STUDYING ELECTRONS ON HELIUM VIA SURFACE ACOUSTIC

WAVE TECHNIQUES

By

Heejun Byeon

A DISSERTATION

Submitted to

Michigan State University

in partial fulfillment of the requirements

for the degree of

Physics - Doctor of Philosophy

2020

ABSTRACT

STUDYING ELECTRONS ON HELIUM VIA SURFACE ACOUSTIC WAVE

TECHNIQUES

By

Heejun Byeon

This thesis describes novel methods and experiments for investigating electrons floating

on the surface of liquid helium by utilizing surface acoustic wave (SAW) techniques. Electrons

on helium are an ideal system for studying a non-degenerate two-dimensional electron system

(2DES), Wigner cyrstallization, and the coupling of a 2DES to a bosonic field. The electrons

in this system are also interesting with regard to quantum information science due to their

predicted long coherence time. An evanescent coupling of electrons to high-frequency SAWs

submerged beneath the liquid helium provides a probe of the high-frequency conductivity of

the 2DES and can be used to create SAW-driven charge transport of the electron system and

opens the door for resonant interaction between SAWs and collective modes of the electron

system.

Before employing SAW techniques, we first examined the low-frequency transport proper-

ties of electrons on helium at several tens of kHz using a lock-in method, which confirms the

realization of a stable and reproducible 2D electron system on liquid helium. Furthermore,

taking advantage of the capactive coupling of the electron layer and submerged electrodes,

we are able to precisely control AC current flows in the 2DES and achieve the analog of

a field-effect transistor with electrons on helium and demonstrate the applicability of long

wavelength transport at high temperature and low electron density.

By coupling SAWs to the system, the SAW-liquid helium interaction is investigated by

measuring the loss of SAW energy into the liquid. We find that the attenuation of the SAW

in bulk liquid helium is dominated by the radiation of compressional wave into the liquid, in a

good agreement with theory and previous experiments. However, for sufficiently thin helium

coverage (several tens of nm), the attenuation anomalously increases beyond that measured

in a thick helium layer. This novel phenomenon is likely associated with the electrostrictive

actuation of the helium surface. Furthermore, this study is a prerequisite for quantitatively

analyzing the SAW measurement results obtained for the 2DES on the liquid helium surface.

In last two chapters, we present a detailed study of the SAW coupling to surface electrons

floating above liquid helium in both the weak and the strong coupling regimes. These

experiments demonstrate find acoustoelectric charge transport and the SAW attenuation

in this system of electrons on helium for the first time. These experiments open the door

to a new class of studies with this system. These include quantized charge pumping, the

possibility of electrical metrology, and ultimately single electron state transfer with electrons

floating on the helium. Also, this coupled system can be utilized to study a commensurate-

incommensurate transition of the 2DES under a time-dependent periodic electric potential

(i.e. a standing piezoelectric SAW). Furthermore, the high-sensitivity of the SAW to the

spatial structure of the 2DES can provide a new method of studying the 2D phase transition

in the electrons on helium system.

Copyright by
HEEJUN BYEON
2020

ACKNOWLEDGMENTS

First of all, I would like to thank my thesis advisor, Johannes Pollanen, who offered a

research opportunity to me five years ago at Michigan State University. He has taught me

all about the low temperature experimental physics through numerous discussions and by

working closely with me in the lab. I deeply appreciate Johannes for sharing his expertise

and deep insight of physics whenever encountering problems. But most importantly, his

positive mindset and enthusiasm for physics always motivates me not to be afraid of failure

and take a step forward from it.

I also would like to thank my thesis committee, especially Mark Dykman for helpful

discussions about electrons on helium early in my Ph.D. Also, his course on quantum fluid

and quantum transport has been one of the most influential classes of my academic career

and greatly helped widen my knowledge about quantum physics.

Many people at Michigan State have helped me complete my Ph.D. work. Among the

faculty, I would like to thank Vladimir Zelevinsky and Scott Bogner for their outstanding

graduate lectures.

I am also grateful to Norman Birge for advice about nano-fabrication

processes. Among my fellow group members, there were Liangji, Justin, Anna, Joshua,

Kostya, Stephen, Niyaz, Trayn, Evan, Brennan, Joe, and Camille. I would especially like to

thank Liangji and Justin for their assistance with SAW device fabrication and measurements.

Also, thanks to Kostya and Niyaz, who have provided constructive and meaningful discussion

about electrons on helium. I am indebted to Reza Loloee for his technical assistance and

keeping the CMP labs running smoothly. Thanks to Baokang Bi for the training and great

advice about fabrication processes.

Lastly, I am particularly indebted to my wife and family for their unfailing support and

v

encouragement during my Ph.D. study at Michigan State.

The projects described in this thesis were partially supported by the National Science

Foundation under grant numbers DMR-1708331 and DMR-2003815.

vi

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

x

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

Chapter 1 Brief Introduction to Electrons on Helium . . . . . . . . . . . .

1

Chapter 2 Electrons on Helium . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 Surface Electron States on Liquid Helium . . . . . . . . . . . . . . . . . . . .
2.1.1 Rydberg Surface State . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.2 External Field Effects
. . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.3 Thin Film Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Wigner Solid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Phase Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2
2D Melting Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Collective Modes of a Two Dimensional Electron System (2DES) . . . . . . .
2.3.1 Plasmons
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 Phonons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.3 Phonon-Ripplon Coupling . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Transport Properties of Electrons on Helium . . . . . . . . . . . . . . . . . .
4He Vapor Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1
2.4.2 Ripplon Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Superfluid Liquid Helium . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.1
. . . . . . . . . . . . . . . . . . . . . . .
2.5.2 Helium Film Formation . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.3 Charged Helium Film and Stability . . . . . . . . . . . . . . . . . . .

Superfluidity of Liquid 4He

Chapter 3

Surface Acoustic Waves (SAWs) . . . . . . . . . . . . . . . . . . .
3.1 Piezoelectric SAWs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 Wave Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.3
SAWs Along [011] Cut of GaAs (100) . . . . . . . . . . . . . . . . . .
3.2 SAW Coupling to a 2DES . . . . . . . . . . . . . . . . . . . . . . . . . . . .
SAW–2DES Interaction Inside a Piezoelectric Substrate . . . . . . . .
SAW–2DES Interaction Above a Piezoelectric Surface . . . . . . . . .
3.3 Acoustoelectric Charge Transport . . . . . . . . . . . . . . . . . . . . . . . .

3.2.1
3.2.2

Chapter 4 Experimental Setup and Devices

. . . . . . . . . . . . . . . . . .
4.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1
Setup Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.2 Cryostat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.3 Experimental Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii

3
4
5
8
10
12
16
19
24
24
25
28
33
35
37
42
42
47
49

53
54
56
61
63
68
69
72
77

81
82
82
83
84

4.1.4 Helium Fill Line and Gas Supply System . . . . . . . . . . . . . . .
Surface Electron Generation . . . . . . . . . . . . . . . . . . . . . . .
4.1.5
4.1.6 Helium Level Sensor
. . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.7 Measurement Principle and Circuit for Transport of Electrons on He-
lium and SAW Experiments . . . . . . . . . . . . . . . . . . . . . . .

96
4.2 Device Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
4.2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
4.2.2 PCB Etching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
4.2.3 Cleanroom Fabrication Processes
. . . . . . . . . . . . . . . . . . . . 121
4.2.4 Assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

87
89
94

Chapter 5

Low-Frequency Transport of Electrons on Helium . . . . . . . . 127
5.1 Experiment and Finite Element Method (FEM) Modeling
. . . . . . . . . . 128
5.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
5.2.1 Electrons on Helium Field Effective Transistor (FET)-IV Characteristics130
5.2.2 Electron Density Estimation . . . . . . . . . . . . . . . . . . . . . . . 131
5.2.3 Observation of Negative Phase of the Electron Layer Current . . . . . 132
5.2.4 Transmission Line Mapping . . . . . . . . . . . . . . . . . . . . . . . 134
5.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

Chapter 6

SAW Coupling to Liquid Helium . . . . . . . . . . . . . . . . . . 138
6.1 Experiment
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
6.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
SAW Attenuation by Bulk Liquid Helium . . . . . . . . . . . . . . . . 141
. . . 143
6.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

6.2.1
6.2.2 Anomalous Attenuation of SAWs by Thin Liquid Helium Films

Chapter 7

SAW Coupling to Electrons on Helium . . . . . . . . . . . . . . 146
7.1 SAW Attenuation by Electrons on Helium . . . . . . . . . . . . . . . . . . . 148
7.1.1 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
7.1.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
7.1.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 155
7.1.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
7.2 Acoustoelectric Transport in Electrons on Helium . . . . . . . . . . . . . . . 163
7.2.1 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
7.2.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 164
7.2.2.1 Frequency Response of the Acoustoelectric Current . . . . . 164
7.2.2.2
. . . . . . . . . . . 166
7.2.2.3 Pulsed measurements of Acoustoelectric Current from Elec-

SAW Power and Gate Bias Dependence

trons on Helium . . . . . . . . . . . . . . . . . . . . . . . . 167
7.2.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

Chapter 8 Conclusions and Future Directions . . . . . . . . . . . . . . . . . 172

APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

viii

APPENDIX A Studies of Electrons on Helium Using a GaAs SAW Device . . . . 176
APPENDIX B Studies of Electrons on Helium Using a LiNbO3 SAW Device . . . 185

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

ix

LIST OF TABLES

Table 2.1: Experimental results for the critical value of the plasma parameter Γ

(c)
pc at
which the liquid-to-solid transition occurs. . . . . . . . . . . . . . . . . . .

Table 4.1: Typical tungsten filament biasing parameters for electron emission using
. . . . . . . . . . . . . . . . . . . . . .

the HP8116A waveform generator.

16

90

Table 4.2: Parts list corresponding to the measurement circuit in Fig. 4.14 . . . . . . 111

Table B.1: Values of the time constants obtained from the exponential curve fits in

Fig. B.3.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

x

LIST OF FIGURES

Figure 2.1: Single electron above the surface of the liquid helium along with an image
charge below the surface. The helium surface supports thermally excited
quantized capillary waves called ripplons, which weakly interact with the
electron.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 2.2: Energy spectrum and the probability for wave functions of an electron
bound to the helium surface. . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 2.3:

(a) Potential energy Ve change at different external fields. (b) Linearly
Stark-shifted transition frequency for 1 → 2 and 1 → 3 Rydberg transitions.

Figure 2.4: Transition frequencies versus voltage across the experimental cell. The
voltage between the top and bottom plates inside the cell is swept by a
triangular wave form applied in a way that maintains the helium surface
at the same positive potential. This process allows for varying the pressing
field acting on the electrons without change in the number of electrons.
The crosses are measured data points while the solid curves are the result
of the variational calculation (from Ref. [1]). . . . . . . . . . . . . . . . .

Figure 2.5: Schematic of electrons on helium film on a dielectric substrate. Major
image charges are represented by + symbols in the dielectric substrate.
The role of the metal depends on the thickness of the dielectric as described
in the text.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 2.6: Phase diagram of a 2D Coulomb liquid (from Ref.

(c)

[2]). Temperature T
and electron density n are expressed in a unit of nc and T c, respectively.
Here nc = 4π−1(aBΓ
(c)
p )2. . . . . . . . . .

p )−2 and kBTc = 2mee4/¯h2(Γ

Figure 2.7: Topological defects in a triangular lattice. Isolated disclinations with (a)
seven folder and (b) five folder symmetry rather than six folder. An
isolated dislocation (c) is composed of a bound pair of such disclinations,
showing undisrupted orientational order at long distance. . . . . . . . . .

Figure 2.8: Computed diffraction pattern of three different phases in 2D: Isotropic
fluid phase (A), hexatic phase (B), and crystalline phase (C). Images are
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
taken from Ref. [3]

xi

3

6

8

10

11

17

21

24

Figure 2.9:

(a) First Brillouin zone for the two-dimensional, hexagonal lattice. (b)
Phonon dispersion curve along the boundary of the first Brillouin zone
(red line in (a)). The frequency ω0 is defined by ω0 = 8e2m∗a3
0, a0 is the
. . . . . . . . . . . . . . . .
lattice constant. Figure taken from Ref. [4]

Figure 2.10: Coupled plasmon-ripplon resonances. The resonance only appear below
0.457 K where the sheet of electrons has crystallized into a triangular lat-
tice (from Ref. [5]). dR/dNs and F denote the derivative of the impedance
of the electron layer with respect to the electron density and the frequency
of the RF signal applied to the underlying electrode used to generate hor-
izontal electron motion. A detailed description of a measurement setup is
presented in the text. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 2.11: Dispersion relation of the longitudinal phonon-ripplon coupled modes (solid
curves). The dashed curve is the bare longitudinal phonon spectrum of
the electron crystal and the horizontal dashed lines represent the bare
ripplon resonance frequencies when ignoring the phonon-ripplon coupling.
The vertical lines are the wave vectors excited in the experiment. The
resonances are labeled as in Ref. [5]. Figure taken from Ref. [6]. . . . . .

Figure 2.12: Mobility of electrons on helium versus the temperature and 4He vapor
density. Experimental data from Grimes and Adams [7], Rybalko et al.
[8], and Iye [9] are shown together with a theoretical analysis from Saitoh
[10]. The dashed line is the mobility considering vapor atom scattering
alone. Figure taken from Ref. [9]. . . . . . . . . . . . . . . . . . . . . . .

Figure 2.13: Collision time τ as a function of temperature for various holding electric
. . . . . . . . . . . . . . . . . . . . . . . . . .

fields Fz (from Ref. [10]).

Figure 2.14: Phase diagram of 4He (from Ref. [11]). . . . . . . . . . . . . . . . . . . .

Figure 2.15: Specific heat of liquid 4He as a function of temperature at saturated vapor
pressure. The specific heat has a sharp peak as T → Tλ. Figure taken
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
from Ref. [12].

Figure 2.16: Helium film formation: (a) Van der Waals film on a vertical wall of the
substrate. µ0 and µf are the chemical potential of the bulk helium and the
thin film of helium located at height h above the bulk level. (b) Helium
film formation inside the channel of the micro-channel array by capillary
force. Here w, d0, d, and h represent channel with, channel depth, helium
film thickness in the channel, and the distance between the channel top
and the bulk helium level in the reservoir.
. . . . . . . . . . . . . . . . .

26

30

34

36

41

43

44

48

xii

Figure 2.17: Thickness d of charged saturated 4He films wetting a glass substrate
(T = 1.6 K) versus the electron density of the films. The thickness of d0
of the uncharged films was 220 ˚A and 420 ˚A, respectively. Figure taken
from Ref. [13].
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 3.1: Schematic of the SAW-2DES on helium experiment. A SAW is launched
from the left interdigitated transducer (IDT) and can, via an evanescent
coupling to the electrons, transport electrons in the direction of the trav-
elling wave. A second IDT to the right can act as a SAW detector or
acoustic Bragg reflector.
. . . . . . . . . . . . . . . . . . . . . . . . . . .

52

53

Figure 3.2: Coordinate system for a piezoelectric SAW.

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

58

Figure 3.3: Schematic of the SAW propagation along GaAs (100) plane. The direction
of the propagation is [011]. . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 3.4: The real and imaginary components of the determinant of B as a func-
tion of the velocity c for GaAs (100) cut [011] propagation direction. At
c (cid:39) 2880 m/s, both components are equal to zero. . . . . . . . . . . . . .

Figure 3.5: Atomic displacements (a) and electric potential (b) of a Rayleigh SAW
versus the distance from the surface of GaAs (100) cut [011] propagation
direction. u1(cid:48) and u3 denote longitudinal and transverse components of
the displacements, respectively. A positive value of z indicates a location
inside the GaAs substrate while a negative value represents the vacuum
. . . . . . . . . . . . . . . . . . . . . . . . . . .
above the GaAs surface.

Figure 3.6: SAW attenuation and the velocity shift as a function of the 2DES con-
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ductivity.

Figure 3.7: Schematic of the SAW propagation along the (100) surface of AlGaAs
containing a 2DES. For the calculation of K2/2 and σm in Ref. [14] at a
depth d, the bottom GaAs layer is approximated by a AlGaAs layer due
to their similar material properties.
. . . . . . . . . . . . . . . . . . . . .

Figure 3.8: Characteristic conductivity σm (a) and coupling constant K2/2 (b) as a
function of kd are calculated using the approximated model geometry of
Fig. 3.7. Here k and d represent the in-plane wave vector and the depth
of the 2DES from the piezoelectric surface, respectively. The inset of
(a) shows a change in effective dielectric constant with increasing 2DES
distance from the surface. The blue trace represents the position of 2DES
in vacuum above the surface while the red trace stands for the 2DES
. . . . . . . . . . . . . . . . . .
position inside the piezoelectric medium.

64

66

67

69

70

72

xiii

Figure 4.1: A superfluid leak-tight sample cell connected to the 1K flange of a closed
cycle cryostat. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81

Figure 4.2: Schematic showing operation of the closed-cycle 1K cryostat. The 1K
cryostat is precooled by a 4K pulse tube cryocooler (Cryomech PT410-
RM) providing 1.0 W cooling power at T = 4.2 K. In the 1K circulation
loop (blue color), the inlet 4He gas at the pressure of ∼1 atm is precooled
by a heat exchanger on the 50K stage and then condenses at the 4K
stage. The condensed helium flows through a counter-flow heat exchanger
wrapped around the tube below the 4K stage to be cooled ∼2 K. It then
passes through a needle-valve into the 1K pot. To produce evaporation,
the 1K pot is pumped by a dry vacuum pump through a pumping line. A
buffer volume installed at the pump discharge stabilizes the pressure. If
additional helium gas needs to be added, a standard helium gas cylinder
is connected to the discharge side of the pump. The base temperature of
the 1K system strongly depends on a flow through the needle valve and
the pumping rate as well as any additional heat load from measurement
wires and the helium fill line inside the fridge.
. . . . . . . . . . . . . . .

Figure 4.3: A simplified illustration of a closed cycle 1K refrigerator in which a helium
. . . . . . . . . . .

fill line, wires, and a sample cell are shown in detail.

Figure 4.4:

(a) Hermetically sealed Cu sample cell. (b) bottom and (c) top part of
the cell. (d) 3D drawing of the cell assembly process. A SAW chip and
a helium interdigitated capacitor (IDC) level sensor mounted to the PCB
are connected to GPO launchers via gold wire bondings. GPO bullets are
then inserted on the GPO launcher to link them to the hermetic GPO
feedthroughs. An indium wire oring is placed onto the cell bottom and
the top part of the cell is evenly pressed against the bottom via tightening
screws. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 4.5:

(b) Schematic of the 4He gas
(a) Helium fill line in the 1K cyrostat.
handling system (GHS). Once the unit volume is filled with 4He gas, the
valve behind it is closed and then helium gas is provided into the cell from
only this volume.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 4.6: Simplified illustration of the process of charging a liquid helium surface
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

with electrons.

Figure 4.7:

(a) Sketch of electrons floating on the surface of liquid helium inside the
sample cell and (b) the simple capacitance model for this geometry.
. . .

85

86

87

88

91

91

xiv

Figure 4.8: Schematic of Sommer-Tanner electrode configuration. A set of three pla-
nar electrodes with typical width w = 8 − 10 mm and length l = 3 − 5 mm
are submerged underneath the charged liquid helium surface. The 2DES is
excited by applying an AC voltage of Vex ∼100 mV to the drive electrode,
which induces a current through the electron layer on the sense electrode.
This current is measured using standard lock-in techniques. In addition
to Vex, a DC voltage of VDC = 30 − 60 V is simultaneously applied to the
underlying electrodes to vertically hold surface electrons on helium. A
bias tee composed of a resistance of R = 10 MΩ and a capacitance of
C = 1 µF, marked by the red-dashed square, is used to apply both a DC
an AC voltage.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 4.9:

(a) Schematic of the IDC helium level sensor inside the cell partially filled
with liquid helium. Inset shows an actual image of the IDC sensor verti-
cally mounted on a copper block attached to the PCB. (b) IDC capacitance
measurement as a function of time while helium is continuously provided
into the cell from a helium gas cylinder at room temperature.
. . . . . .

Figure 4.10: Lumped element equivalent circuit model of the Sommer-Tanner mea-
surement configuration for (a) a neutral and (b) charged surface of liquid
helium.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93

95

99

Figure 4.11: (a) Transmission line model of the Sommer-Tanner geometry. (b) Empiri-
cal circuit model [15] of the lock-in input impedance ZLI, which consists of
cable capacitance Ccab (cid:39) 600 pF, a low-pass current divider stage with a
capacitance Cdiv = 31 pF and resistance Rdiv = 6.5 kΩ, and the nominal
lock-in input impedance of Rin = 10 MΩ in parallel with the capacitance
Cin = 25 pF. (c) A simplified equivalent circuit of (b). Both Z1 and Z2 are
composed of two elements in parallel, given by Z1 = [jω(Ccab + Cdiv)]−1
and Z2 = [jωCin + 1/Rin]−1.

. . . . . . . . . . . . . . . . . . . . . . . . 103

Figure 4.12: Simplified PLL circuit for SAW velocity shift measurements.

. . . . . . . 106

Figure 4.13: Convergence of the SAW PLL. The blue trace is the amplified and filtered
output signal from the mixer Vout = βcos(φ) where φ = 2πxf /v, shown in
Fig. 4.12. The red line represents the dynamic VCO frequency f = fbase +
(df /dV )Vout. Initially before the PLL loop is closed (Vout is connected to
the VCO control frequency input), Vout has a value of Vout = 2πxfbase/v0
at the frequency of f = fbase. In the closed PLL loop, Vout moves along
the dashed dot trajectory and Vout eventually converges at the equilibrium
point marked by the solid circle. . . . . . . . . . . . . . . . . . . . . . . . 109

Figure 4.14: SAW ∆v/v measurement circuit diagram.

. . . . . . . . . . . . . . . . . 110

xv

Figure 4.15: A sketch of the S-matrix elements measurement of a SAW delay line using

a 2-port network analyzer. . . . . . . . . . . . . . . . . . . . . . . . . . . 112

Figure 4.16: SAW attenuation measurement circuit diagram.

. . . . . . . . . . . . . . 115

Figure 4.17: (a) Simplified circuit diagram of the RF power detector (b) Sketch of
voltage waveforms at different positions on the circuit marked by the green
solid circles in Fig. 4.16.

. . . . . . . . . . . . . . . . . . . . . . . . . . . 116

Figure 4.18: Calibration measurement of the output voltage Vout of the boxcar in-
tegrator at variable input power level Pin to the RF power detector.
RF pulses with a width of 20 µs are directly applied to the input of
the power detector and the input power level is quantified by measur-
ing the RMS voltage within the pulse duration, based on the formula

(cid:1) where Vrms,0 = 224 mV. Experimen-

Pin (dBm) = 20log(cid:0)Vrms/Vrms,0

tal data shows Pin is proportional to Vout, which well fits a function of
Pin = a0 (Vout)a1 + a2. Here a0 = 28.59 ± 1.02, a1 = 0.12, and a2 = −55.49 ± 1.13.
Pin is estimated from this function and the measured value of Vout. The
shape of the curve is strongly affected by a different DC offset of the
boxcar integrator and thus must be kept the same between different mea-
surements by adjusting the offset level properly.

. . . . . . . . . . . . . . 118

Figure 4.19: Electrons on helium SAW experiment fabrication. (a) Assembly process of
main part of the cell PCB. A set of three rectangular electrodes (trapping
electrodes) with a width of 8 − 10 mm, a total length of 9 − 15 mm, and
a thickness of ∼35 µm are patterned on a 2 inch diameter and 1.5 mm
thick PCB. A rectangular guard electrode made of a few mm thick Cu
wire is attached around the three planar electrodes for lateral electron
confinement. An interdigitated capacitor (IDC) is fabricated from ∼5 µm
wide and ∼100 nm thick aluminum patterned on a 3.5 mm wide × 8 mm
long × 0.5 mm thick quartz substrate and then vertically mounted to the
PCB to measure the bulk liquid helium level. A SAW chip containing two
identical interdigitated transducers (IDTs) for generation and detection
of the SAW is lastly mounted on the trapping electrodes. (b) Picture of
the electrons on helium LiNbO3 device after assembly.
(c),(d) Optical
microscope image of the IDC helium level sensor made of ∼100 nm thick
aluminium on the quartz substrate. The metal line width and spacing are
equal (∼5 µm each) as shown in the magnified image ((d)). (e),(f) Optical
microscope image of the SAW chip. The zoom-in image shows an inter-
digitated transducer (IDT) with 3/3 µm metal line/spacing, which creates
a SAW having a wavelength of λsaw = 24 µm when the RF excitation is
applied on resonance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

xvi

Figure 4.20: (a) Picture of the custom printed transparent sheet used as the mask for
the PCB patterning process. Two copies are placed on top of each other
with good alignment so that the dark area effectively blocks light during
the exposure process. (b) Actual image of a patterned PCB board after
completing photolithography (1. trapping electrodes, 2. guard electrode
pad, and 3. GPO launcher pad).

. . . . . . . . . . . . . . . . . . . . . . 121

Figure 4.21: Semiconductor lift-off process for sample fabrication. (a) Photoresist (PR)
is coated on the substrate using a spin-coater. (b) The PR is patterned
with UV exposure followed by a developing process.
(c) A thin metal
film is deposited onto the patterned PR via thermal evaporation. (d) The
PR/metal bi-layer is washed away in acetone or PR stripper, leading to a
patterned metal film on the substrate.

. . . . . . . . . . . . . . . . . . . 122

Figure 5.1: Schematic of the experimental setup and circuit modeling of electrons on
helium. (a) 3D CAD rendering of the experimental cell with the array
of FET electrodes and a liquid helium level sensor.
(b) Sketch of the
macroscopic FET device composed of electrons on liquid helium. The
dimensions of each electrode are L × W = 5 × 10 mm and the gap between
them is 0.2 mm. (c) The equivalent lumped RC circuit of the FET device.
Re is the resistance of the electron layer above gate electrode and Ce is
the capacitance of electron system to the source and drain electrodes. (d)
Transmission line equivalent circuit model of the electron system on liquid
helium. In this transmission line mapping, the resistance of the electron
layer and its capacitance to the FET electrodes is spatially distributed.
Figure 5.2: Amplitude of the source-to-drain current, |I|, as a function of gate voltage
at T = 1.35 K for the electrons on helium FET device. For these measure-
ments Vs = Vd = 40 V and the guard electrode was biased with −50 V.
The insets illustrate the electron charge profile over the FET electrodes
obtained by finite element simulation of the device. A uniform electron
density is achieved when the source, drain and gate electrodes have equal
value (dashed vertical line).

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

. 129

Figure 5.3: Real (a) and imaginary (b) components of the FET source-drain cur-
rent as a function of gate voltage measured at T = 1.35 K (blue curve),
T = 1.6 K (green curve), and T = 1.95 K (red curve) at f = 60 kHz. For
these measurements Vs = Vd = 40 V and the guard electrode was biased
with −50 V. The negative values of Im(I), indicated by the vertical ar-
rows and shaded regions, represent a unique departure from the lumped
RC circuit model for an FET made of electrons on helium. . . . . . . . . 133

xvii

Figure 5.4: Frequency dependence of the real Re(I) (a) and imaginary Im(I) (b) com-
ponents of the source-drain current with a uniform electron density above
all three FET electrodes corresponding to Vs = Vd = Vg = 40 V. For
these measurements the guard electrode was biased with −5 V. The solid
traces show the experimental data obtained at T = 1.35 K (blue) and
T = 1.95 K (red). The correspondingly colored dashed lines are calcu-
lated using the transmission line model described in the text.

. . . . . . 136

Figure 6.1: Pulsed surface acoustic wave experimental setup. (a) Schematic of the
lithium niobate chip used for exciting and detecting SAW via aluminum
interdigitated transducers (IDT), which were directly fabricated onto the
surface of the substrate. (b) Block diagram of the circuit used for mea-
suring the attenuation of pulsed SAWs. As described in the text, by
measuring the time-of-flight signal we are able to disentangle direct ca-
pacitive crosstalk between the two interdigitated transducers as well as
multiply reflected SAW pulses.

. . . . . . . . . . . . . . . . . . . . . . . 139

Figure 6.2: Frequency dependence of the transmission coefficient (S12) of the SAW
delay line at T = 1.55 K in vacuum. The resonant peak at ν = 296 MHz
is associated with the generation of surface acoustic waves in the IDT
delay line.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

Figure 6.3: Attenuation, α, of Rayleigh waves on LiNbO3 in contact with bulk liquid
helium. Our measurements (solid red curves) at ν = 296 MHz are in good
agreement with the theory (dashed red curve) of Dransfeld and Salzmann
[22] for energy loss due to radiation of longitudinal compressional waves
into the liquid (Inset). The local minimum of the SAW attenuation at
Tλ = 2.17 K is associated with the transition from normal to superfluid
4He.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

Figure 6.4: Attenuation, α, of Rayleigh waves on LiNbO3 in contact with thin films
of liquid helium at T= 1.55 K and ν = 296 MHz. The measurements
are made by incrementally admitting small amounts of helium gas into
the cell from a standard volume at room temperature (lower horizontal
axis). The vertical dashed line indicates the estimated thickness of the
helium film on the LiNbO3 surface based on equation 6.1 and the height
H from the substrate to the bulk helium reservoir level in the cell (top
horizontal axis). For comparison, the horizontal solid red line indicates
the SAW attenuation due to bulk liquid helium at the same temperature.
The calculated thicknesses of the helium film for the final two data points
above the bulk value are 73 nm and 76 nm. Finally, we note that the IDT
fingers are expected to fill completely with helium via capillary action
for very small amounts of helium introduced into the cell, i.e. already at
H = 0.46 mm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

xviii

Figure 7.1: Experimental setup for measuring the attenuation of SAWs by electrons
on helium.
(a),(b) Top and cross-section view of the SAW device for
studying electrons on helium. Electrons on the helium films formed in
microchannel arrays are held in place by applying a positive DC bias to

three undelying electrodes beneath the GaAs substrate at T ∼= 1.55 K.

Two pairs of interdigitated transducers (IDTs) are located around the mi-
crochannel structure to excite and receive SAW signal. (c) Block diagram
of the circuit used for measuring SAW attenuation. The exciter IDT1 and
the detector IDT1 are operated so that pulsed SAWs can interact with
2D electron sheet above the gate electrode. In this region of the sample,
the electron density can be varied from zero to its maximum value.

. . . 149

Figure 7.2: Approximate model geometry for SAWs experiments. SAWs propagate
along the positive x-axis and the positive z-axis points out of the piezo-
electric plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

Figure 7.3: Attenuation of Rayleigh SAWs in contact with liquid helium. These

measurements are done at T ∼= 1.55 K by incrementally providing small

amounts of helium gas into the sample cell. The volume of the liquid
helium in the cell was determined from the pressure in a standard volume
of 260 cc at room temperature. The corresponding bulk helium level is
estimated from 3D CAD modeling of the liquid helium filling the cell,
which determines the distance between the GaAs substrate and the level
of the helium reservoir, H. The SAW attenuation shown is the relative
value with respect to empty cell. The vertical dashed line indicates when
the reservoir helium level meets the substrate surface. Negative values of
H represent a helium level that exceeds the piezoelectric surface (GaAs).

156

Figure 7.4:

(a) Electron density ne estimation in the 2D electron region above the
gate electrode during a sweep of Vg at Vs = Vd = 60 V. Based on this
ne estimation, we have determined the thickness of the charged helium
film in the channel (b) and the photoresist area (c) with H ∼= 207 µm.
Here dch and dvan denote the liquid helium thickness within the channels
and on top of the photoresist while H is the distance between the GaAs
surface and the level of helium in a reservoir adjacent to the substrate.

. 157

xix

Figure 7.5: SAW attenuation during a gate voltage sweep. Blue closed circles show rel-
ative measurements of the attenuation during a Vg sweep at Vs = Vd = 60 V
in which the electron density ne is linearly proportional to Vg as seen in
Fig. 7.4(a). The blue line shows the theoretically calculated average of
the attenuation in the channel and photoresist regions. This theory only
includes the piezoelectric coupling to electrons floating on helium film.
After removing electrons, no noticeable change in SAW signal was ob-

served (blue open circles). With H ∼= 207 µm, the initial helium film

∼= 1.18 µm and dvan ∼= 80 nm.

thickness with no electrons in the microchannel array is estimated to be
In the presence of electrons, the Vg
dch
dependence of the film thickness is given in Fig. 7.4 (b) and (c).

. . . . 158

Figure 7.6: SAW attenuation versus helium level in the cell. Liquid helium was slowly
and incrementally added to the charged helium film by supplying helium

gas into the cell at T ∼= 1.55 K. This film thickening raised the 2D elec-

tron layer away from the piezoelectric surface (GaAs) and reduced the
piezoelectric coupling. The black closed circles and dashed line show the
measured values and the calculation of the maximum attenuation αm at
different H. Here H < 0 means the helium level is higher than the GaAs
surface. The left inset shows the measurements of the attenuation versus
Vg at various H while the right inset shows the calculation of maximum
attenuation αm as a function of the distance d from the 2DES to the
piezoelectric surface. Note that d is equivalent to |H| for bulk region
(H < 0). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

xx

Figure 7.7: Schematic of the experimental setup and demonstration of acoustoelectric
transport of electrons on helium. (a) Top and (b) cross-section views of
the device for measuring SAW-driven transport of electrons on helium.
Two opposing interdigitated transducers (IDTs) are used to excite and
receive SAWs. A saturated superfluid 4He film is formed on the surface
of the LiNbO3 piezo-substrate at T = 1.55 K. Thermionically emitted
electrons are trapped above the surface of the superfluid film by applying
a positive bias voltages to three underlying electrodes arranged in a field-
effect transistor (FET) configuration with a source (s), gate (g) and drain
(d) [16]. Lateral confinement of the electron layer is achieved with a
negative bias to guard electrode positioned on the outside of the LiNbO3
substrate. (c) Frequency dependence of the transmission coefficient (S12)
of the SAW device demonstrating an expected resonance at 296 MHz.
(d) Measured acoustoelectric current Iae of electrons on helium driven
by a piezo-SAW as a function of frequency. For these measurements the
FET electrode voltages were Vs = Vg = Vd = 40 V, corresponding to an
electron density of n ∼= 0.8 × 109 cm-2, and the guard was biased with
−3.2 V. Inset: Inverse Fourier transform of the acoustoelectric current
signal which reveals a peak at t ≈ 0.9 µs. This time scale corresponds
to a SAW propagation distance of 3.2 mm, roughly the same as twice
the distance between the launching IDT center and the near-edge of the
LiNbO3 substrate, which implicates SAW reflections as responsible for the
superimposed oscillations present on the acoustoelectric current peak. . . 165

Figure 7.8: Power dependence and gate-tunability of the acoustoelectric effect with
electrons on helium. Acoustoelectric current Iae measured as a function
of the SAW driving frequency and power for electrons floating on a helium
thin film on a (a) LiNbO3 and a (b) GaAs piezo-substrate. Each inset
shows that Iae is linear in the driving power when the frequency corre-
sponds to the SAW resonance. (c) Demonstration of an acoustoelectric
field effect transducer (aFET) with electrons on helium on the LiNbO3
substrate. The inset shows line-cuts of Iae both on- and off-resonance
with the SAW. The gate voltage (Vg) sweep is performed at fixed source
and drain voltages (Vs = Vd = 40 V). . . . . . . . . . . . . . . . . . . . . 166

xxi

Figure 7.9: Pulsed acoustoelectric measurements of electrons on helium. (a) Time of
flight measurement of Iae (red curve) using a gated pulse (gray line) of
SAWs at a fixed RF power (0 dBm) and at the SAW resonance frequency
(296 MHz).
In these measurements the number of electrons above the

detection electrode is N ∼= 0.4 × 109. The blue trace is the calculated
change in the equilibrium number of electrons ∆N , which is obtained
from the time integral of Iae (red trace). (b) ∆N and (c) |∆Nsat| for a
In
steady state SAW pulse at various values of the driving RF power.
these measurements the equilibrium number of electrons is N ∼= 0.8× 109.
pulses at different RF power with N ∼= 0.8×109. The time constants τpump
(τrel) are those associated with the rising (falling) edge of the SAWs pulse
and are determined from fitting Iae(t). The time constant τRC (dashed
line) is that determined from from low frequency transport of electrons on
helium.

(d) Dynamical response of two-dimensional electrons on helium to SAWs

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

Figure A.1: The experimental sample cell and helium level estimation. (a) Pictures
of the copper cell used in our experiments. The cell contains a GaAs
device (shiny black) located inside the guard electrode on the printed
circuit board (PCB). Beneath this substrate, three rectangular electrodes
are patterned on the PCB, which are not shown in the picture. (b),(c)
He level estimation as a function of the liquid helium volume using 3D
CAD simulation. The gray volume represents the liquid helium in the
cell. (d) Level sensor capacitance change while condensing liquid helium
into the sample cell. This was measured using the interdigitated capacitor
(IDC) sensor as shown in the inset of the plot. A linear increase of the

capacitance starts happening when the liquid volume ∼= 0.32 cc, which

indicates that the liquid helium reaches the bottom of the IDC.. . . . . . 180

Figure A.2: (a) Transmission line (TL) circuit modeling of electrons on helium (b) Cur-
rent of electrons on a helium film as a function of Vg inthe low-frequency
regime. The current was induced by applying a 0.1 mV, 60 kHz excitation
Vex and detected using standard lock-in techniques. The source and drain
voltage were fixed at 60V. (c) Output voltage Vo as a function of the fre-
quency of Vex. The output voltage Vo corresponds to the voltage drop at
the internal impedance of the signal detector (Lock-in amplifier) due to
current flow in 2D electrons floating helium film. The blue and red solid
traces represent experimental values with and without SAW excitation
while the dashed trace is the theoretical curve. The frequency response
is independent of SAW excitation, which implies that the low frequency
transport is not affected by the SAW. For these measurement, all three
electrodes were biased with 60 V.

. . . . . . . . . . . . . . . . . . . . . . 182

xxii

Figure A.3: Characteristic electron density nm as a function of the mobility transverse
to the channel µt at different H in the helium film (a) and bulk (b) regimes.
The parameter H denotes the distance between the GaAs substrate and
the level of the helium reservoir right next to the substrate. When H > 0,
the level of the helium reservoir is below the surface of the GaAs while
the helium level exceeds the GaAs surface when H < 0 .

. . . . . . . . . 184

Figure B.1: The experimental sample cell and helium film thickness determination.
(a) Photograph of the copper cell used for the SAW-based measurements
of electrons on helium. The bottom portion of the cell contains a LiNbO3
device (semi-transparent white) mounted on a printed circuit board (b)
3D CAD modeling of the liquid helium volume inside the experimental cell
during filling with helium. (c) Calculation of H, the distance between the
piezo-substrate top and the liquid helium level in the cell reservoir volume
from the 3D CAD modeling of the open cell volume. The calculated value
of H in the experiments is 0.2 mm for a helium vol. = 0.44 cc (vertical
dashed black line). (d) Helium film thickness versus H before charging the
film (see equation B.1). For an uncharged helium film d0 ≈ 77 nm (vertical
dashed red line). (e) Charged helium film thickness versus electron density
from equation B.2. The charged helium film thickness is calculated to be

d ∼= 72 nm for an electron density of n ∼= 0.8 × 109 cm-2 (vertical dashed

blue line).

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

Figure B.2: Experimental setup, circuit model, and measurement data for low fre-
quency transport of electrons on helium. (a) Sketch of the electrons on
helium device. A DC bias voltage is applied to the three underlying elec-
trodes to trap electrons above helium surface. An AC excitation voltage
(Vex = 0.1 V) applied to the source electrode induces a current through
electrons on helium, which is capacitively detected using the drain elec-
trode. (b) The equivalent transmission line circuit model for this device.
The resistance of the electron sheet and the capacitance between this sheet
and the three electrodes are spatially distributed per unit length, r∆L and
cl∆L, where r and cl represent the resistance per unit length and the ca-
pacitance per unit length. (c) Current-voltage characteristics for FET op-
eration of the electrons on helium device. For this representative data, the
source-drain current amplitude was measured during a gate voltage sweep
with Vs = Vd = 80 V. The insets show the corresponding electron density
profiles over the electrodes for different values of Vg. A uniform areal den-
sity distribution is achieved at Vg = Vs = Vd. (d) Frequency dependence
of the transport characteristics of the system of electrons on helium. The
red and blue traces represent the experimental data and the calculation
based on the transmission line model respectively. For these measurement,
all three electrodes were biased with 80 V (Vg = Vs = Vd = 80 V). . . . . 189

xxiii

Figure B.3: Determination of acoustoelectric time constants. These time constants
were determined from exponential fits to the acoustoelectric current data
during (a) SAW excitation (charge pumping) and (b) relaxation after the
SAW drive was removed. . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

xxiv

Chapter 1

Brief Introduction to Electrons on

Helium

Electrons on helium are a unique two-dimensional electron system (2DES), where electrons

are able to float several nanometers above the liquid helium surface due to Pauli repulsion

at short ranges and attraction at long range caused by induced positive image charges inside

the helium. Because of the strong Coulomb interaction between electrons, the system of

electrons floating on helium is an ideal model for experimental realization of an electronic

crystal phase of strongly correlated electrons predicted originally by Wigner in 1934 [5, 6, 17].

In the liquid phase, electrons on helium exhibit the features of the strongly correlated non-

degenerate liquid with unusual transport properties at low frequency [18, 19, 20, 21]. The

absence of defect in this system leads to the highest mobility in all of condensed matter

(µ > 108 cm2/Vs) [22], which is several orders of magnitude larger than two-dimensional

electron systems (2DESs) in semiconductor materials (µSi, µGaAs < 106 cm2/Vs). In ad-

dition, the ultra clean physical environment in which electrons reside in vacuum above the

defect-less liquid helium substrate, can exclude many potential source of interaction with the

electrons leading to slow electron decoherence. Thus, as a candidate for quantum information

processing, this system has been actively investigated in recent years [23, 24, 25, 26, 27, 28].

A surface acoustic wave (SAW) on a peizoelectric substrates is accompanied by dynamic

1

electric fields and an electric potential wave, which can interact with a 2D electron system

located within approximately one wavelength above or below the piezo-substrate surface.

Such an interaction enables SAWs to be versatile tools for exploring wavelength and fre-

quency dependent electronic properties of 2DESs. In semiconductor 2DESs, surface acoustic

wave techniques have been successful for studying the integer [29, 30, 31] and fractional [32]

quantum Hall regimes, demonstrating the gate-controlled acoustoelectric transport [33, 34],

as well as for investigating the 2D metal-to-insulator transition [35].

In particular, short

wavelengths of high-frequency SAWs ranging from tens of nanometer to several micrometer

allow for the study of commensurability effects when the SAW wavelength becomes compa-

rable to some length scale within the 2DES [36, 37]. However, SAW methods has not been

previously employed to investigate electrons on helium. The extension of SAW techniques

to electrons on helium would allow for the study not only of the wave-vector and frequency

dependent conductivity in a high-frequency regime but also collective dynamics and resonant

response of the strongly correlated quantum system to high-frequency SAW excitation. In

addition, the high-sensitivity of SAW to the spatial structure of 2DESs can be utilized to

develop powerful tools for studying 2D phase transition driven by unbinding of topological

defects. Furthermore, acoustoelectric charge transport driven by a travelling SAW potential

has recently attracted particular attention as a means of controlling and transporting sin-

gle electrons. Employing micro-/nano-structures (quasi 1D channels and 0D quantum dots)

and the lack of disorder can make the system of electrons on helium an excellent platform

for realizing SAW-driven single-electron devices, or potentially utilized as a novel method

to trap single electrons for quantum information science or quantum metrology application

[38].

2

Chapter 2

Electrons on Helium

Figure 2.1: Single electron above the surface of the liquid helium along with an image charge
below the surface. The helium surface supports thermally excited quantized capillary waves
called ripplons, which weakly interact with the electron.

A two-dimensional electron system (2DES) floating above the surface of superfluid he-

lium was first theoretically proposed by Cole and Cohen [39] and by Shikin [40] and first

achieved experimentally Williams et al. [41]. Since then, it has attracted great interest in

condensed matter physics due to a rich variety of physics including Wigner crystallization

[5], 2D plasmons [7], and unique polaronic effects [42], and the formation of a strongly cor-

related nondegenerate 2D electron fluid [18, 43]. The system is also interesting in Quantum

Information Science (QIS) with the possibility of long-lived qubits with electrons on he-

lium, arising from ultra-clean environment. In this chapter, the basic properties of surface

electrons formed on a liquid helium substrate are introduced and discussed.

3

2.1 Surface Electron States on Liquid Helium

An electron in the vicinity of the surface of liquid 4He is attracted to the surface by the

electrostatic image force arising from the polarization of the dielectric liquid as shown in

Fig. 2.1. As surface electrons (SEs) are far distant from the helium surface, the image

potential energy for z > 0 can be written as [44]

with

Vim(z) = − Λe2
z + z0

≈ −Λe2
z


 − 1
4(
 + 1)

,

Λ =

(2.1)

(2.2)

where 
 is the dielectric constant of liquid 4He, e is the elementary charge, and z0 is an
offset parameter to avoid the singularity at the interface with a typical value of z0 (cid:39) 1˚A.

In inert gases such as helium, the last electronic shell of the atoms is completely filled with

no available states for additional electrons. This means liquids made from the noble gases

usually have a strong repulsive barrier to electron injection. For 4He, the height of this barrier
is approximately Vb ∼ 1 eV [45, 46]. Since z0 is much smaller than average distance (cid:104)z(cid:105) of

the electrons from the liquid surface and the barrier energy largely exceeds the attractive

binding energy (< 1 meV), the total potential energy Ve can be simplified as follows

Ve(z) = −Λe2

z

(z > 0) or + ∞ (z ≤ 0).

(2.3)

Therefore, at low temperature, the electrons are confined in a potential well produced by

Ve with quantized motion normal to the surface (Fig. 2.2). Parallel to the surface, electrons

4

can be considered as nearly free particles.

2.1.1 Rydberg Surface State

The wave function of the electrons above the surface of liquid helium can be separated

into a plane wave representing in-plane motion and a one-dimensional (1D) quantized wave

function χn for vertical motion:

Ψn((cid:126)r, z) =

ei(cid:126)k·(cid:126)rχn(z),

1√
SA

(2.4)

where (cid:126)k is 2D in-plane wave vector, (cid:126)r is the position vector in the plane, n is the quantum

number describing vertical motion of the electron, and SA is the surface area containing

electrons. Assuming perfectly flat helium surface, the motion of SEs perpendicular to the

liquid resembles a one-dimensional hydrogen atom with a reduced nuclear charge Λe. Thus,

the energy En of the vertical motional eigenstate is given by [40]

En = −Λ2mee4
2¯h2n2

= −R∗

y

n2 , n = 1, 2, 3, ...

(2.5)

y is the effective Rydberg energy and me is electron mass. For liquid 4He, R∗

where R∗
y is
estimated to be ∼ 0.658 meV from 
 = 1.05723 and Λ (cid:39) 6.95× 10−3. The energy eigenstates
of this motion in the position basis χn(z) = (cid:104)z|n(cid:105) are determined by solving one-dimensional

Schr¨odinger equation:

χn = Enχn.

(2.6)

(cid:34)

(cid:35)

− ¯h2
2me

∂2

∂z2 − Λe2

z

5

The wave functions of two lowest states χ1(z) and χ2(z) are

2

ze−z/a,

z(1 − z/2a)e−z/2a,

χ1(z) =

χ2(z) =

a3/2
1√
2a3/2

where a is an effective Bohr radius of the electron state defined by

a =

1
Λ

¯h2
mee2 =

a0
Λ

(2.7)

(2.8)

(2.9)

with a0 = ¯h2/mee2 (cid:39) 0.53 ˚A. The effective Bohr radius is about 76 ˚A for liquid 4He.

Fig. 2.2 shows the lowest energy levels En and the probability, of vertical motional wave
function |χn|2 alongside the potential energy Ve. The average distance of the electron from

Figure 2.2: Energy spectrum and the probability for wave functions of an electron bound to
the helium surface.

the surface of liquid helium (cid:104)z(cid:105) ∼ a0 (cid:39) 76˚A is much larger than the spacing of the helium
atoms r0 ∼ 2a0 (cid:39) 1 ˚A, which leads to negligible scattering from the corrugation of the

6

surface due to individual atoms. Thus, electrons staying relatively far away from the surface

are able to move like free particles. The average distance from the surface for three lowest

electron energy levels is given by

(cid:104)1|z|1(cid:105) (cid:39) 141 ˚A,

(cid:104)2|z|2(cid:105) (cid:39) 456 ˚A,

(cid:104)3|z|3(cid:105) (cid:39) 1027 ˚A.

(2.10)

The ground state energy is E1 = −0.66 meV (-7.63 K) and the first and the second excited
state energies are E2 = −0.16 meV (-1.91 K) and E3 = −0.07 meV (−0.85 K). The energy

required for a transition of the electron from the ground state to the first excited state
is E2 − E1 = 5.72 K, which is significantly larger than the working temperature of our
experiments T ∼ 1 K, meaning that most of electrons are in the lowest Rydberg energy

level E1. The transition frequency between these states νn→m can be estimated from the

hydrogen atom-like spectral series:

νn→m =

Em − En

h

R∗
y
h

=

(cid:16)

1/n2 − 1/m2(cid:17)

.

(2.11)

This yields ν1→2 (cid:39) 119.3 GHz and ν1→3 (cid:39) 141.3 GHz, which is in good agreement with
spectroscopic measurement data ν1→2 (cid:39) 125.9 ± 0.2 GHz and ν1→3 (cid:39) 148.6 ± 0.3 GHz [1].

Note that these estimations were made based on the infinite potential barrier approximation

at z = 0 (equation 2.3). In reality, the potential barrier at the surface has a finite energy
Vb ∼ 1 eV and a not completely flat helium surface leads to a small disturbance in the
barrier position at z = 0. Thus, they are responsible for a small deviation of ∼5 % between

theory and experiment for the energy levels [47].

7

2.1.2 External Field Effects

In most experiments, an external electric field E⊥ normal to the electron layer is typically

applied, which modifies the asymptotic form of the potential Ve:

Ve = Vim + Vb + eE⊥z.

(2.12)

The barrier Vb can still be approximated by an infinite barrier Vb = ∞ for z ≤ 0 and
Vb = 0 for z > 0 while Vim(z) = −Λe2/z for z > 0 and Vim(z) = 0 for z ≤ 0 as described
in the previous section. A positive field E⊥ produces a linearly increasing potential Ve for

large z leading to a more stable bound state of electrons by pressing them into the surface

more strongly. For the case of a negative pressing field (E⊥ < 0), the trapping potential

has a triangular shape with increasing z allowing electrons to escape the surface via the

tunneling as shown in Fig. 2.3 (a).

Figure 2.3: (a) Potential energy Ve change at different external fields. (b) Linearly Stark-
shifted transition frequency for 1 → 2 and 1 → 3 Rydberg transitions.

The wave function of the ground state can be found with a variational method using a

8

trial wave:

χ1(z) =

ze−z/b.

2
b3/2

(2.13)

This has the same shape as the ground state wave function at E⊥ = 0 in equation 2.7 except

that the zero-field effective Bohr radius a is replaced with the pressing field dependent one

b = b(E⊥). The relationship between b(E⊥) and a is given by [10].

(cid:18)1

3

(cid:18)9λ

(cid:19)(cid:19)

4

b
a

4
3λ

=

sinh

arcsinh

(2.14)

with λ =(cid:112)E⊥/Ec where a characteristic field Ec = ¯h2/2meea3. Here λ is a parameter that

represents the strength of the holding field compared to Ec (cid:39) 0.9 kV/cm. At weak holding
fields (λ (cid:28) 1), the first order correction to the energy levels can be found via perturbation

theory, which produces a result that is equivalent to the linear Stark shift:

∆E1

n = eE⊥(cid:104)n|z|n(cid:105).

(2.15)

This linear shift has been observed in spectroscopic measurement of transitions between

bound electron states above the liquid helium surface [48]. Fig. 2.3 (b) shows the cal-

culation of the Stark shifted transition frequencies in the limit of small electric fields ob-
tained from equation 2.15, revealing the slope 0.83 and 2.21 GHz/Vcm-1 for 1→2 and 1→3
transitions, respectively. At stronger pressing fields (λ (cid:29) 1) not satisfying the condition
n (cid:28) En+1 − En, the perturbation approach is not applicable and the correction term
∆En becomes nonlinear to E⊥. In this regime, the variational calculation of the Stark shift

∆E1

matches the experiment [1] as shown in Fig. 2.4. One can neglect the image potential term

Vim in equation 2.12 in the limit of strong field. In this case, wave function χn(z) is reduced

9

to the Airy equation [49].

Figure 2.4: Transition frequencies versus voltage across the experimental cell. The voltage
between the top and bottom plates inside the cell is swept by a triangular wave form applied
in a way that maintains the helium surface at the same positive potential. This process
allows for varying the pressing field acting on the electrons without change in the number of
electrons. The crosses are measured data points while the solid curves are the result of the
variational calculation (from Ref. [1]).

2.1.3 Thin Film Effects

A thin superfluid helium film can cover a dielectric substrate due to the van der Waals

interaction between helium atom and the substrate. For surface electrons on such a helium

film, the potential energy of an electron is affected by image charges located in the dielectric

as well as in the film (see Fig. 2.5). If we neglect image charges in the metal for a thick

dielectric layer, the potential energy Ve in the presence of an external electric field may be

approximately written as [50]

Ve(z) (cid:39) −Λe2

z

− Λse2
z + d

+ eE⊥z,

(2.16)

10

Figure 2.5: Schematic of electrons on helium film on a dielectric substrate. Major image
charges are represented by + symbols in the dielectric substrate. The role of the metal
depends on the thickness of the dielectric as described in the text.

where Λ =


 − 1
4(
 + 1)

is the image charge factor for the helium film, Λs =


s − 

4(
s + 
)

is that for

the substrate, d and 
s are the thickness of the film and dielectric constant of the substrate.
For a relatively thick helium film ((cid:104)z(cid:105)/d (cid:28) 1), the potential is further approximated by

Ve (cid:39) −Λse2

d

− Λe2
z

+ (Fd + eE⊥) z,

(2.17)

where Fd = Λse2/d2. The main correction to the binding energy of the surface electron

is revealed in the first term in equation 2.17.
If the substrate is a metal covered by a
300˚A thick helium film, Λse2/d ≈ 140 K meaning highly populated ground state even at no

external field. The second term represents an attractive potential due to the polarization of

the liquid helium and the last term is associated with the image force induced by the solid
substrate leading to an enhanced pressing field (i.e. E⊥ → E⊥ + Fd/e). For instance, a

metal substrate covered by a 100 nm thick charged helium film yields an effective pressing
field Ed = Fd/e (cid:39) 36 kV/cm due to the substrate, which is significantly bigger than Ec in
equation 2.14. Also, the average distance of the electron from the film surface (cid:104)z(cid:105) = 3b/2 for
this case can be estimated from equation 2.14 and leads to (cid:104)z(cid:105) ≈ 40 ˚A, which is substantially

11

decreased compared to the bulk helium case at weak field ((cid:104)z(cid:105) ≈ 140 ˚A).

The electron-electron interaction in the 2DES above helium film is screened by image
charges induced inside the substrate. For a metallic substrate (
s → ∞), the Coulomb

interaction V (r) is given by

(cid:32)

(cid:33)

,

1(cid:112)r2 + (2d)2

−

1
r

V (r) = e2

(2.18)

where r is the mean inter-electron distance. For d (cid:28) r the Coulomb interaction is modified
into a dipole interaction V (r) → 2e2d2/r3 [51]. This suppressed interaction affects the dis-

persion relation for collective modes of the electron system [52] and leads to the modification

of the phase diagram. In particular, Peeters and Platzman evaluated the melting curve in the

presence of the screening effect from the substrate [53], which predicts that the fluid phase

persists down to T = 0 at low density for a metallic substrate in contrast to the Wigner

crystal state for bulk helium.

2.2 Wigner Solid

In 1934, Eugene Wigner predicted an ordered phase of a 2D or 3D electron gas in a positive

background when the potential energy dominates the average kinetic energy at low density

[17]. To minimize their relative potential energy, the electrons form a bcc (body-centered-

cubic) lattice in 3D and a triangular lattice in 2D. This electron crystal is called Wigner solid

(WS). Clear examples of physical systems realizing the WS were not found until Crandall and

Williams proposed electrons on liquid helium system in 1971 [54]. This is because the system

fulfills the requirement for Wigner lattice including an uniform positive potential provided by

12

underlying metal plate for bulk helium and lower electron density (ns ≈ 106 − 109 cm-2). In

1979, the 2D Wigner solid transition was indeed observed in the system of electrons on helium

by Grimes and Adams [5]. They discovered a sudden appearance of the resonance below

T = 0.457 K associated with coupled plasom-ripplon modes. This is the first experimental

demonstration of the WS.

For the case of a degenerate semiconductor 2DES, no direct observation of the WS has

been achieved so far due to difficulty in achieving low density. However, positive evidence

of a magnetically induced Wigner solid in GaAs/AlGaAs heterojuctions was reported by

Andrei et al. [55]. Recently, bilayer semiconductor heterostructures have attracted interest

for direct detection of the WS since the mutual interaction of parallel layers of two 2DESs

could allow for the WS transition at much higher electron density in the quantum regime even

without external fields [56]. More recently, twisted bilayer graphene (TBLG), well-known

for a superconducting state at magic angle [57], also has been gaining the interest due to

the possible existence of the WS in this system [58]. However, experimental confirmation is

required.

The transition from an electron liquid to a Wigner solid can be characterized by the

dimensionless plasma parameter Γp

Γp =

(cid:104)V (cid:105)
(cid:104)K(cid:105),

(2.19)

where (cid:104)V (cid:105) and (cid:104)K(cid:105) represent mean electron-electron Coulomb potential and mean kinetic

energy respectively, which are given by

(cid:104)V (cid:105) =

(cid:104)K(cid:105) =

e2
r0
1
ns

= e2√
(cid:90) ∞

πns,

dεν(ε)f (ε)ε.

0

13

(2.20)

(2.21)

√
Here, r0 = 1/

πns is inter-electron spacing, ν(ε) =

me
π¯h2 is the two-dimensional density of

states, and f (ε) =

is Fermi-Dirac distribution function.

For a degenerate 2DES when T → 0, the mean kinetic energy is given by

1

1 + e(ε−µ)/τ

(cid:82) εF
(cid:82) εF

(cid:104)K(cid:105) =

0 dεν(ε)ε
0 dεν(ε)

=

εF
2

,

(2.22)

where εF =

¯h2πns
me

is the Fermi energy. Introducing the Bruckner parameter rs = r0/aB [44],

where the mean interparticle distance r0 is measured in units of the Bohr radius aB = ¯h2/mee2,
simplifies (cid:104)V (cid:105) and (cid:104)K(cid:105):

(cid:104)V (cid:105) =

(cid:104)K(cid:105) =

=

e2
r0
¯h2πns
2me

aB
r0

=

e2
aB

=

¯h2

2
rs

Ry,

= aB

e2
2r2
0

=

1
rs

Ry,

2mer2
0

(2.23)

(2.24)

e2
2aB

me4
2¯h2 = 13.6 eV is the Rydberg constant. Thus, the plasma parameter

where Ry =
for degenerate case (εF (cid:29) kBT) has the compact form:

=

Γpq = 2rs.

(2.25)

When rs (cid:29) 1 at lower density, the potential energy between electrons far exceeds their kinetic

energy leading to electron crystallization. For GaAs/AlGaAs heterostructures, quantum
Monte Carlo simulations predict that the electron system can crystallize when rs ≥ r
or Γpq ≥ Γ
and the relation rs = r0/aB = mee2/¯h2√

(c)
pq = 62 [59]. This system has a typical density ranging from 1011 to 1012 cm-2
πns yields rs ∼ 0.5 – 2, which is much smaller

(c)
s = 31

than the critical value r

(c)
s = 31. Here, me and e were replaced with the effective mass

14

m∗
e (cid:39) 0.066 me and screened electron charge e/

√
√

 (cid:39) e/

12.5 for GaAs. Therefore, it is

extremely challenging to directly observe a Wigner solid in a semiconductor 2DES.

For a non-degenerate 2DES in the low density limit where kBT (cid:29) εF, the mean kinetic

energy is (cid:104)K(cid:105) = kBT and thus we obtain the plasma parameter in the classical regime:

e2√
kBT

πns

=

e2
r0

1

kBT

.

Γpc =

(2.26)

For 1 < Γpc < 100 (low density and high temperature), electrons behave like a correlated

fluid system while for Γpc > 100 the Coulomb potential dominates thermal fluctuations of

the kinetic energy and leads to phase transition to an ordered state (i.e.

the WS) [50].

Regarding the critical value Γ

(c)

pc for the Wigner transition when Γpc ≥ Γ

(c)
pc , there has been

extensive research based on a variety of techniques to experimentally determine this value,

which are shown in the following table. Note that specification of the system is omitted for

the case of electron on helium in the Table 2.1. Most of these experimental values are in

(c)
reasonable agreement with theoretical estimation in the range of Γ
pc 120 - 130 [66]. At the
maximum attainable density ns ≈ 2 × 109 cm-2 for electrons above a bulk helium surface,
the melting temperature Tm (cid:39) 1 K, obtained from equation 2.26 with Γpc = 130, is much
higher than εF = ¯h2πns/me (cid:39) 56 mK, indicating classical solid-to-liquid transition. For
ns > 2 × 109 cm−2 on bulk liquid helium, the restoring forces in the liquid that counteract

a surface deformation resulting from electron pressure are no longer sufficient to confine the

charge at the surface (electrohydrodynamic instability) [67]. Peeters and Platzman pointed

out that a quantum liquid transition can be achievable on a helium film supported by the
metal substrate [53] due to the higher attainable density (ns ∼ 1011 cm-2) [68] and the
screened Coulomb interaction by the substrate (dipolar interaction ∼ 1/r3

0). Particularly,

15

Reference

Grimes and Adams (1979) [5]

Marty et al. (1980) [60]

Gallet et al. (1982) [61]

Mehrota et al. (1982) [62]

Kajita (1985) [63]

Mellor and Vinen (1990) [64]

(c)
pc

Γ

137 ± 15

118 ± 10

140 ± 10
124 ± 4
138 ± 25

130 ± 3

T(K)

Experiments/Method

0.4 - 0.7 Longitudinal coupled

electron-ripplon mode

0.3 - 0.6 Longitudinal coupled

electron-ripplon mode

0.3 - 0.7 Transverse sound mode

0.2 - 0.7 Mobility

1.4 - 2.5 Conductivity

(electrons on neon)

0.1 - 0.2 Shikin modes

(positive ions on helium)

Shirahama and Kono (1995) [65]

127

0.1 - 1 Conductivity

Table 2.1: Experimental results for the critical value of the plasma parameter Γ
the liquid-to-solid transition occurs.

(c)
pc at which

their calculation reveals a quantum fluid which persists down to T = 0 K as ns → 0. This
is because the potential energy (∼ 1/r3
kinetic energy (∼ 1/r2

) decreases with density ns faster than the

0 ∼ ns).

0 ∼ n

3/2
s

2.2.1 Phase Diagram

Platzman and Fukuyama calculated the phase diagram for electrons on bulk helium with

the assumption of a constant plasma parameter (i.e. Γ

(c)
pq = Γ

(c)
pc = Γ

(c)
p = 3) along the

melting curve (see Fig. 2.6) [2]. In the classical regime at high temperate and low density,

the melting curve in Fig. 2.6 is given by equation 2.26:

ns =

k2
BT 2.

(2.27)

(cid:32)

(cid:33)2

(c)

p√
Γ
πe2

16

Figure 2.6: Phase diagram of a 2D Coulomb liquid (from Ref. [2]). Temperature T and elec-
p )−2

tron density n are expressed in a unit of nc and T c, respectively. Here nc = 4π−1(aBΓ
and kBTc = 2mee4/¯h2(Γ

(c)
p )2.

(c)

In the zero temperature limit, the critical density nc for melting is given by equation 2.25

with rs = r0/aB = 1/aB

√

nsπ:

(cid:16)

(c)
aBΓ
p

(cid:17)−2

.

nc =

4
π

(2.28)

(c)

p = 137 is taken [5], the critical density is nc (cid:39) 2.4 × 1012 cm-2. In the general case at
If Γ
any density and temperature, the kinetic energy (cid:104)K(cid:105) and electron density ns may have the

analytic form:

ns =

1
π

(cid:104)K(cid:105) =

(cid:90) ∞
(cid:90) ∞

kdk

0
1
π

1
ns

0

ln (1 + 1/z)
−Li2(−1/z)
ln(1 + 1/z)

,

(2.29)

(2.30)

1

eβ(εk−µ) + 1

=

1
π

mekBT

¯h2

kdk

εk

eβ(εk−µ) + 1

= kBT

17

where z = e−βµ, µ is the chemical potential, β = 1/kBT , and Li2(z) is called the dilogarithm

(cid:90) z

0

function:

Li2(z) = −

t−1ln(1 − t)dt for z ≤ 1.

(2.31)

Inserting equations 2.29 and 2.30 into equation 2.19, we can obtain the parametric equations

for the phase boundary:

ns/nc =

T /Tc =

1
4

1
2

ln4(1 + 1/z)
2(−1/z)
−Li2
ln3(1 + 1/z)
2(−1/z)
−Li2

,

,

where Tc is the characteristic temperature defined by

kBTc =

¯h2(cid:16)

2mee4
(c)
Γ
p

(cid:17)2 .

(2.32)

(2.33)

(2.34)

From Landen’s identity Li2(z) = −Li2(
function Li2(−1/z) can be expanded to

z
z − 1

) − 1
2

ln2(1 − z) for z < 1, the dilogarithm

Li2(−1/z) = −1
2

ln2(1 +

) − Li2(

1
z

1

1 + z

).

(2.35)

Substituting equation 2.35 into equations 2.32 and 2.33, the parametric equations for ns/nc

and T /Tc are now equivalent to the original equations written in [2], which are used for the

numerical plot in Fig. 2.6. Note that this phase diagram gives only a qualitative picture

since the plasma parameter along the phase boundary might not be the same in classical

and quantum regime.

In reality, the experimental value of Γ

(c)
pc for the classical case is

approximately 130, which is roughly two times larger than Γ

(c)
pq = 62 for the quantum case

18

obtained from quantum Monte Carlo simulation as discussed in the section 2.2.

2.2.2 2D Melting Mechanism

The existence of 2D crystals has long been a fascinating topic in condensed matter physics

and was first theoretically explored by Peierls [69] and Landau [70] in 1930. They pointed

out that long-range order in 2D should be destroyed by thermal fluctuation at any finite

temperature. Mermin and Wagner further demonstrated that magnetic and crystalline long-

range order could not exist in 1D or 2D [71, 72]. However, although there is no true long-range

order in 2D, spatial correlations decay algebraically according to a power law and can be

extended over a finite system size (i.e. quasi-long-range translational order) according to

KTHNY (Kosterlitz-Thouless-Halperin-Nelson-Young) theory [73, 74, 75, 76]. Furthermore,

this predicted that 2D crystal also has long-range bond orientational order as well as quasi-

long-range positional order. Based on KTHNY theory, the 2D melting mechanism is briefly

summarized below.

In a crystalline phase, the structure factor S((cid:126)q) can be defined by [75]

S((cid:126)q) =

ei(cid:126)q· (cid:126)R(cid:104)ei(cid:126)q·[(cid:126)u( (cid:126)R)−(cid:126)u((cid:126)0)](cid:105),

(2.36)

(cid:88)

(cid:126)R

where (cid:126)q is the scattering vector, (cid:126)R the lattice point, (cid:126)u( (cid:126)R) displacement vector at lattice site
(cid:126)R, and the summation is over all lattice sites { (cid:126)R}. In a 3D solid, S((cid:126)q) can be approximated
by considering the case |(cid:126)u|(cid:28) | (cid:126)R|:

S((cid:126)q) (cid:39)(cid:88)

ei(cid:126)q· (cid:126)R = (2π)3(cid:88)

(cid:126)R

(cid:126)G

19

δ3((cid:126)q − (cid:126)G),

(2.37)

where (cid:126)G is a reciprocal lattice vector. Thus, Bragg peaks are delta function at (cid:126)q = (cid:126)G in

three dimension. Also, a 3D crystal has a non-zero correlation function C (cid:126)G

at large R, which

represents long-range positional order. This function is defined by [75]

( (cid:126)R)ρ∗
(cid:126)G

≡ (cid:104)ρ (cid:126)G
((cid:126)0)(cid:105),
≡ ei (cid:126)G·[ (cid:126)R+(cid:126)u( (cid:126)R)].

C (cid:126)G

ρ (cid:126)G

(2.38)

(2.39)

However, in 2D crystals, Jancovici pointed out that fluctuation in long-wavelength phonon

modes leads to the displacement (cid:126)u( (cid:126)R) diverging logartithmatically and the spatial correlation

[77]. Thus, when the sample size is infinite
function decays algebraically: C (cid:126)G
(R → ∞), no long-range translational order exists in 2D. Nevertheless, for a finite sample

(T)

∼ R

−η (cid:126)G

size, the correlation function can survive due to the algebraic decay. This quasi-long-range

positional order can be destroyed by an unbinding of dislocation pairs (formation of iso-

lated dislocations) at the melting temperature Tm. A dislocation stands for an irregularity

within the crystal structure showing an sudden change of atomic arrangement as shown in
Fig. 2.7 (c). Thus, at T ≥ Tm, dislocation-mediated melting of the 2D crystal occurs. In

this case, the correlation function of 2D fluid has exponential decay:

( (cid:126)R) ∼ e−| (cid:126)R|/ξ+(T)

C (cid:126)G

for T > Tm

with the correlation length

ξ+ ∼ exp(C/|t|0.36963...),

(2.40)

(2.41)

where C is a positive constant and t = (T − Tm)/Tm. This exponential decay implies

only short-range positional order and thus the absence of Bragg peaks in the fluid phase.

20

Figure 2.7: Topological defects in a triangular lattice. Isolated disclinations with (a) seven
folder and (b) five folder symmetry rather than six folder. An isolated dislocation (c) is
composed of a bound pair of such disclinations, showing undisrupted orientational order at
long distance.

Another feature distinguishing 2D crystals from 3D crystals is the structure factor, which

can be measured in diffraction-type measurements. In contrast to delta function Bragg peaks

for 3D as mentioned above, the structure factor in 2D has power law singularity:

S((cid:126)q) ∼ |(cid:126)q − (cid:126)G|−2+η (cid:126)G

(T)

.

The exponent η (cid:126)G

is related to the Lam´e elastic constants µ and λ by

= kBT| (cid:126)G|2

η (cid:126)G

3µ + λ

4πµ(2µ + λ)

.

(2.42)

(2.43)

Here λ and µ are the incompressibility and the shear modulus. Lastly, the solid to liquid

transition temperature Tm in the 2D system is given by

Tm =

a2
0

4πkB

µ(µ + λ)
2µ + λ

.

(2.44)

21

From equation 2.26, corresponding critical value Γ

(c)
p is given by

(c)
p =

Γ

4πe2√
a2
0

πns

2µ + λ
µ(µ + λ)

.

(2.45)

Considering the incompressibility of the electron crystal (λ = ∞) [78], one can simplify Tm

and Γ

(c)
p as

Tm =

(c)
p =

Γ

µ,

a2
0
4πkB
4πe2√
a2
0

πns

1
µ

,

(2.46)

(2.47)

√
3ns)1/2 is a lattice constant of the 2D triangular electron lattice, which
where a0 = (2/

is energetically favored. Thus, if the shear modulus µ is given, one can estimate Γ

(c)
p

from

equation 2.47. One way to do this is to use µ obtained from the transverse sound velocity

ct in a 2D electron crystal: ct = (cid:112)µ/ρ where ρ = mens is a mass density of the crystal.

In the long wavelength limit (low temperature), a transverse phonon has a linear dispersion

(cid:115)

e2

mer0

relation [4]: ωt = ctq with ct =

0.138

, where ωt(q) is a frequency of the trans-

√
πns is the inter-electron spacing. This
verse phonon mode at a wavevector q and r0 = 1/

√
πe2n
yields µ = 0.138

3/2
s

. Using this calculated value of µ, Thouless theoretically obtained

(c)
Γ
p = 78.71 [78], which does not agree well with experimental result of Grimes and Adams
p = 137 ± 15 [5]. Morf calculated the temperature dependence of the shear modulus µ
Γ

(c)

using molecular dynamics simulations, revealing a linear decrease of µ with increasing T (see

equation 2.63) and for further increasing T a sharp drop of µ along with the onset of electron

diffusion [79]. This estimation of µ yields Γ

(c)
p = 128.2, which fits well the experiments.

Beside the positional order described above, another type of the ordering characterizes the

22

2D crystal: long-range order in the nearest neighbor bond-orientation. Nelson and Halperin

found that unbinding of dislocation pairs leads to melting into a new type of liquid crystal
at T → Tm, characterized by power-law decay of orientational order. This “hexatic phase”

[75] corresponds to an orientational correlation function given by

with

g6(r) ∼ r−η6(T),

η6 =

18kBT
πKA

.

(2.48)

(2.49)

Here the exponent η6 is proportional to T and KA is the Frank constant [80]. Due to

this quasi-long-range orientational order, a diffraction pattern for an infinite size sample

would show an isotropic ring just like isotropic liquid phase.

In real experiments (finite

size sample), this pattern is modified into the sixfold modulation (see Fig. 2.8). By further
increasing temperature (T → Ti), the orientational order is disrupted by another type of the

defect, the disclination. Dissociation of disclination pairs leads to isotropic fluid phase. This

topological defect can be viewed as a particle having the wrong number of nearest neighbors

as shown in Fig. 2.7. Also, the dislocation mentioned above can be regarded as a bound pair

of disclinations. Now, the corresponding correlation function shows an exponential decay

(i.e. short range orientation order):

g6(r) ∼ e−r/ξ6(T),

(2.50)

where ξ6(T) ∼ eb/|T−Ti|1/2

and b is a constant.

23

Figure 2.8: Computed diffraction pattern of three different phases in 2D: Isotropic fluid
phase (A), hexatic phase (B), and crystalline phase (C). Images are taken from Ref. [3]

2.3 Collective Modes of a Two Dimensional Electron

System (2DES)

2.3.1 Plasmons

Plasmons are collective excitations formed in the electron gas or liquid, which propagate as

longitudinal charge density oscillation through the bulk or the surface. For the 3D case, the

dispersion relation for plasmons is given by the Bohm-Gross equation [81] in the presence of

random thermal motion:

ω2
3D(q) = ω2

p +

3kBT
me

q2,

(cid:115)

4πnee2

(2.51)

where ω3D(q) is the plasma frequency at wave-number q and ωp =
is a finite
angular frequency independent of q. In the long-wavelength limit (q → 0), plasma frequency

me

ω3D(q) = ωp has non-zero value. In contrast to bulk plasmons, the plasma frequency of a

2D electron system (2DES) goes to zero in this limit with the dispersion relation:

ω2D(q) =

2πnee2

me

q1/2.

(2.52)

(cid:115)

24

This square-root dependence on q is a universal features of the 2DES with unscreened

Coulomb interactions. In actual experiments, metal electrodes can screen the electric fields

and thus results in the modification of the dispersion relation of the 2DES [52]:

ω2
2D(q) =

4πnee2

q

me

coth(qd) + 
scoth(qds)

,

(2.53)

where d and ds denote the distance of the 2DES from the electrodes above and beneath the
2D electron layer, 
s (cid:39) 1 is the dielectric constant of the liquid helium. From equation 2.53,

one can easily show that a long-wavelength plasmon mode (qd < 1 and qds < 1) has a linear

dispersion relation:

ω2D(q) (cid:39) q

4πnee2
meαd

,

(2.54)

with αd = 1/d+
s/d. In the short-wavelength limit, one can find the square-root dependence

(cid:115)

(cid:115)

of ω2D on q which is a typical feature of 2D plasmons:

ω2D(q) (cid:39) q1/2

4πnee2

me(1 + 
s)

.

(2.55)

This 2D plasmon dispersion relation was experimentally verified by Grimes and Adams using

standing-wave resonance in plasma of electrons floating above helium surface [7].

2.3.2 Phonons

A 2D solid can exhibit another type of the collective excitation, normal modes of the lattice,

so-called phonons. Bonsall and Maradudin calculated the phonon dispersion relation for the

hexagonal lattice of a 2D electron crystal by solving the eigenvalues of the dynamical matrix

[4]. The corresponding dispersion curve is shown in Fig. 2.9.

25

Figure 2.9: (a) First Brillouin zone for the two-dimensional, hexagonal lattice. (b) Phonon
dispersion curve along the boundary of the first Brillouin zone (red line in (a)). The frequency
ω0 is defined by ω0 = 8e2m∗a3

0, a0 is the lattice constant. Figure taken from Ref. [4]

In the long-wavelength limit, the frequency of phonon modes for 2D triangular electron

lattice has the asymptotic form [82]:

(cid:115)

ωl =

2πe2ne

me

ωt = ctq,

c2
t =

me

q1/2,
0.138e2√

πne

,

(2.56)

(2.57)

where ωl and ωt represent longitudinal and transverse modes and ct is the transverse sound

velocity. The longitudinal phonon in this limit in the 2D electron crystal has the same

dispersion relation as plasmons in the 2D electron gas (i.e. see equation 2.52), indicating

that this phonon mode by itself is irrelevant for observing a phase transition to the electron

solid. However, transverse phonons arise from a resistance to shear stress which does not

exist in a fluid or gas phase, so they are good indicator of a Wigner solid transition. This

shear mode (transverse phonons) was experimentally demonstrated in an electron lattice on

helium by Deville et al. [83]. Shear stress was created by a transverse Lorentz force in this

experiment. One can relate the longitudinal and transverse sound velocity to Lam´e elastic

26

constant λ and µ:

cl =

ct =

(cid:114) λ + 2µ
(cid:114) µ

mene

mene

,

(2.58)

(2.59)

,

where mene is the mass density of Coulomb (electron) crystal. From the above equations,

the incompressibility λ and the shear modulus µ are given by

(cid:16)

(cid:17)

λ = mene

l − 2c2
c2

t

µ = menec2
t .

,

(2.60)

(2.61)

In the limit of small wave numbers (q → 0 at T → 0), the longitudinal velocity cl =
diverges and thus the Coulomb crystal can be thought to be incompressible with λ → ∞

∂ωl
∂q

∼ q−1/2

from equation 2.60. Also, from equations 2.57 and 2.61 the shear modulus in this limit is

given by

√
µ = 0.138

πe2n

3/2
e

for T → 0.

(2.62)

For finite temperature below the melting point Tm, Morf [79] showed that shear modulus
linearly decreases with increasing T up to T (cid:39) 0.9Tm, which has the following form

(cid:18)

1 − 30.8

kBT√
πnee2

(cid:19)

.

(2.63)

µ(T) = µ(0)

Here µ(0) is equivalent to equation 2.62.

Up to now, we have described two branches of phonon modes in zero magnetic field and

assumed that the 2D electron lattice is formed on a smooth substrate. In reality, they are

27

strongly affected by both magnetic fields and surface deformation of the substrate. The mag-

netic field normal to the crystal mixes longitudinal and transverse phonons, leading to new

modes ω±(q). Moreover, a new mode appears at low frequency ω− decreases with increasing
magnetic field B as ω− ∼ 1/ωc (i.e. this branch softens with applied B-field) [82]. Here

ωc = eB/mec is the cyclotron frequency. Also, for 2D electrons on a soft substrate, surface

deformation is inevitable and it can alter the effective mass of the system. A representative

example of this latter case is electrons on liquid helium. In this system, each electron in-
dents the helium surface and creates a small dimple beneath them with a depth of ∼ 1˚A.

Especially, in the Wigner solid regime, electrons in the lattice sites forms a periodic array of

dimples on the helium surface. Fisher and Halperin showed that this dimple array can follow

electrons at low frequency, resulting in increase of effective mass (me → m∗ = me + md) and

thus reduced phonon frequency(cid:112)me/m∗ωl,t [6], where md is effective mass of the dimple

and ωl,t is the frequency of two phonon branches at no surface depression described above.

In the presence of oscillatory motion of electron crystal, the disturbances of superfluid helium

surface generates a quantized capillary wave called a ripplon, which can couple to vibrational

modes of a 2D solid. This coupled mode will be discussed in section 2.3.3.

2.3.3 Phonon-Ripplon Coupling

Shikin suggested that driving the 2D electron crystal against the helium surface with a

perpendicular RF electric field should produce a series of resonances due to the excitation

of standing capillary waves (ripplons) whose wavelength is commensurate with the electron

lattice constant [84]. However, Grimes and Adams experimentally demonstrated ripplons

can be resonantly excited using an alternating electric field parallel to the charged helium

surface [5]. It turns out that the resonant excitation of ripplons by horizontal motion of the

28

WS is more effective than by the WS moving normal to the plane (Shikin modes) due to

the strong electron binding to the helium surface [44]. To achieve horizontal motion of each

electron, they applied a RF signal to a lower circular plate submerged beneath the liquid

helium. When ripplons are resonantly excited at certain RF frequencies in the presence of an

electron crystal, they affect the impedance of the electron layer. Instead of measuring this

impedance directly, they detected the derivative of the impedance with respect to the electron

density dR/dNs to minimize noise, where R and Ns are the real part of the impedance and

the areal electron density. To do this, they applied an audio-frequency signal to the electron

confinement ring to modulate Ns and synchronously detected dR/dNs at the modulation

frequency. Fig. 2.10 shows their experimental result in which resonant features in dR/dNs

derivative suddenly appear below T = 0.457 K, arising from the crystallization of the electron

system.

The two lowest resonance frequencies labeled W and X in Fig. 2.10 at T ∼= 0.42 K do

not match the expected values from the dispersion relation of resonantly excited ripplons

(equation 2.64). An explanation for this discrepancy was given by Fisher, Halperin, and

Platzman [6]. In this work, they showed that the phonon spectrum of a 2D electron crystal

is strongly affected by the interaction with ripplons, and as a result a coupled longitudinal

phonon-ripplon mode appears providing good agreement with all resonance frequencies in

the experiments of Grimes and Adams. If there is no interaction between the 2D electron

lattice vibration and ripplons, the frequency spectrum of phonons at long wavelength is given

by equations 2.56 and 2.57. Also, the ripplons resonantly excited by electron displacements

in the crystal have the following dispersion relation:

Ω2

n =

σ
ρ

G3
n,

29

(2.64)

Figure 2.10: Coupled plasmon-ripplon resonances. The resonance only appear below 0.457 K
where the sheet of electrons has crystallized into a triangular lattice (from Ref. [5]). dR/dNs
and F denote the derivative of the impedance of the electron layer with respect to the
electron density and the frequency of the RF signal applied to the underlying electrode used
to generate horizontal electron motion. A detailed description of a measurement setup is
presented in the text.

30

where σ and ρ are the surface tension and density of the liquid helium while Gn is a reciprocal

lattice vector of the electron lattice. For a triangular lattice, Gn is given by [85]

Gn =

23/2π
31/4

√
ne

√
n, n = l2 + m2 + lm,

(2.65)

where l, m = 0,±1,±2,±3, ... and thus integer number n = 1, 3, 4, 7, .... In the presence of

the phonon-ripplon interaction, the frequency spectrum of the coupled modes is determined

by the secular equation [6]. For coupled longitudinal phonon-ripplon modes such as those

observed in the Grimes and Adams experiment, the corresponding frequency spectrum can

be obtained from [6]:

ω2 − ω2

l (q) − 1

2

(cid:88)

(cid:126)Gn

V 2
(cid:126)Gn

ω2
ω2 − Ω2

n(G)

= 0,

(2.66)

where the coupling constant V (cid:126)G

has the form

VGn = V 0

Gnexp(−nW1) ≡ Vn with 0.5 < W1 < 0.7 near Tm.

(2.67)

Here ω represents the frequency of coupled modes, ωl and Ω denote the frequency of the

longitudinal phonon in equation 2.56 and the ripplon frquency in equation 2.64 respectively.

For the lowest mode (n = 1) and long wavelength, the frequency ω is far below the

ripplon frequency Ω1 so that the dimple underneath each electron can follow the motion
of the electron. Disregarding VGn for n ≥ 3 due to their exponential decay (shown in

31

equation 2.67), we can obtain an analytical solution from the equation 2.66

ωl(cid:113)

ω(q) (cid:39)

.

(2.68)

Note that (cid:80)

V 2
(cid:126)Gn

(cid:126)Gn

1 + 3V 2

1 Ω2
1

in equation 2.66 is equivalent to (cid:80)

Gn 6V 2
Gn

for n < 7 due to a six-

fold degeneracy [44]. Equation 2.68 indicates that the frequency of coupled longitudinal

phonon-ripplon mode is equivalent to reduced bare longitudinal phonon frequency with a

renormalized mass. In other words, me in equation 2.56 is replaced with me(1 + 3V 2

1 Ω2

1) and

me3V 2

1 Ω2

1 can be considered as mass contribution from the ripplon. This feature can be seen

by comparing the dashed curve and the lowest solid curve in Fig. 2.11. With an increase in
wave number q (ωl (cid:29) ω ∼ Ω1), we can additionally ignore the first term ω2 on the left-hand

side of the secular equation 2.66, leading to

ω2(q) (cid:39)

Ω2
1
1 + 3V 2
1 /ω2
l

.

(2.69)

This illustrates that the frequency ω approaches an upper-limit Ω1 with increasing q as

shown by the lowest solid line and the lowest dashed horizontal line in Fig. 2.11. Near other
ripplon frequencies ω ∼ Ω3,4, we have the solution:

(cid:32)

(cid:33)

ω2(q) (cid:39) Ω2

n

1 − 3V 2

n

ω2
l + 3V 2
1

for n = 3, 4.

(2.70)

Since V 2

3,4 (cid:28) V 2

1 , the frequency ω is less affected by the phonon frequency ωl and quite

close to the frequency of the bare ripplon mode as shown by the two solid curves in the
middle of Fig. 2.11. At higher frequency ω (cid:29) Ω1, the dimple underneath each electron can

be regarded as motionless and the corresponding solution of the secular equation gives the

32

dispersion relation:

d = 1/2(cid:80)

(cid:126)Gn

where ω2

(cid:113)

d + ω2
ω2
l ,

ω(q) =

(2.71)

V 2
(cid:126)G

is the frequency of the single electron oscillation in the static dimple.

The non-zero value of the frequency spectrum at q = 0 resembles an optical-like mode as

shown by the top solid curve in Fig. 2.11 . So far, we have discussed how the dispersion

relation is altered due to longitudinal phonon-ripplon coupling. As Fisher, Halperin, and

Platzman pointed out, the dispersion relation of coupled modes of transverse phonons can

be qualitatively similar to the longitudinal phonons case because it can be obtained from

the same secular equation 2.66 with ωl replacing ωt.

2.4 Transport Properties of Electrons on Helium

The charge transport properties are usually characterized by the mobility of charge carri-

ers. At low electron density, in which electron-electron interaction can be neglected, the

mobility of electrons on helium surface is determined by two different scattering mechanisms

depending on temperature [86, 87]. 4He vapor atoms are the dominant electron scatters
at T ≥ 1 K, whereas electron-ripplon scattering dominates at T ≤ 0.8 K. Since Sommer

and Tanner experimentally determined the mobility of surface electrons on helium for the

first time [88], many groups have reported experimental values of the mobility in various

temperature ranges using different techniques [7, 8, 9, 89, 90, 91]. Also, it has been studied

theoretically by Saitoh [10], Shikin and Monarkha [49]. Fig. 2.12 shows experimental data

of the mobility from some of groups together with a theoretical calculation from Saitoh [10].

In this figure, the electron mobility increases exponentially with decreasing temperature T
down to ∼ 1 K due to the exponential decay of the 4He vapor pressure. The moderate

33

Figure 2.11: Dispersion relation of the longitudinal phonon-ripplon coupled modes (solid
curves). The dashed curve is the bare longitudinal phonon spectrum of the electron crystal
and the horizontal dashed lines represent the bare ripplon resonance frequencies when ig-
noring the phonon-ripplon coupling. The vertical lines are the wave vectors excited in the
experiment. The resonances are labeled as in Ref. [5]. Figure taken from Ref. [6].

34

change of the mobility below T (cid:39) 0.8 K results from the slow temperature dependence of

the electron-ripplon scattering rate. At arbitrary temperature, the scattering rate 1/τ is

given by

1

τ (εk)

=

1

τv(εk)

+

1

τR(εk)

,

(2.72)

where τv and τR denote the energy, εk, dependent relaxation time for electron-4He scatter-

ing and electron-ripplon scattering, respectively. From the total relaxation time τ (εk), the

electron mobility is defined as

e(cid:104)τ (εk)(cid:105)

m∗

µ =

(2.73)

where m∗ is the electron effective mass. In the next two subsections 2.4.1 and 2.4.2, a more

detailed discussion of these two scattering mechanisms will be presented.

2.4.1 4He Vapor Scattering

When an electron strikes an atom, the scattering is determined by a competition between

the short-range repulsion arising from the Pauli principle and the long-range polarization

attraction. For the helium atom, the repulsion dominates the scattering due to the very low
atomic polarizability of helium, leading to a positive scattering length as = 1.19a0 (cid:39) 0.63 ˚A
s (cid:39) 4.98 ˚A2 for low-energy electrons [92]. Using the
and a scattering cross section σs = 4πa2

Boltzmann transport equation for electron-He gas scattering, Saitoh calculated the relevant

scattering rate 1/τv, which is given by

1
τv

=

3π¯hσsnv

8meb

,

(2.74)

35

Figure 2.12: Mobility of electrons on helium versus the temperature and 4He vapor den-
sity. Experimental data from Grimes and Adams [7], Rybalko et al. [8], and Iye [9] are
shown together with a theoretical analysis from Saitoh [10]. The dashed line is the mobility
considering vapor atom scattering alone. Figure taken from Ref. [9].

36

where b is the effective Bohr radius in the presence of a vertical holding field from equa-

tion 2.14 and nv is the saturated vapor density of the 4He gas, which is given by [10]

(cid:18) M kBT

(cid:19)3/2

2π¯h2

(cid:18)−Qv

(cid:19)

kBT

nv(T) =

exp

.

(2.75)

Here M = 6.646 × 10−27 kg is the mass of a 4He atom and Qv denotes the vaporization
energy of 4He (Qv/kB (cid:39) 7.17 K from [10]). Since the scattering time τv does not depend

on the electron energy, the mobility due to helium vapor scattering can be obtained from

equation 2.73

µv =

eτv
m∗ =

8eb

3π¯hσsnv

with m∗ (cid:39) me for 4He.

(2.76)

In this expression, it is obvious that the mobility µv depends on both the vertical holding field

dependent effective Bohr radius b(E⊥) and the vapor density nv(T). Particularly, we can

expect an exponential decay of the mobility as T increases originating from the temperature

dependence of the vapor density in equation 2.75, which agrees with experimental data for
T ≥ 1 K as seen in Fig. 2.12.

2.4.2 Ripplon Scattering

As the temperature decreases below 0.8 K, the 4He vapor density becomes very low and

the electron mobility is limited by the electron interaction with thermal capillary waves on

the helium surface (ripplons). These ripplons are characterized by the following dispersion

relation [93]:

Ω2

q =

(cid:18)

gq +

tanh(qd),

(2.77)

(cid:19)

σq3
ρ

37

where g = 9.81 m/s2 is the gravitational acceleration, σ the surface tension of helium,

ρ the mass density of the fluid, and d the thickness of the fluid layer. The first term

describes a gravity wave and the second term is associated with the capillary wave. When

q (cid:29) qc =(cid:112)gρ/σ, the capillary wave term predominantly determines the frequency spectrum

of the ripplon and the gravity waves are neglected. For liquid 4He, the surface tension σ

and the mass density ρ are 0.37 dyn/cm and 0.145 g/cm3 respectively [50] and therefore the
corresponding qc is ∼ 20 cm-1, which means that the capillary waves dominate the dispersion
relation for the wave number q (cid:29) 20 cm-1. The ripplon wave number q involved in electron

scattering is of the order of thermal electron wave number q ∼ 2kT, where kT =(cid:112)2mekBT/¯h

is thermal deBrogile wave number [44]. In the temperature region 0.1 K ≤ T < 1 K, the
range of relevant q is approximately between ∼ 105 and ∼ 106 cm-1, which far exceeds
qc = 20 cm-1. Therefore, for thick helium film (d > 100 nm → tanh(qd) ∼ 1), the dispersion

relation in equation 2.77 reduces to

Ω2
q = σq3/ρ.

(2.78)

In this reduced form, the energy of ripplons taking part in electron scattering is given by

(cid:114) σ

ρ¯h

¯hΩq =

(8mekBT)3/4 ,

(2.79)

which is much smaller than thermal energy kBT. For instance, ¯hΩq is ∼1×10−2 K at
T = 1 K and ∼4×10−4 K at T = 0.01 K. This implies that only long wavelength ripplons

with an energy far smaller than the thermal energy can scatter with electrons. The occupa-

38

tion number Nq for ripplons is given by [49]

(cid:18) ¯hΩq

kBT

(cid:19)

− 1}−1,

(2.80)

Nq = {exp

which is formally equivalent to the occupation number of surface-bound bosonic quasipar-

ticles i.e. harmonic oscillator excitations. By solving the Boltzmann transport equation,

Saitoh obtained the relaxation time τR(εk) for electron-ripplon scattering [10]:

(cid:34)

1

τR(εk)

=

kBT
4¯hσa2

(cid:18)

(cid:19)
(cid:18) π2

− 3

3

16Eb

εk

(cid:19)2

+

(cid:40)(cid:18)

(eE⊥a)2

εk

+

3εk
2

+ 2(eE⊥a)

ln

ln

16Eb

εk

− 19
6

(cid:19)(cid:41)(cid:35)

,

− 115
36

(2.81)

where a = a0/Λ is the effective Bohr radius with zero electric field normal to the plane and

Eb is defined as

Eb =

¯h2
2meb2 .

(2.82)

The relaxation time approximation gives a non-equilibrium electron distribution function in

the presence of a drift field (cid:126)E parallel to the electron plane, which is given by

f (εk) (cid:39) f0(εk) − e τR(εk) (cid:126)vk · (cid:126)E

∂f0
∂εk

,

(2.83)

where (cid:126)vk is the electron velocity and equilibrium electron distribution function f0(εk) for

the non-degenerate system is

f0(εk) =

π¯h2ne
mekBT

exp

(cid:18)

− εk
kBT

(cid:19)

.

(2.84)

39

From the drift current density (cid:126)j = e(cid:82) ∞
(cid:90) ∞

from Ohm’s law:

0 dεk (cid:126)vk D(εk) f (εk), the conductivity is obtained

(cid:20)

(cid:21)

σ0 =

dε D(ε)

e2
me

0

−τR(ε) ε

∂f0
∂ε

,

(2.85)

where D(ε) = me/π¯h2 is the electron density of states in 2D. The DC electron mobility

µ = σ0/nee is given by

where the average of τR(ε) is

(cid:90) ∞

0

(cid:104)τR(ε)(cid:105) =

1
ne

dε D(ε)

µ =

(cid:104)τR(ε)(cid:105),

e
me

(cid:20)

−τR(ε) ε

∂f0
∂ε

(cid:21)

=

(cid:90) ∞

0

(2.86)

dε

ε τR(ε)
k2

BT2 e−ε/kBT.

(2.87)

Using the approximation (cid:104)τR(ε)(cid:105) = τR(ε = kBT) in equation 2.81, the average collision time
due to electron-ripplon scattering at (cid:126)E⊥ = 0 can be approximately given by

(cid:18)

(cid:19)−2

τR(T) ≡ (cid:104)τR(ε)(cid:105) (cid:39) 8¯hσa2
BT2

3k2

ln0.67

E0
T

(2.88)

with E0 = ¯h2/2mea2, which has a much weaker temperature dependence compared to

the collision time for helium vapor scattering in equation 2.74. At non-zero pressing field
( (cid:126)E⊥ (cid:54)= 0), the ripplon-limited collision time for T < 0.8 K is suppressed with increasing (cid:126)E⊥
under the same approximation, as shown in Fig. 2.13. This is because a larger value of (cid:126)E⊥

presses the electrons more strongly against the helium surface leading to an enhancement of

the electron-ripplon interaction [7]. However, for high temperature where the vapor density

dominates the electron scattering, the effect of the holding field on scattering becomes weaker.

40

In the limit of the strong field, equation 2.81 is simply given by

1

τR(ε)

(cid:39) kBT
4¯hσa2

(eE⊥a)2

ε

,

(2.89)

and therefore the integral equation 2.87 yields (cid:104)τR(ε)(cid:105) = 8¯hα/(eE⊥)2 which depends only

on the holding field.

Figure 2.13: Collision time τ as a function of temperature for various holding electric fields
Fz (from Ref. [10]).

41

2.5 Superfluid Liquid Helium

To understand the unique properties of 2D electrons on helium, it is crucial to understand

the physics of the underlying liquid 4He substrate supporting the surface electrons.

In

the experiments described in this thesis, the surface electrons are formed on liquid 4He at
T (cid:39) 1.55 K. At this temperature, the liquid helium is composed of mostly superfluid. Thus,

we first start with a brief review of superfluid 4He in subsection 2.5.1. In subsection 2.5.2,

we will discuss helium film formation in the different geometries used in this Ph.D. project.

This is important since a liquid helium thickness smaller than a SAW wavelength is required

to observe the SAW-electrons coupling described in chapter 7 . The last subsection 2.5.3

will describe the decrease in the helium film thickness by surface electrons and the critical

electron density at which the hydrodynamic instability of the film appears.

2.5.1 Superfluidity of Liquid 4He

Helium exhibits remarkable properties as a quantum substance. Unlike all other liquids, it

remains a quantum liquid even at T = 0 K at ambient pressure. To solidify the helium, one

needs to apply a large pressure as shown in Fig. 2.14. This unique feature originates from the

very weak binding force between helium atoms and the large quantum mechanical zero-point

energy [11]. In particular, the neutral 4He atom, composed of 2 protons and 2 neutrons in

the nucleus and 2 electrons orbiting, has zero net spin and forms a bosonic liquid below the

boiling point Tb = 4.21 K, which is governed by Bose-Einstein statics. When cooled down

to the “lambda point” Tλ = 2.17 K, a remarkable transition takes place (Fig. 2.14) related

to Bose-Einstein condensation and some portion of the liquid becomes a superfluid with
vanishing viscosity as T → 0. In addition to zero viscosity, superfluids have several unusual

42

Figure 2.14: Phase diagram of 4He (from Ref. [11]).

properties, which are summarized below.

Specific Heat

Fig. 2.15 shows an anomalous increase of the specific heat of liquid 4He at the transition

to the superfluid near 2.18 K, which resembles the Greek letter “λ” and is thus called the

λ-transition. This astonishing behavior was first measured by L.J. Dana and H. Kamerlingh

Onnes in 1920 although they didn’t publish the result. Later, this observation was explained

with two-fluid theory [94] and subsequently by Landau introduced the concept of quasi-

particles rotons [95]. Above the λ-transition, 4He behaves like a classical fluid due to very

low density. Below the λ-point, it becomes a bosonic quantum liquid exhibiting a rapidly

decreasing entropy and specific heat with decreasing temperature. In particular, the fast
decrease of the specific heat in the temperature range 1 − 2.18 K is associated with higher

43

energy rotons excitation. Below T = 0.6 K, the heat capacity decreases as T3 due to phonon

excitation.

Figure 2.15: Specific heat of liquid 4He as a function of temperature at saturated vapor
pressure. The specific heat has a sharp peak as T → Tλ. Figure taken from Ref. [12].

Thermal Conductivity

Below the λ-transition, the superfluid has an extremely high thermal conductivity hundreds

times larger than that of copper under the same conditions [96].

It is known that heat

transport in this regime occurs through unique thermal counterflow mode existing only in

superfluid helium, where the normal fluid moves away from the heat source and superfluid

moves toward it to maintain zero net mass flow. This leads to very effective thermal transport

although superfluid turbulence limits the heat flow [97]. This high thermal conductivity

makes superfluid helium a very efficient medium for creating a homogeneous temperature as

well as transporting heat at low temperatures.

44

Viscosity

In the superfluid state, the wave functions of each atom overlap and the fluid act as if they

were one macroscopic quantum system. Consequently, the atoms do not collide with each

other, which leads to zero friction or a vanishing viscosity. This allows for persistent flow

of superfluid helium similar to a persistent current in a superconducting state. Particularly,

frictionless superfluid can flow through any pore or hole in a container unless the fluid

velocity exceeds a critical value. Thus, the container is impermeable to normal fluid helium
for T ≥ 2.18 K but may show a superfluid helium leak below ∼2.18 K. Also, if two containers

having different superfluid level are connected, frictionless flow of superfluid film between

them is driven by the difference in gravitational potential and equalizes their levels. For an

open container, a film of superfluid will creep out and escape.

Sound Waves

Neglecting viscosity term at low temperature, the two-fluid model gives two wave equations

[94]:

∂2ρ
∂t2 = ∇2P
∂2S
∂t2 =

ρs
ρn

S2∇2T,

(2.90)

(2.91)

where the total density ρ of the liquid helium is the sum of the densities of the normal fluid

ρn and superfluid component ρs, P the pressure, and S the entropy. These equations yield

two solutions with two different sound velocities:

(cid:18) ∂P

(cid:19)1/2

(cid:19)1/2

(cid:18) ρs

ρn

TS2
Cp

(2.92)

u1 (cid:39)

∂ρ

S

u2 (cid:39)

,

45

Here P and Cp are the pressure and the specific heat of the fluid at constant pressure,

respectively. u1 is the velocity of ordinary density waves in a fluid driven by changes in

pressure, which is referred to first sound. In this wave, the two components of the fluid move

in phase with each other (i.e. the velocities of the two components are the same (cid:126)vs = (cid:126)vn).

Below the lambda point (T < Tλ), there are entropy waves driven by temperature differences

with velocity u2, which are called second sound. This is a quantum mechanical phenomena

in which the two components are out of phase with constant total density ∂ρ/∂t = 0. The
second sound velocity u2 is zero at the lambda point and reaches ∼20 m/s at T (cid:39) 1.8 K,
which is approximately ten times slower than first sound velocity u1 (cid:39) 2.3 × 102 m/s [12].

In 1959 Kenneth Atkins predicted two more sound waves in liquid helium below the lambda

point, which he called third sound and fourth sound [98]. Few years later, both third and

fourth sound were experimentally observed by Everitt et al. [99] and Rudnick and Shapiro

[100]. Third sound is a long-wavelength surface wave on a thin film of the liquid helium,

in which the superfluid component oscillates but the normal component remains locked

to the substrate. Fourth sound is a pressure and density wave similar to the first sound.

However, this compressional wave only propagates in the superfluid component while the

normal component remains stationary. Such a condition can be achieved in porous media

such packed powders [100]. The velocities of the third and the fourth sound are given by

[12, 100]

3 (cid:39) 3α/d3 for a thin film (d < 30˚A),
u2
1 (u1 (cid:29) u2),
u2

u2
4 =

u2
1 +

ρs
ρ

ρn
ρ

2 (cid:39) ρs
u2
ρ

(2.93)

(2.94)

where α is the van der Waals constant.

46

2.5.2 Helium Film Formation

Inside a container partially filled with liquid helium, adsorption of helium atoms from the

vapor phase leads to formation of vertical liquid helium film on the wall. Finite viscosity

usually prevents movement of such a film. However, in the superfluid state, the vanishing

viscosity allows the film to crawl up the wall as mentioned before. The film thickness is

closely related to the van der Waals potential (between the helium and the walls of the

container or substrate surface) which depends on the height above a bulk liquid helium

reservoir. At a height h above bulk helium (Fig. 2.16 (a)), the chemical potential of the film

can be written as [11]

µf = µ0 + ρgh − α/dn,

(2.95)

where µ0 is the chemical potential of the bulk helium and ρgh is gravitational potential. The

last term in equation 2.95 represents the van der Waals potential of the substrate, which

attracts the liquid helium film. The thickness of the film d can be determined using the

thermal equilibrium condition (µf = µ0):

d =

(cid:18) α

(cid:19)1/n

ρgh

.

(2.96)

The exponent n depends on 4He film thickness. For large film thickness, n is 4, whereas n is

3 for the thin film case. For a metal substrate, the characteristic thickness of the transition
region between n = 3 and n = 4 is theoretically predicted to be ∼50 nm [101].

In a micro-channel geometry, 1 − 2 µm thick liquid helium film can be formed in the

channel by capillary action as shown in Fig. 2.16 (b). This type of device has several merits

for studying surface electrons on helium, including a very uniform helium thickness, high

47

Figure 2.16: Helium film formation: (a) Van der Waals film on a vertical wall of the substrate.
µ0 and µf are the chemical potential of the bulk helium and the thin film of helium located
at height h above the bulk level. (b) Helium film formation inside the channel of the micro-
channel array by capillary force. Here w, d0, d, and h represent channel with, channel depth,
helium film thickness in the channel, and the distance between the channel top and the bulk
helium level in the reservoir.

mobility similar to that on bulk helium [102], increasing hydrodynamic stability [103], and

well-confined surface electrons regions. Assuming that the profile of the liquid in the channel

has a semicircular form with the curvature radius R, the thickness of the liquid film d(x) at

an arbitrary position x in the channel (see Fig. 2.16 (b)) is given by

d(x) = d0 − z(w/2) + z(x)

(2.97)

(cid:16)

1 −(cid:112)1 − (x/R)2(cid:17)

and R = σ/ρgh. When h = 1 mm, the radius R ≈ 270 µm
with z(x) = R
using σ = 0.378× 10−3 N/m and ρ = 0.145 g/cm3. Generally, R is much larger than a chan-

nel width w of a few tens of micrometers, forming a self-stablized helium film in the gap.
The parameter z(x) has the approximate form z(x) (cid:39) x2/2R for this case. This simplifies

equation 2.97 to

d(x) = d0 − w2

8R

+

x2
2R

48

(2.98)

When h → 0 and R → ∞, the channel is fully filled with the film having thickness d(x) = d0.
At the center of the channel (x = 0), the film thickness is d(x = 0) = d0 − w2/8R.

2.5.3 Charged Helium Film and Stability

When electrons are deposited onto the surface of the helium film, the charged helium surface

becomes more unstable and the film thickness is reduced by the electrostatic pressure of

the electrons. In particular, an electrohydrodynamic (EHD) instability can be caused by

surface electrons and places a limit on the maximum surface electron density that can be

achieved. This will be briefly reviewed first, followed by a discussion of the charged helium

film thickness.

EHD Instability and Critical Electron Density

In the presence of electrons above a bulk helium surface, the dispersion relation of the

ripplons (capillary waves) in equation 2.77 is modified into [68]

(cid:18)

(cid:19)

Ω2

q =

gq +

σq3
ρ

− 4πe2n2

e

ρ

q2

tanh(qd),

(2.99)

This indicates a softening of the capillary waves spectrum due to the presence of electrons.

By solving equation 2.99, one can find the wave number qc at which the instability first

occurs (Ω2

q < 0), which is given by

qc =

(cid:114) gρ

σ

The corresponding critical electron density is given by

nc =

(gρσ)1/4
(2πe2)1/2

,

49

(2.100)

(2.101)

which theoretically predicts the maximum attainable density on the surface of bulk helium.
For liquid 4He, a surface tension σ (cid:39) 0.37 dyn/cm and mass density ρ (cid:39) 0.145 g/cm3 yield
qc (cid:39) 20 cm−1 and nc (cid:39) 2.2 × 109 /cm2.

For a superfluid thin film formed by the Van der Waals interaction, the attraction between

the helium and substrate stabilizes the charged surface and thus the capillary waves spectrum

may be written as [68]

Ω2

q =

(cid:19)

(cid:20)(cid:18) 3α

ρd4 + g

q +

σ
ρ

q3 − 4πe2n2

e

ρ

(cid:21)

εsq2

tanh(qd)

(2.102)

when qd (cid:28) 1. Here α and εs denote the Van der Waals constant and the dielectric constant

of the solid substrate, respectively. The electrohydrodynamic instability occurring at the

wave number qc =(cid:112)3α/σ/d2 leads to the maximum attainable electron density, which is

given by

(cid:115)

nc = 1/4

3ασ

(2πe2εsd2)2

(2.103)

For a glass substrate covered with 10 nm thick helium film, εs (cid:39) 5 and 3α/ρgd4 (cid:39) 2 × 108
[68] yield nc (cid:39) 1.2 × 1011/ cm2 and qc (cid:39) 2.7 × 105 cm−1 respectively. Furthermore, Hu and

Dahm theoretically showed that a charged helium film can be stable at any electron density

and that the maximum density is only restricted by tunneling to the substrate [104]. This

is expected to become important when d < 35 ˚A. The corresponding maximum density is
∼1.5 × 1011 cm−2 for a metal substrate [104], in an agreement with experiment [13].

For a charged micro-channel geometry, the small surface area of liquid helium in each
channel leads to a minimum value of the wave vector qm (cid:39) π/w much larger than qc = 20 cm−1

near which the instability occurs for the bulk helium case. For instance, a channel width
w = 10 µm in Fig. 2.16 (b) yields qm (cid:39) 3.1 × 103 cm−1 (cid:29) qc = 20 cm−1. This inequality

50

q ≥ 0 in equation 2.99 when q ≥ qm and thus the EHD instability is forbidden in
can make Ω2
this type of geometry. When neglecting the gravity term in equation 2.99 since q ≥ qm (cid:29) qc,
e ≥ 0, which results in a critical density
the condition for stability reduces to σq − 4πe2n2

nc =(cid:112)σqm/4πe2. For w = 10 µm, nc (cid:39) 2× 1010 cm−2 is an order of magnitude larger than

the above bulk helium case (nc (cid:39) 2.2× 109 cm−2). In fact, this maximum density is slightly

reduced by the depression of the helium film, which will be discussed in the next subsection.

Film Thickness Depression From Electron Charging

Etz et. al measured the maximum density of surface electrons on a liquid 4He film formed by

the Van der Waals interaction and found that their experimental value was significantly larger

than theoretical value determined by equation 2.103 [13]. This discrepancy was ultimately

resolved by taking into account the film thickness depression due to electron pressure.

For the uncharged Van der Waals helium film with a thickness di ranging from 200 to

400 ˚A, the equilibrium thickness after charging is given by [13]

(cid:18)

(cid:19)−1/3

d = di

1 +

ee2
2πn2
ρgh

,

(2.104)

with the assumption that di = (σ/ρgh)1/3 before charging. Fig. 2.17 shows the experimental

confirmation of the change in charged film thickness governed by equation 2.104.

For the micro-channel geometry, the electron pressure affects the curvature of the helium

film leading to [103]:

R =

σ

ρgh + 2πn2

ee2 .

(2.105)

Inserting equation 2.105 into equation 2.98, the charged film thickness with electron density

51

Figure 2.17: Thickness d of charged saturated 4He films wetting a glass substrate (T = 1.6 K)
versus the electron density of the films. The thickness of d0 of the uncharged films was 220 ˚A
and 420 ˚A, respectively. Figure taken from Ref. [13].

ne is given by

d(ne) = d0 − w2

8σ

(ρgh + 2πn2

ee2) at the center (x = 0).

(2.106)

The maximum electron density nc can then be obtained from the condition d(nc) = 0 for
the region where ρgh (cid:28) 2πn2

ee2. This is given by

(cid:114)

nc =

d0

8σ
w2

1
2πe2 .

(2.107)

For w = 10 µm and d0 = 1 µm, this yields nc (cid:39) 1.4 × 1010 cm−2, which is slightly smaller
than the critical density determined by the EHD instability nc (cid:39) 2 × 1010 cm−2 discussed

above.

52

Chapter 3

Surface Acoustic Waves (SAWs)

Figure 3.1: Schematic of the SAW-2DES on helium experiment. A SAW is launched from
the left interdigitated transducer (IDT) and can, via an evanescent coupling to the electrons,
transport electrons in the direction of the travelling wave. A second IDT to the right can
act as a SAW detector or acoustic Bragg reflector.

A surface acoustic wave (SAW) is an elastic wave propagating along the surface of a

substrate. If the substrate has piezoelectricity, the SAW is accompanied by a co-propagating

electric field, which can interact with a 2D electron system (2DES) located within about

one wavelength above or below the surface (see Fig. 3.1). Not only is this interaction a

versatile tool for study interesting physics in 2D materials [29, 30, 31, 32, 35] but it also can

potentially form the basis of single electron devices for metrological current standards [105]

or quantum information processing systems [106, 107]. Recently, SAW coupling to quantum

systems has also attracted great interest in quantum acoustics with superconducting circuits

53

[108, 109, 110].

In this chapter we introduce the relevant properties of surface acoustic waves needed to

understand our SAW experiments with electrons on helium discussed in chapters 6 and 7. In

section 3.1, we start by describing SAW propagation on a piezoelectric substrate. Specifically,

we will solve the relevant wave equation and obtain the solution for surface acoustic waves

taking into account the appropriate boundary conditions.

It should be pointed out that

we will only deal with Rayleigh SAW, where the particle motion in the substrate lies in the

sagittal plane (partially normal and partially perpendicular to the surface), since this mode is

employed in our experiments. In section 3.2, we will describe the coupling of the SAW with a

2DES nearby the surface. Here our interest is restricted to how the 2DES electrically affects

the SAW propagation and the mechanical effect on the SAW by an adjoining medium (in

our case liquid helium) will be discussed later in chapter 6. Strong coupling leads a fraction

of electrons in the 2DES to be trapped by the SAW potential minima and transported at

speed of the sound in the substrate, which leads to acoustoelectric charge transport. This

latter effect will be described in detail in section 3.3.

3.1 Piezoelectric SAWs

Piezoelectricity is an electric voltage generated in certain solid materials when mechanical

stress is applied to them. Reciprocally, the application of the electric voltage can also

produces strain in these piezoelectric materials. In these cases, energy is transferred from

mechanical to electrical energy or vice versa. The constitutive equations of piezoelectricity

54

are given by

Tij = cijklSkl − ekijEk,

Dk = ekijSij + εkiEi.

(3.1)

(3.2)

Here cijkl is the fourth-order stiffness tensor, ekij the piezoelectric tensor, and εki the dielec-

tric tensor while Tij, Dk, Skl, and Ek represent the stress tensor, displacement fields, the

strain tensor, and electric fields, respectively. For a cubic crystal structure, these constitutive

equations have the following matrix form:



0

0

0

0

e14

0

0

0

0

0

0

e14



E1

E2

E3



 .



E1

E2

E3

ε 0 0

0 ε 0

0 0 ε

 ,

(3.3)

(3.4)



0

0






−



S11

S22

S33

2S23

2S13

2S12



S11

S22

S33

2S23

2S13

2S12



+

e14

0

0

0

0

0



e14

55





T11

T22

T33

T23

T13

T12

=




c11

c12

c12

c12

c12

c11

c12

c12

c11

0

0

0

0

0

0

0

0

0

0

0

0

c44

0

0

0

0

0

0

c44

0

0

0

0

0

0

c44

 =



0 0 0 e14

0 0 0

0 0 0

0

0

0

e14

0

D1

D2

D3

In matrix notation the stress tensor and displacement fields can be reduced to the following

form for a material possessing cubic symmetry



,

 =



D1

D2

D3



2e14S23 + εE1

2e14S13 + εE2

2e14S12 + εE3.



(3.5)





T11

T22

T33

T23

T13

T12



c11S11 + c12S22 + c12S33

c12S11 + c11S22 + c12S33

c12S11 + c12S22 + c11S33

=

2c44S23 − e14E1
2c44S13 − e14E2
2c44S12 − e14E3

3.1.1 Wave Equations

In the presence of the piezoelectric effect and no external forces, wave equations in the

piezoelectric medium can be written as [111]

ρ¨ui = ∂jTij = ∂jTji,

∂kDk = 0,

(3.6)

where ρ is the mass density, ui is the displacement component parallel to the Cartesian axis

xi (i.e. x1 = x, x2 = y, x3 = z), and we have used the notations ∂jf = ∂f /∂xj,

˙f = ∂f /∂t.

Displacement components ui,j have a relationship with the strain tensor component Sij as

follows:

(cid:0)∂iuj + ∂jui

(cid:1) .

1
2

Sij = Sji =

(3.7)

56

Inserting equations 3.5 and 3.7 into equation 3.6, one obtains three partial wave equations

in the cubic crystal:

ρ¨u1 = c11∂2

1u1 + c44(∂2

2u1 + ∂2

3u1) + (c12 + c44)(∂1∂2u2 + ∂1∂3u3) + 2e14∂2∂3φ,

ρ¨u2 = c44∂2

1u2 + c11∂2

2u2 + c44∂2

3u2 + (c12 + c44)(∂1∂2u1 + ∂2∂3u3) + 2e14∂1∂3φ,

(3.8)

1u3 + c44∂2

ρ¨u3 = c44∂2
ε∇2φ = 2e14(∂2∂3u1 + ∂1∂3u2 + ∂1∂2u3),

2u3 + c11∂2

3u3 + (c12 + c44)(∂1∂3u1 + ∂2∂3u2) + 2e14∂1∂2φ,

where φ is the electric potential such that Ei = −∂iφ. The plane waves solutions ui and φ

have the form:

ui = u0i e−k αz eik (lx+my−ct),
φ = φ0 e−k αz eik (lx+my−ct),

(3.9)

with l = cos(θ) and m = sin(θ). Here (cid:126)k = (kx, ky) is an in-plane wave vector, α is the decay

constant associated with loss of SAW energy, θ is the angle between the x-axis and (cid:126)k , the

positive z-axis points into the piezoelectric medium, and c is the phase velocity of the wave

(see Fig. 3.2). With solutions of this form, the differential operators in equation 3.8 can be

written as

∂1 → ik l,

, ∂2 → ik m,

∂3 → −k α,

∂t → −ik c.

(3.10)

Substituting equation 3.9 into the partial wave equations 3.8 and applying the equivalent

forms of the operators in equation 3.10, one can get the following matrix form of the wave

57

Figure 3.2: Coordinate system for a piezoelectric SAW.

equations:

W



u01

u02

u03

φ0

 =

 ,



0

0

0

0

where

W =

 c11l2+c44m2−ρc2−c44α2

(c12+c44)lm
(c12+c44)lα
−2e14imα

(c12+c44)lm

(c12+c44)lα −2e14imα
c44l2+c11m2−ρc2−c44α2 (c12+c44)mα −2e14ilα
ρc2+c11α2−c44 −2e14lm
iε(α2−1)

(c12+c44)mα

−2e14lα

2e14ml

(3.11)

 .

(3.12)

For a non-trivial solution, the determinant of the matrix W should be zero (|W|= 0). This

is equivalent to an eighth order equation for the decay constant α with real coefficients.
Therefore, the eight roots αj of the secular equation |W|= 0 are real or occur in conjugate

pairs. Our interest is the surface wave mode which exponentially decays into the bulk of the

substrate with increasing z (Fig. 3.2) and thus we choose only four of them αj (j = 1, 2, 3, 4)

which have a positive real part. For a given propagation direction θ, αj depends on the phase

58

velocity c as shown in equation 3.12. This velocity can be determined from the application

of appropriate boundary conditions, which will be discussed later. Now the general solution

for the displacements ui and the potential φ can be written as a linear combination of

equation 3.9 in terms of different αj’s and is given by

4(cid:88)
4(cid:88)

j=1

ui =

φ =

−k αj z eik (lx+my−ct),

u

(j)
0i e

−k αj z eik (lx+my−ct).

(j)
0 e

φ

(3.13)

j=1

For each αj, the ratios of amplitude u

(j)
01 , u

(j)
02 , u

(j)
(j)
03 , and φ
0

are related to each other as

follows.

(j)
u
01
Aj

=

(j)
u
02
Bj

=

(j)
u
03
Cj

=

(j)
φ
0
Fj

= Kj,

(3.14)

59

where

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

Aj =

Bj =

Cj =

Fj =

c44l2 + c11m2 − ρc2 − c44α2

j

(c12 + c44)mαj

−2e14lαj

(c12 + c44)mαj −2e14ilαj
j − c44 −2e14lm
ρc2 + c11α2
j − 1)

2e14ml

iε(α2

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

(c12 + c44)lm (c12 + c44)mαj −2e14ilαj
j − c44 −2e14lm
(c12 + c44)lαj ρc2 + c11α2
−2e14imαj
j − 1)
(c12 + c44)lm c44l2 + c11m2 − ρc2 − c44α2

2e14ml

iε(α2

(c12 + c44)mαj

(c12 + c44)lαj
−2e14imαj
(c12 + c44)lm c44l2 + c11m2 − ρc2 − c44α2

−2e14lαj

j

(c12 + c44)lαj
−2e14imαj

(c12 + c44)mαj

−2e14lαj

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

j −2e14ilαj
−2e14lm
j − 1)
iε(α2

(c12 + c44)mαj
j − c44

ρc2 + c11α2

2e14ml

(3.15)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) .

Then the general solution for the substrate atomic displacements and potential is given by

4(cid:88)

(u1, u2, u3, φ) =

(Aj, Bj, Cj, Fj) Kj e

−k αj z eik (lx+my−ct),

(3.16)

j=1

where Kj is only unknown amplitude factors. Their ratios will also be determined by bound-

ary conditions discussed below in section 3.1.2.

60

3.1.2 Boundary Conditions

In order to proceed with our analysis of the solutions in the previous section, we must apply

appropriate boundary conditions. A mechanical boundary condition arises from the fact that

the stress on the free surface (z = 0) must be zero. Such a stress free condition is equivalent

to setting T3j = 0 at z = 0. From equations 3.5 and 3.7, this boundary condition can be

written as:

T31(z = 0) = c44(∂1u3 + ∂3u1) + e14∂2φ = 0

T32(z = 0) = c44(∂2u3 + ∂3u2) + e14∂1φ = 0

(3.17)

T33(z = 0) = c12∂1u1 + c12∂2u2 + c11∂3u3 = 0.

Additionally, in the vacuum above the piezoelectric medium, the potential φ(cid:48) must satisfy

Laplace’s equations and must be continuous across the interface at z = 0 if no surface charges
exist. Thus, the potential in the vacuum φ(cid:48) can be written as

4(cid:88)

φ(cid:48)(x, y, z, t) = φ(x, y, 0, t) exp(k z) =

Fj Kj ek z eik (lx+my−ct)

(z ≤ 0).

(3.18)

j=1

In addition, the normal component of the displacement fields must also be continuous at the

interface (z = 0):

(vacuum),

D3|z<0 = −ε0∂3φ(cid:48)
D3|z>0 = e14(∂1u2 + ∂2u1) − ε∂3φ (piezo-medium),
−ε0∂3φ(cid:48) = e14(∂1u2 + ∂2u1) − ε∂3φ at z = 0 (interface),

(3.19)

61

where ε0 and ε denote the dielectric constant of the vacuum and the piezoelectric medium,
respectively. Inserting ∂3φ(cid:48) = k φ(cid:48) and φ(cid:48)|z=0 = φ|z=0 into equation 3.19, one can rewrite

the electrical boundary condition as

e14(∂1u2 + ∂2u1) + ε0k φ − ε∂3φ = 0 at z = 0.

(3.20)

If the assumed solution given in equation 3.16 is substituted into the boundary conditions

equations 3.17 and 3.20, we can obtain the matrix form of the combined mechanical and

electrical boundary conditions:



T

(1)
31

T

(1)
32

T

(1)
33

T

(2)
31

T

(2)
32

T

(2)
33

T

(3)
31

T

(3)
32

T

(3)
33

T

(4)
31

T

(4)
32

T

(4)
33

D

(1)
3 D

(2)
3 D

(3)
3 D

(4)
3

 , B



K1

K2

K3

K4

 = 0,

B =

where

(3.21)

(3.22)

(j)

31 = −c44αjAj + c44lCj + e14imFj, T

(j)

32 = −c44αjBj + c44mCj + e14ilFj,

T

(j)
33 = c12lAj + c12mBj + c11αjCj, D
T

(j)
3 = e14imAj + e14ilBj + εαj + ε0.

For non-trivial solutions of equation 3.21, the determinant of the matrix B must vanish.

Since the amplitudes Aj, Bj, Cj, and Fj can be complex numbers, the determinant is

in general a complex number. Therefore, the wave velocity can be determined from the

condition where the both real and imaginary parts of the determinant vanish independently.

Substituting the determined velocity into equations 3.12 and 3.15 for a given propagation

angle θ, the decay constant αj and amplitudes Aj, Bj, Cj, and Fj can be calculated. Only

62

the amplitude cofactor Kj remains to be found to completely specify the plane wave solution

given in equation 3.16. However, we can only calculate the relative value of each Kj rather

than their absolute value, i.e. only the ratios of Kj can be calculated based on the following

relations.

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

K1

=K3

T

(2)
32

T

(2)
33

T

(3)
32

T

(3)
33

T

(4)
32

T

(4)
33

D

(2)
3 D

(4)
(3)
3 D
3

T

(1)
32

T

(1)
33

T

(2)
32

T

(2)
33

T

(4)
32

T

(4)
33

D

(1)
3 D

(2)
(4)
3 D
3

−1

−1

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

= K2

= K4

T

(1)
32

T

(1)
33

T

(3)
32

T

(3)
33

T

(4)
32

T

(4)
33

D

(1)
3 D

(4)
(3)
3 D
3

T

(1)
32

T

(1)
33

T

(2)
32

T

(2)
33

T

(3)
32

T

(3)
33

D

(1)
3 D

(2)
(3)
3 D
3

−1

−1

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

(3.23)

= K.

Note that the determinant of each 3 × 3 matrix in equation 3.23 can be also calculable if

the velocity and the angle are given. Therefore, the wave solution ui and φ have only one

unknown amplitude factor K.

3.1.3 SAWs Along [011] Cut of GaAs (100)

When the SAW propagates on the (100) plane of GaAs along the [011] direction (θ = 45◦

√
and l = m = 1/
2 in Fig. 3.3), it is convenient to solve the wave equation using a 45 degree
rotated coordinate system, where rotated axes x(cid:48) and y(cid:48) make angles π/4 and 3π/4 with

the original x axis. This propagation direction is the same as in our SAW experiments on a

GaAs substrate, which will be discussed in a chapter 7. For the Rayleigh mode, the atomic

displacements are confined to the sagittal plane with one transverse (u3) and one longitudinal
component (u(cid:48)

1) and thus the matrix form of the wave equations given in equations 3.11 and

63

Figure 3.3: Schematic of the SAW propagation along GaAs (100) plane. The direction of
the propagation is [011].

3.12 reduce to:

W =



ρc2 − c11 + c12

− c44 + c44α2

2

−(c12 + c44)iα

−(c12 + c44)iα
−c44 + c11α2 + ρc2

−2e14iα
−e14

 ,

ε(α2 − 1)

(3.24)

e14

2e14iα

 =



u01(cid:48)

u03

φ0

W





0

0

0

Here u01(cid:48) represents the amplitude of the displacement along x(cid:48)-axis, u1(cid:48). Here we have

performed the transformation into the rotated frame using the below relation [112]:

y(cid:48) = 0,

u2(cid:48) = 0,

∂2(cid:48) = 0,

√
u1 = u2 = u1(cid:48)/
2,

√
2,
∂1 = ∂2 = ik /

with 1(cid:48) → x(cid:48) and 2(cid:48) → y(cid:48). From the condition that |W| = 0, we can obtain the appropriate

solution of equation 3.24, αj = αj(c), j = 1, 2, 3, which have a positive real part as discussed

64

previously. The corresponding general solution is given by

3(cid:88)

(u1(cid:48), u3, φ) =

(u

(j)

01(cid:48), u

(j)
03 , φ

(j)
0 ) e

−k αj z eik (x(cid:48)−ct),

j=1

where

u

(j)

01(cid:48) = AjKj, u

(j)
03 = CjKj, φ

(j)
0 = FjKj.

(3.25)

(3.26)

(cid:35)

,

(3.27)

Similarly, the matrix form of the boundary conditions in equation 3.21 is modified and

becomes

(cid:34) −c44α1A1+c44iC1+ie14F1 −c44α2A2+c44iC2+ie14F2 −c44α3A3+c44iC3+ie14F3

B =

ic12A1−c11α1C1
ie14A1+(εα1+ε0)F1

ic12A3−c11α3C3
ie14A3+(εα3+ε0)F3

ic12A2−c11α2C2
ie14A2+(εα2+ε0)F2

 = 0.



K1

K2

K3

B

For GaAs, the mass density is ρ = 5.307× 103 kg/m3, the elastic constants c11, c12, and c44
given by 12.26 × 1010, 5.71 × 1010, and 6.0 × 1010 N/m2, the dielectric constant is ε (cid:39) 13,

and the piezoelectric constant is e14 = 0.16 C/m2 [14]. As shown in Fig. 3.4, the condition
for a non-trivial solution i.e. Re(|B|) = Im(|B|) = 0, yields a phase velocity of the SAW
c (cid:39) 2880 m/s for αj(c) obtained from equation 3.24 with the above material parameters.
This is in a good agreement with the experimental value of c (cid:39) 2863 m/s [113]. With
c (cid:39) 2880 m/s, the vanishing determinant of the matrix W in equation 3.24 yields a numerical

65

Figure 3.4: The real and imaginary components of the determinant of B as a function of the
velocity c for GaAs (100) cut [011] propagation direction. At c (cid:39) 2880 m/s, both components
are equal to zero.

value of decay constants:

α1,2 = 0.497292 ± 0.481904i,

α3 = 1.

j + ρc2

ε(α2

−e14
j − 1)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , Cj =
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)−(c12 + c44)iα −c44 + c11α2

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)−(c12 + c44)iαj
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

j + ρc2

2e14iαj

2e14iαj

e14

Also, from |W|= 0, the amplitude factors Aj, Bj, and Fj are given by

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)−c44 + c11α2

e14

Aj =

Fj =

(3.28)

(3.29)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

−e14
j − 1)

ε(α2

66

And, from |B|= 0, another amplitude factor Kj is obtained as

K1 =

K2 =

K3 =

ic12A2 − c11α2C2

ic12A3 − c11α3C3

ie14A2 + (εα2 + ε0)F2

ie14A3 + (εα3 + ε0)F3

ic12A1 − c11α1C1

ic12A3 − c11α3C3

ie14A1 + (εα1 + ε0)F1

ie14A3 + (εα1 + ε0)F3

ic12A1 − c11α1C1

ic12A2 − c11α2C2

ie14A1 + (εα1 + ε0)F1

ie14A2 + (εα2 + ε0)F2

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) K,
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) K,
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) K.

(3.30)

After the amplitude ratios are calculated using equations 3.29, 3.30, and 3.26 with c (cid:39) 2880 m/s

and the value of αj given in equation 3.28, we can numerically estimate the spatial depen-

dence of the plane wave solution of equation 3.25 with one amplitude variable (K). Fig. 3.5

shows the calculation of displacements u1(cid:48), u3 and the potential φ as a function of k z. In this

Figure 3.5: Atomic displacements (a) and electric potential (b) of a Rayleigh SAW versus the
distance from the surface of GaAs (100) cut [011] propagation direction. u1(cid:48) and u3 denote
longitudinal and transverse components of the displacements, respectively. A positive value
of z indicates a location inside the GaAs substrate while a negative value represents the
vacuum above the GaAs surface.

figure, both displacements and the potential survive within approximately one wavelength

(k z = 2π) below the GaAs surface, indicating that the SAW is localized near the surface.

67

In particular, the SAW potential (red trace in Fig. 3.5 (b)) exhibits sinusoidal oscillation

and decay into bulk. This is a direct consequence of the complex decay constants αj inside

the piezoelectric material. Above the GaAs surface with no free charge, the decay constant

must be equal to 1 to satisfy Laplace’s equations, which causes the SAW potential to exhibit

monotonic exponential decay into vacuum (blue trace in Fig. 3.5 (b)).

3.2 SAW Coupling to a 2DES

In the section 3.1, we have discussed SAW propagation along a piezoelectric surface in contact

with vacuum. When the adjoining medium is a 2DES or a thin conducting layer, the SAW

can be coupled to it through the piezoelectric field. This leads to a velocity shift and an

attenuation of the SAW, given by the following relation [14]:

K2
2
K2
2

Γ = k

∆v
v

=

σ/σm

1 + (σ/σm)2 ,

1

1 + (σ/σm)2 ,

(3.31)

(3.32)

where Γ is the SAW attenuation per unit length, v is the SAW velocity, k is the SAW

wave number, K2 is the piezoelectric coupling constant, σ is the sheet conductivity of the

2D layer, and σm = v(ε0 + ε)/4π is a characteristic conductivity. Here ε is the dielectric

constant of the piezoelectric substrate and ε0 is the dielectric constant of the medium above
the surface. The velocity change ∆v is defined as ∆v ≡ v − v0, where v0 is the velocity
when the surface is metallized (i.e. σ → ∞). Fig. 3.6 shows the attenuation and the

velocity shift versus the sheet conductivity of the 2DES calculated from equations 3.31 and

3.32. For σ far below σm, both the attenuation and the velocity change are negligibly

68

Figure 3.6: SAW attenuation and the velocity shift as a function of the 2DES conductivity.

affected by the 2DES. As σ approaches σm, both Γ and δv/v are strongly modified. The

maximum attenuation happens when σ = σm and the velocity changes in a steplike fashion

near σ = σm. This behavior results from the screening of the piezoelectric potential by

mobile charges in the 2DES and the physical meaning of σm is well described in terms of

a transmission line model. In this model, σm represents the wave impedance of the SAW.

When the impedance is matched (σ = σm), the energy transfer from the SAW to the 2DES is

maximized. For the (100) GaAs surface with a SAW propagating along the [011] direction,
K2/2 = 3.2 × 10−4 and σm = 3.3 × 10−7 Ω−1 [30]. For YZ-cut LiNbO3 where the SAW
travels on the (010) surface along the [001] direction, K2/2 is two orders of magnitude larger
than GaAs, K2/2 = 4.82 × 10−2 [114], and σm = 2.1 × 10−6 Ω−1 [115].

3.2.1 SAW–2DES Interaction Inside a Piezoelectric Substrate

The strength of the SAW-2DES coupling is strongly modified depending on the location of

the 2DES below or above a piezoelectric surface and its distance d from that surface. In

this section, we describe how K2/2 and σm change as a function of d when the 2DES is

buried below the surface of a piezoelectric substrate such as those formed in GaAs/AlGaAs

69

heterostructures. The SAW coupling to the 2DES located at or above the surface (in our

case electrons on helium) will be described in next section 3.2.2.

In case of SAW propagation along the (100) surface of Al0.37 Ga0.63 As with [011] direc-
tion, experimental values of K2/2 = 3.2 × 10−4 and σm = 4 × 10−7 Ω−1 [29] have been

measured very similar to pure GaAs case. Generally, a two-dimensional electron system

(2DES) at the GaAs/Alx Ga1−x As interface is buried below the surface at a depth d as

shown in Fig. 3.7. Both K2/2 and σm depend on the location d. Simon [14] theoretically

Figure 3.7: Schematic of the SAW propagation along the (100) surface of AlGaAs containing
a 2DES. For the calculation of K2/2 and σm in Ref.
[14] at a depth d, the bottom GaAs
layer is approximated by a AlGaAs layer due to their similar material properties.

estimated their depth dependence using a simplified model geometry where the actual 2DES

in a GaAs/Alx Ga1−x As heterostructure is approximated as existing in a homogeneous Al-

GaAs slab. This is a reasonable approximation since GaAs and Alx Ga1−x As have very

similar material properties. He showed that the depth dependence of K2/2 and σm are

given by

K2
2

=

εeff(kd)

4πH|C|2|φ(kd)|2,

σm =

vεeff(kd)

2π

,

70

(3.33)

(3.34)

with

(cid:18)

1 +

(cid:19)

,

ε − ε0
ε + ε0

e−2kd

1
εeff

=

1
ε

(3.35)

where C is the amplitude of the SAW and H is a material dependent constant which is

equivalent to H = U/kC2 [14] where U represents the energy of the SAW per unit surface
area. For Al0.3 Ga0.7 As, the elastic constants c11, c12, and c44 are given by 11.922 × 1010,
5.476×1010, and 5.925×1010 N/m2, the mass density ρ = 4.852×103 kg/m3, the piezoelectric

constant e14 = 0.1795 C/m2, and the dielectric constant ε = 12.048 [116]. Following the

same procedure we applied to GaAs (100) cut [011] propagation in the previous section with

slightly different material parameters listed above, one can obtain v = 2989 m/s and the

electric potential as

φ(kd) =

(cid:16)

with

(1)

0 e−α1kd + φ

(2)

0 e−α2kd + φ

φ

0 e−α3kd(cid:17)

(3)

eik(x−vt),

(3.36)

(1)

0 (cid:39) (−4.75 − 4.07i)A, φ

(2)

0 (cid:39) (−4.75 + 4.07i)A, φ

(3)

0 (cid:39) 12.47A

φ

α1 (cid:39) 0.50 − 0.48i, α2 (cid:39) 0.50 + 0.48i, α3 = 1,

For the case of the vacuum above the surface (i.e. ε0 = 1), inserting v = 2989 m/s in equa-

tion 3.35, we can calculate the characteristic conductivity σm as a function of kd as shown
in Fig. 3.8 (a). Using φ(kd) in equation 3.36 along with C (cid:39) (−2.37 × 102 + 1.36 × 102i)A
and H (cid:39) 2.4 × 1011 N/m2, the piezoelectric coupling constant K2/2 at any depth d can

be obtained as shown in Fig. 3.8 (b). The coupling constant K2/2 has a nonmonotonic

dependence on kd, which is attributable to complex decay constant αj of the potential. In

71

the limit of kd (cid:39) 0, the calculated values of σm and K2/2 are 3.5× 10−7 Ω−1 and 3.6× 10−4
respectively, in good agreement with experiment [29, 32]. With the depth-dependent K2/2

Figure 3.8: Characteristic conductivity σm (a) and coupling constant K2/2 (b) as a function
of kd are calculated using the approximated model geometry of Fig. 3.7. Here k and d
represent the in-plane wave vector and the depth of the 2DES from the piezoelectric surface,
respectively. The inset of (a) shows a change in effective dielectric constant with increasing
2DES distance from the surface. The blue trace represents the position of 2DES in vacuum
above the surface while the red trace stands for the 2DES position inside the piezoelectric
medium.

and σm as shown in Fig. 3.8, one can estimate the attenuation and the velocity shift of the

SAW for a 2DES at any depth in the GaAs/AlGaAs heterostructure using equations 3.31

and 3.32.

3.2.2 SAW–2DES Interaction Above a Piezoelectric Surface

For electrons floating on a thin helium film supported by a piezoelectric substrate, the 2DES

is located above the piezoelectric surface. For this case, the SAW attenuation Γ and the

velocity shift ∆v/v due to the nearby 2DES can be estimated using an electrical surface

perturbation method. We will limit our discussion to the weak coupling approximation

where the stress field is unchanged by the perturbation produced by the 2DES since this

technique shows the best agreement with exact numerical calculations [117]. In this method,

72

it is assumed that the surface of the piezoelectric substrate is mechanically free, reflecting

the fact that the SAW is not mechanically affected by the adjoining medium and is only
affected through the electric field. According to B.A. Auld [117], the perturbed potential φ(cid:48)
and perturbed displacement D(cid:48)

y due to the electrical surface perturbation are related to the

unperturbed fields φ and Dy:

φ(cid:48)|y=0 = φ|y=0 + A,
D(cid:48)
y|y=0 = Dy|y=0 + kεT

p A,

(3.37)

where εT

p is the dielectric constant of the piezoelectric medium under constant stress and is

given by

εT
p =

εT
yyεT

zz − (εT

yz)2.

(cid:113)

The parameter εT
ij is the dielectric tensor component, k is the unperturbed wave number,
and A is the amplitude. Here the positive y−axis points into the piezoelectric medium with
the surface located at y = 0 and the SAW propagates along z−axis. Also, the approximation
k(cid:48) = k has been made. The electrical boundary conditions can be conveniently expressed in

terms of the electrical surface impedance, which is given by

(cid:18) φ

(cid:19)

ZE(0) =

iωDy

y=0

,

(3.38)

where ω is the angular frequency of the SAW. For the unperturbed case, the region above
the substrate (y < 0) is a semi-infinite vacuum and thus the Laplace equation ∇φ = 0 yields

73

the unperturbed fields as

φ = ekyeikz,
Dy = −kε0ekyeikz.

Therefore, the unperturbed surface impedance ZE(0) is given by

ZE(0) = − 1
ikωε0

,

(3.39)

(3.40)

where ε0 is the dielectric constant of the vacuum. If we introduce the normalized perturbed

surface impedance defined by

z(cid:48)
E(0) =

Z(cid:48)
E(0)
|ZE(0)| = −ikε0

(cid:33)

(cid:32)

φ(cid:48)
D(cid:48)

y

,

y=0

(3.41)

equations 3.40 and 3.41 allow us to express the displacement fields Dy(0) and D(cid:48)

y(0) in terms

of potentials as follows:

D(cid:48)
y(0) = −ikε0φ(cid:48)(0)/z(cid:48)
Dy(0) = −kε0φ(0).

E(0),

(3.42)

Substitution of equation 3.42 into equation 3.37 and elimination of A give

φ(cid:48)(0) = −iz(cid:48)

E(0)

ε0 + εT
p

ε0 − iz(cid:48)

E(0)εT
p

φ(0).

(3.43)

74

1 + iz(cid:48)
E(0)
1 − iz(cid:48)
EεT
p /ε0

,

(3.46)

(cid:18)∆v

(cid:19)

v

SC

= −

∆k
k

(cid:18)∆v

(cid:19)

= −ω(ε0 + εT
p )

4P

|φ|2.

v

SC

equation 3.40 we obtain

where

(cid:18)∆v

(cid:19)

v

SC

The quantity

Inserting equation 3.43 into equation 3.41 yields

D(cid:48)
y(0) = −kε0

ε0 + εT
p

ε0 − iz(cid:48)

E(0)εT
p

φ(0).

(3.44)

The variation of the wave-vector due to electrical perturbation cant be written as [117]

(cid:32)

(cid:33)

∆k ≡ k(cid:48) − k (cid:39) ω

φ(cid:48)D∗

y − φ∗D(cid:48)

y

4P

,

y=0

(3.45)

where P is the average power flow in the z direction and the asterisk denotes complex

conjugation. By the substitution of equations 3.43 and 3.44 into equation 3.45 and use of

is the relative change in the SAW phase velocity due to a short

circuit at the surface (e.g. metalized surface) compared to the metal free surface case. This
has a negative value since v(cid:48) < v and thus ∆v ≡ v(cid:48) − v < 0. In this context, one can relate

the velocity change (∆v/v)SC to the piezoelectric coupling constant [114]:

(cid:18)∆v

(cid:19)

= −K2
2

v

SC

Accordingly, the perturbation can be written as

∆k
k

=

K2
2

1 + iz(cid:48)
E(0)
1 − iz(cid:48)
EεT
p /ε0

.

75

(3.47)

This is the Ingebrigtsen formula for electrical surface perturbations of the piezoelectric SAW.

In some limiting cases, the above perturbation is easily estimated. For instance, for a free
electrical boundary condition the unperturbed fields φ(0) and Dy(0) yield z(cid:48)
In the limit of a highly conducting surface, insertion of z(cid:48)

E(0) = i, leading

to ∆k/k = 0.

E(0) = 0 into

equation 3.47 yields

∆v
v

(= −∆k
k

) = −K2
2

.

In general, a perturbed surface impedance z(cid:48)
is also complex. If we let k(cid:48) = k + β + iα, the perturbed phase velocity is

E(0) is a complex number and accordingly ∆k/k

(cid:18)

(cid:19)−1 (cid:39) v

(cid:18)

(cid:19)

1 − β
k

v(cid:48) =

ω

k + β

=

ω
k

1 +

β
k

since β/k (cid:28) 1.

(3.48)

Here β and α are an additional phase shift and the attenuation constant for the perturbed

wave. These lead to the relation:

= 1 − v(cid:48)

v

β
k

v − v(cid:48)

v

=

= −∆v
v

,

where v is the unperturbed phase velocity. Now one can express ∆k/k as

∆k
k

=

k(cid:48) − k

k

=

β + iα

k

= −∆v
v

+ i

α
k

(3.49)

Therefore, the velocity shift and the attenuation of the SAW have the following relation:

(cid:19)

(cid:18)∆k
(cid:18)∆k
(cid:19)

k

∆v
v

α
k

= −Re

= Im

k

76

,

.

(3.50)

Note that a purely imaginary z(cid:48)

E(0) causes ∆k/k to be a real number in equation 3.47,

indicating no attenuation and only velocity shift. This is the case when the adjoining material

is lossless, i.e. an electrical insulator. However, when a thin insulating helium film covers

the piezoelectric surface, we observed a strong SAW attenuation. This suggests that a

mechanical effect from the liquid helium film on the SAW must be taken into account. A

possible mechanism for this mechanical effect will be discussed in chapter 6. In the presence

of surface electrons above the helium film, the SAW attenuation measurement is in good

agreement with this perturbation theory, which will be discussed in detail in chapter 7.

3.3 Acoustoelectric Charge Transport

The ability of a SAW potential to trap and transport charges at the speed of sound can

generate a local DC current density via acoustoelectric charge transport. In this section, we

will briefly describe the origin of this effect and how to experimentally measure it. When a
SAW propagates on the piezoelectric surface along the x−axis, the piezoelectric field can be

written as [118]

Ep(x) = − ∂
∂x

φ(x, t) = Emei(kx−wt).

(3.51)

Mobile carriers in a 2DES having a diagonal conductivity σxx can screen this field and

consequently the effective field is given by [30]

Eeff(x, t) =

Ep(x, t)

1 + i σxx/σm

,

(3.52)

77

where σm is the characteristic conductivity. This can be used to drive a local acoustoelectric

current density:

jα(x, t) = σαxEeff(x, t), α = (x, y).

(3.53)

Since a small amount of charge carriers can be dragged by the traveling SAW potential, the

modulated carrier density can be expressed as

Ns(x, t) = N 0

s + ∆Nsei(kx−wt),

(3.54)

where an amplitude ∆Ns is significantly smaller than the equilibrium density N 0

s . This leads

to a modulated conductivity tensor, which is given by

σ(x, t) = σ0 +

∂σ
∂Ns

∆Ns(x, t),

(3.55)

where σ0 is the conductivity tensor in the equilibrium. From the continuity equation
∇ · (cid:126)j + ∂/∂t(−eNs) = 0, one can relate the oscillating charge density ∆Ns(x, t) to the ef-

fective field Eeff(x, t):

∆Ns(x, t) = ∆Nsei(kx−wt) = −σxxEeff(x, t)

ec

,

(3.56)

78

where c is the SAW velocity. From equations 3.53, 3.55, and 3.56 the time average of the

acoustoelectric current density can be written as

(cid:104)jα(x)(cid:105)t = (cid:104)

(cid:18)

= (cid:104)σ0

(cid:19)

1
ec

∆Ns(x, t)

∂σαx
∂Ns

σ0
αx +
αxEeff(x, t)(cid:105)t − ∂σαx
∂Ns

Eeff(x, t)(cid:105)t
eff(cid:105)t
(cid:104)σxxE2

(3.57)

= −∂σαx
∂Ns

1
ec

(cid:104)σxxE2

eff(cid:105)t

since (cid:104)σ0

αxEeff(x, t)(cid:105)t = 0. This indicates that the acoustoelectric current is a DC current

proportional to E2

eff arising from SAW-2DES interaction through the piezoelectric field. Ad-

ditionally, the time averaged SAW attenuation is given by

(cid:104)Γ(cid:105)t = − 1
I

∂I
∂x

=

1
I

(cid:104)σxxE2

eff(cid:105)t,

(3.58)

where I = I0e−Γx denotes the intensity of the SAW along the propagation direction. Sub-

stituting equation 3.58 into equation 3.57, the time averaged acoustoelectric current density

can be expressed in terms of the attenuation:

(cid:104)jα(x)(cid:105)t = −∂σxx
∂Ns

1
ce

(cid:104)I · Γ(cid:105)t = −µ
c

(cid:104)I · Γ(cid:105)t,

(3.59)

where µ is the mobility of the 2DES.

In an experiment, this type of acoustoelectric effect can be measured using an external

circuit. For example, when the 2D electron layer has a length L and a width b that is larger

than the SAW beam width W , the terminal current between measurement electrodes at

79

x = 0 and x = L is given by

(cid:34)

Itot = W

−σxx

∂V
∂x

− 1
L

(cid:90) L

0

µIΓ

c

(cid:35)

(cid:20)

dx

= W

−σxx

∂V
∂x

+

µ
cL

I0(1 − e−ΓL)

(cid:21)

,

(3.60)

where V is the voltage difference between the electrodes. Here the time average notation (cid:104)(cid:105)t

is dropped. For a short-circuit geometry (V = 0), we find the total current [119]:

Itot =

W µ
cL

I0(1 − e−ΓL) =

µ
cL

Pd,

(3.61)

where Pd = W I0(1 − e−ΓL) is the total dissipated power of the SAW. In the case of an

open-circuit (Itot = 0 in equation 3.60), the acoustoelectric voltage difference between the

electrodes is given by [115]

(cid:90) L

µ

Vae =

σxxcL

0

I0(1 − e−ΓL) dx =

I0(1 − e−ΓL).

µ

σxxc

(3.62)

As we will show in chapter 7, these type of measurements can be used to demonstrate

acoustoelectric charge transport in the system of electrons on helium.

80

Chapter 4

Experimental Setup and Devices

Figure 4.1: A superfluid leak-tight sample cell connected to the 1K flange of a closed cycle
cryostat.

Employing surface acoustic waves to electrons on helium requires various experimental

techniques and methods such as radio frequency (RF) and DC electrical measurements,

nano- and micro- fabrication, and low temperature physics technique as well as cryogenic.

A SAW device is composed of a piezoelectric substrate onto which pairs of inter-digitated

transducer (IDT) are patterned using optical or e-beam lithography. In our setup, this device

is then mounted to a patterned printed circuit board (PCB) inside a superfulid leak-tight

copper cell (Fig. 4.1) and covered with a liquid 4He film at cryogenic temperature. Then

81

electrons are produced via thermionic emission from a tungsten filament mounted at a top

of the cell and they then float on the top of the liquid 4He film. A set of three rectangular

electrodes are patterned on the PCB via conventional photolithography before mounting the

SAW chip. These electrodes are used to trap electrons on the helium and simultaneously to

detect low-frequency electron transport phenomena. A SAW launched from the input IDT

interacts with electrons on helium and its amplitude and velocity change are measured at

the receiver IDT. All of the data acquisition and the control of measuring-instruments are

done by a python data acquisition software, LabGUI [120].

This chapter describes the details of the experimental techniques used to perform exper-

iments presented in this thesis. In section 4.1, we start with an overview of the cryogenic

experimental setup and then discuss cryostat, experimental cell, helium fill line system, elec-

tron generation, and measurement principle with circuits. Section 4.2 provides the detail of

device fabrication for the experiments.

4.1 Experimental Setup

4.1.1 Setup Overview

All the measurements in this thesis were performed in a continuously operating 4He evapo-

ration cryostat manufactured by Cryomech Inc. This cryostat reaches the base temperature
T (cid:39) 1.5 K with a cooling power of ∼200 mW. To form liquid helium inside the experimen-

tal cell at this temperature, we fabricated a hermetically sealed copper sample cell, which

was then attached to 1K flange of the cryostat as shown in Fig. 4.1. A custom helium gas

supply system was built at room temperature to precisely control the amount of helium gas

admitted into the cell through a stainless still capillary fill line. A tungsten filament was

82

attached inside the cell to generate electrons thermionically and thus charge the surface of

helium with a 2DES. In order to trap and laterally confine electrons floating on the helium,

trapping and guard electrodes were patterned on a submerged PCB and they were biased

with positive or negative voltages. To access the stability of the surface electrons immedi-

ately after charging the helium surface, low-frequency capacitively coupled transport must

be continuously monitored. Therefore, we built a low-frequency transport measurement cir-

cuit that operates via a standard lock-in technique. Also, both the SAW attenuation and

velocity shift readout circuits were set up to characterize the interaction between electrons

on helium and an evanescently coupled surface acoustic wave.

4.1.2 Cryostat

The operation of the Cryomech closed-cycle 1K cyrostat is based on two separate operating

cycles, pulse tube precooling and liquid 4He evaporative cooling. The precooling process is

achieved by a continuously operating pulse tube cold head as shown in Fig 4.2. This process

implements oscillatory compression and expansion of 4He gas within a closed volume and
cools the cold head of the pulse tube and attached cryostat from room temperature to ∼4 K.
At this temperature, ultra pure 4He gas is provided for approximately 2 ∼ 3 hours to fill the

liquid helium container (1K pot) inside the fridge. After filling the cryostat with helium, con-

tinuous operation is achieved by 4He evaporation from the 1K pot with a base temperature
of ∼1.5 K (Please see Fig. 4.2 for additional details). In order to control the temperature of

the cooling stages, heaters are mounted on the 4K and 1K flanges respectively. Also, a SI-410

silicon diode, a SI-RO600 RuO2 resistor, and a CX-1050-CU-HT-1.4L zirconium oxynitride

thin-film temperature sensor are mounted on on the 4K stage, 1K stage, and sample cell to

monitor temperature real-time. Both heating control and temperature reading are performed

83

using a commercial temperature controller (model number SRS CTC100). For measurement

and control of surface acoustic wave experiments with electrons on helium, 10 DC and 12

superconducting copper-nickel coaxial lines were installed in the fridge connecting the room-

temperature electronics to the sample cell at 1K fridge. All lines were thermally anchored

at each stage of the fridge to minimize heat flow as shown in Fig. 4.3. To turn on the

tungsten filament inside the cell with a low bias voltage, two of the DC lines are made of

formvar insulated single filament copper clad superconducting wire (superconducting wire
core OD = 0.01(cid:48)(cid:48), copper cladding OD = 0.012”) to connect to the filament. The remaining

8 DC lines are made of stainless-steel shielded copper wire in twisted pairs.

4.1.3 Experimental Cell

Superfluid liquid helium can flow through very narrow constrictions, even those having

nanometer size due to its vanishing viscosity. This makes it experimentally challenging

to construct a leak-tight cell to contain superfluid helium. A convenient and reliable tech-

nique for creating a superfluid leak-tight seal is to use wire of indium metal to make a O-ring.

Because of its softness, indium can completely fill the gap between two solid surfaces to be

sealed together when they are squeezed along with indium in the gap. This leads to reliable

cryogenic seal even at mK temperatures. Applying this sealing method, we built a 2.5 inch

outer diameter (OD) hermetic copper sample cell with 2 inch inner diameter (ID) and an

internal height of 6.5 as shown in Fig. 4.4, following a similar cell design as that used by

David Schuster’s group at University of Chicago [26, 121]. At the surface of the top of

the cell, there are twelve hermetic GPO microwave feedthroughs (GPO Male to GPO Male

Thread-in connector, Gilbert 0119-783-1), each having their own 0.02 inch diameter indium

seal. It should be pointed out that these hermetic GPO connectors are so robust to tem-

84

Figure 4.2: Schematic showing operation of the closed-cycle 1K cryostat. The 1K cryostat
is precooled by a 4K pulse tube cryocooler (Cryomech PT410-RM) providing 1.0 W cooling
power at T = 4.2 K. In the 1K circulation loop (blue color), the inlet 4He gas at the pressure
of ∼1 atm is precooled by a heat exchanger on the 50K stage and then condenses at the 4K
stage. The condensed helium flows through a counter-flow heat exchanger wrapped around
the tube below the 4K stage to be cooled ∼2 K. It then passes through a needle-valve into
the 1K pot. To produce evaporation, the 1K pot is pumped by a dry vacuum pump through
a pumping line. A buffer volume installed at the pump discharge stabilizes the pressure. If
additional helium gas needs to be added, a standard helium gas cylinder is connected to the
discharge side of the pump. The base temperature of the 1K system strongly depends on a
flow through the needle valve and the pumping rate as well as any additional heat load from
measurement wires and the helium fill line inside the fridge.

85

Figure 4.3: A simplified illustration of a closed cycle 1K refrigerator in which a helium fill
line, wires, and a sample cell are shown in detail.

86

perature cycling that we have not found any superleak since they were first mounted years

earlier. Also, to introduce helium into the cell, a stainless steel capillary fill line (1/16 inch

OD and 0.04 inch ID) is connected to the top of the cell using hard silver solder. Lastly,

the superfluid seal between top and bottom part of the cell is made by inserting a 0.03 inch

diameter indium wire ring between them and subsequently pressing them against each other
evenly with #4−40 screws.

Figure 4.4: (a) Hermetically sealed Cu sample cell.
(b) bottom and (c) top part of the
cell. (d) 3D drawing of the cell assembly process. A SAW chip and a helium interdigitated
capacitor (IDC) level sensor mounted to the PCB are connected to GPO launchers via gold
wire bondings. GPO bullets are then inserted on the GPO launcher to link them to the
hermetic GPO feedthroughs. An indium wire oring is placed onto the cell bottom and the
top part of the cell is evenly pressed against the bottom via tightening screws.

4.1.4 Helium Fill Line and Gas Supply System

Precise regulation of the amount of 4He gas supplied into the sample cell is essential for

forming a partial volume of liquid helium with the desired thickness in the cell. To realized

this, we built a 4He gas supply system at room temperature, consisting of a gas handling

unit (GHU) and a calibrated standard volume of 260 cc as shown in Fig. 4.5 (b). Room tem-

87

Figure 4.5: (a) Helium fill line in the 1K cyrostat. (b) Schematic of the 4He gas handling
system (GHS). Once the unit volume is filled with 4He gas, the valve behind it is closed and
then helium gas is provided into the cell from only this volume.

perature 4He gas first passes through liquid a nitrogen cold trap in the GHU for purification

and then fills up the standard volume of 260 cc until a desired pressure is achieved. Then
the 4He gas is provided into the sample cell at T ≈ 1.5 K through a 1/16 inch OD and 0.04

inch ID stainless capillary fill line installed inside the fridge. The helium fill line is thermally

anchored to the 50K and 4K stage by copper braid and a flow impedance (coil structure) is
employed near 1K stage to reduce the heat flow into the sample cell at T ≈ 1.5 K. Different

sections of the capillary tube are connected using custom brass adapter (inset of Fig. 4.5 (a)).

In order to minimize low-frequency mechanical vibration mainly due to the pulse tube, the fill

line is tightly anchored to the cryostat body using copper tape and dental floss. By varying

the pressure in the standard volume, we can determine the liquid helium volume admitted
into the cell at T ≈ 1.5 K and its thickness is measured in real-time using a interdigitated

capacitor (IDC) level sensor (Fig. 4.4 (b),(d)). A more detailed description of the IDC level

88

sensor will be presented in section 4.2.

4.1.5 Surface Electron Generation

Electron Source

There are many ways to generate large numbers of electrons including photoemission [122],

field emission [123], corona discharge [124], and thermionic emission from a heated tungsten

filament [125]. However, charging the surface of liquid helium requires a source of low-energy
electrons as the helium surface has ∼1 eV potential energy barrier to penetration of electrons

into the liquid from the vacuum. A the well-known method to generate low-energy electrons

at liquid helium temperature is to use a tungsten filament covered by superfluid helium, which

has been used as a durable and reliable electron source in electrons on helium experiments for

many years [126]. Employing this thermionic emission from a tungsten filament allows us to

spray electrons onto the liquid helium surface inside the cell. Fig 4.4 (c) shows the tungsten

filament (model number, T-3/4 wire terminal by CML) used in our experiments, which is

attached to the ceiling of the top portion of the cell with epoxy. Low resistance wires are

connected to two of the hermetic GPO feedthroughs via GPO bullets (GPO Female to GPO

Female connector, Fariview Microwave SM2608). The other side of these feedthroughs is

linked to a room temperature pulse generator through superconducting wires in the cryostat

described in subsection 4.1.2. At room temperature, the typical resistance of the filament is
∼25 Ω while the total resistance including the distributed wiring is ∼27 Ω. This ensures that
the external applied voltage is mostly dropped across the filament in the cell. At T ≈ 1.5 K,

the total resistance of the filament plus wires has substantially reduced value of RT < 10 Ω.

In order to minimize excessive heating during the electron emission process, the filament

89

is operated in pulsed voltage mode using a waveform generator (HP 8116A) to reduce the

duty cycle. Table 4.1 shows the typical pulse parameters for thermionic electron emission.

Filament operation with these parameters allows for reliable low-energy electron generation

Parameter

Peak-to-peak applied voltage

Pulse frequency

Pulse duration

DC offset

Value
2 ∼ 2.5 V
0.5 ∼ 1 Hz
50 ∼ 90 %
−2 ∼ −2.5 V

Table 4.1: Typical tungsten filament biasing parameters for electron emission using the
HP8116A waveform generator.

without any significant degradation of the filament. In fact, we never observed the filament

to burn out under these settings. Although this operation temporarily increases the base
temperature of the 1K stage T ≈ 1.5 K by 100 ∼ 300 mK, the system generally cools back

to its initial value within a few minutes.

Surface Electron Confinement

To trap surface electrons on the liquid helium, a set of three rectangular electrodes (trapping

electrodes) submerged beneath the liquid helium are used as illustrated in Fig. 4.6, which

are patterned onto a 1.5 mm thick PCB mounted in the cell. In addition to these trapping

electrodes, a rectangular guard electrode made of 1.63 mm thick square cross section copper

wire is located outside of the three coplanar electrodes to laterally confine electrons. The

trapping electrodes are biased with a positive voltage typically ranging from 20 to 60 V
while the guard electrode is biased with a negative voltage ranging from −1 to −3 V. When

the surface of the liquid helium is fully charged with electrons generated from the tungsten

90

Figure 4.6: Simplified illustration of the process of charging a liquid helium surface with
electrons.

filament, the saturation condition of the surface charging requires a vanishing electric field

between the top plate and 2DES shown in Fig. 4.7 (a),(b).

In this case, the saturated

surface density can be calculated from a simple capacitance model (Fig. 4.7 (b)).

If we

Figure 4.7: (a) Sketch of electrons floating on the surface of liquid helium inside the sample
cell and (b) the simple capacitance model for this geometry.

ignore the guard electrode geometry, the electric fields above and below the 2D electron

Ve − Vtp

h

Vbt − Ve
εHed

layer are given by (cid:126)E+ =

ˆz and (cid:126)E− =

ˆz, respectively. Here Ve, Vtp, and

Vbt are the electric potential at electron layer, the top plate and the bottom plate (a set

91

of three trapping electrodes in our device) as shown in Fig 4.7 (b). The distance between

the top plate and the 2DES is denoted h and d is the effective distance between the bottom

plate and the 2DES, defined as d = dHe/εHe + ds/εs, where εHe and εs represent dielectric

constant of liquid helium and the underlying substrate while dHe and ds are the thickness of

the liquid helium and the substrate. The boundary condition for the normal component of

the displacement field requires the following relation:

−ne
ε0

= (cid:126)E+ · ˆz − εHe

(cid:126)E− · ˆz,

(4.1)

where n, e, and ε0 denote the surface electron density, the elementary charge, and the

vacuum permittivity. When (cid:126)E+ = 0 and Ve = Vtp, the electron density is saturated and has

a maximum value. With this condition, we can calculate the saturated electron density ns

from equation 4.1:

nse = ε0

Vbt − Vtp

dHe/εHe + ds/εs

.

(4.2)

Thus, the maximum saturated density ns can be determined by geometrical capacitance

between the 2DES and bottom plate, and voltage difference between the top and bottom

plate. Equation 4.2 is not applicable when the electron density is not saturated. For this

general case, the surface electron density is estimated from the low-frequency transport

measurement, which will be described in chapter 5.

Surface Electron Detection

One of the first ways to check for the presence of electrons in the system is to try and measure

their transport properties. However, the measurement of the transport of surface electrons

on the liquid helium substrate presents an experimental challenging due to the impossibility

92

of attaching electrical leads directly to the electron layer. Pioneering experiments in this

field were performed by Sommer and Tanner [88] using capacitive coupling between under-

lying electrodes and the surface electron layer. Based on the technique developed by them,

surface electron detection in our experiments is realized by measuring a current through the
electron layer at T ≈ 1.5 K. Fig. 4.8 shows the Sommer-Tanner configuration we used in our

experiments, which consists of an array of three planar electrodes (drive, gate, and sense)

patterned on the PCB inside the cell. They are submerged under the liquid helium and

Figure 4.8: Schematic of Sommer-Tanner electrode configuration. A set of three planar
electrodes with typical width w = 8 − 10 mm and length l = 3 − 5 mm are submerged un-
derneath the charged liquid helium surface. The 2DES is excited by applying an AC voltage
of Vex ∼100 mV to the drive electrode, which induces a current through the electron layer
on the sense electrode. This current is measured using standard lock-in techniques. In ad-
dition to Vex, a DC voltage of VDC = 30 − 60 V is simultaneously applied to the underlying
electrodes to vertically hold surface electrons on helium. A bias tee composed of a resistance
of R = 10 MΩ and a capacitance of C = 1 µF, marked by the red-dashed square, is used to
apply both a DC an AC voltage.

separated by a very narrow gap. The typical width of the gap g is approximately 0.2 mm
while the electrode width w and length l are usually 8 − 10 mm and 3 − 5 mm respectively.
An AC excitation voltage Vex at the frequency of 60 − 100 kHz with an amplitude of a few

93

hundred mV is applied to the drive electrode. The sense electrode is connected to the in-

put of a lock-in amplifier (model SR830). Simultaneously, a constant positive DC voltage
VDC = 30 − 60 V is applied to all electrodes to vertically hold surface electrons. To prevent

the DC signal from being disturbed by the RF content or vice versa, a custom bias tee

composed of a 10 MΩ resistor and a 1 µF capacitor is added to both the drive and sense

electrode (red dashed box in Fig. 4.8). The underlying gate electrode at the center is only

connected to the DC bias VDC (i.e. this electrode is at AC ground), which substantially

reduces the direct geometrical capacitance between the drive and sense electrode. A simple

description of the measurement principle is as follows: When surface electrons are confined

on the liquid helium area above the three electrodes, applying an AC excitation voltage Vex

to the drive electrode gives a rise to a local charge density modulation above the electrode

due to the capacitive coupling to electrons floating on the helium surface. This induced mod-

ulation in charge density propagates to the 2D electron region above the sensing electrode

and produces a current into the sense electrode. In particular, the phase difference between

the output and input signals depends on the change in resistance of the surface electrons and

hence contains information of the electrical conductivity of the 2DES, as will be described

in subsection 4.1.7.

4.1.6 Helium Level Sensor

As depicted in Fig. 4.9 (a), the helium level sensor we used in our experiments is an interdig-

itated capacitor (IDC) vertically mounted on a copper block attached to the PCB. The IDC
is fabricated on on a 3.5 × 8 mm2 quartz substrate and consists of 125 interlaced aluminum
fingers that are 10 µm wide, 10 mm long, and ∼100 nm thick. These fingers are separated
from each other by 10 µm. Once the sample cell is cooled down to T ∼= 1.5 K, it is slowly

94

filled with liquid helium from the unit volume at room temperature. During this condensa-

tion process, the change in capacitance CIDC is measured and is proportional to the level of

bulk liquid helium in the cell. From the change of CIDC between the empty and fully filled

cell, one can relate CIDC to the bulk helium level by linear interpolation [127]. To figure out

the maximum of CIDC when the cell is full of the liquid, we continuously provided helium

into the cell and simultaneously measure the capacitance change as a function of time using

a capacitance bridge (model number, AH2550A) as shown in Fig. 4.9 (b). The maximum

Figure 4.9: (a) Schematic of the IDC helium level sensor inside the cell partially filled with
liquid helium. Inset shows an actual image of the IDC sensor vertically mounted on a copper
block attached to the PCB. (b) IDC capacitance measurement as a function of time while
helium is continuously provided into the cell from a helium gas cylinder at room temperature.

capacitance Cmax for completely filled case can be then obtained from the capacitance value

measured when helium level touches the top of the IDC (Fig. 4.9 (a)). Also, the minimum

capacitance Cmin for the empty case coincides with the measured capacitance value when

the helium level reaches the bottom of the IDC. A tiny increase in CIDC before the helium

touches the bottom of the IDC is attributable to capillary filling of liquid helium in the space

between the fingers of the IDC. The expected capacitance change between Cmin and Cmax

95

is given by [128]

∆C
C

=

Cmax − Cmin

Cmin

∼=

εHe − ε0
εs + ε0

∼= 1.18 %,

(4.3)

where εHe

∼= 1.057, εs ∼= 3.82, and ε0 = 1 are the dielectric constant of liquid 4He, the

quartz substrate, and the vacuum, respectively. The measured capacitance change we find
is ∼1.16 %, in a good agreement with the calculated value from equation 4.3. With the

measured values of Cmin at helium level h = h0 and Cmax at h = h0 + l, a helium level at

an intermediate value of CIDC between Cmin and Cmax can be estimated from the linear

relationship given below:

h =

l

Cmax − Cmin

(CIDC − Cmin) + h0

∼= 18.66 mm/pF × (CIDC − 11.564pF) + 0.1 mm,

(4.4)

where Cmin = 11.564 pF, Cmin = 11.698 pF, h0 = 100 µm is the distance between the IDC

bottom and the bottom edge of the substrate (Fig. 4.9 (a)), and l = 2.5 mm is the height of

the IDC.

4.1.7 Measurement Principle and Circuit for Transport of Elec-

trons on Helium and SAW Experiments

Lock-in Detection for Low-Frequency Measurements of Electrons on Helium

The Sommer-Tanner technique for transport measurements described in subsection 4.1.5

requires phase-sensitive detection of a very small AC current signal for the electron layer

ranging from a few tens of pA to several nA in the presence of noise. Lock-in amplifiers are

designed for this type of detection and the extremely small bandwith used in this type of mea-

96

surement enables us to measure only the signal of interest. Thus, all transport measurements

presented in this thesis are performed using a lock-in amplifier (Stanford Research SR830).

In a standard lock-in measurement, a reference frequency is needed. A device or system (in

our case the electrodes in the Sommer-Tanner experiment) is excited at a fixed frequency

from a function generator and the lock-in amplifier detects the response of the system at

the reference frequency. If a sinusoidal output from the function generator at the reference

frequency ωR is used to excite the system, the signal from the system will be VS sin(ωRt+θS).

The lock-in amplifier generates its own internal reference signal VL sin(ωRt + θR) by a phase-

locked-loop locked to the external reference. These two sine waves are multiplied using a

phase-sensitive detector (PSD) and their product is given by

VPSD = VSVL sin(ωRt + θS) sin(ωRt + θR)

=

1
2

VSVL{cos(θS − θR) − cos(2ωRt + θS + θR)}.

(4.5)

Once the AC component is removed with a low pass filter, the filtered PSD output has only

a DC component:

VPSD =

1
2

VSVL cos(θS − θR) ∝ VS cos θ,

(4.6)

where θ is defined as the phase difference between the output signal and the lock-in reference
oscillator. In the case of noise at frequencies different from reference frequency (ωnoise (cid:54)= ωR),

the corresponding PSD output has only AC component:

VPSD ∝ cos ([ωnoise − ωR]t + θnoise − θR) − cos ([ωnoise + ωR]t + θnoise + θR).

(4.7)

97

Such a PSD output from the noise source will be removed by the low pass filter and thus

the signal at the reference frequency ωR resulting in a DC output in equation 4.6 is only

detected. If a second PSD multiplies the signal with the lock-in internal reference shifted by

90 degrees, the low pass filtered PSD output will be given by

VPSD2 =

VSVL sin θ ∝ VS sin θ

1
2

(4.8)

Therefore, we get two output signals as shown in equations 4.6 and 4.8: one proportional to

cos θ and the other proportional to sin θ. If we call the first output X and the second Y:

X = VS cos θ

,

Y = VS sin θ,

(4.9)

where X and Y are referred to as the in-phase and quadrature components, respectively.

The amplitude and phase of the signal are related to X and Y by

(cid:112)

R =

X2 + Y 2

θ = tan−1 (Y /X).

,

(4.10)

In the Sommer-Tanner configuration described in subsection 4.1.5, the resistance of sur-

face electrons on liquid helium is generally much smaller than capacitive reactance (i.e.
Re (cid:28) Xc = 1/jωC), indicating an almost purely capacitive response from the electrodes.

Here C is the capacitance between an underlying electrode and 2D electrons. Thus, most of
the measured output current lives in the quadrature component with θ ≈ π/2.

98

Circuit Modeling of the Low-Frequency Transport of Electrons on Helium

With the Sommer-Tanner geometry illustrated in Fig. 4.8, low-frequency transport proper-

ties at T ∼= 1.5 K are analyzed using either a lumped element circuit or transmission line

(distributed element) model. We first start with simple lumped element circuit model and

extract the electric conductivity σ of the surface electrons form the phase information of the

measured signal. Then, we will describes the transmission line analysis to estimate σ via

fitting this model to our experimental data. This latter is primarily used in our experiments.

Fig. 4.10 shows the equivalent lumped element circuit model for the Sommer-Tanner

configuration with or without electrons on the liquid helium surface. In the absence of elec-

Figure 4.10: Lumped element equivalent circuit model of the Sommer-Tanner measurement
configuration for (a) a neutral and (b) charged surface of liquid helium.

trons (Fig. 4.10 (a)), typical value of the capacitance between two neighbor electrodes with
thickness (cid:39) 35 µm is C12 = C23 (cid:39) 0.5 pF while the capacitance between the drive and sense
electrode is C13 (cid:39) 5 × 10−3 pF because of the shielding provided by the AC-grounded center

electrode. No resistive component in this case indicates a purely capacitive circuit. Once

surface electrons are formed on the liquid helium, the equivalent circuit model is modified to

include a resistor in series with two capacitors as shown in Fig 4.10 (b), where the resistive

component Re comes from the surface electrons and the capacitances Ce are formed be-

99

tween the electron layer and the drive/sense electrodes. Here, the direct capacitive current

flowing between the drive and sense electrode is neglected due to the very small capaci-

∼= 5 × 10−3 pF between these electrodes as well as the low-frequency excitation

tance C13
(60 − 100 kHz).

From a 2D Drude model for the system of electrons and electrodes in our experiment,

each of the components in the RC series circuit can be written as

Re = RL · L , Ce = CL · L

(4.11)

with a resistance and a capacitance per unit length: RL =

1
σw

and CL =

ε0 εHe w

d

, where

σ, w, L, and d denote the electric conductivity of the 2DES, the width of the electrode, the

length of three electrodes, and the effective distance of the electron layer from the electrodes

defined as d = dHe + (εHe/εs) ds. Here dHe and ds are the thickness of the liquid helium and

the submerged substrate while εHe and εs represent their dielectric constants. Even if there

are no inductive elements in this circuit, the inertia of surface electrons in an AC driving

field can lead to a kinetic inductance Lk, which is given by

1
2

LkI2 =

1
2

(nevw)2 =

nm∗v2

2

→ Lk =

m∗
ne2w2 ,

(4.12)

where I, n, v, and m∗ are the current in the 2DES, the surface electron density, the drift

velocity of the electron, and the effective mass of electron. One can see the contribution of

Lk to electron transport from the following complex Drude conductivity:

σ(ω) =

σ0

1 − jωτ

=

σ0

1 + ω2τ 2 + jωτ

σ0

1 + ω2τ 2 ,

(4.13)

100

where σ0 =

ne2τ
m∗ is the DC conductivity. In equation 4.13, ω is the AC excitation frequency
and τ is the relaxation time. The imaginary part of equation 4.13 represents the kinetic

inductance term while the real part describes the resistive contribution. Electrons on helium

have a relaxation time τ ≤ τR = µR m∗/e ∼= 10−10 s depending on the temperature, where
τR and µR represent the scattering time and the mobility in the ripplon scattering regime.
In the frequency range of our transport experiments (ω = 0.04 − 0.06 MHz), ωτ (cid:28) 1 yields a

vanishing imaginary component in equation 4.13, indicating a negligible effect of the kinetic

inductance in our experiments. Thus, the total admittance of the circuit can be written in

terms of only Re and Ce:

Y =

I
Vin

=

Re +

(cid:18)

(cid:19)−1

2

jωCe

Re − 1/jωC
Re + 1/ω2C2 with C =

Ce
2

,

=

(4.14)

where Vin is the input voltage used to excite the 2DES. The input impedance 1/jωCs of
the lock-in detector is usually very small compared to the impedance Y −1 of the electron

layer. Thus, the ratio of the output to the input voltage is Vo/Vi = Y /jωCs and their phase

difference is equivalent to

∆φ = tan−1Re(Y )/Im(Y ).

(4.15)

As discussed earlier, the capacitive circuit (Xc = 1/jωCe (cid:29) Re) in the Sommer-Tanner
configuration leads to a small phase difference ∆φ (cid:28) 1 between the input and output

voltages and thus equation 4.15 can be simplified as

∆φ ∼= Re(Y )/Im(Y ) = Re ω C.

(4.16)

The values of ∆φ and ω are experimentally determined while C = Ce/2 is easily calculated

101

by considering the parallel plate capacitor composed of the electron layer and the under-

lying electrode. Therefore, the electrical conductivity of the 2DES can be extracted using

equation 4.16:

Re = σ−1 L

w

≈ ∆φ
ωC

→ σ (cid:39) L
w

ωC
∆φ

.

Furthermore, if the saturated surface charge density is given by

ne =

CeVdc
w(L/3)

=

6CVdc

wL

,

substituting equation 4.18 into equation 4.17 yields the electron mobility

µ (cid:39) L2ω
6Vdc∆φ

.

(4.17)

(4.18)

(4.19)

Here Vdc is the trapping voltage applied to each of the underlying electrodes. This is very
close to the mobility expression µ ∼=

derived by Sommer and Tanner using a

7
54

L2ω
Vdc∆φ

distributed circuit model [88]. The lumped element approximation is only valid when the

phase of the current or voltage wave does not significantly vary over the length of the circuit

(i.e. the wavelength of the excitation must be larger compared to the length of the electrodes,
λ (cid:29) L). The wavelength of the voltage wave propagating in the surface electrons on liquid

helium can be obtained by solving the two-dimensional wave equation with appropriate

boundary conditions, which produces [129]

λ (cid:39)

(cid:114) 2σ

ωCe

.

(4.20)

Therefore, the lumped element circuit model will show a discrepancy from the measurements

102

at high frequency and low electron conductivity. This case will be described in detail in

chapter 5.

Electrical transport of electrons on helium at arbitrary frequencies and wavelengths can

be analyzed using a transmission line (distributed element) model shown in Fig. 4.11. One

Figure 4.11: (a) Transmission line model of the Sommer-Tanner geometry. (b) Empirical
circuit model [15] of the lock-in input impedance ZLI, which consists of cable capacitance
Ccab (cid:39) 600 pF, a low-pass current divider stage with a capacitance Cdiv = 31 pF and
resistance Rdiv = 6.5 kΩ, and the nominal lock-in input impedance of Rin = 10 MΩ in
parallel with the capacitance Cin = 25 pF. (c) A simplified equivalent circuit of (b). Both
Z1 and Z2 are composed of two elements in parallel, given by Z1 = [jω(Ccab + Cdiv)]−1 and
Z2 = [jωCin + 1/Rin]−1.

can relate the output voltage and current to the input voltage and current over a length x of

the transmission line formed by the coupled electron-electrode system by a scattering matrix

[130]

 =

Vin

Iin

 cosh γx

sinh γx

Z−1

c



 ,

Vout

Iout

Zc sinh γx

cosh γx

(4.21)

103

where Zc is the characteristic impedance of the transmission line

(cid:115)

Zc =

RL + jωLL

jωCL

and γ is given by

(cid:113)

γ =

jωRLCL − ω2LLCL.

(4.22)

(4.23)

The quantities RL, CL, and LL denote the resistance, capacitance, and inductance per unit

length of the line, respectively. Neglecting the small inductive component LL, one can find

the output current Iout at the sense electrode from equation 4.21 [129]:

Iout = I0 (1 + j)

9σ
2L

sinh2(jkL/3)

sinh(jkL)

with a current amplitude

I0 = ωCA

wL
9

Vin,

where k is the complex wave vector of the current

1 − j
σ

,

k =

(4.24)

(4.25)

(4.26)

and δ is a two-dimensional skin depth representing the length scale over which charge density

fluctuation propagates in the electron layer,

(cid:115)

δ =

2σ
ωCA

,

(4.27)

and CA is the capacitance per unit area between the electron layer and the top and bottom

104

electrodes, and σ = neµ is the DC conductivity of the 2DES. The voltage Vdet detected by

the lock-in amplifier is determined by the output current Iout and its input impedance ZLI

Vdet = ZLI Iout.

(4.28)

The nominal input impedance specified by the manufacture for the Standard Research 830

(SR830) lock-in is Rin = 10 MΩ in parallel with an input capacitance of Cin = 25 pF. This
is not adequate for accurate circuit modeling over the frequency range 0.01 − 100 kHz [15].

Therefore, Vdet is estimated based on an empirical lock-in input impedance model developed

by Kim et al. [15], which is illustrated in Fig. 4.11 (b). In its simplified equivalent circuit form

(Fig. 4.11 (c)), the current I2 flowing through impedance Z2 is determined by Kirchhoff’s

law,

where

and

I2 =

Z1

Z1 + Rdiv + Z2

Iout, Rdiv = 6.5 kΩ,

(4.29)

Z1 =

1

jω(Ccab + Cdiv)

Z2 =

1

jωCin + R−1

in

.

(4.30)

(4.31)

Here the impedance Z1 consists of the coaxial cable capacitance Ccab

∼= 600 pF to ground in

parallel with the capacitance of Cdiv = 31 pF while the impedance Z2 is the nominal input

impedance described above. Finally, Vdet is equivalent to the voltage applied to Z2,

Vdet = Vdet(σ, CA) = Z2I2.

(4.32)

105

Since the capacitance per unit area CA can be obtained from the parallel-plate capacitor

geometry, the only unknown parameter in equation 4.32 is the conductivity σ of the 2DES,

which can be determined by fitting the in-phase and quadrature components of the voltage

measured by the lock-in. Once the areal density n of the 2DES is known, the mobility µ of

the 2DES can then be estimated from the relation σ = neµ.

SAW Velocity Readout

To measure SAW velocity shifts, we built a custom phase-locked loop (PLL) instrument.

Fig. 4.12 shows its simplified circuit diagram. A signal generator is used as a voltage-

Figure 4.12: Simplified PLL circuit for SAW velocity shift measurements.

controlled oscillator (VCO), which drives both the SAW device and the LO (local oscillator)
port of a mixer at a frequency f = fbase + df /dV · Vout. The RF port of the mixer is driven

by the signal coming from the SAW device, which has the phase difference φ but the same

frequency f as the LO signal. Thus, the IF (intermediate frequency) output signal of the

106

mixer can be determined by multiplying the two input signals:

VIF = VLOVRF = A1 sin(ωt) A2 sin(ωt + φ) =

A1A2

2

{cos(2ωt + φ) − cos(φ)},

(4.33)

where ω = 2πf , VLO = A1 sin(ωt), and VRF = A2 sin(ωt + φ). The IF output has an
oscillating component ∝ cos(2ωt + φ) and a DC component ∝ cos (φ). A low-pass filter

isolates the DC component and thus Vout is determined by only the cos(φ) term. A simple

description of how the PLL circuit works is as follows: If we suppose that the phase difference

φ between the LO and RF signals is φ = (n+1/2) π, we have Vout = 0 and thus the frequency
of the VCO is fixed at f = fbase. For the more general case where φ (cid:54)= (n + 1/2) π and
Vout (cid:54)= 0, the PLL loop will attempt to minimize the frequency control voltage (= Vout),

which eventually brings φ to an equilibrium value of φ0 near (n + 1/2) π. For small change

in the phase, the loop will adjust the VCO frequency to maintain a constant phase φ0. For

example, a change in the areal density of surface electrons or liquid helium thickness leads

to a SAW velocity change v0 + ∆v and hence a change in φ. This leads to a time delay

between the RF and LO input of the mixer and their corresponding phase difference can

be defined as φ =

2π
λ

x = 2π (f /v) x. Here λ, v, and x are the wavelength, velocity, and

propagation distance of the SAW while f is the SAW driving frequency, which is equivalent

to the VCO output frequency. Again, the PLL tries to maintain a constant phase difference

φ by adjusting the frequency f so that (f /v) x is held constant. A negligible change of
x yields f /v = constant or ∆f /f0 = ∆v/v0, where ∆f = f − f0 and ∆v = v − v0 are

the small changes of the SAW frequency and SAW velocity (i.e. f0 is the frequency when

∆v = 0). Recall that the SAW frequency f in this PLL circuit is affected by Vout, i.e.

107

f = fbase + df /dV · Vout, and the velocity shift of the SAW at constant phase is given by

∆v
v0

=

∆f
f0

=

f − f0
f0

=

fbase − f0

f0

+

df
dV

Vout
f0

.

(4.34)

If initially v = v0 and ∆v = 0, one can write the frequency f0 in terms of the initial output

voltage V

(0)
out :

f0 = fbase +

df
dV

(0)
out .
V

(4.35)

Substituting equation 4.35 into equation 4.34, a subsequent velocity shift ∆v/v can be de-
termined from ∆Vout = Vout − V

(0)
out :

∆v
v

=

1
f0

df
dV

∆Vout ∼=

1

fbase

df
dV

∆Vout.

(4.36)

Note that the frequency variation df /dV · Vout of the VCO is generally much smaller than

the set frequency fbase of the signal generator and thus f0 can be approximately replaced

by fbase.

Fig. 4.13 shows a more detailed description of the operation principle of the PLL circuit.

The red line is the frequency of the VCO and the blue curve indicates the amplified and

filtered IF output of the mixer. At the beginning when the frequency control input is not

yet connected, f = fbase and Vout = βcos(2πxfbase/v0), as marked by the open circles,

where β = αA1A2 g (df /dV ). Here α is a constant related to the mixer specifications, A1

and A2 are the amplitude of the voltage applied to the LO and RF port and g is the gain of

the amplifier. Once the PLL loop is closed with connection to the frequency controller, the

VCO moves to a new frequency due to the feed-back signal Vout. This process is indicated

by the dashed-line phase space trajectory in Fig. 4.13. The PLL adjusts the frequency until

108

reaching the equilibrium values of f and Vout in constant phase mode.

Figure 4.13: Convergence of the SAW PLL. The blue trace is the amplified and filtered
output signal from the mixer Vout = βcos(φ) where φ = 2πxf /v, shown in Fig. 4.12. The
red line represents the dynamic VCO frequency f = fbase + (df /dV )Vout. Initially before
the PLL loop is closed (Vout is connected to the VCO control frequency input), Vout has a
value of Vout = 2πxfbase/v0 at the frequency of f = fbase. In the closed PLL loop, Vout
moves along the dashed dot trajectory and Vout eventually converges at the equilibrium point
marked by the solid circle.

As shown in Fig. 4.14, the actual SAW velocity shift measurement circuit is slightly more

complicated compared to the version shown in Fig. 4.12 due to several additional components.

To avoid the signal arising from the direct capacitive coupling between the input and output

SAW transducers, we employ pulsed SAWs rather than a continuous wave (CW) excitation.
The switch “C” pulses the RF input signal (100 − 300 MHz) close to the SAW resonance

frequency (f = v/λ) at a repetition rate of 1 kHz. The typical pulse length ranges from 1 to

3 µs and to maximize signal one can use the longest pulse without overlapping between the

instantaneous capacitive cross-talk and the SAW signal. Mixer “F” in Fig. 4.14 is a level 7

109

Figure 4.14: SAW ∆v/v measurement circuit diagram.

mixer, meaning that +7 dBm is supposed to be applied to the LO port. Also, a power level
between 0 − 7 dBm is recommended for driving the RF port of the mixer to avoid DC offsets

as well as added noise. It should be noted that the low-pass filter in Fig. 4.12 is replaced by

a boxcar integrator in the actual measurement circuit shown in Fig. 4.14. This is a gated

integrator enabling the collection of the signal of interest during a specific time period set

by the user and to reject any signals from other times. The accepted signal is integrated and

then time-averaged over the length of the gate with a repetition rate of 1 kHz. Typically,

1000 samples (cycles) are averaged in our experiments, which leads to a measurement time

constant of 1 sec. The protective diodes “J” prevent excessive voltage from being applied to

the FM input of the signal generator (model number, HP8648B). In particular, 1 V at the

FM input can damage the generator.

110

Part

Manufacturer / Model number

A

Agilent 8648B with frequency modulation (FM) control

B Mini-Circuits ZFSC - 2 - 2500 - S power splitter

C Mini-Circuits ZASWA - 2 -50DR PIN diode switch

D Mini-Circuits ZKL-2-1 30 dB RF preamp

E

Keysight 33500B waveform generator

F Mini-Circuits ZEM-4300 mixer

G

H

I

J

SRS 250 gated integrators and boxcar averagers

Agilent 34401A digit multimeter

Tektronix DPO7054 digital oscilloscope

Overvoltage protection diodes (1N4148)

Table 4.2: Parts list corresponding to the measurement circuit in Fig. 4.14

SAW Attenuation Readout

As described previously, a typical SAW delay line device is composed of a pair of inter-

digitated transducers (IDTs), which is susceptible to noise arising from the almost instan-

taneous capacitive crosstalk between the IDTs. Thus, accurate SAW measurements require

the separation of this crosstalk from the SAW signal of interest. This is achieved by gating

on the time delayed SAW signal as described at the end of this section. The attenuation of

the SAW signal can be determined from the scattering parameters of the SAW device using

a 2-port network analyzer. The scattering matrix is defined only in terms of voltage waves

as shown in Fig. 4.15:

V −

V −

1

2

 ,

(4.37)

V +

1

V +
2

 =

S11 S12

S21 S22



111

where V +

1 and V −

1 are the voltage waves travelling toward and away from IDT1 with wavevec-

tor k and the propagation length z. These waves are given by

1 = V01e−kz,
V +

V −
1 = V01ekz,

Similarly, V +

2 and V −

2 are the voltage waves toward and away from the IDT2 (see Fig. 4.15)

2 = V02e−kz,
V +

V −
2 = V02ekz.

Figure 4.15: A sketch of the S-matrix elements measurement of a SAW delay line using a
2-port network analyzer.

If the voltage wave V +

1 is applied to IDT1 and no voltage (V +

2 = 0) to IDT2, the elements

of the scattering matrix are

S11 =

V −
1
V +
1

, S21 =

V −
2
V +
1

.

(4.38)

Here S11 and S21 indicate the voltages reflected from IDT1 and transmitted from port 1 to

port 2, respectively. To quantify the SAW power PSAW leaving IDT1 toward IDT2, we need

to know the acousto-electric conversion efficiency α1 of the RF-power Pin applied to IDT1.

112

This can be estimated from the following expression [131]:

(cid:32)

α1 =

PSAW
Pin

=

1
2

10S

(cid:33)

,

(4.39)

(nr)

11 /10 − 10S

(r)
11 /10

where S

(nr)
11

and S

(r)
11 are the S11 elements measured at the SAW resonance frequency and

far from it, respectively. The factor 1/2 arises from the bidirectional nature of the IDT

launching two SAW beams in opposite direction. In vacuum, the SAW power arriving at

IDT2 is transformed back into a RF signal with power Pout = α2PSAW. Since an IDT delay

line is a passive and symmetric device, the conversion efficiency α2 of IDT2 transforming

SAW to RF is the same as α1 from RF to the SAW. Therefore, the SAW device in vacuum

will have a transmission coefficient given by

S21,vac = 10 log(Pout/Pin) = 10 log(α1α2) = 10 log(α2)

(4.40)

In the presence of liquid helium, the piezoelectric surface is covered with liquid helium

and leads to SAW attenuation. The corresponding transmission coefficient can be written

as

S21,He = 10 log(α2e−ΓHe L),

(4.41)

where ΓHe is the attenuation coefficient of the SAW due to liquid helium and L is the length

of the SAW delay line usually defined as the center-to-center distance between IDTs. For

the case of charged helium film, both the surface electrons and the liquid helium film result

in SAW attenuation, leading to the transmission coefficient:

S21,EoH = 10 log(α2e−ΓEoHL),

(4.42)

113

where ΓEoH denotes the attenuation coefficient owing to the charged film of liquid helium.

From equations 4.40 and 4.41, ΓHe is given by

S21,He − S21,vac = 10 log(e−ΓHeL) → ΓHe =

|S21,He − S21,vac|

10 log(e)

cm−1.

1
L

(4.43)

If expressed in dB/cm, ΓHe in equation 4.43 is multiplied by a factor of 10 log(e) ∼= 4.34.
Similarly, the attenuation coefficient ΓEoH of the SAW interacting with electrons on helium

can be written as

ΓEoH =

|S21,EoH − S21,vac|

10 log(e)

cm−1.

1
L

(4.44)

If ΓHe is a constant independent of surface electron density, ΓEoH can be expressed in terms

of ΓEoH = Γel + ΓHe, where Γel denotes the attenuation caused by 2D electron layer alone.

Substituting equation 4.43 into equation 4.44, Γel can be determined from

Γel = ΓEoH − ΓHe =

|S21,EoH − S21,He|

10 log(e)

cm−1.

1
L

(4.45)

In fact, charging the helium film with electrons exerts an electronic pressure in addition to

gravity leading to a reduced thickness of the charged helium film [132]. Thus, a change in

the film thickness due to this effect can affect the SAW attenuation by the liquid helium

ΓHe, which will be discussed in detail in chapter 6. Therefore, ΓHe has a different value when

electrons float above the helium surface and equation 4.45 is no longer valid.

Although measuring the scattering elements with the network analyzer (NWA) is a simple

and straightforward method to estimate the SAW attenuation, the direct electrical crosstalk

between IDTs adds a parasitic signal to the SAW signal of interest.

In particular, this

makes it challenging to measure the S-parameters for SAW devices on weak piezoelectric

114

materials such as GaAs. Fortunately, due to significant difference between the velocities of
light (c = 3 × 108 m/s) and sound (vsaw ≈ 103 m/s), the crosstalk signal is usually received

by the second IDT within nanoseconds while the SAW signal arrives after a few microseconds

depending on the length of the SAW delay line. Thus, employing a timing gate, it is possible

to exclude this crosstalk as well as some other noise coming from the environment. Another

possible way to do this is to use a vector network analyzer (VNA) offering a time domain

option. With this approach, the S-parameters are measured in the frequency domain as

usual and then analyzed in time domain via an inverse Fourier transform. By setting a

band-pass time filter, only the signal arising from the SAW remains and is transformed back

into the frequency domain with a Fourier transformation, which enables one to measure the

attenuation of the SAW signal [133].

Figure 4.16: SAW attenuation measurement circuit diagram.

Another way to solve this issue is to use a pulsed-SAW excitation with a gated boxcar

integrator, which will be discussed in detail below. Basically, the SAW attenuation measure-

ment circuit using pulsed-SAWs is similar to the PLL circuit (Fig. 4.14) described earlier.

115

A Schottky diode-based RF power detector (Krytar Model 203AK S/N 00277) is added as

shown in Fig. 4.16. Thus, all instruments and parts other than the RF power detector are

the same as the ones previously described for the SAW read-out circuit. The principle of

the SAW attenuation measurement using the RF power detector is as follows: A RF switch
gates a high-frequency (∼100 MHz) sinusoidal signal for SAW excitation with a periodic

pulse train (TTL) and resultantly produces a pulsed RF signal Vpul at its output as shown

in Fig. 4.16. This signal excites a pulsed SAW at IDT1 in the device with the waveform

shown in Fig. 4.17 (b). The SAW pulse transits the chip and arrives at a IDT2 where it is

Figure 4.17: (a) Simplified circuit diagram of the RF power detector (b) Sketch of voltage
waveforms at different positions on the circuit marked by the green solid circles in Fig. 4.16.

converted back to an electrical signal and then is amplified with a preamp. Fig. 4.17 (b)

shows the amplified SAW signal Vin coming from IDT2 along with the instantaneous crosstalk

and they are well separated in time by setting the TTL pulse width properly. When this

amplified signal is applied to the input of the power detector, the positive half cycles of Vin

forward bias the diode, which in turn charges the capacitor C as shown in Fig. 4.17 (a).

116

On the negative half cycles, the diode bias reverses and the voltage on the capacitor C

yields a DC output proportional to Vin. This results in Vdet (green trace in Fig. 4.17 (b))

with time constant on the rising (falling) edge arising from diode resistance (RL) and ca-

pacitance(C) at the output of the detector. Since Vdet includes IDT crosstalk, the signal

associated with the pulsed SAW is accepted using the time gate (dark yellow in Fig. 4.17 (b))

of the boxcar integrator. Just like the PLL circuit described above, this gated signal is then

time-averaged over the length of the timing gate with 1 ms pulse period, producing a DC

voltage Vout in Fig. 4.16 after taking average over 1,000 samples. From the measured Vout

containing the SAW amplitude information, the corresponding power level of Vin (Pin) is

determined using the power-voltage calibration curve shown in Fig. 4.18. The attenuation

Γsaw of the SAW is then determined with respect to the power level P
Γsaw (dB) = Pin (dBm) − P

(0)
in (dBm).

(0)
in for the empty case.

4.2 Device Fabrication

4.2.1 Overview

Fig. 4.19 illustrates the experimental cell and associated devices described in this thesis.

A set of three coplanar electrodes for trapping surface electrons on liquid helium are first

patterned on a PCB and a rectangular guard electrode made of thick copper wire is then

attached around the trapping electrodes for lateral electron confinement. A compact inter-

digitated capactior (IDC) level sensor is vertically mounted to the PCB to measure the level

of the helium reservoir outside of the guard electrode. Lastly, a patterned SAW chip, where

SAWs are generated and detected, is separately prepared and mounted to a top of trapping

electrodes (for low-frequency transport measurements, a SAW chip is not needed). All device

117

Figure 4.18: Calibration measurement of the output voltage Vout of the boxcar integrator
at variable input power level Pin to the RF power detector. RF pulses with a width of
20 µs are directly applied to the input of the power detector and the input power level
is quantified by measuring the RMS voltage within the pulse duration, based on the for-

(cid:1) where Vrms,0 = 224 mV. Experimental data shows

mula Pin (dBm) = 20log(cid:0)Vrms/Vrms,0

Pin is proportional to Vout, which well fits a function of Pin = a0 (Vout)a1 + a2. Here
a0 = 28.59 ± 1.02, a1 = 0.12, and a2 = −55.49 ± 1.13. Pin is estimated from this function
and the measured value of Vout. The shape of the curve is strongly affected by a differ-
ent DC offset of the boxcar integrator and thus must be kept the same between different
measurements by adjusting the offset level properly.

118

patterning process discussed here are carried out with conventional optical photolithography.

A detailed description of these various fabrication processes will be presented in the next

section.

4.2.2 PCB Etching

To form the trapping electrodes and GPO launcher pads on the PCB board, we used a pre-

sensitized single-sided copper clad circuit board from Jameco. Since the minimum pattern

size is a few hundred micrometers, a custom printed transparent sheet is used as a mask

(Fig. 4.20). To perform the exposure step, in a dark room the light-block film covering

the sensitized PCB is peeled off and the mask is then placed on the photoresist face of the
board. An incandescent desk lamp is placed 6 − 8 inch away and the light is exposed to

the board for twenty minutes. The exposed PCB is then placed into a positive developer

solution, where one part 418 positive developer from MG chemical is diluted with 10 parts

room-temperature water and the solution is agitated around one minute. This removes

the photoresist that was exposed to light during the exposure setup. Thus, the only the

photoresist covered by the black area of the mask during exposure remains (Fig. 4.20). If no

issues arise during or after development, we move forward to the etching process. Using a

ferric chloride solution, the developed PCB is etched for about 20 minutes which removes the

exposed copper. Trapping electrodes are formed on the board along with additional electrical

leads and pads for connecting the guard electrode and GPO launchers (Fig 4.20 (b)). After

etching the remaining photoresist is removed using acetone and followed by a DI water rinse

to remove acetone.

119

(a) Assembly process of
Figure 4.19: Electrons on helium SAW experiment fabrication.
main part of the cell PCB. A set of three rectangular electrodes (trapping electrodes) with a
width of 8 − 10 mm, a total length of 9 − 15 mm, and a thickness of ∼35 µm are patterned
on a 2 inch diameter and 1.5 mm thick PCB. A rectangular guard electrode made of a
few mm thick Cu wire is attached around the three planar electrodes for lateral electron
confinement. An interdigitated capacitor (IDC) is fabricated from ∼5 µm wide and ∼100 nm
thick aluminum patterned on a 3.5 mm wide × 8 mm long × 0.5 mm thick quartz substrate
and then vertically mounted to the PCB to measure the bulk liquid helium level. A SAW
chip containing two identical interdigitated transducers (IDTs) for generation and detection
of the SAW is lastly mounted on the trapping electrodes. (b) Picture of the electrons on
helium LiNbO3 device after assembly. (c),(d) Optical microscope image of the IDC helium
level sensor made of ∼100 nm thick aluminium on the quartz substrate. The metal line
width and spacing are equal (∼5 µm each) as shown in the magnified image ((d)). (e),(f)
Optical microscope image of the SAW chip. The zoom-in image shows an interdigitated
transducer (IDT) with 3/3 µm metal line/spacing, which creates a SAW having a wavelength
of λsaw = 24 µm when the RF excitation is applied on resonance.

120

Figure 4.20: (a) Picture of the custom printed transparent sheet used as the mask for the
PCB patterning process. Two copies are placed on top of each other with good alignment so
that the dark area effectively blocks light during the exposure process. (b) Actual image of
a patterned PCB board after completing photolithography (1. trapping electrodes, 2. guard
electrode pad, and 3. GPO launcher pad).

4.2.3 Cleanroom Fabrication Processes

A conventional semiconductor lift-off process as shown in Fig. 4.21 is used to fabricate the

IDC level sensor and SAW devices. A detailed description of each step is as follows.

Photolithography Mask

All masks were custom made by Photo Science Inc.. These masks are made of chrome on

sodalime glass with a 3 µm limit for the minimum pattern size. They are located in the

LHQS drawers in the KMF clean room.

Wafers

The insulating substrate for the IDC helium level sensor was a 100 mm diameter and 0.5 mm

thick fused quartz (non-crystalline silica, dielectric constant ε ∼= 3.82) wafer. After pattern-

ing with this wafer, it is diced into various sizes of IDC chips. To fabricate SAW samples, two

different kinds of piezoelectric wafers are used. One is a 4 inch YZ-Cut LiNbO3 wafer with a

121

Figure 4.21: Semiconductor lift-off process for sample fabrication. (a) Photoresist (PR) is
coated on the substrate using a spin-coater. (b) The PR is patterned with UV exposure
followed by a developing process.
(c) A thin metal film is deposited onto the patterned
PR via thermal evaporation. (d) The PR/metal bi-layer is washed away in acetone or PR
stripper, leading to a patterned metal film on the substrate.

thickness of 0.5 mm. Its large piezoelectric coupling constant K2/2 = 4.82× 10−2 allows for

strong coupling of the SAW to electrons on helium for acoustoelectric measurements. SAW

chips diced from this wafer have a (010) normal crystal plane and [001] SAW propagation

direction. Additionally, SAW devices were also fabricated on 2 inch diameter and 0.35 mm

thick undoped single crystal GaAs wafers with (100) normal crystal plane. SAW samples

made of this wafer have a [011] SAW propagation direction and relatively weak piezoelectric
coupling constant (K2/2 = 3.2× 10−4). These devices were used to study SAW attenuation

arising from the screening affects the 2DES on the surface of liquid helium.

Wafer Dicing

The initial step for sample fabrication starts with the cutting of a 2 ∼ 4 inch wafer into

individual chips. We used a wafer dicing saw to do this. Once the desired dimension and

122

material information are entered into the machine, the wafer dicing with a diamond blade

(Part No. 2.187-8A-64RU7-3 from Thermocarbon) automatically proceeds with position
precision within to 40 µm. Typical dimensions of the SAW chip are 10 × 20 mm for the
LiNbO3 substrate and 17 × 17 mm for the GaAs substrate. For the IDC level sensor,

chips have various dimensions with height ranges between 3 and 4 mm at a fixed width of
∼13 mm. If the fabrication process is performed on a complete wafer, this dicing step is

employed after the patterning process is completed. In this case, a photoresist coating is

required before wafer dicing to protect the pattern on the chip from contamination from

wafer debris produced during dicing.

Wafer/Chip Cleaning

To remove contaminants from the surface of a wafer or a chip before photoresist coating, a

cleaning process is performed. The wafer or chip is sonciated for 5 min in three different
solvents in the following order: acetone → IPA → DI water. After this, sample is pre-baked

at 100 oC for 10 min to remove any moisture. Additionally, plasma cleaning with 58 sccm of

oxygen at 100 W is employed for 30 seconds to remove photoresist scum left on the developed

area right before depositing metal film.

Photolithography

The patterning of samples is performed using standard photolithography techniques at the

Michigan State University W.M. Keck microfabriction facility (KMF). The sample surface

is spin-coated with photosensitive organic material (photoresist) and the area exposed to

UV is then selectively removed through a developing process. The following is a recipe for

photolithography using the ABM mask aligner in the KMF.

123

1. Spincoat 3 ∼ 4 drops of hexamethyldisilazane (HMDS) to a diced sample with a

micro-pipette at 5000 rpm for 50 sec. This improves photoresist adhesion by turning
hydrophilic surfaces into hydrophobic surfaces. If the substrate is a 2 ∼ 4 inch wafer,
5 ∼ 10 ml of HMDS is dispensed onto the wafer and it is spun under the same condition.

2. Put 3 ∼ 4 drops of S1813 photoresist onto the HMDS-coated sample and then spin

it at 5000 rpm for 50 sec. Photoresist edge beads usually form during this process

with a square substrate. In particular, this problem becomes more pronounced when

using a rectangular substrate and results in poor patternability. To solve this issue, we

typically run all processes including spin-coating on a full wafer and then cut it into

rectangular chips at the end using the dicing saw. For the wafer photoresist coating,
after 5 ∼ 10 ml of the resist is dispensed onto the wafer, it is rotated with the same
speed and time listed above. This gives a ∼1.2 µm thick photoresist layer on the

substrate.

3. Soft bake the chip or wafer on a hot plate at 110 oC for 60 sec to remove the residual

solvent in photoresist film.

4. Place the photoresist coated chip or wafer onto the sample stage vacuum chuck of

the mask aligner and align the mask and sample. Raise the sample until the edge beads

barely touch the mask, which can be seen by observing the formation of an interference

pattern (Newton’s rings) through the transparent region of the masks. Then, expose

the chip/wafer to UV light for 6 sec at an intensity of 15.5 mW/cm2.

5. Put the exposed sample in chlorobenzene for 5 min to harden the top of the pho-

toresist. This produces a larger undercut in the photoresist during development, which

124

is required for a clean lift-off processing.

6. For photoresist developing, dip the sample into 352 developer for 25 sec and then

rinse it with DI water to completely remove any residual developer. This allows the

UV-unexposed area of photoresist to remain while the UV-exposed areas are washed

away in the developer.

Metallization

A 50 ∼ 100 nm thick aluminium film is formed over the entire area of the developed wafer/chip

using a thermal evaporator (EDWARDS AUTO 306) in the KMF. The sample is mounted

at the top of the chamber and a small aluminium pellet is put into the evaporation boat,
the chamber is closed and pumped out until the pressure reaches 10−6 ∼ 10−7 torr. Then,
the power supply input current for the source boat is slowly increased up to ∼2 A until the

aluminium pellet starts to melt. At this period, the evaporation source shutter is closed so

that aluminium vapor cannot reach the sample. After waiting for a stable deposition rate
of 2 ∼ 3 ˚A, the shutter is opened and metal film deposition starts. Once a target thickness

is achieved, the shutter is closed and the power is decreased into zero. The thickness is

measured using a quartz crystal monitor that provides real-time information on the metal

thickness during deposition.

Lift-off

To leave behind only the metal film in the patterned area (i.e. the non-photoresist), the sam-
ple is dipped into acetone or PG remover (N-methyl pyrrolidinone) at ∼60 oC for 30 − 60 min

and then cleaned with IPA (isopropanol).

125

4.2.4 Assembly

Once an IDC level sensor and a surface acoustic wave device have been fabricated, they are
attached to the PCB as illustrated in Fig. 4.19 (a). First, a copper block (10 × 5 × 3 mm)

is firmly attached to the PCB with rubber cement glue. The IDC chip is then vertically

mounted to the side of the block by applying high vacuum grease (DOW CORNING) thinly

and evenly to the back of the quartz substrate. Similarly, the SAW chip is mounted to the

PCB and held fixed by putting one tiny drop of rubber cement glue to each corner of its

back after good alignment with the trapping electrodes. As shown in Fig. 4.19 (b), the SAW

beam area has a rectangular shape where the short side coincides with the height of the

interalced electrodes of each IDT, which should maximally overlap with three underlying

trapping electrodes to achieve a wide-area of interaction between the SAW and the 2DES.

126

Chapter 5

Low-Frequency Transport of Electrons

on Helium

As discussed in subsection 4.1.5, a typical and direct way to demonstrate the formation of

surface electrons floating on liquid helium is to measure their low-frequency electrical trans-

port with a homogeneous areal electron density using the Sommer-Tanner measurement

technique. When electrons thermionically emitted from the filament are trapped and re-

main stable above the liquid helium surface, the amplitude of the electrical current typically
increases by a factor of 30 ∼ 50 compared to the case before charging the surface with elec-

trons. Such a big difference is a good indicator of a stable 2DES in our experiments. More

interestingly, the Sommer-Tanner geometry, where a set of three coplanar electrodes are sep-

arated by tiny gaps, allows for precise control of an inhomogeneous spatial distribution of

electrons by applying different DC biases to each electrode. In fact, these non-uniform charge

devices with electrons on helium have revealed a rich variety of physical phenomena such

as the existence of long-lived inter-edge magnetoplasmons [134], dynamic reorganization of

the electron density under microwave excitation [135, 136, 137], and electron crystallization

[138, 139] and quasi-1D transport [140] under strong spatial confinement. In this chapter, we

present the results of electrical transport measurements in this highly non-uniform electron

density regime by sweeping the voltage Vg at the center electrode at fixed voltages Vs = Vd

127

for the other electrodes. In this mode of operation, the device functions as a macroscopic

field effective transistor (FET) of electrons on helium and reveals novel transport phenomena

in the system of electrons on helium at elevated temperature and low electron density [16].

5.1 Experiment and Finite Element Method (FEM)

Modeling

Fig. 5.1 (a) shows an array of three co-linear rectangular electrodes patterned on a PCB

mounted in the cell, which constitutes the source, gate and drain of a FET device (Fig. 5.1 (b)).
The device is covered by layer of liquid helium of thickness d ≈ 0.5 mm. In contrast to typical

transport experiments with this type of electrode configuration, where electrons are evenly

distributed over the entire area of the device, we can use differing DC voltages on each

electrode to create a spatially inhomogeneous electron density. It is in this way that we can

operate the device as a FET where the source and drain biases are equal while the gate

electrode voltage is varied. In this mode of operation we apply an AC excitation voltage,

Vex = 0.1 V, to the source electrode and detect the gate tunable AC current on the drain

electrode using standard phase-sensitive lock-in techniques described previously. We have
carried out these measurements in the frequency range f = 1 ∼ 100 kHz and in the temper-
ature range T = 1.35 ∼ 2.0 K, which corresponds to the regime where the electron transport

mobility is limited by helium vapor atom scattering [9, 10].

We are able to gain physical insight into our experimental results through the use of

finite element method (FEM) of the electrostatic potential experienced by the 2DES and the

resulting electron density profile ns on the device. Specifically, we solve Poisson’s equation

using FEM to find the density distribution of the 2DES, ns(x), along the length L of the FET

128

Figure 5.1: Schematic of the experimental setup and circuit modeling of electrons on helium.
(a) 3D CAD rendering of the experimental cell with the array of FET electrodes and a liquid
helium level sensor. (b) Sketch of the macroscopic FET device composed of electrons on
liquid helium. The dimensions of each electrode are L × W = 5 × 10 mm and the gap
between them is 0.2 mm. (c) The equivalent lumped RC circuit of the FET device. Re is
the resistance of the electron layer above gate electrode and Ce is the capacitance of electron
system to the source and drain electrodes. (d) Transmission line equivalent circuit model of
the electron system on liquid helium. In this transmission line mapping, the resistance of
the electron layer and its capacitance to the FET electrodes is spatially distributed.

electrodes. The electron system is modelled as a charge continuum on the helium surface

with effective length Le, which is determined by imposing the condition of electrostatic

equilibrium. We determine the total number of electrons in the simulation by allowing

the electrochemical potential of the charge sheet to change in response to the gate voltage.
The areal density distribution is then given by ns(x) = −(ε0/e)(E+(x) − εE+(x)), where
E+,−(x) are electrical field distribution above and below the charge sheet, ε0 is the vacuum

permittivity and ε = 1.057 is the dielectric constant of helium.

129

5.2 Results and Discussion

5.2.1 Electrons on Helium Field Effective Transistor (FET)-IV

Characteristics

A typical source-drain current–voltage (I–V) characteristic is shown in Fig. 5.2 for FET

operation of the device. For small values of the gate voltage, Vg, no current flows through

electron layer because all of the electrons are localized above the source and drain electrodes

and the area over the gate is depleted. This is illustrated in left inset of Fig. 5.2 where we

Figure 5.2: Amplitude of the source-to-drain current, |I|, as a function of gate voltage at
T = 1.35 K for the electrons on helium FET device. For these measurements Vs = Vd = 40 V
and the guard electrode was biased with −50 V. The insets illustrate the electron charge
profile over the FET electrodes obtained by finite element simulation of the device. A
uniform electron density is achieved when the source, drain and gate electrodes have equal
value (dashed vertical line).

show the electron density profile simulated using FEM. Upon increasing Vg electrons are

attracted to the region above the gate leading to the onset of source-to-drain current flow

at a threshold value of the gate voltage, Vth. This behavior mimics that of a conventional

semiconductor FET. However, further increasing Vg reveals the first of several phenomena

unique to a FET of electrons floating on helium. Unlike a conventional FET, the source-drain

current reaches a maximum in the vicinity of uniform areal electron density (Vs = Vd = Vg),

130

as illustrated by the central inset. After reaching a maximum value, the source-drain current

begins to decrease and eventually vanishes with increasing Vg. For sufficiently large Vg all
of the electrons will be located above the gate, leading to |I| = 0, since the charge sheet is

not connected to a ground reservoir of electrons, but rather operates with a fixed number

of particles. This effect is highlighted in the right inset of Fig. 5.2 where we show the FEM

simulated density profile for large gate voltage.

5.2.2 Electron Density Estimation

In the subsection 4.1.5, we showed that the maximum attainable areal electron density

nm can be determined by voltage difference between the top and bottom electrodes using
equation 4.2. For the general case where ns ≤ nm, an uniform areal density ns when

Vs = Vd = Vg can be easily estimated from the above FET I-V measurement where Vg is

swept at constant Vs = Vd. At Vg = Vth, one can assume that electrochemical potential

Ve is in equilibrium state with constant value of Vth over all electron regions [141] and

thus the number of electrons N above each electrodes can be calculated using the following

capacitance model:

Ng = C (Vth − Ve) W L/e = 0 (above the gate)
Ns = Nd = C (Vs,d − Vth) W L/e

(above the source/drain),

(5.1)

where C = εε0/d is geometrical capacitance per unit-area of the electron layer to each

electrode, e is the elementary charge, W = 10 mm and L = 5 mm denote the width and

length of the electrode as shown in Fig. 5.1 (b). Supposing that the total number of electrons

is conserved (Nt = 2 Ns = 2 Nd), the uniform areal electron density at Vs = Vd = Vg is given

131

by

ns =

Nt
3W L

=

2
3e

C∆V, where ∆V = Vs,d − Vth.

(5.2)

For the measurement shown in Fig. 5.2, where Vs = Vd = 40 V, the areal density for our
electrode dimensions is ns ∼= 7 × 107 cm−2, which is consistent with that obtained by FEM

simulations.

5.2.3 Observation of Negative Phase of the Electron Layer Cur-

rent

As described in the subsection 4.1.7, Lock-in measurements allow us to simultaneously mea-

sure the in-phase (real) and quadrature (imaginary) components Re(I) and Im(I) of the

complex source-drain current. Knowledge of both components is necessary to accurately

model the impedance of the system as a function of frequency since it contains both resis-

tive and reactive elements. In Fig. 5.3 we show the gate voltage dependence of the real (a)

and imaginary (b) components of the source-drain current measured at T = 1.35 K (blue

trace), T = 1.6 K (green trace), and T = 1.95 K (red trace). While at low temperature both

Re(I) and Im(I) are positive as expected from the lumped circuit model, in contrast we

find that at high temperature the data exhibit an anomalous gate voltage dependence where

the imaginary component of the current Im(I) is negative at sufficiently high temperature

(shaded red region). This negative current implies that the relative phase angle φ between

the source-drain current and the AC excitation voltage is also negative at high temperature.

Furthermore, negative values of Im(I) are not restricted only to high temperature, but rather
also manifest in the vicinity of FET depletion when Vg ≥ Vth shown as the shaded green

and blue regions in Fig. 5.3 (b).

132

Figure 5.3: Real (a) and imaginary (b) components of the FET source-drain current as
a function of gate voltage measured at T = 1.35 K (blue curve), T = 1.6 K (green curve),
and T = 1.95 K (red curve) at f = 60 kHz. For these measurements Vs = Vd = 40 V and
the guard electrode was biased with −50 V. The negative values of Im(I), indicated by the
vertical arrows and shaded regions, represent a unique departure from the lumped RC circuit
model for an FET made of electrons on helium.

133

5.2.4 Transmission Line Mapping

Both the geometric and the kinetic inductance in electrons on helium are negligible in the

range of temperature studied here and thus the lumped RC model for AC transport cannot

explain this negative value of the quadrature source-drain current (i.e. both Re(I) and

Im(I) are expected to be strictly positive for all values of Re and Ce). To understand

values of Im(I) < 0 the wave nature of the propagating electrical signal must be taken into

account and the 2DES on helium should be considered as a transmission line with spatially

distributed resistance and capacitance (Fig. 5.1 (d)). Assuming that no variation in the

voltage and current along the width W of the electrodes, modeling of the system reduces to

a one-dimensional RC transmission line, where a damped voltage wave propagates from the

source to the drain with the boundary condition that the current density is zero at both of

these electrodes. The complex current I = Re(I) + j Im(I) produced by this voltage wave
can have the same form as the equation 4.24 in the chapter 4 with L → 3L. This is because

L in equation 4.24 denotes the total length of the three electrodes.

The wave nature of the current itself already provides a qualitative understanding of the

data depicted in Fig. 5.3. As expressed in equation 4.26, the complex wavevector of the

current wave is given by

1 − j
δ

k =

with δ =

(cid:114) 2σ

ωC

.

(5.3)

Here σ = nseµ is the Drude conductivity of the 2DES with the electron mobility µ, ω = 2πf

is the angular excitation frequency, and δ is a two-dimensional AC skin depth [142]. When
δ is comparable or smaller than the total length of the FET electrodes, δ ≤ 3L, the imag-

inary component of the source-drain current reverses sign since the phase angle between
the current and driving voltage φ = π/2 − 3kL becomes negative. At low temperature

134

the electron mobility, and corresponding conductivity, are relatively high. As a result δ is

larger than the overall length of the FET electrodes and Re(I) and Im(I) remain positive.

Increasing the temperature leads to an exponential increase of the density of helium vapor

atoms and hence to a marked decrease of the electron mobility. In the range of tempera-
tures T = 1.35 ∼ 1.95 K the reduction of mobility is approximately an order of magnitude,

which reduces δ to a value much smaller than the total length of the electrodes leading to

the negative values of Im(I) we observe in our measurement. Furthermore, our finding that

Im(I) < 0 at low temperature (T = 1.35 K) but only in the vicinity of FET turn-on can

be understood in a similar fashion. When Vg ∼= Vth the number of electrons present in the

gate region is small and their resistivity is relatively high leading to a decrease in δ and

Im(I) < 0. As Vg is further increased more electrons are attracted to the region above the

gate electrode, which increases the 2DES conductivity and tunes the sign of Im(I) to positive

values.

We have also performed a quantitative analysis based on the transmission line model

for the case where the electron density is homogeneously distributed over the device. This
model is defined by the following expression, which relate the mobility µ = σn−1

s e−1 to the

frequency dependent complex source-drain current:

I = I0 (1 + j)

3σ
2L

sinh2(jkL)
sinh(3jkL)

with I0 = ωC

W L

3

Vex.

(5.4)

The comparison between a fit based on this model (dashed curves), with the transport

mobility µ as a fitting parameter, and the experimental data (solid curves) is shown in

Fig. 5.4, where the frequency dependence of both components of the measured current is

plotted at low (T = 1.35 K) and high (T = 1.95 K) temperatures. For the data in Fig. 5.4

135

Figure 5.4: Frequency dependence of the real Re(I) (a) and imaginary Im(I) (b) components
of the source-drain current with a uniform electron density above all three FET electrodes
corresponding to Vs = Vd = Vg = 40 V. For these measurements the guard electrode was
biased with −5 V. The solid traces show the experimental data obtained at T = 1.35 K
(blue) and T = 1.95 K (red). The correspondingly colored dashed lines are calculated using
the transmission line model described in the text.

136

the DC bias potential is the same for all FET electrodes Vs = Vd = Vg = 40 V and the
electron density is ns = 7 × 107 cm−2. The value of the mobility obtained from this fitting
is 2.8 × 104 cm2 V−1 s−1 at T = 1.35 K and 0.45 × 104 cm2 V−1 s−1 at T = 1.95 K. These

values are in reasonable agreement with previous measurements [9] and the theoretical values

given by Saitoh [10].

5.3 Conclusion

We have investigated the operation of a macroscopic field effect transistor composed of

electrons on liquid helium in a regime where electron scattering from helium vapor atoms

is the dominant form of disorder. In this FET mode of operation the electron density can

be made highly non-uniform and we find that the imaginary component of the source-drain

current changes sign from positive to negative when the systems is subjected to sufficiently

high temperatures or when the electron density is made maximally inhomogeneous. These

transport regimes can be understood by considering the system of electrons as a voltage

wave propagating in a transmission line composed of the electrons floating above the metallic

electrodes under the helium surface. We anticipate that an FET mode of operation could

find application in studying other nonequilibrium phenomena such as ultrahot electron on

helium [143, 144, 145, 146], electrons strongly confined in helium microchannel devices or

for future high frequency surface acoustic wave experiments.

137

Chapter 6

SAW Coupling to Liquid Helium

The physics of the underlying liquid helium substrate is pivotal to better understanding

the unique properties of 2D electrons on helium. In particular, when piezoelectric SAWs

propagate along the surface of a piezoelectric crystal underneath surface electrons on helium,

they can interact with not only the 2D electrons but also the liquid helium film, leading to

SAW attenuation and velocity shift. Thus, understanding SAW coupling to the helium film is

a prerequisite for the SAW measurements of the 2DES created on the surface of the liquid. In

this chapter, we present pulsed measurements of the absolute attenuation of high-frequency

SAWs on the surface of lithium niobate caused by bulk layers and thin films of liquid helium

[147].

6.1 Experiment

The SAW attenuation experiments reported here were performed using a YZ-cut lithium

niobate (LiNbO3) single crystalline wafer as a substrate for SAW propagation. The lithium

niobate substrate was diced into a rectangle with a length of 20 mm and a width of 10 mm.

To excite and detect SAWs, two identical interdigitated transducers (IDT) having 40 pairs

of 3 µm wide fingers were patterned on the surface of the LiNbO3 chip in the form of a

delay line, having a length of 16.5 mm, along the crystallographic x-axis using conventional

photo-lithography (see Fig. 6.1 (a)). The transducers were fabricated from aluminum and

138

Figure 6.1: Pulsed surface acoustic wave experimental setup. (a) Schematic of the lithium
niobate chip used for exciting and detecting SAW via aluminum interdigitated transducers
(IDT), which were directly fabricated onto the surface of the substrate. (b) Block diagram
of the circuit used for measuring the attenuation of pulsed SAWs. As described in the text,
by measuring the time-of-flight signal we are able to disentangle direct capacitive crosstalk
between the two interdigitated transducers as well as multiply reflected SAW pulses.

had a thickness of approximately 70 nm and a width of 4 mm defining the width of the

SAW beam within the delay line. We note that in this geometry the IDTs are positioned

1.75 mm from the diced edge of the chip. Resonant SAWs are launched by applying a high-

frequency signal between the transducer fingers of the exciter to create an elastic distortion

of the piezoelectric substrate beneath the IDT. In this configuration, Rayleigh mode surface

acoustic waves with components parallel (longitudinal) and perpendicular (transverse) to

the wave propagation are launched along the LiNbO3 surface toward the detector IDT. The

fundamental resonant frequency of our Rayleigh wave SAW device is ν = vs/λ = 291 MHz,

dictated by the IDT finger periodicity, λ = 12 µm, and the speed of sound in YZ cut

LiNbO3, vs = 3488 m/s. The frequency response of the SAW delay used in our attenuation

measurements was characterized using an Agilent N5230A vector network analyzer. Fig. 6.2

shows the measured transmission coefficient, S12, of the delay line as a function of frequency
at T = 1.55 K in vacuum. The resonance in the transmitted power at ν ∼= 296 MHz

139

is associated with the generation of SAW in the substrate. The measured resonance is a

few percent larger than the expected value, likely due to an increase in the elastic moduli of

LiNbO3 at cryogenic temperature. Fig. 6.1 (b) shows a diagram of the circuit used to measure

Figure 6.2: Frequency dependence of the transmission coefficient (S12) of the SAW delay
line at T = 1.55 K in vacuum. The resonant peak at ν = 296 MHz is associated with the
generation of surface acoustic waves in the IDT delay line.

the attenuation of the piezoelectric waves. Pulsed SAWs are created by gating a continuous

wave signal using a fast solid-state switch. For the measurements reported here, the pulses

had a width of 3 µs and were repeated at a frequency of 1 kHz. The received SAW signal

at the detector IDT was amplified by 30 dB and measured using a calibrated crystal diode

detector (Krytar Model 203AK S/N 00277). A boxcar integrator was used for time-of-flight

measurements. This mode of operation allows us to disentangle the piezoacoustic signal from

direct capacitive crosstalk between the two IDTs as well as multiply reflected surface acoustic

wave pulses within the delay line. The SAW device was mounted inside of a hermetically

sealed copper cell attached to a closed cycle 1K cryostat. Helium gas was supplied into the

cell at T ∼= 1.55 K through a capillary fill line. A resistive thermometer located outside of

140

 !" #" $" %" &""&"'&%()*+,$-"$#"$%"$""%."%-"%#"(/()012,the cell and calibrated relative to the known superfluid transition temperature of 4He was

used to measure the temperature of the liquid helium. The liquid helium volume admitted

into the cell was determined by varying the pressure in a calibrated standard volume of

260 cc at room temperature. As expressed in equation 2.96 of the chapter 2, the thickness

of the helium film can be estimated as follows.

(cid:18) γ

(cid:19)1/4

,

(6.1)

d =

ρgH

where γ is the van der Waals constant, ρ is the mass density of 4He, g is the acceleration

due to gravity, and H is the distance from the LiNbO3 surface down to the liquid helium

level in the reservoir volume in the cell. Three-dimensional modelling of the experimental

cell open volume was used to calculate H from the volume of helium admitted into the cell

from the calibrated volume at room temperature.

6.2 Results and Discussion

6.2.1 SAW Attenuation by Bulk Liquid Helium

Before presenting our results on thin liquid helium films, we first demonstrate that we are

able to use our measurement setup to reproduce the known temperature dependence of

Rayleigh wave attenuation by bulk liquid helium. Rayleigh mode surface acoustic waves are

attenuated by contact with a bulk liquid due to the surface-normal component of the par-

ticle motion and that of the co-propagating piezoelectric field. These components generate

longitudinal compressional waves upward into the liquid, dissipating most of the acoustic

energy [148, 149]. This method of energy loss is closely related to the problem of the Kap-

141

tiza resistance based on the acoustic mismatch between liquid helium and a solid substrate

[11, 150], whereby longitudinal waves having a velocity v are emitted into the liquid from the

substrate surface at an angle φ = arcsin(v/vs). In addition to this mechanism, the in-plane

shear component of the SAW can also radiate energy into a fluid over the viscous penetration

depth,

L =

(cid:114) η

πνρ

(6.2)

where η and ρ are the viscosity and mass density of the fluid. However, SAW energy lost

to shear is two to three orders of magnitude smaller than longitudinal absorption in our

temperature range and is negligible even for our measurements in thin helium films. Our

results for the temperature dependence of the SAW attenuation produced by bulk liquid

helium are shown in Fig. 6.3 as the solid red curve. For comparison, we also show the

Figure 6.3: Attenuation, α, of Rayleigh waves on LiNbO3 in contact with bulk liquid helium.
Our measurements (solid red curves) at ν = 296 MHz are in good agreement with the
theory (dashed red curve) of Dransfeld and Salzmann [22] for energy loss due to radiation
of longitudinal compressional waves into the liquid (Inset). The local minimum of the SAW
attenuation at Tλ = 2.17 K is associated with the transition from normal to superfluid 4He.

142

 !"#"!#$%&'()*+, -. -/ -0 -  -!"-."-/1%234.-!5-.5-/5-05- 5-!/-.$%&'()*+, -0 -6 -  -" -!1%23417prediction for the attenuation (dashed red curve) based on the loss of SAW energy in the

form of longitudinal compressional waves [149]. As expected, our measurements are in good

correspondence with theoretical prediction. Moreover, our results for bulk helium are also

in good agreement with previous measurements of the attenuation of piezoelectric Rayleigh

waves by liquid helium [148, 151, 152, 153].

6.2.2 Anomalous Attenuation of SAWs by Thin Liquid Helium

Films

Unexpectedly, we find that when the thickness of the liquid helium layer is sufficiently reduced

an anomalously large attenuation can be induced. In Fig. 6.4, we show measurements of the

SAW attenuation (solid blue data) at T = 1.55 K while increasing the volume of helium in the

cell by small increments. With an increasing amount of helium, we observe a rapid increase
in the SAW attenuation once the thickness of the superfluid film reaches ≈ 60 ∼ 70 nm
(H ∼= 0.28 mm) and continues to increase and quickly exceeds the attenuation produced

by bulk helium. For reference, we plot the SAW attenuation measured in bulk helium

at the same temperature as the red horizontal line in Fig. 6.4. We emphasize that this

anomalously large attenuation is reproducible on multiple fillings of the experimental cell

and additionally is insensitive to the tilt of the cryostat with respect to vertical. With further

increasing film thickness, the SAW attenuation will eventually recover the value measured
for bulk helium. However in this crossover regime (d ∼ 1 − 10 µm) the film thickness on

the substrate is exceedingly sensitive to fluctuations in H and also to the tilt of the cryostat

leading to non-reproducibility. Future experiments, using microchannel geometries could

allow for controlled measurement in this regime of film thickness. Moreover, we find that

143

this phenomenon is not restricted to superfluid helium but rather persists into the normal

state.

Figure 6.4: Attenuation, α, of Rayleigh waves on LiNbO3 in contact with thin films of liquid
helium at T= 1.55 K and ν = 296 MHz. The measurements are made by incrementally
admitting small amounts of helium gas into the cell from a standard volume at room tem-
perature (lower horizontal axis). The vertical dashed line indicates the estimated thickness
of the helium film on the LiNbO3 surface based on equation 6.1 and the height H from the
substrate to the bulk helium reservoir level in the cell (top horizontal axis). For comparison,
the horizontal solid red line indicates the SAW attenuation due to bulk liquid helium at
the same temperature. The calculated thicknesses of the helium film for the final two data
points above the bulk value are 73 nm and 76 nm. Finally, we note that the IDT fingers are
expected to fill completely with helium via capillary action for very small amounts of helium
introduced into the cell, i.e. already at H = 0.46 mm.

To our knowledge, no prior measurements of the attenuation of high-frequency SAW have

been made in this regime of helium thickness nor is there a theory describing the coupling of

SAWs to such films. We speculate that the occurrence of the attenuation anomaly reported

here for thin helium films can qualitatively be understood in terms of a coupling between the

piezoelectric surface wave on the LiNbO3 substrate and excitations in the helium surface.

144

 ! " # $%!"#$&'()*+,-.$/0$$/"1$/12$/1 $/#03'4--5'"$$1$$#$$ $$$36789-:;<7'4--15&=97>''?'2/@)*+,-While the physical displacement of atoms associated with the SAW is at the angstrom scale,

the SAW piezoelectric field has a spatial extent on the order of the SAW wavelength, which
in our case is ∼12 µm. In fact, it is well-known that an electric field gradient can exert a

force to move liquid helium via electrostriction, an effect which has even been utilized in

developing a number of recent superfluid optomechanical systems [154, 155, 156, 157]. It is

possible that an electrostrictive coupling between the LiNbO3 surface wave and the liquid

helium surface conspire to produce a new mode of energy loss not present in bulk helium.

Finally, we note that our observation of enhanced attenuation in thin normal helium films

is able to rule out the coupling of SAWs to third sound as the sole mechanism for increased

SAW energy loss.

6.3 Conclusion

In conclusion, we have performed high-frequency SAW attenuation measurements in thin

films of superfluid and normal 4He where we find an anomalously large loss of energy from

the piezoelectric surface wave into the liquid. We suspect that this increased attenuation is

associated with electromechanical excitation of the helium film surface via a coupling of the

SAW electric field to the dielectric constant of liquid helium.

145

Chapter 7

SAW Coupling to Electrons on

Helium

We have investigated the coupling of SAWs to the 2DES on liquid helium surface. We

show that this coupling is mediated through the piezoelectric field of the SAW. In the weak

piezoelectric field regime, the screening of the piezoelectric potential by the mobile 2DES

results in energy and momentum transfer from the SAW to the 2DES. This leads to a

change in the attenuation and the velocity of SAWs as described in section 3.2. Measure-

ments of these quantities provide information about electrons on helium films. In particular,

the SAW detects the conductivity of the 2DES with high sensitivity as described in sec-

tion 3.2. This ability allows for the measurement of the conductivity of electrons on helium

at much higher frequencies and shorter wavelength than in conventional transport experi-

ments. The coupling of SAWs to electrons on helium were originally theoretically studied

by Lawrence A. Wilen [122]. Based on an electrical surface perturbation method described

in subsection 3.2.2, he showed that an evanescent piezoelectric coupling of the SAW to the

2DES yields the SAW velocity shift ∆v/v ∼= 2 ppm (parts per million) and the SAW atten-
uation Γ ∼= 1.5 dB/cm for a SAW frequency of f = 180 MHz and an areal electron density
ne = 1010 cm−2. To measure the SAW attenuation in the presence of electrons on helium,

we used a [011] cut of GaAs (100) as the piezoelectric substrate due to its relatively weak

146

piezoelectric coupling constant K2/2 ∼= 3.2 × 10−4. With this piezoelectric substrate cov-

ered by a thin film of liquid helium, we observe that the SAW attenuation is strongly affected

by surface charge density and is in a reasonable agreement with Wilen’s calculation. With

an increasing distance d from the 2D electrons to the surface of the piezo-substrate (GaAs)

along which the SAW propagates, the SAW attenuation decreases and eventually disappears
when d (cid:29) SAW wavelength λ, reflecting the evanescent coupling of electrons on helium and

the SAW potential. These results and discussion are presented in section 7.1.

In the strong piezoelectric field regime, the SAW potential is able to trap and transport

surface electrons and leads to a local DC current so-called acoustoelectric charge transport

as introduced in the section 3.3. This effect has been observed in many low-dimensional

electron systems ranging from [33, 34, 115, 158, 159, 160].

In particular, it has recently

attracted particular attention as a means of controlling and transporting single electron

for metrology [38], quantum optics [161], and quantum information processing [106, 107].

To date, considerable progress in these fields has been made with GaAs 2DESs. However,

crystalline imperfections and foreign impurities (chemical donors or acceptors) in this system

make it challenging to reach long electron coherence times [162], which is required for a

practical quantum computing hardware. Also, these defects have limited the performance

of a single electron transport devices [163, 164, 165]. A SAW-electrons on helium coupling

device has the potential to naturally solve these problems due to the lack of such defects in

electrons on helium. To demonstrate the acoustoelectric effect in electrons on helium, we

have fabricated a SAW device on a YZ cut lithium niobate (LiNbO3, crystal plane (010)

with SAW propagation direction [001]), which has a nearly two orders of magnitude higher

piezoelectric coupling constant K2 ∼= 4.82 × 10−2, compared to GaAs. In section 7.2 we

report the first observation of this effect in electrons on the surface of thin liquid helium film

147

[166].

7.1 SAW Attenuation by Electrons on Helium

7.1.1 Experiment

The SAW device for studying electrons on liquid helium is locted inside a superfluid leak-tight

copper cell. Fig. 7.1 (a) and (b) illustrate the schematic diagram of the device. A 17 mm

square GaAs (100) chip with a thickness of 350 µm is used as the piezo-substrate on which

surface acoustic waves propagates. A layer of 1.2 µm thick hard-baked photoresist (S1813G2)

is patterned on the GaAs substrate to produce monolayer microchannel arrays consisting of

approximately two hundred identical channels having a width and spacing of 15 µm. 4He gas
is supplied into the cell at T ∼= 1.55 K from a standard volume of 260 cc at room temperature

to condense liquid helium in the cell. The liquid helium film is formed in the microchan-
nel region by capillary action. Its thickness is estimated to be dch = 1.16 ∼ 1.18 µm from

equation 7.9 depending on an areal electron density ne. At the same time, the top of the
photoresist is also covered with a very thin liquid film dvan = 60 ∼ 80 nm due to the Van

der Waals interaction. Electrons generated from a filament are trapped on the surface of

the helium film by applying a DC bias to a set of three electrodes (the source, the gate, and

the drain electrode) underneath the GaAs as shown in Fig. 7.1 (b). Each electrode has a

dimension of 5.5 mm length, 3 mm width, and 0.18 mm gap. Two pairs of interdigitated

transducers (IDTs) made of 100 nm thick aluminum are deposited around the microchannel

geometry to create and detect Rayleigh mode SAWs. In this study, only the set of opposite

IDTs near the gate electrode (exciter IDT1 - detector IDT1) are operated. This simplifies the

analysis of the SAW attenuation measurement since only electrons above the gate electrode

148

contribute to the attenuation as shown in Fig. 7.1 (c). Note that if we use the other set of

IDTs (exciter IDT2 - detector IDT2), SAWs interact with electrons over each electrode. This

complicates the analysis of the attenuation of SAWs due to an inhomogeneous distributions

of electrons over the electrodes. Each of the IDTs we used, consists of 22 pairs of 6 µm wide

fingers. The width of the IDT is 2.96 mm defining the width of the SAW beam to be slightly

smaller than 3 mm wide gate electrode so that the SAW can interact with electrons above

the gate electrode with a propagation direction perpendicular to the channel (Fig. 7.1 (c)).

Fig. 7.1 (c) shows the circuit diagram used to measure the attenuation of pulsed Rayleigh

Figure 7.1: Experimental setup for measuring the attenuation of SAWs by electrons on he-
lium. (a),(b) Top and cross-section view of the SAW device for studying electrons on helium.
Electrons on the helium films formed in microchannel arrays are held in place by applying a

positive DC bias to three undelying electrodes beneath the GaAs substrate at T ∼= 1.55 K.

Two pairs of interdigitated transducers (IDTs) are located around the microchannel struc-
ture to excite and receive SAW signal. (c) Block diagram of the circuit used for measuring
SAW attenuation. The exciter IDT1 and the detector IDT1 are operated so that pulsed
SAWs can interact with 2D electron sheet above the gate electrode. In this region of the
sample, the electron density can be varied from zero to its maximum value.

SAWs. An RF power of 5 dBm at 119.2 MHz was applied to the exciter IDT1 via a fast

solid-state switch to generate pulsed SAWs having a wavelength λ = 24 µm, typical pulse

width of 2.5 µs, and a period of the pulse T = 1 ms. The received SAW signal at the detector

149

IDT1 is amplified by 30 dB and converted to a measurement of power using a calibrated

crystal diode detector (Krytar Model 203AK S/N 00277). A boxcar integrator is used to

improve the signal to noise ratio by excluding the instantaneous capacitive cross-talk signal

between the IDTs. The boxcar integrator also averages the SAW pulses over typically 1,000

samples. Simultaneous low-frequency transport measurements of the electrons on helium is

performed with the underlying electrodes. An AC excitation voltage Vex = 0.1 V is applied

to the source electrode and the current is detected with a lock-in amplifier connected to the

drain electrode (Fig. 7.1 (b)). We find that this current does not interfere with the Rayleigh

SAW attenuation by the 2D electrons above the gate and thus they can be simultaneously

measured with the electron density varied during a gate voltage Vg sweep.

7.1.2 Theory

Geometric Effect on SAW attenuation

Piezoelectric SAWs propagating beneath the surface of electrons on helium interact with not

only the 2D electrons but also the underlying liquid helium substrate and thus both can result

in attenuation of the SAW amplitude. In addition to that, a periodic array of inhomogenities

like photoresist trenches in our device, on the surface of the piezoelectric substrate (Fig. 7.7)

can cause attenuation too. In particular, when the SAW wavelength λ is comparable to the

period of irregularties l, the SAW amplitude is reduced by Bragg reflection when λ = 2l,

or the SAW loses energy by emitting phonons into the bulk due to its scattering with the
periodic array structure when λ ≈ l [167]. Since λ = 24 µm is approximately similar to

l = 30 µm in our device, the geometric effect on the SAW attenuation is mainly associated

with the latter. More specifically, the periodic array structure on the piezoelectric surface

150

scatters the SAW into a bulk wave having the same wavelength as the SAW. This bulk

wave propagates into both the piezo-substrate and the adjoining medium (liquid helium in

our case). However, for a thin helium film in our device, where the film thickness is much

smaller than the wavelength of the SAW, bulk waves generated by SAW-periodic structure

scattering mostly propagate into the piezoelectric medium. This leads to approximately

constant attenuation over the film thickness change, which is not our interest in this work.

Therefore, we remove it by measuring relative attenuation in electrons on helium in the

presence of the periodic structure.

SAW Interaction with Surface Electrons on Helium

The interaction between SAWs and 2D electrons on liquid helium through the SAW piezo-

electric field was theoretically investigated by Wilen [168] and we derive an expression for the

SAWs attenuation applicable for our device geometry following his method. For weak piezo-

materials such as GaAs and quartz, the attenuation by a perturbing 2DES can be calculated

using the Ingebrigtsen formula (electrical surface perturbation) [169]. This perturbation ap-

proach allows to estimate a ratio of the perturbed electric potential to the perturbed electric

field in the presence of 2D electrons above the piezoelectric surface. One can relate this ratio

to the attenuation and the velocity shift of SAWs as described in subsection 3.2.2. In this

formulation, the attenuation is given by

where

α
k

=

(cid:48)

z

(0) = −ik

(cid:18) K2

(cid:32)

2

(cid:48)
φ
(cid:48)
z

D

(cid:33)

(cid:19)

(cid:32)

Im

(cid:33)

,

(cid:48)
1 + iz
1 − i
pz

(0)
(cid:48)

(0)

(cid:113)


zz
xx − 
2
zx.

,


p =

z=0

151

(7.1)

(7.2)

Here, Rayleigh SAWs propagates in the direction of the x axis and the positive z axis points

out of piezoelectric plane located at z = 0. The normalized surface impedance z

(cid:48)

(0) is

(cid:48)
defined by a ratio of perturbed potential φ

(0) and the normal component of the perturbed

displacement D

(cid:48)
z(0) at the piezoelectric surface. The effective dielectric constant for the

piezo-substrate 
p is determined by dielectric tensor element 
ij. The parameters α, k, and
K2
2 denote the SAW attenuation constant, the SAW wave number, and piezo-electric coupling

constant of the substrate material. Note that equation 7.1 does not include mechanical losses

of SAW energy and only describes losses from coupling to a mobile 2DES through the electric

field. This is a good approximation for the present device where both the thickness of the
charged helium film and the photoresist (dch ≤ 1.18 µm, dvan ≤ 80 nm, and dPR

∼= 1.2 µm,

see Fig. 7.7) is significantly smaller than the SAW wavelength of 24 µm [119]. Since the

only unknown parameter in equation 7.1 is z

(cid:48)

(0), estimation of the attenuation is a matter

of solving for the electric potentials using appropriate electrical boundary conditions. From

Poisson’s equation, the potential inside the different regions in Fig. 7.2 can be expressed as

φ1 = eikx−iωt(cid:16)
φ2 = eikx−iωt(cid:16)
φ3 = eikx−iωt(cid:16)

Ae−kz + Bekz(cid:17)
Ce−kz + Dekz(cid:17)
Ee−kz(cid:17) − 4πσ0z,

,

,

(7.3)

where σ0 and ω = 2πf are the equilibrium charge density of the 2DES and the angular

frequency of the SAW. There are five unknown amplitude constants (A - E) and they can be

expressed in terms of only one unknown from the electrical boundary conditions, a continuity

equation, and relaxation time approximation (see appendix A.1 for detail). This yields a

formula for the normalized surface impedance z

(cid:48)

(0) and the SAW attenuation divided by the

152

Figure 7.2: Approximate model geometry for SAWs experiments. SAWs propagate along
the positive x-axis and the positive z-axis points out of the piezoelectric plane.

wavenumber α/k as follows:

(cid:48)

z

(0) = −ik

(cid:18) K2

(cid:19)

2

Im

α
k

=

(cid:32)
(cid:32)

(cid:33)

(cid:48)
φ
(cid:48)
z

D

= i
−1

1

rA + rB
rA − rB

,

z=0

rA − rB − 
−1
1 (rA + rB)
rA − rB + 
p/
1(rA + rB)

(cid:33)

,

(7.4)

(7.5)

where

(cid:18) 
1 + 
2

2
1

rA =

rC +


1 − 
2
2
1

e2kh1

(cid:19)

,

rB =

(cid:18) 
1 − 
2

2
1

e−2kh1rC +


1 + 
2

2
1

(cid:19)

,

(7.6)

and

rC =

[ (
2 + 1)ω2 − ω2
[ (
2 − 1)ω2 + ω2

p] τ + (
2 + 1)iω
p] τ + (
2 − 1)iω

e2k(h1+h2).

(7.7)

Here h1,2 and ε1,2 denote the thickness and dielectric constant of dielectric 1 and 2 (see

Fig. 7.2) while ω and ωp are the angular frequency of the SAW and the plasma frequency
p = 4π|σ0|ek/me), and σ0, e, and me are the equilibrium charge den-

in the 2D case (i.e. ω2

sity of the 2DES, the elementary charge, and the electron mass, respectively. A derivation

of equations 7.4 and 7.5 is presented in appendix A. Note that a minus-sign is dropped

on the right-hand side of equation 7.4 due to the fact that in Ingebrigtsen formula (equa-

153

tions 7.1 and 7.2) the positive z-axis points into the piezoelectric medium while the negative

z-axis does so in our derivation. For the simple case of one dielectric film (
1) between the

piezo-substrate and the 2DES, the expression for the SAW attenuation reduces to

(cid:32)

(cid:18) K2

(cid:19)

Im

2

rC − 1 − 
−1
1 (rC + 1)
rC − 1 + 
p/
1(rC + 1)

(cid:33)

,

(7.8)

α
k

=

with 
2 = 
1 and h2 = 0.

In the region of the microchannel, the SAW attenuation can be calculated using equa-

tions 7.7 and 7.8 with 
1 = 
He and h1

∼= dch. 
He denotes the dielectric constant of liquid
4He and dch is the thickness of 4He film at the center of the channel, which can be calculated

from [170]

dch = dPR − w2
8σs

ρgH + 2π

n2
ee2

He

(cid:18)

(cid:19)

,

(7.9)

where dPR, w, and σs are the thickness of the photoresist, the channel width, and the

surface tension of liquid 4He while ρ, g, H, and ne denote the mass density of liquid 4He,

the gravitational acceleration, the distance from the piezoelectric surface down to the liquid

helium level in a lower-lying reservoir, and the areal electron density.

For the photoresist region of our device as shown in Fig. 7.2 (b), equations 7.5, 7.6, and 7.7

can be used to estimate the attenuation with 
1 = 
PR, 
2 = 
He, h1 = dPR, and h2 = dvan.

The parameter 
PR is dielectric constant of the photoresist and Van-der-Waals helium film

above the photoresist can be obtained from [50]

(cid:16)

(cid:17)−1/3

dvan = d0

1 + 2πn2

e/ρgH

,

d0 = (a/ρgH)1/3 ,

(7.10)

where a is van-der-Waals constant and d0 is the film thickness before charging electrons onto

154

the surface of the liquid helium. For both regions (i.e. within the microchannel and on top

of the photoresist), ωp and τ in equation 7.7 are determined by the electron mobility µe and

(cid:113)

electron density ne using ωp =

4π nee2
me

k and τ = µeme

e

.

7.1.3 Results and Discussion

SAW attenuation by liquid helium

Before studying the attenuation of Rayleigh SAWs caused by the electrons floating above the

helium surface, we first investigated the effect of liquid helium by measuring the attenuation

as a function of the liquid helium volume inside the sample cell as shown in Fig. 7.3. In the

helium film regime where the helium level is below the surface of 350 µm thick GaAs substrate

(H ≥ 0.2 mm), a strong attenuation is observed with a maximum value αmax ∼= 9 dB/cm
at H ∼= 0.24 mm as described in chapter 6 for a thin liquid helium film on the LiNbO3

substrate. For the helium level far exceeding the substrate surface, the attenuation shows
nearly constant value (∼10 dB/cm) indicating a bulk helium coverage. Note in these mea-

surements we have chosen the distance from the GaAs surface down to the liquid helium
level (H ≤ 0.2 mm) to avoid the massive attenuation in the thin film region (see chapter 6 ).

This is required in order to observe the relatively small SAW attenuation produced by the
2D electrons (∼1 dB/cm).

SAW Attenuation by 2D Electrons on a Helium Thin Film

To characterize the SAW attenuation produced by the system of electrons on helium, we

measured the SAW attenuation as the electron density ne was increased from zero during a

positive sweep of the gate bias Vg. Note the SAWs only interact with electrons in the region

155

Figure 7.3: Attenuation of Rayleigh SAWs in contact with liquid helium. These measure-

ments are done at T ∼= 1.55 K by incrementally providing small amounts of helium gas into

the sample cell. The volume of the liquid helium in the cell was determined from the pressure
in a standard volume of 260 cc at room temperature. The corresponding bulk helium level
is estimated from 3D CAD modeling of the liquid helium filling the cell, which determines
the distance between the GaAs substrate and the level of the helium reservoir, H. The SAW
attenuation shown is the relative value with respect to empty cell. The vertical dashed line
indicates when the reservoir helium level meets the substrate surface. Negative values of H
represent a helium level that exceeds the piezoelectric surface (GaAs).

156

over the gate electrode (Fig 7.1 (c)). Thus, ne here indicates the electron density in this

region unless otherwise stated. Simultaneous measurement of the low-frequency transport as

well as finite element method (FEM) simulation allows for the estimation of ne at different

value of Vg (see appendix A.3 for details). The contribution of the microchannel structure

to the capacitance between the electron layer and the underlying electrodes is negligible and

we assume that electrons are evenly distributed inside of and outside of the channels above

the gate. Once ne is estimated, we can calculate the helium film thicknesses dch and dvan

above the gate electrode as a function of Vg using equations 7.9 and 7.10. This calculation
∼= 1.18 µm and dvan ∼= 80 nm for the neutral film, and a significant change in

produces dch

the charged film thickness at higher Vg due to the electron pressure (Fig. 7.4 (b) and (c)).

The electron density ne versus Vg in seen Fig. 7.4 (a) shows that for values of Vg < 35 V no

electrons exist above the gate since all of them are attracted to the regions above the source

and drain electrode (Vs = Vd = 60 V) and ne is a linearly increasing function of Vg when
Vg ≥ 35 V.

Figure 7.4: (a) Electron density ne estimation in the 2D electron region above the gate
electrode during a sweep of Vg at Vs = Vd = 60 V. Based on this ne estimation, we have
determined the thickness of the charged helium film in the channel (b) and the photoresist
area (c) with H ∼= 207 µm. Here dch and dvan denote the liquid helium thickness within the

channels and on top of the photoresist while H is the distance between the GaAs surface
and the level of helium in a reservoir adjacent to the substrate.

In the following discussion the SAW attenuation is measured relative to the case with no

157

electrons. Experimental data of the attenuation (the blue closed circles in Fig. 7.5) reveals

that the onset of SAW attenuation matches the value of Vg when electrons start moving to

the region above the gate electrode. We can confirm that this signal arises from the electrons

by repeating the experiment without electrons on the helium film as shown in the blue open

circles of Fig. 7.5. In this case, the attenuation is zero for all values of Vg. In addition, we
note that the calculation of dch and dvan in Fig. 7.4 (b) and (c) show that at Vg ∼= 40 V
there is a negligible change of both film thicknesses of less than ∼1 %, which suggests that

the peak of the attenuation curve is not strongly affected by a film thickness change.

Figure 7.5: SAW attenuation during a gate voltage sweep. Blue closed circles show relative
measurements of the attenuation during a Vg sweep at Vs = Vd = 60 V in which the electron
density ne is linearly proportional to Vg as seen in Fig. 7.4(a). The blue line shows the
theoretically calculated average of the attenuation in the channel and photoresist regions.
This theory only includes the piezoelectric coupling to electrons floating on helium film. After
removing electrons, no noticeable change in SAW signal was observed (blue open circles).

With H ∼= 207 µm, the initial helium film thickness with no electrons in the microchannel
∼= 1.18 µm and dvan ∼= 80 nm. In the presence of electrons, the
array is estimated to be dch
Vg dependence of the film thickness is given in Fig. 7.4 (b) and (c).

We have also performed a quantitative analysis based on the well-developed theory dis-

cussed previously, which only considers the piezoelectric coupling between the SAW and

158

2DES and neglects the mechanical loading of the microchannel array and the helium film.

The SAW attenuation versus Vg in the microchannel was calculated using equations 7.7
∼= dch. For the photoresist region,
equations 7.7, 7.6, and 7.5 were used to calculate the attenuation with 
1 = 
PR = 4,

and 7.8 with 
1 = 
He = 1.057, 
p = 
GaAs = 13, and h1


2 = 
He = 1.057, h1 = dPR = 1.2 µm, and h2 = dvan. Here dch and dvan are calculated

values shown in Fig. 7.4 (b) and (c). For both cases, we have used the coupling constant
K2/2 = 3.2 × 10−4 for GaAs [115]. The calculation of ne in Fig. 7.4 (a) was used to

determine the plasma frequency ωp at different Vg which is needed in equation 7.7. The

relaxation time τ in equation 7.7 can be obtained from the electron mobility transverse to

the channel µt. However, with the present sample, we were only able able to estimate the

mobility µl in the direction along the channel under a homogeneous distribution of electrons

over all these electrodes at Vs = Vg = Vd (see appendix A.3). Neglecting the small vertical
holding field effect on µl with changing Vg in the helium vapor scattering at T ∼= 1.55K [10]
µlme
(i.e. µl = constant) with the assumption of µt < µl, we have obtained τ using τ = f
where µl = 1.4 × 104 cm2/Vs and f < 1 is a scale factor. This value of τ was inserted into

e

equation 7.7 to determine α in equation 7.5. With these parameters, we have calculated the

SAW attenuation resulting from the electrons in both the channel and photoresist regions.

The blue line in Fig. 7.5 shows their average with f = 0.44, which is in good correspondence

with the experimental data for Vg < 40 V where dch and dvan negligibly change with gate

voltage. A deviation appears when the film thickness is strongly reduced for Vg > 40 V

(Fig. 7.4 (b) and (c)). This discrepancy could be associated with enhanced attenuation pro-

duced by a decrease of the film thickness. Despite this discrepancy when taken on balance

the SAW attenuation peak we observe with increasing ne is strong evidence for SAW energy

loss to the 2DES via the piezoelectric coupling to the sheet of the 2D electrons above the

159

helium film.

In order to systematically investigate the piezoelectric coupling strength with increasing

distance d from the 2DES to the piezoelectric surface, we have measured the SAW attenuation

versus Vg with increasing charged helium film thickness by incrementally adding liquid helium

(left inset of Fig. 7.6). The maximum attenuation measured along with the theoretical

Figure 7.6: SAW attenuation versus helium level in the cell. Liquid helium was slowly and
incrementally added to the charged helium film by supplying helium gas into the cell at

T ∼= 1.55 K. This film thickening raised the 2D electron layer away from the piezoelectric

surface (GaAs) and reduced the piezoelectric coupling. The black closed circles and dashed
line show the measured values and the calculation of the maximum attenuation αm at dif-
ferent H. Here H < 0 means the helium level is higher than the GaAs surface. The left
inset shows the measurements of the attenuation versus Vg at various H while the right inset
shows the calculation of maximum attenuation αm as a function of the distance d from the
2DES to the piezoelectric surface. Note that d is equivalent to |H| for bulk region (H < 0).

prediction is shown in Fig. 7.6. The theoretical curve agrees fairly well with the experiments

160

but shows some notable deviations. First, the experimental data of the maximum attenuation

αm is approximately 30 % larger than theoretical prediction in the thin film region where

H > 0. This could be associated with a modification of piezoelecric coupling constant from

the combined effects of mechanical contact of the photoresist as well as liquid helium with

the GaAs substrate [119] and a rising background signal due to a helium film thickness

decrease with increasing electron pressure. Second, when the film is transitioning to a bulk
layer (H ≤ 0), the measured values of αm are larger than zero in contrast to the exponential

decay predicted in the theory (upper right inset of Fig. 7.6), which arises from the evanescent

waves-like nature of the SAW potential [111] in which αm exponentially decays as a function

of kd. A probable reason for this discrepancy can be slight inaccuracy in determining helium

level from the 3D CAD simulation or a small tilt angle of the device relative to gravity, which

could lead to a portion of the device is still in contact with the helium film.

To understand the positive shift of the attenuation peak position with increasing d,

we have also theoretically estimated the characteristic electron density nm at which the

attenuation has a maximum value (α = αm) as a function of the electron mobility at different

liquid helium levels in the cell (see appendix A.4). Fig. A.3 in appendix A reveals that the

liquid level does not strongly affect nm in either the thin film or bulk liquid helium regimes
and only influences nm in the transition region when H ∼= 0 if the mobility is constant

versus d. Thus, a liquid helium level change at constant mobility alone cannot explain the

continuous shift of the peak we observe. However, if µt decreases with increasing helium

level, our calculations predict that nm increase with the rising level and thus a more positive

Vg is required to reach nm and could lead to the positive shift of Vg at ne = nm. This scenario

seems unlikely since the mobility for electrons on bulk helium is much larger compared to

electrons on a helium film due to the surface roughness of the underlying substrate [171]. A

161

more reasonable explanation for our observation might be a small amount of heating of the

sample or weaker holding field from the underlying trapping electrodes each time helium was

added to the cell, resulting in a loss of some electrons from the microchannel arrays. In this

case, a more positive Vg would be required to compensate the electron loss as the liquid level

rises, which can give a rise to the positive shift of the attenuation peak position we observe.

7.1.4 Conclusion

Using a GaAs microchannel arrays device, we have measured the attenuation of Rayleigh

SAWs as a function of the surface charge density in the system of electrons on helium for

the first time. The electron density is controlled by varying a bias voltage on an underlying

gate electrode and estimated from simultaneous low-frequency transport measurement. An

increase of the attenuation is observed when electrons exist on the surface of the helium film

and our experiments suggest only a minor effect of the film thickness on the attenuation peak.

We have also performed numerical comparison of the measured attenuation induced by the

2DES using Wilen’s method, which is in reasonable agreement with our experiments. Thus,

the appearance of the SAW attenuation peak during the gate voltage weep is attributable

to the piezoelectric coupling to electrons floating on helium film.

The attenuation measurements with increasing charged film thickness show the gradual

decrease and then disappearance of the SAW attenuation as the distance d from 2DES to

piezoelectric surface increases. This behavior, which is qualitatively correct, does not match

the fast exponential decay expected from our calculation. This disagreement could be due

to either slight inaccuracy in the helium level estimation or a small tilt of the device, or a

combination of both. Regardless, the measurements described in this section represent the

first successful measurements of the attenuation of a SAW by electrons on helium and open

162

the door to using the wavevector and frequency dependent attenuation α(k, ω) as a probe of

the physics of this non-degenerate 2DES.

7.2 Acoustoelectric Transport in Electrons on Helium

7.2.1 Experiment

We have also developed a device for measuring acoustoelectric transport of electrons on he-

lium, which was measured inside a superfluid leak-tight copper cell, where a sheet of electrons
produced from a tungsten filament were floating on top of a ∼70 nm thick liquid helium film

(see appendix B.1 for the description of the helium film thickness determination). The same

YZ cut lithium niobate (LiNbO3) single crystalline chip used for the SAW measurements

of liquid helium described in chapter 6 was utilized as the substrate for the experiments

described in this chapter as illustrated in Fig. 7.7 (a) and (b). The SAWs in this device were

designed with a fundamental frequency ν = λ/v = 291 MHz, a wavelength λ = 12 µm, and

a speed of the sound v = 3488 m/s. The measured value of the SAW resonance frequency

was ν = 296 MHz (see Fig. 7.7 (c)). The slight difference compared to the expected value

(ν = 291 MHz) is attributable to contraction of the substrate at cryogenic temperatures.

With this experimental setup, the acoustoelectric (ae) transport of electrons on helium was

measured via capacitive coupling between electrons floating on the helium surface and a de-

tection electrode beneath the lithium niobate at T ∼= 1.55 K (see Fig. 7.7 (b)). For frequency

domain analysis of ae transport, an amplitude modulated continuous wave signal was applied

to the exciter IDT using an Agilent 8648B RF signal generator and the generated ae-current

was measured using standard lock-in technique. In order to study the time response of the

ae-current, time-resolved measurements employing SAW pulses at 296 MHz were carried

163

out using the same RF signal generator. The real-time ae current data was collected via a

SR570 low noise current preamplifier using a Tektronix DPO7054 digital oscilloscope with

waveform averaging over 10,000 samples to improve signal to noise ratio. Also, an Agilent

N5230A vector network analyzer was used to measure the SAW resonance frequency and to

estimate SAW intensity.

7.2.2 Results and Discussion

7.2.2.1 Frequency Response of the Acoustoelectric Current

The CW acoustoelectric response is shown in Fig. 7.7 (d) and coincides with the indepen-

dently measured SAW resonance at 296 MHz. This acoustoelectric current Iae is equivalent

to the flux per unit time of electrons passing through the region above the underlying de-

tection electrodes. No acoustoelectric current is observed without electrons present on the

superfluid film (dashed blue trace in Fig. 7.7 (d)) confirming that the signal arises from the

electron layer floating on the helium surface. Moreover, we find that Iae vanishes when the

electrons are moved far away from the surface of the piezo-substrate by increasing the thick-

ness of the superfluid layer (solid red trace in Fig. 7.7 (d)) as would be expected given the

evanescent character of the SAW potential above the piezoelectric surface. The acoustoelec-

tric current signal shown in Fig. 7.7 (d) exhibits a periodic superimposed oscillation on top

of the the main resonance. This corrugation in the acoustoelectric current is attributable to

reflections of the SAW from the edge of the substrate, as is evident from the Fourier trans-

form of the signal into the time domain (inset Fig. 7.7 (d)). Together these experiments

confirm that the acoustoelectric current is generated by the transport of electrons on the

superfluid surface via the SAW electric field extending up from the underlying substrate.

164

Figure 7.7: Schematic of the experimental setup and demonstration of acoustoelectric trans-
port of electrons on helium. (a) Top and (b) cross-section views of the device for measur-
ing SAW-driven transport of electrons on helium. Two opposing interdigitated transducers
(IDTs) are used to excite and receive SAWs. A saturated superfluid 4He film is formed on
the surface of the LiNbO3 piezo-substrate at T = 1.55 K. Thermionically emitted electrons
are trapped above the surface of the superfluid film by applying a positive bias voltages to
three underlying electrodes arranged in a field-effect transistor (FET) configuration with a
source (s), gate (g) and drain (d) [16]. Lateral confinement of the electron layer is achieved
with a negative bias to guard electrode positioned on the outside of the LiNbO3 substrate.
(c) Frequency dependence of the transmission coefficient (S12) of the SAW device demon-
strating an expected resonance at 296 MHz. (d) Measured acoustoelectric current Iae of
electrons on helium driven by a piezo-SAW as a function of frequency. For these measure-
ments the FET electrode voltages were Vs = Vg = Vd = 40 V, corresponding to an electron
density of n ∼= 0.8× 109 cm-2, and the guard was biased with −3.2 V. Inset: Inverse Fourier
transform of the acoustoelectric current signal which reveals a peak at t ≈ 0.9 µs. This
time scale corresponds to a SAW propagation distance of 3.2 mm, roughly the same as twice
the distance between the launching IDT center and the near-edge of the LiNbO3 substrate,
which implicates SAW reflections as responsible for the superimposed oscillations present on
the acoustoelectric current peak.

165

7.2.2.2 SAW Power and Gate Bias Dependence

A characteristic feature of acoustoelectric transport in 2DESs is a linear dependence of the

measured signal, in this case Iae, on the SAW intensity and hence excitation power (see

equation 3.59 in chapter 3). Fig. 7.8 (a) and (b) show the SAW power dependence of Iae for

electrons on helium supported by a LiNbO3 and a GaAs piezoelectric substrate, respectively.

At the SAW resonance Iae increases linearly with the RF input power consistent with this

expected linear response (i.e. see inset Fig. 7.8 (a) and (b)), which serves to further confirm

the acoustoelectric origin of the measured signal from the system of electrons on helium.

The measured value of Iae with the GaAs substrate is two orders of magnitude smaller than

that with the LiNbO3 substrate. This significant difference of Iae is mainly due to the much
smaller piezoelectric coupling constant of GaAs, K2/2 = 3.2 × 10−4, compared to LiNbO3,
K2/2 = 4.82 × 10−2, (i.e. see equations 3.31 and 3.59 in chapter 3).

Figure 7.8: Power dependence and gate-tunability of the acoustoelectric effect with electrons
on helium. Acoustoelectric current Iae measured as a function of the SAW driving frequency
and power for electrons floating on a helium thin film on a (a) LiNbO3 and a (b) GaAs
piezo-substrate. Each inset shows that Iae is linear in the driving power when the frequency
corresponds to the SAW resonance.
(c) Demonstration of an acoustoelectric field effect
transducer (aFET) with electrons on helium on the LiNbO3 substrate. The inset shows
line-cuts of Iae both on- and off-resonance with the SAW. The gate voltage (Vg) sweep is
performed at fixed source and drain voltages (Vs = Vd = 40 V).

Field effect control is crucial for the development of acoustoelectric device for precise

166

electron control using SAWs. In Fig. 7.8 (b) we show a novel form of gate controlled SAW-

driven electron transport. By tuning the gate bias voltage Vg we can turn Iae ON and

OFF, in effect creating an acoustoelectric field effect transistor (aFET). When Vg is different

from the source voltage Vs, electrons dragged by the travelling SAW encounter an effective
potential energy barrier Ueff = −e(Vg − Vs) in the region above the gate. As shown in the

orange trace of the Fig. 7.8 (b) inset, for sufficiently small values of Vg, acoustoelectric charge

transport is blocked by a large positive Ueff, which results in zero current. Upon increasing

Vg, electrons transported by the SAW are allowed to enter the region above the gate due to

a decrease in Ueff, which leads to an increase in Iae at a threshold value of Vg determined by

the overall areal electron density. Further increasing Vg eventually leads to a suppression of

Iae once the region above the source has been depleted of electrons (see Fig. 7.8 (b) inset).

7.2.2.3 Pulsed measurements of Acoustoelectric Current from Electrons on He-

lium

An important step for future SAW-based metrological or quantum information experiments

with electrons on helium is the ability to precisely control the number of SAW-transported

electrons. This can be accomplished by gating the SAW in time, i.e. by performing pulsed

acoustoelectric measurements, which we show in Fig. 7.9. In these pulsed measurements the

SAW IDT is excited on resonance for a fixed period of time, tp, which launches a surface

acoustic wave packet having a duration in time equal to tp. The envelope of the SAW-packet

contains the high-frequency acoustoelectric field, which picks up electrons and carries them

in the propagation direction of the SAW-pulse (see subsection 7.2.1). For an areal electron

167

Figure 7.9: Pulsed acoustoelectric measurements of electrons on helium. (a) Time of flight
measurement of Iae (red curve) using a gated pulse (gray line) of SAWs at a fixed RF
power (0 dBm) and at the SAW resonance frequency (296 MHz). In these measurements the
number of electrons above the detection electrode is N ∼= 0.4 × 109. The blue trace is the
calculated change in the equilibrium number of electrons ∆N , which is obtained from the
time integral of Iae (red trace). (b) ∆N and (c) |∆Nsat| for a steady state SAW pulse at
various values of the driving RF power. In these measurements the equilibrium number of
electrons is N ∼= 0.8 × 109. (d) Dynamical response of two-dimensional electrons on helium
to SAWs pulses at different RF power with N ∼= 0.8 × 109. The time constants τpump (τrel)

are those associated with the rising (falling) edge of the SAWs pulse and are determined
from fitting Iae(t). The time constant τRC (dashed line) is that determined from from low
frequency transport of electrons on helium.

168

density n the continuity equation for the acoustoelectric current density jae,

∂n/∂t + ∇ · jae = 0

(7.11)

predicts that an acoustoelectric current should appear once SAW-driven electrons flow past

the boundary between the detecting electrodes. This behavior is shown in Fig. 7.9 (a) for

the case where tp = 30 µs, where Iae starts sharply increasing at a time td

∼= 1.7 µs after

the SAW is launched. This delay in the onset of Iae corresponds to the arrival time of the

leading-edge of the SAW pulse at the boundary above the detection electrode. The pulse

continues to drive electrons across the boundary, progressively building up an increasing

charge imbalance in the electron layer. This charge imbalance produces an electric field that

opposes the SAW-induced electron flow and leads to a decrease in Iae as shown in Fig. 7.9 (a)

(red trace). Once the trailing edge of the SAW pulse transits past the edge of the detection

electrode (i.e. once t = tp + td) a current reappears but having the opposite polarity. In

contrast to other 2DESs, these features are unique to electrons on helium and arise from

the fact that this 2DES has a fixed total number of electrons. The SAW pulse dynamical

redistributes the electrons above the surface of the superfluid creating a non-equilibrium

density, which then relaxes back to equilibrium after the passage of the SAW pulse.

We can quantitatively analyze this SAW-induced charge imbalance by calculating the
time integral of Iae to extract the change in the number of electrons |∆N| above the detection
electrode (blue trace Fig. 7.9 (a)), which reaches a saturated steady state value, |∆Nsat|, in

the limit of long SAW pulses. Increasing the SAW power can be used to tune the magnitude
of |∆Nsat| as shown in Fig. 7.9 (b), and in the regime of linear response this power dependence

allows us to estimate the minimum number of transported electrons that can be detected

169

per SAW pulse (see Fig. 7.9 (c)). For these measurements where tp = 100 µs we find that
for the lowest power as little as |∆N/N|≤ 0.01 % of the number of electrons N ∼= 0.8 × 109

can be transported by each SAW pulse (i.e. see appendix B.2). These initial acoustoelectric

transport proof-of-concept experiments are promising and we anticipate that optimization

of device geometry to include a combination of microchannel lateral confinement [138] and

single electron transistor charge detection [172] is a fruitful path forward for future single

electron acoustoelectric measurements.

Finally, pulsed SAW measurements allow us to extract information about the dynamical

response of the many-electron system on helium by analyzing the build-up and relaxation

of the SAW-induced charge pumping. Specifically, we fit the rising and falling edges of the

pulsed acoustoelectric signal to an exponential and extract time constants τpump and τrel,

which are shown in Fig. 7.9 (d). In both cases, the time constants do not depend on the
SAW power and we find τpump = 5.7± 0.1 µs and τrel = 4.9± 0.1 µs (i.e. see appendix B.3).

These time constants are ultimately determined by the underlying mechanisms that lead

to electron scattering in the system.

In the case of the present experiments, which were

performed at T ∼= 1.55 K, a strong scattering mechanism is the collision of electrons with

helium vapor atoms above the superfluid surface. These scattering events occur at a fre-
quency (∼300 GHz, i.e. see appendix B.2) much larger than that of the SAW electric field

(296 MHz). Therefore the time constants extracted from the SAW measurements should

be similar in magnitude to the RC time constant determined from conventional transport

measurements, which we shown in Fig. 7.9 (d). We find that τRC = 7.7 µs (i.e. see ap-

pendix B.2), which is in reasonable agreement with our SAW measurements. In addition to

demonstrating controlled charge pumping, these measurements show that SAWs are a ver-

satile new tool for interrogating the dynamical processes in electrons on helium including,

170

when extended to lower temperature, investigations of plasmon modes and ripplo-polaronic

excitations of the Wigner solid [173].

7.2.3 Conclusion

Here we have reported on the coupling of electrons on helium to an evanescent piezoelectric

SAW. The electrons surf with the piezoelectric wave and we demonstrate high-frequency

charge pumping in this system for the first time. With this piezoacoustic method we can
precisely transport as little as ∼0.01 % of the total number of electrons, opening the door

to quantized charge pumping, the possibility of electrical metrology, and ultimately single

electron state transfer with electrons on helium. We also show that SAWs are a route to

directly investigating the high-frequency dynamical response, and relaxational processes, of

collective excitations of the electronic liquid and solid phases of electrons on helium.

171

Chapter 8

Conclusions and Future Directions

By employing the surface acoustic waves technique used to study semiconduct-based 2D

electron systems, we have explored electrons floating on the surface of liquid helium. For the

first time ever, we have measured SAW attenuation and acoustoelectric charge transport in

this system. These results represent a first demonstration of the coupling of SAWs to surface

electrons on liquid helium and pave the way for a number of exciting new experimental

possibilities. In this chapter, we briefly discuss a few particularly exciting future directions.

2D Phase Transition

One interesting feature of the electron crystal (Wigner solid, WS) in the electron on helium

system is the strongly nonlinear conductivity. Each electron emits helium surface capillary

waves (ripplons) when their velocity exceeds the ripplon phase velocity, a process analogous

to the emission of Cherenkov radiation. Moreover, the ripplons radiated from each electron

can interfere constructively if the ripplon wave number is equal to the reciprocal lattice

vector of the WS. This resonant Bragg-Cherenkov emission of ripplons leads to a limitation

on the maximum electron velocity [174]. As a result, a sharp decrease of the nonlinear

conductivity takes place during the phase transition from the electron liquid to the WS

[175]. The high level of sensitivity of SAWs in detecting conductivity changes of a 2DES

could possibly provide a novel method to examine this phase transition in electrons on

172

helium. Furthermore, the SAW technique is also sensitive to the presence of inhomogenity

of the 2D electron system when the average conductivity is very low [35]. This aspect might

possibly allow for exploring inhomogeneous spatial characteristics in the vicinity of the 2D

melting point, such as the existence of a hexatic phase [75].

Commensurate Incommensurate Transitions

When a periodic potential is imposed on a given crystal, under certain conditions where the

period of the potential is close to the crystal periodicity, the crystal structure can adjust

itself to follow the periodicity of the potential, leading to a commensurate state [176]. Such

commensurate-incommensurate transitions are ubiquitous phenomena in condensed matter

physics. Very recently, molecular dynamics simulations show that for electrons on helium a

1D periodic potential submerged below the helium surface can significantly affect not only

the electron crystal but also the strong correlated electron liquid [177]. Particularly, for the

maximum commensurability case in the electron liquid phase, where the periodicity of the 1D

potential is equal to the interelectron distance of the WS, even a very weak potential strongly

affects the mobility, leading to a Wigner solid transition at lower temperature. In particular,
the typical electron-electron distance ae for electrons on helium is ∼ 1 µm and thus a 1D
potential with a period ∼ 1 µm can be easily implemented with conventional lithographic

patterning technology for future experiments of commensurability effects. Also, the ability

to precisely control the electron density allows for the ability to change ae to experimentally

study a incommensurate state as well. Furthermore, a standing wave field [178] established

from the interference of two counter-propagating SAWs could be utilized to investigate not

only the effect of spatial but also temporal periodic potentials on the transport properties

173

of electrons on helium.

SAW-driven Quantized Charge Pumping

A SAW can transport a well-defined small number of electrons along its propagation direc-

tion. In semiconductors, this has been extended down to a single electron transport through

a gate-defined quasi-1D channel [105]. These types of experiments attracted interest for the

exploration of SAW-driven flying qubits [106] as well as quantum acoustodynamics experi-

ments with single electrons [109]. In particular, when 2D electrons are transported through

the quasi-1D channel by SAWs, the plateau in this SAW-driven current (I = N ef ) as a func-

tion of the voltage applied to the side gates on each side of the channel reveals an integer

number N of electrons trapped in each SAW minima. Here e and f represent the elemen-

tary charge and the frequency of the SAW. This feature has been proposed to be utilized

to develop metrological standards of electrical current [38]. Although considerable progress

in these quantum metrology applications has been made using semiconductor 2D electron

systems such as those in AlGaAs/GaAs heterostructures, issues arising from impurities and

disorder in these systems still remain and have limited performance [163, 164, 165]. A SAW-

driven single electron device with electrons on helium could be a possible alternative path

due to the ultra-clean environment provided by the superfluid. In addition to that, tunable

electron confinement in a microchannel device filled with liquid helium [179] allows for the

realization of well-controlled quasi-1D channels. Furthermore, a single electron transistor

(SET) electrometer immersed in liquid helium can be utilized to precisely detect and count

the number of electrons in the potential minimum of the moving SAW down to the single

electron level [180, 181, 182].

174

APPENDICES

175

APPENDIX A

Studies of Electrons on Helium Using

a GaAs SAW Device

A.1 Perturbation Formula for SAW Attenuation

Since the GaAs substrate used in the experiments described in chapter 7 has a weak piezo-

electricity, the attenuation produced by a 2DES can be estimated using the Ingebrigtsen

formula [169] as follows,

where

α
k

=

(cid:48)

z

(0) = −ik

(cid:33)

(cid:48)
1 + iz
1 − i
pz

(0)
(cid:48)

(0)

2

Im

(cid:32)
(cid:18) K2
(cid:19)
(cid:33)(cid:12)(cid:12)(cid:12)(cid:12)z=0
(cid:32)

(cid:48)
φ
(cid:48)
D
z

,

(A.1)

(cid:113)


zz
xx − 
2
zx.


p =

(A.2)

,

Here, Rayleigh SAWs propagate along the x-axis and the positive z-axis points out of piezo-

electric plane located at z = 0. The effective dielectric constant 
p for the piezo-substrate
is determined from the dielectric tensor element 
ij. In equation A.1 α, k, and K2

2 denote

the attenuation constant, the wave number, and the piezoelectric coupling constant. In our

GaAs microchannel array device, the 2DES is located above a helium contained microchan-

nel or on the photoresist regions covered with a thin helium film between microchannels.

These two different regions are represented by one model geometry as seen in Fig. 7.2. For

176

the former region both dielectric 1 and 2 are liquid helium while dielectric 1 and 2 are a

photoresist and a helium thin film for the latter region. We derive the SAW attenuation

formula applicable for both regions using this model geometry. Poisson’s equation gives the

electric potential inside dielectric 1 and 2 and in vacuum as seen in Fig. 7.2, which is given

by

φ1 = eikx−iωt(cid:16)
φ2 = eikx−iωt(cid:16)
φ3 = eikx−iωt(cid:16)

Ae−kz + Bekz(cid:17)
Ce−kz + Dekz(cid:17)
Ee−kz(cid:17) − 4πσ0z,

,

,

(A.3)

where σ0 and ω = 2πf are the equilibrium charge density of the 2DES and the angular

frequency of the SAW. The electric potential and normal component of the displacement

field must be continuous at z = h1. Thus, φ1(h1) = φ2(h1) and 
1

∂φ1
∂z

yield

(cid:12)(cid:12)(cid:12)(cid:12)z=h1

(cid:12)(cid:12)(cid:12)(cid:12)z=h1

= 
2

∂φ2
∂z

A =

B =


1 + 
2
2
1

1 − 
2
2
1


1 − 
2
C +
2
1
e−2kh1C +

e2kh1D,


1 + 
2

2
1

D.

(A.4)

Here the interface between two dielectric films (ε1, ε2) above the piezoelectric substrate is

located at z = h1. The ability of the SAW potential to trap and drag charges at the speed

of sound (i.e. acoustoelectric transport) leads to a small fluctuation of the charge density δσ

at z = h1 + h2, which satisfies the relation

(cid:12)(cid:12)(cid:12)(cid:12)z=h1+h2

− ∂φ3
∂z

(cid:12)(cid:12)(cid:12)(cid:12)z=h1+h2

= 4π (σ0 + δσ) .

(A.5)

+ 
2

∂φ2
∂z

177

δσ =

k
4π

eikx−iωt[ e−k(h1+h2)E − 
2e−k(h1+h2)C + 
2ek(h1+h2)D]

(A.6)

Momentum transfer between the SAW and the charges in the 2DES causes a small momen-

tum fluctuation δP = meδv of the electron in the 2D sheet. The corresponding velocity

fluctuation δv can be obtained from the acoustoelectric equation of motion using the relax-

ation time approximation

(cid:18)

(δP ) =

e

d
dt

∂φ
∂x

− δP
τ

Taking d

dt → −iω and φ → φ3 gives,

δv = ikeikx−iωt(cid:16)

Ee−kz(cid:17) eτ

me

.

(cid:19)(cid:12)(cid:12)(cid:12)(cid:12)z=h1+h2
(cid:12)(cid:12)(cid:12)(cid:12)z=h1+h2

1 − iωτ

1

(A.7)

,

(A.8)

where τ and me represent the momentum relaxation time and the electron mass. The

following continuity equation must hold at z = h1 + h2

∂δσ
∂t

+

∂
∂x

(σ0δv) = −iωδσ + ikδvσ0 = 0.

(A.9)

Substituting for δσ and δv from equations A.4 and A.8 leads to

iω[ Ee−k(h1+h2) − 
2Ce−k(h1+h2) + 
2Dek(h1+h2)] =

ω2
pτ
1 − iωτ

Ee−k(h1+h2),

(A.10)

where ω2

p = 4π

|σ0|ek
me

is the 2D plasma frequency. The electric potential must be continuous

at z = h1 + h2 as well

Ce−k(h1+h2) + Dek(h1+h2) = Ee−k(h1+h2).

(A.11)

178

Solving equations A.10 and A.11 determines the amplitude ratio of φ2 as follows

C
D

= rC,

(A.12)

with

rC =

[ (
2 + 1)ω2 − ω2
[ (
2 − 1)ω2 + ω2

p] τ + (
2 + 1)iω
p] τ + (
2 − 1)iω

e2k(h1+h2).

(A.13)

Similarly, from equations A.4 and A.12, the amplitude of φ1 can be expressed in terms of

the amplitude ratio rc in region 2:

A = rAD, B = rBD,

(A.14)

where

rA =

(cid:18) 
1 + 
2

2
1

rC +


1 − 
2
2
1

e2kh1

(cid:19)

,

rB =

(cid:18) 
1 − 
2

2
1

(cid:48)
Inserting φ

(0) = φ1(0) = Deikx−iωt (rA + rB) and D

(cid:48)
z(0) = − ∂φ1

∂z

into equation A.2 gives the normalized surface impedance

(cid:19)


1 + 
2

2
1

.

(A.15)

= D
1eikx−iωt (krA − krB)

e−2kh1rC +

(cid:12)(cid:12)(cid:12)(cid:12)z=0

(cid:33)(cid:12)(cid:12)(cid:12)(cid:12)z=0

(cid:48)
φ
(cid:48)
D
z

(cid:32)

(cid:32)

(cid:33)

rA − rB − 
−1
1 (rA + rB)
rA − rB + 
p/
1(rA + rB)

179

(cid:48)

z

(0) = −ik

= i
−1

1

rA + rB
rA − rB

.

(A.16)

This yields the formula for finding the SAW attenuation arising from the 2DES above two

dielectric films:

α
k

=

(cid:18) K2

(cid:19)

Im

2

.

(A.17)

Note that the minus-sign is dropped on the right-hand side of equation A.16 due to using a

different coordinate system in which the z-axis points out of the piezoelectric surface whereas

it points in the opposite direction in the Ingebrigtsen formula (equations A.1 and A.2).

A.2 Helium Level Estimation

Figure A.1: The experimental sample cell and helium level estimation. (a) Pictures of the
copper cell used in our experiments. The cell contains a GaAs device (shiny black) located
inside the guard electrode on the printed circuit board (PCB). Beneath this substrate, three
rectangular electrodes are patterned on the PCB, which are not shown in the picture. (b),(c)
He level estimation as a function of the liquid helium volume using 3D CAD simulation.
The gray volume represents the liquid helium in the cell.
(d) Level sensor capacitance
change while condensing liquid helium into the sample cell. This was measured using the
interdigitated capacitor (IDC) sensor as shown in the inset of the plot. A linear increase of

the capacitance starts happening when the liquid volume ∼= 0.32 cc, which indicates that the

liquid helium reaches the bottom of the IDC..

To form liquid helium inside the cell at T ∼= 1.55 K (Fig. A.1 (a)), helium gas was provided

from a standard volume of 260 cc at room temperature. The volume of liquid helium admitted

180

into the cell was determined by the pressure in the standard volume and corresponding

helium level was estimated using the 3D CAD simulation as seen in Fig. A.1 (b)(c). We

have consistently observed that liquid helium starts collecting in the cell after injecting a

considerable amount of helium gas into the cryostat through the fill line. This amount of

helium corresponds to approximately 0.32 cc of liquid (Fig. A.1 (d)). This dead volume is

likely associated with liquid in the fill line. In fact, the volume of the fill line below the 4k-

stage is approximately 0.4 cc. Therefore, the helium volume determination from the pressure

was calibrated with this dead volume taken into account.

A.3 Electron Density and Mobility Estimation

For the low-frequency transport measurements described in this thesis, the capacitive redis-

tribution of electrons during a Vg sweep at fixed Vs = Vd can be used to estimate the electron

density ne versus Vg using the relation

ne =

2
3

c(Vg − Vth)e−1,

(A.18)

where c is the capacitance per unit area between the electron layer and the underlying

electrodes and Vth is the threshold voltage in the low-frequency IV curve (Fig. A.2 (b)).

This capacitance c in the channel and in the photoresist regions can be determined from

geometry:

cch = cPR

∼=


0

dg/
He + dGaAs/
p

.

(A.19)

Here dg ∼= 50 µm denotes the gap between the transport electrodes and the GaAs sub-

strate resulting from the rubber cement glue used to mount the GaAs chip to the PCB

181

Figure A.2: (a) Transmission line (TL) circuit modeling of electrons on helium (b) Current
of electrons on a helium film as a function of Vg inthe low-frequency regime. The current
was induced by applying a 0.1 mV, 60 kHz excitation Vex and detected using standard
lock-in techniques. The source and drain voltage were fixed at 60V. (c) Output voltage Vo
as a function of the frequency of Vex. The output voltage Vo corresponds to the voltage
drop at the internal impedance of the signal detector (Lock-in amplifier) due to current flow
in 2D electrons floating helium film. The blue and red solid traces represent experimental
values with and without SAW excitation while the dashed trace is the theoretical curve. The
frequency response is independent of SAW excitation, which implies that the low frequency
transport is not affected by the SAW. For these measurement, all three electrodes were biased
with 60 V.

182

board (Fig. A.2 (a)) and dGaAs = 350 µm represents the thickness of the GaAs substrate.

It should be noted that the helium film and the photoresist on the GaAs negligibly affect

c since dch/
He, dvan/
He, dPR/
PR << dg/
He, dGaAs/
p. Thus, it is assumed that the
∼= 35 V and
∼= 1.14 × 10−7 F/m2, ne versus Vg in Fig. 7.4 (a) in section 7.1 was calcu-

channel and photoresist regions have the same electron density. With Vth

cch = cPR

lated using equation A.18, which is in good agreement with finite element method (FEM)

simulations.

Using the transmission line circuit analysis [129, 130], we have estimated the mobility

µl in the direction along the channel for a homogeneous electron density distribution (i.e.
Vs = Vg = Vd = 60 V). The conductivity of electrons on helium σ = 2.6 × 10−6 Ω−1 was

obtained by matching the frequency response measurement to a fit based on the transmission
line model (Fig. A.2 (c)). Using the Drude model (σ = neeµl) with ne = 1.16 × 109/cm2
yields µl = 1.4 × 104 cm2/Vs.

A.4 Characteristic Electron Density Estimation

The characteristic electron density nm, corresponding to the maximum attenuation shown

in Fig. 7.5 and 7.6, is calculated from the condition

(cid:18) ∂α

(cid:19)

∂ωp

ωp=ωm

= 0, where ω2

m =

4πe2k

me

nm.

(A.20)

Fig. A.3 shows the mobility dependence of nm at different helium levels in the cell. For both

the helium thin film and bulk layers, the characteristic electron density nm is dominated

by the electron mobility transverse to the channel µt and shows negligible variation with

183

increasing liquid helium level. A significant change of nm due to the helium level is only

expected in the transition region between the thin film and a bulk liquid layer.

Figure A.3: Characteristic electron density nm as a function of the mobility transverse to
the channel µt at different H in the helium film (a) and bulk (b) regimes. The parameter H
denotes the distance between the GaAs substrate and the level of the helium reservoir right
next to the substrate. When H > 0, the level of the helium reservoir is below the surface of
the GaAs while the helium level exceeds the GaAs surface when H < 0 .

184

APPENDIX B

Studies of Electrons on Helium Using

a LiNbO3 SAW Device

B.1 Superfluid Film Thickness

Fig. B.1 (a) shows the hermetically sealed copper cell used for the measurements of electrons

on helium supported on top of a LiNbO3 substrate. To fill the cell with superfluid 4He, helium
gas was supplied into the cell at T ∼= 1.55 K through a capillary line at room temperature.

The liquid helium volume admitted into the cell was determined by varying the pressure in

a calibrated standard volume (V = 260 cc) located at room temperature. The thickness of

a saturated helium film, d0, can be estimated from [11],

α
4 = ρgH,
d0

(B.1)

where α is the van der Waals constant, g is the gravitational acceleration, ρ is the mass

density of helium, and H is the distance from the LiNbO3 substrate top surface down to the

liquid helium surface in the reservoir volume inside the cell. We calculated H as a function

of the volume of helium admitted into the cell using 3D modeling of the cell open volume

(see Fig. B.1 (b),(c)). For the thin film measurements in section 7.2, H = 0.2 mm, which

185

Figure B.1: The experimental sample cell and helium film thickness determination. (a) Pho-
tograph of the copper cell used for the SAW-based measurements of electrons on helium.
The bottom portion of the cell contains a LiNbO3 device (semi-transparent white) mounted
on a printed circuit board (b) 3D CAD modeling of the liquid helium volume inside the
experimental cell during filling with helium. (c) Calculation of H, the distance between the
piezo-substrate top and the liquid helium level in the cell reservoir volume from the 3D CAD
modeling of the open cell volume. The calculated value of H in the experiments is 0.2 mm
for a helium vol. = 0.44 cc (vertical dashed black line). (d) Helium film thickness versus
H before charging the film (see equation B.1). For an uncharged helium film d0 ≈ 77 nm
(vertical dashed red line). (e) Charged helium film thickness versus electron density from
equation B.2. The charged helium film thickness is calculated to be d ∼= 72 nm for an electron
density of n ∼= 0.8 × 109 cm-2 (vertical dashed blue line).

186

corresponds to a thickness of the superfluid film, d0

∼= 77 nm, from equation B.1. Charging

the helium film with electrons exerts an electronic pressure film [132] in addition to gravity

and thus reduces the thickness of the charged helium film, d, which can be calculated from,

α
d4 = ρgH + pel =

α
4 + 2πn2e2,
d0

(B.2)

where pel is the electronic pressure, n is the electron density and e is the electron charge.

We note that due to quantum electrodynamic effects, for thick films (> 60 nm) the van der

Waals potential energy is proportional to 1/d4 rather than 1/d3 [183, 184]. In most of the

acoustoelectric measurements reported in section 7.2, the areal electron density is estimated
to be n (cid:39) 0.8×109 cm-2 from field effect transistor (FET) measurement, which are described
in appendix B.2. This yield a charged helium film thickness of d ∼= 72 nm from equation B.2.

B.2 Low-Freqeuncy Characterization of The Electron

System on Helium

Field Effect Transistor Transport Measurements

Fig. B.2 (a) and (b) illustrate the measurement set-up and the equivalent transmission

line circuit model for the low-frequency transport properties of electrons on helium on a

LiNbO3 substrate. After charging a helium film with electrons, we characterize the electron

distribution with low-frequency field effect transistor (FET) operation of the device [16].

These measurements are done with a gate voltage Vg sweep at fixed source and drain voltage

at a frequency of 60 kHz as shown in Fig. B.2 (c). For sufficiently small values of the gate

187

voltage, Vg < Vth, all of the electrons are localized over the source and drain electrodes

creating a depletion region above the gate, which leads to zero current through electron

layer. At a threshold value of the gate Vg = Vth an AC current onsets from electrons being

attracted to the region above the gate. As Vg increases beyond Vth, source-drain current

quickly rises and reaches a maximum in a vicinity of uniform electron density (Vs = Vd = Vg).

After reaching its maximum value, the current begins decreasing and eventually vanishes as

electron depletion over the source and drain electrode occurs. This ability to change the

electron density above the individual electrodes with the application of a bias voltage enables

the electrically switching ON and OFF of the acoustoelectric current in electrons on helium

system (see Fig. 7.8 (b) in section 7.2).

Electron Density

A homogeneous areal electron density n above all electrodes is achieved when Vs = Vg = Vd

and the electron density can be calculated from the FET transport data by measuring the

difference ∆V between the case where the electron density is uniform and Vth (see Fig. B.2

(c)) [16],

n =

2 cl
3 e

∆V.

(B.3)

Here, cl is the capacitance per unit area between electron layer and underlying electrodes

from geometrical configuration, which is given by



 ∼= 
0

 1

 (for d (cid:28) dg),

(B.4)

cl = 
0

1
dg

 + ds

s

+ d



dg

 + ds

s

188

Figure B.2: Experimental setup, circuit model, and measurement data for low frequency
transport of electrons on helium. (a) Sketch of the electrons on helium device. A DC bias
voltage is applied to the three underlying electrodes to trap electrons above helium surface.
An AC excitation voltage (Vex = 0.1 V) applied to the source electrode induces a current
through electrons on helium, which is capacitively detected using the drain electrode. (b)
The equivalent transmission line circuit model for this device. The resistance of the electron
sheet and the capacitance between this sheet and the three electrodes are spatially distributed
per unit length, r∆L and cl∆L, where r and cl represent the resistance per unit length
and the capacitance per unit length. (c) Current-voltage characteristics for FET operation
of the electrons on helium device. For this representative data, the source-drain current
amplitude was measured during a gate voltage sweep with Vs = Vd = 80 V. The insets show
the corresponding electron density profiles over the electrodes for different values of Vg. A
uniform areal density distribution is achieved at Vg = Vs = Vd. (d) Frequency dependence
of the transport characteristics of the system of electrons on helium. The red and blue
traces represent the experimental data and the calculation based on the transmission line
model respectively. For these measurement, all three electrodes were biased with 80 V
(Vg = Vs = Vd = 80 V).

189

where 
0 is the vacuum permittivity, 
 = 1.057 is the dielectric constant of helium, 
s ∼= 35
is the effective dielectric constant of the LiNbO3, dg ∼= 70µm is the gap between the LiNbO3

substrate and the bottom electrodes, ds = 0.5 mm is the thickness of the LiNbO3 substrate,

(cid:113)

and d is the charged helium film thickness. The effective dielectric constant 
s for LiNbO3

is given by 
s =


11
33 − 
2

13 where 
11, 
33, and 
13 are the dielectric tensor elements

of LiNbO3 at constant stress [133]. These dielectric elements are given in Ref. [185] as
∼= 1.08×10−7 F/m2 obtained from equation B.4,

11 = 44.3, 
33 = 27.6, and 
13 = 0. With cl
the uniform areal electron density is found to be n (cid:39) 0.8×109 cm-2 for most of acoustoelectric

measurements reported on LiNbO3. For the power dependent pulsed SAW measurements
(Fig. 7.9 (b)-(d)), the value of n is about 1.9 ×109 cm-2. We note that in the gate-tunable

acoustoelectric measurements shown in Fig. 7.8 (b), n is no longer a uniform density but

rather varies as a function of Vg, however the total number of electrons over the device is

fixed.

Electron Mobility

The conductivity σ of electrons on helium in the low-frequency transport measurements can

be determined by fitting the frequency response measurement to a RC transmission line

model [129, 130] with σ as fitting parameter [16]. Fig. B.2 (d) shows this frequency response

for the representative case where Vs = Vg = Vd = 80 V, which shows good agreement with
this model fit. The value of the conductivity from this fit is σ ∼= 1.58 × 10−6 Ω−1. The

mobility µ of electrons is then calculated based on a Drude model analysis (σ = neµ). For an

electron density of n = 1.9× 109 cm-2 the mobility is µ ∼= 5.3× 103 cm2/Vs with d ∼= 60 nm.

This mobility is roughly two orders of magnitude less than the mobility of electrons on bulk

4He at the same temperature [9]. Such a low mobility for electrons on a thin superfluid film

190

is consistent with previous measurements and can be explained by the close proximity of the

electron layer to the underlying substrate [171].

RC Delay Time Constant

The RC delay time constant, τRC, introduced in section 7.2 is calculated based on the trans-

mission line modeling described above (see Fig. B.2 (b)) and the Elmore delay model [186],

which is encapsulated by the following equation

τRC = rclL2 1 + N
2N

∼=

RtotCtot

2

(for large N),

(B.5)

where r and cl are the resistance per unit length and the capacitance per unit length of the

electron system and L, Rtot, Ctot, and N represent the total length, the total resistance, the
total capacitance, and the number of nodes in the entire transmission line. Rtot = σ−1W−1L

and Ctot = cW L are approximately 1.06 MΩ and 14.6 pF for W = 9 mm (the width of
the electrodes in the SAW device) and L = 15 mm, which yields τRC ≈ 7.7 µs in the low-

frequency transport regime.

Electron Collision Rate with Helium Vapor Atoms

At a temperature T = 1.55 K, 4He vapor atoms are a strong source of electron scattering.

The collision rate 1/τHe is determined by

(B.6)

1
τHe

=

e
µm∗

191

with the effective electron mass m∗ = me (bare electron mass) and the electron mobility
µ ≈ 1 × 105 cm2/Vs [9]. This yields 1/τHe ≈ 15 GHz.

B.3 Determination of SAW Charge Pumping Time Con-

stants

The time constants for SAW-driven charge pumping (τpump) and subsequent relaxation

(τrel) for different RF power levels was determined from an exponential curve fitting of the

acoustoelectric current (Iae) data as shown in Fig. B.3 and the resulting values are tabulated

in the table below.

Figure B.3: Determination of acoustoelectric time constants. These time constants were de-
termined from exponential fits to the acoustoelectric current data during (a) SAW excitation
(charge pumping) and (b) relaxation after the SAW drive was removed.

192

RF (mW)

τpump (µs)

τrel (µs)

0.25
0.32
0.40
0.50
0.63
0.79
1.00
1.26
1.58
2.00

5.9
5.6
6.5
5.2
5.2
6.3
5.5
5.9
5.6
5.8

4.6
5.3
5.2
4.7
5.0
5.0
4.7
4.7
4.7
4.8

Table B.1: Values of the time constants obtained from the exponential curve fits in Fig. B.3.

193

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