Quantum Transport in Finite Disordered Electron Systems

A Dissertation Presented

Branislav Nikoli´c

The Graduate School

in Partial Fulﬁllment of the Requirements

for the Degree of

Doctor of Philosophy

Physics

State University of New York at Stony Brook

August 2000 State University of New York at Stony Brook The Graduate School

Branislav Nikoli´c

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, Professor, Department of Physics and Astronomy, Stony Brook

Vladimir J. Goldman, Professor, Department of Physics and Astronomy, Stony Brook

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

Branislav Nikoli´c

Doctor of Philosophy

Physics

State University of New York at Stony Brook

2000

The thesis presents a theoretical study of electron transport in various dis-

ordered 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 meso-

scopic 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 finite- ness 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 in- tention 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 na¨ıve 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 Nikoli´c Contents

List of Figures ...... xiv

Acknowledgements ...... xv

1 INTRODUCTION ...... 1

I Diﬀusive Transport Regime 19

2 Linear Transport Theories ...... 20

2.1Introduction...... 20

2.2Ohm’slawandcurrentconservation...... 24

2.3Semiclassicalformalism:Boltzmannequation...... 32

2.4Quantumtransportformalisms...... 36

2.4.1 Linearresponsetheory:Kuboformula...... 36

2.4.2 Scatteringapproach:Landauerformula...... 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 Latticemodelforthetwo-probemeasuringgeometry...... 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-dimensionalexample:singleimpurityinacleanwire...... 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

3.1Introduction...... 70

3.2SemiclassicalResistivity...... 73

3.3Quantumresistivity...... 78

3.4 Conductance vs. Conductivity in mesoscopic

physics...... 87

4 Quantum Transport in Disordered Macroscopically Inhomogeneous Con- ductors ...... 91

4.1Introduction...... 91

4.2Transportthroughdisorderedmetaljunctions...... 92

4.3Transportthroughstronglydisorderedinterfaces...... 105

4.4 Transport through metallic multilayers ...... 109

II 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

5.1Introduction...... 116

5.2Model:Nanocrystal...... 119

5.3Model:Nanowire...... 126

5.4Conclusion...... 127

ix 6 Electron Transport Through a Classical Point Contact ...... 131

6.1Introduction...... 131

6.2Semiclassicaltransporttheoryintheoriﬁcegeometry...... 134

6.3 The conductance of the oriﬁce ...... 140

6.4Conclusion...... 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 Disor- dered Systems ...... 155

8.1Introduction...... 155

8.2 Exact diagonalization study of eigenstates in disordered conductors ..... 161

8.3 Connections of eigenstate statistics to static quantum transport properties . 172

8.4Conclusion...... 175

9 Infrared studies of the Onset of Conductivity in Ultrathin Pb Films .. 177

9.1Introduction...... 177

9.2TheExperiment...... 180

9.3Theoreticalanalysisoftheexperimentalresults...... 181

9.4Conclusion...... 190

References ...... 190

x List of Figures

2.1 A two-dimensional version of our actual 3D model of a two-probe measuring

geometry...... 56

2.2 Local density of states at an arbitrary site of a 1D chain, described by a

tight-bindingHamiltonian...... 65

3.1 Resistivity at diﬀerent values of EF , normalized to the semiclassical Boltz-

mann resistivity ρB calculatedintheBornapproximation...... 72

3.2 The density of states of the clean and dirty metal and the clean metal Boltz-

mannresistivity...... 76

2 3.3 Linear ﬁt R = C1 + ρ/A L,(A = 225 a ) for the disorder averaged resistance

R inthebandcenteranddiﬀerentdisorderstrengths...... 81

3.4 The conductance ﬂuctuations from weak to strong scattering regime in the

disorderedcubicsamples...... 83

3.5 The conductance and resistance ﬂuctuations, at EF = 0, from weak to strong

scatteringregimeindisorderedsamplesofdiﬀerentgeometry...... 85

3.6 The deviation between disorder averaged resistance and inverse of disordered

average conductance, evaluated at EF = 0, as a function of disordered strength

intheAndersonmodelonacubiclattice...... 86

4.1 The diﬀusivity of a disordered binary alloy modeled by the tight-binding

Hamiltonian...... 97

4.2 The diﬀusivity of the diagonally disordered Anderson model...... 98

xi 4.3 The diﬀusivity of a metal junction composed of two disordered binary alloys,

modeledwiththeTBHonalattice...... 100

4.4 Local density of states integrated over the y and z coordinates for the metal

junctioncomposedoftwodisorderedbinaryalloys...... 101

4.5 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,

modeledwithTBH,fordiﬀerentattachedleads...... 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 =6andW = 3) and quantum point contact conductance

ofacleansampleonthesamelattice...... 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

couplingparameters...... 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

andcouplingparameters...... 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

andcouplingparameters...... 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

rarityofthe“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

andDDAndersonmodelsonthecubiclattice...... 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. ω2 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 encour-

agement 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

1Transcript 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 collab- orating with Sergei Maslov from BNL Condensed Matter Theory Group. Thanks also goes to professor Jainendra Jain for his generous contribution to the Condensed Matter The- ory 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 Verg´es, and Dr. Igor E. Smol- yarenko 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. Cirkovi´´ c (whom I followed to Stony Brook), a collaborator on disordered projects Viktor Cerovski, frequent visitor to Long Island Robert Lakatoˇs,andanincisive critic Dario Cupi´ˇ c.

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. Nikoli´c 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. Nikoli´c, and M.

Strongin, Infrared studies of the onset of conductivity in ultrathin Pb ﬁlms,Physical

Review Letters 83, 4880 (1999).

[3] B. K. Nikoli´c 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. Nikoli´c, Statistical properties of eigenstates in three-dimensional mesoscopic sys-

tems with oﬀ-diagonal or diagonal disorder, cond-mat/0003057 (2000), submitted for

publication in Physical Review B.

[5] B. K. Nikoli´c and Philip B. Allen, Resistivity of a metal between the Boltzmann transport

regime and the Anderson transition, cond-mat/0005389 (2000), accepted for publica-

tion in Physical Review B Rapid Communications.

[6] B. K. Nikoli´c and P. B. Allen, Quantum transport in dirty metallic junctions and multi-

layers, unpublished.

xviii List of Symbols and Abbreviations

a lattice constant

A Area

Aˆ 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

2 ETh Thouless energy (=h ¯D/L ) 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

2 GQ =2e /h conductance quantum g dimensionless conductance (= G/GQ)

Gˆr retarded Green operator

Gˆa advanced Green operator

G>, G< non-equilibrium Green functions for particle distribution properties

xix h Planck constant hh/¯ 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 = NNyNz)

Oˆ linear operator (or matrix)

R resistance

Re Real part of a complex number

SS-matrix t time or hopping parameter in the Anderson model t transmission matrix

T temperature

Tn transmission eigenvalue

TT-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

Γˆ lead-sample coupling operator [= i(Σˆ r − Σˆ a)]

δ(x) delta function

δ¯(x) broadened delta function (Box, Lorentzian, etc.)

∆ single-particle level spacing

µ chemical potential

λF Fermi wavelength

Ωvolume

Σˆ r retarded self-energy

Σˆ a advanced self-energy

ρ resistivity

ρB semiclassical resistivity in Born approximation

ρT semiclassical resistivity in T-matrix approximation

ρˆ 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) nonlocal conductivity tensor ¯ τ transport mean free time

2 τD classical diﬀusion time ( L /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 Poincar´e

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

1Below two dimensions and for arbitrary weakdisorder 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 non- interacting 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

2In 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]. 3Aside 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. 4The 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

2 enough wires whose conductance is of the order of one conductance quantum GQ =2e /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 func-

tions 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 lev-

els in the quantum systems whose classical analogs are chaotic (the chaos should be “hard”, as in

either K or ergodic systems [10]). 5It 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

2 spacing ∆ and the Thouless energy ETh =¯hD/L =¯h/τD,whereD 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

6For 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 (localiza-

tion) 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:

7In the open systems (surrounded by an inﬁnitely conducting medium) the single-particle states

−1 are smeared; the magnitude of “smearing” is of the order ∼ min {ETh, ¯hτesc },whereτesc is a characteristic time for the electron escape through the attached leads. 8More 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

10 (∆ ETh), and ETh h/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

−1 −1 free path L , or equivalently τ τf . Here the time of ﬂight τf = L/vF deﬁnes

ETh =¯h/τf in the ballistic system. Using energy scales one can further discern [11]

between the “ballistic” (∆ h/τ¯ h/τ¯ f , where the disorder is strong enough to

thoroughly mix many energy levels) and the “nearly clean” regime (¯h/τ ∆, h/τ¯ f ,

9The 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. 10This is emphasized by using the phrase “diﬀusive metallic” [16], where “metallic” implies weak disorder, as quantiﬁed by g 1orkF 1. 11A 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

2 scale of the potential arp, between the quantum disorder regime arp <λF and quantum

12 2 chaos arp >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 |lnh ¯|/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,

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 fre-

13Universality 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→theappliedfieldandmotionof 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

14Another 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

15The 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

2 2 fraction of trajectories which are aﬀected by the shift, L / Nimp,whereNimp is the total number

2 2 of impurities in the sample. Changing the position of Nimp /L 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 quantum- mechanical treatment of transport is necessary. The ﬁnite-size sample conductances g corresponding to this na¨ıve 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

16In 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 ap-

17The 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

Diﬀusive Transport Regime 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 trans-

port 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

1Before 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 −p disordered conductors Lφ = Dτφ is expressed through the phase-relaxation time τφ ∝ T

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/h¯ ∼ 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 electron- electron 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 kBTτφ h¯ [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). 2The 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 kBT thermalizes with all other

−3/2 electrons). The times τφ and τe−e coincide in 3D samples [16], τφ ∼ τe−e ∝ T . 22 with T → 0 because of the decreasing space of states available for the inelastic scattering. Another length scale, the thermal diffusion length LT = hD/k¯ BT, 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 kBT 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 quantum- mechanical 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. There-

fore, 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 intro-

ducing 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 real-

space 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

Andtheendofallourexploring

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 = GV, (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 kBT . 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

4For 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)onthescaleLφ, but there is also classical nonlocality [56] enforced ¯ by current conservation (cf. Sec. 2.3) which extends throughout the entire sample, irrespec-

tive 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 spe- ciﬁc 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

5It 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 operatorρ ˆ j(r)=Tr ρˆˆj(r) , (2.4) or in general non-equilibrium situation, where kinetic properties are embodied in the double- time correlation function G< (cf. Sec. 2.4.3), 1 eh¯ ie2 j(r)= dE (∇ −∇)G<(r, r; E)+ A(r)G<(r, r; E) , (2.5) 2π 2m m 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

e e ˆj(r,t)= [ˆn(r)vˆ + vˆnˆ(r)] = [ˆn(r)(pˆ − eA(r,t)) + (pˆ − eA(r,t))ˆn(r)] 2 2m

= ˆj0(r,t)+ˆjd(r,t), (2.6) e ˆj0(r,t)= [ˆn(r)pˆ + pˆnˆ(r)], (2.7) 2m nˆ(r)e2 ˆjd(r,t)=− A(r,t), (2.8) m wheren ˆ(r)=|rr| is the particle density operator, vˆ = pˆ/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 = drnS(r) · j(r)= dr dr σ(r, r ) · E(r), (2.13) ¯ S S Ω and the conductance from (2.1). This formula assumes that the electric field E(r)isthelo- cal 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 1 1 G = drE(r) · j(r)= dr dr E(r) · σ(r, r) · E(r). (2.14) V 2 V 2 ¯ Ω Ω It the ﬁeld E(r) is taken to be homogeneous (E = V/L) we obtain the volume averaged conductance tensor 1 G = dr dr σ(r, r). (2.15) L2 ¯ Ω 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 = drj(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]. 7In 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 = − dS1 · σ(r, r ) · dS2, (2.19) ¯ S1 S2 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 =0andµR = eV .Itis important to point out that this formula can be generalized [71] to arbitrary multi-probe

8The 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)andofσ(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 self- consistent 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

2 by τD L/vF in the ballistic regime (L<)orτD L /D in the diﬀusive ( L ξ) 32 regime. For ξ

Then the expression for NLCT does not obey the current conservation conditions (2.17) and (2.18) because of missing higher order corrections inh ¯. 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 formulah ¯k˙ = eE where k is the

9For example, Büttiker [78] has emphasized that gauge-invariant description of nonlinear transport 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 dF + k˙ · + r˙ · = , (2.20) ∂t ∂k ∂r dt 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

10Like 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)deviationδF(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

3 σD(r − r )i(r − r )j |r − r | σCh(r, r) = exp(− ), (2.21) ¯ij 4π |r − r|4 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 (r, r ) = σD[δijδ¯(r − r ) −∇i∇ d(r, r )], (2.22) ¯ij j was emphasized in the study of UCF for complicated geometry of the sample and multiprobe measurements [69]. Here δ¯(r − r) is a sharply peaked function of the width ,whichcan be virtually taken as the Dirac δ function, and stems from σCh(r, r) (2.21). The rescaled ¯ij diﬀusion propagator12 d(r, r), satisﬁes the equation

−∇2d(r, r)=δ(r − r), (2.23)

subject to the boundary conditions d(r, r ) = 0 on a conducting boundary and ∇nd(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). 12The diffusion propagator (or “diffuson”) is the solution of equation −Dτ∇2D(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

13 cl “classical” ﬁeld [56], ∇αEα = 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 GD = = σD , (2.24) h 3π L L ne2τ σD = , (2.25) m

2 where M = kF 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

−1 surrounding medium. This means that the parameter (kF ) 1 should be small (as well as the similar parameter (∆Eτ)−1 1where∆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 energyh/τ ¯ , i.e., (∂ε(k)/∂k)∆k ∼ h/τ¯ .

Fully quantum-mechanical theories, like Kubo linear response theory (cf. Sec. 2.4.1) or non- equilibrium Green function formalism (cf. Sec. 2.4.3), produce Boltzmann theory as a lowest

13This 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 un- polluted 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 Quantum transport formalisms

2.4.1 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 me-

14The 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 lackof 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 non- equilibrium 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 operatorρ ˆ(t) (or density matrixρ ˆ(k, k,t)=k|ρˆ(t)|k when some representation is chosen).

This is obviously a generalization of the distribution function F (k,t)intheBoltzmann theory (cf. Sec. 2.3) which includes phase-relationship between diﬀerent states (oﬀ-diagonal elements ofρ ˆ(k, k,t)), besides the occupation of the states (given by the diagonal elements 38 of the density matrix). In thermodynamic equilibrium

ˆ e−H0/kB T ρˆ0 = , (2.26) Z where Hˆ0 is the Hamiltonian of the unperturbed system and Z =Trexp(−Hˆ0/kBT )isthe partition function (in grand canonical ensemble Hˆ0 should be replaced by Hˆ0 − µNˆ). When a fixed external electric field (in a gauge15 with only vector potential A beingnon-zeroand with small η to turn the field off at t →−∞)

∂A E(r,t)=E(r)e−i(ω+iη)t = − , (2.27) ∂t is imposed the evolution of the statistical operator is generated by the perturbed Hamiltonian16

Hˆ = Hˆ0 + Hˆ , ∂ρˆ ih¯ =[H,ˆ ρˆ(t)]. (2.28) ∂t 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 Hˆ0 should include only EEI in the equilibrium (e.g., scattering

15From Maxwell equations one can get curl E = 0 to ﬁrst order in small ω and |E|,sothatﬁeld can be treated as conservative E(r)=−∇Φ(r) [87] in the limit relevant for DC transport. 16For example, in the non-interacting system, which are mostly considered in the thesis, each

2 particle is described by the Hamiltonian Hˆ0 = pˆ /2m + U(r) in a random potential U(r). When electric ﬁeld is turned on the relevant Hamiltonian is Hˆ =(pˆ − eA)2/2m + U(r), where Hˆ

Hˆ0 − (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 Hˆ is cut on the ﬁrst order (linear in ﬁeld E), so that system in this

approximation is described by the statistical operator

2 ρˆ(t)=ˆρ0 + δρˆ(t)+O(E ). (2.29)

Thetimeevolutionofˆρ(t) deﬁnes the time evolution of the ﬁrst order correction δρˆ(t)tothe

statistical operator

∂ ih¯ δρˆ =[Hˆ0,δρˆ]+[Hˆ , ρˆ0]. (2.30) ∂t

Using the solution for δρˆ the expectation value of the current density is obtained (to ﬁrst order in E) j(r,t)=Tr ρˆ0ˆj0(r,t)+ˆρ0ˆjd(r,t)+δρˆˆj0(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 rewrit- ing (2.31) in the form of local Ohm’s law (2.3)

2 ∞ ie Tr (ρ ˆ0nˆ(r)) 1 iωt σ(r, r ; ω)= δ(r − r )+ dt e Tr ρˆ0 [ˆj0(r,t),ˆj0(r ,t)] , (2.32) ¯ mω hω¯ 0 where the ﬁrst (“diamagnetic”) term is generated by Tr (ˆρ0ˆjd) from Eq. (2.31). In the DC limit ω → 0 (but ωt ﬁnite), which is usually taken before the limit T → 0, diamagnetic term

17Magnetic 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) many- body 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)=−ih¯ lim . (2.33) η→0+ ¯ α,β Eβ − Eα Eβ − Eα + ihη¯

Here Pβ =[ˆρ]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)forsingle 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 de- terminants of single particle states. The expectation value of any single particle operator is given as a trace O =Tr(ˆρ(t)Oˆ) with (now) single particle statistical operatorρ ˆ(t). In equilibrium this operator is given by

ρˆ0 = 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

Hˆ0|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

18This 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 = −ihπδ¯ (Eα − Eα)+P ( ). (2.36) Eα − Eα + ihη¯ Eα − Eα

The delta function here generates the term in NLCT which depends only on the states within kBT 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) 3 ∞ 2 ↔ ↔ e h¯ π ∂f ∗ ∗ − − σ(r, r )= 2 dE [Ψα (r) ∇ Ψα(r)] [Ψα(r ) ∇ Ψα (r )] ¯ 4m −∞ ∂E α,α

×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), (2.38) ∂r ∂r ↔ 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)

2 2πhe¯ 2 σxx = |α|vˆx|α | δ(Eα − EF )δ(Eα − EF ), (2.39) Ω α,α where −∂f/∂E δ(EF − E) at low temperatures. The velocity operator is deﬁned by the commutator dˆr ih¯vˆ = ih¯ =[ˆr, Hˆ0], (2.40) dt 19It 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

d involving Hˆ0. Here the thermodynamic limit Ω = L →∞should be assumed, therefore generating continuous spectrum and conductivity as a continuous function of Fermi energy

EF . We emphasize that Hˆ0 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

20 geometry. Although we started from the (continuous) coordinate representation, 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

eh¯ r |j(ˆr) |α = [δ(r − r)∇ + ∇ δ(r − r)]Ψα(r ), (2.41) 0 2im so that its matrix elements, which appear in the evaluation of the thermal averages, are eh¯ eh¯ ∗ α |j0(r)|α = = dr α |r r |ˆj0(r)|α = Ψ ∇Ψα(r). (2.42) 2im 2im α

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 Hˆ , while having in mind the “equilibrium” Hamiltonian Hˆ0 of this section)

Gˆr,a =(E − Hˆ ± 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, respec-

tively. This deﬁnes the Green operator close to the branch cut (i.e., continuous spectrum of

20The coupling of the vector potential is unambiguously deﬁned only in the coordinate representation. 43

Hˆ associated with extended states [33]) on the real axis. If exact eigenstates (2.35) of the

Hamiltonian Hˆ are known, the Green operator can be expressed in the form

|αα| Gˆr,a = . (2.44) α E − Eα ± iη

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 r,a in the coordinate representation GE (r, r )=r|Gˆ (E)|r gives response at r for the unit

(delta function) excitation at r . It replaces the following expression involving wave functions

∗ 1 r a 1 Ψα(r)Ψα(r )δ(E − Eα)=− [GE(r, r ) − GE(r, r )] = − Im GE(r, r ). (2.45) α 2πi π

Therefore, the Green function expression for NLCT, 3 ∞ 2 ↔ ↔ e h¯ ∂f σ(r, r )= dE − Im GE(r, r ) ∇∇ Im GE(r , r), (2.46) ¯ 2 4πm −∞ ∂E 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)

4e2 1 Gxx = Tr h¯vˆxIm Gˆ h¯vˆxIm Gˆ , (2.47) h L2 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 2 g(r, r) ∇∇ h(r, r)=r|gˆ|r ∇ ∇ r|hˆ|r = − Tr ˆj(r)ˆgˆj(r)hˆ , (2.48) eh¯ dr Tr ˆj(r)ˆg = e Tr (vˆgˆ) , (2.49)

valid for arbitrary operatorsg ˆ and hˆ. This allows us to replace the integration over the

volume with trace over the velocity operator. These identities are easily proven by inserting

21In discrete representations operators act as matrices. We simplify notation by using “hat” (Oˆ) 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 talkabout 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 ˆj(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], B¨uttiker et al. [94]) as well as extended to multiprobe conductance measurement

by B¨uttiker [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 quanti-

zation 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 ±ikx Ψn (r)=φn (y,z)e . (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).

2 2 2 For example, in the case of parabolic subbands kF = kn + k , 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                                  O   I   rt  I            = S   =   ·   . (2.51)                         O I tr I

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].

r,a While the one-particle Green function GE (r, r ), introduced in Sec. 2.4.1, connects the response at any point r with excitation at point r,theS-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 † I¯ = dE [fL(E) − fR(E)] Tr t(E)t (E), (2.52) h 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

∞ I¯ 2e2 ∂f G = lim = dE − Tr t(E)t†(E). (2.53) V →0 V h ∂E 0

At zero temperature this simpliﬁes to the two-probe Landauer formula for conductance

2 M M M 2e † 2 G = Tr t(EF )t (EF )=GQ |tmm (EF )| = GQ Tn(EF ), (2.54) h m=1 m=1 n=1 where all quantities are computed at the Fermi energy EF .HereTn(EF ) are the transmission eigenvalues (or transmission coeﬃcients23), i.e., the eigenvalues of tt†.Thus,theknowledge 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

2 quantum is GQ = e /h, but the number of transmission eigenvalues is doubled [30].

It is insightful to demonstrate the diﬀerence between the quantum-mechanical transmis-

2 FP 2 2 sion probability |tmm | = | FP Zmm | and its semiclassical approximation (|tmm | )SCA =

22We denote explicitly the time-averaging of the steady state current I¯ taking into account the intrinsic ﬂuctuations (shot noise) present in mesoscopic transport [30]. 23The 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 2 FP |Zmm | in the framework of scattering approach. Here we use the picture of Feynman

FP paths (labeled by FP) which are characterized by the complex amplitude Zmm. Each path 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

M conductance or any so-called linear statistics A = n=1 a(Tn) of the transmission eigenvalues

1 M A = a(Tn) = dT a(T )ρ(T ). (2.55) n=1 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 na¨ıve 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 G 1 M ρ(T )= √ , cosh−2

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 )andisgivenbyP ∼ 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 Equi- librium 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 equiv-

alently 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 matrixρ ˆ(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 Non-

Equilibrium Green function formalism (NEGF).24 The central quantity of NEGF is double- time correlation function25

< † G (r1,t1; r2,t2)=iΨˆ (r2,t2)Ψ(ˆ r1,t1), (2.58) where Ψ(ˆ 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)=−iΨ(ˆ r1,t1)Ψˆ (r2,t2). (2.59)

Using the sum and diﬀerence coordinates

r = r1 − r2, (2.60) 1 R = (r1 + r2), (2.61) 2 and times,

t = t1 − t2, (2.62) 1 T = (t1 + t2), (2.63) 2 24This 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.

25We assume here ¯h = 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.,

r † G (r1,t1; r2,t2)=−iθ(t1 − t2){Ψ(ˆ r1,t1), Ψˆ (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].

26One can also take the Fourier transform of ρ(r, R,T)overr, 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. 27In general interacting system these functions are not “true” Green functions in a strict mathematical 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 tight-

binding 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 = dE Tr {[ΓˆL(E) − ΓˆR(E)]Gˆ (E) h r a +[fL(E)ΓˆL(E) − fR(E)ΓˆR(E)] (Gˆ (E) − Gˆ (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

† † † Hˆ = Hˆint({dn}, {dn})+ εkckck + (Vknckdn +H.c.), (2.69) L=L,R k∈L L=L,R k∈L n

† † where {dn} creates a complete set of single-particle states in the sample, ck∈L creates an

† electron in state k of a lead L, and Hˆint is a polynomial in {dn}, {dn} which commutes

† with the electron number Nˆ = n dndn. Our subsequent studies will be conﬁned to non- interacting systems described by the Anderson model (explained in detail in the next section)

† † Hˆnint = εmcmcm + tmncmcn, (2.70) m m,n

† where cm 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”

28Caroli 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 tech-

nique, thus bypassing various problems in the eﬀective tunneling Hamiltonian approach. 53

L ∗ function Γnm(E)=2π k∈L VknVkm(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 Gnm(E) is a Fourier transform of Gnm(t)=

† r r † idm(0)dn(t), while Gnm(E) is a Fourier transform of Gnm(t)=−iθ(t){dn(t),dm(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 Gˆ< (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 Gˆ<

< r a r a Gˆ = ifL(E)Gˆ ΓˆLGˆ + ifRGˆ ΓˆRGˆ , (2.71)

r a r a where Gˆ − Gˆ = iGˆ (ΓˆL + ΓˆR)Gˆ . This leads to the following expression for current

2e a r I = dE [fL(E) − fR(E)] Tr Gˆ ΓˆR Gˆ ΓˆL . (2.72) h

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))

2 2e r a G = Tr ΓˆL Gˆ ΓˆR Gˆ . (2.73) h

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

29In 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 conduc-

tance 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 rep-

resentation (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 rele-

vant 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)

Hˆ = εm|mm| + tmn|mn|. (2.74) m 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,orboth,tobe 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

Hˆ = −h¯2∇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 HˆL,withεm =0andtmn = 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, HˆC, 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 char-

31Our 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 SAMPLE LEAD

µ µ L R t t t C L

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,

Gˆr,a(m, n)=(E − Hˆ (m, n) ± iη)−1, (2.75) would lead into a trouble since Hˆ (m, n) is an inﬁnite matrix (Hˆ = HˆS + HˆL + HˆC). The site representation of the Green operator Gˆr,a is a Green function matrix

Gˆr,a(m, n)=m|Gˆr,a|n, (2.76) and the matrix of Hamiltonian in this representation is a band diagonal matrix [104] of the bandwidth 2NyNz +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 ﬁnite- size 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]

   −1          ˆr ˆr   − ˆ ˆ   GL GL−S   E + iη HL HC  r     Gˆ =   =   , (2.77)          r r   †  GˆS−L GˆS HˆC E + iη + HˆS 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.

r The diagonal blocks are: inﬁnite matrix GˆL, connecting the sites in the left lead; and ﬁnite

r GˆS connecting the states on the lattice sites inside the conductor. The oﬀ-diagonal blocks,

r r GˆL−S and GˆS−L, connect the states in the lead and the sample. The matrix of the coupling

Hamiltonian HˆC(mL, mS)=tC is non-zero only for the adjacent sites in the lead mL and

r r the sample mS. The set of matrix equations for GˆS follows from Hˆ · Gˆ = Iˆ

r r [E + iη − HˆL] · GˆL−S + HˆC · GˆS =0, (2.78)

† r r HˆC · GˆL−S +[E + iη − HˆS] · GˆS = I.ˆ (2.79)

The Equation (2.78) can be solved for GˆL−S

r r r GˆL−S = −gˆL · HˆC · GˆS, (2.80)

r −1 gˆL =(E + iη − HˆL) , (2.81)

32To 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

r where we recognizeg ˆL 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

r GˆL−S (2.80) in Eq. (2.79) we get

r † r −1 GˆS =(E − HˆS − HˆC · gˆL · HˆC) . (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

r 2 r Σˆ (mS, nS)=tCgˆL(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 self-

r energy Σˆ (mS, nS) is non-zero only between the sites (mS, nS)ontheedgelayerofthe 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 HˆC. 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

r r −1 GˆS =(E − HˆS − Σˆ ) , (2.84)

r r r a r † where Σˆ = Σˆ L + Σˆ R. Advanced functions are obtained in a standard way: Gˆ =[Gˆ ] ,and

a r † r Σˆ =[Σˆ ] . In the following Section we give a derivation of a Green functiong ˆL(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

33Analogous 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,

r eff [HˆS + Σˆ ]|αeff = Eα |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,

eﬀ ζα E = Eα − ζ − i , (2.86) α α 2 then its imaginary part ζα gives the “lifetime” of an electron in state α before escaping

eﬀ 2 into the leads. The probability to stay in the state |α decays as | exp(−iEα t/h¯)| = exp(−2ζαt/h¯), and the escape time into the leads is τesc =¯h/2ζα. The “loss” of electrons into the leads is also illustrated by the following identity [43] 1 ∇·j(r)= dr dr Ψ∗(r)Γ(r, r)Ψ(r), (2.87) h¯ where Γ=ˆ −2ImΣ=ˆ i(Σˆ r − Σˆ a), and the evolution of wave functions Ψ(r) is determined

r by the eﬀective Hamiltonian HˆS + Σˆ .

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 Gˆr (m, m; E). (2.88) π S 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

2 ρ(r,E)= |Ψα(r)| δ(E − Eα), (2.89) α deﬁned in terms of exact eigenstates of a closed system [5]. 61 2.5.3 The Green function for an isolated semi-inﬁnite ideal lead

r In the previous Section we learned that the Green function matrix GˆS(m, n) (2.84) at a

r continuous energy E can be computed numerically by inverting the ﬁnite matrix E−HˆS −Σˆ .

r This requires to know the matrix elementsg ˆ (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) N 2 2 y Nz |ky,kz = sin(kynya)sin(kznza) |ny,nz, (2.90) Ny +1 Nz +1 ny=1 nz=1 where |ny,nz denotes the orbitals of the arbitrary 2D layer. We choose a hard wall boundary

conditions iny ˆ andz ˆ 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(kya)+cos(kza)], (2.91) deﬁned by discrete ky(i)=iπ/(Ny +1)a,andkz(j)=jπ/(Nz +1)a.Herei runs from 1 to

Ny and j runs from 1 to Nz. The longitudinal eigenstates |kx (i.e., on the 1D chains) are 2 nx|kx = sin(kxnxa), (2.92) Ninf with eigenvalues ε(kx)=2tL cos(kxa). This states vanish at the open end on which nx =0.

r The Green functiong ˆ (mB, nB) can be expanded in terms of the exact eigenstates |k,

r mB|kk|nB mB|gˆ |nB = E − 2tL cos(kxa) − ε(ky,kz)+iη k = my,mz|ky,kzky,kz|ny,nz ky,kz 2 2 sin kxa × , (2.93) inf − y z − L x N kx E ε(k ,k )+iη 2t cos(k a) 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 can be replaced by the integral

2 2 sin kxa J(ky,kz)= inf − y z − L x N kx E ε(k ,k )+iη 2t cos(k a) π/a a 2 − e2ikxa − e−2ikxa = dkx , (2.94) 4πtL (EJ + iη)/2tL − cos(kxa) 0 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

1 1 − w2 J(ky,kz)=− . (2.95) 4iπt |w|=1 w2/2+1/2 − Yw

Here Y denotes the expression Y =(EJ + iη)/2tL. The poles of the integrand are at √ 2 w1,2 = Y ∓ Y − 1 and have the following properties: (a) w1w2 = 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 − 2 − 1 1 w 1 − 2 − 2 J(ky,kz)= Res = 2 EJ i 4tL EJ . (2.96) L − 1 − 2 t (w w )(w w ) w=w1 2tL

If |Y | > 1, then 1 − 2 − 2 J(ky,kz)= 2 EJ sgn EJ EJ 4tL , (2.97) 2tL 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)

r 2 2 Σˆ L(m, n)= sin(kymya)sin(kzmza) y z N +1N +1ky,kz 2 × tC − 2 − 2 2 EJ i 4tL EJ sin(kynya)sin(kznza), (2.98) 2tL 63 for |EJ | < 2tL,and

r 2 2 Σˆ L(m, n)= sin(kymya)sin(kzmza) y z N +1N +1ky,kz 2 × tC − 2 − 2 2 EJ sgn EJ EJ 4tL sin(kynya)sin(kznza), (2.99) 2tL 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) √ r 2 2 −1 Gin(E)=[E − ε − (E − i 4t − E )] , (2.100) for |E| < 2t. This gives the local density of states (2.88), which is independent of the lattice site, inside the band √ 2 2 1 r 1 4t − E ρin(E)=− Im G (E)= . (2.101) π in π ε2 +4t2 − E2

For energies outside the band, e.g., E>0 > 2t the Green function is √ r 2 2 −1 Gout(E)=[E − ε − (E − E − 4t )+iη] , (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

+ √ 1 r 1 η η→0 2 2 ρout(E)=− Im Gout(E)= √ → δ(−ε + E − 4t ), (2.103) π π ( E2 − 4t2 − ε)2 + η2 64 where delta function properties lead to the following simpliﬁcation 2 2 √ Ep − 4t 2 2 δ(−ε + E − 4t )= δ(E − Ep). (2.104) Ep

Thus, the delta function singularity in LDOS appears outside the band of a 1D chain. This √ 2 2 is signaling the appearance of a bound state at the energy Ep =sgnε ε +4t . 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

2 2 2e r a 2e † G = Tr ΓˆL Gˆ1N ΓˆR GˆN1 = Tr (tt ), (2.105) h h r t = ΓˆL Gˆ1N ΓˆR. (2.106)

r a Here Gˆ1N ,andGˆN1 are matrices whose elements are the Green functions connecting the

34 layer 1 and N of the sample. Thus, only the block NyNz × NyNz of the whole matrix

Gˆr(n, m) (2.84) is needed to compute the conductance. The Hermitian operator

r a ΓˆL = i(Σˆ L − Σˆ L)=−2ImΣˆ L > 0, (2.107)

34We 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

1.0 (a) (c) 0.8

0.6

0.4

0.2 (b) LDOS (at arbitrary site) 0.0 -4 -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, Aˆ = i(Gˆr −Gˆa). Therefore, it is related to the imaginary part of the self-energy Σˆ L introduced by the left lead. The operator ΓˆL “measures” the coupling of an open sample to the left lead (ΓˆR 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 ΓˆL, like the self-energy Σˆ L, 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 ΓˆL means that its square root is well deﬁned

1/2 ΓˆL = γn Pˆn. (2.108) n

Here the operator Pˆn 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 non-

r a Hermitian matrix ΓˆL Gˆ1N ΓˆR GˆN1. 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 Gˆr 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 isgiveninterms

35After disorder averaging the symmetries of the systems will be restored and all diagonal components 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 2 4e 1 ˆ ˆ G = 2 Tr h¯vˆxIm G h¯vˆxIm G . (2.109) h Lx

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) i m|vˆx|n = tmn (mx − nx) . (2.110) h¯

The length of rectangular sample in thex ˆ direction is denoted by Lx = Na. The use of the formula (2.109), together with the Green function Gˆr,a =(E − Hˆ ± 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 2NyNz + 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

2 matrix. In the Landauer-type formula (2.105) we need only (NzNy) elements of the whole

Green function (2.84). They can be obtained from the LU decomposition36 of the band diagonal matrix E − Hˆ − Σˆ r by a forward-backward substitution [104]. The trace operation in formula (2.105) is also performed only over matrices of size NyNz × NyNz. 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

1 1 G = drE(r) · j(r)= dr dr E(r) · σ(r, r) · E(r). (2.111) V 2 V 2 ¯ Ω Ω

Here we have the freedom to choose any electric ﬁeld factors E(r)andE(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

36The most advanced numerical linear algebra routines are provided by the LAPACK package

(available at http://www.netlib.org). 37 r In the recursive Green function method the self-energy from the left lead Σˆ L is iterated through the sample, using the appropriate matrix Dyson equation [105], and ﬁnally matched with the self-

r energy coming from the right lead Σˆ R. In this procedure matrices of dimension NyNz are inverted

N times. 38The 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

a r vˆx Gˆ vˆx Gˆ (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 operatorv ˆx (2.110) has non-zero matrix element only between two adjacent layers, the minimal extension of the ﬁeld is two layers in thex ˆ direction. The layers are arbitrary (can be cho-

sen 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

2 cross section. Thus, one needs to ﬁnd 4(NzNy) elements of the Green function (2.84) and trace over the matrices of size 2NzNy × 2NzNy. 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 con-

ceptual 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.

39In 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

Hˆ = εm|mm| + t |mn|, (3.1) m 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 εmεm =

2 δmm W /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. 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 ∼ 4and ≈ 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 = vkk/3. The diffusivity

−2 Dk diminishes as (W/t) in Boltzmann theory. As /a approaches a minimum value (∼ 1),

2 Dα decreases toward Dmin = ta /h¯, 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 kBT ,andscales smoothly with W/t.

The traditional tool for computation of ρ has been the Kubo formula [85] (cf. Sec. 2.4.1), 72

102 E =0 1 F 10 T 100 B 10-1 Mean Free Path (a) 2 EF=2.4t B

ρ ρ ρ / 1 T/ B ρ

0 2 EF=0

B ρ ρ ρ / / 1 T B ρ

0 0 20406080100120 Disorder Strength (W/t)2

Figure 3.1: Resistivity ρ at EF = 0 (lower panel) and EF =2.4t (middle panel), from a

2 sample of cross section A = 225 a , 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 2 1 e ∂f 2 σ = = − Dα = e N(EF )D,¯ (3.2) ρ Ω α ∂Eα whereΩisthesamplevolume,N(EF ) the density of states (DOS) at EF , D¯ the mean diﬀusivity, and state diﬀusivity is given by

2 Dα = πh¯ |α|vˆx|α | δ(Eα − Eα ). (3.3) α Here vˆ 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-Bu¨ttiker 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., eﬀec- tively 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.AtEF =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 dFk −eE · vk = , (3.4) ∂k dt scatt serves as a reference theory. Here k is the energy band for W =0,namelyk =2t i cos kia, hv¯ ki is ∂k/∂ki,andFk is the non-equilibrium distribution function. The collision integral is dF 2π 2 = − |Ukk | (Fk − Fk )δ(k − k ). (3.5) dt scatt h¯ k

2 The mean squared matrix element of the random potential |Ukk | , in Born approximation,

2 2 is εm = W /12, where 1 W (...)= dεm (...)P (εm)= dεm (...) θ( −|εm|), (3.6) W 2 denotes average over the probability distribution of the random variable εm.TheBoltzmann equation assumes that quasiparticles propagate with mean free path a between isolated collision events. The equation is exactly solvable, yielding (for kBT t) 1 n = e2τ , (3.7) ρB m eﬀ

2 with (n/m)eﬀ = vkxδ(k − 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 ),

2 h¯ 2 W =2π |k|U|k | δ(k − k )=2π N(EF ). (3.8) τ k 12

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 |k = √ eik·m|m, (3.9) Ns m 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

πha¯ W 2 1 ρB = , (3.10) 2 2 e 4t vk E=EF where the velocity squared, 2 2 2 1 ∂k 2ta 2 2 2 v = = [sin kxa +sin kya +sin kza], (3.11) k h¯ ∂k h¯

2 is averaged over the Fermi surface, vk 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 =3A,˚ ρB is 125 µΩcm, typical of dirty transition metal alloys, and close to the largest resistivity normally seen in

2 dirty “good” metals. Figure 3.1 plots ρ/ρB versus (W/t) .EvenforW =10t there is less than a factor of 2 deviation from the (unwarranted) extrapolation of the Boltzmann theory into the regime W/t > 1.

3 3 If Born criterion, pF V (p)=0 h¯ 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

1The perturbative quantum analysis, based on the selection of some class of diagrams in expansion 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]). 76 2.0 2

1.5 )(W/4t) 2 1.0

0.5 (ha/2e B ρ 0.0 0.3 (a)

0.2 (b) (c) 0.1 (d) Density of States 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

r 1 G (k,E)= r . (3.12) E − k − Σ (E,k) 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, h¯ − =ImΣr(k,E), (3.13) 2τ(k,E) 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 ofh/ ¯ .TheT-matrix is given implicitly in terms of the following inhomogeneous integral equation (in operator form) [33]

Tr = U + UGˆrTr. (3.14)

This equation contains the impurity-averaged single particle Green functions (“dressed prop- agators”) introduced by Eq. (3.12). This reﬂects the presence of other impurities (instead

2This would be the most comprehensive semiclassical approach (also called single-site Coherent

Potential Approximation [115]). It is accomplished in the frameworkof diagrammatic impurity- averaged 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. 3Impurity-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)

1 1 G0(m, m; z)= , (3.15) Ns k z − k

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 εm Tmm(z)= . (3.16) [1 − εmG0(m, m; z)] To next order the mean square T-matrix is

W/ 2 2 1 εm |Tmm(z)| = dεm = W (1 − εmG0(m,E)) −W/2 2 2 W 3W ∗ ∗ ∗ = 1+ (G0G + G0G0 + G G )+... , (3.17) 12 20t2 0 0 0 where the ﬁrst term is the Born approximation and the coeﬃcient of the correction (∼

4 O(W )) changes sign from negative to positive as EF moves from 0 to 2.4t.Asshownon

2 Fig. 3.1, the resistivity does not behave like |Tmm(z)| ; 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 re-

sistivity 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)

2 where M ∼ kF 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

4To 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 A˚ has nearly M =106 channels [43], the mean free path of only 10 A˚ still generates ξ ≈ 1 mm. Thus, ξ is much bigger than typical Lφ,evenat 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 they ˆ and zˆ 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]

2 2 2 NyNz 4e r a e † e G = Tr Im Σˆ L Gˆ Im Σˆ R Gˆ = Tr (tt )= Tn, (3.19) πh¯ 1N N1 πh¯ πh¯ n=1 r t =2−Im Σˆ L Gˆ1N −Im Σˆ R, (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)isgivenby

T = 0/(0 +L)[43],with0 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 W=5 PL(R) 0.7 L=15a

) W=11 2 0.6 W=10 0.5 W=8 0.01 0.1 0.4 R (h/2e2) W=9

0.3 W=8 W=7 0.2 W=6 W=5

Resistance (h/2e W=4 0.1 W=3 W=2 0.0 0 2 4 6 8 10 12 14 16 18 20 22 24 Length of the sample L (a)

2 Figure 3.3: Linear ﬁt R = C1 +ρ/A L,(A = 225 a ) 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 πh/¯ 147e2)and becomes negative for around W>7t. The inset shows examples of the distribution of resistances PL(R)(forL =15a)versuslogR. 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/πh¯) MT . (3.21)

Thus, for not too strong scattering, conductance should have the form

−1 L G = RC + ρ . (3.22) A

It describes the (classical) series addition of two resistors. The “contact” resistance [122]

2 RC = πh/e¯ M is non-zero, even in the case of ballistic transport when the second term

2 containing the resistivity ρ =(πh/e¯ ) A/0M 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 + C2L, 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

2 of W (W ≤ 2) the constant C1 is approximately equal to RC = πh/e¯ M,whereM = 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/πh.¯ (3.24)

This well know result [37] has been derived in the semiclassical transport regime G e2/πh¯.

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

1.0

0.8 /h) 2

(2e 0.6 1/2 0.4 15x15x15

(Var G) 0.2 10x10x10

0.0 0246810121416 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, (3.25) G2 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 afactor≈ 3asW 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 paral- lelepiped) 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/πh¯ 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/πh¯ [117]).

Finally, we oﬀer a tentative explanation for the deviation of C1 (3.23) from the quantum

2 point contact resistance RC . In the semiclassical regime G e /πh¯ there are corrections 85

15x15x5 )

2 2 10 15x15x10 15x15x15 15x15x20 (h/2e 100 1/2

10-2 (Var R) 10-4 1.6 /h) 2 1.2 (2e

1/2 0.8

15x15x5 0.4 15x15x10 15x15x15 (Var G) 15x15x20 0.0 0246810121416 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 10

) -1 2 3.5 / /-1 3.0 1 2.5

2.0 -1 0.1

Resistance (h/2e 1.5

0.01 1.0 024681012141618 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

2 2 e √ 1 − e 1 σ(L)=σ + 2 3 . (3.26) π h¯2 2 L π h¯ 0

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 C1in 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−dG(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−dG(L). (3.28) L→∞

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

5Our disorder-averaging procedure can be thought as describing the real sample at finite temperature (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) → 0asL →∞.

8 The correlation length ξc of the LD transition is deﬁned as the size of the conductor

9 (d-dimensional hypercube )forwhichg(ξc) ∼O(1), or equivalently ETh ∼O(∆(ξc)) [5]. For

L ξc the scaling of conductance characterizes a metal

g ∝ Ld−2, (3.29) or an insulator

g ∝ e−L/ξ. (3.30)

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

6Conductance 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]. 7In 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. 9To 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]. This means that

d−1 d−1 g(L)=gq1D(L, Lt)(L/Lt) ,whereLt is the cross section. 90 the scale dependent conductivity [20],

d−2 ξc σ(L)=σL→∞ . (3.31) L

Fortunately, in metallic conductors (g 1) the length ξc is microscopic (in d>2)

1/(d−2) λF ξc ∼ , (3.32) i.e., of the order of Fermi wavelength λF in 3D, as follows from (2.24). The same is true for

−1/(d−2) multichannel wires (quasi-1D systems), ξc ∼ M . 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.Thisis 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].

1Upon applying weakmagnetic 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 Hˆ = εm|mm| + t |mn|. (4.1) m 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 =4andεB = 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)

2 2πhe¯ 2 σxx = |α|vˆx|α | δ(Eα − EF )δ(Eα − EF ). (4.2) Ω α,α

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

2 2πhe¯ 2 f(Eα) − f(Eα ) σxx(ω)= |α|vˆx|α | δ(Eα − Eα − hω¯ ), (4.3) Ω α,α hω¯

2The 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

1 (η/2)2 δ(x) → δ¯(x)= , (4.4) π x2 +(η/2)2 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

δ¯(x).

To simplify the calculation, we compute the diﬀusivity3

x 2 Dα = πh¯ |α|vˆx|α | δ¯(Eα − Eα ). (4.5) α

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 δ¯(Eα − Eα ) in (4.5) is chosen as some multiple of the local average level spacing ∆(Eα)inasmallenergyinterval 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 ×

4 x Ny ×Nz) is obtained by numerical diagonalization; for each eigenstate we compute Dα (4.5),

where summation is going over all states |α “picked” by the Lorentzian δ¯(Eα −Eα) (centered on Eα)inanenergyintervalof3η 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|vˆx|α appearing in the

† deﬁnition of eigenstate diﬀusivity (4.5), is to multiply three matricesα ˆ · vˆx · αˆ,whereˆα is a

matrix containing eigenvectors |α as columns, and then take modulus squared of each matrix

3This 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]). 4For 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 operatorv ˆx 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) D¯ x, which is related to the conductivity through the Einstein relation

2 σxx = e N(EF ) D¯ x(EF ). (4.6)

This formula emphasizes that transport in a degenerate electron gas is a Fermi surface property.6

We ﬁrst calculate D¯(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).

5The number of operations in the na¨ıve calculation of the expectation values, where each of

4 them is calculated separately, scales as ∼ Ns , while in the method presented above it scales as

3 ∼ Ns (Ns × Ns is the dimension of operator matrix). 6Conductivity 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. 7The 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,

4e2 1 σxx = Tr h¯vˆxIm Gˆ h¯vˆxIm Gˆ , (4.7) h ALx 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 cannotbeusedinanopen 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)

1 r N(EF )= − Im GˆS(m, m; EF ). (4.8) m π

Thus, the average diﬀusivity D¯(EF ) is obtained easily from the Einstein relation (4.6)

2 where we divide the conductivity σxx by e 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].

8When compared to binary alloy, Anderson model looks like a conductor with infinite number of different impurities. 9In some sense non-zero η simulates the effect of inelastic scattering as an uncorrelated random event [106]. 97

0.007 12 (b) 10 Diffusivity (ta 0.006 (a) 8 0.005 6 2

4 /h) 0.004 (c) 2 Inverse Participation Ratio 0.003 0 -6 -4 -2 0 2 4 6 Fermi Energy

Figure 4.1: The diﬀusivity D¯(EF ) of a disordered binary alloy modeled by the tight-binding

Hamiltonian (εA = −2andεB = 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. 98

0.1 4

(b) Diffusivity (ta 3

(a) 2 2 /h) 0.01 (c) 1

Inverse Participation Ratio 0 -8-6-4-202468 Fermi Energy

Figure 4.2: The diﬀusivity D¯(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 δ¯(x) equal to one in some energy interval)

2 ρ(mx,E)= ρ(m,E)= |Ψα(m)| δ¯(E − Eα). (4.9) my,mz 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 inter-

10Intricacies 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]. 100

0.007 (b) 8 ε ε m={-4,0} m={0,4} 0.006 Diffusivity (ta 6 0.005 (c) 4 0.004 2

(a) /h) 2 0.003 (d) (e)

Inverse Participation Ratio 0.002 0 -8-6-4-202468 Fermi Energy

Figure 4.3: The diﬀusivity D¯(EF ) of a metal junction composed of two disordered binary alloys, left (εA = −4, εB =0)andright(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. 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 0.002 x={6,7,8,9,10,11,12,13} 0.000 0.006 x={2, 17} 0.004 0.002 0.000 0.006 x={1,18} 0.004

LDOS integrated over y and z coordinates 0.002 0.000 -10-8-6-4-20246810 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})andright

(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 /h) 2

tL=1.5t tL=t 4

2 Conductance (2e 0 -6-4-20246 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

1.2

tL=1, tC=1

1.0 tL=1.5, tC=0.1 t =1.5, t =1 /h) L C 2 0.8 tL=3, tC=1 t =3, t =3 0.6 L C

0.4

0.2 Conductance (2e 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)andright(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 exper-

iments 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.

11The importance of interface scattering in many areas of metal and semiconductor physics has been realized in the plethora of research papers since the seminal workof 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 (two- dimensional) 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)

2 2 2e r a 2e † G = Tr ΓˆL Gˆ1N ΓˆR GˆN1 = Tr (tt ), (4.10) h h r t = ΓˆL Gˆ1N ΓˆR, (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 disorder-

averaged conductance).

Mesoscopic transport methods give the possibility not only to compute the conductance,

12Even disorder-free interface can have a non-zero resistance, e.g., because of mismatch of crystal potential and band structures [130] (cf. Ch. 5). 107 10 /h)

2 t =1.5t 8 L t =t L N =1 6 x

4 N =3 N =2 x 2 x

Conductance (2e 0 -8 -6 -4 -2 0 2 4 6 8 Fermi Energy 103 (a) (b) Nx=1 102 (T) ρ 101 Nx=2 Nx=3 100 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)orthree(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 √ √ 3/2 −1 −1 (a) ρ(T )=(G/2GQ)(T 1 − T ) and (b) Dorokhov’s ρ(T )=(G/2GQ)(T 1 − T ) . 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 δ¯(x) equal to one inside the bin). The lower panel of Figure 4.7 plots ρ(T )forthe

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

G √1 ρ(T )= 3/2 . (4.12) 2GQ T 1 − T

The second formula is, up to a factor, the same as the analytical prediction of Ref. [115] for √ 3/2 a single dirty interface ρ(T )=(G/πGQ)1/(T 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.

13From 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 14Universality 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,wherea 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[ρAdA + ρBdB +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,andWB = 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 na¨ıve 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

15The 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) W =6 W =0 W =6 A B A (b) (c)

/h) 2 2

0 (a) W =0 W =0 W =0 6 A B A (b)

Conductance (2e (c) 4

0 -6-4-20246 Fermi Energy

Figure 4.8: The disorder-averaged (over 200 conﬁgurations) conductance of a multilayer com-

posed 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

14 80 12 Conductance (2e /h) 2 10 (a) (b) 60 8 40 6 4 (c) 20 2 /h) 2 Conductance (2e 0 0 -6-4-20246 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) W =6 W =6 W =6 A B A (b) 2 (c) /h) 2

0 4 (a) W =6 W =3 W =6 A B A (b)

Conductance (2e (c) 2

0 -6-4-20246 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

Diﬀusive Transport Regime 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 one- atom contact, since λF is much smaller in metals than in semiconductors.

The multichannel Landauer formula [4] for the two-probe conductance (2.54)

2 2e † G = Tr (tt )=GQ Tn, (5.1) h n 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

1The 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. 2In 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 proba-

bility 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 na¨ıve 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

2 then gives ∆G (Gbal/Gdiff )∆Gdiff 2e /h,sinceGbal 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) semi-

inﬁnite “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 tight- binding Hamiltonian with nearest-neighbor hopping parameters tmn

Hˆ = tmn|mn|, (5.2) m,n where |m is the orbital ψ(r − m)onthesitem. 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 they ˆ andz ˆ 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,

ikxmx which comprise the scattering basis, have the form Ψk ∝ sin(kymy)sin(kzmz)e at atom m,withenergyE =2tL[cos(kxa)+cos(kya)+cos(kza)], where a is the lattice constant. The 121

(a) 6 /h) 2 (d) 4 (b) 2 (c) Conductance (2e 0 -6-4-20246 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

1.0 (a) (3,1), (1,3), (2,2)

0.5 (b) (d) (c) 0.0 1.0 (3,2), (2,3) (1,2), (2,1) 0.5

0.0 1.0 (3,3) (1,1) Transmission eigenvalues 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)isopenifEF lies between the bottom of the subband, 2tL[−1+cos(kya)+cos(kza)], and the top of the subband, 2tL[1+cos(kya)+cos(kza)]. Because of the degeneracy of diﬀerent transverse modes in 3D, several channels (ky,kz)openorcloseatthesameenergy.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

2 2 NyNz 2e r a 2e † G = Tr ΓˆL Gˆ ΓˆR Gˆ = Tr (tt )=GQ Tn, (5.3) h 1N N1 h n=1 r t = ΓˆL Gˆ1N ΓˆR (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

6 (a)

4 (b) /h) 2 2 (c) 0 -6 -4 -2 0 2 4 6

6 (a)

Conductance (2e (b) 4 (c) 2

0 -6 -4 -2 0 2 4 6 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(kxa)+cos(kya)+cos(kza)], 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

2 4e 1 ˆ ˆ G = 2 Tr h¯vˆxIm G h¯vˆxIm G , (5.5) h Lx 1 Im Gˆ = (Gˆr − Gâ), (5.6) 2i 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

2 ETh =¯h/τD h¯D/L ,(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 flighthv ¯ F /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 =h/τ ¯ dwell 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 128

6 (a) /h) 2 (b)

4 (c) 2 Conductance (2e 0 -6 -4 -2 0 2 4 6 Fermi Energy

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. 129

(a) 6 /h) 2 (b) 4

2 (c) Conductance (2e 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 ρ RM = , (6.1) 2a 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) 2 2 −1 4ρ 2e kF A RS = = , (6.2) 3A h 4π 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

+V -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 aa(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

1It is interesting to note that optimal length [26] for the observation of conductance quanti- √ zation is Lopt ≈ 0.4 WλF (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 → 0andRS/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 Padéfit

1+0.83 l/a γﬁt(l/a)= , (6.4) 1+1.33 l/a 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 geom-

etry

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

∂F(k, r) e∇Φ(r) ∂F(k, r) F (k, r) − fLE(k, r) r˙ · − · = − , (6.5) ∂r h¯ ∂k τ eδn(r) ∇2Φ(r)=− , (6.6) ε 1 δn(r)= (F (k, r) − f(k)), (6.7) Ω k 1 0= (F (k, r) − fLE(k, r)), (6.8) Ω k e j(r)= vkF (k, r). (6.9) Ω k

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)atz =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)usingδµ(r) (local change of the chemical potential) and Ψ(k, r) (deviation function, i.e., energy shift of the altered distribution)

∂f(k) fLE(k, r)=f(k − δµ(r)) ≈ f(k) − δµ(r), (6.13) ∂k ∂f(k) F (k, r)=f(k − Ψ(k, r)) ≈ f(k) − Ψ(k, r). (6.14) ∂k 136

1.0 γ 0.9

0.8 γ Wex γ

0.7

0.6 0.01 0.1 1 10 100 l / a

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)

1 ∂f(k) δn(r)= − Ψ(k, r)=N(EF )Ψ(r) = N(EF )δµ(r), (6.15) Ω k ∂k where N(EF ) is the density of states at the Fermi energy F . In the case of a spherical Fermi surface, 1 Ψ(r) = dΩkΨ(k, r). (6.16) 4π 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)

∂u(k, r) τvk · = u(r)−u(k, r). (6.18) ∂r To solve this equation we need to know only boundary conditions satisﬁed by u(k, r)andthen

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)andthefact that far away from the oriﬁce we can expect local charge neutrality entailing

Φ(r) u(r) = . (6.21) V

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)

R k =(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. Tech- nically 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)

R u(k, rD − vkdt)=u(k , rD − vkdt)=−u(k, rD + vkdt), (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

∂w(k, r) k · + w(k, r) −w(r) = s(k, r)δ(z)θ(a − r), (6.29) ∂r where we have introduced the function

s(k, r)=2kzu(k, r), (6.30) 139 which is confined to the orifice region. It can be related to w(k, r)attheorificeinthe following way:

s(k, r0)=2|kz|(1 − w(k, r0 − vkdt)). (6.31)

It plays the role of a “source of particles” in Eq. (6.29). The notation r0 refers to a vector

2 2 2 lying on the oriﬁce, that is r0 =(x,y,0) with x + y ≤ a . 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)

∂ k · +1− Oˆ GB(k, r; k , r )=δ(Ωk − Ωk )δ(r − r ), (6.32) ∂r and Oˆ is the angular average operator

1 Ofˆ (k)= dΩk f(k)=f. (6.33) 4π

The Green’s function for the Boltzmann equation gives the possibility to express w(k, r0 − vkdt) in the form of a four-dimensional integral equation over the surface of the oriﬁce

w(k, r0 − vkdt)= dΩk dr 0 GB(k, r0 − vkdt; 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

iq·(r−r) −1 1 e q(q − arctan q) GB(k, r; k , r )= δ(Ωk − Ωk )+ . (6.35) Ω q 1+iq·k 4π(1 + iq·k )

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 dr0 jz(r0) G = = = , (6.36) R 2V 2V where the z-component of the current at the surface of the orifice is

2 N(EF )e V jz(r0)= dΩk s(k, r ). (6.37) 8πτ 0

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)overthesurfaceoftheoriﬁce s(k, r0) 1= + dΩk dr 0 G(k, r0; k , r 0)s(k , r 0), (6.38) 2|kz|

where G(k, r0; k , r 0) is non-singular part of the Green’s function (6.35)

iq·(r −r 0) 1 q e 0 G(k, r0; k , r 0)= 4 dq . (6.39) 32π (1 + iq·k)(q − arctan q)(1 + iq·k )

The distribution function s(k, r0)hastwok-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

2Normal operators satisfy condition OÔˆ† = Oˆ†Oˆ. This is sufficient and necessary to make 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 Oˆ = n enPˆn and projectors provide the decomposition of unity operator, n Pˆn = 1,inH.

The standard method to find the Green operator (i.e., the inverse, including relevant boundary conditions) of a linear operator Oˆ in H is GÔ =1/Oˆ = n Pˆn/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

−iMφ0 s(k, r0)= sLM(r0)YLM(θk,φk)e , (6.40) LM and Eq. (6.38) can now be rewritten as

−iM φ0 2 cos θk = sLM (r0)YLM (θk,φk)e sgn (cos θk)+2 dΩk dr 0 G(k, r0; k , r 0) LM −iM φ0 × cos θk sLM (r0)YLM (θk ,φk )e . (6.41) LM

This four dimensional integral equation can be reduced to a system of coupled one dimen-

∗ iMφ0 sional Fredholm integral equations of the second kind after it is multiplied by YLM(θk,φk)e

and integrated over θk, φk and φ0. We also use the following identities

YLM(θ, φ)cosθ = g1YL+1,M(θ, φ)+g2YL−1,M(θ, φ), (6.42) (L − M +1)(L + M +1) g1 = , (6.43) (2L + 1)(2L +3) (L − M)(L + M) g2 = , (6.44) (2L − 1)(2L +1)

1 YLM(θk,φk) L dΩk = i fL(q)YLM(θq,φq), (6.45) 4π 1+iq·k and

2π 2π iqr −iMφ0 iq⊥r0 cos(φ0−φq) −iMφ0 M −iMφq e 0 e dφ0 = e e dφ0 =2πi JM (q⊥r0)e , (6.46) 0 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

∞ −L L −x (−i) 1 fL(q)=(−1) e jL(qx) dx = QL( ), (6.47) iq iq 0 142 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 f0(x)= , (6.48) x −x + arctan x f1(x)= , (6.49) x2 −3x +(x2 + 3) arctan x f2(x)= , (6.50) 2x3 − 4 3 − 2 3 x 5x +(5+3x ) arctan x f3(x)= , (6.51) 2x4 − 55 3 − 2 4 3 x 35x +(35+30x +3x ) arctan x f4(x)= . (6.52) 8x5

The ﬁnal form of the integral equation for sLM(r0) in the expansion of s(k, r0)is a π 4 δL1δM0 = cLM,LM δMM sLM (r0)+4 r0 dr0 KLM,LM (r0,r0)sLM (r0), (6.53) 3 L M L M 0 where the kernel KLM,LM (r0,r0)isgivenby

∞ π 2 M −M M+M 2 q fL (q)YLM (θq) KLM,LM (r0,r )=i (−1) q dq sin θq dθq 0 q − arctan q 0 0 L+L+1 L+1 ×[i (−1) g1fL+1(q)YL+1,M(θq)

L+L−1 L−1 +i (−1) g2fL−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,LM (r0,r0)=LMM|2 cos θG(k, r0; k , r 0)|L M M , (6.55)

−iMφ0 θkφkφ0|LMM = YLM(θk,φk)e (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 non- zero sLM(r0) are real with the property

M sLM(r0)=(−1) 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,LM = dθk dφk sin θkYLM(θk,φk)YLM (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 sLM (r0) in terms of the polynomials in r0 sLM(r0)= anLMpn(r0), (6.60) n

n i and performing integrations numerically. The polynomials pn(r0)= i=0 cir0 are orthogonal with respect to the scalar product

a r0 dr0 pn(r0)pm(r0)=δnm. (6.61) 0 The ﬁrst three polynomials are √ 2 p0(r0)= , (6.62) a 6r0 − 4 p1(r0)= √ , (6.63) 2 − a 9a 16√a +9 2 − 6 3 10 6 r0 5 r0 + 10 p2(r0)= √ . (6.64) a 100a4 − 288a3 + 306a2 − 144a +27 144

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

π nLM 4a δL1δM0δn0 = cLM,LM anLM +4 KnLM anLM , (6.65) 6 L nLM

∞ π 2 nLM M −M M+M 2 q fL (q)YLM (θq) K = i (−1) q dq sin θq dθq nLM q − arctan q 0 0 n n ×jM (qa sin θq)jM (qa sin θq)

L+L+1 L+1 ×[i (−1) g1fL+1(q)YL+1,M(θq)

L+L−1 L−1 +i (−1) g2fL−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 2+M+i M M i M i − 1 2 n a (q sin θq) 1F2(1 + 2 + 2 ;2+ 2 + 2 , 1+M; 4 (qa sin θq) ) j (qa sin θq)= ci , M 1+M M i i=0 2 1+ 2 + 2 Γ(1 + M) (6.68) where 1F2(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) 2 2 N(EF )e a π Glo = 000 , (6.69) τ(3 + K010 )

000 where the lowest order part of the kernel K010 depends on /a,

∞ π 000 4 arctan q K = dq dθq 010 π q − arctan q 0 0 2 2 −3q +(q + 3) arctan q 2 arctan q × (1 − 3cos θq)+ 2q33 q 2 [J1(qa sin θq)] × . (6.70) sin θq 145

1 20 G 16 G I

12( G-G S (G-GI)/G 0.1

8 I )/ G / G

4 (%) (G-G )/G 0 0 0.01 0.01 0.1 1 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 na¨ıve 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 basis sets. The matrix elements KnLM (6.66) are tedious to compute, but the conductance converges rapidly for large n and L. On the other hand, the matrix elements cLM,LM (6.59) are easy to compute and the conductance converges slowly in the ballistic limit determined

nLM by these matrix elements. We keep only low order matrix elements KnLM but go to high 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,forwhich

G(k, r; k, r) → 0,

s(k, r)=2|kz|. (6.71)

In the opposite (Maxwell) limit, when a,wehave

q 3 = +9/5+o((q)2), (6.72) q − arctan q (q)2 3 eiq·(r−r ) 3 G(k, r; k, r) → dq = , (6.73) 32π4 (q)2 16π22|r − r| 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 na¨ıve interpolation formula which approximates resistance of the oriﬁce by the sum of Sharvin and Maxwell resistances 1 3π a = RI = RS 1+ . (6.74) GI 8

Somewhat unexpectedly, the na¨ıve 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) 1 3 32 3π a = RS + 2 γ . (6.75) Glo 4 3π 8

The numerical coeﬃcients in Eq. (6.75) are not accurate in this simplest approximation.

Replacement of 3/4by1and32/(3π2) 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 1 3π a = RS 1+γ , (6.76) G0 8 π γ = K000. (6.77) 16a 010 147

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 na¨ıve

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. (7.1) T →0 ω→0 |q|→0

For a system with ﬁnite metallic conductivity, we typically observe Drude behavior (discussed further in Ch. 9) at small frequencies

ij τ Re σij(T =0,ω → 0) = D , (7.2) c π(1 + ω2τ 2)

ij 2 ij where Dc = δijπe (n/m)eﬀ is the Drude weight. The expression (7.2) goes into Dc 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 An- derson localized phase in the non-interacting disordered electron systems or partially filled bands of strongly correlated electron systems, can be drastically different from the “simple- to-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 elec-

trons (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

1The band insulators were the ﬁrst ones discovered in the early days of quantum mechanics of solids. In a na¨ıve 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 wave-

2 length λF and leads to a transition from metallic to activated conductivity. This is a result of quantum-mechanical effects: the single particle interference effects, which lead to An- derson 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

2At non-zero temperature the insulating phase has a non-zero conductivity because of the hopping mechanism [40]. It increases with the temperature as a result of the assistance of inelastic processes. 3The 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

4 correct deﬁnitions [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 spin-

orbit 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

4The 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. 5The 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 antilocaliza- tion” [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

6When 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

7Strictly 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 na¨ıve 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 Qξa−yl−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 zero- temperature 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

2 GQ =2e /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., self-

similar) 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 uni- fying 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)=E0δ(t). The semiclassical response function (i.e., zeroth order in

the expansion of Diﬀuson-Cooperon diagrammatic perturbation theory for disorder-averaged

quantities [15]), σD t σ0(t)= exp(− ), (8.2) τ τ is valid only on time scales of the order of elastic mean free time, t ∼ τ.Fort 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

2 e 1 1−d/2 −t/τD σ1(t)=− (4πD) e . (8.3) πh tD/2

At very long times t τD the decay of the relaxation current is determined by the higher- order 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 σ 1 2 t σ2(t) ∝ exp − ln , (8.4) τ 4u τ 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 pre- localized 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 pre- localized 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], pre- localized states have anomalously short localization radius, when compared to “ordinary” 159 40

30 "pre-localized" state 20

10 V 2 |

ψ 0 40 t=|

30 extended state 20

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. 2The localization ξ length in 2D is not inﬁnite (as for truly critical systems), but it is exponentially 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 dis-

ordered 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 Hˆ = εm|mm| + tmn|mn|, (8.5) m 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

Hamiltonian is a real symmetric matrix because time-reversal symmetry is assumed.

2 In this Chapter numerical results for the statistics of wave function “intensities” |Ψα(r)|

in 3D disordered electron systems are presented. The statistical properties of eigenstates are

3For 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 2 f(t)= δ(t −|Ψα(r)| V )δ(E − Eα) , (8.6) ρ(E)N r,α

on N discrete points r inside a sample of volume V .Hereρ(E)= α δ(E−Eα) is the mean level density at energy E. Averaging over disorder is denoted by .... Normalization of eigenstates gives t¯ = dttf(t)=1.Theresultsforf(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

1 fPT(t)=√ exp(−t/2). (8.7) 2πt

4In 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. 5In some of the literature [32] only this speciﬁc function is associated with the names of Porter and Thomas. 163

10-2 (c) (d) 10-4 RMT

10-6 (b) -8 (a) 10 6 100 200 300 400 500

f(t) 10 104 (c) 102 (d) 100 10-2 -4 (a) 10 -6 (b) 10 RMT -8 10 -8 -6 -4 -2 0 2 10 10 10 10 10 10 t=|ψ|2V

Figure 8.2: Statistics of wave function intensities in the RH Anderson model, with WRH =1,

3 on a cubic lattice with Ns =16 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

10-2 (a) (b) 10-4 (c) RMT (d) 10-6

10-8 100 200 300 400 500 600 700 f(t)

-2 (a) 10 (b) (c) 10-4 RMT (d) 10-6

-8 10 50 100 150 200 250 300 t=|ψ|2V

Figure 8.3: Statistics of wave function intensities in the DD Anderson model on a cubic

3 lattice with Ns =16 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,wheretH is Heisenberg

2 time tH =¯h/∆, ∆ = 1/ρ(E) is mean energy level spacing and tD L /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) , (8.8) i

6In the case of weakly broken time-reversal symmetry the distribution of eigenfunction amplitudes is complicated even in the frameworkof RMT [186]. 166

In a ﬁnite-size sample the level statistics follow RMT predictions in the ergodic regime,

7 i.e., on the energy separation scale smaller than the Thouless energy ETh. Aremarkable 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

2 D∇ φµ(r)=−ωµφµ(r), ∇φ|B =0. (8.10)

2 Here ωµ is the spectrum of the classical diffusion operator D∇ with eigenstates φµ(r)satis- fying 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.Herep(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]

πξ2 ln(c2V/t) fξ(t)= , (8.11) 2V t −1 2 L c2 = 1 − (1 + )exp(−L/ξ) , (8.12) πξ2 ξ where radially symmetric sample of radius L/2 is assumed.

The distribution function f(t) is equivalently determined in terms of its moments q PT q −q+1 bq = dt t f(t). For GOE, the PT distribution (8.7) has moments bq =2V Γ(q + 2q 1/2)/Γ(1/2). They are related to the moments Iα(q)= dr |Ψα(r)| of the wave function

2 intensity |Ψα(r)| . 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)andΨα(r2). This means that the integration in the deﬁnition

PT of Iα(q) provides self-averaging, and Iα(q) does not ﬂuctuate, i.e., Iα(q)=bq . Following

Wegner [189] we characterize the individual states by an ensemble average of Iα(q) 2q I¯(q)=∆ |Ψα(r)| δ(E − Eα) . (8.13) r,α

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 moments I¯(q) with respect to system size   −d(q−1)  L metal,  I¯(q) ∝ 0 (8.14)  L insulator,   ∗ L−d (q)(q−1) critical. Here d∗(q)

of critical eigenfunctions—they are delocalized but in the thermodynamic limit occupy only

2 an inﬁnitesimal fraction of the sample. The ﬂuctuations of Iα(2) scale [32] as δIα(2) ∼ 1/g ∝ 168

L4−2d. At the critical point (g ∼ 1) the average value is of the same size as ﬂuctuations.

Therefore, I¯(q) is not enough to characterize the critical eigenstates.

We use I¯(2) (Fig. 8.4) as a rough guide in selecting eigenstates with diﬀerent properties

(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).Thereal 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 disorder where the whole band becomes localized [109] for WDD > 16.5. 169

10-1 (b)

10-2

-3 (a) 10 -8-6-4-202468 10-1 (c) 0.0018

10-2 (b) 0.0010 Inverse Participation Ratio (a) -3 10 0.0002 -4 -2 0 2 4 Energy

Figure 8.4: Ensemble averaged Inverse Participation Ratio, I¯(2), of eigenstates in the RH

3 and DD Anderson models on the cubic lattice with Ns =16 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,aswellasforDDmodels, 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)

2 2e † G(EF )= Tr (t(EF )t (EF )), (8.15) h 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)

ρ(E) N(E)=2 , (8.16) V 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 12 0.7 10 (b) 0.6 0.5 8 (a) 0.4 6 (d) 0.3 4 (c) 0.2 Density of States /h) 2 2 0.1 0 0.0 -6-4-20246 80 0.5 60 0.4

Conductance (2e 0.3 40 0.2 20 0.1 0 0.0 -6 -4 -2 0 2 4 6 Fermi Energy

Figure 8.5: Conductance and DOS in the RH and DD Anderson models on the cubic lattice

3 with Ns =16 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 =40for

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 |Ec|.

8.3 Connections of eigenstate statistics to static quan-

tum transport properties

The distribution f(t) of eigenfunction intensities has been studied analytically for dif-

fusive 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 di-

rect 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 δ¯(x). The width of δ¯(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

2 deﬁned by δ¯(t −|Ψα(r)| 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)atspeciﬁc energies Ei in samples characterized by diﬀerent conductance g(EF = Ei) (controlled by

WDD). The conductance g(E =0)ofTBHwithWDD = 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]

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 NLσMforf(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

3 Anderson Hamiltonian on the cubic lattice with Ns =16 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 3 The smaller prefactor C3 in f(t) ∼ exp(−C3 ln 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

UltrathinPbFilms

9.1 Introduction

Measurements of DC transport in ultra-thin ﬁlms have been a subject of active inter-

est 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

1These 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 high-

frequency electric ﬁeld is avoided. If the criterion derived by Altshuler et al. [207] is satisﬁed,

2 then only the intrinsic dephasing introduced by Lω

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 = σE) 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 = −iωεE) 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 high- brightness U12IR beamline of the BNL National Synchrotron Light Source. The substrates were covered with a 5 A˚ 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 A˚ average thickness to 543 Ω/✷ at 98 A.˚ The 98 A˚ 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✷ =20MΩ/✷ at 26 Aand˚

R✷ = 1000 Ω/✷ at 123 A.˚ Finally, the ﬁlms from the set 3, deposited on a Ge-coated glass substrate, range from 13 to 231 A,˚ while R✷ changes between 5.6MΩand22.8Ω. 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]

1 T (ω)= 2 2 . (9.1) [1 + Z0σ✷(ω)/(n +1)] +(Z0σ✷(ω)/(n +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)andσ✷(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.Inthis case, the contribution of the imaginary part of conductance to the transmission coeﬃcient is negligible, and Eq. (9.1) can be approximated by

2 1 T (ω) . (9.2) 1+Z0σ✷(ω)/(n +1)

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), (9.3)

3In 2D conductivity and sheet conductance (conductance per square) have the same dimensions. 182

Fig. 1

108

7 d=231 A −2 10 10 106 105

4 ) 10 Ω (

[] d=131 A 103 R

) 2

−1 10 101 −3 (ω) (Ω 10[] 0 σ 10 10 100 1000 d (Angstroms)

d=24 A

d=13.2 A 10−4 1000 2000 3000 40006000 8000 ω (cm−1)

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) iωt σ✷(ω)= σ✷(t)e dω, (9.4) which determines the current response j(t) to a spatially homogeneous ﬁeld in the form of sharp electric pulse E(t)=E0δ(t) (cf. Ch. 8). The conductivity (9.3) is Fourier transform of the semiclassical response function

σD t σ✷(t)= exp(− ), (9.5) τ τ 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 corre-

sponding 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 A.˚

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

T (ω) (1 + ω2τ 2) = 2 , (9.6) 1 − T (ω) (σD/σ0) +2σD/σ0 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. ω2 (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 6 ) 1 1.2 − 4 cm 4 (10 p 2

1 ω

T) 0.8 0 100 200 300 400 − Ω Rsq ac ( ) T/(1 0.6

0.4

0.2

0 024681012 ω2 (107 cm−2)

Figure 9.2: T (ω)/[1 − T (ω)] plotted vs. ω2 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

−1 plasma frequency of ωp = 59 400 cm [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 = hD/k¯ BT (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 self- averaging 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

2 regime), GQ =2e /h. For AC conductivity the influence of the coherent backscattering is restricted to a spatial region of size Lω = D/ω.HereD 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 e2 ∆σ2D(ω)= ln ωτ, (9.7) ✷ 2π2h¯

in the 2D limit (d

−1 in the 3D limit (d>Lω) [36]. Using the lower end of our frequency range ω = 500 cm ,

2 and a realistic value of D =5cm/s, we can estimate Lω ≤ 20A˚ ≤ 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 √ of the slope of the conductivity vs. ω on the thickness of the film and the DC sheet 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 ≈ 1kΩ. √ Furthermore, the 3D localization theory predicts that the ω dependence of WL theory crosses over to ω(D−1)/D = ω1/3 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

t thickness dc of the ﬁlm. The DC conductivity above the transition point scales as (d − dc) , where t =1.3in2Dandt =1.9 in 3D [211]. Just below the percolation transition the

−s dielectric constant of the medium diverges as (d) ∼ (dc − d) ,wheres =1.3in2Dand s =0.7 in 3D. The diverging dielectric constant is manifested as the imaginary part of

−s the AC conductivity σ(ω) ∼−iω(dc − d) . 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

t −(t+s) σ(ω,d)=|d − dc| F±(−iω|d − dc| ). (9.9)

Here F+(x)andF−(x) are the scaling functions above and below the transition point, respec-

tively. 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, pro-

vided that

(0) (1) (2) 2 F+(x)=F+ + F+ x + F+ x + ..., (9.10)

(1) (2) 2 F−(x)=F− x + F− x + ... (9.11)

One should mention that the predictions of the EMT can also be written in the analogous 188 scaling form where D2 +4(D − 1)x ± D F ±(x)= , (9.12) 2(D − 1) 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 t/(t+s) iω 1/2 σ(ω,dc)=A = A . (9.13) ω0 ω0

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

1.3 the limiting value of σ(ω,d)(dc/|d−dc|) at small values of the scaling variable x = ω(dc/|d−

2.6 dc|) 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,whereA 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.ThisgivesC 189

10−2

−3 10 1.3 1| − c )/|d/d 1 − Ω (

σ(ω) 10−4

10−5 101 102 103 104 105 106 107 ω −1 2.6 (cm )/|d/dc−1|

Figure 9.3: The “data collapse” of the rescaled conductivity of 10 ﬁlms from the set 1

(dc =35.4A,˚ ×), 9 ﬁlms from the set 2 (dc =48.4A,˚ open circles) and 18 form the set 3

(dc =34.2A,˚ solid line). 190

2.6 × 10−19 F, which is in agreement with a very rough estimate of the capacitance between two islands 200 A˚ × 200 A˚ × 30 A˚ separated by a vacuum gap of approximately 20 A,˚ 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 conduc-

tivity 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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