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TUNNELING BETWEEN TWO DIMENSIONAL ELECTRON SYSTEMS IN A HIGH MAGNETIC FIELD AND CRYSTALLINE PHASES OF A TWO DIMENSIONAL ELECTRON SYSTEM IN A MAGNETIC FIELD By FILIPPOS KLIRONOMOS A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2005 ACKNOWLEDGMENTS I would like to thank all of my friends and family who supported me through the difficult years of research and all of my colleagues and professors in the Department of Physics at the University of Florida who helped through the process as well. I would like to specially thank my supervisor, Alan Dorsey, for his mentorship and support and Mouneim Ettouhami for his contribution to this work and for showing me the way independent research is conducted. TABLE OF CONTENTS Page ACKNOWLEDGMENTS . . . . . . ii ABSTRACT . . . . . . . .. v CHAPTER 1 INTRODUCTION TO THE QUANTUM HALL SYSTEM . . 1 1.1 History of the Quantum Hall Effect ................... 1 1.2 Bilv, r Quantum Hall Physics: Experiment and Theory ......... 3 1.3 Crystalline Phases of the 2D Electron System: Experiment and Theory 9 2 TUNNELING CURRENT OF COUPLED BILAYER WIGNER CRYSTALS 12 2.1 Single L ,r Eigenm odes .......................... 12 2.2 Bil ,r Eigenm odes . . . . . . . .. 15 2.3 Coupling the Bili, r to External Electrons: Tunneling Current ..... 18 2.3.1 Analytic Solution . . . . . . 23 2.3.2 Numerical Solution . . . . . . 29 3 QUANTUM HALL SYSTEM IN THE HARTREEFOCK APPROXIMATION 32 3.1 Electron Dynamics in a Perpendicular Magnetic Field ......... 32 3.2 HartreeFock Approximation . . . . . . 37 4 ISOTROPIC CRYSTALLINE PHASES . . . 39 4.1 Stability Analysis of Isotropic Melectron Bubble Crystals . .... 39 4.1.1 Classical Order Parameter Approach ... . . 44 4.1.2 Microscopic Approach . . . . . 48 4.1.3 New State: Bubble Crystal with Basis . . ... 53 4.1.4 Normal Modes and Zero Point Energy . . ... 56 4.2 Energetics of Isotropic Crystalline Phases . . . ... 59 4.2.1 Cohesive Energy of Modified Coulomb Interaction: Classical M odel .... . . . . . . 61 4.2.2 Cohesive Energy of Modified Coulomb Interaction: Microscopic M odel . . . . . . . . 62 5 ANISOTROPIC CRYSTALLINE PHASES . . 5.1 Solving the Static HartreeFock Equation ..... 5.2 Introducing Anisotropy into the Crystalline States 5.3 Elastic Properties of Anisotropic Crystals . . 5.4 Analysis of Experimental Results . . . 6 CONCLUSIONS . . . . . . . . 66 . . . 66 . . . 69 . . . 77 . . . 80 . . . 83 APPENDIX A BILAYER SYSTEM EIGENMODES . ... A.1 Single L,ivr Eigenmodes . . . . A.2 Bil vr Eigenmodes . . . . . A.3 Tunneling Current . . . . . A.3.1 Correlation Function . . . A.3.2 Properties of the Correlation Function . B ISOTROPIC CRYSTALS . . . . B.1 B.2 B.3 B.4 Fock Term Calculation . . . . Microscopic Potential . . . . Bubble with Basis Dynamical Matrix . . Normal Modes . . . . . . . . 86 . . . 86 . . . 9 3 . . . 98 . . . 100 . . . 102 . . . 105 . . . 105 . . . 108 . . . 108 . . . 110 C BUILDING THE STATIC HARTREEFOCK EQUATION . REFERENCES . . . . . . . . 114 . . 119 BIOGRAPHICAL SKETCH . . . . . 124 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy TUNNELING BETWEEN TWO DIMENSIONAL ELECTRON SYSTEMS IN A HIGH MAGNETIC FIELD AND CRYSTALLINE PHASES OF A TWO DIMENSIONAL ELECTRON SYSTEM IN A MAGNETIC FIELD By Filippos Klironomos May 2005 Chi iii 'i,: Alan T. Dorsey Major Department: Physics We study the bilayer quantum Hall system in the incoherent regime. We model the two 1l.. ri as correlated Wigner crystals due to the presence of interlayer interactions and take the continuum limit that treats the system as an elastic medium. Using this approach, we find an analytic solution for the collective modes of the system and calculate the tunneling current associated with external electrons coupled to these modes, reproducing experimental results. Investigation of the role of interlayer inter actions into the response of the system reveals a dual nature: they introduce an excitation gap in the collective modes and also soften the effect of intir'w r inter actions. We further study the collective states formed by the 2D electrons at low Landau levels by working from a semiclassical and microscopic perspective and evaluating the elastic moduli, normal modes, and zeropoint and cohesive energies of the dif ferent < il i11!ii. structures. The effects of screening from filled Landau levels and finite thickness of the sample are found not to influence the overall interplay of the phases. When probing the internal degrees of the crystalline structures, the energy is lowered considerably (which signifies that these degrees have a prominent physical importance). Finally, the static HartreeFock equation for the triangular lattice symmetry subset is numerically solved and anisotropy effects are taken into consideration. The emerging picture is that the isotropic Wigner i *I J1 is favored for small values of the partial filling factor but at higher values the system undergoes a first order transition to an anisotropic Wigner crystal never crossing any other crystalline state for the rest of the filling factor range. The anisotropic Wigner crystal shows a channellike configuration for the guiding centers of the electrons but translation invariance along the channels is never restored. As a result we find that the anisotropic Wigner crystal is more favorable even from the traditional stripe state close to halffilling, and that shear deformations along the channels become costfree due to the vanishing shear modulus along that direction. CHAPTER 1 INTRODUCTION TO THE QUANTUM HALL SYSTEM 1.1 History of the Quantum Hall Effect A hundred and one years after Edwin Hall discovered the Hall effect in 1879, Klaus von Klitzing [1] discovered the Integer Quantum Hall Effect (IQHE), intro ducing the physics community to a new and remarkable class of condensed matter phenomena. A disordered 2D electron system (2DES) at low temperatures and high magnetic fields can exhibit sharp dips in the dissipative resistivity (pxx) and sharp plateaus in the Hall resistivity (pxy) at certain values of the magnetic field B. These plateaus happen at integer values of the quantum of resistivity h/e2. The sample used by von Klitzing was a silicon metal oxide semiconductor field effect transistor. Disorder is induced by the roughness of the insulatorsemiconductor interface in these structures. Two years later in 1982, Tsui et al. [2] performed the same experiment but with a GaAs sample of higher mobility and at lower temperature and discovered that the Hall resistivity piy can take fractional values of h/e2 which are of the form p/q, where p is an odd integer and q can be even or odd. This was the discovery of the Fractional Quantum Hall Effect (FQHE). In Fig. (1.1) we can see this remarkable behavior. Theoretical work has explained in a satisfying manner most of the pronounced features of Fig. (1.1) where all of the hierarchy of quantum Hall states is shown [3]. The single particle gap that opens up in the bulk of the material and the charged edge excitations give rise to the transport phenomena responsible for the IQHE. This gap is attributed to the single particle localized states lying between the spread out (due to disorder) Landau levels. At the edges of the sample, the confining potential distorts 10 20 30 MAGNETIC FIELD (Tesla) Figure 1.1: Dissipative and Hall resistivities of a GaAs sample. Reprinted from St6rmer, P,.I:. ,.a B177, 401 (1992). Copyright (1992), with permission from Elsevier. the Landau level splitting, giving rise to gapless single particle excitations. In the FQHE, the physics is of a many body nature. The electrons form an incompressible ground state with a gap (which is smaller than the IQHE) due to their interactions that become dominant when their kinetic energy is quenched by the applied magnetic field. The ground state can be accurately described (in the symmetric gauge) by Laughlin's wave function [4] N N t *z, ZN)= (z z e 4Z i (1.1) i>j 1 where v = N is the filling factor for a single spin, N is the total number of electrons, + is the total flux penetrating the sample, 4o = h/e is the flux quantum and zi = (xi + :,/ )/1B is the complex position of each electron, where 1B = h/eB is the magnetic length. Going back to Fig. (1.1) we notice that the FQHE happens for filling factors v = q/p where p is odd. Eq. (1.1) gives a first explanation for that, assuming the spin degree of freedom of the electrons is frozen: Fermi statistics are obeyed only when 1/v is odd. For the rest of the FQHE states different theories conduction band conduction band Si donor Si donor * AlGaAs AlGaAs GaAs GaAs d valence band valence band Figure 1.2: Symbolic graph of a bilayer structure of interlayer distance d. The electrons coming from the Si donors are trapped in the GaAsAlGaAs interface and form the 2D electron gas. have been developed based on Laughlinlike wavefunctions [5, 6], microscopic field theoretical treatments [7], or composite fermion theory [8], which have been quite successful. 1.2 Bilayer Quantum Hall Physics: Experiment and Theory Highly interesting and intriguing physics arises if two 2DES are brought within a nonzero separation distance d. Experimentally these structures can be grown by molecularbeam epitaxy where two semiconductors (usually GaAs and doped AlGaAs) form a quantum well at their interface (~1001000 A wide) where electrons, coming from Sidoped 1l, iS occupy, form the 2D electron gas. When an undoped AlGaAs interface separates the two quantum wells by a distance d, then the ratio d/B becomes a measure of the interlayer interaction strength. We show in Fig. (1.2) a schematic graph of the vilv.i r structure. What makes these heterostructures so interesting is that they exhibit "forbidden" QH plateaus. In reported experiments by Suen et al. [9] and Eisenstein et al. [10], the ilv.i r system exhibits plateaus at total filling factors VT = 1 and VT = 1/2. This is a "violation" of the odd denominator constraint for the 4.0 NO QHE 3.0 o o 0 o QHE 2.0 * * 1.0 0.00 0.02 0.04 0.06 0.08 0.10 ASAS/(e 2/ B) Figure 1.3: Phase diagram at VT = 1 of interlayer Coulomb interaction strength vs single particle tunneling strength. Energy measures in units of the initrlv. r Coulomb interaction. Solid symbols indicate samples showing QHE behavior, open symbols denote those that do not. Reprinted inset of figure 1 with permission from S. Q. Murphy, J. P. Eisenstein, G. S. Boebinger, L. N. Pfeiffer, and K. W. West, Phys. Rev. Lett. 72, 728 (1994). Copyright (1994) by the American Physical Society. FQHE. These new FQHE states are attributed to the extra degree of freedom (the l1 vr index) each electron possesses; and to the fact that the spinpolarization of the 2DES is relaxed, leading to spintextures [11]. Yoshioka, MacDonald and Girvin [12] have proposed the so called T3,3,1 state [13] as a candidate ground state for a double l1,r FQHE at VT = This is a Laughlinlike state which introduces correlations among the electrons in the two 1V,iS and keeps them from occupying the same position in the 2D planes, as if they were lying on the same plane. As experimental data indicate, the bil.I r structures have an interesting physical behavior that orig inates from the interplay of the in itr il r Coulomb electron interaction (within the same li r) and the interlayer Coulomb electron interaction (between the two Il, rz). A phase diagram for the QHE, experimentally produced by Murphy ct al. [14], is shown in Fig. (1.3). The strength of interlayer Coulomb interaction, relative to the initr i .r one, is plotted as a function of the tunneling strength ASAs (measured in the same units) that determines the energy difference of a single electron associated with 5 A) N,,.10.9 D) AN=5.4 1.2 V N, 42 v = 1 B) NI (x200) 004 l Z5 0.4 b. N) N 6.4 ^ 1 0\. 0 0.0 0.5 1.0 1.5 2.0 5 0 5 5 0 5 Interlayer Voltage (mV) . (K) Figure 1.4: Zero bias peak anomaly in the tunneling conductance of a .ili ,r system. Left panel: Tunneling conductance vs interlayer voltage V at VT 1. Right panel: Temperature dependence of the zero bias tunneling conductance at VT 1 at high and low densities. Reprinted figures 1 and 3 with permission from I. B. Spielman, J. P. Eisenstein, L. N. Pfeiffer, and K. W. West Phys. Rev. Lett. 84, 5808 (2000). Copyright (2000) by the American Physical Society. symmetric or antisymmetric occupation of the '.ilv .. r quantum well system. What is astonishing is that the QHE persists even when ASAs approaches zero, in other words when tunneling is turned off. The phase boundary intersects the vertical axis at a nonzero value d/lB  2. This signifies the onset of correlation effects between the two 2DES. On the other hand, when strong tunneling is present, electrons tunnel back and forth rapidly, assuming symmetric states with respect to the two 1lV. r. Correlations are not important in this limit and the system behaves as a singlel r 2DES, where the electrons are confined in a wider quantum well. If d/lB becomes large though, the antisymmetric (and more localized in individual wells) state becomes favorable again d. li ',' ii'; ASAS and the QHE altogether. This ,. i a quantum phase transition from an incompressible to a compressible state. Further investigation into the bilayer structures has revealed direct evidence of this interlayer coherence. In an experiment conducted by Spielman et al. [15] measuring the tunneling conductance in GaAs/AlGal_As double quantum well heterostructures, a well pronounced feature appears at total carrier densities of NT < 5.4 x 101cm2 at VT = 1. The two GaAs wells of width 180 A are separated by a 99 A A10.gGao.1As barrier 1V r. The low temperature (~40 mK) mobility of the samples is 2.5 x 105 cm/Vs. At greater carrier concentrations or equivalently greater d/lB ratio, this feature is completely absent due to the energy penalty associated with the rapid injection or extraction of an electron into the strongly correlated electron system, as we discussed above. What is interesting is that at filling factors where the 2DES must be thermodynamically compressible, it appears incompressible because the charge defects acquire a large relaxation time scale at high magnetic field, due to tunneling. Thermal fluctuations are expected to bridge the IQHE gap and similarly destroy the FQHE state by producing more tunneling events. In this case the relax ation time of the charge defects that tunneling electrons create is very low at high magnetic field. Figure (1.4) shows the resonance peak in the tunneling conductance and its temperature and carrier density dependence. The height of the peak continues to grow as the density is reduced and exceeds even the zero magnetic field tunneling conductance peak (3 x 108 Q1) by more than a factor of 10 [16]. It should be noted that VT = 1 is held constant in all the traces, so the magnetic field is varied accordingly. The appearance of this resonance peak , r.. I the existence of a soft collective mode of the ril r system which enhances the electron's ability to tunnel. If we assign a pseudospin quantum number to each electron, indicating the 1. r index which it lies in (up or down just like spin1/2), then the ,yplane" pseudospinferromagnetic state the bilayer system assumes is a symmetry breaking state. The Goldstone mode associated with this broken symmetry, as predicted [17 20], could be the collective mode identified above. Spielman et al. [21] directly observed this linearly dispersing collective mode in the 'il I r 2DES. The best candidate state describing the system's ground state responsible for this high coherence peak is the Halperin '1,1,1 state. In this state, an electron in one lr is abv opposite to a hole in the other 1.Cr in the same 1Ir in which the voltage drop measurements are taken. Figure (1.5) shows these results and the crucial dependence the response of the 'ili r system has on d/lB At d/lB > 1.83 the interlayer coherence is gradually destroyed and the iii i!" singlei Vr behavior is restored. Previous theoretical work has addressed the coherence peak feature the bilayer system exhibits. Balents and Radzihovsky [23] studied it from the quantum Hall ferromagnet point of view. They found rich variations of the tunneling conductance as a function of bias voltage, tunneling strength, disorder, temperature, and an applied magnetic field parallel to the I vr. They provided a scaling theory where disorder effects, based on an idealized pure system with parallel side contacts, were discussed as well. From the same viewpoint Ady Stern et al. [24] studied the finite Josephson effect and argued, based on disorder effects, that it is not a true Josephson effect (infinite tunneling conductance). They also predicted the rich characteristics the tunneling conductance should exhibit and they attributed the finite peak in it to topological defects in the order parameter m (pseudospin magnetization) caused by deviations of the total density from VT 1. Finally, Fogler and Wilczek [25], studying the system as a "classical" Josephson junction, applied perturbation theory and derived a tunneling current formula showing the general characteristics described above. In all the above theoretical approaches, phenomenological arguments and scaling techniques attempt to shed some light on the qualitative physics of this interesting `iliv r phenomenon. Nevertheless, a coherent picture with some microscopic insight to the characteristics of the system is still lacking. This is the direction we have taken. We have tried to shed some light on the effect of the interlayer Coulomb interaction and its importance to the collective physics of the liliv.vr system. We have modeled this system as two Wigner (< i 1 according to the original work of Johansson and Kinaret [26], based on the independent boson model; but we have introduced a coupling between the 1 ., r as well, arising from the existence of the 03 04 05 06 07 0.8 0.9 1.0 ' 4. 05 0.6 0.4A 0.5 10 15 20 25 3.0 4.10 4.15 4.20 4.25 4.30 4.35 f(GHz) v Figure 1.6: Microwave resonance response of a quantum Hall system. Left panel: Real part of crx vs frequency f for different filling factor values (offset for clarity). The inset is reproduction of selective filling factor values and at an expanded scale. Right panel: Peak frequency vs filling factor for the two resonances shown to coexist on the left panel. Reprinted figures 1 and 3 with permission from R. M. Lewis, Yong C!. i, L. W. Engel, D. C. Tsui, P. D. Ye, L. N. Pfeiffer, and K. W. West Phys. Rev. Lett. 93, 176808 (2004). Copyright (2004) by the American Physical Society. interlayer interactions. Our approach has been rewarding because it has provided insight into the dual role of the interlayer interactions We have studied only the incoherent regime of the system, but this kind of systematic modeling has paved the way for later attempts to include coherence and reproduce the fascinating IV response shown earlier. 1.3 Crystalline Phases of the 2D Electron System: Experiment and Theory The proceeding introduction into the physics of quantum Hall systems (whether Real part or single vs freq) should have convinced the reader that systems like these have a rich insei is of states or phases that can potentially manifest themselves, depending on the different parameter values associated with such systems (such as disorder, carrier the different parameter values associated with such systems (such as disorder, carrier density, applied magnetic field or temperature). Two i '.i"r classes of experiments can be conducted with the quantum Hall systems to explore the different phases they can realize. One class consists of transport experiments (such as the ones presented in the previous section) where features in the conductivity (or absence thereof) can lead to conclusions about the different possible phases. This class is subdivided into DC and AC transport experiments; which further specialize in capturing different characteristics of the system (such as the pinning threshold [27] or reentrant insulating states around given IQHE or FQHE states [28] or even anisotropic behavior [29, 30]). An insulating phase usually indicates crystallization in the system. The other large class of experiments consists of resonance absorption experiments where the sample is irradiated (usually microwave radiation) and from the resonance response it exhibits, one can derive valuable information about the collective modes and the actual state of the system [3133]. The latter method is able to capture phase coexistances, since (in principle) different phases will leave different traces in the absorption signal. A typical microwave resonance experiment for a 2D electron system in the n = 2 Landau level is shown in Fig. (1.6). What we see is the microwave absorption response of the system, traced in the real part of the longitudinal conductivity axx, for different applied magnetic field values or filling factors. The traces are displaced for clarity. According to data i, iJ1 ,ii on these measurements [33], the resonance curve can be fitted by two Lorentzians indicating a twophase coexistence. This is also shown in Fig. (1.6) where the peak frequencies of these two phases are plotted for different filling factor values. Disorder and pinning 1 i', a crucial role in the collective response of this system [34, 35] since the pinned domains around impurities resonate to the external stimulus the alternating electric field provides. On the other hand, the effect of disorder and the extent of pinning in the system is determined by the actual state ( i i,i ,ii.w or liquid) that the system occupies. As a result for one to aspire to describe the collective behavior of a quantum Hall system, one is forced first to study the different phases such a system is capable of realizing (for the whole range of filling factors) and then to develop a dynamic response theory for these states. Quantum Hall system states have been studied extensively using the Hartree Fock approximation [3639] or the density matrix renormalization group method (DMRG) [40, 41]. The prevailing picture, also pertaining to the experimental results shown in Fig. (1.6), is that at low filling factors, the system is in a Wigner crystal (WC) state; and for filling factors close to halffilling, the traditional stripe state (charge density wave) becomes favorable. At intermediate filling factors, a variety of bubble < i I J1 (BC) states emerge. A bubble < i i I J1 is a Wigner crystallike structure (where instead of having one electron guiding center occupying a given site, there are more; creating a hierarchy of Melectrons per bubble crystalline structures where M is an integer [36, 42]). According to HartreeFock results, the last possible BC state that can become favorable (as the filling factor is increased) is the M n+ 1 electrons per bubble i il I 1 for the nth Landau level. On the contrary, this last BC state is not observed using the DMRG technique and only up to M = n electrons per bubble SiI are realized. However, all of these studies focus on the density profile (order parameter) of the system in an ad hoc way that does not explore the microscopic physics involved; this is the direction we have pursued. We have attempted to shed some light on the microscopic of the tum Hall system and explore how the internal degrees of freedom react to external stimuli such as magnetic field changes. This improves our understanding of the physics involved and can serve as a stepping stone to describe the dynamic response of such states in light of experimental results (such as those shown in Fig. (1.6)). CHAPTER 2 TUNNELING CURRENT OF COUPLED BILAYER WIGNER CRYSTALS 2.1 Single Layer Eigenmodes Our next task is to formulate a model of a bilayer 2D electron system where tun neling is a quantum mechanical process between the two Il. ri separated by a distance d, in order to capture the incoherent behavior of the system. The bulk of the electrons in such a system provides the collective modes that couple to an independent tunneling electron. Implicitly assumed is that tunneling events are uncorrelated; and most of all, the electrons involved in the tunneling processes are uncorrelated with the bulk of the electrons comprising the hil. r system. This is a justified assumption, since a typical value of the tunneling current passing through the system is in nA, which corresponds to one tunneling event every 10 ps; while a typical period of oscillation for the collective modes of the 'ili r system is one order of magnitude lower. This means that any local excitation caused by a tunneling event is dissipated away much faster than the time it takes for another tunneling event to occur. Additionally, the collective modes dissipate their own energy through the emission of lattice acoustic phonons generated by the underlying substrate (GaAs) with propagating speeds of 5200 m/s, much less than the collective mode phase speed. The typical thermal activation time is 1 ns (which translates to 100 tunneling events throughout the sample). So the small number of tunneling events (and the rapidity with which their local charge defects relax) justifies our treatment of the tunneling electron as uncorrelated and independent from the bulk. Let us start by introducing the single lvr model in which the 2D electron system is assumed to be in a Wigner ( i ,I 1 state. This is not the true experimental realization of the system for the filling factor considered [27, 28] but serves as an accurate starting point if one wants to incorporate short range correlations among the electrons present in the real, liquidlike state of the system. Additionally, we work in the continuous approximation limit, treating the Wigner < i i~I J1 as an elastic medium and imposing a momentum cutoff qgo 2 2 o, where no is the lvr density. This way we are able to capture the long wavelength physics of the correlated electron gas, and retain some information of the short range correlations. Introducing the appropriate Lam6 coefficients [43] A, p to describe elastic deformations of a configuration with hexagonal symmetry (the triangular lattice is the Wigner < i I ,1 ground state [44]), we can write the following Lagrangian describing the system dynamics in the continuum limit L = no d2r 2 ln2 eu A(u) A(89U)2 (aUl + 1iur)2 2 2no 'I + no d2r'[V u(r) ][V u(r')] r'l (2.1) where u is the displacement field, c the dielectric constant of the host material (GaAs in our case of study), and A(u) the applied vector potential. As seen in the last term we have included an intrlv. r Coulomb interaction term in the continuum approximation where local charge variations are given by 6n/no = V u. This is correct in the absence of vacancies and interstitials. For the vector potential we choose to work in the symmetric gauge so that A(u) = (Buy/2, Bu1/2, 0) where B is the applied magnetic field. In the absence of the perpendicular magnetic field, the normal modes of the elastic system are the transverse and longitudinal acoustic modes where their corresponding acoustic speeds are related to the elastic parameters according to L = /(A + 2p)/mno, (2.2) CT = ./mno. (2.3) Since these are the normal modes of the system in the absence of the magnetic field, we would like to decompose the displacement field in terms of them, and then Fourier transform Eq. (2.1) to obtain L no Toit2 } r^T+ ^rL + Mc [iUTUL UfUT] MU) 2 2_ mUT 2 (2.4) J J(27)2 2 2 L 2 2 2 T T where we use the real field property u*(q) = u(q) and the convention u2 u(q) u(q). We see that the magnetic field enters in the dynamics only through the third term which mixes the transverse and longitudinal modes as expected. For the cyclotron frequency we have uc = eB/m while the longitudinal and transverse zero magnetic field eigenfrequencies are respectively given by S q2 + 2q, (2.5) WL cq + 2Tn c WT = CTq. (2.6) The expected effect of the intvr 1 r Coulomb interaction is to introduce incompres sibility (which is realized by the long wavelength divergence of the longitudinal mode velocity). As a result, the quadratic term involving CL becomes negligibly small and for all practical purposes [44, 45] we can set CL = 0; but for the sake of completeness, we will retain it until the last moment. The analytic expression for the transverse velocity is [44] CT 0.0363 (2.7) 3emao where ao is the Wigner J vI ,1 lattice parameter; and we have assumed triangular lattice configuration that seems to be the ground state. Appendix A details the eigenmode calculation. Here, we present only the eigenfrequencies of the single 1v.r system U) 2 t 2 + + L \/(c2 + cU4 + U))2 4U2 U. (2.8) We notice that in the zero magnetic field limit (c = 0) the above modes decouple into the pure longitudinal and transverse ones as expected. Also, in the high magnetic field limit (i.e., to lowest order in 1/we) we obtain + = + L (2.9) 2c L_ = T (2.10) Uwc according to Kohn's theorem [46] which predicts cyclotron frequency absorption for a translationally invariant system. We have recovered these plasmon modes in the continuum limit (w+); but since we have assumed a Wigner crystal state for the electronic system, we have also maintained gapless excitations (w_). The above treatment completes the single 1ivr study of the 2D electron system treated in the continuum elastic limit. We can decompose the transverse and longitu dinal fields involved in Eq. (2.4) in terms of the eigenmodes of the system and couple them to tunneling electrons, in the same way that phonons couple to electrons. That is why the modes of Eq. (2.8) are called magnetophonons (w_) and magnetoplasmons (a;), respectively. 2.2 Bilayer Eigenmodes Having completed the single 1 v.r treatment of a 2D electron system we can turn on the ilv,r problem where two 2D electron systems are separated a finite distance d from one another and interact through the interlayer Coulomb interaction. For simplicity we can assume that the 1lv. rS have the same density no (which can be arranged experimentally) and write down the following Lagrangian L LA+LB+no 2r (UA )2 2 no no d 2/ [V uA(r)][" uB(r)] (2.11)  7x')2+y ( y' y)2+ d2 where LA, LB are the independent single v1vr Lagrangians similar to Eq. (2.4). The term involving the K parameter (associated with the short range physics of the inter IV,r Coulomb interaction) is expected to arise when the two Wigner crystals prefer to lock their positions (and move inphase) by penalizing outofphase fluctuations. Since we are working in the continuum long wavelength limit it is impossible to capture that physics unless we explicitly add this extra term into the system dynamics. By construction, K/no is a measure of the energy density per electron associated with the short range correlations induced by the Coulomb interaction and can be assumed to scale accordingly as K e2 /47ed = K  (2.12) no 7T12 where K is dimensionless and I = h/eB. The dimensionless parameter K can be extracted from magnetophonon experimental measurements or theoretical calculations associated with this rilv r system. For the second term (the long range part of the Coulomb interaction) we have used the usual 3D form applied for the two 2D electron systems and we have employ, ,1 (as in the intirylv. r Coulomb interaction case) the continuum linear approximation in order to describe local charge density fluctuations. Diagonalizing this coupled 'iliv r system involves introducing inphase and outofphase modes given by 1 UA V U, (2.13) 2 1 UB v + U, (2.14) 2 which turn out to be the eigenmodes of it. As a result, the two coupled single l1v r dynamics of Eq. (2.11) decompose to uncoupled "effective single 1 l, i dynamics of inphase and outofphase nature. We started with a system of a total of four modes, so we expect two inphase and two outofphase modes as a result. Details of the calculation are given in the Appendix. The result for the two outofphase eigenfrequencies is S=1 [K + 2 + Q2 /( + ? + )2 4Q (2.15) where the "effective transverse and longitudinal acoustic mode frequencies are given by T2= cq2 + o (2.16) 2 2 2 2K 2mno Sc q2 + q(1 e qd). (2.17) mno 2m  Notice that the single l .r form of these results is preserved but the acoustic modes have acquired a gap relating to the short range correlation physics introduced by the K parameter. For the inphase eigenfrequencies we obtain similarly O = t + O + O + + O + O~)2 40 0 (2.18) where the effective transverse and longitudinal acoustic mode frequencies are given by O= c q2, (2.19) Sc q2 + 2 q( + eqd). (2.20) As is expected for the inphase modes, there is no gap introduced by the short range physics, since these type of modes respect the lockedin position of the two Wigner (i,I l1 Since we have solved the single 1Ivr problem and have found analytic expressions for the creation and annihilation operators of its eigenmodes, we can apply those results to the "effective single 1 ,. il cases here, after we transform the appropriate parameters involved. For example, in order to obtain analytic results for the outofphase operator modes we have to perform the following changes to the parameters of the single 1vr case: m  2, 4  Q2, w%  2 For the inphase case, the changes become: m > 2m, 4 O w 0 Final results are presented in full in the Appendix. This completes the treatment of the il r system. We now have analytic expressions for the eigenmode operators of the coupled 'il r quantum Hall system and a way to describe lattice field displacements in terms of those. What remains is coupling those modes to tunneling electrons introducing the electronmagnetophonon and electronmagnetoplasmon interaction. This will open up an excitation channel for the injected tunneling electrons to dissipate their energy and for the bulk electrons to relax the charge defect associated with the tunneling event. This is the topic of the next section. 2.3 Coupling the Bilayer to External Electrons: Tunneling Current Now that we have accomplished the task of calculating the eigenmodes of a iili r 2D electron system in the continuum approximation, we must complete the picture by introducing a coupling of those modes to an independent tunneling electron injected into the system through a steady current. To do that, we must distinguish between the bulk electrons (and their operators) associated with the displacement field u, and an independent electron tunneling from one 1, r to the other. For the latter, we will use ce, CA and ctB, CB as the creation and annihilation operators for the two 1I.~. rs, respectively. Assuming for simplicity that the tunneling electron is at the origin of li. r A, and couples through the unscreened Coulomb interaction to charge density fluctuations of the Wigner crystals in both 1,. rs A and B, we can express the interaction energy associated with that coupling as follows Hee 472 d2 2rA rA + 2rdrB (2.21) 47c r rAl 47 rB dl In the continuum linear approximation the charge fluctuations in the two Il.. 1r will be given by MnA = noV UA and 6nB = noV UB, respectively. Placing the independent electron at the origin corresponds to ne(r) = (2)(r). Combining all of the above the coupling term assumes the following form e28 0d2 VA UA 62n fd2 BB UB e20 d 2q 2d iq 2 UA (q) UB(q) .(2.22) 47e j (27)2 q I In other words, the coupling term in the i.Ii r system associated with an independent electron injected into the bulk of either of the two quantum Hall systems has the form coupling CACA (27)2q LA2 LB) cq(UL + e d A) .] (2.23) Introducing at this point the inphase and outofphase displacement fields given by Eqs. (2.13, 2.14) we can transform the above to oupng CAA (27 ( + d d)ULj CBB 2 2 + e )v + (1 e)uL (2.24) Using the analytic expressions, derived in the previous section, for the operator form of the inphase and outofphase modes we show in the Appendix that the above coupling term can be written in terms of creation and annihilation operators of the iil,i r quantum Hall eigenmodes as 4 4 Coupling CACA i 1. {(f t a,) +C BCB {iZ .[. as) (2.25) 8" s 1 I s= 1a where at and a8 are the corresponding creation and annihilation operators for the bulk electrons and the coupling matrix elements are given by ( edq) f, s 1,2, s, 1,2, if. ,= < (2.26) (1+ e dq)f, s 3,4, 1. i, s 3,4, Notice that we have included in the s summation the integration in q as well to avoid cluttering the symbolism. The above completes our treatment of the bilayer quantum Hall system since we have an analytic expression for the eigenmodes of the system and the way these modes couple to an independent electron injected in the bulk of the system. We can write the following independent boson Hamiltonian, similar to the one used by Johansson and Kinaret (JK) [26], to describe the bilv r system energetic involved in the tunneling processes H Ho + H+ + HT rCA + iz (at as)j CtCA + CB + i i(a as)jCCB S S + YhUata, + TCtCB + TCeCA. (2.27) We have a system of two 2D electron gases under the presence of a perpendicular magnetic field in the elastic continuum approximation, producing a collective mode bath to which an external tunneling electron couples in order to dissipate its energy. The same channel is used by the bulk electrons to "smoothout" the local charge defect created by tunneling events. These tunneling events are independent quantum mechanical processes with finite tunneling matrix elements T, calculated in a similar manner as JK report [47]. The collective mode operators at, a8 obey boson statistics and cA(B) CA(B) obey fermion statistics. In the above CA and CB are the Madelung energies of the two Wigner crystal lattices. We follow JK in evaluating the tunneling current associated with this model. Their approach involves the application of Fermi's Golden Rule that can be rewritten in the following form C / d+O / I(V) dievd ^ ([Hj(t), Hf+(O)]) C dt e2VI(t) e (t) (2.28) The correlation functions associated with this expression are given by IT(t) = (H(t)H+(0)), (2.29) I:(t) (H+(t)Hj(0)), (2.30) where the timedependence is meant in the interaction picture representation. For the calculation of the above correlation functions we use the same approach as JK. Since the tunneling process is statistically independent from the collective mode propagation it can be averaged independently. The statistical averaging involves the linked cluster expansion method [48]. In this particular case there is only one independent link associated with the exponential resummation. In the Appendix we show in more detail how the calculation proceeds. We obtain for the correlation function I(t)= v(1 v)T2C(t), (2.31) where v (= (cCA) and 1 v = (cBct) and for the timedependent part we get C(t) exp { 2( 2 L 2) N)+ ( ^) (2.32) where N, is the boson thermal occupation number for the magnetophonons and magnetoplasmons. To obtain the form of Eq. (2.30) it suffices to interchange A and B in Eq. (2.32). The experimental temperature range in the tunneling current is of the order of 0.1 K~105 eV while the bias voltage is in the range of mV, so a zero temperature calculation is appropriate which simplifies things considerably since the bosonic occupation numbers Ns = 0. In addition, due to the high magnetic field, the magnetoplasmon modes will have a large gap and will not contribute to the electron coupling so we can drop them. Notice that we are still left with the inphase magneto phonons along with the outofphase ones but as we can see from Eq. (2.32) they enter as a difference in the correlation function and since they are exactly equal for both of the l'zvi [Eq. (2.26)] they cancel out. This is to be expected since a tunneling event is associated with an outofphase motion of the two Wigner crystals. The one that the tunneling electron leaves from will try to close the "hole" left behind while the other, receiving the tunneling electron, will try to "openup" and create an available posi tion for it. This corresponds to an outofphase motion. If we gather all of the above together and switch to dimensionless units for the momentum integration (x = q/qo) we find the following result for the time dependent correlation function C(t) exp J df(x) (eiw(x)t 1, (2.33) where the weight function and the magnetophonon frequency can be approximated in the high magnetic field limit as cx x(1 e7)2 1 f(x) W ( C qo +x2 + 2a + jx(1 e7) 2+ a 1 1 X [a + 3x(1 c)]3/2 6 a (2.34) a)x ) a V/+3x(1 e )Vx2+a. (2.35) wc In the above we have defined the parameters no 2 2 c h c(2.36) c 87hm e ' 2K K 1 e2 a 2 22c ( qo) 2' (2.37) P  o (2.38) 2emcTqo 7 =dqo, (2.39) 6= ( (2.40) and have taken CL = 0. The parameter a is dimensionless and gives a measure of the magnetophonon gap. For the second equation associated with it we have used the definition of Eq. (2.12). The /3 parameter is dimensionless as well and does not depend on a0o if the dependence on it from CT is taken into consideration using Eq. (2.7). The parameter 7 gives a measure of the relative strength of the intrylv. r and interlayer Coulomb interaction. In order to proceed with the derivation of the correlation function we can differentiate Eq. (2.33) and take the Fourier transform to obtain the following equation wC(U) = dxf (x)w(x)C(w u(x)). (2.41) As we show in the Appendix, this correlation function is zero for w < 0. The above integral equation for the correlation function is very hard to solve exactly. In the following subsection we show how we can derive important information to build an Ansatz solution for the IV response of the bilr system. 2.3.1 Analytic Solution In order to try and approximate an analytic solution for the integral equation of the correlation function given by Eq. (2.41) it is important to derive as much information as possible from it. What turns out to be particularly useful is the derivation of the .,i~:, l, I tic behavior for large frequency values (large bias). In that case we can expand C(u uw(x)) in u(x) and obtain to lowest order a first order differential equation with the solution C(w) exp r 21)j, (2.42) where 0/1 ci j dxf(x)w(x) U d4 1x x(1 %x)2 1 1 (243) Jo 6 + X + 2a + Ox(1 e%) a + Ox(1 e') 6 a 2' 0/1 C2 dxf(x)w2(x) L3 1 x(t_ y)2 2 t + a =c( dx1 .x (2.44) 4 q o d + X 2 + 2a + Ox(1 e') a + 3x(1 e_) a X2 a . We see that there is exponential suppression in the tunneling current for very large bias values, something to be expected since the system is unable to cope with the large inflow of energy the tunneling electrons carry and need to dissipate. Any attempt to dissipate these large amounts of energy creates large number of magnetophonons which in turn cause large quantum fluctuations in the system potentially destabilizing it. We are in a position now to investigate different correspondence limits associated with Eq. (2.43). We are interested in the limiting behavior of the above .,vmptotic solution for different 1vr separation values d. For the case where the two 1v.,.riS are far apart and can be considered uncorrelated (d > ao), we can ignore the ex ponentials in the integrand of Eqs. (2.432.44) and it turns out that cl ~ 1/ao and /C c~ 1/a B are the corresponding limits. This is the same scaling behavior JK produce using phenomenological arguments. In the opposite limit, where the inter lr separation is much smaller than the intraelectron distance (d < ao) we can expand the exponentials in the integrand of Eqs. (2.432.44) and find ci ~ d2/a and /2 ~ d/a B. This limit is absent in the JK model. In this regime correlation effects become important. Coherence significantly modifies the actual behavior of the system but this is expected only in the region close to zerobias [15]. For the remaining bias voltage region, correlation effects have the prominent role and the Coulomb barrier peak survives but is "redshifted" significantly [15]. As we see our model is able to reproduce such a limiting behavior. The above ., ii il .1 ic expansion provides a useful starting point to apply a trial solution of Eq. (2.41) by assuming a powerlaw combined with a Gaussian exponential behavior for the correlation function according to C(w) = Nwe 2. (2.45) The above Ansatz captures the essential .,,iii! .ltic behavior and if the parameters are evaluated selfconsistently it should qualitatively reproduce a solution. In particular, if we multiplydifferentiate Eq. (2.33) we end up to the following moment equations (associated with C(u)) that we can use to derive values for N, r and A: j dwC(uw) = 27, (2.46) 2rOO j dwC(w) =27Cl, (2.47) j dw2C() = 2(c2 + ). (2.48) JoO For the above equations the following general integral becomes useful 1 ury2 1 r+2 (2.49) Also, it is convenient to switch frequency (w) to bias voltage (V) (measuring in mV) by introducing the change w = eV/lOOOh. That way the argument of C(Uj) will measure in mV while the corresponding Ansatz parameter A in the exponent of Eq. (2.45) will acquire the form A A (10) 2. Using the general integral result above we can perform the integration in Eqs. (2.462.48) and obtain the results 1 (e r+l \ 1 N( A 21 27, (2.50) 2 1OOOh 2 ) N ) A 2 +) 27cl, (2.51) 2 1OOOh 2 ' 1"e r+3 +3 2 2 N Oh 2 r 2 2 (c2 + c). (2.52) We can divide Eq. (2.51) and Eq. (2.52) by Eq. (2.50) to get C 2 F ( r+2 ) A ( 1000 h) [ 4i ] ( A=( e 2)PT 3 (2.54) or equating the two p2(r+2) (r+1) r3) + /c' (2.55) we get a selfconsistent equation for r that we can solve numerically. The usual range of r for the magnetic field values considered is 1/2 < r < 2. This can be regarded as an estimate for the low bias current powerlow behavior. Having r at hand we can go back and evaluate the rest of the parameters. As a final result we find that the S/ ..11 T 6  975T I \ " UO 4, 4  : 8.25T \ \ 0 5 10 15 20 25 Interlayer Voltage (mV) Figure 2.1: Tunneling current curves for different magnetic field values using the moment expansion solution of Eq. (2.41). The legend shows the peak bias values calculated by Eq. (2.57). tunneling current correlation function assumes the form C(V) O N)e V'reAV2, (2.56) where V measures in mV. To obtain the peak bias value we need to find the root of the first derivative of the above equation Vo O h (2.57) e 2A' where we have converted it to mV. With the last piece of the puzzle in place we are ready to test our theory with a realistic experimental setup. We choose the same system JK used as their reference [49]. The riliv r sample area is S 0.0625mm2 and the single 1, r electron density is no 1.6 x 1011 cm2. The perpendicular magnetic field varies from 8 T to 13.75 T and the Wigner i iI 1 lattice parameter has the value ao 270 A which corresponds X E 05 d=115 04 03 d=145 02 01 d=85 4 0 2 4 6 8 10 12 14 16 18 20 V (mV) Figure 2.2: N., in i,1,. .1 tunneling current curves for different interlayer separation distances d measured in A. to the stable hexagonal lattice configuration (no = 2/V3a ) [44]. The double well separation distance is d=175 A and the dielectric constant of GaAs is e=12.9co _1.14 x 1011 F/m. The transverse sound velocity for the electron gas given by Eq. (2.7) is CT 53552 m/s. As we mentioned earlier we have to take the longitudinal sound velocity cL = 0 in order for our results to correspond correctly to the physics of the experimental system. For the electron mass we use the electron effective mass value in the GaAs background m 0.067me. For the K parameter we use Eq. (2.12) where the value of K is extracted by timedependent HartreeFock calculations investigating the magnetophonon dispersion relation for the .il i vr system, performed by C6t6 et al.. They were able to provide us with a K 0.0085 value. Our IV results based on this model are shown in Fig. (2.1) with the corresponding peak bias values given by Eq. (2.57). At this point we are in a position to investigate the effect of the interlayer Coulomb interaction in the 'il v.r system. We notice that the strength of this inter action is controlled by the interlayer distance d which is introduced into the model in two places. One is in the tunneling matrix elements T in the independent boson model Hamiltonian (with an exponentially suppressive behavior) and the other is through the long range part of the interlayer Coulomb interaction term. Since we are inter ested only in the latter, we will normalize the tunneling current for different interlayer separation values d. What we expect to reproduce is the experimental behavior shown by Eisenstein et al. [50] where the peak bias values are "redshifted" by an amount proportional to e2/ed as the interlayer spacing is reduced. This behavior is due to the attraction between the hole left behind in a tunneling event and the tunneling electron itself which is of the order of e2/ed. In other words the creation of excitons associated with tunneling events are expected to "soften" the effect of their ntrilv, r Coulomb interaction and consequently lower the energy barrier imposed to tunneling. In Fig. (2.2) we show our results for the normalized tunneling current solution for different interlayer separation distances d measured in A. As we see our model is able to capture this important physical behavior of the system. As a result of our theoretical analysis we can conclude that the effect of the interlayer interactions in the bilayer system is twofold. First, in the short range physics it introduces a gap in the long wavelength excitations that contributes to the small bias suppression of the tunneling current. And second, in the long range physics it "softens" the effect of the intr liv r Coulomb interactions through the excitonic creation associated with tunneling events and as a result it "redshifts" the tunneling current peak bias values. 2.3.2 Numerical Solution We have numerically integrated the integral equation for the correlation function in two different vv, first by a direct integration of the integral equation, and then by introducing the density of states (similar to the JK method). Both methods give the same results of course so we will present the latter one only. We can write Eq. (2.41) .**1 S 6 a :/ 9.75 T \ 0 / I ~ 8.25 T \ ' 0 5 10 15 20 25 Interlayer Voltage (mV) Figure 2.3: Tunneling current curves for different magnetic field values produced by numerically integrating Eq. (2.41). The legend shows the peak bias values obtained with this approach. They are in strong agreement with the analytic results. in the following dimensionless form zC(z) dxf(x) C(z ()0(z )), (2.58) Jo 7o 7o 7o where we have introduced the parameter o = to convert the frequency argument of the correlation function into mV. Before we proceed we should notice that the magnetophonon frequency is bounded in a region or < '(x) < a2 which means that the density of states is non zero only in that range. The values of a1 and a2 are given by substituting x = 0 and x 1 into Eq. (2.35) respectively. The upper bound a2 appears due to the momentum cutoff we have introduced. The resulting form of the correlation function integral equation is (1 =l'nIy)C(z y), at < Z < a2, C(z) Z (2.59) fl _r(y)C(z y), > a2, 31 where the definition for the density of states is g(y) = Y f i}g) (2.60) ( f (x) o dx (y) and x(y) is the root of the equation u(x) = 7oy. In this approach one has to "jump start" the algorithm with an assumption for the low bias points. We use a linear approximation since we can show that for z > a1 values C(z) ~ za1. Our numerical solution is presented in Fig. (2.3). As it is clearly shown the qualitative behavior of our analytic solution is verified and the peak bias values are similar as well. CHAPTER 3 QUANTUM HALL SYSTEM IN THE HARTREEFOCK APPROXIMATION 3.1 Electron Dynamics in a Perpendicular Magnetic Field We would like to focus our attention here on a single 1vr quantum Hall system and try and shed some light on the microscopic physics involved in this highly correlated electronic system. We would like to investigate the competition between different crystalline states and their stability for different applied magnetic field values. This kind of work requires some attention to be paid to the microscopic involved in such a system. The quantum nature of the electrons incorporated in the physics of wavefunction overlaps and associated with the electronelectron Coulomb interaction has to be considered, in order to investigate, as accurately as possible, the energetic and stability of the different phases associated with the quantum Hall system. We start this work by finding the noninteracting electronic wavefunction in the presence of a perpendicular magnetic field. For that task, we introduce the Landau gauge A = (By, 0, 0) and write down Schr6dinger's equation for the 2D electron, given by 2( eBy)2 + P (x, y)= E (x,y). (3.1) Since in this gauge choice, the magnetic field does not fully couple the two directions, a plane wave solution is expected in one of them (xdirection) resulting in a wavefunction decoupling of the form: j(x, y) = erkx x(y). This kind of decoupling produces a displaced harmonic oscillator equation for the ydirection given by [ + mwf (y )2] ) hc (n+ t (y). (3.2) 2m 2 2 2 In the above, we have defined Y = kjS2 as the ycoordinate center of mass (C\ 1) variable, V/h /B as the magnetic length, and c = eB/m is the cyclotron frequency. The normalized solution for the displaced harmonic oscillator is given by 1, 2 (yY 0(. y) 1e '), (3.3) T 1/4f1/2 2>! where n is the so called Landau level index, associated with kinetic energy excitations of the noninteracting electrons, and H,(x) is the usual Hermite polynomials of order n. The effect of the applied magnetic field is to quench the kinetic energy of the electrons in the 2D system, which results (in the real system) in an enhancement of the role of interactions among electrons. This can become prominent at high magnetic field values, where kinetic energy excitations (of the order of ha;) might exceed the thermal energy range (of the order of kBT), and as a result become inaccessible. This is the magnetic field range that the kinetic energy becomes irrelevant, and only interelectron interactions affect the energetic of the system and introduce a large class of hierarchical states, as the magnetic field is varied. The degeneracy associated with the plane wave eigenstates in the xdirection allows a macroscopic number of electrons (fermionic particles) to occupy the same kinetic energy eigenstate (even in the noninteracting limit). The spin degree of freedom is assumed to be frozen at these high magnetic field values. For a system with finite length Lx in the xdirection the degeneracy g can be found to be L ko 0 _4 xw = dkiv = dY= (3.4) 27 o 27f2 0 27f2 o0 where Q is the total area of the system, and 0 = h/e is the flux quantum (associated with the quantum Hall system which is twice the value of the superconducting flux quantum). The quantum mechanical operator expressions for the C\ I coordinates X, Y (that enter into the dynamics of the electrons) can be derived from their classical counterparts, and are found to be X = (3.5) mwjc Y=y (3.6) where the dynamical moment 7r, Ty are given by the following expressions in the Landau gauge = ih [xH]H px eBy, (3.7) 7Y =my [y,H] py. (3.8) Combining the above definitions together we can derive the C \ coordinate forms in terms of the usual quantum mechanical operators: X = (3.9) S P= (3.10) We see the effect of the magnetic field and the Landau gauge choice partially mix the dynamics of the two directions. The C\ I coordinates are constants of the motion since they commute with the noninteracting Hamiltonian H, introduced in Eq. (3.1), something to be expected since the cyclotron motion does not drift. Additionally, they are conjugates since [X, Y] = i2. The dynamical moment are conjugates as well since [7x, y] = ih2/ W and they additionally obey [X,7k r] [Y, y] = 0. In other words, the C\ I coordinate operators along with the dynamical momentum operators represent different parts of the degrees of freedom of the electrons. One can use these four operators to define appropriate creation and annihilation operators (associated with these degrees of freedom) to fully describe the electronic field. Additionally, from the following commutation relation [I,7] = [x,p eBy] = ih, (3.11) in the limit of high magnetic field we find [x,y]= 2, (3.12) which implies that high magnetic fields radically change the electron dynamics. In that limit the position operators (that usually commute with one another) become conjugates. What this entails, is that special care needs to be taken when we define physical observables if we want to correctly incorporate the physics of high magnetic fields in such a system [51]. The method that has been developed to address this, involves the projection of all physical observables onto given Landau levels [52]. In principle, a subset of Landau levels needs to be retained for general magnetic field values. However, for the case where interLandau level excitations are not important (high magnetic fields) we can restrict the projection space onto only one Landau level. The mechanism to project onto a given Landau level involves the restriction into a subset of the available Hilbert space of the wavefunction basis used to define the electronic field operator. The noninteracting wavefunction basis (given by Eq. (3.3)) is usually used to construct the electronic field operator. Projecting onto the nth Landau level we find S(r) (r)c.,y, (3.13) Y where c ,y, c,,y are the creation and annihilation operators associated with the non interacting eigenstates. All physical observables involve the above electronic operator. One can show that the electron density operator n(q) when written in terms of the projected onto the nth Landau level operator p(q) acquires a structure factor as can be shown from n(q)= i2, '(r, (r)e, qr = f L, q iqy Y Y Y F,(q)p(q), Cn,yCn,Y+q.g2 (3.14) where L,(x) is the nth order Laguerre polynomial. These structure factor has the form q2,2 q202 F,(q) e 4 L (3.15) and the analytic expression for the projected density operator is given by p(q) eiqY /2. (3.16) Y The following commutation relation holds for these projected density operators [53] [p(q), p(k)] 2i sin (q2 k)2)p(q + k). (3.17) The Landau gauge is useful in introducing the physics of electrons in high magnetic fields but the noninteracting electron wavefunctions associated with such a basis are not very simple to use due to the presence of the continuous quan tum number associated with the CM\1 coordinate position. A much more useful basis of noninteracting electrons arises out of the symmetric gauge choice: A = (By/2, Bx/2, 0). In this basis the good quantum number becomes the zcomponent of angular momentum and the wavefunction form is given by [38] ( r) (nm (r \ i(nm)OL nml ( r2/2 ' (n+m nm)/2 2f2 f f^ (Z r (3.18) where n is the Landau level index, m the zcomponent angular momentum index, and L'(x) are the associated Laguerre polynomials. The normalization constant is given by Cnnm 2 (3.19) 27rn! Parity is determined from the exponent n m (whether it is even or odd). 3.2 HartreeFock Approximation To build a realistic model of 2D electrons we need to include the Coulomb interaction among them. Since the interaction is a fourfermion operator there is not much hope for us to develop analytic results unless we approximate it. The best, and most widely used, way of doing that [38, 39,52] is through the Hartree Fock approximation which captures the necessary long and short range effects of the Coulomb interaction. Additionally, as we have explained previously, we need to project this operator onto a given Landau level, in order to take into consideration the peculiar dynamics that arise due to the presence of the high magnetic field. This task is performed simply in the Landau gauge, where we can derive analytic expressions for all the terms involved. We start by projecting the fourfermion operator of the Coulomb interaction by using the result of Eq. (4.32). What we find is H = d2t d2r'rj(r) j(r')V(r r') (r') (r) 1 fd2q 1 27re2 (23.2 2q t 27 2 [F.(q)]2p (q)p(q), (3.20) 2 (27)2 47e q where c is the background dielectric constant. At this point we treat the 2D electron system as ideal by ignoring the finite thickness in the third direction, which is present in a real system. Additionally, we do not include screening effects arising from the presence of electrons in the filled Landau levels. Later on we will be able to relax that constraint and investigate a more realistic model and conclude on the validity of this simple approach. The HartreeFock approximation consists of pairing the four fermion operators in groups of two, averaging on one of the groups as follows P(q)p(q) ( e ,Y_ t,,Y+ 2) ,y_. ,: Y,Y S(cYqt2/2Cny' '/2)CnY'+qy2/2CY+n '/2 (3.21) What this entails is that the interaction potential VHF = VH + VF is composed of two parts, the Hartree part associated with long range physics (classical Coulomb interaction) VH (q) 272 [F(q)]2, (3.22) and the Fock (exchange) part associated with short range physics, and as we show in the Appendix is of the form OO Vp(q) dxxVH(x/f Jo('.'), (3.23) where Jo(x) is the zeroth order Bessel function and the x integration is dimensionless. In the Appendix we provide analytic expressions for both of the terms above for the n = 0, 1, 2, 3 Landau levels. The final expression for the energy associated with the projected Coulomb interaction of a 2D electron gas in the HartreeFock approximation becomes HHF (2 2VHF(q)(p(q))p(q). (3.24) This will be our starting point for treating the 2D electron system and investigating the energetic and the stability of the different quantum states associated with it. CHAPTER 4 ISOTROPIC CRYSTALLINE PHASES 4.1 Stability Analysis of Isotropic Melectron Bubble Crystals We would like to investigate the stability of the different crystalline states the 2D electron gas is capable of realizing at higher Landau levels. As we mentioned in the introduction the crystalline states can be characterized in general as Melectron bubble crystals. These < il I 1 have the same structure (and triangular symmetry) as the Wigner M electrons; and according to previous theoretical HartreeFock investigations these Melectron bubble <( ,I 1 succeed each other in increasing M order as we approach halffilling in a given Landau level [3639]. The last state to win the energetic race is the charge density wave state (CDW), termed stripe state, that is realized close to halffilling. Before we proceed with the stability calculation let us investigate how the dif ferent parameters of the < i ,I ,liii. structures are interrelated. The total filling factor of the system is given by v 271 (4.1) where f is the magnetic length, and N and A are the total number of electrons and area of the sample, respectively. As we mentioned previously, the electrons that belong to the filled Landau levels are considered inert (they don't participate in the < i, 1 11i. 0i,,ii process) and at most they provide screening effects for the Coulomb interaction. As a result, it is useful to distinguish between the total filling factor (pertaining to the whole system) and the partial filling factor (pertaining to the active electrons in the partially filled Landau level). Depending up on the type of Melectron BC configuration of the system, the partial filling factor is defined as N* v* = 2= 2 (4.2) 2 , (4.3) ABC where n is the Landau level index, N* is the macroscopic number of electrons in the partially filled Landau level, and A = V3/2at is the Melectron BC unit cell area (aB is the lattice constant and as usual we assume triangular lattice configuration). A typical Melectron bubble has a radius rB. Since the local filling factor on each bubble is one, while the density is M/TrrB, if we apply Eq. (4.2) we are lead to the relation rB = 2M for the bubble radius. Additionally, applying Eq. (4.3) for the WC case (M = 1) we find that aB = avM, where a is the WC lattice constant. One can consider Eq. (4.3) as an alternative definition for the unit cell area, which traced back to the lattice constant, produces the useful result aB M (4.4) Finally, it is easy to find the ratio between the Melectron bubble radius and the lattice parameter to be r 3 (4.5) aB 27 The above definitions will prove useful since they allow us to fix the sample density (as is the case in a real sample) and determine changes in the lattice configuration, when the applied magnetic field is varied. In order to avoid cluttering the symbolism too much we will drop the specific BC subscript from the above definitions and introduce it only when necessary. In order to investigate the stability of these structures we need to calculate the shear modulus associated with any given Melectron bubble crystal and discover the region where it becomes zero, which signifies the onset of instability. The shear moduli are evaluated by expanding the cohesive energy given by the general formula Ecoh = U(RR'), (4.6) to second order in the electron displacements around the lattice sites R. Our basic task is to define the electron interaction potential U(r), coming from the Coulomb repulsion among electrons but modified due to the special dynamics the high magnetic field introduces, the HartreeFock approximation, and the quantum corrections arising from the microscopic physics of the system. Having accomplished that task, it is easy to show that the elastic energy associated with Eq. (4.6) is of the form Elastic 2 t U(q) Y Ciq'(RR1) [(R)u(R') U,(R) u(R)], (4.7) q R#R' where Q2 is the total sample area coming from the Fourier transform of U(r). We can introduce at this point, the Fourier transform of the discrete displacement fields according to the following definitions U (R) (q)eiqR (4.8) U,(q)= Ac > u(R)eiq'R. (4.9) R The discrete and continuous transformations are mixed, and one needs to be careful with the units. For that reason, we introduced Ac (the unit cell area). This maintains the proper units for io,(q), which according to Eq. (4.8) are L3. If we substitute the above in Eq. (4.7) and use the following definition iq.R (q Q) Q ,,Q, (4.10) R c c we find that the general elastic energy expression becomes Eelastic =A JU(Q + q)(Qa + q,)(Q + ) U(Q)Qa} x u (q)ua(Q' q) Q/ J ( q {U(Q + q)(Q, + q,)(Q + %q) U(Q)QaQ} x ia(q)Ui3(q). (4.11) In other words we have brought the elastic energy equation into the general form given by Eelastic 2j ( (q)3 (q)q)(q), (4.12) where ,"(q) = ui(q) and 03o(q) is the dynamical matrix defined as t(q) = >Q {U(Q + q)(Q, + q)(Qa + qa) U(Q)QaQa}. (4.13) In the high magnetic field limit (and at low temperatures) the electronic wavefunction extent (of the order of ) can be assumed to be much smaller than the lattice parameter of any given crystalline structure, and as a result we can expand the dynamical matrix given by Eq. (4.13) up to second order around q = 0. Additionally, by assuming isotropic interactions (which is true for the Coulomb interaction) we end up to the following result for the dynamical matrix { U r(Q) ( + '+ U'(Q) 1 F q +2 QQ q] Q U 2C QL40 Q iqJQ3 (4.t14) where the divergent term Q = 0 is removed from the sum if the properties of the positive background are taken into consideration. We should notice at this point that the nonsingular Fock term associated with U(Q = 0) is maintained in the sums, since the positive background cancels only the singular Hartree term. This is observed throughout this work. According to the classical theory of elasticity [43], the elastic energy density of any 2D medium is given by the general formula Selastic = A ., ," UkUl, (4.15) where Aijk are elastic constants with only a certain number of them being independent or nonzero (depending on the given symmetry of the elastic medium). For the tri angular lattice configuration the above expression simplifies to +elastic IAUiUUUj + ( aiUj + jUi 2 L 1 (A + 2p) [(0au)2 + (9yUy)2] + 2A\u9 yuy + Pu [(aY)2 + (9YU)2 + 2,9uy9yu,] (4.16) In the above, the elastic constants A and p are the Lam6 coefficients that are deter mined by the following relations that hold due to the triangular lattice configuration Axyxy Ayxyx Axyy Ayxxy p, (4.1 7) Axxyy Ayyxx A, (4.1 8) xxxx yyyy A + 2p/, (4.19) Axxyy Axxxx 2"Axyxy, (4.20) and the rest of the possible Aijk's is zero. The shear modulus c66 (associated with the energy cost of shear deformations) is given by Ayxy = p and the bulk modulus ci (associated with the compressibility of the system) is given by Axxxx = A + 2p. Our result for the dynamical matrix can be related with the above definition of the energy density for the triangular configuration if we introduce the Fourier transforms of the displacement fields and write 1 Elastic = 2Ab/3g lt3Q/,(q)i *(q). (4.21) Comparing the above with the expression for the elastic energy of Eq. (4.12), we find that the elastic constants and the dynamical matrix are related with the definition \avq^ v 3(q). (4.22) In other words the bulk modulus will be defined as c1 = xz(qzx)/qx, and the shear modulus will be given by c66 'xx(qy)/q2 or if we use Eq. (4.14) we end up to the general expressions for the bulk and shear moduli c 2fM (q) + ( ) [U(Q) + U'(Q) + Q U"(Q) U'(Q) (4.23) c 266 2 U'(Q) + QU (IQ) u'(Q) (4.24) In the above expressions we have used Eq. (4.4) for the Melectron BC lattice constant. These are the most general results one can find by Taylor expanding Eq. (4.6) to second order in the displacements and assuming an isotropic interaction potential. 4.1.1 Classical Order Parameter Approach In this approach we treat the 2D electron system as an isotropic Melectron BC of triangular symmetry (which is proved to be the stable ground state for the Wigner S 10 x 103 3 8 2 6 1 0 4 0 2 1 0C 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 v v Figure 4.1: Shear modulus (in units of e2/4 c3) as a function of partial filling factor v* for the lowest Landau level and for the WC (left panel) and 2e BC (right panel) using the order parameter approach. (ivl I1 [44]). We treat the Melectron bubbles as pointlike particles fluctuating around their lattice site positions and the electrons inside a bubble are treated as classical interacting particles. This allows us to define the local filling factor, in accordance with Goerbig et al., as [39] v*(r) (rB r R u(R)), (4.25) R where 0(r) is the Heaviside step function, rB the Melectron bubble radius, and u(R) the Melectron bubble displacement around the lattice site R. The above choice of local filling factor produces a crude steplike approximation for the density profile of the crystalline structure. The direct and reciprocal lattice vectors for the hexagonal lattice symmetry are defined as [44] Rjj' y + j, (4.26) Q 27 2j (j' Q a / j' (4.27) X103 x 103 6 2 5 4 1 2 1 21 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 v v Figure 4.2: Shear modulus (in units of e2/47 c3) as a function of partial filling factor v* for the n = 1 Landau level and for the WC (left panel) and 2e BC (right panel) using the order parameter approach. where we have used Eq. (4.4) for the Melectron BC lattice parameter. For a 2D sample of total area Q the interaction energy associated with the bubble Ji ~1I 1 con figuration in the HartreeFock approximation is given by [39] E (22 VHF(q) A(q)2, (4.28) 2 (7 q where, VHF(q) is the HartreeFock potential given by Eq. (3.24) and A(q) is the Fourier transform of the local filling factor which is found from Eq. (4.25) to be A(q) M2 J (qrB) eiq(R+(R)) (4.29) Q2 qrB R Ji(x) is the first order Bessel function. If we substitute the above in Eq. (4.28) it is easy to show that it assumes the general form of Eq. (4.6) where U(q) is given by U(q) VHF(q) ( 2MJ(qB) 2. (4.30) We can investigate the stability of this structure by calculating the shear modulus, given by Eq. (4.24), for different magnetic field values, or partial filling factors Notice X 103 x 103 4 1 2 2  3 4 4 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 V V Figure 4.3: Shear modulus (in units of e2/47c 3) as a function of partial filling factor v* for the n = 2 Landau level and for the WC (left panel) and 2e BC (right panel) using the order parameter approach. that for the bulk modulus we find the typical long wavelength singularity coming from the first term of Eq. (4.23) if we take into account the form of the potential energy from Eq. (4.30). This behavior is in accordance with well known results for the classical Wigner ( iil I1 [44]. What we have achieved so far is produce an analytic expression for the elastic moduli in the semiclassical HartreeFock approximation where the electron gas is treated as point particles fluctuating around their lattice equilibrium positions. In Figs. (4.1 4.4) we plot the shear modulus versus partial filling factor v* for Landau levels n = 0, 1, 2, 3 and for the isotropic WC (M = 1) and the isotropic 2electron per bubble (< i ,1 (2eBC) (M = 2) cases, where the interaction energy is given by Eq. (4.30). We notice that in Fig. (4.1) (which corresponds to a WC in the lowest Landau level) we reproduce well known results by Maki and Zotos, where the isotropic WC state becomes unstable around filling factor v* 0.48 [54]. Additionally, we find that for the n = 2 and n = 3 Landau levels the isotropic WC destabilizes around v* ~ 0.24 and v* ~ 0.18 respectively, but the M = 2 isotropic BC can live up to v* 0.39 and v* 0.31, respectively. This is in accordance with known results [38, 39] where the WC becomes unfavorable eventually to the hierarchy of many electron BC's. 48 x 103 x 103 6 4 4 2 22 0 0 2  2 4 4 6 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 V V Figure 4.4: Shear modulus (in units of e2/47f3) as a function of partial filling factor v* for the n = 3 Landau level and for the WC (left panel) and 2e BC (right panel) using the order parameter approach. The above classical approach manages to reproduce general properties for the 2D electron gas and show an indication of stability interplay between the different states, but it is unable to adequately capture the quantum physics associated with the correlated electron system and in particular the fact that the electron wavefunctions extend a considerable distance (of the order of the magnetic length) around their lattice site positions, which radically alters their short range interactions (captured by the Fock term in our model). This semiclassical picture ignores that fact by assuming a stepfunction density pattern, localizing the pointlike electrons around their equilibrium sites. As a result, any further attempt to investigate the energetic of the different phases will not be accurate enough in reproducing realistically the short range interactions associated with electron wavefunction overlaps. In what follows we attempt to improve on this approximation. 4.1.2 Microscopic Approach In order to incorporate the quantum physics of 2D electrons more faithfully in our model we have to build a microscopic theory of the electron wavefunctions and derive, to HartreeFock level, information about the energetic and stability of the system. For the microscopic theory we will use the noninteracting electron wave functions in the symmetric gauge given by Eq. (3.18). The difference with the semi classical approach applied earlier, is in the Ansatz for the local charge density which we can improve by assuming that for the general Melectron isotropic BC the real space approximation of it becomes [36, 38, 55] M1 n,(r) E '' (r Ri)l2. (4.31) i m= 0 In other words, we assume that the electrons in the Melectron isotropic BC con figuration are in their noninteracting eigenstates (characterized by the zcomponent of angular momentum quantum number m, and the Landau level index n) and by Pauli's exclusion principle are forbidden to occupy identical states. The spin degree of freedom is assumed to be frozen by the high magnetic field and does not contribute. For strictly perpendicular magnetic fields this is accurate, but the existence of an inplane component will change that, since it will couple with the electronic spin and force it to become relevant. This microscopic approximation is better than the semi classical one, since the important short range physics (coming from the electronic wavefunction overlaps) is taken into consideration. The Fourier transformation of the projected electronic density is defined according to Eq. (3.14) as p (q) q 2 L(q / (4.32) Ceq22/4Ln(q2f2/2) ' and the generalized HartreeFock cohesive energy similar to Eq. (4.28) assumes the form EH t q 2 EH 1 2 (22 VH(q) (4.33) Since we are interested in the local density per bubble, it will prove useful to separate the lattice summation from the density by defining the projected density at a given bubble as pn(q) = Z nm(q) so that pn(q) p(q) q. (4.34) and the corresponding electron density at a given bubble nn(q) is given by an equation similar to Eq. (4.32), namely M1 hfn(q) / dr'. (r) 2e qr (4.35) mO 0 The projected density for a given Landau level and given angular momentum m assumes the form n(q) f dr'. (r) (236qr Sq22/4L,(q2f2/2) (4.36) If we perform the above integration we find identical results for both n = 2 and n = 3 cases (independent of n) rendering the n index unnecessary, and allowing us to drop it whenever possible to simplify the notation. Below we list the results we find for the projected electron densities per bubble (for the two Landau levels n =2, 3) and for the first three angular momentum cases po(q) e24, (4.37) p(q) ( 2) 24, (4.38) p2(q) q2f2 + q )q 22 /4. (4.39) Using the above analytic expressions we find the following result (depending on angular momentum index m) for the 2D interaction of the electrons U jmm,(r) (272 pnm(q)VHF (q) pm nq)eiq) (4.40) x 10 x 10 3  2 5 3 Figure 4.5: Shear modulus (in units of e2/47c3) as a function of partial filling factor v* for the n = 2 Landau level and for the WC (left panel) and 2e BC (right panel) using the microscopic approach. We show in Appendix B the Fourier transforms of the above interaction potential for different m values and for the 2e BC case. The above general expression can be used to find the cohesive energy associated with such a system, namely EHF mm' Rj) + U (0), (4.41) i4j T7,T7' i Tm which has the general form of Eq. (4.6), besides the i i i, !" term associated with the interaction of electrons within the same bubble. This term does not contribute to the elastic properties of the system, since these degrees of freedom are associated with deformations of the internal structure of the bubble, which we consider higher order corrections in this kind of elastic approximation that we apply to the electronic system. Additionally, we see that the above expression does not allow for selfinteractions among the electrons that lie on the same bubble. The modified interaction potential defined in Eq. (4.40) incorporates quantum effects among electrons and presents a better approximation to the bare HartreeFock interaction since it averages on the spatial effect of the presence of other electrons. Using this interaction potential we can reevaluate the shear modulus from Eq. (4.24). x 10 x 10 005 01 015 02 025 03 035 4 045 5 0 005 01 015 22 025 03 035 04 045 05 V V Figure 4.6: Shear modulus (in units of e2/4wc3) as a function of partial filling factor v* for the n = 3 Landau level and for the WC (left panel) and 2e BC (right panel) using the microscopic approach. Our results for the WC and 2e BC in the n = 2, 3 Landau levels are shown in Figs. (4.5 4.6). We notice that the partial filling factor region of stability for a given case remained the same as in our semiclassical results, but approaching halffilling the behavior has become different. Notice that by increasing the filling factor amounts to decreasing the magnetic field or increasing the density of electrons in the system (with the effect of reducing the crystal lattice parameter value, as can be clearly seen by Eq. (4.4)). This parameter change renders the short range physics more relevant and their effects more well pronounced. Consequently, we see for example in Fig. (4.3) that the n = 2 isotropic WC will become reentrant close to half filling according to the semiclassical model but in Fig. (4.5) (where short range physics is accounted for by the microscopic model) this never happens. This is to be expected since as we mentioned earlier, i< l i,11i. Ii, ii is implemented by the direct long range term in the Coulomb interaction while conglomeration is due to the short range exchange term. The semi classical model favors (by construction) the former, while the current microscopic model attempts to incorporate the quantum physics of wavefunction overlaps, which affect dramatically the importance of the latter term. Another exhibition of this, is 0 021  0 019 0017 0015 0 013 0011 0 009  0 007 0 005  0 2 4 6 8 10 12 14 16 18 20 22 24 r Figure 4.7: Interaction potential Uoi(r) (in units of e2/4wc) vs. r (in units of f). the characteristic nonmonotonic behavior that is observed in the shear moduli which signals the onset of dominance of the exchange term. 4.1.3 New State: Bubble Crystal with Basis As we mentioned earlier the internal degrees of freedom in a bubble have been considered higher order corrections to the physics investigated in this work. This might not necessarily be true, since we have not systematically probed on those degrees of freedom due to the difficulty such a task presents when treated in a microscopic level. Nevertheless, as a preliminary attempt of investigation, we can offer the fol lowing special case which can be easily incorporated into the current model. We can focus our attention on the isotropic 2e BC case, and allow the two electrons to assume a finite distance from one another within the same bubble. This can serve as a rudimentary approximation for internal structure. The possibility of such a state arises, if one plots the interaction potential between these two electrons within a bubble (Uoi(r) given by Eq. (4.40) for the m = 0, m' = 1 case). The Fourier transform of Uoi(r) is shown in Eq. (B.21) of the Appendix. Surprisingly, there is a well pronounced local minimum at an interelectron distance of ro 1.48 (as shown in Fig. (4.7)) that I'', I the possibility of bubble deformations mediated by the minimization of the electronic repulsion. In other words the two electrons in the bubble might adjust their guiding center coordinates at this optimum distance ro in order to minimize their repulsion. The difficulty of describing a general lattice with basis of this sort is considerable (and out of scope for this work) so we would prefer to investigate a limiting case that constraints the electrons to displace their guiding centers along one direction only. This limiting case should be able to indicate if these internal degrees of freedom have a prominent role in the physics of the 2D electron system. The way we can implement a finite distance between the two electrons in a bubble is by redefining the lattice vectors associated with them as rm, = Ri + (m )rox, m =0,1, (4.42) 2 where Ri are given by Eq. (4.26) and m distinguishes between the two electrons. In order to study the stability of this < ii 1li iii structure we need to derive an expression for the dynamical matrix starting from the cohesive energy given by Eq. (4.41) and Taylorexpanding to second order in the displacements. As alv,v, we defer to the Appendix all the cumbersome details and present here the final result A2) = eiQro( in') me')Un,(Q + q)(Qa + qj(Qf + q3) C Q UTmm(Q)QaQaj. (4.43) Notice that in the limit ro 0 we reproduce the usual result of Eq. (4.13). For the shear modulus calculation, we follow the usual procedure of expanding the total dynamical matrix S(q)1 (4.44) (q () 3x10 x.10 005 01 015 02 025 03 035 04 045 05 0 005 01 015 02 025 03 035 04 045 05 V V Figure 4.8: Shear modulus (in units of e2/47c3) as a function of partial filling factor v* for the n = 2 (left panel) and n = 3 (right panel) Landau level for the bubble < i I J1 with basis. in small q and we reproduce the same result of Eq. (4.24) but with the interaction potential U(q) given by U(q) = eiq.ro(mmUmm'(q) = VHFEq)e q2/2 1 + cos(,o) 1 + cos(qro))q22 + q 4] (4.45) Lq2 + cos(q8ro) 2 ( In the Appendix we show analytic expressions for the first and second derivative of the above potential. We present our results in Fig. (4.8) for the shear modulus of the BC with basis for the n = 2, 3 Landau levels. We notice that the (il I 11ii. structure appears quite stable. This is a first indication that the internal degrees of freedom might p1 i, an important role in the physics of the 2D electron system. This BC with basis state is not a solution of the HartreeFock equation (contrary to the rest of the M electron states [38]). It arises as the variational solution though of the HartreeFock Hamiltonian satisfying aE0h(r 0, (4.46) Or where ro 1.48f. In other words, this BC with basis state is the best approximation we can build at this point that probes the internal degrees of freedom associated with bubble deformations, and minimizes at the same time the HartreeFock energy. Below we evaluate the normal modes associated with all the above < iI ,ii:,,w structures as a final test of stability. 4.1.4 Normal Modes and Zero Point Energy In order to study more systematically the stability of a < i 1 ,iii,. configuration we need to calculate the normal modes (for different filling factors) showing that they have real dispersive nature instead of a diffusive one, which is characteristic of instabilities. This kind of calculation is based on the elastic matrices (evaluated earlier for the different crystalline states) since it investigates the dynamics of an electron under the presence of the perpendicular magnetic field and in the vicinity of elastic forces coming from the rest of the electrons in the system. Our most accurate model is the microscopic one so we would like to use the elastic matrices associated with it to perform the normal mode calculation. The elastic force associated with electronic interactions is given by the real space dynamical matrix which starting from the cohesive energy formula Eq. (4.41) and expanding to second order in the displacements is found to be Z Z'(Ri) "(R,) omm' (4.47) where 67 (r) 0aT U~Tn, (r), (4.48) and the interaction potential of the different electrons inside a bubble is given by Eq. (4.40). Notice that another path of approach could be to Fourier transform Eq. (4.13) back to real space but there is a multiplicative constant associated with that as we comment in the Appendix. Having written the dynamical matrix in real space the equation of motion for an electron of mass m and charge e > 0 in the presence of a perpendicular magnetic field B and elastic forces associated with interactions from the rest of the electrons becomes m dt2 E a3 (Ri Rj) lj eBa (4.49) This equation covers both the WC and 2e BC cases since the indices m, m' distinguish between electrons in the same bubble. For the BC with basis one has to repeat the procedure from the beginning starting from Eq. (4.41) but using the lattice vectors of Eq. (4.42) only to find the following expression for the dynamical matrix KIf'(R, Rj + ro(m m')x) >II >I ,w (Rk + ro(m m")x) T" k Q '(R( Rj + ro(m mx), (4.50) where one of Eq. (4.48) in the ro  0 limit. The equation of motion for the two electrons in this kind of BC becomes md fu 4 .'(R R, + ro(m m')x)u eB (4.51) jm' We show in the Appendix in great detail how to solve the above equations of motion and derive the normal modes for all the ( We show our results for the three different isotropic (< iii configurations evaluated on the irreducible element of the first Brillouin zone in Fig. (4.9). In general, there is a gapless mode (magnetophonons) for long wavelength excitations and there is a gapped mode (magnetoplasmons) (of the order of wc) associated with interLandau 00717 . 727 0072 074 0 0627 0 6567 0063 0 71 .: . 0 0538 0 5926 0054 068 S0448 0 528 0 45 0 6s 0o0358 ... 0 4645 0036 .. .062 0 0269 ..' 4 004 0027 059 00179 0 3363 0018 056 0 009 0 2723 0009 053 0 2082 00 5 rq/(f/a) rq/(f/a) 0 09 O 8 0 081 077 0 072.20. 0 74 0and 063a is the lattice paramet 07 0 054 states by picking points inside the irreducible first Brillouin zon68 0 045 0 65 0 036 0 62 0 027 0 59 00/8 056 0 009 053 1 q/(ff/a) I Figure 4.9: Normal modes for the triangular crystalline structures in the n 2 Landau level and on the irreducible first Brillouin zone element. Top left panel: WC at v 0.18. Top right panel: 2e BC at 0.30. Bottom panel: BC with basis at v* 0.20. The left axes correspond to magnetophonons (lower curves) and the right axes to magnetoplasmons (upper curves). Frequency measures in UJ0 e2 /47rWM units and a is the lattice ( i vI ,J parameter. level excitations. For the WC case we reproduce well known results [56], and both the 2e BC and BC with basis show similar structure in their modes. Their degrees of freedom are doubled (due to the presence of an extra electron) which doubles their magnetophonon and magnetoplasmon modes as well. We comment in the Appendix in great detail on the graphical peculiarities associated with plotting elements of a Brillouin zone. Finally, one can evaluate the zero point energy associated with the normal modes of these states by picking I., : X!" points inside the irreducible first Brillouin zone Table 4.1: Zero point energy of WC, 2e BC and BC with basis (BCb) for different partial filling factor values v* within their range of stability. v* Ewc EBC EBcb 0.05 0.359785* 0.360259 0.360394* 0.10 0.362771 0.357701 0.358184 0.15 0.372628 0 :",7il 0.359024 0.20 0.375746 0.360150 0.361797 0.25 0.362931* 0.362661 0.361640* element that have the largest weight and then writing the zero point energy (in units of e2 /4wc) as [57] 1 4wcth Nmodes 6 EZP 2M e2 E aiLj(qi), (4.52) j= 1 i 1 where Nmodes is the number of modes for the given crystalline structure (two for WC, and four for the 2e BC) while ai is the corresponding weight of the given special point. Following Cunningham [57] the special points for the WC hexagonal lattice configuration and their corresponding weights are 21 3 2 1 4, 1 8 1, 3 (4.53) q46: {(,(1, ), (2, ), 2(5, )} a4, 6 2, (4.54) where all reciprocal lattice vectors measure in 7/a units. In Table (4.1) we present our results for the zero point energy associated with the different isotropic crystalline structures investigated so far. The ones marked with an asterisk indicate the onset of instability for the given structure and the corresponding filling factor. 4.2 Energetics of Isotropic Crystalline Phases At this point we would like to consider the energetic of the states whose stability we calculated in the previous sections and in particular investigate the possibility of the inert filled Landau levels altering the energy interplay between different crystalline phases. Also, another point of concern is the finite thickness of any real sample of 2D electrons and how the extent of the wavefunctions in the zdirection can potentially influence the energetic of the system. We would like to investigate both of the models developed earlier, the classical approach and the microscopic approach, by evaluating the cohesive energies, for the different crystalline configurations, given by Eqs. (4.28, 4.41), respectively. In order to incorporate finite thickness effects and screening from filled Landau levels we follow the general path which consists of modifying the dielectric constant so that it acquires a qdependent structure of the following form [58, 59] e(q) e 1 + ( J2 (qRc)) exp (Aqf), (4.55) qaB where c is the bare dielectric constant of the substrate material (for GaAs it is 12.9co), Re = 2n + 1 is the cyclotron radius, aB the Bohr radius, Jo(x) the zeroth order Bessel function, and A a finitethickness parameter involving the finite zdirection extend of the electron wavefunction. The result of the above modification is to add an extra qdependence on the Hartree term of the Coulomb interaction (given by Eq. (3.22)) but overall the expression remains of the same form. On the contrary, for the Fock term (given by Eq. (3.23)) the expressions are no longer analytic (as the ones shown in Appendix B) due to the complicated structure of the integral involved. Below we evaluate the cohesive energies associated with the two models (the semiclassical and the microscopic approximation) and for the different crystalline phases investi gated so far. For the sake of completeness we will include in our energetic comparison for the n = 2 and n = 3 Landau levels the classical stripe state (charge density wave) which consists of continuous stripe areas of the sample that are completely filled by a t .. 01i 1 .. (M1)bare (M1) bare (M=2) bare (M=2) bare (M=1)screened  (M=1)screened 0 0  (M=2) screened 0 05 =2) screened .005 \. (M=1) screened,finite thickness ...... .. (M=2) screened,finite thickness  LU 015 0 15 02 02 0 25 0 25 0 005 01 015 02 025 03 035 04 045 05 0 005 01 015 02 025 03 035 04 045 05 V V Figure 4.10: Cohesive energy for the isotropic WC (red) and 2e BC (blue) in units of e2 /47c for the n 2 Landau level (left panel) and the n 3 Landau level (right panel). Solid lines correspond to bare Coulomb interaction. Dashed lines correspond to Coulomb interaction where screening effects are accounted for, and dotted lines correspond to the latter case where finite thickness effects are included as well. electrons (v* 1= ) separated by empty areas (* = 0) of finite width. The cohesive energy associated with this phase will be shown below. 4.2.1 Cohesive Energy of Modified Coulomb Interaction: Classical Model We numerically calculate the bubble crystal cohesive energy per electron based on Eq. (4.28) (in units of e2/47rc) and given by nC 1 v* 47r 2 J (2 Q) Ecoh 1 VHF 4Q JrBQ) (4.56) h 27f2 M e2 (rBQ)2 where as usual, M is the number of electrons per bubble, v* the partial filling factor given by Eq. (4.2), and rB the bubble radius. The neutralizing background cancels the singular Hartree term involved in the Q = 0 case, but the nonsingular Fock term is maintained (the weight factor involving Ji(x) is evaluated at the Q = 0 limit and it is easy to show it gives 1/2). The general Melectron BC we investigate here incorporates the WC case as well and for the dielectric constant we use the modified expression given by Eq. (4.55) to incorporate finite thickness effects and screening from the filled Landau levels. Our results are shown in Fig. (4.10) for the n = 2 and n = 3 Landau levels, respectively. We see that the screening effects from the inert Landau levels along with the finite thickness effect from the electron wavefunction extend in the zdirection only shift the associated cohesive energy scale, but do not alter the interplay between the phases. At approximately the same partial filling factor (v* _ 0.22 for n = 2 and v* ~ 0.17 for n = 3) the 2e BC < il I becomes more favorable compared with the WC, irrespective of the type of modification applied to the Coulomb interaction. As a result we can conclude that ignoring these corrections in our model we are not missing out on important physics besides some quantitative adjustments. 4.2.2 Cohesive Energy of Modified Coulomb Interaction: Microscopic Model The microscopic model we developed earlier provides a much more accurate approximation for the energetic interplay between the different crystalline states. We evaluate the cohesive energy per electron (in units of e2/47cf) for a general M electron BC state by starting with Eq. (4.41) and using the Fourier transform of Umm,(r) (given by Eq. (4.40)) to find co f e2 y / : U (Q) (4.57) Q As we mentioned in the classical approximation, for the summation in Q we retain the Q = 0 term only for the Fock part. The above expression incorporates the M 1 WC case as well. For the BC with basis we have to start again from Eq. (4.41) and use the lattice vectors associated with this structure (given by Eq. (4.42)). We then Fourier transform and use the by now familiar Eq. (4.10) to find the following expression for S 0Wigner crystal l 0~ ",0 02 0 1 006 LIJ 015 LU 008 02 01 025   ~ 0 12 03 0 14 0 35 0 16 0 005 01 015 02 025 03 035 04 045 05 0 005 01 015 02 025 03 035 04 045 05 V V gner crystal 0 bubble rO=00 002 1 0 04 008 0 005 01 015 02 025 03 035 04 045 05 V Figure 4.11: Cohesive energy in units of e2/47c for the isotropic WC, 2e BC, BC with basis, and stripe state for different modifications of the Coulomb interaction and for the n = 2 Landau level using the microscopic model. Top left panel: bare Coulomb interaction. Top right panel: screened Coulomb interaction with no finite thickness effects. Bottom panel: screened Coulomb interaction with finite thickness effects included. the cohesive energy per electron (in units of e2/4K7E): E BCb )iQ)ro(mm') (4.58) cob.YY Z U 1mm1(Q ) (4.58) mm' Q The above needs to be applied for the specific 2e BC with basis case we have developed in this work. The expression simplifies to Eohb Uoo(Q) + U11(Q) + 2 cos(Qxro) Uoi(Q)], (4.59) 4 C  wg... clysal *"bubble r0_0 0 0 05k 64 015 0 0  005 0 02 0  S025 0 LU" LL\ M006 0 02 08 0 005 01 015 02 025 03 035 04 045 05 (V Figure 4.12: Cohesive energy in units of e2/ 4w for the isotropic WC, 2e BC, BC with basis, and stripe state for different modifications of the Coulomb interaction and for the n = 2 Landau level using the microscopic model. Top left panel: bare Coulomb interaction. Top right panel: screened Coulomb interaction with no finite thickness effects. Bottom panel: screened Coulomb interaction with finite thickness effects included. where we used the fact that Uoi(q) = Uio(q). As we mentioned earlier we would like to evaluate the cohesive energy for the stripe state as well and compare it with the rest of the crystalline structures. The stripe state cohesive energy is constructed by starting from Eq. (4.28), and assuming for the local filling factor [36, 39] v*(r02 0(a/2 x x (4.60) v* (r) 0 (a/2 Ix xjl) (4.60) where a = v*as is the width of one stripe (determined by the partial filling factor), while xj = jas, and as is the stripe periodicity. By Fourier transforming the above we find that the cohesive energy of the stripe phase (in units of e2 /47c) is of the form S 1 e H 27W sin2(TQ *j) coh w2 *e2 > v Ka) .2 (4.61) The optimal stripe periodicity is obtained by minimizing the above with respect to as. Following Goerbig et al. [39], we use as = 2.76Rc for n = 2 and as = 2.74Rc for n = 3. As ahvi the Q = 0 term is retained in the Fock part of the potential and the limit of the weight factor (at j = 0 it gives 7v*) is taken. We present our results for the bare Coulomb interaction, and for modifications associated with finite thickness and screening from filled Landau levels in Figs. (4.11 4.12). Our conclusions from the classical approximation analysis remain intact for both Landau levels, which leads us to the safe generalization that finite thickness effects, and the inert Landau levels can be ignored in any further investigation of the energetic among the (< i i 11iiw. phases. However, another surprising result has emerged. The BC with basis state has become the undisputed winner in the energetic interplay with a very distinct energy difference from any other state for a wide range of partial filling factors. This is another strong indication that the internal degrees of freedom pl i, a crucial role in the physics of the system. What this result infers is that different ( one investigated here) might p1l v a strong role into the physics of the quantum Hall system. This can be a future direction for our research to further investigate the possibility of structural transitions in these kinds of systems. Finally, we should draw attention to the conventional stripe state winning over the conventional WC and 2e BC states when halffilling is approached, something that is to be expected according to previous investigations [36, 38, 39]. CHAPTER 5 ANISOTROPIC CRYSTALLINE PHASES 5.1 Solving the Static HartreeFock Equation In our previous treatment of the 2D electron system we approximated the Coulomb interaction in the HartreeFock level and used an Ansatz for the electron density (being either a classical order parameter or a microscopic approximation based on noninteracting electron wavefunctions). We would like to progress further in that direction in faithfully capturing the electronic density characteristics of the system by solving for the eigenfunctions of the HartreeFock equation and finding the quasi particle states associated with the 2D electrons. This constitutes an improvement on the microscopic model and allows us to systematically study the different ( i i ,1 line phases in the set of lattice symmetry associated with the triangular lattice configuration. We plan to include anisotropy into the ii iii., structures and investigate how the energetic are affected. Our starting point is the HartreeFock Hamiltonian (similar to Eq. (3.24)) that is used in studies of the 2D electron system under the presence of a high perpendicular magnetic field and given by [36, 38, 39] 1 f d2q 2 HHF VHF(q) n(q) 2, (5.1) where n(q) is the electronic density and VHF(q) is the modified HartreeFock inter action potential given by Eq. (C.5). To avoid overloading our presentation of the model we will place all the cumbersome definitions and calculations in Appendix C and retain only the essential ones necessary for the presentation. We define the electronic quasiparticle density for an Melectron BC according to M n(r) E (r R) 2 (5.2) i a=1 where, ,(r) is the aeigenstate of the above HartreeFock Hamiltonian. Notice that this is a similar definition to Eq. (4.31) only we use the more accurate quasi particle wavefunctions instead of the noninteracting electron wavefunctions. Of course, we have no prior knowledge of the former (and in fact we need to find them selfconsistently) so it is necessary to approximate their form by expanding them on the latter basis of noninteracting wavefunctions y(r) according to Nsl *' ,(r) = Cmam(r), (5.3) m=0 where, N, is the dimensionality of the truncated Hilbert space used. We show in the Appendix in detail how extremizing the above Hamiltonian with respect to the quasiparticle wavefunctions and then projecting the result onto the noninteracting particle wavefunction basis we obtain the following eigenvalue equation for the M electrons associated with each bubble in the BC N., > Gm 2mCm EaCmra, (5.4) rn1 0 where Cma are the expansion coefficients associated with Eq. (5.3), GmimT is a 2nd rank tensor given by Eq. (C.12) in the Appendix, while gmTnmn3m4 is a 4th rank tensor associated with the overlap integrals of the quasiparticles and given by Eq. (C.13) in the Appendix. We numerically diagonalize the above equation until our solutions for the expansion coefficients converge within a 104 accuracy. We comment in the Appendix on the details of the algorithm. Once the algorithm has converged, we place each of the M electrons of a bubble on the quasiparticle states associated with 68 0 01 0I 05 0 05 \ 0 05 0 15 LU..... LU1 o \ 02 015 025 0 '02' 0 005 01 015 02 025 03 035 04 045 05 0 005 01 015 02 025 03 035 04 045 05 V* v Figure 5.1: Ground state eigenvalue energies in units of e2 /47c as a function of the partial filling factor v* for the HartreeFock equation. Left panel: WC and 2e BC cases for the n = 2 Landau level. Right panel: WC, 2e BC and 3e BC cases for the n = 3 Landau level. their eigenvalues E, in ascending order. As a result the cohesive energy of the general Melectron BC assumes the form Ecoh= 1 E, (5.5) where the factor of 1/2 compensates for counting each pair of electrons twice in the general Hamiltonian given by Eq. (C.8) [60]. The above is a general result for an Melectron BC so it can be easily applied to our cases of interest for the WC and the 2e BC. Additionally, we can study the 3e BC as well, which according to previous studies [38, 39] can become energetically favorable for the n = 2 and above Landau levels at a certain range of partial filling factors. We show our results in Fig. (5.1) and as we see they are identical to the ones developed earlier within the more simplified microscopic model. This is a verification that the microscopic model used earlier for the Melectron BC configuration is a solution of the HartreeFock equation, in agreement with previous studies [38] where the solution of the timedependent HartreeFock equation produces the hierarchy of Melectron BC's. It should be noted that for the above results we have used a minimum expansion 69 08 02 07 005  06 01 05 015 04 02 03 025 02 0 3  0 005 01 015 02 025 03 035 04 045 05 0 005 01 015 02 025 03 035 04 045 05 V* V* Figure 5.2: Left panel: Values of anisotropy that minimize the cohesive energies for WC and 2e BC for different values of v*. Right panel: Ground state energies associated with the minimizing values of anisotropy for WC and 2e BC. We also plot the traditional stripe state for comparison. basis dimension (Ns), disallowing the electrons to form hybrids by constraining them to lie on their noninteracting ground states, for all the < ii 111ii,, cases studied. So the fact the we get identical results with the simplified microscopic model developed earlier is not a surprise but it serves as a consistency check for this kind of improved method before we employ it for the much harder task which we discuss below. 5.2 Introducing Anisotropy into the Crystalline States As we explained in the introduction the contemporary theoretical results for the 2D electron system under the presence of a perpendicular magnetic field predict that close to halffilling, the stripe state (charge density wave) becomes favorable to all the (< i iii.,,i phases. This stripe state is described in the continuum order parameter language (discussed earlier) by introducing the partial filling factor of Eq. (4.60). One would expect though that the transition from a crystalline to a liquid phase would be less abrupt (especially at fixed low temperatures) allowing for the < l,,I ,11,i, system to explore internal degrees of freedom before finally melting into a liquid. Also, to a certain extent, one would expect reversibility (no hysteresis) associated with the decrease or increase of the applied magnetic field around the region that the stripe state becomes favorable. All of the above point to the importance of the internal degrees of freedom associated with the crystalline phases, which for the subset of triangular symmetry discussed in this work, translates into introducing anisotropy in the (ii iI Iii, configurations. Escaping out of this subset will allow us in the future to investigate a more complete group of structural phase transitions which will potentially involve reorientation of the unit cell due to local straining forces arising from the electron correlations. In order to incorporate anisotropy within the triangular lattice symmetry we have to redefine the lattice vectors given by Eqs. (4.264.27) by introducing the anisotropy parameter E (not to be confused with the dielectric constant e) which is zero for no anisotropy and one for complete anisotropy. The new direct and reciprocal lattice definitions become R3 t' 1 2 v)) (5.6) Q1T \( a (5.7) where the Melectron BC lattice parameter is still given by a M (5.8) Incorporating the above new definitions into our code is straight forward and one has to pick different anisotropy values to investigate (for given partial filling factor) which one minimizes the cohesive energy of the given < i ,~I iii., structure. We show our results in Fig. (5.2) for the simplified microscopic case, where we do not allow interelectron excitations within the WC and 2e BC by choosing the expansion basis dimension to equal the number of electrons per bubble. The left panel shows the specific minimizing values of anisotropy for given value of v* for . . i . . 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'" ". "". "". "".". ". "".." """". ". "'". ""."."".""" ..".""" ." "."'"." '" "."".""".""" .." ." .. .. .. ... .. ... .. .. ... .. .. ... .. ... .. .. ... .. .. ... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ...=O.. 7 5.. .. . Figure 5.3: Reciprocal lattice points for different values of anisotropy. We see how the the WC and 2e BC, while the right panel shows the ground state cohesive energies associated with these minimizing values of anisotropy. We also plot the traditional stripe state cohesive energy (given by Eq. (4.61)) and as we see a surprising result emerges since it does not become energetically favorable over the anisotropic WC state even close to halffilling. In fact, the anisotropic WC state increases considerably its energy difference from the rest of the states, as halffilling is approached. Also, the overall cohesive energies have dropped in value, compared to the isotropic ones from Fig. (5.1). In view of these results one is obliged to reconsider the definition of stripes in these < i i iii. systems in terms of anisotropic X< i 1 and investigate further the effect of anisotropy in the system. Before we proceed, we would like to show how anisotropy deforms the reciprocal lattice vectors by plotting a finite number of them given by Eq. (5.7). Our results are shown in Fig. (5.3), where reciprocal lattice vectors associated with the cohesive 09 WC WC 2o BC 'BC 0B 02 0 8 0 1  07 02 o 06 0 3 05  04 0 6  028 0 7  0 1 0 005 01 5 015 02 025 03 035 04 045 05 0 005 01 015 02 025 03 035 04 045 05 07 02 S03 e 0   04 ;' m 031 06 02 ' 0 7 \ 0 005 01 0 15 02 025 03 035 04 045 05 005 0 1 05 2 025 03 35 04 045 0 V* V* Figure 5.4: Solutions of the HartreeFock equation for the anisotropic triangular lat tice configuration. Top left panel: Values of anisotropy that minimize the cohesive energies for WC, 2e BC and 3e BC for different values of v* and for the n = 2 Landau level. Top right panel: Ground state energies associated with the minimizing values of anisotropy for WC, 2e BC and 3e BC. Bottom left panel: Same as top left panel but for n = 3 Landau level. Bottom right panel: Same as top right panel but for n = 3 Landau level. energy lattice sum and the overlap integrals (given by Eq. (C.13)) are plotted for representative values of anisotropy. We see a channellike structure emerges consisting of onedimensional periodic chains of electron guiding centers. In other words, the < i *l 1iiw., discreteness of broken translational invariance is alv, maintained in this novel "stripe" configuration. This is contrary to the traditional stripe state properties, where translational invariance along the stripes is restored. As we will see below, this is a crucial difference that radically alters the elastic properties of the system as well. We should also mention here that in the HartreeFock approximation, the electronic wavefunction overlap is greatly favored by the Fock term. When anisotropy is introduced into the system the electronic guiding center lattice points are brought into closer proximity enhancing wavefunction overlaps and thus improving on the effect of the Fock term. That is the reason why the overall cohesive energy values are reduced compared to the isotropic ones. One should also mention at this point that a hidden degree of freedom emerges in view of this analysis. When the electronic guiding centers are brought within proximity, the dimensionality of the truncated space used as a basis to describe the electron wavefunctions becomes crucially important. The reason is best understood through the interplay of the Hartree and Fock terms. The more available states exist for electrons within a bubble to occupy and form hybrids, the more the Hartree term is optimized since the electronic charge will be spread out the most. On the other hand, the Fock term, according to our analysis above, is optimized as well since electronic overlaps will be inevitable (but less concentrated) within a bubble. Additionally, the more noninteracting electron wavefunctions (eigenfunctions of the zcomponent of angular momentum) are used in the expansion of Eq. (5.3) the more they will extend around the BC sites causing "seo, ,i,1 i y interbubble overlaps, enhancing even more the effect of the Fock term. This is of course a computational byproduct that one cannot get rid off unless one goes beyond the HartreeFock approximation which is beyond the scope of this work. For practical purposes, if a reasonable number of states is used in the truncated quasiparticle wavefunction space, and convergence is assured through the algorithm, that should be enough to capture the essential physics of the system. We should also notice at this point that the zcomponent of angular momentum eigenstates change their parity with m, meaning that an electron within a bubble will only form same parity hybrid states, using either even or odd values of m. This is verified numerically independent of the dimensionality of the basis space. As a result, for the WC case for example, if one wants the electron to form a hybrid state using five noninteracting electron states one needs to use a dimensionality of N, = 9. For the 2e BC one needs to add the extra odd parity state and use N 10. We show our results for the n = 2 and n = 3 Landau levels in Fig. (5.4) where we have used appropriate dimensionality for the different crystalline structures so that a total number of five noninteracting electron wavefunctions participate in the hybrids. For the M = 3 states though, we have not increased to N 15 since from our experience that state never becomes energetically favorable anyway. As it is clearly shown in these results, the 2e BC becomes irrelevant as well, contrary to our Fig. (5.2) results, and the anisotropic WC seems to be the natural way that the internal degrees of freedom in the crystalline system optimize the ground state as the applied magnetic field is changed. C!i ,i: i5i the truncated quasiparticle wavefunction space does not seem to alter the energetic between the states but as we mentioned earlier all the cohesive energies suffer a downward shift due to the dominance of the negative Fock term into the numerics. Another point of interest is on the transitions from isotropic to anisotropic WC that appear to be of first order and taking place around v* 0.1 for both Landau levels. These transitions appear in the BC cases as well but are not as strong. In light of our discussion earlier about the dominance of the Fock term in the short range physics, along with the existence of the hidden degree of freedom (the truncated quasiparticle wavefunction dimensionality) this is to be expected. Going from the isotropic state to an anisotropic one radically affects the shortrange physics which is controlled by the value of anisotropy in the crystalline structures. In the real system, a similar behavior can be expected as well for similar reasons, but the degree of freedom associated with the truncated space dimensionality is absent, or better stated, optimized. This can become plausible if one prints the actual electron wavefunctions on a unit cell. In order to do that we can focus our attention on the 0 6 4 2 0 2 4 6 8 10 12 14 16 18 20 x/I 8 8 6 6 6 4 2 0 2 4 6 8 10 12 14 16 18 20 6 4 2 0 2 4 6 8 10 12 14 16 18 20 x/I x/I Figure 5.5: Density profiles on the unit cell for the n = 2 Landau level and v* = 0.11 for truncated basis dimensionality of Ns = 9. Length unit is the magnetic length. Top left panel: Isotropic state. Top right panel: Anisotropic F = 0.8 case using only the m = 0 noninteracting state. Bottom left panel: Same as top right panel but for a hybrid of m = 0, m = 2 noninteracting states. Bottom right panel: Optimized hybrid state solution of the HartreeFock equation for E = 0.8. WC case and the n = 2 Landau level for simplicity. We place one electron on each of the three sites of the unit cell and write the electronic wavefunction as an expansion of the noninteracting wavefunctions 0m(r) given by Eq. (3.18). What we find is Ns 1 (r) (r), (5.9) m=o0 where am are normalized to unity coefficients truncated in a Hilbert space of N, dimensions. It is useful to rewrite the above in cartesian coordinates. For the specific case of the WC considered here one can easily show that the above written analytically in cartesian coordinates becomes b v(x, y) a r22x+iY2+ a, ( x+y2) (x+ ey 2/2 N 3 a m x 2 \ + am+2 Y L( 2 2(5.10) o v2 ( + 2)!f f 22 In order to build the density profile on the unit cell we need to define the appropriate lattice vectors (coming from Eq. (5.6) and associated with a unit cell placed at the origin) as R1 (0, 0), (5.11) 3 3a(,2(1 E). R2 1 02 5.2) R 1 2 ( '( R3 1= 2 1 ), (5.13) and then numerically evaluate for any different set of a,'s we want the probability density given by Itot(r) 2 (r R) 2 + i(r R2) 2 +  _(r R3) 2. (5.14) We show our results for the above density profile in Fig. (5.5) for different anisotropy values and hybrid states. Hybridization and the Hilbert space dimensionality have a dramatic effect in shaping the electron wavefunctions and consequently affecting their interactions. On the first two graphs, the electrons are prevented to form hybrid states, and are constrained on the m = 0 states. On the third graph, the electrons are allowed to from m = 0, m = 2 hybrids and on the last graph, we have used the optimization result from our code for the actual hybrid state solution of the Hartree Fock equation for the given values of v* and anisotropy. We notice that the latter is a radically different density profile from all the rest. 5.3 Elastic Properties of Anisotropic Crystals We can proceed further into studying the stability of the This might pose a difficulty since (as we discussed earlier) the short range correlations that exist in these anisotropic (< i I ,! force electrons to extend their wavefunctions at multiples of the magnetic length and render any perturbative expansion around fixed equilibrium positions invalid or inadequate. One has to employ a better method that avoids Taylorexpanding around the real electronic displacements. This can be achieved by following the Miranovi6 and Kogan approach [61]. We start by writing the general elastic energy density associated with a 2D anisotropic < i vI as el = Cll,(tu)2 + C,y(yUy)2 + C66, (OyU)2 + C66,y ( y )2 + 2cil,9y(.9u 9)( + 2c66,xy yU xUy) (5.15) The elastic moduli ci,x, cn,y are associated with uniform compressions along the x:, y directions, respectively. To describe shear deformations along the same directions we use C66,x, C66,y, respectively. The cross term c1,xy, introduces the interaction energy associated with the mixing of the compression modes directed along x and y and the same applies for the shear mode mixing associated with c66,xy. In the isotropic case, the above expression for the elastic energy density assumes the form of Eq. (4.16) but in the present case, the only symmetry left to impose a constraint on the elastic constants is rotational invariance, which imposes the following interrelation C66,xy C66, + C66,y (5.16) 2 This can be easily proved if one applies a uniform rotation u = ",,i, uy = uox (where u0 is a dimensionless constant) and demands invariance of the elastic energy. The method that Miranovit and Kogan have developed is based on these kinds of uniform deformations in the direct lattice that are traced back into deformations of the reciprocal lattice vectors. That way, one avoids expanding around direct lattice site fluctuations in order to calculate the elastic properties of different crystalline structures. The Miranovi6 and Kogan method relates a general uniform deformation of the form U = :, (5.17) where the coefficient Ua,3 is a dimensionless constant, to uniform reciprocal lattice deformations, which to first order in Ua,s can be written as [61] Qa = Qa U3,aQ3. (5.18) We have defined Q and Q to represent the deformed and undeformed reciprocal lattice vectors, respectively. Evaluating the elastic energy on the deformed reciprocal lattice vector set and subtracting the value associated with the undeformed lattice, suffices to reproduce the elastic constant associated with the given deformation. The general expression becomes c 2 [E(Q) EZ(Q)]. (5.19) For a shear deformation along the x direction given by ux = /, Uy = 0, the cor responding deformation in the reciprocal lattice vectors becomes Qx = QX, (5.20) Qy = Qy uoQ,. (5.21) For a shear along the y direction given by uy = uox, ux = 0, the corresponding deformation in the reciprocal lattice vectors becomes Qy = Qy, (5.22) Qx = Q1 uoQy. (5.23) A uniform rotation given by u = u,,, U, uox, induces the deformation Qx = QX uoQy, (5.24) Q Q + ,,, (5.25) and finally, a squash deformation given by ux uox, u = ",,/, induces the reciprocal lattice vector deformation QX (1 uo)QX, (5.26) Qy ( + uo)Qy. (5.27) In all of the above u0 is a small dimensionless constant. In this work we are interested in the shear deformations associated with the anisotropic crystals investigated, so we only focus our attention on C66x and c66y. We present our results for the shear moduli of the isotropic WC and 2e BC for the n = 2 Landau level in Fig. (5.6). They serve as a benchmark for this method of approach since our older results of Fig. (4.5) are reproduced (involving Taylor expansion on the lattice displacements). Our new results, pertaining to the anisotropic (< i iii. structures studied earlier, are presented in Fig. (5.7) where we plot c66x and c66y for the ground state of the anisotropic WC for the n = 2 Landau level. For the partial filling factor values where the isotropic WC is favorable, the two shear moduli are equal (it does not show in the figure due to different scales used) but when 012 / 01  0 08  o o06  004  002  004 006 Figure 5.6: Shear moduli (in units of e2 /4Ec) for the isotropic WC and 2e BC for the n = 2 Landau level produced using the Miranovic and Kogan approach [61]. the transition point to anisotropic WC is crossed (around v* 0.1) the c66y becomes vanishingly small. This is because this kind of shear deformation is along the direction of the channels shown in Fig. (5.3) which does not cost any energy (a characteristic property of smectics). The striking difference with a conventional smectic is that C66, becomes zero as well. The term in the elastic energy density of Eq. (5.15) associated with that elastic constant is replaced by a bending term K(0 u2)2 [62]. This is due to the fundamental difference between the conventional stripe state and the anisotropic WC: in the former, translational invariance is restored along the direction of the stripes but in the latter, this is no longer true. The periodic channellike structure persists at any value of anisotropy or filling factor. As a result, a deformation of the form u = cinx corresponds to a rigid rotation for the stripe state (with no energy cost associated with it) but it corresponds to a compression along the direction of the channels for the anisotropic WC case, with a finite energy cost associated with it. 5.4 Analysis of Experimental Results We are in a position now to discuss the experimental findings shown in Fig. (1.6). According to previous theoretical treatments of the dynamical response of an isotropic iO 025 ' 02 0 0 015 0 0 005  005 01 015 02 025 03 035 04 045 05 0 005 01 0 15 02 025 03 035 04 045 05 V* V* Figure 5.7: Left panel: Shear modulus C66, for the ground state of the anisotropic WC for the n = 2 Landau level. Right panel: Shear modulus c66y for the same crystalline structure. Notice that they both coincide in the range below v* 0.1, where the ground state is the isotropic WC, but it does not clearly show in the graphs due to the different scales used. The shear moduli measure in e2 /4cf units. WC under microwave irradiation [35], the resonance pinning frequency is given by >p  (5.28) PmLc where, pm = m/(7a2) is the mass density, uc the cyclotron frequency, and E is the quasiparticle selfenergy associated with the presence of disorder into the system and given by A ^, (5.29) ('( ,I so where, A is the variance of the random pinning potential, and o is the smallest correlation length between (the magnetic length), and Qd (the disorder correlation length). If we assume that the strong magnetic field imposes < d then we end up with the following result for the resonance pinning frequency A ~ MC6 (5.30) This result is generalized for the Melectron BC case and we specifically show the dependence on the partial filling factor that comes from the shear modulus c66 (the dependence on is weak for n > 2). In light of the experimental findings of Fig. (1.6) (right panel), one expects that the shear modulus associated with the isotropic WC state for the n = 2 Landau level will increase until around v* 0.19 where it should begin to decrease until the crystal becomes unstable. This is the exact behavior we find for the isotropic WC shown in Figs. (4.3, 4.5). On the other hand, the second coexisting phase shown in Fig. (1.6) starting around v* 0.15 follows the opposite trend, if compared with the Melectron BC shear modulus. For example, the 2e BC shear modulus is shown to go up and then decrease in Figs. (4.3, 4.5) for the region of interest (0.15 < v* < 0.35), while according to Eq. (5.30) and the experimental results of Fig. (1.6) it should have the opposite behavior. According to our energetic analysis the anisotropic WC becomes favorable around v* 0.1, which is reasonably close to the region that this coexisting phase reveals itself. All of the above point to the conclusion that this second peak appearing in the experimental data is not due to any isotropic Melectron BC but to an anisotropic crystal (WC or 2e BC). This seems to be supported by the results of Li et al. [63] for the AC response of a quantum Hall smectic which resemble the experimental ones. We expect the same conclusion to hold true for the n = 3 Landau level as well. CHAPTER 6 CONCLUSIONS We have studied different aspects of the quantum Hall system in high magnetic fields. At first we considered a bilayer system and studied the tunneling current characteristics associated with it in the incoherent regime. We found that the inter IV, interactions modify the tunneling current in two v,v. At first, due to the short range part of those interactions, the collective modes of the rilivr system become gapped. This leads to a suppression at low bias values of the tunneling current. Secondly, we found that the long range part of the interlayer interactions soften the effect of the Coulomb interaction among electrons in the same livr and as a result they shift the tunneling current curve to lower bias values. This is attributed to the excitonic attraction that a tunneling event creates between the tunneling electron and the hole that is left behind resulting into an overall reduction of the energy associated with such a process. Our study was analytical and systematic and was able to capture different properties of the experimental system, such as scaling behavior of different para meters or tunneling current response to interlayer separation change. Although we worked in the incoherent regime, we have set a foundation to develop this model fur ther, and incorporate coherence effects as well trying to reproduce the most recent experimental results. Further on, we studied the < i ii,. phases of the quantum Hall system by analyzing their stability. We built our theory successively starting from contemporary treatments of the problem using the classical order parameter approach, that averages on the electronic density neglecting the important microscopic physics, and evolved to the microscopic approach, that uses a more accurate Ansatz for the electronic density which incorporates, to a certain degree, the microscopic physics involved in the real system. We derived a general theory for the elastic moduli of such systems and studied for specific Landau levels the stability of the < i i 1 iii. structures finding that for different ranges of the partial filling factor these structures are stable. We also studied the normal modes associated with the above structures and in the process probed the internal degrees of a bubble finding that these degrees of freedom pl i" an important role into the physics of the system. Additionally, we investigated on the effect of the filled Landau levels and the pos sibility that screening arising from them might contribute significantly into the physics of the system and we also incorporated finite thickness effects, coming from the finite extent of the real system in the third direction. What we found is that although the actual cohesive energies shift in value, the interrelation among the different crystalline states remained the same, and consequently concluded that by omitting the inert filled Landau levels and finite thickness effects we are not missing out on important physics. Finally, we further improved on our model by solving the static HartreeFock equation associated with an electron in these quantum Hall systems and were able that way to find the quasiparticle wavefunctions for different < i lIi i ,' structures. In that part of the work, we were able to include anisotropy into the system and solve for the ground state, finding that there is a first order transition between the isotropic Wigner crystal and the anisotropic one that renders the latter the undisputed winner in the energetic race. We found that the anisotropic Wigner crystal for strong values of anisotropy resembles a smectic with much lower energy (for the whole range of filling factors) than the traditional stripe state. Additionally, we showed that for the anisotropic Wigner crystal, translational invariance is never restored along the smectic direction but shear deformations along that direction cost negligible energy. Our work was limited only in the subset of crystalline symmetry associated with the triangular lattice but we have set the foundation for a more detailed study on the possible structural transitions associated with reorientation and deformation of the unit cell due to straining forces developing among the electrons in these systems. This is part of our future direction, where we hope to find the ground states this kind of system evolves into when the applied magnetic field is varied. These ground states will provide to us realistic elastic constants, associated with the energy cost of deformations, which we plan to couple to a dynamical response theory where disorder effects and thermal noise will be incorporated as well. As a result we expect to be able to reproduce the experimental findings of microwave resonance response associated with these kind of systems. Additionally, we would like to investigate the excitonic condensation problem associated with the quantum Hall bilayer structures and attempt to shed some light into the physics involved in such a system where coherence among electrons in both I rv. i p, i, a dominant role. Having achieved the necessary understanding on that state we would like to couple the modes associated with it to tunneling electrons and reproduce that way the prominent zero bias coherence peak in the tunneling current found in experiments. APPENDIX A BILAYER SYSTEM EIGENMODES A.1 Single Layer Eigenmodes Here we provide an analytic derivation for the diagonalization procedure of the hili r quantum Hall system. We essentially repeat the procedure highlighted in the main text, including all the details, for reasons of completeness and in order to provide a coherent treatment of our theoretical model without having to reference formulas at the beginning chapters, which will result in a somewhat disjoint presentation. We start by introducing the single 1vr 2D system of electrons in the presence of a perpendicular magnetic field in the continuum elastic approximation. For a system of density no, the dynamics are described by the following Lagrangian L = no d2r r 2 efu A(u) A(Au)2 _ a (m'u, + OIU")2 JL 2 2no 'I + no d2r[V. u(r)][V'. u(r) 2 (A.1) +2 nr[0 u(r')] 47lr r'l  where the intir 1 r Coulomb interaction is treated in the continuum linear approx imation (charge fluctuations are given by 6n/no = V u). This approximation is correct in the absence of vacancies and interstitials. For the vector potential we choose to work in the symmetric gauge so that A(u) = (Buy/2, Bu1/2, 0) and B is the applied magnetic field. We decompose the displacement field u into transverse (UT = z (iq x u)/q) and longitudinal (UL = (iq u)/q) components after we Fourier transform Eq. (A.1) to find 1 d 2 2q2 1 2 2 / \ L no Tnit T Tni2 + MLL+^ c [UL 'LULr MU) 2 2 _MU)2 2 m (A.2) The fact that the displacement field is real introduces the property: u*(q) = u(q). Additionally, for simplicity we introduced the compact symbolism: u2 = u(q) u(q). The process of finding the eigenmodes of the above Lagrangian consists in identifying the canonical moment of the transverse and longitudinal displacement fields. It is easy to show that the corresponding results are a 1 PT O nomrT( + tmnoWtUL, (A.3) NUT 2 8 1 PL O nomTL mnoucUT. (A.4) OUL 2 The next step involves building the equations of motion associated with these fields that requires using the wellknown formula Sd V 0, i T, L. (A.5) Oui dt Oti O(Vuj) As a result we find the following two coupled dynamical equations UT + ciUL + TUT = 0, (A.6) 1UL cttT + LUL = 0. (A.7) The presence of the magnetic field couples the two acoustic modes and one needs to diagonalize the system of equations to find the new eigenmodes. It is easy to show that the eigenfrequencies of such a system are given by 1 2 + U2 + U)2 + (2 + U( + U)22 4U] (A.8) In the limit where the magnetic field is turned off we notice that the expected acoustic modes emerge out. Having written the canonical moment of the displacement fields we can switch to the Hamiltonian representation and write down for the Hamiltonian of the 2D electron system H no J(d {ULPL +UTPT L n (o 2 2m 2 11 2 Sno j(2)2 2m + + WC[UTPL ULPTJ + 2m + ()UL + t (4 + ) U 2 (A.9) Decoupling Eqs. (A.6, A.7) produces the following L' + [w~ + a4 + w)2iui + 4WU2 = 0, i T,L. (A.10) A general solution for both the displacement fields whose dynamics are described by the above equation consists in identifying all four different components of them corresponding to all four eigenmodes of the system. We write such a solution as ui = Aiei+t + Ae+t + Bet + B'e,t, i = L,T, (A.11) and using Eq. (A.6) or Eq. (A.7) we solve only for the four independent coefficients we want to keep. In our case we have chosen the following UT = AT'+t + A'e e'+t+BreC t + B e t, (A.12) UL = Z Are [Aiw + AT' 2 Bre' + Be (A.13) 1+  UU L2 T This is the complete analytic solution of the equations of motion for the transverse and longitudinal part of the displacement field in the presence of a perpendicular magnetic field. Next, we canonically quantize the above fields by properly defining creation and annihilation operators. For the sake of clarity and to avoid cluttering the symbolism we will suppress no (which multiplies m) and include it only in the final results. To canonically quantize we need to find the relation between all time derivatives of the fields and their canonical variables. The equations of motion Eqs. (A.6, A.7) and the canonical momentum equations Eqs. (A.3, A.4) combined together provide us with these, given by 2 2 UL PT (L + )UL, m 2 1 2 2 UT = c(c + T+ 2 )UL T +PT, 2 cPLM 1 /) p2 + U) 2 )u)2 p 2 m~T (A.14) (A.15) (A.16) (A.17) We pick out the four nonredundant (out of the eight possible) equations for the coefficients to end up with the 4x4 linear system: AT + A' + BT + B'T UTo, iw[AT A'] + iw _[BT B= PO Wu [A + A] _[BT + B'] = po )ro, i A 3 t + T T ~~2 CT+2Do UC + UT PT Pro, where ULo, UTo and PLo, Pro are the displacements and canonical moment, respectively evaluated at time t = 0. After solving the system, we obtain the following relations between the field coefficients and the canonical variables 1 + U+ 2 2 2 L+ 2 2 2 U _T c/" + PLo + To (A.22) (A.18) (A.19) (A.20) (A.21) LO I , 1 22 22 c . A 2 T w/2]uT, + PLo I FPT + 2 U o (A.23) 2 "u BT U) a) [U2 U2 U2 Ui )a)_ + T , w(2 +w}) 2 2 J B' 2) {_ [w w w /2]uTo  ,W(2 + )U  U)C (a)2 2 + 1 2 Lo . cw UTc m 2 PLo (A.24) c_ (2 _ 2 ) PLo + i Po (A.25) Notice that A = AT, Bt = BT. After some tedious calculations we can prove to ourselves using [uio(q),pji(q')] = :, (27r)22 (q q') that [Ar, BTR] =[A, B ] [A, AT]r 1/ni 2mn + u _ 7 ='1 2mnnow w h 2 ' n2= S W7 We have reintroduced no in the formulas at this point. Also, the same q is assumed in all the operators and we have omitted for clarity the 2D delta functions. Next, we normalize the operators using the following new definitions al(q) AT(q) AT(q) V1'! a2(9) BT (q) = BT(q) V/'n2 at(q) 'n (A.29) (A.30) .0 LL " LL) (A.26) (A.27) (A.28) and that way we restore the usual commutation relation [ai(q), at (q')] (27)22 (q q')ij. (A.31) As a result, the final form of the annihilation eigenmode operators for the single 1 iv.r quantum Hall system in the continuum elastic approximation where the effect of the initr ,i r Coulomb interaction is included (to linear order in the elastic displacement field) is a(q) )(U2 + U2 /2]u(q) + po(q) 2_ U2 W2 +2 + I PTO (q) i +2 o (q) (A.32) a2(q) / (U;27 U;2)(77 w2 [ [ P wf/2]ur0(q) "'7' PLcq) no L + loI 22 w2 w2 _+ 2(2 + 2L PTo (q) + ic 2ULo(q) (A.33) nom 2 Going back to Eqs. (A.12, A.13) we can reexpress them in terms of our new operators t(q) w al(q) + t() +a2(q) t UT(q, t)= a" ei + ei + a(qei + e (A.34) VU1 VU1! V/U2 V/U2 c(q )L+ at(q) +L L+ al (q )e= + c (q ) c_ i2ot a;c a _q) + 2 a ) (A.35) 0 0L Vm2 Our task has been completed because going back to the Hamiltonian of Eq. (A.9) and substituting Eqs. (A.34, A.35) we get the diagonal form H j fj2j [hdaq+a (q)al(q)+ _a(q)a2(q) + +2 (A.36) We have found the eigenmodes and eigenfrequencies of the single 1. r quantum Hall system including the effect of the intrvil.. v Coulomb interaction. It is interesting to see how our solutions behave to different limiting cases. In the low magnetic field limit (wc + 0) we discover that 2 wC2 2 L 2, (A.37) L2 2 L 2 (A.38) W WT 2 (A.38) while Eqs. (A.32, A.33) become ai(q, c  0) 1 i Ult) + 1 Lo(q), (A.39) 2h V2mnoL a2(q,c  0) 2 UTo (q) + i / pro(q), (A.40) r2 h V2mno hT which is the entirely expected behavior of the system in zero magnetic field to decouple into the original transverse and longitudinal eigenmodes. On the other hand, taking the high magnetic field limit (wjc + oc) we discover expanding the square root in Eq. (A.8) that + + + + O(wL 3), (A.41) 2wc L o+0(w3), (A.42) wLc and using the above in Eqs. (A.32, A.33) we discover Imno UTo ULo PLo + ipTo aI (q, uc ) 2h 2 + 2mn +' (A.43) V 2:fhT 2 +'2mnohu"c 0 flo~c WT .L 1 a2(q, c  oc) 2 UTo +1 ULo (A.44) V 2hLLLT z2 2z Notice that Eq. (A.44) is singular in uc. This is not a physical instability though, because from Eq. (A.42) we see that the u_ mode it represents has died out in the high magnetic field limit. A.2 Bilayer Eigenmodes Next, we consider the bilayer case for which we include the effect of the inter Ir Coulomb interaction. The dynamics of the system is described by the following Lagrangian L = no jd2r mnfi + Mfu CIA A(uA) CUB A(uB) A A 2 + 2 1 2 (A)2 + (L\LUL2A illr(amUlA + OIU2A)2 + am^UlB + OiUm" 2no no d2r/[V 7 uA(r) [V UA(r')]  2c r r no Jd2r'[V* uB(r)][V7. uB(r')]  no d22 [ A ( B ( ) ] u(r) [V' uB(r/(x x')2(y y)2 + d2 SK (UA B 2 (A.45) 2 no The effect of the interlayer Coulomb interaction is introduced through the short range and the long range part described by the last two terms, respectively. Decomposing the displacement fields into transverse and longitudinal components (as we did for the single 1 vr case) we find 1 2 l .2 1 .2 1 1 2 2 2 m LA + 2t LB 2 + 2mUT + [..' ULB ULB UT] 2 M UT 1 2 2 2e2no0 qd_ K)2 L U LB UB LAULB qe (A B U (A.46) 2 e 2 no Next, we decouple the two single 1v.,.r Lagrangians by switching to the inphase v = (up + UA)/2 and outofphase u = uB UA modes that produces L = no >(2m)i + (2m)i + (2m)uc[vTVL iLVT] (2m)(crq)2V (2m) (Lq)2 + q(1 + d)] v + l(Tn) + 2(T)i + t(T) [(T)2 2K 2 2 2x)2 2 22 2 2 Mno + t( ) + ( )6 + ()cAUT ULUT] (2) (2rK) 2 q 2 () (cLq)2 + + q(1 e) U (A.47) If we compare the structure of the inphase and outofphase Lagrangians to the single I,,r one given by Eq. (A.2), we realize that we can define two "effectsil single 1'.r dynamics whose eigenmodes and eigenfrequencies can be read off if we change the definitions for the parameters involved in each case. For the outofphase modes we need to apply the following changes m _, (A.48) 2K w [c q2 + o] = ), (A.49) Tmno S cq + + 27e2 q(1 qd)] (A.50) For the inphase modes the parameters need to change according to m + 2m, (A.51) W4 W = O, (A.52) S cq2 + 27e20 q(1 + eqd)] 02 (A.53) The cyclotron frequency a, stays unchanged in both of the cases. After performing the above redefinitions for the parameters involved in the single 1v.r eigenmode 