Quantum transport; introduction to nanoscience

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Quantum Transport
Introduction to Nanoscience

Quantum transport is a diverse field, sometimes combining seemingly contradicting
concepts – quantum and classical, conducting and insulating – within a single nano-device.
Quantum transport is an essential and challenging part of nanoscience, and understanding
its concepts and methods is vital to the successful design of devices at the nano-scale.
This textbook is a comprehensive introduction to the rapidly developing field of quantum transport. The authors present the comprehensive theoretical background, and explore
the groundbreaking experiments that laid the foundations of the field. Ideal for graduate
students, each section contains control questions and exercises to check the reader’s understanding of the topics covered. Its broad scope and in-depth analysis of selected topics will
appeal to researchers and professionals working in nanoscience.
Yuli V. Nazarov is a theorist at the Kavli Institute of Nanoscience, Delft University of Tech-

nology. He obtained his Ph.D. from the Landau Institute for Theoretical Physics in 1985,
and has worked in the field of quantum transport since the late 1980s.
Yaroslav M. Blanter is an Associate Professor in the Kavli Institute of Neuroscience, Delft

University of Technology. Previous to this, he was a Humboldt Fellow at the University of
Karlsruhe and a Senior Assistant at the University of Geneva.

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Quantum Transport
Introduction to Nanoscience

YULI V. NAZAROV
Delft University of Technology

YAROSLAV M. BLANTER
Delft University of Technology

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CAMBRIDGE UNIVERSITY PRESS

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
The Edinburgh Building, Cambridge CB2 8RU, UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521832465
© Y. Nazarov and Y. Blanter 2009
This publication is in copyright. Subject to statutory exception and to the
provision of relevant collective licensing agreements, no reproduction of any part
may take place without the written permission of Cambridge University Press.
First published in print format 2009

ISBN-13

978-0-511-54024-0

eBook (EBL)

ISBN-13

978-0-521-83246-5

hardback

Cambridge University Press has no responsibility for the persistence or accuracy
of urls for external or third-party internet websites referred to in this publication,
and does not guarantee that any content on such websites is, or will remain,
accurate or appropriate.

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Contents

Preface

page vii

Introduction

1

1 Scattering
1.1 Wave properties of electrons
1.2 Quantum contacts

1.3 Scattering matrix and the Landauer formula
1.4 Counting electrons
1.5 Multi-terminal circuits
1.6 Quantum interference
1.7 Time-dependent transport
1.8 Andreev scattering
1.9 Spin-dependent scattering

7
7
17
29
41
49
63
81
98
114

2 Classical and semiclassical transport
2.1 Disorder, averaging, and Ohm’s law
2.2 Electron transport in solids
2.3 Semiclassical coherent transport
2.4 Current conservation and Kirchhoff rules
2.5 Reservoirs, nodes, and connectors
2.6 Ohm’s law for transmission distribution
2.7 Spin transport
2.8 Circuit theory of superconductivity
2.9 Full counting statistics

124
125
130
137
155
165
175
187
193
205

3 Coulomb blockade
3.1 Charge quantization and charging energy
3.2 Single-electron transfers
3.3 Single-electron transport and manipulation
3.4 Co-tunneling
3.5 Macroscopic quantum mechanics
3.6 Josephson arrays
3.7 Superconducting islands beyond the Josephson limit

211
212
223
237
248
264
278
287

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Contents

4 Randomness and interference
4.1 Random matrices
4.2 Energy-level statistics
4.3 Statistics of transmission eigenvalues
4.4 Interference corrections
4.5 Strong localization

299
299
309
324
336
363

5 Qubits and quantum dots
5.1 Quantum computers
5.2 Quantum goodies
5.3 Quantum manipulation
5.4 Quantum dots
5.5 Charge qubits
5.6 Phase and flux qubits
5.7 Spin qubits

374
375
386
397
406
427
436
445

6 Interaction, relaxation, and decoherence
6.1 Quantization of electric excitations
6.2 Dissipative quantum mechanics
6.3 Tunneling in an electromagnetic environment
6.4 Electrons moving in an environment
6.5 Weak interaction
6.6 Fermionic environment
6.7 Relaxation and decoherence of qubits
6.8 Relaxation and dephasing of electrons

457
458
470
487
499
513
523
538
549

Appendix A Survival kit for advanced quantum mechanics
Appendix B Survival kit for superconductivity
Appendix C Unit conversion
References
Index

562
566
569
570
577

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Preface

This book provides an introduction to the rapidly developing field of quantum transport.
Quantum transport is an essential and intellectually challenging part of nanoscience; it
comprises a major research and technological effort aimed at the control of matter and
device fabrication at small spatial scales. The book is based on the master course that has
been given by the authors at Delft University of Technology since 2002. Most of the material is at master student level (comparable to the first years of graduate studies in the
USA). The book can be used as a textbook: it contains exercises and control questions.
The program of the course, reading schemes, and education-related practical information
can be found at our website www.hbar-transport.org.
We believe that the field is mature enough to have its concepts – the key principles
that are equally important for theorists and for experimentalists – taught. We present at a
comprehensive level a number of experiments that have laid the foundations of the field,
skipping the details of the experimental techniques, however interesting and important
they are. To draw an analogy with a modern course in electromagnetism, it will discuss

the notions of electric and magnetic field rather than the techniques of coil winding and
electric isolation.
We also intended to make the book useful for Ph.D. students and researchers, including experts in the field. We can liken the vast and diverse field of quantum transport to a
mountain range with several high peaks, a number of smaller mountains in between, and
many hills filling the space around the mountains. There are currently many good reviews
concentrating on one mountain, a group of hills, or the face of a peak. There are several
books giving a view of a couple of peaks visible from a particular point. With this book, we
attempt to perform an overview of the whole mountain range. This comes at the expense
of detail: our book is not at a monograph level and omits some tough derivations. The level
of detail varies from topic to topic, mostly reflecting our tastes and experiences rather than
the importance of the topic.
We provide a significant number of references to current research literature: more than a
common textbook does. We do not give a representative bibliography of the field. Nor do
the references given indicate scientific precedences, priorities, and relative importance of
the contributions. The presence or absence of certain citations does not necessarily reflect
our views on these precedences and their relative importance.
This book results from a collective effort of thousands of researchers and students
involved in the field of quantum transport, and we are pleased to acknowledge them here.
We are deeply and personally indebted to our Ph.D. supervisors and to distinguished senior
colleagues who introduced us to quantum transport and guided and helped us, and to
comrades-in-research working in universities and research institutions all over the world.

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viii

Preface

This book would never have got underway without fruitful interactions with our students.
Parts of the book were written during our extended stays at Weizmann Institute of Science,
Argonne National Laboratory, Aspen Center of Physics, and Institute of Advanced Studies,
Oslo.
It is inevitable that, despite our efforts, this book contains typos, errors, and less comprehensive discourses. We would be happy to have your feedback, which can be submitted
via the website www.hbar-transport.org. We hope that it will be possible thereby to provide
some limited “technical” support.

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Introduction

It is an interesting intellectual game to compress an essence of a science, or a given
scientific field, to a single sentence. For natural sciences in general, this sentence would
probably read: Everything consists of atoms. This idea seems evident to us. We tend to
forget that the idea is rather old: it was put forward in Ancient Greece by Leucippus and
Democritus, and developed by Epicurus, more than 2000 years ago. For most of this time,
the idea remained a theoretical suggestion. It was experimentally confirmed and established
as a common point of view only about 150 years ago.
Those 150 years of research in atoms have recently brought about the field of
nanoscience, aiming at establishing control and making useful things at the atomic scale.
It represents the common effort of researchers with backgrounds in physics, chemistry,
biology, material science, and engineering, and contains a significant technological component. It is technology that allows us to work at small spatial scales. The ultimate goal of
nanoscience is to find means to build up useful artificial devices – nanostructures – atom by
atom. The benefits and great prospects of this goal would be obvious even to Democritus
and Epicurus.
This book is devoted to quantum transport, which is a distinct field of science. It is
also a part of nanoscience. However, it is a very unusual part. If we try to play the same

game of putting the essence of quantum transport into one sentence, it would read: It is not
important whether a nanostructure consists of atoms. The research in quantum transport
focuses on the properties and behavior regimes of nanostructures, which do not immediately depend on the material and atomic composition of the structure, and which cannot
be explained starting by classical (that is, non-quantum) physics. Most importantly, it has
been experimentally demonstrated that these features do not even have to depend on the
size of the nanostructure. For instance, the transport properties of quantum dots made of
a handful of atoms may be almost identical to those of micrometer-size semiconductor
devices that encompass billions of atoms.
The two most important scales of quantum transport are conductance and energy scale.
The measure of conductance, G, is the conductance quantum G Q ≡ e2 /π , the scale made
of fundamental constants: electron charge e (most of quantum transport is the transport of
electrons) and the Planck constant (this indicates the role of quantum mechanics). The
energy scale is determined by flexible experimental conditions: by the temperature, kB T ,
and/or the bias voltage applied to a nanostructure, eV . The behavior regime is determined
by the relation of this scale to internal energy scales of the nanostructure. Whereas physical
principles, as stressed, do not depend on the size of the nanostructure, the internal scales
do. In general, they are bigger for smaller nanostructures.

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2

Introduction

This implies that the important effects of quantum transport, which could have been seen
at room temperature in atomic-scale devices, would require helium temperatures (4.2 K),
or even sub-kelvin temperatures, to be seen in devices of micrometer scale. This is not

a real problem, but rather a minor inconvenience both for research and potential applications. Refrigeration techniques are currently widely available. One can achieve kelvin
temperatures in a desktop installation that is comparable in price to a computer. The cost
of creating even lower temperatures can be paid off using innovative applications, such as
quantum computers (see Chapter 5).
Research in quantum transport relies on the nanostructures fabricated using nanotechnologies. These nanostuctures can be of atomic scale, but also can be significantly bigger
due to the aforementioned scale independence. The study of bigger devices that are relatively easy to fabricate and control helps to understand the quantum effects and their
possible utilization before actually going to atomic scale. This is why quantum transport
tells what can be achieved if the ultimate goal of nanoscience – shaping the world atom by
atom – is realized. This is why quantum transport presents an indispensable “Introduction
to nanoscience.”
Historically, quantum transport inherits much from a field that emerged in the early
1980s known as mesoscopic physics. The main focus of this field was on quantum signatures in semiclassical transport (see, e.g., Refs. [1] and [2], and Chapter 4). The name
mesoscopic came about to emphasize the importance of intermediate (meso) spatial scales
that lie between micro-(atomic) and macroscales. The idea was that quantum mechanics reigns at microscales, whereas classical science does so at macroscale. The mesoscale
would be a separate kingdom governed by separate laws that are neither purely quantum
nor purely classical; rather, a synthesis of the two. The mesoscopic physics depends on
the effective dimensionality of the system; the results in one, two, and three dimensions
are different. The effective dimensionality may change upon changing the energy scale. In
these terms, quantum transport mostly concentrates on a zero-dimensional situation where
the whole nanostructure is regarded as a single object characterized by a handful of parameters; the geometry is not essential. Mesoscopics used to be a very popular term in the
1990s and used to be the name of the field reviewed in this book. However, intensive
experimental activity in the late 1980s and 1990s did not reveal any sharp border between
meso- and microscales. For instance, metallic contacts consisting of one or a few atoms
were shown to exhibit the same transport properties and regimes as micron-scale contacts
in semiconductor heterostructures. This is why the field is called now quantum transport,
while the term mesoscopic is now most commonly used to refer to a cross-over regime
between quantum and classical transport.
The objects, regimes, and phenomena of quantum transport are various and may seem
unlinked. The book comprises six chapters that are devoted to essentially different physical
situations. Before moving on to the main part of the book, let us present an overview of

the whole field (see the two-dimensional map, Fig. 1). For the sake of presentation, this
map is rather Procrustean: we had to squeeze and stretch things to fit them on the figure.
For instance, it does not give important distinctions between normal and superconducting
systems. Still, it suffices for the overview.

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3

t

Introduction

G
gy
er

semiclassical
coherent
transport

es

ul

o
Th

n

se

2

me

UCF

scattering

er

ord

ic b

op
sosc

8
3

classical
incoherent
transport
GQ

1
4
7

qubits

6

5

quantum dots

Kond
o

single-electron
tunneling

incoherent
tunneling

Coulomb
blockade
elastic
inelastic
co-tunneling co-tunneling

strong
localization

level statistics

t

Fig. 1.

δs

Ec

E

Map of quantum transport. Various important regimes are given here in a log–log plot. The
numbered diamonds show the locations of some experiments described in the book (see the end
of this Introduction for a list).

The axes represent the conductance of a nanostructure and the energy scale at which the
nanostructure is operated; i.e. that set by temperature and/or voltage. This is a log–log plot,
and allows us to present in the same plot scales that differ by several orders of magnitude.
There is a single universal measure for the conductance – the conductance quantum G Q .
If G
G Q , the electron conductance is easy: many electrons traverse a nanostructure
simultaneously and they can do this in many ways, known as transport channels. For G
G Q , the transport takes place in rare discrete events: electrons tunnel one-by-one. The
regions around the cross-over line G G Q attract the most experimental interest and are
usually difficult to comprehend theoretically.
There are several internal energy scales characterizing the nanostructure. To understand
them, let us consider an example nanostructure that is of the same (by order of magnitude)
size in all three dimensions and is connected to two leads that are much bigger than the
nanostructure proper. If we isolated the nanostructure from the leads, the electron energies
become discrete, as we know from quantum mechanics. Precise positions of the energy levels would depend on the details of the nanostructure. The energy measure of such quantum
discreteness is the mean level spacing δS – a typical energy distance between the adjacent

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4

Introduction

levels. Another energy scale comes about from the fact that electrons are charged particles
carrying an elementary change e. It costs finite energy – the charging energy E C – to add
an extra electron to the nanostructure. This charging energy characterizes the interactions
of electrons. At atomic scale, δS 1 eV and E C 10 eV. These internal scales are smaller
for bigger structures, and E C is typically much bigger than δS .
As seen in Fig. 1, these scales separate different regimes at low conductance G
GQ.
At high conductance, G
G Q , the electrons do not stay in the nanostructure long enough
to feel E C or δS . New scales emerge. The time the electron spends in the nanostructure gives rise to an energy scale: the Thouless energy, E Th . This is due to the quantum
uncertainty principle, which relates any time scale to any energy scale by ( E)( t) ∼ .
The Thouless energy is proportional to the conductance of the nanostructure, E Th
δS G/G Q , and this is why the corresponding line in the figure is at an angle in the log–log
plot.
Another slanted line in the upper part of Fig. 1 is due to the electron–electron interaction, which works destructively. It provides intensive energy relaxation of the electron
distribution in a nanostructure and/or limits the quantum-mechanical coherence. On the
right of the line, the quantum effects in transport disappear: we are dealing with classical incoherent transport. At the line, the inelastic time, τin , equals the time the electron
spends in the nanostructure, that is, /τin E Th . The corresponding energy scale can
E Th . In the context of mesoscopics, Thouless has sugbe estimated as δS (G/G Q )2
gested that extended conductors are best understood by subdividing a big conductor into
smaller nanostructures. The size of such nanostructure is chosen to satisfy the condition

/τin E Th . This is why all experiments where mesoscopic effects are addressed are
actually located in the vicinity of the line; we call it the mesoscopic border.
Once we have drawn the borders, we position the material contained in each chapters
on the map. Chapter 1 is devoted to the scattering approach to electron transport. It is an
important concept of the field that at sufficiently low energies any nanostructure can be
regarded as a (huge) scatterer for electron waves coming from the leads. At G
G Q , the
validity of the scattering approach extends to the mesoscopic border. At energies exceeding
the Thouless energy, the energy dependence of the scattering matrix becomes important.
In Chapter 1, we explain how the scattering approach works in various circumstances,
including a discussion of superconductors and time-dependent and spin-dependent phenomena. We relate the transport properties to the set of transmission eigenvalues of a
nanostructure – its “pin-code.” The basics explained in Chapter 1 relate, in one way or
another, to all chapters.
If we move up along the conductance axis, G
G Q , the scattering theory becomes progressively impractical owing to a large number of transport channels resulting in a bigger
scattering matrix. Fortunately, there is an alternative way to comprehend this semi-classical
coherent regime outlined in Chapter 2. We show that the properties of nanostructures are
determined by self-averaging over the quantum phases of the scattering matrix elements.
Because of this, the laws governing this regime, being essentially quantum, are similar to
the laws of transport in classical electric circuits. We explain the machinery necessary to
apply these laws – quantum circuit theory. The quantum effects are frequently concealed
in this regime; for instance, the conductance is given by the classical Ohm’s law. Their

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5

Introduction

manifestations are most remarkable in superconductivity, the statistics of electron transfers, and spin transport. Remarkably, there is no limitation to quantum mechanics at high
conductances as soon as one remains above the mesoscopic border.
Chapter 3 brings us to the lower part of the map – to conductances much lower than G Q .
There, the charging energy scale E C becomes relevant, manifesting a strong interaction
between the electrons (the Coulomb blockade). This is why we concentrate on the energies
of the order of E C , disregarding the mean level spacing δS . Transport in this single-electron
tunneling regime proceeds via incoherent transfer of single electrons. However, the transfers are strongly correlated and can be precisely controlled – one can manipulate electrons
one-by-one. The quantum correction to single-electron transport is co-tunneling, i.e. coop√
erative tunneling of two electrons. The energy scale E C δS separates inelastic and elastic
co-tunneling. In the elastic co-tunneling regime, the nanostructure can be regarded as a
scatterer in accordance with the general principles outlined in Chapter 1. The combination of the Coulomb blockade and superconductivity restores the quantum coherence of
elementary electron transfers and provides the opportunity to build quantum devices of
almost macroscopic size.
The material discussed in Chapter 4 is spread over several areas of the map. In this
chapter, we address the statistics of persistent fluctuations of transport properties. We start
with the statistics of discrete electron levels – this is the domain of low conductances,
G
G Q , and low energies, of the order of the mean level spacing. Then we go to the
different corner, to G
G Q and the energies on the left from the mesoscopic border,
to discuss fluctuations of transmission eigenvalues – the universal conductance fluctuations (UCF) – and the interference correction to transport, weak localization. The closing
section of Chapter 4 is devoted to strong localization in disordered media, where electron hopping is the dominant mechanism of conduction. This implies G
G Q and high
energies.
A fascinating development of the field is the use of nanostructures for quantum information purposes. Here, we do not need a flow of quantum electrons, but rather a flow of
quantum information. Chapter 5 presents qubits and quantum dots, perhaps the most popular devices of quantum transport. For both devices, the discrete nature of energy levels is
essential. This is why they occupy the energy area left of the level spacing δS on the map.
We also present in Chapter 5 a comprehensive introduction to quantum information and

manipulation.
In Chapter 6 we discuss interaction effects that do not fit into the simple framework of the
Coulomb blockade. Such phenomena are found in various areas of the map. We start this
chapter with a discussion of the underlying theory, called dissipative quantum mechanics.
We study the effects of an electromagnetic environment on electron tunneling, remaining
in the area of the Coulomb blockade. We go up in conductance to understand the fate of the
Coulomb blockade at G G Q and the role of interaction effects at higher conductances.
The electrons in the leads provide a specific (fermionic) environment responsible for the
Kondo effect in quantum dots. The Kondo energy scale depends exponentially on the conductance and is given by the curve on the left side of the map. Finally, we discuss energy
dissipation and dephasing separately for qubits and electrons. In the latter case, we are at
the mesoscopic border.

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6

Introduction

At high energies one leaves the field of quantum transport: transport proceeds as
commonly taught in courses of solid-state physics.
We have not yet mentioned the numbered diamonds in the map. These denote the
location of several experiments presented in various chapters of the book.
(1)
(2)
(3)
(4)
(5)

(6)
(7)
(8)

Discovery of conductance quantization (Section 1.2);
interference nature of the weak localization (Section 1.6);
universal conductance fluctuations (Section 1.6);
single-electron transistor (Section 3.2);
discrete states in quantum dots (Section 5.4);
early qubit (Section 5.5);
Kondo effect in quantum dots (Section 6.6);
energy relaxation in diffusive wires (Section 6.8).

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1

Scattering

1.1 Wave properties of electrons
Quantum mechanics teaches us that each and every particle also exists as a wave. Wave
properties of macroscopic particles, such as brickstones, sand grains, and even DNA
molecules, are hardly noticeable to us; we deal with them at a spatial scale much bigger
than their wavelength. Electrons are remarkable exceptions. Their wavelength is a fraction
of a nanometer in metals and can reach a fraction of a micrometer in semiconductors. We
cannot ignore the wave properties of electrons in nanostructures of this size. This is the central issue in quantum transport, and we start the book with a short summary of elementary
results concerning electron waves.
A quantum electron is characterized by its wave function, (r, t). The squared absolute
value, | (r, t)|2 , gives the probability of finding the electron at a given point r at time t.

Quantum states available for an electron in a vacuum are those with a certain wave vector
k. The wave function of this state is a plane wave,
1
k (r, t) = √ exp (ik · r − iE(k)t/ ) ,
V

(1.1)

E(k) = 2 k2 /2m being the corresponding energy. The electron in this state is spread over
the whole space of a very big volume V; the squared absolute value of does not depend
on coordinates. The prefactor in Eq. (1.1) ensures that there is precisely one electron in this
big volume. There are many electrons in nanostructures. Electrons are spin 1/2 fermions,
and the Pauli principle ensures that each one-particle state is either empty or filled with
one fermion. Let us consider a cube in k-space centered around k with the sides dk x , dk y ,
|k|. The number of available states in this cube is 2s V dk x dk y dk z /(2π )3 . The
dk z
factor of 2s comes from the fact that there are two possible spin directions. The fraction
of states filled in this cube is called an electron filling factor, f (k). The particle density n,
energy density E, and density of electric current j are contributed to by all electrons and
are given by
⎡ ⎤
⎡
⎤
n
1
3k
d
⎣ E ⎦ = 2s
⎣ E(k) ⎦ f (k).
(1.2)

(2π )3
j
ev(k)
Here we introduce the electron charge e and the velocity v(k) = k/m. Quantum mechanics puts no restriction on f (k). However, the filling factor of electrons in an equilibrium
state at a given electrochemical potential μ and temperature T is set by Fermi–Dirac
statistics:

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8

t

Scattering
(a)

(c)
4kBT

e

μ
E

ReΨ, ImΨ

(b)

1

0
0
–1
–3

t

Fig. 1.1.

–2

–1

0
kx/2π

1

2

|k|

kF 0

f

1

3

Electrons as waves. (a) An electron in a vacuum is in the plane wave state with the wave vector k.
(b) The profile of its wave function . (c) At zero temperature, the electrons fill the states with
energies below the chemical potential μ (|k| < kF ). At a given temperature, the filling factor f is a
smoothed step-like function of energy.

f eq (k) = f F (E(k) − μ) ≡

1
.
1 + exp((E − μ)/kB T )

(1.3)

The chemical potential at zero temperature is known as the Fermi energy, E F .
Control question 1.1. What is the limit of f F (E) at T → 0? Hint: see Fig. 1.1.
Next, we consider electrons in the field of electrostatic potential, U (r, t)/e. The
wave function (r, t) of an electron is no longer a plane wave. Instead, it obeys the
time-dependent Schrödinger equation, given by
i

2
∂ (r, t)
= Hˆ (r, t); Hˆ ≡ −
∇ 2 + U (r, t).
∂t
2m

(1.4)

This is an evolutionary equation: it determines in the future given its instant value. The
evolution operator Hˆ is called the Hamiltonian. For the time being, we concentrate on the
stationary potential, U (r, t) ≡ U (r). The wave functions become stationary, with their time
dependence given by the energy
(r, t) = exp(−iEt/ )ψ E (r).
The Schrödinger equation reduces to
Eψ E (r) = Hˆ ψ E (r) = −

2

2m

∇ 2 + U (r) ψ E (r).

(1.5)

The Hamiltonian becomes the operator of energy, while the equation becomes a linear
algebra relation defining the eigenvalues E and the corresponding eigenfunctions E of
this operator. These eigenfunctions form a basis in the Hilbert space of all possible wave

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9

1.1 Wave properties of electrons

functions, so that an arbitrary wave function can be expanded, or represented, in this basis.

The first (gradient) term in the Hamiltonian describes the kinetic energy; the second term,
U (r), represents the potential energy.
A substantial part of quantum mechanics deals with the above equation. It cannot be
readily solved for an arbitrary potential, and our qualitative understanding of quantum
mechanics is built upon several simple cases when this solution can be obtained explicitly.
Following many good textbooks, we will concentrate on the one-dimensional motion, in
which the potential and the wave functions depend on a single coordinate x. However, we
pause to introduce a key concept that makes this one-dimensional motion more physical.

1.1.1 Transmission and reflection
Let us confine electrons in a tube – a waveguide – of rectangular cross-section that is
infinitely long in the x direction. We can do this by setting the potential U to zero for
|y| < a/2, |z| < b/2 and to +∞ otherwise. We thus create walls that are impenetrable to
the electron and are perpendicular to the y and z axes. We expect a wave to be reflected
from these walls, changing the sign of the corresponding component of the wave vector,
k y → −k y or k z → −k z . This suggests that the solution of the Schrödinger equation is a
superposition of incident and reflected waves of the following kind:
ψ(x, y, z) = exp(ik x x)

Cs y sz exp(s y ik y y) exp(sz ik z z).

(1.6)

s y ,sz =+,−

Since the infinite potential repels the electron efficiently, the wave function must vanish at the walls, ψ(x, y = ±a/2, z) = ψ(x, y, z = ±b/2) = 0. This gives a linear relation
between Cs y sz that determines these superposition coefficients. To put it simply, the walls
have to be in the nodes of a standing wave in both y and z directions. This can only happen
if k y,z assume quantized values k ny = π n y /a, k zn = π n z /b, with integers n y , n z > 0 corresponding to the number of half-wavelengths that fit between the walls. The notation we
use throughout the book here we introduce for the compound index n = (n y , n z ). The wave

function reads as follows:
ψk x ,n (x, y, z) = ψk x (x)

n (y, z);

ψk x (x) = exp(ik x x);
2
sin(k ny (y − a/2)) sin(k zn (z − b/2)).
n (y, z) = √
ab

(1.7)

The transverse motion of the electron is thus quantized. The electron in a state with
the given n (these states are called modes in wave theory and transport channels in
nanophysics) has only one degree of freedom corresponding to one-dimensional motion.
The energy spectrum consists of one-dimensional branches shifted by a channel-dependent
energy E n (see Fig. 1.2), given by
E n (k x ) =

( k x )2
π2 2
+ En ; En =
2m
2m

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n 2y
a2

+

n 2z
b2

.

(1.8)


10

t

Scattering
12

(d)

(a)

(3,1) (2,2)

10

b
a

y

(b)

2

π2

2ma2

z

x

(1,2)

E

8

y

6

(2,1)

4
(1,1)

Ψ
(c)

2
y
0
–1

x

t

Fig. 1.2.

0

1

2

πkx/a

Waveguide. (a) Electrons are confined in a long tube of rectangular cross-section. (b) Wave
function profiles of the modes (1,1), (2,1), and (3,1). (c) Corresponding trajectories of a classical
particle reflecting from the waveguide walls. (d) Energy spectrum of electron states in the
waveguide (b/a = 0.7). At the chemical potential shown by the dashed line, the electrons are
present only in the modes (1,1) and (2,1).

Let us add some more design to our waveguide. We cross it with a potential barrier of
simple form,
U (x) =

U0 , 0 < x < d
0, otherwise.

(1.9)

The possible solutions outside the barrier for a given n and energy are plane waves of the
form of Eqs. (1.7). It is important to note that there are two possible solutions with k x =
√
±k = ± 2m(E − E n )/ , corresponding to the waves propagating to the right (positive
sign) or to the left. A wave sent from the left is scattered at the barrier, part of it being
reflected back, another part being transmitted. We have
⎧
x <0
⎨ exp(ikx) + r exp(−ikx),
ψ(x) =
(1.10)
B exp(iκ x) + C exp(−iκ x), 0 < x < d
⎩
t exp(ikx),
x > d,
√
where κ = 2m(E − E n − U0 )/ = k 2 − 2mU0 / 2 . The wave function and its xderivative must be continuous at x = 0 and x = d. These four conditions give four linear
equations for the unknown coefficients r , B, C, and t. The most important for us are
the transmission amplitude t and the reflection amplitude r . The transmission coefficient,
T (E) = |t|2 , determines which fraction of the wave is transmitted through the obstacle.
The reflection coefficient, R(E) = |r |2 = 1 − T (E), determines the fraction reflected back.
We find
4k 2 κ 2
T (E) =
.

(1.11)
(k 2 − κ 2 )2 sin2 κd + 4k 2 κ 2

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11

t

1.1 Wave properties of electrons
(a)

(b)

E

1
1

t

T (E )

r
U0

0

d

0

t

Fig. 1.3.

x

0

1

2

3

E/U0

Potential barrier. (a) Scattering of an electron wave at a rectangular potential barrier.
(b) Transmission coefficient (see Eq. (1.11)) of the barrier for two different thicknesses,
√
d 2mU0 / = 3 (solid) and 5 (dashed). For the thicker barrier the transmission coefficient is
close to the classical one, T(E) = 1 at E > U0 .

Control question 1.2. Find the coefficients r , t, B, and C in terms of κ, k, and d.
In classical physics, particles with energies below the barrier (E < U0 ) would be totally
reflected (T = 0), while particles with energies above the barrier would be fully transmitted
(T = 1). Quantum mechanics changes this: electrons are transmitted and reflected at any
energy (Fig. 1.3). Even an electron with an energy well below the barrier (corresponding
to imaginary κ) has a finite, albeit an exponentially small, chance of being transmitted,

√
1. This is called tunneling.
T (E) ∝ exp(−2d 2m(U0 + E n − E)/ )
The above consideration is not limited to barriers localized within a certain interval
of x. For any barrier, the solution very far to the left, x → −∞, can be regarded as a
superposition of incoming and reflected waves, ψ = exp(ikx) + r exp(−ikx). Very far to
the right, x → ∞, the solution is a transmitted wave, ψ = t exp(ikx). To calculate t and
r , we have to solve the Schrödinger equation everywhere and match these two asymptotic
solutions.

1.1.2 Electrons in solids
The above discussion concerns electrons in a vacuum. The electrons in nanostructures are
not in a vacuum, rather they are in a solid state medium such as a metal or a semiconductor.
What does this change? Surprisingly, not much. A crystalline lattice of a solid state medium
provides a periodic potential relief. The solutions of the Schrödinger equation for such a
potential are no longer plane waves as in Eq. (1.1), but rather are Bloch waves,
ψk,P (r) = exp (ikr) u k,P (r),

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(1.12)


t

12

Scattering

where u k,P is a periodic function with the same periods as the lattice. The vector of

quasimomentum, k, is defined up to a period of a reciprocal lattice, and the index P
labels different energy bands. The energy E P (k) is a periodic function of quasimomentum.
This implies that it is bounded. Therefore, the spectrum at a given k consists of discrete
values corresponding to energy bands. The electron velocity in the given state (k, P) is
given by
1 ∂ E P (k)
.
v P (k) =
∂k
With these notations, Eq. (1.2) remains valid. The integration over d3 k must be replaced
by the summation over the energy band index P and integration over the quasimomentum
within the reciprocal lattice unit cell (or the first Brillouin zone).
This summarizes the differences between the descriptions of an electron in a vacuum
and in a crystalline lattice.
Note that the above discussion disregards the interaction between electrons. However,
there are many electrons in a solid state medium, they are charged, and they interact
with each other. One would have to deal with the Schrödinger equation for a many-body
wave function that depends on coordinates of all electrons in the nanostructure, which is a
formidable task. What makes the above discussion relevant?
This was a Nobel Prize question (awarded to Lev Landau in 1962). The above description is relevant because we “cheat.” We do not describe the real interacting electrons.
Indeed, we cannot, nor do we have to. Rather, we implicitly consider the quantum transport of quasielectrons (or quasiparticles), elementary charged excitations above the ground
state of all the electrons present in the solid state. The interaction between these excitations
is weak and in many instances can be safely disregarded.
Let us give a short summary of the arguments that justify this implicit substitution for
the important case of a metal. By definition, a metal is a material that can be charged with
no energy cost. This means that the energy required to add some charge Q into a piece of
metal is μQ/e, where μ is the chemical potential.
We now describe this quantum mechanically. Before the charge was added, the piece of
metal was in its ground state. Let us add one elementary charge. This drives the system to
an excited state, which corresponds to creating precisely one quasielectron. By symmetry

consideration, this state should have a certain quasimomentum and spin 1/2. To conform
to the definition of the metal, the energy of this state has to be equal to μ, E(k) = μ.
This condition defines a surface in three-dimensional space of quasimomentum, the Fermi
surface. Fermi surfaces can look rather complicated. For example, the Fermi surface of
gallium looks like the fossil of a dinosaur – to this end, a very symmetric dinosaur. The
Fermi surface of free electrons is a sphere: noble metals provide a good approximation to
it (see Fig. 1.4). In the following, we count the energy of quasiparticles from the Fermi
level μ.
Let us concentrate on the situation when the temperature and applied voltage are
μ available for
much smaller than μ. This sets the energy scale E max(eV , kB T )
quasiparticles, which are therefore all located close to the Fermi surface. The important
parameter is the density of states ν at the Fermi surface, defined as the number of states per
energy interval in a unit volume. The density of the quasiparticles is therefore ν E, much

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13

t

1.1 Wave properties of electrons
(a)

(b)

z

L

U

Γ

X
y
K

x

(c)

W

1

0.6
E

0.3

0

–0.3

t

Fig. 1.4.

Γ

X

W

L

Γ
q

K X

W K

L

A realistic metal: silver. (a) Brillouin zone with symmetry points , X, W, L, and K and lines.
(b) Fermi surface. (c) Energy bands plotted along the symmetry lines.

smaller than the density of the original electrons in the metal. The smaller E is, the bigger the distance between the quasiparticles. This explains why the interaction is negligible:
the quasiparticles just do not come together to interact.
The original electrons interact according to Coulomb’s law. The quasiparticles are not
original electrons, and the residual interaction between them is strongly modified. First
of all, the electric field around each quasiparticle is screened by electrons forming the
ground state since they redistribute to compensate the quasiparticle charge. This quenches
the long-range repulsion between the quasiparticles. The interaction may be mediated by
phonons (vibrations of the crystalline lattice) and is not even always repulsive. This may
drive the metal to a superconducting state.
The above arguments allow us to start the discussion of quantum transport with the
notion of non-interacting (quasi)electrons. We will see that the interactions may not always

be disregarded in the context of quantum transport. The above arguments do not work if
interaction occurs at mesoscopic rather than at microscopic scales.

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14

t

Scattering
(a)

(b)

conduction

(c)

μ
Δs

E

μ
μ

valence

(d)

conduction
a
Δs

GaAs
GaAlAs

E

Δ1
+ + +

–

–
b

t

Fig. 1.5.

0

valence

–

holes

2DEG

z

Energy bands in a semiconductor. Black-filled regions in (b) and (c) indicate carriers: electrons or
holes. (a) No doping; (b) n-doping; (c) p-doping. (d) Band edges in GaAlAs–GaAs heterostructure
versus the depth z. A two-dimensional electron gas (2DEG) is formed close to the GaAlAs–GaAs
interface.

1.1.3 Two-dimensional electron gas
There is a long way to go from metal solids to practical nanostructures, and this way
has been found during the technological developments of the second half of the twentieth
century. It started with semiconductors: insulators with a relatively small gap separating
conduction (empty) and valence (occupied) bands. Of all the rich variety of semiconductor
applications, one is of particular importance for quantum transport, and that is the making
of an artificial and easily controllable metal from a semiconductor. This is achieved by a
process called doping, in which a small controllable number of impurities are added to a
chemically pure semiconductor. Depending on the chemical valence of the impurity atom,
it either gives an electron to the semiconductor (the atom works as an n-dopant) or extracts
one, leaving a hole in the semiconductor (p-dopant). Even a small density of the dopants
(say, 10−4 per atom) brings the chemical potential either to the edge of the conduction
band (n-type semiconductor) or to the edge of the valence band (p-type semiconductor);
see Fig. 1.5. In both cases, the semiconductor becomes a metal with a small carrier concentration. A rather simple trick of doping different areas of a semiconductor with p- and
n-type dopants creates p-n junctions, transistors (for which W. Shockley, J. Bardeen, and
W. H. Brattain received the Nobel Prize in 1956), and most of the power of semiconductor
electronics.

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t

15

1.1 Wave properties of electrons

A disadvantage of the resulting metal is that it is rather dirty. Indeed, it is made by
impurities, so that the number of scattering centers approximately equals the number of
carriers. It is advantageous to separate spatially the dopants and the carriers induced. In
the course of these attempts, the two-dimensional electron gas (2DEG) has been put into
practice.
The most convenient way to make a 2DEG involves a selectively doped GaAlAs–GaAs
heterostructure, a layer of n-doped GaAlAs on the surface of a p-doped GaAs crystal.
The lattice constants of the two materials match, providing a clean, defect-free interface
between them. The semiconducting energy gap in GaAlAs is bigger than in GaAs, and
the expectation is that the electrons from n-dopants in GaAlAs eventually reach the GaAs.
Why would these carriers stay near the surface? To understand this, let us consider the
electrostatics of the whole structure (see Fig. 1.5). In the one-dimensional (1d) geometry
given, the potential energy of the electrons is U (z) = e (z). The electrostatic potential
(z) and charge density ρ(z) are related by the Poisson equation:
d2 (z)
= 4πρ(z)/ ,
dz 2
where we assume the same dielectric constant in both materials. If no carriers are present
in GaAlAs, the dopants with volume density n 1 make a parabolic potential profile in the
material, U (z) = U (0) + (2πe2 / )n 1 z 2 , 0 < z < a, a being thickness of the layer.1 If we
cross the interface, there is a drop in potential energy that equals the energy mismatch 1 ≈
0.2 eV between the conduction bands of the materials. Let us assume that the electrons are
concentrated close to the interface at the GaAs side and figure out the conditions at which
it actually happens. If the surface density of the electrons equals n 0 , the electric field in the

z direction jumps at the interface, i.e.
d
dz

−
z=a+0

d
dz

= −(4π e/ )n 0 .
z=a−0

The bulk GaAs is p-doped, so there are supposed to be holes. However, the holes are separated from the interface and the electrons by a depletion layer of thickness b. The negatively
charged dopants (with volume density n 2 ) in this layer form an inverse parabolic profile,
U (z) = U (a + 0) + (dU (z = a + 0)/dz)(z − a) − (2π e2 / )n 2 (z − a)2 , a < z < b.
This allows us to determine conditions for the stability of this charge distribution. Since
electrons at the interface and holes in the valence band share the same chemical potential,
the difference of the potential energies just equals the semiconducting gap, s = 1.42 eV
in GaAs, U (a + b) − U (a + 0) = s .2 Further, the holes are in equilibrium, so the electrostatic force −dU (z)/dz vanishes at the edge of the depletion layer, z = a + b. To ensure
that there are no carriers in the GaAlAs layer, one requires U (0) > U (a + b). Solving for
everything, we obtain
n0 = n1a −

sn2
2π e2

.

1 Typically, a = 50 nm. A simple technique to reduce the disorder is not to dope GaAlAs in a spacer layer

adjacent to the interface.
2 To write this, we disregard the kinetic energy of both electrons and holes in comparison with

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s.