•1 gostou•292 visualizações

Denunciar

Mreža zemalja sa niskom prevalencijom hiv a u centralnoj i jugoistočnoj evrop...PinHealth

166 visualizações•15 slides

- 1. PHASE DIAGRAM OF A 2-DIMENSIONAL ELECTRON SYSTEM ON THE SURFACE OF LIQUID HELIUM by IVAN SKACHKO A Dissertation submitted to the Graduate School-New Brunswick Rutgers, The State University of New Jersey in partial fulfillment of the requirements for the degree of Doctor of Philosophy Graduate Program in Physics and Astronomy written under the direction of Prof. Eva Andrei and approved by ________________________ ________________________ ________________________ ________________________ ________________________ New Brunswick, New Jersey [May 2006]
- 2. ABSTRACT OF THE DISSERTATION PHASE DIAGRAM OF A 2-DIMENSIONAL ELECTRON SYSTEM ON THE SURFACE OF LIQUID HELIUM By IVAN SKACHKO Dissertation Director: Prof. Eva Andrei This work is a contribution to the study of the system of 2-dimensional electrons (2DES) bound to the surface of liquid helium-4. The physical properties of the 2DES are probed through the excitation of normal modes in the radio frequency range 10 MHz-1GHz. The normal mode spectra are expected to undergo a radical change at the transition from liquid to crystal, the so-called Wigner transition. This difference is due to coupling between 2DES modes and those of liquid helium surface. The experiments are performed for a range of helium film thicknesses 1mm -100Å. For thin films the influence of the underlying substrate is of great importance: it allows charging the film to high surface densities, which are impossible to reach on bulk helium. This lets us explore a region of the phase diagram where quantum effects arising from zero-point fluctuations play a significant role. Slow-wave structures were developed to excite and detect the normal mode spectra including a meander line and an interdigital capacitor. ii
- 3. Acknowledgment and Dedication This work is dedicated to my parents Valeria and Mikhail and to my wife Rodica. I would like to express the most sincere gratitude to my research adviser Professor Eva Andrei – the leading authority in the field of 2D electrons on helium, for her support and encouragement during the years it took me to complete my research. I received much spiritual and technical help from my colleagues Ross Newsome, Guohong Lee, Ozgur Dogru, Xu Du, Zhili Xiao. I greatly appreciate the direct participation in the project of Kurt Ketola, Gerard Deville, Adam Hauser, Gail Schneider, Sylvain Benazet. The machine and electronics shops at Rutgers Dept. of Physics were extremely skillful in helping me to build the experimental apparatus. iii
- 4. Table of Contents ABSTRACT OF THE DISSERTATION........................................................................ii Lists of tables.................................................................................................................... vi List of illustrations.......................................................................................................... vii 1 Introduction: 2D Electrons on the Surface of Liquid Helium.............................. 1 1.1 THE ORIGIN OF REDUCED DIMENSIONALITY........................................................ 1 1.2 CONFINEMENT OF ELECTRON TO THE SURFACE OF LIQUID HELIUM .................... 3 1.2.1 Influence of Externally Applied Electric Field ........................................... 6 1.2.2 Binding to a Thin Film................................................................................ 7 1.2.3 Hartree Field (Mutual Repulsion of Electrons).......................................... 8 1.3 WIGNER CRYSTAL ............................................................................................... 8 1.4 COLLECTIVE MODES OF 2DES........................................................................... 11 1.5 THE DYNAMICS OF LIQUID HELIUM AND ITS EFFECT ON 2DES......................... 13 1.5.1 The Surface Waves - Ripplons .................................................................. 13 1.5.2 Interaction Between 2DES and the Surface of Liquid Helium ................. 14 1.5.3 Thin Film Effects....................................................................................... 15 1.6 SUMMARY.......................................................................................................... 17 2 Phase Diagram of 2DES on Liquid Helium.......................................................... 19 2.1 ELECTRONS ON BULK HELIUM........................................................................... 19 2.2 WIGNER SOLID AND 2D MELTING MECHANISMS............................................... 21 2.3 ELECTROHYDRODYNAMIC (EHD) INSTABILITY OF A CHARGED LIQUID SURFACE 26 2.4 INFLUENCE OF A SUBSTRATE ON PHASE DIAGRAM ............................................ 29 3 Spectrum of 2DES................................................................................................... 32 3.1 2D DRUDE MODEL ............................................................................................ 32 3.2 ELECTRON SOLID MODES .................................................................................. 39 3.3 2DES MOBILITY................................................................................................ 39 3.4 COUPLED PHONON-RIPPLON MODES ................................................................. 43 3.5 SPECTRUM OF 2DES IN MAGNETIC FIELD ......................................................... 46 4 Measurement of 2DES Spectrum Using Slow-Wave Structure.......................... 50 4.1 2DES IN ELECTRO-MAGNETIC FIELD ................................................................ 50 4.2 DISTRIBUTED CIRCUIT MODEL .......................................................................... 54 4.3 SLOW-WAVE STRUCTURES................................................................................ 58 4.4 COUPLING BETWEEN MEANDER LINE AND 2DES................................................ 65 5 Other Methods for Investigation of Phase Transition in 2DES on Liquid Helium Film..................................................................................................................... 66 5.1 MICROWAVE CAVITY TECHNIQUE ..................................................................... 66 5.2 SOMMER-TANNER METHOD (LOW-FREQUENCY TRANSPORT)........................... 68 iv
- 5. 6 Experimental Apparatus and Cell......................................................................... 70 6.1 CRYOSTAT ......................................................................................................... 70 6.1.1 Vibration Isolation.................................................................................... 72 6.2 DILUTION REFRIGERATOR.................................................................................. 73 6.3 THERMOMETRY.................................................................................................. 74 6.4 MAGNET ............................................................................................................ 74 6.5 EXPERIMENTAL CELL ........................................................................................ 75 6.5.1 Production of Electrons............................................................................ 76 6.5.2 2DES Confinement.................................................................................... 77 6.5.3 Measurement and Control of Liquid Helium Level and Film Thickness.. 79 6.6 MEANDER LINE.................................................................................................. 82 6.7 SUBSTRATE SMOOTHNESS CONTROL ................................................................. 83 6.8 ELECTRONICS..................................................................................................... 86 7 Analysis of Experimental Data .............................................................................. 90 7.1 IDENTIFICATION OF THE INDIVIDUAL RESONANCES ........................................... 93 7.2 EVOLUTION OF 2DES RESONANCES WITH TEMPERATURE AND PHASE DIAGRAM 97 2DES Heating ........................................................................................................... 99 Sliding Transition ..................................................................................................... 99 7.3 2DES MOBILITY MEASUREMENT .................................................................... 102 7.4 OBSERVATION OF MAGNETOPLASMON ............................................................ 104 8 Conclusion ............................................................................................................. 107 Appendices..................................................................................................................... 110 A Ripplon Dispersion Law....................................................................................... 111 B Roughness of Liquid Helium Surface Due to Thermally Excited Ripplons.... 114 C Parametric Resonance in HeII............................................................................. 117 D Heat Exchangers for Liquid He Bath.................................................................. 119 E Experimental Procedure ...................................................................................... 121 PREPARATION .............................................................................................................. 121 COOLDOWN ................................................................................................................. 124 DILUTION FRIDGE OPERATION..................................................................................... 125 FILLING THE CELL WITH HELIUM................................................................................. 126 CHARGING THE HELIUM SURFACE ............................................................................... 126 DATA COLLECTION...................................................................................................... 127 References...................................................................................................................... 128 v
- 6. Lists of tables TABLE 1-1 BASIC PROPERTIES OF 2DES.............................................................................. 3 TABLE 3-1 2D PLASMA DISPERSION LAW IN VARIOUS LIMITS.......................................... 36 TABLE 3-2 BESSEL EXTREMA. ........................................................................................... 38 TABLE 3-3 MOBILITY MEASUREMENTS AND THEORY. ...................................................... 41 TABLE 4-1 ELECTROMAGNETIC RESPONSE COMPARISON. ................................................. 54 TABLE 4-2 SLOW-WAVE STRUCTURES. ............................................................................. 60 vi
- 7. List of illustrations FIGURE 1-1 SCREENING MECHANISM GIVING RISE TO ELECTRON CONFINEMENT TO A LIQUID HELIUM SURFACE............................................................................................ 3 FIGURE 1-2 ENERGY DIAGRAM OF AN ELECTRON BOUND TO HE SURFACE.......................... 5 FIGURE 1-3 SCHEMATIC 2DES PHASE DIAGRAM. THE STRAIGHT LINE SEPARATES THE REGION WHERE CLASSICAL FLUCTUATIONS DOMINATE (LOWER PART) FROM THE ONE WHERE QUANTUM FLUCTUATIONS DOMINATE (THE UPPER PART)................................. 9 FIGURE 1-4 2D HEXAGONAL LATTICE - LOWEST ENERGY CONFIGURATION OF 2D COULOMB CRYSTAL................................................................................................... 10 FIGURE 2-1 2DES PHASE DIAGRAM ON HELIUM BULK..................................................... 21 FIGURE 2-2 2DES DIFFRACTION PATTERNS (T. WILLIAMS 1995)..................................... 25 FIGURE 2-3 CHARGING INSTABILITY. DASHED LINE IS THE PLOT OF N VS. D FROM EQ.(2-7). DOT-DASHED LINE IS THE PLOT OF N VS. D FROM EQ.(2-8). ONE OF THEIR INTERSECTIONS CORRESPONDS TO THE STABLE CHARGE DENSITY. SOLID LINE IS THE EHD LIMIT OF 2.2 109 CM -2 . ....................................................................................... 28 FIGURE 2-4 2DES PHASE DIAGRAM ON DIELECTRIC SUBSTRATE VS. HELIUM FILM THICKNESS................................................................................................................. 30 FIGURE 2-5 2DES PHASE DIAGRAM ON METALLIC SUBSTRATE VS. HELIUM FILM THICKNESS................................................................................................................. 31 FIGURE 3-1 2DES BOUNDARY CONDITIONS: TWO DIELECTRIC MEDIA. ........................... 33 FIGURE 3-2 2DES BOUNDARY CONDITIONS IN OUR CELL: THREE DIELECTRIC MEDIA. ... 34 FIGURE 3-3 POLARONIC TRANSITION................................................................................. 41 FIGURE 3-4 WIGNER TRANSITION...................................................................................... 41 FIGURE 3-5 2D ELECTRON LIQUID MOBILITY FOR A RANGE OF PRESSING FIELDS. ........... 42 FIGURE 3-6 VARIOUS METHODS OF MOBILITY MEASUREMENT......................................... 42 FIGURE 3-7 COUPLED (SOLID) AND UNCOUPLED (DASHED) MODES OF WIGNER SOLID AND SURFACE WAVES OF HE (FROM (FISHER, HALPERIN ET AL.))...................................... 45 FIGURE 3-8 EVOLUTION OF MAGNETOPLASMONS (FROM (GLATTLI, ANDREI ET AL. 1985)). ................................................................................................................................... 48 FIGURE 4-1 REFLECTION BETWEEN TWO DISCONTINUITIES IN TRANSMISSION LINE.......... 53 FIGURE 4-2 EQUIVALENT CIRCUIT OF 2DES AND EXCITATION LINE. STAR DENOTES THE VALUE PER UNIT LENGTH. .......................................................................................... 56 FIGURE 4-3 MEANDER LINE............................................................................................... 60 FIGURE 4-4 COPLANAR MEANDER LINE. ........................................................................... 60 FIGURE 4-5 INTERDIGITAL CAPACITOR (IDC).................................................................... 60 FIGURE 4-6 CROSS-SECTION OF A SLOW-WAVE STRUCTURE IN INHOMOGENEOUS ENVIRONMENT........................................................................................................... 61 FIGURE 4-7 MEANDER LINE DISPERSION LAW FROM (CRAMPAGNE AND AHMADPANAH 1977). θ = PQ−2πJ, P IS THE PERIOD OF THE LINE, OTHER PARAMETERS ARE DEFINED IN FIGURE 4-6. ........................................................................................................... 62 FIGURE 4-8 IDC DISPERSION LAW FROM (CRAMPAGNE AND AHMADPANAH 1977). SEE FIGURE 4-7 FOR MORE DETAILS.................................................................................. 64 vii
- 8. FIGURE 5-1 SOMMER-TANNER TECHNIQUE. ...................................................................... 69 FIGURE 6-1 EXPERIMENTAL CELL MOUNTED ON DILUTION FRIDGE.................................. 70 FIGURE 6-2 CRYOSTAT AND DILUTION UNIT. .................................................................... 71 FIGURE 6-3 CRYOSTAT ALIGNMENT SYSTEM. ................................................................... 72 FIGURE 6-4 CROSS-BELLOWS DESIGN ISOLATES A PUMPING LINE FROM THE CRYOSTAT. 73 FIGURE 6-5 SPLIT-COIL CROSSED-FIELD MAGNET............................................................ 75 FIGURE 6-6 EXPERIMENTAL CELL...................................................................................... 76 FIGURE 6-7 ELECTROSTATIC BOUNDARY CONDITIONS IN THE CELL. THE CELL HAS A CYLINDRICAL SYMMETRY. RGR IS THE GUARD RING RADIUS; RE IS THE RADIUS OF 2DES POOL................................................................................................................ 77 FIGURE 6-8 THIN FILM MEASUREMENT. ............................................................................ 80 FIGURE 6-9 THICK FILM MEASUREMENT........................................................................... 80 FIGURE 6-10 FILLING CURVES. THE MEASURED CAPACITORS ARE SHOWN IN FIGURE 6-9 AND IN FIGURE 6-8. THE MEASUREMENTS ARE TAKEN WHILE FILLING THE CELL WITH HELIUM. ..................................................................................................................... 81 FIGURE 6-11 SPRING CONTACT. MANUFACTURED BY EVERETT CHARLES TECHNOLOGIES (PART # MEP-30U)................................................................................................... 83 FIGURE 6-12 AFM IMAGE OF PASSIVATED SI (111) SURFACE: TRIANGULAR PITS ARE APPROXIMATELY 150Å DEEP. RMS ROUGHNESS IS 55Å. .......................................... 85 FIGURE 6-13 DIRECTLY TRANSMITTED SIGNAL VS. MODULATED ONE. ............................ 86 FIGURE 6-14 LOCK-IN OUTPUT CORRESPONDING TO MODULATION OF THE RESONANCE FREQUENCY............................................................................................................... 87 FIGURE 6-15 LOCK-IN OUTPUT CORRESPONDING TO MODULATION OF THE LINEWIDTH. .. 88 FIGURE 6-16 LOCK-IN OUTPUT CORRESPONDING TO MODULATION OF BOTH THE RESONANCE FREQUENCY AND THE LINEWIDTH......................................................... 89 FIGURE 6-17 MEASUREMENT SETUP.................................................................................. 89 FIGURE 7-1 VARIATION OF THE 2DES SPECTRA WITH SURFACE DENSITY: THE SPECTRA ARE SHIFTED VERTICALLY TO FACILITATE COMPARISON. THE LEGEND SHOWS CHARGING VOLTAGE AS WELL AS THE VALUE OF SURFACE DENSITY DERIVED FROM IT. T=110MK. ................................................................................................................. 90 FIGURE 7-2 THE SPECTRA CORRESPONDING TO DIFFERENT DENSITIES ARE PLOTTED ON LOG F SCALE AND SHIFTED BY –0.5 LOG N. ................................................................ 91 FIGURE 7-3 DETERMINATION OF RESONANCE FREQUENCY. .............................................. 92 FIGURE 7-4 THE EVOLUTION OF RESONANCE FREQUENCIES FOR THE SPECTRA OF FIGURE 7-1 WITH CHARGING VOLTAGE. THE SLOPE OF LINES ON THIS LOG-LOG PLOT IS ½, AS EXPECTED FROM 2D PLASMA DISPERSION ω ~ N 0.5 ..................................................... 92 FIGURE 7-5 MATCHING OF AN OBSERVED SPECTRUM (THICK LINE) TO 2D PLASMA RESONANCES (THIN LINES). HERE (ν,µ) ARE THE AZIMUTHAL AND RADIAL NUMBERS RESPECTIVELY. THE AMPLITUDE OF EACH RESONANCE IN THEORETICAL SPECTRUM REFLECTS THE COUPLING OF THAT MODE TO THE EXCITATION STRUCTURE (SEE 4.4) AS DISCUSSED IN THE TEXT. ............................................................................................ 94 FIGURE 7-6 CORRECTED SPECTRUM MATCHING................................................................ 95 FIGURE 7-7 THE SEQUENCE OF RESONANCE FREQUENCIES VS. THEIR NUMBER I: ASYMPTOTIC BEHAVIOR IS I 0.5 . ................................................................................... 96 FIGURE 7-8 EVOLUTION OF THE 2DES SPECTRUM WITH TEMPERATURE. THE SPECTRA ARE SHIFTED VERTICALLY TO FACILITATE COMPARISON. TM DENOTES THE EXPECTED viii
- 9. WIGNER SOLID MELTING TEMPERATURE FOR N = 6 107 CM -2 . THE INCIDENT RF POWER IS -44DBM.................................................................................................................. 98 FIGURE 7-9 NONLINEARITY OF 2D ELECTRON CRYSTAL RESPONSE................................ 101 FIGURE 7-10 CLASSICAL PART OF 2DES PHASE DIAGRAM. ............................................ 101 FIGURE 7-11 2DES MOBILITY VS. TEMPERATURE. THE LEGEND SPECIFIES PRESSING FIELD AND DENSITY FOR EACH DATASET. MELTING POINTS ARE INDICATED BY ARROWS. THE SOLID LINES CORRESPOND TO THEORETICAL MOBILITY OF 2D ELECTRON LIQUID. THE INSET SHOWS MOBILITY VS. PRESSING FIELD FOR A CONSTANT DENSITY.................. 103 FIGURE 7-12 2DES SPECTRUM IN MAGNETIC FIELD. THE SPECTRA ARE SHIFTED VERTICALLY TO FACILITATE COMPARISON............................................................... 105 FIGURE 7-13 CYCLOTRON FREQUENCY VS. MAGNETIC FIELD FOR THE SPECTRA OF FIGURE 7-12. ........................................................................................................................ 106 FIGURE A-1 SURFACE WAVE........................................................................................... 111 FIGURE B-1 RMS ROUGHNESS OF LIQUID 4 HE SURFACE (FROM (COLE 1970))............... 116 FIGURE C-1 PARAMETRIC RESONANCE STABILITY DIAGRAM. HORIZONTAL AXIS: RIPPLON FREQUENCY. VERTICAL AXIS: VIBRATION ACCELERATION AMPLITUDE. BOTH ARE NORMALIZED BY VIBRATION FREQUENCY. FROM HTTP://MONET.PHYSIK.UNIBAS.CH/~ELMER/PENDULUM/PARRES.HTM. .......... 117 FIGURE D-1 HEAT EXCHANGER MOUNTED ON CRYOSTAT'S BAFFLE............................... 120 FIGURE E-1 TOP OF THE CRYOSTAT................................................................................. 122 FIGURE E-2 50MK RADIATION SHIELD............................................................................ 123 FIGURE E-3 STILL RADIATION SHIELD............................................................................. 123 FIGURE E-4 DILUTION FRIDGE CONTROL PANEL............................................................. 125 ix
- 10. 1 1 Introduction: 2D Electrons on the Surface of Liquid Helium 1.1 The Origin of Reduced Dimensionality The study of low-dimensional structures is central to modern condensed-matter physics (March and Tosi 1984; Butcher, March et al. 1993). Examples include quantum wires, 2D electrons in semiconductor heterostructures (von Klitzing 1987), quantum dots (Khoury, Gunther et al. 2000), electrons floating on the surfaces of inert cryoliquids and cryosolids (Andrei 1997), etc. Recently low-dimensional structures (and 2D electrons on helium is one of the most promising candidates - (Dykman, Platzman et al.)) are being used as a basis for the growing field of quantum computing (DiVincenzo 2000; Nielsen and Chuang 2000). The subject of this work is 2D electrons on liquid 4 He. First I discuss the conditions leading to reduced dimentionality. Every physical system resides in the 3D world and is subject to 3D interactions. Occasionally however the nature of the interactions makes it possible to separate variables in the Hamiltonian. One such system is an electron above a perfectly smooth surface: its motion parallel to the surface is completely independent from the motion normal to the surface, which in terms of the hamiltonian implies: H(x,y,z) = H||(x,y) + H⊥(z). The other necessary condition for the system to be considered 2-dimensional is that H⊥(z) contains a confining potential. The energy of motion in the z- direction is quantized in this case. Let E1, E2 be the energies of the ground and the first
- 11. 2 excited state of the system for such motion respectively. If the system is cooled to a sufficiently low temperature so that 12 EETkB −<< , it will then mostly remain in the ground state. The motion in the z direction is “frozen out”. Such system becomes two- dimensional. Since quantization of energy is essential here, the reduction of dimentionality is therefore a purely quantum effect. It is important to note that in two dimensions the density of states of a free particle with nonzero mass as a function of energy is constant ( ) 2 hπ ρ m EE = (1-1) as opposed to 3D ( ( ) EEE ~ρ ) or 1D ( ( ) E EE 1 ~ρ ) cases, where m is the effective mass of the 2D particle. Even though the physics of reduced dimensionality applies to all 2DES, they will differ quantitatively depending on the environment where electrons are confined. The main factors are the dielectric constant of the medium, its band structure and fabrication technique (for semiconductors). Table 1-1 lists the values of basic parameters describing 2DES on helium versus analogous quantities for 2DES in GaAlAs heterostructures and in silicon MOS structures. Unmodified electron mass, low densities and high mobilities are the distinguishing features of 2DES on helium as opposed to 2DES in semiconductors. I will argue in section 1.3 that such combination of properties makes 2DES on helium the likeliest candidate for observing an ordering transition known as Wigner crystallization. For a review of the basic properties of 2D electronic systems refer to (Ando, Fowler et al. 1982).
- 12. 3 Table 1-1 Basic Properties of 2DES. On Helium In Si-MOS In GaAs/AlxGa1-xAs Binding Energy, K 7 50-500 200-400 Wavefunction Extent out of 2D plane, Å 114 30 50-100 Surface Density, cm-2 10 6 -10 9 -? 10 11 -10 13 10 11 -10 12 Fermi Energy, K 10 -5 -10 -2 10-500 200-1000 Effective Mass, m*/me 1.0 0.19 0.066 Mobility, cm2/V s 10 7 10 3 10 5 -10 6 In the next section we will consider the confining potential experienced by an electron above the surface of liquid helium. 1.2 Confinement of Electron to the Surface of Liquid Helium Bound states of electrons on the surface of liquid 4 He were predicted (Shikin; Cole and Cohen 1969) in 1969. An electron is bound to the surface of liquid 4 He (or any other dielectric) (Andrei 1997) by inducing an electrostatic screening charge as shown in Electron Image Charge Induced Charge 4 He 1 1 + − He He e ε ε Figure 1-1 Screening Mechanism Giving Rise to Electron Confinement to a Liquid Helium Surface.
- 13. 4 Figure 1-1. This forms the attractive part of the binding potential (Figure 1-2). The repulsive part comes from the fact that the supporting surface is made of inert atoms having complete electronic shells and thus no states available for an external electron. More exactly it takes a sufficiently high energy for an electron to enter the bulk of liquid helium and to form a so called bubble state (Onn and Silver). This poses a barrier for penetration into the liquid helium whose magnitude is roughly 1 eV (Fetter 1976). However, since the typical binding energy for an electron on liquid helium, as will be shown soon, is a fraction of a meV it can be approximated by an infinite barrier. Also the interface between liquid helium and its vapor is considered abrupt and perfectly flat in this approximation (the actual transition layer is about 6 Å (Penanen, Fukuto et al. 2000). In addition the helium surface is rough as there are thermally excited surface waves – the mean square amplitude of the roughness is ~1 Å, see appendix B and also (Edwards and Saam 1978)). The total potential experienced by an electron in the direction normal to the liquid surface along with the resulting energy levels is shown in Figure 1-2. The assumptions are: • Simple Coulomb attraction between an electron and its image in the substrate. • The electron’s wavefunction vanishes at the interface. • The Hamiltonian is separable if the surface is completely smooth. The motion in the z direction is then described by a 1D Hamiltonian <∞ >−+ ∂ ∂ −=⊥ 0 0 42 2 2 22 z z z e zm H δh , 1 1 + − = ε ε δ which is identical to the radial part of Schrödinger equation for the hydrogen atom with zero angular momentum. Thus the energy spectrum of a 2D electron is:
- 14. 5 2 42 2 32 , h em Ry n Ry En δ =∗ ∗ −= (1-2) where ε is the dielectric constant of the substrate, m and e are the effective mass and charge of the electron respectively (a cyclotron resonance measurement confirmed that the effective mass is equal to that of a free electron m = me (Edel'man 1976)). The ground state wavefunction for motion in the z direction is 02 3 01 2)( a z ezaz −− =φ where δ ba a 4 0 = is the effective Bohr radius. Liquid 4 He 1eV z e zV 4 )( 2 δ −= z 2 1 n E En −= Å76 4 0 == δ Ba a 028.0 1 1 = + − = He He ε ε δ KRyE 5.7 16 2 1 −=−= δ Figure 1-2 Energy Diagram of an Electron Bound to He surface. The spectrum can be investigated experimentally for instance by recording spectra of mm-wave absorption corresponding to transitions between the energy levels (Grimes, Brown et al.; Zipfel, Brown et al.). For an electron bound to the surface of liquid 4 He the
- 15. 6 binding energy is Ry* = 0.65 meV = 7.5 K. Therefore for the temperatures used in the presented experiment (< 1K), electrons are always found in the ground state. The effective Bohr radius a0 = 76 Å is greater than both the interatomic distance in liquid helium (~1Å) and the roughness of the helium surface. This in turn justifies the use of a macroscopic dielectric constant to describe screening effects in the substrate as well as the assumption of an ideally smooth helium surface. 1.2.1 Influence of Externally Applied Electric Field Usually the experiments are conducted with electrons bound not only by attraction to their images but also by an external clamping electric field E⊥. This is necessary in order to counter the mutual repulsion of electrons and to reduce their thermally activated escape rate from the 2D layer. The probability of escape to the unbound states is proportional to the density of states there and the latter becomes infinite for the 1D hydrogen potential as the energy approaches zero. In the presence of a clamping field the approximate ground state wavefunction is (Saitoh 1977): h ⊥− −− = == Eeam abwithezbz b z 3 01 0 2 3 1 2 , 4 9 sinh 3 1 sinh 3 4 2)( λ λ λ φ (1-3) whereas the ground state energy is −−= 4 32 0 2 2 1 a b bm E h . Therefore applying an external field allows controlling the out-of-plane extent of the wavefunction b (as opposed to a0 without pressing field). This will affect the interaction between electrons and the helium surface – see 1.5.2. The parameter λ describes the strength of the external field as compared to a characteristic field:
- 16. 7 cm V mea Ec 864 2 3 0 2 == h (1-4) which is roughly the field due to the image charge. In the limiting cases of weak (λ << 1) and strong (λ >> 1) field, b is given by 3 2 0 2 0 3 4 4 3 1 λ λ =−= a b and a b respectively. 1.2.2 Binding to a Thin Film When the helium film thickness d becomes comparable to the distance of the electrons from the helium surface (given by a0 = 76 Å) the total potential must include the field of the image charge in the underlying substrate as well as the image in helium. Hence the total potential is (including the external pressing field): ( ) ( ) zeE dz e z e zU sh ⊥+ + −−= 44 22 δδ Here δh = (εh- 1)/( εh+ 1) and δs = (εs- εh)/( εs+ εh), εh, εs are the dielectric constants of helium and the substrate respectively. In the limit when d > a0 = 76 Å the second term can be expanded in z/d resulting in: ( ) zeE z e zU h ⊥+−≈ 4 2 δ where we made the substitution 2 4d e EE hδ +→ ⊥⊥ and dropped the constant term. Therefore in this situation the effect of the image charge in the substrate is equivalent to an enhancement of the pressing field. For d = 100 Å this enhancement is 1 kV/cm – exceeding Ec in Eq.(1-4), so that the 2DES on thin films is always in the regime of strong pressing fields.
- 17. 8 Using the strong field approximation formulas of 1.2.1 one can get an estimate for the variation of the average distance between an electron and the helium surface caused by the substrate: <z> = 3/2 b ranges from 30 Å for thin film to 110 Å for bulk. 1.2.3 Hartree Field (Mutual Repulsion of Electrons) So far, the fields acting on a single surface electron have been considered. In a typical experiment the surface of helium is charged with an almost uniform surface density n whose typical values range from 107 to 109 electrons per cm2 . Each electron is then creating its own image charge in helium. This range of densities corresponds to interelectronic distances of order of mm, which is much larger than 2 a0 = 152 Å – the distance between an electron and its own image while its distance to all other images is larger. Therefore the total electrical field an electron sees is a superposition of the Coulomb field from its image and a uniform background, which includes other images. The vertical stability of the 2D electron is further discussed in 6.5.2. Beside liquid 4 He, other substances used to support 2D electrons include: solid 4 He, solid hydrogen (Adams and Paalanen 1988; Kono, Albrecht et al. 1991; Kono, Albrecht et al. 1991; Mugele, Albrecht et al. 1992), solid neon (Kajita), liquid 3 He. The work that was recently carried out on 0D and 1D electrons on helium is reviewed in (Kovdrya Yu). 1.3 Wigner Crystal The most striking phenomenon caused by the interactions between electrons bound to the surface of liquid helium is the appearance of an ordered phase – Wigner crystal (Crandall and Williams; Wigner 1934).
- 18. 9 The phase of a 2D electron system is determined by the competition of three energy scales: • Coulomb energy of interaction between electrons Ve-e = n e π ε 2 . • Thermal energy kBT (or classical kinetic energy). • 2D Fermi energy m n EF 2 hπ= (or quantum kinetic energy). Here n is the electron surface density, T – temperature, m and e – effective mass of electron and charge respectively, ε is the dielectric constant of the substrate (ε = 1.057 for liquid helium-4). A qualitative phase diagram of the 2D electron system resulting from this competition is presented in Figure 1-3 (Fukuyama; Platzman and Fukuyama 1974). Figure 1-3 Schematic 2DES Phase Diagram. The straight line separates the region where classical fluctuations dominate (lower part) from the one where quantum fluctuations dominate (the upper part).
- 19. 10 The range of parameters for which the potential energy term dominates corresponds to the ordered part of the phase diagram. Outside this regime, the kinetic energy (either classical or quantum) dominates and the system is disordered. At high densities and low temperatures the quantum fluctuations are more important than classical ones and we have a quantum phase transition at a critical density nw determined from the condition Ve−e(nw ) ~ EF(nw ): 42 42 ~ hε em nw (1-5) For 2DES on helium bulk nw ~1013 cm-2 . A thorough discussion of the phase diagram is deferred to Chapter 2. The theoretical description of the properties of Wigner crystal (it is sometimes referred to as Coulomb crystal in the classical portion of the phase diagram) was given in (Bonsall and Maradudin 1977). In particular they estimated the energies of all possible 2D lattices, having found the lowest value for a hexagonal lattice. The magnitudes of reciprocal lattice (which is also hexagonal) vectors form the following sequence: Figure 1-4 2D Hexagonal Lattice - lowest energy configuration of 2D Coulomb crystal. ....16,13,12,9,7,4,3,1, 3 8 , 22 11 === pnGGpGp π (1-6) Theoretical work (Tanatar and Hakioglu) has revealed the possibility of a superconducting region in 2DES phase diagram forming at temperatures of order of
- 20. 11 millikelvin. However it is not possible to cool the 2D electrons to this temperature due to their weak coupling to thermal bath, consisting mainly of ripplons (surface phonons) on liquid helium. Due to the limit on electron surface density imposed by the collapse of a bulk charged liquid surface - electrohydrodynamic (EHD) instability (discussed in 2.3 of this work and also (Ikezi and Platzman 1981)) only a small portion of the phase diagram (classical one) is experimentally accessible as shown schematically in Figure 1-3. One of the goals of this work is to investigate methods for reaching the areas of the phase diagram where quantum effects are substantial – in particular to be able to observe a quantum phase transition. The transition is in principle achievable for 2DES on liquid helium films but not in 2DES in Si-MOS or GaAlAs heterostructures. This is because the latter have lower effective carrier mass (and therefore larger Fermi energy) and lower electron- electron interaction energy due to higher dielectric constant (see Table 1-1). Therefore, according to Eq.(1-5) the critical density nw is lower than that of 2DES on helium. It is actually lower than the minimum density achievable by present day fabrication techniques of Si-MOS or GaAlAs heterostructures (~1011 cm-2 ). This explains the choice of 2DES on helium as a candidate for exhibiting the Wigner crystallization. To date there is no convincing experimental evidence for Wigner crystallization in 2DES in semiconductors. 1.4 Collective Modes of 2DES In this work, the physical properties of the 2DES are deduced from RF absorption spectra corresponding to normal modes of the electron sample. The excitation branch common to
- 21. 12 all charged plasmas is a longitudinal plasmon. In a 3D plasma this is a dispersionless oscillation of frequency m en D 0 2 3 ε ω = (1-7) In 2D the dispersion relation is: q m en D 0 2 2 2ε ω = (1-8) This particular dependence on q is the consequence of ~ 1/r interaction potential between electrons (r is the interelectronic distance). However since in a realistic experiment metal electrodes are present in the vicinity of the 2DES (at a distance d from 2DES), the interaction is screened so that the potential is ~ 1/r3 when r > d. This makes the dispersion law approximately linear near q = 0 so that long wavelength plasmons have a finite propagation velocity: m edn cp 0 2 ε = (1-9) This velocity corresponds to 1.8 108 cm/s for typical values, n = 108 cm-2 , d = 1mm. These formulas are derived in section 3.1. The longitudinal plasmon dispersion is practically the same for either electron liquid or solid (exact dispersion laws for both are in section 3.2). Therefore, it is not possible to use this mode alone to study crystallization. However there are two alternatives: • Solids (2D as well as 3D) unlike liquids possess a finite shear modulus and therefore permit propagation of a transverse mode if the dissipation is not too large. See 3.2
- 22. 13 • As will be explained later when the 2DES is coupled to the underlying liquid its normal mode spectrum is very sensitive to the phase of the 2D electrons. Specifically the correlated nature of electron motion leads to strong electron- ripplon coupling which opens a gap in the spectrum. The size and geometry of the 2D electron pool determine the wavevectors q of the normal modes. Excitation of these resonances is attained by sweeping the frequency of the external electromagnetic field through the range of interest. Chapter 3 contains a discussion of the 2DES spectra. The detection technique is the subject of Chapter 4. 1.5 The Dynamics of Liquid Helium and Its Effect on 2DES One has to consider the properties of the liquid helium supporting the 2DES since the interaction between them affects both their excitation spectrum and the phase diagram. 1.5.1 The Surface Waves - Ripplons Because the electrons reside on the helium surface of liquid, their motion is coupled to surface excitations. These excitations – capillary waves - are governed by gravity and surface tension leading to a dispersion relation of the form (appendix A): kdkkgr tanh32 += ρ σ ω (1-10) where g – is the gravitational acceleration, σ – surface tension, ρ – density of liquid, d – depth of liquid. The characteristic wavevector determining the relative strength of terms in Eq.(1-10) is referred to as capillary constant: 1 20 − == cm g kc σ ρ (1-11)
- 23. 14 The numeric value is for liquid 4 He at 0K (see appendix B). The quanta of capillary waves are referred to as ripplons. Thermally excited ripplons determine the roughness of helium surface - appendix B. 1.5.2 Interaction Between 2DES and the Surface of Liquid Helium Electrons with surface density n bound to liquid helium in the presence of a clamping field E⊥ exert a pressure on the surface: ⊥= EenPe . This pressure is a result of the counteraction corresponding to the repulsive barrier at the helium surface – see discussion in 1.2. As a result, the helium surface under the electrons is depressed whereas the uncharged surface is elevated. The difference in level is: g Een h ρ ⊥ =∆ The absolute depression of the surface depends on the ratio of the uncharged area to the total area of the helium surface: total edunch LHe S S g Een d arg ρ ⊥ =∆ For typical parameter values (n = 5 108 cm-2 , =1000V/cm, and the ratio of areas ¼) the helium surface is depressed by 14 microns. This effect is even more significant if the experiment is performed on a thin film of liquid helium (section 1.5.3). ⊥E Another effect of the electron pressure is that each electron produces a small depression in the helium surface – the so-called single electron dimple. If the helium thickness is sufficiently large (~ 1 mm) the typical depth of the dimple is ~0.1Å for a pressing field of 500 V/cm. The binding energy for such a dimple is small – of order of
- 24. 15 ( ) K eE 3 2 10 4 −⊥ ≅ πσ (Jackson and Platzman). Therefore no bound state of electron – so- called ripplonic polaron (Shikin), which is an electron dressed by ripplons - is expected for realistic experimental temperatures T > 50mK on bulk helium (Jackson and Platzman). Yet, even on bulk the dimple effect is important when the motion of electrons is correlated as in the case for the Wigner crystal. The effect of this dimple is to open a gap in the spectrum of the Wigner crystal and will be discussed in section 3.4. The ripplons also scatter 2D electrons thus affecting their mobility. This is discussed in section 3.3. 1.5.3 Thin Film Effects As the thickness of the helium film becomes small (<1000Å), a number of new factors have to be taken into account. First the solid substrate supporting the film alters (usually increasing) the binding energy (section 1.2) and also screens the interaction between electrons. The latter effect is important when the film thickness becomes comparable to the distance between the electrons and leads to a reduction in the size of the Wigner crystal “bubble” in the 2DES phase diagram as will be explained in 2.4. The polaronic transition now becomes a reality since an electron is subjected to a strong field from its image in the substrate in addition to the external pressing field. Signatures of this transition were reported in early experiments (Andrei 1984; Tress, Monarkha et al. 1996). The second important consequence of the film becoming thin concerns the film itself. The helium surface becomes stiffer as a result of the Van der Waals forces (I. E. Dzyaloshinskii 1961; Sabisky and Anderson 1973) from the underlying substrate
- 25. 16 (see 2.3). Formally this can be accounted for by using a modified gravitational constant g~ in Eq.(1-10) which turns out to be four orders of magnitude larger than g = 9.8m/s2 for a 100 Å helium film. Because of this stiffening, thin films are stable against the EHD instability (see 2.3). This allows charging to high surface densities. In addition the two- fluid nature of superfluid 4 He (Putterman 1974) starts playing a role since the motion of the viscous normal component is restricted by the substrate giving rise to third sound propagation (Komuro, Kawashima et al. 1996). This happens when the film is thinner than the boundary layer thickness ωρ η n 2 , where η and ρn are the normal fluid viscosity and density respectively, ω is the sound frequency. For typical values of the parameters at 1K (η = 27 µP, ρn = 2 10-3 g/cm3 (Donnelly, Glaberson et al. 1967)) and a frequency of 10 MHz the boundary layer is 2000 Å thick. The most widely used method for obtaining a submicron helium film relies on the fact that in the presence of helium gas the solid walls of a container are coated with an adsorbed film of helium atoms. If the film is superfluid, it has a thickness profile according to its vertical position above the surface of helium bulk. Therefore, by adjusting the height of the helium bulk with respect to a solid substrate one can control the film thickness adsorbed on the substrate. As will be argued in 2.3 it is also possible to arrive at a 2DES deposited on a thin film by starting from a relatively thick film such as might be obtained by directly adjusting the level of helium bulk.
- 26. 17 1.6 Summary The main contributions of this thesis are: • A new technique is proposed for measuring the thickness of a helium film by using the spectral splitting that arises when the 2DES is strongly coupled to a transmission line. • The experiment described in this work uses two novel transmission line structures to couple electromagnetic waves into a 2D electron layer: an interdigital capacitor and a coplanar meander line. Their properties are reviewed in Chapter 4. • As discussed in section 2.3 a method was developed to obtain a charged thin film of helium starting with an uncharged thick film. This is the first step toward achieving the more ambitious goal of accessing the quantum regime of 2DES phase diagram – namely the region of high densities and low temperatures, and to observe the quantum melting of the Wigner crystal. As was briefly mentioned in section 1.5.3 it is the use of thin (~100 Å) helium films that makes it possible to charge the helium surface to the desired densities. In order to achieve this goal it will be necessary to overcome a number of technical difficulties, which were encountered in the course of this study: • Using thin helium film requires preparation of nearly atomically smooth substrate to support the liquid. Even though it is quite straightforward to prepare such a substrate (6.7) it turned out to be impossible to keep it clean and smooth during the subsequent assembly of the experimental cell and cooldown. • There seems to be no measurable coupling between 2DES and helium film such as required to insure different spectra in liquid vs. solid state of 2DES. This is
- 27. 18 probably due to a stiffening of the helium surface and to an increase of the associated frequencies as the film thickness becomes small.
- 28. 19 2 Phase Diagram of 2DES on Liquid Helium Electrons floating on the surface of liquid 4 He can be (with regards to their in-plane motion) in a spatially disordered state (2D electron liquid) or they can form a regular structure known as Wigner crystal. The range of parameters for which each phase is stable and the resulting phase diagram are outlined bellow. For a more extensive discussion see (Devreese, Peeters et al. 1987). 2.1 Electrons on Bulk Helium The simpler case is when the depth of the He layer is not too small – above 100µm and much larger than the interelectronic distance (~µm) so that there is little influence from the substrate underneath the helium – this is referred to as 2DES on helium bulk. The plasma parameter describes the relative importance of interaction over fluctuation and is defined as the ratio between the average potential and kinetic energy (thermal or quantum – whichever is greater): K V ee− =Γ For 2D electrons with surface density n neV ee π2 =− The kinetic energy (for either degenerate or nondegenerate 2DES) is given by ( ) ∫ ∞ − + = 0 2 2 1Tk x B B e dxx n Tkm K µ π h (2-1)
- 29. 20 where ( ) −= 1ln 2 Tkm n B B eTk hπ µ is the chemical potential. This expression reduces to <K>classical = kBT for nondegenerate electrons (EF << kBT) or to <K>quantum = EF = m n2 hπ (2-2) for degenerate electrons (EF >> kBT). It is obvious that for sufficiently low surface densities and temperatures the interaction would dominate giving rise to spatial order. The phase diagram can be obtained by setting the plasma parameter equal to its critical value – also known as the Lindemann melting criterion (Lindeman 1910): mΓ=Γ (2-3) In other words, the ratio of average potential to average kinetic energy has to drop below a certain value Γm for melting to occur or equivalently the amplitude of fluctuations has to become comparable to the lattice constant. A value for the classical case most widely confirmed both by theory (Gann, Chakravarty et al. 1979; Morf 1979) and experiment is Γm ~130 (Deville, Gallet et al.; Grimes and Adams; Kajita; Marty and Poitrenaud; Rybalko, Esel'son et al.; Shirahama and Kono; Mehrotra, Guenin et al. 1982; Mellor and Vinen 1990). The phase diagram resulting from the Lindemann criterion with Γm =137 in n, T variables is presented in Figure 2-1 (Peeters 1984). Note that it is a qualitative phase diagram since Γm = 137 is only valid for nondegenerate 2DES and would certainly have a different value for a transition driven by quantum fluctuations. This value (33 ± 5) was obtained in (Tanatar and Ceperley 1989) using Green’s-function Monte Carlo calculations.
- 30. 21 0 2 4 6 8 10 12 14 0 1x10 11 2x10 11 3x10 11 4x10 11 1x10 13 2x10 13 QUANTAL LIQUID CLASSICAL LIQUIDTHERMAL FLUCTUATIONS n[cm -2 ] T[K] QUANTUM SOLID QUANTUM FLUCTUATIONS n W Figure 2-1 2DES Phase Diagram on Helium Bulk. The “bubble” defined by this criterion is the Wigner solid. The area outside the bubble corresponds to a disorder phase – electron liquid. The surface density values in the classical part of solid phase are less then 1011 cm-2 , which corresponds to a lattice constant of order of 10-2 µm – a large value compared to its counterpart in regular 3D solids. 2.2 Wigner Solid and 2D Melting Mechanisms The existence of an ordered electron phase was first predicted (for 3D electrons) by E. Wigner (Wigner 1934) in 1934. However 3D Wigner solid was never unambiguously realized experimentally. One of the reasons is that in the 3D electron plasma screening of interaction is much more significant than in lower dimensions. However in 3D there exist
- 31. 22 a number of phenomena similar in nature to Wigner crystallization such as Mott transition (Mott 1961). Theoretically it was shown that no true long-range order can exist in 2D (N. D. Mermin 1966; Peierls 1979) which is expressed by stating that density-density correlation function decays with distance. This is due to a stronger impact of fluctuations (thermal or quantum) in lower dimensions (and fewer nearest neighbors). However as shown by KTNHY theory (see below) the correlations in 2D decay only as a power law with the distance R between particles: ( )T G G Rg η− ~ (2-4) (G – is a reciprocal lattice vector). This decay corresponds to quasi-long-range order in contrast to the exponential decay in the liquid phase where the order is lost. In the harmonic approximation the exponent in Eq.(2-4) is ( )T TG TG µπ η 2 )( 2 = where m(T) is the shear modulus in the Wigner crystal (m(0)=0.245 e2 n3/2 (Bonsall and Maradudin 1977)). Among the possible 2D lattice structures the hexagonal one was shown to be the most stable (Haque, Paul et al. 2003). The theory of melting in 2D was developed in the 1970’s and is known as KTHNY theory (Kosterlitz and Thouless 1972; Halperin and Nelson 1978; Young 1979). In this theory the solid is destroyed by unbinding of topological defects – dislocations - in the crystal. In some cases, the melting is a two-stage process. During the first stage, occurring at temperature Tm, paired dislocations become unbound and the solid is
- 32. 23 transformed into a hexatic phase characterized by the absence of long-range positional order while retaining bond orientation order. The critical exponent ηG (Eq.(2-4)) and plasma parameter can be expressed via elastic Lame: coefficients (µ is the shear modulus, B = µ+λ is the bulk modulus (Landau and Lifshits 1970)) as follows: ( )λµµ λµ π η + + = 2 3 4 )( 2 GTk T B G ( )λµµ λµππ + + =Γ 24 2 a n a is a lattice constant here. The KTHNY gives the following value for Tm: ( )λµ λµµ π + + = 2 )(4 16 2 B m k a T In the approach to the hexatic phase the shear modulus µ behaves as: ( ) ( ) ))(1( ν µµ TTconstTT mm −+= ν = 0.36963 (Deville, Valdes et al. 1984) Finally in a second step at a temperature above Tm disclination pairs unbind and the bond orientation order is destroyed as well resulting in a liquid phase. The most direct way of studying the lattice structure is by measuring the Bragg diffraction patterns that result from scattering of some kind of waves on the lattice. The wavelength should be close to the lattice constant. The order in 2DES on liquid helium cannot be investigated with visible light or neutron scattering – the cross-sections for these processes are miniscule. It is in principle possible to study the scattering of helium surface waves – ripplons (section 1.5) off 2DES. Such experiment has been proposed in (T. Williams 1995; Deville 1997). To generate a ripplon with the wavelength in
- 33. 24 micrometric range one can use an interdigital capacitor structure (IDC) described in 4.3. IDC should be positioned outside of 2DES pool, but in such a way that it is covered with liquid helium. The voltage is applied between the two sets of IDC’s fingers. The resulting electrostrictive force excites a surface wave whose wavelength is related to the period of the IDC λIDC: kr = 4πn/λIDC. If the excitation frequency matches that of a ripplon ωr(kr) as given by Eq.(1-10), a resonant surface wave is generated. It then travels through the 2DES, coherently scattering from electrons. The detection can be performed with an identical IDC located on the opposite side of the 2DES pool: the incoming surface waves cause variation of capacitance between the IDC fingers. The mean square of induced current is proportional to the structure factor of 2DES: ( ) ( ) ∑ − = jii eS rrk k π2 1 where ri is the coordinate of a 2D electron. In this experiment, the direction of the ripplon wavevector k is kept constant, while reciprocal vectors of 2D lattice can be swept either by squeezing it with an external electric field and therefore changing the 2DES size or by turning it with changing vertical magnetic field. The expected results are shown in Figure 2-2 for three possible phases of 2DES. The structure factor is plotted for two different orientation of the 2DES.
- 34. 25 kx |k| S(k) ky kx |k| S(k) ky kx |k| S(k) k) = ∑<exp i(k.(ri-rj))> /2π SOLID HEXATIC LIQUID Figure 2-2 2DES Diffraction Patterns (T. Williams 1995). Structure Factor: S(
- 35. 26 2.3 Electrohydrodynamic (EHD) Instability of a Charged Liquid Surface One of the main limitations on the experimental investigation of the 2DES phase diagram with electrons on helium bulk comes from the electrohydrodynamical instability and is related to the well-known Rayleigh instability (Rayleigh 1882). The instability sets a limit on the maximum electron density that can be supported by a surface of liquid helium or any other liquid (Gor'kov and Chernikova 1973; Gor'kov and Chernikova 1976). After exceeding this limit the surface would collapse. This happens because of the softening of liquid surface modes – capillary waves (see 1.5.1). More specifically the charged liquid surface has a dispersion relation, which differs from Eq.(1-10) by the electron pressure term: kdk ne kkgr tanh 4 2 22 32 −+= ρ π ρ σ ω (2-5) It is obvious that for a sufficiently high electron density the frequency vanishes, which corresponds to the development of an instability. This critical density is 29 4 2 102..2 )2( 1 − == cm g e nc π ρσ and the wavevector corresponding to the instability is equal to the capillary constant of liquid He 1 20 − == cm g c σ ρ k . Theory and experiments studying the nonlinearities developing at the onset of this instability (referred to as multielectron dimples as opposed to single electron dimple 1.5.2) are presented in (Ikezi 1979; Ikezi, Giannetta et al. 1982).
- 36. 27 However the instability can be avoided if a thin film of liquid helium is used instead of bulk. On the thin film there is an additional term in the dispersion formula of surface waves (Ikezi and Platzman 1981) arising from the Van der Waals interaction between helium atoms and the underlying substrate: kdk ne kk d gr tanh 4 ) 3 ( 2 22 3 4 2 −++= ρ π ρ σ ρ α ω (2-6) Here α is the Van der Waals constant (~ 10-15 erg for liquid helium on glass). This effective enhancement of the gravitational constant 4 3~ d gg ρ α += postpones the EHD instability to much higher densities (as a matter of fact for d in the submicron range g} >> g). The critical wavevector and density become σ α31 2 d kc = and 4 22 )2( 31 de nc π σα = respectively. In fact, the experiments have shown that the film is stable for practically any n. This enhanced stability can be understood by taking into account the reduction in film thickness caused by the electron pressure. The new film thickness can be found by the equating chemical potential over the charged area of the surface to that of the uncharged area: 2 30 )(2 en d dgdg π α ρρ +−= (2-7) The electron density in our experiments is directly related to the charging voltage V (described in 6.5.2). The above equation has to be satisfied along with the relationship between charging voltage and density:
- 37. 28 ssHe dde V n εεπ // 1 4 + = (2-8) here d, εHe – are the thickness and dielectric constant of liquid helium respectively, ds, εs – thickness and dielectric constant of substrate if it is used. These two equations have to 15 35 55 75 95 115 135 155 175 195 d@mmD 9 19 29 n@108cm-2D FilmStabilityfor V=20 Volt 15 35 55 75 95 115 135 155 175 195 d@mmD 9 19 29 39 n@108cm-2D FilmStabilityforV=40 Volt Figure 2-3 Charging Instability. Dashed line is the plot of n vs. d from Eq.(2-7). Dot-dashed line is the plot of n vs. d from Eq.(2-8). One of their intersections corresponds to the stable charge density. Solid line is the EHD limit of 2.2 109 cm-2 . (b) (a)
- 38. 29 be solved simultaneously since the density depends on the final film thickness. Graphical solutions for two values of charging voltage are shown in Figure 2-3 for d0 = 0.3mm, ds = 0.3mm, εs = 11. (EHD stability limit is shown as well). The lower density solution in Figure 2-3(a) is stable while the other one is unstable. For sufficiently high charging voltages no “bulk” solution would exist which is illustrated in Figure 2-3(b). This happens when gddV ssHe ρπεε 4))//( 3 2 ( 2 3 0 +> (2-9) Comparing Figure 2-3(a) and (b) one can see that for any charging voltage lower than this value the stable density would be below the EHD limit. Therefore by starting with a film, which is a fraction of a millimeter thick and using sufficiently high charging voltage one can end up with 100Å film bypassing EHD. This method has important practical implication because it is very difficult to charge a helium film since for electrons it is easier to shoot through a thin film at the beginning of charging when there is no repulsion due to already deposited electrons. 2.4 Influence of a Substrate on Phase Diagram When electrons are deposited on a film of liquid helium with thickness comparable to the inter-electronic distance screening due to the substrate supporting the film leads to a modified phase diagram. Consequently the Wigner crystal melts at a lower temperature. To obtain the phase diagram of 2DES on liquid He film of thickness d supported by a dielectric substrate of permittivity Sε one solves Eq.(2-3) with
- 39. 30 1 1 41 1 2 2 + − = + −=− S S s s ee where dn neV ε ε δ π δ π (2-10) The phase diagram for several thicknesses of liquid helium film supported by a substrate with δs=0.9 is shown in Figure 2-4. 0 1 2 3 4 5 6 7 8 9 10111213 0 1 2 T[K] n[1012 cm-2 ] Bulk 300Å 100Å 70Å Figure 2-4 2DES Phase Diagram on Dielectric Substrate vs. Helium Film Thickness. The case of a metallic substrate corresponds to 1, =∞= ss δε in Eq.(2-10) and the phase diagram is shown in Figure 2-5. The difference from the diagram for a dielectric substrate is the appearance of a liquid dipole phase at T = 0 for thin films. This is because on metallic substrate in the limit n d2 << 1 the Coulomb term in the electron-electron energy vanishes and the interaction is dipolar: 2 2eV ee ππ=− . Comparing to22 3 dn
- 40. 31 Eq.(2-2) for Fermi energy, one can see that for densities below 2 22 2 ≅ emd n h quantum fluctuations will dominate the electron-electron interactions promoting the liquid phase. Figure 2-5 2DES Phase Diagram on Metallic Substrate vs. Helium Film Thickness.
- 41. 32 3 Spectrum of 2DES This chapter discusses the normal modes of 2DES on helium. First the normal modes will be considered disregarding their coupling to the helium. Section 3.3 reviews the dissipative processes affecting the spectrum. The coupling between 2DES and helium substrate, presented in 3.4, provides a way to detect the Wigner crystallization and is used in 7.2. Finally 3.5 discusses the spectrum in a magnetic field. 3.1 2D Drude Model Here I will derive the dispersion law for classical (kBT >> EF) 2D plasma normal modes in the longwave limit. In what follows 2D plasma will be confined to the z = 0 plane. The electrodes surrounding 2DES are assumed to be properly DC biased to ensure confinement (section 6.5.2) so that they can be considered as grounded with regards to AC voltage induced due to the 2D plasma oscillations. First Poisson equation supplemented by the boundary conditions allow to obtain the relationship between the distribution of an excess surface charge density σe in the plane and the potential it creates Ve : ( ) ( )z z VV eee δ ε σ 0 2 2 2 2 r r −= ∂ ∂ + ∂ ∂ (3-1) In this equation r is the 2D coordinate vector in the electron plane. For the sake of simplicity we first solve this equation with boundary condition Ve = 0 as ±∞→z . The ultimate goal is to obtain the formula that describes the response of 2D plasma to an
- 42. 33 incident electromagnetic wave. Therefore one should look for a solution that has the following form: zqtie q e eeVV −− = )( qrω (3-2) The choice of z dependence is set by the requirement that Ve has to satisfy the Laplace equation outside the 2D plasma and its derivative02 =∇ e V z V E e z ∂ ∂ −= has to be discontinuous at z = 0. Substituting Eq.(3-2) into Eq.(3-1) one gets the formula relating the charge density and potential in the 2D plasma plane: 02 ε σ q V e qe q = (3-3) One might generalize this result for various different boundary conditions other than Ve = 0 as ∞→z . In each case the solution to Poisson’s equation would have the form )(~ )( zqzqzqzqtie eDCeeBAeeV +++ −−−qrω where the choice of constants A,B,C,D satisfies particular boundary conditions. One example is when there are two metallic electrodes present – one at the distance d below the 2D electron layer and the other at a distance h above it, as shown in Figure 3-1. h d Top Plate 2DES Ground Plane eh ed Figure 3-1 2DES Boundary Conditions: Two Dielectric Media.
- 43. 34 Then Eq.(3-3) should be replaced by: ( )qhqdq hd e qe q cothcoth0 εεε σ + =V . Figure 3-2 shows the most relevant case for our experiment: a layered arrangement where h is the distance between 2DES and top plate, dHe is liquid helium thickness and ds is semiconductor wafer thickness. Eq.(3-3) then should be modified to become: −−+++ −−+++ + = )(sinh)()(sinh)( )(cosh)()(cosh)( )coth(0 sHeHessHeHes sHeHessHeHes He e qe q ddqddq ddqddq hqq V εεεε εεεε εε σ h ds Top Plate 2DES Ground Plane eHe es dHe Figure 3-2 2DES Boundary Conditions in our Cell: Three Dielectric Media. Very important situation is when plasmon wavelength is much longer than the distance to the nearest electrode (let’s assume it happens to be d): qd<<1. The formula relating excess charge and voltage is then: d e qe q d εε σ 0 =V . Later I will use cylindrical coordinates r,θ, z for which Poisson equation solution should be written in terms of cylindrical harmonics:
- 44. 35 ( ) zqitie q e eeqrJeVV − = νθ ν ω ν, (3-4) instead of Eq.(3-2). Eq.(3-3) is still valid as it stands. Having analyzed the electrostatics of 2DES, we proceed to deriving the 2D plasma dispersion law by writing down the equation describing the motion of 2D electrons in the presence of an external electromagnetic wave E = E0exp(i(ωt-q0r)) propagating in the 2D plane of electrons: )( 1 E r uu − ∂ ∂ =+ e V m e &&& τ (3-5) Here u is the 2D displacement of an electron from its equilibrium position, τ describes dissipation (see 3.3) and the choice of signs is determined by the negative value of the electron charge. The wave is presumed longitudinal (E0||q0) – see the discussion at the end of this section. Electron displacement is related in the linear approximation to an excess charge density distribution by a material equation: r u ∂ ∂ = nee σ (3-6) where n – is the equilibrium electron density. Combining Eq.(3-5) with Eq.(3-6) one gets: )( 1 2 22 r E r ∂ ∂ − ∂ ∂ =+ e ee V m ne σ τ σ &&& Switching to harmonics, (both temporal and spatial) and substituting Eq.(3-3) for σe , one obtains: ( ) ( ) 00 0 0 2 22 , 2 Eqqe q q m ne i i qV p e q ετ ω ωω −= −− (3-7) Here e(q0, q) are the projections of the external wave onto the chosen set of normal modes of the 2D system, which represent the coupling between the 2DES and the
- 45. 36 excitation line. For example, if the normal modes are plane waves (V ), then e(q( )(exp~ qr−∑ tiV q e q e ω ) 0, q) appear as Fourier coefficients in the expansion of the external wave into plane waves: ( ) ( ) ( )qrErqEE iqqei q −=−= ∑ exp,exp 0000 (3-8) and turn out to be simply Kronecker’s deltas: e(q0, q) = δq0,q. The choice of the normal modes is dictated by geometry of the experiment. In our case cylindrical harmonics are appropriate. The form of coupling coefficients e(q0, q) for cylindrical geometry is to be discussed in section 4.4. Returning to Eq.(3-7) one can define 2D plasma susceptibility χ: ( ) ( ) ( ) ( ) τ ω ωωε ωχ ωχ i q m ne q q q q qqei E V p e q −− ≡ −= 220 2 0 0 0 1 2 , ,, (3-9) The imaginary unit in front of the formula is due to AC nature of coupling (no response to DC voltage). The susceptibility has a pole, whose real part ωp(q) corresponds to 2D plasma resonance frequency at wavevector q. Since, as discussed before, the electrostatics of 2D plasma varies with the boundary conditions, the dispersion law will reflect these changes. The 2D plasma dispersion law in different limits is shown in Table 3-1: Table 3-1 2D Plasma Dispersion Law in Various Limits. Unscreened (short wave) case: qd >>1 q m ne p 0 2 2 ε ω = (3-10)
- 46. 37 Screened (long wave) case: q d<<1 q m edn eff p 0 2 ε ω = where hd eff hd d εε += (3-11) Intermediate case, as in Figure 3-1. )coth()coth(0 2 hqdq q m en hd p εεε ω + = (3-12) If the wavelength is much longer than the distance to the nearest metallic electrode, as is the case in this experiment, the dispersion will be linear for small q (Dahl and Sham 1977). 2D plasmons in 2DES on helium in its liquid phase were first observed by (Grimes and Adams) as a sequence of standing wave resonances excited in a rectangular electron pool using a technique similar to ours. It follows from Eq.(3-9) that relaxation time τ determines the quality factor of 2D plasma resonances: in the spectrum the observed 2D plasma resonances will appear to have finite linewidth equal to 1/τ. τ reflects various scattering processes (see 3.3) that electrons undergo in their motion. Therefore the scattering time of the dominant scattering mechanisms can be extracted by measuring the width of resonance lines in the spectrum. One should bear in mind however that there is also an instrumental contribution to the linewidth due to finite coupling between 2DES and the excitation structure (Chapter 4). The formulas just derived give the normal frequencies of the 2D plasma. When 2D plasma is bound laterally only certain wavevectors q are allowed. We consider relevant case of 2D electron system having circular boundary as is realized in our experiment. The normal modes are best expressed via cylindrical harmonics (Eq.(3-4)). In the absence of
- 47. 38 external field the normal mode should be a standing wave therefore one can assume the radial component of electron velocity to vanish at the edge of the 2DES pool (further discussion of 2DES size and lateral confinement is deferred to 6.5.2). In this case one obtains a set of drum modes. As follows from Eq.(3-5) after setting E = 0, this condition is equivalent to 0= ∂ ∂ =Rr e r V 0), =Rµν . Substitution of Eq.(3-4) leads to a condition on the allowed q : where R is the radius of 2DES pool. Therefore the allowed wavevectors are: (′ qJν R q νµ µν ξ′ =, (3-13) where νµξ′ are the extrema of Bessel functions: Table 3-2 Bessel Extrema. νµξ′ 1.84 3.05 3.83 4.20 5.32 5.33 6.42 6.71 ν 1 2 0 3 4 1 5 2 µ 1 1 1 1 1 2 5 2 Some general remarks follow. The chosen shape of Ve (r) and the equation of motion suggest that • The electrons move in the direction of propagation given by q – the oscillations are longitudinal • The electric field vector ( r∂ ∂ − e V ) is in the same direction
- 48. 39 One then expects that in order to excite 2D plasmons the external field should be longitudinal. However since divεE = 0 the plane electromagnetic waves are transverse. The use of specially shaped transmission lines makes it possible to synthesize longitudinal waves. Chapter 4 contains a discussion of the degree of coupling between the 2DES modes and the external electromagnetic field necessary to excite these modes. 3.2 Electron Solid Modes The mode previously considered is longitudinal and is the only mode present in 2DES in its liquid state. Upon solidification a new transverse mode appears – the so-called shear mode. This mode can be used to detect the Wigner solid (Deville, Valdes et al. 1984). A detailed study of the electron solid spectrum is found in (Bonsall and Maradudin 1977). Here we only quote the long wavelength limit dispersion law for a hexagonal lattice: ( ) ( ) ( ) ( ) ( ) ( ) n m e cqcqq qqqq ttt ptp π ε ωω ωωωω 0 2 22222 2 2222 1 138.0 , 5 === ≈−= (3-14) where ( )qpω is the 2D plasma dispersion given by the formulas in section 3.1. In the long wave limit (q->0) the term quadratic in q in the first formula can be dropped and ω1(q) coincides with 2D plasma dispersion law. It will be shown later that the excitation of the transverse mode in our experiment requires the presence of a strong magnetic field perpendicular to the plane of 2DES (the formula above is for B = 0). 3.3 2DES Mobility Three main processes limit the mobility of 2D electrons on liquid helium:
- 49. 40 1. Electron scattering off helium vapor atoms. 2. Electron scattering off surface waves of liquid – ripplon scattering. 3. For thin helium films there is also scattering from the surface roughness of the underlying substrate. While the first mechanism dominates for temperatures higher than 0.8 K, low temperature mobility is determined by ripplons. Table 3-3 summarizes various theoretical and experimental data on mobility of 2DES and specifies the phenomenon (right column) that each set of data is meant to elucidate. In Figure 3-5 and Figure 3-6 one can see crossover between the regimes where scattering from vapor dominates (high temperatures) to the ripplon limited mobility at low temperature. The surface density is such that at all temperatures electrons remain in the liquid phase. The vapor scattering time is: gasHe gas nA mb hπ τ 3 8 = (3-15) where b is the z-extent of 2D electron wavefunction given by Eq.(1-3), AHe – is the He atom cross-section, ngas – is the helium vapor density (Saitoh 1977). The expression specific for mobility limited by the scattering from 4 He gas is: sV cm T T a b gas 2 2 3 0 17.7 exp1300 = − µ The ripplon scattering rate is: ( ) ( ) −+ −++= ⊥ ⊥ 36 115 36 1 3exp 16 ln 2 3 3exp 16 ln 2 4 11 22 2 0 2 0 22 π στ T E a T T E a EeT Ee bb r h (3-16)
- 50. 41 Table 3-3 Mobility Measurements and Theory. Figure 3-3 Polaronic Transition. Low freq. transport Polaron transition (Andrei 1984) Figure 3-4 Wigner Transition. Low freq. Transport Wigner transition (Mehrotra, Guenin et al. 1982)
- 51. 42 2D plasma resonances (Grimes and Adams) Figure 3-5 2D Electron Liquid Mobility for a Range of Pressing Fields. Low freq. Transport in 2D liquid (Sommer and Tanner 1971) Corbino geometry (Iye 1980) 2D plasma resonances (Grimes and Adams) Microwave cavity (Rybalko and Kovdrya Yu) Figure 3-6 Various Methods of Mobility Measurement. Boltzmann equation (Saitoh 1977)
- 52. 43 where T is in units of energy and 2 2 2 bm Eb h = . In the regime of strong holding field (E⊥ > Ec see Eq.(1-4)) the scattering time becomes temperature independent (Shikin and Monarkha Yu): ( )2 8 ⊥ = Ee r hσ τ where σ is the helium surface tension. In the limit of strong holding field the DC scattering time is twice as high as the one measured in RF or microwave experiment: τdc = 2τac (Platzman and Beni 1976). For localized electrons, one ripplon scattering becomes inefficient. This is because a ripplon is slow (section 1.5.1) and therefore cannot supply enough energy for electron to make a transition between its localized states while conserving momentum. 2-ripplon scattering mechanism dominates then with a scattering time given by (Dykman 1978): ( ) 3 0 4 2 2 4 = lmb TkBHe r σπ ρ τ h Here l0 is the electron’s localization length. The total scattering time due to all independent mechanisms is: rrgas 2 1111 ττττ ++= 3.4 Coupled Phonon-Ripplon Modes The coupling between the Wigner crystal phonons and the He surface ripplons leads to different spectra for the 2DES liquid and solid phases because the coupling is only efficient for solid due to the correlated electron motion. It is this difference that allows
- 53. 44 investigation of melting by measuring spectra. The theory of these coupled phonon- ripplon (CPR) modes (Fisher, Halperin et al.) centers on the coupling that involves the longitudinal phonon, which is also a focus of our experiment. However a similar theory may be developed describing coupling between transverse phonon and ripplon modes. The theory divides electron’s motion into slow component, which is treated perturbatively, and a fast one, which is averaged over temperature fluctuations and represented by the Debye-Waller factor, which describes the effect of smearing of electron wavefunction: )ln( 3 1 21 ct q G mc T W = (3-17) In the above equation qc is a low frequency cutoff for fast component, and is essentially a fitting parameter of the theory. ct (Eq.(3-14)) enters this equation because the transverse phonons are the lowest lying excitations. CPR frequencies are then obtained by solving the following secular equations: 0 2 1 22 2 222 = −Ω −− ∑ ω ω ωω GG Gl V (3-18) where ΩG - are ripplon frequencies corresponding to reciprocal vectors of Wigner lattice Gp (Eq.(1-6)) for a hexagonal lattice, ωl – are longitudinal phonon frequencies in the absence of coupling and )exp( 1pW m n eEVG −= ⊥ σ (3-19) are coupling constants. Here is the pressing field.⊥E
- 54. 45 Figure 3-7 is the reduced Brillouin zone plot, that illustrates the hybridization between the longitudinal mode of 2D electron crystal and He surface modes, that leads to formation of CPR. Horizontal dashed lines are ripplon frequencies corresponding to Wigner crystal’s reciprocal lattice vectors. Solid lines represent resulting CPR modes. Figure 3-7 Coupled (solid) and uncoupled (dashed) modes of Wigner solid and surface waves of He (from (Fisher, Halperin et al.)).
- 55. 46 Wavevectors are determined by the geometry of the cell (cf section 3.1) and are much smaller than WC reciprocal vectors. The uppermost optical branch is described by 2 0 22 ωωω += l (3-20) which is the solution of Eq.(3-18) when ω >> ΩG. It is associated with a formation of a “dimple” depression of the liquid He surface caused by the electron pressure. ∑= G GV 2 0 2 1 ω is the frequency of electron’s oscillation in the dimple. We attribute the resonances of 2D electron solid observed in our experiment to this branch of the spectrum, which will be referred to as optical plasmon. 3.5 Spectrum of 2DES in Magnetic Field First we derive dispersion law for unbound 2D plasma placed in magnetic field which is normal to the 2DES plane. The equation of motion for 2D electron in magnetic filed (without external electric field) is: zu r uu ˆ 1 ×− ∂ ∂ =+ &&&& c e V m e ω τ where ωc is the cyclotron frequency: ][6.17 GB G MHz m Be c ==ω (3-21) and is the unit vector normal to the plane. After performing spatial Fourier transformation (V zˆ e ~ exp(-iqr)) we write down the equation of motion in terms of velocity components parallel and perpendicular to wavevector q:
- 56. 47 || |||| 1 1 uuu uqVi m e uu c c e q &&&& &&&& ω τ ω τ =+ −−=+ ⊥⊥ ⊥ (3-22) Eq.(3-6) becomes: σq e = -inequ||. Substituting it into Eq.(3-3) and then into Eq.(3-22), one gets the following secular equation determining magnetoplasmon dispersion law: ( ) 0 2 22 = − −−− τ ω ωωω ωω τ ω ωω ii iiq c cp (3-23) where ωp(q) is the 2D plasmons frequency in zero field (Table 3-1)). When dissipation is negligible, Eq.(3-23) becomes: ( ) ( ) 222 cpmp qq ωωω += (3-24) An interesting consequence of Eq.(3-23) is the linewidth doubling occurring as one goes from the limit of weak magnetic field (ωc << ωp) to that of strong field – cyclotron resonance (ωc >> ωp). For weak field the linewidth (defined as the width of resonance at half height) is 1/τ, whereas for strong field it is 2/τ as might be easily verified by solving Eq.(3-23) in these limits. When 2D plasma with cylindrical boundary conditions as described in section 3.1 is placed in a magnetic field the dispersion equation and boundary condition become interdependent: )()( )( 222 RqJRqJRq q c cp νν ω ω ν ωωω ±=′ += (3-25) Solutions of Eq.(3-25) )( ,, µνµνω ±± q determine the spectrum of bound 2DES.
- 57. 48 It is obvious that unlike the zero field case the modes differing only by signs of azimuthal number ν will no longer be degenerate – magnetic field removes inversion symmetry. The radial modes – the ones with ν = 0, will have wavevectors defined by the same Eq.(3-13) as without magnetic field and their frequencies will evolve as: 2 0 22 )( cBp ωωω += = (3-26) As shown in Figure 3-8 all other modes would split in magnetic field although most of them approach ωc asymptotically with increasing field except for those with µ =1,ν<0. Figure 3-8 Evolution of magnetoplasmons (from (Glattli, Andrei et al. 1985)).
- 58. 49 The latter have imaginary radial wavevector q so they are attenuated away from the edge of 2DES and therefore called perimeter modes or edge magnetoplasmons. Their frequencies and wavevectors behave asymptotically as: p c p c q R c ω νω ν ν → →− 1, (3-27) cp – is long wavelength plasmons velocity given by Eq.(1-9). Next chapter discusses the techniques used to excite and measure 2DES resonances.
- 59. 50 4 Measurement of 2DES Spectrum Using Slow-Wave Structure The normal modes of 2DES is the subject of Chapter 3. Here I discuss the experimental methods for exciting and detecting these modes. First the general ideas concerning this type of measurement are discussed. Next the coupling between external electro-magnetic field and 2DES is considered. 4.1 2DES in Electro-Magnetic Field Unlike 2D electrons in semiconductor heterostructures the properties of 2D electrons on liquid helium can not be studied with DC fields (for a clever way to set up a DC measurement see however (Klier, Doicescu et al.)). Therefore 2DES is usually coupled to a structure that generates AC field such as a planar capacitor (Sommer and Tanner 1971), transmission line (Grimes and Adams), microwave cavity (Kovdrya Yu and Buntar; Mistura, Gunzler et al. 1997), optical (Lambert and Richards) or resonance circuit (Kovdrya Yu and Buntar). The method used in this work is excitation of resonances using a transmission line. The analysis is as follows: the 2DES is thought of as a resonator with its set of normal modes (Chapter 3). The external transmission line is weakly coupled to 2DES. A wave of a constant power is sent to the input of the transmission line and the output power is measured as a function of frequency, which is slowly swept through the range of interest. The peaks in the derivative transmission spectrum are attributed to the absorption by the 2DES resonances.
- 60. 51 It follows from a general theory of resonators (Pozar 1997) that coupling between the resonator (2DES in this case) and the excitation line has to have the right value – so called critical coupling. A very weak coupling would result in a small signal to noise ratio whereas overcoupling would lead to spurious broadening and shift of the resonance lines. However due to the need to broadly vary the helium film thickness in this experiment the case of an arbitrary coupling has to be incorporated and will be considered in 4.2. Part of the following discussion follows (Valdes 1982). In section 3.1 we derived the dispersion law and susceptibility χ of a 2D plasma. The response of the 2DES to an applied field is given by Eq.(3-9). Based on this formula we can express the influence of 2DES on the wave propagating in excitation line as follows: ( )χαiTVV lineinout −= 1 (4-1) where Vin, Vout are voltages at the input and output of the transmission line respectively. Here Tline is the line’s complex transmission coefficient and α includes the coupling between 2DES and the line as well as some other factors. The assumption that coupling is weak, i.e. αχ << 1 allows us to expand this formula: ( ) ( )"1'"1 222 χαχαχα δδ +≈++= i linein i lineinout eTVeTVV Here δ=Arg(χ), χ’ = Re(χ) χ” = Im(χ). The measurement is made in such a way (homodyne detection with phase matching delay line – see 6.8) that the phase information (δ + Arg(Tline)) is dropped from the measured signal Vs: ( )"1 χα+= lineins TVV The above signal is the sum of apparatus transmission spectrum in the absence of 2DES and a weak absorption due to 2DES. In order to eliminate this large background
- 61. 52 phase-sensitive detector PSD is used (see Figure 6-17 and section 6.8). In essence, in order to achieve phase sensitive detection a low frequency modulation is imposed onto χ: ( ) tt rωχχχ cos0)( ∆+= The signal after PSD is: "χα ∆= lineins TVV This signal may still include some spurious information arising from the transmission coefficient of the line without 2DES. This coefficient contains information about the losses in the line and reflections. The reflections are caused by variations of characteristic impedance of the line - impedance mismatches. For instance if at a certain location in the line the impedance abruptly changes its value from Z1 to Z2 the propagating wave will experience a reflection at that point with the voltage reflection coefficient given by: 21 21 ZZ ZZ r + − = However since χ”(ω) is expected to have (Chapter 3) the shape of a sequence of narrow resonance peaks the identification of resonances is not obscured by Tline as long as the latter varies slowly enough with frequency. To quantify this statement lets consider how Tline depends on the frequency for a simplified transmission line with only two impedance mismatches having voltage transmission coefficients t1, t2 respectively and attenuation constant κ. The reflection coefficients of these discontinuities (from the inside of the region between discontinuities) are r1 = 1 - t1, r2 = t1 - 1 (these relations follow from the boundary conditions for voltage and current).
- 62. 53 To find the total transmission coefficient one sums all partial waves reflected from discontinuities: t1 t2 Vin Vout l Figure 4-1 Reflection between Two Discontinuities in Transmission Line. iql iql iqliqliql in out err ett ettrrettrrett V V T 2 21 215 21 2 21 3 212121 1 )( − =+++== K where q = k+iκ is the wavevector. For simplicity r’s are assumed to be <<1. Then ikll eerrT 2 211 κ− +≈ The plot of transmission coefficient vs. frequency looks like a series of peaks whose width is: )1( 21 l line err l v κ ω − −=∆ (4-2) where v is the propagation velocity in the line. As discussed above the shape of Tline will not hinder observation of 2DES resonances if the width of the latter is smaller than ∆ωline: ∆ω2DES < ∆ωline The qualitative generalization of Eq.(4-2) for the case of arbitrary number impedance mismatches leads to the final criterion for mismatch tolerance: KK ++=−−−<∆ −− 1132212 ),1( 21 llLerrerr L v ll DES κκ ω total length of the line This implies the following:
- 63. 54 • Reflections have to be sufficiently small – impedance of the line should be well matched. • The line should be as short as possible. • The attenuation between impedance mismatches should be large. Therefore the attenuators should be inserted into the line in many places. In the experiment the direct spectrum (without PSD) is first obtained in order to estimate the quality of the transmission line. 4.2 Distributed Circuit Model The task of measuring the 2DES spectrum leads to a natural question: how can the technique distinguish the response of ~108 surface electrons from that of ~1024 electrons contained in the metal electrodes present nearby? The key to answering this question is in the qualitative difference of 2DES electro-magnetic properties as contrasted to that of 3D objects. To illustrate this difference we compare electro-magnetic responses for apparently similar cases of 2DES and that of a thin sheet of metal in Table 4-1. Table 4-1 Electromagnetic Response Comparison. 2DES 3D plasmon Thin sheet of metal (guiding action) Dispersion equation q m en D 0 2 2 ε ω = m en D 0 2 3 ε ω = ε ω c k= What determines the allowed wavevectors Boundary conditions Boundary conditions Boundary conditions Frequency shift when placed in a microwave cavity Up Down Down In this experiment the field was created by a TEM transmission line (section 4.3) – meander line or IDC - above which 2DES was situated. Here I discuss the
- 64. 55 electromagnetic coupling between a generic transmission line and 2DES using distributed circuit theory. In this approach both 2DES and excitation line are treated as transmission lines having a uniformly distributed capacitance, inductance and resistance. It is complementary to the analysis in 4.1: for example, it allows calculating the coefficient α in Eq.(4-1). A similar analysis (applied to the arrangement of electrodes known as Sommer-Tanner techniques (Sommer and Tanner 1971)) is found in (Mehrotra and Dahm 1987). Notice that unlike the case of a metal sheet the coupling between 2DES and the excitation line is predominantly capacitive – there is no mutual inductance. This is because the current density in 2DES is related to surface charge density as j = σe v, where v is the velocity of electrons. The electric field E created by electrons is E ~ σe , whereas the magnetic field B is B ~ j/c. Therefore B ~ (v/c)E and, since typical electron velocities are much smaller than speed of light, the magnetic field is negligible compared to the electric one. For the same reason one can neglect the magnetic self-inductance of 2DES. However there is a self-inductance due to inertia of electrons. The equivalent circuit for a meander line is shown in Figure 4-2. The following values should be assigned to the distributed parameters of the circuit: gr eff e l eff le e e d w C d w C wen R wen m L 1* * * 2 * ; ; 1 ; ε ε µ = = = = (4-3)
- 65. 56 0 0 Excitation Line 2DES Capacitive Coupling Z=50 00 1 0 L*e R*e L*l C*e C*le C*l Figure 4-2 Equivalent Circuit of 2DES and Excitation Line. Star denotes the value per unit length. where m is effective mass of electron; e - electron’s charge; µ – electron’s mobility; w – “width” of 2DES pool; n – 2DES surface density; dl – the distance between 2DES plane and excitation line; dgr – the distance between 2DES plane and ground; εeff, εeff1 – effective dielectric constants. Usual arrangement of dielectric layers is: dielectric wafer such as Si supporting a film of liquid helium-4. A single transmission line (Pozar 1997) is described in terms of its dispersion law ( ) ( ) ( )ωωω ** YZiq −= and characteristic impedance ( ) ( )ω ω * * Y Z Zc = where Z* (ω) and Y* (ω) are longitudinal impedance and transverse admittance per unit length of the line respectively. For example for 2DES ( ) ; ; 1 2 1 22* ** 1 2 12*** eff gr eff gr e ee c gr eff gr eff eee wen d i wen dm Ci RLi Z den i den m iCiRLiiq εµωεω ω µ ε ω ε ωωω −= + = +−−=+−= If 2DES losses are small (µ is large) the dispersion law becomes:
- 66. 57 gr eff den m q 2 1ε ω= (4-4) which is the same as 2D plasmons dispersion Eq.(1-9). This expected result confirms the validity of modeling 2DES with a distributed circuit. In the case of coupled transmission lines Z* and Y* are matrices whose rank is equal to the number of lines, two in our case. Transmission line equations for voltages and currents are as follows: +− −+ −= −= e l leele lelel e l e l e l e l V V YYY YYY I I dx d I I Z Z V V dx d *** *** * * 0 0 (4-5) where are longitudinal impedances and transverse admittances per unit length for the excitation line (subscript l) and 2DES (subscript e). Details of the theory of coupled lines can be found in (Faria 1993). By diagonalizing the system of Eq.(4-5) one finds the dispersion laws for the normal modes of excitation line + 2DES system: *** , * , * , * , * , leleelelelelel CiYCiYLiRZ ωωω ==+= 2 4)( 422222 2 2,1 leelel qqqqq q +−±+ = (4-6) where ql, qe are the dispersion laws of uncoupled lines and ***2 ellele ZZYq = (4-7) As is explained below it is always preferable to match the lines so that ql = qe = q. Therefore only this case will be considered here. Then Eq.(4-6) is simplified to: 222 2,1 leqqq ±=
- 67. 58 When boundary conditions are imposed on one of the transmission lines (in our case this is the condition of having zero current on the edge of 2DES disk whose size is R) there will be a resonance whenever either q1 or q2 are multiple of π/R thus giving rise to two peaks per each resonance of uncoupled 2DES. The splitting is therefore: ** * el le CC C = ∆ ω ω The film thickness is the parameter that controls the mutual capacitance between 2DES and the excitation line and therefore the splitting that corresponds to coupling between 2DES and excitation line. Thus the splitting can be used to measure the helium film thickness. 4.3 Slow-Wave Structures As shown in section 1.4 the plasmons have phase velocities that are several orders of magnitude lower (depending on the surface density) than the speed of light. However in order to be observable 2DES has to be critically coupled to the excitation line as was explained in 4.1. This implies that the propagation velocity of the line should be as close as possible to that of the 2D plasma. It is generally the case that in order to have the most efficient energy transfer between two objects their dispersions must be matched. A simple example is the collision between two particles: the largest energy transfer happens if their masses are equal. To match an excitation line to slow plasmons one can use a transmission line with a special geometry that results in “slow” effective propagation of the electromagnetic wave. Two such structures used in this experiment are the meander line and the
- 68. 59 interdigital capacitor. These structures are shown in Table 4-2. The clear areas correspond to non-metalized surface. The basic idea of these structures is to create a convoluted path of the traveling wave so as to effectively slow down the electromagnetic wave. It follows from this argument that the effective propagation speed of the electromagnetic wave is reduced with respect to its speed in vacuum by the aspect ratio of the structure. The aspect ratio of the structure is defined as the ratio of shortest distance between terminals to total length of the line, which is eff aL a ε 1 + in terms of the line parameters. A rigorous treatment of these structures is given in (Weiss 1974; Crampagne and Ahmadpanah 1977). It makes use of the symmetries of the line while solving the Helmholtz equation for the wavefunction Φ: eff c kk ε ω 2 2 222 ,0 ==Φ+Φ∇ (4-8) Since the media above the line (silicon in this experiment) and below (sapphire) have different permittivities, the wave propagates in an inhomogeneous environment (see Figure 4-6). Strictly speaking, no TEM (transverse electromagnetic) wave can propagate in such a layered medium. However a quasi-TEM solution is a good approximation for low frequencies. According to transmission line theory, the wave impedance is given by ∗ Cv 11 where C is the capacitance per unit length of the line and∗ eff c v ε = – the propagation velocity.
- 69. 60 Table 4-2 Slow-Wave Structures. Effective propagation velocity eff aL a cv ε 1 + = Reduces crosstalk between fingers. More sensitive to 2DES located above than non-coplanar meander line. The dispersion law has a gap – no propagation at low frequencies. Figure 4-4 Coplanar Meander Line. a L y x Figure 4-3 Meander Line. Figure 4-5 Interdigital Capacitor (IDC).
- 70. 61 H d Top Plate Slow-wave Structure Bottom Plate Vacuum er w p s Figure 4-6 Cross-section of a Slow-Wave Structure in Inhomogeneous Environment. The effective dielectric constant effε can be calculated as * 1 * C C eff =ε where C1 * is the capacitance per unit length with all dielectric media replaced by vacuum. A solution of Eq.(4-8) is sought as a combination of TEM waves propagating along the strips: which after substitution in the above equation yields a Laplace equation in the plane perpendicular to the strips of the line: iky ezxUzyx ± =Φ ),(),,( 02 2 2 2 = ∂ ∂ + ∂ ∂ z U x U The choice of x,y coordinates is indicated in Figure 4-3, z – is perpendicular to the plane of the meander line. Solving this equation subject to the boundary conditions in the xz plane for the distribution of voltages and charges allows one to calculate the capacitances necessary for determining effε . So far, the solution is the same for both meander line and the IDC. The last step is to apply the boundary conditions at the ends of the fingers appropriate for each structure. For example for IDC, these conditions are: zero current at one end of a finger, and a voltage V at the other end, with V being the same for the fingers belonging to the same side of the IDC. Below is a short discussion of the results for the meander line.
- 71. 62 The choice of normal modes for the solution is dictated by the fact that the meander line is periodic with the unit cell comprising two lines: Since the unit cell also has inversion symmetry the solution is chosen as linear combinations of odd and even Bloch waves: ∑ − = j zqxiq j jj eeAzxU ),( where qj for low frequencies are given by: K,1,0 ±=+ + = jwith a j a aL c q effj π ε ω (4-9) The absolute value of x and z wavevector components are equal because U(x,z) has to satisfy Laplace equation. 2DES is normally located high above meander line so that Figure 4-7 Meander Line Dispersion Law from (Crampagne and Ahmadpanah 1977). θ = pq−2πj, p is the period of the line, other parameters are defined in Figure 4-6.
- 72. 63 z2DES >> d/εr referring to the cell geometry shown in Figure 4-6. In this case we can assume zero voltage value in the gaps between meander line fingers. The potential will then be ( ) ( ) ( )∑ ∞ −∞= − − −− = j xiq j j j e dHq zdHq j a w j UzxU sinh sinhsin , 0 π π above the meander line (z > 0). The x component of the electric field is: ( ) ( ) ( )∑ ∞ −∞= − − −− = j xiq j j jx j e dHq zdHq j a w j qiUzxE sinh sinhsin , 0 π π (4-10) The full dispersion law for meander line is shown in Figure 4-7. The plot corresponds to the reduced zone scheme. The first term in Eq.(4-9) (j = 0) corresponds to a plane wave with a large period (slow wave) unattenuated even at the heights above the line larger than its period. The rest of the sum is a wave modulated by the period of meander line and strongly attenuated at the heights larger than the period of the line. 2DES should be positioned higher above the meander line than its period but lower than the slow wave wavelength a aL f c eff + ε . In this case the x component of electric field acting on 2D electrons is determined by j = 0 term in Eq.(4-10): ( ) ( ) xiqDES x e dHq zdHq a w UiqE 0 0 20 00 sinh sinh − − −− = (4-11) The effective voltage is U0 = (Z0P0)0.5 where Z0 is the impedance of the slow-wave structure and P0 is the incident RF power. The speed of the line can now be adjusted to match that of 2DES by choosing the appropriate aspect ration of the structure. However since 2DES wave velocity depends on
- 73. 64 its surface density the matching is achieved for just this value of density. Further discussion of matching will be provided in section 4.4. Another choice of structure is a coplanar meander line depicted in Figure 4-4. Such line has both the ground and the meander in the same plane. The advantage of this configuration is increased sensitivity to a charge above the line – 2DES in particular. In the case of the IDC structure the low-frequency dispersion has a low frequency gap as shown in Figure 4-8. This is not surprising since IDC is a capacitor so DC current cannot flow through it. Figure 4-8 IDC Dispersion Law from (Crampagne and Ahmadpanah 1977). See Figure 4-7 for more details.
- 74. 65 4.4 Coupling between meander line and 2DES The coupling strength e(q, qp) between a plane electromagnetic wave and the 2DES appeared in Eq.(3-8). Here the formula specific to the coupling between meander line and circular 2DES is discussed. Coupling between 2DES modes (cylindrical waves – Eq.(3-4))) and meander line modes (plane waves) is given by their dot product: dSe R e rqi θ µνµν ϕ π cos ,2, 1 ∫∫= (4-12) Here ϕν,µ – is the eigenfunction of 2DES with cylindrical boundary conditions, R – is the radius of 2DES pool, q – is the wavevector of an external plane wave, r,θ – cylindrical coordinates. The evaluation of this integral (Glattli 1986) produces the following result: 22' , ' 2' , 2 , )( )( 1 2 qR qRJRqi e − − = µν ν µν ν µν ξ ξ ν (4-13) When the line is matched to 2DES R q νµξ′ =Re (Eq.(3-13)) and the denominator of Eq.(4-13) is kept from diverging at this value of q by the losses in the line. It is these losses that permit coupling if the line is not perfectly matched to 2DES as might take place in measurements with broad range of surface densities. Therefore there is always a window of frequencies where there is non-zero coupling between the meander line and 2DES.