The Henryk Niewodniczanski Institute of Nuclear Physics Polish

Transcription

The Henryk Niewodniczański Institute
of Nuclear Physics Polish Academy of
Sciences
Properties Of Insulating Materials From Quantum
Calculations
by
Yuriy Natanzon
PhD Thesis
was done at the Department of Structural Research
under supervision of Dr. hab. Zbigniew Lodziana
Kraków, 2010
iii
Acknowledgements
This thesis is submitted in candidacy for PhD Degree in Physics at The Henryk
Niewodniczański Institute of Nuclear Physics Polish Academy of Sciences in Kraków.
The work presented in this thesis was done in the Department of Structural Research
under supervision of Dr. hab. Zbigniew Lodziana, from October, 2006 until May,
2010.
Here I would like to express a great gratitude to my PhD supervisor, Dr. hab. Zbigniew Lodziana for his enormous help, patience and support during all the four years
of the PhD programme. Thanks to Dr. Lodziana, I’ve gained a lot of knowledge in
the area of quantum calculations as well as access to the computation facilities which
allowed me to prepare this thesis.
I thank Prof. Tadeusz Lesiak (the head of International PhD studies) and Ms.
Danuta Filipiak (PhD Studies Secretary) for being helpful in the issues related to the
PhD Programme, and Ms. Dorota Erbel for the regular help regarding administrative
issues related to the residence permission for non-EU citizens.
Many thanks to the scientists, PhD students and post-docs of the Department for
Structural Research for creating a friendly atmosphere and support.
I thank Academic Computer Centre CYFRONET AGH (project No. MNiSW/
SGI3700/ IFJ/ 139/ 2006) and Interdisciplinary Centre for Modelling of Warsaw University (project No. G28–22) for providing computer resources. The financial support
from the Ministry of Science and Higher Education of Poland is highly acknowledged
(grant No. N N202 240437).
I also thank Prof. Petro Holod from the National University of Kyiv-Mohyla
Academy for inspiring me to become a physicist. I acknowledge Olena Kucheryava
from Nantes for providing me the nice photo of Minoan pottery (Fig. 2.1 in Chapter 2).
Many thanks to my Kraków friends, especially Olesya Klymenko, Ztanislaw Zajac,
Alexander Zaleski and Natalia Borodai for excellent atmosphere and support. I also
thank Hanna Dobrynska (Dobrzyńska) for the nice friendship during the years 2007–
2008.
List of Abbreviations
AMF
APW
BASE
CAMD
DFT
DoE
DOS
FLL
FP-LAPW
FSS
GGA
GGA+U
GW
HF
HOMO
HSRS
ICSD
LAPW+lo
LCAO
LDA
LSDA
LUMO
OFDFT
PBE
PBE0
PEGS
RMT
Around Mean Field
Augmented Plane-Wave method
β-aluminia Solid Electrolyte
Center for Atomic Scale Materials Design
Density Functional Theory
The United States Department of Energy
Density of States
Fully Localised Limit
Full Potential Linear Augmented Plane-Wave method
Fine Structure Superplasticity
Generalised Gradient Approximation
Generalised Gradient Approximation + Hubbard U
Green function (G) + screened Coulomb interaction (W )
Hartree–Fock
Highest Occupied Molecular Orbital
High Strain Rate Superplasticity
Inorganic Crystal Structure Database
Linear Augmented Plane-Wave plus local orbitals method
Linear Combination of Atomic Orbitals
Local Density Approximation
Local Spin Density Approximation
Lowest Unoccupied Molecular Orbital
Orbital Free Kinetic-Energy Density Functional Theory
Perdew–Burke–Ernzerhof parametrisation
PBE functional + exact exchange with α = 0.25 fraction
Prototype Electrostatic Ground State
Muffin-tin radius
v
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RPA
SIESTA
SOFC
YZP
Random Phase Approximation
Spanish Initiative for Electronic Simulations of Thousands of Atoms
Solid Oxide Fuel Cells
Yttria-stabilised zirconium dioxide
Contents
Acknowledgements
iii
List of Abbreviations
v
Introduction
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1 Density Functional Theory
1.1 Fundamentals of DFT . . . . . . . . . . . . .
1.1.1 Many-particle Shrödinger equation . .
1.1.2 Hohenberg-Kohn theorem . . . . . . .
1.1.3 Kohn-Sham equations . . . . . . . . .
1.1.4 Exchange-correlation functionals . . . .
1.2 Implementations of Density Functional Theory
1.2.1 The SIESTA method . . . . . . . . . .
1.2.2 (L)APW methods . . . . . . . . . . . .
1.3 Calculation of selected physical properties . .
1.3.1 Geometric optimisation . . . . . . . . .
1.3.2 Calculation of elastic properties . . . .
1.3.3 Bulk modulus . . . . . . . . . . . . . .
1.3.4 Elastic constants . . . . . . . . . . . .
1.3.5 Defects: vacancies and dopants . . . .
1.3.6 Calculation of atomic charges . . . . .
1.3.7 Band structure and density of states .
1.3.8 Optical properties . . . . . . . . . . . .
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2 Superplasticity in tetragonal zirconium dioxide
2.1 Ceramic materials . . . . . . . . . . . . . . . . .
2.2 Superplasticity . . . . . . . . . . . . . . . . . .
2.2.1 Superplasticity in ceramic materials . . .
2.2.2 Superplasticity mechanisms . . . . . . .
2.3 Zirconium dioxide . . . . . . . . . . . . . . . . .
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ceramics
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2.3.1 Superplastic properties of ZrO2 ceramics .
2.3.2 Calculation of elastic properties of ZrO2 .
2.3.3 Model of the superplasticity in doped YZP
2.3.4 Electronic effects in YZP superplasticity .
2.4 Summary . . . . . . . . . . . . . . . . . . . . . .
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3 Structural and thermodynamic properties of metal borohydrides
3.1 Hydrogen energy storage and metal hydrides . . . . . . . . . . . . . .
3.2 Crystal structure prediction methods . . . . . . . . . . . . . . . . . .
3.2.1 Simulated annealing . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 Genetic algorithm . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.3 Database searching methods . . . . . . . . . . . . . . . . . . .
3.2.4 Prototype electrostatic ground state approach (PEGS) . . . .
3.2.5 Coordination screening . . . . . . . . . . . . . . . . . . . . . .
3.3 Application of coordination screening to complex metal borohydrides
3.4 Details of calculations for NaX(BH4 )3,4 . . . . . . . . . . . . . . . . .
3.5 General trends in borohydride stability . . . . . . . . . . . . . . . . .
3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4 Electronic structure of alkali and alkaline earth metal hydrides
4.1 Selected properties of hydrides . . . . . . . . . . . . . . . . . . . . .
4.2 Recent studies of alkali and alkaline earth metal hydrides . . . . . .
4.3 Calculation method . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Alkali hydrides . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.1 Crystal structure and formation energy . . . . . . . . . . . .
4.4.2 Electronic structure . . . . . . . . . . . . . . . . . . . . . . .
4.4.3 Electron density and atomic charges . . . . . . . . . . . . .
4.4.4 Optical properties . . . . . . . . . . . . . . . . . . . . . . . .
4.5 Alkaline earth hydrides . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.1 Crystal structure and formation energy . . . . . . . . . . . .
4.5.2 Electronic structure . . . . . . . . . . . . . . . . . . . . . . .
4.5.3 Electron density and atomic charges . . . . . . . . . . . . .
4.5.4 Optical properties . . . . . . . . . . . . . . . . . . . . . . . .
4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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5 Summary and Outlook
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Bibliography
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Index
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Introduction
From Schrödinger equation to Computational Materials Science
The fundamental laws necessary for the mathematical treatment of a large
part of physics and the whole of chemistry are thus completely known, and
the difficulty lies only in the fact that application of these laws leads to
equations that are too complex to be solved.
Paul Dirac
The development of quantum mechanics in early 20-ies of the XX century has
opened the new horizons for studying the properties of matter from first principles.
Based on the assumption of de Broglie, in 1926 Erwin Schr odinger has introduced
his famous non-relativistic wave equation, which describes correctly the spectrum of
the hydrogen atom [1]. In the same year Heitler and London have solved this equation for the hydrogen molecule [2] and explained how the wave-functions of the two
hydrogen atoms are linked together to form a covalent bond. Their ideas were further
expanded by Linus Pauling who introduced the concept of orbital hydridization and
gave a detailed explanation of the nature of chemical bonding [3].
At the same time, the challenges of finding the solutions of Schrödinger equation
for many particles, forced the development of various methods which allow to apply
the quantum mechanics to the complex systems. In 1927 Thomas [4] and Fermi [5]
in 1927 presented a semiclassical model, were the electrons were considered as a free
electron gas distributed in the region around the nuclei. This model is based on the
Fermi distribution of electrons and the uncertainty principle and for the first time
uses the concept of electron density instead of wave-functions. Although, this model
did not correctly describe the chemical bonding (e.g. the energy of separated atoms is
larger than the energy of the molecule [6]), it has become a basis for more sofisticated
methods.
In the 30-ies Hartree, Fock and Slater have created the self-consistent field method,
3
4
where the wave-function of the system in this method is represented by the singleparticle orbitals. Antisymmetrical properties of the wave-function are handled via
Slater determinants [7]. The Hartree-Fock equations are solved iteratively, but the
electron correlations in these equations is not taken into account.
In 1964, another method of solving Schrödinger equation for many electron system
has been developed by Hohenberg and Kohn [8] and later by Kohn and Sham [9].
This method is called “Density Functional Theory”. Using the concept of density
from the Thomas-Fermi model, they have introduced the one-electron Schrödingerlike equations which describe the electron gas in the external potential such that the
electron density of this gas equals to the density of the real system. The advantage of
this mean-field method over the Hartree-Fock method is that the Kohn-Sham external
potential can describe electron interactions more accurately.
Hartree-Fock, Density Functional Theory and other methods for many-electron
systems require advanced computational resources. The development of computer
hardware, which is satisfying the Moor’s law from the late 80-ies till now, has made
these quantum methods available for practical use. During the last two decades, a new
branch of physics called “Computational Materials Science” has been established. It
uses the development of theoretical concepts and the power of computers to calculate
and predict properties of condensed matter and nanomaterials.
Thesis outline
In the present thesis the results of computer simulations of solid insulating materials
from first principles are presented. The calculations were performed within Density
Functional Theory approach implemented in various software packages (SIESTA [10],
Wien2k [11], Dacapo [12]). As the calculations of systems with hundreds of electrons
require large amount CPU time and auxiliary resources (memory, disk space, number
of cores), they were performed on the supercomputer facilities at Academic Computer
Centre CYFRONET AGH in Kraków (project No MNiSW/ SGI3700/ IFJ/ 139/
2006 ), Interdisciplinary Centre of Modelling of Warsaw University (ICM) (project
No G28–22), and Swiss Supercomputer Centre (CSCS).
The research subject presented in the present PhD Thesis covers a broad range of
physical phenomena that can be divided into three major topics:
First subject is related to studies the mechanical and electronic properties in
doped yttria stabilised zirconium dioxide ceramics. This work was done in close collaboration with Dr. Marek Boniecki from the Institute of Electronic Materials Technology (Warsaw) who has performed experimental measurements of grain boundary
diffusion in this compound.
Second subject is associated with theoretical predictions of the stable ground
5
state crystal structures of complex metal borohydrides and analysis of their thermodynamic properties. This work is a collaborative project under of the more than
100 scientists, the participants of the CAMD Summer School“Electronic Structure
Theory and Materials Design”, which took place In Centre of Atomic Scale Materials
Design (CAMD) of Danish Technical University in August 2008, Lyngby, Denmark.
Third subject is devoted to the calculation of the electronic structure of alkali
and alkaline- earth metal hydrides. Here, new approach for calculations of the band
structure and optical properties of these compounds is proposed, that allows significantly better, than standard DFT, description of these properties. This approach is
based on the Hubbard model applied for exchange-correlation functional for hydrogen
atoms, even though the metal hydrides are not strongly correlated materials.
The results of research are summarised in the following four main chapters of the
Thesis:
In Chapter 1 a brief overview of the methods used in calculations is presented.
Special attention is focused on the Density Functional Theory (DFT) method
and its practical implementation in software packages used for calculations is
briefly described. Application of DFT for calculations of such measurable quantities as elastic and optical properties, atomic charges, electronic structure is
also presented.
Chapter 2 is devoted to computational studies of superplasticity in zirconium
dioxide ceramics. Elastic constants for doped zirconia are calculated for several
dopant concentrations. A simple model based on calculated elastic properties
of ZrO2 is proposed. This model explains the enhancement of the superplastic properties in fine grained ZrO2 ceramics after addition of such dopants via
softening of the surface of zirconia grains that affects intergranular sliding. Relevant electronic properties of ZrO2 are calculated. The results of calculations
are supported by the experimental measurement of Dr. M. Boniecki.
Chapter 3 is related to hydrogen storage in complex hydrides and the crystal
structure prediction methods based on quantum mechanical calculations. The
coordination screening method is used for prediction of the stable ground state
crystal structures of complex borohydrides (M1 M2 (BH4 )2 with M1 , M2 =Li, Na,
K and M1 M2 (BH4 )n , where n=3,4, M1 =Li, Na, K and M2 =Li, Na, K, Mg,
Al, Ca, Sc-Zn, Y-Mo,Ru-Cd). Several new compounds are predicted to fulfil
requirements for practical hydrogen storage for mobile applications.
Chapter 4 contains the results of calculations of structural, electronic and optical
properties of alkali (MH, M=Li, Na, K, Rb, Cs) and alkaline earth metal hydrides (MH2 , M=Mg, Be, Ca, Sr, Ba). A new method based on Hubbard model,
which describes more correctly the optical properties in these compounds, is
6
proposed and compared with other theoretical calculations and experimental
measurements. Basic electronic and structural properties are calculated for all
alkali and alkaline earth metal hydrides.
Chapters 2–4 contain brief introduction to studied materials and their physical
phenomena.
Chapter 1
Density Functional Theory
Among all the existing quantum-mechanical methods the density functional theory
(DFT) has gained wide applications, ranging from molecular systems to complex solid
structures. The reason for this is extreme robustness and efficiency of this method.
The central idea of DFT was published in 1965 in two fundamental works by Hohenberg and Kohn [8], and Kohn and Sham [9]. However, in that times the application
of the method was limited due to the lack of sufficient computational facilities. The
method gained its second life in the late 80-es and became the most popular quantum mechanical method nowadays. ISI Web of Knowledge gives more than 60,000
publications, which use this method since 1996 [13]. Thus, it was not a surprise when
Walter Kohn got a Nobel Prize in Chemistry in 1999 “for his development of the
density-functional theory“ [14]. In this chapter we present the basic principles of this
method which was used for the calculations in this thesis.
1.1
1.1.1
Fundamentals of DFT
Many-particle Shrödinger equation
In a non-relativistic case a many-particle system (e.g. a solid) can be described by
the Schrödinger equation [15]:
NelX
+Nnucl
Nel
Nnucl
X
1X
Zj
1 X
1
2
2
−
∇i −
∇j −
+
+
2 i
2 i
|ri − Rj | i<j |ri − rj |
i,j
!
X Zj Zk
+
Ψ(ri , Rj ) = EΨ(ri , Rj )
|R
j − Rk |
j<k
(1.1)
Here, Nel and Nnucl are the number of electrons and nuclei in the system respectively,
ri and Rj are the positions of the i-th electron and j-th nucleus, and Ψ(ri , Rj ) is a
8
1. Density Functional Theory
many-particle wave-function, which depends on 3Nel + 3Nnucl variables. The computation time required to solve such an equation grows with the number of particles
drastically. One can reduce the number degrees of freedom to 3Nel by introducing Born-Oppenheimer approximation [16, 17]. Since electrons are ≈ 1836 times
lighter [18] than nuclei, according to this approximation one can ”freeze” the latter
and treat the atomic nuclei classically. Thus, kinetic energy operator for nuclei will
vanish, which leads to the following equation:
!
NelX
+Nnucl
Nel
X
X
1
1
Zj
−
∇2i −
+
Ψ(ri ) = EΨ(ri )
(1.2)
2 i
|ri − Rj | i<j |ri − rj |
i,j
Here the nuclei positions are now considered as constant parameters, thus the wavefunction depends only on the electron positions and has 3Nel degrees of freedom.
1.1.2
Hohenberg-Kohn theorem
One may ask a question, if it is possible to reduce the number of degrees of freedom to
just three and somehow to get the one-electron equation? That would be theoretically
possible, if one would be able to describe the Coulomb repulsion between the electrons
by the effective external potential in non-interacting electrons system. The idea was
originally proposed by Hohenberg an Kohn in Ref. [8], where they replace the many
electron wave-function by electron density which depends only on 3 degrees of freedom
and describes the charge density of all electrons in the system:
Z
ρ(r) = Nel Ψ∗ (r, r2 , r3 ...rNel )Ψ(r, r2 , r3 ...rNel )dr2 dr3 ...drNel
(1.3)
Having considered the interacting electron gas in an external potential V (r), Hohenberg and Kohn have proved
that there exists a universal potential F [ρ(r)] such
R
that the minimum of E = V (R)ρ(r)dr + F [ρ(r)] corresponds to the ground state
energy of this gas [8, 19]. This potential, as well as the external potential V (r) are
unique functionals of the electron density ρ(r), thus instead of solving Schrödinger
equation one can find the ground state energy from the electron density by calculating
the universal potential F [ρ] which is defined by the following equation [8, 20]:
F [ρ] = hΨ(r)|T (r) + Vee (r)|Ψ(r)i ,
(1.4)
where T and Vee are kinetic energy and electron-electron interaction energy operators,
respectively. The ground state energy is then calculated by solving Thomas-FermiHohenberg-Kohn equation [20]:
δE
δF [ρ] δVne [ρ]
=
+
= λ,
δρ
δρ
δρ
(1.5)
1.1. Fundamentals of DFT
9
where Vne is the electron-nuclei interaction energy functional and λ is the Lagrange
multiplier, which is used to keep the electron density normalised to the number of
electrons, Nel . The method of solving this equation solely in terms of electron density
without introducing the concept of electron orbitals is called orbital free kinetic-energy
density functional theory (OFDFT) [20].
1.1.3
Kohn-Sham equations
However, it turns out that solving the equation 1.5 in terms of the electron density is
quite complicated. Thus, instead of this Kohn and Sham have proposed a simpler way:
to solve the set of one-electron Schrödinger-like equations for the fictitious system of
non-interaction electrons which have exactly the same electron density as the real
system [9, 19]:
1
(1.6)
− ∇ + v[ρ](r) φi (r) = ǫi φi (r), i = 1..Nel
2
Here, φi (r) are Kohn-Sham orbitals, which have no physical meaning, but the sum of
the charge density related to these orbitals equals to the electron density of the real
system:
ρ(r) =
Nel
X
i
|φi (r)|2
(1.7)
Kohn-Sham equations are one-electron equations, which are easy to solve but
contain the unknown external potential V (r) that mimics the electron-electron interaction of the real system leading to the correct electron density [15]. It consists from
three main parts:
V [ρ](r) = −
N
X
i,j
Zj
+
|ri − Rj |
Z
ρ(r′ )
dr′ + Vxc (r)
|r − r′ |
(1.8)
The first part of the Eq.1.8 is an external Coulomb potential, which originates from
the interaction of the electrons of the real system with the nuclei, the second term is
called the Hartree potential, that describes Coulomb interaction between electrons,
and the remaining part is the exchange-correlation potential Vxc (r), which contains
the non-Coulomb part of the electron–electron interactions. This potential depends
on the electron density in a complicated way and this exact dependence is unknown.
However, several approximations for Vxc (r) exist and they lead to surprisingly good
results. In the next section we will briefly describe such approximations.
10
1.1.4
Exchange-correlation functionals
Existing approximations to the exchange-correlation potential can be arranged in
the hierarchy sometimes called Jacob’s ladder [21, 22]. It starts from the simplest
potentials, which account only for the charge density ρ(r) through those which include
the dependence on density gradients ∇ρ(r), kinetic energy density |∇ρ(r)2 |/8ρ(r) and
other terms, which finally lead to the exact Kohn-Sham functional [21].
The exchange and correlation part of Vxc in these approximations can be separated
into exchange Vx and correlation Vc , respectively [15]:
Vxc [ρ](r) = Vx (ρ(r)) + Vc (ρ(r))
(1.9)
The first rung in the Jacob’s ladder is local density approximation (LDA), where
the electron density is considered to be uniformly distributed in space (like in the homogeneous electron gas) and the exchange-correlation potential Vxc [ρ](r) in the point
r = r0 depends only on the density in the same point. Second rung of the Jacob’s
ladder is to consider the dependence of Vxc on both the density at r = r0 and density
gradient at this point, this is called generalised gradient approximation (GGA) [21].
The third rung adds the kinetic energy density |∇ρσ (r)2 |/8ρσ (r) of the selected orbitals σ to the exchange-correlation functional (meta-GGA functionals [22]).
One of the main problems of DFT is the presence of the electron self-interaction error in the external potential V (r) (see Eq.1.8). The second term of Eq. 1.8 VHartree =
R ρ(r′ ) ′
dr describes the Coulomb interaction of each electron in the system with the
|r−r ′ |
total electron density ρ(r). This means that this term also contains the interaction of
each electron with itself because each electron contributes to the total electron density. In density functional theory this effect is cancelled by the exchange-correlation
potential if the exact expression for this potential is known. However, this is not the
case in existing approximations of this functional and the self-interaction is cancelled
only partly [23].
So, the fourth rung of the Jacob’s ladder is to calculate the exchange part exactly
as it is done in the Hartree-Fock method in order to eliminate the self-interaction
error at least partially. The fifth rung should include the exact exchange and at least
the part of the correlation term calculated exactly. Finally, this should lead to the
calculation of the exact functional where both exchange and the correlation part are
known. However, at present the exact functional is only known for the small systems
(e.g., a Ne atom [24]) where Vxc was calculated by means of Quantum Monte-Carlo
method (see [24, 25].
11
Local Density Approximation (LDA)
This is the first and the simplest approximation to the exchange-correlation functional
which considers the electron density as uniformly distributed in the space that corresponds to the density of the uniform electron gas. This approximation is thus exact
only for a uniform electronic system, but in fact it gives surprisingly good results for
the real systems. The exchange part has the form [15]:
4
Vx (ρ) = Aρ 3 ,
(1.10)
where A = 34 ( π3 )1/3 .
The correlation part is determined from the following expression [15]:
Vc (ρ) = −
0.896 1.325
+ 3/2 ,
rs
rs
(1.11)
where rs is Wigner-Seitz radius which is the radius of a sphere around the electron
with a volume which equals to the volume per electron of the system Nel /V :
4 3
πr =
3 s
Nel
V
−1
(1.12)
The exact value of rs was calculated by using quantum Monte-Carlo method [26, 27].
The most popular parametrisation of the LDA functional are Ceperley-Adler [26]
and Perdew-Wang [27] parametrisation which are implemented in variety of software
packages [10, 11].
Local Spin Density Approximation (LSDA)
Although Kohn-Sham equations do not take into account the spin of the electron,
one can do it by including the spin into definition of electron density:
Z
ρσ (r) = Nel Ψ∗ (r, σ, r2 , r3 ...rNel )Ψ(r, σ, r2 , r3 ...rNel )dr2 dr3 ...drNel ,
(1.13)
where σ = ↑, ↓ is electron spin. The Kohn-Sham equations for the electron density of
the spin-up ρ↑ and spin-down ρ↓ electrons should be solved separately. The exchangecorrelation potential in LDA approximation for this case looks more complicated, than
for spinless case. The exchange part has the following form [15]:
(1 + ζ(r))4/3 + (1 − ζ(r))4/3
,
Vx = Aρ
2
4
3
(1.14)
12
where A is defined as in Eq. 1.10, and ζ is the spin polarisation of the system:
ζ=
ρ↑ (r) − ρ↓ (r)
ρ(r)
(1.15)
The case of ζ = 0 corresponds to the system which is not spin polarised, where
actually there is no need to use LSDA approximation. The correlation part of Vxc is
also calculated by means of quantum Monte-Carlo method [15].
Generalised Gradient Approximation (GGA)
The Generalised Gradient Approximation (GGA) is the second rung in “Jacob’s ladder” of approximations where the dependence of exchange-correlation potential on
both the electron density and its gradient are considered. The exchange part of this
potential is enhanced by a factor Fx (s) [28]:
vx (ρ) = Fx (s)vx(LDA)
(1.16)
Where Fx (s) is an enhancement factor and s is dimensionless density gradient defined
by the expression:
1
4 3 |∇ρ|
s=
(1.17)
6πrs ρ
As in LDA, several parametrisation of the functional exist which define the form
of Fx (s). The first one was proposed by Perdew et alter [29] and has the form:
2
1 + 0.19645s sinh−1 (7.7956s) + (0.2743 − 0.1508e−100s )
Fx (s) =
1 + 0.19645s sinh−1 (7.7956s) + 0.004s4
(1.18)
Another well-known parametrisation was proposed by Perdew, Burke and Ernzerhof [28]:
κ
Fx (s) = 1 + κ −
,
(1.19)
1 + µs2 κ
where µ = 0.21951, κ = 0.804.
The correlation potential in this approximation has the form [28]:
1
3
,
Vc = −γf ln 1 + 2 2
χs /f + (χs2 /f 2 )2
(1.20)
where χ = 0.72161, f = (1 + ζ)2/3 + (1 − ζ)2/3 is a spin scaling factor.
In order to illustrate the difference between the LDA and GGA approximation
we have calculated the exchange-correlation potential and the corresponding electron
density of the hydrogen atom in magnesium hydride MgH2 . The calculations were
13
6
LDA CA
GGA PBE
3
Electron density (e/Å )
0
Vxc (Ry)
-5
-10
-15
LDA CA
GGA PBE
-0.5
0
r (Å)
(a)
0.5
5
4
3
2
1
0
-0.8 -0.6 -0.4 -0.2
0 0.2 0.4 0.6 0.8
r (Å)
(b)
Figure 1.1: Exchange-correlation potential (a) and the corresponding electron density (b)
in LDA (black line) and GGA (red dashed line) approximations as the functions of distance
from the centre of the hydrogen atom (r = 0) in MgH2 .
performed using Wien2k software package [11] which will be described in Section 1.2.2.
The results are shown on Fig.1.1.
The black line on Fig.1.1 corresponds to LDA exchange-correlation potential and
electron density in Ceperley-Adler parametrisation, and red line is for GGA exchangecorrelation potential in Perdew-Burke-Ernzerhof parametrisation. One can observe
that due to the presence of enhancement factor, GGA potential is more localised and
leads to the more localised density of the s orbital of the hydrogen atom.
The band gap problem and a need for orbital dependent functionals
One of the major challenges of DFT is the fact that it is ground state theory and
in general the Kohn-Sham eigenvalues cannot be assigned to the excitation energies
of the system. The Koopmans’ theorem [30] states that the first ionization energy
of the system equals to the energy of the highest occupied molecular orbital. This
theorem is valid in Hartree-Fock method and allows to assign the HF eigenvalues the
corresponding excitation energies [30, 23]. In DFT theory only the highest occupied
eigenvalue can be assigned the physical interpretation: it is exactly the chemical
potential of the system [23].
However, the interpretation of Kohn-Sham eigenvalues as the excitation energies
(like in Hartree-Fock method) leads to the surprisingly good results. But one of the
most important issue which comes from this interpretation is underestimation of the
band gap in DFT. One of the reason of this the self-interaction error, which was
14
discussed in the previous sections.
The solution to this problem proceeds to the usage of the orbital dependent semiempirical functionals for the selected orbitals. Such functionals partly cancel this
error and allow improvement of the calculations of the band gaps [23].
There exist variety of implementations of such methods. Below we discuss three
of them, which are commonly used.
The self-interaction correction: One of the first methods of correcting selfinteraction error was proposed by Perdew and Zunger [31]. These authors have
introduced the following modification to the DFT total energy EGGA [ρ(r)]:
X 1 Z ρnlσ (r)ρnlσ (r′ )
′
dr + EGGA [ρnlσ (r)] ,
(1.21)
E = EGGA [ρ(r)] −
2
|r − r′ |
nlσ
where ρnlσ (r) is an electron density of the selected nl orbital in the spin state
σ. The second term of Eq. 1.21 describes the Coulomb self-interaction of the
electrons of this orbital and the last term EGGA [ρnlσ (r)] is the GGA exchangecorrelation energy for calculated for the density ρnlσ (r).
However, the efficiency and accuracy of this method highly depends on the
selection of the orbitals and requires an additional minimisation of the total
energy with respect to the orbitals.
Exact exchange and hybrid functionals: Another possibility to cancel selfinteraction error is to add an exact exchange term to the exchange-correlation
functional. Thus, the total energy will have the following form [32, 33]:
E = EGGA [ρ(r)] + α(ExHF [Ψsel ] − ExGGA [ρsel ])
(1.22)
where ρsel and Ψsel are the density and the wave-function of the selected orbitals,
ExHF [Ψsel ] and ExGGA [ρsel ] are the Hartree-Fock and GGA exchange energy for
the selected orbitals. The second term of Eq.1.22 is the term, which eliminates
the error of double counting of the exchange term in GGA and HF. The α
parameter defines the fraction of exact exchange that is taken into account in
exchange-correlation potential. The value of α can be from 0 to 1. The case
of α = 1 corresponds to exact exchange functional, and the other cases are
called hybrid functionals. A special case of α = 0.25 and GGA-PBE exchange
correlation functional is commonly denoted as PBE0.
The GGA+U exchange-correlation method relies on the combination of the
LDA or GGA functional with the Hubbard Hamiltonian which describes the
repulsion of the electrons in the nearest neighbour approximation. The total
ground state energy is corrected in the following way [34, 35, 36]:
EGGA+U [ρ(r), {ρσ }] = EGGA [ρ(r)] + ∆EGGA+U (ρσ , U, J),
(1.23)
15
where ρσ is an occupation matrix of the selected orbital with the occupancies
nmσ (m is a momentum quantum number and σ is a spin), U and J are averaged
Coulomb and exchange parameters. The first term of the Eq.1.23 is the GGAtotal energy and the second one is the Hubbard correction and is determined
from the following expression [34, 35]:
J X
U X
nmσ nm′ σ
(1.24)
nmσ nm′ σ′ −
∆EGGA+U =
2 mσ6=m′ σ′
2 m6=m′ ,σ
One can rewrite this equation in terms of the orbital occupation matrix ρσ as
follows [34]:
U X
U − J X
∆EGGA+U = −
Tr(ρσ · ρσ ) − (Tr(ρσ ))2 +
Trρσ Trρσ′ , (1.25)
2
2 σ6=σ′
σ
P
where Trρ
=
σ
m nmσ . The total number of electrons on the selected orbital
P
is Nel = σ Trρσ [34].
The Eq. 1.24–1.25 contain double counting of electrons, and the additional
correction should be add. Depending on the implementation of the double
counting corrections, several modifications of GGA+U exist. One of them is
called around mean field (AMF) [34, 37]. In this method the average electron
occupancy nav , σ is introduced:
P
nmσ
Trρσ
Nel,σ
=
= m
(1.26)
nav,σ =
2l + 1
2l + 1
2l + 1
This formula describes a uniform orbital occupancy limit where the selected orbitals have the same occupancy and the fluctuations from this limit is described
by making the following replacement in Eqs. 1.24–1.25 [34, 37]:
nmσ → nmσ − nav,σ , ρσ → ρσ − δmm′ nav,σ
Then Eq. 1.24 will have form:
U X
AM F
(nmσ − nav,σ )(nm′ σ′ − nav,σ′ ) −
=
∆EGGA+U
2 mσ6=m′ σ′
J X
−
(nmσ − nav,σ )(nm′ σ − nav,σ )
2 m6=m′ ,σ
(1.27)
(1.28)
And the corresponding equation for in terms of the occupation matrix ρσ
reads [34]:
U −J X
AM F
∆EGGA+U
=−
(1.29)
Tr [(ρσ − δmm′ nav,σ )(ρσ − δmm′ nav,σ )]
2
σ
16
However, this approximation is not valid for strongly localised electrons [34],
and another approximation can be used in this case. In the fully localised limit
one implies that occupancies of either nmσ = 0 or nmσ = 1 are possible. For
this case the double counting correction can be expressend in terms of the total
number of electrons [36], and the Eq. 1.24 will have form:
J X
U X
F LL
nmσ nm′ σ −
nmσ nm′ σ′ −
∆EGGA+U
=
2 mσ6=m′ σ′
2 m6=m′ ,σ
!
JX
U
Nel (Nel − 1) −
Nel,σ (Nel,σ − 1) ,
(1.30)
−
2
2 σ
P
where Nel,σ = Trρσ and Nel = σ Nel,σ . In this approximation Eq. 1.25 will
have form [34, 38]:
U −J X
F LL
[T r(ρσ · ρσ ) − T r(ρσ )]
(1.31)
=−
∆EGGA+U
2
σ
In order to use this approximation one may need to know the values of U and
J parameters (or their difference Uef f = U − J), which are related to energies
required to move the electron between neighbouring orbitals. These parameters
can be determined by series of calculations with one electron added and removed
to the system [37]:
n 1 n
n 1 n
Uef f = ǫnl↑
+ ,
− ǫnl↑
+ , −1 −
2 2 2
2 2 2
n 1 n 1
n 1 n 1
+ , −
− ǫnl↓
+ , −
,
(1.32)
− ǫnl↓
2 2 2 2
2 2 2 2
where ǫnlΣ (nl ↑, nl ↓) is the Kohn-Sham eigenvalue of nl orbital with the occupancy of (nl ↑, nl ↓) and n = nl ↑ +nl ↓ is a total occupancy of this orbital
before adding or removing electrons.
Approximations described above not only improve the total energies for the ground
state, but they also affect the energy levels related to orbitals of interest. In this way
they improve calculations of the electronic structure of the system.
1.2
Implementations of Density Functional Theory
Density functional theory is implemented in the variety of software packages including
but not limited to VASP [39], SIESTA [10], Wien2k [11], Dacapo [12], Gaussian [40]
and many others.
1.2. Implementations of Density Functional Theory
17
Depending on the implementation of the basis set which is used in the expansion
of Kohn-Sham orbitals, the the programs can be divided into three main groups:
those which use plane-wave basis [39, 12];
those which use the linear combination of localised atomic orbitals (LCAO) [10,
40];
and those with mixed plane-wave and atomic basis set [11].
The basis set also influences the performance, application and the accuracy of
the method. The plane-wave basis provides good accuracy, but the scaling with the
system size is usually poor and is about Nel3 . In contrast, selected implementations
of the localised basis set method can provide an order Nel scaling of the computation
time, but they are less accurate and can be used only for selected systems. The
selection between the DFT implementation is based on the analysis of the desired
accuracy and the computation time.
The existing implementations can be divided into two main groups: all-electron
implementation (Wien2k [11], Gaussian [40]) and pseudopotential implementations
(SIESTA [10], Dacapo [12], VASP [39]). The last ones treat only valence electrons and
replaces the core electrons by an effective pseudopotential. This allows to speed up
the calculations but leads to less accurate solutions in comparison to the all electron
methods. In this section we describe two main methods and software which were
applied for the calculations on this thesis, the SIESTA method and LAPW+lo method
implemented in Wien2k program.
1.2.1
The SIESTA method
The SIESTA (abbreviated from Spanish Initiative of Electronic Simulations for Thousands of Atoms) is a Fortran-based program that uses the specific implementation of
DFT which is designed to both efficiency of the calculations and the balance between the accuracy and computation time [10]. In this method only the valence
electrons, which take part in bond formation, are considered and the other electrons
(so called core electrons) are replaced by an effective norm-conserving pseudopotentials of Kleinmann-Bylander form with Troullier and Martins parametrisation [10].
All the integrals in SIESTA are calculated on the finite real space grid and the
wave functions are expanded into strictly localised atomic orbitals, which consist of
the product of numerical radial function and spherical harmonics [10].
18
Basis set and k point sampling
The basis functions are calculated as a linear combination of atomic orbitals:
X
ψi (k, r) =
eikRj,lmn φj,lmn (r)cj,lmn;i (k),
(1.33)
j,lmn
where k is a point in the first Brillouin zone and cj,lmn;i (k) is an expansion coefficient.
The electron density is obtained by direct integration of the basis functions over the
whole Brillouin zone:
XZ
ni (k)|ψi (k, r)|2 dk,
(1.34)
ρ(r) =
i
BZ
where ni (k) is the occupation number of the basis orbital. In fact, the integration is
done on a finite k-point grid. If the unit cell is large enough, a Γ-point calculation
is sufficient to obtain the reasonable accuracy, while for the electronic structure calculations (i.e. density of states and the band structure) a dense mesh of k points is
required.
The basis set expansion of the atomic orbital of the atom Rj has the form [10]:
φj,lmn (r) = φj,ln (r − Rj )Ylm (r − Rj ),
(1.35)
where Ylm is the spherical harmonics and φj,ln (r − Rj ) is the radial function which is
called multiple-ζ basis. The number of ζs in this basis determine its size and accuracy.
The first and the most simple single-ζ basis φ1ζ
l (r) is a linear combination of Gaussians
which is fit to the eigenvalues of the equation for the atom Rj in the spherical box.
The more accurate double-ζ basis is then defined by the expression [10]:
l
2
s
φ2ζ
l (r) = r (al − bl r ), if r < rl
s
φ1ζ
l (r), if r ≥ rl
(1.36)
Here rls is a split radius which is defined manually, and the coefficients al and bl can
be calculated from the continuity condition at r = rls .
Pseudopotentials
The main reason for the use of pseudopotentials is to speed up the calculations by
replacing some of the electrons in the core by effective potentials, which give the core
electron density that corresponds to the real one as close as possible. Note, that the
choice of valence and core electrons is not strict and depends on the type of the atom
and on the system. The selection is based on the simple tests based on the calculation
of the lattice constants, elastic and electronic properties of simple systems of atoms
19
(for example, if we want to test pseudopotential of Mg planned to be used in MgO,
we can do the calculation for Mg crystal).
In many cases including all electrons in the core except those which take part
in the bond formation, is an optimal solution. However, for atoms with d and f
valence electrons additional s and p electrons, which are located just below the valence
orbitals have to be treated as the valence. One can also treat the other core electrons
as the valence or even refuse from using the pseudopotentials and make an all-electron
calculation.
The pseudopotential is calculated by solving Kohn-Sham equations for the core
electrons [10]:
l(l + 1)
1 d2
r+
+ Vxc (r) + VHartree (r) + Vl (r) ψln (r) = ǫln ψln (r)
(1.37)
−
2r dr2
2r2
Here VHartree and Vxc are Hartree and exchange-correlation potentials, respectively,
Vl is a part of pseudopotential called “semilocal potential”, l is angular momentum.
The pseudopotential and the eigenfunctions ψln are calculated by the following expressions [10]:
φln (r) = ψln (r) −
1.2.2
n−1
X
n′ =1
Vlnpseudo (r) = hφln |Vl (r)|φln i ,
(1.38)
hφln′ |Vl (r) − Vlocal (r)|ψln i
hφln′ |Vl (r) − Vlocal (r)|φln i
(1.39)
φln′ (r)
(L)APW methods
The Linearised Augmented Plane-Wave Method (LAPW) is an all-electron method
implemented among others in Wien2k software package [11]. It is the improvement of
the original Augmented Plane-Wave Method (APW), which was used in this package
before. The idea of this method lies in the fact that in the region far away from
nucleus the electrons behave like free electrons and can therefore be described by
plane waves. In contrast, the electrons close to the nucleus behave like they were in
the free atom and can be described by spherical functions which are the solutions of
Schrödinger equation for a single atom. So, following this idea, the unit cell of the
system is partitioned into atom-centred spheres of selected radii and the interstitial
region (see. Fig.1.2). The atomic spheres are called muffin-tin spheres and their radii
are called muffin-tin radii (RMT).
In these two regions two different basis sets are used for solving Kohn-Sham equations.
In the interstitial region the plane-wave expansion of Kohn-Sham orbitals is used [41]:
1
φkG (r, E) = √ ei(k+G)r
V
(1.40)
20
Figure 1.2: Partitioning of the MgH2 unit cell into muffin-tin spheres and the interstitial
region. The spheres of different atoms have different muffin-tin radii (RMT). Larger green
spheres correspond to the muffin-tin regions of Mg atoms and smaller white spheres are
muffin-tin regions of the hydrogen atoms. The empty space outside the spheres represent
interstitial region described by the plane waves.
Here k is the wave vector in the first Brillouin zone, G is a reciprocal lattice vector,
V is a unit cell volume. Inside the atomic sphere an expansion of spherical harmonics
times radial functions is used:
X j j
φkG (r, E) =
Alm ul (r′ , E)Ylm (r)
(1.41)
lm
Here Ylm (r) are spherical harmonics, Ajlm are expansion coefficients (unknown at the
beginning), E is the parameter with the dimension of energy, ujl (r′ , E) is the solution
of the Schrödinger equation for the atom j at the energy E.
Of course, these two parts of the basis set should match together at rj = RjM T ,
where RjM T is j-th atom muffin-tin radius (RMT). Expanding the plane-wave part
(Eq. 1.40) into the basis of spherical harmonics and comparing it to the atomic part
of the basis set (Eq. 1.41) at rj = RjM T , gives the expression for Ajlm [41]:
4πil ei(k+G)ṙj
jl (|k + G|RjM T )Ylm (k + G),
Ajlm = √ j M T
V ul (Rj , E)
(1.42)
here jl is a Bessel function of order l.
LAPW method. In order to solve the Kohn-Sham equations with such a basis one
needs to “guess” the value of the parameter E, then solve the equations and determine
E from them again. That’s time consuming and that is the reason why APW method
is replaced by more efficient one. Unlike this method, the LAPW method sets this
parameter fixed to some value E = E0 (it is called linearisation energy), and the
21
Schrödinger equation is solved to determine ujl (r′ , E0 ). Then one can make a Taylor
expansion of ujl (r′ , E) near the point E = E0 :
ujl (r′ , E) = ujl (r′ , E0 ) + (E0 − ǫnk )u̇jl (r′ , E0 ) + O(E0 − ǫnk )2
(1.43)
where E = ǫnk is the searched eigenstate of the Kohn-Sham orbital. As this state is
unknown (we are actually solving the equations to determine it), one have to introduce
j
it as a new parameter Blm
= Ajlm (E0 − ǫnk ):
X j j
j
φkG (r, E) =
(1.44)
u̇jl (r′ , E0 ) Ylm (r′ )
Alm ul (r′ , E0 ) + Blm
lm
By matching the basis at the boundary, the system of equations is obtained where
j
and Ajlm can be determined. As we now have a fixed value E0 , the basis can
both Blm
be determined uniquely from the linearisation energy E0 . However, the eigenstate
of different atoms have different dominant orbitals, which requires choosing different
linearisation energies for different atoms. For example, for atoms with d states the
E0 can be chosen near the centre of d orbital. Thus, the definition of basis requires
j
:
to fix different linearisation energies for different atoms and orbitals E1,l
X j
j
j
j
Alm ual (r′ , E1,l
) + Blm
u̇jl (r′ , E1,l
) Ylm (r′ )
(1.45)
φkG (r, E) =
lm
LAPW+lo method. In addition to all of this, one can treat some electrons close
to the nuclei (e.g. 1s electrons) as “core states” and use a different basis for them,
which is called local orbitals (LO) basis. Local orbital is defined in a way, that is it
zero everywhere except inside the sphere of the particular atom a (so, it is zero in the
interstitial and in the spheres of the other atoms). The nonzero part has the form:
X j,LO j
j
j,LO j ′
j
j,LO j ′
j
′
))
Ylm (r′ )
(r
,
E
u
+
C
(r
,
E
u̇
)
+
B
(r
,
E
u
(r)
=
A
φlm
j,LO
2,l
l
lm
1,l
l
lm
1,l
l
lm
lm
(1.46)
The reason why this basis is useful is that is does not need to be matched with plane
wave basis and thus does not depend on k. All three parameters of the basis can be
calculated by assuming that it should be zero at the sphere boundary r = Ra .
Regardless of the basis used, the resulting wave-functions are expanded in terms
of the basis functions:
X
ψk (r) =
Ck φkG
(1.47)
k
The accuracy of the basis is defined by the parameter k = Kmax . In Wien2k program,
this parameter is not set directly, but is obtained from the expression:
α
Kmax
=
RKmax
,
Rα
(1.48)
22
where RKmax is a dimensionless parameter, which is defined by the user and depends
on the RMT radii Rα of atoms. The smaller the radii are, the smaller RKmax is
needed for the accurate calculations. If Ra = 0, then there is only interstitial region
left and the basis set will be fully plane wave.
As in the SIESTA method, in Wien2k program the Kohn-Sham equations are
solved here by the self-consistent iterative minimisation of the Hamiltonian matrix
hψk (r)|H|ψk (r)i.
1.3
Calculation of selected physical properties
Two main quantities which are computed from Kohn-Sham equations are the ground
state energy (total energy) of the system and the electron density. Most properties
of the materials can be calculated from these quantities. In the following section
we describe selected methods for the calculation of the electronic and mechanical
properties.
1.3.1
Geometric optimisation
Once the Kohn-Sham equations for the electrons are solved one should find the ground
GS
state configuration of the system Etot
by minimising Kohn-Sham total energy Etot
with respect to atomic positions Rj :
GS
= min{Etot ; Rj }
Etot
(1.49)
tot
in the
This is done by minimisation of the forces excerpted on atoms Fj = ∂E
∂Rj
iterative way. Each time the total energy is calculated, the forces are computed and
the atoms are displaced to reduce them. Several geometric optimisation methods
exist to determine such directions efficiently, such as the method of steepest descent,
conjugate gradient or variety of methods based on Newtonian dynamics [42].
The geometric optimisation not only includes the optimisation of atomic positions inside the unit cell, but the optimisation of the unit cell volume, the lattice
constants and angles (depending on the symmetry). So, in most cases the geometric
optimisation is performed in the following steps:
1. Starting from the experimentally determined structure. Keeping the unit cell
fixed, the optimisation of the atomic positions inside the cell is performed.
2. For the cubic systems, the atomic positions are kept fixed and several calculations with varying the unit cell volume by ±x %, where x usually equals to
1–5 %, are performed. After finding the energy minimum, go to step 1 util the
required accuracy is achieved.
1.3. Calculation of selected physical properties
23
3. For non-cubic systems there are additional degrees of freedom, which have to
be varied and optimised in the same way (e.g. c/a ratio for tetragonal systems,
angles for monoclinic lattices etc).
In this work we have considered systems with various symmetries where the atomic
positions, unit cell volume, lattice constant ratio, and lattice angles were optimised
in such a way.
1.3.2
Calculation of elastic properties
The basic elastic properties describe the homogeneous (bulk modulus) and inhomogeneous (tensor of the elastic constants) deformations. These quantities can be derived
from the total energy of the system as will be described below.
1.3.3
Bulk modulus
The bulk modulus describes the resistance of the solid to the uniform volume deformation (i.e. hydrostatic pressure, see Fig.1.3) and it is described by the following
equation:
∂P
(1.50)
B = −V
∂V
Here P is a hydrostatic pressure and V is a unit cell volume. It is possible to calculate
bulk modulus by performing series of calculations for the volume of the unit cell
ranging from −m% to m%, where usually m < 10. Such calculations provide the
dependence of the total energy on the unit cell volume Etot (V ), which can be fit with
Murnaghan equation of state [43, 44]:
′
B0 V (V0 /V )B0
B0 V0
Etot (V ) = Etot,0 +
+1 − ′
(1.51)
′
′
B0
B0 − 1
B0 − 1
Here Etot,0 and V0 are equilibrium total energy and volume, respectively, and B0 is
the bulk modulus at the equilibrium (P = 0).
An example of dependence of the total energy on the unit cell volume calculated
for ZrO2 is shown in Fig. 1.4. In the linear regime of elastic deformation the calculated
data give an excellent fit to the Murnaghan equation (see Fig. 1.4).
1.3.4
Elastic constants
In the linear regime elastic constants describe the relation between the stress tensor
of the system and the applied strain (generalised Hooke’s law):
σi =
6
X
k=1
Cik ǫk ,
(1.52)
24
Figure 1.3: An example of the homogeneous volume expansion of the cubic crystal from the
initial volume V0 (green dashed line) to the final volume V . The bulk modulus describes
the resistance of the crystal to such deformation.
Total energy (eV)
-8630.4
-8630.6
-8630.8
-8631
-8631.2
125
130 135 140 145
3
Unit cell volume (Å )
150
Figure 1.4: The dependence of the total energy on the unit cell volume for ZrO2 (red
squares). The solid line is for the fit by Murnaghan equation of state (Eq. 1.51). The value
of bulk modulus extracted from this graph is B0 =205.085 GPa
where i, k = 1..6 (here, the Voight notation is used for all tensors).
Thus, elastic constants can be calculated by series of finite deformations of the
supercell as described in the work of Nielsen and Martin [45] and Hongzhi Yao et
alter [46]. For each constant the fixed strain of about 1 % is applied (as shown on
Fig.1.5) and the components of the stress tensor are calculated as the derivatives
of Kohn-Sham total energy with respect to the strain tensor. Elastic constants are
obtained by dividing the components of stress tensor by the components of strain
tensor:
σn
,
(1.53)
Cnn =
ǫn
where n = 1..6 and ǫi = x (x = ±0.01 for ±1% deformation) for i = n and all the
other components are zero. The case of n = 1..3 correspond to the uniform expansion
25
and contraction in the directions of x, y and z axes, respectively (see Fig. 1.3), while
n = 4..6 describe shear deformations as shown on Fig. 1.5.
For each deformation a full relaxation of the atomic positions in the unit cell is
performed. To minimise the numerical errors, each elastic constant is obtained as the
average of two calculations with the values of stress of +x % and −x %.
Figure 1.5: The cross section of the unit cell along (a, b) crystallographic plane before (black
solid lines) and after (green dashed lines) the shear deformation α, where tan α = ∆a
b . The
a crystallographic axis is oriented along x Cartesian direction and b along y. The resistance
of the crystal to such deformation is described by an elastic constant C66
In addition, the accuracy of calculations can be improved by providing zero-stress
correction. Due to the accuracy and time limitation, it is not possible to relax the unit
cell ideally leading to zero forces on all the atoms, thus in reality the cell is considered
to be optimised when the forces on atoms reach a treshold value (about 0.04 Å in
this work). Thus, some residual stress tensor σ 0 exist and influences the accuracy of
elastic constants calculations. If, it is small enough, then the elastic constant Cnn can
be calculated as follows:
Cnn± =
σn± − σn0
Cnn+ + Cnn−
; Cnn =
±
ǫn
2
(1.54)
Here “+” index corresponds to elongation and “−” to contraction.
1.3.5
Defects: vacancies and dopants
The DFT implementations, which were described in this chapter, use periodic boundary conditions for solving Kohn-Sham equations. Thus, if we want to introduce any
kind of defect to our material, a special supercell technique is needed. This includes
but not limited to removing atom from the cell (vacancy), replacing one of the atoms
or substitution the atoms in non-symmetric positions (impurity).
26
Figure 1.6: 2 × 2 × 2 supercell of MgH2 which can be used for the calculations of vacancies
or impurities. Rose spheres represent magnesium ions and white ones are for hydrogens.
The c crystallographic direction is vertical.
If, one wants to introduce a defect in the unit cell, then it will be repeated in
all directions because of periodic boundary conditions. The distance between neighbouring defects will be equal to the shortest lattice constant. This results in the large
defect concentration, which is not usually the case needed. If, one wants to study the
influence of one single defect, one can imitate it by creating a large supercell, which
are made by simple multiplication of the unit cell. For example, one can repeat the
unit cell twice in x, y and z directions, and create a 2 × 2 × 2 supercell, which consists
of 8 original unit cells (see Fig. 1.6). If, one adds a defect to this new unit cell, the
distance between the defects will be equal to 2 times the shortest lattice constant,
which reduces the defect concentration. If, we want to reduce the defect concentration
further, we can repeat the original unit cell more than two times and create 3 × 3 × 3
supercells, 4 × 4 × 4 etc. Mind, that defect calculations are usually time consuming,
because the unit cell contains many more atoms than in the 1 × 1 × 1 unit cell (e.g.
8 times more atoms in 2 × 2 × 2 cell). In the Chapter 2 we show the results of such
calculations where the defect concentration is varied in a way described above.
1.3.6
Calculation of atomic charges
Net atomic charges are important quantities, which describe the real ionicity of ions.
There are different methods of calculation of atomic charges. They can be classified
into the following groups [47]:
1. The methods, which are based on the calculation of atomic charges from the
electron wave-functions. One of the first and the most known method is Mulliken population analysis [48]. It is very sensitive to the choice of the basis set
27
in the used implementation of the DFT because atomic charges are calculated
directly from the expansion of the basis functions:
X
Qj = Zj −
Dµ,ν Sµ,ν ,
(1.55)
µ,ν
where Sµ,ν is the overlap matrix of the basis functions and Dµ,ν is the density
matrix, which is a function of the coefficients of the basis functions Cµi :
X
∗
Dµ,ν = 2
Cµi Cνi
(1.56)
i
2. The methods which are based on the partitioning of the electron density. These
include Voronoi deformation density method, Hirschfield charges and Bader
charges [49]. As the electron density maxima are located on the centra of atoms
and the density minima are between atoms, the Bader charge decomposition is
actually a method of searching for the atoms and partitioning the unit cell into
regions where the density reaches its maximum. The atomic charges, as in the
first method, are obtained by direct integration of the electron density inside
the Bader region [49].
3. The methods, which derive the charge directly from the density or densityrelated properties. The simplest one integrates the electron density over the
atomic sphere of selected radius:
Z r0
Qj =
ρ(r)dr
(1.57)
0
The choice of the integration radius r0 is arbitrary and is usually selected as
approximately the half of the bond length. The practical issues of selection of
this radius will be described in the next chapter. As the electron density is
normalised to the number of electrons Nel in the system, it is obvious that the
sum of all atomic charges should also be equal to the number of electrons:
X
Qj = Nel
(1.58)
j
Also, it is possible to derive the atomic charges from different experimental data
and compare with the results of DFT calculations.
1.3.7
Band structure and density of states
In quantum mechanics an electron is allowed to occupy only selected discrete energy
levels, which are the solutions of the Schrödinger equation (Eq. 1.1). For example,
28
an electron in the hydrogen atom can occupy the following energy levels [17]:
En = −
R
,
2n2
(1.59)
where R is a Rydberg constant and n = 1, 2, 3.. is the principal quantum number.
In contrast, in a solid with the lattice vector R~L = {a, b, c} the allowed electron
energies depend on the periodical boundary conditions for the many-electron wavefunction:
Ψ(~r) = Ψ(~r + R~L )
(1.60)
To satisfy this condition, the eigenvectors Ψm are described by the Bloch functions:
X
~ ~
(1.61)
Ψm (k, ~r) =
um (RJ , ~r)eikRJ ,
J
where ~k is the wave vector. The energies Em (k), which correspond to these eigenvectors are functions of the wave vector k and are called bands.
Due to the periodicity of the lattice the band
structure is calculated for the first Brillouin zone
only. Also, the symmetry of the lattice allows
to calculate all the possible non-equivalent bands
by selecting a one-dimensional path in this zone,
which includes all the important high symmetry points. These points are marked by capital Greek and Latin letters (e.g. Γ = (0, 0, 0),
X = (0, 2π/a, 0), W = (π/a, 2π/a, 0) etc for cubic lattice) and are shown on the band structure
plots. For example, for cubic face-centred lattice
the first Brillouin zone and its symmetry points
are shown on Fig. 1.7.
The electronic band structure calculated in
density functional theory is calculated from the Figure 1.7: First Brillouin zone for
electron density by solving Kohn-Sham equations the cubic face centred lattice. The
and calculating the energy eigenvalues. The ex- Greek letters denote special symmeample of the band structures of TiC and MgO try points which are used for the calcalculated with Wien2k software [11] using the culation of the band structure.
GGA approximation is shown in Figs. 1.8(a)-(b).
Both compounds have NaCl cubic fcc structure, where titanium carbide is a metal
and MgO is an insulator. This is illustrated by the difference in their band structures:
the bands of the states occupied by electrons and the unoccupied bands of TiC overlap,
so the material is conducting. In contrast, insulating magnesium oxide has a gap
5
4
3
1
0
EF
-1
-2
-3
-4
-5W
L
G
X
W K
Energy (eV)
Energy (eV)
2
10
9
8
7
6
5
4
3
2
1
0
-1
-2
-3
-4
-5
-6
-7
-8
W
a.) TiC
29
EF
L
G
X
W K
b.) MgO
Figure 1.8: Electronic band structure of (a) metallic titanium carbide and (b) insulating
magnesium oxide calculated with Wien2k software [11]. The Fermi energy is at zero, represented by horizontal dashed line. For MgO the unoccupied bands are located above the
Fermi level and they are represented by blue lines, and occupied bands are black. The
empty space between the unoccupied bands and the Fermi energy represents the energy
gap. For metallic TiC the bands are crossing the Fermi level.
between the occupied and unoccupied states (the band gap). The energy, which
corresponds to the highest occupied orbital is called Fermi level which is set to zero
on Figs. 1.8(a)-(b). The energy levels just below the Fermi level are called valence
band as they correspond to the electrons, which take part in the bond formation.
The unoccupied states are called conduction band. Another possibility to analyse the
electronic structure is to calculate the density of states (DOS), which is the number
of electrons occupying the same energy level in the first Brillouin zone:
g(E) =
1 dNel
V dE
(1.62)
The total density of states describes the distribution of energies for all electrons in
the system, while the partial DOS shows the energy distribution of the electrons of
particular atom or its orbital (s, p, d, f ). Examples of the calculated DOS for MgO are
shown on Fig. 1.9. One can observe the band gap, and the occupied and unoccupied
states. The red and blue lines shows the contribution of the oxygen atom electrons
and p electrons of Mg, respectively.
Density of States (arb. units)
30
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
Total
Mg p
Total O
-4
-2
0
2
4
8
6
10
12
14
16
E-EF (eV)
Figure 1.9: Total (solid black line) and partial density of states for oxygen (dotted red line)
and magnesium (dashed blue line) for MgO calculated with Wien2k software [11].
As both band structure and density of states are calculated from the electron
density, it is natural that if the system is spin polarised (the spin-up and spin-down
densities differ ρ ↑6= ρ ↓), these quantities are calculated separately for the spin-up
and spin-down energy eigenvalues.
1.3.8
Optical properties
The relation between the optical properties and conductivity can be understood in
terms of the Drude-Lorentz model, which describes the optical response by a damping
harmonic oscillator with mass m, charge q, damping
coefficient γ, the restoring force
q
F and undamped angular frequency ωT = 2π
function of a system of N oscillators reads [50]:
ǫ(ω) = 1 +
q2 N
(ω02 − ω 2 )
m
(ωT2 − ω 2 ) + ω 2 γ 2
+
F
.
m
The expression for the dielectric
q2 N
ωγ
i 2 m2
(ωT − ω ) +
ω2γ 2
(1.63)
The dielectric function is the basic property, from which the optical transition and
reflection can be calculated. The peaks in the imaginary part of this functional
corresponds to the same peaks of the reflectivity [50].
On the contrary, the behaviour of the electrons in metal is different from the
insulator, as the electrons are free (so called electron gas), and thus the restoring
force is zero F = 0, and the dielectric function is described by a Drude model with
ωp =
p
N e2 /m being plasma frequency:
ωp2
ω2γ
+
i
ǫ(ω) = 1 −
ω(ω 2 + γ 2 )
ω(ω 2 + γ 2 )
31
(1.64)
When the oscillating frequency ω is zero, than the dielectric function in Eq.1.63
becomes real and positive, which indicates the insulator. Unlike this, for the metal ǫ
in Eq.1.64 is infinite for ω = 0 which corresponds to electric conductivity and 100%
reflectivity.
Optical properties can be derived from the dielectric function ǫij (ω) calculated by
DFT. This function is a three-dimensional tensor, which depends on the symmetry of
the crystal. For example, for the orthorhombic unit cell only diagonal components of
this tensor are non-zero. For the cubic cell there is only one non-equivalent component
Im ǫxx , while tetragonal and hexagonal cells have two independent components [51].
The dielectric function is calculated directly from the Kohn-Sham energy eigenvalues εk . In the random phase approximation (RPA) the expression for ǫij reads [51]:
4π X
∂F (ε)
ǫij (ω) = δij −
n, k −
pi;n,n,k pj;n,n,k −
V ω2
∂ε εn,k
4π X
pi;c,v,k pj;c,v,k
,
(1.65)
2
V ω k;c,v (εc,k − εv,k − ω)(εc,k − εv,k )2
where V is a unit cell volume, pn,m,k are momentum matrix elements between the
bands n and m, for the the point k of the crystal. F (ε) is a Fermi-Dirac distribution
function:
1
,
(1.66)
F (ε) =
F
exp ǫ−ǫ
+
1
kB T
where εF is a Fermi level.
The other important optical properties can be calculated from the tensor of dielectric function. For example, refractive index nii (ω) and extinction coefficient kii (ω)
are given by the expressions [51]:
r
|ǫii (ω)| + Re ǫii (ω)
nii (ω) =
,
(1.67)
2
r
|ǫii (ω)| − Re ǫii (ω)
(1.68)
kii (ω) =
2
From this quantities the reflectivity can be calculated:
Rii (ω) =
(nii (ω) − 1)2 + kii (ω)
(nii (ω) + 1)2 + kii (ω)
(1.69)
32
And the expression for the optical conductivity reads:
1
Re σij (ω) = −Im
ǫ(ω) ij
(1.70)
Chapter 2
Superplasticity in tetragonal
zirconium dioxide ceramics
2.1
Ceramic materials
The word “ceramics” origins from Greek κǫραµικσ which means pottery. Ceramics
are made by heating the clay minerals (e.g. kaolinite Al2 Si2 O5 (OH)4 ) to high temperatures in order to remove the water. The result of the process is a creation of hard
material which consists of the mixture of metal oxides (see Fig. 2.1.) Nowadays the
Figure 2.1: The ceramic pottery which dates back to the late period of Bronze Age Minoan
civilisation located at the island of Crete, approximately 15th century BC. From Pre-historic
museum of Santorini Island, Greece [52].
term “ceramics” is customary used for any inorganic insulating heat-resisting solids.
The ceramic materials are one of the most widespread compounds in nature. Most of
them are either minerals or sands. Thus, the application of these materials is wide,
they are actually used in all aspects of our life. For example, silicon dioxide (SiO2 ) is
one of the major components of the sea sands and is used in the glass production. It
34
2. Superplasticity in tetragonal zirconium dioxide ceramics
can also exist as the single crystal of quartz (Fig. 2.2(a)).
Due to their hardness (usually 7–9 by Mohs scale) and chemical stability ceramic
(a)
(b)
Figure 2.2: Examples of minerals which are ceramic materials: (a) quartz (SiO2 ) is a
widespread ceramic which is used in the glass production or piezoelectric, (b) a 1.41-carat
oval ruby (Al2 O3 ) from Sartor Hamann Jewelry in downtown Lincoln, Nebraska, as an
example of gemstone cut from the natural mineral.
materials are used as the refractory elements in various industrial processes, which
take place at high temperatures and in corrosive environment (e.g. acids). Ceramic
membranes are used for gas separation (e.g. removal of CO2 , H2 S and water vapour
from natural gas by silica membranes [53]) or filtration of liquids at molecular level
like water purification [54, 55].
Ceramic materials are also traditionally used in the building industry as roof tiles,
cements and bricks.
The variety of jewels are also made from the ceramic minerals, e.g. ruby that is
Al2 O3 with chromium inpurities, which gives the red colour (see Fig. 2.2(b)), sapphire
(Al2 O3 with Va, Ti, Cr or Fe impurities which can give blue, yellow, pink, purple,
orange, or green colour), amethyst, which is quartz (SiO2 ) with iron and aluminium
impurities which are responsible for its violet colour, moissanite (SiC), tanzanite
(Ca2 Al3 (SiO4 )(Si2 O7 )O(OH)) or emerald (Be3 Al2 (SiO3 )6 ). The ion conducting ceramics are used as solid electrolytes in the solid oxide fuel cells and batteries. The
first ionic conductor was based on β-aluminia (β-Al2 O3 doped with 10% Na2 O). It
was discovered in 1916 [56]. Further research have shown, that addition of magnesium
or lithium oxide impurities enhances the sodium conductivity of this compound to
the values which are suitable for the industrial applications [56]. This lead to the
2.1. Ceramic materials
35
production of the first sodium-sulphur (Na/S) battery by Ford Motor Company in
1960. In 1992, the first Ford Ecostar electric vehicle equipped with the sodium-based
battery with β-aluminia solid electrolytes (BASE), was presented [57]. However, due
to the development of more efficient type of batteries, this vehicle never went into
mass production. BASE batteries are now used in load levelling storage systems [56].
The alternatives have also been developed based on the Li ion conductivity (e.g. Li/S
batteries [58]).
Another prospective ion conductor is zirconium dioxide ZrO2 which is the main component of the baddeleyite mineral (see Fig. 2.3).
The solid oxide fuel cells (SOFC) make use
of the oxygen ion conductivity, where yttriastabilised zirconia (ZrO2 +Y2 O3 ) serves as an
electrolyte [59]. Another oxygen ion conductors,
which are prospective candidates for SOFC include ceria (CeO2 ), bithmuth rare-earth doped
oxides [59], and perovskites (e.g. Sr or Mg doped
LaGaO3 [60]). In 1997, the first 100 kW power
station based on solid oxide fuel cells began its
operation in the Netherlands [59].
In medicine ceramic materials, mainly hydroxyapatites [61] and calcium phosphates [61],
are used as coating for different prostheses [54]
and bone substitutes [61, 62, 63]. In the dentistry, the applications of ceramics include fillings, crowns, veneers, implants and dental brack- Figure 2.3: Baddeleyite of dimenets [64]. Such wide application of ceramics in sions 2.5 × 2.2 × 1 cm which origins from Phalaborwa, Limpopo
medicine results from its chemical inertness.
Province, South Africa, as an examCeramic materials are also used [54]:
ple of natural monoclinic ZrO2
in the capacitors,
electrical insulators,
piezoelectric transducers and actuators,
monolithic or fibre-composite brakes used in the automotive industry,
waste treatment, especially used for the storage of nuclear waste;
The ceramic materials are either oxides (e.g. aluminium oxide Al2 O3 , zirconium
dioxide ZrO2 or silicate SiO2 ) or non-oxides (carbides, borides, nitrides, silicides) and
36
most of them are hard and break if a strong plastic deformation is applied. However,
there is a group of ceramic materials with superplastic properties, which can undergo
plastic deformations for more than 100% of their initial length [65]. This allows
their wide applications as a “glue” for joining different elements. Most superplastic
ceramics are polycrystalline fine grained materials, and the superplasticity can be
reached by intergranular sliding. Among them the zirconium dioxide ceramics focus
particular interest due to the extremely high elongations before the fracture (up to
1000% of its initial length.) However, the nature of superplasticity in such ceramics
is still not understood on the atomic level.
In this section we discuss the results of application of the density functional theory
for the calculation of mechanical properties of tetragonal zirconium dioxide ceramics.
2.2
Superplasticity
Most materials can be deformed either elastically or plastically, and the fracture
usually occurs when the strain reaches a value which is below 100% of the material’s
initial length. While elastic deformations show linear stress-strain dependence and the
material restores its original shape after the deformation, the plastic deformations are
non-linear and irreversible, after deformation the material doesn’t return its original
shape.
The plastic deformations are described by the following stress-strain relation:
σ(t) = K ǫ̇(t)m
(2.1)
Here, K is a constant and σ(t) is a flow stress, which is the stress required to deform
the material at the particular strain ǫ. This time-dependent quantity describes the
material deformation flow as it is undergoing the dynamically changed strain at the
strain rate ǫ̇(t). The exponent m is called strain-rate-sensitivity exponent and it is
material dependent. For ideal viscous materials like glass at melting temperature
it equals to m = 1, while for normal (non-superplastic) metals and alloys usually
m < 0.2. Superplasticity corresponds to 0.3 < m < 0.8 [66]. The superplasticity is the
possibility to undergo extremely large deformations without a fracture. Although, the
extensive research in the phenomenon has started in XX century, it could be possible
that superplastic materials were used in old times. For example, the composition of
the famous Damascus steels is similar to superplastic carbon steels used nowadays [66].
The modern studies of superplasticity began as early as in the 1912 when Bengough reported 163% elongation for the brass at 700
[66]. Further reports about
superplasticity are also related to metallic alloys. One of them is the famous report
by Pearson were the Author achieved 1950% elongation for the Pb-Sn and Bi-Sn alloys(see Fig. 2.4 [67]). For that reason Pearson sometimes is credited as a person

2.2. Superplasticity
37
who first demonstrated the superplasticity. At present time, the world record of the
highest possible elongation is 8000% and corresponds to Cu-Al alloy [66].
Figure 2.4: The first report (after Pearson, 1934 [67]) of the extremely large 1950% elongation in Bi-Sn superplastic alloy.
2.2.1
Superplasticity in ceramic materials
The study of the superplasticity in grained ceramic materials began in late 80-es after
the discovery of the superplasticity in yttria-stabilised zirconium dioxide ceramics by
Wakai with almost 1000% elongation [68]. At present time the zirconium dioxide is
still the best ceramic material with superplastic properties. Other ones are silicon
nitride Si3 N4 , silicon carbide SiC and their composites [65].
The superplastic properties in ceramic materials are either activated or enhanced
by the addition of impurities. For silicon carbide doped with 1wt% boron and 3.5%
free carbon the elongation of 140% is reported [69], while addition of aluminium
dopant results in the elongation of the order of 300% [70].
For pure Si3 N4 the elongation of up to 180% was observed [71]. Addition of yttria
Y2 O3 and aluminia Al2 O3 dopants results in much higher elongation up to 470% [72].
The composites Si3 N4 /SiC with the same dopant metal oxide have lower elongations
of 100% [73].
For zirconia ceramics addition of metal oxide dopants significantly affects its superplasticity [65]. The elongation is about 140% for yttria-doped ZrO2 (YZP), which
increases to 180% for YZP doped with TiO2 and to 420% for YZP doped with GeO2 .
Combining TiO2 with GeO2 dopants gives the elongation of 1000% [74]. The effect
is illustrated on Fig. 2.5.
38
Figure 2.5: Illustration of the dependence of the superplastic properties of ZrO2 on the type
of the dopant. Left figure shows schematic representation of the superplasticity increase
from pure ZrO2 to yttria-stabilised zirconia with metal oxide dopants. Figure on the right
show experimentally determined stress strain curves for 3mol% yttria-stabilised tetragonal
ZrO2 (solid line), YZP with 2mol% TiO2 (△) or GeO2 () dopants. Vertical line corresponds
to fracture of the material. From A. Kuwabara et al. Acta Mat. 52, 5563 (2004) [74]
The nature of influence of dopants on the superplasticity in ceramics is still not
completely understood on the atomic level. Several superplasticity mechanisms which
will be described below, do not give the clear picture.
2.2.2
Superplasticity mechanisms
One may ask a question what makes such large elongations of hard and brittle materials possible? The answer is the grain diffusion in fine-grained polycrystalline ceramics,
which takes place during the deformation. Addition of dopant oxides enhances such
process and makes the higher elongation possible. Depending on the value of exponent m in Eq. 2.1 one distinguishes fine structure superplasticity (FSS) and high
strain rate superplasticity (HSRS). For metallic alloys m ≈ 0.5 and the strain rate is
proportional to the second or third power of the grain size d:
1
,
(2.2)
dn
where n = 2, 3. The typical grain sizes, which allow superplasticity are d = 1–10 µm.
The fine structure superplasticity corresponds to relatively big grains and thus low
strain rates of the order of 10−3 ..10−4 s−1 [66]. For example, from the microstructure
of the zirconia polycrystals (Fig. 2.6), one can see, that the grain sizes of zirconia are
of the order of several µm, which corresponds to the fine-structure superplasticity.
ǫ̇(t) =
2.3. Zirconium dioxide
39
Figure 2.6: Scanning electron micrograph of micro-structure of undoped yttria stabilised
ZrO2 ceramics before (left) and after (right) annealing at 1510 ◦ C for 72 h [75].
In contrast, the high strain rate superplasticity corresponds to the cases of ǫ̇(t) =
10 ..10 s−1 and the small grain size is not sufficient to explain this mechanism. It
is believed, that in this case the glass liquid phase between grains is created and
promotes intergranular sliding due to the reduced friction. For example, this type of
superplasticity is responsible for the superplastic properties of silicon nitride [65].
−1
2.3
Zirconium dioxide
Zirconium dioxide ZrO2 (zirconia) is an insulating solid ceramic material with a wide
range of applications. This mixed iono-covalent oxide that has an electronic band
gap ranging from 4.2 eV to 4.6 eV depending on the crystal structure [76], however it
exhibits ionic conductivity when doped with other elements. Zirconia is also rather
hard material, pure cubic zirconia has the hardness 8 on the Mohr scale.
These properties make ZrO2 widely used, for example as a refractory material in
various high temperature or harsh environment processes; for joint material for various
ceramic elements; as a low-cost substitute for diamond in the industrial processes; as
in ionic conductor in the solid oxide fuel cells and many others [59].
Pure ZrO2 exist in three main phases depending on the temperature:
cubic phase (F m3m) exists above 2650 K and below the melting point at 2990 K;
tetragonal phase (P 43 /nmc) which is stable between 1400 K and 2650 K;
monoclinic phase (P 21 /c) is the most stable below 1400 K with the density of
5.68 g/cm3 .
40
Schematic structures of these phases are presented in Fig. 2.7, one can see that size
of the unit cell increases with increasing temperature. Zr atoms are coordinated
to eight oxygen atoms in tetragonal and cubic phases. In the monoclinic phase Zr
has strongly distorted octahedral coordination, with the seventh oxygen in the close
proximity. Thus, this phase can be percieved with unusual coordination of metal to
seven oxygens.
Doping of ZrO2 stabilises the high temperature cubic and tetragonal phases even
Figure 2.7: Structural phases of ZrO2 (from left to right): cubic, tetragonal and monoclinic.
Oxygen atoms are represented by red spheres and zirconium atoms are light blue.
close to ambient temperatures. The most important dopant is yttrium oxide Y2 O3
(yttria) (yttrium stabilised zirconia is usually denoted as YZP ceramic) [77], but the
stabilisation can also be achieved with other oxides such as Ce2 O3 , MgO and CaO [78].
The phase diagram of yttria-stabilised zirconium dioxide is presented in Fig. 2.8.
Doping with 5–50 mol% of yttria stabilises the cubic phase in the wide range of
temperatures. Such composition has also one of the best known oxygen ionic conductivity [59].
2.3.1
Superplastic properties of ZrO2 ceramics
Among all the phases of zirconium dioxide we select the tetragonal phase, which
exists at a temperature range 600–2650 K and for yttria concentration range of 0–6.7
mol%. This phase is particularly interesting because its superplastic properties. It
was discovered that superplasticity in this material can be drastically increased by the
addition of the small amount of metal oxide dopants as can be seen from Fig. 2.5 [74].
Although the superplasticity in ZrO2 is usually explained by the grain boundary
sliding [79], there is lack of understanding this phenomenon on the atomic level. In
order to gain more insight into this phenomenon we have performed DFT calculations
41
Figure 2.8: Structural phases of ZrO2 as a function of temperature and yttria concentration.
“M” and “T” one the left side of the graph correspond to monoclinic and tetragonal phases,
respectively. From A. Bogicevic et al., Phys. Rev. B 64, 014106 [77]
of the elastic properties of doped ZrO2 and proposed the simple explanation for the
relation between the changes of elastic properties of single zirconia grains and overall
superplasticity.
Superplasticity in zirconium dioxide
The main and widely accepted explanation of the superplasticity mechanism is that
the superplastic strain rate ǫ̇(t) increases due to the increase of the grain boundary
diffusion as described by the following empirical equation [79]:
µb
ǫ̇(t) = A
kT
σ − σth
µ
2 2
b
Dgb ,
d
(2.3)
where A is a constant, µ is shear modulus, σ and σth are the applied and critical
stresses, respectively, d is the grain size, b is the Burgers vector modulus, kT has its
usual meaning and Dgb is the diffusion coefficient, which is given by the expression:
QA
Dgb = D0 e− kT
Here QA is an activation energy and D0 is the frequency factor.
(2.4)
42
One can observe that three main quantities on which the strain rate depends
are: the grain size, diffusion coefficient and shear modulus. While most experimental
studies of the superplasticity in ZrO2 concentrate mainly on the grains and their
diffusion, here we take attention to the mechanical properties of the singular grains,
which have an influence on the shear modulus.
Dopant dependence
One of the challenges in understanding the superplastic properties in ZrO2 is its
enhancement after the addition of small amounts of dopant metal oxides. From the
stress-strain dependence of tetragonal yttria-stabilised zirconia (Fig. 2.5 one can see,
that addition of 2 mol% TiO2 results in the elongation of 180%, while pure YZP can
be elongated by 160% only. GeO2 dopant gives more elongation, up to 420%, and the
mixture of these metal oxides results in the elongation of more than 1000%.
The explanation of these phenomenon was proposed by Kuwabara et alter [74].
They have analysed the changes in ionicities of the oxygen atoms after replacement of
Zr ion (electronegativity 1.32 [80]) by Ti and Ge atoms with larger electronegativities
of 1.38 and 1.99, respectively. They have found the linear correlation between the
reduction of the average ionic charge of the system and the flow stress, and psoposed,
that the increase in superplasticity is related to the changes of the bond strengths in
the region close to the dopant atoms.
However, their study does not include the dependence of superplasticity on the
dopant concentration. Below we calculate the dependence of elastic and electronic
properties of doped YZP and analyse the dependence on both the dopant concentration and the type of the dopant. As a result, the model which explains this
phenomenon on the atomic level, is proposed.
2.3.2
Calculation of elastic properties of ZrO2
The tensor of elastic constants of
notation):

C11

C12


C13
Cij = 

 0

 0

0
tetragonal phase looks as follows (in the Voight
C12 C13
0
0
C11 C13
0
0
C13 C33
0
0
0
0
C44
0
0
0
0
C44
0
0
0
0
0


0 


0 


0 

0 

C66
(2.5)
43
Here C44 and C66 are two non-equivalent elastic constants which correspond to shear
deformation and the other constants correspond to the uniform deformation along
principale crystallographic axes. As an effect of the last ones are actually “hidden”
in bulk modulus, so only shear elastic constants C44 and C66 and the bulk modulus B
were calculated for all the cases. The dependence of bulk modulus and shear elastic
constants on the concentration of dopant and yttria was determined by means of
Density Functional Theory calculations.
All the calculations were performed using the SIESTA program [10] as described
in Section 1.2.1. The calculations involved testing the appropriate pseudopotentials
for all the atoms which correctly describe the structural and electronic properties of
the corresponding metals and metal oxides. Then, the elastic properties of doped
zirconia were calculated.
Doping process
The SIESTA implementation of the Density Functional Theory uses periodical boundary conditions to solve Kohn-Sham equations. That is why added impurity in the
unit cell will be uniformly distributed in the whole space with a distance equal to the
corresponding lattice constants. So, to vary the concentration of the defect one needs
a supercell approach, which is described in Section 1.3.5. As presented in Fig. 2.7,
the 2 × 1 × 1 supercell of tetragonal zirconia contains 12 atoms – 4 Zr atoms and 8 O
atoms. To introduce the dopant, two Zr atoms are replaced by Y atoms and another
Zr atom is replaced by Ti. To preserve the charge neutrality, one oxygen atom should
be removed:
4ZrO2 + Y2 O3 + TiO2 → ZrY2 TiO8−1 + 3ZrO2
For this case both the yttria and Ti dopant concentration will be 33 mol%. The
crystal structures which correspond to this case are shown in Fig. 2.9.
As seen from phase diagram on Fig. 2.8, such a large concentration is unphysical
and will destroy the tetragonal symmetry. However, we did use such a cell for testing
purposes because it is small and its calculation do not requite much computation
time. For the realistic cases the larger supercells were constructed, they correspond
to smaller yttria/dopant concentrations: 2×2×2 with the concentration of 6.67 mol%
(47 atoms), 3 × 2 × 2 with 4.3 mol% (71 atoms, 568 valence electrons), 3 × 3 × 2 with
2.8 mol% (107 atoms, 856 valence electrons) and the largest 3 × 3 × 3 supercell with
1.9 mol% (161 atoms, 1288 valence electrons). The case of 6.67 mol% correspond to
the phase transition from tetragonal to the cubic phase and was also used for the
testing purposes. Example of the supercell with 6.67 mol% dopant concentration
is shown in Fig. 2.10. Several mutual locations of host Zr atoms and dopant Y
atoms were tested. The energetically most stable configuration is for Y atoms located
in (a, b) plane with the oxygen vacancy located in the first coordination sphere of
44
Figure 2.9: Crystal structure of tetragonal ZrO2 stabilised with 33 mol% of yttria (left);
the same structure with addition of 33 mol% of TiO2 dopant (right). Oxygen atoms are
red, grey, green and light blue balls correspond to Ti, Y and Zr atoms, respectively.
dopants. Such configuration is in agreement with previous calculations by Bogicevic
et alter [77] and Eichler [81].
2.3.3
Model of the superplasticity in doped YZP
Calculated lattice parameters for ZrO2 are a = b = 3.61 Å and c = 5.20 Å, the
bulk modulus is 209.5 GPa, which is in good agreement with literature [82]. Elastic
constants for this phase are presented in Table 2.1. They compare well with other
theoretical calculations. Bulk modulus and shear elastic constants (C44 and C66 ) were
calculated for all considered dopant concentrations.
No systematic changes of the Table 2.1: Elastic constants of pure tetragonal phase
bulk modulus with Y/dopant of ZrO2 . Values are given in GPa
concentration were observed.
C11 C12 C13 C33 C44 C66
Only for the concentration of
293 248 111 385 51 187
6.67 mol% large fluctuations of This thesis
the bulk modulus are observed Ref. [83], LDA 382 221 72 346 42 167
due to the phase transition from
Ref. [84], exp. 327 100 62 264 59 64
tetragonal to the cubic phase.
Similar behaviour is observed Ref. [85], exp. 340 33 160 325 66 95
for C44 elastic constant. On the
contrary, the systematic softening of C66 elastic constant is observed for increasing
45
Figure 2.10: Crystal structure of 2 × 2 × 2 supercell of pure zirconia (left) and the same
supercell of yttria-stabilised ZrO2 doped with titania (right). This supercell corresponds to
6.67 mol% dopant concentration. Oxygen atoms are red, grey, green and light blue balls
correspond to Ti, Y and Zr atoms, respectively.
dopant cocnentration as shown in Fig. 2.11(a).
Based on this results, a simple model of the grain sliding is proposed and presented
in Fig 2.7(b). It is well known that the distribution of the dopant atoms in zirconia
grains is not homogeneous, and that the dopants tend to segregate toward the grain
boundary [79]. Thus, the concentration of dopants in the subsurface region will be
larger than the average concentration in the bulk. Since, the concentration influences
the shear modulus, the softening of the grain surface will be observed.
As a result, the decrease of C66 shear elastic constant induces the increase of
strain rate, which is proportional to the inverse of shear modulus. In the limiting
case when the C66 is zero, the grain surface will act as a liquid lubricant allowing the
easiest sliding and the huge superplasticity increase. The grain sliding is promoted
by the decrease of the friction between the grains, which contributes to the overall
superplasticity besides the electrostatic and kinetic effects. More details are presented
in Ref. [82] (see Paper I attached to this thesis).
2.3.4
Electronic effects in YZP superplasticity
Superplasticity in ZrO2 is also related to its electronic structure. It was shown by
Kuwabara et alter [74], that the addition of metal oxides to YZP induces the local
changes in the net ionic charges of the system. These authors defined the “average
46
190
C66 (GPa)
180
170
160
150
No dopant
Ti
Ge
Si
140
1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7
Yttria concentration (mol%)
(a)
(b)
Figure 2.11: (a) Dependence of C66 elastic constant on the yttria/dopant concentration for
different dopant atoms. Filled circles correspond to undoped YZP and empty symbols are
for YZP with Ti, Ge or Si dopant. Solid line is a linear fit of the average values for each
concentration. (b) A schematic model showing the sliding of ZrO2 grains. The d denotes
the size of the layer where the dopant concentration varies, arrows show the direction of
the applied force F and the direction of sliding. The grain surface layers undergo shear
deformation during sliding.
of products of net charges”, that is proportional to the average cation–oxygen bond
strength in the region around the metal dopant atom:
θ=
1
Nbonds
X
QC QO ,
(2.6)
Nbonds
where Nbonds is the number of cation–anion bonds in the first coordination sphere of
the metal dopant atom, QC and QO are the net ionic charges of the cation (Zr, Y
or dopant) and oxygen, respectively. This quantity (θ) is reduced by addition of the
dopant metal oxides.
We have calculated the average of products of net charges for all dopant concentrations as described in the previous sections. No systematic changes of the QC and
QO ionic charges were observed with the increase of the dopant concentration, which
indicates that the changes of the atomic ionicities are a local phenomenon.
Our calculations have shown significant decrease of the net charges in the region
around the dopant. This results in the corresponding decrease of the average of
products of net charges θ [75] (see Paper II, attached to this thesis).
2.4. Summary
47
Calculated θ is compared with experimentally determined grain boundary diffusion coefficient Dgb and presented in Fig. 2.12. For
pure dopants the decrease of θ is linearly correlated with the increase of Dgb . For the case
of the mixed dopants (Al2 O3 and MgO) the
change of the θ is much smaller and the value
grain boundary diffusion coeffient is comparable to the case of pure dopants.
2.4
Summary
Figure 2.12: Correlation between experimentally determined grain boundary difThe dependence of the elastic properties fusion coefficient D and calculated avgb
of doped tetragonal yttria-stabilised zirconia erage of products of net charges θ in elec(YZP) on the type of the dopant and its con- trons.
centration was calculated by means of Density Functional Theory implemented in the SIESTA program [10]. While, the homogeneous elastic deformations shows no significant change with the change of the dopant
concentration, the linear decrease of C66 elastic constant with the dopant concentration increase is observed. Such decrease induces the enchancement of the strain rate
in polycrystalline zirconia and promotes intergranular sliding. This effect contributes
to the superplasticity increase in YZP after addition of metal oxides dopant [82](see
Paper I attached to this thesis).
The relation between the superplasticity in YZP and its electronic structure was
shown by calculating the dependence of the net ionic charges in the region around
the dopant metal atom on the dopant type and its concentration. No systematic
dependence of the atoms ionicities on the dopant concentration was observed, which
indicates that this is a local phenomenon which depends on the type of dopant only.
The decrease of net charges of both metal cations and oxygen anions was observed in
the first coordination sphere around the dopant atom. Calculated average of products
of net charges θ (Eq. 2.6) decrease in comparison to pure YZP is observed for all the
dopants. Such descrease is linearly correlated with the increase of grain boundary
diffusion coefficient Dgb , which was experimentally measured by Boniecki et alter [75]
(see Paper II attached to this thesis).
48
Chapter 3
Structural and thermodynamic
properties of metal borohydrides
3.1
Hydrogen energy storage and metal hydrides
Energy supply for the future is one of the most important dilemma to be faced by
the mankind. Answer to this problem requires research and solution of problems
related to novel energy sources, energy storage and transportation. It is difficult to
imagine that any significant breakthrough can be done without inventions of new
materials with strictly designed properties. In particular new solutions are required
for energy storage for mobile applications, like electronic devices (portable electronics)
or road transportation (automobiles). Also, a new solutions are needed for energy
load levelling from power stations that use renewable energy sources like wind or
solar [86].
At present, majority of energy for transportation comes from crude oil (gasoline or
diesel fuels) and several means of energy storage are available, they include: batteries,
capacitors or hydrogen fuel cells for electric energy.
Electric capacitors are widely used in electronic devices as a transitory energy
storage. The standard capacitor consists of two conducting metal plates separated by
a dielectric material. Application of the direct electric field induces opposite charges
on the electrodes. Amount of the charge accumulated is proportional to the surface
area of the plates (electrodes) and properties of dielectric material determine potential
between them. Both quantities define amount of the energy stored in capacitor. This
energy, however is rather small (below 0.1 Wh/kg) and cannot be useful for practical
applications for transportation purposes. In recent years new type of capacitors, so
called supercapacitors, has attracted research attention. Unlike in ordinary capacitors
the metal plates are not separated by dielectric, but layers of material in which electric
field induces electric double layer in similar manner as in electrochemical systems.
50
3. Structural and thermodynamic properties of metal borohydrides
Low voltage is a disadvantage of supercapacitors, but they allow to store orders of
magnitude more energy than conventional ones and the energy density can reach up
to 30Wh/kg [86, 87]. Advantage of the capacitors is an immediate storage of energy
that can be further used in short pluses with high current densities.
Electric batteries are nowadays the most important electric energy storage solution that finds application from miniature applications as in watches to the megawatt
installations in load-levelling of the power stations [86]. A battery (or rechargeable
battery) usually consists of several electrochemical cells. Each cell is made by two electrodes immersed in the ion conducting electrolyte (liquid or solid). Positive charges
form the electrolyte migrate to the cathode, while negative ones migrate toward the
anode. Batteries can be divided with respect to type of electrolyte or electrode materials. Low energy density ( 40 Wh/kg) lead-acid batteries are still used, however
the most advanced lithium-ion batteries can reach energy density of 800 Wh/kg [86].
For comparison energy density of gasoline is 6000 Wh/kg. The major drawback of
batteries is relatively slow charging cycle, safety and costs. Batteries often contain
toxic metals (like Cd, Hg etc) and improper recycling can result in the irreversible environmental pollution. In many developing countries there are no regulations obliging
battery recycling. As a result batteries are considered as standard waste that leads
to serious soil and water contamination.
There exists variety of large scale electric energy load-levelling storage methods.
For example hydro power stations, where water is pumped uphill in low grid load and
energy is recovered during peak load of the grid. This method is characterised by
very low energy density, as the gravity is responsible for energy storage. The excess
electric energy on smaller scale can also be used for air compression, which can be
converted back to the electricity by allowing the compressed air to drive turbines [86].
Novel and promising energy storage method is to convert energy into hydrogen.
This can be done by water electrolysis or reforming of fossil fuels [88]. Electrolysis
of water is particularly appealing since there is no emission of greenhouse gases if
renewable energy sources are used. Burning of the hydrogen leads to water. Also,
unlike batteries, the hydrogen itself is non-toxic and when obtained by electrolysis
it does not contain any toxic contaminations. One should also mention that the
water is the most widespread compound on the Earth that makes the hydrogen easily
accessible for the mass production. The energy density output from the hydrogen
fuel cell is around 1100 Wh/kg [86], and this can be significantly increased once more
efficient methods of hydrogen storage are known.
The US Department of Energy (DoE) has formulated a roadmap guiding the
desired hydrogen storage properties of materials, which allow the practical use of the
hydrogen fuel cells in automotive industry [89]. According to DoE roadmap the target
storage capabilities are 3000 Wh/kg energy density (9 wt% H2 ) by the year 2015.
This corresponds to the half of the energy density of gasoline (6000 Wh/kg [86]).
3.1. Hydrogen energy storage and metal hydrides
51
Additionally the storage material has to be operational within temperature range
of −40 to +60 . The possible candidates for achieving this goal are shown in on
Fig. 3.1 [90].

Properties of hydrogen
Hydrogen is the lightest element in the Periodic Table thus it has the highest possible
energy-to-mass ratio [91]. Application of hydrogen as an energy storage media faces
unsolved challenges due to physicochemical properties of this element.
Hydrogen is a gas with the density of 0.08988 kg/m3 at standard thermodynamic
conditions and energy density per unit volume stored in this gas is very low. Liquid
hydrogen has density 70.8 kg/m3 but it exist as a liquid only below 20.28 K. Density
of solid hydrogen H2 is 70.6 kg/m3 and solid phase of this element exists below
14.01 K at normal pressure. Unfortunately critical point of hydrogen is at 32.97
K, and 1.293 MPa, thus there is no possibility to preserve liquid phase at room
temperature. The phase diagram of hydrogen can be seen in Fig. 1 in Ref. [92].
On the atomic level hydrogen forms di-atomic H2 molecules below 2000 K. Dissociation energy of H2 equals 435.9 kJ/mol [93]. Separation between hydrogen atoms
is 0.75 Å [80]. Hydrogen molecules exist as para and ortho hydrogen due to spin
of the proton (anti-parallel and parallel, respectively). In standard thermodynamic
conditions hydrogen gas contains 75% of ortho and 25% of para hydrogen [94].
Thermodynamic properties of para and ortho hydrogen are different at low temperatures.
In chemical compounds hydrogen usually exists in atomic form as an ion. It can
carry both positive (NH3 ) or negative (B2 H6 ) charge, however in organic compounds
the charge of hydrogen is close to neutral (CH4 ) [93, 94].
Storage of hydrogen in reasonable volumetric densities requires advanced solutions,
as compression, liquefaction or variety of physicochemical processes [92] that will be
briefly presented below. Compression: at present this is widely used hydrogen
storage method in steel cylinders at pressures 100–350 bar. Since, hydrogen is a light
element, compression of this gas is energy intensive and usually 10–30 % energy stored
in hydrogen is lost for compression. There are problems related to high diffusivity
of H2 , and embitterment of metals related to it. Despite that, compressed hydrogen
storage with perspective pressures of 700 bars in composite tanks still is considered
as safe storage method. Unfortunately this does not fulfil requirements for the mass
density to be applicable for large scale transportation [92, 89].
Liquefaction of hydrogen consumes up to 50% of energy stored in this gas, it
requires ultra-low cryogenic temperatures, and is related to inevitable losses due to
52
evaporation (the loss can be as high as 0.4% per day for the tanks with a storage
volume of 50 m3 [92]). For practical applications advanced cryogenic systems are required that already provide mass density superior to the compressed ones adding extra
costs to the storage system, and they also require energy supply to keep temperature
below 20 K.
Physisorption was intensively studied storage method few years ago due to
perspective of hydrogen storage on high surface materials like carbon nanotubes or
clathrates. This method is based on a weak interaction of molecular hydrogen with
the surface of the solid (i.e. surface of a carbon nanotube) via Van der Waals interactions described by the Lennard-Jones potential:
VLJ (r) =
1
1
− 6,
12
r
r
(3.1)
where r is a distance between the hydrogen molecule and the interacting atoms of the
substrate. The first term of this potential describes the short range Pauli repulsion
due the overlap of the electron orbitals, and the second term is the Van der Waals
attractive force. This weak interaction requires, however low temperatures, usually
below 200 K and provides low volumetric and gravimetric hydrogen density (2 wt%
at 100 bar and 193 K) that limit the applicability of this method [92]. Physisorption
on carbon nanotubes was officially abandoned as practical hydrogen storage method
by US Department of Energy [95].
Chemical reactions of hydrogen and metals could also serve as hydrogen storage
method. For example alkali metals reacting with water transform to metal hydroxides
and release hydrogen (6.3 mass% H2 ):
Li + H2 O → LiOH + 21 H2
This very safe and efficient method suffers reversibility problems and cannot be,
at present, applied for large scale applications [92]. Metal hydrides: are reversibly
formed for selected metals or alloys exposed to high pressures of hydrogen. The
equilibrium formation of metal hydrides is described by Van ’t Hoff relation:
P
∆H ∆S
ln
=
−
(3.2)
P0
RT
R
where P refers to hydrogen pressure, ∆H and ∆S are respectively enthalpy and
entropy of metal hydride formation, R is gas constant, T – temperature. For light
metals this method provides relatively good storage density, however catalysts are
requires for the full reversibility of storage in Mg, Li or similar metals [93]. More
details about metal hydrides will be presented in Chapter 4.
Complex hydrides is a new class of materials that are seriously considered
as practical hydrogen storage media. Complex hydrides can be perceived as ionic
3.1. Hydrogen energy storage and metal hydrides
53
Figure 3.1: Gravimetric and volumetric hydrogen density for metal hydrides, pure hydrogen and other hydrogen storage materials. Pure liquid hydrogen is coloured in light blue,
pressurized gas storage (dark blue dotted lines) is shown for steel with tensile strength of
460 MPa and density 6500 kg·m−3 ) and a hypothetical composite material (tensile strength
1500 MPa, density 3000 kg·m−3 ); hydrocarbons are marked in green, and metal hydrides
with ionic or metallic bonding are marked with red and hydrides with covalent bonding are
marked with violet. After A. Züttel. Materials Today 24:24 (2003) [90].
−
molecular crystals, consisting of rather stable BH−
4 (borohydrides); AlH4 (alanates) or
NH−
2 (amides) anionic groups and the metal cations. Once light metals are embedded
in a complex hydride it provides very appealing hydrogen density up to 18.51 mass%
for LiBH4 (see, Fig. 3.1 and Table 3.1).
However, in these compounds hydrogen is semi-covalently bounded in a molecule,
thus storage reversibility pose some practical problems. As strongly ionic systems
these compounds are rather stable, and decompose usually above 300 (see Table 3.1)
that makes additional problems with their practical applications. It was shown, for
sodium alanate that Ti catalyst can provide full reversibility of hydrogen storage
and lower decomposition temperature [96]. Unfortunately for other compounds such

54
catalysts are not known, but decomposition temperature can be tuned by appropriate composition of the cationic sublattice. For example, Nickels et alter [97] have
observed the inverse linear dependence between the decomposition temperature and
the Pauling electronegativity of the metal cations in alkali metal borohydrides. This
observation is not general, and information about the stability and the formation
energy of new compositions with metals at different oxidation states is required for
rational design of compound with desired properties.
Such information can be provided on theoretical basis, and it requires extensive search for the
ground state structures and their properties by Table 3.1: Decomposition temperaquantum methods. As the search for the ground tures of selected borohydrides and
their hydrogen weight content
state structure of compounds is a complex problem in the next section we provide brief overview Compound Tdecomp , K wt%
of methods that can be used for that purposes.
LiBH4
553a
18.5
3.2 Crystal structure prediction methods
NaBH4
680a
10.7
KBH4
580a
7.5
Be(BH4 )2
396b
20.67
Mg(BH4 )2
593b
14.9
533b
11.6
One of the greatest challenges in theoretical solid Ca(BH4 )2
state physics is prediction of the crystal struc- Al(BH4 )3
208.5b
16.9
tures by knowing only the composition of the a
Ref. [90], b Ref. [98]
solid. Such method must be based on quantum
calculations, as the interactions between atoms in
the solid are not known a priori. So far, existing
implementations of the density functional theory
and other quantum-mechanical methods still require, beside chemical composition,
some empirical knowledge about the structure of the material (e.g. atomic positions
and/or lattice constants) in order to properly describe its ground state configuration [99]. For example, in the previous chapter we used experimentally available
structural information of zirconium dioxide as a starting point for modelling the influence of defects on crystal structure and ZrO2 mechanical properties.
For novel materials or structures of nanoparticles experimental data are unavailable or far from being complete and atomic-level insight into these structures is often
possible only via theoretical models. These models are later verified experimentally.
Also, for materials containing light elements with small scattering cross-section for
neutrons or X-rays experimental determination of the structure is often difficult or
impossible. Complex borohydrides are one of the examples of these materials, as both
hydrogen and boron pose problems in scattering experiments. Hydrogen atoms have
3.2. Crystal structure prediction methods
55
to be substituted by deuterium also isotope of boron has to be used for neutron experiments. X-ray scattering requires synchrotron sources with high luminosity. Also
single crystals of complex hydrides are often unavailable. For that reason crystal
structure prediction has gained importance in determination of the structure of these
compounds. Knowledge of the crystal structure is a starting point for any thermodynamic considerations about complex hydrides. However, the energy landscape for
these compounds is rather flat that create additional complication in structural predictions. Below selected methods for crystal structure prediction are presented with
particular focus on complex hydrides. They include simulated annealing [99], genetic
algorithm [99], database search methods [100], and prototype electrostatic ground
state (PEGS) methods [101].
3.2.1
Simulated annealing
The simulated annealing method is based on the physical annealing process [102].
Annealing is the process where the system is slowly cooled from a liquid state to
temperature below melting point. At the phase transition from disordered liquid
phase to the crystalline structure atoms arrange in ordered manner representative for
the symmetry of the solid. However, if the initial temperature is too low, or the rate
of temperature change is too high it is possible to achieve structures different from
the ground state. For example, during a quench (the temperature change ratio is very
high), the system is transformed into a glass phase. In metallurgy, annealing with
slow temperature drop is used to remove defects or to vary the hardness and ductility
of the material for larger temperature change ratios (see Ref. [103]).
In simulated annealing method one starts from the temperature with the kinetic
energy high enough to go across several local minima [99]. Depending on implementation at each temperature the atoms are perturbed randomly and the minimum
is searched by means of Monte Carlo method or molecular dynamics method. In
molecular dynamics method the system is cooled at constant temperature rate until
predefined temperature is reached. Than the standard optimisation is used to achieve
the ground state configuration. In the Monte Carlo method standard Metropolis algorithm [104] can be used. For each atom a random displacement is applied that
results in the total energy difference of ∆E. If the total energy of annealed system is
lowered (∆E < 0), then the displacements are accepted as a new system configuration, in other case (∆E > 0) the acceptance probability is exp(− ∆E
). This criterion
kT
simulates the Boltzmann distribution which describes the probability of the solid to
reach the energy E at the temperature T :
E
1
exp −
P (E) =
Z(T )
kT
(3.3)
56
where Z(T ) is the partition function, E is the total energy of the system. Within this
approach the system either can reach global energy minimum or it can be trapped in
a local minima. Thus more advanced techniques as basin hoping (relaxation of the
structure after each Monte Carlo step) are often used [99]. Simulated annealing was
used for complex hydrides [105, 106], however this method is time consuming, and
other methods perform better for flat energy landscape systems.
3.2.2
Genetic algorithm
This class of methods mimics the Darwinian theory of evolution. In original Darwin
theory only organisms, which better fit to the environments survive. In computer implementation of genetic (evolutionary) method for structural search the initial “population” consists of random arrangement of atoms in the structure. These populations
evolve via applying two main operations: a “crossover”, which combines several features of pair structures into the new structure, and a “mutation”, which introduce
new structural features [99]. The mutation is simulated by several random Monte
Carlo moves of atoms. For the crossover, structural elements from several parent
structures are integrated into a new child structure. The crossovers, which result in
the lowering of total energy of the system are accepted implementing the evolutionary
principle of “survival of the species with better genes”. There are numerous implementations of genetic algorithms which differ in realisation of crossover and mutations
between populations of the candidate crystal structures. For example, one can use
these operators separately for creating new child structure or combine them in one
evolutionary step. The “natural selection” is simulated by selecting the new structure that is the best candidate (lowest total energy) of old set of structures (so called
elitism) [99]. Another possibility is called a roulette wheel where evolution depends
on the population slots that are proportional to the quality of the whole population
in the slot [99].
Evolutionary method was applied for simple metal hydrides [107, 108], however
this method is also computationally ineffective for this purposes.
3.2.3
Database searching methods
In this method, the candidate structures of compounds with the same stoichiometry are selected from structural database (for example, Inorganic Crystal Structure
Database(ICSD)) [100, 101]. As, such databases contain crystal structures of large
majority of compounds known to the mankind the set of initial candidates can be
as large as few hundreds. Within this method each candidate structure is optimised
(atomic positions and unit cell parameters are relaxed) and structures with the lowest energy after such optimisation are considered as a preliminary ground state. For
3.2. Crystal structure prediction methods
57
this smaller set of structures lattice vibrations are calculated, and for structures that
are dynamically unstable a simulated annealing is applied to achieve a stable ground
state. This method was successfully applied to prediction of the crystal structure of
calcium alanate (Ca(AlH4 )2 or CaAlH5 ) [100]; for calcium borohydride Ca(BH4 )2 the
number of initial structures with the formula AB2 X8 was 271 [101]. However, the
stable structure was not found until it was determined by other methods [109].
The major drawback of the database searching method is a large number of candidates to be calculated, which requires large computation resources. The data mining
method [110] helps to reduce the number of possible structures by using linear correlations between DFT total energies of different compounds of similar structure. This
is particularly useful for alloys, where one can plot the dependence of the total energy
on the alloy composition and its structure (e.g. concentration of metals in the M1 M2
alloy) and use linear regression to obtain the total energies and possible structures of
yet unknown alloy compositions [110].
3.2.4
Prototype electrostatic ground state approach (PEGS)
This method is a sophisticated modification of the simulated annealing method where
the electrostatic interaction energy is used as a minimisation criterion instead of DFT
total energy. The method is based on the fact that electrostatic interaction between
cations and anions play the major role that determine the crystal structure of complex
hydrides, and most anion complexes have the similar tetrahedral shape in different
−
crystal structures of alanates (AlH−
4 ) or tetrahydroboranes (BH4 ) [101]. Method was
especially designed for complex hydrides [101]. The electrostatic interaction energy
is given by the formula [101]:
Eel =
X Qi Qj
i>j
Rij
+
X 1
12
Rij
i>j
(3.4)
The first term of this formula is the Coulomb interaction energy, where Qi and Qj
are ionic charges of cations and anions and Rij is the distance between the ions. The
second part represents the repulsive part of the Lennard-Jones potential and it contributes only when the ions overlap [101]. The Metropolis Monte Carlo minimisation
is performed with respect to this electrostatic energy Hamiltonian, and prototypical
ground state structures are relaxed by means of density functional theory methods.
3.2.5
Coordination screening
This method is similar to the database search method as is also based on the calculations of the total energies of structurally similar crystals in order to find the energy
minimum. However, unlike in the database search method, the number of initial
58
structures is lower since the local coordination of the ligands in complex compounds
is taken into account. The initial structures are formed by combinations of building
blocks, that are made of predefined coordination around each local structure in the
crystal. The method was successfully applied to the prediction of the crystal structure
of complex hydrides and metal hydrides mixtures [111].
The coordination screening method was used for prediction of possible crystal
structures of the borohydrides mixtures [111]. The local coordination screening is
based on the information about possible coordination of the BH−
4 group [112]. The
results of applying this method will be described in the following section.
3.3
Application of coordination screening to complex metal borohydrides
Coordination screening was performed for group of alloys with composition M1 M2 (BH4 )2
with M1 and M2 =Li, Na, K. For the composition M1 M2 (BH4 )n , where n = 3, 4, the
following metals were used: M1 =Li, Na, K and M2 =Li, Na, K, Mg, Al, Ca, Cr, Mn,
Fe, Ni, Co, Pd, Rh, Sc-Zn, Y-Mo,Ru-Cd, Cu, Ag, V, Nb, Ti, Zr.
Generation of the initial structures was based on the fact that in borohydrides of
alkali and alkaline earth metals cations are arranged either in trihedral, tetrahedral
or octahedral coordination to BH−
4 group [112]. Initial structures contained the following combinations to M1 and M2 atoms, respectively: tetrahedral/tetrahedral, octahedral/octahedral, tetrahedral/octahedral, octahedral/ tetrahedral and for selected
metals (M2 =Mg, Al, Ca, Sc-Zn,Y-Mo,Ru-Cd) tetrahedral/trihedral coordination (see
Fig. 3.2). The coordination polyhedra were either corner, edge sharing or combination of both. All initial unit cells were designed to contain only one stoichiometric
unit, and proper stoichiometry was imposed. This resulted in 757 initial structures.
Density functional theory calculations were performed within plane-wave basis set
implemented in Dacapo program [12]. The plane-wave cutoff energy of 350 eV was
applied. The GGA revised PBE functional [113] was used to describe the exchangecorrelation and the Brillouin zone was sampled with 64 k-points. For the minimisation
of the atomic positions the quasi-Newton method was used.
The structural optimisation of these compounds was performed in similar way
as it was done for the zirconium dioxide as described in the Section 1.3.1. However,
several modifications of the relaxation procedure were introduced. At a first step only
positions of hydrogen atoms were relaxed this allows to preserve initial coordination
of metal as the initial forces excerpted on hydrogen might be large. In the next step
atomic positions were fixed and the unit cell volume was optimised by means of the
Murnaghan equation of state [43]. This step was followed by additional optimisation
of the positions of hydrogen atoms. Following this optimisation step the most stable
3.3. Application of coordination screening to complex metal borohydrides
(a)
(b)
59
(c)
Figure 3.2: Initial structures for M1 M2 (BH4 )4 corresponding to the tetrahedral (a), octahedral (b) and tetra/octahedral (c) coordination of the (BH4 )− group (blue tetrahedra) around
metal atoms (red and yellow polyhedra). The octa/tetrahedral coordination is obtained by
switching the M1 and M2 atoms in the corresponding tetra/octahedral structure [112].
structures were relaxed without any constraints of subsequent relaxation of atomic
positions and the unit cell with fixed internal positions of atoms.
Once the lowest energy structures were obtained the following two quantities were
calculated in order to determine their stability (here we refer to NaCo(BH)4 )4 for
illustration):
∆Ealloy = ENaCo(BH4 )4 − (ENaBH4 + ECo(BH4 )3 )
(3.5)
This energy describes alloy stability with respect to separation into two original binary
borohydrides, i.e. it shows whether the alloy is more stable than its components. The
second quantity illustrates the stability of the alloy with respect to decomposition into
elements (or stable metal hydrides, if they exist). It is called decomposition energy:
∆Edecomp = ENaCo(BH4 )4 − (ENaH + ECo + 3EB + 5.5EH2 )
(3.6)
For the purpose of being the sufficient candidate for the hydrogen storage, the structures should satisfy the conditions of ∆Ealloy < 0 eV/formula unit and −0.5 <
∆Edecomp < 0 eV/H2 .
After thermodynamically most stable configuration was identified for each alloy,
the next optimisation run was done where both atomic positions and the unit cell
volume were allowed to change. That was done in order to be sure if our starting
guess was good enough or if the most stable configuration corresponds to completely
another structure.
The work published in [112] was a part of the joined Computational Materials
Design Project at the CAMD Summer School 2008, where the author investigated
60
the structural and thermodynamical properties of NaCo(BH4 )3,4 and NaRh(BH4 )3,4
(see Paper III attached to this thesis).
3.4
Details of calculations for NaX(BH4)3,4
Application of coordination screening to a broad class of complex hydrides requires
significant human attention. For example, for compounds NaX(BH4 )3 ; 4 (X=Co,Rh)
with the hydrogen storage capacities of 11.41 wt% for NaCo(BH4 )4 and 8.70 wt%
of hydrogen for NaRh(BH4 )4 (that meets the 2015 target of the US Department of
Energy of 9 wt% [89]) the each proceeds as follows.
The ground state optimisation of these compounds reveals that the most stable stoichiometry are NaX(BH4 )4 , where the metal atom X prefers the tetrahedral
coordination to the BH4 group.
For this stoichiometry the structural optimisation of the metal–boron distances
(both Na-B and X-B) of NaX(BH4 )4 showed that the most stable configuration has
both Co and Rh metal atoms in their tetrahedral position, while the separation
between sodium and boron shrinks by 5%. In addition, minor distortion of B-Na-B
angle were observed.
NaCo(BH4 )4 is unstable with respect to decomposition to its components (NaBH4
and Co(BH4 )3 ), with ∆Ealloy > 0. The alloy stability of NaRh(BH4 )4 is negative ∆Ealloy = −0.033 eV/formula unit, that indicate this borohydride may exist
as non-segregated phase. The decomposition energy of this compound is ∆Edecomp =
−0.016 eV/H2 . This minor stability is far from the target value of −0.2 eV/H2 , what
discriminate this borohydride from potential to practical applications [112].
3.5
General trends in borohydride stability
For the majority of hydrides studied in this project, the structure optimisation indicate strong preference of tetrahedral coordination of the metal by borohydride groups
(BH4 ). However, for some alloys (indicated as “other” in Fig. 3.3) the most stable
structure was slightly different from the tetrahedral, with minor distortions of the
M1 –B–M1 angle.
Out of 700 hundred compounds studied more than twenty are stable with respect to the decomposition into binary borohydrides (∆Ealloy < 0) and with respect
to decomposition into elements (∆Edecomp < 0). Among them about 10 structures
were recognised as highly promising candidates for the purpose of hydrogen storage
(Edecomp = −0.4..−0.1 eV/H2 ) (see Fig. 3.3). The gravimetric hydrogen density in
these compounds ranges between 8 and 17 wt%, thus most of them meet the 2015
DoE target of 9wt% [89].
3.5. General trends in borohydride stability
61
Figure 3.3: The dependence of alloy stability ∆Ealloy on decomposition energy ∆Edecomp
for all investigated structures. Colours indicate the type of the main cation: Li is red,
Na is blue and K is green; shape of the points indicate the coordination type: tetrahedral
(), octahedral(◦), octa-tetra(△), tetra-octa(+), tetra-tri(g), other(⊳). The candidate
structures which meet the criteria for hydrogen storage, should have ∆Ealloy < 0 and
∆Edecomp = −0.4..−0.1 [112].
Stability and electronegativity The linear correlation between the average Pauling electronegativity of metal cations and decomposition temperature which was discovered by Miwa et alter [114], is also supported by the present results, Fig. 3.4. The
trends on this figure indicate linear dependence of the decomposition energy on the
average electronegativity of metals involved. Although, some deviations of about 1020% from this dependence are observed, one can safely conclude, that the replacement
of the metal cation by the one with smaller Pauling electronegativity results in the
decrease of the decomposition energy of the compound. For example, LiNa(BH4 )2
has the lowest decomposition energy of ∆Edecomp = −0.6 eV/H2 . Replacement of
the sodium atom with electronegativity 0.93 [80] by the transition metal like V or Fe
with electronegativities 1.63 and 1.83, respectively results in significant increase of
decomposition energy (−0.113 and −0.104 eV/H2 , respectively).
However, in view of recent studies [115] segregation of compounds and overestimation of the local interaction between BH−
4 groups and metal cations might be a
reason for oversimplifications, thus linear dependence of electronegativity and stability requires further verification.
62
Figure 3.4: The dependence of decomposition energy ∆Edecomp on the average Pauling electronegativity of the stable (with ∆Ealloy < 0) structures of complex borohydrides. Colours
indicate the type of the main cation: Li is red, Na is blue and K is green; shape of the
points indicate the coordination type: tetrahedral (), octahedral(◦), octa-tetra(△), tetraocta(+), tetra-tri(g), other(⊳) [112].
3.6
Summary
The coordination screening approach combined with density functional theory is a
powerful tool which allows prediction of the possible crystal structures of complex
metal borohydrides. The ground state stability of more than 700 structures and compounds were analysed, and 10 were found to be promising candidates for the purpose
of hydrogen storage. Some of structures are already confirmed experimentally, i.e.
LiFe(BH4 )3 , LiAl(BH4 )4 , (Li/Na)Mn(BH4 )3; 4, (Li/Na)Zn(BH4 )3 [112], while other
await verification.
Chapter 4
Electronic structure of alkali and
alkaline earth metal hydrides
4.1
Selected properties of hydrides
The hydrogen forms binary compounds with majority of elements in the periodic
table. Only noble gases and some of transition metals do not form any stable binary
compounds with hydrogen. Binary compounds of hydrogen can range from ionic,
through metallic to the covalent ones, as presented in Table 4.1.
Starting from the right hand side of the periodic table, Fig. 4.1, compounds shaded
in cyan are covalent compounds that can range from the gas phase (e.g. CH4 ) through
liquid (H2 O) to the solid state (BiH3 ) at standard thermodynamic conditions. Compounds of carbon, oxygen or nitrogen are of great importance for humans. Hydrocarbons may form longer chains or cyclic molecules.
Elements of the group 12 (zinc) and the group 13 (boron) (shaded in violet in
Fig. 4.1) form polymeric hydrides, that are covalently bounded. Especially borohydrides form broad range of structures, from simple diborane to very complex structures of higher boranes, however in general those compounds do not occur in nature
as frequently as those of the previous group.
For transition metals ranging from the group 6 (chromium) to the group 11 (copper) metal hydrides are unstable (shaded yellow in Fig. 4.1), with exception of Cr,
Ni, Cu and Pd (red in Fig. 4.1). Hydrides of chromium and copper can be formed but
they are poorly defined; NiH<1 is very unstable. Hydrogen in palladium has very high
mobility and stoichiometry of this compound never reaches PdH. Palladium hydride
is superconductor below 9 K.
Transition metals to the left of group 6 form metallic hydrides and they are
shaded in red in Fig. 4.1. In these compounds hydrogen may occupy octahedral,
tetrahedral or combination of both empty interstitial sites. For that reason they are
64
1
4. Electronic structure of alkali and alkaline earth metal hydrides
2
13
14
15
16
17
H
18
He
2.2
LiH BeH2
Ionic hydrides
Covalent hydrides
Covalent polymeric hydrides
Metallic hydrides
0.97 1.47
NaH MgH2
1.01 1.23
3
4
5
KH CaH2 ScH2 TiH2
VH
VH2
0.91 1.04
1.45
1.2
1.32
RbH SrH2 YH2 ZrH2 (NbH2 )
6
7
CrH Mn
(CrH2)
CH4
NH3 H2O
HF
2.01
2.5
3.07
3.5
4.1
AlH3 SiH4 PH3
H2S
HCl
8
9
10
11
Fe
Co
NiH
CuH
ZnH2 GaH3 GeH4 AsH3 H2Se HBr
1.66
1.56
1.6
1.64
1.7
1.75
1.75
Mo
Tc
Rb
Rh
PdH2
Ag
12
BH3
Ne
Ar
1.47 1.74 2.06 2.44 2.83
1.82 2.02
2.2
Cr
2.48 2.74
CdH2 InH3 SnH4 SbH3 H2Te
HI
Xe
YH3
0.89 0.99 1.11 1.22
1.23
1.3
CsH BaH2 LaH2 HfH3
TaH
W
1.33
1.4
1.36 1.42 1.45 1.35
Re
Os
Ir
Pt
1.42
1.46
1.49 1.72 1.82 2.01 2.21
(AuH) (HgH2) TlH3 PbH4 BiH3 H2Po HAt
Rn
LaH3
0.86 0.97 1.08 1.23
Fr
Ra
1.46 1.52 1.55 1.44
1.42
1.44
1.44 1.55 1.67 1.76 1.90
AcH2
1.0
Figure 4.1: Table of the binary hydrides and Allred-Rochow electronegativity. From J. E.
Huheey, Inorganic Chemistry, Harper & Row, New York (1983) [116]
often named interstitial hydrides. The stoichiometry ranges between MH, MH2 , and
MH3 depending on the occupancy of the empty lattice sites. Interstitial hydrides
are good electric conductors, and for most cases they can be reversibly formed under
hydrogen pressure. Metal surface catalyse H2 dissociation and hydrogen diffuses inside
the metal framework continuously forming new phase. Formation of transition metal
hydrides is related to successive occupation of interstitial sites by hydrogen atoms.
This process leads to large lattice strains, therefore these compounds usually exist in
powder form having grey colour.
Rare Earth metals form either di- or tri- hydrides, depending on applied load
of hydrogen. For these elements, especially for yttrium and lanthanum change of
stoichiometry is related to metal insulator transition that offers interesting physical
properties, that are presented in the frame below.
Finally, alkali and alkaline earth metals form ionic structures with physical
4.1. Selected properties of hydrides
Switchable mirrors
Rare-earth and yttrium or lanthanum hydrides attracted interest in the late nineties
of XX century after discovery of their switchable optical properties [117].
Left panel illustrate reflectance and transmittance of Y film as deposited, Y in
the dihydride and trihydride phase (from left to right, respectively). Variation
of the optical transmittance and electrical resistivity of yttrium thin film capped
by a 20 nm Pd layer is shown on the right panel. The system is brought into
contact with H2 gas at 105 Pa at room temperature. The hydrogen concentration is approximately proportional to the elapsed time t, except after 1600s,
where both electrical resistivity and optical transmittance have reached saturation. Reproduced from J. N. Huiberts et alter, Nature(London) 380, 231
(1996) [117].
Already in 1969 metal–insulator transition was reported for CeHx . Transition occurs around x=2.8 [118]. Discovery of similar transition in YHx by Huiberts et
alter [117] has triggered constant interest in these materials. One mole of yttrium
can absorb 3 moles of hydrogen. During this absorption, initially shiny, metallic
yttrium transforms to YH2 , that is also metallic. Further load of hydrogen results
in optically transparent insulating YH3 . Loading of hydrogen into yttrium results in
the decrease of electrical resistivity, that is 5 times lower than in dihydride phase. In
parallel, the optical transmittance of sample is low. For x 2.86 optical transmittance
suddenly rises and it stays high for YH3 ; this process comes with sharp increase of
electrical resistivity of the system. Such behaviour is an indicator of metal–insulator
phase transition. Structural change between cubic YH2 and insulating YH3 is not
directly related to electronic transition, as YHx already possesses hexagonal structure, when conductivity change occurs. Despite intensive theoretical research, at
present, there is no general consensus on the mechanism of this transition.
Since materials that may convert from translucent and opaque appearance are of
potential technological interest, many new materials with switchable mirror properties were discovered. Besides variety of earth metals, alloys that contain magnesium
are widely studied. Alloys of Gd, Y, La and Mg show behaviour that is superior
to the pure yttrium, and in fully hydride form they are transparent and switching
is faster [50]. Also alloys (thin films) containing nickel, cobalt, iron or calcium
were shown to posses switchable optical properties [119]. This is very appealing
technologically as they do not contain scarce elements.
Theoretical studies of these materials require accurate and efficient method suitable
for description of optical properties.
65
66
properties that are very different from the metallic hydrides. They are transparent
insulators with large band gap. Stoichiometry reflects valency of metals, i.e. univalent
alkali metals form monohydrides (LiH, NaH, KH, RbH, CsH), and alkali earth metal
hydrides contain two hydrogen atoms per metal (BeH2 , MgH2 , CaH2 , SrH2 , BaH2 ).
Both types of hydrides are very stable and thay decompose at melting temperature
(with exception of LiH, CaH2 and BaH2 ). For alkali metal hydrides melting temperature decrease with increasing mass of metal, while for alkaline earth metal hydrides
melting temperature increases moving down to the periodic table, see Table 4.1.
Table 4.1: Decomposition temperatures of alkali and
are given in K.
LiH
NaH
KH
993a 962b
698a,b,d
690b 483c
823c
483c
693d
d
1245
BeH2
MgH2
CaH2
523a,b
603a 600b
873a
463d
358c 573d
1089b
1158c
1273d
a
Ref. [120],
b
Ref. [80],
c
alkaline earth metal hydrides. Values
RbH
443b,c
637d
CsH
443b,c
662d
SrH2
1323b
858c
1273d
BaH2
1473b
503c
1273d
Ref. [93],
d
Ref. [94]
From the structural point of view alkali metals hydrides form simple rock
salt structures with Fm3̄m symmetry.
Metal atoms are octahedrally coordinated to hydrogen that occupies octahedral interstitial sites [93], see Fig. 4.2.
There are no phase transitions reported
for alkali metal hydrides, however it was
recently reported that hydrogen enriched
lithium hydride becomes metallic under
high pressure [121].
Alkali earth metal hydrides are Figure 4.2: Rock salt crystal structure of alstrongly ionic systems, except BeH2 that kali metal hydrides. Larger green spheres cordue to small atomic radius of beryllium respond to metal atoms (Li, Na, K, Rb, Cs),
form a covalent solid. They form variety small blue spheres are for hydrogen.
of structures ranging from rather simple
orthorhombic Co2 Si-type (space group Pnma) crystals for heavier elements (Ca, Sr,
and Ba) through tetragonal rutile structure (P42 /mnm) for α-MgH2 to complex body
4.2. Recent studies of alkali and alkaline earth metal hydrides
67
centred orthorhombic (Ibam) for α-BeH2 . The latter two elements may exist in a variety of polymorphs at higher pressures. Theoretical calculations suggest that at
pressure around 7.07 GPa α-BeH2 transform to β-BeH2 with symmetry I41 /amd
(that is the symmetry of anatase, polymorph of TiO2 ). Further compression should
induce transformation to γ-BeH2 (Pmnm) at pressure 51.41 GPa, than to δ-BeH2
(P42 /mnm, the symmetry of rutile structure of TiO2 ) at 86.56 GPa, and finally to
ǫ-BeH2 (P21 3 ) above 97.55 GPa [122]. Within all structural polymorphs beryllium
hydride remains insulating, however the band gap shrinks significantly in β-BeH2 to
recover initial width of α-BeH2 at highest pressures.
(a)
(b)
(c)
Figure 4.3: Crystal structure of alkaline earth metal hydrides: (a) α-BeH2 (Ibam); (b)
α-MgH2 (P42 /mnm) and (c) Pnma structure representative for CaH2 , SrH2 , and BaH2 .
Metal atoms are green and hydrogens are white.
Magnesium hydride transforms from α phase into orthorhombic γ-MgH2 (α-PbO2
type with Pbcn symmetry) at 0.39 GPa and the subsequent phase transformation
from γ to β-MgH2 (a modified CaF2 -type structure, Pnma symmetry) occurs at
3.84 GPa [123]. All these phases were observed as by-products during high-pressure
synthesis or in ball-milled samples (the nanostructured) MgH2 . Upon compression
the electronic band gap decreases and this compound remains insulating up to the
highest pressures. Structures of alkali earth metal hydrides are illustrated in Fig. 4.3.
4.2
Recent studies of alkali and alkaline earth metal
hydrides
Ionic metal hydrides were studied theoretically since very early development of electronic structure calculation methods. However, early calculations often failed to agree
68
with experimental data. For example, Krasko [124] employed a perturbation approach
of covalent bonding for MgH2 , calculated binding energy was positive, that means
magnesium hydride cannot be formed, contrary to the experimental evidence. Yu
and Lam [125] used pseudopotential method for MgH2 . They were first to report ab
initio calculated structural and elastic parameters, formation energy, basic electronic
and phonon structure. Their LDA calculations reasonably agree with the experiments
and the authors concluded that magnesium hydride is strongly ionic system.
In recent years number of theoretical papers on alkali and alkaline earth metal
hydrides was published and they focus on both structural/thermodynamic and electronic properties.
Van Setten et alter [126] use DFT and GW [127] methods to study electronic
properties of lightweight metal hydrides. Among binary metal hydrides authors show
that GGA approximation of DFT underestimated the principal band gap by around
2 eV for LiH, NaH and MgH2 . GW method was also used by Moyses Araujo et alter [123] for MgH2 . They report electronic band gaps for three polymorphs of this
compound, also indicating significant underestimation of the band gap by standard
DFT method. Another set of reports concerns thermodynamic stability and structural properties of hydrides. Smithson et alter [128] report general studies of a broad
range of metal hydrides that allow them identification of main factors that define
formation energy of these compounds. They are: lattice energy required to convert
structure of the metal to this of hydride; loss of metal cohesive energy due to lattice expansion; chemical bonding between metal and hydrogen. Thorough studies
of structural and thermodynamical properties of alkaline earth metal hydrides was
reported by Hector et alter [129], and accurate quantum Monte Carlo calculations by
Pozzo and Alfè [130] provide excellent agreement of the calculated formation energy
with the experiment. Also, Hartree-Fock method was applied for MgH2 [131] and
CaH2 [132]. While the structural properties of these calculations are in reasonable
agreement with experimental data, the electronic properties, especially the band gap
is severely overestimated by several eV. Also, alkali metal hydrides were studied in
details by FP-LAPW method [133], were authors report structural, elastic parameters, and optical phonon vibrations. Analysis of the electronic structure of alkali
metal series lead them to conclusion that in LiH enhanced bonding between Li and H
together with resonant bonding among H ions is responsible for large stability of this
compound. This effect is absent in other hydrides due to larger ionic sizes of metal.
X-ray synchrotron radiation was used to reveal the charge distribution in MgH2 [134].
The room temperature measurements indicate weaker ionicity than it was believed.
According to these studies, the ions exist as Mg1.91+ and H0.26− , and the excess
charge is uniformly distributed within the empty space in the crystal. Optical studies
of MgH2 [135] revealed the band gap of 5.6 eV and optical transparency of 80%.
4.3. Calculation method
4.3
69
Calculation method
The calculations were performed with Linear Augmented Plane-Wave Method (LAPW)
using Wien2k software package [11] as described in Section 1.2.2. GeneralisedGradient Approximation (GGA) to the exchange-correlation functional in PerdewBurke-Ernzerhof parametrisation [28] was applied for all calculations.
Let us remind that in this method the unit cell is partitioned into atomic (muffintin) spheres and the interstitial region where the different basis sets are used to
describe the Kohn-Sham orbitals: plane-wave basis is used in the interstitial region
and an expansion of spherical harmonics times radial functions is used inside atomic
sphere. Hubbard U correction in fully-localised limit (will be further referred as
GGA+U, see Section 1.1.4) was applied for the empty p orbital of the hydrogen
atom inside the muffin-tin sphere and hybrid functional PBE0 which corresponds
to the case of α = 0.25 in Eq. 1.22 was applied for the s orbital of the hydrogen.
For GGA+U the value of Uef f = U − J = 0.66 Ry (8.98 eV) was determined by
adding and removing one electron from the system of two hydrogen atoms separated
by 2 Å(see Eq. 1.32).
The key parameter which defines the size of the atomic spheres and influences the
size of the region where GGA+U is applied is the muffin-tin radius (RMT) which was
selected individually for each type of atom. The preliminary selection was based on
the information about the H− and metal ionic radii and then for GGA+U the tests
were performed with respect to band gap and structural parameters to determine the
most optimal RMT for the hydrogen. The RMT for the metal was set as the difference
between the shortest M–H bond length and the RMT radius of the hydrogen. For
alkali metals the RMT was 1.8 a.u for Li, 2.38 a.u. for Na, K, Rb, Cs. The radii
of hydrogen atoms were 2.0 a.u. for LiH and 2.2 a.u. for the rest of hydrides. For
beryllium hydride BeH2 the radii were 1.38 and 1.4 a.u. for Be and H, respectively,
while in MgH2 the radius of Mg was 1.5 a.u. and the RMT of H was 1.8 a.u. For Ca
and Sr the RMT was 1.9 and 2.0 a.u., respectively, and the radius of the hydrogen
atom equals to 2.2 a.u. The radii of Ba and H in barium dihydride were 2.2 a.u. and
2.3 a.u., respectively. The same RMT radii of atoms and other parameters were used
both for GGA and GGA+U functionals. Structural parameters and energies were
compared with recommended values for GGA and no significant discrepancies were
found.
500 k points in the unit cell were used for alkali hydrides and MgH2 , 100 k points for
BeH2 and 300 k points for CaH2 , SrH2 , BaH2 . For calculation of optical properties a
more dense mesh of 1000 k points was used. The magnitude of the largest vector in the
Fourier expansion of charge density Gmax was set to 20. The energy of separation of
core and valence electrons states was set at −6 Ry. The states below this energy were
described by local orbitals basis (LO) while other states were describe by LAPW basis
70
(see Section 1.2.2 for details).
The cutoff parameter R · Kmax which limits the expansion of the basis set for the
sphere of the radius R to a maximum value of Kmax , was different for different values
of the RMT radii, and varied from 7.5 to 9. The implementation of orbital dependent
functionals in Wien2k requires the calculations to be spin-polarised, although the
resulting spin-up and spin-down electron density was the same. For this reason we
show the results of the band structure, density of states etc as if the calculations were
done for the spin non-polarised case.
For alkaline earth hydrides the minimisation of atomic positions inside the unit
cell was performed using PORT method [136]. Formation energy of MHx (x = 1, 2)
is the reverse of the decomposition energy and is obtained from the following equation:
MHx ↔ M + x2 H2
The expression for this quantity reads:
x
∆Ef orm (MHx ) = (3 − x) Etot (MHx ) − Etot (M) − Etot (H2 ) ,
2
(4.1)
where Etot (MHx ) is a DFT total energy of MHx hydride, Etot (M) stands for the total
energy of metal M and Etot (H2 ) is a total energy of the hydrogen molecule. The latter
two quantities need to be calculated with a same set of parameters, which were used
for the metal hydride MH. As Wien2k software does not support the calculations of
isolated molecules, we have created a pseudo-crystal, which was a cubic box with a
lattice constant 10 Å, were two hydrogen atoms located in its centre. The distance
between the neighbouring hydrogen molecules were 10 Å, which is enough to neglect
the possible errors due to the interaction between periodic images.
4.4
4.4.1
Alkali hydrides
Crystal structure and formation energy
At the first step of calculations the structure of all MH (M=Li, Na, K, Rb, Cs) were
optimised with respect to lattice parameters and internal atomic positions. For all
structures atoms remain in their high symmetry positions. Structural parameters
and formation energy for alkali metal hydrides are presented in Table 4.2. In general
there is very good agreement of the present GGA calculations and those reported
previously [133], small discrepancies are present for Rb and Cs. Our results also
compare well with the plane wave calculations for LiH and NaH [126]. Compared to
the experimental data GGA results are underestimated for light metals (Li, Na), while
for heavier metals agreement is very good, with tendency to underestimation. For
4.4. Alkali hydrides
71
calculation with corrected potential for hydrogen lattice parameters are systematically
overestimated by 6% for LiH or 1% for NaH. Taking into account that reported
lattice parameters are calculated without the influence of zero point vibration, and a
general tendency of GGA calculations for overestimation of lattice parameters, one
has to conclude that correction proposed here is consistent with known literature [137].
Underestimation of the lattice parameters by GGA functional indicate overbinding
of the structures that has electronic origin.
Table 4.2: Lattice constant a and formation
LiH
NaH
Lattice constant a (Å)
This thesis, GGA
4.01
4.84
This thesis, GGA+U
4.36
4.93
Ref. [126], GGA
4.02
4.83
Ref. [133], GGA
4.01
4.84
Ref. [138], exp.
4.07
4.88
Ref. [139], exp.
4.08
–
∆Ef orm (kJ/mol H2 )
This thesis, GGA
-174.28
-84.78
This thesis, GGA+U
-152.62
-64.04
Ref. [133], GGA
-184.2
-118.00
Ref. [140], GGA
-174
-172
Ref. [141], exp.
-181.0
-56.30
energy ∆Ef orm of alkali hydrides.
KH
RbH
CsH
5.71
5.75
–
5.71
5.70
–
6.05
6.11
–
6.08
6.04
–
6.36
6.46
–
6.41
6.39
–
-101.10
-95.78
-140.80
-82
-115.40
-88.58
-83.04
-99.80
–
–
-117.70
-109.70
-91.00
–
–
The formation energies of the present calculations slightly differ from those reported for the previous calculations [133]. This can be explained by much more
precise sampling of the Brillouin zone in the present approach. In fact, test calculations with k point sampling as in Ref. [133], give the formation energy -174.24 kJ/mol
for LiH.
The largest formation energy (by absolute value) is for LiH that is in agreement
with highest decomposition temperature of this compound in the series of alkali metal
hydrides. The lowest formation energy is for sodium hydride for which low decomposition temperature is also reported [94]. Between remaining hydrides the formation
energy varies by tens of kJ/mol, but reported decomposition temperature also varies,
see Table 4.1.
Comparison with experimental data indicates that calculated formation energies
are overestimated for NaH and KH and underestimated for LiH. However, application
of corrected functional lowers formation energy by 21.66 kJ/mol for LiH, by 20 kJ/mol
for NaH and below 10 kJ/mol for other metals. Reduction of the formation energy has
the electronic origin. Good numerical agreement between experimental and calculated
72
formation energy for KH is fortuitous, as one has to keep in mind that formation
energies reported in Table 4.2 do not contain zero point energy contribution.
4.4.2
Electronic structure
0.6
LiH
0.4
0.2
0
0.8
0.6
0.4
0.2
0
4
NaH
KH
2
0
2.25
1.5
0.75
0
RbH
2
CsH
1
0
-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
E-EF (eV)
Figure 4.4: Density of states of alkali hydrides. Black solid lines correspond to GGA
exchange-correlation functional and red dashed lines are for GGA+U. Vertical dashed line
indicates the Fermi level.
The total density of states for all alkaline hydrides is presented in Fig. 4.4. For
all compounds the valence bands consist of hydrogen 1s orbitals, while conduction
states consist of empty metal s,p orbitals (for Rb and Cs additionally d states).
This features together with a large band gap indicates fairly ionic structure of all
compounds. However, while for K, Rb, and Cs the valence band is narrow, for Na
73
and especially for Li valence band has more dispersed character. This dispersion
is an indication for non-ionic contribution to the atomic bonding, as discussed in
Ref. [133]. Namely, for LiH direct interaction between hydrogen ions takes place
due to small separation between H and additionally orbital overlap between metal
and hydrogen is present below the Fermi level. The calculated band gaps for alkali
Table 4.3: Direct band gaps Eg of alkali metal hydrides.
LiH
Eg (eV)
This thesis, GGA
This thesis, GGA+U
Ref. [142], LDA
Ref. [143], GGA
Ref. [133], GGA
Ref. [126], GGA/GW
Ref. [133], exp.
Ref. [144], exp.
NaH
KH
RbH
CsH
3.03
3.87
3.53
3.85
4.44
3.83
–
3.50
–
–
4.90
–
3.03
3.77
3.44
3.00/4.54 3.79/5.50
–
4.99, 4.40
–
–
9.20
–
–
3.01
3.25
–
–
2.92
–
–
–
2.36
2.60
–
–
2.34
–
–
–
hydrides are presented on Table 4.3, and compared with available literature data. The
general trend of the band gap reduction with increasing mass of the metal cation can
be observed. Moreover, the band gaps of the present calculations are comparabale
with similar ones [133] (for GGA approximation). Application of PBE+U method
enlarge the band gap and the correction has the largest influence for LiH (0.82 eV)
and NaH (0.87 eV). For KH, RbH, and CsH the band gap is enlarged approximately
by 0.25 eV. The band gap for LiH compares well with the experimental data [144],
however it is smaller than this calculated by GW method [126]. Taking into account
usual overestimation of the bands gaps by GW, one can conclude, that our value is
reasonable. Also, for sodium hydride the GGA+U band gap is 0.8 eV smaller than
this obtained by GW method [126].
In Figs. 4.5–4.6 the band structures for alkali metal hydrides are presented, and one
can observe that for each element the band structure remains “rigid” with conduction
bands shifted upwards for GGA+U calculations. Unfortunately, there are no GW
calculations of the band structure for alkali hydrides known to author, but such
constant shift is similar to this reported for MgH2 [123] and it will be discussed in
more details in the following sections.
4.4.3
Electron density and atomic charges
The real space charge distribution provides a simple insight into bonding of the crystal. For ionic structures the atomic charges are of primary interest. Atomic charges
74
10
10
8
8
6
6
Energy (eV)
Energy (eV)
4
2
EF
0
Na2H
4
2
EF
0
-2
-2
-4
-4
W
L
Γ
X
W K
W
Γ
L
(a)
X
W K
(b)
8
8
6
6
6
4
4
4
2
EF
0
2
EF
0
-2
-2
W
Energy (eV)
8
Energy (eV)
Energy (eV)
Figure 4.5: Band structure of LiH (a) and NaH (b) for GGA-PBE (black solid lines) and
GGA+U (red dashed lines) exchange-correlation functionals.
L
Γ
KH
X WK
W
2
EF
0
-2
L
Γ
RbH
X WK
W
L
Γ
X WK
CsH
Figure 4.6: Band structure of KH, RbH and CsH for GGA-PBE (black solid lines) and
GGA+U (red dashed lines) exchange-correlation functionals.
75
were calculated by means of Bader electron density decomposition method [49]. This
method partitions the total electron density into basins uniquely assigned to atoms.
Basins are separated by zero flux regions, thus space partitioning is unique and conserves total charge, see also Chapter 1.3.6. Binary structure of alkali hydrides guarantees that the charges on metal and the hydrogen are the same but with the opposite
sign. Calculated Bader charges are presented in Table 4.4. The charge transfer is
largest for LiH (0.86 e) and lowers for heavier elements, however even for CsH it is of
the order of 0.74 e, that indicates these compounds are strongly ionic. For calculations with corrected functional the charge transfer do not change within the accuracy
of calculations, see Table 4.4. Metal hydrogen distance and Pauling electronegativity
for all hydrides in the series are also presented in Table 4.4. The trend of charge
transfer reduction toward heavier elements is correlated with both quantities.
Table 4.4: Bader atomic charges calculated for alkali metal hydrides. Pauling electronegativities for metals, bond lengths and hydrogen ionic radii for alkali metal hydrides.
LiH
NaH
KH
RbH
CsH
+
M cation
GGA
0.86 0.80 0.77 0.76 0.74
GGA+U
0.87 0.80 0.77 0.76 0.74
−
H anion
GGA
-0.86 -0.80 -0.77 -0.76 -0.74
GGA+U
-0.86 -0.80 -0.77 -0.76 -0.74
a
Electronegativity
0.91 0.87 0.73 0.71 0.66
b
M–H bond length (Å) 2.04 2.45 2.86 3.03 3.20
H− radius (Å)
This thesis
0.88 0.80 0.75 0.75 0.60
c
Ref. [94]
1.14 1.29 1.34 1.37 1.39
a
Ref. [80],
Ref. [94]. Calculated bond lengths equal to the half of the lattice constant given in
Table 4.2.
Calculated as a difference between the M–H bond length and Pauling ionic radius of
metal atom.
b
c
To reveal details of the charge distribution that result from GGA and GGA+U
calculations we compare the difference of the charge distribution. To do that a crosssection of the charge density in the middle of the unit cell along (a, b) was cut.
For this comparison the lattice constants of GGA calculations were used to allow
straightforward subtraction of resulting charges. Differences of the charge density
between GGA and GGA+U calculations are presented in Fig. 4.7 and 4.8. For all
compounds this difference is positive, that means the charge is more localised on s
orbital of hydrogen for GGA+U calculations. The region of charge localisation is
76
0.050
0.050
5
6
0.025
0.025
H
H
H
0.012
0.012
5
4
0.006
0.006
Na
Na
0.003
0.003
4
y (Å)
y (Å)
3
0.002
0.002
3
2
H
1
1
H
H
2
H
Na
H
H
Na
0
0
0
1
2
3
4
0
5
1
2
3
4
x (Å)
x (Å)
(a)
(b)
5
6
Figure 4.7: Electron density difference between the densities of GGA+U and GGA-PBE
functionals for LiH (a) and NaH (b). For NaH the dashed line shows the cross section of the
(a,b) plane of the unit cell. The atomic positions are indicated in (b), for (a) they remain
the same. Only regions of charge accumulation are presented for clarity.
8
0.050
8
0.050
0.012
0.012
6
0.006
0.006
0.002
4
3
3
2
2
1
1
0
1
2
3
4
5
x (Å)
KH
6
7
8
0.003
5
0.002
4
3
2
1
0
0
0.006
0.003
5
y (Å)
y (Å)
0.002
0.012
7
6
0.003
4
0.025
7
6
5
0.050
8
0.025
0.025
y (Å)
7
0
0
1
2
3
4
5
x (Å)
RbH
6
7
8
0
1
2
3
4
5
6
7
8
x (Å)
CsH
functionals for KH, RbH and CsH. For all compounds the atomic positions are the same as
shown in Fig. 4.7(b). Only regions of charge accumulation are presented for clarity.
77
largest for LiH, additionally it does not have spherical symmetry. Namely, the charge
density is elongated toward lithium ions, that is an indication of non ionic interaction.
Such effect was also reported in Ref. [133].
For other alkali metals the region of charge localisation decreases for heavier atoms,
in accordance with the charge transfer, Table 4.4. One has to remember that the
charge density plots illustrate only localisation enhancement of the charge due to
application of GGA+U functional.
Based on the region of this localisation one can estimate the ionic radius of hydrogen, that is presented in Table. 4.4 and compared with Pauling radius. Calculated
H− radius ranges between 0.875 Å for LiH and 0.6 Å for CsH in similar manner as
the charge transfer. On the other hand it is smaller than the radius calculated from
the structural data [94] which additionally has the opposite trend.
4.4.4
Optical properties
The calculated complex dielectric function for LiH and NaH are presented in Fig. 4.9.
The common procedure in the GGA calculations is based on the application of scissor
operation. This is to shift empty conduction bands (imaginary part of dielectric
function) upwards to match experimental (or calculated) band gap. Example of
scissor operation is presented for LiH in Fig. 4.9(a), the conduction bands were shifted
by 0.85 eV to match the corrected calculations. The first absorption edge for GGA and
GGA+U calculations match together, however the fine structure at higher energies
slightly differ. The uniform shift of the conduction bands for LiH (see, Fig. 4.9)
15
6
4
2
0
Re ε
Re ε
10
0
6
Im ε
12
Im ε
4
2
0
0
5
8
4
1 2
3 4
5 6 7 8 9 10 11 12
Energy (eV)
(a)
0
0
1 2
3 4
5 6 7 8 9 10 11 12
Energy (eV)
(b)
Figure 4.9: Real and imaginary part of dielectric function for LiH (a) and NaH (b). The
black solid lines are for GGA-PBE calculations and dashed red lines are for GGA+U calculations. For LiH additional blue dotted line is for GGA-PBE calculations where the scissor
operation was applied to correct the band gap error of this exchange-correlation functional.
78
4
3
2
1
0
3
Re ε
4
Re ε
Re ε
2
0
4
Im ε
Im ε
2
Im ε
5
4
3
2
1
0
4
3
2
1
0
0
2
1
1 2
3 4
5 6 7 8 9 10 11 12
Energy (eV)
KH
0
0
1 2
3 4
5 6 7 8 9 10 11 12
Energy (eV)
0
0
1 2
RbH
3 4
5 6 7 8 9 10 11 12
Energy (eV)
CsH
Figure 4.10: Real and imaginary part of dielectric function for KH, RbH, and CsH. The
black solid lines are for GGA-PBE calculations and dashed red lines are for GGA+U calculations.
results in good match of the first absorption peak (transition at point X). Transitions
at K, W, or L contribute to the high energy structure of the spectrum and shift
of the absorption peaks, between those of scissor operation and method proposed
here, by 0.2 eV can be observed at 7 eV. For sodium hydride the optical absorption
has less structure and follows the uniform shift of the bands. Dielectric function for
potassium, rubidium, and caesium hydrides are presented in Fig. 4.10. The more
structured absorption spectra results from complexity of empty orbitals of heavier
atoms.
Application of the correction proposed in the present thesis has very simple consequence due to simplicity of the band structure of alkali metal hydrides. However, for
systems that contain defects or cannot be considered as pure hydrides, the induced
gap states affect optical properties. For such systems application of the scissor operation is far from being straightforward, while method proposed can be applied here
without any limitations.
4.5
4.5.1
Alkaline earth hydrides
Crystal structure and formation energy
Lattice parameters and formation energies of alkaline earth metal hydrides were optimized for all compounds in the series, and they are presented in Table 4.5. In
general they are in good agreement with the experiment, however not as good as
for alkali metal hydrides. For these compounds comparison is made with the plane
wave calculations [129]. As compared to the experimental lattice parameters for light
metals there is tendency for underestimation of the unit cell volume for GGA calculations, while for SrH2 and BaH2 this volume is slightly overestimated. As for alkali
hydrides application of GGA+U functional affects calculated lattice parameters and
for all compounds the volume of the unit cell is systematically larger than for GGA.
4.5. Alkaline earth hydrides
79
Table 4.5: Lattice constants and formation energy ∆Ef orm of
MgH2 lattice constants are given in a sequence a, c, for others
BeH2
MgH2
CaH2
Lattice constants (Å)
This thesis, GGA
9.00,
4.54, 3.02
5.92,
4.16, 7.64
3.58, 6.77
This thesis, GGA+U
9.17,
4.66, 3.08
6.00,
4.24, 7.78
3.62, 6.94
Ref. [126], GGA
–
4.52, 3.01
–
Ref. [125], LDA
–
4.55, 3.04
–
Ref. [129], GGA
8.63,
4.43, 2.97
5.58,
3.91, 7.43
3.56, 6.75
Ref. [145], exp.
9.08,
–
–
4.16, 7.71
Ref. [146], exp.
–
4.50, 3.01
–
Ref. [147], exp.
–
–
6.00,
3.60, 6.81
Ref. [148], exp.
–
–
–
∆Ef orm (kJ/mol H2 )
This thesis, GGA
This thesis, GGA+U
Ref. [129], GGA
Ref. [140], GGA
Ref. [141], exp.
Ref. [149], exp.
Ref. [129], exp.
-31.44
-16.82
-25.10
–
–
–
-18.86
-71.45
-49.03
-62.20
-64
-75.2
–
–
-175.34
-134.20
-173.80
-172
-183.60
-188.80
–
alkaline earth hydrides. For
as a, b, c.
SrH2
BaH2
6.26,
3.87, 7.31
6.46,
3.91, 7.44
–
–
6.35,
3.85, 7.29
–
6.82,
4.16, 7.90
7.06,
4.24, 8.20
–
–
6.84,
4.16, 7.87
–
–
6.36,
3.86, 7.30
–
–
–
-175.25
-158.27
-165.20
-168
-180.40
-177
–
6.79,
4.17, 7.85
-172.47
-149.28
-149.30
-142
-187
-171.50
–
Also for this functional the lattice parameters are larger than experimental ones, in
agreement with general tendency of GGA calculations for overestimation of the lattice constants [137]. For MgH2 we have performed additional calculations with PBE0
exchange correlation functional in order to check applicability of this functional for
calculations of electronic properties in this compound. The calculated lattice parameters are located between those for GGA and GGA+U (a = 4.60 Å, b = 3.04 Å,
not shown in Table 4.5), that seems reasonable, however major discrepancies for this
calculations are explained below.
Electronic contribution to the formation energy as shown in Table4.5 indicates that
for GGA calculations the present values are lower (larger by absolute value) than for
plane wave calculations. Deviations from the experimental values are not larger than
∼ 15 kJ/mol for GGA calculations. Addition of the Hubbard term for GGA functional
80
lowers (by the absolute value) the formation energies below experimental ones. This
is similar trend as for alkali metal hydrides, however while the magnitude of this
change is similar for light metals, for heavier ones (Ca, Sr, Ba) the change is larger.
The formation energy for MgH2 calculated with PBE0 functional is off by two orders
of magnitude, thus due to strongly unrealistic calculations of the energetic properties
one can conclude that application for this functional for MgH2 is unjustified. Test
performed for other compounds confirm this conclusion.
4.5.2
Electronic structure
6
BeH2
4
2
0
2
1.5
1
0.5
0
6
4
2
0
8
6
4
2
0
15
10
MgH2
CaH2
SrH2
BaH2
5
0
-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
E-EF (eV)
Figure 4.11: Total density of states of alkaline earth hydrides. Black solid lines correspond
to GGA exchange-correlation functional and red dashed lines are for GGA+U. Vertical
dashed line indicates the Fermi level.
81
Total density of states of alkaline earth hydrides is shown in Fig. 4.11. All compounds are insulating with the band gap exceeding 3 eV. The valence band consists
mainly of the occupied s states of the hydrogen, however for all compounds in the
series some structural features of the valence band can be seen in Fig. 4.11. They
indicate more complex band structure of these compounds than in alkali earth counterparts. Complexity is a reflection of complex crystalline structures, and differences
between BeH2 , MgH2 and heavier hydrides is related to the great extend to differences in the crystal structure. Additionally non-equivalent atoms of the same kind
have different energy levels. The conduction band consists mostly of the empty s, p
and d states of the metal atoms. Application of the corrected functional shifts conduction states toward higher energies, similarly as for compounds described in the
previous section. The character of the valence band does not change, in general,
with GGA+U functional. The calculated direct and indirect band gaps are shown
Table 4.6: Direct (Eg ) and indirect (Eig ) band gaps of alkaline earth metal hydrides.
Eg (eV)
This thesis, GGA
This thesis, GGA+U
Ref. [126], GW
Ref. [123], GW
Ref. [135], exp.
Ref. [150], exp.
Ref. [151], exp.
Ref. [152], exp.
Eig (eV)
This thesis, GGA
This thesis, GGA+U
Ref. [126], GGA/GW
Ref. [123], GW
Ref. [125], LDA
Ref. [122], GGA
Ref. [132], H–F
Ref. [153], GW
Ref. [150], exp.
Ref. [151], exp.
BeH2
MgH2
CaH2
SrH2
BaH2
5.63
6.05
–
–
–
–
–
–
4.70
5.27
6.19
6.52
5.60
5.40
–
–
3.86
4.17
–
–
–
–
5.00
5.00
4.22
4.56
–
–
–
–
–
5.00
3.32
3.01
–
–
–
4.30
–
5.00
5.63
6.05
–
–
–
5.51
–
8.27
–
–
3.84
4.45
3.72/5.32
5.58
5.00
–
–
–
3.40
–
3.04
3.54
–
–
–
–
10.00
–
–
2.50
3.21
3.63
–
–
–
–
–
–
–
–
2.59
3.38
–
–
–
–
–
–
2.30
–
in Table 4.6 and compared with calculated theoretical and experimental data. Here
other calculations are available only for selected compounds, and the present GGA
data agree reasonably with the them, however they are smaller than experimental
82
data, where available. Calculations with GGA+U functional enlarge the band gaps
by ∼ 0.3 to 0.69 eV to values close to the experimental ones. On the other hand, band
gaps are systematically smaller than those available and calculated in GW method
that can be attributed to overestimation of band gaps in GW method. The band
gaps for MgH2 calculated with PBE0 functional are Eg = 4.84 eV and Eig = 3.91 eV
thus they are larger than for GGA approximation, however this might by accidental
as this approach fails to describe properly thermodynamics of metal hydrides. Band
10
8
Energy (eV)
6
4
2
EF
0
-2
-4
-6
-8R
(a)
Γ
X
M
Γ
(b)
Figure 4.12: Band structure of BeH2 (a) and MgH2 (b). Black solid lines indicate the band
structure for GGA functional and red dashed lines are for GGA+U functional.
structure of alkaline earth hydrides is shown in Figs. 4.12–4.13. In general GGA+U
approach shifts the conduction bands toward higher energies, as observed as it is for
alkali hydrides. For magnesium hydride the band structure calculated for GGA+U
is consistent with calculated in GW approximation [123, 126], but the shift of the
conduction band is smaller for GGA+U. Additionally the width of the valence band
narrows for GGA+U calculations.
83
6
6
4
4
4
2
EF
0
-2
-4
R
Γ
X
CaH2
M
Γ
2
EF
0
Energy (eV)
6
Energy (eV)
Energy (eV)
2
-2
-2
-4
-4
R
Γ
X
M
Γ
SrH2
EF
0
R
Γ
X
M
Γ
BaH2
Figure 4.13: Band structure of CaH2 , SrH2 and BaH2 .Black solid lines indicate the band
structure for GGA functional and red dashed lines are for GGA+U functional.
4.5.3
Electron density and atomic charges
Atomic charges of alkaline earth hydrides calculated with Bader decomposition method
[49] are presented on Table 4.7. Much more complex crystal structure of alkaline earth
hydrides results in larger number of non-equivalent charges (that correspond to nonequivalent crystallographic atomic positions) than in alkali hydrides. For BeH2 two
non-equivalent beryllium atoms and two hydrogen atoms are present, while for other
compounds only one symmetry non-equivalent metal atom exist. The charge of two
hydrogen anions differ usually by ∼ 0.05 e. For all compounds in the series the charge
transfer is large, however only for BeH2 and MgH2 it is comparable to this in alkali
metal hydrides, and it is decreasing with increasing mass of metal. Such an effect is
related to decreasing Pauling electronegativities that are additionally much smaller
Ca, Sr and Ba than that of Be and Mg, see Table 4.7. In general GGA+U functional does not significantly change the charge state of ions. The cross-sections of
the difference of the charge density distribution for GGA+U and GGA calculations
(prepared in a similar way as for alkali hydrides) made along the crystallographic
plane passing through hydrogen atoms are presented in Fig. 4.14 and Fig. 4.15. For
all compounds the changes of the charge distribution are more complex than for alkali metals. This can be seen foe BeH2 in Fig. 4.14(a). Here hydrogen atoms do
not posses spherical symmetry, and the charge distribution is also changed on metal
atoms. Both effects indicate that direct bonding between atoms exist and this bonding has covalent features. For BeH2 change in the charge distribution in the region
between hydrogen atoms also indicated direct interaction of H in this compound. For
magnesium hydride the difference in the charge distribution is quite large and H ions
84
0.100
H
7
0.025
6
0.006
H
5
0.002
y (Å)
3.981E-4
4
H
Be
1.000E-4
3
Be
H
2
1
0
0
1
2
3
4
x (Å)
(a)
(b)
functionals for BeH2 (a) and MgH2 (b). For BeH2 x and y axes correspond to the b and
c crystallographic directions, respectively, and for MgH2 x and y axes correspond to the a
and b crystallographic axes, respectively. The atomic positions are shown on the plots for
both compounds. Only regions of charge accumulation are presented for clarity.
7
0.100
6
0.026
6
0.006
5
0.002
0.100
7
0.025
0.006
5
Ba
0.100
H
0.025
0.006
6
0.002
0.002
5
1.000E-4
3
3.981E-4
4
1.000E-4
3
2
2
1
1
0
1
2
3
x (Å)
CaH2
4
1.000E-4
3
2
H
1
0
0
3.981E-4
Ba
y (Å)
4.000E-4
y (Å)
y (Å)
4
0
0
1
2
3
x (Å)
SrH2
0
1
2
3
4
x (Å)
BaH2
functionals for CaH2 , SrH2 and BaH2 . x and y axes correspond to the b and c crystallographic directions, respectively. The atomic positions are shown for BaH2 , for CaH2 and
SrH2 they remain the same. Only regions of charge accumulation are presented for clarity.
85
Table 4.7: Bader atomic charges calculated for alkaline earth metal hydrides. Pauling
electronegativities for metals, bond lengths and hydrogen ionic radii for alkaline earth metal
hydrides. For BeH2 the charges of two non-equivalent Be atoms are given.
Property
M+ cation
GGA
GGA+U
H−
1 anion
GGA
GGA+U
H−
2 anion
GGA
GGA+U
Electronegativitya
Bond lengthb (Å)
H− radius (Å)
This thesis
Ref. [94]c
BeH2
MgH2
CaH2
SrH2
BaH2
1.62,
1.63
1.61,
1.63
1.64
1.44
1.48
1.43
1.65
1.44
1.49
1.46
-0.86
-0.86
-0.85
-0.85
-0.73
-0.74
-0.74
-0.75
-0.68
-0.73
-0.79
-0.79
1.58
1.33
-0.79
-0.80
1.29
1.95
-0.69
-0.70
1.03
2.32
-0.70
-0.70
0.96
2.49
-0.64
-0.68
0.88
2.67
0.88
0.88
0.93
1.09
0.75
1.06
0.75
1.09
0.88
1.11
a
c
Ref. [80], b Ref. [94]
Ref. [94]. Calculated as a difference between the average M–H bond length and Pauling
ionic radius of the metal atom.
are not spherical, however there is no direct charge density flow between metal and
hydrogen, at least on the plane shown in Fig. 4.14(b). For hydrides of Ca, Sr, and
Ba the cross-section looks complex but this is due to the fact that it was made at
the position of hydrogen located in the lower right corner in Fig. 4.15. The charge
density of the other atoms shown is only partial cut of their orbitals, as none of these
atoms is located on the cross-section plane (they are below or above that plane). Also
for these compounds hydrogen 1s orbital is distorted, that indicate non-ionic effects
in metal-hydrogen interaction. The charge distribution on atomic orbitals of metal is
also changed upon application of GGA+U functional for hydrogen.
4.5.4
Optical properties
Calculated complex dielectric function for BeH2 and MgH2 are shown in Fig. 4.16.
For both compounds the application of the GGA+U functional results in the shift
of the first absorption peak due to the shift of the conduction band in a similar way
as for the alkali hydrides (see Section 4.4.4). For magnesium hydride the results are
86
10
15
Re ε
Re ε
10
5
0
0
10
Im ε
10
Im ε
5
5
0
0
1 2
3 4
5 6 7 8 9 10 11 12
Energy (eV)
5
0
0
1 2
3 4
(a)
5 6 7 8 9 10 11 12
Energy (eV)
(b)
Figure 4.16: Real and imaginary part of dielectric function of BeH2 (a) and MgH2 (b)
for GGA-PBE and GGA+U exchange-correlation functionals. The black solid lines are for
GGA-PBE calculations and dashed red lines are for GGA+U calculations.
comparable to those obtained by Moysés Araújo et alter [123]. These authors have
used the GW approximation for calculations of the value of the direct optical gap of
MgH2 . Than the dielectric function was calculated in GGA with scissor operation
to match the band gap form GW calculations. According to their calculations, the
absorption starts at the energy 6.6 eV and ends at 8.2 eV, while our calculations shows
the absorption range of 5.2-8.8 eV. The calculated imaginary part of the dielectric
function is more broad than in Ref. [123] and shows two main absorption peaks at
6.4 and 7.2 eV, while Moysés Araújo et alter [123] observe one main peak at 7.9 eV.
Calculated dielectric functions for CaH2 , SrH2 and BaH2 are shown on Fig. 4.17.
Im ε
8
6
4
2
0
0
Re ε
4
Im ε
0
Im ε
Re ε
Re ε
8
4
8
6
4
2
0
0
8
6
4
2
0
6
10
8
6
4
2
0
12
1 2
3 4
5 6 7 8 9 10 11 12
Energy (eV)
CaH2
2
1 2
3 4
5 6 7 8 9 10 11 12
Energy (eV)
SrH2
0
0
1 2
3 4
5 6 7 8 9 10 11 12
Energy (eV)
BaH2
Figure 4.17: Real and imaginary part of dielectric function of CaH2 , SrH2 and BaH2 for
GGA-PBE and GGA+U exchange-correlation functionals. The black solid lines are for
GGA-PBE calculations and dashed red lines are for GGA+U calculations.
The absorption spectra are more broad than observed for BeH2 and MgH2 due to
more complex structure of the empty orbitals in these compounds. The shift of
4.6. Summary
87
the main absorption peak due to the band gap correction is also observed for these
compounds. They have the similar crystal structure and have the same absorption
edge of approximately 4 eV. CaH2 has one main absorption peak at 5.6 eV and one
smaller peak at 4.5 eV. For SrH2 the absorption spectrum is more localised and two
main peaks at 5 eV and 5.4 eV are observed. For BaH2 there is one main peak at
4.8 eV.
4.6
Summary
Structural parameters, formation energies, electronic and optical properties were calculated for alkali and alkali earth metal hydrides. While for alkali metal hydrides
the present calculations agree well with existing data, for all alkali earth hydrides
the electronic and structural parameters were determined for the first time. Our
approach is focused on electronic properties related to optical reflectivity or to the
nature of the band gap in general. While accurate calculations of the band gap require computationally extensive approach as GW method [127] or Time-Dependent
DFT [154], we propose here new, more efficient, method inspired by the Hubbard
interaction Hamiltonian and denote this method as GGA+U. Since this approach is
of the computational cost of standard GGA calculations it can be efficiently applied
for extended systems (with up to hundred of atoms) where GW method is practically
unavailable.
Approach proposed here is based on application of the Hubbard correction to unoccupied p orbitals (LUMO) of hydrogen: projector operators are equal 1 for lowest
unoccupied orbitals for hydrogen, and 0 for 1s single particle states. Such formulation
leaves calculated ground state energies of hydrogen unchanged. However, electronic
and energetic properties of hydrogen embedded in alkali or alkaline earth hydrides
change substantially, especially the band gaps are enlarged. For exact formulation of
DFT the total energy shall be set of straight lines connecting energies with discontinuity of the first derivative of energy with respect to density at integer occupancies. For
local (multiplicative) exchange correlation functionals partial occupancies or mixed
states are favoured and they are they yield a line below these straight lines. We have
shown that forcing LUMO states to move toward higher energies can cure the band
gap problem partially, while the application of exact exchange corrected functional
fail for MgH2 , and performance of the energy dependent functionals remain to be
studied.
88
Chapter 5
Summary and Outlook
Density Functional Theory has proved to be fast and efficient method of calculating
the various properties of solid insulating materials. The results of the calculations
presented in this thesis illustrate scope of application of this method for elastic, structural, and electronic properties of such materials as ZrO2 , LiH, MgH2 or complex
hydrides.
In particular the model of superplasticity in doped ZrO2 ceramic was proposed.
This model is based on DFT calculations of the elastic properties of zirconia. Additionally, the relation between changes in the charge distribution of cation-anion bonds
with experimentally determined grain boundary diffusion coefficient was proposed.
These calculations show that it is possible to obtain direct link between materials
properties calculated on the atomic scale and those measured on the macroscopic
level. As the present approach is a first step toward understanding of the superplasticity on the microscopic level further development of our model toward realistic
dopant concentration distribution, surface modification or direct insight into diffusion
processes is possible.
The application of DFT calculations for crystal structure prediction is presented
for complex hydrides, that are materials for hydrogen storage. Coordination screening method applied to more than 700 candidate structures and stoichiometries of
complex hydrides has delivered ten promising compounds with favourable thermodynamic properties. Many of these structures remain to be synthesized and tested
experimentally as well further theoretical analysis with respect to dynamical stability
is possible.
Calculation of the structural and electronic properties for alkali metal hydrides
(LiH, NaH, KH, RbH, CsH) and alkali earth hydrides (BeH2 , MgH2 , CaH2 , SrH2 ,
BaH2 ) reveal the problem with correct description of the band gap (optical properties) for these compounds within PBE formulation of generalised gradient approximation. We propose a new method of description of the electron correlations in these
90
5. Summary and Outlook
compounds inspired by Hubbard U model, where we apply the GGA+U correction
in fully-localised limit for the empty p states of hydrogen. Our method significantly
improves the description of the electronic structure and optical properties in Density
Functional Theory without using time-consuming approximations like GW [127] or
TDDFT [154], which allows each applications for more complex systems with up to
hundreds of atoms.
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Index
alkali hydrides, see hydrides
alkaline earth hydrides, see hydrides
atomic charge
calculation of, 26
band structure, 27
battery, 50
Born-Oppenheimer approximation, 8
bulk modulus, 23
Murnaghan equation fit, 23
capacitor, 49
ceramics
application of, 33
solid oxide fuel cells, 34, 35
superplastic properties, 37
charge
atomic, 26
coordination screening
method, 57
Coulomb
interaction, 10
repulsion, 8
database search method, 56
decomposition energy, 59
density
electron, 8, 12, 76
Density Functional Theory, 7
exact exchange functional, 14
exchange-correlation functionals, 10
Fundamentals of, 7
GGA, 12
GGA+U, 14
Hohenberg-Kohn theorem, 8
hybrid functional, 14
implementations, 16
Kohn-Sham equations, 9
LDA, 11
LSDA, 11
orbital free kinetic energy DFT, 8
self-interaction correction, 14
density of states, 29
dielectric function, 30
elastic constants, 23
electron
density, 8, 11
gas, 8
self-interaction, 10, 14
spin, 11
energy
formation, 70
linearisation, 20
of decomposition, 59
energy storage, 49
batteries, 50
capacitors, 49
complex hydrides, 52
hydrogen, 50
load-levelling, 50
metal hydrides, 52
exact exchange functional, 14
formation energy, 70
Generalised Gradient Approximation, 12
Perdew-Burke-Ernzerhof (PBE), 12
genetic algorithm, 56
GGA+U, 14
around mean field (AMF), 15
fully-localised limit, 16
Hohenberg-Kohn theorem, 8
hybrid functional, 14
hydrides
alkali
atomic charges, 73
crystal structure, 70
electron density, 76
electronic structure, 72
optical properties, 77
properties, 64
alkaline earth
atomic charges, 83
crystal structure, 78
electron density, 83
electronic structure, 80
optical properties, 85
properties, 64
complex, 52
energy storage, 52
properties of, 63
switchable mirrors, 65
hydrogen
energy storage, 50
properties, 51
storage methods, 51
ICSD, 56
Jacob’s ladder, 10
Kohn-Sham equations, 9
Koopmans’ theorem, 13
LAPW, 19
LAPW+lo method, 21
Local Density Approximation, 11
Local Spin Density Approximation, 11
metal hydrides, see hydrides
Mulliken population analysis, 26
Murnaghan equation, 23
OFDFT, 8
PEGS, 57
potential
Coulomb, 9
exchange-correlation, 9, 10, 13, 19
external, 8, 9
Hartree, 9, 19
Lennard-Jones, 52
semilocal, 19
universal, 8
pseudopotential, 18
radius
muffin-tin (RMT), 19, 20
Wigner-Seitz, 11
self-interaction error, 10
Shrödinger equation, 7
Born-Oppenheimer approximation, 8
SIESTA, 17
basis set, 18
pseudopotentials, 18
simulated annealing, 55
solid oxide fuel cells, 34, 35
superplasticity
definition of, 36
in ceramics, 37
in ZrO2 , 40, 42, 44
mechanism, 38
switchable mirrors, 65
Van ’t Hoff equation, 52
wave-function
many-electron, 8
Wien2k, 19
zirconium dioxide
elastic properties of, 44
physical properties of, 39
superplastic properties of, 40, 42
List of publications
Paper I
Y. Natanzon, M. Boniecki, Z. Lodziana. Influence of elastic properties on superplasticity in doped yttria-stabilized zirconia. Journal of Physics and Chemistry of Solids.
70:15 (2009).
In this paper we present the results of quantum-mechanical calculations of elastic
properties of tetragonal phase of Y2 O3 -stabilised zirconium dioxide doped with various
metal oxides such as GeO2 , TiO2 and GeO2 . The author of this thesis has done all
the quantum-mechanical calculations as well as participated in the creation of the
simple model which explains the enhancement of the superplasticity in fine-grained
zirconia ceramics.
Paper II
M. Boniecki, Y. Natanzon, Z. Lodziana. Effect of cation doping on lattice and grain
boundary diffusion in superplastic yttria-stabilised tetragonal zirconia. Journal of
Journal of the European Ceramic Society 30:657 (2010).
This paper is the continuation of the Paper I and contains the results of experimental measurements of the grain boundary diffusion coefficients performed by
Dr. Boniecki which are supported by the results of quantum mechanical calculations
of ionic charges of cations and anions in doped ZrO2 , performed by Y. Natanzon.
The results show the correlation between the increase of the grain boundary diffusion
coefficient and the decrease of the ionic charges in the region close to the dopant
atom. This gives additional insight on the phenomenon of superplasticity in zirconia
ceramics and supports the ideas expressed in Paper I.
Paper III
J. S. Hummelshøj, D. D. Landis, J. Voss, T. Jiang, A. Tekin, N. Bork, M. Dulak,
J. J. Mortensen, L. Adamska, J. Andersin, J. D. Baran, G. D. Barmparis, F. Bell,
A. L. Bezanilla, J. Bjork, M. E. Björketun, F. Bleken, F. Buchter, M. Bürkle, P.
D. Burton, B. B. Buus, A. Calborean, F. Calle-Vallejo, S. Casolo, B. D. Chandler,
D. H. Chi, I Czekaj, S. Datta, A. Datye, A. DeLaRiva, V Despoja, S. Dobrin, M.
Engelund, L. Ferrighi, P. Frondelius, Q. Fu, A. Fuentes, J. Frst, A. Garca-Fuente,
J. Gavnholt, R. Goeke, S. Gudmundsdottir, K. D. Hammond, H. A. Hansen, D.
Hibbitts, E. Hobi, Jr., J. G. Howalt, S. L. Hruby, A. Huth, L. Isaeva, J. Jelic, I. J.
T. Jensen, K. A. Kacprzak, A. Kelkkanen, D. Kelsey, D. S. Kesanakurthi, J. Kleis,
P. J. Klüpfel, I. Konstantinov, R. Korytar, P. Koskinen, C. Krishna, E. Kunkes, A.
H. Larsen, J. M. G. Lastra, H. Lin, O. Lopez-Acevedo, M. Mantega, J. I. Martı́nez,
I. N. Mesa, D. J. Mowbray, J. S. G. Mýrdal, Y. Natanzon, A. Nistor, T. Olsen, H.
Park, L. S. Pedroza, V. Petzold, C. Plaisance, J. A. Rasmussen, H. Ren, M. Rizzi, A.
S. Ronco, C. Rostgaard, S. Saadi, L. A. Salguero, E. J. G. Santos, A. L. Schoenhalz,
J. Shen, M. Smedemand, O. J. Stausholm-Møller, M. Stibius, M. Strange, H. B. Su,
B. Temel, A. Toftelund, V. Tripkovic, M. Vanin, V. Viswanathan, A. Vojvodic, S.
Wang, J. Wellendorff, K. S. Thygesen, J. Rossmeisl, T. Bligaard, K. W. Jacobsen,
J. K. Nørskov, and T. Vegge. Density functional theory based screening of ternary
alkali-transition metal borohydrides: A computational material design project. The
Journal of Chemical Physics 131:014101 (2009).
In this paper the density functional theory based screening method is for the prediction of the stable crystal structure of ternary alkali-transition metal borohydrides.
Here a group of more than 100 scientists have performed the geometric optimisation
of more than 720 structures of various complex borohydrides with a general formula
of M1 M2 (BH4 )2 with M1 , M2 =Li, Na, K and M1 M2 (BH4 )n , where n = 3, 4, M1 =Li,
Na, K and M2 =Li, Na, K, Mg, Al, Ca, Sc-Zn, Y-Mo,Ru-Cd. The author of this
thesis was responsible for finding the most stable configuration of NaCo(BH4 )3,4 and
NaRh(BH4 )3,4 . For this structures, the tetrahedral configuration of the metal atom
with respect to the borohydride group was found to be be the most stable.
Other papers
List of papers which were not included in this thesis:
1. Y. Natanzon, Z. Lodziana. Quantum mechanical calculations of elastic properties of doped tetragonal yttria-stabilized zirconium dioxide. Computer Science:
Annual of AGH University of Science and Technology 9:87 (2008).
2. R. Bujakiewicz-Korońska, Y. Natanzon. Determination of elastic constants of
Na0.5 Bi0.5 TiO3 from ab initio calculations. Phase Transitions 81:1117 (2008).
3. R. Bujakiewicz-Korońska, Y. Natanzon. First principles calculations of elastic
constants for defected Na1/2 Bi1/2 TiO3 . Integrated Ferroelectrics 108:21 (2009).
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THE JOURNAL OF CHEMICAL PHYSICS 131, 014101 共2009兲
Density functional theory based screening of ternary alkali-transition metal
borohydrides: A computational material design project
J. S. Hummelshøj, D. D. Landis, J. Voss, T. Jiang, A. Tekin, N. Bork, M. Dułak,
J. J. Mortensen, L. Adamska, J. Andersin, J. D. Baran, G. D. Barmparis, F. Bell,
A. L. Bezanilla, J. Bjork, M. E. Björketun, F. Bleken, F. Buchter, M. Bürkle, P. D. Burton,
B. B. Buus, A. Calborean, F. Calle-Vallejo, S. Casolo, B. D. Chandler, D. H. Chi,
I Czekaj, S. Datta, A. Datye, A. DeLaRiva, V Despoja, S. Dobrin, M. Engelund, L. Ferrighi,
P. Frondelius, Q. Fu, A. Fuentes, J. Fürst, A. García-Fuente, J. Gavnholt, R. Goeke,
S. Gudmundsdottir, K. D. Hammond, H. A. Hansen, D. Hibbitts, E. Hobi, Jr., J. G. Howalt,
S. L. Hruby, A. Huth, L. Isaeva, J. Jelic, I. J. T. Jensen, K. A. Kacprzak, A. Kelkkanen,
D. Kelsey, D. S. Kesanakurthi, J. Kleis, P. J. Klüpfel, I Konstantinov, R. Korytar,
P. Koskinen, C. Krishna, E. Kunkes, A. H. Larsen, J. M. G. Lastra, H. Lin,
O. Lopez-Acevedo, M. Mantega, J. I. Martínez, I. N. Mesa, D. J. Mowbray, J. S. G. Mýrdal,
Y. Natanzon, A. Nistor, T. Olsen, H. Park, L. S. Pedroza, V Petzold, C. Plaisance,
J. A. Rasmussen, H. Ren, M. Rizzi, A. S. Ronco, C. Rostgaard, S. Saadi, L. A. Salguero,
E. J. G. Santos, A. L. Schoenhalz, J. Shen, M. Smedemand, O. J. Stausholm-Møller,
M. Stibius, M. Strange, H. B. Su, B. Temel, A. Toftelund, V Tripkovic, M. Vanin,
V Viswanathan, A. Vojvodic, S. Wang, J. Wellendorff, K. S. Thygesen, J. Rossmeisl,
T. Bligaard, K. W. Jacobsen, J. K. Nørskov, and T. Veggea兲
The 2008 CAMD Summer School in Electronic Structure Theory and Materials Design, Center
for Atomic-scale Materials Design, Department of Physics, Technical University of Denmark,
DK-2800 Kgs. Lyngby, Denmarkb兲
共Received 20 November 2008; accepted 8 May 2009; published online 1 July 2009兲
We present a computational screening study of ternary metal borohydrides for reversible hydrogen
storage based on density functional theory. We investigate the stability and decomposition of alloys
containing 1 alkali metal atom, Li, Na, or K 共M 1兲; and 1 alkali, alkaline earth or 3d / 4d transition
metal atom 共M 2兲 plus two to five 共BH4兲− groups, i.e., M 1M 2共BH4兲2–5, using a number of model
structures with trigonal, tetrahedral, octahedral, and free coordination of the metal borohydride
complexes. Of the over 700 investigated structures, about 20 were predicted to form potentially
stable alloys with promising decomposition energies. The M 1共Al/ Mn/ Fe兲共BH4兲4,
共Li/ Na兲Zn共BH4兲3, and 共Na/ K兲共Ni/ Co兲共BH4兲3 alloys are found to be the most promising, followed
by selected M 1共Nb/ Rh兲共BH4兲4 alloys. © 2009 American Institute of Physics.
关DOI: 10.1063/1.3148892兴
I. INTRODUCTION
The development of sustainable energy solutions for the
future requires new and improved materials. Specifically designed material properties are needed to solve the grand challenges in energy production, storage, and conversion. Within
energy storage, hydrogen has been investigated extensively
over the past decade1 as one of the few promising energy
carriers which can provide a high energy density without
resulting in CO2 emission by the end user. Finding materials
for efficient, reversible hydrogen storage, however, remains
challenging. Here, the specific requirements of the rapidly
growing transportation sector coupled with complex engineering challenges2 have directed research toward complex
materials with extreme hydrogen storage capacities3 such as
metal borohydrides4 and metal ammines.5 Finding materials
with high reversible hydrogen content and optimal thermodynamic stability is essential if hydrogen is going to be used
a兲
Electronic mail: [email protected].
For a full list of affiliations, see Ref. 41.
b兲
0021-9606/2009/131共1兲/014101/9/$25.00
as a commercial fuel in the transport sector. The binary metal
borohydrides have been studied extensively: the alkali based
compounds, e.g., LiBH4,6–8 are too thermodynamically
stable, the alkaline earth compounds are kinetically too slow
and practically irreversible,9 and the transition metal borohydrides are either unstable or irreversible.10 This leaves hope
that mixed metal 共“alloyed”兲 systems might provide new
opportunities.
The use of computational screening techniques has
proved a valuable tool in narrowing the phase space of potential candidate materials for hydrogen storage.11,12 Recent
density functional theory 共DFT兲 calculations have shown that
the thermodynamic properties of even highly complex borohydride superstructures can be estimated by DFT using
simple model structures, if the primary coordination polyhedra are correctly accounted for.13 These findings enable faster
screening studies of thermodynamic stability and decomposition temperatures for, e.g., ternary and quaternary borohydride systems; not only in terms of reduced computational
effort due to smaller system sizes but also with the advantage
that the exact space group does not need to be known
131, 014101-1
© 2009 American Institute of Physics
Downloaded 06 Jul 2009 to 149.156.47.33. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp
014101-2
Hummelshøj et al.
a priori.
In the present paper, we apply a “local coordination
screening” 共LCS兲 approach to search for novel metal borohydrides. The vast majority of the calculations were performed as part of the 2008 CAMD summer school in electronic structure theory and materials design, where more than
100 scientists combined DFT calculations, database methods,
and screening techniques to investigate the structure and stability of promising ternary borohydrides. A few additional
calculations were subsequently performed based on the insight gained from the initial screening.
Out of 757 investigated M 1M 2共BH4兲2–5 共M 1 = alkali
metal and M 2 = alkali, alkaline earth or 3d / 4d transition
metal兲 compositions and structures, a total of 22 were found
to form potentially stable alloys with promising decomposition energies, which should subsequently be subjected to
more detailed theoretical and experimental verification.
II. COMPUTATIONAL SETUP
Groups of alloy compositions and structures were divided among different groups of scientists, each of which
was responsible for its own subset of the alloy configuration
space. A number of predefined structural templates and optimization procedures had been prepared to assist the groups
in setting up structures and calculations for the initial optimization 共see Sec. II B兲. This was done to ensure a sufficient
accuracy in all calculations 共i.e., convergence with respect to
plane wave cutoff, k-point sampling, etc.兲.
To ensure reliability of the generated results, an automated checking procedure was enforced before a result could
be included in the database 共see Sec. III兲 to ensure the presence of the required output 共total energies, lattice constants,
etc.兲.
A. Computational parameters
The total energies and gradients were calculated within
density functional theory14 as implemented by the software
package Dacapo.15 A plane wave basis set with a cutoff energy of 350 eV 共density grid cutoff of 700 eV兲 and the RPBE
exchange-correlation functional15 were used for all calculations. Dacapo uses ultrasoft pseudopotentials16 for a description of the ionic cores. The coordinate optimization was
implemented and performed within the atomic simulation
environment.17 The electronic Brillouin zones were sampled
with 共4 ⫻ 4 ⫻ 4兲 k-points 共spacings of ⬃0.05 Å−1兲. A quasiNewton method18 was used for all relaxations.
B. Configuration space and template structures
The alloys which were initially screened have the general formula M 1M 2共BH4兲x, where M 1 苸兵Li, Na, K其 and
x = 2 – 4. The x = 2 alloys were investigated for
M 2 苸兵Li, Na, K其, and x = 3 , 4 for M 2 苸兵Li, Na, K , Mg,
Al, Ca, Sc– Zn, Y – Mo, Ru– Cd其.
In order to limit the total number of calculations, only
template structures with tetrahedral and octahedral coordination of the 共BH4兲− groups to the metal atoms were used.
Most metals prefer an octahedral coordination of their
ligands, but for the metal borohydrides the ligand-ligand re-
J. Chem. Phys. 131, 014101 共2009兲
pulsion between the relatively large 共BH4兲− ions often forces
a lower coordination number. The primary structures observed and reported in literature for the alkali and alkaline
earth borohydrides are either tetrahedral 共for the smallest Li
and Mg兲 or octahedral 共for the larger Na, K, and Ca兲, while
a trigonal planar ligand arrangement is observed for
Al共BH4兲3. However, Al can also have a tetrahedral coordination as is the case of the LiAl共BH4兲4 alloy obtained here 共see
Sec. V兲, and since the radii of the considered ions lie between the radius for K and the radius for Al, the tetrahedral
and octahedral primary structures are expected to be
representative.
For each alloy composition, four different template
structures were used to sample the tetrahedral and octahedral
primary structures in the combinations: tetrahedral/
tetrahedral, octahedral/octahedral, tetrahedral/octahedral, and
octahedral/tetrahedral, referring to the coordination of the
共BH4兲− groups to the M 1 and M 2 atoms, respectively. The
coordination polyhedra were either corner sharing, edge
sharing, or a combination to yield the required stoichiometric
ratio of 共BH4兲− groups 共see Fig. 1兲. All structures were designed to have a unit cell containing only one formula unit
共see Sec. II D兲. It has previously been shown that these
simple template structures can be within ⬃0.1 eV 共10 kJ/
mol H2兲 of the true ground state energy if the local coordination is correctly accounted for; e.g., M 1M 2共BH4兲2-tetra for
LiBH4,7 M 1M 2共BH4兲4-octa for Ca共BH4兲2,19 and even
M 1M 2共BH4兲4-tetra for the free energy of Mg共BH4兲2
superstructures.13
The initial optimization of the structures only relaxed the
hydrogen positions and the unit cell volume while keeping
the metal-boron coordination polyhedra fixed. For a given set
of 共M 1 , M 2兲, the most stable structure was then used as the
starting point for a calculation in which all atomic positions
and the unit cell were relaxed. Even though many of the
structures did not change significantly during the final relaxation, it added, in principle, an additional structure to the
phase space for each set of 共M 1 , M 2兲. These are included as
“other” structures in the results 共Figs. 3–12兲 to distinguish
them from the structures with fixed metal-boron coordination
polyhedra, even though the original coordination polyhedra
are only slightly distorted in many of them.
A number of structures were subsequently added based
on the knowledge gained from the initial screening and the
reference binary borohydride structures 共see Secs. IV and
VI兲. In some of these structures, the metal ions had the same
valence as in the reference structures, which meant that
the four x = 2 templates were also applied to
M 2 苸兵Ni, Pd, Cu, Ag其, while a new template for x = 5 was
investigated for M 2 苸兵Ti, Zr其 in the two combinations
tetrahedral/octahedral and octahedral/tetrahedral. An alternative x = 3 tetragonal/trigonal template was applied to
M 2 苸兵Mg, Al, Ca, Sc– Zn, Y – Mo, Ru– Cd其 to investigate
possible size effects. In this structure, the M 1 ion has a tetrahedral coordination while the M 2 atom is surrounded by
three 共BH4兲− groups in a trigonal planar arrangement 共see
Fig. 2兲. This enabled the metal-boron distances for the two
metals to be optimized independently, which was not pos-
014101-3
DFT screening of ternary metal borohydrides
M1 M2 (BH4 )2 -tetra
M1 M2 (BH4 )2 -octa
M1 M2 (BH4 )2 -tetra/octa
M1 M2 (BH4 )3 -octa
M1 M2 (BH4 )4 -octa
M1 M2 (BH4 )3 -tetra/tri
J. Chem. Phys. 131, 014101 共2009兲
TABLE I. The calculated reference energies for the binary borohydrides in
their most stable template structures 共see Fig. 2兲.
K共BH4兲
Na共BH4兲
Li共BH4兲
Ag共BH4兲
Cu共BH4兲
Pd共BH4兲
Ni共BH4兲
Ca共BH4兲2
Mg共BH4兲2
Zn共BH4兲2
Cd共BH4兲2
V共BH4兲2
Nb共BH4兲2
Fe共BH4兲2
Cr共BH4兲2
Mn共BH4兲2
Co共BH4兲2
Mo共BH4兲2
Rh共BH4兲2
Ru共BH4兲2
Y共BH4兲3
Sc共BH4兲3
Al共BH4兲3
Zr共BH4兲4
Ti共BH4兲4
wt %
共kg H2 / kg material兲
⌬Edecomp
共eV/ H2兲
7.5
10.7
18.5
3.3
5.1
3.3
5.5
11.6
14.9
8.5
5.7
10.0
6.6
9.4
9.9
9.5
9.1
6.4
6.1
6.2
9.1
13.5
16.9
10.7
15.0
⫺0.968
⫺0.729
⫺0.422
0.278
0.352
0.661
0.680
⫺0.636
⫺0.467
⫺0.063
⫺0.043
⫺0.031
0.066
0.090
0.162
0.174
0.264
0.280
0.340
0.351
⫺0.676
⫺0.595
⫺0.209
⫺0.429
⫺0.252
FIG. 1. The template structures of M 1M 2共BH4兲2–5. Red and yellow polyhedra show the coordination of the B atoms around the M 1 and M 2 atoms,
respectively; blue tetrahedra represent the 共BH4兲− groups. The octa/tetra
structures are obtained by switching M 1 and M 2 in the tetra/octa structures.
sible in the original x = 3 templates, but found to be required
to obtain the preferred local coordination of certain alloys.
In total, 757 structures have been simulated and are reported herein.
C. Group calculations
The 69 sets of 共M 1 , M 2兲 combinations investigated in
this study were divided among 32 groups of scientists for the
initial screening. Each group followed step I of the calculational procedure outlined below for each alloy containing M 1
and M 2 and step II for the most stable resulting structure.
D. Calculational procedure
1. Step I
An initial structure was set up by calling a function that
populates one of the four template structures with two supplied metal ions, e.g., Li and Sc. The function utilizes the
ionic radii obtained from the calculations of binary reference
borohydrides, i.e., individual metal atom borohydrides, to
calculate metal-boron distances, where the ionic radius used
for a 共BH4兲− group depends on whether a face, edge or corner of the H-tetrahedron points toward the metal atom. In
general, this ensured that the effective lattice constant and
the c / a ratio were close to the optimum. The initial structure
was used as the initial guess for the first iteration of the
following procedure.
All hydrogen positions were relaxed until the maximum
force on the atoms reached 0.05 eV/Å or, alternatively, a
maximum of 50 quasi-Newton steps had been performed.
The resulting structure was then contracted and expanded to
90%, 95%, 105%, and 110% of the unit cell volume by a
proportional scaling of the unit cell, while keeping the B–H
distances in each 共BH4兲− group fixed; a single total energy
calculation was performed for each volume. A Murnaghan
equation-of-state was fitted to the calculated five points to
estimate the optimal unit cell volume, to which the unit cell
was then scaled 共again while conserving B–H distances兲, followed by a relaxation of the hydrogen positions to a force
convergence of 0.05 eV/Å.
After each iteration, an energy versus unit cell volume
plot was inspected visually to decide whether the minimum
had been sufficiently sampled or an additional iteration of the
procedure should be performed; in the latter case, a structure
resulting from the first iteration was used as the starting
guess for the next iteration.
014101-4
Hummelshøj et al.
J. Chem. Phys. 131, 014101 共2009兲
Cr,Mo
Mn
Fe,Ru,Co
Rh
Li,Ni,Pd,Cu,Ag
Na,K
Mg,Ca,V,Nb,Zn,Cd
Al
Sc,Y
Ti,Zr
FIG. 2. The structures used for calculating the binary reference energies. For Cr, Mo, Mn, Fe, Ru, Co, Rh, Li, Ni, Pd, Cu, and Ag, the polyhedra show the
coordination of the H atoms; the coordination of the 共BH4兲− groups are tetrahedral in these structures. For the remaining metals the coordination polyhedra
show the coordination of the 共BH4兲− groups.
2. Step II
B. Back end
When all template structures for each of the 共M 1 , M 2兲
alloys had been optimized in step I, the most stable structure
was relaxed without constraints by repeating the procedure
that first relaxes all atomic positions for a fixed cell and then
the unit cell for fixed internal positions. To limit the computational time used by this algorithm, the number of iterations
was limited to 5, and the number of steps per iteration was
limited to 12 for the internal relaxation and 5 for the unit cell
relaxation.
A second Python script was used to extract the relevant
parameters, i.e., the total energy, unit cell volume, chemical
symbols, structure, and the calculational parameters such as
k-points, number of bands, density wave cutoff, and to select
the best structure 共at any given time兲 for every borohydride
to create/update the intermediate result plots, which were
accessible to all participants. Python, in combination with
Matplotlib,22 was used to ensure a flexible user interface and
to generate the plots. A special Python class managed the
resulting data, consisting of approximately 5500 calculations. This class provided basic database operations such as
selecting, sorting, and filtering of data and facilitated the
creation of the plots considerably.
The overall construction of the database and data retrieval procedures will also facilitate screening for possible
correlations between combinations of a number of different
values in future projects.
3. Procedure for the additional structures
For the structures calculated later, the x = 2 structures
共monovalent transition metals兲 followed the same procedure
mentioned above, whereas only a single free optimization
was performed on the extra x = 3 and x = 5 structures, in
which all atoms were allowed to relax.
III. DATA COLLECTION AND STORAGE
Every group executed the calculation procedures for
steps I and II. After each step, the validity of the results was
checked by the group and the results were checked in 共stored
in a global location for indexing兲 to the common database.
A. Front end
A Python20 script took care of checking in all relevant
files that were needed for subsequent checking. This included the calculation script and the output files containing
the atoms, energies and the calculational parameters. A subversion 共svn兲 version control system21 assisted to manage
groups and users, storing results and assuring transaction
consistency.
IV. DATA ANALYSIS
The initial screening procedure presented here is performed to reduce the number of potential alloys for further
investigation, and two simple selection criteria were set up to
assess the stability of the investigated alloy structures against
phase separation/disproportionation and decomposition. The
stabilities were first analyzed against phase separation into
the original binary borohydrides as illustrated for
LiSc共BH4兲4:
⌬Ealloy = ELiSc共BH4兲4 − 共ELiBH4 + ESc共BH4兲3兲.
共1兲
Reference energies for the 3 alkali, 2 alkaline earth,
Al共BH4兲3 plus 19 transition metal borohydrides were ob-
014101-5
TABLE II. Structures with alloying energies ⌬Ealloy ⬍ 0.0 eV/ f.u. 共formula
unit兲 and decomposition energies ⌬Edecomp ⬍ 0.0 eV/ H2.
LiNa共BH4兲2
KZn共BH4兲3
KAl共BH4兲4
NaAl共BH4兲4
KCd共BH4兲3
NaZn共BH4兲3
LiAl共BH4兲4
KFe共BH4兲3
LiZn共BH4兲3
NaFe共BH4兲3
KMn共BH4兲4
NaNb共BH4兲4
KCo共BH4兲3
NaMn共BH4兲4
KNi共BH4兲3
LiFe共BH4兲3
LiNb共BH4兲4
NaCo共BH4兲3
KRh共BH4兲4
LiMn共BH4兲4
NaNi共BH4兲3
NaRh共BH4兲4
wt %
⌬Ealloy
共eV/f.u.兲
⌬Edecomp
共eV/ H2兲
13.5
8.1
12.9
14.7
6.2
9.1
17.3
8.7
10.4
9.8
10.5
9.2
8.5
11.7
8.5
11.3
10.1
9.6
8.0
13.3
9.6
8.7
⫺0.020
⫺0.349
⫺0.138
⫺0.279
⫺0.005
⫺0.358
⫺0.391
⫺0.116
⫺0.362
⫺0.141
⫺0.148
⫺0.128
⫺0.089
⫺0.284
⫺0.120
⫺0.141
⫺0.194
⫺0.143
⫺0.058
⫺0.358
⫺0.164
⫺0.033
⫺0.581
⫺0.423
⫺0.416
⫺0.373
⫺0.352
⫺0.344
⫺0.311
⫺0.282
⫺0.243
⫺0.206
⫺0.174
⫺0.165
⫺0.161
⫺0.131
⫺0.116
⫺0.104
⫺0.097
⫺0.090
⫺0.079
⫺0.063
⫺0.043
⫺0.016
J. Chem. Phys. 131, 014101 共2009兲
TABLE III. Structures with alloying energies 0 ⬍ ⌬Ealloy ⬍ 0.2 eV/ f.u. 共formula unit兲 with decomposition energies ⌬Edecomp ⬍ 0.0 eV/ H2.
KNa共BH4兲2
NaY共BH4兲4
NaCa共BH4兲3
LiY共BH4兲4
LiCa共BH4兲3
LiSc共BH4兲4
NaCd共BH4兲3
KNb共BH4兲4
NaV共BH4兲4
NaAg共BH4兲2
LiCd共BH4兲3
KCr共BH4兲4
LiV共BH4兲4
NaCr共BH4兲4
KPd共BH4兲3
KMo共BH4兲4
KRu共BH4兲3
NaMo共BH4兲4
LiCr共BH4兲4
NaPd共BH4兲3
wt %
⌬Ealloy
共eV/f.u.兲
⌬Edecomp
共eV/ H2兲
8.8
9.4
11.2
10.4
13.2
14.5
6.7
8.4
12.1
5.0
7.4
10.7
13.8
12.0
6.4
8.3
6.5
9.0
13.6
7.0
0.095
0.115
0.129
0.033
0.052
0.143
0.003
0.016
0.076
0.193
0.102
0.199
0.061
0.050
0.047
0.185
0.168
0.056
0.029
0.052
⫺0.825
⫺0.675
⫺0.645
⫺0.609
⫺0.556
⫺0.534
⫺0.271
⫺0.207
⫺0.188
⫺0.177
⫺0.152
⫺0.136
⫺0.113
⫺0.095
⫺0.095
⫺0.079
⫺0.061
⫺0.035
⫺0.021
⫺0.014
⌬Edecomp = ELiMn共BH4兲3 − 共ELiH + EMn + 3EB + 5.5EH2兲.
tained using the most stable structures among the applied
M 2共BH4兲1–4 model templates 共see Table I兲. Due to computational constraints, the performed calculations are not spin
polarized, which causes certain reference structures, e.g.,
Mn共BH4兲2, to become unstable. In order not to exclude potentially stable candidates, the assessment in Eq. 共1兲 was
used for all reference structures 共see Table I兲.
For assessing the stability of alloys with a potentially
less favorable stoichiometry, like LiSc共BH4兲3, an effective
reference value for ESc共BH4兲2 was determined from the stable
as
ESc共BH4兲2 ⴱ = ESc共BH4兲3 − 2EH2 − EB.
Using
ESc共BH4兲3
1 / 2共B2H6 + H2兲 as a reference only shifts the energy by
0.07 eV/H2 and does not result in a new preferred coordination for any of the stable alloys.
The decomposition pathways of binary and ternary metal
borohydrides are often highly complex and differ significantly from one system to the next, e.g., LiBH4,23
Mg共BH4兲2,24 and LiZn共BH4兲3,25 and the formed products can
even depend on the details of the desorption conditions. Certain compounds form transition metal hydrides,26 others form
transition metal borides,27 di-,10 or dodeca-boranes,28 and
others again, e.g., Cr, Cd, Mn, and Zn共BH4兲2 decompose to
the elements.29,30 Given the inclusive nature of this initial
screening study and the fact that the true decomposition
pathways in most of the investigated alloys are not well
known, a simple and generic decomposition pathway was
selected, which all interesting mixed borohydrides must be
stable against 共as a minimum兲. Here, the alloys decompose
into the highly stable alkali- and alkaline earth hydrides,
transition metals, boron and H2, e.g.:
共2兲
In this definition, ⌬Edecomp estimates the stability of the alloy
against decomposition. Transition metal hydrides, metal
borides, higher order boranates and diborane, which may potentially form, are thus not taken into consideration in this
first screening.
The analysis is based on the ground state energies only.
Although the difference in vibrational entropy between hydrogen in an alkali metal borohydride and in the gas phase is
often significantly smaller than in conventional metal
hydrides,31 the contributions to the free energy from the vibrational entropy may be significant.
A stability range of ⌬Ealloy ⱕ 0.0 eV/ f.u. 共formula unit兲
and ⌬Edecomp 苸兵−0.5; 0.0其 eV/ H2 is used to select the most
interesting alloys with ⌬Edecomp = −0.2 eV/ H2 as the target
value 共see Table II兲, but given the idealized screening criteria
in Eqs. 共1兲 and 共2兲, alloys with only small instabilities, i.e.,
⌬Ealloy ⱕ 0.2 eV/ f.u and ⌬Edecomp ⱕ 0.0 eV/ H2 should not
be discarded a priori 共see Table III兲.
V. RESULTS
As the first step of the stability screening, we have plotted the alloying energy against the decomposition energy of
the 757 investigated alloys 共see Fig. 3兲. Most of the alloys
are found to be stable against decomposition, but the majority are found to be unstable against separation into their binary components 共⌬Ealloy ⬎ 0.0 eV/ f.u.兲. Many are still
within the 0.2 eV/f.u. boundary regime. The lithiumcontaining alloys 共red兲 are less stable against decomposition
than those containing sodium 共blue兲 and potassium 共green兲.
Restricting the plot to only the most stable structure for each
014101-6
Hummelshøj et al.
J. Chem. Phys. 131, 014101 共2009兲
100
LiNa(BH4 )2
8
LiZn(BH4 )3
*
@
LiFe(BH4 )3
90
6
*
4
70
NaNi(BH4 )3
NaCo(BH4 )3
LiAl(BH4 )4
NaZn(BH4 )3
LiNb(BH4 )4
&
NaMn(BH4 )4
NaAl(BH4 )4
∆ E
a llo y
( e V / f.u .)
!
LiMn(BH4 )4
&
NaFe(BH4 )3
80
NaRh(BH4 )4
NaNb(BH4 )4
60
2
KFe(BH4 )3
KZn(BH4 )3
KAl(BH4 )4
KCd(BH4 )3
50
KCo(BH4 )3
KNi(BH4 )3
KMn(BH4 )4
KRh(BH4 )4
0
-1 .0
-0 .5
0 .0
∆ E
d e c o m p
(e V / H
0 .5
)
2
1 .0
40
-0.6
1 .5
FIG. 3. The alloying energy, ⌬Ealloy, as a function of the decomposition
energy, ⌬Edecomp, for all alloy compositions. Colors: Li 共red兲, Na 共blue兲, and
K 共green兲. Investigated coordinations: tetra 共䊐兲, octa 共䊊兲, octa-tetra 共䉭兲,
tetra-octa 共+兲, tetra-tri 共丫兲, other 共䉰兲. Total number of structures: 757.
M 1M 2 system 共see Fig. 4兲 seems to support this observation,
and yields a total of 22 stable alloys 共see Table II兲. Figure 4
is dominated by alloys where 共a兲 both metal atoms are tetrahedrally coordinated to the borohydride groups 共䊐兲, 共b兲 one
is tetrahedral the other trigonal 共丫兲, and 共c兲 so-called other
共䉰兲, where all constraints have been lifted. Some octa-tetra
共䉭兲 and tetra-octa 共+兲 are also observed.
Plotting the hydrogen density of the stable alloys,
⌬Ealloy ⱕ 0.0 eV/ f.u. and ⌬Edecomp ⱕ 0.0 eV/ H2, Fig. 5
shows that alloys containing potassium 共in green兲 are found
to have the lowest density, followed by sodium 共in blue兲 and
lithium 共in red兲, as expected. The overall density is found to
be around that of liquid hydrogen, which is largely due to the
choice of simple template structures; higher densities are expected for real systems as previously observed for
Mg共BH4兲2.9 Alloys containing Al, Mn, Fe, and Zn are found
to be stable for all alkali metals screened, whereas those
-0.5
-0.4
-0.3
-0.2
-0.1
based on Co, Ni, Nb, and Rh are stable for two out of three
alkali metals. The only other stable alloys are KCd共BH4兲3
and LiNa共BH4兲2 共see Table II兲.
The storage capacity 共wt % hydrogen兲 of the stable alloys is plotted as a function of the decomposition energy,
⌬Edecomp, in Fig. 6. Here, the data from the binary reference
structures have also been included, and it is clearly seen that
the stability has been reduced significantly compared to
the highly stable binary borohydrides. Most alloys have storage capacities above the DOE 2015 system target of 9 wt %
共Ref. 3兲 and several also have favorable stabilities. A number
of these ternary borohydrides have been synthesized either
20
Li2B2
!
1 .0
LiAlB4
Al4B12
16
Mg2B4 NaAlB
4
0 .8
wt.% H
14
0 .6
( e V / f.u .)
0.1
FIG. 5. The hydrogen density 共kg H2 m−3 as a function of the decomposition energy for the 22 alloys with ⌬Ealloy ⱕ 0.0 eV/ f.u. and ⌬Edecomp
ⱕ 0.0 eV/ H2. References to experimental observations: !: Ref. 25, @: Ref.
37, &: Ref. 38, *: Ref. 39. Colors: Li 共red兲, Na 共blue兲, and K 共green兲.
Preferred local coordination: tetra 共䊐兲, octa 共䊊兲, octa-tetra 共䉭兲, tetra-octa
共+兲, tetra-tri 共丫兲, other 共䉰兲.
18
Sc2B6
LiNaB2
12
Y 2B6
8
0 .0
&
ZrB4
10
0 .2
K 2B2
4
-1.0
LiZnB3 KMnB
4
KCdB3
-0.8
-0.6
@
LiMnB4
NaMnB4
LiFeB3
LiNbB4 V B
*
2 4
NaZnB3 NaFeB3 NaCoB3
NaNiB3
NaNbB
KFeB
*
4
3
NaRhB4
KCoB3
KZnB3
Zn2B4
KNiB3
KRhB4
6
-0 .2
&
Ca2B4
Na2B2
0 .4
TiB4
KAlB4
∆ E
a llo y
0.0
-0.4
Cd2B4
-0.2
0.0
-0 .4
-0 .8
-0 .6
∆ E
-0 .4
d e c o m p
(e V / H
2
)
-0 .2
0 .0
0 .2
FIG. 4. The alloying energy, ⌬Ealloy, as a function of the decomposition
energy, ⌬Edecomp, for all preferred alloy systems. Colors: Li 共red兲, Na 共blue兲
and K 共green兲. Preferred local coordination: tetra 共䊐兲, octa 共䊊兲, octa-tetra
共䉭兲, tetra-octa 共+兲, tetra-tri 共丫兲, other 共䉰兲.
FIG. 6. The weight percent of hydrogen 共wt. %兲 as a function of the decomposition energy, ⌬Edecomp 关Eq. 共2兲兴, for all 22 stable alloys and 13 binary
reference structures 共⌬Edecomp ⱕ 0.0 eV/ H2 and ⌬Ealloy ⱕ 0.0 eV/ f.u.兲. References to experimental observations: !: Ref. 25, @: Ref. 37, &: Ref. 38, *:
Ref. 39. Colors: Li 共red兲, Na 共blue兲, K 共green兲, and reference structures
共black兲. Labels: “MBx” refers to “M共BH4兲x.” Preferred local coordination:
tetra 共䊐兲, octa 共䊊兲, octa-tetra 共䉭兲, tetra-octa 共+兲, tetra-tri 共丫兲, other 共䉰兲.
014101-7
J. Chem. Phys. 131, 014101 共2009兲
4 .0
5
II
III
II
( e V / f.u .)
II
II
II
II
II
II
II
II
II
II
II
II
∆ E
II
II
1 .0
-0 .5
II
II
II
II
II
III
III
III III
III
III
III
III
III
III
M g
A l
II
S c
T i
V
C r
II
III
II
III
II
3
II
II
II
2
III
II
III
III
III
II
II
II
II
II
II
III
II
III
III
III
0
III
III
III
III
III
III
II
III
III
II
III
III
III
II
II
III
II
1
II
III
III
III
II
II
II
II
II
II
III
III
III
III
III
C a
III
III
III
III
III
III
III
III
III
III
III
III
III
II
II
0 .0
II
III
III
II
II
II
0 .5
II
4
III
II
II
II
III
II
II
2 .0
a llo y
III
III
III
II
2 .5
1 .5
II
II
II
II
( e V / f.u .)
3 .0
III
III
a llo y
III
III
∆ E
3 .5
II
III
M n
F e
C o
N i
C u
-1
Z n
FIG. 7. The alloying energy, ⌬Ealloy, for the 3d-metals 共plus Mg, Al, and Ca兲
in their preferred M 1M 2共BH4兲x template structures with M 1, M 2, and B fixed
for both x = 3 and x = 4. Colors: Li 共red兲, Na 共blue兲, and K 共green兲. Preferred
local coordination: tetra 共䊐兲, octa 共䊊兲, octa-tetra 共䉭兲, tetra-octa 共+兲. The
labels indicate the oxidation state of M 2.
very recently or historically 共circled in Figs. 5 and 6兲. Of the
experimentally observed stable/metastable structures,
LiSc共BH4兲4,10 KNa共BH4兲2,32 and Li2Cd共BH4兲4 共Ref. 30兲
show a weak preference for phase separation, but are all
found to be potentially stable 共see Table III兲; only LiK共BH4兲2
共Ref. 33兲共⌬Ealloy = 0.202 eV/ f.u. and ⌬Edecomp =
−0.645 eV/ H2兲 and LiNi共BH4兲3 共Ref. 30兲共⌬Ealloy =
−0.104 eV/ f.u. and ⌬Edecomp = 0.069 eV/ H2兲 fall marginally outside the selection criteria. Furthermore, LiMn共BH4兲3
and NaMn共BH4兲3 are found experimentally to decompose at
⬃100 and 110 ° C,34 and LiZn共BH4兲3 and LiAl共BH4兲4 are
found to disproportionate at ⬃130 ° C.25 These are all structures that are located near the optimal stability in the figure
共the nonshaded region兲.
VI. TRENDS
Z r
M o
N b
T c
R u
R h
A g
P d
C d
FIG. 8. The alloying energy, ⌬Ealloy, for the 4d-metals in their preferred
M 1M 2共BH4兲x template structures with M 1, M 2, and B fixed for both x = 3 and
4. Colors: Li 共red兲, Na 共blue兲, and K 共green兲. Preferred local coordination:
tetra 共䊐兲, octa 共䊊兲, octa-tetra 共䉭兲, tetra-octa 共+兲. The labels indicate the
oxidation state of M 2.
Size effects also become apparent here since the
M 1M 2共BH4兲4-tetra template is the only template structure
that allows the coordination polyhedra of M 1 and M 2 to be
relaxed independently. This is supported by the larger spacing between most of the Li, Na, and K alloy energies produced by the other template structures 共see Figs. 7 and 8兲.
To investigate this further, the M 1M 2共BH4兲3tetra/tri-template was applied to all alloys, and in Figs. 9 and
10, the final alloy stabilities are presented; these also include
the free relaxation and the additional x = 2 and x = 5 calculations. It is seen that the M 1M 2共BH4兲3-tetra/tri-structures now
become the most stable for a number of alloys and that the
Li, Na, and K points lie closer indicating a reduction in the
size effects.
There is a general agreement between valencies in the
reference calculations and the alloys; divalent metals are
Given the systematic approach to the screening study it
is also possible to extract information from the database
about possible trends and correlations, in order to search for
predictors and descriptors35 for the design of future quaternary alloys or alloys with different cation stoichiometries.
0 .8
0 .6
IV
I
III
IV
0 .4
A. 3d and 4d transition metals
I
II
0 .2
II
II
II
III
III
III
III
II
0 .0
I
III
III
II
a llo y
( e V / f.u .)
III
∆ E
The stability of the alloys, as produced by the most
stable x = 3 and x = 4 initial template structures before the free
relaxation, is presented for all 3d transition metals 共plus Mg,
Ca, and Al兲 in Fig. 7, and for the 4d transition metals in Fig.
8. A clear preference for the M 1M 2共BH4兲4-tetra template is
observed, which is somewhat surprising, because many of
the transition metals have an oxidation state of II in the reference calculations 共see Table I兲. This apparent discrepancy
could result from partially non-ionic bonding in these structures, meaning that the coordination of the hydrogen atoms
to the metal is the determining factor, not whether the metal
has the “correct” valence. For instance, we find no significant
energy difference between Fe2共BH4兲3 and Fe共BH4兲2 as long
as the H atoms are octahedrally coordinated to the Fe atom.
Y
III
III
III
III II
III
III
-0 .2
M g
A l
II
II
II
II
II
II
III
III
-0 .4
II
II
C a
S c
T i
V
C r
M n
II
F e
C o
N i
C u
II
II
Z n
FIG. 9. The alloying energy, ⌬Ealloy, for the 3d-metals using only the energy
of the preferred M 1M 2共BH4兲x, x = 2 – 5 structure. Colors: Li 共red兲, Na 共blue兲,
and K 共green兲. Preferred local coordination: tetra 共䊐兲, octa 共䊊兲, octa-tetra
共䉭兲, tetra-octa 共+兲, tetra-tri 共丫兲, other 共䉰兲. The labels indicate the oxidation state of M 2.
014101-8
Hummelshøj et al.
J. Chem. Phys. 131, 014101 共2009兲
0 .2
0 .8
IV
IV
0 .6
0 .0
IV
2
d e c o m p
I
III
0 .2
III
II
III
III
0 .0
I
II
II
III
III
II
II
II
II
III
-0 .6
II
II
III
III
III
-0 .4
∆ E
∆ E
(e V / H
I
a llo y
( e V / f.u .)
)
-0 .2
0 .4
-0 .8
III
-0 .2
Y
Z r
III
N b
M o
T c
R u
R h
P d
A g
C d
FIG. 10. The alloying energy, ⌬Ealloy, for the 4d-metals using only the
energy of the preferred M 1M 2共BH4兲x, x = 2 – 5 structure. Colors: Li 共red兲, Na
共blue兲, and K 共green兲. Preferred local coordination: tetra 共䊐兲, octa 共䊊兲,
octa-tetra 共䉭兲, tetra-octa 共+兲, tetra-tri 共丫兲, other 共䉰兲. The labels indicate
the oxidation state of M 2.
found to prefer a M 1M 2共BH4兲3 configuration, whereas trivalent metals prefer M 1M 2共BH4兲4, tetravalent metals prefer
M 1M 2共BH4兲5 and the monovalent Cu and Ag prefer
M 1M 2共BH4兲2. Some deviations are found, but given the
simple model structures used for both alloys and reference
calculations, and given the fact that some of the metals are
found by experiments to form ternary borohydrides in different oxidation states, the agreement is good.
The most stable alloys are found for the half-filled
d-bands, but interesting alloys are also found for the empty
and fully occupied d-bands with the addition of Al, where the
M 1Al共BH4兲4 are found to be promising 共see Figs. 9 and 10兲.
Lithium-based alloys 共red兲 are generally found to be the
most stable, followed by sodium 共blue兲 and potassium
共green兲, although significant deviations are observed. This
follows the observed trend for the storage capacities.
B. Stability versus electronegativity
-1 .0
0 .8
1 .4
1 .2
E le c t r o n e g a t iv it y
1 .0
FIG. 11. The decomposition energy, ⌬Edecomp, as a function of the average
Pauling electronegativity for all alloys in their preferred M 1M 2共BH4兲x coordination: tetra 共䊐兲, octa 共䊊兲, octa-tetra 共䉭兲, tetra-octa 共+兲, tetra-tri 共丫兲,
other 共䉰兲. Colors: Li 共red兲, Na 共blue兲, and K 共green兲.
promising with ⌬Edecomp ⯝ −0.1 eV/ H2 for Mn and Nb and
particularly promising for Al, Zn, and Fe with ⌬Edecomp ⯝
−0.3 eV/ H2. The Mo and Rh alloys at electronegativities
around 1.6 are found to border on decomposition, but experimental work by Nikels et al.33 estimates the decomposition
temperature of such compounds to be around 150 ° C.
VII. CONCLUSIONS
We have analyzed the thermodynamic properties of possible alkali-transition metal borohydride systems, finding a
number of candidates showing favorable properties.
The M 1共Al/ Mn/ Fe兲共BH4兲4, 共Li/ Na兲Zn共BH4兲3, and
共Na/ K兲共Ni/ Co兲共BH4兲3 alloys are found to be the most promising, followed by selected M 1共Nb/ Rh兲共BH4兲4 alloys. These
findings are in good agreement with experimental observations for LiFe共BH兲3,37 LiAl共BH4兲4,25 共Li/ Na兲Mn共BH兲3,4,38
0.2
33,36
A number of recent publications
have shown an apparent linear correlation between the decomposition temperature and the average cation Pauling electronegativity. Although this might be expected, given the definition of
Pauling’s electronegativity, it also indicates that the kinetic
barriers—if any—do not appear to be particularly system
dependent.
Plotting the calculated decomposition energy as a function of the average cation electronegativity for all alloys in
their most stable local coordination 共see Fig. 11兲 appears to
support this observation. The scatter of the data points
around the “line” 共which would have a slope that agrees with
Ref. 36 to within 10%–15%兲 is, however, significant and
deviations of ⫾0.1 eV/ H2 can be sufficient to shift a material from interesting to irrelevant for storage applications, or
vice versa.
The stable alloys 共⌬Ealloy ⱕ 0.0 eV/ f.u.兲 are seen to
cluster around certain average electronegativities of 1.3–1.4
and 1.6 共see Fig. 12兲. The cluster around 1.3–1.4 is highly
1 .6
LiRh(BH4 )4
LiNi(BH4 )3
LiCo(BH4 )3
0.0
NaNi(BH4 )3
LiMn(BH4 )4
NaCo(BH4 )3
LiNb(BH4 )4
LiFe(BH4 )3
NaMn(BH4 )4
KNi(BH4 )3
KMn(BH4 )4
NaNb(BH4 )4 KCo(BH4 )3
NaFe(BH4 )3
LiZn(BH )
-0.2
LiMo(BH4 )4
NaRh(BH4 )4
KRh(BH4 )4
4 3
LiAl(BH4 )4
KCd(BH4 )3
-0.4
-0.6
KAl(BH4 )4
KFe(BH4 )3
NaZn(BH4 )3
NaAl(BH4 )4
KZn(BH4 )3
LiNa(BH4 )2
-0.8
-1.0
0.8
1.0
1.2
1.4
1.6
FIG. 12. The decomposition energy, ⌬Edecomp, as a function of the average
Pauling electronegativity for alloys with ⌬Ealloy ⱕ 0.0 eV/ H2. Colors: Li
共red兲, Na 共blue兲, and K 共green兲. Preferred M 1M 2共BH4兲x, x = 2 – 5, coordination: tetra 共䊏兲, octa 共䊊兲, octa-tetra 共䉭兲, tetra-octa 共+兲, tetra-tri 共丫兲, other
共䉰兲.
014101-9
and 共Li/ Na兲Zn共BH4兲3,39 whereas the Co, Cd, Nb, and Rh
and alloys still remain to be synthesized and tested. Although
some structures can be observed experimentally in different
metal-metal stoichiometries than those used in the screening
study, e.g., the Li–Zn system,39 the alloy systems were still
identified as promising candidates in this screening study.
Some of the nearly stable compounds in Table III, e.g.,
LiSc共BH4兲4 共Ref. 10兲 and KNa共BH4兲2 共Ref. 32兲 have recently been found to be metastable, while LiNi共BH4兲3
共Ref. 30兲 was found to be marginally unstable here. The LCS
approach was found to limit the 757 potential alloys to 22
promising candidates of which ⬃10 are highly promising.
These structures can now be pursued further, analyzing their
detailed decomposition pathways, both theoretically40 and
experimentally.
ACKNOWLEDGMENTS
The authors acknowledge financial support by the European Commission DG Research 共Contract No. SES6-200651827/NESSHy兲, the Nordic Energy Research Council 共Contract No. 06-HYDRO-C15兲 and the Danish Center for
Scientific Computing 共DCSC兲 for computer time 共Grant No.
HDW-1103-06兲. The Center for Atomic-Scale Materials Design is funded by the Lundbeck Foundation.
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