZ
CBM
CNL
D
DFT
DOS
E
Ed
EC
ECNL
EF
EV
ε
ε0
ε∞
FE
φn , φp
φp
GGA
LDA
LDOS
λeff
M
MIGS
Nx
P
PDOS
VBM
Z∗
Antiferrodistortive (mode)
B1-Wu-Cohen approximation
Brillouin zone
Conduction band minimum
Charge neutrality level
Electric displacement field
Density functional theory
Density of states
Electric field
Depolarizing field
Energy of the bottom of the conduction band
Charge neutrality level
Fermi energy
Energy of the top of the valence band
Relative permittivity
Vacuum permittivity
Electronic permittivity
Strain
Ferroelectric
Schottky barrier for electrons
Schottky barrier for holes
Generalized gradient approximation
Local density approximation
Local density of states
Effective screening length
Metal
Metal-induced gap states
Domain structure periodicity
Polarization
Projected density of states
Valence band maximum
Born effective charge
iii
Contents
Glossary
iii
Introduction
1
1 Ferroelectric thin films
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . .
1.2 Ferroelectricity in bulk . . . . . . . . . . . . . . . . .
1.2.1 Soft modes and double well energy . . . . . .
1.2.2 Anomalous dynamical charges . . . . . . . . .
1.2.3 Origin of the ferroelectricity . . . . . . . . . .
1.2.4 Non-polar instabilities . . . . . . . . . . . . .
1.3 Ferroelectric thin films . . . . . . . . . . . . . . . . .
1.3.1 Mechanical boundary condition . . . . . . . .
1.3.2 Electrical boundary condition . . . . . . . . .
1.4 Convergence of experiments and theoretical methods
1.5 References . . . . . . . . . . . . . . . . . . . . . . . .
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2 Methodology
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . .
2.2 Overview of approximations . . . . . . . . . . . . .
2.3 Born-Oppenheimer approximation . . . . . . . . .
2.4 Density functional theory . . . . . . . . . . . . . .
2.4.1 Exchange and correlation functional . . . .
2.5 Pseudopotentials . . . . . . . . . . . . . . . . . . .
2.6 Periodic boundary conditions . . . . . . . . . . . .
2.7 Brillouin zone sampling and electronic temperature
2.8 Basis sets . . . . . . . . . . . . . . . . . . . . . . .
2.8.1 Plane waves . . . . . . . . . . . . . . . . . .
2.8.2 Atomic orbitals . . . . . . . . . . . . . . . .
2.8.3 Atomic spheres . . . . . . . . . . . . . . . .
2.9 References . . . . . . . . . . . . . . . . . . . . . . .
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vi
Contents
3 Band alignment issues in the ab initio simulation of ferroelectric
pacitors
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 General theory of the band offset . . . . . . . . . . . . . . . . . . . . .
3.2.1 Schottky barriers at metal/insulator interfaces . . . . . . . . .
3.2.2 Theory of Schottky barriers in ferroelectric capacitors . . . . .
3.2.3 Ferroelectric capacitors in a pathological regime . . . . . . . .
3.2.4 Implications for the analysis of the ab-initio results . . . . . . .
3.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Schottky barriers from ab initio simulations . . . . . . . . . . .
3.3.2 Electrical analysis of the charge spill-out . . . . . . . . . . . . .
3.3.3 Constrained-D calculations . . . . . . . . . . . . . . . . . . . .
3.3.4 Computational parameters . . . . . . . . . . . . . . . . . . . .
3.4 Results: Non polar capacitors . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 Non-pathological cases . . . . . . . . . . . . . . . . . . . . . . .
3.4.2 Pathological cases . . . . . . . . . . . . . . . . . . . . . . . . .
3.5 Results: Polar capacitors . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.1 Short-circuit calculations . . . . . . . . . . . . . . . . . . . . .
3.5.2 Open-circuit calculations . . . . . . . . . . . . . . . . . . . . .
3.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6.1 Structural properties of the film . . . . . . . . . . . . . . . . .
3.6.2 Stability of the ferroelectric state . . . . . . . . . . . . . . . . .
3.6.3 Transport properties in the tunneling regime . . . . . . . . . .
3.6.4 Interface magnetoelectric effects . . . . . . . . . . . . . . . . .
3.6.5 Schottky barriers . . . . . . . . . . . . . . . . . . . . . . . . . .
3.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 Metal-induced gap states in ferroelectric capacitors
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Metal-induced gap states and complex band structure .
4.2.1 Complex band structure: a simple example . . .
4.2.2 Connection with Schottky barriers . . . . . . . .
4.3 Computational details . . . . . . . . . . . . . . . . . . .
4.3.1 Compatibility tests . . . . . . . . . . . . . . . . .
4.4 Complex band structure of bulk PbTiO3 . . . . . . . . .
4.5 MIGS in ab initio simulations of ferroelectric capacitors
4.6 Discussion and perspectives . . . . . . . . . . . . . . . .
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5 Ferromagnetic-like closure domains in ferroelectric capacitors
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 System and computational details . . . . . . . . . . . . . . . . .
5.3 Structure of polarization domains in ferroelectric thin films . . .
5.4 Role of the electrodes on the formation of polarization domains .
5.5 Screening of the depolarizing field . . . . . . . . . . . . . . . . . .
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vii
5.6
5.7
Theoretical prediction and experimental observation of closure domains
in ferroelectric thin films . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
6 PbTiO3 /SrTiO3 superlattices
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Structure and computational details . . . . . . . . . . . . . . . . . . . .
6.3 Mixed ferroelectric-antiferrodistortive-strain coupling in the monodomain
configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3.1 Periodicity dependence of FE-AFD coupling . . . . . . . . . . . .
6.3.2 Emergence of an r-phase in PbTiO3 /SrTiO3 superlattices . . . .
6.3.3 Covalent model for the polarization-octahedra rotation coupling
6.3.4 Piezoelectric response of the system . . . . . . . . . . . . . . . .
6.4 Polydomain structures . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Conclusions
A Ocupation function and energy smearing of
A.1 Convolutions . . . . . . . . . . . . . . . . .
A.2 Local density of states . . . . . . . . . . . .
A.3 Gaussian vs. Fermi-Dirac smearing . . . . .
A.4 On the optimal choice of g . . . . . . . . . .
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B Local polarization via Born effective charges
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C Complex band structure within the nearly-free electron model
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D Resumen
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E Conclusiones
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Bibliography
211
Introduction
The research of transition metal oxides is today in a momentous stage. The situation is
indeed so exciting that it has been compared to that of semiconductor physics sixty years
ago [1]. That is quite a serious comparison, since our lives today are overwhelmingly
dependent on devices developed from basic material science research taking the first
steps at that period.
This comparison however is not gratuitous. The last decades of research on transition metal oxides have been tremendously exciting, with the discovery of an enormous
number of different functionalities in these materials, from superconductivity to colossal
magnetoresistance, just to cite a couple of examples.
One particular family of transition metal oxides have attracted a lot of attention
in the last years. This family of materials share the relatively simple perovskite parent
crystal structure. The simplicity of the crystal structure, with only five atoms in the
unit cell in the high-symmetry cubic reference structure, hides an enormous amount of
subtle physics. These materials, all sharing a very similar atomic structure, display an
extremely wide range of properties: superconductivity, ferromagnetism, colossal magnetoresistance, multiferroism, non-linear optics ...
More interestingly, the wide range of properties arising from what can be considered
relative materials suggest that the emergence of one particular property must be the consequence of delicate a balance between multiple interactions that are probably common
to many of the members of this family of materials. Indeed, this diverse phase diagram
emerges from the close competition of different interactions that takes place in these
materials. While in other kinds of materials such as semiconductors or metals, one of
the different interactions involved – Coulomb repulsions, strain, exchange, etc – clearly
dominates over the other and determines the macroscopic properties of the system, in
transition metal oxides this effects are very often of the same magnitude. As a result
these materials usually display multiple competing phases and a strong susceptibility to
external perturbations [2, 3]. This makes these materials ideal candidates for the design
of artificial devices with tailored functionalities.
One of the properties displayed by some perovskite oxides is ferroelectricity. A ferroelectric material is an insulator exhibiting at least two different states of nonzero polarization, being possible to switch between them by applying an external electric field
[4]. The term “ferroelectric” was coined due to the analogy of these materials with the
ferromagnets, as they both exhibit a hysteresis loop when polarization (magnetization
1
2
Introduction
in the case of ferromagnet) is measured as a function of the applied electric (magnetic)
field.
Ferroelectrics are materials with a great applied interest [5, 6]. The property of ferroelectricity itself, i.e. the capacity of switching between two or more polarization states
with the application of an external field, can be exploited for example for the fabrication
of memory devices, where each state of the polarization can be assigned to the values
0 and 1 of a bit of information. This is the basic working principle of ferroelectric random access memories (FeRAM). Besides, ferroelectricity is usually associated with other
properties of great interest. All ferroelectric materials, for instance, are also piezoelectric
(a deformation can be induced by the application of an electric field and vice versa) and
pyroelectric (temperature changes of the sample modifies the polarization), properties
that are used for the fabrication of transducers, actuators or infrared detectors. One of
the most successful examples is the family of PbZr1−x Tix O3 ceramics which today are
present in a great variety of devices, from ultrasound imaging equipments, to fuel injectors in automobile engines or atomic-force microscopes. Ferroelectric materials posses
a very large dielectric constant which is the reason for their use in the fabrication of
dynamic random access memory (DRAM) capacitors.
The steady miniaturization of electronic devices imposed by electronic industry and
impulsed by the need of faster, but at the same time smaller and more energy-efficient
electronic devices, have motivated the study of properties of ferroelectric materials at
the nanoscale. It is well known that bulk properties of ferroelectric materials and most
perovskite oxides are strongly affected by boundary conditions, which became specially
relevant as devices became smaller. Ferroelectricity, for instance, was suggested to have
a critical size of about 10 nm, below which modifications of the balance between driving interactions together with the electrostatic coupling with depolarizing fields would
provoke the loss of an spontaneous polarization. Nevertheless as the synthesis and experimental characterization of ultra-thin film developed, thinner and thinner ferroelectric
films were observed to preserve a remnant polarization, bringing the critical thickness
down to just a few monolayers.
On the other hand, the improvement of growth techniques allowed to take advantage
of the subtle balance of instabilities and the strong sensitivity of these materials to the
boundary conditions to tune the properties of perovskite oxide thin films. The seek of
a route to design and realize artificial materials with tailored functionalities brought a
great excitement to the field and boosted the research activity on these systems. However
the close competition between different interactions and phases makes very difficult if
not impossible to predict the properties of the artificial structure in terms of simple rules
and given the known properties of the bulk constituents.
This is one of the reasons for the fundamental role played by first-principles simulations in the outstanding improvements of the field during the last years [7]. The rapid
evolution in the atomistic modeling of materials, driven both by the fast and steady
increase of the computational power (hardware), and by important progresses in the
development of more efficient algorithms (software) makes possible to describe very accurately the properties of materials using methods directly based on the fundamental
3
laws of quantum mechanics and electrostatics. Even if the study of complex systems
requires some practical approximations, these methods are free of empirically adjustable
parameters. For this reason, they are referred to as “first-principles” or “ab-initio”
techniques.
The current situation of the field is particularly exciting. On the one hand, recent breakthroughs on materials synthesis and characterization techniques have allowed
the growth of ferroelectric thin films with a control at the atomic scale and the local
measurement of the ferroelectric properties [8]. On the other, the steady increase in computational power and improvements in the efficiency of the algorithms permit accurate
first-principles study of larger and more complex systems, overlapping in size with those
grown epitaxially. This allows a continuous feedback between experiments and theoretical models. This combined effort has led, in the last couple of decades, to very significant
advances in the microscopic understanding of the properties of perovskite ferroelectrics
and related compounds. Nevertheless every step forward gives rise to the discovery of
new phenomena (an excellent example is the discovery of conducting interfaces at superlattices of LaAlO3 and SrTiO3 , two insulating materials [9]; or the conductivity of
domain walls in BiFeO3 [10]), generating new opportunities for practical exploitation of
functionalities and, more importantly, rising new questions and motivating the further
investigation of these materials.
One problem under active investigation due to its wide implications both in the basic
physical properties exhibited by ferroelectric thin films and in the potential applications,
is the understanding of the screening mechanisms in such systems. The termination of
the ferroelectric polarization at the surface of a film or its discontinuity across an interface with an electrode or a different insulating material generates a polarization charge
which gives rise to a depolarizing field tending to suppress the polarization. Multiple
mechanisms can take place in order to compensate the polarization charges: accumulation of screening charge at the electrodes, ionic adsorbates at a free surface or the
breaking up of the system into polarization domains. In this thesis we have performed
a first-principles study of some of these mechanisms. We have paid special attention to
the methodological issues associated to the study of screening properties in ferroelectrics
which can be important for the first-principles research of other interfacial properties.
Then we focus on two particularly relevant systems: (i) ferroelectric/metal interfaces,
ubiquitous in ferroelectric capacitors and (ii) ferroelectric/incipient ferroelectric interfaces, such as PbTiO3 /SrTiO3 superlattices, a system which is attracting a lot of interest
due to the appearance of an improper ferroelectricity in the ultrathin limit. In the first
case we study charge rearrangements at metal/ferroelectric junctions associated with
the formation of gap evanescent states, and the formation and properties of polarization
domains in these kind of devices. In the case of PbTiO3 /SrTiO3 superlattices, the discovery of an interface-intrinsic coupling of ferroelectricity with non polar instabilities,
absent in the parent bulk materials, have attracted a lot of attention in the last years.
We explore the phase diagram with strain, its effect on the coupling between instabilities
and the properties of polydomain phases.
This manuscript is organized as follows. In Chapter 1 we introduce the general
4
Introduction
properties of bulk ABO3 ferroelectric materials. We discuss the different instabilities
present in these compounds, their competition and the connection with the arising of
ferroelectricity. We then consider how size effects and boundary conditions affect these
properties, and how this can be used to our advantage allowing to fine tune perovskite
oxides functionalities. Special attention is devoted in this Chapter to discuss the electrostatic boundary conditions and the different screening mechanisms that can take part
in a ferroelectric thin film, which is the subject of this thesis work. In Chapter 2 we
will describe the basic theoretical details of first-principles methods used to carry out
the research reported in this memory. Some issues for the simulation of heterostructures
associated to the theoretical approach are investigated in Chapter 3. Here we provide a
clear procedure to detect spurious results consequence of the misuse of the most extended
theoretical methodology, namely the density functional theory (DFT), and suggest paths
to avoid them. Results of this Chapter are more than just methodological, since some
screening mechanisms detected in pathological heterostructures might actually be relevant for real interfaces. Chapter 4 is devoted to the study of evanescent states in the gap
of a ferroelectric in a metal/ferroelectric junction, the so called metal-induced gap states
(MIGS). This states play a fundamental role in tunneling phenomena and Schottky barrier formation. In this Chapter we discuss to what extent the characteristics of these
states can be predicted from bulk properties of the ferroelectric, and which are related to
intrinsic interface effects. In Chapter 5 we discuss the formation of polarization domains
in ferroelectric capacitors. For the first time, closure domains are predicted from first
principles to form in ferroelectric thin films. Despite being considered unlikely to occur
in ferroelectric thin films due to the large coupling between polarization and strain, this
structure is found to be surprisingly general and to provide an extremely good screening. In Chapter 6 we study the effect of strain on the coupling between polarization
and oxigen octahedra rotation in PbTiO3 /SrTiO3 superlattices. The strong coupling
of these two instabilities in this system is explained in terms of a covalent model. In
view of results of Chapter 5, we also consider the formation of polarization domains in
the superlattices. Structures similar to those observed in the ferroelectric capacitors are
found, forming vortex-like dipole arrangements at the domain walls. Finally the results
of this work are summarized in the Conclusions.
Chapter 1
Ferroelectric thin films
1.1
Introduction
As it was mentioned before, to be a ferroelectric, a material must satisfy two conditions
(i) it must exhibit at least two states of different finite polarization, and (ii) it must be
possible to switch from one of these states to the other by applying an external electric
field. This condition of switching shows up experimentally as a hysteresis loop when the
polarization of the materials is measured as a function of the applied electric field. In
the ideal hysteresis loop depicted schematically in Fig. 1.1 the two opposite values of
the polarization at zero field correspond to the spontaneous polarization of the material
and the coercive field is the threshold electric field to switch the polarization. In a real
crystal, polarization domains and depolarizing field can make the average polarization of
the sample at zero field (remnant polarization) to be smaller than the ideal spontaneous
polarization Also in analogy with ferromagnetic materials, all ferroelectrics undergo a
phase transition to a paraelectric phase above the Curie temperature TC .
Ferroelectricity was first discovery by J. Valasek in 1920 in Rochelle salt [11] For almost three decades after these pioneering works, all known ferroelectrics were hydrogenbonded materials, and for some time this was thought to be a requirement for a given
material to exhibit this property. The discovery in 1949 of ferroelectricity in BaTiO3 , a
much more robust material, with a much simpler crystal structure, stimulated the use of
ferroelectric compounds for practical applications. This, in turn encouraged theoretical
research of these materials beyond simple scientific curiosity and, thanks in part to the
simplicity of the crystal structure of BaTiO3 , the understanding of the physics behind
the ferroelectricity evolved rapidly.
BaTiO3 became then the prototype of a whole family of materials, the perovskite
oxides, which still today are providing huge amounts of new physics thanks to the tunability of their numerous properties. This family of materials have in common a crystal
structure derived from the ideal cubic perovskite, with a formula unit usually denoted
as ABO3 , where A is a mono-, di- or tri-valent cation and and B is a penta-, tetra- or
tri-valent cation respectively. In the centrosymmetric cubic structure these all materials
derive from, A atoms are located at the corners of the unit cell and oxygen atoms are at
5
6
Chapter 1. Ferroelectric thin films
P
+P0
Ec
E
-P0
Figure 1.1: Idealized hysteresis loop that would correspond, for instance to the switching
process of a single unit cells of a ferroelectric material. The values of the spontaneous
polarization P0 and the coercive fields Ec are shown.
the center of the cube faces, forming an octahedra at which center is the B atom. This
structure corresponds for instance to the high temperature, cubic, non polar phase of
BaTiO3 , which unit cell is represented in Fig. 1.2(a). In ferroelectric materials, below
the Curie temperature, this non polar ideal cubic structure is unstable and present distortions compatible with an spontaneous polarization. In the most simple cases – like
for instance in the temperature-driven cubic-to-tetragonal phase transition in BaTiO3 at
130 ◦ C – this distortion consist of an opposing shift of B cation and the oxygen octahedra, as in Fig. 1.2(b). The atomic displacements are accompanied by a shape distortion
of the unit cell caused by the symmetry reduction.
1.2
Ferroelectricity in bulk
Most perovskite oxides exhibit different instabilities that distort the crystal structure
from the ideal cubic one. The stability of the high symmetry structure is very often
discussed in terms of a steric model, in which the size of A and B cations determine the
possible symmetry-lowering distortions that might emerge. The tendency of ABO3 to
undergo a given phase transition to a state with lower symmetry is usually quantified
through the Goldschmidt tolerance factor
RA + RO
τ=√
,
2RB + RO
(1.1)
which is equal to 1 when the atomic radii of A (RA ) and B (RB ) cations, and oxygens
(RO ) are such that the latter just touch the cations. For τ > 1, the B cation has free space
around and tend to move off-center, generating dipoles that might align cooperatively
between neighboring unit cells and giving rise to ferroelectric order. If on the contrary
7
1.2. Ferroelectricity in bulk
(a)
(b)
Figure 1.2: Unit cell of the prototypical ferroelectric BaTiO3 in the (a) non-polar cubic
configuration and (b) the tetragonal ferroelectric phase. The polar mode in the ferroelectric phase is associated in most materials with an elongation of the unit cell in the
polarization direction, as sketched in (b).
τ < 1 the larger B cation “pushes” the oxygens out of the center of the faces favoring
rotations of the oxygen octahedra.
This model provides a simplistic yet intuitive tool to understand the relative stability
of some common distortions in ABO3 compounds, but we need a more detailed analysis
of the underlying physical effects to be able to understand some subtle phenomena that
take place at the interfaces of these materials, which is the final goal of this work.
1.2.1
Soft modes and double well energy
These instabilities are associated with the existence of “soft modes” in the phonon band
structure of the ideal cubic crystal [12]. A vibrational normal mode is said to soften
when the condensation of the atomic displacements associated to its eigenmode leads to
a decrease in energy of the system. There is not, in this case, a restoring force bringing
the atoms back to the – unstable – reference equilibrium position, but a driving force that
distort the system into a lower symmetry, stable configuration, with an energy below
that of the centrosymmetric one. The curvature of the energy surface with respect to
the amplitude of the soft mode, or conversely the associated force constant within the
harmonic approximation, is negative and its frequency, ω, imaginary.
In some cases these instabilities lead to a polarization of the system. In the particular
case of BaTiO3 , it exhibits a mode at Γ which eigenmode, depicted schematically in
Fig. 1.2(b), involves a displacement of the Ti atoms in opposition to the oxygen cage.
The opposite shift of positive and negative charges gives rise to the polarization of
the system. The value of the imaginary frequency ω associated with the soft mode
measures to some extent the strength of the instability in the harmonic approximation.
However, for the distortion from the cubic reference structure to lead to a different stable
structure we need to consider also the effects of anharmonicity. In many of the most
8
Figure 1.3: Double well potential energy curve with respect to the normalized amplitude
of the distortion ξ, for bulk tetragonal BaTiO3 . Bottom row show schematically the
structure at the energy minima (left and right for a downward and an upward pointing
polarization respectively) and at the reference unstable configuration (central).
common perovskite ferroelectrics the final polar structure can be actually obtained from
the eigendisplacements of one single mode (typically a polar mode at Γ). Assuming a
relatively smooth variation of the energy with respect to the distortion, the energy of
the system can be expressed as a Taylor expansion in the amplitude of the distortion
with respect to the cubic reference structure. If ξ is the amplitude of the distortion,
proportional to a given eigenmode, the potential energy of the system can be expanded
as
1
1
1
U(ξ) = α2 ξ 2 + α4 ξ 4 + α6 ξ 6 ,
2
4
6
(1.2)
where we have truncated to sixth order. Only even terms are allowed in the expansion
as a consequence of the cubic symmetry of the reference structure. The quadratic term
is the harmonic contribution and is related to the curvature of the energy at ξ = 0. It
is proportional to the square of the imaginary frequency associated to the soft mode,
ω 2 , so it must be negative for the polar distortion in a ferroelectric. Nevertheless, this
term alone would lead to infinite displacements of the atoms from the high symmetry
positions. Higher order terms are positive and take into account anharmonic effects that
are responsible of the stabilization of the distortion.
If the potential energy is plotted as a function of the amplitude of the soft mode
9
Table 1.1: Born effective charges of some ABO3 perovskites. O1 denotes the apical
oxygen and O2 the equatorial one (see Fig. 1.2). All charges have been calculated with
the Siesta code.
BaTiO3
PbTiO3
SrTiO3
A
2.640
3.870
2.527
B
7.370
7.030
7.555
O1
-5.700
-5.760
-5.951
O2
-2.155
-2.570
-2.066
one obtains the double well potential energy, characteristic of ferroelectrics and shown
in Fig. 1.3. The amplitude of this distortion can be associated with the polarization of
the system and thus, the two energy minima of the potential energy curve correspond
to the two equivalent states with P equal to the spontaneous polarization
1.2.2
Anomalous dynamical charges
The contribution of a lattice distortion to the polarization of the material can be quantified introducing the concept of Born effective charges [13, 14]. For each atom i in the
unit cell, the Born effective charge tensor is defined as the change to linear order in the
polarization along direction α induced by a small displacement of the atomic sublattice
along direction β,
∂Pα ∗
Zi,αβ = Ω
,
(1.3)
∂xi,β E=0
where Ω is the volume of the unit cell, and the polarization is calculated at zero electric
field. Due to the cubic symmetry of the reference structure in perovskites, the tensor
becomes diagonal. For A and B atoms, the tensor only displays one independent entry.
Then the tensor can be considered as an scalar and a single Born effective charge is
associated to A and B atoms. On the contrary, in tetragonal ferroelectrics the polarization direction differentiates oxygen atoms into apical (usually denoted O1, at the unit
cell face perpendicular to the polarization direction) and equatorial (O2, at the faces
parallel to polarization direction, see Fig. 1.2). Consequently two different Born charges
are associated to oxygen atoms depending on the direction of the polarization (see Table
1.1). The Born effective charges are defined for a change induced in the polarization at
zero electric field. Similarly effective charges for zero electric displacement field can be
defined, which are called Callen charges.
In Table 1.1 we report the Born effective charges of some common perovskite oxides
that are relevant to this work. All these materials are characterized by effective charges
significantly larger than the nominal ones (+2 for the A atoms, +4 for B atoms and -2
for oxygens for these three compounds). The anomalously large effective charges reflect
the fact that the electronic cloud of a given ion does not rigidly follow the core as it
displaces out of the reference configuration, but the displacement triggers a polarization
10
Partial
O 2p - B d
hybridization
Harrison's model
Anomalously large Z*
associated to
macroscopic electronic currents
For the specific displacement patter
associated with the ferroelectric mode
Giant destabilizing
dipole-dipole interaction
Cochran's model
Ferroelectric instability
Figure 1.4: Flowchart summarizing the origin of ferroelectricity in ABO3 compounds in
connection with the hybridization between oxygen 2p and B cation 3d orbitals. (Adapted
from Ref. [13])
of the atomic orbitals and a transfer of charge through the cation-oxygen bond that
contributes to the polarization of the material. A large anisotropy of these charges,
with the effective charge of apical oxygens (oxygens in the faces perpendicular to the
displacement direction) also reflects the strong hybridization that takes place along the
B-O chains, in contrast with the much weaker interaction with the equatorial oxygens.
1.2.3
Origin of the ferroelectricity
Following the classical interpretation of Cochran [12], the stabilization of a polar distortion, and consequently the origin of ferroelectricity, relies on the competition between
long range dipole-dipole interaction, which favor the development of a polarization, and
short range forces that tend to destabilize polar configurations. In this context long
range interactions contribution is enhanced by large values of the Born effective charges
that give rise to giant dipolar interactions and contributes to the softening of the phonon
with a negative contribution to ω 2 , destabilizing the reference cubic structure. The polar
distortion becomes then an instability of the reference cubic structure.
In the case of BaTiO3 and SrTiO3 , Born charges of A atoms and equatorial oxygens
are relatively close to the nominal ones and main contribution to polarization comes
from the atomic chains formed by Ti atoms and apical O. This reflects the fact that
11
AFD mode at R:
AFD mode at M:
Figure 1.5: Schematic view of AFD modes at M [(a) side and (b) top view] and R
[(c) side and (d) top view], consisting respectively in in-phase and out-of-phase oxygen
octahedra rotations. Using Glazer notation these two modes can be denoted by a0 a0 c+
and a0 a0 c− respectively.
polar distortions in this compounds are driven by the hybridization of the empty Ti 3d
orbitals with the occupied O 2p orbitals. Meanwhile A atoms remain essentially inert
during the condensation of the polar instability. In other cases, like PbTiO3 , covalency
of Pb-O bonds play a significant role in the ferroelectric instability, and there is a sizable
contribution to the spontaneous polarization of the material from the opposite shift of
Pb and O apical atoms in the same plane, which reflects in the large Born effective
charges of these atoms.
1.2.4
Non-polar instabilities
Many of these materials also exhibit non polar, Brillouin zone boundary antiferrodistortive (AFD) instabilities, consisting in rotations of the oxygen octahedra. These instabilities also lower the symmetry of the system from the cubic structure, but involving
only rotations of the negative charges around the inversion center they do not give rise to
any polarization. Interestingly the energy balance between long and short range forces
is, in this case, the opposite than for the polar modes, with short range repulsion favoring
12
rotations. As a consequence, even though both ferroelectric and AFD instabilities might
be present in the phonon dispersion curves of the cubic structure, these two instabilities
usually compete and the ground state of these perovskites typically display either one
or the other, but rarely both type of distortions. SrTiO3 for instance displays both
ferroelectric and AFD modes with imaginary frequencies in the cubic phase. Freezing
any of these instabilities individually leads to a decrease in the energy of the crystal.
However the coupling term between the polar and AFD modes is positive and both
modes compete to suppress each other [15]. As a result, and despite the large effective
charges of this material that softens the polar mode at Γ, the ground state of SrTiO3
only displays AFD distortions. Nevertheless, the presence of a polar soft mode shows
up in this material as a great polarizability and dielectric constant.
Oxygen octahedra rotation patterns can be described in a compact way using Glazer
notation [16]. According to this notation three indexes are used referring to the rotation
along directions parallel to the three unit cell vectors respectively (the first one describes
the rotations around the [100] direction and so on). Each index is in turn composed
by a symbol describing if the rotations of successive octahedra along a given direction
(denoted by a, b or c) are in-phase (+), out of phase (−) or no rotation takes place
(0). For example, ground state of bulk SrTiO3 can be expressed within this notation as
a0 a0 c− , meaning that oxygen octahedra rotate out-of-phase around the [001] axis, see
Fig. 1.5.
1.3
Ferroelectric thin films
As discussed above, ferroelectricity in most common ABO3 perovskites is linked to spontaneous atomic off-center displacements, resulting from a delicate balance between longrange dipole-dipole Coulomb interaction and short-range covalent repulsions. In ultrathin films and nanostructures, both interactions are modified with respect to the bulk.
Size effects on short range interactions are conceptually simpler to understand. They
are modified by the presence of surfaces and interfaces which alter the chemical environment. They are also affected by changes in the unit cell size and shape induced
by pressure and homogeneous or inhomogeneous strains determined by the mechanical
boundary conditions.
Effects on the dipole-dipole interactions are more subtle. To understand some of the
properties of ferroelectrics and particularly the size effect of this materials it is important
to note that the driving force for ferroelectricity, namely the dipole-dipole interaction,
posses an intrinsic collective character. While short-range forces act essentially on each
individual atom repelling any polar distortion, long range dipole-dipole destabilizing interaction buildup from the alignment of localized dipoles within a correlation volume
[17]. Furthermore, dipole-dipole forces are extremely anisotropic. They favor the parallel alignment of dipoles along the polarization axis, forming chains of dipoles aligned
longitudinally. But at the same time, interactions within the plane perpendicular to
the polarization of the dipoles favor anti-parallel arrangement. Depending on the geometric arrangement of the dipoles in the lattice the overall balance will favor either the
1.3. Ferroelectric thin films
13
parallel or antiparallel alignment of dipole chains, to form respectively a ferroelectric
or antiferroelectric material. This anisotropic character of the dipole-dipole interaction
implies an anisotropic correlation volume, meaning that the energy penalty due to the
loss of dipole-dipole interactions is considerably higher in a perpendicularly polarized
thin-film geometry than in other geometries. This, with the development of depolarizing fields as a result of polarization discontinuities (discussed in Sec. 1.3.2), suggest a
finite-size effect on the ferroelectric properties of ferroelectric thin films. The occurrence
of a size effect would have important implications for applications, since it would limit
the minimum useful thickness of these materials. In fact, until the late 1990’s, it was
widely accepted that ferroelectricity in perovkite oxides would disappear below a critical
thickness of about 10 nm as a consequence of the lost of long range interactions, and
the arising of depolarizing electrostatic fields. This assumption proved to be erroneous
and revealed that the delicate balance between all these variables and the boundary
conditions hampers the prediction of the behavior of finite-sized ferroelectric samples.
The complex coupling between instabilities and the different boundary conditions
can actually be taken to our advantage. A deep understanding on how boundary conditions affect ferroelectricity and other properties of perovskite oxides in finite samples,
together with the atomic control achieved in the growth of this materials, allow to build
artificial systems in which we can play with the boundary conditions to engineer new
functionalities. However the fact that many of the most spectacular examples of new
functionalities in artificial systems have been found unexpectedly is a good prove that
there is still a lot of research to be done in this direction. This was for instance the
case of the appearance of metallic [9] (or even superconducting [18]) interfaces at the
boundary between two band insulators LaAlO3 and SrTiO3 , or the discovery of improper
ferroelectricity in PbTiO3 /SrTiO3 superlattices driven by a coupling between polar and
non-polar instabilities not present in the parent materials [19]. The pertinent question
is then, what new properties will we be able to engineer once a better understanding of
the physical processes involved at oxide interfaces is achieved?
1.3.1
Mechanical boundary condition
The coupling between different instabilities – polar and non polar – with strain is well
known to be specially strong in ferroelectric perovskite oxides, and can have a substantial
impact on the structure, transition temperatures, dielectric and piezoelectric responses.
This opened the door to the possibility of tunning functionalities in these compounds
by playing with the strain, what led to the coinage of the term “strain-engineering”.
In ferroelectric thin films, homogeneous strain can be achieved by means of the
epitaxial growth of the film on a substrate with a different lattice parameter. Thanks to
the availability of multiple perovskite substrates with a wide variety of in-plane lattice
parameters, we can now fine tune the ferroelectric and related properties in thin films by
using the homogeneous strain almost as a continuous knob. One key factor to achieve the
exceptionally wide range of stains that can be obtained with today’s growth techniques
is the fact that both the ferroelectric and the substrate share a very similar crystal
structure. This facilitates the coherency of in-plane crystal structure at the interface.
14
Assuming perfect coherency, the strain is defined as a function of the bulk lattice
a −a0
parameters of the film material, a0 , and of the substrate, ak , as = ka0 . The clamping
between the film and the substrate onto which it is deposited can be maintained only
in ultrathin films, where the elastic energy stored in the overlayer is still relatively
small. For thicker films a progressive relaxation and lost of coherency with the substrate
will occur via formation of misfit dislocations, which generally cause a degradation in
film quality. Note that, in general, the strain state of the film will also depend on the
differences in the thermal evolution of the lattice parameters of the substrate and the film
material. The different thermal expansion coefficients of substrate and films material
also opens a door for the design of pyroelectric devices with enhanced performance.
In order to understand the effect of the “polarization-strain” coupling, let us generalize the expansion of the potential energy of Eq. (1.2) in terms of additional strain
ij (where i and j are cartesian directions) degrees of freedom. In the paradigmatic example of a tetragonal ferroelectric film (e.g. PbTiO3 or BaTiO3 at room temperature)
epitaxially grown on a (001) cubic substrate (like SrTiO3 ) we have mixed strain/stress
boundary conditions: on the one hand the in-plane strains xx = yy are fixed by the
lattice mismatch between the ferroelectric and the substrate, while xy = 0. On the
other hand, the out-of-plane strain zz and the shear strains xz and yz are free to relax (condition of zero stress: σzz = σxz = σyz = 0). Assuming for simplicity only an
homogeneous polarization along z-direction, vanishing shear strains, and restricting the
expansion to leading orders in ξ and , the free energy functional to be minimized now
reads [20, 21]
U(ξ, ) =
1
1
1
α2 ξz2 + α4 ξz4 + α6 ξz6
2
4
6
1
1
+ C11 (22xx + 2zz ) + C12 (22xx + 4xx zz )
2
2
+2g0 xx ξz2 + (g0 + g1 )zz ξz2 .
(1.4)
The terms in the first line correspond to the double-well energy of Eq. (1.2). The terms
in the second line are the elastic energy while the terms in the third line arise from the
coupling between ionic and strain degrees of freedom. They correspond to the so-called
“polarization-strain coupling” and are at the origin of the piezoelectric response. It is
clear from Eq. (1.4) that the polarization-strain coupling terms are responsible for a
renormalization of the quadratic part of U that now takes the form
1
α2 + 2g0 xx + (g0 + g1 ) zz ξz2 .
2
(1.5)
Depending on the value of the parameters g0 and g1 , and of xx and zz (deduced from the
relation ∂U/∂zz = 0 which follows from the boundary condition σzz = 0), we see that,
playing properly with the epitaxial strain conditions, it is possible to make the coefficient
α2 more negative (i.e. induce a ferroelectric material to be even more ferroelectric, or
15
even induce a non-ferroelectric material to become ferroelectric [22, 23]), or to make α2
positive, thus suppressing the ferroelectric character of the film.
The first milestone theoretical work on the influence of the strain in ferroelectric
polarization is due to Pertsev et al. [20], who identified the right “mixed” mechanical
boundary conditions of the problem (fixed in-plane strains, and vanishing out-of-plane
stresses), and computed the corresponding Legendre transformation of the standard
elastic Gibbs function to produce the correct phenomenological free-energy functional
to be minimized. Then, they introduced the concept now known as “Pertsev phase
diagram”, of mapping the equilibrium structure as a function of temperature and misfit
strain, which has proven of enormous value to experimentalists seeking to interpret the
behaviour of epitaxial thin films and heterostructures.
These kind of diagrams were produced for the most standard perovskite oxides, either
fitting the parameters of the energy expansion to the experiment (usually near the bulk
ferroelectric transition) (see for instance Ref. [20]), or performing first-principles studies
(see Ref. [24] for full sequences of epitaxially-induced phase transitions for some of the
most common oxide perovskites).
From all these theoretical studies, a general trend of the strain-induced phase transitions emerged for perovskite oxides on a (001) substrate [24]: sufficiently large epitaxial
compressive strains tend to favor a ferroelectric c-phase with an out-of-plane polarization
along the [001] direction; conversely, tensile strains usually lead to an aa-phase, with an
in-plane P oriented along the [110]direction. The behavior at an intermediate regime is
material-dependent, but the general trend is that the polarization rotates continuously
from aa to c passing through the [111]-oriented r-phase. In non-ferroelectric perovskites
like SrTiO3 and BaZrO3 the intermediate regime is non-polar, while in PbTiO3 the
formation of mixed domains of c and aa phases could be favorable.
From the experimental side, there have been impressive advances as well. For instance, dramatic effects were observed experimentally by Haeni et al. [22] were room
temperature ferroelectricity was obtained in otherwise paraelectric SrTiO3 . The substrate used was DyScO3 which produces a +1% strain leading to an in-plane polarization
and bringing TC close to room temperature. Another example of strain-engineering was
demonstrated by Choi et al. [25], with a large enhancement of ferroelectricity induced
in strained BaTiO3 thin films by using a biaxial compressive strain imposed by coherent
epitaxy on single-crystal substrates of GdScO3 and DyScO3 . The strain resulted in a
ferroelectric transition temperature nearly 500 K higher than the bulk one and a remanent polarization at least 250% higher than bulk BaTiO3 single crystals. Very recently,
another spectacular strain effect was demonstrated both experimentally and theoretically [26, 27]: multiferroic BiFeO3 films undergo an isosymmetric phase transition to a
tetragonal-like structure with a giant axial ratio [28] when grown on a highly compressive substrate such as LaAlO3 . Furthermore, both phases appear to coexist [26] in some
conditions, with a boundary that can be shifted upon application of an electric field.
This appears to be by far the largest experimentally realized epitaxial strain to date, of
the order of 4-5 %; the existence of this new phase of BiFeO3 was also predicted to be
promising for enhancing the magnetoelectric response of this material [29].
16
As mentioned above, coherency between the substrate and the thin films lattice
constants can be maintained only up to a limiting thickness, after which defects and
misfit dislocations start to form. Strain relaxation leads to inhomogeneous strain fields
(or strain gradients), which can have profound consequences on the properties of the thin
film. A strain gradient intrinsically breaks the spatial inversion symmetry and hence acts
as an effective field, generating electrical polarization even in centrosymmetric materials.
This phenomenon became known as flexoelectricity, by analogy with a similar effect in
liquid crystals, and is allowed in materials of any symmetry. Strain-gradient-induced
polarization has, for instance, been measured in single crystals of SrTiO3 , an incipient
ferroelectric (but non-polar in bulk) material [30].
Flexoelectric effects can play an important role in the degradation of ferroelectric
properties [31, 32] and therefore proper management of strain gradients is crucial to
the performance of ferroelectric devices. At the same time, an increasing amount of
research is now aimed at exploiting flexoelectricity for novel piezoelectric devices. The
possibility of generating a piezoelectric response in any dielectric material [33], irrespective of its symmetry, by carefully engineering strain gradients has generated a lot
of excitement in the field (see, for instance, the review by L. E. Cross [34]). At the
same time, fundamental questions about flexoelectricity are being revisited, and modern first-principles-based approaches [35, 36] are being devised to go beyond existing
phenomenological theories [37].
Exhaustive discussions on strain effects on ferroelectric thin films can be found in
Ref. [38] (combined experimental and theoretical report), and in Refs. [39] and [40] (more
focused on the theoretical point of view)
1.3.2
Electrical boundary condition
From basic electrostatics we known that any discontinuity of the polarization at a surface
or an interface gives rise to an accumulation of bound charges [41],
∇P = −ρb .
(1.6)
This is true for any geometry of a finite ferroelectric sample, but has dramatic consequences in the case of thin-films made of a uniaxial ferroelectric material with an
out-of-plane polarization, which is the desired configuration for many practical implementations such as ferroelectric memories. The presence of unscreened bound charges
at the surfaces or interfaces leads to the arising of a depolarizing field that is generally
strong enough to suppress completely the spontaneous polarization of the film.
The limit case is a free-standing slab with an out-of-plane polarization Pz and under
open circuit boundary conditions. In this situation the condition of the continuity of the
electric displacement, Dz , across the surface translates into
Dzslab = ε0 E + Pz = Dzvacuum = 0,
which gives an electric field inside the slab of
(1.7)
17
Ed = −
Pz
.
ε0
(1.8)
The field in Eq. (1.8) points opposite to the polarization and thus the energy term
corresponding to the interaction of the polarization with the electric field, proportional
to −EP , is positive, opposing to the polarization of the slab. This field is called the
depolarizing field. It is important to note that Pz in Eq. (1.7) and (1.8) is the total
polarization relaxed in the presence of the depolarizing field, meaning that both sides of
the identity in Eq. (1.8) must be self-consistent. We can obtain an idea of the magnitude
of this field introducing the spontaneous polarization of BaTiO3 , about 30 µC/cm2 , in
Eq. (1.8). This yields a value of ∼ 30 GV/m, that might be compared with the coercive
fields of bulk ferroelectric materials, that typically are of the order of several tens of
MV/m. These considerations suggest that the presence of an unscreened depolarizing
field will completely suppress the out-of-plane polarization of a ferroelectric slab. For this
reason, much of the theoretical and experimental research in ferroelectric thin films is
connected directly or indirectly with the study of screening mechanism in these systems.
Arguments discussed above suggest that a ferroelectric thin film with a free surface
should not be able to sustain an out-of-plain polarization [irrespective of its thickness,
since Eq. (1.8) is independent of this variable]. Nevertheless, this conclusion contrast
with the experimental evidence. Lots of experiments concerns measurements on ferroelectric films with open surfaces that can, for instance, be switched locally applying a
voltage with a piezo-force microscope tip. Different models and first-principles simulations suggest that in ambient conditions the compensating charges necessary to stabilize
an out-of plane polarization in a film with an open surface could be provided mostly by
chemical adsorbates (water molecules, OH groups, CH, etc.) [42, 43, 44]. Screening efficiency of the molecular adsorption was demonstrated by Wang et al. [45], who achieved
the polarization switching of a PbTiO3 film by varying the partial oxygen pressure at
the open surface.
Different mechanisms can be invoked to provide the screening of the depolarizing
fields in finite-sized ferroelectric samples, including the adsoption of molecualr groups
just discussed. Most of them are summarized in Fig. 1.6. In the next sections we discuss
some of the most relevant.
Imperfect screening by metallic electrodes
The most evident solution for the screening of the depolarizing field is to sandwich the
ferroelectric film between two metallic electrodes in short circuit. Assuming ideal electrodes, the free charges would overlap with the bound charges at the surface, providing
a perfect screening of the depolarizing field.
However, this is not what happens in real electrodes, even in structurally perfect ones.
Two models are typically invoked to explain the origin of the imperfect screening: (i) the
finite screening length, (ii) and the appearance of a “dead layer” at the metal/insulator
interface. According to the first one, the screening charges in the metal distribute over a
region of finite depth. The key parameter in this case is the distance from the interface to
18
+–+–+–+– +– +– +
–
Ps
–+–+–+– +– +– +– +
Finite conductivity
λeff
V
λeff
Suppression of
polarization
–––––––
+++++++
Ps
–––––––
+++++++
Ps
Polarization rotation
Metallic electrodes
–
–
+
Ps=0
–––––––
+++++++
Ps
–––––––
+++++++
+++++++
Ps
Edep
–––––––
Ps
PE
Unstable
Ps
FE
Ps
PE
+
+`
–
Atmospheric adsorbates
Continuity of
polarization
Polarization vortices
180°
Closure
Domain formation
Figure 1.6: The depolarization field arising from unscreened bound charges on the surface
of the ferroelectric is generally strong enough to suppress the polarization completely and
hence must be reduced in one of a number of ways if the polar state is to be preserved.
Much of the research on ultrathin ferroelectrics thus deals directly or indirectly with
the question of how to manage the depolarization fields. The left part of the diagram
illustrates screening by free charges from metallic electrodes, ions from the atmosphere
or mobile charges from within the semiconducting ferroelectric itself. Note that even in
structurally perfect metallic electrodes, the screening charges will spread over a small
but finite length, giving rise to a non-zero effective screening length λeff that will dramatically alter the properties of an ultrathin film. Even in the absence of sufficient
free charges, however, the ferroelectric has several ways of preserving its polar state, as
shown in the right part of the diagram. One possibility is to form polarization domains
that lead to overall charge neutrality on the surfaces. Typical domain structures discussed in the literature are 180◦ (or Kittel domains) and closure domains (also refered
as Landau-Lifshitz domains). Under suitable mechanical boundary conditions, another
alternative is to rotate the polarization into the plane of the thin ferroelectric slab. In
nanoscale ferroelectrics polarization rotation can lead to vortex-like states generating
“toroidal” order. In heterostructures such as ferroelectric-paraelectric superlattices, the
non-ferroelectric layers may polarize in order to preserve the uniform polarization state
and hence eliminate the depolarization fields. If all else fails, the ferroelectric polarization will be suppressed.(Reprinted from Ref. [46], courtesy of P. Zubko from Université
de Genève.)
19
P
ୈ୐
଴ ஽௅
௘௙௙
଴
V(z)
Metal
FE
Metal
t
ୣ୤୤
஽௅
ୣ୤୤
஽௅
஽௅
஽௅
Figure 1.7: Schematic representation of a symmetric ferroelectric capacitor under short
circuit. t is the thickness of the ferroelectric (FE) layer, P is the polarization and Ed is
the residual depolarizing field. The ferroelectric film is assumed to be separated from the
electrodes by a vacuum (within the imperfect screening model) or a dielectric (within
the dead layer model) layer, with a thickness of λeff or λDL respectively. The thick line
represents the electrostatic potential.
the center of mass of the screening charge distribution, usually denoted as λeff (see Fig.
1.7) [7]. The value of λeff measures the degree of screening provided by the electrodes,
being zero the limit of ideal metallicity. It is tempting to relate this effective screening
length with the Thomas-Fermi screening length, λTF , used in macroscopic models, but
the latter is a bulk parameter of the electrode while the former is an interface-intrinsic
property, dependent on the chemical details of the interface, such as its orientation or
the combination of materials [47].
The dead layer, on the other hand, is a region at the metal/insulator interface with
degraded ferroelectric properties that behaves as a linear dielectric with a low permittivity [48, 49, 50]. In this case, the the thickness of the dead layer λDL and its permittivity
εDL determine the level of screening.
In fact both models are perfectly equivalent and, regardless of the interpretation,
the deviation from the ideal screening can be quantified by the ratio λeff = λDL /εDL ,
which is indeed the only relevant magnitude. If we consider, for instance, the dead
layer model, we know that at the interface between the ferroelectric and the dielectric
layer, the normal component of the electric displacement field, D, must be preserved.
Therefore, and homogeneous electric field appears inside the dead layer, of magnitude
EDL =
D
,
ε0 εDL
(1.9)
where D is the electric displacement field within the ferroelectric. The electric field EDL
20
causes a potential drop, ∆VDL = EDL λDL , at each interface. The short circuit boundary
condition requires that the potential across the whole capacitor vanishes, so
2
D
λDL + ∆VFE = 0,
ε0 εDL
(1.10)
where ∆VFE is the potential drop across the ferroelectric layer. As a consequence, an
electric field arises inside the ferroelectric (see Fig. 1.7), with a magnitude of
Ed =
∆VFE
2λDL D
=−
.
t
ε0 εDL t
(1.11)
As mentioned above, this is the same expression one gets from the finite screening length
model, simply substituting λeff = λDL /εDL
Ed = −
2λeff D
.
ε0 t
(1.12)
Eq. (1.11) can also be obtained from the internal energy of the system. This derivation
is particularly interesting because it connects the electrostatics of the system with the
internal energy profile of the ferroelectric material. We will make use of it in Chapter 3.
The relation between the polar soft mode of a ferroelectric [which is in most cases the
order parameter, see Eq. (1.2)] and the polarization, often makes it useful to expand the
internal energy of a ferroelectric in terms of the polarization, as in Devonshire-GinzburgLandau theories. However, the parametrization in P does not reflect a realistic setup.
In an experiment or a first principles simulation we usually do not have direct control
over the value of P . Instead the polarization of the material reacts in the presence
of an internal electric field determined by the electrostatic boundary conditions. In
first principles calculations, these might be a fixed electric field (equivalent to a fixed
voltage in a capacitor in closed-circuit; short-circuit boundary condition is a particular
case where the potential drop across the system is zero) or electric field displacement
(equivalent to a capacitor in open-circuit with fixed free charges on the electrodes)
[51, 52]. From a fundamental point of view, it is more appropriate to expand the internal
energy per unit cell of the ferroelectric in terms of D as
Ub (D) = A0 + A2 D2 + A4 D4 + O(D6 ).
(1.13)
Here A0 is an arbitrary reference energy, A2 is negative and the higher expansion coefficients are positive. The internal energy of Eq. (1.13) implicitly contain all the complexity
of the microscopic physics, including the internal ionic and electronic coordinates, and
the electrostatic energy due to macroscopic electric fields [51].
For a ferroelectric capacitor within the dead layer model, as depicted in Fig. 1.7, we
make use of the continuity of D again and, knowing that the internal energy density of
a linear dielectric is 12 ED, we write the internal energy density of the interface regions
as D2 /(2ε0 εDL ). The total internal energy of a capacitor made of an N -unit-cells-thick
ferroelectric film between two symmetric dead layers is thus
21
D2
,
(1.14)
2ε0 εDL
where S is the surface cell area. In short circuit the potential drop across the whole
system must be zero. It follows from elementary electrostatics [51] that the internal
electric field, E(D), is the derivative of U (D) with respect to D,
UN (D) = N Ub (D) + 2SλDL
1 dU (D)
.
(1.15)
Ω dD
Combining Eq. (1.15) and (1.14), the short circuit electrostatic boundary condition can
be written as
E(D) =
D
dUb (D)
= 0.
+ 2SλDL
dD
ε0 εDL
The electric field inside the ferroelectric layer is
N
(1.16)
1 dUb (D)
1 dUb (D)
=
,
(1.17)
Ω dD
Sc dD
where c is the bulk out-of-plane lattice constant of the ferroelectric. Introducing Eq.
(1.17) into (1.16), and using t = N c as the thickness of the ferroelectric layer, we end
up with the following expression for the residual depolarizing field
Ed =
2λDL D
,
(1.18)
ε0 εDL t
which, as we anticipated, is the same expression we obtained before from purely electrostatic arguments.
Even though the polarization is not the control parameter in most of the cases, it is
the order parameter (or at least a magnitude we are interested in monitoring) in typical
theoretical or experimental studies of ferroelectric systems. For this reason it is useful
to have an expression for the residual depolarizing field in terms of the polarization.
Substituting D = ε0 Ed + P in Eq. (1.12) we get the well known expression for the
depolarizing field in terms of the polarization of the ferroelectric film
Ed = −
Ed = −
2P λeff
.
ε0 t 1 + 2λteff
(1.19)
We should emphasize here that P in Eq. (1.19) is not the spontaneous bulk polarization
nor the polarization calculated from the Born effective charges (which does not take into
account the polarization of the electronic cloud, recall that the Born effective charges are
obtained, by definition, at zero electric field), but the total polarization in the presence
of the field Ed . Often, the approximation λeff t is assumed [53, 54], transforming Eq.
(1.19) into
Ed = −
2P λeff
.
ε0 t
(1.20)
22
CDL CN
CDL
Figure 1.8: Series of capacitors modeling the influence of the imperfect screening in a
ferroelectric capacitor. The device as a whole behaves like a series of capacitors with CN
being the “ideal” capacitance of the ferroelectric (FE) film and CDL being the capacitance
of the interface regions.
The formation of a layer with degraded metallic properties at the interfaces of a capacitor also affects another characteristic property, its capacitance. The presence of the
interfacial layer causes a significant reduction of the capacitance of the system as a consequence of the interface region with degraded permittivity. The total capacitance of
the device can be calculated a as a series of capacitors (see Fig. 1.8)
1
2
1
=
+
,
C
CDL CN
(1.21)
where CN is the expected capacitance of the insulator/ferroelectric condenser assuming
perfect screening of the electrodes, and CDL is the capacitance intrinsic to the interfacial
layer. The interfacial capacitance is, of course, linked to the properties the dead layer as
Ci =
εDL ε0 S
ε0 S
=
λDL
λeff
(1.22)
Regardless of the model used to quantify the screening in a capacitor, the physical origin
of the imperfect screening can be due to different factors, depending on the system, its
component materials and growth conditions. It might be due to extrinsic effects such
as the damage of the surface of the electrode during deposition of the ferroelectric film,
impurities or oxygen vacancies, that deteriorates the interface and creates a layer with
degraded properties where neither the electrodes nor the ferroelectric behave as the bulk
material. In atomically perfect interfaces the origin of the imperfect screening lies on
the intrinsic finite screening length of the metal, the modified chemical environment at
the interface or the penetration of conduction states into the insulating film.
The effect of the depolarizing field arising from the imperfect screening of the polarization charge is sizable for films up to 10 nm [54], causing the monotonic reduction of the
spontaneous polarization and Curie temperature of ferroelectric films as the thickness
is reduced. Ultimately the electrostatic energy term becomes too large, a monodomain
configuration is now longer supported and ferroelectricity is lost. Below this critical
thickness the film either becomes paraelectric or breaks into domains of polarization.
Although the use of realistic metallic electrodes with finite screening length is usually
assumed to be linked with a detriment of the ferroelectric properties of the capacitor,
23
this might not be always the case. We have seen above that the only parameter determining the ability of a ferroelectric film of retaining a monodomain polarization is
the interface-intrinsic effective screening length, and not the bulk screening length of
the metal. The former parameter contains all the microscopic details of the interface,
such as the local electronic and ionic structure and the chemical bonding. The detailed
understanding of the relationship between the microscopic phenomena involved in the
charge rearrangements at the interface and its macroscopic manifestation could actually
lead to the design of interfaces with enhanced screening properties. The main effect of
the interface is to alter the short-range interactions which tend to suppress the ferroelectricity. A careful choice of metallic electrode and interface termination could cause
an effective decrease in the magnitude of short range repulsions, enhancing locally the
ferroelectricity of the film. The proof of concept of this mechanism was first proposed
for BaTiO3 /Pt capacitors by M. Stengel and coworkers [47]. First-principles simulation
on the realistic capacitor, with explicit treatment of the Pt metallic electrodes, showed
that at the BaO3 /Pt interface the bonding between Pt and oxygen and the repulsion
of the Ba atoms lead to a dipole at the interface that contributes to enhance the local
polarization of the ferroelectric film. In the limit of ultrathin films the contribution of
the “interfacial ferroelectricity” could increase the polarization of the ferroelectric films
even beyond the spontaneous polarization of bulk BaTiO3
Polarization domains
The formation of polarization domains is an alternative mechanism to avoid the accumulation of polarization charge at the surface or interface of a ferroelectric layer. In this
case the system breaks into multiple domains with opposite sign of the polarization normal to the interface so the average bound charge at the surface vanishes and the electric
field in the ferroelectric layer is greatly reduced. This mechanism does not need for the
participation of any material outside the ferroelectric film, but macroscopic models and
first-principles simulations show that the formation of domains can actually be assisted
or facilitated by the other material across the interface, either if it is a metallic electrode
[55] or a highly polarizable material [56].
Although the overall neutrality of the surface or interface of the ferroelectric reduces
very significantly the electric fields inside the domains (they should decay rapidly at the
center of each domain), stray fields still contribute to the energy balance that ultimately
determines the domain structure. In a polydomain structure as the one in Fig. 1.9(a)
electric fields are confined to a very small region near the interface and their magnitude
decay exponentially over a length-scale comparable to the domain width w [56]. The
electrostatic energy of such structure thus increases with the domain size as
Felec ∝ wP 2 .
(1.23)
Minimization of the electrostatic energy is favored by the reduction of the domains size,
however the size of the domains of polarization is a consequence of the balance between
the minimization of the electrostatic energy of the system, which tend to decrease the
24
Figure 1.9: Schematic representation of typical domain arrangements in ferroelectric
thin films. (a) 180◦ (Kittel) and (b) closure (Landau-Lifshitz) domains. (c) Effect of the
electrostriction in closure domains. The dashed line represents on exaggerated scale the
volume which would be occupied by the domain of closure if the constrain exerted by
the rest of the crystal were removed. Domains of closure were thought unlikely to occur
in ferroelectrics due to the elastic penalty associated with the large electrostriction in
typical ferroelectrics.
lateral size of domains, and the domain wall energy, which tend to increase it. The
domain wall energy has in turn several contributions. Firstly at the domain wall, the
structure of the ferroelectric is microscopically different from its bulk ground state and
thus the internal energy density at those regions is larger. Secondly as a consequence
of the different dipoles orientation when passing through the domain wall, dipole-dipole
interaction is modified. Finally the polarization is strongly coupled with strain thus,
either if the polarization switching across the domain wall occurs via a reduction of the
polarization as we approach the wall or via rotation of the polarization, formation of a
domain wall is associated to a strain field. As a result of all these factors a domain wall
has an energy cost in the form of an energy per unit area of the wall σW . The associated
energy, thus scales with the domain wall density as
t
,
(1.24)
w
where t is the thickness of the ferroelectric film. The optimum balance for a given
thickness is obtained minimizing Felec + FW , yielding
FW ∝ σW
w∝
√
t.
(1.25)
This formula, stating that as the thickness of the film reduces, so does the lateral size
of the domains, is know as the Kittel relation. This formula was first obtained in the
context of ferromagnetic materials, for which the theory of domain structures was first
developed [57, 58, 59], and it has been found to be valid, also for ferroelectric materials,
over a remarkably wide range of sizes [60].
Although the theory for domain structures for ferroelectrics is formally the same
as the one developed for ferromagnets, the different physical origin of both phenomena
25
leads to significant differences between domains structures in these two types of materials. In ferromagnets the exchange energy (which favors parallel alignment of spins)
and the magnetocrystaline energy (analogous to the short-range interactions and the
polarization-strain coupling, favoring sharp domain wall and hindering spin rotations)
is usually dominated by the former and magnetic domain walls usually consist of a
gradual rotation of spins over lengths of several nanometers (Bloch-type domain wall).
Conversely, in ferroelectric materials – but also in some ferromagnets like cobalt – the
coupling of the polarization (magnetization in ferromagnets) with the strain is much
larger, and atomically sharp domain walls where polarization switch occurs over a few
unit cells, are observed in these materials. This is also favored by the anisotropy of
dipole-dipole interaction, favoring parallel alignment along the polarization axis but antiparallel in the perpendicular direction. First principles simulations on either bulk [61]
or in thin films [55, 62] find that in typical perovskite ferroelectrics the polarization flips
over a single unit cell.
Not only the domain wall in ferroelectric materials differ from that typically observed
in ferromagnets, the domain morphology and arrangement itself presents dissimilarities.
Domain structures closing the magnetic flux have been observed for a long time in ferromagnetic materials [57, 58, 59]. In these structures, depicted schematically in Fig,
1.9(b), the normal component of the magnetization in always continuous across the
domain walls and no poles are formed anywhere in the sample. Analogously closure domains in ferroelectric materials would minimize the electrostatic energy preventing the
accumulation of bound charges. Nevertheless this kind of domain structure, usually referred as “Landau-Lifshitz domains”, were thought to be unlikely to form in ferroelectric
materials due to the very large crystalline anisotropy that makes polarization rotation
difficult: the polarization in ferroelectric materials is strongly coupled with strain, thus
such domain would cause large stresses, as schematically depicted in Fig, 1.9(c). The
unlikeness of closure domains is not due to the absence of aa-phases or to an intrinsic
difficulty of polarization rotation, but more to the electrostriction, that imposes a great
penalty in terms of elastic energy.
Interestingly, recent theoretical works pointed to the formation of domains of closure
in nanometric ferroelectric capacitors (see Chapter 5 and references therein) nanodots
and nanorods [63, 64]; and other nanometric structures [65]. The theoretical predictions
have been recently confirmed by the first experimental observations of such structures
in PZT [66] and BiFeO3 [67] films.
Some author also interpret Landau-Lifshitz domains in ferroelectric films as mixed
“Bloch-Ising-Néel” domains, in analogy with domains in ferromagnets where the polarization rotates gradually [68].
Recent discoveries of exotic phenomena intrinsic to domain walls open new routes for
the development of new functionalities exploiting the properties of these structures [2].
Examples of such phenomena are the conductivity observed in domain walls in BiFeO3
[10] and PZT [69] thin films, or the possibly polar ferroelastic domain walls in CaTiO3
[70] or SrTiO3 [30].
26
Eletrostatic coupling in superlattices
Superlattices are artificially layered structures grown in an attempt to combine or tune
properties of different materials. These seemingly simple systems have been found to
display a rich spectrum of functionalities arising from complex interaction between mechanical and electrical boundary conditions, and the coupling of structural instabilities
in the reduced symmetry environment of the interfaces [2, 46].
A notable type of superlattices are those consisting of ferroelectric films separated by
paraelectric or incipient ferroelectric slabs. In these systems the development of depolarizaing fields is typically avoided adopting a a uniform component of the out-of-plane
polarization throughout the structure. The continuity of the polarization is achieved
at the cost of poling the paraelectric material and reducing the spontaneous polarization of the ferroelectric. In the most simple cases, the value of the polarization will
be determined by the competition between the ferroelectric willing to polarize and the
paraelectric opposing, the volume fraction of each material and the mechanical boundary
conditions [71]. In some cases interfacial-intrinsic effects might play a significant role,
like the in case of improper ferroelectricity observed in short-period PbTiO3 /SrTiO3
superlattices [19, 15], which behavior departs from that predicted by electrostatic coupling models. The improper ferroelectricity in this structure arises a consequence of a
coupling between polar instabilities and AFD modes which is not present in the parent
materials. In more complex systems, like tricolor superlattices, the inversion symmetry
breaking have been proposed to lead to a built-in bias that would produce a self-poling
of the heterostructure [72].
1.4
Convergence of experiments and theoretical methods
This is a particularly exciting time for nanoscale material science, as the experimental
advances in materials preparation and characterization have come together with great
progress in theoretical modeling of ferroelectrics, and both theorists and experimentalists
can finally work on the same length and time scales. This allows real time feedback
between theory and experiment, with new discoveries now routinely made both in the
laboratory and on the computer.
The rapid evolution in the atomistic modeling of materials, driven by the fast and
steady increase of the computational power, the important progresses in the implementation of more efficient algorithms and the development of more and more accurate theoretical models, allows today the simulation of realistic systems, which can be often directly
compared with actual samples. Computationally demanding first-principles simulations
can today be used in systems large enough to investigate the complex physical processes
that takes place at interfaces, isolated nanoparticles or very complicated crystal structures in bulk samples. Relevant degrees of freedom can be extracted and parametrized
from first-principles calculations that are then introduced in model Hamiltonian or shell
model methods. Reducing the number of degrees of freedom extends the size and time
scales of the simulated system. These methods have also be a great importance to bridge
1.4. Convergence of experiments and theoretical methods
27
Figure 1.10: Sketch with the different length and time scales affordable with the variety of theoretical schemes typically discussed in the literature. Arrows indicate the
interconnection between the methods. First-principles methods with atomic resolution
(represented by the balls and sticks cartoon), feed second-principles models where only
some degrees of freedom are considered (for instance, the soft mode in every unit cell,
represented by the arrows or the springs in the cartoons). Parameters for the phenomenological Devonshire-Ginzburg-Landau methods can be determined from atomistic
methods. The arrows in red stress the interconnection between experiments and theories
at the different levels.
the atomic magnitudes with the continuum models like the Devonshire-Ginzburg-Landau
theory.
On the experimental side, many groups are today capable of routinely grow singlecrystalline thin films with atomic control. The key for such achievement is the steady
improvement of epitaxial growth techniques [73]. Traditionally, sputtering and pulsed
laser deposition (PLD) were used for the growth of oxide thin films. These techniques
28
consist in ejecting the component materials from solid source targets (typically a ceramic
of the material one wants to grow in thin film) that then deposit on the free surface of a
substrate. In the first case a high voltage is applied between the target (cathode) and the
substrate (anode), the gas between the electrodes get ionized and the plasma bombards
the target ejecting the source material. The ejected atoms are then transfered to the
substrate. In the case of PLD, a high energy laser beam is used to ablate the targets
instead of a plasma. The ejected material is transfered in gas phase (forming a “plume”
emerging from the sources) to the substrate. Part of the success of both techniques
has been the availability high-quality substrates with atomically smooth surfaces that
facilitate the high-quality, defect-free deposition of the films.
An alternative technique for the growth of thin films is the molecular beam epitaxy.
Here, instead of ceramic targets, the sources are evaporated beams of the constituent
atomic elements of the film. The process takes place in vacuum, not allowing the use of
background atmospheres (in contrast with previous techniques), but in turn, this allows
the use of reflection high-energy electron diffraction (RHEED) to monitor the growth
of the film. The use of a real-time characterization techniques of the free surface of
the sample during growth process opened up the ability to have submonolayer precision
during film deposition. This fundamental achievement allows complete control over the
growth of the interfaces in heterostructures.
Characterization techniques have also undergone great improvements. Different varieties of atomic force microscopy (AFM) methods permit the sampling of various properties of thin films: from the topography of the surface (AFM itself) to the magnetic
properties of the film (magnetic force microscopy, MFM). Particularly relevant in the
case of ferroelectric thin films is the piezo-force microscopy (PFM) that allows to test
the ferroelectricity of samples measuring its piezoelectric response upon applying an
electric voltage between the AFM tip and the film. This technique is able to perform
characterizations of either the overall film by the deposition of a top electrode or local
measurements in free surfaces. The later case is specially interesting for the study of
domains walls, allowing to obtain in-plane and out-of-plane piezoelectric responses (from
which local orientation of the polarization can be inferred) or local hysteresis loops.
Recently, great advances have been achieved in transmission electron microscopy
(TEM) imaging as well. The recent implementation of spherical aberration correction in
TEM improved the resolution of the TEM experiments from the Raleigh limit (typically
around 1 Å) to the picometer range[74], allowing great precision in the measurement of
inter-atomic distances and to extract relevant physical magnitudes with atomic resolution. Local polarization orientation, for instance can be extracted just from the relative
displacements of the atoms. This technique can be combined with others, like electronenergy loss spectroscopy (EELS) capable of measuring atomic composition, chemical
bonding properties, and electronic structure properties of individual atoms. However,
very often TEM and associated techniques provides such amount of information that
understanding the results is generally not straightforward and only possible through the
comparison with extensive quantum-mechanical computer calculations.
Pictures in Fig. 1.11 are a good example of the current situation of the research
1.5. References
29
of oxide thin films. This figure illustrate an ongoing project aiming to characterize the
properties of PbTiO3 /SrTiO3 superlattices. Fig. 1.11(a) and (b) are TEM images of
actual samples being grown and investigated by P. Zubko and coworkers at the University of Geneva. Fig. 1.11(c), on the other hand, is the relaxed structure of exactly
the same system in a polydomain phase, as obtained by means of a first-principles simulation. This research, that is discussed in Chapter 6, is benefiting from a continuous
interaction between experimental and theoretical groups that is being determinant for
the understanding of the physical properties of this heterostructure.
Figure 1.11: Comparison between (a, b) an actual image, obtained by transmission electron microscopy (TEM), of a PbTiO3 /SrTiO3 superlattice and (c) a real first-principles
simulation of the same system. (TEM images courtesy of A. Torres-Pardo, from Université Paris Sud.)
1.5
References
In this Chapter we have tried to provide a very brief review of the basic physics of
ferroelectric materials as well as of the state of the art of this research field as far as we
are concerned, aiming to provide in every case the appropriated references. However,
all the topics discussed in this Chapter have been thoroughly reviewed before and much
more detailed analysis can be found in the following works and references therein:
30
• K. M. Rabe, C. H. Ahn, and J.-M. Triscone, editors, Physics of Ferroelectrics: A
Modern Perspective. Springer-Verlag, Berlin Heidelberg, 2007
• Ph. Ghosez and J. Junquera. Handbook of theoretical and computational nanotechnology, vol. 7, ch. 134. American Scientific Publishers, 2006.
• C. Lichtensteiger, P. Zubko, M. Stengel, P. Aguado-Puente, J.-M Triscone and
J. Junquera, Ferroelectricity in ultrathin film capacitors. In G. Pacchioni and S.
Valera, editors, Oxide ultrathin films: science and technology. John Wiley & Sons,
2011.
Chapter 2
Methodology
2.1
Introduction
The aim of this work is the theoretical study of screening properties in nanometric ferroelectric devices, systems where interfaces between ferroelectric and non ferroelectric
materials play a key role. Overall properties of such systems are the result of the combination of intrinsic bulk properties of the materials composing the device with properties
specific to the interfaces. In most of the systems we are interested in, which usually
display a planar geometry, the region of interest is only a few nanometers thick, but the
system is extended in the plane.
Take for instance a typical ferroelectric thin film sample, the ferroelectric layer is
only a few atomic monolayers thick, but the area of the device can be as large as 1 cm2
and in practice, given the in-plane to out-of-plane dimensions ratio, can be assumed
to extend to infinity in the plane. This is a complicated situation since we want to
perform simulations with atomic detail – to capture the subtle physics of the interface
– in macroscopically large systems. We will see throughout this Chapter how these
difficulties can be overcome.
At the atomic scale classical continuum models of condensed matter are no longer
valid, its explicit atomic nature must be taken into account and laws of quantum mechanics apply. Methods based exclusively on the equations of quantum mechanics and
electromagnetism, not making use of any parameter fitted to experimental results, are
called first-principles methods. These methods provide, in principle, an unbiased probe
to study properties of condensed matter systems in conditions difficult or impossible
to achieve in a laboratory and allow to isolate different contributions from the various
effects.
2.2
Overview of approximations
Properties of materials at the nanometric scale are obtained studying the behavior of
their basic components at the atomic level, i.e. nuclei and electrons. At this scale,
according to quantum mechanics the expected value of any physical quantity can be
31
32
Chapter 2. Methodology
obtained theoretically if one knows the wave function Φ for all the particles of the system.
This object, in turn, is the solution of the Schrödinger equation of the system, that
assuming a time-independent problem (i.e. that the potential entering the Hamiltonian
of the system does not depend on time) is the following
H(R, r)Φ(R, r) = EΦ(R, r),
(2.1)
where H is the Hamiltonian of the system, which is a function of the position of all nuclei,
{R}, and electrons, {r}; and E is the total energy of the system. The Hamiltonian of
a system of interacting nuclei and electrons can be written as (unless otherwise stated
throughout the chapter we will use atomic units, i.e. e = h̄ = me = 1)
H(R, r) = TN (R) + VN N (R) + Te (r) + Vee (r) + VN e (R, r) =
(2.2)
P
P
P
P
2
2
Z
Z
Z
1
1
1
1 ∂
∂
1
I
I J
I,i |RI −ri | .
I MI ∂R2 + 2
I6=J |RI −RJ | − 2
i ∂r 2 + 2
i6=j |ri −rj | −
1P
= −2
I
i
In Eq. (2.2) TN and Tn are, respectively, the kinetic energy of nuclei and electrons; and
VN N , Vee and VN e are, respectively, the nucleus-nucleus, electron-electron and nucleuselectron electrostatic interactions. The latter is the only attractive interaction among all
of them and thus can be thought as the “glue” of condensed matter systems. Solution
of Eq. (2.1) would provide all possible information about the studied system, however
its complexity makes it impossible to solve but for the simplest molecules. Several
approximations must be done to transform this into a solvable problem in practical
systems:
• Born-Oppenheimer approximation allows to decouple the movement of electrons and nuclei.
• Density functional theory provide a very efficient method to solve the electronic
problem substituting the electronic wave function by the electronic charge density
as the fundamental variable of the problem. This formalism is exact but needs
some approximation to the exchange-correlation energy functional.
• Using pseudopotentials we can get rid of a large number of chemically inert
electrons and avoid sharp oscillations of the wave functions near the core region,
reducing the number of required basis functions.
• Periodic boundary conditions allows the simulation of infinite periodic crystals
reducing the problem to calculation on a periodically repeated simulation box. By
means of the supercell technique also non-periodic systems can be studied within
these boundary conditions.
• Calculation of physical magnitudes that require integrations over the first Brillouin
zone are typically performed by means of a finite sampling of reciprocal space.
2.3. Born-Oppenheimer approximation
33
• Expanding one-electron wave functions into a linear combination of basis functions our search for an unknown function transforms into a search for a set of
coefficients.
It is beyond the scope of this work to provide a detailed derivation of all these approximations, we will instead try to provide a concise overview of how we take advantage
of them to solve our problem. We will emphasize their strong and weak points trying to
clarify their applicability limits. Extensive reviews of all these approximations can be
found in the references provided in Sec. 2.9.
2.3
Born-Oppenheimer approximation
The very first approximation we use to transform Eq. (2.1) into a solvable problem
is the Born-Oppenheimer or adiabatic approximation. Taking into account that nuclei
mass is thousands of times larger than electrons mass – the mass ratio between an
electron and a proton is comparable to that of a human baby and an adult elephant
–, we can assume that the velocity of electrons would be much larger than that of the
nuclei (typical velocities of nuclei and electrons are of the order of 103 m/s and 106 m/s
respectively). For this reason the nuclei kinetic energy TN in Eq. (2.2) can be considered
as a perturbation of an electronic Hamiltonian
He (r; R) = Te (r) + Vee (r) + VN e (r; R) + VN N (R).
(2.3)
where nuclear positions enter as parameters [this is denoted in Eq.(2.3) separating parameters and variable with a semicolon] and are kept constant while solving the electronic
problem. Notice that now in Eq. (2.3) VN e (R, r) is the energy of the electrons in a
fixed potential created by nuclei and VN N (R) is just a constant. Now, after solving the
electronic Schrödinger equation
He (r; R)Ψ(r) = Ee Ψ(r),
(2.4)
its eigenvalues Ee act as the potential energy for the nuclei, which movement can be
usually solved using classical mechanics.
The solution of the electronic Schrödinger equation provides the quantum description
of electronic properties for any given ionic positions. Moreover, forces and stresses can
be obtained which can be used to perform structural relaxations molecular dynamics
simulations or phonon analysis. This way the Born-Oppenheimer approximation decouples nuclei and electrons movement greatly simplifying the problem, but allowing, at
the same time, for efficient characterization of both electronic an structural properties
without loss of accuracy.
2.4
Density functional theory
Solving Eq. (2.4) for a set of interacting electrons requires searching for a very complex object, the many body electronic wave function Ψ(r), a function of 3Ne coupled
34
variables (where Ne is the number of electrons in the system), which – since electrons
are fermions – has to be antisymmetric with respect to an interchange of any pair of
electrons. This is still an impossible task in most practical cases. Most methods in
quantum chemistry approach this issue constructing the all-electron wave function as
a combination of Slater determinants, which in turn are build from one-particle wave
functions. These methods can provide extremely precise results incorporating more and
more configurational determinants in the expansion, but the computational cost scales
extremely fast and restrict their usefulness to molecular systems or small atomic clusters.
Density functional theory (DFT) provide an alternative and elegant approach to
the electronic problem. DFT was born in the 1960’s from the works of Hohenberg
and Kohn [75] and Kohn and Sham [76]. In the first paper, Hohenberg and Kohn
proved that the external potential vext (r) in the electronic Hamiltonian (the electrostatic
potential created by the nuclei, for instance) is univocally determined by the ground state
electronic density ρ0 (r). It trivially follows that the whole electronic Hamiltonian, and
thus also the ground state energy and wave function, are defined by the ground state
electronic density. Consequently an energy functional of the density can be defined and
the minimization of such functional would provide the ground state density of the system
for a given external potential.
Using the groundbreaking work by Hohenberg and Kohn [75] we can reformulate the
problem of searching for the solution of the many-body Schrödinger equation in terms of
the electronic density, which is a function of just three variables. However, although the
ground state electronic charge density is in principle sufficient to obtain any property
of a given material, the paper by Hohenberg and Kohn doesn’t provide any clue about
how to extract such properties from it.
At this point is where the work by Kohn and Sham [76] comes into play providing precisely a practical method to take advantage of the density functional theory. The
approach consists in replacing the many-body problem of interacting electrons by a fictitious system of non interacting particles moving in an effective potential. The electronic
charge density of this auxiliary system can be calculated from the one-electron wave
functions as
ρ(r) =
occ
X
i
|ψi (r)|2 .
(2.5)
If we build the system of non-interacting particles so that their electronic charge density
equals the real one, the energy functional of the real system can be written in terms of
the one-electron auxiliary functions and the electronic density as
Z
1X
1
ρ(r)ρ(r 0 )
2
EKS [ψi ] = −
< ψi |∇ |ψi > +
drdr 0
2
2
|r − r 0 |
i
Z
+Exc [ρ(r)] + vext (r)ρ(r)dr + EN N
(2.6)
where the first term is the kinetic energy of a collection of non interacting electrons. The
35
2.4. Density functional theory
second is the Coulomb electrostatic energy, which in the context of quantum mechanics
of condensed matter systems is also referred as Hartree term. The third term in the
equation is the so called exchange-correlation term which contains the difference between
the kinetic energy of the real electrons and the non interacting particles as well as any
electron-electron interaction beyond the classic Coulomb repulsion. The fourth term
is the potential energy of the electrons on the external potential (usually the potential
created by the nuclei) and the last term is the nucleus-nucleus electrostatic energy.
The variational problem on the electronic density transforms into the variational
problem over the set of one-electron functions. The minimization of the energy functional
of Eq. (2.6) should give rise to the set of {ψi } which through Eq. (2.5) would lead
to the ground state electronic density of the real system of interacting electrons. This
minimization is performed under the constraint of orthonormalization of the one-electron
wave functions
< ψi |ψj >= δij ,
and gives rise to the following Schrödinger-like equations
1 2
Heff ψi (r) = − ∇ + veff (r) ψi (r) = Ei ψi (r),
2
(2.7)
(2.8)
where the effective potential takes the form
Z
veff (r) =
ρ(r 0 )
dr 0 + vxc (r) + vext (r).
|r − r 0 |
(2.9)
The exchange-correlation potential is defined as
vxc (r) =
δExc [ρ(r)]
.
δρ(r)
(2.10)
Equations (2.5),(2.8) and (2.9) constitute the so called Kohn and Sham equations, which
when solved provide the one-electron eigenstates which electronic density coincide with
the ground state density of the actual system. In practice this problem is solved selfconsistently: equations are solved for a trial density ρin (r) (coming from a trial set of
{ψi (r)}) and a new density ρout (r) is obtained as an output. The input and output
densities are compared and a new input density is created as a mix between ρin (r) and
ρout (r). The procedure is repeated until self consistency is achieved and ρout (r) = ρin (r).
At this point several aspects of the density functional theory should be highlighted.
Firstly, leaving aside the Born-Oppenheimer approximation on top of which most theoretical methods for condensed matter physics are build, the DFT is an exact theory. The
density computed with the one-electron wave functions obtained from the Kohn-Sham
equations would be equal to the actual ground state electronic density and any ground
state physical property of the system could be derived from it. This is true as long as the
energy functional of Eq. (2.6) is the exact one. Unfortunately the exchange-correlation
term is not known so in practice approximations to this term must be made.
36
2.4.1
Exchange and correlation functional
As we have just remarked, the formalism of the density functional theory is exact but
the form of the universal exchange-correlation energy functional to be used is unknown
and, in practice, it must be approximated.
In principle Exc [ρ(r)] depends on the charge density in every point of space. However
we can assume that for slow varying charge densities, as in metals, this functional could
be approximated by a function of the density at each point. This is the local density
approximation (LDA). Within this approximation the exchange and correlation energy
at a given point is that of an homogeneous electron gas, hom
xc , with the same electronic
density of the system at that point
LDA
Exc
[ρ(r)]
Z
=
hom
ρ(r) ρ(r)dr.
xc
(2.11)
The exchange-correlation potential of an homogeneous gas can be calculated very precisely by means of Quantum Montecarlo simulations [77].
This approximation can be improved including in the functional terms depending
on the gradients of the charge density. Under this approximation, named generalized
gradient gpproximation (GGA), the exchange-correlation term is still a “semi-local”
function and not a functional:
GGA
Exc
[ρ(r)]
Z
=
xc ρ(r), |∇ρ(r)|, ... ρ(r)dr.
(2.12)
More precise approximations exist that go beyond these two examples. One example is
the hybrid functionals which typically consist of a linear combination of one of the previous local functionals (LDA or GGA) with the Hartree-Fock exact exchange calculated
from the one-electron wave functions. Of course, non local approximations like these do
not come cheap and comparatively have a great computational cost. Nevertheless, some
of these hybrid functionals, in particular the B1-WC [78], have been proved to provide
good ground state electronic and atomic structure and would be very reasonable options
to go beyond the LDA for some of the results of the present work.
Despite being crude approximations, the local LDA and GGA functional are widely
used due to their efficiency and remarkably performance. In many cases they have been
probed to be surprisingly reliable, specially for the study of structural properties (see
Table 2.1). However they also have some limitations that should be noticed. The most
serious error is the electronic band gap estimation, for which values of around 50% of the
experimental gap are typically obtained. Usually in calculations of bulk materials this
limitation doesn’t have any further consequence but in Chapter 3 we will see how this
can lead to unphysical results in the case of ferroelectric capacitors and we will provide
the correct procedure to detect such pathological behaviors.
Besides the approximation required to the exchange-correlation functional some further approximations help to provide a practical implementation of the density functional
theory, allowing for very efficient yet precise and reliable quantum simulations of con-
37
2.5. Pseudopotentials
Table 2.1: Typical errors introduced in the computation of structural, energetic and
electronic properties by LDA and GGA aproximations to the exchange-correlation functional. For comparative studies see [79], [80] and [81].
Lattice constants
Bulk modulus
Cohesion energy
Band gap
LDA
-1% – -3%
+10% – +40%
+15%
-50%
GGA
+1%
-20% – +10%
-5%
-50%
densed matter systems. In the following sections we will discuss some of the most
common and important of these approximations.
2.5
Pseudopotentials
Taking into account their participation in the bonding formation and chemical processes,
electrons in an atom can be classified into core and valence electrons.
The core electrons are inner electrons with deeper energy levels and are extremely
localized close to the nucleus. They do not overlap with orbitals from neighboring atoms
and thus they are essentially insensitive to the chemical environment. In fact most of
the physical and chemical properties of a condensed matter system can be perfectly
explained in terms of the valence electrons and the main effect of core electrons is to
screen the nucleus potential felt by the former.
Besides increasing unnecessarily the number of electrons in the system (and thus the
number of Kohn-Sham eigenfunctions to compute) the core electrons introduce another
undesirable feature: exclusion principle forces the orthogonality between core and valence
wave functions and this causes strong oscillations of the latter in the inner region of the
electronic cloud. Many DFT methods expand the Kohn-Sham eigenfunctions in plane
waves or make use of real space grids (see Sec. 2.8). In that case the description of
valence states require the use of a huge number of basis functions or very fine grids to
accurately reproduce the strong oscillations of the wave functions.
Pseudopotentials were developed to get rid of these issues. The pseudopotential
approximation is carried out in two steps: (i) since core electrons are not affected by
chemical environment they can be considered as frozen into their isolated atom configuration (frozen-core approximation), and then (ii) the frozen core electrons together with
the nucleus potential can be substituted by a screened and smoothed fictitious potential,
the pseudopotential. Once core wave functions have been removed and nucleus potential
screened, valence wave functions are allowed to vary smoothly in the inner regions.
In practice a pseudopotential generation starts with an all-electron simulation of an
isolated atom in a reference electronic configuration. The nucleus potential together
with the core electrons are then substituted by the pseudopotential, forcing the valence
38
wave functions to coincide with the valence wave functions of the all-electron atom in
the outer region (in practice a cutoff radius is used). The pseudopotential is fitted to
reproduce not only the valence wave functions but also the eigenvalues of the reference
all-electron simulation.
A good pseudopotential is expected to have two qualities. The pseudopotential is
generated for a reference electronic configuration, but to be useful it must be transferable
to different chemical environments, being flexible enough to reproduce reliably the results
one would obtain in an all electron simulation. At the same time it is desired to be
sufficiently smooth so the number of plane waves and/or grid points can be reduced.
These two qualities however are competing, the more the pseudo-wave functions resemble
the real ones the more transferable and less smooth the pseudopotential is. At the end
of the day, and as in any other approximation made, one must obtain a suitable balance
between reliability and feasibility.
Sometimes the division in core or valence states is not as strict as stated before
and there is a significant overlap between core and valence change densities. The so
called semi-core states are chemically inert but introduce some exchange-correlation
interactions with valence electrons that must be taken into account either including
these states in the valence set or using non-linear core corrections [82].
Thanks to the pseudopotential approximation the number of electrons in the simulation is greatly reduced and shorter basis sets can be used while keeping a good description of the valence wave functions, decreasing enormously the number of variables
of our problem.
2.6
Periodic boundary conditions
In the study of periodic solids it is usual to consider Born-von Karman periodic boundary
conditions. Within these boundary conditions a periodic lattice is defined as an infinite
repetition of a given unit cell in every direction. Any observable of the system, and of
course the effective potential, must be periodic, or in other words it must be invariant
with respect to a translation of a lattice vector a,
veff (r) = veff (r + a).
(2.13)
For such infinite periodic systems, the Bloch theorem states that the eigenfunctions of
the one-electron Hamiltonian in a periodic potential may be written as the product of a
plane wave envelope function and a Bloch function un,k (r) that has the same periodicity
as the lattice [83]
ψn,k (r) = eik·r un,k (r),
(2.14)
un,k (r) = un,k (r + a).
(2.15)
where
2.6. Periodic boundary conditions
39
The one-electron Kohn-Sham wave functions (ψi in previous Sections, ψn,k here) can
thus be classified in terms of a wave vector k inside the first Brillouin zone and a band
index n.
A crystal is precisely a material which structure consists of a unit cell periodically
repeated along the three dimensions. Although the ideal perfectly periodic and infinite
crystal does not exist in Nature, many properties of materials depend mostly on the bulk
and not on the surfaces (there are notable exceptions to this rule though, and this work
is mostly concern with them, but we shall see in a moment that the implementation
of periodic boundary conditions doesn’t preclude at all the study of those phenomena).
Thus, periodic boundary conditions are the natural choice for the simulation of bulk
crystalline materials and are the boundary conditions implemented in most DFT-based
methods for the study of solid state systems.
Figure 2.1: Schematic view of appropiate supercells for the simulations under periodic
boundary conditions of a bulk crystal (a), a superlattice (b), an isolated cluster (c) and
a slab (d).
Despite being specially suited for the study of bulk systems [Fig. 2.1(a)], more
complex structures can be efficiently and reliably simulated within periodic boundary
40
conditions making use of the supercell technique [84]. This technique consists in building
a simulation box, containing more than just a primitive unit cell of a given material,
that is repeated periodically over all space. This technique is regularly applied for the
simulation of:
• Complex structures within bulk crystals, like domains (of polarization, magnetic,
elastic, etc.). In this case several unit cells of the material are included in the
simulation box so different arrangements of the order parameter are allowed.
• Superlattices can also be simulated using supercells [Fig. 2.1(b)]. These are structures consisting of a periodic stacking along one dimension of alternating layers of
different materials.
Capacitors, which are one of the systems that constitutes the main subject of the
present thesis, might be considered as a particular case of superlattice. A real
thin film capacitor consists of an insulating layer between relatively thick (typically assumed to be semi-infinite) metallic electrodes. In practice however, due
to the periodic boundary conditions, what we are simulating is an infinite series
of alternating metallic and insulating layers. Thus, Figure 2.1(b) might also depict this case, with the darker atoms corresponding to the metallic region and
the lighter ones to the insulator/ferroelectric. To mimic the properties of a realistic capacitor with semi-infinite electrodes the metallic region included in the
simulation box should be thick enough to avoid interactions of the two periodically repeated interfaces. For this particular kind of systems it is worth remarking
that periodic boundary conditions affect any measurable property of the system,
and in particular the electrostatic potential must obey the periodicity of the supercell. This is particularly important for the simulation of capacitors since it is
equivalent to assume short-circuit electrostatic boundary conditions between the
electrodes. Recently strategies to performed constrained electric field E [85, 86]
and electric displacement D [87, 52] calculations under periodic boundary conditions have been implemented, allowing to overcome this limitation and to perform
simulations under virtually any electrostatic boundary condition.
• Zero-dimensional systems like molecules or nanoparticles can be simulated with
this technique as well, embedding the object in vacuum inside a large simulation
box, as in Fig. 2.1(c).
• For the simulation of slabs and surfaces [systems periodic only along two dimensions, see Fig. 2.1(d)] a vacuum region should be included in the simulation box
to avoid spurious interactions between periodic images.
Every time supercell technique is used, a careful convergence study of the dependence
of the properties of the system with respect to the supercell size should be carried out.
41
2.7. Brillouin zone sampling and electronic temperature
2.7
Brillouin zone sampling and electronic temperature
We have seen in the previous section that as a consequence of the periodic boundary conditions eigenfunction of the one-particle Kohn-Sham equations can be written as Bloch
states, characterized by the quantum number k and the band index n. Many physical
properties of condensed matter systems do not depend on one particular eigenstate, but
on the integrated contribution of all of them. The obvious example is the electronic
charge density, that is calculated as the sum over bands of an integral over the first
Brillouin zone
XZ
|ψn,k (r)|2 fn (k)dk,
(2.16)
ρ(r) =
n
k∈1BZ
where 1BZ denotes the first Brillouin zone and fn (k) is the occupation function. At zero
temperature energy levels are filled following the Aufbau rule and accordingly fn (k) is
a step function where one-electron levels can only have an occupation of two (assuming
spin degeneracy) or zero electrons. In an infinite system, as those we are dealing with,
the integral in Eq. (2.16) implies a continuous sum over an infinite number of k-points.
Obviously such infinite sum is impossible to treat in practice. The usual approach to
compute physical magnitudes that require integration over the first Brillouin zone is to
replace the integral by a discrete sum over a finite selection of k-points
ρ(r) =
X X
n k∈1BZ
|ψn,k (r)|2 wk fn (k),
(2.17)
where wk are the weight factors of k-points that depend on the way the sampling is
performed. Fortunately eigenfunctions change smoothly with k so it is possible to obtain
a good representation of the continuum of states with a finite sample over an appropriate
grid of k-points. Several schemes have been proposed in the past for efficient samplings
of the Brillouin zone [88, 89] but today the most widely used is the one proposed by
Monkhorst and Pack [90]. Within this method a number of divisions Ni along each
reciprocal lattice vector of the simulation box bi is chosen. Then the sampled k-points
are obtained as
k = q1 b 1 + q2 b 2 + q3 b 3 ,
(2.18)
2j − Ni − 1
,
2Ni
(2.19)
where
qi =
j = 1, 2, ..., Ni .
Very often the study of a particular device involves the comparison of different calculations performed on different simulation boxes (or supercells). In those cases, and
specially in metallic systems, it is strongly advisable to make sure that the density (and
not the total number) of k-points used for the discretization of the reciprocal space is
the same in all calculations.
42
Table 2.2: Convergence with respect to the Monkhorst-Pack mesh of structural parameters and energy difference between tetragonal and cubic phases of bulk BaTiO3 . ∆zα
is the off-center displacement of atom α in units of the lattice vector c. (For a 2 × 2 × 2
Monkhorst-Pack mesh the cubic phase is found as ground state.)
MP mesh
2×2×2
4×4×4
6×6×6
8×8×8
Experiment
a
(Å )
3.947
3.942
3.938
3.939
3.986
c
(Å )
3.947
3.974
3.997
3.991
4.026
c/a
∆zT i
∆zO1
∆zO2
1.000
1.008
1.015
1.013
1.010
0.000
0.012
0.016
0.015
0.015
0.000
-0.015
-0.022
-0.020
-0.023
0.000
-0.010
-0.015
-0.014
-0.016
∆ (Etetra − Ecubic )
(meV)
-3.41
-9.77
-10.86
How fine this sampling should be would strongly depends on the nature of our system
and the property we are interested in. Metallic systems where many properties depend
on a few states around the Fermi surface require finer grids than insulators in which the
occupied states are well defined thanks to the presence of the band gap. Ferroelectric
perovskites are exception to this rule, since despite being insulators they require fairly
fine k-point mesh. For instance, in Table 2.2 structural and energetic properties of bulk
BaTiO3 obtained with different Monkhorst-Pack samplings show that a mesh of at least
6 × 6 × 6 is required to achieve converged results [91].
In metallic systems a finite k-point sampling can lead to convergence problems as
some states can cross the Fermi level during the course of the self-consistency process,
leading to large oscillations of the electronic density. To avoid the use of huge samplings,
mitigate the sensitivity with respect to the k-points, and improve convergence rate in
metals, a smearing is often applied to the Fermi surface. This smearing is implemented
introducing a smooth occupation distribution fn (k) = f (Enk ) in Eq. (2.17) (substituting
the step function that would correspond to an occupation function at zero temperature).
Typical choices for the occupation function are a Gaussian distribution [92]
En,k − EF
1
fG (En,k ) = erfc
,
(2.20)
2
σ
where erfc is the complementary error function, or Fermi-Dirac distribution
fFD (En,k ) =
1
e(En,k −EF )/σ
+1
,
(2.21)
where σ is the broadening energy parameter. A smooth occupation function leads to a
partial population of the energy levels around the Fermi energy, improving performance
during the simulation of metallic systems. The smearing parameter can be understood
as a fictitious electronic temperature Te = σ/kB , but the minimization of the KohnSham energy functional only leads to the ground state electronic density for a system of
43
2.8. Basis sets
electrons at zero temperature so, when a finite electronic temperature is used to improve
the stability of a simulation it must be gradually reduced until convergence is achieved.
2.8
Basis sets
Let’s go back to the Kohn-Sham Eqs. (2.5), (2.8) and (2.9). Equation (2.8) is a set of
differential equations which solution is a set of functions ψi (r). This problem can be
greatly simplified expanding the solutions as linear combinations of basis functions
ψi (r) =
X
cij φj (r).
(2.22)
j
With this trick Eq. (2.8) transforms into a set of algebraic equations, for which very
efficient numerical methods are available. Essentially what we have just done is to replace
the problem of solving Eq. (2.8) for a function in every point of space to solve it for a
series of coefficients (a “few” complex numbers).
Several types of basis functions can be used, each with its own advantages and
disadvantages. Among all, most common methods implement one of the following: plane
waves, atomic orbitals or atomic spheres.
2.8.1
Plane waves
This basis emerges naturally from the periodic boundary conditions and the Bloch theorem. Since, as we have seen in Sec. 2.6, the solutions of the Kohn-Sham equations can
be expressed as products of plane waves with periodic functions, decomposing ψn,k in a
Fourier expansion provide intuitive understanding of solid physics in terms of the band
structure. If Ω is the volume of the simulation box and G is a reciprocal lattice vector
this expansion can be written as
1 X
ψn,k (r) = √
Cn,k,G ei(k+G)r .
Ω G
(2.23)
In principle previous sum involves an infinite number of plane waves, but in practice a
cutoff is imposed to the maximum energy of the plane wave in the expansion.
Plane waves as basis functions present several advantages. Firstly, they are asymptotically complete and, most importantly, the completeness of the basis is trivial to
approach increasing the number of elements in the expansion. Being completely non
localized they provide unbiased, homogeneous and isotropic description of the electronic
structure of the system. Moreover they allow a trivial use of fast Fourier transform
techniques for which extremely efficient subroutines are available simplifying the implementation of plane waves codes.
Of course this approach has also some pitfalls. While being able to represent any
function, they are at the same time not particularly well suited for any. Besides, their
homogeneity turns into a disadvantage in simulations with supercells containing large
44
vacuum regions, since plane waves extend over the whole space regardless of the presence
or absence of material near a particular point.
2.8.2
Atomic orbitals
This type of basis sets captures the essence of the atomic-like features of solids and
molecules expanding the one-electron eigenfunctions in a set of functions centered on each
atom of the system. This allows extremely efficient expansions, requiring, comparatively,
a much smaller number of basis functions with respect to a plane wave calculation to
obtain the same accuracy. Localized functions typically consist of a spherical harmonic
Ylm times a radial function Rln (r) centered at an atomic position RI ,
d
φInlm (r) = Rln (|r − RI |)Ylm (r −
RI ),
(2.24)
d
where r −
RI is the direction vector. Different choices are possible for the radial function,
either analytical or numerical functions have been implemented. Among the former, the
most common choice are Gaussian functions [93] for their easy implementation, and
Slater orbitals [94] due to their similarity with the actual shape of the wave functions of
an isolated atom. Numerical atomic orbitals [95] on the other hand, are more difficult
to implement due to the lack of analytic integrals, but are the most flexible and can be
optimized for each particular element and chemical environment [96, 97].
Atomic orbitals methods can also take advantage of locality principle to implement
order-N methods which computational cost scales linearly with the number of electrons
of the system. In contrast to plane waves, which provide an intuitive vision of band
structure, atomic orbitals are suitable tools for the analysis of bonding formation and
hybridizations. They are specially suited to represent atomic wave functions and therefore they are much more efficient, typically requiring just a few dozens of basis functions
to achieve an accuracy comparable to a calculation with several hundreds of plane waves
[95].
Despite being able to achieve accuracies perfectly comparable to plane waves calculations, they are much harder to improve and there is not an easy recipe to do it
systematically. Atomic basis sets are improved increasing their radial and angular flexibility [98]. Radial flexibility of basis functions is improved including more than one
different radial functions for some selected orbitals. The number of radial functions of
a given orbital is denoted by its ζ (Zeta) number. Radial flexibility is important for instance to reproduce correctly different ionization states of the atom, where the extension
of the electronic cloud would depend on the greater or smaller inter-electronic repulsion.
An extra shell corresponding to the first unoccupied angular momentum channel
is usually added to polarize the most extended atomic valence orbitals, giving angular
freedom to the valence electrons in order to improve the description of bond formation
and hybridization.
A basis set including, for instance, two radial functions per atomic orbital plus polarization functions is usually denoted as double-ζ plus polarization or DZP. The correct
size (ζ) and polarization of a basis set can only be determine by means of tests.
2.9. References
45
It is already evident that a great cost in terms of human effort is the main drawback
of the use of atomic orbitals as basis functions, and the price to pay if we want to take
advantage of their great efficiency.
2.8.3
Atomic spheres
This method combines the best of both previous worlds. Within this method, spheres
centered at every atomic position are defined. Inside these spheres, atomic-like orbitals
are used to expand the wave functions, while plane waves are used for the description
of the wave functions in the interstitial regions. As in the case of plane waves, these
basis sets are asymptotically complete but they converge faster with the number of basis
functions and they are able to provide a good description of the electronic structure
of real solids with rapidly varying atomic-like functions near the nuclei and smoothly
varying functions between the atoms. The use of sharp functions in the atomic core
regions make unnecessary the use of pseudopotentials, applying only the frozen-core
approximation to avoid the calculation of the core levels.
Their use is, however, limited by their difficult implementation and the computational
cost, much larger than for the previously explained methods. Also, as they do not rely on
the use of pseudopotentials and explicitly incorporate the core levels, the total energies
obtained with these methods are usually huge, requiring very converged simulations
when energy differences between different phases are being calculated.
2.9
References
All the approximations described in this Chapter are discussed in great detail in many
books and reviews. Some particularly good ones are:
• R. M. Martin, Electronic Structure, Basic Theory and Practical Methods, Cambridge University Press (2004).
• J, Kohanoff, Electronic Structure, Calculations for Solids and Molecules, Cambridge University Press (2006).
• R. G. Parr and W. Yang, Density Functional Theory of atoms and molecules,
Oxford University Press (1989).
• M. C. Payne, M. P. Teter, D. C. Allan, T. A. Arias and J. D. Joannopoulos, Rev.
Mod. Phys. 64, 1045 (1992).
Chapter 3
Band alignment issues in the ab
initio simulation of ferroelectric
capacitors
3.1
Introduction
As pointed out in the Introduction of this thesis, the outstanding degree of development
achieved in the last years in thin film growth and characterization techniques has lead to
the fabrication of many novel oxide-based metal-insulator heterostructures with a dizzying range of functionalities. Density functional theory (DFT) methods, either within the
local density (LDA) or generalized gradient (GGA) approximation, have been an invaluable tool in achieving a fundamental understanding of this class of systems [99, 7, 21],
particularly with recent developments which allow the application of finite electric fields
to periodic solids or layered heterostructures [85, 100, 86, 87, 52]. However, since this
domain of research is relatively new, it is important to identify, in addition to the virtues,
also the limitations of DFT that are specific to metal/ferroelectric interfaces, and that
when overlooked might lead to erroneous physical conclusions.
For most practical applications, a capacitor must be insulating to direct current (DC);
transmission of electrons via non-zero conductivity and/or direct tunneling (leakage) is
generally an undesirable source of heating and power consumption. At the quantum
mechanical level, the insulating properties of a capacitor are guaranteed by the presence
of a dielectric film with a finite band gap at the Fermi level, where propagation of the
metallic conduction electrons is forbidden. In the language of semiconductor physics, we
can alternatively say that both Schottky barrier heights (SBH), respectively φn and φp
for electrons and holes, need to be positive for the device to behave as a capacitor [see
Fig. 3.1(a)]. (By convention we assume that, if the Fermi level of the metal lies in the
gap, both φn and φp are positive.)
If, on the contrary, either φp or φn is negative, injection of holes or electrons into the
dielectric becomes energetically favorable and the device behaves instead as an Ohmic
contact. Most importantly, at such a junction there is necessarily (at thermodynamic
47
48 Chapter 3. Band alignment issues in the ab initio simulations of FE capacitors
(a)
φn
φp
EF
(d)
(b)
(c)
(e)
(f)
EC
EV
(g)
Figure 3.1: Schematic representation of the common band-alignment scenarios that can
be found in short-circuited ferroelectric capacitors. Left panels (a,d,g) refer to unpolarized devices (this configuration is accessible experimentally by increasing the temperature above the Curie point, or in a first-principles calculation by imposing a mirror
symmetry plane). Central panels (b,e) refer to thin-film capacitors in a polarized monodomain state (note the linear slope of the bands that is due to the depolarizing field).
Right panels (c,f) refer to macroscopically thick capacitors where the depolarizing field
is negligible (although there is a potential drop across the capacitor, the dotted lines
in these panels) and the polarization converges to the bulk spontaneous value. The
first row (a-c) pictures a hypothetical well-behaved system, where the Schottky barrier
heights are positive at any polarization state. The second row (d-f) refers to a situation where the paraelectric configuration (d) is well-behaved, but the band alignment
becomes pathological once the system goes ferroelectric (e-f). The third row (g) shows
a capacitor that is pathological already in the paraelectric reference structure; in such a
case, it is meaningless to analyze the ferroelectric state as the system is metallic throughout its thickness. EV and EC stand, respectively, for the valence band maximum and
the conduction band minimum of the insulator. EF is the Fermi level of the metal. φn
and φp represent, respectively, the n-type and p-type Schottky barriers.
3.1. Introduction
49
equilibrium) a spill-out of charge from the metal to the insulator, as the system reequilibrates the chemical potential of the free carriers on either side. Such intrinsic space
charge induces metallicity (by intrinsic doping) in the dielectric film, and overall profoundly alters the electronic and structural properties of the interface.
While in principle the charge spillage might be a real physical feature of a given
system, there are several arguments that advise caution in the interpretation of DFT
calculations where this effect is found. As we indicated in Sec. 2.4.1, the main limitation
of the most widely used approximation to the exchange-correlation functional, namely
the LDA and GGA, generally produce a severe underestimation of the fundamental band
gap in insulators. This error directly propagates to the values of φp and φn that are
extracted from a DFT calculation. In the best scenario, DFT yields values of φp and
φn that are still positive. In such a case, it is commonly assumed that the interfacial
charge density (which is a ground-state property) and, consequently, the associated
dipole moment are well described by the theory, even if both φp and φn (an excitedstate property) are significantly smaller than the experimentally measured Schottky
barrier heights. Then, a relatively simple quasiparticle correction to the band edges of
the insulator is commonly used to correct φp and φn , and obtain a reliable estimate of
the Schottky barrier heights (SBH) [101].
There are situations, however, where the DFT simulation yields a negative value
of either φp or φn [see Fig.3.1(g) for the case where φn is negative in an unpolarizaed
capacitor]. In such cases, as we mentioned before, there is a spill-out of charge carriers
that populate (or deplete) the bands of the insulator. Obviously, the amount of spilledout charge directly depends on the DFT values of φp and φn (the more negative the
SBH, the larger the number of states of the insulator that cross the Fermi level). While
it is widely known that electronic excited states are not necessarily well described within
the usual exchange-correlation approximations, the possible influence of these on some
ground-state magnitudes is usually overlooked. In this case, negative values of φp and
φn , which are likely to be an artifact of LDA or GGA, can directly affect the ground-state
charge density, and potentially produce a number of unphysical features in the relaxed
electronic and atomic structure of the capacitor. As we shall see in the following, in
such a pathological regime the interface dipole is poorly described by the theory, and
the physical properties of the dielectric film are strongly altered by the presence of space
charge in the system. It is therefore crucial to clearly identify whether this scenario
applies to a given interface, by performing an accurate analysis of the band alignment
and of the distribution of the metallic carriers in the relaxed structure.
Such an analysis is not entirely straightforward as the distinction between “nonpathological” and “patological” band alignment can be subtle in a DFT calculation. In
capacitors with thicknesses of a few nanometers, quantum phenomena that can be disregarded in Schottky barrier measurements of macroscopic capacitors must be considered.
The quantum nature of the electrons, for instance, implies that the metallic wave functions do penetrate in the insulator up to a distance from the interface, even in cases where
both φp and φn are well positive. The evanescent tails of these Bloch states, which are
exponentially damped with increasing distance from the interface, are commonly known
as “metal-induced gap states” (MIGS) [102]. Their presence is normal and should not
raise any concern when detected in a calculation. In order to identify a pathological case
it is therefore crucial to assess the physical nature of the charged carriers that diffuse into
the dielectric film, i.e. whether they originate from MIGS (non-pathological) or from
population/depletion of the bands of the insulator (pathological). A clear picture only
emerges once one combines several postprocessing tools, as we shall explain in detail in
the remainder of this Chapter. Properties of the physically meaningful charge transfers
due to the MIGS are explored in Chapter 4 of this manuscript.
The previous issues are general and apply not only to the Schottky barrier heights
at metal/oxide junctions but to the band offsets at semiconductor/semiconductor and
semiconductor/metal interfaces as well. However, the case of metal/perovskite oxide
interfaces deserves special attention, as here the band-alignment issue appears to be
particularly serious. Indeed, in most of these compounds the bottom of the conduction
band and the top of the valence band have a marked localized orbital character (respectively, cation d and oxygen p), which makes the effects of the self-interaction error [103]
particularly pernicious. Very often, it is so strong that the band lineup calculated from
first-principles is pathological [e.g. the conduction band of the insulator is near or below
the Fermi level, as schematically represented in Fig. 3.1(g)], even in an unpolarized
centrosymmetric structure.
Furthermore, many perovskites are ferroelectric, and this introduces additional complications that directly affect the band alignment at the electrode interface. One of the
most delicate issues is that the band offset generally has a strong dependence on the
macroscopic electric displacement field (i.e. on the polarization of the insulator) [52].
As soon as the system becomes polarized (spontaneously in a ferroelectric or under the
application of an external field), the imperfect screening at the electrode interface (which
can be quantified, as dicussed in Sec. 1.3.2, by means of an interfacial capacitance [50]
or, equivalently, by an effective screening length [21]) produces a potential drop [50] that
is roughly linear with the magnitude of the polarization [53], and modifies the lineup
between the bands of the insulator and the Fermi level of the metal. This phenomenon is
central to the physics of ferroelectric capacitors, and has important implications for the
stability of a monodomain polar state [21], and for devices based on the tunneling electroresistance effect [104]. In a well-behaved system, the offsets between the metal Fermi
level and the band edges of the insulator in close proximity of the interface stay positive
upon condensation of the ferroelectric instability [Fig. 3.1(b)], even for a macroscopically
thick capacitor where the depolarizing field is negligible and the polarization attains the
bulk spontaneous value [Fig. 3.1(c)]. However, a large polarization, combined with a
relatively small interface capacitance, can lead to changes in the Schottky barriers that
are as large as few electron volts and therefore, to negative values of φn or φp [Fig.
3.1(e-f)] even if they are both positive in the centrosymmetric reference structure [Fig.
3.1(d)].
Throughout this Chapter we revise usual methods for the computation of Schottky
barriers in order to test their validity and limits in the case of ferroelectric nanocapacitors. We shall make clear distinction between two qualitatively regimes, corresponding
3.2. General theory of the band offset
51
to (i) that of a normal Schottky alignment and (ii) that of a pathological Ohmic junction. We demonstrate the artifacts typically associated with (ii) by performing extensive
calculations of technologically relevant ferroelectric/metal interfaces. We suggest clear
criteria to avoid ambiguities in the determination of Schottky barriers and provide recipes
for the analysis of results suspicious of being pathological. We discuss the relevant literature works, pointing the attention to those where our results suggest a revision of the
currently accepted interpretation. Finally, we discuss a number of viable methodological
perspectives to overcome the limitations of DFT illustrated in this work.
The research appearing in this Chapter, that has also been published in Ref. [105],
was carried out in collaboration with Massimiliano Stengel and Nicola Spaldin. Although
for many of the discussions that follow it is difficult to disentangle our original contribution to the work, some parts are clearly due to M. Stengel. We keep those contributions
here for the sake of completeness and clarity. However, we will try to mark them clearly
all along the Chapter.
3.2
General theory of the band offset
The theory of Schottky barriers in metal/semiconductor interfaces is today well established after the huge amount of works devoted to these systems over the last half century.
However, the physics governing the band alignment in ferroelectric capacitors significantly departs from the well-established concepts of semiconductor physics. Recently,
a theory that intends to account for the particularities that arise when the dielectric
material is substituted with a ferroelectric was developed by M. Stengel [105]. Details
of this theory will be essential for the analysis of the results reported in Sec. 3.4 and 3.5
so we will make a thorough review of its fundamental points.
3.2.1
Schottky barriers at metal/insulator interfaces
The Schottky barrier, a rectifying barrier for electrical conduction across a metal/insulator
junction, is of vital importance for the operation of any modern electronic device. For
the case of an n-type semiconductor, the Schottky barrier height is the energy difference between the conduction band minimum (EC ) and the Fermi level (EF ) across the
interface, and we indicate it as φn .
The nature of the microscopic mechanisms governing the magnitude of φn has troubled scientists for several decades. In spite of the ongoing debates, it seems to be widely
accepted now that, while bulk materials properties play certainly a substantial role, φn
is best understood as a genuine interface property (further discussions on this point can
be found in Chapter 4). As a consequence Schottky barriers can not be obtained from
a mere comparison the bulk band structure of the two constituting materials, but the
explicit role of the interface must be taken into account. The charge rearrangement
due to chemical bonding at the interface produces an interface dipole, which translates
into a step in the electrostatic potential that acts as the origin of energies shifting the
eigenvalues of the materials and uniquely determines the band alignment at the interface
φn
EF
metal
<VH
φp
EC
EV
>
∆<V>
dielectric
<VH
>
Figure 3.2: Schematic representation of the band offset at a metal/insulator junction,
illustrating the main quantities discussed in the text.
(see Fig. 3.2). The interface dipole is, of course, a function of the constituent materials
and the termination and orientation of the interface. For this reason the simulation of
the actual interface is required for the characterization of Schottky barriers in a given
system.
The electrostatic Hartree potential at the interface between two semi-infinite solids,
Z
VH (r) =
ρ(r 0 ) 3 0
d r,
|r − r 0 |
(3.1)
where ρ(r) is the total charge density (including electrons and nuclei), is a rapidly varying
function of the position, reflecting the underlying atomic structure. In order to filter out
the large oscillations and preserve only those features that are relevant on a macroscopic
scale, it is convenient to apply an averaging procedure [106, 107]. This consists in
(i) performing a global average of VH (r) over planes parallel to the interface, and (ii)
convoluting the resulting one-dimensional function with a Fourier filter to suppress the
high spatial frequency components. (See Ref. [108] for a detailed description of the
method, and Ref. [109] for an extensive review of its applications to Schottky barriers
calculations.) After this “nanosmoothing” [108] procedure, and in the absence of a
depolarizing field, the doubly-averaged V H (z) reduces to a step function, from which we
can extract the electrostatic lineup term [106, 107],
∆hV i = hVHdielectric i − hVHmetal i,
(3.2)
which includes all the physics of the interface dipole formation. [hVHdielectric i and hVHmetal i
are the asymptotic values of V H (z) far from the interface.] To determine the band offsets
from ∆hV i it is then necessary to know how the bulk energy bands of the insulator and
53
the Fermi level of the metal are related to their respective average electrostatic potential.
In full generality, one can write
φp = −EV + EF − ∆hV i,
φn = EC − EF + ∆hV i.
(3.3a)
(3.3b)
EV , EC and EF are usually referred to as the band structure term [106, 107], and are
bulk properties of the two materials. They are defined as the energy positions of the
valence (EV ) and conduction (EC ) band edges of the insulator, and the Fermi level of
the metal (EF ), all referred to the average hVH i in the respective bulk (see Fig. 3.2).
In Sec. 3.3 we provide further details of the standard computational procedures used
to calculate these quantities in practice. In the remainder of Sec. 3.2 we discuss how
the above theory needs to be revised and extended in the case of metal/ferroelectric
interfaces. This discussion is mostly due to M. Stengel, to whom we acknowledge the
permission to reproduce it here.
3.2.2
Theory of Schottky barriers in ferroelectric capacitors
Ferroelectric materials entail a new degree of freedom, the macroscopic polarization P ,
which is absent in the semiconductor case. It is natural then to expect that the above
picture of the band offset at metal/insulator interfaces may need to be extended to take
this new variable into account. In the following, we discuss how P affects both the lineup
and the band-structure terms in Eqs. (3.3a) and (3.3b).
Lineup term
We represent a simple ferroelectric material as a non-linear dielectric. We have seen in
Sec. 1.3.2 that in bulk, a ferroelectric is characterized by an internal energy Ub per unit
cell of the form
Ub (D) = A0 + A2 D2 + A4 D4 + O(D6 ),
(3.4)
where D is the electric displacement field, A0 is an arbitrary reference energy, A2 is
negative and the higher expansion coefficients are positive. (As we are concerned with
the essentially one-dimensional case of a parallel-plate capacitor, we can replace all
the vector interfacial quantities of interest by one component only, that corresponds to
the direction along which the interface is oriented.) The A0,2,4,... coefficients implicitly
contain all the complexity of the microscopic physics, and can be calculated from first
principles using the methods of Ref. [51]. It follows from elementary electrostatics [51]
that the internal electric field, E(D), is the derivative of U (D),
E(D) =
where Ω is the cell volume.
1 dU (D)
,
Ω dD
(3.5)
P
D
ε 0 λ eff
−V(z)
M
FE
λ eff
M
t
λ eff
Figure 3.3: Schematic representation of a symmetric short-circuited ferroelectric capacitor in a polarized configuration within the imperfect-screening model. t is the thickness
of the ferroelectric film. M and FE represent, respectively, the metal electrode and the
ferroelectric film. Both materials are assumed to be separated by a vacuum layer of
thickness λeff . Full line represent the electrostatic energy felt by the electrons, i.e. the
electrostatic potential times the electron charge.
The electrostatics of a parallel-plate capacitor configuration, sketched in Fig. 3.3,
was discussed in Sec. 1.3.2, where it was shown that within the “finite screening length”
model, the N -layer thick ferroelectric film can be thought as separated from the ideal
metal electrode by a thin layer of vacuum, of thickness λeff [50, 47, 53]. As a consequence of the imperfect screening, electric fields develop inside the ferroelectric and the
interfacial regions. In Sec. 1.3.2 we found that within this model, the energy of the
N -layer thick ferroelectric film can be written as
UN (D) = N Ub (D) + 2Sλeff
D2
,
2ε0
(3.6)
where D is the electric diplacement field, that must be preserved at the interface between
the ferroelectric and the vacuum layer, and S is the surface cell area. [Note that two
symmetric electrodes of equal λeff are considered in Eq. (3.6).]
The second important consequence of a non-zero λeff is that the lineup term, Eq. (3.2),
now linearly depends on the external parameter D, due to the additional potential drop
at the interface, that can be computed as the product of the electric field within the
vacuum layer times its width,
∆hV i(D) = ∆hV i(0) + λeff
D
.
ε0
(3.7)
[It is worth noting that, whenever Eb (D) 6= 0, the linear variation of V H (z) introduces
an arbitrariness in the determination of ∆hV i(D). Techniques to deal with these issues
in practical calculations are described in Ref. [52].]
55
EC
2λeff DS /ε0
EF
11111111111111111
00000000000000000
00000000000000000
11111111111111111
EV
00000000000000000
11111111111111111
00000000000000000
11111111111111111
00000000000000000
11111111111111111
00000000000000000
11111111111111111
00000000000000000
11111111111111111
00000000000000000
11111111111111111
00000000000000000
11111111111111111
00000000000000000
11111111111111111
00000000000000000
11111111111111111
00000000000000000
11111111111111111
00000000000000000
11111111111111111
00000000000000000
11111111111111111
00000000000000000
11111111111111111
00000000000000000
11111111111111111
00000000000000000
11111111111111111
00000000000000000
11111111111111111
00000000000000000
11111111111111111
00000000000000000
11111111111111111
00000000000000000
11111111111111111
00000000000000000
11111111111111111
Polarization
reversal
111111111111111111
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111111111111111111
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111111111111111111
EC
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EV
000000000000000000
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111111111111111111
∆ <V>(−DS)
−V(z)
2λeff DS /ε0
∆ <V>(D S)
−DS
DS
Figure 3.4: Schematic representation of the band alignment change induced by polarization reversal in a ferroelectric capacitor. Here we assume a ferroelectric layer of infinite
thickness (t → ∞), so the depolarizing field vanishes and therefore the polarization attains for the bulk spontaneous value, PS . Since the field is zero, D = PS . Also, because
of the vanishing field in the ferroelectric we can represent the macroscopic potential as
flat. In the scheme we assume that the band-alignment is not pathological when the polarization points to the left. But, upon polarization reversal, that change in the potential
lineup is larger than the fundamental band-gap and the system becomes pathological.
Table 3.1: Estimation of the change in the lineup term ∆φ of typical ferroelectric
capacitors upon polarization reversal. DS is the bulk spontaneous polarization of the
ferroelectric material. λeff were calculated in Ref. [47] for capacitors with SrRuO3
electrodes.
BaTiO3
PbTiO3
DS (C/m2 )
0.39
0.75
λeff (Å)
0.20
0.15
∆φ (V)
1.8
2.6
Non polar PbTiO3
Polar PbTiO3
5
4
3
Energy (eV)
2
1
Eg = 1.45 eV
Eg = 1.35 eV
0
-1
-2
-3
-4
-5
Γ
Z
R
A
Γ
X
Γ
Z
R
A
Γ
X
Figure 3.5: Band structure of centrosymmetric and polar tetragonal bulk PbTiO3 obtained within the Siesta code and the computational parameters described in Sec. 3.3.4.
In-plane lattice constant is constrained to the theoretical one of SrTiO3 (3.874 Å). The
polar structure is fully relaxed both the atomic coordinates and out-of-plane cell vector.
To give a more quantitative flavor of the impact of this D-dependence in real systems, we can use the values of λeff reported in the literature for PbTiO3 /SrRuO3 and
BaTiO3 /SrRuO3 capacitors. As is represented squematically in Fig. 3.4, upon polarization reversal, the interface lineup term ∆hV i will undergo a variation corresponding
to
DS
∆φ = ∆hV i(DS ) − ∆hV i(−DS ) = 2λeff
,
(3.8)
ε0
where DS is the spontaneous polarization of the ferroelectric material (in the spontaneous
configuration the internal electric field within the ferroelectric vanishes and DS equals
the spontaneous polarization). The values reported in Table 3.1 indicate that this effect
can be rather large, of the order of 2 – 3 eV. Those values are larger than the fundamental
LDA band gaps of BaTiO3 and PbTiO3 (1.58 eV and 1.35 eV respectively). Therefore,
even if the band alignment is non-pathological in the paraelectric configuration, in both
cases it will indeed become pathological once the ferroelectric instability is fully relaxed.
Band-structure term
The polar displacements in the ferroelectric film modify not only the lineup term, but
also the bulk band-structure term. This is most easily understood by recalling the role
played by covalent bonding in the ferroelectric instability of perovskite titanates. This
implies that the polar distortions significantly modify both the conduction and valence
band structure. For example, in both BaTiO3 and PbTiO3 the fundamental gap typically
increases when going from the centrosymmetric structure to the polar tetragonal phase
(see Fig. 3.5 for the case of PbTiO3 ). Using the arguments of Ref. [52], we can think of
a continuous dependence of both EV and EC , respectively in Eq. (3.3a) and Eq. (3.3b),
57
on the electric displacement D. The Fermi level EF , of course, remains fixed as the
electric displacement does not affect the bulk of the metallic electrode. In summary, the
general expression for the n-type Schottky barrier at a metal/ferroelectric interface (an
analogous expression follows for the p-type one) is
φn (D) = EC (D) − EF + ∆hV i(D),
(3.9)
where at the lowest order EC is quadratic in D (the linear order is forbidden by symmetry), and in most cases of interest ∆hV i(D) can be approximated by a linear function
as in Eq. (3.7). In the following, we shall elaborate on this expression and identify a
new, qualitatively different regime, with important implications for the physics of the
interface.
3.2.3
Ferroelectric capacitors in a pathological regime
Equation (3.9) implies that φn (D) might become negative for some values of D. From the
point of view of first-principles calculations, already by looking at the values of Table 3.1
we can be reasonably sure that this will happen at the PbTiO3 /SrRuO3 interface: 2.6
eV is already larger than the LDA gap of PbTiO3 in the ferroelectric phase (∼1.45 eV).
This possibility has been almost systematically overlooked in the literature. As this is a
central point of this work, we shall illustrate in detail the consequences of such a regime,
and explain why we regard it as “pathological”. We discuss in the following two possible
occurrences of this scenario: (i) φn is negative already in the paraelectric configuration at
D = 0 and (ii) φn is positive at D = 0 but becomes negative at some value of |D| < DS .
The centrosymmetric case
We start with a capacitor in the reference paraelectric structure with two symmetric
electrodes, and we hypothesize that, for whatever physical reason, the interface dipole
that forms between the metal and the film leads to a negative φn . (Similar arguments
apply to the case, not explicitly discussed here, of a negative φp .) As the quantum states
of the conduction band of the film lie at lower energy than the Fermi level of the metal,
the former will be filled up to EF , leading to a nonzero free charge density, ρfree , in the
film. Neglecting quantum confinement effects, sufficiently far from the interfaces we can
approximate the density of states of the insulating material in the capacitor with its
bulk density of states ρb (E). Within this approximation, ρfree is exactly given in terms
of φn ,
Z
e EC −eφn
ρfree = −
ρb (E)dE.
(3.10)
Ω EC
Notice that in the pathological regime, φn < 0 and EC − eφn > EC . This additional
charge density, superimposed to an otherwise charge-neutral insulating film, will produce a strong electrostatic perturbation in the system. For example, if such a charge
rearrangement occurred in vacuum, the Poisson equation
d2 V (z)
ρfree
=−
,
dz 2
ε0
(3.11)
would imply a parabolic potential of the form
V (z) = −
ρfree 2
z .
2ε0
(3.12)
(We assume that z = 0 corresponds to the center of the ferroelectric film.) Throughout
this work, we shall assume that the interface is oriented along the z axis, and each material is periodic in the plane parallel to the interface, referred to as the (x, y) plane. As
typical ferroelectric materials are exceptionally good dielectrics, in a first approximation
we can assume that V (z) will be perfectly screened by the polar displacements of the
lattice. However, this does not mean that electrostatics has no consequences – quite the
contrary. Macroscopic Maxwell equations in materials indeed dictate that
dD(z)
= ρfree .
dz
(3.13)
Hence, if we assume perfect bulk screening, we have E(z) = 0, D(z) = ε0 E(z) + P (z) =
P (z) and, after integrating Eq. (3.13), P (z) = ρfree z. So, since the sign of the electronic
charge and ρfree is negative within our convention, we have a non-uniform and linearly
decreasing polarization in the ferroelectric film [see Fig. 3.6(d)]. This means that, at the
film boundaries (z = ±t/2, where t is the thickness), the local electric displacement has
now opposite values, proportional to the total amount of free charge that was transferred,
t
t
D(− ) = − ρfree ,
2
2
t
t
D( ) = ρfree .
2
2
(3.14)
Of course, the band offset at the interface depends on the local value of D in the film
region adjacent to the interface, so φn will be consequently shifted in energy according
to Eq. (3.9). We can expect that for small D values the (quadratic) polarization effects
on the band structure will be less important than the (linear) dependence of the lineup
term on D. (Note that the presence of additional charge in the conduction band might
also alter the bandstructure term, e.g. through on-site Coulomb repulsions or other
exchange and correlation effects; in the limit of weak correlations we expect these to be
even smaller and essentially irrelevant for this discussion.) Therefore, we approximate
Eq. (3.9) with Eq. (3.7), and write
φn = φ0n +
λeff D
tλeff ρfree
= φ0n −
.
ε0
2ε0
(3.15)
[The minus sign comes from the fact that at the z < 0 interface, which is the one for
which Eq. (3.7) is valid within our conventions, D is positive.] In turn, the new φn will
modify ρfree through Eq. (3.10). For some value of φn , Eq. (3.10) and Eq. (3.15) will be
mutually self-consistent and the system will reach electrostatic equilibrium. This can be
expressed through an integral equation where we have eliminated ρfree ,
Z
e EC −eφn
2ε0 (φn − φ0n )
ρb (E)dE =
.
(3.16)
Ω EC
tλeff
59
(a)
(b)
EF
φ 0n
EF
φn
CBM
(c)
(d)
D
−t/2
0
D
t/2
z
z
Figure 3.6: Schematic representation of the effect of free-charge redistribution onto the
band diagram of a paraelectric capacitor with a negative φn . (a) band alignment under
perfect interface screening (i. e. when ρfree vanishes), and (b) after charge spill-out
and electrostatic reequilibration. The corresponding profile of the electric displacement
field within the ferroelectric films are displayed in panels (c) and (d), obtained after
integrating Eq. (3.13).
To qualitatively appreciate the physical implications of this expression, we can explicitly
solve it by using a constant ρb (E) = α. (Note that this assumption is not completely
unrealistic as the t2g bands forming the bottom of the conduction band in many ferroelectric perovskites have a marked two-dimensional character – recall that the density
of states of a free electron band in two dimensions is a step function.) This leads to
φn − φ0n
e2 tλeff α
,
=−
φn
2ε0 Ω
(3.17)
and with a few rearrangements to
φn =
φ0n
,
Ctλeff α̃ + 1
(3.18)
where C = e2 /2ε0 is a constant, and α̃ = α/Ω is the density of states per unit energy and
volume of the bulk. In spite of the drastic simplifications, Eq. (3.18) already contains
most of the relevant ingredients for our analysis. A few notable ones are missing –
we shall come back to those below. Before going into more detailed considerations,
however, it is important to spell out the direct implications of Eq. (3.18), which we shall
be concerned with in the following.
First, note that all quantities appearing at the denominator at the right-hand side
of Eq. (3.18) are positive. Since we are assuming a pathological regime (φ0n < 0), this
means that φn will be negative, and will satisfy φ0n < φn < 0. The lower limit corresponds
to the perfect interface screening case: λeff = 0 and changes in D in the ferroelectric
are perfectly screened in the interface. The upper limit corresponds to no screening:
λeff → ∞ and bound charges at the interface due to changes in D directly affects
the interfacial dipole that determine the band-offset. The situation is schematically
represented in Fig. 3.6(a) and Fig. 3.6(b). Given a negative φ0n [Fig. 3.6(a)], the charge
redistribution will induce an upward energy shift of the conduction band minimum,
bringing φn closer to the Fermi level [Fig. 3.6(b)]. Second, in the limit of t → ∞
(infinite thickness) φn will tend to zero from below as φn ∝ −1/t. This means that
the self-consistent band offset φn is not determined by the local physical properties of
the junction, i.e. it is no longer an interface property – the spilled-out charge will
redistribute over the whole film thickness as t is varied. Third, the density of states of
the conduction band, represented in Eq. (3.18) by the parameter α, will also affect the
value of φn : the larger α, the strongest the reduction in φn upon charge spill-out and
electrostatic reequilibration. (To avoid confusion, note that in the above paragraphs,
we used the word “screening” in two different contexts. By “perfect bulk screening” we
mean Eb (D) = 0. By “perfect interface screening” we mean λeff = 0.)
We can attempt a semiquantitative assessment of Eq. (3.18) in a representative capacitor of thickness t = 50 Å (comparable to those that are typically simulated within
DFT). In atomic units, we use λeff = 0.3 (of the order of the values reported in Table 3.1),
C = 2π, and α̃ = 0.05 (appropriate for the conduction band of SrTiO3 , a prototypical
perovskite material). We obtain
φ0
φn ∼ n .
(3.19)
10
This implies that the effect is quite strong – even if φ0n is a rather large negative value (e.g.
of the order of -1 eV), in most practical cases the conduction charge redistribution will
reduce it to a value that lies just below the Fermi level. Most importantly, this implies
that, when φ0n < 0, the physical parameters, φ0n and λeff , governing the band offset at
the interface are neither accessible in a simulation, nor are they directly measurable in
an experiment – only φn might be. Note, however, that the “self-consistent” φn value
is generally not a well-defined physical quantity – this is only true within the many
approximations used in the above derivations. In particular, we have neglected bandbending effects: in general the electrostatic potential will be non-uniform in the film (see
rubric Towards a quantum model below) and φn will be a function of the distance from
the interface. But even if we put this caveat aside for a moment, the reader should keep
in mind that φn is determined here by space-charge effects through several independent
contributions. Furthermore, the film is no longer insulating but becomes a metal. This is
a substantial, qualitative departure from the physical concepts that were developed in the
context of semiconductor/metal interfaces, and that led to the consensus understanding
of φn as a genuine interface property.
Given this situation, one needs to revisit the very foundations of the methodological
61
(a)
(b)
P
CBM
EF
EF
M
FE
M
M
FE
M
Figure 3.7: (a) Paraelectric capacitor with a Schottky-like band alignment in the paraelectric structure. (b) When the polar instability sets in, the band alignment becomes
pathological, the conduction band is locally populated (red shaded area) and the film
becomes partially metallic (light shaded area bounded by the dashed line).
ab-initio approaches that have been used with great success in the past to compute Schottky barriers heights. This success has critically relied on a key observation: the interface
dipole, that one identifies with the lineup term Eq. (3.2) is a ground-state property,
i.e. is not directly affected by the well-known limitations of the Kohn-Sham eigenvalue
spectrum. This would be excellent news: one could efficiently (and accurately) calculate
∆hV i within DFT, and combine it with a band-structure term (EV or EC ) calculated
at a higher level of theory (e.g. GW); within this formally sound procedure, theoretical
calculations have shown remarkable agreement with the experimental observations in
the past. In the spill-out regime (i.e. φ0n < 0) described in this Section the above key
observation no longer holds – the erroneous DFT value of φ0n plays a direct and dominant
role in the interface dipole formation, as it is apparent from Eq. (3.18). Furthermore,
as φ0n is systematically underestimated within LDA or GGA, there is the concrete possibility that the spill-out regime itself (φ0n < 0) might be an artifact of the band-gap
problem. Thus, the ground-state properties of the system found in a simulation might
be qualitatively wrong due to this issue, in loose analogy to, e.g., the erroneous LDA
prediction of metallicity in many transition metal compounds. It goes without saying
that the results of a simulation where significant spill-out of charge is found because of
the mechanism described in this Section should be regarded with great suspicion.
The broken-symmetry case
Even if the band alignment is Schottky-like in the reference paraelectric structure of
the capacitor, Eq. (3.9) entails the possibility that it might become pathological in the
ferroelectric regime (i.e. when a polar instability is allowed to fully relax). Unfortunately,
for this case many of the simplifying assumptions used above are no longer valid, and
for a detailed description one would need to take into account the more refined physical
ingredients discussed in the next rubric. At the qualitative level, however, we can already
draw some important conclusions, as we shall briefly illustrate in the following.
Equation (3.9) predicts that, if φ0n is positive and the capacitor is compositionally
symmetric [as in Fig. 3.7(a)], at finite D at most one of the two opposite interfaces
will have a negative φn . This implies that only part of the ferroelectric film, i.e. the
region adjacent to this “pathological” interface, will become metallic, while the rest
of the film will stay insulating [Fig. 3.7(b)]. (To understand this point, note that at
difference with the previous case one has now a finite “depolarizing” electric field in
the insulating part of the capacitor. This wedge-like potential will keep the conduction
electrons electrostatically confined to the pathological side.) In the insulating region,
the polarization will be macroscopically constant, as in a well-behaved capacitor [recall
Eq. (3.13)]. According to the same Eq. (3.13), D(z) [and hence P (z)] will be nonhomogeneous, with a negative slope, in the metallic region. In this context it is worth
pointing out an important physical consequence of such a peculiar electronic ground
state. This concerns the response of the capacitor to an applied bias potential. In wellbehaved cases, the polarization of the capacitor will respond uniformly to a bias, i.e.
all the perovskite cells up to the electrode interface will undergo roughly the same polar
distortion. In the present “ferroelectric-pathological” regime, part of the ferroelectric
film has become metallic, i.e. the metal/insulator interface has moved to a place that lies
somewhere in the film. This means that, if one tries to switch the device with a potential,
the electric field won’t affect the dipoles that lie closest to the pathological interface –
they will be screened by the spilled-out free charge. A consequence is that the dipoles
near a pathological interface will appear as if they were pinned to a fixed distortion, that
is almost unsensitive to the electrical boundary conditions. This pinning phenomenon
has been studied in earlier theoretical works, and was ascribed to chemical bonding
effects. In Sec. 3.5 we shall substantiate with practical examples that “dipole pinning”
is instead a direct consequence of the problematic band-alignment regime described here.
In Sec. 3.6 we shall come back to this point and put it in the context of the relevant
literature.
Towards a quantum model
In order to draw a closer connection between the semiclassical arguments of the previous sections and the quantum-mechanical results that we present in Sec. 3.4 and 3.5, we
briefly discuss here how to improve our physical understanding of the charge spill-out
process by lifting some of the simplifying approximations used so far. As a detailed treatment goes beyond the scopes of the present work, we shall limit ourselves to qualitative
considerations.
The most drastic approximation of our model appears to be the assumption of perfect dielectric screening within the ferroelectric material, where the spill-out charge is
perfectly compensated by the polar displacements of the lattice. This implies that the
electric field in the film vanishes, and the excess conduction charge can spread itself spatially at essentially no cost. In this scenario, the macroscopically uniform distribution
of ρfree postulated in Sec. 3.2.3 appears very reasonable. In reality, the internal E field
in the bulk ferroelectric material does not vanish, but is a non-linear function of D, that
can be written by combining Eq. (3.4) and Eq. (3.5),
Eb (D) ∼
1
2A2 D + 4A4 D3 .
Ω
(3.20)
63
(a)
(b)
CBM
EF
M
CBM
EF
FE
M
M
Dielectric
M
Figure 3.8: Schematic representation of the effects of dielectric nonlinearity onto the
band diagram of a centrosymmetric capacitor. The effective potential felt by the conduction electrons is −V H (z). (a): Ferroelectric material (in a non-polar configuration).
(b): Dielectric material.
Of course, solving for the self-consistent ρfree (z) in a non-linear medium would require a
numerical treatment. Still, we can gain some insight about qualitative trends by starting,
for example, from the linearly decreasing P (z) found in the centrosymmetric case. Even
though we are treating now a non vanishing electric field, the large dielectric constant
of ferroelectric materials makes reasonable to approximate D(z) ∼ P (z) = ρfree z. Using
Eq. (3.20) we can write
E(z) =
1
2A2 ρfree z + 4A4 ρ3free z 3 .
Ω
(3.21)
Since
E(z) = −
dV H (z)
,
dz
(3.22)
The electrostatic potential is then given by integrating E(z) from the interface to a
position z within the ferroelectric
V H (z) = −
Z
0
z
E(z 0 )dz 0 = −
A2 ρfree z 2 A4 ρ3free z 4
−
.
Ω
Ω
(3.23)
[Here we are assuming the interface to be located at z = 0, using the previous reference
position of the interface at −t/2 only adds a constant to Eq. (3.23) and does not affect
this discussion.] Using again D(z) ∼ ρfree z, we get
V H (z) ∼ −
A2 D2 (z) A4 D4 (z)
−
,
Ωρfree
Ωρfree
(3.24)
which, comparing with Eq. (3.4), essentially leads to
V H (z) = Ub [D(z)]/Q0 ,
(3.25)
where Ub is the internal electrostatic energy of the bulk ferroelectric, and Q0 = −Ωρfree is
a (positive) constant of the dimension of a charge. This means that the spatial variation
in V H (z) reflects the energy landscape of the bulk material: V H (z) will be a double-well
potential in a ferroelectric material (A2 < 0), and a single-well potential in a dielectric
material (A2 > 0). Remarkably, the double-well potential accounts for the possibility of
free-charge accumulation in the middle of the centrosymmetric film [Fig. 3.8(a), notice
that the band deformation is given by −eV H (z), and thus it displays the shape of an
inverted double well]. This excess of negative charge in the middle of capacitor would
help to stabilize even further the head-to-head domain wall in the polarization P (z).
Conversely, for a paraelectric material one would expect the free charge to be loosely
bound to the interface, and have a minimum in the middle of the film [Fig. 3.8(b)]. Of
course, these considerations are valid for a centrosymmetric capacitor, and are presented
just to give the reader an idea of the complications involved in the analysis of this
phenomenon.
A second important approximation is the neglect of (i) quantum confinement effects beyond the simple Thomas-Fermi filling of the bulk-like density of states and (ii)
the band-structure changes due to the polar distortions, which we briefly mentioned in
Sec. 3.2.2. These will further modify the equilibrium distribution of the free charge,
and we expect them to be important to gain a truly microscopic understanding of the
system, although not essential for the main scopes of this work. Remarkably, a promising model taking all these ingredients into account (dielectric non-linearity and bandstructure effects) was recently proposed in the context of the (at first sight unrelated)
LaAlO3 /SrTiO3 interface. [110] This indicates that the physics of a ferroelectric capacitor in the pathological band-alignment regime described here is essentially analogous to
that of the “electronic reconstruction” [111] in oxide superlattices.
3.2.4
Implications for the analysis of the ab-initio results
The above derivations show that there are two qualitatively dissimilar regimes in the
physics of a metal/insulator interface, Ohmic-like and Schottky-like. During the derivation, we have evidenced some distinct physical features that we expect to be intimately
associated to the “pathological” Ohmic case. As these are of central importance to help
distinguish one scenario from the other, we shall briefly summarize them in the following, mentioning also how each of these “alarm flags” can be detected in a first-principles
simulation.
First, even after the electron reequilibration takes place, the band edges cross the
Fermi level of the metal, i.e. the apparent Schottky barrier is negative. Therefore, the
analysis of the local electronic structure and of the Schottky barriers appears to be the
primary tool to identify a pathological case. However, as the “self-consistent” φn tends
to stay very close to the Fermi level, this analysis should be performed with unusual
accuracy – techniques to do this will be discussed in Sec. 3.3.1.
Second, the presence of a substantial density of free charge populating the conduction
band of the insulator is another important consequence of the pathological regime. In
65
3.3. Methods
Sec. 3.3.2 we illustrate how to rigorously define ρfree in a ferroelectric heterostructure.
Finally, a remarkable consequence of charge spill out is the presence of an inhomogeneous polarization in the system. Note that this feature has been ascribed in earlier
works to phenomena of completely different physical origin. We shall devote special attention in Sections 3.4 and 3.5 to demonstrating the intimate relationship between ρfree
and spatial variations in P .
3.3
Methods
In this Section we spell out the practical techniques that we use to extract the Schottky barriers’ height from first-principles calculations, the operational definitions of free
charge and bound charge, and the methods we use to control the electrical boundary
conditions in supercell calculations. We also summarize the other relevant computational
parameters used in Sec. 3.4 and 3.5.
3.3.1
Schottky barriers from ab initio simulations
First, we briefly review the methods that were used in earlier works to compute Schottky
barriers at metal/semiconductor interfaces, pointing out advantages and limitations of
each of them. Then, we illustrate potential complications that might arise, with special
focus on ferroelectric oxide systems and the issues discussed in Sec. 3.2.
From the local density of states
In order to calculate the band offset at a metal/insulator interface, one needs to identify
the location of the band edges deep in the insulating region, with the Fermi level of the
metal taken as a reference. To that end, it has become common practice[112] to define
a spatially-resolved density of states,
XZ
ρ(i, E) =
dk |hi|ψnk i|2 δ(E − Enk ),
(3.26)
n
BZ
where |ii is a normalized function, localized in space around the region of interest. When
|ii = |ri is an eigenstate of the position operator, the resulting ρ(r, E) is commonly
known as local density of states (LDOS). Conversely, when |ii = |φnlm i is an atomic orbital of specified quantum numbers (n, l, m), we call it instead projected density of states
(PDOS). The integral is performed over the first Brillouin zone (BZ) of the supercell
and the sum runs over all the bands n. Enk stands for the eigenvalue of the one-particle
wave function ψnk .
The LDOS defined in Eq. (3.26), that depends on the position in real space as well
as in the energy, gives a very intuitive picture of the band offset: “sufficiently far” away
from the interface, the LDOS converges to the bulk one of the corresponding crystal
[112], and in principle the location of the band edges (and hence the Schottky barriers)
can be easily extracted following the next steps: (i) we integrate Eq. (3.26) over the
central unit cell of the dielectric (again assuming that we are sufficiently far away from
the interface so that the LDOS is bulk-like) and (ii) we compare the position of the
Fermi level of the whole heterostructure with the band edges obtained from the previous
integration.
However, several approximations are used in practice to make the calculation tractable,
and these can introduce significant deviations in the Schottky barriers computed by
means of either the LDOS or PDOS.
First, all studies are done on a finite supercell, usually with a symmetric capacitor
geometry. This implies that the LDOS of the most dispersive bands will be altered
by quantum confinement effects, which might produce a spurious gap opening. Also,
the LDOS associated to the evanescent metal-induced gap states (MIGS) might be still
important at the center of an insulating film that is not thick enough, thus preventing
an accurate identification of the band edge.
Second, as we explained in Sec. 2.7, a discrete k-point mesh is used instead of
the continuous one implicitly assumed in Eq. (3.26). Such a k-point mesh is generally
optimized for efficiency, which means that high-symmetry points are often excluded 1 .
As the edges of the valence and conduction band manifolds are usually located at the
high symmetry points 2 , extracting those features from the calculated LDOS might lead
to substantial inaccuracies. For materials that display a very dispersive band structure
(see e.g. Ref. [114]) it is not unusual to have deviations of the order of several tenths of
an eV.
Third, a fictitious electronic temperature (or Fermi surface smearing) is commonly
used, in order to alleviate the errors introduced by the k-mesh discretization. This
implies that the Dirac delta function in Eq. (3.26) is actually replaced by a normalized
smearing function with finite width. This is a again potential source of inaccuracies,
because the apparent edges of the smeared LDOS/PDOS actually might not correspond
to the physical band edges but to the (artificial) tail of the smearing function used.
Summarizing the above, we get to the following operational definition of the smeared
LDOS,
X
ρ̃(r, E) =
wk |ψnk (r)|2 g(E − Enk ),
(3.27)
nk
where the BZ integral has been replaced with a sum over a discrete set of special points k
with corresponding weights wk , and the Dirac delta has been replaced with a smearing
function g. As it will become clear shortly, it is very important to use in Eq. (3.27)
a g function that is minus the analytical derivative of the occupation function f (E)
defined in Sec. 2.7, and used in the actual calculations to determine the population of
the one-particle eigenfunctions during the selfconsistency procedure. Only in that case
the position of the Fermi level with respect to the LDOS obtained from the integration
1
In practical simulations, the origin of the k-point grid may be displaced from k = 0 in order to
decrease the number of inequivalent k-points [90, 113]. This shift usually prevents the appearance
of high-symmetry points from the list of k-points used during the self-consistent procedure or in the
calculations of the density of states.
2
The band gap of both BaTiO3 and PbTiO3 is indirect, with the top of the valence band located at
R in BaTiO3 and at X in PbTiO3 , and the bottom of the conduction band at Γ in both materials.
67
3.3. Methods
of Eq. (3.27) is consistent with the actual population of the conduction or valence bands
in the calculation (a detailed analysis is provided in Appendix A). Consequently the
corresponding Gaussian (G) and the Fermi-Dirac (FD) smearing functions are
E − EF
σ
1
fFD (E) = (E−E )/σ
F
e
+1
1
fG (E) = erfc
2
⇒
⇒
1
2
2
gG (E) = √ e−(E−EF ) /σ ,
πσ
σ −1
gFD (E) =
,
2 + e(E−EF )/σ + e−(E−EF )/σ
(3.28a)
(3.28b)
where σ is the smearing energy (or electronic temperature in the case of the Fermi-Dirac
distribution, with σ = kB T ) used during self-consistent minimization of the electronic
ground state.
From the electrostatic potential
To work around these difficulties, it is in most cases preferable to avoid the direct estimation of the Schottky barriers based on the LDOS/PDOS, and use instead the indirect procedure, based on the nanosmoothed electrostatic potential, V H , described in
Sec. 3.2.1. The lineup term ∆hV i – that can only be obtained from the simulation of
the actual interface, and thus represents the computationally most expensive part of the
Schottky barrier calculation – generally converges much faster than the LDOS/PDOS
with respect to all the computational parameters described above (slab thickness, kmesh, Fermi surface smearing). (A notable exception is the pathological spill-out regime
described in Sec. 3.2 – for further details see below.) The band-structure terms, EV
and EC , can be then accurately and economically evaluated in the bulk, without the
complications inherent to MIGS and quantum confinement effects. Finally, as depicted
schematically in Fig. 3.2, the band structure terms are shifted according to the spatially varying energy reference given by the lineup term ∆hV i to obtain the values of
the Schottky barriers of the interface.
While this is in principle a very convenient and robust methodological framework it
is, however, also prone to systematic errors. In particular, great care must be used when
performing the reference bulk calculations. In the vast majority of cases these must not
be performed on the equilibrium structure of the bulk solid, but will be constructed
to accurately match (i) the mechanical and (ii) the electrical boundary conditions of
the insulating film in the supercell. The issue (i) is well known: in a coherent heterostructure the insulating film is strained to match the substrate lattice parameter,
and for consistency the “bulk” calculation should be performed at the same in-plane
strain. (The dependence of the band-structure term on the lattice strain is well known
in the literature, and referred to as “deformation potentials” [115].) Issue (ii) concerns
ferroelectric systems, and is therefore not widely appreciated within the semiconductor community. Whenever the symmetry of the capacitor is broken and there is a net
macroscopic polarization in the ferroelectric film, the structural distortions may alter
the band structure significantly, often more than purely elastic effects do [110]. Note
that in most capacitor calculations the film is only partially polarized (i.e. it has neither
the centrosymmetric non-polar structure, nor the fully polarized ferroelectric structure
because of the depolarizing effects described in Sec. 3.2.2). The “bulk” reference calculation should then accurately match the polar distortions of the film, extracted in a
region where the interface-related short-range perturbations have healed into a regular
pattern.
The “best of both worlds”
In order to minimize the drawbacks associated with either of the two methods described
above, we find it very convenient to combine them in the following procedure.
1. We compute the LDOS in the supercell at an atomic site (or layer) located far
away from the interfaces, where the relaxed atomic structure has converged into a
regular pattern.
2. We extract the relaxed atomic coordinates from the same region of the supercell, and build a periodic bulk calculation based on them, by preserving identical
structural distortions and strains, and by using an equivalent k-mesh. (An approximation is made here, since the periodic bulk simulation is carried out at zero
macroscopic field while the LDOS in the supercell might be computed at a nonzero depolarizing field. The problem of computing the bulk layer by layer LDOS
under a finite electric field remains an open question.)
3. We extract the LDOS from the bulk for the same atomic site or layer; we construct
the bulk LDOS using Eq. (3.27) and an identical g function to that used in the
supercell.
4. Finally, we superimpose the bulk LDOS to the supercell LDOS at each layer j;
we align them by matching the sharp peaks of a selected deep semicore band,
supercell
bulk . The deep semicore states are
(j) and Esc
which are located at energies Esc
insensitive to the chemical environment and have negligible band dispersion; this
means that they provide an excellent, spatially localized reference energy for the
estimation of the lineup term.
At this point, we look at either LDOS curve in a vicinity of the Fermi level. If it is
non-zero we are probably facing a pathological spill-out case (see the following Section).
If it is zero, then we can go one step further and accurately estimate the position of the
bulk and E bulk ,
local band edges. To this end, we compute from the bulk calculation EC
V
bulk . (A further non-selfconsistent run might be needed if the original ktogether with Esc
mesh did not include the high-symmetry k-points where the band edges are located.) In
all cases these values should be directly extracted from the actual eigenvalues, and never
supercell
from the tails of the smeared LDOS. Finally, assuming that Esc
(j) are all referred
to an energy zero corresponding to the self consistent Fermi level of the supercell, we
define the local position of the band edges as
69
3.3. Methods
supercell
supercell
bulk
bulk
EC,V
(j) = Esc
(j) + (EC,V
− Esc
).
(3.29)
This procedure avoids the (often inaccurate) estimate of the band edges based on the tails
of the smeared LDOS, and at the same times preserves the advantages of the “lineup +
supercell
band structure” technique (where Esc
(j) is playing the role of ∆hV i). In principle,
the latter method should accurately match the results of Eq. (3.29), except for quantum
confinement effects in the metallic slab used to represent the semi-infinite electrode, as
discussed in Ref. [116].
Note that this technique is not only useful to detect pathological band alignments
and extract accurate band offsets in the non-pathological cases. Given that we are superimposing two LDOS calculated with identical computational parameters and structures,
their direct comparison can be very insightful. Most importantly, one expects all the
features to closely match unless there are MIGS or confinement effects. Therefore, one
has also a powerful tool to directly assess the impact of the latter physical ingredients
in the supercell electronic structure. This procedure, therefore, yields far more physical information than the separate use of either the PDOS/LDOS or the nanosmoothing
method.
We systematically applied this analysis in our results presented in Sections 3.4 and
3.5.
Pathological regime
In the pathological regime described in Sec. 3.2, many of the conditions that formally
justify application of the above methods to the estimation of the SBH break down.
First, the presence of a non-uniform electric displacement D(z) implies that the
polar distortions are also non-uniform, and they may not converge to a regular bulk-like
pattern anywhere in the film.
Second, electrostatic and exchange-correlation effects due to the partial filling of the
conduction band imply that the band structure may significantly depart from what one
computes in the insulating bulk (note that this has nothing to do with the effect of the
structural distortions discussed in the previous Section).
Third, the usual assumption of fast convergence of the interface dipole with respect
to slab thickness, k-mesh resolution and smearing energy also breaks down, as the conduction band DOS (which converges slowly with respect to these parameters) is now
directly involved in the electrostatic reequilibration process.
Based on this, the reader should keep in mind that there is an intrinsic arbitrariness,
of physical more than methodological nature, in the definition of the band edges in spillout cases. This arbitrariness reflects itself in the fact, already pointed out in Sec. 3.2,
that the band alignment at a pathological interface is no longer a well-defined interface
property, nor it is directly measurable in an experiment. The position of the bands is
essentially the result of a complex electron redistribution process that may occur on a
scale that is almost macroscopic, and is driven by different factors than those usually
involved in the Schottky barrier formation.
Of course, by using all the precautions that are valid at well-behaved interfaces one
might still gain some qualitative insight into the local electronic properties of the system.
However, the data must be interpreted with some caution, and it is most appropriate
to combine the analysis with other post-processing tools before drawing any conclusion.
We shall discuss some of these further analysis tools in the following Sections.
3.3.2
Electrical analysis of the charge spill-out
In this Section we introduce the methodological tools that we use to analyze in practice
the spill-out regime, in light of the theory developed in Sec. 3.2. In particular, we illustrate how to rigorously define the “local electric displacement” D(z) and the “conduction
charge” ρfree . To evaluate the former, we discuss two approaches. The first one is based
on a Wannier decomposition of the bound charges. The second one is an approximate
formula in terms of the ionic distortions and the Born effective charges. This simplified
formula is very practical for a quick analysis, but is generally affected by systematic
errors. This issue is addressed using a simple correction proposed by M. Stengel that
significantly improves the accuracy of the estimation based on the Born charges [105].
Definition of bound charge and conduction charge
In a typical metal, it is difficult to rigorously identify conduction electrons and bound
charges, as usually the respective energy bands intersect each other in at least some
regions of the Brillouin zone. (This is true, for example, in all transition metals, where
the delocalized sp bands cross the more localized d bands.) By contrast, in all perovskite
materials considered here, even upon charge spill out and metallization a well-defined
energy gap persists between the bound electrons and the partially filled conduction
bands. Therefore, it is straightforward to separate the two types of charge densities, free
and bound, simply by integrating the local density of states, defined in Eq. (3.26), over
two distinct energy windows. For example, for the conduction charge ρfree we have
Z EF
X
ρ̃(r, E)dE =
ρfree (r) =
wk fnk |ψnk (r)|2 ,
(3.30)
E0
εnk >E0
where E0 is an energy corresponding to the center of the gap between valence and
conduction band, ρ̃ is the smeared density of states of Eq. (3.27), fnk are the occupation
numbers and the sum is restricted to the states with eigenvalue Enk higher than E0 .
[Note that Eq. (3.30) only holds if the g-smearing of ρ̃ is compatible with the definition
of fnk , see more details about this discussion in Sec. 3.3.1 and Appendix A.] Since we
are working with layered systems that are perfectly periodic in plane, we will be mostly
concerned with the planar average of ρfree ,
Z
1
ρfree (z) =
ρfree (r)dxdy,
(3.31)
S S
where S is the area of the interface unit cell. In some cases, it is also useful to consider
the nanosmoothened function, [108] which we indicate by a double bar symbol, ρfree (z).
71
3.3. Methods
Of course, we could use a similar strategy to extract ρbound , but in practice we find
it more convenient to work with the local polarization P (z), as it is more closely related
to the way that ferroelectric materials and superlattices are usually discussed in the
literature. Techniques to extract P (z) are described in the following sections.
Local polarization via Wannier functions
A very useful tool to describe the local polarization properties of layered oxide superlattice are the “layer polarizations” introduced by Wu et al. [117] First, the electronic
ground state is transformed into a set of “hermaphrodite” Wannier orbitals [117, 118]
by means of the parallel-transport [119] procedure. Note that the parallel-transport procedure is restricted only to the orbitals that we consider as “bound charge”, i.e. those
with an energy eigenvalue lower than E0 . Then, the Wannier centers and the ion cores
are grouped into individual oxide layers, and the dipole density of layer j is defined as
pj =
X 1X
Zα Rαz − 2e
zi ,
S
α∈j
(3.32)
i∈j
where Zα is now the bare valence charge of the atom α, whose position along z is Rαz ,
and zi is the location of the Wannier orbital i.
Note that individual oxide layers in II-IV perovskites like BaTiO3 or PbTiO3 are
charge-neutral and pj are well-defined; however, in I-V perovskites like KNbO3 , individual layers are charged, and pj become meaningless as they are origin-dependent. To
circumvent this problem, one can either combine the layers two by two as was done
in Ref. [71], or perform some averaging with the neighboring layers, as for example in
Ref. [118]. It is important to keep in mind that, depending on the specific averaging
procedure, one might end up with the formal or with the effective local polarization
[120]; in this work we find it more convenient to work with the latter. As we do not
need, for the purpose of our discussion, to resolve P into contributions from individual
AO and BO2 oxide layers, at variance with Ref. [118] we perform a simple average
1
1
1
p̄j = pj−1 + pj + pj+1 .
(3.33)
4
2
4
We then define the local polarization by scaling this surface dipole density by the average
out-of-plane lattice parameter, c, of the oxide film, and by taking into account that every
individual oxide layer occupies only half the cell. We thus define the local polarization
as
2
Pj = p̄j .
(3.34)
c
The local polarization Pj is, of course, a discrete set of values, but we can think of it
as a continuous function of the z coordinate, P (z), which is sampled at the oxide plane
locations. In the remainder of this work, we will write Pj or P (z) depending on the
context, but the reader should bear in mind that these two notations refer to the same
object.
Approximate formula via Born effective charges
While the above definition of Pj in terms of Wannier functions is accurate and rigorous,
it is not immediately available in most electronic structure codes. An approximate
estimate of the local polarization can be simply inferred from the bulk Born effective
charges Zα∗ , defined in Sec. 1.2.2, and the local atomic displacements.
Analogously to the above formulation, we can write the Zα∗ -based approximate layer
dipole density, pZ
j , as
pZ
j =
1X ∗
Zα Rαz ,
S
(3.35)
α∈j
where Zα∗ is now the bulk Born effective charge associated with the atom α. Again,
pZ
j are ill-defined in perovskite materials, as typically individual oxide layers do not
satisfy the acoustic sum rule separately. To address this issue, we perform an analogous
averaging procedure and define
1 Z
1 Z 1 Z
p̄Z
j = pj−1 + pj + pj+1 .
4
2
4
(3.36)
The approximate local polarization then immediately follows,
2
.
PjZ = p̄Z
c j
(3.37)
Such an approximation provides an exact estimate, in the linear limit, of the polarization
induced by a small polar distortion under short-circuit electrical boundary conditions,
i.e. assuming that the macroscopic electric field vanishes throughout the structural
transformation. Neither of these conditions is respected in a ferroelectric capacitor,
where the polar distortion is generally large (close to the spontaneous polarization of
the ferroelectric insulator), and where there is generally an imperfect screening regime,
with a macroscopic “depolarizing field” [53]. Both issues have been investigated by M.
Stengel in Ref. [105] (discussion included in Appendix B), where it is found that a
simple scaling factor corrects, to a large extent, the discrepancy between Pj and PjZ . In
particular, the “corrected” P̃jZ can be written as
P̃jZ
χ∞
= 1+
PjZ ,
χion
(3.38)
where χ∞ and χion are, respectively, the electronic and ionic susceptibilities of the bulk
material in the centrosymmetric reference structure, calculated at the same in-plane
strain as the capacitor heterostructure. Note that for a ferroelectric material in the
centrosymmetric reference structure, χion is negative, which is a consequence of the
polar unstable mode in the phonon spectrum. This means that the scaling factor will
be smaller than 1 (∼0.9 for the materials considered in this work). Practical methods
to calculate χ∞ and χion are reported in Appendix B.
3.3. Methods
3.3.3
73
Constrained-D calculations
In Sec. 3.2 we have shown that a pathological spill-out regime can be triggered by the
ferroelectric displacement D of the film, as the band offset generally strongly depends on
D. It is therefore important, in order to perform the analysis described in the previous
Sections, to calculate the electronic and structural ground state of a metal/ferroelectric
interface at different values of D. To this end, two different approaches can be used in
first-principles calculations.
The first, and more “traditional” approach, involves the construction of capacitor
of varying thicknesses t, and the relaxation of the corresponding ferroelectric ground
states within short-circuit boundary conditions. Due to the interface-related depolarizing
effects mentioned in Sec. 3.2 (these are strongest in thinner films and tend to reduce P
from the bulk value Ps ), the polarization will increase from P = 0 (for t < tcrit , where tcrit
is the “critical thickness” [47, 53]) to P ∼ Ps , in the limit of very large thicknesses. Due
to the small value of the vacuum permittivity ε0 , for typical values of the polarization
in a ferroelectric material, ε0 E P , and D ∼ P , meaning that the variation of the
polarization can be identified with a variation of D. (Throughout this Chapter we shall
make use of this identity, and talk about either P or D.) This might be cumbersome in
practice: thicker capacitor heterostructures imply a substantial computational cost, due
to the larger size of the system; this severely limits the range of P (D) values that can
be studied within short-circuit boundary conditions.
An alternative, more efficient methodology to explore the electrical properties of the
interface as a function of polarization, is to use the recently developed techniques to constrain the macroscopic electric displacement to a fixed value. [51, 52] With this method,
one is able, in principle, to access at the same computational cost the structural and
electronic polarization of the capacitor for an arbitrary polarization state. In the specific
context of the present work, however, there are two drawbacks related to the use of the
constrained-D method as implemented in Refs. [51] and [52]. First, fixed-D strategies
make use of applied electric fields to control the polarization of the system. This is a
problem here, where the metallicity associated with the space charge which populates
the ferroelectric film makes such a solution problematic. (If a capacitor becomes metallic, it is a conductor and no metastable polarized state can be defined at any given bias.)
Second, our philosophy in this work is to adopt “standard” computational techniques,
i.e. those that are in principle available in any standard electronic structure package.
To this end, an alternative way of performing constrained-D calculations for a metalinsulator interface has been suggested by M. Stengel, which does not rely on the direct
application of macroscopic electric fields or on the calculation of the macroscopic Berryphase polarization. According to this method a vacuum/ferroelectric/metal geometry
is adopted, as sketched in Fig. 3.9(b). To induce a given value of the polarization in
the ferroelectric film, a layer of bound charges (Q per surface unit cell S) is introduced
at its free surface. If this is done in such a way that the surface region remains locally
insulating, at electrostatic equilibrium, the difference in the macroscopic displacement D
on the left and on the right side of the surface will exactly correspond to the additional
surface charge density Q/S. By applying a dipole correction in the vacuum region, we
(a)
(b)
σ =P
+
+
+
+
+
+
+
+
+
+
+
+
+
+
Conduc
tion ban
d
E F (1)
E F (2)
Valence
Metal
band
P
Ferroelectric
E F (2)
P
Metal
Vacuum
Ferroelectric
Metal
Figure 3.9: Two alternative ways of simulating a metal/insulator interface at a given
value of the macroscopic electric displacement D. (a) Capacitor heterostructure with
a finite applied field using the methods of Ref. [87]. (b) Vacuum/Insulator/Metal heterostructure with an external bound charge density σ = D applied to the open surface
of the insulating film; the electrostatic potential in the vacuum region is assumed to be
flat (in practice this is achieved by applying a dipole correction) .
ensure that D = 0 in the region near the surface on the vacuum side; then on the
insulator side we have exactly
Q
D= .
(3.39)
S
In practice, the additional charge density is introduced by substituting a cation at the
ferroelectric surface by a fictitious cation of different formal valence. For example, by
terminating a PbTiO3 film with a NbO2 layer (of formal charge +1), we would induce a
polarization in the PbTiO3 film of about 1 C/m2 ; a termination with ZrO2 , on the other
hand, would enforce D = 0 everywhere. As we are interested in exploring intermediate
values of D, the virtual crystal approximation is used to effectively induce a fractional
nuclear charge.
The reader might have noted that this method to control D is just a generalization
of Eq. 3.13 to consider other forms of “external” charge that are not “free” in nature.
Indeed, in the most general case, one can state
∇D = ρext (r),
(3.40)
where D encompasses all bound-charge effects that can be referred to the properties of a
periodically repeated primitive bulk unit, and ρext contains all the rest (e.g., delta-doping
layers, metallic free charges, charged adsorbates, variations in the local stoichiometry,
etc.). In Eq. 3.39 we simply applied Eq. 3.40 to the vacuum/ferroelectric interface,
where the “bound” nature of the external charge allows us to control it as an external
parameter.
3.3. Methods
3.3.4
75
Computational parameters
To demonstrate the generality of our arguments, which are largely independent of the
fine details of the calculation (except for the choice of the density functional), we compare
simulations performed with two different DFT-based electronic structure codes, Siesta
and Lautrec. In both cases, the interfaces where simulated by using a supercell approximation with periodic boundary conditions. [84] A (1×1) periodicity of the supercell
perpendicular to the interface is assumed. This inhibits the appearance of ferroelectric
domains and/or tiltings and rotations of the O octahedra. A reference ionic configuration was defined by piling up m unit cells of the perovskite oxide (PbTiO3 , BaTiO3 or
KNbO3 ), and n unit cells of the metal electrode (a conductive oxide such as SrRuO3 or a
transition metal such as Pt). In order to simulate the effect of the mechanical boundary
conditions due to the strain imposed by the substrate, the in-plane lattice constant was
fixed to the theoretical equilibrium lattice constant of bulk SrTiO3 (a0 = 3.874 Å for
Siesta and a0 = 3.85 Å for Lautrec).
To simulate the capacitors in an unpolarized configuration in Sec. 3.4, we imposed a
mirror symmetry plane at the central BO2 layer, where B stands for Ti or Nb, and relaxed
all the atomic coordinates and out-of-plane strain of the tetragonal supercells within
P 4/mmm symmetry. For the ferroelectric capacitors described in Sec. 3.5 a second
minimization was carried out, with the constraint of the mirror symmetry plane lifted.
Tolerances for the forces and stresses are 0.01 eV/Å and 0.0001 eV/Å3 , respectively.
Other computational parameters, specific to each code, are summarized below.
Siesta
Computations in Sec. 3.4.1 and 3.5.1 on short-circuited SrRuO3 /PbTiO3 and SrRuO3 /
BaTiO3 capacitors have been performed within a numerical atomic orbital method, as
implemented in the Siesta code [121]. Core electrons were replaced by fully-separable
[122] norm-conserving pseudopotentials, generated following the recipe given by Troullier
and Martins [123]. Further details on the pseudopotentials and basis sets can be found
in Ref. [124].
A 6 × 6 × 1 Monkhorst-Pack [90, 113] mesh was used for the sampling of the reciprocal space. A Fermi-Dirac distribution was chosen for the occupation of the one-particle
Kohn-Sham electronic eigenstates, with a smearing temperature of 0.075 eV (870 K).
The electronic density, Hartree, and exchange-correlation potentials, as well as the corresponding matrix elements between the basis orbitals, are computed in a uniform real
space grid, with an equivalent plane-wave cutoff of 400 Ry in the representation of the
charge density.
Lautrec
Calculations in Sec. 3.4.2 and 3.5.2 were performed by Massimiliano Stengel with
Lautrec, an “in-house” plane-wave code based on the projector-augmented wave method
[125]. He used a plane-wave cutoff of 40 Ry and a 6 × 6 × 1 Monkhorst-Pack [90, 113]
mesh. As the systems considered here are metallic, a Gaussian smearing of 0.15 eV was
adopted to perform the Brillouin-zone integrations.
3.4
3.4.1
Results: Non polar capacitors
Non-pathological cases
In the centrosymmetric unpolarized reference structure, some metal/ferroelectric interfaces such as BaTiO3 /SrRuO3 or PbTiO3 /SrRuO3 with a TiO2 /SrO termination
[the properties of the alternative (Ba,Pb)O/RuO2 termination might differ] are “wellbehaved” within LDA. This conclusion emerges from the analysis shown in Fig. 3.10
and 3.11 for the PbTiO3 - and BaTiO3 -based capacitors respectively. Figures 3.10(a)
and 3.11(a) represents schematically the Schottky barriers for electrons (φn ) and holes
(φp ) at the ferroelectric/metal interfaces, computed using the nanosmoothed electrostatic potential method described in Sec. 3.3.1. The bottom of the conduction band
of the ferroelectric lies, in both cases, above the Fermi level of the metal (φn amounts
to 0.38 eV for the PbTiO3 -based capacitor, and only to 0.19 eV in the BaTiO3 -based
case). Note that, if the experimental band gap could be reproduced in our simulations,
φn would be much larger [dashed lines in Fig. 3.10(a) and 3.11(a); we have taken the
experimental indirect gap of the cubic phase of PbTiO3 , 3.40 eV [126], and BaTiO3 ,
3.20 eV [127]; and assumed that the quasiparticle correction on the valence band edge
is negligible].
The results summarized in Table 3.2 illustrates some important issues commented
in 3.3.1. Results based on the nanosmoothed electrostatic potential and on the method
combining supercell and bulk LDOS calculations, yield Schottky barrier values that are
consistent within few hundredths of an eV, in accordance with the fundamental equivalence of these two approaches. Estimations based on the PDOS alone, however, results
in larger values due to the absence of the high symmetry k-points in the calculations.
The flatness of the profile of the nanosmoothed electrostatic potential at the central layers of the ferroelectric in Fig. 3.10(a) and 3.11(a) confirms the absence of any
macroscopic electric field, as expected from a locally charge-neutral and centrosymmetric
system.
Figures 3.10(b) and 3.11(b) displays ρ̄free (z), as defined in Sec. 3.3.2. As expected,
ρ̄free (z) has a rapid decay in the insulating layers, consistent with the evanescent character of the metallic states (MIGS): these cannot propagate in the insulator as their energy
eigenvalue fall within the forbidden band gap. In agreement with the positive values of
the Schottky barriers, except at the region very close to the interface where MIGS still
have some weight, an absolute abscence of free charge is found in the insulating region.
Figures 3.10(c) and 3.11(c) shows the layer-by-layer polarization, PjZ , computed
using Eqs. (3.35)-(3.37). Consistently with the absence of space charge, the PjZ profile
is remarkably flat. Due to the imposed mirror-symmetry constraint, PjZ also vanishes
inside the ferroelectric materials.
Finally, Fig. 3.12 shows, for the PbTiO3 /SrRuO3 capacitor, the layer-resolved PDOS
77
3.4. Results: Non polar capacitors
1.0
8
0.8
-3
EF = +8.91 eV
0.4
(b)
0.4
2
(a)
0.2
0
-0.2
-0.4
(c)
TiO2
PbO
TiO2
PbO
TiO2
PbO
TiO2
PbO
TiO2
PbO
TiO2
SrO
RuO2
SrO
RuO2
SrO
RuO2
SrO
-2
0.6
0.2
P (C/m )
0
EC = +5.94 eV
2
∆<V> = 3.34 eV
4
φp = 0.97 eV
EV = +4.59 eV
Energy (eV)
6
-3
φn = 0.38 eV
ρfree (10 e Bohr )
10
Figure 3.10: (a) Schematic representation of φn and φp in an unpolarized
SrRuO3 /PbTiO3 /SrRuO3 capacitor. EV , EC , EF and ∆hV i were defined in Sec. 3.2.
The calculated values are also indicated in the Figure. The black solid curve represents
−V H (z). The dashed line represents the hypothetical position of the CBM, if EC were
shifted as to reproduce the experimental band gap. (b) Profile of ρ̄free as defined in
Eq. (3.31). (c) Profile of the layer-by-layer polarization PjZ . The size of the capacitor
corresponds to n = 5.5 unit cells of SrRuO3 and m = 12.5 unit cells of PbTiO3 . Only
the top half of the symmetric supercell is shown.
1.0
10
φn= 0.19 eV
0.8
-3
ρfree (10 e Bohr )
8
φp= 1.39 eV
0.6
(b)
0.4
0
-0.2
SrO
SrO
RuO2
SrO
RuO2
BaO
TiO2
SrO
RuO2
BaO
TiO2
(c)
BaO
TiO2
-0.4
BaO
TiO2
2
(a)
0.2
BaO
TiO2
-2
0.4
0.2
P (C/m )
0
EF= +8.90 eV
2
EC= +7.29 eV
EV= +5.71 eV
4
∆V= 1.79 eV
Energy (eV)
-3
6
Figure 3.11: (a) Schematic representation of φn and φp in an unpolarized
SrRuO3 /BaTiO3 /SrRuO3 capacitor. EV , EC , EF and ∆hV i were defined in Sec. 3.2.
The calculated values are also indicated in the Figure. The black solid curve represents
−V H (z). The dashed line represents the hypothetical position of the CBM, if EC were
shifted as to reproduce the experimental band gap. (b) Profile of ρ̄free as defined in
Eq. (3.31). (c) Profile of the layer-by-layer polarization PjZ . The size of the capacitor
corresponds to n = 5.5 unit cells of SrRuO3 and m = 8.5 unit cells of BaTiO3 . Only
the top half of the symmetric supercell is shown.
79
Table 3.2: LDA values of φn and φp , obtained with the three different methods described
in Sec. 3.3.1: using the edges of the PDOS calculated over the central TiO2 layer
(PDOS), using the decomposition into EV,C and ∆hV i (BS + Lineup), and using Ti
(3s) semicore states to align bulk band structure to the density of states of the capacitor
(Semicore). Methods including high-symmetry (HS) k-points are indicated.
Capacitor
SrRuO3 /PbTiO3 /SrRuO3
φp
φn
SrRuO3 /BaTiO3 /SrRuO3
φp
φn
PDOS
(no HS)
BS + Lineup
(HS)
Semicore
(HS)
1.28
0.53
0.97
0.38
0.99
0.37
1.55
0.42
1.39
0.19
1.40
0.19
of the Ti(3s) semicore peaks, the O(2s) peak, the upper valence band and the lower
conduction band (black curves, shaded in gray). On top of the heterostructure PDOS
we superimpose the bulk PbTiO3 PDOS, calculated with an equivalent k-point sampling
and aligned with the Ti(3s) peak (dashed red curves). Note that all PDOS curves were
calculated using Eq. (3.27), and the smearing function gFD of Eq. (3.28b) with σ = 0.075
eV, consistent with the parameters used in the calculation. The PDOS of the conduction
and valence bands converges fairly quickly to the bulk curve when moving away from
the interface – they are practically indistinguishable already at the fourth layer. The
estimated energy location of the conduction and valence bands converge even faster [these
are directly related to the shifts of the Ti(3s) state, which are less affected by quantum
confinement effects]. All curves except those adjacent to the electrode interface vanish
at the Fermi level, confirming the absence of charge spill-out in this system.
As a summary of this Section we can conclude that, when a centrosymmetric unpolarized interface is non-pathological in the sense that the bottom of the conduction band
of the ferroelectric is above the Fermi level of the metal, (i) the free charge, as defined
in Sec. 3.3.2, vanishes due to the absence of charge spill-out; (ii) the local polarization
profile (Sec. 3.3.2) is perfectly flat as the interface-induced polar lattice distortions heal
rapidly (within the first unit cell); and (iii) the LDOS/PDOS vanishes at the Fermi level,
except for one or two interface layers where the signatures of the MIGS are still present.
3.4.2
Pathological cases
We analyze now two examples of capacitors that are characterized by a pathological
band alignment already in their centrosymmetric reference structure: NbO2 -terminated
KNbO3 /SrRuO3 , and TiO2 -terminated BaTiO3 /Pt. This choice of materials is motivated by the fact that there exist recent theoretical works on these systems, [128, 129]
PDOS (arb. units)
-57.6 -57.2 -56.8
-20
-18
-16
-4 -3 -2 -1 0
1
2
3
Energy (eV)
Figure 3.12: PDOS of the inequivalent TiO2 layers in the unpolarized PbTiO3 /SrRuO3
capacitor (solid curves with gray shading). The bottom curve lies next to the electrode,
the top one lies in the center of the PbTiO3 film. Only the PDOS on half of the symmetric
supercell are shown. Panels show from left to right, Ti (3s) states, O (2s) states and
band gap energy region. The bulk PDOS curves (red dashed) are aligned to match the
Ti(3s) peak at E ∼ −57 eV. The Fermi level is located at zero energy and marked as a
dotted line.
where the consequences of the pathological band alignment were neglected.
Calculations presented in this Section were carried out by M. Stengel and are included
here for the completeness of the discussion.
KNbO3 /SrRuO3
We construct a heterostructure consisting of m=6.5 KNbO3 unit cells and n=7.5 SrRuO3
cells, for a total of 14 perovskite units; we use symmetrical NbO2 (SrO) terminations of
the KNbO3 (SrRuO3 ) film.
After full relaxation with a mirror symmetry constraint at the central NbO2 layer,
we perform the analysis of the LDOS, the conduction charge and the local polarization
as explained in Sec. 3.3. In Fig. 3.13 we show the local density of states integrated over
the NbO2 layers
The unphysical Ohmic band alignment is evident from the location of the conduction
band bottom – the whole film is clearly metallic. This points to the pathological situation
that is sketched in Fig. 3.1(g). Note that the LDOS does not converge to the bulk curve
anywhere in the heterostructure. There are non-trivial shifts of all peaks that make it
81
LDOS (arb. units)
-56 -20 -19 -18 -17
-8 -7 -6 -5 -4 -3 -2 -1 0
1
2
3
Energy (eV)
Figure 3.13: LDOS integrated over the NbO2 layers of the KNbO3 /SrRuO3 heterostructure (solid curves with gray shade). The bottom curve lies next to the electrode, the
top one lies in the middle of the KNbO3 film. Only the LDOS on half of the symmetric
supercell are shown. Panels show from left to right, Nb (4s) states, O (2s) states and
band gap energy region. The bulk LDOS (red dashed curves) are aligned as to match
the valence and conduction band edges. The Fermi level is located at zero energy and
marked with a dotted line
difficult to identify a well-defined alignment with the bulk curves. In Fig. 3.13 we choose
to align the O(2s)-derived feature at E ∼ −19 eV. In this specific system, aligning the
O(2s) peaks appears to yield a reasonably good match of the conduction and valence
band edges (the most relevant features from a physical point of view); this, however,
leads to a marked mismatch, e.g. in the position of the semicore Nb(4s) state. We show
in the following that these effects stem from a number of (rather dramatic) electrostatic
and structural perturbations acting on the KNbO3 film, which are a direct consequence
of the pathological band alignment.
First we show that this misalignment results in a sizeable spill-out of conduction
charge into the ferroelectric film. To that end, we plot ρfree (z), which represents the
planar average of the artificially populated part of the KNbO3 conduction band, and
the corresponding nanosmoothened version, ρfree (z) in Fig. 3.14 respectively as black
continuous and red dashed lines. The additional electron density in the ferroelectric
region is apparent [compare the scale of the plot with that of Fig. 3.10(b)], and reaches
a maximum of about 0.15 electrons in the central perovskite unit cell. Such a density
is significant – it can be thought as resulting from an unrealistically large doping of,
1.2
-3
0.8
-3
ρfree [10 e bohr ]
1.0
0.6
0.4
NbO2
KO
NbO2
KO
NbO2
KO
NbO2
KO
NbO2
KO
NbO2
NbO2
0.0
KO
0.2
Figure 3.14: Calculated free charge for paraelectric SrRuO3 /KNbO3 /SrRuO3 heterostructure. Black curve: planar-averaged ρfree . Red dashed: ρfree , nanosmoothened
using a Gaussian filter. Blue symbols: finite differences of the local Pj (shown as a black
curve in Fig. 3.15), calculated using the Wannier-based layer polarization described in
Sec. 3.3.2.
e.g. one Sr2+ cation every six or seven K+ ions. However, unlike in a doped perovskite,
the spurious electron spill-out here is not compensated by an appropriate density of
heterovalent cations. The system is therefore not locally charge neutral, and as a consequence strong, non uniform electric fields arise in the insulating film that act on the
ionic lattice.
In order to elucidate how the underlying polarizable material responds to such an
electrostatic perturbation, we plot in Fig. 3.15 the effective polarization profile in the
KNbO3 film calculated in two ways, (i) the rigorous Wannier-function analysis of the
layer polarizations and (ii) the approximate expression based on the renormalized bulk
dynamical charges. The matching between the curves is excellent, indicating that the
approximate Z ∗ -based formula provides a reliable estimate of P (z); this suggests that the
electrostatic screening is indeed dominated by structural relaxations [see also Fig. 3.16],
as anticipated in Sec. 3.2.2, and as expected in a ferroelectric material.
The polarization profile Pj is characterized by a uniform, negative slope. This nicely
confirms the prediction of our semiclassical analysis in Sec. 3.2.2 of a uniform linear
decrease of D(z) throughout the film. [Recall that the difference between P and D is
negligible in KNbO3 , of the order of 1% or less, which justifies our use of P (z) in place
of D(z) in the differentiation.] Pj varies from 0.3 to -0.3 C/m2 when moving from the
bottom to the top interface Note that such spatial variation is completely absent in, e.g.,
isostructural paraelectric BaTiO3 /SrRuO3 (Fig. 3.11(c) and diamonds in Fig. 3.15), and
PbTiO3 /SrRuO3 [Fig. 3.10(c)] capacitors, where the profile is remarkably flat with P
83
0.2
2
Polarization [C/m ]
0.4
0
AO
BO2
AO
BO2
AO
BO2
AO
BO2
AO
BO2
-0.4
AO
-0.2
Figure 3.15: Local polarization profile in the SrRuO3 /KNbO3 /SrRuO3 capacitor. Black
circles: polarization from Wannier-based layer polarizations. Red squares: approximate
polarization from “renormalized” Born effective charges (see Sec. 3.3.2). Analogous
results for a paraelectric SrRuO3 /BaTiO3 /SrRuO3 capacitor are shown for comparison
(blue diamonds).
vanishing throughout the film.
To demonstrate that the spatial variation in P (z) is directly related to ρfree according
to Eq. (3.13), we perform a numerical differentiation of the polarization profile derived
from the Wannier-based layer polarizations (again making use of the small difference
between P and D).
The result, plotted in Fig. 3.14 as a blue line, shows an essentially perfect match
between dP/dz and −ρfree illustrating the fact that the polarization profile is really a
consequence of KNbO3 responding to the spurious population of the conduction band,
rather than of interface bonding effects [128].
Finally, to illustrate the role of the lattice relaxation in the screening of the excess
charge, we also compare in Fig. 3.16 the relaxed layer rumplings in KNbO3 /SrRuO3 to
those of the non-pathological case, PbTiO3 /SrRuO3 , discussed in Sec. 3.4.1.
The KNbO3 film is characterized by strong non-homogeneous distortions, which are
at the origin of the polarization pattern shown in Fig. 3.15. Conversely, the distortions
are negligible in the PbTiO3 /SrRuO3 capacitor, where all the oxide layers are essentially flat. This striking qualitative difference is a strong evidence that the perturbation
experienced by KNbO3 /SrRuO3 , which affects the whole volume of the film, is of different physical nature than the localized interface bonding effects in BaTiO3 /SrRuO3
or PbTiO3 /SrRuO3 . We note that this behavior is also qualitatively different from a
ferroelectric distortion, which involves a rigid displacement of the ionic sublattices, and
preserves a macroscopically uniform rumpling pattern across the film [52].
Layer rumpling (Å)
0.2
0.1
0
-0.1
-0.2
-10
-5
0
Layer index
5
10
Figure 3.16: Layer rumplings (cation-oxygen vertical relaxations) in the centrosymmetric KNbO3 /SrRuO3 (black line, empty circles) and PbTiO3 /SrRuO3 (red line, filled
circles) capacitors. Dashed vertical lines indicate the location of the BO2 planes. The
shaded areas correspond to the SrRuO3 electrode region.
BaTiO3 /Pt
We next present results of an analogous investigation for a paraelectric (BaTiO3 )m /(Pt)n
capacitor, with m = 8.5 and n = 11. We consider symmetric TiO2 terminations, with the
interfacial O atoms in the on-top positions. (Note that this interface structure is different
than the AO-terminated films simulated, e.g. in Refs. [52] and [47], where a Schottky-like
band offset was found.) We find this interface to have a pathological band alignment,
similar to the KNbO3 /SrRuO3 case discussed above. The comparative analysis of the
bound-charge polarization profile and of the excess conduction charge, shown in Fig. 3.17,
again shows excellent agreement between ρ̄¯free (z) and the compensating bound charge.
The effect is analogous to KNbO3 /SrRuO3 , with an overall magnitude which is smaller
by roughly a factor of two; the polarizations at the two extremes of the film reach values
of about ±0.15 C/m2 .
The almost perfect similarity in behavior between these two chemically dissimilar
systems is further proof that the unusual effects described here and in Ref. [128] – the
apparent head-to-head domain wall in the ferroelectric film – have little to do with the
bonding at the interface, but are merely a consequence of the artificial charge spill out,
as discussed in Sec. 3.2.
85
0.6
0.2
(a)
(b)
0.1
2
Polarization [C/m ]
-3
0.4
-3
ρfree [10 e bohr ]
0.5
0.3
0.2
0.0
-0.1
TiO2
TiO2
TiO2
TiO2
TiO2
TiO2
-0.2
TiO2
TiO2
TiO2
TiO2
TiO2
TiO2
TiO2
0.0
TiO2
0.1
Figure 3.17: (a) Calculated free charge and (b) local polarization profile for a paraelectric
Pt/BaTiO3 /Pt capacitor with TiO2 -type interfaces. All symbols have the same meaning
as in Fig. 3.14 and Fig. 3.15.
Physical nature of the conduction charge in pathological band alignments
Before moving on to the next Section we briefly comment on the physical nature of the
conduction charge that spills into the ferroelectric film. In particular, it is important to
clarify that the charge densities plotted in Fig. 3.14 and Fig. 3.17(a) indeed originate
from population of the conduction band of the insulator, and not from metal-induced gap
states (MIGS) as some authors have recently argued [130]. All our data unambiguously
point to the former hypothesis. First, all charge density plots show a maximum in the
middle of the ferroelectric layer, rather than a minimum, which one would expect if the
former hypothesis were true, given the evanescent character of the MIGS. Second, the
individual wavefunctions that contribute most to ρfree in the ferroelectric film appear
to be confined within the insulator, that is they have almost no weight in the electrode
region. Therefore, we must conclude that these are genuine conduction band states,
and not MIGS. The maximum of ρ̄¯free in the middle of the ferroelectric film can be
interpreted either as a quantum confinement effect [the lowest-energy solution of the
electron-in-a-box problem is indeed a sine function with a shape reminiscent of the ρ̄¯free
plots of Fig. 3.14 and Fig. 3.17(a)], and/or as a result of the dielectric nonlinearities
discussed in Sec. 3.2.3 and depicted schematically in Fig. 3.8. This picture is fully
consistent with the LDOS of Fig. 3.13, where it is clear that there are no occupied states
(i.e. MIGS) within the gap below the conduction band edge of KNbO3 .
0.4
φn (eV)
0.2
0
-0.2
-0.4
0
0.2
0.4
0.8
0.6
Interface doping x
1
Figure 3.18: n-type Schottky barrier as a function of interface doping in KNbO3 /AOterminated SrRuO3 , where A is a fictitious atom with atomic number Z = 37 + x. Only
the Sr atoms at the interfacial layer are replaced by this fictitious atom. The dashed
line is a linear regresion of the data between x = 0 and x = 0.3, where the interface is
non-pathological from the band alignment point of view. Blue and red empty symbols
represent, respectively, the results for x = 0.5 and x = 1.0, where the interface is already
pathological. All values were obtained from Eq. (3.29), using either the Nb(4s) (squares)
or the O(2s) (circles) semicore peaks of the central NbO2 layer as a reference.
Estimating the “pre-spill” band offset
We mentioned in Sec. 3.2 that, whenever an electrode/ferroelectric interface enters the
pathological spill-out regime, the transfer of charge into the conduction band of the insulator produces an upward shift of the CBM. This effect prevents a direct, unambiguous
determination of the interface parameter φ0n .
In Ref. [105], M. Stengel suggest an approach, inspired by a recent work [131], to
circumvent this problem, and obtain an approximate estimate of the negative “pre-spill”
Schottky barrier φ0n . The authors of Ref. [131] show that the Schottky barrier at the interface between a perovskite insulator (SrTiO3 ) and a perovskite electrode (La0.7 A0.3 MnO3 ,
where A is Ca, Sr, or Ba) evolves linearly as a function of the compositional charge of
the interface layer. (Such interface layer is of the type Lax Sr1−x O, where x interpolates
between a +3 and a +2 cation.)
Of course, this linear behavior refers to a range of x values where the interface is
non-pathological; our arguments indicate that as soon as the system enters the spill-out
regime, the value of φn saturates to a nearly constant value. Based on this observation,
if one knows the linear behavior of φn in a range of x values for which the interface is
non-pathological, one can extrapolate this straight line to the values of x which cannot
be directly calculated, and obtain an estimate for φ0n .
87
(a)
0.3
(b)
0.2
1.0
2
Polarization (C/m )
-3
-3
Charge density (10 e bohr )
1.2
0.8
0.6
0.4
0.1
0
-0.1
-0.2
0.2
-0.3
0.0
NbO2
NbO2
NbO2
NbO2
NbO2 NbO2 NbO2 NbO2 NbO2
Figure 3.19: (a) Conduction charge density, and (b) local (Wannier-based) polarization
profiles extracted from the calculations with x =0.0, 0.1, 0.2 and 0.3 (filled circles, thin
black curves); 0.5 (empty blue circles, dashed blue curve) and 1.0 (empty red circles,
solid red curve). In (a) only half of the KNbO3 film is shown.
This strategy was applied to the same KNbO3 /SrRuO3 capacitor system described
Sec. 3.4.2. To tune the interface charge, the Sr cation in the interface SrO layer was
replaced with a fictitious atom of fractional atomic number Z = 37+x. x = 1 corresponds
to the example already shown in Sec. 3.4.2, with a charge-neutral SrO interface layer, and
x = 0 corresponds to a RbO layer of net formal charge -1. The results for the Schottky
barrier are plotted in Fig. 3.18. The region from x = 0.0 to x = 0.3 is non-pathological
and shows an almost perfectly linear evolution of φn (dashed line). By extrapolating
this linear trend to x = 1 we obtain φn ∼ −1.2 eV, which is about 1 eV lower than
the value calculated from first principles. This confirms the remarkable efficiency of
the self-healing mechanism. Assuming a polarization of ∼ 0.3 C/m2 for KNbO3 near
the interface, a potential drop of 1 eV would be accounted for by an effective screening
length of 0.3 Å at the electrode interface. This value is quite reasonable, and similar in
magnitude to those reported in Table 3.1.
In order to examine the crossover between the Schottky (non-pathological) and the
Ohmic (pathological) regimes in terms of the analysis tools developed in this work, we
plot in Fig. 3.19 the polarization profiles and the density of conduction electrons for
each of the calculations summarized in Fig. 3.18. These plots confirm that from x = 0
to x = 0.3 the capacitors are non-pathological, with absence of conduction charge in
the insulating region [Fig. 3.19(a), thinner black lines] and a flat polarization profile
[Fig. 3.19(b), filled black circles – all these curves overlap on this scale]. Conversely,
at x = 0.5 the conduction band starts populating significantly [thicker dashed blue line
in Fig. 3.19(a), empty blue circles in Fig. 3.19(b), note that the point in Fig. 3.18
starts to depart from the linear regime]. At x = 1.0 population of the conduction band
has become dramatic, and so is the corresponding slope in the polarization profile. The
departure from linearity in Fig. 3.18 is correspondingly large.
Note that the use of either the Nb(4s) or the O(2s) semicore peaks in Eq. (3.29) yields
identical results in the non-pathological regime (the filled squares and circles overlap in
Fig. 3.18). Conversely, the result depends significantly on this (completely arbitrary)
choice at x = 0.5, and even more so at x = 1.0 (the circles and squares split). This is
another proof that in the pathological regime the band lineup is ill-defined – due to the
electrostatic effects discussed throughout this work, the LDOS does not converge to a
bulk-like value in the center of the KNbO3 film (see Fig. 3.13), and there is no obvious
reference energy to determine the offset.
3.5
Results: Polar capacitors
As discussed in the Introduction of this chapter, although some of the unpolarized reference structures (e.g. the PbTiO3 /SrRuO3 interface) appear artifact-free within LDA,
because of the strong dependence of the Schottky barrier on D they might become problematic when the constraint of mirror symmetry is lifted and the system is polarized. In
the spirit of the “traditional” approach to explore the band offset in ferroelectric capacitors as a function of D, we address this issue performing simulations of short-circuited
capacitors for selected thicknesses of the ferroelectric layer. Results of the standard approach have been further extended by M. Stengel using the fixed-D strategy described
in Sec. 3.3.3 to explore the behavior of the ferroelectric/metal interface over a wide and
continuous range of polarization states.
3.5.1
Short-circuit calculations
PbTiO3 /SrRuO3
We have performed simulations on [PbTiO3 ]m /[SrRuO3 ]n heterostructures, with m =
12.5 and n = 5.5 unit cells. A soft-mode distortion of the bulk tetragonal phase, inducing
a polarization perpendicular to the interface, is superimposed on the PbTiO3 layers of the
previous unpolarized configurations discussed in Sec. 3.4.1. Then the atomic positions of
all the ions, both in the ferroelectric and in the metallic electrodes, and the out-of-plane
stress are re-relaxed with the same convergence criteria as before.
By means of the approximate Eq. (3.38), derived in Sec. 3.3.2, we computed the
local layer-by-layer polarization, P̃jZ , plotted in Fig. 3.20 (a). Far enough from the
interface, the polarization profile is rather uniform, with a polarization that amounts
to 0.53 C/m2 in PbTiO3 (64 % of the strained bulk polarization), which we identify as
the macroscopic P of the PbTiO3 film. The uniform value of the polarization inside
the ferroelectric layer breaks up in the region adjacent to the top electrode [right end of
curve in Fig. 3.20 (a)]. As we will se below, the origin of this inhomogeneous distortion
can be traced back to a pathological charge injection, in analogy to the centrosymetric
case discussed in previous section.
89
3.5. Results: Polar capacitors
(a)
2
P [C/m ]
0.8
0.6
RuO2
SrO
RuO2
SrO
TiO2
PbO
TiO2
PbO
TiO2
PbO
TiO2
PbO
TiO2
PbO
TiO2
PbO
TiO2
PbO
TiO2
PbO
TiO2
PbO
TiO2
PbO
TiO2
PbO
TiO2
PbO
TiO2
SrO
RuO2
SrO
RuO2
SrO
RuO2
SrO
0.4
0.5
(b)
-3
ρfree [10 e Bohr ]
0.4
-3
0.3
0.2
RuO2
SrO
TiO2
PbO
TiO2
PbO
TiO2
PbO
TiO2
PbO
TiO2
TiO2
0.0
PbO
0.1
Figure 3.20: (a) Profile of the layer-by-layer polarization P̃jZ , defined in Eq. (3.38), in
the relaxed polar configuration of a short-circuited SrRuO3 /PbTiO3 /SrRuO3 capacitor.
The dashed line represents the bulk spontaneous polarization under the same in-plane
strain as in the capacitor. (b) ρ̄free (z) as defined in Eq. (3.31) (black solid line), and its
nanosmoothened average ρ̄¯free (z) (red dashed line). The blue line represents the profile
of the bound charge, computed as a finite-difference derivative of P̃jZ .
After transforming the dipole density of the layers to local polarization by means of
Eq. (3.37), we can infer a value of d ∼ 0.5, where d = DS is the reduced macroscopic
displacement field [51].
The band alignment for the relaxed polar capacitor is obtained plotting the layerby-layer PDOS, shown in Fig. 3.21. The curves were constructed exactly as in Fig. 3.12,
except that (i) the capacitor is now polarized; and (ii) consistent with the discussion of
Sec. 3.3.1 we set up the bulk reference calculation by using the PbTiO3 structure extracted from the central layer of the polarized supercell (i.e. with atomic distortions and
out-of-plane strain consistent with a polarization of 0.53 C/m2 ). The agreement is again
very good, showing that our approximation of neglecting the macroscopic depolarizing
field in the bulk reference calculation is reasonable, and that the most important effect
PDOS (arb. units)
-58
-57
-56 -20
-18
-16
-4 -3 -2 -1 0
1
2
3
Energy (eV)
Figure 3.21:
Layer by layer PDOS on the TiO2 layers of the polar
SrRuO3 /PbTiO3 /SrRuO3 ferroelectric capacitor. Meaning of the lines corresponding
to the PDOS curves as in Fig. 3.12, but now the PDOS on all the TiO2 layers are plotted (there is no mirror symmetry plane any more). The conduction and valence band
edges are plotted as empty squares, and the dashed lines correspond to the extrapolation
of such edges into the pathological region.
on the PDOS originate from the lattice distortions.
In the capacitor we clearly distinguish two regions. In the lower part of the PbTiO3
film, the PDOS at the Fermi level vanishes (bottom panels in Fig. 3.21), which implies
that the system is locally insulating. Furthermore, the PDOS in each layer appears
rigidly shifted with respect to the neighboring two layers, consistent with a tilting of the
PbTiO3 Kohn-Sham eigenstates due to the the presence of a depolarizing field. In this
region the sloping of the bands (-0.0286 eV/Å) is found to match reasonably well the
value of the electric field obtained from the nanosmoothed electrostatic potential V H (z)
(-0.0278 eV/Å, shown in Fig. 3.22).
In the upper region, however, close to the top electrode, the PDOS crosses the Fermi
level and the system is locally metallic. All these features are in full agreement with the
scheme drawn in Fig. 3.7.
In Fig. 3.21 we also plot the estimated band edges for each layer, EV,C (j) (empty
91
Energy (eV)
1.0
0.5
0.0
-0.5
SrO
RuO2
SrO
RuO2
SrO
TiO2
PbO
TiO2
PbO
TiO2
PbO
TiO2
PbO
TiO2
PbO
TiO2
PbO
TiO2
PbO
TiO2
PbO
TiO2
PbO
TiO2
PbO
TiO2
PbO
TiO2
PbO
TiO2
SrO
RuO2
SrO
RuO2
SrO
RuO2
SrO
-1.0
Figure 3.22:
Nanosmoothed electrostatic potential across the polar
SrRuO3 /PbTiO3 /SrRuO3 ferroelectric capacitor.
The depolarizing field can be
extracted from the slope of the nanosmoothed potential in the central layer of the
PbTiO3 .
squares in Fig. 3.21), obtained from Eq. (3.29). We used the semicore Ti(3s) peak
at each layer as Esc (j), and we calculated the bulk contribution in Eq. (3.29) from a
non-self-consistent bulk calculation (based on the ground state charge density of the
bulk reference calculation at P = 0.53 C/m2 described above) that included the highsymmetry k-points. The band edges calculated by this procedure are only plotted for
those atomic layers where the calculation of the Schottky barriers are meaningful, namely
those layers out of the pathological region and sufficiently far from the interface. The
resulting data points lie very accurately on a straight line. By extrapolating this straight
line (black dashed line in Fig. 3.21), we see that it crosses the Fermi level near the fourth
PbTiO3 cell from the top electrode interface. This illustrates the pathological character
of the band alignment in this system, consistent with the model of Fig. 3.7.
The population of the conduction band in the region close to the upper interface
in Fig. 3.21 is confirmed when we analyze the spatial distribution of the free charge
density. The planar average of ρfree (r) for the relaxed polar configuration is plotted in
Fig. 3.20(b). The existence of a charge populating the Ti 3d orbitals is evident from the
peaks of ρ̄free (z) at the TiO2 layers, which are detectable up to five unit cells away from
the interface. In contrast with the result for the pathological centrosymmetric capacitors
where the conduction charge concentrated in the middle of the “insulating” layer, for
the polar capacitors ρfree is confined in a region close to the interface due to the internal
field in PbTiO3 .
As we already anticipated in the previous Sections, spurious population of the conduction band and is responsible for non-trivial lattice relaxations, which act as to screen
the electrostatic perturbation. Fig. 3.20(a) indeed shows a small bending of the local
polarization profile, starting roughly four unit cells away from the top interface and with
a negative slope of the local polarization, P̃jZ . From classical electrostatics [Eq. (3.13)], a
divergence of P̃jZ produces net bound charges, whose magnitude can be estimated by numerical differentiation of the polarization profile. The resulting profiles are shown in the
red curves of Fig. 3.20(b). As in the SrRuO3 /KNbO3 /SrRuO3 unpolarized case (see Fig.
3.14), the bound charge almost perfectly cancels the conduction charge (nanosmoothed
profile of ρfree ).
BaTiO3 /SrRuO3
We have performed similar calculations for a polarized BaTiO3 /SrRuO3 capacitor, and
found a very similar scenario. In this case the capacitor consisted of a layer of the
ferroelectric material with a thickness of 8.5 unit cells sandwiched between a electrodes
5.5 unit cells thick. As in the previous case, the polar ground state is found after breaking
the symmetry of a non polar configuration and relaxing. Computational parameters are
the same as in previous calculations.
Polarization inside the ferroelectric layer amounts to 0.32 C/m2 (80% of bulk value
for BaTiO3 strained to the SrTiO3 in-plane lattice constant). The absolute value of the
polarization in this system is significantly smaller than the one of PTO/SRO capacitor,
nevertheless, due to the small value of φn found in the paralectric configuration (see
Table 3.2), it is sufficient to cause the artificial breakdown of the capacitor.
Analysis of the layer-by-layer PDOS (Fig. 3.23) analogous to that performed in the
case of a PbTiO3 /SrRuO3 capacitor, shows that the conduction band crosses the Fermi
level at about three unit cells from the top interface. A zoom over the curve corresponding to the TiO2 layer closest to the interface [Fig. 3.23(i)] reveals that the interface
PDOS is significantly larger than the bulk one, indicating that the main contributions
to the PDOS at this position and in this energy window is due to the MIGS. On the
second TiO2 layer, on the contrary, the PDOS of the capacitor lies below the bulk one
reflecting the fact that not only the weight of the MIGS is already negligible but that
the conduction band is being pushed upwards by the spurious charge spillage (see Fig.
3.13, where the bulk semicore peaks are lowered with respect to those of the supercell
when the LDOS is aligned using the band edges, and recall that in 3.23 the bulk LDOS is
being aligned with the supercell semicore states). Panel (iii), corresponding to the third
TiO2 layer from the interface, still shows some density of states from the conduction
band below the Fermi level. At the fourth TiO2 layer from the interface [panel (iv)] the
effect is barely noticeable and the edge of the CB can already be considered above the
Fermi level.
93
(a)
(i)
(c)
(b)
(ii)
PDOS (arb. units)
(i)
(ii)
(iii)
(iv)
(iii)
(iv)
-58
-57
-20
-18
-16
-4 -3 -2 -1 0
1
2
3
-0.1
0
0.1
Energy (eV)
Figure 3.23:
Layer by layer PDOS on the TiO2 layers of the polar
SrRuO3 /BaTiO3 /SrRuO3 ferroelectric capacitor. Meaning of the lines corresponding
to the PDOS curves as in Fig. 3.12. The conduction and valence band edges are plotted
as empty squares and the dashed lines correspond to the extrapolation of such edges
into the pathological region. Regions close to the Fermi level are zoomed in panels (i)
to (iv). Each panel (i) to (iv) corresponds to the layer with the same label in panel (c),
and all are plotted with the same scale.
3.5.2
Open-circuit calculations
Calculations reported in previous section provides a snapshot of the dependence of the
band alignment with respect to the electric displacement D in ferroelectric capacitors.
The approach followed in Sec. 3.5.1, however, would be impractical in order to obtain the whole picture, since it would require the simulation of several capacitors with
increasing thicknesses in order to explore a reasonable range of values of D. The tremendous computational and human effort this would involve can be greatly reduced using
the method proposed by M. Stengel and discussed in Sec. 3.3.3. Within this method
a vacuum/PbTiO3 /SrRuO3 heterostructure is build, as explained in Sec. 3.3.3 and depicted schematically in Fig. 3.9. The reduced macroscopic displacement field, d = DS,
is controlled by substituting the Ti at the PbTiO3 /vacuum interface with a fictitious
cation of atomic number Zlef t = 40 + d (i.e. Zr for d=0). To keep an integer total
number of electrons, the free end of the SrRuO3 lattice is also terminated by replacing
the surface Sr atom with a cation of Zright = 20 − d (i.e. Ca for d=0). This is done out
of convenience, and does not have an unphysical influence on the results. As explained
1.0
0.20
Layer polarization [10 C/m]
0.15
-9
-3
-3
0.6
0.4
0.2
0.10
0.05
PbO
TiO2
PbO
TiO2
PbO
TiO2
PbO
TiO2
0.00
PbO
(b)
SrO
RuO2
SrO
PbO
TiO2
PbO
TiO2
PbO
TiO2
PbO
TiO2
PbO
TiO2
TiO2
(a)
0.0
TiO2
ρfree [10 e bohr ]
0.8
Figure 3.24: Results for the polarized PbTiO3 /SrRuO3 interface for increasing polarization of the film. (a) planar averaged ρfree . Black, red, green and blue curves correspond
to the results for d =0.20, 0.40, 0.60 and 0.74 e, respectively. The sharp peaks in ρfree
correspond to the Ti ions in the PbTiO3 film. (b) layer polarizations from the Wannierbased analysis. Same color code as in (a)
in Sec. 3.3.3, a dipole is applied correction in the middle of the vacuum layer to ensure
that the electric field vanishes outside the material.
The thickness of the PbTiO3 slab is set to 5 unit cells, and that of SrRuO3 to 4;
other computational parameters for caclulations with Lautrec have been reported in
Sec. 3.3.4.
Four different values of d were considered: 0.2, 0.4, 0.6 and 0.74, the latter one
corresponding to the ferroelectric ground state of PbTiO3 at the SrTiO3 in-plane lattice
constant. In each case, we verify by examining the LDOS that the free surface remains
locally insulating; therefore, the macroscopic D = d/S in the film corresponds exactly
to the value enforced by the artificial pseudopotential.
The evolution of ρfree and of the Wannier-based layer polarization profiles for 0.2 ≤
d ≤ 0.74 is shown in Fig. 3.24. It is apparent from the plots of ρfree that already for
the smallest value of the polarization [d = 0.20, black curve in Fig. 3.24(a)] the TiO2
layer closest to the electrode has an important density of conduction electrons. This is
expected, as the evanescent tails of the metal-induced gap states (MIGS) penetrate into
the insulating region for some distance at any metal/insulator junction. However, these
states do not propagate very far, and already at the second TiO2 layer they are barely
noticeable on the scale of the plot. At d = 0.4 [red curve in Fig. 3.24(a)] the peak on the
second TiO2 layer significantly increases in magnitude, and a new small peak appears at
the third TiO2 layer. Analysis of the local density of states equivalent to that performed
for the short-circuited capacitors (not shown) shows that these new peaks are conduction
band states of PbTiO3 , rather than evanescent SrRuO3 states. The progressive increase
of d stresses out the fundamental differences between the confinement of the conduction
charge and the quantum-mechanical damping of the MIGS that fall in a forbidden energy
95
SrO
PbO
PbO
PbO
PbO
SrO
PbO
PbO
PbO
PbO
PbO
PbO
PbO
2
Polarization [C/m ]
0.8
0.75
(a)
(c)
0.7
0.65
0.6
0.55
0.5
(b)
(d)
0.6
0.4
RuO2
TiO2
TiO2
TiO2
TiO2
TiO2
RuO2
TiO2
TiO2
TiO2
TiO2
TiO2
TiO2
0.0
TiO2
0.2
TiO2
-3
-3
ρfree [10 e bohr ]
0.8
Figure 3.25: Calculated results for the fully polarized PbTiO3 /SrRuO3 interface at
d = 0.74. (a) local polarization from the Wannier-based layer polarizations, and (b)
planar averaged ρfree (black curve), macroscopically averaged ρfree (red dashed curve),
and finite differences of the polarization shown in the panel (a) (blue squares). for a m
= 8-unit cell thick PbTiO3 film. Panels (c) and (d) are the corresponding figures for a
m = 5-unit cell thick PbTiO3 film. The sharp peaks in ρfree correspond to the Ti ions
in the PbTiO3 film.
window of the insulator. We identify confinement of ρfree with the onset of the Schottky
breakdown, which becomes increasingly apparent if the polarization of the film is further
increased to d = 0.60 [green curve in Fig. 3.24(a).]
At d = 0.74, the population of the conduction band becomes rather dramatic, and
the charge distributes over the whole film. Here, the space charge is no longer confined
by the depolarizing field: in the fully polarized ferroelectric state the internal field of
PbTiO3 is zero. Therefore, the intrinsic carriers are only loosely bound to the interface
by the band bending effect, in a way which is entirely analogous to the well-studied
case of doped semiconductor interfaces. Since the dielectric permittivity of PbTiO3 is
rather large, the band bending is very efficiently screened, and the distribution of charge
can reach quite far into the insulator. To demonstrate this fact, we have repeated the
simulation with the same value of d=0.74, but with a thicker PbTiO3 film of 8 unit cells
[Fig. 3.25 (b)]; indeed, the conduction electrons redistribute over the whole volume of
the film to minimize their kinetic energy. The metalization of the fully polarized PbTiO3
film at d=0.74 can be thought as a form of “electrostatic doping” induced by spill-out
of electrons from the electrode to the PbTiO3 conduction band.
As in the discussion of the paraelectric capacitors, the presence of the space charge is
reflected in the progressive “bending” of the layer polarization profile [Fig. 3.24(b)]. The
value of d ∼ 0.5 estimated for the short-circuited PbTiO3 /SrRuO3 capacitor analyzed
in previous section would yield a polarization profile intermediate between the red and
green curves in Fig. 3.24(b). The similarities between spatial distribution of free charge
for d = 0.4 and d = 0.6 with respect to that found in the fully relaxed short-circuited
PbTiO3 /SrRuO3 capacitor (Fig. 3.21) reinforces the validity of this method.
Fig. 3.24(b) illustrates a further important consequence of the charge spill-out regime,
which was mentioned already in Sec. 3.2.3: in the pathological regime the dipoles that
lie closest to the electrode interface may appear “pinned” to a fixed value. This is indeed
the case for the TiO2 layer adjacent to the electrode, which seems to saturate at ∼ 0.08
nC/m for increasing values of D. Again, we caution against interpreting this dipole
pinning effect as a robust physical result.
In general, the onset of such a pathological regime has important consequences on
many physical properties of the capacitor. In the following Section we shall discuss some
of them.
3.6
Discussion
In this Section we discuss the important aspects of our work in the context of the
existing literature. The discussion is organized in several categories, corresponding to
the different properties of a ferroelectric/electrode interface (or, more generally, of a
perovskite material) that might be affected by the (more or less spurious) presence of
free charges in the system.
3.6.1
Structural properties of the film
The authors of Ref. [128] studied KNbO3 thin films placed between symmetric metallic
electrodes (either SrRuO3 or Pt) under short-circuit electrical boundary conditions. In
the SrRuO3 case, the layer by layer polarization pointed in opposite directions at the
top and bottom interfaces for all thicknesses, creating 180◦ head to head domains walls,
which were denominated interface domain walls (IDW). The physical origin of the IDW
was attributed to a strong bonding between interfacial Nb and O atoms, which would
induce a “pinning” of the interface dipoles to opposite values at the top and bottom
electrode interfaces.
Here we have demonstrated with analytical derivations and practical examples that
both the inhomogeneous polarization and the “dipole pinning” effect are clear signatures
3.6. Discussion
97
of a pathological band alignment. In particular, in an unpolarized KNbO3 /SrRuO3
capacitor analogous to those simulated by Duan et al. [128], we obtain a monotonously
decreasing polarization profile, from (∼0.3 C/m2 ) at the bottom interface to an opposite
value of ∼-0.3 C/m2 at the top, in excellent agreement with the results of Duan and
coworkers [128]. In contrast with the conclusion of Ref. [128], however, here we find that
the microscopic origin of this strong inhomogeneous polarization is the spillage of charge
from the metallic electrode to the bottom of the conduction band of KNbO3 , rather than
a bonding effect.
These findings have important consequences concerning the physical understanding
of the system with regard to the relevant observables. First, the ferroelectric material
becomes in fact a metal, and such a device would respond Ohmically with a large direct DC current that would make switching difficult or impossible. This questions the
appropriateness of interpreting the “average” polarization of the film as a macroscopic
physical quantity that can be measured in an experiment (see next Section). Second, our
arguments indicate that one of the essential factors governing the equilibrium free charge
distribution (and hence the spatial variation of P ) is the conduction band structure of
the ferroelectric material. This ingredient is missing in the traditional Landau-Ginzburg
models, e.g. those used in Ref. [128] to interpret the above data on KNbO3 /SrRuO3
capacitors, or in Ref. [130] to interpret qualitatively similar results for a electron-doped
BaTiO3 /SrRuO3 interface appear unjustified. A more promising route to capturing
the essential physics of the charge equilibration mechanisms appears to be the model
Hamiltonian approach proposed in Ref. [110]. Extending that strategy to the case of a
metal/ferroelectric interface will be an interesting subject of further research.
3.6.2
Stability of the ferroelectric state
The pathological spill-out of charge has important consequences on the spontaneous
polarization of a ferroelectric capacitor. To give a qualitative flavor of such an effect, we
consider the case of a capacitor that is only partially metallic, i.e. there is a depolarizing
field that keeps the carriers confined to the pathological side as sketched in Fig. 3.26(a).
We further consider two symmetric electrodes, i.e. characterized by identical values of
φ0n (that we assume positive) and λeff . Assuming a monodomain state, there are then two
stable configurations, related by a mirror symmetry operation. As φ0n is positive, upon
application of an electric field there will be always an insulating region in the middle
of the film, i.e. the polarization can be switched without passing through a globally
metallic state.
To appreciate the impact of the charge spill-out on the spontaneous polarization of
the film, it is useful to look at the schematic band diagram of Fig. 3.26(a), where the
conduction band bottom goes below the Fermi level in proximity of the right electrode
(red area). This induces metallicity in a significant portion of the film (light grey shaded
area, up to the dashed line). Based on our arguments of Sec. 3.2, the charge spill-out is
associated with a spatially decreasing D(z) [Fig. 3.26(b)]. This, in turn, modifies the interface potential barrier by producing a strong upward shift in energy of the conduction
band edge from what one would have if D(z) were uniform and equal to the “physical”
(a)
P
EF
φ (D2 )
φ (D1 )
M
(b)
FE
M
D
D1
D2
z
Figure 3.26: Schematic representation of the impact of the charge spillage on the ferroelectric stability. M are the metal electrodes, and FE is the ferroelectric film. The
polarization points to the right.
value D1 . This implies that the charge spill out generally reduces the depolarizing field
[the “pre-spill” estimate is sketched as a thick dashed line in Fig. 3.26(a)], and hence
overstabilizes the ferroelectric state. This is what one intuitively expects – population
of the conduction band constitutes an additional channel for screening the polarization
charge, and this cooperates with the metallic carriers of the electrode. This, however,
contrasts with the conclusions of Ref. [130], where it was argued that charge leakage suppresses P by producing a ferroelectrically “dead” layer. These conclusions are based on
the assumption that the physically measurable P is the average polarization, hP i, taken
over the whole volume film. As the polarization is locally reduced near a pathological
interface, charge spill-out indeed results in a reduced hP i.
Is it justified, though, to assume that hP i is the physically relevant quantity in the
capacitor? Does hP i, in other words, reflect what is experimentally measured? In an
experiment one measures the time integral of the transient current density, ∆j, that flows
through the capacitor during the switching process. ∆j does not relate to hP i but to the
free charge accumulated at the interfaces, which in turn is the discontinuity of D at the
interfaces. Under the hypothesis that at least a portion of the film remains insulating
throughout switching, it rigorously follows from the modern theory of polarization [132]
that ∆j = ∆D = 2|D|; D is the value of the (locally uniform) electric displacement
deep in the insulating region. (We assume for simplicity that D = 0 in the paraelectric
reference state.) Therefore, observing that hP i is reduced upon charge leakage does
not reflect the true effect of the pathological band alignment, which is an artificial
enhancement of the spontaneous P via the reduction of the depolarizing field illustrated
3.6. Discussion
99
above.
A large number of works [133, 134, 135, 136] have investigated the stability of
PbTiO3 -based capacitors, and it is impossible here to discuss in detail whether and
how the above band-alignment issues might have affected each of them (for instance,
regarding the polarization enhancements reported in Ref. [133]). We limit ourselves to
observe that, due to the large spontaneous polarization of PbTiO3 , the possible consequences of having a pathological ferroelectric state need to be taken seriously into
account in the analysis, as we showed for the example of SrRuO3 electrodes in Sec. 3.5.
3.6.3
Transport properties in the tunneling regime
Ferroelectric capacitors have been explored as potential tunneling electroresistance devices [104], and many recent calculations focused on the calculation of the conductance
by means of first-principles methods. Metallicity and spill-out of electrons is a serious
potential issue in this context, as the calculated conductance can potentially be affected
by the presence of space charge in the system, in a way which is difficult to predict.
The recent work of Velev et al. [129] appears to be concerned by these worries, as
it focuses on TiO2 -terminated Pt/BaTiO3 /Pt capacitors. Indeed, we have showed in
Sec. 3.4.2 that this interface is problematic already in the centrosymmetric paraelectric
case. While we have not explored the ferroelectric regime in this system, based on the
imperfect screening arguments of Sec. 3.2 (the lineup depends linearly on P around the
paraelectric reference phase) we expect the spill-out effect to become worse at least at
one of the two interfaces when the capacitor is polarized. In fact, the metallicity of the
ferroelectric film seems to be confirmed by the data presented by the authors: In Fig.
2(a-b) of Ref. [129] the conduction band minimum (CBM) of the central BaTiO3 cell
appears to be degenerate or lower than the Fermi level, and in Fig. 1 of the same paper
the atomic displacements of the ferroelectric phase seems to be strongly asymmetric,
consistent with our speculations. While we cannot draw a definitive conclusion, our
analysis highlights the crucial importance of the band alignment issue, and the necessity
of performing an adequate and convincing assessment of its impact on the results (e.g.
the conductance) in each case.
3.6.4
Interface magnetoelectric effects
Magnetoelectricity is one of the emerging topics in oxide research. Despite the intense
efforts, one of the main limiting factors still persists: bulk materials displaying a robust
magnetoelectric effect are notoriously difficult to find. To work around this problem,
several researchers have been looking for alternative solutions by exploring heterostructures and composite materials. An interface has a lower symmetry than either of the
constituent bulk materials, and might therefore allow for physical response properties
that are absent in the parent compounds. A promising route to interfacial magnetoelectric coupling that has been proposed recently [137] is mediated by charge. The
polarization of the ferroelectric (or dielectric) lattice produces a bound charge at the interface, that is screened by the carriers of the metal. If these carriers are spin-polarized,
as in a ferromagnet, there will be a net change in the magnetization.
It is easy to see that the band-alignment issues that we discuss in this work have direct
and important implications for the calculation of the carrier-mediated interface magnetoelectric coefficient. In the pathological regime, the calculated (magnetic) response will
most likely be suppressed, as the spill-out charge, rather than the spin-polarized carriers
in the electrode, will screen the applied bias potential (or the ferroelectric polarization).
This speculation is directly relevant for interpreting the results of Yamauchi et al. [138]
on BaTiO3 films sandwiched between Co2 MnSi (Heusler alloy) electrodes. Depending
on the termination, two qualitatively different behaviors were reported: the MnSi/TiO2
interface results in a pathological band alignment and a strongly non-homogeneous local
polarization profile; conversely, neither is present in the capacitor with the other type
of termination, which has symmetric Co/TiO2 interfaces. A very small magnetoelectric response was reported for the MnSi/TiO2 case (contrary to the Co/TiO2 case), in
qualitative agreement with our arguments above.
Other recent studies,[139, 140] focusing on magnetoelectric effects in thin Fe film
deposited on ATiO3 (A=Ba,Pb,Sr), also reported strongly non-uniform polarization
profiles in the ferroelectric film (e.g. Fig. 3 of Ref. [140]). This suggests that also
the ATiO3 /Fe interface might be concerned by the band-alignment issues discussed in
this work, with potential impact on the physical observables. Our analysis tools should
help clarify these issues in the above systems and in the Fe/BaTiO3 /Fe capacitors of
Ref. [141].
3.6.5
Schottky barriers
Direct calculations of Schottky barriers at metal/ferroelectric interfaces are, among the
many useful physical properties of these junctions, those that are most directly affected
by the issues we discuss here. The consequence of a pathological band alignment is that
the estimated Schottky barrier is no longer a physically meaningful interface property,
but is influenced by macroscopic space-charge phenomena.
A rather comprehensive work on the SrTiO3 /transition metal interface was recently
reported in Ref. [142]. Without going into a too detailed analysis of the results, we limit
ourselves to noting that many of the reported p-type SBH for TiO2 - or SrO-terminated
interfaces are very close to, or sometimes well in excess of 1.8 eV. Considering that the
LDA/GGA fundamental gap of SrTiO3 is around 1.8 eV, the actual n-type SBH of the
calculation (i.e. not the value corrected with the experimental band gap) is close to zero
or negative. Therefore, charge spill out is a concrete and likely possibility for many of
the investigated structures.
Note that, contrary to the case of oxide electrodes, ideal interfaces between SrTiO3
and simple metals tend to have a smaller λeff [50]. This implies that the effects of the
electrostatic reequilibration described in Sec. 3.2 might be somewhat less dramatic, and
the values of the self-consistent φn closer to φ0n . This suggests that the trends and
the conclusions reported in Ref. [142] are likely to be robust with respect to the issues
described in this work. However, a more detailed analysis would be certainly interesting
in order to assess their impact at the quantitative level.
3.7. Conclusions
3.7
101
Conclusions
Due to its accuracy and efficiency, density functional theory has emerged as the method
of choice for studying ferroelectric oxides from first-principles. This predominance has
been reinforced since the early 1990s by the many successes achieved in the determination
of the structural, energetic, piezoelectric, and dielectric properties at the bulk level. In
the last few years, those efforts have evolved to address the behaviour of the functional
properties in thin films and superlattices, including in same cases (for instance, in the
study of ferroelectric capacitors) the presence of metal/insulator interfaces.
For a reliable prediction of the functional properties of these devices, the atomic
displacements, distortions of the unit cell, the electronic structure and the band gap
have to be accurately described simultaneously. However, the proper DFT treatment of
such interfaces is complicated by the so-called “band-gap problem”, (significant understimation of the band gaps of the Kohn-Sham electronic band structure by the standard
implementations of the LDA and GGA functionals), which might produce a pathological
alignment between the Fermi level of the metal and the conduction band of the insulator, thus precluding explicit DFT investigation of many systems of practical interest. In
this Chapter we have provided useful guidelines to identify such a pathological scenario
in a calculation by examining its main physical consequences: (i) an inhomogeneous
polar distortion propagating into the bulk of the film, (ii) the film becoming partially or
totally metallic due to a non-vanishing free charge, and (iii) the local conduction band
edge crossing the Fermi level. The above three effects are intimately linked, and should
be considered as potential artifacts of the aforementioned band-gap problem. Whenever
one of these “alarm flags” is raised in a calculation, the results should be examined with
great caution.
A route to overcoming this limitation involves correcting the LDA/GGA bandgap
while preserving the excellent accuracy of these functionals in the prediction of groundstate properties. Traditional methods to increase the band gap of insulators, like the
inclusion of a Hubbard U term in the Hamiltonian, are not satisfactory in the case of
a ferroelectric capacitor with a B-cation driven ferroelectricity: the application of a U
to the B-cation d orbitals opens the gap, but reduces the B cation d- O p hybridization
that is responsible for the ferroelectric distortion.
A more promising avenue has been recently opened by Bilc et al. [78] and Wahl
and coworkers, [143] using the so-called “hybrid” functionals that combine Hartree-Fock
exchange and DFT. In these works the problems of previous hybrid functionals, such
as the B3LYP and B1 functionals (which overestimate the volumes and atomic distortions of ferroelectric oxides even leading to supertetragonal structures, mainly due to
the GGA exchange part) are overcome. In particular the B1-WC functional proposed
in Ref. [78] (a combination of GGA functional by Wu and Cohen [144] with a small
percentage of exact exchange) have shown to provide good structural, electronic and
ferroelectric properties as compared to experimental data for BaTiO3 and PbTiO3 . Verifying the accuracy of B1-WC in interface studies appears as an interesting subject of
future research. Unfortunately, the price to pay for this accuracy is the substantially
higher computational cost of B1-WC compared to LDA/GGA.
In addition to the purely technical issues, our work also opens interesting avenues
regarding fundamental physical concepts. For example, ferroelectricity is usually understood within the modern theory of polarization, which is only applicable in the absence
of conduction electrons (i.e. in pure insulators at zero electronic temperature). It is an
important fundamental question, therefore, to assess whether our understanding of ferroelectrics in terms of bound charges, polarization and macroscopic electrical quantities
still applies (and to what extent) in a regime where a sizable amount of space charge
is present in the system. This issue is of crucial importance also for other systems, e.g.
electrostatically doped perovskites, which bear many analogies to the physical mechanisms discussed in this work. The first-principles-based modeling approach proposed in
Ref. [110] appears to be a promising route to further exploring this interesting topic.
Chapter 4
Metal-induced gap states in
ferroelectric capacitors
4.1
Introduction
When a junction between two different materials is formed, different charge rearrangements take place in order to equilibrate the chemical potential across the interface. This
is a well known problem in semiconductors interfaces physics. For instance, in any textbook [145, 83] we can find discussions about PN junctions of doped semiconductors,
where this charge rearrangement consists mainly in a transfer of the corresponding conducting charge carriers (holes and electrons in P and N semiconductors respectively)
from one side of the interface to the other. In interfaces between non doped perfect
materials different charge rearrangements become more relevant, in particular the ionic
relaxations at the interface [142], compensation (screening) charge in the electrode, and
charge injection via metal-induced gap states (MIGS) [102]. The latter of these mechanisms consist in the appearance of states with energies in the band gap of forbidden
energies of the insulator in a metal/insulator junction, and which decay exponentially
in the insulating side of the interface.
All these effects together determine the interfacial dipole in heterostructures which
greatly affect – and in some cases even govern – some of the functional properties of
electronic devices. Typical examples of these properties are breakdown fields in capacitors or diode effect in semiconductor junctions. For this reason, theory of MIGS have
received a lot of attention in the past, specially in the field of semiconductors physics
[102, 146, 147, 148]. In the field of oxides, and in particular of perovskite oxides, the
recent exploitation of their functionalities in heterostructures has increased the interest
of MIGS in such materials. In this context MIGS have been studied specially in connection with tunnel junctions and the recent prediction [149] and experimental measurement [150] of giant magnetoresistance in magnetic tunnel junctions using ferroelectric
materials for the barriers. MIGS were even used to explain the spatial distribution of
2-dimensional electron gases at the interface between two bulk insulating materials [151].
Given the leitmotiv of this thesis, namely the study of screening properties in fer103
104
Chapter 4. Metal-induced gap states in ferroelectric capacitors
roelectric thin films, we deal very often with MIGS in our simulations. Some criteria
to recognize these states, which in most cases can be applied to any metal/insulator
interface, were discussed for metal/ferroelectric capacitors in previous Chapter.
In this Chapter we aim to characterize the MIGS in ferroelectric capacitors and
their connection with the complex band structure of the ferroelectric material. We
study the mechanisms involved in the Schottky barrier formation, in which MIGS play
a fundamental role. More importantly we emphasize the influence of the real interface
beyond the complex band structure of bulk materials.
4.2
Metal-induced gap states and complex band structure
In the presence of an interface between a metal and an insulator, those electronic states
which are propagating states in the former but have energies inside the gap of the latter,
do not vanish right at the interface between the two materials, but decay exponentially
inside the insulator. This situation is, to some extent, analogous to the problem of a
free electron traveling across a potential barrier of finite height. Quantum mechanics
predicts that those electrons with a kinetic energy below the barrier still have a non-zero
probability of tunneling through it, with this probability decaying exponentially with the
barrier thickness. This dependence of the tunneling probability is due to the exponential
decay of the wave function inside the potential barrier.
Similarly, electrons in a metal encountering an interface with an insulator will decay
exponentially when their energies lie inside the band gap of the second material, as
shown schematically in Fig. 4.1(a). Of course, at the interface, the decaying states in
the insulator side should match the propagating states from the metal, and properties
associated to the MIGS would depend also on the matching conditions of the wave
function.
The exponentially decaying wave functions can be actually regarded as Bloch functions with an associated complex wave vector. We can directly take the 1-dimensional
expression of a Bloch wave function as introduced in Sec. 2.6
ψ(z) = eikz un,k (z),
(4.1)
and introduce a complex wave vector k̂ = <(k̂) + i=(k̂)
ψ(z) = ei<(k̂)z e−=(k̂)z un,k̂ (z).
(4.2)
It becomes inmediately evident that the wave function is now a regular Bloch function
under an exponential envelope, as depicted in Fig. 4.1(a). Usually only real values
of the wave vectors are discussed in textbooks, since infinite periodicity is assumed
and, in that situation, wave functions growing exponentially in any direction would not
be physically valid. However, localized wave functions with an exponential decay are
indeed perfectly valid solutions of the Schrödinger equation in the presence of defects,
surfaces or interfaces (in the remainder of this Chapter we will refer only to the case of
4.2. Metal-induced gap states and complex band structure
105
interfaces, but some of the discussions, and in particular those regarding the properties
of the complex band structure, could be extended to other non-periodic systems).
Our main interest in this Chapter is to compare the properties of MIGS obtained
from the simulation of a realistic ferroelectric capacitor with those derived from the bulk
complex band structure of the ferroelectric material. MIGS in these systems will decay
in the direction perpendicular to the interface, which we choose as the z direction, but
they still must be periodic in the (x, y) plane (due to periodic boundary conditions used
in our calculations). As a consequence, this becomes a 1-dimensional problem in which
we would search for eigenfunctions characterized by real wave vectors parallel to the
interface, kk , and by a complex ones k⊥ = kz + iq perpendicular to it.
Furthermore, the in-plane periodicity of the system makes the wave vector kk a good
quantum number that should be preserved across the interface. Thus wave functions
from the electrode side must match vanishing function on the insulator with the same
kk . This is true in atomically coherent interfaces where the lack of disorder precludes
the mixing of states with different kk . This was not a realistic assumption a few years
ago, but current experimental techniques have achieved such control in the growth of
thin films that, in practice, experimental samples can be well compared with our perfect
defect-free systems and the hypothesis coherent and non dispersive interface appears
reasonable.
We shall see in the next Section that, even under the simplest theoretical model,
allowing for complex values of the wave vector in the solution of the Schrödinger equation
gives rise to vanishing states in the presence of a surface or an interface in an otherwise
periodic system.
4.2.1
Complex band structure: a simple example
Interesting insights on the properties of MIGS can already be achieved at the level of
the nearly-free electron model [145]. This model for the electronic structure of a solid
departs from the free-electron approach introducing, as a perturbation, a potential with
the periodicity of the lattice. The effect of this perturbation is the mixing of wave
functions at the k-points where different bands cross, causing the opening of energy
gaps where solutions of the Schrödinger equation with real k-vectors are not allowed.
We shall see, however, that wave functions with complex k-vectors are perfectly valid
solutions for energies within the band gap. While these solutions are unphysical for an
infinite system they can describe electronic states localized at surfaces or interfaces,
Let us consider a 1-dimensional system as the one depicted in Fig. 4.1(b), consisting
of two semi-infinite materials in close contact. On the left side we have a metallic region
with a potential that we model as constant. In the insulator, on the right hand side in
Fig. 4.1(b), the potential is a oscillating function with the periodicity of the lattice.
Given its periodicity, the potential in the insulator can be expressed as a Fourier
series. If, for the sake of simplicity, we restrict the Fourier expansion to the first term
(besides the Γ term which is a constant) we get a potential in the insulator with the
form
106
ψ(z)
(a)
V (z)
(b)
2Vg
V0
Metal
Insulator
a
z
0
Figure 4.1: (a) Typical wave function at a metal/insulator interface for an eigenstate
with an energy in the insulator band gap, consisting in a propagating Bloch-like wave on
the metallic side which decays exponentially on the insulator side. (b) Potential profile
at the same interface.
V (z) = V0 + Vg cos(gz),
(4.3)
where g = 2π/a is the shortest reciprocal lattice vector. Within the nearly-free electron
model, the perturbative potential mixes eigenstates from different bands at the crossing
k-points. Taking for instance the first crossing that takes place for at the Brillouin zone
boundary (k = g/2), the eigenfunctions are a mixing of two plane waves, which are the
eigenfunctions of the unperturbed system
ψk (z) = Aeikz + Bei(k−g)z .
(4.4)
Inserting previous trial function into the time-independent Schrödinger equation
d2
− 2 + V (z) ψ = Eψ,
dz
(4.5)
leads to the following expression for the eigenvalues of the system (see Appendix C for
the detailed derivation)
E = κ2 + (g/2)2 + V0 ± κ2 g 2 + Vg2
1/2
,
(4.6)
107
a
a
Figure 4.2: Complex band structure at the Brillouin zone boundary within the nearlyfree electron model.
where we have introduced a new variable κ = k − g/2, which is the deviation of the wave
vector k from the Brillouin zone boundary.
Notice that scanning over real values of κ (i.e. of k), one would necessarily find
the real bands depicted as solid lines in Fig. 4.2, which display an energy gap at the
Brillouin zone boundary where no states with real wave vectors are allowed.
One could alternatively try imaginary values of κ (k̂ = π/a + iκ) in Eq. (4.6) and
imaginary bands would be found connecting the real ones (dashed lines in Fig. 4.2).
This very simple model is particularly interesting because it is useful to illustrate some
of the fundamental properties of complex band structures:
• Complex bands connect extrema of real bands through the imaginary part of the
complex plane. This means that a high density of complex bands is usually found
at high symmetry points of the Brillouin zone, where extrema of real bands are
typically located. This will be discussed in more detail in Sec. 4.4 where complex
the band structure of a realistic material is analyzed.
• The imaginary part of the wave vector, =(k̂) = q is associated with a penetration
length, δ = 1/2q, of the probability density associated to the evanescent state. The
smaller the q (κ in this model), the larger the penetration length. If q vanishes,
then the penetration length diverges and the corresponding wave function spread
over the whole system, as a standard Bloch wave function.
Therefore, and knowing that complex bands depart from extrema of real bands, the
penetration length of the evanescent states grows (the imaginary part of the wave
108
vector drops) as the energy of the state gets close to those extrema. This has some
similarities with the case of a free electron tunneling through a potential barrier,
where the tunnel probability increases as the electron kinetic energy approaches
the height of the barrier. Conversely, if a complex band connects the top of the
valence band and the bottom of the conduction band, the penetration of this band
has its minimum around the middle of the gap (see Fig. 4.2).
• The shape of a complex band is closely related with (i) the energy difference
between the real bands it emerges from, and (ii) their curvature [152] (which in
turn is related to the effective masses at the departing k-point).
4.2.2
Connection with Schottky barriers
Since the works by Heine [102], properties of MIGS have been known to be closely linked
with the formation of Schottky barriers in metal/insulator interfaces. The transfer of
charge that takes place at the interface of an heterostructure and the resulting band
alignment in connection with the bulk real and complex band structures can be better
understood introducing the concept of charge neutrality level (CNL).
Definition of the charge neutrality level
Let us first take a step back and recall that, in bulk, only solutions of the one-electron
Hamiltonian described in Sec. 2.4 with real k-vector are physically valid. Those solutions
in a periodic system can be expressed as Bloch functions. As we have seen in Sec. 4.2.1,
if the material is an insulator, its energy spectrum is characterized by the presence
of a region of forbidden energies (gap) separating (at zero temperature and in non
doped systems) occupied (valence band) and unoccupied states (conduction band). If we
compute the density of states (DOS), using for instance Eq. (3.27), we obtain something
resembling the sketch of Fig. 4.3(a).
However, as demonstrated in Sec. 4.2.1, formally, the Schrödinger equation admits
solutions with complex k-vectors, solutions that became physically meaningful in the
presence of an interface. The eigenenergies of these evanescent states might lie within the
energy gap of the insulator, as in the simplified model corresponding to Fig. 4.2. Now,
if the local density of states is computed including the evanescent states a non vanishing
DOS emerges in the gap [146], as shown schematically in Fig. 4.3(b). Normalization
of the total DOS of the insulator material requires that the DOS carried by the MIGS
must be compensated by a decrease of the DOS at the valence and conduction bands
with respect to those in the unperturbed system [153, 154], as schematically depicted in
Fig. 4.3(b).
In order to discuss the character of the MIGS we should recall that, since the Hamiltonian of a system is an hermitian operator, its eigenfunctions constitute a complete
basis of the Hilbert space. Then, an evanescent state can be expanded in terms of the
eigenstates with real k-vector (physically valid solution of the Schrödinger equation of
the periodic crystal). If we restrict ourselves to a 1-dimensional case, where evanescent
109
Figure 4.3: (a) Schematics of the local density of states (LDOS) of a bulk, infinitely
periodic system, showing the valence and conduction band. (b) Local density of states
modified by the presence of a interface (shadowed area). MIGS are formed in the gap
of the bulk material (darker region). The dashed lines in (b) represent the LDOS of the
bulk material, far from the interface. The charge neutrality level is defined from the
normalization condition of the DOS.
states penetrate along the z direction perpendicular to the interface, this expansion can
be expressed as
ψ̂k̂ (z) =
X
n,k
Cnkk̂ ψnk (r) =
VB
X
n,k
Cnkk̂ ψnk (z) +
CB
X
Cnkk̂ ψnk (z)
(4.7)
n,k
where VB and CB denote sums over eigenstates contained in the valence and conduction
band respectively, and ψ̂(z) represents an evanescent state. From perturbation theory
arguments it follows that the contribution of a given eigenstate with real k-vector, given
by the coefficient Cnkk̂ , is proportional to 1/(E − Enk ) [155], where E is the eigenenergy
of an evanescent state within the gap and En,k is an eigenvalue for a real k-vector. As
a consequence, gap states take their weight primarily from those real bands that are
nearest in energy [154] (allowing for wave function matching and symmetry rules). In
the case of complex bands connecting states from the valence band to states within the
conduction band, there must be a gradual transition across the gap of the character of
110
the MIGS, from valence-like to conduction-like.
Of course, the bulk material is neutral when all states below the top of the valence
band (EV ) are filled, and those above the bottom of the conduction band (EC ) are empty.
However, since evanescent states take their DOS from the “real” DOS (real in the sense
that it comes real bands) filling up to EV would still leave empty part of the valence-like
DOS that has been drawn into the gap. Conversely, filling up to EC would populate
conduction-like evanescent states resulting in a local doping with electrons. Therefore
there must exist an energy inside the gap region such that a filling of all states right up
to it would result in a locally neutral system. The charge neutrality level is precisely this
energy inside the gap of the insulator, marking the transition from mostly valence-like
to mostly conduction-like (see Fig. 4.3). Accordingly, the charge neutrality level can be
defined as the energy ECNL satisfying the following expression
Z
EV
Z
ECNL
ρ̂(E) dE,
ρ(E) dE =
(4.8)
−∞
−∞
where EV denotes the valence band maximum, and ρ̂(E) is the disturbed density of
states at the interface, including both propagating and evanescent states. From Eq.
(4.8) it immediately follows that
Z
EV
−∞
ρ(E) dE −
Z
EV
Z
ECNL
ρ̂(E) dE =
−∞
ρ̂(E) dE.
(4.9)
EV
The left hand side of Eq. (4.9) corresponds to the area between the dashed line and
the shadowed area in Fig. 4.3(b) for energies below EV , while the right hand side is the
integral of the MIGS from EV to ECNL . An equivalent condition can be defined in terms
of the conduction band. Eq. (4.9) reflects the fact, mentioned above, that the DOS
of the MIGS is taken from the valence and conduction band DOS of the ideal infinite
material.
Numerical estimation of the charge neutrality level
Since ρ̂(E) emerges exclusively in the presence of an interface, the use of Eq. (4.8)
to compute the charge neutrality level would require the calculation of a a very well
converged DOS of the actual interface. Nevertheless, more practical ways to calculate
the charge neutrality level in terms of the bulk spectrum of eigenstates with real k-vector
can be defined. In particular, as a consequence of the 1/(E − Enk ) dependence of the
coefficients in expansion of Eq. (4.7), the change of the character of the MIGS from
dominantly valence-like to conduction-like can be linked to the change of sign of the
Green function of the material [156, 154] The cell-averaged Green function is defined as
G(R, E) =
X
n,k
eikR
.
E − Enk
(4.10)
111
Figure 4.4: Complex band structure of an alkane chain. The right panel shows the
conventional band structure, for real k-vectors. The left panel shows β, the imaginary
part of the complex solutions. The units of β are such that e−β is the reduction of the
tunneling probability from one carbon to the next along the alkane chain. The red line
indicates the dominant complex band, with the shortest β. Reprinted with permission
from Ref. [152] (DOI: 10.1103/PhysRevB.65.245105).
where R is a lattice vector, k is a Bloch wave vector in the first Brillouin zone, n is the
band index, and Enk is the set of eigenvalues of the system. For a sufficiently large R,
G(R, E) changes its sign at the charge neutrality level [154].
The MIGS in metal/insulator interfaces will decay in one particular direction, the
direction perpendicular to the interface (which we choose as the z direction) while still
obeying the in-plane periodicity. Given the 1-dimensionality of our problem lattice
vectors R are chosen along the z axis. This transforms Eq. (4.10) into a integration
over the in-plane Brillouin zone of a series of Green functions in one dimension [154]
G(m, E) =
X X eik⊥ mc
.
E − Enk
(4.11)
kk n,k⊥
This function however converges slowly with respect to the energy, typically requiring
the sum over hundreds of bands to obtain a converged value of ECNL .
Alternatively, it can be proved that the charge neutrality level of an individual complex band (the energy marking the transition along the band from mostly valence-like
to mostly conduction-like) coincides with its branch point (the energy of infinite slope,
dEk̂ /dk̂, of the band) [156]. For systems displaying one complex band clearly dominating MIGS formation in the band gap region (i.e a complex band with an imaginary part
significantly smaller than the rest and that does not cross with any other), searching for
the branch point might be a relatively simple method to calculate the charge neutrality
112
level. This is the case of some molecular systems like alkane chains [152] which complex
band structure is shown in Fig 4.4. Complex band structure of realistic materials are
explained in detail in next section, but here we can already notice that for this system
there is a complex band (in red in Fig 4.4), connecting the edges of valence and conduction bands, and with an imaginary part in the middle of the gap clearly smaller (larger
penetration length) than the rest of the bands. For this particular case, calculating the
branch point of this individual band is a simple way to estimate the charge neutrality
level [152].
Unfortunately, in general many bands will contribute to the formation of the MIGS
and this analysis is no longer straightforward. In those cases an “effective” branch point
should be calculated. Since the branch point of an individual complex band corresponds
to the maximum imaginary part of the k-vector, q, of the band (minimum penetration
length), the “effective” branch point can be calculated as the energy of the maximally
localized MIGS local density [155], or in other words, the energy of minimum penetration
of the local density considering the overall contribution of all evanescent states. From
the dependence with the energy of the coefficients Cnkk̂ in the expansion given by Eq.
(4.7) it can be deduced that
ρ̂(E) ∝
X
n,k
1
,
(E − En,k )2
(4.12)
then, the maximum localization of MIGS local density is found at the energy satisfying
the condition [155]
X
nk
1
= 0,
(ECNL − Enk )3
(4.13)
This approximation has proved to provide reliable estimations of ECNL for a great variety
of materials [155] and the third power of the denominator results in a much faster
convergence than Eq. 4.11 with respect to the energy.
Determination of Schottky barriers
As pointed out in Sec. 4.1 and discussed extensively in Chapter 3 the magnitudes
governing many of the processes in metal/insulator junctions are the Schottky barriers
(denoted by φn and φp for electrons and holes respectively). The magnitude of these
energy barriers is, in turn, determined by the charge rearrangements that take place
every time a junction between two different materials is formed. We devoted Chapter
3 to the development of rigorous procedures for the analysis of Schottky barriers from
first-principles and the associated limitations. Here we aim to discuss the physical origin
of Schottky barriers and the different effects involved, with especial attention to MIGS,
the topic of this chapter, and the intrinsic role of interfaces.
We have seen that when a metal/insulator junction is formed, propagating wave
functions in the metal with energies in the insulator band gap, penetrate in the latter,
decaying exponentially. The matching of the wave function requires that its symmetry
113
Figure 4.5: Schematic of the band alignment at a metal/insulator interface and the
associated charge transfers.
and the kk vector have to be preserved across the interface. At the insulator side of
the interfaces, all evanescent states below the Fermi level became occupied, populating
states above the charge neutrality level or leaving empty states below it. By definition
of the charge neutrality level, the relative position of EF and ECNL determines the local
net charge of the system and its contribution to the interface dipole.
In Fig. 4.5 an schematic view of a band alignment in a metal/insulator junction
is shown, indicating the position of the valence and conduction bands (EV and EC
respectively), the Fermi level (EF ) and the charge neutrality level (ECNL ).
The alignment of the band structures across the heterojunction takes place by means
of charge rearrangement at the interface. The exchange of charge gives rise to a dipole
which shifts the bands until equilibrium is achieved. In the end, the Schottky barrier,
φn (for electrons, analogous expressions can be derived for holes), is obtained as
φn = EC + ∆V − EF .
(4.14)
As mentioned above, as a consequence of the relative position of EF and ECNL , a net
charge at the interface, σ, gives rise to a dipole σδ, where δ is the average penetration
of the charge density due to evanescent states within the gap. The potential drop across
the interface can be split into two contributions
∆V = ∆VMIGS + ∆V ∗ = −e
σδ
+ ∆V ∗ ,
εε0
(4.15)
114
where we have multiplied the electrostatic potential by the electronic charge to get the
right sign of the energy bands shift. The first term in Eq. (4.15) is the potential step
due to the electronic charge transfer to the MIGS. The term ∆V ∗ includes any other
contribution, in particular the ionic and electronic relaxations that take place when
the junction is formed and that are due to changes with respect to bulk in the chemical
bonding at the interface. Notice that ε in the MIGS term takes into account the screening
of the transferred charge provided by the lattice, so this expression implicitly contain
also some contribution of the lattice relaxations to the formation of the interface dipole.
On the other hand the calculation of the charge transfer σ involves a two-step process.
First, the integral in space from the interface (z = 0) to a position deep in the insulator
(z = ∞) must be performed for the LDOS of MIGS, ρ̂(z, E). This results in a surface
density of MIGS, Ds (E), that then must be integrated between the charge neutrality
level and the Fermi level
σ = −e
Z
EF
Z
dE
ECNL +∆V
0
∞
dz ρ̂(z, E) = −e
Z
EF
dE Ds (E).
(4.16)
ECNL +∆V
Typically, in this kind of models of Schottky barrier formation, Ds (E) is assumed to be
relatively homogeneous in energy within the gap so Ds (E) ∼ N , with N constant [157].
Within this approximation the charge extracted (injected) from (into) the gap states
states below (above) the charge neutrality level is
σ = −e (EF − ECNL − ∆V ) N.
(4.17)
Comparing Eq. (4.15) and (4.17) one notices that the equilibration of the Fermi energy
across the interface involves a self-consistent process where the band alignment causes
a transfer of charge that, in turn, shifts the bands altering the band alignment. The
combination of Eq. (4.15) and (4.17) yields the potential step at equilibrium, ∆V .
Finally, the Schottky barrier, φn , can be obtained from Eq.(4.14) as
φn = S(ECNL − EF + ∆V ∗ ) + (EC − ECNL ),
(4.18)
−1
e2 N δ
S = 1+
.
εε0
(4.19)
with
S is a factor that provides a quantitative estimation of the screening provided by the
MIGS and is discussed in more detail below. But before, let us try to make the connection
with the expressions typically used in empirical models of Schottky barriers.
In Eq. (4.18) and (4.19), all magnitudes except ∆V ∗ can be, in principle, extracted
from bulk calculations of the materials constituting the interface (either from the real
eigenstate spectrum or the complex band structure). However, empirical models usually
rely on different magnitudes, namely the work function of the metal (Wf ) and the
electron affinity of the insulator (χ), which constitute indirect measurements of the
Fermi level and the conduction band bottom respectively (for the determination of the
115
Figure 4.6: Schematic of the band alignment at (a) a metal/vacuum and (b) an insulator/vacuum surfaces.
116
Schottky barrier for holes φp , the ionization potential is used instead of the electron
affinity). The work function is the energy required to extract an electron from the a
metallic surface [see Fig. 4.6(a)]
Wf = Evac + ∆VM − EF .
(4.20)
Analogously, the electron affinity of the insulator is the energy of the conduction band
deep inside the material with respect to the vacuum level [see Fig. 4.6(b)]
χ = Evac + ∆VI − EC .
(4.21)
As a consequence of these two magnitudes being surface measurements, two new potential steps (∆VM and ∆VI ) due to surface dipoles must be considered. Introducing this
expression into Eq. 4.18, and referring also the charge neutrality level in the insulator
vac = E
to the vacuum level, ECNL
vac + ∆VI − ECNL , we end up with
vac
vac
φn = S(Wf − ECNL
− ∆VI − ∆VM + ∆V ∗ ) + (ECNL
− χ).
(4.22)
It is clear from Eq. (4.22) that the modeling of the formation of Schottky barriers from
bulk properties of the interface constituent materials is far from straightforward. ∆VM
and ∆VI are magnitudes intrinsic to the metal and insulator surfaces respectively, and
might depend in a rather complex way on the surface orientation and termination, and
on the strain state of the sample. The term −∆VI − ∆VM + ∆V ∗ , on the other hand, is
necessarily intrinsic to the interface, accounting for the change in the chemical environment from having to separate surfaces to having a close contact between the insulator
and the electrodes, and all the associated atomic and electronic relaxations. It might
seem from equation (4.22) that the role of the MIGS in the Schottky barrier determination (which effect is collected by the factor S) should be, in most cases, overwhelmed
by the interface-intrinsic terms. However, the usual approach is precisely the opposite,
and model for Schottky barrier formation tend to neglect the effect of changes in the
chemical bonding and model the formation of Schottky barriers relying purely on the
contribution from the MIGS. This transforms Eq. (4.22) into
vac
vac
φn = S(Wf − ECNL
) + (ECNL
− χ).
(4.23)
Despite the crude approximation this expression is implicitly assuming, this simple model
for the prediction of Schottky barriers from bulk properties has proved to be surprisingly
predictive in the case of metal/semiconductor contacts [148, 158, 146]. In this context,
the factor S is often referred as the “slope parameter” because when Schottky barriers
of a semiconductor are plotted as a function of the work function of different metallic
electrodes, a linear dependence is often found, with the slope being S. Extreme values
of S within MIGS model describe two limit regimes. Either if the DOS of gap states is
very low or their penetration is very small and the transfer of charge that takes place is
negligible; or if there is indeed a transfer of charge but it is efficiently screened by the
lattice, S → 1 and the Schottky barrier for electrons is simply φn = ΦM − χ. This is
usually called the “Schottky limit”. If on the contrary, the DOS is large, any deviation
4.3. Computational details
117
of the Fermi level from the charge neutrality level results in a large transfer of charge
that in turn, produces a dipole that pushes the Fermi level back towards ECNL . In this
limit case, called “Bardeen limit”, S → 0 and φn = ECNL − χ, and the Fermi energy on
the insulator side is effectively pinned at the charge neutrality level.
Unfortunately it remains unclear whether S in Eq. (4.23) is really moslty due to
formation of evanescent states, as it is in Eq. (4.22), or if interface-intrinsic effects
could also account for a dependence of the Schottky barriers given by Eq. (4.23). With
the improve in quality of interfaces and the first principles studies, larger deviations
from the MIGS model are being observed, highlighting the importance bonding effects
due to details of the chemical environment at the interface [159, 142]. Despite showing
some signs of possible pathological band alignment in some of the studied interfaces
(see Sec. 3.6.5 for a more detailed discussion on this point), Ref. [142] provide and
insightful analysis of the role of interface-intrinsic effects, confronted with the parameters
extracted from surface experimental measurements or obtained from bulk first-principles
simulations. However, in this work the contribution from the MIGS was not explicitly
treated. We will try in Sec. 4.5 to analyze how common assumptions in MIGS formation
and their participation in the Schottky barrier hold when the real interface is explicitly
described within a first principles simulation.
4.3
Computational details
Two codes have been used in this work, Siesta [121, 160] for the simulation of the
complete interfaces, and the Quantum-espresso [161, 162] package for the complex band
structure calculations [163]. The use of two different codes is a delicate issue and thus
keeping strict convergence criteria becomes critical in order to ensure the compatibility
of all the simulations.
In order to mimic the effect of the mechanical boundary conditions due to the strain
imposed by the substrate, the in-plane lattice constant was fixed to the theoretical
equilibrium lattice constant of bulk SrTiO3 (aSTO = 3.874 Å for Siesta and aSTO =
3.850 Å for Quantum-espresso).
Quantum-ESPRESSO
Calculations in Sec. 4.4 on bulk PbTiO3 were performed using different codes included
in the Quantum-Espresso package. Relaxations were carried out with pwscf using a
plane-wave cutoff of 40 Ry, a 12 × 12 × 12 Monkhorst-Pack [90, 113] mesh and ultrasoft
pseudopotentials [164]. Froce and stress thresholds during the relaxations were 10−4 a.u.
and 10−3 kbar respectively.
Complex band structures were obtained with the code pwcond [163] included in the
Quantum-Espresso package.
118
SIESTA
Computations in Sec. 4.5 on short-circuited SrRuO3 /PbTiO3 capacitors have been performed within a numerical atomic orbital method, as implemented in the Siesta code.
Core electrons were replaced by fully-separable [122] norm-conserving pseudopotentials,
generated following the recipe given by Troullier and Martins [123]. Further details on
the pseudopotentials and basis sets can be found in Ref. [124].
A 12 × 12 × 2 Monkhorst-Pack [90, 113] mesh was used for the sampling of the
reciprocal space of the capacitor, equivalent to a 12 × 12 × 2 Monkhorst-Pack mesh in
the bulk primitive cell. A Fermi-Dirac distribution was chosen for the occupation of the
one-particle Kohn-Sham electronic eigenstates, with a smearing temperature of 8 meV
(100 K). The electronic density, Hartree, and exchange-correlation potentials, as well as
the corresponding matrix elements between the basis orbitals, are computed in a uniform
real space grid, with an equivalent plane-wave cutoff of 1200 Ry in the representation of
the charge density.
To simulate the capacitors in a non polar configuration, we impose a mirror symmetry
plane at the central TiO2 layer, and relax the out-of-plane stress and the internal forces
of the resulting centrosymmetric tetragonal phases until they are smaller than 10−4
eV/Å3 (= 1.602 · 10−3 kbar = 1.602 · 10−4 GPa) and 0.01 eV/Å(= 1.945 · 10−4 a.u.),
respectively.
4.3.1
Compatibility tests
The tests we have performed have shown that the parameters above provide converged
values of the atomic (Table 4.1) and band structures (Fig.4.7) of the bulk material.
Given the sensitivity of complex bands to the size of the gap and the curvature of real
bands at the connecting k-points [165], a good agreement between the band structures
obtained with both codes is required to obtain comparable results. Fig. 4.7 shows that
the band structures obtained with both codes are essentially indistinguishable in shape,
although the pwscf calculations display a slightly larger gap. This must be taken into
account since larger gap is going to translate in slightly larger values of the imaginary
part of complex wave vectors.
It is also important to note that the bulk paraelectric phase we are discussing here
is not the cubic phase, but a tetragonal centrosymetric P 4/mmm phase. We compare
complex band structure calculations on bulk PbTiO3 with properties of MIGS in a
capacitor where the out-of-plane cell vector is allowed to relax. As required by the
strong sensitivity of complex bands on the real band structure, bulk calculations must
be performed under the same symmetry constrains applied to the PbTiO3 layer in the
capacitor.
4.4
Complex band structure of bulk PbTiO3
We have seen in Sec. 4.2 how imaginary bands appear naturally even at the level of the
nearly-free electron model. Such a simplistic model is useful to illustrate the origin of
119
4.4. Complex band structure of bulk PbTiO3
Table 4.1: Lattice vectors of PbTiO3 and SrTiO3 with Siesta and pwscf. For SrTiO3
u = a while for PbTiO3 u = c calculated imposing an in-plane lattice constant equal to
that of SrTiO3 (aPTO
= aSTO ). Experimentally, at room temperature, in-plane lattice
k
constant of bulk PbTiO3 is virtually the same than aSTO , this justifies the comparison with the experimental value of c/a in bulk PbTiO3 . Values in brackets are the
tetragonality c/a.
cubic
non polar
polar
upwscf (Å)
3.850
3.901 (1.013)
4.012 (1.042)
5
5
4
4
3
3
2
2
1
1.558 eV
Eg = 1.462 eV
2.660 eV
0
Energy (eV)
Energy (eV)
SrTiO3
PbTiO3
usiesta (Å)
3.874
3.907 (1.009)
4.030 (1.040)
-1
1
1.412 eV
2.544 eV
Eg = 1.354 eV
0
-1
-2
-2
-3
-3
-4
-4
-5
uExp (Å)
3.905
4.156 (1.064)
Γ
Z
R
A
Γ
X
-5
Γ
Z
R
A
Γ
X
Figure 4.7: Band structures of bulk non-polar PbTiO3 obtained with pwscf (left) and
Siesta (right). The symmetry of the unit cell is P4/mmm, since the in-plane lattice
constant was fixed to the theoretical one of SrTiO3 and the out-of plane lattice constant
was allowed to relax, while atoms were kept in the centrosymmetric positions. A direct
gap is found at X.
the evanescent states characterized by the complex wave vectors, but as for the usual
band structure calculation, real systems are much more complicated. The idea behind a
complex band structure calculation within DFT though, can be perfectly extrapolated
from the example detailed in previous Section.
Usually, conventional band structure calculations involve solving the energy eigenvalue problem scanning over real values of k-vectors along high symmetry directions in
k-space. This procedure produces the well known band structure E(k). This method is
convenient because very often interesting features of the real band structure are located
at high symmetry k-points, and a scan along high symmetry directions usually provide
the information we need for the interpretation of multiple physical phenomena.
Similarly one could also perform a scan over complex values of the wave vectors
120
2
Energy (eV)
1
0
-
-1
-2
1
0.5
q (2π/c)
0
0.5
kz (2π/c)
0.5
q (2π/c)
1
Figure 4.8: Complex band structure of centrosymmetric P4/mmm PbTiO3 at kk = Γ̄.
Central panel correspond to <(k⊥ ) and side panels to =(k⊥ ). The dashed lines delimit
the band gap. Color code is described in the main body of Sec. 4.4.
to obtain the complex band structure. Nevertheless, given the kind of information
provided by imaginary wave vectors (evanescent states, tunneling, etc.), when we perform
a complex band structure calculation we are usually interested in small energy ranges
(the band gap) and large areas of the Brillouin zone, so previous approach turns out
to be quite inefficient. An alternative approach is typically used instead: the scan is
performed over the energies and the spectrum of k-vectors producing real values of the
energy is obtained. Notice, looking at Fig. 4.2 for instance, that under this approach
imaginary bands emerge in a completely transparent way.
As pointed out in Sec. 4.2.1 imaginary bands always connect extrema of the real
bands. These extrema are most of the times located at the high symmetry points in the
Brillouin zone, which means that at those special k-points there will always be a great
density of imaginary bands. Besides, at the energies of interest, i.e. the energy gap,
the band with the shortest q (largest penetration length) is typically one that connects
the edges of the valence and conduction bands. The analysis of the real band structure
depicted in Fig. 4.7 reveals that, in the non polar P4/mmmm phase, PbTiO3 posses
a direct gap located at X, and a relatively narrow gap, slightly larger than the direct
one at Z. Therefore we can presume that in non-polar PbTiO3 , complex bands with
the shortest imaginary wave vector q are going to depart from the those high symmetry
points of the Brillouin zone. Such complex wave vectors will have the form (π/a, 0, iq)
and (0, 0, π/a + iq) respectively, which correspond to complex bands at kk = X̄ and Γ̄
in the 2-dimensional Brillouin zone parallel to the interface.
Complex band structures of bulk non polar PbTiO3 was obtained performing a sampling of 48 × 48 k-points in the 2-dimensional Brillouin zone. Since important features
121
2
Energy (eV)
1
0
-1
-2
1
0.5
q (2π/c)
0
0.5
kz (2π/c)
0.5
q (2π/c)
1
Figure 4.9: Complex band structure of centrosymmetric P4/mmm PbTiO3 at kk = X̄.
Central panel correspond to <(k⊥ ) and side panels to =(k⊥ ). The dashed lines delimit
the band gap. Color code is the same as in Fig. 4.8 and is described in the main body
of Sec. 4.4.
of the complex band structure take place at high symmetry k-points, a centered k-point
mesh was used to include the special points. Complex band structure at Γ̄ (Γ → Z path
in the 3-dimensional Brillouin zone) and X̄ (X → R path in the 3-dimensional Brillouin
zone) are shown in Fig. 4.8 and 4.9 respectively.
Some of the features described in Sec. 4.2.1 for the simple model are also observed in
the DFT complex band structure of PbTiO3 . Imaginary bands always connect extrema of
the real bands (and also of complex bands), usually located at the high symmetry points
in the Brillouin zone. Taking this into account, and following the notation established
by C.-Y Chang [165], complex bands at every kk can be classified into four different
categories:
(i) Real bands, with q = 0 (black lines in the central panel of Fig. 4.8 and 4.9).
(ii) Imaginary bands with q 6= 0 and kz = 0 (black lines in the left panel), that we will
label as “of the first kind” after the notation introduced by C.-Y Chang [165] .
(iii) Imaginary bands of the second kind, with q 6= 0 and kz = π/a (black lines in the
right panel).
(iv) Complex bands with q 6= 0 and kz 6= 0 or π/a (real part of these bands is plotted
as a red line in the central panel and imaginary part as a red line in the left or
right panel depending if they connect imaginary bands of the first or second kind
respectively). Also complex bands connecting imaginary bands of different kind
can exist but we don’t find any at these k-points and energy ranges.
122
Figure 4.10: Minimum value of q (corresponding to maximum penetration lengths) for
values of kk over a quadrant of the 2D Brillouin zone.
We observe in Fig. 4.8 and 4.9 that at the kk defining the gap of the material
(delimited with dashed lines in Fig. 4.8 and 4.9) , and for energies around the middle of
the gap, the bands with the shortest q (largest penetration length) take values from 0.10
to 0.15 in units of (2π/c). Bands around these two high symmetry k-points in the 2-D
Brillouin zone are expected to dominate tunnel conductivity phenomena [166]. However
these k-points represent a very small fraction of the area of the 2-D Brillouin zone. In
Fig. 4.10 we plot the surface of minimal q over one whole quadrant of the 2-D Brillouin
zone for an energy in the middle of the gap. We can see that bands with the lowest
decay rates are located at Γ̄ (corresponding to the branch departing from the top of
the valence band at kz = π/c in Fig. 4.8, i.e. the Z point in the 3-D Brillouin zone).
¯ direction, from Γ̄ to X̄.
There is also an important contribution coming from kk in the ∆
However, besides the contribution from high symmetry points and paths, there is also a
¯ path with still small values of
large, relatively flat area out of the axes defined by the ∆
q (q ∼ 0.2, red plateau in Fig. 4.10). These states could also play an important role, not
only in Schottky barrier formation, but also in tunneling phenomena [166], depending
on the shape of the Fermi surface of the metal [129].
The complex band structures at Γ̄ and X̄ plotted in Fig, 4.8 and 4.9 respectively,
might not be representative of the whole complex band structure of the system, since
they represent such a small fraction of the total 2-D Brillouin zone. The integrated
contribution from the complete 2-D Brillouin zone to the complex band structure is
analyzed in Fig. 4.11(a), where we plot the the density of states, Nq (E), due to all kk ,
123
Figure 4.11: (a) Density of states with respect to the the energy and the imaginary
wave vector q. This histogram in two variables is generated considering the complex
bands at every kk over the 2-D Brillouin zone. Then, the density of complex bands [the
density of q(E) points] is plotted as a gray scale, where darker regions represent a larger
density of bands. The zero of energies corresponds to the top of the valence band. (b)
A slice of the density of states plotted in (a) for and energy E = 0.7 eV above the top
of the valence band [dashed line in (a)].
124
Nqdecay (a.u.)
m=4
m=8
m=16
0
0.05
0.1
m=32
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
q (2π/c)
Figure 4.12: q-resolved density of states weigthed with the decay rate e−2qm , measuring
the tunnel probability of states with imaginary wave vector q across a barrier with a
thickness of m unit cells.
with respect to the energy E and the imaginary part of the wave vector q. Fig. 4.11(a) is
actually a 2-D histogram in which the gray scale indicate the number of complex bands
contributing to each q(E) point. Darker regions indicate a larger density of complex
bands. A careful analysis of this plot reveals that those bands at Γ̄ with q ∼ 0.1 (2π/c)
(see Fig. 4.8) are completely inappreciable here, since only one kk of the 2-D Brillouin
zone is contributing. On the other hand, the total complex band structure is clearly
dominated by bands at q ∼ 0.2 and q ∼ 0.3 in units of (2π/c).
The relative weight of different values of q in the density of states can be better
quantified plotting a slice of the Nq (E) of Fig. 4.11(a). In Fig. 4.11(b) we plot the
profile of Nq (E) for and energy in the middle of the gap [E = 0.7 eV above the top of
the valence band, dashed line in Fig. 4.11(b)]. Here we see that the weight of the bands
at q ∼ 0.2(2π/c) and q ∼ 0.3(2π/c) is about 100 times larger than the contribution from
Γ̄ [showing up in Fig. 4.11(b) as a weak band at q ∼ 0.1(2π/c)]
In order to get a qualitative hint about the relative importance of states with larger
q in the tunneling phenomena we have plotted in Fig. 4.12 the density of states relative
to q, Nq , weighted with the tunnel probability e−2qm 1 , for various barrier thicknesses
m. This quantity represent the relative contribution to the tunneling of states with
different values of q [166]. We observe that the large band of Nq around 0.2 in units
of (2π/c) decreases rapidly its weight as the barrier thickness increases. Nevertheless,
for thicknesses up to 16 unit cells the contribution to the tunneling of these states are
still as important as that of the most penetrating band at Γ̄. The contribution from Γ̄
becomes dominant for m ∼ 32. This result highlights the necessity of considering higher
1
The decay rate of an evanescent wave function is given by q. The decay of the probability density
(or equivalently, the charge density) is given by the square of the wave function, explaining the factor 2
in the exponential.
125
order imaginary bands and non high-symmetry points in the 2-D Brillouin zone in the
analysis of tunneling phenomena.
The polarization of PbTiO3 modifies its band structure, and in particular the band
gap is increased. Even though the indirect gap (from X to Γ) changes little within LDA
(about 5%), there is very significant opening of the direct gap at the X high-symmetry
point. Most of the complex bands only connect real bands departing at the same kpoint, thus the development of a polarization in the ferroelectric layer should cause an
increase of the minimal imaginary wave vector at this particular point of the Brillouin
zone, reducing the tunneling probability across the barrier. If analogous calculations
of the complex band structure were to be done for the ferroelectric configuration of
the capacitor, they should be performed at the same electric displacement field as the
PbTiO3 layer in the capacitor.
The analysis of the complex band structure also provides relevant information about
the Schottky barrier formation, for instance, allowing the estimation of the effective
penetration of MIGS, δ in Eq. (4.19). This quantity represents the center of mass of the
charge contained in the MIGS. This argumentation comes with a caveat: not all complex
bands (as calculated for the bulk material) contribute to the formation of MIGS at the
interface; matching arguments (including symmetry of the bands and preservation of
the kk across the interface) ultimately determine their distribution. Taking this into
account requires a realistic simulation of the interface, as will be done in Sec. 4.5.
Keeping this consideration aside, we can still obtain a crude estimation of δ from the
complex band structure. If we choose the interface to be located at z = 0, the center of
mass of the probability density of one individual exponentially-decaying state is located
as z = 1/2q. Then the individual decay length of each band should be weighted by
the integral of the corresponding exponential decay, 1/2q as well (here we are again
disregarding any prefactor that might arise from the matching of the wave function at
the interface). Under these approximation, the collective penetration of MIGS can be
estimated from the complex band structure as
1
X
δ(E) =
ni ,kk
1
2qni kk (E) 2qni kk (E)
X
ni ,kk
1
2qni kk (E)
X
ni ,kk
= X
ni ,kk
1
4qn2 i kk (E)
1
,
(4.24)
2qni kk (E)
where ni denotes the imaginary band index. In practice sums in Eq. (4.24) are found
to be well converged for q < 3, yielding a value of δ = 0.20 unit cells (or equivalently
1.47 Bohrs) for an energy in the middle of the gap. This quantity will be compared in
next section with the effective decay of MIGS obtained from the simulation of a whole
metal/PbTiO3 interface.
Also important for the study of Schottky barriers is the charge neutrality level, which
can be obtained from the spectrum eigenstates with real k-vector of bulk PbTiO3 , as
explained in Sec. 4.2.2. The eigenstate spectrum was obtained performing a non-selfconsistent calculation with a centered 72 × 72 × 72 k-point sampling. Using Eq. (4.10)
126
convergence of the charge neutrality was not achieved using up to 80 bands. Eq. (4.11)
on the other hand, converged with just about 20 bands to a value of ECNL = 0.66 eV
above the top of the valence band, close to the mid-gap energy, located at 0.69 eV with
respect to the VBM. As a consequence of the well known band-gap underestimation by
LDA, if this value were to be compared with experimental measurements of Schottky
barriers it should be scaled with the experimental band gap of PbTiO3 of 3.40 eV [126],
yielding a value of 1.66 eV above the top of the valence band.
4.5
MIGS in ab initio simulations of ferroelectric capacitors
A more rigorous approach for the study of the properties of MIGS in the metal/ferroelectric
interface is the simulation of an actual capacitor from first principles. Results from the
full simulation not only would account for properties of MIGS derived from the complex
band structure of the bulk ferroelectric but also for any intrinsic interfacial property
that could affect the gap states.
We have simulated a SrRuO3 /PbTiO3 capacitor as model system. The ferroelectric
film is 8.5 unit cells thick, terminated in a TiO2 atomic layer. The metal electrode is 5.5
unit cells thick, terminated in SrO. Test simulations with electrodes up to 9.5 unit cell
thick were performed to confirm that interfacial effects are perfectly screened within the
metal but results showed no difference with respect to simulations with 5.5 unit-cell-thick
electrodes so only results on the latter are reported.
The analysis of MIGS in full capacitor simulations requires to work with some sort of
energy-resolved probability density. For this, the LDOS defined as in Eq. (3.26) could
be used. However, strong oscillations of this function due to the underlying atomic
structure could difficult the analysis. Alternatively, the nanosmoothed version of this
function might be used, but some interfacial properties, like precisely the decay length
of the MIGS charge in the band gap, are sensitive to the specific convolution function
used for the nanosmoothing procedure.
A reasonable choice is to work with the layer-by-layer (z-resolved) PDOS (E-resolved).
In this case the bias of the method lies in the choice of the basis of atomic orbitals.
However, a sufficiently converged basis should minimize its effect and atoms of the same
species at different sites are equally described, so the z dependence might be considered
as less biased than with previous methods. In Fig. 4.13 we plot the layer-by-layer PDOS
of the simulated capacitor. The energy distribution of the charge density (the PDOS in
this particular case, the DOS in general) converges much slower with the k-point sampling than its spatial distribution, for this reason the PDOS was calculated performing
an extra non-self consistent calculation with a finer k-point grid of 54 × 54 × 9. Only
half of the atomic layers of the ferroelectric are plotted in Fig. 4.13 due to the mirror
symmetry plane present in the non-polar configuration. The fast decay of the MIGS can
be clearly observed, being negligible for atomic layer further than 2 unit cells from the
interface.
We can obtain a measure of the spatial distribution of the probability density at a
4.5. MIGS in ab initio simulations of ferroelectric capacitors
127
Figure 4.13: Layer-by-layer projected density of states of the ferroelectric inside the
capacitor. The plot at the bottom corresponds to the atomic layer at the interface with
the electrode, while the plot at the top is the PDOS corresponding to the layer in the
middle of the ferroelectric film. The dashed lines delimit one of the energy windows used
to perform the integrations in Eq. (4.25).
128
given energy integrating the PDOS in small energy windows centered at different energies
inside the band gap
Z
QPDOS (Ei , zj ) =
Ei + ∆E
2
Ei − ∆E
2
PDOSj (E)dE,
(4.25)
where subindex j denotes the atomic layer. The decay of this quantity into the ferroelectric provide a direct way to obtain the effective decay length δ(E) of the gap evanescent
states. Alternatively we can talk about an effective imaginary wave vector qeff (E), describing the effective decay of the wave functions. Since δ(E) is obtained from the decay
of PDOS, connected with the probability density, it is related with qeff (E) (the decay
rate of the wave function) as
δ(E) = 1/2qeff (E).
(4.26)
To obtain the dependence of the decay length, or conversely the effective imaginary wave
vector qeff (E), with the energy inside the gap, we have integrated the PDOS using Eq.
∆E
(4.25) inside energy windows [Ei − ∆E
2 , Ei + 2 ] with ∆E = 0.02 eV (one of this windows
is indicated in Fig. 4.13 as the energy range between the dashed lines). An example of
QPDOS (Ei , zj ) is plotted as circles in Fig. 4.14 for an energy window in the middle of
the band gap. Then the effective value of the imaginary wave vector is obtained fitting
those points to a function
QPDOS (E, z) ' cosh(2qeff (E)z),
(4.27)
which is plotted in Fig. 4.14 as a black dashed line. A hyperbolic cosine was used to
account for the presence of two interfaces, although the analysis in Fig. 4.13 and 4.14
shows that there is very little overlap of MIGS coming from opposite interfaces.
Repeating this procedure for energy windows covering the whole energy gap of the
ferroelectric we obtain the energy dependence of the effective imaginary wave vector,
qeff (E). This effective value of the imaginary wave vector is plotted as a solid black line
in Fig. 4.15 as a function of the energy. It is legit to assume that this qeff (E) reflects the
collective contribution of all complex bands in the gap of PbTiO3 . In the background of
Fig. 4.15 we plot a zoomed region of the density of complex bands Nq (E) with respect
to the energy and the imaginary wave vector for bulk PbTiO3 , as previously plotted
in Fig. 4.11(a) and discussed in Sec. 4.4. In systems with one complex band with a
imaginary part notably smaller than the rest (see Fig. 4.4), effective decay rates obtained
from the LDOS of the actual interface has been found to overlap very accurately with
that (dominant) complex band [152]. In the present case there are clearly two groups
of bands clustered at q ∼ 0.2 and ∼ 0.3 in units of (2π/c) that dominate the complex
band structure of bulk PbTiO3 . This is reflected in the qeff obtained from the PDOS of
the actual interface, which oscillates between those two values throughout the gap. The
contribution of the Γ̄ point to smaller values of q (invisible in Fig. 4.15 due to its low
weight) is overwhelmed by the much larger area of the 2-D Brillouin zone contributing
to larger values of q. As a final remark regarding Fig. 4.15, we should mention that the
129
3.5
3
log10(QPDOS) (a.u.)
2.5
2
1.5
1
0.5
0
−0.5
TiO2 PbO TiO PbO TiO2 PbO TiO PbO TiO PbO TiO PbO TiO PbO TiO PbO TiO2
2
2
2
2
2
2
Figure 4.14: Charge (in arbitrary units) obtained integrating the layer-by-layer PDOS
in an energy window in the middle of the band gap (black circles). The decay of the
charge inside the ferroelectric material is fitted to a curve following Eq. (4.27) centered
at the mirror symmetry plane located in the middle of the capacitor (red curve).
larger gap obtained within pwscf (see Fig, 4.7) is responsible for the imaginary bands
dropping to q → 0 at about 0.1 eV above the bottom of the conduction band of the
capacitors (simulated with Siesta).
Another quantity which is relevant to the MIGS and in connection with the Schottky
barrier formation is the surface density of metal-induced gap states Ds (E). This is can
be computed as
Z
1 ∞
Ds (E) =
ρ(r, E)dz,
(4.28)
A 0
where A is the interface area [(3.874 Å)2 in our case], ρ(r, E) is the LDOS as defined in
Eq. (3.26), and the integral over z is to be performed from the interface to deep into the
ferroelectric layer [in Eq. (4.28) we are assuming a semi-infinite insulator]. Performing
the integral of Eq. (4.28) is equivalent to sum the PDOS plotted in Fig. 4.13 over all
atomic layers in the ferroelectric. The resulting surface density of states is plotted in
Fig. 4.16.
With the knowledge of the effective decay rate of gap states and their density of states,
shown in Fig. 4.15 and Fig. 4.16 respectively, an estimation of the slope parameter can
be obtained. The definition of the slope parameter S in Sec. 4.2.2 assumes a constant
value of both the penetration length and the surface density of states within the gap.
In particular, the integral in Eq. (4.16) is approximated by an average density of states,
N , times the energy difference between the Fermi level and the charge neutrality level,
130
Figure 4.15: Effective imaginary wave vector qeff obtained fitting the decay of the PDOS
inside the ferroelectric material of the capacitor (black line) and imaginary wave vectors
at high symmetry points of the 2-dimensional Brillouin zone. In the background we
plotted the density of complex bands of bulk PbTiO3 integrated over the whole 2-D
Brillouin, as was previously plotted in Fig. 4.11(a) and discussed in Sec. 4.4. The
energy of the bulk complex band structure has been shifted to align its valence band
maximum with that of the PbTiO3 layer in the capacitor.
assuming a slow variation of Ds (E). Curves in Fig. 4.15 and Fig. 4.16, however, display
a rather complex dependence of both the penetration length and the surface density of
states with the energy in the gap. A relatively strong pinning of the Fermi level would
make the estimation of the slope parameter very sensitive to the particular values of the
density of states and penetration length of the MIGS around (and close to) the charge
neutrality level. If, on the contrary, the Fermi level is significantly shifted from the
charge neutrality level, the details of the curves in Fig. 4.15 and Fig. 4.16 are averaged
out and the use of constant values of the effective decay hδi and the surface density of
states inside the gap hDs i [N in Eq. (4.19)] is justified. Here we find a large shift of the
Fermi level with respect to the charge neutrality energy (EF ' ECNL + 0.31 eV, using
ECNL obtained in Sec. 4.4), supporting the use of averaged values.
The effective decay hδi and the surface density of states inside the gap hDs i, as
obtained from the simulation of the actual capacitor, are gathered in Tab. 4.2, together
with an approximated value of the dielectric constant of PbTiO3 . These magnitudes
131
0.20
Ds (states/eV)
0.15
0.10
0.05
0.00
-2
-1
-1.5
0
-0.5
0.5
1
Energy (eV)
Figure 4.16: Surface density of metal-induced gaps states in the PbTiO3 /SrRuO3 capacitor.
Table 4.2: Parameters for the calculation of the slope parameter. Averaged values of
the penetration length (in atomic units) and surface density of MIGS (in states/eV),
relative permittivity of PbTiO3 and estimated value of the slope parameter.
hδi
2.65
hDs i
0.0512
ε
100
S
0.99
yield an estimated value of S = 0.99, very close indeed to the Schottky limit (weak
pinning).
The value for the slope parameter we obtain from the MIGS study in the whole
capacitor is much larger than that obtained by Robertson et. al. by means of a tight
binding model fitted to experimental photoemission and optical data [157]. These authors report a value of S = 0.31 for PbTiO3 , far from the Schottky limit we found. Our
large value of the slope parameter is mainly a consequence of the relatively small density
of states in the gap. Unfortunately this method for the calculation of the slope parameter S has the limitation of the intrinsic arbitrariness of the position of the interface.
Calculation of the surface density of states of the MIGS relies on the integration of the
DOS of gap states from the interface to a position deep in the insulator layer, as in Eq.
(4.28). Given the exponential decay of MIGS, small changes in the chosen position of
the interface can modify severely the estimated value of the surface density of states.
PDOS provides an intuitive criterion for the calculation of the surface density of states,
132
performing the integration over the DOS projected over all the atoms in the insulator
layer. Nevertheless the arbitrariness persist, since interfacial atomic layers (SrO on the
electrode side and TiO2 on the ferroelectric) might be regarded as partially metallic or
insulating. However, the dependence of S obtained from Eq. (4.19) on the density of
states is very weak: increasing the value of hDs i by hand in one order of magnitude
only reduces S to about 0.92. This suggest that, even if the result of S ∼ 1 is probably
unrealistic due, mainly, to limitations in the determination of N , the formation of the
interface dipole (and as a result of the Schottky barriers) in this system is probably
dominated by interface-intrinsic atomic relaxations and not by the contribution of the
MIGS.
4.6
Discussion and perspectives
The research reported in this Chapter is still a work in progress. Here we have made
an effort to compile many different aspects related to the MIGS that arise often in the
literature, and we have perform a detailed characterization for the case of a ferroelectric
capacitor. These properties are often discussed in different contexts (tunneling, Schottky
barriers, screening, etc) but are rarely connected, making difficult to have a broad picture
of the MIGS properties and the various phenomena they are involved in.
Here, we have seen that basic characteristics of MIGS can be traced back to properties
of the complex band structure in the bulk insulator material. We have seen that even
disregarding the matching of the wave function at the interface, the penetration of the
evanescent states clearly resemble the complex bands. Less has been discussed about
the DOS of MIGS, for this the matching of the wave function is expected to play a
more important role than in the case of the penetration length. How this affects derived
properties should be investigated in more detail.
At the same time, our study highlights the importance of a careful analysis of the
complex band structure: common approximations, like considering only bands at high
symmetry k-point in the 2D-Brillouin zone must be performed with great care, specially
in perovskite oxides where many kk , and many bands at each kk , might contribute
similarly. This consideration directly affects the energetic and spatial distribution of
the evanescent states, which is, of course, of great importance for the analysis of the
tunneling conductivity of the junctions or the role of the MIGS on the interface dipole.
We have seen that the MIGS model for the analysis of Schottky barriers only reflects
part of the process that takes place during charge rearrangement at an interface. Previous works have already highlighted the importance of atomic relaxations and chemical
bonding in the formation of the interfacial dipole [159, 142]. Separate different contributions to this problem is tremendously complicated. The best effort so far has been, to our
knowledge, the work by Mrovec et. al. [142]. This work demonstrates the importance of
the interface details (termination of the surfaces, atomic relaxation or type of electrode)
in the Schottky barrier formation, separating the atomic relaxations with the electronic
transfer of charge. However, we have seen in the derivation and breakdown of Eq. (4.18)
that both atomic an electronic rearrangements are a mix of contributions intrinsic to the
4.6. Discussion and perspectives
133
constituting materials and to the interface. Here we are trying to decouple, at least, the
part due to the MIGS by correlating their properties to the bulk complex band structure;
pointing out, at the same time, the extent of validity of the approximations.
For instance, interface-intrinsic effects like the matching of the wave function or the
changes in the chemical environment with respect to bulk or the surface, is likely to
modify significantly, not only the DOS of MIGS, but also the charge neutrality level.
Charge neutrality level has always be considered a bulk property of the insulator, however
it is a property derived from evanescent states that only emerge in the presence of a
surface or an interface. If we admit that the particular details of the interface are
playing a role in the energetic and spatial distribution of the evanescent states, it follows
that the charge neutrality level itself can vary from one particular interface to another.
Nevertheless, MIGS are not relevant only in the context of Schottky barriers, even
beyond the discussion about its relative importance in the formation of the interface
dipole, it is important to ascertain to what extent the MIGS properties themselves are
properties of the bulk material or of the interface. Some more insight in this direction
could be gained improving the connection between the complex band structure and the
MIGS distribution in the actual interface. Considerations about the matching conditions
at the interface are very likely to be important. For this, introducing new ingredients in
the model, like the k-resolved DOS of the electrodes, is desirable.
A definite test about the intrinsic character of MIGS (bulk-like or interface-like) could
be carried out doing simulations on different metal/insulator for the same insulating
material and then, performing the same analysis as in Sec. 4.5. The problem here relies
on the band alignment issues discussed in Chapter 3, since very few metallic electrodes
provide non pathological interfaces.
This is one study that would certainly benefit from the use of improved functionals,
like the one developed by D. Bilc and coworkers [78]. This move would not only improve
the comparison between first-principles simulations and experiments thanks to the better
description of the electronic structure, but it would also increase the number of possible
electrodes to be used in the investigation.
This whole thesis is devoted to the study of ferroelectric thin films. In this Chapter,
although we have focused on a paraelectric capacitor and most of the discussion is quite
general, it is natural to extend the study to the polar case. We have seen in Chapter
3 the dramatic effects the polarization of a capacitor has on its band alignment. The
band structure term is altered, displaying a reduction of the band gap that should affect
the penetration of the MIGS and, by extension, interface dipole and tunneling effects.
At the same time, the bands are tilted as a consequence of the depolarizing field arising
in the ferroelectric film. The effect of the electric field on the MIGS properties will be
explored performing analyses analogous to those of Sec. 4.5 on ferroelectric capacitors
in a polar configuration. The results of this research would be particularly relevant for
the study of ferroelectric tunnel junctions and electroresistance.
Chapter 5
Ferromagnetic-like closure
domains in ferroelectric
capacitors
5.1
Introduction
Previous Chapters focused on interfacial properties in monodomain capacitors, where
polarization distribution – whether when it was zero (as in non-polar configurations
where polarization was not allowed to develop), or not (as in polar configurations) – is
homogeneous throughout the ferroelectric layer. Monodomain phases of ferroelectric thin
films where the polarization points perpendicular to the interfaces/surfaces are known
to be destabilized by the depolarizing field [53], as discussed in Sec. 1.3.2. Previous first
principles local density calculations on realistic short-circuited ferroelectric capacitors
suggested a critical thickness for monodomain phase destabilization that ranged between
m = 2 and m = 6 layers [53, 167, 128] of ferroelectric, depending on the perovskite, the
electrode, and the termination at the interface. In all these approaches the electrode was
the only source of screening, providing free charges that accumulate at the interface on
the metallic side and even decay exponentially into the first few layers of the ferroelectric
(the MIGS discussed in previous Chapter), or sharing the ionic displacements responsible
for the polarization in the ferroelectric [168]. In any case, the mechanism is ineffective
below this critical thickness where the paraelectric phase was stabilized.
It is expected thus, the existence of a transition thickness below which the breaking
up into polarization domains is preferred over the monotonic reduction of the polarization
in order to avoid the increase of electrostatic energy. Exhaustive experimental studies
on the critical size of ferroelectricity have recently observed this transition [169, 170,
171], although it has been found not to happen for some combinations of metal and
ferroelectric material [169]. Thus the question of whether the transition, as the thickness
of the ferroelectric film decreases, takes place as a gradual reduction of the polarization
in a monodomain phase until paraelectric phase is reached below the critical thickness,
or if a breaking up into domains takes place remaines unclear.
135
136
Chapter 5. Ferromagnetic-like closure domains in ferroelectric capacitors
Recently, Landau theory models developed by Bratkovsky and Levanyuk [172, 173]
suggest that polydomain phases always exist in ferroelectric thin films. These authors
even claim that polydomain phases are the most stable phase for any thickness with
monodomain phases being just metastable configurations above the critical thickness for
ferroelectricity.
Model Hamiltonian simulations on PbZr1−x Tix O3 (PZT) alloys also support the
formation of polarization domains for ferroelectricity thin films only a few unit cells thick
[174, 175]. These two theoretical approaches, however, lack the atomic resolution that
might be fundamental at the nanoscale range. First-principles simulations would provide
invaluable information about the domain-wall structure, and the energy balance at the
atomic level, helping to sort out the discussion about the generality of the formation of
polydomain phases and the different effects involved.
In order to study from first principles the breaking up into domains of ferroelectric
ultrathin films we have performed simulations on typical BaTiO3 and PbTiO3 capacitors
with SrRuO3 electrodes. With this study we aim to provide some insights, not only on
which factors favor or hinder the formation of polydomain phases, but also about the
existence of a particular structure of nanosized ferroelectric domains. Results reported
in this Chapter have been published in Ref. [55].
5.2
System and computational details
We have performed first principles simulations within the local density approximation to
the density functional theory and the numerical atomic orbital method as implemented
in the Siesta code [121]. Our starting point is a reference non polar heterostructure
built sandwiching a ferroelectric layer (BaTiO3 or PbTiO3 ) with a thickness of m unit
cells, between 5.5-unit-cells-thick SrRuO3 electrodes. We assumed a TiO2 /SrO interface
since the volatility of Ru makes this the most likely interface to form during the growth
process in experiments. The reference non-polar structure is obtained relaxing both
the atomic coordinates and out-of-plane stress of the capacitor while imposing a mirror
symmetry plane located at the TiO2 layer in the middle of the ferroelectric layer.
Polydomain supercells are build replicating the reference non-polar structure Nx
times along the [100] direction, where Nx ranges from 2 to 8. Due to the periodic
boundary conditions Nx determines the periodicity of the domain structure and we will
refer to it either way throughout the following sections. Then, a soft mode distortion
of the bulk tetragonal phase is superimposed to the BaTiO3 or PbTiO3 layers of the
previous non-polar configuration, so the polarization points upwards in half of the superlattice and downwards in the other half [see inset of Fig. 5.1(a)]. For the BaTiO3
capacitors, the twinning on both the AO (Ba-centred), and TiO2 (Ti-centred) planes is
considered. In the case of PbTiO3 , calculations on bulk show that PbO domain walls
are clearly preferred [61] and are the only option considered here.
The most stable polydomain configuration is obtained relaxing the atomic positions
of all the ions, both in the electrode and in the ferroelectric thin film, until the maximum
component of the force on any atom is smaller than 0.01 eV/Å (for BaTiO3 capacitors
5.3. Structure of polarization domains in ferroelectric thin films
137
with m = 2 and PbTiO3 capacitors with m = 4) or 0.04 eV/Å (for BaTiO3 capacitors
with m = 4). In these system, differences in energy between relevant phases are tiny
(eight orders of magnitude smaller than the absolute value of the energy), requiring very
accurate computations to resolve reliably the relative phase ordering. In order to achieve
the required accuracy, electronic density, Hartree, and exchange-correlation potentials
are computed in a uniform real space grid, with an equivalent plane-wave cutoff of 400
Ry.
We used a Nkx × 12 × 1 Monkhorst-Pack mesh for all the Brillouin zone integrations,
where Nkx = N12x except for the interface with Nx =8, where Nkx = 2. Details on
pseudopotentials and basis set used can be found in Ref. [124].
5.3
Structure of polarization domains in ferroelectric thin
films
Our calculations support the stabilization of a polydomain phase in films with a thickness
below the critical one for a monodomain configuration. This phases display an exceptionally small periodicity below the previous critical thickness [see Fig. 5.1(a) for the
case of a BaTiO3 capacitor], in good agreement with the results obtained with Landau
theory [172].
Within our computational parameters the critical thicknesses for the stability of a
monodomain polarization in the BaTiO3 and PbTiO3 capacitors are m = 4 and m = 8
respectively. For the same ferroelectric material and electrodes our simulations prove
that for a two-unit-cell-thick film (m = 2) in the case of BaTiO3 and a four-unit-cell
thick (m = 4) in the case of PbTiO3 , the extra source of screening due to the formation
of polarization domains is efficient provided that the domain period is between two and
three times the thickness of the film. Landau theory models are indeed in remarkable
agreement with our first-principles simulations, predicting a critical thickness for BaTiO3
capacitors of m = 2 for polydomain phases and m = 6 for monodomain metaestability
[172].
Within this regime, the energy cost of forming the domain wall is compensated by
reduction of the net polarization charge at the interfaces. In the particular case of
PbTiO3 , both the absence of polydomain structures for m = 2 and its existence for
m = 4 is in perfect agreement with x-ray scattering experiments in PbTiO3 thin films
on a SrTiO3 substrate [170].
Figure 5.1(a) shows the relative energy of the polydomain phases in the BaTiO3
capacitors with respect to the position of the domain wall and the domain periodicity.
For the BaTiO3 two possible domain walls where simulated, displaying twining on the
BaO or TiO2 planes. As in 180◦ stripe-domains in bulk [61], the Ba-centered wall
configuration is preferred. The energy difference between the most stable polydomain
and the paraelectric phase for a BaTiO3 capacitor with m = 2 is very small, of the order
of 1.5 meV (' 16 K) for the whole supercell. For this thickness there is essentially no
energy difference between domains of lateral periods Nx = 4 and 6, suggesting that both
might be equally present in a sample. Model Hamiltonian simulations on ferroelectric
138
Figure 5.1: Difference in energy between polydomain and paraelectric phases as a function of (a) the domain period Nx for a BaTiO3 thin film two unit cells thick (m = 2),
and (b) the thickness of the ferroelectric film for a capacitor with Nx = 4. The energy
of the paraelectric phase (dotted line) is taken as reference. First-principles results for
both Ba-centred (circles, solid line) and Ti-centred (squares, dashed) domain walls are
shown. In (a) differences in energies between local minima of the polydomain phase are
represented by error bars. Inset, structure of the ferroelectric capacitor considered. Nx
is the stripe period and m is the thickness of the ferroelectric thin film, in number of
unit cells of the ferroelectric perovskite oxide. In (b) the result for the most stable monodomain configuration is also shown (triangle). Full symbols correspond to constrained
relaxations where no in-plane displacements are allowed.
thin films with larger thicknesses demonstrate that the energy versus domain periodicity
landscape is indeed very flat [176]. Heating or cooling processes might help the system to
overcome potential energy barriers and activate the transition between them. Although
the conductive nature of the substrate is different, this fact might provide an extra source
of explanation [175] for the intriguing richness in behavior of the stripe domain patterns
observed experimentally in PbTiO3 thin films grown on SrTiO3 , where two different
periods coexisted [170]. (Note that our ratio between domain periods, 1.5, is close to
the experimental factor 1.4 for the so-called α and β phases in Ref. [170].)
The energy differences between polydomain and paraelectric phases increase very
quickly with thickness [Fig. 5.1(b)] and amounts to 120 (80) meV for a m = 4 BaTiO3
capacitor with a domain periodicity of Nx = 4 and Ba-centered (Ti-centered) domain
walls (note these energies are per domain period, while values reported in Fig. 5.1(b)
139
Figure 5.2: Schematic representation of the atomic relaxations in patterns of domains of
closure in BaTiO3 /SrRuO3 capacitors with domain period of Nx = 4 (a), and Nx = 6 (b).
Balls, representing atoms, are located at the positions of the reference paraelectric phase.
Atomic displacements for the polydomain configuration after relaxation are represented
by arrows, whose magnitude can be gauged with respect to the displacements in the
bulk tetragonal phase of BaTiO3 at the scale on the left. Only displacements of cations
are displayed for clarity. Dotted lines indicate the position of the domain wall. Only
Ba-centred domains are shown. Similar results are obtained for Ti-centred domains.
have all been divided by Nx in order to compare with monodomain and paraelectric configurations). For this size, the polydomain phases are more stable than the monodomain
configuration, itself more stable than the paraelectric phase by 20 meV.
The minimum energy structures of these ferroelectric capacitors, shown in Fig. 5.2
for BaTiO3 capacitors and Fig. 5.3 for PbTiO3 capacitors, display the closure domain
configuration proposed by Landau and Lifshitz [57] and Kittel [58] for some magnetic
systems. At the center of the ferroelectric layer, the displacement of the atoms and
therefore the corresponding local dipoles, point normal to the interface (coordinate z), as
expected for 180◦ stripe domains. However, upon approaching the ferroelectric/electrode
interface a tilt towards [100] is observed. In PbTiO3 capacitors, the tilt of Pb atoms is
as large as to close the polarization flux within the ferroelectric layer, resembling very
much the closure domains in ferromagnets. Similar domain patterns have been found
using a first-principles effective Hamiltonian for PZT ceramics [174, 175]. However, in
these simulations composition of PZT is very close to its morphotropic phase boundary
140
Figure 5.3: Schematic representation of the atomic relaxations in patterns of domains
of closure in PbTiO3 /SrRuO3 capacitors with domain period of Nx = 6 and m = 4.
Balls, representing atoms, are located at the positions of the reference non polar phase.
Atomic displacements for the polydomain configuration after relaxation are represented
by arrows. Only displacements of cations are displayed for clarity. Dotted lines indicate
the position of the domain wall.
(x = 0.5 in Ref. [174] and x = 0.6 in Ref. [175], while morphotropic phase boundary
lies at x = 0.52). This makes rotation of the polarization less costly than in PbTiO3 ,
which in bulk displays a tetragonal c phase and for which the strain imposed by the
SrTiO3 substrate is even slightly compressive (-0.5% within our simulations). On the
other hand, domains of closure in the PbTiO3 capacitors can be regarded as formed by
90◦ domain walls, which are found to be more stable than 180◦ domain walls in bulk
[61].
Remarkable differences are found between the domains of closure in the PbTiO3 and
BaTiO3 capacitors. In the latter, the domains are not closed by the surface layer of the
ferroelectric, as in PbTiO3 or PZT [174, 175], but by the in-plane displacements of the Sr
and O atoms at the first layer of the electrode, which yield a closure domain pattern, with
90◦ domain walls with the z-oriented domains inside the film. The in-plane displacements
of the atoms at the interfacial SrO layer, although small in magnitude, stabilize the
domain structure. If a constrained relaxation of the BaTiO3 capacitors is performed in
which the in-plane forces on all the atoms are artificially eliminated, the atoms move
141
Figure 5.4: Measurement of the polarization in the ferroelectric layer as a function of
position along the [100] direction of the capacitor. (a) Definition of the average change
in distance ∆ between Ti and apical O in a chain along [001]. In every case, the atomic
positions correspond to the lowest energy structure. A positive value of ∆ means a
polarization pointing upward. Profile of the normalized averaged change in distance
along z as a function of the position of the chain for a BaTiO3 /SrRuO3 capacitor of
domain period Nx = 4 (b) and a PbTiO3 /SrRuO3 capacitor of domain period Nx = 6
(c). The chains are numbered as indicated in Fig. 5.2 and 5.3. For the BaTiO3 -based
capacitor (b), results are shown for both m = 2 (dashed line) and m = 4 (dot-dashed).
Dotted lines represent the position of the domain walls.
back to the paraelectric positions for m = 2, or to a structure comparable in energy to
the most stable monodomain configuration for m = 4 [Fig. 5.1(b)]. Whether the in-plane
displacement is allowed or not might partially explain the very different configurations
found experimentally in related heterostructures: Lichtensteiger et al., using the same
experimental setup, have observed how high-quality ultrathin films of PbTiO3 grown
on Nb-SrTiO3 electrodes remain in a monodomain configuration [54] (although with
reduced polarization and tetragonality) whereas they form domains when the electrode
is replaced by La0.67 Sr0.33 MnO3 [169]. The same domain formation is suggested for
Pb(Zr0.2 Ti0.8 )O3 on SrRuO3 [177].
In contrast to the metallic relaxations in monodomain configurations, where ionic
displacements penetrate into the metal over a distance of two or three unit cells [168, 50],
in polydomain capacitors the displacements beyond the second RuO2 layer are negligible,
an indication of more effective screening produced by the domains of closure.
The polarization profile of polydomain structures can be estimated from the structural calculations. Figure 5.4 displays how much the polar distortion along z is changed
by the presence of a domain pattern. We define ∆ as the average of the change of
distance, with respect the most stable non polar configuration, between a Ti atom and
the nearest O atom lying on top along the z direction (cf. Ref. [61]), normalized with
142
respect to the short Ti-O distance in the bulk tetragonal phase. ∆norm is a very sensitive indicator of the polar order: it is zero as long as the atoms lie in the non polar
position and tends to unity as the full bulk polar distortion is attained. Figure 5.4
shows a very narrow 180◦ domain wall for the BaTiO3 capacitor, about a lattice constant wide, across which the polar distortion symmetrically reverses its sign. A smoother
domain wall is observed in the case of PbTiO3 , very much in the fashion predicted from
continuum Landau-Ginzburg models for ferroelectric thin films close to the monodomainpolydomain transition [178].
In contrast to 180◦ domains in bulk [61], where the ferroelectric distortion fully
recovers its bulk value by the second atomic plane far away from the domain wall,
here the finite size of the ferroelectric layer limits the value of the polarization at the
center of the domains. Polarization in BaTiO3 capacitors only amounts to 13% of the
bulk value at the center of each domain for a m = 2 thickness, about one order of
magnitude smaller than in bulk. The mean polarization increases with thickness, and
already amounts to 60% for a four-unit-cells-thick (m = 4) thin film. In the case of
PbTiO3 capacitors, already for the smallest thickness supporting polydomain structures
(m = 4), polarization in the center of the domains amounts more than 70% of the bulk
value, suggesting an extremely efficient screening by the domains of closure.
5.4
Role of the electrodes on the formation of polarization
domains
Domains of closure are a very favorable configuration. So favorable indeed, that when
this structure cannot be formed by in-plane atomic displacements within the ferroelectric
layer (due to in-plane strain, very uniaxial material, etc.) the formation of domains of
closure can be assisted by interfacial atoms belonging to the electrodes.
A direct way to prove the polarizability of the interfacial SrO layer is the calculation
of the dynamical charges of the atoms lying within this layer. We saw in Chapter 1 that
the effective charge of a given atom i is defined as
∂Pα ∗
Zi,αβ
=Ω
,
(5.1)
∂xi,β E=0
where α and β indicate Cartesian directions. Typically, an effective charge calculation
involves calculations of changes in the polarization via its Berry phase definition [179],
when the atom of interest is displaced with respect to a reference configuration. However
measurements of the polarization via the Berry phase is only possible in insulating
materials, which is not the case of our capacitors. Alternatively, the classic procedure
developed by R. Martin and K. Kunc [180] for the calculation of dynamical charges
can be applied. Within this approach effective charges of bulk materials are obtained
building a supercell containing several unit cells of the material. Then symmetric atomic
displacements, δxi,α , are induced in two equivalent atoms sufficiently far away one from
the other. The symmetric displacements create opposite dipole densities di,β at each
atomic plane,
5.4. Role of the electrodes on the formation of polarization domains
143
Table 5.1: (3,3) component of the effective charges of the SrO layer at the
PbTiO3 /SrRuO3 interface in comparison with those in bulk SrTiO3 . Values between
brackets are the Born effective charges calculated by means of the Berry phase method.
bulk SrTiO3
Interfacial SrO
∗
ZSr
2.41 (2.52)
1.62
di,β =
−2.49
∗
ZO1
(−5.95)
∗
Zi,αβ
δxi,α
,
(5.2)
S
where S is the area of the supercell perpendicular to its elongated axis. The opposite
dipole densities, in turn, induce two potential steps of opposite sign [see Fig. 5.5(a)] at
the sites of each displaced atom
di,β
.
(5.3)
∞ 0
The symmetric distortion avoids the development of depolarizing fields inside the supercell. The use of a sufficiently large supercell is required in order to recover a flat
electrostatic potential to accurately measure the potential step. The effective charge of
the displaced atom can then be calculated as
∆V =
∆V S∞ 0
,
(5.4)
δxα
where, in this case, β is necessarily the direction parallel to the elongated axis of the
supercell. Since ions are not allowed to relax after the atoms of interest have been
displaced by hand, the electronic relative permitivity of the material ∞ should be used
in Eq. 5.4. We have used the electronic permitivity of SrTiO3 (∞ = 6.15) for the
estimation of the effective charges of the interfacial SrO layer.
Nanosmoothed profiles of the electrostatic potential obtained using this method, after
a displacement of δz = 0.05 Bohr of the Sr atom in bulk SrTiO3 and SrRuO3 are shown
in Fig. 5.5(a) as a solid and a dashed line respectively. The corresponding values of
the (3,3) component of the effective charges are reported in Table 5.1 for SrTiO3 (for
SrRuO3 the potential drop, and correspondingly, the effective charges are negligible, as
expected for a metallic material).
Applying this method for the PbTiO3 /SrRuO3 ferroelectric capacitor (using the
centrosymmetric configuration as the reference structure) we obtain the potential profile
shown in Fig. 5.5(b). The comparison of the corresponding effective charges with the
bulk ones, all gathered in Table 5.1, shows that atoms at the interfacial SrO layer display
sizable effective charges. Although the dynamical charges of the SrO layer are not far
from the nominal ones, they are clearly not zero, as would corresponds to a metal, and
constitute an indication of the polarizability of the interfacial region.
∗
=
Zαβ
144
0.2
SrTiO3
SrRuO3
VH (V)
0.1
∆V < 1 mV
0.0
-0.1
∆V = 125 mV
-0.2
SrO
BO2
SrO
BO2
SrO
BO2
SrO
BO2
SrO
BO2
SrO
BO2
SrO
BO2
BO2
SrO
BO2
(a)
-0.3
0.10
∆V = 129 mV
VH (V)
0.05
∆V = 84 mV
0.00
-0.05
(b)
Sr displacement
O displacement
RuO2
SrO
RuO2
SrO
RuO2
SrO
TiO2
PbO
TiO2
PbO
TiO2
PbO
TiO2
PbO
TiO2
PbO
TiO2
PbO
TiO2
PbO
TiO2
PbO
TiO2
SrO
RuO2
SrO
RuO2
SrO
RuO2
-0.10
Figure 5.5: Nanosmoothed electrostatic potential of (a) SrTiO3 and SrRuO3 supercells and (b) SrRuO3 /PbTiO3 capacitor, after two Sr or O atoms have been displaced
symmetrically 0.05 Bohr out of their reference positions.
The resemblance of this SrO layer to SrO in bulk SrTiO3 can also be appreciated
from the analysis of the projected density of states displayed in Fig. 5.6. The SrO layer
closest to the interface displays a dramatic drop of the DOS at the Fermi level and a
simultaneous rise of a peak at about -5 eV, with the overall PDOS resembling that of
bulk SrTiO3 .
The enhanced polarizability of Sr in interfaces with ferroelectric materials has already
been observed before. Similar behaviour was found in AO/ATiO3 heterostructures [124],
where A = Ba or Sr. Both first-principles computations [71] and experimental measurements [181] have shown that SrTiO3 is highly polarizable when combined with BaTiO3
in heterostructures.
5.4. Role of the electrodes on the formation of polarization domains
145
Figure 5.6: Projected density of states on all the atoms as a function of the distance to
the interface for the SrRuO3 /BaTiO3 paraelectric heterostructure. (a) Schematic view
of the basic unit cell considered in our simulations. The reflection planes at the central
RuO2 layer in the electrode and at the central TiO2 layer in the ferroelectric are shown.
The layer by layer [numbered as in (a)] PDOS for the unpolarized interface is shown for
the cations in (b) and for the O in (c). PDOS of bulk SrRuO3 , BaTiO3 , and SrTiO3
are also plotted for comparison. All the energies have been rigidly displaced in order
to align the Fermi energy (vertical dashed line) with zero. Only the projection on the
atoms in the bottom half of the supercell is shown. The projection on the atoms in the
top half is identical due to the existence of the reflection planes shown in (a).
146
5.5
Screening of the depolarizing field
Figure 5.7: Map of the nanosmoothed electrostatic potential in a two-unit-cells-thick
BaTiO3 /SrRuO3 ferroelectric capacitor with a stripe period of Nx = 4 (a), and Nx = 6
(b). The arrows represent the atomic displacements with respect the paraelectric phase
as in Fig. 5.2. Only the displacements of the cations are shown for simplicity. Full lines
are a schematic representation of the domains of closure, while dashed lines mark the
position of the BaTiO3 /SrRuO3 interface.
Ideally, closure domains do not produce any polarization charge anywhere since the
normal component of the polarization is preserved across any domain wall. Therefore
the depolarizing field should vanish everywhere [58], and a nearly constant electrostatic
potential is expected. To further check this point we plot in Fig. 5.7 the electrostatic
potential for the BaTiO3 capacitor, nanosmoothed [106, 108] along z and as a function
of the position along the [100] direction. No nanosmoothing is performed along x. For
a stripe of thickness m = 2 and period Nx = 4, the potential is essentially flat at the
center of the domain, in contrast to the depolarizing field reported for monodomain
configurations [53].
A direct comparison between the electrostatic potential in the center of a domain and
that in a monodomain capacitor is shown in Fig. 5.8 for a BaTiO3 /SrRuO3 capacitor
with a thickness of the ferroelectric of m = 4 unit cells and a domain periodicity of
Nx = 4 unit cells in the case of the polydomain structure. The field in the center
of the domains is significantly smaller in the later – about one third – than that in
5.6. Theoretical prediction and experimental observation of closure domains
147
0.2
monodomain
polydomain
VH (V)
0.1
0.0
-0.1
SrO
RuO2
SrO
RuO2
SrO
TiO2
BaO
TiO2
BaO
TiO2
BaO
TiO2
BaO
TiO2
SrO
RuO2
SrO
RuO2
SrO
RuO2
SrO
-0.2
Figure 5.8: Profile of the nanosmoothed electrostatic potential for
BaTiO3 /SrRuO3 in a monodomain
phase (solid line) and in the center of a
domain in a polydomain configuration
(dashed line).
the monodomain capacitor, despite the larger polarization attained in the polydomain
structure (29 µC/cm2 , in contrast to the 13 µC/cm2 in the monodomain case).
In Fig. 5.7 we also observe a large microscopic field along [100] inside the domains
of closure at the metal-ferroelectric interface. The origin of this field is the variation
of the magnitude of P as the dipoles rotate from the center of the ferroelectric film
to the closure domains in the interface region. Besides, after nanosmoothing in z a
residual depolarizing field along [001] is identified in the neighborhood of the domain
wall, decaying rapidly away from it. This last field might be responsible for the lowering
of the polarization with respect to bulk shown in Fig. 5.4. Both fields might play an
important role in the fatigue of ferroelectric capacitors, the most serious device problem
in ferroelectric thin films [99]. In particular we identify at the ferroelectric/electrode
interface the preferred points of migration of charged defects, which pin the domain
walls and inhibit their motion [182]. The depolarizing field at the center of the domain
increases with the domain period; it starts to be appreciable for Nx = 6 [Fig. 5.7(b)]
and finally destabilizes the ferroelectric distortions for Nx = 8, as shown in Fig. 5.1(a).
5.6
Theoretical prediction and experimental observation
of closure domains in ferroelectric thin films
Domains of closure in ferroelectric thin films were unexpected to form due to the large
elastic coupling of the polarization with the strain, which in these materials play the
role analogous to the magnetocrystaline anisotropy in ferromagnets [183]. We have seen
throughout this Chapter that despite this common assumption, in the last years this
closure domains in ferroelectric thin films have been predicted to exist independently
of the theoretical approach, for ferroelectric thin films made of different materials and
regardless of the electrostatic boundary conditions. Domains of closure have been found
using a first-principles-based effective Hamiltonian for PZT symmetrically [174] and
asymmetrically [175] screened (grown on a nonconducting substrate and with a metal
with a dead layer as top electrode), and using a Landau-Ginzburg phenomenological
148
approach for a PbTiO3 thin film [56], both asymmetrically and symmetrically coated
with insulating SrTiO3 . Here, we observe the formation of domains of closure even for a
symmetrical metal/ferroelectric/metal capacitor, with uniaxial ferroelectrics that were
expected to profoundly dislike rotating the polarization and formation of in-plane dipoles
[174], and where the metallic plates should provide significant screening.
Besides, the formation of ferromagnetic-like closure domains in two very different
ways – by rotation towards the in-plane direction of the polarization within the ferroelectric layer (in PbTiO3 ) or assisted by in-plane displacements of the interfacial electrode layer (in BaTiO3 ) – suggest that this kind of structures, previously thought to
be unlikely to occur in ferroelectric materials, could be a much more general result.
This conclusion has been further reinforced after the observation of closure domains in
other systems, such as ferroelectric/insulator superlattices (see Chapter 6) or even free
standing slabs [184].
In spite of all the theoretical predictions, the experimental observation of domains
of closure in ferroelectric thin films have been elusive. Very recently, the development
of the spherical aberration correction in transmission electron microscopy (TEM) [74]
have allowed the determination of atomic structure and composition with subangstrom
resolution. With this technique the ferroelectric polarization can be directly determined
from the atomic displacements measured in atomic-resolution TEM images. This new
technique finally led to the direct observation of closure domains in PbZr0.2 Ti0.8 O3 thin
film grown on SrTiO3 [66]. Fig. 5.9 shows TEM images of the sample close to the
boundary between two 180◦ domains. Upon approaching the interface local dipoles
(measured as the relative displacement of Ti and Sr atoms with respect to the oxygen
cages) rotate in order to close the polarization flux and avoid the accumulation of bound
charges. Although scaled to the much larger thickness of the experimental sample, the
flux-closure structure bears a great similarity with the gradual dipole rotation we observe
in the capacitor of the parent ferroelectric material PbTiO3 .
Similar structures have also been found between 109◦ domain walls in BiFeO3 on
insulating TbScO3 substrates [67]. All these researches provide an illustrative example of
the predictive power of today’s theoretical methods and the feasibility of first-principles
simulations on systems which are directly comparable with experiments.
5.7
Conclusions
The perfect screening provided by the formation of polarization domains in ferroelectric
capacitors has been something given for granted for a long time. The energetic balance
and particular structure at the atomic level of domain walls for ferroelectric thin films,
however, was often disregarded.
Our simulations prove that the screening is so good indeed that it brings the limit
thickness of ferroelectricity virtually to zero. Strictly speaking, although we have demonstrated that the domains are stable, it is not clear whether in this limit the capacitor as
a whole can be called ferroelectric since, for this, the polarization has to be switchable
under external electric fields [177, 174]. However the great stability of the structures
5.7. Conclusions
149
Figure 5.9: (Left) Atomic-resolution TEM image of a closure domain with continuous
dipole rotation in PbZr0.2 Ti0.8 O3 close to the interface to the SrTiO3 substrate. The
interface is marked by a horizontal dashed line (I), which is determined on the basis of
a SrRuO3 marker layer with a nominal thickness of 1.5 unit cells at the interface. The
RuO2 marker layer is also indicated. Two larger domains with out-of-plane polarization
(indicated by arrows) can be identified. In the lower part, close to the interface, a triangular domain with in-plane polarization can be seen. The inset at the lower right shows
a calculated image demonstrating the excellent match between the atomic model and the
specimen structure. (Right) Map of the atomic displacement vectors. The displacement
of the Zr-Ti atoms (arrows) from the center of the projected oxygen octahedra is shown
here superposed on the TEM image. To enhance contrast, the gray scale is converted
into a false-color representation. The length of the arrows represents the modulus of
the displacements with respect to the yellow scale bar in the lower left corner. Note
the continuous rotation of the dipole directions from “down” to “up”, which closes the
electric flux of the two 180◦ domains. Reprinted with permission from Ref. [66].
150
obtained in this work suggest that polarization of domains would still be the most stable
configuration for thicknesses well above the critical thickness of monodomain ferroelectricity.
Our calculations provide insightful results on the energetic, structural and electronic
properties of ferromagnetic-like closure domains in ultrathin capacitors. This kind of
domain structure, that was first predicted theoretically but had not been detected experimentally, has just been observed for the first time in PbZr0.2 Ti0.8 O3 [66] and BiFeO3
thin films [67]. The agreement between different theoretical approaches and experimental measurements in this case is remarkable and strongly suggest that closure domains
in ferroelectric thin films might be a much more general result than previously thought.
We have also provided some hints to explain why some systems break into domains
while others remain in a monodomain configuration. We also predict the preferential
sites for pinning charged defects, important for understanding the fatigue of thin films.
Both issues are of vital importance in the integration of ferroelectric in practical devices.
Chapter 6
PbTiO3/SrTiO3 superlattices
6.1
Introduction
As it was already discussed in Chapter 1 the growth of superlattices composed of thin
layers of ABO3 perovskites with different physical properties has become one the the
most promising paths to exploit the coupling between instabilities in these compounds
in order to engineer new functionalities.
The fascination for layered system comes from the fact that the properties of epitaxial
structures, made by stacking different perovskites, are not a simple combination of the
properties of the constituent materials, but exotic phenomena might emerge that fully
rely on interfacial effects. These phenomena arise as a consequence of the delicate
balance between instabilities in perovskite oxides, that is altered by the presence of
an interface, the interplay between the effects of strain and electrostatic interactions
between the layers, and the coupling of different structural instabilities in the reduced
symmetry environment of the interface [46]. The intrinsic interfacial character of these
properties allow their tunning, no only by external electrical and strain [38] fields, but
also by changing the chemical environment through the use of different materials and
periodicities in the stack.
Of course, the combination of ferroelectric perovskites with other materials in layered
structures have attracted a lot of interest during the last few years [99, 7]. If ferroelectricity is preserved in the layered system it would permit, for instance, the use of electric
fields to control other functional properties of the superlattice. In the pursuit of this
kind of functionalities and the improvement of the performance of ferroelectric thin films
[23], ferroelectric materials are being combined with different paraelectric (or incipient
ferroelectric) ABO3 perovskites.
For a long time the focus was on the electrostatic coupling between the layers of
the superlattice, and the interplay with the epitaxial strain. The most studied case
was BaTiO3 /SrTiO3 , where the structure adopts a uniformly polarized state, with the
polarization value determined by the competition between the energy cost of polarizing
the SrTiO3 layers and the energy gain in preserving the polarization in the BaTiO3
(BTO) layers [71]. More recently the interest has evolved to include also the interaction
151
152
Chapter 6. PbTiO3 /SrTiO3 superlattices
between FE and AFD modes in perovskite related systems [185].
A startling system in the recent literature is the (PbTiO3 )m /(SrTiO3 )n [(PTO)m /(STO)n ]
superlattice. It was theoretically predicted and experimentally observed that the polarization, tetragonality and phase transition temperature of the system can be monitored
with the number of PTO layers, m. As the volume fraction of PTO is reduced, the
penalty imposed by the poling of the STO becomes dominant in the energy balance of
the system and the polarization of the superlattice decreases [186, 187]. This trend, that
can be accounted by simple electrostatic models, breaks down in the limit of ultrashort
periods, for which PTO/STO superlattices exhibit an unexpected recovery of ferroelectricity that cannot be accounted for by simple electrostatic considerations alone [19].
First-principles calculations found that the ground state of the system is not purely
ferroelectric, but involves a trilinear coupling term between two AFD modes, that correspond to in-phase (AFDzi ) and out-of-phase (AFDzo ) rotations of the oxygen octahedra
around the z axis, that induce a polar FE distortion (Pz ) in a way compatible with
improper ferroelectricity [19].
In recent experiments, Zubko and coworkers [188] have studied the dependence of
the structural and dielectic properties of PbTiO3 /SrTiO3 superlattices as a function of
the volume fraction of PbTiO3 , the electrodes, and the applied electric fields. While
the results of the superlattices asymmetrically sandwiched between Nb-doped SrTiO3
(bottom) and gold (top) electrodes were consistent with a monodomain configuration,
those corresponding to the use of symmetrically coated SrRuO3 electrodes (both top
and bottom) suggested a polydomain phase with domain wall motion.
Previously, 180◦ stripe domains were also detected in PbTiO3 thin films grown on
SrTiO3 (001) substrates. The domain structure was characterized both in reciprocal
space (strong satellites around PbTiO3 Bragg peaks in synchrotron x-ray scattering
measurements) [189, 190, 170], and in real space (images recorded by atomic force
microscopy) [171]. In these works, the equilibrium polarization structure of epitaxial
films undergo a sequence of changes as a function of the thickness of the PbTiO3 layer
and temperature. The phase transitions involve (i) polydomain phases with different
stripe periods (the so-called Fα phase with a short period, and the Fβ phase with a period
√
2 longer), (ii) the monodomain phase, Fγ , for thicker films near room temperature,
(iii) and the paraelectric phase at high temperatures.
It is clear, from this discussion and in view of the results of Chapter 5, that different
phases are in close competition, and a small change in one of the parameters (cooling
temperature, thickness, choice of electrode, etc.) might help to incline the balance
towards one of the local minima displayed in the energy landscape, yielding a different
polarization structure.
In this Chapter we explore the phase diagram of the PTO/STO superlattices systems to gain further insight about the different instabilities present in the superlattices
and their coupling. First we study the mixed coupling between the polar, the AFD
instabilities and the strain in monodomain phases in order to find new paths to engineer functional properties in this system. In a second step we explore the formation
of polarization domains and the structure of domain walls in PTO/STO superlattices,
6.2. Structure and computational details
153
considering the influence of the lateral domain size, orientation and energy of domain
walls, and the influence of AFD instabilities in these structures.
6.2
Structure and computational details
For this study we perform first principles simulations of (PTO)n /(STO)n superlattices,
within the local density approximation (LDA) to the density functional theory (DFT)
using the Siesta code [121]. Real space integrations are computed in a uniform grid,
with an equivalent plane-wave cutoff of 1200 Ry. For the Brillouin zone integrations we
use a Monkhorst-Pack sampling equivalent to 12 × 12 × 12 in a five atom perovskite
unit cell. Details on the norm-conserving pseudopotentials and the basis set used can
be found in Ref. [124].
The superlattices are simulated by means of a supercell approach, where we repeat
periodically in space a simulation box, as was explained in Sec. 2.6. The basic building
block of our structures is built stacking alternating layers of PTO and STO along the
[001] direction (z axis) with a thickness of n unit cells for a global periodicity of (n/n)
[Fig. 6.1(a)]. Monodomain superlattices have been simulated with n ranging from one
to three. For polydomain superlattices a (3/3) periodicity was chosen because it has
been experimentally proved that this is a critical periodicity beyond which the PbTiO3
layers in the superlattices are electrostatically decoupled [65]. Besides, first works on
these systems suggested that, for short-period superlattices, a ratio between the number
of layers of PbTiO3 and SrTiO3 close to 1 represents the borderline case between a
“normal” behaviour and the appearance of the improper ferroelectricity [99].
For each periodicity, a reference structure is obtained imposing a mirror symemtry
plane in the middle of the PbTiO3 layer and relaxing both atomic positions and the
out-of-plane lattice vector until the value of the Hellmann-Feynman forces and zz stress
tensor components fall below 0.01 eV/Å and 0.0001 eV/Å3 respectively. From this
starting structure different configurations have been constructed and then relaxed.
The mechanical boundary conditions imposed by the substrate are implicitly treated
by fixing the in-plane lattice constant, ak . The use of periodic boundary conditions
imposes short-circuit electrical condition across the whole unit cell.
For the simulation of monodomain configurations, in-plane lattice vectors should be
doubled to account for the condensation of AFD instabilities. With the (2 × 2) in-plane
periodicity, TiO6 octahedra are allowed both to rotate an angle φ around the z-axis or
to tilt an angle θ around an axis contained in the (x, y) plane [Fig. 6.1(b)]. After the
in-plane doubling of the building block cell, symmetry is broken displacing the atoms by
hand and the whole structure (atoms and out-of-plane lattice vector) is relaxed again.
Non-polar structures are obtained constraining the atomic displacements to avoid the
development of a polarization in any direction.
For the simulation of polydomain configurations the reference structure in replicated
Nx times in the [100] direction (x axis) and Ny times in the [010] direction (y axis).
Due to the periodic boundary conditions used in the simulations, Nx determines the
domain periodicity (or, equivalently the domain lateral size), while Ny allows to switch
154
Figure 6.1: (a) Squematic representation of a (2/2) PTO/STO superlattice. TiO6
octahedra are labeled according to the chemical identity of the first two neighbor layers
of TiO2 planes, and the direction of the polarization. (b) Definition of the angles of
rotation along the z axis, φ, and tilting along an axis in the (x, y) plane, θ, of the O
octahedron.
on (Ny = 2) and off (Ny = 1) the AFD instabilities. As it was done before for the study
of ferroelectric capacitors in Chapter 5, an initial distortion is induced displacing atoms
by hand, so the polarization points upwards in half of the superlattice and downwards
in the other half. Initial AFD-like displacements are induced by hand for supercells
with Ny = 2. Then the most stable polydomain structure for each domain lateral size is
obtained relaxing the forces and out-of-plane stress.
6.3
Mixed ferroelectric-antiferrodistortive-strain coupling
in the monodomain configuration
For the (2/2) superlattice we have performed structural relaxations under different inplane strains, while keeping the square surface symmetry, to mimic the effect of some of
the most common cubic substrates. The misfit strain is defined as
ε=
ak − a0
,
a0
(6.1)
where a0 is our LDA theoretical lattice constant of cubic bulk cubic PTO (3.892 Å).
(Unless otherwise stated, throughout this Chapter strain is always given relative to the
PTO layer.)
The dependence of the polarization (inferred from the bulk Born effective charges
and the local atomic displacements) with the epitaxial strain, Fig. 6.2(a), has some
6.3. Mixed FE-AFD-strain coupling in the monodomain configuration
155
Figure 6.2: (a) Polarization and (b) absolute value of the oxygen octahedra rotations
and tiltings in monodomain (2/2) superlattices under different epitaxial strains, corresponding to the in-plane lattice constants of representative substrates. In (a), the insets
represent schematically the orientation of polarization in the PTO (darker arrow) and
STO (lighter arrow) layers. In (b), the rotation, φ, and tilting, θ, angles of the O octahedra (labeled as in Fig. 6.1) are represented as star-dotted and cross-dashed lines,
respectively.
156
resemblances with the one already observed in strained bulk BTO [191], PTO [192], or
PTO/PbZrO3 (1/1) superlattices [192]. For large compressive strains a homogeneous
polarization (including the naturally paraelectric STO layer) is stabilized along the zdirection (the c-phase in Refs. [20, 191]). The polarization mismatch at the interface is
always smaller than 0.5 µC/cm2 , highlighting the large electrostatic cost of a polarization
discontinuity between the layers [71, 186, 193]. The price to pay for poling the STO
layer is a reduction in the polarization of PTO with respect the bulk spontaneous value
(90 µC/cm2 ). This value of the polarization, 90.35 µC/cm2 corresponds to the bulk
tetragonal phase, with a = 3.855 Å and c = 4.071 Å. Imposing STO in-plane lattice
constant a = 3.874 Å, then c = 4.028 Å and Pz = 82.52 µC/cm2 . On the opposite
limit, for large tensile strains, the polarization in the most stable configuration lies in
the plane along the [110] direction (aa-phase [20, 191]). Note that, in this case, there
is no electrostatic restriction to keep the in-plane polarization at the same value in the
PTO and STO layers. (Neither ∂Px /∂z nor ∂Py /∂z contribute to the divergence of the
polarization.)
Interestingly, there is an intermediate region of strains (around ε ≈ 0) where the
polarization rotates continuously from the c to the aa-phase. In this range of strains,
around that imposed by a STO substrate, in-plane and out-of-plane polarization coexist,
giving rise to an r-phase [20, 191]. Both, the decrease of out-of-plane and the onset of
the in-plane polarization, display the typical shape of a second-order phase transition.
The appearance of a r-phase is a rather unexpected result since PTO is a tetragonal
ferroelectric in bulk and the strain imposed by the STO substrate is even slightly negative
(-0.5% within our simulations). Moreover, the existence of an r-phase in strained bulk
PTO has not been reported in previous first-principles simulations at any value of the
strain [192]. Our simulations suggest that, in order to reduce Pz , the rotation of the
polarization is energetically less coslty than a monotonic reduction along [001]. This
result is in agreement with previous simulations on BaTiO3 , which showed that rotation
of the polarization was prefered over a change in its magnitud under the application of
an external applied field [194].
We observe also a strong coupling between FE and AFD modes, that can be tuned by
epitaxial strain [Fig. 6.2(b)]. For a fine analysis of the coupling, it is useful to label the O
octahedra depending on the identity of the first two neighboring layers along the z-axis,
and on the direction of the polarization, as is done in Fig. 6.1. In this way, we can define
octahedra with three different enviroments: those between two PbO layers (labeled as
PTO), those between two SrO layers (labeled as STO), and those at the interfaces, with
SrO at one side and PbO at the other. As polarization of the superlattice breaks the
inversion symmetry we should distinguish between the two different interfaces, thus we
label as P+ (P− ) to the top (bottom) interface of the PTO layer with respect to the
polarization direction.
In the previous Figures we find that, similar to the case of FE distortions, AFD
ones are strongly coupled with strain. Compressive strains favor the rotations of the
octahedra and suppress tiltings, while tensile strains produce the opposite effect. This
trend can be understood if we consider that the Ti-O bond is very rigid. Then, as strains
157
Table 6.1: Relative energies of different monodomain configurations as a function of the
periodicity of the supercell. GS stands for ground state. Energies per 5-atoms unit cell
in meV.
Paraelectric
[110]
[001]
[111]
(1/1)
+15.3
+4.6
+1.2
GS
(2/2)
+12.8
+3.5
+3.1
GS
(3/3)
+9.7
+3.5
+4.0
GS
Table 6.2: Polarization of the different monodomain configurations for superlattices as
a function of the periodicity of the supercell. In-plane strain corresponding to a STO
substrate (3.874 Å). AFD modes are allowed in the paraelectric phase. Polarizations in
µC/cm2 .
(1/1)
(2/2)
(3/3)
PSTO
PPTO
PSTO
PPTO
PSTO
PPTO
Para.
(0.0, 0.0, 0.0)
(0.0, 0.0, 0.0)
(0.0, 0.0, 0.0)
(0.0, 0.0, 0.0)
(0.0, 0.0, 0.0)
(0.0, 0.0, 0.0)
[110]
[001]
[111]
(20.7, 20.7, 0.0)
(31.4, 31.4, 0.0)
(16.0, 16.0, 0.0)
(34.5, 34.5, 0.0)
(14.2, 14.2, 0.0)
(35.9, 35.9, 0.0)
(0.0, 0.0, 35.5)
(0.0, 0.0, 35.0)
(0.0, 0.0, 34.4)
(0.0, 0.0, 34.8)
(0.0, 0.0, 33.6)
(0.0, 0.0, 34.3)
(14.2, 14.2, 31.5)
(23.3, 23.3, 31.1)
(9.2, 9.2, 29.8)
(29.5, 29.5, 30.4)
(6.9, 6.9, 29.0)
(31.8, 31.8, 30.3)
are applied, the system allows the TiO6 octahedra to change orientation to maintain the
Ti-O distance constant (see Fig. 3 of Ref. [3]). This is consistent with experimental
results [195] where it is found that in most cases the AFD distortions are stabilized
under hydrostatic pressure.
6.3.1
Periodicity dependence of FE-AFD coupling
We have also carried out simulations for different periodicities, while keeping constant
the PTO/STO volume fraction (n/m = 1) and fixing ak to the LDA theoretical one of
SrTiO3 . Unconstrained and constrained structural optimizations were performed, where
we imposed a purely out-of-plane or in-plane polarization on the superlattice, otherwise
oxygen octahedra are allowed to rotate freely. Relative energies and polarizations of
the PTO and STO layers are gathered in Tables 6.1 and 6.2 respectively. The ground
state monodomain configuration displays both in-plane and out of plane polarizations,
independently of n. Within the PTO layer, P lies close to the diagonal of the perovskite
unit cell, especially for n ≥ 2 (configuration labeled as [111] in Table 6.2). The ground
state structure can be considered as a condensation of FEz + FExy + AFDz + AFDxy
158
modes.
For n = 1, the c and r-phase are essentially degenerated (the difference in energy, only
1.2 meV per 5-atoms unit cell, is within the accuracy of our simulations). The delicate
competition was already observed by Bousquet et al., who found that at the ground
state for (1/1) superlattices in Ref. [19] the phonon frequency of the mode involving
in-plane distortions is only 6 cm−1 , very close indeed to become unstable. The small
difference between the results in Table 6.1 and those in Ref. [19] can be ascribed to small
changes in the methodology. In particular ak in Ref. [19] was fixed to the bulk LDA of
SrTiO3 within a plane-wave (PW) simulation (3.84 Å [186]), slightly smaller than the
corresponding value obtained with a numerical atomic orbital (NAO) code (3.874 Å), as
we are doing here.
Larger periodicities of the superlattice seems to increase the range of stability of
the r-region, as the difference in energies between this phase and the rest increases,
nevertheless examination of Fig. 6.2 shows that even for the (2/2) superlattices a small
change in the mechanical boundary conditions can significantly alter the ground state
of the system.
In every case, Pz is nicely preserved at the PTO/STO interface, with a value between
30 and 35 µC/cm2 . On the other hand, the in-plane polarization increases with the
reduction of n.
Oxygen octahedra rotations, shown in Fig. 6.3, display a strong coupling with the
polarization direction for all the studied periodicities. Tilting is suppressed in almost
any case but for the ground state, where polarization direction is not constrained and lies
close to the [111] direction. The strong coupling of polarization with oxygen octahedra
around the z axis at the interfaces is clearly observed here: angle φ is clearly enhanced
at the P+ interfaces and damped in the P− in [111] and [001] configurations (in [110]
and paraelectric configuration there is no difference between P+ and P+ interfaces), in
agreement with the covalent model described Sec. 6.3.3.
6.3.2
Emergence of an r-phase in PbTiO3 /SrTiO3 superlattices
We observe that the coupling between different distortions together with the electrostatics of the superlattice gives rise to the appearance of a r-phase at intermediate strains.
In fact our simulations suggest that in order to reduce the out-of-plane component of
the polarization, Pz , to a value of around 30 µC/cm2 (the typical order of magnitude
shown in Table 6.2), PbTiO3 prefers to rotate the direction of the polarization towards
the [111] direction over a monotonic reduction along the [001] axis. This point was
tested performing simulations of bulk PbTiO3 where polarization was constrained to lie
along [111]. When forces and out-of-plane stress were relaxed while fixing the in-plane
lattice constant to that of SrTiO3 (3.874 Å within our simulations), PbTiO3 was found
to display a polarization along [111] of 53.1 µC/cm2 (the out-of-plane component being
31.0 µC/cm2 ).
Calculations of the double well energies shown in Fig. 6.4 reveal that below an outof-plane component of the polarization of 31.8 µC/cm2 , a poling of PbTiO3 along the
159
Figure 6.3: Oxygen octahedra rotation (solid lines, squares) and tilting (dashed lines,
diamonds) angles. On the leftmost part of the figure, schematic view of the atomic
structure of the superlattices are shown. Direction of the polarization is indicated on
the top of each panel. The same notation as in the main manuscript is used.
160
[111] direction is preferred over the monotonic reduction along [001].
20
P along [111]
Energy (meV)
0
-20
-40
-60
P along [001]
0
20
31.8 40
80
60
100
120
2
Pz (µC/cm )
Figure 6.4: Energy as a function of the out-of-plane component of the polarization for
bulk tetragonal PbTiO3 displaying a polarization along [001] (red line) or [111] (black
line). Dashed line indicates the threshold out-of-plane polarization below which a rotation of P towards the [111] direction is preferred over a monotonic reduction along [001].
The square (circle) correspond to a single point calculation of bulk PTO in which the
atomic coordinates and unit cell are fixed to those of the PTO layer in the (3/3) [001]
([111]) superlattice. Last two data points (and only those) include AFD distortions. In
all cases, in-plane lattice constant is fixed to the theoretical SrTiO3 one. All energies
are given per 5-atoms unit cell.
However the energy difference between the [001] and the [111] phases are too small
to explain the energy differences between these phases in the superlattices, reported in
Table 6.1. The increased stability of the [111] phase can be accounted for by including the
AFD distortions in the analysis. Bulk tetragonal PbTiO3 ground state displays only the
Γ point ferroelectric distortion. Nevertheless, constrained simulations show that when
the polarization in bulk PTO is forced to decrease or rotate, AFD distortions emerge,
decreasing the energy of the system. We have performed single point calculations on
bulk PbTiO3 , but doubling the size of the unit cell along the three directions of space
(i.e. using a 2 × 2 × 2 simulation cell with 40 atoms) freezing the coordinates to those
found in the middle of the PbTiO3 layer in the (3/3) [001] and [111] superlattices. These
two configurations, represented respectively by an square and a circle in Fig. 6.4, display
oxygen octahedra rotations and their energies are below the double well curves of [001]
and [111] phases, for which AFD instabilities were forbidden. The energy difference
between these two configurations of bulk PTO is about 7 meV, in good agreement with
the energies reported for the superlattices in Table 6.1.
6.3.3
161
Covalent model for the polarization-octahedra rotation coupling
We have seen that the coupling of the oxygen octahedra rotations with strain can be
understood in terms of steric effects (TiO6 octahedra rotate in order to accomodate inside
the distorted cell), however, superimposed to the main strain effect there is an extra
coupling with the polarization which is distinct for each of the octahedra types defined
before. The largest difference is observed when strong compressive strains are applied:
here the P+ octahedra rotate more than PTO and STO ones, while P− octahedra rotate
much less. In order to understand these results let us discuss the origin of the coupling
between between FE and AFD modes.
Figure 6.5: (a) Diagram showing the change of distances between the Pb and O ions
at the PTO/STO interface under compressive strain for a P+ TiO6 octahedra. Only
one TiO2 plane and the Pb atoms directly underneath are represented. For the leftmost
Ti, the full TiO6 octahedra is depicted by dashed blue lines. (b) same as in panel
(a) but for a P− octahedra. Now, only the Pb atoms direcly on top are represented.
Reduction of distance and reinforcement of the Pb-O bond is shown by full red lines
while an increase in the Pb-O distance and weakening of the bond is shown by dotted
lines. Green arrows represent out-of-plane displacement of Pb. Yellow arrows represent
the in-plane displacement of O, consistent with an extra convalent contribution to TiO6
rotation.
AFD distortions are usually regarded as purely steric phenomena, where the rotation
of the octahedra takes place if the A-ion is small enough to let the B-O-B bond bend [196].
A polar distortion of the A cation along the positive z-axis for the P+ octahedra reduces
the distance between the metal and the oxygen ions of the TiO2 plane immediately above
[Fig. 6.5(a)]. Thus, according to a steric model we would expect that the P+ octahedron
would rotate less than the P− one where the Pb ion moves away [Fig. 6.5(b)], as the
free space around the oxygen ions is reduced in the former. However, Fig. 6.2(b) shows
precisely the opposite trend: P+ displays a much increased rotation with respect to P− .
Besides steric effects, we propose that as a consequence of the symmetry reduction at
the interfaces the mixed AFD-FE-strain coupling in PTO/STO superlattices is driven by
162
a force of covalent nature. It is well known that a chemically active lone pair on the Pb
ion, that allows for strong covalent hybridization with O, lies at the origin of FE in bulk
PTO. Due to the coupling between FE and AFD distortions, not all the Pb-O bonds are
equivalent leading to a reduction of energy. In particular, having two shorter (2.447 Å)
and two longer (2.928 Å) bonds per Pb atom increases covalency with respect to having
four equivalent bonds. For the P− octahedra, the polar distortion increases the Pb-O
distances and the previous mechanism does not apply [Fig. 6.5(b)]. These results agree
with recent ab-initio calculations that emphasize the role of covalent interactions in the
origin of AFD distortions [197].
As the in-plane strain is increased, the polarization rotates away from the z axis and
this coupling is reduced making the in-plane rotations small and similar for all octahedratypes when the values of the strain are larger than +1%. Under these tensile strains, the
Pb displace in-plane and both P+ and P− become equivalent. From the point of view of
symmetry this can be demonstrated considering the polar displacement of the Pb ions
along the z-axis and the xy directions. In Fig. 6.6 we see that a horizontal reflection
plane coinciding with the TiO2 plane transforms the P+ and P− octahedra into each
other in the aa-phase, probing their equivalence. On the contrary no symmetry operation
is found to relate the P+ and P− octahedra in the c-phase showing their differences in
physical behavior.
Figure 6.6: Diagrams showing the equivalence of the P+ and P− octahedra in the aaphase owing to the horizontal reflection plane associated to the TiO2 plane. In contrast
no symmetry operation relates P+ and P− in the c-phase.
6.3.4
163
Piezoelectric response of the system
σ (GPa)
-0.08 -0.06 -0.04 -0.02
80
0
0.02
0.04
0.06
0.08
(a)
2
P (µC/cm )
60
40
PTO
Pz
PTO
20
Pxy
STO
Pxy
0
20
15
(b)
d (nC/N)
10
d11
5
d11
0
-5
PTO
STO
d31
-10
-15
-20
-0.08 -0.06 -0.04 -0.02
0
0.02
0.04
0.06
0.08
σ (GPa)
Figure 6.7: Polarization (a) and piezoelectric constants (b) as a function of the in-plane
stress for the PTO/STO (2/2) superlattice.
From the analysis of the dependence of the polarization with respect to the strain,
shown in Fig. 6.2, it follows that close to the c ↔ r ↔ aa phase transition series the
electromechanical response of the system (d31 piezoelectric constant) should be greatly
enhanced. The d31 and d11 piezoelectric constants were computed using the polarization,
calculated from the Born effective charges and atomic displacemnts, and the in-plane
stress, routinely available as an output of most common DFT codes.
For the calculation of the piezoelectric constants, in-plane and out-of-plane polarizations were plotted as a function of the in-plane stress, σ1 in Voigt notation. The available
points P (σ) were fitted to the characteristic curve of a ferroelectric-paraelectric second
order phase transition [20]
P =
p
A(σ − σ0 ),
(6.2)
where σ0 is the stress at which the polarization drops to zero. In-plane and out of plane
164
Figure 6.8: Piezoelectric coeficients of the PTO/STO (2/2) superlattice as a function of
PTO strain.
components of the polarization are fitted separately despite being naturally coupled,
with all the coupling terms being hidden in the fitting constants A and σ0 . Despite this
approximation the separate fittings match very accurately the calculated polarizations,
as can be seen in Fig. 6.7(a). The piezoelectric constants were then computed as analytical derivatives of the fitting curves [shown in Fig. 6.7(b)]. Since in-plane polarization
is mesured for both the PTO and STO layers, two d11 piezoelectric coeficients can be
computed, being the effective d11 coefficient of the superlattice the average of them.
Piezoelectric constants diverge at the phase transition strains (see Fig. 6.8), reaching
values of more than 10 nC/N (around one hundred times larger than typical piezoelectric
constants of Pb(Zr0.5 Ti0.5 )O3 ceramics [198]) at strains easily achievable in a laboratory.
6.4
Polydomain structures
The continuity of the component of the polarization perpendicular to the PTO/STO
interface in monodomain phases (see overlap of PzSTO and PzPTO curves in Fig. 6.2 or
polarization values in Table 6.2) reflects the huge energy cost that would be associated
with the formation of bound charges at the interface and the development of a depolarizing field. Minimization of the electrostatic energy in the monodomain phase is achieved
poling the SrTiO3 layer and reducing and rotating the polarization in the PbTiO3 layer.
We have seen in Chapter 1 how accumulation of bound charges can alternatively be
screened by the formation of polarization domains. On the other hand one of the main
conclusions of Chapter 5 was the the idea that closure domains are a very favorable
structure in ferroelectric thin films, predicted to be formed in a great variety of ferroelectric materials and under a great range of screening degrees. It is thus reasonable to
expect similar arrangements in the PTO/STO superlattices. It is the aim of this Section
to investigate the particular structure of domains in the PTO/STO superlattices as well
as their coupling with the different distortions detected in the monodomain phase.
165
6.4. Polydomain structures
70
60 Non-polar
Energy (meV)
50
40
30
Monodomain [0,0,1]
20
10
Monodomain [1,1,1]
0
0
2
4
6
8
10
12
14
16
18
Nx
Figure 6.9: Differences in energies between polydomain, monodomain and non-polar
configurations in (3/3) PbTiO3 /SrTiO3 superlattices, as a function of the domain period
Nx . Total energies of supercells are divided by Nx × Ny to make them comparable.
Circles represent the configurations where the AFD modes are not allowed (Ny = 1),
while squares represent configuration with AFD modes condensed (Ny = 2). Diamond
indicates a configuration where the domain wall lies along the (1,1,0) direction, also
allowing for the condensation of AFD modes. The monodomain phases have been labeled
as in Sec. 6.3. In the non-polar configuration, the AFD distortions have been considered.
All energies are given with respect to the most stable monodomain configuration.
The relative energies of the different polydomain, monodomain and non-polar configurations as a function of the domain periodicity are shown in Fig. 6.9. The most
important conclusions that can be drawn are:
• The energy of the polydomain structures decreases with the increase of the domain period. The balance between the electrostatic energy (which tends to reduce
the domain lateral size), and the domain wall energy density (which tends to increase it) results in an optimum domain periodicity of about 12 unit cells (46.5 Å)
(the energy for Nx equal 12 and 16 might be considered as equivalent within the
accuracy of our simulations).
• For a given domain periodicity, the energy is systematically lowered if rotations of
the oxygen octahedra are allowed, with reductions ranging between 22 meV per
30 atom supercell building block (for Nx = 6) to 12 meV (for Nx = 12). This fact
highlights the importance of the FE-AFD coupling in these heterostructures, as
already discussed in previous section.
166
• The effect of the domain wall orientation is small: a change in the orientation
of the domain wall from [100] to [110] does not affect significantly the energy of
the superlattice, pointing to a rather isotropic domain wall structure, with the
energy of the domains depending very weakly on the stripe orientation, in good
agreement with experimental results [199], phenomenological Landau-GinzburgDevonshire theory [173], and model Hamiltonian [62] simulations.
• Finally, the most stable phase found in our simulations correspond to a monodomain structure, with the polarization in the PbTiO3 layer pointing close to
the unit cell diagonal of the perovskite unit cell (configuration labeled as [111] in
previous section and Fig. 6.9). Nevertheless the differences in energies between
the monodomain and the most stable polydomain configuration are not very large
(of the order of 9 meV/30-atom-supercell), suggesting a close competition between
them and a strong dependence of their relative stability with, for instance, the
electrostatic boundary conditions or the chemical environment during growth process. This energy differences mean that, in practice, at room temperature these
two possible configurations are degenerated. Under the light of our results, it
sounds plausible that, under certain conditions, the system could be stabilized in
a metastable domain structure, specially if one takes into account that due to the
finite thickness of actual samples, they might suffer from a depolarizaing effect
that would favor domain formation. These considerations would explain why both
configurations have been observed experimentally [188].
Figure 6.10 shows the most stable polydomain configuration found for a (3/3) superlattice with Nx = 12 and Ny = 2 (similar patterns are obtained for other domain sizes).
The pattern of polarization clearly displays the typical vortices of the closure domain
configurations, where the polarization rotates continuously forming a closed flux structure connecting two 180◦ domains. The domain walls are one unit cell thick, as already
suggested by first-principles simulations on bulk PbTiO3 [61] and nanocapacitors (see
Chapter 5 and Refs. [55] and [46]). As it was already observed in the PbTiO3 /SrRuO3
ferroelectric capacitors, the closing of polarization flux in PbTiO3 /SrTiO3 superlattices
is due to large in-plane displacement of the Pb atoms at the PbO layers in the vicinity of
the interface. This behavior contrast with that of Ba-based heterostructures, were the
rotation of the polarization in BaTiO3 is much more costly. In-plane displacements of
Pb are of the order of 0.2 Å, large enough to be visible in a high-resolution transmission
electron microscopy (HRTEM) image. However, the experimental observation of the
domains of closure in these superlattices remains to be done.
Within this vortex configuration, there is no need anymore to keep constant the
normal component of the polarization at the interface, since the electric fields that arise
from its discontinuity is efficiently screened by the domain structure. This reflects in
the out-of-plane polarization obtained at the center of the domains, reaching values in
the PbTiO3 layer between 50 and 60 µC/cm2 , depending on the domain periodicity. In
some cases we observe that the symmetry lowering due to the AFD distortions allows for
the development of a component of the polarization along y. The values for the different
167
Figure 6.10: Polydomain structure of a PbTiO3 /SrTiO3 superlattice showing the pattern of a domain of closure. Balls, representing atoms, are located at the positions of the
relaxed structure. The arrows represent local polarization around A atoms (either Pb
or Sr) calculated by means of Born effective charges and atomic displacements.. Dotted
lines indicate the position of the domain wall.
components of the polarization at the center of each domain, inferred from the bulk Born
effective charges and the local atomic displacements, are summarized in Table 6.3. For
the calculation of values in Table 6.3 a layer by layer polarization is first computed for a
unit cell surrounding each cation, and then an average is performed for all the unit cells
within the PbTiO3 and SrTiO3 layers. The layer-by-layer polarization of Fig. 6.11(b)
shows convergence to an homogeneous well defined value (around 30 µC/cm2 ) within
SrTiO3 . On the other hand, at the center of the PbTiO3 layer a progressive reduction
is observed upon approaching the interface. This behavior is in good agreement with
the t2g -eg splitting experimentally measured by using electron energy loss spectroscopy
[199].
Competition between FE and AFD modes is also manifested in polydomain phases.
Under the same mechanical boundary conditions, oxygen octahedra rotations and tiltings
in the PbTiO3 layer in polydomain structures, where out of plane polarization (50-60
µC/cm2 ) is much larger than in monodomain configurations (30 µC/cm2 ), are smaller
than in monodomain phases. Rotations, displayed in Fig. 6.11(c) for a section in the
middle of a domain, become approximately one half in polydomain than those in monodomain superlattices (around 2◦ the former and 4◦ in the latter), while tilting angles in
polydomain phases are indeed almost negligible. The rotations around the z-axis display
the same behavior as in the ground state monodomain case, with an enhancement at
168
Figure 6.11:
(a) Schematic representation of the center of a domain in a (3/3)
PbTiO3 /SrTiO3 superlattice (see region embodied in a dashed box in Fig. 6.10). Color
scheme is the same as in Fig. 6.10. (b) Layer by layer polarization inferred from the
Born effective charges and the atomic displacements. Each point corresponds to the
layer at the same height in panel (a). (c) Amplitude of the rotations (squares) and
tiltings (diamonds) of each TiO6 octahedra.
the P+ interface due to the covalent bonding between Pb and O, in agreement with the
mechanism described in Sec. 6.3.3.
Another interesting feature of the polydomain structure is the distortion induced by
the opposite shift of up and down domains in the PbTiO3 layer, described in Fig. 6.12.
The offset between 180◦ domains was already predicted from first-principles calculations
to occur in bulk PbTiO3 [61]. The offset of the Pb sublattice in bulk was found to amount
0.6 Å. In the case of domains in superlattices, the relative shift is about on half of the
bulk value, 0.32 Å, for the domain structures with Nx = 12 and 16.
In bulk, the distortion of the unit cell induced by the offset between domains with
opposite polarization decay rapidly, and the PbTiO3 unit cells recover the mondomain
bulk-like shape just two unit cells away from the domain wall. On the contrary, inside
the PbTiO3 layer of the superlattices the small distance between domain walls (6 to 8
unit cells) prevents the relaxation to an homogeneously strained region. This induces an
inhomogeneous distortion of the unit cells in the PbTiO3 layer, as schematically depicted
in Fig. 6.12(b). The effect of this distortion can be quantified by the change in the inplane lattice constant at the center of a domain, when going from the top to the bottom
interface of the PbTiO3 layer. This difference, shown in Fig. 6.12(b) amounts a sizable
169
Table 6.3: Components of the polarization along y and z direction at the center of the
domains (see dashed box in Fig. 6.10) in (3/3) PbTiO3 /SrTiO3 superlattices for different
domain sizes. Polarization in µC/cm2 .
Nx
6
6
12
12
Ny
1
2
1
2
PyPTO
0
25
0
0
(a)
PzPTO
68
56
62
55
(b)
3.87 Å
3.91 Å
3.85 Å
4.00 Å
3.91 Å
4.00 Å
4.00 Å
3.91 Å
4.00 Å
3.85 Å
3.91 Å
3.87 Å
PxSTO
0
1
0
0
PzSTO
25
23
32
30
3.92 Å
0.32 Å
3.81 Å
Figure 6.12: Elastic distortion induced by the domain structure in PbTiO3 /SrTiO3
superlattices. (a) Inter-layer distance at the center of the domains for a periodicity of
Nx = 12 (only half of each domain is shown), here balls representing atoms are located at
their positions in a reference paraelectric structure and arrows represent displacements
of the cations in the relaxed configuration with respect to that phase. (b) Schematic
picture of the distortion where the relative shift of up and down domains is indicated
together with the change in the in-plane lattice constant across the PbTiO3 layer.
value of 0.1 Å.
Notice that, as a consequence of the periodic boundary conditions, domains are
aligned along the [001] direction and, while the inhomogeneous in-plain strain is forced
to be of opposite sign in the SrTiO3 layer, the offset could, in principle, propagate across
the SrTiO3 layer. Instead, despite the very small periodicity of the simulated supercell,
we find the “wiggle” induced by the domains to propagate little into the SrTiO3 layer,
with the offset being “absorbed” mostly by the interfaces. As a consequence of this,
very different interfacial inter-layer distance at P + (3.85 Å) and P − (4.00 Å) interfaces
of each domain are observed. For comparison, the interfacial inter-layer distances in the
170
monodomain configuration for the (3/3) periodicity are found to be 3.91 Å and 3.92 Å
at P + and P + interfaces respectively. As a result the offset of the Sr sublattice is 0.16
Å, significantly smaller than that of the Pb sublattice.
Interestingly, the distortion displayed in Fig. 6.12(b) resembles the kind of deformation associated to the flexoelectric effect [200, 201]. This phenomenon consist in an
inhomogeneous deformation of a material under applied electric field, or conversely, a
polarization of a sample under inhomogeneous strain. This effect has been suggested as a
possible mechanism to enhance piezoelectric properties or even induce piezoelectricity in
non-piezoelectric materials through strain engineering [34, 202, 30], and is known to be
particularly strong in ferroelectrics [203, 204]. In the case of domain structures in these
superlattices, this result suggest that the coupling between polarization and strain gradients through the flexoelectric effect might be strong and could be playing a significant
role in the elastic coupling of domain structures in thin films. Investigating flexoelectric
effects from first principles is hindered by the used periodic boundary condition that
limit the number of tensor components that can be calculated [36], but results of this
work should encourage further studies on this subject.
6.5
Conclusions
Our first-principles simulations show how the FE-AFD-strain coupling in PTO/STO
superlattices produces a phase diagram much richer than initially envisaged. The driving
force of the coupling is a combination of electrostatic and covalent effects triggered by
the symmetry reduction at the interface.
In monodomain PTO/STO superlattices we report a new phase (r-phase), with the
polarization lying along the [111] direction, for strain values around that corresponding
to a SrTiO3 substrate. The new phases might contribute to the stabilization of the monodomain phases over the recently observed and competing polydomain structures [188].
The experimental observation of the in-plane component of the polarization in the superlattices remains to be confirmed. The studied ferroelectric-AFD-strain couplings are
not restricted to PTO/STO, and are a promising way of generate novel magnetoelectric
couplings in interfaces involving magnetic materials [185].
We have also proved how the formation of domains in short-period PbTiO3 /SrTiO3
superlattices might compete in energy with monodomain configurations. The domains
are rather isotropic and display a domain of closure shape, similar to the one theoretically predicted in ferroelectric nanocapacitors and recently observed in various ferroelectric ultrathin films. The formation of domains help to screen the electrostatic energy
arising from the polarization discontinuity at the PbTiO3 /SrTiO3 interface. Both the
homogeneous out-of-plane polarization of SrTiO3 and the evolution of the layer by layer
polarization of PbTiO3 at the center of each domain are in good agreement with the
t2g -eg splitting inferred from electron energy loss spectroscopy with unit cell resolution.
Our results suggest that by controlling the superlattice periodicity, the monodomainpolydomain phase transition could be engineered to give a broad spectrum of enhanced
functional properties.
Conclusions
As it has been discussed extensively throughout this thesis, perovskite oxide thin films
offer a fantastic possibility for the design of multifunctional systems. One particular issue
of great importance for the performance of such devices are the electrostatic boundary
conditions, which can dramatically affect the structural and electronic properties of the
materials. The first principles study of the screening properties in perovskite oxide thin
films, has been precisely the leitmotif of our work throughout these years.
We have pointed out several times throughout this report that, in the field of complex
oxide thin films, the degree of development of first principles simulations and experimental techniques have advanced to the extent that both theorist and experimentalist can investigate essentially the same kind of systems. In this context, the role of first-principles
calculations is two-folded: it constitutes a tool that allows us to explore material combinations and configurations even before they are grown or detected in the laboratory,
and to complement the experimental results providing physical interpretations, helping
to optimize the functionalities of the systems.
A very important fraction of our work in the last years has been devoted to the
study of the applicability limits of the most common first-principles approach for the
study of oxides thin films, namely the DFT and more particularly, the LDA and GGA
approximations. These methods have provided in the last years invaluable insights on
the properties of these materials but have some limitations. As more and more complex
systems are being studied from first principles, issues derived from the misuse of these
approximations have started to arise. We have made an important effort to provide a
comprehensive, but at the same time clear and self-contained, guide aiming to clarify
the limitations of this theoretical method. We have tried to point out the typical fingerprints of unphysical results which are a consequence of those limitations, focusing in a
particularly relevant kind of systems: the ferroelectric capacitors.
These limitation are specially important for the study of the electronic structure.
This does not mean that they affect exclusively the electronic structure, as we have seen
the pathological description of the electronic properties have dramatic consequences on
the structural properties of the system, which is something that has been systematically
overlooked in the past.
Some of the pathological charge transfers that we have attributed to a wrong description of the electronic structure by the LDA or GGA methods have been sometimes
identify with metal-induced gap states (MIGS). In Chapter 3 we already suggest some
171
172
Conclusions
tools to clearly distinguish these physically meaningful and relevant charge transfers
and the pathological ones. Chapter 4 is entirely devoted to the characterization of the
evanescent states in ferroelectric capacitors. Though this is still a work in progress here
we tried to put in context many properties associated to MIGS that are often discussed
in the literature, but which connection is rarely explicitly treated. These states, are
propagating states in the metallic side of a metal/insulator junction and have energies
within the band gap of the latter. As an electron encountering a finite potential barrier,
this states penetrate in the insulator and decay exponentially. We have seen that the
charge rearrangements associated to the MIGS play an important role in the properties
of capacitors, particularly in the band alignment. Properties of these states can be connected to the characteristics of the complex band structure of the bulk insulator but, as
has been discussed in Chapter 4, interface intrinsic properties ultimately determine their
spatial an energetic distribution. This stresses out the importance of an explicit treatment of the actual interface, taking into account details with atomic resolution which
are very relevant for thin-film devices.
Chapters 3 and 4, which are mainly devoted to properties of ferroelectric capacitors
related with their electronic structure, highlight the importance of moving towards improved functionals. LDA and GGA approximations showed to be sufficient to achieve
a good description of structural properties in oxide thin films, but the discovery of
new functionalities and couplings between structural and electronic properties demands
taking a step forward and start working with more accurate methods. GW method
constitutes a precise alternative for the research of the electronic structure, and has the
advantage of being developed on top of DFT. Although it is true that these methods
provide very accurate description of the electronic structure of materials, they are still
too computationally-expensive to be applicable in realistic complex systems like those
studied in this report. Hybrid functionals, on the other hand, provide a very promising
way to improve the current knowledge of the physics of oxide thin films. “Traditional”
hybrid functionals, like B3LYP or B1, were developed in the context of quantum chemistry and have been shown to provide accurate description of the atomization energies,
bond lengths, and vibrational frequencies, together with good energy spectra for most
molecules. However when applied to some solids, and to ferroelectrics in particular,
they were found to fail to reproduce the correct structural properties of the materials.
Fortunately, there have been important advances in this direction. Specifically, the development of the B1-WC functional (a mix of exact exchange with the Wu-Cohen GGA
functional that has shown an exceptional description of the atomic structure of oxide
perovskites) opens the door to the study of many phenomena in complex oxide systems
that were completely beyond the scope of the most common approximations used today.
The use of a hybrid functional is, of course, much more computationally demanding than
LDA or GGA, but given the the fast evolution of computer science (both on the software
and hardware fronts) we should expect them to become the “standard” calculation relatively soon. And today, given the limitations of common DFT approaches discussed in
this thesis, a careful choice of the systems to be simulated can already provide enormous
amounts of information.
173
We have also investigated polarization domains in ferroelectric thin films, both in
capacitors and in superlattices. In both kind of systems, we have observed formation
of closure domains, a structure that was thought unlikely to form in ferroelectric materials due to the associated elastic energy. At odds with this common assumption, first
principles and model Hamiltonian simulations predicted the formation of domains of
closure in a several different systems, from BaTiO3 /SrRuO3 capacitors (as reported in
Chapter 5) to PbTiO3 free standing slabs. Theoretical studies have found this domain
wall structure to be surprisingly robust with respect to the ferroelectric material, the
level of screening provided by electrodes (if they exist), and the theoretical approach.
Very recently, the development of subatomic-resolution transmission electron microscopy
have finally allowed the experimental observation of such structures, confirming the theoretical predictions proposed just a few years before. This is one beautiful example
of the level of predictability that first-principles simulations have achieved in the last
years. Now that details about the precise structure of domain walls in ultra-thin films
are being determined, it is time to analyze other associated properties. For instance,
very recently, conductivity has been observed in domain walls in ferroelectric thin films.
It was first observed in BiFeO3 thin films and attributed to a reduction of the band gap
at the BiFeO3 domain walls, but the predicted change in the electronic structure at the
studied domain walls was too small to account for such a conductivity. It has also been
suggested that head-to-head or tail-to-tail domain walls and the associated electric fields
could explain the conductivity, but then, conductivity was observed in tetragonal PZT
as well, where spontaneous formation of polar domain walls is more unlikely. The most
reasonable explanation is, to our understanding, the pinning of domain walls at vacancies sites, something that has already been studied from first principles in bulk PbTiO3 .
The migration of vacancies to domain walls and the pinning of the latter suggest a relatively large concentration of defects at those regions of the material. The large electric
fields that we have found at the boundaries between domains of closure reinforces this
hypothesis. The deep understanding of properties of domain boundaries beyond their
structure could help to optimize some of these functionalities and lead to a potential
use of domains walls as active pieces in ultra-thin films devices. This would open new
avenues for the development of new multifunctional materials, where both the bulk-like
and domain-wall properties could be exploited. The investigation from first-principles
of such functionalities would, again, benefit from the use of improved functional that
provide a better description of the electronic structure of the system.
The number of possible functionalities that can be engineered in superlattices grows
every day. There is a great number of different materials, with similar structures but
completely different properties, that can be combined in the laboratory with atomic
control on the interface quality, giving rise to unimaginable properties. We have seen
how new couplings and phases emerge as a consequence of the interfacial effects in superlattices, properties that can be exploited to design synthetic functional materials.
The breaking of the system in domains of polarization increases complexity to the problem. Our simulations have show that, even thought the domain structure provide a
very effective screening, for small periodicities there is an electrostatic coupling between
174
Conclusions
the PbTiO3 layers, which shows up through the polarization of the SrTiO3 in between.
The energy penalty due to the polarization of the SrTiO3 suggest that the electrostatic
coupling should weaken as the staking periodicity increases. However, superimposed
to the electrostatic effect we have observed an inhomogeneous strain field associated to
the polydomain structure that might enhance the coupling between the ferroelectric layers. To our knowledge, this mechanism of elastic coupling has never been discussed and
could be crucial in the determination of the domain structure in superlattices. We plan
to perform simulations on polydomain superlattices with increasing periodicity aiming
to observe a transition from electrostatically-coupled to elastically-coupled ferroelectric
layers.
In this context, not only new combinations of materials should be explored, but
also new geometries that might optimize a given particular property. The stability of
vortex structures in ferroelectric nanostructures, for instance, can be used to design
nanoparticles able to sustain a toroidal moment or chiral nanorods with switchable
optical activity.
Some of these systems are still beyond the possibilities of current experimental techniques, and of course, some experimental devices are still too complicated to be simulated
fully from first-principles. Therefore, the feedback with model Hamiltonian simulations
is fundamental, bridging the size and energy scale between first principles simulations
and experiments. Fortunately, the collaboration between theoretical and experimental
groups in this field is very close, providing an extremely motivating work environment.
This definitely translate into more effective interactions and fruitful projects, which constitutes one of keys reasons for the fast development that this field has experienced in
the last years.
Appendix A
Ocupation function and energy
smearing of the local density of
states
The following analysis is entirely due to M. Stengel. We include it here because it came
up after discussions during the research reported Chapter 3 and because it is critical
for the understanding of some key points of that work. We aknowledge M. Stengel the
permission to reproduce it here.
A.1
Convolutions
Convolution is a mathematical operation on two functions f and g, producing a third
function that is typically viewed as a modified version of one of the original functions.
For the purpose of the present notes, it is useful to think of f as a data curve containing
the relevant physical information, and g as a rapidly-decaying “smoothing” function that
produces a local weighted average of f . We define the convolution of f and g, f ∗ g, as
the following integral transform,
(f ∗ g)(x) =
Z
+∞
−∞
f (y)g(x − y)dy
(A.1)
Convolutions have many properties, including commutativity and associativity. Furthermore, the Dirac delta can be thought as the identity under the convoluton operation,
(f ∗ δ)(x) = f (x),
(A.2)
and under certain assumptions an inverse operation can also be defined. In other words,
the set of invertible distributions forms an abelian group under the convolution.
A particularly useful property holds in relationship to the Fourier transform,
F(f ∗ g) = k · F(f ) · F(g)
175
(A.3)
176
Appendix B. Ocupation function and energy smearing of the LDOS
where F(f ) denotes the Fourier transform of f , and k is a constant that depends on
the normalization convention for the Fourier transform. Thus, in reciprocal space the
convolution becomes a simple product. This naturally provides an efficient convolution
algorithm: the workload is reduced from O(N 2 ) to O[N log(N )].
A.2
Local density of states
In this work we use [Eq. (3.27)] the following formula to compute the smeared local
density of states (LDOS),
X
ρ̃(r, E) =
wk |ψnk (r)|2 g(E − Enk ).
(A.4)
nk
We shall see that this is indeed a convolution. We first get rid of the spatial cordinates.
To this end, it is customary to integrate the LDOS in real space over a given volume V ,
X
ρV (E) =
wk ρnk (V )g(E − Enk ),
(A.5)
nk
where
Z
ρnk (V ) =
V
d3 r |ψnk (r)|2 .
(A.6)
(Note that sometimes it might be more convenient to use a projected density of states,
rather than a local density of states. In such cases it is sufficient to replace the realspace integral in the above equation with an appropriate sum over angular momentum
components. The following discussion remains unchanged.) Now the LDOS is a function
of a single energy variable. If we write
X
fV (E) =
wk ρnk (V )δ(E − Enk ),
(A.7)
nk
we can easily see that ρV = fV ∗g. This leads to a simple reciprocal-space expression. We
first define an energy window, [Elow , Ehigh ], that contains the entire eigenvalue spectrum
Enk . We actually take a window which is slightly larger, where this “slightly” depends
on the decay properties of g,
Elow = min(Enk ) − ,
Ehigh = max(Enk ) + .
(A.8)
The width of this window is Ehigh − Elow = ∆E. We represent ρV (E) in reciprocal space
as a discrete Fourier transform,
X
ρV (E) =
eiωE ρV (ω),
(A.9)
ω
where ω = 2πn/∆E and n is an integer. By using Eq. (A.3) we have
ρV (ω) = ∆E · fV (ω) · g(ω).
(A.10)
177
A.3. Gaussian vs. Fermi-Dirac smearing
a
1
c
0.5
fG(E)
fFD(E)
0.4
0.3
0.2
0.1
0
-0.5
0
0
-30
0.5
4
-20
-10
0
10
20
30
0.05
b
gG(E)
gFD(E)
3
d
0.04
gG(ω)
gFD(ω)
0.03
2
0.02
1
0.01
0
0
-0.5
0
Energy (eV)
0
0.5
10
20
30
40
-1
Frequency (eV )
50
Figure A.1: (a) Gaussian (σ = 0.15 eV) and Fermi-Dirac (σ = 0.075 eV) occupation
functions. (b) Kernel of the occupation functions as defined in the text. (c-d) Fourier
transform of the smearing kernels g, assuming an energy window of [−1, 1].
The Fourier transform of a Dirac delta centered in the origin is a constant. Eq. (A.10)
then decomposes the local density of states into a structure factor,
fV (ω) =
1 X
wk ρnk (V )e−iωEnk ,
∆E
(A.11)
nk
and a form factor g(ω). Obviously, this formulation is only convenient if the function g
has a fast decay in both real and reciprocal space, so that the sum in Eq. (A.9) can be
truncated. This is indeed the case for the most widely used smoothing functions g, as
we shall see in the following.
A.3
Gaussian vs. Fermi-Dirac smearing
The Gaussian smearing (G) and the Fermi-Dirac (FD) smearing are by far the most
popular choices for the occupation function in first-principles calculations of metallic
systems. If we define the occupation function f as the integral of a “kernel” function g,
f (E) = 1 −
Z
E
g(x)dx,
−∞
(A.12)
178
one can verify that the Gaussian or Fermi-Dirac occupation are, respectively, reproduced
by the following choices of g,
gG (x) =
gFD (x) =
1
2
2
√ e−x /σ ,
πσ
σ −1
,
2 + ex/σ + e−x/σ
(A.13)
(A.14)
where σ is the smearing energy [these correspond to Eq. (3.28a) and Eq. (3.28b)]. It is
easy to see that, by combining Eq. (A.13) or Eq. (A.14) with Eq. (A.12) one obtains
the standard definitions of the occupation function (we assume that the complementary
error function, erfc, values 2 at −∞),
fG (x) =
fFD (x)
1
2
erfc (x/σ),
=
1
ex/σ +1
.
(A.15)
(A.16)
It is useful to spell out the explicit formulas for the Fourier transforms of both smearing
functions,
2 2
gG (ω) =
gFD (ω) =
e−ω σ /4
,
∆E
πωσ
.
∆E sinh(πωσ)
(A.17)
(A.18)
Note that the above formulas are normalized according to the conventions on the Fourier
transforms that we used in the previous section. The functions f and g defined above
are shown in Fig. A.1. Note that a different choice of σ was used in the Fermi-Dirac and
in the Gaussian case. A FD distribution is roughly equivalent to a G distribution with
a σ value that is twice as large.
In the main text and here we have assumed that it is a good idea to use the same
g kernel in the calculation and in the construction of the LDOS. We shall substantiate
this point in the following Section.
A.4
On the optimal choice of g
In many cases, the specific choice of the g function to be used in Eq. (A.4) is largely
arbitrary. Typically, the goal is to filter out the unphysical wiggles due to the discretization of the k-mesh, but at the same time to preserve the main physical features, without
blurring them out completely. This calls for a smearing function that is neither too sharp
nor too broad. Since a “slightly too broad” or a “slightly too sharp” smearing function
usually does not influence the physical conclusions, in many cases one has the freedom
of choosing whatever yields the clearest visual aid to support the discussion.
There are cases, however, where this choice is not just a matter of aesthetics, and
using the “wrong” g function can qualitatively and quantitatively influence the interpretation of the results. More specifically, the issue concerns cases where the analysis of the
179
A.4. On the optimal choice of g
4
1
0.8
3
0.6
2
0.4
1
0.2
0
-1
-0.5
0
0.5
Energy (eV)
1
0
-1
-0.5
0
0.5
Energy (eV)
1
Figure A.2: Left: Fermi-Dirac occupation function, identical to that of Fig. A.1(a) (solid
curve); hypothetical orbital located at an energy of 0.15 eV above the Fermi level (dashed
line); the thermal occupation of this state yields a total charge of 0.119 electrons (red
dot). Right: density of states corresponding to the single isolated orbital at an energy
of 0.15 eV above the Fermi level, smeared by using the gFD kernel of Fig. A.1(b); the
integral of the DOS up to the Fermi level (shaded area) yields the exact same charge of
0.119 electrons.
LDOS (or DOS or PDOS) is used to detect and quantify the population of orbitals that
lie close in energy to the Fermi level. As we focus on charge spill-out phenomena that
concern the conduction band of a dielectric/ferroelectric film in contact with a metallic
electrode, this is a central point of our work. The problem is most easily appreciated
by looking at the left panel of Fig. A.2. There is a single orbital lying at an energy
of 0.15 eV above the Fermi level. As this orbital lies above the Fermi level, one might
be tempted to think that the orbital is empty, and that charge spill-out does not occur
at all. However, calculations in metallic systems are routinely performed by using an
occupation function that is artificially broadened, in order to improve convergence of the
ground-state properties; in Fig. A.2 we assume a Fermi-Dirac occupation with a fictitious electronic temperature of 0.075 eV. It is easy to see that with such an occupation
function, the orbital lying at 0.15 eV won’t be empty, but will be “thermally” populated
by tail of the Fermi-Dirac distribution. The final result is a charge transfer of 0.119
electrons into this orbital.
Now, is there a “right” way to construct the DOS curve, such that the abovementioned charge transfer could be qualitatively and quantitatively inferred from the
DOS, without knowing any further detail of the calculation? The answer is yes, and
consists in constructing the DOS by using a broadening g function which is consistent
with the occupation function used by the code. In this case, this is gFD , with a σ identical
to that used to calculate the electronic ground state. To demonstrate this point, we plot
in the right panel of Fig. A.2 the DOS of this isolated orbital at 0.15 eV, appropriately
convoluted with gFD . Eq. (A.12) guarantees that, by doing this, one recovers the very
intuitive result that the total amount of electron charge, Q, present in the volume V
(over which the LDOS was integrated) exactly corresponds to the integral of the DOS
180
up to the Fermi level,
Z
EF
Q=
ρV (E)dE.
(A.19)
−∞
Then, a simple look at the DOS curve is sufficient to ascertain whether a significant
transfer of charge has occurred into a specific group of bands. As this rigorous sum rule
can be very practical in the analysis of the results, we encourage a systematic use of the
“internally consistent” LDOS construction described above.
Appendix B
Local polarization via Born
effective charges
In this Appendix we include a discussion due to M. Stengel about the approach, used
in several parts in this manuscript and ubiquitously in the recent literature, of associating the local value of the “effective” polarization (i.e. the induced P with respect to
the reference centrosymmetric configuration [132]) in capacitor heterostructures with an
approximate formula, based on the Born effective charges, Z ∗ . In particular, M. Stengel
[105] provide formal justification for an improved formula, still based on the Z ∗ , that
we introduced in Sec. 3.3.2 [Eq. (3.38)]. We aknowledge M. Stengel the permision to
reproduce here the discussion.
Recall the definition of the approximate effective polarization in terms of the Born
effective charges in a bulk solid,
PZ =
e X ∗
Z Rαz .
Ω α α
(B.1)
It is easy to verify that the layer-resolved expression PjZ of Eq. (3.37) reduces to P Z
in the case of a periodic crystal, where PjZ is a constant function of the layer index j.
P Z does not reduce to the “correct” polarization P (D) at any value of D, as it does not
take into account the additional polarization of the electronic cloud due to the internal
field E(D) (recall that the Born effective charges are defined under the condition of zero
macroscopic electric field. [14])
Taking the Taylor expansion of the polarization as a function of D (we assume for
simplicity that D, P and P Z all vanish in the reference centrosymmetric structure), we
can write
P Z (D) =
dP Z
dP Z dE
D + ... =
D + ...
dD
dE dD
(B.2)
For small values of D, we can truncate the previous expansion at the linear order term.
Now, by definition
181
182
Table B.1: Values of the susceptibilities χ and scaling factors χTOT /χION for the ferroelectric materials considered in Chapter 3.
BaTiO3
PbTiO3
KNbO3
where χION
TOT
-48.87
-96.54
-34.92
∞
6.48
8.33
6.27
χTOT /χION
0.90
0.93
0.87
dP Z
= χION ,
dE
is the lattice-mediated susceptibility, and
(B.3)
dE
= (0 TOT )−1 ,
(B.4)
dD
where TOT is the total dielectric constant of the insulator (relative to the vacuum
permittivity 0 ). Substituting Eq. (B.3) and Eq. (B.4) into Eq. (B.2)
P Z (D) ∼ D
χION
.
0 TOT
(B.5)
The same kind of arguments applied to the total polarization yield
P (D) ∼ D
χTOT
,
0 TOT
(B.6)
where χTOT is the sum of the lattice-mediated susceptibility, χION , and the purely
electronic (frozen-ion) susceptibility, χ∞ . Note that χION is not bound to be positive. In
a ferroelectric material, for example, the centrosymmetric reference structure is unstable
and therefore yields a negative χION (and hence TOT ), as discussed in Ref. [51]. The
present derivation is general and encompasses those cases.
From the above considerations it immediately follows that an estimate of the total
polarization, which is exact in the linear limit, can be given as
P (D) ∼
χTOT Z
P (D).
χION
(B.7)
This is essentially Eq. (3.38). In practice, χION and χ∞ are calculated in the reference
phase according to the standard definitions, [205]
χION = 0 (TOT − ∞ ) =
∗ )2
0 e2 X (Z̃m
,
2
M0 Ω m ωm
(B.8)
∗ are the normal mode charges and ω 2 are the eigenvalues
where M0 is a unit mass, Z̃m
m
of the dynamical matrix, and
183
0.8
1.5
PbTiO3 (bulk)
2
Polarization (C/m )
0.6
2
Polarization (C/m )
BaTiO3 (film)
0.4
0.2
1
0.5
*
Bare Z
Berry phase
Rescaled Z
0
0
0 0.1 0.2 0.3 0.4 0.5
0 0.2 0.4 0.6 0.8
Reduced electric displacement d
*
1
Figure B.1: Polarization P in a BaTiO3 film and PbTiO3 bulk as a function of the
reduced electric displacement field d = DS. Data are taken from Ref. [52] (see Section
III.C.1) and Ref. [51].
χ∞ = 0 (∞ − 1),
−1
∞ = 0
dE .
dD fixed−ions
(B.9)
The values of these physical constants that are relevant for the results presented in this
manuscript are reported in Table B.1.
We proceed in the following to test this approximation on two representative bulk
ferroelectric materials, PbTiO3 and BaTiO3 . We take the relevant data (linear susceptibilities, Born charges and relaxed structures as a function of D) from the calculations
of Ref. [51] and Ref. [52]. Note that the BaTiO3 calculation was performed at a fixed
value of the in-plane lattice parameter (indicated as “film” in the figure) while in the
PbTiO3 calculation both a and c parameters were relaxed for each value of D. The
results are presented in Fig. B.1. In both cases, the “bare” value P Z is systematically
overestimated compared to the Berry-phase polarization. With the correction described
above, i.e. by rescaling all values by the factor χTOT /χION , the approximate value of P
accurately matches the Berry-phase one. The accuracy is surprisingly good in BaTiO3 ,
where the maximum deviation is of the order of 1%. In PbTiO3 , for large values of d the
rescaled-Z ∗ value of P presents significant deviations. Note that these deviations mostly
concern values of d that are larger than that of the ferroelectric ground state (d ∼ 0.74),
and therefore are not of concern in this manuscript. We ascribe these deviations to the
field-induced structural transition that was described in Ref. [51].
In conclusion, this simple rescaling factor appears to be an effective way to obtain
184
a relatively accurate value of the local P in heterostructure calculations, based only on
the local atomic positions and a few ingredients that can be easily computed in the
bulk reference structure. From the results of our tests, we expect the agreement to
be best in cases where the polarization is small (closer to the linear limit where the
approximation becomes exact). Furthermore, cases where the ferroelectric polarization
can be represented in terms of a single “soft mode” such as BaTiO3 seem to work
better than cases, like PbTiO3 , where significant mode mixing and non-trivial structural
transitions occur at higher D values.
Appendix C
Complex band structure within
the nearly-free electron model
Given its periodicity, the potential in the insulator can be expressed as a Fourier series.
If, for the sake of simplicity, we restricting the Fourier expansion to the first term (besides
the Gamma term which is a constant) we get a potential in the insulator
V (z) = V0 + Vg cos(gz),
(C.1)
where g = 2π/a is the shortest reciprocal lattice vector. Within the nearly-free electron
model, at the Brillouin zone boundaries, the perturbative potential mixes eigenstates
from different bands. At the Brillouin zone boundaries the eigenfunctions are thus a
mixing of two plane waves (which are the eigenfunctions of the unperturbed system)
ψIk (z) = Aeikz + Bei(k−g)z .
(C.2)
Inserting previous trial function into the time-independent Schrödinger equation
−
d2
+
V
(z)
ψI = EψI ,
dz 2
(C.3)
we get
2
k + V0 + Vg cos(gz) − E Aeikz + (k − g)2 + V0 + Vg cos(gz) − E Bei(k−g)z = 0,
(C.4)
which must be satisfied at every point in space. Evaluating in z = 0 and z = a/2 we
get, respectively
2
k + V0 + Vg − E A + (k − g)2 + V0 + Vg − E B = 0,
2
k + V0 − Vg − E A − (k − g)2 + V0 − Vg − E B = 0.
185
(C.5a)
(C.5b)
186
Appendix C. Complex band structure within the nearly-free electron model
Adding and subtracting previous equations, and writing in matrix form, they transform
into
2
k + V0 − E
Vg
A
= 0.
(C.6)
Vg
(k − g)2 + V0 − E
B
Previous system of equations can be more easily solved using a new variable κ = k − g/2,
which is the deviation of k from the Brillouin zone boundary. Equation C.6 transforms
into
(κ + g/2)2 + V0 − E
Vg
A
= 0.
(C.7)
Vg
(κ − g/2)2 + V0 − E
B
Eigenvalues of the Schrödinger equation for the system are obtained finding the energy
values E that makes 0 the determinant of previous matrix equation
(κ + g/2)2 + V0 − E
Vg
= 0,
2
Vg
(κ − g/2) + V0 − E 2
κ + κg + (g/2)2 + V0 − E
Vg
Vg
κ2 − κg + (g/2)2 + V0 − E
= 0,
κ4 − κ2 κg + κ2 (g/2)2 + κ2 V0 − κ2 E
+κ2 κg − κ2 g 2 + κg(g/2)2 + κgV0 − κgE
+κ2 (g/2)2 − κg(g/2)2 + (g/2)4 + (g/2)2 V0 − (g/2)2 E
+κ2 V0 − κgV0 + (g/2)2 V0 + V02 − V0 E
−κ2 E + κgE − (g/2)2 E − V0 E + E 2 − Vg2 = 0,
κ4 + (g/2)4 + V02 + E 2
+ 2κ2 (g/2)2 + 2κ2 V0 − 2κ2 E
+ 2(g/2)2 V0 − 2(g/2)2 E
− 2V0 E
− κ2 g 2 − Vg2 = 0,
2
2
κ + (g/2)2 + V0 − E − κ2 g 2 − Vg2 = 0,
(C.8)
which finally leads to
E = κ2 + (g/2)2 + V0 ± κ2 g 2 + Vg2
1/2
.
(C.9)
Appendix D
Resumen
La investigación de óxidos de metales de transición se encuentra en un momento crucial.
La situación es tan interesante que ha sido comparada con la de la fı́sica de semiconductores de hace sesenta años [1]. Se trata ésta de una comparación muy seria ya que hoy
en dı́a nustras vidas dependen enormemente de multitud de dispositivos que han podido
desarrollarse gracias a la investigación básica en ciencia de materiales que se realizaba
en aquella época. Sin embargo esta comparación no es gratuita. Las última décadas de
investigación en óxidos de metales de transición han sido tremendamente excitantes y ha
llevado al descubierto un gran número de funcionalidades en estos materiales, como la
superconductividad o la magnetoresistencia colosal por citar un par de ejemplos. En los
últimos años se ha prestado una particular atención a una familia de óxidos de metales
de transición en particular, que comparten una estructura cristalina de tipo perovskita.
Bajo la sencillez de esta estructura, con tan solo cinco átomos en la celda unidad para
la estructura cúbica de referencia de alta simetrı́a, se esconde un gran número de sutiles propiedades fı́sicas. Estos materiales, a pesar de compartir una estructura atómica
muy similar, presentan una amplia gama de propiedades: superconductividad, ferromagnetismo, magnetorresistencia colosal, multiferroicidad, propiedades ópticas no lineales
... Más interesante aún, la amplia gama de propiedades que surgen de estos materiales
sugiere que la aparición de una propiedad en particular ha de ser la consecuencia de
un delicado equilibrio entre múltiples interacciones que probablemente sean comunes
a muchos de los miembros de esta familia de materiales. De hecho, este diagrama de
fases tan diverso surge de la estrecha competencia entre las diferentes las interacciones
que tienen lugar en estos materiales. Mientras que en otros tipos de materiales tales
como semiconductores o algunos metales, alguna de las interacciones involucradas – repulsiones coulombianas, deformaciones, intercambio, etc – domina claramente sobre las
otras y determina las propiedades de volumen del sistema, en los óxidos de metales de
transición estos efectos son muy a menudo de la misma magnitud. De tal forma que
estos materiales suelen presentar una compleja competición de varias fases y una fuerte
sensibilidad hacia perturbaciones externas [188, 3]. Esto hace que estos materiales sean
los candidatos ideales para el diseño de dispositivos artificiales con funcionalidades a la
carta.
187
188
Resumen
Una de las propiedades que presentan algunas de estas perovskitas es la ferroelectricidad. Un material ferroeléctrico es un aislante que presenta al menos dos estados
diferentes de la polarización e nausencia de campo eléctrico, pudiendose inducir la transición de uno a otro mediante la aplicación de un campo eléctrico externo [4]. El término
“ferroeléctrico” fue acuñado en analogı́a a los materiales ferromagnéticos, ya que ambos
presentan un ciclo de histéresis cuando la polarización (magnetización en el caso de un
material ferromagnético) se mide en función del campo eléctrico (magnético) aplicado.
Los ferroeléctricos son materiales con un gran interés aplicado [5, 6]. La ferroelectricidad, es decir, la capacidad de transitar entre dos o más estados de polarización al
aplicar un campo externo, puede ser explotada por ejemplo para la fabricación de dispositivos de memoria, donde cada estado de la polarización puede ser asignado a los
valores 0 y 1 de un bit de información. ste es el principio básico de funcionamiento de
las memorias ferroeléctricas de acceso aleatorio (FeRAM). Además la ferroelectricidad
suele estar asociada a otras propiedades de gran interés. Por ejemplo, todos los materiales ferroeléctricos son también piezoeléctricos (la aplicación de un campo eléctrico
puede inducir una deformación y viceversa) y piroeléctricos (los cambios de temperatura
de la muestra modifican la polarización). Estas propiedades se explotan ya hoy en dı́a
en la fabricación de transductores, actuadores o detectores de infrarrojos. Uno de los
ejemplos más exitoso son las cerámicas de PbZr1−x Tix O3 que están presentes en gran
variedad de dispositivos, desde equipos de ecografı́a hasta los inyectores de los motores de
automóvil o los microscopios de fuerza atómica. Además, los materiales ferroeléctricos
poseen una constante dieléctrica muy grande que permite su uso para la fabricación de
condensadores de memorias dinámicas de acceso aleatorio (DRAM).
La constante miniaturización de los dispositivos electrónicos impuesta por la industria electrónica e impulsada por la necesidad de dispositivos electrónicos más rápidos y
al mismo tiempo más pequeños y más eficientes energéticamente, ha motivado el estudio de las propiedades de los materiales ferroeléctricos a escala nanométrica. Es bien
sabido que las propiedades de volumen los ferroeléctricos y de la mayorı́a de las perovskitas se ven fuertemente afectadas por las condiciones de contorno, que se hacen
especialmente relevantes al disminuir el tamaño de los dispositivos. Se sugirió, por ejemplo, que la ferroelectricidad tenı́a un tamaño crı́tico de unos 10 nm, por debajo del
cual la alteración en el balance entre las interacciones que gobiernan la ferroelectricidad
y la aparición de campos de depolarización, provocarı́an la pérdida de la polarización
espontánea. Sin embargo, con los avances en la sı́ntesis y caracterización experimental
de láminas ultradelgadas se ha observado ferroelectricidad en pelı́culas cada vez más
delgadas, encontrando que el espesor crı́tico para la ferroelectricidad es de tan sólo unas
pocas monocapas.
Por otro lado, la mejora de las técnicas de crecimiento ha permitido aprovechar el sutil
equilibrio entre las diferentes inestabilidades y la fuerte sensibilidad de estos materiales
a las condiciones de contorno para modificar a voluntad las propiedades de las láminas
delgadas de perovskitas. La búsqueda de una ruta para diseñar y sintetizar materiales
artificiales con funcionalidades a la carta ha impulsado enormemente la investigación en
estos sistemas. Sin embargo, la estrecha competencia entre las diferentes interacciones
189
y fases hace muy difı́cil, si no imposible, predecir las propiedades de las estructuras artificiales en términos de reglas simples y a partir de las propiedades de volumen de los
materiales constituyentes. Ésta es una de las razones por las que las simulaciones desde
primeros principios juegan un papel fundamental en las notables avances que el campo
ha experimentado en los últimos años [7]. La rápida evolución de los modelos atómicos,
impulsada por el rápido y constante aumento de la potencia de cálculo (hardware) y por
importantes avances en el desarrollo de algoritmos más eficientes (software), hace posible describir las propiedades de los materiales de forma muy precisa utilizando métodos
basados en las leyes fundamentales de la mecánica cuántica y de la electrostática. Incluso si el estudio de sistemas complejos requiere algunas aproximaciones prácticas, estos
métodos no emplean ningún parámetro ajustado empı́ricamente. Por ello se les conoce
como métodos desde “primeros-principios” o “ab-initio”.
La situación actual es particularmente interesante. Por un lado, los recientes avances
permiten controlar la sı́ntesis de láminas delgadas a escala atómica y medir localmente
sus propiedades ferroeléctricas [8]. Por otro lado, el aumento constante de la potencia
de cálculo y las mejoras en la eficiencia de los algoritmos permiten el estudio preciso
desde primeros principios de sistemas cada vez más grandes y complejos que coinciden
en tamaño con aquéllos crecidos en los laboratorios, lo que permite una retroalimentación
continua entre los experimentos y los modelos teóricos. Este esfuerzo conjunto ha llevado,
en el último par de décadas, a avances muy significativos en el conocimiento a nivel
microscópico de las propiedades ferroeléctricas de las perovskitas y de los compuestos
relacionados. Cada paso adelante da lugar al descubrimiento de nuevos fenómenos que
no hace más que incrementar el número de interrogantes, genera nuevas oportunidades
para la explotación práctica de las funcionalidades y motiva aún más la investigación
de estos materiales. Un buen ejemplo es el descubrimiento de interfases conductoras en
superredes de LaAlO3 y SrTiO3 , dos materiales aislantes [9], o la conductividad de las
paredes de dominio en BiFeO3 [10].
Un aspecto de gran interés debido a sus implicaciones tanto en las propiedades fı́sicas
básicas de las láminas delgadas ferroeléctricas como en sus posibles aplicaciones, es la
comprensión de los mecanismos de apantallado en dichos sistemas. El valor de la polarización en la superficie de una lámina delgada o su discontinuidad a través de una
interfase con un electrodo o con otro material aislante genera una carga que da lugar a
un campo de depolarización que tiende a suprimir la polarización. Pueden darse varios mecanismos para compensar estas cargas de polarización: acumulación de carga de
apantallado en los electrodos, adsorbentes iónicos en una superficie libre o la ruptura del
sistema de dominios en la polarización. En esta tesis se han estudiado algunos de estos
mecanismos desde primeros principios. Hemos prestado una especial atención a los aspectos metodológicos relacionados con las propiedades de apantallado en ferroeléctricos que
pueden ser importantes para el estudio desde primeros principios de propiedades de interfases. Para ello nos hemos centrado en dos sistemas particularmente importantes: (i) las
interfases ferroeléctrico/metal presentes en los condensadores ferroeléctricos y (ii) las interfases ferroeléctrico/ferroeléctrico incipiente tales como las superredes PbTiO3 /SrTiO3
(sistema que está atrayendo un gran interés debido a la aparición de una ferroelectrici-
190
Resumen
dad impropia en el lı́mite de pelı́culas ultradelgadas). En el primer caso estudiaremos el
reordenamiento de carga en las uniones metal/ferroeléctrico, asociado a la formación de
estados evanescentes en el gap, y la formación y las propiedades de los dominios de polarización en este tipo de dispositivos. En el caso de las superredes de PbTiO3 /SrTiO3 ,
el descubrimiento de un acoplamiento propio de la interfase entre inestabilidades polares
y no polares, que no se da en los materiales en volumen, ha atraı́do mucha atención
en los últimos años. En estos sistemas exploramos cómo varı́a el diagrama de fases con
la deformación epitaxial, su efecto sobre el acoplamiento entre las inestabilidades y las
propiedades de las fases polidominio.
Esta memoria se encuentra organizada de la siguiente forma. En el Capı́tulo 1 se
introducen las propiedades generales de los materiales ferroeléctricos ABO3 , se discuten
las distintas inestabilidades presentes en estos compuestos, su competición y la conexión
con la aparición de la ferroelectricidad. Estudiamos cómo los efectos del tamaño y las
condiciones de contorno afectan a estas propiedades, y cómo esto puede utilizarse en
la obtención de estructuras con nuevas funcionalidades. En este capı́tulo se presta una
especial atención a las condiciones de contorno electrostático y a los distintos mecanismos de apantallado que pueden tener lugar en una lámina ferroeléctrica delgada, siendo
éste el tema principal de este trabajo de tesis. En el capı́tulo 2 se describen los detalles teóricos básicos de los métodos de primeros principios utilizados para llevar a cabo
la investigación que se presenta en esta memoria. Algunas problemas asociados con
la simulación de heteroestructuras se tratan en el capı́tulo 3. Ofreceremos un procedimiento claro para detectar resultados patológicos como consecuencia de un mal uso del
método teórico más empleado, la teorı́a funcional de la densidad (DFT), y sugeriremos
mecanismos para evitar tales errores. Los resultados de este capı́tulo no son tan solo
metodológicos, ya que algunos mecanismos de apantallado que han sido detectados en
heteroestructuras patológicas pueden ser relevantes para algunas interfases reales. El
capı́tulo 4 se centra en estudio de los estados evanescentes en el gap de un ferroeléctrico,
llamados estados de gap inducidos por el metal (MIGS, metal-induced gap states), que
se forman en la unión metal/ferroeléctrico. Estos estados juegan un papel fundamental
en los fenómenos de efecto túnel y en la formación de las barreras Schottky. En este
capı́tulo discutimos hasta qué punto las caracterı́sticas de estos estados pueden predecirse a partir de las propiedades de volumen del ferroeléctrico, y cuáles están relacionadas
con efectos intrı́nsecos de la interfase. En el capı́tulo 5 hablamos de la formación de dominios de polarización en condensadores ferroeléctricos. Hemos predicho por primera
vez la formación de dominios de cierre en láminas delgadas ferroeléctricas a partir de
primeros principios. A pesar de que su formación en láminas delgadas ferroeléctricas
era considerada poco probable debido al gran acoplamiento entre la polarización y la
deformación, esta estructura resulta ser increı́blemente general y proporciona un apantallado extremadamente eficiente. En el capı́tulo 6 se estudia el efecto de la deformación
epitaxial sobre el acoplamiento entre la polarización y la rotación de los octaedros de
oxı́geno en las superredes de PbTiO3 /SrTiO3 . El fuerte acoplamiento de estas dos inestabilidades en este sistema se explica en términos de un modelo covalente. En vista
de los resultados del capı́tulo 5, también tenemos en cuenta la formación de dominios
191
de la polarización en las superredes. En estas superredes se han encontrado estructuras
similares a las observadas en los condensadores ferroeléctricos, formandose vórtices de
dipolos en las paredes de dominio. Por último, los resultados de este trabajo se resumen
en las conclusiones.
Appendix E
Conclusiones
Como se ha discutido ampliamente a lo largo de esta tesis, las láminas delgadas de perovskitas ofrecen una posibilidad fantástica para el diseño de sistemas multifuncionales.
Un tema de gran importancia para el funcionamiento de dichos dispositivos son las
condiciones de contorno electrostáticas, que pueden afectar de manera dramática a las
propiedades estructurales y electrónicas de los materiales. El tema central de nuestro
trabajo a lo largo de estos años ha sido precisamente el estudio desde primeros principios
de las propiedades de apantallado de láminas delgadas de perovskitas.
En esta memoria se ha señalado en varias ocasiones que el grado de desarrollo de las
simulaciones desde primeros principios y de las técnicas experimentales en este campo
han convergido de tal manera que, hoy en dı́a, tanto teóricos como experimentales podemos investigar en esencia el mismo tipo de sistemas. En este contexto, los cálculos
desde primeros principios tienen dos papeles: constituyen una herramienta que nos permite explorar combinaciones de materiales y configuraciones, incluso antes de que éstas
se sinteticen o detecten en el laboratorio, y sirven para complementan los resultados
experimentales proporcionando interpretaciones fı́sicas que ayudan a optimizar las funcionalidades de los sistemas. Una fracción muy importante de nuestro trabajo en los
últimos años se ha centrado en el estudio de los lı́mites de aplicabilidad de la DFT, que
es método desde primeros principios más comúnmente empleado en el estudio de óxidos
en láminas delgadas. Las aproximaciones LDA y GGA han proporcionado en los últimos
años una información inestimable sobre las propiedades de estos materiales, pero tienen
algunas limitaciones. Al aumentar la complejidad de los sistemas estudiados a partir
de primeros principios han empezado a surgir problemas derivados del mal uso de estas
aproximaciones. Hemos realizado un importante esfuerzo para proporcionar una guı́a
completa y clara para aclarar las limitaciones de este método teórico. Hemos tratado
de destacar las tı́picas huellas que permiten detectar resultados no fı́sicos consecuencia
de esas limitaciones, centrándonos en un tipo de sistemas de especial relevancia: los
condensadores ferroeléctricos.
Estas limitaciones son especialmente importantes para el estudio de la estructura
electrónica. Esto no quiere decir que afecten exclusivamente a la estructura electrónica,
como hemos visto, la descripción patológica de las propiedades electrónicas tiene conse193
194
Resumen
cuencias dramáticas sobre las propiedades estructurales del sistema, que es algo que ha
sido sistemáticamente ignorado en el pasado.
Algunas de las transferencias de carga patológicas que hemos atribuido a una deficiente descripción de la estructura electrónica debid a las aproximaciones LDA y GGA se
han identificado en algunas ocasiones con estados de gap inducidos por el metal (MIGS:
metal-induced gap states). En el capı́tulo 3 sugerimos algunas herramientas para distinguir claramente estas transferencias de carga, con significado fı́sico significativo y relevante, de las patológicas. El capı́tulo 4 está totamente dedicado a la caracterización de
los estados evanescentes en condensadores ferroeléctricos. Aunque aún nos encontramos
trabajando en este tema, hemos tratado de poner en contexto muchas propiedades que
a menudo se discuten en la literatura y que a pesar de estar todas asociadas a los MIGS
rara vez se relacionan con estos estado de forma explı́cita. Estos estados son estados de
propagación en el lado metálico de una unión metal/aislante y sus energı́as se encuentran dentro del gap de este último. Cuando un electrón se encunetra con una barrera de
potencial finita estos estados penetran en el aislante y decaen exponencialmente. Hemos
visto que los reordenamientos de carga asociados a los MIGS juegan un papel importante en las propiedades de los condensadores y en particular en el alineamiento de la
estructura de bandas. Las propiedades de estos estados están relacionadas con las caracterı́sticas de la estructura de banda compleja del aislante en volumen, aunque como
se ha discutido en el capı́tulo 4, las propieddades intrı́nsecas de la interfase determinan
en última instancia, sus distribuciones espacial y energética. Esto pone en relieve la
importancia de un tratamiento explı́cito de la interfase teniendo en cuenta los detalles a
nivel atómico, fundamentales en los dispositivos de láminas delgadas.
A lo largo de los capı́tulos 3 y 4, que se dedican principalmente a las propiedades de
los condensadores ferroeléctricos relacionadas con su estructura electrónica, se destaca
la importancia de la mejora de los funcionales. Las aproximaciones LDA y GGA han demostrado ser válidas para describir las propiedades estructurales de las pelı́culas delgadas
de óxidos, pero el descubrimiento de nuevas funcionalidades y acoplamientos entre las
propiedades estructurales y electrónicas exige dar un paso más allá y empezar a trabajar
con métodos más precisos. El método GW constituye una alternativa para la investigación de la estructura electrónica y tiene la ventaja de estar basado en la DFT. Si bien
es cierto que estos métodos proporcionan una descripción muy precisa de la estructura
electrónica de los materiales, siguen siendo demasiado costosos computacionalmente para
ser aplicados a sistemas complejos reales, como los que se estudian en esta memoria. Por
otro lado, los funcionales hı́bridos constituyen una opción prometedora para mejorar el
actual conocimiento acerca de la fı́sica de láminas delgadas de óxidos. Los funcionales
hı́bridos “tradicionales”, como el B3LYP o el B1, fueron desarrollados en el contexto
de la quı́mica cuántica y han demostrado proporcionar una muy buena descripción de
las energı́as de disociación, de las distancias de enlace y de las frecuencias de vibración,
ası́ de los espectros de energı́a para la mayorı́a de las moléculas. Sin embargo, se ha
encontrado que fallan al intentar reproducir las propiedades estructurales de algunos
sólidos y de, en particular, de los ferroeléctricos. Afortunadamente ha habido importantes avances en esta dirección. En concreto, el desarrollo del funcional B1-WC (una
195
mezcla de intercambio exacto con el funcional GGA de Wu y Cohen, que proporciona
una muy buena descripción de la estructura atómica de las perovskitas) abre la puerta
al estudio de muchos fenómenos de sistemas complejos de óxidos que estaban fuera del
alcance de la mayorı́a de las aproximaciones utilizadas en la actualidad. Aunque el uso
de funcionales hı́bridos es mucho más exigente computacionalmente que el de LDA o
GGA, dada la rápida evolución de la informática (tanto en el software como en el hardware) es de esperar que las diferencias se reduzcan en poco tiempo. Ya en la actualidad,
dadas las limitaciones de los enfoques DFT más comunes, que han sido discutidos en
esta memoria, una cuidada selección de los sistemas a simular puede proporcionar una
enorme cantidad de información.
También hemos investigado los dominios de polarización en láminas delgadas ferroeléctricas, tanto en condensadores como en superredes. En ambos tipos de sistemas,
hemos observado la formación de de dominios de cierre, una estructura que se creı́a poco
probable en los materiales ferroeléctricos debido a la energı́a elástica asociada. Tanto
los cálculos desde primeros principios como las simulaciones de Hamiltoniano modelo
han predicho la formación de los dominios de cierre en varios sistemas diferentes, desde
condensadores de BaTiO3 /SrRuO3 (como se muestra en el capı́tulo 5 y la Ref. [55])
hasta láminas aisladas de PbTiO3 [184]. Estudios teóricos han encontrado que la estructura de paredes de dominio es sorprendentemente robusta independientemente material
ferroeléctrico, el nivel de apantallado proporcionado por los electrodos (si existen), y
el enfoque teórico utilizado para el estudio. El reciente desarrollo de la microscopı́a
electrónica de transmisión con resolución subatómica ha permitido finalmente la observación experimental de tales estructuras [66], lo que confirma las predicciones teóricas
propuestas pocos años antes. Se trata un buen ejemplo del nivel de predicción que han
alcanzado las simulaciones desde primeros principios en los últimos tiempos. Ahora que
los detalles acerca de la estructura de las paredes de dominio en estos sistemas está siendo
caracterizada, es el momento de analizar otras propiedades asociadas. Por ejemplo, muy
recientemente, se ha observado la conductividad de las paredes de dominio en láminas
delgadas ferroeléctricas. Se observó por primera vez en láminas delgadas de BiFeO3 [10]
y se atribuyó a una reducción del gap en las paredes de dominio, sin embargo, el cambio
previsto en la estructura electrónica en las paredes estudiadas era demasiado pequeño
para explicar la conductividad observada. También se sugirió que las paredes de dominio con polarizaciones enfrentadas y los campos eléctricos asociados podrı́an explicar
la conductividad, pero recientemente se ha observado conductividad en PZT tetragonal
donde la formación espontánea de las paredes de dominio polares es más improbable. En
nuestra opinión, la explicación más razonable es la fijación de vacantes de oxı́geno en las
paredes de dominio, algo que ya ha sido estudiado desde primeros principios en PbTiO3
en volumen [182]. La migración de las vacantes a las paredes de dominio y la fijación de
estas últimas sugieren una concentración relativamente alta de defectos en estas zonas
del material. Esta hipótesis se ve reforzada por los altos campos eléctricos que hemos
encontrado las fronteras entre los dominios de cierre. Algunas de estas funcionalidades
podrı́an optimizarse con un conocimiento detallado de las propiedades de la paredes
de dominio más allá de su estructura, lo que podrı́a dar lugar al uso de las paredes
196
Resumen
de dominio como elementos funcionales en los dispositivos de láminas ultra-delgadas.
Naturalmente, la investigación desde primeros principios de tales propiedades, también
saldrı́a beneficida de la utilización de un mejor funcional que proporcionase una mejor
descripción de la estructura electrónica del sistema.
El número de posibles funcionalidades que pueden inducirse en las superredes aumenta cada dı́a. Existe un gran número de materiales diferentes con una estructura
similar, pero con propiedades completamente distintas, que pueden combinarse con un
control atómico sobre la calidad de la interfase, dando lugar a propiedades absolutamente inimaginables. Hemos visto cómo surgen nuevos acoplamientos y fases como
consecuencia de los efectos de las interfases en las superredes, propiedades que pueden
ser explotadas para el diseño a medida de nuevos materiales artificiales. En este contexto, no sólo nuevas combinaciones de materiales deben ser exploradas, sino también
nuevas geometrı́as que podrı́an reforzar una determinada propiedad en particular. Por
ejemplo la estabilidad de los vórtices de polarización en nanoestructuras ferroeléctricas
puede ser utilizada para diseñar nanopartı́culas con un momento toroidal o nanopilares
quirales con actividad óptica sintonizable.
La sı́ntesis de algunos de estos sistemas está todavı́a lejos de las posibilidades de las
técnicas de experimentales actuales y algunos dispositivos experimentales siguen siendo
demasiado complicados como para ser simulados por completo desde primeros principios. Por lo tanto, es fundamental la continua retroalimentación con las simulación con
Hamiltonianos modelo, que conectan las dimensiones y las escalas de energı́a entre las
simulaciones desde primeros principios y los experimentos. Afortunadamente, la colaboración entre los grupos teóricos y experimentales en este campo es muy estrecha, lo
que proporciona un entorno de trabajo enormemente motivador. En definitiva esto se
traduce en colaboraciones y proyectos más eficaces y fructı́feros, lo que constituye una
de las razones claves para el rápido desarrollo que este campo ha experimentado en los
últimos años.
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Estudio desde primeros principios de
mecanismos de apantallado del campo de
depolarización en condensadores nanométricos.
TESIS DOCTORAL
Pablo Aguado Puente

Estudio desde primeros principios de mecanismos de apantallado

Transcription

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