1. 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]
2. 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
3. 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
4. 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 ELECTRONS ON BULK HELIUM........................................................................... 19
2.2 WIGNER SOLID AND 2D MELTING MECHANISMS............................................... 21
2.3 ELECTROHYDRODYNAMIC (EHD) INSTABILITY OF A CHARGED LIQUID SURFACE
26
2.4 INFLUENCE OF A SUBSTRATE ON PHASE DIAGRAM ............................................ 29
3 Spectrum of 2DES................................................................................................... 32
3.1 2D DRUDE MODEL ............................................................................................ 32
3.2 ELECTRON SOLID MODES .................................................................................. 39
3.3 2DES MOBILITY................................................................................................ 39
3.4 COUPLED PHONON-RIPPLON MODES ................................................................. 43
3.5 SPECTRUM OF 2DES IN MAGNETIC FIELD ......................................................... 46
4 Measurement of 2DES Spectrum Using Slow-Wave Structure.......................... 50
4.1 2DES IN ELECTRO-MAGNETIC FIELD ................................................................ 50
4.2 DISTRIBUTED CIRCUIT MODEL .......................................................................... 54
4.3 SLOW-WAVE STRUCTURES................................................................................ 58
4.4 COUPLING BETWEEN MEANDER LINE AND 2DES................................................ 65
5 Other Methods for Investigation of Phase Transition in 2DES on Liquid
Helium Film..................................................................................................................... 66
5.1 MICROWAVE CAVITY TECHNIQUE ..................................................................... 66
5.2 SOMMER-TANNER METHOD (LOW-FREQUENCY TRANSPORT)........................... 68
iv
5. 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 IDENTIFICATION OF THE INDIVIDUAL RESONANCES ........................................... 93
7.2 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
6. 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. ............................................................................. 60
vi
7. 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 2.2 109
CM
-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). θ = PQ−2πJ, 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
8. 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 ω ~ N
0.5
..................................................... 92
FIGURE 7-5 MATCHING OF AN OBSERVED SPECTRUM (THICK LINE) TO 2D PLASMA
RESONANCES (THIN LINES). HERE (ν,µ) 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 I
0.5
. ................................................................................... 96
FIGURE 7-8 EVOLUTION OF THE 2DES SPECTRUM WITH TEMPERATURE. THE SPECTRA ARE
SHIFTED VERTICALLY TO FACILITATE COMPARISON. TM DENOTES THE EXPECTED
viii
9. WIGNER SOLID MELTING TEMPERATURE FOR N = 6 107
CM
-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
4
HE 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
10. 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 4
He. 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 z-
direction is quantized in this case. Let E1, E2 be the energies of the ground and the first
11. 2
excited state of the system for such motion respectively. If the system is cooled to a
sufficiently low temperature so that 12 EETkB −<< , it will then mostly remain in the
ground state. The motion in the z direction is “frozen out”. Such system becomes two-
dimensional. 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
( ) 2
hπ
ρ
m
EE = (1-1)
as opposed to 3D ( ( ) EEE ~ρ ) or 1D ( ( )
E
EE
1
~ρ ) 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).
12. 3
Table 1-1 Basic Properties of 2DES.
On Helium In Si-MOS In GaAs/AlxGa1-xAs
Binding Energy, K 7 50-500 200-400
Wavefunction Extent
out of 2D plane, Å
114 30 50-100
Surface Density, cm-2 10
6
-10
9
-? 10
11
-10
13
10
11
-10
12
Fermi Energy, K 10
-5
-10
-2
10-500 200-1000
Effective Mass, m*/me 1.0 0.19 0.066
Mobility, cm2/V s 10
7
10
3
10
5
-10
6
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 4
He were predicted (Shikin; Cole and
Cohen 1969) in 1969. An electron is bound to the surface of liquid 4
He (or any other
dielectric) (Andrei 1997) by inducing an electrostatic screening charge as shown in
Electron
Image
Charge
Induced
Charge 4
He
1
1
+
−
He
He
e
ε
ε
Figure 1-1 Screening Mechanism Giving Rise to Electron Confinement to a Liquid Helium
Surface.
13. 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
<∞
>−+
∂
∂
−=⊥
0
0
42
2
2
22
z
z
z
e
zm
H
δh
,
1
1
+
−
=
ε
ε
δ which is identical to the radial part of
Schrödinger equation for the hydrogen atom with zero angular momentum. Thus the
energy spectrum of a 2D electron is:
14. 5
2
42
2
32
,
h
em
Ry
n
Ry
En
δ
=∗
∗
−= (1-2)
where ε 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)).
The ground state wavefunction for motion in the z direction is 02
3
01 2)( a
z
ezaz
−−
=φ where
δ
ba
a
4
0 = is the effective Bohr radius.
Liquid
4
He
1eV
z
e
zV
4
)(
2
δ
−=
z
2
1
n
E
En −=
Å76
4
0 ==
δ
Ba
a
028.0
1
1
=
+
−
=
He
He
ε
ε
δ
KRyE 5.7
16
2
1 −=−=
δ
Figure 1-2 Energy Diagram of an Electron Bound to He surface.
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 4
He the
15. 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):
h
⊥−
−−
=
==
Eeam
abwithezbz b
z 3
01
0
2
3
1
2
,
4
9
sinh
3
1
sinh
3
4
2)( λ
λ
λ
φ (1-3)
whereas the ground state energy is
−−=
4
32
0
2
2
1
a
b
bm
E
h
.
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 λ describes the
strength of the external field as compared to a characteristic field:
16. 7
cm
V
mea
Ec 864
2 3
0
2
==
h
(1-4)
which is roughly the field due to the image charge. In the limiting cases of weak (λ << 1)
and strong (λ >> 1) field, b is given by 3
2
0
2
0 3
4
4
3
1
λ
λ
=−=
a
b
and
a
b
respectively.
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):
( )
( )
zeE
dz
e
z
e
zU sh
⊥+
+
−−=
44
22
δδ
Here δh = (εh- 1)/( εh+ 1) and δs = (εs- εh)/( εs+ εh), εh, εs 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:
( ) zeE
z
e
zU h
⊥+−≈
4
2
δ
where we made the substitution 2
4d
e
EE hδ
+→ ⊥⊥ 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.
17. 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 4
He, other substances used to support 2D electrons include: solid 4
He, 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 3
He. 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).
18. 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 = n
e
π
ε
2
.
• Thermal energy kBT (or classical kinetic energy).
• 2D Fermi energy
m
n
EF
2
hπ= (or quantum kinetic energy).
Here n is the electron surface density, T – temperature, m and e – effective mass of
electron and charge respectively, ε is the dielectric constant of the substrate (ε = 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).
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).
19. 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
):
42
42
~
hε
em
nw
(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,
having found the lowest value for a hexagonal lattice. The magnitudes of reciprocal
lattice (which is also hexagonal) vectors form the following sequence:
Figure 1-4 2D Hexagonal Lattice - lowest energy configuration of 2D Coulomb crystal.
....16,13,12,9,7,4,3,1,
3
8
, 22
11 === pnGGpGp π (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
20. 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 electron-
electron 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 (~1011
cm-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
21. 12
all charged plasmas is a longitudinal plasmon. In a 3D plasma this is a dispersionless
oscillation of frequency
m
en
D
0
2
3
ε
ω = (1-7)
In 2D the dispersion relation is:
q
m
en
D
0
2
2
2ε
ω = (1-8)
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:
m
edn
cp
0
2
ε
= (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
22. 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 electron-
ripplon 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):
kdkkgr tanh32
+=
ρ
σ
ω (1-10)
where g – is the gravitational acceleration, σ – surface tension, ρ – 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:
1
20 −
== cm
g
kc
σ
ρ
(1-11)
23. 14
The numeric value is for liquid 4
He 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: ⊥= EenPe . 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:
g
Een
h
ρ
⊥
=∆
The absolute depression of the surface depends on the ratio of the uncharged area to the
total area of the helium surface:
total
edunch
LHe S
S
g
Een
d
arg
ρ
⊥
=∆
For typical parameter values (n = 5 108
cm-2
, =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).
⊥E
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
24. 15
( ) K
eE 3
2
10
4
−⊥
≅
πσ
(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
25. 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 two-
fluid nature of superfluid 4
He (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
ωρ
η
n
2
, where η and ρn are the normal fluid viscosity
and density respectively, ω is the sound frequency. For typical values of the parameters at
1K (η = 27 µP, ρn = 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.
26. 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
27. 18
probably due to a stiffening of the helium surface and to an increase of the
associated frequencies as the film thickness becomes small.
28. 19
2 Phase Diagram of 2DES on Liquid Helium
Electrons floating on the surface of liquid 4
He 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 100µm and
much larger than the interelectronic distance (~µm) 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):
K
V ee−
=Γ
For 2D electrons with surface density n
neV ee π2
=−
The kinetic energy (for either degenerate or nondegenerate 2DES) is given by
( )
∫
∞
−
+
=
0
2
2
1Tk
x
B
B
e
dxx
n
Tkm
K µ
π h (2-1)
29. 20
where ( )
−= 1ln
2
Tkm
n
B
B
eTk
hπ
µ is the chemical potential.
This expression reduces to <K>classical = kBT for nondegenerate electrons (EF << kBT) or to
<K>quantum = EF =
m
n2
hπ (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):
mΓ=Γ (2-3)
In other words, the ratio of average potential to average kinetic energy has to drop below
a certain value Γm 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
Γm ~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 Γm =137 in
n, T variables is presented in Figure 2-1 (Peeters 1984). Note that it is a qualitative phase
diagram since Γm = 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.
30. 21
0 2 4 6 8 10 12 14
0
1x10
11
2x10
11
3x10
11
4x10
11
1x10
13
2x10
13
QUANTAL
LIQUID
CLASSICAL
LIQUIDTHERMAL FLUCTUATIONS
n[cm
-2
]
T[K]
QUANTUM SOLID
QUANTUM FLUCTUATIONS
n
W
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
µm – 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
31. 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:
( )T
G
G
Rg η−
~ (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
( )T
TG
TG
µπ
η
2
)(
2
=
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
32. 23
transformed into a hexatic phase characterized by the absence of long-range positional
order while retaining bond orientation order.
The critical exponent ηG (Eq.(2-4)) and plasma parameter can be expressed via elastic
Lame: coefficients (µ is the shear modulus, B = µ+λ is the bulk modulus (Landau and
Lifshits 1970)) as follows:
( )λµµ
λµ
π
η
+
+
=
2
3
4
)(
2
GTk
T
B
G
( )λµµ
λµππ
+
+
=Γ
24
2
a
n
a is a lattice constant here. The KTHNY gives the following value for Tm:
( )λµ
λµµ
π +
+
=
2
)(4
16
2
B
m
k
a
T
In the approach to the hexatic phase the shear modulus µ behaves as:
( ) ( ) ))(1( ν
µµ TTconstTT mm −+= ν = 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
33. 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 λIDC: kr = 4πn/λIDC. If the excitation frequency matches that of a ripplon ωr(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: ( ) ( )
∑
−
= jii
eS
rrk
k
π2
1
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.
35. 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:
kdk
ne
kkgr tanh
4 2
22
32
−+=
ρ
π
ρ
σ
ω (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
29
4
2
102..2
)2(
1 −
== cm
g
e
nc
π
ρσ
and the wavevector corresponding to the instability is equal to the capillary constant of
liquid He 1
20 −
== cm
g
c
σ
ρ
k .
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).
36. 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:
kdk
ne
kk
d
gr tanh
4
)
3
( 2
22
3
4
2
−++=
ρ
π
ρ
σ
ρ
α
ω (2-6)
Here α is the Van der Waals constant (~ 10-15
erg for liquid helium on glass). This
effective enhancement of the gravitational constant 4
3~
d
gg
ρ
α
+= postpones the EHD
instability to much higher densities (as a matter of fact for d in the submicron range
g} >> g). The critical wavevector and density become
σ
α31
2
d
kc = and
4
22
)2(
31
de
nc
π
σα
= respectively. In fact, the experiments have shown that the film is
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:
2
30 )(2 en
d
dgdg π
α
ρρ +−= (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:
37. 28
ssHe dde
V
n
εεπ //
1
4 +
= (2-8)
here d, εHe – are the thickness and dielectric constant of liquid helium respectively, ds, εs
– thickness and dielectric constant of substrate if it is used. These two equations have to
15 35 55 75 95 115 135 155 175 195
d@mmD
9
19
29
n@108cm-2D
FilmStabilityfor V=20 Volt
15 35 55 75 95 115 135 155 175 195
d@mmD
9
19
29
39
n@108cm-2D
FilmStabilityforV=40 Volt
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 2.2 109
cm-2
.
(b)
(a)
38. 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, εs = 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
gddV ssHe ρπεε 4))//(
3
2
( 2
3
0 +> (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 Sε one solves Eq.(2-3) with
39. 30
1
1
41
1
2
2
+
−
=
+
−=−
S
S
s
s
ee where
dn
neV
ε
ε
δ
π
δ
π (2-10)
The phase diagram for several thicknesses of liquid helium film supported by a substrate
with δs=0.9 is shown in Figure 2-4.
0 1 2 3 4 5 6 7 8 9 10111213
0
1
2
T[K]
n[1012
cm-2
]
Bulk
300Å
100Å
70Å
Figure 2-4 2DES Phase Diagram on Dielectric Substrate vs. Helium Film Thickness.
The case of a metallic substrate corresponds to 1, =∞= ss δε 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
energy vanishes and the interaction is dipolar: 2
2eV ee ππ=− . Comparing to22
3
dn
40. 31
Eq.(2-2) for Fermi energy, one can see that for densities below
2
22
2
≅
emd
n
h
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.
41. 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 σe
in the plane
and the potential it creates Ve
:
( ) ( )z
z
VV eee
δ
ε
σ
0
2
2
2
2
r
r
−=
∂
∂
+
∂
∂
(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
42. 33
incident electromagnetic wave. Therefore one should look for a solution that has the
following form:
zqtie
q
e
eeVV
−−
= )( qrω
(3-2)
The choice of z dependence is set by the requirement that Ve
has to satisfy the Laplace
equation outside the 2D plasma and its derivative02
=∇ e
V
z
V
E
e
z
∂
∂
−= has to be
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:
02 ε
σ
q
V
e
qe
q = (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
)(~ )( zqzqzqzqtie
eDCeeBAeeV +++ −−−qrω
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.
h
d
Top Plate
2DES
Ground Plane
eh
ed
Figure 3-1 2DES Boundary Conditions: Two Dielectric Media.
43. 34
Then Eq.(3-3) should be replaced by:
( )qhqdq hd
e
qe
q
cothcoth0 εεε
σ
+
=V .
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
semiconductor wafer thickness. Eq.(3-3) then should be modified to
become:
−−+++
−−+++
+
=
)(sinh)()(sinh)(
)(cosh)()(cosh)(
)coth(0
sHeHessHeHes
sHeHessHeHes
He
e
qe
q
ddqddq
ddqddq
hqq
V
εεεε
εεεε
εε
σ
h
ds
Top Plate
2DES
Ground Plane
eHe
es
dHe
Figure 3-2 2DES Boundary Conditions in our Cell: Three Dielectric Media.
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:
d
e
qe
q
d
εε
σ
0
=V .
Later I will use cylindrical coordinates r,θ, z for which Poisson equation solution should
be written in terms of cylindrical harmonics:
44. 35
( ) zqitie
q
e
eeqrJeVV
−
= νθ
ν
ω
ν, (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(ωt-q0r)) propagating in the 2D
plane of electrons:
)(
1
E
r
uu −
∂
∂
=+
e
V
m
e
&&&
τ
(3-5)
Here u is the 2D displacement of an electron from its equilibrium position, τ 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:
r
u
∂
∂
= nee
σ (3-6)
where n – is the equilibrium electron density. Combining Eq.(3-5) with Eq.(3-6) one gets:
)(
1
2
22
r
E
r ∂
∂
−
∂
∂
=+
e
ee V
m
ne
σ
τ
σ &&&
Switching to harmonics, (both temporal and spatial) and substituting Eq.(3-3) for σe
, one
obtains:
( ) ( ) 00
0
0
2
22
,
2
Eqqe
q
q
m
ne
i
i
qV p
e
q
ετ
ω
ωω −=
−− (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
45. 36
excitation line. For example, if the normal modes are plane waves
(V ), then e(q( )(exp~ qr−∑ tiV
q
e
q
e
ω ) 0, q) appear as Fourier coefficients in the expansion
of the external wave into plane waves:
( ) ( ) ( )qrErqEE iqqei
q
−=−= ∑ exp,exp 0000 (3-8)
and turn out to be simply Kronecker’s deltas: e(q0, q) = δq0,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 χ:
( ) ( )
( )
( )
τ
ω
ωωε
ωχ
ωχ
i
q
m
ne
q
q
q
q
qqei
E
V
p
e
q
−−
≡
−=
220
2
0
0
0
1
2
,
,,
(3-9)
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 ωp(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
q
m
ne
p
0
2
2 ε
ω = (3-10)
46. 37
Screened
(long wave)
case: q d<<1
q
m
edn eff
p
0
2
ε
ω =
where
hd
eff
hd
d
εε
+=
(3-11)
Intermediate
case, as in
Figure 3-1.
)coth()coth(0
2
hqdq
q
m
en
hd
p
εεε
ω
+
= (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 τ 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/τ. τ 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
47. 38
external field the normal mode should be a standing wave therefore one can assume the
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
is equivalent to 0=
∂
∂
=Rr
e
r
V
0), =Rµν
. Substitution of Eq.(3-4) leads to a condition on the
allowed q : where R is the radius of 2DES pool. Therefore the allowed
wavevectors are:
(′ qJν
R
q
νµ
µν
ξ′
=, (3-13)
where νµξ′ are the extrema of Bessel functions:
Table 3-2 Bessel Extrema.
νµξ′ 1.84 3.05 3.83 4.20 5.32 5.33 6.42 6.71
ν 1 2 0 3 4 1 5 2
µ 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
• The electric field vector (
r∂
∂
−
e
V
) is in the same direction
48. 39
One then expects that in order to excite 2D plasmons the external field should be
longitudinal. However since divεE = 0 the plane electromagnetic waves are transverse.
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:
( ) ( ) ( ) ( )
( ) ( ) n
m
e
cqcqq
qqqq
ttt
ptp
π
ε
ωω
ωωωω
0
2
22222
2
2222
1
138.0
,
5
===
≈−=
(3-14)
where ( )qpω 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
ω1(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:
49. 40
1. Electron scattering off helium vapor atoms.
2. Electron scattering off surface waves of liquid – ripplon scattering.
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:
gasHe
gas
nA
mb
hπ
τ
3
8
= (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 4
He gas is:
sV
cm
T
T
a
b
gas
2
2
3
0
17.7
exp1300
=
−
µ
The ripplon scattering rate is:
( ) ( )
−+
−++= ⊥
⊥
36
115
36
1
3exp
16
ln
2
3
3exp
16
ln
2
4
11 22
2
0
2
0
22 π
στ T
E
a
T
T
E
a
EeT
Ee bb
r h
(3-16)
50. 41
Table 3-3 Mobility Measurements and Theory.
Figure 3-3 Polaronic Transition.
Low freq.
transport
Polaron transition
(Andrei 1984)
Figure 3-4 Wigner Transition.
Low freq.
Transport
Wigner transition
(Mehrotra,
Guenin et al.
1982)
51. 42
2D plasma
resonances
(Grimes and
Adams)
Figure 3-5 2D Electron Liquid Mobility for a Range of Pressing Fields.
Low freq.
Transport in 2D
liquid
(Sommer and
Tanner 1971)
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)
52. 43
where T is in units of energy and 2
2
2 bm
Eb
h
= .
In the regime of strong holding field (E⊥ > Ec see Eq.(1-4)) the scattering time becomes
temperature independent (Shikin and Monarkha Yu):
( )2
8
⊥
=
Ee
r
hσ
τ
where σ 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:
τdc = 2τac (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):
( )
3
0
4
2
2
4
=
lmb
TkBHe
r
σπ
ρ
τ
h
Here l0 is the electron’s localization length.
The total scattering time due to all independent mechanisms is:
rrgas 2
1111
ττττ
++=
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
53. 44
investigation of melting by measuring spectra. The theory of these coupled phonon-
ripplon (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:
)ln(
3
1
21
ct
q
G
mc
T
W = (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:
0
2
1
22
2
222
=
−Ω
−− ∑ ω
ω
ωω
GG
Gl V (3-18)
where ΩG - are ripplon frequencies corresponding to reciprocal vectors of Wigner lattice
Gp (Eq.(1-6)) for a hexagonal lattice, ωl – are longitudinal phonon frequencies in the
absence of coupling and
)exp( 1pW
m
n
eEVG −= ⊥
σ
(3-19)
are coupling constants. Here is the pressing field.⊥E
54. 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
formation of CPR. Horizontal dashed lines are ripplon frequencies corresponding to
Wigner crystal’s reciprocal lattice vectors. Solid lines represent resulting CPR modes.
Figure 3-7 Coupled (solid) and uncoupled (dashed) modes of Wigner solid and surface
waves of He (from (Fisher, Halperin et al.)).
55. 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
0
22
ωωω += l (3-20)
which is the solution of Eq.(3-18) when ω >> ΩG. It is associated with a formation of a
“dimple” depression of the liquid He surface caused by the electron pressure.
∑=
G
GV 2
0
2
1
ω is the frequency of electron’s oscillation in the dimple. We attribute the
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:
zu
r
uu ˆ
1
×−
∂
∂
=+ &&&& c
e
V
m
e
ω
τ
where ωc is the cyclotron frequency:
][6.17 GB
G
MHz
m
Be
c ==ω (3-21)
and is the unit vector normal to the plane. After performing spatial Fourier
transformation (V
zˆ
e
~ exp(-iqr)) we write down the equation of motion in terms of
velocity components parallel and perpendicular to wavevector q:
56. 47
||
||||
1
1
uuu
uqVi
m
e
uu
c
c
e
q
&&&&
&&&&
ω
τ
ω
τ
=+
−−=+
⊥⊥
⊥
(3-22)
Eq.(3-6) becomes: σq
e
= -inequ||. Substituting it into Eq.(3-3) and then into Eq.(3-22), one
gets the following secular equation determining magnetoplasmon dispersion law:
( )
0
2
22
=
−
−−−
τ
ω
ωωω
ωω
τ
ω
ωω
ii
iiq
c
cp
(3-23)
where ωp(q) is the 2D plasmons frequency in zero field (Table 3-1)). When dissipation is
negligible, Eq.(3-23) becomes:
( ) ( ) 222
cpmp qq ωωω += (3-24)
An interesting consequence of Eq.(3-23) is the linewidth doubling occurring as one goes
from the limit of weak magnetic field (ωc << ωp) to that of strong field – cyclotron
resonance (ωc >> ωp). For weak field the linewidth (defined as the width of resonance at
half height) is 1/τ, whereas for strong field it is 2/τ 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:
)()(
)( 222
RqJRqJRq
q
c
cp
νν
ω
ω
ν
ωωω
±=′
+=
(3-25)
Solutions of Eq.(3-25) )( ,, µνµνω ±± q determine the spectrum of bound 2DES.
57. 48
It is obvious that unlike the zero field case the modes differing only by signs of azimuthal
number ν will no longer be degenerate – magnetic field removes inversion symmetry.
The radial modes – the ones with ν = 0, will have wavevectors defined by the same
Eq.(3-13) as without magnetic field and their frequencies will evolve as:
2
0
22
)( cBp ωωω += = (3-26)
As shown in Figure 3-8 all other modes would split in magnetic field although most of
them approach ωc asymptotically with increasing field except for those with µ =1,ν<0.
Figure 3-8 Evolution of magnetoplasmons (from (Glattli, Andrei et al.
1985)).
58. 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:
p
c
p
c
q
R
c
ω
νω
ν
ν
→
→− 1,
(3-27)
cp – is long wavelength plasmons velocity given by Eq.(1-9).
Next chapter discusses the techniques used to excite and measure 2DES resonances.
59. 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.
60. 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 χ 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:
( )χαiTVV lineinout −= 1 (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 α includes the coupling
between 2DES and the line as well as some other factors. The assumption that coupling
is weak, i.e. αχ << 1 allows us to expand this formula:
( ) ( )"1'"1 222
χαχαχα δδ
+≈++= i
linein
i
lineinout eTVeTVV
Here δ=Arg(χ), χ’ = Re(χ) χ” = Im(χ). The measurement is made in such a way
(homodyne detection with phase matching delay line – see 6.8) that the phase information
(δ + Arg(Tline)) is dropped from the measured signal Vs:
( )"1 χα+= lineins TVV
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
61. 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 χ:
( ) tt rωχχχ cos0)( ∆+=
The signal after PSD is:
"χα ∆= lineins TVV
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:
21
21
ZZ
ZZ
r
+
−
=
However since χ”(ω) 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 κ. 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).
62. 53
To find the total transmission coefficient one sums all partial waves reflected from
discontinuities:
t1 t2
Vin Vout
l
Figure 4-1 Reflection between Two Discontinuities in Transmission Line.
iql
iql
iqliqliql
in
out
err
ett
ettrrettrrett
V
V
T 2
21
215
21
2
21
3
212121
1
)(
−
=+++== K
where q = k+iκ is the wavevector. For simplicity r’s are assumed to be <<1. Then
ikll
eerrT 2
211 κ−
+≈
The plot of transmission coefficient vs. frequency looks like a series of peaks whose
width is:
)1( 21
l
line err
l
v κ
ω −
−=∆ (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 ∆ωline:
∆ω2DES < ∆ωline
The qualitative generalization of Eq.(4-2) for the case of arbitrary number impedance
mismatches leads to the final criterion for mismatch tolerance:
KK ++=−−−<∆ −−
1132212 ),1( 21
llLerrerr
L
v ll
DES
κκ
ω total length of the line
This implies the following:
63. 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.
2DES 3D plasmon
Thin sheet of metal
(guiding action)
Dispersion equation q
m
en
D
0
2
2
ε
ω =
m
en
D
0
2
3
ε
ω =
ε
ω
c
k=
What determines
the allowed wavevectors
Boundary
conditions
Boundary
conditions
Boundary conditions
Frequency shift when placed
in a microwave cavity
Up Down 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
64. 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 α
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 = σe
v, where
v is the velocity of electrons. The electric field E created by electrons is E ~ σe
, 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:
gr
eff
e
l
eff
le
e
e
d
w
C
d
w
C
wen
R
wen
m
L
1*
*
*
2
*
;
;
1
;
ε
ε
µ
=
=
=
=
(4-3)
65. 56
0
0
Excitation Line
2DES
Capacitive Coupling
Z=50 00
1
0
L*e R*e
L*l
C*e
C*le
C*l
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; µ – 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; εeff, εeff1 –
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 ( ) ( ) ( )ωωω **
YZiq −= and characteristic impedance
( )
( )ω
ω
*
*
Y
Z
Zc = where Z*
(ω) and
Y*
(ω) are longitudinal impedance and transverse admittance per unit length of the line
respectively. For example for 2DES
( )
;
;
1
2
1
22*
**
1
2
12***
eff
gr
eff
gr
e
ee
c
gr
eff
gr
eff
eee
wen
d
i
wen
dm
Ci
RLi
Z
den
i
den
m
iCiRLiiq
εµωεω
ω
µ
ε
ω
ε
ωωω
−=
+
=
+−−=+−=
If 2DES losses are small (µ is large) the dispersion law becomes:
66. 57
gr
eff
den
m
q 2
1ε
ω= (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:
+−
−+
−=
−=
e
l
leele
lelel
e
l
e
l
e
l
e
l
V
V
YYY
YYY
I
I
dx
d
I
I
Z
Z
V
V
dx
d
***
***
*
*
0
0
(4-5)
where 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:
***
,
*
,
*
,
*
,
*
, leleelelelelel CiYCiYLiRZ ωωω ==+=
2
4)( 422222
2
2,1
leelel qqqqq
q
+−±+
= (4-6)
where ql, qe are the dispersion laws of uncoupled lines and
***2
ellele ZZYq = (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:
222
2,1 leqqq ±=
67. 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 π/R thus giving rise to two
peaks per each resonance of uncoupled 2DES. The splitting is therefore:
**
*
el
le
CC
C
=
∆
ω
ω
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
68. 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
eff
aL
a
ε
1
+
in terms of
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 Φ:
eff
c
kk ε
ω
2
2
222
,0 ==Φ+Φ∇ (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 ∗
Cv
11
where C is the capacitance per unit length of the line and∗
eff
c
v
ε
= – the propagation velocity.
69. 60
Table 4-2 Slow-Wave Structures.
Effective propagation velocity
eff
aL
a
cv
ε
1
+
=
Reduces crosstalk between fingers. More
sensitive to 2DES located above than
non-coplanar meander line.
The dispersion law has a gap – no
propagation at low frequencies.
Figure 4-4 Coplanar Meander Line.
a
L
y
x
Figure 4-3 Meander Line.
Figure 4-5 Interdigital Capacitor (IDC).
70. 61
H
d
Top Plate
Slow-wave
Structure
Bottom Plate
Vacuum
er
w
p
s
Figure 4-6 Cross-section of a Slow-Wave Structure in Inhomogeneous Environment.
The effective dielectric constant effε can be calculated as *
1
*
C
C
eff =ε where C1
*
is the
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:
which after substitution in the above equation yields a Laplace
equation in the plane perpendicular to the strips of the line:
iky
ezxUzyx ±
=Φ ),(),,(
02
2
2
2
=
∂
∂
+
∂
∂
z
U
x
U
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 effε . 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.
71. 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:
∑
−
=
j
zqxiq
j
jj
eeAzxU ),(
where qj for low frequencies are given by:
K,1,0 ±=+
+
= jwith
a
j
a
aL
c
q effj
π
ε
ω
(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).
θ = pq−2πj, p is the period of the line, other parameters are defined in Figure 4-6.
72. 63
z2DES >> d/εr 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
( )
( )
( )∑
∞
−∞=
−
−
−−
=
j
xiq
j
j j
e
dHq
zdHq
j
a
w
j
UzxU
sinh
sinhsin
, 0
π
π
above the meander line (z > 0). The x
component of the electric field is:
( )
( )
( )∑
∞
−∞=
−
−
−−
=
j
xiq
j
j
jx
j
e
dHq
zdHq
j
a
w
j
qiUzxE
sinh
sinhsin
, 0
π
π
(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
a
aL
f
c
eff
+
ε
. In this case the x component of electric field acting on 2D
electrons is determined by j = 0 term in Eq.(4-10):
( )
( )
xiqDES
x e
dHq
zdHq
a
w
UiqE 0
0
20
00
sinh
sinh −
−
−−
= (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
73. 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
flow through it.
Figure 4-8 IDC Dispersion Law from (Crampagne and Ahmadpanah 1977). See Figure 4-7 for
more details.
74. 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:
dSe
R
e rqi θ
µνµν ϕ
π
cos
,2,
1
∫∫= (4-12)
Here ϕν,µ – 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,θ – cylindrical
coordinates.
The evaluation of this integral (Glattli 1986) produces the following result:
22'
,
'
2'
,
2
,
)(
)(
1
2
qR
qRJRqi
e
−
−
=
µν
ν
µν
ν
µν
ξ
ξ
ν (4-13)
When the line is matched to 2DES
R
q
νµξ′
=Re (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.
![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]](https://image.slidesharecdn.com/8f8f8b97-c821-402c-9ffc-0a01c4383161-150618230031-lva1-app6892/85/skachkothesis-1-320.jpg)
