PHASE DIAGRAM OF A 2-DIMENSIONAL ELECTRON SYSTEM ON THE
SURFACE OF LIQUID HELIUM
by
IVAN SKACHKO
A Dissertation submitted to the
Graduate School-New Brunswick
Rutgers, The State University of New Jersey
in partial fulfillment of the requirements
for the degree of
Doctor of Philosophy
Graduate Program in Physics and Astronomy
written under the direction of
Prof. Eva Andrei
and approved by
________________________
________________________
________________________
________________________
________________________
New Brunswick, New Jersey
[May 2006]
ABSTRACT OF THE DISSERTATION
PHASE DIAGRAM OF A 2-DIMENSIONAL ELECTRON SYSTEM ON THE
SURFACE OF LIQUID HELIUM
By IVAN SKACHKO
Dissertation Director:
Prof. Eva Andrei
This work is a contribution to the study of the system of 2-dimensional electrons (2DES)
bound to the surface of liquid helium-4. The physical properties of the 2DES are probed
through the excitation of normal modes in the radio frequency range 10 MHz-1GHz. The
normal mode spectra are expected to undergo a radical change at the transition from
liquid to crystal, the so-called Wigner transition. This difference is due to coupling
between 2DES modes and those of liquid helium surface. The experiments are performed
for a range of helium film thicknesses 1mm -100Å. For thin films the influence of the
underlying substrate is of great importance: it allows charging the film to high surface
densities, which are impossible to reach on bulk helium. This lets us explore a region of
the phase diagram where quantum effects arising from zero-point fluctuations play a
significant role.
Slow-wave structures were developed to excite and detect the normal mode spectra
including a meander line and an interdigital capacitor.
ii
Acknowledgment and Dedication
This work is dedicated to my parents Valeria and Mikhail and to my wife Rodica.
I would like to express the most sincere gratitude to my research adviser Professor Eva
Andrei – the leading authority in the field of 2D electrons on helium, for her support and
encouragement during the years it took me to complete my research.
I received much spiritual and technical help from my colleagues Ross Newsome,
Guohong Lee, Ozgur Dogru, Xu Du, Zhili Xiao.
I greatly appreciate the direct participation in the project of Kurt Ketola, Gerard Deville,
Adam Hauser, Gail Schneider, Sylvain Benazet.
The machine and electronics shops at Rutgers Dept. of Physics were extremely skillful in
helping me to build the experimental apparatus.
iii
Table of Contents
ABSTRACT OF THE DISSERTATION ....................................................................... ii
Lists of tables .................................................................................................................... vi
List of illustrations .......................................................................................................... vii
1
Introduction: 2D Electrons on the Surface of Liquid Helium .............................. 1
1.1
THE ORIGIN OF REDUCED DIMENSIONALITY ........................................................ 1
1.2
CONFINEMENT OF ELECTRON TO THE SURFACE OF LIQUID HELIUM .................... 3
1.2.1
Influence of Externally Applied Electric Field ........................................... 6
1.2.2
Binding to a Thin Film ................................................................................ 7
1.2.3
Hartree Field (Mutual Repulsion of Electrons) .......................................... 8
1.3
WIGNER CRYSTAL ............................................................................................... 8
1.4
COLLECTIVE MODES OF 2DES ........................................................................... 11
1.5
THE DYNAMICS OF LIQUID HELIUM AND ITS EFFECT ON 2DES ......................... 13
1.5.1
The Surface Waves - Ripplons .................................................................. 13
1.5.2
Interaction Between 2DES and the Surface of Liquid Helium ................. 14
1.5.3
Thin Film Effects ....................................................................................... 15
1.6
SUMMARY .......................................................................................................... 17
2
Phase Diagram of 2DES on Liquid Helium .......................................................... 19
2.1
2.2
2.3
2.4
3
Spectrum of 2DES ................................................................................................... 32
3.1
3.2
3.3
3.4
3.5
4
ELECTRONS ON BULK HELIUM ........................................................................... 19
WIGNER SOLID AND 2D MELTING MECHANISMS ............................................... 21
ELECTROHYDRODYNAMIC (EHD) INSTABILITY OF A CHARGED LIQUID SURFACE
26
INFLUENCE OF A SUBSTRATE ON PHASE DIAGRAM ............................................ 29
2D DRUDE MODEL............................................................................................. 32
ELECTRON SOLID MODES .................................................................................. 39
2DES MOBILITY ................................................................................................ 39
COUPLED PHONON-RIPPLON MODES ................................................................. 43
SPECTRUM OF 2DES IN MAGNETIC FIELD.......................................................... 46
Measurement of 2DES Spectrum Using Slow-Wave Structure .......................... 50
4.1
4.2
4.3
4.4
2DES IN ELECTRO-MAGNETIC FIELD ................................................................ 50
DISTRIBUTED CIRCUIT MODEL .......................................................................... 54
SLOW-WAVE STRUCTURES ................................................................................ 58
COUPLING BETWEEN MEANDER LINE AND 2DES................................................ 65
5
Other Methods for Investigation of Phase Transition in 2DES on Liquid
Helium Film ..................................................................................................................... 66
5.1
5.2
MICROWAVE CAVITY TECHNIQUE ..................................................................... 66
SOMMER-TANNER METHOD (LOW-FREQUENCY TRANSPORT) ........................... 68
iv
6
Experimental Apparatus and Cell ......................................................................... 70
6.1
CRYOSTAT ......................................................................................................... 70
6.1.1
Vibration Isolation .................................................................................... 72
6.2
DILUTION REFRIGERATOR.................................................................................. 73
6.3
THERMOMETRY.................................................................................................. 74
6.4
MAGNET ............................................................................................................ 74
6.5
EXPERIMENTAL CELL ........................................................................................ 75
6.5.1
Production of Electrons ............................................................................ 76
6.5.2
2DES Confinement .................................................................................... 77
6.5.3
Measurement and Control of Liquid Helium Level and Film Thickness .. 79
6.6
MEANDER LINE.................................................................................................. 82
6.7
SUBSTRATE SMOOTHNESS CONTROL ................................................................. 83
6.8
ELECTRONICS..................................................................................................... 86
7
Analysis of Experimental Data .............................................................................. 90
7.1
7.2
IDENTIFICATION OF THE INDIVIDUAL RESONANCES ........................................... 93
EVOLUTION OF 2DES RESONANCES WITH TEMPERATURE AND PHASE DIAGRAM
97
2DES Heating ........................................................................................................... 99
Sliding Transition...................................................................................................... 99
7.3
2DES MOBILITY MEASUREMENT .................................................................... 102
7.4
OBSERVATION OF MAGNETOPLASMON ............................................................ 104
8
Conclusion ............................................................................................................. 107
Appendices ..................................................................................................................... 110
A
Ripplon Dispersion Law ....................................................................................... 111
B
Roughness of Liquid Helium Surface Due to Thermally Excited Ripplons .... 114
C
Parametric Resonance in HeII ............................................................................. 117
D
Heat Exchangers for Liquid He Bath.................................................................. 119
E
Experimental Procedure ...................................................................................... 121
PREPARATION .............................................................................................................. 121
COOLDOWN ................................................................................................................. 124
DILUTION FRIDGE OPERATION ..................................................................................... 125
FILLING THE CELL WITH HELIUM ................................................................................. 126
CHARGING THE HELIUM SURFACE ............................................................................... 126
DATA COLLECTION ...................................................................................................... 127
References ...................................................................................................................... 128
v
Lists of tables
TABLE 1-1 BASIC PROPERTIES OF 2DES. ............................................................................. 3
TABLE 3-1 2D PLASMA DISPERSION LAW IN VARIOUS LIMITS. ......................................... 36
TABLE 3-2 BESSEL EXTREMA. ........................................................................................... 38
TABLE 3-3 MOBILITY MEASUREMENTS AND THEORY. ...................................................... 41
TABLE 4-1 ELECTROMAGNETIC RESPONSE COMPARISON. ................................................. 54
TABLE 4-2 SLOW-WAVE STRUCTURES. ............................................................................. 61
vi
List of illustrations
FIGURE 1-1 SCREENING MECHANISM GIVING RISE TO ELECTRON CONFINEMENT TO A
LIQUID HELIUM SURFACE. ........................................................................................... 3
FIGURE 1-2 ENERGY DIAGRAM OF AN ELECTRON BOUND TO HE SURFACE.......................... 5
FIGURE 1-3 SCHEMATIC 2DES PHASE DIAGRAM. THE STRAIGHT LINE SEPARATES THE
REGION WHERE CLASSICAL FLUCTUATIONS DOMINATE (LOWER PART) FROM THE ONE
WHERE QUANTUM FLUCTUATIONS DOMINATE (THE UPPER PART). ................................ 9
FIGURE 1-4 2D HEXAGONAL LATTICE - LOWEST ENERGY CONFIGURATION OF 2D COULOMB
CRYSTAL. ................................................................................................................... 10
FIGURE 2-1 2DES PHASE DIAGRAM ON HELIUM BULK. .................................................... 21
FIGURE 2-2 2DES DIFFRACTION PATTERNS (T. WILLIAMS 1995). .................................... 25
FIGURE 2-3 CHARGING INSTABILITY. DASHED LINE IS THE PLOT OF N VS. D FROM EQ.(2-7).
DOT-DASHED LINE IS THE PLOT OF N VS. D FROM EQ.(2-8). ONE OF THEIR
INTERSECTIONS CORRESPONDS TO THE STABLE CHARGE DENSITY. SOLID LINE IS THE
EHD LIMIT OF 2.2 109CM-2......................................................................................... 28
FIGURE 2-4 2DES PHASE DIAGRAM ON DIELECTRIC SUBSTRATE VS. HELIUM FILM
THICKNESS. ................................................................................................................ 30
FIGURE 2-5 2DES PHASE DIAGRAM ON METALLIC SUBSTRATE VS. HELIUM FILM
THICKNESS. ................................................................................................................ 31
FIGURE 3-1 2DES BOUNDARY CONDITIONS: TWO DIELECTRIC MEDIA. ............................ 33
FIGURE 3-2 2DES BOUNDARY CONDITIONS IN OUR CELL: THREE DIELECTRIC MEDIA. .... 34
FIGURE 3-3 POLARONIC TRANSITION. ................................................................................ 41
FIGURE 3-4 WIGNER TRANSITION. ..................................................................................... 41
FIGURE 3-5 2D ELECTRON LIQUID MOBILITY FOR A RANGE OF PRESSING FIELDS. ........... 42
FIGURE 3-6 VARIOUS METHODS OF MOBILITY MEASUREMENT. ........................................ 42
FIGURE 3-7 COUPLED (SOLID) AND UNCOUPLED (DASHED) MODES OF WIGNER SOLID AND
SURFACE WAVES OF HE (FROM (FISHER, HALPERIN ET AL.)). ..................................... 45
FIGURE 3-8 EVOLUTION OF MAGNETOPLASMONS (FROM (GLATTLI, ANDREI ET AL. 1985)).
................................................................................................................................... 48
FIGURE 4-1 REFLECTION BETWEEN TWO DISCONTINUITIES IN TRANSMISSION LINE.......... 53
FIGURE 4-2 EQUIVALENT CIRCUIT OF 2DES AND EXCITATION LINE. STAR DENOTES THE
VALUE PER UNIT LENGTH. .......................................................................................... 56
FIGURE 4-3 MEANDER LINE. .............................................................................................. 60
FIGURE 4-4 COPLANAR MEANDER LINE. ........................................................................... 60
FIGURE 4-5 INTERDIGITAL CAPACITOR (IDC). ................................................................... 60
FIGURE 4-6 CROSS-SECTION OF A SLOW-WAVE STRUCTURE IN INHOMOGENEOUS
ENVIRONMENT. .......................................................................................................... 61
FIGURE 4-7 MEANDER LINE DISPERSION LAW FROM (CRAMPAGNE AND AHMADPANAH
1977). = PQ2J, P IS THE PERIOD OF THE LINE, OTHER PARAMETERS ARE DEFINED IN
FIGURE 4-6. ............................................................................................................... 62
FIGURE 4-8 IDC DISPERSION LAW FROM (CRAMPAGNE AND AHMADPANAH 1977). SEE
FIGURE 4-7 FOR MORE DETAILS. ................................................................................. 64
vii
FIGURE 5-1 SOMMER-TANNER TECHNIQUE. ...................................................................... 69
FIGURE 6-1 EXPERIMENTAL CELL MOUNTED ON DILUTION FRIDGE. ................................. 70
FIGURE 6-2 CRYOSTAT AND DILUTION UNIT. .................................................................... 71
FIGURE 6-3 CRYOSTAT ALIGNMENT SYSTEM. ................................................................... 72
FIGURE 6-4 CROSS-BELLOWS DESIGN ISOLATES A PUMPING LINE FROM THE CRYOSTAT. 73
FIGURE 6-5 SPLIT-COIL CROSSED-FIELD MAGNET. ........................................................... 75
FIGURE 6-6 EXPERIMENTAL CELL. ..................................................................................... 76
FIGURE 6-7 ELECTROSTATIC BOUNDARY CONDITIONS IN THE CELL. THE CELL HAS A
CYLINDRICAL SYMMETRY. RGR IS THE GUARD RING RADIUS; RE IS THE RADIUS OF 2DES
POOL. ......................................................................................................................... 77
FIGURE 6-8 THIN FILM MEASUREMENT. ............................................................................ 80
FIGURE 6-9 THICK FILM MEASUREMENT. .......................................................................... 80
FIGURE 6-10 FILLING CURVES. THE MEASURED CAPACITORS ARE SHOWN IN FIGURE 6-9
AND IN FIGURE 6-8. THE MEASUREMENTS ARE TAKEN WHILE FILLING THE CELL WITH
HELIUM. ..................................................................................................................... 81
FIGURE 6-11 SPRING CONTACT. MANUFACTURED BY EVERETT CHARLES TECHNOLOGIES
(PART # MEP-30U). .................................................................................................. 83
FIGURE 6-12 AFM IMAGE OF PASSIVATED SI (111) SURFACE: TRIANGULAR PITS ARE
APPROXIMATELY 150Å DEEP. RMS ROUGHNESS IS 55Å. ........................................... 85
FIGURE 6-13 DIRECTLY TRANSMITTED SIGNAL VS. MODULATED ONE. ............................. 86
FIGURE 6-14 LOCK-IN OUTPUT CORRESPONDING TO MODULATION OF THE RESONANCE
FREQUENCY. .............................................................................................................. 87
FIGURE 6-15 LOCK-IN OUTPUT CORRESPONDING TO MODULATION OF THE LINEWIDTH. .. 88
FIGURE 6-16 LOCK-IN OUTPUT CORRESPONDING TO MODULATION OF BOTH THE
RESONANCE FREQUENCY AND THE LINEWIDTH. ........................................................ 89
FIGURE 6-17 MEASUREMENT SETUP. ................................................................................. 89
FIGURE 7-1 VARIATION OF THE 2DES SPECTRA WITH SURFACE DENSITY: THE SPECTRA ARE
SHIFTED VERTICALLY TO FACILITATE COMPARISON. THE LEGEND SHOWS CHARGING
VOLTAGE AS WELL AS THE VALUE OF SURFACE DENSITY DERIVED FROM IT. T=110MK.
................................................................................................................................... 90
FIGURE 7-2 THE SPECTRA CORRESPONDING TO DIFFERENT DENSITIES ARE PLOTTED ON
LOG F SCALE AND SHIFTED BY –0.5 LOG N. ................................................................ 91
FIGURE 7-3 DETERMINATION OF RESONANCE FREQUENCY. .............................................. 92
FIGURE 7-4 THE EVOLUTION OF RESONANCE FREQUENCIES FOR THE SPECTRA OF FIGURE
7-1 WITH CHARGING VOLTAGE. THE SLOPE OF LINES ON THIS LOG-LOG PLOT IS ½, AS
EXPECTED FROM 2D PLASMA DISPERSION ~ N0.5. .................................................... 92
FIGURE 7-5 MATCHING OF AN OBSERVED SPECTRUM (THICK LINE) TO 2D PLASMA
RESONANCES (THIN LINES). HERE () ARE THE AZIMUTHAL AND RADIAL NUMBERS
RESPECTIVELY. THE AMPLITUDE OF EACH RESONANCE IN THEORETICAL SPECTRUM
REFLECTS THE COUPLING OF THAT MODE TO THE EXCITATION STRUCTURE (SEE 4.4) AS
DISCUSSED IN THE TEXT. ............................................................................................ 94
FIGURE 7-6 CORRECTED SPECTRUM MATCHING. ............................................................... 95
FIGURE 7-7 THE SEQUENCE OF RESONANCE FREQUENCIES VS. THEIR NUMBER I:
ASYMPTOTIC BEHAVIOR IS I0.5. ................................................................................... 96
FIGURE 7-8 EVOLUTION OF THE 2DES SPECTRUM WITH TEMPERATURE. THE SPECTRA ARE
SHIFTED VERTICALLY TO FACILITATE COMPARISON. TM DENOTES THE EXPECTED
viii
WIGNER SOLID MELTING TEMPERATURE FOR N = 6 107CM-2. THE INCIDENT RF POWER
IS -44DBM. ................................................................................................................. 98
FIGURE 7-9 NONLINEARITY OF 2D ELECTRON CRYSTAL RESPONSE. ............................... 101
FIGURE 7-10 CLASSICAL PART OF 2DES PHASE DIAGRAM.............................................. 101
FIGURE 7-11 2DES MOBILITY VS. TEMPERATURE. THE LEGEND SPECIFIES PRESSING FIELD
AND DENSITY FOR EACH DATASET. MELTING POINTS ARE INDICATED BY ARROWS. THE
SOLID LINES CORRESPOND TO THEORETICAL MOBILITY OF 2D ELECTRON LIQUID. THE
INSET SHOWS MOBILITY VS. PRESSING FIELD FOR A CONSTANT DENSITY. ................. 103
FIGURE 7-12 2DES SPECTRUM IN MAGNETIC FIELD. THE SPECTRA ARE SHIFTED
VERTICALLY TO FACILITATE COMPARISON. .............................................................. 105
FIGURE 7-13 CYCLOTRON FREQUENCY VS. MAGNETIC FIELD FOR THE SPECTRA OF FIGURE
7-12. ........................................................................................................................ 106
FIGURE A-1 SURFACE WAVE. .......................................................................................... 111
FIGURE B-1 RMS ROUGHNESS OF LIQUID 4HE SURFACE (FROM (COLE 1970)). .............. 116
FIGURE C-1 PARAMETRIC RESONANCE STABILITY DIAGRAM. HORIZONTAL AXIS: RIPPLON
FREQUENCY. VERTICAL AXIS: VIBRATION ACCELERATION AMPLITUDE. BOTH ARE
NORMALIZED BY VIBRATION FREQUENCY.
FROM HTTP://MONET.PHYSIK.UNIBAS.CH/~ELMER/PENDULUM/PARRES.HTM. .......... 117
FIGURE D-1 HEAT EXCHANGER MOUNTED ON CRYOSTAT'S BAFFLE............................... 120
FIGURE E-1 TOP OF THE CRYOSTAT. ................................................................................ 122
FIGURE E-2 50MK RADIATION SHIELD. ........................................................................... 123
FIGURE E-3 STILL RADIATION SHIELD. ............................................................................ 123
FIGURE E-4 DILUTION FRIDGE CONTROL PANEL. ............................................................ 125
ix
1
1 Introduction: 2D Electrons on the Surface of Liquid
Helium
1.1 The Origin of Reduced Dimensionality
The study of low-dimensional structures is central to modern condensed-matter physics
(March and Tosi 1984; Butcher, March et al. 1993). Examples include quantum wires,
2D electrons in semiconductor heterostructures (von Klitzing 1987), quantum dots
(Khoury, Gunther et al. 2000), electrons floating on the surfaces of inert cryoliquids and
cryosolids (Andrei 1997), etc. Recently low-dimensional structures (and 2D electrons on
helium is one of the most promising candidates - (Dykman, Platzman et al.)) are being
used as a basis for the growing field of quantum computing (DiVincenzo 2000; Nielsen
and Chuang 2000).
The subject of this work is 2D electrons on liquid 4He. First I discuss the conditions
leading to reduced dimentionality.
Every physical system resides in the 3D world and is subject to 3D interactions.
Occasionally however the nature of the interactions makes it possible to separate
variables in the Hamiltonian. One such system is an electron above a perfectly smooth
surface: its motion parallel to the surface is completely independent from the motion
normal
to
the
surface,
which
in
terms
of
the
hamiltonian
implies:
H(x,y,z) = H||(x,y) + H(z). The other necessary condition for the system to be considered
2-dimensional is that H(z) contains a confining potential. The energy of motion in the zdirection is quantized in this case. Let E1, E2 be the energies of the ground and the first
2
excited state of the system for such motion respectively. If the system is cooled to a
sufficiently low temperature so that k B T  E2  E1 , it will then mostly remain in the
ground state. The motion in the z direction is “frozen out”. Such system becomes twodimensional. Since quantization of energy is essential here, the reduction of
dimentionality is therefore a purely quantum effect.
It is important to note that in two dimensions the density of states of a free particle with
nonzero mass as a function of energy is constant
 E E  
m
 2
as opposed to 3D (  E E  ~ E ) or 1D (  E E  ~
(1-1)
1
E
) cases, where m is the effective
mass of the 2D particle.
Even though the physics of reduced dimensionality applies to all 2DES, they will differ
quantitatively depending on the environment where electrons are confined. The main
factors are the dielectric constant of the medium, its band structure and fabrication
technique (for semiconductors). Table 1-1 lists the values of basic parameters describing
2DES on helium versus analogous quantities for 2DES in GaAlAs heterostructures and in
silicon MOS structures. Unmodified electron mass, low densities and high mobilities are
the distinguishing features of 2DES on helium as opposed to 2DES in semiconductors. I
will argue in section 1.3 that such combination of properties makes 2DES on helium the
likeliest candidate for observing an ordering transition known as Wigner crystallization.
For a review of the basic properties of 2D electronic systems refer to (Ando, Fowler et al.
1982).
3
Table 1-1 Basic Properties of 2DES.
Binding Energy, K
Wavefunction Extent
out of 2D plane, Å
Surface Density, cm-2
Fermi Energy, K
Effective Mass, m*/me
Mobility, cm2/V s
On Helium
7
In Si-MOS
50-500
In GaAs/AlxGa1-xAs
200-400
114
30
50-100
6
9
10 -10 -?
-5
-2
10 -10
1.0
7
10
11
13
10 -10
10-500
0.19
3
10
11
12
10 -10
200-1000
0.066
5
6
10 -10
In the next section we will consider the confining potential experienced by an electron
above the surface of liquid helium.
1.2 Confinement of Electron to the Surface of Liquid Helium
Bound states of electrons on the surface of liquid 4He were predicted (Shikin; Cole and
Cohen 1969) in 1969. An electron is bound to the surface of liquid 4He (or any other
dielectric) (Andrei 1997) by inducing an electrostatic screening charge as shown in
Figure 1-1 Screening Mechanism Giving Rise to Electron Confinement to a Liquid Helium
Surface.
4
Figure 1-1. This forms the attractive part of the binding potential (Figure 1-2). The
repulsive part comes from the fact that the supporting surface is made of inert atoms
having complete electronic shells and thus no states available for an external electron.
More exactly it takes a sufficiently high energy for an electron to enter the bulk of liquid
helium and to form a so called bubble state (Onn and Silver). This poses a barrier for
penetration into the liquid helium whose magnitude is roughly 1 eV (Fetter 1976).
However, since the typical binding energy for an electron on liquid helium, as will be
shown soon, is a fraction of a meV it can be approximated by an infinite barrier. Also the
interface between liquid helium and its vapor is considered abrupt and perfectly flat in
this approximation (the actual transition layer is about 6 Å (Penanen, Fukuto et al. 2000).
In addition the helium surface is rough as there are thermally excited surface waves – the
mean square amplitude of the roughness is ~1 Å, see appendix B and also (Edwards and
Saam 1978)). The total potential experienced by an electron in the direction normal to the
liquid surface along with the resulting energy levels is shown in Figure 1-2. The
assumptions are:

Simple Coulomb attraction between an electron and its image in the substrate.

The electron’s wavefunction vanishes at the interface.

The Hamiltonian is separable if the surface is completely smooth.
The motion in the z direction is then described by a 1D Hamiltonian
2

 1
 2  2   e
H  
  4z z  0 ,  
which is identical to the radial part of
2


1
2m  z
 
z0
Schrödinger equation for the hydrogen atom with zero angular momentum. Thus the
energy spectrum of a 2D electron is:
5
m 2 e 4
Ry 
E n   2 , Ry  
n
32 2
(1-2)
where  is the dielectric constant of the substrate, m and e are the effective mass and
charge of the electron respectively (a cyclotron resonance measurement confirmed that
the effective mass is equal to that of a free electron m = me (Edel'man 1976)).

3
2
The ground state wavefunction for motion in the z direction is 1 ( z )  2 a0 z e

z
a0
where
Figure 1-2 Energy Diagram of an Electron Bound to He surface.
a0 
4a b

is the effective Bohr radius.
The spectrum can be investigated experimentally for instance by recording spectra of
mm-wave absorption corresponding to transitions between the energy levels (Grimes,
Brown et al.; Zipfel, Brown et al.). For an electron bound to the surface of liquid 4He the
6
binding energy is Ry* = 0.65 meV = 7.5 K. Therefore for the temperatures used in the
presented experiment (< 1K), electrons are always found in the ground state. The
effective Bohr radius a0 = 76 Å is greater than both the interatomic distance in liquid
helium (~1Å) and the roughness of the helium surface. This in turn justifies the use of a
macroscopic dielectric constant to describe screening effects in the substrate as well as
the assumption of an ideally smooth helium surface.
1.2.1 Influence of Externally Applied Electric Field
Usually the experiments are conducted with electrons bound not only by attraction to
their images but also by an external clamping electric field E. This is necessary in order
to counter the mutual repulsion of electrons and to reduce their thermally activated
escape rate from the 2D layer. The probability of escape to the unbound states is
proportional to the density of states there and the latter becomes infinite for the 1D
hydrogen potential as the energy approaches zero. In the presence of a clamping field the
approximate ground state wavefunction is (Saitoh 1977):

3
1 ( z )  2 b 2 z e

z
b
with b 
4
9 
1
a0 sinh  sinh 1
,  
3
4 
3
whereas the ground state energy is E1  
2 2
mb 2
2mea03 E 

(1-3)
 b 3
   .
 a0 4 
Therefore applying an external field allows controlling the out-of-plane extent of the
wavefunction b (as opposed to a0 without pressing field). This will affect the interaction
between electrons and the helium surface – see 1.5.2. The parameter  describes the
strength of the external field as compared to a characteristic field:
7
Ec 
2
V
 864
3
cm
2mea0
(1-4)
which is roughly the field due to the image charge. In the limiting cases of weak (<< 1)
and strong ( >> 1) field, b is given by
b
32
 1
a0
4
and
b 3 4

respectively.
a0
32
1.2.2 Binding to a Thin Film
When the helium film thickness d becomes comparable to the distance of the electrons
from the helium surface (given by a0 = 76 Å) the total potential must include the field of
the image charge in the underlying substrate as well as the image in helium. Hence the
total potential is (including the external pressing field):
U z   
 he2
4z

 se2
4z  d 
 eE z
Here h = (h-)/( h+) and s = (s-h)/( s+h), h, s are the dielectric constants of
helium and the substrate respectively. In the limit when d > a0 = 76 Å the second term
can be expanded in z/d resulting in:
U z   
 he2
4z
 eE  z
where we made the substitution E  E 
 he
4d 2
and dropped the constant term.
Therefore in this situation the effect of the image charge in the substrate is equivalent to
an enhancement of the pressing field. For d = 100 Å this enhancement is 1 kV/cm –
exceeding Ec in Eq.(1-4), so that the 2DES on thin films is always in the regime of strong
pressing fields.
8
Using the strong field approximation formulas of 1.2.1 one can get an estimate for the
variation of the average distance between an electron and the helium surface caused by
the substrate: <z> = 3/2 b ranges from 30 Å for thin film to 110 Å for bulk.
1.2.3 Hartree Field (Mutual Repulsion of Electrons)
So far, the fields acting on a single surface electron have been considered. In a typical
experiment the surface of helium is charged with an almost uniform surface density n
whose typical values range from 107 to 109 electrons per cm2. Each electron is then
creating its own image charge in helium. This range of densities corresponds to
interelectronic distances of order of mm, which is much larger than 2 a0 = 152 Å – the
distance between an electron and its own image while its distance to all other images is
larger. Therefore the total electrical field an electron sees is a superposition of the
Coulomb field from its image and a uniform background, which includes other images.
The vertical stability of the 2D electron is further discussed in 6.5.2.
Beside liquid 4He, other substances used to support 2D electrons include: solid 4He, solid
hydrogen (Adams and Paalanen 1988; Kono, Albrecht et al. 1991; Kono, Albrecht et al.
1991; Mugele, Albrecht et al. 1992), solid neon (Kajita), liquid 3He. The work that was
recently carried out on 0D and 1D electrons on helium is reviewed in (Kovdrya Yu).
1.3 Wigner Crystal
The most striking phenomenon caused by the interactions between electrons bound to the
surface of liquid helium is the appearance of an ordered phase – Wigner crystal (Crandall
and Williams; Wigner 1934).
9
The phase of a 2D electron system is determined by the competition of three energy
scales:

Coulomb energy of interaction between electrons Ve-e =

Thermal energy kBT (or classical kinetic energy).

2D Fermi energy E F    2
e2

n .
n
(or quantum kinetic energy).
m
Here n is the electron surface density, T – temperature, m and e – effective mass of
Figure 1-3 Schematic 2DES Phase Diagram. The straight line separates the region
where classical fluctuations dominate (lower part) from the one where quantum
fluctuations dominate (the upper part).
electron and charge respectively,  is the dielectric constant of the substrate (= 1.057 for
liquid helium-4). A qualitative phase diagram of the 2D electron system resulting from
this competition is presented in Figure 1-3 (Fukuyama; Platzman and Fukuyama 1974).
10
The range of parameters for which the potential energy term dominates corresponds to
the ordered part of the phase diagram. Outside this regime, the kinetic energy (either
classical or quantum) dominates and the system is disordered. At high densities and low
temperatures the quantum fluctuations are more important than classical ones and we
have a quantum phase transition at a critical density nw determined from the condition
Vee(nw) ~ EF(nw):
m2e4
n ~ 2 4
 
w
(1-5)
For 2DES on helium bulk nw ~1013 cm-2. A thorough discussion of the phase diagram is
deferred to Chapter 2.
The theoretical description of the properties of Wigner crystal (it is sometimes referred to
as Coulomb crystal in the classical portion of the phase diagram) was given in (Bonsall
and Maradudin 1977). In particular they estimated the energies of all possible 2D lattices,
Figure 1-4 2D Hexagonal Lattice - lowest energy configuration of 2D Coulomb crystal.
having found the lowest value for a hexagonal lattice. The magnitudes of reciprocal
lattice (which is also hexagonal) vectors form the following sequence:
Gp 
p G1 ,
G12 
8
3
 2 n , p  1, 3, 4, 7, 9,12,13,16....
(1-6)
Theoretical work (Tanatar and Hakioglu) has revealed the possibility of a
superconducting region in 2DES phase diagram forming at temperatures of order of
11
millikelvin. However it is not possible to cool the 2D electrons to this temperature due to
their weak coupling to thermal bath, consisting mainly of ripplons (surface phonons) on
liquid helium.
Due to the limit on electron surface density imposed by the collapse of a bulk charged
liquid surface - electrohydrodynamic (EHD) instability (discussed in 2.3 of this work and
also (Ikezi and Platzman 1981)) only a small portion of the phase diagram (classical one)
is experimentally accessible as shown schematically in Figure 1-3. One of the goals of
this work is to investigate methods for reaching the areas of the phase diagram where
quantum effects are substantial – in particular to be able to observe a quantum phase
transition. The transition is in principle achievable for 2DES on liquid helium films but
not in 2DES in Si-MOS or GaAlAs heterostructures. This is because the latter have
lower effective carrier mass (and therefore larger Fermi energy) and lower electronelectron interaction energy due to higher dielectric constant (see Table 1-1). Therefore,
according to Eq.(1-5) the critical density nw is lower than that of 2DES on helium. It is
actually lower than the minimum density achievable by present day fabrication
techniques of Si-MOS or GaAlAs heterostructures (~1011cm-2). This explains the choice
of 2DES on helium as a candidate for exhibiting the Wigner crystallization. To date there
is no convincing experimental evidence for Wigner crystallization in 2DES in
semiconductors.
1.4 Collective Modes of 2DES
In this work, the physical properties of the 2DES are deduced from RF absorption spectra
corresponding to normal modes of the electron sample. The excitation branch common to
12
all charged plasmas is a longitudinal plasmon. In a 3D plasma this is a dispersionless
oscillation of frequency
n e2
0 m
(1-7)
n e2
q
2 0 m
(1-8)
3D 
In 2D the dispersion relation is:
2D 
This particular dependence on q is the consequence of ~ 1/r interaction potential between
electrons (r is the interelectronic distance). However since in a realistic experiment metal
electrodes are present in the vicinity of the 2DES (at a distance d from 2DES), the
interaction is screened so that the potential is ~ 1/r3 when r > d. This makes the
dispersion law approximately linear near q = 0 so that long wavelength plasmons have a
finite propagation velocity:
cp 
n d e2
 0m
(1-9)
This velocity corresponds to 1.8 108 cm/s for typical values, n = 108 cm-2, d = 1mm.
These formulas are derived in section 3.1.
The longitudinal plasmon dispersion is practically the same for either electron liquid or
solid (exact dispersion laws for both are in section 3.2). Therefore, it is not possible to
use this mode alone to study crystallization. However there are two alternatives:

Solids (2D as well as 3D) unlike liquids possess a finite shear modulus and
therefore permit propagation of a transverse mode if the dissipation is not too
large. See 3.2
13

As will be explained later when the 2DES is coupled to the underlying liquid its
normal mode spectrum is very sensitive to the phase of the 2D electrons.
Specifically the correlated nature of electron motion leads to strong electronripplon coupling which opens a gap in the spectrum.
The size and geometry of the 2D electron pool determine the wavevectors q of the normal
modes. Excitation of these resonances is attained by sweeping the frequency of the
external electromagnetic field through the range of interest. Chapter 3 contains a
discussion of the 2DES spectra. The detection technique is the subject of Chapter 4.
1.5 The Dynamics of Liquid Helium and Its Effect on 2DES
One has to consider the properties of the liquid helium supporting the 2DES since the
interaction between them affects both their excitation spectrum and the phase diagram.
1.5.1 The Surface Waves - Ripplons
Because the electrons reside on the helium surface of liquid, their motion is coupled to
surface excitations. These excitations – capillary waves - are governed by gravity and
surface tension leading to a dispersion relation of the form (appendix A):

 
 r 2   g k  k 3  tanh kd
 

(1-10)
where g – is the gravitational acceleration,  – surface tension,  – density of liquid, d –
depth of liquid. The characteristic wavevector determining the relative strength of terms
in Eq.(1-10) is referred to as capillary constant:
kc 
g

 20 cm 1
(1-11)
14
The numeric value is for liquid 4He at 0K (see appendix B).
The quanta of capillary waves are referred to as ripplons. Thermally excited ripplons
determine the roughness of helium surface - appendix B.
1.5.2 Interaction Between 2DES and the Surface of Liquid Helium
Electrons with surface density n bound to liquid helium in the presence of a clamping
field E exert a pressure on the surface: Pe  n e E . This pressure is a result of the
counteraction corresponding to the repulsive barrier at the helium surface – see
discussion in 1.2. As a result, the helium surface under the electrons is depressed whereas
the uncharged surface is elevated. The difference in level is:
h 
n e E
g
The absolute depression of the surface depends on the ratio of the uncharged area to the
total area of the helium surface:
d 
n e E  S unch arg ed
 LHe g S total
For typical parameter values (n = 5 108cm-2, E  =1000V/cm, and the ratio of areas ¼) the
helium surface is depressed by 14 microns. This effect is even more significant if the
experiment is performed on a thin film of liquid helium (section 1.5.3).
Another effect of the electron pressure is that each electron produces a small depression
in the helium surface – the so-called single electron dimple. If the helium thickness is
sufficiently large (~ 1 mm) the typical depth of the dimple is ~0.1Å for a pressing field of
500 V/cm. The binding energy for such a dimple is small – of order of
15
eE  2
4
 10 3 K (Jackson and Platzman). Therefore no bound state of electron – so-
called ripplonic polaron (Shikin), which is an electron dressed by ripplons - is expected
for realistic experimental temperatures T > 50mK on bulk helium (Jackson and
Platzman). Yet, even on bulk the dimple effect is important when the motion of electrons
is correlated as in the case for the Wigner crystal. The effect of this dimple is to open a
gap in the spectrum of the Wigner crystal and will be discussed in section 3.4.
The ripplons also scatter 2D electrons thus affecting their mobility. This is discussed in
section 3.3.
1.5.3 Thin Film Effects
As the thickness of the helium film becomes small (<1000Å), a number of new factors
have to be taken into account. First the solid substrate supporting the film alters (usually
increasing) the binding energy (section 1.2) and also screens the interaction between
electrons. The latter effect is important when the film thickness becomes comparable to
the distance between the electrons and leads to a reduction in the size of the Wigner
crystal “bubble” in the 2DES phase diagram as will be explained in 2.4.
The polaronic transition now becomes a reality since an electron is subjected to a strong
field from its image in the substrate in addition to the external pressing field. Signatures
of this transition were reported in early experiments (Andrei 1984; Tress, Monarkha et al.
1996).
The second important consequence of the film becoming thin concerns the film itself.
The helium surface becomes stiffer as a result of the Van der Waals forces (I. E.
Dzyaloshinskii 1961; Sabisky and Anderson 1973) from the underlying substrate
16
(see 2.3). Formally this can be accounted for by using a modified gravitational constant
g~ in Eq.(1-10) which turns out to be four orders of magnitude larger than g = 9.8m/s2 for
a 100 Å helium film. Because of this stiffening, thin films are stable against the EHD
instability (see 2.3). This allows charging to high surface densities. In addition the twofluid nature of superfluid 4He (Putterman 1974) starts playing a role since the motion of
the viscous normal component is restricted by the substrate giving rise to third sound
propagation (Komuro, Kawashima et al. 1996). This happens when the film is thinner
than the boundary layer thickness
2
 n
, where  and n are the normal fluid viscosity
and density respectively,  is the sound frequency. For typical values of the parameters at
1K (= 27 P, n = 2 10-3 g/cm3 (Donnelly, Glaberson et al. 1967)) and a frequency of
10 MHz the boundary layer is 2000 Å thick.
The most widely used method for obtaining a submicron helium film relies on the fact
that in the presence of helium gas the solid walls of a container are coated with an
adsorbed film of helium atoms. If the film is superfluid, it has a thickness profile
according to its vertical position above the surface of helium bulk. Therefore, by
adjusting the height of the helium bulk with respect to a solid substrate one can control
the film thickness adsorbed on the substrate.
As will be argued in 2.3 it is also possible to arrive at a 2DES deposited on a thin film by
starting from a relatively thick film such as might be obtained by directly adjusting the
level of helium bulk.
17
1.6 Summary
The main contributions of this thesis are:

A new technique is proposed for measuring the thickness of a helium film by
using the spectral splitting that arises when the 2DES is strongly coupled to a
transmission line.

The experiment described in this work uses two novel transmission line structures
to couple electromagnetic waves into a 2D electron layer: an interdigital capacitor
and a coplanar meander line. Their properties are reviewed in Chapter 4.

As discussed in section 2.3 a method was developed to obtain a charged thin film
of helium starting with an uncharged thick film.
This is the first step toward achieving the more ambitious goal of accessing the quantum
regime of 2DES phase diagram – namely the region of high densities and low
temperatures, and to observe the quantum melting of the Wigner crystal. As was briefly
mentioned in section 1.5.3 it is the use of thin (~100 Å) helium films that makes it
possible to charge the helium surface to the desired densities. In order to achieve this goal
it will be necessary to overcome a number of technical difficulties, which were
encountered in the course of this study:

Using thin helium film requires preparation of nearly atomically smooth substrate
to support the liquid. Even though it is quite straightforward to prepare such a
substrate (6.7) it turned out to be impossible to keep it clean and smooth during
the subsequent assembly of the experimental cell and cooldown.

There seems to be no measurable coupling between 2DES and helium film such
as required to insure different spectra in liquid vs. solid state of 2DES. This is
18
probably due to a stiffening of the helium surface and to an increase of the
associated frequencies as the film thickness becomes small.
19
2 Phase Diagram of 2DES on Liquid Helium
Electrons floating on the surface of liquid 4He can be (with regards to their in-plane
motion) in a spatially disordered state (2D electron liquid) or they can form a regular
structure known as Wigner crystal. The range of parameters for which each phase is
stable and the resulting phase diagram are outlined bellow. For a more extensive
discussion see (Devreese, Peeters et al. 1987).
2.1 Electrons on Bulk Helium
The simpler case is when the depth of the He layer is not too small – above 100m and
much larger than the interelectronic distance (~m) so that there is little influence from
the substrate underneath the helium – this is referred to as 2DES on helium bulk.
The plasma parameter describes the relative importance of interaction over fluctuation
and is defined as the ratio between the average potential and kinetic energy (thermal or
quantum – whichever is greater):

Vee
K
For 2D electrons with surface density n
Vee  e 2  n
The kinetic energy (for either degenerate or nondegenerate 2DES) is given by
K 
mk B T 
 n 2
2 
x dx

0
x
e
(2-1)

kB T
1
20
  n

where   k B T ln  e m k B T   1 is the chemical potential.




2
This expression reduces to <K>classical = kBT for nondegenerate electrons (EF << kBT) or to
<K>quantum = EF =   2
n
m
(2-2)
for degenerate electrons (EF >> kBT). It is obvious that for sufficiently low surface
densities and temperatures the interaction would dominate giving rise to spatial order.
The phase diagram can be obtained by setting the plasma parameter equal to its critical
value – also known as the Lindemann melting criterion (Lindeman 1910):
  m
(2-3)
In other words, the ratio of average potential to average kinetic energy has to drop below
a certain value m for melting to occur or equivalently the amplitude of fluctuations has
to become comparable to the lattice constant. A value for the classical case most widely
confirmed both by theory (Gann, Chakravarty et al. 1979; Morf 1979) and experiment is
m ~130 (Deville, Gallet et al.; Grimes and Adams; Kajita; Marty and Poitrenaud;
Rybalko, Esel'son et al.; Shirahama and Kono; Mehrotra, Guenin et al. 1982; Mellor and
Vinen 1990). The phase diagram resulting from the Lindemann criterion with m =137 in
n, T variables is presented in Figure 2-1 (Peeters 1984). Note that it is a qualitative phase
diagram since m = 137 is only valid for nondegenerate 2DES and would certainly have a
different value for a transition driven by quantum fluctuations. This value (33  5) was
obtained in (Tanatar and Ceperley 1989) using Green’s-function Monte Carlo
calculations.
21
13
2x10
n
QUANTAL
LIQUID
W
QUANTUM SOLID
13
-2
n[cm ]
1x10
NS
TIO
TUA
C
NS
FLU
TIO
M
A
U
U
T
T
AN
LUC
QU
LF
A
CLASSICAL
RM
THE
LIQUID
11
4x10
11
3x10
11
2x10
11
1x10
0
0
2
4
6
8
10
12
14
T[K]
Figure 2-1 2DES Phase Diagram on Helium Bulk.
The “bubble” defined by this criterion is the Wigner solid. The area outside the bubble
corresponds to a disorder phase – electron liquid. The surface density values in the
classical part of solid phase are less then 1011 cm-2, which corresponds to a lattice
constant of order of 10-2 m – a large value compared to its counterpart in regular 3D
solids.
2.2 Wigner Solid and 2D Melting Mechanisms
The existence of an ordered electron phase was first predicted (for 3D electrons) by
E. Wigner (Wigner 1934) in 1934. However 3D Wigner solid was never unambiguously
realized experimentally. One of the reasons is that in the 3D electron plasma screening of
interaction is much more significant than in lower dimensions. However in 3D there exist
22
a number of phenomena similar in nature to Wigner crystallization such as Mott
transition (Mott 1961).
Theoretically it was shown that no true long-range order can exist in 2D (N. D. Mermin
1966; Peierls 1979) which is expressed by stating that density-density correlation
function decays with distance. This is due to a stronger impact of fluctuations (thermal or
quantum) in lower dimensions (and fewer nearest neighbors). However as shown by
KTNHY theory (see below) the correlations in 2D decay only as a power law with the
distance R between particles:
g G ~ R G T 
(2-4)
(G – is a reciprocal lattice vector). This decay corresponds to quasi-long-range order in
contrast to the exponential decay in the liquid phase where the order is lost. In the
harmonic approximation the exponent in Eq.(2-4) is
 G (T ) 
G 2T
2  T 
where m(T) is the shear modulus in the Wigner crystal (m(0)=0.245 e2 n3/2 (Bonsall and
Maradudin 1977)). Among the possible 2D lattice structures the hexagonal one was
shown to be the most stable (Haque, Paul et al. 2003).
The theory of melting in 2D was developed in the 1970’s and is known as KTHNY
theory (Kosterlitz and Thouless 1972; Halperin and Nelson 1978; Young 1979). In this
theory the solid is destroyed by unbinding of topological defects – dislocations - in the
crystal. In some cases, the melting is a two-stage process. During the first stage,
occurring at temperature Tm, paired dislocations become unbound and the solid is
23
transformed into a hexatic phase characterized by the absence of long-range positional
order while retaining bond orientation order.
The critical exponent G (Eq.(2-4)) and plasma parameter can be expressed via elastic
Lame coefficients ( is the shear modulus, B = + is the bulk modulus (Landau and
Lifshits 1970)) as follows:
2
 G (T ) 

k BT G
4
3  
 2    
4  n 2   
    
a2
a is a lattice constant here. The KTHNY gives the following value for Tm:
a 2 4 (    )
Tm 
16 k B 2   
In the approach to the hexatic phase the shear modulus  behaves as:
 T    Tm (1  const (Tm  T ) )
 = 0.36963 (Deville, Valdes et al. 1984)
Finally in a second step at a temperature above Tm disclination pairs unbind and the bond
orientation order is destroyed as well resulting in a liquid phase.
The most direct way of studying the lattice structure is by measuring the Bragg
diffraction patterns that result from scattering of some kind of waves on the lattice. The
wavelength should be close to the lattice constant. The order in 2DES on liquid helium
cannot be investigated with visible light or neutron scattering – the cross-sections for
these processes are miniscule. It is in principle possible to study the scattering of helium
surface waves – ripplons (section 1.5) off 2DES. Such experiment has been proposed in
(T. Williams 1995; Deville 1997). To generate a ripplon with the wavelength in
24
micrometric range one can use an interdigital capacitor structure (IDC) described in 4.3.
IDC should be positioned outside of 2DES pool, but in such a way that it is covered with
liquid helium. The voltage is applied between the two sets of IDC’s fingers. The resulting
electrostrictive force excites a surface wave whose wavelength is related to the period of
the IDC IDC: kr = 4n/IDC. If the excitation frequency matches that of a ripplon r(kr) as
given by Eq.(1-10), a resonant surface wave is generated. It then travels through the
2DES, coherently scattering from electrons. The detection can be performed with an
identical IDC located on the opposite side of the 2DES pool: the incoming surface waves
cause variation of capacitance between the IDC fingers. The mean square of induced
current is proportional to the structure factor of 2DES: S k  
1
2

e

ik ri r j

where ri is
the coordinate of a 2D electron. In this experiment, the direction of the ripplon
wavevector k is kept constant, while reciprocal vectors of 2D lattice can be swept either
by squeezing it with an external electric field and therefore changing the 2DES size or by
turning it with changing vertical magnetic field. The expected results are shown in Figure
2-2 for three possible phases of 2DES. The structure factor is plotted for two different
orientation of the 2DES.
25
Structure Factor:
SOLID
S(k) = <exp i(k.(ri-rj))> /2
S(k)
ky
kx
HEXATIC
ky
|k|
S(k)
kx
|k|
LIQUID
S(k)
kx
|k|
Figure 2-2 2DES Diffraction Patterns (T. Williams 1995).
26
2.3 Electrohydrodynamic (EHD) Instability of a Charged Liquid
Surface
One of the main limitations on the experimental investigation of the 2DES phase diagram
with electrons on helium bulk comes from the electrohydrodynamical instability and is
related to the well-known Rayleigh instability (Rayleigh 1882). The instability sets a
limit on the maximum electron density that can be supported by a surface of liquid
helium or any other liquid (Gor'kov and Chernikova 1973; Gor'kov and Chernikova
1976). After exceeding this limit the surface would collapse. This happens because of the
softening of liquid surface modes – capillary waves (see 1.5.1).
More specifically the charged liquid surface has a dispersion relation, which differs from
Eq.(1-10) by the electron pressure term:
r
2

 3 4 e 2 n 2 2 
  g k  k 
k  tanh kd




(2-5)
It is obvious that for a sufficiently high electron density the frequency vanishes, which
corresponds to the development of an instability. This critical density is
nc 
1  g
4
 2..2 10 9 cm 2
2
e (2 )
and the wavevector corresponding to the instability is equal to the capillary constant of
liquid He k c 
g

 20 cm 1 .
Theory and experiments studying the nonlinearities developing at the onset of this
instability (referred to as multielectron dimples as opposed to single electron dimple
1.5.2) are presented in (Ikezi 1979; Ikezi, Giannetta et al. 1982).
27
However the instability can be avoided if a thin film of liquid helium is used instead of
bulk. On the thin film there is an additional term in the dispersion formula of surface
waves (Ikezi and Platzman 1981) arising from the Van der Waals interaction between
helium atoms and the underlying substrate:
r
2

3
 3 4 e 2 n 2 2 

  (g 
)k  k 
k  tanh kd
4



d


(2-6)
Here  is the Van der Waals constant (~ 10-15 erg for liquid helium on glass). This
3
effective enhancement of the gravitational constant g~  g 
postpones the EHD
d 4
instability to much higher densities (as a matter of fact for d in the submicron range
g >> g).
nc 
The
critical
wavevector
and
density
become
kc 
1
d2
3

and
3 
1
4
respectively. In fact, the experiments have shown that the film is
e (2 d 2 ) 2
stable for practically any n. This enhanced stability can be understood by taking into
account the reduction in film thickness caused by the electron pressure. The new film
thickness can be found by the equating chemical potential over the charged area of the
surface to that of the uncharged area:
 gd0   gd 

d3
 2 (ne) 2
(2-7)
The electron density in our experiments is directly related to the charging voltage V
(described in 6.5.2). The above equation has to be satisfied along with the relationship
between charging voltage and density:
28
n
V
1
4 e d /  He  d s /  s
(2-8)
here d, He – are the thickness and dielectric constant of liquid helium respectively, ds, s
– thickness and dielectric constant of substrate if it is used. These two equations have to
Film S tabilityfor V 20 Volt
29
19
8
n 10 cm
2
(a)
9
15
35
55
75
95 115 135 155
d m
Film S tabilityfor V 40 Volt
175
195
39
29
n 108 cm
2
19
(b)
9
15
35
55
75
95 115
d m
135
155
175
195
Figure 2-3 Charging Instability. Dashed line is the plot of n vs. d from Eq.(2-7). Dot-dashed line is
the plot of n vs. d from Eq.(2-8). One of their intersections corresponds to the stable charge density.
Solid line is the EHD limit of 2.2 109cm-2.
29
be solved simultaneously since the density depends on the final film thickness. Graphical
solutions for two values of charging voltage are shown in Figure 2-3 for d0 = 0.3mm,
ds = 0.3mm, s = 11. (EHD stability limit is shown as well). The lower density solution in
Figure 2-3(a) is stable while the other one is unstable. For sufficiently high charging
voltages no “bulk” solution would exist which is illustrated in Figure 2-3(b). This
happens when
3
2
V  ( (d 0 /  He  d s /  s )) 2 4  g
3
(2-9)
Comparing Figure 2-3(a) and (b) one can see that for any charging voltage lower than
this value the stable density would be below the EHD limit. Therefore by starting with a
film, which is a fraction of a millimeter thick and using sufficiently high charging voltage
one can end up with 100Å film bypassing EHD. This method has important practical
implication because it is very difficult to charge a helium film since for electrons it is
easier to shoot through a thin film at the beginning of charging when there is no repulsion
due to already deposited electrons.
2.4 Influence of a Substrate on Phase Diagram
When electrons are deposited on a film of liquid helium with thickness comparable to the
inter-electronic distance screening due to the substrate supporting the film leads to a
modified phase diagram. Consequently the Wigner crystal melts at a lower temperature.
To obtain the phase diagram of 2DES on liquid He film of thickness d supported by a
dielectric substrate of permittivity  S one solves Eq.(2-3) with
30

s
Vee  e 2  n 1 

1  4 nd 2


 where    S  1
s

 S 1

(2-10)
The phase diagram for several thicknesses of liquid helium film supported by a substrate
Bulk
2
12
300Å
-2
n[10 cm ]
70Å
100Å
1
0
0 1 2 3 4 5 6 7 8 9 10 11 12 13
T[K]
Figure 2-4 2DES Phase Diagram on Dielectric Substrate vs. Helium Film Thickness.
with s=0.9 is shown in Figure 2-4.
The case of a metallic substrate corresponds to  s  ,  s  1 in Eq.(2-10) and the
phase diagram is shown in Figure 2-5. The difference from the diagram for a dielectric
substrate is the appearance of a liquid dipole phase at T = 0 for thin films. This is because
on metallic substrate in the limit n d2 << 1 the Coulomb term in the electron-electron
3
energy vanishes and the interaction is dipolar: Vee  e 2  2 n 2 d 2 . Comparing to
31
 2

n

Eq.(2-2) for Fermi energy, one can see that for densities below
2 2
 md e
2

 quantum

fluctuations will dominate the electron-electron interactions promoting the liquid phase.
Figure 2-5 2DES Phase Diagram on Metallic Substrate vs. Helium Film Thickness.
32
3 Spectrum of 2DES
This chapter discusses the normal modes of 2DES on helium. First the normal modes will
be considered disregarding their coupling to the helium. Section 3.3 reviews the
dissipative processes affecting the spectrum. The coupling between 2DES and helium
substrate, presented in 3.4, provides a way to detect the Wigner crystallization and is used
in 7.2. Finally 3.5 discusses the spectrum in a magnetic field.
3.1 2D Drude Model
Here I will derive the dispersion law for classical (kBT >> EF) 2D plasma normal modes
in the longwave limit. In what follows 2D plasma will be confined to the z = 0 plane. The
electrodes surrounding 2DES are assumed to be properly DC biased to ensure
confinement (section 6.5.2) so that they can be considered as grounded with regards to
AC voltage induced due to the 2D plasma oscillations.
First Poisson equation supplemented by the boundary conditions allow to obtain the
relationship between the distribution of an excess surface charge density e in the plane
and the potential it creates Ve:
 2V e  2V e
 e r 



 z 
0
r 2
z 2
(3-1)
In this equation r is the 2D coordinate vector in the electron plane. For the sake of
simplicity we first solve this equation with boundary condition Ve = 0 as z   . The
ultimate goal is to obtain the formula that describes the response of 2D plasma to an
33
incident electromagnetic wave. Therefore one should look for a solution that has the
following form:
V e  Vqe e i (t qr ) e
q z
(3-2)
The choice of z dependence is set by the requirement that Ve has to satisfy the Laplace
equation  2V e  0 outside the 2D plasma and its derivative E z  
V e
has to be
z
discontinuous at z = 0. Substituting Eq.(3-2) into Eq.(3-1) one gets the formula relating
the charge density and potential in the 2D plasma plane:
Vqe 
 qe
2q 0
(3-3)
One might generalize this result for various different boundary conditions other than
Ve = 0 as z   . In each case the solution to Poisson’s equation would have the form
V e ~ e i (t qr ) ( Ae
q z
 Be
qz
 Ce  q z  De q z ) where the choice of constants A,B,C,D
satisfies particular boundary conditions. One example is when there are two metallic
electrodes present – one at the distance d below the 2D electron layer and the other at a
distance h above it, as shown in Figure 3-1.
Figure 3-1 2DES Boundary Conditions: Two Dielectric Media.
34
Then Eq.(3-3) should be replaced by: Vqe 
 qe
q 0  d coth qd   h coth qh 
.
Figure 3-2 shows the most relevant case for our experiment: a layered arrangement where
h is the distance between 2DES and top plate, dHe is liquid helium thickness and ds is
Figure 3-2 2DES Boundary Conditions in our Cell: Three Dielectric Media.
semiconductor
wafer
thickness.
Eq.(3-3)
then
should
be
modified
to
become:
V 
e
q
 qe

(   He ) cosh q(d He  d s )  ( s   He ) cosh q(d He  d s ) 

q 0  coth( q h)   He s
(



)
sinh
q
(
d

d
)

(



)
sinh
q
(
d

d
)
s
He
He
s
s
He
He
s


Very important situation is when plasmon wavelength is much longer than the distance to
the nearest electrode (let’s assume it happens to be d): qd<<1. The formula relating
excess charge and voltage is then: Vqe 
 qe d
.
0 d
Later I will use cylindrical coordinates r,z for which Poisson equation solution should
be written in terms of cylindrical harmonics:
35
V e  Vqe, e it J qr e i e
q z
(3-4)
instead of Eq.(3-2). Eq.(3-3) is still valid as it stands.
Having analyzed the electrostatics of 2DES, we proceed to deriving the 2D plasma
dispersion law by writing down the equation describing the motion of 2D electrons in the
presence of an external electromagnetic wave E = E0exp(i(t-q0r)) propagating in the 2D
plane of electrons:
1
e V e
u  u  (
 E)

m r
(3-5)
Here u is the 2D displacement of an electron from its equilibrium position,  describes
dissipation (see 3.3) and the choice of signs is determined by the negative value of the
electron charge. The wave is presumed longitudinal (E0||q0) – see the discussion at the
end of this section. Electron displacement is related in the linear approximation to an
excess charge density distribution by a material equation:
 e  ne
u
r
(3-6)
where n – is the equilibrium electron density. Combining Eq.(3-5) with Eq.(3-6) one gets:
1
ne 2  2V e
E
 e   e 
(
 )

m r 2
r
Switching to harmonics, (both temporal and spatial) and substituting Eq.(3-3) for e, one
obtains:
i 
ne 2 q0

Vqe   2   p2 q     i
eq 0 , q E0
 
2m 0 q

(3-7)
Here e(q0, q) are the projections of the external wave onto the chosen set of normal
modes of the 2D system, which represent the coupling between the 2DES and the
36
excitation
line.
For
example,
if
the
normal
modes
are
plane
waves
( V e ~  Vqe exp i (t  qr )  ), then e(q0, q) appear as Fourier coefficients in the expansion
q
of the external wave into plane waves:
E  E 0 exp  iq 0 r   E 0  eq 0 , q  exp  iqr 
q
(3-8)
and turn out to be simply Kronecker’s deltas: e(q0, q) = q0,q. The choice of the normal
modes is dictated by geometry of the experiment. In our case cylindrical harmonics are
appropriate. The form of coupling coefficients e(q0, q) for cylindrical geometry is to be
discussed in section 4.4.
Returning to Eq.(3-7) one can define 2D plasma susceptibility :
Vqe
E0
 i eq 0 , q 
ne 2
  , q  
2m 0
q0
  , q 
q
(3-9)
1
 2   p2 q  
i

The imaginary unit in front of the formula is due to AC nature of coupling (no response
to DC voltage). The susceptibility has a pole, whose real part p(q) corresponds to 2D
plasma resonance frequency at wavevector q. Since, as discussed before, the
electrostatics of 2D plasma varies with the boundary conditions, the dispersion law will
reflect these changes. The 2D plasma dispersion law in different limits is shown in
Table 3-1:
Table 3-1 2D Plasma Dispersion Law in Various Limits.
Unscreened
(short wave)
case: qd >>1
p 
ne 2
q
2m 0
(3-10)
37
p 
Screened
(long wave)
case: q d<<1
Intermediate
case, as in
Figure 3-1.
n d eff e 2
 0m
q
(3-11)
where d eff 
p 
d
d

h
h
n e2
q
 0 m  d coth( q d )   h coth( q h)
(3-12)
If the wavelength is much longer than the distance to the nearest metallic electrode, as is
the case in this experiment, the dispersion will be linear for small q (Dahl and Sham
1977).
2D plasmons in 2DES on helium in its liquid phase were first observed by (Grimes and
Adams) as a sequence of standing wave resonances excited in a rectangular electron pool
using a technique similar to ours.
It follows from Eq.(3-9) that relaxation time  determines the quality factor of 2D plasma
resonances: in the spectrum the observed 2D plasma resonances will appear to have finite
linewidth equal to 1/.  reflects various scattering processes (see 3.3) that electrons
undergo in their motion. Therefore the scattering time of the dominant scattering
mechanisms can be extracted by measuring the width of resonance lines in the spectrum.
One should bear in mind however that there is also an instrumental contribution to the
linewidth due to finite coupling between 2DES and the excitation structure (Chapter 4).
The formulas just derived give the normal frequencies of the 2D plasma. When 2D
plasma is bound laterally only certain wavevectors q are allowed. We consider relevant
case of 2D electron system having circular boundary as is realized in our experiment. The
normal modes are best expressed via cylindrical harmonics (Eq.(3-4)). In the absence of
external field the normal mode should be a standing wave therefore one can assume the
38
radial component of electron velocity to vanish at the edge of the 2DES pool (further
discussion of 2DES size and lateral confinement is deferred to 6.5.2). In this case one
obtains a set of drum modes. As follows from Eq.(3-5) after setting E = 0, this condition
V e
is equivalent to
r
 0 . Substitution of Eq.(3-4) leads to a condition on the
r R
allowed q : J (q ,  R)  0 where R is the radius of 2DES pool. Therefore the allowed
wavevectors are:
q ,  

(3-13)
R
where  are the extrema of Bessel functions:
Table 3-2 Bessel Extrema.

1.84 3.05 3.83 4.20 5.32 5.33 6.42 6.71

1
2
0
3
4
1
5
2

1
1
1
1
1
2
5
2
Some general remarks follow.
The chosen shape of Ve(r) and the equation of motion suggest that

The electrons move in the direction of propagation given by q – the oscillations
are longitudinal

V e
The electric field vector ( 
) is in the same direction
r
One then expects that in order to excite 2D plasmons the external field should be
longitudinal. However since divE = 0 the plane electromagnetic waves are transverse.
39
The use of specially shaped transmission lines makes it possible to synthesize
longitudinal waves. Chapter 4 contains a discussion of the degree of coupling between
the 2DES modes and the external electromagnetic field necessary to excite these modes.
3.2 Electron Solid Modes
The mode previously considered is longitudinal and is the only mode present in 2DES in
its liquid state. Upon solidification a new transverse mode appears – the so-called shear
mode. This mode can be used to detect the Wigner solid (Deville, Valdes et al. 1984). A
detailed study of the electron solid spectrum is found in (Bonsall and Maradudin 1977).
Here we only quote the long wavelength limit dispersion law for a hexagonal lattice:
12 q    p2 q   5t2 q    p2 q 
0.138e 2
 q    q   c q , c 
n
m 0
2
2
2
t
2
t
2
2
t
(3-14)
where  p q  is the 2D plasma dispersion given by the formulas in section 3.1. In the
long wave limit (q->0) the term quadratic in q in the first formula can be dropped and
1(q) coincides with 2D plasma dispersion law. It will be shown later that the excitation
of the transverse mode in our experiment requires the presence of a strong magnetic field
perpendicular to the plane of 2DES (the formula above is for B = 0).
3.3 2DES Mobility
Three main processes limit the mobility of 2D electrons on liquid helium:
1. Electron scattering off helium vapor atoms.
2. Electron scattering off surface waves of liquid – ripplon scattering.
40
3. For thin helium films there is also scattering from the surface roughness of the
underlying substrate.
While the first mechanism dominates for temperatures higher than 0.8 K, low
temperature mobility is determined by ripplons. Table 3-3 summarizes various theoretical
and experimental data on mobility of 2DES and specifies the phenomenon (right column)
that each set of data is meant to elucidate. In Figure 3-5 and Figure 3-6 one can see
crossover between the regimes where scattering from vapor dominates (high
temperatures) to the ripplon limited mobility at low temperature. The surface density is
such that at all temperatures electrons remain in the liquid phase. The vapor scattering
time is:
 gas 
8mb
3 AHe n gas
(3-15)
where b is the z-extent of 2D electron wavefunction given by Eq.(1-3), AHe – is the He
atom cross-section, ngas – is the helium vapor density (Saitoh 1977). The expression
specific for mobility limited by the scattering from 4He gas is:
 gas
2

b
 7.17  cm
 1300 T 2 exp 

a0
 T  Vs
3
The ripplon scattering rate is:
16 Eb
1  2 2 2eT E 
3T 2

e E 
ln
 2
 r 4  
a0
T exp 3 2a 0

1
2

16 Eb
1    2 115  

   

 ln
36  
 T exp 3 6   3

(3-16)
41
Table 3-3 Mobility Measurements and Theory.
Low freq.
transport
Polaron transition
(Andrei 1984)
Figure 3-3 Polaronic Transition.
Low freq.
Transport
Wigner transition
(Mehrotra,
Guenin et al.
1982)
Figure 3-4 Wigner Transition.
42
2D plasma
resonances
(Grimes and
Adams)
Low freq.
Transport in 2D
liquid
(Sommer and
Tanner 1971)
Figure 3-5 2D Electron Liquid Mobility for a Range of Pressing Fields.
Corbino
geometry
(Iye 1980)
2D plasma
resonances
(Grimes and
Adams)
Microwave
cavity
(Rybalko and
Kovdrya Yu)
Figure 3-6 Various Methods of Mobility Measurement.
Boltzmann
equation
(Saitoh 1977)
43
where T is in units of energy and Eb 
2
.
2 mb 2
In the regime of strong holding field (E > Ec see Eq.(1-4)) the scattering time becomes
temperature independent (Shikin and Monarkha Yu):
r 
8 
e E 2
where  is the helium surface tension. In the limit of strong holding field the DC
scattering time is twice as high as the one measured in RF or microwave experiment:
dc = 2ac (Platzman and Beni 1976). For localized electrons, one ripplon scattering
becomes inefficient. This is because a ripplon is slow (section 1.5.1) and therefore cannot
supply enough energy for electron to make a transition between its localized states while
conserving momentum. 2-ripplon scattering mechanism dominates then with a scattering
time given by (Dykman 1978):
 2r
 He k B T 2

4b 4
 

  ml 0



3
Here l0 is the electron’s localization length.
The total scattering time due to all independent mechanisms is:
1


1
 gas

1
r

1
 2r
3.4 Coupled Phonon-Ripplon Modes
The coupling between the Wigner crystal phonons and the He surface ripplons leads to
different spectra for the 2DES liquid and solid phases because the coupling is only
efficient for solid due to the correlated electron motion. It is this difference that allows
44
investigation of melting by measuring spectra. The theory of these coupled phononripplon (CPR) modes (Fisher, Halperin et al.) centers on the coupling that involves the
longitudinal phonon, which is also a focus of our experiment. However a similar theory
may be developed describing coupling between transverse phonon and ripplon modes.
The theory divides electron’s motion into slow component, which is treated
perturbatively, and a fast one, which is averaged over temperature fluctuations and
represented by the Debye-Waller factor, which describes the effect of smearing of
electron wavefunction:
W1 
T
2
t
3 mc
ln(
G1
)
qc
(3-17)
In the above equation qc is a low frequency cutoff for fast component, and is essentially a
fitting parameter of the theory. ct (Eq.(3-14)) enters this equation because the transverse
phonons are the lowest lying excitations. CPR frequencies are then obtained by solving
the following secular equations:
 2  l2 
1
2
2
V
 G 2   2  0
2 G
G
(3-18)
where G - are ripplon frequencies corresponding to reciprocal vectors of Wigner lattice
Gp (Eq.(1-6)) for a hexagonal lattice, l – are longitudinal phonon frequencies in the
absence of coupling and
VG  eE
n
exp(  pW1 )
m
are coupling constants. Here E  is the pressing field.
(3-19)
45
Figure 3-7 is the reduced Brillouin zone plot, that illustrates the hybridization between
the longitudinal mode of 2D electron crystal and He surface modes, that leads to
Figure 3-7 Coupled (solid) and uncoupled (dashed) modes of Wigner solid and surface
waves of He (from (Fisher, Halperin et al.)).
formation of CPR. Horizontal dashed lines are ripplon frequencies corresponding to
Wigner crystal’s reciprocal lattice vectors. Solid lines represent resulting CPR modes.
46
Wavevectors are determined by the geometry of the cell (cf section 3.1) and are much
smaller than WC reciprocal vectors. The uppermost optical branch is described by
 2   l2   02
(3-20)
which is the solution of Eq.(3-18) when  >> G. It is associated with a formation of a
“dimple” depression of the liquid He surface caused by the electron pressure.
0 
1
VG2 is the frequency of electron’s oscillation in the dimple. We attribute the

2 G
resonances of 2D electron solid observed in our experiment to this branch of the
spectrum, which will be referred to as optical plasmon.
3.5 Spectrum of 2DES in Magnetic Field
First we derive dispersion law for unbound 2D plasma placed in magnetic field which is
normal to the 2DES plane. The equation of motion for 2D electron in magnetic filed
(without external electric field) is:
e V e
  u 
u
  c u  zˆ

m r
1
where c is the cyclotron frequency:
c 
eB
MHz
 17.6
B[G ]
m
G
(3-21)
and ẑ is the unit vector normal to the plane. After performing spatial Fourier
transformation (Ve ~ exp(-iqr)) we write down the equation of motion in terms of
velocity components parallel and perpendicular to wavevector q:
47
1
e
u||  u||   i qVqe   c u 

m
1
u  u    c u||
(3-22)

Eq.(3-6) becomes: qe = -inequ||. Substituting it into Eq.(3-3) and then into Eq.(3-22), one
gets the following secular equation determining magnetoplasmon dispersion law:
 2   p2 q   i


 i c
i c
2 i


0
(3-23)
where p(q) is the 2D plasmons frequency in zero field (Table 3-1)). When dissipation is
negligible, Eq.(3-23) becomes:
2
q   p2 q  c2
mp
(3-24)
An interesting consequence of Eq.(3-23) is the linewidth doubling occurring as one goes
from the limit of weak magnetic field (c << p) to that of strong field – cyclotron
resonance (c >> p). For weak field the linewidth (defined as the width of resonance at
half height) is 1/, whereas for strong field it is 2/ as might be easily verified by solving
Eq.(3-23) in these limits.
When 2D plasma with cylindrical boundary conditions as described in section 3.1 is
placed in a magnetic field the dispersion equation and boundary condition become
interdependent:
 2   p2 (q )   c2
q R J (q R )  
c
J (q R)

Solutions of Eq.(3-25)   ,  (q  ,  ) determine the spectrum of bound 2DES.
(3-25)
48
It is obvious that unlike the zero field case the modes differing only by signs of azimuthal
number  will no longer be degenerate – magnetic field removes inversion symmetry.
The radial modes – the ones with  = 0, will have wavevectors defined by the same
Eq.(3-13) as without magnetic field and their frequencies will evolve as:
 2   p2 ( B0 )  c2
(3-26)
As shown in Figure 3-8 all other modes would split in magnetic field although most of
Figure 3-8 Evolution of magnetoplasmons (from (Glattli, Andrei et al.
1985)).
them approach c asymptotically with increasing field except for those with  =1,<0.
49
The latter have imaginary radial wavevector q so they are attenuated away from the edge
of 2DES and therefore called perimeter modes or edge magnetoplasmons.
Their frequencies and wavevectors behave asymptotically as:
  ,1  
q 
c
cp
R
(3-27)
cp
cp – is long wavelength plasmons velocity given by Eq.(1-9).
Next chapter discusses the techniques used to excite and measure 2DES resonances.
50
4 Measurement of 2DES Spectrum Using Slow-Wave
Structure
The normal modes of 2DES is the subject of Chapter 3. Here I discuss the experimental
methods for exciting and detecting these modes. First the general ideas concerning this
type of measurement are discussed. Next the coupling between external electro-magnetic
field and 2DES is considered.
4.1 2DES in Electro-Magnetic Field
Unlike 2D electrons in semiconductor heterostructures the properties of 2D electrons on
liquid helium can not be studied with DC fields (for a clever way to set up a DC
measurement see however (Klier, Doicescu et al.)). Therefore 2DES is usually coupled to
a structure that generates AC field such as a planar capacitor (Sommer and Tanner 1971),
transmission line (Grimes and Adams), microwave cavity (Kovdrya Yu and Buntar;
Mistura, Gunzler et al. 1997), optical (Lambert and Richards) or resonance circuit
(Kovdrya Yu and Buntar).
The method used in this work is excitation of resonances using a transmission line. The
analysis is as follows: the 2DES is thought of as a resonator with its set of normal modes
(Chapter 3). The external transmission line is weakly coupled to 2DES. A wave of a
constant power is sent to the input of the transmission line and the output power is
measured as a function of frequency, which is slowly swept through the range of interest.
The peaks in the derivative transmission spectrum are attributed to the absorption by the
2DES resonances.
51
It follows from a general theory of resonators (Pozar 1997) that coupling between the
resonator (2DES in this case) and the excitation line has to have the right value – so
called critical coupling. A very weak coupling would result in a small signal to noise ratio
whereas overcoupling would lead to spurious broadening and shift of the resonance lines.
However due to the need to broadly vary the helium film thickness in this experiment the
case of an arbitrary coupling has to be incorporated and will be considered in 4.2.
Part of the following discussion follows (Valdes 1982). In section 3.1 we derived the
dispersion law and susceptibility  of a 2D plasma. The response of the 2DES to an
applied field is given by Eq.(3-9). Based on this formula we can express the influence of
2DES on the wave propagating in excitation line as follows:
Vout  Vin Tline 1  i   
(4-1)
where Vin, Vout are voltages at the input and output of the transmission line respectively.
Here Tline is the line’s complex transmission coefficient and  includes the coupling
between 2DES and the line as well as some other factors. The assumption that coupling
is weak, i.e.  << 1 allows us to expand this formula:
Vout  Vin Tline e i
Here =Arg(),
1   "2   2  '2
 Vin Tline e i 1    "
’ = Re() ” = Im(). The measurement is made in such a way
(homodyne detection with phase matching delay line – see 6.8) that the phase information
( + Arg(Tline)) is dropped from the measured signal Vs:
Vs  Vin Tline 1    "
The above signal is the sum of apparatus transmission spectrum in the absence of 2DES
and a weak absorption due to 2DES. In order to eliminate this large background
52
phase-sensitive detector PSD is used (see Figure 6-17 and section 6.8). In essence, in
order to achieve phase sensitive detection a low frequency modulation is imposed onto :
 (t )   0   cos r t
The signal after PSD is:
Vs  Vin Tline  "
This signal may still include some spurious information arising from the transmission
coefficient of the line without 2DES. This coefficient contains information about the
losses in the line and reflections. The reflections are caused by variations of characteristic
impedance of the line - impedance mismatches. For instance if at a certain location in the
line the impedance abruptly changes its value from Z1 to Z2 the propagating wave will
experience a reflection at that point with the voltage reflection coefficient given by:
r
Z1  Z 2
Z1  Z 2
However since ”() is expected to have (Chapter 3) the shape of a sequence of narrow
resonance peaks the identification of resonances is not obscured by Tline as long as the
latter varies slowly enough with frequency. To quantify this statement lets consider how
Tline depends on the frequency for a simplified transmission line with only two impedance
mismatches having voltage transmission coefficients t1, t2 respectively and attenuation
constant . The reflection coefficients of these discontinuities (from the inside of the
region between discontinuities) are r1 = 1 - t1, r2 = t1 - 1 (these relations follow from the
boundary conditions for voltage and current).
53
l
t1
t2
Vin
Vout
Figure 4-1 Reflection between Two Discontinuities in Transmission Line.
To find the total transmission coefficient one sums all partial waves reflected from
discontinuities:
T
Vout
t t e iql
 t1t 2 e iql  r1 r2 t1t 2 e 3iql  (r1 r2 ) 2 t1t 2 e 5iql    1 2 2iql
Vin
1  r1 r2 e
where q = k+i is the wavevector. For simplicity r’s are assumed to be <<1. Then
T  1  r1 r2 e l e 2ikl
The plot of transmission coefficient vs. frequency looks like a series of peaks whose
width is:
 line 
v
(1  r1 r2 e l )
l
(4-2)
where v is the propagation velocity in the line. As discussed above the shape of Tline will
not hinder observation of 2DES resonances if the width of the latter is smaller than line:
2DES  line
The qualitative generalization of Eq.(4-2) for the case of arbitrary number impedance
mismatches leads to the final criterion for mismatch tolerance:
 2 DES 
v
(1  r1 r2 e l1  r2 r3 e l2  ), L  l1  l1   total length of the line
L
This implies the following:
54

Reflections have to be sufficiently small – impedance of the line should be well
matched.

The line should be as short as possible.

The attenuation between impedance mismatches should be large. Therefore the
attenuators should be inserted into the line in many places.
In the experiment the direct spectrum (without PSD) is first obtained in order to estimate
the quality of the transmission line.
4.2 Distributed Circuit Model
The task of measuring the 2DES spectrum leads to a natural question: how can the
technique distinguish the response of ~108 surface electrons from that of ~1024 electrons
contained in the metal electrodes present nearby? The key to answering this question is in
the qualitative difference of 2DES electro-magnetic properties as contrasted to that of 3D
objects. To illustrate this difference we compare electro-magnetic responses for
apparently similar cases of 2DES and that of a thin sheet of metal in Table 4-1.
Table 4-1 Electromagnetic Response Comparison.
Dispersion equation
What determines
the allowed wavevectors
Frequency shift when placed
in a microwave cavity
2DES
3D plasmon
n e2
2 D 
q
0 m
Boundary
conditions
n e2
3D 
0 m
Boundary
conditions
Up
Down
Thin sheet of metal
(guiding action)
c
k

Boundary conditions
Down
In this experiment the field was created by a TEM transmission line (section 4.3) –
meander line or IDC - above which 2DES was situated. Here I discuss the
55
electromagnetic coupling between a generic transmission line and 2DES using distributed
circuit theory. In this approach both 2DES and excitation line are treated as transmission
lines having a uniformly distributed capacitance, inductance and resistance. It is
complementary to the analysis in 4.1: for example, it allows calculating the coefficient 
in Eq.(4-1). A similar analysis (applied to the arrangement of electrodes known as
Sommer-Tanner techniques (Sommer and Tanner 1971)) is found in (Mehrotra and Dahm
1987). Notice that unlike the case of a metal sheet the coupling between 2DES and the
excitation line is predominantly capacitive – there is no mutual inductance. This is
because the current density in 2DES is related to surface charge density as j = e v, where
v is the velocity of electrons. The electric field E created by electrons is E ~ e, whereas
the magnetic field B is B ~ j/c. Therefore B ~ (v/c)E and, since typical electron velocities
are much smaller than speed of light, the magnetic field is negligible compared to the
electric one. For the same reason one can neglect the magnetic self-inductance of 2DES.
However there is a self-inductance due to inertia of electrons. The equivalent circuit for a
meander line is shown in Figure 4-2. The following values should be assigned to the
distributed parameters of the circuit:
m
;
ne 2 w
1
Re* 
;
ne w
 eff w
C le* 
;
dl
L*e 
C e* 
 eff 1 w
d gr
(4-3)
56
2DES
L*e
C*e
R*e
C*le
L*l
1
0 Z=50
0
0
Capacitive Coupling
C*l
0
Excitation Line
0
Figure 4-2 Equivalent Circuit of 2DES and Excitation Line. Star denotes the value per unit length.
where m is effective mass of electron; e - electron’s charge; – electron’s mobility; w –
“width” of 2DES pool; n – 2DES surface density; dl – the distance between 2DES plane
and excitation line; dgr – the distance between 2DES plane and ground; eff, eff1 –
effective dielectric constants. Usual arrangement of dielectric layers is: dielectric wafer
such as Si supporting a film of liquid helium-4.
A single transmission line (Pozar 1997) is described in terms of its dispersion
law q   i Z *  Y *   and characteristic impedance Z c 
Z *  
where Z*() and
*
Y  
Y*() are longitudinal impedance and transverse admittance per unit length of the line
respectively. For example for 2DES


q  i iL*e  Re* iC e*  i   2
Zc 
iL*e  Re*

iC e*
m d gr
ne w  eff 1
2
2
i
m eff 1
2
ne d gr
 i
 eff 1
;
n e  d gr
d gr
 n e  w 2  eff 1
;
If 2DES losses are small ( is large) the dispersion law becomes:
57
q 
m eff 1
ne 2 d gr
(4-4)
which is the same as 2D plasmons dispersion Eq.(1-9). This expected result confirms the
validity of modeling 2DES with a distributed circuit.
In the case of coupled transmission lines Z* and Y* are matrices whose rank is equal to the
number of lines, two in our case. Transmission line equations for voltages and currents
are as follows:
 Z l*
d Vl 



dx Ve 
0
0  Il 
 
Z e*   I e 
Yl*  Yle*
 Yle*  Vl 
d Il 



 
*
dx  I e 
Ye*  Yle*  Ve 
  Yle
where Z l*,e  Rl*,e  iL*l ,e
(4-5)
Yl*,e  iCl*,e Yle*  iCle* are longitudinal impedances and
transverse admittances per unit length for the excitation line (subscript l) and 2DES
(subscript e). Details of the theory of coupled lines can be found in (Faria 1993). By
diagonalizing the system of Eq.(4-5) one finds the dispersion laws for the normal modes
of excitation line + 2DES system:
q12, 2 
ql2  q e2  (ql2  q e2 ) 2  4qle4
(4-6)
2
where ql, qe are the dispersion laws of uncoupled lines and
qle2  Yle* Z l* Z e*
(4-7)
As is explained below it is always preferable to match the lines so that ql = qe = q.
Therefore only this case will be considered here. Then Eq.(4-6) is simplified to:
q12, 2  q 2  qle2
58
When boundary conditions are imposed on one of the transmission lines (in our case this
is the condition of having zero current on the edge of 2DES disk whose size is R) there
will be a resonance whenever either q1 or q2 are multiple of /R thus giving rise to two
peaks per each resonance of uncoupled 2DES. The splitting is therefore:



C le*
C l*C e*
The film thickness is the parameter that controls the mutual capacitance between 2DES
and the excitation line and therefore the splitting that corresponds to coupling between
2DES and excitation line. Thus the splitting can be used to measure the helium film
thickness.
4.3 Slow-Wave Structures
As shown in section 1.4 the plasmons have phase velocities that are several orders of
magnitude lower (depending on the surface density) than the speed of light. However in
order to be observable 2DES has to be critically coupled to the excitation line as was
explained in 4.1. This implies that the propagation velocity of the line should be as close
as possible to that of the 2D plasma. It is generally the case that in order to have the most
efficient energy transfer between two objects their dispersions must be matched.
A simple example is the collision between two particles: the largest energy transfer
happens if their masses are equal.
To match an excitation line to slow plasmons one can use a transmission line with a
special geometry that results in “slow” effective propagation of the electromagnetic
wave. Two such structures used in this experiment are the meander line and the
59
interdigital capacitor. These structures are shown in Table 4-2. The clear areas
correspond to non-metalized surface. The basic idea of these structures is to create a
convoluted path of the traveling wave so as to effectively slow down the electromagnetic
wave. It follows from this argument that the effective propagation speed of the
electromagnetic wave is reduced with respect to its speed in vacuum by the aspect ratio
of the structure. The aspect ratio of the structure is defined as the ratio of shortest
distance between terminals to total length of the line, which is
a
1
in terms of
L  a  eff
the line parameters.
A rigorous treatment of these structures is given in (Weiss 1974; Crampagne and
Ahmadpanah 1977). It makes use of the symmetries of the line while solving the
Helmholtz equation for the wavefunction :
2  k 2  0 , k 2 
2
c2
 eff
(4-8)
Since the media above the line (silicon in this experiment) and below (sapphire) have
different permittivities, the wave propagates in an inhomogeneous environment
(see Figure 4-6). Strictly speaking, no TEM (transverse electromagnetic) wave can
propagate in such a layered medium. However a quasi-TEM solution is a good
approximation for low frequencies. According to transmission line theory, the wave
impedance is given by
v
c
 eff
1 1
where C  is the capacitance per unit length of the line and

vC
– the propagation velocity.
60
a
L
y
x
Effective propagation velocity
a
1
vc
L  a  eff
Figure 4-3 Meander Line.
Reduces crosstalk between fingers. More
sensitive to 2DES located above than
non-coplanar meander line.
Figure 4-4 Coplanar Meander Line.
The dispersion law has a gap – no
propagation at low frequencies.
Figure 4-5 Interdigital Capacitor (IDC).
61
Table 4-2 Slow-Wave Structures.
Top Plate
Vacuum
p
s
w
H
d
er
Slow -wave
Structure
Bottom Plat e
Figure 4-6 Cross-section of a Slow-Wave Structure in Inhomogeneous Environment.
The effective dielectric constant  eff can be calculated as  eff 
C*
where C1* is the
*
C1
capacitance per unit length with all dielectric media replaced by vacuum. A solution of
Eq.(4-8) is sought as a combination of TEM waves propagating along the strips:
( x, y, z )  U ( x, z ) e  iky which after substitution in the above equation yields a Laplace
equation in the plane perpendicular to the strips of the line:
 2U  2U

0
2x 2z
The choice of x,y coordinates is indicated in Figure 4-3, z – is perpendicular to the plane
of the meander line. Solving this equation subject to the boundary conditions in the xz
plane for the distribution of voltages and charges allows one to calculate the capacitances
necessary for determining  eff . So far, the solution is the same for both meander line and
the IDC. The last step is to apply the boundary conditions at the ends of the fingers
appropriate for each structure. For example for IDC, these conditions are: zero current at
one end of a finger, and a voltage V at the other end, with V being the same for the fingers
belonging to the same side of the IDC. Below is a short discussion of the results for the
meander line.
62
The choice of normal modes for the solution is dictated by the fact that the meander line
is periodic with the unit cell comprising two lines:
Since the unit cell also has
inversion symmetry the solution is chosen as linear combinations of odd and even Bloch
waves:
U ( x, z )   A j e
iq j x
e
q j z
j
where qj for low frequencies are given by:
qj 
 La
c
a
 eff  j

a
with j  0,1, 
(4-9)
The absolute value of x and z wavevector components are equal because U(x,z) has to
satisfy Laplace equation. 2DES is normally located high above meander line so that
Figure 4-7 Meander Line Dispersion Law from (Crampagne and Ahmadpanah 1977).
= pq2j, p is the period of the line, other parameters are defined in Figure 4-6.
63
z2DES >> d/r referring to the cell geometry shown in Figure 4-6. In this case we can assume
zero voltage value in the gaps between meander line fingers. The potential will then be
U  x, z   U 0

w
a sinh q j H  d  z  e iq j x above the meander line (z > 0). The x
j
sinh q j H  d 
sin j

j  
component of the electric field is:
E x x, z   iU 0
q
j  
w
a sinh q j H  d  z  e iq j x
j
sinh q j H  d 
sin j

j
(4-10)
The full dispersion law for meander line is shown in Figure 4-7. The plot corresponds to
the reduced zone scheme. The first term in Eq.(4-9) (j = 0) corresponds to a plane wave
with a large period (slow wave) unattenuated even at the heights above the line larger
than its period. The rest of the sum is a wave modulated by the period of meander line
and strongly attenuated at the heights larger than the period of the line. 2DES should be
positioned higher above the meander line than its period but lower than the slow wave
wavelength
c
f  eff
La
. In this case the x component of electric field acting on 2D
a
electrons is determined by j = 0 term in Eq.(4-10):
E x  iq 0U 0
w sinh q0 H  d  z 2 DES  iq0 x
e
a
sinh q0 H  d 
(4-11)
The effective voltage is U0 = (Z0P0)0.5 where Z0 is the impedance of the slow-wave
structure and P0 is the incident RF power.
The speed of the line can now be adjusted to match that of 2DES by choosing the
appropriate aspect ration of the structure. However since 2DES wave velocity depends on
64
its surface density the matching is achieved for just this value of density. Further
discussion of matching will be provided in section 4.4.
Another choice of structure is a coplanar meander line depicted in Figure 4-4. Such line
has both the ground and the meander in the same plane. The advantage of this
configuration is increased sensitivity to a charge above the line – 2DES in particular.
In the case of the IDC structure the low-frequency dispersion has a low frequency gap as
shown in Figure 4-8. This is not surprising since IDC is a capacitor so DC current cannot
Figure 4-8 IDC Dispersion Law from (Crampagne and Ahmadpanah 1977). See Figure 4-7 for
more details.
flow through it.
65
4.4 Coupling between meander line and 2DES
The coupling strength e(q, qp) between a plane electromagnetic wave and the 2DES
appeared in Eq.(3-8). Here the formula specific to the coupling between meander line and
circular 2DES is discussed. Coupling between 2DES modes (cylindrical waves –
Eq.(3-4))) and meander line modes (plane waves) is given by their dot product:
e ,  
1
 R2
   e
i q r cos
,
dS
(4-12)
Here  – is the eigenfunction of 2DES with cylindrical boundary conditions, R – is the
radius of 2DES pool, q – is the wavevector of an external plane wave, r, – cylindrical
coordinates.
The evaluation of this integral (Glattli 1986) produces the following result:
e ,  
2i
q R J' (qR )
 2 ' ,   (qR) 2
2
1
 , 
'
When the line is matched to 2DES Re q 
(4-13)
2

R
(Eq.(3-13)) and the denominator of
Eq.(4-13) is kept from diverging at this value of q by the losses in the line. It is these
losses that permit coupling if the line is not perfectly matched to 2DES as might take
place in measurements with broad range of surface densities. Therefore there is always a
window of frequencies where there is non-zero coupling between the meander line and
2DES.
66
5 Other Methods for Investigation of Phase Transition
in 2DES on Liquid Helium Film
Our primary experimental method for studying 2DES is the use of the normal modes,
which are excited via coupling to a transmission line. This chapter will provide a brief
overview of other methods.
5.1 Microwave Cavity Technique
In this method, 2DES is placed inside a microwave cavity and the changes in resonance
frequency  and quality factor Q caused by the presence of electrons are measured.
These changes are related to the real and imaginary parts of 2DES susceptibility
(Eq.(3-9)) as follows:
2
1    E dS
 
Q   E 2 dV
2

   E dS


2  E 2 dV
(5-1)
In these formulas E – is the electric field in the cavity. The integration is performed over
the 2DES plane in the numerator and over the entire volume of the cavity in the
denominator. The resonator is usually excited in such a way that only one resonant mode
is present. The level of liquid helium can also be measured by the shift of the resonance
frequency as the resonator is filled prior to the deposition of electrons.
67
The natural frequency of the cavity is usually much higher than the frequencies of 2D
plasma normal modes: >> p. Under this condition the real and imaginary parts of
2DES susceptibility (Eq.(3-9)) are:
 
ne 2
 2
2m 0 1   2 2
ne 2

  
2m 0 1   2 2
(5-2)
Measuring Q and  one can deduce the scattering time from Eqs.(5-1)(5-2) as
follows:


2

   2  1
Q
The mobility is then calculated as: = e/m.
The Konstanz group (Mistura, Gunzler et al. 1997) used this technique with a 10GHz
cylindrical cavity excited in TM010 mode. The helium film was adsorbed on a conducting
silicon platelet coated with various polymers such as PMMA and PFPE. The thickness of
the latter ranged from a few m to 10nm. The helium film thickness was between 300Å
and 800Å (uncharged). The surface density was swept between 0 and 1010cm-2 by
continuously charging at a constant temperature (~1K) while recording the real and
imaginary part of the susceptibility thus producing vs. n curves. The authors observed a
kink in these curves occurring at a certain density (around 4 109cm-2), which they
attributed to the pinning of 2DES by substrate imperfections associated with Wigner
crystallization. This density was measured for several temperatures and the result
compared against theoretical phase diagram (Saitoh 1989). The plasma parameter at
melting was m = 117 compared to the expected value of 131.
68
This technique has the advantage of probing the electron motion over very small
distances in 2D plane: eE/(m2)~10Å, for typical driving field E = 20 V/m. This is
essential for thin helium film where the influence of the substrate roughness is large.
However the measurement is performed at a fixed frequency. One still has an option of
sweeping the surface density, so that 2DES normal modes could be observed this way. In
terms of measuring mobility, this method provides higher accuracy than the spectral
measurement using a transmission line, since the quality of the unloaded resonator can be
made quite high (~10000), so that tiny changes can be detected.
5.2 Sommer-Tanner Method (Low-Frequency Transport)
This technique was pioneered by (Sommer and Tanner 1971). The measurement scheme,
illustrated on the drawing below, consists of three electrodes situated below 2DES. The
generator is connected to the excitation electrode while the amplifier collects the current
from the signal electrode. The intermediate electrode (isolation) serves to eliminate cross
talk between the other two when no electrons are present. A thorough analysis of this
technique is found in (Mehrotra and Dahm 1987). The quantity measured is the phase
shift between Vin and Vout. It is related to the resistance per unit square of 2DES as
7
follows:    Rcl 2 where c is the capacitance between 2DES and the electrodes per
6
unit area, l is the length of each electrode and R is related to 2DES mobility:
R neTherefore by measuring the phase shift one can deduce the mobility.
The Case-Western university group applied this technique to measure the mobility of
2DES on 150Å to 600Å helium film deposited on a glass substrate (Dahm and Jiang).
69
They measured a substrate roughness of 20Å peak-to-peak with a typical wavelength of
3 - 5 m.
2DES
+
Vin
-
Vout
Figure 5-1 Sommer-Tanner Technique.
The excitation frequency varied from 100 Hz to 100 kHz. As the electrons were cooled,
they observed a metal-insulator transition. The dependence of the resistivity on frequency
and driving voltage as well as the noise spectrum were similar to a sliding charge density
wave transition (Gruner and Zettl 1985). The metal-insulator transition was explained by
the pinning of the Wigner crystal, which forms as the temperature reaches the critical
value.
The Sommer-Tanner setup is somewhat similar to that used in measuring the normal
modes with the IDC structure (refer to section 4.3), except that the excitation frequency is
usually far below 2D plasma resonances. The coupling doesn’t have to be weak unlike in
the normal mode measurement.
70
6 Experimental Apparatus and Cell
Figure 6-1 Experimental Cell Mounted on Dilution Fridge.
This chapter describes the experimental setup including the layout of the cell and
electronics.
6.1 Cryostat
The schematic drawing of the cryostat and dilution unit is presented in Figure 6-2.
71
Figure 6-2 Cryostat and Dilution Unit.
72
6.1.1 Vibration Isolation
For experiments with superfluid liquid helium it is important to provide sufficient
isolation from mechanical vibrations which otherwise may cause parametric resonance
(described in appendix C). Because the motion of liquid HeII is undamped due to its
vanishing viscosity, it is easy to excite large amplitude oscillations in helium such as the
surface modes described in section 1.5.1. This would lead to a loss of 2DES deposited on
the surface of liquid. Therefore the cryostat is suspended on a vibration isolation table
consisting of four pneumatic pistons that also serves to align the system. The angular
Ceiling
Wall2
Wall1
Mirrors
Laser
Cryostat
Figure 6-3 Cryostat Alignment System.
alignment and tilt control is performed using a laser beam reflected from a series of
mirrors mounted on the cryostat and on the ceiling as shown in Figure 6-3.
All pumping lines necessary for the operation of the cryostat are connected to it via a
system of cross-bellows (Richardson and Smith 1988) allowing decoupling from
vibrations fed along those lines. One such connection (still pumping line) is shown in
Figure 6-4. Here six threaded rods control the tension of four bellows and prevent them
from collapse. The liquid 4He cryostat has no liquid nitrogen jacket but instead uses
layers of highly reflective material - superinsulationTM to achieve adequate shielding
against thermal radiation (50 mW/cm2 between room temperature and liquid helium),
73
which is the main source of heat leaks into the system. This measure is necessary to avoid
vibration associated with boiling of liquid N2.
Figure 6-4 Cross-Bellows Design Isolates a Pumping Line from the Cryostat.
6.2 Dilution Refrigerator
In order to achieve the low temperatures required by the experiment (1K to 50 mK) a
3
He-4He dilution refrigerator is used. Its operating principles are described in (Pobell
1992). The cooling is achieved by using the enthalpy of mixing liquid 3He and 4He. The
latter decreases with reducing temperature as a power law as opposed to the exponential
decrease for other mechanisms such as evaporation. The mixing is performed in a
continuous cycle (Pobell 1992). The system used in this experiment is Oxford
Instruments Dilution Refrigerator Model 200s. The nominal cooling power at 100mK is
200uW.
74
6.3 Thermometry
1K RuO2 resistance thermometer mounted outside the cell is used to measure the
temperature. Liquid 4He below -point equalizes the cell thermally. This is because the
heat exchange in HeII is mediated by the frictionless flow of superfluid component
(thermal superconductivity). It flows along the gradient of temperature, diluting the
normal component (which carries all the entropy) in hotter places and thus balancing the
temperature. In order to minimize the heat flowing into the cryostat, thermometer leads
are made of superconducting wire (NbTi) between the cell and 1K stage of the fridge and
Cu upwards. They are anchored at several places between room temperature and the
experiment. The resistance is measured with a low-level RV-Elektroniikka AVS-46 AC
resistance bridge using 4-lead technique.
The temperature can be stabilized with a heater on the mixing chamber. It is powered by
the analog output of a data acquisition board and the temperature is maintained using PID
(proportional, integral, derivative) algorithm program.
6.4 Magnet
A superconducting magnet (Oxford Instruments) capable of producing fields up to 0.5 T
is used in experiments involving magnetic field. The magnet has two coils – one for
vertical field, one for horizontal. The split-coil design shown in Figure 6-5 allows access
to the center of the magnet.
75
Figure 6-5 Split-Coil Crossed-Field Magnet.
The magnet is powered by LakeShore Cryotronics MPS-622 magnet power supply.
6.5 Experimental Cell
Figure 6-6 shows the experimental cell.
It is made of stainless steel in order to withstand pressures due to helium gas. The flanges
are ConflatTM. A multipin hermetically sealed electrical connector capable of functioning
at low temperature is used for all connections except the transmission line. The later
connects to hermetically sealed SMA connectors, which are handpicked according to
their performance at low temperatures. These connectors are soldered to the bottom
flange using In-Sn solder.
76
The cell is mounted on gold-plated Cu rods that couple it to the mixing chamber of
Figure 6-6 Experimental Cell.
dilution refrigerator (section 6.2).
6.5.1 Production of Electrons
Electrons are produced by applying ~300V between the tungsten filament and the Cu grid
that covers an opening in the top plate electrode (see also Figure 6-17) thus initiating a
glow discharge. To produce and maintain the discharge He vapor pressure is kept at
0.1 torr, which is achieved by keeping the temperature of the cell at 1K. Pulsing the
filament with ~100 ms long pulses carrying a few tens mJ of energy initiates the glow
77
discharge. By making the pulse duration shorter than the thermal relaxation time of the
filament efficient heating of the filament is achieved with minimal heating of the cell.
Filament leads are superconducting NbTi wires between the cell and 1 K pot and Cu
between 1 K pot and room temperature.
6.5.2 2DES Confinement
To hold electrons on the surface of liquid helium a system of two electrodes (Figure
6-17) is used: top plate and guard ring. The former provides the vertical confinement
field while the latter holds the 2DES laterally. Because the meander line is part of
impedance matched transmission line, which is connected to ground via a 50 Ohm
Top Plate, Vtp
Rgr
h
Re
2DES, Ve
Guard Ring, Vgr
d
Ground Plane, 0
Figure 6-7 Electrostatic Boundary Conditions in the Cell. The cell has a
cylindrical symmetry. Rgr is the guard ring radius; Re is the radius of 2DES
pool.
resistor at the source as well as at the detector, it can be considered as the ground plane
for the purpose of controlling DC fields in the cell. The resulting electrostatic boundary
conditions for the 2DES are schematically shown in Figure 6-7.
Let h denote the distance between 2DES and top plate and d is the effective distance
between the 2DES and the ground plane defined as d 
d He
 He

ds
s
where the subscript s
refers to the dielectric substrate supporting the He film (this can be generalized for an
78
arbitrary number of dielectric layers). The electric fields above and below the 2DES
are E  
Ve  Vtp
h
, E 
 Ve
ne
 E    He E 
respectively. Continuity requires that 
0
 He d
whereas the magnitude of the field in the plane of 2DES is E 
equations 2DES potential is related to the charge density as Ve 
E  E
. From these
2
hd  ne Vtp 

.

h  d   0
h 
Varying the top plate voltage during the charging process allows one to control the 2DES
density: the latter saturates when E+ = 0, Ve = Vtp. We use an infinite plane geometry as a
first order approximation (corrections will be of order of the aspect ratio of the
cell ~ 1/6). The saturated surface density is:
n  Vtp
0
1
e d He

 He
ds
(6-1)
s
The usual procedure for depositing electrons is to start with Vtp = 0 and then keep
increasing it until the desired 2DES density is reached. The rate of increase should not be
too fast in order to prevent electrons from acquiring the energy sufficient to penetrate the
liquid – around 1eV (see Figure 1-2).
The guard ring is normally kept negative with respect to the top plate to prevent 2D
electrons from spreading out. The guard ring voltage can be used to squeeze the 2DES
pool and therefore to change surface density. The amount of change depends on the exact
geometry of the guard ring. For the simplest case of cylindrical shape and the cell half
filled with liquid He the problem of relating 2DES size to the voltages in the cell can be
solved using conformal mapping (Glattli 1986):
79
sinh 2 (
 ( R gr  Re )
2H
)
Ve  V gr
Vtp  2Ve
(6-2)
here H = h + d – is the total height of the cell. It follows from this formula that guard ring
voltage Vgr has to be changed at half the rate of change of top plate voltage in order to
maintain the constant size of 2DES (presuming no loss of electrons while doing this).
6.5.3 Measurement and Control of Liquid Helium Level and Film
Thickness
A controlled amount of helium gas from a container at room temperature is passed
through liquid a N2 trap for filtering and then condensed into the cell via a thin stainless
steel capillary that is thermally anchored at various stages in the fridge. The volume
reduction ratio between He gas at STP and liquid He below 1K is 812.
The filling capillary has to be very narrow (0.1mm ID is used) in order to restrict HeII
film flow effects (fountain effect (Donnelly, Glaberson et al. 1967)), which would
otherwise lead to uncontrolled variations in the helium film thickness in the cell.
A common technique is to place an orifice inside the capillary. It would limit the film
flow velocity on the one hand while not increasing the overall impedance of the capillary.
A capacitive technique is employed to measure the level of liquid helium in the cell.
Thick films are detected by measuring the capacitance between the meander line and the
top plate as displayed in Figure 6-9.
80
The capacitance measured includes both the He film (dielectric permittivity He,
Figure 6-9 Thick Film Measurement.
Figure 6-8 Thin Film Measurement.
thickness dHe) and the dielectric wafer (dielectric permittivity S, thickness dS). It is given
by the following expression:
d He
C  C0
H  dS /  S
H  dS /  S  d
 He  1
 He
 C 0 (1 
 He
( He  1)
H  dS /  S
)  C 0 (1 
d He ( He  1)
)
H
where C0 –total capacitance before filling, H – distance between the top of the wafer and
top plate. In practice one measures both C0 and Cf (full capacitance). Then dHe is
determined from
C  C0 (1 
d He C f  C0
)
H
C0
For measuring the thickness of a thin film (~100 Å) we use a pair of vertical parallel
plates - level capacitor – placed a few millimeters below the wafer surface and partially
81
immersed in helium - Figure 6-8. This capacitor measures the level of bulk helium in the
cell. The film condenses from He vapor coating the rest of the cell. Its thickness is related
to the height h above the surface of bulk liquid:
d 3

(6-3)
 He gh
with  the Van-der-Waals constant of interaction between liquid He and solid surface
(typical value 10-15 erg for glass substrate), g – gravitational acceleration. The filling
curves – values of these capacitances vs. amount of helium condensed, are shown in
level capacitance [pF]
meander line to top plate capacitance [pF]
0.341
0.339
0.137
0.338
Level Capacitance[pF]
0.337
0.136
0.336
0.135
0.335
0.334
0.134
0.333
0.332
0.133
0.331
0.132
0.330
0.329
0.131
0.328
0.327
Meander Line to Top Plate Capacitance[pF]
0.138
0.340
0.130
0.326
-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4
He gas condensed[ltr stp]
Figure 6-10 Filling Curves. The measured capacitors are shown in Figure 6-9
and in Figure 6-8. The measurements are taken while filling the cell with
helium.
Figure 6-10.
We used He = 1.0572 the dielectric constant of liquid helium at 4.2K to extract the film
thickness. The measurement has to be conducted at temperatures T < 1K to avoid the
82
error associated with thermal contraction of liquid helium (its density changes by 15%
between 4.2K and 2K) although this effect will be partially compensated by increase
in He.
Capacitances are measured using General Radio 1615-A AC capacitance bridge. The
typical S/N ratio for such a measurement under the conditions of the experiment is
~1000. There is however a much larger systematic error in determining He thickness due
to capacitor edge effect and stray capacitance. The overall error is estimated to be as high
as 20%.
When the film is charged its thickness might change substantially under electron pressure
so it has to be determined from 2DES spectrum.
6.6 Meander Line
A detailed discussion of the meander line transmission and electromagnetic properties
was given in section 4.3. It is patterned in silver or gold on a polished dielectric wafer
(glass or sapphire, 0.005 - 0.010” thick) using optical lithography (typical width of the
line is about 50 m). The preferred method for performing lithography is lift-off using
nLOF type of photoresist. The total length of meander line might be of order of meters
and a single break will render it useless. Therefore it’s important to start with a clean
(unused) substrate, which should be treated with either UV or O2 plasma both before
spinning the photoresist and also after developing.
Geometrical dimensions of the
meander line are chosen so that the wave impedance is close to 50 at low temperatures.
The ground plane is either coplanar with the line or on the opposite side of the wafer
where the contacts for coaxial connectors are located as well. Electrical connection
83
between both sides of the wafer is made using vias drilled in the wafer or simply painting
conducting epoxy over the edge of the wafer.
Contacts to the meander line are made using commercial spring contacts.
Figure 6-11 Spring Contact. Manufactured by Everett Charles Technologies (Part # MEP-30U).
Figure 6-11 shows their design. All parts are gold-plated. The spring is made of beryllium
copper. This is the only material retaining springiness at low temperatures.
6.7 Substrate Smoothness Control
In order to ensure uniformity of the potential in the plane of 2DES, the upper surface of
the substrate supporting the He film has to be reasonably smooth (the characteristic
roughness scale is not to exceed either the helium film thickness or the capillary
length (1.5.1)). In addition it has to be electrically neutral – i.e. free of trapped charge.
The latter condition is only possible to meet if the substrate surface has no dangling
bonds – potential sources of attraction for external ions. Passivation of Si is a chemical
procedure developed (Higashi, Chabal et al. 1990) to terminate such bonds with
hydrogen. The recipe followed in the present work is given below:

Prepare a (1:10) HF:H2O (5ml:50ml) solution (etch solution). Dip the sample
into for 5 min.

Prepare a (4:1:1) solution of H2O:H2O2:HCl by first mixing the H2O2 and H2O
(25ml:100ml) and heating.
84

Rinse the sample thoroughly with water and transfer to a beaker of water. Add the
HCl (25ml) to the hot H2O2:H2O solution.

Grow oxide (approx. 20) by dipping into (4:1:1) H20:H2O2:HCl solution. Heat for
about 10 min.

Rinse the sample and transfer into a beaker of water. Remove oxide with HF
solution for 5 min.

Rinse the sample and transfer back into the hot H2O:H2O2:HCl solution to grow
oxide for 10 min.

Rinse the sample and transfer into a beaker of water. Prepare a final buffered (7:1)
NH4F:HF (35g:5ml) (pH = 15.0) and transfer the sample to remove oxide.

Prepare a final acid solution (4:1:1) of H2O:H2O2:HCl by first mixing H2O2 and
H2O (25ml:100ml) and heating.

Rinse the sample with water and transfer into a beaker of water. Add the
HCl
(25ml) to the H2O2:H2O solution. Transfer the sample to grow oxide. Heat for 10
min.

Rinse the sample with water. Remove oxide by (4:10) NH4F:H2O solution
(20g:50ml) (pH = 7.8) for 6.5 min.

Rinse the sample in a number of beakers of water to thoroughly wash away any
trace of NH4F.
After passivation, the substrate is mounted in the cell, which is evacuated using
cryopump to avoid contamination with oil.
An AFM scan of Si-(111) surface is shown in Figure 6-12. It shows that the procedure
above failed to produce the required result in terms of smoothness of the surface.
85
We therefore had our Si wafers passivated in a specialized lab (Dept. of Chemistry,
Cornell University, Prof. M. Hines, Dr. S. Garcia). However, we still observed no signal
on thin film of liquid helium. Given the success of some other experiments (Chapter 5)
Figure 6-12 AFM Image of Passivated Si (111) Surface: Triangular pits are
approximately 150Å deep. RMS roughness is 55Å.
performed with much less care about surface preparation we believe that it is the
sensitivity of the chosen method (measurement of RF absorption spectra) that precluded
us from detecting a signal from 2DES on a thin helium film.
86
6.8 Electronics
The measuring setup (shown in Figure 6-17) consists of a homodyne detection circuit,
DC voltage control and phase sensitive detector. RF lines are coaxial outside of the
cryostat and strip lines inside in order to ensure proper thermal anchoring: the usual
problem with coaxial cable is thermal inaccessibility of the center conductor so that heat
from outside propagates along into the cryostat. It is matched to 50. A swept frequency
RF generator (Hewlett-Packard 8341A) supplies the input power. Variable attenuators
(Wavetek, 50 series) control the power entering the cryostat. A broadband (10MHz1GHz) low-noise (1.4 dB noise figure) amplifier (MITEQ AM-4A-000110) amplifies the
output signal. The lengths of both shoulders of the homodyne circuit are carefully
0.3
-5
-1.5x10
0.2
Modulated Signal
0.1
Direct Signal
-5
Direct Signal[V]
-5
0.0
-2.5x10
-0.1
-3.0x10
-0.2
-3.5x10
-5
-5
-0.3
-5
-4.0x10
Modulated Signal[V]
-2.0x10
-0.4
-5
-4.5x10
-0.5
-5
0
200
400
600
800
1000
-5.0x10
1200
Frequency[MHz]
Figure 6-13 Directly Transmitted Signal vs. Modulated One.
matched to avoid phase differences, which is necessary as discussed in section 4.1.
Due to the small amount of power (1 pW for critical coupling condition with 100 nW
incident power) absorbed by 2DES the recovery of the signal (discussed in 4.1) from the
87
background requires using phase sensitive detection (Horowitz and Hill 1989). The
reference signal from the lock-in amplifier (PAR 124A) is applied to the guard ring or top
plate so that 2DES parameters are modulated. The signal at the output of the lock-in is
the derivative of the directly transmitted signal with respect to the modulated parameter.
This allows recovering the 2DES signal even though signal to noise ratio might be
significantly less than unity. Reference signal frequency used is in the range of
100-300Hz.
Figure 6-13 shows comparison between the signal corresponding to detection with and
without lock-in amplifier. 2DES resonances are buried in the background of the directly
transmitted signal.
Section 6.5.2 described the dependence of 2DES parameters on voltages of the top plate
and guard ring. Both electrodes affect the radius of 2DES pool (and electron density
along with it) according to Eq.(6-2). Applying modulation to the radius is equivalent to
0
0.7
0.8
0.9
1.0
1.1
1.2
1.3
f/fp
Figure 6-14 Lock-in Output Corresponding to Modulation of
the Resonance Frequency.
88
modulating the resonance frequency since in accordance with Eqs.(3-10)(3-13) the latter
depends on both the density and 2DES radius. Therefore when reference signal from the
lock-in is applied to the guard ring, one expects to observe the shape of a resonance line
shown in Figure 6-14 at the output. This shape is antisymmetrical with respect to the
0
0.7
0.8
0.9
1.0
1.1
1.2
1.3
f/fp
Figure 6-15 Lock-in Output Corresponding to Modulation of
the Linewidth.
resonance frequency.
If instead of a resonance frequency one modulates the losses (i.e. the linewidth), the line
has a symmetric shape as shown in Figure 6-15. We can achieve this situation by
applying the reference signal to the top plate. However doing this affects both the size of
2DES pool and the pressing field so the line has a hybrid shape shown in Figure 6-16.
The RF frequency sweep rate has to be set slow enough to match the time constant  of
the lock-in amplifier. Specifically to resolve features of the measured spectrum, which
have width f (resonances i.e.) the sweep rate should not exceed f/.
89
The demodulated signal from the lock-in amplifier is fed to a PC data acquisition board
(National Instruments 6070E, 12 bit resolution, 1 Ms/s acquisition rate). The sampling
rate of acquisition should be at least 2/ to avoid aliasing and to provide sufficient
averaging of the signal.
0
0.7
0.8
0.9
1.0
1.1
1.2
1.3
f/fp
Figure 6-16 Lock-in Output Corresponding to Modulation of
both the Resonance Frequency and the Linewidth.
Filament
Top Plate
Guard Ring
RF
Generator
Meander Line
Splitter
RF amplifier
Mixer
Low-pass
50W
Compensating Line
Figure 6-17 Measurement Setup.
Lock-in Detector
90
7 Analysis of Experimental Data
RF SPECTRA OF 2DES ON LIQUID HELIUM
Spectra for range of charging voltages
7
-2
Vt=-10V, n=7.6 10 cm
0.0
7
-2
Vt=-12V, n=9 10 cm
8
-2
Vt=-20V, n=1.5 10 cm
-4
-2.0x10
8
-2
Vt=-30V, n=2.3 10 cm
-4
-4.0x10
8
-2
Vt=-40V, n=3 10 cm
-4
-6.0x10
-4
Signal[V]
-8.0x10
8
-2
Vt=-50V, n=3.8 10 cm
-3
-1.0x10
-3
-1.2x10
8
-2
Vt=-60V, n=4.6 10 cm
-3
-1.4x10
-3
-1.6x10
8
-2
Vt=-70V, n=5.3 10 cm
-3
-1.8x10
0
100
200
300
400
500
600
Frequency[MHz]
Figure 7-1 Variation of the 2DES Spectra with Surface Density: the spectra are shifted vertically
to facilitate comparison. The legend shows charging voltage as well as the value of surface density
derived from it. T=110mK.
91
The series of spectra measured for a range of different surface densities n (but constant
helium film thickness d) is shown in Figure 7-1. Since the only difference between these
spectra is the value of surface density, it is more convenient to plot them using
logarithmic frequency scale, and shift each spectrum by ½ log n to the left. This is done
in Figure 7-2.
-4
2.0x10
0.0
-4
-2.0x10
-4
-4.0x10
-4
-6.0x10
-4
-8.0x10
-3
-1.0x10
-3
-1.2x10
-3
-1.4x10
-3
-1.6x10
-3
-1.8x10
-3
-2.0x10
50
100
150
200
250
Frequency[MHz]
Figure 7-2 The Spectra corresponding to different densities are plotted on log f
scale and shifted by –0.5 log n.
The spectra display the expected characteristic derivative shape, obtained with the
modulated technique as discussed in 6.8. The asymmetric shape of some peaks suggests
that, in addition to modulating the surface density, the attenuation is modulated too. This
effect is neglected however in the forthcoming analysis and the resonance frequency
corresponding to each derivative peak is calculated as the average of negative and
positive peak frequencies as shown in Figure 7-3.
92
fres=(fpos+fneg)/2
fpos
126
135
144
fneg
Frequency[MHz]
Figure 7-3 Determination of Resonance Frequency.
According to the dispersion law summarized in Table 3-1 the frequency of 2D plasma
450
Resonance Frequencies for
Various Charging Voltages
400
350
Frequency[MHz]
300
250
200
150
y=1.81937+0.49336x
20
30
40
50
60
70
80 90
Top Plate Voltage[V]
Figure 7-4 The Evolution of Resonance Frequencies for the Spectra of Figure 7-1 with
Charging Voltage. The slope of lines on this log-log plot is ½, as expected from 2D
plasma dispersion ~ n0.5.
93
normal modes should scale as
n . Verifying this dependence would serve as an
unambiguous evidence that the observed resonances are 2D plasmons. The resonance
frequencies from Figure 7-1 are plotted vs. the charging voltage in Figure 7-4. Each of
the spectra in Figure 7-1 corresponds to saturation charging (section 6.5.1), therefore the
surface density is proportional to the charging voltage according to Eq.(6-1). The linear
fit of the data in Figure 7-4 confirms the
n behavior for all observed resonances.
7.1 Identification of the Individual Resonances
For sufficiently low densities and/or high temperatures the 2DES should be in a liquid
state and therefore its spectrum is that of the bounded 2D plasma described in section 3.1.
The sequence of resonances in such a spectrum is determined by the dispersion law and
the perimeter boundary conditions of the 2DES. The latter have cylindrical symmetry in
our experiment. Therefore it is natural to associate each resonance with Bessel function
order numbers , (section 3.1) In this section we will discuss the procedure for
identifying the resonances. This allows extraction of the 2DES parameters, most
importantly the density, using the 2D plasmon dispersion curve. However, the effective
thickness deff given by Eq.(3-11) cannot be deduced simultaneously because the saturated
charge density achieved for a given charging voltage itself is inversely proportional to deff
as given by Eq.(6-1).
94
The matching between observed and theoretical spectra is demonstrated in Figure 7-5 for
50
100
-5
-5.0x10
(1,1)
150
200
250 300
0.4
T=90mK, P=-44dBm, Vt=-12V, d=.7mm
7
-2
2D Plasma Spectrum, n=9 10 cm
0.2
(0,1)
(3,1)
(0,2)
(2,2)
(4,1)
(1,2) (5,1)(6,1)
Signal[V]
0.0
-4
-1.0x10
-0.2
Coupling[a.u]
(2,1)
-0.4
-4
-1.5x10
50
100
150
200
-0.6
250 300
Frequency[MHz]
Figure 7-5 Matching of an Observed Spectrum (thick line) to 2D Plasma Resonances (thin lines).
Here () are the Azimuthal and Radial Numbers Respectively. The amplitude of each resonance
in theoretical spectrum reflects the coupling of that mode to the excitation structure (see 4.4) as
discussed in the text.
a low surface density spectrum. The theoretical frequencies (solid circles) are calculated
using the full dispersion law (Eq.(3-12)). The wavevectors used (Eq.(3-13)) correspond to
a range of Bessel extrema starting from the lowest. The density is determined from the
charging voltage. As an additional guide, we try to match the relative strength of the
resonances, which is determined by the coupling of a particular normal mode to the
excitation field (see section 4.4). These are represented by the height of theoretical
resonance lines.
95
We note a clear mismatch between theory and experiment. This might arise from the
error in one of the multiplicative factors in the dispersion law – most likely the density or
the film thickness. One may correct for these errors by using a logarithmic frequency
scale and then shifting the two spectra to achieve the best match. In addition, in the long
50
100
-5
-5.0x10
(1,1)
150
200
250 300
0.4
T=90mK, P=-44dBm, Vt=-12V, d=.7mm
7
-2
2D Plasma Spectrum, n=6 10 cm
0.2
(0,1)
(3,1)
(0,2)
(2,2)
(4,1)
(1,2) (5,1)(6,1)
Signal[V]
0.0
-4
-1.0x10
-0.2
Coupling[a.u]
(2,1)
-0.4
-4
-1.5x10
50
100
150
200
-0.6
250 300
Frequency[MHz]
Figure 7-6 Corrected Spectrum Matching.
wave limit, the plasma frequency is ~ n, whereas for short wavelengths it is ~ n .
Therefore the error due to an incorrect determination of the He film thickness or the
effective dielectric constant in the parts of spectrum corresponding to these limits will be
corrected as well. Figure 7-6 shows the result of applying this procedure to the spectra of
Figure 7-5. Here the theoretical lines are in better agreement with the experiment. If we
assume that the error comes solely from the density, the right value of the density turns
96
out to be 30% lower than the one calculated from the charging voltage. This could imply
Resonance Frequencies [MHZ]
100
70 i
0.5
10
1
0.1
1
10
Number of a resonance, i
Figure 7-7 The Sequence of Resonance Frequencies vs. their Number i: asymptotic behavior is i 0.5.
that the helium surface got charged with fewer electrons than what might be expected
from the charging voltage. Indeed the field near the helium surface is reduced by the
guard ring, whose voltage is more negative than that of the top plate as discussed in 6.5.2.
The matching of strength for the first few resonances however is still poor. This is
probably due to a reduced transmission of the whole measuring circuit below 100MHz
(see direct spectrum in Figure 6-13).
We also note that higher order measured resonances appear to follow a more regular
pattern than Bessel extrema. Namely, the interval between successive resonances is
becoming monotonically smaller. In fact instead of Bessel sequence one may suppose
that the observed wavevectors are simply multiples of some base wavevector, so given
97
the  ~ q dispersion relation at large q, the frequency of a resonance behaves as a
square root of a monotonically increasing integer number.
A log-log plot of resonance frequencies vs. their sequence number in Figure 7-7 confirms
this. This observation is probably related to the fact that the overall boundary conditions
on the electron pool have an effective rectangular symmetry rather than the assumed
cylindrical one, possibly arising from the field of the excitation structure (having the
rectangular symmetry).
7.2 Evolution of 2DES Resonances with Temperature and Phase
Diagram
The observed evolution of 2DES spectrum is shown in Figure 7-8. The difference
between solid and liquid spectra is apparently small. This is consistent with the spectrum
evolving from that of plasmon in the liquid phase to optical plasmon in the solid phase as
predicted by CPR theory (section 3.4). However examination of the data in Figure 7-8, as
well as other datasets, reveals no upward frequency shift, that would correspond to the
formation of 0 gap (Figure 3-7) in the spectrum upon solidification.
We suggest two mechanisms that can explain the absence of the shift. Both are of
nonlinear nature and can therefore be studied by varying the driving RF power.
1.1K
880mK
735mK
625mK
547mK
483mK
434mK
392mK
303mK
283mK
0.0
-5
-5.0x10
-4
-1.0x10
262mK
-4
-1.5x10
LIQUID
98
235mK
-4
Signal[V]
-2.0x10
203mK
Tm=174mK
-4
-2.5x10
-4
-3.0x10
158mK
-4
-3.5x10
SOLID
171mK
-4
-4.0x10
90mK
-4
-4.5x10
0
50
100
150
200
250
300
350
400
Frequency[MHz]
Figure 7-8 Evolution of the 2DES Spectrum with Temperature. The spectra are shifted vertically to
facilitate comparison. Tm denotes the expected Wigner solid melting temperature for n = 6 107cm-2.
The incident RF power is -44dBm.
99
2DES Heating
As was mentioned in section 3.3 scattering processes that determine RF absorption by
2DES are quasielastic. Consequently electron energy relaxation is inefficient – it is easy
to overheat the 2DES. Below is a simple estimate of the difference between electron’s
temperature and that of the bath based on a measured value of thermal conductivity 
(Glattli, Andrei et al.) between 2DES and underlying helium. In a steady state the power
absorbed by 2DES (Pabs ~ N  eE2, N – total number of electrons,  – mobility,
E - driving field) is equal to the one flowing from 2DES to helium (Te - T). The value of
 appropriate for our working temperature (100 mK), pressing field (75 V/cm) and
number of electrons (108) is ~ 10 pW/K. Estimated mobility is  ~ 106 cm2/Vs (section
3.3). For a frequency of 100 MHz and incident power of –44dBm (the one used to excite
the spectra in Figure 7-8) Eq.(4-11) gives an estimate of driving electric field of
E ~ 5 10-4 V/cm. With this numbers Te - T is as high as a few hundred millikelvin.
Therefore applying sufficiently high RF power results in melting of 2D crystal despite the
bath temperature being below melting point. This is one possible explanation why solid
and liquid spectra appear similar in Figure 7-8.
Sliding Transition
The other explanation involves a nonlinear effect similar to the depinning of CDW
(Gruner 1988). The authors of CPR theory (Fisher, Halperin et al.), which was outlined in
section 3.4, were first to suggest that when electron’s velocity exceeds phase velocity of a
ripplon
r(G1)
G1 , where G1 is basic reciprocal lattice vector of electron crystal, electron
“leaves the dimple in helium behind” – the coupling is no longer efficient. W.F. Vinen
developed a more detailed model of this sliding transition (Vinen 1999). The experiments
100
(Shirahama and Kono) were conducted using Corbino geometry version of SommerTanner method (section 5.2). These authors used a sliding criterion different from that of
Vinen. According to it, 2D electron crystal would slide when the driving force exceeds
the pinning force due to static depression in helium. However it seems inappropriate to
treat the dimples in liquid surface as a static potential, and therefore we will use velocity
criterion to estimate the possibility of sliding in our experiment. Using the same values as
in the evaluation of heating, we obtain the order of magnitude electron velocity
ve ~  E ~ 500 cm/s. This should be compared with the phase velocity of ripplon whose
wavelength is of order of interelectron spacing (
1
n
): vr ~ 200 cm/s – see appendix A.
Thus sliding is possible for the used driving power, even though the above value of
electron velocity is an overestimate due to the issues of coupling (section 4.4). Upon
sliding electron crystal becomes uncoupled from the helium surface and the measured
spectrum is then due to longitudinal phonon (Eq.(3-14)), which has the same dispersion
as liquid plasmon. An important difference from 2DES heating is that the electron system
remains in a crystal state. It is also worth noting that the corresponding electron
displacement is ~ 100Å which is much less that a lattice constant for the density in
question ~ 1m.
Figure 7-9 shows that 2DES response is indeed nonlinear when the temperature and the
density are such that it is expected to be in a crystal state in equilibrium. The upward shift
for smaller power is consistent with either mechanisms of nonlinearity described above.
This distinction of a crystal state can be used to measure the melting temperature. For
every value of temperature we can compare the spectra corresponding to low and high
101
driving power. If the upward shift for smaller power is observed, 2DES is presumed to be
in a crystal phase while in equilibrium. Otherwise it is in a liquid phase.
-50dBm(melted or uncoupled)
-0.00014
T = 90mK
Tm = 460mK
n = 4.5 108 cm-2
-56dBm (coupled solid)
160
180
200
220
240
260
280
Frequency[MHz]
Figure 7-9 Nonlinearity of 2D Electron Crystal Response.
The phase boundary measured using this technique is shown in Figure 7-10. The
Classical Part of 2DES Phase Diagram
9
1x10
Experimental Points:2DES on 1mm He film
Theretical Curve m=130
Theretical Curve m=160
8
-2
n[cm ]
8x10
8
6x10
8
4x10
8
2x10
0
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60
T[K]
Figure 7-10 Classical Part of 2DES Phase Diagram.
102
measured value of critical plasma parameter m was 160. The discrepancy with
theoretical value is probably caused by the loss of electrons during the measurement,
which would lead to an overestimate in the surface density.
However there remains a question: which of the two nonlinear mechanisms is more
important in each situation? In both cases the coupling between 2DES and helium surface
is weak. It was remarked earlier that sliding, unlike heating, leaves the electron system in
a crystal state. The transport properties of electron crystal (even when it is unpinned) are
different from that of the electron liquid. In the next section I will describes mobility
measurement as a tool for investigation of 2D crystal nonlinear dynamics.
7.3 2DES Mobility Measurement
The momentum relaxation time  can be extracted from the width of the observed
resonances using the formula  
1
2 3 pp
, where pp is the linewidth obtained from the
RF spectra. The linewidth is taken to be the frequency interval between the extrema of
the derivative shape as discussed in 6.8. This formula assumes a Lorentian line shape as
expected from the 2D Drude model. The linewidth used is the value averaged over
several (most pronounced) detected resonances. We do not observe any frequency
dependence of the linewidth within the range of frequencies used in the experiment. The
free electron mass is used in calculating the mobility  = e/m (the validity of doing this
will be verified in section 7.4). The mobility data is presented in Figure 7-11. The solid
lines represent theoretical mobility determined by vapor and ripplon scattering according
to Eqs.(3-15)(3-16) (discussed in section 3.3). The pressing field for each dataset in the
103
main plot has the same value as at the end of charging the surface (saturated value).
However for the data shown in the inset additional pressing field was applied, while
keeping the density constant.
The experimental values of mobility deviate from theoretical ones below melting point
for each set of data. The ones corresponding to lower pressing fields are below their
respective theoretical curve. The opposite situation takes place for the points measured
with a high pressing field. Theoretical and experimental data elsewhere suggest the
following sequence of electron mobilities in liquid, coupled solid and uncoupled solid
-2
E[V/cm]n[cm ]
T = 95mK
n = const
8
8
10
2
Mobility[cm /Vs]
7
434.510
7
86 910
8
143 1.510
8
250 2.710
8
357 3.810
8
500 5.310
9
10
7
10
30
40
50
60 70 80 90
200
300
400
500 600
E[V/cm]
2
Mobility[cm /V s]
10
7
10
6
10
0.1
1
T[K]
Figure 7-11 2DES Mobility vs. Temperature. The legend specifies pressing field and density for
each dataset. Melting points are indicated by arrows. The solid lines correspond to theoretical
mobility of 2D electron liquid. The inset shows mobility vs. pressing field for a constant density.
104
phases: uncoupled
solid
> liquid > coupled
solid.
We propose the following interpretation:
1. Low pressing field: the electron-ripplon coupling (and therefore the energy relaxation
time) is poor, whereas mobility is high, so that 2DES is easily overheated by RF power.
Since the electrons have higher effective temperature than that of the bath, their mobility
is lower than the theoretical one calculated using bath temperature. 2. High pressing field:
2DES is well thermalized, so that the crystal is not melted by the RF power but
undergoes a sliding transition, gaining higher mobility than the liquid at the same
temperature. The mobility data collected by varying the pressing field while keeping
density constant confirms this interpretation.
7.4 Observation of Magnetoplasmon
In order to prove that electrons do loose their coupling to helium surface when
sufficiently high driving RF field is applied, we measure the effective mass of electron by
studying the spectra in a vertical magnetic field. The effective mass is deduced from the
cyclotron frequency fc =
eB
. The theoretical description of magnetoplasmon was given
2m
in section 3.5. The evolution of 2DES resonances with magnetic field in our experiment
is shown in Figure 7-12. The usable range of the field is limited by the fact that the
resonances become weak and difficult to detect for B > 100G. We are dealing therefore
with nonquantizing magnetic field: c ~ 13mK < Tmin ~ 100mK where c is the
cyclotron frequency. This is consistent with magnetotransport theory including
interactions between electrons according to which the classical (Drude) model will apply
well into the regime of classically strong magnetic fields: c >>1, where  is the
scattering time (Dykman, Fang-Yen et al. 1997).
105
We have observed no new resonance lines appearing as magnetic field increases. This is
T=112mK, P=-50dBm
0G
5G
10 G
15 G
20 G
25 G
30 G
35 G
40 G
45 G
50 G
55 G
60 G
65 G
100 G
0
50 100 150 200 250 300 350 400 450
Frequency[MHz]
Figure 7-12 2DES Spectrum in Magnetic Field. The spectra are shifted vertically to facilitate
comparison.
106
either because the modes are already split before the application of magnetic field (the
system is not rotationally invariant), or the instrumental linewidth does not allow
detecting the split. The absence of edge magnetoplasmons is explained by the same
factors.
The cyclotron frequency for spectra in Figure 7-12 is plotted versus magnetic field in
Figure 7-13. The slope of the fitting line is within the experimental error from its
theoretical value e/2m = 2.799 MHz/G for the free electron mass. Therefore the
magnetoplasmon measurement indicates no renormalization of electron mass, which
300
fc=(2.8  0.1) B
250
fc[MHz]
200
150
100
50
0
0
20
40
60
80
100
B[G]
Figure 7-13 Cyclotron frequency vs. magnetic field for the spectra of Figure 7-12.
could have been caused by the coupling of electron to the dimple in helium. This proves
our previous assumption about 2DES being either melted by power or undergoing the
sliding transition: in both cases the electron system becomes decoupled from the dimples
in helium.
107
8 Conclusion
This work is concerned with investigating the properties of 2D electron systems on liquid
helium by means of radio-frequency absorption in the normal modes of a circular pool.
The absorption spectra were used to monitor the electron density, dissipation
mechanisms, the effects of magnetic fields as well as to determine whether they are in a
liquid or solid phase. In order to investigate the regime where quantum fluctuations
dominate the physical properties the electron density has to be increased beyond the
critical value n = 109 cm-2 at which the helium surface become unstable. To achieve this
goal we have devised a method of producing 2D electron layers that can attain much
higher densities. In this method the electrons are supported by a thin He film (~ 100 Å)
covering the (111) surface of a passivated Si crystal. We detected no signal on the film,
which was <1m thin before charging. In order to make further progress on this problem
one should first determine the reason for the disappearance of RF absorption on thin
films. Several factors need to be considered:

The signal to noise ratio obtained with this technique is too low. Due to the use of
swept frequency measurement the bandwidth has to be broad by necessity
(~1GHz). Despite the use of homodyne detection, a lock-in amplifier and a high
quality narrow band output filter a further improvement might be required. One
such possibility is to place a part of the detection circuit inside the cryostat as
close to the cell as possible. This would eliminate pickup due to the portion of the
leads running at room temperature.
108

The mobility should be high enough so that the resonances do not merge.
Mobility measurements on thin helium film e.g. (Andrei, Grimes et al. 1984) have
shown significant decrease in mobility as the film thickness is reduced below
~1000Å. These experiments also suggest the major influence of surface roughness
on the mobility. STM and AFM images of the substrate supporting the helium
film indicate that in spite of passivation the substrate was not sufficiently smooth
as compared to that required for use of ~100Å helium film. It seems hardly
possible to resolve such difficulty because the wafer’s surface is not immune to
handling during its installation into the cell and cooling down after the
passivation. One should use an in-situ preparation method. It should be noted that
some groups (Kovdrya Yu) turned the unavoidable roughness of the substrate to
their advantage by studying 1D and 0D electron system that might form over the
grooves and imperfections of the wafer.

The thickness of He film that can be achieved by absorption from the saturated
vapor ranges from ~3000Å downwards; directly controlling the level of helium
surface results in films that are 100m or thicker. It would be interesting to
explore the region of intermediate thicknesses. Then one could start with thicker
films where 2DES spectra are easily detectable and gradually crossover to a thin
film.
This work put significant effort into developing the analysis and fabrication of slowwave structures – meander line and interdigital capacitor. One of the lessons is the
paramount importance of controlling the coupling between 2DES and the excitation
structure. This includes a) condition of critical coupling: the amount of power
109
exchanged between the structure and 2DES should be the maximum one that doesn’t
yet lead to the broadening of 2DES resonances b) the propagation velocities of 2DES
mode and the excitation line should be matched.
110
Appendices
111
A Ripplon Dispersion Law
Here we derive the equations of motion and dispersion law for the surface of nonviscous, incompressible and irrotational liquid. The hydrodynamical equations describing
such liquid are:
v
P
 v v  
g
t

v  0
 v  0
Euler Equation
Incompressibility Condition 
 Absence of Rotation 
In these formulas v is fluid velocity, P – pressure,  is the density of liquid and g –
modified gravity constant, which includes the van der Waals attraction between the liquid
and supporting substrate. The last equation allows introducing velocity potential :
v   .
The
above
system
of
equations
combined
with
the
identity
Vapor
z

x
Liquid
d
Substrate
Figure A-1 Surface Wave.
vv   v
2
2
 v    v  leads to Bernoulli law, which in term of velocity potential is:
112
P

v2

  gz  const
t
2
(A-1)
where z is the direction opposite g (up).
Lets apply this law to the perturbed free surface of liquid described by the equation
z = (r), where(r) is the vertical displacement of the surface at location r in xy plane as
illustrated in Figure A-1 (y axis is perpendicular to the plane of the drawing). The vapor
pressure can be neglected at the temperatures of interest (T < 2K), and the pressure right
under the surface is proportional to its curvature: P z  r   
 2 r 
where,  is the
r 2
surface tension. Substituting this into Eq.(A-1), we get:


 2 r 
  g r   
t
r 2
(A-2)
The nonlinear term was dropped. In addition, we have for the free surface of liquid:
 r, t   r, z, t 

t
z
z 0
(A-3)
Combining these equations, we get:

 2

  2


g


z
z r 2
t 2
0
(A-4)
z 0
Because the liquid is incompressible (v = 0), velocity potential satisfies Laplace
equation: 2 or (k2 + kz2)k = 0 in terms of Fourier harmonics (k is the component
of the wavevector in the xy plane, kz is its z component)Therefore we will look for a
solution of (A-4) having the form: = exp(-ikr)(Ak(t) exp(kz) + Bk(t) exp(-kz)). The
choice of functions Ak(t) and Bk(t) should satisfy the boundary condition of normal
component of velocity being zero at the surface of the solid substrate located at z = - d:
113

z
 0 . The suitable solution is = k(t) exp(-ikr) cosh(k(z+d)). Upon substitution
z  d
into Eq.(A-4) we obtain the harmonic oscillator equation for the harmonic amplitudes:
k   r2 k  k  0
(A-5)
where the dispersion law of the liquid surface wave is:

 r 2 k    g k 

 3
k  tanh kd
 
Some limiting cases:
 3
k


Deep liquid kd >> 1:  r  g k 

Short wavelength k >> kc = (g/)1/2:  r 

Long wavelength (gravity waves) k << kc: r  gk tanh kd

Shallow liquid k << kc, kd << 1: r  gd k
2
 3
k tanh kd

(A-6)
114
B Roughness of Liquid Helium Surface Due to
Thermally Excited Ripplons
The derivation below follows (Cole 1970). Most of the quantities are defined in
appendix A. The rms roughness of helium surface is the sum of contributions from all
surface modes:
 2   2 r  
1
  k  k
A k
(B-1)
where k is the Fourier component of the vertical displacement of the surface (r),
corresponding to the mode with wavevectors k, A is the area of the surface. In order to
evaluate this sum we will express the average potential energy contained in a mode via
k. To begin with, the potential energy is the sum of surface energy and gravitational
energy:

d   r 2
2
P   dxdy   1   r   1   g

2
A
 

 . The first term can be expanded


provided |(r)| << 1, i.e. the amplitude of perturbation is much smaller that its
wavelength: P   dxdy
A


1
2
  r    g 2d r    2 r  . The constant term has been
2
dropped. According to the plan, we need to evaluate this expression for a single mode:
 r  
1
A

k

e ikr   k e ikr . Substituting this into the above formula and averaging
over the wave period, we get: Pk 


1
 k 2   g  k  k . The average total energy in
2
the mode (potential plus kinetic) is twice this value: <Ek> = 2<Pk> (virial theorem). On
115
the other hand, the average energy is equal to the ripplon energy times the occupation
number of chosen mode plus zero-motion term:
Ek 
 r k 
  k  
  1
exp  r
 k BT 

 r k 
2
Combining with the previous equation gives:
 k  k





 r k 
 r k  
1



 . After the substitution into Eq.(B-1)
2 
k2  g 
  r k  


 exp  k T   1

 B



and converting the sum to the integral, we obtain the rms roughness:





 r k 
 r k  
1
1
2
   2



A k k  g 
2 
  r k  


 exp  k T   1

B









 r k 
 r k   d 2 k
1
k  k 2   g    k    2  2 2
 exp  r   1

 k T 


 B 


where r(k) is given by Eq.(A-6). For deep liquid (kd >> 1) this becomes:
2




k 2 dk 

1
1

    T2   02 where we divided the integral

2 k  r k  
2
  r k  

  1
exp


 k BT 


into the temperature dependent and zero-motion parts. One has to impose a cutoff km on
the upper limit of integration, so that the total number of surface modes is equal to the
number of atoms. Then  02 
 k m3 / 2
6 
~ 1Å 2 for 0 K values of liquid 4He parameters
116
(= 0.38 erg/cm2,  = 0.146 g/cm3) and km = 108cm-1. The total rms roughness is shown
in Figure B-1 (Cole 1970).
Figure B-1 RMS Roughness of Liquid 4He Surface (from
(Cole 1970)).
117
C Parametric Resonance in HeII
In this appendix I will discuss excitation of parametric resonances on the surface of
superfluid helium (a.k.a. Faraday waves for normal fluids) as might be caused by
mechanical vibrations. Similar discussion can be found in appendix I of (Glattli 1986).
The dispersion of the surface waves is given by Eq.(A-6). The effect of mechanical
vibration with an angular frequency v is equivalent to adding an oscillating term to the
gravitational acceleration: g  g 1  v cosv t  where v is the relative vibration
acceleration amplitude. This will affect the ripplon frequency through dependence of the
dispersion on g. The Eq.(A-5) now becomes: k    k   r2 1   v cos v t  k  0 where
we also introduced a dissipative term including all lossy processes affecting the surface
waves. If the following condition is met:
d v(w r/ w v )2
g / w v = 0.1
=0
g= 0
( w r/ w v )2
Figure C-1 Parametric Resonance Stability Diagram. Horizontal axis: ripplon frequency.
Vertical axis: vibration acceleration amplitude. Both are normalized by vibration frequency.
From http://monet.physik.unibas.ch/~elmer/pendulum/parres.htm.
118
v 
2 r
s
(C-1)
where s is integer and attenuation  is small enough, the surface of liquid becomes
unstable – a parametric resonance sets in. The stability diagram in terms of ripplon
frequency and vibration amplitude is shown in Figure C-1 for two values of . The
lowermost point of the instability region corresponding to first order resonance (s = 1) is
given by (Landau and Lifshits 2002):
 vmax 
4
(C-2)
r
When the vibration frequency is close to the first order resonance, the Eq.(C-2) gives the
maximum amplitude that will not excite a parametric resonance:  v 
Av v2 4
where

g
r
Av is the vibration displacement amplitude. Substituting Eq.(C-1) with s = 1 we get:
Av 
g
 r3
(C-3)
The attenuation for surface waves in superfluid 4He is extremely small due to its
vanishing viscosity and is limited by induced electrical losses in nearby electrodes. I will
take  ~ 0.05 Hz (Glattli 1986). The geometry of the liquid helium pool determines the
allowed surface mode wavevectors whose magnitude is roughly kr ~ 1/RHe where RHe is
the radius of the helium pool. In our experiment RHe ~1cm, therefore kr ~ 2cm-1.The
depth of our helium pool is d ~1mm. Plugging these values into the dispersion Eq.(A-6)
on gets r ~ 24 Hz for liquid helium4. Now Eq.(C-3) gives the following estimate for
the limit that has to be imposed on vibration amplitude in order to prevent excitation of
parametric resonance on the surface of superfluid 4He: Av < 50 m.
119
D Heat Exchangers for Liquid He Bath
Due to the low latent heat of liquid He (21 J/g) one must use cold He vapors to cool the
experiment below liquid N2 temperature 77K. Given the specific heat cp = 5 J/g K of the
He gas, the efficiency of operating a cryostat is drastically improved by making He vapor
travel through a series of heat exchangers – at least one per each radiation baffle. Their
design follows Chapter 3 of (White 1993). To ensure large heat flow between the cold
gas and the exchanger a turbulent flow is preferable. In the latter case the gas temperature
is uniform in a cross section of a tube except for a thin layer adjacent to the walls. Thus
the Reynolds number Re 
4m
should be kept higher than 2300. On the other hand the
 d
buildup of large pressures inside the cryostat is undesirable because it endangers the
mechanical integrity of the latter and/or might promote Taconis oscillations (Richardson
and Smith 1988), wasting extra liquid helium. Thus we don’t allow the pressure drop
across the exchanger to exceed: p  0.316
helium
gas
(initially
T  Texch  (Texch  Ti )e

L
L0
Ti)
approaches
L  0.25 m 1.75
 2 psi . The temperature of
2 d 4.75
that
of
the
exchanger
according
as it travels a distance L along the exchanger with L0 
to
m c p
hS
.
0.023 (4m ) 0.8 0.2
cp
Here S is the perimeter of the tube, h – heat transfer coefficient h 
,
Pr 0.6
 0.8 d 1.8
Pr 
c p
- Prandtl number. In the above formulae cp is specific heat of gas,  - its

 - mass flow rate.
viscosity,  - thermal conductivity,  – density, m
120
The heat exchanger was implemented as a length of 1/16” ID copper tubing coiled and
soldered to a copper sheet that was mounted onto each radiation baffle of the cryostat.
Teflon tubing connects heat exchangers on different baffles.
Figure D-1 Heat Exchanger Mounted on Cryostat's Baffle.
As a result of the heat exchanger installation the liquid helium boil-off was reduced
significantly from 30 ltr/day to 20.
121
E Experimental Procedure
In this Appendix, I will describe a typical experimental procedure including the
preparation for the run, operation of the dilution fridge and data collection.
Preparation
First assemble the cell and test it for leaks. Then perform an electrical test. Here is the
description of a typical electrical test, which will have to be repeated at several stages
later:
1. Check that the filament is continuous and is not shorted to the top plate or ground.
2. Perform continuity test and a reflection test of the transmission line being used.
3. Measure capacitances between the following pairs of electrodes in the cell: a) level
capacitor plates b) top plate and transmission line c) top plate and guard ring
Install the cell on the fridge using a coupler between the cell and the mixing chamber
(Figure 6-1). Check the throughput of the helium fill line then connect the line to the cell
by sealing the flanges with an In gasket. Leaktest the connector by pumping with a leak
detector on the fill line port on top of the cryostat. Also check the throughput of the fill
line by monitoring pressure drop while pumping on the cell. Leave the cell under
vacuum.
Check for radiation leaks through the baffles of the dilution unit. Plug all unused
openings with metallic objects (screws); do not use tape – it has high emissivity at low
temperatures. Make all electrical connections to the cell. Install the cell thermometer and
make sure it is well coupled to the cell.
122
Check that all leads are thermally anchored at a few stages in the fridge. It is important to
ensure now that the cell (including its lower part) is well thermalized to the mixing
1K Pot Needle
Valve Control
Cell Fill Line
Valve
Figure E-1 Top of the Cryostat.
chamber. If necessary perform the angular alignment of the cell. Make a note on the
position of the cell with respect to the IVC top flange: this is required for adjusting
magnet’s position so that the cell is in its center. Check the condenser capillary
throughput as described in the dilution fridge manual. Tie all extruding leads to the fridge
posts or other steady objects. Close the thermal radiation shields. There should be no
touches between the shields and the cell.
Repeat the electrical test from the top of the cryostat: from this point it should include
both the cell wiring and that of the fridge – thermometers, heaters, etc. Check the
throughput of 1 K pot needle valves. Close the internal vacuum chamber (IVC) can (In
gasket seal) and check that there are no touches between the can and the radiation shields
by gently pushing the lower end of the IVC can around.
123
Figure E-3 Still Radiation Shield.
Figure E-2 50mK Radiation Shield.
If suspicious remove the bottom cap of the IVC can (In gasket seal) and check visually.
Connect the leak detector to the IVC. Check for leaks by filling the dilution unit, the cell
and the 1 K pot with He gas. From this point forward the cell pressure should not exceed
that of the IVC by more than ~ 100 mbar: this will prevent damage to soft-soldered
electrical connectors in the cell. Install the magnet and its leads. It is important to position
the magnet so that the helium surface in the cell would be leveled with the center of the
magnet. Check for radiation leaks through the baffles of the cryostat. Test the magnet and
its leads for continuity and absence of shorts. The outer vacuum chamber (OVC) of the
dewar should be pumped with a high vacuum pump to achieve the stable pressure reading
of < 10-5 mbar. Slowly raise the dewar using the winch, the steal cables and the two
pulleys on the floating plate of the vibration isolation table. Make sure the inner surface
of the dewar doesn’t catch any objects extruding from the cryostat: 1 K pot intake tube,
baffles, magnet leads, etc. The dewar is connected to the top plate of the cryostat by 6
screws. Tighten them and then release the cables of the winch. Leaktest the IVC by
filling the bath space with He gas. It is advisable to repeat the electrical test now.
124
Cooldown
Fill the IVC with ~ 10 torr of N2 exchange gas. The 1 K pot should be purged and then
pressurized with helium gas to a few psi above 1 atm. This overpressure should be
maintained until the liquid nitrogen is removed from the bath at a later stage. Leave the
dilution unit under vacuum. The cell should preferably be filled with He gas to a pressure
slightly above that of the IVC. Fill half of the bath space with liquid N2 to precool the
system overnight. When 77K is reached, connect the leak detector to the IVC and leaktest
the cell and 1K pot by filling and evacuating them with He gas a few times. Then check
condenser capillary throughput watching the leak detector at the same time. Perform
electrical checks. If all tests are satisfactory, remove liquid N2 from the bath space
completely by pressurizing the bath exhaust port with N2 gas or by heating the liquid at
the bottom with a dipstick heater. The fill port should be connected to a suitable liquid
nitrogen storage container.
Fill IVC and the cell with a few torr of He gas. Transfer liquid He into the bath. Proceed
slowly until accumulation of liquid in the bath begins. At this time pump out the IVC.
The pumping should be performed with a diffusion pimp for ~20 hours. The 1K pot
should not be used during that time.
125
Dilution Fridge Operation
First 1K pot should be filled with liquid helium using the needle valves and then pumped
on constantly with the dedicated mechanical pump. Needle valve can be adjusted so that
Figure E-4 Dilution Fridge Control Panel.
the pot fills continuously. At the same time open the dumps storing dilution mixture to
liquid N2 traps. Then slowly open the valves isolating the traps from the still. Condenser
126
line valve should be closed to avoid a blockage of the capillary! After the mixture is fully
condensed the condenser line can be open. The mixture can be circulated using rotary and
later booster pump. The circulation rate can be change by applying power to the still
heater.
Filling the Cell with Helium
There should be a liquid N2 trap between the cell and external helium container. The
condensation starts with opening the cell fill valve (Figure E-1). Record the volume of He
gas condensed (using pressure gauge), level capacitance and top-plate to meander line
capacitance. The capacitances can be measured with a capacitance bridge. It is advisable
to use gas tanks of different size in case a fine adjustment to the amount of liquid in the
cell is needed. To calibrate the capacitance one should completely fill the space up to the
top plate with helium noticing empty and full values of capacitance. This has to be done
in the first fill.
Charging the Helium Surface
Inflate vibration isolation table. Disconnect all pumping lines not in use from the
cryostat. The top plate voltage should be set to ~ +100V. The glow discharge would start
when filament voltage reaches ~ -300V. To facilitate the discharge pulse the filament
using the pulser circuit. Observe both the current to the filament and the top plate: they
should be roughly equal. Otherwise the discharge is happening in the wrong place. After
the discharge begins, quickly ramp the top plate voltage to 0V and then slowly to the
desired negative value. Terminate the discharge by setting the filament voltage to zero.
127
Data Collection
Make sure the circuit is connected as in Figure 6-17. Turn on the RF generator. It should
be operated in the swept mode. Incident RF power is controlled using a series of variable
attenuators. Make sure RF amplifier is energized. Adjust the phase and frequency settings
of the lock-in amplifier for the maximum signal. Collect the spectra using data
acquisition program.
128
References
Adams, P. W. and M. A. Paalanen (1988). "High-field magnetoconductivity of electrons
on hydrogen." Physical Review B (Condensed Matter) 37(7): 3805-7.
Ando, T., A. B. Fowler, et al. (1982). "Electronic properties of two-dimensional
systems." Reviews of Modern Physics 54(2): 437-672.
Andrei, E. Y. (1984). "Observation of the polaronic transition in a two-dimensional
electron system." Physical Review Letters 52(16): 1449-52.
Andrei, E. Y. (1997). Two-dimensional electron systems on helium and other cryogenic
substrates. Dordrecht ; Boston, Kluwer Academic Publishers.
Andrei, E. Y., C. C. Grimes, et al. (1984). "Electrons on helium-the `polaron' transition."
Surface Science 142(1-3): 104-6.
Bonsall, L. and A. A. Maradudin (1977). "Some static and dynamical properties of a twodimensional Wigner crystal." Physical Review B (Solid State) 15(4): 1959-73.
Butcher, P. N., N. H. March, et al. (1993). Physics of low-dimensional semiconductor
structures. New York, Plenum Press.
Cole, M. W. (1970). "Width of the surface layer of liquid He/sup 4." Physical Review A
(General Physics) 1(6): 1838-40.
Cole, M. W. and M. H. Cohen (1969). "Image-potential-induced surface bands in
insulators." Physical Review Letters 23(21): 1238-41.
Crampagne, R. and M. Ahmadpanah (1977). "Meander and interdigital lines as periodic
slow wave structure. II. Applications to slow-wave structures." International
Journal of Electronics 43(1): 33-9.
Crampagne, R. and M. Ahmadpanah (1977). "Meander and interdigital lines as periodic
slow wave structures. I. Characteristics of waves propagating along an infinite
array." International Journal of Electronics 43(1): 19-32.
Crandall, R. S. and R. Williams Crystallization of electrons on the surface of liquid
helium, Physics Letters A 34a(7) 19 April 1971 p. 404-5.
Dahl, D. A. and L. J. Sham (1977). "Electrodynamics of quasi-two-dimensional
electrons." Physical Review B (Solid State) 16(2): 651-61.
Dahm, A. J. and H. W. Jiang Properties of a pinned electron solid on a liquid helium film,
Journal de Physique IV. 3(C2) July 1993 p. 39-43.
Deville, G. (1997).
Deville, G., F. Gallet, et al. Detection of ordering of electrons on liquid helium, NorthHolland 1981 p. 309-12 Amsterdam Netherlands.
Deville, G., A. Valdes, et al. (1984). "Propagation of shear in a two-dimensional electron
solid." Physical Review Letters 53(6): 588-91.
Devreese, J. T., F. Peeters, et al. (1987). The physics of the two-dimensional electron gas.
New York, Plenum Press.
DiVincenzo, D. (2000). "Quantum computing with quantum dots." Abstracts of Papers of
the American Chemical Society 220: U190-U190.
129
Donnelly, R. J., W. I. Glaberson, et al. (1967). Experimental superfluidity. Chicago,,
University of Chicago Press.
Dykman, M. I. (1978). "Theory of cyclotron resonance of two-dimensional electrons
interacting with surface and volume phonons." Physica Status Solidi B 88(2):
463-75.
Dykman, M. I., C. Fang-Yen, et al. (1997). "Many-electron transport in strongly
correlated nondegenerate two-dimensional electron systems." Physical Review B
(Condensed Matter) 55(24): 16249-71.
Dykman, M. I., P. M. Platzman, et al. Qubits with electrons on liquid helium, Physical
Review B-Condensed Matter. 67(15)APS through AIP 15 April 2003 p. 1554021-15.
Edel'man, V. S. (1976). "Effective mass of the electrons, localised above the liquid
helium surface." Zhurnal Eksperimental'noi i Teoreticheskoi Fiziki, Pis'ma v
Redaktsiyu 24(9): 510-13.
Edwards, D. O. and W. F. Saam (1978). The free surface of liquid helium. Progress in
low temperature physics, vol. VIIa, North-Holland: 283-369.
Faria, J. A. B. (1993). Multiconductor transmission-line structures : modal analysis
techniques. New York, Wiley.
Fetter, A. L. (1976). Vortices and ions in helium. Physics of liquid and solid helium, pt.1,
Wiley: 207-305.
Fisher, D. S., B. I. Halperin, et al. Phonon-ripplon coupling and the two-dimensional
electron solid on a liquid-helium surface, Physical Review Letters. 42(12) 19
March 1979 p. 798-801.
Fukuyama, H. Two-dimensional electron systems, Solid State Physics. 12(12) Dec. 1977
p. 727-36.
Gann, R. C., S. Chakravarty, et al. (1979). "Monte Carlo simulation of the classical twodimensional one-component plasma." Physical Review B (Condensed Matter)
20(1): 326-44.
Glattli, D. C. (1986). Dynamique de cristaux bidimensionnels classiques d'electrons,
L'Université de Paris-Sud, Centre d'Orsay.
Glattli, D. C., E. Y. Andrei, et al. (1985). "Dynamical Hall effect in a two-dimensional
classical plasma." Physical Review Letters 54(15): 1710-13.
Glattli, D. C., E. Y. Andrei, et al. Specific and latent heat of the 2D electron solid,
Surface Science. 196(1-3) March 1988 p. 17-23.
Gor'kov, L. P. and D. M. Chernikova (1973). "Concerning the structure of a charged
surface of liquid helium." Zhurnal Eksperimental'Noi i Teoreticheskoi Fiziki,
Pis'Ma V Redaktsiyu 18(2): 119-22.
Gor'kov, L. P. and D. M. Chernikova (1976). "Mode of stability loss of charged helium
surface." Doklady Akademii Nauk SSSR 228(4-6): 829-32.
Grimes, C. C. and G. Adams Evidence of a liquid-to-crystal phase transition in a
classical, two-dimensional sheet of electrons, Physical Review Letters. 42(12) 19
March 1979 p. 795-8.
Grimes, C. C. and G. Adams Observation of two-dimensional plasmons and electronripplon scattering in a sheet of electrons on liquid helium, Physical Review
Letters. 36(3) 19 Jan. 1976 p. 145-8.
130
Grimes, C. C. and G. Adams Plasmons in a sheet of electrons in liquid helium, Surface
Science. 58(1) Aug. 1976 p. 292-4.
Grimes, C. C., T. R. Brown, et al. Spectroscopy of electrons in surface states outside
liquid helium, (I)North-Holland 1975 p. 443-6 Amsterdam Netherlands.
Gruner, G. (1988). "The dynamics of charge-density waves." Reviews of Modern Physics
60(4): 1129-81.
Gruner, G. and A. Zettl (1985). "Charge density wave conduction: a novel collective
transport phenomenon in solids." Physics Reports 119(3): 117-232.
Halperin, B. I. and D. R. Nelson (1978). "Theory of two-dimensional melting." Physical
Review Letters 41(2): 121-4.
Haque, M., I. Paul, et al. (2003). "Structural transition of a Wigner crystal on a liquid
substrate." Physical Review B 68(4): -.
Higashi, G. S., Y. J. Chabal, et al. (1990). "Ideal hydrogen termination of the Si(111)
surface." Applied Physics Letters 56(7): 656-8.
Horowitz, P. and W. Hill (1989). The art of electronics. Cambridge [England] ; New
York, Cambridge University Press.
I. E. Dzyaloshinskii, E. M. L., L.P. Pitaevskii (1961). Adv. Physics 10: 165.
Ikezi, H. (1979). "Macroscopic electron lattice on the surface of liquid helium." Physical
Review Letters 42(25): 1688-90.
Ikezi, H., R. W. Giannetta, et al. (1982). "Nonlinear equilibria of the electron-charged
surface of liquid helium." Physical Review B-Condensed Matter 25(7): 4488-94.
Ikezi, H. and P. M. Platzman (1981). "Stability of helium films charged with electrons."
Physical Review B-Condensed Matter 23(3): 1145-8.
Iye, Y. (1980). "Mobility of electrons in the surface state of liquid helium." Journal of
Low Temperature Physics 40(5-6): 441-51.
Jackson, S. A. and P. M. Platzman Temperature-dependent effective mass of a selftrapped electron on the surface of a liquid-helium film, Physical Review BCondensed Matter. 25(7) 1 April 1982 p. 4886-9.
Kajita, K. Wigner crystallization of two dimensional electrons formed on the surface of
solid neon, Journal of the Physical Society of Japan. 54(11) Nov. 1985 p. 4092-5.
Khoury, M., A. Gunther, et al. (2000). "Single-electron quantum dots in silicon MOS
structures." Applied Physics a-Materials Science & Processing 71(4): 415-421.
Klier, J., I. Doicescu, et al. First dc measurements of electrons on liquid helium: the
helium-FET, Journal of Low Temperature Physics. 121(5-6)Kluwer
Academic/Plenum Publishers Dec. 2000 p. 603-8.
Komuro, T., H. Kawashima, et al. (1996). "Third sound in one and two dimensional
modulated structures." Journal of Low Temperature Physics 102(3-4): 359-364.
Kono, K., U. Albrecht, et al. (1991). "Surface-state electrons on a hydrogen film. I.
Annealing of the film." Journal of Low Temperature Physics 82(5-6): 279-93.
Kono, K., U. Albrecht, et al. (1991). "Surface state electrons on a hydrogen film. II.
Influence of adsorbed helium films." Journal of Low Temperature Physics 85(56): 423-44.
Kosterlitz, J. M. and D. J. Thouless (1972). "Long range order and metastability in two
dimensional solids and superfluids. [Application of dislocation theory]." Journal
of Physics C (Solid State Physics) 5(11): 124-6.
131
Kovdrya Yu, Z. One-dimensional and zero-dimensional electron systems on liquid
helium, Fizika Nizkikh Temperatur. 29(2)AIP Feb. 2003 p. 107-44.
Kovdrya Yu, Z. and V. A. Buntar Effect of optic plasma modes on the RF power
absorption by the electron crystal over liquid helium, Fizika Nizkikh Temperatur.
9(11) Nov. 1983 p. 1125-8.
Lambert, D. K. and P. L. Richards Measurement of local disorder in a two-dimensional
electron fluid, Physical Review Letters. 44(21) 26 May 1980 p. 1427-9.
Landau, L. D. and E. M. Lifshits (1970). Theory of elasticity. Oxford, New York,,
Pergamon Press.
Landau, L. D. and E. M. Lifshits (2002). Mechanics. Oxford, Butterworth-Heinemann.
Lindeman, F. (1910). Z. Phys. 11: 609.
March, N. H. and M. P. Tosi (1984). Polymers, liquid crystals, and low-dimensional
solids. New York, Plenum Press.
Marty, D. and J. Poitrenaud Experimental study of coupled electron-ripplon vibrations of
the 2D electron crystal at the surface of liquid helium, Journal de Physique I.
45(7) July 1984 p. 1243-55.
Mehrotra, R. and A. J. Dahm (1987). "Analysis of the Sommer technique for
measurement of the mobility for charges in two dimensions." Journal of Low
Temperature Physics 67(1-2): 115-21.
Mehrotra, R., B. M. Guenin, et al. (1982). "Ripplon-limited mobility of a twodimensional crystal of electrons: experiment." Physical Review Letters 48(9):
641-4.
Mellor, C. J. and W. F. Vinen (1990). "Experimental observation of crystallization and
ripplon generation in a two-dimensional pool of helium ions." Surface Science
229(1-3): 368-70.
Mistura, G., T. Gunzler, et al. (1997). "Microwave study of screened two-dimensional
electron crystals on helium films." Physical Review B (Condensed Matter)
56(13): 8360-6.
Morf, R. H. (1979). "Temperature dependence of the shear modulus and melting of the
two-dimensional electron solid." Physical Review Letters 46(13): 931-5.
Mott, N. F. (1961). " THE TRANSITION TO THE METALLIC STATE." Philosophical
Magazine 6: 287.
Mugele, F., U. Albrecht, et al. (1992). "Possible correlation effects of surface state
electrons on a solid hydrogen film." Journal of Low Temperature Physics 89(3-4):
743-6.
N. D. Mermin, H. W. (1966). Phys. Rev. Lett. 17: 1133.
Nielsen, M. A. and I. L. Chuang (2000). Quantum computation and quantum information.
Cambridge ; New York, Cambridge University Press.
Onn, D. G. and M. Silver Injection and thermalization of hot electrons in solid, liquid,
and gaseous helium at low temperatures, Physical Review a (General Physics).
3(5) May 1971 p. 1773-9.
Peeters, F. M. (1984). "Two-dimensional Wigner crystal of electrons on a helium film:
static and dynamical properties." Physical Review B (Condensed Matter) 30(1):
159-65.
Peierls, R. E. (1979). Surprises in theoretical physics. Princeton, N.J., Princeton
University Press.
132
Penanen, K., M. Fukuto, et al. (2000). "He-4 liquid-vapor interface below 1 K studied
using x-ray reflectivity." Physical Review B 62(14): 9621-9640.
Platzman, P. M. and G. Beni (1976). "Comment on plasma linewidth experiments for
electrons on a helium surface." Physical Review Letters 36(11): 626-8.
Platzman, P. M. and H. Fukuyama (1974). "Phase diagram of the two-dimensional
electron liquid." Physical Review B (Solid State) 10(8): 3150-8.
Pobell, F. (1992). Matter and methods at low temperatures. Berlin ; New York, SpringerVerlag.
Pozar, D. M. (1997). Microwave engineering. New York, Wiley.
Putterman, S. J. (1974). Superfluid hydrodynamics. Amsterdam
New York, North-Holland Pub. Co. ;
American Elsevier Pub. Co.
Rayleigh (1882).
Richardson, R. C. and E. N. Smith (1988). Experimental techniques in condensed matter
physics at low temperatures. Redwood City, Calif., Addison-Wesley Pub. Co.
Rybalko, A. S., B. N. Esel'son, et al. Liquid-crystal phase transition in the system of
surface electrons at temperatures below 0.3K, Fizika Nizkikh Temperatur. 5(8)
Aug. 1979 p. 947-9.
Rybalko, A. S. and Z. Kovdrya Yu Mobility of electrons along the liquid helium surface
in strong retarding electric fields, Journal of Low Temperature Physics. 18(3-4)
Feb. 1975 p. 219-27.
Sabisky, E. S. and C. H. Anderson (1973). "Verification of the Lifshitz theory of the van
der Waals potential using liquid-helium films." Physical Review A (General
Physics) 7(2): 790-806.
Saitoh, M. (1977). "Warm electrons on the liquid /sup 4/He surface." Journal of the
Physical Society of Japan 42(1): 201-9.
Saitoh, M. (1989). "Melting temperature of two-dimensional electron crystals trapped on
thin-film liquid He." Physical Review B (Condensed Matter) 40(1): 810-12.
Shikin, V. B. Some properties of surface electrons in liquid helium, Zhurnal
Eksperimentalnoi i Teoreticheskoi Fiziki. 60(2) 1971 p. 713-23.
Shikin, V. B. and P. Monarkha Yu On the interaction of surface electrons in liquid
helium with oscillations of the vapor-liquid interface, Journal of Low
Temperature Physics. 16(1-2) July 1974 p. 193-208.
Shirahama, K. and K. Kono Dynamical transition in the Wigner solid on a liquid helium
surface, Physical Review Letters. 74(5) 30 Jan. 1995 p. 781-4.
Sommer, W. T. and D. J. Tanner (1971). "Mobility of electrons on the surface of liquid
/sup 4/He." Physical Review Letters 27(20): 1345-9.
T. Williams, P. R., N. Appleyard, G. Deville, K. Keshishev (1995). Human Capital
Mobility Meeting, Grenoble.
Tanatar, B. and D. M. Ceperley (1989). "Ground state of the two-dimensional electron
gas." Physical Review B (Condensed Matter) 39(8): 5005-16.
Tanatar, B. and T. Hakioglu Possibility of superconductivity of two-dimensional
electrons on the surface of liquid helium films, Solid State Communications.
88(2) Oct. 1993 p. 115-18.
133
Tress, O., Y. P. Monarkha, et al. (1996). "Observation of 2D polarons and
magnetopolarons on superfluid helium films." Physical Review Letters 77(12):
2511-2513.
Valdes, A. (1982). Contribution a l'etude du solide electronique a deux dimensions.
Centre D'orsay. Paris, L'Universite de Paris-sud.
Vinen, W. F. (1999). "Non-linear electrical conductivity and sliding in a two-dimensional
electron crystal on liquid helium." Journal of Physics: Condensed Matter 11(48):
9709-13.
von Klitzing, K. (1987). The quantum Hall effect, Oostduinkerke, Belgium, Plenum.
Weiss, J. A. (1974). "Dispersion and field analysis of a microstrip meander-line slowwave structure." IEEE Transactions on Microwave Theory and Techniques MTT22(12, pt.2): 1194-201.
White, G. K. (1993). Experimental techniques in low-temperature physics. Oxford
New York, Clarendon Press ;
Oxford University Press.
Wigner, E. P. (1934). Physical Review 46: 1002.
Young, A. P. (1979). "Melting and the vector Coulomb gas in two dimensions." Physical
Review B (Condensed Matter) 19(4): 1855-66.
Zipfel, C. L., T. R. Brown, et al. Spectroscopic studies of electron surface states on liquid
helium, Surface Science. 58(1) Aug. 1976 p. 283-8.
134
Curriculum Vitae
Ivan Skachko
Sept.1989-June1995 Moscow Institute of Physics and Technology, Moscow, Russia
M.S., Physics.
Sept.1996-May2006 Rutgers, The State University of New Jersey, New Brunswick, NJ
Ph.D., Physics and Astronomy