Two-Dimensional Coulomb Liquids and Solids

356 Pages·2004·10.371 MB·English

by  Professor Yuriy Monarkha, Professor Kimitoshi Kono (auth.)

Springer Series in
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Yuriy Monarkha Kimitoshi Kono
Two-Dimensional
Coulomb Liquids
and Solids
With 109 Figures
i
Springer
Professor Yuriy Monarkha Professor Kimitoshi Kono
Institute for Low Temperature Physics Low Temperature Physics Laboratory
and Engineering RlKEN
National Academy of Sdences of Ukraine Hirosawa 2.-1
47 Lenin Ave. Wako-shi 351-0198
Kharkov 61103 Japan
Ukraine
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Professor Dr., Dres. h. c. Hans-Joachim Queisser
Max-Planck-Institut rur Festkorperforschung, Heisenbergstrasse 1, D-70569 Stuttgart, Germany
• Max-Planck-Institut rur Physik komplexer Systeme, Nothnitzer Strasse 38
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Dept. Phys. and Dept. Appl. Physics, Columbia University, New York, NY 1002.7 and
Bell Labs., Lucent Technologies, Murray Hill, NJ 07974. USA
ISSN 0171-1873
ISBN 978-3-642-05858-5 ISBN 978-3-662-10639-6 (eBook)
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Preface
This book is about quantum phenomena in two-dimensional (2D) electron
systems with extremely strong internal interactions. The central objects of
interest are Coulomb liquids, in which the average Coulomb interaction en
ergy per electron is much higher than the mean kinetic energy, and Wigner
solids. The main themes are quantum transport in two dimensions and the
dynamics of highly correlated electrons in the regime of strong coupling with
medium excitations.
In typical solids, the mutual interaction energy of charge carriers is of the
same order of magnitude as their kinetic energy, and the Fermi-liquid ap
proach appears to be quite satisfactory. However, in 1970, a broad research
began to investigate a remarkable model 2D electron system formed on the
free surface of superfluid helium. In this system, complementary to the 2D
electronic systems formed in semiconductor interface structures, the ratio of
the mean Coulomb energy of electrons to their kinetic energy can reach ap
proximately a hundred before it undergoes the Wigner solid (WS) transition.
Under such conditions, the Fermi-liquid description is doubtful and one needs
to introduce alternative treatments. Similar interface electron systems form
on other cryogenic substrates like neon and solid hydrogen.
It might be concluded that the Coulomb liquid is a pure classical system
because its Fermi energy is much smaller than the interaction energy and
(often) even the temperature. But this is not true, especially if the system
is subject to a strong quantizing magnetic field. For example, the Coulomb
liquid reveals an unconventional Hall effect with behavior quite opposite to
the classical picture and complementary to the quantum Hall effect observed
in semiconductor 2D electron systems.
This book has been written to provide a new and comprehensive review
of the remarkable properties of interface Coulomb liquids and solids. In spite
of the broad range covered by the reviewed properties (from free 2D electron
gases and single-electron polarons to electron crystals), it was not difficult
to maintain the uniformity and coherence required in a single book because
of the emphasis placed on the quantum transport framework formulated for
electrons with extremely strong mutual interactions. This transport frame
work combines the memory function formulation of electron conductivity and
the momentum-balance equation method. It is used throughout this book and
VI Preface
provides a good description of the unconventional Hall effect, quantum cy
clotron resonance, WS phonon-medium excitation coupling and conductivity
of the electron solid. The universal conductivity form is obtained as a quan
tum extension of the Drude form with a relaxation kernel containing the
real and imaginary parts. The simplest approximation for the relaxation ker
nel establishes its relationship with the electron dynamical structure factor
(DSF), which is proven to provide quite accurate conductivity equations in
the whole frequency range, if electron-electron collisions dominate the mo
mentum distribution within the electron layer. For practical purposes, the
main advantage of this approach is that one can avoid summing an infinite
series of Feynman diagrams, which is impossible in the conventional conduc
tivity treatment, or solving the kinetic equations. The final equations for the
effective collision frequency (the imaginary part of the conductivity relax
ation kernel) have a very simple integral form applicable to any state of the
electron system, liquid or solid, which is very important for Coulomb liquids.
In a 2D Coulomb liquid subject to a magnetic field, mutual interactions
are so strong that it is impossible to introduce a universal energy excita
tion spectrum in a single reference frame. Therefore, to go beyond the usual
Fermi-liquid approach, instead of the energy excitation spectrum, one has to
consider more global properties of the strongly interacting electrons, namely
the quantum correlation functions, such as the dynamical structure factor.
We pay special attention to this particular property of Coulomb liquids and
solids because it plays an essential role in the quantum transport theory. For
a 2D electron gas subject to a magnetic field, it is the single-electron density
of states, broadened due to interaction with scatterers, which plays the key
role in quantum transport phenomena. For the Coulomb liquid, it is more
convenient and useful to operate with the DSF of the whole liquid. As a
function of frequency it has a series of maxima which are broadened owing to
both the interaction with scatterers and the electron--electron interaction. In
terms of density-density correlation functions it is possible to obtain remark
able similarities between an ideal 2D electron gas and the strongly interacting
electrons. For example, in some instances the DSF of the Coulomb liquid and
even the Wigner crystal can be found as a simple double integral of the
single-electron self-correlation function, or the DSF of an ideal gas.
This text is intended for graduate students and research physicists work
ing in the field of quantum liquids, electronic properties of 2D electron sys
tems, and solid-state physics. It has different levels of sophistication and will
hopefully be useful for both theorists and experimentalists. The text con
tains some instructive examples of general interest, showing that a careless
utilization of conventional methods and well-known formulae from solid-state
physics can sometimes lead to conclusions that are quite opposite to the cor
rect answers. This especially concerns electron interactions with interface
irregularities and the behavior of the electron crystal under normal magnetic
fields.
Preface VII
We start this book with a broad overview (Chap. 1) ofthe single-electron
properties of the 2D electron system formed on the surface of quantum liquids
(4He and 3He) and on the solid cryogenic substrates (H2 and Ne). In Chap. 2,
we discuss those properties of the 2D electron liquid which originate from
the strong mutual Coulomb interaction. Among new and interesting themes
presented here, we would like to emphasize the electric field effect on the
single-electron density of states, the Coulomb narrowing of Landau levels
and the Coulomb broadening of the electron DSF.
Chapter 3 concerns the quantum transport framework for practical eval
uations applying to the highly correlated Coulomb liquid and Wigner solid.
The general relations obtained here are then applied to a description of
the unconventional Hall effect (Chap. 4) and quantum cyclotron resonance
(Chap. 5). Theoretical results are compared with a large body of experimental
data displaying many-electron effects on quantum magnetotransport.
In Chap. 6, we discuss the polaronic effect for 2D electrons strongly inter
acting with interface displacements. Besides an interesting relationship with
the general problem of polaron self-trapping, this text introduces important
physical ideas needed to understand the behavior of the 2D electron crystal
on 'soft' interfaces. Many different problems related to the non-dissipative
dynamics of the 2D Wigner solid are discussed in Chap. 7. The quantum
transport framework helps here to describe the properties of the WS which
originate from the strong electron coupling to interface excitations. The re
markable transport properties of the Wigner solid formed on the free surfaces
of quantum liquids are discussed in Chap. 8.
One of us wishes to express special thanks to the Institute of Physical and
Chemical Research (RIKEN, Japan) for its international invitation program
(RESIP), visiting professorship and hospitality. We have benefited from col
laboration with our colleagues R.W. van der Heijden, P. Leiderer, V.B. Shikin,
E. Teske, W.F. Vinen, F.I.B. Williams, and P. Wyder.
Tokyo, Japan, Yuriy Manarkha
December 2003 Kimitashi Kana
Contents
1 Two-Dimensional Interface Electron Systems. . .... . ... . .. 1
1.1 Introduction........................................... 1
1.2 Hovering Electrons Above Superfiuid Helium 4He. . . . . . . . . . . 3
1.3 Electrons Bound to a Helium Film. . . . . . . . . . . . . . . . . . . . . . .. 13
1.4 Scattering by Vapor Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 17
1.5 Electron Scattering at an Uneven Interface ................ 21
1.5.1 Capillary Wave Quanta (Ripplons) ................. 21
1.5.2 Bloch Approach for Bound Electrons. . . . . . . . . . . . . . .. 23
1.5.3 Adiabatic Approximation. . . . . . . . . . . . . . . . . . . . . . . . .. 30
1.6 Mobility Along the Helium Surface. . . . . . . . . . . . . . . . . . . . . .. 35
1.7 Other Cryogenic Interfaces .............................. 39
1.7.1 Electrons on the Surface of Fermi Liquid 3He ........ 39
1.7.2 Solid Interfaces (H2' Ne) .......................... 42
1.8 Retrapping Transition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 47
1.9 Cyclotron Motion: Quantization and Collision Broadening. .. 53
1.9.1 Self-Consistent Approximation. . . . . . . . . . . . . . . . . . . .. 56
1.9.2 Cumulant Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 59
2 Strongly Correlated Coulomb Liquid ..................... 65
2.1 Fundamental Correlation Functions. . . . . . . . . . . . . . . . . . . . . .. 65
2.1.1 General Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 66
2.1.2 Density-Density Correlation Function. . . . . . . . . . . . . .. 68
2.1.3 Dynamical Structure Factor. . . . . . . . . . . . . . . . . . . . . .. 70
2.2 Fluctuational Electric Field Concept. . . . . . . . . . . . . . . . . . . . .. 78
2.3 Coulomb Narrowing of Landau Levels. . . . . . . . . . . . . . . . . . . .. 80
2.3.1 Electric Field Effect on the Density of States. . . . . . . .. 80
2.3.2 Ensemble of Electrons with Ultra-Fast Orbit Centers.. 83
2.4 Coulomb Broadening of the Dynamical Structure Factor. . . .. 88
2.5 Plasmons and Magnetoplasmons in Reduced Dimensions. . .. 91
2.5.1 Interior Excitations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 92
2.5.2 Edge Waves ..................................... 95
2.6 Electron Correlations and Binding Energy ................. 103
X Contents
3 Quantum Transport Framework
for Highly Correlated Electrons ........................... 115
3.1 An Approach to Universality ............................. 115
3.2 Phenomenological Analysis .............................. 119
3.3 Force-Balance Method (DC Case) ........................ 123
3.4 Memory Function Formulation (AC Case) ................. 128
3.5 Comparison with the Kinetic Equation Method ............ 136
3.6 Energy Relaxation Rate ................................. 143
4 Unconventional Hall Effect ............................... 149
4.1 Transport of Electrons with Discrete Energy Spectrum ...... 149
4.2 Experimental Techniques ................................ 154
4.3 Quantization-Induced Decrease in the Hall Angle ........... 157
4.4 Many-Electron Effects .................................. 159
4.5 Inelastic Magnetotransport .............................. 171
4.6 Cold Nonlinear Effect ................................... 176
5 Quantum Cyclotron Resonance ........................... 183
5.1 Early Achievements ..................................... 183
5.2 Single-Electron Approaches .............................. 188
5.3 Cyclotron Resonance and Internal Forces .................. 193
5.3.1 Many-Electron Effects in the Linear Regime ......... 199
5.3.2 Power-Induced Coulomb Narrowing ................. 205
5.4 Peak Shift ............................................. 206
6 Interface Polarons ........................................ 213
6.1 Relation to the General Polaron Problem .................. 213
6.2 Ground-State Properties ................................ 217
6.2.1 Strong Coupling Theory ........................... 217
6.2.2 Detrapping Transition ............................ 221
6.3 Transport Along the Interface ............................ 227
6.3.1 Effective Mass ................................... 227
6.3.2 Viscosity Drag of Self-Trapped Electrons ............ 230
6.3.3 Long Mean-Free-Path Regime ...................... 232
6.3.4 Ripplon-Limited Mobility ......................... 234
7 Wigner Solid. I. Dynamics on Rigid and Soft Interfaces ... 237
7.1 Contemporary Practice of an Old Hypothesis .............. 237
7.2 Phase Diagram ......................................... 241
7.2.1 Boundary Shape ................................. 242
7.2.2 Dislocation Melting in Two Dimensions ............. 244
7.2.3 Quantum Melting Regime ......................... 246
7.3 Normal Modes and Quantization Under a Magnetic Field .... 249
7.3.1 Phonon Mode Mixing Induced by a Magnetic Field ... 252
7.3.2 Mean-Square Displacement ........................ 255
Contents XI
7.4 Coupling with Medium Vibrations ........................ 256
7.4.1 Dimple Lattice and Medium Response Force ......... 257
7.4.2 Coupled Modes .................................. 261
7.4.3 Self-Consistent Debye-Waller Factor ................ 267
7.4.4 Coupling Under a Magnetic Field .................. 272
7.5 Dynamical Structure Factor ............................. 274
7.5.1 Conventional Approximations ...................... 276
7.5.2 Correlations in Two Dimensions .................... 277
7.5.3 Strong Coupling and Consistency Requirements ...... 280
7.5.4 High Magnetic Field Case ......................... 282
7.6 Shear Mode Excitation and Specific Heat Measurements ..... 284
7.7 Bilayer Electron Crystals ................................ 286
8 Wigner Solid. II. Transport Properties ................... 293
8.1 Solid Current .......................................... 293
8.2 AC Conductivity and Phonon Damping ................... 296
8.2.1 Basic Relations .................................. 296
8.2.2 Spectrum-Splitting Reduction of Phonon Damping ... 306
8.2.3 Resonance Structure of the Collision Rate ........... 308
8.3 Mobility over Normal and Superfluid 3He .................. 313
8.3.1 Viscous Drag of the Dimple Lattice ................. 314
8.3.2 Long Mean-Free-Path Regime ...................... 316
8.4 Nonlinear Transport .................................... 321
8.4.1 Bragg-Cherenkov Scattering ....................... 326
8.4.2 Sliding Wigner Solid .............................. 330
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335
Index ......................................................... 345