Quantum Transport in Finite Disordered Electron Systems

Transcription

Quantum Transport in Finite Disordered Electron
Systems
A Dissertation Presented
by
Branislav Nikolić
to
The Graduate School
in Partial Fulfillment of the Requirements
for the Degree of
Doctor of Philosophy
in
Physics
State University of New York
at
Stony Brook
August 2000
State University of New York
at Stony Brook
The Graduate School
Branislav Nikolić
We, the dissertation committee for the above candidate for the Doctor of Philosophy degree,
hereby recommend acceptance of the dissertation.
Philip B. Allen,
Professor, Department of Physics and Astronomy, Stony Brook
Gerald E. Brown,
Vladimir J. Goldman,
Myron Strongin,
Research Staff Member, Brookhaven National Laboratory, Upton
This dissertation is accepted by the Graduate School.
Graduate School
ii
Abstract of the Dissertation
Quantum Transport in Finite Disordered Electron
Systems
by
Branislav Nikolić
Doctor of Philosophy
in
Physics
State University of New York at Stony Brook
2000
The thesis presents a theoretical study of electron transport in various disordered conductors. Both macroscopically homogeneous (nanoscale conductors
and point contacts) and inhomogeneous (metal junctions, disordered interfaces,
metallic multilayers, and granular metal films) samples have been studied using
different mesoscopic as well as semiclassical (Bloch-Boltzmann and percolation in
random resistor networks) transport formalisms. The main method employed is a
real-space Green function technique and related Landauer-type or Kubo formula
for the exact static quantum (zero temperature) conductance of a finite-size mesoscopic sample in a two-probe measuring geometry. The finite size of the sample
makes is possible to treat the scattering on impurities exactly and thereby study
all transport regimes. Special attention has been given to the transitional regions
connecting diffusive, ballistic and localized transport regimes. Thorough analysis
iii
of the proper implementation of different formulas for the linear conductance has
been provided.
The thesis has three parts. In the first Chapter of Part I the quantum transport methods have been used to extract the bulk resistivity of a three-dimensional
conductor, modeled by an Anderson model on an nanoscale lattice (composed of
several thousands of atoms), from the linear scaling of disorder-averaged resistance with the length of the conductor. The deviations from the corresponding
semiclassical Boltzmann theory have been investigated to show how quantum
effects evolve eventually leading to the localization-delocalization transition in
strongly disordered systems. The main result is discovery of a regime where
semiclassical concepts, like mean free path, loose their meaning and quantum
states carrying the current are “intrinsically diffusive”. Nevertheless, scaling of
disorder-averaged resistance with the sample length is still approximately linear and “quantum” resistivity can be extracted. Different mesoscopic effects,
like fluctuations of transport coefficients, are explored in the regime of strong
disorder where the concept of universality (independence on the sample size or
the degree of disorder—within certain limits), introduced in the framework of
perturbation theory, breaks down. The usual interpretation of a semiclassical
limit of the disorder-averaged Landauer formula in terms of the sum of contact
resistance and resistance of a disordered region was found to be violated even for
low disorder. The “contact resistance” (i.e., the term independent of the sample
length) diminishes with increasing disorder and eventually turns negative.
The second Chapter of Part I investigates transport in metal junctions, strongly
disordered interfaces and metallic multilayers. The Kubo formula in exact state
representation fails to describe adequately the junction formed between two conductors of different disorder, to be contrasted with the mesoscopic methods (in
iv
the Landauer or Kubo linear response formulation) which take care of the finiteness of a sample by attaching the ideal leads to it. Transmission properties of a
single strongly disordered interface are computed. The conductance of different
nanoscale metallic multilayers, composed of homogeneous disordered conductors
coupled through disordered interfaces, is calculated. In the presence of clean
conductors the multilayer conductance oscillates as a function of Fermi energy,
even after disorder averaging. This stems from the size quantization caused by
quantum interference effects of electron reflection from the strongly disordered
interfaces. The effect is slowly destroyed by introducing disorder in the layer
between the interfaces, while keeping the mean free path larger than the length
of the that layer. If all components of the multilayer are disordered enough, the
conductance oscillations are absent and applicability of the resistor model (multilayer resistance understood as the sum of resistances of individual layers and
interfaces) is analyzed.
In Part II an atomic-scale quantum point contact was studied with the intention to investigate the effect of the attached leads on its conductance (i.e.,
the effect of “measuring apparatus” on the “result of measurement”, in the sense
of quantum measurement theory). The practical merit of this study is for the
analogous effects one has to be aware of when studying the disordered case. The
transitional region between conductance quantization and resonant tunneling has
been observed. The other problem of this Part is a classical point contact modeled as an orifice between two metallic half-spaces. The exact solution for the
conductance is found by transforming the Boltzmann equation in the infinite
space into an integral equation over the finite surface of the orifice. Such conductance interpolates between the Sharvin (ballistic) conductance and the Maxwell
(diffusive) conductance. It deviates by less than 11% from the naı̈ve interpolation
v
formula obtained by adding the corresponding resistances.
The third Part is focused on the transport close to the metal-insulator transition in disordered systems and effects which generate this transition in the
non-interacting electron system. Eigenstate statistics are obtained by exact diagonalization of the 3D Anderson Hamiltonians with either diagonal or off-diagonal
disorder. Special attention has been given to the so-called pre-localized states
which exhibit unusually high amplitudes of the wave function. The formation
of such states should illustrate the quantum interference effects responsible for
the localization-delocalization transition. The connection between the eigenstate
statistics and quantum transport properties has been established showing that
deviations (i.e., asymptotic tails of the corresponding distribution function in
finite-size conductors) from the universal predictions of Random Matrix Theory
are strongly dependent on the microscopic details of disorder. The mobility edge
is located at the minimum energy at which exact quantum conductance is still
non-zero.
The second problem of Part III is a theoretical explanation of the infrared
conductivity measurement on ultrathin quench-condensed Pb films. It was shown
that quantum effects do not play as important a role as classical electromagnetic
effects in a random network of resistors (grains in the film) and capacitors (capacitively coupled grains). The experimental results exhibit scaling determined
by the critical phenomena at the classical percolation transition point.
vi
Dedicated to the memory of
my late grandfather Petronije Nikolić
Contents
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv
Acknowledgements
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xv
1 INTRODUCTION
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
I
Diffusive Transport Regime
2 Linear Transport Theories
19
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
2.2
Ohm’s law and current conservation . . . . . . . . . . . . . . . . . . . . . . .
24
2.3
Semiclassical formalism: Boltzmann equation . . . . . . . . . . . . . . . . . .
32
2.4
Quantum transport formalisms . . . . . . . . . . . . . . . . . . . . . . . . .
36
2.4.1
Linear response theory: Kubo formula . . . . . . . . . . . . . . . . .
36
2.4.2
Scattering approach: Landauer formula . . . . . . . . . . . . . . . . .
44
2.4.3
Non-equilibrium Green function formalism . . . . . . . . . . . . . . .
49
2.5
Quantum expressions for conductance: Real-space Green function technique
54
2.5.1
Lattice model for the two-probe measuring geometry . . . . . . . . .
54
2.5.2
Green function inside the disordered conductor
. . . . . . . . . . . .
57
2.5.3
The Green function for an isolated semi-infinite ideal lead . . . . . .
61
2.5.4
One-dimensional example: single impurity in a clean wire . . . . . . .
63
viii
2.5.5
Equivalent quantum conductance formulas for the two-probe geometry 64
3 Residual Resistivity of a Metal between the Boltzmann Transport Regime
and the Anderson Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
3.2
Semiclassical Resistivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
3.3
Quantum resistivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
3.4
Conductance vs. Conductivity in mesoscopic
physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
87
4 Quantum Transport in Disordered Macroscopically Inhomogeneous Conductors
II
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
91
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
91
4.2
Transport through disordered metal junctions . . . . . . . . . . . . . . . . .
92
4.3
Transport through strongly disordered interfaces . . . . . . . . . . . . . . . . 105
4.4
Transport through metallic multilayers . . . . . . . . . . . . . . . . . . . . . 109
Ballistic Transport and Transition from Ballistic to Diffusive
Transport Regime
115
5 Quantum Transport in Ballistic Conductors: Evolution From Conductance
Quantization to Resonant Tunneling . . . . . . . . . . . . . . . . . . . . . . . . 116
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.2
Model: Nanocrystal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.3
Model: Nanowire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.4
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
ix
6 Electron Transport Through a Classical Point Contact
. . . . . . . . . . 131
6.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.2
Semiclassical transport theory in the orifice geometry . . . . . . . . . . . . . 134
6.3
The conductance of the orifice . . . . . . . . . . . . . . . . . . . . . . . . . . 140
6.4
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
III
Transport Near a Metal-Insulator Transition in Disordered
Systems
148
7 Introduction to Metal-Insulator Transitions
. . . . . . . . . . . . . . . . . 149
8 Statistical Properties of Eigenstates in three-dimensional Quantum Disordered Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
8.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
8.2
Exact diagonalization study of eigenstates in disordered conductors . . . . . 161
8.3
Connections of eigenstate statistics to static quantum transport properties . 172
8.4
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
9 Infrared studies of the Onset of Conductivity in Ultrathin Pb Films . . 177
9.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
9.2
The Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
9.3
Theoretical analysis of the experimental results . . . . . . . . . . . . . . . . 181
9.4
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
References
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
x
List of Figures
2.1
A two-dimensional version of our actual 3D model of a two-probe measuring
geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2
Local density of states at an arbitrary site of a 1D chain, described by a
tight-binding Hamiltonian. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1
83
The conductance and resistance fluctuations, at EF = 0, from weak to strong
scattering regime in disordered samples of different geometry. . . . . . . . . .
3.6
81
The conductance fluctuations from weak to strong scattering regime in the
disordered cubic samples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5
76
Linear fit R = C1 + ρ/A L, (A = 225 a2 ) for the disorder averaged resistance
R in the band center and different disorder strengths. . . . . . . . . . . . .
3.4
72
The density of states of the clean and dirty metal and the clean metal Boltzmann resistivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3
65
Resistivity at different values of EF , normalized to the semiclassical Boltzmann resistivity ρB calculated in the Born approximation. . . . . . . . . . .
3.2
56
85
The deviation between disorder averaged resistance and inverse of disordered
average conductance, evaluated at EF = 0, as a function of disordered strength
in the Anderson model on a cubic lattice. . . . . . . . . . . . . . . . . . . . .
4.1
4.2
86
The diffusivity of a disordered binary alloy modeled by the tight-binding
Hamiltonian. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
97
The diffusivity of the diagonally disordered Anderson model. . . . . . . . . .
98
xi
4.3
The diffusivity of a metal junction composed of two disordered binary alloys,
modeled with the TBH on a lattice. . . . . . . . . . . . . . . . . . . . . . . . 100
4.4
Local density of states integrated over the y and z coordinates for the metal
junction composed of two disordered binary alloys.
4.5
. . . . . . . . . . . . . . 101
Conductance of a disordered conductor modeled by the Anderson model on a
lattice 10 × 10 × 10 for two different values of the hopping parameter in the
leads. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.6
Conductance of a metal junction composed of two disordered binary alloys,
modeled with TBH, for different attached leads. . . . . . . . . . . . . . . . . 104
4.7
Conductance of a single disordered interface and thin layers composed of two
or three interfaces, modeled by the Anderson model, as well as numerically
obtained distribution of transmission eigenvalues ρ(T ) in the band center. . . 107
4.8
The disorder-averaged (over 200 configurations) conductance of a multilayer
composed of strongly disordered interfaces and clean bulk conductors (lower
panel) or clean and disordered bulk conductors (upper panel) on a lattice
17 × 10 × 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
4.9
Conductance of a disordered conductor modeled by the Anderson model a
lattice 5×10×10 (W = 6 and W = 3) and quantum point contact conductance
of a clean sample on the same lattice. . . . . . . . . . . . . . . . . . . . . . . 112
4.10 The disorder-averaged (over 200 configurations) conductance of a multilayer
composed of strongly disordered interfaces and disordered bulk conductors
17 × 10 × 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.1
Conductance of an atomic-scale ballistic contact 3 × 3 × 3 for various lead and
coupling parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.2
Transmission eigenvalues of an atomic-scale ballistic contact 3 × 3 × 3. . . . 122
xii
5.3
Conductance of an atomic-scale ballistic conductor 3 × 3 × 3 for various lead
and coupling parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.4
Conductance of a ballistic quantum wire 12×3×3 for various lead and coupling
parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
5.5
Conductance of a ballistic quantum wire 12 × 3 × 3 for the different set of lead
and coupling parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6.1
Electron transport through the circular constriction in an insulating diaphragm
separating two conducting half-spaces. . . . . . . . . . . . . . . . . . . . . . 132
6.2
The dependence of factor γ on the ratio /a. . . . . . . . . . . . . . . . . . . 136
6.3
The conductance G, normalized by the Sharvin conductance GS , plotted
against the ratio /a. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
8.1
An example of eigenstates in the band center of a delocalized phase. The
average conductance at half filling is g(EF = 0) ≈ 17, entailing anomalous
rarity of the “pre-localized” states. . . . . . . . . . . . . . . . . . . . . . . . 159
8.2
Statistics of wave function intensities in the RH Anderson model on a cubic
lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
8.3
Statistics of wave function intensities in the DD Anderson model on a cubic
lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
8.4
¯
Ensemble averaged Inverse Participation Ratio, I(2),
of eigenstates in the RH
and DD Anderson models on the cubic lattice. . . . . . . . . . . . . . . . . . 169
8.5
Conductance and DOS in the RH and DD Anderson models on the cubic lattice.171
9.1
Sheet conductance vs. frequency for set 3. . . . . . . . . . . . . . . . . . . . 182
xiii
9.2
T (ω)/[1 − T (ω)] plotted vs. ω 2 for the seven thickest films from the set 3
(dots), and two annealed films form set 1 (solid circles). The solid lines are
Drude model fits (9.3). The inset shows the plasma frequency extracted from
these fits with solid line representing the plasma frequency of bulk lead from
Ref. [215]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
9.3
The “data collapse” of the rescaled conductivity. . . . . . . . . . . . . . . . . 189
xiv
Acknowledgements
There will always be a lot of reasons
for avoiding what we really want to do.
— Swami Janakananda Saraswati
The last five years, spent in Stony Brook, have been the time of immense personal
growth in many realms of human existence: intellectual, scientific, social, spiritual... Many
have contributed on this path. In his acceptance speech for the 1991 Oersted Medal, Freeman
J. Dyson1 lists six faces of science a neophyte is given to explore: three beautiful (“science
as subversion of authority, science as an art form, and science as an international club”)
and three ugly (“rigid and authoritarian discipline, tied to mercenary and utilitarian ends,
and tainted by its association with weapons and mass murder”). In Stony Brook, I have
encountered mostly the beautiful faces thanks to the following people. First and foremost I
would like to thank my adviser, professor Philip B. Allen, for support, patience and encouragement he has offered throughout the years of grappling with physics (and life) problems
leading to this thesis. His striving for simplicity, physical intuition, and guidance (with a
sometimes stringent, but helpful, attitude toward assignments) were the most valuable. I
also learned the importance of keeping in mind experiments when conducting research in
Condensed Matter Physics. The choice of problems from the field of disordered physics,
which I enjoyed a lot, commenced a personal scientific revolution since in undergraduate
days we worshipped symmetries as the final answer to all questions. Thus, the traditional
1 Transcript
published in American Journal of Physics 59, 491 (1991).
adviser-disciple relationship is enduring one and it will hardly be surpassed by any technological advances (like omnipresent Internet, for example). The person whom I admired most
in terms of mastery of physics is professor Igor L. Aleiner. This has led to frequent bothering
him with all kind of question and answers he provided in private communications, as well as
in the superb courses on disordered physics and superconductivity, helped me in many cases
to extract physics from the complicated mathematical models of Condensed Matter Theory.
I had a great opportunity to discuss the “real” problems with the experimental group
from BNL lead by Myron Strongin, and provide some simple theory for them while collaborating with Sergei Maslov from BNL Condensed Matter Theory Group. Thanks also goes
to professor Jainendra Jain for his generous contribution to the Condensed Matter Theory Group at Stony Brook in the form of state of the art Alpha stations (with powerful 1
Gb of RAM) which have dramatically shortened the time to complete the computationally
demanding research, when computers were sine qua non to accomplish the task.
Daily conversations with colleagues from B-127, Kwon Park and Vasili Perebeinos, as
well as those from the same generation, Gianluca Oderda and Kunal Das, helped to solve
various graduate student problems and enjoy→“science as an international club”←In this
sense, I also thank prof. Boris Shapiro, Dr. Jose Antonio Vergés, and Dr. Igor E. Smolyarenko for taking the time to read some of my manuscripts posted on the cond-mat preprint
server at http://xxx.lanl.gov and provide useful suggestion for their improvement.
Looking back, I could say that coming to Stony Brook, strangely enough on the first
sight (sic !), was the proper choice. What will stay in my (photographic) memory are endless
conversations and adventures, directing the future life paths, with some of the most interesting people I have met thus far. Without them it seems to be impossible to survive physical
and (mental) distance from home (in alphabetical order): Adil Atari, Athanassios Bardas,
Alec Maassen van den Brink, Daniel Burley, Kunal Das, Stacy Dermont, Jaroslav Fabian,
Alok Gambhir, Sergio Angelim O. Silva, and Pavel Sumazin. Also, “old” friends (dating
xvi
back to undergraduate or high school days) have provided the traditionally indispensable
support—Milan M. Ćirković (whom I followed to Stony Brook), a collaborator on disordered
projects Viktor Cerovski, frequent visitor to Long Island Robert Lakatoš, and an incisive
critic Dario Čupić.
Valuable tips and constant help, that made the side effects of bureaucracy less of a
burden to my life in Stony Brook, have been provided by the secretary of Condensed Matter
Group Sara Lutterbie and assistant director of the graduate program Pat Peiliker. The
aggressive approach of prof. Peter W. Stephens (director of the graduate studies) in dealing
with various offices around the campus was crucial in some daunting situations. He has also
shown a great care in fostering student progress toward the doctoral degree, among other
things, by appointing the Ph.D. supervising committee. I also thank the members of my
committee, prof. Gerald E. Brown and Vladimir J. Goldman, who were happy to allocate
some of their valuable time to follow my progress.
Last, but not least, my gratitude goes to my parents, Jelena and Konstantin, my brother
Predrag, who perpetually sacrificed for my well-being, Jean Baudrillard for providing me
with impetus for (facetious) intellectual adventures, and to Paramahamsa Satyananda and
Paramahamsa Niranjanananda who have been teaching me the meaning of life through the
means known only to them.
xvii
List of Publications
[1] B. Nikolić and P. B. Allen, Electron transport through a circular constriction, Physical
Review B 60, 3963 (1999).
[2] P. F. Henning, C. C. Homes, S. Maslov, G. L. Carr, D. N. Basov, B. Nikolić, and M.
Strongin, Infrared studies of the onset of conductivity in ultrathin Pb films, Physical
Review Letters 83, 4880 (1999).
[3] B. K. Nikolić and P. B. Allen, Quantum transport in ballistic conductors: transition from
conductance quantization to resonant tunneling, Journal of Physics: Condensed Matter
12, 9629 (2000).
[4] B. K. Nikolić, Statistical properties of eigenstates in three-dimensional mesoscopic systems with off-diagonal or diagonal disorder, cond-mat/0003057 (2000), submitted for
publication in Physical Review B.
[5] B. K. Nikolić and Philip B. Allen, Resistivity of a metal between the Boltzmann transport
regime and the Anderson transition, cond-mat/0005389 (2000), accepted for publication in Physical Review B Rapid Communications.
[6] B. K. Nikolić and P. B. Allen, Quantum transport in dirty metallic junctions and multilayers, unpublished.
xviii
List of Symbols and Abbreviations
a
lattice constant
A
Area
Â
spectral function
A
vector potential
B
magnetic field
d
spatial dimensionality
D
diffusion constant
e
electron charge
E
energy
E
electric field
Eb
band edge energy
Ec
mobility edge
EF
Fermi energy
ETh
Thouless energy (= h̄D/L2 )
f (k )
equilibrium Fermi-Dirac distribution function
f (t)
distribution of wave function intensities |Ψ(r)|2
fLE (k, r, t)
local equilibrium distribution function
F (k, r, t)
non-equilibrium distribution function
G
conductance
GQ = 2e2 /h
conductance quantum
g
dimensionless conductance (= G/GQ )
Ĝr
retarded Green operator
Ĝa
advanced Green operator
G> , G<
non-equilibrium Green functions for particle distribution properties
xix
h
Planck constant
h̄
h/2π
I
current
Im
Imaginary part of a complex number
j
current density
k
wavevector
kB
Boltzmann constant
kF
Fermi wavevector
mean free path
L
length (of the sample)
LT
thermal diffusion length
Lφ
phase-coherence length
m
effective mass
N(EF )
density of states at the Fermi level
N, Ny , Nz
number of lattice sites along x, y and z axis, respectively
NLCT
nonlocal conductivity tensor
Ns
total number of lattice sites (Ns = NNy Nz )
Ô
linear operator (or matrix)
R
resistance
Re
Real part of a complex number
S
S-matrix
t
time or hopping parameter in the Anderson model
t
transmission matrix
T
temperature
Tn
transmission eigenvalue
T
T-matrix
xx
Tr
trace
U(r)
random potential
V
voltage
v
velocity
vF
Fermi velocity
. . .
averaging over disorder (impurity ensemble)
(. . .)
averaging over probability distribution
|α
eigenstate of a single-particle Hamiltonian
β
symmetry index (β ∈ {1, 2, 4}) in Random Matrix Theory
Γ
energy level broadening
Γ̂
lead-sample coupling operator [= i(Σ̂r − Σ̂a )]
δ(x)
delta function
δ̄(x)
broadened delta function (Box, Lorentzian, etc.)
∆
single-particle level spacing
µ
chemical potential
λF
Fermi wavelength
Ω
volume
Σ̂r
retarded self-energy
Σ̂a
advanced self-energy
ρ
resistivity
ρB
semiclassical resistivity in Born approximation
ρT
semiclassical resistivity in T-matrix approximation
ρ̂
statistical operator
ρ(r, E)
local density of states
ρ(T )
distribution function of transmission eigenvalues
xxi
σ
conductivity
σD
semiclassical (Drude-Boltzmann) conductivity
σ(L)
quantum conductivity of a cube of size L
σ (r, r )
¯
τ
nonlocal conductivity tensor
transport mean free time
τD
classical diffusion time ( L2 /D)
τesc
electron escape time into the leads
τf
time of flight in ballistic systems (= L/vF )
τφ
phase-coherence time
Φ
electric potential
Ψ(r)
wave function (= r|Ψ)
ω
frequency
Ω
volume of the sample
AC
Alternating Current
CQ
Conductance Quantization
CPP
Current Perpendicular to the Plane
DC
Direct Current
DOS
Density of States
EEI
Electron-Electron Interaction
EMT
Effective Medium Theory
FDT
Fluctuation-Dissipation Theorem
FLT
Fermi Liquid Theory
GMR
Giant Magnetoresistance
GOE
Gaussian Orthogonal Ensemble
IPR
Inverse Participation Ratio
xxii
KLRT
Kubo Linear Response Theory
LD
Localization-Delocalization
LDOS
Local Density of States
NLCT
Nonlocal conductivity tensor
QPC
Quantum Point Contact
QPT
Quantum Phase Transition
RMT
Random Matrix Theory
SCA
Semiclassical Approximation
SUSY NLσM Supersymmetric Nonlinear σ-Model
TBH
Tight-Binding Hamiltonian
WL
Weak Localization
UCF
Universal Conductance Fluctuation
xxiii
1
Chapter 1
INTRODUCTION
It is with logic that one proves;
it is with intuition that one invents.
— Henri Poincaré
The study of electron (or phonon) transport in solids is one of the most fundamental
problems in Condensed Matter Physics. Transport measurements are a powerful tool for
the investigation of electronic properties of materials. In particular, electron transport in
disordered conductors [1] has been a popular playground for a plethora of ideas from the
non-equilibrium statistical mechanics. This has led to efficient computational schemes for
obtaining the kinetic coefficients. A lot can be learned about disordered conductors (dirty
metals and doped semiconductors) using the simple non-interacting (quasi)particle approach.
Thus the development of the quantum intuition is facilitated since one particle quantum
mechanics is formally similar to the theory of classical wave propagation. The major impetus
for the several decades of the exploration of quantum dynamics of electrons in disordered
systems came with the seminal paper of Anderson [2] (preceded in some respect by Landauer
and Helland [3] or Landauer [4]) who showed that strong enough disorder can localize all
states.1 This renders the zero temperature conductivity (in the thermodynamic limit) equal
1 Below
two dimensions and for arbitrary weak disorder a quantum particle is always localized,
except for some special types of disorder or presence of spin-orbit scattering, cf. Part III.
2
to zero (or, equivalently, conductance decays exponentially with the system size) even though
the density of states is non-zero. The disorder induced metal-insulator transition in noninteracting electron systems is called Anderson localization, or in modern terminology [5]
the localization-delocalization (LD) transition. This is one of several types of metal-insulator
transitions (MIT) encountered in Condensed Matter Physics. Its discovery was a bit of
surprise since quantum mechanics was known to delocalize particles by tunneling effects, a
standard example being the Bloch states in a perfect crystal which give infinite conductivity.2
Over the course of time it has been realized that the phenomenon of localization is one of
the major manifestations of quantum mechanics in solids. The localization theory (i.e., the
theory of disordered solids) received a major boost by the work of Mott [7], the application of
scaling concepts [8] borrowed from the critical phenomena theory, and the recent development
of mesoscopic physics [9].
The “standard model” to begin with in disordered electron physics is the Hamiltonian of
a single particle in a random potential.3 An astonishingly rich physics has arisen from such
“simple” problem. The random potential simulates the disorder. Similar non-integrable
models have been encountered in other realms of physics, like e.g., Quantum Chaos [10]
(quantum behavior of systems which are classically chaotic), and are connected recently
through the statistical approach akin to that of Random Matrix Theory (RMT) [11]. The
usual fruitful exchange of ideas between apparently different fields, which use very different
techniques to analyze their respective systems, has ensued.4 The electron-electron interaction
2 In
the early days of localization theory it was expected that localization of wave functions is not
important since an electron tunneling far enough could find a state with the same energy. Although
this is possible, it does not prevent the formation of the Anderson insulator [6].
3 Aside
from the random potential, there is also the potential which confines the particle inside
the sample. It is usually taken into account through appropriate boundary conditions.
4 The
RMT, originally discovered in the realm of nuclear many-body physics, and localization
theory were developing quite independently until the beginning of 80s. The mutual interaction,
3
(EEI) is also important. Interesting phenomena emerge as a result of the interplay between
disorder and the Coulomb interaction [14]. Nevertheless, it is obvious that before embarking
on a full problem one should follow the route of simplification, indigenous to the thinking in
physics, and understand first the pure “disorder part”.5
Despite years of vigorous pursuits and tantalizing simplicity, a complete understanding
of disordered electron physics is still not reached. This is especially true in the “sectors” of the
theory not amenable to perturbative techniques (similarly to other fields of physics, like QCD,
string theories, strongly correlated electron systems, etc., where interesting problems [18] still
await future generations of physicists). From a point of view of real world materials, this
means that one is dealing with very dirty metals. However, the same regime is entered for long
enough wires whose conductance is of the order of one conductance quantum GQ = 2e2 /h,
even if they are made of good metal [19]. The prime example of the non-perturbative sector
is the LD transition itself. It can occur with increasing disorder at any energy, or for fixed
disorder at energies |E| > |Ec |. The mobility edge Ec separates extended and localized
leading eventually to their coalescence [11], came with the emergence of Efetov’s supersymmetric
technique [12], developed as an efficient tool for calculating the disorder-averaged correlation functions which can produce the spectral correlations of RMT. Around the same time Bohigas et al. [13]
conjectured, using substantial numerical evidence, that RMT describes the statistics of energy levels in the quantum systems whose classical analogs are chaotic (the chaos should be “hard”, as in
either K or ergodic systems [10]).
5 It
has been known since the work of Altshuler and Aronov [14] that effects EEI in disordered
systems are small as the inverse dimensionless conductance of a system on a relevant linear scale.
That is why the regime where those effects become strong is impeded with the developing localization (signaled by the diminishing conductance). Such results of perturbative (diagrammatic)
calculations are intuitively interpreted either in terms of the interaction time being much longer in
disordered conductors due to diffusive (instead of ballistic in clean samples) motion of electrons [15],
or by invoking the statistical properties of exact one-electron wave functions [16, 17].
4
states inside an energy band. An intuitive argument of Mott [6] suggests that extended and
localized states cannot coexist at the same energy. This argument is not rigorous [20] and
mixing of states can in fact occur in very inhomogeneous systems, like in the samples modeled
by quantum percolation [21]. The LD transition is a “strange” transition (when compared to
familiar critical phenomena) with no obvious order parameter [5] or upper critical dimension
(needed for standard perturbative techniques in 4 + dimensions [22] which provide the
computational realization of the renormalization group scheme).
The Chapters in the thesis are loosely grouped into three Parts corresponding to the
different transport regimes encountered in disordered conductors. The basic concepts of
disordered electron physics, which characterize different transport regimes, are introduced
below, and will serve as a guide for a reader in following the rest of the thesis. A systematic
approach to the properties of disordered conductors at zero temperature requires consideration of the relationship between various length and energy (or time) scales characterizing the
dynamics of a single quasiparticle. To be precise, the Condensed Matter Physics approach
to disordered electron problem assumes a non-interacting gas of quasielectrons in a random
potential (instead of just one particle). This brings the Fermi energy EF as the largest energy
scale in the problem and simplifies many computational algorithms6 for transport in metals. The finite size of a system generates two relevant energy scales: the single-particle level
spacing ∆ and the Thouless energy ETh = h̄D/L2 = h̄/τD , where D is the diffusion constant.
The central energy scale ETh , which unifies many concepts in disordered electron physics,
is determined by the classical diffusion time τD across the sample of size L (the largest size
of a system). Nevertheless, ETh is proportional to various quantum energy scales [24] in
disordered conductors. For example, it represent the finite width of the energy levels of
6 For
example, in many-body physics the confinement of electronic momenta to the neighborhood
of the Fermi surface, which leads to linear theory in terms of external driving field, has been
exploited in the powerful quasi-classical Green function approach [23].
5
an open system.7 In particular, the Thouless energy plays an essential role in the purely
quantum phenomenon of LD transition. While ETh is a transport-related energy scale, the
other important energy scale ∆ is thermodynamically determined. The development of the
scaling theory of localization [8] has elevated the dimensionless conductance g = G/GQ of
a d-dimensional hypercube Ld to a fundamental parameter in disordered electron physics.
The dimensionless conductance was originally introduced by Thouless [25] as the ratio of
two energy scales, g = ETh /∆. The arguments of the scaling theory assume that g is the
zero temperature conductance of a d-dimensional macroscopically homogeneous hypercube
(i.e., impurity concentration is spatially uniform).
The relevant length scales for electron systems in a static potential are: the geometrical size of the system L; the (elastic) transport mean free path = vF τ as a characteristic
distance a particle can travel before the direction of its momentum is randomized (after transport mean free time τ ); the characteristic scale arp for the change of (random) potential; the
lattice constant a of a crystal; and the Fermi wavelength λF , which is de Broglie wavelength
λF = h/pF = 2π/kF (pF = mvF ) characterizing the degenerate Fermi gas. When disorder
is strong enough a new type of state is formed by quantum interference effects. Anderson
localized states have an envelope which decays strongly, exponentially in the typical case
Ψ ∼ exp(−r/ξ), at large distances from the localization center with characteristic (localization) length ξ.8 The mean free path is always much smaller than the localization length,
except for strictly one-dimensional samples where ξ = 4 [5].
Using these energy and length scales, as well as g, three different transport regimes can
be clearly distinguished:
7 In
the open systems (surrounded by an infinitely conducting medium) the single-particle states
−1 }, where τ
are smeared; the magnitude of “smearing” is of the order ∼ min {ETh , h̄τesc
esc is a
characteristic time for the electron escape through the attached leads.
8 More
general types of localized states in random potential have been found, e.g., Ψ(r) extends
over a length ξ and then oscillates after being small for a while [20].
6
• Diffusive regime The standard definition of this transport regime in the literature
is λF L ξ. Almost all states are extended and Ohm’s law is applicable
g ∝ Ld−2 . The dimensionless conductance9 is usually assumed to be large g 1
(∆ ETh ),10 and ETh h̄/τ . This means that τD is too short to resolve individual
levels, but long enough for the electron to be multiply scattered.
• Localized regime In the localized regime the system size exceeds the localization
length L ξ. The conductance g 1 of finite samples in the (“strongly”) localized
regime is small but non-zero, exponentially decaying (“scaling”) with the size of the
system, g ∝ exp(−L/ξ). The scaling of g in the localized phase serves as a more
convenient definition of the localization length than the one defined from the envelope
of a single wave function, since it implies averaging over many states around EF . This
means that most states are localized, but a fraction of them, exponentially diminishing
with length, extend to the boundaries and carry the current in finite-size systems.
• Ballistic regime11 In this transport regime the sample size is smaller than the mean
free path L , or equivalently τ −1 τf−1 . Here the time of flight τf = L/vF defines
ETh = h̄/τf in the ballistic system. Using energy scales one can further discern [11]
between the “ballistic” (∆ h̄/τ h̄/τf , where the disorder is strong enough to
thoroughly mix many energy levels) and the “nearly clean” regime (h̄/τ ∆, h̄/τf ,
9 The
conductance here and in most of the thesis is the so called residual conductance. At low
temperature transport properties are determined by the elastic scattering on impurities.
10 This
is emphasized by using the phrase “diffusive metallic” [16], where “metallic” implies weak
disorder, as quantified by g 1 or kF 1.
11 A
special case of ballistic quantum transport, denoted adiabatic [26], occurs in the Quantum
Hall Effect (QHE) or in some Quantum Point Contacts (QPC). In such systems scattering between
different transport channels (defined in Sec. 2.4.2), e.g., interedge channel scattering in QHE, is
suppressed. In Ch. 5 an example of adiabatic transport in QPC is presented.
7
the disorder is very weak and cannot be taken into account by low-order perturbation
theory).
In the diffusive metallic samples a further distinction can be made, according to the
scale of the potential arp , between the quantum disorder regime a2rp < λF and quantum
chaos12 a2rp > λF regime [27]. In the quantum disorder regime many physical quantities are
universal, i.e., independent of the details of the scattering. Since in this thesis disorder is
usually simulated by the on-site impurity potential (sharp on the scale of λF ), the samples
are always in the regime of “quantum disorder”.
At finite temperatures a new important length scale for the localization problem is
the dephasing length Lφ below which the transport is phase-coherent. Thus, the study of
quantum transport effects (T = 0) is confined to the regions inside the sample which are of
the size Lφ . For example, the scaling theory criterion for localization g(L) ∼ 1 should be
replaced by g(Lφ ) ∼ 1 at finite temperatures. This means that the conductance of a whole
system, composed of many phase-coherent resistors stacked classically, is not limited by the
value of g(Lφ ) which is used to characterize different transport regimes. Samples smaller
than Lφ are called mesoscopic conductors.
The vernacular language of the transport community is obviously not exhaustive in
covering the full spectrum of possible electron dynamics in disordered systems. One has
to worry about crossover regimes between these “clearly” defined transport regimes. This
12 The
condition arises by looking at the quantum uncertainty δθ λF arp in the direction of
particle momentum after scattering event which entails the uncertainty in the position of the
particle δx δθ λF /arp . In the quantum chaos regime δx is unimportant and semiclassical
methods can be used, examples being antidot (arp λF ) arrays and ballistic cavities (arp L).
Furthermore, this analysis introduces another time scale, the so called Ehrenfest time tE | ln h̄|/λ
(λ being the Lyapunov exponent), above which quantum indeterminacy combined with classical
chaos washes out completely the concept of trajectory and classical predictability [28].
8
is one of the major tasks accomplished in this thesis. Namely, in each Part of the thesis
we start from one of these three regimes and then usually move continuously into another
one. Therefore, the reader should track the exposition in the thesis in the same way—
by following the transition which electron experiences when the disorder (or EF for fixed
disorder) is changed. Sometimes we enrich the terminology, like in Ch. 3 where we start
from the metallic regime and follow the transition into “intrinsically diffusive” regime in
which mean free path looses its meaning, < a, but the conductor is still away from the
LD transition. In that Chapter we apply quantum transport methods to extract the bulk
resistivity of a homogeneous conductor not only in the diffusive regime, but also in the
transitional regime, as long as the scaling of disorder-averaged resistance with the length
of the sample (at fixed cross sample cross section) is approximately linear. We observe
by computation the build-up of localization effects—from perturbative weak localization to
non-perturbative effects which eventually lead to the LD transition. In the course of study
we face typical fluctuation effects. However, they were traditionally studied in good metals
(g 1). Thus, we find several novel and interesting results on conductance fluctuations in
the non-perturbative regime. We also study carefully the properties of the disorder-averaged
Landauer formula. Our findings could be termed as mesoscopic effects in very dirty metals.
In the second Chapter (4) of Part I we use the same computational methods to study
macroscopically inhomogeneous disordered structures: junctions composed of two different
disordered conductors, a single strongly disordered interface, and a multilayer composed of
bulk disordered conductors and interfaces. Some of these models are used to analyze the
relationship between different formulas for conductance. We find that the Kubo formula in
exact state representation differs only quantitatively from the Kubo formula (or, equivalent,
Landauer formula) in terms of Green functions for the finite-size homogeneous sample with
attached leads, but fails to describe the inhomogeneous structures properly. For example,
it gives the non-zero conductivity of the metal junction (composed of two conductors with
9
different types of disorder) for Fermi energies at which there are no states which can carry
the current on one side of the junctions.
In the first Chapter (5) of Part II we study quantum transport in nanoscale ballistic
conductors (three-dimensional quantum point contacts) focusing on the effects which leads
(“measuring apparatus”) impart on the results of measurement (conductance). The study
explains pedagogically the conductance quantization phenomenon (in the adiabatic regime
of ballistic transport), resonant tunneling conductance and the wide crossover regime in between. Aside from these conceptual issues (borrowed from the quantum measurement theory), the results clarify some practical questions related to the transport method introduced
in Part I and employed throughout the thesis on disorder problems. The second Chapter (6)
of Part II contains a study of a classical point contact where the exact solution for the semiclassical Boltzmann conductance has been found, after providing some contribution to the
mathematical physics of integro-differential equations. It interpolates between the well-know
Sharvin (ballistic) and Maxwell (diffusive) conductance. This theoretical description of the
contact starts with an infinite conductor, but the final equations are formulated only over
the finite surface of the orifice which connects two metallic half-spaces.
The applicability of RMT concepts [29] in localization theory (and vice versa), initially
to spectral fluctuations [11] and later to quantum-mechanical transmission properties [30],
has elucidated further different transport regimes. This has been achieved by classifying
the energy level statistics (clean, ballistic, ergodic, diffusive, critical and localized), or by
looking at the evolution of transmission properties of the sample with the systems size.
In fact, the universal predictions of RMT for the statistical properties of energy levels and
eigenstates is directly applicable only in the systems with infinite g, or in the energy intervals
smaller than the Thouless energy (the so called ergodic regime where the entire phase space
of a system is explored). Some of the major advances in disorder physics (influencing the
RMT approach itself) have been achieved by looking (and developing relevant tools) at
10
the deviations [31] of the RMT spectral statistics for conductors characterized by finite g,
i.e., in the non-ergodic sector of the diffusive regime. Only recently [32] the same project has
been undertaken for the statistics of eigenstate amplitudes, motivated partly by the unusual
relaxation properties of transport in disordered samples (even in good metals characterized
by large conductance) as well as the development of asymptotic tails of distribution functions
of other mesoscopic quantities (like conductance). An interesting contribution to this newly
open direction of research is given in Ch. 8. The eigenstate statistics (usually studied in less
than three dimensions) and quantum transport properties of three-dimensional (3D) samples,
characterized by different type of microscopic disorder, have been connected both in diffusive
and “intrinsically diffusive” regimes of transport. It is shown that fluctuation properties of
those wave functions disagree with the notions of universality13 which have been the major
paradigm in many aspects of the localization theory. Namely, the statistics of eigenfunction
amplitudes show deviation from the RMT predictions (states with uniform amplitude up
to inevitable Gaussian fluctuations) which cannot be parametrized just by conductance,
shape of the sample and dimensionality of the system. The formation of localized states has
presumably some, not yet fully understood, similarities with the bound state formation [33].
Thus, the study of peculiar states in the metallic regime, which in 3D systems exhibit
huge amplitude spikes on the top of homogeneous background, should help to comprehend
completely the quantum mechanisms which evolve extended into localized states. In 3D
the simple quantum interference picture, like that of weak localization introduced below,
is not exhaustive (unlike in the two-dimensional systems where it provides the complete
explanation of Anderson localization through its divergence in the thermodynamic limit).
In the last Chapter (9) of Part III a theory for the experimentally measured finite fre13 Universality
in disordered electron physics usually refers to independence on the details of
disorder, but in some cases also included in the term is the independence on the size and shape of
the system (or the degree of disorder in certain limits) [30].
11
quency conductivity of ultra-thin quench-condensed Pb films is presented. The experiments
on the infrared beam were performed at Brookhaven National Laboratory. Comprehensive
analysis of the interplay between quantum effects and classical electromagnetic effects on
small grains has favored an explanation based on classical percolation in an AC random
resistor network.
The Chapters are mostly self-contained since they are derived from the research publications. The common calculational methods and concepts are explained in detail in the
introductory Chapter of Part I, so that they can be referred when used for solving the specific
problems of other Chapters (some general ideas on MITs are given in the introduction of the
Part III). To steer the interest of a reader, we would like to highlight that some of the most
interesting results listed above are actually very transparent, elucidating the well established
paradigms in the field. But once the mathematical formalism (or analogously experimental
techniques) are mastered, the “only” task left is to ask the right questions. The results are
usually not a definitive answer but, being open-ended, pose new questions. Frequently, the
examples dealt with are in the regime of strong disorder. In general, our approach follows
the typical way of attacking problems which is favored by the theoretical community—use
whatever tool is necessary to sort out the problem. In the work presented here this means
employment of both analytical and numerical techniques, quantum as well as semiclassical
formalism to compute conductance, statistical approach to non-integrable (quantum chaotic)
systems etc. We were “forced” to tackle most of the major tenets of the disordered electron
systems physics: Anderson localization and its precursors (like weak localization), percolation, critical phenomena at Metal-Insulator transitions, statistical distribution of physical
quantities (brought about by quantum coherence and randomness which induce large fluctuations of physical quantities) in finite-size systems (as well as their scaling with increasing
systems size), etc.
We complete this introduction by giving an overview of the main methods of non-
12
equilibrium quantum statistical mechanics which are used to explore the transport regimes
explicated (or alluded) above. The mathematical details are reserved for Ch. 2. The boundaries of the sectors of the theory can be delineated by looking at the relevant parameters:
kF , the product of the Fermi wavevector kF and a mean free path , or alternatively, the
dimensionless conductance g. Using the mean free path to account for the impurity scattering means that averaging over the ensemble of all possible impurity configurations is
implied, thus restoring various symmetries on average. The impurity ensemble is defined as
a collection of systems having the same macroscopic parameters (like the average impurity
concentration) but differing in the detailed arrangement of disorder. For kF 1 the impurity and temperature dependence of the (average) transport quantities can be obtained in the
framework of the Bloch-Boltzmann theory [34]. It is highly successful for lightly disordered
conductors and represents an example of the semiclassical14 approaches with the meaning:
semi→some part of the theory deals with quantum mechanics—like Bloch waves which take
into account rapidly varying periodic potential of the average ion arrangement (i.e., band
structure effects on the effective mass), quantum collision integral in the Boltzmann equation and the Fermi-Dirac statistics for electrons; classical→the applied field and motion of
the electron in response to it is treated in a classical manner (e.g., the quantum interference effects from the scattering on successive impurities are not taken into account). The
Boltzmann equation is used in Ch. 3 (as a reference theory compared to the more involved
quantum transport methods) and Ch. 6 (where quantum corrections in the classical point
14 Another
usage of the term semiclassical (with a different meaning) is common in the picturesque
treatment of interference effects in disorder physics: the so called semiclassical approximation (SCA)
uses intuitively appealing picture of Feynman path integral formulation of quantum physics (which
is the closest one can get to quantum world while moving from classical concepts). In SCA one
adds the amplitudes for the motion along the classical trajectories with appropriate phases and
then squares the amplitude to get the probability [35]. This phase information included in SCA is
totally neglected in the Boltzmann semiclassical theory.
13
contact problem are small).
The quantum effects in the weakly scattering regime (another synonym for the diffusive
metallic regime introduced before) are revealed through diagrammatic perturbation theory
where the small parameter for systematic expansion is 1/kF (or 1/g). The celebrated
examples are weak localization (WL) [36] (quantum correction to the average Boltzmann
conductance of the order of GQ ) or sample to sample conductance fluctuations [37] (with
variance of conductance of the order of GQ ). One should be aware that criterion for the
validity of Boltzmann equation, kF 1, is applicable only in 3D. The two-dimensional
(2D) “metal” (i.e., non-interacting electron gas in a random potential outside of the magnetic
field and without spin-orbit scattering) is not a conductor but for arbitrary small amount
of disorder it is an insulator. This is one of the astonishing results of the scaling theory
of localization as well as its microscopic (in the perturbative regime) justification, namely
WL theory. In 2D a “small” WL correction, which arises from the interference of coherent
quantum-mechanical amplitudes along the time reversed closed paths (therefore unaffected
by the averaging over disorder which otherwise cancels out random interference effects),
diverges with the system size L. Thus, WL in 2D drives the system into an Anderson
insulator (which exists only in the thermodynamic limit L → ∞ ⇒ g → 0). The lower
critical dimension for the LD transition is two.
The development of mesoscopic physics [9] has unearthed the fluctuations effects in
physical quantities generated by the sensitivity of quantum transport to specific arrangement
of impurities. This has entailed the shift of the research in disordered physics toward the
studies of full statistical distributions [5] of physical quantities (which is called the mesoscopic
approach in the folklore of the community) which characterize the finite-size samples and
contain the seed of emerging localization even in the case of good metals (cf. Ch. 8). Also,
the weirdness of quantum non-locality and quantum measurement theory was encountered
in matter on a much bigger scale than previously reserved for the atomic-size systems. For
14
example, the conductance of mesoscopic samples contains nonlocal terms since carrier wave
functions are not classical local objects, but instead probe the whole phase-coherent region.
Therefore, the conductance is non-zero far from the classical current paths throughout the
sample and is not symmetric under the reversal of magnetic field. This leads to surprising
effects (at least for a “classical mind”). For instance, it is enough to shift a single impurity
to observe the conductance fluctuations [9] of the same magnitude ∼ e2 /h as if the whole
impurity configuration has been changed.15
Mesoscopic physics has not been driven just by the inquisitive theoretical mind but
most importantly by the experiments [9] brought about by the technological advances in
nanotechnology. The fabrication of small samples (typical dimension L < 1 µm) at low
temperatures (typically T < 1K 0.09 meV) has allowed quantum coherence to extend
throughout a disordered conductor. These conductors are still much bigger than a molecule,
but smaller than macroscopic samples traditionally studied in the Condensed Matter Physics.
The motion of an electron in such samples is coherent since it propagates across the whole
sample without inelastic scattering, thereby retaining a definite phase of its wave function.
The other quantum effects arise from the discreteness of electronic energy levels. However,
the interaction with the outside world can broadened the levels enough to make these effects
less relevant.
We are accustomed to macroscopic samples which are self-averaging and thus amenable
to a statistical approach aimed at bulk properties, which assumes the thermodynamic limit at
the end of computation. In mesoscopic physics one deals with finite-size samples coupled to
the environment (like that of transport measurement circuits). The meaning of the statistical
15 The
shift of a single impurity in a diffusive sample will affect the phase of all Feynman paths
which passed through it. The resulting change in conductance is of the order e2 /h times the
fraction of trajectories which are affected by the shift, L2 /2 Nimp , where Nimp is the total number
of impurities in the sample. Changing the position of Nimp 2 /L2 Nimp impurities will completely
change the interference pattern and thus generate new member of the impurity ensemble [9].
15
approach, applied still to many (e.g., 1019 ) elementary objects like electrons and atoms, has to
be adjusted accordingly. The resistivity to applied voltage arises from the degrees of freedom
such as: the static disorder potential created by impurities, defects and the inhomogeneous
electric field caused by the surrounding media. These microscopic details influence global
quantities like conductance, revealed in the transport experiments as a specific fingerprint of
the mesoscopic conductors [38]. Therefore, the properties of the whole ensemble of disordered
conductors are studied in mesoscopic physics together with the quantum statistical treatment
for an individual conductor. Even in the diffusive regime (characterized by Ohmic behavior
of conductance) the conductor can no longer be described just by the bulk material constants,
like conductivity σ which is related to the conductance G = σA/L (A is the cross sectional
area of the conductor). This has entailed the development of a (mesoscopic) transport theory
(or appropriate revisions of “old” approaches) based on sample-specific quantities which are
meaningful for a given sample measured in a given manner.
Mesoscopic samples are natural realization of the systems studied in the context of
Anderson localization. They were previously encountered only as theoretical constructs
(with size limited by the computer power in the research based on numerical simulations).
One can say that “mesoscopic physics” has extended and encompassed all of the previous
research in disordered electron physics. The LD transition is a generic continuous quantum
(T = 0) phase transition [39]. True insulators, characterized by zero conductivity, exist only
at zero temperature (and in the thermodynamic limit), since at finite temperatures inelastic
processes foster hopping conduction [40]. Hopping conduction is not considered in this
thesis, so what we mean by “transport in the strongly localized regime” is zero-temperature
transport in finite-size samples (for which conductance is non-zero, but exponentially falling
with the system size). Thus, inelastic scattering (e.g., with phonons) is introduced only
phenomenologically through the cutoff on the coherent propagation. When the parameter
kF ∼ 1 becomes close to one (the so called Ioffe-Regel criterion [41]) the semiclassical
16
theory (as well as the notion of ) breaks down, thus signaling that a fully quantummechanical treatment of transport is necessary. The finite-size sample conductances
g corresponding to this naı̈ve criterion (which we show explicitly in Ch. 3) can still be
above g ∼ 1, which is the “modern”, scaling theory [8], condition for entering the regime of
strong localization. In the localized phase the intuitively appealing picture of semiclassical
theory does not exist and the picture of Anderson localized states takes over. In the samples
with strong disorder (or close to the mobility edge), one has to use the non-perturbative
quantum methods, like numerical simulations employed in this thesis. The other available
non-perturbative methods (analytical and useful in low-dimensional systems) include the
recently developed formalism of supersymmetric nonlinear σ-model (SUSY NLσM) [29],
which is a field theoretical formulation of the localization problem,16 and RMT of quantum
transport [30] in quasi-1D disordered wires.
As stressed in the thesis title, most of the systems studied here are of finite size. This
makes it possible to treat exactly the scattering on impurities and, therefore, access all transport regimes. In such pursuit we use the appropriate lattice models, like the tight-binding
Hamiltonian [2]. The lattices are typically composed of ∼ 1000 atoms (which allows us
to use a fashionable term “nanoscale” conductors), the size being limited by the available
computer memory and computational complexity [42] of numerical algorithms used to invert
or diagonalize matrices. In the spirit of mesoscopic transport methods [43], the conductors
are usually placed between two semi-infinite disorder-free leads. This two-probe geometry
is naturally related to the circuits encountered in the real world of transport experiments
(although experimentalist favor multiprobe geometries). The mesoscopic methods provide
16 In
the context of SUSY NLσM Efetov denotes all transport amenable to perturbative quantum
techniques (like WL or UCF), including the Boltzmann theory as the lowest order approximation,
a semiclassical theory. Lacking a better language, we use this definition in some Chapters of the
thesis.
17
the efficient means for finding the mobility edge, as utilized in Ch. 8, or even getting the
localization length from the scaling of conductance [44]. Despite the fact that localization
theory is in essence the theory of transport in disordered solids, the conductance based calculations used to be a “dream” in the “old”⇔pre-mesoscopic times. Since all computational
schemes for transport properties, utilized before mid 80s, were crammed with arbitrary small
parameters [45] (like broadening of the delta functions in the Kubo formula in exact state
representations, cf. Sec. 4.2, or small imaginary part added to the energy in the Green
function based expression, cf. Sec. 2.4.1), numerical tricks were required to reach the static
limit for the conductance of a finite-size sample. Therefore, the exact conductance of such
sample was, for practical purposes, out of reach.
Research in Condensed Matter Theory is inextricably tied to experiments, which provide
guidance and the ultimate test of theory.17 The relation of the thesis to experimental research
is multifaceted. In Ch. 9 a theory has been provided for the transport measurements on
ultrathin Pb films. The exact result for the conductance of the classical point contact,
presented in Ch. 6, has also been simplified into a useful formula for the experimentalist
who find these elements regularly in various circuits. Although most of the thesis deals
with basic issues of transport theory and its proper applications in specific systems, the
interesting results from the application of these formalisms to specific disordered conductors
should serve as a predictions on what one should observe in experiments.
Aside from the connections inside the field of Solid State Physics and scientific curiosity, one should bear in mind that theoretical modeling played a key role in the invention
of the transistor and later development of integrated circuits. Thus, fundamental research
has always been important in opening new frontiers for technological applications. Device
modeling for the present day Si-based microelectronics is founded on the semiclassical ap17 The
other two pillars of the scientific method, which is followed when concocting new theories,
are simplicity and generality.
18
proximation that considers dynamics of electrons and holes to be those of classical particles,
except that their kinetic energy is determined by the semiconductor bands. This is usually done by employing the effective-mass approximation. Technically, modeling involves
grappling with the Boltzmann equation using drift-diffusion approximation, higher-order
hydrodynamic approximation or a direct approach using Monte Carlo techniques. At the
limits of conventional electronics (below 100 nm) classically minded human beings (including
present device engineers) are faced with electronics living in the strange world of quantum
mechanics (like tunneling, quantum interference, etc.). Thus, the so called nanodevices (a
recent example being single molecules [46]) will require quantum modeling of transport. This
is a new frontier for the continuation of research and application of techniques developed in
this thesis.
19
Part I
20
Chapter 2
Linear Transport Theories
I understand what an equation means
if I have a way of figuring out
the characteristics of its solution
without actually solving it.
— Paul A. M. Dirac
2.1
Introduction
The experimental and theoretical advances in our understanding of mesoscopic transport have shed a new light on various conceptual issues in transport theory and, in fact,
“enforced” the major revisions in the theory of electrical conduction [47]. The discovery
of various mesoscopic effects (brought about by the progress in nanostructure technology),
such as universal conductance fluctuations (UCF) [37], conductance quantization [48, 49],
the effect of a Aharonov-Bohm flux on the conductance [50] and on the thermodynamic
properties (persistent currents [51]) in mesoscopic rings, etc. has led to reconsider the role of
quantum coherence of electron wave functions in disordered electron systems. This coherence
was studied earlier1 in the guise of Anderson (“strong”) localization [2] or weak localization
1 Before
the emergence of mesoscopic physics, Anderson localization (as the major quantum
interference effect in disordered electron systems) was approached from the viewpoint of critical
21
(WL) [36]. Mesoscopic conductors are smaller than the dephasing (or coherence) length Lφ .
The length Lφ (usually ≤ 1 µm in the present experimental techniques) is determined at
low temperatures by the electron-electron inelastic2 scattering [17] (i.e., scattering on the
fluctuating potential generated by other electrons). The important insight of mesoscopic
physics is that elastic scattering on impurities does not destroy the phase coherence [53]. In
disordered conductors Lφ =
Dτφ is expressed through the phase-relaxation time τφ ∝ T −p
with p = 2 for the case of electron-electron scattering according to the Landau Fermi liquid
theory, or p < 2 in the presence of strong disorder (for scattering on phonons p > 2). The
dephasing time τφ is defined as a time after which the mean squared spread in the phase δφ
of the electronic wave function is of the order of one, δφ ∼ τφ δ/h̄ ∼ 1 (δ is the energy
exchanged in the particular collision processes). It can be orders of magnitude longer than
the momentum relaxation time, thereby giving rise to mesoscopic effects in disordered conductors. Although dephasing rate 1/τφ can be expressed as the sum of contributions arising
from the electron-electron and electron-phonon interactions, at low temperatures electronelectron interaction (EEI) dominates and it is strongly enhanced by the static disorder [15]
due to the long-range diffusive correlations of single electron wave functions.3 The parameter
τφ is also of fundamental interest for the Fermi liquid type behavior: the single particle states
are well defined for kB T τφ h̄ [17]. The commonly accepted view is that τφ should diverge
phenomena theory, together with the other critical phenomena where disorder plays important role
(like percolation or spin glasses).
2 The
term “inelastic”, in the sense of general theory of decoherence, would imply just changing
the quantum state of the environment [52]. For example, this includes even zero energy transfer
processes where environment is flipped into a degenerate state.
3 In
general, the EEI generates three different scattering times: the outscattering time τe−e
appearing in the kinetic equation formalism [15], the dephasing time τφ , and the energy relaxation
time [17] τ (during which a “hot” quasiparticle of energy kB T thermalizes with all other
electrons). The times τφ and τe−e coincide in 3D samples [16], τφ ∼ τe−e ∝ T −3/2 .
22
with T → 0 because of the decreasing space of states available for the inelastic scattering.
Another length scale, the thermal diffusion length LT =
h̄D/kB T , is important for
some mesoscopic phenomena. At length scale LT quantum-mechanical coherence effects are
cut by thermal smearing effects generated by energy of the particle being in the interval of
order of kB T around EF . While both Lφ and LT are relevant for UCF [54], the interaction
correction depends only on LT and WL at finite temperature is determined by Lφ (even
surviving the self-averaging in macroscopic samples which are bigger than Lφ ).
In mesoscopic systems the electron wave function retains a memory of its initial quantummechanical phase even though it can experience elastic scattering from impurities or the
sample boundaries. This makes the quantum interference effects (i.e., linear superpositions
in the Hilbert space of quantum states) observable in transport experiments. Transport in
such systems has to be treated as a fully quantum-mechanical process with the appropriate dynamical equation being the Schrödinger equation. Thus, the mesoscopic conductor
is viewed as being effectively at zero temperature. In low temperature and low bias measurements only electrons at the Fermi energy carry the current which is analogous to doing
optical experiments with a monochromatic light source [43].
In this Chapter we survey different approaches to linear transport, in the spirit of
mesoscopic physics. We emphasize their mutual connections and domains of validity. The
linear(ized) quantum transport methods provide, as an end product, the expressions for the
quantum transport coefficients in terms of the equilibrium quantities. This is a consequence
of the fluctuation-dissipation theorem (FDT) which connects non-equilibrium properties in
the systems close to equilibrium (where response, like current, is proportional to a “small”
driving field) with thermal fluctuations in equilibrium. The Kubo linear response theory is
a prime example of such thinking (Sec. 2.4.1). The Landauer-Büttiker scattering formalism
(Sec. 2.4.2) is particularly suited for transport in mesoscopic (i.e., phase-coherent) conductors
of finite size. In such conductors a single wave function throughout the sample can be defined
23
and a complicated problem, such as quantum transport of degenerate Fermi gas in a random
potential, can be studied using just one-particle quantum mechanics. In both the Kubo
and Landauer formulas for the linear conductance one is using conservative Hamiltonians
(which generate reversible quantum dynamics), and proper application of such schemes,
as well as the connection of two formalism, turns out to be related to such eternal issues
as the understanding of the appearance of irreversibility (i.e., dissipation) [55] from the
reversible microscopic underlying dynamics. The Non-Equilibrium Green Function (NEGF)
formalism (Sec. 2.4.3) is the most general (and technically most demanding) approach to
quantum kinetics, i.e., applicable to both non-coherent and non-linear problems. Therefore,
it encompasses both Kubo and scattering formalisms in the limits of their validity. When
quantum interference effects are not important one can use the Boltzmann equation which,
in its linearized form, gives an expression for the semiclassical conductivity. All quantum
formalisms listed above reproduce the semiclassical Bloch-Boltzmann equation (Sec. 2.3) to
leading order in 1/kF .
It is assumed that conduction can be described in terms of a gas of non-interacting
(quasi)particles. This becomes a subtle point when one starts to think about the role of
EEI [15]. In a single band metal and in the absence of umklapp processes total electron momentum is conserved and electron-electron collisions do not affect conductivity [57]. However,
this argument requires translational invariance, while in disordered conductors interaction
gets modified from the standard picture of screening in a translationally invariant Fermi
gas. Therefore, disorder-dependent effective interaction induces quantum corrections to the
semiclassical conductivity [14] of the same order as WL (which arises from interference effects). This immediately leads to the questions on the boundaries of validity of the Fermi
liquid concepts in disordered systems [15]. We will assume that interacting system is replaced by a set of non-interacting quasielectrons (at low T and for small perturbation) with
mass renormalized by interactions as well as by the band structure effects. This implies that
24
transition rates for scattering of quasiparticles on charged impurities are to be evaluated
for the screened interaction. At T = 0 quasielectrons fill energies up to the Fermi energy
EF (electrochemical potential). In non-equilibrium situations the electrochemical potential
is not well-defined since the electron distribution function is not a Fermi function. Therefore, the only meaning ascribed to quasi-Fermi level is that corresponding Fermi distribution
integrated over the energy should give the correct number of electrons [58].
The following Section (2.2) prepares the ground for subsequent developments by introducing the basic linear response quantities. It provides some general remarks on the Ohm’s
law and constraint which current conservation in the steady state (DC) transport imposes
on the formulas for conductance (or the nonlocal conductivity tensor introduced below). We
give several examples (both elementary and research results) of the importance of keeping
in mind current conservation when computing transport properties. This Chapter should
serve as a reference when a particular method is invoked in the rest of the thesis (which
then saves the space and avoids unnecessary repetitions). This is especially true of the realspace one-particle Green function technique and the related Landauer-type formula for the
conductance, which we study in Sec. 2.5.
2.2
Ohm’s law and current conservation
We shall not cease from exploration
And the end of all our exploring
Will be to arrive where we started
And know the place for the first time.
— T. S. Eliot
The basic global transport property, for small applied voltages, is the (linear) conductance or, equivalently, the resistance R = 1/G. The conductance G is introduced by the
25
Ohm’s law
I = G V,
(2.1)
as a proportionality factor relating the total current I to the voltage drop V across the
conductor. This relation is valid for any conductor in the linear transport regime and quite
plausible (or even “trivial”). Linearity4 is ensured when bias V → 0 is small compared
to kB T . In a modern language of mesoscopics, Eq. (2.1) corresponds to a finite conductor
placed between two ideal semi-infinite leads (at least in the view of a theoretician). This
is elaborated further in Sec. 2.5 and illustrated there on Fig. 2.1. Experimentalists often
favor more complicated situations than the one depicted in Fig. 2.1. The standard example
is the four-probe measurement [59] in which (typically a low frequency AC) current is fed
through two current leads while the voltage is measured using two auxiliary voltage probes
attached at some points along the current carrying conductor. If all leads are treated on the
same footing, one arrives at the generalization of Ohm’s law for the multi-probe measuring
geometry [60]
Ip =
Gpq (Vq − V0 ),
(2.2)
q
where linearity is ensured in the case of small currents. Here Ip is the total current through
lead p and Vq − V0 is the difference between the voltage measured at probe q and a reference
voltage V0 (which is usually taken to be zero, at least in theoretical analysis). This formula
introduces the conductance coefficients Gpq (independent of voltage in the linear regime)
between lead p and lead q instead of the simple conductance G in Eq. (2.1).
The generalization of a measurement geometry becomes especially important for the
mesoscopic samples. In the standard lore of quantum mechanics the observation conditions
strongly influence the result of a measurement [61]. As a consequence of quantum nonlocality,
the transport measurements with probes spaced less than Lφ give results for the whole sample
4 For
exhaustive and elucidating analysis of the conditions for linearity of transport see Ref. [43]
p. 88-92.
26
plus probes [62], instead of just depending on the part of the sample between the probes (like
in the standard electrical engineering circuit theory). In what follows we will focus on the
two-probe geometry where voltage is measured between the same leads through which the
current is passed. In other words, the two-probe conductance would be measured between
the points deep inside the reservoirs. Inasmuch as the phase of an electron entering the leads
is randomized before reinjection into the disordered region, the dephasing length Lφ at T = 0
is, by definition, equal to the distance L between the leads in the two-probe configuration.
The local form of Ohm’s law contains substantially more information than (2.1). It
gives the local current density j(r) in terms of the local electric field E(r) = −∇µ(r)/e (in
the noninteracting picture electrochemical potential is identified with voltages eV and serves
to parametrize carrier population, as explained above) inside the sample5
j(r, ω) =
dr σ (r, r ; ω) · E(r, ω).
¯
(2.3)
This relation defines the nonlocal conductivity tensor σ (r, r; ω) as the fundamental micro¯
scopic quantity in the linear response theory. Its meaning is obvious—it gives the current
response at r due to an electric field at r . It turns out that quantum mechanics generates
nonlocality of σ (r, r ; ω) on the scale Lφ , but there is also classical nonlocality [56] enforced
¯
by current conservation (cf. Sec. 2.3) which extends throughout the entire sample, irrespective of Lφ . Thus, nonlocal conductivity tensor (NLCT) depends on both r and r (it cannot
be made local by a Fourier transform [57]) and is not translationally invariant for a specific sample, unless thermal averaging or dephasing effectively makes it possible to average
over the impurity ensemble. In most of the discussion to follow we will be analyzing the
5 It
is possible to treat E(r) as an externally applied electric field and then include the effects
of Coulomb interaction between electrons as a contribution to the vertex correction. However, the
usual approach is to use local electric field Eloc (r), which is the sum of external field plus the
field due to the charge redistribution from the system response, and treat electrons as independent
particles [63].
27
properties of transport in the zero-frequency (DC) limit ω → 0. The quantum-mechanical
description requires j(r) to be the expectation value of the current density operator (we will
avoid another bracket notation and assume that in the quantum context classical labels mean
quantum-mechanical expectation values). For example, in the case of a system described by
a statistical operator ρ̂
j(r) = Tr ρ̂ ĵ(r) ,
(2.4)
or in general non-equilibrium situation, where kinetic properties are embodied in the doubletime correlation function G< (cf. Sec. 2.4.3),
1
j(r) =
2π
dE
eh̄ ie2
(∇ − ∇)G< (r, r ; E) +
A(r)G< (r, r; E)
2m
m
,
(2.5)
r=r
per single spin component. The quantum-mechanical current density operator (for a single
particle) is defined by replacing the classical quantities in the current density definition by
respective operators and symmetrizing the products of Hermitian operators
ĵ(r, t) =
e
e
[n̂(r)v̂ + v̂n̂(r)] =
[n̂(r)(p̂ − eA(r, t)) + (p̂ − eA(r, t))n̂(r)]
2
2m
= ĵ0 (r, t) + ĵd (r, t),
(2.6)
e
[n̂(r)p̂ + p̂n̂(r)],
2m
n̂(r)e2
A(r, t),
ĵd (r, t) = −
m
ĵ0 (r, t) =
(2.7)
(2.8)
where n̂(r) = |rr| is the particle density operator, v̂ = p̂/m is the velocity operator, m is
the effective mass of a particle, and A(r, t) is the vector potential. For many-particle system
expression (2.6) should be summed over all particles. The conservation of current in the DC
transport implies that
∇ · j(r) = 0.
(2.9)
This, together with (2.3) and boundary conditions at infinity and at insulating surfaces
V (r) = 0, x → −∞; V (r) = V, x → ∞;
(2.10)
nS (r) · j(r) = 0, r ⊂ boundary,
(2.11)
28
forms a closed set of equation for determining the (conservative) electric field E(r). The
vector n(r)S is the unit vector normal to the surface of interest at a given point r. When
there are interfaces in the conductor, an extra boundary condition at the interface should
be added [64]
nS (r) · j(r) = gS (r)(V1 (r) − V2 (r)), r ⊂ interface,
(2.12)
where gS (r) is the unit area conductance of the interface.
Transport experiments do not measure explicitly NLCT σ (r, r). Instead they measure
¯
(macroscopic) conductance and theory should provide an expression for this experimentally
available quantity. By integrating (2.3) over the cross-section of the conductor a total current
is obtained in the finite sample of volume Ω
I=
S
dr nS (r) · j(r) =
dr
S
Ω
dr σ (r, r) · E(r),
¯
(2.13)
and the conductance from (2.1). This formula assumes that the electric field E(r) is the local self-consistent field (determined by current conservation and self-consistency between the
potentials and charge density) inside the conductor. The electric field inside the disordered
sample is a very complicated function of the position due to the local charge imbalances . It
depends on the precise location of the impurities which give rise to highly localized fields (the
so called residual resistivity dipoles [4]) centered on the impurity sites. The residual resistivity dipoles result from the difference in spatial variation of electrochemical and electrostatic6
6 The
bottom of the band Es follows [65] the electrostatic potential energy eV . Therefore, a
measurement of the difference between absolute values of Es at two points gives the change in
the electrostatic potential. As emphasized before, electrochemical potential is equilibrium concept,
and in non-equilibrium is defined conventionally as the absolute position of the Fermi level which
would produce the local electron number density. The change of such µ (i.e., weighted average of
the occupancy of electronic energy levels) would be measured by a voltmeter which has a constant
weighting factor [65]. In mesoscopic considerations it is usually assumed that carriers moving in a
particular direction are in equilibrium and can be assigned an electrochemical potential [65], which
29
potentials across an impurity—sharply or over the screening length, respectively. The problem of localized fields when coherent multiple scattering takes place on random scatterers
is still an open question [67]. Numerical simulations of a single disordered sample show
extremely inhomogeneous current flow on a microscopic scale [68].
The conductance can be expressed by dividing the dissipated power by the voltage
squared
G=
1 1 dr
E(r)
·
j(r)
=
dr dr E(r) · σ (r, r) · E(r ).
¯
V2
V2
Ω
(2.14)
Ω
It the field E(r) is taken to be homogeneous (E = V /L) we obtain the volume averaged
conductance tensor
G=
1
L2
Ω
dr dr σ (r, r).
¯
(2.15)
In a general non-isotropic case Eqs. (2.14), (2.15) are to be understood as the relation between
the conductance tensor7 and the volume integrated tensor σ (r, r). For a rectangular sample
¯
the conductance can be expressed in terms of the conductivity, G = σA/L where A is the
cross sectional area and L is the length of the sample. Macroscopic conductivity σ (limit
Ω → ∞ assumed, while keeping the impurity concentration finite) relates the spatially
averaged current j =
dr j(r)/Ω to the spatially-averaged electric field,
j = σE.
(2.16)
For ballistic systems or restricted geometries only conductance is a meaningful characteristic
because conductivity as a local quantity, defined by (2.16), does not exist. In addition to
the conductance, knowledge of NLCT opens up the possibility to calculate local properties,
such as the distribution of current densities inside the conductor.
then clarifies the difference between the two-probe and four-probe conductances [66].
7 In
homogenously disordered conductors averaging over the disorder will restore the symmetries
(translational and rotational), so that conductance becomes a scalar quantity, e.g., G = 1/3Gxx +
Gyy + Gzz .
30
Theoretical studies of UCF have given a strong impetus to reexamine the properties
of σ (r, r). This approach was invoked since the calculations using the bulk conductance
¯
G of a rectangular sample do not contain enough information to account for the measuring geometry effects [69], or to investigate the current density fluctuations [70] and related
voltage fluctuations in the multi-probe devices [56]. Surprisingly enough, it was shown only
recently [69] that current conservation, ∇ · j(r) = 0, imposes stringent requirement on any
microscopic expression for NLCT
∇ · σ (r, r ) · ∇ = 0.
¯
(2.17)
In the presence of time-reversal invariance (magnetic field absent, B = 0) the requirement
becomes even stronger
∇ · σ (r, r) = σ (r, r ) · ∇ = 0.
¯
¯
(2.18)
The condition (2.18) is sufficient, while (2.17) is necessary, to show that [71]
G=−
S1 S2
dS1 · σ (r, r ) · dS2 ,
¯
(2.19)
by using the divergence theorem to push the integration (2.14) from the bulk onto the
boundary surface8 going through the leads and around the disordered sample (the integration
over this insulating boundary obviously gives zero contribution because no current flows out
of it). The surface integration in the two-probe conductance formula (2.19) is over surfaces S1
and S2 separating the leads from the disordered sample. The vectors dS1 and dS2 are normal
to the cross sections of the leads, and are directed outwards from the region encompassed
by the overall surface (composed of S1 , S2 and insulating boundaries of the sample). It
is assumed that voltage in one of the leads is zero, e.g., µL = 0 and µR = eV . It is
important to point out that this formula can be generalized [71] to arbitrary multi-probe
8 The
mathematical subtleties (like proper order of non-commuting limits) in finding zero and
non-zero surface terms (forgotten even by Kubo!) when formulating linear transport microscopically
are accounted in [72, 73].
31
geometry, while the volume-averaged conductance (2.15) is meaningful only for the two-probe
measurement. Also, the expression (2.19) is generally valid in the presence of interactions,
where many-body effects can be introduced using Kubo formalism (cf. Sec. 2.4.1) to get
σ (r, r ) microscopically. However, this route is tractable and useful especially in the case of
¯
non-interacting quasiparticle systems, thereby providing the link [71] between two different
linear response formulations—Kubo and Landauer.
The only information about the electric field needed to derive the formula (2.19) is
fixed potentials in the leads [72]—the current is uniquely determined by the asymptotic
voltages (Landauer-Büttiker, Eq. (2.2)), instead of being a complicated nonlocal function
of the field (Kubo, Eq. (2.3)). This corresponds to the experimental situation where only
applied voltage is known. Thus, DC conductance can be computed without the knowledge
of detailed distribution of charges and electrical fields generated by them. In fact, instead
of the true self-consistent field E(r) one can use any electric field distribution [69] Ecl (r)
which gives the voltage V when integrated along arbitrary path connecting two leads. The
boundary conditions require that the components of Ecl (r) and of σ (r, r ) perpendicular to
¯
the insulating boundary vanish. Moreover, the two factors of E(r) in (2.15) can be chosen to
differ from each other. This then leads to (2.19) when the electric field is concentrated in the
left lead for one factor and in the right lead for the other factor. Such freedom in choosing
electrical field becomes advantageous when devising the most effective computational scheme
for conductance (cf. Sec. 2.5).
The outlined procedure remains applicable for electrons interacting through a selfconsistent field. In the case of finite frequency transport, charge and current conservation require consideration of the long-range Coulomb interaction [74]. Nonetheless, it was
shown [75] that these features of the static limit remain valid for transport at finite frequencies which are smaller than the inverse passage time across the sample τD . This time is given
by τD L/vF in the ballistic regime (L < ) or τD L2 /D in the diffusive ( L ξ)
32
regime. For ξ < L < Lφ the sample is in the insulating phase. The independence of the
linear conductance on the field distribution can be crudely understood as follows [76]. The
incorrect field (which only gives the right potential in the leads) is compensated by the
concentration gradients which together provide the necessary spatially varying driving force
and ensure the continuity of current. Taking into account the Coulomb interaction between
the electrons will immediately generate the genuine field distribution inside the sample. The
self-consistent field becomes important [77] in the nonlinear transport.9 It is also important
for linear transport in the so called semi-classical approximation (SCA) [80] to the quantum
expressions for conductance (i.e., Kubo or Landauer formulas in Sec. 2.4.1 or Sec. 2.4.2,
respectively). In SCA [35] each classical trajectory corresponds to a quantum-mechanical
amplitude. The simple qualitative picture of quantum interference phenomena arises after
adding these amplitudes for the motion along classical trajectories (with appropriate phases)
and then squaring the sum. The current conservation in quantum theory is a consequence
of the unitarity of quantum evolution. The evolution in SCA is only approximately unitary.
Then the expression for NLCT does not obey the current conservation conditions (2.17)
and (2.18) because of missing higher order corrections in h̄. Thus, SCA expression for the
conductance will depend on the electric field distribution inside the sample.
2.3
Semiclassical formalism: Boltzmann equation
The development of quantum mechanics has brought up the first quantum theories of
electrical transport. In a perfect lattice the eigenstates of the Hamiltonian are Bloch states
which span the irreducible representations of a translational group. The wave packets of
Bloch states are accelerated according to semiclassical formula h̄k̇ = eE where k is the
9 For
example, Büttiker [78] has emphasized that gauge-invariant description of nonlinear trans-
port requires a proper treatment of the long-range Coulomb interaction [79] which explicitly includes
the external gates and reservoirs.
33
central wavevector. This is valid for a single band and it would lead to an infinite conductivity. Thus the acceleration must be balanced by the scattering due to phonons and defects
which restores the distribution in k space towards the equilibrium state. Quantum mechanics enters through the cross section for scattering and band structure, but the balancing
processes are taken through occupation probabilities thus neglecting the coherent superpositions of probability amplitudes at a single scattering center or from different scatterers.
Such description of transport widens the application of Boltzmann equation, originally derived for dilute gases, to electronic transport. The Boltzmann equation follows directly [34]
from the Landau Fermi liquid theory (FLT) which views conductor as a gas of nearly free
(quasi)electrons. This is an effective theory10 which gives low energy and long wavelength
dynamics in terms of the quasiparticle distribution function F (k, r, t). The “quasiparticles”
are dressed electrons where the neglected interaction is absorbed in “dressing” (i.e., renormalized physical parameters of quasielectron). The distribution function gives the ensemble
average occupancy of the state with wave vector k in a “smeared” region (because of quantum uncertainty) of space time near (r, t). The evolution of F (k, r, t), the central quantity
of FLT, is actually given by the Boltzmann equation
∂F
∂F
∂F
+ k̇ ·
+ ṙ ·
=
∂t
∂k
∂r
dF
dt
,
(2.20)
scatt
where (dF/dt)scatt is the collision integral (a non-linear functional of the distribution function) which takes into account scattering processes responsible for changing the occupancy
10 Like
other effective (field) theories, FLT can be derived by coarse graining (“integrating out” the
short wavelength modes) the microscopic Hamiltonian. This is done in the spirit of renormalization
group procedure [81] using the special kinematics of the Fermi surface. Thus, FLT is able to treat
those correlations, induced by electron-electron interaction, that can be described by the continuous
and one-to-one correspondence between the eigenstates (ground state and low energy excitations) of
the non-interacting and interacting system (where interactions do not lead to any phase transition
or symmetry-broken ground state).
34
of state k.
The solution of a linearized Bloch-Boltzmann equation provides (linear in the electric
field E) deviation δF (k, r, t) from the equilibrium (Fermi-Dirac) distribution function f (k ).
This approach can be used to get NLCT and conductance, introduced as general concepts
in Sec. 2.2. The so called Chambers formula,11
|r − r |
3 σD (r − r )i (r − r )j
exp(−
),
σ Ch
(r,
r
)
=
¯ ij
4π
|r − r |4 (2.21)
occurs frequently in the literature [70]. It implies that an electron loses the memory of an
initial direction of motion on a distance of the order of mean free path (i.e., Chambers
NLCT is localized on the scale ). In (2.21) the ensemble average is taken through the mean
free path as a single parameter characterizing the distribution of impurities. However, this
expression does not conserve the current. The complete form of the semiclassical NLCT,
σ ij (r, r ) = σD [δij δ̄(r − r ) − ∇i ∇j d(r, r)],
¯
(2.22)
was emphasized in the study of UCF for complicated geometry of the sample and multiprobe
measurements [69]. Here δ̄(r − r ) is a sharply peaked function of the width , which can
(r, r) (2.21). The rescaled
be virtually taken as the Dirac δ function, and stems from σCh
¯ ij
diffusion propagator12 d(r, r), satisfies the equation
−∇2 d(r, r) = δ(r − r ),
(2.23)
subject to the boundary conditions d(r, r) = 0 on a conducting boundary and ∇n d(r, r) = 0
on an insulating boundary. Thus, the expression (2.22) is nonlocal without taking into
11 If
we use (dF/dt)scatt → (F − fLE )/τ (fLE is local equilibrium distribution function) in the
Boltzmann equation then: (1) current is not conserved, and (2) the exact solution for NLCT is
given by (2.21).
12 The
diffusion propagator (or “diffuson”) is the solution of equation −Dτ ∇2 D(r, r ) = δ(r − r ),
which is the long wavelength approximation [1] to the equation for the sum of ladder diagrams in
disorder-averaged perturbation theory.
35
account any quantum interference effects, and can be derived solely from the Boltzmann
equation [82]. However, using the possibility to choose arbitrary electric field, like e.g., the
“classical” field13 [56], ∇α Eαcl = 0, the volume integral in Eq. (2.14) of the nonlocal part
d(r, r) of NLCT (2.22) vanishes (which then does not invalidate UCF studies [61] using just
the local part (2.21)). The disorder-averaged conductance of a three-dimensional rectangular
sample of length L and cross section A is given by the semiclassical Boltzmann formula
2e2 4 M
A
= σD ,
h 3π L
L
ne2 τ
=
,
m
GD =
(2.24)
σD
(2.25)
where M = kF2 S/4π, n is the electron density, τ mean free time, m is the effective mass
of (quasi)electrons, and the simplest (spherical) Fermi surface is assumed. This formula is
also known as the Drude formula, although the Drude expression historically predates the
quantum-mechanical calculation of τ in the Bloch-Boltzmann formalism and the understanding of n/m as an effective parameter in FLT. In fact, effective parameters are provided by
experiments, and FLT gains predictive power only in non-equilibrium situations when it is
used in conjunction with the Boltzmann equation (2.20).
The picture of Bloch waves scattered occasionally, as implied in the Boltzmann formalism, requires that a particle freely propagates far enough to see the periodicity of the
surrounding medium. This means that the parameter (kF )−1 1 should be small (as well
as the similar parameter (∆E τ )−1 1 where ∆E is the interband transition energy [34]).
In disordered conductors this corresponds to a weak scattering limit. The real states in
disordered conductor are not plane waves because scattering broadens wavevector k into
∆k ∼ 1/. The broadening corresponds to the energy h̄/τ , i.e., (∂ε(k)/∂k)∆k ∼ h̄/τ .
Fully quantum-mechanical theories, like Kubo linear response theory (cf. Sec. 2.4.1) or nonequilibrium Green function formalism (cf. Sec. 2.4.3), produce Boltzmann theory as a lowest
13 This
electric field would exist if there were no charge and resembles the true field on the length
longer than the screening length.
36
order term14 when expanding their respective formulas for the conductivity in terms of the
small parameter 1/kF . In fact, for a long time it seemed that these rigorous (quantum)
formulations of transport were merely serving to justify the intuitively appealing Boltzmann
approach. The shift came with the first explicit calculation of quantum corrections like
weak localization [36]—a quantum interference effect which adds a term to the Boltzmann
result, and is responsible at low T for all of the temperature and magnetic field dependence
(“anomalous magnetoresistance” [84]) of the conductivity. Therefore, the “extreme” accord
between the theory and subsequent experimental activity has been achieved since WL is unpolluted by other phenomena happening at the same time. For strong disorder a continuous
quantum phase transition takes place and states undergo Anderson localization [2] due to the
multiple interference of electron wave functions. Also, for strong scattering on impurities a
complete quantum-mechanical description is required. This is clearly demonstrated in Ch. 3
where such calculations, in the transport regime in which putative mean free path would be
smaller than the lattice spacing (or ∼ 1/kF ), are compared to the Boltzmann result.
2.4
2.4.1
Quantum transport formalisms
Linear response theory: Kubo formula
The first fully quantum-mechanical theories of transport appeared in the mid fifties.
Particularly important, and widely accepted, has been Kubo’s formulation [85] of the linear
response theory (KLRT). This is an approach to non-equilibrium quantum statistical me14 The
success of the Boltzmann equation, e.g., in semiconductor systems, is sometimes far from
obvious. The same is true even in the case of some metals, like the strongly interacting ones,
example being Pb [34]. The pertinent expansions of the quantum kinetic equation in the case of
semiconductors are formal [83] because of the lack of small parameter or, equivalently, the largest
energy scale provided by EF in metals.
37
chanics based on the fluctuation-dissipation theorem (FDT): irreversible processes in nonequilibrium are connected to the thermal fluctuations in equilibrium. The use of FDT limits
the Kubo formalism to non-equilibrium states close to equilibrium. KLRT has its origins [86]
in the Einstein relation for the diffusion constant and mobility of a Brownian particle.
When KLRT is applied to the problem of electrical conduction, an isolated system is
subjected to an electromagnetic plane wave at frequency ω. By looking at the scattering of
the wave by the system one can deduce its conductance. The absorption is given through the
outgoing wave amplitude, while its phase gives the reactive type of information. KLRT uses
Schrödinger equation, which “does not know” about dissipation or openness of the sample,
and is essentially an extension of the theory of polarizability [67]. No stationary regime
can be reached if the system is neither infinite nor coupled to some thermostat. Thus, the
question of dissipation in the finite sample with boundaries, as well as general question
of the appearance of irreversibility from microscopic reversible dynamics, were always a
great concern of Landauer [55] (who felt that KLRT hides them under the carpet by its
computational pragmatism and efficacy). It is shown in Sec. 2.5 and Sec. 4.2 that mulling
over such deep problems in physics can also have a practical merit for those oriented toward
the calculational aspect of physics. The proper application of the Kubo formula on finite-size
systems is equivalent to choosing the corresponding Landauer formula, and requires to keep
in mind where the randomization is coming from.
In KLRT the current is viewed as a response to an electric field. The current density is
proportional to the field strength, i.e., it is linear in field for systems which are not driven far
away from thermodynamic equilibrium. The state of the system is described by the statistical
operator ρ̂(t) (or density matrix ρ̂(k, k , t) = k|ρ̂(t)|k when some representation is chosen).
This is obviously a generalization of the distribution function F (k, t) in the Boltzmann
theory (cf. Sec. 2.3) which includes phase-relationship between different states (off-diagonal
elements of ρ̂(k, k , t)), besides the occupation of the states (given by the diagonal elements
38
of the density matrix). In thermodynamic equilibrium
ρ̂0 =
e−Ĥ0 /kB T
,
Z
(2.26)
where Ĥ0 is the Hamiltonian of the unperturbed system and Z = Tr exp(−Ĥ0 /kB T ) is the
partition function (in grand canonical ensemble Ĥ0 should be replaced by Ĥ0 − µN̂). When
a fixed external electric field (in a gauge15 with only vector potential A being non-zero and
with small η to turn the field off at t → −∞)
E(r, t) = E(r)e−i(ω+iη)t = −
∂A
,
∂t
(2.27)
is imposed the evolution of the statistical operator is generated by the perturbed Hamiltonian16
Ĥ = Ĥ0 + Ĥ ,
ih̄
∂ ρ̂
= [Ĥ, ρ̂(t)].
∂t
(2.28)
The NLCT is extracted from the response to this external field ones the current expectation
value is expressed in the form (2.3). It is not necessary to use the total electric field (external
plus induced by EEI) in such derivation, as sometimes claimed in the textbook literature [88].
This stems from the fact that current induced by the external field is already linear in the field
and does not have any corrections due to induced charges [89], as long as we are interested in
the linear response. A simple example of this general feature of linear transport is given in
Ch. 6. There we start from the Boltzmann equation coupled to the Poisson equation for the
local induced potential, only to find out that, upon linearization, these equations decouple.
Therefore, the Hamiltonian Ĥ0 should include only EEI in the equilibrium (e.g., scattering
15 From
Maxwell equations one can get curl E = 0 to first order in small ω and |E|, so that field
can be treated as conservative E(r) = −∇Φ(r) [87] in the limit relevant for DC transport.
16 For
example, in the non-interacting system, which are mostly considered in the thesis, each
particle is described by the Hamiltonian Ĥ0 = p̂2 /2m + U (r) in a random potential U (r). When
electric field is turned on the relevant Hamiltonian is Ĥ = (p̂ − eA)2 /2m + U (r), where Ĥ Ĥ0 − (e/2m)(p · A + A · p), to linear order in E.
39
cross section of impurities should take into account self-consistent screening), while linear
currents are determined by external field or potential in the leads (cf. Sec. 2.2).
We only sketch a route to the quantum-mechanical expression for the nonlocal conductivity tensor below since this is a well covered subject in both research literature [71]
and lecturing notes [90]. Solution of the Liouville equation (2.28) by iteration in powers
of the perturbation Ĥ is cut on the first order (linear in field E), so that system in this
approximation is described by the statistical operator
ρ̂(t) = ρ̂0 + δ ρ̂(t) + O(E2 ).
(2.29)
The time evolution of ρ̂(t) defines the time evolution of the first order correction δ ρ̂(t) to the
statistical operator
ih̄
∂
δ ρ̂ = [Ĥ0 , δ ρ̂] + [Ĥ , ρ̂0 ].
∂t
(2.30)
Using the solution for δ ρ̂ the expectation value of the current density is obtained (to first
order in E)
j(r, t) = Tr ρ̂0 ĵ0 (r, t) + ρ̂0 ĵd (r, t) + δ ρ̂ĵ0 (r, t) .
(2.31)
The first term vanishes in equilibrium as a consequence of the time-reversal symmetry (i.e., in
the absence of magnetic field17 ). The Kubo answer for NLCT is obtained after rewriting (2.31) in the form of local Ohm’s law (2.3)
ie2 Tr (ρ̂0 n̂(r))
1
δ(r − r ) +
σ (r, r ; ω) =
¯
mω
h̄ω
∞
dt eiωt Tr ρˆ0 [ĵ0 (r, t), ĵ0 (r , t)] ,
(2.32)
0
where the first (“diamagnetic”) term is generated by Tr (ρ̂0 ĵd ) from Eq. (2.31). In the DC
limit ω → 0 (but ωt finite), which is usually taken before the limit T → 0, diamagnetic term
17 Magnetic
field generates closed current loops in translationally non-invariant system, making
the first term in (2.31) non-zero even in equilibrium. However, this term does not contribute to the
transport current [71], see also discussion below.
40
diverges,18 but is canceled by another divergent term in the second part of the formula. The
mathematical intricacies of separating NLCT into dissipative (oscillating in phase with the
field) and reactive part (oscillating out of phase), as well as taking different limits (like DC
limit) are treated meticulously in Ref. [71].
The expression (2.32) can be rewritten [90, 89] in terms of the (usually unknown) manybody eigenstates after the statistical operator and related thermal averages are expanded in
terms of the complete set of these eigenstates
Pβ − Pα β|j(r)|αα|j(r)|β
.
σ (r, r ) = −ih̄ lim+
η→0
¯
Eβ − Eα + ih̄η
α,β Eβ − Eα
(2.33)
Here Pβ = [ρ̂]ββ is the thermodynamic occupation probability of a many-body state |β. In
the non-interacting limit statistical weights Pβ become the Fermi function f (Eα ) for single
particle states replacing many-body eigenstates (we denote the eigenstates of a single-particle
Hamiltonian by |α throughout the thesis).
We focus now on the non-interacting Fermi gas in a random potential. The exact
many-body states of non-interacting systems are trivially expressible in terms of Slater determinants of single particle states. The expectation value of any single particle operator
is given as a trace O = Tr (ρ̂(t)Ô) with (now) single particle statistical operator ρ̂(t). In
equilibrium this operator is given by
ρ̂0 =
f (Eα )|αα|,
(2.34)
α
with Fermi-Dirac function f (Eα ) determining the occupation of the single particle exact
eigenstates in the impurity potential
Ĥ0 |α = Eα |α.
(2.35)
In the limit T → 0 the Fermi-Dirac function becomes f (Eα ) θ(EF − Eα ). The most
general NLCT for non-interacting systems (i.e., when magnetic field is present) consists of
18 This
divergence is formal and stems from using the vector potential to describe the electric
field instead of some gauge invariant form needed to describe the physical field.
41
two different terms. This becomes transparent after applying the Cauchy principal value
identity to the denominator of a non-interacting version of (2.33)
1
1
= −ih̄πδ(Eα − Eα ) + P (
).
Eα − Eα + ih̄η
Eα − Eα
(2.36)
The delta function here generates the term in NLCT which depends only on the states
within kB T of the Fermi surface, and is symmetric in magnetic field (without interchanging
r → r ). The other term, which stems from the principal value in (2.36), is determined by all
states below the Fermi surface (however, the conductance can be expressed solely in terms
of Fermi surface properties, at low temperatures [71, 89]). This term is antisymmetric under
the change B → −B [71]. Only the first part is of interest in our studies (where magnetic
field is absent)
e2 h̄3 π
σ (r, r ) = −
¯
4m2
∞
dE
−∞
∂f
−
∂E
α, α
↔
↔
[Ψ∗α (r) ∇ Ψα (r)] [Ψ∗α (r ) ∇ Ψα (r )]
×δ(E − Eα )δ(E − Eα ).
(2.37)
↔
Here we use ∇ to denote the double sided derivative
↔
g(r) ∇ h(r) = g(r)
∂
∂
h(r) − h(r) g(r),
∂r
∂r
(2.38)
↔
and ∇ denotes the same derivative over r . When this expression19 is averaged over a hypercubic sample with uniform electric field in Eq. (2.14), the Kubo (longitudinal) conductivity
at zero temperature is extracted from σ = GL2−d (we include the factor of two for the twofold
spin degeneracy)
σxx
2πh̄e2 =
|α|v̂x |α|2 δ(Eα − EF )δ(Eα − EF ),
Ω α, α
(2.39)
where −∂f /∂E δ(EF − E) at low temperatures. The velocity operator is defined by the
commutator
ih̄v̂ = ih̄
19 It
dr̂
= [r̂, Ĥ0],
dt
(2.40)
is straightforward to show [69] that microscopic expression for the NLCT (2.37) satisfy both
conditions (2.17), (2.18) which stem from current conservation.
42
involving Ĥ0 . Here the thermodynamic limit Ω = Ld → ∞ should be assumed, therefore
generating continuous spectrum and conductivity as a continuous function of Fermi energy
EF . We emphasize that Ĥ0 is, in the spirit of FDT, the Hamiltonian before an electric field
is turned on.
The final goal of this section is to get the Green function expression for the Kubo
conductance (or conductivity (2.39)), which will be important tool in application of KLRT
to finite-size samples (cf. Sections. 2.5 and 4.2). The Kubo NLCT for finite-size system is a
sample specific quantity—it depends on impurity configuration, sample shape and measuring
geometry. Although we started from the (continuous) coordinate representation,20 Ψα (r) =
r|α, the expressions below are given in terms of the trace over abstract operators. The
traces can be evaluated in any representation, in particular, the one defined by the lattice
models. The action of the current density operator in the coordinate representation is
ˆ |α = eh̄ [δ(r − r)∇ + ∇ δ(r − r)]Ψα (r ),
r |j(r)
0
2im
(2.41)
so that its matrix elements, which appear in the evaluation of the thermal averages, are
α |j0 (r)|α =
eh̄
=
2im
dr α |r r|ĵ0 (r)|α =
eh̄ ∗
Ψ ∇Ψα (r).
2im α
(2.42)
This is the origin of the respective terms in Eq. (2.37).
The one-particle Green operator is defined as the inverse of Hamiltonian (for generality,
we use label Ĥ, while having in mind the “equilibrium” Hamiltonian Ĥ0 of this section)
Ĝr,a = (E − Ĥ ± iη)−1 ,
(2.43)
where appropriate boundary conditions, introduced by adding the small imaginary part ±iη
(η → 0+) to energy, select r-retarded or a-advanced operator for plus or minus sign, respectively. This defines the Green operator close to the branch cut (i.e., continuous spectrum of
20 The
coupling of the vector potential is unambiguously defined only in the coordinate
representation.
43
Ĥ associated with extended states [33]) on the real axis. If exact eigenstates (2.35) of the
Hamiltonian Ĥ are known, the Green operator can be expressed in the form
Ĝr,a =
α
|αα|
.
E − Eα ± iη
(2.44)
Thus, the single-particle Green operator contains the same information as encoded in the
wave function (to be contrasted to the many-body Green functions). The Green function
r,a
in the coordinate representation Gr,a
E (r, r ) = r|Ĝ (E)|r gives response at r for the unit
(delta function) excitation at r . It replaces the following expression involving wave functions
α
Ψα (r)Ψ∗α (r )δ(E − Eα ) = −
1
1
[GrE (r, r) − GaE (r, r )] = − Im GE (r, r).
2πi
π
(2.45)
Therefore, the Green function expression for NLCT,
e2h̄3
σ (r, r ) =
¯
4πm2
∞
dE
−∞
∂f
−
∂E
↔ ↔
Im GE (r, r ) ∇∇ Im GE (r , r),
(2.46)
integrated over the volume of a sample of length L, as in Eq. (2.15), gives the following Kubo
formula for conductance [91] (with factor two for spin degeneracy)
Gxx
4e2 1
=
Tr
h̄v̂
Im
Ĝ
h̄v̂
Im
Ĝ
,
x
x
h L2
(2.47)
where all energy-dependent quantities are evaluated at EF . To perform the trace one can
choose any representation for the operators.21 To obtain this result we used integration by
parts and the following quantum-mechanical identities
↔ ↔
↔ ↔
2mi
g(r, r ) ∇∇ h(r , r) = r|ĝ|r ∇ ∇ r |ĥ|r = −
eh̄
dr Tr ĵ(r)ĝ
2
Tr ĵ(r)ĝ ĵ(r )ĥ , (2.48)
= e Tr (v̂ĝ) ,
(2.49)
valid for arbitrary operators ĝ and ĥ. This allows us to replace the integration over the
volume with trace over the velocity operator. These identities are easily proven by inserting
21 In
discrete representations operators act as matrices. We simplify notation by using “hat” (Ô)
to denote both operators in the abstract Hilbert space as well as matrices acting on a space of
column. In the continuous representation we remove hats and talk about functions [1].
44
the unity operator Iˆ = dr |rr| and following the rules of Dirac bra(c)ket notation. In
Eq. (2.49) we also used the coordinate representation of ĵ(r) (2.41).
2.4.2
Scattering approach: Landauer formula
The main features of transport in disordered conductors are captured by studying the
problem of just one (quasi)particle in a random potential (generated by some impurities).
The interactions in the disordered region are neglected. The scattering formalism follows
directly from this picture once the conduction is viewed as a result of incoming flux being
scattered by a disordered conductor. It was pioneered through subtle physical arguments in
one-dimensional systems and two or four-probe geometry by Landauer [4, 92] (long before
the birth of mesoscopic physics) and later generalized to multichannel case (Fisher and
Lee [93], Büttiker et al. [94]) as well as extended to multiprobe conductance measurement
by Büttiker [60]. Thus, the complicated quantum-mechanical scattering processes build
charges and fields inside the sample. The conductance is obtained from the probability for
injected carriers at one end to reach the other end of the sample. Landauer has perpetually
emphasized [55] the role of the local electric field viewed as the response to an incoming
current. This is an alternate view to that of KLRT where currents are found as the response
to a given (external) electric field. The approach mimics closely the experimental point of
view where one usually imposes an external current and measures the resulting potential
drop due to the scattering. This paradigm has become an important tool in guiding the
intuition (as well as calculations) when studying the mesoscopic transport.
In a two-probe case the conductor is placed between the two semi-infinite leads (Fig. 2.1)
which define the basis states for the scattering matrix (S-matrix). Because of the quantization of the transverse wavevector kn in a lead of a finite width, the wave function of an
electron at EF factorizes into a product of transverse and longitudinal part
trans
(y, z)e±ikx .
Ψ±
n (r) = φn
(2.50)
45
Therefore, the leads (to simplify, we assume that two leads are identical) define the complete
orthonormal set [96], i.e., a basis of scattering states. The integer n = 1, 2, . . . , M labels
the transverse propagating modes, also know as the scattering or conducting “channels”.
The mode is characterized by a real wavevector k and transverse wave function φn (y, z).
For example, in the case of parabolic subbands kF2 = kn2 + k 2 , so that propagating modes
are labeled by the transverse wavevectors which give real k > 0 in this equation. Each
channel can carry two waves traveling in the opposite direction, denoted by ± in (2.50), and
is normalized to unit flux in the direction of propagation. This means that a wave function
on either side of the disordered region (i.e., inside the lead) is specified as a 2M-component
vector. The scattering S-matrix is a 2M × 2M matrix which relates the amplitudes of the
incoming waves to the amplitudes of the outgoing waves

















S




O 







O
=












I 







I
=
 

 

 
 
 
r



I


r t 
  I 
 

·
.
 

t
(2.51)
Here I, O are M-component vectors (in the basis spanned by the eigenstates (2.50) describing the wave amplitudes in the left lead, and I , O are contain the coefficient of the
same expansion in the right lead. The S-matrix has a block structure with t and t being
M × M transmission matrices from left to right and from right to left, respectively. The
matrices r and r describe reflection from left to left and from right to right, respectively.
Current conservation implies unitarity of the S-matrix, S† = S−1 . The scattering matrix
of a disordered system is a random matrix which can be classified, in the same fashion as
random Hamiltonian of RMT, using appropriate symmetries. However, the distribution of
S-matrices depends on the type of conducting structure to which it is applied [30].
While the one-particle Green function Gr,a
E (r, r ), introduced in Sec. 2.4.1, connects the
response at any point r with excitation at point r , the S-matrix give the response in one
lead due to the excitation in another lead (in the space of conducting channels). Once the
46
scattering matrix of a disordered sample is known the (time-averaged22 ) current at the cross
section S1 in the left lead is given by
∞
2e dE [fL (E) − fR (E)] Tr t(E)t†(E),
I¯ =
h
(2.52)
0
where t is the transmission matrix. The incident flux concentrated in the channel |n will
give the wave function in the opposite lead
m tnm |m.
From here the linear conductance
follows in the limit of vanishingly small voltage difference V between the reservoirs
∞
∂f
I¯
2e2
dE −
Tr t(E)t† (E).
G = lim
=
V →0 V
h
∂E
(2.53)
0
At zero temperature this simplifies to the two-probe Landauer formula for conductance
G=
M
M M
2e2
Tr t(EF )t† (EF ) = GQ
|tmm (EF )|2 = GQ
Tn (EF ),
h
m=1 m =1
n=1
(2.54)
where all quantities are computed at the Fermi energy EF . Here Tn (EF ) are the transmission
eigenvalues (or transmission coefficients23 ), i.e., the eigenvalues of tt† . Thus, the knowledge
of the transmission eigenstates, each of which is a complicated superposition of incoming
modes (2.50), is not required to get the conductance. The factor of two in the conductance
quantum GQ is due to the two-fold spin degeneracy in the absence of spin-orbit scattering.
In the presence of spin-orbit interaction it stems from the Kramers degeneracy in zero magnetic field. When both magnetic field and spin-orbit scattering are present the conductance
quantum is GQ = e2 /h, but the number of transmission eigenvalues is doubled [30].
It is insightful to demonstrate the difference between the quantum-mechanical transmission probability |tmm |2 = |
22 We
FP
FP 2
Zmm
and its semiclassical approximation (|tmm |2 )SCA =
|
denote explicitly the time-averaging of the steady state current I¯ taking into account the
intrinsic fluctuations (shot noise) present in mesoscopic transport [30].
23 The
attempts to generalize Landauer formula to interacting systems, while retaining the simple
picture where each channel carries a current e/hδµ, lead to “transmission coefficients” which have
no simple physical interpretation [89] like in the case of non-interacting fermions elaborated above.
47
FP
FP 2
|Zmm
| in the framework of scattering approach. Here we use the picture of Feynman
FP
. Each path
paths (labeled by FP) which are characterized by the complex amplitude Zmm
originates in some “channel” m in the left lead, ending in one of the “channel” m of the
right lead. The semiclassical approximation neglects the interference between scatterers (i.e.,
different Feynman paths). In practical calculations, which effectively perform the complicated summation over the denumerably infinite number of Feynman paths, the difference
between quantum and semiclassical conductance can be studied by concatenating scattering
matrices of the successive disordered regions to get the former and combining the “probability scattering matrices” (obtained by replacing each element of the S-matrix by its squared
module) to get the latter [97]. The difference between two conductances obtained in this
way then shows the effects of quantum interference on the transport properties of disordered
conductors.
For computational purposes the Landauer formula is frequently used in a phenomenological way. The conductor is treated as a black box described by some stochastic scattering
matrix drawn from the appropriate random matrix distribution [30]. In this formalism it is
possible to get global transport properties (but not the local, or truly microscopic ones) like
conductance or any so-called linear statistics A =
A =
M
M
n=1
1
a(Tn ) =
n=1
a(Tn ) of the transmission eigenvalues
dT a(T )ρ(T ).
(2.55)
0
Here a(Tn ) is an arbitrary function of Tn . The Equation (2.55) introduces the distribution
function of transmission eigenvalues Tn
ρ(T ) =
δ(T − Tn ) ,
(2.56)
n
where an average over all possible realization of disorder . . . is performed. While specific Tn
are sensitive to a particular configuration of impurities, the distribution ρ(T ) allows us to get
various disorder-averaged transport properties (e.g., shot noise power, Andreev conductance
of normal metal-superconductor junctions, etc. [30]).
48
Contrary to the naı̈ve expectation that Tn /L for all channels, which would follow
by comparing (2.54) to the Boltzmann conductance (2.24), it was shown by Dorokhov [98]
that in a uniform quasi one-dimensional conductor
1
M
G
√
, cosh−2
ρ(T ) =
2GQ T 1 − T
g
The cutoff at small T is such that
1
0
< T < 1.
(2.57)
dT ρ(T ) = M, which for M G/GQ does not affect
the averages of the first and higher order moments of T . Thus, ρ(T ) is “bimodal” meaning:
most of Tn are either Tn = 0 (“closed” channels) or Tn = 1 (“open” channels). This
has important consequences when calculating linear statistics other than the conductance
since we can get conductance (first moment of the distribution) without really knowing the
details of ρ(T ). For example, the shot noise power spectrum in the zero frequency limit [30]
depends on the variance of ρ(T ) and is given by P ∼
n
Tn (1 − Tn ). The universal validity
of the distribution ρ(T ) was (claimed to be) extended to the diffusive conductor of arbitrary
shape, dimensionality and spatial resistivity distribution in [64, 99]. Universality means
that it depends only on the global characteristic of the conductor, like the dimensionless
conductance g = G/GQ . This form of distribution breaks down close to the Anderson
localization regime (g ∼ 1) or ballistic regime [100] (g ≤ N). Even in the metallic regime,
universality can be broken [64] by the presence of extended defects in the conductor, such
as tunneling barriers, grain boundaries, or interfaces (cf. Sec. 4.3).
It was proven rigorously [71, 93] that Landauer formula can be derived from the Kubo
formula. This requires to use the Kubo NLCT for a finite-size system connected to ideal
leads (which stem from the “sample-specific linear response theory” [89]). The proof goes
through the integration of NLCT over the surfaces, as in Eq. (2.19). The surfaces should be
positioned deep inside the leads so that all evanescent modes have “died out” and do not
contribute to the conductance. The equivalence shows that transmission properties can be
calculated from the Kubo NLCT (2.46). It also confirms the independence of linear transport properties on the local current and field distribution (cf. Sec. 2.2), i.e., non-equilibrium
49
charge redistribution, since no such quantities enter into the Landauer formula for conductance. We explore further the practical meaning of equivalence between the Landauer-type
formula and Kubo formula, expressed in terms of real-space Green functions on a lattice, in
Sec. 2.5.
The scattering approach is conceptually simple, but it is difficult to use it directly (by
solving the Schrödinger equation and computing transmission amplitudes) in complicated
geometries. The difficulty arises also when one wants to include arbitrary spatial variation
and band structure. The root of the problem stems from the need to calculate the precise
eigenstate spectrum in the leads. Therefore, one usually resorts to some Green function
method. The most general treatment of electronic transport is provided by the Non Equilibrium Green Function (NEGF) formalism, which is surveyed in the next section. It is
equivalent to the Landauer formalism in the absence of dephasing processes [43]. The technical advantage of the Green function approaches is that it does not require the existence of
well defined asymptotic conducting channels.
2.4.3
Non-equilibrium Green function formalism
The central quantity in the Landauer formula (5.1) is transmission matrix t (or equivalently transmission eigenvalues Tn ). The transmission matrix t is a block of the whole
S-matrix which connects states in the leads. The internal state of the conductor, expressed
in terms of some quantities which depend on the position vector r inside the conductor, is
irrelevant in the scattering approach. Nevertheless, it is possible to derive the formula for
conductance, which can be cast in the form of (5.1), containing Green functions Gr,a (r, r)
defined inside the conductor. Thus, the formalism based on Green functions is more general,
because the inclusion of electron-phonon or electron-electron interaction in the disordered
region cannot be described by the S-matrix (which keeps track only of the states in the
leads). In non-interacting cases the primary reason for the employment of Green function
50
techniques is computational efficacy in obtaining essentially the S-matrix of an arbitrarily
shaped conductor (as discussed at the end of previous Section).
In the Kubo formalism of Sec. 2.4.1 density matrix ρ̂(k, k , t) was used to include the
quantum information (phase-correlations) not contained in the distribution function F (k, t).
However, this quantity depends only on one time coordinate and is not the most general
description of (many-body) quantum systems out of equilibrium. The most comprehensive quantum generalization of the semiclassical distribution function is based on the NonEquilibrium Green function formalism (NEGF).24 The central quantity of NEGF is doubletime correlation function25
G< (r1 , t1 ; r2, t2 ) = iΨ̂† (r2 , t2 )Ψ̂(r1 , t1 ),
(2.58)
where Ψ̂(r1 , t1 ) is electron field operator in the Heisenberg picture. The brackets . . . denote
the non-equilibrium quantum expectation values [83]. The other double-time correlation
function is defined as
G> (r1 , t1 ; r2 , t2 ) = −iΨ̂(r1 , t1 )Ψ̂† (r2 , t2 ).
(2.59)
Using the sum and difference coordinates
r = r1 − r2 ,
R =
1
(r1 + r2 ),
2
(2.60)
(2.61)
and times,
t = t1 − t2 ,
T =
24 This
1
(t1 + t2 ),
2
(2.62)
(2.63)
formalism is also known as the Keldysh formalism. In order to give the proper credit,
we mention that there are two equivalent formulation of NEGF [83], i.e., equations for its central
quantity G< : Kadanoff-Baym and Keldysh. Their relationship is the same as that of ordinary
differential equation with boundary conditions to corresponding integral equation.
25 We
assume here h̄ = 1, but restore it in the final formulas for current and conductance.
51
the density matrix is obtained26 from ρ(r, R, T ) = −iG< (r, t = 0; R, T ). Any observable,
such as particle and current densities, can be computed by taking the moments of G< [43].
The other two functions used in NEGF are retarded and advanced,27 e.g.,
Gr (r1 , t1 ; r2 , t2 ) = −iθ(t1 − t2 ){Ψ̂(r1 , t1 ), Ψ̂† (r2 , t2 )} = −iθ(t1 − t2 )A(t1 , t2 ).
(2.64)
They describe the propagation of an extra particle added to the system (i.e., the dynamics
of electron inside the conductor) and cannot give the distribution of particles (which is
determined by G<,> ). Here A(t1 , t2 ) is the spectral function which connects all four Green
functions A = i(Gr − Ga ) = i(G> − G< ). If we use the Fourier transform
<
G (p, E; R, T ) =
dr dt e−i(p·r−Et)G< (r, t; R, T ),
(2.65)
then only one function remains independent (e.g., Gr ) in equilibrium (FDT theorem)
G< (r, E; R) = iA(p, E; R)f (E),
(2.66)
G> (r, E; R) = −iA(p, E; R)(1 − f (E)),
(2.67)
where f (E) is the Fermi-Dirac distribution function. Obviously, in equilibrium situations
there is no dependence on time T . For general, non-equilibrium, system one needs to solve
both Dyson equations for Gr,a and coupled to them quantum kinetic equations for G<,> [83].
26 One
can also take the Fourier transform of ρ(r, R, T ) over r, the so-called Wigner function
fW (k, R, T ), which serves as a quantum analog of the Boltzmann distribution function F (k, r, t)
(in the sense that expression for kinetic properties, such as particle and current densities, look
the same). However, fW (k, R, T ) is not a positive-definite function since momentum and coordinate do not commute and cannot be defined simultaneously in quantum mechanics. Taking a
Gaussian smoothing in both position and momentum of the Winger function leads to the Husimi
distribution [101], which is non-negative and can be interpreted as a probability distribution.
27 In
general interacting system these functions are not “true” Green functions in a strict math-
ematical sense, i.e., the inverse of some operator (like the Green function (2.43)).
52
The use of NEGF technique in the problems we are interested in, i.e., the transport
in disordered conductor placed between two ideal semi-infinite leads as on Fig. 2.1, was
pioneered by Caroli28 et al. [102] for systems modeled on a lattice described by a tightbinding Hamiltonian (cf. next Section). Through the use of similar procedure, Meir and
Wingreen [103] derived the following general expression for the steady state (DC) electronic
current through an interacting sample attached to ideal semi-infinite leads (pedagogical
derivation is reproduced in Ref. [83])
ie
I =
h
dE Tr {[Γ̂L (E) − Γ̂R (E)]Ĝ< (E)
+[fL (E)Γ̂L (E) − fR (E)Γ̂R (E)] (Ĝr (E) − Ĝa (E))},
(2.68)
where fL,R (E) are the Fermi-Dirac distributions in the leads, determined by electrochemical
potentials µL and µR in the reservoirs. The interacting region is described by the Hamiltonian
Ĥ = Ĥint ({d†n }, {dn }) +
L=L,R k∈L
εk c†k ck +
L=L,R k∈L
n
(Vkn c†k dn + H.c.),
(2.69)
where {d†n } creates a complete set of single-particle states in the sample, c†k∈L creates an
electron in state k of a lead L, and Ĥint is a polynomial in {d†n }, {dn } which commutes
with the electron number N̂ =
n
d†n dn . Our subsequent studies will be confined to non-
interacting systems described by the Anderson model (explained in detail in the next section)
Ĥnint =
m
εm c†m cm +
m,n
tmn c†m cn ,
(2.70)
where c†m denotes the creation operator of an electron at the site m of a three-dimensional
simple cubic lattice. The coupling of a lead L to the sample is described by the “level-width”
28 Caroli
et al. [102] were interested in a tunneling current through a metal-insulator-metal junc-
tion. By describing this system on a lattice, similar to our calculation based on tight-binding
Hamiltonian, they got a natural decomposition of the junction into sample connected to the leads.
This allowed them to calculate the current to all orders in the applied voltage using Keldysh technique, thus bypassing various problems in the effective tunneling Hamiltonian approach.
53
function ΓLnm (E) = 2π
∗
k∈L Vkn Vkm (E
− εk ). The coupling constants Vkn between the leads
and the central region depend, in general, on the charge density and should, therefore, be
<
determined self-consistently. The Green function G<
nm (E) is a Fourier transform of Gnm (t) =
id†m (0)dn (t), while Grnm (E) is a Fourier transform of Grnm (t) = −iθ(t){dn (t), d†m (0)}.
All functions in the equation (2.68) are in fact matrices in the central region indices m, n
(which we denote by the usual hats). For the equilibrium correlation function Ĝ< (2.66) the
current (2.68) vanishes.
When there are no interactions in the central region it is possible to solve [102] the
quantum kinetic equation for Ĝ<
Ĝ< = ifL (E)Ĝr Γ̂L Ĝa + ifR Ĝr Γ̂R Ĝa ,
(2.71)
where Ĝr − Ĝa = iĜr (Γ̂L + Γ̂R )Ĝa . This leads to the following expression for current
2e
I=
h
dE [fL (E) − fR (E)] Tr Ĝa Γ̂R Ĝr Γ̂L .
(2.72)
In the linear transport regime, µL − µR → 0, current is proportional to the bias V =
(µL − µR )/e and δ[fL (E) − fR (E)] ≈ (−∂f /∂E) (µL − µR ). Thus, we obtain the formula for
the linear-response conductance at low temperatures (f (E) ≈ θ(EF − E))
G=
2e2 Tr Γ̂L Ĝr Γ̂R Ĝa .
h
(2.73)
The final expression contains only equilibrium quantities, in the spirit of FDT. Therefore,
it can be related to the Landauer or Kubo linear response theory. General features of the
formula (2.73) are studied in the next Section. Armed with this knowledge, we apply this
formula various non-interacting disordered electronic systems throughout the thesis.29
29 In
the thesis we follow the deductive route of exposition, while in real life the understanding of
the proper use of different expressions for conductance comes also from the experience in applying
the formalism to concrete problems.
54
2.5
Quantum expressions for conductance: Real-space
Green function technique
The iconolaters of Byzantium were subtle folk,
who claimed to represent God to his great glory,
but who, simulating God in images, thereby
dissimulated the problem of his existence.
— Jean Baudrillard, The Perfect Crime
2.5.1
Lattice model for the two-probe measuring geometry
In this Section we give practical meaning to different quantum expressions for conductance introduced thus far (Kubo or Landauer-like) by: starting from the Hamiltonian of a
single electron in a random potential → finding the Green functions in a real-space representation (i.e., corresponding matrices) for the sample placed between two semi-infinite
disorder-free leads → showing how to plug in efficiently these Green functions into the relevant conductance formulas.
In most of the problems studied in the thesis a disordered electron sample is modeled
microscopically by a tight-binding Hamiltonian (TBH) on a finite hypercubic lattice30 N ×
Ny × Nz (lattice constant is denoted by a)
Ĥ =
m
εm |mm| +
tmn |mn|.
(2.74)
m,n
This model is widely used in the localization theory. Here tmn are nearest-neighbor hopping
matrix element between s-orbitals r|m = ψ(r−m) on adjacent atoms located on sites m of
the lattice. The disorder is simulated by taking either the on-site potential (diagonal elements
in the Hamiltonian matrix) εm or the hopping (off-diagonal elements) tmn , or both, to be
a random variable characterized by some probability distribution. The on-site energies εm
30 We
simplify notation by using N ≡ Nx for the number of sites along the x-axis.
55
correspond to the potential energy, while hopping matrix elements tmn are the kinetic energy
(and depend on the effective mass of an electron). The hopping defines the unit of energy.
In the Chapters to follow, specific random variable distributions will be employed. Here we
are interested only in the generic methods applicable to any Hamiltonian. The TBH is a
matrix (in site-representation) of dimension ∼ (L/a)d , which is sparse since nearest-neighbor
condition means that most of the elements are zero. It can be considered as a model of either
a nanoscale conductor,31 or a discretized version of a continuous one-particle hamiltonian
Ĥ = −h̄2 ∇2 /2m + U(r). In a discretized interpretation the continuous position vector r is
replaced by the position of a point m on a discrete lattice, and derivatives are approximated
by finite differences [43].
The standard theoretical view of our two-probe measurement circuits is shown on
Fig. 2.1. The sample is placed between two semi-infinite ideal leads. Each lead is modeled by the same clean TBH ĤL , with εm = 0 and tmn = tL , which is defined on an infinite
Hilbert space of site states |m. The coupling between the end layer sites in the lead and
corresponding sites in the sample are taken into account through TBH, ĤC , which describes
only hopping tmn = tC between these sites. The leads are connected at infinity to a particle
reservoirs through smooth contacts. Left and right reservoirs (large conductors) are at a
constant chemical potential µL and µR , respectively. Thus they are biased relative to each
other by a battery of voltage V = (µL − µR )/e. Each reservoir injects the fully thermalized
carriers into the lead. The distribution function of electrons to be injected is equilibrium
Fermi-Dirac with chemical potential of the reservoir. It is assumed that reservoirs are large
enough conductors such that passage of current does not disturb these equilibrium char31 Our
lattices will be small, containing typically several thousands of atoms. This is the limitation
imposed by the available computer memory and computational complexity [42] of matrix operations.
For example, less than 20 atoms are placed along the length of the conductor. This is why we use
the term nanoscale (or atomic-scale) conductor.
56
z
y
PARTICLE RESERVOIRS
x
LEAD
µL
SAMPLE
tC
111
000
000
111
000
111
000
111
11
00
00
11
00
11
00
11
11
00
00
11
00
11
00
11
111
000
000
111
000
111
000
111
000
111
000
111
000
111
11
00
00
11
00
11
00
11
00
11
00
11
00
11
11
00
00
11
00
11
00
11
00
11
00
11
00
11
t
LEAD
tL
µR
V
Figure 2.1: A two-dimensional version of our actual 3D model of a two-probe measuring
geometry. Each site hosts a single s-orbital which hops to six (or fewer for surface atoms)
nearest neighbors. The hopping matrix element is t (within the sample), tL (within the
leads), and tC (coupling of the sample to the leads). The leads are semi-infinite and connected
smoothly at ±∞ to reservoirs biased by the chemical potential difference µL − µR = eV .
57
acteristics (i.e., chemical potential can be defined and stays the same as in the reservoir
decoupled from the conductor). The transport in the central part is phase-coherent. Thus
the reservoirs account for the dissipation necessary to establish the steady state. They accept
non-equilibrium distribution of electrons from the non-dissipative conductor and provide the
thermalization. Even though resistance is related to the dissipation, its value is determined
solely by the momentum relaxation processes caused by the scatterers inside the disordered
region. However, only the leads at a fixed potential are explicitly taken into account when
calculating transport properties. The leads provide the boundary condition for the relevant
equations. Since electron leaving the central mesoscopic sample looses the phase-coherence,
leads, in a practical way for theoretical calculations, introduce the heuristic construction of
the perfect macroscopic reservoirs.
2.5.2
Green function inside the disordered conductor
The direct inversion (2.43) of TBH for the whole system, consisting of semi-infinite leads
and the sample,
Ĝr,a (m, n) = (E − Ĥ(m, n) ± iη)−1 ,
(2.75)
would lead into a trouble since Ĥ(m, n) is an infinite matrix (Ĥ = ĤS + ĤL + ĤC ). The site
representation of the Green operator Ĝr,a is a Green function matrix
Ĝr,a (m, n) = m|Ĝr,a |n,
(2.76)
and the matrix of Hamiltonian in this representation is a band diagonal matrix [104] of the
bandwidth 2Ny Nz + 1. The usual method in the literature to avoid this is to use the periodic
boundary conditions [45]. However, this would generate a discrete energy spectrum, instead
of continuous one of our open system, and is plagued with problems which we explicitly
demonstrate in Sec. 4.2. The correct handling of the leads and openness of the system was
initiated by Caroli et al. [102], as discussed in general terms in Sec. 2.4.3. Instead of just
58
truncating the matrix (2.75), which would lead to a conductor with reflecting boundaries
instead of open one where electrons can enter and leave the conductor, the leads are taken
into account through the exact “self-energy” terms describing the “interaction” of the finitesize conductor with the leads.
If we consider just the sample and one lead32 then Green function for this system can
be written in the form of a block matrix [43]

Ĝr =









ĜrL
ĜrS−L












ĜrL−S 

ĜrS






=
−1
E + iη − ĤL
ĤC†
ĤC
E + iη + ĤS









,
(2.77)
where we have shorten the notation by using operator labels without respective matrix
indices. The partition above follows from the intrinsic separation of the Hilbert space of
states, brought about by the physical separation of lead and the sample in the lattice space.
The diagonal blocks are: infinite matrix ĜrL , connecting the sites in the left lead; and finite
ĜrS connecting the states on the lattice sites inside the conductor. The off-diagonal blocks,
ĜrL−S and ĜrS−L, connect the states in the lead and the sample. The matrix of the coupling
Hamiltonian ĤC (mL , mS ) = tC is non-zero only for the adjacent sites in the lead mL and
the sample mS . The set of matrix equations for ĜrS follows from Ĥ · Ĝr = Iˆ
[E + iη − ĤL ] · ĜrL−S + ĤC · ĜrS = 0,
(2.78)
ˆ
ĤC† · ĜrL−S + [E + iη − ĤS ] · ĜrS = I.
(2.79)
The Equation (2.78) can be solved for ĜL−S
ĜrL−S = −ĝLr · ĤC · ĜrS ,
ĝLr = (E + iη − ĤL )−1 ,
32 To
(2.80)
(2.81)
clarify notation, we use the subscript L for a general lead and subscript L for the left lead
or reservoir in a two-probe geometry.
59
where we recognize ĝLr as a Green function of a bare semi-infinite lead. This is still an
infinite matrix, but can be found exactly as demonstrated in the following Section. Using
ĜrL−S (2.80) in Eq. (2.79) we get
ĜrS = (E − ĤS − ĤC† · ĝLr · ĤC )−1 .
(2.82)
The final result is a Green function inside a finite-size disordered region which “knows” about
the semi-infinite leads, and relevant boundary conditions at infinity they provide, through
the “self-energy” function33
Σ̂r (mS , nS ) = t2C ĝLr (mL , nL ).
(2.83)
Since the self-energy provides a well defined imaginary part (which then “helps” the Green
function to become retarded or advanced), we drop the small iη in Eq. (2.82). The selfenergy Σ̂r (mS , nS ) is non-zero only between the sites (mS , nS ) on the edge layer of the
sample which are adjacent to the sites (mL , nL ) lying on the edge layer of the lead. This
follows from the structure of lead-sample coupling matrix ĤC . If the sample is attached to
many leads (multi-probe geometry) then one should add the self-energy terms generated by
each lead, i.e., in our two-probe case
ĜrS = (E − ĤS − Σ̂r )−1 ,
(2.84)
where Σ̂r = Σ̂rL + Σ̂rR . Advanced functions are obtained in a standard way: Ĝa = [Ĝr ]† , and
Σ̂a = [Σ̂r ]† . In the following Section we give a derivation of a Green function ĝLr (mL , nL ) on
the end layer of the lead.
The self-energies “measuring” the coupling of the sample to the leads can be related to
the average time an electron spends inside the sample before escaping into the leads. This
33 Analogous
terms in Green functions appear when solving the Dyson equation in diagrammatic
perturbation theory [1]. Here we use the same name, following Ref. [43], keeping in mind that
no approximation is taken for the self-energy (as is usually done when discussing self-energies in
perturbation theory by summing only a specific set of diagrams).
60
can be understood from the following simple arguments. The open system is surrounded by
an ideal conducting medium. In that case we cannot talk about eigenstates. Nevertheless,
we can formally use an effective Hamiltonian, which is inverted to get the Green function,
[ĤS + Σ̂r ]|αeff = Eαeff |αeff .
(2.85)
This is not a Hermitian operator, and total probability is not conserved. If we write the
formal eigenenergy [43] using the eigenvalue of the corresponding isolated system Eα ,
Eαeff = Eα − ζα − i
ζα
,
2
(2.86)
then its imaginary part ζα gives the “lifetime” of an electron in state α before escaping
into the leads. The probability to stay in the state |α decays as | exp(−iEαeff t/h̄)|2 =
exp(−2ζα t/h̄), and the escape time into the leads is τesc = h̄/2ζα . The “loss” of electrons
into the leads is also illustrated by the following identity [43]
∇ · j(r) =
1
h̄
dr dr Ψ∗ (r)Γ(r, r)Ψ(r ),
(2.87)
where Γ̂ = −2 Im Σ̂ = i(Σ̂r − Σ̂a ), and the evolution of wave functions Ψ(r) is determined
by the effective Hamiltonian ĤS + Σ̂r .
Even though the eigenstates are not defined in the standard quantum-mechanical sense,
one can still use the local density of states (LDOS) given by the imaginary part of the Green
function
1
ρ(m, E) = − Im ĜrS (m, m; E).
π
(2.88)
It turns out that this LDOS is qualitatively similar to the LDOS of 2D and 3D closed
system. We check this explicitly by comparing (2.88) to LDOS of a closed system obtained
from exact diagonalization studies, cf. Fig. 3.2. However, in quasi-1D conductors LDOS
computed from (2.88) is quite different from LDOS
ρ(r, E) =
|Ψα (r)|2 δ(E − Eα ),
α
defined in terms of exact eigenstates of a closed system [5].
(2.89)
61
2.5.3
The Green function for an isolated semi-infinite ideal lead
In the previous Section we learned that the Green function matrix ĜrS (m, n) (2.84) at a
continuous energy E can be computed numerically by inverting the finite matrix E − ĤS − Σ̂r .
This requires to know the matrix elements ĝ r (mB , nB ) of the Green operator for (each)
isolated semi-infinite lead between the states |mB located on the sites mB at the open
boundary of the lead. The lead is modeled by TBH on a rectangular lattice Ninf × Ny × Nz ,
where Ninf → ∞ to make the lead semi-infinite. The exact eigenstates of such lead (which
has uniform cross section) are separable into a tensor product |k = |kx ⊗ |ky , kz . Here
|ky , kz are transverse eigenstates (i.e., eigenstates of each isolated transverse layer)
|ky , kz =
2
Ny + 1
Ny
Nz
2
sin(ky ny a) sin(kz nz a) |ny , nz ,
Nz + 1 ny =1 nz =1
(2.90)
where |ny , nz denotes the orbitals of the arbitrary 2D layer. We choose a hard wall boundary
conditions in ŷ and ẑ direction, so the state m|ky , kz vanishes at the sites |m belonging to
the transverse boundary surfaces. This makes the transverse states |ky , kz quantized with
eigenvalues (dispersion relation)
ε(ky , kz ) = 2tL [cos(ky a) + cos(kz a)],
(2.91)
defined by discrete ky (i) = iπ/(Ny + 1)a, and kz (j) = jπ/(Nz + 1)a. Here i runs from 1 to
Ny and j runs from 1 to Nz . The longitudinal eigenstates |kx (i.e., on the 1D chains) are
nx |kx =
2
sin(kx nx a),
Ninf
(2.92)
with eigenvalues ε(kx ) = 2tL cos(kx a). This states vanish at the open end on which nx = 0.
The Green function ĝ r (mB , nB ) can be expanded in terms of the exact eigenstates |k,
mB |ĝ r |nB =
=
k
mB |kk|nB E − 2tL cos(kx a) − ε(ky , kz ) + iη
my , mz |ky , kz ky , kz |ny , nz ky ,kz
×
sin2 kx a
2 ,
Ninf kx E − ε(ky , kz ) + iη − 2tL cos(kx a)
(2.93)
62
where only sites at the edge nx = 1 are needed (|nB ≡ |nx = 1, ny , nz ). When Ninf → ∞,
kx is continuous and the sum
kx
J(ky , kz ) =
can be replaced by the integral
sin2 kx a
2 Ninf kx E − ε(ky , kz ) + iη − 2tL cos(kx a)
a
=
4πtL
π/a
dkx
0
2 − e2ikx a − e−2ikx a
,
(EJ + iη)/2tL − cos(kx a)
(2.94)
where we shorten the notation with EJ = E − ε(ky , kz ). This integral can be solved by
converting it into a complex integral over the unit circle and finding the residues at the poles
lying inside the circle
J(ky , kz ) = −
1 − w2
1 .
4iπt |w|=1 w 2 /2 + 1/2 − Y w
(2.95)
Here Y denotes the expression Y = (EJ + iη)/2tL . The poles of the integrand are at
√
w1,2 = Y ∓ Y 2 − 1 and have the following properties: (a) w1 w2 = 1, for any |Y |; (b)
|w1 | < 1, |w2 | > 1, for |Y | > 1; and (c) |Y | ≤ 1, both poles lie on the unit circle. If (c) is
satisfied, then +iη (η → 0+ ) is needed to define the retarded Green function
1
J(ky , kz ) = − Res
tL
1 − w2
(w − w1 )(w − w2 )
If |Y | > 1, then
w=w1
1
= 2 EJ − i 4t2L − EJ2 .
2tL
(2.96)
1
J(ky , kz ) = 2 EJ − sgn EJ EJ2 − 4t2L ,
2tL
(2.97)
because one pole is always inside the circle, and the small imaginary term iη is not required
to define the Green function.
We summarize the results of this section by giving the complete expression for the
self-energies introduced by each lead L (in a two-probe case left L and right R)
Σ̂rL (m, n) =
2
2
sin(ky my a) sin(kz mz a)
Ny + 1 Nz + 1 ky ,kz
t2
× C2 EJ − i 4t2L − EJ2 sin(ky ny a) sin(kz nz a),
2tL
(2.98)
63
for |EJ | < 2tL , and
Σ̂rL (m, n) =
2
2
sin(ky my a) sin(kz mz a)
Ny + 1 Nz + 1 ky ,kz
t2
× C2 EJ − sgn EJ EJ2 − 4t2L sin(ky ny a) sin(kz nz a),
2tL
(2.99)
for |EJ | > 2tL . In these expression it is assumed that n and m are the sites on the edge
layers (first or Nth) of a conductor.
2.5.4
One-dimensional example: single impurity in a clean wire
To illustrate the power of concepts introduced above, we provide a “back of the envelope”
calculation for the single impurity, modeled by an on-site potential ε, in a clean infinite 1D
chain (εm = 0 on all other site). The same problem is solved using T-matrix in a lengthy
calculation elaborated in Ref. [33]. Our derivation assumes that impurity is the “sample”
from Fig 2.1 and the rest of the chain are the “leads” with hopping parameter t throughout
the system. The Green function of the “sample” is just a number Gr (E) (i.e., 1 × 1 matrix)
√
Grin (E) = [E − ε − (E − i 4t2 − E 2 )]−1 ,
(2.100)
for |E| < 2t. This gives the local density of states (2.88), which is independent of the lattice
site, inside the band
√
4t2 − E 2
1
1
r
.
ρin (E) = − Im Gin (E) =
π
π ε2 + 4t2 − E 2
(2.101)
For energies outside the band, e.g., E > 0 > 2t the Green function is
Grout (E) = [E − ε − (E −
√
E 2 − 4t2 ) + iη]−1 ,
(2.102)
where a small imaginary part is added to E because the “self-energy” generated by the
“leads” is real. The corresponding LDOS is
√
η
1
1
η→0+
√
→
δ(−ε
+
E 2 − 4t2 ),
ρout (E) = − Im Grout (E) =
π
π ( E 2 − 4t2 − ε)2 + η 2
(2.103)
64
where delta function properties lead to the following simplification
δ(−ε +
√
E2
−
4t2 )
=
Ep2 − 4t2
Ep
δ(E − Ep ).
(2.104)
Thus, the delta function singularity in LDOS appears outside the band of a 1D chain. This
√
is signaling the appearance of a bound state at the energy Ep = sgn ε ε2 + 4t2 . In a clean
chain (ε = 0) LDOS is singular at the band edges (Fig. 2.2). Thus, the introduction of a
single impurity is enough to smooth out the band edge singularities in 1D. These proceeds
in accordance with the sum rule: LDOS summed over all sites and energies is constant,
meaning that weight is transferred from the continuous spectrum at each site n into the
discrete level LDOS, proportional to the overlap of the discrete state with |n (Fig. 2.2).
2.5.5
Equivalent quantum conductance formulas for the two-probe
geometry
Finally, we employ the Green function for the open finite-size conductor (2.84) in the
computation of linear quantum (i.e., zero-temperature) conductance. The Landauer-type
formula (2.73) is obtained from the Keldysh technique of Sec. 2.4.3
G =
t =
2e2
2e2 Tr Γ̂L Ĝr1N Γ̂R ĜaN 1 =
Tr (tt† ),
h
h
Γ̂L Ĝr1N Γ̂R .
(2.105)
(2.106)
Here Ĝr1N , and ĜaN 1 are matrices whose elements are the Green functions connecting the
layer 1 and N of the sample.34 Thus, only the block Ny Nz × Ny Nz of the whole matrix
Ĝr (n, m) (2.84) is needed to compute the conductance. The Hermitian operator
Γ̂L = i(Σ̂rL − Σ̂aL ) = −2 Im Σ̂L > 0,
34 We
(2.107)
avoid using subscript S here since it is clear from the discussion above that all Green
functions which we are going to use are defined inside the sample.
65
LDOS (at arbitrary site)
1.0
(c)
(a)
0.8
0.6
0.4
0.2
0.0
-4
(b)
-2
0
2
4
Fermi Energy
Figure 2.2: Local density of states (LDOS) at an arbitrary site of a 1D chain, described by a
tight-binding Hamiltonian, for: (a) energies inside the band of a clean 1D chain, (b) energies
inside the band of a 1D chain with one impurity of on-site energy ε = 1, and (c) outside the
band of a 1D chain with one impurity of on-site energy ε = 1.
66
is the counterpart of the spectral function for the Green operator, Â = i(Ĝr − Ĝa ). Therefore,
it is related to the imaginary part of the self-energy Σ̂L introduced by the left lead. The
operator Γ̂L “measures” the coupling of an open sample to the left lead (Γ̂R is equivalent
for the right lead). Although the product of full matrices, connecting the sites of the whole
sample, is more complicated than what is shown in Eq. (2.105), the trace is the same. This
follows from the fact that Γ̂L , like the self-energy Σ̂L , has non-zero elements between the
orbitals on the sites of layer 1 and N of the conductor. Thus, the expression under the
trace in Eq. (2.105) is evaluated only in the Hilbert space spanned by the orbitals located on
the edge layers of the sample. This is in the same spirit as the computation of Landauer’s
S-matrix (cf. Sec. 2.4.2), i.e., without worrying about the “internal state of the conductor”.
The positive definiteness of Γ̂L means that its square root is well defined
Γ̂L =
n
γn1/2 P̂n .
(2.108)
Here the operator P̂n is the spectral projector onto eigensubspace corresponding to the
eigenvalue γn . By “reshuffling” the matrices under the trace (using its cyclical properties)
we can get the Hermitian matrix tt† . The matrix tt† has the same trace as the initial nonHermitian matrix Γ̂L Ĝr1N Γ̂R ĜaN 1 . We recognize in this Hermitian product the transmission
matrix t from the Landauer formula (2.54). The Green function expression for t will allow us
to find the transmission eigenvalues Tn by diagonalizing tt† . The corresponding eigenvectors
define a way in which atomic orbitals in the definition of TBH contribute to each conducting
channel. Therefore, the computation of Ĝr makes it possible to study both conductance and
more detailed mesoscopic transmission characteristics of the sample.
An equivalent formula for the quantum conductance follows from the Kubo formalism
(cf. Sec. 2.4.1). The Kubo formula for the static quantum conductance35 is given in terms
35 After
disorder averaging the symmetries of the systems will be restored and all diagonal com-
ponents of the, in general conductance tensor are approximately equal. Therefore, we denote the
conductance as a scalar.
67
of the Green functions (2.47) as
G=
4e2 1
Tr
h̄v̂
Im
Ĝ
h̄v̂
Im
Ĝ
.
x
x
h L2x
(2.109)
In this formula we will use the site representation of the velocity operator vx which is obtained
from the commutator in Eq. (2.40) with the tight-binding Hamiltonian (2.74)
m|v̂x |n =
i
tmn (mx − nx ) .
h̄
(2.110)
The length of rectangular sample in the x̂ direction is denoted by Lx = Na. The use of
the formula (2.109), together with the Green function Ĝr,a = (E − Ĥ ± iη)−1 for finite-size
system (without attaching the leads), would lead into ambiguity requiring some numerical
trick to handle iη (as was done historically in the literature [106]). However, if we employ
the Green function (2.84), the Kubo formula (2.109) produces a result completely equivalent
to the Landauer-type conductance formula (2.105) introduced above. As emphasized before,
the Green function (2.84) takes into account leads and corresponding boundary conditions,
i.e., the presence of reservoirs. The leads effectively destroy the phase memory of electrons
which is the same what realistic modeling of reservoirs (i.e., inelastic processes occurring in
them) would do. This type of discussion, brought about by mesoscopic physics [67], can help
us also to understand some experiments. For example, a current passed through a carbon
nanotube [107] would heat the sample to 20 000 K (and obviously melt it completely) if the
dissipation occurred across the sample and not in some “reservoirs”.
What is the most efficient way to use these formulas for conductance? Optimization of
computations is essential because of the limited memory and speed of computers. Thus, the
formulas should not be employed in a way which requires more operations than required.
Careful analysis of all physical properties of the conduction process is the best guidance in
achieving efficient algorithms. It also helps to differentiate the real computational complexity [42] of the problem from the apparent one. Since nearest-neighbor TBH of the sample is
a band diagonal matrix of bandwidth 2Ny Nz + 1, one can shorten the time needed to compute the Green function (2.84) by finding the LU decomposition [104] of a band diagonal
68
matrix. In the Landauer-type formula (2.105) we need only (Nz Ny )2 elements of the whole
Green function (2.84). They can be obtained from the LU decomposition36 of the band
diagonal matrix E − Ĥ − Σ̂r by a forward-backward substitution [104]. The trace operation
in formula (2.105) is also performed only over matrices of size Ny Nz × Ny Nz . This procedures require the same computational effort as the recursive Green function method [93, 105]
usually found in the literature.37
It might appear at the first sight that the trace in the Kubo formula (2.109) should
be performed over the whole Green function matrix (i.e., the space of states inside the
conductor). A better answer is obtained once we invoke the results of the discussion on
current conservation in Sec. 2.2 and the derivation of this formula from Sec. 2.4.1. Namely,
the formula is derived by volume integrating the Kubo NLCT
G=
1 1 dr
E(r)
·
j(r)
=
dr dr E(r) · σ (r, r) · E(r ).
¯
V2
V2
Ω
(2.111)
Ω
Here we have the freedom to choose any electric field factors E(r) and E(r ) because of the
DC current conservation.38 The electric field can be taken as homogeneous and non-zero in
some region of the conductor. Therefore, the trace operation in formula (2.109) is reduced
36 The
most advanced numerical linear algebra routines are provided by the LAPACK package
(available at http://www.netlib.org).
37 In
the recursive Green function method the self-energy from the left lead Σ̂rL is iterated through
the sample, using the appropriate matrix Dyson equation [105], and finally matched with the selfenergy coming from the right lead Σ̂rR . In this procedure matrices of dimension Ny Nz are inverted
N times.
38 The
current conservation was essential in arriving at the Kubo formula (2.109). Therefore, the
claims, sometimes found in the literature [64], that conductance can be computed by tracing over
v̂x Ĝa v̂x Ĝr (instead of the expression in Eq. (2.109)) are incorrect because such operator products
do not conserve the current inside the disordered region [82] (and its trace is in fact negative in
some energy interval).
69
to the Hilbert space spanned by the states in that part of the conductor. Since velocity operator v̂x (2.110) has non-zero matrix element only between two adjacent layers, the minimal
extension of the field is two layers in the x̂ direction. The layers are arbitrary (can be chosen either inside the conductor or on the boundary). That the conductance computed from
tracing over any two layers is the same is a consequence of current I being constant on each
cross section. Thus, one needs to find 4(Nz Ny )2 elements of the Green function (2.84) and
trace over the matrices of size 2Nz Ny × 2Nz Ny . This is a bit more complicated than tracing
in the Landauer-type formula (2.105). It is interesting that to get the proper conductance in
this way one should replace Lx in the denominator of Eq. (2.109) with the lattice constant
a. So, if one traces “blindly” over the whole conductor the denominator should contain the
number of pairs of adjacent layers (N − 1)a instead of Lx = Na. In the rest of the thesis
we mostly prefer the Landauer-type formula because of the less time consuming evaluation
of Green functions and the trace.39
We complete the discussion of conductance formulas with some remarks on the conceptual issues which arise when applying them to finite-size conducting systems. In both
Eqs. (2.105) and (2.109) the transport coefficients are computed using the Hamiltonian of
an isolated system (although the dissipation occurs in the reservoirs). The connection of the
sample to the reservoirs changes the boundary conditions for the eigenstates, transforms the
discrete spectrum of the finite sample into a continuous one, and modifies the way electrons
loose energy and phase coherence. Nevertheless when the coupling between the system and
the reservoirs is strong (sic !) it is assumed that is has no influence on the conductance. We
study such “counterintuitive” (for the quantum intuition) feature in Sec. 4.2 and Ch.5 by
looking at the influence of leads on the conductance of our model. It is shown there that
these requires to consider carefully the relationship between relevant energy scales.
39 In
order to reduce the time needed to compute the trace of four matrices one should multiply
them inside the trace in the following way: Tr [A · B · C · D] = Tr[(A · B) · (C · D)].
70
Chapter 3
Residual Resistivity of a Metal between the Boltzmann
Transport Regime and the Anderson Transition
3.1
Introduction
Ever since Anderson’s seminal paper [2], a prime model for the theories of the disorder
induced metal-insulator, or localization-delocalization (LD) [5], transition in non-interacting
electron systems has been the tight-binding Hamiltonian on the hypercubic lattice
Ĥ =
m
εm |mm| + t
|mn|,
(3.1)
m,n
with nearest-neighbor hopping matrix element t between s-orbitals r|m = ψ(r − m) on
adjacent atoms located at sites m of the lattice. The disorder is simulated by taking random
on-site potential such that εm is uniformly distributed in the interval [-W/2,W/2]. Thus,
the on-site potential εm is uncorrelated white noise with zero mean and variance εm εm =
δmm W 2 /12. This is commonly called the “Anderson model”. There are many numerical
studies [108] of the LD transition, which occurs in three-dimensions (3D) for a half-filled
band at the critical disorder strength Wc ≈ 16.5t [109]. Experiments on real metals with
strong scattering or strong correlations often yield resistivities which are hard to analyze.
Theory gives guidance in two extreme regimes: (a) the semiclassical case where quasiparticles
with definite k vector justify a Boltzmann approach and “weak localization” correction [36],
71
and (b) a scaling regime [8] near the LD transition to “strong localization”. Lacking a
complete theory it is often assumed that the two limits join smoothly with nothing between.
Experiments, however, are very often in neither extreme limit. The middle is wide and needs
more attention.
In this chapter a 3D numerical analysis is presented, focused not on the transition itself
but instead on the resistivity for 1 < W/t < Wc /t; specifically we ask how rapidly does the
resistivity ρ(W ) deviate from the values predicted by the usual Boltzmann theory valid when
W t. It has long been assumed that “Ioffe-Regel condition” ∼ 1/kF ∼ a [41] (a being the
lattice constant) gives the criterion for sufficient disorder to drive the metal into an Anderson
insulator. Figure 3.1 shows that this is wrong. For W/t ∼ 4 and ≈ 2a, Boltzmann theory is
no longer justifiable. At larger W/t one cannot properly speak of quasiparticles or mean free
paths. However, Kubo theory permits discussion of the diffusivity Dα of an eigenstate |α,
defined below in Eq. (3.3). In the semiclassical regime, Dα → Dk = vk k /3. The diffusivity
Dk diminishes as (W/t)−2 in Boltzmann theory. As /a approaches a minimum value (∼ 1),
Dα decreases toward Dmin = ta2 /h̄, which can be regarded as a minimum metallic diffusivity
below which localization sets in. But there is a wide range of W/t over which Dα ≤ Dmin and
yet the Boltzmann scaling D ∼ (t/W )2 is approximately right. In this regime single particle
eigenstates |α are neither ballistically propagating nor are they localized. There is a third
category: “intrinsically diffusive” [110]. A wave packet built from such states has zero range
of ballistic motion but an infinite range of diffusive propagation. Such states are not found
only in a narrow crossover regime but over a wide range of parameters physically accessible
in real materials and mathematically accessible in models like the Anderson model, as shown
in Ch. 8. In this regime, there is not a simple scaling parameter nor a universal behavior.
But the behavior is quite insensitive to a changes in Fermi energy EF or kB T , and scales
smoothly with W/t.
The traditional tool for computation of ρ has been the Kubo formula [85] (cf. Sec. 2.4.1),
2
10
1
10
EF=0
T
0
10
EF=2.4t
ρT/ρB
1
0
2
ρ/ρB
B
-1
10
2
ρ/ρB
Mean Free Path (a)
72
EF=0
ρT/ρB
1
0
0
20
40
60
80
100 120
2
Disorder Strength (W/t)
Figure 3.1: Resistivity ρ at EF = 0 (lower panel) and EF = 2.4t (middle panel), from a
sample of cross section A = 225 a2, normalized to the semiclassical Boltzmann resistivity ρB
calculated in the Born approximation. Also plotted are the ratios of ρB to the Boltzmann
resistivity ρT obtained using a T-matrix for multiple scattering on a single impurity. The
upper panel shows putative mean free paths /a obtained from ρB (labeled by B) or ρT
(labeled by T). Error bars at small W/t are smaller than the size of the dot.
73
originally derived for the system in thermodynamic limit. In a basis of exact single particle
electron state |α of energy Eα , the Eq. (2.39) can be written as
e2 ∂f
1
−
Dα = e2 N(EF )D̄,
σ= =
ρ
Ω α
∂Eα
(3.2)
where Ω is the sample volume, N(EF ) the density of states (DOS) at EF , D̄ the mean
diffusivity, and state diffusivity is given by
Dα = πh̄
α
|α|v̂x |α|2 δ(Eα − Eα ).
(3.3)
Here v̂ is the velocity operator which was defined in Eq. (2.40). These formulas, while
correct, are hard to use numerically. We demonstrate explicitly in Sec. 4.2 some of the
problems arising in application of the Kubo formula in exact single particle state representation (3.2). Thanks to the recent advances in mesoscopic physics [43], it is now apparent
that the Landauer-Buẗtiker scattering approach [4, 92] provides superior numerical efficiency
when computing the transport properties of finite [111] disordered conductors. Here we also
have in mind the Kubo formula, which, when applied properly to the finite-size systems (e.g.,
calculations on 3D samples in Ref. [95]), amounts to choosing the appropriate multi-channel
Landauer formula [71]. This was pointed out in Sections 2.4.1 and 2.5, which are devoted to
detailed explanation and comparison of different transport formalisms. The Landauer formula relates the conductance of a sample to its quantum-mechanical transmission properties.
This formalism emphasizes the importance of taking into account the interfaces between the
sample and the rest of the circuit [67]. Transport in the sample is phase-coherent (i.e., effectively occurring at zero temperature); the dissipation and thus thermalization of electrons
(necessary for the establishment of steady state) takes place in other parts of the circuit.
3.2
Semiclassical Resistivity
The principal result for the (quantum) resistivity of the Anderson model, obtained here
from the Landauer-type formula, is shown on Fig. 3.1 for two different Fermi energies EF = 0
74
(half-filled band) and EF = 2.4t. At EF = 2.4 the band is approximately 70% filled but the
filling decreases somewhat as W , and thus the band-width, increases. The widening of the
energy band of a disordered sample is shown on Fig. 3.2.
The linearized Boltzmann equation
∂f
=
−eE · vk
∂k
dFk
dt
,
(3.4)
scatt
serves as a reference theory. Here k is the energy band for W = 0, namely k = 2t
i
cos ki a,
h̄vki is ∂k /∂ki , and Fk is the non-equilibrium distribution function. The collision integral is
dF
dt
=−
scatt
2π |U |2 (Fk − Fk )δ(k − k ).
h̄ k kk
(3.5)
The mean squared matrix element of the random potential |Ukk |2 , in Born approximation,
is ε2m = W 2 /12, where
(. . .) =
dεm (. . .)P (εm) =
dεm (. . .)
1 W
θ( − |εm |),
W 2
(3.6)
denotes average over the probability distribution of the random variable εm . The Boltzmann
equation assumes that quasiparticles propagate with mean free path a between isolated
collision events. The equation is exactly solvable, yielding (for kB T t)
n
1
= e2 τ
ρB
m
with (n/m)eff =
2
v
kx δ(k
,
(3.7)
eff
− EF )/Ω. The exact solution of Eqs. (3.4, 3.5) using the Born
approximation for Ukk gives a ‘Fermi golden rule’ for the momentum lifetime τ (at EF ),
h̄
W2
|k|U|k |2 δ(k − k ) = 2π
= 2π
N(EF ).
τ
12
k
(3.8)
This is equal to the transport mean free time since the scattering is isotropic on the point
scatterers of the Anderson model (3.1) (i.e., no factor of [1 − cos θ] is needed). Thus, the
isotropic scattering eliminates the vertex correction in the linear response formalism, or
equivalently, the scattering in term in the Boltzmann equation. Here the matrix element of
75
the impurity potential is taken between the eigenstates of TBH (Ns is the number of lattice
site)
1 ik·m
e
|m,
|k = √
Ns m
(3.9)
and the final result is averaged over the probability distribution P (εm ). We evaluate (n/m)eff
and N(EF ) numerically. The Boltzmann-Born answer for the semiclassical resistivity is
ρB =
πh̄a W
e2
4t
2
1
,
(3.10)
[sin2 kx a + sin2 ky a + sin2 kz a],
(3.11)
vk2 E=EF
where the velocity squared,
vk2
=
1 ∂k
h̄ ∂k
2
2ta
=
h̄
2
is averaged over the Fermi surface, vk2 E=EF . The clean metal DOS, dirty metal DOS
(obtained from the exact diagonalization of diagonally disordered TBH), and ρB are plotted
as a function of EF on Fig. 3.2. Evaluation of ρB close to the band edges or for strong disorder
is unwarranted.1 In this energy intervals or for large W/t an accurate calculation requires a
complete quantum description. Nevertheless, it is instructive to follow the deviation between
the semiclassical and the quantum calculations. When W = 3t and a = 3 Å, ρB is 125 µΩcm,
typical of dirty transition metal alloys, and close to the largest resistivity normally seen in
dirty “good” metals. Figure 3.1 plots ρ/ρB versus (W/t)2 . Even for W = 10t there is less
than a factor of 2 deviation from the (unwarranted) extrapolation of the Boltzmann theory
into the regime W/t > 1.
If Born criterion, p3F V (p) = 0 h̄3 EF (EF is the largest energy scale in the problem) [114], is relaxed, then summation of all diagrams for the multiple scattering on a single
1 The
perturbative quantum analysis, based on the selection of some class of diagrams in expan-
sion in disorder strength, is not enough to account for such non-perturbative phenomena like the
exponentially small tails in the DOS near the band edges of normal metals [112] (instead, one has
to use the instanton analysis, also known as the “optimal fluctuation method” [113]).
2.0
ρB (ha/2e )(W/4t)
2
76
2
1.5
1.0
Density of States
0.5
0.0
0.3
(a)
0.2
(b)
(c)
(d)
0.1
0.0
-6
-4
-2
0
2
4
6
Fermi Energy
Figure 3.2: The density of states of (lower panel): (a) clean metal (W = 0); (b) dirty metal
(W = 6 on a lattice 15 × 15 × 15 averaged over 50 impurity configurations), obtained by
exact diagonalization of a closed sample Hamiltonian; (c) dirty metal (W = 6 on a lattice
10×10×10 averaged over 50 impurity configurations), obtained from the imaginary part (4.8)
of the Green function (2.84) of an open system; (d) is the same as (c) except for the smaller
lattice, 4 × 4 × 4. The upper panel shows the Boltzmann resistivity ρB (3.10), evaluated in
the Born approximation, at all EF throughout the clean metal energy band.
77
impurity should be performed. This gives the disorder-averaged Green function (r-retarded)
in the “non-crossing”2 approximation
Gr (k, E) =
1
.
E − k − Σr (E, k)
(3.12)
The expression for the scattering time, which can always be expressed in terms of partially
summed diagrams for the self-energy Σr (E, k) in the perturbation theory3 generating the
disorder-averaged quantities,
−
h̄
= Im Σr (k, E),
2τ (k, E)
(3.13)
is then the same as Eq. (3.8) except that Born amplitude Ukk is changed into T-matrix
element Tkk for the scattering on a single impurity [1]. The validity of this substitution
requires the absence of resonances, making the T-matrix a slowly varying function of momentum on the scale of h̄/. The T-matrix is given implicitly in terms of the following
inhomogeneous integral equation (in operator form) [33]
Tr = U + U Ĝr Tr .
(3.14)
This equation contains the impurity-averaged single particle Green functions (“dressed propagators”) introduced by Eq. (3.12). This reflects the presence of other impurities (instead
2 This
would be the most comprehensive semiclassical approach (also called single-site Coherent
Potential Approximation [115]). It is accomplished in the framework of diagrammatic impurityaveraged perturbation theory [1] by summing all diagrams in which lines representing potential
scattering do not cross. This means that scattering from a single impurity is treated exactly, but
scattering from all other impurities is taken into account in a mean-field approximation. It is clear
that this method neglects quantum interference effects on the electron wave function scattered from
different impurities. In the strong scattering regime crossed diagrams (lowest order of which generates WL) become of the same order of magnitude as the non-crossed diagrams. The semiclassical
part of our study deals only with a subset of the non-crossed diagrams.
3 Impurity-averaged
perturbation theory is equivalent to the perturbation theory for electrons
interacting with static interaction, except that closed fermion loops are absent.
78
of just a single impurity in vacuum). By taking the site representation (e.g., m|T|m ) we
solve Eq. (3.14) for the T-matrix of the Anderson model in a lowest order approximation
(using the free particle Green function)
G0 (m, m; z) =
1 1
,
Ns k z − k
(3.15)
Therefore, the attempt to “improve” Boltzmann theory, by including multiple scattering
from single impurities, technically leads to the replacement of the impurity potential εm in
Eq. (3.8) with
Tmm (z) =
εm
.
[1 − εm G0 (m, m; z)]
(3.16)
To next order the mean square T-matrix is
|Tmm (z)|2
1
=
W
W/2
dεm
−W/2
εm
=
(1 − εm G0 (m, E))
W2
3W 2
=
1+
(G0 G∗0 + G0 G0 + G∗0 G∗0 ) + . . . ,
2
12
20t
(3.17)
where the first term is the Born approximation and the coefficient of the correction (∼
O(W 4 )) changes sign from negative to positive as EF moves from 0 to 2.4t. As shown on
Fig. 3.1, the resistivity does not behave like |Tmm (z)|2 ; multiple scattering with interference
from pairs of impurities is at least equally important, and the “exact” ρ(W ) is less sensitive
to details like EF than is the T-matrix approximation. The rest of the Chapter presents the
method of calculation and describes a bit of mesoscopic physics of very dirty metals.
3.3
Quantum resistivity
We use a Landauer-type formula, introduce in detail in Sec. 2.5, to get the exact quantum
conductance G of finite samples with disorder configurations chosen by a random number
generator. Finite samples permit exact solutions for any strength of disorder. The bulk resistivity is extracted from the disorder-averaged resistance R by finding the linear (Ohmic)
79
scaling of R versus the length of the sample L at fixed cross section A (Fig. 3.3). This brute
force method has been used recently to extract resistivities for the liquid and amorphous
transition metals [116] or “3D quantum wires” [117]. The drawbacks of the finiteness of the
sample are faced when trying to elevate these results to the true bulk values. Two kinds of
errors [117] may arise: (a) The transition from the Ohmic regime to the localized regime
occurs for length of the sample L ∼ ξ which happens when G ∼ O(2e2 /h). If L is made
large enough, G will always diminish to this magnitude, no matter that the material of which
the sample is made may not be strongly disordered. This is shown for the first time in the
landmark paper of Thouless [19] by finding the localization length in quasi one-dimensional
samples4
ξ = (βM + 2 − β) ≈ βM, when M 1,
(3.18)
where M ∼ kF2 A is the number of propagating transverse modes at the Fermi energy EF
(referred to as “channels”, in the spirit of Landauer scattering approach, cf. Sec. 2.4.2) and
β ∈ {1, 2, 4} is the symmetry index (defined by the presence or absence of time-reversal
and/or spin-rotation symmetry) which delineates the universality classes in the localization
theory or random matrix theory, as explained in Ch. 7. The label in Eq. (3.18) differs
from the transport mean free path of kinetic theory by some numerical coefficient which
depends on the Fermi surface. Therefore, we avoid using the sample sizes with too small
G. (b) Finite-size boundary conditions and non-specular reflection [118] cause the density
4 To
satisfy the curiosity of a reader, who might wonder about e.g. copper wires becoming
localized when they are long enough to have conductance of around h/2e2 ≈ 12.5 kΩ, we underline
that this analysis is a zero-temperature one. The same argument at finite temperatures require
that dephasing length Lφ (which replaces L in the scaling analysis) has to be bigger than ξ. Since
metallic wire with a cross section of 2000 × 2000 Å has nearly M = 106 channels [43], the mean
free path of only 10 Å still generates ξ ≈ 1 mm. Thus, ξ is much bigger than typical Lφ , even at
very low temperature.
80
of states [119] and scattering properties of the sample to be slightly altered as compared to
the true bulk (cf. Fig. 3.2). We expect these effects to be small for our samples where is
√
smaller than the transverse size A. In fact, it is demonstrated on Fig. 3.2 that even DOS
computed from a very small sample exhibits minuscule deviations from the one computed in
a large system limit. The observed deviation is mostly pronounced close to the band edges,
while our result are confined to the fillings around the band center.
A two-probe measuring configuration is used for computation. The sample is placed
between two disorder-free (εm = 0) semi-infinite leads connected to macroscopic reservoirs
which inject thermalized electrons at electrochemical potential µL (from the left) or µR
(from the right) into the system, as shown on the “standard” example of Fig. 2.1. The
electrochemical potential difference eV = µL − µR is measured between the reservoirs. The
leads have the same cross section as the sample. The hopping parameter in the lead tL and
the one which couples the lead to the sample tC are equal to the hopping parameter t in
the sample. Thus, extra scattering (and resistance) at the sample-lead interface is avoided
(cf. Ch. 5), but transport at Fermi energies |EF | greater than the clean-metal band edge
|Eb | = 6t cannot be studied (Fig. 3.2). Hard wall boundary conditions are used in the ŷ and
ẑ directions. The sample is modeled on a 3D simple cubic lattice with N × Ny × Nz sites,
where Ny = Nz = 15 and lengths L = Na are taken from the set N ∈ {5, 10, 15, 20}.
The linear conductance is calculated using an expression obtained from the Keldysh
technique [102]
Ny Nz
4e2 e2
e2 r
a
†
Tr Im Σ̂L Ĝ1N Im Σ̂R ĜN 1 =
Tr (tt ) =
G =
Tn ,
πh̄
πh̄
πh̄ n=1
t = 2
−Im Σ̂L Ĝr1N
(3.19)
−Im Σ̂R ,
(3.20)
which is our standard Landauer-type formula (2.105). In the case of two-probe geometry,
the average transmission in the semiclassical transport regime (a < L ξ) is given by
T = 0 /(0 +L) [43], with 0 being of the order of . The semiclassical limit of the Landauer
formula for conductance, obtained e.g., from the stationary-phase approximation [121] of the
81
0.8
W=2
PL(R)
L=15a
W=5
2
Resistance (h/2e )
0.7
W=11
0.6
W=8
0.5
0.01
0.4
R (h/2e )
0.3
0.1
W=9
W=8
W=7
W=6
W=5
W=4
W=3
W=2
0.2
0.1
0.0
2
W=10
0 2 4 6 8 10 12 14 16 18 20 22 24
Length of the sample L (a)
Figure 3.3: Linear fit R = C1 +ρ/A L, (A = 225 a2 ) for the disorder averaged resistance R
in the band center EF = 0 and different disorder strengths W . The intercept C1 is decreasing
with increasing W (i.e., it is not determined just by the contact resistance πh̄/147e2 ) and
becomes negative for around W > 7t. The inset shows examples of the distribution of
resistances PL (R) (for L = 15a) versus log R. The distribution broadens either by increasing
W or the length of the sample (the units on y-axis are arbitrary and different for each
distribution).
82
Green function expression (2.106) for the transmission amplitude, is given by
G = (e2 /πh̄) MT .
(3.21)
Thus, for not too strong scattering, conductance should have the form
L
G−1 = RC + ρ .
A
(3.22)
It describes the (classical) series addition of two resistors. The “contact” resistance [122]
RC = πh̄/e2 M is non-zero, even in the case of ballistic transport when the second term
containing the resistivity ρ = (πh̄/e2 ) A/0 M vanishes. A ballistic conductor with a finite
cross section can carry only finite currents (the voltage drop occurs at the lead-reservoir
interface), cf. Ch. 6. Using this simple analysis for guidance, we plot average resistances
(taken over Nconf = 200 realization of disorder) versus L in Fig. 3.3, and fit with the linear
function
R = C1 + C2 L, C2 = ρ/A.
(3.23)
The resistivity ρ on Fig. 3.1 is obtained from the fitted value of C2 . Only for very small values
of W (W ≤ 2) the constant C1 is approximately equal to RC = πh̄/e2 M, where M = 147 is
the number of open channels in the band center [120] (the opening of the channels of TBH,
as a function of EF , is explained thoroughly in Ch. 5. To our surprise, C1 diminishes steadily
with increasing W , and even turns negative around W > 7t.
The quantum conductance G fluctuates from sample to sample exhibiting universal
conductance fluctuations (UCF) [37],
∆G =
√
Var G =
(G2 − G)2 e2 /πh̄.
(3.24)
This well know result [37] has been derived in the semiclassical transport regime G e2 /πh̄.
The amplitude of the UCF in this regime does not depend on the microscopic details of
disorder but only on the symmetry properties of the Hamiltonian, and can be thus classified
into three universality classes discussed in Ch. 7. Due to conductance fluctuations, generated
83
0.8
1/2
2
(Var G) (2e /h)
1.0
0.6
0.4
15x15x15
10x10x10
0.2
0.0
0
2
4
6
8
10
12
14
16
Disorder Strength W/t
Figure 3.4: The conductance fluctuations (∆G =
√
Var G at EF = 0) from weak to strong
scattering regime in the disordered cubic samples 10 × 10 × 10 and 15 × 15 × 15.
84
by quantum interference, individual mesoscopic conductors do not add in series. Therefore,
the conductance (or resistance) are not self-averaging quantities [54] as a function of the
sample length. Only the combination of decoherence and multiple scattering provides for
the ubiquity of the Ohm’s law found in (weakly disordered) macroscopic sample. Even in
the metallic hypercubic samples Ld the relative fluctuations scale as
Var G
∼ L4−2d ,
2
G
(3.25)
which means that there is no self-averaging in one and two dimensions. In 3D relative variance decays slower than the classically expected inverse volume dependence. The proper handling of fluctuations effects in our calculations is essential, especially when entering the regime
of strongly disordered (finite-size) conductor. Only disorder-averaged value are supposed to
exhibit the Ohmic scaling in the appropriate transport regime. The inset on Fig. 3.3 shows
the distribution of resistance PL (R) [123] for our numerically generated impurity ensemble.
The error bars, used as weights in the fit (3.23), are computed as δR =
VarR/Nconf (which
is the statistical error estimating the standard deviation of the average values). We find that
∆G is indeed independent of the size L (of cubic samples), but decreases systematically by
a factor ≈ 3 as W increases to the critical value Wc (Fig. 3.4).
On the other hand, ∆R, being similar to ∆G/G2 , depends on the sample size. The
evolution of ∆G and ∆R with disorder, and for different sample geometries (cubic or parallelepiped) is shown on Fig. 3.5.
As W approaches Wc , G gets smaller until (for our finite
samples) ∆G/G ∼ 1. At this point the distribution of resistances R = 1/G becomes very
broad and R begins to rise above 1/G (Fig. 3.6). For L = 15 this happens when W ≥ 12t.
At large W the conductance of long samples (N = 20) becomes close to e2 /πh̄ and deviations from Ohmic scaling are expected. Therefore, we do not use these points in the fitting
procedure when W > 10t (keeping the conductance of the fitted samples G > 2e2 /πh̄ [117]).
Finally, we offer a tentative explanation for the deviation of C1 (3.23) from the quantum
point contact resistance RC . In the semiclassical regime G e2 /πh̄ there are corrections
10
10
(Var R)
2
2
1/2
(h/2e )
85
0
15x15x5
15x15x10
15x15x15
15x15x20
-2
10
-4
2
0.8
(Var G)
1.2
1/2
(2e /h)
10
1.6
15x15x5
15x15x10
15x15x15
15x15x20
0.4
0.0
0
2
4
6
8 10 12 14 16
W/t
√
Figure 3.5: The conductance fluctuations, ∆G = Var G (lower panel), and resistance
√
fluctuations, ∆R = Var R (upper panel), at EF = 0, from weak to strong scattering
regime in disordered samples of different geometry.
86
4.0
-1
10
3.5
-1
2
<G>
3.0
-1
<R>/<G>
1
2.5
0.1
0.01
-1
2.0
<R>/<G>
Resistance (h/2e )
<R>=<G >
1.5
1.0
0
2
4
6
8
10 12 14 16 18
W/t
Figure 3.6: The deviation between disorder averaged resistance R = 1/G and inverse
of disordered average conductance 1/G, evaluated at EF = 0, as a function of disordered
strength W in the Anderson model on a cubic lattice 15 × 15 × 15.
87
to the Ohmic scaling G ∝ Ld−2 . The Diffuson-Cooperon diagrammatic perturbation [15]
theory produces a (negative) WL correction [36], which is given in 3D by
σ(L) = σ +
1
e2 1
e2
√
−
.
π 2 h̄2 2 L π 3 h̄ 0
(3.26)
Here 0 is a length of order . The precise value of 0 does not lead to observable consequences
in the experiments studying WL (as long as it is unaffected by the temperature and the
magnetic field). The positive 1/L term in Eq. (3.26) provides a possible picture for our
finding that C1 in Eq. (3.23) goes negative as W increases. However, this picture is an
extrapolation from the semiclassical into the “middle” regime of intrinsically diffusive states,
and therefore should be given little weight. The negative values of C1 is better regarded as
a new numerical result from the mesoscopic dirty metal theory.
3.4
Conductance vs. Conductivity in mesoscopic
physics
This section is a brief discourse on the mesoscopic view of conductance and conductivity
which is closely tied to the computation of transport properties in finite-size disordered
systems. Inasmuch as mesoscopic transport methods are concerned with samples where
electrons have a totally quantum-mechanical coherent history within the sample, they must
treat explicitly surfaces through which electron leaves the conductor and, thereby, loses the
memory of its phase. It is obvious that these procedures naturally take into account the finite
size of the sample. Thus, the central linear transport quantity in the mesoscopic methods is
conductance [5, 26], rather than the conductivity
σ(L) = L2−d G(L).
(3.27)
In fact, the length scale necessary to characterize conductivity is Lφ , and not as usual
in macroscopic samples, because of the intrinsic non-locality of quantum mechanics [26].
88
The importance of conductance, emphasized by mesoscopics, is also transparent in the experiments in which measure the conductance. In fact, the mesoscopic experiments have
directed the development of the theory of phase-coherent transport toward sample-specific
quantities, i.e., those which describe a single sample measured in a given manner (where
quantum-mechanical features of transport “violate” the standard rules of electrical engineering circuit approach [125]). This is to be contrasted with the notions of traditional condensed
matter physics of macroscopic systems where only quantities which are just the average over
impurity ensemble were studied. Nonetheless, the efficiency of mesoscopic transport methods is too appealing to be abandoned, and in this Chapter we have employed them5 to get
the intensive quantity (resistivity) at the price of having to deal with quantum coherence
fluctuation effects in the finite-size samples (which act as a nuisance on this path).
The bulk conductivity is a material constant defined only in the thermodynamic limit [124]
σ = lim L2−d G(L).
L→∞
(3.28)
The computation of conductance is exemplified by either the Landauer formula or the
Kubo [105] formula (cf. Sec. 2.5) which is properly applied to the finite-size samples (i.e.,
on the setup from Fig. 2.1). The scaling theory [8] of Anderson localization also stresses
the role of the conductance in disordered systems. The conductance is a single scaling
5 Our
disorder-averaging procedure can be thought as describing the real sample at finite tem-
perature (but low enough that transport coefficients are determined by the scattering on quenched
disorder) where inelastic effects enter phenomenologically through dephasing length Lφ . Such sample can be viewed as a classical stack (where rules of combining parallel and series resistors apply)
of quantum resistors. Inside each quantum resistor of the size Lφ quantum diffusion takes place but
the whole sample has an intrinsic self-averaging which then “kills” the observability of mesoscopic
fluctuations [54] but leaves the effects of quantum coherence on localization (like WL and higher
order, particularly non-perturbative in our case, corrections) untouched.
89
variable6 for the localization-delocalization (LD) transition viewed as a critical phenomenon.
Strictly speaking, scaling theory teaches7 us that conductivity σ(L) of a disordered conductor (d-dimensional hypercube of volume Ld ) depends on its size L. At the critical point
the dimensionless conductance g(L) = G/GQ = gc is length-scale independent, therefore
conductivity scales to zero σ(L) → 0 as L → ∞.
The correlation length ξc of the LD transition8 is defined as the size of the conductor
(d-dimensional hypercube9 ) for which g(ξc ) ∼ O(1), or equivalently ETh ∼ O(∆(ξc )) [5]. For
L ξc the scaling of conductance characterizes a metal
g ∝ Ld−2 ,
(3.29)
g ∝ e−L/ξ .
(3.30)
or an insulator
In the localized phase the correlation length is ξc = ξ. The change of σ(L) is substantial
for the case L ξc where localized and delocalized phases are not discernable. For
example, assuming that g does not change by more than an order of magnitude, we get for
6 Conductance
of a disorder system is a fluctuating quantity [37] and one should scale the whole
distribution function or some typical value which can characterize this distribution, see Ref. [124].
7 In
2D systems (in the absence of magnetic field or spin-orbit scattering) one can say that
conductivity is an ill-defined quantity since it is non-zero for conductor size L < ξ, even though
one deals with an insulator in the limit L → ∞ for any disorder strength.
8 The
correlation length ξc is analogous to the correlation length of the order parameter φ(r) in
the theory of critical phenomena, χ(r) = φ(0)φ(r) ∝ exp(−r/ξc ). At the critical point ξc diverges
and the correlation function obeys a power law χ(r) ∝ r −η .
9 To
define the correlation length of a quasi-1D system one can use the conductance of a
hypercubic conductor which is a parallel stacking of quasi-1D samples [5].
is the cross section.
g(L) = gq1D (L, Lt ) (L/Lt )d−1 , where Ld−1
t
This means that
90
the scale dependent conductivity [20],
σ(L) = σL→∞
ξc
L
d−2
.
(3.31)
Fortunately, in metallic conductors (g 1) the length ξc is microscopic (in d > 2)
ξc ∼
λF
1/(d−2)
,
(3.32)
i.e., of the order of Fermi wavelength λF in 3D, as follows from (2.24). The same is true for
multichannel wires (quasi-1D systems), ξc ∼ M −1/(d−2) . So, one does not have to worry, in a
pragmatical sense, about the proper definition of conductivity from the finite-size sample (at
least in the semiclassical transport regime). Nevertheless, even in the semiclassical regime
with large conductance g 1 there are corrections to the Ohmic scaling g ∝ Ld−2 . This is
what is essentially given by the microscopic (perturbation) theory to first order, namely WL
correction in Eq. (3.26). Thus, the disorder-averaged two-probe Landauer formula (3.22),
reproduces Ohm’s law up to the corrections of the order of /L. It is plausible that this
effect become more important as disorder increases, as pointed out at the end of previous
Section.
91
Chapter 4
Quantum Transport in Disordered Macroscopically
Inhomogeneous Conductors
4.1
Introduction
In Chapter 3 the quantum transport methods were employed to study the resistance
of homogeneous samples with disorder (i.e., inhomogeneity) introduced on the microscopic
scale (∼ λF ). This Chapter investigates some of the transport properties of macroscopically
inhomogeneous conductors. Although the problem of transport through the contact of two
metals is an old one [126] in the solid state physics, the impetus to study metal junctions [127],
metallic multilayers [128], and even single disordered interfaces [115] has arisen only recently
in connection with the discovery (and potential applications) of giant magnetoresistance1
(GMR) [129] in antiferromagnetically coupled Fe/Cr multilayers. To understand the full
problem of spin dependent transport one should first clarify the effects of non-magnetic
inhomogeneous structures (with sometimes strong disorder) on conduction. For example, it
was pointed out that scattering on the interface roughness plays an important role in the
GMR effects [130].
1 Upon
applying weak magnetic field the resistance of a magnetic multilayer can drop to less
than a half of its value outside of the field.
92
Our goal in this Chapter is twofold:
• Most mesoscopic studies have been confined to bulk conductors in the weak scattering
(or “weak localization”) regime. Here we use non-perturbative methods from Sec. 2.5
to access the strongly disordered metal junctions, single strongly disordered interfaces
(when stacked together into a bulk conductor our interfaces would form an Anderson
insulator), and multilayers composed of interfaces and bulk disordered conductors. In
all three cases we study the transport perpendicular to the layers. This is the so-called
current perpendicular to the plane (CPP) geometry. [130] Once the quantum resistance
is computed, we investigate if it can be described by some resistor model, i.e., as a sum
of bulk and interface resistances which would form a corresponding classical circuit.
• Using some of the inhomogeneous models listed above, as well as homogeneous samples
as a reference, we compare the transport properties computed from the Kubo formula
in exact single particle state representation (2.39) to the ones obtained from the Kubo
formula for an open system surrounded by ideal leads (2.109). In the first case the
system is closed and we solve the Hamiltonian exactly by exact diagonalization. In the
second case the energy levels of the disordered region are broadened by the coupling to
the leads and we use real-space Green functions (from Sec. 2.5) to describe the system.
Also, we look at the change of conductance induced by varying the hopping parameters
in the leads or the ones characterizing the lead-sample coupling (this problem is similar
to the analysis undertaken in Ch. 5).
4.2
Transport through disordered metal junctions
In this section we study the static (DC) transport properties of a metal junction composed of two disordered conductors with different type of disorder on each side of an interface which halves the whole structure. Both conductors are modeled as binary alloys
93
(i.e., composed of two types of atoms) using tight-binding Hamiltonian on a hypercubic
lattice N × Ny × Nz
Ĥ =
m
εm |mm| + t
|mn|.
(4.1)
m,n
The disorder in the binary alloy is simulated by taking the random on-site potential such
that εm is either εA or εB with equal probability. Specifically, we take the lattice 18 × 8 × 10
on each side of the junction and for the binary disorder: εA = −4, εB = 0 on the left; and
εA = 4 and εB = 0 on the right. This junction has an “intrinsic” rough interface [131]
modeled by the random positions of three different types of atoms around it.
The conductivity2 of a disordered conductor can be calculated from the Kubo formula
in exact single-particle state representation (2.39)
σxx =
2πh̄e2 |α|v̂x |α|2 δ(Eα − EF )δ(Eα − EF ).
Ω α, α
(4.2)
The computation of transport properties from exact single particle eigenstates, obtained by
the numerical diagonalization of Hamiltonian, has been frequently employed throughout the
history of disordered electron physics [45]. However, direct application of the formula (4.2)
leads to a trouble since eigenvalues are discrete when the sample is finite and isolated.
Therefore, the conductivity is a sum of delta function. There are two numerical tricks which
can be used to circumvent this problem: (1) One can start from the Kubo formula for the
frequency dependent conductivity
2πh̄e2 f (Eα ) − f (Eα )
|α|v̂x |α |2
δ(Eα − Eα − h̄ω),
σxx (ω) =
Ω α, α
h̄ω
2 The
(4.3)
conductivity is a tensor in general case, but since symmetries are restored after disorder-
averaging, one can use for the scalar conductivity σ = (σxx + σyy + σzz )/3. This is valid only in
the case of homogenously disordered sample. For our metal junction it is clear that σxx is different
from σyy and σzz .
94
average the result over finite ω values, and finally extrapolate [132] to the static limit ω → 0.
(2) The delta functions in (4.2) can be broadened into a Lorentzian
δ(x) → δ̄(x) =
1 (η/2)2
,
π x2 + (η/2)2
(4.4)
where η is the width (at half maximum) of the Lorentzian. We find that both methods
produce similar results. The calculation presented below uses the broadened delta function
δ̄(x).
To simplify the calculation, we compute the diffusivity3
Dαx = πh̄
α
|α|v̂x |α|2 δ̄(Eα − Eα ).
(4.5)
The eigenstate diffusivity was introduced in Ch. 3. It can be extracted from the Kubo
formula (4.2), as shown in Eq. (3.2). The width η of the Lorentzian δ̄(Eα − Eα ) in (4.5) is
chosen as some multiple of the local average level spacing ∆(Eα ) in a small energy interval
around the eigenstate |α. The method of computing the eigenstate diffusivity is as follows: a
set of eigenstates (the number of eigenstates is equal to the number of lattice sites Ns = N ×
Ny ×Nz ) is obtained by numerical diagonalization;4 for each eigenstate we compute Dαx (4.5),
where summation is going over all states |α “picked” by the Lorentzian δ̄(Eα −Eα ) (centered
on Eα ) in an energy interval of 3η around Eα ; finally, we average over the disorder and bin
the diffusivities in an energy bin of the size ∆E = 0.0225. The smart way of computing
the quantum-mechanical average values of some operator, like α|v̂x |α appearing in the
definition of eigenstate diffusivity (4.5), is to multiply three matrices α̂† · v̂x · α̂, where α̂ is a
matrix containing eigenvectors |α as columns, and then take modulus squared of each matrix
3 This
is an additional transport information, related to conductivity, which is not usually seen
in the literature on disordered electron physics, but was studied in the physics of glasses (e.g.,
thermal conductivity in amorphous silicon [132]).
4 For
numerical diagonalization we use the latest generation of the linear algebra packages, LA-
PACK, available at http://www.netlib.org.
95
element in such product.5 This procedure becomes a natural choice once we understand
that it actually transforms the matrix of the operator v̂x from defining representation to the
representation of eigenstates |α. The end result of the calculation is the average diffusivity
(averaged over both disorder and energy interval) D̄x , which is related to the conductivity
through the Einstein relation
σxx = e2 N(EF ) D̄x (EF ).
(4.6)
This formula emphasizes that transport in a degenerate electron gas is a Fermi surface
property.6
We first calculate D̄(EF ) for the homogeneously disordered sample, with binary disorder εA = −2, εB = 2, modeled on a lattice 18 × 8 × 10. This is shown on Fig. 4.1. To
get an insight into the microscopic features of the eigenstates, a fraction of which around
EF determines the transport properties at EF , we also plot on this figure (averaged over
disorder and energy) Inverse Participation Ratio (IPR), defined and studied in more detail
in Ch. 8. The IPR is a simple one-number measure of the degree of localization (the bigger
the IPR the more localized the states is, e.g., IPR= Ns corresponds to a completely localized
states on one lattice site). IPR is also connected to the dynamics.7 The second calculation
plotted on Fig. 4.2 is for a homogeneous sample described by the Anderson model where
εm ∈ [−W/2, W/2] is a random variable in the TBH (equivalent to the samples from Ch. 3).
5 The
number of operations in the naı̈ve calculation of the expectation values, where each of
them is calculated separately, scales as ∼ Ns4 , while in the method presented above it scales as
∼ Ns3 (Ns × Ns is the dimension of operator matrix).
6 Conductivity
is a Fermi surface property at low temperatures only for conductors outside of
magnetic field [71]. On the other hand, conductance, measured in experiments between two voltage
terminals, depends only on the states at the Fermi surface, even in the presence of magnetic field.
7 The
IPR can be related to the average return probability [108] that particle, initially launched
in a state |m localized on a lattice site m, will return to the same site after a very long time.
96
The reference calculations on Figs. 4.1 and 4.2 are obtained from the Kubo formula (2.109)
expressed in terms of Green functions (2.84) for a sample attached to ideal semi-infinite
leads. This method gives the exact static conductivity, as discussed in Sec. 2.5,
σxx
4e2 1
=
Tr h̄v̂x Im Ĝ h̄v̂x Im Ĝ ,
h ALx
(4.7)
for a cubic sample of cross sectional area A. Its optimal application was elaborated in
Sec. 2.5.5. The concept of eigenstates and related diffusivity Dα cannot be used in an open
system sample+leads. Nonetheless, we can still get the density of states (cf. Sec. 2.5.2) from
the imaginary part of the Green function (2.84)
N(EF ) =
m
1
− Im ĜrS (m, m; EF ).
π
(4.8)
Thus, the average diffusivity D̄(EF ) is obtained easily from the Einstein relation (4.6)
where we divide the conductivity σxx by e2 N(EF ) and average over the results obtained
from samples with different disorder configurations. This clarifies the meaning of “diffusivity”
extracted from the Kubo formula (4.7) for an open finite-size sample.
In both calculations for the homogeneous samples it appears that discrepancy between
the Kubo formula in single particle representation (4.2) and the exact method, based on the
formula (4.7) for sample+lead system, is only numerical. In fact, the numerical discrepancy
is very small in the disordered binary alloy and a bit larger in the Anderson model.8 It
originates from the ambiguity in using the width η of the broadened delta function.9 The
increase of the diffusivity close to the band edges of diagonally disordered Anderson model
(Fig. 4.2) was seen in direct simulations of the wave function diffusion, performed in the
early days of localization theory [133].
8 When
compared to binary alloy, Anderson model looks like a conductor with infinite number
of different impurities.
9 In
some sense non-zero η simulates the effect of inelastic scattering as an uncorrelated random
event [106].
0.007
12
(b)
10
0.006
8
(a)
0.005
6
0.004
0.003
(c)
-6
-4
-2
0
2
4
2
6
2
4
Diffusivity (ta /h)
97
0
Fermi Energy
Figure 4.1: The diffusivity D̄(EF ) of a disordered binary alloy modeled by the tight-binding
Hamiltonian (εA = −2 and εB = 2) on a lattice 18 × 8 × 10: (a) computed using the Kubo
formula (4.7) in terms of the Green function for the sample with attached leads; (b) computed
from the Kubo formula in exact single particle eigenstate representation (4.5) using the width
of the Lorentzian broadened delta function, η = 25∆(EF ). Also plotted (c) is the Inverse
Participation Ratio (8.13) which measures the degree of localization of eigenstates. Disorder
averaging is performed over 50 different realization.
0.1
4
(b)
3
(a)
2
2
Diffusivity (ta /h)
98
0.01
1
(c)
-8
-6
-4
-2
0
2
4
6
8
0
Fermi Energy
Figure 4.2: The diffusivity D̄(EF ) of the diagonally disordered Anderson model (disorder
strength W = 10) on a lattice 18 × 8 × 10: (a) computed using the Kubo formula (4.7)
in terms of the Green function for the sample with attached leads; (b) computed from
the Kubo formula in exact single particle eigenstate representation (4.5) using the width
of the Lorentzian broadened delta function, η = 25∆(EF ). Also plotted (c) is the Inverse
Participation Ratio (8.13) which measures the degree of localization of eigenstates. Disorder
averaging is performed over 50 different realization.
99
We now repeat the same computation for a junction (introduced at the beginning of this
section) which is composed of two disordered binary alloys on each side of an interface. The
result is shown on Fig. 4.3. Large fluctuations of the diffusivity are caused by the conductance
being of the order of 2e2 /h (Fig. 4.6), i.e., the property of the strongly localized transport
regime at this level of disorder in the junction (cf. Ch. 3). Here the discrepancy between
the two different methods is not only quantitative, but the Kubo formula in single particle
exact eigenstate representation (4.2) shows non-zero diffusivity (and thereby conductivity)
at Fermi energies at which there are no states on one side of the junction which can carry
the current (it falls to zero only at the band edges).10 The result persist with decreasing
of the width η of the Lorentz broadened delta function. Therefore, it is not an artifact of
this numerical trick (because of which we were unable to get the exact value of diffusivity in
the homogenous sample above). The states which have non-zero amplitude throughout the
junction cease to exist at |E| ∼ 4.7. This is clearly seen by looking at the local density of
states ρ(m, E) (2.89) integrated over y and z coordinates (we broaden the delta function in
the definition of LDOS into a box function δ̄(x) equal to one in some energy interval)
ρ(mx , E) =
my ,mz
ρ(m, E) =
|Ψα (m)|2 δ̄(E − Eα ).
(4.9)
my ,mz α
This “LDOS in the planes” along the x-axis is plotted on Fig. 4.4. It changes abruptly while
going from one side of the junction to the other side (except for the small tails near the
interface).
It is clearly demonstrated on Fig. 4.3, where diffusivity vanishes at the same point at
which LDOS goes to zero, that Kubo formula (4.7) for an open finite-size sample, plugged
between ideal semi-infinite leads, correctly describes the junction. This is the primary result
of this section. It should be emphasized that, once the leads are attached, two new inter10 Intricacies
in the application of Kubo formula on the finite-size samples, “extended” through
the use of periodic boundary conditions, were discovered also in some other condensed matter
problems, e.g., in the conduction in 1D Hubbard model [111].
0.007
8
(b)
εm={-4,0}
0.006
εm={0,4}
6
(c)
0.005
4
0.004
0.003
2
(d)
2
(a)
Diffusivity (ta /h)
100
(e)
0.002
-8
-6
-4
-2
0
2
4
6
8
0
Fermi Energy
Figure 4.3: The diffusivity D̄(EF ) of a metal junction composed of two disordered binary
alloys, left (εA = −4, εB = 0) and right (εA = 0, εB = 4), modeled with the TBH on
a lattice 36 × 8 × 10: (a) computed using the Kubo formula (4.7) in terms of the Green
function for the sample with attached leads; (b) computed from the Kubo formula in exact
single particle eigenstate representation (4.5) using the width of the Lorentzian broadened
delta function, η = 25∆(EF ); (c) same formula as for (b) with η = 10∆(EF ); (d) same as (b)
with η = 5∆(EF ). Also plotted (e) is the Inverse Participation Ratio (8.13) which measures
the degree of localization of eigenstates. Disorder averaging is performed over 50 different
realization.
LDOS integrated over y and z coordinates
101
0.006
0.004
0.002
x={24,25,26,27,28,29,30,31}
0.000
0.006
x={20,35}
0.004
0.002
0.000
0.006
x={19,36}
0.004
0.002
0.000
0.006
0.004
x={6,7,8,9,10,11,12,13}
0.002
0.000
0.006
x={2, 17}
0.004
0.002
0.000
0.006
x={1,18}
0.004
0.002
0.000
-10
-8
-6
-4
-2
0
2
4
6
8
10
Energy
Figure 4.4: Local density of states (LDOS) integrated over the y and z coordinates for the
metal junction composed of two disordered binary alloys, left (εm ∈ {−4, 0}) and right
(εm ∈ {0, 4}). This “LDOS in the planes” along the x-axis is computed (4.9) from the
exact eigenstates of TBH. The result is plotted after averaging over several planes along the
x-axis (the planes used in this procedure are labeled on each panel). Disorder averaging is
performed over 50 different samples.
102
faces in the problem arise. They separate the sample from the leads. Landauer’s picture
of transport (which has motivated a proper application of the Kubo formula to finite-size
samples, and gave us some comfort in dealing with the puzzle of dissipation in such systems)
naturally takes care of these boundaries (cf. Sec. 2.5). Thus, it describes a real system
where electrons can leave or enter through the boundaries (furthermore, it emphasizes that
current is the response to gradient of the electrochemical potential and not to an electric
field). For example, this means that conductance will go to zero at the band edge of the
clean lead |Eb | = 6t if we use the same hopping parameter in the lead tL = t as in the
disordered sample (because there are no states in the lead which can propagate the current
for Fermi energies |EF | > 6t). Thus, the conductance of the whole band of disordered sample
cannot be computed unless we increase tL in the leads. This is illustrated on Fig. 4.5 for the
homogeneous sample described by the Anderson model where disordered extends the band,
|Eb | > 6t.
When we take tL != t, the natural question arises: how sensitive is the conductance on
the properties of leads? Some general remarks on this problem in mesoscopic physics (which
resembles “quantum measurement problem”, since leads also play the role of a macroscopic
apparatus necessary for the measuring of transport properties) are provided in Ch. 5 where
we study the same issue in the absence of disorder. It is understood [29] that if broadening
of the energy levels due to the leads is greater than the Thouless energy ETh , then level
discreteness is unimportant and conductance will be independent of the properties of leads
(i.e., of the level width they introduce). This limit corresponds to the “intrinsic conductance”
of a sample being much smaller than the conductance generated by the lead-sample contact.
We study this dependence by looking at the conductance for our model of junction as a
function of the hopping in the leads tL and coupling hopping parameter tC (these parameters
were introduced in Sec 2.5). The result is shown on Fig. 4.6. The conductance is virtually
independent of tL , which is a consequence of the smallness of the conductance of disordered
103
6
2
Conductance (2e /h)
8
tL=1.5t
tL=t
4
2
0
-6
-4
-2
0
2
4
6
Fermi Energy
Figure 4.5: Conductance of a disordered conductor modeled by the Anderson model with
W = 6 on a lattice 10 × 10 × 10 for two different values of the hopping parameter tL in
the leads. The computation is done using the Kubo formula (4.7) for the finite sample with
semi-infinite leads attached. Note that conductance vanishes at |E| = 6t (band edge in a
clean sample) when tL equals to the hopping t in the disordered sample. Disorder averaging
is performed over 50 different samples.
104
2
Conductance (2e /h)
1.2
tL=1, tC=1
tL=1.5, tC=0.1
tL=1.5, tC=1
tL=3, tC=1
tL=3, tC=3
1.0
0.8
0.6
0.4
0.2
0.0
-8
-6
-4
-2
0
2
4
6
8
Fermi Energy
Figure 4.6: Conductance of a metal junction composed of two disordered binary alloys, left
(εA = −4, εB = 0) and right (εA = 0, εB = 4), modeled with the TBH on a lattice 36×8×10.
The computation is based the Kubo formula (4.7) for the finite sample with semi-infinite leads
attached where different hopping parameters in the lead tL and the lead-sample coupling tC
are used. Disorder averaging is performed over 50 different samples.
105
junction. It goes down drastically with decreasing of the coupling tC , as suggested above
(the same behavior is anticipated when tL is increased substantially because of the increased
reflection at the lead-sample interface).
4.3
Transport through strongly disordered interfaces
This section present the study of transport properties of a single dirty interface. The
problems is not only a “theoretical” one, namely to understand the difference between the
transport in the bulk and through the interfaces, but has been brought about by the experiments on transport through metallic interfaces11 which are parts of magnetic multilayers
exhibiting giant magnetoresistance [135]. It seems that interface scattering is crucial for the
understanding of transport through more complex inhomogeneous systems, such as multilayers composed of bulk conductors separated by the interfaces (which is pursued in the next
section). These are the conductors typically encountered in the theoretical and experimental
studies of GMR phenomenon (with the added complication of spin-dependent interface resistance [136], which can dominate the magnetoresistance of magnetic multilayers). Theories
also show how interface resistances can be extracted from experiments. Since the nature
of transport relaxation time in inhomogeneous systems is not well understood [130], it is
wise to treat first single interfaces, and then study them as elements of more complicated
circuits (e.g., in the semiclassical theories interfaces are viewed as elements of some resistor
network [137]). For example, the properties of a single interface cannot be described in
terms of the Boltzmann conductivity (2.24), i.e., using mean free path (or transport mean
free time) familiar from the bulk conductors.
11 The
importance of interface scattering in many areas of metal and semiconductor physics has
been realized in the plethora of research papers since the seminal work of Fuchs [134]. They are
mainly concerned with the transport parallel to impenetrable rough interface, while we study the
transport normal to an interface (i.e., CPP geometry from the GMR studies).
106
It is conjectured in the literature [137] that resistance of a disordered interface12 results from defects (interfacial roughness) or interdiffused atoms. We model the short range
scattering potential generated by the impurities in the plane of interface using our usual
description in terms of the Anderson model (3.1), with strong disorder W = 30, on a (twodimensional) lattice of atoms 1 × Ny × Nz . The bulk conductor composed of such interfaces
(stacked in parallel and coupled with nearest-neighbor hopping t) is an Anderson insulator,
because all states are localized already for Wc ≈ 16.5 [109]. In order to apply the quantum
transport method based on the Landauer-type formula (2.105)
G =
t =
2e2
2e2 Tr Γ̂L Ĝr1N Γ̂R ĜaN 1 =
Tr (tt† ),
h
h
Γ̂L Ĝr1N Γ̂R ,
(4.10)
(4.11)
we place the interface between two semi-infinite disorder free leads. This can be viewed
as a conductance of a single sheet of the disordered material. Thus, such calculation will
demonstrate the difference between the (perpendicular) transport through the interface and
the transport in the bulk. Also computed is the conductance of a thin layer composed of
two (2 × Ny × Nz ) or three sheets (3 × Ny × Nz ) of the same bulk disordered material.
In this way we can follow the emergence of the Anderson insulator (G → 0) in the bulk
conductor. Both types of calculations are shown on the upper panel of Fig. 4.7. Also studied
is the influence of the leads on the conductance, undertaken in the same fashion as in the
previous section (compare to Fig. 4.5). It appears that hopping parameter in the leads tL
affects the conductance of the interface to a much grater extent than in the case of the
conductance of a bulk disordered conductor (characterized by a similar value of disorderaveraged conductance).
Mesoscopic transport methods give the possibility not only to compute the conductance,
12 Even
disorder-free interface can have a non-zero resistance, e.g., because of mismatch of crystal
potential and band structures [130] (cf. Ch. 5).
2
Conductance (2e /h)
107
10
tL=1.5t
tL=t
8
Nx=1
6
4
Nx=2
2
0
Nx=3
-8 -6 -4 -2 0 2 4
Fermi Energy
6
8
3
10
ρ(T)
2
Nx=1
(b)
(a)
10
1
10
Nx=2
Nx=3
0
10
0.0
0.2
0.4
0.6
0.8
1.0
Transmission eigenvalues T
Figure 4.7: Conductance of a single disordered interface (N = 1) and thin layers composed
of two (N = 2) or three (N = 3) interfaces, modeled by the Anderson model with W = 30
on a lattice N × Ny × Nz (upper panel). The calculation is for different values of the hopping
parameter tL in the attached leads and G is averaged over 200 impurity configurations.
Lower panel: Numerically obtained distribution of transmission eigenvalues ρ(T ) in the band
center, averaged over 1000 disordered configurations. The analytical functions plotted are
√
√
(a) ρ(T ) = (G/2GQ ) (T 3/2 1 − T )−1 and (b) Dorokhov’s ρ(T ) = (G/2GQ ) (T 1 − T )−1 .
108
but also to use the picture of conducting channels and transmission properties they entail.13
This information is more comprehensive than the one provided by conductance itself (cf.
Sec. 2.4.2). Digonalization of tt† in formula (4.10) gives a set of transmission eigenvalues Tn
for each realization of disorder. Counting the number of Tn in each bin along the interval [0, 1]
gives the numerical estimate for the distribution function ρ(T ) = n
δ(T − Tn ) (where
numerical procedure effectively mens that delta function has been broadened into a box
function δ̄(x) equal to one inside the bin). The lower panel of Figure 4.7 plots ρ(T ) for the
interface (and two thin layers introduced above). The result is compared to the Dorokhov’s
√
distribution for bulk conductor ρ(T ) = (G/GQ ) 1/(T 1 − T ) (2.57) and the one which fits
the numerical data
ρ(T ) =
1
G
√
.
3/2
2GQ T
1−T
(4.12)
The second formula is, up to a factor, the same as the analytical prediction of Ref. [115] for
√
a single dirty interface ρ(T ) = (G/πGQ ) 1/(T 3/2 1 − T ). Thus, our numerical computation
confirms the universality14 of ρ(T ) for a single interface. However, this universality class
differs from that of the bulk conductors.
13 From
a technical point of view, one does not need mesoscopic transport methods to study
the transport in macroscopic conductors (dominated by semiclassical features). Nevertheless, the
study of transmission probabilities (which requires phase-coherent transport) obviously enhances
our knowledge of the conduction in condensed matter systems
14 Universality
here means that ρ(T ) scales only with the sample conductance G, and thereby
does not depend on microscopic details of disorder. While being intriguing concept in disorder
electron physics, universality can be frustrating for the device engineers. Not all features of the
transport through dirty interface are universal [115].
109
4.4
Transport through metallic multilayers
Here we continue the study of inhomogeneous conductors by analyzing some examples
of (mesoscopic) metallic multilayers (while relying on the introduction and results exposed
in the previous two sections). The multilayer is composed of three bulk conductors joined
through two dirty interfaces. The whole structure is modeled by the Anderson model on a
lattice 17 × 10 × 10, where layers 6 and 12 contain the same interface as the one studied in
Sec. 4.3. The disorder strength in the interface atomic monolayer is fixed at W = 30, while
disorder inside the bulk layers (composed of five atomic monolayers) is varied. We take the
disorder strength to be the same in two outer layers where diffusive bulk scattering takes
place. This type of multilayer can be viewed as a period of an infinite A/B multilayer [137]:
layer of material A on the outside (of resistivity ρA and total thickness dA = 10a, where a
is the lattice spacing) and material B between the interfaces (of resistivity ρB and thickness
dB = 5a). We neglect any potential step at the interface (caused by the conduction band
shift at the interface [130]). Such multilayers are usually described in terms of the resistor
model [138]
ART = Mb [ρA dA + ρB dB + 2ARA/B ],
(4.13)
where RT is the total multilayer resistance, Mb is the number of bilayers (we study below just
one multilayer period, i.e., Mb = 1), A is the cross sectional area, and RA/B is the interface
resistance. Thus, resistor model treats both bulk and interface resistances as semiclassical
elements of a circuit in which resistors add in series. From the measurement of RT as a
function of layer thickness, the bulk and interface resistances can be extracted experimentally.
If quantum interference effects are important in the CPP transport, this picture breaks down.
Our goal in this section is to probe such effects in a mesoscopic (small) multilayer.
The conductance is computed from the Landauer-type formula (4.10) which intrinsically
takes into account all finite-size effects in the problem (cf. Ch. 3). In all calculations the
hopping throughout the disordered sample and the leads is the same (tL = tC = t). We
110
first study the multilayer with ballistic propagation in the layers outside of the interfaces,
i.e., WA = WB = 0. The disorder-averaged results are plotted on Fig. 4.8. The same figure
plots the conductance of a multilayer with ballistic propagation confined to the layer which
separates the interfaces, i.e., WA = 6, and WB = 0. Both calculations exhibit the oscillating
conductance, even after disorder-averaging, which is obviously a quantum effect. It is a
consequence of the size quantization caused by a coherent interference of electrons reflected
back and fort at the strongly disordered interface. The middle layer is composed of only few
atomic monolayers (i.e., its length is of the order of λF ) and it would be interesting to check
the dependence of the oscillating conductance on the thickness of this layer.
In order to compare these and subsequent results to the resistor model (4.13), we need
the conductances of an individual bulk conductors appearing in the multilayer. They are
plotted on Fig. 4.9, together with the quantum point contact conductance corresponding
to a lead-reservoir contact accommodating the maximum of 100 channels.15
The QPC
conductance is needed because disorder-averaged Landauer formula for resistance (3.22) can
be expressed, in the semiclassical transport regime, as a sum of this conductance and the
conductance of disordered sample attached to ideal leads. Therefore, the naı̈ve application
of the resistor model, where we use the average resistances computed for the sample+leads,
requires to subtract (NR − 1) QPC resistances. Here NR is the number of bulk and interface
resistances summed to get RT (4.13). For specific disordered conductors this procedure
becomes tricky since our calculation from Ch. 3 shows that QPC resistance appears in (3.22)
only for very small disorder and steadily decreases as W increases. Therefore, we plot both
the resistor model result with and without subtracted QPC resistance. This should serve as
a reference to be compared with quantum calculations for the whole multilayer. In the cases
shown on Fig. 4.8 resistor model is clearly incapable to take into account quantum effects
15 The
number of open channels carrying the current at Fermi energy is defined by the cross
section of a lead and EF , cf. Ch. 5.
111
4
(a)
(b)
(c)
2
Conductance (2e /h)
WA=6 WB=0 WA=6
2
0
6
(a)
(b)
(c)
WA=0 WB=0 WA=0
4
2
0
-6
-4
-2
0
2
4
6
Fermi Energy
Figure 4.8: The disorder-averaged (over 200 configurations) conductance of a multilayer composed of strongly disordered interfaces and clean bulk conductors (lower panel) or clean and
disordered bulk conductors (upper panel) on a lattice 17 × 10 × 10. The results are obtained
from: (a) Landauer-type formula (4.10) applied to the whole multilayer, (b) summing the
individual bulk and interface resistances, and (c) summing the individual bulk and interface
resistances and subtracting the extraneous 100 channel quantum point contact resistances
(RQPC from Fig. 4.9), following the resistor model (4.13). We subtract 2RQPC in the lower
panel and 3RQPC in the upper panel.
112
2
80
12
10
60
(b)
(a)
8
40
6
4
20
2
0
-6
-4
-2
0
2
4
6
2
(c)
Conductance (2e /h)
Conductance (2e /h)
14
0
Fermi Energy
Figure 4.9: Conductance of a disordered conductor modeled by the Anderson model on a
lattice 5 ×10 ×10 with disorder strength: (a) W = 6, (b) W = 3. Also shown is the quantum
point contact conductance (1/RQPC ) of a clean sample modeled on the same lattice (i.e., with
maximum of 100 channels on the cross section), cf. Ch. 5.
113
(a)
(b)
(c)
2
2
Conductance (2e /h)
WA=6 WB=6 WA=6
0
4
(a)
(b)
(c)
WA=6 WB=3 WA=6
2
0
-6
-4
-2
0
2
4
6
Fermi Energy
Figure 4.10: The disorder-averaged (over 200 configurations) conductance of a multilayer
composed of strongly disordered interfaces and disordered bulk conductors 17 × 10 × 10 (
is bigger than the thickness of the layer for W = 3). The results are obtained from: (a)
Landauer-type formula (4.10) applied to the whole multilayer, (b) summing the individual
bulk and interface resistances, and (c) summing the individual bulk and interface resistances
and subtracting the extraneous 100 channel quantum point contact resistances, RQPC from
Fig. 4.9, following the resistor model (4.13). We subtract 4RQPC on both panels.
114
which generate the oscillating conductance.
In further endeavors we use two multilayers where ballistic layers are removed by either
adding enough disorder to get the bulk diffusive layer (W = 6) or a “quasiballistic” layer
(for W = 3 the mean free path is bigger than 5 lattice spacings, as shown on Fig. 3.1). The
disorder-averaged conductance of such multilayers is plotted on Fig. 4.10. The oscillating
conductance has vanished in both cases. However, the application of the resistor model, following the procedure described above, is unable to explain the conductance of the multilayer
treated as a single conductor attached to the ideal leads. Here we face again the problem of
interpretation of the disorder-averaged Landauer formula, encountered previously in Ch. 3,
probably intertwined with some quantum effects which cannot be accounted by the resistor
model, even if proper subtraction (instead of the plain QPC resistance) would be made. This
seems to be an interesting project for the future investigation, based on the findings of this
Chapter and Ch. 3.
115
Part II
Ballistic Transport and Transition from Ballistic to
116
Chapter 5
Quantum Transport in Ballistic Conductors: Evolution
From Conductance Quantization to Resonant
Tunneling
The aim of science and technology would seem to be
much more that of presenting us with a definitively
unreal world, beyond all criteria of truth and reality.
— Jean Baudrillard, The Transparency of Evil
5.1
Introduction
The advent of mesoscopic physics [43] has profoundly influenced our understanding
of transport in condensed matter systems. In this spirit, quite interesting thesis results
are reached after critical reexamination of some of the transport “dogmas” (in the sense
that impromptu answers to those questions are usually given or found in the literature)
while exploring the mesoscopic methods to calculate transport properties. One of the most
spectacular discoveries of mesoscopics is that of conductance quantization (CQ) [48, 49]
in short and narrow constrictions connecting two high-mobility (ballistic) two-dimensional
electron gases. When the sample size is reduced below the elastic mean free path , a
ballistic regime is entered. In ballistic transport the electron traverses the conductor without
117
experiencing any scattering on defects. The conductance as a function of constriction width
W has steps of magnitude 2e2 /h. These constrictions are the simplest example of ballistic
conductors and are usually called quantum point contacts (QPC). The QPC differs from
the classical point contact [139] in having the width W comparable to the Fermi wavelength
λF . The conductance of a classical point contacts, modeled as an orifice in an insulating
diaphragm separating two metallic electrodes, is studied in Ch. 6. The development of
experimental techniques has given the possibility to observe similar phenomena [140] in
metallic nanocontacts and nanowires. These conductors are of atomic-size, even just oneatom contact, since λF is much smaller in metals than in semiconductors.
The multichannel Landauer formula [4] for the two-probe conductance (2.54)
G=
2e2
Tn ,
Tr (tt† ) = GQ
h
n
(5.1)
has provided an explanation of the stepwise conductance in terms of the integer number
M ∼ kF W of transverse propagating modes (“channels”) at the Fermi energy EF which
are populated in the constriction. In the ballistic case (tt† )ij is δij , or equivalently Tn is
1. This means that changing W opens new transport channels in discrete steps.1 The
possibility to see actual systems where conductance is related to the quantum-mechanical
transmission probability has taken by surprise both theoretical and experimental community.
Thus, the study of QPC and (quasi)ballistic structures in general, has given impetus for the
exploration of various quantum transport concepts and sharpening of the quantum intuition.
Particularly important was the clarification of the physically relevant Landauer formula.2
1 The
discreteness of the conductance steps is not observable if the width is much bigger than
λF since then fractional change of W would open many channels at the same time.
2 In
the beginning of 80s the controversy surrounding various Landauer-type formulas has pro-
duced, among other things, a debate on whether a ballistic conductor can have a finite conductance,
as predicted by the so called two-probe (chemical potential difference measure between macroscopic
reservoirs) multichannel Landauer formula (5.1). The original Landauer formula in one dimension
118
Further studies have unveiled the realistic conditions [141] for CQ as well as the mechanisms [142] which lead to its disappearance. They include geometry [143, 144], scattering
on impurities and boundaries [147], temperature effects, and magnetic field. For example,
in the adiabatic limit of a smoothly tapered constriction, the correction to the θ-function
steps is exponentially small [143]. The adiabatic geometry enables independent passage of
different transverse modes through QPC (“no-mode-mixing” regime), which corresponds to
the diagonal transmission matrix t in the representation of incident modes from the leads.
It provides a sufficient, but not necessary, condition for CQ. This is clearly demonstrated by
the results presented below. It was pointed out [145] that necessary condition is the absence
of backscattering (direct numerical calculation [146] shows that conductance is quantized
even if the channel mixing is significant). Numerical simulations [144] have demonstrated
that an electron can exit from a narrow conductor into wide reservoir with negligible probability of reflection if its energy is not too close to the bottom of the band. Even the opposite
limit to adiabatic, of abrupt wide-narrow geometry (and all interpolation between the two
limits [141]), generates stepwise conductance, but with resonant structures superimposed
onto the plateaus [144].
How is it possible to observe the conductance quantization when ballistic region is
inevitably strongly coupled to the diffusive structures exhibiting conductance fluctuations?
It was shown that this is a result of filtering [147, 148] properties of the constriction. The QPC
between two disordered leads (i.e., the reservoirs) suppresses the fluctuations and recovers
CQ. This suppression is less effective than the prediction of a naı̈ve analysis based on the
Ohm’s law for two classical resistors in series, one ballistic Gbal and one Gdiff . Ohm’s law
G = (2e2 /h)T /R (which hints at localization phenomenon [92] by generating the exponential scaling of 1D sample resistance with the sample length) gives infinite conductance (reflection R = 0)
of ballistic systems since it stems from taking the local chemical potential difference inside the
sample (the four-probe measurement in modern terminology [59]). The history of such debates is
recounted in Ref. [87].
119
then gives ∆G (Gbal /Gdiff )∆Gdiff 2e2 /h, since Gbal Gdiff . One should bear in mind
that application of the Ohm’s law is not justified when the coherence length Lφ is big enough
to encompass both the ballistic subregion and the disordered subregion. In such cases one
has to use the quantum transport theory.
In this Chapter we study the influence of the attached leads on ballistic transport ( > L)
in a nanocrystal (or “nanowire”). We assume that in the two-probe theory an electron
leaving the sample does not reenter the sample in a phase-coherent way. This means that
at zero temperature phase coherence length Lφ is equal to the length of the sample L. In
the jargon of quantum measurement theory, the leads act as a “macroscopic measurement
apparatus”. Our concern with the influence of the leads on conductance is therefore also a
concern of quantum measurement theory. Recently, the effects of a lead-sample contact on
quantum transport in molecular devices have received increased attention in the developing
field of “nanoelectronics” [46]. Also, the simplest lattice model and related real-space Green
function technique are chosen here in order to address some practical issues which appear
in the frequent use of these methods [43] to study transport in disordered samples. We
emphasize that the relevant formulas for transport coefficients contain three different energy
scales (corresponding to the lead, the sample, and the lead-sample contact), as discussed
below.
5.2
Model: Nanocrystal
In order to isolate only the effects of the attached leads on the ballistic transport we
pick the simplest geometry, namely that of a strip, in the two-probe measuring setup shown
on Fig. 2.1. The nanocrystal (“sample”) is placed between two ideal (disorder-free) semiinfinite “leads” which are connected to macroscopic reservoirs. The electrochemical potential
difference eV = µL − µR is measured between the reservoirs. The leads have the same
cross section as the sample. This eliminates scattering induced by the wide to narrow
120
geometry [144] of the sample-lead interface. The whole system is described by a clean tightbinding Hamiltonian with nearest-neighbor hopping parameters tmn
Ĥ =
tmn |mn|,
(5.2)
m,n
where |m is the orbital ψ(r − m) on the site m. The “sample” is the central section with
N × Ny × Nz sites. The “sample” is perfectly ordered with tmn = t. The leads are the
same except tmn = tL . Finally, the hopping parameter (coupling) between the sample and
the lead is tmn = tC . We use hard wall boundary conditions in the ŷ and ẑ directions.
Different hopping parameters introduced are useful when studying the conductance at Fermi
energies throughout the whole band extended by the disorder (Fig. 3.2). In order to have the
bandwidth 12tL of the leads as wide as that of the disordered sample one needs tL > t (cf.
Sec. 4.2). Thus, the conductances computed in this Chapter are relevant for such studies,
where the semiclassical approximation of the Landauer formula (3.22) ceases to be just a
sum of contact resistance and the disordered region resistance.
Our toy model shows exact conductance steps in multiples of GQ when tC = tL = t.
This is a consequence of infinitely smooth (“ideally adiabatic” [143]) sample-lead geometry.
Then we study the evolution of quantized conductance into resonant tunneling conductance
while changing the parameter tL of the leads as well as the coupling between the leads and
the conductor tC . An example of this evolution is given on Fig. 5.1. The equivalent evolution
of the transmission eigenvalues Tn of channels is shown on Fig. 5.2. A similar evolution has
been studied recently in one-atom point contacts [149].
The non-zero resistance when tL = tC = t is a purely geometrical effect [150] caused
by reflection when the large number of channels in the macroscopic reservoirs matches the
small number of channels in the lead. The sequence of steps (1, 3, 6, 5, 7, 5, 6, 3, 1 multiples
of GQ as the Fermi energy EF is varied) is explained as follows. The eigenstates in the leads,
which comprise the scattering basis, have the form Ψk ∝ sin(ky my ) sin(kz mz )eikx mx at atom
m, with energy E = 2tL [cos(kx a) + cos(ky a) + cos(kz a)], where a is the lattice constant. The
121
6
2
Conductance (2e /h)
(a)
(d)
4
(b)
(c)
2
0
-6
-4
-2
0
2
4
6
Fermi Energy
Figure 5.1: Conductance of an atomic-scale ballistic contact 3 × 3 × 3 for the following values
of lead and coupling parameters: (a) tC = 1, tL = 1, (b) tC = 1.5, tL = 1 (c) tC = 3, tL = 1,
and (d) tC = 0.1, tC = 1. In the case (d) the conductance peaks are connected by the smooth
curves of G < 0.004e2 /h.
122
Transmission eigenvalues
1.0
(a)
(3,1), (1,3), (2,2)
(b)
(c)
0.5
(d)
0.0
1.0
(1,2), (2,1)
(3,2), (2,3)
0.5
0.0
1.0
(1,1)
(3,3)
0.5
0.0
-6
-4
-2
0
2
4
6
Fermi Energy
Figure 5.2: Transmission eigenvalues of an atomic-scale ballistic contact 3 × 3 × 3. The
parameters tL and tC are the same as in Fig. 5.1. All channels (i, j) ≡ (ky (i), kz (j)) whose
subbands are identical have the same Tn . This gives the degeneracy of Tn : three (upper
panel), two (middle panel), and one (bottom panel). In the middle panel the lower two
subbands have an energy interval of overlap with the upper two subbands.
123
discrete values ky (i) = iπ/(Ny + 1)a and kz (j) = jπ/(Nz + 1)a define subbands or “channels”
labeled by (ky , kz ) ≡ (i, j), where i runs from 1 to Ny and j runs from 1 to Nz . The channel
(ky , kz ) is open if EF lies between the bottom of the subband, 2tL [−1 + cos(ky a) + cos(kz a)],
and the top of the subband, 2tL [1+cos(ky a)+cos(kz a)]. Because of the degeneracy of different
transverse modes in 3D, several channels (ky , kz ) open or close at the same energy. Each
channel contributes one conductance quantum GQ . This is shown on Fig. 5.1 for a sample
with 3 × 3 cross section where the number of transverse propagating modes is M = 9. In
the adiabatic geometry, channels do not mix, and the transmission matrix is diagonal in the
basis of channels defined by the leads.
We compute the conductance using the expression obtained in the framework of Keldysh
technique by treating the coupling between the central region and the lead as a perturbation [102]. This Landauer-type formula (2.105) for the conductance in a non-interacting
system
Ny Nz
2e2
2e2 r
a
†
Tr Γ̂L Ĝ1N Γ̂R ĜN 1 =
Tr (tt ) = GQ
Tn ,
G =
h
h
n=1
t =
Γ̂L Ĝr1N
(5.3)
Γ̂R
(5.4)
is discussed in detail in Sec. 2.5. In order to study the conductance as a function of two
parameters tL and tC we change either one of them while holding the other fixed (at the
unit of energy specified by t), or both at the same time. The first case is shown on Fig. 5.1
and Fig. 5.3 (upper panel), while the second one on Fig. 5.3 (lower panel). The conductance
is depressed in all cases since these configurations of hopping parameters tmn effectively
act as a barriers. There is a reflection at the sample-lead interface due to the mismatch
of the subbands in the lead and in the sample when tL differs from t. This demonstrates
that adiabaticity is not a necessary condition for CQ (since our model is in the adiabatic
transport regime for any values of tL and tC ). In the general case, each set of channels,
which have the same energy subband, is characterized by its own transmission function
Tn (EF ). When the coupling tC = 0.1 is small a double-barrier structure is obtained which
124
(a)
(b)
4
2
Conductance (2e /h)
6
2
0
-6
(c)
-4
-2
0
4
6
4
6
(a)
6
(b)
4
(c)
2
0
-6
2
-4
-2
0
2
Fermi Energy
Figure 5.3: Conductance of an atomic-scale ballistic conductor 3 × 3 × 3 for the following
values of lead and coupling parameters: Upper panel — (a) tC = 1, tL = 1, (b) tC = 1,
tL = 1.5, and (c) tC = 1, tL = 3; Lower panel — (a) tC = 1, tL = 1, (b) tC = 1.5, tL = 1.5,
and (c) tC = 3, tL = 3.
125
has a resonant tunneling conductance. The electron tunnels from one lead to the other
via discrete eigenstates. The transmission function is composed of peaks centered at Er =
2t[cos(kx a) + cos(ky a) + cos(kz a)], where kx = kπ/(N + 1)a is now quantized inside the
sample, i.e., k runs from 1 to N. The magnitude and width of peaks is defined by the rate at
which an electron placed between barriers leaks out into the lead. These rates are defined by
the level widths generated through the coupling to the leads. In our model they are energy
(or mode) dependent. For example at EF = 0 seven transmission eigenvalues are non-zero
(in accordance with open channels on Fig. 5.2) and exactly at EF = 0 three of them have
T = 1 and four T = 0.5. Upon decreasing tC further all conductance peaks, except the one
at EF = 0, become negligible. Singular behavior of G(EF ) at subband edges of the leads
was observed before [147].
It is worth mentioning that the same results are obtained using a non-standard version
of Kubo-Greenwood formula [91] for the volume averaged conductance
4e2 1
Tr
h̄v̂
Im
Ĝ
h̄v̂
Im
Ĝ
,
x
x
h L2x
1 r
(Ĝ − Ĝa ),
Im Ĝ =
2i
G =
(5.5)
(5.6)
where vx is the x component of the velocity operator. This was originally derived for an
infinite system without any notion of leads and reservoirs. The crucial non-standard aspect
is use of the Green function (2.84) in formula (5.5). This takes into account, through the
lead self-energy (2.98), (2.99), the boundary conditions at the reservoirs. The reservoirs
are necessary in both Landauer and Kubo formulations of linear transport for open finite
systems. They provide thermalization and thereby steady state of the transport in the central
region. Semi-infinite leads [93] are a convenient method which takes into account electrons
entering of leaving the phase-coherent sample, and therefore bypasses the explicit modeling
of the thermodynamics of macroscopic reservoirs. When employing the Kubo formula (5.5)
one can (and should) use current conservation and compute the trace only on two adjacent
126
layers inside the sample (cf. Sec. 2.5). To get the correct results in this scheme Lx in Eq. (5.5)
should be replaced by a lattice constant a.
In the quantum transport theory of disordered systems the influence of the leads on
the conductance of the sample is understood as follows [152]. An isolated sample has a
discrete energy spectrum. Attaching leads for transport measurements will broaden energy
levels. If the level width Γ due to the coupling to leads is larger than the Thouless energy
ETh = h̄/τD h̄D/L2 , (D = vF /d being the diffusion constant) the level discreteness is
unimportant for transport. For our case of ballistic conduction, ETh is replaced by the inverse
time of flight h̄vF /L. In the disordered sample where Γ ETh , varying the strength of the
coupling to the leads will not change the transport coefficients. In other words, the intrinsic
resistance of the sample is much larger than the resistance of the lead-sample contact [29]. In
the opposite case, discreteness of levels becomes important and the strength of the coupling
defines the conductance. This is the realm of quantum dots [30] where weak enough coupling
can make the charging energy e2 /2C of a single electron important as well. Changing the
properties of the dot-lead contact affects the conductance and the result of measurement
depends on the measuring process. The decay width Γ = h̄/τdwell of the electron emission
into one of the leads is determined by transmission probabilities of channels through the
contact and mean level spacing [152] (Γ = αM∆/2π, where ∆ is the average level spacing,
and 0 ≤ α ≤ 1 measures the quality of the contact). This means that mean dwell time τdwell
inside our sample depends on both tC and tL . Changing the hopping parameters will make
τdwell greater than the time of flight τf = L/vF . Thus we find that ballistic conductance
sensitively depends on the parameters of the dephasing environment (i.e., the leads).
5.3
Model: Nanowire
To complete the study in this Chapter, we also show the conductance of a ballistic
sample modeled on the lattice 12 × 3 × 3 (which we call “nanowire” since length is greater
127
than the other two dimensions). The study is performed for the same variations of tC and
tL as in the case of the 3 × 3 × 3 sample. The results are shown on Fig. 5.4 and Fig. 5.5.
Here the conductance in the tunneling limit has more peaks corresponding to the different
spectrum of eigenstates through which the tunneling proceeds. On the other hand, in all
other cases similar oscillatory structure is observed and the difference between changing just
one parameter or both is much less pronounced.
Increasing the length of the wire (ratio length to width) would just increase the frequency of the ripples. They were accounted in the previous studies [153] as being due to
the multiple reflection at the interface between the wire and the semi-infinite leads. Here
the oscillatory structure is the dominant feature and completely washes out the stepwise
conductance. Similar resonant structures appear in the abrupt (wide-narrow) constriction
geometry as a result of alternatively constructive and destructive internal reflections within
the constrictions [144]. At certain energies electrons in one or more subbands can form the
quasi-standing waves [154]. Thus, they become partially trapped in the wire region and the
conductance is lowered. Since one particle quantum mechanics is analogous to the wave
propagation, the insight into these phenomenon can be obtained by studying the properties
of the corresponding waveguides.
5.4
Conclusion
In this Chapter a study of the transport properties of a nanoscale contact in the ballistic
regime was presented. The results for the conductance and related transmission eigenvalues
show how the properties of the ideal semi-infinite leads (“measuring device”), as well as
the coupling between the leads and the conductor, influence the transport in a two-probe
geometry. The evolution from conductance quantization to resonant tunneling conductance
peaks was observed upon changing the hopping parameter in the disorder-free TBH which
describes the leads and the coupling to the sample. This result could have been anticipated
(a)
6
(b)
2
Conductance (2e /h)
128
4
(c)
2
0
-6
-4
-2
0
2
Fermi Energy
4
6
Figure 5.4: Conductance G of a ballistic quantum wire 12 × 3 × 3 for the following values
of lead and coupling parameters: (a) tC = 1, tL = 1, (b) tL = 1, tC = 1.5, and (c) tL = 1,
tC = 0.1. In the case (c) the conductance peaks are connected by the smooth curves with
G < 0.004e2 /h.
(a)
6
2
Conductance (2e /h)
129
(b)
4
2
(c)
0
-6
-4
-2
0
2
4
6
Fermi Energy
Figure 5.5: Conductance G of a ballistic quantum wire 12 × 3 × 3 for the following values
of lead and coupling parameters: (a) tC = 1, tL = 1, (b) tC = 1, tL = 1.5, and (c) tC = 1,
tL = 3.
130
when from the quantum transport intuition. Nevertheless, it is quite amusing that vastly
different G(EF ) are obtained between these two limits (e.g., Fig. 5.3). The crossover region
is much less distinctive for the case of “nanowire” than in the case of “nanocrystal”. Thus,
these systems exhibit extreme sensitivity of the conductance to the changes in the hopping
parameter inside the leads or in the coupling between the leads and the sample. The results
are of relevance for the analogous theoretical studies in disordered conductors presented in
Part 1, as well as in the experiments using clean metal junctions with different effective
electron mass throughout the circuit.
131
Chapter 6
Electron Transport Through a Classical Point Contact
All classical physics is boring.
— Amsterdams theoreticus
6.1
Introduction
The problem of electron transport through an orifice (also known as point contact) in
an insulating diaphragm separating two large conductors (Fig. 6.1) has been studied for
more than a century. Maxwell [155] found the resistance in the diffusive regime when the
characteristic size a (radius of the orifice) is much larger than the mean free path . Maxwell’s
answer, obtained from the solution of Poisson equation and Ohm’s law, is
RM =
ρ
,
2a
(6.1)
where ρ is resistivity of the conductor on each side of the diaphragm. Later on, Sharvin [122]
calculated the resistance in the ballistic regime ( a)
4ρ
=
RS =
3A
2e2 kF2 A
h 4π
−1
,
(6.2)
where A is the area of the orifice. This “contact resistance” persists even for ideal conductors
(no scattering) and has a purely geometrical origin, because only a finite current can flow
through a finite-size orifice for a given voltage. In the Landauer-Büttiker transmission formalism [92, 60] of Sec. 2.4.2, we can think of a reflection when a large number of transverse
132
a
+V
z
-V
Figure 6.1: Electron transport through the circular constriction in an insulating diaphragm
separating two conducting half-spaces (each characterized by the mean free path ).
133
propagating modes in the reservoirs matches a small number of propagating modes in the
orifice. In the intermediate regime, when a , the crossover from RM to RS was studied
by Wexler [156] using the Boltzmann equation in a relaxation time approximation. The
complete potential distribution in the 2D classical point contact geometry (λF a < L, L
being the length of the constriction) was found in [157] for the ballistic transport regime. The
influence of electron-phonon collisions on the orifice current-voltage characteristics was studied using classical kinetic equations in Ref. [158] and quantum kinetic equations (Keldysh
formalism) in Ref. [159]. This provides a theoretical basis for an experimental technique
allowing extraction of the phonon density of states from the nonlinear current-voltage characteristics (point contact spectroscopy [160]). The analogous problem for the conductance
of a wire of length L > a (a is the width of the wire) for all ratios /L was solved by de
Jong [161] using a semiclassical treatment of the Landauer formula (cf. Eq. (3.21)). De
Jong makes a connection between his approach and semiclassical Boltzmann theory used in
Wexler’s work. Recently, the size of orifice has been shrunk to a λF leading to the observation of quantum-size effects on the conductance [48, 49]. In the case of a tapered orifice
on each side of a short constriction between reservoirs, discrete transverse states (“quantum
channels”) below the Fermi energy which can propagate through the orifice give rise to a
quantum version of Eq. (6.2). The quantum point contact conductance1 is equal to an integer
number of conductance quanta 2e2 /h, as discussed in detail in Ch. 5.
Here we give a semiclassical treatment using the Boltzmann equation. Bloch-wave
propagation and Fermi-Dirac statistics are included, but quantum interference effects are
1 It
is interesting to note that optimal length [26] for the observation of conductance quanti√
zation is Lopt ≈ 0.4 W λF (W is the width of two-dimensional constriction), which separates a
short constriction regime (transmission via evanescent modes cannot be ignored), from a long constriction regime (transmission resonances superimposed on the plateaux). For shorter constrictions
the plateaus acquire a finite slope but do not disappear completely even at zero length (which
corresponds to the model studied here).
134
neglected. Electrons are scattered specularly and elastically at the diaphragm separating
the electrodes made of material with a spherical Fermi surface. Collisions are taken into
account through the mean free path . A peculiar feature is that the driving force can change
rapidly on the length scale of a mean free path around the orifice region. The local current
density depends on the driving force at all other points. Our approach follows Wexler’s [156]
study. We find an explicit form of the Green’s function for the integro-differential Boltzmann
operator. The Green’s function becomes the kernel of an integral equation defined on the
compact domain of the orifice. Solution of this integral equation gives the deviation from
the equilibrium distribution function on the orifice. Therefore, it defines the current through
the orifice and its resistance.
The exact answer can be written as
R(/a) = RS + γ(/a)RM ,
(6.3)
where γ(/a) has the limiting value 1 as /a → 0 and RS /RM → 0. We are able to compute
γ(/a) numerically to an accuracy of better than 1%. Our calculation is shown on Fig. 6.2.
We also find the first order Padé fit
γfit (l/a) =
1 + 0.83 l/a
,
1 + 1.33 l/a
(6.4)
which is accurate to about 1%. Our answer for γ differs little from the approximate answer of
Wexler [156], also shown on Fig. 6.2 as γWex . Section 6.2 formulates the algebra and Sec. 6.3
explains the solution.
6.2
Semiclassical transport theory in the orifice geometry
In order to find the current density j(r) through the orifice, in the semiclassical approach,
we have to solve simultaneously the stationary Boltzmann equation (2.20) in the presence
135
of an electric field (cf. Sec. 2.3) and the Poisson equation for the electric potential
ṙ ·
∂F (k, r) e∇Φ(r) ∂F (k, r)
F (k, r) − fLE (k, r)
−
·
= −
,
∂r
h̄
∂k
τ
eδn(r)
,
∇2 Φ(r) = −
ε
1
(F (k, r) − f (k )),
δn(r) =
Ω k
1
(F (k, r) − fLE (k, r)),
0 =
Ω k
e
vk F (k, r).
j(r) =
Ω k
(6.5)
(6.6)
(6.7)
(6.8)
(6.9)
Here F (k, r) is the distribution function, f (k ) is the equilibrium Fermi-Dirac function, Φ(r)
is electric potential, Ω is the volume of the sample and fLE (k, r) is a Fermi-Dirac function
with spatially varying chemical potential µ(r) which has the same local charge density as
F (k, r). In general, we have to deal with the local deviation δn(r) of electron density
from its equilibrium value self-consistently. The collision integral is written in the standard
relaxation time approximation with scattering time τ = l/vF . This system of equations
should be supplemented with boundary conditions on the left electrode (LE) at z = −∞,
right electrode (RE) at z = ∞, and on the impermeable diaphragm (D) at z = 0:
Φ(rLE ) = V,
(6.10)
Φ(rRE ) = −V,
(6.11)
jz (rD ) = 0,
(6.12)
where the z-axis is taken to be perpendicular to the orifice. In linear approximation we
can express the distribution function F (k, r) and the local equilibrium distribution function
fLE (k, r) using δµ(r) (local change of the chemical potential) and Ψ(k, r) (deviation function,
i.e., energy shift of the altered distribution)
∂f (k )
δµ(r),
∂k
∂f (k )
F (k, r) = f (k − Ψ(k, r)) ≈ f (k ) −
Ψ(k, r).
∂k
fLE (k, r) = f (k − δµ(r)) ≈ f (k ) −
(6.13)
(6.14)
136
1.0
γ
0.9
γ
0.8
γWex
0.7
0.6
0.01
0.1
1
l/a
10
100
Figure 6.2: The dependence of factor γ in Eqs. (6.3), (6.76) on the ratio /a. Also shown is
the variational calculation of γWex from Ref. 4.
137
These equations imply that δµ(r) is identical to the angular average of Ψ(k, r)
δn(r) =
1 ∂f (k )
−
Ψ(k, r) = N(EF )Ψ(r) = N(EF )δµ(r),
Ω k
∂k
(6.15)
where N(EF ) is the density of states at the Fermi energy F . In the case of a spherical Fermi
surface,
1 Ψ(r) =
dΩk Ψ(k, r).
4π
(6.16)
Following Wexler [156], we introduce a function u(k, r) by writing Ψ(k, r) as
Ψ(k, r) = eV u(k, r) − eΦ(r).
(6.17)
Thereby, the linearized Boltzmann equation (6.5) becomes an integro-differential equation
for the function u(k, r)
τ vk ·
∂u(k, r)
= u(r) − u(k, r).
∂r
(6.18)
To solve this equation we need to know only boundary conditions satisfied by u(k, r) and then
we can use this solution to find the potential Φ(r). Thus the calculation of the conductance
from u(k, r) is decoupled from the Poisson equation. Here we encounter again this intrinsic
property of linear transport theories [26] which was discussed in general terms in Ch. 2. The
boundary conditions for (6.18) are:
u(rLE) = 1,
(6.19)
u(rRE) = −1.
(6.20)
They follow from the boundary conditions (6.10)-(6.11) for the potential Φ(r) and the fact
that far away from the orifice we can expect local charge neutrality entailing
u(r) =
Φ(r)
.
V
(6.21)
The driving force does not explicitly appear in (6.18), but it enters the problem through
these boundary conditions. Since Eq. (6.18) is invariant under the reflection in the plane of
138
the diaphragm
(k, r) → (kR , rR ),
(6.22)
rR
= (x, y, −z),
(6.23)
kR
= (kx , ky , −kz ),
(6.24)
the boundary conditions imply that u(k, r) has reflection antisymmetry
u(k, r) = −u(kR , rR ).
(6.25)
Wexler’s solution [156] to the equation (6.18) relied on the equivalence between the problem
of orifice resistance and spreading resistance of a disk electrode in place of the orifice. Technically this is achieved by switching from the equation for function u(k, r) to the equation
for function
w(k, r) = 1 + sgn (z)u(k, r).
(6.26)
The beauty of this transformation is that the new function allows us to replace the discontinuous behavior of u(k, r) on the diaphragm (which is the mathematical formulation of
specular scattering)
u(k, rD − vk dt) = u(kR , rD − vk dt) = −u(k, rD + vk dt),
(6.27)
with continuous behavior of w(k, r) over the diaphragm, discontinuous behavior over the
orifice and simpler boundary conditions on the electrodes
w(rLE) = w(rRE) = 0.
(6.28)
The Boltzmann equation (6.18) now becomes
k ·
∂w(k, r)
+ w(k, r) − w(r) = s(k, r)δ(z)θ(a − r),
∂r
(6.29)
where we have introduced the function
s(k, r) = 2kz u(k, r),
(6.30)
139
which is confined to the orifice region. It can be related to w(k, r) at the orifice in the
following way:
s(k, r0 ) = 2|kz |(1 − w(k, r0 − vk dt)).
(6.31)
It plays the role of a “source of particles” in Eq. (6.29). The notation r0 refers to a vector
lying on the orifice, that is r0 = (x, y, 0) with x2 + y 2 ≤ a2 . The discontinuity of w(k, r)
on the orifice is handled by replacing it by the disk electrode which spreads particles into a
scattering medium.
The Green’s function for Eq. (6.29) is the inverse Boltzmann operator (including boundary conditions)
∂
+ 1 − Ô GB (k, r; k , r ) = δ(Ωk − Ωk )δ(r − r ),
k ·
∂r
(6.32)
and Ô is the angular average operator
Ôf (k) =
1
4π
dΩk f (k) = f .
(6.33)
The Green’s function for the Boltzmann equation gives the possibility to express w(k, r0 −
vk dt) in the form of a four-dimensional integral equation over the surface of the orifice
w(k, r0 − vk dt) =
dΩk dr 0 GB (k, r0 − vk dt; k , r 0 + vk dt)s(k , r 0 ).
(6.34)
The function w(k, r) is discontinuous over the orifice, so we formulate the equation for this
function at points infinitesimally close (dt → +0) to the orifice. We find the following explicit
expression for the Green’s function
1 eiq·(r−r )
q(q − arctan q)−1
δ(Ωk − Ωk ) +
.
GB (k, r; k , r ) =
Ω q 1 + iq·k
4π(1 + iq·k )
(6.35)
Its form reflects the separable structure of Boltzmann operator, i.e., the sum of operators
whose factors act in the space of functions of either r or k. However it is nontrivial because
140
the factors acting in k-space do not commute and the Boltzmann operator is not normal2 —
it does not have a complete set of eigenvectors and the standard procedure for constructing
the Green’s function from the projectors on these states fails. The first term in (6.35) is
singular and generates the discontinuity of w(k, r) over the orifice.
6.3
The conductance of the orifice
The conductance of the orifice is defined by
1
I
G=
=
=
R
2V
dr0 jz (r0 )
,
2V
(6.36)
where the z-component of the current at the surface of the orifice is
N(EF )e2 V
jz (r0 ) =
8πτ
dΩk s(k, r0 ).
(6.37)
The Green’s function result (6.35) allows us to rewrite Eq. (6.34) in the following integral
equation for the smooth function s(k, r0 ) over the surface of the orifice
1=
s(k, r0 ) + dΩk dr 0 G(k, r0 ; k , r 0 )s(k , r 0 ),
2|kz |
(6.38)
where G(k, r0 ; k , r 0 ) is non-singular part of the Green’s function (6.35)
1
G(k, r0 ; k , r 0 ) =
32π 4
q eiq·(r0 −r 0 )
.
dq
(1 + iq·k )(q − arctan q)(1 + iq·k )
(6.39)
The distribution function s(k, r0 ) has two k-space variables, the polar and azimuthal angles
(θk , φk ) of the vector k on the Fermi surface, and the radius r0 and azimuthal angle φ0 of
operators satisfy condition ÔÔ† = Ô† Ô. This is sufficient and necessary to make
2 Normal
them the largest class of completely diagonalizable operators in the complex Hilbert space H. This
means that one can find the set of eigenvalues en and projectors onto the eigensubspaces Pn such
than Ô =
n en P̂n
and projectors provide the decomposition of unity operator,
n
P̂n = 1, in H.
The standard method to find the Green operator (i.e., the inverse, including relevant boundary
conditions) of a linear operator Ô in H is ĜO = 1/Ô =
n P̂n /en .
141
the point r0 on the orifice. Because of the cylindrical symmetry, s(k, r0 ) does not depend
separately on φk , φ0 , but only on their difference φk − φ0 . This makes possible the expansion
s(k, r0 ) =
sLM (r0 )YLM (θk , φk )e−iM φ0 ,
(6.40)
LM
and Eq. (6.38) can now be rewritten as
2 cos θk =
L M −iM φ0
sL M (r0 )YL M (θk , φk )e
× cos θk
sgn (cos θk ) + 2
dΩk dr 0 G(k, r0 ; k , r 0 )
L M sL M (r0 )YL M (θk , φk )e−iM φ0 .
(6.41)
This four dimensional integral equation can be reduced to a system of coupled one dimen∗
(θk , φk )eiM φ0
sional Fredholm integral equations of the second kind after it is multiplied by YLM
and integrated over θk , φk and φ0 . We also use the following identities
YLM (θ, φ) cos θ = g1 YL+1,M (θ, φ) + g2 YL−1,M (θ, φ),
g1 =
g2 =
1
4π
(L − M
+ 1)(L + M + 1)
,
(2L + 1)(2L + 3)
(L − M)(L + M)
,
(2L − 1)(2L + 1)
YLM (θk , φk )
dΩk = iL fL (q)YLM (θq , φq ),
1 + iq·k
(6.42)
(6.43)
(6.44)
(6.45)
and
2π
iqr0 −iM φ0
e
0
e
2π
dφ0 =
eiq⊥ r0 cos(φ0 −φq ) e−iM φ0 dφ0 = 2πiM JM (q⊥ r0 )e−iM φq ,
(6.46)
0
where q⊥ is projection of q = qz + q⊥ in the plane of orifice and JM (z) is the Bessel function
of the first kind. For the function fL (q) in (6.45) we get the following expression
∞
fL (q) = (−1)
L
0
(−i)−L
1
QL ( ),
e jL (qx) dx =
iq
iq
−x
(6.47)
142
where jL (x) is spherical Bessel function and QL (x) is Legendre function of the second kind.
Explicit formulas for fL (x) are
arctan x
,
x
−x + arctan x
,
x2
−3x + (x2 + 3) arctan x
,
2x3
− 43 x3 − 5x + (5 + 3x2 ) arctan x
,
2x4
− 55
x3 − 35x + (35 + 30x2 + 3x4 ) arctan x
3
.
8x5
f0 (x) =
f1 (x) =
f2 (x) =
f3 (x) =
f4 (x) =
(6.48)
(6.49)
(6.50)
(6.51)
(6.52)
The final form of the integral equation for sLM (r0 ) in the expansion of s(k, r0 ) is
π
δL1 δM 0 =
cLM,L M δMM sL M (r0 ) + 4
r0 dr0 KLM,L M (r0 , r0 )sL M (r0 ), (6.53)
4
3
LM
LM
a
0
where the kernel
KLM,L M (r0 , r0 )
KLM,L M (r0 , r0 )
is given by
M −M
= i
∞
π
2
(−1)
L +L+1
×[i
M +M q dq
0
L+1
(−1)
sin θq dθq
0
q2 fL (q)YLM (θq )
q − arctan q
g1 fL+1 (q)YL+1,M (θq )
+iL +L−1 (−1)L−1 g2 fL−1 (q)YL−1,M (θq )]
×JM (qr0 sin θq )JM (qr0 sin θq ).
(6.54)
The kernel (6.54) does not depend on φq so that only the part of spherical harmonic dependent on θq , YLM (θq ), is integrated (which is, up to a factor, associated Legendre polynomial).
The kernel differs from zero only if L + M has parity different from L + M . This follows
from the fact that the kernel is the expectation value
KLM,L M (r0 , r0 ) = LMM|2 cos θ G(k, r0 ; k , r 0 )|L M M ,
(6.55)
θk φk φ0 |LMM = YLM (θk , φk )e−iM φ0
(6.56)
of an operator which is odd under inversion. The basis functions |LMM have parity given
by
P |LMM = (−1)L+M |LMM.
(6.57)
143
Exactly under this condition the kernel becomes a real quantity. This means that the nonzero sLM (r0 ) are real with the property
sLM (r0 ) = (−1)M sL,−M (r0 ),
(6.58)
ensuring that s(k, r0 ) is real. The conductance is determined by the (L, M) = (0, 0) function
s00 (r0 ). The non-zero sLM (r0 ) coupled to it are selected by the condition that L + M is even.
This follows from s(k, r0 ) being even under reflection in the plane of orifice. Under this
operation, cos θk → − cos θk , but φk , φ0 are unchanged; this means that the expansion (6.40)
contains only terms with L + M even.
The first term on the right hand side in (6.41) is determined by the matrix element
cLM,L M =
∗
dθk dφk sin θk YLM
(θk , φk )YL M (θk , φk ) sgn (cos θk ),
(6.59)
which is the expectation value of sgn (cos θk ) in the basis of spherical harmonics. It is different
from zero if M = M and L − L is odd. The states must be of different parity, as determined
by L, because sgn (cos θk ) is odd under inversion.
The system of equations (6.53) can be solved for all possible ratios of /a by either
discretizing the variable r0 or by expanding sL M (r0 ) in terms of the polynomials in r0
sLM (r0 ) =
anLM pn (r0 ),
(6.60)
n
and performing integrations numerically. The polynomials pn (r0 ) =
n
i
i=0 ci r0
are orthogonal
with respect to the scalar product
a
r0 dr0 pn (r0 )pm (r0 ) = δnm .
(6.61)
0
The first three polynomials are
p0 (r0 ) =
√
2
,
a
6r0 − 4
√
,
a 9a2 − 16a + 9
√ 3
10 6 r02 − 65 r0 + 10
.
p2 (r0 ) = √
a 100a4 − 288a3 + 306a2 − 144a + 27
p1 (r0 ) =
(6.62)
(6.63)
(6.64)
144
The system of integral equations (6.53) then becomes a matrix equation for either sLM (r0 )
at discretized r0 or expansion coefficients anLM . The latter version is
4a
π
n L M
δL1 δM 0 δn0 =
cLM,L M anL M + 4
KnLM
an L M ,
6
L
n L M n L M
KnLM
M −M
= i
M +M (−1)
n
(qa
×jM
∞
π
2
q dq
0
n
sin θq )jM (qa
sin θq dθq
0
(6.65)
q2 fL (q)YL M (θq )
q − arctan q
sin θq )
×[iL +L+1 (−1)L+1 g1 fL+1 (q)YL+1,M (θq )
+iL +L−1 (−1)L−1 g2 fL−1 (q)YL−1,M(θq )],
(6.66)
a
n
jM
(qa sin θq ) =
r0 dr0 pn (r0 )JM (qr0 sin θq ),
(6.67)
0
which simplifies using the following result
n
(qa sin θq ) =
jM
n
ci
a2+M +i (q sin θq )M 1 F2 (1 +
M
2
+ 2i ; 2 +
21+M 1 +
i=0
M
2
+
i
2
M
2
+ 2i , 1 + M; − 14 (qa sin θq )2 )
Γ(1 + M)
,
(6.68)
where 1 F2 (α; β1 , β2 ; z) is a hypergeometric function. The lowest order approximation for
s(k, r0 ) is obtained by truncating the expansion in pn (r0 ) to zeroth order (i.e., constant—
which is the space dependence of the Sharvin limit) and the expansion in YLM (θk , φk ) to
order L = 0. Then the conductance is determined only by the constant a000 following
trivially from (6.65)
Glo =
N(EF )e2 a2 π
,
000
τ (3 + K010
)
(6.69)
000
where the lowest order part of the kernel K010
depends on /a,
000
K010
∞
π
4 arctan q
=
dq dθq
π
q − arctan q
0
0
arctan q
−3q + (q 2 2 + 3) arctan q
(1 − 3 cos2 θq ) +
×
3
3
2q q
2
[J1 (qa sin θq )]
×
.
sin θq
(6.70)
145
20
1
0.1
GI
16
(G-GI)/G
12
(G-GI)/G (%)
G / GS
G
8
4
(G-G0)/G
0.01
0.01
0.1
1
0
10
100
l/a
Figure 6.3: The conductance G (L = 2, n = 2), normalized by the Sharvin conductance
GS (6.2), plotted against the ratio /a. It is compared to the naı̈ve interpolation formula
GI (6.74), and the plausible interpolation formula G0 (6.76).
Further corrections are obtained by solving the matrix equation (6.65) with larger truncated
nLM
(6.66) are tedious to compute, but the conductance
basis sets. The matrix elements KnLM
converges rapidly for large n and L. On the other hand, the matrix elements cLM,L M (6.59)
are easy to compute and the conductance converges slowly in the ballistic limit determined
nLM
but go to high
by these matrix elements. We keep only low order matrix elements KnLM
order in cLM,LM . In practice we find that for the c-matrix Lmax = 12 is sufficient, whereas
for the K-matrix the approximation Lmax = 2, nmax = 2 gives convergence to 1%. The
conductance as a function of /a is shown on Fig. 6.3. It is normalized to the Sharvin
146
conductance, i.e., the limit a, for which
G(k, r; k, r ) → 0,
s(k, r) = 2|kz |.
(6.71)
In the opposite (Maxwell) limit, when a, we have
q
q − arctan q
3
+ 9/5 + o((q)2 ),
(q)2
3 eiq·(r−r )
3
G(k, r; k , r ) →
dq
=
,
4
2
2
2
32π
(q)
16π |r − r |
=
(6.72)
(6.73)
which is the standard Green’s function for the Poisson equation. The dependence of the
full Green’s function (6.35) on k vector is reflection of non-locality. The conductance in the
transition region from Maxwell to Sharvin limit can be compared with the naı̈ve interpolation
formula which approximates resistance of the orifice by the sum of Sharvin and Maxwell
resistances
3π a
1
= RI = RS 1 +
.
GI
8 (6.74)
Somewhat unexpectedly, the naı̈ve interpolation formula GI deviates from our result for G
at most by 11% when /a → 1 as shown on Fig. 6.3. We can also cast our lowest order
approximation for the conductance (6.69) in an analogous form as (6.74)
32 3π a
3
1
+ 2γ
= RS
.
Glo
4 3π 8 (6.75)
The numerical coefficients in Eq. (6.75) are not accurate in this simplest approximation.
Replacement of 3/4 by 1 and 32/(3π 2 ) by 1 yields correct limiting values of the conductance
and leads to a plausible interpolation formula. It differs from Eq. (6.74) by the introduction
of a factor γ which multiplies the Maxwell resistance
3π a
1
= RS 1 + γ
,
G0
8 γ=
π 000
K .
16a 010
(6.76)
(6.77)
147
This formula is compared to G and GI on Fig. 6.3. It differs from our most accurate
calculation of G by less then 1%. Therefore, for all practical purposes it can be used as an
exact expression for the conductance in this geometry, and it is the main outcome of our
work. The factor γ is of order one and depends on the ratio /a as shown on Fig. 6.2. We
also plot on Fig. 6.2 Wexler’s [156] previous variational calculation, γWex .
6.4
Conclusion
The following is a summary of the main results of this Chapter and their relevance to the
recent experiments on granular metals. The conductance of the orifice has been calculated
in all transport regimes, from the diffusive to the ballistic. The altered version (6.76) of the
simplest approximate solution of our theory (6.69) is already accurate to 1%. The naı̈ve
interpolation formula (sum of Maxwell and Sharvin resistances) agrees to 11% with our
accurate answer. Further corrections converge rapidly to an exact result. Our solution is
not variational and therefore we cannot test its stability with respect to the anisotropy in a
simple manner. This analysis is of interest in any situation where the geometry of the sample
can enhance the resistivity while the physics of conduction stays the same as in the bulk
material. One example is provided by some granular metals above the percolation threshold.
In this system the grains can touch in a way which provides thin, narrow and twisting
conduction paths [162] so that there is no macroscopic anisotropy induced by the special
arrangement of the grains. The microstructure of this random resistor network entails the
geometrical renormalization of resistivity. It is the origin of the anomalously high resistivity
scale found in these materials. The resistances of the contacts between the grains resemble
the type of resistances we have studied, after taking into account the correction to the finite
size of the grains on each side of the contact.
148
Part III
Transport Near a Metal-Insulator Transition in
Disordered Systems
149
Chapter 7
Introduction to Metal-Insulator Transitions
When the horizon disappears, what then
appears is the horizon of disappearance.
— Dietmar Kamper
Metal-Insulator transitions (MIT) [164] are one of the most widely observed and studied phenomena in condensed matter systems. They can exhibit huge resistivity changes,
sometimes going over several orders of magnitude. Different physical mechanisms can lead
to MIT, thus generating different types of insulating phases. The insulator is defined as a
substance at zero temperature characterized by a vanishing conductivity (tensor) in a weak
static electrical field
σij (T = 0) ≡ lim lim lim Re σij (q, ω) = 0.
T →0 ω→0 |q|→0
(7.1)
For a system with finite metallic conductivity, we typically observe Drude behavior (discussed
further in Ch. 9) at small frequencies
Re σij (T = 0, ω → 0) = Dcij
τ
,
π(1 + ω 2 τ 2 )
(7.2)
where Dcij = δij πe2 (n/m)eff is the Drude weight. The expression (7.2) goes into Dcij δ(ω)
when scattering (1/τ → 0) is absent and ideal (translationally invariant) metal is restored.
Since electrons interact, through Coulomb interaction, with both ions and other electrons,
150
the simple classification [163] of insulators starts from either electron-ion interaction, where
ions are static and single-electron theory suffices (band, Peierls and Anderson insulators), or
electron-electron interaction (Mott insulators [163]). The “complicated” insulators, like Anderson localized phase in the non-interacting disordered electron systems or partially filled
bands of strongly correlated electron systems, can be drastically different from the “simpleto-grasp” band insulators with completely filled highest occupied band.1 The metallic phase
near the transition point can also be quite exotic [164] when compared to “ordinary” metals
characterized by (7.2). Experiments reveal the unusual features of these phases as various
anomalous transport, optical and magnetic properties. Although different mechanisms can
influence and couple with each other, the following is an attempt toward a simple classification of the major scenarios behind the observed MITs:
• disorder effects on both non-interacting (Anderson localization) and interacting electrons (Anderson-Mott transition), as well as classical percolation,
• electronic band structure effects (Peierls),
• correlation effects from the electron-electron interaction (Mott-Hubbard),
• excitonic mechanisms,
• self-trapping of electron by self-generated lattice displacement.
When a control parameter of the transition is related to quantum dynamics, the MIT becomes
an example of the quantum phase transition (QPT) [39]. These transitions occur at zero
1 The
band insulators were the first ones discovered in the early days of quantum mechanics of
solids. In a naı̈ve view of noninteracting electron theory, the band formation is totally due to the
translationally invariant lattice of atoms in crystal. In a more sophisticated approach, we know
that systems without long-range order can also exhibit bands (like the disordered bands studied
throughout the thesis).
151
temperature when a change in the ground state of the system is induced by the change of
some parameter in the Hamiltonian.
In this Chapter we give a brief survey of disordered-induced MITs, which are relevant
to the problems studied in different Chapters of the thesis. Increasing disorder (e.g., concentration of impurities) in metallic systems leads to an Anderson MIT. The disorder due to
impurities causes microscopic potential fluctuations on the length scale of the Fermi wavelength λF and leads to a transition from metallic to activated conductivity.2 This is a result
of quantum-mechanical effects: the single particle interference effects, which lead to Anderson localization [2] of noninteracting electrons; and many-body effects of strengthening
the electron-electron interaction by increased disorder (e.g., Altshuler-Aronov correction to
conductivity [14]). The Anderson localization-delocalization transition is a generic continuous quantum (T = 0) phase transition [39]. In this transition disorder plays the role that
temperature plays in the “classical” (i.e., thermal) phase transitions.3 Namely, a system
can go from the insulating (localized) phase to the conducting (delocalized) phase by continuously changing the relevant parameters (such as degree of disorder, electron density or
external fields like pressure, electric or magnetic field). The quantum or zero-temperature
nature of the LD transition (which is not an end point at T = 0 of some line of thermal
phase transitions) is emphasized throughout the thesis since it leads straightforwardly to the
2 At
non-zero temperature the insulating phase has a non-zero conductivity because of the hop-
ping mechanism [40]. It increases with the temperature as a result of the assistance of inelastic
processes.
3 The
critical behavior of any transition happening at a non-zero critical temperature can be
described entirely by the classical physics in the region asymptotically close to the transition point.
This stems from the fact that thermal fluctuations are large close to the critical point and drive
the correlation length to infinity.
152
correct definitions4 [15] of the insulating, ρinsulator (T → 0) → ∞, and the metallic phase,
ρmetal (T → 0) < ∞.
The critical behavior of the LD transition falls into three universality classes delineated
in the field theory of localization5 in the same way as the ensembles of random matrices which
model disorder Hamiltonians: orthogonal (time-reversal symmetry present, β = 1), unitary
(time reversal symmetry broken β = 2, e.g., by magnetic field or magnetic impurities) and
symplectic (time-reversal symmetry present but spin-rotation symmetry broken by the spinorbit interaction, β = 4). These classes are labeled by the symmetry index β. However, some
features, like critical level spacing or critical conductance distribution were recently found
to depend on the boundary conditions [109] employed in numerical simulations. One should
also be aware of the fact that random Hamiltonians which describe real disordered systems
do not satisfy all of the statistical assumptions underlying the ensembles in RMT [168]. For
example, the matrix elements of TBH (2.74) in the coordinate representation are dependent
on the spatial coordinates (e.g., hopping tmm is non-zero only if m, m are nearest neighbors). On the other hand, in the matrices of RMT all matrix elements are non-zero, and
4 The
attempt to identify different phases at finite temperature by the sign of dρ/dT was argued
to be misleading [165], since dρ/dT is negative in both the metallic and insulating phases when the
system is close to the transition point and the temperature is low enough.
5 The
effective field theory approach to localization, which provides a mathematical basis for the
one-parameter scaling theory [8], was pioneered by Wegner [166] using the non-linear σ-model and
enhanced by “supersymmetry” through the development of SUSY NLσM [12]. In these formalisms
initial stochastic problem is mapped onto a deterministic field theoretical model without any random
parameters [29]. Like in other effective field theories, the action of such models can be regarded
as Landau-Ginzburg functional for the low energy, long wavelength density fluctuations which are
governed by diffusion modes [167] (Goldstone modes of NLσM). Diffusion modes (which appear in
the conventional perturbation theory for impurity averaging [167]) behave as particles described by
a propagator N (E)[Dq2 − iω]−1 , and their interaction drives the LD transition.
153
their distribution is independent of the matrix indices. Thus, RMT methods exploiting this
feature are inapplicable on disordered electronic Hamiltonians where one has to deal with
their spatial structure (as realized in the disorder-averaging technique of SUSY NLσM). In
the context of universality classes it is important to point out that standard statement of
the scaling theory [8], “LD transitions in 2D is absent”, is valid only in the orthogonal class
but not in the symplectic class where WL correction is positive. This “weak antilocalization” [169] would lead to an ideal metal in the case of weak disorder; strong enough disorder
always leads to the Anderson localization [165]. The problems studied in the thesis fall into
the orthogonal class. Therefore, the random Hamiltonian matrices of our models are real
and symmetric.
The disorder can also cause large-scale fluctuations giving rise to a MIT due to the
separation of conductor into classically allowed and forbidden region for the motion of electrons. The formation of such structures is described by percolation theory [170]. It can
be formulated as the theory of geometrical properties (connectivity) of random clusters and
their statistics. Especially important in this context is the infinite cluster that spans the
(infinite) system above the transition point. This cluster provides a continuous path for the
conducting electrons and its topology determines the conductivity. In general, both quantum and classical effects can be present, and a crossover from percolation to localization
can occur [171]. Despite the different physical origin of these phenomena, the formalism in
both cases uses the same language of scaling borrowed from the theory of continuous phase
transitions. The origin of the successful transfer of concepts is the appearance of long range
correlations which control the transition and generate divergent length scales6 —localization
6 When
dynamics is important there is also a characteristic frequency which vanishes at the
transition point giving rise to the dynamical scaling in addition to the static scaling generated by
divergent length scale.
154
length or percolation correlation length.7
7 Strictly
speaking, the divergent length scale is not the only important length scale [172]. The
presence of some microscopic length scale in the scaling of physical variables leads to critical exponents which deviate from their mean field values (i.e., only the mean field theory exponents are
compliant with the naı̈ve dimensional analysis used to describe the change of the units of length).
In other words, the physical quantity Q, which has the dimension of length Lx , can appear in the
scale invariant combination Qξ −x , but also as Qξ a−y l−a . When some microscopic length scale l
appears in this form the quantity Q has “acquired” an anomalous dimension.
155
Chapter 8
Statistical Properties of Eigenstates in
three-dimensional Quantum Disordered Systems
All this time the guard was looking at her, first
through a telescope, then through a microscope,
and then through an opera glass.
— Lewis Carroll, Through the Looking Glass
8.1
Introduction
The disorder induced localization-delocalization (LD) transition in solids has been one
of the most vigorously pursued problems in condensed matter physics since the seminal
work of Anderson [2]. In thermodynamic limit, strong enough disorder generates a zerotemperature critical point in d > 2 dimensions [108] as a result of quantum interference
effects. Thus, research in the “pre-mesoscopic” era [173] was mostly directed toward the
viewpoint provided by the theory of critical phenomena [8]. The advent of mesoscopic
quantum physics [9] has unearthed large fluctuations, induced by quantum coherence and
randomness of disorder [5], of various physical quantities [174] (e.g., conductance, local
density of states, current relaxation times, etc.), even well into the delocalized phase. Thus,
complete understanding of the LD transition requires to examine full distribution functions
156
of relevant quantities [124]. Especially interesting are deviations of their asymptotic tails,
caused by the incipient localization, from the (usually) Gaussian distributions expected in the
limit of infinite dimensionless conductance g = G/GQ (in units of the conductance quantum
GQ = 2e2 /h). This Chapter presents the study of such type—numerical computation of
the statistics of eigenfunction amplitudes in finite-size three-dimensional (3D) nanoscale
(composed of ∼ 1000 atoms) mesoscopic disordered conductors. The 3D conductors are
often “neglected” in favor of the more popular (and tractable) playgrounds—two-dimensional
systems (2D), where one can study states resembling 3D critical wave functions in a wide
range of systems sizes and disorder strengths [175], or quasi one-dimensional systems [176]
where analytical techniques can handle even non-perturbative phenomena (like the ones at
small g) [29, 30]. In 3D systems critical eigenfunctions, exhibiting multifractal [5] (i.e., selfsimilar) scaling, appear only at the mobility edge Ec which separates extended and localized
states inside the energy band.
The essential physics of disordered conductors is captured by studying just the quantum dynamics of a non-interacting (quasi)particle in a random and confining potential. This
problem is classically non-integrable, thereby exhibiting quantum chaos. The concepts unifying disordered electron physics with standard examples of quantum chaos [10] come from
the statistical approach to the properties of energy spectrum and corresponding eigenstates,
which cannot be computed analytically. While energy level statistics of disordered systems
have been explored to a great extent [177, 11], investigation of the statistics of eigenfunctions has been initiated only recently [32]. These studies are not only divulging peculiar
spectral properties of random Hamiltonians, but are relevant for the thorough understanding of various unusual features of quantum transport in diffusive metallic samples. The
celebrated examples are long-time tails in the relaxation of current [178] or log-normal tails
(in d = 2 + ) of the distribution function of mesoscopic conductances [174]. Since the goal
of this Chapter (and the thesis, overall) is to elucidate various facets of microscopic picture
157
of transport in disordered conductors, we give a short introduction into the topic of current
relaxation, which will be refereed to in the next Chapter where we study the related concept
of frequency dependent conductivity.
Relaxation properties of disordered conductors are described by the response function
σ(t) (time-dependent conductivity)
∞
j(t) =
dt σ(t)E(t − t ),
(8.1)
0
which determines the current response j(t) to a spatially homogeneous field in the form of
sharp electric pulse E(t) = E0 δ(t). The semiclassical response function (i.e., zeroth order in
the expansion of Diffuson-Cooperon diagrammatic perturbation theory for disorder-averaged
quantities [15]),
σ0 (t) =
σD
t
exp(− ),
τ
τ
(8.2)
is valid only on time scales of the order of elastic mean free time, t ∼ τ . For t τ , quantum
corrections have to be included. This leads to the response on the times scales of the diffusion
time t ∼ τD being determined by the lowest order quantum correction. The WL correction
to σ0 (t), defined by the Cooperon diagram [36], is given in the time domain by
σ1 (t) = −
e2 1
(4πD)1−d/2 e−t/τD .
πh tD/2
(8.3)
At very long times t τD the decay of the relaxation current is determined by the higherorder quantum corrections. It was shown by Altshuler et al. [174] (using the replicated
σ-model) that for t > (tD /4u) ln(tD /τ ) these higher-order quantum correction generate the
logarithmically normal decay law
σ2 (t) ∝
t
σ
1
exp − ln2
,
τ
4u
τ
(8.4)
where u is the parameter defined as u = ln(σD /σ(L)). Similar, and plausibly connected,
log-normal tails (instead of Gaussian in the limit g → ∞) of the distribution function
of conductances have been found in 2D conductors [174]. Such tails signal the onset of
158
localization even in the metallic regime. The decay in (8.4) is far slower than the exponential
decay (8.3), although faster than any power of t−1 . In phase-coherent samples one has to
worry about fluctuations effects accompanying quantum transport: the relaxation times are
dispersed in an ensemble of disordered samples [174]. The appearance of this long time tail in
the relaxation process described by σ(t) (8.1) has been one of the initial motivations to look
for the eigenstates with unusual features. They should explain microscopically this effect,
which appears even in (good) metallic samples characterized by large conductance (g 1).
Connections between correlations in the detailed microscopic structure of eigenstates
and (quantum) transport have been revealed in tunneling experiments on quantum dots.
They probe the coupling of the dot to the external leads, which depends sensitively on
the local features of wave functions near the contact [179]. Experiments which are the
closest to directly delving into the microscopic structure of quantum chaotic or disordered
wave functions exploit the correspondence between the Schrödinger and Maxwell equations
in microwave cavities [180]. The study of fluctuations and correlations of eigenfunction
amplitudes in diffusive mesoscopic systems has lead to the concept of the so-called prelocalized states [178, 181]. The notion refers to anomalously localized states which have
sharp amplitude peaks on top of an extended background (in the 3D delocalized phase).
These kind of states appear even in the diffusive, L < ξ, metallic (g 1) regime, but
are anomalously rare in such samples. In order to get “experimental” feeling for the structure
of states with unusually high amplitude spikes, an example is given on Fig. 8.1; this state is
found in a special realization of quenched disorder (out of many randomly generated impurity
configurations) inside the sample characterized by large average conductance. Thus, the prelocalized states are putative precursors of LD transition and determine asymptotics of some
of the distribution functions [5, 32] studied in open or closed mesoscopic systems, which
are introduced above. In d ≤ 2, where all states are supposed to be localized [8], prelocalized states have anomalously short localization radius, when compared to “ordinary”
159
40
30
"pre-localized"
state
20
2
t=|ψ| V
10
0
40
30 extended state
20
10
0
0
400
800
1200 1600
Lattice Site Number
Figure 8.1: An example of eigenstates in the band center of a delocalized phase. The
average conductance at half filling is g(EF = 0) ≈ 17, entailing anomalous rarity of the
“pre-localized” states. The disordered conductor is modeled by an Anderson model with
diagonal disorder on a simple cubic lattice with 123 sites. For plotting of the eigenfunction
values in 3D, the sites m are mapped onto the lattice site numbers ∈ {1, ..., 1728} in a
lexicographic order, i.e., m ≡ (mx , my , mz ) #→ 144(mx − 1) + 12(my − 1) + mz .
160
localized states in the low-dimensional systems. The parallel development of the concept of
scars [182] in the structure of quantum chaotic wave functions seems to be closely related to
the pre-localized states discovered in the disordered electron physics.1
The localization length ξ plays the role of a correlation length ξc (cf. Sec. 3.4) in d ≤ 2.
Therefore, in 2D systems with g 1 correlation length is much larger than the system size
and all eigenstates exhibit critical like behavior (like multifractal scaling introduced below).2
In 3D this behavior is reserved only for the states close to the mobility edge. Thus, while in 2D
systems the pre-localized states are directly related to the wave function multifractality [5],
the case of similar rare events outside the critical region in 3D conductors is less clear since
the correlation length is microscopic in good (g 1) metals, as demonstrated in Sec. 3.4.
In general, the study of properties of wave functions on a scale smaller than ξ should
probe quantum effects causing evolution of extended into localized states upon approaching
the LD critical point. In the marginal two-dimensional case, the divergent (in the limit
L → ∞) weak localization (WL) correction [36] to the semiclassical Boltzmann conductivity
provides an explanation of localization in terms of the interference between two amplitudes
to return to initial point along the same classical path in the opposite directions [35]. This
simple quantum interference effect leads to a coherent backscattering (i.e., suppression of
conductivity) in a time-reversal invariant systems without spin-orbit interaction. However,
in 3D systems WL correction is not “strong” enough to provide a full microscopic picture
1 “Scarring”
is the anomalous enhancement (or suppression) of the squared amplitude of the
wave function on the unstable periodic orbit of the classical system corresponding to the quantum
chaotic one. The scars demonstrate how quantum dynamics alleviates classical chaos (which erases
the memory of an initial state after long enough time). The appearance of small regions inside
disordered solids where eigenstates can have large amplitudes seems to be a “strongly pronounced”
analog [175, 180] of the phenomenon of scarring.
2 The
localization ξ length in 2D is not infinite (as for truly critical systems), but it is exponen-
tially large and one can study “criticality” in the wide range of systems sizes L < ξ.
161
of complicated quantum interference processes which are responsible for LD transition, and
facilitate the expansion of “quantum intuition”.
8.2
Exact diagonalization study of eigenstates in disordered conductors
A finite-size conductor is described by the appropriate non-interacting Hamiltonian on
a lattice. This makes possible the exact diagonalization by representing Hamiltonian in the
basis of site states and solving the corresponding matrix eigenproblem numerically.3 The
disordered sample is modeled by a tight-binding Hamiltonian with nearest-neighbor hopping
tmn
Ĥ =
m
εm |mm| +
tmn |mn|,
(8.5)
m,n
on the simple cubic lattice 16 × 16 × 16. Each site m contains a single orbital r|m =
ψ(r − m). Periodic boundary conditions are chosen in all directions. In the random hopping
(RH) model the disorder is introduced by taking the off-diagonal matrix elements 1−2WRH <
tmn < 1 to be a uniformly distributed random variable (diagonal elements are zero, εm = 0).
The strength of the disorder is measured by WRH . We also use the standard diagonally
disordered (DD) Anderson model with on-site (potential) energy εm on site m drawn from
the uniform distribution −WDD /2 < εm < WDD /2 and tmn = 1 as the unit of energy. The
Hamiltonian is a real symmetric matrix because time-reversal symmetry is assumed.
In this Chapter numerical results for the statistics of wave function “intensities” |Ψα (r)|2
in 3D disordered electron systems are presented. The statistical properties of eigenstates are
3 For
this purpose we use the latest generation of the Linear Algebra routines known as the
LAPACK package (available at http://www.netlib.org).
162
usually characterized by the following impurity-averaged distribution function [181]
1
f (t) =
ρ(E)N
r,α
δ(t − |Ψα (r)|2 V )δ(E − Eα ) ,
on N discrete points r inside a sample of volume V . Here ρ(E) = α
(8.6)
δ(E −Eα ) is the mean
level density at energy E. Averaging over disorder is denoted by . . .. Normalization of
eigenstates gives t̄ = dt t f (t) = 1. The results for f (t) in the samples described by the RH
and DD Anderson models are shown on Fig. 8.2 and Fig. 8.3, respectively. Although some
of the samples are characterized by similar values of conductance, the eigenstates in two
models show different statistical behavior. In what follows the meaning of these findings is
explained in the context of the statistical approach to quantum systems with non-integrable
classical dynamics. In particular, the results are contrasted with the universal predictions of
the random matrix theory.
In the statistical approach of random matrix theory (RMT) [11] the Hamiltonian of a
disordered (or general quantum chaotic system) is replaced by a random matrix drawn from
an ensemble defined by its symmetry under time-reversal and spin-rotation. This leads to
Wigner-Dyson (WD) statistics for eigenvalues4 and a Porter-Thomas (PT) distribution for
the eigenfunction intensities. For the Gaussian orthogonal ensemble (GOE), relevant to the
study of the time-reversal invariant Hamiltonian (8.5), the PT distribution5 is given by
fPT (t) = √
4 In
1
exp(−t/2).
2πt
(8.7)
general, the WD statistics is applicable to quantum chaotic systems whole classical analog
exhibit “hard” chaos (K or ergodic systems) [13]. This requires that each classical trajectory
uniformly explores the whole phase space on a time scale of ergodic time τD (which is Thouless
time in localization theory). The trajectories diverge exponentially in time ∝ exp(−t/τD ). The
“soft” chaos [30] has a phase space containing both regions of integrable and non-integrable motion.
5 In
some of the literature [32] only this specific function is associated with the names of Porter
and Thomas.
163
-2
(c)
(d)
RMT
10
-4
10
-6
10
-8
(b)
(a)
f(t)
10
6
100
10
4
10
2
10
0
10
-2
10
-4
10
-6
RMT
10
-8
10 -8
-6
10
10
200
300
400
500
(c)
(d)
(a)
(b)
-4
10
-2
10
2
0
10
2
10
t=|ψ| V
Figure 8.2: Statistics of wave function intensities in the RH Anderson model, with WRH = 1,
on a cubic lattice with Ns = 163 sites. The distribution function f (t), Eq. (8.6), is computed
for the states around the following energies: (a) E = 0, (b) E = 1.5, (c) E = 2.6, and
(d) E = 2.75. Disorder averaging is performed over NEns = 40 different samples. The
Porter-Thomas distribution (8.7) is labeled by RMT.
164
-2
(a)
(b)
(c)
RMT
10
-4
10
(d)
-6
10
-8
f(t)
10
100 200 300 400 500 600 700
(a)
(b)
(c)
RMT
-2
10
-4
10
(d)
-6
10
-8
10
50
100 150 200 250 300
2
t=|ψ| V
Figure 8.3: Statistics of wave function intensities in the DD Anderson model on a cubic
lattice with Ns = 163 sites. The distribution function f (t), Eq. (8.6), is computed for the
states around following energies. Upper panel, WDD = 10: (a) E = 0, (b) E = 6.0, (c)
E = 7.6, and (d) E = 7.85. Lower panel, WDD = 6: (a) E = 0, (b) E = 4.1, (c) E = 6.56,
and (d) E = 6.7. Disorder averaging is performed over NEns = 40 different samples. The
Porter-Thomas distribution (8.7) is labeled by RMT.
165
The function fPT (t) is plotted as a reference on both Fig. 8.2 and Fig. 8.3. The RMT
answer (8.7) for the distribution function (8.6) was derived by Porter and Thomas [185] by
assuming that the coordinate-representation eigenstate r|Ψα in a disordered (or chaotic in
a classical limit) system is a Gaussian random variable when the time-reversal symmetry is
unbroken or completely broken.6 Thus, RMT assumes statistical equivalence of eigenstates
which equally test the random potential all over the sample—typical wave function has a
uniform amplitude, up to the inevitable Gaussian fluctuations.
The predictions of RMT are universal—they depend only on the symmetry properties
of an ensemble. They apply to the statistics of real disordered systems [183] in the limit
g → ∞ with g being the dimensionless conductance (g = tH /tD , where tH is Heisenberg
time tH = h̄/∆, ∆ = 1/ρ(E) is mean energy level spacing and tD L2 /D is Thouless time
for the classical diffusion with diffusion constant D). The spectral correlations in RMT are
determined by the logarithmic level repulsion which gives rise to the universality. This stems
from the form of a probability distribution of eigenvalues P (E1 , E2 , ..., En )

P (E1 , E2 , ..., En ) =


C exp −β  u(Ei , Ej ) +
U(Ei ) ,
i<j
u(E, E ) = − ln |E − E |,
(8.8)
i
(8.9)
where spectral correlations are generated by a purely geometrical effect due to the Jacobian
which relates volume elements in the matrix and eigenvalues spaces. Microscopic details
of the system are contained in the potential U(Ei ), which does not by itself create any
correlations between the eigenvalues. Therefore the correlations in the spectrum described
by RMT are universal and dependent only on the symmetry index β, while the system
specific properties are absorbed in the mean level spacing ∆. Also, the level correlations
are independent of the eigenstate correlations. For the Poisson statistics, applicable in the
localized regime, levels do not “interact”, u(E, E ) = 0.
6 In
the case of weakly broken time-reversal symmetry the distribution of eigenfunction ampli-
tudes is complicated even in the framework of RMT [186].
166
In a finite-size sample the level statistics follow RMT predictions in the ergodic regime,
i.e., on the energy separation scale smaller than the Thouless energy ETh .7 A remarkable
feature of the spectral statistics at finite g is the possibility to express all non-universal
corrections to RMT picture through the spectral determinant of a single classical differential
operator. For the disordered metallic sample it turns out to be the diffusion operator for the
corresponding geometry
D∇2 φµ (r) = −ωµ φµ (r), ∇φ|B = 0.
(8.10)
Here ωµ is the spectrum of the classical diffusion operator D∇2 with eigenstates φµ (r) satisfying the Neumann boundary conditions on the sample boundary B. The eigenvalues ωµ are
not universal since they depend on both g and the shape of the disordered sample. Thus, the
non-universal corrections to spectral statistics [31], or eigenfunction statistics (which describe
the long-range correlations of wave functions [187]), depend on dimensionality, shape of the
sample, and conductance g. These deviations from RMT predictions grow with increasing
disorder (i.e., lowering of g). At the LD transition wave functions acquire multifractal properties while the critical level statistics become scale-independent [188]. For strong disorder,
or energies |E| above the mobility edge |Ec |, wave functions are exponentially localized. A
typical wave function decays as Ψ(r) = p(r) exp(−r/ξ) from its maximum centered at an
arbitrary point inside the sample of size L > ξ. Here p(r) is a random function and approximately radial symmetry of decay is assumed. Since two states close in energy are localized
at different points in space, there is almost no overlap between them. Therefore, the levels
become uncorrelated and obey Poisson statistics. If p(r) = c is simplified as a normalization
7 For
|E − E | ETh the logarithmic level repulsion goes into the power law and eventually
becomes weakly attractive in 3D [184].
167
constant, then the distribution function of intensities is given by [175]
πξ 2 ln(c2 V /t)
,
2V
t
−1
2
L
=
1 − (1 + ) exp(−L/ξ)
,
πξ 2
ξ
fξ (t) =
(8.11)
c2
(8.12)
where radially symmetric sample of radius L/2 is assumed.
The distribution function f (t) is equivalently determined in terms of its moments
bq =
dt tq f (t). For GOE, the PT distribution (8.7) has moments bPT
= 2q V −q+1 Γ(q +
q
1/2)/Γ(1/2). They are related to the moments Iα (q) =
dr |Ψα(r)|2q of the wave function
intensity |Ψα (r)|2 . In the finite g case, the spatial correlations of wave function amplitudes at
distances comparable to the system size are non-negligible. Therefore, Iα (q) fluctuates from
state to state and from sample to sample [32]. In the universal regime g → ∞ wave functions cover the whole volume with only short-range correlations (on the scale |r1 − r2 | ≤ )
persisting between Ψα (r1 ) and Ψα (r2 ). This means that the integration in the definition
of Iα (q) provides self-averaging, and Iα (q) does not fluctuate, i.e., Iα (q) = bPT
q . Following
Wegner [189] we characterize the individual states by an ensemble average of Iα (q)
¯ =∆
I(q)
r,α
|Ψα (r)| δ(E − Eα ) .
2q
(8.13)
The moment Iα (2) is usually called inverse participation ratio (IPR). It is a one-number measure of the degree of localization (i.e., it measures the portion of space where the amplitude
of the wave function differs markedly from zero). This is seen from the scaling properties of
¯ with respect to system size
moments I(q)
¯ ∝
I(q)


L−d(q−1)




0
L




 −d∗ (q)(q−1)
L
metal,
insulator,
(8.14)
critical.
Here d∗ (q) < d is the fractal dimension. Its dependence on q is the hallmark of multifractality
of critical eigenfunctions—they are delocalized but in the thermodynamic limit occupy only
an infinitesimal fraction of the sample. The fluctuations of Iα (2) scale [32] as δIα (2) ∼ 1/g 2 ∝
168
L4−2d . At the critical point (g ∼ 1) the average value is of the same size as fluctuations.
¯ is not enough to characterize the critical eigenstates.
Therefore, I(q)
¯ (Fig. 8.4) as a rough guide in selecting eigenstates with different properties
We use I(2)
(especially in the delocalized phase). The second parameter used in the “selection procedure”
is the conductance g(EF ) (see below, Fig. 8.5) computed for a band filled up to the Fermi
energy EF equal to the state eigenenergy. The conductance as a function of band filling
allows us to delineate delocalized from localized phase as well as to narrow down the critical
region around LD transition point (which is defined by Ec when disorder strengths WRH or
WDD are fixed). Upon inspection of these two parameters, a small window is placed around
chosen energy, and f (t) is computed for all eigenenergies whose eigenvalues fall inside that
window. This provides a detailed information on the structure of eigenstates.
The models with random hopping [190] have attracted recently considerable attention
inasmuch as they show a disorder induced quantum critical point in less than three dimensions [191, 192], where delocalization occurs in the band center (E = 0). The real
system which correspond to TBH (8.5) with off-diagonal disorder include doped semiconductors [190], such as P-doped Si, where hopping matrix elements tmn vary exponentially
with the distances between the orbitals they connect, while diagonal on-site energies εm
are nearly constant. The behavior of low-dimensional RH Anderson model goes against the
standard mantra of the scaling theory of localization [8] that all states in d ≤ 2 are localized.
This is actually known since the work of Dyson [193] on glasses. Also, the scaling theory
for quantum wires with off-diagonal disorder requires two parameters [194] which depend
on the microscopic model, thus breaking the celebrated universality in disordered electron
problems. In 3D case explored here, the states in the band center are less extended than
other delocalized states inside the band (Fig. 8.4). The off-diagonal disorder is not strong
enough [195] to localize all states in the band, in contrast to the usual case of diagonal
c
> 16.5.
disorder where the whole band becomes localized [109] for WDD
169
-1
10
(b)
-2
10
(a)
-3
10
-1
-8 -6 -4 -2 0
2
4
6
8
10
0.0018
(c)
-2
10
(b)
0.0010
(a)
-3
10
-4
-2
0
2
4
0.0002
Energy
¯
Figure 8.4: Ensemble averaged Inverse Participation Ratio, I(2),
of eigenstates in the RH
and DD Anderson models on the cubic lattice with Ns = 163 sites. Top: diagonal disorder
with (a) WDD = 6, and (b) WDD = 10. Bottom: off-diagonal disorder with (a) WRH = 0.25,
(b) WRH = 0.375, and (c) WRH = 1; right axis is for (a) and (b).
170
The mobility edge for the strongest RH disorder WRH = 1, as well as for DD models,
is found by looking at an exact zero-temperature static conductance. This quantity (which
is a Fermi surface property) is computed from the Landauer-type formula [92] (the factor of
two here and in the density of states (8.16) is for spin degeneracy)
2e2
G(EF ) =
Tr (t(EF )t† (EF )),
h
(8.15)
where transmission matrix t(EF ) is expressed in terms of the real-space (lattice) Green
functions (cf. Sec. 2.5) for the sample attached to two clean semi-infinite leads. To study
the conductance in the whole band of the DD model, tmn = 1.5 is used [120] for the hopping parameter in the leads. This mesoscopic computational technique “opens” the sample,
thereby smearing the discrete levels of initially isolated system. Therefore, the spectrum
of sample+leads=infinite system becomes continuous, and conductance can be calculated at
any EF inside the band. However, the computed conductance, for not too small disorder or
coupling to the leads (of the same transverse width as the sample) [152, 196], is virtually
equal to the “intrinsic” conductance g = ETh /∆ expressed in terms of the spectral properties
of a closed sample.
The conductance and density of states (DoS)
N(E) = 2
ρ(E)
,
V
(8.16)
are plotted on Fig. 8.5. The DoS is obtained from the histogram of the number of eigenstates
which fall into equally spaced energy bins along the band. The conductance and DoS of the
RH model have a peak at E = 0, which becomes a logarithmic singularity in the limit of
infinite system size [193]. For weak off-diagonal disorder (WRH = 0.25), N(E) still resembles
the DoS of a clean system, even after ensemble averaging (lower panel of Fig. 8.5). On the
other hand, the conductance is a smooth function of energy since discrete levels of an isolated
sample are broadened by the coupling to leads. The same is true for DoS computed from
the imaginary part of the Green function for an open system. The mobility edge is absent at
171
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
12
(b)
10
(a)
8
(d)
6
(c)
2
Density of States
Conductance (2e /h)
4
2
0
80
-6
-4
-2
0
2
4
6
0.5
60
0.4
0.3
40
0.2
20
0
-6
0.1
-4
-2
0
2
4
6
0.0
Fermi Energy
Figure 8.5: Conductance and DOS in the RH and DD Anderson models on the cubic lattice
with Ns = 163 sites. Top: off-diagonal disorder with (a) and (d) WRH = 1 (mobility edge
is at |Ec | 2.6), (b) WDD = 6 (|Ec | 6.6), and (c) WDD = 10 (|Ec | 7.7). Disorder
averaging is performed over NEns = 20 different samples for conductance and NEns = 40 for
DOS. Bottom: off-diagonal disorder WRH = 0.25; sharp lines correspond to the DOS of a
clean system (scaled by 1/10 for clarity).
172
low RH disorder (WRH = 0.25 and WRH = 0.375) for system sizes L ≤ 16a. This means that
localization length ξ is greater than 16a (lattice spacings is denoted by a) for all energies
inside the band of these systems. For other samples on Fig. 8.5 the mobility edge appears
inside the band. This is clearly shown for WRH = 1 case where band edge Eb (N(Eb ) = 0)
differs from Ec . We locate the mobility edge at the minimum energy |Ec | for which g(Ec )
is still different from zero. The conductance of finite samples is always finite, although
exponentially small at |EF | > |Ec |. The approximate values of |Ec | listed on Fig. 8.5 are
such that conductance satisfies: g(EF ) < 0.1, for |EF | > |Ec |; typically g(Ec ) ∈ (0.2, 0.5)
is obtained, like in the recent detailed studies [197] of conductance properties at Ec . Thus
found Ec is virtually equal to the true mobility edge, defined only in thermodynamic limit
(and is usually obtained from some numerical finite-size scaling procedure [108]). Namely,
the position of mobility edge extracted in this way will not change [95] when going to larger
system sizes if ξ < L for all energies |E| > |Ec |.
8.3
Connections of eigenstate statistics to static quantum transport properties
The distribution f (t) of eigenfunction intensities has been studied analytically for diffusive conductors close to the universal limit (where conductance is large and localization
effects are small) in Refs. [32, 181] using the supermatrix σ-model [29], or by means of a direct optimal fluctuations method in Ref. [198]. Numerical studies [175, 199] were conducted
in 2D and 3D for all disorder strengths. Here we show how f (t) evolves in 3D disordered
samples where a genuine LD transition occurs. The complete eigenproblem of a single particle disordered Hamiltonian is solved numerically, and f (t) is computed as a histogram
of intensities for the chosen eigenstates in: metallic phase (|E| < |Ec |), insulating phase
(|E| > |Ec |), and close to the mobility edge |Ec |. The two delta functions in Eq. (8.6) are
173
approximated by a box function δ̄(x). The width of δ̄(E − Eα ) is small enough at a specific
energy that ρ(E) is constant inside that interval. For each sample, 5–10 states are picked by
the energy bin, which effectively provides additional averaging over the disorder (according
to ergodicity [30, 11] in RMT). The amplitudes of wave functions are sorted in the bins
defined by δ̄(t − |Ψα (r)|2 V ) whose width is constant on a logarithmic scale. The function
f (t) is computed at all points inside the sample, i.e., N = 163 in Eq. (8.6).
The evolution of f (t), when sweeping the band through the “interesting” states, is
plotted on Fig. 8.2 for the RH disordered sample. Since pre-localized states generate slow
decay of f (t) at high wave function intensities (where PT distribution is negligible) [29], this
region is enlarged on Fig. 8.2. This is obvious from the “pre-localized” example in Fig. 8.1
where state with large amplitude spikes, highly unlikely in the framework of RMT, was found
in a very good metal. The same is trivially true for the localized states which determine
extremely long tails of fξ (t) (8.11). Thus, the asymptotic tails of f (t), appreciably deviating
from PT distribution, are signaling the onset of localization. It is interesting that states in the
band center of RH model, which define the largest zero-temperature conductance [g(EF =
0) ≈ 10.2, Var g(EF = 0) ≈ 0.63], are mostly pre-localized. Namely, both the frequency
of their appearance and high amplitude splashes resembles the situation at criticality. It
might be conjectured that these pre-localized states would generate multifractal scaling of
IPR in the band center. This result, together with the DoS and conductance from Fig. 8.5,
shows that phenomena in the band center of 3D conductors with off-diagonal disorder are
as intriguing as their much studied counterparts in low-dimensional systems [191, 192]. The
origin of these phenomena can be traced back to a special sublattice, or “chiral” [190, 200],
symmetry (leading to an eigenspectrum which for Eα contains −Eα as well, Fig. 8.4) obeyed
by TBH (2.74) with random hopping (and constant on-site energy). In the cases with
WRH = 0.25 or WRH = 0.375 all states are extended. Their f (t) overlaps with the distribution
function for the delocalized states at E = 1.5 in the sample characterized by WRH = 1. The
174
distribution function fξ (t) in Eq. (8.11), obtained from the simple parameterization of a
localized state, does not fit precisely the numerical result for the states corresponding to
E = 2.75. An estimate of the localization length, ξ 5.5a, would generate a distribution
with a similar tail to that of the analyzed states.
The same statistical analysis is performed for the eigenstates of DD Anderson model—a
“standard model” in the localization theory [108, 109, 202]. Figure 8.3 plots f (t) at specific
energies Ei in samples characterized by different conductance g(EF = Ei ) (controlled by
WDD ). The conductance g(E = 0) of TBH with WDD = 6 is numerically close to the
conductance of RH disordered samples with WRH = 1. Nevertheless, comparison of the
corresponding distribution functions reveals model dependent features [32] which are beyond
corrections [187, 201] accounted by the eigenmodes of the classical diffusion operator (8.10).
In both models, all computed f (t) intersect PT distribution (from below) around 6 ≤ t ≤ 10,
and then develop tails far above PT values. The length of the tails is defined by the largest
amplitude exhibited in the pre-localized state, e.g., Fig. 8.1. For strong DD (WDD = 10)
the conductance g(EF ) is smaller than 3.5. In this regime transport becomes “intrinsically
diffusive”, as discussed in Ch. 3, but one can still extract resistivity from the approximate
Ohmic scaling of disorder-averaged resistance [202] (for those fillings where [117] g(EF ) >
2). However, the close proximity to the critical region g ∼ 1 induces long tails of f (t) at
all energies throughout the band—a sign of increased frequency of appearance of highly
inhomogeneous states. This provides an insight into the microscopic structure of eigenstates
which carry the current in a non-perturbative transport regime [29, 167] (characterized by
the lack of semiclassical concepts, like mean free path , where unwarranted use of the
Boltzmann theory would give [202] < a).
Using better statistics (i.e., more realization of disorder configurations) would allow us
to focus on the rare events (big spikes in the eigenstate intensity on the top of homogeneous
background, like that on Fig. 8.1) in diffusive metallic samples (g 1), and compare the
175
predictions of SUSY NLσM for f (t) (exponential of the log-cube) [32] to that of the optimal
fluctuation method [198] (exponential of the log-cube × smaller prefactor8 ). These analytical
predictions for the asymptotic behavior of the distribution function f (t), as well as for the
envelope of pre-localized states, are applicable only in a weakly disordered conductor.
8.4
Conclusion
This Chapter reports on the statistics of eigenstates in 3D samples, modeled by the
Anderson Hamiltonian on the cubic lattice with Ns = 163 sites. The disorder is introduced
either in the potential energy (diagonal) or in the hopping (off-diagonal) matrix elements.
Also calculated are the average inverse participation ratio of eigenfunctions as well the conductance of different samples as a function of energy. This comprehensive set of parameters
makes it possible to compare the eigenstates in 3D nanoscale mesoscopic conductors with different types of disorder, but characterized by similar values of conductance. Sample-specific
details, which are not parameterized by the conductance alone, are found. This is in spite
of the fact that dimensionality, shape of the sample, and conductance (i.e., the eigenvectors
and eigenvalues of the classical diffusion operator) are expected to determine the finite-size
(non-universal) corrections to the universal (sample-independent) predictions of random matrix theory. The appearance of states with large amplitude spikes on the of top of RMT like
background is clearly demonstrated even in good metals. At criticality, such “pre-localized”
states are directly related to the extensively studied multifractal scaling of IPR. However,
even in the delocalized metallic (g 1) phase, where the correlation length [5] ξc expected
from the sample conductance g(ξc ) = O(1) is microscopic (L < ξc would naturally account
for the multifractal scaling [5], like in 2D), pre-localized state are found in the band center
8 The
smaller prefactor C3 in f (t) ∼ exp(−C3 ln3 t) (prediction of the optimal fluctuation
method [198]) would substantially increase probability to observe a rare event, when compared
to form provided by SUSY NLσM calculation.
176
of the random hopping disordered systems. They are inhomogeneous enough to generate
extremely long (critical like) tails of the distribution of eigenfunction amplitudes.
177
Chapter 9
Infrared studies of the Onset of Conductivity in
Ultrathin Pb Films
9.1
Introduction
Measurements of DC transport in ultra-thin films have been a subject of active interest for many years [204]. Such systems, consisting of a thin layer of metal deposited onto
a substrate held at liquid Helium temperatures, provide a relatively simple way to study
the interplay between localization, electron-electron interactions, and superconductivity1 in
disordered quasi-2D metals. These experiments are in quantitative agreement with predictions of weak localization theory [36, 173] combined with the effects of diffusion-enhanced
electron-electron interactions [15]. The reason why these theories, developed for homogeneous materials, work so well in the case of granular, (i.e., inhomogeneous) films is that in
DC transport experiments the relevant length scale is usually much larger than the characteristic size of inhomogeneities (grains, themselves as well as the percolation clusters that
form from them) in the film.
For the AC conductivity one can modify the characteristic transport length scale Lω =
D/ω by simply changing the probing frequency (here D denotes the electron diffusion
1 These
phenomena were also actively studied in disordered layered oxide superconductors [205].
178
constant). In the frequency range, where Lω =
D/ω is smaller than all relevant DC
length scales, one has the frequency-dependent WL correction to the conductivity [36]. This
theory can account for a slow increase of AC conductivity with frequency [206], in the
region ωτ 1 where Drude theory predicts a plateau. The observation of this frequency
dependence requires the electric field not to be too strong, so that dephasing by the highfrequency electric field is avoided. If the criterion derived by Altshuler et al. [207] is satisfied,
then only the intrinsic dephasing introduced2 by Lω < Lφ will be observed. However, in our
system these quantum effects constitute only one source of the frequency dependence of
the conductivity. In the region where material is strongly inhomogeneous the frequency
dependence of the conductivity is dominated by the purely classical effects due to charge
dynamics in a network of capacitively and resistively coupled clusters of grains.
The physics of small metallic particles [209] and their composites [210] was initiated
at the beginning of the century, but wider interest has been attracted only in the last few
decades. Small particles are usually treated as a bulk solids, with properly defined boundary
conditions, using standard techniques and ideas of quasielectronic excitations. But their
size (of the order of nm) can be smaller than some of the characteristic lengths, like the
wavelength of light, electron mean free path, superconducting coherence length, etc. The
finite size of particles introduces qualitatively new features when compared to the bulk material. They arise both from the realm of classical (e.g., surface plasmon collective excitation
mode) and quantum physics (e.g, discreteness of the energy levels which is observed if the
relevant energies are comparable to the level spacing). Thus, the behavior of these systems
at finite frequencies is drastically different from the predictions of elementary Drude theory
valid homogeneous bulk materials. The electromagnetic response [210] of granular systems
can be described in terms of the phenomenological complex functions: complex dielectric
2 For
example, Lω regularizes divergent WL correction in 2D, which was historically the first
dephasing length introduced in the theory of WL [36].
179
constant ε(ω) = ε (ω) + ε (ω), or complex conductivity σ(ω) = σ (ω) + σ (ω). They are
related to each other through ε(ω) = 4πiσ(ω)/ω. The real part of the conductivity (i.e.,
“optical conductivity”) or imaginary part of the dielectric constant are direct measures of
the spectrum of dissipative processes. At low-frequencies (|ε | ε or σ σ ) the response
of the conducting electrons is Ohmic (j = σ E) current flowing through the connected clusters formed by grains. The electromagnetic properties of these systems can be modeled as
electrical percolation in a random resistor network [211]. In particular, various scaling properties are expected around MIT, occurring at the percolation threshold where an incipient
cluster of connected resistors, spanning the whole system, appears. In the high-frequency
region (|ε | ε ) the displacement current from the Maxwell equations (j = −iωε E) starts
to dominate. It generates a non-compensated surface charge on small particles (free electron
displacement becomes less than atomic dimension) which can be characterized by the dipole
moment. Thus, the field of polarized particles leads to a long range dipole-dipole interactions between the particles inside the clusters as well as between the clusters. This type of
response is the subject of various mean-field-like theories known as effective medium theories [212], as well as more involved theories dealing with extended and localized collective
dipolar modes [210].
The suitable theoretical framework to describe the relevant classical electromagnetic
effects in our system is provided by percolation theory [211]. In this approach the AC
conductivity is shown to increase with frequency. Indeed, since capacitive coupling between
the grains is proportional to the frequency, grains become more and more connected as
ω is increased. While experimental data on the frequency dependence of conductivity are
virtually non-existent for ultra-thin quenched-condensed films, classical charge dynamics is
known to play a dominant role in frequency dependence of the AC conductivity in thicker,
more granular films, deposited onto a warm substrate [208]. It is also known that quantum
corrections themselves become profoundly modified on length scales where the material can
180
no longer be treated as homogeneous [171, 213].
9.2
The Experiment
In this Section the first measurement of conductivity of ultra-thin films at infrared
frequencies is explained in detail as an overture for the subsequent theoretical account of
these results [203]. The films used in this experiment were made in situ by evaporating Pb
onto Si(111) (sets 1 and 2) and glass (set 3) substrates, mounted in an optical cryostat, held
at 10 K. Ag tabs, pre-deposited onto the substrate, were used to monitor the DC resistance
of the film. Infrared transmission measurements from 500 to 5000 cm−1 (set 1), and 2000
to 8000 cm−1 (sets 2 and 3) were made using a Bruker 113v spectrometer at the new highbrightness U12IR beamline of the BNL National Synchrotron Light Source. The substrates
were covered with a 5 Å thick layer of Ge to promote two-dimensional thin-film growth,
rather than the agglomeration of the deposited Pb in larger grains. For different depositions
a variation in the thickness where continuity first occurs was observed. However, the optical
properties of the films show rather similar behavior. The salient feature of this behavior is
frequency-dependent conductivity that can be understood by classical arguments assuming
an inhomogeneous structure on a nanoscale level. The films were evaporated at pressures
ranging from the low 10−8 to the mid 10−9 Torr range. The transmission spectra were
obtained after successive in-situ Pb depositions. The DC resistances in set 1 on Si range
from 64 MΩ/✷ at 24.4 Å average thickness to 543 Ω/✷ at 98 Å. The 98 Å sample was then
annealed twice, first to 80 K, and then to 300 K. As a result its resistance at 10 K became
166 Ω/✷ after the first annealing, and 100 Ω/✷ after the second annealing. The films from
the set 2 (also on Si) are similar to set 1: it was observed R✷ = 20 MΩ/✷ at 26 Å and
R✷ = 1000 Ω/✷ at 123 Å. Finally, the films from the set 3, deposited on a Ge-coated glass
substrate, range from 13 to 231 Å, while R✷ changes between 5.6 MΩ and 22.8 Ω.
181
The transmission coefficient of a film deposited on the substrate, measured relative to
the transmission of the substrate itself, is related to the real and imaginary parts of the sheet
conductance of the film as [214]
T (ω) =
1
[1 +
Z0 σ✷ (ω)/(n
+
1)]2
+ (Z0 σ✷ (ω)/(n + 1))2
.
(9.1)
Here Z0 = 377 Ω is the impedance of free space, n is the index of refraction of the substrate
(nSi = 3.315 for silicon and nG = 1.44 for glass), and σ✷ (ω) and σ✷ (ω) are, respectively,
the real and imaginary parts of the sheet conductance of the film. It is common in such
experiments to have the following condition satisfied, σ✷ (ω), σ✷ (ω) (n + 1)/Z0. In this
case, the contribution of the imaginary part of conductance to the transmission coefficient
is negligible, and Eq. (9.1) can be approximated by
T (ω) 1
1 + Z0 σ✷ (ω)/(n + 1)
2
.
(9.2)
Even for the thickest films, where σ✷ (ω) ≈ (n + 1)/Z0, the error in calculating σ✷ (ω) in
this way is less than 10% over our frequency range. Therefore, throughout the Chapter this
approximation will be used to extract the real part of the sheet conductance of the film from
its transmission coefficient. Only for our thickest films the exact formula (9.1) has to be
used in order to extract the parameters of Drude fits.
9.3
Theoretical analysis of the experimental results
Figure 9.1 plots the frequency-dependent conductance, extracted from the transmission
data for the films from set 3 with the help of the above approximation to Eq. (9.1). The seven
thickest films from this set exhibit characteristic behavior of the Drude (or semiclassical)
sheet conductance3 which falls at high frequencies in a characteristic fashion
σ✷ (ω) = σD /(1 − iωτ ),
3 In
(9.3)
2D conductivity and sheet conductance (conductance per square) have the same dimensions.
182
Fig. 1
8
10
10
−2
d=231 A
7
10
6
10
5
4
10
3
d=131 A
2
10
−1
σ (ω) (Ω )
10
[]
R (Ω)
10
1
[]
10
−3
10
0
10
10
100
1000
d (Angstroms)
d=24 A
10
d=13.2 A
−4
1000
2000
3000
−1
ω (cm )
4000
6000
8000
Figure 9.1: Sheet conductance vs. frequency for set 3. The dashed lines plotted between
3000 and 4000 cm−1 (where the glass substrate is opaque) are a guide to the eye. The inset
shows the inverse average AC conductance in this frequency range (solid circles) and the DC
sheet resistance (open symbols) as a function of the film thickness.
183
where σD is DC semiclassical Drude (2.25) sheet conductance and τ is the transport mean
free time. The frequency dependent conductivity is Fourier transform of the time dependent
conductivity σ✷ (t) (8.1)
σ✷ (t)eiωt dω,
σ✷ (ω) =
(9.4)
which determines the current response j(t) to a spatially homogeneous field in the form of
sharp electric pulse E(t) = E0 δ(t) (cf. Ch. 8). The conductivity (9.3) is Fourier transform
of the semiclassical response function
σ✷ (t) =
t
σD
exp(− ),
τ
τ
(9.5)
which is valid only on a time scales of the elastic mean free path, t ∼ τ . At longer time scales
one has to include the quantum corrections discussed in Ch. 8 (which then give the corresponding frequency dependent WL). For films other than the seven thickest ones mentioned
above, the conductivity systematically increases with frequency throughout our frequency
range. The inset in Fig. 9.1 shows the average AC conductance as well as the DC sheet
conductance for the set 3 as a function of its thickness. Note the curves start to deviate
significantly from each other at around 50 Å.
In order to fit the conductance of the thickest films with the Drude formula, one needs to
use the untruncated Eq. (9.1) for the transmission coefficient. Inserting the Drude expression
for the sheet conductance (9.3) directly into Eq. (9.1) one gets
(1 + ω 2 τ 2 )
T (ω)
=
,
1 − T (ω)
(σD /σ0 )2 + 2σD /σ0
(9.6)
where σ0 = (n+1)/Z0. Therefore, the transmission data which are consistent with the Drude
formula can be fitted with a straight line on a plot of T (ω)/[1 − T (ω)] vs. ω 2 (Fig. 9.2).
The knowledge of the average thickness of films, along with parameters σD and τ of the
Drude formula, allows us to calculate the plasma frequencies of the films. They are shown
on the inset of Fig. 9.2 as a function of 1/σD (σD was extracted from the Drude fits explained
above). These results are in excellent agreement with the experimentally determined lead
184
Fig. 2
8
1.4
−1
ωp (10 cm )
6
1.2
4
4
2
T/(1−T)
1
0
0.8
0
100 200 300
Rsq ac (Ω)
400
0.6
0.4
0.2
0
0
2
4
6
2
7
−2
ω (10 cm )
8
10
12
Figure 9.2: T (ω)/[1 − T (ω)] plotted vs. ω 2 for the seven thickest films from the set 3 (dots),
and two annealed films form set 1 (solid circles). The solid lines are Drude model fits (9.3).
The inset shows the plasma frequency extracted from these fits with solid line representing
the plasma frequency of bulk lead from Ref. [215].
185
plasma frequency of ωp = 59 400 cm−1 [215]. In the remainder of this Section we discuss
possible interpretations of the increase of the conductance with frequency, which are observed
in the measurements on thinner films.
One mechanism which is known to cause a frequency dependence of the conductivity
within a Drude plateau (ωτ 1) originates from purely quantum-mechanical effects in
transport. The conductivity is known to be reduced due to the increased back scattering
of phase-coherent electrons (the so-called weak localization [36, 173]), as well as diffusion
enhanced electron-electron interactions (EEI) [15]. The magnitude of the (negative) WL
correction depends on the dephasing length Lφ =
Dτφ over which an electron maintains
the memory of its phase, while EEI correction depends on the thermal coherence length
LT =
h̄D/kB T (cf. Sec. 2.1). A sample much larger than Lφ can be viewed as a classical
resistor network of phase-coherent units. Thus, inside this resistor of size Lφ the transport
is essentially quantum. However, this resistors are independent of each other and can be
stacked according to the Ohm’s law. Therefore, WL as a quantum effect survives the selfaveraging in such network, and the conductivity of the entire sample is the same as that of
a single resistor. The quantum features of transport inside each phase-coherent unit lower
down its semiclassical conductance by one conductance quantum (in the weakly scattering
regime), GQ = 2e2 /h. For AC conductivity the influence of the coherent backscattering
is restricted to a spatial region of size Lω =
D/ω. Here D is a “constant” (i.e., length
scale independent) only if the probe sees macroscopically homogeneous sample [213] and
the sample is far away from the LD transition [173]. If Lω is the shortest characteristic
scale in the problem, then it enters as a cutoff in all WL formulas. In general, the effective
dimensionality of a (quasi-2D) sample is decided by comparing the characteristic length scale
(Lω in this case) to the film thickness d.
The frequency-dependent WL corrections to the sheet conductance of the film are given
186
by
∆σ✷2D (ω) =
e2
ln ωτ,
2π 2 h̄
(9.7)
in the 2D limit (d < Lω ) [173], and
√
∆σ✷3D (ω)
2e2
ω
d
,
=
2
4π h̄
D
(9.8)
in the 3D limit (d > Lω ) [36]. Using the lower end of our frequency range ω = 500 cm−1 ,
and a realistic value of D = 5 cm2 /s, we can estimate Lω ≤ 20Å ≤ d. Therefore, for our
films one should use the formulas of three-dimensional WL theory. The frequency-dependent
√
sheet conductance in most of our films is consistent, on the first sight, with ω dependence
of 3D WL. However, a more detailed look reveals several problems in ascribing the observed
frequency dependence of conductivity solely to WL and EEI effects: (i) The dependence
√
ω on the thickness of the film and the DC sheet
of the slope of the conductivity vs.
conductance, which determines the diffusion coefficient D, does not agree with predictions of
the 3D WL. (ii) The WL theory is only supposed to work in the limit where its corrections
are much smaller than the DC conductivity. In experimental data presented in Sec. 9.2 there
is no change of behavior as the corrections to the conductivity become bigger than the DC
√
conductivity. In fact the ω fit works very well and gives roughly the same slope even for
films with DC sheet resistance of ≈ 100 kΩ, while the AC sheet resistance is only ≈ 1 kΩ.
√
Furthermore, the 3D localization theory predicts that the ω dependence of WL theory
crosses over to ω (D−1)/D = ω 1/3 dependence at or near the 3D metal-insulator transition
[173] (we use D to denote spatial dimensionality in this Chapter). In analyzed experimental
data there is no evidence for such a crossover.
There exists yet another, purely classical electromagnetic effect that gives rise to the
frequency dependence of the conductivity. It is relevant in strongly inhomogeneous, granular
films. There is ample experimental evidence that even ultra-thin quenched-condensed films
have a microscopic granular structure [216, 217]. In order to describe the AC response of a
film with such a granular microstructure, one needs to know the geometry and conductivity
187
of individual grains as well as the resistive and capacitive couplings between grains. The
disorder, which is inevitably present in the placement of individual grains, makes this problem
even more complicated. However, there exist two very successful approaches to the analytical
treatment of such systems. One of them, known as the effective-medium theory (EMT) [218],
takes into account only the concentrations of metallic grains and of the voids between the
grains, disregarding any spatial correlations. A more refined approach also considers the
geometrical properties of the mixture of metallic grains and voids. The insulator-to-metal
transition in this approach is nothing else but the (geometrical) percolation transition, in
which metallic grains first form a macroscopic connected path at a certain critical average
thickness dc of the film. The DC conductivity above the transition point scales as (d − dc )t ,
where t = 1.3 in 2D and t = 1.9 in 3D [211]. Just below the percolation transition the
dielectric constant of the medium diverges as (d) ∼ (dc − d)−s , where s = 1.3 in 2D and
s = 0.7 in 3D. The diverging dielectric constant is manifested as the imaginary part of
the AC conductivity σ(ω) ∼ −iω(dc − d)−s . In general, the complex AC conductivity of
the metal-dielectric mixture, where void represents the dielectric, close to the percolation
transition is known [211] to have the following scaling form
σ(ω, d) = |d − dc |t F± (−iω|d − dc |−(t+s) ).
(9.9)
Here F+ (x) and F− (x) are the scaling functions above and below the transition point, respectively. Note that this scaling form correctly reproduces the scaling of the DC conductivity
above the transition and the divergence of the dielectric constant below the transition, provided that
(0)
(1)
(2)
F+ (x) = F+ + F+ x + F+ x2 + . . . ,
(1)
(2)
F− (x) = F− x + F− x2 + . . .
(9.10)
(9.11)
One should mention that the predictions of the EMT can also be written in the analogous
188
scaling form where
D 2 + 4(D − 1)x ± D
F ±(x) =
2(D − 1)
,
(9.12)
and with mean-field values for the exponents t = s = 1.
Since the metallic grains in the experimentally studied films form not more than two
layers, the data should be interpreted in terms of the two-dimensional percolation theory.
In 2D t = s = 1.3 [211], and according to Eq. (9.9) the AC conductivity precisely at the
transition point d = dc is given by
iω
σ(ω, dc ) = A
ω0
t/(t+s)
iω
=A
ω0
1/2
.
(9.13)
This prediction is in agreement with the experimental data. In Fig. 9.3 we attempt to rescale
the data according to Eq. (9.9). The critical thickness dc is determined as the point where
√
the AC conductivity divided by ω is frequency independent. The experimental uncertainty
in the data points does not allow us to determine values of exponents t and s which would
provide the best data collapse [219]. Also, in almost all scaling phenomena outside the
realm of temperature driven classical phase transitions it is hard to have a large number of
decades (on logarithmic scale) where convincing scaling holds. Nevertheless, as can be seen
from Fig. 9.3 the analyzed data are consistent with the scaling picture of the 2D percolation
theory.
Finally, we use Fig. 9.3 to estimate basic physical parameters, such as typical resistance
R of an individual grain or typical capacitance C between nearest-neighboring grains. From
the limiting value of σ(ω, d)(dc/|d−dc |)1.3 at small values of the scaling variable x = ω(dc/|d−
dc |)2.6 for d > dc , one estimates the resistance of an individual grain to be of the order of
R ∼ 1000 Ω. In the simplest RC model, where the fraction of the bonds on the square
lattice are occupied by resistors of resistance R, while the rest of the bonds are capacitors
with capacitance C, the AC conductivity exactly at the percolation threshold is given by
√
A/R(iωRC)1/2 , where A is a constant of the order of one. Therefore, the slope ∂σ/∂ ω
√
in our system should be of the same order of magnitude as RC/R. This gives C 189
10
−2
−3
−1
σ(ω) (Ω )/|d/dc−1|
1.3
10
10
10
−4
−5
10
1
10
2
10
3
4
10
−1
2.6
ω (cm )/|d/dc−1|
10
5
10
6
10
7
Figure 9.3: The “data collapse” of the rescaled conductivity of 10 films from the set 1
(dc = 35.4Å, ×), 9 films from the set 2 (dc = 48.4Å, open circles) and 18 form the set 3
(dc = 34.2Å, solid line).
190
2.6 × 10−19 F, which is in agreement with a very rough estimate of the capacitance between
two islands 200 Å × 200 Å × 30 Å separated by a vacuum gap of approximately 20 Å, thus
giving C 2.7 × 10−19 F. This order of magnitude estimate confirms the importance of
taking into account interisland capacitive coupling when one interprets the AC conductivity
measured in experiments such as the one elaborated in this Chapter. Indeed, R = 1000 Ω,
and C = 3 × 10−19 F define a characteristic frequency 1/RC 17000 cm−1 comparable to
the frequency range accessed in these experiments.
9.4
Conclusion
The results of the first measurement of low-temperature, normal-state infrared conductivity of ultra-thin quenched-condensed Pb films in the frequency range 500-8000 cm−1 are
presented in this Chapter, together with our theoretical account in which we emphasize
classical electromagnetic effects dominating over “more interesting” quantum mechanical
“usual suspects”. For DC sheet resistances, such that ωτ 1, the AC conductance increases with frequency, but in disagreement with the predictions of WL theory at finite
frequency (i.e., where the two-dimensional WL correction is regularized by the length scale
Lω introduced by the AC frequency probe). This behavior is attributed to the effects of
an inhomogeneous granular structure of these films when they are first formed. It is manifested at the very small probing scale of infrared measurements. The evolution of σ(ω)
with DC sheet resistance can be explained using the scaling argument from the classical 2D
percolation in a random network of resistors (grains) and capacitors (charged capacitively
coupled grains). At lower probing frequencies, where Lω becomes larger than the scale of
inhomogeneities in these films, we expect that the effects of WL will become more prevalent.
191
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Quantum Transport in Finite Disordered Electron Systems

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