TUNNELING BETWEEN TWO DIMENSIONAL ELECTRON SYSTEMS IN AHIGH MAGNETIC FIELD AND CRYSTALLINE PHASES OF A TWO

DIMENSIONAL ELECTRON SYSTEM IN A MAGNETIC FIELD

By

FILIPPOS KLIRONOMOS

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOLOF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREE OFDOCTOR 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.

ii

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 . . . . . . . . . . . . . . . . . . . . 11.2 Bilayer Quantum Hall Physics: Experiment and Theory . . . . . . . . . 31.3 Crystalline Phases of the 2D Electron System: Experiment and Theory 9

2 TUNNELING CURRENT OF COUPLED BILAYER WIGNER CRYSTALS 12

2.1 Single Layer Eigenmodes . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2 Bilayer Eigenmodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3 Coupling the Bilayer to External Electrons: Tunneling Current . . . . . 18

2.3.1 Analytic Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 232.3.2 Numerical Solution . . . . . . . . . . . . . . . . . . . . . . . . . 29

3 QUANTUM HALL SYSTEM IN THE HARTREE-FOCK APPROXIMATION 32

3.1 Electron Dynamics in a Perpendicular Magnetic Field . . . . . . . . . . 323.2 Hartree-Fock Approximation . . . . . . . . . . . . . . . . . . . . . . . . 37

4 ISOTROPIC CRYSTALLINE PHASES . . . . . . . . . . . . . . . . 39

4.1 Stability Analysis of Isotropic M -electron Bubble Crystals . . . . . . . 394.1.1 Classical Order Parameter Approach . . . . . . . . . . . . . . . 444.1.2 Microscopic Approach . . . . . . . . . . . . . . . . . . . . . . . 484.1.3 New State: Bubble Crystal with Basis . . . . . . . . . . . . . . 534.1.4 Normal Modes and Zero Point Energy . . . . . . . . . . . . . . 56

4.2 Energetics of Isotropic Crystalline Phases . . . . . . . . . . . . . . . . . 594.2.1 Cohesive Energy of Modified Coulomb Interaction: Classical

Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.2.2 Cohesive Energy of Modified Coulomb Interaction: Microscopic

Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

iii

5 ANISOTROPIC CRYSTALLINE PHASES . . . . . . . . . . . . . . 66

5.1 Solving the Static Hartree-Fock Equation . . . . . . . . . . . . . . . . . 665.2 Introducing Anisotropy into the Crystalline States . . . . . . . . . . . . 695.3 Elastic Properties of Anisotropic Crystals . . . . . . . . . . . . . . . . . 775.4 Analysis of Experimental Results . . . . . . . . . . . . . . . . . . . . . 80

6 CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . 83

APPENDIX

A BILAYER SYSTEM EIGENMODES . . . . . . . . . . . . . . . . . 86

A.1 Single Layer Eigenmodes . . . . . . . . . . . . . . . . . . . . . . . . . . 86A.2 Bilayer Eigenmodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93A.3 Tunneling Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

A.3.1 Correlation Function . . . . . . . . . . . . . . . . . . . . . . . . 100A.3.2 Properties of the Correlation Function . . . . . . . . . . . . . . 102

B ISOTROPIC CRYSTALS . . . . . . . . . . . . . . . . . . . . . . 105

B.1 Fock Term Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . 105B.2 Microscopic Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108B.3 Bubble with Basis Dynamical Matrix . . . . . . . . . . . . . . . . . . . 108B.4 Normal Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

C BUILDING THE STATIC HARTREE-FOCK EQUATION . . . . . . . 114

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . 124

iv

Abstract of Dissertation Presented to the Graduate Schoolof the University of Florida in Partial Fulfillment of theRequirements for the Degree of Doctor of Philosophy

TUNNELING BETWEEN TWO DIMENSIONAL ELECTRON SYSTEMS IN AHIGH MAGNETIC FIELD AND CRYSTALLINE PHASES OF A TWO

DIMENSIONAL ELECTRON SYSTEM IN A MAGNETIC FIELD

By

Filippos Klironomos

May 2005

Chairman: Alan T. DorseyMajor Department: Physics

We study the bilayer quantum Hall system in the incoherent regime. We

model the two layers 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 intralayer inter-

actions.

We further study the collective states formed by the 2D electrons at low Landau

levels by working from a semi-classical and microscopic perspective and evaluating

the elastic moduli, normal modes, and zero-point and cohesive energies of the dif-

ferent crystalline 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

v

is lowered considerably (which signifies that these degrees have a prominent physical

importance).

Finally, the static Hartree-Fock 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 crystal 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 channel-like

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 half-filling, and that

shear deformations along the channels become cost-free due to the vanishing shear

modulus along that direction.

vi

CHAPTER 1INTRODUCTION 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 (ρxx) and sharp

plateaus in the Hall resistivity (ρxy) 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 insulator-semiconductor 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 ρxy 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

1

2

Figure 1.1: Dissipative and Hall resistivities of a GaAs sample. Reprinted fromStormer, Physica 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]

ψ(z1, · · · , zN) =N∏

i>j=1

(zi − zj)1ν e− 1

4

NPi=1

|zi|2, (1.1)

where ν = NΦ/Φ0

is the filling factor for a single spin, N is the total number of electrons,

Φ is the total flux penetrating the sample, Φ0 = h/e is the flux quantum and zi =

(xi + iyi)/lB is the complex position of each electron, where lB =√~/eB is the

magnetic length. Going back to Fig. (1.1) we notice that the FQHE happens for

filling factors ν = 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/ν is odd. For the rest of the FQHE states different theories

3

Figure 1.2: Symbolic graph of a bilayer structure of interlayer distance d. Theelectrons coming from the Si donors are trapped in the GaAs-AlGaAs interface andform the 2D electron gas.

have been developed based on Laughlin-like 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

molecular-beam epitaxy where two semiconductors (usually GaAs and doped AlGaAs)

form a quantum well at their interface (∼100-1000 A wide) where electrons, coming

from Si-doped layers 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 bilayer 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 bilayer system exhibits plateaus at total filling factors

νT = 1 and νT = 1/2. This is a “violation” of the odd denominator constraint for the

4

0.00 0.02 0.04 0.06 0.08 0.10∆SAS/(e

2/εlB)

1.0

2.0

3.0

4.0

d/l B

NO QHE

QHE

Figure 1.3: Phase diagram at νT = 1 of interlayer Coulomb interaction strength vssingle particle tunneling strength. Energy measures in units of the intralayer Coulombinteraction. Solid symbols indicate samples showing QHE behavior, open symbolsdenote 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

layer index) each electron possesses; and to the fact that the spin-polarization of the

2DES is relaxed, leading to spin-textures [11]. Yoshioka, MacDonald and Girvin [12]

have proposed the so called Ψ3,3,1 state [13] as a candidate ground state for a double

layer FQHE at νT = 12. This is a Laughlin-like state which introduces correlations

among the electrons in the two layers 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 bilayer structures have an interesting physical behavior that orig-

inates from the interplay of the intralayer Coulomb electron interaction (within the

same layer) and the interlayer Coulomb electron interaction (between the two layers).

A phase diagram for the QHE, experimentally produced by Murphy et al. [14], is

shown in Fig. (1.3). The strength of interlayer Coulomb interaction, relative to the

intralayer one, is plotted as a function of the tunneling strength ∆SAS (measured in the

same units) that determines the energy difference of a single electron associated with

5

-5 0 5-5 0 5

Interlayer Voltage (mV)

Tu

nn

elin

gC

ondu

ctan

ceat

nT=

1 (

10-7

W-1

)

0.5

A)

B)

C)

D)

NT=6.9

NT=6.4

NT=5.4N

T=10.9

2.01.51.00.50.0

Temperature (K)

1.2

1.0

0.8

0.6

0.4

0.2

0.0

G0

(10

-6W

-1)

nT

= 1

NT

= 10.9

(x200)

NT

= 4.2

Figure 1.4: Zero bias peak anomaly in the tunneling conductance of a bilayer system.Left panel: Tunneling conductance dI

dVvs interlayer voltage V at νT = 1. Right panel:

Temperature dependence of the zero bias tunneling conductance at νT = 1 at highand 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 bilayer quantum well system. What is

astonishing is that the QHE persists even when ∆SAS 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 layers. Correlations

are not important in this limit and the system behaves as a single-layer 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

destroying ∆SAS and the QHE altogether. This suggests 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/AlxGa1−xAs double quantum well

heterostructures, a well pronounced feature appears at total carrier densities of NT ≤

6

5.4 × 1010cm−2 at νT = 1. The two GaAs wells of width 180 A are separated by

a 99 A Al0.9Ga0.1As barrier layer. The low temperature (∼40 mK) mobility of the

samples is 2.5× 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× 10−8 Ω−1) by more than a factor of 10 [16]. It should be noted

that νT = 1 is held constant in all the traces, so the magnetic field is varied accordingly.

The appearance of this resonance peak suggests the existence of a soft collective mode

of the bilayer system which enhances the electron’s ability to tunnel. If we assign a

pseudospin quantum number to each electron, indicating the layer index which it lies

in (up or down just like spin-1/2), then the “easy-plane” pseudospin-ferromagnetic

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 bilayer 2DES. The best candidate state describing the system’s

ground state responsible for this high coherence peak is the Halperin Ψ1,1,1 state. In

this state, an electron in one layer is always opposite to a hole in the other layer

7

Figure 1.5: Evidence of coherence in a bilayer quantum Hall drag experiment. Leftpanel: Conventional and Coulomb drag resistances of a low density double layer 2DES.Trace A is the conventional longitudinal resistance Rxx measured with a current inboth layers. Trace B is the Hall drag resistance Rxy,D. Trace C is the longitudinaldrag resistance Rxx,D, and trace D is the Hall resistance R∗

xy of the single current-carrying layer. Right panel: Collapse of νT = 1 Hall drag quantization and secondh/e2 plateau in R∗

xy. Reprinted figures 1 and 2 with permission from M. Kellogg, I.B. Spielman, J. P Eisenstein, L. N. Pfeiffer, and K. W. West Phys. Rev. Lett. 88,126804 (2002). Copyright (2002) by the American Physical Society.

introducing the possibility that way for electrons to form a coherent state where both

quantum wells will be symmetrically occupied.

Most recently a quantum Hall drag experiment conducted by Kellogg et al. [22]

gave direct evidence of coherence in the transport properties of the bilayer system.

In that experiment, an excitation current was driven through one layer, and a Hall

voltage drop was reported on the other layer. The same quantization of Hall drag was

observed, even when the roles of the two layers were interchanged. The sign of the

Hall drag was the same as in the conventional Hall effect. However, the longitudinal

drag voltage was opposite in sign to the longitudinal resistive voltage drop in the

current-carrying layer. This sign difference signifies the force balance between the

layers, resulting from the constraint that no current flow is allowed in the drag layer.

Remarkably these phenomena are observed at total filling factor νT = 1. There is a

quantum Hall plateau and a corresponding voltage drop even when a current is driven

8

in the same layer in which the voltage drop measurements are taken. Figure (1.5)

shows these results and the crucial dependence the response of the bilayer system has

on d/lB . At d/lB > 1.83 the interlayer coherence is gradually destroyed and the

“normal” single-layer 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 layers. 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 νT = 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

bilayer 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 bilayer system. We

have modeled this system as two Wigner crystals, according to the original work of

Johansson and Kinaret [26], based on the independent boson model; but we have

introduced a coupling between the layers as well, arising from the existence of the

9

Figure 1.6: Microwave resonance response of a quantum Hall system. Left panel:Real part of σxx 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 coexiston the left panel. Reprinted figures 1 and 3 with permission from R. M. Lewis, YongChen, 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 I-V response

shown earlier.

1.3 Crystalline Phases of the 2D Electron System: Experiment andTheory

The preceeding introduction into the physics of quantum Hall systems (whether

bilayer or single layer) should have convinced the reader that systems like these have a

rich variety of states or phases that can potentially manifest themselves, depending on

the different parameter values associated with such systems (such as disorder, carrier

10

density, applied magnetic field or temperature). Two major 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 [31–33]. 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 σxx, for different

applied magnetic field values or filling factors. The traces are displaced for clarity.

According to data analysis on these measurements [33], the resonance curve can be

fitted by two Lorentzians indicating a two-phase 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 play 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

(crystalline 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

11

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 [36–39] 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 half-filling, the traditional stripe state

(charge density wave) becomes favorable. At intermediate filling factors, a variety of

bubble crystal (BC) states emerge. A bubble crystal is a Wigner crystal-like structure

(where instead of having one electron guiding center occupying a given site, there are

more; creating a hierarchy of M -electrons per bubble crystalline structures where M

is an integer [36, 42]). According to Hartree-Fock results, the last possible BC state

that can become favorable (as the filling factor is increased) is the M = n+1 electrons

per bubble crystal for the n-th 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

crystals 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 microscopics of the crystalline phases associated with the quan-

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 2TUNNELING 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 layers 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 bilayer 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 bilayer 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 layer model in which the 2D electron

system is assumed to be in a Wigner crystal state. This is not the true experimental

12

13

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, liquid-like state of the system. Additionally, we work in

the continuous approximation limit, treating the Wigner crystal as an elastic medium

and imposing a momentum cut-off q0 = 2√

πn0, where n0 is the layer 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

Lame coefficients [43] λ, µ to describe elastic deformations of a configuration with

hexagonal symmetry (the triangular lattice is the Wigner crystal ground state [44]), we

can write the following Lagrangian describing the system dynamics in the continuum

limit

L = n0

∫d2r

1

2mu2 − eu ·A(u)− λ

2n0

(∂iui)2 − µ

4n0

(∂mul + ∂lum)2

+1

2n0

∫d2r′[∇ · u(r)][∇′ · u(r′)]

e2

4πε|r− r′|

, (2.1)

where u is the displacement field, ε 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 intralayer Coulomb interaction term in the continuum

approximation where local charge variations are given by δn/n0 = −∇ · 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, Bux/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

cL =√

(λ + 2µ)/mn0, (2.2)

cT =√

µ/mn0. (2.3)

14

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 = n0

∫d2q

(2π)2

1

2mu2

T +1

2mu2

L+1

2mωc[uT uL− uLuT ]− 1

2mω2

Lu2L−

1

2mω2

T u2T

, (2.4)

where we use the real field property u∗(q) = u(−q) and the convention |u|2 ≡ 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 ωc = eB/m while the longitudinal and transverse zero

magnetic field eigenfrequencies are respectively given by

ωL =

√c2Lq2 +

e2n0

2mεq, (2.5)

ωT = cT q. (2.6)

The expected effect of the intralayer 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 '√

0.0363e2

√3εma0

, (2.7)

where a0 is the Wigner crystal 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 layer

system

ω2± =

1

2

[ω2

c + ω2T + ω2

L ±√

(ω2c + ω2

T + ω2L)2 − 4ω2

T ω2L

]. (2.8)

15

We notice that in the zero magnetic field limit (ωc = 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/ωc) we obtain

ω+ = ωc +ω2

L + ω2T

2ωc

, (2.9)

ω− =ωLωT

ωc

, (2.10)

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 (ω+); but since we have assumed a Wigner crystal state for the

electronic system, we have also maintained gapless excitations (ω−).

The above treatment completes the single layer 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 (ω−) and magnetoplasmons

(ω+), respectively.

2.2 Bilayer Eigenmodes

Having completed the single layer treatment of a 2D electron system we can

turn on the bilayer 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 layers have the same density n0 (which can be

arranged experimentally) and write down the following Lagrangian

L = LA + LB + n0

∫d2r

[− 1

2

K

n0

(uA − uB)2

− n0

∫d2r′

e2[∇ · uA(r)][∇′ · uB(r′)]

4πε√

(x− x′)2 + (y − y′)2 + d2

], (2.11)

16

where LA, LB are the independent single layer Lagrangians similar to Eq. (2.4). The

term involving the K parameter (associated with the short range physics of the inter-

layer Coulomb interaction) is expected to arise when the two Wigner crystals prefer

to lock their positions (and move in-phase) by penalizing out-of-phase 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/n0 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

n0

= κe2/4πεd

πl2, (2.12)

where κ is dimensionless and l =√~/eB. The dimensionless parameter κ can be

extracted from magnetophonon experimental measurements or theoretical calculations

associated with this bilayer 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 employed (as in the intralayer Coulomb interaction

case) the continuum linear approximation in order to describe local charge density

fluctuations. Diagonalizing this coupled bilayer system involves introducing in-phase

and out-of-phase modes given by

uA = v − 1

2u, (2.13)

uB = v +1

2u, (2.14)

which turn out to be the eigenmodes of it. As a result, the two coupled single layer

dynamics of Eq. (2.11) decompose to uncoupled “effective single layer” dynamics

of in-phase and out-of-phase nature. We started with a system of a total of four

modes, so we expect two in-phase and two out-of-phase modes as a result. Details

of the calculation are given in the Appendix. The result for the two out-of-phase

17

eigenfrequencies is

Ω2± =

1

2

[ω2

c + Ω2T + Ω2

L ±√

(ω2c + Ω2

T + Ω2L)2 − 4Ω2

LΩ2T

], (2.15)

where the “effective” transverse and longitudinal acoustic mode frequencies are given

by

Ω2T = c2

T q2 +2K

mn0

, (2.16)

Ω2L = c2

Lq2 +2K

mn0

+e2n0

2mεq(1− e−qd). (2.17)

Notice that the single layer 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 in-phase eigenfrequencies we obtain similarly

O2± =

1

2

[ω2

c + O2T + O2

L ±√

(ω2c + O2

T + O2L)2 − 4O2

LO2T

], (2.18)

where the “effective” transverse and longitudinal acoustic mode frequencies are given

by

O2T = c2

T q2, (2.19)

O2L = c2

Lq2 +e2n0

2mεq(1 + e−qd). (2.20)

As is expected for the in-phase modes, there is no gap introduced by the short range

physics, since these type of modes respect the locked-in position of the two Wigner

crystals. Since we have solved the single layer problem and have found analytic

expressions for the creation and annihilation operators of its eigenmodes, we can

apply those results to the “effective single layer” cases here, after we transform the

appropriate parameters involved. For example, in order to obtain analytic results

18

for the out-of-phase operator modes we have to perform the following changes to the

parameters of the single layer case: m → m2, ω2

T → Ω2T , ω2

L → Ω2L. For the in-phase

case, the changes become: m → 2m, ω2T → O2

T , ω2L → O2

L. Final results are presented

in full in the Appendix.

This completes the treatment of the bilayer system. We now have analytic

expressions for the eigenmode operators of the coupled bilayer 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 electron-magnetophonon

and electron-magnetoplasmon 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

bilayer 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 layer to the other. For the

latter, we will use c†A, cA and c†B, cB as the creation and annihilation operators for the

two layers, respectively. Assuming for simplicity that the tunneling electron is at the

origin of layer A, and couples through the unscreened Coulomb interaction to charge

density fluctuations of the Wigner crystals in both layers A and B, we can express

the interaction energy associated with that coupling as follows

He−e =e2

4πε

∫d2r

∫d2rA

ne(r)δnA(rA)

|r− rA| +e2

4πε

∫d2r

∫d2rB

ne(r)δnB(rB)

|r− rB − d| . (2.21)

19

In the continuum linear approximation the charge fluctuations in the two layers will

be given by δnA = −n0∇ · uA and δnB = −n0∇ · uB, respectively. Placing the

independent electron at the origin corresponds to ne(r) = δ(2)(r). Combining all of

the above the coupling term assumes the following form

He−e = −e2n0

4πε

∫d2rA

∇A · uA

|rA| − e2n0

4πε

∫d2rB

∇B · uB

|rB + d |= −e2n0

4πε

∫d2q

(2π)2

2π

qiq ·

[uA(q)− e−qduB(q)

]. (2.22)

In other words, the coupling term in the bilayer system associated with an independent

electron injected into the bulk of either of the two quantum Hall systems has the form

Hcoupling =− c†AcA

[ ∫d2q

(2π)2

e2n0

2εqq(uLA

+ e−dquLB

)]

− c†BcB

[ ∫d2q

(2π)2

e2n0

2εqq(uLB

+ e−dquLA

)]. (2.23)

Introducing at this point the in-phase and out-of-phase displacement fields given by

Eqs. (2.13, 2.14) we can transform the above to

Hcoupling =− c†AcA

∫d2q

(2π)2

e2n0

2ε

[(1 + e−dq

)vL − 1

2

(1− e−dq

)uL

]

− c†BcB

∫d2q

(2π)2

e2n0

2ε

[(1 + e−dq

)vL +

1

2

(1− e−dq

)uL

]. (2.24)

Using the analytic expressions, derived in the previous section, for the operator form

of the in-phase and out-of-phase modes we show in the Appendix that the above

coupling term can be written in terms of creation and annihilation operators of the

bilayer quantum Hall eigenmodes as

Hcoupling = c†AcA

i

4∑s=1

MsA

(a†s − as

)+ c†BcB

i

4∑s=1

MsB

(a†s − as

), (2.25)

20

where a†s and as are the corresponding creation and annihilation operators for the

bulk electrons and the coupling matrix elements are given by

MsA =

− e2n0

4ε

(1− e−dq

)fs, s = 1, 2,

e2n0

2ε

(1 + e−dq

)fs, s = 3, 4,

, MsB =

−MsA, s = 1, 2,

MsA, s = 3, 4,

. (2.26)

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 bilayer system energetics involved

in the tunneling processes

H =H0 + H−T + H+

T

=

[εA + i

∑s

MsA

(a†s − as

)]c†AcA +

[εB + i

∑s

MsB

(a†s − as

)]c†BcB

+∑

s

~Ωsa†sas + Tc†AcB + Tc†BcA. (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 “smooth-out” 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 a†s, as obey boson statistics

and c†A(B), cA(B) obey fermion statistics. In the above εA and εB are the Madelung

21

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

I(V ) =e

~2

∫ +∞

−∞dteieV t/~〈[H−

T (t), H+T (0)

]〉

=e

~2

∫ +∞

−∞dt

[e

ieV t~ I∓(t)− e−

ieV t~ I±(t)

]. (2.28)

The correlation functions associated with this expression are given by

I∓(t) = 〈H−T (t)H+

T (0)〉, (2.29)

I±(t) = 〈H+T (t)H−

T (0)〉, (2.30)

where the time-dependence 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) = ν(1− ν)T 2C(t), (2.31)

where ν = 〈c†AcA〉 and 1− ν = 〈cBc†B〉 and for the time-dependent part we get

C(t) = exp

−

∑s

(MsB −MsA)2

(~Ωs)2

[(Ns + 1

)(1− e−iΩst

)+ Ns

(1− eiΩst

)], (2.32)

where Ns 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

22

B in Eq. (2.32). The experimental temperature range in the tunneling current is of

the order of 0.1 K∼10−5 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 in-phase magneto-

phonons along with the out-of-phase 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 layers [Eq. (2.26)] they cancel out. This is to be expected since a tunneling event is

associated with an out-of-phase 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 “open-up” and create an available posi-

tion for it. This corresponds to an out-of-phase motion. If we gather all of the above

together and switch to dimensionless units for the momentum integration (x = q/q0)

we find the following result for the time dependent correlation function

C(t) = exp

[ ∫ 1

0

dxf(x)(e−iω(x)t − 1

)], (2.33)

where the weight function and the magnetophonon frequency can be approximated in

the high magnetic field limit as

f(x) =cω5

c

c8T q6

0

x(1− e−γx)2

δ + x2 + 2α + βx(1− e−γx)

1√x2 + α

× 1

[α + βx(1− e−γx)]3/2

1

δ − α− x2, (2.34)

ω(x) =c2T q2

0

ωc

√α + βx(1− e−γx)

√x2 + α. (2.35)

23

In the above we have defined the parameters

c =n0

8π~m

(e2

ε

)2

, (2.36)

α =2K

mn0c2T q2

0

=κ

2π2mc2T

1

(lq0)2

e2

εd, (2.37)

β =e2n0

2εmc2T q0

, (2.38)

γ = dq0, (2.39)

δ =

(ωc

cT q0

)2

, (2.40)

and have taken cL = 0. The parameter α 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 β parameter is dimensionless as well and does

not depend on a0 if the dependence on it from cT is taken into consideration using

Eq. (2.7). The parameter γ gives a measure of the relative strength of the intralayer

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

ωC(ω) =

∫ 1

0

dxf(x)ω(x)C(ω − ω(x)). (2.41)

As we show in the Appendix, this correlation function is zero for ω ≤ 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 I-V response of the bilayer 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

24

information as possible from it. What turns out to be particularly useful is the

derivation of the asymptotic behavior for large frequency values (large bias). In that

case we can expand C(ω − ω(x)) in ω(x) and obtain to lowest order a first order

differential equation with the solution

C(ω) ∼ exp

[− (ω − c1)

2

2c2

], (2.42)

where

c1 =

∫ 1

0

dxf(x)ω(x)

= cω4

c

c6T q4

0

∫ 1

0

dxx(1− e−γx)2

δ + x2 + 2α + βx(1− e−γx)

1

α + βx(1− e−γx)

1

δ − α− x2, (2.43)

c2 =

∫ 1

0

dxf(x)ω2(x)

= cω3

c

c4T q2

0

∫ 1

0

dxx(1− e−γx)2

δ + x2 + 2α + βx(1− e−γx)

√x2 + α

α + βx(1− e−γx)

1

δ − α− x2. (2.44)

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 asymptotic

solution for different layer separation values d. For the case where the two layers

are far apart and can be considered uncorrelated (d À a0), we can ignore the ex-

ponentials in the integrand of Eqs. (2.43-2.44) and it turns out that c1 ∼ 1/a0 and

25

√c2 ∼ 1/a2

0B are the corresponding limits. This is the same scaling behavior JK

produce using phenomenological arguments. In the opposite limit, where the inter-

layer separation is much smaller than the intra-electron distance (d ¿ a0) we can

expand the exponentials in the integrand of Eqs. (2.43-2.44) and find c1 ∼ d2/a30 and

√c2 ∼ d/a3

0B. 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 zero-bias [15]. For the remaining bias

voltage region, correlation effects have the prominent role and the Coulomb barrier

peak survives but is “red-shifted” significantly [15]. As we see our model is able to

reproduce such a limiting behavior.

The above asymptotic expansion provides a useful starting point to apply a trial

solution of Eq. (2.41) by assuming a power-law combined with a Gaussian exponential

behavior for the correlation function according to

C(ω) = Nωre−λω2

. (2.45)

The above Ansatz captures the essential asymptotic behavior and if the parameters are

evaluated self-consistently it should qualitatively reproduce a solution. In particular,

if we multiply-differentiate Eq. (2.33) we end up to the following moment equations

(associated with C(ω)) that we can use to derive values for N , r and λ:

∫ ∞

0

dωC(ω) = 2π, (2.46)

∫ ∞

0

dωωC(ω) = 2πc1, (2.47)

∫ ∞

0

dωω2C(ω) = 2π(c2 + c2

1

). (2.48)

26

For the above equations the following general integral becomes useful

∫ ∞

0

dyyre−y2

=1

2Γ

(r + 2

2

). (2.49)

Also, it is convenient to switch frequency (ω) to bias voltage (V ) (measuring in mV) by

introducing the change ω = eV/1000~. That way the argument of C(ω) will measure

in mV while the corresponding Ansatz parameter λ in the exponent of Eq. (2.45)

will acquire the form Λ = λ(

e1000~

)2. Using the general integral result above we can

perform the integrations in Eqs. (2.46-2.48) and obtain the results

1

2N

(e

1000~

)r+1

Λ−r+12 Γ

(r + 1

2

)= 2π, (2.50)

1

2N

(e

1000~

)r+2

Λ−r+22 Γ

(r + 2

2

)= 2πc1, (2.51)

1

2N

(e

1000~

)r+3

Λ−r+32 Γ

(r + 3

2

)= 2π

(c2 + c2

1

). (2.52)

We can divide Eq. (2.51) and Eq. (2.52) by Eq. (2.50) to get

Λ =

(e

1000~c1

)2[Γ(

r+22

)

Γ(

r+12

)]2

, (2.53)

Λ =

(e

1000~√

c2 + c21

)2 Γ(

r+32

)

Γ(

r+12

) , (2.54)

or equating the two

Γ2(

r+22

)

Γ(

r+32

) =Γ(

r+12

)

1 + c2/c21

, (2.55)

we get a self-consistent 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 power-low behavior. Having r at hand we can

go back and evaluate the rest of the parameters. As a final result we find that the

27

0 5 10 15 20 25

Interlayer Voltage (mV)

0

2

4

6

8

10

Tunnel

ing C

urr

ent

(nA

)

Peak = 5.62 mVPeak = 6.43 mVPeak = 6.92 mVPeak = 7.63 mV

13.75 T

11 T

9.75 T

8.25 T

Figure 2.1: Tunneling current curves for different magnetic field values using themoment expansion solution of Eq. (2.41). The legend shows the peak bias valuescalculated by Eq. (2.57).

tunneling current correlation function assumes the form

C(V ) = N

(e

1000~

)r

V re−ΛV 2

, (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

V0 =1000~

e

√r

2λ, (2.57)

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 bilayer sample area is S=0.0625mm2 and the single layer electron density

is n0=1.6 × 1011 cm−2. The perpendicular magnetic field varies from 8 T to 13.75 T

and the Wigner crystal lattice parameter has the value a0 ' 270 A which corresponds

28

0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

I/Im

ax

V (mV)

← d=175

← d=145

d=85 →

d=115 →

Figure 2.2: Normalized tunneling current curves for different interlayer separationdistances d measured in A.

to the stable hexagonal lattice configuration (n0 = 2/√

3a20) [44]. The double well

separation distance is d=175 A and the dielectric constant of GaAs is ε=12.9ε0 '1.14

× 10−11 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 κ is extracted by time-dependent Hartree-Fock calculations investigating

the magnetophonon dispersion relation for the bilayer system, performed by Cote et

al.. They were able to provide us with a κ=0.0085 value. Our I-V 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 bilayer system. We notice that the strength of this inter-

action is controlled by the interlayer distance d which is introduced into the model in

29

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 “red-shifted” by an amount

proportional to e2/εd 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/εd. In other words the creation of excitons

associated with tunneling events are expected to “soften” the effect of the intralayer

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 two-fold. 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 intralayer Coulomb interactions through the

excitonic creation associated with tunneling events and as a result it “red-shifts” 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 ways, 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)

30

0 5 10 15 20 25

Interlayer Voltage (mV)

0

2

4

6

8

10

Tunnel

ing C

urr

ent

(nA

)

Peak = 5.59 mVPeak = 6.32 mVPeak = 6.84 mVPeak = 7.52 mV

13.75 T

11 T

9.75 T

8.25 T

Figure 2.3: Tunneling current curves for different magnetic field values produced bynumerically integrating Eq. (2.41). The legend shows the peak bias values obtainedwith this approach. They are in strong agreement with the analytic results.

in the following dimensionless form

zC(z) =

∫ 1

0

dxf(x)ω(x)

γ0

C(z − ω(x)

γ0

)θ(z − ω(x)

γ0

), (2.58)

where we have introduced the parameter γ0 = e1000~ 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 α1 ≤ ω(x)γ0

≤ α2 which means that

the density of states is non zero only in that range. The values of α1 and α2 are given

by substituting x = 0 and x = 1 into Eq. (2.35) respectively. The upper bound α2

appears due to the momentum cutoff we have introduced. The resulting form of the

correlation function integral equation is

C(z) =

1z

∫ z

α1dyg(y)C(z − y), α1 ≤ z ≤ α2,

1z

∫ α2

α1dyg(y)C(z − y), z ≥ α2,

(2.59)

31

where the definition for the density of states is

g(y) = y

(f(x)

1γ0|dω(x)

dx|

)

x(y)

, (2.60)

and x(y) is the root of the equation ω(x) = γ0y. 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 ≥ α1 values C(z) ∼ z−α1. 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 3QUANTUM HALL SYSTEM IN THE HARTREE-FOCK APPROXIMATION

3.1 Electron Dynamics in a Perpendicular Magnetic Field

We would like to focus our attention here on a single layer 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 microscopics involved in

such a system. The quantum nature of the electrons incorporated in the physics of

wavefunction overlaps and associated with the electron-electron Coulomb interaction

has to be considered, in order to investigate, as accurately as possible, the energetics

and stability of the different phases associated with the quantum Hall system.

We start this work by finding the non-interacting 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 Schrodinger’s equation for the 2D electron,

given by [1

2m(px − eBy)2 +

1

2mp2

y

]ψ(x, y) = Eψ(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 (x-direction) resulting in a wavefunction

decoupling of the form: ψ(x, y) = eikxxφ(y). This kind of decoupling produces a

displaced harmonic oscillator equation for the y-direction given by

[p2

y

2m+

1

2mω2

c (y − Y )2

]φ(y) = ~ωc

(n +

1

2

)φ(y). (3.2)

32

33

In the above, we have defined Y = kx`2 as the y-coordinate center of mass (CM)

variable, ` =√~/eB as the magnetic length, and ωc = eB/m is the cyclotron

frequency. The normalized solution for the displaced harmonic oscillator is given

by

φn(y) =1

π1/4`1/2√

2nn!e−

(y−Y )2

2`2 Hn

(y − Y

`

), (3.3)

where n is the so called Landau level index, associated with kinetic energy excitations

of the non-interacting electrons, and Hn(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 ~ωc) 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

inter-electron interactions affect the energetics 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 x-direction allows a macroscopic number of

electrons (fermionic particles) to occupy the same kinetic energy eigenstate (even in

the non-interacting 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 x-direction

the degeneracy g can be found to be

g =Lx

2π

∫ k0

0

dkx =Lx

2π`2

∫ Ly

0

dY =Ω

2π`2=

Φ

Φ0

, (3.4)

where Ω is the total area of the system, and Φ0 = 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 CM coordinates X,

Y (that enter into the dynamics of the electrons) can be derived from their classical

34

counterparts, and are found to be

X = x− πy

mωc

, (3.5)

Y = y − πx

mωc

, (3.6)

where the dynamical momenta πx, πy are given by the following expressions in the

Landau gauge

πx = mˆx =m

i~[x,H] = px − eBy, (3.7)

πy = mˆy =m

i~[y, H] = py. (3.8)

Combining the above definitions together we can derive the CM coordinate forms in

terms of the usual quantum mechanical operators:

X = x− py

mωc

, (3.9)

Y =px

mωc

. (3.10)

We see the effect of the magnetic field and the Landau gauge choice partially mix

the dynamics of the two directions. The CM coordinates are constants of the motion

since they commute with the non-interacting 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 ] = i`2. The dynamical momenta are conjugates as well

since [πx, πy] = −i~2/`2 and they additionally obey [X, πx] = [Y , πy] = 0. In other

words, the CM 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

35

the following commutation relation

[x, πx] = [x, px − eBy] = i~, (3.11)

in the limit of high magnetic field we find

[x, y] = −i`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 inter-Landau 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 non-interacting wavefunction basis (given by Eq. (3.3))

is usually used to construct the electronic field operator. Projecting onto the n-th

Landau level we find

ψn(r) =∑Y

φn(r)cn,Y , (3.13)

where c†n,Y , cn,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 n-th Landau level operator ρ(q) acquires a structure factor as can

36

be shown from

n(q) =

∫d2rψ†n(r)ψn(r)e−iq·r

= e−q2`2

4 Ln

(q2`2

2

) ∑Y

e−iqyY e−iqxqy`2

2 c†n,Y cn,Y +qx`2

= Fn(q)ρ(q), (3.14)

where Ln(x) is the n-th order Laguerre polynomial. These structure factor has the

form

Fn(q) = e−q2`2

4 Ln

(q2`2

2

), (3.15)

and the analytic expression for the projected density operator is given by

ρ(q) =∑Y

e−iqyY c†n,Y−qx`2/2cn,Y +qx`2/2. (3.16)

The following commutation relation holds for these projected density operators [53]

[ρ(q), ρ(k)] = −2i sin

((q× k)`2

2

)ρ(q + k). (3.17)

The Landau gauge is useful in introducing the physics of electrons in high

magnetic fields but the non-interacting 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 coordinate position. A much more useful

basis of non-interacting electrons arises out of the symmetric gauge choice: A =

(−By/2, Bx/2, 0). In this basis the good quantum number becomes the z-component

of angular momentum and the wavefunction form is given by [38]

φnm(r) =Cnm

`

(r

`

)|n−m|ei(n−m)θL

|n−m|(n+m−|n−m|)/2

(r2

2`2

)e−r2/4`2 , (3.18)

37

where n is the Landau level index, m the z-component angular momentum index, and

Lkn(x) are the associated Laguerre polynomials. The normalization constant is given

by

Cnm =

√2n−mn!2πm!

m ≥ n,√

2m−nm!2πn!

m ≤ n,

(3.19)

Parity is determined from the exponent n−m (whether it is even or odd).

3.2 Hartree-Fock Approximation

To build a realistic model of 2D electrons we need to include the Coulomb

interaction among them. Since the interaction is a four-fermion 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 four-fermion operator of the

Coulomb interaction by using the result of Eq. (4.32). What we find is

H =1

2

∫d2r

∫d2r′ψ†(r)ψ†(r′)V (r− r′)ψ(r′)ψ(r)

=1

2

∫d2q

(2π)2

1

4πε

2πe2

q

[Fn(q)

]2ρ(q)ρ(−q), (3.20)

where ε 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

38

that constraint and investigate a more realistic model and conclude on the validity

of this simple approach. The Hartree-Fock approximation consists of pairing the four

fermion operators in groups of two, averaging on one of the groups as follows

ρ(q)ρ(−q) '∑

Y,Y ′e−iqy(Y−Y ′)

[〈c†n,Y−qx`2/2cn,Y +qx`2/2〉c†n,Y ′+qx`2/2cn,Y ′−qx`2/2

− 〈c†n,Y−qx`2/2cn,Y ′−qx`2/2〉c†n,Y ′+qx`2/2cn,Y +qx`2/2

]. (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) =1

4πε

2πe2

q

[Fn(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

VF (q) = −∫ ∞

0

dxxVH(x/`)J0(xq`), (3.23)

where J0(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 Hartree-Fock approximation

becomes

HHF =1

2

∫d2q

(2π)2VHF (q)〈ρ(−q)〉ρ(q). (3.24)

This will be our starting point for treating the 2D electron system and investigating

the energetics and the stability of the different quantum states associated with it.

CHAPTER 4ISOTROPIC CRYSTALLINE PHASES

4.1 Stability Analysis of Isotropic M-electron 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 M -electron

bubble crystals. These crystals have the same structure (and triangular symmetry)

as the Wigner crystal, but instead of having one electron per given site, there are

M electrons; and according to previous theoretical Hartree-Fock investigations these

M -electron bubble crystals succeed each other in increasing M order as we approach

half-filling in a given Landau level [36–39]. The last state to win the energetics race

is the charge density wave state (CDW), termed stripe state, that is realized close to

half-filling.

Before we proceed with the stability calculation let us investigate how the dif-

ferent parameters of the crystalline structures are interrelated. The total filling factor

of the system is given by

ν = 2π`2N

A, (4.1)

where ` 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

crystallization 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 M -electron

39

40

BC configuration of the system, the partial filling factor is defined as

ν∗ = ν − 2n = 2π`2N∗

A(4.2)

= 2π`2 M

ABc

, (4.3)

where n is the Landau level index, N∗ is the macroscopic number of electrons in the

partially filled Landau level, and ABc =√

3/2a2B is the M -electron BC unit cell area

(aB is the lattice constant and as usual we assume triangular lattice configuration).

A typical M -electron bubble has a radius rB. Since the local filling factor on each

bubble is one, while the density is M/πr2B, 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 = a√

M , 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 = `

√4πM√

3ν∗. (4.4)

Finally, it is easy to find the ratio between the M -electron bubble radius and the

lattice parameter to be

rB

aB

=

√√3ν∗

2π. (4.5)

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 M -electron bubble crystal and discover the

41

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 =1

2

∑

R6=R′U(R−R′), (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 Hartree-Fock 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

Eelastic =1

2Ω

∑q

U(q)∑

R6=R′eiq·(R−R′)[uα(R)uβ(R′)− uα(R)uβ(R)

], (4.7)

where Ω 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

uα(R) =

∫d2q

(2π)2uα(q)eiq·R, (4.8)

uα(q) = Ac

∑R

uα(R)e−iq·R. (4.9)

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 uα(q), which according to Eq. (4.8) are L3. If we substitute the

above in Eq. (4.7) and use the following definition

∑R

eiq·R =(2π)2

Ac

∑Q

δ(q−Q) =Ω

Ac

∑Q

δq,Q, (4.10)

42

we find that the general elastic energy expression becomes

Eelastic =1

2A2c

∑Q

∫d2q

(2π)2

U(Q + q)(Qα + qα)(Qβ + qβ)− U(Q)QαQβ

×∑

Q′uα(q)uβ(Q′ − q)

' 1

2A2c

∑Q

∫d2q

(2π)2

U(Q + q)(Qα + qα)(Qβ + qβ)− U(Q)QαQβ

× uα(q)uβ(−q). (4.11)

In other words we have brought the elastic energy equation into the general form

given by

Eelastic =1

2

∫d2q

(2π)2Φαβ(q)uα(q)u∗β(q), (4.12)

where u∗β(q) = uβ(−q) and Φαβ(q) is the dynamical matrix defined as

Φαβ(q) =1

A2c

∑Q

U(Q + q)(Qα + qα)(Qβ + qβ)− U(Q)QαQβ

. (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

Φαβ(q) =1

A2c

∑

Q6=0

qαqβ

[U(Q) + (Q2

α + Q2β)

U ′(Q)

Q+ Q2

αQ2β

U ′′(Q)Q− U ′(Q)

Q3

]

+1

2δαβ

[q2Q2

α

U ′(Q)

Q+ Q2

α(Q2xq

2x + Q2

yq2y)

U ′′(Q)Q− U ′(Q)

Q3

− 2q2αQ4

α

U ′′(Q)Q− U ′(Q)

Q3

], (4.14)

43

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 non-singular 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

Eelastic =1

2λijkl∂iuj∂kul, (4.15)

where λijkl are elastic constants with only a certain number of them being independent

or non-zero (depending on the given symmetry of the elastic medium). For the tri-

angular lattice configuration the above expression simplifies to

Eelastic =1

2

[λ∂iui∂juj +

µ

2(∂iuj + ∂jui)

2

]

=1

2

[(λ + 2µ)

[(∂xux)

2 + (∂yuy)2]+ 2λ∂xux∂yuy

+ µ[(∂xuy)

2 + (∂yux)2 + 2∂xuy∂yux

]]. (4.16)

In the above, the elastic constants λ and µ are the Lame coefficients that are deter-

mined by the following relations that hold due to the triangular lattice configuration

λxyxy = λyxyx = λxyyx = λyxxy = µ, (4.17)

λxxyy = λyyxx = λ, (4.18)

λxxxx = λyyyy = λ + 2µ, (4.19)

λxxyy = λxxxx − 2λxyxy, (4.20)

and the rest of the possible λijkl’s is zero. The shear modulus c66 (associated with

the energy cost of shear deformations) is given by λxyxy = µ and the bulk modulus c1

44

(associated with the compressibility of the system) is given by λxxxx = λ + 2µ. 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

Eelastic =1

2λµανβqµqν uα(q)u∗β(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

λµανβqµqν = Φαβ(q). (4.22)

In other words the bulk modulus will be defined as c1 = Φxx(qxx)/q2x, and the shear

modulus will be given by c66 = Φxx(qyy)/q2y or if we use Eq. (4.14) we end up to the

general expressions for the bulk and shear moduli

c1 =

(ν∗

2π`2M

)2

U(q)

+

(ν∗

2π`2M

)2 ∑

Q6=0

[U(Q) +

5Q2x

2QU ′(Q) +

Q4x

2Q3

(QU ′′(Q)− U ′(Q)

)]], (4.23)

c66 =1

2

(ν∗

2π`2M

)2 ∑

Q 6=0

[Q2

x

QU ′(Q) +

Q2xQ

2y

Q3

(QU ′′(Q)− U ′(Q)

)]. (4.24)

In the above expressions we have used Eq. (4.4) for the M -electron 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 M -electron BC

of triangular symmetry (which is proved to be the stable ground state for the Wigner

45

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−2

−1

0

1

2

3

4x 10

−3

c6

6

ν*

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−2

0

2

4

6

8

10x 10

−3

c6

6

ν*

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−2

−1

0

1

2

3

4x 10

−3

c6

6

ν*

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−2

0

2

4

6

8

10x 10

−3

c6

6

ν*

Figure 4.1: Shear modulus (in units of e2/4πε`3) as a function of partial filling factorν∗ for the lowest Landau level and for the WC (left panel) and 2e BC (right panel)using the order parameter approach.

crystal [44]). We treat the M -electron bubbles as point-like 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]

ν∗(r) =∑R

θ(rB − |r−R− u(R)|), (4.25)

where θ(r) is the Heaviside step function, rB the M -electron bubble radius, and u(R)

the M -electron bubble displacement around the lattice site R. The above choice of

local filling factor produces a crude step-like 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′ =

√3a

2

(j,

2j′ + j√3

), (4.26)

Qjj′ =2π

a

(2j − j′√

3, j′

), (4.27)

46

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−3

−2

−1

0

1

2

3x 10

−3

c6

6

ν*

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−1

0

1

2

3

4

5

6

7x 10

−3

c6

6

ν*

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−3

−2

−1

0

1

2

3x 10

−3

c6

6

ν*

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−1

0

1

2

3

4

5

6

7x 10

−3

c6

6

ν*

Figure 4.2: Shear modulus (in units of e2/4πε`3) as a function of partial filling factorν∗ 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 M -electron BC lattice parameter. For a 2D

sample of total area Ω the interaction energy associated with the bubble crystal con-

figuration in the Hartree-Fock approximation is given by [39]

E =1

2

Ω

(2π`2)2

∑q

VHF(q)|∆(q)|2, (4.28)

where, VHF(q) is the Hartree-Fock potential given by Eq. (3.24) and ∆(q) is the

Fourier transform of the local filling factor which is found from Eq. (4.25) to be

∆(q) =4πM`2

Ω

J1(qrB)

qrB

∑R

e−iq·(R+u(R)). (4.29)

J1(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)

(2MJ1(qrB)

qrB

)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

47

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−5

−4

−3

−2

−1

0

1

2

3x 10

−3

c6

6

ν*

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−6

−4

−2

0

2

4

6x 10

−3

c6

6

ν*

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−5

−4

−3

−2

−1

0

1

2

3x 10

−3

c6

6

ν*

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−6

−4

−2

0

2

4

6x 10

−3

c6

6

ν*

Figure 4.3: Shear modulus (in units of e2/4πε`3) as a function of partial filling factorν∗ 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 crystal [44].

What we have achieved so far is produce an analytic expression for the elastic

moduli in the semiclassical Hartree-Fock 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 ν∗ for Landau

levels n = 0, 1, 2, 3 and for the isotropic WC (M = 1) and the isotropic 2-electron

per bubble crystal (2e-BC) (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 ν∗ ' 0.48 [54]. Additionally, we find

that for the n = 2 and n = 3 Landau levels the isotropic WC destabilizes around

ν∗ ' 0.24 and ν∗ ' 0.18 respectively, but the M = 2 isotropic BC can live up to

ν∗ ' 0.39 and ν∗ ' 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

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

−4

−2

0

2

4

6

x 10−3

c6

6

ν*

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

−6

−4

−2

0

2

4

x 10−3

c6

6

ν*

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

−4

−2

0

2

4

6

x 10−3

c6

6

ν*

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

−6

−4

−2

0

2

4

x 10−3

c6

6

ν*

Figure 4.4: Shear modulus (in units of e2/4πε`3) as a function of partial filling factorν∗ 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 step-function density pattern, localizing the point-like electrons around

their equilibrium sites. As a result, any further attempt to investigate the energetics 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 Hartree-Fock level, information about the energetics and stability of the

49

system. For the microscopic theory we will use the non-interacting 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 M -electron isotropic BC the real

space approximation of it becomes [36, 38, 55]

nn(r) =∑

i

M−1∑m=0

|φnm(r−Ri)|2. (4.31)

In other words, we assume that the electrons in the M -electron isotropic BC con-

figuration are in their non-interacting eigenstates (characterized by the z-component

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

in-plane 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

ρn(q) =nn(q)

e−q2`2/4Ln(q2`2/2), (4.32)

and the generalized Hartree-Fock cohesive energy similar to Eq. (4.28) assumes the

form

EHF =1

2

∫d2q

(2π)2VHF(q)|ρn(q)|2. (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

50

bubble as ρn(q) =∑

m ρnm(q) so that

ρn(q) = ρn(q)∑

i

e−iq·Ri , (4.34)

and the corresponding electron density at a given bubble nn(q) is given by an equation

similar to Eq. (4.32), namely

nn(q) =M−1∑m=0

∫dr|φnm(r)|2e−iq·r. (4.35)

The projected density for a given Landau level and given angular momentum m

assumes the form

ρnm(q) =

∫dr|φnm(r)|2e−iq·r

e−q2`2/4Ln(q2`2/2). (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

ρ0(q) = e−q2`2/4, (4.37)

ρ1(q) =(1− q2`2

2

)e−q2`2/4, (4.38)

ρ2(q) =(1− q2`2 +

q4`4

8

)e−q2`2/4. (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

Umm′(r) =

∫d2q

(2π)2ρm(q)VHF(q)ρm′(q)eiq·r. (4.40)

51

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

−5

−4

−3

−2

−1

0

1

2

x 10−3

ν*

c 66

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

−3

−2

−1

0

1

2

3

x 10−3

ν*

c 66

Figure 4.5: Shear modulus (in units of e2/4πε`3) as a function of partial filling factorν∗ 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 =1

2

∑

i6=j

∑

m,m′Umm′(Ri −Rj) +

∑i

∑

m<m′Umm′(0), (4.41)

which has the general form of Eq. (4.6), besides the “internal” 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 self-interactions

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 Hartree-Fock

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).

52

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

−5

−4

−3

−2

−1

0

1

2

3

4

5

x 10−3

ν*

c 66

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

−3

−2

−1

0

1

2

x 10−3

ν*

c 66

Figure 4.6: Shear modulus (in units of e2/4πε`3) as a function of partial filling factorν∗ 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 semi-classical results, but approaching half-filling 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

semi-classical 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, crystallization 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

53

0 2 4 6 8 10 12 14 16 18 20 22 240.003

0.005

0.007

0.009

0.011

0.013

0.015

0.017

0.019

0.021

0.023

0.025

0.027

0.029

U01

r

Figure 4.7: Interaction potential U01(r) (in units of e2/4πε`) vs. r (in units of `).

the characteristic non-monotonic 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 (U01(r) given by Eq. (4.40) for the m = 0, m′ = 1 case). The Fourier

transform of U01(r) is shown in Eq. (B.21) of the Appendix. Surprisingly, there is a

well pronounced local minimum at an inter-electron distance of r0 ' 1.48` (as shown

54

in Fig. (4.7)) that suggests 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 r0 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

rmi = Ri + (m− 1

2)r0x, m = 0, 1, (4.42)

where Ri are given by Eq. (4.26) and m distinguishes between the two electrons. In

order to study the stability of this crystalline structure we need to derive an expression

for the dynamical matrix starting from the cohesive energy given by Eq. (4.41) and

Taylor-expanding to second order in the displacements. As always, we defer to the

Appendix all the cumbersome details and present here the final result

Φmm′αβ (q) =

1

A2c

∑Q

eiQxr0(m−m′)[eiqxr0(m−m′)Umm′(Q + q)(Qα + qα)(Qβ + qβ)

− Umm′(Q)QαQβ

]. (4.43)

Notice that in the limit r0 → 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

Φxx(q) =1

2

∑

mm′Φmm′

xx (q), (4.44)

55

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

−2

−1

0

1

2

3x 10

−3

ν*

c 66

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

−4

−3

−2

−1

0

1

2

x 10−3

ν*

c 66

Figure 4.8: Shear modulus (in units of e2/4πε`3) as a function of partial filling factorν∗ for the n = 2 (left panel) and n = 3 (right panel) Landau level for the bubblecrystal 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) =1

2

∑

mm′eiqxr0(m−m′)Umm′(q)

= VHF(q)e−q2`2/2

[1 + cos(qxr0)− 1

2

(1 + cos(qxr0)

)q2`2 +

1

8q4`4

]. (4.45)

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 crystalline structure appears

quite stable. This is a first indication that the internal degrees of freedom might

play an important role in the physics of the 2D electron system. This BC with basis

state is not a solution of the Hartree-Fock equation (contrary to the rest of the M -

electron states [38]). It arises as the variational solution though of the Hartree-Fock

Hamiltonian satisfying

∂Ecoh(r0)

∂r= 0, (4.46)

56

where r0 = 1.48`. 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 Hartree-Fock energy. Below

we evaluate the normal modes associated with all the above crystalline 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 crystalline 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

Φmm′αβ (Ri) = δmm′δRi,0

∑

m′′

∑j

φmm′′αβ (Rj)− φmm′

αβ (Ri), (4.47)

where

φmm′αβ (r) = ∂α∂βUmm′(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

57

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

md2

dt2uα

mi = −∑

jm′Φmm′

αβ (Ri −Rj)uβm′j − eBεαβ

d

dtuβ

mi. (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

Φmm′αβ (Ri −Rj + r0(m−m′)x) = δmm′δij

∑

m′′

∑

k

φmm′′αβ (Rk + r0(m−m′′)x)

− φmm′αβ (Ri −Rj + r0(m−m′)x), (4.50)

where φαβ is given by Eq. (4.48), and as we see the expression reduces to the general

one of Eq. (4.48) in the r0 → 0 limit. The equation of motion for the two electrons in

this kind of BC becomes

md2

dt2uα

mi = −∑

jm′Φmm′

αβ (Ri −Rj + r0(m−m′)x)uβm′j − eBεαβ

d

dtuβ

mi. (4.51)

We show in the Appendix in great detail how to solve the above equations of motion

and derive the normal modes for all the crystalline structures of interest.

We show our results for the three different isotropic crystalline 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 ωc) associated with inter-Landau

58

J X0

0.009

0.0179

0.0269

0.0358

0.0448

0.0538

0.0627

0.0717

0.0807

0.0896

ω/ω

0

J X0.2082

0.2723

0.3363

0.4004

0.4645

0.5285

0.5926

0.6567

0.7207

0.7848

0.8489

ω/ω

0

q/(π/a)Γ Γ J X

0

0.009

0.018

0.027

0.036

0.045

0.054

0.063

0.072

0.081

0.09

ω/ω

0

J X0.5

0.53

0.56

0.59

0.62

0.65

0.68

0.71

0.74

0.77

0.8

ω/ω

0

q/(π/a)Γ Γ

J X0

0.009

0.018

0.027

0.036

0.045

0.054

0.063

0.072

0.081

0.09

ω/ω

0

J X0.5

0.53

0.56

0.59

0.62

0.65

0.68

0.71

0.74

0.77

0.8

ω/ω

0

q/(π/a)Γ Γ

Figure 4.9: Normal modes for the triangular crystalline structures in the n = 2Landau level and on the irreducible first Brillouin zone element. Top left panel: WCat ν∗ = 0.18. Top right panel: 2e BC at ν∗ = 0.30. Bottom panel: BC with basis atν∗ = 0.20. The left axes correspond to magnetophonons (lower curves) and the rightaxes to magnetoplasmons (upper curves). Frequency measures in ω0 = e2/4πε~` unitsand a is the lattice crystal 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 “special” points inside the irreducible first Brillouin zone

59

Table 4.1: Zero point energy of WC, 2e BC and BC with basis (BCb) for differentpartial filling factor values ν∗ within their range of stability.

ν∗ EWC EBC EBCb

0.05 0.359785∗ 0.360259 0.360394∗

0.10 0.362771 0.357701 0.3581840.15 0.372628 0.357806 0.3590240.20 0.375746 0.360150 0.3617970.25 0.362931∗ 0.362661 0.361640∗

element that have the largest weight and then writing the zero point energy (in units

of e2/4πε`) as [57]

EZP =1

2M

4πε`~e2

Nmodes∑j=1

6∑i=1

αiωj(qi), (4.52)

where Nmodes is the number of modes for the given crystalline structure (two for WC,

and four for the 2e BC) while αi 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

q1→3 :

2

9

(1,

1√3

),

4

9

(1,

1√3

),

8

9

(1,

1√3

), α1→3 =

1

9, (4.53)

q4→6 :

2

3

(1,

1

3√

3

),

4

9

(2,

1√3

),

2

9

(5,

1√3

), α4→6 =

2

9, (4.54)

where all reciprocal lattice vectors measure in π/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 energetics of the states whose stability

we calculated in the previous sections and in particular investigate the possibility of

60

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 z-direction can potentially

influence the energetics 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 q-dependent structure of the following form [58, 59]

ε(q) = ε

[1 +

2

qaB

(1− J2

0 (qRc))]

exp(λq`

), (4.55)

where ε is the bare dielectric constant of the substrate material (for GaAs it is 12.9ε0),

Rc = `√

2n + 1 is the cyclotron radius, aB the Bohr radius, J0(x) the zeroth order

Bessel function, and λ a finite-thickness parameter involving the finite z-direction

extend of the electron wavefunction. The result of the above modification is to add

an extra q-dependence 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 semi-classical

and the microscopic approximation) and for the different crystalline phases investi-

gated so far. For the sake of completeness we will include in our energetics 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

61

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

Eco

h

ν*

(M=1):bare(M=2):bare(M=1):screened(M=2):screened(M=1):screened,finite thickness(M=2):screened,finite thickness

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

Eco

h

ν*

(M=1):bare(M=2):bare(M=1):screened(M=2):screened(M=1):screened,finite thickness(M=2):screened,finite thickness

Figure 4.10: Cohesive energy for the isotropic WC (red) and 2e BC (blue) in unitsof e2/4πε` for the n = 2 Landau level (left panel) and the n = 3 Landau level (rightpanel). Solid lines correspond to bare Coulomb interaction. Dashed lines correspondto Coulomb interaction where screening effects are accounted for, and dotted linescorrespond to the latter case where finite thickness effects are included as well.

electrons (ν∗ = 1) separated by empty areas (ν∗ = 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 e2/4πε`) and given by

EBCcoh =

1

2π`2

ν∗

M

4πr2Bε

e2`

∑Q

VHF(Q)J2

1 (rBQ)

(rBQ)2, (4.56)

where as usual, M is the number of electrons per bubble, ν∗ 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 J1(x) is evaluated at the Q = 0 limit

and it is easy to show it gives 1/2). The general M -electron BC we investigate here

incorporates the WC case as well and for the dielectric constant we use the modified

62

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 z-direction only

shift the associated cohesive energy scale, but do not alter the interplay between the

phases. At approximately the same partial filling factor (ν∗ ' 0.22 for n = 2 and

ν∗ ' 0.17 for n = 3) the 2e BC crystal 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: MicroscopicModel

The microscopic model we developed earlier provides a much more accurate

approximation for the energetics interplay between the different crystalline states.

We evaluate the cohesive energy per electron (in units of e2/4πε`) 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

EBCcoh =

ε

`e2

ν∗

M2

∑

mm′

∑Q

Umm′(Q). (4.57)

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

63

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

−0.35

−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

Eco

h

ν*

wigner crystalbubble: r0=0.0bubble: r0=1.48lstripe

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

−0.16

−0.14

−0.12

−0.1

−0.08

−0.06

−0.04

−0.02

0

Eco

h

ν*

wigner crystalbubble: r0=0.0bubble: r0=1.48lstripe

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−0.12

−0.1

−0.08

−0.06

−0.04

−0.02

0

Eco

h

ν*

wigner crystalbubble: r0=0.0bubble: r0=1.48lstripe

Figure 4.11: Cohesive energy in units of e2/4πε` for the isotropic WC, 2e BC, BCwith basis, and stripe state for different modifications of the Coulomb interactionand for the n = 2 Landau level using the microscopic model. Top left panel: bareCoulomb interaction. Top right panel: screened Coulomb interaction with no finitethickness effects. Bottom panel: screened Coulomb interaction with finite thicknesseffects included.

the cohesive energy per electron (in units of e2/4πε`):

EBCbcoh =

ε

`e2

ν∗

M2

∑

mm′

∑Q

Umm′(Q)eiQxr0(m−m′). (4.58)

The above needs to be applied for the specific 2e BC with basis case we have developed

in this work. The expression simplifies to

EBCbcoh =

εν∗

4`e2

∑Q

[U00(Q) + U11(Q) + 2 cos(Qxr0)U01(Q)

], (4.59)

64

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

Eco

h

ν*

wigner crystalbubble: r0=0.0,M=2bubble: r0=0.0,M=3bubble: r0=1.48l,M=2stripe

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

−0.12

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

Eco

h

ν*

wigner crystalbubble: r0=0.0,M=2bubble: r0=0.0,M=3bubble: r0=1.48l,M=2stripe

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

−0.08

−0.06

−0.04

−0.02

0

0.02

Eco

h

ν*

wigner crystalbubble: r0=0.0,M=2bubble: r0=0.0,M=3bubble: r0=1.48l,M=2stripe

Figure 4.12: Cohesive energy in units of e2/4πε` for the isotropic WC, 2e BC, BCwith basis, and stripe state for different modifications of the Coulomb interactionand for the n = 2 Landau level using the microscopic model. Top left panel: bareCoulomb interaction. Top right panel: screened Coulomb interaction with no finitethickness effects. Bottom panel: screened Coulomb interaction with finite thicknesseffects included.

where we used the fact that U01(q) = U10(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]

ν∗(r) =∑

j

θ(a/2− |x− xj|), (4.60)

65

where a = ν∗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 e2/4πε`) is of the form

EScoh =

1

π2`

ε

ν∗e2

∑j

VHF

(2π

aS

j

)sin2(πν∗j)

j2. (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 always 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 πν∗) 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 energetics among the crystalline phases. However, another surprising result has

emerged. The BC with basis state has become the undisputed winner in the energetics

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 play a crucial role in the physics of the system. What this result infers

is that different crystalline structures or lattice orientations (besides the triangular

one investigated here) might play 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 half-filling is approached, something that is to be expected according

to previous investigations [36, 38, 39].

CHAPTER 5ANISOTROPIC CRYSTALLINE PHASES

5.1 Solving the Static Hartree-Fock Equation

In our previous treatment of the 2D electron system we approximated the

Coulomb interaction in the Hartree-Fock level and used an Ansatz for the electron

density (being either a classical order parameter or a microscopic approximation based

on non-interacting 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 Hartree-Fock 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 crystal-

line phases in the set of lattice symmetry associated with the triangular lattice

configuration. We plan to include anisotropy into the crystalline structures and

investigate how the energetics are affected.

Our starting point is the Hartree-Fock 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]

HHF =1

2

∫d2q

(2π)2VHF (q)|n(q)|2, (5.1)

where n(q) is the electronic density and VHF (q) is the modified Hartree-Fock 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

66

67

electronic quasiparticle density for an M -electron BC according to

n(r) =∑

i

M∑α=1

|ψα(r−Ri)|2, (5.2)

where, ψα(r) is the α-eigenstate of the above Hartree-Fock Hamiltonian. Notice

that this is a similar definition to Eq. (4.31) only we use the more accurate quasi-

particle wavefunctions instead of the non-interacting electron wavefunctions. Of

course, we have no prior knowledge of the former (and in fact we need to find them

self-consistently) so it is necessary to approximate their form by expanding them on

the latter basis of non-interacting wavefunctions ϕm(r) according to

ψα(r) =Ns−1∑m=0

Cmαϕm(r), (5.3)

where, Ns 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 non-interacting

particle wavefunction basis we obtain the following eigenvalue equation for the M -

electrons associated with each bubble in the BC

Ns−1∑m1=0

Gm1m2Cm1α = EαCm2α, (5.4)

where Cmα are the expansion coefficients associated with Eq. (5.3), Gm1m2 is a 2nd

rank tensor given by Eq. (C.12) in the Appendix, while gm1m2m3m4 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 10−4 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 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

−0.25

−0.2

−0.15

−0.1

−0.05

0

ν*

EH

F

WCBC: M=2

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

ν*

EH

F

WCBC: M=2BC: M=3

Figure 5.1: Ground state eigenvalue energies in units of e2/4πε` as a function of thepartial filling factor ν∗ for the Hartree-Fock equation. Left panel: WC and 2e BCcases for the n = 2 Landau level. Right panel: WC, 2e BC and 3e BC cases for then = 3 Landau level.

their eigenvalues Eα in ascending order. As a result the cohesive energy of the general

M -electron BC assumes the form

Ecoh =1

2M

M∑α=1

Eα, (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 M -electron 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 M -electron BC configuration is a solution of

the Hartree-Fock equation, in agreement with previous studies [38] where the solution

of the time-dependent Hartree-Fock equation produces the hierarchy of M -electron

BC’s. It should be noted that for the above results we have used a minimum expansion

69

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

ν*

ε

WC2e BC

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−0.4

−0.35

−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0

ν*

Eg

WC2e BCstripe

Figure 5.2: Left panel: Values of anisotropy that minimize the cohesive energiesfor WC and 2e BC for different values of ν∗. Right panel: Ground state energiesassociated with the minimizing values of anisotropy for WC and 2e BC. We also plotthe traditional stripe state for comparison.

basis dimension (Ns), disallowing the electrons to form hybrids by constraining them

to lie on their non-interacting ground states, for all the crystalline 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 half-filling, the stripe state (charge density wave) becomes favorable to all the

crystalline 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 crystalline 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

70

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 crystalline 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. (4.26-4.27) by introducing the

anisotropy parameter ε (not to be confused with the dielectric constant ε) which is

zero for no anisotropy and one for complete anisotropy. The new direct and reciprocal

lattice definitions become

Rjj′ =

√3

1− ε

a

2

(j,

2j′ + j√3

(1− ε)

), (5.6)

Qjj′ =√

1− ε2π

a

(2j − j′√

3,

j′

1− ε

), (5.7)

where the M -electron BC lattice parameter is still given by

a = `

√4πM√

3ν∗. (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 crystalline structure.

We show our results in Fig. (5.2) for the simplified microscopic case, where we

do not allow inter-electron 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 ν∗ for

71

−25 −20 −15 −10 −5 0 5 10 15 20 25

−20

−15

−10

−5

0

5

10

15

20

ε=0−25 −20 −15 −10 −5 0 5 10 15 20 25

−20

−15

−10

−5

0

5

10

15

20

ε=0.25

−25 −20 −15 −10 −5 0 5 10 15 20 25

−20

−15

−10

−5

0

5

10

15

20

ε=0.5−25 −20 −15 −10 −5 0 5 10 15 20 25

−20

−15

−10

−5

0

5

10

15

20

ε=0.75

Figure 5.3: Reciprocal lattice points for different values of anisotropy. We see howthe crystal gradually evolves a channel-like structure.

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 half-filling. In fact, the anisotropic WC state increases considerably its

energy difference from the rest of the states, as half-filling 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 crystalline systems in terms of anisotropic crystals, 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

72

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

ν*

ε

WC2e BC3e BC

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

ν*

Eg

WC2e BC3e BC

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

ν*

ε

WC2e BC3e BC

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

ν*

Eg

WC2e BC3e BC

Figure 5.4: Solutions of the Hartree-Fock equation for the anisotropic triangular lat-tice configuration. Top left panel: Values of anisotropy that minimize the cohesiveenergies for WC, 2e BC and 3e BC for different values of ν∗ and for the n = 2 Landaulevel. Top right panel: Ground state energies associated with the minimizing valuesof anisotropy for WC, 2e BC and 3e BC. Bottom left panel: Same as top left panelbut for n = 3 Landau level. Bottom right panel: Same as top right panel but forn = 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 channel-like structure emerges consisting

of one-dimensional periodic chains of electron guiding centers. In other words, the

crystalline discreteness of broken translational invariance is always 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

73

as well. We should also mention here that in the Hartree-Fock 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 non-interacting electron wavefunctions (eigenfunctions of the z-component of

angular momentum) are used in the expansion of Eq. (5.3) the more they will extend

around the BC sites causing “secondary” inter-bubble overlaps, enhancing even more

the effect of the Fock term. This is of course a computational by-product that one

cannot get rid off unless one goes beyond the Hartree-Fock 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 z-component 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.

74

As a result, for the WC case for example, if one wants the electron to form a hybrid

state using five non-interacting electron states one needs to use a dimensionality of

Ns = 9. For the 2e BC one needs to add the extra odd parity state and use Ns = 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 non-interacting electron wavefunctions participate in the

hybrids. For the M = 3 states though, we have not increased to Ns = 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. Changing the truncated quasiparticle wavefunction

space does not seem to alter the energetics 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 ν∗ ' 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 short-range 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

75

−6 −4 −2 0 2 4 6 8 10 12−6

−4

−2

0

2

4

6

8

10

12

14

x/l

y/l

−6 −4 −2 0 2 4 6 8 10 12 14 16 18 20−6

−4

−2

0

2

4

6

8

x/l

y/l

−6 −4 −2 0 2 4 6 8 10 12 14 16 18 20−6

−4

−2

0

2

4

6

8

x/l

y/l

−6 −4 −2 0 2 4 6 8 10 12 14 16 18 20−6

−4

−2

0

2

4

6

8

x/l

y/l

Figure 5.5: Density profiles on the unit cell for the n = 2 Landau level and ν∗ = 0.11for truncated basis dimensionality of Ns = 9. Length unit is the magnetic length.Top left panel: Isotropic state. Top right panel: Anisotropic ε = 0.8 case using onlythe m = 0 non-interacting state. Bottom left panel: Same as top right panel but fora hybrid of m = 0, m = 2 non-interacting states. Bottom right panel: Optimizedhybrid state solution of the Hartree-Fock equation for ε = 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 non-interacting wavefunctions φm(r) given by Eq. (3.18). What we find is

ψ(r) =Ns−1∑m=0

amφm(r), (5.9)

where am are normalized to unity coefficients truncated in a Hilbert space of Ns

dimensions. It is useful to rewrite the above in cartesian coordinates. For the specific

76

case of the WC considered here one can easily show that the above written analytically

in cartesian coordinates becomes

ψ(x, y) =a0

4√

π`e−r2/4`2

(x + iy

`

)2

+a1√2π`

(1− x2 + y2

4`2

)(x + iy

`

)e−r2/4`2

+Ns−3∑m=0

am+2√2mπ(m + 2)!`

(x− iy

`

)m

Lm2

(r2

2`2

)e−r2/4`2 . (5.10)

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)

R2 =

√3

1− ε

a

2

(0,

2(1− ε)√3

), (5.12)

R3 =

√3

1− ε

a

2

(1,

1− ε√3

), (5.13)

and then numerically evaluate for any different set of am’s we want the probability

density given by

|ψtot(r)|2 = |ψ(r−R1)|2 + |ψ(r−R2)|2 + |ψ(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-

77

Fock equation for the given values of ν∗ 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 crystalline structures.

This might pose a difficulty since (as we discussed earlier) the short range correlations

that exist in these anisotropic crystals 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 Taylor-expanding around the real electronic displacements. This can be

achieved by following the Miranovic and Kogan approach [61]. We start by writing

the general elastic energy density associated with a 2D anisotropic crystal as

Eel =1

2

[c11,x(∂xux)

2 + c11,y(∂yuy)2 + c66,x(∂yux)

2 + c66,y(∂xuy)2 + 2c11,xy(∂xux)(∂yuy)

+ 2c66,xy(∂yux)(∂xuy)

]. (5.15)

The elastic moduli c11,x, c11,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 c11,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,x + c66,y

2. (5.16)

78

This can be easily proved if one applies a uniform rotation ux = −u0y, uy = u0x

(where u0 is a dimensionless constant) and demands invariance of the elastic energy.

The method that Miranovic 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 Miranovic and Kogan method relates a general uniform deformation

of the form

uα = uα,βxβ, (5.17)

where the coefficient uα,β is a dimensionless constant, to uniform reciprocal lattice

deformations, which to first order in uα,β can be written as [61]

Qα = Qα − uβ,αQβ. (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

u20

[Eel(Q)− Eel(Q)

]. (5.19)

For a shear deformation along the x direction given by ux = u0y, uy = 0, the cor-

responding deformation in the reciprocal lattice vectors becomes

Qx = Qx, (5.20)

Qy = Qy − u0Qx. (5.21)

79

For a shear along the y direction given by uy = u0x, ux = 0, the corresponding

deformation in the reciprocal lattice vectors becomes

Qy = Qy, (5.22)

Qx = Qx − u0Qy. (5.23)

A uniform rotation given by ux = −u0y, uy = u0x, induces the deformation

Qx = Qx − u0Qy, (5.24)

Qy = Qy + u0Qx, (5.25)

and finally, a squash deformation given by ux = u0x, uy = −u0y, induces the reciprocal

lattice vector deformation

Qx = (1− u0)Qx, (5.26)

Qy = (1 + u0)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

crystalline 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

80

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

c 66

ν*

WC2e BC

Figure 5.6: Shear moduli (in units of e2/4πε`) for the isotropic WC and 2e BC forthe n = 2 Landau level produced using the Miranovic and Kogan approach [61].

the transition point to anisotropic WC is crossed (around ν∗ ' 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 c66x 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(∂2yux)

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 channel-like structure persists at any

value of anisotropy or filling factor. As a result, a deformation of the form u = u0yx

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

81

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

5

10

15

20

25

30

35

c 66x

ν*

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

c 66y

ν*

Figure 5.7: Left panel: Shear modulus c66x for the ground state of the anisotropic WCfor the n = 2 Landau level. Right panel: Shear modulus c66y for the same crystallinestructure. Notice that they both coincide in the range below ν∗ ' 0.1, where theground state is the isotropic WC, but it does not clearly show in the graphs due tothe different scales used. The shear moduli measure in e2/4πε` units.

WC under microwave irradiation [35], the resonance pinning frequency is given by

ωp ' Σ

ρmωc

, (5.28)

where, ρm = m/(πa2) is the mass density, ωc the cyclotron frequency, and Σ is the

quasiparticle self-energy associated with the presence of disorder into the system and

given by

Σ ∼ ∆

c66a2ξ20

, (5.29)

where, ∆ is the variance of the random pinning potential, and ξ0 is the smallest

correlation length between ` (the magnetic length), and ξd (the disorder correlation

length). If we assume that the strong magnetic field imposes ` ¿ ξd then we end up

with the following result for the resonance pinning frequency

ωp ∼ ∆

Mc66(ν∗)`4. (5.30)

82

This result is generalized for the M -electron 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 ν∗ ' 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 ν∗ ' 0.15 follows the opposite

trend, if compared with the M -electron 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 ≤ ν∗ ≤ 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 energetics

analysis the anisotropic WC becomes favorable around ν∗ ' 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 M -electron 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 6CONCLUSIONS

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-

layer interactions modify the tunneling current in two ways. At first, due to the short

range part of those interactions, the collective modes of the bilayer 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 layer 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 crystalline 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

83

84

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 crystalline 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 play 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 Hartree-Fock

equation associated with an electron in these quantum Hall systems and were able

that way to find the quasiparticle wavefunctions for different crystalline 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 energetics 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

85

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

layers plays 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 ABILAYER SYSTEM EIGENMODES

A.1 Single Layer Eigenmodes

Here we provide an analytic derivation for the diagonalization procedure of the

bilayer 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 layer 2D system of electrons in the presence of

a perpendicular magnetic field in the continuum elastic approximation. For a system

of density n0, the dynamics are described by the following Lagrangian

L = n0

∫d2r

1

2mu2 − eu ·A(u)− λ

2n0

(∂iui)2 − µ

4n0

(∂mul + ∂lum)2

+1

2n0

∫d2r′[∇ · u(r)][∇′ · u(r′)]

e2

4πε|r− r′|

, (A.1)

where the intralayer Coulomb interaction is treated in the continuum linear approx-

imation (charge fluctuations are given by δn/n0 = −∇ · 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, Bux/2, 0) and B

is the applied magnetic field. We decompose the displacement field u into transverse

(uT = z · (iq× u)/q) and longitudinal (uL = (iq ·u)/q) components after we Fourier-

transform Eq. (A.1) to find

L = n0

∫d2q

(2π)2

1

2mu2

T +1

2mu2

L+1

2mωc[uT uL−uLuT ]− 1

2mω2

Lu2L−

1

2mω2

T u2T

. (A.2)

86

87

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 momenta of the transverse and longitudinal displacement fields. It is

easy to show that the corresponding results are

pT =∂L∂uT

= n0muT +1

2mn0ωcuL, (A.3)

pL =∂L∂uL

= n0muL − 1

2mn0ωcuT . (A.4)

The next step involves building the equations of motion associated with these fields

that requires using the well-known formula

∂L∂ui

− d

dt

∂L∂ui

−∇ ∂L∂(∇ui)

= 0, i = T, L. (A.5)

As a result we find the following two coupled dynamical equations

uT + ωcuL + ω2T uT = 0, (A.6)

uL − ωcuT + ω2LuL = 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

ω2± =

1

2

[ω2

c + ω2T + ω2

L ±√

(ω2c + ω2

T + ω2L)2 − 4ω2

T ω2L

]. (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 momenta of the displacement fields

we can switch to the Hamiltonian representation and write down for the Hamiltonian

88

of the 2D electron system

H = n0

∫d2q

(2π)2

uLpL + uT pT − L

= n0

∫d2q

(2π)2

p2

T

2m+

p2L

2m+

1

2ωc[uT pL − uLpT ] +

1

2m

(ω2

L +ω2

c

4

)u2

L

+1

2m

(ω2

T +ω2

c

4

)u2

T

. (A.9)

Decoupling Eqs. (A.6, A.7) produces the following

....u i + [ω2

c + ω2T + ω2

L]ui + ω2T ω2

Lui = 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 = Aieiω+t + A′

ie−iω+t + Bie

iω−t + B′ie−iω−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 eiω+t + A′T e−iω+t+BT eiω−t + B′

T e−iω−t, (A.12)

uL = iωcω+

ω2+ − ω2

L

[−AT eiω+t + A′T e−iω+t]+i

ωcω−ω2− − ω2

L

[−BT eiω−t + B′T e−iω−t]. (A.13)

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

89

we will suppress n0 (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

uT = −ωc

mpL − (ω2

T +ω2

c

2)uT , (A.14)

uL =ωc

mpT − (ω2

L +ω2

c

2)uL, (A.15)

...u T =

1

2ωc(ω

2c + ω2

T + 2ω2L)uL − ω2

c + ω2T

mpT , (A.16)

...uL = −1

2ωc(ω

2c + ω2

L + 2ω2T )uL − ω2

c + ω2L

mpL. (A.17)

We pick out the four non-redundant (out of the eight possible) equations for the

coefficients to end up with the 4x4 linear system:

AT + A′T + BT + B′

T = uT0 , (A.18)

iω+[AT − A′T ] + iω−[BT −B′

T ] =pT0

m− ωc

2uL0 , (A.19)

−ω2+[AT + A′

T ]− ω2−[BT + B′

T ] = −ωc

mpL0 − (ω2

T +ω2

c

2)uT0 , (A.20)

−iω3+[AT − A′

T ]− iω3−[BT −B′

T ] =1

2ωc(ω

2c + ω2

T + 2ω2L)uL0 −

ω2c + ω2

T

mpT0 , (A.21)

where uL0 , uT0 and pL0 , pT0 are the displacements and canonical momenta, respectively

evaluated at time t = 0. After solving the system, we obtain the following relations

between the field coefficients and the canonical variables

AT =1

2(ω2+ − ω2−)ω+

−ω+[ω2

− − ω2T − ω2

c/2]uT0 +ωcω+

mpL0 + i

ω2− − ω2

c − ω2T

mpT0

+ iωc(ω

2+ + ω2

L)

2uL0

, (A.22)

90

A′T =

1

2(ω2+ − ω2−)ω+

−ω+[ω2

− − ω2T − ω2

c/2]uT0 +ωcω+

mpL0 − i

ω2− − ω2

c − ω2T

mpT0

− iωc(ω

2+ + ω2

L)

2uL0

, (A.23)

BT =1

2(ω2+ − ω2−)ω−

ω−[ω2

+ − ω2T − ω2

c/2]uT0 −ωcω−m

pL0 − iω2

+ − ω2c − ω2

T

mpT0

− iωc(ω

2− + ω2

L)

2uL0

, (A.24)

B′T =

1

2(ω2+ − ω2−)ω−

ω−[ω2

+ − ω2T − ω2

c/2]uT0 −ωcω−m

pL0 + i(ω2

+ − ω2c − ω2

T )

mpT0

+ iωc(ω

2− + ω2

L)

2uL0

. (A.25)

Notice that A†T = AT , B†

T = BT . After some tedious calculations we can prove to

ourselves using [ui0(q), pj0(q′)] = i~δij(2π)2δ2(q− q′) that

[AT , BT ] = [A′T , B′

T ] = [AT , B′T ] = [A′

T , BT ] = 0 (A.26)

[A′T , AT ] = 1/n1 , [B′

T , BT ] = 1/n2 (A.27)

n1 =2mn0ω+

~ω2

+ − ω2−

ω2+ − ω2

L

, n2 =2mn0ω−~

ω2+ − ω2

−ω2

L − ω2−. (A.28)

We have reintroduced n0 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

A′T (q) =

a1(q)√n1

, AT (q) =a†1(q)√

n1

, (A.29)

B′T (q) =

a2(q)√n2

, BT (q) =a†2(q)√

n2

, (A.30)

91

and that way we restore the usual commutation relation

[ai(q), a†j(q′)] = (2π)2δ2(q− q′)δij. (A.31)

As a result, the final form of the annihilation eigenmode operators for the single layer

quantum Hall system in the continuum elastic approximation where the effect of the

intralayer Coulomb interaction is included (to linear order in the elastic displacement

field) is

a1(q) =

√mn0

2~ω+(ω2+ − ω2

L)(ω2+ − ω2−)

[ω+[ω2

+ − ω2L − ω2

c/2]uT0(q) +ωcω+

n0mpL0(q)

+ iω2

+ − ω2L

n0mpT0(q)− iωc

ω2+ + ω2

L

2uL0(q)

], (A.32)

a2(q) =

√mn0

2~ω−(ω2L − ω2−)(ω2

+ − ω2−)

[ω−[ω2

L − ω2− + ω2

c/2]uT0(q)− ωcω−n0m

pL0(q)

+ iω2

L − ω2−

n0mpT0(q) + iωc

ω2− + ω2

L

2uL0(q)

]. (A.33)

Going back to Eqs. (A.12, A.13) we can re-express them in terms of our new operators

uT (q, t) =a†1(q)√

n1

eiω+t +a1(q)√

n1

e−iω+t +a†2(q)√

n2

eiω−t +a2(q)√

n2

e−iω−t, (A.34)

uL(q, t) = −iωcω+

ω2+ − ω2

L

a†1(q)√n1

eiω+t + iωcω+

ω2+ − ω2

L

a1(q)√n1

e−iω+t − iωcω−

ω2− − ω2L

a†2(q)√n2

eiω−t

+ iωcω−

ω2− − ω2L

a2(q)√n2

e−iω−t, (A.35)

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 =

∫d2q

(2π)2

[~ω+a†1(q)a1(q) + ~ω−a†2(q)a2(q) +

~ω+ + ~ω−2

]. (A.36)

92

We have found the eigenmodes and eigenfrequencies of the single layer quantum

Hall system including the effect of the intralayer Coulomb interaction. It is interesting

to see how our solutions behave to different limiting cases. In the low magnetic field

limit (ωc → 0) we discover that

ω2+ ' ω2

L +ω2

L

ω2L − ω2

T

ω2c , (A.37)

ω2− ' ω2

T −ω2

L

ω2L − ω2

T

ω2c , (A.38)

while Eqs. (A.32, A.33) become

a1(q, ωc → 0) ' −i

√n0mωL

2~uL0(q) +

1√2mn0~ωL

pL0(q), (A.39)

a2(q, ωc → 0) '√

n0mωT

2~uT0(q) + i

1√2mn0~ωT

pT0(q), (A.40)

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 (ωc → ∞) we discover expanding the square root in

Eq. (A.8) that

ω+ ' ωc +ω2

L + ω2T

2ωc

+ O(ω−3c ), (A.41)

ω− ' ωLωT

ωc

+ O(ω−3c ), (A.42)

and using the above in Eqs. (A.32, A.33) we discover

a1(q, ωc →∞) =

√mn0ωc

2~uT0 − iuL0

2+

pL0 + ipT0√2mn0~ωc

, (A.43)

a2(q, ωc →∞) =

√mn0ωc

2~ωLωT

[ωT

2uT0 + i

ωL

2uL0

]. (A.44)

93

Notice that Eq. (A.44) is singular in ωc. This is not a physical instability though,

because from Eq. (A.42) we see that the ω− 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-

layer Coulomb interaction. The dynamics of the system is described by the following

Lagrangian

L = n0

∫d2r

1

2mu2

A +1

2mu2

B − euA ·A(uA)− euB ·A(uB)

− λ

2n0

[(∂iuiA)2 + (∂iuiB)2

]− µ

4n0

[(∂mulA + ∂lumA

)2 + (∂mulB + ∂lumB)2

]

− 1

2εn0

∫d2r′[∇ · uA(r)][∇′ · uA(r′)]

e2

|r− r′|− 1

2εn0

∫d2r′[∇ · uB(r)][∇′ · uB(r′)]

e2

|r− r′|− n0

∫d2r′[∇ · uA(r)][∇′ · uB(r′)]

e2

ε√

(x− x′)2 + (y − y′)2 + d2

− 1

2

K

n0

(uA − uB)2

. (A.45)

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 layer case) we find

L = n0

∫d2q

(2π)2

1

2mu2

LA+

1

2mu2

TA+

1

2mωc[uTA

uLA− uLA

uTA]− 1

2mω2

T u2TA

− 1

2mω2

Lu2LA

+1

2mu2

LB+

1

2mu2

TB+

1

2mωc[uTB

uLB− uLB

uTB]− 1

2mω2

T u2TB

− 1

2mω2

Lu2LB− uLA

uLB

2πe2n0

εqe−qd − 1

2

K

n0

(uA − uB)2

. (A.46)

94

Next, we decouple the two single layer Lagrangians by switching to the in-phase

v = (uB + uA)/2 and out-of-phase u = uB − uA modes that produces

L = n0

∫d2q

(2π)2

1

2(2m)v2

L +1

2(2m)v2

T +1

2(2m)ωc[vT vL − vLvT ]− 1

2(2m)(cT q)2v2

T

− 1

2(2m)

[(cLq)2 +

2πe2n0

mεq(1 + e−qd)

]v2

L

+1

2(m

2)u2

L +1

2(m

2)u2

T +1

2(m

2)ωc[uT uL − uLuT ]− 1

2(m

2)

[(cT q)2 +

2K

mn0

]u2

T

− 1

2(m

2)

[(cLq)2 +

2K

mn0

+2πe2n0

mεq(1− e−dq)

]u2

L

. (A.47)

If we compare the structure of the in-phase and out-of-phase Lagrangians to the single

layer one given by Eq. (A.2), we realize that we can define two “effective” single layer

dynamics whose eigenmodes and eigenfrequencies can be read off if we change the

definitions for the parameters involved in each case. For the out-of-phase modes we

need to apply the following changes

m → m

2, (A.48)

ω2T →

[c2T q2 +

2K

mn0

]= Ω2

T , (A.49)

ω2L →

[c2Lq2 +

2K

mn0

+2πe2n0

mεq(1− e−qd)

]= Ω2

L, (A.50)

For the in-phase modes the parameters need to change according to

m → 2m, (A.51)

ω2T → ω2

T = O2T , (A.52)

ω2L →

[c2Lq2 +

2πe2n0

mεq(1 + e−qd)

]= O2

L, (A.53)

The cyclotron frequency ωc stays unchanged in both of the cases. After performing

the above redefinitions for the parameters involved in the single layer eigenmode

95

and eigenfrequency definitions we can directly write down the final results for the

corresponding bilayer in-phase and out-of-phase “effective” single layer ones. For the

out-of-phase displacement field operators we have

a+(q) =

√mn0

4~Ω+(Ω2+ − Ω2

L)(Ω2+ − Ω2−)

Ω+[Ω2

+ − Ω2L − ω2

c/2]uT0(q) + 2ωcΩ+

n0mpL0(q)

+ 2iΩ2

+ − Ω2L

n0mpT0(q)− iωc

Ω2+ + Ω2

L

2uL0(q)

, (A.54)

a−(q) =

√mn0

4~Ω−(Ω2L − Ω2−)(Ω2

+ − Ω2−)

Ω−[Ω2

L − Ω2− + ω2

c/2]uT0(q)− 2ωcΩ−n0m

pL0(q)

+ 2iΩ2

L − Ω2−

n0mpT0(q) + iωc

Ω2− + Ω2

L

2uL0(q)

, (A.55)

and the displacement field transverse and longitudinal parts are given by

uT (q, t) =a†+(q)√

N+

eiΩ+t +a+(q)√

N+

e−iΩ+t +a†−(q)√

N−eiΩ−t +

a−(q)√N−

e−iΩ−t, (A.56)

uL(q, t) = −iωcΩ+

Ω2+ − Ω2

L

a†+(q)√N+

eiΩ+t + iωcΩ+

Ω2+ − Ω2

L

a+(q)√N+

e−iΩ+t − iωcΩ−

Ω2− − Ω2L

a†−(q)√N−

eiΩ−t

+ iωcΩ−

Ω2− − Ω2L

a−(q)√N−

e−iΩ−t, (A.57)

while for the eigenfrequencies the following formula applies

Ω2± =

1

2

[ω2

c + Ω2T + Ω2

L ±√

(ω2c + Ω2

T + Ω2L)2 − 4Ω2

LΩ2T

], (A.58)

and the normalization coefficients are of the form

N+ =n0mΩ+

~Ω2

+ − Ω2−

Ω2+ − Ω2

L

, (A.59)

N− =n0mΩ−~

Ω2+ − Ω2

−Ω2

L − Ω2−. (A.60)

In the zero magnetic field limit we find (as expected) the same decoupling of the

acoustic modes as encountered previously for the pure single layer case. These out-

96

of-phase modes (as we have mentioned in the main text) play a prominent role in

the response physics of the bilayer system that involve coupling with independent

tunneling electrons injected into the system. For the in-phase displacement field

operators we find

b+(q) =

√mn0

~O+(O2+ −O2

L)(O2+ −O2−)

O+[O2

+ −O2L − ω2

c/2]vT0(q) +ωcO+

2n0mpvL0

(q)

+ iO2

+ −O2L

2n0mpvT0

(q)− iωc

O2+ + O2

L

2vL0(q)

, (A.61)

b−(q) =

√mn0

~O−(O2L −O2−)(O2

+ −O2−)

O−[O2

L −O2− + ω2

c/2]vT0(q)− ωcO−2n0m

pvL0(q)

+ iO2

L −O2−

2n0mpvT0

(q) + iωc

O2− + O2

L

2vL0(q)

, (A.62)

and the displacement field transverse and longitudinal parts are given by

vT (q, t) =b†+(q)√

S+

eiO+t +b+(q)√

S+

e−iO+t +b†−(q)√

S−eiO−t +

b−(q)√S−

e−iO−t, (A.63)

vL(q, t) = −iωcO+

O2+ −O2

L

b†+(q)√S+

eiO+t + iωcO+

O2+ −O2

L

b+(q)√S+

e−iO+t − iωcO−

O2− −O2L

b†−(q)√S−

eiO−t

+ iωcO−

O2− −O2L

b−(q)√S−

e−iO−t, (A.64)

while the eigenfrequencies have the usual form

O2± =

1

2

[ω2

c + O2T + O2

L ±√

(ω2c + O2

T + O2L)2 − 4O2

LO2T

], (A.65)

and the normalization coefficients have become

S+ =4n0mO+

~O2

+ −O2−

O2+ −O2

L

, (A.66)

S− =4n0mO−

~O2

+ −O2−

O2L −O2−

. (A.67)

97

The in-phase modes are related to a translation of the bilayer system as a whole and

as we have mentioned in the main text do not couple with independent tunneling

electrons injected into the bilayer system and naturally drop out from the tunneling

current calculation.

In the limit where the two layers are taken far apart (d → ∞) their coupling

is expected to diminish (K → 0) and the above redefinitions for the acoustic modes

become

Ω2T ' ω2

T , (A.68)

Ω2L ' ω2

L +2πe2n0

mεq, (A.69)

O2T = ω2

T , (A.70)

O2L ' ω2

L +2πe2n0

mεq. (A.71)

This means that the four modes become two-fold degenerate (but still orthogonal)

and there is no difference between in-phase or out-of-phase fluctuations. On the other

hand, when the two layers are brought very close together (d → 0) which naturally

enhances their coupling as well we find

Ω2T ' ω2

T +2K

mn0

, (A.72)

Ω2L ' ω2

L +2K

mn0

, (A.73)

O2T = ω2

T , (A.74)

O2L ' ω2

L +2πe2(2n0)

mεq. (A.75)

In this case the in-phase modes describe the correct physics of one layer of electrons

with density 2n0. We can see this clearly from Eq. (A.47) as well. The out-of-

phase modes are unphysical. They are produced out of the independent way the

98

displacement fields of the two different layers are treated, something that persists all

the way. It can be removed by hand starting from Eq. (A.47) and assuming what is

physically correct: uA(d → 0) = uB(d → 0).

A.3 Tunneling Current

Here we provide all the details for the tunneling current calculation. We start

by deriving the matrix elements associated with the independent electrons injected

into the system and coupling with the bulk electron modes. As we showed in the text,

the coupling is given by

Hcoupling = −c†AcA

∫d2q

(2π)2

e2n0

2ε

[(1 + e−dq

)vL − 1

2

(1− e−dq

)uL

]

− c†BcB

∫d2q

(2π)2

e2n0

2ε

[(1 + e−dq

)vL +

1

2

(1− e−dq

)uL

]. (A.76)

If we use the definitions for the displacement fields in terms of their creation and

annihilation operators rewritten in more compact form

uL = −if1a†1 + if1a1 − if2a

†2 + if2a2, (A.77)

vL = −if3a†3 + if3a3 − if4a

†4 + if4a4, (A.78)

where

f1 = ωc

√~

n0m

Ω+

(Ω2+ − Ω2

L)(Ω2+ − Ω2−)

, (A.79)

f2 = −ωc

√~

n0m

Ω−(Ω2

L − Ω2−)(Ω2+ − Ω2−)

, (A.80)

f3 = ωc

√~

4n0m

O+

(O2+ −O2

L)(O2+ −O2−)

, (A.81)

f4 = −ωc

√~

4n0m

O−(O2

L −O2−)(O2+ −O2−)

, (A.82)

99

and a1, a2 correspond to a+, a− given by Eqs. (A.54, A.55), while a3, a4 correspond

to b+, b− given by Eqs. (A.61, A.62), we can insert them back into Eq. (A.76) to find

Hcoupling = −c†AcA

∫d2q

(2π)2

−ie2n0

2ε

[f3

(a†3 − a3

)+ f4

(a†4 − a4

)](1 + e−dq

)

− 1− e−dq

2

[f1

(a†1 − a1

)+ f2

(a†2 − a2

)]

− c†BcB

∫d2q

(2π)2

−ie2n0

2ε

[f3

(a†3 − a3

)+ f4

(a†4 − a4

)](1 + e−dq

)

+1− e−dq

2

[f1

(a†1 − a1

)+ f2

(a†2 − a2

)]

= ic†AcA

∫d2q

(2π)2

e2n0

2ε

(1 + e−dq

) 4∑i=3

fi

[a†i − ai

]

−∫

d2q

(2π)2

e2n0

4ε

(1− e−dq

) 2∑i=1

fi

[a†i − ai

]

+ ic†BcB

∫d2q

(2π)2

e2n0

2ε

(1 + e−dq

) 4∑i=3

fi

[a†i − ai

]

+

∫d2q

(2π)2

e2n0

4ε

(1− e−dq

) 2∑i=1

fi

[a†i − ai

]

= c†AcA

[i

4∑s=1

MsA

(a†s − as

)]+ c†BcB

[i

4∑s=1

MsB

(a†s − as

)], (A.83)

where

MsA =

− e2n0

4ε

(1− e−dq

)fs, s = 1, 2,

e2n0

2ε

(1 + e−dq

)fs, s = 3, 4,

, MsB =

−MsA, s = 1, 2,

MsA, s = 3, 4,

. (A.84)

We have included in the summation s an integration in q as well. The above result

provides the proof of Eq. (2.26) for the matrix elements of the coupling between an

independent tunneling electron injected into the bilayer system and the collective

modes of this system produced by the bulk electrons. In what follows, we use these

100

matrix elements to build the independent boson model associated with the tunneling

current of such a system.

A.3.1 Correlation Function

The independent boson model, introduced originally by Johansson and Kinaret

[26] for the bilayer system is described by the following Hamiltonian

H = H0 + H−T + H+

T

=

[εA + i

∑s

MsA

(a†s − as

)]c†AcA +

[εB + i

∑s

MsB

(a†s − as

)]c†BcB

+∑

s

~Ωsa†sas + Tc†AcB + Tc†BcA. (A.85)

As we showed in the main text the tunneling current correlation function associated

with the above model is given by

I∓(t) = 〈H−T (t)H+

T (0)〉, (A.86)

where the time-dependence is meant in the interaction picture representation. The

statistical averaging can be independently performed for the bulk collective modes

and the injected tunneling electron according to

I∓(t) = 〈H−T (t)H+

T (0)〉

= T 2〈e i~H0tc†AcBe−

i~H0tc†BcA〉

= T 2〈e i~Hitc†AcBe−

i~Hf tc†BcA〉

= T 2ν(1− ν)〈e iHit

~ e−iHf t

~ 〉

= T 2ν(1− ν)〈Tt exp

− i

~

∫ t

0

dt′eiHit′~

(Hf −Hi

)e−

iHit′~

〉. (A.87)

101

In the above we have used 〈c†AcAcBc†B〉 = (1 − ν)ν to average on the injected tun-

neling electron operators (ν is the filling factor). Also, we have defined the following

operators associated with the state that the bulk of the system is left before and after

a tunneling event

Hi = εA + i∑

s

MsA

(a†s − as

)+

∑s

~Ωsa†sas

= εA −∆A +∑

s

~Ωs

(a†s −

iMsA

~Ωs

)(as +

iMsA

~Ωs

), (A.88)

Hf = εB + i∑

s

MsB

(a†s − as

)+

∑s

~Ωsa†sas

= εB −∆B +∑

s

~Ωs

(a†s −

iMsB

~Ωs

)(as +

iMsB

~Ωs

). (A.89)

Performing the calculation using the bosonic operator algebra we find

eiHit

~(Hf −Hi

)e−

iHit

~ = εB −∆B − εA + ∆A +∑

s

|MsB −MsA|2~Ωs

+∑

s

(iMsB − iMsA

)[eiΩst

(a†s −

iMsA

~Ωs

)− e−iΩst(as +

iMsA

~Ωs

)]. (A.90)

In the above εA, εB are the corresponding Madelung energies while

∆A(B) =∑

s

|MsA(B)|2~Ωs

, (A.91)

are the polaron shifts [26]. In order to calculate the time-ordered expression of

Eq. (A.87) we use the linked cluster expansion that involves only one link in the

exponential resummation. We apply the results of Mahan [48] and we find

U(t) = 〈Tt exp

− i

~

∫ t

0

dt′[λa†eiωt′ + h.c.

]〉

= exp

− 1

2~2φ(t)

, (A.92)

102

where

φ(t) =

2

ω2

[(N + 1

)(1− e−iωt

)+ N

(1− eiωt

)]+

2t

iω

|λ|2, (A.93)

and N = 1eβω−1

is the bosonic occupation number for a finite temperature. The

resultant correlation function becomes

I∓(t) = ν(1− ν)T 2e−it~

(εB−εA−∆B+∆A

)exp

[−

∑s

(MsB −MsA)2

(~Ωs)2

[(Ns + 1

)

× (1− e−iΩst

)+ Ns

(1− eiΩst

)]]. (A.94)

Interchanging A, B produces I±(t) which is identical.

A.3.2 Properties of the Correlation Function

Here we explore the properties that the correlation function given by Eq. (2.32)

has. We can isolate the time-dependent part and write it in the general form

C(t) = e−P

αλα(1+2Nα)

e

Pα

λα[(1+Nα)e−iωαt+Nαeiωαt], (A.95)

where α involves a summation in the collective modes of the bilayer system and an

integration in q as well, while

λα =

(MαA −MαB

~Ωα

)2

, (A.96)

are the weights of those modes and Nα are their corresponding bosonic thermal factors.

Notice that the correlation function evaluated at time t = 0 produces C(t = 0) = 1.

We can Fourier transform it according to

2πC(t) =

∫ +∞

−∞dωe−iωtC(ω), (A.97)

103

and keeping in mind the property C(t = 0) = 1, we can multiply differentiate it to

get the following set of equations

∫ +∞

−∞dωC(ω) = 2π, (A.98)

∫ +∞

−∞dωωC(ω) = 2π

∑α

λαωα, (A.99)

∫ +∞

−∞dωω2C(ω) = 2π

(∑α

λα

(1 + 2Nα

)ω2

α +(∑

α

λαωα

)2)

, (A.100)

∫ +∞

−∞dωω3C(ω) = 2π

(∑α

λαω3α + 3

∑α

λαωα

∑α

λα

(1 + 2Nα

)ω2

α

+(∑

α

λαωα

)3)

.

(A.101)

Since we are only interested in the out-of-phase magnetophonons (Ω−), and as we

explained in the main text, a zero temperature calculation suffices, we can set Nα = 0

in the above and find for the high magnetic field limit the following expression for λ−

λ− =

(e2n0

2ε

)2(1− e−dq

)2 ω3c

~mn0

1

ω2c + Ω2

L + Ω2T

1

Ω3LΩT

1

1− Ω2T

ω2c

. (A.102)

At this point if we define dimensionless units in the momentum integration (x = q/q0)

such that

dxf(x) =

[dq

2πqλ−

]

q0x

, (A.103)

ω(x) =[Ω−

]q0x

=

[ΩLΩT

ωc

]

q0x

, (A.104)

we reproduce Eqs. (2.34, 2.35) and end up to a simpler formula for the correlation

function

C(t) = exp

−

∫ 1

0

dxf(x)

exp

∫ 1

0

dxf(x)e−iω(x)t

. (A.105)

104

The last property we would like to prove for the correlation function is that it is

zero if the argument is negative or zero. Starting from the inverse Fourier transform

definition and using Eq. (A.105) we have

C(ω) =

∫ +∞

−∞dteiωt exp

−

∫ 1

0

dxf(x)

exp

∫ 1

0

dxf(x)e−iω(x)t

= e−R 10 dxf(x)

∫ +∞

−∞dteiωt

∞∑n=0

1

n!

[ ∫ 1

0

dxf(x)e−iω(x)t

]n

= e−R 10 dxf(x)

∞∑n=0

1

n!

∫ 1

0

dx1f(x1) · · ·∫ 1

0

dxnf(xn)

∫ +∞

−∞dtei[ω−ω(x1)−···−ω(xn)]t

= 2πe−R 10 dxf(x)

∞∑n=0

1

n!

∫ 1

0

dx1f(x1) · · ·∫ 1

0

dxnf(xn)δ(ω − ω(x1)− · · · − ω(xn)

)

= 0, ω < 0. (A.106)

The above clearly proves that the correlation function not only is zero for negative

frequency values but also for values lower than the magnetophonon gap value ω(0).

APPENDIX BISOTROPIC CRYSTALS

B.1 Fock Term Calculation

Here, we develop the Hartree-Fock approximation and show in details how

analytic expressions are obtained even for the complicated Fock term. As it was

shown in the introduction of the method, the Hartree term is given by

HH =1

2

∫d2q

(2π)2VH(q)〈ρ(−q)〉ρ(q), (B.1)

where, the four fermion operator is rearranged by averaging on one of the densities,

and the bare Coulomb interaction associated with the above definition is modified by

structure factors and given by

VH(q) =1

4πε

2πe2

q

[Fn(q)

]2. (B.2)

This is the same interaction potential that enters in the Fock term definition according

to

HF = −1

2

∫d2q

(2π)2VH(q)

∑

Y,Y ′eiqy(Y ′−Y )〈c†n,Y−qx`2/2cn,Y ′−qx`2/2〉

× c†n,Y ′+qx`2/2cn,Y +qx`2/2, (B.3)

where the four fermion operator is rearranged by averaging on one of the off-diagonal

pairs (which is the reason why the extra minus sign is introduced by Fermi statistics).

In order to bring the above term in an analytic form, we need to apply the following

105

106

change of variables [39]

Y − qx`2/2 = Y0 − px`

2/2, (B.4)

Y ′ − qx`2/2 = Y0 + px`

2/2. (B.5)

Additionally, if we introduce Y ′0 = Y0+qx`

2 the expression for the Fock term transforms

accordingly as

HF = −1

2

∫d2q

(2π)2VH(q)

∑

Y0,Y ′0

δY ′0 ,Y0+qx`2

∑px

eiqypx`2〈c†n,Y0−px`2/2cn,Y0+px`2/2〉

× c†n,Y ′0+px`2/2cn,Y ′0−px`2/2. (B.6)

The final step consists of introducing the following definition for the Kronecker delta

δY ′0 ,Y0+qx`2 =1

g

∑py

eipy(Y ′0−Y0−qx`2). (B.7)

It is easy to show that the final expression for the Fock term, if this last step is

performed, becomes

HF =1

2

∑p

VF (p)〈ρ(−p)〉ρ(p). (B.8)

This is a Hartree-like form but the potential term associated with the Coulomb inter-

action is no longer given by Eq. (B.2). It is defined instead as

VF (p) = −1

g

∑q

VH(q)eip×q`2 = −2π`2

∫d2q

(2π)2VH(q)eip×q`2 , (B.9)

where g is the degeneracy of the Landau level given by Eq. (3.4). In the last expression

we have transformed the summation into an integration. If we perform the azimuthal

integration and switch to dimensionless units (x = r/`) we end up to the final result

shown in Eq. (3.24).

107

In what follows we present analytic results for the Hartree and Fock interaction

potential energy for the first four Landau levels. For the lowest n = 0 Landau level

we find

VH(q) =1

4πε

2πe2

qe−q2`2/2, (B.10)

VF(q) = − 1

4πεπ3/2e2

√2`e−q2`2/4I0(q

2`2/4), (B.11)

for the n = 1 Landau level we have

VH(q) =1

4πε

2πe2

qe−q2`2/2

(1− 1

2q2`2

)2, (B.12)

VF(q) = − 1

4πε

π3/2e2√

2`

8e−q2`2/4

[(6− 2q2`2 + q4`4

)I0(q

2`2/4)

− q4`4I1(q2`2/4)

], (B.13)

for the n = 2 Landau level we find

VH(q) =1

4πε

2πe2

qe−q2`2/2

(1− q2`2 +

1

8q4`4

)2, (B.14)

VF(q) = − 1

4πε

π3/2e2√

2`

128e−q2`2/4

[(82− 52q2`2 + 44q4`4 − 10q6`6

+ q8`8)I0(q

2`2/4) +(− 30q4`4 + 8q6`6 − q8`8

)I1(q

2`2/4)

], (B.15)

and finally for the n = 3 Landau level we get

VH(q) =1

4πε

2πe2

qe−q2`2/2

(1− 3

2q2`2 +

3

8q4`4 − 1

48q6`6

)2, (B.16)

VF(q) = − 1

4πε

π3/2e2√

2`

4608e−q2`2/4

[(2646− 2430q2`2 + 2889q4`4 − 1236q6`6

+ 270q8`8 − 26q10`10 + q12`12)I0(q

2`2/4) +(− 1539q4`4 + 828q6`6

− 224q8`8 + 24q10`10 − q12`12)I1(q

2`2/4)

], (B.17)

108

where, I0(x) and I1(x) are the zero-order and first-order modified Bessel functions,

respectively. Notice that the above results are based on the fact that the dielectric

constant of the host material does not have any wavevector dependence (which is the

case when screening or finite thickness effects are included). If that were the case the

integral in Eq. (B.9) would become very complicated and analyticity would be lost.

B.2 Microscopic Potential

As we showed in the main text Eq. (4.40) defines the microscopic interaction

potential among electrons in the 2D system in the Hartree-Fock approximation. If we

use the definitions for the projected electron densities per bubble given by Eqs. (4.37-

4.39) it is easy to show that the total potential U(q) for the 2e BC is given by

U(q) =1

2

1∑m=0

1∑

m′=0

Umm′(q), (B.18)

and since U01(q) = U10(q) we find

U(q) =1

2

(U00(q) + U11(q) + 2U01(q)

), (B.19)

with the following definitions

U00(q) = e−q2`2/2VHF(q), (B.20)

U01(q) =(1− 1

2q2`2

)e−q2`2/2VHF(q), (B.21)

U11(q) =(1− 1

2q2`2

)2e−q2`2/2VHF(q). (B.22)

B.3 Bubble with Basis Dynamical Matrix

The dynamical matrix for the bubble with basis crystalline structure is evaluated

by starting from Eq. (4.41) and expanding to second order in the displacements. We

109

use the discrete Fourier transform definition of Eq. (4.8) along with

∑Q

eiQ·r = Ac

∑R

δ(r−R) =∑R

δr,R, (B.23)

and we find

Eel =1

2

1

A2c

∫d2q

(2π)2

∑Q

∑

mm′eiQxr0(m−m′)

[Umm′(Q + q)(Qα + qα)(Qβ + qβ)eiqxr0(m−m′)

− Umm′(Q)QαQβ

] ∑

Q′uα

m(q)uβm′(Q

′ − q)

' 1

2

∫d2q

(2π)2

1

A2c

∑Q

∑

mm′eiQxr0(m−m′)

[Umm′(Q + q)(Qα + qα)(Qβ + qβ)eiqxr0(m−m′)

− Umm′(Q)QαQβ

]uα

m(q)uβm′(−q), (B.24)

which has the general form of Eq. (4.12) where the dynamical matrix is given by

Eq. (4.43). The interaction potential given by Eq. (4.45) is easy to show that has the

following first and second derivatives

U ′(q) =∑

mm′eiqxr0(m−m′)U ′

mm′(q)

= V ′HF(q)e−q2`2/2

[1 + cos(qxr0)− 1

2

(1 + cos(qxr0)

)q2`2 +

1

8q4`4

]

− 2q`2VHF(q)e−q2`2/2

[1 + cos(qxr0)− 1

2

(1 +

1

2cos(qxr0)

)q2`2

+1

16q4`4

], (B.25)

U ′′(q) = V ′′HF(q)e−q2`2/2

[1 + cos(qxr0)− 1

2

(1 + cos(qxr0)

)q2`2 +

1

8q4`4

]

− 4q`2V ′HF(q)e−q2`2/2

[1 + cos(qxr0)− 1

2

(1 +

1

2cos(qxr0)

)q2`2 +

1

16q4`4

]

− 2`2VHF(q)e−q2`2/2

[1 + cos(qxr0)− 1

2

(5 +

7

2cos(qxr0)

)q2`2

+1

4q4`4

(13

4+ cos(qxr0)

)− 1

16q6`6

]. (B.26)

110

B.4 Normal Modes

Here, we discuss in more detail the normal mode calculation for the different

crystalline states considered in this work. As we showed in the main text the real

space representation of the dynamical matrix for a general BC is given by Eq. (4.47)

and the equation of motion for an electron is of the form given by Eq. (4.49). For the

WC there is only one electron associated with each bubble so the dynamical matrix

equation becomes

Φ00αβ(Ri −Rj) = δij

∑

k

φ00αβ(Rk)− φ00

αβ(Ri −Rj), (B.27)

and the equation of motion obtains the form

md2

dt2uα

i = −∑

j

Φ00αβ(Ri −Rj)u

βj − eBεαβ

d

dtuβ

i . (B.28)

If we introduce for the displacements the form uαi = Aαei(q·Ri−ωt) we find that the

system of Ax, Ay has a finite solution when

det

Φ00xx − ω2 Φ00

xy − iωcω

Φ00xy + iωcω Φ00

yy − ω2

= 0, (B.29)

where ωc = eB/m and we have defined

Φ00αβ(q) =

1

m

∑j

Φ00αβ(Rj)e

−iq·Rj . (B.30)

As we commented in the main text, the above is not quite the Fourier transform

of the dynamical matrix given by Eq. (4.13). What is missing in the latter, is the

proper prefactor of Ac/m to transform its units into [T−2]. We can introduce that

111

multiplicative constant into Eq. (4.13) and rewrite the dynamical matrix expression

in wavevector space as

Φ00αβ(q) =

1

mAc

∑Q

(U00(Q + q)(Qα + qα)(Qβ + qβ)− U00QαQβ

). (B.31)

We are in a position now to use the above definition for the dynamical matrix, and

evaluate it for wavevectors lying on the irreducible element of the first Brillouin zone.

Notice, that for this simple case the normal mode equation can be written as

ω4 − (Φ00

xx + Φ00yy + ω2

c

)ω2 + Φ00

xxΦ00yy − (Φ00

xy)2 = 0, (B.32)

which is very easy to solve, but nevertheless one still needs to evaluate numerically

the dynamical matrix. Our results are shown in the upper left panel of Fig. (4.9).

The 2e BC normal mode calculation is along the same lines and one finds the

following equation

det

Φ00xx − ω2 Φ01

xx Φ00xy − iωcω Φ01

xy

Φ01xx Φ11

xx − ω2 Φ01xy Φ11

xy − iωcω

Φ00xy + iωcω Φ01

xy Φ00yy − ω2 Φ01

yy

Φ01xy Φ11

xy + iωcω Φ01yy Φ11

yy − ω2

= 0, (B.33)

where we introduced again

Φmm′αβ (q) =

1

m

∑j

Φmm′αβ (Rj)e

−q·Rj

=1

mAc

∑Q

(Umm′(Q + q)(Qα + qα)(Qβ + qβ)− Umm′(Q)QαQβ

),

and used the fact that Φ10αβ(q) = Φ01

αβ(q). The above has the solution shown in the

upper right panel of Fig. (4.9).

112

The BC with basis proceeds along the same lines as well. The dynamical matrix

is given by Eq. (4.50) and the equations of motion for the two electrons in the bubble

are given by Eq. (4.51). As a result we find that the normal modes have to obey

det

Φ00xx − ω2 Φ01

xx Φ00xy − iωcω Φ01

xy

Φ10xx Φ11

xx − ω2 Φ10xy Φ11

xy − iωcω

Φ00xy + iωcω Φ01

xy Φ00yy − ω2 Φ01

yy

Φ10xy Φ11

xy + iωcω Φ10yy Φ11

yy − ω2

= 0, (B.34)

where we introduced again

Φmm′αβ (q) =

1

m

∑j

Φmm′αβ (Rj + r0(m−m′)x)e−q·Rj

=1

mAc

∑Q

eiQxr0(m−m′)(

Umm′(Q + q)(Qα + qα)(Qβ + qβ)eiqxr0(m−m′)

− Umm′(Q)QαQβ

), (B.35)

and we used the fact that [Φ01αβ(q)]∗ = Φ10

αβ(q). The solution for the normal modes in

this case is shown in the bottom panel of Fig. (4.9).

In these kinds of calculations one has to constrain the wavevectors to lie in the

first Brillouin zone. As it turns out it is easier to use only the irreducible element of

the first Brillouin zone, which for the hexagonal lattice configuration is consisted of

the points [57]

Γ :(0, 0

), J :

4π

3a

(1, 0

), X :

π

a

(1,

1√3

), (B.36)

where a is the crystal lattice parameter. The corresponding lengths that we will need

to use later on are given by

(ΓJ) =4π

3a, (JX) =

2π

3a, (XΓ) =

2π√3a

. (B.37)

113

The general convention on how to plot the Brillouin zone element is to start from

point Γ go to point J , and then measuring lengths from J go to point X, and finally

measuring lengths from X go back to point Γ. Notice that this convention is only for

the plotting of the first Brillouin zone, not for the evaluation of the physical quantities

that sum on Q’s. For the latter we have to use the proper Q’s that originate from

point Γ and their tips lie on the irreducible Brillouin zone contour. It is easy to show

that from the latter (the proper wavevectors) we can find the former (wavevectors

used for graphical purposes) if we apply the following transformations

qJXx → qJX

x − Jx, (B.38)

qXΓx → qXΓ

x −Xx, (B.39)

qXΓy → qXΓ

y −Xy, (B.40)

where qJXx is the x-component of the wavevectors associated with branch JX of the

irreducible element of the first Brillouin zone (and so on), while Jx is the x-component

of point J (and so on). Using the above, we must recalculate the lengths q =√

q2x + q2

y

and then finally transform the branches again according to

qJX → qJX + q(J), (B.41)

qXΓ → qXΓ + q(X). (B.42)

The above transformation creates a continuous mapping of all the proper wavevectors

originating from point Γ with their tips on the contour of the irreducible element of

the first Brillouin zone to the graphical ones which trace around the contour. This

is the mapping that we use to plot the normal mode solutions for all crystalline

configurations shown in Fig. (4.9).

APPENDIX CBUILDING THE STATIC HARTREE-FOCK EQUATION

Here, we show how to develop the static Hartree-Fock equation and diagonalize

it, to find the eigenstates and eigenvalues associated with the correlated 2D electron

system under the presence of a perpendicular magnetic field. Our starting point is

the Hartree-Fock Hamiltonian (similar to Eq. (3.24) developed earlier) that is used in

similar studies of the 2D electron system and is given by [36, 38, 39]

HHF =1

2

∫d2q

(2π)2VHF (q)|ρ(q)|2, (C.1)

where ρ(q) is the projection of the electronic density n(q) onto the n-th Landau level

(given by Eq. (4.32)) that we rewrite here for the sake of completeness (and remove

the implicit Landau level index)

ρ(q) =n(q)

e−q2`2/4Ln(q2`2/2). (C.2)

The Hartree-Fock potential VHF (q) is given by the sum of the Hartree and Fock terms

developed in our earlier treatment of the 2D electron gas and given by

VH(q) =1

4πε

2πe2

qe−q2`2/2

[Ln(q2`2/2)

]2, (C.3)

VF (q) = −∫ ∞

0

dxxVH(x/`)J0(xq`). (C.4)

Since we want to approximate the electronic density in terms of wavefunctions (eigen-

functions of the Hartree-Fock equation) we have to rewrite the above Hamiltonian in

terms of the electronic density (instead of its projection onto a given Landau level). In

114

115

order to do that, we use Eq. (C.2) and if we define the following interaction potential

VHF (q) =VHF (q)

e−q2`2/2[Ln(q2`2/2)

]2 , (C.5)

then the Hamiltonian assumes the form

HHF =1

2

∫d2q

(2π)2VHF (q)|n(q)|2, (C.6)

that we can rewrite in real space as

HHF =1

2

∫d2r

∫d2r′n(r)VHF (r− r′)n(r′). (C.7)

If we use the Ansatz of Eq. (5.2) the above Hamiltonian becomes

HHF =1

2

∑ij

M∑

α,α′=1

∫d2r

∫d2r′|ψα(r−Ri)|2VHF (r− r′)|ψα′(r

′ −Rj)|2, (C.8)

and by extremizing it with respect to ψα(r−Ri) we obtain the Hartree-Fock equation

M∑

α′=1

∑j

∫d2r′VHF (r− r′)|ψα′(r

′ −Rj)|2ψα(r) = Eαψα(r), (C.9)

where α = 1, · · · ,M distinguishes among electrons inside a bubble and different

eigenstates as well. Using the expansion of Eq. (5.3) for the quasiparticle states, we

can project the above equation onto the non-interacting electron wavefunction basis.

Additionally, using the orthonormal properties of the latter, we can obtain for the

expansion coefficients Cmα the following equation

M∑

α′=1

Ns−1∑m3=0

Ns−1∑m4=0

Ns−1∑m1=0

Cm3α′C∗m4α′Cm1α

∑j

∫d2r

∫d2r′VHF (r− r′)ϕm3(r

′ −Rj)

× ϕ∗m4(r′ −Rj)ϕm1(r)ϕ

∗m2

(r) = EαCm2α. (C.10)

116

This is a non-linear equation in the expansion coefficients. Nevertheless, we can

improve on it further by grouping appropriately the separate terms according to

Ns−1∑m1=0

Gm1m2Cm1α = EαCm2α, (C.11)

where we have defined the 2nd rank tensor

Gm1m2 =M∑

α′=1

Ns−1∑m3=0

Ns−1∑m4=0

gm1m2m3m4Cm3α′C∗m4α′ , (C.12)

and the 4th rank tensor gm1m2m3m4 involves the actual reciprocal lattice summation

of the modified Coulomb interaction, weighted by overlap integrals. The expression

we obtain for it is

gm1m2m3m4 =1

Ac

∑Q

VHF (Q)

∫d2q

(2π)2ϕm1(q)ϕ∗m2

(q−Q)

∫d2q

(2π)2ϕm3(q)ϕ∗m4

(q + Q)

=1

Ac

∑Q

VHF (Q)e−Q2`2/2Γm1m2(Q)Γm3m4(−Q). (C.13)

In the last equation we have defined the overlap integrals multiplied by the exponential

factor according to

ΓM1M2(Q) = eQ2`2/4

∫d2rϕM1(r)ϕ

∗M2

(r)e−iQ·r = eQ2`2/4

∫d2q

(2π)2ϕM1(q)ϕ∗m1

(q−Q)

=

√M2!M1!

((−iQx+Qy

)`√

2

)M1−M2

LM1−M2M2

(Q2`2/2) ,M1 ≥ M2

√M1!M2!

((−iQx−Qy

)`√

2

)M2−M1

LM2−M1M1

(Q2`2/2) ,M1 < M2

(C.14)

where LM1−M2M2

(x) are the associated Laguerre polynomials given by

LM1−M2M2

(x) =

M2∑

l=0

M1!

(M2 − l)!(M1 −M2 + l)!

(−x)l

l!. (C.15)

117

Using the above analytic expression (coming from Murthy et al. [64]) we can show

that the overlap integrals have the following properties

Γm1m2(Q) = Γm2m1(−Q)∗, (C.16)

Γm1m2(Qx, Qy) = Γm2m1(Qx,−Qy), (C.17)

Γm1m2(Q) = (−1)m1−m2Γ∗m2m1(Q), (C.18)

Γm1m2(Qx, Qy) = Γ∗m1m2(−Qx, Qy), (C.19)

Γm1m2(Q) = (−1)m1−m2Γm1m2(−Q), (C.20)

which in turn they define specific symmetries for the 4th rank tensor gm1m2m3m4 .

The only symmetry that is useful to take advantage of (since it alleviates on the

computational burden of evaluating redundant elements of the tensor) is the following

gm1m2m3m4 =

0, m1 + m2 + m3 + m4 = odd

gm3m4m1m2 , m1 + m2 + m3 + m4 = even

. (C.21)

The computational algorithm of diagonalizing Eq. (C.11) starts by placing all

the M electrons of a bubble on their non-interacting eigenstates by defining the

expansion coefficients as: Cmα = δm+1,α. Using these values we compute the 2nd

rank tensor Gm1m2 (given by Eq. (C.12)) and we numerically diagonalize it, to find a

better estimate for the expansion coefficients. Having these better estimates at hand

we repeat the process again, reevaluating Gm1m2 until it converges. Notice, that by the

way we have setup the problem, we need to evaluate the 4th rank tensor gm1m2m3m4

only once, saving an enormous amount of computational time. For the convergence

criterion we use [65]

δ =1

MNs

√√√√Ns−1∑m=0

Ns−1∑

m′=0

[ M∑α=1

(Cnew

mα Cnewm′α

∗ − ColdmαCold

m′α∗)]2

< 10−4, (C.22)

118

where, Cnew correspond to the newly calculated expansion coefficients, while Cold

correspond to the old ones coming from the previous run of the code. An accuracy of

10−4 in the convergence criterion translates usually to an accuracy of roughly 10−5 in

the eigenvalues. In the end when the algorithm has converged we place each of the

M electrons in ascending order on the quasiparticle eigenstates Eα and define their

associated cohesive energy from Eq. (5.5).

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BIOGRAPHICAL SKETCH

I was born in Athens, Greece in 1975. After high school I enrolled in the Physics

Department of the University of Crete in 1993, where I obtained a B.S. and a M.S.

in physics in 1999. My area of interest at that time was computational physics and

high energy and condensed matter theory.

In 1999 I joined the graduate school of the Physics Department of the University

of Florida. After a year of core course preparation, I successfully passed the qualifiying

exams for the Ph.D degree. At that time, the research area of theoretical physics that

appeared the most compelling and facinating for me was condensed matter theory. I

pursued research in that area which has rewarded me ever since.

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