quantum transport. introduction to nanoscience, 2009, p.591

The two most important scales of quantum transport are conductance and energy scale.. The measure of conductance, G, is the conductance quantum GQ ≡ e2/π, the scale made of fundamental c

Introduction to Nanoscience

Quantum transport is a diverse field, sometimes combining seemingly contradictingconcepts – quantum and classical, conducting and insulating – within a single nano-device.Quantum transport is an essential and challenging part of nanoscience, and understandingits 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 quan-tum transport The authors present the comprehensive theoretical background, and explorethe groundbreaking experiments that laid the foundations of the field Ideal for graduatestudents, each section contains control questions and exercises to check the reader’s under-standing of the topics covered Its broad scope and in-depth analysis of selected topics willappeal 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 ofKarlsruhe and a Senior Assistant at the University of Geneva

Quantum TransportIntroduction to Nanoscience

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© Y Nazarov and Y Blanter 2009

2009

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Preface pagevii

4 Randomness and interference 299

This book provides an introduction to the rapidly developing field of quantum transport.Quantum transport is an essential and intellectually challenging part of nanoscience; itcomprises a major research and technological effort aimed at the control of matter anddevice fabrication at small spatial scales The book is based on the master course that hasbeen given by the authors at Delft University of Technology since 2002 Most of the mat-erial is at master student level (comparable to the first years of graduate studies in theUSA) 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 informationcan be found at our website www.hbar-transport.org.

We believe that the field is mature enough to have its concepts – the key principlesthat are equally important for theorists and for experimentalists – taught We present at acomprehensive level a number of experiments that have laid the foundations of the field,skipping the details of the experimental techniques, however interesting and importantthey are To draw an analogy with a modern course in electromagnetism, it will discussthe notions of electric and magnetic field rather than the techniques of coil winding andelectric isolation

We also intended to make the book useful for Ph.D students and researchers, ing experts in the field We can liken the vast and diverse field of quantum transport to amountain range with several high peaks, a number of smaller mountains in between, andmany hills filling the space around the mountains There are currently many good reviewsconcentrating on one mountain, a group of hills, or the face of a peak There are severalbooks giving a view of a couple of peaks visible from a particular point With this book, weattempt to perform an overview of the whole mountain range This comes at the expense

includ-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 thanthe importance of the topic

We provide a significant number of references to current research literature: more than acommon textbook does We do not give a representative bibliography of the field Nor dothe references given indicate scientific precedences, priorities, and relative importance ofthe contributions The presence or absence of certain citations does not necessarily reflectour views on these precedences and their relative importance

This book results from a collective effort of thousands of researchers and studentsinvolved 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 seniorcolleagues who introduced us to quantum transport and guided and helped us, and tocomrades-in-research working in universities and research institutions all over the world

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 prehensive discourses We would be happy to have your feedback, which can be submittedvia the website www.hbar-transport.org We hope that it will be possible thereby to providesome limited “technical” support

com-It is an interesting intellectual game to compress an essence of a science, or a givenscientific 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 andDemocritus, 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 com-ponent 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 Democritusand 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 ately depend on the material and atomic composition of the structure, and which cannot

immedi-be explained starting by classical (that is, non-quantum) physics Most importantly, it hasbeen experimentally demonstrated that these features do not even have to depend on thesize 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 semiconductordevices 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 GQ ≡ 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, kBT , 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 physicalprinciples, as stressed, do not depend on the size of the nanostructure, the internal scales

do In general, they are bigger for smaller nanostructures.

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 cations Refrigeration techniques are currently widely available One can achieve kelvintemperatures in a desktop installation that is comparable in price to a computer The cost

appli-of creating even lower temperatures can be paid appli-off using innovative applications, such asquantum computers (see Chapter 5)

Research in quantum transport relies on the nanostructures fabricated using nologies These nanostuctures can be of atomic scale, but also can be significantly biggerdue to the aforementioned scale independence The study of bigger devices that are rel-atively easy to fabricate and control helps to understand the quantum effects and theirpossible utilization before actually going to atomic scale This is why quantum transporttells what can be achieved if the ultimate goal of nanoscience – shaping the world atom by

nanotech-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

sig-natures 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 ics reigns at microscales, whereas classical science does so at macroscale The mesoscalewould be a separate kingdom governed by separate laws that are neither purely quantumnor purely classical; rather, a synthesis of the two The mesoscopic physics depends onthe effective dimensionality of the system; the results in one, two, and three dimensionsare different The effective dimensionality may change upon changing the energy scale Inthese terms, quantum transport mostly concentrates on a zero-dimensional situation wherethe whole nanostructure is regarded as a single object characterized by a handful of param-eters; the geometry is not essential Mesoscopics used to be a very popular term in the1990s and used to be the name of the field reviewed in this book However, intensiveexperimental activity in the late 1980s and 1990s did not reveal any sharp border betweenmeso- and microscales For instance, metallic contacts consisting of one or a few atomswere shown to exhibit the same transport properties and regimes as micron-scale contacts

mechan-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 seemunlinked The book comprises six chapters that are devoted to essentially different physicalsituations Before moving on to the main part of the book, let us present an overview ofthe whole field (see the two-dimensional map, Fig.1) For the sake of presentation, thismap 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 superconductingsystems Still, it suffices for the overview

semiclassical coherent transport

mesoscopic border

Tho uless ener gy

quantum dots

single-electron tunneling

elastic co-tunneling

level statistics

inelastic co-tunneling

strong localization

incoherent tunneling

Coulomb blockade

3 8

2

4

5 6

7 1

tFig 1. 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 thenanostructure 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 GQ

If G  GQ, 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 

GQ, the transport takes place in rare discrete events: electrons tunnel one-by-one The

regions around the cross-over line G  GQattract the most experimental interest and areusually difficult to comprehend theoretically

There are several internal energy scales characterizing the nanostructure To understandthem, 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 thenanostructure proper If we isolated the nanostructure from the leads, the electron energiesbecome discrete, as we know from quantum mechanics Precise positions of the energy lev-els 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

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 EC– to add

an extra electron to the nanostructure This charging energy characterizes the interactions

of electrons At atomic scale,δS 1 eV and EC 10 eV These internal scales are smaller

for bigger structures, and ECis typically much bigger thanδS

As seen in Fig.1, these scales separate different regimes at low conductance G  GQ

At high conductance, G  GQ, the electrons do not stay in the nanostructure long enough

to feel EC or δS New scales emerge The time the electron spends in the

nanostruc-ture gives rise to an energy scale: the Thouless energy, ETh This is due to the quantumuncertainty principle, which relates any time scale to any energy scale by (E)(t) ∼  The Thouless energy is proportional to the conductance of the nanostructure, ETh

δSG/GQ, and this is why the corresponding line in the figure is at an angle in the log–logplot

Another slanted line in the upper part of Fig.1 is due to the electron–electron action, which works destructively It provides intensive energy relaxation of the electron

inter-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 cal incoherent transport At the line, the inelastic time,τin, equals the time the electronspends in the nanostructure, that is, /τin ETh The corresponding energy scale can

classi-be estimated as δS(G /GQ)2 ETh In the context of mesoscopics, Thouless has gested that extended conductors are best understood by subdividing a big conductor intosmaller nanostructures The size of such nanostructure is chosen to satisfy the condition

sug-/τin ETh 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  GQ, thevalidity of the scattering approach extends to the mesoscopic border At energies exceedingthe 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 phe-nomena We relate the transport properties to the set of transmission eigenvalues of ananostructure – its “pin-code.” The basics explained in Chapter 1 relate, in one way oranother, to all chapters

If we move up along the conductance axis, G  GQ, the scattering theory becomes gressively impractical owing to a large number of transport channels resulting in a bigger

pro-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 tothe 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

manifestations are most remarkable in superconductivity, the statistics of electron fers, and spin transport Remarkably, there is no limitation to quantum mechanics at highconductances as soon as one remains above the mesoscopic border.

trans-Chapter 3 brings us to the lower part of the map – to conductances much lower than GQ

There, the charging energy scale EC becomes relevant, manifesting a strong interaction

between the electrons (the Coulomb blockade) This is why we concentrate on the energies

of the order of EC, disregarding the mean level spacingδS Transport in this single-electron tunneling regime proceeds via incoherent transfer of single electrons However, the trans-

fers 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√

ECδSseparates inelastic and elasticco-tunneling In the elastic co-tunneling regime, the nanostructure can be regarded as ascatterer in accordance with the general principles outlined in Chapter 1 The combina-tion of the Coulomb blockade and superconductivity restores the quantum coherence ofelementary electron transfers and provides the opportunity to build quantum devices ofalmost macroscopic size

The material discussed in Chapter 4 is spread over several areas of the map In thischapter, we address the statistics of persistent fluctuations of transport properties We startwith the statistics of discrete electron levels – this is the domain of low conductances,

G  GQ, and low energies, of the order of the mean level spacing Then we go to the

different corner, to G  GQ and the energies on the left from the mesoscopic border,

to discuss fluctuations of transmission eigenvalues – the universal conductance tions (UCF) – and the interference correction to transport, weak localization The closing

fluctua-section of Chapter 4 is devoted to strong localization in disordered media, where

elec-tron hopping is the dominant mechanism of conduction This implies G  GQ and highenergies

A fascinating development of the field is the use of nanostructures for quantum mation purposes Here, we do not need a flow of quantum electrons, but rather a flow ofquantum information Chapter 5 presents qubits and quantum dots, perhaps the most pop-ular devices of quantum transport For both devices, the discrete nature of energy levels isessential This is why they occupy the energy area left of the level spacingδSon the map

infor-We also present in Chapter 5 a comprehensive introduction to quantum information andmanipulation

In Chapter 6 we discuss interaction effects that do not fit into the simple framework of theCoulomb 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  GQand 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

con-ductance and is given by the curve on the left side of the map Finally, we discuss energydissipation and dephasing separately for qubits and electrons In the latter case, we are atthe mesoscopic border

At high energies one leaves the field of quantum transport: transport proceeds ascommonly taught in courses of solid-state physics.

We have not yet mentioned the numbered diamonds in the map These denote thelocation of several experiments presented in various chapters of the book

(1) Discovery of conductance quantization (Section1.2);

(2) interference nature of the weak localization (Section1.6);

(3) universal conductance fluctuations (Section1.6);

(4) single-electron transistor (Section3.2);

(5) discrete states in quantum dots (Section5.4);

(6) early qubit (Section5.5);

(7) Kondo effect in quantum dots (Section6.6);

(8) energy relaxation in diffusive wires (Section6.8)

1.1 Wave properties of electrons

Quantum mechanics teaches us that each and every particle also exists as a wave Waveproperties of macroscopic particles, such as brickstones, sand grains, and even DNAmolecules, are hardly noticeable to us; we deal with them at a spatial scale much biggerthan 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 Wecannot ignore the wave properties of electrons in nanostructures of this size This is the cen-tral issue in quantum transport, and we start the book with a short summary of elementaryresults 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,

k(r, t) = √1

E(k)= 2k2/2m being the corresponding energy The electron in this state is spread over

the whole space of a very big volumeV; 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,

dk z  |k| The number of available states in this cube is 2sV dkx dk y dk z /(2π)3 Thefactor 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 densityE, and density of electric current j are contributed to by all electrons and

–1 0 1

tFig 1.1. Electrons as waves (a) An electron in a vacuum is in the plane wave state with the wave vectork.

(b) The profile of its wave function (c) At zero temperature, the electrons fill the states with

energies below the chemical potentialμ (|k| < kF) At a given temperature, the filling factor f is a

smoothed step-like function of energy.

feq(k) = fF(E(k) − μ) ≡ 1

1+ exp((E − μ)/kBT ). (1.3)The chemical potential at zero temperature is known as the Fermi energy, EF

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∂(r, t) ∂t = ˆH (r, t); Hˆ ≡ −2

This is an evolutionary equation: it determines in the future given its instant value The

evolution operator ˆH 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

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

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 bereadily solved for an arbitrary potential, and our qualitative understanding of quantummechanics 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,

ky → −k y or k z → −k z This suggests that the solution of the Schrödinger equation is asuperposition of incident and reflected waves of the following kind:

van-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 n y = πn y /a, k n

z = πn z /b, with integers ny , n z > 0

cor-responding 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 wavefunction reads as follows:

y z

a b

y

Ψ

y x

(a)

(b)

(c)

0 2 4 6 8 10 12

(1,1) (2,1) (1,2) (3,1) (2,2)

tFig 1.2. 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

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 beingreflected back, another part being transmitted We have

x-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

(k2− κ2)2sin2κd + 4k2κ2 (1.11)

0 d

U0E

tFig 1.3. 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 anyenergy (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,

T (E) ∝ exp(−2d√2m(U0+ E n − E)/)  1 This is called tunneling.

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 asuperposition 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 arenot 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 mediumprovides a periodic potential relief The solutions of the Schrödinger equation for such apotential are no longer plane waves as in Eq (1.1), but rather are Bloch waves,

ψk ,P (r) = exp (ikr) uk ,P (r), (1.12)

where uk ,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

With these notations, Eq (1.2) remains valid The integration over d3k 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 vacuumand 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 interactwith each other One would have to deal with the Schrödinger equation for a many-bodywave function that depends on coordinates of all electrons in the nanostructure, which is aformidable task What makes the above discussion relevant?

This was a Nobel Prize question (awarded to Lev Landau in 1962) The above tion 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 trans-

descrip-port 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 forthe 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 isμQ/e, where μ is the chemical potential.

We now describe this quantum mechanically Before the charge was added, the piece ofmetal 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 toμ, 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 TheFermi 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 Fermilevelμ.

Let us concentrate on the situation when the temperature and applied voltage aremuch smaller thanμ This sets the energy scale E  max(eV , kBT )  μ available for

quasiparticles, which are therefore all located close to the Fermi surface The importantparameter is the density of statesν at the Fermi surface, defined as the number of states per

energy interval in a unit volume The density of the quasiparticles is thereforeνE, much

tFig 1.4. 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 smallerE is, the

big-ger 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 notoriginal 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 quenchesthe long-range repulsion between the quasiparticles The interaction may be mediated byphonons (vibrations of the crystalline lattice) and is not even always repulsive This maydrive the metal to a superconducting state

The above arguments allow us to start the discussion of quantum transport with thenotion 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 ifinteraction occurs at mesoscopic rather than at microscopic scales

valence

μ μ

μ E

tFig 1.5. 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 wayhas been found during the technological developments of the second half of the twentiethcentury It started with semiconductors: insulators with a relatively small gap separatingconduction (empty) and valence (occupied) bands Of all the rich variety of semiconductorapplications, 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 extractsone, leaving a hole in the semiconductor (p-dopant) Even a small density of the dopants(say, 10−4per atom) brings the chemical potential either to the edge of the conductionband (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 con-centration A rather simple trick of doping different areas of a semiconductor with p- andn-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 semiconductorelectronics

A disadvantage of the resulting metal is that it is rather dirty Indeed, it is made byimpurities, so that the number of scattering centers approximately equals the number ofcarriers It is advantageous to separate spatially the dopants and the carriers induced Inthe course of these attempts, the two-dimensional electron gas (2DEG) has been put intopractice.

The most convenient way to make a 2DEG involves a selectively doped GaAlAs–GaAsheterostructure, 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 interfacebetween them The semiconducting energy gap in GaAlAs is bigger than in GaAs, andthe 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 theelectrostatics 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)

dz2where we assume the same dielectric constant

in GaAlAs, the dopants with volume density n1make a parabolic potential profile in the

material, U (z) = U(0) + (2πe2

1z2, 0< z < a, a being thickness of the layer.1If wecross the interface, there is a drop in potential energy that equals the energy mismatch1≈0.2 eV between the conduction bands of the materials Let us assume that the electrons areconcentrated 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 n0, the electric field in the

z direction jumps at the interface, i.e.



d dz



z =a+0−



d dz



.The bulk GaAs is p-doped, so there are supposed to be holes However, the holes are sepa-

rated from the interface and the electrons by a depletion layer of thickness b The negatively charged dopants (with volume density n2) in this layer form an inverse parabolic profile,

U (z) = U(a + 0) + (dU(z = a + 0)/dz)(z − a) − (2πe2

2(z − a)2, a < z < b.

This allows us to determine conditions for the stability of this charge distribution Sinceelectrons 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.2Further, the holes are in equilibrium, so the trostatic 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

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.

Control question 1.3. What is the thickness of the depletion layer in terms of n2ands?

The carriers in GaAlAs are absent only if the dopant density is sufficiently low, n1<

1( 2e2) Since n0> 0, the desired charge distribution occurs in a certain interval of

the dopant density:1( 2e2)> n1>sn2 2e2 The 2DEG structure is also

stable in the limit of vanishing p-dopant density, n2→ 0 In this case, the surface density

of the 2DEG just equals that of the n-dopants, n0= n1a.

Now we turn to the details of electron wave functions and spectrum in the 2DEG Theelectrons concentrated near the interface experience the potential that depends only on

z As for waveguides, we can separate the electron motion in the z-direction from that

in the x y plane The motion in the x, y directions is free, whereas it is finite in the z direction: The potential U (z) takes the form of a triangular-shaped well So, the motion

in the z direction is quantized, giving rise to a series of discrete energy levels E n withcorresponding wave functions n (z) of the localized states (Fig. 1.6) The wave func-

tions of the electron states are plane waves in the x, y directions and can be presented as

ψk x ,k y ,n (x, y, z)∝ ei(k x x +k y y) n (z) Each n thus gives rise to a subband of two-dimensional

states with energies given by (k ≡ (k x , k y))

En (k) = E n+2k2

2m .Here, m is the effective mass of electrons in GaAs, equal to 0.067 of the electron mass in

a vacuum If we count energy from the bottom of the infinitely deep triangular well, the

energy levels read E n = c n ((U)22/(2m))1/3 , where U≡ dU(z = a + 0)/dz, and c1=

tFig 1.6. Energy spectrum of a 2DEG (a) Localized wave functions and energy levels in a triangular well

potential near the interface (b) All electrons are accommodated in the lowest subband.

Control question 1.4. What is U in terms of n0, n1, and a? Why can E0 notexceed1?

Let us put electrons into the levels We note that the density of states in each subbanddoes not depend on energy and equals

modated in the lowest subband This implies that the corresponding Fermi energy, EF,

counted from the subband edge, EF = n0/ν, does not reach the edge of the second band, EF < E2− E1 This Fermi energy is smaller by two orders of magnitude than atypical Fermi energy in metals The Fermi wavelength,λ =√2π/n0, ranges from 40 to

sub-80 nm and also exceeds the typical Fermi wavelength in metals by two orders of tude This means that the quantum effects in a 2DEG can be seen at much larger spacescales than in common metals

magni-1.2 Quantum contacts

A common nanostructure does not even remotely resemble an infinitely long waveguide.However, the physics of quantum transport is surprisingly similar to that of a waveguide.The recognition of this fact and its experimental verification was, and still is, one of themain events in the history of the field We introduce this important idea in two steps In

this section, we consider in detail the quantum point contact (QPC) – a system without potential barriers – and show that it is equivalent to a waveguide with a potential barrier In

Section 1.3, we turn to the more complicated case of a generic nanostructure

1.2.1 Adiabatic quantum transport

We start by looking at a waveguide of variable cross-section The waveguide is extended

along the x axis, bounded by impenetrable potential walls, and has a rectangular

cross-section,|y| < a(x)/2, |z| < b(x)/2, with the dimensions varying as one moves along the contact Far to the right and the left, x→ ±∞, these dimensions assume constant val-

ues a∞and b∞ In the middle, the walls come closer, forming a constriction (Fig 1.7).The solutions, Eqs (1.7), found for the ideal waveguide do not apply to this case, andsolving the Schrödinger equation is cumbersome (The variables in the three-dimensionalSchrödinger equation do not separate and the motion does not become one-dimensional.)

We obtain, however, a general understanding of the quantum waves in the system by

looking at an adiabatic waveguide [3] Its dimensions are assumed to vary smoothly sothat the length scale at which they change is much longer than the dimensions themselves:

a = const (x)

a(x) y

tFig 1.7. (a) From a waveguide of constant cross-section to an adiabatic waveguide (b) Effective potential

energy for the transport channels At the given energy E, only three transport channels (solid

curves) are open.

|a(x)|, |b(x)|  1, a(x)|a(x)|, b(x)|b(x)|  1 Under these conditions, the walls are locally flat and parallel, so that locally the wave functions can be approximated by those

of the ideal waveguide (Eqs (1.7)) The variables are locally separated, so that

ψn (x, y, z) = ψ(x) n (a(x), b(x), y, z), (1.13)where the transverse wave functions (a, b, y, z) are given by Eqs (1.7) The function

ψ(x) corresponds to one-dimensional motion and satisfies

Here, E n presents a channel-dependent energy introduced by Eq (1.8) This energy

depends on x via the waveguide dimensions a(x), b(x):

We note that this term plays the role of potential energy for one-dimensional motion.

Strangely, this potential energy depends on the channel index Let us plot these energies

versus x (see Fig.1.7) For each channel we see a potential barrier forming in the narrowest

part of the constriction The bigger the numbers n y , n z, the higher the barrier

Let us concentrate on a given energy E In a given channel, we compare it with the maximal barrier height assuming an impenetrable barrier If E exceeds the height, the

electrons coming to the constriction traverse it; otherwise, they are reflected back Since

the barrier height increases with the channel index, there are only a finite number of open channels where electrons can pass the constriction All other channels are closed.

Therefore, the adiabatic waveguide of variable cross-section without a potential barrierappears to be the same as for an ideal waveguide with a potential barrier, already considered

in Section1.1 For each channel, we define transmission and reflection amplitudes in theusual way (Fig.1.3) and we end up with the channel-dependent transmission coefficient

Tn (E) It appears that the adiabaticity also implies an almost classical potential barrier, so

that T = 1 for open channels and T = 0 for closed ones The only exception is a narrow

energy interval where the energy almost aligns with the top of the barrier Remarkably, theelectrons in the closed channels are almost perfectly reflected in spite of the absence ofpotential barriers in the system

Control question 1.5. Assume the waveguide dimensions depend on x as follows: a(x) = b(x) = a∞− a0/(1 + (x/ξ)2), a0< a∞ At which energy does the first openchannel appear? At which energy are there three open channels?

Now we are ready to turn to our core business: the quantum transport First, we mine the electric current in the constriction The first step is to adapt the expression forthe current given by Eq (1.2) to the case of quantized transverse motion We do this by

deter-replacing the integration over k y and k z by the summation over their discrete quantized

values k n y and k z n, as follows:

This procedure describes an ideal waveguide, in which case k x stands for the wave vector

In our case, the waveguide is not ideal and the wave vector depends on x However, we have chosen the shape in such a way that for x → ±∞ the waveguide is ideal, and we can use

Eq (1.16) to evaluate the full current I via the cross-section located infinitely far to the left

from the constriction Note that we are free to choose the cross-section in an arbitrary waysince the charge conservation law implies that the stationary full current flowing through

any cross-section is the same We get the full current by multiplying the current density j x

by the cross-section area a∞b∞, thus absorbing the factors in Eq (1.16),

different for open (T = 1) and closed (T = 0) channels (Fig.1.8) If the channel is closed,all electrons passing the cross-section from the left are reflected from the barrier and sub-sequently pass the same cross-section from the right Therefore, in a closed channel there

is the same amount of right- and left-going electrons, and the filling factors are the same

for the two momentum directions, f n (k x)= f n(−kx) Since their velocities are opposite,the contribution of the closed channels to the net current vanishes Thus, we concentrate

on open channels

For open channels, the filling factors for the two momentum directions are not the same

To realize this fact, we have to understand how the electrons get to the waveguide This

leads us to the concept of a reservoir Any nanostructure taking part in quantum transport

is part of an electric circuit This means that it is connected to large, macroscopic electric

pads each kept at a certain voltage (electrochemical potential) These pads contain a largenumber of electrons at thermal equilibrium These electrons are characterized by the fill-ing factor, Eq (1.3), which depends only on the energy and the chemical potential of thecorresponding reservoir In our setup, the waveguide is connected to two such reservoirs:

0 f 1 0 f 1

E

μL

μR

tFig 1.8. Filling factors in a quantum paint contact for open and closed channels.

left (x → −∞) and right (x → ∞) Electrons with k x > 0 come from the left reservoir and have the filling factor fL(E) ≡ fF(E − μL) Electrons with k x < 0 come to the cross-

section having passed the constriction Therefore, they carry the filling factor of the right

reservoir, fR(E) ≡ fF(E − μR)

Since the filling factors depend only on the energy, it is natural to replace k x in favor of

the total energy E for each momentum direction Since the velocity is vx = −1(∂ E/∂kx),

we have dE = v(k x )dk x, and this cancels the velocity in Eq (1.17) Thus, we end up withthe remarkably simple expression,

We have integrated over energy; this yields a factorμL− μR The simplest way to

inte-grate is to assume vanishing temperature Then the filling factors fL,R= (μL,R− E)

differ only within an energy strip min(μL,R)< E < max(μL,R) and are constant within thestrip The width of the strip is given by|μL− μR|

Exercise 1.1. (a) Making use of the explicit form of the Fermi distribution fF, showthat the integral



dE

fL(E) − fR(E)equalsμL− μRat any temperature

(b) Prove that the integral retains the same value for any function f (E) expressing the filling factors, fL,R= f (E − μL,R), provided f → 0 at E → ∞ and f → 1 at

E → −∞

The difference in the chemical potentials corresponds to the voltage difference applied,

V = (μL− μR)/e The voltage difference drives the current; there is no current at V = 0

since that corresponds to the state of thermodynamic equilibrium The factorμL− μRisthe same for all open channels Therefore the current is proportional to the number of open

channels, Nopen, and the voltage The proportionality coefficient is called the conductance quantum and conventionally defined as3 GQ = 2se2/(2π) The conductance of the sys- tem, I /V , appears to be quantized in units of GQ This factor is made up from fundamentalconstants The conductance quantum does not depend on material properties, nanostruc-ture size, and geometry, or from a concrete theoretical model used to evaluate the transportproperties

Equation (1.18) is a specific case of the celebrated Landauer formula We have derived

the formula for the case when T (E) can be either zero or one The general case is treated

in Section 1.3

Let us remark that we have obtained this very general relation in the framework of aspecific model of a constricted adiabatic waveguide with impenetrable walls and rectangu-lar cross-section Now we discuss why this result holds in a far more general setup First,let us get rid of the assumption of impenetrable walls and rectangular cross-section, and

introduce a waveguide with an arbitrary confining potential U x (y, z) Provided the

wave-guide is adiabatic, we can still separate the variables in the Schrödinger equation and writethe solution in the form of Eq (1.13), where the transverse wave functions now obey thefollowing equation:

where the discrete index n labels the transverse states, E n (x) being the channel-dependent

potential energy (see Eq (1.15)) This is the only change compared with the previousmodel

Next, let us note that the number of open channels, and, consequently, the conductance

of our system, are determined only by the narrowest part of the waveguide Therefore wecan change the shape of the waveguide without changing its transport properties, providedthe narrowest cross-section stays the same Let us see what happens if we change it in theway shown in Fig.1.9, sending a∞and b∞to infinity The structure we end up with – a

quantum point contact (QPC) – is not a waveguide at all In particular, in a waveguide we

have a finite number of channels at each energy and the spectrum consists of discrete energybranches (Fig.1.2) In contrast to this, the number of transport channels approaching the

QPC is infinite, and the energy spectrum is continuous Of all these channels, only a finite

number are transmitted through the constriction

3 Eventually, it could be more logical to incorporate the number of spin directions 2sinto the number of open

channels, so that one has two transport channels for each n.

tFig 1.9. From a model to real life (a) Adiabatic waveguide with a finite number of transport channels.

(b) QPC: infinite number of transport channels, finite number of open channels (c) Scattering

around the QPC is negligible provided the resistance of scattering region RQPC is much smaller (d) Experimental realization of QPC in a 2DEG with a split gate (adapted from Ref [ 4 ]).

1.2.2 Experimental evidence of conductance quantization

Quantization of conductance was first observed in GaAlAs–GaAs semiconductor erostructures [4,5] In these heterostructures, the electrons are confined near the surfaceforming a 2D electron gas (2DEG); see Section1.1 In terms of our waveguide, this means

het-that one of the dimensions b → 0 Then only the lowest subband (n z = 1) is relevant Inaddition, two gate electrodes were imposed on the top of the heterostructure These elec-trodes were electrically isolated from the 2DEG However, they were used to shape this2DEG; the potential applied to these electrodes repels the electrons from them (Fig.1.9),creating surrounding impenetrable walls The constriction is formed by the walls in the gap

between the electrodes, the width corresponding to the dimension a of our model

wave-guide Increasingly negative voltage created greater repulsion, shifting the walls outwards

and therefore making the constriction narrower The minimum width aminis thus controlled

by the gate voltage

The number of open channels, in its turn, is determined by the minimum width Let uswrite down this dependence explicitly for a model that disregards the potential inside the2DEG and assumes infinite potential outside (a more realistic model is treated later in this

section) A new channel with index n = (n y, 1) opens when the energy position of the top

of the barrier, W n , passes the Fermi energy as we change amin:

tFig 1.10. Experimental evidence of conductance quantization: discrete transport channels are visible.

Adapted from Ref [ 4 ].

and thus

where the brackets denote the integer part Thus, by varying the gate voltage, one changesthe number of open channels Therefore, the gate voltage dependence of the conductance

is expected to look like a set of stairs with step height GQ, and this is what was measured

in the experiment discussed in Ref [4] (see Fig.1.10)

Control question 1.6. Comparing Figs.1.9and1.10, estimate the upper bound for kF

in the 2DEG used in the experiment discussed in Ref [4]

At the time of the experiment, it was expected that such quantization would be observedfor an ideal waveguide, but it was a complete surprise to observe it in a relatively shortconstriction in a far from ideal system We cite Ref [4]:

We propose an explanation of the observed quantization of the conductance, based on the assumption of quantized transverse momentum in the contact constriction In prin- ciple this assumption requires a constriction much longer than wide, but presumably the quantization is conserved in the short and narrow constriction of the experiment.

This quote, written a year before a theoretical understanding of quantum point contactwas achieved, shows the essence of this section: discrete transport channels do not needwaveguides to persist

The conductance steps observed in the experiment are not very sharp In reality, thetransmission coefficient of a given channel does not change abruptly from zero (closed)

to one (open) This change is only abrupt if the reflection from the barrier is classical As

we have learned in the example of a simple barrier considered in Section1.1, quantum

mechanics makes the transmission coefficient a continuous function of energy This willalso be true for a QPC.

To illustrate this, let us concentrate on the energy E that lies close to the top of the barrier

Wn at certain n y , E = W n + δE, |δE|  W n In this case, the potential close to the top of

the barrier can be expanded in Taylor series, E n (x) = W n − |E

n(0)|x2/2 (we assume that the top of the barrier is located at x = 0) Thus, electrons with E ≈ W n are transmitted

through a parabolic potential This well known quantum-mechanical problem was solved

by Kemble in 1935 The transmission coefficient obtained from the solution,

smoothly joins the two classical values, T n = 1 at δE   n and T n = 0 for |δE|   n,

δE < 0, thus providing the smearing of the steps at energy scale n This energy scale isgiven by

with a, a being taken at x = 0 We see that this energy scale is much smaller than

Wn provided the adiabaticity condition aa 1 is met Moreover, this smearing scale

is much smaller than the energy distance between the opening of consecutive channels,

Wn+1− W n This makes the cross-over sharp, even for moderately adiabatic constrictions

(aa∼ 0.3 for the experiment mentioned)

To evaluate the conductance, we cannot use Eq (1.18), since it only applies for T = 0

or T = 1 We need the full Landauer formula (to be derived in Section1.3):

At temperatures and voltages much smaller than n, the integration over energy is

essen-tially multiplication of the integrand with eV , with the substitution E → μ The opening

of the nth channel occurs at |μ − W n |   nand is described by

1.2.3 Electrostatic shaping of 2DEG

The QPC in the experiment considered was shaped electrostatically with the gate trodes We have understood that the conductance is determined by the number of openchannels, this number being changed by the gate voltage Let us now explore the details ofthe electrostatics of the shaping

elec-In the following we present a piece of classical rather than quantum physics; we discuss

it here because of practical importance and because it is a piece of interesting physics We

tFig 1.11. Depletion of 2d gas by a side gate.

loosely follow Refs [7] and [8] in this discussion It turns out that plain classical statics is more important for the 2DEG shaping than any quantum effects related to themotion of electrons in the 2DEG To demonstrate this, let us first make a simple estimation

electro-of the electrostatic potential involved In the heterostructures used, electrons are confined

to a plane Let us look at one semi-infinite gate electrode repelling a semi-infinite 2DEG(Fig.1.11), both lying in the plane z = 0 and separated by distance l The 2DEG density is defined by the fixed density n0of donors located beneath the heterostructure surface This

means that far from the gate (y → ∞) the electron density equals n0to compensate for thispositive-charge background The same compensation takes place in the gate far from the

2DEG (y→ −∞) The charge density is not compensated for at distances of the order of

l This resembles a capacitor with the metal plates wrenched to be in the same plane The

voltage across the capacitor can be still estimated using the formula for a planar capacitor,

Vg

the vacuum, due to the large dielectric constant

the order of en0, so that Vg∼ eln0

of electrons in the 2DEG, EF= (kF)2/(2m) ∼ 2n0/m We see that the potential energy dominates provided l  aB≡ 2 2m), aB being the Bohr radius in the semiconduc-

tor (aBis 10 nm for GaAs) This condition is thus fulfilled for l in the 100 nm range In

this parameter range, the 2DEG can be considered as an ideal conductor that screens theelectric field, very much like a metallic gate electrode

A surprising result of this approximation is that at Vg→ 0, l → 0, and it looks like the

2DEG is depleted from under the gate This is because, in the above reasoning, we have

disregarded the distance d from the top metallic gate to the 2DEG proper in comparison with l If we take d into account, the 2DEG stays under the gate at zero gate voltage, but

is depleted at relatively small gate voltages Vg= 4πen0d

reference for further discussion

In what follows, we consider electrostatics to quantify the gate voltage dependence ofthe 2DEG shape

Although the charge density in the 2DEG and the gate electrode is concentrated in the

z= 0 plane, the distribution of electrostatic potential experienced and created by trons is essentially three-dimensional; i.e., one has to consider the potential in the whole

elec-three-dimensional space to find it in the (x, y) plane in which the electrons are located In

principle, one has to solve the electrostatic problem separately in two half-spaces: in the

axis, the geometry that corresponds to an ideal waveguide considered previously Thus,

the electric field only has two components, E y and E z The positively charged donors

pro-duce electric field in the z direction, E z(d)= −(4πn0e

this from the total field and consider the complex-valued fieldE ≡ E y + iE zproduced byall other charges This allows us to incorporate a common trick that enables us to solve

a variety of electrostatic problems: ifE is an analytic function of the complex coordinate

u ≡ y + iz, it automatically satisfies the Poisson equation for z = 0.

This solution must obey the boundary conditions that we now describe Both the metallic

gate (y < 0) and the 2DEG (y > l) are ideal conductors, so the potential is constant along each conductor The in-plane component of the field, E y , thus vanishes at y < 0 and y > l The component E z should experience a jump at z= 0 proportional to the charge density

(with the density of donors subtracted) Symmetry requires E z (y + i0) = −E z (y− i0) We

conclude that E z(0)= 0 for 0 < y < l, whereas far from the capacitor, y → ±∞,

Ez (y + i0) = −E z (y− i0) = 4πn0e

In addition, the 2DEG must be in mechanical equilibrium; no net force is acting on

elec-trons close to its boundary This implies that E y (l)= 0 This does not apply to the gateelectrode; the electrostatic force acting on the charge in there may be compensated for by

elastic forces keeping the gate at the substrate so that E y(0)= 0

The above conditions are easy to reformulate in terms of the complex functionE; it is real

and single-valued at 0< y < l and z = 0, it is imaginary, and has branch cuts elsewhere

at z = 0, E(y + i0) = −E(y − i0) This is the characteristic property of the square root

function! We conjecture thatE is a square root of a product or a ratio of two polynomials The simplest guess satisfying the boundary conditions at y → ∞, y < 0, and y = l reads

Control question 1.7. Check that all boundary conditions are indeed satisfied To this

end, explicitly evaluate the imaginary and real parts of the above expression (E z , E y)

tFig 1.12. Split gate: electrostatic shaping of a 2DEG.

Now we relate the distance l to the gate voltage:

Vg=

 l0

Exercise 1.2. Consider a thin long gate hanging at a distance a from the 2DEG plane.

If the width of the gate is much smaller than a, it can be viewed as an infinitely thin wire

with uniform charge density ˜q per unit length Upon increasing ˜q, the electron density

underneath the gate decreases Evaluate the critical value of ˜q at which the 2DEG is

completely depleted at a certain position, i.e its density approaches zero there

Let us return to the shaping of the QPC We now have two gates with the same

poten-tial Vg separated by distance W A strip of 2DEG of width a is formed in between (see

Fig.1.12) We begin with a naive and straightforward model Assume that the 2DEG is

depleted by both gates independently Then a = W − 2l At a certain pinch-off voltage



=



kFW π

Vp− Vg

Vp



Thus, the conductance decreases linearly with the gate voltage

Let us try to improve on the model First, let us improve our consideration of

electrostat-ics There are two gates, left (y < −W/2) and right (y > W/2), and the 2DEG is confined

in the area|y| < a/2, with a to be determined The boundary conditions are similar to

0 1

tFig 1.13. Split gate (a) Voltage versus a/W (b) The number of open channels versus a/W (c) The resulting

voltage dependence of the number of channels Dashed lines show estimates based on

of the strip width on the gate voltage is not linear (see Fig.1.13)

Next, we improve on the relation between the number of open channels and the width

of the strip a Naively, we have assumed that all potential in the 2DEG is screened This is indeed true for a large potential of the order of Vg In fact, there is a residual potential in the

2DEG which is of the order EF/e  Vg One can see this from the fact that the electron

density in the 2DEG differs from the bare value of n0, i.e

2s kF2/(2π)2, was derived in Section1.1 The sum of the potential and the kinetic energy

of an electron equals EF, thus EF= Ur+ 2kF2/2m We can therefore express the residual

There is a shortcut, however In the ideal waveguide, we saw that transverse wave tions obey the condition that an integer number of half-wavelengths is contained betweenthe walls In other words, the total phase change of the wave function between the wallsshould beπny , n y being a positive integer The maximum n y allowed at a given energyyields the number of open channels at this energy If the wave vector varies smoothly with

func-y, the phase change equals 

dy k y (y) The maximum n y at energy EF is given by the

substitution k y (y) = kF(y), and the number of open channels is thus given by

Let us now complete the work and plot the number of open channels as a function of the

gate voltage normalized to Vp What a sad irony! The function evaluated hardly differs fromthe naive linear estimate of Eq (1.26) This is intrinsic for all detailed studies of quantumtransport: harder work that is intended to include all possible parameters characterizing

a nanostructure yields very little improvement in comparison with the “naive” reasoning,

provided the reasoning is correctly based on general laws of quantum transport.

1.3 Scattering matrix and the Landauer formula

In Sections 1.1 and 1.2, we studied electron transport in idealized waveguides with orwithout a potential barrier These waveguides not only illustrate the concepts of quantumtransport, but also model concrete experimental situations A waveguide with no poten-tial barrier models a QPC, a constriction created by gates in a 2DEG A waveguide with

a potential barrier models electron propagation through an insulating layer between twometals

Real nanostructures can be made in a variety of ways, and can be more complicated.Modern fabrication technology allows for sophisticated semiconductor heterostructures,combining and shaping different metals, using nanotubes, molecules, and even singleatoms as elements of an electron transport circuit Various means can be used to controlthe transport properties of a fabricated nanostructure It is only possible to describe all this

in a single book because all these systems obey the general laws of quantum transport that

we formulate in this section

There is a common feature of all fabrication methods: two nanostructures that are

intended to be identical, that is, are made with the same design and technology, are never

identical (see Fig.1.14) Beside the artificial features brought by design, there is also order originating from defects of different kind inevitably present in the structure The

dis-position of and/or potential created by such defects is random, and in most cases can beneither controlled nor measured It is unlikely that this situation will change with fur-ther technological developments; even if one achieves a perfect control of every atom in

tFig 1.14. Nanostructures of an identical design are never identical.

a nanostructure, one would not be able to control all the atoms in the macroscopic tact leads, which cannot be separated from the nanostructure The defects scatter electrons,affecting the transport properties Conductance of the structure is thus random, depending

con-on a specific realizaticon-on of disorder in the structure and in the leads; this means there is aformidable number of uncontrollable parameters

Fortunately, the transport properties of any nanostructure can be expressed through asmaller set of parameters The condition for this is that electrons traverse the structurewithout energy loss, so they experience only elastic scattering These conditions for a givenstructure are always achieved at sufficiently low temperature and applied voltage The scat-

tering is characterized by a scattering matrix that contains information about electron wave functions far from the structure The transport is described by a set of transmission eigen- values derived from this scattering matrix A great deal of literature on quantum transport,

and a great deal of this book, is in fact devoted to evaluation of the transmission ues and establishing their general properties In this section, we derive the relation betweenconductance and the transmission eigenvalues and thus demonstrate that understandingthe transmission properties of a system automatically means understanding its transportproperties

eigenval-1.3.1 Scattering matrix

We have mentioned in Section1.2that any nanostructure taking part in quantum transport

is part of an electric circuit It is connected to several reservoirs, which are in thermal

equilibrium and are characterized by a fixed voltage In this section, we only considerthe case when there are two reservoirs (referred to as left and right) Generalization tomany reservoirs is given in Section1.5 Between the reservoirs is the scattering region –

the nanostructure proper Let us start with a feature borrowed from the QPC model ofSection 1.2: ideal waveguides connect the reservoirs and the scattering region (Fig.1.15).This is convenient since the scattering only takes place in a finite region, the reservoirsbeing far from this region The wave functions may have very complicated forms in thescattering region, but in the waveguides they are always combinations of plane waves Theleft and right waveguides do not have to have the same axis and the same cross-section

This is why it is convenient to introduce the separate coordinates xL< 0, yL, zLand xR>

0, yR, zRfor the left and right waveguides, respectively Generally, a wave function at fixed

energy E can be presented as a linear combination of the plane waves