SURFACE OF LIQUID
4HELIUM AND OTHER CRYOGENIC SUBSTRATES
DISSERTATION
zur Erlangung des akademischen Grades des Doktors der Naturwissenschaften (Dr. rer. nat.)
an der Universität Konstanz,
Mathematisch-Naturwissenschaftliche Sektion, Fachbereich für Physik,
Konstanz, Germany vorgelegt von
Ram-Krishna Thakur
2006
Tag der mündlichen Prüfung: 24. November 2006
1. Referent: Prof. Dr. Paul Leiderer 2. Referent: Prof. Dr. Peter Nielaba
Konstanzer Online-Publikations-System (KOPS) URL: http://www.ub.uni-konstanz.de/kops/volltexte/2007/3798/
URN: http://nbn-resolving.de/urn:nbn:de:bsz:352-opus-37988
Acknowledgement
I am extremely indebted to my supervisor Hon’ble Prof. Dr. Paul Leiderer for the opportunity he gave me, his sincere support, inspiring guidance and affection throughout the past years. In this context, I take vicarious pleasure and feel very lucky to have such a lively person as my doctor father.
I am highly grateful to PD Dr. J. Klier for kind cooperation, very careful training in the field of low temperature physics, stimulating inspiration along with original comments related to experimental technique.
I take pleasure to extend special thanks to Prof. Dr J. Bonneberg, Prof.
Dr. G. Schatz, Prof. Dr. Valeri Shikin for their constructive comments, and Prof. Dr.
H. Dehnen for ace good time during work in Konstanz, and Dr. B-U Runge for suggestion and help.
I am indebted to my colleague and friend Dr Ismail Karakurt, for his creative advice, enlightening discussions related to the aspects of LTP.
I am highly thankful to my research mates; Dr Anni Wakata, Dr. Gaza Seezeli, Mr.
S. Koutsoumpos, Dr. Masud Soheili, Dr. Valentin Iov, Dr. (Ms.) Irena Doicesscu, Jörg, Michael, Dr. Conrad Mangold, Armin-Martin duo, and Mark for their help, and pleasant time during the work.
I extend heartfelt thanks to the University Mechanical and Electronic work shop crews, who have made all the LT components under high preference:
Especially Mr. Michael Weihland, Mr. D’Imperio, Mr. Straus and co-workers along with Mr. Louis Kukk and Mr. Christoph Goldbach for an excellent technical aids and essential training. Many thanks to liquid helium laboratory co-workers, Mr Ralf Sieber and Mr Hartmut Görig, for providing liquid helium in time, taking care of the system and also for numerous valuable suggestions during the experimental work.
I acknowledge Deutsche Forschungsgemeinschaft, for, the research work presented in this thesis has been possible with the financial assistance.
America, Canada, and India: Mr. Helmut Jager, Dr. Anuj, Dr. V.R. Saha, Dr. Pinaki R. Manjhi, Prof. Dr. Narayan P. Adikari, Dr. K. Muttu, Dr. Alka ji, Prof. Dr. O.P.S.
Negi, Prof. Dr. T.P. Pareek (Dear dada), Mr. R.K. Tewari, Dr. Phaneeswara Rao, Dr Sudarshan Rao ‘Suddu’, Dr. (Mrs.) Mona Rahelker + Dr. Rahul Bahuliker, Ms.
Sanmukh, Dr. Hem chandra, Dr. Hemwati Nandan, and Dr Rajesh Kumar Singh for their help and special effects throughout.
Any accomplishment requires the effort of many people and resources, where scientific work is one of the most important activity among them. In this context, many examples, stories, anecdotes are the result of a collection of knowledge from various sources, such as seminar participants, several speakers, journals, news papers, magazines over decades. Unfortunately, sources were not always noted or available. Hence, it become impractical to provide an accurate acknowledgement.
Source of inspiration: Linus Carl Pauling
My wife, Divya and son, Abhinav whose support is instrumental in accomplishing the task, thanks for sharing some of the surreal times during the work.
Watching the tides of Bodan Sea of Konstaz always gave me new zeal throughout the work.
Every effort has been made to give credit where it is due herein. I apologize if inadvertently it might have crept in.
Dedicated
To all my teachers, with whom I learnt much
Hon’ble Prof. Dr. Paul Leiderer, Prof. Dr. V. D. Gupta, Prof. Dr. C. K.
Misra, Prof. Dr. Poonam Tandon, Prof. Dr. M.S. Sodha, (Late) Dr. R.L. Shukla, Prof.
Dr. L.M. Bali, Prof. Dr. U..D. Misra, Prof. Dr. S.N. Shukla, Prof. Dr. P.K. Rath, Dr.
Lal Singh, Prof. Dr. S.M. Kamil Rizvi, Dr. S.M. Sagar Zaidi, Sri Baidyanath Prasad Verma, Sri Krishna Bahadur Pandey, Sri Ravindra Jha, Sri Raghunath Prasad, and my Mom-Dad.
.…Without whose inspiration and faith it would not have been completed!
K Kelvin VC Vacuum chamber OVC Outer vacuum chamber UHV Ultra high vacuum IVC Inner vacuum chamber LN2 Liquid nitrogen
L 4He Liquid 4helium T Temperature Cu Copper h Height d Diameter do Outer diameter di Inner diameter C Capacitance
S-T Sommer-Tanner (electrodes) TP Top plate
BP Bottom plate nG Gas-density
Figure1. Phase diagram of liquid 4helium [ Fritz London (‘64)]... 3 Figure2. Normal and super fluid component vs. temperature [ Andronikashvili’s
experiment (Tilley and Tilley)] ... 4 Figure3. A charge approaching dielectric material at z distance above surface
polarizes the atoms of liquid 4He... 11 Figure 4. The image potential for motion in z-direction(first two states are shown),
and potential barrier of 1 eV together with the ground state and first excited state of the electron on liquid 4He are shown. ... 12 Figure 5. Ground and first excited state wave function. Distance from the surface ( in
units of beff=7.6 nm)... 13 Figure 6 : Experimental observation of mm-wave absorption derivative as a function
of potential difference across the experimental cell with a layer of electrons on the surface of liquid helium taken at 220 GHz. The cell height is 0.32 cm and the bath temperature is 1.2 K. (After C.C. Grimes et al, PR B, 1976). ... 17 Figure 7 : Dependence of transition frequency as a function of voltage across the
experimental cell. Transitions are plotted as a solid curves, which are the result of the variational calculation based on simple model potential, and crosses are measured data points (After C.C. Grimes et al, PR B, 1976). ... 17 Figure8. Surface electron mobility on liquid 4He (After Sommer-Tanner, PRL ’71)
... 19 Figure9. Mobility of electrons in the 4He. Solid curve shows theoretical calculation
of mobility given by Saitoh taking into account both the gas-atom scattering and the ripplon scattering. ... 21 Figure10: Experimental traces showing sheet of electrons has crystallized suddenly
into a triangular lattice with decreasing temperature of 0.457 K on the surface of liquid helium ( After Grimes and Adams, PRL, 1979)... 23 Figure11: Crystallized triangular electron lattice for Γ ≥127 ... 24 Figure12. Phase diagram of a 2DES for bulk helium ( After Peeters et al ’83). ... 25 Figure13. Maximum electron density as a function of helium film thickness for a
metallic and non-metallic substrate for 2DES is shown. Inset shows the critical electron density that corresponds to wave vector of the electro-hydrodynamic
substrate at T=1.6 K as a function of charge density. Uncharged film thickness d0
was 22 and 42 nm respectively. The film thickness on charging has been
confirmed by an ellipsometric determination of d for two values of an uncharged film (After Leiderer PRL’84). ... 29 Figure15. Proposal to trap each electron above a patterned substrate. The shape of
field lines and the gate voltages are included. The average distance of the electron from the helium surface are ~11.4 nm and ~45 nm respectively that can be used as a qubit ( After Dykman et al science ’99) ... 30 Figure16: Schematic drawing of Oxford Cryostat... 34 Figure17. An overview (camera Image) of 1K-pot (top) along with Experimental cell
(bottom). ... 36 Figure18. Schematic drawing of experimental cell where all the experiments are
performed... 38 Figure19. Camera image of the experimental cell used in all measurements... 39 Figure20. The calibration curve of RuO2 for experimental cell. ... 40 Figure21. Stability of high temperature for similar measurements on Neon surface.
Cell resistance 3,140 Ohm corresponds to ~22 K. Straight line indicates that cell temperature is stable for many hours... 42 Figure22 Observed small amount of charge after each pulse at liquid helium
thickness,d~0.8 mm,Vdc =12 ,V Vg = −8V , cell temperature~1.68 K,
7 2
8.76*10
ne= cm− ... 48 Figure23. Bottom part of experimental cell (left figure) along with Sommer-Tanner
electrodes (Right figure) arrangement. ... 49 Figure24 a. Top plate; b. Guard ring. ... 50 Figure25. Electronic circuit diagram for Sommer-Tanner electrode structure used
inside experimental set up to set in measurement of escape of electrons over L
4He surface... 52 Figure26. Observation of stable signal at minimum liquid helium thickness over bare ST electrode. ... 56 Figure27. A typical recorded Lock-in trace of signals (amplitude and phase) vs time
that was charged up to saturation. The stability of charged helium surface is illustrated for the helium thickness, dLHe~ 0.85 mm, and Vdc= 3 V, VG=-2 V, Vtop-plate at ground potential, ... 57
function of time shows the stability of charged helium surface for dLHe = 1mm thickness, and Vdc= 6 V, VG=-4 V, Vtop-plate at ground potential, surface electron density, se=3.5*107 /cm2. ... 58 Figure29. Calibration of experimental observation of LI Amplitude and phase as a
function of electron density for liquid helium thickness, d=1.0 mm... 62 Figure30. Amplitude of fully charged surface taken from the recorded Lock-in trace
vs no. of pulses. All the negative pulses were applied through the bottom electrode at saturated electron density and fixed temperature. After each pulse surface was charged. Back-ground signal was 60 nV (i.e. without charging the surface) ... 63 Figure31. The plot illustrates situation of imposed Top plate positive voltage pulses,
which is used to expel electrons from the surface of super fluid 4helium,
corresponding electrostatic barrier of different extracting voltage, and saturated electron density as a function of time. ... 64 Figure32. Amplitude of escaping electrons taken from the recorded Lock-in trace as
a function of imposed pulse duration through top-plate. (A) at Vdc=12 V, VG=-8 V, dLHe=1 mm, ne= ×7 107e cm/ 2and VTP=14 V. (B) at Vdc=6 V, VG=-4 V, dLHe=0.8 mm, ne = ×7 107e cm/ 2and VTP=13 V... 65 Figure33. Equipotential lines for two different holding and top-plate voltages
corresponding to two different electron densities. Top- figure (A) is calculated for Vdc=12 V, VG=-8 V, and VTP=14 V. Bottom-figure (B) is at Vdc=6 V, VG=-4 V, and VTP=13 . In both the case liquid helium thickness has not been taken into account... 66 Figure34. Dependence of top plate pulses at 0,4, 0.6, 0.8, and 1 mm L 4He.Vdc=12
V; VG= -8 V. Figure shows 50% of transition time of escaped electrons from the surface as a function of positive top plate voltage pulse. Data are taken from the recorded Lock-in trace. For example, one original data of red square showing transition time 200µs for VExt=14 V, at dLHe= 1.0 mm is taken from Figure 32-A.
... 67 Figure35. The figure illustrates the transition time of escaped electrons vs L 4He
thickness at Vdc=12 V, VG= -8, and a series of extracting voltage pulse to top- plate... 69 Figure36. Escape rate as a function of surface charge density at, Vdc=12 V, VG=-8
V, D=6.5 mm, Tcell~1.7 K, VExt in the range of 14V <VExt <22V , and helium thickness0.4<dL He4 <1.0mm (L 4He thickness decreases from left to right),.... 70
Figure shows 50% of transition time of escaping electrons from the surface as a function of positive top plate voltage pulse... 71 Figure38. Escape rate as a function of surface charge density for Vdc=9 V, VG=-6 V,
D=6.5 mm, Tcell~1.7 K, VExt in the range of12V <VExt <16V
,ne =5.25 10× 7cm−2, and dL He4 =1.0mm... 71 Figure39. Dependence of top plate pulse at 1 mm and 0.8 mm L 4He. Tcell= 1.7 K.
Vdc=6 V; VG= -4 V. Figure shows 50% of transition time of escaping electrons from the surface as a function of positive top plate voltage pulse... 72 Figure40. The plot demonstrates Escape rate 1/τ , as a function of surface charge
density. At fixed temperature, Tcell= 1.7 K. Vdc=6 V; VG= -4 V. Measurements were done at 1 mm and 0.8 mm L 4He thickness that gives two different electron densities. ... 73 Figure41. Transition time of escaped electrons vs extracting voltage imposed to the
top plate for two different thickness, 0.8 and 1 mm L 4He. At Vdc=3 V; VG= -2 V.
Figure shows 50% of transition time of escaping electrons from the surface as a function of positive top plate voltage pulse. Data are taken at two different
electron densities and temperature... 73 Figure42. Escape rate, W as a function of surface charge density. Measurements
were done at 1 mm and 0.8 mm L 4He thickness. At fixed temperature, Tcell= 1.74, and 1.9 K. Vdc=3 V; VG= -2 V. Figure shows reciprocal of 50% of transition time of escaping electrons from the surface as a function of two
different electron densities... 74 Figure43. A set of summarized measurements of escape rate , 1/τ (sec-1) vs Surface
electron density are plotted . All four graphs illustrate escape measurements at four different holding voltage, and corresponding guard voltage in proportion in the decreasing order. From the above measurements we conclude that the escape rate is high at low electron density and decreases toward high surface electron density regime. Details about all the measurements were given before... 75 Figure44. The plotted data illustrates Escape rate, 1/τ as a function of Surface
charge density. All the data are taken from previous graphs. Squares are taken from Figure 42, at V dc=3 V, and VExt=11 V, Circle data are taken from Figure 40, at V dc=6 V, and VExt=15 V, and trangle data are taken from Figure 36, at V
dc=12 V, and VExt=14 V. Tcell~ 1.7 K... 76 Figure45. Theoretical calculation of escape rate as a function of extracting voltage
applied through the top plate, V dc=12 V, and 2V ≤VExt ≤10V . ... 77
voltage applied through the top plate, D= 5.5 mm, d~1.0 mm, Tcell~ 1.7 K, V
dc=12 V, and 2V ≤VExt ≤10V ... 78 Figure47. Amplitude of escaping electrons vs short multi-pulse for Vt.p.= 20 V,
60 t µs
δ = ,Tcell= 1.75K, dLHe~ 0,8 mm, Vdc = 12 V, and VG = - 8 V ... 79 Figure48. Electron density (calculated from amplitude) vs no of pulse at saturated
charge density. ... 79 Figure49. Escape of electrons vs no. of pulse. In Fig. A Phase of escaping electrons
taken from the recorded Lock-in trace as a function of no. of pulses that is
imposed at fully charged surface. Fig. A’ Electron density calculated from phase Vs no of pulse at saturated density. For VTP.= 20 V, Tcell~ 1,75K, dLHe~ 0,8 mm, Vdc = 12 V,VG = - 8 V ... 80 Figure50. This figure illustrates Influence of short multi-pulse, at extracting voltage
Vt.p.= 20 V, Tcell= 1.75K, dLHe= 0.8 mm, Vdc = 12 V, VG = - 8 V. Fig (A) is Lock- in measurement of amplitude of signal vs no of pulse imposed of definite pulse length, 60µs. Fig (A’) is the electron density (calculated from amplitude of Fig A) vs no of pulse. Fig (B) is the measured phase at saturated density vs no of pulse, while Fig (B’) is the electron density (calculated from phase of Fig B’) vs no of pulse. Fig. (A’’) and (B’’) are electron density ( calculated from amplitude and phase) shown in log scale vs no. of pulse. In Fig. (A) N≤ 1.69*106 e/cm2 (i.e., remaining electron density over the surface after the last extracting voltage pulse)... 81 Figure51. Electron density of escaping electrons vs no of pulse for VTP=22 V, Vdc=
12 V, VG=-8V. and δt=5,12,and28µs, dLHe= 1.0 mm, Tcell =1.66 K. ... 82 Figure52. In Figure A. The plot is illustrated for amplitude as a function of
different top plate voltage pulses at fixed temperature, Tcell = 1.74 K, and electron density. dLHe =1,0 mm, Vdc = 12 V, VG = - 8 V. In Figure B, the plot is illustrated for amplitude as a function of pulse duration of different top plate voltage at fixed temperature, Tcell = 1.74 K, and electron density. Tcell~ 1,7 K, dLHe~ 1,0 mm, Vdc = 12 V, VG = - 8 V. ... 83 Figure53: Amplitude vs pulse duration for two different top plate pulses are shown
in this plot for definite temperature, Tcell=1.8 K, and electron density 7*107
e/cm2. At Vdc=12V, VG=-8V, and dLHe= 1.0 mm, extracting voltage VTP=14 V. 85 Figure54: Two different extracting voltage pulses, one is reduced voltage amplitude,
while other is higher extracting voltage amplitude vs two different pulse
duration is shown... 86 Figure55 Two different extracting voltage pulses A, and B as a function of pulse
reduced voltage amplitude, with fixed long pulse duration that does not expels electrons. In Fig A+B, both the extracting voltage amplitude are superimposed which results as shown in Fig. C . Their corresponding cause and effect
explained above. ... 87 Figure56. All the measurements were done at the thickness of 0.6 mm L 4He, and
pulses of top plate were 18, 20 and 22 V for different cell temperatures... 88 Figure57. Escape rate vs cell temperature plotted from the data of Figure56. All the
measurements were done at the thickness of 0.6 mm L 4He, and pulses of top plate were 18, 20 and 22 V for different cell temperatures. ... 89 Figure58: All the measurements were done by imposing pulses through top plate,
data are at VTP= 22 V and 20 V, and the thickness of 1.0 mm L 4He, for different cell temperatures... 90 Figure59. In this plot transition time (50%) vs no. of pulse is illustrated for fixed
electron density and pulse duration for two different temperature. Vdc=12 V, VG=-8 V, dLHe=1.0 mm, and VTP= 22... 90 Figure60: Plot of comparison of experimental data with Iye et al at two different
temperatures. Iye et al, T~1.52 K; D=4.0 mm, Vext=9.5 V; Vdc=6.7 V. Ram et al, T≥1.7K; D=6.5 mm; Vext=14V; Vdc=12V. ... 91 Figure61. Mobility of electrons over surface of solid hydrogen ( After Troyanovskii
and Khaikin Sov. Phys. JETP, 1981)... 95 Figure62. Mobility of electrons on bulk Hydrogen. Sharp drop of solidification at
the triple point temperature Tt is shown which occurred due to the electron scattering from surface irregularities (After Cieslikowski, Leiderer and Dham, Can. J. Phys. 65, 1987). ... 96 Figure63. Schematic of the potential of hydrogen in external dc field, showing
potential barrier for an electron at hydrogen surface (F= 14.54 kV/cm)... 99 Figure64. Transmission probability as a function of electric field. ... 99 Figure 65. The transmission probability of electron as a function of the barrier height for hydrogen surface ... 100
Acknowledgement...III Acronyms ... VII List of Figures...VIII
Chapter 1 Foundations ... 1
Chapter 2 Two-dimensional electron system... 9
2.1 Introduction:... 9
2.2 Bound States- Electrons on liquid Helium ... 10
2.3 Effect of external field-Stark effect ... 15
2.4 State transition-Spectroscopic technique ... 15
2.5 Mobility of Electron- Scattering mechanism... 18
2.6 Phase transition- Wigner crystallisation ... 22
2.7 Instability of a charged liquid 4He surface... 25
2.8 Applications ... 30
Chapter 3: Experimental Techniques... 32
3.1 General Introduction: ... 32
3.2 Experimental Set up... 33
3.2.1 The Oxford Cryostat : ... 33
3.3 The low temperature equipments... 35
3.3.1 1-K Pot ... 35
3.3.2 The experimental cell... 37
3.3.3 Thermometer... 39
3.3.4 Stabilization of high temperature for hydrogen and neon ... 41
3.3.5 Cylindrical capacitor-L4He level meter ... 42
3.3.6 Electrical cable connections to the cell... 43
3.4 Cool down process... 44
3.5 Filling the cell ... 45
3.6 Charging technique ... 47
3.7 Measuring procedure ... 49
3.8 Conclusions... 52
Chapter 4: Measurement of Escape of Electrons from the 4Helium Surface. 53 4.1 Introduction: Escape of Electrons from the surface of Liquid 4helium. ... 53
4.2 The Experimental Results... 55
4.2.1 Stability of charged bulk surface ... 56
4.2.2 Thermal Activation to the escape of surface electrons: ... 58
4.2.3 Influence of pulses:... 78
4.3 Comparison with earlier work ... 91
4.4 Conclusions... 92
Chapter 5: Tunnelling probability of Electron from the Hydrogen surface: A theoretical study ... 93
5.1 General Introduction ... 93
5.2. Theoretical aspects... 94
5.2.1 Electrons on hydrogen surface... 94
5.2.2 Mobility of electrons... 95
5.3 Mathematical methods and tools ... 96
5.4 An estimate of tunnelling of electron ... 98
5.5 Conclusions... 100
Chapter 6 Summary... 101
Future out look... 103
Kapitel 6 Zusammenfassung ... 104
Ausblick ... 106
Publication... 107
References... 108
Lebenslauf... 114
Chapter 1 Foundations
General Introduction:
The processes of science, which help in searching the truth about nature and its phenomena are characterized by the given values. Curiosity, quest for knowledge, objectivity, honesty and truthfulness, courage to question, systematic reasoning, acceptance after verification, open-mindedness, search for perfection and team spirit are some of the basic values related to science. Science’s goal and aim is to explain things and events. Science is an international enterprise, beyond cultural, religious and political structures. In the field of science, the world’s diverse human groups have been unified. For many centuries physicists have been trying to understand the world around them both at an intuitive and at a quantitative level. In the influence of many philosophers, earlier it was believed that knowledge of nature could be acquired only by means of observation and introspection. In the era of dark age, experiments were considered useless, nature would not allow to be interrogated.
Relinquishing that belief in the modern era, had led to an unparalleled success of empirical science, that significantly changed not only our view of nature but also our way of living. Interrogating nature became the highly preferred way of doing science. Physicist constantly try to arrange situation far from the ordinary where subtle effects becomes dominant.
Attempt to answer it have led to discovery of completely new phenomena, and equally new ideas necessary to make sense of observations. The revolution, in context of knowledge about electrons which started with the experimental study of electrical discharges in gases, and the discovery of the electrons, is broadly called modern physics. Since then these small particles have been subject of detail study for different prospects. Electrons play a key role in all chemical properties of atoms and molecules, and in the physical properties of condensed matter physics.
Modern low temperature physics began with the liquefaction of helium by Kamerlingh Onnes and the discovery of superconductivity at the university of Leiden
in the early twentieth century. Helium exhibits very peculiar properties at low temperatures. In the early work there were two most remarkable facts came out. One was that essentially all of the electrical resistance of metals like lead, tin, and mercury abruptly vanished at definite transition temperatures that was the first evidence for superconductivity. The other fact was that, in contrast to other known liquids, liquid helium never solidified under its own vapour pressure. Helium is an inert gas so that the interactions between the helium atoms are very weak. Hence, the liquid phase itself is very weakly bound and the normal boiling point (4.2 K) is very low, i.e. it is found that helium gas at atmospheric pressure condenses at 4.2 K (its critical temperature being 5.2 K) into a liquid of very low density. Further cooling does not freeze it and it is believed that it remains liquid all the way right down to absolute zero. In fact, a combination of two factors, small atomic masses and weak interaction between them lead to large amplitude quantum mechanical zero point vibrations that do not permit the liquid to freeze into crystalline state. The solid state of helium does not form unless it is subjected to an external pressure of at least 25 atmosphere. It is in principle, possible to study liquid helium all the way down to very close of absolute zero. For 4He in liquid phase, there is an another remarkable phase transition was discovered under its saturated vapour pressure at 2.17 K so called λ-transition, which divides the liquid state into two phases He-I and He-II [70]. It is shown in Figure1 . As the liquid cooled through lambda temperature, all boiling ceased and the liquid became perfectly quiescent. This effect occurs because the liquid helium becomes an enormously good heat conductor so that thermal in homogeneities which can give rise to bubble nucleation are absent. The specific heat vs. temperature curve of liquid 4helium was shaped like Greek letter lambda, characteristic of second order phase transition at 2.17 K. Hence, this temperature is known as lambda point that represents a new state of matter known as liquid He-II.
Figure1. Phase diagram of liquid 4helium [ Fritz London (‘64)]
In liquid He-II it was found that heat conductivity is very large, coefficient of viscosity gradually diminishes as the temperature is lowered, and appears to be approaching zero at absolute zero temperature, and the specific heat measurements by Keesom shows that the specific heat curve is discontinuous at 2.18 K. Furthermore, liquid helium could flow freely through the tiniest pores and cracks. Kapitzy performed a number of ingenious experiments involving flow properties of super fluid helium. Later on, the two fluid hypothesis was introduced. A phenomenological model, accounting for many of the properties of helium II was developed by Landau and Tisza that is called Two-Fluid Model. It is shown in Figure2. The basic assumption according to this model is that the liquid 4He below lambda temperature behaves as it is a mixture of two inseparable fluids with each other without any viscous interaction [71, [72], the normal and super fluid components. The total densityρ of the fluid is the sum of the densities of the components,
n s
ρ ρ= +ρ , [1.1]
where the indices n and s denotes the normal and super fluid component respectively.
The normal fluid has energy in excess of the zero point energy, and behaves as an ordinary liquid and possesses no unusual properties except that it is liquid at low temperature. The normal and super fluid densities are temperature dependent. At lambda point there is only normal fluid density whereas decreasing temperature normal fluid density decreases and consequently, the super fluid density increased becoming dominant at the lowest temperatures. Hence, in He-II the normal fraction is very less and almost entirely super fluid below 1 K [73]. The normal fluid component carries heat away from the heat source and is replaced by the super fluid component toward a source of heat is spectacularly manifested in the fountain effect discovered by Allen and Jones. The total density is approximately constant for temperatures below lambda temperature.
Figure2. Normal and super fluid component vs. temperature [ Andronikashvili’s experiment (Tilley and Tilley)]
However, the super fluid possesses only zero point energy, and also shows some queer properties; it has zero viscosity and does not carry entropy and flows without resistance through channels- the narrower the base, freer is its flow through them. Due to the later reason it is named as super fluid. It consists of
percentage goes on increasing till at absolute zero, all the atoms go into super fluid state. Furthermore, it does not interact with the walls of a vessel containing the fluid in a dissipative fashion.
He-II is often referred to as a quantum fluid, as quantum mechanics plays an important role in determining the macroscopic properties of liquid 4He [70].
Moreover, super fluid helium has provided an enormous amount of enthusiasm to the physicists working in the field. Electrons on liquid helium have proved a very suitable candidate for studying two-dimensional samples, electrons are used to investigate the properties of the substrates [29]. The simplicity of the inter particle interaction, the freedom from impurities, and a substrate which does not impose a regular potential on the lattice makes this system an ideal prototype for the study of two-dimensional electron system [2]. Surface state electrons held a major position in the forefront of progress in low temperature physics. Quantum mechanics owes a debt to LTP. STM is an important tool towards nano technology analysis.
One can cite, in this way, a number of other important disciplines have their origin in LTP. Physicist soon realized that they had a new and beautiful tool which was simple in its basic idea, requiring minimum of equipment and likely to have ingenious applications. Since then enormous effort was devoted to the study of 2DES by the physicist world-wide. We consider two-dimensional electron system that constitutes only a part of the large class of dynamical 2D system. By dynamically two- dimensional that mean the components of the system are free to move in two spatial dimensions but their motion is restricted in the third dimension. In principle, most of the work on liquid helium has been carried out on 4He. However, there has been some work on electrons bound to the surface of liquid 3helium as well. So, here we mainly consider the system of electrons on the liquid 4helium surface.
Sommer and Tanner [1] were first produced free electrons on liquid helium surface experimentally in 1971. These surface state electrons on liquid helium have proved a very suitable candidate for studying two-dimensional behaviour, electrons are used to investigate the properties of the substrates [2]. The simplicity of the inter particle interaction, the freedom from impurities, and a substrate which does
not impose a regular potential on the lattice makes this system an ideal prototype for the study of two-dimensional lattices [3]. Studying 2DES at the surface of liquid helium or at semiconductor hetero junctions physicists found huge possibilities and many new ways to probe the condense matter physics theoretically and experimentally. The further study in this direction has given rise to a number of novel and very unusual phenomena including the discovery of the Wigner Crystal lattice [4], magneto-edge plasmons, and the study of elementary excitations (i.e. ripplons) of the helium surface, through measurements of electron ripplon scattering [5]. In this chain, one of the most significant research in the field of 2DES made possible by the discovery of Quantum hall effect in hetero-structures in 1980s [6]. The discovery permits more precise measurement of electrical resistance and more accurate testing of theories about electronic movements within atoms. The discovery established precise steps in the behaviour of electrons under certain applications in semiconductor electronics, a field of importance in computer and other modern technology.
In context of two-dimensional electron system on the cryogenic surfaces, a number of experiments of fundamental interest have been carrying out intensively by the experimental group of Leiderer et al and theoretical collaborators [7], [8], [9], [13], [14] for past four decades. Some of the most notable in this context are;
Application of a pressing field can lead to softening of coupled electron-ripplon modes and to an instability which causes the electrons to form hexagonal dimple lattice, in which each dimple holds about 107 electrons/cm2 [42.c]. Ebner and Leiderer have studied the development of the dimple instability on a time scale of few seconds for a range of electron densities [42.b]. Mode softening and dimple crystals also occur for ions trapped at the boundary between liquid 3He and 4He mixture [2], [42.a]. Particularly, electrons on helium has led physicists to new scientific insights. Experimental proliferation is also accompanied by theoretical interpretations, which have prompted a considerable amount of experimental as well as theoretical work in the nature of two-dimensional electron system.
The research work described in this thesis has been done in the department of low temperature physics, University of Konstanz.
In this doctoral work we consider the system of two-dimensional electrons on the bulk liquid 4helium surface, and liquid hydrogen and neon surfaces as well. More precisely, this thesis deals with the thermally activated escape of electrons from the surface of L 4He, along with an estimate of tunnelling probability of escape of electrons through the potential barrier from the surface of hydrogen for high extracting field. In essence, a method is described that makes it possible to trap and expel electrons from electrostatic trap on the cryogenic surface employing different parameters at lowest possible temperature that cannot be found anywhere else in the universe except in the laboratory of physicist.
2DES has greatly extended our knowledge of LTP and solid state physics.
Significant advances have been made to understand the complex processes occurring in the transport properties of electrons at low temperature. The remarkable properties of electrons on helium surface open ways for unprecedented scientific and technological applications.
Even though, one can think that what is so special about 2D electrons on helium surface? What are the characteristics which distinguish them?
The significant feature of such electron system on dielectric substrate is scaled down in many respects; It has been studied extensively both theoretically and experimentally and well understood for more than four decades. Electrons have extremely long relaxation time: highest mobility in the plane known in a condensed matter system: m ~ 104 - 105 m2/Vs ( τ0 ~ 10-7 s ), allowing spectroscopy to be carried out with greater precision than for space charge layers in semiconductors.
Inter-electron distance is comparatively large, ~1 μm, which makes fabrication technologically feasible. Most notably the energy separations and attainable electron densities, and observation of crystallization of electrons on the surface of liquid helium at the temperatures of mK range.
The dissertation presents six chapters.
Chapter one is introductory. A very general overview of scientific journey that led towards two-dimensional electron system on liquid helium surface has been presented.
In chapter two, a good overview of many aspects of the physics of two- dimensional system is given. A brief description of theoretical and experimental approaches (i.e., developments, achievements and applications) in the field of two dimensional electron system that results in the more general context of escape of electrons from the liquid 4helium surface are mentioned.
In chapter three, an experimental set-up of the Oxford Cryostat, which is improved and assembled for this framework, along with experimental procedure are explained. Since Oxford cryostat is one of the standard tools in the low temperature laboratories to keep the lowest possible temperature a brief description about construction and working of the cryostat, involved in low temperature components and systematic experimental procedure are given.
In chapter four, we present the thermal activation of escape of electrons from the surface of liquid 4helium by imposing external electric field through top plate. That was further observed by changing some parameters, such as liquid helium thickness, extracting field, electron density and temperature.
In chapter five, a theoretical estimate of probability of tunnelling of electrons through the potential barrier from the surface of molecular hydrogen for high extracting field is given using WKB approximation. The JWKB methodology represents one of the powerful approximate methods that is extremely used not only in quantum mechanics but also in many other areas.
Chapter six includes the summary of main achievements of this work and possibilities of future research activity.
Chapter 2 Two-dimensional electron system
2.1 Introduction:
Scientific interest toward theoretical and experimental investigation in the field of Two-Dimensional Electron System has been accelerating world-wide after the pioneering work proposed by WT Sommer [10], Cole et al [11], [12] and also independently by Shikin [13], [14]. In earlier work, electrons that could form stable state above the surface of liquid helium was introduced by Sommer, and such an observation of electron system along with mobility of electrons were measured [1].
Later on, in this chain, the life time of electrons were measured by R. Williams et al [15] and Ostermeier et al [16]. However, their results did not match with each other, i.e., 10-4 sec had been reported by former while later claimed less than a few µsec.
Since then, a number of experiments of fundamental interest, which were hitherto completely out of range, have now become possible. The theoretical and experimental work has been contributed by several scientists [17], [18], [21], [38], [39]. The most notable development in this field has been reported by Grimes and Adams [35], who reported the observation of crystallization of electrons on L 4He surface at very low temperatures, 440 mK. The changes produced by 2DES in LTP are revolutionary in achieving order of magnitude improvement in resolution and accuracy.
As it has been already mentioned that our work is mainly concerned about the study of escape of electrons from the helium surface. From the study there are two important information could be extracted. One is that the motion of electrons could be studied in the plane of two-dimension and other is escape out of all those electrons into three-dimensional free space.
In this chapter we give a brief description of several of the properties of the electrons on liquid helium that covers some formal aspects of the physics of ideal two-dimensional system. A theoretical and experimental approaches, in which
electrons on helium surface play a starring role, that is like a new door that has opened opportunities to extend studies for better understanding of two-dimensional electron system. Especially that has been used to study the escape phenomena of electrons over the liquid 4helium surface is mentioned. In this context, first we begin with the model for the binding of the electron to the liquid helium surface with respect to the attraction arising between an electron and its image charge in the helium. Then this model will be modified to include major corrections from external electric field. Thereafter, spectroscopy of the bond states (i.e., energy level) is given that elegantly refines the system. Later on, motion within the plane of electrons is considered, and discussion about Wigner crystallization, and instability of charged helium surface are mentioned. At the end of the chapter, we have included the applications of two-dimensional electrons towards basic scientific research and information technology as well.
2.2 Bound States- Electrons on liquid Helium
In order to study the escape of electrons from the helium surface it is very necessary to understand electrostatic barrier which is responsible for binding electrons to the surface. Here, we will discuss key mechanism i.e., binding potential, electronic states etc.
There are combination of three forces that confines electrons in the potential well. These are:
[I] An attractive force due to induced image charges in liquid helium: In principle, as soon as a charge q approaches close to the dielectric substrate like helium it polarizes the material i.e., helium atoms in the liquid, which results an attractive polarization force towards helium surface. This polarization can be explained as an image charge at a distance z under the surface as shown in Figure3.
Mathematically, the image potential (Coulomb potential) is as follows,
2
( ) Qe , 0
V z z
= − πε > ; Q 1 εh εv
ε ε ε
= −
+ (2.1)
( ) 0, 0 V z =V z≤
where z is the distance normal to the surface. Q represents the strength of image charge, which is equal to 0.0069 (e). The size of the image charge depends on the material used through its dielectric constant [19], for 4He=1.057 (at 4.2K) and dielectric constant of vapour, εv=1. Where as ε0 represents dielectric constant of vacuum. Moreover, the inter-electron spacing is approximately two order of magnitude higher than the distance between electron and image charge. So that each electron finds itself in an externally created potential well whose arising from its own image charge [20].
+ eQ - e Z Vapor
z> 0
Liquid 4Helium z< 0
Figure3. A charge approaching dielectric material at z distance above surface polarizes the atoms of liquid 4He.
[II] A short range repulsive force, the potential barrier, which arises from the Pauli’s Exclusion principle. The image charge is positive and opposite in sign to the electron so that the result is an attractive to the helium surface, and it can be imagined that such an attractive force could pull the electron down to the liquid, but electron experiences a large potential barrier close to the surface. Due to the fact that the s- shell of the helium atoms of the liquid is completely filled, and the wave function of any excess electron must oscillate in the vicinity of each atom which requires minimum energy of the order of 1eV [11].That is the energy needed by an electron to enter into the substrate(liquid 4helium), and binding energy of the liquid is very small in comparison to the potential barrier of 1eV. Hence the size of V0 is much
more larger than any other energy in the system, such as potential energy due to image charge and the applied field. That’s why to a first approximation it is taken to be ∞, and the penetration of the electron wave function into the helium is zero. The image potential for motion in z-direction (first two states are shown) and potential barrier, V0 =1 eV are shown in Figure 4.
Figure 4. The image potential for motion in z-direction(first two states are shown), and potential barrier of 1 eV together with the ground state and first excited state of the electron on liquid 4He are shown.
[III] External field: An applied vertical electric field E⊥ gives a linear potential variation eE⊥z.
Bound states of the system is expected to be localized outside, very close to the surface, due to the fact that an electron is repelled when inside the dielectric surface and attracted when outside the dielectric surface.
Furthermore, we can explore the change, considering that the assumption is made (caused by V0) that the wave function vanishes at the surface of liquid 4He, an electron then behaves like a one-dimensional hydrogen atom with a reduced nuclear charge Q=0.007e, in the z-direction. The energy for the motion in z- direction will be quantized in the Rydberg series,
n 2
E R
= −n (2.2) Where, n=1,2, and R is the Rydberg constant given as,
2
2 7.6
R 2 K
= mbh = (2.3)
For F=0, the ground state and first excited state wave function for the potential well were given
ψ1(z) 3 / 22 zexp z b/ b
= − (2.4)
ψ2(z) 13/ 2 2 exp / 2 2
z b
z z
b b
⎛ ⎞ −
= ⎜⎝ − ⎟⎠ (2.5)
Here, b=7.64 nm for 4He and 9.9 nm for 3He, and denotes the effective Bohr radius of the problem, and explains the length scale of the wave functions. Wave functions for ground and first excited state are shown in Figure 5. The wave functions are without electric field. However, by applying electric fieldE⊥, the average distance to the liquid helium will be shifted closer to the surface.
Figure 5. Ground and first excited state wave function. Distance from the surface ( in units of beff=7.6 nm).
The form of the energy of the ground and first excited state for the vertical motion are given as,
E1=-0.66 meV = -7.6 K (2.6) E2=-0.17 meV =-1.9 K
⇒ ΔE = E2 - E1 = 5.7 K (2.7)
Due to the fact that the smallness of ground state energy justifies the assumption of infinite potential barrier (V0 =1 eV) at the surface.
The average distance of the electrons from the liquid helium surface is, ψ1(z)= 11.4 nm (2.8) ψ2(z)= 45.6 nm (2.9) The fermi temperature is given by,
f f ,
B
T E
= k (2.10)
where,
2 e f
E n
m
=πh
Here, 2
h
= π
h , m is the effective mass of electron, ne is electron density, and kB is Boltzman constant.
The maximum electron density on bulk liquid helium, n<2*109 cm-2; Tf is less than 50 mK. At the experimental temperature T >100mK, the Fermi temperature is always much smaller than the average thermal energy, k TB of the electrons.
In general, the electrons on liquid 4He form a two-dimensional, classical electron gas at a distance about 11.4 nm from the helium surface obeying Boltzmann statistics in energy distribution. Since all measurements performed here are in temperature range between 1.59 K and 1.95 K, therefore almost all electrons are in ground state. Ground state of the system is important for most of the experimental
and drags attention, like as direct spectroscopic observation of bound states by Grimes et al [21], where they observed transition between the ground state, n=1 and excited states up to n=7 at a driving frequency of 220 GHz (see article 2.4).
2.3 Effect of external field-Stark effect
There has been a resurgence of interest in the effect of externally applied electric field to study escape of electron over cryogenic surface. An asymptotic form of the potential barrier can be modified by applying uniform electric field. Which is one of the major corrections to the system. Hence, pressing field adds as an additional component, eF z⊥ to the potential Vz . By applying vertical field to the helium surface, the potential above the liquid helium comes in the form,
Mathematically,
2
0
( ) 4
V z Qe eF z
πε z ⊥
= − + , z > 0 (2.11)
There are mainly two cases which can be observed when the electric field is switched on towards image coulomb potential. First, in the case where it presses the electrons closer to the surface and the barrier will rise linearly for large z-distance that results stable two-dimensional system. However, in second case, where the field pulls out the electrons from the surface, due to increasing field applied to the surface will lead to a linear drop of potential barrier that cause escape of electron by means of thermal activation or tunnelling depends on temperature. For the experimental consideration, pressing electric field produces a small change to the potential of the bound states and shift energy level closer to surface known as Stark shift, with respect to the zero field solution.
2.4 State transition- Spectroscopic technique
Spectroscopy that held a major position in the forefront progress in physics in the nineteenth and early twentieth century. Raman spectroscopy is an
important tool in molecular analysis. A number of other important disciplines having their origin in spectroscopy. Bound state energies can be measured using spectroscopic technique, system being studied is prepared in an excited state and the energy of the radiation emitted, upon decay of the system is detected by using bolo meter output. The energy difference between bond state can be calculated from the fixed energies of the emitted radiation. Such a technique is very useful for the investigation of atomic energy levels, as a matter of fact that it produces a continuous emission of radiation by passing an electric discharge through a gas of atomic species being investigated. It is proved a beautiful tool which is simple in its basic ideas, requiring minimum of equipment and likely to have ingenious applications. An application of the technique, in context of electrons on helium is impractical due to the fragility of the system. In this way, the successful work done by Grimes and Brown at Bell laboratories [22], who first presented the direct spectroscopic study of surface electrons.
After theoretical predictions [11], [13] and experimental observation of surface electrons [15], [23], Grimes and Brown [22] investigated the inter-band transitions of bound state over the helium surface, by monitoring absorption of microwaves frequency while changing the external pressing field for the different surface charge densities in the range 106 to 107 electrons cm-2. There results are illustrated in Figure 6 and in Figure 7, respectively. They also confirmed that the predicted sub band-structure is with a good approximation to the hydrogen atom. The Rydberg states were tuned to resonance by sweeping a modest electric field across the cell at temperature 1.2 K. They observed transitions between frequencies of 130 to 220 GHz. Transitions were measured between ground state and excited states up to n=7. They measured the linear stark tuning between ground state and first two excited states to be 0.8 and 2.1 GHz /V/cm, respectively. This is ~10% less than the actual solution from stark shift arising from external electric field, thus the difference between energy levels disagreed with the expected from a simple hydrogenic model.
Figure 6 : Experimental observation of mm-wave absorption derivative as a function of potential difference across the experimental cell with a layer of electrons on the surface of liquid helium taken at 220 GHz. The cell height is 0.32 cm and the bath temperature is 1.2 K. (After C.C. Grimes et al, PR B, 1976).
Figure 7 : Dependence of transition frequency as a function of voltage across the experimental cell. Transitions are plotted as a solid curves, which are the result of the variational calculation based on simple model potential, and crosses are measured data points (After C.C. Grimes et al, PR B, 1976).
As a matter of fact that hydrogenic approximation treating the barrier as infinite, hence such a discrepancy occurred. That is the main result of this experiment.
They matched their experimental data for 1→2 splitting at 125.9 GHz and 1→3
transition at experimental value of 148.6 GHz by taking the potential near the origin in the form 1
(z β)
− + , with the adjustable parameter,
0
β =1.04Α. Hence, the calculated value agrees with experimental value.
2.5 Mobility of Electron- Scattering mechanism
Taking into account some important needs and reasons, mobility of the electrons is one of the most important quantity among transport properties of the 2DEG that associates in the observation of two dimensional electron system. In fact, it is an indirect measure of the amount of scattering an electron experiences during the motion along the surface of liquid helium.
There are mainly two factors [12], [24], which determine the mobility,
- Interactions with the helium gas atoms above substrate, that dominates at temperature above 0.8 K. The random collisions occur between electron and gas atoms. The collision is elastic and is caused by short range repulsion arising from the Pauli principle. Since there is a large mass difference between gas atom and an electron, scattering results in small fluctuation in the electron energy but causes huge change in the in-plane momentum.
- Next is from interactions of the electrons with the helium surface waves, ripplons, if the surface is rough, that dominates below 0.8 K where the density of gas atom becomes negligible. From ripplon decay, contribution depends on the electric field that presses the two-dimensional electron gas closer to the helium surface.
And also from electron-electron interactions for high electron density.
However, such a scattering can be neglected for low electron density. In order to measure the properties of the electron system, it is very necessary that the interaction with the substrate should be smaller than the electron-electron interaction. If this is not the case then the phase diagram for the electrons could not be calculated by considering only the properties of the electrons.
Furthermore, roughness of the substrate would complicate calculations of the escape rates due to thermal activation and tunnelling because the binding energy depend on the shape of the dielectric substrate, hence important properties of the electrons will than depend on the surface we used.
Previous experimental results confirm the expectation that the mobility in this system is extremely high. At first, Sommer-Tanner [1] determined the zero field mobility, and observed the response characteristics of surface electrons to an AC driving field. Mobility Vs temperature was observed between 0.9 to 3 K , in the region of scattering mechanism is electron-gas atom dominates, which is shown in Figure8 below. Later on Brown and Grimes [25] observed mobility from Cyclotron resonance line widths, and time of flight measurements were performed by Bridges and McGill [26].
Figure8. Surface electron mobility on liquid 4He (After Sommer-Tanner, PRL ’71)
Levine and Sanders [27] measured mobility at high temperature between 2.6 to 4.2 K, and concluded variation of mobility over a narrow range of pressure and
temperature in each case, also showed a sharp decrease of mobility for gas density, nG >1021 atoms/cm3, which is caused by the formation of bubbles in the helium gas.
In the region of electron-ripplon interaction mechanism, and temperature range between 0.4 to 0.9 K, mobility was measured by Grimes and Adams [28] using plasmon resonance width. While Mehrotra et al, observed mobility using Sommer- Tanner technique, temperatures between 200 mK to 1 K.
However, Leiderer [29] measured mobility of surface electrons on bulk
4He taking into account both the ripplon scattering and the gas-atom scattering, between the temperatures 0.4 to 2 K. In spite of, mobility measurement in the region of electron-ripplon scattering is still remains an interesting field of research.
At first, Shikin and Monarkha [30] explained correct electron-ripplon interaction, later on Monarkha gave an integral expression for zero pressing field, and Mehrotra et al [31] extended their one-electron theories at low densities by measuring mobility as a function of density at fixed pressing fieldE⊥ and temperature, then compared this with their observations, which agreed only at low charge density,n≤10−8 cm-2 but did not give information about the discrepancy between experiment and theory for higher density.
Theoretical calculations of mobility by driving an integral expression, in the region of scattering with helium atom were done by Saitoh [32] that agree well with the measurements of Iye [33] and also the measurements of Grimes and Adams [28] over the whole temperature range considered. Iye observed the high mobility 2*107 cm2V-1sec-1 and found temperature dependence, and showed a cross over from a regime where scattering with the gas atom dominates at temperature above 0.8K to one where scattering with the ripplons is the limiting factor, also mentioned that the clear transition from gas to ripplon limited mobility for gas densities below 2*1017 cm-2. However, experimental data of Rybalko et al [34] did not agree with the theoretically calculated value of Saitoh [32]. A summary of all above mentioned observations is shown in Figure9.
Figure9. Mobility of electrons in the 4He. Solid curve shows theoretical calculation of mobility given by Saitoh taking into account both the gas-atom scattering and the ripplon scattering.
The scattering rate depends on the vapour density above helium surface, which drops exponentially with temperature. Saitoh calculated temperature dependent scattering time due to scattering with the He gas, which is given below,
Mathematically, 3 1
8
G G
An mb π
τ = h (2.12)
where, A is cross section of 4He gas atom, b is the effective Bohr radius (b=7.6 nm) in an applied electric field E⊥ perpendicular to the liquid 4He surface, m is the mass of electron, and nG is the vapour density above the liquid helium surface, that drops exponentially with temperature,
3/ 2 / 2
) . 2
Q k TB
He B G
M k T
n e
π −
⎛ ⎞
= ⎜⎝ h ⎟⎠ (2.13)
where, MHe=6.6*10-27 kg, and denotes mass of the 4Helium atom.
Q= 7.2 K, and denotes vaporization energy of 4He.
The total scattering rate will be the sum of the rate of two separate scattering mechanism, given as below:
1 1 1
G R
τ− =τ− +τ− (2.14) where, τG is scattering time for 4He gas, andτR is scattering with ripplons.
2.6 Phase transition- Wigner crystallisation
At first, an experimental evidence for the classical Wigner crystal of two- dimensional electrons at high areal density and low temperature was demonstrated by Grimes and Adams [35] on liquid helium surface at 0.44 K ( shown in Figure10 ) with the experiments on plasma oscillations, that is one of the most significant feature proposed by Wigner [4] for three dimensional electron gas. No clear observation of such a crystal in three-dimensions has been made up to the time of writing this thesis.
So, in 3D it lacks existence but electrons on liquid helium surface undergo phase transition in 2D. In short, Wigner crystal can be explained as, electrons in two dimensions that can organize themselves into a regular array at very low temperatures. Glasson et al [54] were created a ‘Wigner Wire’, where the electrons move along in organised groups, instead of randomly and alone. Physically, such a tendency of Wigner crystal occurs to the fact that the ratio of potential energy to the kinetic energy denoted by Γ can be made large, as a result the crystalline state is energetically favoured over the disordered state. Platzman and Fukuyama [36] were the first who studied the phase diagram displaying the region of ne and T plane where crystal should occur. They employed an analytic calculation based on a self- consistent harmonic approximation to the phonon spectrum of the crystal which yielded a melting transition near the parameter, Γ =m 3.
In detail, an explanation of phase transition and phase diagram can be
the electron-electron Coulomb repulsion, keeping in mind one of the experimental interest, i.e. electrons on helium where the interaction with the substrate is sufficiently low. In the situation where two-dimensional order in the ground state is expected at high temperature and low density, Fermi energy can be neglected in the classical regime with respect to thermal energy, i.e. k TB >>EF by considering dimensionless plasma parameterΓ, as given below.
2/ s
c
B
e r V
K K T
Γ = 〈 〉 =
〈 〉 (2.15)
which is the ratio of potential (i.e. electron pair Coulomb energy) over the thermal energy per electron, where rs denotes average separation of the electrons.
Figure10: Experimental traces showing sheet of electrons has crystallized suddenly into a triangular lattice with decreasing temperature of 0.457 K on the surface of liquid helium ( After Grimes and Adams, PRL, 1979).
Note that, most important for the phase transition is plasma parameterΓ, which is independent of temperature near T=0, that defines three thermodynamic states per particle as follows [37]; In a rough estimate when,
Γ <<1 : Kinetic energy dominates, thus system behaves as 2D electron gas.
1<< Γ <<100: Electronic motion will become more highly correlated and system should behaves like 2D electron liquid ( i.e. fluid like behaviour).
Γ >>100: The Coulomb potential dominates. Electrons should form a triangular two-dimensional crystal, solidification should take place for some critical value of plasma parameter. In an infinite system, each electron has six electrons in their neighbour at distance of ~1µm (inter electron separation), and bond angle of sixty degree as shown in Figure11.
Moreover, when the density of the system is increased further, a point will be reached where quantum fluctuations will dominate the thermal energy (i.e.
thermal energy will be replaced by two-dimensional Fermi energy EF ) and the correct expression for Γ would be as follows,
2 2
2 1/ 2 1/ 2
/ s e
q
F
e r m e
V
K E π n
Γ = 〈 〉 = =
〈 〉 h (2.16)
Figure11: Crystallized triangular electron lattice for Γ ≥127
In non-degenerate electron system, melting (liquid solid transition) of two- dimensional Wigner crystal takes place at the value of 127.
1/ 2 2 e m
B m
T n e
= k
Γ (2.17)
In order to probable explanation for melting mechanism, The KTHNY theory is
phase transition. Dislocation, i.e., a simple defects in the regular lattice could cause destruction of the long range positional order which occurs during melting.
The phase diagram [38] is shown in Figure12, for the system using two independent variables ne and temperature that consists expression of Γ. Figure shows for bulk helium and for a d=10 nm film lying on two different insulating substrates, sapphire (δ =0.9) and metallic substrate (δ =1). For nc=2.4*1012 cm-2 and Tc=33 K.
Figure12. Phase diagram of a 2DES for bulk helium ( After Peeters et al ’83).
At high electron density, crystallization do not occur in the degenerate two-dimensional electron system. However, Saitoh [39] has pointed out that the application of magnetic field perpendicular to the 2DES, can enhance the tendency of crystallization at high electron density. At first, such a crystallization was experimentally measured by Andrei et al [40] in a GaAs/AlGaAs heterostructure.
2.7 Instability of a charged liquid
4He surface
An exotic surface of liquid helium need to charge properly in order to obtain higher supportable charge density. However, it would be the most significant to investigate the cause of instability over the bulk surface for the experimental and
