Films in Confined Geometry
Mohamed Abdelaziz Galal Ashari
Fachbereich Physik Universit¨at Konstanz
A thesis submitted for the degree of PhD
Konstanz, 2011
2. Reviewer: Prof. Dr. Elke Scheer
Day of the defense: 02.09.2011
Head of PhD committee: Prof. Dr. Peter Nielaba
In this work, investigations on the transport of electrons on liquid helium films through narrow channels using suitable substrate structures, micro- fabricated on a silicon wafer which resembles Field Effect Transistors (we call it He-FET) have been presented. The sample has Source and Drain regions, separated by a Gate structure, which consists of 2 gold electrodes with a narrow gap (channel) through which the electron transport takes place. The electron densities on the source and drain are determined directly by electrical method. For time-resolved measurements, a pulse of electrons from a small filament is first collected on the source area, and then the passage of this pulse through the channel of the split gate towards the drain is monitored. This allows determining the electron transport of surface state electrons in channels of various dimensions and for a wide range of electron densities.
The study of the potential distribution across the He-FET sample results in a new model of number of saved electrons as a function of gate voltage.
Therefore, this model helped much in understanding the whole 2-DES using He-FET and makes it easy to study the electron transport of such sample.
This work can be summarized in the following points:
a) we improved the maximum electron densities which have been observed for the He-FETs by making new samples with more defined structure dimensions,
b) we did time resolved measurements which guided us to study the po- tential distribution across the sample,
c) we studied the potential distribution across the sample which we used and which has been used by the people who work in this experimental set-up before in more detail,
model to study the transport of surface state electrons,
e) we used a new method to study the transport of the electrons on the liquid helium films by pulsing (opening the gate for a short time) the gate.
f) some improvement in the experimental set-up were made and used for the first time, for example the step-motor which calibrated with the cylindrical capacitor to determine the helium level. so this work is done under defined conditions better than before.
In general we are getting more insight into the system by studying the electron transport on helium films using such geometry.
I would like to acknowledge all the people who have helped me in this work.
My first thanks go to my supervisor Prof.Dr. Paul Leiderer who has helped and taught me a lot. Also I would like to thank my second supervisor Prof.Dr. Elke Scheer, Prof.Dr. Kimitoshi Kono, David Rees, my colleagues in the group and our secretary Nicole Frederick. I would like also thank the DAAD for financial support. It is due to their efforts that this work is finished.
List of Figures v
List of Tables xv
1 Introduction 1
2 Theoretical considerations and state of the art 3
2.1 Electrons on bulk helium . . . 3
2.1.1 Liquid helium . . . 3
2.1.2 Electron-helium interaction . . . 4
2.1.3 Surface state electrons . . . 7
2.1.4 Distribution at finite temperature . . . 11
2.1.5 Scattering mechanisms and mobility . . . 13
2.1.6 Stability of surface state electron . . . 14
2.1.7 Phase diagram of two-dimensional electron systems . . . 16
2.2 Electrons on helium films . . . 20
2.2.1 Influence of substrate below the helium film . . . 21
2.2.2 Stability of electrons on helium films . . . 22
2.2.3 Phase diagram of the 2-DES on helium films . . . 23
2.2.4 Mobility of electrons on helium films . . . 25
2.3 State of the art . . . 27
3 Experimental set-up 31 3.1 Cryostat . . . 31
3.2 Sample overview . . . 32
3.3 Filament . . . 34
3.4 Substrates . . . 35
3.4.1 Different used samples . . . 35
3.4.2 Stability of the contacts . . . 38
3.4.3 Sample construction . . . 40
3.4.4 Helium level capacitor . . . 40
3.4.5 Measuring electronics . . . 42
4 Preliminary results 43 4.1 Measurements of pick-up current as a function of helium film thickness . 44 4.1.1 Pump-runs and the behavior of the helium level . . . 44
4.1.2 Step motor calibration . . . 46
4.2 Number of saved electrons determination . . . 47
4.3 Various types of electrons . . . 52
5 Time resolved measurements, potential distribution and electron dis- tribution model 55 5.1 The first time resolved measurements . . . 55
5.2 Potential distribution . . . 57
5.3 Electron distribution model . . . 64
6 Electron transport measurements 71 6.1 Passed and rest electrons . . . 72
6.2 Mobility calculations . . . 75
6.2.1 Transit time for different channel lengths . . . 77
6.2.2 Transit time for different driving potentials . . . 78
6.2.3 Transit time for different film thicknesses . . . 80
7 Wigner crystal observations 83
8 Zusammenfassung 87
References 89
2.1 Phase diagram for4He. 4He−I is the normal heliumfluid, 4He−II is the superfluid helium [EH00]. . . 5 2.2 Vapor pressure curve of liquid 4Heand 3He[Fau04]. . . 6 2.3 The electron-He interaction V +VR as a function of separation. The
attractive long-range tail is too small to discern. The apparent attraction at close approach is an artifact of the calculation. Units are Bohr radius and Hartree(= 2 Rydbergs) [JKRC65]. . . 6 2.4 The potential V(z) and the wave function ψ0(z) for the surface state
electron [Col70]. . . 8 2.5 Derivative absorption curves of surface state electrons as a function of
potential difference across the interface, at T = 1.2K. The first two absorption lines are seen[GB74]. . . 10 2.6 Tuning of the levels by means of an applied E-field. . . 10 2.7 The density of states N(E) for the external electronic surface states.
A step occurs at each eigenvalue of perpendicular motion. For large quantum numbers, N(E) lies far off scale. . . 11 2.8 The fraction of electrons in the ground state of perpendicular motion as
a function of temperature, in the absence of an applied field. . . 12 2.9 Electron mobility on bulk4He. The dominant electron scattering mech-
anisms are scattering from ripplons and gas atoms at low and high tem- perature respectively [Lei92] . . . 13 2.10 Schematic of apparatus used to measure electron surface mobility. Width
of electrode array was about 2.5cm and length (3l) was about 3cm. Spac- ing between electrode 2 and set of submerged electrodes was 1cm.[ST71] 14
2.11 Dispersion relation of ripplons at the interface of a phase separated3He-
4Hemixture completely charged with positive ions atT = 0.665K. The interface becomes unstable at Ec = 875V /cm, corresponding to an ion density of nc = 4.8·108cm−2. •, E/Ec = 0.12;N,0.71;,0.995. The dashed curves are calculated according to Eq. 2.20. The dispersion of the uncharged interface is given by the solid line[WL79]. . . 15 2.12 2DES Phase diagram on bulk helium [Pee84]. . . 18 2.13 Phase diagram of the 2DES without substrate. The areas are distin-
guished with the help of the plasma parameter [Wil82]. The area is in the hexatic KTHNY theory introduced . . . 19 2.14 2D Hexagonal Lattice . . . 19 2.15 Thickness of the uncharged helium film as a function of the vertical
distance from the substrate to the helium bath level [Wr06]. The retar- dation of the van der Waals potential α is not considered . . . 21 2.16 Schematic representation of electrons on thin helium films. Image charges
are now produced in both, the helium film and the substrate.[Sch07] . . 22 2.17 Thickness d of charged saturated 4He films wetting a glass substrate
(atT = 1.6K) as a function of electron density on the film [Lei92]. The thicknessd0 of the uncharged films was 220 and 420˚A. . . 24 2.18 Phase Diagram of 2-DES on a helium film for different helium film thick-
nesses for a dielectric substrate[Ska06]. . . 25 2.19 Phase Diagram of 2-DES on a helium film for different helium film thick-
nesses for a metalic substrate[Ska06] . . . 25 2.20 The mobility of SSE as a function of helium film thickness d at two
different temperatures. triangles denote increasing d; circles denote de- creasing d.[And84] . . . 27 2.21 Current of SSE through a small channel as a function of increasing
gate voltage. Measured at 1.4K. The geometrical channel width is 20µm.[Doi04b] . . . 28 2.22 The current I through the microchannel versus split-gate voltage VSG.
The current is completely suppressed at a threshold voltage VT which depends on the applied driving voltage Vin.[RK10] . . . 29
3.1 Schematically the structure of the used4Hebath cryostat. The cryostat was manufactured by KGW Isotherm. The inner diameter of the helium Dewar is 150mm, the height is 1000mm and the capacity is about 17liter. 32 3.2 Photo for the arrangement of the step-motor which is connected to the
outer part of the insert. . . 33 3.3 Schematic of the experimental arrangement with the corresponding elec-
trodes (shown in red) [Doi04b, Sch07]. The electrons with the assistance of the applied potential on the electrodes from the filament to the helium surface (shown in blue) over the source electrode are then transported along the surface direction to the pick-up electrode. . . 33 3.4 (a) Schematic structure of the filament plate [Sch07]. (b) Photo recording
the filament plate. The filaments are attached to the white plastic screw. 34 3.5 (a) View of He-FET design which has a triangular source area in front of
the channel from PSI. Non-doped silicon wafer is used. The evaporated gold electrodes are 130nmthick. The isolated layer is 300nmthick. The left hand side is a scanning electron microscope image for the channel which is 250µm long and 10µm wide. [Doi04b]. (b) Roughness of such a substrate.[Doi04b] . . . 36 3.6 SEM images of a Long-channel sample . . . 36 3.7 View of rectangular source design of the RIKEN He-FET sample: doped
silicon wafer used. The evaporated gold electrodes are 100nmthick. The isolated layer is 500nm thick. The left hand side is a scanning electron microscope image for the channel which is 100µm long and 10µm wide. 37 3.8 Arrangement for determining the resistance between the electrodes. To
avoid damaging the sensitive gold electrodes a conductive rubber were used [Sch07]. . . 39 3.9 Schematic view of the plexi-glass substrate holder: a) View from the
front side b) View from the back side . . . 39
3.10 Real arrangement of the sample construction. The top metal plate is used to fix the sample holder to the insert. At the second plexiglass sheet from the top , the three filaments and the collimator were mounted.
The substrate is located on the third plexi-glass sheet from the top. At the lowest copper plate are the connections for the respective electrodes attached. With this method it is possible to minimize the mechanical strain or shear forces on the contacts on the substrate. To the right of the three lowest discs is the cylindrical capacitor. . . 41 3.11 Schematic arrangement of the applied potential and the current mea-
surement . . . 42 3.12 RC circuit used for pulsing the gate. . . 42 4.1 Temperature overview: a) A typical temperature profile on the first day
of the experiment, which is characterized by a steep decline during the pumping. After reaching the minimum pressure the system is approach- ing an equilibrium value and is saturated for long periods. b) Temper- ature profile on the second day of the experiment. The temperature increases by the first immersion of the insert and the saturation can be seen, but this is overshadowed by steep inclines and declines which are due to the immersion process. . . 45 4.2 Forsample No.2: One pump-run shows the dependence of the pick-up
current on the helium level dropping with time. The transition region from bulk helium to helium film is supposed to be the region shown in red. . . 45 4.3 The dependence of the capacitance on the height between the helium
level and the substrate. . . 46 4.4 Forsample No.2: Two pump-runs (pick-up current (Ip) as a function of
Helium Level Capacitance). The normal pump-run shown in black. The pump-run which is measured using a step motor speed of 13step/s up- wards shown in red. The duration of the first pump-run was 30 minutes, and the duration of the second pump-run was only 5 minutes. . . 47
4.5 Method of charging the source. tc is the charging time and ts is the saving time. The total charge is calculated from the area under the Ip
curve (area highlighted in blue) . . . 48 4.6 Forsample No.1: a) The number of saved electrons at the source as a
function of charging time. b) The stability of electrons at the source by measuring the remaining electrons with the saving time for two different intial number of saved electrons. . . 49 4.7 Forsample No.2: a) The number of saved electrons at the source as a
function of charging time. b) The stability of electrons at the source by measuring the remaining electrons with the saving time. . . 50 4.8 Forsample No.3: a) The number of saved electrons at the source as a
function of charging time. The saturation happened at 3×109. b) The stability of electrons at the source by measuring the remaining electrons with the saving time. . . 50 4.9 For sample No.5: a) The number of saved electrons at the source as
a function of charging time. The saturation happened at a number of saved electrons of 4.2×1010. b) The stability of electrons at the source by measuring the remaining electrons with the saving time. . . 51 4.10 Forsample No.8: a) The number of saved electrons at the source as a
function of charging time. The saturation happened at electron density of 1.4×1010. b) The stability of electrons at the source by measuring the remaining electrons with the saving time. The saved electrons is more stable until 10 minutes saving time. . . 52 4.11 Forsample No.3: The number of loosely bound electrons as a function
of charging time after shaking process. . . 53 5.1 For sample No.2: First time resolved measurements for a) Positive
pulse applied to operate the filament. b) Negative pulse applied to op- erate the filament. . . 56 5.2 Forsample No.2: Time resolved measurement which shows the depen-
dence of the pick-up signal on the external illumination. . . 56 5.3 Forsample No.1: Time resolved measurements for the pick-up current
as a function of time for different filament voltages with illumination. . . 57
5.4 I-V characteristics of a rectangular piece of non-doped silicon at different intensity of external light at 1.3K. . . 58 5.5 The driving potential gradient for non-doped silicon substrate: a) With
illumination. b) Without illumination. For different voltages between source and drain at 1.3K. The electrometer which used to measure the potentials has a relatively low input resistance. . . 59 5.6 The driving potential gradient for non-doped silicon substrate with illu-
mination for different voltages between source and drain at 1.3K. . . 60 5.7 a) Modelled potential distribution across the sample from the simula-
tion computer program. b) The potential barrier at the gate region.
Simulations carried out by David Rees from RIKEN institute. . . 61 5.8 a) Modelled potential distribution for the gate channel from the sim-
ulation computer program. b) Potential profile across the channel for different Vgate. Simulations carried out by David Rees from RIKEN in- stitute. . . 62 5.9 a) The dependance of barrier height on Vgate at different helium films
according to the model. Here the slope is the coupling constant k = 0.0241. b) Coupling constantkvs helium depth . . . 63 5.10 Schematically, the electron distribution at the source region. Here the
source has a triangular area infront of the channel until xmax = 2mm.
The electron density increases from left to right. . . 65 5.11 Measurement of filling the resulting well of the gate barrier for Vgate =
-2V. t1 is the charging time andt2 is the saving time. SO is spilled over electrons and SE is saved electrons. . . 67 5.12 Forsample No.8: a)Pick-up current as a function of time for different
gate voltages. b)The number of saved electrons as a function of gate voltage. This graph has two functions which contain the two contribu- tions of the triangular and the rectangular area. The number of saved electrons as a function of gate voltage is in agreement with our model. . 68
5.13 Forsample No.1: a) Pick-up current as a function of time by ramping the gate voltage. b)The total number of saved electron as a function of gate voltage. As mentioned before this graph has two functions which contain the two contributions of the triangular and the rectangular area.
The number of saved electrons as a function of gate voltage is in agree- ment with our model. . . 68 5.14 For sample No.4: a)Pick-up current as a function of time by ramping
the gate voltage. b) The total number of saved electrons as a function of gate voltage. The number of saved electrons for this sample is directly proportional to (Vgate−1.5)2 also in agreement with our model. . . 69 6.1 Forsample No.6: a) The pick-up current (Ip) during pulsing the gate
measurement which show the passed and rest electrons. b) The number of passed and rest electrons at the source as a function of pulse width.
Here the the channel is long (100µm) and the helium film thickness
≈65nm and the average of the intial number of saved electrons before pulsing the gate is 4.0∗108electrons. . . 73 6.2 The logarithmic dependence of the number of rest electrons on the pulse
width. . . 74 6.3 Schematically RC circuit which illustrate the model of the relaxation
time on the source. . . 74 6.4 Forsample No.7: The number of passed and rest electrons at the source
as a function of pulse width. Here the the channel is short (2µm), the helium film thickness ≈ 50nm and the average of the intial number of saved electrons before pulsing the gate is 7.0∗108electrons. . . 75 6.5 For sample No.10: The number of rest electron at the source as a
function of pulse width: a) forVs−d= 1.5V. b) forVs−d= 3V. Here the the channel is short (2µm), the helium film thickness≈ 50nm and the average of the intial number of saved electrons before pulsing the gate is 4.5∗107electrons. . . 75
6.6 Forsample No.1: The number of rest electrons as a function of number of pulses for different intensity of external light. Here the channel is long (100µm) and the film thickness ≈ 50nm. One can see that without illumination the transport of the electrons through the channel is much slower, due to the bad conductivity of the substrate, which leads to an ill-defined driving potential. . . 76 6.7 For sample No.1: a) The pick-up current as a function of time for
different pulse widths for a long channel sample. b) The total passed electrons calculated from the area under pick-up current graph for each pulse width. Here the channel is long (100µm) and the film thichness
≈50nmand Vs−d= 1V . . . 78 6.8 Forsample No.2: The number of passed electrons as a function of pulse
width for a short channel sample (2µm). The film thichness is ≈50nm and Vs−d= 2V . . . 79 6.9 For sample No.4: a) The number of passed electrons as a function
of pulse width. Here the channel is long (200µm), the film thichness
≈50nmandVs−d= 2V. b) The number of passed electrons as a function of pulse width for different driving potentials. . . 80 6.10 Schematically the transport mechanism of the electrons through the
channel. The red spot represents the electrons which will go back to the source because for pulse widths <400ns the time is not enough for the electrons to cross at least more than L/2. The green spot represents the electrons which will pass through the channel to the drain because for pulse widths ≥ 400ns the time is enough for the electrons to cross more than L/2. . . 81 6.11 For sample No.3: a) The number of passed electrons as a function
of pulse width for a film thickness of 30nm. b) The number of passed electrons as a function of pulse width for a film thickness of 40nm. Here the channel is short (2µm) and Vs−d= 1V. . . 82
7.1 Forsample No.5: Pick-up current (Ip) as a function of time during sav- ing electrons measurements which shows the Wigner crystal and classical gas regimes. a) for 20 minutes charging time. b) for 60 minutes charging time. . . 84 7.2 For sample No.5: The number of saved electrons at the source as a
function of charging time and t at high numbers of saved electrons. Here t is the time needed for the total number of saved electrons to pass from the source to drain through the channel. . . 85
2.1 properties of liquid helium. The superfluid transition temperature is at saturated vapor pressure. . . 4 3.1 Different samples which are used in this study. Samples number 4 and
5 have rectangular source geometry in front of the channel. The others have triangular source geometry in front of the channel. . . 38
Introduction
Electrons in low dimension are an important topic of modern electronics, and they are relevant for applications in many systems, like in Metal Oxide Semiconductor Field Ef- fect Transistors (MOSFETs). Most of these electron systems have been investigated in semiconductors. Apart from application, interesting fundamental phenomena appear in such low-dimensional electron systems, like the Quantum Hall Effect (in 2D), or effects related to quantum wires (1D) and quantum dots (0D). Parallel to the electrons in semi- conductors, also 2-dimensional electrons on the surface of liquid helium were proven as a nearly ideal substrate for surface state electrons (SSE) [And97, MK04, Gri78]. In contrast to electrons in bulk solids, surface electrons on helium move in a dilute helium gas or, at sufficiently low temperature, nearly in vacuum, hence complications due to band structure effects do not appear. The most important advantage of SSE is that the electron density can easily be varied continuously from very low values, where the system behaves like a classical gas, via a Wigner crystal, and maybe up to the degener- ate Fermi gas because the surface can be charged deliberately from a separate electron source (e.g. a tiny heated filament) in a very dedicated way . In the early investigations of these systems, which started nearly forty years ago, the focus was on the infinite 2D systems, and many fundamental phenomena have been observed there for the first time.
For example, the first experiment was done by Sommer [Som64] which shows the liquid helium as a barrier to electrons, then the existence of a 2DES on the surface of liq- uid helium by Cole and Cohen [Col74], also predicted by Shikin [Shi70]. But the most prominent one is possibly the discovery of the 2D Wigner crystal by Grimes and Adams in 1979 [GA79]. In addition to electron liquid solidification, the stability and instability
of the helium surface has been studied [EGIL84, IP81]. The experiments show that for thick helium layers known asbulkhelium, the surface is stable only for a limited number of electrons, but thin films of helium offer very interesting possibilities for stabilizing high densities of mobile electrons. Since then, one has tried also for electrons on helium to realize lower-dimensional systems, and there are indeed several experiments in which quasi-one-dimensional behavior has been observed [SHS95, RKY07]. Now many open questions remain, and there are interesting aspects where investigations of electrons on helium in confined geometry could provide crucial insight, even in the context of qubits for quantum computing [PD99]. In the group of Prof. Leiderer at the University of Konstanz, where I did this work, there is a long experience with electrons on liquid helium [LES82, AL87, CDL87]. In addition to the experiments of electrons on bulk liquid also studies of electrons on thin helium films have been made. The group deter- mined the stability limit of charged films again at high electron densities [EGIL84], has demonstrated Wigner crystallization on these films [ML93] and for the first time also melting of the Wigner crystal towards a degenerate 2D Fermi gas [MGNL97]. Recent advances in microfabrication technology have allowed the study of SSE in confined ge- ometry using devices such as microchannel arrays[RK10, IAK09, IAK10, IK07], and so our group has also developed a set-up which resembles the geometry of a MOSFET (Source-Gate-Drain), and have studied the characteristics of this ”He-FET” [KDL00].
The topic of this thesis is electron transport parallel to the surface of liquid helium and the confined geometry due to the specific substrate structure which resembles a Field Effect Transistor. The first dc-measurements using such a geometry were done by Irena Doicescu [Doi04a, Doi04b]. After that, Marc Schmid has continued with the same set-up and also studied the electron transport on helium films [Sch07] but not in details. In this work I will present some calculations and determination for the poten- tial distribution across the He-FET, new model of total saved electrons as a function of gate barrier and some investigations on the electron mobility on helium films. The theoretical considerations and state of the art of this work will be discussed in chapter 2. Chapter 3 discusses the experimental set-up. The results and discussion of this work will be discussed in chapters 4, 5, 6 and 7. Finally in chapter 8 I will give some conclusion to what has been done in this work.
Theoretical considerations and state of the art
The transport of surface state electrons in this work was done on helium films. In section 2.1 some basics of electrons on bulk helium are discussed. The influence of the immersed substrate increases when the helium layer becomes thinner and thinner. For thin helium films with a thickness in the order of magnitude less than 10nm the electronic system properties develop differently from the bulk helium. Section 2.2 addresses this in more details.
2.1 Electrons on bulk helium
Liquid helium is ideally suited as a supporting substrate for 2-DES and there are many investigations of this topic for electrons on bulk helium in quasi-infinite geometry. Bulk helium is defined as a macroscopically thick layer of liquid helium where the electrons can move on. In many cases, an electrode or substrate should be used under this thick layer to create some electric field.
2.1.1 Liquid helium
Helium exists in liquid form only at low temperatures. The boiling point and critical point depend on the isotope of the helium; see table 2.1 below for values. The density of liquid helium-4(4He) which was used as a substrate for the electrons in this work is approximately 0.125 g/mL at its boiling point and 1 atmosphere (atm). 4He was
first liquefied in 1908 by the Dutch physicist Heike Kamerlingh Onnes [Onn13]. It is used for many purposes, for example, as a cryogenic refrigerant; it is produced commercially for use in superconducting magnets such as those used in MRI or NMR.
4Heis liquefied using the Hampson-Linde cycle [SP99]. The temperatures required to liquefy helium are low because of the weakness of the attraction between helium atoms.
The interatomic forces are weak in the first place because helium is a noble gas, but the interatomic attraction is reduced even further by quantum effects, which are important in helium because of its low atomic mass. The zero point energy of the liquid is less if the atoms are less confined by their neighbors; thus the liquid can lower its ground state energy by increasing the interatomic distance. But at this greater distance, the effect of interatomic forces is even weaker [WF68]. Because of the weak interatomic
Properties of liquid helium Helium-4 Helium-3
Critical temperature 5.2 K 3.3 K
Boiling point at 1 atm 4.2 K 3.2 K
Minimum melting pressure 25 atm 29 atm at 0.3 K
Superfluid transition temperature 2.17 K 1 m K in zero magnetic field Table 2.1: properties of liquid helium. The superfluid transition temperature is at satu- rated vapor pressure.
forces, helium remains liquid down to absolute zero; helium solidifies only under great pressure. At sufficiently low temperature, 3He and 4He undergo a transition to a superfluid phase, see Fig. 2.1 and Fig. 2.2. The experiments in this thesis were done using 4Hein the superfluid state.
2.1.2 Electron-helium interaction
It is important to understand how the electron-liquid interaction results from the electron-atom interaction. The results of a pseudopotential calculation by Kestner [JKRC65] of the electron-He atom interaction is shown in Fig. 2.3. Because the exclu- sion principle requires that the scattering electron wave function be orthogonal to the bound atomic states, there is a strong repulsion at small separation r. This repulsion has a range comparable to the Hartree-Fock radius ea of He. At larger separation an attractive interaction becomes dominant, but the magnitude is too small for it to ap- pear in Fig. 2.3. This weak attraction is entirely responsible for the states discussed
Figure 2.1: Phase diagram for4He. 4He−Iis the normal heliumfluid,4He−II is the superfluid helium [EH00].
here. The attractive tail of the potential is derived from the force between the electron and the instantaneous dipole moment of the atom. In the adiabatic approximation, valid for frequencies small compared to ∆E12
~ , where ∆E12 is the first atomic excitation energy, the moment is proportional to the polarization α. In this limit the interaction is of the form:
Vpol(r) = −αe2
2r4 , r >>ea (2.1) The net e-He interaction in the limit of zero energy is represented by the scattering length as, related to the electron-atom cross section σ = 4πa2s. The sign of as is determined by the competition between the repulsive core and attractive tail of the interaction. Helium has the smallest polarizability and the largest ∆E12 of all atoms.
As a result, its scattering length is relatively large and positive. The key parameter characterizing the electron-liquid interaction is the conduction band minimum energy V0. The experimental value ofV0is (1.3±0.3)eV found by Sommer via electron injection [Som64].
Figure 2.2: Vapor pressure curve of liquid4Heand3He[Fau04].
Figure 2.3: The electron-He interactionV+VRas a function of separation. The attractive long-range tail is too small to discern. The apparent attraction at close approach is an artifact of the calculation. Units are Bohr radius and Hartree(= 2 Rydbergs) [JKRC65].
2.1.3 Surface state electrons
Electrons near the surface of any dielectric are attracted due to the image potential Vimage(z) = −Qe2
z , z >0 (2.2)
Q= (−1)
(4+ 1) (2.3)
where e is the elementary charge, is the dielectric constant of the material and z is the distance of the electron from the surface. For helium, this image potential attracts the electrons to the liquid-vapor interface. This attraction is rather weak because of the low polarizability of helium.
In addition, as mentioned before, at atomic distance the negative electron affinity of helium leads to a potential barrier, which is≈1eV [Som64]. As a result the electrons are trapped in a potential well perpendicular to the surface, forming a state similar to the electronic radial distribution in hydrogen atom, however with a much smaller binding energy in the ground state (8K) and much larger effective Bohr radiusaB=76˚A(see Fig.
2.4).
There are a number of review articles which have been devoted to the SSE on liquid helium [Gri78], and some of the most important results are:
a) the excited electronic states are hydrogen-like, and spectroscopic measurements in the range of a hundred GHz agree well with calculation.
b) the mass of the SSE is close to the free electron mass, as measured, for example, by cyclotron resonance.
c) Coulomb interaction in the 2-D layer of electrons leads to collective excitations, i.e. 2-D longitudinal plasmons.
d) one of the most intriguing effects in this system is the formation of an electron solid, the so-called Wigner crystal, which appears when the plasma parameter (the ratio of Coulomb and thermal energy) rises above about 130 [GA79].
With these limitations in mind, we write the potential for the wave function ψ in the effective mass approximation
V(z) =V0, z≤0 (2.4)
Figure 2.4: The potential V(z) and the wave functionψ0(z) for the surface state electron [Col70].
V(z) = −Qe2
z , z >0 (2.5)
The quantity Q, defined in Eq. 2.3, depends on temperature T through the dielectric constant of the liquid and negligibly on that of the vapor, until one gets to relatively highT.
The eigenfunctions and the eigenvalues of the Hamiltonian of Eq. 2.4 and Eq. 2.5 are of the form
Ψk(r) =A−1/2exp(ik.ξ)φ(z) (2.6)
Ek=E⊥+~2k2
2m (2.7)
where A is the surface area, k the wave vector, ξ the component of r parallel to the surface, and z the perpendicular component of r. For states near the conduction band minimum in He the effective mass equals the bare electron mass m, as follows from the effective mass sum rule. Thus the boundary conditions on ψ, the envelope function in the effective mass approximation, reduce to those on an ordinary wave function- continuity of ψ and its derivative [BD66, BD67].
The Schr¨odinger equations for perpendicular motion are
φ”−γ2φ= 0, z≤0 (2.8)
φ”+ (2m/~2)(E⊥+Qe2/z)φ= 0, z >0 (2.9) where the prime denotes differentiation with respect to z andγ2 = (2m/~2)(V0−E⊥).
The other condition on φ is that it vanishes as |z| → ∞. The solution of Eq. 2.8 in the liquid is a decaying exponential with a characteristic decay length γ−1 ≈ 2.5
˚A. The solution for z > 0 is the Whittaker function [Whi63], a form of confluent hypergeometric function:
φn =Wn,1/2(2Qz/na0) (2.10)
wheren is a quantum number related to the energy by
E⊥n =−(Q2/n2)R0 (2.11)
HereR0 =e2/2a0 is the Rydberg constant,a0 is the Bohr radius. Cole has studied the properties of these eigenfunctions in some detail [Col70]. The essential result is that the small magnitude of Q for He results in a binding energy which is of the order of one percent ofV0. The problem therefore reduces approximately to the solution of Eq.
2.5 with the boundary condition that φ vanishes at the origin. The spectrum in this limit is exactly Eq. 2.11 with integral n, and the wave functions are products of an exponential and an associated Laguerre polynomial.
For example, the energy and normalized wave function of the ground state are
E⊥0=−Q2R0≈ −0.65meV ≈ −7.5K, (2.12)
φ0 = 2a−3/2zexp(−z/a), (2.13)
a=a0/Q (2.14)
The Bohr radius for this state is a≈ 76˚A from the surface; the expectation value of z is 3/2 this value, or 114˚A. The electron is most likely to be found a distance of about 20 times the interparticle spacing of He from the surface. Its wave function becomes small within a few angstroms of the interface. The binding energy is infinitesimal on the scale ofV0 (≈1eV), but fortunately large enough to have some probability of being bound at typical experimental temperatures.
As mentioned before, Grimes and Brown have performed an extremely precise mi- crowave absorption experiment which confirms the existence of image potential induced surface states [GB74]. Microwave radiation of frequency ν > 125 GHz is absorbed by the surface electrons at one or more values of an electric fieldE normal to the surface.
Varying E shifts the levels linearly and the resonance ν value shifts accordingly, see Fig. 2.5 and Fig. 2.6.
Figure 2.5: Derivative absorption curves of surface state electrons as a function of po- tential difference across the interface, at T = 1.2K. The first two absorption lines are seen[GB74].
Figure 2.6: Tuning of the levels by means of an applied E-field.
2.1.4 Distribution at finite temperature
In contrast to the quantization perpendicular to the surface, the electrons are free to move parallel to the surface of liquid helium if the substrate is smooth enough. For each bandn of perpendicular motion, the two-dimensional free-particle motion of the electron parallel to the surface gives rise to a density of states per unit area Nn(E) which is constant above the threshold E⊥n including spin degeneracy.
Nn(E) = (m/π~2)θ(E−E⊥) (2.15) Here θ is the usual step function. The total density of the states N(E) is then simply (m/π~2) times the number of states of perpendicular motion which haveE⊥ < E, see Fig. 2.7. The result shows that there is a discontinuity at each eigenvalueE⊥nbecause the nth perpendicular state contributes only above that energy. One consequence of
Figure 2.7: The density of states N(E) for the external electronic surface states. A step occurs at each eigenvalue of perpendicular motion. For large quantum numbers, N(E) lies far off scale.
this large density of weakly bound states is that entropy considerations result in their dominating the thermal distribution of occupied states at finite temperatureT[CW72].
In fact these states contribution to the partition function Z =X
n
exp(−En/kBT) (2.16)
is divergent, because of the infinite density of states near zero energy. For a simple case let us place a wall (infinite barrier) at a distance L from the surface, where L might be of order an electron mean free path, or an experimental chamber dimension. The new spectrum of perpendicular states will differ little from Eq. 2.11, apart from elimination of the divergence. The density of quasicontinuum states having E⊥ > Qe2/L can be approximated by that of free particle in a box
ν⊥(E) =L(m/(2π2~2E))1/2. (2.17) With this form for ν⊥(E), the equilibrium ratio at temperature T of ground state to continuum state populations is:
N0/Nc=~L−1exp(E⊥0/kBT)(8π/mkBT)1/2 (2.18) As we can see from Fig. 2.8, the fraction of electrons in the ground state drops rapidly from unity to nearly zero between 0.5K and 1K. In our experiment the temperature is around 1.3K and we have an applied electric field in this case all the electrons should be in the ground state.
Figure 2.8: The fraction of electrons in the ground state of perpendicular motion as a function of temperature, in the absence of an applied field.
2.1.5 Scattering mechanisms and mobility
As mentioned before, in addition to the quantization perpendicular to the surface, the electron can move freely parallel to the surface. Fig. 2.9 shows that electrons on liquid helium have a high mobility esptially at low temperature where the motion of individ- ual electrons is only limited by scattering from quantized surface waves (ripplons).
The two-dimensional (2-D)electron systems which can be generated on liquid helium are a classical counterpart to the degenerate 2-DES in MOSFETs and related semicon- ductor structures [Lei92].
In addition to ripplons, there is also gas atom scattering which is proportional to the
Figure 2.9: Electron mobility on bulk4He. The dominant electron scattering mechanisms are scattering from ripplons and gas atoms at low and high temperature respectively [Lei92]
density of helium gas atoms. On the basis of the vapor pressure curve in Fig. 2.2 one can see the exponential dependence of the gas pressure on the temperature. Therefore the density of the helium gas also depends exponentially on the temperature and so the mobility of the electrons also. As shown in Fig. 2.9, the gas atom scattering for T<0.6K can be completely neglected [MISK97]. but in the temperature range of our experiment (T ≈1.3) it is the dominating process.
Mobility measurements
One of the most popular experiment to measure the mobility of the electrons on the surface of liquid helium is the experiment of Sommer and Tanner [ST71]. They mea- sured the mobility of electrons on the surface of liquid4Hefrom 0.9 to 3.2oK. Although scattering from atoms in the vapor appears to be the dominant scattering mechanism, the mobility of the surface electrons is significantly lower than that of free electrons in the vapor. Fig. 2.10 shows schematically the experimental set up which they used.
Figure 2.10: Schematic of apparatus used to measure electron surface mobility. Width of electrode array was about 2.5cm and length (3l) was about 3cm. Spacing between electrode 2 and set of submerged electrodes was 1cm.[ST71]
2.1.6 Stability of surface state electron
To work with electrons at the surface of quantum systems under well defined conditions it is necessary to study the stability of the surface state electrons on such a system.
When charging the helium surface with electrons this leads to a surface deformation of the liquid helium, which can lead to an instability at certain circumstances. To understand the instability it is necessary to use the ripplon dispersion relation.
Uncharged surface
The first investigations of the instability of a charged helium surface have been carried out for charges at the interface of a phase-separated 3He-4He mixtures [WL79], but the effect is similar also for the charged gas-liquid interface of helium. Without the
presence of charges, the dispersion relation of ripplons at the 3He-4He interface has been found to be well described by [LPW77]
ω02= (ρl−ρu)
(ρl+ρu)gk+ σi ρlρu
k3, (2.19)
where the terms on the right-hand side are due to gravity and interfacial tension σi, respectively. Here ρl and ρu are the densities of 3He, 4He and g is acceleration of gravity. In Eq. 2.19 it is assumed that the depth of the liquid is large compared with 1/k, and that the dissipation is negligible. For long wavelengths, the first term on the right-hand side of Eq. 2.19 dominates, giving rise to the gravitional waves with a dispersion relation ω0 ≈k1/2. The capillary waves at short wavelengths, on the other hand, obey the relation ω0 ≈ k3/2. The transition between the two regimes occurs around a characteristic wavelength λrmc = 2πa, where a= (ρg/σ)1/2 is the capillary length, which is typically on the order of millimetres. The resulting dispersion relation for the interface of phase-separated liquid3He-4Hemixtures is shown as a solid line in Fig. 2.11.
Charged surface
Figure 2.11: Dispersion relation of ripplons at the interface of a phase separated3He-
4Hemixture completely charged with positive ions atT = 0.665K. The interface becomes unstable at Ec = 875V /cm, corresponding to an ion density of nc = 4.8·108cm−2. •, E/Ec= 0.12;N,0.71;,0.995. The dashed curves are calculated according to Eq. 2.20. The dispersion of the uncharged interface is given by the solid line[WL79].
When the interface is charged with electrons (or positive or negative ions respectively), an additional contribution has to be taken into account, which reduces the frequency of the excitations [L.P, MI78].
ω2 =ω02− El2+Eu2
4π(ρl+ρu)k2 (2.20)
ElandEu are the electric fields below and above the interface(related to the 2-D charge densityρ byEu−El= 2πρ), and the charged carrier mobility was assumed to be high enough so that the interface is always an equipotential. The physical origin for the reduction in ω is given by the fact that a local elongation of the surface leads to a redistribution of charges and consequently to a nonuniform electrostatic force which counteracts the restoring force due to gravity and interfacial tension. Eq. 2.20 is interesting because it does not only predict a softening of interfacial waves, but also an instability of the interface when a critical electric field (or a critical charge density) is reached. The condensed phases of the helium isotopes, which provide liquid-gas, liquid- liquid, and liquid-solid interfaces, have proven to be a unique system for studying these effects of charges below, near, and above the instability threshold [Lei92].
2.1.7 Phase diagram of two-dimensional electron systems
Electrons moving on the surface of liquid 4He can be (with regards to their in-plane motion) in a spatially disordered state or they can form a regular structure known as Wigner crystal. The range of parameters for which each phase is stable and the result- ing phase diagram are discussed below.
The simpler case is when the helium depth layer is not very thin (bulk Helium), above 100µm and much larger than interelectronic distance (≈µm)so that there is little in- fluence from the substrate underneath the helium. In this case, the plasma parameter which describes the interaction is defined as the ratio between the average potential and the kinetic energy (thermal or quantum, whichever is greater):
Γ = hVi
hKi (2.21)
For 2DES with surface electron densityne the Coulomb energy, hVi=e2√ πne.
The kinetic energy for either degenerate or nondegenerate 2DES is given by:
hKi= m(kBT)2 πne~2
Z ∞
0
x ex−
µ kB T + 1
.dx (2.22)
whereµ=kBT ln(eπne~
2
mkB T −1) is the chemical potential.
This expression reduces tohKiclassical=kBT for nondegenrate electrons (EF << kBT) and to
hKiquantum=EF = π~2ne
me (2.23)
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 [Lin10]:
Γ = Γm (2.24)
Or we can say that 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)[GCC79, KKM79]and experiment (Deville, Gallet et al.; Grimes and Adams; Shirahama and Kono; Mehrotra, Guenin et al. 1982; Mellor and Vinen 1990)[GAD+85, GA79, SK95, GMM+83, MV90] is Γm ≈ 130. The phase diagram resulting from the Lindemann criterion with Γm = 137 inne, T variables is presented in Fig. 2.12[Pee84]. This phase diagram is qualitative because Γm = 137 is only valid for nondegenerate 2-DES and would probably have a different value for a transition driven by quantum fluctuations.
Wigner Solid and 2D Melting Mechanisms
E. Wigner was the first who predicted the existence of an ordered electron phase (for 3D electrons)in 1934 [Wig34]. 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 a number of phenomena similar in nature to Wigner crystallization such as Mott transition [Mot61].
Theoretically it was shown that no true long-range order can exist in 2D [MW66, Pei79]
which is expressed by stating that density-density correlation function decays with
Figure 2.12: 2DES Phase diagram on bulk helium [Pee84].
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.
gG≈R−ηG(T) (2.25)
Here G is a reciprocal lattice vector. In the harmonic approximation the exponent in Eq. 2.25 is ηG(T) = 2πµ(TG2T) where µ(T) is the shear modulus in the Wigner crystal (Bonsall and Maradudin in 1977 found thatµ(0) = 0.245e2n3/2e )[BM77].
Among the possible 2D lattice structures the hexagonal one was shown to be the most stable (Haque, Paul et al. 2003)[HPP03]. The theory of melting in 2D was developed in the 1970s and is known as KTHNY theory (Kosterlitz and Thouless 1972; Halperin and Nelson 1978; Young 1979)[KT72, HN78, You79]. 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
(see Fig. 2.13), paired dislocations become unbound and the solid is transformed into a hexatic phase (see Fig. 2.14)characterized by the absence of long-range positional order while retaining bond orientation order.
Finally in a second step at a temperature aboveTm disclination pairs unbind and the
Figure 2.13: Phase diagram of the 2DES without substrate. The areas are distinguished with the help of the plasma parameter [Wil82]. The area is in the hexatic KTHNY theory introduced
Figure 2.14: 2D Hexagonal Lattice
bond orientation order is destroyed as well resulting in a liquid phase. The direct way
to study 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 in the same order of magnitude of the lattice constant.
Degenerate Fermi gas
At very high electron densities, the Fermi energy dominates the Coulomb energy. The corresponding plasma parameters in this case is:
Γ = hVi
hKi = mee2
~2√ πne
(2.26) Therefore the phase boundary curve does not depend on the temperature .
The Wigner crystal can melts and turns into an uncorrelated state of degenerate Fermi gas. This region in the phase diagram is not able to be done with experiments on bulk helium. Even with a maximum achievable electron density of bulk helium 109cm−2(see section 2.1.6) and a temperature of 10−2K, the degenerate Fermi gas is not reached, and the crystal would be just cool [PP83]. In helium films, as it will be discussed in the next section, one could reach much higher electron densities with a good stability, so that one can move through all areas of the phase diagram.
2.2 Electrons on helium films
The maximum electron density on bulk helium is limited to nc = 2.4×109cm−2. At this density, the melting temperature is about 1K and the Fermi energy compared to approximately 50mK. The system of electrons on bulk helium therefore is limited in the phase diagram only on the classical regime. Ikezi and Platzman realized that with thin helium films d≈ 100˚A higher electron densities can be reached than in bulk helium, because the van der Waals forces between the helium atoms and the substrate for this small film thickness are very large and thus stabilize the film [IP81]. As shown by H.
Etz et al. a very high electron densities can be stable on helium films, which is limited only by tunneling of electrons in the substrate[EGIL84]. This is also shown theoretically by Hu and Dahm [HD90]. In order to study the electrons on the helium surface, one should consider the super-fluidity behavior of the helium at low temperatures below the lambda point. The film thickness on the substrated0, which is located at a vertical
distance of h from the helium bath level, results from the balance of the energies of the helium atoms and the atoms at the film surface given by:
d0 = α
ρlgh 1/3
(2.27) It should be noted that this equation is only valid for d ≤ 100nm. For d > 100nm d∝ h−1/4 [CC88]. For example for h = 1cm, using Eq. 2.27 one should have a film thickness of about 30nm(see Fig. 2.15).
Figure 2.15: Thickness of the uncharged helium film as a function of the vertical distance from the substrate to the helium bath level [Wr06]. The retardation of the van der Waals potentialαis not considered
2.2.1 Influence of substrate below the helium film
When the thickness of the film (d) is smaller compared to the distance between the electrons (r = 1000˚A) , the force between the electrons changes because of the presence of the image charge in the surface under the helium film [PP83]. For a film thickness of 65˚Athe energy gap between the first and the second state is E12= 1200K [AGP81].
F.M.Peeters in 1987 found that the highest average probability densityhzi shifts from hzi= 114˚A ford= 104˚Atohzi<15˚Afor d= 10˚A [Pee87].
The potential of electrons on a thin helium films changed to [PP83]:
Vd(r) =e2(1
r − ∆
p(r2+ 4d2)) (2.28)
here ∆ = r−1
r+1 and d = d+hzi. For bulk helium d → ∞ and so Vd(r) ∝ er2 as expected. The additional image charges in the substrate cause a screening effect so that with increasing dielectric constant (i.e. increasing of ∆), the substrate material is
important. For electrons on helium films for r ¿¿ d the interaction of electrons with each other still has a Coulomb character, and the potential due to the strong polarizability of the metal is assumed to have a dipole form [Sch07]:
Vd(r)∝ 1
r3 (2.29)
here r =∞ andr >> d.
Fig. 2.16 shows a schematic representation of electrons on thin helium films.
Figure 2.16: Schematic representation of electrons on thin helium films. Image charges are now produced in both, the helium film and the substrate.[Sch07]
2.2.2 Stability of electrons on helium films
As already discussed for bulk helium, there is also for helium films at a critical electron density nc where the helium surface becomes unstable. Using the ripplon dispersion relation, one can show that nc on helium films is higher than on bulk helium [IP81]:
ω2= ρs ρ
3α ρd4 +g
k+σ
ρk3− 4πn2e2
ρ F(k, )
tanh(kd) (2.30) By comparing this equation with the dispersion relation for charged bulk helium (Eq.
2.20) one finds that there is an additional term which includes the van der Waals con- stant α. So the stability of a film is increased due to van der Waals forces. The last factor on the rhs of Eq. 2.30 arises from the finite depth d of the liquid film, the function F(k, ) take the effect of image charges in the solid substrate into account, and the factor ρρs indicates that in a thin film only the superfluid component can move.
With Eq. 2.30 one could find that charging a film reduces the ripplon frequency in a
similar way as for bulk helium, but as I said before, the critical charge density where the instability occur is higher.
Film thickness depression
Experimentally it was shown that it is possible to reach very high electron densities [EGIL84]. Saturated helium films on a hostaphan polymer foil e.g. could be charged with electrons up to 9.8×1010electron/cm2 before breaking through to the solid sub- strate. Also, this instability threshold appeared to be independent of the thickness d0 of the uncharged films in the investigated range 200˚A < d0 <400˚A[Lei92].
This apparent discrepancy can be resolved when the electronic pressurePel exerted on the film is taken into account. In the chemical potential,Pel adds to the gravitational and van der Waals terms. Therefore the equilibrium thickness of a charged film will be reduced to:
d=d0
1 +2πn2e2 ρgh
−1/3
(2.31) the film thickness d0 before charging given in Eq. 2.27 (here retardation effects were neglected). The value of d from Eq. 2.31 is an average, around which the local film thickness is expected to vary because of the formation of a dimple underneath each electron [And84, PJ85]. As a result of the reduction of d the instability threshold, calculated from Eq. 2.30, is raised, and a higher charge density can be reached[Lei92].
The change in film thickness during charging the film was confirmed experimentally by an ellipsometric determination of d [EGIL84]. In Fig. 2.17, some data are plotted for two values of the intial thicknessd0. The two curves converge for high electron densities as expected from Eq. 2.31. As a result the stability limit for high electron densities on helium films is not affected by the initial film thickness, as observed in the experiment.
On the other hand, tunneling of the electrons through the helium into surface states of the substrate could happen when the film is very thin (d ≤ 50˚A)which limits the charge density on the helium film.
2.2.3 Phase diagram of the 2-DES on helium films
The phase diagram for electrons on helium films was investigated by Peeters and Platz- man [PP83]. To obtain the phase diagram of 2-DES on helium film of thickness d supported by a dielectric substrate of permittivity r, Eq. 2.24 has been solved with
Figure 2.17: Thickness d of charged saturated 4He films wetting a glass substrate (atT = 1.6K) as a function of electron density on the film [Lei92]. The thickness d0 of the uncharged films was 220 and 420˚A.
Eq. 2.28. In Fig. 2.18 the phase diagram for several thicknesses of a liquid helium film supported by a substrate with ∆ = 0.9 is shown.
For a metallic substrater=∞, ∆ = 1 by using Eq. 2.28 the phase diagram in this case shown in Fig. 2.19. The difference between these two diagrams is the appearance of a liquid dipole phase at T = 0. This is because on the metallic substrate the Coulomb term in the electron-electron energy vanishes and the interaction is diploar. By using Eq. 2.23 for the Fermi energy, one can see that for charge densitiesne≤
~2 md2e2
2
the quantum fluctuations will dominate the electron-electron interactions promoting the liquid phase.
For a metallic substrate r =∞, ∆ = 1 by using Eq. 2.28 the phase diagram in this case shown in Fig. 2.19. The difference between these two diagrams is the appearance of a liquid dipole phase atT = 0. This is because on the metallic substrate the Coulomb term in the electron-electron energy vanishes and the interaction is diploar. By using Eq. 2.23 for the Fermi energy, one can see that for charge densitiesne≤
~2 md2e2
2
the quantum fluctuations will dominate the electron-electron interactions promoting the liquid phase.
Figure 2.18: Phase Diagram of 2-DES on a helium film for different helium film thick- nesses for a dielectric substrate[Ska06].
Figure 2.19: Phase Diagram of 2-DES on a helium film for different helium film thick- nesses for a metalic substrate[Ska06]
2.2.4 Mobility of electrons on helium films
In helium films, the mobility of the SSE not only depends on the gas atom scattering, and the electron-ripplon coupling [JP81], but also on the roughness of the substrate.
As above-mentioned the mobility depends on the temperature (see section 2.1.5). For a helium film thickness of 350˚A and at 1.3K a mobility of 200cmV s2 has been obtained
[JSD88]. This value is 5 times less than the mobility on bulk helium at the same tem- perature (1000cmV s2). It is shown that for very thin helium films≈100˚Aone should take into account the deformation of a single electron due to the strong electrostatic attrac- tion of the helium surface [Shi70, Shi71]. L.M. Sander found that for a thin helium film
≈ 100˚A there is a localization or self-trapping of the motion of SSE and he observed a trapping energy of about 8K [San75]. This polaronic effect with a strong electron- ripplon coupling can be interpreted as an increase in the effective mass. The mobility of the electrons moving such as a dimple is≈100cmV s2 [TMP+96](also see[RD99]). A num- ber of review articles has been devoted to the electron mobility in helium films. They found that the mobility can change strongly due to substrate roughness. For example Kono measured the conductivity of SSE on a quench-condensed hydrogen film. He found that the surface has very high roughness just after the quench condensation, the SSE conductivity is unobservably small, and annealing of the hydrogen film improved the surface quality. For smoother substrates, however, the movement is based on ther- mally excited jumps and there are strong changes of the mobility in the growth of the helium film. This behavior was explained quantitatively by a re-trapping structural transition within the disordered system of localized charges [MAKL93]. The SSE bind to the roughness and are no longer free to move. For even thinner films, however, the electrons are in the valleys between the peaks from the lowest energy state. Between these states a transition must take place which can actually explain the behavior of the mobility [KAL91]. Also to study the mobility of SSE on helium films, Andrei used sap- phire substrate, which appears to have a smooth scale of 100˚A(see Fig. 2.20)[And84]
which showed a marked decline in mobility (dip problem).
Shikin suggested distinguishing between the SSE which can move freely on the helium film and the electrons at asperities (active tops) which are tightly bound, and so the total number of the electrons should be the summation of these two. This so-called two-fraction model, which can derive the position of the dips on the roughness of the substrate. This also depends on the equilibrium density of electrons. The model can be confirmed by additional experimental data from I.Doicescu [SKD+01, KGW+01].
Figure 2.20: The mobility of SSE as a function of helium film thickness d at two different temperatures. triangles denote increasing d; circles denote decreasing d.[And84]
2.3 State of the art
In this section I present the recent experiments of the electron transport on liquid helium in our system (or in a system which resembles ours). As I said before, the first dc-measurements using He-FET geometry was done by I.Doicescu [KDL00]. The most recent results of the electron transport using He-FET is the measurement of the current of SSE through the channel as a function of gate voltage (Vgate)and also the dependence of the driving voltage (Vs−d)[Doi04b]. Fig. 2.21 shows the current of the surface bound electrons through the split-gate constriction as a function of increasing gate
voltage(Vgate). The dependence is shown for three differents driving voltage between
Figure 2.21: Current of SSE through a small channel as a function of increasing gate voltage. Measured at 1.4K. The geometrical channel width is 20µm.[Doi04b]
source and drain(Vs−d). As we can see, for (Vgate<−1.5V) the gate is closed, but opens for higher gate voltages and for smallest driving voltage (0.25V) a step-like dependence is observed, which smears out at higher driving voltage (1.0V). This behavior is also observed by D. Rees [RK10] (see Fig. 2.22).
The results described above illustrate that the electrons on liquid helium are well suited not only for investigations of quasi-infinite 2-dimensional systems, but also for studies of electrons in nanostructured confinement. On the basis of these results more systematic measurements will be carried out in order to study the electron transport through nanostructures in more detail.
Figure 2.22: The current I through the microchannel versus split-gate voltageVSG. The current is completely suppressed at a threshold voltageVT which depends on the applied driving voltageVin.[RK10]
