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Order Number 9238200
Crystal states in two-dimensional quantum Hall systems
Kahng, Jonghyun, Ph.D.
The Ohio State University, 1992
UMI 300 N. Zeeb Rd. Ann Arbor, MI 48106
C r y s t a l S t a t e s
in Two. Dimensional Quantum H all Systems
DISSERTATION
Presented in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in the Graduate School of the Ohio State University
By
Jonghyun Kahng, B.S., M.S.
)fc $ jfc jfc $
The Ohio State University 1992
Dissertation Committee: Approved by Prof. Charles A. Ebner Prof. Ciriyam Jayaprakash f l X a o U s Advisor Prof. Charles H. Pennington Department of Physics To my mom, Chung Wha, and Hisoo
ii ACKNOWLEDGEMENTS
I thank Prof. Ho for his guidance throughout the research and for the preparation of this thesis. In particular, his concern about the thesis even in his very critical moment is greatly appreciated. It is unfortunate that he could not be in my dissertation committee in the last moment. However, with no doubt, he has everything to do with this thesis. I also thank Prof. Ebner for his generosity to lead the committee as a chairman in Prof. Ho’s absence. In addition, his advises on computational techniques during my early work in this graduate program have been indispensable for the rest of my graduate study as well as for my future career. This thesis would not have been complete without Prof. Jayaprakash’s critical reading of and comments on the manuscript. His kindness to do so is much appreciated. I am indebted to Prof. Patton for his continuous and warm encouragement for the last four years. I thank Prof. Pennington for serving in the committee in spite of his compact schedule. My special thanks goes to my mom and my wife Chung Wha for their patient support and encouragement throughout the period of my graduate work. I like to attribute much credit to my daughter Hisoo for the tremendous amount of joy that she has brought me during the time of pressure. Without her, the last two years would have been even more painful. VITA
February 2, 1959 ...... Born - Seoul, Korea
February, 1982 ...... B.S., Physics, Seoul National University, Seoul, Korea
1982-1985 ...... System Programmer, Korean Air Lines, Seoul, Korea
1985-1988 ...... University/Graduate Fellow, The Ohio State University, Columbus, Ohio
1986-1992 ...... Graduate Teaching/Research Associates, Department of Physics, The Ohio State University, Columbus, Ohio
August, 1991 ...... M.S., Physics, The Ohio State University, Columbus, Ohio
PUBLICATIONS J. Kahng, C. Ebner, Melting of multilayer films: Further studies of a Potts lattice- gas model, Phys. Rev. B 40, 11269 (1989) T.-L. Ho, J. Kahng, Macroscopic angular momentum of the crystal states in two- dimensional quantum Hall systems, Phys. Rev. B 45, 9481 (1992)
FIELD OF STUDY Major Field: Physics Theoretical Condensed Matter Physics TABLE OF CONTENTS
ACKNOWLEDGEMENTS ...... iii VITA ...... iv LIST OF TABLES ...... vii LIST OF FIGURES ...... viii CHAPTER PAGE
I. Introduction 1 1.1 Overview of Crystal States in Two Dimensional Electron Systems 1 1.2 Fundamentals of Two Dimensional Electron System s ...... 4
II. Review of Experiments and Theories 11 2.1 Experiments on Quantum Hall Systems ...... 11 2.1.1 Two Dimensional Electron System s ...... 11 2.1.2 Quantum Hall E ffe c ts ...... 15 2.1.3 Crystal States ...... 17 2.2 Theories of Quantum Hall S y stem s ...... 22 2.2.1 Integer Quantum Hall E ffe c ts ...... 22 2.2.2 Incompressible Quantum Fluids at v — 1 / m ...... 25 2.2.3 Hierarchical Schemes ...... 33 2.2.4 Crystal States ...... 40 III. Crystal States 47 3.1 Crystals of Q u asip articles ...... 48 3.2 Energy of Crystals ...... 52 3.3 Angular Momentum and Induced Magnetic Field in Crystals . . 60 3.4 Monte Carlo M e th o d s ...... 69 3.5 Particle-Hole Symmetry ...... 74 3.5.1 Characteristics of Particle-Hole Conjugate States .... 76 3.5.2 Quasihole Crystal vs. Wigner Crystal ...... 80 3.6 Quasiparticle Crystals in Hierarchical S ta te s ...... 81
IV. Conclusions 91
APPENDICES
A. Lists of Computer Programs 95 A.l Program for the Density of Wigner Crystals ...... 95 A.2 Program for the Pair Correlation Function of Quasihole Crystals 99
B IB L IO G R A PH Y ...... 110
vi LIST OF TABLES
Monte Carlo data for quasihole crystals LIST OF FIGURES
1.1 The density of states and the eigenstates of a two-dimensional electron in a magnetic field ...... 7 1.2 A schematic diagram of the Hall effect ...... 9
2.1 Electron energy level diagrams for a Si MOSFET and a GaAs-AlGaAs heteroj unction ...... 12 2.2 A typical geometry of samples in magnetotransport experiments.. . . 14 2.3 Magnetotransport measurements in a GaAs-AlGaAs heterojunction. . 18 2.4 A schematic phase diagram of two-dimensional electron systems. . . . 21 2.5 The density of states for two-dimensional electrons and the geometry of a metal strip in Laughlin’s thought experiment ...... 23 2.6 The electron density profile of quasiparticle excitations at v — 1/3. . 32 2.7 The estimated energy of hierarchical quantum Hall states ...... 36 2.8 The relative stability of fractional quantum Hall states...... 41 2.9 The comparison of energies of fractional quantum Hall states and Wigner crystals ...... 45
3.1 Pair correlation functions of quasihole crystals ...... 54 3.2 The energy of quasiparticle crystals near v = 1/5...... 55 3.3 The energy of quasiparticle crystals near v = 1/3...... 56 3.4 The energy of quasiparticle crystals near v — 1/2...... 57
viii 3.5 The angular momentum in Wigner crystals and quasihole crystals 67 3.6 The angular momentum in quasihole crystals ...... 68 3.7 The induced magnetic field in quasihole crystals and Wigner crystals. 70 3.8 The density contour plot of the quasihole crystal at v = 1/2. . . . 73 3.9 The extrapolation of the energy of a quasihole crystal ...... 75
ix CH A PTER I
Introduction
1.1 Overview of Crystal States in Two Dimensional Elec tron Systems
In 1934, Wigner [1] predicted that the electron gas in a uniform neutralizing positive background crystallizes at sufficiently low densities. His argument is based on the comparison of the kinetic and potential energies of the electrons. At low temperatures, the kinetic energy is proportional to 1/r^ while the potential energy is proportional to l/rs, where r3 is the radius of the sphere whose volume is equal to the volume per electron. At sufficiently low densities, the potential energy dominates the kinetic energy. The electron gas therefore crystallizes to minimize the potential energy. Such an electron crystal has been called the Wigner crystal. However, observation of the Wigner crystal has not been successful until very recently. The reason is that it is generally difficult to achieve an electron density low enough for crystallization to occur in real experiments. Grimes and Adams [2] were the first to observe the Wigner crystal on the liquid helium surface. The two-dimensional Wigner crystal they observed is in the classical limit because the Fermi temperature of the electrons was much lower than their experimental temperature.
1 2
Since highly mobile two-dimensional electron systems (2DES) became available in inversion layers in Si-metal-oxide-semiconductor-field-effect-transistors (MOS- FET) and GaAs-AlGaAs heterojunctions, there have been intensive experimental investigations on 2DES in a strong magnetic field in search of the Wigner crystal. W hat was found, instead, is the quantum Hall effect (QHE) [3-5]. At low temper atures, the Hall conductivity shows a series of steps at quantized values ( e2/h)i each of which is centered around an integer filling factor v = i [6-8] {v is related to the electron density p and the magnetic field B by the relation v = hcp/eB). Moreover, the magnetoresistance drops sharply at those integer filling factors. It presumably vanishes at zero temperature. This remarkable observation of the quantized Hall conductivity has since been called the integer quantum Hall ef fect (IQHE). Soon after the existence of the IQHE was well confirmed, the same behaviors of the Hall conductivity and the magnetoresistance were also observed at fractional filling factors such as u = 1/3,2/5,2/7,... [9-21] (the fractional quantum Hall effect: FQHE). While the IQHE could be explained in the framework of independent elec trons [22-24], the FQHE was believed to result from the interaction between electrons [25-29]. It was first attempted to interpret the FQHE in terms of the Wigner crystal [9]. But the attempt turned out to be not successful. Laugh- lin [26] later proposed that an incompressible quantum fluid state is responsible for the FQHE at v = 1/m with an odd integer m. The highly correlated state of electrons that he proposed was shown to be fluid like and have lower energy than the Wigner crystal. He also showed that the excited state is a collection of fractionally charged quasiparticles on top of the incompressible quantum fluid. As more FQHE’s were observed at several other rational filling factors with odd denominators, a number of hierarchical schemes have been proposed to extend Laughlin’s idea to other filling factors [30-39]. At present, it is not clear whether 3 all these schemes are identical. The understanding of their difference is still a subject under active research [40,41]. However, it is believed that the general picture of the FQHE can be accounted for by Laughlin’s incompressible quantum fluid state together with some hierarchical schemes. Recently, the search for the Wigner crystal has become active again because further improvement in the technology of producing cleaner samples made it pos sible to study 2DES at lower filling factors. Since the Wigner crystal is believed to exist at sufficiently low filling factors [1,42-51], one might expect a phase tran sition from the quantum Hall state to the Wigner crystal at some critical filling factor uc [52-54], Some years ago, Lam and Girvin [52] had estimated vc 1/7 by comparing the energy of Wigner crystals with the energy of quantum Hall states. On the experimental side, there have been reports of solid-like phases at a number of filling factors [55-63]. Although there are some disagreements among exper iments on the ranges of filling factors where these phases are observed, most of them agree that the solid-like phases exist at filling factors just above and below v = 1/5. Based on measurements of nonlinear I-V relations and diverging mag netoresistance at low temperature, there is a tendency among experimentalists to interpret these phases as the Wigner crystal. However, concrete evidence for the Wigner crystal is still lacking. For example, there is no direct measurement of basic characteristics such as the lattice constant of these phases. If these solid-like phases are in fact crystal states, they contradict the general belief that there is a well-defined phase transition to crystal states at some critical filling factor vc: Crystal states exist at filling factors much higher than the theoretically estimated vc k, 1/7 and, moreover, there are many such transitions. If there are series of phase transitions between the quantum Hall state and the crystal state, there is a possibility that those two states are closely related to each other and therefore a new interpretation of crystal states in 2DES is necessary. 4
Together with T.-L. Ho [64], I have studied crystallization of quasiparticles as a possible explanation of the crystal phases just above and below v = 1/5 as well as at other filling factors. The reason for studying these crystals is the following. Close to any stable quantum Hall state, a 2DES can be described as a dilute gas of fractionally charged quasiparticles floating on Laughlin’s quantum fluid state. When the density of the fractional charges is sufficiently low, they are expected to crystallize for the same reason that electrons form a Wigner crystal at low densities. If the quantum Hall state is more stable than the Wigner crystal at a given filling factor v, the crystals of quasiparticles are guaranteed to be more stable than the Wigner crystal at least sufficiently close to u. In this picture, every stable quantum Hall state is surrounded by crystals of quasiparticles. This prediction is consistent with the experimental claim of crystal states near v = 1/5. Our first step was to study the energetics of these crystals. (By the time we finished our calculation, we found that the possibility of these crystal states was proposed by Halperin [31] a number of years ago. Halperin, however, did not carry out any calculation.) Later we studied some macroscopic properties of these crystals cis a means to distinguish them from other electron crystals. The study of the energetics and the macroscopic properties of quasiparticle crystals forms the bulk of this thesis.
1.2 Fundamentals of Two Dimensional Electron Systems
Since most of our discussions will be based on the properties of 2DES in a strong magnetic field, we shall describe some of the well known properties of 2DES in this section. The notations set up in this section will be used throughout the entire thesis. The Hamiltonian for an electron in the xy plane under a magnetic field B in 5 the z direction is
H = ir-{iv+icA)2’ ™ where V = xdx + ydy, m* is the effective mass of the electron, and —e (e > 0) is the charge of the electron. For typical samples that will be introduced later, m* « 0.067 rae, where me is the mass of the electron. It is assumed that the magnetic field is strong enough to freeze out the spin degree of freedom of the electron. As will be seen later for the case of interacting electrons, it is useful to choose the symmetric gauge
A=|(j,x-xy) (1.2) for the magnetic field B = V x A = — Bz. (The reason for choosing the magnetic field in the negative z direction is purely for notational convenience. If one chooses B = J9z, all complex numbers in the following discussion should be replaced by their complex conjugates, which will be inconvenient as one can see shortly.) For two-dimensional systems, it is convenient to use complex number notations: z = x + iy, z* = x — iy, dz = dx — idy, and dz * = dx + idy. Note that dzz = dz*z* = 2 and dzz* = dz*z = 0. In terms of complex coordinates, the Hamiltonian is H - c ) (3“ +1). (1.3) where u)c = eB/m*c is the cyclotron frequency and all lengths are in the unit of the magnetic length l0 = yhc/eB. (From now on, all lengths will be given in the unit of /0 unless otherwise mentioned.) /o is the radius of the circular motion of a classical electron in a magnetic field with the frequency ojc- For a typical o magnetic field B = 10 T in experiments, the magnetic length lo « 80 A. It is straightforward to show that eigenstates of the Hamiltonian are [26]
$m,n(*)= (2m+n+17rm!n!)"1/2el"|2/4^ a ;e -l2|2/2, (1.4) 6 where m and n are nonnegative integers. The corresponding energy eigenvalues are En = (n + ^ hu>c. (1.5)
The index n is referred to as the Landau level index. The integer m labels the degeneracies of the nth Landau level (Fig. 1.1(a)). Due to the choice of the sym metric gauge, the energy eigenstate is also the eigenstate of the angular momentum
Lz = (r x p)2 (1.6)
= ^(zd.-z'd,.) ( 1.7) with the eigenvalue Lm = mh. (1.8)
The unusual nature of the 2DES in the high field limit is best illustrated by comparing its density of states with that of the 2DES in the absence of the magnetic field [3]. Without the magnetic field, the density of states D(E) is independent of E (Fig. 1.1(b)). In the presence of a magnetic field, D(E) reduces to a series of infinitely degenerate Landau levels (Fig. 1.1(c)). Each Landau level is separated from the next higher Landau level by the gap hu>c. The degeneracy of each Landau level is known to be N# = £?v4/0, where A is the area of the two dimensional plane and <- = ^ = 2 Wjp, (1.9) where N is the number of electrons and p = N/A is the average areal density of electrons. Since we will be mostly concerned with the states in the lowest Landau level, it will be useful to discuss some characteristics of these states. The eigenstates in 7 (a) (b) n D(E) 3 2 1 ...... 0 0 1 m Figure 1.1: (a) Solid circles represent eigenstates of the 2DES in a magnetic field; n and m are the energy and angular momentum quantum numbers respectively. The states on the same horizontal line are degenerate. The density of states D(E) of 2DES (b) without and (c) with a magnetic field. Ep is the Fermi energy of the system, (d) The probability density of $m (Eq. (1.10)). Concentric circles represent the trace of peaks of the probability density. The radius of the circle corresponding to is y/2m. 8 the lowest Landau level are * „ (* ) = * m,o(z) = (1.10) The probability density of the eigenstate 4>m has its peak centered at \z\ = y/2m (Fig. 1.1(d)). In particular, that of 3>o is a Gaussian centered at \z\ = 0. Eq. (1.10) implies that the general single-electron state in the lowest Landau level is f(z)e~\z\2/4, where f(z) is an analytic function of z. As we will see later when the wavefunction of the Wigner crystal is discussed, one sometimes needs a state whose probability density is a Gaussian centered at an arbitrary point on the plane. Such a state can be found to be $ a(*) = - U e za*/2 e- (l"|2+|a|2)/4 (1.11) \Z2n = —-L=e(2a*~z*°)/4 e- l*-Bl2/4, (1.12) V27T where a is the center of the Gaussian. Note that e^za*_z*a^4 in Eq. (1.12) is just a phase factor so that the probability density |$a(z)|2 is the Gaussian centered at a as desired. As the last part of this introduction, the basic phenomena of the Hall effect will be introduced. When crossed magnetic and electric fields are applied to a 2DES, a Hall current is induced in the direction perpendicular to the crossed fields (Fig. 1.2) [3]. Suppose the 2DES is placed in the xy plane and the magnetic field B and electric field E are applied in the z and y directions respectively. The Hall current induced in the x direction can be written as j = pev where p is the areal density of electrons and v is the average velocity of electrons in x direction. Since the Hall current consists of the electrons not deflected by the magnetic and electric fields, the velocity of electrons is determined from e(v/c)B = eE. The Hall resistivity is then pxy = E /j — B/(pec). In terms of the filling factor (Eq. (1.9)), P ,y = 4e1 ~u - (113) Figure 1.2: A schematic diagram of the Hall effect. The resistivity in the direction of the applied electric field, that is, the magnetore- sistivity is pxx = m*/(pe2r), where r is the average time between scattering. It is known that the magnetoresistance is nearly independent of the magnetic field. The rest of this thesis is organized in the following way. Recent experiments and theories on the Wigner crystal as well as those on the QHE are reviewed in chapter 2. The results of our studies on crystal states in 2DES are presented in chapter 3. The summary and conclusions follow in chapter 4. C H A PTER II Review of Experiments and Theories 2.1 Experiments on Quantum Hall Systems 2.1.1 Two Dimensional Electron Systems Recently, the development, of technologies to make high quality inversion layers in semiconductors has inspired studies of the 2DES formed in the inversion layers. Si metal-oxide-semiconductor-field-effect-transistors (MOSFET) and GaAs-AlGaAs heterojunctions are the devices most widely used to make high quality inversion layers. The principles of inversion layers are the following [4]. An electric field is generated perpendicular to the interface Si/Si02 (Si MOSFET) or GaAs/AlGaAs (the heterojunction). The electric field and the interface create a quantum well at the interface which attracts electrons (Fig. 2.1). The motion of electrons per pendicular to the interface is quantized and so frozen out at low temperatures. The resulting electrons in the inversion layer effectively form a 2DES. The source of the electric field at the interface is different for the two devices. In a Si MOS FET, “a gate voltage” is applied perpendicular to the interface between Si and Si02- The electron density in the inversion layer is determined by the applied gate voltage. In the case of a GaAs-AlGaAs heterojunction, AlGaAs has a wider gap between the valence and conduction bands (« 2.2eV) than GaAs (~ 1.5eV). 11 12 CONDUCTION ;BANO OjO-O-O-0-0-0 I VALENCE ! BAND A lx GohX A$ G i As CONDUCTION BANO CONDUCTION BANO eF - g>—o—o-o-o (*>) v a l e n c e BANO Figure 2.1: Electron energy level diagrams (a) for a Si MOSFET and (b) for a GaAs-AlGaAs heterojunction. Conduction bands of Si and GaAs, respectively, that are lowered below the Fermi level £p near the interface create inversion layers of thickness 100 A or less, (from Ref. [4]) 13 The difference between the gaps causes the electric field at the interface. In this device, the electron density is fixed by the amount of donors in AlGaAs if no gate voltage is applied. In both Si MOSFET and GaAs-AlGaAs heterojunctions, the filling factor v = hcp/eB (Eq. (1.9)) is controlled by the electron density p and/or the applied magnetic field B. Of the two devices, the GaAs-AlGaAs heterojunction has better parameters relevant to studies of the QHE and the Wigner crystal. Since experiments are to be performed in the quantum limit hu>c ksT, smaller effective mass m* of electrons is desirable (recall that ujc = eB/m*c). The effective mass in a GaAs- AlGaAs heterojunction is about three times smaller than that in a Si MOSFET [4]. For typical samples of the GaAs-AlGaAs heterojunction, m* « 0.067me and u>c « 0.02 eV/h « 200 K/h for a typical magnetic field 10 T. Another advantage of the heterojunction is high mobility of electrons (10 6 cm2/Vs compared to 10 4 cm2/Vs for a Si MOSFET) [4]. Since the mobility of electrons is p, = er/m* where r is the average scattering time, the high mobility is attained by maximizing r, that is, by reducing defects and impurities in the inversion layer as well as at the interface. A clean interface in a GaAs-AlGaAs heterojunction is possible because GaAs and AlGaAs have the same crystal structure and their lattice constants are nearly the same. Using molecular beam epitaxy, nearly perfect crystalline heterojunction can be grown with an interface confined within an atomic layer. Impurities at the interface can be further reduced by implanting donors in AlGaAs far away from the interface. GaAs-AlGaAs heteroj unctions have been almost exclusively used in recent experiments on 2DES. Measurements of magnetotransport properties are the most common experi ments on 2DES in a magnetic field. A typical geometry of a sample is shown in Fig. 2.2 [4]. The length of the sample L (usually a few mm) is much larger than the width W to reduce edge effects. Electrons flow from a source S to a drain D 14 S mm Figure 2.2: A typical geometry of samples in magnetotransport experiments. A magnetic field is applied in the z direction, (from Ref. [4]) 15 due to the longitudinal voltage Vl applied in the x direction. A magnetic field B is applied in the z direction which is perpendicular to the plane where the sample is placed. Then a Hall voltage Vh is induced in the y direction. Vl is measured between contacts A and B while Vh is measured between A and C. Since the longitudinal current density in two dimensions is j = I/W and the longitudinal electric field E l = V l/L , the magnetoresistance is Rl = VL/I = (EL/j)(L/W ) = pxx(L/W), (2.1) where pxx is the magnetoresistivity. Similarly, the Hall resistance is Rh = VH/I = pXy, (2.2) where pxy is the Hall resistivity. Note that the Hall resistance is independent of the geometry of the sample, i. e. independent of L and W. This is an important consequence of dimensionality of the sample. 2.1.2 Quantum Hall Effects The QHE at integer filling factors was first discovered by von Klitzing et al. [6 ]. They performed magnetotransport measurements for the 2DES in a Si MOSFET placed in a strong perpendicular magnetic field. When the temperature is suf ficiently low, it was observed that pxy is flat with quantized values (h/e2)(l/i) near integer filling factors u = i. Each plateau is centered around v = i. As u deviates further from integers, pxy changes linearly with 1 ju (Eq. (1.13)). The quantization of pxy seemed to be exact within the accuracy of the measurements (one part in 105). It was also observed that pxx shows minima at those integer filling factors. pxx appeared to vanish at zero temperature. It is interesting to note that the macroscopic quantity R h isquantized in the unit of /i/e 2, which contains only fundamental constants. In their first report 16 of the IQHE, von Klitzing at a1. [6 ] pointed out that the quantized Rh can be utilized to determine the fine structure constant a. Rh and a are closely related by Ho ce2 hqc 1 ” = T I = ~2R^ (l” SI UMts)’ (2'3) where i is the integer filling factor, the permeability of free space fio is 4 7 T x 10~7 H/m , and the speed of light c was recently defined as 299, 792,458 m /s [ 6 8 ]. Therefore, the measurement of Rh in any precision leads to determination of a in the same precision. On the other hand, if a is determined by some other method, R h = (h/e2)(l/i) « 25,813 0/* can be used as a standard of resistance. Soon after the discovery of IQHE, similar behaviors of pxx and pxy were also observed at fractional filling factors v — 1/3 and 2/3 in the 2DES in GaAs- AlGaAs heteroj unctions [9-13]. The width of plateaus in pxy and the depth of minima in pxx depend on the sample quality and the temperature. Subsequent measurements showed that the quantization of Rh — (h/e2)(l/u) at v — 1/3 is extremely accurate [11,17]. Initially, observed fractional quantum Hall states were thought to be Wigner crystals [9] since it was believed that the ground state of the 2DES at low filling factors is likely to be a Wigner crystal. However, such an interpretation was shortly ruled out for the following reasons. The I-V relation for pxy at v = 2/3 is linear down to very low Hall voltage [13], which indicates that the ground state is not likely to be the Wigner crystal because the Wigner crystal is believed to be easily pinned by impurities below some threshold electric field. Another piece of evidence against the Wigner crystal is the activated behavior of pxx at low temperatures [11,13,19]. It was found that at v = 1/3, pxx oc exp(—A/271), where A is the activation energy. This activated behavior implies the presence of a gap in the excitation spectrum of the system. If the ground state is the 17 Wigner crystal, phonon excitations will result in a continuous excitation spectrum without a gap. A breakthrough of understanding the nature of the FQHE was made when Laughlin [26] proposed that fractional quantum Hall states at u = 1/m with odd integers m are incompressible quantum fluids with the structure largely determined by the symmetry. As the sample quality was further improved, the FQHE was found at many rational filling factors with odd denominators, v — 1/3, 2/3, 4/3, 5/3, 7/3, 8/3, 1/5, 2/5, 3/5, 4/5, 7/5, 8/5, 1/7, 2/7, 3/7, 4/7, 10/7, 11/7, 1/9, 2/9, 4/9, 5/9, 13/9, 3/11, 4/11, 5/11, 6/11, 7/11, 4/13, 6/13, and 7/13 (compiled from Ref. [9- 21] and others). A typical magnetotransport measurement is shown in Fig. 2.3. Fractional quantum Hall states at v ^ 1 /m are believed to be closely related to Laughlin’s 1/m states. There are many proposals about the nature of these states. They will be discussed in a later section. These states are now referred to as “hierarchical states”. A deep minimum of pxx has also been observed at u — 1/2 [69,70]. How ever, pxx doesn’t drop to zero as T —» 0 and this state doesn’t resemble any observed fractional quantum Hall state. The nature of this state is still being investigated [71]. The only fractional quantum Hall state with even denominator that has been observed is the one at v = 5/2 [72]. This state is believed to be made up of electrons of different spins and is generally described as a spin singlet state [73,74]. Since the structure of this state appears to be rather intricate and is not relevant to our study of crystal states, it will not be discussed in this thesis. 2.1.3 Crystal States Although the search for the Wigner crystal has unexpectedly resulted in discov ering quantum Hall states at various integer and fractional filling factors, it is still believed that the Wigner crystal will be eventually found at sufficiently low 18 v 2 0 1.5 N o.x 1.0 0.5 a v . a •X x 50 100 B (kG) Figure 2.3: Magnetotransport measurements at 100 mK for the 2DES in a GaAs- AlGaAs heterojunction. The upper curve is the Hall resistivity pxy and the lower one is the magnetoresistivity pxx. (from Ref. [18]) 19 filling factors. However, if the Wigner crystal indeed exists, it may be difficult to detect for a number of reasons. Firstly, one needs a 2DES of very high quality. Since the occurrence of crystallization is due to the Coulomb interaction between electrons and the electron density under consideration is very low, the system will be easily affected by even a moderate amount of impurities. Secondly, it is not practicable to use X-ray or other scattering methods to study basic proper ties of crystals. This is because the binding energy of the Wigner crystal is so low that the scattering might destroy the whole system. Thirdly, the 2DES in the inversion layers is buried deeply inside the device, which makes it difficult to perform many conventional surface experiments such as LEED (low energy elec tron diffraction). Finally, a yet more delicate problem is to distinguish between different kinds of crystalline states in addition to distinguishing them from the localization of electrons by random potentials. As cleaner samples of the 2DES with high mobility became available in GaAs- AlGaAs heterojunctions, it has been possible to study 2DES at low filling factors. Recently, there have been a number of experiments that observed highly insulating phases at various filling factors. Since then, a good deal of experimental effort has been devoted to identify the nature of these phases. There are reports of the temperature dependence of pxx in these insulating states at low temperatures. It was found that pxx rises exponentially on lowering the temperature at filling factors just above v — 1/5 as well as below v — 1/5 down to very low v [57, 58]. This behavior of pxx is just the opposite of that at v = 1/5 where pxx approaches zero at low temperatures. The insulating phases were interpreted as Wigner crystals pinned at impurities at low temperatures. Nonlinear I-V relations for pxx in a narrow region of filling factors above v — 1/5 and at filling factors below v = 1/5 have also been reported [59,61]. The conduction of electrons remains very low at low applied voltages. But it suddenly grows as the voltage 20 increases above some threshold. In addition, noise like ac conduction is detected at high applied voltages [59]. The observed conduction threshold was interpreted again in terms of the pinned Wigner crystal. The ac conduction might come from sliding of the crystal across the impurities. At high temperatures, the I-V relation becomes linear due to the melting of the crystal. More recently, magneto-optical measurements have been used to study 2DES. After the sample is illuminated by photons with energy slightly larger than the band gap of GaAs, the spectrum of light emitted by the sample is recorded. As v is lowered below 1/3, a new luminescence line begins to develop at a lower energy and finally dominates the spectrum except at 1/5, 1/7 and 1/9 [21,60]. The data indicate that a different electron state with energy lower than that of the fluid states develops below v = 1/3. Formation of the Wigner crystal was considered as a possible explanation. There is still some disagreement among experiments on the filling factors where the insulating phases are observed. Such discrepancies are likely to come from dif ferences of sample quality and the method used to identify these phases. However all of them agree that these phases exist both above and below v = 1/5. It is also the common feature among experiments that these phases begin to appear at a filling factor somewhat below 1 /3 and persist down to the lowest filling fac tor attainable in their experiments. On the other hand, in spite of much effort to identify the insulating phase as the Wigner crystal, none of the observations could be considered as a direct evidence for crystalline states. If it turns out that these phases are in fact crystals, then the experimental observations imply a series of phase transitions between quantum fluid states and crystal states. Based on this conjecture, Buhmann et a 1. [60] suggested a schematic phase diagram of the 2DES as shown in Fig. 2.4. (The insulating states observed in experiments will be referred to as crystal states from time to time when the meaning of the phrase is clear from the context.) 21 crLU z> cr< LU CL Z LU t- 0 0.1 0.2 0.3 FILLING FACTOR Figure 2.4: A schematic phase diagram of the 2DES. The shaded region is for crystal phases and the white region is for fluid phases, (from Ref. [60]) 22 2.2 Theories of Quantum Hall Systems 2.2.1 Integer Quantum Hall Effects Soon after the IQHE was discovered, the unusual behaviors of pxy and pxx, that is, pxy is quantized at filling factors close to integers and pxx at integer filling factors vanishes at zero temperature, were well accounted for by theories based on the picture of independent electrons [22-24]. The dissipationless current flow at v = i can be understood in terms of the large gap in the excitation spectrum of the electrons. The strong perpendicular magnetic field generates a series of degenerate Landau levels separated by large gaps hu>c (Fig. 2.5(a)). In the limit i hojc ksT, at v = i, the i lowest Landau levels are completely full while all the upper levels are empty. At low temperatures, electrons scattered by other electrons or by impurities are hardly excited to higher Landau levels because of the large gap between levels. Electrons therefore don’t suffer from dissipation. The quantization of pxy is quite unusual in that, independent of the geometry of samples and the amount of impurities present in the samples, the quantization appears to be exact. To address the question of exactness of the quantized pxy, Prange [ 2 2 ] studied the Hall conductivity of a 2DES in the presence of a (5-function impurity. He showed that electrons localized by the impurity don’t carry any current, while the mobile electrons carry extra current to compensate for the loss of current due to the trapped electrons. The Hall conductivity is therefore not affected by the (5-function impurity. Even though his argument is not readily applicable to more realistic situations, it gave an insight into exactness of the quantization. A more general proof for the quantized Hall resistance was later given by Laughlin using gauge invariance arguments. In a thought experiment, Laugh- 23 Magnetic field Test flux Applied voltage V 6 Hall current Figure 2.5: The density of states for 2DES (a) without and (b) with impurities in a magnetic field. In the presence of impurities, each Landau level is broadened into bands of extended states (the shaded region). Localized states are present in the mobility gap between the bands, (c) The geometry of a metal strip in Laughlin’s thought experiment, (from Ref. [3]) 24 lin [23] considered a metal strip bent into a loop of length L (Fig. 2.5(c)). The potential drop across the strip is V and a uniform magnetic field B is applied nor mal to the surface of the metal. The coordinates are chosen as shown in Fig. 2.5(c). The Hall current / produces the magnetic moment fj, = IS/c where S is the area enclosed by the loop. If a flux 8 where 8 A is the change in the uniform vector potential A pointing around the loop. After introducing the gauge potential A, eigenstates of an electron are multiplied by the factor exp(zeA:r/ftc). If the eigenstate is extended along the loop, the wavefunction is not valid unless A is a multiple of /ic/eT, or equivalently, 8 2.2.2 Incompressible Quantum Fluids at v = 1/m After the FQHE was first observed at v = 1/3, it was shortly realized that a highly correlated electron state other than the Wigner crystal is responsible for the FQHE. Yoshioka et a 1. [25] studied the 2DES using the exact diagonalization of the Hamiltonian with up to six electrons. New features of the 2DES near v = 1/3 were found in their study. The ground state energy at v = 1/3 is much lower than the energy of the Wigner crystal calculated by the Hartree-Fock 26 approximation. And there is a downward cusp in the energy at 1/ = 1/3. Moreover, the pair correlation function at v = 1/3 is quite different from that of the Wigner crystal. In an important paper, Laughlin [26] proposed that ground states at v — 1 /m with odd integers m are incompressible quantum fluid states. In the following, we show how Laughlin’s wavefunction can be derived from symmetry arguments. Our derivation is different from Laughlin’s. In fact, the description below displays the general structure of many-electron wavefunctions in the lowest Landau level, regardless of the nature of the state, i. e., fluids, crystals, or others. The reason that the derivation of this general property is presented in this review section rather than integrated into the next chapter where the thesis research is presented is because it is so fundamental to the quantum Hall states that it should be pointed out at the outset. As far as we know, the derivation of the general property (Eq. (2.10)) below has not appeared in the literature. The many-electron Hamiltonian in a uniform neutralizing positive background subject to a magnetic field is 1 / ^ ____ C . \ XT/ \ e2 -Vj + -A j +V(rj) + T~ TT’ (2-5) 2 m* \i c i v{T) = -e* l^ -? fT' (2 -6) is the potential energy generated by the positive background of density po. In the strong magnetic field limit, the single electron state is given by the orbitals of the lowest Landau level whose wavefunction in the most general form is f(z)e~hi2/4, where f(z) is an entire function. The ground state of the many-electron Hamil tonian constructed from these orbitals is therefore $[z] = .P[z]e-£‘ (2 .7) 27 where [z] = (z\,..., z^) and P[z] is an entire antisymmetric function of zlf..., zjv. Since any entire function f(z) can be approximated arbitrarily accurately by a finite series, we can, without loss of generality, take P[z] as an antisymmetric polynomial. If P{z] is considered to be a polynomial of z\ with z2,... ,zjv param eters, then by the fundamental theorem of algebra, P[z] must be factorizable in the form P[z] = ria^i — &(*) where aa are functions of z2, • • •, 2 n - However, by the Pauli principle, P[z] = 0 when z\ = Zk for k = 2,... ,N. This means that P[z\ must contain a factor FIa^i^i “ zk)• Similarly, considering P[z] as a function of zj and treating all other z’s as parameters, we can conclude that P{z\ must contain a factor Ylk^j(zj — Zk). The only way for P[z] to have all these factors while maintaining the antisymmetry is for P[z\ = - zk) Q[z], (2.8) 3 Q[z] = I I (ZJ ~ Zk) P'iz1 (2-9) 3 3 M = I I (zj - z„r (2.11) 3 | $ m[z] |2 = exp(-^[z]), ( 2 .1 2 ) where the fictitious temperature /3 = 1 /m and 4>[z] = - 2 m 2 J 2 In \zj ~ zk\ + y E N 2 • (2>13) j » = 2 xm ' <2'14> Since the wavefunction 'I'm is made up of single electron orbitals in the low est Landau level from z°e~ld 2 / 4 to 2rm(^v—i)e— the density vanishes for \z\ > m(N — 1), which is the radius of the outermost orbital. \&m therefore describes a uniform droplet of radius « N/irpo. The pair correlation function is g{z,z') = ^ (p(z)p(z')) (2.15) Po N(N - 1) / | 'fym{z,z',zz,...,z N)\2 = 1 , - T . I + I f (2.17) 2 &k\zi - zk\ i J \s~ zi\ 2 ' N-s'l where the last term is the self energy of the positive background. (Since the kinetic energy per electron is hu>c/ 2 for any state in the lowest Landau level, it will be omitted from the energy from now on.) The first term in the energy can be written as <"•> = = ( 2 - i 8 ) 30 The remaining terms are (U,) = -P°N J ^ + \l>lA J j t f = - \ e ° A S y v <2'19> where A is the area covered by the electrons and the uniform background. From Eqs. (2.18) and (2.19), the energy Etot = ( U) is „ 1 , f Note that this equation is valid for both fluid and crystal states. For fluid states, since g(z,z') depends only on | z — z'|, the energy per electron can be written as * = ( 2 -2 D This expression for E is particularly useful for numeric calculations. Since it is expected that < 7 (|z|) — 1 approaches zero within a few multiples of the magnetic length /o, the energy of the infinite system can be extracted by calculating g(\z\) for small \z\ with a finite system (see section 3.4 for details). Based on extensive Monte Carlo simulations, Levesque et a 1. [53] found the energy per electron of Laughlin’s v = 1/m states to be represented by an “interpolation formula” \ e2 E l = (-0.782133 1/0-5 + 0.165 v1M - 0.009 1/2'2) —. (2.22) ' * lQ Note that the first term is the classical energy of the electron lattice [44]. The exponent of the second term was chosen from the plot of the energy correction to the classical lattice energy vs. Ini'. The exact results known at v = 1, ^(l^l) = 1 — e- lzl2 / 2 and E = — ^J^v/S (e2/10) [75], were used in the fitting of this formula. In addition to the ground state, Laughlin [26] also proposed elementary excita tions of the 2DES at v — 1 /m. He considered an infinitely thin magnetic solenoid 31 piercing the ground state at zq. If a flux is introduced through the solenoid adia- batically, the system remains an eigenstate of the changing Hamiltonian. After the flux quantum = e-E'l''|2/4n(5--.-zS )IIte-^r (2-23) « j The charge of the quasihole in the state v ? " 20 can be found using the plasma analogy. As before (Eqs. (2.12) and (2.13)), I'f " 2 0 !2 can be written as e~^\ where /3 = 1/m and (/>' = (/ — 2m In | Z{ — zq\ with 30 N*42 x > tf) C ft) T3 0.5 0 0 4 .0 6.0 6.0 R/R0 2.0 x 4J ft in (b) c T30) 0.0 2.0 4 .0 6.0 R /R n Figure 2.6: The electron density profile of (a) the quasielectron and (b) the quasi hole excitations at v = 1/3. N is the number of electrons used in Monte Carlo calculations and Rq = \/2m Iq. (from Ref. [28]) 33 quasiparticle excitations is now believed to account for the FQHE at v — 1/m. Consistent with the experimental observation of the linear I-V relation for pxy at v — 1/3 [4], Laughlin’s 1/m sta.te is a fluid state. Furthermore, the existence of quasiparticle excitations implies a gap in the excitation spectrum. And the gap is responsible for the activated behavior of pxx at low temperatures observed at v = 1/3 [11,13,19]. On the other hand, it is still not well known how quasiparti cles interact with each other. If the interaction is like the Coulomb interaction, the FQHE can be understood much the same way as the IQHE with electrons replaced by fractionally charged quasiparticles [26]. Quasiparticles form their own Landau levels and the role of impurities is the same as that in the IQHE. Moreover, the existence of a gap in the excitation spectrum is responsible for the vanishing mag- netoresistance. This picture is the basis of hierarchical schemes of FQHE which shall be discussed in the next section. 2.2.3 Hierarchical Schemes As the FQHE has been observed at many filling factors, it is now generally be lieved that quantum Hall states exist at all rational filling factors v = p/q with odd integers q. The observation of p/q states is however limited by the sample quality and possibly by the formation of the Wigner crystal at low filling fac tors. Since Laughlin’s theory of the FQHE was proposed, there have been a lot of theoretical efforts to extend his idea of quantum fluid states to other rational filling factors. Currently, there are essentially two different but related schemes. The first is the hierarchical schemes proposed by Haldane [30], Halperin [32], and Laughlin [33] independently. The idea is that when quasiparticles are added to a known quantum Hall state (a parent state), at certain quasiparticle densities, they condense to Laughlin-like incompressible fluid states, which are the next higher level quantum Hall states (daughter states) in the hierarchy. The daughter state 34 in turn generates a set of its own daughter states. Such constructions continue up the hierarchy. The second scheme is a theory based on the composite-fermion approach proposed by Jain [37-39]. In his theory, the IQHE and the FQHE are incorporated into a single framework. However, in this scheme, it is necessary to involve higher Landau levels to account for the FQHE at rational filling factors. Halperin’s scheme [32] is the only one so far where approximate evaluation of the energy is possible for arbitrary p/q states. This approximation will also be used for our subsequent discussion of crystal states. Halperin’s scheme is the following. Suppose the 2DES is in a quantum Hall state at the 5 th level of the hierarchy. The state can be characterized by three parameters i/3, q„, and m 3, where ua is the filling factor of the state with quasiparticle charges ±qse. The quasiparticles are assumed to obey fractional statistics such that the exchange of two quasiparticles generates a nontrivial phase factor (—l)*1/"1* depending on the sense of rotation in the process of switching [77]. Quasiparticles at the sth level condense to a new incompressible fluid state at the (s + l)th level whose pseudowavefunction in terms of quasiparticle coordinates can be written in the form y[z) = n (zk - z,)2p'+i n (zk - zo-a‘+i/m‘i K MI*fc|2/4> (2-25) k [Z] = (Zi,..., Zm,)- The state at the ( s + l)th level is determined by two addi tional parameters: a5+i = + 1 / — 1 for quasielectrons/quasiholes and a positive integer ps+i. Using the plasma analogy mentioned before, the number of quasi particles per flux quantum is found to be n3 = |<7s|/ra 4+i, where — 2 pa+i a3+i/m 3. (2.26) 35 Since the charge of quasiparticles is a3+iq 3, the filling factor of the new state is u3+1 = u3 + as+iq3\q3\/ma+i. (2.27) By an argument similar to Laughlin’s [26], the charge of quasiparticle excitations in the newly condensed state can be found to be Qs+l ~ ®3 +l Qa/^a+l' (2.28) Starting with uq = 0 and qo = m 0 = 1, the first level of the hierarchy recovers Laughlin’s states at u — 1,1/3,1/5,... by adding quasielectrons (c*i = +1). From v\ = 1/3, quasielectrons (a 2 = +1) generate daughter states at v2 = 2/5,4/11,... and quasiholes (a2 = ~1) generate those at v2 = 2/7,4/13,— It can be shown that every rational filling factor with an odd denominator is generated in this way. Halperin [32] estimated the energy of hierarchical states assuming that the interaction between quasiparticles is Coulombic. In his estimation, the energy per flux quantum is ^ (^ 3 +1 ) « E (v3) + n .e* + n3\q3\5/2Epi(l/m3+i), (2.29) where c* is the energy to add one quasiparticle together with the neutralizing uniform background and Epi(l/ma+i) is the energy per electron of Laughlin’s state at u = l/m s+i (Eq. (2.22)). To use Eq. (2.29), e± should be calculated at all quantum Hall states. e± is currently not well known except at v = 1/3 [28]. For the purpose of illustration, assuming that the ratio of e+ to e~ is the same for all quantum Hall states, the estimated energy of hierarchical states is plotted in Fig. 2.7 [32], Although the energy diagram Fig. 2.7 serves a useful guide for understanding the stability of various phases, it should be stressed that it is based on a number of approximations that remain to be justified. At this time, there is no rigorous justification of Halperin’s picture. 36 0.03 c* 0.02 a HI i LU 0.01 0 0 0.5 1.0 v Figure 2.7: The estimated energy per flux quantum of hierarchical quantum Hall states. The plasma energy Epi = vEvi(v) is subtracted off. (from Ref. [32]) 37 While Halperin’s theory based on the pseudowavefunction of quasiparticles is appealing, it is desirable to express the wavefunction of hierarchical states in terms of electron coordinates. To find a new stable state from a known one, the particle-hole symmetry within the lowest Landau level turns out to be very useful. If a 2DES at v is in a stable state ’L, its particle-hole conjugate state which has holes described by is also stable at the filling factor 1 — v. Girvin [34] showed that the wavefunction of the state ^ can be written in terms of electron coordinates as M n dzN+k^*(zN+-i, . . . , Zn+m)$N+m (zi, ■ ■ •,ZN+M), (2.30) fc=l where N is the number of electrons, M is the number of holes, and <£jv+m is the wavefunction of the fully filled Landau level with N + M electrons. MacDonald et a 1. [35,36] have made use of this idea to construct a hierarchical scheme. As mentioned before, a many-electron wavefunction entirely within the lowest Lan dau level can be written as ^[z] = P[z]e~£^ 12,l2 / 4 5 where P[z] is an antisymmetric polynomial of z,-’s. Since the quantum Hall states which one is interested in are isotropic, P[z] should be homogeneous. Recall that any homogeneous antisym metric polynomial P[z] can be written as P[z] = Q[z\Pv[z\, where Q[z] is a ho mogeneous symmetric polynomial and Pv[z\ = — zk) IS the Vandermonde determinant. For simplicity of notation, C(P[z]) is defined to be the polynomial part of the particle-hole conjugate of ^[z] and C{Q[z\) = C{Q[z] Pv[z])/ Pv[z). It was proposed that if P[z] is a good approximation to the ground state polynomial for the inverse filling factor v~l, P[z](Py[z])2p with a positive integer p is also a good approximation for the inverse filling factor v~x + 2p. Based on these obser vations together with the particle-hole symmetry, MacDonald et a I. considered recurrence relations between trial wavefunctions of hierarchical states: QiW = C(Qi^\z])(Pv[Af,‘ (2-31) 38 Qi[z] = CiQi-rlztfiPviz})2", (2.32) where p,- is a positive integer and C{Qi-\[z})1( is the adjoint of C{Qi-\[z\) obtained by replacing z’s in C(Qi-\[z\) by d2's [27]. The first one is interpreted as quasi holes at the parent filling factor v — ( 1 + 2 p , ) - 1 condensed into a quantum fluid state described by Qi~\\z\. The second one is similar to the first one except that it is quasielectrons that condense to Qi-\[z\. Simple counting of the degree of polynomials leads to v 1 = 1 + 2pi H—ri (2.33) where a,_i = +1 (quasiholes) or —1 (quasielectrons). In this construction, Vi = 2/7,2/5 are obtained from the parent state at = 1/3 with a,_i = ±1. MacDonald et a 1. also studied the pair correlation function and the energy of the states at v = 2/5,2/7 using their trial wavefunctions. More extensive study of hierarchical states is necessary at this time to acquire better understanding of hierarchical schemes. The standard hierarchical scheme described so far has been successful in pro viding a general picture of the FQHE at rational filling factors with odd denom inators. It also predicts reasonably well, even though not perfectly, the rela tive stability of the FQHE at various filling factors. However Jain [39] pointed out a few problems with the standard hierarchy. In this scheme, by the time a daughter state is reached starting from a parent state, the density of quasi particles approaches that of electrons. As one goes further to the next level daughter state in the hierarchy, the total number of quasiparticles will exceed the number of electrons. Under such circumstances, it will be more appropriate to describe the system in terms of electrons instead of quasiparticles. As an ex ample, the experimentally observed FQHE at v = 6/13 can be reached through 1/3 —> 2/5 —► 3/7 —► 4/9 —> 5/11 —■> 6/13 in the standard hierarchy. However it is 39 hard to image that the state at v = 6/13 results from the delicate balance of five different kinds of quasiparticles created in each level of the hierarchy starting from the state at v = 1 /3. Another point he made is that experimental observations of the IQHE and FQHE are phenomenologically very much similar to each other but there are two separate theories for the IQHE and FQHE respectively. It would be nice if one theory can account for all QHE on the same footing. Jain [37-39] proposed a different scheme which is claimed to be non-hierarchical and to incorporate the FQHE and IQHE in the same framework. The proposed trial wavefunction is X*=II(*i-**r~,X>.. (2.34) i u = L . (2.35) (m - l)p + 1 It is more convenient to write the wavefunction as X* = X pX ?'1 = [?,!>• ••,!]• (2-36) In this notation, a new state x* Is constructed by the product of an integer quan tum Hall state X p and an even number of Xi’s- To make this notation consistent, the exponential factor of each wavefunction Xa involved in the construction should be defined to be e~9a^ M 2 / 4 with qQ = v/a. The construction can be further generalized. From the state one can build the state [n ± v, 1,..., 1] with an 40 integer n. It was shown that all odd denominator rational fractions can be ob tained in this way. The main difference between this theory and the standard hierarchy is the role of higher Landau levels, i. e. when p>2orn±i/>2. Note that Jain’s trial wavefunction for the filling factor v < 1 is not guaranteed to lie entirely within the lowest Landau level. The projection into the lowest Landau level should be taken to get the desired wavefunction. Jain [39] predicted the relative stability of quantum Hall states by applying a few simple rules: (1) If Xu is more stable than X u then [v, 1 , • • • > 1 ] is more stable than [v', 1 ,..., 1 ]. (2 ) [z/, 1,..., 1] is more stable than [v + 1,1,..., 1]. (3) [i/,1 ,..., 1] is more stable than [n, 1,..., 1 ,1], The predictions are in excellent agreement with experiments (Fig. 2 .8 ). 2.2.4 Crystal States The crystallization of the electron gas was first predicted by Wigner [ 1 ] sixty years ago. When the electron density is sufficiently low, the potential energy of electrons becomes more important than the kinetic energy. The electrons will be therefore arranged in a lattice to minimize the potential energy. The Wigner crystal was first observed on the liquid helium surface [ 2 ] in the low density classical limit. Since high quality 2DES was realized in inversion layers in semiconductors, there has been much discussion on the possible formation of the Wigner crystal in 2DES subject to a strong magnetic field. If the magnetic field is strong and the temperature is low, most electrons in the 2DES stay in the lowest Landau levels. Because the kinetic energy of a 2DES entirely within the lowest Landau levels is the same for any configuration of the 2DES, which is essentially zero if the energy scale is reset, the energy of such a system is completely determined by the potential energy. By Wigner’s arguments, one might expect the 2DES to crystallize. Since the strong magnetic field will help electrons to be localized within the magnetic 41 L 1 L 5 5 9 13 _L 1 1 i. A 1 1 3 3 5 7 9 11 13 15 (b) Figure 2.8: The relative stability of quantum Hall states predicted in Jain’s theory of FQHE. (a) va's constructed by [p, 1,1,...]. (b) Vb = 1 — va. (c) i/c’s constructed by [1 — va, 1 , 1 ]. (d) Vd's constructed by [1 + i/a, 1,1]. (e) ve = 1 — Vd- The relative stability decreases to the right column and to the upper row in each diagram. The FQHE at all filling factors except the rightmost one in each row was observed in experiments, (from Ref. [39]) 42 length /o, the crystallization might occur at electron densities where it does not take place at zero magnetic field [42,43]. At electron densities 10lo-10n cm - 2 accessible in GaAs-AlGaAs heterojunctions, the Fermi temperature of the 2DES is of the order of one Kelvin. Since experiments can be performed well below the Fermi temperature, such 2DES as found in GaAs-AlGaAs heterojunctions are considered as good candidates to observe the Wigner crystal in the quantum regime. The energy and the melting temperature of the Wigner crystal in 2DES were extensively studied in the late 70s since at the time the Wigner crystal was believed to be a favorable state at low electron densities. Studies of the Wigner crystal were mostly done in the Hartree-Fock approximation [45-47,50]. More recently, Maki and Zotos [51] proposed a trial wavefunction for the Wigner crystal, which is an antisymmetrized product of Gaussians centered at the lattice sites of the hexagonal lattice. That is, 1 V[z] = det (2.37) y/Nl where a/s are coordinates of the lattice sites and 1 < M 2«) exp ~ \4 \ '- z i ~ « j | 2 +■ l4 ( ’ ziaj ~ ziai) (2.38) is the Gaussian centered at aj (Eq. (1.12)). The interaction energy of the electrons is calculated from ,2 E = ( V (2.39) ^2^1*- j^k \ZJ Zk\ Using this trial wavefunction, Maki and Zotos [51] obtained essentially the same energy as the Hartree-Fock calculations. However, the advantage of introducing the trial wavefunction is that the calculation is much easier than the Hartree- Fock approximation. Integrals involved in the calculation of the energy are all 43 Gaussian integrals and most of integrals involving two or more Gaussians centered at different sites can be ignored because the overlap of the Gaussians is minimal at the low electron densities under consideration. Since the wavefunction is relatively easy to handle, it enabled studying other properties of the Wigner crystal such as lattice dynamics [51]. Lam and Girvin [52] further improved the energy of the Wigner crystal by including correlations in the trial wavefunction. Their wavefunction for the correlated Wigner crystal is V[z] = n **(*•■)» (2-4°) t where £, = Z{ — a, and Bij are variational parameters. Note that the antisym- metrization of the wavefunction is omitted for the purpose of easy calculation. For 0 < v < 1/2, they fitted the energy per electron of the correlated Wigner crystal to the equation / \ e2 Ewc = (~0.782133i/1/2 + 0.2410i/3/2 + 0.16i/5/2) —. (2.41) v y »o The energy of the Wigner crystal at 1/2 < v < 1 can be obtained by applying particle-hole symmetry. For the Wigner crystal wavefunction Eq. (2.37), Maki and Zotos [51] showed that particle-hole symmetry implies vE(v) = uE{v) - ~ ")> (2-42) where v = 1 — v. We will show later (section 3.4) that this relation is true for any state in the lowest Landau level. When the FQHE was discovered at u = 1/3, experimental findings were very different from the theoretical expectation at the time. The 2DES is in a fluid rather than a crystal state. Now that incompressible quantum fluid states are known to exist at many rational filling factors, one might expect a phase transition from quantum Hall states to crystal states at some critical filling factor vc since 44 the Wigner crystal is believed to exist at sufficiently low filling factors. Lam and Girvin [52] predicted vc « 1/7 by comparing energies of Wigner crystals (Eq. (2.41)) and Laughlin’s 1 /m states (Eq. (2.22)) (Fig. 2.9). However, there are a number of discrepancies between this prediction and experimental observations of highly insulating phases if these phases are really crystal states. Not only is the FQHE observed at v = 1/9 [21] which is lower than vc but also crystal states exist at filling factors much higher than vc. Moreover, crystal states exist both below and above v = 1/5 which implies that there are more than one phase transition between quantum Hall states and crystal states. These discrepancies have called for more studies of crystal states in 2DES. There are other proposed crystal states that are different from the Wigner crystal discussed so far. Tesanovic et a1. [65,66] proposed “the Hall crystal” that gives the quantized Hall conductivity while maintaining crystalline order. They argued that the Hall crystal exists not with the pure Coulomb interaction but with a four-body interaction in addition. However such a crystal has not been found in experiments. Another attempt to explain the FQHE in terms of a crystal state was made by Chui et a I. [67]. Their trial wavefunction is basically the product of Gaussians multiplied by the factor ~ Zk)m- They claim that their crystal state at v = 1/3 has a lower energy than Laughlin’s fluid state. Such a claim, however, has raised much skepticism. From the structure of their crystal state, it is probable that the energy falls between that of the Wigner crystal (Eq. (2.37)) and Laughlin’s fluid state. In the rest of this thesis, we will discuss crystals due to quasiparticle crys tallization. As we shall see, crystallization of quasiparticles will result whenever the filling factor deviates from those of stable quantum Hall fluids. This feature is consistent with experimental observations of insulating states at various filling factors, especially above and below v = 1/5. The properties of these crystals, 45 V 1/2 1/3 1/4 1/5 1/6 1/7 1/8 1/9 1/10 (0 0 “ 1 10 V wc J2 ^ V HF o WC m • " iu 5 — 8.28 16.56 24.84 33.12 41.40 B (Tesla) Figure 2.9: E^c is the energy of the Wigner crystals obtained from Eq. (2.41) and E is from the Hartree-Fock approximation. E l is the energy of Laughlin’s 1/m states obtained from Eq. (2.22). Ec/aastco/ is the classical energy of the electron lattice [44]. (from Ref. [52]) 46 however, are very different from those of Wigner crystals. In the next chapter, we shall show that crystals carry macroscopic angular momentum, which is different for different crystals. Before we proceed any further, we want to point out some ambiguity of nomen clature in the literature. On the one hand, the term “Wigner crystal” has been implicitly used from time to time to refer to any kind of crystal states that might exist in the electron gas. On the other hand, the term has also been used to refer to the specific Gaussian product (Eq. (2.37)) or the modification of such a product (Eq. (2.40)). To avoid ambiguity, the term “Wigner crystal” will be used from now on to refer to the electron crystal given by Eq. (2.37) or Eq. (2.40). When it is necessary to distinguish between them, the former will be referred to as the “uncorrelated Wigner crystal” while the latter is the “correlated Wigner crystal”. CHAPTER III Crystal States The current status of experiments and theories on crystal states in 2DES has been reviewed in chapter 2. The following observations, in particular, motivated us to study crystallization of quasiparticles. It has been conventional belief that there is a single phase transition from the quantum Hall state to the Wigner crystal at some critical filling factor vc. By comparing the energy of quantum Hall states with that of Wigner crystals, vc was estimated to be around 1/7 [52,53]. On the other hand, there have been recent experimental observations of highly insulating states at many filling factors including those just above and below v = 1/5 [56-63]. If the observed insulating states turn out to be crystals, the theoretical picture of crystal states in a 2DES needs to be modified. This is because the experimental observations imply that not only do crystal states exist well above the estimated vc but there is also more than one phase transition between quantum Hall states and crystal states. Note that crystal states in the 2DES have been interpreted mostly in terms of the Wigner crystal, which is the product of Gaussians centered at hexagonal lattice sites (Eq. (2.37)) or the improved version of it (Eq. (2.40)). We have studied different kinds of crystals, that is, quasiparticle crystals which might be responsible for the observed insulating states near u = 1/5 as well as at other filling factors. 47 48 In this chapter, we present our studies of crystal states in the 2DES. The wave function of quasiparticle crystals are built largely based on symmetry properties of the 2DES in the lowest Landau level. From the general analysis of the energetics and the structure of the states in the 2DES, it will be shown that quasiparticle crystals are stable near every stable quantum Hall state. Monte Carlo simulations were carried out to determine the range of filling factors where quasiparticle crys tals are more stable than the Wigner crystal and the results are compared with experimental observations of the insulating phases. We also show that crystals in the 2DES have interesting characteristic properties that they carry an array of circular currents which will result in induced magnetic fields. The magnitude of these fields and their extent in the direction perpendicular to the plane where the electrons reside are estimated. 3.1 Crystals of Quasiparticles Given that the nature of the observed insulating states is not well known, at present, it is useful to study the symmetry properties of crystal states and identify their characteristic properties. The understanding of these properties will help us to identify the nature of the observed states. As we shall see, these symmetry properties led us to a new class of crystals which we later understood from energy considerations. We showed earlier that any many-electron wavefunction in the lowest Landau level must be of the form = e-E^'IV n(*i - **)” must have a lattice of quasiholes to satisfy the crystal symmetry. Then Q\z] can be written as fit a ( z i ~ a a ) 5[z], where aa are coordinates of the lattice sites and .Sfz] is another symmetric polynomial. 5[z], in turn, may either have the factor JlU*i ~ a") or ^ave no quasihole at all. By repeating similar arguments, we find that Q[z] — lli'.aC-2 !- aa)1 ‘5,/[2r]> where I is a positive integer and 5'[z] is yet another symmetric polynomial. The minimal structure of such a crystal is obtained by choosing ^'[z] = 1. The resulting state is a quasihole crystal with *;oJW = " “»>' n(*i - z*>”\ (3.2) i,Q j< k where v0 = 1 /m , m is a positive odd integer, and I is a positive integer. Similar arguments lead to quasielectron crystals, = e -£ 'Wi/1II(& .-O' n>, - «)”• (3.3) «,£» j< k (In subsequent discussions, + /— will refer to quasielectron/quasihole crystals un less otherwise specified. The filling factor vq and the quantum Hall state at uo will be referred to as the parent filling factor and the parent state respectively.) Note that the crystals t obtained in this way are the lattices of quasiparticles added on Laughlin’s 1/m fluid states. It is more complicated to construct wave- functions of quasiparticle crystals derived from hierarchical states partly because it is still controversial how to determine wavefunctions of the hierarchical states. Such quasiparticle crystals will be introduced in the last section of this chapter. Both types of crystals , are specified by four parameters: the parent filling factor Vq = 1 /m, the multiplicity of quasiparticles /, the lattice type, and the lattice constant. We have considered three lattice types: hexagonal, square, and honeycomb lattices. Although / = 1 is the most interesting case, arbitrary positive integer 1 is maintained in / to keep the general form of wavefunctions that satisfy required symmetry properties. As we will see later, while quasihole crystals 50 are well manageable in numerical studies, quasielectron crystals are much more difficult to study because of differential operators in their wavefunctions. However, many characteristics of both kinds of crystals can be deduced by the general analysis without rigorous calculations. First of all, we considered the stability of the quasiparticle crystals against other competing states such as the Wigner crystal or the quantum Hall state. At the filling factor v close to the parent filling factor vq, the 2DES consists of a dilute gas of fractionally charged quasiparticles on top of the parent fluid state. The density of quasiparticles decreases as v approaches v q . When the density is sufficiently low, that is, if v is sufficiently close to vo, the quasiparticles crystallize for the same reason that electrons form a Wigner crystal at low densities [1]. Since these crystals smoothly reduce to the parent state as v approaches Vo, if the parent state is stable, the quasiparticle crystals are also stable at least sufficiently close to Vq. From these observations, it follows that every stable quantum Hall state is surrounded by crystals of quasielectrons/quasiholes respectively on either side. This prediction is especially interesting because it agrees with the experimental observation of crystal states above and below v = 1/5. At v far away from vq, the density of quasiparticles becomes higher and the crystallization of quasiparticles is therefore energetically not favorable any more. At such filling factors, the Wigner crystal or other fractional quantum Hall states may be more stable. In view of the hierarchical scheme of the FQHE, at certain densities, quasiparticles condense to a new quantum Hall state which is expected to more stable than quasiparticle crystals derived from the old quantum Hall state. By arguments similar to Arovas et aV s [77], the charge of quasiparticles in the states , is found to be dclv0e. Since the electron density of the parent state is i/o/27t (Eq. (1.9)), the average electron density of the crystals 'k* , can be written 51 as Pe = ^±lvoP±, (3.4) where p± is the density of quasiparticles. For quasihole crystals derived from Laughlin’s 1/m states, the average density can be also derived using the plasma analogy [26] as follows. The probability density to find electrons at [z] is 1 / 7 7 0 _ __ exp - 2 m 2 In | Zj - zk\ - 2 ml ^ In \zi - aQ\ + — ^ |z;| (3.5) The exponent of | \ & " ' | 2 is, apart from the factor — 1 /m, the potential energy of a two-dimensional classical plasma of charges m immersed in a uniform background charge of density — 1/27T and the lattice of charges I fixed at aa. The energy is minimized and thus |\I , - | 2 is maximized if charges m are arranged to achieve charge neutrality everywhere. Therefore, . mpe + (3.6) which reduces to Eq. (3.4) since vQ = 1/m. In terms of the filling factor v = 2irpe of the state the density of quasiparticles can be written as * - ± v ± ^ r > °- <3-7) Note that as v —> Vo, p± —^► 0 and the lattice constant of the crystal goes to infinity. At v = vq , the lattice of quasiparticles disappears and therefore the parent fluid state is recovered. Eq. (3.7) also implies that quasielectron crystals exist only in the range v > vq , which is as expected because quasielectrons add fractions of electrons to the parent state at v0. For similar reasons, quasihole crystals exist only for v < v0. 52 From Eq. (3.7), the number of electrons in a unit cell is Tle 71 j. Pe i . , (3.8) P± v0 - V~x where n± is the number of quasiparticles in a unit cell. n± is fixed by the lattice type, e. g. n± = 1/1/2 for hexagonal/square/honeycomb lattices. Note that ne can be any real number from 0 to oo. This is quite different from the Wigner crystal that always has an integral number of electrons in a unit cell. Nonintegral values for ne come from the quantum nature of quasiparticle crystals. 3.2 Energy of Crystals The potential energy of electrons in a uniform positive background po is + <3 ' 9 > For fluid states, the pair correlation function < 7 (rx,r2) = (p(ri)p(r2)) Ip\ was in troduced to nicely handle finite size effects [26]. A similar procedure with some modifications can be used to calculate the energy of crystals. The pair correlation function of crystals, unlike fluids, is not simply a function of |r — r'|. However, the energy per electron of crystals (Eq. (2.20)) can be still written in the form E = 7Tpo J dr (g(r) - i) (3.10) where g(r) is the pair correlation function averaged over the center of mass and the angular direction of the relative coordinate. That is, sW = ^ / l / ^ ^ (riWr2)>’ (3-n) where R = (ri + r2)/2 is the center of mass, r = (r, 6) = ri — r 2 is the relative coordinate, and A is the total area covered by the electrons and the positive background. 53 We have carried out Monte Carlo simulations to study the energetics of quasi hole crystals. The results of our studies will be presented in this section and technical details of Monte Carlo calculations will be given in section 3.4. Among the three types of quasihole lattices considered (hexagonal, square, and honey comb lattices), the hexagonal lattice turned out to be the most stable. It was also verified that crystals t with larger I are energetically less favorable than those with / = 1. For the rest of this chapter, the crystals vk* j will be denoted as 'k*. Fig. 3.1 shows typical # (r)’s of $ 7 / 5 ’ ^ i/ 3 » an^ ^ 7 close to their parent fill ing factors. It is clear from the figure that the density-density correlation decays within a few magnetic lengths; g(r) asymptotically approaches 1 while, unlike for quantum Hall states [28], oscillates with the period of the lattice constant. It can be deduced from the figure that the amplitude of oscillation is smaller if the filling factor v of the crystal is closer to its parent filling factor vq. This is because the crystal at v closer to vq is more like its parent fluid state. These behaviors of g{r ) reflect the structure of the underlying quasihole lattice. The results for the energy of quasihole crystals calculated from pair correlation functions are summarized in Figs. 3.2-3.4 and Table 3.1. We have not done similar calculations for quasielec tron crystals because they need different Monte Carlo techniques and much more computing time due to the presence of differential operators in their wavefunc tions (Eq. (3.3)). However, based on general arguments and results for quasihole crystals, the energy of quasielectron crystals is also sketched schematically in the figures. The crystals $ 7 / 5 near vq = 1/5 are currently the most interesting because of the experimental observations of insulating phases at both sides of v = 1/5. Our results show that the crystals $7 / 5 are more stable than the Wigner crystal in the range —0.003 < v — 1/5 < 0 (Fig. 3.2). We therefore suggest that the crystal states that might be observed above and below v = 1/5 should rather be 54 LO CNJ 0 01 LO o in o o o 0.0 5.0 10.0 15.0 20.0 r Figure 3.1: Pair correlation functions of quasihole crystals. From the top, g(r) +1 of at v = 0.198, g(r) + 0.5 of at u = 0.3, and g(r) of at v = 0.5; r is in units of 4/0, 2/0, and Iq respectively. The data is taken from 500,000 Monte Carlo steps with 192-832 electrons. 55 CM 1=2 1=2 o / 1= 1 CM I I 0.1950.200 0.2050.190 v Figure 3.2: The energy per electron of quasiparticle crystals \f,*0, near vq = 1/5. Solid lines are quasihole crystals and dashed lines are quasielectron crystals. Ewc is the energy of the correlated Wigner crystal obtained from Eq. (2.41). Error bars show statistical errors. Only the energy of quasihole crystals with / = 1 is from Monte Carlo calculations. All the others are schematic drawings based on our conjectures. 56 O 1=2 o 1=2 o o CM II 0.20 0.25 0.30 0.35 0.40 Figure 3.3: The energy per electron of quasiparticle crystals l near uQ = 1/3. Notation is the same as in Fig. 3.2. 57 LO O LO I 1= 1 o LO I 0.45 0.50 0.55 Figure 3.4: The energy per electron of quasiparticle crystals j with Vq = 1 near u = 1/2. Notation is the same as in Fig. 3.2. 58 Table 3.1: Monte Carlo data for quasihole crystals ’J^/nr & *s the energy Per electron of the crystals and Ewc is the energy per electron of the Wigner crystal obtained from Eq. (2.41). S is the statistical error. N is the number of electrons used in the Monte Carlo calculation. m u E E - Ewc 6 N 1 0.48 -0.4349 0.0013 ± 0 . 0 0 1 1 64 1 0.49 -0.4387 -0.0008 ± 0 . 0 0 1 1 128 1 0.50 -0.4435 -0.0039 ±0.0009 128 1 0.52 -0.4518 -0.0094 ± 0 . 0 0 1 0 128 3 0 . 2 2 -0.3342 0.0055 ±0.0005 128 3 0.25 -0.3552 0.0007 ±0.0005 128 3 0.28 -0.3742 -0.0027 ±0.0005 192 3 0.30 -0.3857 -0.0048 ±0.0003 192 3 0.32 -0.3984 -0.0089 ±0.0003 256 5 0.195 -0.3217 0.0003 ±0.0004 512 5 0.198 -0.3243 -0.0003 ±0.0005 832 59 interpreted as quasiparticle crystals than Wigner crystals. The stability range of \& 7 / 3 against the Wigner crystal is rather wide: —0.08 < v — 1/3 < 0 (Fig. 3.3). However one cannot conclude that the ground state in this range of filling factors is the quasihole crystal because there are, among other possible states that may exist, stable quantum Hall states observed in this range [20]. The one closest to v0 = 1/3 observed so far is at v = 2/7 « 0.286. The crystals $ 7/3 beyond v = 2/7 are probably physically irrelevant because they are expected to have higher energy than those quasiparticle crystals derived from the quantum Hall state at vq — 2/7. So far, there is no experimental report of crystal phases in the range 2/7 < v < 1/3 even though such a possibility cannot be excluded as the sample quality is further improved in future experiments. It appears at this time that in most of the range 2/7 < v < 1/3 the ground state is some kind of fluid state other than the quantum Hall state. On the other hand, the crystals $ 7/3 must be more stable than any other states at filling factors sufficient close to Vq = 1/3 because it is now well established that their parent state is the ground state at v = 1/3. However, it is possible that these crystals are very hard to detect if they are stable only in a very narrow range of filling factors near VQ = 1/3. The crystals $7 are also interesting because these crystals are the particle-hole conjugates of the uncorrelated Wigner crystals (Eq. (2.37)) and both crystals have the same energy at v = 1/2 (see section 3.5). Since we believe that our Monte Carlo data is reliable, this particle-hole relation can be used to check the accuracy of the energy of Wigner crystals estimated by other methods. In the estimation of the energy Ewc (Eq. (2.41)), Lam and Girvin [52] used the correlated Wigner crystal (Eq. (2.40)), which is expected to have lower energy than the uncorrelated Wigner crystal. Their estimated Ewc is therefore also expected to be lower than the energy of $ 7 at v = 1/2. Fig. 3.4 indicates that their estimate is not accurate 60 near u = 1/2. We have also utilized particle-hole conjugation to check our Monte Carlo calculations by comparing the energy of ’IT with that of the Wigner crystal at filling factors where the latter is relatively well known. In summary, we have reached the following conclusions for the stability of quasiparticle crystals. The range of filling factors where quasiparticle crystals are more stable than the Wigner crystal is wider for higher vo . Since such a range for is already very narrow, crystal states that may be observed below v = 1/5 down to v = 0 are more likely to be Wigner crystals whereas quasiparticle crystals are expected to be observed at around or higher than v = 1/5. On the other hand, the stability of quasiparticle crystals at high filling factors, such as around 1/3, may be possibly limited by the existence of other more stable states. From our results together with the experimental observations, it appears that around v — 1/5 is the best place to look for quasiparticle crystals. 3.3 Angular Momentum and Induced Magnetic Field in C rystals As discussed earlier, the nature of the insulating phases experimentally observed at many filling factors below u = 1/3 is still not well known. In addition to finding out whether they are in fact crystals, it will be interesting to determine whether they are Wigner crystals or quasiparticle crystals. To distinguish between different crystal states, it is useful to identify some of their characteristic properties. As we shall see, quasiparticle crystals in general have a number of unusual properties. In contrast to fluid states, density variations in crystals will result in an array of circular currents and magnetic moments, which will be different for different crystals. If the induced magnetic field due to the circular currents can be detected in experiments, it will help to identify the nature of the crystal states. In this 61 section, we will discuss formulations and numerical results for these properties of quasihole crystals as well as the Wigner crystal. The mass current density g(r) of a 2DES in a uniform perpendicular magnetic field is, by definition, g(r ) = \ [V,t(r)TI(r)V,(r) + (n(r)*^t(r)) ^>(r)] , (3.12) where II(r) = — iKS7 + (e/c)A(r), A(r) = (B/2)(yx — xy) is the vector potential in the symmetric gauge for a uniform magnetic field B = —B z, and ip(r)/ip^(r) are the field annihilation/creation operators in two dimensions. It can be easily shown that, in complex notation, n n+ = Ux + illy — -7 (dz* + — ^ (3.13) n. = nx - lU, = 7 (& - y) ■ (3.14) Using these identities, 0+(*) Qx i9y 1 J (dg* + + | - y (d2. - 0 | rf>(z) 2 1 2 ( dz . + | j %!>(z) - j d z» (3.15) 2 If only the states in the lowest Landau level are considered, x(>(z) = Ylm where $ m(z) oc zme~lzl2 / 4 and cm is the annihilation operator that destroys the state $ m(z). Then the first term in the bracket vanishes because ( d z• 4 - zf 2) V’(z) = 0. Therefore, (3.16) Similar arguments lead to (3.17) 62 From Eqs. (3.16) and (3.17), the mass current density reduces to g(r ) = x Vp(r), (3.18) where p(r) = V,t(r)V’(r) is the density operator. Note that Eq. (3.18) is valid for both fluids and crystals as long as they lie entirely in the lowest Landau level. In fluid states, /3(r) is a constant and we recover the expected result #(r) = 0. On the other hand, in crystal states, density maxima or minima form a lattice and the density variation around them induces the lattice of circular mass currents. Once the density profile of the crystal state is found, the mass current can be calculated from Eq. (3.18). The angular momentum contained in a unit cell of the crystal state is Lceii = I r x (g(r)) dr J cell = 4 L dr{T-v)p{r) = & ./cellf „ dr tv ‘ M 1’)) ~ f,(r)(v • r)l = Zh [ dr (pb(r) - p(r)) (3.19) ./cell = Zh(AN), (3.20) where p(r) = {p(r)) and />&(r) is the electron density on the boundary of the unit cell. Since the second term of the integrand in Eq. (3.19) integrates to the number of charges in the unit cell, which is fixed by the lattice type, Lceii is determined by the electron density on the boundary. In the limit of large lattice constants (the dilute limit), pb(r) for the Wigner crystal is zero while that for quasiparticle crystals is equal to the density of the parent state. It therefore follows that e(AN) is the total charge of electrons and quasiparticles respectively inside a unit cell. It is interesting that Lceii for quasiparticle crystals reflects the underlying fractional charge of quasiparticles. 63 The circular current Eq. (3.18) in the xy plane induces a magnetic field in the z direction. Since we now consider three-dimensional space for the purpose of studying this induced magnetic field, it is necessary to change some of the notation. Let us define r = (rj.,:r3), V = itdx + ydy + Sc3dX3 = V x + x3dX3. With this new notation, the electric current density is J(r) = —(e/m*)g(r) = —(e^/2m *)x 3 x Vj_n(r). And the vector potential for the induced magnetic field due to J(r) is J(r') eh ,x 3 x V'xn(r') dr (3.21) 2 m*cJI " |r — r'| ’ where n(r) is the volume density of electrons. For a three-dimensional system with finite but small thickness, it is convenient to write n(r) = f(x3)p{ r±) = f(x3)J2pGe'Gr±, (3.22) where f(x3) is the density in the z direction that is localized around x3 = 0 , p(rx ) is the areal density in the xy plane, G is the reciprocal lattice vector, and pc is the Fourier coefficient of p(rx) defined as PG = - j — f drxp(rx)e ,Gr\ (3.23) A cell J -'cell cell where Aceii is the area of the unit cell. It is straightforward to show that irek A ind(r) = ix 3 x G pQ e'Gr± ha{x 3), (3.24) nrrc g? o where ho(x3) = f^°00dxl3f(x3)e Glr 3 x 3 l. The induced magnetic field obtained from V x A‘nd(r) can be written as G/t-rx B ind(r) = Y, B(G) ~ ( * 3 ~ * sgn(x 3 )Gjt) e* ha(x3), (3.25) G^O PG |G fc |= G 64 where pa is the number of G’s such that |G| = G and B(G) = ~ ~ G p g p g - (3.26) m e Note that it is used in this derivation that pq depends only on G = |G|. Since elec trons are highly localized near the xy plane, f(x3) « ^(a;3) and ha(x3 ) « e_G?lX3L Moreover, since ha(x3 ) decays rapidly for large G, to a good approximation, only the shortest reciprocal lattice vectors Go need to be kept in the summations. From h.G{x3), one can see that the induced magnetic field extends over a distance in the z direction on the order of the lattice constant of the crystal in the xy plane. Eqs. (3.19) and (3.26) show that the electron density profile of crystals is the key to our calculations. The density of quasihole crystals was evaluated using Monte Carlo simulations. For the Wigner crystal, we find that the density reduces to a simple form, which makes it possible to calculate it to high accuracy. The wavefunction of the Wigner crystal (Eq. (2.37)) can be written as (3.27) where P is the permutation of (1,2,..., iV) and \z — a | 2 za* — z*a 1 ' + (3.28) F^z) = (3.29) Ajj = (o.|aj). (3.30) The electron density of the Wigner crystal is p(z) = (f",l^ (|~^)l$wc>. (3.31) V (®wc I tfwc) 65 Consider the denominator of the density, (*WC I tfwc) = 4 j 5 I ( - l ) P+<3 A p 1,Q1 ••• A pNiQN TV1 1 • p ,Q Tn ' ' ' &Pn,PSN iV ■ P,S SN s = det(A), (3.32) where P, Q = PS, and S are permutations of (1,2,..., N). And the numerator of the density, $ w c J 2 6(z - Z i ) ^ w c) = ■ F p .,q ,(z ) • • ■ & p n ,Qn / iV - i P,Q = ^2 ^ (-if^ .P S , • • • FPl,PS%(z) ■ ■• A p n ,p s n • i P,S = !C]C(-1 )5Ai,Si ■ ■ ■ Fi,s,(z ) ■ ■ • & n ,sn i S = (&-')„ det(A), (3.33) where P, Q, and S are the same as before and (—l)'+JC,j is a cofactor of the matrix A. The third equation follows because for any permutation P the sum mation over i can be reduced to the same expression by relabling the first indices of Fi'j(z) and AThe summation over S in that equation isthe determinant of the matrix whose zth row is F ij(z )’s. The next equation is obtained when the determinant is expanded as the sum of products of the elements in the zth row and the corresponding cofactors. The identity (A-1)^,- = (—l),+JC,ij/ det(A) was used to obtain the last equation. From Eqs. (3.32) and (3.33), the electron density reduces to p(*) = 5Z F Together with Eqs. (3.29) and (3.30), the electron density of the Wigner crystal can be calculated to any desired accuracy. Note that both matrices F(z) and A involve the product of Gaussians centered at different lattice sites. Since the Gaussian decays rapidly away from the center, only a small number of lattice sites need to be included in the calculation of F(z) and A. The density of the hexagonal Wigner crystal is calculated from Eq. (3.34) with 20 to 100 lattice sites depending on the filling factor (see appendix A.l for the computer program). The angular momentum per unit cell LCeii is then obtained by numerically integrating Eq. (3.19). The angular momentum per unit area L/A = (v/2ir)Lcen as well as Lceii of the Wigner crystal are shown in Fig. 3.5. These essentially exact results for the Wigner crystal are utilized to check our Monte Carlo calculations for hexagonal quasihole crystals as follows. Since \IT is the particle-hole conjugate of the Wigner crystal, the angular momenta of can be accurately determined from relations Lceu(i>) = — Tcen(^) and ( LfA){y) = —(L/A)(u), where v = 1 — v (see section 3.5). Fig. 3.5 indicates that angular momenta of 'Pf calculated from Monte Carlo simulations agree well with the results obtained from particle-hole conjugation of the Wigner crystal. Note that if the ground state at v — 1/2 is nondegenerate, the particle-hole conjugate of the state should be mapped into itself and therefore Lcen = L/A = 0. Nonzero angular momenta at v — 1/2 are the consequence of degenerate ground states. The Monte Carlo results for ^7/3) ^i/s* anc^ ^ 1 / 7 are shown in Fig. 3.6. The data is taken from up to 500,000 Monte Carlo steps with 64 to 128 electrons. At filling factors close to i/0, the density of quasiholes in is very low. In that limit, L/A = hu0(u0 — v )/{2ki>o), which is basically the product of the charge of quasiholes and the density of quasiholes (see Eqs. (3.19) and (3.7)). In Fig. 3.6, the dotted lines emerging from parent filling factors vq = 1/3, 1/5, and 1/7 are L/A in the dilute limit. It is apparent from the figure that there is a large fluctuation 67 LD CNJ o o oo CM o o m o CN o o wc o o o o o 0.0 0.2 0.4 0.6 0.8 1.0 v Figure 3.5: The solid and dashed lines are L/A and Lceu of the Wigner crystal. The circles and squares are —L/A and —Lceu of quasihole crystals calculated from Monte Carlo simulations. For comparison, exact results for obtained by particle-hole conjugate of the Wigner crystal are also shown as dotted and dot-dashed lines. 68 LO o o o 1/7 1/S 1/3 o CM LO £ q CN o I o o I m o 1 0.0 0.1 0.2 0.3 0.4 V Figure 3.6: The angular momentum per unit area L/A for quasihole crystals ^7/3’ and ^7/7- The straight dotted lines emerging from vQ = 1/3, 1/5, and 1/7 are L/A in the dilute limit. 69 when L/A begins to deviate from the dilute limit. The fluctuation is larger for smaller parent filling factor. It was verified that the fluctuation does not originate from the statistical error. When quasiholes begin to overlap, the high density region surrounding quasiholes (Fig. 2.6) falls on the boundary of the unit cell. According to the definition of Lceii (Eq. (3.19)), high densities on the boundary result in small or even positive Z/ceii- Fig. 3.7 shows the induced magnetic fields B(G) (Eq. (3.26)) for the shortest reciprocal vector Go in quasihole crystals and Wigner crystals. For illustration, B = 20 T and m* = 0.067 me were chosen. To calculate B(G), the Fourier coeffi cient pa of the density p(r) was obtained by numerical integration of Eq. (3.23). Near v = 1/5, the induced field in the state ^ wc is on the order of one Gauss while that in is much smaller. The extension of the induced fields in the direction normal to the plane is on the order of Gq1 = 134 A and 45 A for and 'I'wc respectively at u = 0.195. It might be very difficult to detect these in duced magnetic fields because their magnitude is very small compared to the huge background magnetic field and they reside rather close to the plane. 3.4 Monte Carlo Methods In this section, some details of Monte Carlo calculations for the electron density p(r) and the pair correlation function g(r ) in quasihole crystals be discussed. These quantities are used to calculate the induced magnetic field B(G) and the energy E of the crystals respectively. While p(r) is straightforward to evaluate, one needs to follow a number of nontrivial steps to obtain < 7 (r). One urgent problem in numerical calculations is to extract results for the infinite system from the data for a finite sample. As mentioned before, Laughlin’s 1/m state with N electrons describes a fluid state of a uniform density po = crystals. For illustration, illustration, For crystals. Figure 3.7: The induced magnetic fields fields magnetic induced The 3.7: Figure B(Go) (Gauss) o o o CN o o CD o oo . 01 . 03 . 0.5 0.4 0.3 0.2 0.1 0.0 B = B 20 T and and 20 T wc 1/5 m B{G * = 0.067 0.067 * = q ) of quasihole crystals and Wigner Wigner and crystals quasihole of 1/3 me are chosen. are 70 71 l/(27rm) in a disk of radius sa ^JN|^Tpo. It is therefore expected that the system near the center of the disk looks pretty much the same as that in the infinite system. One can apply similar arguments to see that quasihole crystals also form a disk of radius « yjN/-Kp0 and the system near the center resembles that of the infinite system as long as the size of the disk is large enough. To find the density profile of the crystal, it is sufficient to evaluate p(r) inside one unit cell since it is periodic. We therefore need sample systems with only a small number of unit cells for the purpose of calculating p(r). However, much larger samples are necessary for g(r) because we need to evaluate g(r) up to more than a few magnetic length to estimate the energy of the infinite system. Any measurable physical quantity is the expectation value of the corresponding operator in quantum mechanics. If the operator is a simple function of coordinates, its expectation value in the state '5 is J¥r]/[r]|*[r]f /[ ^ (0 = (3.36) (The angular bracket will be understood, in this section, as the expectation value in the quasihole crystal state (3-2)).) The unit cell is divided into 72 rectangular bins and we evaluate p(r)Ar = (Nr), (3.37) where Ar is the area of the bin located at r and Nr is the number of electrons inside the bin. A typical contour plot of the density evaluated in this way is shown in Fig. 3.8. The plot is for 'I'i" at u = 1/2 obtained from 106 Monte Carlo steps with 20 electrons. It can be clearly seen in the plot that the density minima form a hexagonal lattice while the maxima form a honeycomb lattice. The angular momentum and the induced magnetic field are calculated from the density p(r) using Eq. (3.19) and Eq. (3.26) respectively. To compute g(r) (Eq. (3.11)), we divide the disk into circular bins of uniform width and we evaluate s W A„A, = ± / ^ / f (W, (3.38) where Ar is the area of the circular bin located at r and Nr is the number of electrons inside the bin. Note that circular bins already insure the average over the angular direction of the relative coordinate 6. To achieve the average over the center of mass R, measurements are taken from several sets of circular bins in each Monte Carlo step. The centers of the bin are randomly chosen within the unit cell near the center of the disk. This method is not a perfect way to achieve the desired average, but satisfactory enough for the purpose of our calculation. At low electron densities, many sets of bins are necessary to improve the statistical error because the bin at the center is very likely to be empty. The computer program is listed in appendix A.2 for reference. Typical g(r ) computed in this way are shown in Fig. 3.1. We confirmed that the average electron density up to near the disk boundary is the same as that of the infinite system, which is an indication that the sample system inside the disk 73 .0.7 >" X Figure 3.8: The contour plot of 2'Kp(v) for the quasihole crystal ’Pf at u = 1/2. The lattice constant is 3.8 Iq. The data is taken from 106 Monte Carlo steps with 20 electrons. 74 resembles the infinite system. g(r) is expected to asymptotically approach 1 while oscillating because of the lattice of quasiholes. The evaluation of g(r) is reliable only up to r where the edge effect sets in. However, the rest of the tail of g(r) can be extrapolated to high accuracy from the expected asymptotic behavior of g(r). Once g(r) for the whole range of r is determined, the energy per electron of the infinite system can be obtained by numerically integrating Eq. (3.10). But we found the following alternative procedure is more convenient. The energy E(r) is calculated by integrating Eq. (3.10) up to r and is plotted against r (Fig. 3.9). Due to the presence of the lattice, E(r) also oscillates and gradually settles down to the desired energy. Since the amplitude of oscillation is still appreciable near the disk boundary, a fitting function is used. The oscillating part of E(r) well away from the center and the disk boundary is fitted to the function f(r) = A e sin (D(r - C)) + E, (3.39) with five adjustable parameters A-E. Then E is the energy per electron extrap olated for the infinite system. As the final step, the statistical error is estimated from a number of inde pendent Monte Carlo calculations. The number of Monte Carlo steps in each calculation is typically 5000. It was chosen to optimize the total computing time. If it is too small, the statistical fluctuation between runs is too large. On the other hand, if it is too large, the cost increase of each run outruns the improve ment of the statistical error. The average is taken from 100 independent runs. The statistical error is typically less than 0.1 %. 3.5 Particle-Hole Symmetry In this section, we shall discuss the so-called particle-hole symmetry of quantum Hall systems, which enables one to relate the energies and densities of states at h lt fteftigfnto E. (3.39). Eq. at function fitting the of plot the is crystal line quasihole the of dotted electron per energy The 3.9: Figure 0.45 -0.40 -0.35 0.0 E(r) band y nertn E. 31) p o . h sld ie is line solid The r. to up (3.10) Eq. integrating by obtained 10.0 20.0 r 30.0 40.0 v = 0.3. The The 0.3. = 75 76 filling factors v and 1 — v. It will also be shown from energy considerations that if a state at v is stable, its particle-hole conjugate state at the filling factor 1 — v is also stable. It is necessary to study quantum Hall systems only in the range of filling factors 0 < v < 1/2 since those in the other half can be readily understood using the particle-hole symmetry. As we discussed in section 2.3, the particle-hole symmetry has also played an important role in the construction of hierarchical schemes of the FQHE. 3.5.1 Characteristics of Particle-Hole Conjugate States The particle-hole symmetry has been considered by Girvin [34] and has been used by Maki and Zotos [51] in their study of the Wigner crystal. Here, we give a more general discussion that applies to arbitrary states in quantum Hall systems. Any many-body state in the lowest Landau level can be written as lx) = £ am1-mLclL ■•■c\ni |0), (3.40) mi where |0) is the vacuum, L is the number of electrons, and m i,... ,m,L are non negative integers. Note that is the creation operator that creates the single electron state in the lowest Landau level described by the wavefunction <&m(z) = (7rm! 2m+1)-1/2zm exp( — |z|2/4). Its counterpart is the annihilation operator Cm that destroys the state $m. The particle-hole conjugate state of |x) is defined as lx) = Y , aml...mLcm1---CmL\l) , (3.41) where |1) is the fully filled Landau level. In the state |x), holes in the fully filled Landau level are in the state given by |x). This definition coincides with Girvin’s that was introduced earlier. In the lowest Landau level, field operators are defined as V>(z) — Ylm$m{z) cm and tf>\z) = 4>m(z) cln> which satisfy commutation 77 relations z — z zz'* — z*z' y>(z),^(z'j\+ = exp + A(z,z') (3.42) [^(z),^(z,)]+ = [^(z),^{z')\+ = 0. (3.43) It is straightforward to show that 0 |C«z, ' ' ’ 4 i Cmi * ' ' CrnL | ^ |CmL ‘ " ' Cml Cnj ‘ ‘ ‘ c nx | (3*44) (x|4i •••41Cmi •••CmL|x) = (x|cmL--,Cmi4r'-4x.|x)» (3-45) where n.-’s and m,-’s are non-negative integers. Then it readily follows that (x \ ^ { h ) • • • • • • *l>{aL)| x ) = (x |^(«l) • • • (3.46) for any complex coordinates a.’s and 6;’s. The electron density in the state |x) is therefore p(z) = (x\^(z)*l>(z)\x) (3.47) = (x|V,(^)V,t(^)|x) = 7^ - (x|^fW («)|x) (3.48) (3.49) = ^ - p{z)’ using Eqs. (3.42) and (3.46). The energy of the state |x) can be also derived in a similar way. The pair correlation function g(z,z') of the state |x) is given by pl = Pi 9(z, z ') + 0 “^) ~ J ~P^ ~ A^’ ^ |^t(2r')^(2r)| x) + A{z, z') (x |V’t(2)V’(2')| x) i (3.50) 78 where po and po are the average electron densities of the states |x) and |x) respec tively. Note that the second equality follows after applying Eqs. (3.42), (3.43), and (3.46) repeatedly. Then the total energy of the state |x) (Eq. (2.20)) is - 1 _2 f cPzcPz' , e2 E m = 2h J V ^ \ Cg{Z'Z)- = E,„ + ^ J (-A (z ', z) (X |0’(2')lH*)| x) + A (z,2 /)(x |V ’t(^)V’(^ )|x ) ) (3-51) To evaluate the integral in this equation, we need to calculate the density matrix (x |^,t(-2r)V’(-2r/)| x) different points z and z'. To do that, we note that the density of |x) is given by . . . . _*m_n„-z*2/2 < x |^ ( * W W |x ) = g W m , „ |2 .+ . ( * M x ) (3-52) = (3-53) 9 where pq is the Fourier component of the density p(z). If z and z* are considered as two independent variables, Eq. (3.52) is analytic in both z and z*. Therefore, the analytic continuation of Eqs. (3.52) and (3.53) by replacing z by z' leads to = Z > « « * (,V+"* )- (3-54) m,„ 27rVm! nl 2m+n x 1 1 ' q Then the density matrix is *m /n < x |0 W M |x ) = e-(|4i+l»'l,)/- g 9TVm!n,2m+n (x|4,<=.|x) (3-55) m^n 27r\/m ! n\ 2m+n = e-(M2+bT)/4+***'/2 yX - „ > „*(*•*'+,*•) ,e i< ’v + ’z'> (3-56) 7 and A(*, z‘) ( X \*'{z)i>{z')\X) = (3.57) 79 It can be easily shown that after integrating the term that includes Eq. (3.57) over z and z' in Eq. (3.51), all Fourier components except pq=o vanish. Therefore, under the integral, A(z, z') (x \^)i> W )\x) = (3.58) Similarly, A(z', z) (x \^(z'Wz)\X) = (3.59) Using Eqs. (3.58) and (3.59), Eq. (3.51) gives EtotW _ E t M = __L / |£ ! (p _ *,), (3.60) A A 4tt V 2 /0 V ' V ' where u and v — 1 — v are the filling factor of the states |x) and |x) respectively. Then the equation for the energy per electron E = (Etot/A)pQ can be written as vE{u)-vE{v) = E{l){v-v), (3.61) where E( 1) = — y/,7r/8(e2//o) is the energy per electron at v = 1. If a given state at v is stable, Eq. (3.61) guarantees that its particle-hole conjugate state is also stable at 1 — u. If it is not, there is another state at 1 — v with lower energy whose particle-hole conjugate state, in turn, has lower energy than the given state at u. This is not possible because the given state is the stable state at v. Finally, consider the angular momentum in |x). From Eq. (3.19), the angular momentum per unit cell is Tceii(^) = zh f dr (pb(r) - p{r)) (3.62) Jcell = zh f dr (-pb(r) + p(r)) (3.63) Jcell = -L cen(u). (3.64) Since the angular momentum per area L/A = TceiiMceii, where z4ceu is the area of the unit cell, and particle-hole conjugation doesn’t change the unit cell, (!)<’) = - (l) (3 -65) 80 3.5.2 Quasihole Crystal vs. Wigner Crystal We now discuss particle-hole conjugation of the Wigner crystal. Girvin [34] showed that the particle-hole conjugate state of the uncorrelated Wigner crystal \PWC (Eq. (2.37)) is given by ^[z} = e - ^ ^ /4H(zi-a a) Uizj-Zk), (3.66) t,a j tf[z] = {Zl ■ ■ ■ zNai ■ ■ ■ aL\l) je |oq|2/4 f j (aa - ap) (3.67) \ a<0 The particle-hole conjugate state of V& can be written as (Eq. (2.30)) = j Y [ dzi{^\zi - " zNa\---aL) {sx---sLzl---zN\l) X oc (l |V,t(aL)---V,t(aiM si)---V,(sL)|l) (s ja i) ••• (si\aL) oc (3.68) (slIax )(slIol ) where (s|a) = (l l) 1 s — a sa sa exp — _J_ + — (3.69) 27T 81 which is a Gaussian centered at a. Note that 'Pfs] is the properly symmetrized wavefunction of a Wigner crystal which has peaks at cti,...,a£. Therefore, the particle-hole conjugate of 'I'j" at the filling factor v is the Wigner crystal at 1 — v. This relation has been used to test our Monte Carlo programs throughout the study of quasihole crystals. An interesting consequence of this relation between ^ wc and 'kj' is that the ground state of the 2DES at v = 1/2 is degenerate. This is because ^ wc and have the same energy at v = 1/2 (Eq. (3.61)) but they are different crystals, which can be seen from the following observation, ^wc is the hexagonal lattice of electrons. On the other hand, is the hexagonal lattice of holes, which is nothing but the honeycomb lattice of electrons. A crystal cannot be a hexagonal lattice and a honeycomb lattice at the same time. 3.6 Quasiparticle Crystals in Hierarchical States We introduce, in this section, quasiparticle crystals derived from hierarchical states. The wavefunction of these crystals is explicitly constructed. To do that, the wavefunction of hierarchical states should be determined first. Basically, those wavefunctions implied in the hierarchical scheme proposed by MacDonald et ai. [35,36] (see section 2.2.3) are used for our construction of quasiparticle crys tals. However, we will build those wavefunctions based on symmetry arguments which have been discussed earlier in this thesis. In this construction, a hierarchical state is understood as a multi-component plasma of quasiparticles. Because wave functions of hierarchical states and quasiparticle crystals derived from them are extremely complex, we have not carried out any explicit calculations. However, based on general arguments, their internal structure will be illustrated. Read [40] proposed similar wavefunctions for hierarchical states, but they have some unde 82 sirable properties such as singularities occurring when two quasiparticles overlap. Nevertheless, his analysis of the structure of hierarchical states produces results similar to those discussed in this section. Before we proceed, a number of notational conventions are introduced, which will be used for the rest of this section. Any capital letter is defined as the vector of complex coordinates, e. g. Z = (zx,..., 2jv*), where Nz is the number of z’s. Also define the complex conjugate as Z* = (z \,..., z*Nz), the norm as \Zj2 = \zi\2, the differential operator as dz = (dZl,..., dZNz), and the integral as Jz — n, ^ - The binary operation ” between two vectors is defined as Z — Z = E M * . — zj) f°r fwo identical vectors and Z — W = fli.y(^t — wj) f°r two distinct ones. Between a vector Z and a complex coordinate a, the operation ” is defined as Z — a = rii^ i ~ a)- In fhe following discussions, a few equations will be written out explicitly without this simplified notation as a reminder. As discussed earlier, the state ^ that is the particle-hole conjugate of the state $ can be written as V(Z) = f V*{W)VX{Z,W), (3.70) Jw where \&i is the fully filled Landau level. Since both $ and ^ are in the low est Landau level, one may write 'f’(Z) = e~^2fA{Z — Z)S(Z) and 'I'(VE) = e-|W|2/4^W/ — W )ni+1,S(W), where S and S are symmetric polynomials and nx is a positive even integer (see Eq. (2.10)). Then Eq. (3.70) reduces to e-\z\2!\Z - Z)S(Z) = f e -|z|2/V |MT/2£*(VE)(Vr - W m)n'\W - W\2 Jw x(Z — W)(Z — Z). (3.71) This equation implies that any symmetric polynomial S(Z) can be expanded in the manner S{Z) = f e-'w?/2S’{W)(W* - W*)n'\W - W \ 2(Z -W ) (3.72) Jw 83 = I J J dwi e lw,'l2/2S*(W) - wk)n1 n Iwi “ w*\2 I i S(Z) = f e~m2/2S(W)(W - W)n' \W - W\2(dz - W*) (3.74) Jw — f ~ w k)ni n \w i ~ ^ l 2 » j V(Z) = e~]zl2/4S(Z)(Z - Z)m. (3.76) Any hierarchical quantum Hall state at a rational filling factor with an odd denom inator can be constructed from Eq. (3.76) by expanding symmetric polynomials recursively using Eqs. (3.72) and (3.74). Before we get to a general hierarchical state, a few examples of the states in the low levels of the hierarchy will be given in order to illustrate mechanisms of the construction. The states in the zero-th level of the hierarchy, which are Laughlin’s 1/m states, are obtained from Eq. (3.76) with S(Z) = 1. If S(Z) is expanded using Eqs. (3.72) or (3.74) with 5(VE) = 1, one obtains the states in the first level. These states result from the condensation of quasiholes and quasiparticles, respectively, that are created in Laughlin’s 1/m states. As an example, we consider the former 84 case. The state is given by the wavefunction, 9(Z) = e- ^ |2/4 f e~\w? 12 S* {W){W - W ) n'\W -W \2(Z -W )(Z - Z)m (3.77) Jw with ^(W ) = 1. The filling factor of this state is found as follows. The con figuration of electrons that maximizes the probability density |\H|2 is obtained from (3.78) where <^>’s are such that ty(Z) = fw (f>(Z,W). We postulate that the most signifi cant contribution of <^>’s to the integral in Eq. (3.78) occurs when the quasiparticle coordinates W and U coincide. Under such an assumption, Eq. (3.78) reduces to E ^ 7 + £ ^ - 4 e < = °- (3.79) Z* Z3 j Zl W3 * i This is equivalent to minimizing the potential energy of a classical two-dimensional two-component plasma, rjo _ 4>(Z) = —2m2 ^2In \zi - Zj\ - 2m]TUn \zt - wj \ + — ^ \zi\2. (3.80) » mvz -f iv, = 1, (3.81) where uz and vw are numbers of z’s and w’’s per flux quantum respectively. They are basically 2tt times the density of corresponding particles. The filling factor we 85 wish to find is vz. Similarly, £|\&|2/6t«i = ^|1®,|2/5tw* = 0 gives 1 n i -L = 0 , (3.82) j wi ~ zi & w< ~ W3 which leads to vz — n\vw = 0. (3.83) It is useful to write Eqs. (3.81) and (3.83) in a matrix form \ m 1 \ 1 (3.84) 1 — Til vw J 0 From this equation, the filling factor is found to be 1 (3.85) 1 • m — -n i To understand how the expansion in Eq. (3.74) works, we consider the state in the second level of the hierarchy, V(Z) = e~|z|2/4 J e- |vv|2/2e -|c;|2/2(t/* - t /*)”2\U - U\2 x (d w - U){W - W*)ni IW - W\2(Z - W){Z - Z)m. (3.86) This state is obtained from Eq. (3.77) with 5,(kF) expanded using Eq. (3.74) and the remaining symmetric polynomial set to 1. To work with this state, the differential operator dw in the integrand should be properly handled. It can be easily shown that / e~m2/2dwP[W] = [ e"lw?l2WmP[W] (3.87) Jw Jw for any polynomial P[W]. Therefore, for any wavefunction we are discussing in this section, whenever the differential operator of the form dw is under the integral, 86 it can be replaced by W*. As before, 8\^l^/8zi = 6|\I,|2/6itft- = 8\^!\2/8w* = 6 |^ |2/<5ut- =.<5|^|2/<5u* = 0 give m 1 1 = 0 (3.88) E + E' Zi 2, — Wj - t 4 E < J t 1 nx 1 = 0 (3.89) Ej Wi - zj j# w' ~ W3 + E j w' - ui V 1 _ _ n2 = 0. (3.90) j u« " wi u« ~ ui By interpreting these equations in terms of “the force” again, one finds that the matrix equation for the densities of particles is / n N ( \ ( i \ m 1 0 Vz 1 1 —rii 1 vw = 0 (3.91) ^ 0 1 —n2 } K Uu ) 1 0 ! where i/’s are interpreted as before, and the filling factor of the state is 1 (3.92) 1 m — T -n i -n2 Now we are ready to write down the wavefunction of any hierarchical state. For convenience, let us define the quasihole operator Q+{W\, W2) = W\ — W2 and the quasielectron operator Q-(W\,W2) = dwx — W£. Also define (IF,- — JF,)+ = IF* — W* for quasiholes and (IF,- — VF,-)_ = IF,- — Wi for quasielectrons. With these definitions, the wavefunction of a general hierarchical state can be written as V(Z) = e~|2r|2/4n f e-|iy,|2/2|IF,- - IF,-|2 • • • (W2 - W2)n^Q ±(Wu W2) i JWi x(W'i-W,,)2;,Q±(Z,lVi)(Z-Zr, (3.93) where m is a positive odd integer and n,-’s are positive even integers. From the experience with previous examples, the interpretation of the state f as a multi- 87 component plasma is intuitively simple; starting from Laughlin’s 1/m state, quasi particles created by Q± in each level of the hierarchy condense themselves to Laughlin’s 1 /(n,- + 1) state. It is straightforward to find / m ±1 \ ( i \ f * 1 ±1 jpni ±1 Vun 0 (3.94) ±1 =F«2 Vw2 0 V / { 1 } \ : / where uz is the filling factor (the number of electrons per flux quantum) and uWi is the number of quasiparticles per flux quantum in the zth level of the hierarchy. The off-diagonal element +1/ — 1 is for adding quasiholes/quasielectrons in the corresponding parent state. The sign attached to n, is opposite to that of the off-diagonal element located to the left or above. Then the filling factor of the state is 1 . (3.95) 1 m T T n i — T =F”2 “ It is necessary to understand quasiparticle excitations in hierarchical states to construct quasiparticle crystals derived from those states. We first consider the quasihole excitation located at z0 in the state given by Eq. (3.77). Following Read’s proposal [40], we insert the quasihole in every level of the hierarchy. The resulting wavefunction is $ +2°(Z) = e ~ W 4 jw e~mil2{W - zQ)h{W* -W *)ni\W - W \ 2 x(Z -W )(Z - z 0),0(Z - Z)m, (3.96) where Iq and li are non-negative integers. To determine the charge of the quasihole excitation, we consider 8\ty+z°\2/8z{ = 0, which gives m _ 1 E + £ (3.97) &>z>~ Zi j Z i-W j 88 This equation can be interpreted as before except that the third term implies the presence of a phantom charge lo at zq. In a stable configuration, the phantom charge will be screened by the plasma of charges m and 1. Since the sum of charges around z0 is zero, one obtains mqz + qw + l0 = 0, (3.98) where qz and qw are the numbers of charges m and 1 respectively accumulated around z0. They can be considered as the charges of the quasihole excitations in the zero-th and first level of the hierarchy respectively. Note that only qz is for the real physical charge. From 6|'I'+i:o \2/8wi = 6|^ +2°\2 j Sw* = 0, 1 n j h £ - £ + 0, (3.99) j wi - zi jfr W' - wi wi - z0 which leads to qz - nxqw + /i = 0. (3.100) Eqs. (3.98) and (3.100) give the matrix equation for the charge of the quasihole excitation, \ ( 1 ' l o ' (3.101) -ni \ Qv \ ll ) The results can be readily generalized. The quasiparticle excitation in the general hierarchical state (Eq. (3.93)) is given by the wavefunction tf±20(Z) = e->z '2/4n / e-'w'f/2\Wi -W i\2---Q±(W2,z0)h(W2-W 2)¥ , JWi xQ±(Wu W2)Q±(Wu z0)h(W1 - xQ±(Z,Wi)Q±(Z,z0)lo(Z - Z)m, (3.102) where s are non-negative integers and all other symbols are the same as before. 89 The equation for the charge of quasiparticle excitations can be found to be / . , \ ( \ m ±1 Qz ( I‘0 \ ± 1 ±1 Qw i h = q: (3.103) ±1 =F n2 Qw 2 h \ V : ) \ ) where —/+ on the right hand side of the equation corresponds to the quasi hole/quasielectron excitation. Note again that the physical charge of quasipar ticle excitations is — eqz. Eq. (3.103) implies that there is more than one quasi hole/quasielectron excitation in a given hierarchical state. Each excitation is determined by the set of /’s. However, the excitation with the lowest energy is expected to be the one with the smallest charge qz. Finally, quasiparticle crystals derived from a hierarchical state can be obtained by adding a lattice of quasiparticles to the state. The wavefunction of these crystals is, in the general form, ^ ( Z ) = e~|zl2/4n f e-m2/2\W i-W i\2---Q±(W2,A)l*(W2- W 2)l2 x Q i W , W2)Q±(Wu Aj'(W x - WOi* xQ±(Z, W,)Q±(Z, A)'°(Z - Z)m, (3.104) where A is the vector whose elements are the lattice sites. The equation for the density of particles can be written as / \ / \ m q:l ' l ' ( *0 1 ^ ^1 ±ni ^1 0 /l T I'a (3.105) Tl ± n 2 0 h \ / \ / \ : / \ : / where ua is the number of quasiparticles per flux quantum and —/+ in front of va is for the quasihole/quasielectron lattice. In summary, one can find the wavefunction 90 of quasiparticle crystals from the following steps: (1) determine the wavefunction of its parent state using Eqs. (3.93) and (3.95). (2) choose the set of Vs that gives the smallest charge of quasiparticle excitations using Eq. (3.103). (3) add the lattice of quasiparticles on top of the parent state as in Eq. (3.104). Some basic properties of these crystal states such as the filling factor, the quasiparticle density, and the number of electrons in a unit cell can be determined from Eq. (3.105) in the same way as we did in section 3.1. By the same arguments as before, these crystals are stable at filling factors sufficiently close to their parent filling factors as far as their parent quantum Hall states are stable. However, it is extremely difficult to determine the range of stability against other known states because of the complexity of their wavefunctions. We wish to conclude this section by presenting a specific example. The quan tum Hall state at = 2/7 is obtained from Eq. (3.93) with m = 3 and n\ = 2, that is, by adding a certain set of quasiholes to Laughlin’s 1/3 state. To find the charge of quasiparticle excitations, consider 1 \ / Qz N - =F (3.106) -2 \ Qw ) which solves to { \ Qz 2fo + h (3.107) \ Qw j = T 7 ^ lo — 3/i j The charge qz will be a minimum with the choice of lo = 0 and /j = 1, which gives Qz = Tl/7. The wavefunction of the quasielectron crystal, for instance, derived from the parent quantum Hall state at uQ = 2/7 can be written as V-(Z) = e"|z|2/4 / e-'w?/2(dw - A*)(VT - W*)2\W - W\2(Z - W)(Z - Zf. Jw (3.108) C H A PTER IV Conclusions The crystalline phase of electron systems has been sought in experiments ever since Wigner [1] predicted that electrons crystallize at low densities. In order to observe crystal states in the quantum regime, the electron density should be high enough for the Fermi level to be well above experimentally accessible temperatures but still low enough for the potential energy to dominate the kinetic energy. Since such environments became available for the 2DES in GaAs-AlGaAs heterojunctions, there have been experimental observations of highly insulating phases at many filling factors below v — 1/3. Without clear understanding of the nature of these phases, it was attempted to interpret them in terms of the Wigner crystal. However, if these phases turn out to be indeed crystals, the phase diagram of the 2DES implied by the experimental observations is much different from what has been generally believed based on the picture of the Wigner crystal. Our studies of crystal states in 2DES show that quasiparticle crystals, rather than the Wigner crystal, might be responsible for experimentally-claimed crystal states above and below v = 1/5 as well as at many other filling factors. It is quite clear from the consideration of the energetics that quasiparticle crystals derived from a parent quantum Hall state are more stable than Wigner crystals at filling factors sufficiently close to the parent filling factor. In this conjecture, every stable 91 92 quantum Hall state is surrounded by pairs of quasihole and quasielectron crystals on either side of its filling factor. This picture of quasiparticle crystals agrees with experimental reports that the insulating phases exist below v = 1/3 down to very low filling factors excluding v — 1/5 and 1/7 where the FQHE is observed. One of the interesting properties of quasiparticle crystals is that the number of electrons per unit cell can be any real number in contrast to the Wigner crystal, which is an indication of the quantum nature of these crystals. For numerical studies of quasihole crystals, we have developed Monte Carlo techniques to calculate the pair correlation function of crystal states, which makes it possible to extrapolate the energy of the infinite system. The range of filling factors where quasihole crystals derived from Laughlin’s 1/m states are more stable than the Wigner crystal was determined by the Monte Carlo calculations. It was found that this range is wider for higher parent filling factor. The Wigner crystal is therefore expected to be found only at low filling factors while crystals at higher filling factors such as those near 1/5 and above are more likely to be quasiparticle crystals. From the general analysis of the 2DES in the lowest Landau level, we have shown that the structure of crystal states as well as quantum Hall states can be understood by studying their symmetry properties. The wavefunctions of Laugh lin’s 1/m states and quasiparticle crystals from them were derived largely based on their symmetry properties. We have also constructed the wavefunctions of the general hierarchical state at v ^ 1/m and the corresponding quasiparticle crystals. Although there are generally-agreed structures of wavefunctions for hierarchical states in the literature, there are differences between different hierarchical schemes and the wavefunctions are usually not given explicitly. We felt that it will be use ful for future studies of the FQHE to give those wavefunctions explicitly in terms of electron coordinates. Both hierarchical states and quasiparticle crystals de 93 rived from them have the structure which is simple conceptually but very difficult to study numerically. However, some basic properties of these crystals could be deduced from the general analysis of their structure. We also showed that particle-hole conjugation for the 2DES in the lowest Landau level implies a number of useful relations between basic properties of the particle-hole conjugate states such as electron densities and energies. And it was demonstrated that the particle-hole conjugate of the Wigner crystal is the quasihole crystal derived from the fully filled Landau level. Since there is more than one possible crystal state in the 2DES, it is useful to identify their characteristic properties. Measuring basic properties of crystals such as lattice constants will be the most direct method to identify the crystal. Such measurements are, however, very impracticable for the 2DES. We found interesting properties of crystals in the 2DES, that is, the density variation in crystal states results in an array of circular currents and induced magnetic fields. These properties are different for different crystals. To estimate the magnitude of these magnetic fields, the density profile in quasihole crystals was evaluated by Monte Carlo calculations. For Wigner crystals, we were able to obtain essentially exact results from the simplified expression that we derived for the density. For typical experimental environments, it was found that the induced magnetic field in Wigner crystals near v = 1/5 is on the order of one Gauss while that of quasihole crystals at the same filling factors is much smaller. We suggest that measurements of these magnetic fields might serve as criteria in experiments to distinguish be tween different crystals. However, even though their magnitude is appreciable, the detection of them may not be easy because of the huge background magnetic field. It is interesting to note that quasiparticle crystals are closely related to the hierarchical construction of the FQHE; quasiparticles created on top of the parent 94 quantum Hall state tend to crystallize at low densities while they condense to a new quantum Hall state at certain higher densities. It will be therefore useful to have better understanding of hierarchical schemes of the FQHE for the purpose of studying quasiparticle crystals as well as other properties of the 2DES. The hierarchical schemes are based on the assumption that the interaction between quasiparticles is very much like the Coulomb interaction. Although this assump tion is intuitively plausible, the nature of the interaction still needs more investi gation. It will be useful also to verify the structure of hierarchical states. While the construction of hierarchical states is conceptually appealing, these states have hardly been tested because of their complex structure. Some numerical studies for the energetics of these states, for instance, will help us to understand them better. On the experimental side, the most urgent question that needs to be answered at this time is about the nature of the observed insulating phases. It is necessary for further studies of crystal states in the 2DES to confirm or disprove the claim among experimentalists that these phases are crystals. If confirmed, it will be useful to determine whether they are Wigner crystals or quasiparticle crystals. We hope our study of the induced magnetic field in crystal states will be useful for this purpose. A PPE N D IX A Lists of Computer Programs A .l Program for the Density of Wigner Crystals The program to evaluate the density profile of Wigner crystals given by Eq. (2.37) is listed below. Programs for the angular momenta and the induced magnetic fields are very similar to this program. The angular momentum (Eq. (3.19)) is obtained by numerically integrating the density along the boundary of the unit cell. For the induced magnetic field (Eq. (3.23)), we need Fourier components of the density (Eq. (3.23)), which can be calculated by integrating the density inside a unit cell multiplied by some factor. c c Evaluate the density profile of Wigner Crystals. c c c Parameters to be fixed before compilation c - maxe : maximum no of electrons c - maxr : maximum no of rings (electrons at the same distance c from the origin are grouped into rings) c c Input parameters from the file 'in' c - ff : filling factor c - nr : no of rings 95 96 c - ni : no of meshes c - ib : density is evaluated in the area -ib*aa COMMON /WORKSP/ RWKSP REAL RWKSP(70524) CALL IWKIN(70524) open(10 ,f ile= ’ in' ,status='old') read(10,input) close(lO) open(10,file='out',status='unknown') call lattice(ff,aa,ua,maxr,maxe,nr,ne,zc) write(10,*) 1 nr=',nr,' ne=',ne,' ni=',ni call invert(ne,zc,zm) ul=aa/ni do 150 j= -ib * n i, ib*ni 97 y=j*ul do 150 i=-ib*ni,ib*ni x=i*ul z=dcmplx(x,y) dd=dreal(zden(z,ne,zc,zm)) write(10,'(2f10.5,f15.10)*) x,y,dd 150 continue stop end c****************************************************************** c c Calculate the density at zz and return (2 pi * density). c c****************************************************************** function zden(zz,ne,zc,zm) implicit real*8(a-h,o-y).complex*16(z) dimension zc(ne),zm(ne,ne) zg(zl)=cdexp(-(cdabs(zz-zl)**2+dconjg(zz)*zl-zz*dconjg(zl)) * /4.0d0) zl=(0.0d0,0.0d0) z2=(0.0d0,0.OdO) do 100 j=l,ne zl=zl+dconjg(zg(zc(j)))*zg(zc(j))*zm(j,j) do 100 i=j+l,ne z2=z2+dconjg(zg(zc(i)))*zg(zc(j))*zm(j,i) 100 continue zden=zl+z2+dconjg(z2) return end C:tc$9|c3fc$3fc3|c3f(3ic)t(3|c:{e)|e£$$)4c)|c3tc)|c)|'9|e£3|c)te:|e)tcjfc)|c3tc9|e3|c3|c$)|c2ica|e3|c9jc3ic3|e)|c:|e3|c3fc]fc3tc$$)|c3|e3tc:|e3|e3|c3|e3|c$j|c)ie$)|c)|es|c$j|c C c Invert the matrix zm. c c******************************** ********************************** subroutine invert(ne,zc,zm) implicit real*8(a-h,o-y),complex*16(z) dimension zc(ne),zm(ne,ne) 98 do 100 j=l,ne do 100 i=l,ne zl=cdabs(zc(i)-zc(j))**2 z2=dconj g(zc(i))*zc(j)-zc(i)*dconj g(zc(j)) zm(i,j)=cdexp(-(zl+z2)/4.OdO) 100 continue call dlincg(ne,zm,ne,zm,ne) return end c****************************************************************** c c Fix lattice related parameters. c subroutine lattice(ff,aa,ua,maxr,maxe,nr,ne,zc) implicit real*8(a-h,o-y).complex*16(z) parameter (mr=21) dimension zc(maxe),ir(mr),il(mr),i2(mr) data ir/1,1,1,2,1, 1,2,1,2,2, 1,1,2,2,1,2,2,2,1,1, 2/ data il/1,1,2,2,3, 2,3,4,3,4, 5,3,4,5,6,4,5,6,4,7, 5/ data i2/0,1,0,1,0, 2,1,0,2,1, 0,3,2,1,0, 3,2,1,4,0, 3/ data pi/3.1415926535897932385d0/ aa=dsqrt(4.0d0*pi/(ff*dsqrt(3.OdO))) ua=2.0d0*pi/ff if((nr.gt.maxr).or.(nr.gt.mr)) stop 'error...lattice 1' xl=l.OdO y 1=0.OdO x2=0.5d0 y2=dsqrt(3.OdO)/2.OdO an=pi/3.OdO ne=l zc(l)=(0.0d0,0.0d0) do 100 k=l,nr do 100 j=0,5 do 100 i=l,ir(k) 99 if(i.eq.l) then xO=il(k)*xl+i2(k)*x2 yO=il(k)*yl+i2(k)*y2 else x0=i2(k)*xl+il(k)*x2 y0=i2(k)*yl+il(k)*y2 end if ne=ne+l if(ne.gt.maxe) stop ’error...lattice 2 ’ x=dcos ( j *an) *xO-dsin(j *an) *y0 y=dsin(j *an)*xO+dcos(j *an)*y0 zc(ne)=dcmplx(aa*x,aa*y) 100 continue return end A.2 Program for the Pair Correlation Function of Quasi hole Crystals The following is the Monte Carlo program to calculate the pair correlation function of quasihole crystals averaged over the center of mass and the angular direction of the relative coordinate (Eq. (3.11)). This program produces sets of raw data for the density and the correlation function after each of a fixed number of Monte Carlo steps. A number of other programs are used to analyze the data and cal culate the energy of the crystals. The program listed below is however the key program that takes most of the CPU time. c c Calculate the pair correlation function of quasihole crystals. c c c Features c - takes average over one or more sets c of non-fixed circular bins 100 c - uses circular bins of fixed width c - stores the raw data of each MC interval c Random number generator c - use ranset & ranf on Cray c - use srand & rand on DecStations c c Parameters to be fixed before compilation c - maxe maximum no of electrons c - maxh maximum no of holes c - maxcb maximum no of circular bins c - maxpb maximum no of parallel bins c Input parameters from the file ’i.en’ c - ffl,ff2: ffl/ff2 is the filling factor c - mm 1/mm is the parent filling factor c - 11 multiplicity of quasiholes c - ltype 1 (hexagonal), 2 (square), 3 (honeycomb) c - ne no of electrons c - dmax maximum moving distance of electrons to generate c a new configuration in each MC step c - eset probability that the central circular bin is c not empty (used to calculate 'nset') c - wide width of circular bins c - widp width of parallel bins c - mcs total no of MC steps c - mint no of MC steps in each interval c - mskip no of MC steps to be skipped initially c - mout no of MC steps between temporary outputs c - iseed seed of the random number generator c Calculated parameters c - ff filling factor c - eden density of electrons c - hden density of holes c - ecell no of electrons in a unit cell c - hcell no of holes in a unit cell c - ua area of a unit cell c - rdisk radius of the disk where electrons mostly stay c - rlatt radius of the disk that covers the hole lattice 101 c - aa : lattice constant c - ax(2),ay(2): primitive vectors c - nset : no of sets of non-fixed bins c - nhole : no of holes within rdisk c - nhole2: no of holes within rlatt c - xh(maxh),yh(maxh): hole coordinates c - numcb : no of circular bins c - numpb : no of parallel bins c - heip : height of parallel bins c c Output files c - 'o.o' : input and calculated parameters c - 'o.t' : Coulomb energies of electrons calculated after each c 'mout' MC steps, used to determine 'mint' and 'mskip' c - 'o.u' : raw data of each interval written after each 'mint' c MC steps c c Variables c - xx(maxe),yy(maxe) : electron coordinates c - cden(0:maxcb) : densities of circular bins c for one interval c - corr(0:maxcb) : correlations for one interval c - pden(-maxpb:maxpb) : densities of parallel bins c - xn(maxe),yn(maxe) : temporary storage for coordinates c - rn(maixe) : temporary storage for random numbers c - acc : acceptance ratio c (used to determine dmax) c parameter (maxe=1280,maxh=1000,maxcb=6000,maxpb=6000) implicit real*8(a-h,o-z) common ff1,ff2,ff,dmax,eset,wide,widp,heip, * eden,hden,ecell,hcell,ua,rdisk,rlatt, * aa,ax(2),ay(2),xh(maxh),yh(maxh), * m m ,11,Itype,ne ,nset,nhole,nhole2,numcb,numpb dimension xx(maxe),yy(maxe) dimension cden(0:maxcb),corr(0:maxcb),pden(-maxpb:maxpb) dimension xn(maxe) ,yn(maxe) ,m(maxe) data pi2/6.283185307179586477d0/ namelist /input/ffl,ff2,mm,ll,ltype, ne,dmax,eset, 102 * widc.widp, mcs,mint,mskip,mout, iseed namelist /output/f f,eden,hden,ecell,hcell,ua,rdisk,rlatt,aa, * nset,nhole,nhole2,numcb,numpb,heip,acc c c read input parameters c open(10,file='i.en',status='old’) read(10,input) close(lO) c c set up tables, initialize variables, etc. c call srand(iseed) call setup(xx,yy) do 50 ii=0,numcb cden(ii)=0.OdO corr(ii)=0.OdO 50 continue do 60 ii=-numpb,numpb pden(ii)=0.0d0 60 continue icnt=0 c c perform MC calculation c open(21,file=’o.t ’,status=’unknown *) open(22,file='o.u',status-'unknown',form='unformatted’) do 200 ii=l,mskip+mcs CDIR$ IVDEP do 100 kk=l,ne d=rand()*dmax a=rand()*pi2 xn(kk)=xx(kk)+d*dcos(a) yn(kk)=yy(kk)+d*dsin(a) rn(kk)=rand() 100 continue do 120 kk=l,ne 103 ww=trans(xx,yy,kk,xn(kk),yn(kk)) if(rn(kk).It.ww) then xx(kk)=xn(kk) yy(kk)=yn(kk) icnt=icnt+l end if 120 continue if(mod(ii,mout).eq.O) call printout(ii,xx,yy) if(ii.le.mskip) goto 200 call cumbin(xx,yy ,cden,corr,pden) if(mod(ii-mskip,mint).ne.O) goto 200 write(22) (cden(kk),kk=0,numcb) write(22) (corr(kk),kk=0,numcb) do 150 kk=0,numcb cden(kk)=0.OdO corr(kk)=0.OdO 150 continue 200 continue write(22) (pden(kk).kk^-numpb,numpb) close(21) close(22) c c write input and calculated parameters c acc=dfloat(icnt)/((mcs+mskip)*ne) open(20,file='o.o',status='unknown') write(20,input) write(20,output) close(20) stop end c c Calculate the transition probability when kk'th electron moves c from (xx(kk),yy(kk)) to (xn,yn). 104 c function trans(xx,yy,kk,xn,yn) parameter (maxe=1280,maxh=1000,maxcb=6000,raaxpb=6000) implicit real*8(a-h,o-z) common ff1,ff2,ff,dmax,eset,wide,widp,heip, * eden,hden,ecell,hcell,ua,rdisk,rlatt, * aa,ax(2) ,ay(2) ,xh(maxh) ,yh(maxh) , * mm,11,Itype,ne,nset,nhole,nhole2,numcb,numpb dimension xx(maxe),yy(maxe),ppl(maxe) ,pp2(maxe) trans=dexp(-0.5d0*(xn**2+yn**2-xx(kk)**2-yy(kk)**2)) do 100 ii=l,ne ppl(ii)=(xn-xx(ii))**2+(yn-yy(ii))**2 pp2(ii)=(xx(kk)-xx(ii))**2+(yy(kk)-yy(ii))**2 100 continue ppl(kk)=l.OdO pp2(kk)=l.OdO tt=l.OdO do 120 ii=l,ne tt=tt*ppl(ii)/pp2(ii) 120 continue trans=trans*(tt**mm) if(ll.ne.O) then tt=l.0d0 do 150 ii=l,nhole2 tt=tt*((xn-xh(ii))**2+(yn-yh(ii))**2) * / ((xx(kk)-xh(ii))**2+(yy(kk)-yh(ii))**2) 150 continue trans=trans*(tt**ll) endif return end £ £ + $ £ $ % $ $ * $ $ $ sfc $ * $ $ % £ * * $ $ % $ * * % * $ * * * * * £ $ * $ * afe $ * $ $ * % $ * $ * * * $ $ * jfe * £ $ * * $ jfc $ * C c Accumulate data on cden, corr and pden calculated c from electron coordinates xx and yy. c 105 C 4c * * lie * * * * * * 4c * * * * * * * * * * * * * 4c * * * * * * * * * * * * * * * * * 4c * * * * * * * * * * * * * * * * * * * 3|C * * subrout ine cumbin(xx,yy,cden,corr,pden) parameter (maxe=1280,maxh=1000,maxcb=6000,maxpb=6000) implicit real*8(a-h,o-z) common f f 1, f f 2, f f,dmax,eset,wide,widp,heip, * eden,hden,ecell,hcell,ua,rdisk,rlatt, * aa,ax(2),ay(2),xh(maxh),yh(maxh), * m m ,11,Itype,ne,nset,nhole,nhole2,numcb,numpb dimension xx(maxe),yy(maxe) dimension cden(0:maxcb),corr(0:maxcb),pden(-maxpb:maxpb) dimension den(0:maxcb),nbin(maxe) c c circular bins c do 200 jj=l,nset do 110 ii=0,numcb den(ii)=0.0d0 110 continue rl=rand()-0.5d0 r2=rand()-0.5d0 xc=rl*ax(l)+r2*ax(2) yc=rl*ay(l)+r2*ay(2) do 120 ii=l,ne nbin(ii)=dsqrt((xx(ii)-xc)**2+(yy(ii)-yc)**2)/widc 120 continue do 140 ii=l,ne if(nbin(ii).gt.numcb) goto 140 den(nbin(ii))=den(nbin(ii))+l.OdO 140 continue do 160 ii=0,numcb cden(ii)=cden(ii)+den(ii) 160 continue if(den(0).eq.O.OdO) goto 200 do 180 ii=l,numcb corr(ii)=corr(ii)+den(0)*den(ii) 180 continue 200 continue c c parallel bins c 106 do 300 ii=l,ne if(abs(yy(ii)).gt.heip) goto 300 ix=nint(xx(ii)/widp) if(abs(ix).gt.numpb) goto 300 pden(ix)=pden(ix)+l.0d0 300 continue return end C ***** * *********************************** **** * *** *** ****** ** * * * ** * c c Write Coulomb energy calculated from xx and yy. c c****************************************************** ************ subroutine printout(ii,xx,yy) parameter (maxe=1280,maxh=1000,maxcb=6000,maxpb=6000) implicit real*8(a-h,o-z) common ff1,ff2,ff,dmax,eset.wide,widp,heip, * eden,hden,ecell,hcell.ua,rdisk,rlatt, * aa,ax(2) ,ay(2) ,xh(maxh) ,yh(raaxh) , * mm,11,ltype,ne,nset,nhole,nhole2,numcb,numpb dimension xx(maxe),yy(maxe) ee=0.OdO do 100 kl=l,ne do 100 k2=kl+l,ne ee=ee+((xx(kl)-xx(k2))**2+(yy(kl)-yy(k2))**2)**(-0.5d0) 100 continue write(21,' (i9,el5.7)') ii,ee/ne return end C$)|e3|c3ic3te3tc)te3|c3tc;|e3|c$3fc9tc:|c3te$$$3fe3|c>|cj|ej|c)|e$)te$3|e:t:)|e)tc£$$$)tc>|c3|e$$)|e34e3te3|c9|c3|c4cs|c3tc3tej|e$3|c3|c3te$3tc)|c3{cji()ic3|e)|e$)|c C c Initialize tables and variables, c c****************************************************************** subroutine setup(xx,yy) parameter (maxe=1280,maxh=1000,maxcb=6000,maxpb=6000) implicit real*8(a-h,o-z) common ff1,ff2,ff,dmax,eset,wide,widp,heip, * eden,hden,ecell,hcell,ua,rdisk,rlatt, * aa,ax(2),ay(2),xh(maxh),yh(maxh), * mm,11,Itype,ne,nset,nhole,nhole2,numcb,numpb dimension xx(maxe),yy(maxe),hh(3) data hh/1.OdO,1.OdO,2.OdO/ data pi/3.1415926535897932385d0/ calculated parameters and tables if(ne.gt.maxe) stop 'setup 1' ff=ffl/ff2 eden=ff/(2.OdO*pi) hden=(l.OdO-mm*ff)/(2.OdO*pi*ll) zz=hden/eden hcell=hh(ltype) ecell=hcell/zz ua=hcell/hden rdisk=dsqrt(ne/(pi*eden)) numcb=l.2d0*rdisk/widc if(numcb.gt.maxcb) stop 'setup 2' numpb=l.2d0*rdisk/widp if(numpb.gt.maxpb) stop 'setup 3' nset=eset/(eden*pi*(widc**2)) hexagonal lattice if(ltype.eq.l) then aa=dsqrt(ua/dsin(pi/3.OdO)) rlatt=rdisk+2.OdO*aa heip=aa*dsin(pi/3.0d0)/6.OdO ax(l)=aa*dcos(pi/3.OdO) ay(l)=aa*dsin(pi/3.0d0) ax(2)=aa ay(2)=0.0d0 ii=3.OdO*rdisk/aa nhole=0 nhole2=0 108 do 320 i2=-ii,ii do 320 il=-ii,ii x=il*ax(l)+i2*ax(2) y=il*ay(l)+i2*ay(2) r=dsqrt(x**2+y**2) if(r.gt.rlatt) goto 320 nhole2=nhole2+l if(nhole2.gt.maxh) stop 'setup 4' xh(nhole2)=x yh(nhole2)=y if(r.le.rdisk) nhole=nhole+l 320 continue endif c square lattice if(ltype.eq.2) then aa=dsqrt(ua) rlatt=rdisk+2.0d0*aa heip=0.5d0*aa*dsin(pi/4.OdO) ax(l)=aa*dcos(pi/4.0d0) ay(l)=aa*dsin(pi/4.0d0) ax(2)=aa*dcos(-pi/4.OdO) ay(2)=aa*dsin(-pi/4.0d0) ii=3.0d0*rdisk/aa nhole=0 nhole2=0 do 220 i2=-ii,ii do 220 il=-ii,ii x=il*ax(l)+i2*ax(2) y=il*ay(l)+i2*ay(2) r=dsqrt(x**2+y**2) if(r.gt.rlatt) goto 220 nhole2=nhole2+l if(nhole2.gt.maxh) stop 'setup 5' xh(nhole2)=x yh(nhole2)=y if(r.le.rdisk) nhole=nhole+l 220 continue endif 109 c honeycomb lattice if(ltype.eq.3) then aa=dsqrt(ua/dsin(pi/3.OdO)) rlatt=rdisk+2.OdO*aa heip=0.5d0*aa*dsin(pi/6.OdO) ax(l)=aa*dcos (pi/6. OdO) ay(l)=aa*dsin(pi/6.OdO) ax(2)=aa*dcos(-pi/6.0d0) ay(2)=aa*dsin(-pi/6.OdO) ii=3.OdO*rdisk/aa nhole=0 nhole2=0 do 100 i2=-ii,ii do 100 il=-ii,ii x0=il*ax(l)+i2*ax(2) yO=il*ay(l)+i2*ay(2) do 100 kk=-l,1,2 x=xO+kk*(ax(1)+ax(2))/3.OdO y=y0+kk*(ay(1)+ay(2))/3.OdO r=dsqrt(x**2+y**2) if(r.gt.rlatt) goto 100 nhole2=nhole2+l if(nhole2.gt.maxh) stop 'setup 6' xh(nhole2)=x yh(nhole2)=y if(r.le.rdisk) nhole=nhole+l 100 continue endif c c electron coordinates c do 500 ii=l,ne r=rand()*rdisk a=rand 0* 2 . 0d0*pi xx(ii)=r*dcos(a) yy(ii)=r*dsin(a) 500 continue return end BIBLIOGRAPHY [1] E. Wigner, Phys. Rev. 46, 1002 (1934). [2] C. C. Grimes and G. Adams, Phys. Rev. Lett. 42, 795 (1979). [3] K. Huang, Statistical Mechanics, 2nd ed. (John Wiley &: Sons, New York, 1987). [4] The Quantum Hall Effect, edited by R. E. Prange and S. M. Girvin (Springer- Verlag, New York, 1987). [5] T. Chakraborty and P. Pietilainen, The Fractional Quantum Hall Effect (Springer-Verlag, New York, 1988). [6] K. v. 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