PHASE DIAGRAM OF A 2-DIMENSIONAL ELECTRON SYSTEM ON THE
SURFACE OF LIQUID
HELIUM
by
IVAN SKACHKO
A Dissertation submitted to the
Graduate School-New Brunswick
Rutgers, The State University of New Jersey
in partial fulfillment of the requirements
for the degree of
Doctor of Philosophy
Graduate Program in Physics and Astronomy
written under the direction of
Prof. Eva Andrei
and approved by
________________________
________________________
________________________
________________________
________________________
New Brunswick, New Jersey
[May 2006]
ABSTRACT OF THE DISSERTATION
PHASE DIAGRAM OF A 2-DIMENSIONAL ELECTRON SYSTEM ON THE
SURFACE OF LIQUID
HELIUM
By IVAN SKACHKO
Dissertation Director:
Prof. Eva Andrei
This work is a contribution to the study of the system of 2-dimensional electrons (2DES) bound to the
surface of liquid helium-4. The physical properties of the 2DES are probed through the excitation of normal
modes in the radio frequency range 10 MHz-1GHz. The normal mode spectra are expected to undergo a
radical change at the transition from liquid to crystal, the so-called Wigner transition. This difference is due
to coupling between 2DES modes and those of liquid helium surface. The experiments are performed for a
range of helium film thicknesses 1mm -100Å. For thin films the influence of the underlying substrate is of
great importance: it allows charging the film to high surface densities, which are impossible to reach on
bulk helium. This lets us explore a region of the phase diagram where quantum effects arising from zeropoint fluctuations play a significant role.
Slow-wave structures were developed to excite and detect the normal mode spectra including a meander
line and an interdigital capacitor.
ii
Acknowledgment and Dedication
This work is dedicated to my parents Valeria and Mikhail and to my wife Rodica.
I would like to express the most sincere gratitude to my research adviser Professor Eva
Andrei – the leading authority in the field of 2D electrons on helium, for her support and
encouragement during the years it took me to complete my research.
I received much spiritual and technical help from my colleagues Ross Newsome,
Guohong Lee, Ozgur Dogru, Xu Du, Zhili Xiao.
I greatly appreciate the direct participation in the project of Kurt Ketola, Gerard Deville,
Adam Hauser, Gail Schneider, Sylvain Benazet.
The machine and electronics shops at Rutgers Dept. of Physics were extremely skillful in
helping me to build the experimental apparatus.
iii
Table of Contents
ABSTRACT OF THE DISSERTATION ...................................................................... II
LISTS OF TABLES ........................................................................................................ VI
LIST OF ILLUSTRATIONS ....................................................................................... VII
1
INTRODUCTION: 2D ELECTRONS ON THE SURFACE OF LIQUID
HELIUM ........................................................... ERROR! BOOKMARK NOT DEFINED.
1.1
THE ORIGIN OF REDUCED DIMENSIONALITY ERROR! BOOKMARK NOT DEFINED.
1.2
CONFINEMENT OF ELECTRON TO THE SURFACE OF LIQUID HELIUM ......... ERROR!
BOOKMARK NOT DEFINED.
1.2.1
Influence of Externally Applied Electric Field .......... Error! Bookmark not
defined.
1.2.2
Binding to a Thin Film .................................Error! Bookmark not defined.
1.2.3
Hartree Field (Mutual Repulsion of Electrons) ......... Error! Bookmark not
defined.
1.3
WIGNER CRYSTAL ....................................... ERROR! BOOKMARK NOT DEFINED.
1.4
COLLECTIVE MODES OF 2DES ..................... ERROR! BOOKMARK NOT DEFINED.
1.5
THE DYNAMICS OF LIQUID HELIUM AND ITS EFFECT ON 2DES ................ ERROR!
BOOKMARK NOT DEFINED.
1.5.1
The Surface Waves - Ripplons .....................Error! Bookmark not defined.
1.5.2
Interaction Between 2DES and the Surface of Liquid Helium .......... Error!
Bookmark not defined.
1.5.3
Thin Film Effects ..........................................Error! Bookmark not defined.
1.6
SUMMARY .................................................... ERROR! BOOKMARK NOT DEFINED.
2
PHASE DIAGRAM OF 2DES ON LIQUID HELIUM.... ERROR! BOOKMARK
NOT DEFINED.
2.1
2.2
ELECTRONS ON BULK HELIUM ..................... ERROR! BOOKMARK NOT DEFINED.
WIGNER SOLID AND 2D MELTING MECHANISMS ......... ERROR! BOOKMARK NOT
DEFINED.
2.3
ELECTROHYDRODYNAMIC (EHD) INSTABILITY OF A CHARGED LIQUID SURFACE
ERROR! BOOKMARK NOT DEFINED.
2.4
INFLUENCE OF A SUBSTRATE ON PHASE DIAGRAM ...... ERROR! BOOKMARK NOT
DEFINED.
3
SPECTRUM OF 2DES ............................ ERROR! BOOKMARK NOT DEFINED.
3.1
3.2
3.3
3.4
3.5
2D DRUDE MODEL....................................... ERROR! BOOKMARK NOT DEFINED.
ELECTRON SOLID MODES ............................ ERROR! BOOKMARK NOT DEFINED.
2DES MOBILITY .......................................... ERROR! BOOKMARK NOT DEFINED.
COUPLED PHONON-RIPPLON MODES ........... ERROR! BOOKMARK NOT DEFINED.
SPECTRUM OF 2DES IN MAGNETIC FIELD.... ERROR! BOOKMARK NOT DEFINED.
iv
4
MEASUREMENT OF 2DES SPECTRUM USING SLOW-WAVE
STRUCTURE ................................................... ERROR! BOOKMARK NOT DEFINED.
4.1
4.2
4.3
4.4
2DES IN ELECTRO-MAGNETIC FIELD .......... ERROR! BOOKMARK NOT DEFINED.
DISTRIBUTED CIRCUIT MODEL .................... ERROR! BOOKMARK NOT DEFINED.
SLOW-WAVE STRUCTURES .......................... ERROR! BOOKMARK NOT DEFINED.
COUPLING BETWEEN MEANDER LINE AND 2DES.......... ERROR! BOOKMARK NOT
DEFINED.
5
OTHER METHODS FOR INVESTIGATION OF PHASE TRANSITION IN
2DES ON LIQUID HELIUM FILM .............. ERROR! BOOKMARK NOT DEFINED.
5.1
MICROWAVE CAVITY TECHNIQUE ............... ERROR! BOOKMARK NOT DEFINED.
5.2
SOMMER-TANNER METHOD (LOW-FREQUENCY TRANSPORT) .................. ERROR!
BOOKMARK NOT DEFINED.
6
EXPERIMENTAL APPARATUS AND CELL ....... ERROR! BOOKMARK NOT
DEFINED.
6.1
CRYOSTAT ................................................... ERROR! BOOKMARK NOT DEFINED.
6.1.1
Vibration Isolation .......................................Error! Bookmark not defined.
6.2
DILUTION REFRIGERATOR............................ ERROR! BOOKMARK NOT DEFINED.
6.3
THERMOMETRY............................................ ERROR! BOOKMARK NOT DEFINED.
6.4
MAGNET ...................................................... ERROR! BOOKMARK NOT DEFINED.
6.5
EXPERIMENTAL CELL .................................. ERROR! BOOKMARK NOT DEFINED.
6.5.1
Production of Electrons ...............................Error! Bookmark not defined.
6.5.2
2DES Confinement .......................................Error! Bookmark not defined.
6.5.3
Measurement and Control of Liquid Helium Level and Film Thickness
Error! Bookmark not defined.
6.6
MEANDER LINE............................................ ERROR! BOOKMARK NOT DEFINED.
6.7
SUBSTRATE SMOOTHNESS CONTROL ........... ERROR! BOOKMARK NOT DEFINED.
6.8
ELECTRONICS............................................... ERROR! BOOKMARK NOT DEFINED.
7
ANALYSIS OF EXPERIMENTAL DATA .............. ERROR! BOOKMARK NOT
DEFINED.
7.1
IDENTIFICATION OF THE INDIVIDUAL RESONANCES ..... ERROR! BOOKMARK NOT
DEFINED.
7.2
EVOLUTION OF 2DES RESONANCES WITH TEMPERATURE AND PHASE DIAGRAM
ERROR! BOOKMARK NOT DEFINED.
2DES Heating ..............................................................Error! Bookmark not defined.
Sliding Transition.........................................................Error! Bookmark not defined.
7.3
2DES MOBILITY MEASUREMENT ................ ERROR! BOOKMARK NOT DEFINED.
7.4
OBSERVATION OF MAGNETOPLASMON ........ ERROR! BOOKMARK NOT DEFINED.
8
CONCLUSION ........................................ ERROR! BOOKMARK NOT DEFINED.
APPENDICES .................................................. ERROR! BOOKMARK NOT DEFINED.
A
RIPPLON DISPERSION LAW.............. ERROR! BOOKMARK NOT DEFINED.
v
B ROUGHNESS OF LIQUID HELIUM SURFACE DUE TO THERMALLY
EXCITED RIPPLONS .................................... ERROR! BOOKMARK NOT DEFINED.
C PARAMETRIC RESONANCE IN HEII.................. ERROR! BOOKMARK NOT
DEFINED.
D HEAT EXCHANGERS FOR LIQUID HE BATH .. ERROR! BOOKMARK NOT
DEFINED.
E
EXPERIMENTAL PROCEDURE......... ERROR! BOOKMARK NOT DEFINED.
PREPARATION .......................................................... ERROR! BOOKMARK NOT DEFINED.
COOLDOWN ............................................................. ERROR! BOOKMARK NOT DEFINED.
DILUTION FRIDGE OPERATION ................................. ERROR! BOOKMARK NOT DEFINED.
FILLING THE CELL WITH HELIUM ............................. ERROR! BOOKMARK NOT DEFINED.
CHARGING THE HELIUM SURFACE ........................... ERROR! BOOKMARK NOT DEFINED.
DATA COLLECTION .................................................. ERROR! BOOKMARK NOT DEFINED.
REFERENCES ................................................. ERROR! BOOKMARK NOT DEFINED.
vi
Lists of tables
TABLE 1-1 BASIC PROPERTIES OF 2DES. ..................... ERROR! BOOKMARK NOT DEFINED.
TABLE 3-1 2D PLASMA DISPERSION LAW IN VARIOUS LIMITS. ... ERROR! BOOKMARK NOT
DEFINED.
TABLE 3-2 BESSEL EXTREMA. ..................................... ERROR! BOOKMARK NOT DEFINED.
TABLE 3-3 MOBILITY MEASUREMENTS AND THEORY. ERROR! BOOKMARK NOT DEFINED.
TABLE 4-1 ELECTROMAGNETIC RESPONSE COMPARISON. ........... ERROR! BOOKMARK NOT
DEFINED.
TABLE 4-2 SLOW-WAVE STRUCTURES. ....................... ERROR! BOOKMARK NOT DEFINED.
vii
List of illustrations
FIGURE 1-1 SCREENING MECHANISM GIVING RISE TO ELECTRON CONFINEMENT TO A
LIQUID HELIUM SURFACE. ................................... ERROR! BOOKMARK NOT DEFINED.
FIGURE 1-2 ENERGY DIAGRAM OF AN ELECTRON BOUND TO HE SURFACE............... ERROR!
BOOKMARK NOT DEFINED.
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). ERROR! BOOKMARK
NOT DEFINED.
FIGURE 1-4 2D HEXAGONAL LATTICE - LOWEST ENERGY CONFIGURATION OF 2D COULOMB
CRYSTAL. ............................................................. ERROR! BOOKMARK NOT DEFINED.
FIGURE 2-1 2DES PHASE DIAGRAM ON HELIUM BULK. .............. ERROR! BOOKMARK NOT
DEFINED.
FIGURE 2-2 2DES DIFFRACTION PATTERNS (T. WILLIAMS 1995).ERROR! BOOKMARK NOT
DEFINED.
FIGURE 2-3 CHARGING INSTABILITY. DASHED LINE IS THE PLOT OF N VS. D FROM EQ.(2-7).
DOT-DASHED LINE IS THE PLOT OF N VS. D FROM EQ.(2-8). ONE OF THEIR
INTERSECTIONS CORRESPONDS TO THE STABLE CHARGE DENSITY. SOLID LINE IS THE
EHD LIMIT OF 2.2 109CM-2................................... ERROR! BOOKMARK NOT DEFINED.
FIGURE 2-4 2DES PHASE DIAGRAM ON DIELECTRIC SUBSTRATE VS. HELIUM FILM
THICKNESS. .......................................................... ERROR! BOOKMARK NOT DEFINED.
FIGURE 2-5 2DES PHASE DIAGRAM ON METALLIC SUBSTRATE VS. HELIUM FILM
THICKNESS. .......................................................... ERROR! BOOKMARK NOT DEFINED.
FIGURE 3-1 2DES BOUNDARY CONDITIONS: TWO DIELECTRIC MEDIA. ................... ERROR!
BOOKMARK NOT DEFINED.
FIGURE 3-2 2DES BOUNDARY CONDITIONS IN OUR CELL: THREE DIELECTRIC MEDIA.
............................................................................. ERROR! BOOKMARK NOT DEFINED.
FIGURE 3-3 POLARONIC TRANSITION. .......................... ERROR! BOOKMARK NOT DEFINED.
FIGURE 3-4 WIGNER TRANSITION. ............................... ERROR! BOOKMARK NOT DEFINED.
FIGURE 3-5 2D ELECTRON LIQUID MOBILITY FOR A RANGE OF PRESSING FIELDS. .. ERROR!
BOOKMARK NOT DEFINED.
FIGURE 3-6 VARIOUS METHODS OF MOBILITY MEASUREMENT. .. ERROR! BOOKMARK NOT
DEFINED.
FIGURE 3-7 COUPLED (SOLID) AND UNCOUPLED (DASHED) MODES OF WIGNER SOLID AND
SURFACE WAVES OF HE (FROM (FISHER, HALPERIN ET AL.)). ERROR! BOOKMARK NOT
DEFINED.
FIGURE 3-8 EVOLUTION OF MAGNETOPLASMONS (FROM (GLATTLI, ANDREI ET AL. 1985)).
............................................................................. ERROR! BOOKMARK NOT DEFINED.
FIGURE 4-1 REFLECTION BETWEEN TWO DISCONTINUITIES IN TRANSMISSION LINE. ERROR!
BOOKMARK NOT DEFINED.
FIGURE 4-2 EQUIVALENT CIRCUIT OF 2DES AND EXCITATION LINE. STAR DENOTES THE
VALUE PER UNIT LENGTH. .................................... ERROR! BOOKMARK NOT DEFINED.
viii
FIGURE 4-3 MEANDER LINE. ........................................ ERROR! BOOKMARK NOT DEFINED.
FIGURE 4-4 COPLANAR MEANDER LINE. ..................... ERROR! BOOKMARK NOT DEFINED.
FIGURE 4-5 INTERDIGITAL CAPACITOR (IDC). ............. ERROR! BOOKMARK NOT DEFINED.
FIGURE 4-6 CROSS-SECTION OF A SLOW-WAVE STRUCTURE IN INHOMOGENEOUS
ENVIRONMENT. .................................................... ERROR! BOOKMARK NOT DEFINED.
FIGURE 4-7 MEANDER LINE DISPERSION LAW FROM (CRAMPAGNE AND AHMADPANAH
1977). = PQ2J, P IS THE PERIOD OF THE LINE, OTHER PARAMETERS ARE DEFINED IN
FIGURE 4-6. ......................................................... ERROR! BOOKMARK NOT DEFINED.
FIGURE 4-8 IDC DISPERSION LAW FROM (CRAMPAGNE AND AHMADPANAH 1977). SEE
FIGURE 4-7 FOR MORE DETAILS. ........................... ERROR! BOOKMARK NOT DEFINED.
FIGURE 5-1 SOMMER-TANNER TECHNIQUE. ................ ERROR! BOOKMARK NOT DEFINED.
FIGURE 6-1 EXPERIMENTAL CELL MOUNTED ON DILUTION FRIDGE. ... ERROR! BOOKMARK
NOT DEFINED.
FIGURE 6-2 CRYOSTAT AND DILUTION UNIT. .............. ERROR! BOOKMARK NOT DEFINED.
FIGURE 6-3 CRYOSTAT ALIGNMENT SYSTEM. ............. ERROR! BOOKMARK NOT DEFINED.
FIGURE 6-4 CROSS-BELLOWS DESIGN ISOLATES A PUMPING LINE FROM THE CRYOSTAT.
............................................................................. ERROR! BOOKMARK NOT DEFINED.
FIGURE 6-5 SPLIT-COIL CROSSED-FIELD MAGNET. ..... ERROR! BOOKMARK NOT DEFINED.
FIGURE 6-6 EXPERIMENTAL CELL. ............................... ERROR! BOOKMARK NOT DEFINED.
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. ................................................................... ERROR! BOOKMARK NOT DEFINED.
FIGURE 6-8 THIN FILM MEASUREMENT. ...................... ERROR! BOOKMARK NOT DEFINED.
FIGURE 6-9 THICK FILM MEASUREMENT. .................... ERROR! BOOKMARK NOT DEFINED.
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. ............................................................... ERROR! BOOKMARK NOT DEFINED.
FIGURE 6-11 SPRING CONTACT. MANUFACTURED BY EVERETT CHARLES TECHNOLOGIES
(PART # MEP-30U). ............................................ ERROR! BOOKMARK NOT DEFINED.
FIGURE 6-12 AFM IMAGE OF PASSIVATED SI (111) SURFACE: TRIANGULAR PITS ARE
APPROXIMATELY 150Å DEEP. RMS ROUGHNESS IS 55Å. ..... ERROR! BOOKMARK NOT
DEFINED.
FIGURE 6-13 DIRECTLY TRANSMITTED SIGNAL VS. MODULATED ONE. .................... ERROR!
BOOKMARK NOT DEFINED.
FIGURE 6-14 LOCK-IN OUTPUT CORRESPONDING TO MODULATION OF THE RESONANCE
FREQUENCY. ........................................................ ERROR! BOOKMARK NOT DEFINED.
FIGURE 6-15 LOCK-IN OUTPUT CORRESPONDING TO MODULATION OF THE LINEWIDTH.
............................................................................. ERROR! BOOKMARK NOT DEFINED.
FIGURE 6-16 LOCK-IN OUTPUT CORRESPONDING TO MODULATION OF BOTH THE
RESONANCE FREQUENCY AND THE LINEWIDTH. .. ERROR! BOOKMARK NOT DEFINED.
FIGURE 6-17 MEASUREMENT SETUP. ........................... ERROR! BOOKMARK NOT DEFINED.
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.
............................................................................. ERROR! BOOKMARK NOT DEFINED.
ix
FIGURE 7-2 THE SPECTRA CORRESPONDING TO DIFFERENT DENSITIES ARE PLOTTED ON
LOG F SCALE AND SHIFTED BY –0.5 LOG N. .......... ERROR! BOOKMARK NOT DEFINED.
FIGURE 7-3 DETERMINATION OF RESONANCE FREQUENCY. ........ ERROR! BOOKMARK NOT
DEFINED.
FIGURE 7-4 THE EVOLUTION OF RESONANCE FREQUENCIES FOR THE SPECTRA OF FIGURE
7-1 WITH CHARGING VOLTAGE. THE SLOPE OF LINES ON THIS LOG-LOG PLOT IS ½, AS
EXPECTED FROM 2D PLASMA DISPERSION ~ N0.5. .............. ERROR! BOOKMARK NOT
DEFINED.
FIGURE 7-5 MATCHING OF AN OBSERVED SPECTRUM (THICK LINE) TO 2D PLASMA
RESONANCES (THIN LINES). HERE () ARE THE AZIMUTHAL AND RADIAL NUMBERS
RESPECTIVELY. THE AMPLITUDE OF EACH RESONANCE IN THEORETICAL SPECTRUM
REFLECTS THE COUPLING OF THAT MODE TO THE EXCITATION STRUCTURE (SEE 4.4) AS
DISCUSSED IN THE TEXT. ...................................... ERROR! BOOKMARK NOT DEFINED.
FIGURE 7-6 CORRECTED SPECTRUM MATCHING. ......... ERROR! BOOKMARK NOT DEFINED.
FIGURE 7-7 THE SEQUENCE OF RESONANCE FREQUENCIES VS. THEIR NUMBER I:
ASYMPTOTIC BEHAVIOR IS I0.5. ............................. ERROR! BOOKMARK NOT DEFINED.
FIGURE 7-8 EVOLUTION OF THE 2DES SPECTRUM WITH TEMPERATURE. THE SPECTRA ARE
SHIFTED VERTICALLY TO FACILITATE COMPARISON. TM DENOTES THE EXPECTED
WIGNER SOLID MELTING TEMPERATURE FOR N = 6 107CM-2. THE INCIDENT RF POWER
IS -44DBM. ........................................................... ERROR! BOOKMARK NOT DEFINED.
FIGURE 7-9 NONLINEARITY OF 2D ELECTRON CRYSTAL RESPONSE. ... ERROR! BOOKMARK
NOT DEFINED.
FIGURE 7-10 CLASSICAL PART OF 2DES PHASE DIAGRAM.......... ERROR! BOOKMARK NOT
DEFINED.
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. .......... ERROR!
BOOKMARK NOT DEFINED.
FIGURE 7-12 2DES SPECTRUM IN MAGNETIC FIELD. THE SPECTRA ARE SHIFTED
VERTICALLY TO FACILITATE COMPARISON. .......... ERROR! BOOKMARK NOT DEFINED.
FIGURE 7-13 CYCLOTRON FREQUENCY VS. MAGNETIC FIELD FOR THE SPECTRA OF FIGURE
7-12. .................................................................... ERROR! BOOKMARK NOT DEFINED.
FIGURE A-1 SURFACE WAVE. ...................................... ERROR! BOOKMARK NOT DEFINED.
FIGURE B-1 RMS ROUGHNESS OF LIQUID 4HE SURFACE (FROM (COLE 1970)). ....... ERROR!
BOOKMARK NOT DEFINED.
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. ... ERROR!
BOOKMARK NOT DEFINED.
FIGURE D-1 HEAT EXCHANGER MOUNTED ON CRYOSTAT'S BAFFLE... ERROR! BOOKMARK
NOT DEFINED.
FIGURE E-1 TOP OF THE CRYOSTAT. ............................ ERROR! BOOKMARK NOT DEFINED.
FIGURE E-2 50MK RADIATION SHIELD. ....................... ERROR! BOOKMARK NOT DEFINED.
FIGURE E-3 STILL RADIATION SHIELD. ........................ ERROR! BOOKMARK NOT DEFINED.
x
FIGURE E-4 DILUTION FRIDGE CONTROL PANEL. ........ ERROR! BOOKMARK NOT DEFINED.
xi