NONLINEAR DYNAMICS OF COUPLED ELECTRON-VIBRATIONAL SYSTEMS

By

Kirill Moskovtsev

A DISSERTATION

Submitted to

Michigan State University

in partial fulfillment of the requirements

for the degree of

Physics – Doctor of Philosophy

2018

ABSTRACT

NONLINEAR DYNAMICS OF COUPLED ELECTRON-VIBRATIONAL SYSTEMS

By

Kirill Moskovtsev

We study coupled electron-vibrational condensed-matter systems. Of primary interest are many-

electron effects in two types of systems: nano- and micromechanical semiconductor resonators and

electrons on the surface of liquid helium. Both types of systems are well characterized and display

rich and unusual behavior, which makes them particularly attractive for revealing manifestations

of the electron-vibrational coupling. We show that, in nano- and micro-mechanical resonators, the

coupling leads to strong nonlinearity of the modes and a specific temperature dependence of their

frequencies. The mechanism is the lifting of the degeneracy of the multi-valley electron energy

spectrum by the vibration-induced strain. The redistribution of the electrons between the valleys

is controlled by a large ratio of the electron-phonon coupling constant to the electron chemical

potential or temperature. We find unusually large quartic in the strain terms in the electron free

energy, which result in an unusually strong amplitude dependence of the mode frequencies. This

dependence is calculated for silicon micro-systems, which at present are most broadly used in ap-

plications. It is significantly different for different modes and crystal orientations, and can vary

nonmonotonously with the electron density and temperature. The proposed mechanism leads also

to a very strong temperature dependence of the mode frequencies. These results explain the ex-

perimental observations and suggest ways of improving the stability and sensitivity of modern

micromechanical devices. The other system studied, the electrons floating above the helium sur-

face and coupled to the vibrations in the liquid helium, is special at least in two respects. First,

this system is free from defects, which makes it the best known conductor and a perfect system

for studying electron-vibrational coupling unmasked by disorder. Second, the electron-electron

interaction is strong, so that the electrons form a strongly correlated liquid or a solid. We have

developed an algorithm that allowed us to carry out extensive molecular dynamics simulations of

the system with the account taken of the microscopic mechanisms of the electron scattering by

the excitations in helium. The emphasis was made on calculating the experimentally observable

characteristic, the many-electron mobility. The mobility displays particularly interesting features

when the electrons are placed in a periodic potential. These features are a direct consequence of the

strongly correlated electron motion. As we found, they enable a direct characterization of the cor-

relations in the electron liquid and a direct measurement of the correlation length through its effect

on the activated transport. They also provide a means for studying commensurate-incommensurate

transitions through mobility measurements. We show that the crystallization in the electron system

can be identified from the disappearance of self-diffusion. The way the diffusion coefficient van-

ishes is qualitatively different in the cases of incommensurate and commensurate electron systems.

The mobility transverse to the periodic potential displays an activation dependence on the potential

strength, with the exponent that strongly depends on the extent of the incommensurability.

Copyright by
KIRILL MOSKOVTSEV
2018

ACKNOWLEDGEMENTS

First of all, I would like to thank my advisor, Dr. Mark Dykman. It is a privilege to have Mark as

my guide into the beautiful world of physics. In addition to the invaluable scientific knowledge,

I also learned from him how to thoroughly study large and complicated problems. Mark taught

me to be patient in understanding every possible detail of a problem at hand before claiming any

results. Only with this in mind was I able to produce the results presented in this thesis. I am also

thankful to him for encouraging me to do difficult calculations that I would hesitate to do otherwise.

Mark has always been caring and supportive beyond academic matters as well for which I greatly

appreciate.

I would also like to thank my Thesis Committee members, Dr. Norman Birge, Dr. Carlo

Piermarocchi, Dr. Scott Pratt, and Dr. Carl Schmidt, for the valuable discussions of my research

and their useful comments. I also learned a lot of condensed matter physics from Norman and

Carlo during the seminar and journal club discussions. I would like to thank Carl who taught me

Electrodynamics which is useful in all of my studies. Special thanks go to Scott who helped me

on multiple occasions during my PhD. In fact, Scott was the first to recommend me to meet Mark.

I am very grateful to Dr. Thomas Kenny and his graduate students Yushi Yang and David Heinz

for providing ideas for my research and sharing their experimental results.

I would also like to thank Dr. Steven Shaw for useful and enjoyable discussions and for sharing

his perspective on microelectromechanical systems.

I am thankful to Dr. Johannes Pollanen and Dr. Kostyantyn Nasyedkin for stimulating discus-

sions and sharing their fascinating research ideas about the electrons on helium.

Many thanks go to my colleagues and officemates at MSU, Yaxing Zhang, Juan Atalaya, Pavel

Pulunin, Ori Shoshani, and Nathan Satchell, with whom I enjoyed discussing physics and every-

thing else.

I would also like to thank the staff at the Physics and Astronomy Department, who made my

stay at MSU smooth and pleasant.

v

Last but not least, I would like to thank my family for their support and encouragement during

my long journey at MSU.

vi

TABLE OF CONTENTS

LIST OF TABLES .

LIST OF FIGURES .

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ix

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x

1

CHAPTER 1

INTRODUCTION .

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. .

Introduction .

CHAPTER 2 STRONG VIBRATION NONLINEARITY IN SEMICONDUCTOR-BASED
NANOMECHANICAL SYSTEMS . . . . . . . . . . . . . . . . . . . . . .
5
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5
2.1
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. 12
2.2 Nonlinear frequency shift of low-frequency eigenmodes . . . . . . . . . . . . . .
. 14
2.3 The nonlinearity due to the electron-phonon coupling . . . . . . . . . . . . . . .
2.4 Explicit form of the tensors of nonlinear elasticity . . . . . . . . . . . . . . . . . . 17
2.4.1 Nonlinear elasticity in the limiting cases . . . . . . . . . . . . . . . . . . . 21
. 22

2.5 Doping-induced nonlinearity of simple vibrational modes . . . . . . . . . . . . .

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2.5.1 Temperature and electron density dependence of the scaled nonlinear

frequency shift

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2.6 Linear and nonlinear elasticity in polysilicon . . . . . . . . . . . . . . . . . . . . . 26
2.7 Conclusions .
. 32

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3.1
3.2 Motivation .
3.3 Method . .

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Introduction .
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CHAPTER 3 ELECTRONS ON HELIUM IN A PERIODIC POTENTIAL . . . . . . . . 35
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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.3.1
Integration of the equations of motion . . . . . . . . . . . . . . . . . . . . 53
3.3.2 Ewald summation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.3.3
Scattering of electrons by ripplons and energy relaxation . . . . . . . . . . 56
3.3.4 The simulation unit cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.3.5 Temperature definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.4 Dynamical properties of a spatially uniform electron system . . . . . . . . . . . . . 62
3.4.1 Thermalization in a Wigner solid and a strongly correlated electron liquid . 62
3.4.2 Diffusion in a spatially uniform electron system . . . . . . . . . . . . . . . 66
3.4.3 Mobility in a uniform system . . . . . . . . . . . . . . . . . . . . . . . . . 73
. . . . 78
Single-electron approximation . . . . . . . . . . . . . . . . . . . . . . . . 80
3.5.1
3.5.2 Case p (cid:28) 1 .
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3.5.3 The case of the maximal commensurability, p = 1 . . . . . . . . . . . . . . 82
Structure and crystallization for p < 1 close to p = 1 . . . . . . . . . . . . 90
3.5.4
3.5.5 Mobility in a liquid phase for p < 1 close to p = 1 . . . . . . . . . . . . . 101
3.5.6 Case p > 1 .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
3.5.7 Mobility µ
⊥ as a function of p . . . . . . . . . . . . . . . . . . . . . . . . 107
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
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3.5 Strongly correlated electron system in a one-dimensional periodic potential

3.6 Conclusions .

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vii

CHAPTER 4 CONCLUSIONS .
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4.1 Outlook .

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APPENDIX .

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BIBLIOGRAPHY .

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. 115
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viii

LIST OF TABLES

Table 2.1 The change of the components of the nonlinear elasticity tensors due to the
strain-induced electron redistribution between equivalent energy valleys in
doped silicon. The coordinate axes are chosen along the (cid:104)100(cid:105) axes. Pa-
rameter Ξsh characterizes the effectively two-phonon coupling to shear strain.
This parameter as well as function F1/2(x) are defined in Appendix A.1;
x = µ0/kBT and n is the electron density.

. . . . . . . . . . . . . . . . . . . . . 20

Table 2.2 Elastic constants of pure Si at room temperature in units of 1012 dyn/cm2 . . . . 25

Table A.1 Characters of group elements for various representations of D4h with basis
2 (xy +yx) and

functions up to third order for each representation. Here [xy] ≡ 1
{xy} ≡ 1

2 (xy− yx).

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

Table A.2 Representations of D4h with basis sets x, y, z, and Jx, Jy, Jz. . . . . . . . . . . . . 129

Table A.3 Representation of D4h with the basis set x2, y2, xy, yx.

. . . . . . . . . . . . . . 130

Table A.4 Fourth-order basis sets for some reducible and irreducible representations of D4h. 131

Table A.5 Characters of X2 and X4 representations. The first column contains the ele-
ments of D4h group, while the second contains the corresponding elements of
the little group of the wave vector k = kX. Other columns show characters of
X2, X4, X2 × X∗2 , and X4 × X∗4 representations. . . . . . . . . . . . . . . . . . . . 132

Table A.6 Irreducible representations contained in X2× X∗2 and basis functions of differ-

ent orders, as well as matrices that transform according to the representations. . . 133

ix

LIST OF FIGURES

Figure 2.1 Left panel: energy levels of a nonlinear oscillator in the lowest order pertur-
bation theory. Right panel: the simplest and most common type of relaxation
processes of an oscillator. A transition between the neighboring oscillator
states is accompanied by creation or annihilation of excitations in a thermal
reservoir. The relative transition probabilities are determined by the oscil-
lator Planck number ¯n = [exp(¯hω0/kBT )− 1]−1. In the phenomenological
description of the oscillator dynamics, the sketched processes are responsible
for the friction force proportional to the oscillator velocity.

. . . . . . . . . . .

Figure 2.2 Corrections to some elastic constants as functions of electron density, at T =
300 K. (a) δ c12 and δ c44; (b) δ c111 and δ c144; (c) δ c1111, δ c1144, and
δ c4444. Corrections to other constants are proportional to one of those depicted.

7

18

Figure 2.3 Corrections to some elastic constants as functions of temperature, at electron
density n = 5 × 1019 cm−3. (a) δ c12 and δ c44; (b) δ c111 and δ c144; (c)
δ c1111, δ c1144, and δ c4444. Corrections to other constants are proportional
to one of those depicted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Figure 2.4 Relative change δων /ων of the vibration frequency of a mode with the vi-
bration amplitude ην scaled by the relevant size of the system, cf. Eq. (2.10).
The results refer to single crystal silicon resonators. Curves 1 and 2 refer to
the first Lamé mode in square plates cut in (cid:104)100(cid:105) and (cid:104)110(cid:105) directions, re-
spectively. In this case, the size of the resonator is the length of the side of
the square. Curves 3 and 4 refer to the first extension mode in beams cut in
(cid:104)100(cid:105) and (cid:104)110(cid:105) directions, respectively. In this case, the size of the resonator
is the length of the beam.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Figure 2.5 Mode shapes, displacement shown by color from blue to red, dark blue de-
noting zero and red maximum displacement. (a) first Lamé mode, (b) length
extension mode .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

. .

Figure 2.6 Corrections to elastic constants of polycrystalline silicon as functions of elec-
tron density, at T = 300 K. (a) δ c11; (b) δ c111; (c) δ c1111. Corrections to
other constants are proportional to one of those depicted.

. . . . . . . . . . . . 29

Figure 2.7 Corrections to elastic constants of polycrystalline silicon as functions of tem-
perature, at electron density n = 5 × 1019 cm−3. (a) δ c11; (b) δ c111; (c)
δ c1111. Corrections to other constants are proportional to one of those depicted.

30

x

Figure 3.1

(a) The geometry of typical experiments on a spatially uniform electron sys-
tem on the helium surface (schematically). Electrons reside in vacuum above
the interface and are pressed to the surface by image potentials and the ca-
pacitor electric field. The capacitor electrode creates neutralizing background
for electrons and determines the maximum electron density. (b) The hydro-
genic energy levels of motion transverse to the helium surface in the image
potential V (z) = −Λe2/z for z > 0. Also sketched is the wave function of the
ground state (m = 1) of motion normal to the surface.

. . . . . . . . . . . . . . 35

Figure 3.2 Matrix elements (cid:104)m|z|n(cid:105) of the electron coordinate z and the energy differ-
ence E2 − E1 between the first excited state |2(cid:105) and the ground state |1(cid:105) of
the z motion vs the pressing field E⊥. eE⊥rB/R = 1 for E⊥ ≈ 0.91 kV/cm. . . . 36

Figure 3.3 Phase diagram of a nondegenerate strongly correlated electron system. The
system is a Wigner solid below the line Γ ≈ 130. If the temperature exceeds
the short-wavelength plasma frequency ωp (in energy units), the electron
dynamics is classical.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Figure 3.4 Portion of the liquid-solid phase boundary for a classical 2D electron system
on the helium surface [1]. The data points denote the melting temperatures
measured at different electron densities. The transition is detected from the
sudden appearance of resonant electron absorption at radio-frequencies, an
example of which is shown in the inset. The solid line corresponds to Γ =
137.

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Figure 3.5 Measured magnetoconductivity σ for electron density n = 0.55×108cm−2 at
T = 0.7 K in the 2D electron fluid [2]. Theory lines show: d: Drude model, s:
self-consistent Born approximation, m: many-electron theory [3] and t: phe-
nomenological calculation that combines the results from the many-electron
theory and the SCBA for strong magnetic fields.

. . . . . . . . . . . . . . . . . 41

Figure 3.6 Contour plots of the averaged density distribution ρ(x,y) and pair correlation
function g(x,y) obtained in the experiment [4] for two values of the potential
strength: 2.1kBT(a,b) for which the crystal forms due to modulation-induced
freezing, and 6.3kBT (c,d) above the reentrant-melting threshold, for which
the system is a modulated liquid. In both cases the temperature is above the
melting temperature in the absence of modulation.

. . . . . . . . . . . . . . . . 44

xi

Figure 3.7

a-c, Snapshots of particle velocities in a colloidal monolayer in an incom-
mensurate external potential with a triangular structure [5]. The colloidal lat-
tice spacing is a = 5.7 µm, while the potential lattice spacing is s = 5.2 µm.
Fast and slow particles are marked in dark and light blue depending on
whether their velocities are above or below 70 % of the maximum particle
velocity. (d-f), Voronoi tessellation with color-coded areas of the Voronoi
cells: light (large) to dark (small) green. The following driving forces were
applied: 0 fN in (a,d), 19 fN in (b,e), and 82 fN in (c,f).

. . . . . . . . . . . . . 47

Figure 3.8 Vortex lattice structures observed in the simulation [6] in a superconductor in
the presence of a square lattice of pinning sites. The vortices are represented
by black dots, while the pinning sites are shown as empty circles. (a-i) rep-
resent different magnetic field magnitudes, corresponding to the number of
vortices per pinning site from 1 to 9 in (a) through (i).

. . . . . . . . . . . . . . 48

Figure 3.9 Dynamic phase diagram of a vortex lattice driven across a 1D external po-
tential [7] as a function of the drive field Fd and the ratio w/a of the substrate
lattice constant w to the vortex lattice constant a. P: pinned phase. Plas-
tic: plastic flow regime. ML: moving modulated lattice state. MFS: moving
flowing solid state .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

Figure 3.10 Electrons on the surface of liquid helium with a periodic substrate consisting

of an array of electrodes (schematically).

. . . . . . . . . . . . . . . . . . . . . 50

Figure 3.11 The histograms show the scattering angle distributions obtained in the sim-
ulation of 2 × 106 free non-interacting electrons over 100 time steps with
all electrons having momentum corresponding to (a) k = 2× 105 cm−1, (b)
k = 2 × 106 cm−1. Elastic ripplon scattering is the only scattering in the
simulation, and only particles that scattered during the simulation period con-
tribute to the histograms. The y-axis shows the total count of scattering events
in each of the 40 bins covering angles from 0 to 2π. Solid lines correspond
to the values obtained directly from Eq. (3.20), normalized appropriately for
comparison with the total count of scattering events. . . . . . . . . . . . . . .

. 57

Figure 3.12 Scattering rate w(ph)

kkk(cid:48)kkk as a function of kkk(cid:48) for: (a) kkk = (10,0) µm−1, (b) kkk =
(15,0) µm−1, (c) kkk = (25,0) µm−1. Axes kx and ky have dimensions µm−1.
The scattering rate vanishes at points where |kkk(cid:48)| = |kkk|.

. . . . . . . . . . . . . . 59

xii

Figure 3.13 (a) Averaged over the particles kinetic energy Ekin = (2N)−1 ∑i mev2

i of elec-
trons as a function of time. At t = 0, the system of N = 400 electrons is
prepared in a triangular lattice with zero velocities for all but four electrons.
The velocities of the four moving electrons are set to a large magnitude of
3.4×106 cm/s and four different directions along the coordinate axes, so that
the total momentum vanishes. The system then evolves with no scattering by
the excitations in helium, remaining a crystal (with defects). The energy is
measured in the units of the average Coulomb energy e2(πns)1/2. Time t is
measured in units of 2π/ωp. The inset shows the initial part of the evolution,
with the same units on both axes. (b) The histogram of the distribution of the
electrons over their kinetic energy E(k) after 800 time steps (tωp/2π ≈ 15).
The instantaneous averaged over the particles kinetic energy at this point is
Ekin/kB = 0.175 K (for the mean interelectron distance a = 1 µm). The solid
line depicts the Boltzmann distribution at this temperature (properly scaled
to compare with the histogram). . . . . . . . . . . . . . . . . . . . . . . . . . . 63

Figure 3.14 (a) Averaged over the particles kinetic energy Ekin = (2N)−1 ∑i mev2

i of elec-
trons as a function of time. At t = 0, the system of N = 400 electrons is
prepared in a disordered liquid-like configuration with zero velocities. The
system then evolves with no scattering by helium excitations and remains
a liquid. The energy is measured in units of the average Coulomb energy
e2(πns)1/2. Time t is measured in the units of 2π/ωp. The inset shows the
initial part of the plot, with the same units on both axes. (b) The histogram
of the distribution of the electrons over their kinetic energy E(k) after 600
time steps (tωp/2π ≈ 11). The instantaneous averaged over the particles ki-
netic energy at this point is Ekin/kB = 0.362 K (for the mean interelectron
distance a = 1 µm). The solid line depicts the Boltzmann distribution at this
temperature (properly scaled to compare with the histogram).

. . . . . . . . . . 64

Figure 3.15 Power spectrum Qω of the fluctuations of the kinetic energy per particle Ekin
(arbitrary units). (a) corresponds to the crystal, Fig. 3.13. (b) corresponds
to the liquid, Fig. 3.14. Evolution of Ekin over a longer period of time was
taken to produce the Fourier transforms. The constant component (ω = 0) is
removed in both plots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. 65

Figure 3.16 Delaunay triangulation of the electron positions for a portion of the system
with N = 1600 electrons. Electrons are represented by the vertices. The red
vertices are electrons with five nearest neighbors, while the green vertices
are electrons with seven nearest neighbors. Pairs of red and green vertices
represent dislocations. (a) Liquid phase with Γ = 133 close to the freezing
(cid:107) ≈ 0.6 cm2/s. There are unbound
transition, the diffusion constant D⊥ = D
dislocations, pairs of bound dislocations, and domain walls. (b) Solid phase
with Γ = 140 and D⊥ = D
(cid:107) = 0. Only bound pairs of dislocations are present
in the solid. The two snapshots refer to the two points from Fig. 3.17 (b) for
the corresponding values of Γ.

. . . . . . . . . . . . . . . . . . . . . . . . . . 67

xiii

Figure 3.17 Diffusion constants as a function of Γ with: (a) no scattering, (b) elastic and
inelastic scattering. For each point Γ, the system is prepared as a triangular
lattice with the same density but with different random velocities. Then the
system is equilibrated for 106 time steps before collecting data. The final
temperature, and thus the value of Γ, are measured after the equilibration
period. N = 1600 particles were used in both simulations. We plot both
Dx and Dy to facilitate comparison with the results for the non-zero external
periodic potential.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

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. 69

Figure 3.18 Diffusion constant as a function of Γ for the number of particles (a) N = 100,
(b) N = 252. For each point Γ, the system is prepared as a triangular lattice
at the temperature determined by Γ and then equilibrated for 106 time steps
before collecting data. The final temperature is measured after the equilibra-
tion period. The transition point is slightly below Γ = 130 for N = 100 and
at Γ ≈ 130 for N = 252. The results are obtained without scattering.

. . . . . . 70

Figure 3.19 Low-frequency part of the power spectrum Qω of the temperature fluctua-
tions close to the Wigner crystallization (arbitrary units). Qω is calculated
using Eq. (3.34). The data correspond to three points in Fig. 3.17 (a). (a)
Liquid at Γ ≈ 130. (b) Liquid in the transition regime at Γ ≈ 137. (b) Crystal
at Γ ≈ 140. The constant component (ω = 0) is removed.

. . . . . . . . . . . . 71

Figure 3.20 Diffusion constants as a function of Γ with: (a) no scattering, (b) elastic
and inelastic scattering. For each point Γ, the system is prepared as a liquid
at Γi = 45 and then cooled down to the temperature corresponding to Γ, as
described in the text. N = 1600 particles were used in both simulations.

. . . . 72

Figure 3.21 Theoretical calculation for the single-electron mobility µse and the elec-
tron liquid mobility µse vs temperature. The electron density ns = 2/√3×
108 cm−2 is assumed in the calculation for the strongly correlated system.
The freezing transition is at T ≈ 0.23 K. Above T ≈ 0.7 K, the mobility is
primarily limited by the scattering by the helium atoms in the vapor phase.

. . . 74

Figure 3.22 (a) Drift velocity vd vs drive field Ed and (b) temperature vs drive field for
a uniform system without electron-electron interaction. The filled circles
are the simulation data, the dashed red line is the least-squares fit, and the
solid black line is the prediction by Eqs. (3.39)–(3.40). The temperature T is
computed using Eqs. (3.29)–(3.31). Error bars are not shown since they are
smaller than the marker size.

. . . . . . . . . . . . . . . . . . . . . . . . . . . 76

xiv

Figure 3.23 Drift velocity vd vs drive field Ed for Ed < 0.07 mV/cm. (a) corresponds to
the regular rate of inelastic scattering (∼ 106 s−1). (b) corresponds to the rate
of inelastic scattering reduced by a factor of 10 (to ∼ 105 s−1). The filled
circles are the simulation data, the dashed red line is the least-squares fit by
a straight line. The solid black line is the prediction by Eqs. (3.39)–(3.40).
The error bars are one standard error. These simulations used 1.4× 108 time
steps and ∆t was increased to ∆t = 2.57× 10−10 s. . . . . . . . . . . . . . . . . 77

Figure 3.24 (a) The drift velocity vd vs the drive field Ed and (b) the temperature T vs
the drive field Ed for a uniform strongly correlated electron liquid initially
at Γ = 90. The filled circles are the simulation data, the red dashed line is
the least-squares fit, and the black solid line is the prediction by Eqs. (3.41)–
(3.42). The effective temperature T is computed using Eqs. (3.29)–(3.31).
The error bars correspond to one standard error.

. . . . . . . . . . . . . . . .
Figure 3.25 Triangular lattice with the distance between the vertical rows √3a/2 = 2π/G.
The dashed lines display the maxima of the periodic potential, with the pe-
riod 2π/K. The commensurability parameter is defined as p = K/G.

. . . . . . 79

. 77

Figure 3.26 Diffusion constants D⊥ and D
(cid:107)

vs Γ in the periodic potential with A = 0.5 K
and p = 0.3. The simulation includes both elastic and inelastic scattering.
For every point Γ, the system is prepared in a liquid state with Γ = 45 and
then cooled down, as described in Sec. 3.4.2. electrons. The potential has a
negligible effect on the diffusion and on the freezing transition for p = 0.3. . . . 81

Figure 3.27 The mobility µ

⊥ transverse to the modulation troughs as a function of the
periodic potential amplitude A in a non-interacting electron gas and in a
strongly correlated electron liquid for p = 10/32 = 0.3125. In both cases,
T = 0.354 K which corresponds to Γ = 90 for the liquid. Error bars are not
shown for clarity since most of them are smaller than the marker size. The
potential with p = 0.3125 significantly changes the mobility in the gas, but
only slightly in the liquid.

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

. 82

Figure 3.28 Diffusion constants D⊥ and D
(cid:107)

vs Γ in the periodic potential with p = 1 and
(a) A = 0.5 K, (b) A = 0.05 K. The simulations use N = 1600 electrons and
include both elastic and inelastic scattering. For every point Γ, the system is
prepared in a liquid state with Γi = 45 and then cooled down in the periodic
potential, as described in Sec. 3.4.2.

. . . . . . . . . . . . . . . . . . . . . . . 83

Figure 3.29 Pair correlation function g(2)(x,y) of the electron system in the external po-
tential with A = 0.05 K and p = 1. (a) Γ = 100, (b) Γ = 105, and (c) Γ = 110.
The correlation function is dimensionless. The red cross indicates the loca-
tion of the origin.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

.

xv

Figure 3.30 (a) The transverse mobility µ

⊥ in the electron liquid as a function of temper-
ature. (b) The logarithm of the transverse mobility µ
⊥/µ0 as a function of
the inverse temperature. Both panels are for the electron liquid in the peri-
⊥/µ0)
odic potential with A = 0.05 for p = 1. The hollow circles show log(µ
for points with T < 0.3 K. These points correspond to the crystallized system
(Γ (cid:38) 110), which is pinned. The nonzero values of µ
⊥ here give the measure-
ment error. The number of electrons in this and other mobility calculations
is N = 1024.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

Figure 3.31 (a) The transverse mobility µ

⊥ in the electron liquid as a function of the
periodic potential amplitude A for p = 1. (b) The logarithm of the transverse
mobility µ
⊥ as a function of the periodic potential amplitude A for p = 1. The
hollow circles represent the points where the calculation error is comparable
to the value of µ
⊥. The solid line is the least-squares fit, its slope is 100±
4 K−1. The hollow points were excluded from the fitting procedure. Γ = 90
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. .
in both plots. . .

. 86

Figure 3.32 Logarithm of the transverse mobility µ

⊥ as a function of the periodic po-
tential amplitude A with p = 1. Three sets of points correspond to the elec-
tron liquid with Γ = 40 (red circles), Γ = 60 (green squares), and Γ = 90
(blue triangles). Error bars are not shown for clarity. The system consists
of N = 1024 electrons. The solid lines are the least-squares fit. For each Γ,
we excluded points with µ
⊥ smaller than 5 % of the maximum value of µ
⊥,
since the calculation error for them is comparable with their value.

. . . . . . . 87

Figure 3.33 (a) The radial distribution function g(r) (solid lines) of the electron system
without modulation for Γ = 30 (top), Γ = 60 (middle), and Γ = 100 (bottom).
The number of electrons is N = 1024. The dashed lines show the least-
squares fit of the peaks gp with a function of the form γe−rp/ξc + 1, see the
text for details. The found values of ξc are: ξc = (0.76± 0.05)a for Γ = 30,
ξc = (1.05 ± 0.05)a for Γ = 60, and ξc = (1.47 ± 0.09) for Γ = 100. (b)
The exponential decay rate s plotted vs 2π(ξc + a)2/a2T . The values of s
are found from the plots such as that in Fig. 3.32. Different points represent
systems with Γ from 30 to 110. The correlation length is computed without
the external potential or drive, as in panel (a). The solid line is the least-
squares fit by a straight line passing through the origin.
It has the slope
1.06± 0.02.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

Figure 3.34 Diffusion constants D⊥ and D
(cid:107)

tude A, with p = 1. The system consists of N = 1024 electrons and Γ = 90. . . . 90

as a function of the periodic potential ampli-

xvi

Figure 3.35 Diffusion constants D⊥ and D
(cid:107)

vs Γ in the periodic potential with A = 0.5 K
and p = 30/40 = 0.75, N = 1600. The simulation includes both elastic and
inelastic scattering. For every point Γ, the system is prepared as a liquid at
Γi = 45 in the periodic potential and then cooled down to the temperature
corresponding to Γ, as described in Sec. 3.4.2. After cooling, the system is
equilibrated for 106 more steps. The freezing transition occurs at Γ ≈ 145.
Figure 3.36 Voronoi tessellation of the Wigner crystal with the electron number N = 1600
in the incommensurate periodic potential with A = 0.5 K and p = 30/40 =
0.75. (a) Γ ≈ 156, (b) and Γ ≈ 158. Voronoi cells are colored depending on
their surface area: from dark purple for smaller cells to light yellow for larger
cells. The variation of the cell areas is from 0.84a2 to 0.88a2. The simula-
tion procedure is the same as in Fig. 3.35. The electron positions are averaged
over time to average over small-amplitude thermal fluctuations. Clear rows
of increased density appear at a nonzero angle with the modulation direc-
tion, which correspond to the incommensuration solitons. The two different
soliton structures correspond to slightly different orientation angles that the
system crystallized into. Groups of black cells are a result of disorder and
averaging over time. .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

. . . 91

Figure 3.37 The system of N = 1024 electrons at Γ ≈ 131 in the periodic potential with
A = 0.5 K and p = 30/40 = 0.75. (a) Pair correlation function g(2)(rrr). (b)
Delaunay triangulation of a snapshot of the system with electrons represented
by vertices of the triangulation. The background color shows Re [ψ6(rrr)]
which essentially means the local orientation of the nearest-neighbor links
in the ordered parts of the liquid.

. . . . . . . . . . . . . . . . . . . . . . . . . 94

Figure 3.38 Diffusion constants D⊥ and D
(cid:107)

vs Γ in the periodic potential with A = 0.5 K
and p = 32/40 = 0.8 for N = 1600 electrons. The procedure is outlined in
Fig. 3.35. The phase transition occurs at Γ ≈ 160.

. . . . . . . . . . . . . . . . 94

Figure 3.39 Soliton superstructures corresponding to different rotation angles of the crys-
tal in the incommensurate periodic potential with A = 0.5 K and p = 32/40 =
0.8. Voronoi tessellation for the Wigner crystal with N = 1600 electrons is
displayed for two very close values of Γ: (a) Γ = 176.5, (b) Γ = 177.1.
Voronoi cells are colored depending on their surface area: from dark purple
for smaller cells to light yellow for larger cells. The variation of the cell areas
is from 0.84a2 to 0.88a2. The electron positions are averaged over time to
average over small-amplitude thermal fluctuations. The simulation procedure
is the same as in Fig. 3.35.

. . . . . . . . . . . . . . . . . . . . . . . . . . .

. 95

Figure 3.40 Diffusion constants D⊥ and D

in the periodic potential with A = 0.5 K and
p = 34/40 = 0.85. The procedure is outlined in the caption to Fig. 3.35. The
transverse diffusion ceases at Γ ≈ 180.

. . . . . . . . . . . . . . . . . . . . . . 96

(cid:107)

xvii

Figure 3.41 The system at Γ ≈ 200 in the periodic potential with A = 0.5 K and p =
34/40 = 0.85. (a) Pair correlation function g(2)(rrr). (b) Delaunay triangula-
tion of a snapshot of the system with electrons represented by vertices of the
triangulation. The background color shows Re [ψ6(rrr)]. . . . . . . . . . . . . . . 97

Figure 3.42 Diffusion constants D⊥ and D

in the periodic potential with A = 0.5 K and
p = 37/40 = 0.925. The procedure of obtaining the data is outlined in the
caption to Fig. 3.35. The freezing transition occurs at Γ ≈ 130.

(cid:107)

. . . . . . . . . 98

Figure 3.43 The system at Γ ≈ 130 in the periodic potential with A = 0.5 K and p =
37/40 = 0.925. (a) Pair correlation function g(2)(rrr). (b) Delaunay triangu-
lation of a snapshot of the system with electrons represented by vertices of
the triangulation. The background color shows Re [ψ6(rrr)]. . . . . . . . . . . . . 99

Figure 3.44 The system at Γ ≈ 60 in the periodic potential with A = 0.5 K and p =
37/40 = 0.925. (a) Pair correlation function g(2)(rrr). (b) Delaunay trian-
gulation of a snapshot of the system with electrons represented by vertices of
the triangulation. The background color shows Re [ψ6(rrr)]. . . . . . . . . . . . . 99

Figure 3.45 Snapshots of the system in the potential with p = 37/40 = 0.925 and A =
0.5 K taken at the beginning of the simulation (left) and 1.9× 104 time steps
later (right). Data for the point Γ ≈ 115 from the simulation depicted in
Fig. 3.42 is used. We reconstructed the electrons’ trajectories to eliminate the
effects of the periodic boundary conditions (details in the text). Comparing
the two snapshots, one can see that the entire electron channels along the y
axis (along potential troughs) are displaced. Only a few electrons were able
to hop across the potential troughs, i.e.along x. . . . . . . . . . . . . . . . . . . 100

Figure 3.46 (a) The transverse mobility µ

⊥ in the electron liquid as a function of the
periodic potential amplitude A for p = 23/32 ≈ 0.72. (b) The logarithm of
the transverse mobility µ
⊥ as a function of the periodic potential amplitude
A for p = 23/32 ≈ 0.72. The solid line is the least-squares fit, with the slope
−5.6± 0.3 K−1. Γ = 90 in the both plots.
. . . . . . . . . . . . . . . . . . .

. 101

Figure 3.47 (a) The transverse mobility µ

⊥ in the electron liquid as a function of the
periodic potential amplitude A for p = 25/32 ≈ 0.78. (b) The logarithm of
the transverse mobility µ
⊥ as a function of the periodic potential amplitude
A for p = 25/32 ≈ 0.78. The solid line is the least-squares fit, with the
slope −10.3± 0.5 K−1. The hollow circles were excluded from the fit, since
the calculation error is comparable in size to the value µ
⊥ for these points.
Γ = 90 in the both plots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

xviii

Figure 3.48 (a) The transverse mobility µ

⊥ in the electron liquid as a function of the
periodic potential amplitude A for p = 27/32 ≈ 0.84. (b) The logarithm of
the transverse mobility µ
⊥ as a function of the periodic potential amplitude
A for p = 27/32 ≈ 0.84. The solid line is the least-squares fit, with the
slope −21 ± 1 K−1. The hollow circles were excluded from the fit, since
the calculation error is comparable in size to the value µ
⊥ for these points.
Γ = 90 in the both plots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

Figure 3.49 (a) The transverse mobility µ

⊥ in the electron liquid as a function of the
periodic potential amplitude A for p = 30/32 ≈ 0.94. (b) The logarithm of
the transverse mobility µ
⊥ as a function of the periodic potential amplitude
A for p = 30/32 ≈ 0.94. The solid line is the least-squares fit, with the
slope −74 ± 3 K−1. The hollow circles were excluded from the fit, since
the calculation error is comparable in size to the value µ
⊥ for these points.
Γ = 90 in the both plots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

Figure 3.50 (a) The transverse mobility µ

⊥ in the electron liquid as a function of the
periodic potential amplitude A for p = 35/32 ≈ 1.09. (b) The logarithm of
the transverse mobility µ
⊥ as a function of the periodic potential amplitude
A for p = 35/32 ≈ 1.09. The solid line is the least-squares fit, with the
slope −48 ± 2 K−1. The hollow circles were excluded from the fit, since
the calculation error is comparable in size to the value µ
⊥ for these points.
Γ = 90 in the both panels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

Figure 3.51 (a) The transverse mobility µ

⊥ in the electron liquid as a function of the
periodic potential amplitude A for p = 45/32 ≈ 1.4. (b) The logarithm of
the transverse mobility µ
⊥ as a function of the periodic potential amplitude
A for p = 45/32 ≈ 1.4. The solid line is the least-squares fit, with the slope
−29± 1 K−1. Γ = 90 in the both plots. . . . . . . . . . . . . . . . . . . . . . . 105

Figure 3.52 (a) The transverse mobility µ

⊥ in the electron liquid as a function of the
periodic potential amplitude A for p = 2. (b) The logarithm of the transverse
mobility µ
⊥ as a function of the periodic potential amplitude A for p = 2.
The solid line is the least-squares fit, with the slope −50± 3 K−1. Γ = 90 in
the both plots.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

.

.

.

.

Figure 3.53 Diffusion constants D⊥ and D
(cid:107)

vs Γ in the periodic potential with A = 0.5 K
and p = 56/40 ≈ 1.4. The procedure is outlined in the caption for Fig. 3.35.
The diffusion constants are very small in the considered range of Γ, however
the freezing transition does not occur for Γ < 170.

. . . . . . . . . . . . . . . . 106

xix

Figure 3.54 The higher order commensurate configuration, where KKK = GGG1 +GGG2. Here GGG1
and GGG2 are two elementary reciprocal lattice vectors of the triangular lattice,
|G1| = |G2| = 4π/√3a. The vertical dashed lines show the periodic potential

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

maxima.

.

.

.

Figure 3.55 The Mobility µ

⊥ of the strongly correlated electron system as a function of
the commensurability ratio p for Γ = 50, 90, and 130. A = 0.05 K for all
.
.
three curves.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

.

.

xx

CHAPTER 1

INTRODUCTION

The dynamics of an electron system interacting with a quantized vibrational field is one of the

central problems of physics. This problem has been studied in quantum electrodynamics, but the

results there are limited to the very weak coupling and largely to the single-electron formulation.

In condensed matter systems, it is well-known that the electron-phonon interaction leads to a rich

variety of phenomena, from superconductivity to small and large polarons. New aspects of the

problem have come forth with the recent progress in technology. Two important breakthroughs in

this respects is the emergence of nanomechanics and the studies of the many-electron physics in a

quanitzed layer of electrons localized on the surface of liquid helium. The major feature of these

systems, which makes them particularly attractive, is that the effects of the electron-vibrational

coupling are not masked by defects. In nanomechanical systems, the vibrational modes are well

characterized and can be accessed directly, whereas the electron density can be often independently

controlled. For the electrons on helium, on the other hand, the electrons are floating in free space,

and their coupling with each other and with the vibrations of the helium surface are the only

couplings that are of importance.

Another important common feature of nanomechanical systems and the electrons on the helium

surface is the substantial nonlinearity of the dynamics. The best-understood nonlinear system is a

nonlinear oscillator. Even in a simplest system like this, the nonlinearity leads to an overwhelm-

ingly rich set of phenomena, especially when the system is coupled to a thermal bath or other

degrees of freedom. Nanomechanical systems provide an opportunity to explore all aspects of this

rich phenomenology. Electrons on helium, on the other hand, are known to display remarkable

nonlinear transport behavior that is due to coupling of strongly correlated electrons to the helium

surface vibrations. It is the goal of this thesis to theoretically study some effects of the electron-

phonon coupling and the electron-electron correlations in relation to the nonlinear behavior of

these two systems.

1

A recently developed new type of vibrational systems, mechanical nanoresonators (MNRs),

also known as nano-electro-mechanical systems (NEMS), are mesoscopic by nature. Because

they are small, they display many features of microscopic systems. At the same time, they are

sufficiently large to make it possible to study the dynamics of an individual vibrational system.

Because of their small size, these devices become nonlinear even at comparatively small vibration

amplitudes. Another universal feature arising from the small size is the invariable presence of fluc-

tuations. Moreover, the system often displays nonlinear behavior already for thermal fluctuations.

It is the interplay of fluctuations, dissipation, and nonlinearity that makes these systems fascinating

objects for fundamental research and potential applications.

Interaction of vibrational modes with other degrees of freedom brings even more complex

phenomenology and new opportunities. It may be the interaction with other vibrational or optical

modes, a qubit, or an electronic system. Interaction with other vibrational modes leads to a variety

of nonlinear resonance effects [8, 9, 10], while coupling to optical modes and qubits may lead to

a new way of manipulating quantum information [11, 12, 13]. Interaction with electrons leads to

a variety of effects as well, Coulomb blockade being a simple example [14, 15, 16, 17, 18, 19, 20,

21, 22, 23].

In Chapter 2, we present a theory of a recently observed effect [24] that arises because of the

coupling between electronic and vibrational degrees of freedom in nano- and micro-mechanical

systems. We explain strong mechanical nonlinearity that appears in semiconductor-based nano-

and micro-mechanical systems with high concentrations of free carriers. We show that the mech-

anism is the lifting of the valley degeneracy by strain. Redistribution of the electrons between

the valleys is controlled by a large ratio of the electron-phonon coupling constant to the electron

chemical potential or temperature. The described effect is unusually strong and is the main source

of nonlinearity in resonators made of doped multi-valley semiconductors. The presented theory

provides tools for understanding and controlling the vibration nonlinearity in these systems. The

results are also very important for applications, they are relevant for the majority of silicon MEMS

devices, which are broadly used in modern electronics.

2

Electrons on the surface of liquid helium provide another well-characterized system where

strongly nonlinear aspects of the electron dynamics and the electron-vibrational coupling can be

investigated. This is one of the most perfect electronic systems, with the highest measured mo-

bility [25]. A characteristic feature of this system is a strong electron-electron interaction. This

interaction affects, and sometimes completely modifies the electron scattering by surface waves

(ripplons) and phonons in helium. In typical experiments, the electron system is non-degenerate

yet strongly correlated. The electrons crystallize into a Wigner solid at low temperatures, while

they form a non-degenerate strongly correlated liquid for higher temperatures. Correlation effects

should acquire new features and lead to pronounced consequences when the 2D electronic system

is brought in contact with a substrate. The substrate may create quenched disorder or periodic

modulation depending on its nature. So far, in other systems, two groups of substrate effects have

been studied. One area is the effects of the substrate on the structure of a two-dimensional crystal

deposited onto it, including the phase transitions related to restructuring of the crystal [26, 27].

The second area is depinning in response to a strong drive and the emerging dynamical phases, see

Ref. [28] and papers cited therein. The dynamical effects have been studied primarily for vortices

and skyrmions in superconductors and for colloidal particles in a periodic optical potential. In

all studies we are aware of the motion was overdamped, inertial effects played no role. This is

qualitatively different from the system of electrons on helium, where the electron relaxation time

is extremely long and inertia is a major factor.

In Chapter 3, we present the results of a numerical study of the strongly correlated electron

system on helium in an external periodic 1D potential. We find the effects of the modulation

on the freezing transition and the structure of the ground state of the electron system. However,

of the central interest to us are transport properties of the strongly correlated electron liquid.

Such properties are accessible to direct experimental measurements. This makes them particularly

interesting. We note that the presence of a long range order in the electron system can be revealed

through transport measurements [29, 30]. In contrast, it is not known how to directly reveal strong

electron correlations in the liquid phase.

3

Traditionally, commensurability effects have been considered for ordered phases. We show,

however, that effective commensurability plays a crucial role in the transport even when no long-

range order is present in the electron system. In this regime, no classical pinning can occur since

the system remains disordered with dynamically changing structure due to thermal motion. Never-

theless, as we demonstrate, a periodic potential can dramatically suppress conductivity across the

modulation troughs when the commensurability ratio is close to integer values. We also find an

exponential dependence of the mobility on the modulation amplitude in the “commensurate” liq-

uid phase, indicating activated behavior. This behavior is highly sensitive to the correlation length

in the liquid, which can provide a direct means for measuring this length, an issue of significant

broad interest for an electron liquid. The commensurability effects we observe persist in a wide

range of regimes: from highly underdamped to overdamped dynamics. However, highly under-

damped dynamics, such as that of the electrons on helium, allows one to measure the effects of

particle-particle correlations in the cleanest way, not modified by friction.

To study the effect of the electron coupling to the quantized filed of excitations in helium on

the dynamics of a strongly correlated electron system, we extend the conventional molecular dy-

namics method to include electron scattering by ripplons and phonons. This scattering is described

microscopically in terms of the momentum-dependent scattering rate for the actual coupling to the

excitations in helium.

4

CHAPTER 2

STRONG VIBRATION NONLINEARITY IN SEMICONDUCTOR-BASED

NANOMECHANICAL SYSTEMS

2.1 Introduction

Understanding vibrational motion has always been one of the major motifs in physics. Histor-

ically, such motion was first studied for macroscopic systems. The progress in optics in the 20th

century made it possible to study microscopic vibrational systems, in particular molecular systems

and vibrational modes in solids. However, these studies relied on ensemble measurements, where

the signal came from a large number of individual systems, e.g., a large number of molecules. Re-

cent progress in nanotechnology has made it possible to study a new type of vibrational systems,

mechanical nanoresonators (MNRs). Such systems are mesoscopic by nature. Because they are

small, they display many features of microscopic systems. At the same time, they are sufficiently

large to make it possible to study the dynamics of an individual vibrational system.

The interest in the MNRs is threefold. On the one hand, the vibrational excitations result

from the collective motion of a large number of atoms, which leads to rich dynamics that involves

various many-body effects. On the other hand, MNRs are one of the best systems for studying,

in a well-characterized environment, statistical physics far from thermal equilibrium and quantum

physics at the macroscale. Last but not least, they are of significant interest for various applications,

from ultrasensitive mass, charge, and force detection [31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42,

43, 44, 45, 46, 47, 48, 49, 50, 51, 52] to magnetometers with a single-spin sensitivity [53, 54, 55,

56, 57, 58] to applications in electronics [59] to mention but a few.

The MNRs were developed in the 90(cid:48)s [60] and quickly attracted considerable interest. It was
realized that these systems are extremely versatile. The vibration frequency can be tuned not only

through the dimensions and the shape of a device, but also in situ by electrostatic and optical means.

In addition, the lifetime of vibrations has now been increased to milliseconds and above thanks to

5

the progresses in nanofabrication. These advances have opened new possibilities in controlling the

interaction of nanomechanical vibrations with electrons, photons, and propagating phonons and

studying the effects of this interaction.

A consequence of the small size of MNRs expected from the general arguments of statistical

physics [61] is the occurrence of comparatively strong quantum and classical fluctuations. These

fluctuations play a significant role in the vibration dynamics. A feature of MNRs is that, along

with the invariably present thermal fluctuations, they are subject to nonequilibrium fluctuations.

Characterizing and understanding the nature of these fluctuations, as well as revealing their conse-

quences is a major challenge.

Another basic feature of the MNRs is the vibration nonlinearity. Because of the small system

size, vibrations with even comparatively small amplitudes become nonlinear. The nonlinearity

often comes into play already for thermal fluctuations. It leads to the broadening of the spectra

of the response to an external field, which is not related to energy dissipation. As a result, the

shape of the spectra is determined by the interplay of dissipation, fluctuations, and nonlinearity.

A consequence of the vibration nonlinearity is resonant and nonresonant coupling of different

vibrational modes. This specific feature of MNRs leads to nonlinear inter-mode energy exchange

and also to cross-modulation of the vibration frequencies.

The interplay of fluctuations and nonlinearity in the vibrational spectra can be understood from

the following argument. The energy spectrum of a harmonic oscillator is equidistant, with the
level spacing ¯hω0, where ω0 is the oscillator eigenfrequency. The response of the oscillator to a
periodic field displays a resonant peak at ω0. Because transitions between different neighboring
energy levels are resonant at the same time, the response is nontrivial and has been a subject of

interest from the early days of quantum mechanics [62]. It is somewhat surprising, from this point

of view, that the resulting peak in the oscillator absorption spectrum often has a simple Lorentzian

shape with a halfwidth determined by the oscillator decay rate Γ. Phenomenologically, this decay
rate corresponds to a friction force −2Γm ˙q, where q is the oscillator coordinate and m is the
effective mass. In MNRs, the ratio Q = ω0/2Γ, which is called the Q factor, can be extremely

6

high, approaching 107 − 108 [63].

Figure 2.1 Left panel: energy levels of a nonlinear oscillator in the lowest order perturbation the-
ory. Right panel: the simplest and most common type of relaxation processes of an oscillator.
A transition between the neighboring oscillator states is accompanied by creation or annihilation
of excitations in a thermal reservoir. The relative transition probabilities are determined by the
oscillator Planck number ¯n = [exp(¯hω0/kBT )− 1]−1. In the phenomenological description of the
oscillator dynamics, the sketched processes are responsible for the friction force proportional to
the oscillator velocity.

Even a comparatively small vibration nonlinearity can strongly change the shape of the spec-

trum. Indeed, for a nonlinear oscillator transitions between neighboring energy levels occur at

different frequencies, see Fig. 2.1. The energy spectrum in this figure corresponds to the lowest
order perturbation theory in the oscillator nonlinearity. If the frequency difference |V| significantly
exceeds the decay rate Γ, the response of the oscillator will display spectral peaks at frequencies
ω0 + nV with integer n. The overall spectral width then is ∼ |V|(2 ¯n + 1), where ¯n is the typi-
cal number of excited states. In thermal equilibrium, the populations of the levels fluctuate, and

therefore such spectral broadening is indeed fluctuational by nature.

The above mechanism is an example of non-dissipative spectral broadening due to fluctuations
of the vibration eigenfrequency ω0. There are different sources of such fluctuations. An example
of another important source is attachment/detachment of molecules to an NMR. The resonator

mass, and thus the frequency, depend on the number of attached molecules, which underlies the

MNR-based mass spectroscopy [31, 32, 34, 38, 39, 46, 50]. However, frequency fluctuations

associated with attachment, detachment, or diffusion of molecules along the resonator lead to

spectral broadening [64]. It resembles [65, 66] the broadening due to the mode nonlinearity, with

7

2ω0ω0 + 2Vω0 + V|0>|1>|2>|3>|m+1>|m>nn+1FIG.1Figuretitle.captionitselftionsbetweendifferentneighboringenergylevelsareres-onantatthesametime,theresponseisnontrivialandhasbeenasubjectofinterestfromtheearlydaysofquantummechanics(WeisskopfandWigner,1930).Itissomewhatsurprising,fromthispointofview,thattheresultingpeakintheoscillatorabsorptionspectrumoftenhasasimpleLorentzianshapewithahalfwidthdeterminedbytheoscillatordecayrateΓ.Phenomenologically,thisde-cayratecorrespondstoafrictionforce−2Γm˙q,whereqistheoscillatorcoordinateandmistheeffectivemass.InMNRs,theratioQ=ω0/2Γ,whichiscalledtheQfactor,canbeextremelyhigh,approaching107−108,seetables??andSec.below.Evenacomparativelysmallvibrationnonlinearitycanstronglychangetheshapeofthespectrum.Indeed,foranonlinearoscillatortransitionsbetweenneighboringen-ergylevelsoccuratdifferentfrequencies,cf.Fig.1.Theenergyspectruminthisfigurecorrespondstothelowestorderperturbationtheoryintheoscillatornonlinearity.Ifthefrequencydifference|V|significantlyexceedsthedecayrateΓ,theresponseoftheoscillatorwilldisplayspectralpeaksatfrequenciesω0+nVwithintegern.Theoverallspectralwidththenis∼|V|(2¯n+1),where¯nisthetypicalnumberofexcitedstates.Inthermalequilibrium,thepopulationsofthelevelsfluctuate,andthereforesuchspectralbroadeningisindeedfluctuationalbynature.Theabovemechanismisanexampleofnon-dissipativespectralbroadeningduetofluctuationsofthevibrationeigenfrequencyω0.Therearedifferentsourcesofsuchfluctuations.Anexampleofanotherimportantsourceisattachment/detachmentofmoleculestoanNMR.Theresonatormass,andthusthefrequency,dependonthenumberofattachedmolecules,whichunderliestheMNR-basedmassspectroscopy(Sazonovaetal.,2004;EkinciandRoukes,2005;Cleland,2005;Jensenetal.,2008;De-Martinietal.,2008;Moseretal.,2013;Hanayetal.,2015).However,frequencyfluctuationsassociatedwithattachment,detachment,ordiffusionofmoleculesalongtheresonatorleadtospectralbroadening(Yangetal.,2011).Itresembles(Dykmanetal.,2010;Atalayaetal.,2011)thebroadeningduetothemodenonlinearity,withthedifferentfrequenciesinFig.1nowcorrespondingtothedifferentnumberorpositionsofattachedmolecules.AprofoundconsequenceofthenonlinearityofMNRsisbi-(andmulti-)stabilityofforcedvibrations.Itemergesalreadyforcomparativelyweakdrivingforceprovidedtheforceresonateswiththemodefrequencyoritsover-tones.PeriodicallydrivenMNRsareamongthebestsys-temsforstudyingtheFloquetdynamicsinthepresenceofdissipationandquantumandclassicalnoise.Theac-cessiblephenomenarangefromthefluctuationsqueezingtothestatisticaldistributionandtherateofswitchingbetweencoexistingmetastablestatesfarfromthermalequilibrium,whenasystemhasnodetailedbalance.Inthisreviewwewillprovideanoverviewofthetypesofnanomechanicalsystemsandthetypicalparametersthatcharacterizethesesystems.Majorwaysofcontrol-lingMNRsbyexternalfieldsandmeasuringtheirvi-brationalstateswillbeoutlined.Themechanismsofthenonlinearityandrelaxation,resonantandnonreso-nantcouplingofMNRsmodes,andthecouplingofthesemodestoelectrons,phonons,andphotonswillbedis-cussed.WewilldescribevariousaspectsoffluctuationsinMNRsandthewaysofrevealingandcharacterizingtheirunderlyingmechanisms.WewillalsodiscussthephysicsofperiodicallydrivenMNRsandthenonequilib-riumquantumandclassicalfluctuationphenomenathathavebeenstudiedwiththesewell-characterizedsystems.I.FLUCTUATION-DISSIPATIONTHEOREMANDTHEREACTIONFORCEFROMTHETHERMALBATHThesimplesttheoreticalmodelemployedinthedesaltothevelocityandalsoathermalnoise,whichisassumedtobeGaussianandδ-correlatedintime.ThismodeldescribesBrownianmotionofalinearoscillator(Risken,1996).Iftheoscillatorcoordinateisqandthemassism,themotionisdescribedbytheLangevinequationm¨q+2mΓ˙q+mω20q=fT(t),hfT(t)fT(t0)i=4mΓkBTδ(t−t0)(1)Boththefrictionandthenoisecomefromthecouplingoftheoscillatortoathermalreservoir(thermalbath).Historically,theanalysisofsuchcouplingwasstartedbyEinsteinandHopf(1910).AdetailedclassicalstudywasperformedbyBogolyubov(1945),whereasthestudiesofthequantumdynamicswerestartedinthelate50s-early60s,seeToda(1958),Senitzky(1961),Schwinger(1961),Krivoglaz(1961),Louisell(1990),Fordetal.(1965),andUllersma(1966);morereferencescanbefoundinFordetal.(1988).Inmanyofthesepaperstherewasusedthemodelinwhichthethermalbathwasdescribedbyasetofharmonicoscillatorsthatwerelinearlycoupledtotheconsideredoscillator.Overtheyears,suchmodelofthebath,oftencalled“bosonicbath”,hasbeenoneofthemostfrequentlyusedinthestudyofquantumrelaxation,cf.FeynmanandVernonJr.(1963);CaldeiraandLeggett(1981);andGrabertetal.(1988).MNRsareoneofthebestexamplesofsystemswheredissipationdescribedbylinearcouplingtoabosonicbathcanplayanimportantrole.Inthiscase,thebathisthe different frequencies in Fig. 2.1 now corresponding to the different number or positions of

attached molecules.

The interplay between fluctuations and nonlinearity remains an active topic of research. An

example of such interplay is the recently observed nonlinear frequency transduction of nanome-

chanical Brownian motion [67, 68]. In that case, the Duffing nonlinearity transduces the fluctu-

ational motion of the observed eigenmode and other nonlinearly coupled modes into frequency

noise. Yet, even though there has been substantial progress in understanding of frequency fluctu-

ations and nonlinearities, the fundamental limits of the frequency noise in nanomechanics are still

not understood [69].

A profound consequence of the nonlinearity of MNRs is bi- (and multi-) stability of forced

vibrations. It emerges already for comparatively weak driving force provided the force resonates

with the mode frequency or its overtones. Periodically driven MNRs are among the best systems

for studying the Floquet dynamics in the presence of dissipation and quantum and classical noise.

The accessible phenomena range from the fluctuation squeezing to the statistical distribution and

the rate of switching between coexisting metastable states far from thermal equilibrium, when a

system has no detailed balance.

In addition to non-dissipative phenomena described above, MNRs often display dissipative

nonlinear effects also known as nonlinear damping. One of the most important characteristics of

nano-scale vibrational modes is their decay rate, which can be much smaller than the vibration

frequency [70, 71, 72, 73, 53, 46, 49, 38, 44, 50]. The relaxation mechanisms include radiation

into the bulk modes of the medium surrounding the nanosystem [74, 75, 76, 77, 78], nonlinear

coupling between the modes localized inside the system [79, 80, 81, 82, 83, 84], and coupling

with electronic excitations[18, 20, 21, 85] and two-level defects [86, 87]. Conventionally, decay

of a vibrational mode is associated with an exponential fall-off of the vibration amplitude in time.

This behavior is often observed in the experiment. In the phenomenological description of the
mode dynamics, it comes from the friction force −2Γ ˙q, where q is the vibration coordinate and
Γ is the friction coefficient. Such friction is called linear. Recent experiments have shown that,

8

already for moderate vibration amplitudes, in various nano- and micromechanical systems the

amplitude decay is nonexponential in time and/or the maximal amplitude of forced vibrations

nonlinearly depends on the force amplitude [88, 89, 90, 91, 83]; respectively, the friction force is

a nonlinear function of the mode coordinate and velocity. Nonlinear friction (nonlinear damping)

has recently attracted considerable attention also in the context of quantum information processing

with microwave cavity modes, where it was engineered to be strong [92, 93].

Nonlinear damping is well understood in quantum optics, where this is a major mechanism that

limits the laser intensity [94]. In nanomechanical systems, the mechanism is scattering of thermally

excited vibrational modes off the considered eigenmode [95]. The scattering is accompanied by
energy transfer of 2¯hω0 for nonlinear damping. Along with damping, scattering of phonons off
the considered mode leads to its dephasing, i.e., loss of coherence, as was first discussed for high-

frequency vibrations localized near defects in solids [96, 97]. In contrast to damping, the scattering

that leads to dephasing is quasi-elastic, the energy of the mode is not changed. To the lowest order,

the process comes from the quartic nonlinearity in the mode coupling, although the parameters are

renormalized by the cubic nonlinearity, which is also the case for nonlinear friction.

Another important consequence of nonlinearity in vibrational systems is the appearance of non-

linear resonances [9]. Nonlinear resonance, where the energy-dependent vibrational frequencies

are commensurate, has a long history in quantum and classical mechanics. It goes back at least to

Laplace and Poincare on the classical side and to the Fermi resonance on the quantum side [98, 99],

and occurs in a broad range of systems, from celestial bodies to ecological systems to molecules

[100, 101, 102, 103]. Recently, nonlinear resonance has attracted particular interest in the context

of nano- and micro-mechanical vibrational systems [8, 104, 10, 105, 106, 107, 108, 109, 110] and

microwave cavities used in quantum information [92, 111]. These mesoscopic systems provide

unprecedented access to studying, using, and controlling this complicated phenomenon.

In conservative classical systems, nonlinear resonance leads to energy oscillations between the

resonating modes. This is reminiscent of the energy oscillations between two coupled harmonic

oscillators with close frequencies. However, the actual picture in nonlinear resonance is more

9

complicated, extending to dynamical chaos. On the quantum side, nonlinear resonance is in some

sense simpler in the absence of dissipation, as its primary signature is the familiar level repulsion.

The quantum situation changes if the resonating modes are dissipative. If the modes have very

different decay rates, one of them can serve as a thermal reservoir for another, cf. [112]. This

effect has been used to drive a slowly decaying microwave cavity mode to a coherent quantum

state [92]; it extends to driven modes [113], and such extension has attracted much attention in

cavity optomechanics [72].

An important advantageous feature of mesoscopic oscillators is the possibility to tune them in

and out of nonlinear resonance. This can be done by directly controlling their frequencies [8] or

dynamically, using the dependence of the frequency on the vibration amplitude. Here, by driving

a mode, one brings its overtone into resonance with an overtone of another mode, which is then

also excited. The ensuing backaction significantly changes the dynamics of the driven mode and,

in particular, its decay after the driving is switched off. Such decay is a major means of studying

mesoscopic vibrational systems [69].

Electron-phonon coupling strongly affects vibrational modes of nano- and micro-electro-mechanical

systems. Much interest have attracted the effects of this coupling related to the reduced dimen-

sionality of the electron system, as they make it possible to reveal interesting consequences of the

electron correlations at the nanoscale, the Coulomb blockade being a simple example, and their

strong effect on the vibrational dynamics, cf. [14, 15, 16, 17, 18, 19, 20, 21, 22, 23] and references

therein.

Much less attention has been paid to the consequences of the electron-phonon coupling, which

are related to the discreteness of the vibrational spectrum of a nanosystem, but emerge in the

absence of size quantization of the electron motion. One of such consequences, which we study

in this paper, is the coupling-induced change of the vibration nonlinearity. Strong nonlinearity is a

generic feature of vibrations in small systems [114, 115]. Its easily accessible manifestation is the

dependence of the mode frequencies on the vibration amplitudes. This dependence corresponds to

the “self-action" of the mode, and its familiar analog in bulk crystals are acoustic solitons [116,

10

117]; however, the nonlinearity required for observing such solitons usually is sufficiently strong

only for high-frequency phonons. Also, the change of the eigenfrequency with the mode amplitude

is of interest for modes with a discrete frequency spectrum, such as standing waves in mesoscopic

systems, but not for propagating waves with a quasi-continuous spectrum.

Much attention have been recently attracting Si-based nano- and micromechanical systems,

see [118, 69] and references therein. In such systems there was observed an unexpectedly large

change of the amplitude dependence of the vibration frequency with the varying electron density
[119, 24]. When the doping level was increased from 2.8× 1018 cm−3 to 5.9× 1019 cm−3, the
nonlinearity parameter increased by more than an order of magnitude. Moreover, the nonlinearity

change was different for the vibrational modes with different spatial structure.

In this paper we develop a theory of the nonlinearity of vibrational modes in semiconductor

nano- and micro-mechanical systems with high electron density. We show that the electron-phonon

coupling can lead to a strong self-action of the vibrational modes, which in turn significantly

modifies the amplitude dependence of the mode frequencies. We find the dependence of the effect

on the electron density and temperature.

For bulk semiconductors, the effect of the electron-phonon coupling on the elastic properties,

including the three-phonon coupling, was first analyzed by Keyes [120]. The analysis referred to n-

Ge and was based on the deformation potential approximation. The idea was that deformation lifts

the degeneracy of the equivalent electron valleys, which leads to a redistribution of the electrons

over the valleys. In turn, such redistribution changes the speed of sound depending on the direction

and polarization of the sound waves and also affects the sound speed in the presence of uniaxial

stress. This theory was extended to silicon and the corresponding measurements were done by

Hall [121]. However, Hall also observed the change of the speed of transverse sound waves and

the effect of stress on sound propagation in the geometries, where these effects are due to shear

deformation and do not arise in the deformation potential model. A theory of the change of the

linear shear elastic constant in silicon due to the intervalley redistribution of the electrons was

developed by Cerdeira and Cardona [122].

11

As we show, in mesoscopic systems the strain-induced redistribution of the electrons over the

valleys of the conduction band leads to the previously unexplored strong fourth-order nonlinearity

of the vibrational modes. This nonlinearity gives a major contribution to the amplitude dependence

of the vibration frequency. The redistribution also leads to a temperature dependence of the fre-

quencies. The magnitudes of the effects sensitively depends on the mode structure. We describe

them for several types of modes, including those studied in the experiment [119, 24] and qualita-

tively compare the results with the observations. The theoretical results refer to both degenerate

and nondegenerate electron systems. Specific calculations are done for silicon resonators.

In Sec. 2.2 we give, for completeness, the expressions for the mode normalization and the

amplitude-dependent frequency shift of coupled nonlinear modes in a nano- or micro-system.

In Sec. 2.3 and Appendix A.1 we provide expressions for the electron-phonon coupling induced

change of the elasticity parameters, including the parameters of quartic nonlinearity. In Sec. 2.4

we discuss the asymptotic behavior of the parameters of quartic nonlinearity for low and high

electron density and give their explicit form for silicon. In Sec. 2.5 we calculate the nonlinear

frequency shift for several frequently used vibrational modes in single-crystal silicon systems and

show the dependence of this shift on the electron density and temperature. The explicit analytical

expressions are given in Appendices A.4 and A.5. Sec. 2.7 contains concluding remarks.

2.2 Nonlinear frequency shift of low-frequency eigenmodes

Of primary interest for nano- and micro-mechanical systems are comparatively low-frequency

modes with wavelength on the order of the maximal size of the system. Examples are provided

by long-wavelength flexural modes of nanotubes, nanobeams, and nano/micro-membranes, or

acoustic-type modes in microplates or beams. These modes are easy to excite and detect. We

will enumerate them by index ν. Their dynamics is described by the elasticity theory [123]. The
spatial structure of the displacement field of a mode uuu(ν)(rrr) in the harmonic approximation is
determined by the boundary conditions. We will choose uuu(ν)(rrr) dimensionless, so that in our

12

finite-size system

(cid:90)

drrr uuu(ν)(rrr)· uuu(ν(cid:48))(rrr) = Vδνν(cid:48).

(2.1)

Here, V is the volume of the system. We assumed that the mode eigenfrequencies ων are non-
degenerate; including degenerate modes is straightforward. For simplicity, we also assumed that

the system is spatially uniform; an extension to spatially nonuniform systems is straightforward as

well.

We emphasize the distinction of the normalization (2.1) from the conventional normalization

for bulk crystals, where ν corresponds to the wave vector and the branch number, and the nor-

malization integral is independent of the volume. The normalization (2.1) is convenient for the

analysis of low-frequency modes with the discrete spectrum characteristic of mesoscopic systems.
Such modes are standing waves, and therefore vectors uuu(ν) can be chosen real.

The low-frequency part of the displacement can be written as

uuu(rrr,t) = ∑
ν

Qν (t)uuu(ν)(rrr).

(2.2)

Functions Qν (t) give the mode amplitudes. In the harmonic approximation the dynamics of the
standing waves is described by the Hamiltonian

Hh =

1
2 ∑

ν

(M−1P2

ν + Mω2

νQ2
ν ),

(2.3)

where Pν is the momentum of mode ν and M is the mass of the system.

The anharmonicity of the crystal leads to mode-mode coupling. Within the elasticity theory

this coupling is described by the terms in the Hamiltonian, which are cubic and quartic in the

strain tensor. We will not consider higher-order terms, which are small for the mode amplitudes of

interest. From the expansion (2.2), we obtain the nonlinear part of the Hamiltonian in the form

Hnl =

1
3 ∑βν1ν2ν3Qν1Qν2Qν3
1
4 ∑γν1ν2ν3ν4Qν1Qν2Qν3Qν4.

+

(2.4)

13

Equation (2.4) is essentially an expansion in the ratio of the mode amplitudes to their characteristic

wavelength, which is of the order of the appropriate linear dimension of the system. This is why

mesoscopic systems are of particular interest, as here vibrations of low-frequency eigenmodes

become nonlinear for already small vibration amplitudes.

A familiar consequence of nonlinearity in nano- and micromechanical systems is the depen-

dence of the vibration frequency of a mode on its own amplitude and on the amplitudes of other

modes, see Ref. [115] for a review. In particular, the change δων of the mode frequency due to the
vibrations of the mode itself, Qν (t) = Aν cosωνt, is [124, 125]

(cid:35)

(cid:34)

δων ≈

3γν
8Mων −∑
ν(cid:48)

ννν(cid:48)(3ω2
β 2
4M2ω3
ν (ω2

ν(cid:48) − 8ω2
ν )
ν(cid:48) − 4ω2
ν )

A2
ν ,

(2.5)

where γν ≡ γνννν and we kept the terms of the first order in γ and the second order in β .

The nonlinear mode coupling (2.4) leads also to the frequency shift due to thermal vibrations

of the modes. The dominating contribution to this shift for low-frequency modes comes from their
coupling to modes with frequencies ∼ kBT /¯h, which have a much higher density of states. This
shift is described by an expression that is similar to Eq. (2.5) with A2
ν(cid:48) ∼ kBT /Mω2
ν(cid:48)
and placed under the sum over ν(cid:48), in the classical limit.

ν replaced by A2

2.3 The nonlinearity due to the electron-phonon coupling

We will consider the vibration nonlinearity due to the electron-phonon coupling in multi-valley

semiconductors with cubic symmetry, silicon and germanium being the best known examples. In

such semiconductors, the energy valleys of the conduction band are located at high-symmetry axes

of the Brillouin zone. Strain lifts the symmetry and thus the degeneracy of the valleys.

The simplest mechanism of the electron-phonon coupling is the deformation potential. Here,
the energy shift δ Eα of valley α is determined by the deformation potential parameters Ξu and
Ξd. Parameter Ξu gives the valley shift due to the strain along the symmetry axis of the valley. In
contrast, parameter Ξd gives the shift related to dilatation; such shift is the same for all valleys.
These parameters are well known for various semiconductors [126]. In terms of the strain tensor
εi j, we have δ Eα = ∑i j Ξ(α)
i j ε ji, where ˆΞ(α) = Ξd ˆI +Ξueee(α)⊗eee(α), with eee(α) being the unit vector

14

along the symmetry axis of the valley. We use the hat symbol to indicate tensors and symbol “⊗"
to indicate tensor products (eee(α) ⊗ eee(α) singles out the strain along the axis of valley α). The
analysis below is not limited to the deformation potential approximation. An important extension

will be discussed using silicon as an example.

We assume that the strain varies in time and space slowly compared to the reciprocal rate of

intervalley electron scattering and the intervalley scattering length, respectively. Then the electron
system follows the strain adiabatically. The electron density n(α)(rrr) in valley α is decreased or

increased depending on whether the bottom of the valley goes up or down. In the single-electron

approximation and for the deformation potential coupling, the electron free energy density for a
given strain is Fe = ∑α{ fe[n(α)(rrr)] +n(α)(rrr)Ξ(α)
i j ε ji(rrr)} where fe[n(rrr)] is the free energy density
for electrons with density n(rrr) in a valley in the absence of coupling to phonons.

The electro-neutrality requires that the total electron density summed over the valleys be con-
stant. The free energy density Fe has to be minimized over n(α)(rrr) to meet this constraint. This
gives the change of the electron chemical potential δ µ due to strain ˆε. The resulting increment of

the electron free energy density has the form of a series expansion in the strain tensor,

δ Fe =(cid:98)Λ1 · ˆε +

2(cid:98)Λ2 · ˆε ⊗ ˆε +

1

6(cid:98)Λ3 · ˆε ⊗ ˆε ⊗ ˆε

1

24(cid:98)Λ4 · ˆε ⊗ ˆε ⊗ ˆε ⊗ ˆε + ...

1

+

Here(cid:98)Λ1,(cid:98)Λ2,(cid:98)Λ3, and(cid:98)Λ4 are tensors of ranks 2, 4, 6, and 8, respectively. They are contracted with
the tensor products of the strain tensor ˆε. Respectively,(cid:98)Λk are the electronic contributions to the

(2.6)

linear (for k = 2) and nonlinear (for k > 2) elasticity parameters of the crystal. These contributions

are isothermal, but since the change of the mode frequencies from the electron-phonon coupling

is small and the nonlinearity is also small, the difference with the adiabatic expressions can be

disregarded.

To the third order in ˆε the expression for δ Fe in terms of the shift of the valleys was found by

Keyes [120] in the analysis of sound wave propagation. However, to find the parameters of the

quartic nonlinearity of resonant modes in small systems, which is of primary interest to us, we also

15

need to keep quartic terms in Eq. (2.6).

As seen from the explicit form of the parameters of the expansion (2.6) given in Appendix A.1,

(cid:98)Λk ∝ Ξu[Ξu/max(µ0,kBT )]k−1 (k = 1,2, ...), where µ0 is the electron chemical potential in the

absence of strain; it is determined by the total (summed over the valleys) electron density n. Of
central importance for the analysis is that parameter Ξu/max(µ0,kBT ) ∼ 103 for electron densities
n ∼ 1019 cm−3 and room temperatures, i.e.

Ξu/max(µ0,kBT ) (cid:29) 1.

(2.7)

As a consequence, the coefficients at the nonlinear in ˆε terms in Eq. (2.6) quickly increase with

the increasing order of the nonlinearity [the overall series (2.6) is converging fast because of the

smallness of the strain tensor].

The increase of(cid:98)Λk with k allows us to keep in ˆε only the terms linear in the lattice displacement,

i.e., to set εi j = (1/2)(∂ ui/∂ x j +∂ u j/∂ xi), where ui and xi are the components of the displacement
and the coordinates, respectively. Indeed, in this case a kth term of the series (2.6) is of order k

in the displacement. If we included the quadratic in ∂ ui/∂ x j term into one of the ˆε tensors in the
kth term, this term would become of order k + 1 in the displacement. However, for linear ˆε the

(k + 1)th term in the series (2.6) is also of the (k + 1)th order in the displacement, but is larger
by factor Ξu/max(µ0,kBT ). We note that, for the modes that involve rotation, one should use the
relative lattice displacement uuu(rrr) rather than the overall displacement [123].

For linear ˆε, the total strain is a sum of partial contributions of strain from individual modes.
For mode ν, such partial contribution is expressed in terms of the scaled displacement uuu(ν)(rrr)
[see Eq. (2.2)] as ˆε = Qν ˆε(ν), where ε(ν)
(rrr)/∂ xi]. We note that, in
contrast to the dimensionless strain tensor ˆε, tensor ˆε(ν) has dimension [length]−1.

(rrr)/∂ x j + ∂ u(ν)
j

i j (rrr) = 1

2 [∂ u(ν)
i

From Eq. (2.6) we find the electronic contributions to the nonlinearity parameters β (e)

in Hamiltonian (2.4) ,

(cid:90)
(cid:90)

1
2
1
6

drrr(cid:98)Λ3 · ˆε(ν1) ⊗ ˆε(ν2) ⊗ ˆε(ν3),
drrr(cid:98)Λ4 · ˆε(ν1) ⊗ ˆε(ν2) ⊗ ˆε(ν3) ⊗ ˆε(ν4),

β (e)
ν1ν2ν3 =

γ(e)
ν1ν2ν3ν4 =

16

ν1ν2ν3,γ(e)

ν1ν2ν3ν4

(2.8)

where ˆε(ν) ≡ ˆε(ν)(rrr); tensors(cid:98)Λk are independent of rrr.

Similarly, the electronic contribution to the eigenfrequency is

(cid:90)

drrr(cid:98)Λ2 · ˆε(ν) ⊗ ˆε(ν).

∆ω(e)

ν =

1

2Mων

erate, this mixing is weak and can be disregarded, to the leading order in the electron-phonon

Generally, the term ∝(cid:98)Λ2 leads to mode mixing; however, if the mode frequencies are nondegen-
coupling. One can see that the effect of the static stress ∝(cid:98)Λ1 can be disregarded as well.
(cid:98)Λ2. The nonlinearity (2.8) also leads to a temperature dependence of the mode eigenfrequency.

The frequency change (2.9) depends on temperature because of the temperature dependence of

(2.9)

Together they modify the temperature dependence of the mode eigenfrequencies compared to that

of undoped crystals. This modification often weakens the temperature dependence of the eigen-

frequencies, which proves very important for applications of micro-mechanical systems in devices

that work in a broad temperature range [127].

Equations (2.6) - (2.9) are generic and apply beyond the deformation potential approximation.
This is of particular importance for silicon. Here, the electron band valleys lie on the (cid:104)100(cid:105)-axes
close to the X-points on the zone boundaries where two electron energy bands cross. Lattice

strain can lead to a band splitting at X-points and a shift of the valleys [128, 129]. Importantly,

this shift results from a shear strain, which does not lead to a linear in the strain shift in the

deformation potential approximation. The valley shift is quadratic in ˆε in this case, as explained in

Appendix A.1, which corresponds to an effectively two-phonon coupling. The coupling parameter
Ξsh is quadratic in the strain-induced band splitting, see Eq. (7). It is large, much larger than the
constant Ξu. Therefore the arguments given below Eq. (2.7) apply in this case as well. For purely
shear strain in silicon, terms of odd order in ˆε in δ Fe, Eq. (2.6), vanish.

2.4 Explicit form of the tensors of nonlinear elasticity

Tensors(cid:98)Λn can be obtained by minimizing the free energy density of the electron system for a

given strain and expanding the result in a series in ˆε. A general procedure that allows one to find

17

(a)

(b)

(c)

Figure 2.2 Corrections to some elastic constants as functions of electron density, at T = 300 K.
(a) δ c12 and δ c44; (b) δ c111 and δ c144; (c) δ c1111, δ c1144, and δ c4444. Corrections to other
constants are proportional to one of those depicted.

18

101810191020n [cm−3]0.070.060.050.040.030.020.010.00δc [1012dyn/cm2]δc11δc44101810191020n [cm−3]0.00.51.01.52.02.53.0δc [1012dyn/cm2]δc111δc144101810191020n [cm−3]2001000100200300400500δc [1012dyn/cm2]δc1111δc1144δc4444(a)

(b)

(c)

Figure 2.3 Corrections to some elastic constants as functions of temperature, at electron density
n = 5 × 1019 cm−3. (a) δ c12 and δ c44; (b) δ c111 and δ c144; (c) δ c1111, δ c1144, and δ c4444.
Corrections to other constants are proportional to one of those depicted.

19

50100150200250300350400450T [K]0.0450.0400.0350.0300.0250.0200.0150.0100.005δc [1012dyn/cm2]δc11δc4450100150200250300350400450T [K]1.01.11.21.31.41.51.61.7δc [1012dyn/cm2]δc111δc14450100150200250300350400450T [K]2001000100200300400500600δc [1012dyn/cm2]δc1111δc1144δc4444Table 2.1 The change of the components of the nonlinear elasticity tensors due to the strain-induced
electron redistribution between equivalent energy valleys in doped silicon. The coordinate axes are
chosen along the (cid:104)100(cid:105) axes. Parameter Ξsh characterizes the effectively two-phonon coupling to
shear strain. This parameter as well as function F1/2(x) are defined in Appendix A.1; x = µ0/kBT
and n is the electron density.

1
9nΞuΞshC1
2
27nΞ4
uC2

− 1
27nΞ2

uΞshC3

C1 = F(cid:48)1/2/F1/2kBT = d lnn/dµ0
C2 = (kBT )−3F(cid:48)2
1/2
d2(dµ0/dn)/dµ2
0

= (dn/dµ0)2(cid:104)

d2(1/F(cid:48)1/2)/dx2(cid:105)
(cid:104)
(cid:105)

/n
C3 = F(cid:48)(cid:48)1/2/F1/2(kBT )2 = n−1d2n/dµ2
0

/F1/2

−1
6nΞ2

shC4

C4 = F(cid:48)1/2/F1/2kBT = d lnn/dµ0

δ c144 = −2δ c155
δ c1111 = −2δ c1112

= 2δ c1122

δ c1144 = −2δ c1155
= −2δ c1244 = δ c1266
δ c4444 = −6δ c4455

elasticity tensors are conveniently written in the contracted (Voigt) notation where the symmetric

the components(cid:98)Λn for n ≤ 4 is described in Appendix A.1. Using the symmetry arguments, the
strain tensor is associated with a six-component vector. Then the nonlinear elasticity tensors (cid:98)Λ3
and(cid:98)Λ4 become tensors of rank three and four in the corresponding vector space. We use notation
δ ˆc for tensors(cid:98)Λ in these notations to emphasize that we are calculating corrections to the nonlinear

elasticity tensors due to the electron-phonon coupling.

The explicit expressions for the nonlinear elasticity tensors δ ˆc are given in Table 2.1. They

refer to silicon and include the contributions that come from both the deformation potential cou-

pling and from the splitting of the electron bands due to shear strain. In the deformation potential

approximation, the components of the third-rank tensor δ ˆc, which determine the cubic in the strain

terms in the free energy, were found earlier [121]. Therefore we give only the components that

contain a contribution from shear strain. Numerically estimated concentration and temperature

behavior of some elastic constants corrections are shown in Figs. 2.2 and 2.3.

The fourth-rank tensor δ ˆc determines the quartic in the strain terms in the free energy and has

not been discussed before, to the best of our knowledge. We give all independent components of

this tensor. It is expressed in terms of the derivative of the electron density n over the chemical po-
tential in the absence of strain µ0, which is a familiar thermodynamic characteristic. It is intuitively

20

clear that the considered effect of the change of the electron density in different valleys in response
to strain should be related to the derivative dn/dµ0. Interestingly, because we consider nonlinear
response to strain, the expressions in Table 2.1 contain also higher-order derivatives of n over µ0.
As we will see, this leads to a nontrivial behavior of the nonlinear frequency shift with varying

temperature and density. The considered mechanism of the strain-induced inter-valley electron re-
distribution does not contribute to the components c1123 and c1456, therefore δ c1123 = δ c1456 = 0.

2.4.1 Nonlinear elasticity in the limiting cases

The expressions for δ ˆc simplify in the case of low doping (or high temperature), where the electron

gas is strongly nondegenerate, and in the opposite case of a strongly degenerate electron gas. For a
nondegenerate gas, where the chemical potential in the absence of strain is µ0 < 0,|µ0| (cid:29) kBT , we
have in Table 2.1 F1/2(x) = 1
4π1/2ex with x = µ0/kBT . The µ0-dependent factors exp(µ0/kBT )
in F1/2 and its derivatives cancel each other in the expressions for δ ˆc and drop out from these
expressions. The dependence of δ ˆc on density is then just linear, δ ˆc ∝ n. Parameters C1,...,4 in
Table 2.1 depend only on temperature, C1 ∝ T−1,C2 ∝ T−3,C3 ∝ T−2 and C4 ∝ T−1.

The decrease of the nonlinear elasticity parameters with increasing temperature in a nondegen-

erate electron gas is easy to understand. The effect we consider is determined by the competition

between the energetically favorable unequal population of the electron energy valleys in a strained

crystal and the entropically more favorable equal valley population. With increasing temperature

the entropic factor becomes stronger, leading to a smaller population difference and thus smaller

effect of the electron system on the vibrations.

For strong doping, where µ0/kBT (cid:29) 1, we have µ0 ∝ n2/3, and then F1/2(x) ≈ 2

3x3/2 with
x = µ0/kBT . Therefore parameters C1,...,4 in Table 2.1 become temperature independent, with
nC1 ∝ n1/3,nC2 ∝ n−1,nC3 ∝ n−1/3, and C4 ∝ n1/3.

The results on the asymptotic behavior of the corrections to nonlinear elasticity are not limited

to silicon. Since parameters C1,2,3,4 are given by the coefficients in the general expansion of
the free energy in strain, (6), these results can be applied to the nonlinear elasticity induced by

21

the electron-phonon coupling in other multi-valley semiconductors. To illustrate this point, in

Appendix A.2 we give δ ˆc tensor in germanium.

The difference between the asymptotic behavior of the tensors δ ˆc in the limits of nondegenerate

and strongly degenerate electron gas can lead to a peculiar density and temperature dependence

of the nonlinear frequency shift of the vibrational modes. It comes from the coefficients C1,...,4
containing higher-order derivatives of n with respect to µ0. In the transition region µ0 ∼ kBT ,
thinking of the competition between the entropic and energetic factors does not provide a simple

insight into the behavior of δ ˆc, as both the energy and the entropy are complicated functions of

density and temperature.

2.5 Doping-induced nonlinearity of simple vibrational modes

The nonlinear elasticity tensors in Table 2.1 give the doping-induced contributions to the non-

linearity parameters of the eigenmodes of micro- and nanomechanical systems. These contribu-

tions are described by Eq. (2.8). As mentioned before, an important characteristic of the mode

nonlinearity is the dependence of the mode frequency on the vibration amplitude. To the leading

order, it is given by Eq. (2.5). This dependence has a contribution from the nonlinearity of an

undoped crystal, which is quadratic in the parameters of the cubic nonlinearity; for example, if the

latter is described by the Grüneisen constant, the corresponding contribution is quadratic in this

constant. It is typically small. There is also a contribution from the quartic nonlinearity; the pa-

rameters of such nonlinearity are not known in undoped crystals and are not expected to be large.

Respectively, the amplitude dependence of the vibration frequency for low-frequency modes in

weakly doped single-crystal micro-mechanical systems is relatively weak [24].

A feature of the doping-induced nonlinearity described by Table 2.1 is that the quartic in the

strain term in the free energy has a large coefficient compared to the cubic term, cf. Eq. (2.7)

and the discussion below this equation. Therefore, in Eq. (2.5) for the amplitude dependence of

vibration frequency one can keep only the Duffing nonlinearity constant γν. The contribution
from the cubic nonlinearity terms ∝ β 2
ννν(cid:48) can be disregarded. For a mode ν, the doping-induced

22

Figure 2.4 Relative change δων /ων of the vibration frequency of a mode with the vibration ampli-
tude ην scaled by the relevant size of the system, cf. Eq. (2.10). The results refer to single crystal
silicon resonators. Curves 1 and 2 refer to the first Lamé mode in square plates cut in (cid:104)100(cid:105) and
(cid:104)110(cid:105) directions, respectively. In this case, the size of the resonator is the length of the side of the
square. Curves 3 and 4 refer to the first extension mode in beams cut in (cid:104)100(cid:105) and (cid:104)110(cid:105) directions,
respectively. In this case, the size of the resonator is the length of the beam.

(b)

(a)

Figure 2.5 Mode shapes, displacement shown by color from blue to red, dark blue denoting zero
and red maximum displacement. (a) first Lamé mode, (b) length extension mode

contribution to γν is equal to γ(e)

νννν in Eq. (2.8).

To find the dependence of the mode frequency on the vibration amplitude we go through the

following steps. First, we find the normal modes of interest for the given geometry of the system,
with account taken of the boundary conditions, and normalize the displacements uuu(ν)(rrr) as indi-
cated in Eq. (2.1). We use uuu(ν)(rrr) to find the strain tensor ˆε(ν)(rrr). The result is substituted into

Eq. (2.8) and is convoluted with tensor(cid:98)Λ4, giving the value of γν, which is then used in Eq. (2.5)

23

101810191020n (cm−3)04008001200δων/ωνη2ν3412T = 300 K100200300400T (K)04008001200δων/ωνη2ν3412n=5.9×1019 cm−3to find the frequency dependence on the vibration amplitude δων.

Of particular interest is the relative frequency shift δων /ων. To find this shift to the lead-
ing order, one can disregard nonlinearity when calculating the eigenfrequency ων. Then, from
Eq. (2.5),

δων
ων

3γνA2
ν
· ˆε(ν) ⊗ ˆε(ν)
where(cid:98)Λ( f )
ticity of the undoped crystal and the doping-induced correction(cid:98)Λ2.

=

2

is the full tensor of linear elasticity, which includes the major term of the linear elas-

8(cid:82) drrr(cid:98)Λ( f )

2

,

(2.10)

An important feature of the relative shift δων /ων is its scaling with the size of the system.
The vibration amplitude Aν in Eq. (2.10) can be scaled by the lateral dimension L, for example the
length of a nanobeam or a nanowire for an extension mode, or the size of the square for a Lamé

mode, or the diameter of a disk for a breathing mode in a disk. Respectively, we write Aν = ηνL.
Then, if one takes into account the explicit form (2.8) of the parameter γν = γ(e)
νννν, one finds from
Eq. (2.10) that the ratio δων /(η2
νων ) is independent of the system size for the aforementioned

modes. In this estimate we used that the tensors(cid:98)Λ are material parameters and are independent of
the geometry. We also used that the modes of interest have typical wavelength ∼ L, and therefore
ˆε(ν) scales as L−1.

Most of the experiments in nano- and micromechanics are done with nanobeams, nanowires,

membranes, or thin plates. In such systems the thickness is much smaller than the length or, in

the case of membranes or plates, the lateral dimensions. Then, from the boundary condition of

the absence of tangential stress on free surfaces [123], it follows that the strain tensor ˆε weakly

depends on the coordinate normal to the surface. This simplifies the denominator in Eq. (2.10),

making it proportional to the thickness. Similarly, from Eq. (2.8) γν is also proportional to the
thickness, and the thickness drops out of Eq. (2.10).

The explicit expressions for Mω2

ν and γν that determine the denominator and the numerator
in Eq. (2.10), respectively, are given in Appendices A.4 and A.5 for Lamé and extension modes.

These expressions are cumbersome, and it is convenient to use symbolic programming to obtain

24

Table 2.2 Elastic constants of pure Si at room temperature in units of 1012 dyn/cm2

c112

c111

c44
c123
0.795 −7.95 −4.45 −0.75
0.796 −8.25 −4.51 −0.64

c166

c144
c456
0.15 −3.10 −0.86
0.12 −3.10 −0.64

Ref.
[121]
[130]

c11
1.656
1.658

c12
0.639
0.639

them. 1

2.5.1 Temperature and electron density dependence of the scaled nonlinear frequency shift

The scaled ratio δων /(η2
νων ) that characterizes the relative nonlinear frequency shift is shown in
Fig. 2.4 for several modes that are often used in single-crystal silicon MEMS. This ratio depends

on the type of the mode and the crystal orientation. Figure 2.4 refers to high-symmetry crystal

orientations, in which case the modes have a comparatively simple spatial structure and the surfaces
can be made smooth. We used the values Ξu = 8.8 eV [126], Ξsh = 300 eV, the effective mass for
density of states me f f = 0.32me [129], and the temperature-dependent linear elasticity parameters
given in Ref. [131]. We also used experimental values of third-order elastic constants of pure

silicon measured in [121] (see Table 2.2).

Figure 2.4 shows that the electron-redistribution induced nonlinearity of vibrational modes
is very strong. For the ratio of the vibration amplitude to the system size η ∼ 10−4 and the
mode eigenfrequency ων /2π ∼ 10 MHz, the frequency change can be as a large as δων /2π ∼
0.1 kHz. This explains, qualitatively, the observations [24]. A quantitative comparison with the

experiment [24] is complicated, as the observations refer to different samples. Our preliminary

results show an excellent quantitative agreement with the data obtained for the same sample at

different temperatures and for different types of modes [132].

The nonlinear frequency shift displays several characteristic features, as seen from Fig. 2.4.

One of them is the strong dependence of the shift on the type of the mode and the crystal orienta-

tion. For both the Lamé and the extension mode, the shift is much stronger for crystals cut out in

1The program that performs the analytical calculations and evaluates the numerical val-
ues of the parameters is available at http://www.pa.msu.edu/people/dykman/nonlinear_
elasticity.

25

(cid:104)100(cid:105) direction than in (cid:104)110(cid:105) direction. This is a consequence of the electron energy valleys lying
along the (cid:104)100(cid:105) axes, making the system more “responsive" to the lattice displacement along these
axes. Interestingly, in the both configurations the shifts for the Lamé modes are larger than for the

extension modes.

A somewhat unexpected feature is the nonmonotonic dependence of the nonlinear frequency

shift on the electron density and temperature. The nonmonotoncity occurs in the range where
the electron system is close to degeneracy, µ0/kBT ∼ 1, and it strongly depends on the crystal
orientation. It is much stronger for crystals cut in (cid:104)100(cid:105) than (cid:104)110(cid:105) directions. For a crystal cut
in (cid:104)110(cid:105) direction, both the density and temperature dependence of the shift are monotonic in the
case of the Lamé mode, whereas for the extension mode the nonmonotonicity is weak.

The nonmonotonicity of the frequency shift stems from the behavior of the parameters nC2,3,4
in the range µ0 ∼ kBT . As seen from Table 2.1, parameter nC2 exponentially increases with the in-
creasing µ0/kBT for negative µ0/kBT , but for large positive µ0/kBT it falls off as (µ0/kBT )−3/2.
It has a pronounced maximum for µ0/kBT ≈ 0.6. Parameter nC3 also displays a maximum, which
occurs for µ0/kBT ≈ 1.1. In contrast, parameters nC1,4 depend on µ0/kBT monotonically.

The results of Appendices A.4 and A.5 show that, for the Lamé and extension modes in crystals
cut in (cid:104)100(cid:105) direction, the relative shift δων /ων is determined by coefficient nC2, which explains
the nonmonotonicity of the shift. For crystals cut in (cid:104)110(cid:105), the shift of the Lamé mode is fully
determined by coefficient nC4 and is monotonic, whereas for the extension mode the expression
for the shift has contributions from nC2, nC3, and nC4 that partly compensate each other, leading
to a comparatively small shift all together and its weak nonmonotonicity.

2.6 Linear and nonlinear elasticity in polysilicon

In polycrystalline silicon, grains have random orientations. We will assume each grain to be

a perfect single crystal, while on average the material is isotropic. In this section, we estimate

effects of electron-phonon interaction on linear and nonlinear elasticity of polysilicon. We disre-

gard effects of grain boundaries, which are not necessarily small, since there is no reliable way

26

to account for them. Since so much depends on the actual material structure, the calculation that

follows should be regarded as an order of magnitude estimate only. If the grains are small enough,

one can assume that the chemical potential is the same in all grains under any spatially uniform

strain. Then the calculation of corrections to tensors of elasticity proceeds in a way similar to that

presented in Appendix A.1, but we must calculate the chemical potential and the free energy taking

into account the polycrystalline nature of the material.

Each grain is characterized by its orientation. We will describe orientation of each grain with

respect to some fixed orientation using Euler angles: rotation by φ around axis z, followed by a
rotation by θ around x(cid:48), and finally rotation by ψ around z(cid:48)(cid:48). The matrix of such rotation is

R = Z(φ )X(θ )Z(ψ)

where




X(θ ) =

Z(φ ) =




0

0

1
0 cos (θ ) −sin (θ )
cos (θ )
0 sin (θ )

cos (φ )
sin (φ ) 0
−sin (φ ) cos (φ ) 0
1

0

0

All possible orientations of a body are uniquely defined by φ ∈ [0,2π), θ ∈ [0, pi), and ψ ∈ [0,2π).
Valley bottom of valley x in a grain with orientation defined by R is shifted under strain by

(cid:16)

(cid:17)

(cid:16)

(cid:17)2

(cid:16)

(cid:17)4

δ Ex = Ξ(x)
kl

R−1
kp R−1

lq εpq

− Ξsh

R−1
yp R−1

zq εpq

+ Ξs4

R−1
yp R−1

zq εpq

,

(2.11)

where Ξ(x) is the deformation potential tensor in valley x, and R−1 ≡ Z(−φ )X(−θ )Z(−ψ). Sum-
mation over repeated indices is assumed. Chemical potential in a single crystal is calculated

through Eqs. (2) and (3). In a polycrystal, the calculation proceeds the same way, but we need
to average terms of the form ∆m
ε over all possible orientations. If we denote this averaging by (cid:104)(cid:105),
ε

(cid:11) = (cid:104)(δ Ex/kBT )m(cid:105) since all valleys become

we obtain terms of the form(cid:10)∆m

(cid:11). We argue that(cid:10)∆m

ε

27

equivalent upon averaging over all grain orientations. The same averaging procedure should be

applied to the free energy calculation, Eq. (6). Thus, the free energy calculation in a polycrystal
should be done in the way described in Appendix A.1, but with ∆m

ε replaced with

(cid:104)(δ Ex/kBT )m(cid:105) =

1

(kBT )m

R−1
kp R−1

lq εpq

R−1
yp R−1

zq εpq

R−1
yp R−1

zq εpq

(cid:42)(cid:18)

(cid:16)

Ξ(x)
kl

+ Ξs4

(cid:16)
3Ξs2 −
(cid:33)

8
3

(cid:17)

− Ξsh

(cid:16)
(cid:17)4(cid:19)m(cid:43)


16
9

δ c(p)
456

(cid:17)2

(2.12)

(2.13)

2

uF(cid:48)(cid:48)1/2

16nΞ4
675F1/2T 3F(cid:48)1/2

Then the corrections to tensors of elasticity are found by Eq. (8):

2Ξ2
uF(cid:48)1/2
F1/2T
δ c(p)

4
3

δ c(p)

44 =

2n
45

δ c(p)
11 = −2δ c(p)
12 =
δ c(p)
111 = −2δ c(p)
8nΞu

(cid:32)

=

112 = δ c(p)

123 = −
9Ξs2F(cid:48)1/2 +

4
3

Ξ2
u

155 =

144 =

δ c(p)
2F(cid:48)(cid:48)1/2
T
1144 = 4δ c(p)
1122 = 4δ c(p)
1155
16
16
δ c(p)
4444 =
1266 =
9
3
uΞs2F(cid:48)(cid:48)1/2

32nΞ2
1575F1/2T 2 −

945F1/2T
1112 = 2δ c(p)
1111 = −2δ c(p)
δ c(p)
1244 = −8δ c(p)
= −8δ c(p)
s2F(cid:48)1/2
8n
Ξs4 −
35

32nΞ2
525F1/2T −

=

δ c(p)
4455
16nΞ4
945F1/2T 3 +

uF(cid:48)(cid:48)(cid:48)1/2

δ c(p)
δ c(p)

1123 = 0
1456 = 0

Numerical values of these corrections as functions of electron density and temperature are shown

in Figs. 2.6 and 2.7 respectively. The predicted corrections in polysilicon are generally smaller

than those in the single crystal, but still significant.

Above, we presented corrections to tensors of elasticity in polysilicon, but did not touch the cal-

culation of intrinsic values of the tensors without electronic effects. The latter is done by averaging

of single-crystal tensors for pure silicon over all possible orientations. Consider a single-crystal

grain with orientation defined by R relative to the lab frame. In the lab frame, the stiffness tensor

has a new value c(cid:48)i jkl that is related to its value in crystal coordinates ci jkl through some tensor

28

(a)

(b)

(c)

Figure 2.6 Corrections to elastic constants of polycrystalline silicon as functions of electron den-
sity, at T = 300 K. (a) δ c11; (b) δ c111; (c) δ c1111. Corrections to other constants are proportional
to one of those depicted.

29

101810191020n [cm−3]0.050.040.030.020.010.00δc [1012dyn/cm2]δc11101810191020n [cm−3]1.21.00.80.60.40.20.00.2δc [1012dyn/cm2]δc111101810191020n [cm−3]01020304050607080δc [1012dyn/cm2]δc1111(a)

(b)

(c)

Figure 2.7 Corrections to elastic constants of polycrystalline silicon as functions of temperature, at
electron density n = 5×1019 cm−3. (a) δ c11; (b) δ c111; (c) δ c1111. Corrections to other constants
are proportional to one of those depicted.

30

50100150200250300350400450T [K]0.0240.0230.0220.0210.0200.0190.0180.0170.016δc [1012dyn/cm2]δc1150100150200250300350400450T [K]0.600.550.500.450.400.350.30δc [1012dyn/cm2]δc11150100150200250300350400450T [K]30405060708090100δc [1012dyn/cm2]δc1111Amnpqi jkl:

c(cid:48)i jkl = Amnpqi jklcmnpq.

To find tensor A, we consider the elastic free energy that should be a scalar independent of the

coordinate system choice:

1
2

c(cid:48)i jklεi jεkl =

1
2

Amnpqi jklcmnpqεi jεkl =

1
2

cmnpq ˆεmn ˆεpq,

(2.14)

where the first and last expressions are written in the lab and crystal frames respectively. We know

that in the crystal frame,

Therefore,

ˆεpq = R−1

pk R−1

ql εkl.

Amnpqi jklcmnpqεi jεkl = R−1

mi R−1

n j R−1

pk R−1

ql cmnpqεi jεkl,

that means Amnpqi jkl ≡ R−1
mi R−1
(cid:90) π
tensor. It means that the average tensor component in a polycrystal would be

n j R−1
pk R−1
(cid:90) 2π

(cid:90) 2π

ql , which is of course the transformation for a fourth-order

c(poly)
i jkl = (cid:104)c(cid:48)i jkl(cid:105) =

0

dθ

dφ

0

0

dψ sinθρ(ααα)Amnpqi jkl(φ ,θ ,ψ)cmnpq

(2.15)

where ρ(ααα) = 1/8π2 for the uniform on [0,2π)× [0,π)× [0,2π) distribution of angles. Third- and
fourth- order elastic constants are found in a similar way, using appropriate transformation tensors.

31

The resulting values for second- and third- order elastic constants are given below:

c(p)
11 =
c(p)
12 =
c(p)
44 =
c(p)
111 =
c(p)
112 =
c(p)
123 =
c(p)
144 =
c(p)
155 =
c(p)
456 =

1
5(3c11 + 2c12 + 4c44),
1
5(c11 + 4c12 − 2c44),
1
5(c11 − c12 + 3c44),
1
35 (15c111 + 18c112 + 2c123 + 12c144 + 72c155 + 16c456) ,
1
35 (3c111 + 26c112 + 6c123 + 8c144 − 8c155 − 8c456) ,
1
35 (c111 + 18c112 + 16c123 − 30c144 − 12c155 + 16c456) ,
1
35 (c111 + 4c112 − 5c123 + 19c144 + 2c155 − 12c456) ,
1
35 (3c111 − 2c112 − c123 + c144 + 20c155 + 6c456) ,
1
35 (c111 − 3c112 + 2c123 − 9c144 + 9c155 + 9c456) ,

(2.16)

These values are not all linearly independent because of the isotropic symmetry, but we give them

all here for completeness. Fourth-order elastic constants are not shown since the expressions are

too lengthy and their values for pure silicon are unknown. One may notice that we can apply

equations of the type (2.16) to electronic corrections δ cIJKL given in Table 2.1. That would give us
δ c(p)
IJKL calculated in the opposite limit of isolated grains, where chemical potential is independent
in each grain. We note that numerical values obtained in this limit are remarkably close to those

given in Eqs. (2.13), within 20%, and display similar dependence on temperature and electron

density.

2.7 Conclusions

The results of this chapter show that the electron-phonon coupling strongly affects the non-

linearity of vibrational modes in semiconductor-based nano- and micromechanical systems. The

mechanism of the effect is the strain-induced redistribution of the electrons between the valleys of

the conduction band. The redistribution results from lifting the degeneracy of the electron energy

spectrum by the strain from a vibrational mode. The analysis refers to the range of temperatures

32

where the rate of intervalley scattering strongly exceeds the frequencies of the considered modes.

In this case the valley populations follow the strain adiabatically.

The change of the valley populations is a strongly nonlinear function of the strain tensor. The

respective expansion of the free energy in the strain is an expansion in the strain multiplied by

the ratio of the electron-phonon coupling energy (in particular, the deformation potential) to the
chemical potential of the electron system or the temperature. This ratio is large, (cid:38) 103. It is this
parameter that makes the nonlinearity of the vibrational modes in doped semiconductor structures

strong.

Of special interest in nano- and micromechanical systems is the amplitude dependence of the

vibration frequency. To the leading order, it is determined by the quartic terms in the expansion of

the free energy in strain. These terms are comparatively large in doped crystals.

We have calculated the nonlinear elasticity tensor that describes the electron contribution to

the terms in the free energy, which are quartic in the strain. The explicit expressions for the
tensor components refer to semiconductors with the valleys on (cid:104)100(cid:105) axes, in particular, to silicon.
We have also found this tensor for germanium. In silicon, along with the deformation potential

coupling, an important role is played by the coupling to shear strain. Such strain lifts the band

degeneracy at the zone boundary and is effectively described by a two-phonon coupling. We show

that this coupling also leads to strong nonlinearity of vibrational modes.

The parameter of the electron coupling to shear strain in silicon is not easy to access in the

experiment [129, 133]. Measurements of the nonlinear frequency shift provide a direct means for

determining this parameter. In particular, the nonlinear frequency shift of the fundamental Lamé
mode in a silicon plate cut along (cid:104)110(cid:105) axes is determined by this parameter only, except for small
corrections from the nonlinearity of the undoped crystal.

We found that the nonlinear frequency shift strongly depends on the type of a vibrational mode

and the crystal orientation. We also found that the ratio of the frequency shift to the squared vibra-

tion amplitude can be profoundly nonmonotonic as a function of electron density and temperature.

The results provide an insight into the experimentally observed strong mode nonlinearity in doped

33

crystals [24]. In terms of applications, they enable choosing the appropriate range of doping and

the temperature regime to optimize the operation of nano- and micromechanical resonators.

34

CHAPTER 3

ELECTRONS ON HELIUM IN A PERIODIC POTENTIAL

3.1 Introduction

Electrons on the surface of liquid helium have been studied extensively experimentally and

theoretically [134]. They constitute one of the most perfect two-dimensional electron systems.

The highest mobility among all known condensed matter systems has been measured in electrons

on helium [25]. Such high mobility is a consequence of the absence of defects inherent to any solid

state system. Therefore, electrons on helium surface are ideally suited for studying effects of the

interaction of the electrons with each other as well as their interaction with the quantized fields of

surface and bulk excitations in helium and with substrate-induced potentials.

A typical experimental geometry and the energy spectrum of electrons floating on the helium

surface are shown schematically on Fig. 3.1. Liquid helium covers one plate of a parallel plate

Figure 3.1 (a) The geometry of typical experiments on a spatially uniform electron system on the
helium surface (schematically). Electrons reside in vacuum above the interface and are pressed
to the surface by image potentials and the capacitor electric field. The capacitor electrode creates
neutralizing background for electrons and determines the maximum electron density.
(b) The
hydrogenic energy levels of motion transverse to the helium surface in the image potential V (z) =
−Λe2/z for z > 0. Also sketched is the wave function of the ground state (m = 1) of motion normal
to the surface.

35

Liquid heliumElectronsm=2rE = 6KHeliumz<0Vacuumz>0m=1∆B= 76 A(a)(b)Figure 3.2 Matrix elements (cid:104)m|z|n(cid:105) of the electron coordinate z and the energy difference E2 − E1
between the first excited state |2(cid:105) and the ground state |1(cid:105) of the z motion vs the pressing field E⊥.
eE⊥rB/R = 1 for E⊥ ≈ 0.91 kV/cm.
capacitor, which is connected to a voltage source. The thickness of the helium film is typically
d ≥ 0.5 µm. Electrons are supplied to the surface by briefly heating a filament (cathode) located
near the top capacitor plate, while negative voltage is applied to the filament. The voltage applied

across the capacitor determines the ultimate charge density on the surface and creates a uniform

neutralizing background in the lower plate for the surface electrons. A variation of this geometry

has been used recently where the electrons are confined in a narrow channel above the back gate

[30], but the physics is essentially the same.

An electron is confined to the surface by a relatively weak image potential and by the voltage

that can be applied to the capacitor. To a good approximation, the image potential is of the form
V (z) = −Λe2/z for z > 0 [134], where z is the direction normal to the surface and Λ = (ε −
1)/4(ε + 1). Λ is quite small, Λ ≈ 0.01, since the dielectric constant of liquid helium is ε ≈ 1.057.
There is a barrier of ∼ 1 eV to penetrate into the helium, therefore the electron’ motion normal to
the surface (the z-direction) is quantized. The energy spectrum of the motion in z-direction has the
form of the energy spectrum of a hydrogen atom, that is the m-th state has energy Em = −R/m2,
with an effective Rydberg constant R = Λ2e4me/2¯h2 ≈ 8 K. The effective Bohr radius is rB =
¯h2/mee2Λ ≈ 76 Å. In the presence of an external pressing field E⊥, the states and their energies
are modified (see Fig. 3.2), although the qualitative picture remains the same.

36

The quantization described above was observed in the experiment by Grimes et al.

in 1976

[135]. Grimes et al. observed a series of peaks corresponding to transitions between energy levels

of motion transverse to the helium surface. The level spacing was controlled by an external electric
field E⊥ perpendicular to the helium surface, which shifts the energy levels through the Stark effect.
The transitions were observed at frequencies from 130 to 220 GHz by measuring the derivative of

the microwave absorption at a fixed frequency while tuning the splitting between the states by
modulating the field E⊥. The Stark shift of the levels is large, 0.3 GHz per V/cm for m = 1, and
1.1 GHz per V/cm for m = 2, and thus allows one to change the level spacing appreciably even

with moderate fields.

While the electron’s motion is quantized in the transverse direction, it is essentially free along
the surface. For T (cid:38) 0.7, the interaction of an electron with the outside world is dominated by scat-
tering by the He atoms in the gas phase, while for T < 0.7 K it becomes limited by the coupling to

thermally excited capillary waves on the helium surface (ripplons), and to bulk phonons in helium.

At low temperatures, electron-ripplon scattering is the fastest process and mostly determines trans-
port properties. The dispersion relation for the ripplons has the form ω2(k) = gk + (σ /ρ)k3 [134],
where ρ = 0.145 g/cm3 and σ = 0.37 erg/cm2 are helium density and surface tension respectively,
and g is the free fall acceleration. The height variation ξ (rrr,t) of the surface caused by ripplons

leads to the electron-ripplon coupling. To the first order in ξ (rrr,t) the coupling Hamiltonian reads

[136]:

with

H(1)
i = ∑
qqq

ξqqqeiqqqrrr ˆVqqq,

i
m(qqq· ˆppp) ˆpz −

i¯h
ˆVqqq = −
2m
vpol(x) = x−2 [1− xK1(x)]

q2 ˆpz + eE⊥ + Λq2vpol(qz),

(3.1)

(3.2)

Here, rrr is the in-plane electron coordinate, ˆppp = −i¯h∂rrr is the 2D electron momentum, ˆpz =
−i¯h∂z, and ξ (rrr) is expanded in Fourier components ξ (rrr) = ∑qqq ξqqqeiqqqrrr with wave vectors qqq. The
first two terms in the operator ˆVqqq describe a kinematic interaction, which arises because the electron

37

wave function is set equal to zero on a non-flat surface. The third term is due to the external
pressing electric field E⊥. The last term Λq2vpol(qz) describes the change of the polarization
energy due to surface curvature [137, 138] (K1(x) is the modified Bessel function). The scattering
rates due to electron-ripplon coupling have been carefully tested experimentally by measuring the

in-plane electron mobility at low frequencies [139]. Good agreement has been found between

the simple single-electron theory and experiment in a broad range of temperatures and electron

densities with no magnetic field [134].

It has been pointed out that there are limitations on the wave numbers q of ripplons for which

the electron-ripplon coupling has the form (3.2)–(3.2). According to Ref. [138], qrB should be
(cid:46) 1, while another argument [140] is that q should certainly be small compared to the reciprocal
width of the helium liquid-vapor interface and the reciprocal decay length of the electron wave

function into helium. Both lengths are of order of a few angstroms, which means that the large-q
cutoff qmax should be ∼ 107 cm−1.

While the quasielastic electron-ripplon scattering determines transport properties, it does not

lead to energy relaxation. Energy relaxation in this system is governed by absorbing or emitting

bulk phonons or pairs of short-wavelength ripplons [140, 141]. Experimentally the energy relax-

ation measurements are challenging. To our knowledge, there have only been measurements of
total energy relaxation rate [142], which turned out to be of the order ∼ 106 s−1 for T = 0.57 K.
This is the average decay rate of states over several subbands, and it is impossible to resolve indi-

vidual contributions of the two scattering processes mentioned above.

In typical experiments, the electron system is nondegenerate yet strongly correlated. The cor-

relation strength in a classical low-density plasma is conventionally quantified by the ratio Γ of
the average Coulomb energy to the average kinetic energy: Γ = e2(πns)1/2/kBT which usually
exceeds 20 (ns is the electron density, typically ns ∼ 108 cm−2). One of the most striking effects
of the electron-electron interaction is Wigner crystallization, in which electrons minimize the en-

ergy of mutual repulsion by forming a triangular crystal. The crystallization happens for Γ > 130

[1, 143, 144, 145, 146], the phase diagram is shown in Fig. 3.3. The phase diagram also contains a

38

Figure 3.3 Phase diagram of a nondegenerate strongly correlated electron system. The system is
a Wigner solid below the line Γ ≈ 130. If the temperature exceeds the short-wavelength plasma
frequency ωp (in energy units), the electron dynamics is classical.

line corresponding to kBT = ¯hωp, where ωp = (2πe2n3/2
frequency. Above this line all oscillations in the system are classical.

s /me)1/2 is the short-wavelength plasma

The crystallization was first experimentally observed in a beautiful experiment by Grimes and

Adams [1]. In that experiment, the in-plane microwave conductivity as a function of frequency

showed a complicated pattern of resonances below a certain temperature (see inset in Fig. 3.4,
which refers to the electron density n = 4.5× 108 cm−2; for this density the transition happens
at T = 0.457 K). These resonances were explained by Fisher, Halperin and Platzman [147] and

were attributed to the onset of long-range order in the electron system. Therefore, the resonance

detection allowed Grimes and Adams to trace a portion of the phase boundary shown in Fig. 3.4.

Recently Wigner crystallization was observed in quasi-one-dimensional channels on the surface of

helium [30], and in a carbon nanotube [148]. There is also evidence that Wigner crystallization

occurs in strong magnetic fields in 2D electron systems in semiconductor quantum wells [149,

150].

Wigner crystallization is known to make transport of electrons on helium remarkably nonlinear
even at small driving fields [2, 30]. The velocity vvv of a Wigner crystal saturates when vvv·GGG = ω(GGG),
where GGG is a reciprocal lattice vector of the Wigner crystal. This effect is known as the Bragg-

Cherenkov scattering [29]. Ripplons emitted by individual electrons that form a crystal moving

39

Figure 3.4 Portion of the liquid-solid phase boundary for a classical 2D electron system on the
helium surface [1]. The data points denote the melting temperatures measured at different electron
densities. The transition is detected from the sudden appearance of resonant electron absorption
at radio-frequencies, an example of which is shown in the inset. The solid line corresponds to
Γ = 137.

with this velocity constructively interfere, thereby increasing the momentum transferred to ripplons

per unit time dramatically. Essentially, the reaction force becomes so strong, that the crystal can

not accelerate above the Bragg-Cherenkov velocity unless the driving field is extremely strong

[30].

Even for higher temperature, when electrons do not form a Wigner crystal, the electron-electron
interaction is still strong, Γ (cid:29) 1. The electron system is a strongly correlated non-degenerate
electron fluid in this case. One might expect that the electron-electron interaction may substantially

change the in-plane mobility. However, as has been shown [3], the mobility in a strongly correlated

liquid remains close to that predicted by single-electron theory for a broad range of temperatures

and electron densities, in a spatially uniform system without a magnetic field. This is explained by
the short duration of electron-ripplon collisions, ∼ ¯h(kBT )−1, during which the interaction with
other electrons only slightly modifies the electron motion. The main difference comes from the

fact that in a liquid electrons exchange momenta with each other much faster than with ripplons.

40

T, K0.60.40.212300403020, MHz100.46K0.440.42wxyz(n /108 cm-2)1/2dA/dnLiquidCrystalFigure 3.5 Measured magnetoconductivity σ for electron density n = 0.55 × 108cm−2 at T =
0.7 K in the 2D electron fluid [2]. Theory lines show: d: Drude model, s: self-consistent Born
approximation, m: many-electron theory [3] and t: phenomenological calculation that combines
the results from the many-electron theory and the SCBA for strong magnetic fields.

This leads to a change of the expression for the mobility. However, numerically this change turns

out to be comparatively small in the range studied in the early experiments.

The situation is very different in the presence of a magnetic field. In this case, the electron-

electron interaction lifts the degeneracy of the single-electron energy spectrum and makes the

spectrum continuous. It turns out that the magnetotransport can be described in terms of an easily

calculated force on an electron from other electrons [3, 151]. This many-electron theory predicts

a non-monotonic magnetoconductivity for moderate magnetic fields, which is in good agreement

with the experimental data obtained by Lea, et al. [152, 2], see Fig. 3.5. The large difference

between one-electron self-consistent Born approximation (SCBA) and the many-electron theory in

Fig. 3.5 shows the importance of the electron-electron interaction in this system.

As mentioned before, a significant advantage of the electrons on helium over other two-dimensional

strongly correlated systems is the absence of defects. One can study effects of the electron-electron

interaction in this system unobstructed by disorder (which plays a very significant role for elec-

trons in semiconductors or vortices in superconductors). Also, in contrast to macroscopic classical

41

0.111010-1010-910-8B (T)s(B)smtdsystems of colloidal particles, the coupling of electrons on helium to the substrate is weak, the

electron motion is underdamped.

The role of correlations becomes particularly clear when the system is a crystal in the ab-

sence of the external potential. The role of a random potential in this case has been attracting

much attention in various contexts, from superconductors to charge/spin density waves to Wigner

crystals [153, 154, 155, 156, 157, 158]. It is well-known that a random potential in one- and two-

dimensional systems causes pinning, with no threshold in the potential strength. An external field

can lead to depinning and formation of dynamical phases, see [28] for a review.

Electrons on helium can be used to study the effect of quenched disorder by using thin helium

films, in which case the electrons “feel” the random potential of the substrate. However, electrons

on helium also provide a unique tool for investigating the role of correlations in the case where

the external potential is periodic. Such potential can be created by creating a periodic potential on
the substrate while keeping it further away from the electrons, at a distance (cid:38) 1µm. In fact, as we
discuss below, the study of the electrons on helium in the presence of such a potential can provide

a direct way to reveal and characterize the correlations in the electron system.

Various phenomena that emerge in correlated systems placed into a periodic external potential

have been broadly discussed in the literature [159, 26, 160, 27]. In analyzing the ensuing dynamics

one should consider separately atomic systems, in which the pair potential has a minimum and the

structure of the crystal in the absence of the potential is defined by the form of the interparticle

potential. The situation for a Wigner crystal or crystals formed by the dipolar interparticle repulsion

is different. In this case the pair potential does not have a minimum. The structure of the crystal

is determined just by the particle density and usually corresponds, in two-dimensional systems, to

the closely packed triangualr.

The simplest example of an externally imposed periodic potential is the potential of a crys-

talline substrate on which a monolayer film of a different crystal is deposited [161, 162, 158]. Of

primary interest here have been the effects of commensurability. Two periodic systems are con-

sidered to be commensurate (C) if one or more of their reciprocal lattice vectors coincide. They

42

are said to be incommensurate (I) otherwise. Some parameter is usually introduced to describe

the degree or commensurability, the choice of which depends on the system in question. Exam-

ples of commensurate-incommensurate (CI) transitions, besides those in monolayer films, include

the transitions in magnetic materials [163], charge-density waves [164, 165], colloidal crystals in

light-induced potentials [5, 166], vortex lattices in type-II superconductors [167, 168], metallic dot

arrays [169]. Recently CI effects in stacked 2D materials, such as graphene on boron nitride [170],

have also attracted substantial interest.

Colloidal crystals have proven to be especially useful in studying phase transitions and the

effects of external potentials in strongly interacting systems, see [171] for a review. Typically, these

systems consist of small dielectric particles, e.g. polystyrene spheres or silica spheres, in a liquid

suspension. The particle size may vary from a few nanometers to a few microns. The interaction

between the particles may have a variety of origins. Colloidal particles often become charged in the

suspension through accumulation of ions on their surface [172], which makes the particles interact

through the screened Coulomb potential. The charge of the particles is fluctuating, but often can

be made reasonably uiniform, with a variation of a few percent. The concentration of ions in

the solution determines the screening length. The inter-particle interaction can also arise due to

polarization of the dielectric particles in an external electric field [173, 174], magnetic moments

of the particles [175], etc. In most cases, the inter-particle potential does not have a minimum, in

contrast to monolayer solid-state films. An advantageous feature of the colloidal systems is that

the interaction is often tunable. The system can be made effectively two-dimensional by confining

the particles to a thin layer by flat walls, or by pressing them to the bottom of the experimental

cell by gravity. An external periodic potential is typically created by the interference pattern of

laser beams [176]. Another important advantageous feature of colloidal systems is the possibility

to resolve the structure of the system in real time directly by optical imaging.

It is clear from qualitative arguments that a 2D solid placed on a maximally commensurate sub-

strate becomes pinned to the substrate. Another consequence of the fully commensurate substrate

is the so-called modulation-induced freezing (also known as light induced freezing for potentials

43

Figure 3.6 Contour plots of the averaged density distribution ρ(x,y) and pair correlation function
g(x,y) obtained in the experiment [4] for two values of the potential strength: 2.1kBT(a,b) for
which the crystal forms due to modulation-induced freezing, and 6.3kBT (c,d) above the reentrant-
melting threshold, for which the system is a modulated liquid. In both cases the temperature is
above the melting temperature in the absence of modulation.

created by laser radiation) [172, 4, 27], where the melting temperature increases because of the

commensurate potential. A less obvious effect may occur in colloidal systems of charged parti-

cles when the potential strength is increased even further beyond the modulation-induced freezing

point. In a certain range of parameters above the freezing temperature in the absence of the mod-

ulation, the further increase of the potential amplitude leads to a reentrant modulation-induced

melting transition, when the system becomes a modulated liquid again. Fig. 3.6 shows the ex-

perimental observation of this transition, the upper row corresponding to the crystal state induced

by the modulation, and the lower row to the liquid state for the modulation strength above the

modulation-induced melting threshold. Qualitatively, both effects are related to the suppression of

thermal fluctuations of particles transverse to the imposed periodic potential [4, 166, 27].

Quantitatively, the effects of the commensurate potential can be understood using the free

44

energy of the solid phase “locked” in the commensurate potential. For a one-dimensional periodic

potential in the continuum approximation the elastic energy can be written as [27]:

(cid:90)

Keff(∂yuy)2 + µeff(∂xuy)2(cid:111)
(cid:110)

HLFS =

1
2

d2r

,

(3.3)

where uy is the y-component of the local displacement, and Keff and µeff are the effective bulk
and shear moduli respectively. The modulation troughs are oriented along the y axis. Keff and
µeff depend on both temperature and the external potential amplitude A. This model predicts the
melting temperature for small A to be

Tm(A) ∼ Tm(0) + [ln(kBT /A)]−1/ ¯ν ,

(3.4)

with ¯ν ≈ 0.37. The melting temperature is increasing with A, as expected for the modulation-
induced freezing. For large A, on the other hand,

1 +

5

(cid:104)
(cid:105)
(κa)2 − 31
64π2

(cid:18)

1 +

13
3κa

kBT ∞
m
p2A

(cid:19) ,

Tm(A) = T ∞
m

(3.5)

where a is the lattice spacing, κ is the inverse of the Debye screening length, and T ∞
m = Tm(A →
∞) > Tm(0) can be estimated in the model. This is a decreasing function of A for sufficiently large
κ, corresponding to the modulation-induced melting described above.

For incommensurate 2D systems, the competition of periodicities leads to rich phenomenology.

A typical feature of incommensurate solids is the instability against formation of defects that elim-

inate long-range order, or can themselves form ordered phases. Two types of topological defects

are usually identified: solitons (or kinks) [160, 26] that enable formation of crystalline domains of

uniform density, and a system of dislocations. Proliferation of dislocations can lead to a hexatic

phase [177, 178, 159, 179]. It is the relative stability of the two types of defects that essentially

determines the phase diagram. Another prominent feature of incommensurate phases is that often

they can “float” without pinning to the substrate.

In the continuous limit, the free energy of a weakly incommensurate phase can be written

in a way similar to Eq. (3.3), but with an additional term accounting for the potential energy of

45

the particles in the external potential due to incommensurability. Consider a crystal in a weak
external potential V (rrr) = ∑qqqVqqqeiqqqrrr, where qqq are the reciprocal vectors of the external potential.
The particles in the crystal without modulation are assumed to form a lattice with positions RRR and

reciprocal lattice vectors bbb. Let us assume that only one pair of vectors qqq and bbb are close, so that

for a small vector sss we can write qqq = bbb + sss (weak incommensurability). Then, at zero temperature,

one can write the total potential energy in this system as [26]:

(cid:1)2

+(cid:0)∂yux + ∂xuy

(cid:1)2(cid:105)

(cid:27)

+ f (φ )

,

(3.6)

(cid:90)

U =

d2r

(cid:26)K

2

(cid:0)∂xux + ∂yuy

(cid:1)2

(cid:104)(cid:0)∂xux − ∂yuy

+

µ
2

where K and µ are the bulk and shear moduli respectively, and

f (φ ) =

∞
∑
n=−∞

b2Vn(bbb+sss)einφ ,

(3.7)

with φ = uuu +sssRRR. The term f (φ ) represents the interaction of the lattice with the external potential.

This interaction tends to place particles at the minima of the periodic potential. The elastic terms,

however, tend to place the particles in their regular lattice, with a periodicity different from that of

the potential. For s smaller than some critical value sc, which depends on the periodic potential

strength, the system will minimize its energy by becoming commensurate with the external poten-

tial. For s > sc, however, the commensurate phase is not favorable and the particles are predicted

to assume some compromise arrangement that minimizes the energy. For sufficiently small s, the

system is divided into domains of commensurate structure separated by narrow sections where the

inter-particle distances are substantially different from the lattice spacing in the rest of the system.

These structural features are the solitons (kinks) mentioned in the previous paragraph. Further

increasing of s leads to a superstructure better described as a periodic density modulation in the

crystal. The density modulation is directed at some angle with respect to the reciprocal lattice vec-

tor bbb in a general case. The period of the superstructure continuously depends on parameters s and

Vqqq. The described superstructure, in a somewhat more complicated case of the external potential

with two reciprocal lattice vectors being close to two reciprocal lattice vectors of the lattice, was

experimentally observed in the system of charged colloidal particles [5]. The solitons can be seen

46

Figure 3.7 a-c, Snapshots of particle velocities in a colloidal monolayer in an incommensurate
external potential with a triangular structure [5]. The colloidal lattice spacing is a = 5.7 µm, while
the potential lattice spacing is s = 5.2 µm. Fast and slow particles are marked in dark and light blue
depending on whether their velocities are above or below 70 % of the maximum particle velocity.
(d-f), Voronoi tessellation with color-coded areas of the Voronoi cells: light (large) to dark (small)
green. The following driving forces were applied: 0 fN in (a,d), 19 fN in (b,e), and 82 fN in (c,f).

in Fig. 3.7 (d,e) as narrow regions of small density. Because of the 2D nature of the potential in

that experiment, the solitons form a complicated 2D structure instead of a series of 1D rows.

Although the continuum elasticity theories are powerful, they are limited to weak incommen-

surability. Also, it is very hard to describe transport using these theories. That is why numerical

simulations are often used to study structural [6, 180, 181, 182, 183] and dynamical properties

[184, 7] of commensurate and incommensurate phases. Molecular dynamics simulations for over-

damped dynamics have been extensively performed to study vortex lattices in superconductors and

colloidal systems. In this framework, for example, simulated annealing has been used to study a

rich variety of ground state structures of vortex lattices with square and triangular pinning arrays

[6] (Fig. 3.8). By applying a driving force to the particles, one can study depinning and dynamical

phases as a function of commensurability. Fig. 3.9 shows the rich dynamic phase diagram obtained

in the numerical experiment [7] with a vortex lattice driven across a 1D periodic potential. Monte

Carlo simulations [181, 182, 183] have also been used to study structural and thermodynamic

47

Figure 3.8 Vortex lattice structures observed in the simulation [6] in a superconductor in the pres-
ence of a square lattice of pinning sites. The vortices are represented by black dots, while the
pinning sites are shown as empty circles. (a-i) represent different magnetic field magnitudes, cor-
responding to the number of vortices per pinning site from 1 to 9 in (a) through (i).

Figure 3.9 Dynamic phase diagram of a vortex lattice driven across a 1D external potential [7] as
a function of the drive field Fd and the ratio w/a of the substrate lattice constant w to the vortex
lattice constant a. P: pinned phase. Plastic: plastic flow regime. ML: moving modulated lattice
state. MFS: moving flowing solid state

properties of atomic and molecular systems on periodic substrates, showing good agreement with

experimental observations.

Simulations have also been used to study the effects of external (non-periodic) potentials in the

electrons on helium. Reichhardt et al. [185] studied the depinning and subsequent dynamics for

48

ORIENTATIONALORDERING,BUCKLING,ANDDYNAMIC...PHYSICALREVIEWB93,014504(2016)12345w/a01234Fd1234567ForcPPlasticMLMFS123(a)FIG.13.(a)Forc,thedriveatwhichthesystemtransitionsfromadensitymodulatedmovingcrystaltoanorderedmovingfloatingsolid,vsw/aatFp=0.5(redcircles),1.0(greensquares),1.5(orangediamonds),and2.0(blacktriangles).(b)DynamicphasediagramasafunctionofFdandw/aforasystemwithFp=1.5.P:pinnedphase.Plastic:plasticflowregime.ML:movingmodulatedlatticestate.MFS:movingflowingsolidstate.Dashedlinesareguidestotheeyethatindicatethetransitionfromr1tor2,r2tor3,andr3toadisorderedpinnedstate.intheregimewhereFd<Fp,thesystemisinaplasticflowstateinwhichthereisacoexistenceofmovingvorticesandimmobilevortices.Inthisregime,thestructurefactorgenerallyshowsdisorderedfeatures.ForFd>Fp,allthevorticesaremovingandthesystemiseitherinamodulatedlattice(ML)stateoramovingflowingsolid(MFS)state.WefindsimilardynamicalphasediagramsforothervaluesofFp>1.0,whilefortheweakersubstrates,thesizeoftheplasticflowregionisreducedandtheMLphaseisreplacedwithasmecticmovingstatesimilartothatshowninFigs.11(c)and11(d).VII.DISCUSSIONThedynamicphasesweobservehavecertainsimilaritiestothedynamicstatesobservedforvorticesmovingoverrandompinningarraysinthattherecanbepinned,plastic,anddynamicallyorderedphasesasafunctionofexternaldrive[86–91].Therearesomedifferences,includingthefactthatthemovingmodulatedlatticeweobservedoesnotformasmecticstatethatisfullyalignedinthedrivedirection,asfoundforvorticesmovingoverrandompinningarrays[88–91].Experimentalobservationsofthesmectictosmecticphasetransitioncouldbeachievedusingdirectdynamicalimagingtechniques[13,85]orneutronscatteringtechniques[96].Voltagenoisemeasurementscouldalsorevealthetransitionsbetweenthedifferentdynamicalphases,asinexperimentsonsampleswithrandompinning[91,97,98].Thedynamicaltransitionscanbeobservedifthedepinningcurrentissufficientlyfarbelowthedepairingcurrent,whichwilldependonthemagneticfield,thestrengthoftheintrinsicpinning,andhowclosethesystemistoTc.AccordingtoFig.13,thedynamicalreorderingcurrentsareexpectedtobe2.5to10timeshigherthanthedepinningcurrent,sotheaccessiblecurrentrangeneedstobeatleastthiswideinordertoexperimentallyobservethedynamicaltransitions.Simulationswithrandompinningarraysindicatethatforincreasingvortexdensity,thedriveatwhichthetransitiontotheorderedstateoccursdecreasesduetotheincreaseinthestrengthofthevortex-vortexinteractions[91].Fortheq1Dsubstrate,thelocationoftheorderingtransitionfluctuatesstronglyduetotheabilityofthemovinglatticetobecomedynamicallycommensuratewiththeperiodicityofthesubstrate.Futurestudiesmightconsidercombinationsofrandomdisorderwithperiodicdisorder,whichwouldintroduceacompetitioninthemovingphasebetweenthesmecticorderingimposedbythesubstrateandthesmecticorderinginducedbythedrive.Wefocusonvorticesdrivenalongthedirectionofthesubstrateperiodicity,butitwouldalsobeinterestingtostudytheresponseofthesystemwhenthevorticesaredrivenperpendicularoratanarbitraryangletothesubstrateperiodicity.Previousstudiesofvorticesdrivenatvariedanglesthrough2Dperiodicpinningarraysshowthatthesubstratecaninduceanumberofdynamicaleffects,includingdirectionallockingofthevortexmotionaswellasvortexchannelingeffects[67,99–102].Foraq1Dsubstrate,vorticesdrivenatarbitraryanglesshouldexhibitchannelingalongthesubstratetroughs,andthedriveatwhichthischannelingisovercomeandthevorticesstarttoflowinthedirectionofdriveshouldbeafunctionofdrivingangle.Measurementsofthetransverseandlongitudinalvortexvelocitiesinthiscaseshouldshowzerovelocityinthedirectionofthesubstrateperiodicityatlowdrives,andfinitevelocitybothparallelandperpendiculartothedrivingdirectionforintermediatedrives.Studiesofvorticesdrivenovera1Dlinepotentialshowchannelingeffectsofthistype[102,103].Weconsideraparticlebasedmodel,butsimulationsbasedontime-dependentGinzburg-Landau(TDGL)theorymayrevealotherinterestingbehaviorforhigherfieldsorcurrentsthanwestudy,suchastheelongationofthevorticesalongcertaindirectionsortheformationofmultiple-quantavortices.TDGLstudiesofvorticesin2Dperiodicpinningarraysshowthatarichvarietyofcommensurateandincommensuratestatescanformwithmulti-quantavorticesorcombinationsofmultiquantaandinterstitialvortices[19–22].Similareffectscouldariseforq1Dsubstrates.ATDGLapproachcouldcaptureadditionalfeaturesinthedepinningbehaviorandsubsequentflowsuchasmerging,splitting,elongation,orlocalheatingeffectsofthevortices[104–107].Itcanalsotreattheeffectofthesubstratethicknessinthelimitofathinfilm.Ourparticle-basedmodelassumesthatthevorticesexperiencethesamedrivingforceregardlessoftheirpositioninthesample;however,recentTDGLsimulationsshowthattheappliedcurrentcanbemodifiedbythethicknessofthesample,andthattheeffectiveviscosityalsovarieswiththicknessduetothechanginglengthofthevortex[108–111].SucheffectscouldbeexploredusingTDGLsimulationsormodifiedparticle-basedmodelsinwhichadditionaltermsareintroducedtorepresentthespatialdependenceoftheviscosityortheapplieddrivingforce.014504-11a Wigner solid in the quenched disorder, observing a transition from a plastic flow to a moving

Wigner smectic. Damasceno et al. [186] considered a Wigner solid pinned to a single pinning

site and its depinning with the subsequent plastic flow. Piacente and Peeters [187] simulated the

transport of a quasi-1D electron system in a channel across a single constriction, observing elastic

and quasielastic depinning. These studies use Langevin dynamics with the phenomenological

friction and random noise to model the coupling to the helium surface. As it will be discussed later,

a more rigorous treatment of scattering on the surface and bulk quantized excitations is required to

adequately describe the dynamics in this system.

3.2 Motivation

We study the effects of periodic potentials in strongly correlated 2D systems using the electrons

on helium. One cannot directly observe the spatial structure in this system. However, this system

allows one to perform careful transport measurements. Such measurements are of particular in-

terest for strongly correlated systems, as transport depends on the interplay of strong correlations,

scattering by defects or a quantum field to which the system is coupled, and a periodic potential.

We will concentrate on the liquid state. So far, for this state the effect of correlations on the elec-

tron transport have been identified only through the measurements of the magnetoconductivity. A

periodic potential brings in a new tool which, we expect, should allow one to directly characterize

the correlations in the electron liquid. The periodic potential can be created by an array electrodes

in the substrate with a period on the order of one micron, see Fig. 3.10. The advantageous fea-

ture of electrons on helium over other 2D correlated systems is that all electrons are identical (for

example, no difference in charge, in contrast to charged colloidal systems) and there are no de-

fects, whereas scattering by the excitations in helium is extremely weak. Therefore the effects of

the electron-electron interactions in a periodic potentials are not masked by disorder or by strong

coupling to the helium excitations.

Our results are obtained by numerically simulating the electron system with the account taken

of the long-range Coulomb forces, the microscopic mechanism of the electron scattering by the ex-

49

Figure 3.10 Electrons on the surface of liquid helium with a periodic substrate consisting of an
array of electrodes (schematically).

citations in helium, and the periodic potential. We concentrate on the regime where the electrons

form a classical liquid, but we also briefly study the behavior of the electron solid in a periodic po-

tential as well as the dynamics in the regime where the electron-electron interaction is disregarded;

the latter regime is studied primarily as a means to demonstrate the role of the correlations where

the electron-electron interaction is present and is strong.

In a way, the system of electrons on helium is an ideal system for studying the effects of a

periodic potential on transport of a strongly correlated system. The characteristic inter-electron

distance is in the range of 1 µm. With modern technology, it is straightforward to create periodic

structures with a comparable period. These structures can create an electrostatic potential, but they

can also create a periodic magnetic field that will induce spin-orbit coupling in the electron system

and will allow one to study the spin dynamics, a topic that so far has not been explored for electrons

on helium at all. However, in this work we concentrate on the electrostatic potential.

The effects of a periodic potential on a solid and an ideal gas have been studied in much depth

for other systems than the electrons on helium. In particular, the role of commensurability has been

explored, as described above. In a classical ideal gas, the effect of the periodic potential becomes

significant only when the potential is comparable to the temperature. For a strong potential, the

mobility displays a simple activation behavior.

In contrast, very little is known about the effects of the periodic potential on the strongly corre-

50

lated liquid phase, where the role of interactions is important but there is no long-range order. We

expect that the periodic potential significantly affects all transport characteristics. We consider a

one-dimensional periodic potential. It makes the system strongly anisotropic. The major transport

characteristics we are interested in are many-electron diffusion and the current in response to an

additional spatially uniform driving field.

We should emphasize that, in a correlated system, the very term “diffusion” has to be taken

with care. There is diffusion associated with a smooth (on the scale of the correlation length)

gradient of the particle density. This diffusion is described by Fick’s law, with the current propor-

tional to the density gradient. In noninteracting systems, as shown by Einstein, Fick’s diffusion is

simply related to fluctuations of particle positions, with the mean square displacement of a particle

being proportional to time with the coefficient that enters Fick’s law. This is because diffusion and

mobility are determined by the same scattering processes. In systems of interacting particles, the

relation between the Fick diffusion coefficient and the mean-square particle displacement gener-

ally does not hold. In a correlated liquid, particles are wondering around, but their mean square

displacement is determined, in the first place, by their interaction with each other. Such an inter-

action does not change the total momentum of the system and therefore does not contribute to the

mobility. Interestingly, we find that in the electron liquid the mean-square particle displacement

is still linear in time. We use the term diffusion coefficient for the proportionality coefficient be-

tween the mean square displacement and the time over which it occurs. As we show, the diffusion

coefficient defined this way displays a nontrivial behavior and provides an insight into the phase

transition in the electron system.

Even in the absence of long-range order, an important characteristic of the system in a periodic

potential is the commensurability. For a one-dimensional potential, we characterize it by the pa-

rameter p, which is equal to the ratio p = K/G of the smallest reciprocal lattice vector K of the

potential to the smallest reciprocal lattice vector G of the triangular lattice that would exist at the

given density of the liquid. In the liquid phase, no conventional pinning can occur since the system

remains disordered with dynamically changing structure. However, one may expect that, because

51

of the strong electron correlations, the transport will change significantly if the electron system can

perfectly “fit into” the potential.

Another property that can be expected is a strong effect of the periodic potential even where

it is weak. One can think that the barriers imposed by the potential are “amplified" by the strong

correlations, as the electrons cannot overcome them independently.

A topic of significant interest is thermalization in many-body systems. It has been studied for a

long time, starting with the famous Fermi-Pasta-Ulam numerical experiment. That experiment was

done on nonlinear vibrations in a chain which, in the case of the electron system, could be extended

to vibrations in the electron solid. On the opposite side of the parameter range, where the electrons

form a weakly nonideal plasma, thermalization has been well understood since the Landau work

[188] on such a plasma. However, to the best of our knowledge, the question of thermalization

in a strongly correlated liquid has not been addressed. The electron system on helium provides a

natural platform for studying this effect.

To guide the experiment, it is necessary to include into the analysis the full microscopic cou-

pling of the electrons to the quantum field of excitations in liquid helium. This coupling is known.

Incorporating it requires a significant modification of the conventional means of numerical simu-

lations of the dynamics of correlated systems. The conventional analysis proceeds by assuming

that the motion of a particle is described by a Langevin equation with a phenomenologically added

friction force and a random force that mimics the noise from the thermal reservoir. This has no

relation to the system that we consider, it does not adequately describe the electron relaxation.

A distinct feature of our method is the direct simulation of both elastic and inelastic scattering

processes. It is known that the electrons scatter elastically off the ripplons and inelastically off

the phonons in helium; there is also inelastic two-ripplon scattering. We use realistic scattering

parameters. It is known that the scattering rate is notoriously low. This refers to both elastic and

inelastic scattering. In addition, the inelastic scattering rate is an order of magnitude smaller than

the elastic one. This leads to exceptionally slow energy relaxation. In contrast to the Langevin

approach, not only are the momentum and energy scattering rates very different, but they also

52

depend on the electron energy, which should lead to a qualitatively different transport.

In Sec. 3.3, we present the methods that we use to study the electrons on the helium surface.

The results of the simulations of a uniform system are described in Sec. 3.4. Sec. 3.5 is the central

part of our study, focusing on the effects of a periodic potential on the strongly correlated electron

system. The conclusions are presented in Sec. 3.6.

3.3 Method

3.3.1

Integration of the equations of motion

To study properties of the electrons on helium we integrate classical equations of motion numeri-

cally. We use HOOMD-Blue [189, 190] as the base code for our simulations, with the integrator,
interaction potentials, and external forces developed specifically for our study1. The code is de-

signed to be used on a graphics processing unit (GPU) which gives up to sixteen-fold acceleration

compared with a single CPU core. For integration, we use the standard velocity Verlet algorithm,

which is symplectic [191]. The algorithm is implemented numerically as follows:

vvvi (t + ∆t/2) = vvvi(t) +

1
2

aaai(t)∆t,

rrri (t + ∆t) = rrri(t) + vvvi (t + ∆t/2)∆t,

vvvi(t + ∆t) = vvvi (t + ∆t/2) +

1
2

aaai(t + ∆t)∆t,

(3.8)

where vvvi is the i-th particle’s velocity, and the acceleration aaai(t) is evaluated based on forces derived
from the positions rrr j(t) of all particles at time t. ∆t is the time-discretization step. We modify this
algorithm to accommodate for discrete scattering events that are described by abrupt changes of
velocity vvv → vvv(cid:48) according to some probability distributions. To conserve potential energy in such
scattering to the second order in ∆t, the scattering event must happen at integer time points. To see

that, consider the energy of a single particle in an external potential U(rrr). Assume that the particle
scatters elastically at time t, i.e. its velocity at time t changes by ∆vvv such that |vvv(t) + ∆vvv| = |vvv(t)|.
1Our fork of the HOOMD-Blue code can be found at https://github.com/kmoskovtsev/
Scripts used in simulations are available at https://github.com/

HOOMD-Blue-fork.
kmoskovtsev/Electrons-on-Helium-Scripts

53

Then the velocity and position at time t + ∆t is

vvv(t + ∆t) = vvv(t) + ∆vvv +

rrr(t + ∆t) = rrr(t) + ∆t

(cid:20)

(cid:21)
1
2(aaa(t) + aaa(t + ∆t))∆t,
vvv(t) + ∆vvv +
.

aaa(t)∆t

1
2

(3.9)

Then it is easily shown that the energy difference over one step vanishes in the first order in ∆t:

E(t + ∆t)− E(t) =

m
2

(cid:16)
(cid:17)
vvv2(t + ∆t)− vvv2(t)

+U (rrr(t + ∆t))−U (rrr(t)) = O(∆t2).

(3.10)

Since scattering events in our simulations are rare, about one scattering event per 105 time steps

per particle, this accuracy is enough for our purpose.

We choose ∆t ≈ 2.57×10−12 s for most simulations, which corresponds to ∆t ≈ (2π/ωp)/50,
where ωp is the short-wavelength plasma frequency. Reducing ∆t proved to have no visible effect

in the studied phenomena.

3.3.2 Ewald summation

Electrons on the surface of liquid helium interact through the Coulomb repulsion. In a real system,

the interaction may be screened by the electrode placed under the helium surface. We will neglect

any screening effects in this study for simplicity. These effects will be investigated in future studies.

Since we are considering a system with periodic boundary conditions and long-range interactions,

we use the Ewald summation to properly calculate the interaction forces and energies [144]. We

consider a 2D unit cell that is periodically replicated in two dimensions by the lattice vectors LLLn,
with a charges qi placed at the coordinate ρi in the unit cell. The results in this section are valid
for an arbitrary unit cell, however, only rectangular unit cells are used in our study. The system is
neutralized by a uniform charge background with the charge density −qi/Ω, where Ω is the unit
cell surface area. Then the potential created by all replicas of the charge qi at an arbitrary point rrr
in the unit cell, is

φi(rrr) = φ L

i (rrr) + φ S

i (rrr),

(3.11)

54

where

φ L
i (rrr) =

2πqi
Ω ∑
kkkn

(cid:48) 1
kn

and

erfc(knη)eikkkn(rrr−ρρρi),

(3.12)

(cid:18)

(cid:19)

−

qi

erfc

|rrr− ρρρi − LLLn|

|rrr− ρρρi − LLLn|

i (rrr) = ∑
φ S
LLLn

4√πηqi
Ω .
Here erfc(x) is the complementary error function. The summation in φ L
i (rrr) is carried over all re-
ciprocal lattice vectors of the lattice LLLn, excluding kn = 0. The parameter η has the dimensionality
of distance, and can be chosen arbitrarily. Typically, such η is chosen that it makes both φ L
i (rrr) and
φ S
i (rrr) converge relatively quickly.

(3.13)

2η

For the molecular dynamics simulations, we need forces as well as potentials. Therefore, we

calculate the electric field created by all replicas of qi by differentiating Eq. (3.11) over rrr:

(cid:18)

(cid:19)

rrr− ρρρi − LLLn
EEEi(rrr) = qi∑
|rrr− ρρρi − LLLn|3 erfc
LLLn
rrr− ρρρi − LLLn
qi
√πη ∑
|rrr− ρρρi − LLLn|2 exp
LLLn
2πqi
(cid:48) ikkkn
Ω ∑
kn
kkkn

−

+

erfc(knη)eikkkn(rrr−ρρρi)

2η

|rrr− ρρρi − LLLn|
(cid:32)
−|rrr− ρρρi − LLLn|2

4η2

(cid:33)

(3.14)

Both eφi(rrr) and eEEEi(rrr) are tabulated on a mesh covering a quarter of the rectangular unit cell. The
values in the rest of the unit cell are recovered by symmetry. The bilinear interpolation is used to

obtain the values of these fields at an arbitrary point in the unit cell. Consider some field f (rrr),

tabulated as fm,n on the mesh (xm,yn) with indices m and n. The mesh steps are hx and hy along
the x and y axes respectively. The value of f (rrr) at a point (x,y) within the rectangle with corners

(xm,yn) and (xm+1,yn+1) is interpolated as

where

f (x,y) ≈ fα,n +

δ y
hy

( fα,n+1 − fα,n),

fα,n = fm,n +

δ x
hx

( fm+1,n − fm,n),

(3.15)

(3.16)

and δ x = x− xm, δ y = y− yn. The interpolation scheme is implemented in the new table2D pair
interaction for HOOMD-Blue. The tables are prepared using the ewald_module.py module in

55

our code base, that uses Eqs. (3.11)–(3.14). Grids of the size 2000 × 2000 are used in most of
our simulations. The GPU version of the code provides the acceleration of up to sixteen times

compared with a single CPU core. The electron-electron interaction is the bottle neck of our

computations. The computation time scales as the number of particles squared. For 1600 particles,

on NVIDIA® Tesla® K80 GPU, the computation speed is about 150 time steps per second.

3.3.3 Scattering of electrons by ripplons and energy relaxation

As it was discussed above, the electron-ripploon coupling is described by Eqs. (3.1)–(3.2). For
simplicity we consider coupling in the case with no pressing field E⊥. Then only the last term in
Eq. (3.2) is relevant for the in-plane scattering. The scattering rate from the wave-vector kkk into kkk(cid:48)
is then determined by the square of the matrix element [3]

(cid:12)(cid:12)(cid:12)H(rpl)

qqq

(cid:12)(cid:12)(cid:12)2

φ 2(cid:16)qrB

(cid:17)

2

=

S−1kBT ¯h4q2

4αm2

er4
B

,

(3.17)

(cid:34)
φ (x) = (x2 − 1)−1 − (x2 − 1)−3/2 tan−1(cid:104)

φ (x) = (x2 − 1)−1 + (1− x2)−3/2 ln

1 + (1− x2)1/2

(cid:35)
(x2 − 1)1/2(cid:105)

x

, x < 1,

, x > 1.

(3.18)

where

k

by ripplons from the wave-vector kkk to any other allowed state

The total rate of scattering w(rpl)
is determined by the Fermi golden rule with the matrix element (3.17). The scattering is elastic,
therefore only scattering to states with |kkk(cid:48)| = |kkk| is allowed. Therefore, the total scattering rate is
(3.19)

dθ wkθ ,

w(rpl)
k =

(cid:90) 2π
φ 2(cid:16)qrB

0

2

(cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)q=k√2(1−cosθ )

where

w(rpl)
kθ =

kBT ¯hq2
8παmer2
B

(3.20)

is the distribution of scattering angles θ (normalized to the total scattering rate). The detailed

derivation of this result is reproduced in Appendix A.7. To simulate this scattering process, we
tabulate the total scattering rate w(rpl)
on the grid of 200 k points from kmin = 1 µm−1 to kmax =

k

56

Figure 3.11 The histograms show the scattering angle distributions obtained in the simulation of
2× 106 free non-interacting electrons over 100 time steps with all electrons having momentum
corresponding to (a) k = 2× 105 cm−1, (b) k = 2× 106 cm−1. Elastic ripplon scattering is the
only scattering in the simulation, and only particles that scattered during the simulation period
contribute to the histograms. The y-axis shows the total count of scattering events in each of the
40 bins covering angles from 0 to 2π. Solid lines correspond to the values obtained directly from
Eq. (3.20), normalized appropriately for comparison with the total count of scattering events.

150 µm−1. The angle distribution is tabulated in the form of the inverse cumulative distribution
function W−1
kθ because we use the inverse transform sampling to generate random numbers with
arbitrary distributions. W−1
kθ is tabulated on a 150×1000 grid spanning the rectangle [kmin,kmax]×
[0,1]. The values are then linearly interpolated for w(rpl)
. Interpolation scheme (3.15)–(3.15) is
used for W−1
kθ . One random number needs to be generated at every time step to determine whether
scattering should take place at this step, and another random number is generated if scattering

k

occurs to determine the scattering angle. The scattering rates and distributions measured in the

simulations agree well with Eq. (3.20), see Fig. 3.11.

To have realistic energy relaxation in the system, we consider inelastic scattering. For the elec-

trons on helium, two inelastic processes are important: emission and absorption of bulk phonons

in helium and of pairs of short-wavelength ripplons [141]. Transport properties, on the other hand,

are predominantly determined by quasielastic ripplon scattering. The energy relaxation rate is also

57

02050100150200250wk(a)020100200300400500wk(b)much slower than the electron-ripplon scattering in this system. Therefore only the total energy

relaxation rate is important for our study, and not the particular form of the scattering distributions.

For simplicity, we only consider inelastic scattering by bulk phonons and artificially adjust the total
scattering rate to be of the order of 106 s−1 at T = 500 mK [142] and of the order of 105 s−1 at
T = 58 mK [192].

For the form of the inelastic scattering distribution functions, we choose that of the scattering

of electrons by bulk phonons that is mediated by the polarization interaction hamiltonian (see

Appendix A.9 for details):

with

H(ph) = Λ ∑
qqq,Qz>0
(cid:90) 0
−∞

dz(cid:48)

Ξ(QQQ,z) ≡

QquQz,qqqeiqqqrrrΞ(QQQ,z),

(cid:0)q(z− z(cid:48))(cid:1) .

K1

sin(Qzz(cid:48))
2q|z− z(cid:48)|

(3.21)

(3.22)

Here as before, rrr and z are the in-plane and perpendicular to the plane coordinates of the electron.

QQQ is the total 3D phonon wavevector, qqq is the in-plane part of QQQ, Qz is the z-component of QQQ, and
K1(x) is the modified Bessel function. The QQQ-component uQz,qqq of the displacement field in the
helium bulk has the form

(cid:115)

(cid:16)

(cid:17)

uQz,qqq =

¯h

ρvsV Q

bQz,qqq + b†

Qz,−qqq

,

(3.23)

where ρ and V are the density and the volume of helium respectively, vs is the speed of sound, and
bQz,qqq, b†
Qz,qqq are the annihilation and creation operators for a phonon mode with the wavevector
QQQ. Then by the Fermi golden rule, the rate of scattering from state kkk into the phase volume d2k(cid:48)
through a phonon absorption is

dw(ph1)

kkk(cid:48)kkk = d2k(cid:48)

Λ2

2π2ρ ¯hv2
s

Qq2NQz,qqq|(cid:104)Oz|Ξ|Oz(cid:105)|2

,

(3.24)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) qqq = kkk(cid:48) − kkk

Qz = 1

¯hvs (εk(cid:48) − εk)

where

(cid:104)Oz|Ξ(QQQ,z)|Oz(cid:105) =

(cid:90) ∞

0

dz

4z2
r3
B

58

e−2z/rBΞ(QQQ,z),

(3.25)

Figure 3.12 Scattering rate w(ph)
(a) kkk = (10,0) µm−1, (b) kkk =
kkk(cid:48)kkk
(15,0) µm−1, (c) kkk = (25,0) µm−1. Axes kx and ky have dimensions µm−1. The scattering
rate vanishes at points where |kkk(cid:48)| = |kkk|.

as a function of kkk(cid:48) for:

e¯hωQ/kBT

is the average number of phonons with the wave-vector QQQ and en-
and NQz,qqq = 1/
ergy ¯hωQ = ¯hvsQ in thermal equilibrium. The electron kinetic energy is denoted as εk = ¯h2k2/2me.
The second process, with a phonon emitted, has the rate

(cid:16)

(cid:17)
− 1

dw(ph2)

kkk(cid:48)kkk = d2k(cid:48)

Λ2

2π2ρ ¯hv2
s

Qq2(NQz,qqq + 1)|(cid:104)Oz|Ξ|Oz(cid:105)|2

We can write the combined scattering rate for the two processes as

 w(ph1)

kkk(cid:48)kkk
w(ph2)
kkk(cid:48)kkk

, k(cid:48) > k,
, k(cid:48) < k.

w(ph)
kkk(cid:48)kkk =

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) qqq = kkk(cid:48) − kkk

Qz = 1

¯hvs (εk − εk(cid:48) )

.

(3.26)

(3.27)

The shape of w(ph)
kkk(cid:48)kkk as a function of kkk(cid:48) for three values of kkk is shown in Fig. 3.12. Due to the
ratio e¯hωQ/kBT of (3.26) to (3.24) for the same value of Q, this scattering mechanism supports
the detailed balance condition for the Boltzmann distribution and leads to energy relaxation. As

discussed above, we multiply the scattering rate (3.27) by a constant factor of 20 to have the total

energy relaxation rate close to that obtained in the experiment [142, 192].

In the code, we simulate the inelastic scattering events by changing the electron momentum

instantaneously according to the scattering rates given by Eqs. (3.24) and (3.26). The difference

with the elastic scattering is that the magnitude of the momentum changes as well as the direction.
At every time step, the total scattering rate w(ph)
is calculated. Then the probability for a given electron to scatter at the current time step is deter-

from momentum ¯hkkk into any other momentum

k

59

k0x40040k0y40040wk0k(a)k0x40040k0y40040wk0k(b)k0x40040k0y40040wk0k(c)mined as w(ph)
k ∆t. A random number is generated to determine if scattering should occur at this
step according to the probability to scatter. If the electron is scattered at this time step, the scat-

tering momentum kkk(cid:48) is determined in two steps. First, the absolute value of kkk(cid:48) is drawn from the
distribution fk(cid:48)k of k(cid:48) for the given k, where fk(cid:48)k is an integral of w(ph)
kkk(cid:48)kkk over the scattering angle θ.
Second, θ is generated according to the angle cumulative distribution Yk(cid:48)k. This radial scheme is
appropriate here because the scattering rate varies quickly with the absolute value of k(cid:48) and rather
smoothly with the angle.

All probability distributions are computed beforehand and tabulated in three tables. We use

k

inverse transform sampling to draw random numbers from the distributions, therefore all dis-
tributions except w(ph)
w(ph)
k
two-dimensional, on the grid spanning [kmin,kmax] × [0,1]. Finally, the table for Y−1
dimensional, on the grid spanning [kmin,kmax]× [kmin,kmax]× [0,1].

are stored as inverse cumulative distribution functions. The table for
k(cid:48)k is
k(cid:48)k is three-

is one-dimensional, on the k-grid used in the ripplon scattering. The table for F−1

Similar to the ripplon scattering, w(ph)

k
Eqs. (3.15)–(3.15). For the 3D table Y−1
used.

is linearly interpolated, and F−1

k(cid:48)k is interpolated with
k(cid:48)k a trilinear scheme, in analogy with Eqs. (3.15)–(3.15), is

Both scattering mechanisms are implemented in the new custom_scatter2D integrator in

HOOMD-Blue. The tables for the ripplon and phonon scattering are prepared using ripplon_scat-

tering_module.py and phonon_scattering_module.py modules in the code base.

3.3.4 The simulation unit cell
In all of our simulations, we use a rectangular 2D unit cell with the aspect ratio of √3/2 to 1. The
size of the cell along x, denoted by Lx, depends on the number of electrons. We keep the electron
density ns = 2/√3×108 ≈ 1.15×108 cm−2 constant in all simulations. This means that the lattice
constant of a triangular lattice with this density is a = 1 µm. The system consists of N = 1024

electrons in all mobility calculations and N = 1600 electrons in all diffusion calculations, unless

stated otherwise.

60

We impose periodic boundary conditions in both dimensions. Therefore, an electron is returned

into the unit cell on the opposite side if it crosses the boundary. The periodic boundary conditions

also dictate how we deal with the long-range Coulomb forces, see below.

3.3.5 Temperature definition

Strictly speaking, the temperature is defined only for a system in equilibrium. In this case it can be

computed as

T (t) =

me
2kB

N
∑
i=1

v2
i (t).

(3.28)

However, we also want to have an analog of temperature for a system in the presence of an external

drive field, which establishes the stationary drift velocity vvvd. In this case, we define the temperature
in the frame moving with the velocity vvvd:

T (t) =

me
2kB

N
∑
i=1

(vvvi(t)− vvvd)2,

where the drift velocity is found as

vvvd =

1
NNt

N
∑
i=1

Nt∑

s=1

vvvi(s∆t).

(3.29)

(3.30)

Here s is the index enumerating time steps and Nt is the total number of time steps in the simulation.

This definition makes sense since we use small driving fields, and in the strongly correlated system

with a fast inter-electron energy exchange the velocity distribution remains close to the Maxwell-

Boltzmann distribution but centered around vvv = vvvd instead of vvv = 0. In the finite system with N
particles, the temperature fluctuates with the characteristic fluctuation amplitude ∆T ∝ T /√N (see
below for more details). For this reason, we average the temperature over the simulation time:

T =

1
Nt

Nt∑

s=1

T (s∆t).

(3.31)

The process T (t) has a finite correlation time. Therefore, when we calculate the standard error

of the mean, we must take the autocorrelation into account. The procedure for estimating the

standard error of the mean for a random processes with a finite correlation time is described in

Appendix A.10.

61

3.4 Dynamical properties of a spatially uniform electron system

3.4.1 Thermalization in a Wigner solid and a strongly correlated electron liquid

We study thermalization in the correlated electron system by following its time evolution from

a state far from thermal equilibrium. A natural measure of the thermalization is the distribution

over velocities: if the system is thermalized, this distribution should be of the Maxwell-Boltzmann

form, in the considered classical formulation. The time it takes to approach this distribution is the

characteristic thermalization time. The results of this section refer to the case where the electron

scattering by excitations in helium is disregarded. Therefore the considered thermalization is the

intrinsic property of the electron system itself.

First, we choose the initial state as a triangular lattice with all electrons but four having zero

velocities. The four moving electrons have momenta, which are equal in the absolute value; they

point in the opposite directions along two orthogonal axes, so that the total momentum is zero. The

absolute values of the momenta are large. Therefore, if they are expanded in the normal modes of

the finite-size crystal, the mode amplitudes are large and the dynamics of the crystal is substantially

nonlinear. It is this nonlinearity that ultimately leads to the thermalization.

The evolution of the kinetic energy averaged over particles is shown in Fig. 3.13 (a). After a

short transient, we found that the mean (over the electrons) kinetic energy reaches a steady value

and fluctuates about it. The initial transition period is only a few oscillations with the characteristic

frequency ωp equal to the short-wavelength plasma frequency, Eq. (3.37). After the transition

period the distribution of the electrons over the kinetic energy is close to the Boltzmann distribution

(see Fig. 3.13 (b)). The kinetic energies of individual electrons are oscillating in time, in agreement

with the picture that the electron motion is largely thermal fluctuations about quasiequilibrium

positions in the correlated system.

The mean kinetic energy, or equivalently, the temperature are fluctuating in time. These fluc-

tuations are seen in Fig. 3.13 (a). They result from the standing wave modes always present in the

system. Fluctuations of the kinetic energy in a system of N electrons are easy to describe if the

62

Figure 3.13 (a) Averaged over the particles kinetic energy Ekin = (2N)−1 ∑i mev2
i of electrons as
a function of time. At t = 0, the system of N = 400 electrons is prepared in a triangular lattice
with zero velocities for all but four electrons. The velocities of the four moving electrons are set
to a large magnitude of 3.4× 106 cm/s and four different directions along the coordinate axes, so
that the total momentum vanishes. The system then evolves with no scattering by the excitations
in helium, remaining a crystal (with defects). The energy is measured in the units of the average
Coulomb energy e2(πns)1/2. Time t is measured in units of 2π/ωp. The inset shows the initial
part of the evolution, with the same units on both axes. (b) The histogram of the distribution of
the electrons over their kinetic energy E(k) after 800 time steps (tωp/2π ≈ 15). The instantaneous
averaged over the particles kinetic energy at this point is Ekin/kB = 0.175 K (for the mean inter-
electron distance a = 1 µm). The solid line depicts the Boltzmann distribution at this temperature
(properly scaled to compare with the histogram).

electrons form a crystal and the electron motion can be represented as small-amplitude vibrations

about the equilibrium positions. One can write the kinetic energy Ekin in terms of the creation and
annihilation operators b†
kkkα ,bkkkα of the normal modes (here, kkk is the wave vector and α = 1,2 is the

mode polarization)

Ekin =

¯h
4N ∑

kkkα

ωkkkα (b†

kkkα − b

−kkkα )(b†

From this expression, in the classical limit

(cid:10)[Ekin(t)−(cid:104)Ekin(cid:105)][Ekin(0)−(cid:104)Ekin(cid:105)](cid:11) =

−kkkα − bkkkα ).
(cid:90)

(kBT )2
32π2Nns

∑
α

dkkk cos2ωkkkαt,

(3.32)

(3.33)

where ωkkkα are the mode frequencies. This expression shows that the limiting fluctuation frequency
in the harmonic approximation is ∼ 2ωp.

63

Figure 3.14 (a) Averaged over the particles kinetic energy Ekin = (2N)−1 ∑i mev2
i of electrons as
a function of time. At t = 0, the system of N = 400 electrons is prepared in a disordered liquid-
like configuration with zero velocities. The system then evolves with no scattering by helium
excitations and remains a liquid. The energy is measured in units of the average Coulomb energy
e2(πns)1/2. Time t is measured in the units of 2π/ωp. The inset shows the initial part of the
plot, with the same units on both axes. (b) The histogram of the distribution of the electrons over
their kinetic energy E(k) after 600 time steps (tωp/2π ≈ 11). The instantaneous averaged over
the particles kinetic energy at this point is Ekin/kB = 0.362 K (for the mean interelectron distance
a = 1 µm). The solid line depicts the Boltzmann distribution at this temperature (properly scaled
to compare with the histogram).

The power spectrum of the fluctuations for the crystal is shown in Fig. 3.15 (a). The power

spectrum is calculated as

Qω =

1
ts

(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ts

0

(cid:12)(cid:12)(cid:12)(cid:12)2

dt(cid:48) Q(t(cid:48))e−iωt(cid:48)

,

(3.34)

where ts is the simulation time, and Q(t(cid:48)) = (2N)−1 ∑i mv2
i (t(cid:48))−(cid:104)Ekin(cid:105) is the averaged over the
particles instantaneous value of the fluctuations of the kinetic energy at time t(cid:48). The actual max-
imum frequency is slightly different from 2ωp calculated using Eq. (3.37), because this value of

the limiting frequency is an approximation. More importantly, the overall spectrum differs from

the simple expression (3.33 in the low-frequency part. This is likely related to the presence of

defects generated by the initial “kick”; such defects should lead to higher spectral density of low-

frequency fluctuations. Overall, the results show a dramatic difference from the Fermi-Pasta-Ulam

64

Figure 3.15 Power spectrum Qω of the fluctuations of the kinetic energy per particle Ekin (arbitrary
units). (a) corresponds to the crystal, Fig. 3.13. (b) corresponds to the liquid, Fig. 3.14. Evolution
of Ekin over a longer period of time was taken to produce the Fourier transforms. The constant
component (ω = 0) is removed in both plots.

observation for a weakly nonlinear chain of oscillators. In part, this is a consequence of the strong

anahramonicity, but also of the long-range coupling and higher dimensionalty.

In another experiment, we study thermalization of a strongly correlated liquid. We prepare the

system in a disordered state with zero velocities and then let it evolve. The total energy of the sys-

tem is equal to the potential energy in the initial state. The energy is large enough for the system

to remain in the liquid phase during the subsequent dynamics. The dynamics of the mean kinetic

energy is shown in Fig. 3.14 (a). Fig. 3.14 (b) displays the established distribution of kinetic ener-

gies of individual electrons after the initial transition period. Qualitatively, the behavior is similar

to that in the case of a crystal discussed above. The thermalization time and the characteristic

frequencies of the fluctuations are approximately the same as in the case of a crystal. The Fourier

spectrum of the kinetic energy fluctuations for the liquid is shown in Fig. 3.15 (b). The spectrum is
close to the uniform spectrum from ω = 0 to ω ≈ 2ωp, which is characteristic of chaotic systems
and is a natural consequence of diffusion in the system.

These results demonstrate that the electron system does thermalize in both crystal and liquid

phases. The electrons exchange momenta on the time scale of ω−1

p , which determines the order of

65

0123ω/ωp012Qω×10−3(a)0123ω/ωp012Qω×10−2(b)magnitude of the thermalization time.

3.4.2 Diffusion in a spatially uniform electron system

An important characteristic of a many-particle system is diffusion. It describes the mean squared

particle displacement as a function of time. The diffusion constant Dx in one dimension is conven-

tionally defined as:

2Dxt =

(x(t)− x(0))2(cid:69)
(cid:68)

,

(3.35)

where x(t) is a particle’s coordinate, and the average is the ensemble average over realizations of

the particle’s trajectories. In an ideal electron gas, the diffusion is determined by scattering of elec-

trons off scattering centers, such as impurities, or quantized excitations of fields in the substrate,

such as phonons or ripplons. When the system is a strongly correlated liquid, the electron-electron

interaction limits the diffusion, and scattering by defects plays a less important role. In this case

the diffusion is also referred to as self-diffusion. It quantifies the rate and magnitude of displace-

ment of the particles with respect to each other. Therefore, vanishing of self-diffusion is a useful

indicator of a freezing transition.

The correlation length in the liquid is increasing with decreasing temperature. When correla-

tions become strong, one cannot adequately describe the dynamics of the system in terms of the

motion of individual particles. In this regime, diffusion is better characterized by the motion of

structural defects, in the first place, dislocations. We believe the activated character of this mo-

tion causes the diffusion constant to sharply depend on temperature. Even for a short correlation

length, diffusion in liquids is thermally activated for low temperatures; it is usually associated with

diffusion of individual particles and with creating a vacancy into which the particle can “jump” by

overcoming a relevant barrier. However, in our case, where the correlation length is much larger

than the interparticle distance, we believe the activation dependence is related primarily to diffu-

sion of topological defects. The other factor that contributes to the temperature dependence of the

diffusion coefficient is that the density of defects, in particular, of unbound dislocations, is strongly

temperature dependent.

66

Figure 3.16 Delaunay triangulation of the electron positions for a portion of the system with N =
1600 electrons. Electrons are represented by the vertices. The red vertices are electrons with five
nearest neighbors, while the green vertices are electrons with seven nearest neighbors. Pairs of
red and green vertices represent dislocations. (a) Liquid phase with Γ = 133 close to the freezing
(cid:107) ≈ 0.6 cm2/s. There are unbound dislocations, pairs of
transition, the diffusion constant D⊥ = D
bound dislocations, and domain walls. (b) Solid phase with Γ = 140 and D⊥ = D
(cid:107) = 0. Only
bound pairs of dislocations are present in the solid. The two snapshots refer to the two points from
Fig. 3.17 (b) for the corresponding values of Γ.

According to the Berezinskii-Kosterlitz-Thouless picture, freezing of a liquid into a crystal re-

sults from dislocation binding into pairs. When this happens, the diffusion should sharply decrease,

because diffusion would now require moving a pair of dislocations, which is virtually impossible.

Therefore we expect that, by measuring the diffusion coefficient, one can find the crystallization

transition as a point where the diffusion coefficient becomes equal to zero. As an illustration of this

mechanism of the phase transition, we show two snapshots of of the electron system in Fig. 3.16.

The two snapshots represent the liquid and the solid phases very close to the phase transition. The

liquid phase contains ordered areas as well as unbound and bound dislocations. The solid phase

is mostly ordered and only contains a few bound pairs of dislocations which do not contribute to

diffusion. Bound dislocations in a crystal can spontaneously emerge, but then they annihilate; they

do not unbind in the crystalline phase.

We measured the diffusion constants as a function of Γ in our simulations. The value of Γ was

determined from the data using Eq. (3.28) above. For each value of Γ, which sets the temperature,

67

the system was prepared in the triangular crystal lattice at a given temperature. Then the system
is equilibrated for 106 time steps before collecting data. With scattering, the thermal bath created

by phonons sets the temperature. Without scattering, we prepare the system with nonzero initial

velocities and measure the resulting temperature after the equilibration. The diffusion coefficient

was calculated in the following way. We simulate a trajectory by making Nt steps of duration ∆t, so
that the overall length of the trajectory is 2t = Nt∆t. We then look at the sections of the trajectory of
length t. These sections are shifted in time by ds∆t, with sufficiently large ds, so that the correlation
between the motion in different sections of the trajectory separated by a multiple of ds∆t is small.

We then sum up the squared displacement in different sections and also do the averaging over all

particles. The resulting expression for one-dimensional diffusion along the x-axis is

Dx =

1
2tN

1

(1 + Nt/2ds)

Nt /2ds
∑
s=0

∑
i

[xi(sds∆t +t)− xi(sds∆t)]2

(3.36)

Here, s enumerates different sections of the trajectory: a section starts at sds∆t. In our simulations,
ds is typically 103 or 104. If a particle crosses the simulation cell boundary, its trajectory becomes
discontinuous due to the periodic boundary condition. To correct for this non-physical behavior

when calculating diffusion from the data, we reconstruct the continuous particle trajectories in

such a way that a particle is allowed to continuously move into neighboring virtual cells. We found

that the diffusion constant obtained this way is independent of time t, which indicates the appli-

cability of our method; the method of calculating the diffusion constant described by Eq. (3.36) is

conventionally applied in molecular dynamics simulations [193].

A natural scale for the diffusion constant in our system can be estimated by thinking that the

electron displacement comes from fluctuations about quasi-equilibrium positions of the electron in
the field of other electrons. The average squared displacement is ∼ 1/Γns. The characteristic time
of vibrations in the classical system is the short-wavelength plasma frequency

ωp = (2πe2n3/2

s /me)1/2.

(3.37)

Therefore we expect that the scale of D far from the transition will be ωp/Γns. This agrees with

the simulations, to the order of magnitude. This is essentially the two-dimensional analog of the

68

Figure 3.17 Diffusion constants as a function of Γ with: (a) no scattering, (b) elastic and inelastic
scattering. For each point Γ, the system is prepared as a triangular lattice with the same density
but with different random velocities. Then the system is equilibrated for 106 time steps before
collecting data. The final temperature, and thus the value of Γ, are measured after the equilibration
period. N = 1600 particles were used in both simulations. We plot both Dx and Dy to facilitate
comparison with the results for the non-zero external periodic potential.

De Gennes scaling of the diffusion coefficient in atomic liquids [194] and of the Frenkel picture of

atomic motion in liquids [195].

For convenience, we define D⊥ ≡ Dx and D

(cid:107) ≡ Dy, which will serve as diffusion constants
across and along the periodic potential troughs. We maintain this notation even when there is no
periodic potential to enable direct comparison of all diffusion calculations. D⊥ and D
a uniform system, but they may be different in a periodic potential, when the system is anisotropic.

coincide in

(cid:107)

The results of the diffusion measurement in the absence of the scattering by ripplons or phonons

are shown in Fig. 3.17 (a), whereas Fig. 3.17 (b) shows the results in the presence of the scattering.

In both cases, we find a sharp change of D around Γ = 135–140. It indicates melting of the crystal.

This value of Γ is in good agreement with the experiments [1, 143]. However, it is slightly higher

than the value Γ = 130 found in most of the previous simulations [144, 145, 196, 146]. The

difference may be related to the difference in the size of the simulated systems. The results in

Ref. [144] and most of the results in Ref. [196] refer to the system of 100 electrons. The results in

Ref. [146] were obtained for 256 electrons. We find (see Fig. 3.18) that, if we simulate a system
with N = 100 and N = 252, the transition point is Γ ≈ 130, in agreement with [144, 146]. Our

69

1301401500.00.20.40.60.8D [cm2/s]DD(a)1201301401500.000.501.00D [cm2/s]DD(b)Figure 3.18 Diffusion constant as a function of Γ for the number of particles (a) N = 100, (b) N =
252. For each point Γ, the system is prepared as a triangular lattice at the temperature determined
by Γ and then equilibrated for 106 time steps before collecting data. The final temperature is
measured after the equilibration period. The transition point is slightly below Γ = 130 for N = 100
and at Γ ≈ 130 for N = 252. The results are obtained without scattering.

result for N = 1600 is in agreement with a recent Monte Carlo simulation [197] with N = 8100

particles, where the melting transition was observed at Γ = 138–140.

Without scattering, as in Fig. 3.17 (a), the curve D(Γ) has an unusual shape close to the freezing

point. When it approaches zero, it passes through the point with the vertical slope, as if there were a

bistability in a small range of temperatures close to freezing. We believe this to be a computational

artefact due to the finite system size and the finite simulation time. We find that close to the freezing

point, the temperature has large fluctuations on the time scale comparable with the simulation time.

Note that the total energy is conserved in this simulation. Therefore, along with the temperature,

the potential energy has similar fluctuations on the large time scales. It means that, close to the

freezing transition, the structure of the system is changing on the time scale comparable with the

simulation time. Therefore, the system spends a significant amount of time in non-equilibrium

configurations due to the fluctuations. This is seen also from the power spectrum of the energy

fluctuations in Fig. 3.19 (b), which displays a pronounced peak at zero temperature. Therefore, the

measured temperature averaged over the simulation time may be lower than the temperature in the

true equilibrium state for the same total energy by up to 1 %.

70

1201301400.00.20.40.6D [cm2/s]DD(a)1201301400.00.40.8D [cm2/s]DD(b)Figure 3.19 Low-frequency part of the power spectrum Qω of the temperature fluctuations close to
the Wigner crystallization (arbitrary units). Qω is calculated using Eq. (3.34). The data correspond
to three points in Fig. 3.17 (a). (a) Liquid at Γ ≈ 130. (b) Liquid in the transition regime at Γ ≈ 137.
(b) Crystal at Γ ≈ 140. The constant component (ω = 0) is removed.

The long-lived fluctuations of temperature do not occur far from the freezing transition, as

seen from the power spectrum in Figs. 3.19 (a), (c) for points above and below the crystallization

transition. In a hot liquid, the dynamics is fast and the non-equilibrium states quickly decay. In a

crystal, fluctuations of the kinetic and potential energies are not sufficient to bring the system to a

long-lived non-equilibrium state, since it would require melting. The described slow fluctuations

are more likely to occur in a small system, since temperature fluctuations scale with the system

size as ∆T ∝ T /N. The boundary conditions can also prevent non-equilibrium configurations from

decaying. As Fig. 3.17 (b) shows, electron scattering seems to eliminate this phenomenon. We

believe that the scattering quickly destroys non-equilibrium states, as it imposes on the electron

system the temperature of the thermal reservoir.

In the previous simulations, there was observed hysteresis in the temperature dependence of

the total energy and the diffusion constant [145, 196, 146]. To relate our observations to these
results, we carried out simulations where for each value of Γ we started with a liquid at Γi = 45
and then cooled it down to the desired value of Γ. The cooling is provided by the thermal bath of

phonons when the inelastic scattering is present. To cool the system to the desired temperature, we

simply set the thermal bath temperature to the desired value. The scattering rate is small, therefore
the system is cooled down gradually over 7× 105 time steps. Without scattering, we simulate the

71

0.000.020.04ω/ωp01234Qω×10−4(a)0.000.020.04ω/ωp0.000.250.500.751.00Qω×10−3(b)0.000.020.04ω/ωp012345Qω×10−4(c)Figure 3.20 Diffusion constants as a function of Γ with: (a) no scattering, (b) elastic and inelastic
scattering. For each point Γ, the system is prepared as a liquid at Γi = 45 and then cooled down to
the temperature corresponding to Γ, as described in the text. N = 1600 particles were used in both
simulations.

cooling in the following way: we run the system for consecutive intervals of time. At the end of

each interval, for the given positions of the particles, we proportionately scale down their velocities.

The duration of the intervals is gradually increased upon approaching the final temperature. On
total we use 20 intervals and approximately 7× 105 steps. Then we let the system evolve on its
own for at least 106 time steps before we (i) measure the distribution over the velocities and thus

determine the value of Γ and (ii) start accumulating data on diffusion. We repeatedly measure the

effective Γ, which displays small fluctuations, see below.

By using this procedure, we obtain the Γ-dependence of the diffusion constant shown in Fig. 3.20.

It turns out to be nearly the same as that obtained by starting with a crystal lattice for the same Γ.

We do not observe hysteresis in the melting-freezing transition. The hysteresis observed in the

previous simulations may be related to a shorter simulation time. This time is much longer in the

present study. A long time is needed because, close to freezing, the dynamics is determined by

diffusion of dislocations, which becomes very slow.

We also studied the effect of scattering by ripplons on the electron diffusion. Where the scat-

tering is weak, as is the case for electrons on the helium surface, so that the scattering rate is small

compared to ωp, we do not expect a strong effect of the scattering on the diffusion. Primarily the

72

1251301351400.000.250.500.751.001.25D [cm2/s]DD(a)1201301400.000.501.00D [cm2/s]DD(b)diffusion is determined by the electron-electron interaction. This is indeed seen in the simulations,

see Figs. 3.17 and 3.20. The result should be contrasted with the results obtained by assuming

that the electron motion is overdamped, as it is often done in molecular dynamics simulations of

interacting systems [28]. We carried out simulations of diffusion assuming a phenomenological

friction force with the friction coefficient comparable to ωp. In this case the change of the diffusion

coefficient due to dissipation was very significant.

3.4.3 Mobility in a uniform system

As another benchmark for our setup, we study the mobility of the electron system without a peri-

odic potential. This characteristic, which is of primary interested for the experiment, has not ever

been studied in simulations for electrons on helium, as it requires taking into account the actual

mechanism of the electron scattering rather than using a phenomenological overdamped dynamics.

We use the conventional definition of the mobility µ

vd = µEd,

(3.38)

where vd is the drift velocity induced by the constant uniform in-plane electric field Ed.

In a uniform system without a periodic potential, the mobility is mostly determined by the scat-

tering by ripplons. We expect the mobility in the non-interacting system to follow the expression

(cid:90) ∞
dθ q2φ 2(cid:16)qrB

0

(cid:17)

e

(cid:90) 2π

0

(1− cosθ )

2

where

1

τse(p)

=

¯hkBT

8παmer2
B

µse =

2m3

e(kBT )2

d p p3τse(p)e−p2/2mekBT ,

(3.39)

.

(3.40)

(cid:12)(cid:12)(cid:12)¯hq=p√2(1−cosθ )

This is a standard result obtain using the Boltzmann equation, see Appendix A.8 for details. In

a strongly correlated uniform liquid, the momentum is transferred between electrons much faster

than the time between scattering events. Therefore, the expression for the mobility is different

from that in the non-interacting limit (see Ref. [3] and Appendix A.8.2):

µme =

eτme
me

,

73

(3.41)

Figure 3.21 Theoretical calculation for the single-electron mobility µse and the electron liquid mo-
bility µse vs temperature. The electron density ns = 2/√3×108 cm−2 is assumed in the calculation
for the strongly correlated system. The freezing transition is at T ≈ 0.23 K. Above T ≈ 0.7 K, the
mobility is primarily limited by the scattering by the helium atoms in the vapor phase.

with

τ−1
me =

¯h4

16m2

Bα√2πmekBT
er2

(cid:90) ∞

0

dqq4φ 2(cid:16)qrB

2

(cid:17)

¯h2q2
8mekBT .

e−

(3.42)

The theoretical results for the mobility in the two cases are compared in Fig. 3.21. It turns out

that the two expressions lead to very close numerical values. The difference is decreasing with the

increasing temperature.

In the simulations, we determine the mobility in the following way. The procedure described

below is also applied in the presence of an external periodic potential. In the first stage, the system
is prepared in a liquid state with Γ = 45. Then the system is gradually cooled down for 2× 104
steps without a driving field to bring it to the desired temperature T . If there is a periodic potential,

the cooling is performed in the periodic potential. The cooling is mediated by the coupling to

the thermal bath of phonons through the inelastic scattering. The phonon thermal bath has the

temperature T . During the cooling time, the inelastic scattering rate is accelerated by a factor of 50

to reduce the computation time. The purpose of the cooling stage is to allow the system to adapt to

its equilibrium structure in the presence of the external potential.

During the second stage, the inelastic scattering rate is returned to its regular value. The driving

74

0.20.30.40.50.60.7T [K]0.20.40.60.81.0 [cm2/Vs]×108mesefield EEEd is turned on, and the system is allowed to establish the steady state over 106 time steps. In
the steady state, the temperature may be different from T due to heating by the drive. We measure

the temperature in the steady state using Eq. (3.29).

The third stage is the data collection in the steady state. For this stage, we run the simulation
for 106 time steps, with data recorded every 103 or 104 time steps. It is indicated in the text where

we use a different simulation time. The drift velocity is then computed as a simple average over

all particles and all data snapshots:

vvvd =

ds
NNt ∑

i,s

vvvi(sds∆t),

(3.43)

where s enumerates the data snapshots, and ds is the number of steps between the snapshots. The

standard error of this estimate must take into account the nonzero autocorrelation of the velocity,
(cid:104)(vvvi(t)−vvvd)(vvvi(t(cid:48))−vvvd)(cid:105) (cid:54)= 0, in a finite range of t −t(cid:48). The procedure for calculating the standard
error is described in Appendix A.10, Eq. (129).

The obtained response, i.e., vd vs. Ed for the non-interacting electron gas in the range of
EEEd from 0 to 0.7 mV/cm is shown in Fig. 3.22. The simulation result slightly deviates from the
prediction of Eq. (3.39). There are two reasons for the deviation. First, the transport is not in

the linear regime in the simulation, as can be seen from the small curvature of vd vs Ed. The
nonlinearity is caused by the substantial heating of the electrons from T = 0.354 K to T = 0.4 K.

The difference of the simulations from Eq. (3.39) on the small-drive side is a consequence of the

fact that Eq. (3.39) does not take into account the inelastic scattering that can potentially reduce

the mobility by up to 15 %.

To test this hypothesis, we perform two additional simulations. First, we reduce the width of

the range of Ed from 0.6 mV/cm to 0.06 mV/cm (we remind that the electron density is fixed
in our simulation; therefore the field is measured in dimensional units). The result is shown in

Fig. 3.23 (a). We can assume that the nonlinearity is negligible in this case. Nevertheless, the devi-

ation from the theoretical line is not entirely eliminated. We continue by repeating the simulation

with the 10 times smaller inelastic scattering rate. It makes the inelastic scattering rate two orders

of magnitude smaller than the elastic scattering rate. The result is shown in Fig. 3.23 (b). Now

75

Figure 3.22 (a) Drift velocity vd vs drive field Ed and (b) temperature vs drive field for a uniform
system without electron-electron interaction. The filled circles are the simulation data, the dashed
red line is the least-squares fit, and the solid black line is the prediction by Eqs. (3.39)–(3.40).
The temperature T is computed using Eqs. (3.29)–(3.31). Error bars are not shown since they are
smaller than the marker size.

the data points are very close to the theoretical line, as expected. This result confirms that our

simulation correctly describes the response of the system to an external drive.

The nonlinearity observed already for Ed ∼ 0.7 mV/cm suggests that we should operate at
much smaller driving fields. However, this is computationally too expensive for simulations with

the electron-electron interaction. Due to the low scattering rates, the drift velocity vd fluctuates
significantly on large time scales. Therefore, to obtain acceptable relative accuracy for vd one needs
either long simulation times or large Ed. To reduce the simulation time, we perform all transport
simulations with Ed ∼ 0.7 mV/cm and keep in mind that the transport in weakly nonlinear. The
weak nonlinearity will not alter the phenomena described below in any significant way.

We also study the response of the uniform electron system to the drive with the electron-

electron interaction taken into account. The result of the simulation of a strongly correlated liquid

at Γ = 90 is shown in Fig. 3.24. The theoretical line in this case is calculated using Eqs. (3.41)–

(3.42). As with the non-interacting system, the response curve deviates from the theoretical predic-

tion. The reasons for the deviation are also the weak nonlinearity of the response and the inelastic

scattering, which are not taken into account in Eqs. (3.41)–(3.42). Overall, this result confirms that

76

0.00.20.40.6Ed [mV/cm]0.01.02.03.0vd [cm/s]×1040.00.20.40.6Ed [mV/cm]0.350.370.390.41T [K](a)(b)Figure 3.23 Drift velocity vd vs drive field Ed for Ed < 0.07 mV/cm. (a) corresponds to the regular
rate of inelastic scattering (∼ 106 s−1). (b) corresponds to the rate of inelastic scattering reduced
by a factor of 10 (to ∼ 105 s−1). The filled circles are the simulation data, the dashed red line is
the least-squares fit by a straight line. The solid black line is the prediction by Eqs. (3.39)–(3.40).
The error bars are one standard error. These simulations used 1.4 × 108 time steps and ∆t was
increased to ∆t = 2.57× 10−10 s.

Figure 3.24 (a) The drift velocity vd vs the drive field Ed and (b) the temperature T vs the drive field
Ed for a uniform strongly correlated electron liquid initially at Γ = 90. The filled circles are the
simulation data, the red dashed line is the least-squares fit, and the black solid line is the prediction
by Eqs. (3.41)–(3.42). The effective temperature T is computed using Eqs. (3.29)–(3.31). The
error bars correspond to one standard error.

77

0.000.020.040.06Ed [mV/cm]0.01.02.03.0vd [cm/s]×1030.000.020.040.06Ed[mV/cm]0.3450.3500.3550.360(a)0.000.020.040.06Ed [mV/cm]0.01.02.03.0vd [cm/s]×1030.000.020.040.06Ed[mV/cm]0.3450.3500.3550.3600.3650.370(b)0.00.20.40.6Ed [mV/cm]0.00.51.01.52.02.5vd [cm/s]×1040.00.20.40.6Ed [mV/cm]0.350.360.370.380.39T [K](a)(b)our simulation procedure correctly describes mobility for non-interacting and interacting electrons

on helium close to the linear regime. We note that our simulation is unable to capture the correct

nonlinear transport behavior for a Wigner crystal, when the Bragg-Cherenkov scattering limits the

mobility. However, the simulation should give a correct estimate for the linear mobility of a Wigner

crystal corresponding to very small drive fields.

3.5 Strongly correlated electron system in a one-dimensional periodic po-

tential

The central part of the study of the electron dynamics refers to the analysis of the effect of a

periodic potential. We study this effect by calculating diffusion and mobility. We use diffusion

as an important characteristic of the state of the system in the periodic potential, whereas the

mobility is the parameter directly accessible to the experiment. We consider the simplest type of

the periodicity, where the potential is one-dimensional, and we take it in the simple sinusoidal form

Vext(rrr) = −Acos(Kx),

(3.44)

where A is the potential amplitude, K is the reciprocal lattice vector of the potential, and the origin

is at the center of the cell. For convenience, and keeping in mind that our dimensional electron

density is fixed, we measure A in kelvins, implying the value A/kB. The minus sign is introduced
for convenience, so that no potential minimum lies at the cell boundary. Because of the periodic

boundary conditions, only K = 2πn/Lx with integer n can be simulated this way.

In the periodic potential (3.44), the system becomes anisotropic. We calculate the diffusion

coefficients and the mobility both across and along the periodic potential troughs. As discussed

above, these quantities are not simply related in the correlated system. We will use the notations
⊥ ≡ µx,
D⊥ ≡ Dx, D
µ
(cid:107) ≡ µy for the mobility across and along the periodic potential troughs.
Of particular interest are the effects of commensurability of the potential and the electron sys-

(cid:107) ≡ Dy for the diffusion across and along the periodic potential troughs, and µ

tem. For a Wigner solid, we define the commensurability parameter p as the ratio p = K/G of

the smallest reciprocal lattice vector K of the potential to the smallest reciprocal lattice vector

78

Figure 3.25 Triangular lattice with the distance between the vertical rows √3a/2 = 2π/G. The
dashed lines display the maxima of the periodic potential, with the period 2π/K. The commensu-
rability parameter is defined as p = K/G.

G = 4π/√3a of the triangular lattice (see Fig. 3.25). Here a = (2/√3ns)1/2 is the crystal lattice
constant. We say that the system is commensurate if p is integer and is incommensurate otherwise;

for p = 1 the period of the potential is equal to the minimal distance between the electron rows

in the lattice. We note that, for a one-dimensional potential, there is another option of commen-

surability, where the lattice is rotated by π/2, so that in the fully commensurate configuration the

period of the potential is a/2.

To a large extent, we focus on the properties of a strongly correlated liquid. For a liquid, we

define p in the same way as for a crystal but using the reciprocal lattice vector G of the crystal that

would exist at a given electron density ns. In this case, the lattice constant a is also calculated as
a = (2/√3ns)1/2.

The effect of the modulation on the mobility µ
(cid:107)

along the modulation troughs is most pro-

nounced if the modulation changes the state of the electron system between the liquid and the

primary interest to us. We will concentrate on the results for the transverse mobility µ

ordered phase. We will briefly describe the ordering in the periodic potential, but this is not of
⊥, where
the effect of the periodic potential is often strong even where it is not accompanied by a phase

transition.

79

• 1111( • • • • • • • • • • • • • • • • • • 1111( • • •2π/K2π/G3.5.1 Single-electron approximation

Before we present the results for a strongly correlated system in a periodic potential, let us briefly

discuss the effects of the potential on a non-interacting electron gas. Let us assume that elastic

scattering is the main scattering mechanism that determines diffusion. Only electrons with total

energy above 2A can participate in transport across the modulation troughs. The fraction of such

electrons, according to the Boltzmann distribution, is:

(cid:90)

(cid:18)
d2r d2pe−

(cid:19)

(cid:34)

Vext(rrr)+

/kBT

p2
2me

1
C

where

and

(cid:35)

,

p2
2me − 2A

,

(3.45)

(3.46)

(3.47)

θ

Vext(rrr) +

(cid:19)

p2
2me

/kBT

(cid:90)

C =

(cid:18)
d2r d2pe−

Vext(rrr)+

 0, x ≤ 0,

1, x > 0.

θ (x) =

Both diffusion D⊥ and mobility µ
⊥ across the modulation troughs are roughly proportional to the
expression (3.45). It is significantly different from unity when 2A is comparable to kBT . Therefore,
the effect of the modulation on transport is going to be appreciable only for 2A (cid:38) kBT .

3.5.2 Case p (cid:28) 1
We will divide our analysis of the modulation effects into several parameter ranges. First, we con-
sider the case p (cid:28) 1, where the system is far from commensurability. In this range, the potential
has very little effect on the strongly interacting electron system. Then we consider the case of max-

imal commensurability p = 1 where the effect of the potential is very strong. In the intermediate

regime, where p < 1 but close to 1, the system is undergoing the commensurate-incommensurate

(C-I) transition, and both structural and transport properties are significantly affected by the peri-

odic potential. Finally, we show the results for the case p > 1.

We begin our analysis with p (cid:28) 1. In Fig. 3.26, we show the results of the diffusion simulation
for the electron system in an external potential with A = 0.5 K and p = 0.3. Interestingly, the

80

vs Γ in the periodic potential with A = 0.5 K and
Figure 3.26 Diffusion constants D⊥ and D
(cid:107)
p = 0.3. The simulation includes both elastic and inelastic scattering. For every point Γ, the
system is prepared in a liquid state with Γ = 45 and then cooled down, as described in Sec. 3.4.2.
electrons. The potential has a negligible effect on the diffusion and on the freezing transition for
p = 0.3.

diffusion constants and the phase transition point are virtually unaffected by the external potential

(cf. Fig. 3.17). For the non-interacting electron gas, the expression (3.45) predicts a significant
reduction of D⊥ since 2A > kBT . However, the electron-electron interaction in the strongly in-
teracting system effectively averages out the potential for small p. The average Coulomb energy
per particle is e2(πns)1/2 ≈ 31.827 K, it is much larger than A = 0.5 K. Far from the commen-
surability, the electron-electron interaction is a dominating factor that essentially fully determines

system’s properties.

To further illustrate the difference between the non-interacting and interacting systems, we plot

the transverse mobility for both systems as a function of A for p = 10/32 = 0.3125 in Fig. 3.27. For

both systems, the temperature is T = 0.354 K, which is smaller than the upper limit Amax = 0.5 K
of the potential amplitude. As predicted by Eq. (3.45), the mobility µ
⊥ in the non-interacting gas
is dramatically decreased by the potential when 2A (cid:38) kBT . In contrast, the mobility of the liquid
is only slightly reduced for the same potential strength.

The system displays the effective averaging of the periodic potential in a broad range of the

81

1301401500.000.501.00D [cm2/s]DDFigure 3.27 The mobility µ
⊥ transverse to the modulation troughs as a function of the periodic
potential amplitude A in a non-interacting electron gas and in a strongly correlated electron liquid
for p = 10/32 = 0.3125. In both cases, T = 0.354 K which corresponds to Γ = 90 for the liquid.
Error bars are not shown for clarity since most of them are smaller than the marker size. The
potential with p = 0.3125 significantly changes the mobility in the gas, but only slightly in the
liquid.

ratio of the periods where p (cid:28) 1. This range depends on the relation between the modulation
strength and the melting temperature. For A = 0.5 K, it is p (cid:46) 0.3.

3.5.3 The case of the maximal commensurability, p = 1

The system is most sensitive to the periodic potential when p is integer. It turns out that in this
case the transverse mobility and the both diffusion constants D⊥ and D
are significantly affected
(cid:107)
even by a weak potential with A (cid:28) kBT . The crystallization transition is also very sensitive to the
periodic potential amplitude.

We begin by studying diffusion in a relatively strong potential with A = 0.5 K and p = 1. Here,

2A is larger than the typical temperature in the simulation, but much smaller than the average
Coulomb energy of the electron-electron interaction e2(πns)1/2. The diffusion constants calcu-
lated as a function of Γ for this case are shown in Fig. 3.28 (a). The first striking feature of the

82

0.00.10.20.30.40.5A[K]1234µ⊥[cm2/Vs]×107e-gase-liquidvs Γ in the periodic potential with p = 1 and (a)
Figure 3.28 Diffusion constants D⊥ and D
(cid:107)
A = 0.5 K, (b) A = 0.05 K. The simulations use N = 1600 electrons and include both elastic and
inelastic scattering. For every point Γ, the system is prepared in a liquid state with Γi = 45 and
then cooled down in the periodic potential, as described in Sec. 3.4.2.

data is the increased melting temperature. This effect is similar to the modulation-induced freezing

observed in colloidal systems [172, 4, 27]. Close to the freezing point, the order in the system is

governed by the competition of the potential energy and the entropy. In the commensurate poten-

tial, the system can gain more potential energy by ordering in a triangular lattice. Therefore, the

transition happens at a higher temperature than in a uniform system.

Another notable difference from the case of a uniform system (cf. Fig. 3.17) is that the tran-

sition, as seen in the diffusion, becomes smooth. This can be understood in the case of a strong

potential, if one thinks of the modulated liquid as a series of electron channels in the modulation

troughs. Freezing occurs when the channels become strongly coupled with each other and even-

tually ordered. The nature of this transition is different from the Berezinskii-Kosterlitz-Thouless

mechanism. Therefore, the transition in the presence of the commensurate potential is expected to

have a different form.

The diffusion is also strongly anisotropic since the potential breaks the rotation symmetry.

In the picture of weakly coupled electron channels, the electrons in different channels can move

relative to each other, producing a high value of D
(cid:107)

. Since longitudinal diffusion is mediated by

an intra-channel motion, the results of simulating it in a finite-size system should display strong

83

60801000123D [cm2/s]DD(a)1001101201301400.00.20.40.60.81.0D [cm2/s]DD(b)fluctuations. This is because the electrons within a channel are strongly correlated, and therefore

essentially move all together, in a small system. The averaging over particles becomes less efficient

and the values of D

can be broadly spread. Such spreading of D

is seen in Fig. 3.28 (a).

(cid:107)

(cid:107)

On the other hand, to diffuse in the transverse direction an electron needs to overcome a barrier

of height 2A in addition to the barrier created by other electrons. These “other” electrons also

have to move in a concerted way to create a vacancy for the hopping electron. Given that they

are confined to a channel, this is less likely to happen than in an isotropic case. Therefore the

activation energy of such process is larger than 2A and than the activation energy in the absence of
the modulation, so that D⊥ is significantly reduced.

The effect of the commensurate potential remains strong even for a much weaker potential,
2A (cid:28) kBT . Fig. 3.28 (b) shows the diffusion constants as a function of Γ in the commensurate
potential with A = 0.05 K. The potential amplitude is much smaller than the temperature for all

points in the simulation. As in the case of A = 0.5 K, the diffusion smoothly approaches zero at
Γ ≈ 110. However, in this case there is no strong anisotropy of the diffusion coefficient and no
strong spreading of the data for D
, because the binding into channels and the ordering of electrons

in different channels occur simultaneously.

(cid:107)

To confirm that the system is undergoing a freezing transition at this point, we plot the second-
order correlation function g(2)(x,y) for three values of Γ approaching the freezing transition in
Fig. 3.29. We calculate g(2)(x,y) as the average density of electrons (divided by ns) at the coordi-
nates (x,y) counted off from a given electron,

As seen in Fig. 3.29, the system is gradually changing from a modulated liquid at Γ = 100 to an

ordered crystal with sharp peaks of the triangular lattice at Γ = 110. The transition is accompanied

by ordering within and between the channels simultaneously. The system remains a crystal with a
pronounced long-range order for Γ (cid:38) 110.

Clearly, a crystal in a commensurate potential must be pinned to the potential. A liquid, on

the other hand, should not be pinned and must exhibit nonzero transverse mobility. The nature of

84

g(2)(rrr) =

1

nsN(N − 1)

(cid:42)
∑
i

∑
j(cid:54)=i

(cid:43)
δ (rrr− rrri j)

.

(3.48)

Figure 3.29 Pair correlation function g(2)(x,y) of the electron system in the external potential with
A = 0.05 K and p = 1. (a) Γ = 100, (b) Γ = 105, and (c) Γ = 110. The correlation function is
dimensionless. The red cross indicates the location of the origin.

⊥ in the electron liquid as a function of temperature. (b)
Figure 3.30 (a) The transverse mobility µ
⊥/µ0 as a function of the inverse temperature. Both
The logarithm of the transverse mobility µ
panels are for the electron liquid in the periodic potential with A = 0.05 for p = 1. The hollow
⊥/µ0) for points with T < 0.3 K. These points correspond to the crystallized
circles show log(µ
system (Γ (cid:38) 110), which is pinned. The nonzero values of µ
⊥ here give the measurement error.
The number of electrons in this and other mobility calculations is N = 1024.

85

0.20.30.40.50.6T[K]0123µ⊥[cm2/Vs]×106(a)2341/T[1/K]−9−8−7−6−5−4−3−2log(µ⊥/µ0)(b)Figure 3.31 (a) The transverse mobility µ
⊥ in the electron liquid as a function of the periodic
potential amplitude A for p = 1. (b) The logarithm of the transverse mobility µ
⊥ as a function of
the periodic potential amplitude A for p = 1. The hollow circles represent the points where the
calculation error is comparable to the value of µ
⊥. The solid line is the least-squares fit, its slope
is 100± 4 K−1. The hollow points were excluded from the fitting procedure. Γ = 90 in both plots.

by calculating the transverse mobility µ

the transition between these two distinct states is of considerable interest. We study this transition
⊥ as a function of temperature. The result of the simula-
tion for A = 0.05 K is displayed in Fig. 3.30 (a). As discussed above, the system is pinned for
temperatures T < 0.3 K, which corresponds to the freezing point Γ ≈ 110 for this system. For
higher temperatures, the mobility is smoothly increasing the temperature. The shape of the curve
suggests that the current is activated. Indeed, if we plot log(µ
⊥/µ0) vs 1/T (see Fig. 3.30 (b)), it is
close to a straight line for temperatures above freezing. Here µ0 = 3.5× 107 cm2/Vs is a constant
introduced for normalization; it corresponds to the mobility of the electron liquid at T = 0.354 K

in a uniform system. Such revealing form of µ

⊥(T ) is also reproduced for other values of A.
To confirm that the current is activated, we perform a series of simulations that compute µ

⊥ as a
function of the periodic potential amplitude. The result of the simulation for Γ = 90 (T = 0.354 K)

is shown in Fig. 3.31. The mobility is reduced almost to zero for A as small as 0.05 K. The plot of
⊥ decays

⊥ vs A is nearly a straight line (Fig. 3.31 (b)) with some slope −s, indicating that µ

log µ

86

0.000.020.040.06A[K]01234µ⊥[cm2/Vs]×107(a)0.000.020.040.06A[K]−8−6−4−20log(µ⊥/µ0)(b)Figure 3.32 Logarithm of the transverse mobility µ
⊥ as a function of the periodic potential ampli-
tude A with p = 1. Three sets of points correspond to the electron liquid with Γ = 40 (red circles),
Γ = 60 (green squares), and Γ = 90 (blue triangles). Error bars are not shown for clarity. The
system consists of N = 1024 electrons. The solid lines are the least-squares fit. For each Γ, we
excluded points with µ
⊥, since the calculation error
for them is comparable with their value.

⊥ smaller than 5 % of the maximum value of µ

exponentially with A:

⊥ ∼ e−sA.
µ

(3.49)

The curve deviates from the exponential for very small A, where the scattering limits the mobility.

However, already for A as small as 0.005 K the interplay between the periodic potential and the

electron-electron interactions becomes the major factor that determines the mobility. The rate s of

the exponential decay depends on temperature. This is clearly demonstrated by the plots of log µ
⊥

vs A for three values of Γ in Fig. 3.32.

The exponential decay of µ

⊥ as a function of A or 1/T is an indication of the activated nature of
the current. Simple activation of a single electron over the modulation barrier of height 2A would
give the slope s = 2/kBT , which would be s = 5.66 K−1 for the temperature T = 0.354 K used
in Fig. 3.31. Instead, s ≈ 110 K−1 in the correlated electron liquid at T = 0.354 K. An electron
in the liquid cannot overcome a modulation barrier as a single particle since it moves in the field

87

0.000.050.10A[K]−3−2−10log(µ⊥/µ0)Γ=40Γ=60Γ=90of other electrons. Instead, a collective motion of electrons should be considered. We expect the

current to be mediated by locally ordered groups of particles overcoming the modulation barrier

simultaneously.

To find the natural length scale over which the electrons are ordered, we consider the radial

distribution function g(r) in the absence of a periodic potential. It is defined as (divided by ns) the

average density of electrons at a distance r from a given electron,

.

(3.50)

g(r) =

1

2πrnsN(N − 1)

(cid:42)
∑
i

∑
j(cid:54)=i

(cid:43)
δ (r− ri j)

The function g(r) for several values of Γ is shown in Fig. 3.33 (a). The envelope of g(r) decays

exponentially in a liquid, approaching unity at infinity. If we denote positions of the peaks of
g(r) as rp, and the values at the peaks as gp, then they are approximately related as gp ≈ 1 +
γ exp(−rp/ξc), with some constants γ and ξc. We will call ξc a correlation length. We find γ and
ξc by fitting the peaks of g(r) with the exponential form above.

Strictly speaking, the correlation length describes g(r) at large r/a. In our case the parameter
ξc/a is (cid:46) 1. The typical distance over which the electron positions are correlated is a + ξc. It
means that, for a weak periodic potential, the average number of electrons that have to collec-
tively overcome the barrier in the activation process is ∼ πns(ξc + a)2. In accordance with this
hypothesis, the slope s in Eq. (3.49) should be of the order

s ∼ 2πns(ξc + a)2/kBT

(3.51)

To test the relation (3.51), we compute the correlation length ξc for several values of Γ and com-
pare the obtained values of s with Eq. (3.51). We find the correlation length by fitting the peaks of
g(r) located at points rp with a function of the form γe−rp/ξc +1. The fitting parameters are γ and
ξc. The fitting curves are shown as the the dashed lines in Fig. 3.33.

We plot s vs 2π(ξc + a)2/a2T in Fig. 3.33 (b). The points follow a linear trend with the slope
1.06± 0.02. If the relation Eq. (3.51) is valid, we should expect the slope to be of the order of
a2ns ≈ 1.15, which is indeed the case up to a small factor. It is a strong evidence that the current

88

Figure 3.33 (a) The radial distribution function g(r) (solid lines) of the electron system without
modulation for Γ = 30 (top), Γ = 60 (middle), and Γ = 100 (bottom). The number of electrons
is N = 1024. The dashed lines show the least-squares fit of the peaks gp with a function of the
form γe−rp/ξc + 1, see the text for details. The found values of ξc are: ξc = (0.76± 0.05)a for
Γ = 30, ξc = (1.05± 0.05)a for Γ = 60, and ξc = (1.47± 0.09) for Γ = 100. (b) The exponential
decay rate s plotted vs 2π(ξc + a)2/a2T . The values of s are found from the plots such as that
in Fig. 3.32. Different points represent systems with Γ from 30 to 110. The correlation length is
computed without the external potential or drive, as in panel (a). The solid line is the least-squares
fit by a straight line passing through the origin. It has the slope 1.06± 0.02.

in the commensurate system is mediated by the activated collective motion of electrons within the
radius of ∼ (a + ξc).

The relation (3.51) can potentially provide an opportunity of a direct experimental measure-

ment of the correlation length in strongly correlated electron liquids.

The diffusion constants also depend on A in the system with p = 1, as shown in Fig. 3.34.

However, the diffusion is not as sensitive to the modulation as the mobility. The diffusion constant
D⊥ is only reduced by a factor of ∼ 1/2 by the potential with A = 0.05 K, while the same potential
reduces the mobility by a factor of 50. This is a manifestation of the fact that diffusion and mobility

in a strongly correlated liquid are not directly related. Without modulation, the liquid can easily

slide along the helium surface as a whole even when the diffusion constant is small. The sliding

velocity is limited by the scattering of electrons by ripplons. The weak commensurate potential

89

20601001402π(ξc+a)2/a2T[K−1]2060100140s[K−1](b)Figure 3.34 Diffusion constants D⊥ and D
with p = 1. The system consists of N = 1024 electrons and Γ = 90.

(cid:107)

as a function of the periodic potential amplitude A,

prevents the system from freely sliding across the modulation troughs, thus dramatically reducing

the mobility. The diffusion constants, on the other hand, quantify the motion of electrons relative to

each other, and are mostly limited by the inter-electron interaction. This motion is not as strongly

affected by the potential as the ability to slide along the surface.

3.5.4 Structure and crystallization for p < 1 close to p = 1

In the intermediate range of p, where p < 1 but close to p = 1, the periodicities of the electron sys-

tem and of the external periodic potential compete. In this range, multiple phenomena can occur.

First, if we consider a Wigner crystal, it is expected to make a transition from an incommensurate

crystal to a commensurate crystal. This transition should be accompanied by a structural change.

According to the theory of Pokrovsky and Talapov [26], close to the C-I transition, the crystal

is expected to develop one-dimensional solitons of density variation. As a consequence, at some

point when p becomes close to p = 1, the system is expected to adjust to the potential and become

commensurate. We also expect to find some intermediate structures in the transition region.

The liquid phase is also expected to change its structure in a similar way. Far from commen-

90

0.000.020.040.06A[K]0.60.81.01.21.41.61.82.02.2D[cm2/s]D⊥Dkvs Γ in the periodic potential with A = 0.5 K and
Figure 3.35 Diffusion constants D⊥ and D
(cid:107)
p = 30/40 = 0.75, N = 1600. The simulation includes both elastic and inelastic scattering. For
every point Γ, the system is prepared as a liquid at Γi = 45 in the periodic potential and then cooled
down to the temperature corresponding to Γ, as described in Sec. 3.4.2. After cooling, the system
is equilibrated for 106 more steps. The freezing transition occurs at Γ ≈ 145.

surability, the liquid has a structure similar to that of a uniform system but with some density

modulation. When p is approaching p = 1, for a sufficiently strong potential we expect the liquid

to become a series of semi-ordered decoupled channels residing in the periodic potential minima.

Since the potential in this range of p is expected to significantly affect the system’s structure, it is

also expected to change the freezing transition temperature.

In this section, we will study the structure of both liquid and solid phases of the electron system
in the external potential with A = 0.5 K as we increase p from p = 0.75 to p ≈ 1. For A = 0.5 K,
the properties of the system become significantly different from the case p (cid:28) 1 for p ≈ 0.75. The
diffusion coefficients as functions of Γ for p = 30/40 = 0.75 and A = 0.5 K are shown in Fig. 3.35.
As seen in the figure, the freezing transition occurs at the point Γ ≈ 145 instead of Γ ≈ 137 that was
observed in the uniform system and in the system with the periodic potential with p (cid:28) 1. For Γ
above the freezing point, the system assumes a structure of a triangular crystal with a superstructure

formed by solitons of density. Two of such structures are shown in Fig. 3.36. In the figure, we show

the Voronoi tessellation for the resulting structures. Each cell in the Voronoi tessellation contains

91

1301401500.00.20.40.60.8D [cm2/s]DDFigure 3.36 Voronoi tessellation of the Wigner crystal with the electron number N = 1600 in the
incommensurate periodic potential with A = 0.5 K and p = 30/40 = 0.75. (a) Γ ≈ 156, (b) and Γ ≈
158. Voronoi cells are colored depending on their surface area: from dark purple for smaller cells to
light yellow for larger cells. The variation of the cell areas is from 0.84a2 to 0.88a2. The simulation
procedure is the same as in Fig. 3.35. The electron positions are averaged over time to average
over small-amplitude thermal fluctuations. Clear rows of increased density appear at a nonzero
angle with the modulation direction, which correspond to the incommensuration solitons. The
two different soliton structures correspond to slightly different orientation angles that the system
crystallized into. Groups of black cells are a result of disorder and averaging over time.

points that are closer to a given electron than to any other electrons. Inspired by Ref. [5], we

use color to show the surface area of each Voronoi cell. To reduce the disorder resulting from

thermal motion, we average the electrons’ positions over time. In both panels of Fig. 3.36, one can

see superstructures of density modulation appearing due to the incommensurate periodic potential.

We observe a variety of such superstructures. The structures are different for different orientation

angles at which the crystal is formed. Only two of them are shown in Fig. 3.36.

To study the liquid phase at p = 0.75, we plot the pair correlation function g(2)(rrr) in Fig. 3.37 (a).

The pair correlation function is defined in Eq. (3.48). The oscillations in the correlation function

are rapidly decaying with the increasing distance from the origin, as one would expect for a liq-

uid. The hexagonal shape of the rings means that there are ordered domains in the liquid, and in

92

the relatively small data sample that was used to calculate g(2)(rrr), one orientation in the ordered

domains prevailed. The correlation function also has a non-decaying periodic structure with the

periodicity of the external potential, which reflects the average density modulation by the periodic

potential.

We also plot the Delaunay triangulation for a snapshot of this system in Fig. 3.37 (b). The
colored background displays the real part of the hexagonal bond order parameter ψ6, which is
defined in the following way:

ψ6 =

1
Mnn ∑

m

e6iθm.

(3.52)

Here Mnn is the number of nearest neighbors of a given electron, m enumerates the neighbors, and
θm is the angle between the bond to neighbor m and the x-axis. The real part of ψ6 can be used
as an indicator of the local orientation of the (quasi-)hexagonal structure in ordered parts of the
liquid. To plot ψ6, we interpolate it between the electron positions so that ψ6 is constant within in
each Voronoi cell. As seen in Fig. 3.37 (b), the liquid indeed has ordered domains with different

orientations.

Increasing p to p = 32/40 = 0.8 leads to the shift of the freezing point up to Γ ≈ 160. We plot
vs Γ for this case in Fig. 3.38. Below the freezing temperature,

the diffusion constants D⊥ and D
as in the previous case, the structures that appear have crystalline form with density modulated by

(cid:107)

solitonic superstructures. Two of the typical structures are shown in Fig. 3.39. The superstructure

period is increased as compared with the case of p = 0.75. The structure of the liquid phase in the

potential with p = 0.8 is qualitatively the same as in the case p = 0.75.

The appearance of solitons in the crystal structure is a general feature predicted by the theory of

incommensurate crystals of Pokrovsky and Talapov [26]. The theory predicts that the orientation

of the solitons is generally different from the orientation of the periodic potential troughs. Another

prediction is that the period of the superstructure should increase when p is approaching the C-I

transition threshold value pc. We observe similar behavior in our simulations of the electron sys-

tem. Even though the theory was developed for the atomic crystals with short-range inter-particle

interactions, it appears to correctly describe the qualitative features of the behavior of the electron

93

Figure 3.37 The system of N = 1024 electrons at Γ ≈ 131 in the periodic potential with A = 0.5 K
(b) Delaunay triangulation of a
and p = 30/40 = 0.75.
snapshot of the system with electrons represented by vertices of the triangulation. The background
color shows Re [ψ6(rrr)] which essentially means the local orientation of the nearest-neighbor links
in the ordered parts of the liquid.

(a) Pair correlation function g(2)(rrr).

vs Γ in the periodic potential with A = 0.5 K and
Figure 3.38 Diffusion constants D⊥ and D
(cid:107)
p = 32/40 = 0.8 for N = 1600 electrons. The procedure is outlined in Fig. 3.35. The phase
transition occurs at Γ ≈ 160.

94

1501601701800.00.20.40.6D [cm2/s]DDFigure 3.39 Soliton superstructures corresponding to different rotation angles of the crystal in the
incommensurate periodic potential with A = 0.5 K and p = 32/40 = 0.8. Voronoi tessellation
for the Wigner crystal with N = 1600 electrons is displayed for two very close values of Γ: (a)
Γ = 176.5, (b) Γ = 177.1. Voronoi cells are colored depending on their surface area: from dark
purple for smaller cells to light yellow for larger cells. The variation of the cell areas is from 0.84a2
to 0.88a2. The electron positions are averaged over time to average over small-amplitude thermal
fluctuations. The simulation procedure is the same as in Fig. 3.35.

system with a long-range Coulomb interaction. Formally, the theory is not directly applicable to

the coulombic system since the long-range interaction leads to non-acoustic longitudinal phonon

modes.

If the periodic solitonic superstructure can be reproduced in the experiment with electrons on

helium, it should be detectable with the standard experimental techniques. The additional period-

icity in the crystal should lead to new plasmon-ripplon resonance peaks in the Grimes and Adams

type of experiment. The superstructure may also be detectable by measuring the current along the

modulation troughs. The solitonic superstructure should lead to the Bragg-Cherenkov scattering

at a velocity much smaller than that in the case of a uniform crystal. By measuring the Bragg-

Cherenkov saturation velocity, one can potentially measure the period of the superstructure. The

proposed experimental methods would require fabrication of electrode arrays of high quality to

create the periodic potential. Disorder introduced by imperfections of the electrodes may destroy

95

Figure 3.40 Diffusion constants D⊥ and D
in the periodic potential with A = 0.5 K and p =
(cid:107)
34/40 = 0.85. The procedure is outlined in the caption to Fig. 3.35. The transverse diffusion
ceases at Γ ≈ 180.

the periodic structure and therefore eliminate the effects described above because they critically

depend on periodicity.

The incommensurate structure with arrays of solitons exists for values of p up to some threshold
value pc. For A = 0.5 K, this threshold value turns out to be pc ≈ 0.85. For p (cid:38) 0.85, the structure
of the system is qualitatively different. The diffusion vs Γ for p = 34/40 = 0.85 is shown in

Fig. 3.40. The diffusion is behaving differently than for p = 32/40 = 0.8 in a number of ways.
First, the transverse diffusion ceases for a significantly higher value of Γ ≈ 180. Second, the
has large values for Γ < 180 (taking into account the small temperature) and a very
computed D

(cid:107)

points. Finally, D

(cid:107)

large random error. The random error manifests itself as a wide spread of D

(cid:107)
does not vanish completely even for Γ > 180. These changes suggest that the system is qualitatively

different than the system with p = 32/40 = 0.8.

We plot the pair correlation function and the Delaunay triangulation for the system at Γ ≈ 200
in Fig. 3.41. Instead of a triangular crystal with incommensuration solitons, this system has a

structure consisting of semi-ordered electron channels along the minima of the periodic potential.

The correlated motion of electrons within each channel leads to the large calculation error in our

finite-size systems for D

(cid:107)

. This issue is discussed in greater detail in Sec. 3.5.3 and below. The

96

1601701801902000.000.250.500.751.001.25D [cm2/s]DDFigure 3.41 The system at Γ ≈ 200 in the periodic potential with A = 0.5 K and p = 34/40 = 0.85.
(a) Pair correlation function g(2)(rrr). (b) Delaunay triangulation of a snapshot of the system with
electrons represented by vertices of the triangulation. The background color shows Re [ψ6(rrr)].

electrons never form completely ordered channels. Some displacements of the channels relative

to each other persists even for Γ > 180. That is why D
(cid:107)

never vanishes completely in Fig. 3.40.

Qualitatively, there is no visible difference in the correlation function and the spatial structure of

the system below and above the Γ = 180 threshold.

The structural change at p ≈ 0.85 signifies approaching the C-I transition in the crystal phase.
Due to the competing periodicities, the system assumes some intermediate structure with no long-

range order even at very low temperatures. This structure is neither a triangular lattice nor a

fully commensurate crystal. As p is approaching p = 1, the electron channels are expected to

become more ordered at low temperatures and eventually form a commensurate crystal. At higher

temperatures, the system is expected to remain a modulated liquid for all values of p.

This picture of the C-I transition is in agreement with the data for p = 37/40 = 0.925. The

behavior of the diffusion constants vs Γ for this case is shown in Fig. 3.42. The diffusion is highly

anisotropic with the wide spread of D
(cid:107)

periodic potential troughs.

values, which is a sign of the channel formation in the

97

Figure 3.42 Diffusion constants D⊥ and D
in the periodic potential with A = 0.5 K and p =
(cid:107)
37/40 = 0.925. The procedure of obtaining the data is outlined in the caption to Fig. 3.35. The
freezing transition occurs at Γ ≈ 130.

Below the freezing point, which is at Γ ≈ 128 for p = 37/40 = 0.925, the electrons form a
lattice with a small built-in disorder. The vertical rows of electrons reside in the periodic potential
minima. Fig. 3.43 shows the pair correlation function and a snapshot of the system for Γ ≈ 130.
The lattice is similar to the triangular lattice in the uniform system, but the triangles are not equi-

lateral since they have to accommodate the periodicity of the periodic potential. The disorder is

also necessary to “adjust” the electrons to the incommensurate periodic potential. Therefore, there

is no true long range order in this system.

At temperatures above the melting transition, the electron channels in the potential minima

become more disordered and decouple from each other. At Γ = 60, the system’s pair correlation

function has clear vertical rows, which correspond to the partially-ordered channels in the troughs,

see Fig. 3.44 (a). In the snapshot of the system in Fig. 3.44 (b), the disordered chains of electrons

along the y-axis are visible.

To illustrate the fact that the longitudinal diffusion is governed by the displacements of entire
electron channels, we show two snapshots of the system with Γ ≈ 115 in Fig. 3.45. The snapshot
in the left panel shows the electron positions in the first data frame. The snapshot on the right
shows the electron positions 1.9× 104 time steps later into the simulation. In the second snapshot,

98

60801001201400.00.51.01.52.0D [cm2/s]DDFigure 3.43 The system at Γ ≈ 130 in the periodic potential with A = 0.5 K and p = 37/40 = 0.925.
(a) Pair correlation function g(2)(rrr). (b) Delaunay triangulation of a snapshot of the system with
electrons represented by vertices of the triangulation. The background color shows Re [ψ6(rrr)].

Figure 3.44 The system at Γ ≈ 60 in the periodic potential with A = 0.5 K and p = 37/40 = 0.925.
(a) Pair correlation function g(2)(rrr). (b) Delaunay triangulation of a snapshot of the system with
electrons represented by vertices of the triangulation. The background color shows Re [ψ6(rrr)].

99

Figure 3.45 Snapshots of the system in the potential with p = 37/40 = 0.925 and A = 0.5 K taken
at the beginning of the simulation (left) and 1.9×104 time steps later (right). Data for the point Γ ≈
115 from the simulation depicted in Fig. 3.42 is used. We reconstructed the electrons’ trajectories
to eliminate the effects of the periodic boundary conditions (details in the text). Comparing the two
snapshots, one can see that the entire electron channels along the y axis (along potential troughs)
are displaced. Only a few electrons were able to hop across the potential troughs, i.e.along x.

we reconstructed the positions in such a way that the electrons’ trajectories remain continuous if
an electron crosses the simulation cell boundary during the time from t = 0 to t = 1.9× 104∆t.
This allows us to track the electrons’ positions relative to the initial configuration as if the periodic

boundary conditions did not exist. One should look at the edges of the system in the right panel.

The left and right edges remained clean and straight during the simulation, which indicates that

very few electrons could hop transverse to the periodic potential barriers. In contrast, the top and

bottom edges have a fence-like structure, which allows us to track which vertical channels shifted

up or down relative to one another. This motion of entire channels makes D
large compared
to D⊥. It also makes the averaging of the squared displacement over all electrons not efficient.
Since electrons within a channel move coherently, one cannot assume that their trajectories are

(cid:107)

independent. Therefore, we observe large data spread in D
(cid:107)

system do to give accurate expectation values of the squared displacement.

. The data collected in our finite

100

Figure 3.46 (a) The transverse mobility µ
⊥ in the electron liquid as a function of the periodic
potential amplitude A for p = 23/32 ≈ 0.72. (b) The logarithm of the transverse mobility µ
⊥
as a function of the periodic potential amplitude A for p = 23/32 ≈ 0.72. The solid line is the
least-squares fit, with the slope −5.6± 0.3 K−1. Γ = 90 in the both plots.
3.5.5 Mobility in a liquid phase for p < 1 close to p = 1

The mobility of the electron liquid for p (cid:28) 1 is remarkably insensitive to the periodic potential
strength in a broad range of this strength (see Sec. 3.5.2). For p = 1, on the other hand, the
transverse mobility is exponentially reduced even for A (cid:28) kBT (see Sec. 3.5.3). In the intermediate
range of p between p (cid:28) 1 and p = 1, the system is expected to have a transition between the two
opposite limits.

To probe the mobility in the intermediate range of p, we calculate the transverse mobility µ
⊥
vs A for several values of p from p = 23/32 ≈ 0.72 to p = 15/16 ≈ 0.94. We keep Γ = 90 in all of
these simulations. The results are displayed in Figs. 3.46–3.49. For all p in this range, the shape of
⊥ vs A is close to the exponential, but the decay rate s of the exponential is increasing
⊥) vs A in Figs. 3.46–3.49. The decay

as we increase p. To determine the decay rate, we plot log(µ
rate s is increasing from s ≈ 5.6 for p ≈ 0.72 to s ≈ 78 for p ≈ 0.94.

the curve µ

As discussed in Sec. 3.5.3, the current in the commensurate system is governed by the acti-

101

0.00.20.40.6A[K]01234µ⊥[cm2/Vs]×107(a)0.00.20.40.6A[K]−6−5−4−3−2−10log(µ⊥/µ0)(b)Figure 3.47 (a) The transverse mobility µ
⊥ in the electron liquid as a function of the periodic
potential amplitude A for p = 25/32 ≈ 0.78. (b) The logarithm of the transverse mobility µ
⊥ as
a function of the periodic potential amplitude A for p = 25/32 ≈ 0.78. The solid line is the least-
squares fit, with the slope −10.3± 0.5 K−1. The hollow circles were excluded from the fit, since
⊥ for these points. Γ = 90 in the both
the calculation error is comparable in size to the value µ
plots.

vation of clusters of electrons with the activation energy ∼ 2πAns(ξc + a)2. As p deviates from
1, electrons in the liquid cannot on average occupy the potential trough minima, at least for small
A. Therefore, the average activation energy becomes smaller than 2πAns(ξc + a)2 as |p − 1| is
increasing. In the limit p (cid:28) 1, approximately as many electrons reside in the periodic potential
minima as near the maxima. Therefore, for p (cid:28) 1 the activation energy for the collective motion is
vanishing. This makes the system insensitive to the periodic potential, as discussed in Sec. 3.5.2.
Thus, the activation energy of the collective motion is gradually increasing from zero for p (cid:28) 1 to
2πAns(ξc + a)2 for p = 1.

3.5.6 Case p > 1

As p becomes larger than one but smaller than two, the electrons in the liquid cannot all reside in

the potential minima on average, as for p < 1. Therefore, the activation energy for transport trans-

102

0.00.20.40.6A[K]01234µ⊥[cm2/Vs]×107(a)0.00.20.40.6A[K]−10−8−6−4−20log(µ⊥/µ0)(b)Figure 3.48 (a) The transverse mobility µ
⊥ in the electron liquid as a function of the periodic
potential amplitude A for p = 27/32 ≈ 0.84. (b) The logarithm of the transverse mobility µ
⊥ as
a function of the periodic potential amplitude A for p = 27/32 ≈ 0.84. The solid line is the least-
squares fit, with the slope −21± 1 K−1. The hollow circles were excluded from the fit, since the
⊥ for these points. Γ = 90 in the both plots.
calculation error is comparable in size to the value µ

Figure 3.49 (a) The transverse mobility µ
⊥ in the electron liquid as a function of the periodic
potential amplitude A for p = 30/32 ≈ 0.94. (b) The logarithm of the transverse mobility µ
⊥ as
a function of the periodic potential amplitude A for p = 30/32 ≈ 0.94. The solid line is the least-
squares fit, with the slope −74± 3 K−1. The hollow circles were excluded from the fit, since the
⊥ for these points. Γ = 90 in the both plots.
calculation error is comparable in size to the value µ

103

0.00.10.20.30.4A[K]01234µ⊥[cm2/Vs]×107(a)0.00.10.20.30.4A[K]−8−6−4−20log(µ⊥/µ0)(b)0.000.020.040.06A[K]01234µ⊥[cm2/Vs]×107(a)0.000.020.040.06A[K]−5−4−3−2−10log(µ⊥/µ0)(b)Figure 3.50 (a) The transverse mobility µ
⊥ in the electron liquid as a function of the periodic
potential amplitude A for p = 35/32 ≈ 1.09. (b) The logarithm of the transverse mobility µ
⊥ as
a function of the periodic potential amplitude A for p = 35/32 ≈ 1.09. The solid line is the least-
squares fit, with the slope −48± 2 K−1. The hollow circles were excluded from the fit, since the
⊥ for these points. Γ = 90 in the both panels.
calculation error is comparable in size to the value µ
verse to the potential troughs should decrease and the decay rate s of the exponential dependence

of µ

⊥ on A should decrease compared to the case p = 1.
We observe that s decreases to s = 47 K−1 in the system with p = 35/32 ≈ 1.09, as shown in
Fig. 3.50. As p is increased further to p45/32 ≈ 1.4, s is reduced even more to s = 29 K−1, see
Fig. 3.51. At some point when p is approaching p = 2, s should start increasing again, since the

system becomes commensurate again at p = 2. Indeed, for p = 2, the value of s increases again to
s = 51.4 K−1.

There may be another maximum of s at the point p = √3. At this point, the triangular lattice
becomes commensurate with the potential since the lattice vector K of the potential coincides with

a higher order reciprocal lattice vector of the lattice. This situation is shown in Fig. 3.54. However,

we find that the mobility has a minimum at this special point only at low temperatures close to the

freezing transition, see below.

The details of the structure of the liquid and solid phases for p > 1 remain a subject for future

104

0.000.020.040.060.08A[K]01234µ⊥[cm2/Vs]×107(a)0.000.020.040.060.08A[K]−4−3−2−10log(µ⊥/µ0)(b)Figure 3.51 (a) The transverse mobility µ
⊥ in the electron liquid as a function of the periodic
potential amplitude A for p = 45/32 ≈ 1.4. (b) The logarithm of the transverse mobility µ
⊥ as
a function of the periodic potential amplitude A for p = 45/32 ≈ 1.4. The solid line is the least-
squares fit, with the slope −29± 1 K−1. Γ = 90 in the both plots.

Figure 3.52 (a) The transverse mobility µ
⊥ in the electron liquid as a function of the periodic
potential amplitude A for p = 2. (b) The logarithm of the transverse mobility µ
⊥ as a function of
the periodic potential amplitude A for p = 2. The solid line is the least-squares fit, with the slope
−50± 3 K−1. Γ = 90 in the both plots.

105

0.000.020.040.060.080.10A[K]01234µ⊥[cm2/Vs]×107(a)0.000.020.040.060.080.10A[K]−3.0−2.5−2.0−1.5−1.0−0.50.0log(µ⊥/µ0)(b)0.000.020.040.06A[K]01234µ⊥[cm2/Vs]×107(a)0.000.020.040.06A[K]−5−4−3−2−10log(µ⊥/µ0)(b)vs Γ in the periodic potential with A = 0.5 K and
Figure 3.53 Diffusion constants D⊥ and D
(cid:107)
p = 56/40 ≈ 1.4. The procedure is outlined in the caption for Fig. 3.35. The diffusion constants
are very small in the considered range of Γ, however the freezing transition does not occur for
Γ < 170.

Figure 3.54 The higher order commensurate configuration, where KKK = GGG1 + GGG2. Here GGG1 and GGG2
are two elementary reciprocal lattice vectors of the triangular lattice, |G1| = |G2| = 4π/√3a. The

vertical dashed lines show the periodic potential maxima.

106

1401501601700.0000.0050.0100.0150.0200.025D [cm2/s]DDFigure 3.55 The Mobility µ
mensurability ratio p for Γ = 50, 90, and 130. A = 0.05 K for all three curves.

⊥ of the strongly correlated electron system as a function of the com-

studies.

3.5.7 Mobility µ
⊥

as a function of p

To illustrate the behavior of the mobility as a function of commensurability, we calculated µ

⊥ as
a function of p for the fixed value of A = 0.05 K. Such curves for three values of Γ are shown in

Fig. 3.55. The potential amplitude is small compared to the temperature for all three curves. For

two curves, namely Γ = 50 and 90, the system remains a liquid in all points of the simulation. For
Γ = 130, the system becomes frozen in small intervals around the points p = 1, p = √3, and p = 2.
It remains in the liquid phase in all other points, where it is incommensurate with the potential.

All three curves share several general features already discussed in the previous sections. First,
for small p (cid:28) 1, µ
⊥ has a plateau. At this plateau, the mobility is approximately equal to the
mobility in a uniform system with the same temperature. Then all three curves have a sharp dip

at the point p = 1. Here the system is most sensitive to the periodic potential. For 1 < p < 2, the

107

0.00.51.01.52.02.5p012345µ⊥[cm2/Vs]×107Γ=50Γ=90Γ=130mobility is increasing up to the point p ≈ 1.4, after which it begins to decrease until it reaches a
minimum at p ≈ 2. The curve Γ = 130 has an additional minimum at p ≈ √3, where the system
becomes frozen into the configuration shown in Fig. 3.54. We note that the curve Γ = 130 reaches

zero at its minima, since the system is pinned when it is frozen into a commensurate crystal. The

two other curves have small but nonzero values at their minima because they remain in the liquid

phase for all p.

3.6 Conclusions

In this chapter, we studied the dynamical properties of a strongly correlated 2D electron system

on the surface of liquid helium in an external 1D periodic potential. To study this system, we

developed novel computational methods for microscopic simulation of the scattering of electrons

by the quantized excitations in helium. We also developed a 2D Ewald summation algorithm that

is parallelized for computations on a GPU. This increases the computation speed by a factor up to

16 compared to a single CPU. These are the necessary ingredients for studying transport properties

of strongly correlated electrons on the helium surface.

With the developed tools, we studied structure, diffusion, and mobility in both a spatially uni-

form system and a system in a periodic potential. The diffusion coefficient was determined as the

proportionality coefficient between the mean squared electron displacement and the time which,

as we showed, was independent of time. In the spatially uniform system, we observed the Wigner

crystallization transition. We simulated a large system with N = 1600 electrons, and observed the
Wigner crystallization at Γ ≈ 137, which is slightly larger than the values obtained in previous
simulations. The previous studies were done with significantly smaller numbers of particles. The

weak scattering by ripplons and phonons does not significantly change the diffusion or the phase

transition in our study. We also calculated the mobility in the spatially uniform system with and

without the inter-electron interaction. In both cases, the results are in good agreement with the

theory in Ref. [3].

The central part of our study is devoted to the effects of a periodic potential on the system of

108

strongly correlated electrons. In particular, we were interested in the effects of commensurability

of the potential and the electron system. We found that the degree of commensurability plays an

important role even when the electron system is in the liquid phase. In this case, the degree of

commensurability is defined for the triangular lattice that would exist at a given electron density.

We found that diffusion in the system is insensitive to the periodic potential when the period of

the potential is much larger than the average inter-electron distance.

In this case, the Wigner

crystallization takes place at approximately the same point as it would in the uniform system.

The transverse mobility is also very weakly affected by the potential, as opposed to the case of

a non-interacting electron gas in the same potential. This remarkable insensitivity is a result of

the effective averaging of the potential by the electrons. In this regime, approximately as many

electrons reside in the periodic potential minima as near the maxima, due to the strong electron-

electron interaction and a large mismatch between the period of the potential and the mean inter-

electron spacing.

We found the situation to be dramatically different when the system is maximally commensu-

rate with the potential. In this case, both the diffusion and the mobility are very sensitive to the

periodic potential even for the potential amplitude much smaller than the temperature. For such

a weak potential, there would be no effect in a non-interacting electron gas. However, the effect

is strong in the interacting system because of the strong electron-electron correlations. The posi-

tion and the way the diffusion goes to zero at the freezing transition are significantly altered by

the commensurate potential. The freezing temperature is increased as compared to the uniform

system, while the transition becomes smooth as seen in the diffusion behavior.

Some of the most interesting results were obtained for a strongly correlated liquid in the com-

mensurate potential. We found that the transverse mobility is exponentially decreasing as a func-

tion of the periodic potential amplitude in the commensurate liquid. The strength of the effect is

directly proportional to the correlation length squared in the liquid. One can understand that by

considering the current as a series of collective displacements of many electrons. The activation

energy of this process is proportional to the number of electrons within the radius determined by

109

the correlation length. This relation can provide an opportunity to experimentally measure the

correlation length in an electron liquid via transport measurements.

In an “incommensurate” liquid (i.e., for the periodic potential incommensurate with the elec-

tron crystal of the same density as the considered liquid) the mobility is also found to exponentially

decay with the potential strength. However, the decay rate is significantly smaller than that in the

commensurate case and quickly decreases with the increasing period of the potential. In an incom-

mensurate liquid, the current also results from a strongly correlated electron motion. However,

the electron distribution is less correlated with the potential, and as a result the average activation

energy is reduced.

The commensurate-incommensurate transition in the Wigner solid was also studied. We find

that, qualitatively, the C-I transition follows the scenario outlined by the theory of Pokrovsky and

Talapov for a short-range inter-particle potential [26]. Close to the C-I transition, we observe

density solitons in the Wigner crystal, as predicted by the theory. When the system is approaching

maximal commensurability, the structure changes to a set of ordered electron channels residing

in the periodic potential minima. We also observe some intermediate disordered structures in the

transition region. The phase transition temperature and the temperature dependence of the diffusion

coefficient are significantly modified in the C-I transition region. In the vicinity of the transition,

the freezing temperature is significantly lowered compared to that in the uniform system. This is

in contrast to the commensurate case, where the phase transition temperature is increased by the

periodic potential.

We also studied thermalization in a Wigner crystal and in a strongly correlated electron liquid.

We found that both systems thermalize in our simulations, with the thermalization time on the order

of the short-wavelength plasma oscillation period. This behavior is substantially different from that

observed in the well-known Fermi-Pasta-Ulam problem. The difference may be attributed to the

difference in dimensionality and to the long-range nature of the Coulomb potential in our study.

The results of this work provide an insight into the behavior of the strongly correlated 2D elec-

tron system in a periodic potential. We show the importance of the electron-electron correlations

110

in the liquid phase. The results also suggest that the periodic potential is a promising tool for an

experimental investigation of various properties of strongly correlated electron systems.

111

CHAPTER 4

CONCLUSIONS

Electron-vibrational coupling, which is the topic of this thesis, plays a key role in condensed mat-

ter physics. This coupling has been studied in a variety of contexts, from transport phenomena to

polaronic effects to transitions to a superconducting state. Recent progress in mesoscopic physics,

that has led to a discovery of low-dimensional electron systems and to creation of nano- and micro-

scopic mechanical resonators, has opened a new era in studying effects of the electron-vibrational

coupling. Many aspects of these effects are novel and challenging, which makes investigating them

interesting and important.

In this thesis we have shown that the electron-vibrational coupling can strongly modify the

dynamics of nano- and micromechanical vibrational modes and that it enables revealing and char-

acterizing strong electron correlations in a nondegenerate two-dimensional electron liquid. Even

though these two groups of phenomena are very different, their analysis is based on the common

ideas of the transport phenomena in condensed matter physics, and the results demonstrate the

richness of the many-body phenomena in the nascent areas of condensed matter physics.

One of the most common types of nano- and micromechanical resonators are resonators made

of semiconductors, which is an immediate consequence of the progress in semiconductor tech-

nology. By doping semiconductors with impurities, one can significantly change the density of

electrons and holes. Our results show that such a change has a profound effect on the dynamics

of vibrational modes in the resonators. We have studied the case of n-doping, where it is the elec-

tron density that is changed by doping. The specific calculations referred to the most broadly used

semiconductors, silicon and germanium. These semiconductors have a multi-valley band structure.

In the absence of strain, equivalent valleys are equally populated by the electrons. The strain asso-

ciated with nanomechanical vibrations breaks the symmetry and makes the valleys non-equivalent.

This leads to repopulations of the valleys. In the most cases of interest, the rate of electron redistri-

bution over the valleys is much larger than the vibration frequency, so that the valley populations

112

follow the vibrations adiabatically. In turn, the electron redistribution over the valleys affects the

vibrations, so that overall we have a mechanism of a nonlinear electron-mediated self-action of the

vibrations.

The change of the valley populations is a strongly nonlinear function of the strain tensor. It is

controlled by a very large ratio of the electron-phonon coupling energy to the electron chemical

potential or the temperature. It is this ratio that makes the nonlinearity of the vibrational modes

in doped semiconductor structures strong. The most pronounced effect of the nonlinearity is the

amplitude dependence of the vibration frequency. To the leading order, it is determined by the

quartic terms in the expansion of the free energy in strain. For the reason described above, these

terms are comparatively large.

We have calculated the quartic in the strain components of the nonlinear elasticity tensor in

n-doped semiconductors. For silicon, the calculation required going beyond the standard deforma-

tion potential approximation and taking into account the band degeneracy in the absence of strain.

As an application, we found the amplitude dependence of the vibration frequency for several modes

used in silicon-based micromechanical resonators. This dependence has a characteristic form as

a function of the electron density and temperature. We have also described the temperature de-

pendence of the mode frequencies, which comes from the quadratic terms of the elasticity tensor

induced by the electron-vibrational coupling.

Our results provide an insight into the experimentally observed strong mode nonlinearity in

doped crystals. They also provide a means for measuring the parameters of the electron-phonon

coupling which are not easy to directly access otherwise. In terms of applications, these results are

important for developing thermally stable nanomechanical reosnators with a controllable amplitude

dependence of the vibration frequency.

The electron-vibrational coupling in the other system that we studied, electrons on the helium

surface, is of special interest, since the electron system is strongly correlated. It forms a nondegen-

erate liquid or a solid. If in the case of a solid one can at least characterize elementary excitations

in the electron system, the very language of elementary excitations is not helpful in the case of a

113

liquid. A natural path toward understanding a liquid is molecular dynamics simulations. We have

performed these simulations for a record-large electron system, with the account taken of the mi-

croscopic mechanisms of the electron coupling to vibrational excitations in helium. Along with a

spatially uniform system, we studied the electron system placed into a spatially periodic potential

with the period comparable to the mean inter-electron distance.

Our results for a uniform system showed that self-diffusion in the system is a sensitive tool

that allows one to identify the melting transition. The mobility, on the other hand, is basically

unchanged at the transition and is totally controlled by the electron-vibrational coupling. The

results obtained for a strongly correlated liquid are in agreement with the analytic theory based on

the assumptions of a fast inter-electron momentum exchange and the independence of the scattering

of an individual electron by helium excitations from the electron-electron interaction.

The results have also provided an insight into thermalization in strongly correlated electron

liquids and solids driven away from equilibrium. We have found that the thermalization is fast

and the Fermi-Pasta-Ulam absence of thermalization does not occur in a system with long-range

interaction, irrespective of the presence of long-range order.

The major emphasis of the work was placed on the dynamics of the system in a periodic po-

tential. We found that, even where the electrons form a liquid, the dynamics, and in particular the

electron mobility transverse to the potential troughs, is highly sensitive to the effective commensu-

rability, i.e., the relation between the mean inter-electron distance and the period of the potential.

For a large potential period, the mobility is weakly affected by the potential even where the po-

tential barriers are much higher than temperature, in contrast to a single-electron system. On the

other hand, unexpectedly, in the fully commensurate case, the mobility displays an exponential de-

pendence on the barrier height even where this height is small compared to the temperature. This

dependence allows one to directly measure the correlation length in the electron liquid. This is one

of our central results.

We have also found that a periodic potential can strongly affect the melting transition, leading

to a different transition temperature and a qualitatively different dependence of the way the self-

114

diffusion coefficient goes to zero at the transition. The shift of the transition temperature has

opposite signs close and far from commensurability. In the commensurate solid phase, the mobility

transverse to the potential troughs vanishes. It displays a characteristic dependence on the potential

period close to full commensurability.

Overall, the results on the electrons on helium show that this system can be used to study

commensurate-incommensurate transitions in the regime of an effectively weak electron-vibrational

coupling. This regime has not been explored before. The transitions can be directly accessed by

measuring the electron mobility, and moreover, the measurements can provide a direct means for

determining the correlation length in the electron liquid.

4.1 Outlook

The theory developed in the thesis can be, and hopefully will be, extended in several directions.

It is interesting and important for applications to understand the electron-vibrational coupling in

semiconductor micromechanical resonators in the case of p-doping. A strong effect in this case

comes, in the first place, from the strain-induced splitting of the valence band. Its anisotropy comes

from the warping of the heavy hole band. Another open problem is the effect of the electron-

vibrational coupling on decay of the mechanical modes and the possibility to control this decay in

doped semiconductors using phononic crystals. Yet another problem emerges when the size of the

resonator is reduced and the length associated with intervalley scattering becomes comparable to

the resonator size.

For electrons on helium, one of the most challenging open problems is electron spin relaxation.

The mechanism is not known and the attempts to measure the relaxation rate have not been suc-

cessful so far. By putting the electrons into a spatially periodic magnetic field (with a nonzero

mean), one “turns on" a spin-orbit coupling, which can modulate the electron density and lead

to commensurate-incommensurate transitions. These transitions should be detectable through the

mobility measurements, whereas an extension of the theory to this case is more or less straightfor-

ward. Yet another extension, which is important for a direct comparison with the experiment, is an

115

incorporation of the screening of the electron-electron interaction by the image force in the sub-

strate, which partly reduces the Coulombic tail of the interaction. This extension is also straight-

forward, given that it is the long-range part of the interaction that is most complicated to allow

for.

116

APPENDIX

117

A.1 Expansion of the free energy in terms of the strain-induced shift of the

energy valleys

The major effect of a strain on the electron free energy comes from the shift of the energy
valleys. We will assume that valley α is shifted in energy by δ Eα and the shift is small, |δ Eα| (cid:28)
max(kBT, µ0), where µ0 is the chemical potential in the absence of strain. We further assume
that the vibrations are slow compared to the time it takes the electron system to come, locally, to

thermal equilibrium for given values of δ Eα, i.e., the temperature and the chemical potential are
the same in all valleys. This is true for almost any realistic system, since inter- and intravalley
relaxation times are smaller than 10−11 s [198, 199, 200]. Since for high electron densities the
thermal conductivity is high, the change of the temperature compared to the ambient temperature

can be disregarded; also, as mentioned in the main text, the electron density n summed over all

valleys is constant.

The contribution F (α) of the electrons in valley α to the free energy density can be written as

F (α) = Ω(α) + µn(α) with

Ω(α) = −kBT

(cid:104)
1 + e(µ−E−δ Eα )/kBT(cid:105)

,

dE ρ(E)log

(cid:90) ∞

0

where ρ(E) is the electron density of states in a valley, which is the same for all valleys, µ is
the chemical potential, and n(α) = −∂Ω(α)/∂ µ is the electron density in valley α. The total free
energy can be rewritten in terms of Fermi-Dirac integrals:

(cid:20) µ
We use the standard notation F1/2(x) =(cid:82) ∞

nkBT
NF1/2

N
∑
α=1

F =

F1/2

kBT

(cid:19)

(cid:18)µ − δ Eα

kBT

2
3

−

F3/2

(cid:19)(cid:21)

(cid:18)µ − δ Eα

kBT

(1)

0 dyy1/2/[1 +exp(y−x)]; primes indicate differentiation
over x, for example, F(cid:48)1/2 ≡ dF1/2/dx. Function F1/2 and its derivatives are calculated for x =
µ0/kBT . The chemical potential in the strained crystal is changed by kBTδ (ε), which is found
from conservation of the total electron density:

n =

N
∑
α=1

n(α) = 2π−1/2Nc

N
∑
α=1

F1/2 ((µ0 + kBTδ (ε)− δ Eα ) /kBT ) ,

(2)

118

that must be equal to its value in an unstrained crystal:

n = 2π−1/2V NcNF1/2 (µ0/kBT ) ,

where

Nc =

2m3/2
eff
¯h3

(3)

(4)

(5)

(cid:19)3/2

(cid:18) T

2π

.

(cid:16)

From (2), δ (ε) can be expressed in terms of δ Eα, i.e., in terms of strain. Expanding Fermi integrals
around the unstrained values and iteratively solving (2) for δ to the third order, we obtain:

δ (3)(ε) = ∆ε +

F(cid:48)(cid:48)1/2
2F(cid:48)1/2

(cid:16)
(∆ε )2 − ∆2

ε

(cid:17)

+

F(cid:48)(cid:48)(cid:48)1/2
6F(cid:48)1/2

3(cid:17)

∆3
ε − 3∆2

ε∆ε + 2∆ε

Here, ∆m

ε = N−1 ∑α (δ Eα /kBT )m. Third order terms in δ (ε) are enough to obtain fourth-order

expansion of the free energy.

Using (5) to expand (1) to the 4th order in the strain-induced shifts δ Eα, we find that, in an

N-valley semiconductor, the change δ Fe of the free energy density summed over the valleys is

(cid:105)

(cid:104)
(∆ε )2 − ∆2
(cid:104)

ε

ε )2 − 2∆2
(∆2

(cid:104)

ε ∆ε + 2(∆ε )3(cid:105)

1
6

F(cid:48)(cid:48)1/2
F1/2

+

∆3
ε − 3∆2

ε (∆ε )2 + (∆ε )4(cid:105)
ε (∆ε )2 + 3(∆ε )4(cid:105)

.

δ Fe
nkBT =∆ε +
1
8

+

F(cid:48)1/2
1
F1/2
2
2
F(cid:48)(cid:48)1/2
F1/2F(cid:48)1/2
F(cid:48)(cid:48)(cid:48)1/2
1
F1/2
24

(cid:104)

+

4∆3

ε ∆ε − ∆4

ε − 6∆2

Equation (6) immediately gives the tensors(cid:98)Λn of the expansion of the free energy increment (2.6)

(6)

if one expresses the shift δ Eα of the valleys in terms of the strain tensor.
potential approximation the relation between δ Eα and ˆε is given in the main text, see also Eq. (7)
below.

In the deformation

In the case of Si crystals, which are often used in micromechanical resonators, an important

contribution to δ Eα comes from the shear-strain induced splitting of the electron energy bands
at the zone boundary. Shear strain does not lead to the valley shift in the deformation potential

approximation. The overall shift of valley α, to the lowest order in the coupling that causes it (i.e.,

119

to the first order in the deformation potential where its contribution is nonzero and to the second

order in the band splitting for shear strain) is [129]:

δ Eα = ∑
i j

Ξ(α)
i j εi j − Ξshε2

α , Ξsh =

4Ξ2
u(cid:48)
∆E .

(7)

The derivation of this equation using a symmetry argument is reproduced in Appendix A.6. Here
we use that silicon has six valleys located at the (cid:104)100(cid:105) axes, and we chose the coordinate axes
x,y,z along (cid:104)100(cid:105). Respectively, the valley index α takes on three values that correspond to the
x,y,z axes (the valleys lying on the same axis, but in the opposite directions, are equivalent). The
strain εα, which enters the second term in the right-hand side of Eq. (7), is a component of the
strain tensor εi j with i, j such that i, j (cid:54)= α and i (cid:54)= j. The parameter 2Ξu(cid:48) is the interband matrix
element of the electron-phonon coupling calculated for the electron conduction bands ∆1 and ∆2(cid:48)
at the X point on the boundary of the Brillouin zone, where the bands cross; ∆E is the energy
separation between the bands ∆1 and ∆2(cid:48) at the value of the wave vector kkk that corresponds to the
conduction band minimum. Parameter Ξsh is the effective deformation potential of two-phonon
coupling to shear strain. The numerical value of Ξsh is not well known. The experimental data give
Ξu(cid:48) ≈ 7− 8 eV [129, 133] and the numerical data on the band splitting give ∆E ≈ 0.7 eV [201] so
that Ξsh is in the range of 280− 360 eV; this is essentially an order of magnitude estimate.
(cid:98)Λk in Eq. (2.6) are expressed in terms of δ Fe as

In calculating δ Fe in Eq. (6) we kept terms that are quartic in ˆε. The components of the tensors

∂ kδ Fe

.

(8)

(Λk)i1 j1...ik jk =

Tensors(cid:98)Λ are symmetric with respect to the interchange of indices ik ↔ jk and the pairs (ik jk) ↔
(ik(cid:48) jk(cid:48)). For the considered long-wavelength strain, tensors (cid:98)Λk are independent of coordinates.
The corrections (cid:98)Λ2 to the linear elasticity tensors were found previously [121, 122] and are not

∂εi1 j1...∂εik jk

discussed in this paper.

120

A.2 Nonlinear elastic constants of germanium

In this section we provide the corrections to the nonlinear elastic constants of germanium,

which are due to the redistribution of the electrons over the valleys. Germanium has four equivalent

valleys in the conduction band, which are located on the boundary of the Brillouin zone along
(cid:104)111(cid:105) axes. We use the Voigt notation and write the components of the corrections to the nonlinear
elasticity tensor δ ˆc in the frame where the axes (x,y,z) are along the (cid:104)100(cid:105) directions of the crystal.
Using the results of Appendix A.1, we obtain

nΞ3

uF(cid:48)(cid:48)1/2

27F1/2(kBT )2 ,

δ c456 =

δ c4444 =

δ c4455 =

nΞ4
u

81(kBT )3

nΞ4
u

81(kBT )3

 3(F(cid:48)(cid:48)1/2)2
 (F(cid:48)(cid:48)1/2)2

F1/2F(cid:48)1/2 −

F1/2F(cid:48)1/2 −

 ,
 .

F(cid:48)(cid:48)(cid:48)1/2
F1/2
F(cid:48)(cid:48)(cid:48)1/2
F1/2

(9)

The notations are the same as in Appendix A.1 and in Table 2.1. The electron-phonon coupling

does not contribute to the other third- and fourth-order elastic constants.

Corrections δ c44 and δ c456 for germanium were found by Keyes [120]; however, his final
expression for δ c456 differs from Eq. (9) by a factor of 4 (our expressions for δ c44 coincide
with Ref. [120]). Parameters δ c4444 and δ c4455 have not been found before, to the best of our
knowledge. In the limiting cases, corrections δ c4444 and δ c4455 have the same dependence on
temperature and electron density as constant nC2 discussed in Sec. 2.4.1.
A.3 Lamé and length extension modes in (cid:104)100(cid:105) and (cid:104)110(cid:105) orientations

In this appendix, we find the two modes we are interested in: length extension and Lamé. These

results are well known [202], but it is convenient to have them derived here. First, we write general

equations of motion of an elastic medium in linear approximation:

ρ ¨ui = ci jkl

∂ 2uk
∂ x j∂ xl

.

121

(10)

Here ρ is the material density, ui are components of the displacement tensor, and ci jkl is the
stiffness tensor in explicit coordinate subscript notation. Taking into account that in a cubic crystal

the “off-diagonal” terms like cxxxy or cxzxy vanish, we obtain explicit equations of motion, with

tensors written in Voigt notation:

(cid:32) ∂ 2uy
(cid:32)
(cid:32) ∂ 2uy

∂ x∂ y +
∂ 2ux
∂ x∂ y +

∂ y∂ z +

∂ 2uz
∂ x∂ z
∂ 2uz
∂ y∂ z
∂ 2ux
∂ x∂ z

(cid:33)
(cid:33)
(cid:33)

+ c44

+ c44

+ c44

∂ 2ux
∂ y2 +

(cid:32)
(cid:32)∂ 2uy
(cid:32)

∂ x2 +
∂ 2uz
∂ y2 +

ρ ¨ux = c11

ρ ¨uy = c11

ρ ¨uz = c11

∂ 2ux
∂ x2 + c12
∂ 2uy
∂ y2 + c12
∂ 2uz
∂ z2 + c12

(cid:33)
(cid:33)
(cid:33)

∂ 2uy
∂ x∂ y +
∂ 2ux
∂ x∂ y +
∂ 2uy
∂ z∂ y +

∂ 2ux
∂ z2 +
∂ 2uy
∂ z2 +
∂ 2uz
∂ x2 +

∂ 2uz
∂ x∂ z
∂ 2uz
∂ y∂ z
∂ 2ux
∂ x∂ z

(11)

To obtain vibrational modes, we need to solve these equations for given boundary conditions. Be-

fore we do that, let us show how stiffness tensor is transformed under coordinate system rotations.
This will help us to obtain modes in (cid:104)110(cid:105) geometry. The symmetry of the stiffness tensor in co-
ordinates rotated by an arbitrary angle around z-axis is modified. In general, the stiffness tensor

becomes:

c(cid:48) =



c(cid:48)11
c(cid:48)12
c(cid:48)13
0

c(cid:48)12
c(cid:48)11
c(cid:48)13
0

0
0
c(cid:48)16 −c(cid:48)16

c(cid:48)13
c(cid:48)13
c(cid:48)33
0

0

0

0

0

0

c(cid:48)16
0
0 −c(cid:48)16
0
0

c(cid:48)44
0

0

0

c(cid:48)44
0

0

0
c(cid:48)66



In particular, if coordinates are rotated by π/4 around z-axis, we obtain:



c(cid:48) =

c11
2 +
c11
2 +

c12
2 + c44
c12
2 − c44
c12
0

c11
2 +
c11
2 +

c12
2 − c44 c12
c12
2 + c44 c12
c11
c12
0
0

0

0

0

0

0

0

122

0

0

0

c44
0

0

0

0

0

0

c44
0

0

0

0

0

0

c11
2 −

c12
2

(12)

(13)



In the rotated coordinates, the Eqs. (11) are modified as follows:

ρ ¨ux = c(cid:48)11

ρ ¨uy = c(cid:48)11

ρ ¨uz = c(cid:48)33

∂ 2ux
∂ x2 + c(cid:48)12
∂ 2uy
∂ y2 + c(cid:48)12
∂ 2uz
∂ z2 + c(cid:48)13

∂ 2uy
∂ x∂ y + c(cid:48)13
∂ 2ux
∂ x∂ y + c(cid:48)13

(cid:32) ∂ 2uy

∂ y∂ z +

∂ 2uz
∂ x∂ z + c(cid:48)66
∂ 2uz
(cid:33)
∂ y∂ z + c(cid:48)66
∂ 2ux
∂ x∂ z

+ c(cid:48)44

∂ 2ux
∂ y2 +

(cid:32)
(cid:32)∂ 2uy
(cid:32)

∂ x2 +
∂ 2uz
∂ y2 +

(cid:33)
(cid:33)

∂ 2uy
∂ x∂ y
∂ 2ux
∂ x∂ y
∂ 2uy
∂ z∂ y +

+ c(cid:48)44

+ c(cid:48)44

∂ 2uz
∂ x2 +

(cid:33)
(cid:33)

(14)

∂ 2uz
∂ x∂ z
∂ 2uz
∂ y∂ z

(cid:32)
(cid:32)∂ 2uy

∂ 2ux
∂ z2 +

(cid:33)

∂ z2 +
∂ 2ux
∂ x∂ z

Young modulus and Poisson ratio are also modified in the rotated system. Poisson ratios for y

and z directions are different now [203]. For <110> direction, the Poisson ratios and the Young

modulus become:

σ2 =

σ3 =

E =

2

(cid:1)

12

12

2 =

c11 (c11 + c12 − 2c44)− 2c2
c11 (c11 + c12 + 2c44)− 2c2
2(cid:17)

c(cid:48)13
− c(cid:48)33c(cid:48)12
(cid:0)c(cid:48)12 − c(cid:48)11
c(cid:48)11c(cid:48)33 − c(cid:48)13
c(cid:48)13
(cid:16)
(cid:17)
(cid:16)
c11 (c11 + c12 + 2c44)− 2c2
c(cid:48)11c(cid:48)33 − c(cid:48)13
12
c11(c11 + c12)− 2c2
c(cid:48)33
− c(cid:48)12
+ 2c(cid:48)13
4c44
2
c11(c11 + c12 + 2c44)− 2c2
c(cid:48)11c33 − c(cid:48)13

2(cid:0)c(cid:48)12 − c(cid:48)11

4c12c44

2 =

c(cid:48)11

(cid:1)

12

12

=

2

(15)

In the case of <100> orientation, the expressions are greatly simplified by symmetry, and the

result is well known:

c12

σ2 = σ3 ≡ σ = −
E =

c11 + c12
c11 (c11 + c12)− 2c2
12

c11 + c12

Now we can find solutions to Eqs. (14) and (11) in a square plate with side L. One solution is

cos(πx/L)sin(πy/L)e−iωt
L
sin(πx/L)cos(πy/L)e−iωt
π

ux =

L
π
uy = −
uz = 0

123

(16)

(17)

This solution has the name of first Lamé mode. This solution satisfies both Eqs. (14) and (11), with

frequencies being

ω(Lame)
<100> =

ω(Lame)
<110> =

π
l
π
l

(cid:114)c11 − c12
(cid:115)

ρ
2c44
ρ

,

for plate sides along <100> and <110> directions respectively. Strain field (without time factor)

in <100> case is

εxx = ε1 = −sin(πx/L)sin(πy/L),
εyy = ε2 = sin(πx/L)sin(πy/L),

while in <110> case, the strain tensor has only a shear component:

εxy =

1
2

ε6 = sin(πx/L)sin(πy/L).

Now let us turn to length extension modes. Since the plate extension mode is quite complicated,

we consider a simplified model for a thin beam. In this approximation, the motion can be described

by the Young modulus E only, and transverse strain can be found with the Poisson ratios. We
assume that the resonator dimensions are lx × ly × lz and the center of it is fixed. The equation of
motion is

∂ 2ux
∂ x2 −
For free ends of the beam, the first solution is

ρ
E

∂ 2ux
∂t2 = 0

with k = π/lx and frequency ω = π
lx

1
k

cos(kx)e−iωt,

ux = −
ρ . Using Poisson ratios, one can obtain

(cid:113) E
uy ≈ νyεxxy = σ2ycos(kx)e−iωt
uz ≈ νzεxxz = σ3zcos(kx)e−iωt

(18)

(19)

Directions <100> and <110> are different only by the values of the Young modulus and Poisson

ratios that one should use (Eqs. (15)–(16)). We will not calculate strain components for the length

extension modes here for brevity.

124

A.4 Duffing nonlinearity parameter for a Lamé mode in a square single-

crystal plate

We consider a square plate with side L and thickness h made out of a single crystal with cubic
symmetry. If the crystal is cut out along (cid:104)100(cid:105) or (cid:104)110(cid:105) axes, one of the simplest modes is the first
Lamé mode [202]. The normalized displacement field is

x = √2cos(πx/L)sin(πy/L),
u(ν)
√2sin(πx/L)cos(πy/L).
u(ν)
y = −

(20)

Here, x and y axes are in the lateral plane along the sides of the square, axis z is perpendicular to
the plate and u(ν)
expressions into Eqs. (2.8) and the relation

z = 0. Calculating the strain tensor for the displacement (20) and substituting the

(cid:90)

drrr(cid:98)Λ( f )

2

Mω2

ν =

· ˆε(ν) ⊗ ˆε(ν),

(21)

(22)

for the plate cut out along (cid:104)100(cid:105) axes we obtain, in Voigt notation for the elasticity tensors,

Mω2

γν =

ν = π2h (c11 − c12) ,
3π4h
16L2 (c1111 − 4c1112 + 3c1122) .

If we consider silicon and take into account only the contribution δ ˆc to the nonlinear elasticity
tensor ˆc, with the account taken of Table 2.1, the expression for γν simplifies to

γν = (27π4h/32L2)δ c1111.

(23)

For the Lamé mode cut along the (cid:104)110(cid:105) axis, if the tensors are calculated in the axes (cid:104)100(cid:105), we

have

ν = 2π2hc44,

Mω2
γν = (3π4h/2L2)δ c4444.

(24)

Note that only coupling to shear strain contributes to the nonlinearity parameter γν in this case.

125

A.5 Duffing nonlinearity parameter for an extension mode in a single-crystal

narrow beam

We consider the fundamental extension mode in a thin beam of length L with a rectangular
cross-section of area S (cid:28) L2. The beam is cut along a symmetry axis, and the sides are also along
symmetry planes of a cubic crystal. From the free-surface boundary conditions, the normalized

displacement field is [202]:

u(ν)
x ≈
u(ν)
y ≈
u(ν)
y ≈

√2cos(πx/L),
√2πσ2
√2πσ3

L

ysin(πx/L),

zsin(πx/L).

L

(25)

This expression takes into account transverse compression that accompanies beam extension and
uses the smallness of the beam cross-section; corrections ∼ S/L2 are disregarded. The transverse
compression in a cubic crystal cut in a symmetric direction is described by Poisson’s ratios σ2 and
σ3. Generally, they do not coincide. In Eq. (25) the transverse coordinates y and z are counted off
from the center of the beam

For the longitudinal direction of the beam (cid:104)100(cid:105) and the sides parallel to (100) planes, the
Poisson parameters are equal, σ2 = σ3 and σ ≡ σ2 = σ3 = c12/(c11 + c12). In this case Eqs. (2.8)
and (21) give

(cid:16)
c11 (c11 + c12)− 2c2
L (c11 + c12)

12

(cid:17)

π2S

Mω2

,

ν =
γν = (π4S/4L3) [c1111 − 8σc1112
+ 12σ 2(c1122 + c1123)− 8σ 3(c1112 + 3c1123)
+2σ 4(c1111 + 4c1112 + 3c1122)

(cid:105)

.

(26)

The expression for γν is simplified if in the nonlinear elasticity tensors we take into account
only the contribution from the electron-phonon coupling as given in Table 2.1 and also allow for

126

the interrelation between different components of the tensor δ ˆc. Then for a silicon beam

γν = (π4S/4L3)(1 + σ )4δ c1111.

(27)

For extension along (cid:104)110(cid:105) axis, with one side parallel to (100) plane and the other side parallel to
(1¯10) plane, the Poisson’s ratios σ2 = σ (110,1¯10) and σ3 = σ (110,001) are given in Ref. [203].
Then Eqs. (2.8) and (21) give

4π2S

Mω2

ν =

γν =

(cid:17)

3 + 1

12

,

12

L
π4S
32L3

(cid:17)
(cid:16)
c11(c11 + c12)− 2c2
c44
(cid:16)
(cid:104)
c11(c11 + c12 + 2c44)− 2c2
(cid:16)
2 − 4σ 3
σ 4
2 − 4σ2 + 8σ 4
c1111
+ 3c1122 (σ2 − 1)2(cid:16)
(cid:17)
2 − 4σ2σ3 + 3σ2 + 8σ 3
2 σ3 − 3σ 2
2 + 2σ 2
σ 3
2 − 2σ2 + 8σ 2
σ 2
3 + 1
(cid:21)
(cid:17)2

(cid:16)
3 (σ2 + 1)2 + 96c1244σ3 (σ2 − 1) (σ2 + 1)2 + 24c1155
+ 48c1144σ 2
σ 2
2 − 1

+ 4c1112 (σ2 − 1)

+ 8c4444 (σ2 + 1)4

2 + 6σ 2

+24c1266

.

(cid:17)
3 + 2σ3 − 1
(cid:16)

(cid:17)2
+ 24c1123σ3 (σ2 − 1)2 (σ2 + σ3 − 1)
σ 2
2 − 1

(28)

If in the nonlinear elasticity tensor ˆc we take into account only the contribution δ ˆc from the
electron-phonon coupling, in the case of a silicon beam the expression for γν simplifies to

(cid:16)
(σ2 − 2σ3 − 1)4 δ c1111 + 24 (σ2 + 1)2 (σ2 − 2σ3 − 1)2 δ c1144 + 16 (σ2 + 1)4 δ c4444
(29)

γν =

π4S
32L3

(cid:17)

.

Expressions (26) and (28) were generated using a computer code to calculate the sums and integrals

in Eq. (2.8).

A.6 Strain effects on the conduction band in Si from a symmetry argument

In this appendix, we reproduce the derivation from [128] of electron-phonon coupling func-

tional form in silicon using group theory arguments. It is the most convenient to use the theory

of invariants. Silicon is an Oh crystal, with six minima in the conduction band, which are close
to X points in the Brillouin zone. Point X has symmetry D4h = D4 × Ci. It is known that the
conduction band minimum corresponds to ∆1 representation. Together with ∆(cid:48)2, representation ∆1

127

Table A.1 Characters of group elements for various representations of D4h with basis functions up
to third order for each representation. Here [xy] ≡ 1

2(xy + yx) and {xy} ≡ 1

2 (xy− yx).

Representation

A+
1
A−1
A+
2
A−2
B+
1
B−1
B+
2
B−2
E+
E−

0
0

ie
1

1

1

1

ic2
1

1

2ic4
1

2iu2
1

2u2
1
1

e
1
1
1
1
1
1
1
1
2 −2
2 −2

c2
1
1
1
1
1 −1
1
1 −1
1
1 −1 −1
1 −1 −1
0
0

2iu(cid:48)2
2u(cid:48)2
2c4
1
1
1
1
1 −1 −1 −1 −1 −1
1 −1 −1
1
−1 −1
1 −1 −1 −1 −1 −1
1
1
−1
−1
1
1
−1
−1 −1 −1
1
1
−1
1
1
1
−1 −1
1
1
1 −1 −1
−1
0
0
0
0
2 −2
2
0
0 −2
0
0

Basis

z2; x2 + y2
{xy}z ∼ Jzz
z; (x2 + y2)z; z3

Jz

x2 − y2
[xy]z
xy

(x2 − y2)z
Jx,Jy; xz,yz

x,y

turns into X2 at point X (in notations of [128]). Therefore, one can find the Hamiltonian corre-
sponding to X2 representation of D4h and assume it is valid up to the conduction band minimum
point k0 ≈ 0.85(2π/a) [204, 205]. More specifically, we need to find basis functions of X2 × X∗2
representation of D4h, according to Section 25 in [128].

A.6.1 Representations of D4h

Representations of D4h = D4 × Ci are obtained from the representations of D4 by splitting of
each representation into two representations: ‘−’ and ‘+’. The resulting representations and their
characters are given in Table A.1. Fourth-order basis sets for these representations are given in a

separate Table A.4. In the rest of this subsection we show how the basis functions in Table A.1

are obtained. Characters for D4 representations are found in [128], Table 11.1. Characters of the
elements ig, g ∈ D4 are χ(ig) = ±χ(g), plus and minus corresponding to ‘−’ and ‘+’ represen-
tations. To obtain the basis functions, we first find representations for the basis sets x, y, z, and

Jx, Jy, Jz. These representations are presented in Table A.2. Matrices of these representations for

128

Table A.2 Representations of D4h with basis sets x, y, z, and Jx, Jy, Jz.

Representation

D−1
D+
1

c2
e
3 −1
3 −1

elements g without i are:

2u2

2u(cid:48)2

2c4
ie
1 −1 −1 −3
1 −1 −1

ic2
1
3 −1

2ic4
−1
1

2iu(cid:48)2
2iu2
1
1
−1 −1

Basis
x, y, z
Jx, Jy, Jz

Matrices for elements g with i are

0

0
0 −1 0
1
0

0

 −1
 −1 0

0

0

0

1
0
0 −1

 ,
 ,

 , D(c2) =
 , D(u2) =
 .

 ,
 ,

1 0

0

0 1
0
0 0 −1
1
0
0 −1 0
1
0

0

0




 , D(ic2) =
 , D(iu2) =
;

D(e) = III =

D(c4) =

D(u(cid:48)2) =

D(ie) = −III =

D(ic4) =

D(iu(cid:48)2) =

1 0 0

0 1 0

0 0 1

0
1 0
−1 0 0
0 1
0

0 1

0

1 0
0
0 0 −1




 −1



0

0

0
0 −1
0

0
0 −1
0

0 −1
0
1
0
0 −1
0 −1 0
0
−1
0

0

1

0

129

Table A.3 Representation of D4h with the basis set x2, y2, xy, yx.

Representation

E− × E−

e
4

c2
4

2c4
0

2u2
0

2u(cid:48)2
0

ie
4

ic2
4

2ic4
0

2iu2
0

2iu(cid:48)2
0

Basis

x2, y2, xy, yx

Using equation (8.19) from [128], we find the irreducible representations entering D+

1 and D+
1 :

Nµ =

1
h ∑

g

χ∗µ (g) χ (g) .

(30)

The coefficient Nµ indicates the number of times the irreducible representation with characters
χµ (g) occurs in the given reducible representation D with characters χ(g). This number is called
the multiplicity of the irreducible representation. We find that D−1 = A−2 +E− and D+
2 +E+.
Thus, we find that z is transformed according to A−2 , and x,y are transformed according to E−.
Corresponding components of JJJ are transformed by A+

1 = A+

2 and E+.

Now, we can find how the products such as xy are transformed. The functions x2,y2,xy,yx lead
to a representation E− × E− with characters given in Table A.3. Therefore, we find irreducible
representations contained in E− × E− by means of (30):

E− × E− = A+

1 + A+

2 + B+

1 + B+
2 .

(31)

The basis functions for these irreducible representations are easily guessed. Products xz,yz are
transformed according to E− × A−2 = E+. To find the basis functions, we use Eqs. (9.11) and
(9.12) from [128]:

PE+xz =

PE+yz =

g

2
16 ∑
2
16 ∑

g

χ∗E+(g)D(g)xz =

χ∗E+(g)D(g)yz =

1
8 (2xz + 2xz + 2xz + 2xz) = xz
1
8 (2yz + 2yz + 2yz + 2yz) = yz

We thus find that xz and yz form the basis of E+. Similarly, we can find the third order basis

functions. Functions xxz, xyz, yxz, yyz are transformed as

E− × E− × A−2 = A−1 + A−2 + B−1 + B−2

(32)

130

Table A.4 Fourth-order basis sets for some reducible and irreducible representations of D4h.

Reducible
(E−)4

(E−)3 × A−2
(E−)2 × (A−2 )2

(A−2 )4

Irreducible Basis functions

4A+
1
4A+
2
4B+
1
4B+
2
4E+
A+
1
A+
2
B+
1
B+
2
A+
1

xxxx + yyyy; xxyy + yyxx; xyxy + yxyx; xyyx + yxxy
xyxx− yxyy; xxxy− yyyx; yxxx− xyyy; xxyx− yyxy
xxxx− yyyy; xxyy− yyxx; xyyx− yxxy; xyxy− yxyx
xyxx + yxyy; xxxy + yyyx; yxxx + xyyy; xxyx + yyxy
xxxz,yyyz; xxyz,yyxz; xyxz,yxyz; xyyz,yxxz
xxzz + yyzz
xyzz− yxzz
xxzz− yyzz
xyzz + yxzz
zzzz

Again, to find the basis functions, we act on the mentioned third-order functions by the operator
Pµ. This way we find that {xy}z is another basis set (with one function) of A−1 . Similarly (x2 +y2)z
is another basis function for A−2 ; (xy + yx)z = [xy]z for B−1 ; and (x2 − y2)z for B−2 .

Another third-order set of functions is xxx,yxx,xyx,xxy,yyx,yxy,xyy,yyy. They are transformed
according to E− × E− × E− = 4E−. Their basis sets are easily found: xxx,yyy; xxy,yyx; xyx,yxy;
xyy,yxx. However, do not really need their basis functions because E− does not enter D× D∗. In a
similar manner we find basis sets of the fourth order, that are summarized in Table A.4.

A.6.2 Representations of D4h entering X2 × X∗2
To write the general form of Hamiltonian at point X that would be invariant under all symmetry
operations at this point, we need to find all irreducible representations of D4h entering X2 × X∗2 .
First, we find representations X2 and X4, characters of which are summarized in Table A.5. Char-
acters of X2 × X2 and X4 × X4 are the same, so they contain the same irreducible representations.
Using Eq. (30), we find those representations in terms of D4h = D4 ×Ci:

X2 × X∗2 = A+

1 + A−2 + B−1 + B+
2

(33)

131

Table A.5 Characters of X2 and X4 representations. The first column contains the elements of D4h
group, while the second contains the corresponding elements of the little group of the wave vector
k = kX. Other columns show characters of X2, X4, X2 × X∗2 , and X4 × X∗4 representations.

D4h
e
c2
c4
c4
u2
u2
u(cid:48)2
u(cid:48)2
ie

ic2 = σz

ic4
ic4

iu2 = σx
iu2 = σy
iu(cid:48)2 = σxy
iu(cid:48)2 = σx ¯y

GX
(e|0)
(c2z|0)
(c4z|τ)
(c3
4z|τ)
(c2x|0)
(c2y|0)
(c2xy|τ)
(c2x ¯y|τ)
(ie|τ)
(ic2z|τ)
(ic4z|0)
(ic3
4z|0)
(ic2x|τ)
(ic2y|τ)
(ic2xy)
(ic2x ¯y)

X2 X4 X2 × X∗2 X4 × X∗4
2
2
2
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2 −2
2 −2

4
4
0
0
0
0
0
0
0
0
0
0
0
0
4
4

4
4
0
0
0
0
0
0
0
0
0
0
0
0
4
4

We can also choose matrices III,σz,σy,σx that transform according to these representations. The

resulting representations and and their basis functions of all required orders are summarized in

Table A.6.

132

Table A.6 Irreducible representations contained in X2 × X∗2 and basis functions of different orders,
as well as matrices that transform according to the representations.

X2 × X∗2 : Basis 1

A+
1

Basis 2
z2,x2 + y2

Basis 3

Basis 4
xxxx + yyyy; xxyy + yyxx; xyxy + yxyx;
xyyx + yxxy; xxzz + yyzz; zzzz

z

A−2
B−1
B+
2

(x2 + y2)z
[xy]z ∼ Jzz

xy

xyxx + yxyy; xxxy + yyyx; yxxx + xyyy;
xxyx + yyxy; xyzz + yxzz

Matrix

III

σz
σy
σx

A.6.3 Effects of strain on the spectrum around the valley minimum

Using the basis functions of the irreducible representations of X2 × X∗2 , we write the Hamiltonian
for the z-valley up to the second power in the most general form:

H(z) =III

k2
x + k2
y

+ D1εzz + D2

A1k2

(cid:104)
(cid:16)
 λ + A4kz

z + A2

=

A3kxky + D3εxy

(cid:17)

A3kxky + D3εxy

λ − A4kz

(cid:1)(cid:105)

(cid:0)εxx + εyy
 ,

(cid:2)A3kxky + D3εxy

(cid:3) + A4σzkz

+ σx

(34)

Where A1–A4 and D1–D3 are some constants, we will identify their meaning later. Then, the
spectrum around point X is determined by the equation det (H − IIIE) = 0:
z +(cid:0)A3kxky + D3εxy
(cid:1)2
(cid:1)2(cid:105)1/2
z +(cid:0)A3kxky + D3εxy

(cid:16)
E(z) − λ

(cid:17)2
(cid:104)

From Eq. (35),

(35)

= A2

4k2

(36)

.

.

E(z) = λ ±

A2
4k2

133

We know that there is a minimum on the lower branch of this spectrum along kz at point k0. We
expand the lower branch around the minimum to the fourth order in strain:

E(z) =A1 (kz − k0)2 + 2A1k0 (kz − k0) + A1k2
≈A1 (kz − k0)2 + 2A1k0 (kz − k0) + A1k2
−

A3D3εxykx

0 + A2

0 + A2

ky +
A4k0
xy (kz − k0)2
3ε2
D2
2A4k3
0

−

− A4 (kz − k0)−
3ε4
3D4
xy (kz − k0)
8A3

4k4
0

(cid:0)εxx + εyy
(cid:0)εxx + εyy

(cid:1)
(cid:1)

(cid:16)
(cid:16)

(cid:17)
(cid:17)

+ D1εzz + D2

+ D1εzz + D2

x + k2
k2
y
k2
x + k2
y
3ε2
3ε2
D2
D2
xy (kz − k0)
xy
2A4k2
2A4k0
0
xy (kz − k0)2
3ε4
3D4
4A3

+

+

.

4k5
0

Without strain, point k0 should be a minimum, therefore we find |A4| = 2A1k0. Then, up to a
constant,

+

3ε3
A3D3
xykxky
4k3
2A3
0
3ε4
D4
xy
0 −
4k3
8A3
(cid:17)

k2
x + k2
y
2A3D3

(cid:16)

(37)

(38)

(39)

(40)

+ D1εzz + D2

E(z) =A1 (kz − k0)2 + A2
D4
3ε4
xy
∆E3 −

D2
3ε2
xy
∆E +
3ε2
D2
xy (kz − k0)
∆Ek0

−

−

+

εxykxky +
∆E
3D4
3ε4
xy (kz − k0)
∆E3k0

We will use the following notation:

(cid:1)

(cid:0)εxx + εyy
4A3D3
∆E3 ε3
3
xykxky
xy (kz − k0)2
3ε2
D2
∆Ek2
0

−

+

6D4

xy (kz − k0)2
3ε4
∆E3k2
0

−

D2
3ε2
xy
2|A4|k0
D4
3ε4
xy
8|A4|3k3

0

= −

=

D2
3ε2
xy
4A1k2
0
D4
3ε4
xy
0

(cid:0)4A1k2

= −

(cid:1)3 =

(cid:1)
E(cid:0)∆2(cid:48)
3ε2
D2
xy
− E (∆1) ≡ −Ξshε2
− E (∆1)(cid:3)3 ≡ Ξs4ε4
(cid:1)
(cid:2)E(cid:0)∆2(cid:48)
D4
3ε4
xy

xy

xy

And D3 is identified as D3 ≡ 2Ξu(cid:48) from [129]. We can also identify D1 ≡ Ξu +Ξd, D2 ≡ Ξd, A1 ≡
, A2 ≡ ¯h2/2m⊥, where Ξd and Ξu are the well-known deformation potential constants, and
¯h2/2m
, m⊥ are longitudinal and transverse masses in the valley. Constant A3 is unknown to the best

(cid:107)
of our knowledge. We will not substitute all of theses values in the equations below for brevity.

m

(cid:107)

There are three effects of strain that can be identified from Eq. (38): valley minimum is shifted

in k-space, the effective mass is modified, and the valley minimum is shifted in energy. The shift

of the minimum in k-space (to the second order in εxy) is obtained from:

(cid:34)
A1 (kz − k0)2 +−Ξshε2

xy

(cid:32)

0 =

d
dkz

(cid:33)(cid:35)

,

(kz − k0)2

k2
0

(kz − k0)

k0

−

134

and the result is

The lowest order correction to A1 is:

kmin

z ≈ k0 + −Ξshε2

xy
2k0A1

.

δ A1 = −Ξshε2
k2
0

xy

.

From here, the longitudinal mass becomes

1 +

m

(cid:107) = m0
(cid:107)

−1

,

(41)

2m0

xy

(cid:107) − Ξshε2
¯h2k2
0

where m0
(cid:107)

is the longitudinal mass without strain. We will not comment on the transverse mass

effects since they are of no importance to our calculations. The most relevant result for us is the

shift of the valley bottom in energy by

δ Ez = (Ξu + Ξd)εzz + Ξd(εxx + εyy)− Ξshε2

xy + Ξs4ε4
xy.

(42)

It includes the familiar deformation potential and two new terms. The term −Ξshε2
experimentally observed [129, 133]. The last term Ξs4ε4
the main calculation since it only leads to a very small correction of 1
may become important for very high electron concentrations, above 1020 cm−3.

xy has been
xy has not been observed, and we drop it in
2nΞs4 to δ c4444. However it

A.7 Electron-ripplon scattering

A.7.1 Capillary wave quantization

To quantize the capillary wave field, we first need to write the field Lagrangian, then find the

conjugate variables, and construct the Hamiltonian. Afterwards we second-quantize the field, in-

troducing the creation and annihilation operators. We will describe the system by the displacement

ξ (rrr,t) of the surface along the z axis, and the velocity potential Φ(rrr,z,t) in the bulk. The velocity
is expressed through Φ as vvv(rrr,z,t) = −∇Φ(rrr,z,t). It is convenient to solve the problem in the
Fourier space. We are looking for the solutions Φ(rrr,z,t) that exponentially decay into the bulk, so

135

we can write

Φ(rrr,z,t) = ∑
kkk
ξ (rrr,t) = ∑
kkk

Φkkk(t)ekzeikkkrrr

ξkkk(t)eikkkrrr

(43)

Then the kinetic energy in the Fourier variables is

dz

T =

d2r(cid:48)

ρS
2 ∑
(cid:32)
In a similar way, find the gravitational potential energy:
∑
kkk

ρ
2 (∇Φ)2 =
(cid:90)

dzρgz =

ρg
2

Vg =

d2r(cid:48)

d2r(cid:48)

(cid:90)

(cid:90)

(cid:90)

(cid:90) 0
−∞
(cid:90) ξ (rrr,t)
(cid:118)(cid:117)(cid:117)(cid:116)1 +

0

(cid:32)
∑
kkk

(cid:33)

kΦkkkΦ

kkk

−kkk =

ρS
2 ∑

kkk

1
k

˙ξkkk

˙ξ
−kkk

(cid:33)2

ξkkkeikkkrrr

=

ρgS
2 ∑

kkk

ξkkkξ

−kkk

(44)

To find the surface energy, we first calculate the change of the surface area due to the ripplon field:

S + δ S =

d2r(cid:48)

ikkkξkkk∑
kkk(cid:48)

ikkk(cid:48)ξkkk(cid:48)eirrr(cid:48)(kkk+kkk(cid:48))

≈ S + 1

2Sk2∑
kkk

ξkkkξ

−kkk.

Then the surface energy is expressed through δ S and the surface tension α:

The Lagrangian of the system is

Vs = αδ S =

αS
2 ∑

kkk

k2ξkkkξ

−kkk.

L = T −Vg −Vs.

The conjugate variables are obtained are by differentiating the Lagrangian:

πkkk =

∂ L
∂ ˙ξkkk

=

ρS
k

˙ξ
−kkk

or

With the conjugate variables, we can write the Hamiltonian

(cid:34)

H = ∑
kkk

˙ξkkk =

k
ρS

π
−kkk.

(cid:35)

,

1
2mk

πkkkπ

−kkk +

mkω2
k
2 ξkkkξ

−kkk

136

(45)

(46)

(47)

(48)

(49)

(50)

where we introduced the notations

ρS
k ,

mk ≡
k ≡ S(ρg + αk2).

mkω2

Now we quantize the field ξ by requiring that

[ξkkk,πkkk(cid:48)] = i¯hδkkkkkk(cid:48).
We introduce the second quantization operators akkk and a†
kkk:

akkk =

a†
kkk =

Or equivalently,

π
−kkk
π
−kkk

2¯h

2¯h

ξkkk + i

(cid:114)mkωk
(cid:16)
(cid:114)mkωk
(cid:16)
mkωk
ξkkk − i
mkωk
(cid:115)
(cid:16)
(cid:114) ¯hmkωk
akkk + a†
(cid:16)
−kkk
akkk − a†
−kkk

2mkωk

¯h

2

(cid:17)
(cid:17)

,

ξkkk =

πkkk = i

(cid:16)

(cid:17)

H = ∑
kkk

¯hωk

a†
kkkakkk + 1
2

(51)

(52)

(53)

(54)

(55)

(cid:17)
(cid:17)

,

.

.

(56)

One can check that [akkk,a†
The Hamiltonian in the second-quantized form is

kkk(cid:48)] = δkkk,kkk(cid:48), and other familiar relations for bosonic operators are satisfied.

A.7.2 Polarization coupling to the surface displacement field

We consider a particle with coordinates rrr along the surface and z perpendicular to the surface. We

are computing the energy of the coupling of the electron with the helium surface when the surface

is perturbed by the vertical displacement field ξ (rrr) created by ripplons. The energy of a polarized

body in an electric field created by charge δ q is written as [206]:

δ F = −

PPP· δCCCdV

(57)

(cid:90)

137

Here PPP is the polarization of the medium, and CCC is the electric field that would have been in the

medium if the medium had permittivity of 1 (or there was no medium). Integration is done over

the medium volume only. The polarization is found through the electric field as

The full interaction energy of charge e with the surface is obtained by integrating over δ q:

Then

δ F = −

ε − 1
2π(ε + 1)

F(rrr,z) = −

Λ
π

where

PPP =

ε − 1
4π
(cid:90)

drrr(cid:48)

EEE =

(cid:90) ξ (rrr(cid:48))

(cid:90)

drrr(cid:48)

−∞

(cid:90) ξ (rrr(cid:48))

−∞

ε − 1
2π(ε + 1)

CCC.

qδ q

(cid:2)(rrr(cid:48) − rrr)2 + (z(cid:48) − z)2(cid:3)2 dz,
(cid:2)(rrr(cid:48) − rrr)2 + (z(cid:48) − z)2(cid:3)2

dz(cid:48)

e2(ε − 1)
4(ε + 1)

.

Λ ≡

(58)

(59)

(60)

(61)

(62)

Eq. (60) is the full energy of the electron-helium interaction, including the trivial attraction without

the surface perturbation. For the electron-ripplon scattering, we are interested only in the part that

is induced by the surface displacement, that we will denote as ∆Fqqq for a single Fourier component
of the surface displacement ξqqqeiqqqrrr. We will also make the coordinate transformation ˜z = z− ξ (rrr)
for the reasons that will be explained in the next part. With this in mind,

(cid:90)

∆Fqqq(˜z) = −

≈ −

d2r(cid:48)

Λ
π
Λξqqqeiqqqrrr

π

Λξqqqq2eiqqqrrr

π

=

0

dz(cid:48)

(cid:90) ξqqq(eiqqqrrr(cid:48)−eiqqqrrr)
(cid:2)(rrr(cid:48) − rrr)2 + (z(cid:48) − ˜z)2(cid:3)2
(cid:90) d2r(cid:48)(eiqqqrrr(cid:48) − 1)
(cid:2)r(cid:48)2 + ˜z2(cid:3)2
1− eixcosθ(cid:17)
(cid:16)
(cid:90) dxdθ x
(cid:2)x2 + (q˜z)2(cid:3)2

Here we used the smallness of ξqqq compared to the typical electron coordinate z. The first part of

this integral is

(cid:90)

xdxdθ

(cid:2)x2 + (q˜z)2(cid:3)2 =

π
(q˜z)2 ,

138

(63)

while the second is

(cid:90) dxdθ xeixcosθ
(cid:2)x2 + (q˜z)2(cid:3)2 =

(cid:90) dxdθ x∑∞

(cid:2)x2 + (q˜z)2(cid:3)2
n=−∞ inJn(x)einθ

(cid:90)

= 2π

(cid:2)x2 + (q˜z)2(cid:3)2

dxxJ0(x)

(64)

Integration of terms with n (cid:54)= 0 over θ gives zero. The remaining integral is found using formula
(11.4.44) in Ref. [207]:

(cid:90) ∞

0

2π

(cid:2)x2 + (q˜z)2(cid:3)2 =

dxxJ0(x)

π
q˜z

K1(q˜z)

(65)

Taking all terms in ∆Fqqq together, and summing over qqq, we obtain the coupling energy to the ripplon
field ξ through polarization:

(cid:20)1

(cid:21)
˜z − qK1(q˜z)

(66)

(67)

∆F(rrr, ˜z) = ∑
qqq

∆Fqqq = Λ∑
qqq

ξqqqeiqqqrrr 1
˜z

A.7.3 Total coupling to the surface displacement field

In the presence of a pressing field E⊥, the system is described by the Schrödinger equation

¯h2
2me

∇2ψ + [λ − eE⊥z− F(r,z)]ψ = 0.

where λ is the eigenvalue of the Hamiltonian of the system. In addition to the polarization coupling

outlined in Section A.7.2, the electron is coupled to the surface deformation through the modifi-

cation of the boundary condition on the surface, which is not flat anymore. The wavefunction

vanishes on the surface z = ξ (rrr):

ψ (rrr,z)|z=ξ (rrr) = 0.

(68)

In general, we do not know how to solve the problem with the curved boundary of arbitrary shape.

In the case of small ξ (rrr), however, we can facilitate solving the equation by a coordinate transfor-

mation:

˜z = z− ξ (rrr), ˜r = r.

(69)

In these coordinates, the problem has a flat boundary, and therefore can be solved with conventional

approaches. However, we need to rewrite the equation in the new coordinates. The kinetic energy

139

term in new coordinates is found by transforming ∇2ψ:

¯h2∇2ψ = ¯h2 ˜∇2ψ + 2

˜px +

∂ξ
∂ ˜y

˜py

˜pzψ − i¯h

(cid:18)∂ξ

∂ ˜x

(cid:19)

(cid:33)

(cid:32)

∂ 2ξ
∂ ˜x2 +

∂ 2ξ
∂ ˜y2

˜pz.

(70)

Now if we expand ξ in the Fourier series ξ = ∑qqq ξqqqeiqqqrrr and drop tildes, we obtain

¯h2
2me

∇2ψ →

¯h2
2me

∇2ψ +∑
qqq

iξqqq(qqq· ppp)

me

eiqqqrrr pzψ +∑
qqq

i¯h
2me

q2ξqqqeiqqqrrr pzψ

(71)

We can write the additional terms arising from the transformation of ∇2 as a perturbation term in

energy

V∇ = ∑
qqq
V∇q = −

ξqqqeiqqqrrrV∇q,
i
me

(qqqppp)pz −

i¯h
2me

q2pz

(72)

This term can not lead to transitions between states with the same m quantum number of the

motion transverse to the surface, and its magnitude is small. Since we are considering scattering

only within the first subband, we can ignore this term in our calculation of scattering. The total

potential energy is transformed to

V (rrr,z) = eE⊥z + F0(z) + eE⊥

ξ (rrr) + ∆F(rrr,z).

(73)

We dropped tildes here. The first two terms represent the coupling to helium with unperturbed

surface. Therefore, the electron-ripplon coupling energy that contributes to the scattering within

the first subband is

and ∆F(rrr,z) is given by Eq. (66).

H(rpl) = eE⊥

ξ (rrr) + ∆F(rrr,z),

(74)

A.7.4 Scattering rate of electrons by ripplons

Now that we know the coupling energy of electrons with the ripplon field, we can calculate the

rate of scattering and the distribution of scattering angles. Scattering rate from the state kkk into the

140

volume d2k(cid:48), according to the Fermi golden rule, is

(cid:20)(cid:12)(cid:12)(cid:12)H(rpl)

1,kkk(cid:48)kkk

(cid:12)(cid:12)(cid:12)2

(cid:12)(cid:12)(cid:12)H(rpl)

2,kkk(cid:48)kkk

(cid:12)(cid:12)(cid:12)2(cid:21)

δ (εkkk(cid:48) − εkkk)

Sd2k(cid:48)
(2π)2

+

dw(rpl)

kkk(cid:48)kkk =

2π
¯h

where

H(rpl)
1,kkk(cid:48)kkk = (cid:104)kkk(cid:48),Nqqq − 1,N−qqq|H(rpl)|kkk,Nqqq,N−qqq(cid:105),
H(rpl)
2,kkk(cid:48)kkk = (cid:104)kkk(cid:48),Nqqq,N−qqq + 1|H(rpl)|kkk,Nqqq,N−qqq(cid:105),

Here, the state vectors |kkk,Nqqq,N−qqq(cid:105) are defined by the electron wavevector kkk, the number of rip-
plons Nqqq in the qqq state, and the number of ripplons N−qqq in the −qqq state. The average number of
ripplons in thermal equilibrium is Nqqq = 1/(e¯hωq/kBT − 1). These two matrix elements represent
two possible processes of scattering: with creation and absorption of a ripplon. The delta-function

can be rewritten as:

δ

Then, the scattering rate is

(cid:32)

meδ (k(cid:48) − k)

¯h2k(cid:48)

=

(cid:33)
(cid:12)(cid:12)(cid:12)H(rpl)

2,kkk(cid:48)kkk

+

(cid:12)(cid:12)(cid:12)2(cid:21)

¯h2k(cid:48)2
2me −

¯h2k2
2me

(cid:20)(cid:12)(cid:12)(cid:12)H(rpl)

1,kkk(cid:48)kkk

(cid:12)(cid:12)(cid:12)2

dw(rpl)

kkk(cid:48)kkk =

mS
2π ¯h3

δ (k(cid:48) − k)dk(cid:48)dθ

We first find the matrix elements between states |kkk(cid:105) and |kkk(cid:48)(cid:105), without the pressing field, which we
will include later:

(cid:90)

dz

(cid:90) ∞

0

d2r

4z2
Sr3
B

e−2z/rBeirrr(kkk−kkk(cid:48)+qqq)ξqqq

Λ
z

(cid:20)1

(cid:21)
z − qK1(qz)

Integration over rrr gives δkkk(cid:48)−kkk,qqq, while integrals over z of the two terms give

qqq

(cid:104)kkk(cid:48)|H(rpl)|kkk(cid:105) = ∑
(cid:90) ∞

(cid:90) ∞

0

dze−2z/rB = rB/2,

0

dzze−2z/rBK1(qz) =

(cid:90) ∞

0

1
q2

(75)

(76)

(77)

(78)

(79)

(80)

(81)

The explicit form of Ωr will be introduced later. Taking the above results into account, we obtain

1
q2 Ωr(qrB)

dzze−2z/qrBK1(z) ≡
(cid:20)rB

Ωr(qrB)

1
q

2 −

(cid:21)(cid:12)(cid:12)(cid:12)(cid:12)qqq=kkk(cid:48)−kkk

(cid:104)kkk(cid:48)|H(rpl)|kkk(cid:105) =

4Λ
Bπ2 ξqqq
r3

141

In the full matrix elements, (aqqq + a†

tively. Using the notation

Ωr(x) [3], we obtain:

−qqq) will give(cid:112)Nqqq and(cid:112)N−qqq + 1 in H(rpl)
(cid:12)(cid:12)(cid:12)H(rpl)
(cid:12)(cid:12)(cid:12)H(rpl)
(cid:12)(cid:12)(cid:12)2
(cid:12)(cid:12)(cid:12)H(rpl)
(cid:12)(cid:12)(cid:12)2

(cid:12)(cid:12)(cid:12)2
φ 2(cid:16)qrB

(cid:12)(cid:12)(cid:12)H(rpl)

S−1kBT ¯h4q2

(cid:12)(cid:12)(cid:12)2

1,kkk(cid:48)kkk

2,kkk(cid:48)kkk

(cid:17)

+

=

=

qqq

qqq

,

2

4αm2

er2
B

1,kkk(cid:48)kkk respec-
and the analytical result for the function

1,kkk(cid:48)kkk and H(rpl)

where

Function φ (x) is continuous at x = 1 and is monotonically decreasing everywhere. Therefore, the

scattering rate (78) will take the form

Integrated over k(cid:48), it gives the angle distribution (normalized to the total scattering rate):

(82)

(83)

(84)

(85)

(86)

(88)

, x < 1,

, x > 1.

x

1 + (1− x2)1/2

φ (x) = (x2 − 1)−1 + (1− x2)−3/2 ln

(cid:35)
(cid:34)
(x2 − 1)1/2(cid:105)
φ (x) = (x2 − 1)−1 − (x2 − 1)−3/2 tan−1(cid:104)
(cid:17)(cid:12)(cid:12)(cid:12)q=k√2(1−cosθ )
q2φ 2(cid:16)qrB

q2φ 2(cid:16)qrB

8παmer2
B

kkk(cid:48)kkk =

¯hkBT

¯hkBT

2

w(rpl)
kθ =

2

8παmer2
B

dw(rpl)

δ (k(cid:48) − k)dk(cid:48) dθ .

(cid:17)(cid:12)(cid:12)(cid:12)q=k√2(1−cosθ )
(cid:17)(cid:12)(cid:12)(cid:12)q=k√2(1−cosθ )
(cid:19)

2

The integral of this distribution over θ is the total scattering rate from the state kkk to all other allowed

states:

w(rpl)
k =

¯hkBT

8παmer2
B

dθ q2φ 2(cid:16)qrB

(cid:90) 2π

0

.

(87)

The expressions (87) and (86) are used in the simulations. For completeness, we also give the total
scattering rate with the pressing field E⊥, even though we do not use it in our study:

(cid:34)

(cid:18)rB

(cid:90) 2π

0

w(rpl)
k ≈

mekBT
2πα ¯h3

dθ 1
q2

4Λ
r3
B

1
q

2 −

Ωr(qrB)

+

eE⊥2

(cid:35)2(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)q=k√2(1−cosθ )

In our simulations, we neglect the eE⊥ term for two reasons. First, it is setup-dependent. Second,
it causes the integral over θ to diverge at small angles. That, in principle, can be avoided by

assuming a reasonable threshold on the long-wavelength ripplons that can participate in scattering.

However, we choose not to complicate the problem further since the inclusion of this term is not

expected change the qualitative behavior of the system that we study.

142

A.8 Mobility in a uniform electron system on the helium surface

A.8.1 Single-electron mobility

The mobility in the spatially uniform system of noninteracting electrons can be calculated from

the Boltzmann equation. We assume that a uniform driving field EEEd is applied to the electron gas.
Then the Boltzmann equation in the relaxation time approximation can be written as

eEEEd

∂ f
∂ ppp ≈

g(ppp)
τse(p)

,

(89)

where ppp is the electron momentum, f (ppp) is the equilibrium momentum distribution, and g(ppp) is

the additional term in the distribution that is induced by the driving field. The inverse relaxation

time 1/τse(p) due to the ripplon scattering without a pressing field is calculated by an integral
similar to Eq. (87). We only need to modify the integrand by a factor of (1− cosθ ):

dθ q2φ 2(cid:16)qrB

(cid:17)

(cid:90) 2π

0

(1− cosθ )

2

(cid:12)(cid:12)(cid:12)q=k√2(1−cosθ )

1

τse(p)

=

¯hkBT

8παmer2
B

.

(90)

Here p = ¯hk, and notations from Appendix A.7 are used. From Eq. (89),

g(ppp) =

e
me

(EEEd ppp)

∂ f
∂ε

τse(p).

To find the drift velocity vx in the field EEEd along the x axis, we integrate
∂ f
∂ε

d pdθ p3 cos2(θ )

vx =

d2p

eEd
m2
e

τse(p)

(cid:90)

(cid:90)

(cid:90)

px
g(ppp) =
me
d p p3 ∂ f
∂ε

=

πeEd
m2
e

τse(p).

If we define the single electron mobility µse through

then for µse we have

f must be normalized to give(cid:82) d2p f (ppp) = 1. For the Boltzmann distribution, this means

µse = −

τse(p).

|vx| = µseEd,
(cid:90)

d p p3 ∂ f
∂ε

πe
m2
e

f (ppp) =

1

2πmekBT

p2

2mekBT ,

e−

143

(91)

(92)

(93)

(94)

(95)

(96)

which gives the mobility

µse =

(cid:90) ∞

e

2m3

e(kBT )2

0

A.8.2 Many-electron mobility

d p p3τse(p)e−p2/2mekBT .

(97)

When the electron-electron interaction is strong, the electrons exchange momentum with each

other much faster than with the ripplons. Therefore, the calculation of the mobility should be

different from the non-interacting case. According to Eqs. (38) and (39) in Ref. [3], mobility in the

strongly correlated nondegenerate electron fluid on the sufrace of helium without magnetic field is

given by

where

Here

τ−1
me =

1

4mekBT ∑

qqq

q2|H(rpl)

qqq

µme =

eτme
me

,

(98)

S

4mekBT (2π)2

(cid:90) ∞
eiqqqrrr(t)e−iqqqrrr(0)(cid:69)
(cid:68)

0

dq2πq3|H(rpl)

qqq

|2ζ (qqq).

(99)

(100)

|2ζ (qqq) =
(cid:90) ∞
−∞

dt

ζ (qqq) =

is the dynamical structure factor, and notations from Appendix A.7 are used. The statistical averag-

ing is computed to the zeroth order in the coupling to the scatterers. To compute τ, we must calcu-
1
2 [A,B]
late ζ (qqq) first. We expand rrr(t) to the first order and use the operator identity eA+B = eBeAe

for the case when [A,B] commutes with both A and B:

(cid:43)

(cid:28)

(cid:90) ∞
−∞

dt

(cid:29)

eiqqqpppt/mee−i¯hq2t/2me

=

It is convenient to compute the statistical average with the Boltzmann distribution first:

(cid:42)

dt

(cid:90) ∞
ζ (qqq) ≈
−∞
e−iqqqpppt/me(cid:69)
(cid:68)

eiqqqrrr(0)+

iqqqppp(0)t

me

e−iqqqrrr(0)

(cid:90)

1

=

1

2πmekBT
√2πmekBT

=
= e−kBT q2t2/2me.

144

d px d py e−p2x /2kBT me+iqt px/mee−p2y /2me
(cid:90)
e−kBT q2t2/2me

1
2me (px−iqtkBT )2

d px e−

Therefore,

(cid:90) ∞
−∞
Now we can calculate τ−1:

ζ (qqq) =

¯h2q2
8mkBT .

2πme
kBT q2 e−
(cid:17)

e−

¯h2q2
8mekBT .

(101)

(102)

(cid:115)

dt e−kBT q2t2/2mee−i¯hq2t/2me =

τ−1 =

¯h4

16m2

Bα√2πmekBT
er2

(cid:90) ∞

0

dqq4φ 2(cid:16)qrB

2

A.9 Electron-phonon scattering

A.9.1 Phonon quantization in half-space with free surface

In this appendix, we reproduce the derivation of the electron-phonon scattering rates following

Ref. [141]. Only polarization effects are considered here, without the effects of the surface de-

formation by the phonons. Before we consider the scattering we must quantize the elastic waves

(phonons) in the half-space occupied by the helium with a free surface boundary condition. The

free surface condition implies zero stress at the surface. Since it is a liquid, the only stress that can

exist is pressure. No pressure at the surface means

(cid:18)∂ ux

∂ uy
∂ y +

∂ uz
∂ z

∂ x +

(cid:19)(cid:12)(cid:12)(cid:12)(cid:12)z=0

= 0,

(103)

where ui are the components of the displacement field. The displacement field in a liquid can be
written as a superposition of longitudinal waves:

uuu =

1
2 ∑

QQQ

uQQQ

QQQ
Q

eiQQQRRR,

where QQQ are three-dimensional wavevectors. The free-surface condition (103) then reads as

∇· uuu = −∑

QQQ

uQQQQuQQQeiqqqrrr = 0

(104)

(105)

where rrr and qqq are the coordinate and the wave vector in the plane parallel to the surface. The last

equation must be true at any point rrr, which means

u−Qz,qqq = −uQz,qqq.

145

(106)

(cid:33)

This leads to the following form of the expression (104), where we split the sum into two parts for

(cid:32)

convenience:

uuu =

1
2 ∑

qqq

∑
Qz>0

uQz,qqq

QQQ
Q

eiQzzeiqqqrrr + ∑
Qz<0

uQz,qqq

QQQ
Q

eiQzzeiqqqrrr

(107)

The term with Qz = 0 vanishes due to the condition (106). Using (106) we can write the compo-
nents of uuu as

ux = i∑
qqq
uy = i∑
qqq
uz = ∑
qqq

∑
Qz>0
∑
Qz>0
∑
Qz>0

uQz,qqq

uQz,qqq

uQz,qqq

qx
Q
qy
Q
Qz
Q

sin(Qzz)eiqqqrrr,

sin(Qzz)eiqqqrrr,

cos(Qzz)eiqqqrrr.

(108)

To quantize the phonon field, we express the Lagrangian of the system in terms of the new Fourier
variables uuuQz,qqq, construct the Hamiltonian and then express the displacement and momentum in
terms of ladder operators. The kinetic energy is

(cid:90)

ρ
(cid:32)
2
i∑
(cid:32)
qqq
i∑
(cid:32)
qqq
∑
qqq

(cid:16)

(cid:17)

T =

d3R

˙u2
x + ˙u2

y + ˙u2
z

qqq(cid:48)

(cid:33)i∑
(cid:33)i∑
(cid:33)∑

qqq(cid:48)

ρ
2

ρ
2

ρ
2

qx
Q

qy
Q

˙uQz,qqq

˙uQz,qqq

∑
Qz>0

∑
Qz>0

sin(Qzz)eiqqqrrr

sin(Qzz)eiqqqrrr

∑
Q(cid:48)z>0
∑
Q(cid:48)z>0
∑
Q(cid:48)z>0
Integral over rrr gives Sδqqq(cid:48),−qqq, while the integral over z gives 1
and Lz is the helium depth. Therefore,

cos(Qzz)eiqqqrrr

∑
Qz>0

˙uQz,qqq

Qz
Q

qqq(cid:48)

˙uQ(cid:48)z,qqq(cid:48)

˙uQ(cid:48)z,qqq(cid:48)

q(cid:48)x
Q

q(cid:48)y
Q

sin(Q(cid:48)zz)eiqqq(cid:48)rrr

sin(Q(cid:48)zz)eiqqq(cid:48)rrr

 +
 +
 .

˙uQ(cid:48)z,qqq(cid:48)

Q(cid:48)z
Q

cos(Q(cid:48)zz)eiqqq(cid:48)rrr

(109)

2LzδQz,Q(cid:48)z

, where S is the surface area

T =

ρV
4 ∑
bq,Qz>0

˙uQz,qqq ˙uQz,−qqq

(110)

146

Similarly, the potential energy of the elastic excitations is expressed through the bulk modulus λ :

(cid:90)
(cid:90)

λ

λ

U =

=

=

1
2
1
2
1
4

(cid:32)
d3R (∇· uuu)2
∑
d3R
qqq,Qz>0
Q2uQz,qqquQz,−qqq.

λV ∑
qqq,Qz>0

QuQz,qqq sin(Qzz)eiqqqrrr

(cid:33)2

(111)

Integration in the last step is done in the same way it was done to obtain Eq. (110). Defining the
Lagrangian as L = T −U, we obtain the momentum variables π j through π j = ∂ L
∂ ˙u j

:

πQz,qqq =

˙uQz,qqq =

ρV
2
2
ρV

˙uQz,−qqq,
πQz,−qqq.

This allows us to write the Hamiltonian:

(112)

(113)

H =

1
ρV ∑
qqq,Qz>0
Q = λ

If we define M ≡ 1
and b†

2ρV and ω2
Qz,qqq in the standard way:

πQz,qqqπQz,−qqq +

1
4

λV ∑
qqq,Qz>0

Q2uQz,qqquQz,−qqq

s Q2, then we can introduce the ladder operators bQz,qqq

uQz,qqq =

πQz,qqq =

¯h

2MωQ

ρ Q2 ≡ v2
(cid:115)
(cid:114) ¯hMωQ
(cid:16)
(cid:16)

2MωQ

¯h

2

¯h

2MωQ

(cid:115)
(cid:115)

i

Qz,−qqq

Qz,−qqq

bQz,qqq + b†

(cid:16)
(cid:16)
aQz,qqq − a†
(cid:17) Qz
(cid:17) qx

Qz,−qqq

Q

Qz,−qqq

Q

bQz,qqq + b†

bQz,qqq + b†

(cid:17)
(cid:17)

,

.

(114)

cos(Qzz)eiqqqrrr,

sin(Qzz)eiqqqrrr.

(115)

Then

uz = ∑
qqq,Qz>0
ux = ∑
qqq,Qz>0

147

A.9.2 Electron-phonon coupling

We consider only the polarization coupling to phonons for simplicity. We also consider the dielec-

tric constant ε close to 1. In this model, the coupling Hamiltonian is

(cid:90)

1
8π

(cid:90)

H(ph) ≈ −
Λ
≈
π

d3R(cid:48) δε(RRR)C(RRR(cid:48))2 ≈
∇· uuu
|RRR(cid:48) − RRR|4 .

d3R(cid:48)

(cid:90)

ε − 1
8π

d3R(cid:48) ∇· uuu

e2

|RRR(cid:48) − RRR|4

(116)

Here δε(RRR) ≈ (1− ε)∇· uuu is the local change of the dielectric constant due to the phonon defor-
mation field, C(RRR(cid:48)) is the electric field at point RRR(cid:48) that would be created by an elementary charge
located at position RRR above the surface without a dielectric. We use RRR for the 3D coordinate to

distinguish it from the in-plane coordinate rrr. Integration over RRR(cid:48) is done in the z < 0 half-space
e2(ε−1)
4(ε+1) ≈ e2(ε − 1)/8. We can
that is occupied by helium. We also used the approximation Λ ≡
find ∇· uuu using Eq. (108), therefore

H(ph) =

=

Λ
π ∑
qqq,Qz>0
Λ
π ∑
qqq,Qz>0
= 2Λ ∑
qqq,Qz>0
(cid:90) ∞

QuQz,qqq

(cid:90) 0
(cid:90) 0
−∞
QuQz,qqqq2(cid:90) 0
−∞

QuQz,qqq

eiqqqrrr

d2r(cid:48)

dz(cid:48) sin(Qzz(cid:48))

(cid:90)
(cid:2)(z− z(cid:48))2 + (rrr− rrr(cid:48))2(cid:3)2
dxx(cid:82) dθ ∑n einθ Jn(x)in
dz(cid:48) sin(Qzz(cid:48))q2(cid:90) ∞
(cid:2)q(z− z(cid:48))2 + x2(cid:3)2
(cid:90) ∞
(cid:2)q(z− z(cid:48))2 + x2(cid:3)2
(cid:0)q(z− z(cid:48))(cid:1) ,

dz(cid:48) sin(Qzz(cid:48))

dxxJ0(x)

K1

0

0

1

−∞

2q|z− z(cid:48)|

dxxJ0(x)

(cid:2)q(z− z(cid:48))2 + x2(cid:3)2 =

0

(117)

(118)

It can be shown that

Where K1(x) is a modified Bessel function. Then

QquQz,qqqeiqqqrrr(cid:90) 0

dz(cid:48)

sin(Qzz(cid:48))
2q|z− z(cid:48)|

K1

−∞

(cid:0)q(z− z(cid:48))(cid:1)

≡ Λ ∑

qqq,Qz>0

QquQz,qqqeiqqqrrrΞ(QQQ,z),
(119)

H(ph) = Λ ∑
qqq,Qz>0

where we defined the integral by Ξ(QQQ,z). The function Ξ(QQQ,z) is calculated numerically.

148

A.9.3 Scattering of electrons by phonons

To calculate the rate of scattering of an electron from state kkk to state kkk(cid:48), we need the corresponding
matrix elements:

(cid:104)kkk(cid:48)|H(ph)|kkk(cid:105) = Λ ∑
(cid:90) ∞

Qz,qqq

(cid:104)Oz|Ξ(QQQ,z)|Oz(cid:105) =

0

QquQz,qqqδqqq,kkk(cid:48)−kkk(cid:104)Oz|Ξ(QQQ,z)|Oz(cid:105),
4z2
dz
r3
B

e−2z/rBΞ(QQQ,z).

(120)

(121)

Here rB = ¯h2/meΛ is the “Bohr radius” of the ground state of the electron motion along the z
axis. There are two scattering processes, with creation and absorption of a phonon. The phonon-

while the emission process matrix element is

absorption process has the following matrix element:

¯hNQz,qqq
2MωQ (cid:104)Oz|Ξ|Oz(cid:105)|qqq=kkk(cid:48)−kkk ,

(cid:115)
(cid:104)kkk(cid:48),NQz,qqq − 1,NQz,−qqq|H(ph)|kkk,NQz,qqq,NQz,−qqq(cid:105) = ΛQq
(cid:115)
(cid:104)kkk(cid:48),NQz,qqq,NQz,−qqq +1|H(ph)|kkk,NQz,qqq,NQz,−qqq(cid:105) = ΛQq
(cid:104)Oz|Ξ|Oz(cid:105)|qqq=kkk(cid:48)−kkk , (123)
Here, the state vectors |kkk,NQz,qqq,NQz,−qqq(cid:105) are defined by the electron wavevector kkk, the number of
phonons NQz,qqq in the (qqq,Qz) state, and the number of phonons NQz,−qqq in the (−qqq,Qz) state. In
thermal equilibrium, the average population of phonons is

¯h(NQz,qqq + 1)

2MωQ

(122)

NQz,qqq =

1
e¯hωQ/T

− 1

.

(124)

Finally, the scattering rate corresponding to the first process is

dw(ph1)

kkk(cid:48)kkk =

2π
¯h ∑

Qz>0

Λ2Q2q2 ¯hNQz,qqq

2MωQ |(cid:104)Oz|Ξ|Oz(cid:105)|2 δ (εk(cid:48) − εk − ¯hωQ)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)qqq=kkk(cid:48)−kkk

= d2k(cid:48)

Λ2

2π2ρ ¯hv2
s

Qq2NQz,qqq|(cid:104)Oz|Ξ|Oz(cid:105)|2

.

(125)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) qqq = kkk(cid:48) − kkk

Qz = 1

¯hvs (εk(cid:48) − εk)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) qqq = kkk(cid:48) − kkk

In direct analogy to this, the second process, with a phonon emitted, has the scattering rate

dw(ph2)

kkk(cid:48)kkk = d2k(cid:48)

Λ2

2π2ρ ¯hv2
s

Qq2(NQz,qqq + 1)|(cid:104)Oz|Ξ|Oz(cid:105)|2

149

(126)

Qz = 1

¯hvs (εk − εk(cid:48) )

A.10 Standard error estimation for a correlated process

In this appendix, we show the method used for calculating the standard error of the mean value

estimate for a stochastic process with a finite correlation time. Let us assume that vt is a stochastic

process with the expectation value v and nonzero correlation time. In the simulation, we want to

estimate v from the sample vt of size Nt. The estimate is simply the average:

¯v =

1
Nt ∑

t

vt.

(127)

We want to estimate the standard error σs of this measurement. The definition of the standard error

is

σ 2
s = (cid:104)( ¯v− v)2(cid:105).

Here ¯v is treated as a random variable, and (cid:104)(cid:105) means the expectation value. σs is a measure of the
width of the distribution of ¯v. Therefore, it is a good estimate of the statistical error of v calculated
with Eq. (127) from a sample. It is convenient to introduce δt ≡ vt − v with a zero expectation
value. Then

(cid:42)(cid:32)

(cid:33)2(cid:43)

(cid:42)(cid:32)

(cid:33)2(cid:43)

(cid:10)δtδt(cid:48)

(cid:11) .

=

1
N2
t

∑
tt(cid:48)

(128)

σ 2
s =

1
Nt ∑

t

vt − v

=

1
Nt ∑

t

δt

With no correlation between δt and δt(cid:48) at different time moments, the standard error would be
simply σ 2
t (cid:105)/Nt. The more statistics we have the more precise estimate of v we can obtain.

For the time-correlated case, let us define the time-correlation function as ρ(t − t(cid:48)) =(cid:10)δtδt(cid:48)

s = (cid:104)δ 2

(cid:11),

which is symmetric around zero. Assuming that ρ(t) decays after some short time tρ, we can
approximately rewrite

σ 2
s ≈

1
Nt

tρ
∑
t(cid:48)=−tρ

ρ(t(cid:48))

(129)

We summed over t since ρ depends only on the difference t − t(cid:48). One approximation here is that
we truncate ρ at tρ. Another approximation is that we disregarded the edges t = 0 and t = Nt when
summing over t. These approximations are good as long as tρ (cid:28) Nt, which is usually satisfied in
our simulations.

150

Finally, to estimate σs from the data, we need to estimate ρ(t −t(cid:48)):

ρ(t) =

1

(Nt −t)

Nt−t
∑
t(cid:48)=1

δt(cid:48)δt(cid:48)+t

(130)

Together with Eq. (129), it gives the desired estimate of the standard error.

151

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