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2006

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In this thesis I deal with two effects which are related to the

Quantum Hall systems. In Chapter 2

I study the Coulomb magneto-drag, particularly concentrating on the

question of the so-called anomalous drag recently observed

experimentally. The observed sign reversal of the

drag signal poses an intriguing task for theoretical investigations.

Chapter 3 is an attempt to predict possible a Jahn-Teller

effect in Si inverse layers. There I consider the interplay between

the valley splitting, the electron-phonon and electron-electron

interactions in these systems and their effect on the ground state

and low-lying skyrmionic excitations.

Quantum Hall systems. In Chapter 2

I study the Coulomb magneto-drag, particularly concentrating on the

question of the so-called anomalous drag recently observed

experimentally. The observed sign reversal of the

drag signal poses an intriguing task for theoretical investigations.

Chapter 3 is an attempt to predict possible a Jahn-Teller

effect in Si inverse layers. There I consider the interplay between

the valley splitting, the electron-phonon and electron-electron

interactions in these systems and their effect on the ground state

and low-lying skyrmionic excitations.

In dieser Dissertation beschaeftige ich mich mit zwei mit Quanten-Hall-Systemen verbundenen Effekten. Im Kapitel 2 ist eine Untersuchung des sogenannten anomalen Magneto-Drag, der cor wenigen Jahren experimentell entdeckt wurde. Der beobachtete Vorzeichenwechsel des Drags stellt eine Herausvorderung fuer die theoretische Studien dar.

Kapitel 3 ist ein Versuch, den Jahn-Teller-Effekt in Si-Inversionsschichten vorherzusagen. Dieser ist ein Resultat der Zusammenwirkung von Energietal-Aufspaltung, Elektron-Phonon- und Elektron-Elektron-Wechselwirkungen in diesen Systemen. Der Grundzustand und die Energie eines Skyrmions werden untersucht.

Kapitel 3 ist ein Versuch, den Jahn-Teller-Effekt in Si-Inversionsschichten vorherzusagen. Dieser ist ein Resultat der Zusammenwirkung von Energietal-Aufspaltung, Elektron-Phonon- und Elektron-Elektron-Wechselwirkungen in diesen Systemen. Der Grundzustand und die Energie eines Skyrmions werden untersucht.

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Coulomb Drag and Jahn-Teller Effect in
Two-Dimensional Electron Systems in Strong

Magnetic Fields

Von der Fakulta¨t fu¨r Mathematik und Physik der Universita¨t Stuttgart

zur Erlangung der Wu¨rde eines Doktors der

Naturwissenschaften (Dr. rer. nat.) genehmigte Abhandlung

Vorgelegt von

Sergej Brener

aus Moskau

Hauptberichter: Prof. Dr. W. Metzner

Mitberichter: Prof. Dr. A. Muramatsu

Tag der mu¨ndlichen Pru¨fung: 11. Mai 2006

Max-Planck Institut fu¨r Festko¨rperforschung

2006

Contents

Deutsche Zusammenfassung 5

1 Overview 9

2 Theory of Coulomb Magneto-Drag 11

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 History of the problem. Experiment. . . . . . . . . . . . . . . 12

2.3 History of the problem. Theory. . . . . . . . . . . . . . . . . . 17

2.3.1 Early works. . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3.2 Boltzmann equation approach. . . . . . . . . . . . . . . 20

2.3.3 Diagrammatic approach. . . . . . . . . . . . . . . . . . 22

2.3.4 Magneto-drag. Early diagrammatic approach. . . . . . 29

2.3.5 Negative magneto-drag. First explanation attempt. . . 30

2.3.6 Diagrammatic calculation of magneto-drag in the SCBA. 33

2.4 Semiclassical theory of electron drag in strong magnetic fields 41

2.4.1 Drag between parallel links. . . . . . . . . . . . . . . . 46

2.4.2 Drag between non-parallel links. . . . . . . . . . . . . . 70

2.4.3 Connection to two-dimensional drag. . . . . . . . . . . 80

2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

3 Possible Jahn-Teller effect in Si-inverse layers 83

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

3.2 Hamiltonian of electron gas on Si interface . . . . . . . . . . . 85

3.3 SU(4) Symmetric Case . . . . . . . . . . . . . . . . . . . . . . 88

3

4 CONTENTS

3.4 Jahn-Teller effect . . . . . . . . . . . . . . . . . . . . . . . . . 92

3.5 Phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . . 96

3.6 Anisotropic energy of skyrmion . . . . . . . . . . . . . . . . . 98

3.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

3.8 Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

3.9 Appendix B . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

3.10 Appendix C . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

4 Summary 107

Deutsche Zusammenfassung

Seit Jahrzehnten zogen zwei-dimensionale elektronische Systeme das Inter-

esse vieler Physiker und Technologen an. Der Grund dafu¨r ist nicht nur - wie

man sich naiv vorstellen ko¨nnte - die technologische Anwendung in Rech-

nerprozessoren. Die scheinbar einfache physikalische Konfiguration zwei-

dimensionaler Elektronensysteme fu¨hrt zu unerwartet vielen experimentell

beobachtbaren Effekten und theoretischen Modellen, die eine eigene Bedeu-

tung fu¨r die Entwicklung von Physik, Technologie, Computerrechnungen

usw. haben.

Seit der Entdeckung des Quanten-Hall-Effekts [1] galt zwei-dimensionalen

elektronischen Systemen in starken Magnetfeldern (sogenannte Quanten-Hall-

Systeme) ein besonders starkes Interesse. Die physikalischen Effekte, die

in solchen Systemen zu beobachten sind, sind ein Resultat der Zusamen-

wirkung von Landau-Quantisierung der elektronischen Bewegung, Elektron-

Elektron-Wechselwirkung, Bandstruktur des Materials (normaleweise Si oder

GaAs), Unordnungseffekte usw. Unter den interessanten Aspekten zwei-

dimensionaler elektronischer Systeme kann man folgende Themen nennen:

Quanten-Hall-Effekt, Coulomb-Drag, skyrmionische Phasen, Energietal-Auf-

spaltung (valley splitting) in Si-Inversionsschichten, Auswirkung der Mikro-

wellenstrahlung auf die Transporteigenschaften der Quanten-Hall-Systeme

und viele andere.

In der vorgelegten Dissertation bescha¨ftige ich mich mit zwei mit Quanten-

Hall-Systemen verbundenen Effekten: dem Coulomb-Drag im Kapitel 2 und

dem Jahn-Teller-Effekt in Si-Inversionsschichten im Kapitel 3.

5

6 DEUTSCHE ZUSAMMENFASSUNG

Kapitel 2 ist ein Resultat der durch die Werke [2, 3] motivierten Unter-

suchung des sogenannten anomalen Magneto-Drag, der in den obengenannten

Referenzen experimentell entdeckt wurde. Der Drag (auch Coulomb-Drag

genannt) ist ein physikalischer Effekt, der in elektronischen Doppelschicht-

systemen beobachtet wird. Durch eine Schicht wird Strom gefu¨hrt, der in der

zweiten Schicht eine elektrische Spannung verursacht. Falls das System sich

in einem senkrechten Magnetfeld befindet, wird der Drag als Magneto-Drag

bezeichnet. Der anomale Magneto-Drag besteht darin, dass unter gewissen

Unsta¨nden der Magneto-Drag sein Vorzeichen wechselt und negativ wird.

In diesem Kapitel stelle ich detaillierte Resultate fu¨r ein Modellproblem,

das eng mit dem anomalen Magneto-Drag verbunden ist, dar. Nach einer

kurzen Einfu¨hrung gebe ich im zweiten Teil eine kurze U¨bersicht der Experi-

mente, die in den letzen 15 Jahren durchgefu¨rt wurden. Besondere Aufmerk-

samkeit ist denjenigen Experimenten gewidment, die sich auf das Disserta-

tionsthema beziehen. Der dritte Teil entha¨lt eine detaillierte Zusammenfas-

sung der heutigen theoretischen Ansa¨tze zum Dragproblem. Insbesondere

wird erla¨utert, warum diese Ansa¨tze nicht genu¨gen, um die u¨berraschenden

Eigenschaften des Magneto-Drag zu beschreiben. Der vierte Teil ist der

Hauptteil des Kapitels. Hier fu¨hre ich zuna¨chst das Modell ein, mit dem

ich mich in diesem Teil bescha¨ftigen werde. Dann zeige ich verschiedene

Wege, mit denen das Modell behandelt werden kann, vergleiche die Resul-

tate, bespreche ihre Relevanz und Zuverla¨ssigkeit. Schliesslich zeige ich die

Verbindungen zwischen dem betrachteten Modell und dem Originalproblem

und bringe u¨berzeugende Argumente um zu zeigen, dass diese Verbingungen

stark genug sind, um viele (wenngleich nicht alle) interessante Eigenschaften

des zwei-dimensionalen Magneto-Drags zu erkla¨ren.

Kapitel 3 ist ein Versuch, den Jahn-Teller-Effekt in Si-Inversionsschichten

vorherzusagen. Dieser ist ein Resultat der Zusammenwirkung von Energietal-

Aufspaltung, Elektron-Phonon- und Elektron-Elektron-Wechselwirkungen in

diesen Systemen. Das Kapitel ist wie folgt aufgebaut. Im Teil 3.2 wird der

Hamilton-Operator des Systems eingefu¨hrt, der in einen SU(4)-symmetrischen

DEUTSCHE ZUSAMMENFASSUNG 7

und einen nicht SU(4)-symmetrischen Teil aufgespaltet wird. Im Teil 3.3

wird der SU(4)-symmetrische Fall untersucht. In den Teilen 3.4, 3.5 fu¨hre

ich die anisotropen Terme des Hamilton-Operators ein und untersuche deren

Einfluss auf den Grundzustand des Systems fu¨r verschiedene ganzzahlige

Fu¨llfaktoren. Im Teil 3.5 wird das Phasendiagramm fu¨r den interessan-

teren Fall ν = 2 gegeben. Dabei werden drei Phasen vorhergesagt, na¨mlich

eine ferromagnetische, eine Spin-Singlet und eine antiferromagnetische Phase.

Im Teil 3.6 wird der Effekt der nicht-symmetrischen Terme im Hamilton-

Operator auf die Energie eines Skyrmions untersucht.

Kapitel 4 entha¨lt schließlich eine kurze Zusammenfassung aller Resul-

tate.

8 DEUTSCHE ZUSAMMENFASSUNG

Chapter 1

Overview

Over decades two-dimensional electron systems have drawn the attention of

an extremely large number of physicists and technologists. The reason for

this interest is not only - as one might naively suppose - due to the techno-

logical applications in computer processors. This seemingly simple physical

configuration gives rise to an unexpectedly rich variety of experimentally ob-

servable effects and theoretical models which are themselves important for

the development of physics, technology, computing etc.

Since the discovery of the quantum Hall effect [1] a particular attention

has been payed to two-dimensional electron systems subjected to a strong

magnetic field (the so-called quantum Hall systems). The physical effects

observed in such systems are the result of an interplay between the Landau

quantization of the electron motion, the electron-electron Coulomb interac-

tion, the band structure of the material (usually Si or GaAs), disorder effects

etc. Among the interesting aspects of the two-dimensional electron physics

one can mention the fractional quantum Hall effect, Coulomb drag, skyrmion

phases, valley splitting in Si inverse layers, bilayer quantum Hall systems, ef-

fects of the microwave radiation on transport properties of the quantum Hall

systems and many others.

In this thesis I deal with two effects which are related to the Quantum Hall

systems. In the main part of the work, Chapter 2, I study the Coulomb

9

10 OVERVIEW

magneto-drag, particularly concentrating on the question of the so-called

anomalous drag recently observed experimentally [2, 3]. The observed sign

reversal of the drag signal poses an intriguing task for theoretical investiga-

tions.

Chapter 3 is an attempt to predict possible a Jahn-Teller effect in Si

inverse layers. There I consider the interplay between the valley splitting,

the electron-phonon and electron-electron interactions in these systems and

their effect on the ground state and low-lying skyrmionic excitations.

In Chapter 4 a short summary of the results is given.

Chapter 2

Theory of Coulomb

Magneto-Drag

2.1 Introduction

In this chapter I will present extended results for a model problem closely

related to the well-known and still not closed problem of the two-dimensional

magneto-drag. In the next section I’ll give a brief review of history of the

experiments during the last 15 years. Special attention will be drawn to the

results closely related to the theme of this work. In the third section I’ll

give a detailed overview of the present theoretical approaches to the drag

problem. An important part of that section will be devoted to explaining of

why do these approaches fail to explain the striking features of the magneto-

drag. The forth section is the main part of the work. There I’ll start with

presenting the model, I’ll deal with in that section. Then I’ll show different

approaches to that model, compare the results, discuss there relevance and

reliability. Finally, I’ll show links between the considered model and the

initial problem and will bring convincing arguments to show that these links

are strong enough to explain many (though not all) interesting features of

the two-dimensional drag.

11

12 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

2.2 History of the problem. Experiment.

Electron-electron interactions are responsible for a multitude of fascinating

effects in condensed matter. They play an important role in such different

phenomena as high-temperature superconductivity, fractional quantum Hall

effect, Wigner crystallization or Coulomb gaps in disordered media. The

effects of this interaction on transport properties, however, are difficult to

measure. In the beginning of the 90-ties, a new technique has been proved to

be effective in measuring the scattering rates due to the Coulomb interaction

directly. [4].

This technique is based on an earlier proposal by Pogrebinski˘ı [5, 6]. The

prediction was that for two conducting systems separated by an insulator (a

semiconductor-insulator-semiconductor layer structure in particular) there

will be a drag of carriers in one film due to the direct Coulomb interaction

of the carriers in the other film. This effects resembles conventional friction

that is why Coulomb drag is often referred to as frictional drag. Unlike the

one-layer case, where Coulomb forces alone can not lead to finite conductance

due to momentum conservation, in the double layer setup current in one layer

leads to a finite mean force acting on the carriers in the other layer, which in

turn leads to a finite current there. If the second layer is an open circuit the

interaction with the electrons from the first layer leads to redistributing of

charge density and to building a voltage in the second layer which opposes

the frictional force and exactly cancels it. This voltage is referred to as drag

voltage.

The standard experimental setup can be seen in Fig. 2.1. A constant

low-frequency current (typically 25 Hz) is imposed on one of the layers (drive

layer), and the drift of those electrons creates a frictional drag on the electrons

in the adjacent layer (drag layer). Although for theoretical investigations it is

more convenient to consider both layers as closed circuits, in practice the drag

layer is an open one which allows to measure the drag voltage induced there.

Typical experimental parameters for drag experiments are: the well thickness

δ ranges from 10 to 30 nm, the layer separation d lies typically between 20 and

2.2 HISTORY OF THE PROBLEM. EXPERIMENT. 13

Figure 2.1: (a) Sample geometry, including the mesa and the front and back

gates in gray. The central bar is 20 µm wide. Each mesa arm is terminated

with an indium diffusion contact roughly 1.5 mm from this central region.

(b) Schematic of the measurement configuration [4].

120 nm. This ensures that no tunneling occurs between the layers and that

the only way the electrons from different layers may interact is the direct

(screened) Coulomb force (or, under certain circumstances, quasi particle-

mediated interaction may occur, such as phonon or plasmon interaction).

The measured value is the so-called drag resistance which is defined as

ρD = −VD I . (2.1)

Here VD is the drag voltage and I is the current driven through the drive

layer. The minus sign means that the direction of the drag voltage is normally

opposite to that of the current.

The experimental plots can be seen in Figs. 2.2, 2.3. It can be seen that

the naive phase space arguments that predict T 2 low-temperature behavior

of the drag (only a layer of electrons around the Fermi surface of the order

14 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

Figure 2.2: Temperature dependence of the observed frictional drag between

two 2D electron systems separated by 175-A˚ barrier. Data are plotted as an

equivalent resistance and a momentum-transfer rate [4].

of T in each layer contributes to drag) do not give a complete explanation of

the temperature dependence. The roots of this discrepancy will be discussed

in the next section. Here I just mention that in the field of the frictional drag

in the absence of a magnetic field at the moment there are no open questions

(for a review see [7]). Among other papers on drag worth mentioning is the

work [8] in which electron-hole drag is studied. As one would expect, the

drag signal in this case is negative if defined according to Eq. (2.1).

A boost of interest to the theme of Coulomb drag is due to drag experi-

ments in presence of a finite magnetic field. First such experiments lead back

to 1992 [9]. Many other followed [10, 11, 12].

In 1998 though a new, completely unexpected phenomenon has been

found by X. Feng et al. [2], see Fig. 2.4. The new aspect in this work is the

study of magneto-drag with different electron densities in the layers. The

unusual and somewhat counterintuitive observation is the negative drag, i.e.

electrons are accelerated in the direction opposite to the net transferred mo-

2.2 HISTORY OF THE PROBLEM. EXPERIMENT. 15

Figure 2.3: Temperature dependence of the interwell momentum transfer

rate divided by T 2 for both the 175- and 225-A˚ barrier samples [4].

mentum. A closer look at the data of this paper (see Fig. 2.5), and at further

papers [3, 13] (Figs. 2.6, 2.7) allows us to see that the negative drag occurs

when in one of the layers the top Landau level is more than half filled, while

in the other one it is less than half filled.

An analysis of the experimental data shows that the negative drag is ob-

served only under quite specific conditions. First, the mobility of the samples

should be quite high (more than 2 × 106 cm2V−1s−1), second, the magnetic field should be neither too low, nor too high (the effect was observed for filling

factors ν between 6 and ≈ 70), and finally the temperature should be quite low (for a cleaner sample the negative drag could be observed only for tem-

peratures under 1.8 K, while for a dirtier one the allowed temperature range

extends and at some particular magnetic field even reaches 18 K). Another

important feature revealed recently [13] is the activated low-temperature be-

havior of the drag signal with the activation energy periodically depending

16 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

Figure 2.4: Drag versus magnetic field for matched (c) and unmatched (a)

layer densities at 1.15K. The longitudinal magneto-resistance (b) for the drive

layer in both cases (1.66× 1011/cm2) (solid line), the dotted line is the drag layer with a mismatched density (1.49 × 1011/cm2). A clear difference in filling factors is evident for negative drag (a) [2].

on the magnetic field (see Fig. 2.7). The activation energy reaches its maxi-

mum when the Fermi level lies between two Landau levels and drops to zero

when the Fermi level hits the center of either of the two spin-split Landau

levels. All the above indicates the important role of disorder.

For high-mobility samples the disorder potential mainly comes from the

inhomogeneity of the distribution of donors. They are localized in a layer

∼50nm away from each electron layer. Such setup, supposing uncorrelated distribution of donors, results in a smooth random potential in the electron

layer with the correlation length equal to the distance between the donor and

electron layers. This property of the random potential is crucial for the rest

of the work.

The above-mentioned works of Feng et al. and Lok et al. attracted my

2.3 HISTORY OF THE PROBLEM. THEORY. 17

Figure 2.5: Drag versus drag layer filling factor(density). The drive layer den-

sity is fixed at 1.66× 1011/cm2, with magnetic fields for (a)/(b) of 0.92/0.81 Tesla, and drive layer ν of 7.39/8.47. Nearly periodic behavior is observed

[2].

supervisor’s interest to this question so that he proposed me this problem

as a topic for my doctor thesis. The observed effect being simultaneously

simple and elusive for theoretical understanding for quite a long time (see

next section) presented a challenge for us.

2.3 History of the problem. Theory.

2.3.1 Early works.

As mentioned above, the drag effect was first predicted theoretically [5, 6]

long before it became possible for the experimentalists to produce double-

layer systems with the necessary properties to observe the effect. These

early works did not in fact present any calculations of what we call drag

resistance, but rather gave just the idea that there might be a momentum

transfer between two parallel systems separated by an insulator, only due to

Coulomb interaction, without matter transfer.

18 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

Figure 2.6: Drag resistivity ρT (bottom) and conventional longitudinal re-

sistivity ρxx (top) for two 2DEGs at mismatched densities (n1=2.27 and

n2=2.08·1011 cm−2) as a function of magnetic field at T=0.25 (K). Two sets of oscillations can be distinguished in ρT : i) a quick one resulting from the

overlap of the Landau levels in the 2DEGs and ii) a slow one which causes

(positive) maxima in ρT whenever the filling factor difference between the

2DEGs is even, and (negative) minima whenever this difference is odd. The

inset shows ρT at fixed magnetic field of 0.641 (T) versus filling factor differ-

ence [3].

2.3 HISTORY OF THE PROBLEM. THEORY. 19

Figure 2.7: (a) ρD for matched and mismatched densities with (νF , νB) =

(9.5, 9.5) and (8.5, 9.5). (b) Arrhenius plot of |ρD| at low T , demonstrating the activated behavior. (c) measured activation energies, ∆ (U), of ρD (σxx)

for matched densities vs. ν. Data of U are shown for the front layer only,

in regions where σxx ∝ exp(−U/T ). The bare Zeeman energy in GaAs with |g| = 0.4 is shown as a reference [13].

In the work [5] the author mainly concentrated on solving electrodynamic

equations in a semiconductor-insulator-semiconductor heterostructure, tak-

ing into account the drag effect. The main result of that paper is the current-

voltage characteristic of an SIS structure under drag conditions.

The work [6] is devoted to calculating of the energy transfer between the

neighboring layers in a heterostructure due to Coulomb inter-layer scattering.

Thus, both of these pioneering works have only indirect connection to the

theme of my work, but they should definitely be mentioned, as it is in these

works that the idea of Coulomb inter-layer scattering was first mentioned.

20 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

2.3.2 Boltzmann equation approach.

First theoretical works in which approaching the drag problem has been

set as the main goal were published in 1993. First A.-P. Jauho and H.

Smith published a work where they used a Boltzmann equation approach to

calculate the drag [14] and then L. Zheng and A.H. MacDonald used memory

function formalism for the same goal [15]. Here I will review the former work

not only because of its priority but also as the method elaborated there will

be important for me later (see section 2.4.1).

The starting point is the standard linearized Boltzmann equation

k˙2 ∂f 02 ∂k2

=

[ ∂f2 ∂t

] coll

, (2.2)

where

k˙2 = −eE2, (2.3)

E2 being the electric field induced in the drag layer, and[ ∂f2 ∂t

] coll

= − ∫

dk1dk ′ 2

(2pi)4 w(1, 2; 1′, 2′)(ψ1 + ψ2 − ψ1′ − ψ2′)

× f 01 f 02 (1− f 01′)(1− f 02′)δ(�1 + �2 − �1′ − �2′) (2.4)

is the linearized collision integral. w(1, 2; 1′, 2′) determines the probability

of scattering of two electrons from states 1,2 to 1′, 2′, f is the distribution

function, f 0 is the equilibrium distribution function. Indices 1 and 2 refer

to the initial states in the drive and drag layers respectively, 1′ and 2′ to

the final states. Spin indices are omitted here; k1′ = k2 + k1 − k2′ due to momentum conservation. The deviation function ψ is defined by

f − f 0 = f 0(1− f 0)ψ. (2.5)

The current flowing in the layer 1 is assumed to be limited by impurity

scattering, corresponding to

ψ1 = − 1 T τ1ev1xE1, (2.6)

2.3 HISTORY OF THE PROBLEM. THEORY. 21

where E1 is the electric field in the drive layer, directed along the x axis, and

τ1 is the momentum relaxation time.

The authors set ψ2 = 0 ”since no current is flowing in the drag layer”.

Here I will follow there line of reasoning and discuss the validity of this state-

ment later. Collecting all the above, multiplying the Boltzmann equation by

k2x, integrating over k2, and using the antisymmetry of the integrand on the

RHS with respect to interchange of 2 and 2′ they get

eE2n2 = −eE1τ1 8mT

∫ dqdω

(2pi)2 [=χ(q, ω)]2w(1, 2; 1′, 2′) q

2

sinh2 (ω/2T ) , (2.7)

with ni being the electron density in the ith layer. Energy transfer ω is

introduced by formal substitution

δ(�1 + �2 − �1′ − �2′) = ∫ dωδ(�1 − �1′ + ω)δ(�2 − �2′ − ω), (2.8)

q = k2′ − k2 = −k1′ + k1, and the susceptibility χ is defined as

χ(q, ω) = − ∫

dk2 (2pi)2

f 02 − f 02′ �2 − �2′ + ω + i0 . (2.9)

Further evaluation of this expression will be discussed in the next section.

Here, to round off the reviewing of this work I want to discuss why ψ2 may be

set to zero. In fact, the kinetic equation used in [14] is incomplete. This can

be seen extremely clearly for the closed circuit set-up, i.e. no electric field, but

finite current. Then nothing prevents the current in the drag layer to grow

infinitely. To resolve this problem one must add the collision term for the

electrons in the drag layer. Having done this it is possible to leave out the ψ2-

terms in the inter-layer collision integral i.e. consider it as a functional of the

equilibrium distribution function for the electrons in the drag layer. That is

justified as the correction to this distribution function must be proportional

both to the non-equilibrium part of the distribution function in the drive

layer (i.e. to the electric field E1) and to the inter-layer interaction squared

(as the scattering probability w(q) is proportional to it). That means that

keeping ψ2 in the inter-layer collision term would result in a term containing

the 4th power of the interaction, which is small compared to the intra-layer

22 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

collision integral. Taking some specific (for example τ -approximation) form

of the intra-layer collision term and using the zero-current condition∫ dk2

∂�2 ∂k2

f 02 (1− f 02 )ψ2 = 0, (2.10)

one can in principle find ψ2 which is not equal to zero. (In fact the Boltzmann

equation with the intra-layer collision term gives ψ2 as a function of E2,

the latter is then found from the condition for the current to be 0.) But

to determine E2 it is not needed: it is enough to multiply the Boltzmann

equation by k2 and integrate over k2 — that is exactly what is done in the

paper — then the intra-layer collision term disappears due to the fact that

there is no current in the drag layer. This explains why using a physically

wrong form of the Boltzmann equation the authors still got a physically

consistent and correct result for the drag. I will refer to this arguments later,

in the main part of the work.

In the work [15] the authors derive the same equation (2.7) using the

memory function formalism. Reviewing this work lies beyond the scope of

the present thesis.

2.3.3 Diagrammatic approach.

An important step in the theoretical understanding of drag was made in

1995 by Kamenev and Oreg [16]. They formalized all what was done before

them and got some new results, and, what is more important, showed how

this problem can be handled generally and how, in principle, new results can

be obtained. The authors used the general linear response theory method,

i.e. the Kubo formula in its diagrammatic representation. The calculations

were done in the lowest non-vanishing order in the inter-layer interaction,

all intra-layer effects (electron-electron interaction, impurities etc.) can, in

principle, be included up to arbitrary order of perturbation theory in their

method.

2.3 HISTORY OF THE PROBLEM. THEORY. 23

Figure 2.8: Two diagrams contributing to the transconductance to the second

order in the inter-layer interaction.

The starting point is the general linear response expression of the con-

ductivity via a current-current correlation function.

σijD( ~Q,Ω) =

1

ΩS

∫ ∞ 0

dteiΩt ⟨[ J i†1 ( ~Q, t), J

j 2( ~Q, 0)

]⟩ . (2.11)

Here i, j = xˆ, yˆ; ~Q,Ω are the wave vector and the frequency of the external

field, respectively; S is the area of the sample; and J il is the ith component of

the current operator in the lth layer. Diagrams corresponding to Eq. (2.11)

include two separate electron loops with a vector (current) vertex on each

one of them, coupled only by the inter-layer interaction lines. The first non-

vanishing order in the limit ~Q −→ 0 in the inter-layer interaction is the second order. In this order there are two diagrams, shown in Fig. 2.8.

Analytically, this two diagrams may be written in a symmetric form:

σijD = T

2iΩmS

∑ ~q,ωn

Γi1(~q, ωn +Ωm, ωn)Γ j 2(~q, ωn, ωn +Ωm)U(~q, ωn +Ωm)U(~q, ωn),

(2.12)

where T is the temperature, U(~q, ω) is the inter-layer screened Coulomb

interaction and the vector ~Γ1(2)(~q, ω1, ω2) is the three-legged object given by

the two diagrams depicted in Fig. 2.9 (the factor 1/2 in Eq. (2.12) is included

to prevent double counting). In Eq. (2.12) the usual Matsubara technique

with Ωm = 2piimT is employed. After summing over the boson frequencies

24 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

Figure 2.9: Diagrams defining the three-legged vertex, Γ(~q, ω1, ω2) =

Γ(−~q,−ω1,−ω2)

ωn, one should perform an analytical continuation to real values of Ω, and

finally the limit Ω −→ 0 should be taken. The sum over ωn is done by a contour integration in the complex ω plane along the contour shown in

Fig. 2.10.

After the sum over ωn is taken we can finally perform the analytical

continuation Ωm −→ Ω and take the limit Ω −→ 0. The result is:

σijD = 1

16piTS

∑ ~q

∫ ∞ −∞

dω

sinh2 ω 2T

Γi+−1 (~q, ω, ω)Γ j−+ 2 (~q, ω, ω)

∣∣U+(~q, ω)∣∣2 . (2.13)

The +,− indices indicate the way the analytical continuation is performed, see Fig. 2.11 for explanation.

Eq. (2.13) gives the final expression for the transconductivity. The impor-

tant feature is that the integrand decouples into one part depending only on

the interaction potential and two other factors, that are functions of the lay-

ers’ properties only. The latter factors, i.e. the Γ’s are called the rectification

coefficients. They are in fact nonlinear susceptibilities of the two-dimensional

electron gases in an external ac field, and give the proportionality coefficient

between the square of a time-dependent scalar potential φ(~q, ω) and the dc

2.3 HISTORY OF THE PROBLEM. THEORY. 25

Figure 2.10: The contour of integration in the complex ω plane employed to

perform the sum over the Matsubara boson frequencies in Eq. (2.12).

Figure 2.11: The way in which the analytical continuation of ~Γ(~q, ω1, ω2) −→ ~Γ+−(~q, ω, ω) is performed.

26 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

current in a layer:

~J = ~Γ−+|φ(~q, ω)|2. (2.14)

The next parts of the paper are devoted to evaluating of the general for-

mula (2.13) in different cases. To this end the authors calculate the nonlinear

susceptibility Γ. The case of normal metals without intra-layer interaction is

considered. Then

~Γ(~q, ω1, ω2) = T ∑ �n

Tr { G�nG�n+ω2 ~ˆIG�n+ω1 + G�nG�n−ω1 ~ˆIG�n−ω2

} (2.15)

Here G stands for the one-electron Green function and the trace is taken over exact eigenstates of the disordered system. The sum over fermionic Matsub-

ara frequencies is evaluated similarly to the sum over bosonic frequencies.

After the analytical continuation (as shown in Fig. 2.11) the final result for

the nonlinear susceptibility reads:

~Γ+−(~q, ω, ω) = 1

4pii

∫ d�

( tanh

�+ ω

2T − tanh �

2T

) Tr [(G−� − G+� )G−�+ω ~ˆIG+�+ω]

(2.16)

+{~q, ω −→ −~q,−ω}.

Here G± stands for the retarded (advanced) single-electron Green function. This result is valid in the absence of a magnetic field, when the trace of three

advanced or three retarded Green functions vanishes.

Then the authors evaluate the last formula in special cases of clean and

dirty systems and find that in the absence of intra-layer electron-electron

correlations the main results of previous works [14, 15] (i.e. expressing the

drag signal via the imaginary parts of polarization operators (2.7)) holds,

though not generally, rather only up to the first non-vanishing order in 1/�F τ ,

where �F is the Fermi energy and τ is the elastic mean free time.

In order to do it, they take the simplest model, i.e. delta-correlated impu-

rities, and calculate the diagram for Γ (Fig. 2.9) in the diffuson approximation

(in the next section, also the weak-localization corrections are considered).

The result in the diffusive regime (ω � 1/τ and q � 1/l [l = vF τ is the

2.3 HISTORY OF THE PROBLEM. THEORY. 27

mean free path]) is:

~Γ+−(~q, ω, ω) = e 2D~q

�F

[ 2Sν

ωDq2

(Dq2)2 + ω2

] , (2.17)

here D is the diffusion coefficient and ν is the density of states at the Fermi

level. The expression (2.17) has the form

~Γ+−(~q, ω, ω) = ~Γ−+(~q, ω, ω) = e 2D~q

�F =Π+(~q, ω), (2.18)

with

Π+(~q, ω) = 2Sν Dq2

Dq2 − iω (2.19) being the polarization operator in the diffusive regime.

For the ballistic regime (q > 1/l or ω > 1/τ) Γ turns out to be

~Γ+−(~q, ω, ω) = e 2D~q

�F

[ 2Sν

ω

vF q θ(vF q − ω)

] , (2.20)

where θ(x) is the Heaviside step function. As in the previous case, this

expression has the form (2.18) with

=Π+(~q, ω) = 2Sν ω vF q

θ(vF q − ω). (2.21)

An important observation, that one could make, is that the above results

can be obtained only if one takes into account the non-linearity of the dis-

persion around the Fermi surface. More precisely, two terms in Eq. (2.16)

cancel each other if the calculations are done in a usual way, by linearizing

the dispersion near the Fermi surface. From the physical point of view it

means that the electron-hole asymmetry is crucial for the Coulomb drag.

If we had exact electron-hole symmetry the electron and hole drags would

cancel each other completely.

Ultimately the drag conductivity is calculated. Here I’ll list the results

in different cases. It is distinguished between the results for ballistic, l � d and diffusive, l � d samples. In the former case, the dominant contribution to the transresistance comes from the ballistic part of the plane (ω < vF q,

q > 1/l, see Fig. 2.12) and one can obtain:

ρD = ~

e2 pi2ζ(3)

16

T 2

�F1�F2

1

(κ1d)(κ2d)

1

(kF1l1)(kF2l2) , (2.22)

28 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

d −1

Diffusive

q

ω

T

τ

−1l

−1

PSfrag replacements

ω = vF q

ω = Dq2

Figure 2.12: Domains in the (q, ω) plane. The dashed area represents

(schematically) the regions where =Π+ 6= 0.

where κ1,2 are the Thomas-Fermi momenta in the two layers.

On the contrary, for diffusive samples the entire contribution comes from

the diffusive part of the (q, ω)-plane (ω < 1/τ , q < 1/l). For T � min (τ−1, T0) [where T0 ≡ Dmin (κ1, κ2)/d arises due to divergence of the interaction po- tential at small momenta] one obtains

ρd = ~

e2 pi2

12

T 2

�F1�F2 ln T0 2T

1

(κ1d)(κ2d)

1

(kF1d)(kF2d) . (2.23)

For higher temperatures, T, T0 � τ−1, the energy integration is dominated by the region ω ≈ τ−1, the result is

ρD ≈ ~ e2

lnT0τ 1

(κ1d)(κ2d)

1

(kF1l1)(kF2l2) . (2.24)

Actually in this temperature region the plasmon contribution becomes dom-

inant (see for example [17]), but this goes far beyond the scope of the present

work.

2.3 HISTORY OF THE PROBLEM. THEORY. 29

Table 2.1: Temperature and inter-layer distance dependence of the drag co-

efficient, ρD, for different mobilities.

T � min (T0, τ−1) T � min (T0, τ−1) l � d d−2T 2 lnT d−2T l � d d−4T 2

To summarize, I give the dependence of the drag resistance on the tem-

perature and the inter-layer distance for different parameter ranges in table

2.1.

To finish off the review of this very important paper, I’ll mention that in

section IV the authors calculated the corrections to the drag resistance due

to weak localization and in section VI the drag above the superconductor

transition was considered. In the former case the formula (2.18) was found

to be correct in the lowest order in 1/kF l, while in the latter case, where the

intra-layer electron correlations play an important role, this formula turned

out to be wrong. This shows that although the diagrammatic approach does

reproduce the old results, they are not general enough and under specific

conditions the old approach fails.

2.3.4 Magneto-drag. Early diagrammatic approach.

After the publication of the work discussed in the previous section the next

logical step would be to extend the diagrammatic approach to the drag in

presence of a strong quantizing magnetic field. This was done by M. C. Bøn-

sager et al. in 1997. This section is devoted to a critical review of their work.

The starting point was the same as in the work of Kamenev and Oreg, i.e. the

drag was expressed in terms of a convolution of two rectification coefficients.

The results of the paper are all based on expressing the rectification coeffi-

cient (triangle function) in terms of the imaginary part of the polarization

function (see (2.18)). As we will see, unlike the case without magnetic field

30 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

that was considered in the previous section, here the authors did not show

rigorously that the formula (2.18) is correct, so all their results are based on

an assumption that it is true.

The starting point of their calculations is the formula (2.13). Then they

try to prove the relation (2.18) in presence of magnetic field in the limit

ωcτ � 1 (ωc being the cyclotron frequency). Doing this the authors make some dubious statements (see Appendix B of their work) which turn out to

be not correct, most importantly, the claim that the scalar vertex corrections

are negligible for n 6= n′ is wrong. The direct calculations in work [18] show this. Also only the delta-correlated disorder case is considered, which is

rather far from real experimental conditions.

The results of the work consist in a numerical evaluation of the resulting

expression for the drag resistivity

ρxx12 = − ~

2

4e2n1n2kBT

1

S

∑ q

a2 ∫

dω

2pi |U(q, ω)|2 =Π1(q, ω)=Π2(q, ω)

sinh2 ~ω/2kBT , (2.25)

which is obtained by substituting (2.18) into (2.13) and inverting the transcon-

ductivity tensor. U(q, ω) is taken in the random phase approximation, i.e.

U(q, ω) = V12(q)

[1− Π(q, ω)V11(q)]2 − [Π(q, ω)V12(q)]2 , (2.26)

V11 and V12 being the non-screened intra- and inter-layer Coulomb electron-

electron interactions. The layers were assumed identical i.e. Π1 = Π2 ≡ Π. Further evaluation (particularly the expression for Π(q, ω)) in the work

is also based an the erroneous omitting of the scalar vertex correction for

n 6= n′ and thus does not merit a special discussion.

2.3.5 Negative magneto-drag. First explanation at-

tempt.

The results of the experimental works [2, 3] remained for quite a while com-

pletely ununderstood. Indeed, from the generally accepted formula (2.25) it

2.3 HISTORY OF THE PROBLEM. THEORY. 31

is absolutely impossible to get negative drag due to analytical properties of

the imaginary part of the polarization function. So the obvious decision is

to go one step back, to the expression (2.13) and try to make a more careful

analysis. The first attempt to do this was made by F. von Oppen et al. in

2001 [19]. In this work an evaluation of the triangle function Γ was made.

Unfortunately there was a mistake in the calculations for the experimen-

tally relevant ballistic limit, that led to a completely wrong result: negative

drag occurred for matched densities and not for mismatched as one observes

experimentally.

Still I think it is worth presenting this work here for several reasons,

the most important of which is a very simple semiclassical derivation of the

non-linear susceptibility that gives a simple and clear physical picture of

it, and also, in principle, can give a negative value for the drag resistivity.

This derivation is based on the assumption that the current is related to the

electric field locally in time and space and that the conductivity depends

only on the local density of charge carriers:

J(~r, t) = −σˆ[n(~r, t)]∇φ(~r, t). (2.27)

In addition to the current, the perturbation φ(q, ω) also induces a density

perturbation δn(q, ω) = −Π(q, ω)eφ(q, ω) due to the polarizability Π(q, ω). Up to quadratic order in the applied potential,

J(~r, t) = − [ σˆ(n0) +

dσˆ

dn δn(~r, t)

] ∇φ(~r, t). (2.28)

Taking the time and space average of the second term yields a dc contribution

to the current given by

Jdc = − ∑ q,ω

( dσˆ

dn

) [Π(q, ω)eφ(q, ω)iqφ(−q,−ω)], (2.29)

or, equivalently, the rectification coefficient is given by

Γ(q, ω) = dσˆ

d(en) · q=Π(q, ω). (2.30)

32 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

At zero magnetic field this expression reproduces (2.18) and thus leaves no

possibility for drag sign change. On the contrary, in presence of a magnetic

field, in the quantum Hall and SdH regimes the derivative of the longitudinal

component of the conductance changes sign as a function of the electron

density. This opens the possibility for a sign change in the drag resistivity

as a function of density. Substituting (2.30) into (2.13) and inverting the

transconductivity tensor one gets up to an overall positive pre-factor for the

drag resistivity tensor:

ρˆD ∼ ρˆp dσˆ p

d(en)

dσˆa

d(en) ρˆa, (2.31)

where ρˆp(a) and σˆp(a) are the resistivity and conductivity tensors of the pas-

sive (active) layer. This expression easily reproduces some standard results.

In the absence of a magnetic field, the tensor structure is trivial. Observing

that the conductivity increases (decreases) with increasing electron density

for electron (hole) layers, we recover that coupled layers with the same type

of charge carriers exhibit positive drag, while coupled electron-hole layers

have a negative drag resistivity. If the system is strictly electron-hole sym-

metric, the derivative of the conductivity with respect to density vanishes

and, consequently, there is no drag.

In the presence of strong magnetic fields new effects appears. To see this

we can multiply out the products in (2.31), keeping in mind that already for

quite small magnetic fields ρxy � ρxx. Up to an overall positive pre-factor:

ρDxx ∼ ρpxy { dσˆpyy d(en)

dσˆayy d(en)

+ dσˆpyx d(en)

dσˆaxy d(en)

} ρayx. (2.32)

Generally, the derivative of the longitudinal conductivity σxx changes sign as

the magnetic field or Fermi energy is varied in the SdH and integer quantum

Hall regimes, being positive for less than half filled of the topmost Landau

level and negative for more than half filled one. By contrast, the Hall conduc-

tivity generally increases monotonically with n, and its derivative is therefore

positive.

2.3 HISTORY OF THE PROBLEM. THEORY. 33

The obvious problem with this result is that for equal densities in the two

layers the drag is negative as ρxy = −ρyx in contradiction with the experi- mental results [2, 3]. This is due to the fact that the above treatment can

be justified only in the diffusive limit ωτ,Dq2τ � 1 which is experimentally irrelevant.

In order to calculate the rectification coefficient in the ballistic limit the

authors use the diagrammatic approach, starting from the expression (2.16)

from the work [16]. Here a mistake was made, as we will see later, this equa-

tion is inapplicable in the presence of a magnetic field, as important terms

involving traces of three Green’s functions of the same type are omitted.

Thus further reviewing of this work may be only of calculational interest and

will therefore not be done.

2.3.6 Diagrammatic calculation of magneto-drag in the

SCBA.

The work [19] gave the wrong result in the experimentally relevant ballistic

limit. But the idea of the work was mainly correct. This line was drawn to

its logical end in the work [18] which I will review in this section.

In this work the authors calculated the magneto-drag diagrammatically

in the self-consistent Born approximation i.e. for short-range disorder, which

is rather not the experimentally relevant case. Still, it is quite important to

look at one of the limiting cases (high Landau levels, intermediate magnetic

fields), and as this was done in the considered work and as the results are

similar to the experimentally observed data I can not leave this work out of

consideration.

The starting point is once again the expression (2.12) which is evaluated in

the standard way by going from the summation over Matsubara frequencies

to contour integration (see sec. 2.3.3, particularly Fig. 2.10). The triangle

function Γ(q, ω) is then obtained by analytic continuation like in sec. 2.3.3,

34 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

see Fig. 2.11. and is given by Γ = Γ(a) + Γ(b) with the two contributions:

Γ(a)(q, ω) =

∫ d�

4pii tanh

� + ω − µ 2T

Tr {vG+(�+ ω)eiqrG+(�)e−iqrG+(�+ ω)

− vG−(� + ω)e−iqrG−(�)eiqrG−(� + ω)}+ (ω,q→ −ω,−q), (2.33)

Γ(b)(q, ω) =

∫ d�

4pii

( tanh

� + ω − µ 2T

− tanh � +−µ 2T

)

× Tr {vG−(�+ ω)eiqr[G−(�)− G+(�)]e−iqrG+(�+ ω)}+ (ω,q→ −ω,−q). (2.34)

Here G± denotes the advanced (retarded) Green function for a given real- ization of the random potential. Note that for zero magnetic field only Γ(b)

survives [16]. By contrast, in strong magnetic fields both Γ(a) and Γ(b) should

be retained. In the work [19] Γ(a) was left out which led to wrong results in

the ballistic limit.

For small ω, the expressions for Γ(q, ω) simplify to

Γ(a)(q, ω) = ω

2pii Tr {vG+(�F )eiqrG+(�F )e−iqrG+(�F )− (G+ → G−)}, (2.35)

Γ(b)(q, ω) = ω

pii Tr {vG−(�F )eiqr[G−(�F )− G+(�F )]e−iqrG+(�F )}. (2.36)

For well-separated Landau levels this approximation holds as long as ω is

small compared to the Landau level width ∆. We also mention that Γ(a)(q, ω)

gives only a longitudinal contribution (parallel to q) to Γ(q, ω).

Then the averaging over impurities is done using the SCBA [20]. White-

noise disorder is assumed:

〈U(r)U(r′)〉 = 1 2piντ

δ(r − r′). (2.37)

The SCBA (it neglects the crossing impurity diagrams) gives the leading

contribution in the high Landau levels. Although the authors realize that

this approximation is, strictly speaking, not justified for the experimental

samples, where the disorder potential is correlated on the scale of the distance

between the two-dimensional sample and the donor layer, they still expect

their results to reflect reality adequately.

2.3 HISTORY OF THE PROBLEM. THEORY. 35

q, ω q, ω

n, k

n′, k′

µ

ν

=

q, ω

n, k

n′, k′

µ

ν

+

q, ω x

q, ω

n, k

n′, k′

µ

ν

N

N

Figure 2.13: Diagrammatic representation of the equation for the vertex

corrections γµνnk,n′k′(� + ω, �,q) of the scalar vertices in the SCBA [18].

The impurity averaged Green function (denoted by G±(�)) in the SCBA

is given by the expression

G±n (�) = 1

�− En − Σ±(�) , (2.38)

with the Landau energies En = ~ωc(n+1/2). For energies � within a Landau

level, the self-energy is given by

Σ±n (�) = 1

2 {�− En ± i[∆2 − (�− En)2]1/2}. (2.39)

The broadening of the Landau level ∆ can be expressed in terms of the

zero-field scattering time τ as

∆2 = 2ωc piτ

. (2.40)

For white-noise disorder there is no correction of the vector vertex in

Γ, while the corrections of the scalar vertices include impurity ladders (see

Fig. 2.13).

γµνnk,n′k′(� + ω, �,q) = γ µν(q, ω)〈nk|eiqr|n′k′〉. (2.41)

Here, the indices µ, ν = ± indicate the type of Green functions involved in the vertex; � is set to �F . For ∆� ~ωc the vertex corrections at ω = 0 are

γ++(q, ω = 0) = 1

1− J20 (qRc)[∆/2Σ−]2 , (2.42)

γ+−(q, ω = 0) = 1

1− J20 (qRc) , (2.43)

36 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

where Jn(z) denotes the Bessel functions.

The bare vertex is given by

〈nk|eiqr|n′k′〉 = δqy,k−k′ 2n−n

′ n′!

n! exp

[ −1

4 q2l2H −

i

2 qx(k + k

′)l2H

]

× [(qy + iqx)lH ]n−n′Ln−n′n′ (q2l2H/2), (2.44)

where lH = (~c/eH) 1/2 is the magnetic length and Lnm is the associated

Laguerre polynomial and n ≥ n′. The expression for the matrix element for n < n′ can be obtained from Eq. (2.44) by complex conjugation with

the replacement q −→ −q, nk ←→ n′k′. For high Landau levels, which are considered in [18], n, n′ � 1, |n−n′| and q � kF , Eq. (2.44) can be simplified to

〈nk|eiqr|n′k′〉 = δqy,k−k′in−n ′

e−iφq(n−n ′)e−iqx(k+k

′)l2H/2Jn−n′(qR(m)c ), (2.45)

where φq is the polar angle of q, R (n) c = lH

√ 2n+ 1 is the cyclotron radius of

the nth Landau level, and m + (n+ n′)/2.

Now I sum up the evaluation of the triangle vertex Γ(q, ω). Let us first

look at the leading order. As already mentioned ∆/ωc � 1 and the valence Landau level N � 1. The relevant diagrams are shown on Fig. 2.14. In the lowest order in ∆/ωc two of the three Green functions should be evaluated on

the Nth Landau level. The third one (adjacent to the vector vertex) should

be taken on the (N±1)th Landau level as the matrix elements of the velocity are non-zero only for Landau levels differing by one:

〈nk|vx|n′k′〉 = δkk′ i ml √

2 ( √ nδn.n′+1 −

√ n+ 1δn,n′−1), (2.46)

〈nk|vy|n′k′〉 = δkk′ 1 ml √

2 ( √ nδn.n′+1 +

√ n+ 1δn,n′−1). (2.47)

For ω, T � ∆ one gets, using the simplified expressions (2.35), (2.36),

Γ(a)(q, ω) = −2qˆ ωRc pi2l2H

J0(qRc)J1(qRc) 1

2i [G+Nγ

++]2 − [G−Nγ−−]2, (2.48)

2.3 HISTORY OF THE PROBLEM. THEORY. 37

vi

q, ω

q, ω

N ± 1

N

N vi

q, ω

q, ω

N

N ± 1

N

Figure 2.14: The diagrams, contributing to the triangle vertex in the SCBA

to the leading order, in the limit of well-separated Landau levels and large

N [18].

Γ(b)(q, ω) = 2qˆ ωRc pi2l2H

J0(qRc)J1(qRc)

× 1 2i

[G+Nγ ++γ+− −G−Nγ+−γ−−] · [G+N +G−N ]. (2.49)

Here qˆ ≡ q/q, γµν ≡ γµν(q, 0). The sum of the two contributions gives

Γ(q, ω) = qˆ 4ωRc pi2l2H

J0(qRc)J1(qRc)<[G+N(γ++ − γ+−)]=[G+N ]γ++. (2.50)

For arbitrary T, ω < ωc, this contribution takes the form

Γ(q, ω) = qˆ 8Rc

pi2l2H∆ 2

J1(qRc)

J0(qRc)

∫ ∞ −∞

d�

[ tanh

�+ ω − µ 2T

− tanh �− µ 2T

]

× <[γ−+(q, ω)− γ++(q, ω)]=γ++(q, ω). (2.51)

The q-integration in (2.12) is cut off at q ∼ 1/d (d is the interlayer distance). Thus, depending on the relation between Rc and d, one can dis-

tinguish between ballistic (d � Rc) and diffusive (d � Rc) regimes. In the typical experimental setups Rc > d (for the magnetic fields for which the

results of [18] are applicable), so the authors mainly concentrate on the bal-

listic regime in which both the ballistic (qRc � 1) and diffusive (qRc � 1) momenta are relevant.

For diffusive momenta γ+− has a singular behavior, so γ++, γ−− can be

neglected compared to γ+−. So only Γ(b) contributes to (2.50). The result

38 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

is the equation (2.30) (multiplied by 2, as spin has been taken into account)

with the SCBA conductivity [20]

σxx = e2

pi2 N

[ 1− (µ− EN)

2

∆2

] , (2.52)

and the polarization function in the diffusive limit given by Eq. (2.19), which

is valid also in presence of the magnetic field if the diffusion constant D(�) =

R2c/2τ(�) is taken to be energy dependent with the elastic scattering time in

the SCBA being

τ(�) = 1√

∆2 − (�− En)2 . (2.53)

For ballistic momenta (qRc � 1) γ++(q, ω) ≈ γ+−(q, ω) ≈ 1. Thus, in the leading order Eq. (2.50) yields zero. I emphasize that only the sum

of the two contributions (2.48) and (2.49) vanishes, and not each of them

separately. As in the work [19] the contribution (2.48) was missed, the result

in that work in the ballistic limit is wrong. To calculate Γ in the ballistic

limit it is therefore necessary to look at the next order contributions. There

are three kinds of such contributions: (i) contributions of the order of 1/qRc,

(ii) contributions of the order of ∆/ωc, and (iii) contributions of the order of

q/kF . Here I present only the results.

(i). Keeping the contributions of the order of 1/qRc in (2.42) and (2.43),

and plugging them into (2.50) one gets for ω, T � ∆

Γ(1/qRc)(q, ω) = −qˆ64ωRc pi2l2

(µ− EN)[∆2 − (µ− EN )2]3/2 ∆6

J1(qRc)J 3 0 (qRc).

(2.54)

For arbitrary ω, T < ωc one can also find the considered contribution by

using Eq. (2.51). Here and for the two other contributions I present only the

result for low temperatures.

(ii). Contributions of the order of ∆/ωc come from two sources. First

one can evaluate the diagrams in Fig. 2.14 more accurately, keeping the self-

energy parts of the Green function of the Landau levels N ± 1, and second one can consider the diagrams in Fig. 2.15. The result is

Γ(∆/ωc) = −qˆ× zˆ16ωRc pi2l2

(µ− EN )[∆2 − (µ− EN)2] ωc∆4

J1(qRc)J0(qRc). (2.55)

2.3 HISTORY OF THE PROBLEM. THEORY. 39

n± 1

n

N

n

n± 1

N

Figure 2.15: Diagrams contributing to the corrections of the order ∆/ωc in

the triangle vertex [18].

(iii). The contribution of the order of q/kF arises from a more accurate

treatment of the matrix elements involved in the scalar vertices, particularly,

the dependence of Rc on the Landau level number. This contribution is the

direct consequence of the zero-B electron spectrum curvature. For T, ω � ∆ it reads

Γ(q/kF ) = q× zˆ8ω pi2

∆2 − (µ− EN )2 ∆4

J20 (qRc). (2.56)

Having computed the triangle vertex one can go on to calculate the drag

resistivity. As the Hall conductivity dominates over the longitudinal one, the

drag resistivity can be approximated by

ρDxx ≈ ρ(1)xy σDyyρ(2)yx , (2.57)

or, using Eq. (2.12),

ρDxx = − B

en1

B

en2

1

9pi

∫ ∞ −∞

dω

2T sinh2 (ω/2T )

× ∫

d2q

(2pi2) Γ(1)y (q, ω, B)Γ

(2) y (q, ω,−B)|U12(q, ω)|2. (2.58)

The overall minus sign is due to the relation ρxy = −ρyx. It follows that for identical layers, the longitudinal (Γ ∝ qˆ) component Γ‖ of the triangle vertex gives rise to negative drag, since Γ‖(−B) = Γ‖(B), while the transverse (Γ ∝ zˆ× qˆ) component Γ⊥ yields positive drag as Γ⊥(−B) = −Γ⊥(B).

In what follows the authors mainly concentrate on the ballistic regime

ωc/∆� Rc/d� N∆/ωc. (2.59)

40 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

It turns out that in this regime the (ii) contribution (2.55) dominates over

(2.54) and (2.56); indeed, this is easy to see by comparing the order of mag-

nitude for these contributions. This is valid provided that the Landau level

number is sufficiently high: N > (ωc/∆) 2. Without going into details of the

calculation, I will just mention the result. For T � ∆, the drag resistivity is given by

ρDxx = 32

3pi2e2 1

(kFd)2(κRc)2

( T

∆

2) ln

( Rc∆

dωc

)

× ( µ− EN

∆

[ 1− (µ− EN )

2

∆2

]) 1

( µ− EN

∆

[ 1− (µ− EN )

2

∆2

]) 2

. (2.60)

The subscripts 1 and 2 mean that the whole parenthesis refers to the 1st(2nd)

layer. This yields an oscillatory sign of the drag. For identical layers the drag

is positive.

The authors also computed the drag resistance for other ranges of param-

eters. Without going into details of the calculations, I just show the results

in form of a schematic plot, see Fig. 2.16.

In the section devoted to the comparison with the experiment the authors

optimistically state that their results in the ballistic regime fully explain the

experimental data. I find this statement somewhat exaggerated for the fol-

lowing reasons. First, the theory of the discussed paper predicts T 2 low-

temperature dependence of ρD, while experimentalists claim the behavior is

activated with the activation energy periodically depending on the filling fac-

tor. Second, for the ballistic regime in their terminology to be present in the

first place the condition N � (ωc/∆)2 must be fulfilled. Or in other words N � ωcτ � 1. This condition makes doubtful the presence of the discussed regime, or in any case, restricts it to a quite narrow range of magnetic fields,

while experimentally the negative drag is observed in a broad region of mag-

netic fields and the Landau level numbers for which it is observed range from

N = 4 to N ≈ 40. Indeed, taking the experimental parameters from [2]: ∆ = 1.7K for H = 0.1T, I estimate τ ≈ 0.5K−1. Thus ωcτ ≈ 10 ·H[T]. N for the electron density n0 = 2 · 1011cm−1 is equal to 4/H[T]. Substituting the

2.4 SEMICLASSICAL THEORY OF ELECTRON DRAG... 41

above values into the condition N � ωcτ , one obtains H � 0.6T or N � 7. Then, the definition of the ballistic regime (2.59) reads

4 ·H[T]1/2 � 70 d[nm]B[T]

� 1 H[T]3/2

.

Both these conditions restrict the magnetic field from above and for, say,

d =35nm one getsH �0.25T from the second inequality and for d =100nm — H �0.3T from the first one. The condition ωcτ � 1 yields H � 0.1T.

Thus it seems very desirable to elaborate a theory which is valid for

high magnetic fields (low Landau levels) and accounts for the long-range

disorder and localization which is totally omitted in the approach elaborated

by Gornyi et al. in [18].

D ρ x x

T 1

T 1

T 1

T 1

T 1 T*

T2

T1/2

T1/2 ∆

ba

−

lnTT2−

2

−T 2

(T )

∆

T 3

−

T 3

∆

mismatched

matched

c d

T 2lnTT

T 2

−

T 2

−

T

T 1

1

∆ T **

−

−

1

1

Figure 2.16: Schematic temperature dependence of the low-temperature drag

in different regimes: (a) diffusive, Rc/d � 1; (b) weakly ballistic, 1 � Rc/d � ωc/∆, T∗∗ ≡ ωc(d/Rc); (c) ballistic, ωc/∆ � Rc/d � N∆/ωc, T∗ ≡ ∆ ln1/2 (Rc∆/dωc); (d) ultraballistic, N∆/ωc � Rc/d [18].

2.4 Semiclassical theory of electron drag in

strong magnetic fields

To accomplish the task defined at the end of the previous section it is nec-

essary to work with long range disorder. Let us first consider the relevant

length and energy scales in the double-layer system. The distance between

42 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

the layers varies from 30 to 120 nm. The disorder is due to remote donors,

which leads to a smooth disorder potential with a correlation length ξ of order

50-100 nm. The Landau level broadening, which is related to the amplitude

of the disorder potential, has been estimated from low-field Shubnikov-de-

Haas oscillations to lie in the range 0.5-2 K [2, 3]. The magnetic fields at

which drag minima at odd integer filling and/or negative drag at mismatched

densities are observed vary over a relatively wide range from 0.1 T to 1 T

for the cleanest samples, and up to 5 T for samples with a slightly reduced

mobility. This corresponds to magnetic lengths lH between 15 and 80 nm.

Hence, ξ and lH are generally of the same order of magnitude, which makes

a quantitative theoretical analysis rather difficult.

Since the anomalous drag phenomena are observed over a wide range of

fields, one may hope that qualitative insight can be gained also by analyzing

limiting cases. For ξ � lH a treatment of disorder within the self-consistent Born approximation is possible [21]. This route has been taken by Gornyi

et al. in the work [18] discussed above. Localization is not captured by

the Born approximation, and consequently the resistivities obey power-laws

rather than activated behavior for T → 0. For stronger magnetic fields, on the other hand, localization will become important, and a semiclassical

approximation is a better starting point, which is the route I take in what

follows. Applying a criterion derived by Fogler et al. [22] I estimate that

(classical) localization may set in already at 0.1 T. (According to Eq. (1.7) of

[22] localization occurs for B >

√ mc2EF eξ

( W EF

)2/3 , m being the effective mass

and W the amplitude of the potential; substituting typical experimental

values ξ = 60nm, W = 1.7K, n = 2 · 1011cm−2 [2, 3, 13] one gets the above mentioned value.) The localization is classical in the sense that the

electrons’ classical trajectories are closed. From the quantum mechanical

point of view it means that the electron’s wave function is localized in space.

More precisely, an electron is moving along the contours of equal energy of

the random potential or, in a more general case, along the contours of equal

energy of the averaged random potential defined as [22]:

U0(x, y) =

∮ dφ

2pi U(x +Rc cosφ, y +Rc sinφ). (2.61)

2.4 SEMICLASSICAL THEORY OF ELECTRON DRAG... 43

These contours form closed loops, corresponding to localized states, except

at a single energy �0 in the center of each Landau level, for which there is a

percolating contour through the whole system (Fig. 2.17) [23, 24, 25, 26, 27].

If the Landau level broadening induced by the disorder potential and also

kBT is much smaller than ~ωc, as is the case in the anomalous drag regime,

all Landau levels except one are either fully occupied in the bulk of the whole

sample or completely empty. At zero temperature and for �F < �0 the sample

then consists of islands where the highest (partially) occupied Landau level is

locally full, while it is locally empty in the rest of the sample; for �F > �0 the

empty regions form islands. In reality, the percolating contour at the center of

the Landau level is broadened to a percolation region for two reasons. First,

electrons near the percolating contour can hop across saddle points from one

closed loop to another, by using for a moment some of their cyclotron energy

[22]. Second, electrons near the center of the Landau level can screen the

disorder potential, such that the percolating equipotential line at �0 broadens

to form equipotential terraces [28, 29]. Hence, there is a region around the

percolating path, where states are extended.

The percolating region forms a two-dimensional random network con-

sisting of links and crossing points [30]. For simplicity we assume that the

links are straight lines (Fig. 2.18). The region near the link (except near

the end points) can be parameterized by a Cartesian coordinate system with

a variable y following the equipotential lines parallel to the link and x for

the transverse direction. The disorder potential varies essentially only in

transverse direction, and can thus be represented by a function U(x). In the

Landau gauge A = (0, Hx) the Hamiltonian for electrons on the link is then

translation invariant in y-direction and the Landau states

ψnk(x, y) = Cn e −(x+l2k)2/(2l2H )Hn[(x + l

2k)/lH ] e iky (2.62)

are accurate solutions of the stationary Schro¨dinger equation. Here Hn is the

n-th Hermite polynomial and Cn a normalization constant. The correspond-

44 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

Figure 2.17: Schematic pattern of the contours of equal energy [24].

ing energy is simply

�nk = (n + 1/2)ωc + U(l 2 Hk) . (2.63)

In the following we drop the Landau level index n, because only the highest

occupied level is relevant. The quantum number k is the momentum asso-

ciated with the translation invariance in y-direction. It is proportional to a

transverse shift x0 = l 2 Hk of the wave function. The potential U lifts the de-

generacy of the Landau levels and makes energies depend on the momentum

along the link. The group velocity

vk = d�k dk

= l2H U ′(x0) (2.64)

corresponds to the classical drift velocity of an electron in crossed electric

and magnetic fields. In our simplified straight link approximation U ′(x0)

does not depend on y. In general it will depend slowly on the longitudinal

coordinate, but near percolating paths away from crossing points it has a

fixed sign, and hence the group velocity has a fixed direction, that is the

motion on the links is chiral.

2.4 SEMICLASSICAL THEORY OF ELECTRON DRAG... 45

Figure 2.18: Schematic pattern of the percolating region network.

The drag between two parallel layers is dominated by electrons in the

percolating region, corresponding to states near the center of the Landau

level, since electrons in deeper localized states are not easily dragged along.

In the network picture, macroscopic drag arises as a sum of contributions

from interlayer scattering processes between electrons in different links. If no

current is imposed in the drive layer, both layers are in thermal equilibrium

and the currents on the various links cancel each other on average. Now

assume that a small finite current is switched on in the drive layer such

that the electrons move predominantly in the direction of the positive y-axis

(the electric current moving in the opposite direction). This means that

the current on links oriented in the positive y-direction is typically larger

than the current on links oriented in the negative y-direction. Interlayer

scattering processes lead to momentum transfers between electrons in the

drive and drag layer. The preferred direction of momentum transfers is such

46 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

that the scattering processes tend to reduce the current in the drive layer,

that is the interlayer interaction leads to friction.

Now the crucial point is that electrons moving in the disorder potential

are not necessarily accelerated by gaining momentum in the direction of their

motion, or slowed down by loosing momentum (as for free electrons). The

fastest electrons are those near the center of the Landau level: they have the

highest group velocity on the links of the percolation network and they get

most easily across the saddle points. Electrons in states below the Landau

level center are thus accelerated by gaining some extra momentum in the

direction of their motion, but electrons with an energy above �0 are pushed

to still higher energy by adding momentum and are thus slowed down.

2.4.1 Drag between parallel links.

To make the above speculations more quantitative I consider drag between

two parallel quasi one-dimensional links. Instead of the layers for the 2D

case I take strips with infinite length which have a potential profile with its

value depending only on the transverse coordinate x. The eigenstates of the

electrons subjected to a magnetic field and the potential are similar to the

eigenstates without the potential if the gauge is taken to be (0, Hx, 0) (the y-

axis is directed along the strip, H is the magnetic field). The difference is that

the eigenfunctions have contributions from the adjacent Landau levels that

results in a finite velocity of the electron proportional to the x-derivative of

the potential (2.64). The wave-functions in the zeroth order in the potential

are given by (2.62); the corresponding energies in first order of the potential

are given by (2.63). From now on the magnetic length lH will be set to unity,

unless the contrary is explicitly mentioned.

This shows us that there is an effective dispersion law for the electron.

The electron has got a finite mass if the second derivative of U with re-

spect to x is non-zero. In the following calculations I neglect the intra-layer

electron-electron scattering and consider only the interlayer interaction. The

2.4 SEMICLASSICAL THEORY OF ELECTRON DRAG... 47

corresponding Hamiltonian is of the form

Hˆ = ∑

n,p,α=1,2

[( n +

1

2

) ωH + Uα(p)

] c†α,n,pcα,n,p

+

∫ d3r1d

3r2ψˆ † 1(r1)ψˆ

† 2(r2)V (r1 − r2)ψˆ2(r2)ψˆ1(r1). (2.65)

Here c, c† are the annihilation and creation operators of the electron in Lan-

dau basis, α labels the layers and

ψˆα(r) = ∑ n,p

cα,n,pΨn,p(x, y)δ(z + jd/2) (2.66)

with d being the inter-layer distance and j = 3 − 2α. Finally V (r) is the Coulomb potential. Substituting (2.66) into (2.65) we obtain

Hˆ = ∑

n,p,α=1,2

[( n +

1

2

) ωH + Uα(p)

] c†α,n,pcα,n,p

+ ∑

n1...n4 p1...p4

( c†1,n4,p4c

† 2,n3,p3c2,n2,p2c1,n1,p1

× ∫ d2r1d

2r2Ψ ∗ n4,p4

(r1)Ψ ∗ n3,p3

(r2) e2√|r1 − r2|2 + d2 Ψn2,p2(r2)Ψn1,p1(r1)

) .

(2.67)

I model the distribution function in the drive layer in the following way

(arguments in favor of this assumption will be given later):

n1(�1(n, p)) = n 0 1(�1(n, p− p˜)). (2.68)

Here �i is the energy of the electron in i-th layer. It is given by (2.63); ni

is the distribution function and n0i is the Fermi distribution function in the

i-th layer

n0i (�) = 1

1 + exp ( �−µi T

) ,

µi being the chemical potential in the i-th layer and T the temperature. This

assumption corresponds to the shifted Fermi sea for free electrons for velocity

p˜/m. It can also be regarded as a p-dependence of the chemical potential:

µ1(p) = µ1 + p˜ ∂�1 ∂p

. (2.69)

48 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

This distribution leads to a non-zero total current in the layer as there are

more electrons with positive velocity than those with negative, i.e. more

”right-movers” than ”left-movers”. Expanding n1(�1(n, p)) in p˜ we get

n1(�1(n, p)) ≈ n01(�1(n, p)) + p˜

T

∂�1 ∂p

n01(�1(n, p))(1− n01(�1(n, p))). (2.70)

p˜ can be found knowing the current j1 in the drive layer: p˜ = pij1 evF1

As the electrons in the two layers interact, the electrons in the drag layer

will tend to rearrange, so that the drag current occurs also there. To describe

the process quantitatively I use the kinetic equation approach with the col-

lision integral determined in the two-particle approximation using Fermi’s

golden rule. I restrict myself to the case T � ωH so that I don’t need to take scattering processes that change the Landau level into account. More

over, provided that condition is fulfilled, I only need to consider the upper

Landau levels in both layers as no scattering processes are possible in any

other levels due to Pauli principle. I refer to the Landau levels in drive and

drag layer as n and n′ respectively.

At this point I follow the line of calculations from the work [14] adapted

to my model. A detailed review of this work can be found in the section 2.3.2

of the present work. The Boltzmann equation for my model reads:

p˙2 · ∂n 0 2

∂p2 =

[ ∂n2 ∂t

] coll

, (2.71)

with p˙2 = −eE2, where E2 is the electric field leading to the drag voltage. The interlayer collision term is given by

[ ∂n2(p2)

∂t

]12 coll

= 2pi

L

∫ dp1 2pi

∫ dp4 2pi

∫ dp3 2pi

2pi

~

∣∣∣〈p1, p2|Vˆ |p4, p3〉∣∣∣2[ n1(p1)

( 1− n1(p4)

) n2(p2)

( 1− n2(p3)

) − n1(p4)

( 1− n1(p1)

) n2(p3)

( 1− n2(p2)

)] × δ(�1(p1) + �2(p2)− �1(p4)− �2(p3)). (2.72)

2.4 SEMICLASSICAL THEORY OF ELECTRON DRAG... 49

Here Vˆ is the Coulomb potential operator, 〈p1, p2| and |p4, p3〉 are the initial and final states, L is the length of the sample, and for brevity I write

ni(p) instead of ni(�i(p)). For the drag layer in Eq. (2.72) I will take the

distribution to be equilibrium1:

n2(p) = n 0 2(p).

The expression in the square brackets comes from the creation and annihila-

tion operators and represents the difference between incoming and outgoing

flows in the phase space, the delta-function ensures the energy conservation.

The matrix element is given by

〈p1, p2|Vˆ |p4, p3〉 =∫ d2r1d

2r2Ψ ∗ n,p4

(r1)Ψ ∗ n′,p3

(r2) e2√|r1 − r2|2 + d2 Ψn′,p2(r2)Ψn,p1(r1). (2.73)

Using (2.62) and the well-known Fourier-transform of the Coulomb potential

it can be rewritten in the form:

〈p1, p2|Vˆ |p4, p3〉 =

2pie2 ∫ dqx

e−qd√ q2x + q

2 y

e− q2x+q

2 y

2 Ln

( q2x + q

2 y

2

) Ln′

( q2x + q

2 y

2

)

exp

( iqx

(p1 + p4)− (p2 + p3) 2

) δ(p1 − p4 − p3 + p2). (2.74)

Here qy = p1−p4 = p3−p2 is the momentum transfer between the layers. The delta-function shows that the momentum is conserved. Ln is the Laguerre

polynomial. Substituting (2.74) into (2.72) one has to keep in mind that

squaring the delta functions yields an additional factor of L/2pi.

Now I multiply both sides of the Boltzmann equation by ∂�2/∂p2 and in-

tegrate over p2. The left hand side yields −e ∫

dp2 2pi n02

′ (�p2) v

2 p2E2 which tends

to e 2pi vF2E2 at low temperatures. Using the antisymmetry of the integrand

of Eq. (2.72) under exchange of p2 and p3, and collecting all the above I can

rewrite the result for the drag resistance ρD = E2/j1:

1See section 2.3.2 for a detailed discussion

50 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

ρD = 2pi2

e2vF1vF2T

∫ dp1dp2dqy

(2pi)3 q2y|V (p1, p2, qy)|2

∂2�1 ∂p21

∂2�2 ∂p22

× 1 cosh �1(p1)−µ1

2T

1

cosh �1(p1−qy)−µ1 2T

1

cosh �2(p2)−µ2 2T

1

cosh �2(p2+qy)−µ2 2T

× δ(�1(p1) + �2(p2)− �1(p1 − qy)− �2(p2 + qy)), (2.75)

with

V (p1, p2, qy) = 2pie 2

∫ dqx

e− √

q2x+q 2 yd√

q2x + q 2 y

e− q2x+q

2 y

2 Ln

( q2x + q

2 y

2

) Ln′

( q2x + q

2 y

2

)

× exp [iqx(p1 − p2 − qy)]. (2.76)

To obtain (2.75) I first had to recast the filling factor part of (2.72) using

(2.70), the expression for the Fermi distribution, and the energy conservation

law. Then the p4 integral was evaluated using the delta function and the

variable qy = p3 − p2 was introduced instead of p3. The expression (2.75) turns out to be symmetric with respect to in-

terchanging the layers. The formal reason for it is the choice of the non-

equilibrium distribution (2.68). Indeed, the two second derivatives in (2.75)

are from different origin — the one from the drag layer comes from the veloc-

ity change in the scattering process and is directly connected to the curvature

of the dispersion, while the one in the drive layer results from expanding the

distribution function (2.68). For any type of the non-equilibrium distribu-

tion in the drive layer we get the derivative of the chemical potential over p1

instead of ∂2�1/∂p 2 1; but for the specific form I chose this derivative becomes

the second derivative of the energy (compare with (2.69)). Consequently,

the symmetry between the layers occurs only for the used form of the non-

equilibrium distribution. This is a strong argument in favor of Eq. (2.68).

In the following I describe two different methods of calculating the drag.

They differ from each other in the way the electrons’ dispersion is parame-

terized.

2.4 SEMICLASSICAL THEORY OF ELECTRON DRAG... 51

First method

I expand the electron energy around the Fermi energy up to second order in

p− piF , where piF is the momentum corresponding to the Fermi edge in the i-th layer. This is justified whenever the relevant electron states (i.e. those

states that give the essential contribution to the integral in (2.75)) are close

enough to the Fermi energy. What this precisely means will be clarified later,

now we just mention that this condition might fail in the vicinity of the point

where the second derivative of �i with respect to p vanishes. The expansion

yields:

�i(p)− µi = αi(p− piF ) + βi 2

(p− piF )2. (2.77) From now on, for the sake of brevity, I will write pi instead of pi − piF . It is also important to mention that the first term in (2.77) must be much larger

than the second, in order for the expansion to be valid.

The next step is the introduction of the energy transfer ω. This is done

by including an additional integration:

δ(�1(p1) + �2(p2)− �1(p1 − qy)− �2(p2 + qy)) =∫ dωδ(ω − �1(p1) + �1(p1 − qy))δ(ω + �2(p2)− �2(p2 + qy)), (2.78)

or, after substituting (2.77),

δ(�1(p1) + �2(p2)− �1(p1 − qy)− �2(p2 + qy)) =∫ dωδ(ω − α1qy − β1p1qy + β1

2 q2y)δ(ω − α2qy − β2p2qy −

β2 2 q2y). (2.79)

This allows us to evaluate the p1,2-integrals in (2.75):

ρD = pi

e2vF1vF2T sgn (β1β2)

∫ dqdω

(2pi)2 |V (p1 + p1F , p2 + p2F , q)|2

× 1 cosh

[ 1

4β1T

(( ω q

+ β1q 2

)2 − α21

)] 1 cosh

[ 1

4β1T

(( ω q − β1q

2

)2 − α21

)]

× 1 cosh

[ 1

4β2T

(( ω q

+ β2q 2

)2 − α22

)] 1 cosh

[ 1

4β2T

(( ω q − β2q

2

)2 − α22

)] . (2.80)

52 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

Here the momenta p1,2 in the argument of V are determined by the delta

functions in (2.79).

A significant simplification which allows an analytical analysis arises in

the case d� Rc, where Rc = √

2 max (n, n′)lH is the cyclotron radius of the

electron in the layer with higher Landau level2. In this limit the expression

for V (p1, p2, q) reduces to

V (p1, p2, q) =

2pie2 ∫ dqx

e− √

q2x+q 2d√

q2x + q 2

exp [iqx(p1 + p1F − p2 − p2F − q)] = 4pie2K0(qD),

(2.81)

where K0(x) is the McDonald function and

D = √ d2 + (p1 + p1F − p2 − p2F − q)2.

Now I suppose that p1,2, q � d (this will also be checked later), so that D becomes independent on q and ω and is just the distance between the two

scattering electrons. This leaves the ω-dependence only in the four cosh-

factors in the integrand of (2.80) and thus we are left with the integral∫ dω

1

cosh

[ 1

4β1T

(( ω q

+ β1q 2

)2 − α21

)] 1 cosh

[ 1

4β1T

(( ω q − β1q

2

)2 − α21

)]

× 1 cosh

[ 1

4β2T

(( ω q

+ β2q 2

)2 − α22

)] 1 cosh

[ 1

4β2T

(( ω q − β2q

2

)2 − α22

)] , (2.82)

which will be very important in the next parts of the work. When evaluating

this integral we have to consider only q . 1 D

due to the McDonald function.

It can easily be checked that, if the above mentioned conditions hold, the

integrand in the expression (2.80) is invariant under transformations of the

2In real samples this condition is hardly ever met for the magnetic fields for which

negative drag is observed. However, no principle changes occur in the behavior of the

interaction matrix element: it is still logarithmic for small q’s and falls off exponetially for

large ones.

2.4 SEMICLASSICAL THEORY OF ELECTRON DRAG... 53

q

Ω

IV

VIII

I

V

III

Figure 2.19: The four curves are ω = α1,2q ± β1,2q 2

2 . q∗ is the q-coordinate of

the touching point of III and VI

type αi → −αi, βi → −βi, q → −q and ω → −ω. This allows to consider only positive values of αi and βi and to restrict the q, ω-integration to the first

quadrant, multiplying the final result by 4. Moreover, the factor sgn (β1β2)

shows that for the same signs of curvatures the drag is positive, while for

different signs it is negative.

I first consider small temperatures (we will see what it means later). If the

temperature is small enough, one can replace cosh−1 (A/T ) by 2 exp (−|A|/T ). Then in the integrand of (2.82) I obtain the exponential of the sum of two

expressions of the type |ω−αq+βq2/2|(ω+αq+βq2/2)+|ω−αq−βq2/2|(ω+ αq− βq2/2). That leads to the splitting of the integration region in 6 subre- gions (see Fig. 2.19), so that (2.82) can be rewritten as

16

∫ dωe−A/2T ,

with A given by different expressions in different regions of the (q, ω)-plane

(see Fig. 2.19) (we take α1 > α2):

54 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

A(q, ω) =

ω2

q2 β1 + β2 β1β2

+ β1 + β2

4 q2 −

( α21 β1

+ α22 β2

) for I;

ω + ω2

β2q2 + β2q

2

4 − α

2 2

β2 for II;

ω2

q2 β1 − β2 β1β2

+ β2 − β1

4 q2 − α

2 2

β2 + α21 β1

for III;

ω − ω 2

β1q2 − β1q

2

4 + α21 β1

for IV;

−ω 2

q2 β1 + β2 β1β2

− β1 + β2 4

q2 +

( α21 β1

+ α22 β2

) for V;

2ω for VI.

(2.83)

For a given q < q∗ = 2(α1 − α2)/(β1 + β2), A(ω) reaches its minimum (i.e. the integrand in (2.82) reaches its maximum) on one of the banks of

III, depending on which of the two β’s is bigger. Indeed, as can be seen

from (2.83), A(ω) is decreasing in V and IV, increasing in I and II, and

either decreasing or increasing in III depending on the sign of β1 − β2. For an analytical analysis it is enough to consider only the case β1 > β2, which

means that the minimum of A is reached along the line ω = α2q + β2q2

2 . On

this line A(q) is given by:

A(q, ω = α2q + β2q

2

2 ) =

α21 − α22 β1

− q2 (β1 − β2) 2

4β1 + α2q

β1 − β2 β1

, (2.84)

and the integral in (2.82) is proportional to exp (−A(q, ω = α2q + β2q22 ))/2T . As the McDonald function for small arguments is logarithmic, we conclude

that the q-integral for small T is determined by q ∼ Tβ1 α2(β1−β2) . For such q’s

the q2 term in A is negligible for T � α22 β1

.

To calculate the pre-factor I linearize A(ω) with respect to ω−(α2q+ β2q22 ) and calculate the integral (2.82). Analytical calculation can be done in the

limit |α1 − α2| � α1, |β1 − β2| � β1, so I restrict myself to such values. I denote β ≈ β1 ≈ β2, α ≈ α1 ≈ α2, ∆β = β1 − β2, ∆α = α1 − α2. Now I will show that under the conditions mentioned above, for T � T ∗ = αq∗ = α∆α/β the integral in (2.80) is determined by the region III. Indeed, the

2.4 SEMICLASSICAL THEORY OF ELECTRON DRAG... 55

ω-integral of e−A/2T over III is

exp

[ − 1 T

( α11 − α22

2β1 + αq

∆β

β

)]

× α1q−β1q

2

2∫ α2q+

β2q 2

2

dω exp

( − 1

2T

ω2 − (α2q + β2q22 )2 q2

∆β

β2

) =

exp

[ − 1 T

( α21 − α22

2β1 + αq

∆β

β

)] min

( qTβ2

α∆β ,∆αq − βq2

) . (2.85)

The two options in the last equation correspond to the cases when the upper

limit in (2.85) can be set to infinity or not. We see that for T � T ∗∗ = α∆α∆β β2

and for q . Tβ α∆β � q∗ (which are the only relevant ones) the first option in

the min-function is valid, while for larger T ’s and q < q∗ it is the second one.

This means that for T � T ∗∗ I only have to check whether regions IV and V give comparable contribution to (2.80), while for T ∗∗ � T � T ∗ one can set ∆β = 0 and has to look at all regions surrounding III in order to prove

that the contribution from III dominates.

From (2.83) one can see that summed up contributions to (2.82) from I,

II, IV and V are of the order of

exp

[ − 1 T

( α11 − α22

2β1 + αq

∆β

β

)]( Tβq

α

) ,

which is smaller than (2.85). (For T � T ∗∗ it follows from ∆β � β and for T ∗∗ � T � T ∗ it follows from ∆αq = qT ∗β/α � qTβ/α.3) In turn, for q > q∗ minimum of A(ω) is reached for ω = α1q − β1q22 and the result of ω-integration is of the order of (T � T ∗∗, so we set ∆β = 0)

exp

( −αq −

βq2

2

T

)( βq2 +

Tβq

α

) ,

3In fact, for q’s close enough to q∗ the contribution for III tends to zero, but as the

relevant value arises from the integration over q, we have to compare the integrated values.

Doing so, I mention that the results of the q-integration over I,II, IV and V from 0 to q∗

have an additional factor of T/T ∗ � 1.

56 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

(the first term in the parenthesis comes from VI and the second — from II

and IV). Now it is easy to see that the contribution to (2.80) from q > q∗

also has an extra factor of T/T ∗ � 1 compared to the contribution from III (see also footnote).

Thus I have proved that for T � T ∗ it is sufficient to integrate over III. Evaluating the ω-integral in (2.80) yields

ρD = 64pie2

vF1vF2T sgn (β1β2) exp

[ − 1 T

( α11 − α22

2β1

)]

× ∫ q∗

0

dq [K0(qD)] 2 exp

( −αq∆β

βT

) min

( qTβ2

α∆β ,∆αq − βq2

) . (2.86)

As the McDonald function is logarithmic for small arguments and falls off

exponentially for large ones there are three scales for q: q∗, Tβ/∆βα, and

1/D. Depending on which of the three wave-vectors is smaller three results

can be obtained:

ρD = 64pie2

vF1vF2 sgn (β1β2)

×

exp [ − 1

T

( α21−α22

2β1

)] [ ln ( D Tβ

∆βα

)]2 T 2β4

α3∆β3 , T � T ∗∗, T � α∆β

βD ;

exp [ −α∆α

Tβ

] [ ln ( D∆α

β

)]2 ∆α3

6β2T , T ∗∗ � T � T ∗, q∗ � 1

D ;

exp [ −α∆α

Tβ

] 1

2D2 min

( β2

α∆β , ∆α

T

) , 1

D � q∗, α∆β

βD � T � T ∗.

(2.87)

Now let us consider higher temperatures (T � T ∗). For this temperature region it is possible to neglect the difference between α1 and α2 as well as

the difference between β1 and β2. So (2.82) reduces to

∞∫ 0

dω 1

cosh2 [“

ω−αq−βq2 2

”“ ω+αq−βq2

2

”

4βq2T

] 1 cosh−2

[“ ω−αq+ βq2

2

”“ ω+αq+ βq

2

2

”

4βq2T

] . (2.88)

I will show later that for q � T/α integral (2.88) falls off exponentially, so another temperature scale arises: the one at which T/α ∼ 1/D. For

2.4 SEMICLASSICAL THEORY OF ELECTRON DRAG... 57

smaller temperatures the cut-off in q-integration is T/α and for higher T ’s

it is 1/D. If, as usually is the case in the experiment, D is of the order of

the characteristic size of the potential then 1/D � α/β, so this temperature scale T ′ = α/D � α2/β. For such temperatures we can rewrite (2.88) in the form: (after shifting the integration variable to ω˜ = ω − αq − βq2

2 )

∞∫ −∞

dω˜ cosh−2 [ ω˜α

2βqT

] cosh−2

[ (ω˜ + βq2)α

2βqT

]

= 2βqT

α

∞∫ −∞

dx

cosh2 x cosh2 (x+ qα 2T

) . (2.89)

Indeed, it can easily be shown that for T . T ′ relevant ω˜’s in the integral are

of the order of max (βqT/α, βq2)� αq, which allows to set ω+αq− βq2 2 ≈ 2αq

and extend the lower integration limit to −∞. The integral in (2.89) can be evaluated and yields

8βqT

α

qα 2T

coth qα 2T − 1

sinh2 qα 2T

, (2.90)

which is, as expected, exponentially small for q � T/α. Substituting it into (2.80) one obtains for ρD : (for T � α/D)

ρD = 64piβT 2e2

α3vF1vF2 sgn (β1β2) ln

2

( D T

α

) . (2.91)

Ultimately for T � α/D the answer for the drag resistivity is

ρD = 32pie2β

3vF1vF2D2α sgn (β1β2). (2.92)

The equations (2.87), (2.91) and (2.92) give the value of ρD for all possible

combinations of parameters. It turns out that for smallest temperatures

the behavior is activated, then for T ∼ α∆α/β there is a cross-over to T 2- behavior, and finally for T ∼ α/D quadratic dependence changes to constant. If α/D . α∆α/β the T 2-regime is absent.

Now I have to check the validity of the approximations made. I started

with neglecting the third-order term in (2.77). In order for it to be justified,

58 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

the second-order term must be larger than the third-order one. We will set

�i(p) = −ti sin 2pi(p+piF )l 2 H

l . The second-order term is βip

2/2 and the third-

order one is −αip3(2pil2H/l)2/6, with βi = ti(2pil2H/l)2 sin 2pipiF l 2 H

l and αi =

−t(2pil2H/l) cos 2pipiF l 2 H

l . The ratio of the two terms is 6pipl2H/l cot

2pipiF l 2 H

l and

it should be small. This condition is likely to be violated in the vicinity of

piF = 0, which corresponds to the inflection point of the potential, so we can

replace cotx by 1/x and that leaves us with the inequality to be verified for

relevant p’s:

3|p|/|piF | � 1.

From (2.79) we have

pi = ω

βiq − αi βi ± q

2 . (2.93)

If now one substitutes the typical values of ω/q and q it becomes evident

that for piF . ∆α β

the approximation (2.77) is wrong. Thus for this region

of parameters a new approximation should be applied. This is done in the

next part of the work.

The second approximation that was to be verified is the condition p1,2, q � d which allows to neglect the ω-dependence of the interaction matrix element.

From (2.93) one sees that this condition is always fulfilled if d � Rc which is assumed in all the above calculations.

Second method.

The approach elaborated in the previous section consisting of expanding

�(p) around the Fermi momentum fails in the interesting region of pF ≈ 0 that corresponds to a particle-hole symmetric situation. In particular, using

that technique it is not possible to show that at the particle-hole symmetric

point the drag is identically zero, which is a crucial feature. To account for

this region of parameters I have developed another approach that is not as

transparent as the first one, but more general. As we will see it gives the

same results as the old one for most parameter regions. The exceptions to

the above will be discussed separately.

2.4 SEMICLASSICAL THEORY OF ELECTRON DRAG... 59

Now, instead of expanding �(p) around pF and dealing with the variables

αi, βi which are pFi-dependent, I expand around the inflection point of the

potential (� = 0, p = 0) and carry out the calculations in more natural

parameters i.e. those of the potential profile and the chemical potentials.

Let the dispersion relations (directly connected to the potential profiles)

be

�1(p) = t1 sin(pγ1) and �2(p) = t2 sin(pγ2)

for the drive and drag layer respectively. γi ≈ 2pil 2 H

Li characterizes the size Li of

the potential hill. In the 2D situation it corresponds to the correlation length

of the potential. Even if it is the same in both layers, the local values of γ’s,

characterizing the form of the potential at a given point may be different for

the two layers, so we have to consider the general case. As I will show this

leads to no qualitative changes in the results.

I expand the dispersions around the inflection point up to third order:

�i = tiγip− tiγ 3 i

6 p3. (2.94)

Then, as in the previous section, I solve the energy conservation equation for

a scattering process with given ω and q:

�1(p 0 1 −

q

2 )− �1(p01 +

q

2 ) + ω = 0,

�2(p 0 2 −

q

2 )− �2(p02 +

q

2 ) + ω = 0.

I obtain

p02i = 1

γ2i

( 2

( 1− ω

tiγiq

) − (qγi)

2

12

) . (2.95)

This implies that

ω 6 tiγiq

( 1− (qγi)

2

24

) . (2.96)

Suppose, t1γ1 > t2γ2 (in the opposite case no changes in the final result

occur). I introduce a new variable δ instead of ω:

ω = (1− δ)t2γ2q (

1− (qγ2) 2

24

) . (2.97)

60 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

Thus

p022 = 2δ

γ22

( 1− (qγ2)

2

24

) ;

p021 = 1

γ21

t1γ1

( 1− (qγ1)2

24

) − t2γ2

( 1− (qγ2)2

24

) t1γ1

+ δ t2γ2 t1γ1

( 1− (qγ2)

2

24

) . (2.98)

We see that for every pair (q,ω) satisfying the condition (2.96) we have 4

different pairs of initial states that differ from each other by the signs of the

p0i ’s. So the expression (2.80) turns into

ρD = 4pie2

vF1vF2T

∫ dqdωK20(qD)

× (

1

cosh �1(p01− q2 )−µ1

2T cosh

�1(p01+ q 2 )−µ1

2T

− 1 cosh

�1(p01− q2 )+µ1 2T

cosh �1(p01+

q 2 )+µ1

2T

)

× (

1

cosh �2(p02− q2 )−µ2

2T cosh

�2(p02+ q 2 )−µ2

2T

− 1 cosh

�2(p02− q2 )+µ2 2T

cosh �2(p02+

q 2 )+µ2

2T

) .

(2.99)

After performing the multiplication of the two parenthesis in the integrand

I get a sum of four expressions of the type of (2.80) (with obvious changes

due to different expression for �i used), corresponding to the four possible

initial states mentioned above. If µi � T , than only one of the four terms survives and we return to the old expression (2.80). On the other hand,

if either of the µ’s happens to be zero, the expression (2.99) yields zero

which is the manifestation of the particle-hole symmetry. In the frame of the

consideration in the previous method it is impossible to explicitly obtain this

zero, while in the present one it comes out naturally. If µi are not zero, but

still µi � T one can expand the expressions in the parenthesis in powers of µi/T and carry out the calculations in this region of parameters.

At this stage it is worth mentioning that the overall sign of ρD given by

(2.99) is sgn (µ1µ2), which means that in the situation, when one layer is

2.4 SEMICLASSICAL THEORY OF ELECTRON DRAG... 61

more than half-filled and the other is less than half-filled, the drag signal will

be negative.

As in the previous section I restrict myself to the consideration of almost

identical layers. In this case the calculations can be carried out analytically

and I do not expect any qualitative changes if the parameters of the layers

begin to differ significantly. So I consider t1γ1 ≈ t2γ2 ≈ tγ, γ1 ≈ γ2 ≈ γ and |µ1| ≈ |µ2| ≈ µ. The absolute values of the differences of the parameters in the two layers will be denoted respectively as ∆(tγ), ∆γ and ∆µ.

I start with the case µ� T . As in the previous section I replace cosh ( A 2T

)

by 2 exp (− |A| 2T

). This divides the (q, δ) plane into several regions, within each

of which the arguments of the cosh’s have constant signs. In principle, I

proceed just as in the previous section: I set both µ’s positive and consider

only the region of positive q’s, as the other cases are trivially obtained from

this one due to symmetry properties of the integrand. As both µ’s are positive

and � T , only the first terms in the parenthesis of (2.99) survive. To determine the borders of the mentioned regions I solve the equations

�i(p 0 i ± q2) = µi. If, as I set before, t1γ1 > t2γ2, then these equations read (for

the time being I keep some extensive terms):

µ2 = t2 √

2δ

( 1− 7(qγ2)

2

48

)( 1− δ

3

) ± t2 qγ2

2

( 1− (qγ2)

2

24

) (1− δ), (2.100)

µ1 = t1

√√√√√2 ∆ + δ t2γ2

t1γ1

( 1− (qγ2)2

24

) (1− ∆

3 − δ

3 − (qγ1)

2

8

)

± t2 qγ2 2

( 1− (qγ2)

2

24

) (1− δ), (2.101)

with

∆ = t1γ1

( 1− (qγ1)2

24

) − t2γ2

( 1− (qγ2)2

24

) t1γ1

. (2.102)

For q2 � 12 ∆(tγ) tγ2∆γ

, ∆ ≈ ∆(tγ) tγ

, in the opposite limit ∆ ≈ − q2 12 γ∆γ (∆γ =

γ1 − γ2).

62 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

q

∆

VI IV

II

V

I

III

PSfrag replacements

δ1

δ2

Figure 2.20: Regions of the (δ, q)-plane in which arguments of the cosh’s in

Eq. (2.99) keep their signs. δ1 = 1 2

( µ t

)2 , δ2 =

1 2

( µ t

)2 −∆. Solving the Eqs. (2.100), (2.101) (omitting this time the irrelevant small

corrections, the consistency of this can be easily checked after solving the

equations by direct comparison of the omitted terms with the kept ones), I

get

�2(p 0 2 ±

q

2 ) = µ2 for δ ≈ 1

2

( µ2 t2 ∓ qγ2

2

)2 , (2.103)

�1(p 0 1 ±

q

2 ) = µ1 for δ ≈

[ 1

2

( µ1 t1 ∓ qt2γ2

2t1

) −∆

] t1γ1 t2γ2

.

The notation of the regions (Fig. 2.20) corresponds to the one used in

the previous section in the sense that the Fig. 2.20 is just a replotting of

Fig. 2.19 in the coordinates (δ, q) instead of (ω, q). The region where the

approximation used there is applicable is (

µ t

)2 � ∆ or µ�√ t∆(tγ) γ

. If this

condition is fulfilled, the quadratic term in the expansion of the dispersion

around the Fermi point is much larger than the cubic one.

2.4 SEMICLASSICAL THEORY OF ELECTRON DRAG... 63

The end of region III (q∗) is given by q∗ = ∆(tγ) µγ2

(equivalent to ∆α β

from the

previous section). It can also be easily checked that for µ� √

t∆(tγ) γ

√ ∆γ γ

the

condition q2 � 12 ∆(tγ) tγ2∆γ

holds within III and I can set ∆ = ∆(tγ) tγ

everywhere

in III. We will see that, as in the previous section, it is the region III that gives

the main contribution to the result. Unlike that case though, the maximum

of the integrand is reached on the upper border of III, independent on which

µiγ 2 i (analog of βi in the previous section) is bigger. Indeed, in III the sum of

absolute values of the arguments of cosh’s is (in the first non-vanishing order

in δ)

− 1 T

[ t2 √

2δ

( 1− 7(qγ2)

2

48

) − t1

√ 2

( ∆ + δ

t2γ2 t1γ1

)( 1− 7(qγ2)

2

48

) − µ2 + µ1

] .

(2.104)

It is straightforward to show that apart from the pathological case t∆γ � ∆(tγ), the δ-derivative of this expression is negative and thus the maximum

is reached on the upper border of III.

On the other hand, in IV the sum of the absolute values of the cosh’s

arguments becomes

− 1 T

[ µ1− t1

√ 2

( ∆ + δ

t2γ2 t1γ1

)( 1− 7(qγ2)

2

48

) + t2γ2q

( 1− (qγ2)

2

24

) (1− δ)

] ,

the δ-derivative of which positive and its absolute value is much greater that

that of the δ-derivative in III. That proves the aforementioned statement,

that the integral in the low temperature limit (T � tγ∆(tγ) µγ2

) is determined

by III.

To determine the exponential factor in the answer it is enough to substi-

tute δ = 1 2

( µ t

)2 in (2.104). It turns out that

ρD ∝ exp [ − 1 T

t∆(tγ)

γµ

]

in agreement with the results from the previous section. Hereby the difference

between t1 and t2 in (2.104) turns out to be insignificant. To determine the

pre-factor the δ- and q-integrals must be evaluated. It is done in a similar

64 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

manner as before. For the lowest temperatures the δ-integral is determined

by the derivative of the argument of the exponent, and for higher ones — by

the size of III. What becomes different is the definition of ”low temperatures”.

Before it was determined by the value of ∆β (the analog of which now is ∆µ),

but now, as can be seen from (2.104), the derivative of the argument of the

exponent in the integrand does not depend on ∆µ.

I denote the expression in (2.104) as A(δ). To find the cross-over T one

has to compare A (

1 2

( µ t − qγ

2

)2) − A( 1 2

( µ t

+ qγ 2

)2 −∆) with unity. If it is much greater than unity, then the δ-integral converges within III and the

lower limit can be extended to infinity; in the opposite case, within III the

argument of the exponent can be considered constant and the integration

becomes trivial.

The cross-over temperature is

T ∗ = t2[∆(tγ)]2

γ2µ3 . (2.105)

For T � T ∗ the lower limit of the δ-integral, as already mentioned, can be set to −∞ after expanding the argument of the exponent in powers of[

1 2

( µ t − qγ

2

)2 − δ]: (µt −

qγ 2 )

2∫ −∞

eAdδ

with

A ≈ − t∆ T (

µ t − qγ

2

) + t∆ T (

µ t − qγ

2

)3 [(

1

2

µ

t − qγ

2

)2 − δ ] .

Here, the second term comes from the expansion of the δ-dependent denom-

inator. Other contributions are negligible. Keeping the q-dependence only

in the exponent we get the result of the δ-integration:

T (

µ t

)3 t∆

exp

( − t∆ T (

µ t − qγ

2

) ) .

Expanding the exp’s argument in q and evaluating the q-integral (keeping in

mind that due to the variable change ω → δ one gets an additional factor

2.4 SEMICLASSICAL THEORY OF ELECTRON DRAG... 65

tγq in the integrand) I finally evaluate the q-integral in (2.99) to obtain the

final result:

ρD = 256pie2γ2T 2µ7

vF1vF2t 6[∆(tγ)]3

sgn (µ1µ2) exp

[ −t∆(tγ)

γµT

] ln2 ( D

Tµ2

t2∆(tγ)

) . (2.106)

For T ∗ � T � t∆(tγ) γµ

I easily get the same answer as in the previous

chapter:

ρD = 32pie2[∆(tγ)]3

3vF1vF2T (µγ 2)2

sgn (µ1µ2) exp

[ −t∆(tγ)

γµT

] ln2 ( D

∆(tγ)

γ2µ

) . (2.107)

For T � t∆(tγ) γµ

the result (2.91) holds (of course changes in notation α→ tγ and β → µγ2 should be made).

There is no reason to search for any deep meaning in the fact that two

different approximations give different results in the low temperature limit.

It just means that the pre-factor in (2.106) is sensitive to the precise form of

the potential and thus should not be taken literally. On the other hand, the

exponential factor is universal. The mathematical reason for which the pre-

factor differs in the two approximations (note that I am in the region where

both approximations are self-consistent, i.e. all approximations made are

justified by the final results, so it is not an indication that one approximation

is ”wrong” and the other one is ”correct”) is quite simple. In fact, as can

be checked numerically, in a very broad region of parameters the values of �i

given by the two approximations differ very insignificantly. But the values

of p0i which are determined from the energy conservation relations turn out

to be quite sensitive to the approximations used, the reason for it is that the

second order term in (2.77) is much smaller than the first order one in the

considered range of parameters, and a slight change in it, not having any

visible effect on the value of �i has a much larger effect on p 0 i . Further, the

value that is essential for the calculation is �2(p 0 2) − �1(p01). Its zero order

term in δ happens not to to depend on the approximation, while the first

order term (which accounts for the pre-factor) turns out to depend strongly

on the approximation used.

66 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

q

∆

IV

V

III

PSfrag replacements

δ1

Figure 2.21: Regions of the (δ, q)-plane for µ� √

t∆(tγ) γ

; δ1 = 1 2

( µ t

)2 .

Now I proceed to the case T � µ � √

t∆(tγ) γ

. In this parameter region

the plot from Fig. 2.20 takes the form shown in Fig. 2.21 (I remind that δ is

positive by definition). In this parameter region the approximation from the

previous section is inapplicable, so there will be no comparison of the results

from the two approaches. In this case the exponential factor comes from the

cosh’s with �1 in the argument. The value of the product of these two factors

is approximately constant for relevant values of δ and is equal to

4 exp

( − 1 T

√ 2 t∆(tγ)

γ

) .

Indeed, for µ � √

t∆(tγ) γ

and δ ∼ 1 2

( µ t

)2 , ∆ � δ and �1 ≈ t

√ 2(∆ + δ) ≈

t √

2∆. So I take the cosh factors with �1 out of the integral and consider

only the other two factors which give the pre-exponential factor in the drag

resistivity.

The contributions from III, IV, and V must be taken into account (see

Fig. 2.21). Those from III and V are equivalent and give the same result.

2.4 SEMICLASSICAL THEORY OF ELECTRON DRAG... 67

The contribution from IV turns out to be of the same order of magnitude.

Indeed, A(δ) in these three regions is given by (I leave only the relevant part,

arising from the layer 2):

III: − 1 T

( µ2 − t

√ 2δ ) , δ ≤ 1

2

(µ2 t − qγ

2

)2 ;

IV: − 1 T

( t qγ

2

) ;

V: − 1 T

( −µ2 + t

√ 2δ ) , δ ≥ 1

2

(µ2 t

+ qγ

2

)2 .

Expanding in δ around the borders of the regions III and V we get:

III: − 1 T

( tqγ 2 − t2

µ ∆δ )

;

V: − 1 T

( tqγ 2

+ t 2

µ ∆δ ) ,

where ∆δ is the difference between δ and the border of the corresponding

region. It is evident from the above that for T � µ the δ-integral converges in the bulk of the regions III and V and thus the above statement about the

equivalence of this regions is correct. Within IV, A is independent on δ and

the δ-integration is thus trivial and is determined by the size of the region

IV. Collecting the above one gets:∫ dδeA = exp

( −tqγ

2T

)( 2µT

t2 + qµγ

t

) , (2.108)

from which it is easy to see that the contributions from III, IV, and V are of

the same order as stated above. Finally the q-integral in (2.99) is taken to

get the final result for ρD:

ρD = 384pie2T 2µ

vF1vF2t 3γ

sgn (µ1µ2) exp

( − 1 T

√ 2 t∆(tγ)

γ

) ln2 ( DT

tγ

) . (2.109)

For µ ∼ √

t∆(tγ) γ

it is consistent with the case µ � √

t∆(tγ) γ

, T � T ∗. One can also note that the condition q2 � 12 ∆(tγ)

tγ2∆γ holds for q . T/tγ for

T � µ� √

t∆(tγ) γ

√ γ

∆γ .

68 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

This rounds off the consideration of the case µ � T . Now I turn to the other case, i.e. µ � T . For this I return to the Eq. (2.99) and expand both parenthesis up the first order in µ

T :

ρD = 4pie2µ1µ2 vF1vF2T 3

∫ dq K20 (Dq)tγq

× ∫

dδ 1

cosh �1(p01− q2 )

2T

1

cosh �1(p01+

q 2 )

2T

1

cosh �2(p02− q2 )

2T

1

cosh �2(p02+

q 2 )

2T

× (

tanh �1(p

0 1 − q2) 2T

+ tanh �1(p

0 1 +

q 2 )

2T

)( tanh

�2(p 0 2 − q2) 2T

+ tanh �2(p

0 2 +

q 2 )

2T

) .

(2.110)

Two limiting cases must be considered: T � √

t∆(tγ) γ

and T � √

t∆(tγ) γ

.

1) T � √

t∆(tγ) γ

.

The cosh’s stemming from the first layer, similarly to the case T � µ �

√ t∆(tγ)

γ , account for the exponential factor 4 exp

( − 1

T

√ 2 t∆(tγ)

γ

) . The

parentheses with tanh’s of �1 is equal to 2. So the δ-integral is reduced to∫ dδ

1

cosh t √

2δ+ tqγ 2

2T

1

cosh t √

2δ− tqγ 2

2T

( tanh

t √

2δ + tqγ 2

2T + tanh

t √

2δ − tqγ 2

2T

)

=

( 2T

t

)2 tqγ

2T sinh tqγ 2T

. (2.111)

The q-integral can now also be evaluated and yields the final result for the

drag resistance:

ρD = 1792ζ(3)pie2µ2T

vF1vF2t 3γ

sgn (µ1µ2) exp

( − 1 T

√ 2 t∆(tγ)

γ

) ln2 ( DT

tγ

) ,

(2.112)

with ζ(3) being the Riemann zeta-function.

2) T � √

t∆(tγ) γ

.

For such temperatures the ∆’s in the expressions for �1 can be neglected

and the δ-integral turns to∫ dδ

1

cosh2 t √

2δ+ tqγ 2

2T

1

cosh2 t √

2δ− tqγ 2

2T

( tanh

t √

2δ + tqγ 2

2T + tanh

t √

2δ − tqγ 2

2T

)2 .

(2.113)

2.4 SEMICLASSICAL THEORY OF ELECTRON DRAG... 69

This integral can not be evaluated analytically, but it is easily shown that the

answer for the drag differs from (2.112) only by the absence of the exponent

and by a different numerical factor:

ρD = C e2µ2T

vF1vF2t 3γ

sgn (µ1µ2) ln 2

( DT

tγ

) , (2.114)

with C ≈ 113.

I now summarize all the above. In the schematic T−µ (Fig. 2.22) diagram one can see different regions of parameters in each of which ρD has different

asymptotics:

µ

Τ

I

II III

IV

V

VI

PSfrag replacements

√ t∆(tγ)

γ

√ t∆(tγ)

γ

µ ∼ 1 T

µ ∼ 1 T 1/3

Figure 2.22: Regions of parameters T and µ for which ρD has different be-

havior.

70 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

I: ρD = 256pie2γ2T 2µ7

vF1vF2t 6[∆(tγ)]3

sgn (µ1µ2) exp

[ −t∆(tγ)

γµT

] ln2 ( D

Tµ2

t2∆(tγ)

) ;

II: ρD = 32pie2[∆(tγ)]3

3vF1vF2T (µγ 2)2

sgn (µ1µ2) exp

[ −t∆(tγ)

γµT

] ln2 ( D

∆(tγ)

γ2µ

) ;

III: ρD = 64pie2µT 2

vF1vF2t 3γ

sgn (µ1µ2) ln 2

( DT

tγ

) ;

IV: ρD = C e2µ2T

vF1vF2t 3γ

sgn (µ1µ2) ln 2

( DT

tγ

) ;

V: ρD = 448ζ(3)pie2µ2T

vF1vF2t 3γ

sgn (µ1µ2) exp

( − 1 T

√ 2 t∆(tγ)

γ

) ln2 ( DT

tγ

) ;

VI: ρD = 384pie2T 2µ

vF1vF2t 3γ

sgn (µ1µ2) exp

( − 1 T

√ 2 t∆(tγ)

γ

) ln2 ( DT

tγ

) .

2.4.2 Drag between non-parallel links.

For parallel links, energy and momentum conservation restrict the allowed

scattering processes very strongly. At low temperatures, this leads to an

exponential suppression of the drag between non-equivalent parallel links.

This has nothing to do with the exponential suppression of drag observed in

the experiments, since the links are generally not parallel. For non-parallel

links the sum of momenta on the two links is no longer conserved in the

scattering process. In this section of the work I will compute the drag between

non-parallel links.

Unlike the parallel links case, the interaction matrix element here can be

evaluated analytically. Indeed, the integral

∫ dx1dx2dy1dy2

ψ∗n0(x1, y1)ψnq1(x1, y1)ψ ∗ n′0(x2, y2)ψn′q2(x2, y2)

D(x1, x2, y1, y2)

with

D(x1, x2, y1, y2) = √ d2 + (y1 − y2 cos θ + x2 sin θ)2 + (x1 − y2 sin θ + x2 cos θ)2,

2.4 SEMICLASSICAL THEORY OF ELECTRON DRAG... 71

θ being the angle between the links, can be evaluated and yields:

V (q1, q2) = 2pie 2 √

2pi e−

dQ | sin θ|

Q e−

Q2l2H 2 sin2 θLn

[ Q2l2H

2 sin2 θ

]

× Ln′ [ Q2l2H

2 sin2 θ

] exp

[ i q22 − q21

2 l2H cot θ

] , (2.115)

with Q = √ q21 + q

2 2 − 2q1q2 cos θ. The last exponent in the matrix element

is insignificant as it vanishes after taking the square of the absolute value

of the matrix element. Above x1,2 and y1,2 respectively are the coordinates

perpendicular to and along the links 1 and 2; the zero point for the x1,2-axes

corresponds to the centers of the initial states and the zero-point of the y1,2-

axes corresponds to the crossing of the projections of the center lines of the

intial states along the z-axis (the direction of the magnetic field).

This expression should be inserted into (2.72). Proceeding as in the case

of parallel links we can get the expression for the drag resistance. The main

difference compared to the parallel links case is that instead of δ(q1 − q2) I get

exp (− dQ| sin θ|) Q

which in principle allows any couple (q1, q2) of momentum

transfers. This leads, as I will show, to nontrivial consequences, in particular

to a T 2 behavior for the lowest temperatures for any θ 6= 0. Before going on to the calculations, I will make some qualitative remarks. For any parameters

of the layers it is possible to choose q1,2 such that t1q1γ1 = t2q2γ2. From the

calculations in the previous section it is quite clear that the ω- (or δ-) integral

will have no exponential smallness for such q1,2. On the other hand there

might be an exponentially small factor stemming from the matrix element,

as for q1 6= q2, dQ/| sin θ| can be much larger than unity. So, as now becomes clear, the qualitative analysis will consist in comparing the two exponents,

one stemming from the matrix element, and the other is the one — known

from the previous section — coming from the difference in Fermi velocities

in the two layers.

Another point that should be mentioned is the sign of the drag sig-

nal, which can be determined without any additional calculations. Indeed,

Eq. (2.75) will have q1q2 in the integrand instead of q 2 y. The sign of this

72 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

product has a vast effect on the value of the interaction matrix element: if

cos θ > 0 then Q for q1q2 > 0 is smaller than Q for q1q2 < 0 provided the

absolute values of q’s are the same. And vice versa. As Q enters the matrix

element in the exponent, it is clear that for cos θ > 0 the q’s from the 1st and

the 3rd quadrants dominate, and for cos θ < 0 it is the q’s from the 2nd and

the 4th quadrants. For perpendicular links the drag is, of course, identically

zero. The only other factor affecting the sign of the result is still the factor

sgn (µ1µ2).

The above clearly shows that each pair of links from different layers of a

real 2D sample gives contribution to the drag of the sign determined only by

the relative sign of the local dispersions on the Fermi level. This, in turn,

leads to the experimentally observed overall sign of the drag in a 2D sample.

Now to the quantitative analysis. Let us follow the calculations from the

parallel links case and mention the differences. First in (2.72) I substitute

(2.115) instead of (2.74). It leaves the factor 2pi/L which was canceled for

parallel links. L is the length of the link which corresponds to the correlation

length of the 2D random potential. In (2.75) changes are evident: qy in the

arguments of �i are changed to qi, the integration is done over q1,2 instead of

qy and the factor 2pi/L is still there.

I follow the line of calculations presented as ”second” method. I will

not keep trace of all numerical factors as some of the calculations below are

only estimates and any numerical prefactor would not be reliable, rather I’ll

concentrate on the qualitative features. First I will look at the δ-integral and

will determine its behavior as a function of q1,2, then the q1,2-integrals will

be estimated.

For estimating the δ-integral only some redefinitions should be made,

in principle all the calculations are already done. The crucial point is the

redefinition of ∆, which should now be defined as

∆ = ∆(tγq)

tγq ,

2.4 SEMICLASSICAL THEORY OF ELECTRON DRAG... 73

with ∆(tγq) = |t1γ1q1 − t2γ2q2|, rather than

∆ = ∆(tγ)

tγ

with ∆(tγq) = |t1γ1q1 − t2γ2q2|. This can be seen by rewriting the formulae (2.95)-(2.102) allowing for different q’s.

The obvious consequence is that the result of the δ-integration has the

exponential part

exp

[ − 1 T

t∆(tγq)

γµq

] , (2.116)

and is not exponentially small for

t∆(tγq)

Tγµq ∼ 1. (2.117)

I consider the case t∆(tγ)

Tγµ � 1, (2.118)

as it is more interesting and allows to demonstrate interesting features of the

crossed links. Conditions (2.117),(2.118) imply

∆q

q ≈ −∆(tγ)

tγ , (2.119)

and as can be seen from (2.117), ∆q/q must be close enough to these value:∣∣∣∣∆qq + ∆(tγ)tγ ∣∣∣∣ . µTt2 . (2.120)

The variation of ∆q/q is small compared with its absolute value if T � t∆(tγ) γµ

,

which is the same condition that is necessary for the activated behavior of

ρD for parallel links. The corresponding region is referred to in Fig. 2.23 as

1. The region 2 in this Fig. shows the values q1,2 for which dQ/| sin θ| . 1. The result of the δ-integration (for µ� T ) is obtained as the Eq. (2.90)

with appropriate changes. As I’m not interested in numerical prefactors, I

can write for it µT

t2 e−qtγ/T .

74 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

q

q 2

1

2

1

Figure 2.23: regions of the q1q2-plane which give main contributions to the

drag. Region 1 shows where the ω-integral is not exponentially small and

region 2 shows where the interaction matrix element has a significant value.

For details see the text.

Instead of q1,2 I introduce ∆q = q1 − q2 and q = (q1 + q2)/2. For the drag resistivity I can write:

ρD ∼ e 2

LvF1vF2T

∫ dqd∆q

e−Qd/| sin θ|

Q2 tqγ

µT

t2 e−qtγ/T , (2.121)

with

Q2 = ∆q2 cos2 θ

2 + 4q2 sin2

θ

2 .

For θ � 1 the last expression can be rewritten as Q2 = ∆q2 + q2θ2. (2.122)

The region of (q,∆q)-plane that gives rise to the main contribution to the

drag signal is given by (2.119,2.120). From here one can conclude that for

θ � ∣∣∣∆(tγ)tγ ∣∣∣ it is the first term in (2.122) that dominates, while for larger θ’s

it is the second term.

Looking at Fig. 2.23 one can notice that for sufficiently small temper-

atures the q-integral will converge inside the region 2 due to the exponent

exp (−qtγ)/T . This happens for

T . θ

d

(tγ)2

∆(tγ) .

2.4 SEMICLASSICAL THEORY OF ELECTRON DRAG... 75

For such temperatures and smallest θ’s, (2.121) can be rewritten as

ρD ∼ e 2

LvF1vF2T

∫ dqd∆q

(tγ)2

q2[∆(tγ)]2 tqγ

µT

t2 e−qtγ/T . (2.123)

(∆q has been substituted from (2.119)). The ∆q-integral is determined by

the width of region 1 in Fig. 2.23, which is given by (2.120). The q-integral

afterwards is trivial. The result for the above-mentioned parameters is

ρD ∼ e 2µ2T 2γ2

LvF1vF2t2[∆(tγ)]2 . (2.124)

For µ� T there are no principle changes in the way the calculations are done. As for parallel links I expand the integrand in µ/T . The resulting δ-integral

for q’s obeying (2.119,2.120) is the same as in Eq. (2.113). The result —

as in the case of parallel links — differs from (2.124) only by changing one

power of T to one power of µ:

ρD ∼ e 2µ3Tγ2

LvF1vF2t2[∆(tγ)]2 . (2.125)

For T & θ d

(tγ)2

∆(tγ) , as can be seen from Fig. 2.23 the q-integral is cut off at

θ d

tγ ∆(tγ)

� 1 d . This leads to multiplying the above results by θ

d tγ

∆(tγ) tγ T � 1

d tγ T

:

ρD ∼ e 2µ2Tγ4

LvF1vF2[∆(tγ)]3 θ

d , for µ� T (2.126)

and

ρD ∼ e 2µ3γ4

LvF1vF2[∆(tγ)]3 θ

d , for µ� T. (2.127)

The above results are valid for lowest temperatures, when it is possible

to neglect the contribution from the regions of the (q1, q2)-plane where the

integrand is exponentially small. With growing temperature the contribution

from these regions grows and for high enough T becomes bigger than the

contribution from the region 1 of the Fig. 2.23. Now I will make quantitative

estimates to substantiate these qualitative arguments. As stated above, I

will not keep any numerical pre-factors. I also omit the factor sgn (µ1µ2) as

well as sgn (cos θ) which was discussed above. In particular, I’ll concentrate

76 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

on the case of very small θ which is interesting from the point of view of

theoretical understanding of the drag between non-parallel links. The drag

resistivity is given by

ρD ∼ e 2

vF1vF2TL

∫ dqdkdδ

Q2 tq2γ

× e −Qd/| sin θ|

cosh �1(p01−

q1 2

)−µ1 2T

cosh �1(p01+

q1 2

)−µ1 2T

cosh �2(p02−

q2 2

)−µ2 2T

cosh �2(p02−

q2 2

)−µ2 2T

= e2

vF1vF2TL

∫ dqdk

Q2 tq2e−Qd/| sin θ|γB(q, k). (2.128)

Here q = (q1 + q2)/2, k = (q1 − q2)/q. Then Q2 ≈ q2(k2 + θ2). Using the intermediate results from the previous section we can write for

the δ-integral which I denote B(q, k):

B(q, k) =

1) e− t2∆ Tµ Tµ3

t4∆ e−

t2∆ Tµ

qγt µ , T .

t4

µ3 ∆2;

2) e− t2

Tµ ∆ ( ∆− µqγ

t

) , q <

∆t

µγ , t4

µ3 ∆2 . T .

t2

µ ∆;

3) µT

t2 e−

qtγ T ,

t2

µ ∆ . T.

(2.129)

Here ∆ = ∣∣∣∆(tγ)tγ + ∆qq ∣∣∣ ≡ |∆0 + k|.

Next the q-integration can be evaluated. It is done differently for the

cases 1), 2), and 3) (see above). In fact, these cases correspond to different

values of k (we concentrate on k’s between −∆0 and 0 as it is clear that only these values can give significant contribution):

1) −∆0 + µ t2

√ Tµ . k;

2) −∆0 + Tµ t2

. k . −∆0 + µ t2

√ Tµ;

3) |∆0 + k| . Tµ t2 .

(2.130)

The result is given by:

ρD ∼ e 2

LvF1vF2T

∫ dkY (k) (2.131)

2.4 SEMICLASSICAL THEORY OF ELECTRON DRAG... 77

k-Θ

YHkL 3L 2L 1L

PSfrag replacements

−∆0

Figure 2.24: Typical plot of Y (k). The definition of regions 1)-3) is given

by (2.130); the depicted situation corresponds to the case considered in the

text, when k = 0 belong to the region 2).

with

Y (k) =

1) e− t2

Tµ |∆0+k| Tµ

3

t4|∆0 + k| tγ

k2 + θ2 min

( |θ| d √ k2 + θ2

, µ2T

γt3|∆0 + k| )

;

2) e− t2

Tµ |∆0+k| tγ

k2 + θ2 |∆0 + k|min

( |θ| d √ k2 + θ2

, |∆0 + k|t

γµ

) ;

3) µγT

t∆20 min

( θ

∆0d , T

tγ

) .

For lowest temperatures the region 3) dominates. I will now look at what

happens for growing temperatures. For this I will look at the behavior of the

function Y (k) for different T and compare the integrated contribution to the

drag signal from different regions. First of all I notice that in the case 3) it is

the first option in the min’s argument that must be realized, otherwise it is the

region 3) that dominates, and this has been studied above. So I concentrate

on the case when the q-integration in 3) is limited by the exponent from the

interaction matrix element and not by the temperature. At some value of k

the situation changes and the option in the min-function switches. A typical

plot of Y (k) can be seen on fig. 2.244.

One has to distinguish two cases: k = 0 corresponds to 2) and to 1).

These cases, as can be easily seen from (2.130) are determined by the same

4In fact the shown plot is rather specific. A slight change in the parameters of the plot

immediately changes its character: one of the two peaks becomes dominating. This fact

justifies the logarithmic accuracy calculations presented below.

78 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

restrictions for the temperature as those that limit the regions I and II from

Fig. 2.22: if T . t 4

µ3 ∆20 then k = 0 is in the region 1) and if

t4

µ3 ∆20 . T .

t2

µ ∆0

it is in the region 2). For simplicity I only consider the latter case, as it will be

clear that the difference between them is purely calculative and considering

the former case would not teach us anything new.

For |k| & |θ|, k2 + θ2 in the expression for Y (k) can be replaced by k2 if I do not wish to calculate the numerical pre-factors. For k from 2) one can

write:

Y (k) = e− t2

Tµ |∆0+k| tγ

k2 |∆0 + k|min

( |θ| d|k| ,

|∆0 + k|t γµ

) . (2.132)

For k ∼ θ it crosses over to

Y (k) = e− t2

Tµ ∆0 tγ

θ2 ∆0 min

( 1

d , ∆0t

γµ

) . (2.133)

The minimum of the first expression is reached for |k| = 3µT t2 � ∆0, for

smaller absolute values of k, Y (k) begins to grow and reaches its maximum

at k = 0 and the width of this peak is ∼ θ. So I can estimate the contribution to ρD from all k’s outside of 3) as

ρD ∼ e 2

LvF1vF2T e−

t2

Tµ ∆0 tγ

|θ|∆0 min (

1

d , ∆0t

γµ

) . (2.134)

The contribution from the region 3) is given by the Eq. (2.126) which can be

rewritten as

ρD ∼ e 2

LvF1vF2T

µ2T 2γ

t3∆30

|θ| d .

Comparing the two contributions one can find with logarithmic accuracy that

for

T & t2∆0

µ ln ∆0|θ| (2.135)

the activated behavior stemming from the region where |k| . |θ| dominates, while in the broad region of smaller T ’s one would observe a power law

T -dependence. I do not consider in detail the whole variety of possible pa-

rameter relations as it would not lead to any qualitatively new results. The

2.4 SEMICLASSICAL THEORY OF ELECTRON DRAG... 79

main result: logarithmic dependence of the cross-over temperature on θ does

not change.

Now, for some new insight, I look at the above from a different point of

view: I fix T and consider ρD as a function of θ. Comparing (2.134) with

(2.126) again I find that the first one dominates for

θ . ∆0 t2∆0 µT

e− t2∆0 2µT .

For larger angles, but still θ � ∆0 it is the power dependance contribution (2.126) that dominates. For even larger angles, 1 � θ � ∆0, k � θ every- where in the relevant regions. That means that the expression (2.131) holds

with the change of the first option in all min-functions in the expressions for

Y (k) to 1/d, and, clearly k2 + θ2 ≈ θ2. Thus it becomes evident that Y (k) reaches its maximum for |∆0 + k| = 0 and the width of the peak is ∼ Tµ2 . So, for 1� θ � ∆0:

ρD ∼ e 2

LvF1vF2T

µ2T 2γ

t3∆20 min

( 1

d , T

tγ

) . (2.136)

So, in agreement with (2.135), no exponential behavior is expected in any

temperature range. In other words, already a relatively small non-parallelicity

is sufficient to kill any trace of the activated behavior, characteristic for the

parallel links case. This means that this behavior, as was mentioned above,

is just an artefact of the degenerate case of parallel links and it would have

no effect after averaging over real two-dimensional samples, where θ’s take

all possible values randomly.

Finally I touch the question of the limit θ −→ 0. Here the essential point is the size of the sample L. For parallel links of finite length, q1 and q2 can

differ from one another by the value ∼ 1/L; for nonparallel links this value is θ/d. Thus the cross-over value of θ is d/L. For such angles Eq. (2.115)

is, as can be easily checked, no longer valid and for smaller angles it crosses

over to Eq. (2.74). The last sentence should be understood in the sense

that the expression (2.115) as a function of ∆q tends to the δ-function with

θ −→ 0 and the integrals over ∆q of both expressions coincide for small θ’s.

80 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

As I have shown, for small θ’s (or for large enough T ’s) the exponent in the

expression for ρD is recovered. I failed to find a transparent way to see how

the pre-factors of the expressions for the drag for parallel and non-parallel

links cross-over to each other for θ ∼ d/L. More over I am quite convinced that to do this I need to obtain an analytic expression for the matrix element

of the interaction for these angles which is definitely impossible. Still I’m

sure that the arguments presented in this paragraph give enough insight to

understand how the cross-over works.

2.4.3 Connection to two-dimensional drag.

In the previous parts of the work I have presented a detailed theory of the

drag between two one-dimensional chiral spinless electron systems with an

arbitrary angle between them. Spin can be easily included in the above

picture. Since the interlayer interaction is spin independent, one simply has

to sum over the two spin species (up and down) in both drive and drag

layer, taking the (exchange enhanced) Zeeman spin splitting of the Landau

levels into account. If the Fermi level of one layer lies between the centers

of the highest occupied Landau levels for up and down spins, respectively,

positive and negative contributions to the drag partially cancel each other.

The cancelation is complete due to particle-hole symmetry in the case of odd

integer filling, as observed in experiment.

As was mentioned in the beginning of the section 2.4, to get the result for

the two-dimensional drag, knowing the results for the one-dimensional drag,

one should in principle sum up the contributions from all crossing points

between the links of the two random networks. The full accomplishing of

this task presents a separate challenge and is not yet done. Still, many

things can be seen already from simple arguments that have already to a

large extent been mentioned above and that I recollect here.

Within our semiclassical picture anomalous drag, especially negative drag,

is suppressed at temperatures above the Landau level width, because then

electron- and hole-like states within the highest occupied level are almost

equally populated. This agrees with the results from the Born approximation

[18], and also with experiments.

2.5 CONCLUSION 81

It is plausible that localized states give small — if any — contribution

to the drag. Indeed, such states represent electrons drifting around hills or

valleys of the random potential and drag between two such states is formally

zero as the contributions from the crossing points between them exactly

cancel. The only way such localized states can affect the drag is if an electron

scatters from such a state to a delocalized state, drifts along it away from

its initial position and then scatters back to another localized state spatially

separated from the first one. These processes are not forbidden as there is no

energy conservation for a separate layer, so, unlike the one-layer transport

problem, we can not restrict the role of these states to a reservoir for the

delocalized ones. It is not clear whether the contribution of such processes

to the drag is significant, still, it is clear that the delocalized states play a

special role in the overall drag. Their contribution is largely determined by

the Fermi filling factor at the percolation energy. If the Fermi level does

not hit any extended states (for either spin species), the drag should vanish

exponentially for T −→ 0, since thermal activation or scattering of electrons into extended states is then suppressed by an energy gap. By contrast, for

a Fermi level within the extended states band (for at least one spin species)

the gap vanishes and the drag obeys generally quadratic low temperature

behavior, as obtained for the drag between non-parallel links.

As to the sign of the drag, it is negative if one of the layers is more

than half-filled and the other is less than half-filled, and positive in other

cases. This follows from the results for the sign of the drag between two

one-dimensional links that have been obtained above.

2.5 Conclusion

In summary, I have presented a semiclassical theory for electron drag between

two parallel two-dimensional electron systems in a strong magnetic field,

which provides a transparent picture of the most salient qualitative features

of anomalous drag phenomena observed in recent experiments [2, 3, 13]. Lo-

82 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

calization plays a role in explaining activated low temperature behavior, but

is not crucial for anomalous (especially negative) drag per se. In particular I

have elaborated a detailed theory for the drag between two one-dimensional

links (sections 2.4.1, 2.4.2). The temperature dependence of the drag has

been derived for a whole variety of the parameters of the system as well as

for parallel links and for non-parallel ones. I have also presented qualitative

arguments to show to what consequences for the two-dimensional drag my

results lead (section 2.4.3). A quantitative theory of drag which covers the

whole range from low magnetic fields, where the Born approximation is valid,

[21, 18] to high fields, where localization becomes important, remains an im-

portant challenge for work to be done in the future. A promising path here is

a very general diagrammatic approach for long-range disorder systems which

starts from the equation of the type (2.12). The averaging over disorder is

done not for a single Green’s function as usually ([16, 18]) but for the triangle

function Γ as a whole and the Green’s functions are taken in their exact form

for a given random potential realization. This work is in progress now.

Chapter 3

Possible Jahn-Teller effect in

Si-inverse layers

3.1 Introduction

In a symmetric configuration of atoms, the electronic state may be degenerate

due to the high symmetry of the Hamiltonian. H. A. Jahn and E. Teller

proved that such a symmetric configuration is unstable against a deformation

lowering the overall symmetry of the Hamiltonian. The Jahn-Teller effect is

well accounted for in molecules and crystals [31, 32]. The kinetic energy of

electrons in the 2D electron gas (2DEG) under the conditions of quantum

Hall effect is quenched. Electron states become macroscopically degenerate

forming Landau levels. Interaction may remove some of this degeneracy.

Nevertheless, one finds the global valley degeneracy at integer fillings1. In

this chapter we raise the question whether a lattice deformation — Jahn-

Teller effect — lifts this valley degeneracy.

The theory of the 2DEG at integer ν uses [33, 34] the Hartree-Fock ap-

proximation in the limit of the small parameter: Ec/~ωc, where Ec = e 2/lH

is the energy of the Coulomb interaction and ωc is the frequency of the cy-

clotron resonance. At ν = 1 the theory predicts a ferromagnetic ground

1If we neglect the Zeeman splitting this degeneracy turns to a spin-valley one.

83

84 CHAPTER 3. POSSIBLE JAHN-TELLER EFFECT...

state with degenerate uniform spin orientation. The elementary excitations

are electron-hole pairs or neutral excitons, which correspond to gapless and

noninteracting [35] spin-waves for vanishing momentum and vanishing effec-

tive g-factor. In the limit of large exciton momentum the electron and the

hole become independent charged excitations. A special topological spin tex-

ture in a 2D ferromagnet called ”skyrmion” [36] has unit charge and half of

the quasiparticle energy [37]. In this chapter such textures are considered in

the presence of a Jahn-Teller effect.

In bulk Si there is a six-fold valley degeneracy of low energy electron states

in the conduction band. It is reduced to two-fold degeneracy for the (100)

orientation of the 2D-plane due to the transverse quantization of electron

motion in a quantum well [38]. Even for the (111) orientation of the 2D-plane

with six equivalent valleys only two valleys connected by the time reversal

symmetry are actually occupied in a strong magnetic field [39]. This effect is

governed by the anisotropy of the effective mass of electrons and corresponds

to a spontaneous breaking of valley symmetry. Here we study the effects

related to this valley degeneracy and show that such a bivalley system is

similar to the bilayer systems. The bilayer setup has been extensively studied

theoretically for ν = 1 [40, 41] and ν = 2. [42, 43] The layer degeneracy of

electron states can be described by a four component spinor in combined

spin-layer space. In former works [44, 45] the total 2DEG Hamiltonian was

subdivided into a symmetric part and a small anisotropic part which reduces

the symmetry. The symmetric part is invariant under a SU(4) group of

electron spinor rotations in the combined spin and layer Fock space. This

SU(4) group has the apparent SU(2)⊗SU(2) subgroup of separate rotations in spin and valley spaces.

The electron annihilation operator can be expanded using one electron

orbital functions:

ψα(~ρ) = ∑

τ

ψατ (~r)χ(z)e iτQz/2, (3.1)

where α is the spin index, z is the coordinate perpendicular to the 2D plane

and ~r is the in-plane coordinate vector. The index τ = ±1 numerates the two valleys. Valley wave functions of electrons in a Si inverse layer:

3.2 HAMILTONIAN OF ELECTRON GAS ON SI INTERFACE 85

χ(z) exp(±iQz/2), are normalized and almost orthogonal for smooth, real χ(z) with the negligible overlap

∫ χ2(z) exp(iQz)dz [38]. Q is the shortest

distance between the valley minima in reciprocal space, equals approximately

to 2/aSi, with aSi being the lattice constant. ψατ (~r) is the electron wave func-

tion of in-plane motion in valley τ with spin α and constitutes a four compo-

nent spinor. Electrons in the system will strongly interact with phonons with

momentum ±Q, giving rise to scattering from one valley to another. The Jahn-Teller effect (JTE) in Si-inverse layers corresponds to a displacement

of silicon atoms in z-direction. The new equilibrium is determined from the

balance of electron-phonon and elastic energies. A strong magnetic field is

essential for JTE. Indeed, in its absence a coherent JT state would lead to an

appearance of one Fermi-sphere instead of two smaller ones for two different

valley states. This would lead to an increase of the electrons’ kinetic energy

that makes the JTE energetically unprofitable. In the following sections we

discuss the possible JTE in detail.

This chapter is structured as follows. In section 3.2 we introduce the

Hamiltonian of the system and subdivide it into SU(4)-symmetric and -asym-

metric parts. Section 3.3 is devoted to a discussion of the SU(4)-symmetric

case. In the sections 3.4, 3.5 the anisotropic terms in the Hamiltonian are

included and the ground state of the system for different parameters is found.

In section 3.6 the effect of the anisotropic terms on the energy of the skyrmion

is considered. And finally a brief review of the results of this chapter is

presented in the conclusion.

3.2 Hamiltonian of electron gas on Si inter-

face

The Hamiltonian of the Si inverse layer describes the electron system inter-

acting with phonons:

Hˆ = Hˆe + Hˆe−ph + Hˆph. (3.2)

For a narrow quantum well, the Hamiltonian of the 2DEG in a magnetic field

can be expressed in terms of a spinor ψατ (~r) and reads (we use the system of

86 CHAPTER 3. POSSIBLE JAHN-TELLER EFFECT...

units where: ~ = 1, e = c, B = 1 and the magnetic length lH = √ c~/eB = 1)

Hˆe =

∫ ψ†ατ (~r)

( 1

2m

[ −i~∇ + ~A

]2 + gµB ~B~σ

) αβ

ψτβ(~r)d 2~r

+ e2

2κ

∫ ψ†ατ1(~r1)ψ

† βτ2

(~r2)ψβτ2(~r2)ψατ1(~r1)

|~r1 − ~r2| d 2~r1d

2~r2, (3.3)

where g is the conduction band gyromagnetic ratio, κ is the dielectric con-

stant of Si, µB is Bohr magneton and ~B is the uniform magnetic field in

z-direction. Here and in the following summation over repeated greek in-

dices is assumed. We neglect small terms with different valley indices in the

electron density operator and we use the Debye model for phonons with large

momenta ±Q

Hˆph[uz] = ρSi 2

∫ [ (∂tuz)

2 + c2 ( ~∇uz

)2] d3~ρ, (3.4)

where uz(~ρ) is the lattice displacement in z-direction. We consider only the

valley-mixing electron-phonon interaction:

Hˆe−ph = Θ ∫ ψ†ατ1(~r)τ

± τ1τ2

ψατ2(~r)×(∫ χ2(z)e±iQz∇zuz(~ρ)dz

) d2~r, (3.5)

with Θ and ρSi being the deformation potential and the density of Si re-

spectively. Here and in the following c is the speed of sound. τ µ and σµ,

with µ = x, y and z, are the Pauli matrices in valley and spin spaces and

τ± = (τx ± iτ y)/2. ~A(~r) = (0, Bx) is the vector-potential in Landau gauge. In a magnetic field we expand the electron operator over Landau orbital

states φn,p(~r) in the Landau gauge:

ψˆατ (~r) = ∑ n,p

φn,p(~r)cˆnp,ατ , (3.6)

where c†np,ατ and cnp,ατ are electron creation and annihilation operators in

the τ valley with spin α, n enumerates Landau levels and the continuous

parameter p specifies states within one Landau level.

3.2 HAMILTONIAN OF ELECTRON GAS ON SI INTERFACE 87

We assume that the electrons are confined to the lowest Landau level

n = 0 in the ground state. The electronic part of the Hamiltonian contains

the Coulomb and the kinetic energy in the lowest Landau level:

Hsym = 1

2m

∑ p

c†pατcpατ + 1

2

∫ d2~q

(2pi)2 V (~q)N(~q)N(−~q), (3.7)

where

V (~q) = 2pie2/κq (3.8)

is the 2D Fourier transform of the Coulomb interaction. Notice that in our

system of units 1/m is the cyclotron frequency. The electron density operator

is:

N(~q) = ∑

p

c†pατcp−qy ατe −iqx(p−qy/2)−q2/4. (3.9)

Unitary transformations of electron operators

cpατ1 = Uατ1,βτ2cpβτ2 (3.10)

in the combined spin and valley space leave the Hamiltonian (3.7) invariant

if U is a matrix from the SU(4) Lie group. For the Landau level filling factors

ν = 1, 2, 3 (ν = 4 corresponds to a fully filled zeroth Landau level) we assume

that the ground state is uniform with electrons of spin αi and valley τi filling

every orbital of the lowest Landau level:

Ψ(α1τ1 . . . αντν) =

ν∏ i=1

∏ p

c†pαiτi |vac〉. (3.11)

One can check that any such wave-function (3.11) represents an eigenfunc-

tion of Hsym Eq. (3.7). The state (3.11) is degenerate and a set of related

eigenstates can be generated by applying uniform rotations U .

The remaining terms in the Hamiltonian (3.2) arise from a valley splitting

term due to a singularity of the well potential on the Si/SiO2 interface [38],

Zeeman term, electron-phonon interaction and the phonon energy:

Han = −t ∑

p

c†pατ1τ x τ1τ2cpατ2

− |g|µBH ∑

p

c†pατσ z αβcpβτ +He−ph +Hph, (3.12)

88 CHAPTER 3. POSSIBLE JAHN-TELLER EFFECT...

where t is the phenomenological valley-splitting constant. This part breaks

the SU(4) symmetry and lifts the degeneracy of the eigenstates of SU(4)-

symmetric Hamiltonian (3.2). The splitting of energy levels is determined by

small matrix elements of the anisotropic Hamiltonian (3.12) projected onto

a linear space of the symmetric Hamiltonian (3.7) level degeneracy. There

is no renormalization of the anisotropic Hamiltonian parameters (3.12) due

to electron-electron interaction in the symmetric part Eq. (3.7). Thus the

Hartree-Fock approach is valid up to O(Han/ωH) terms for ν = 1, 2, 3.

The total Hamiltonian Hsym + Han, Eqs. (3.7,3.12), can be treated as

in the theory of magnetism and can be expressed in terms of an order pa-

rameter matrix Qˆ(~r). According to the Goldstone theorem the symmetric

part Eq. (3.7) can be expanded in powers of spatial derivatives of the order

parameter: ~∇Qˆ(~r), whereas the anisotropic part Eq. (3.12) depends on the local value of the order parameter in the leading order.

3.3 SU(4) Symmetric Case

Non-homogeneous states can be generated from the ground state (3.11) by

slow rotations, with the effective action being dependent on the rotation

matrix U(t, ~r):

S[U ] = −i Tr log ∫ DψDψ† exp

( i

∫ Ldt ) , (3.13)

where the spin-valley symmetric Lagrangian of the 2DEG is

L = ∫ ψ†α

[ i ∂

∂t − 1

2m

( −i~∇ + ~A0 + ~Ω

)2] αβ

ψβ d 2~r

− ∫ ψ†αΩ

t αβψβ d

2~r + 1

2

∑ ~q

V (~q)N(~q)N(−~q), (3.14)

where Ωt = −i U †∂tU and ~Ω = −i U † ~∇U . We treat the nonhomogeneous ma- trix U as a classical field and therefore the effective action (3.13) describes a

macroscopic motion of the corresponding spin-texture. The essential points

3.3 SU(4) SYMMETRIC CASE 89

here are the slow variation of U(~r) on the microscopic scale given by the

magnetic length and the use of the gradient expansion assuming a locally

ferromagnetic ground state. The effective action (3.13) for multivalley sys-

tems was found in the work of Arovas et al. [46] We derive it here following

the method of Ref. [47].

The Hamiltonian corresponding to (3.14) with ~Ω = 0 has a uniform

ground state (3.11). We consider three filling factors: ν = 1 with Ψ(↑ +1), ν = 2 with Ψ(↑ +1, ↓ −1) and ν = 3 reducing to the case ν = 1 under the electron-hole transformation. The reference state is alternatively given by an

electron occupation diagonal matrix: Nατ1 ,βτ2 = (1, 0, 0, 0) in the case ν = 1,

and Nατ1 ,βτ2 = (1, 1, 0, 0) in the case ν = 2. The Green function for electron

propagation in the lowest Landau level evaluated over the ground state reads

G0ατ1,βτ2(�, p) = Nατ1,βτ2

� + E0 + µ− i0 + (1ˆ−N)ατ1 ,βτ2 � + µ+ i0

, (3.15)

where µ = −E0/2 is the chemical potential and exchange constants in the limit of vanishing thickness of the electron layer are

E0 = 2E1 =

√ pi

2

e2

κlH . (3.16)

The effective action is S[Ω] = S0[Ω]+S2[Ω], where S0[Ω] = iTr log(G/G0),

whereas S2[Ω] is represented by the two diagrams on Fig. 3.1. The first

order perturbation correction to the action Eq. (3.13) depends on the Green

function of electron propagation in the first excited Landau level [47]:

G1ατ1,βτ2(�) = Nατ1,βτ2

�− ωc + E1 + µ+ i0 + (1ˆ−N)ατ1 ,βτ2 �− ωc + µ+ i0 . (3.17)

The two terms of the Hamiltonian depending on the gradient matrix field

are:

H1 = 1

2m

∫ ψ†ατ1

( Ω+Πˆ− + Πˆ+Ω−

) αβτ1τ2

ψβτ2 d 2~r, (3.18)

H2 = 1

2m

∫ ψ†ατ1

( ~Ω2 − �µν∂µΩν

) αβτ1τ2

ψβτ2 d 2~r, (3.19)

where Ω± = Ωy ∓ iΩx, and the differential operators Πˆ± shift an electron to the adjacent Landau levels: Πˆ−φnp(~r) =

√ 2nφn−1p(~r), Πˆ+φn−1p(~r) =

90 CHAPTER 3. POSSIBLE JAHN-TELLER EFFECT...

1

H1 ba

H1H

Figure 3.1: Second order (a) and the anomalous first order (b) diagrams.

Solid and wavy lines represent the electron Green functions and the Coulomb

interaction respectively.

√ 2nφnp(~r), though only the n = 0 and n = 1 Landau level states are relevant

for this problem. An expansion of the 2DEG action up to the second order

of Hamiltonian (3.18) gives:

S0[Ω] = iTr (H1G0) + i

2 Tr (H1G0H1G0) + iTr (H2G0) . (3.20)

The Hartree-Fock diagram in Fig. 3.1a has been calculated in Ref. [47]

whereas the anomalous diagram in fig. 3.1b is calculated in Appendix A.

The resulting effective Lagrangian is

Leff [Ω] = ∫ (

Tr(Ωt(t, ~r)Nˆ) + E0 + E1

2 �µν∇µΩzν −

− E1 2

Tr [ NΩ+(t, ~r)(1ˆ−N)Ω−(t, ~r)

] )d2~r 2pi

, (3.21)

where Ωzµ = −iTr ( NU †∂µU

) . The matrices N and 1ˆ − N are projecting

operators onto the physical rotations that form a subset of the SU(4) Lie

group. The matrix field Ωµ can be expanded in the basis of fifteen generators

of the SU(4) group: {Γl}, with l = 1..15. They form two complementary sets: the first even set includes those generators that do commute with N ,

whereas the second odd set includes the remaining generators. Generators

of the even set constitute an algebra themselves. This algebra has a normal

3.3 SU(4) SYMMETRIC CASE 91

Abelian subalgebra formed by a single traceless generator: N − ν/4. A Lie group generated by the even set is a stabilizer sub-group S of the SU(4) group. The odd set contains an even number of generators: eight in the case

of ν = 2 and six in the case ν = 1, 3.

The Hamiltonian is invariant under time reversal symmetry. The time

reversal operation transforms the rotation matrix as U → U ∗, the gradient field as ~Ω → −~ΩT and also inverts the magnetic field Bz. Accordingly, we rewrite the energy in the Lagrangian (3.21) in a time reversal symmetric

form:

Eeff [Ω] =

∫ d2~r

2pi

[E1 2

Tr ( NΩµ(1ˆ−N)Ωµ

)− − E0

2 sgn(Bz)�µν∇µΩν

] , (3.22)

where the identity �µνTr (ΩµΩνN) = i�µν∇µΩzν is used. It follows immedi- ately that Eeff ≥ 0. The first term is the gradient energy whereas the second term is proportional to the topological index of the spin texture:

Q = ∫ �µν∇µΩzν

d2~r

2pi = Z, (3.23)

where Z is an integer. The states with Z 6= 0 are called skyrmions after T. H. R. Skyrme who has first considered such textures. The case Q = ±1 corresponds to the simplest spin skyrmions in the first valley which can be

rotated by a SU(4) matrix to become a general bivalley skyrmion. The spin

stiffness in Eeff Eq. (3.22) coincides identically with that of the one-valley

case [47]. This means that the bivalley skyrmion energy is the same as that

found for one valley. The charge in the bivalley skyrmion core is distributed

over the two valleys with long divergent tails: n±1(~r) ∼ ±1/ (R2 + r2), if one neglects anisotropy. But the total charge of the two valleys follows a

convergent distribution identical to the charge density in the one valley case

[37]:

n(~r) = �µν∇µΩzν(~r)

2pi =

R2

pi (R2 + r2)2 , (3.24)

where R is the radius of the skyrmion core.

92 CHAPTER 3. POSSIBLE JAHN-TELLER EFFECT...

The expectation value of any operator A in the ground state 〈A〉 = Tr(AQˆ) can be expressed in terms of the order parameter matrix

Qˆ(~r) = U(~r)NU+(~r). (3.25)

Obviously rotations from the small sub-group S leave the order parameter invariant. Thus, rotations in Eq. (3.25) can be restricted to a physical space

of the bivalley 2DEG which is the complex Grassmannian manifold GC4ν =

U(4)/U(ν)⊗ U(4− ν). We rewrite Eq. (3.22) in terms of Qˆ:

Eeff [Qˆ] = E1 4

∫ Tr ( ~∇Qˆ~∇Qˆ

) d2~r 2pi − E0

2 Q. (3.26)

In this representation the topological index Q is an index of a map from the 2D plane onto the coset space of the order parameter. The rule (3.23) is a

consequence of the homotopy group

pi2 ( GC4ν )

= Z, (3.27)

proven in Appendix B.

Varying the effective Lagrangian consisting of the first kinetic term Eq. (3.21)

(rewritten as the Wess-Zumino term [46]) and the energy Eq. (3.26), we find

the matrix Landau-Lifshitz equation[ Qˆ ∂tQˆ

] −

= E1 2 ~∇2Qˆ. (3.28)

It describes three (ν = 1) and four (ν = 2) degenerate ”spin-valley”-waves

with dispersion ω(~q) = E1q 2/2.

3.4 Jahn-Teller effect

The Jahn-Teller effect is the deformation of a lattice which lowers the energy

He−ph +Hph. Phonons interact with the local electron density. We illustrate

the main points of our approach for ν = 1, keeping the spin of electrons

fixed and up. The electron annihilation operator (3.1) is expanded in the

basis of two mutually orthogonal valley wave functions τ = ±1. The density

3.4 JAHN-TELLER EFFECT 93

operator nˆ(~ρ) = ψˆ†(~ρ)ψˆ(~ρ) has the diagonal element uniform over the 2D

plane: χ2(z)〈ψˆ†τ (~r)ψˆτ (~r)〉 = νχ2(z)/2pil2H , where the average is taken over the ground state (3.11), and the term oscillating in z direction: δnˆ(~ρ) =

χ2(z)(ψˆ†+1(~r)ψˆ−1(~r)e iQz +h.c.). We use the electronic order parameter (3.25)

to express the average oscillating density as

δn(~ρ) = χ2(z)

2pil2H

( Tr(Qˆτ+)eiQz + Tr(Qˆτ−)e−iQz

) . (3.29)

The omitted diagonal density term couples electrons to phonons with in-

plane momentum ~q⊥ = 0 and can be neglected in the thermodynamic limit.

The oscillatory part of the average electron density allows for lowering of the

energy by a static lattice deformation which oscillates with wave vector Q

inside the quantum well:

∇zuzst(~ρ) = Θ

ρSic2 δn(~ρ). (3.30)

This deformation is a phonon ”condensate” or Jahn-Teller effect. The corre-

sponding energy gain is found from the minimization of He−ph +Hph as

Est = −NGTr(Qˆτ+)Tr(Qˆτ−). (3.31)

where N = A/2pil2H is the number of states within Landau level for a sample with area A and G = Θ2

∫ χ4(z)dz /2pil2HρSic

2 is the strength of the electron-

phonon interaction. The integral in the last definition is the inverse width of

the 2D layer in z-direction.

This static deformation causes a new equilibrium position of the lattice.

Assuming that the static deformation is small, we neglect the change of

the phonon spectrum due to anharmonic effects. However we do take into

account the dynamical phonons which produce a polaronic effect in second

order of the electron-phonon interaction. Expanding the phonon field around

the new equilibrium: uz(~ρ) = uzst(~ρ) + δu z(~ρ), we obtain

Hˆe−ph = Θ ∫ ψ†ατ1(~r)τ

± τ1τ2

ψατ2(~r)×(∫ χ2(z)e±iQz∇zδuz(~ρ) dz

) d2~r, (3.32)

94 CHAPTER 3. POSSIBLE JAHN-TELLER EFFECT...

where the lattice deformation

δuz(~ρ) = ∑

~q

√ ~

2V ρSiω(q)

( b~qe

i~q~ρ + b†~qe −i~q~ρ ) ,

can be expanded in phonon creation and annihilation operators b†~q and b~q. In

order to find the ground state energy we sum up diagrams for the thermody-

namic potential expanded in powers of the (assumed to be) weak electron-

phonon and Coulomb interactions using the Matsubara method [48]. We

express the energy in terms of the order parameter matrix Qˆ.

The electron Green function can be expressed in terms of Qˆ as

G0(εn, p) = Qˆ

iεn + E0/2 +

1ˆ− Qˆ iεn − E0/2 . (3.33)

We include the coordinate dependent electron wave-functions in the interac-

tion vertices. The phonon Green function is [48]

D(ωn, q) = − ω 2(q)

ω2n + ω(q) 2 . (3.34)

For small momentum transfers the Coulomb interaction is much larger

than the phonon propagator. In the opposite case of large transferred mo-

mentum with ±Q z-component we neglect the Coulomb interaction. There- fore in the Coulomb vertex the valley index is conserved as well as the spin

index. Due to the identity Qˆ(1 − Qˆ) = 0, all one loop diagrams with only Coulomb lines vanish.

The expression for the electron-phonon vertex in Landau gauge reads

g(pp′, ~q) = Θ√ ρSic2

δp,qy+p′e −q2⊥/4−qx(p+p′)/2χ2(qz −Q),

where ~q is the phonon momentum and the envelope function χ2(qz) =∫ χ2(z)eiqzz dz has a characteristic width l−1 in momentum space. The

polaron contribution to the energy assuming weak Coulomb interaction:

E0/ω(Q)→ 0, is E0e−ph =

∑ εnωnpp′

Tr ( G0(εn, p)τ

+G0(εn + ωn, p ′)τ−

) ∑

~q

g2(pp′, ~q)D(ωn, q) = NG ( Tr(Qˆτ+Qˆτ−)− ν

2

) . (3.35)

3.4 JAHN-TELLER EFFECT 95

Figure 3.2: The two diagrams that give easy-plane anisotropy in the case

ν = 1. Solid, dashed and wavy lines are the electron, the phonon propagators

and the Coulomb interaction respectively.

Summing it up with Eq. (3.31) we get the total electron-phonon energy in

second order of the electron-phonon interaction. Here we neglect small terms

of the order (Ql)−4 due to the discontinuity of the derivative of the wave

function χ(z) at the interface [40]. For ν = 1 the total energy given by the

sum of Eqs. (3.31,3.35) is isotropic in spin-valley space. Thus, if we neglect

the Coulomb interaction corrections to the electron-phonon interaction we

find that the degeneracy in isospin direction is not lifted. Physically, the

polaron energy is a single electron energy and there is no anisotropy operator

for a single electron because of the Pauli matrix identity τ 2i = 1.

The Coulomb interaction creates the easy-plane anisotropy in the case

ν = 1. The essential diagrams are shown in Fig.(3.2), where the wavy line

represents 2D Coulomb potential V (q) whereas the solid and the dashed

lines represent the electron, Eq. (3.33), and the phonon, Eq. (3.34), propa-

gators. Direct calculation neglecting the dependence of ω(Q) on the in-plane

momentum q⊥ shows that the sum of the two diagrams in Fig. 3.2 is

Eδe−ph = −NGδTr ( Qˆτ+Qˆτ−

) , (3.36)

where δ = Eex/ω(Q) is the ratio between the Coulomb exchange energy

Eex = e2

piκ

∫ d2q⊥ dqz q2⊥ + q2z

|χ2(qz)|2e−q2⊥l2H/2 ( 1− e−q2⊥l2H/2

) , (3.37)

to the energy of the valley-mixing phonon. Obviously, δ > 0 (Eex > 0),

and the anisotropy is of the the easy-plane type. The above result is valid if

E0 � ω(Q).

96 CHAPTER 3. POSSIBLE JAHN-TELLER EFFECT...

The sum of the static, polaronic and the Coulomb correction energies,

Eqs. (3.31,3.35,3.36), gives the total anisotropic energy:

Eep = G

∫ [− Tr(Qˆτ+)Tr(Qˆτ−) + +(1− δ)Tr

( Qˆτ+Qˆτ−

) ]d2~r 2pi

. (3.38)

Thus the JTE is energetically favorable and corresponds to the easy-plane

valley anisotropy.

The easy-plane anisotropy changes the dispersion of collective excitations.

Let us consider this effect in the limit of vanishing valley symmetry breaking

t and Zeeman h anisotropies. We add the easy-plane term (3.38) to the

Landau-Lifshitz equation (3.28):

[ Qˆ ∂tQˆ

] −

= E1 2 ~∇2Qˆ +Gδ

( τ+Tr(Qˆτ−) + τ−Tr(Qˆτ+)

) . (3.39)

This equation can be linearized in the vicinity of Q = N . It describes two

”spin”-modes (the same as for the symmetric case) and one acoustic mode

with dispersion ω = √ GδE1q in analogy to the bilayer case [40].

3.5 Phase diagram

The anisotropic bivalley energy in the uniform state is the diagonal matrix

element of the anisotropic Hamiltonian (3.2) expressed in terms of the order

parameter Qˆ:

Ean/N = −t Tr(Qˆτx)− h Tr(Qˆσx) + G [ (1− δ)Tr

( Qˆτ+Qˆτ−

) − Tr(Qˆτ+)Tr(Qˆτ−)

] , (3.40)

where h = |ge|µBH. The order parameter Qˆ can be parameterized by six (ν = 1, 3) or eight (ν = 2) angles. The diagonal matrix elements are real

despite the fact that in external magnetic field there is no time reversal

symmetry. Therefore the ground state can be chosen real, produced from

the reference state by SO(4) sub-group rotations. This sub-group has 6

3.5 PHASE DIAGRAM 97

S

t

h

F C

G Figure 3.3: Phase diagram in the case ν = 2 and δ = 0. F, S and C are

ferromagnetic, spin-singlet and canted antiferromagnetic phases, respectively.

parameters with two of them falling into the denominator sub-group (for

ν = 2). One of the remaining four angles corresponds to global rotation of

all spins and is fixed by the magnetic field direction z. Thus, the ground

state is obtained from the reference state by just three rotations (for ν = 2).

First we consider the case ν = 1. The ground state is UΨ(↑ +1), where U = P (ϑ)R(θ). The matrix R rotates the ↑-spin component in valley space by an angle θ, and then the matrix P rotates the valley components of the

resulting state in spin space by an angle ϑ. Substituting Qˆ(ϑ, θ) Eq. (3.25)

into Eq. (3.40) we obtain the energy per electron:

E1an = −h sin ϑ− t sin θ +Gδ cos2 θ. (3.41)

The minimum of this energy is reached at ϑ = pi/2 and θ = pi/2 corresponding

to the ferromagnet phase in spin and valley spaces. The case ν = 3 is identical

to ν = 1 due to electron-hole symmetry.

In the case ν = 2 the ground state is UΨ(↑ +1, ↓ −1) where U = P (ϑ)R(θ↑, θ↓). The matrix R rotates the ↑, ↓-spin components by angles θ↑,↓ in valley space and then the matrix P rotates the ±1-valley components by angles ±ϑ in spin space. We find θ↑ = θ↓. Substituting the corresponding

98 CHAPTER 3. POSSIBLE JAHN-TELLER EFFECT...

Qˆ(ϑ, θ) into Eq. (3.40) we get:

E2an = −2t cosϑ sin θ − 2h sinϑ cos θ + +G(1− δ)

( 1

2 cos2 θ − cos2 ϑ

) −Gδ cos2 ϑ sin2 θ. (3.42)

The last term represents the easy-plane anisotropy and the antiferromagnetic

exchange interaction between spins in two valleys. The phase diagram in the

case δ = 0 is shown in Fig. 3.3. Here F is the spin ferromagnetic and the

valley singlet phase (ϑ = pi/2, θ = 0), S is the spin singlet and the valley

ferromagnetic phase (ϑ = 0, θ = pi/2), C is the canted antiferromagnetic

phase in spin and valley spaces.

At δ = 0, we find two lines of continuous phase transitions between

F and C phases: (h − G)(h − G/2) = t2, and between S and C phases: (t+G)(t+G/2) = h2. Small positive δ transforms the S-C phase transition

into a discontinuous first order one. The C phase ends at some critical hc(δ),

and the direct S-F first order transition occurs at h > hc(δ) on the line:

t = h − G(3 − δ)/4. For δ > 0.33 the C phase disappears. A detailed examinaning of Eq. (3.42) is given in the Appendix C.

At typical magnetic fields B ∼ 5− 10T the Zeeman energy in Si is a few times larger than the valley splitting t ∼ 2K [49]. To our knowledge, the deformation potential at large phonon momentum is unknown. Therefore

we could only assume the value electron-phonon energy, speculatively: G ∼ 0.2 − 2K. Probably, to observe phases of Fig. 3.3 in Si one would need to lower the effective electron g-factor artificially.

3.6 Anisotropic energy of skyrmion

Many experiments found activation gaps for diagonal conductivity in QHE

systems. For symmetry broken ”ferromagnetic” systems theory predicts the

conductivity to be mediated by charged topological textures — skyrmions,

with activation energies being determined by a large exchange constant E1

from Eq. (3.26). Contrary to this, experiments find much lower gaps [50].

3.6 ANISOTROPIC ENERGY OF SKYRMION 99

One possible explanation is that some intrinsic inhomogeneity, like defects or

a long range slow variation of the electrostatic potential, makes skyrmions of

opposite topological charges to be already present in the system and there-

fore experiments reveal either a depinning activation energy or a skyrmion

mobility activation energy. In both cases the gap is determined by the small

anisotropic part of the skyrmion energy. In this section we compute this

anisotropic energy for a bivalley skyrmion.

Belavin and Polyakov (BP) found the skyrmion solution in the case of a

S2 order parameter [36]. Their solution is readily generalized for a general

Grassmannian order parameter. The non-homogeneous order parameter that

represents one skyrmion in the symmetric case with |Q| = 1 is given by

QˆBP (z, z¯) = Nˆ + 1

R2 + |z|2 ( −R2|vf 〉〈vf | zR|vf 〉〈ve| z¯R|ve〉〈vf | R2|ve〉〈ve|

) (3.43)

where z = x + iy, R is the radius of the skyrmion core and |vf〉 and |ve〉 are two vectors of dimension ν and 4 − ν. We choose |vf,e〉 = (1, 0..0) in (3.43). The skyrmion order parameter has to be rotated by a homogeneous

matrix U calculated in the previous section in such a way that the order pa-

rameter far away from the skyrmion center minimizes the anisotropy energy.

In addition to this rotation, we have to allow the global rotations W from

the denominator sub-group S that transform the skyrmion order parameter (3.43): Qˆ(~r) = UWQˆBP (z, z¯)W

+U+. For the case ν = 2 the block diagonal

matrix W = (Wf ,We) can be parameterized by seven angles:

We,f =

cos

βe,f 2 ei(γe,f +αe,f ) sin

βe,f 2 ei(γe,f−αe,f )

− sin βe,f 2 ei(−γe,f +αe,f ) cos βe,f

2 ei(−γe,f−αe,f )

. (3.44)

The additional seventh parameter angle of the denominator sub-group rotates

the coordinates: z → eiγ7z. We find explicitly that the skyrmion anisotropic energy does not depend on the angles γe, γf and γ7 whereas αe = 0 and

αf = pi correspond to the energy minimum. Thus the order parameter inside

the core of the skyrmion depends on the two angles βe and βf . Beside these

100 CHAPTER 3. POSSIBLE JAHN-TELLER EFFECT...

angles the BP solution depends on arbitrary conformal parameters that do

not change the energy of the skyrmion. We consider here the simplest case

of topological index |Q| = 1 Eq. (3.43), where only one such conformal pa- rameter R is essential. Calculating the energy of the skyrmion we encounter

different spatial integrals with one integral being logarithmically large:

K(R) =

( R

lH

)2 log

( lH R

√ E1 Eskmin

) , (3.45)

and we neglect all the others. In this way we find the Zeeman energy of the

skyrmion to be EskZ = K(R)Z, where

Z = 2h (2 sinϑ cos θ − cosϑ(sin βf + sin βe) sin θ) (3.46)

The bare valley splitting energy of the skyrmion is given by: Eskt = K(R)T , where

T = 2t (2 cosϑ sin θ − sinϑ(sin βf + sin βe) cos θ) (3.47) Finally, the Jahn-Teller energy of the skyrmion is given by: Gsk = K(R)G, where

G = G [ − 1

2 + 3 cos2 ϑ +

1

2 cos2 θ cos(βe + βf)−

−3 2

cos2 θ − (

1

2 − 2 cos2 ϑ

) cos(βe − βf )

] . (3.48)

The total energy of a skyrmion also includes the energy of direct Coulomb

repulsion of an additional charge distribution n(~r) (3.24):

EskC =

∫ e2n(~r)n(~r′)

2κ|~r − ~r′| d 2~rd2~r′ =

3pi2

64

e2

κR = EC lH

R . (3.49)

The minimum of the total anisotropic skyrmion energy, the sum of Eqs.

(3.46)-(3.48): Esk = Z +T +G, with respect to βe and βf was found numeri- cally and is denoted as: Eskmin. Next, we find a minimum of the total skyrmion energy including Eq. (3.49): Esk = K(R)Eskmin + EC/R, with respect to the skyrmion radius R:

∆ = E1|Q| − E0Q

2 +

3

2

( EskminE2C log

E1 Eskmin

)1/3 . (3.50)

3.6 ANISOTROPIC ENERGY OF SKYRMION 101

0 G

h

0

G t

gap

Figure 3.4: Anisotropic part of skyrmion energy gap at ν = 2 and δ = 0

This equation is valid in the limit E skmin � E1. The resulting anisotropic part of the skyrmion gap is shown in the Fig. 3.4. Note that the prominent

minimum of the skyrmion anisotropy energy gap coincides with the phase

transition line C-F from the phase diagram Fig. 3.3.

This skyrmion anisotropic energy is similar to the bilayer case [45] with

t being the hopping constant between the two layers. In an experiment on

a GaAs bilayer [51], a profound reduction of the thermal activation gap has

been found in some interval on the ν = 2 line.

In the case ν = 1 we parameterize general rotations from the denominator

sub-group by four angles:

|ve〉 = (

cos β

2 , sin

β

2 cosαeiλ1 , sin

β

2 sinαeiλ2

) (3.51)

The skyrmion energy does not depend on the angles λ1,2 whereas α = pi/2

corresponds to the energy minimum. We find the Zeeman energy Z = h (1− cos β), the valley splitting energy T = 2t cos2(β/2) and the easy-plane Jahn-Teller energy G = Gδ (1 + cos β). The minimum of the sum of these

102 CHAPTER 3. POSSIBLE JAHN-TELLER EFFECT...

energies with respect to β and α: E skmin = 2 min (t+Gδ, h), determines the charge activation gap ∆ in Eq. (3.50). The case ν = 3 is identical to the case

ν = 1.

3.7 Conclusion

We have shown that the valley degeneracy for a Si(100) MOSFET can lead

to a Jahn-Teller effect in a strong magnetic field with a longitudinal lattice

deformation within the width of a quantum well. This deformation has an

atomic periodicity corresponding to the momentum difference between the

two valley minima in the direction perpendicular to the plain. The integer

fillings ν = 1, 3 and ν = 2 are treated in different ways. For ν = 1, 3 the Jahn-

Teller effect alone can not remove the valley degeneracy and the Coulomb

interaction is crucial. The level splitting here is small if the Coulomb energy

is small compared to the Debye energy. For the ν = 2 case the Jahn-Teller

effect removes the valley degeneracy via an easy-plane anisotropy similar to

that found in the 2D bilayer system in strong magnetic fields. The phase

diagram as a function of different physical parameters was established and

the anisotropic energy of a topological skyrmion-like texture was calculated.

3.8 Appendix A

Here we calculate the second diagram in Fig. 3.1 which was overlooked in

Ref. [47]. It represents the change of Hartree-Fock exchange energy in the

presence of a spin texture. The spatial part of the electron Green function

in Landau gauge is:

G0s(z, z ′) =

(z − z′)s√ 2ss!

e−|z−z ′|2/4+i(x+x′)(y−y′)/2, (3.52)

where z = x + iy, s is the Landau level and Gs0 = G ∗ 0s. The bottom part of

Fig. 3.1b features the G00 Green function whereas the upper part has the G10

APPENDIX B 103

and G00 Green functions. Evaluating the two frequency integrals we find:

δE = 1√ 2

∫ d2z

2pi

d2z′

2pi

d2ξ

2pi G00(z, z

′)V (z − z′) (G00(z

′ξ)Ω+(ξ)G01(ξz) +G10(z′ξ)Ω−(ξ)G00(ξz)) , (3.53)

Expanding Ω(ξ) around the center point of the diagram, z0 = (z + z ′)/2:

Ω±(ξ) = Ω±(z0) ± (ξ − z0) �µν∇µΩzν(z0), we evaluate the energy δE in Eq. (3.53) as (z0 = ~r):∫

d2z

( 1− |z|

2

4

) V (|z|)e−|z|2/2

∫ �µν∇µΩzν(~r)

d2~r

2pi . (3.54)

The front integral is evaluated for the Coulomb interaction and equals: (E1+

E0)/2.

3.9 Appendix B

Eq. (3.27) is proved using a principal bundle of the complex Stiffel manifold

V Cnk = SU(n)/SU(n − k) — that is defined as a manifold of k orthogonal complex vectors in n dimensional complex linear space — over the Grass-

mannian manifold GCnk with a layer U(k). The exact map sequence for this

bundle is

...pi2(V C nk)→ pi2(GCnk)→ pi1(U(k))→ pi1(V Cnk)... (3.55)

The Stiffel manifold has the property pij(V C nk) = 0 for j < 2(n − k) [52],

proven using the principal bundle SU(n) over V Cnk with a layer SU(n−k). Eq. (3.55) means that pi2(G

C nk) = pi1(U(k)). The Lie group U(k) is a product of

SU(k) and U(1) groups with the fundamental homotopy group: pi1(U(k)) =

pi1(SU(k)) + pi1(U(1)) = Z [52].

3.10 Appendix C

In this Appendix we derive the results for the ground state phase diagram

for the case ν = 2 and also present some other formal results which are too

104 CHAPTER 3. POSSIBLE JAHN-TELLER EFFECT...

specific for the main part of the work. We start with the expression (3.42) for

the energy of a homogeneous state as a function of the angles θ and ϑ. First

we derive the expressions for the lines of the phase transitions between F, C,

and S phases for δ = 0. To do this we compute the second derivative matrix

for the energy and check the positiveness of its minors. In this appendix all

energies are measured in units of G.

For the F phase the corresponding condition reads:∣∣∣∣∣ 2h− 2 + 2δ 2t2t 2h− 1 + δ ∣∣∣∣∣ > 0 (3.56)

or

(h− (1− δ)/2) (h− (1− δ)) > t2. (3.57) Generally this expression gives only the region where the F phase is stable

(metastable), but for the 2nd order F-C transition, we get the phase transi-

tion line for δ = 0:

(h− 1)(h− 1/2) = t2. In the same way we get the condition for the (meta)stability of the S

phase:

(t + 1)(t+ (1 + δ)/2) > h2, (3.58)

which for δ = 0 gives the phase transition line

(t+ 1)(t+ 1/2) = h2.

Both the phase transition lines tend to h = t + 3/4 for large t and h.

The line of the direct first order transition between the F and S phases

for large t and h and δ > 0 is obtained by direct comparison of the energies

of the two phases: the energy of the F phase

EF = −2h + 1 2 (1− δ)

and the energy of the S phase

ES = −2t− 1.

APPENDIX C 105

We get

t = h− 3− δ 4

. (3.59)

In order to find the phase transition lines for arbitrary δ we need to find

the values for θ and ϑ in the C phase. This is quite a difficult problem and

here we give an analytical solution for it in some limiting cases. First we find

the C phase for δ = 0. To do it a variable change is convenient: instead of θ

and ϑ we use α = ϑ−θ and β = ϑ+θ. In these variables we get consequently:

E2an = −t(sin β−sinα)−h(sin β+sinα)+ 1

4 (3 sinα sin β−cosα cos β); (3.60)

∂E2an ∂α

= −(h− t) cosα + 1 4 (3 sin β cosα + sinα cos β), (3.61)

∂E2an ∂β

= −(t + h) cos β + 1 4 (sin β cosα + 3 sinα cos β). (3.62)

The conditions ∂E2an/∂α = 0, ∂E 2 an/∂β = 0 yield:

sinα = 3(h+ t)− A(h− t)

2 ,

sin β = 3(h− t)− A−1(h+ t)

2 (3.63)

with

A = cosα

cos β = ±

[ 2(h+ t)2 − 1 2(h− t)2 − 1

]1/2 .

Notice that, as can be seen from Eq. (3.42), θ and ϑ for the ground state

must be in the first quadrant that means that cosα ≥ 0. Calculating the second derivative matrix and checking the stability of the found states we

can show that cos β < 0 corresponds to a maximum of the energy rather

than a minimum.

Finally, substituting (3.63) into (3.60), we find the energy of the C phase:

EC =

√ (2(h− t)2 − 1)(2(h+ t)2 − 1)− 6(h2 − t2)− 1

4 .

Comparing it to EF and ES at the F-C and S-C transition lines calculated

above, we see that at the transition lines the energies are equal, which proves

that the transition is indeed of the second order.

106 CHAPTER 3. POSSIBLE JAHN-TELLER EFFECT...

Now we examine the case of vanishing t and arbitrary δ. The conditions

∂E2an/∂θ = 0 and ∂E 2 an/∂ϑ = 0 for t = 0 read as follows:

2h sinϑ sin θ − 1 2

sin 2θ − δ 2

cos 2ϑ sin 2θ = 0,

−2h cosϑ cos θ + sin 2ϑ− δ sin 2ϑ cos2 θ = 0. (3.64) Apart from the trivial solutions corresponding to the F and S phases we find

the C phase given by

θ = 0, sin ϑ = h

1− δ . (3.65) Other solutions are either non-stable or lose in energy to other states. This

C phase is stable for

(1− δ) √

1 + δ

2 < h < (1− δ).

(The S phase is stable for h < √

(1 + δ)/2 and the F phase for h > 1 − δ.) Comparing the energies of the S and C phases we find the point of the S-C

transition:

hS−C =

√ 1− δ2

2 ;

this is a first order phase transition with regions of metastability for both

phases on either side of it. The C-F transition occurs at hC−F = 1− δ, this transition is of 2nd order.

To find δ at which the C phase disappears we need to set hC−F = hS−C,

this gives δ = 0.33 as mentioned in the main part of the work.

Finally, to find the ending point hc(δ) of the C phase we can find the

intersection of the two known phase transition lines: (3.57) and (3.59). We

get:

hc(δ) = 1 + 10δ − 7δ2

16δ . (3.66)

Chapter 4

Summary

In this thesis I have presented the results on two subjects: the Coulomb

drag and Jahn-Teller effect in two-dimensional electron systems in a strong

magnetic field.

A semiclassical theory for electron magneto-drag between two parallel

two-dimensional electron systems, which provides a transparent picture of

the most salient qualitative features of anomalous drag phenomena observed

in recent experiments, is presented. Localization plays a role in explaining

activated low temperature behavior, but is not crucial for anomalous (es-

pecially negative) drag per se. In particular a detailed theory for the drag

between two one-dimensional links has been elaborated. The temperature

dependence of the drag has been derived for a whole variety of the param-

eters of the system for parallel links as well as for non-parallel ones. Also,

qualitative arguments showing to what consequences for the two-dimensional

drag my results lead are presented.

The Jahn-Teller effect in bivalley Si(100) MOSFET under conditions of

quantum Hall effect at integer filling factors ν = 1, 2, 3 has been studied.

This system is described by an approximate SU(4) symmetry. At ν = 2 static

and dynamic lattice deformations give rise to an easy-plane anisotropy and

antiferromagnetic exchange and lift the valley degeneracy. At ν = 1, 3 the

Coulomb interaction is essential to produce weak easy-plane anisotropy. At

107

108 SUMMARY

ν = 2 three phases: ferromagnetic, canted antiferromagnetic and spin-singlet,

have been found. The anisotropy energy of a charged skyrmion excitation in

every phase has been calculated.

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Acknowledgments.

First I would like to thank Prof. Dr. Walter Metzner for inviting me to do my

PhD at the Max-Planck Institut fu¨r Festko¨rperforschung, for the interesting

topic proposed, and for many helpful discussions during the work.

I thank Prof. Dr. Sergey Iordanskii, the supervisor during my diploma work

and during the first year of my PhD in Institute for Theoretical Physics,

Chernogolovka for his constant support and advice as well in the time of our

collaboration as also after I went to Stuttgart.

I thank Prof. Dr. Alejandro Muramatsu for accepting to be the co-reporter

of my thesis.

My special thanks goes to Prof. Dr. Leonid Glazman, University of Min-

nesota. I always enjoyed the stimulating discussions with him which greatly

influenced this thesis.

I would like to thank Dr. A. Kashuba, Institute for Theoretical Physics, for

the collaboration during the work on the second part of the thesis.

I thank my father, Dr. Efim Brener for his constant interest in my work and

for the idea of the arguments presented in the end of section 2.3.2.

I have the pleasure of thanking all members of the Theorie II department of

115

116

the Max-Planck Institut fu¨r Festko¨rperforschung for the pleasant atmosphere

and all the interesting conversations.

Same goes for the members of the ITP and ISSP in Chernogolovka. In

particular I would like to thank Pavel Ostrovskiy, Mikhail Feigel’man, and

Vladislav Timofeev.

My regards go to Ingrid Knapp for all her organizational support.

I thank Prof. Dr. V. Falko for offering me the opportunities to participate

in very interesting workshops in Dresden and Windsor which gave boosts to

my work.

Finally I would like to thank all the people who were near me during all

these years, in particular my wife Katia, my mother, my brother, and my

two children, for the various support and help.

Publications.

The content of this thesis is partly published in the following journals:

• S. Brener, S. V. Iordanskii, and A. Kashuba. Possible Jahn-Teller effect in Si inverse layers. Phys. Rev. B 67, 125309 (2003).

• S. Brener, W. Metzner. Semiclassical theory of electron drag in strong magnetic fields. Pis’ma Zh. Eksp. Teor. Fiz. 81, 618 (2005) [JETP

Lett. 81, 498 (2005)].

117

118

Lebenslauf

Name Sergej Brener

Geboren am 16. Oktober 1978

Geburtsort Tschernogolowka, Moskauer Gebiet, Russland

Seit Juli 2002 Wissenschaftlicher Mitarbeiter bei Prof. Dr. W. Metz-

ner, Max-Planck-Institut f¨r Festko¨rperforschung in

Stuttgart

Juli 2001 – Juli 2002 Wissenschaftlischer Mitarbeiter bei Prof. Dr. S. Ior-

danskii, Institut fu¨r Theoretische Physik in Tscherno-

golowka

Sep. 1995 – Juni 2001 Studium der Physik an der Moskauer Physikalisch-

Technische Hochschule

1985 – 1995 Schule, Tschernogolowka

Abschluss Juni 1995

Magnetic Fields

Von der Fakulta¨t fu¨r Mathematik und Physik der Universita¨t Stuttgart

zur Erlangung der Wu¨rde eines Doktors der

Naturwissenschaften (Dr. rer. nat.) genehmigte Abhandlung

Vorgelegt von

Sergej Brener

aus Moskau

Hauptberichter: Prof. Dr. W. Metzner

Mitberichter: Prof. Dr. A. Muramatsu

Tag der mu¨ndlichen Pru¨fung: 11. Mai 2006

Max-Planck Institut fu¨r Festko¨rperforschung

2006

Contents

Deutsche Zusammenfassung 5

1 Overview 9

2 Theory of Coulomb Magneto-Drag 11

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 History of the problem. Experiment. . . . . . . . . . . . . . . 12

2.3 History of the problem. Theory. . . . . . . . . . . . . . . . . . 17

2.3.1 Early works. . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3.2 Boltzmann equation approach. . . . . . . . . . . . . . . 20

2.3.3 Diagrammatic approach. . . . . . . . . . . . . . . . . . 22

2.3.4 Magneto-drag. Early diagrammatic approach. . . . . . 29

2.3.5 Negative magneto-drag. First explanation attempt. . . 30

2.3.6 Diagrammatic calculation of magneto-drag in the SCBA. 33

2.4 Semiclassical theory of electron drag in strong magnetic fields 41

2.4.1 Drag between parallel links. . . . . . . . . . . . . . . . 46

2.4.2 Drag between non-parallel links. . . . . . . . . . . . . . 70

2.4.3 Connection to two-dimensional drag. . . . . . . . . . . 80

2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

3 Possible Jahn-Teller effect in Si-inverse layers 83

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

3.2 Hamiltonian of electron gas on Si interface . . . . . . . . . . . 85

3.3 SU(4) Symmetric Case . . . . . . . . . . . . . . . . . . . . . . 88

3

4 CONTENTS

3.4 Jahn-Teller effect . . . . . . . . . . . . . . . . . . . . . . . . . 92

3.5 Phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . . 96

3.6 Anisotropic energy of skyrmion . . . . . . . . . . . . . . . . . 98

3.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

3.8 Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

3.9 Appendix B . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

3.10 Appendix C . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

4 Summary 107

Deutsche Zusammenfassung

Seit Jahrzehnten zogen zwei-dimensionale elektronische Systeme das Inter-

esse vieler Physiker und Technologen an. Der Grund dafu¨r ist nicht nur - wie

man sich naiv vorstellen ko¨nnte - die technologische Anwendung in Rech-

nerprozessoren. Die scheinbar einfache physikalische Konfiguration zwei-

dimensionaler Elektronensysteme fu¨hrt zu unerwartet vielen experimentell

beobachtbaren Effekten und theoretischen Modellen, die eine eigene Bedeu-

tung fu¨r die Entwicklung von Physik, Technologie, Computerrechnungen

usw. haben.

Seit der Entdeckung des Quanten-Hall-Effekts [1] galt zwei-dimensionalen

elektronischen Systemen in starken Magnetfeldern (sogenannte Quanten-Hall-

Systeme) ein besonders starkes Interesse. Die physikalischen Effekte, die

in solchen Systemen zu beobachten sind, sind ein Resultat der Zusamen-

wirkung von Landau-Quantisierung der elektronischen Bewegung, Elektron-

Elektron-Wechselwirkung, Bandstruktur des Materials (normaleweise Si oder

GaAs), Unordnungseffekte usw. Unter den interessanten Aspekten zwei-

dimensionaler elektronischer Systeme kann man folgende Themen nennen:

Quanten-Hall-Effekt, Coulomb-Drag, skyrmionische Phasen, Energietal-Auf-

spaltung (valley splitting) in Si-Inversionsschichten, Auswirkung der Mikro-

wellenstrahlung auf die Transporteigenschaften der Quanten-Hall-Systeme

und viele andere.

In der vorgelegten Dissertation bescha¨ftige ich mich mit zwei mit Quanten-

Hall-Systemen verbundenen Effekten: dem Coulomb-Drag im Kapitel 2 und

dem Jahn-Teller-Effekt in Si-Inversionsschichten im Kapitel 3.

5

6 DEUTSCHE ZUSAMMENFASSUNG

Kapitel 2 ist ein Resultat der durch die Werke [2, 3] motivierten Unter-

suchung des sogenannten anomalen Magneto-Drag, der in den obengenannten

Referenzen experimentell entdeckt wurde. Der Drag (auch Coulomb-Drag

genannt) ist ein physikalischer Effekt, der in elektronischen Doppelschicht-

systemen beobachtet wird. Durch eine Schicht wird Strom gefu¨hrt, der in der

zweiten Schicht eine elektrische Spannung verursacht. Falls das System sich

in einem senkrechten Magnetfeld befindet, wird der Drag als Magneto-Drag

bezeichnet. Der anomale Magneto-Drag besteht darin, dass unter gewissen

Unsta¨nden der Magneto-Drag sein Vorzeichen wechselt und negativ wird.

In diesem Kapitel stelle ich detaillierte Resultate fu¨r ein Modellproblem,

das eng mit dem anomalen Magneto-Drag verbunden ist, dar. Nach einer

kurzen Einfu¨hrung gebe ich im zweiten Teil eine kurze U¨bersicht der Experi-

mente, die in den letzen 15 Jahren durchgefu¨rt wurden. Besondere Aufmerk-

samkeit ist denjenigen Experimenten gewidment, die sich auf das Disserta-

tionsthema beziehen. Der dritte Teil entha¨lt eine detaillierte Zusammenfas-

sung der heutigen theoretischen Ansa¨tze zum Dragproblem. Insbesondere

wird erla¨utert, warum diese Ansa¨tze nicht genu¨gen, um die u¨berraschenden

Eigenschaften des Magneto-Drag zu beschreiben. Der vierte Teil ist der

Hauptteil des Kapitels. Hier fu¨hre ich zuna¨chst das Modell ein, mit dem

ich mich in diesem Teil bescha¨ftigen werde. Dann zeige ich verschiedene

Wege, mit denen das Modell behandelt werden kann, vergleiche die Resul-

tate, bespreche ihre Relevanz und Zuverla¨ssigkeit. Schliesslich zeige ich die

Verbindungen zwischen dem betrachteten Modell und dem Originalproblem

und bringe u¨berzeugende Argumente um zu zeigen, dass diese Verbingungen

stark genug sind, um viele (wenngleich nicht alle) interessante Eigenschaften

des zwei-dimensionalen Magneto-Drags zu erkla¨ren.

Kapitel 3 ist ein Versuch, den Jahn-Teller-Effekt in Si-Inversionsschichten

vorherzusagen. Dieser ist ein Resultat der Zusammenwirkung von Energietal-

Aufspaltung, Elektron-Phonon- und Elektron-Elektron-Wechselwirkungen in

diesen Systemen. Das Kapitel ist wie folgt aufgebaut. Im Teil 3.2 wird der

Hamilton-Operator des Systems eingefu¨hrt, der in einen SU(4)-symmetrischen

DEUTSCHE ZUSAMMENFASSUNG 7

und einen nicht SU(4)-symmetrischen Teil aufgespaltet wird. Im Teil 3.3

wird der SU(4)-symmetrische Fall untersucht. In den Teilen 3.4, 3.5 fu¨hre

ich die anisotropen Terme des Hamilton-Operators ein und untersuche deren

Einfluss auf den Grundzustand des Systems fu¨r verschiedene ganzzahlige

Fu¨llfaktoren. Im Teil 3.5 wird das Phasendiagramm fu¨r den interessan-

teren Fall ν = 2 gegeben. Dabei werden drei Phasen vorhergesagt, na¨mlich

eine ferromagnetische, eine Spin-Singlet und eine antiferromagnetische Phase.

Im Teil 3.6 wird der Effekt der nicht-symmetrischen Terme im Hamilton-

Operator auf die Energie eines Skyrmions untersucht.

Kapitel 4 entha¨lt schließlich eine kurze Zusammenfassung aller Resul-

tate.

8 DEUTSCHE ZUSAMMENFASSUNG

Chapter 1

Overview

Over decades two-dimensional electron systems have drawn the attention of

an extremely large number of physicists and technologists. The reason for

this interest is not only - as one might naively suppose - due to the techno-

logical applications in computer processors. This seemingly simple physical

configuration gives rise to an unexpectedly rich variety of experimentally ob-

servable effects and theoretical models which are themselves important for

the development of physics, technology, computing etc.

Since the discovery of the quantum Hall effect [1] a particular attention

has been payed to two-dimensional electron systems subjected to a strong

magnetic field (the so-called quantum Hall systems). The physical effects

observed in such systems are the result of an interplay between the Landau

quantization of the electron motion, the electron-electron Coulomb interac-

tion, the band structure of the material (usually Si or GaAs), disorder effects

etc. Among the interesting aspects of the two-dimensional electron physics

one can mention the fractional quantum Hall effect, Coulomb drag, skyrmion

phases, valley splitting in Si inverse layers, bilayer quantum Hall systems, ef-

fects of the microwave radiation on transport properties of the quantum Hall

systems and many others.

In this thesis I deal with two effects which are related to the Quantum Hall

systems. In the main part of the work, Chapter 2, I study the Coulomb

9

10 OVERVIEW

magneto-drag, particularly concentrating on the question of the so-called

anomalous drag recently observed experimentally [2, 3]. The observed sign

reversal of the drag signal poses an intriguing task for theoretical investiga-

tions.

Chapter 3 is an attempt to predict possible a Jahn-Teller effect in Si

inverse layers. There I consider the interplay between the valley splitting,

the electron-phonon and electron-electron interactions in these systems and

their effect on the ground state and low-lying skyrmionic excitations.

In Chapter 4 a short summary of the results is given.

Chapter 2

Theory of Coulomb

Magneto-Drag

2.1 Introduction

In this chapter I will present extended results for a model problem closely

related to the well-known and still not closed problem of the two-dimensional

magneto-drag. In the next section I’ll give a brief review of history of the

experiments during the last 15 years. Special attention will be drawn to the

results closely related to the theme of this work. In the third section I’ll

give a detailed overview of the present theoretical approaches to the drag

problem. An important part of that section will be devoted to explaining of

why do these approaches fail to explain the striking features of the magneto-

drag. The forth section is the main part of the work. There I’ll start with

presenting the model, I’ll deal with in that section. Then I’ll show different

approaches to that model, compare the results, discuss there relevance and

reliability. Finally, I’ll show links between the considered model and the

initial problem and will bring convincing arguments to show that these links

are strong enough to explain many (though not all) interesting features of

the two-dimensional drag.

11

12 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

2.2 History of the problem. Experiment.

Electron-electron interactions are responsible for a multitude of fascinating

effects in condensed matter. They play an important role in such different

phenomena as high-temperature superconductivity, fractional quantum Hall

effect, Wigner crystallization or Coulomb gaps in disordered media. The

effects of this interaction on transport properties, however, are difficult to

measure. In the beginning of the 90-ties, a new technique has been proved to

be effective in measuring the scattering rates due to the Coulomb interaction

directly. [4].

This technique is based on an earlier proposal by Pogrebinski˘ı [5, 6]. The

prediction was that for two conducting systems separated by an insulator (a

semiconductor-insulator-semiconductor layer structure in particular) there

will be a drag of carriers in one film due to the direct Coulomb interaction

of the carriers in the other film. This effects resembles conventional friction

that is why Coulomb drag is often referred to as frictional drag. Unlike the

one-layer case, where Coulomb forces alone can not lead to finite conductance

due to momentum conservation, in the double layer setup current in one layer

leads to a finite mean force acting on the carriers in the other layer, which in

turn leads to a finite current there. If the second layer is an open circuit the

interaction with the electrons from the first layer leads to redistributing of

charge density and to building a voltage in the second layer which opposes

the frictional force and exactly cancels it. This voltage is referred to as drag

voltage.

The standard experimental setup can be seen in Fig. 2.1. A constant

low-frequency current (typically 25 Hz) is imposed on one of the layers (drive

layer), and the drift of those electrons creates a frictional drag on the electrons

in the adjacent layer (drag layer). Although for theoretical investigations it is

more convenient to consider both layers as closed circuits, in practice the drag

layer is an open one which allows to measure the drag voltage induced there.

Typical experimental parameters for drag experiments are: the well thickness

δ ranges from 10 to 30 nm, the layer separation d lies typically between 20 and

2.2 HISTORY OF THE PROBLEM. EXPERIMENT. 13

Figure 2.1: (a) Sample geometry, including the mesa and the front and back

gates in gray. The central bar is 20 µm wide. Each mesa arm is terminated

with an indium diffusion contact roughly 1.5 mm from this central region.

(b) Schematic of the measurement configuration [4].

120 nm. This ensures that no tunneling occurs between the layers and that

the only way the electrons from different layers may interact is the direct

(screened) Coulomb force (or, under certain circumstances, quasi particle-

mediated interaction may occur, such as phonon or plasmon interaction).

The measured value is the so-called drag resistance which is defined as

ρD = −VD I . (2.1)

Here VD is the drag voltage and I is the current driven through the drive

layer. The minus sign means that the direction of the drag voltage is normally

opposite to that of the current.

The experimental plots can be seen in Figs. 2.2, 2.3. It can be seen that

the naive phase space arguments that predict T 2 low-temperature behavior

of the drag (only a layer of electrons around the Fermi surface of the order

14 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

Figure 2.2: Temperature dependence of the observed frictional drag between

two 2D electron systems separated by 175-A˚ barrier. Data are plotted as an

equivalent resistance and a momentum-transfer rate [4].

of T in each layer contributes to drag) do not give a complete explanation of

the temperature dependence. The roots of this discrepancy will be discussed

in the next section. Here I just mention that in the field of the frictional drag

in the absence of a magnetic field at the moment there are no open questions

(for a review see [7]). Among other papers on drag worth mentioning is the

work [8] in which electron-hole drag is studied. As one would expect, the

drag signal in this case is negative if defined according to Eq. (2.1).

A boost of interest to the theme of Coulomb drag is due to drag experi-

ments in presence of a finite magnetic field. First such experiments lead back

to 1992 [9]. Many other followed [10, 11, 12].

In 1998 though a new, completely unexpected phenomenon has been

found by X. Feng et al. [2], see Fig. 2.4. The new aspect in this work is the

study of magneto-drag with different electron densities in the layers. The

unusual and somewhat counterintuitive observation is the negative drag, i.e.

electrons are accelerated in the direction opposite to the net transferred mo-

2.2 HISTORY OF THE PROBLEM. EXPERIMENT. 15

Figure 2.3: Temperature dependence of the interwell momentum transfer

rate divided by T 2 for both the 175- and 225-A˚ barrier samples [4].

mentum. A closer look at the data of this paper (see Fig. 2.5), and at further

papers [3, 13] (Figs. 2.6, 2.7) allows us to see that the negative drag occurs

when in one of the layers the top Landau level is more than half filled, while

in the other one it is less than half filled.

An analysis of the experimental data shows that the negative drag is ob-

served only under quite specific conditions. First, the mobility of the samples

should be quite high (more than 2 × 106 cm2V−1s−1), second, the magnetic field should be neither too low, nor too high (the effect was observed for filling

factors ν between 6 and ≈ 70), and finally the temperature should be quite low (for a cleaner sample the negative drag could be observed only for tem-

peratures under 1.8 K, while for a dirtier one the allowed temperature range

extends and at some particular magnetic field even reaches 18 K). Another

important feature revealed recently [13] is the activated low-temperature be-

havior of the drag signal with the activation energy periodically depending

16 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

Figure 2.4: Drag versus magnetic field for matched (c) and unmatched (a)

layer densities at 1.15K. The longitudinal magneto-resistance (b) for the drive

layer in both cases (1.66× 1011/cm2) (solid line), the dotted line is the drag layer with a mismatched density (1.49 × 1011/cm2). A clear difference in filling factors is evident for negative drag (a) [2].

on the magnetic field (see Fig. 2.7). The activation energy reaches its maxi-

mum when the Fermi level lies between two Landau levels and drops to zero

when the Fermi level hits the center of either of the two spin-split Landau

levels. All the above indicates the important role of disorder.

For high-mobility samples the disorder potential mainly comes from the

inhomogeneity of the distribution of donors. They are localized in a layer

∼50nm away from each electron layer. Such setup, supposing uncorrelated distribution of donors, results in a smooth random potential in the electron

layer with the correlation length equal to the distance between the donor and

electron layers. This property of the random potential is crucial for the rest

of the work.

The above-mentioned works of Feng et al. and Lok et al. attracted my

2.3 HISTORY OF THE PROBLEM. THEORY. 17

Figure 2.5: Drag versus drag layer filling factor(density). The drive layer den-

sity is fixed at 1.66× 1011/cm2, with magnetic fields for (a)/(b) of 0.92/0.81 Tesla, and drive layer ν of 7.39/8.47. Nearly periodic behavior is observed

[2].

supervisor’s interest to this question so that he proposed me this problem

as a topic for my doctor thesis. The observed effect being simultaneously

simple and elusive for theoretical understanding for quite a long time (see

next section) presented a challenge for us.

2.3 History of the problem. Theory.

2.3.1 Early works.

As mentioned above, the drag effect was first predicted theoretically [5, 6]

long before it became possible for the experimentalists to produce double-

layer systems with the necessary properties to observe the effect. These

early works did not in fact present any calculations of what we call drag

resistance, but rather gave just the idea that there might be a momentum

transfer between two parallel systems separated by an insulator, only due to

Coulomb interaction, without matter transfer.

18 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

Figure 2.6: Drag resistivity ρT (bottom) and conventional longitudinal re-

sistivity ρxx (top) for two 2DEGs at mismatched densities (n1=2.27 and

n2=2.08·1011 cm−2) as a function of magnetic field at T=0.25 (K). Two sets of oscillations can be distinguished in ρT : i) a quick one resulting from the

overlap of the Landau levels in the 2DEGs and ii) a slow one which causes

(positive) maxima in ρT whenever the filling factor difference between the

2DEGs is even, and (negative) minima whenever this difference is odd. The

inset shows ρT at fixed magnetic field of 0.641 (T) versus filling factor differ-

ence [3].

2.3 HISTORY OF THE PROBLEM. THEORY. 19

Figure 2.7: (a) ρD for matched and mismatched densities with (νF , νB) =

(9.5, 9.5) and (8.5, 9.5). (b) Arrhenius plot of |ρD| at low T , demonstrating the activated behavior. (c) measured activation energies, ∆ (U), of ρD (σxx)

for matched densities vs. ν. Data of U are shown for the front layer only,

in regions where σxx ∝ exp(−U/T ). The bare Zeeman energy in GaAs with |g| = 0.4 is shown as a reference [13].

In the work [5] the author mainly concentrated on solving electrodynamic

equations in a semiconductor-insulator-semiconductor heterostructure, tak-

ing into account the drag effect. The main result of that paper is the current-

voltage characteristic of an SIS structure under drag conditions.

The work [6] is devoted to calculating of the energy transfer between the

neighboring layers in a heterostructure due to Coulomb inter-layer scattering.

Thus, both of these pioneering works have only indirect connection to the

theme of my work, but they should definitely be mentioned, as it is in these

works that the idea of Coulomb inter-layer scattering was first mentioned.

20 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

2.3.2 Boltzmann equation approach.

First theoretical works in which approaching the drag problem has been

set as the main goal were published in 1993. First A.-P. Jauho and H.

Smith published a work where they used a Boltzmann equation approach to

calculate the drag [14] and then L. Zheng and A.H. MacDonald used memory

function formalism for the same goal [15]. Here I will review the former work

not only because of its priority but also as the method elaborated there will

be important for me later (see section 2.4.1).

The starting point is the standard linearized Boltzmann equation

k˙2 ∂f 02 ∂k2

=

[ ∂f2 ∂t

] coll

, (2.2)

where

k˙2 = −eE2, (2.3)

E2 being the electric field induced in the drag layer, and[ ∂f2 ∂t

] coll

= − ∫

dk1dk ′ 2

(2pi)4 w(1, 2; 1′, 2′)(ψ1 + ψ2 − ψ1′ − ψ2′)

× f 01 f 02 (1− f 01′)(1− f 02′)δ(�1 + �2 − �1′ − �2′) (2.4)

is the linearized collision integral. w(1, 2; 1′, 2′) determines the probability

of scattering of two electrons from states 1,2 to 1′, 2′, f is the distribution

function, f 0 is the equilibrium distribution function. Indices 1 and 2 refer

to the initial states in the drive and drag layers respectively, 1′ and 2′ to

the final states. Spin indices are omitted here; k1′ = k2 + k1 − k2′ due to momentum conservation. The deviation function ψ is defined by

f − f 0 = f 0(1− f 0)ψ. (2.5)

The current flowing in the layer 1 is assumed to be limited by impurity

scattering, corresponding to

ψ1 = − 1 T τ1ev1xE1, (2.6)

2.3 HISTORY OF THE PROBLEM. THEORY. 21

where E1 is the electric field in the drive layer, directed along the x axis, and

τ1 is the momentum relaxation time.

The authors set ψ2 = 0 ”since no current is flowing in the drag layer”.

Here I will follow there line of reasoning and discuss the validity of this state-

ment later. Collecting all the above, multiplying the Boltzmann equation by

k2x, integrating over k2, and using the antisymmetry of the integrand on the

RHS with respect to interchange of 2 and 2′ they get

eE2n2 = −eE1τ1 8mT

∫ dqdω

(2pi)2 [=χ(q, ω)]2w(1, 2; 1′, 2′) q

2

sinh2 (ω/2T ) , (2.7)

with ni being the electron density in the ith layer. Energy transfer ω is

introduced by formal substitution

δ(�1 + �2 − �1′ − �2′) = ∫ dωδ(�1 − �1′ + ω)δ(�2 − �2′ − ω), (2.8)

q = k2′ − k2 = −k1′ + k1, and the susceptibility χ is defined as

χ(q, ω) = − ∫

dk2 (2pi)2

f 02 − f 02′ �2 − �2′ + ω + i0 . (2.9)

Further evaluation of this expression will be discussed in the next section.

Here, to round off the reviewing of this work I want to discuss why ψ2 may be

set to zero. In fact, the kinetic equation used in [14] is incomplete. This can

be seen extremely clearly for the closed circuit set-up, i.e. no electric field, but

finite current. Then nothing prevents the current in the drag layer to grow

infinitely. To resolve this problem one must add the collision term for the

electrons in the drag layer. Having done this it is possible to leave out the ψ2-

terms in the inter-layer collision integral i.e. consider it as a functional of the

equilibrium distribution function for the electrons in the drag layer. That is

justified as the correction to this distribution function must be proportional

both to the non-equilibrium part of the distribution function in the drive

layer (i.e. to the electric field E1) and to the inter-layer interaction squared

(as the scattering probability w(q) is proportional to it). That means that

keeping ψ2 in the inter-layer collision term would result in a term containing

the 4th power of the interaction, which is small compared to the intra-layer

22 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

collision integral. Taking some specific (for example τ -approximation) form

of the intra-layer collision term and using the zero-current condition∫ dk2

∂�2 ∂k2

f 02 (1− f 02 )ψ2 = 0, (2.10)

one can in principle find ψ2 which is not equal to zero. (In fact the Boltzmann

equation with the intra-layer collision term gives ψ2 as a function of E2,

the latter is then found from the condition for the current to be 0.) But

to determine E2 it is not needed: it is enough to multiply the Boltzmann

equation by k2 and integrate over k2 — that is exactly what is done in the

paper — then the intra-layer collision term disappears due to the fact that

there is no current in the drag layer. This explains why using a physically

wrong form of the Boltzmann equation the authors still got a physically

consistent and correct result for the drag. I will refer to this arguments later,

in the main part of the work.

In the work [15] the authors derive the same equation (2.7) using the

memory function formalism. Reviewing this work lies beyond the scope of

the present thesis.

2.3.3 Diagrammatic approach.

An important step in the theoretical understanding of drag was made in

1995 by Kamenev and Oreg [16]. They formalized all what was done before

them and got some new results, and, what is more important, showed how

this problem can be handled generally and how, in principle, new results can

be obtained. The authors used the general linear response theory method,

i.e. the Kubo formula in its diagrammatic representation. The calculations

were done in the lowest non-vanishing order in the inter-layer interaction,

all intra-layer effects (electron-electron interaction, impurities etc.) can, in

principle, be included up to arbitrary order of perturbation theory in their

method.

2.3 HISTORY OF THE PROBLEM. THEORY. 23

Figure 2.8: Two diagrams contributing to the transconductance to the second

order in the inter-layer interaction.

The starting point is the general linear response expression of the con-

ductivity via a current-current correlation function.

σijD( ~Q,Ω) =

1

ΩS

∫ ∞ 0

dteiΩt ⟨[ J i†1 ( ~Q, t), J

j 2( ~Q, 0)

]⟩ . (2.11)

Here i, j = xˆ, yˆ; ~Q,Ω are the wave vector and the frequency of the external

field, respectively; S is the area of the sample; and J il is the ith component of

the current operator in the lth layer. Diagrams corresponding to Eq. (2.11)

include two separate electron loops with a vector (current) vertex on each

one of them, coupled only by the inter-layer interaction lines. The first non-

vanishing order in the limit ~Q −→ 0 in the inter-layer interaction is the second order. In this order there are two diagrams, shown in Fig. 2.8.

Analytically, this two diagrams may be written in a symmetric form:

σijD = T

2iΩmS

∑ ~q,ωn

Γi1(~q, ωn +Ωm, ωn)Γ j 2(~q, ωn, ωn +Ωm)U(~q, ωn +Ωm)U(~q, ωn),

(2.12)

where T is the temperature, U(~q, ω) is the inter-layer screened Coulomb

interaction and the vector ~Γ1(2)(~q, ω1, ω2) is the three-legged object given by

the two diagrams depicted in Fig. 2.9 (the factor 1/2 in Eq. (2.12) is included

to prevent double counting). In Eq. (2.12) the usual Matsubara technique

with Ωm = 2piimT is employed. After summing over the boson frequencies

24 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

Figure 2.9: Diagrams defining the three-legged vertex, Γ(~q, ω1, ω2) =

Γ(−~q,−ω1,−ω2)

ωn, one should perform an analytical continuation to real values of Ω, and

finally the limit Ω −→ 0 should be taken. The sum over ωn is done by a contour integration in the complex ω plane along the contour shown in

Fig. 2.10.

After the sum over ωn is taken we can finally perform the analytical

continuation Ωm −→ Ω and take the limit Ω −→ 0. The result is:

σijD = 1

16piTS

∑ ~q

∫ ∞ −∞

dω

sinh2 ω 2T

Γi+−1 (~q, ω, ω)Γ j−+ 2 (~q, ω, ω)

∣∣U+(~q, ω)∣∣2 . (2.13)

The +,− indices indicate the way the analytical continuation is performed, see Fig. 2.11 for explanation.

Eq. (2.13) gives the final expression for the transconductivity. The impor-

tant feature is that the integrand decouples into one part depending only on

the interaction potential and two other factors, that are functions of the lay-

ers’ properties only. The latter factors, i.e. the Γ’s are called the rectification

coefficients. They are in fact nonlinear susceptibilities of the two-dimensional

electron gases in an external ac field, and give the proportionality coefficient

between the square of a time-dependent scalar potential φ(~q, ω) and the dc

2.3 HISTORY OF THE PROBLEM. THEORY. 25

Figure 2.10: The contour of integration in the complex ω plane employed to

perform the sum over the Matsubara boson frequencies in Eq. (2.12).

Figure 2.11: The way in which the analytical continuation of ~Γ(~q, ω1, ω2) −→ ~Γ+−(~q, ω, ω) is performed.

26 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

current in a layer:

~J = ~Γ−+|φ(~q, ω)|2. (2.14)

The next parts of the paper are devoted to evaluating of the general for-

mula (2.13) in different cases. To this end the authors calculate the nonlinear

susceptibility Γ. The case of normal metals without intra-layer interaction is

considered. Then

~Γ(~q, ω1, ω2) = T ∑ �n

Tr { G�nG�n+ω2 ~ˆIG�n+ω1 + G�nG�n−ω1 ~ˆIG�n−ω2

} (2.15)

Here G stands for the one-electron Green function and the trace is taken over exact eigenstates of the disordered system. The sum over fermionic Matsub-

ara frequencies is evaluated similarly to the sum over bosonic frequencies.

After the analytical continuation (as shown in Fig. 2.11) the final result for

the nonlinear susceptibility reads:

~Γ+−(~q, ω, ω) = 1

4pii

∫ d�

( tanh

�+ ω

2T − tanh �

2T

) Tr [(G−� − G+� )G−�+ω ~ˆIG+�+ω]

(2.16)

+{~q, ω −→ −~q,−ω}.

Here G± stands for the retarded (advanced) single-electron Green function. This result is valid in the absence of a magnetic field, when the trace of three

advanced or three retarded Green functions vanishes.

Then the authors evaluate the last formula in special cases of clean and

dirty systems and find that in the absence of intra-layer electron-electron

correlations the main results of previous works [14, 15] (i.e. expressing the

drag signal via the imaginary parts of polarization operators (2.7)) holds,

though not generally, rather only up to the first non-vanishing order in 1/�F τ ,

where �F is the Fermi energy and τ is the elastic mean free time.

In order to do it, they take the simplest model, i.e. delta-correlated impu-

rities, and calculate the diagram for Γ (Fig. 2.9) in the diffuson approximation

(in the next section, also the weak-localization corrections are considered).

The result in the diffusive regime (ω � 1/τ and q � 1/l [l = vF τ is the

2.3 HISTORY OF THE PROBLEM. THEORY. 27

mean free path]) is:

~Γ+−(~q, ω, ω) = e 2D~q

�F

[ 2Sν

ωDq2

(Dq2)2 + ω2

] , (2.17)

here D is the diffusion coefficient and ν is the density of states at the Fermi

level. The expression (2.17) has the form

~Γ+−(~q, ω, ω) = ~Γ−+(~q, ω, ω) = e 2D~q

�F =Π+(~q, ω), (2.18)

with

Π+(~q, ω) = 2Sν Dq2

Dq2 − iω (2.19) being the polarization operator in the diffusive regime.

For the ballistic regime (q > 1/l or ω > 1/τ) Γ turns out to be

~Γ+−(~q, ω, ω) = e 2D~q

�F

[ 2Sν

ω

vF q θ(vF q − ω)

] , (2.20)

where θ(x) is the Heaviside step function. As in the previous case, this

expression has the form (2.18) with

=Π+(~q, ω) = 2Sν ω vF q

θ(vF q − ω). (2.21)

An important observation, that one could make, is that the above results

can be obtained only if one takes into account the non-linearity of the dis-

persion around the Fermi surface. More precisely, two terms in Eq. (2.16)

cancel each other if the calculations are done in a usual way, by linearizing

the dispersion near the Fermi surface. From the physical point of view it

means that the electron-hole asymmetry is crucial for the Coulomb drag.

If we had exact electron-hole symmetry the electron and hole drags would

cancel each other completely.

Ultimately the drag conductivity is calculated. Here I’ll list the results

in different cases. It is distinguished between the results for ballistic, l � d and diffusive, l � d samples. In the former case, the dominant contribution to the transresistance comes from the ballistic part of the plane (ω < vF q,

q > 1/l, see Fig. 2.12) and one can obtain:

ρD = ~

e2 pi2ζ(3)

16

T 2

�F1�F2

1

(κ1d)(κ2d)

1

(kF1l1)(kF2l2) , (2.22)

28 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

d −1

Diffusive

q

ω

T

τ

−1l

−1

PSfrag replacements

ω = vF q

ω = Dq2

Figure 2.12: Domains in the (q, ω) plane. The dashed area represents

(schematically) the regions where =Π+ 6= 0.

where κ1,2 are the Thomas-Fermi momenta in the two layers.

On the contrary, for diffusive samples the entire contribution comes from

the diffusive part of the (q, ω)-plane (ω < 1/τ , q < 1/l). For T � min (τ−1, T0) [where T0 ≡ Dmin (κ1, κ2)/d arises due to divergence of the interaction po- tential at small momenta] one obtains

ρd = ~

e2 pi2

12

T 2

�F1�F2 ln T0 2T

1

(κ1d)(κ2d)

1

(kF1d)(kF2d) . (2.23)

For higher temperatures, T, T0 � τ−1, the energy integration is dominated by the region ω ≈ τ−1, the result is

ρD ≈ ~ e2

lnT0τ 1

(κ1d)(κ2d)

1

(kF1l1)(kF2l2) . (2.24)

Actually in this temperature region the plasmon contribution becomes dom-

inant (see for example [17]), but this goes far beyond the scope of the present

work.

2.3 HISTORY OF THE PROBLEM. THEORY. 29

Table 2.1: Temperature and inter-layer distance dependence of the drag co-

efficient, ρD, for different mobilities.

T � min (T0, τ−1) T � min (T0, τ−1) l � d d−2T 2 lnT d−2T l � d d−4T 2

To summarize, I give the dependence of the drag resistance on the tem-

perature and the inter-layer distance for different parameter ranges in table

2.1.

To finish off the review of this very important paper, I’ll mention that in

section IV the authors calculated the corrections to the drag resistance due

to weak localization and in section VI the drag above the superconductor

transition was considered. In the former case the formula (2.18) was found

to be correct in the lowest order in 1/kF l, while in the latter case, where the

intra-layer electron correlations play an important role, this formula turned

out to be wrong. This shows that although the diagrammatic approach does

reproduce the old results, they are not general enough and under specific

conditions the old approach fails.

2.3.4 Magneto-drag. Early diagrammatic approach.

After the publication of the work discussed in the previous section the next

logical step would be to extend the diagrammatic approach to the drag in

presence of a strong quantizing magnetic field. This was done by M. C. Bøn-

sager et al. in 1997. This section is devoted to a critical review of their work.

The starting point was the same as in the work of Kamenev and Oreg, i.e. the

drag was expressed in terms of a convolution of two rectification coefficients.

The results of the paper are all based on expressing the rectification coeffi-

cient (triangle function) in terms of the imaginary part of the polarization

function (see (2.18)). As we will see, unlike the case without magnetic field

30 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

that was considered in the previous section, here the authors did not show

rigorously that the formula (2.18) is correct, so all their results are based on

an assumption that it is true.

The starting point of their calculations is the formula (2.13). Then they

try to prove the relation (2.18) in presence of magnetic field in the limit

ωcτ � 1 (ωc being the cyclotron frequency). Doing this the authors make some dubious statements (see Appendix B of their work) which turn out to

be not correct, most importantly, the claim that the scalar vertex corrections

are negligible for n 6= n′ is wrong. The direct calculations in work [18] show this. Also only the delta-correlated disorder case is considered, which is

rather far from real experimental conditions.

The results of the work consist in a numerical evaluation of the resulting

expression for the drag resistivity

ρxx12 = − ~

2

4e2n1n2kBT

1

S

∑ q

a2 ∫

dω

2pi |U(q, ω)|2 =Π1(q, ω)=Π2(q, ω)

sinh2 ~ω/2kBT , (2.25)

which is obtained by substituting (2.18) into (2.13) and inverting the transcon-

ductivity tensor. U(q, ω) is taken in the random phase approximation, i.e.

U(q, ω) = V12(q)

[1− Π(q, ω)V11(q)]2 − [Π(q, ω)V12(q)]2 , (2.26)

V11 and V12 being the non-screened intra- and inter-layer Coulomb electron-

electron interactions. The layers were assumed identical i.e. Π1 = Π2 ≡ Π. Further evaluation (particularly the expression for Π(q, ω)) in the work

is also based an the erroneous omitting of the scalar vertex correction for

n 6= n′ and thus does not merit a special discussion.

2.3.5 Negative magneto-drag. First explanation at-

tempt.

The results of the experimental works [2, 3] remained for quite a while com-

pletely ununderstood. Indeed, from the generally accepted formula (2.25) it

2.3 HISTORY OF THE PROBLEM. THEORY. 31

is absolutely impossible to get negative drag due to analytical properties of

the imaginary part of the polarization function. So the obvious decision is

to go one step back, to the expression (2.13) and try to make a more careful

analysis. The first attempt to do this was made by F. von Oppen et al. in

2001 [19]. In this work an evaluation of the triangle function Γ was made.

Unfortunately there was a mistake in the calculations for the experimen-

tally relevant ballistic limit, that led to a completely wrong result: negative

drag occurred for matched densities and not for mismatched as one observes

experimentally.

Still I think it is worth presenting this work here for several reasons,

the most important of which is a very simple semiclassical derivation of the

non-linear susceptibility that gives a simple and clear physical picture of

it, and also, in principle, can give a negative value for the drag resistivity.

This derivation is based on the assumption that the current is related to the

electric field locally in time and space and that the conductivity depends

only on the local density of charge carriers:

J(~r, t) = −σˆ[n(~r, t)]∇φ(~r, t). (2.27)

In addition to the current, the perturbation φ(q, ω) also induces a density

perturbation δn(q, ω) = −Π(q, ω)eφ(q, ω) due to the polarizability Π(q, ω). Up to quadratic order in the applied potential,

J(~r, t) = − [ σˆ(n0) +

dσˆ

dn δn(~r, t)

] ∇φ(~r, t). (2.28)

Taking the time and space average of the second term yields a dc contribution

to the current given by

Jdc = − ∑ q,ω

( dσˆ

dn

) [Π(q, ω)eφ(q, ω)iqφ(−q,−ω)], (2.29)

or, equivalently, the rectification coefficient is given by

Γ(q, ω) = dσˆ

d(en) · q=Π(q, ω). (2.30)

32 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

At zero magnetic field this expression reproduces (2.18) and thus leaves no

possibility for drag sign change. On the contrary, in presence of a magnetic

field, in the quantum Hall and SdH regimes the derivative of the longitudinal

component of the conductance changes sign as a function of the electron

density. This opens the possibility for a sign change in the drag resistivity

as a function of density. Substituting (2.30) into (2.13) and inverting the

transconductivity tensor one gets up to an overall positive pre-factor for the

drag resistivity tensor:

ρˆD ∼ ρˆp dσˆ p

d(en)

dσˆa

d(en) ρˆa, (2.31)

where ρˆp(a) and σˆp(a) are the resistivity and conductivity tensors of the pas-

sive (active) layer. This expression easily reproduces some standard results.

In the absence of a magnetic field, the tensor structure is trivial. Observing

that the conductivity increases (decreases) with increasing electron density

for electron (hole) layers, we recover that coupled layers with the same type

of charge carriers exhibit positive drag, while coupled electron-hole layers

have a negative drag resistivity. If the system is strictly electron-hole sym-

metric, the derivative of the conductivity with respect to density vanishes

and, consequently, there is no drag.

In the presence of strong magnetic fields new effects appears. To see this

we can multiply out the products in (2.31), keeping in mind that already for

quite small magnetic fields ρxy � ρxx. Up to an overall positive pre-factor:

ρDxx ∼ ρpxy { dσˆpyy d(en)

dσˆayy d(en)

+ dσˆpyx d(en)

dσˆaxy d(en)

} ρayx. (2.32)

Generally, the derivative of the longitudinal conductivity σxx changes sign as

the magnetic field or Fermi energy is varied in the SdH and integer quantum

Hall regimes, being positive for less than half filled of the topmost Landau

level and negative for more than half filled one. By contrast, the Hall conduc-

tivity generally increases monotonically with n, and its derivative is therefore

positive.

2.3 HISTORY OF THE PROBLEM. THEORY. 33

The obvious problem with this result is that for equal densities in the two

layers the drag is negative as ρxy = −ρyx in contradiction with the experi- mental results [2, 3]. This is due to the fact that the above treatment can

be justified only in the diffusive limit ωτ,Dq2τ � 1 which is experimentally irrelevant.

In order to calculate the rectification coefficient in the ballistic limit the

authors use the diagrammatic approach, starting from the expression (2.16)

from the work [16]. Here a mistake was made, as we will see later, this equa-

tion is inapplicable in the presence of a magnetic field, as important terms

involving traces of three Green’s functions of the same type are omitted.

Thus further reviewing of this work may be only of calculational interest and

will therefore not be done.

2.3.6 Diagrammatic calculation of magneto-drag in the

SCBA.

The work [19] gave the wrong result in the experimentally relevant ballistic

limit. But the idea of the work was mainly correct. This line was drawn to

its logical end in the work [18] which I will review in this section.

In this work the authors calculated the magneto-drag diagrammatically

in the self-consistent Born approximation i.e. for short-range disorder, which

is rather not the experimentally relevant case. Still, it is quite important to

look at one of the limiting cases (high Landau levels, intermediate magnetic

fields), and as this was done in the considered work and as the results are

similar to the experimentally observed data I can not leave this work out of

consideration.

The starting point is once again the expression (2.12) which is evaluated in

the standard way by going from the summation over Matsubara frequencies

to contour integration (see sec. 2.3.3, particularly Fig. 2.10). The triangle

function Γ(q, ω) is then obtained by analytic continuation like in sec. 2.3.3,

34 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

see Fig. 2.11. and is given by Γ = Γ(a) + Γ(b) with the two contributions:

Γ(a)(q, ω) =

∫ d�

4pii tanh

� + ω − µ 2T

Tr {vG+(�+ ω)eiqrG+(�)e−iqrG+(�+ ω)

− vG−(� + ω)e−iqrG−(�)eiqrG−(� + ω)}+ (ω,q→ −ω,−q), (2.33)

Γ(b)(q, ω) =

∫ d�

4pii

( tanh

� + ω − µ 2T

− tanh � +−µ 2T

)

× Tr {vG−(�+ ω)eiqr[G−(�)− G+(�)]e−iqrG+(�+ ω)}+ (ω,q→ −ω,−q). (2.34)

Here G± denotes the advanced (retarded) Green function for a given real- ization of the random potential. Note that for zero magnetic field only Γ(b)

survives [16]. By contrast, in strong magnetic fields both Γ(a) and Γ(b) should

be retained. In the work [19] Γ(a) was left out which led to wrong results in

the ballistic limit.

For small ω, the expressions for Γ(q, ω) simplify to

Γ(a)(q, ω) = ω

2pii Tr {vG+(�F )eiqrG+(�F )e−iqrG+(�F )− (G+ → G−)}, (2.35)

Γ(b)(q, ω) = ω

pii Tr {vG−(�F )eiqr[G−(�F )− G+(�F )]e−iqrG+(�F )}. (2.36)

For well-separated Landau levels this approximation holds as long as ω is

small compared to the Landau level width ∆. We also mention that Γ(a)(q, ω)

gives only a longitudinal contribution (parallel to q) to Γ(q, ω).

Then the averaging over impurities is done using the SCBA [20]. White-

noise disorder is assumed:

〈U(r)U(r′)〉 = 1 2piντ

δ(r − r′). (2.37)

The SCBA (it neglects the crossing impurity diagrams) gives the leading

contribution in the high Landau levels. Although the authors realize that

this approximation is, strictly speaking, not justified for the experimental

samples, where the disorder potential is correlated on the scale of the distance

between the two-dimensional sample and the donor layer, they still expect

their results to reflect reality adequately.

2.3 HISTORY OF THE PROBLEM. THEORY. 35

q, ω q, ω

n, k

n′, k′

µ

ν

=

q, ω

n, k

n′, k′

µ

ν

+

q, ω x

q, ω

n, k

n′, k′

µ

ν

N

N

Figure 2.13: Diagrammatic representation of the equation for the vertex

corrections γµνnk,n′k′(� + ω, �,q) of the scalar vertices in the SCBA [18].

The impurity averaged Green function (denoted by G±(�)) in the SCBA

is given by the expression

G±n (�) = 1

�− En − Σ±(�) , (2.38)

with the Landau energies En = ~ωc(n+1/2). For energies � within a Landau

level, the self-energy is given by

Σ±n (�) = 1

2 {�− En ± i[∆2 − (�− En)2]1/2}. (2.39)

The broadening of the Landau level ∆ can be expressed in terms of the

zero-field scattering time τ as

∆2 = 2ωc piτ

. (2.40)

For white-noise disorder there is no correction of the vector vertex in

Γ, while the corrections of the scalar vertices include impurity ladders (see

Fig. 2.13).

γµνnk,n′k′(� + ω, �,q) = γ µν(q, ω)〈nk|eiqr|n′k′〉. (2.41)

Here, the indices µ, ν = ± indicate the type of Green functions involved in the vertex; � is set to �F . For ∆� ~ωc the vertex corrections at ω = 0 are

γ++(q, ω = 0) = 1

1− J20 (qRc)[∆/2Σ−]2 , (2.42)

γ+−(q, ω = 0) = 1

1− J20 (qRc) , (2.43)

36 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

where Jn(z) denotes the Bessel functions.

The bare vertex is given by

〈nk|eiqr|n′k′〉 = δqy,k−k′ 2n−n

′ n′!

n! exp

[ −1

4 q2l2H −

i

2 qx(k + k

′)l2H

]

× [(qy + iqx)lH ]n−n′Ln−n′n′ (q2l2H/2), (2.44)

where lH = (~c/eH) 1/2 is the magnetic length and Lnm is the associated

Laguerre polynomial and n ≥ n′. The expression for the matrix element for n < n′ can be obtained from Eq. (2.44) by complex conjugation with

the replacement q −→ −q, nk ←→ n′k′. For high Landau levels, which are considered in [18], n, n′ � 1, |n−n′| and q � kF , Eq. (2.44) can be simplified to

〈nk|eiqr|n′k′〉 = δqy,k−k′in−n ′

e−iφq(n−n ′)e−iqx(k+k

′)l2H/2Jn−n′(qR(m)c ), (2.45)

where φq is the polar angle of q, R (n) c = lH

√ 2n+ 1 is the cyclotron radius of

the nth Landau level, and m + (n+ n′)/2.

Now I sum up the evaluation of the triangle vertex Γ(q, ω). Let us first

look at the leading order. As already mentioned ∆/ωc � 1 and the valence Landau level N � 1. The relevant diagrams are shown on Fig. 2.14. In the lowest order in ∆/ωc two of the three Green functions should be evaluated on

the Nth Landau level. The third one (adjacent to the vector vertex) should

be taken on the (N±1)th Landau level as the matrix elements of the velocity are non-zero only for Landau levels differing by one:

〈nk|vx|n′k′〉 = δkk′ i ml √

2 ( √ nδn.n′+1 −

√ n+ 1δn,n′−1), (2.46)

〈nk|vy|n′k′〉 = δkk′ 1 ml √

2 ( √ nδn.n′+1 +

√ n+ 1δn,n′−1). (2.47)

For ω, T � ∆ one gets, using the simplified expressions (2.35), (2.36),

Γ(a)(q, ω) = −2qˆ ωRc pi2l2H

J0(qRc)J1(qRc) 1

2i [G+Nγ

++]2 − [G−Nγ−−]2, (2.48)

2.3 HISTORY OF THE PROBLEM. THEORY. 37

vi

q, ω

q, ω

N ± 1

N

N vi

q, ω

q, ω

N

N ± 1

N

Figure 2.14: The diagrams, contributing to the triangle vertex in the SCBA

to the leading order, in the limit of well-separated Landau levels and large

N [18].

Γ(b)(q, ω) = 2qˆ ωRc pi2l2H

J0(qRc)J1(qRc)

× 1 2i

[G+Nγ ++γ+− −G−Nγ+−γ−−] · [G+N +G−N ]. (2.49)

Here qˆ ≡ q/q, γµν ≡ γµν(q, 0). The sum of the two contributions gives

Γ(q, ω) = qˆ 4ωRc pi2l2H

J0(qRc)J1(qRc)<[G+N(γ++ − γ+−)]=[G+N ]γ++. (2.50)

For arbitrary T, ω < ωc, this contribution takes the form

Γ(q, ω) = qˆ 8Rc

pi2l2H∆ 2

J1(qRc)

J0(qRc)

∫ ∞ −∞

d�

[ tanh

�+ ω − µ 2T

− tanh �− µ 2T

]

× <[γ−+(q, ω)− γ++(q, ω)]=γ++(q, ω). (2.51)

The q-integration in (2.12) is cut off at q ∼ 1/d (d is the interlayer distance). Thus, depending on the relation between Rc and d, one can dis-

tinguish between ballistic (d � Rc) and diffusive (d � Rc) regimes. In the typical experimental setups Rc > d (for the magnetic fields for which the

results of [18] are applicable), so the authors mainly concentrate on the bal-

listic regime in which both the ballistic (qRc � 1) and diffusive (qRc � 1) momenta are relevant.

For diffusive momenta γ+− has a singular behavior, so γ++, γ−− can be

neglected compared to γ+−. So only Γ(b) contributes to (2.50). The result

38 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

is the equation (2.30) (multiplied by 2, as spin has been taken into account)

with the SCBA conductivity [20]

σxx = e2

pi2 N

[ 1− (µ− EN)

2

∆2

] , (2.52)

and the polarization function in the diffusive limit given by Eq. (2.19), which

is valid also in presence of the magnetic field if the diffusion constant D(�) =

R2c/2τ(�) is taken to be energy dependent with the elastic scattering time in

the SCBA being

τ(�) = 1√

∆2 − (�− En)2 . (2.53)

For ballistic momenta (qRc � 1) γ++(q, ω) ≈ γ+−(q, ω) ≈ 1. Thus, in the leading order Eq. (2.50) yields zero. I emphasize that only the sum

of the two contributions (2.48) and (2.49) vanishes, and not each of them

separately. As in the work [19] the contribution (2.48) was missed, the result

in that work in the ballistic limit is wrong. To calculate Γ in the ballistic

limit it is therefore necessary to look at the next order contributions. There

are three kinds of such contributions: (i) contributions of the order of 1/qRc,

(ii) contributions of the order of ∆/ωc, and (iii) contributions of the order of

q/kF . Here I present only the results.

(i). Keeping the contributions of the order of 1/qRc in (2.42) and (2.43),

and plugging them into (2.50) one gets for ω, T � ∆

Γ(1/qRc)(q, ω) = −qˆ64ωRc pi2l2

(µ− EN)[∆2 − (µ− EN )2]3/2 ∆6

J1(qRc)J 3 0 (qRc).

(2.54)

For arbitrary ω, T < ωc one can also find the considered contribution by

using Eq. (2.51). Here and for the two other contributions I present only the

result for low temperatures.

(ii). Contributions of the order of ∆/ωc come from two sources. First

one can evaluate the diagrams in Fig. 2.14 more accurately, keeping the self-

energy parts of the Green function of the Landau levels N ± 1, and second one can consider the diagrams in Fig. 2.15. The result is

Γ(∆/ωc) = −qˆ× zˆ16ωRc pi2l2

(µ− EN )[∆2 − (µ− EN)2] ωc∆4

J1(qRc)J0(qRc). (2.55)

2.3 HISTORY OF THE PROBLEM. THEORY. 39

n± 1

n

N

n

n± 1

N

Figure 2.15: Diagrams contributing to the corrections of the order ∆/ωc in

the triangle vertex [18].

(iii). The contribution of the order of q/kF arises from a more accurate

treatment of the matrix elements involved in the scalar vertices, particularly,

the dependence of Rc on the Landau level number. This contribution is the

direct consequence of the zero-B electron spectrum curvature. For T, ω � ∆ it reads

Γ(q/kF ) = q× zˆ8ω pi2

∆2 − (µ− EN )2 ∆4

J20 (qRc). (2.56)

Having computed the triangle vertex one can go on to calculate the drag

resistivity. As the Hall conductivity dominates over the longitudinal one, the

drag resistivity can be approximated by

ρDxx ≈ ρ(1)xy σDyyρ(2)yx , (2.57)

or, using Eq. (2.12),

ρDxx = − B

en1

B

en2

1

9pi

∫ ∞ −∞

dω

2T sinh2 (ω/2T )

× ∫

d2q

(2pi2) Γ(1)y (q, ω, B)Γ

(2) y (q, ω,−B)|U12(q, ω)|2. (2.58)

The overall minus sign is due to the relation ρxy = −ρyx. It follows that for identical layers, the longitudinal (Γ ∝ qˆ) component Γ‖ of the triangle vertex gives rise to negative drag, since Γ‖(−B) = Γ‖(B), while the transverse (Γ ∝ zˆ× qˆ) component Γ⊥ yields positive drag as Γ⊥(−B) = −Γ⊥(B).

In what follows the authors mainly concentrate on the ballistic regime

ωc/∆� Rc/d� N∆/ωc. (2.59)

40 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

It turns out that in this regime the (ii) contribution (2.55) dominates over

(2.54) and (2.56); indeed, this is easy to see by comparing the order of mag-

nitude for these contributions. This is valid provided that the Landau level

number is sufficiently high: N > (ωc/∆) 2. Without going into details of the

calculation, I will just mention the result. For T � ∆, the drag resistivity is given by

ρDxx = 32

3pi2e2 1

(kFd)2(κRc)2

( T

∆

2) ln

( Rc∆

dωc

)

× ( µ− EN

∆

[ 1− (µ− EN )

2

∆2

]) 1

( µ− EN

∆

[ 1− (µ− EN )

2

∆2

]) 2

. (2.60)

The subscripts 1 and 2 mean that the whole parenthesis refers to the 1st(2nd)

layer. This yields an oscillatory sign of the drag. For identical layers the drag

is positive.

The authors also computed the drag resistance for other ranges of param-

eters. Without going into details of the calculations, I just show the results

in form of a schematic plot, see Fig. 2.16.

In the section devoted to the comparison with the experiment the authors

optimistically state that their results in the ballistic regime fully explain the

experimental data. I find this statement somewhat exaggerated for the fol-

lowing reasons. First, the theory of the discussed paper predicts T 2 low-

temperature dependence of ρD, while experimentalists claim the behavior is

activated with the activation energy periodically depending on the filling fac-

tor. Second, for the ballistic regime in their terminology to be present in the

first place the condition N � (ωc/∆)2 must be fulfilled. Or in other words N � ωcτ � 1. This condition makes doubtful the presence of the discussed regime, or in any case, restricts it to a quite narrow range of magnetic fields,

while experimentally the negative drag is observed in a broad region of mag-

netic fields and the Landau level numbers for which it is observed range from

N = 4 to N ≈ 40. Indeed, taking the experimental parameters from [2]: ∆ = 1.7K for H = 0.1T, I estimate τ ≈ 0.5K−1. Thus ωcτ ≈ 10 ·H[T]. N for the electron density n0 = 2 · 1011cm−1 is equal to 4/H[T]. Substituting the

2.4 SEMICLASSICAL THEORY OF ELECTRON DRAG... 41

above values into the condition N � ωcτ , one obtains H � 0.6T or N � 7. Then, the definition of the ballistic regime (2.59) reads

4 ·H[T]1/2 � 70 d[nm]B[T]

� 1 H[T]3/2

.

Both these conditions restrict the magnetic field from above and for, say,

d =35nm one getsH �0.25T from the second inequality and for d =100nm — H �0.3T from the first one. The condition ωcτ � 1 yields H � 0.1T.

Thus it seems very desirable to elaborate a theory which is valid for

high magnetic fields (low Landau levels) and accounts for the long-range

disorder and localization which is totally omitted in the approach elaborated

by Gornyi et al. in [18].

D ρ x x

T 1

T 1

T 1

T 1

T 1 T*

T2

T1/2

T1/2 ∆

ba

−

lnTT2−

2

−T 2

(T )

∆

T 3

−

T 3

∆

mismatched

matched

c d

T 2lnTT

T 2

−

T 2

−

T

T 1

1

∆ T **

−

−

1

1

Figure 2.16: Schematic temperature dependence of the low-temperature drag

in different regimes: (a) diffusive, Rc/d � 1; (b) weakly ballistic, 1 � Rc/d � ωc/∆, T∗∗ ≡ ωc(d/Rc); (c) ballistic, ωc/∆ � Rc/d � N∆/ωc, T∗ ≡ ∆ ln1/2 (Rc∆/dωc); (d) ultraballistic, N∆/ωc � Rc/d [18].

2.4 Semiclassical theory of electron drag in

strong magnetic fields

To accomplish the task defined at the end of the previous section it is nec-

essary to work with long range disorder. Let us first consider the relevant

length and energy scales in the double-layer system. The distance between

42 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

the layers varies from 30 to 120 nm. The disorder is due to remote donors,

which leads to a smooth disorder potential with a correlation length ξ of order

50-100 nm. The Landau level broadening, which is related to the amplitude

of the disorder potential, has been estimated from low-field Shubnikov-de-

Haas oscillations to lie in the range 0.5-2 K [2, 3]. The magnetic fields at

which drag minima at odd integer filling and/or negative drag at mismatched

densities are observed vary over a relatively wide range from 0.1 T to 1 T

for the cleanest samples, and up to 5 T for samples with a slightly reduced

mobility. This corresponds to magnetic lengths lH between 15 and 80 nm.

Hence, ξ and lH are generally of the same order of magnitude, which makes

a quantitative theoretical analysis rather difficult.

Since the anomalous drag phenomena are observed over a wide range of

fields, one may hope that qualitative insight can be gained also by analyzing

limiting cases. For ξ � lH a treatment of disorder within the self-consistent Born approximation is possible [21]. This route has been taken by Gornyi

et al. in the work [18] discussed above. Localization is not captured by

the Born approximation, and consequently the resistivities obey power-laws

rather than activated behavior for T → 0. For stronger magnetic fields, on the other hand, localization will become important, and a semiclassical

approximation is a better starting point, which is the route I take in what

follows. Applying a criterion derived by Fogler et al. [22] I estimate that

(classical) localization may set in already at 0.1 T. (According to Eq. (1.7) of

[22] localization occurs for B >

√ mc2EF eξ

( W EF

)2/3 , m being the effective mass

and W the amplitude of the potential; substituting typical experimental

values ξ = 60nm, W = 1.7K, n = 2 · 1011cm−2 [2, 3, 13] one gets the above mentioned value.) The localization is classical in the sense that the

electrons’ classical trajectories are closed. From the quantum mechanical

point of view it means that the electron’s wave function is localized in space.

More precisely, an electron is moving along the contours of equal energy of

the random potential or, in a more general case, along the contours of equal

energy of the averaged random potential defined as [22]:

U0(x, y) =

∮ dφ

2pi U(x +Rc cosφ, y +Rc sinφ). (2.61)

2.4 SEMICLASSICAL THEORY OF ELECTRON DRAG... 43

These contours form closed loops, corresponding to localized states, except

at a single energy �0 in the center of each Landau level, for which there is a

percolating contour through the whole system (Fig. 2.17) [23, 24, 25, 26, 27].

If the Landau level broadening induced by the disorder potential and also

kBT is much smaller than ~ωc, as is the case in the anomalous drag regime,

all Landau levels except one are either fully occupied in the bulk of the whole

sample or completely empty. At zero temperature and for �F < �0 the sample

then consists of islands where the highest (partially) occupied Landau level is

locally full, while it is locally empty in the rest of the sample; for �F > �0 the

empty regions form islands. In reality, the percolating contour at the center of

the Landau level is broadened to a percolation region for two reasons. First,

electrons near the percolating contour can hop across saddle points from one

closed loop to another, by using for a moment some of their cyclotron energy

[22]. Second, electrons near the center of the Landau level can screen the

disorder potential, such that the percolating equipotential line at �0 broadens

to form equipotential terraces [28, 29]. Hence, there is a region around the

percolating path, where states are extended.

The percolating region forms a two-dimensional random network con-

sisting of links and crossing points [30]. For simplicity we assume that the

links are straight lines (Fig. 2.18). The region near the link (except near

the end points) can be parameterized by a Cartesian coordinate system with

a variable y following the equipotential lines parallel to the link and x for

the transverse direction. The disorder potential varies essentially only in

transverse direction, and can thus be represented by a function U(x). In the

Landau gauge A = (0, Hx) the Hamiltonian for electrons on the link is then

translation invariant in y-direction and the Landau states

ψnk(x, y) = Cn e −(x+l2k)2/(2l2H )Hn[(x + l

2k)/lH ] e iky (2.62)

are accurate solutions of the stationary Schro¨dinger equation. Here Hn is the

n-th Hermite polynomial and Cn a normalization constant. The correspond-

44 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

Figure 2.17: Schematic pattern of the contours of equal energy [24].

ing energy is simply

�nk = (n + 1/2)ωc + U(l 2 Hk) . (2.63)

In the following we drop the Landau level index n, because only the highest

occupied level is relevant. The quantum number k is the momentum asso-

ciated with the translation invariance in y-direction. It is proportional to a

transverse shift x0 = l 2 Hk of the wave function. The potential U lifts the de-

generacy of the Landau levels and makes energies depend on the momentum

along the link. The group velocity

vk = d�k dk

= l2H U ′(x0) (2.64)

corresponds to the classical drift velocity of an electron in crossed electric

and magnetic fields. In our simplified straight link approximation U ′(x0)

does not depend on y. In general it will depend slowly on the longitudinal

coordinate, but near percolating paths away from crossing points it has a

fixed sign, and hence the group velocity has a fixed direction, that is the

motion on the links is chiral.

2.4 SEMICLASSICAL THEORY OF ELECTRON DRAG... 45

Figure 2.18: Schematic pattern of the percolating region network.

The drag between two parallel layers is dominated by electrons in the

percolating region, corresponding to states near the center of the Landau

level, since electrons in deeper localized states are not easily dragged along.

In the network picture, macroscopic drag arises as a sum of contributions

from interlayer scattering processes between electrons in different links. If no

current is imposed in the drive layer, both layers are in thermal equilibrium

and the currents on the various links cancel each other on average. Now

assume that a small finite current is switched on in the drive layer such

that the electrons move predominantly in the direction of the positive y-axis

(the electric current moving in the opposite direction). This means that

the current on links oriented in the positive y-direction is typically larger

than the current on links oriented in the negative y-direction. Interlayer

scattering processes lead to momentum transfers between electrons in the

drive and drag layer. The preferred direction of momentum transfers is such

46 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

that the scattering processes tend to reduce the current in the drive layer,

that is the interlayer interaction leads to friction.

Now the crucial point is that electrons moving in the disorder potential

are not necessarily accelerated by gaining momentum in the direction of their

motion, or slowed down by loosing momentum (as for free electrons). The

fastest electrons are those near the center of the Landau level: they have the

highest group velocity on the links of the percolation network and they get

most easily across the saddle points. Electrons in states below the Landau

level center are thus accelerated by gaining some extra momentum in the

direction of their motion, but electrons with an energy above �0 are pushed

to still higher energy by adding momentum and are thus slowed down.

2.4.1 Drag between parallel links.

To make the above speculations more quantitative I consider drag between

two parallel quasi one-dimensional links. Instead of the layers for the 2D

case I take strips with infinite length which have a potential profile with its

value depending only on the transverse coordinate x. The eigenstates of the

electrons subjected to a magnetic field and the potential are similar to the

eigenstates without the potential if the gauge is taken to be (0, Hx, 0) (the y-

axis is directed along the strip, H is the magnetic field). The difference is that

the eigenfunctions have contributions from the adjacent Landau levels that

results in a finite velocity of the electron proportional to the x-derivative of

the potential (2.64). The wave-functions in the zeroth order in the potential

are given by (2.62); the corresponding energies in first order of the potential

are given by (2.63). From now on the magnetic length lH will be set to unity,

unless the contrary is explicitly mentioned.

This shows us that there is an effective dispersion law for the electron.

The electron has got a finite mass if the second derivative of U with re-

spect to x is non-zero. In the following calculations I neglect the intra-layer

electron-electron scattering and consider only the interlayer interaction. The

2.4 SEMICLASSICAL THEORY OF ELECTRON DRAG... 47

corresponding Hamiltonian is of the form

Hˆ = ∑

n,p,α=1,2

[( n +

1

2

) ωH + Uα(p)

] c†α,n,pcα,n,p

+

∫ d3r1d

3r2ψˆ † 1(r1)ψˆ

† 2(r2)V (r1 − r2)ψˆ2(r2)ψˆ1(r1). (2.65)

Here c, c† are the annihilation and creation operators of the electron in Lan-

dau basis, α labels the layers and

ψˆα(r) = ∑ n,p

cα,n,pΨn,p(x, y)δ(z + jd/2) (2.66)

with d being the inter-layer distance and j = 3 − 2α. Finally V (r) is the Coulomb potential. Substituting (2.66) into (2.65) we obtain

Hˆ = ∑

n,p,α=1,2

[( n +

1

2

) ωH + Uα(p)

] c†α,n,pcα,n,p

+ ∑

n1...n4 p1...p4

( c†1,n4,p4c

† 2,n3,p3c2,n2,p2c1,n1,p1

× ∫ d2r1d

2r2Ψ ∗ n4,p4

(r1)Ψ ∗ n3,p3

(r2) e2√|r1 − r2|2 + d2 Ψn2,p2(r2)Ψn1,p1(r1)

) .

(2.67)

I model the distribution function in the drive layer in the following way

(arguments in favor of this assumption will be given later):

n1(�1(n, p)) = n 0 1(�1(n, p− p˜)). (2.68)

Here �i is the energy of the electron in i-th layer. It is given by (2.63); ni

is the distribution function and n0i is the Fermi distribution function in the

i-th layer

n0i (�) = 1

1 + exp ( �−µi T

) ,

µi being the chemical potential in the i-th layer and T the temperature. This

assumption corresponds to the shifted Fermi sea for free electrons for velocity

p˜/m. It can also be regarded as a p-dependence of the chemical potential:

µ1(p) = µ1 + p˜ ∂�1 ∂p

. (2.69)

48 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

This distribution leads to a non-zero total current in the layer as there are

more electrons with positive velocity than those with negative, i.e. more

”right-movers” than ”left-movers”. Expanding n1(�1(n, p)) in p˜ we get

n1(�1(n, p)) ≈ n01(�1(n, p)) + p˜

T

∂�1 ∂p

n01(�1(n, p))(1− n01(�1(n, p))). (2.70)

p˜ can be found knowing the current j1 in the drive layer: p˜ = pij1 evF1

As the electrons in the two layers interact, the electrons in the drag layer

will tend to rearrange, so that the drag current occurs also there. To describe

the process quantitatively I use the kinetic equation approach with the col-

lision integral determined in the two-particle approximation using Fermi’s

golden rule. I restrict myself to the case T � ωH so that I don’t need to take scattering processes that change the Landau level into account. More

over, provided that condition is fulfilled, I only need to consider the upper

Landau levels in both layers as no scattering processes are possible in any

other levels due to Pauli principle. I refer to the Landau levels in drive and

drag layer as n and n′ respectively.

At this point I follow the line of calculations from the work [14] adapted

to my model. A detailed review of this work can be found in the section 2.3.2

of the present work. The Boltzmann equation for my model reads:

p˙2 · ∂n 0 2

∂p2 =

[ ∂n2 ∂t

] coll

, (2.71)

with p˙2 = −eE2, where E2 is the electric field leading to the drag voltage. The interlayer collision term is given by

[ ∂n2(p2)

∂t

]12 coll

= 2pi

L

∫ dp1 2pi

∫ dp4 2pi

∫ dp3 2pi

2pi

~

∣∣∣〈p1, p2|Vˆ |p4, p3〉∣∣∣2[ n1(p1)

( 1− n1(p4)

) n2(p2)

( 1− n2(p3)

) − n1(p4)

( 1− n1(p1)

) n2(p3)

( 1− n2(p2)

)] × δ(�1(p1) + �2(p2)− �1(p4)− �2(p3)). (2.72)

2.4 SEMICLASSICAL THEORY OF ELECTRON DRAG... 49

Here Vˆ is the Coulomb potential operator, 〈p1, p2| and |p4, p3〉 are the initial and final states, L is the length of the sample, and for brevity I write

ni(p) instead of ni(�i(p)). For the drag layer in Eq. (2.72) I will take the

distribution to be equilibrium1:

n2(p) = n 0 2(p).

The expression in the square brackets comes from the creation and annihila-

tion operators and represents the difference between incoming and outgoing

flows in the phase space, the delta-function ensures the energy conservation.

The matrix element is given by

〈p1, p2|Vˆ |p4, p3〉 =∫ d2r1d

2r2Ψ ∗ n,p4

(r1)Ψ ∗ n′,p3

(r2) e2√|r1 − r2|2 + d2 Ψn′,p2(r2)Ψn,p1(r1). (2.73)

Using (2.62) and the well-known Fourier-transform of the Coulomb potential

it can be rewritten in the form:

〈p1, p2|Vˆ |p4, p3〉 =

2pie2 ∫ dqx

e−qd√ q2x + q

2 y

e− q2x+q

2 y

2 Ln

( q2x + q

2 y

2

) Ln′

( q2x + q

2 y

2

)

exp

( iqx

(p1 + p4)− (p2 + p3) 2

) δ(p1 − p4 − p3 + p2). (2.74)

Here qy = p1−p4 = p3−p2 is the momentum transfer between the layers. The delta-function shows that the momentum is conserved. Ln is the Laguerre

polynomial. Substituting (2.74) into (2.72) one has to keep in mind that

squaring the delta functions yields an additional factor of L/2pi.

Now I multiply both sides of the Boltzmann equation by ∂�2/∂p2 and in-

tegrate over p2. The left hand side yields −e ∫

dp2 2pi n02

′ (�p2) v

2 p2E2 which tends

to e 2pi vF2E2 at low temperatures. Using the antisymmetry of the integrand

of Eq. (2.72) under exchange of p2 and p3, and collecting all the above I can

rewrite the result for the drag resistance ρD = E2/j1:

1See section 2.3.2 for a detailed discussion

50 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

ρD = 2pi2

e2vF1vF2T

∫ dp1dp2dqy

(2pi)3 q2y|V (p1, p2, qy)|2

∂2�1 ∂p21

∂2�2 ∂p22

× 1 cosh �1(p1)−µ1

2T

1

cosh �1(p1−qy)−µ1 2T

1

cosh �2(p2)−µ2 2T

1

cosh �2(p2+qy)−µ2 2T

× δ(�1(p1) + �2(p2)− �1(p1 − qy)− �2(p2 + qy)), (2.75)

with

V (p1, p2, qy) = 2pie 2

∫ dqx

e− √

q2x+q 2 yd√

q2x + q 2 y

e− q2x+q

2 y

2 Ln

( q2x + q

2 y

2

) Ln′

( q2x + q

2 y

2

)

× exp [iqx(p1 − p2 − qy)]. (2.76)

To obtain (2.75) I first had to recast the filling factor part of (2.72) using

(2.70), the expression for the Fermi distribution, and the energy conservation

law. Then the p4 integral was evaluated using the delta function and the

variable qy = p3 − p2 was introduced instead of p3. The expression (2.75) turns out to be symmetric with respect to in-

terchanging the layers. The formal reason for it is the choice of the non-

equilibrium distribution (2.68). Indeed, the two second derivatives in (2.75)

are from different origin — the one from the drag layer comes from the veloc-

ity change in the scattering process and is directly connected to the curvature

of the dispersion, while the one in the drive layer results from expanding the

distribution function (2.68). For any type of the non-equilibrium distribu-

tion in the drive layer we get the derivative of the chemical potential over p1

instead of ∂2�1/∂p 2 1; but for the specific form I chose this derivative becomes

the second derivative of the energy (compare with (2.69)). Consequently,

the symmetry between the layers occurs only for the used form of the non-

equilibrium distribution. This is a strong argument in favor of Eq. (2.68).

In the following I describe two different methods of calculating the drag.

They differ from each other in the way the electrons’ dispersion is parame-

terized.

2.4 SEMICLASSICAL THEORY OF ELECTRON DRAG... 51

First method

I expand the electron energy around the Fermi energy up to second order in

p− piF , where piF is the momentum corresponding to the Fermi edge in the i-th layer. This is justified whenever the relevant electron states (i.e. those

states that give the essential contribution to the integral in (2.75)) are close

enough to the Fermi energy. What this precisely means will be clarified later,

now we just mention that this condition might fail in the vicinity of the point

where the second derivative of �i with respect to p vanishes. The expansion

yields:

�i(p)− µi = αi(p− piF ) + βi 2

(p− piF )2. (2.77) From now on, for the sake of brevity, I will write pi instead of pi − piF . It is also important to mention that the first term in (2.77) must be much larger

than the second, in order for the expansion to be valid.

The next step is the introduction of the energy transfer ω. This is done

by including an additional integration:

δ(�1(p1) + �2(p2)− �1(p1 − qy)− �2(p2 + qy)) =∫ dωδ(ω − �1(p1) + �1(p1 − qy))δ(ω + �2(p2)− �2(p2 + qy)), (2.78)

or, after substituting (2.77),

δ(�1(p1) + �2(p2)− �1(p1 − qy)− �2(p2 + qy)) =∫ dωδ(ω − α1qy − β1p1qy + β1

2 q2y)δ(ω − α2qy − β2p2qy −

β2 2 q2y). (2.79)

This allows us to evaluate the p1,2-integrals in (2.75):

ρD = pi

e2vF1vF2T sgn (β1β2)

∫ dqdω

(2pi)2 |V (p1 + p1F , p2 + p2F , q)|2

× 1 cosh

[ 1

4β1T

(( ω q

+ β1q 2

)2 − α21

)] 1 cosh

[ 1

4β1T

(( ω q − β1q

2

)2 − α21

)]

× 1 cosh

[ 1

4β2T

(( ω q

+ β2q 2

)2 − α22

)] 1 cosh

[ 1

4β2T

(( ω q − β2q

2

)2 − α22

)] . (2.80)

52 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

Here the momenta p1,2 in the argument of V are determined by the delta

functions in (2.79).

A significant simplification which allows an analytical analysis arises in

the case d� Rc, where Rc = √

2 max (n, n′)lH is the cyclotron radius of the

electron in the layer with higher Landau level2. In this limit the expression

for V (p1, p2, q) reduces to

V (p1, p2, q) =

2pie2 ∫ dqx

e− √

q2x+q 2d√

q2x + q 2

exp [iqx(p1 + p1F − p2 − p2F − q)] = 4pie2K0(qD),

(2.81)

where K0(x) is the McDonald function and

D = √ d2 + (p1 + p1F − p2 − p2F − q)2.

Now I suppose that p1,2, q � d (this will also be checked later), so that D becomes independent on q and ω and is just the distance between the two

scattering electrons. This leaves the ω-dependence only in the four cosh-

factors in the integrand of (2.80) and thus we are left with the integral∫ dω

1

cosh

[ 1

4β1T

(( ω q

+ β1q 2

)2 − α21

)] 1 cosh

[ 1

4β1T

(( ω q − β1q

2

)2 − α21

)]

× 1 cosh

[ 1

4β2T

(( ω q

+ β2q 2

)2 − α22

)] 1 cosh

[ 1

4β2T

(( ω q − β2q

2

)2 − α22

)] , (2.82)

which will be very important in the next parts of the work. When evaluating

this integral we have to consider only q . 1 D

due to the McDonald function.

It can easily be checked that, if the above mentioned conditions hold, the

integrand in the expression (2.80) is invariant under transformations of the

2In real samples this condition is hardly ever met for the magnetic fields for which

negative drag is observed. However, no principle changes occur in the behavior of the

interaction matrix element: it is still logarithmic for small q’s and falls off exponetially for

large ones.

2.4 SEMICLASSICAL THEORY OF ELECTRON DRAG... 53

q

Ω

IV

VIII

I

V

III

Figure 2.19: The four curves are ω = α1,2q ± β1,2q 2

2 . q∗ is the q-coordinate of

the touching point of III and VI

type αi → −αi, βi → −βi, q → −q and ω → −ω. This allows to consider only positive values of αi and βi and to restrict the q, ω-integration to the first

quadrant, multiplying the final result by 4. Moreover, the factor sgn (β1β2)

shows that for the same signs of curvatures the drag is positive, while for

different signs it is negative.

I first consider small temperatures (we will see what it means later). If the

temperature is small enough, one can replace cosh−1 (A/T ) by 2 exp (−|A|/T ). Then in the integrand of (2.82) I obtain the exponential of the sum of two

expressions of the type |ω−αq+βq2/2|(ω+αq+βq2/2)+|ω−αq−βq2/2|(ω+ αq− βq2/2). That leads to the splitting of the integration region in 6 subre- gions (see Fig. 2.19), so that (2.82) can be rewritten as

16

∫ dωe−A/2T ,

with A given by different expressions in different regions of the (q, ω)-plane

(see Fig. 2.19) (we take α1 > α2):

54 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

A(q, ω) =

ω2

q2 β1 + β2 β1β2

+ β1 + β2

4 q2 −

( α21 β1

+ α22 β2

) for I;

ω + ω2

β2q2 + β2q

2

4 − α

2 2

β2 for II;

ω2

q2 β1 − β2 β1β2

+ β2 − β1

4 q2 − α

2 2

β2 + α21 β1

for III;

ω − ω 2

β1q2 − β1q

2

4 + α21 β1

for IV;

−ω 2

q2 β1 + β2 β1β2

− β1 + β2 4

q2 +

( α21 β1

+ α22 β2

) for V;

2ω for VI.

(2.83)

For a given q < q∗ = 2(α1 − α2)/(β1 + β2), A(ω) reaches its minimum (i.e. the integrand in (2.82) reaches its maximum) on one of the banks of

III, depending on which of the two β’s is bigger. Indeed, as can be seen

from (2.83), A(ω) is decreasing in V and IV, increasing in I and II, and

either decreasing or increasing in III depending on the sign of β1 − β2. For an analytical analysis it is enough to consider only the case β1 > β2, which

means that the minimum of A is reached along the line ω = α2q + β2q2

2 . On

this line A(q) is given by:

A(q, ω = α2q + β2q

2

2 ) =

α21 − α22 β1

− q2 (β1 − β2) 2

4β1 + α2q

β1 − β2 β1

, (2.84)

and the integral in (2.82) is proportional to exp (−A(q, ω = α2q + β2q22 ))/2T . As the McDonald function for small arguments is logarithmic, we conclude

that the q-integral for small T is determined by q ∼ Tβ1 α2(β1−β2) . For such q’s

the q2 term in A is negligible for T � α22 β1

.

To calculate the pre-factor I linearize A(ω) with respect to ω−(α2q+ β2q22 ) and calculate the integral (2.82). Analytical calculation can be done in the

limit |α1 − α2| � α1, |β1 − β2| � β1, so I restrict myself to such values. I denote β ≈ β1 ≈ β2, α ≈ α1 ≈ α2, ∆β = β1 − β2, ∆α = α1 − α2. Now I will show that under the conditions mentioned above, for T � T ∗ = αq∗ = α∆α/β the integral in (2.80) is determined by the region III. Indeed, the

2.4 SEMICLASSICAL THEORY OF ELECTRON DRAG... 55

ω-integral of e−A/2T over III is

exp

[ − 1 T

( α11 − α22

2β1 + αq

∆β

β

)]

× α1q−β1q

2

2∫ α2q+

β2q 2

2

dω exp

( − 1

2T

ω2 − (α2q + β2q22 )2 q2

∆β

β2

) =

exp

[ − 1 T

( α21 − α22

2β1 + αq

∆β

β

)] min

( qTβ2

α∆β ,∆αq − βq2

) . (2.85)

The two options in the last equation correspond to the cases when the upper

limit in (2.85) can be set to infinity or not. We see that for T � T ∗∗ = α∆α∆β β2

and for q . Tβ α∆β � q∗ (which are the only relevant ones) the first option in

the min-function is valid, while for larger T ’s and q < q∗ it is the second one.

This means that for T � T ∗∗ I only have to check whether regions IV and V give comparable contribution to (2.80), while for T ∗∗ � T � T ∗ one can set ∆β = 0 and has to look at all regions surrounding III in order to prove

that the contribution from III dominates.

From (2.83) one can see that summed up contributions to (2.82) from I,

II, IV and V are of the order of

exp

[ − 1 T

( α11 − α22

2β1 + αq

∆β

β

)]( Tβq

α

) ,

which is smaller than (2.85). (For T � T ∗∗ it follows from ∆β � β and for T ∗∗ � T � T ∗ it follows from ∆αq = qT ∗β/α � qTβ/α.3) In turn, for q > q∗ minimum of A(ω) is reached for ω = α1q − β1q22 and the result of ω-integration is of the order of (T � T ∗∗, so we set ∆β = 0)

exp

( −αq −

βq2

2

T

)( βq2 +

Tβq

α

) ,

3In fact, for q’s close enough to q∗ the contribution for III tends to zero, but as the

relevant value arises from the integration over q, we have to compare the integrated values.

Doing so, I mention that the results of the q-integration over I,II, IV and V from 0 to q∗

have an additional factor of T/T ∗ � 1.

56 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

(the first term in the parenthesis comes from VI and the second — from II

and IV). Now it is easy to see that the contribution to (2.80) from q > q∗

also has an extra factor of T/T ∗ � 1 compared to the contribution from III (see also footnote).

Thus I have proved that for T � T ∗ it is sufficient to integrate over III. Evaluating the ω-integral in (2.80) yields

ρD = 64pie2

vF1vF2T sgn (β1β2) exp

[ − 1 T

( α11 − α22

2β1

)]

× ∫ q∗

0

dq [K0(qD)] 2 exp

( −αq∆β

βT

) min

( qTβ2

α∆β ,∆αq − βq2

) . (2.86)

As the McDonald function is logarithmic for small arguments and falls off

exponentially for large ones there are three scales for q: q∗, Tβ/∆βα, and

1/D. Depending on which of the three wave-vectors is smaller three results

can be obtained:

ρD = 64pie2

vF1vF2 sgn (β1β2)

×

exp [ − 1

T

( α21−α22

2β1

)] [ ln ( D Tβ

∆βα

)]2 T 2β4

α3∆β3 , T � T ∗∗, T � α∆β

βD ;

exp [ −α∆α

Tβ

] [ ln ( D∆α

β

)]2 ∆α3

6β2T , T ∗∗ � T � T ∗, q∗ � 1

D ;

exp [ −α∆α

Tβ

] 1

2D2 min

( β2

α∆β , ∆α

T

) , 1

D � q∗, α∆β

βD � T � T ∗.

(2.87)

Now let us consider higher temperatures (T � T ∗). For this temperature region it is possible to neglect the difference between α1 and α2 as well as

the difference between β1 and β2. So (2.82) reduces to

∞∫ 0

dω 1

cosh2 [“

ω−αq−βq2 2

”“ ω+αq−βq2

2

”

4βq2T

] 1 cosh−2

[“ ω−αq+ βq2

2

”“ ω+αq+ βq

2

2

”

4βq2T

] . (2.88)

I will show later that for q � T/α integral (2.88) falls off exponentially, so another temperature scale arises: the one at which T/α ∼ 1/D. For

2.4 SEMICLASSICAL THEORY OF ELECTRON DRAG... 57

smaller temperatures the cut-off in q-integration is T/α and for higher T ’s

it is 1/D. If, as usually is the case in the experiment, D is of the order of

the characteristic size of the potential then 1/D � α/β, so this temperature scale T ′ = α/D � α2/β. For such temperatures we can rewrite (2.88) in the form: (after shifting the integration variable to ω˜ = ω − αq − βq2

2 )

∞∫ −∞

dω˜ cosh−2 [ ω˜α

2βqT

] cosh−2

[ (ω˜ + βq2)α

2βqT

]

= 2βqT

α

∞∫ −∞

dx

cosh2 x cosh2 (x+ qα 2T

) . (2.89)

Indeed, it can easily be shown that for T . T ′ relevant ω˜’s in the integral are

of the order of max (βqT/α, βq2)� αq, which allows to set ω+αq− βq2 2 ≈ 2αq

and extend the lower integration limit to −∞. The integral in (2.89) can be evaluated and yields

8βqT

α

qα 2T

coth qα 2T − 1

sinh2 qα 2T

, (2.90)

which is, as expected, exponentially small for q � T/α. Substituting it into (2.80) one obtains for ρD : (for T � α/D)

ρD = 64piβT 2e2

α3vF1vF2 sgn (β1β2) ln

2

( D T

α

) . (2.91)

Ultimately for T � α/D the answer for the drag resistivity is

ρD = 32pie2β

3vF1vF2D2α sgn (β1β2). (2.92)

The equations (2.87), (2.91) and (2.92) give the value of ρD for all possible

combinations of parameters. It turns out that for smallest temperatures

the behavior is activated, then for T ∼ α∆α/β there is a cross-over to T 2- behavior, and finally for T ∼ α/D quadratic dependence changes to constant. If α/D . α∆α/β the T 2-regime is absent.

Now I have to check the validity of the approximations made. I started

with neglecting the third-order term in (2.77). In order for it to be justified,

58 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

the second-order term must be larger than the third-order one. We will set

�i(p) = −ti sin 2pi(p+piF )l 2 H

l . The second-order term is βip

2/2 and the third-

order one is −αip3(2pil2H/l)2/6, with βi = ti(2pil2H/l)2 sin 2pipiF l 2 H

l and αi =

−t(2pil2H/l) cos 2pipiF l 2 H

l . The ratio of the two terms is 6pipl2H/l cot

2pipiF l 2 H

l and

it should be small. This condition is likely to be violated in the vicinity of

piF = 0, which corresponds to the inflection point of the potential, so we can

replace cotx by 1/x and that leaves us with the inequality to be verified for

relevant p’s:

3|p|/|piF | � 1.

From (2.79) we have

pi = ω

βiq − αi βi ± q

2 . (2.93)

If now one substitutes the typical values of ω/q and q it becomes evident

that for piF . ∆α β

the approximation (2.77) is wrong. Thus for this region

of parameters a new approximation should be applied. This is done in the

next part of the work.

The second approximation that was to be verified is the condition p1,2, q � d which allows to neglect the ω-dependence of the interaction matrix element.

From (2.93) one sees that this condition is always fulfilled if d � Rc which is assumed in all the above calculations.

Second method.

The approach elaborated in the previous section consisting of expanding

�(p) around the Fermi momentum fails in the interesting region of pF ≈ 0 that corresponds to a particle-hole symmetric situation. In particular, using

that technique it is not possible to show that at the particle-hole symmetric

point the drag is identically zero, which is a crucial feature. To account for

this region of parameters I have developed another approach that is not as

transparent as the first one, but more general. As we will see it gives the

same results as the old one for most parameter regions. The exceptions to

the above will be discussed separately.

2.4 SEMICLASSICAL THEORY OF ELECTRON DRAG... 59

Now, instead of expanding �(p) around pF and dealing with the variables

αi, βi which are pFi-dependent, I expand around the inflection point of the

potential (� = 0, p = 0) and carry out the calculations in more natural

parameters i.e. those of the potential profile and the chemical potentials.

Let the dispersion relations (directly connected to the potential profiles)

be

�1(p) = t1 sin(pγ1) and �2(p) = t2 sin(pγ2)

for the drive and drag layer respectively. γi ≈ 2pil 2 H

Li characterizes the size Li of

the potential hill. In the 2D situation it corresponds to the correlation length

of the potential. Even if it is the same in both layers, the local values of γ’s,

characterizing the form of the potential at a given point may be different for

the two layers, so we have to consider the general case. As I will show this

leads to no qualitative changes in the results.

I expand the dispersions around the inflection point up to third order:

�i = tiγip− tiγ 3 i

6 p3. (2.94)

Then, as in the previous section, I solve the energy conservation equation for

a scattering process with given ω and q:

�1(p 0 1 −

q

2 )− �1(p01 +

q

2 ) + ω = 0,

�2(p 0 2 −

q

2 )− �2(p02 +

q

2 ) + ω = 0.

I obtain

p02i = 1

γ2i

( 2

( 1− ω

tiγiq

) − (qγi)

2

12

) . (2.95)

This implies that

ω 6 tiγiq

( 1− (qγi)

2

24

) . (2.96)

Suppose, t1γ1 > t2γ2 (in the opposite case no changes in the final result

occur). I introduce a new variable δ instead of ω:

ω = (1− δ)t2γ2q (

1− (qγ2) 2

24

) . (2.97)

60 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

Thus

p022 = 2δ

γ22

( 1− (qγ2)

2

24

) ;

p021 = 1

γ21

t1γ1

( 1− (qγ1)2

24

) − t2γ2

( 1− (qγ2)2

24

) t1γ1

+ δ t2γ2 t1γ1

( 1− (qγ2)

2

24

) . (2.98)

We see that for every pair (q,ω) satisfying the condition (2.96) we have 4

different pairs of initial states that differ from each other by the signs of the

p0i ’s. So the expression (2.80) turns into

ρD = 4pie2

vF1vF2T

∫ dqdωK20(qD)

× (

1

cosh �1(p01− q2 )−µ1

2T cosh

�1(p01+ q 2 )−µ1

2T

− 1 cosh

�1(p01− q2 )+µ1 2T

cosh �1(p01+

q 2 )+µ1

2T

)

× (

1

cosh �2(p02− q2 )−µ2

2T cosh

�2(p02+ q 2 )−µ2

2T

− 1 cosh

�2(p02− q2 )+µ2 2T

cosh �2(p02+

q 2 )+µ2

2T

) .

(2.99)

After performing the multiplication of the two parenthesis in the integrand

I get a sum of four expressions of the type of (2.80) (with obvious changes

due to different expression for �i used), corresponding to the four possible

initial states mentioned above. If µi � T , than only one of the four terms survives and we return to the old expression (2.80). On the other hand,

if either of the µ’s happens to be zero, the expression (2.99) yields zero

which is the manifestation of the particle-hole symmetry. In the frame of the

consideration in the previous method it is impossible to explicitly obtain this

zero, while in the present one it comes out naturally. If µi are not zero, but

still µi � T one can expand the expressions in the parenthesis in powers of µi/T and carry out the calculations in this region of parameters.

At this stage it is worth mentioning that the overall sign of ρD given by

(2.99) is sgn (µ1µ2), which means that in the situation, when one layer is

2.4 SEMICLASSICAL THEORY OF ELECTRON DRAG... 61

more than half-filled and the other is less than half-filled, the drag signal will

be negative.

As in the previous section I restrict myself to the consideration of almost

identical layers. In this case the calculations can be carried out analytically

and I do not expect any qualitative changes if the parameters of the layers

begin to differ significantly. So I consider t1γ1 ≈ t2γ2 ≈ tγ, γ1 ≈ γ2 ≈ γ and |µ1| ≈ |µ2| ≈ µ. The absolute values of the differences of the parameters in the two layers will be denoted respectively as ∆(tγ), ∆γ and ∆µ.

I start with the case µ� T . As in the previous section I replace cosh ( A 2T

)

by 2 exp (− |A| 2T

). This divides the (q, δ) plane into several regions, within each

of which the arguments of the cosh’s have constant signs. In principle, I

proceed just as in the previous section: I set both µ’s positive and consider

only the region of positive q’s, as the other cases are trivially obtained from

this one due to symmetry properties of the integrand. As both µ’s are positive

and � T , only the first terms in the parenthesis of (2.99) survive. To determine the borders of the mentioned regions I solve the equations

�i(p 0 i ± q2) = µi. If, as I set before, t1γ1 > t2γ2, then these equations read (for

the time being I keep some extensive terms):

µ2 = t2 √

2δ

( 1− 7(qγ2)

2

48

)( 1− δ

3

) ± t2 qγ2

2

( 1− (qγ2)

2

24

) (1− δ), (2.100)

µ1 = t1

√√√√√2 ∆ + δ t2γ2

t1γ1

( 1− (qγ2)2

24

) (1− ∆

3 − δ

3 − (qγ1)

2

8

)

± t2 qγ2 2

( 1− (qγ2)

2

24

) (1− δ), (2.101)

with

∆ = t1γ1

( 1− (qγ1)2

24

) − t2γ2

( 1− (qγ2)2

24

) t1γ1

. (2.102)

For q2 � 12 ∆(tγ) tγ2∆γ

, ∆ ≈ ∆(tγ) tγ

, in the opposite limit ∆ ≈ − q2 12 γ∆γ (∆γ =

γ1 − γ2).

62 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

q

∆

VI IV

II

V

I

III

PSfrag replacements

δ1

δ2

Figure 2.20: Regions of the (δ, q)-plane in which arguments of the cosh’s in

Eq. (2.99) keep their signs. δ1 = 1 2

( µ t

)2 , δ2 =

1 2

( µ t

)2 −∆. Solving the Eqs. (2.100), (2.101) (omitting this time the irrelevant small

corrections, the consistency of this can be easily checked after solving the

equations by direct comparison of the omitted terms with the kept ones), I

get

�2(p 0 2 ±

q

2 ) = µ2 for δ ≈ 1

2

( µ2 t2 ∓ qγ2

2

)2 , (2.103)

�1(p 0 1 ±

q

2 ) = µ1 for δ ≈

[ 1

2

( µ1 t1 ∓ qt2γ2

2t1

) −∆

] t1γ1 t2γ2

.

The notation of the regions (Fig. 2.20) corresponds to the one used in

the previous section in the sense that the Fig. 2.20 is just a replotting of

Fig. 2.19 in the coordinates (δ, q) instead of (ω, q). The region where the

approximation used there is applicable is (

µ t

)2 � ∆ or µ�√ t∆(tγ) γ

. If this

condition is fulfilled, the quadratic term in the expansion of the dispersion

around the Fermi point is much larger than the cubic one.

2.4 SEMICLASSICAL THEORY OF ELECTRON DRAG... 63

The end of region III (q∗) is given by q∗ = ∆(tγ) µγ2

(equivalent to ∆α β

from the

previous section). It can also be easily checked that for µ� √

t∆(tγ) γ

√ ∆γ γ

the

condition q2 � 12 ∆(tγ) tγ2∆γ

holds within III and I can set ∆ = ∆(tγ) tγ

everywhere

in III. We will see that, as in the previous section, it is the region III that gives

the main contribution to the result. Unlike that case though, the maximum

of the integrand is reached on the upper border of III, independent on which

µiγ 2 i (analog of βi in the previous section) is bigger. Indeed, in III the sum of

absolute values of the arguments of cosh’s is (in the first non-vanishing order

in δ)

− 1 T

[ t2 √

2δ

( 1− 7(qγ2)

2

48

) − t1

√ 2

( ∆ + δ

t2γ2 t1γ1

)( 1− 7(qγ2)

2

48

) − µ2 + µ1

] .

(2.104)

It is straightforward to show that apart from the pathological case t∆γ � ∆(tγ), the δ-derivative of this expression is negative and thus the maximum

is reached on the upper border of III.

On the other hand, in IV the sum of the absolute values of the cosh’s

arguments becomes

− 1 T

[ µ1− t1

√ 2

( ∆ + δ

t2γ2 t1γ1

)( 1− 7(qγ2)

2

48

) + t2γ2q

( 1− (qγ2)

2

24

) (1− δ)

] ,

the δ-derivative of which positive and its absolute value is much greater that

that of the δ-derivative in III. That proves the aforementioned statement,

that the integral in the low temperature limit (T � tγ∆(tγ) µγ2

) is determined

by III.

To determine the exponential factor in the answer it is enough to substi-

tute δ = 1 2

( µ t

)2 in (2.104). It turns out that

ρD ∝ exp [ − 1 T

t∆(tγ)

γµ

]

in agreement with the results from the previous section. Hereby the difference

between t1 and t2 in (2.104) turns out to be insignificant. To determine the

pre-factor the δ- and q-integrals must be evaluated. It is done in a similar

64 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

manner as before. For the lowest temperatures the δ-integral is determined

by the derivative of the argument of the exponent, and for higher ones — by

the size of III. What becomes different is the definition of ”low temperatures”.

Before it was determined by the value of ∆β (the analog of which now is ∆µ),

but now, as can be seen from (2.104), the derivative of the argument of the

exponent in the integrand does not depend on ∆µ.

I denote the expression in (2.104) as A(δ). To find the cross-over T one

has to compare A (

1 2

( µ t − qγ

2

)2) − A( 1 2

( µ t

+ qγ 2

)2 −∆) with unity. If it is much greater than unity, then the δ-integral converges within III and the

lower limit can be extended to infinity; in the opposite case, within III the

argument of the exponent can be considered constant and the integration

becomes trivial.

The cross-over temperature is

T ∗ = t2[∆(tγ)]2

γ2µ3 . (2.105)

For T � T ∗ the lower limit of the δ-integral, as already mentioned, can be set to −∞ after expanding the argument of the exponent in powers of[

1 2

( µ t − qγ

2

)2 − δ]: (µt −

qγ 2 )

2∫ −∞

eAdδ

with

A ≈ − t∆ T (

µ t − qγ

2

) + t∆ T (

µ t − qγ

2

)3 [(

1

2

µ

t − qγ

2

)2 − δ ] .

Here, the second term comes from the expansion of the δ-dependent denom-

inator. Other contributions are negligible. Keeping the q-dependence only

in the exponent we get the result of the δ-integration:

T (

µ t

)3 t∆

exp

( − t∆ T (

µ t − qγ

2

) ) .

Expanding the exp’s argument in q and evaluating the q-integral (keeping in

mind that due to the variable change ω → δ one gets an additional factor

2.4 SEMICLASSICAL THEORY OF ELECTRON DRAG... 65

tγq in the integrand) I finally evaluate the q-integral in (2.99) to obtain the

final result:

ρD = 256pie2γ2T 2µ7

vF1vF2t 6[∆(tγ)]3

sgn (µ1µ2) exp

[ −t∆(tγ)

γµT

] ln2 ( D

Tµ2

t2∆(tγ)

) . (2.106)

For T ∗ � T � t∆(tγ) γµ

I easily get the same answer as in the previous

chapter:

ρD = 32pie2[∆(tγ)]3

3vF1vF2T (µγ 2)2

sgn (µ1µ2) exp

[ −t∆(tγ)

γµT

] ln2 ( D

∆(tγ)

γ2µ

) . (2.107)

For T � t∆(tγ) γµ

the result (2.91) holds (of course changes in notation α→ tγ and β → µγ2 should be made).

There is no reason to search for any deep meaning in the fact that two

different approximations give different results in the low temperature limit.

It just means that the pre-factor in (2.106) is sensitive to the precise form of

the potential and thus should not be taken literally. On the other hand, the

exponential factor is universal. The mathematical reason for which the pre-

factor differs in the two approximations (note that I am in the region where

both approximations are self-consistent, i.e. all approximations made are

justified by the final results, so it is not an indication that one approximation

is ”wrong” and the other one is ”correct”) is quite simple. In fact, as can

be checked numerically, in a very broad region of parameters the values of �i

given by the two approximations differ very insignificantly. But the values

of p0i which are determined from the energy conservation relations turn out

to be quite sensitive to the approximations used, the reason for it is that the

second order term in (2.77) is much smaller than the first order one in the

considered range of parameters, and a slight change in it, not having any

visible effect on the value of �i has a much larger effect on p 0 i . Further, the

value that is essential for the calculation is �2(p 0 2) − �1(p01). Its zero order

term in δ happens not to to depend on the approximation, while the first

order term (which accounts for the pre-factor) turns out to depend strongly

on the approximation used.

66 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

q

∆

IV

V

III

PSfrag replacements

δ1

Figure 2.21: Regions of the (δ, q)-plane for µ� √

t∆(tγ) γ

; δ1 = 1 2

( µ t

)2 .

Now I proceed to the case T � µ � √

t∆(tγ) γ

. In this parameter region

the plot from Fig. 2.20 takes the form shown in Fig. 2.21 (I remind that δ is

positive by definition). In this parameter region the approximation from the

previous section is inapplicable, so there will be no comparison of the results

from the two approaches. In this case the exponential factor comes from the

cosh’s with �1 in the argument. The value of the product of these two factors

is approximately constant for relevant values of δ and is equal to

4 exp

( − 1 T

√ 2 t∆(tγ)

γ

) .

Indeed, for µ � √

t∆(tγ) γ

and δ ∼ 1 2

( µ t

)2 , ∆ � δ and �1 ≈ t

√ 2(∆ + δ) ≈

t √

2∆. So I take the cosh factors with �1 out of the integral and consider

only the other two factors which give the pre-exponential factor in the drag

resistivity.

The contributions from III, IV, and V must be taken into account (see

Fig. 2.21). Those from III and V are equivalent and give the same result.

2.4 SEMICLASSICAL THEORY OF ELECTRON DRAG... 67

The contribution from IV turns out to be of the same order of magnitude.

Indeed, A(δ) in these three regions is given by (I leave only the relevant part,

arising from the layer 2):

III: − 1 T

( µ2 − t

√ 2δ ) , δ ≤ 1

2

(µ2 t − qγ

2

)2 ;

IV: − 1 T

( t qγ

2

) ;

V: − 1 T

( −µ2 + t

√ 2δ ) , δ ≥ 1

2

(µ2 t

+ qγ

2

)2 .

Expanding in δ around the borders of the regions III and V we get:

III: − 1 T

( tqγ 2 − t2

µ ∆δ )

;

V: − 1 T

( tqγ 2

+ t 2

µ ∆δ ) ,

where ∆δ is the difference between δ and the border of the corresponding

region. It is evident from the above that for T � µ the δ-integral converges in the bulk of the regions III and V and thus the above statement about the

equivalence of this regions is correct. Within IV, A is independent on δ and

the δ-integration is thus trivial and is determined by the size of the region

IV. Collecting the above one gets:∫ dδeA = exp

( −tqγ

2T

)( 2µT

t2 + qµγ

t

) , (2.108)

from which it is easy to see that the contributions from III, IV, and V are of

the same order as stated above. Finally the q-integral in (2.99) is taken to

get the final result for ρD:

ρD = 384pie2T 2µ

vF1vF2t 3γ

sgn (µ1µ2) exp

( − 1 T

√ 2 t∆(tγ)

γ

) ln2 ( DT

tγ

) . (2.109)

For µ ∼ √

t∆(tγ) γ

it is consistent with the case µ � √

t∆(tγ) γ

, T � T ∗. One can also note that the condition q2 � 12 ∆(tγ)

tγ2∆γ holds for q . T/tγ for

T � µ� √

t∆(tγ) γ

√ γ

∆γ .

68 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

This rounds off the consideration of the case µ � T . Now I turn to the other case, i.e. µ � T . For this I return to the Eq. (2.99) and expand both parenthesis up the first order in µ

T :

ρD = 4pie2µ1µ2 vF1vF2T 3

∫ dq K20 (Dq)tγq

× ∫

dδ 1

cosh �1(p01− q2 )

2T

1

cosh �1(p01+

q 2 )

2T

1

cosh �2(p02− q2 )

2T

1

cosh �2(p02+

q 2 )

2T

× (

tanh �1(p

0 1 − q2) 2T

+ tanh �1(p

0 1 +

q 2 )

2T

)( tanh

�2(p 0 2 − q2) 2T

+ tanh �2(p

0 2 +

q 2 )

2T

) .

(2.110)

Two limiting cases must be considered: T � √

t∆(tγ) γ

and T � √

t∆(tγ) γ

.

1) T � √

t∆(tγ) γ

.

The cosh’s stemming from the first layer, similarly to the case T � µ �

√ t∆(tγ)

γ , account for the exponential factor 4 exp

( − 1

T

√ 2 t∆(tγ)

γ

) . The

parentheses with tanh’s of �1 is equal to 2. So the δ-integral is reduced to∫ dδ

1

cosh t √

2δ+ tqγ 2

2T

1

cosh t √

2δ− tqγ 2

2T

( tanh

t √

2δ + tqγ 2

2T + tanh

t √

2δ − tqγ 2

2T

)

=

( 2T

t

)2 tqγ

2T sinh tqγ 2T

. (2.111)

The q-integral can now also be evaluated and yields the final result for the

drag resistance:

ρD = 1792ζ(3)pie2µ2T

vF1vF2t 3γ

sgn (µ1µ2) exp

( − 1 T

√ 2 t∆(tγ)

γ

) ln2 ( DT

tγ

) ,

(2.112)

with ζ(3) being the Riemann zeta-function.

2) T � √

t∆(tγ) γ

.

For such temperatures the ∆’s in the expressions for �1 can be neglected

and the δ-integral turns to∫ dδ

1

cosh2 t √

2δ+ tqγ 2

2T

1

cosh2 t √

2δ− tqγ 2

2T

( tanh

t √

2δ + tqγ 2

2T + tanh

t √

2δ − tqγ 2

2T

)2 .

(2.113)

2.4 SEMICLASSICAL THEORY OF ELECTRON DRAG... 69

This integral can not be evaluated analytically, but it is easily shown that the

answer for the drag differs from (2.112) only by the absence of the exponent

and by a different numerical factor:

ρD = C e2µ2T

vF1vF2t 3γ

sgn (µ1µ2) ln 2

( DT

tγ

) , (2.114)

with C ≈ 113.

I now summarize all the above. In the schematic T−µ (Fig. 2.22) diagram one can see different regions of parameters in each of which ρD has different

asymptotics:

µ

Τ

I

II III

IV

V

VI

PSfrag replacements

√ t∆(tγ)

γ

√ t∆(tγ)

γ

µ ∼ 1 T

µ ∼ 1 T 1/3

Figure 2.22: Regions of parameters T and µ for which ρD has different be-

havior.

70 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

I: ρD = 256pie2γ2T 2µ7

vF1vF2t 6[∆(tγ)]3

sgn (µ1µ2) exp

[ −t∆(tγ)

γµT

] ln2 ( D

Tµ2

t2∆(tγ)

) ;

II: ρD = 32pie2[∆(tγ)]3

3vF1vF2T (µγ 2)2

sgn (µ1µ2) exp

[ −t∆(tγ)

γµT

] ln2 ( D

∆(tγ)

γ2µ

) ;

III: ρD = 64pie2µT 2

vF1vF2t 3γ

sgn (µ1µ2) ln 2

( DT

tγ

) ;

IV: ρD = C e2µ2T

vF1vF2t 3γ

sgn (µ1µ2) ln 2

( DT

tγ

) ;

V: ρD = 448ζ(3)pie2µ2T

vF1vF2t 3γ

sgn (µ1µ2) exp

( − 1 T

√ 2 t∆(tγ)

γ

) ln2 ( DT

tγ

) ;

VI: ρD = 384pie2T 2µ

vF1vF2t 3γ

sgn (µ1µ2) exp

( − 1 T

√ 2 t∆(tγ)

γ

) ln2 ( DT

tγ

) .

2.4.2 Drag between non-parallel links.

For parallel links, energy and momentum conservation restrict the allowed

scattering processes very strongly. At low temperatures, this leads to an

exponential suppression of the drag between non-equivalent parallel links.

This has nothing to do with the exponential suppression of drag observed in

the experiments, since the links are generally not parallel. For non-parallel

links the sum of momenta on the two links is no longer conserved in the

scattering process. In this section of the work I will compute the drag between

non-parallel links.

Unlike the parallel links case, the interaction matrix element here can be

evaluated analytically. Indeed, the integral

∫ dx1dx2dy1dy2

ψ∗n0(x1, y1)ψnq1(x1, y1)ψ ∗ n′0(x2, y2)ψn′q2(x2, y2)

D(x1, x2, y1, y2)

with

D(x1, x2, y1, y2) = √ d2 + (y1 − y2 cos θ + x2 sin θ)2 + (x1 − y2 sin θ + x2 cos θ)2,

2.4 SEMICLASSICAL THEORY OF ELECTRON DRAG... 71

θ being the angle between the links, can be evaluated and yields:

V (q1, q2) = 2pie 2 √

2pi e−

dQ | sin θ|

Q e−

Q2l2H 2 sin2 θLn

[ Q2l2H

2 sin2 θ

]

× Ln′ [ Q2l2H

2 sin2 θ

] exp

[ i q22 − q21

2 l2H cot θ

] , (2.115)

with Q = √ q21 + q

2 2 − 2q1q2 cos θ. The last exponent in the matrix element

is insignificant as it vanishes after taking the square of the absolute value

of the matrix element. Above x1,2 and y1,2 respectively are the coordinates

perpendicular to and along the links 1 and 2; the zero point for the x1,2-axes

corresponds to the centers of the initial states and the zero-point of the y1,2-

axes corresponds to the crossing of the projections of the center lines of the

intial states along the z-axis (the direction of the magnetic field).

This expression should be inserted into (2.72). Proceeding as in the case

of parallel links we can get the expression for the drag resistance. The main

difference compared to the parallel links case is that instead of δ(q1 − q2) I get

exp (− dQ| sin θ|) Q

which in principle allows any couple (q1, q2) of momentum

transfers. This leads, as I will show, to nontrivial consequences, in particular

to a T 2 behavior for the lowest temperatures for any θ 6= 0. Before going on to the calculations, I will make some qualitative remarks. For any parameters

of the layers it is possible to choose q1,2 such that t1q1γ1 = t2q2γ2. From the

calculations in the previous section it is quite clear that the ω- (or δ-) integral

will have no exponential smallness for such q1,2. On the other hand there

might be an exponentially small factor stemming from the matrix element,

as for q1 6= q2, dQ/| sin θ| can be much larger than unity. So, as now becomes clear, the qualitative analysis will consist in comparing the two exponents,

one stemming from the matrix element, and the other is the one — known

from the previous section — coming from the difference in Fermi velocities

in the two layers.

Another point that should be mentioned is the sign of the drag sig-

nal, which can be determined without any additional calculations. Indeed,

Eq. (2.75) will have q1q2 in the integrand instead of q 2 y. The sign of this

72 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

product has a vast effect on the value of the interaction matrix element: if

cos θ > 0 then Q for q1q2 > 0 is smaller than Q for q1q2 < 0 provided the

absolute values of q’s are the same. And vice versa. As Q enters the matrix

element in the exponent, it is clear that for cos θ > 0 the q’s from the 1st and

the 3rd quadrants dominate, and for cos θ < 0 it is the q’s from the 2nd and

the 4th quadrants. For perpendicular links the drag is, of course, identically

zero. The only other factor affecting the sign of the result is still the factor

sgn (µ1µ2).

The above clearly shows that each pair of links from different layers of a

real 2D sample gives contribution to the drag of the sign determined only by

the relative sign of the local dispersions on the Fermi level. This, in turn,

leads to the experimentally observed overall sign of the drag in a 2D sample.

Now to the quantitative analysis. Let us follow the calculations from the

parallel links case and mention the differences. First in (2.72) I substitute

(2.115) instead of (2.74). It leaves the factor 2pi/L which was canceled for

parallel links. L is the length of the link which corresponds to the correlation

length of the 2D random potential. In (2.75) changes are evident: qy in the

arguments of �i are changed to qi, the integration is done over q1,2 instead of

qy and the factor 2pi/L is still there.

I follow the line of calculations presented as ”second” method. I will

not keep trace of all numerical factors as some of the calculations below are

only estimates and any numerical prefactor would not be reliable, rather I’ll

concentrate on the qualitative features. First I will look at the δ-integral and

will determine its behavior as a function of q1,2, then the q1,2-integrals will

be estimated.

For estimating the δ-integral only some redefinitions should be made,

in principle all the calculations are already done. The crucial point is the

redefinition of ∆, which should now be defined as

∆ = ∆(tγq)

tγq ,

2.4 SEMICLASSICAL THEORY OF ELECTRON DRAG... 73

with ∆(tγq) = |t1γ1q1 − t2γ2q2|, rather than

∆ = ∆(tγ)

tγ

with ∆(tγq) = |t1γ1q1 − t2γ2q2|. This can be seen by rewriting the formulae (2.95)-(2.102) allowing for different q’s.

The obvious consequence is that the result of the δ-integration has the

exponential part

exp

[ − 1 T

t∆(tγq)

γµq

] , (2.116)

and is not exponentially small for

t∆(tγq)

Tγµq ∼ 1. (2.117)

I consider the case t∆(tγ)

Tγµ � 1, (2.118)

as it is more interesting and allows to demonstrate interesting features of the

crossed links. Conditions (2.117),(2.118) imply

∆q

q ≈ −∆(tγ)

tγ , (2.119)

and as can be seen from (2.117), ∆q/q must be close enough to these value:∣∣∣∣∆qq + ∆(tγ)tγ ∣∣∣∣ . µTt2 . (2.120)

The variation of ∆q/q is small compared with its absolute value if T � t∆(tγ) γµ

,

which is the same condition that is necessary for the activated behavior of

ρD for parallel links. The corresponding region is referred to in Fig. 2.23 as

1. The region 2 in this Fig. shows the values q1,2 for which dQ/| sin θ| . 1. The result of the δ-integration (for µ� T ) is obtained as the Eq. (2.90)

with appropriate changes. As I’m not interested in numerical prefactors, I

can write for it µT

t2 e−qtγ/T .

74 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

q

q 2

1

2

1

Figure 2.23: regions of the q1q2-plane which give main contributions to the

drag. Region 1 shows where the ω-integral is not exponentially small and

region 2 shows where the interaction matrix element has a significant value.

For details see the text.

Instead of q1,2 I introduce ∆q = q1 − q2 and q = (q1 + q2)/2. For the drag resistivity I can write:

ρD ∼ e 2

LvF1vF2T

∫ dqd∆q

e−Qd/| sin θ|

Q2 tqγ

µT

t2 e−qtγ/T , (2.121)

with

Q2 = ∆q2 cos2 θ

2 + 4q2 sin2

θ

2 .

For θ � 1 the last expression can be rewritten as Q2 = ∆q2 + q2θ2. (2.122)

The region of (q,∆q)-plane that gives rise to the main contribution to the

drag signal is given by (2.119,2.120). From here one can conclude that for

θ � ∣∣∣∆(tγ)tγ ∣∣∣ it is the first term in (2.122) that dominates, while for larger θ’s

it is the second term.

Looking at Fig. 2.23 one can notice that for sufficiently small temper-

atures the q-integral will converge inside the region 2 due to the exponent

exp (−qtγ)/T . This happens for

T . θ

d

(tγ)2

∆(tγ) .

2.4 SEMICLASSICAL THEORY OF ELECTRON DRAG... 75

For such temperatures and smallest θ’s, (2.121) can be rewritten as

ρD ∼ e 2

LvF1vF2T

∫ dqd∆q

(tγ)2

q2[∆(tγ)]2 tqγ

µT

t2 e−qtγ/T . (2.123)

(∆q has been substituted from (2.119)). The ∆q-integral is determined by

the width of region 1 in Fig. 2.23, which is given by (2.120). The q-integral

afterwards is trivial. The result for the above-mentioned parameters is

ρD ∼ e 2µ2T 2γ2

LvF1vF2t2[∆(tγ)]2 . (2.124)

For µ� T there are no principle changes in the way the calculations are done. As for parallel links I expand the integrand in µ/T . The resulting δ-integral

for q’s obeying (2.119,2.120) is the same as in Eq. (2.113). The result —

as in the case of parallel links — differs from (2.124) only by changing one

power of T to one power of µ:

ρD ∼ e 2µ3Tγ2

LvF1vF2t2[∆(tγ)]2 . (2.125)

For T & θ d

(tγ)2

∆(tγ) , as can be seen from Fig. 2.23 the q-integral is cut off at

θ d

tγ ∆(tγ)

� 1 d . This leads to multiplying the above results by θ

d tγ

∆(tγ) tγ T � 1

d tγ T

:

ρD ∼ e 2µ2Tγ4

LvF1vF2[∆(tγ)]3 θ

d , for µ� T (2.126)

and

ρD ∼ e 2µ3γ4

LvF1vF2[∆(tγ)]3 θ

d , for µ� T. (2.127)

The above results are valid for lowest temperatures, when it is possible

to neglect the contribution from the regions of the (q1, q2)-plane where the

integrand is exponentially small. With growing temperature the contribution

from these regions grows and for high enough T becomes bigger than the

contribution from the region 1 of the Fig. 2.23. Now I will make quantitative

estimates to substantiate these qualitative arguments. As stated above, I

will not keep any numerical pre-factors. I also omit the factor sgn (µ1µ2) as

well as sgn (cos θ) which was discussed above. In particular, I’ll concentrate

76 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

on the case of very small θ which is interesting from the point of view of

theoretical understanding of the drag between non-parallel links. The drag

resistivity is given by

ρD ∼ e 2

vF1vF2TL

∫ dqdkdδ

Q2 tq2γ

× e −Qd/| sin θ|

cosh �1(p01−

q1 2

)−µ1 2T

cosh �1(p01+

q1 2

)−µ1 2T

cosh �2(p02−

q2 2

)−µ2 2T

cosh �2(p02−

q2 2

)−µ2 2T

= e2

vF1vF2TL

∫ dqdk

Q2 tq2e−Qd/| sin θ|γB(q, k). (2.128)

Here q = (q1 + q2)/2, k = (q1 − q2)/q. Then Q2 ≈ q2(k2 + θ2). Using the intermediate results from the previous section we can write for

the δ-integral which I denote B(q, k):

B(q, k) =

1) e− t2∆ Tµ Tµ3

t4∆ e−

t2∆ Tµ

qγt µ , T .

t4

µ3 ∆2;

2) e− t2

Tµ ∆ ( ∆− µqγ

t

) , q <

∆t

µγ , t4

µ3 ∆2 . T .

t2

µ ∆;

3) µT

t2 e−

qtγ T ,

t2

µ ∆ . T.

(2.129)

Here ∆ = ∣∣∣∆(tγ)tγ + ∆qq ∣∣∣ ≡ |∆0 + k|.

Next the q-integration can be evaluated. It is done differently for the

cases 1), 2), and 3) (see above). In fact, these cases correspond to different

values of k (we concentrate on k’s between −∆0 and 0 as it is clear that only these values can give significant contribution):

1) −∆0 + µ t2

√ Tµ . k;

2) −∆0 + Tµ t2

. k . −∆0 + µ t2

√ Tµ;

3) |∆0 + k| . Tµ t2 .

(2.130)

The result is given by:

ρD ∼ e 2

LvF1vF2T

∫ dkY (k) (2.131)

2.4 SEMICLASSICAL THEORY OF ELECTRON DRAG... 77

k-Θ

YHkL 3L 2L 1L

PSfrag replacements

−∆0

Figure 2.24: Typical plot of Y (k). The definition of regions 1)-3) is given

by (2.130); the depicted situation corresponds to the case considered in the

text, when k = 0 belong to the region 2).

with

Y (k) =

1) e− t2

Tµ |∆0+k| Tµ

3

t4|∆0 + k| tγ

k2 + θ2 min

( |θ| d √ k2 + θ2

, µ2T

γt3|∆0 + k| )

;

2) e− t2

Tµ |∆0+k| tγ

k2 + θ2 |∆0 + k|min

( |θ| d √ k2 + θ2

, |∆0 + k|t

γµ

) ;

3) µγT

t∆20 min

( θ

∆0d , T

tγ

) .

For lowest temperatures the region 3) dominates. I will now look at what

happens for growing temperatures. For this I will look at the behavior of the

function Y (k) for different T and compare the integrated contribution to the

drag signal from different regions. First of all I notice that in the case 3) it is

the first option in the min’s argument that must be realized, otherwise it is the

region 3) that dominates, and this has been studied above. So I concentrate

on the case when the q-integration in 3) is limited by the exponent from the

interaction matrix element and not by the temperature. At some value of k

the situation changes and the option in the min-function switches. A typical

plot of Y (k) can be seen on fig. 2.244.

One has to distinguish two cases: k = 0 corresponds to 2) and to 1).

These cases, as can be easily seen from (2.130) are determined by the same

4In fact the shown plot is rather specific. A slight change in the parameters of the plot

immediately changes its character: one of the two peaks becomes dominating. This fact

justifies the logarithmic accuracy calculations presented below.

78 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

restrictions for the temperature as those that limit the regions I and II from

Fig. 2.22: if T . t 4

µ3 ∆20 then k = 0 is in the region 1) and if

t4

µ3 ∆20 . T .

t2

µ ∆0

it is in the region 2). For simplicity I only consider the latter case, as it will be

clear that the difference between them is purely calculative and considering

the former case would not teach us anything new.

For |k| & |θ|, k2 + θ2 in the expression for Y (k) can be replaced by k2 if I do not wish to calculate the numerical pre-factors. For k from 2) one can

write:

Y (k) = e− t2

Tµ |∆0+k| tγ

k2 |∆0 + k|min

( |θ| d|k| ,

|∆0 + k|t γµ

) . (2.132)

For k ∼ θ it crosses over to

Y (k) = e− t2

Tµ ∆0 tγ

θ2 ∆0 min

( 1

d , ∆0t

γµ

) . (2.133)

The minimum of the first expression is reached for |k| = 3µT t2 � ∆0, for

smaller absolute values of k, Y (k) begins to grow and reaches its maximum

at k = 0 and the width of this peak is ∼ θ. So I can estimate the contribution to ρD from all k’s outside of 3) as

ρD ∼ e 2

LvF1vF2T e−

t2

Tµ ∆0 tγ

|θ|∆0 min (

1

d , ∆0t

γµ

) . (2.134)

The contribution from the region 3) is given by the Eq. (2.126) which can be

rewritten as

ρD ∼ e 2

LvF1vF2T

µ2T 2γ

t3∆30

|θ| d .

Comparing the two contributions one can find with logarithmic accuracy that

for

T & t2∆0

µ ln ∆0|θ| (2.135)

the activated behavior stemming from the region where |k| . |θ| dominates, while in the broad region of smaller T ’s one would observe a power law

T -dependence. I do not consider in detail the whole variety of possible pa-

rameter relations as it would not lead to any qualitatively new results. The

2.4 SEMICLASSICAL THEORY OF ELECTRON DRAG... 79

main result: logarithmic dependence of the cross-over temperature on θ does

not change.

Now, for some new insight, I look at the above from a different point of

view: I fix T and consider ρD as a function of θ. Comparing (2.134) with

(2.126) again I find that the first one dominates for

θ . ∆0 t2∆0 µT

e− t2∆0 2µT .

For larger angles, but still θ � ∆0 it is the power dependance contribution (2.126) that dominates. For even larger angles, 1 � θ � ∆0, k � θ every- where in the relevant regions. That means that the expression (2.131) holds

with the change of the first option in all min-functions in the expressions for

Y (k) to 1/d, and, clearly k2 + θ2 ≈ θ2. Thus it becomes evident that Y (k) reaches its maximum for |∆0 + k| = 0 and the width of the peak is ∼ Tµ2 . So, for 1� θ � ∆0:

ρD ∼ e 2

LvF1vF2T

µ2T 2γ

t3∆20 min

( 1

d , T

tγ

) . (2.136)

So, in agreement with (2.135), no exponential behavior is expected in any

temperature range. In other words, already a relatively small non-parallelicity

is sufficient to kill any trace of the activated behavior, characteristic for the

parallel links case. This means that this behavior, as was mentioned above,

is just an artefact of the degenerate case of parallel links and it would have

no effect after averaging over real two-dimensional samples, where θ’s take

all possible values randomly.

Finally I touch the question of the limit θ −→ 0. Here the essential point is the size of the sample L. For parallel links of finite length, q1 and q2 can

differ from one another by the value ∼ 1/L; for nonparallel links this value is θ/d. Thus the cross-over value of θ is d/L. For such angles Eq. (2.115)

is, as can be easily checked, no longer valid and for smaller angles it crosses

over to Eq. (2.74). The last sentence should be understood in the sense

that the expression (2.115) as a function of ∆q tends to the δ-function with

θ −→ 0 and the integrals over ∆q of both expressions coincide for small θ’s.

80 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

As I have shown, for small θ’s (or for large enough T ’s) the exponent in the

expression for ρD is recovered. I failed to find a transparent way to see how

the pre-factors of the expressions for the drag for parallel and non-parallel

links cross-over to each other for θ ∼ d/L. More over I am quite convinced that to do this I need to obtain an analytic expression for the matrix element

of the interaction for these angles which is definitely impossible. Still I’m

sure that the arguments presented in this paragraph give enough insight to

understand how the cross-over works.

2.4.3 Connection to two-dimensional drag.

In the previous parts of the work I have presented a detailed theory of the

drag between two one-dimensional chiral spinless electron systems with an

arbitrary angle between them. Spin can be easily included in the above

picture. Since the interlayer interaction is spin independent, one simply has

to sum over the two spin species (up and down) in both drive and drag

layer, taking the (exchange enhanced) Zeeman spin splitting of the Landau

levels into account. If the Fermi level of one layer lies between the centers

of the highest occupied Landau levels for up and down spins, respectively,

positive and negative contributions to the drag partially cancel each other.

The cancelation is complete due to particle-hole symmetry in the case of odd

integer filling, as observed in experiment.

As was mentioned in the beginning of the section 2.4, to get the result for

the two-dimensional drag, knowing the results for the one-dimensional drag,

one should in principle sum up the contributions from all crossing points

between the links of the two random networks. The full accomplishing of

this task presents a separate challenge and is not yet done. Still, many

things can be seen already from simple arguments that have already to a

large extent been mentioned above and that I recollect here.

Within our semiclassical picture anomalous drag, especially negative drag,

is suppressed at temperatures above the Landau level width, because then

electron- and hole-like states within the highest occupied level are almost

equally populated. This agrees with the results from the Born approximation

[18], and also with experiments.

2.5 CONCLUSION 81

It is plausible that localized states give small — if any — contribution

to the drag. Indeed, such states represent electrons drifting around hills or

valleys of the random potential and drag between two such states is formally

zero as the contributions from the crossing points between them exactly

cancel. The only way such localized states can affect the drag is if an electron

scatters from such a state to a delocalized state, drifts along it away from

its initial position and then scatters back to another localized state spatially

separated from the first one. These processes are not forbidden as there is no

energy conservation for a separate layer, so, unlike the one-layer transport

problem, we can not restrict the role of these states to a reservoir for the

delocalized ones. It is not clear whether the contribution of such processes

to the drag is significant, still, it is clear that the delocalized states play a

special role in the overall drag. Their contribution is largely determined by

the Fermi filling factor at the percolation energy. If the Fermi level does

not hit any extended states (for either spin species), the drag should vanish

exponentially for T −→ 0, since thermal activation or scattering of electrons into extended states is then suppressed by an energy gap. By contrast, for

a Fermi level within the extended states band (for at least one spin species)

the gap vanishes and the drag obeys generally quadratic low temperature

behavior, as obtained for the drag between non-parallel links.

As to the sign of the drag, it is negative if one of the layers is more

than half-filled and the other is less than half-filled, and positive in other

cases. This follows from the results for the sign of the drag between two

one-dimensional links that have been obtained above.

2.5 Conclusion

In summary, I have presented a semiclassical theory for electron drag between

two parallel two-dimensional electron systems in a strong magnetic field,

which provides a transparent picture of the most salient qualitative features

of anomalous drag phenomena observed in recent experiments [2, 3, 13]. Lo-

82 CHAPTER 2. THEORY OF COULOMB MAGNETO-DRAG

calization plays a role in explaining activated low temperature behavior, but

is not crucial for anomalous (especially negative) drag per se. In particular I

have elaborated a detailed theory for the drag between two one-dimensional

links (sections 2.4.1, 2.4.2). The temperature dependence of the drag has

been derived for a whole variety of the parameters of the system as well as

for parallel links and for non-parallel ones. I have also presented qualitative

arguments to show to what consequences for the two-dimensional drag my

results lead (section 2.4.3). A quantitative theory of drag which covers the

whole range from low magnetic fields, where the Born approximation is valid,

[21, 18] to high fields, where localization becomes important, remains an im-

portant challenge for work to be done in the future. A promising path here is

a very general diagrammatic approach for long-range disorder systems which

starts from the equation of the type (2.12). The averaging over disorder is

done not for a single Green’s function as usually ([16, 18]) but for the triangle

function Γ as a whole and the Green’s functions are taken in their exact form

for a given random potential realization. This work is in progress now.

Chapter 3

Possible Jahn-Teller effect in

Si-inverse layers

3.1 Introduction

In a symmetric configuration of atoms, the electronic state may be degenerate

due to the high symmetry of the Hamiltonian. H. A. Jahn and E. Teller

proved that such a symmetric configuration is unstable against a deformation

lowering the overall symmetry of the Hamiltonian. The Jahn-Teller effect is

well accounted for in molecules and crystals [31, 32]. The kinetic energy of

electrons in the 2D electron gas (2DEG) under the conditions of quantum

Hall effect is quenched. Electron states become macroscopically degenerate

forming Landau levels. Interaction may remove some of this degeneracy.

Nevertheless, one finds the global valley degeneracy at integer fillings1. In

this chapter we raise the question whether a lattice deformation — Jahn-

Teller effect — lifts this valley degeneracy.

The theory of the 2DEG at integer ν uses [33, 34] the Hartree-Fock ap-

proximation in the limit of the small parameter: Ec/~ωc, where Ec = e 2/lH

is the energy of the Coulomb interaction and ωc is the frequency of the cy-

clotron resonance. At ν = 1 the theory predicts a ferromagnetic ground

1If we neglect the Zeeman splitting this degeneracy turns to a spin-valley one.

83

84 CHAPTER 3. POSSIBLE JAHN-TELLER EFFECT...

state with degenerate uniform spin orientation. The elementary excitations

are electron-hole pairs or neutral excitons, which correspond to gapless and

noninteracting [35] spin-waves for vanishing momentum and vanishing effec-

tive g-factor. In the limit of large exciton momentum the electron and the

hole become independent charged excitations. A special topological spin tex-

ture in a 2D ferromagnet called ”skyrmion” [36] has unit charge and half of

the quasiparticle energy [37]. In this chapter such textures are considered in

the presence of a Jahn-Teller effect.

In bulk Si there is a six-fold valley degeneracy of low energy electron states

in the conduction band. It is reduced to two-fold degeneracy for the (100)

orientation of the 2D-plane due to the transverse quantization of electron

motion in a quantum well [38]. Even for the (111) orientation of the 2D-plane

with six equivalent valleys only two valleys connected by the time reversal

symmetry are actually occupied in a strong magnetic field [39]. This effect is

governed by the anisotropy of the effective mass of electrons and corresponds

to a spontaneous breaking of valley symmetry. Here we study the effects

related to this valley degeneracy and show that such a bivalley system is

similar to the bilayer systems. The bilayer setup has been extensively studied

theoretically for ν = 1 [40, 41] and ν = 2. [42, 43] The layer degeneracy of

electron states can be described by a four component spinor in combined

spin-layer space. In former works [44, 45] the total 2DEG Hamiltonian was

subdivided into a symmetric part and a small anisotropic part which reduces

the symmetry. The symmetric part is invariant under a SU(4) group of

electron spinor rotations in the combined spin and layer Fock space. This

SU(4) group has the apparent SU(2)⊗SU(2) subgroup of separate rotations in spin and valley spaces.

The electron annihilation operator can be expanded using one electron

orbital functions:

ψα(~ρ) = ∑

τ

ψατ (~r)χ(z)e iτQz/2, (3.1)

where α is the spin index, z is the coordinate perpendicular to the 2D plane

and ~r is the in-plane coordinate vector. The index τ = ±1 numerates the two valleys. Valley wave functions of electrons in a Si inverse layer:

3.2 HAMILTONIAN OF ELECTRON GAS ON SI INTERFACE 85

χ(z) exp(±iQz/2), are normalized and almost orthogonal for smooth, real χ(z) with the negligible overlap

∫ χ2(z) exp(iQz)dz [38]. Q is the shortest

distance between the valley minima in reciprocal space, equals approximately

to 2/aSi, with aSi being the lattice constant. ψατ (~r) is the electron wave func-

tion of in-plane motion in valley τ with spin α and constitutes a four compo-

nent spinor. Electrons in the system will strongly interact with phonons with

momentum ±Q, giving rise to scattering from one valley to another. The Jahn-Teller effect (JTE) in Si-inverse layers corresponds to a displacement

of silicon atoms in z-direction. The new equilibrium is determined from the

balance of electron-phonon and elastic energies. A strong magnetic field is

essential for JTE. Indeed, in its absence a coherent JT state would lead to an

appearance of one Fermi-sphere instead of two smaller ones for two different

valley states. This would lead to an increase of the electrons’ kinetic energy

that makes the JTE energetically unprofitable. In the following sections we

discuss the possible JTE in detail.

This chapter is structured as follows. In section 3.2 we introduce the

Hamiltonian of the system and subdivide it into SU(4)-symmetric and -asym-

metric parts. Section 3.3 is devoted to a discussion of the SU(4)-symmetric

case. In the sections 3.4, 3.5 the anisotropic terms in the Hamiltonian are

included and the ground state of the system for different parameters is found.

In section 3.6 the effect of the anisotropic terms on the energy of the skyrmion

is considered. And finally a brief review of the results of this chapter is

presented in the conclusion.

3.2 Hamiltonian of electron gas on Si inter-

face

The Hamiltonian of the Si inverse layer describes the electron system inter-

acting with phonons:

Hˆ = Hˆe + Hˆe−ph + Hˆph. (3.2)

For a narrow quantum well, the Hamiltonian of the 2DEG in a magnetic field

can be expressed in terms of a spinor ψατ (~r) and reads (we use the system of

86 CHAPTER 3. POSSIBLE JAHN-TELLER EFFECT...

units where: ~ = 1, e = c, B = 1 and the magnetic length lH = √ c~/eB = 1)

Hˆe =

∫ ψ†ατ (~r)

( 1

2m

[ −i~∇ + ~A

]2 + gµB ~B~σ

) αβ

ψτβ(~r)d 2~r

+ e2

2κ

∫ ψ†ατ1(~r1)ψ

† βτ2

(~r2)ψβτ2(~r2)ψατ1(~r1)

|~r1 − ~r2| d 2~r1d

2~r2, (3.3)

where g is the conduction band gyromagnetic ratio, κ is the dielectric con-

stant of Si, µB is Bohr magneton and ~B is the uniform magnetic field in

z-direction. Here and in the following summation over repeated greek in-

dices is assumed. We neglect small terms with different valley indices in the

electron density operator and we use the Debye model for phonons with large

momenta ±Q

Hˆph[uz] = ρSi 2

∫ [ (∂tuz)

2 + c2 ( ~∇uz

)2] d3~ρ, (3.4)

where uz(~ρ) is the lattice displacement in z-direction. We consider only the

valley-mixing electron-phonon interaction:

Hˆe−ph = Θ ∫ ψ†ατ1(~r)τ

± τ1τ2

ψατ2(~r)×(∫ χ2(z)e±iQz∇zuz(~ρ)dz

) d2~r, (3.5)

with Θ and ρSi being the deformation potential and the density of Si re-

spectively. Here and in the following c is the speed of sound. τ µ and σµ,

with µ = x, y and z, are the Pauli matrices in valley and spin spaces and

τ± = (τx ± iτ y)/2. ~A(~r) = (0, Bx) is the vector-potential in Landau gauge. In a magnetic field we expand the electron operator over Landau orbital

states φn,p(~r) in the Landau gauge:

ψˆατ (~r) = ∑ n,p

φn,p(~r)cˆnp,ατ , (3.6)

where c†np,ατ and cnp,ατ are electron creation and annihilation operators in

the τ valley with spin α, n enumerates Landau levels and the continuous

parameter p specifies states within one Landau level.

3.2 HAMILTONIAN OF ELECTRON GAS ON SI INTERFACE 87

We assume that the electrons are confined to the lowest Landau level

n = 0 in the ground state. The electronic part of the Hamiltonian contains

the Coulomb and the kinetic energy in the lowest Landau level:

Hsym = 1

2m

∑ p

c†pατcpατ + 1

2

∫ d2~q

(2pi)2 V (~q)N(~q)N(−~q), (3.7)

where

V (~q) = 2pie2/κq (3.8)

is the 2D Fourier transform of the Coulomb interaction. Notice that in our

system of units 1/m is the cyclotron frequency. The electron density operator

is:

N(~q) = ∑

p

c†pατcp−qy ατe −iqx(p−qy/2)−q2/4. (3.9)

Unitary transformations of electron operators

cpατ1 = Uατ1,βτ2cpβτ2 (3.10)

in the combined spin and valley space leave the Hamiltonian (3.7) invariant

if U is a matrix from the SU(4) Lie group. For the Landau level filling factors

ν = 1, 2, 3 (ν = 4 corresponds to a fully filled zeroth Landau level) we assume

that the ground state is uniform with electrons of spin αi and valley τi filling

every orbital of the lowest Landau level:

Ψ(α1τ1 . . . αντν) =

ν∏ i=1

∏ p

c†pαiτi |vac〉. (3.11)

One can check that any such wave-function (3.11) represents an eigenfunc-

tion of Hsym Eq. (3.7). The state (3.11) is degenerate and a set of related

eigenstates can be generated by applying uniform rotations U .

The remaining terms in the Hamiltonian (3.2) arise from a valley splitting

term due to a singularity of the well potential on the Si/SiO2 interface [38],

Zeeman term, electron-phonon interaction and the phonon energy:

Han = −t ∑

p

c†pατ1τ x τ1τ2cpατ2

− |g|µBH ∑

p

c†pατσ z αβcpβτ +He−ph +Hph, (3.12)

88 CHAPTER 3. POSSIBLE JAHN-TELLER EFFECT...

where t is the phenomenological valley-splitting constant. This part breaks

the SU(4) symmetry and lifts the degeneracy of the eigenstates of SU(4)-

symmetric Hamiltonian (3.2). The splitting of energy levels is determined by

small matrix elements of the anisotropic Hamiltonian (3.12) projected onto

a linear space of the symmetric Hamiltonian (3.7) level degeneracy. There

is no renormalization of the anisotropic Hamiltonian parameters (3.12) due

to electron-electron interaction in the symmetric part Eq. (3.7). Thus the

Hartree-Fock approach is valid up to O(Han/ωH) terms for ν = 1, 2, 3.

The total Hamiltonian Hsym + Han, Eqs. (3.7,3.12), can be treated as

in the theory of magnetism and can be expressed in terms of an order pa-

rameter matrix Qˆ(~r). According to the Goldstone theorem the symmetric

part Eq. (3.7) can be expanded in powers of spatial derivatives of the order

parameter: ~∇Qˆ(~r), whereas the anisotropic part Eq. (3.12) depends on the local value of the order parameter in the leading order.

3.3 SU(4) Symmetric Case

Non-homogeneous states can be generated from the ground state (3.11) by

slow rotations, with the effective action being dependent on the rotation

matrix U(t, ~r):

S[U ] = −i Tr log ∫ DψDψ† exp

( i

∫ Ldt ) , (3.13)

where the spin-valley symmetric Lagrangian of the 2DEG is

L = ∫ ψ†α

[ i ∂

∂t − 1

2m

( −i~∇ + ~A0 + ~Ω

)2] αβ

ψβ d 2~r

− ∫ ψ†αΩ

t αβψβ d

2~r + 1

2

∑ ~q

V (~q)N(~q)N(−~q), (3.14)

where Ωt = −i U †∂tU and ~Ω = −i U † ~∇U . We treat the nonhomogeneous ma- trix U as a classical field and therefore the effective action (3.13) describes a

macroscopic motion of the corresponding spin-texture. The essential points

3.3 SU(4) SYMMETRIC CASE 89

here are the slow variation of U(~r) on the microscopic scale given by the

magnetic length and the use of the gradient expansion assuming a locally

ferromagnetic ground state. The effective action (3.13) for multivalley sys-

tems was found in the work of Arovas et al. [46] We derive it here following

the method of Ref. [47].

The Hamiltonian corresponding to (3.14) with ~Ω = 0 has a uniform

ground state (3.11). We consider three filling factors: ν = 1 with Ψ(↑ +1), ν = 2 with Ψ(↑ +1, ↓ −1) and ν = 3 reducing to the case ν = 1 under the electron-hole transformation. The reference state is alternatively given by an

electron occupation diagonal matrix: Nατ1 ,βτ2 = (1, 0, 0, 0) in the case ν = 1,

and Nατ1 ,βτ2 = (1, 1, 0, 0) in the case ν = 2. The Green function for electron

propagation in the lowest Landau level evaluated over the ground state reads

G0ατ1,βτ2(�, p) = Nατ1,βτ2

� + E0 + µ− i0 + (1ˆ−N)ατ1 ,βτ2 � + µ+ i0

, (3.15)

where µ = −E0/2 is the chemical potential and exchange constants in the limit of vanishing thickness of the electron layer are

E0 = 2E1 =

√ pi

2

e2

κlH . (3.16)

The effective action is S[Ω] = S0[Ω]+S2[Ω], where S0[Ω] = iTr log(G/G0),

whereas S2[Ω] is represented by the two diagrams on Fig. 3.1. The first

order perturbation correction to the action Eq. (3.13) depends on the Green

function of electron propagation in the first excited Landau level [47]:

G1ατ1,βτ2(�) = Nατ1,βτ2

�− ωc + E1 + µ+ i0 + (1ˆ−N)ατ1 ,βτ2 �− ωc + µ+ i0 . (3.17)

The two terms of the Hamiltonian depending on the gradient matrix field

are:

H1 = 1

2m

∫ ψ†ατ1

( Ω+Πˆ− + Πˆ+Ω−

) αβτ1τ2

ψβτ2 d 2~r, (3.18)

H2 = 1

2m

∫ ψ†ατ1

( ~Ω2 − �µν∂µΩν

) αβτ1τ2

ψβτ2 d 2~r, (3.19)

where Ω± = Ωy ∓ iΩx, and the differential operators Πˆ± shift an electron to the adjacent Landau levels: Πˆ−φnp(~r) =

√ 2nφn−1p(~r), Πˆ+φn−1p(~r) =

90 CHAPTER 3. POSSIBLE JAHN-TELLER EFFECT...

1

H1 ba

H1H

Figure 3.1: Second order (a) and the anomalous first order (b) diagrams.

Solid and wavy lines represent the electron Green functions and the Coulomb

interaction respectively.

√ 2nφnp(~r), though only the n = 0 and n = 1 Landau level states are relevant

for this problem. An expansion of the 2DEG action up to the second order

of Hamiltonian (3.18) gives:

S0[Ω] = iTr (H1G0) + i

2 Tr (H1G0H1G0) + iTr (H2G0) . (3.20)

The Hartree-Fock diagram in Fig. 3.1a has been calculated in Ref. [47]

whereas the anomalous diagram in fig. 3.1b is calculated in Appendix A.

The resulting effective Lagrangian is

Leff [Ω] = ∫ (

Tr(Ωt(t, ~r)Nˆ) + E0 + E1

2 �µν∇µΩzν −

− E1 2

Tr [ NΩ+(t, ~r)(1ˆ−N)Ω−(t, ~r)

] )d2~r 2pi

, (3.21)

where Ωzµ = −iTr ( NU †∂µU

) . The matrices N and 1ˆ − N are projecting

operators onto the physical rotations that form a subset of the SU(4) Lie

group. The matrix field Ωµ can be expanded in the basis of fifteen generators

of the SU(4) group: {Γl}, with l = 1..15. They form two complementary sets: the first even set includes those generators that do commute with N ,

whereas the second odd set includes the remaining generators. Generators

of the even set constitute an algebra themselves. This algebra has a normal

3.3 SU(4) SYMMETRIC CASE 91

Abelian subalgebra formed by a single traceless generator: N − ν/4. A Lie group generated by the even set is a stabilizer sub-group S of the SU(4) group. The odd set contains an even number of generators: eight in the case

of ν = 2 and six in the case ν = 1, 3.

The Hamiltonian is invariant under time reversal symmetry. The time

reversal operation transforms the rotation matrix as U → U ∗, the gradient field as ~Ω → −~ΩT and also inverts the magnetic field Bz. Accordingly, we rewrite the energy in the Lagrangian (3.21) in a time reversal symmetric

form:

Eeff [Ω] =

∫ d2~r

2pi

[E1 2

Tr ( NΩµ(1ˆ−N)Ωµ

)− − E0

2 sgn(Bz)�µν∇µΩν

] , (3.22)

where the identity �µνTr (ΩµΩνN) = i�µν∇µΩzν is used. It follows immedi- ately that Eeff ≥ 0. The first term is the gradient energy whereas the second term is proportional to the topological index of the spin texture:

Q = ∫ �µν∇µΩzν

d2~r

2pi = Z, (3.23)

where Z is an integer. The states with Z 6= 0 are called skyrmions after T. H. R. Skyrme who has first considered such textures. The case Q = ±1 corresponds to the simplest spin skyrmions in the first valley which can be

rotated by a SU(4) matrix to become a general bivalley skyrmion. The spin

stiffness in Eeff Eq. (3.22) coincides identically with that of the one-valley

case [47]. This means that the bivalley skyrmion energy is the same as that

found for one valley. The charge in the bivalley skyrmion core is distributed

over the two valleys with long divergent tails: n±1(~r) ∼ ±1/ (R2 + r2), if one neglects anisotropy. But the total charge of the two valleys follows a

convergent distribution identical to the charge density in the one valley case

[37]:

n(~r) = �µν∇µΩzν(~r)

2pi =

R2

pi (R2 + r2)2 , (3.24)

where R is the radius of the skyrmion core.

92 CHAPTER 3. POSSIBLE JAHN-TELLER EFFECT...

The expectation value of any operator A in the ground state 〈A〉 = Tr(AQˆ) can be expressed in terms of the order parameter matrix

Qˆ(~r) = U(~r)NU+(~r). (3.25)

Obviously rotations from the small sub-group S leave the order parameter invariant. Thus, rotations in Eq. (3.25) can be restricted to a physical space

of the bivalley 2DEG which is the complex Grassmannian manifold GC4ν =

U(4)/U(ν)⊗ U(4− ν). We rewrite Eq. (3.22) in terms of Qˆ:

Eeff [Qˆ] = E1 4

∫ Tr ( ~∇Qˆ~∇Qˆ

) d2~r 2pi − E0

2 Q. (3.26)

In this representation the topological index Q is an index of a map from the 2D plane onto the coset space of the order parameter. The rule (3.23) is a

consequence of the homotopy group

pi2 ( GC4ν )

= Z, (3.27)

proven in Appendix B.

Varying the effective Lagrangian consisting of the first kinetic term Eq. (3.21)

(rewritten as the Wess-Zumino term [46]) and the energy Eq. (3.26), we find

the matrix Landau-Lifshitz equation[ Qˆ ∂tQˆ

] −

= E1 2 ~∇2Qˆ. (3.28)

It describes three (ν = 1) and four (ν = 2) degenerate ”spin-valley”-waves

with dispersion ω(~q) = E1q 2/2.

3.4 Jahn-Teller effect

The Jahn-Teller effect is the deformation of a lattice which lowers the energy

He−ph +Hph. Phonons interact with the local electron density. We illustrate

the main points of our approach for ν = 1, keeping the spin of electrons

fixed and up. The electron annihilation operator (3.1) is expanded in the

basis of two mutually orthogonal valley wave functions τ = ±1. The density

3.4 JAHN-TELLER EFFECT 93

operator nˆ(~ρ) = ψˆ†(~ρ)ψˆ(~ρ) has the diagonal element uniform over the 2D

plane: χ2(z)〈ψˆ†τ (~r)ψˆτ (~r)〉 = νχ2(z)/2pil2H , where the average is taken over the ground state (3.11), and the term oscillating in z direction: δnˆ(~ρ) =

χ2(z)(ψˆ†+1(~r)ψˆ−1(~r)e iQz +h.c.). We use the electronic order parameter (3.25)

to express the average oscillating density as

δn(~ρ) = χ2(z)

2pil2H

( Tr(Qˆτ+)eiQz + Tr(Qˆτ−)e−iQz

) . (3.29)

The omitted diagonal density term couples electrons to phonons with in-

plane momentum ~q⊥ = 0 and can be neglected in the thermodynamic limit.

The oscillatory part of the average electron density allows for lowering of the

energy by a static lattice deformation which oscillates with wave vector Q

inside the quantum well:

∇zuzst(~ρ) = Θ

ρSic2 δn(~ρ). (3.30)

This deformation is a phonon ”condensate” or Jahn-Teller effect. The corre-

sponding energy gain is found from the minimization of He−ph +Hph as

Est = −NGTr(Qˆτ+)Tr(Qˆτ−). (3.31)

where N = A/2pil2H is the number of states within Landau level for a sample with area A and G = Θ2

∫ χ4(z)dz /2pil2HρSic

2 is the strength of the electron-

phonon interaction. The integral in the last definition is the inverse width of

the 2D layer in z-direction.

This static deformation causes a new equilibrium position of the lattice.

Assuming that the static deformation is small, we neglect the change of

the phonon spectrum due to anharmonic effects. However we do take into

account the dynamical phonons which produce a polaronic effect in second

order of the electron-phonon interaction. Expanding the phonon field around

the new equilibrium: uz(~ρ) = uzst(~ρ) + δu z(~ρ), we obtain

Hˆe−ph = Θ ∫ ψ†ατ1(~r)τ

± τ1τ2

ψατ2(~r)×(∫ χ2(z)e±iQz∇zδuz(~ρ) dz

) d2~r, (3.32)

94 CHAPTER 3. POSSIBLE JAHN-TELLER EFFECT...

where the lattice deformation

δuz(~ρ) = ∑

~q

√ ~

2V ρSiω(q)

( b~qe

i~q~ρ + b†~qe −i~q~ρ ) ,

can be expanded in phonon creation and annihilation operators b†~q and b~q. In

order to find the ground state energy we sum up diagrams for the thermody-

namic potential expanded in powers of the (assumed to be) weak electron-

phonon and Coulomb interactions using the Matsubara method [48]. We

express the energy in terms of the order parameter matrix Qˆ.

The electron Green function can be expressed in terms of Qˆ as

G0(εn, p) = Qˆ

iεn + E0/2 +

1ˆ− Qˆ iεn − E0/2 . (3.33)

We include the coordinate dependent electron wave-functions in the interac-

tion vertices. The phonon Green function is [48]

D(ωn, q) = − ω 2(q)

ω2n + ω(q) 2 . (3.34)

For small momentum transfers the Coulomb interaction is much larger

than the phonon propagator. In the opposite case of large transferred mo-

mentum with ±Q z-component we neglect the Coulomb interaction. There- fore in the Coulomb vertex the valley index is conserved as well as the spin

index. Due to the identity Qˆ(1 − Qˆ) = 0, all one loop diagrams with only Coulomb lines vanish.

The expression for the electron-phonon vertex in Landau gauge reads

g(pp′, ~q) = Θ√ ρSic2

δp,qy+p′e −q2⊥/4−qx(p+p′)/2χ2(qz −Q),

where ~q is the phonon momentum and the envelope function χ2(qz) =∫ χ2(z)eiqzz dz has a characteristic width l−1 in momentum space. The

polaron contribution to the energy assuming weak Coulomb interaction:

E0/ω(Q)→ 0, is E0e−ph =

∑ εnωnpp′

Tr ( G0(εn, p)τ

+G0(εn + ωn, p ′)τ−

) ∑

~q

g2(pp′, ~q)D(ωn, q) = NG ( Tr(Qˆτ+Qˆτ−)− ν

2

) . (3.35)

3.4 JAHN-TELLER EFFECT 95

Figure 3.2: The two diagrams that give easy-plane anisotropy in the case

ν = 1. Solid, dashed and wavy lines are the electron, the phonon propagators

and the Coulomb interaction respectively.

Summing it up with Eq. (3.31) we get the total electron-phonon energy in

second order of the electron-phonon interaction. Here we neglect small terms

of the order (Ql)−4 due to the discontinuity of the derivative of the wave

function χ(z) at the interface [40]. For ν = 1 the total energy given by the

sum of Eqs. (3.31,3.35) is isotropic in spin-valley space. Thus, if we neglect

the Coulomb interaction corrections to the electron-phonon interaction we

find that the degeneracy in isospin direction is not lifted. Physically, the

polaron energy is a single electron energy and there is no anisotropy operator

for a single electron because of the Pauli matrix identity τ 2i = 1.

The Coulomb interaction creates the easy-plane anisotropy in the case

ν = 1. The essential diagrams are shown in Fig.(3.2), where the wavy line

represents 2D Coulomb potential V (q) whereas the solid and the dashed

lines represent the electron, Eq. (3.33), and the phonon, Eq. (3.34), propa-

gators. Direct calculation neglecting the dependence of ω(Q) on the in-plane

momentum q⊥ shows that the sum of the two diagrams in Fig. 3.2 is

Eδe−ph = −NGδTr ( Qˆτ+Qˆτ−

) , (3.36)

where δ = Eex/ω(Q) is the ratio between the Coulomb exchange energy

Eex = e2

piκ

∫ d2q⊥ dqz q2⊥ + q2z

|χ2(qz)|2e−q2⊥l2H/2 ( 1− e−q2⊥l2H/2

) , (3.37)

to the energy of the valley-mixing phonon. Obviously, δ > 0 (Eex > 0),

and the anisotropy is of the the easy-plane type. The above result is valid if

E0 � ω(Q).

96 CHAPTER 3. POSSIBLE JAHN-TELLER EFFECT...

The sum of the static, polaronic and the Coulomb correction energies,

Eqs. (3.31,3.35,3.36), gives the total anisotropic energy:

Eep = G

∫ [− Tr(Qˆτ+)Tr(Qˆτ−) + +(1− δ)Tr

( Qˆτ+Qˆτ−

) ]d2~r 2pi

. (3.38)

Thus the JTE is energetically favorable and corresponds to the easy-plane

valley anisotropy.

The easy-plane anisotropy changes the dispersion of collective excitations.

Let us consider this effect in the limit of vanishing valley symmetry breaking

t and Zeeman h anisotropies. We add the easy-plane term (3.38) to the

Landau-Lifshitz equation (3.28):

[ Qˆ ∂tQˆ

] −

= E1 2 ~∇2Qˆ +Gδ

( τ+Tr(Qˆτ−) + τ−Tr(Qˆτ+)

) . (3.39)

This equation can be linearized in the vicinity of Q = N . It describes two

”spin”-modes (the same as for the symmetric case) and one acoustic mode

with dispersion ω = √ GδE1q in analogy to the bilayer case [40].

3.5 Phase diagram

The anisotropic bivalley energy in the uniform state is the diagonal matrix

element of the anisotropic Hamiltonian (3.2) expressed in terms of the order

parameter Qˆ:

Ean/N = −t Tr(Qˆτx)− h Tr(Qˆσx) + G [ (1− δ)Tr

( Qˆτ+Qˆτ−

) − Tr(Qˆτ+)Tr(Qˆτ−)

] , (3.40)

where h = |ge|µBH. The order parameter Qˆ can be parameterized by six (ν = 1, 3) or eight (ν = 2) angles. The diagonal matrix elements are real

despite the fact that in external magnetic field there is no time reversal

symmetry. Therefore the ground state can be chosen real, produced from

the reference state by SO(4) sub-group rotations. This sub-group has 6

3.5 PHASE DIAGRAM 97

S

t

h

F C

G Figure 3.3: Phase diagram in the case ν = 2 and δ = 0. F, S and C are

ferromagnetic, spin-singlet and canted antiferromagnetic phases, respectively.

parameters with two of them falling into the denominator sub-group (for

ν = 2). One of the remaining four angles corresponds to global rotation of

all spins and is fixed by the magnetic field direction z. Thus, the ground

state is obtained from the reference state by just three rotations (for ν = 2).

First we consider the case ν = 1. The ground state is UΨ(↑ +1), where U = P (ϑ)R(θ). The matrix R rotates the ↑-spin component in valley space by an angle θ, and then the matrix P rotates the valley components of the

resulting state in spin space by an angle ϑ. Substituting Qˆ(ϑ, θ) Eq. (3.25)

into Eq. (3.40) we obtain the energy per electron:

E1an = −h sin ϑ− t sin θ +Gδ cos2 θ. (3.41)

The minimum of this energy is reached at ϑ = pi/2 and θ = pi/2 corresponding

to the ferromagnet phase in spin and valley spaces. The case ν = 3 is identical

to ν = 1 due to electron-hole symmetry.

In the case ν = 2 the ground state is UΨ(↑ +1, ↓ −1) where U = P (ϑ)R(θ↑, θ↓). The matrix R rotates the ↑, ↓-spin components by angles θ↑,↓ in valley space and then the matrix P rotates the ±1-valley components by angles ±ϑ in spin space. We find θ↑ = θ↓. Substituting the corresponding

98 CHAPTER 3. POSSIBLE JAHN-TELLER EFFECT...

Qˆ(ϑ, θ) into Eq. (3.40) we get:

E2an = −2t cosϑ sin θ − 2h sinϑ cos θ + +G(1− δ)

( 1

2 cos2 θ − cos2 ϑ

) −Gδ cos2 ϑ sin2 θ. (3.42)

The last term represents the easy-plane anisotropy and the antiferromagnetic

exchange interaction between spins in two valleys. The phase diagram in the

case δ = 0 is shown in Fig. 3.3. Here F is the spin ferromagnetic and the

valley singlet phase (ϑ = pi/2, θ = 0), S is the spin singlet and the valley

ferromagnetic phase (ϑ = 0, θ = pi/2), C is the canted antiferromagnetic

phase in spin and valley spaces.

At δ = 0, we find two lines of continuous phase transitions between

F and C phases: (h − G)(h − G/2) = t2, and between S and C phases: (t+G)(t+G/2) = h2. Small positive δ transforms the S-C phase transition

into a discontinuous first order one. The C phase ends at some critical hc(δ),

and the direct S-F first order transition occurs at h > hc(δ) on the line:

t = h − G(3 − δ)/4. For δ > 0.33 the C phase disappears. A detailed examinaning of Eq. (3.42) is given in the Appendix C.

At typical magnetic fields B ∼ 5− 10T the Zeeman energy in Si is a few times larger than the valley splitting t ∼ 2K [49]. To our knowledge, the deformation potential at large phonon momentum is unknown. Therefore

we could only assume the value electron-phonon energy, speculatively: G ∼ 0.2 − 2K. Probably, to observe phases of Fig. 3.3 in Si one would need to lower the effective electron g-factor artificially.

3.6 Anisotropic energy of skyrmion

Many experiments found activation gaps for diagonal conductivity in QHE

systems. For symmetry broken ”ferromagnetic” systems theory predicts the

conductivity to be mediated by charged topological textures — skyrmions,

with activation energies being determined by a large exchange constant E1

from Eq. (3.26). Contrary to this, experiments find much lower gaps [50].

3.6 ANISOTROPIC ENERGY OF SKYRMION 99

One possible explanation is that some intrinsic inhomogeneity, like defects or

a long range slow variation of the electrostatic potential, makes skyrmions of

opposite topological charges to be already present in the system and there-

fore experiments reveal either a depinning activation energy or a skyrmion

mobility activation energy. In both cases the gap is determined by the small

anisotropic part of the skyrmion energy. In this section we compute this

anisotropic energy for a bivalley skyrmion.

Belavin and Polyakov (BP) found the skyrmion solution in the case of a

S2 order parameter [36]. Their solution is readily generalized for a general

Grassmannian order parameter. The non-homogeneous order parameter that

represents one skyrmion in the symmetric case with |Q| = 1 is given by

QˆBP (z, z¯) = Nˆ + 1

R2 + |z|2 ( −R2|vf 〉〈vf | zR|vf 〉〈ve| z¯R|ve〉〈vf | R2|ve〉〈ve|

) (3.43)

where z = x + iy, R is the radius of the skyrmion core and |vf〉 and |ve〉 are two vectors of dimension ν and 4 − ν. We choose |vf,e〉 = (1, 0..0) in (3.43). The skyrmion order parameter has to be rotated by a homogeneous

matrix U calculated in the previous section in such a way that the order pa-

rameter far away from the skyrmion center minimizes the anisotropy energy.

In addition to this rotation, we have to allow the global rotations W from

the denominator sub-group S that transform the skyrmion order parameter (3.43): Qˆ(~r) = UWQˆBP (z, z¯)W

+U+. For the case ν = 2 the block diagonal

matrix W = (Wf ,We) can be parameterized by seven angles:

We,f =

cos

βe,f 2 ei(γe,f +αe,f ) sin

βe,f 2 ei(γe,f−αe,f )

− sin βe,f 2 ei(−γe,f +αe,f ) cos βe,f

2 ei(−γe,f−αe,f )

. (3.44)

The additional seventh parameter angle of the denominator sub-group rotates

the coordinates: z → eiγ7z. We find explicitly that the skyrmion anisotropic energy does not depend on the angles γe, γf and γ7 whereas αe = 0 and

αf = pi correspond to the energy minimum. Thus the order parameter inside

the core of the skyrmion depends on the two angles βe and βf . Beside these

100 CHAPTER 3. POSSIBLE JAHN-TELLER EFFECT...

angles the BP solution depends on arbitrary conformal parameters that do

not change the energy of the skyrmion. We consider here the simplest case

of topological index |Q| = 1 Eq. (3.43), where only one such conformal pa- rameter R is essential. Calculating the energy of the skyrmion we encounter

different spatial integrals with one integral being logarithmically large:

K(R) =

( R

lH

)2 log

( lH R

√ E1 Eskmin

) , (3.45)

and we neglect all the others. In this way we find the Zeeman energy of the

skyrmion to be EskZ = K(R)Z, where

Z = 2h (2 sinϑ cos θ − cosϑ(sin βf + sin βe) sin θ) (3.46)

The bare valley splitting energy of the skyrmion is given by: Eskt = K(R)T , where

T = 2t (2 cosϑ sin θ − sinϑ(sin βf + sin βe) cos θ) (3.47) Finally, the Jahn-Teller energy of the skyrmion is given by: Gsk = K(R)G, where

G = G [ − 1

2 + 3 cos2 ϑ +

1

2 cos2 θ cos(βe + βf)−

−3 2

cos2 θ − (

1

2 − 2 cos2 ϑ

) cos(βe − βf )

] . (3.48)

The total energy of a skyrmion also includes the energy of direct Coulomb

repulsion of an additional charge distribution n(~r) (3.24):

EskC =

∫ e2n(~r)n(~r′)

2κ|~r − ~r′| d 2~rd2~r′ =

3pi2

64

e2

κR = EC lH

R . (3.49)

The minimum of the total anisotropic skyrmion energy, the sum of Eqs.

(3.46)-(3.48): Esk = Z +T +G, with respect to βe and βf was found numeri- cally and is denoted as: Eskmin. Next, we find a minimum of the total skyrmion energy including Eq. (3.49): Esk = K(R)Eskmin + EC/R, with respect to the skyrmion radius R:

∆ = E1|Q| − E0Q

2 +

3

2

( EskminE2C log

E1 Eskmin

)1/3 . (3.50)

3.6 ANISOTROPIC ENERGY OF SKYRMION 101

0 G

h

0

G t

gap

Figure 3.4: Anisotropic part of skyrmion energy gap at ν = 2 and δ = 0

This equation is valid in the limit E skmin � E1. The resulting anisotropic part of the skyrmion gap is shown in the Fig. 3.4. Note that the prominent

minimum of the skyrmion anisotropy energy gap coincides with the phase

transition line C-F from the phase diagram Fig. 3.3.

This skyrmion anisotropic energy is similar to the bilayer case [45] with

t being the hopping constant between the two layers. In an experiment on

a GaAs bilayer [51], a profound reduction of the thermal activation gap has

been found in some interval on the ν = 2 line.

In the case ν = 1 we parameterize general rotations from the denominator

sub-group by four angles:

|ve〉 = (

cos β

2 , sin

β

2 cosαeiλ1 , sin

β

2 sinαeiλ2

) (3.51)

The skyrmion energy does not depend on the angles λ1,2 whereas α = pi/2

corresponds to the energy minimum. We find the Zeeman energy Z = h (1− cos β), the valley splitting energy T = 2t cos2(β/2) and the easy-plane Jahn-Teller energy G = Gδ (1 + cos β). The minimum of the sum of these

102 CHAPTER 3. POSSIBLE JAHN-TELLER EFFECT...

energies with respect to β and α: E skmin = 2 min (t+Gδ, h), determines the charge activation gap ∆ in Eq. (3.50). The case ν = 3 is identical to the case

ν = 1.

3.7 Conclusion

We have shown that the valley degeneracy for a Si(100) MOSFET can lead

to a Jahn-Teller effect in a strong magnetic field with a longitudinal lattice

deformation within the width of a quantum well. This deformation has an

atomic periodicity corresponding to the momentum difference between the

two valley minima in the direction perpendicular to the plain. The integer

fillings ν = 1, 3 and ν = 2 are treated in different ways. For ν = 1, 3 the Jahn-

Teller effect alone can not remove the valley degeneracy and the Coulomb

interaction is crucial. The level splitting here is small if the Coulomb energy

is small compared to the Debye energy. For the ν = 2 case the Jahn-Teller

effect removes the valley degeneracy via an easy-plane anisotropy similar to

that found in the 2D bilayer system in strong magnetic fields. The phase

diagram as a function of different physical parameters was established and

the anisotropic energy of a topological skyrmion-like texture was calculated.

3.8 Appendix A

Here we calculate the second diagram in Fig. 3.1 which was overlooked in

Ref. [47]. It represents the change of Hartree-Fock exchange energy in the

presence of a spin texture. The spatial part of the electron Green function

in Landau gauge is:

G0s(z, z ′) =

(z − z′)s√ 2ss!

e−|z−z ′|2/4+i(x+x′)(y−y′)/2, (3.52)

where z = x + iy, s is the Landau level and Gs0 = G ∗ 0s. The bottom part of

Fig. 3.1b features the G00 Green function whereas the upper part has the G10

APPENDIX B 103

and G00 Green functions. Evaluating the two frequency integrals we find:

δE = 1√ 2

∫ d2z

2pi

d2z′

2pi

d2ξ

2pi G00(z, z

′)V (z − z′) (G00(z

′ξ)Ω+(ξ)G01(ξz) +G10(z′ξ)Ω−(ξ)G00(ξz)) , (3.53)

Expanding Ω(ξ) around the center point of the diagram, z0 = (z + z ′)/2:

Ω±(ξ) = Ω±(z0) ± (ξ − z0) �µν∇µΩzν(z0), we evaluate the energy δE in Eq. (3.53) as (z0 = ~r):∫

d2z

( 1− |z|

2

4

) V (|z|)e−|z|2/2

∫ �µν∇µΩzν(~r)

d2~r

2pi . (3.54)

The front integral is evaluated for the Coulomb interaction and equals: (E1+

E0)/2.

3.9 Appendix B

Eq. (3.27) is proved using a principal bundle of the complex Stiffel manifold

V Cnk = SU(n)/SU(n − k) — that is defined as a manifold of k orthogonal complex vectors in n dimensional complex linear space — over the Grass-

mannian manifold GCnk with a layer U(k). The exact map sequence for this

bundle is

...pi2(V C nk)→ pi2(GCnk)→ pi1(U(k))→ pi1(V Cnk)... (3.55)

The Stiffel manifold has the property pij(V C nk) = 0 for j < 2(n − k) [52],

proven using the principal bundle SU(n) over V Cnk with a layer SU(n−k). Eq. (3.55) means that pi2(G

C nk) = pi1(U(k)). The Lie group U(k) is a product of

SU(k) and U(1) groups with the fundamental homotopy group: pi1(U(k)) =

pi1(SU(k)) + pi1(U(1)) = Z [52].

3.10 Appendix C

In this Appendix we derive the results for the ground state phase diagram

for the case ν = 2 and also present some other formal results which are too

104 CHAPTER 3. POSSIBLE JAHN-TELLER EFFECT...

specific for the main part of the work. We start with the expression (3.42) for

the energy of a homogeneous state as a function of the angles θ and ϑ. First

we derive the expressions for the lines of the phase transitions between F, C,

and S phases for δ = 0. To do this we compute the second derivative matrix

for the energy and check the positiveness of its minors. In this appendix all

energies are measured in units of G.

For the F phase the corresponding condition reads:∣∣∣∣∣ 2h− 2 + 2δ 2t2t 2h− 1 + δ ∣∣∣∣∣ > 0 (3.56)

or

(h− (1− δ)/2) (h− (1− δ)) > t2. (3.57) Generally this expression gives only the region where the F phase is stable

(metastable), but for the 2nd order F-C transition, we get the phase transi-

tion line for δ = 0:

(h− 1)(h− 1/2) = t2. In the same way we get the condition for the (meta)stability of the S

phase:

(t + 1)(t+ (1 + δ)/2) > h2, (3.58)

which for δ = 0 gives the phase transition line

(t+ 1)(t+ 1/2) = h2.

Both the phase transition lines tend to h = t + 3/4 for large t and h.

The line of the direct first order transition between the F and S phases

for large t and h and δ > 0 is obtained by direct comparison of the energies

of the two phases: the energy of the F phase

EF = −2h + 1 2 (1− δ)

and the energy of the S phase

ES = −2t− 1.

APPENDIX C 105

We get

t = h− 3− δ 4

. (3.59)

In order to find the phase transition lines for arbitrary δ we need to find

the values for θ and ϑ in the C phase. This is quite a difficult problem and

here we give an analytical solution for it in some limiting cases. First we find

the C phase for δ = 0. To do it a variable change is convenient: instead of θ

and ϑ we use α = ϑ−θ and β = ϑ+θ. In these variables we get consequently:

E2an = −t(sin β−sinα)−h(sin β+sinα)+ 1

4 (3 sinα sin β−cosα cos β); (3.60)

∂E2an ∂α

= −(h− t) cosα + 1 4 (3 sin β cosα + sinα cos β), (3.61)

∂E2an ∂β

= −(t + h) cos β + 1 4 (sin β cosα + 3 sinα cos β). (3.62)

The conditions ∂E2an/∂α = 0, ∂E 2 an/∂β = 0 yield:

sinα = 3(h+ t)− A(h− t)

2 ,

sin β = 3(h− t)− A−1(h+ t)

2 (3.63)

with

A = cosα

cos β = ±

[ 2(h+ t)2 − 1 2(h− t)2 − 1

]1/2 .

Notice that, as can be seen from Eq. (3.42), θ and ϑ for the ground state

must be in the first quadrant that means that cosα ≥ 0. Calculating the second derivative matrix and checking the stability of the found states we

can show that cos β < 0 corresponds to a maximum of the energy rather

than a minimum.

Finally, substituting (3.63) into (3.60), we find the energy of the C phase:

EC =

√ (2(h− t)2 − 1)(2(h+ t)2 − 1)− 6(h2 − t2)− 1

4 .

Comparing it to EF and ES at the F-C and S-C transition lines calculated

above, we see that at the transition lines the energies are equal, which proves

that the transition is indeed of the second order.

106 CHAPTER 3. POSSIBLE JAHN-TELLER EFFECT...

Now we examine the case of vanishing t and arbitrary δ. The conditions

∂E2an/∂θ = 0 and ∂E 2 an/∂ϑ = 0 for t = 0 read as follows:

2h sinϑ sin θ − 1 2

sin 2θ − δ 2

cos 2ϑ sin 2θ = 0,

−2h cosϑ cos θ + sin 2ϑ− δ sin 2ϑ cos2 θ = 0. (3.64) Apart from the trivial solutions corresponding to the F and S phases we find

the C phase given by

θ = 0, sin ϑ = h

1− δ . (3.65) Other solutions are either non-stable or lose in energy to other states. This

C phase is stable for

(1− δ) √

1 + δ

2 < h < (1− δ).

(The S phase is stable for h < √

(1 + δ)/2 and the F phase for h > 1 − δ.) Comparing the energies of the S and C phases we find the point of the S-C

transition:

hS−C =

√ 1− δ2

2 ;

this is a first order phase transition with regions of metastability for both

phases on either side of it. The C-F transition occurs at hC−F = 1− δ, this transition is of 2nd order.

To find δ at which the C phase disappears we need to set hC−F = hS−C,

this gives δ = 0.33 as mentioned in the main part of the work.

Finally, to find the ending point hc(δ) of the C phase we can find the

intersection of the two known phase transition lines: (3.57) and (3.59). We

get:

hc(δ) = 1 + 10δ − 7δ2

16δ . (3.66)

Chapter 4

Summary

In this thesis I have presented the results on two subjects: the Coulomb

drag and Jahn-Teller effect in two-dimensional electron systems in a strong

magnetic field.

A semiclassical theory for electron magneto-drag between two parallel

two-dimensional electron systems, which provides a transparent picture of

the most salient qualitative features of anomalous drag phenomena observed

in recent experiments, is presented. Localization plays a role in explaining

activated low temperature behavior, but is not crucial for anomalous (es-

pecially negative) drag per se. In particular a detailed theory for the drag

between two one-dimensional links has been elaborated. The temperature

dependence of the drag has been derived for a whole variety of the param-

eters of the system for parallel links as well as for non-parallel ones. Also,

qualitative arguments showing to what consequences for the two-dimensional

drag my results lead are presented.

The Jahn-Teller effect in bivalley Si(100) MOSFET under conditions of

quantum Hall effect at integer filling factors ν = 1, 2, 3 has been studied.

This system is described by an approximate SU(4) symmetry. At ν = 2 static

and dynamic lattice deformations give rise to an easy-plane anisotropy and

antiferromagnetic exchange and lift the valley degeneracy. At ν = 1, 3 the

Coulomb interaction is essential to produce weak easy-plane anisotropy. At

107

108 SUMMARY

ν = 2 three phases: ferromagnetic, canted antiferromagnetic and spin-singlet,

have been found. The anisotropy energy of a charged skyrmion excitation in

every phase has been calculated.

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Acknowledgments.

First I would like to thank Prof. Dr. Walter Metzner for inviting me to do my

PhD at the Max-Planck Institut fu¨r Festko¨rperforschung, for the interesting

topic proposed, and for many helpful discussions during the work.

I thank Prof. Dr. Sergey Iordanskii, the supervisor during my diploma work

and during the first year of my PhD in Institute for Theoretical Physics,

Chernogolovka for his constant support and advice as well in the time of our

collaboration as also after I went to Stuttgart.

I thank Prof. Dr. Alejandro Muramatsu for accepting to be the co-reporter

of my thesis.

My special thanks goes to Prof. Dr. Leonid Glazman, University of Min-

nesota. I always enjoyed the stimulating discussions with him which greatly

influenced this thesis.

I would like to thank Dr. A. Kashuba, Institute for Theoretical Physics, for

the collaboration during the work on the second part of the thesis.

I thank my father, Dr. Efim Brener for his constant interest in my work and

for the idea of the arguments presented in the end of section 2.3.2.

I have the pleasure of thanking all members of the Theorie II department of

115

116

the Max-Planck Institut fu¨r Festko¨rperforschung for the pleasant atmosphere

and all the interesting conversations.

Same goes for the members of the ITP and ISSP in Chernogolovka. In

particular I would like to thank Pavel Ostrovskiy, Mikhail Feigel’man, and

Vladislav Timofeev.

My regards go to Ingrid Knapp for all her organizational support.

I thank Prof. Dr. V. Falko for offering me the opportunities to participate

in very interesting workshops in Dresden and Windsor which gave boosts to

my work.

Finally I would like to thank all the people who were near me during all

these years, in particular my wife Katia, my mother, my brother, and my

two children, for the various support and help.

Publications.

The content of this thesis is partly published in the following journals:

• S. Brener, S. V. Iordanskii, and A. Kashuba. Possible Jahn-Teller effect in Si inverse layers. Phys. Rev. B 67, 125309 (2003).

• S. Brener, W. Metzner. Semiclassical theory of electron drag in strong magnetic fields. Pis’ma Zh. Eksp. Teor. Fiz. 81, 618 (2005) [JETP

Lett. 81, 498 (2005)].

117

118

Lebenslauf

Name Sergej Brener

Geboren am 16. Oktober 1978

Geburtsort Tschernogolowka, Moskauer Gebiet, Russland

Seit Juli 2002 Wissenschaftlicher Mitarbeiter bei Prof. Dr. W. Metz-

ner, Max-Planck-Institut f¨r Festko¨rperforschung in

Stuttgart

Juli 2001 – Juli 2002 Wissenschaftlischer Mitarbeiter bei Prof. Dr. S. Ior-

danskii, Institut fu¨r Theoretische Physik in Tscherno-

golowka

Sep. 1995 – Juni 2001 Studium der Physik an der Moskauer Physikalisch-

Technische Hochschule

1985 – 1995 Schule, Tschernogolowka

Abschluss Juni 1995

Comments