c© 2012 Youngseok Kim

PSEUDOSPIN TRANSFER TORQUES AND THE EFFECT OFDISORDER IN SEMICONDUCTOR ELECTRON BILAYERS

BY

YOUNGSEOK KIM

THESIS

Submitted in partial fulfillment of the requirementsfor the degree of Master of Science in Electrical and Computer Engineering

in the Graduate College of theUniversity of Illinois at Urbana-Champaign, 2012

Urbana, Illinois

Adviser:

Assistant Professor Matthew J. Gilbert

ABSTRACT

We use self-consistent quantum transport theory to investigate the influ-

ence of electron-electron interactions on interlayer transport in semiconduc-

tor electron bilayers in the absence of an external magnetic field. We con-

clude that, even though spontaneous pseudospin order does not occur at

zero field, interaction-enhanced quasiparticle tunneling amplitudes and pseu-

dospin transfer torques do alter tunneling I-V characteristics, and can lead

to time-dependent response to a dc bias voltage.

In addition, we study the effects of disorder on the interlayer transport

properties of disordered semiconductor bilayers. We find that the addition

of material disorder to the system affects interlayer interactions leading to

significant deviations in the interlayer transfer characteristics. In particular,

we find that disorder decreases and broadens the tunneling peak, effectively

reducing the interacting system to a non-interacting system. Our results

suggest that the experimental observation of interaction-enhanced interlayer

transport in semiconductor bilayers requires materials with mean-free paths

larger than the spatial extent of the system.

ii

To my parents, for their love and support

iii

ACKNOWLEDGMENTS

I would like to express my deep gratitude to my thesis adviser Prof. Matthew

J. Gilbert who made the whole work possible. I appreciate his strong encour-

agement and insightful guidance in my graduate study. I thank him also for

his patience on my slow progress and giving me opportunities to interact

with great researchers outside the university.

I would also like to thank Prof. Allan H. MacDonald (University of Texas-

Austin) for many useful insights on the theoretical aspects of this particular

project. I also thank Brian Dellabetta for his initial setup on the simulation

program and many useful discussions.

I am extremely grateful to the Fulbright Foundation for financial support

on my study as well as for several conference trips. The Fulbright Foundation

provided me a unique opportunity to interact with invaluable individuals

from all over the world.

Finally, I would like to thank my parents and my sister, whose love and

support throughout my academic career has been invaluable.

This work was supported by the Army Research Office.

iv

TABLE OF CONTENTS

CHAPTER 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . 11.1 Two-Dimensional Electron Gas . . . . . . . . . . . . . . . . . 11.2 Self-consistency in the Interacting System . . . . . . . . . . . 21.3 Interacting Bilayer System with Zero Field . . . . . . . . . . . 41.4 Figure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

CHAPTER 2 NON-EQUILIBRIUM GREEN’S FUNCTION . . . . . 62.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Retarded Green’s Function . . . . . . . . . . . . . . . . . . . . 72.3 Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.4 Figure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

CHAPTER 3 PSEUDOSPIN TORQUE THEORY IN THE SEMI-CONDUCTOR BILAYER SYSTEM . . . . . . . . . . . . . . . . . 113.1 System Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . 113.2 Pseudospin Transfer Torque Theory . . . . . . . . . . . . . . . 143.3 Simulation Details . . . . . . . . . . . . . . . . . . . . . . . . 163.4 Results and Discussions . . . . . . . . . . . . . . . . . . . . . 173.5 Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

CHAPTER 4 THE EFFECT OF DISORDER IN THE SEMI-CONDUCTOR BILAYER SYSTEM . . . . . . . . . . . . . . . . . 274.1 Modeling of Disorder . . . . . . . . . . . . . . . . . . . . . . . 274.2 Simulation Details . . . . . . . . . . . . . . . . . . . . . . . . 284.3 Results and Discussions . . . . . . . . . . . . . . . . . . . . . 294.4 Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

CHAPTER 5 CONCLUSION . . . . . . . . . . . . . . . . . . . . . . 35

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

v

CHAPTER 1

INTRODUCTION

1.1 Two-Dimensional Electron Gas

The first seminal paper of Fowler et al. [1] reported the low-temperature

electron transport measurement on a silicon metal-oxide-semiconductor field-

effect transistor (MOSFET). In their experiment, the electron is confined at

the interface between a silicon substrate and a gate oxide. Further experiment

[2] gave a first evidence on the existence of the 2D states at the semiconductor

interface. Aided by the development of the modulation doping [3] with δ-

doping method [4], it is found that the high quality sample of GaAs/AlGaAs

supports two-dimensional electron gas (2DEG) whose mobility is as high as

106 cm2/V s [5]. The finding of the “flatland” in the GaAs/AlGaAs system

has opened abundant possibilities of finding interesting phenomena. One of

the most exciting findings came from a simple low-temperature transport

measurement under large magnetic field, which is the quantum Hall effect

(QHE). Figure 1.1 (all figures are placed at the end of chapters) shows a

longitudinal (ρxx) and Hall (ρxy) resistivity at a particular Landau level filling

factor ν (vertical marking in the graph) [6]. Surprisingly, the ρxx drops

exponentially and approaches zero as temperature goes to zero at a certain

magnetic field (marked as A). Meanwhile, the system exhibits exponentially

increasing resistance with decreasing temperature at a certain magnetic field

(marked as B). As a result, simply by changing the field, one can tune the

system to have ground states of either metallic or insulating nature. This

astonishing phenomenon led to two physics Nobel prizes, one for the integer

QHE (IQHE), and the other for the fractional QHE (FQHE).

In addition to the IQHE and FQHE, an interesting feature at ν = 1/2

has attracted a number of researchers’ attention. In general, it is possible

to transform empty states as holes using the particle-hole transformation

1

[7, 8] and keep track of the empty states of a Landau level. Interestingly, at

half filling factor ν = 1/2, the electron layer can be transformed as the hole

layer with the same density. The same argument is applied to the bilayer

electron system which supports two adjoint 2DEGs registered between a thin

barrier. The bilayer system possesses both interlayer and intralayer electron-

electron interaction and this additional layer degrees of freedom leads to

unique characteristics. When the system has equal densities of two 2DEGs

with ν = 1/2, the strong interlayer interaction induces spontaneous interlayer

phase coherence. As a result, the system has exotic insulating states called

exciton condensation [9, 10]. The exciton condensate transition occurs where

the interlayer as well as intralayer interaction are dominant and is signaled

by a huge enhancement in the tunneling rate [9, 11]. The bilayer system leads

to a number of interesting experimental results such as interlayer transport

anomalies [11, 12] and a signature of dissipationless transport [13, 14]. As

a consequence, the bilayer system has served as a test bed for many-body

physics for the past two decades.

1.2 Self-consistency in the Interacting System

As we discussed in the previous section, interesting phenomena are found

in the presence of strong interactions. Thus, how to model and evaluate

the interacting system properly is an important question. When interaction

effects are neglected a nanoscale conductor always reaches a steady state

[15] in which current increases smoothly with bias voltage. This problem is

efficiently solved using Green’s function techniques, for example, by using

the non-equilibrium Green’s function (NEGF) method [15]. Real electrons

interact, of course, and the free-fermion degrees of freedom which appear

in this type of theory should always be thought of as Fermi liquid theory

[16, 17, 18] quasiparticles. The effective single-particle Hamiltonian there-

fore depends on the microscopic configuration of the system. In practice the

quasiparticle Hamiltonian is often [19, 20] calculated from a self-consistent

mean-field theory like Kohn-Sham density functional theory (DFT). DFT,

spin-density functional theory, current-density functional theory, Hartree the-

ory, and Hartree-Fock theory all function as useful fermion self-consistent-

field theories. Since a bias voltage changes the system density matrix, it in-

2

evitably changes the quasiparticle Hamiltonian. Determination of the steady

state density matrix therefore requires a self-consistent calculation.

Self-consistency is included routinely in NEGF simulations [21, 22, 23] of

transport at the Hartree theory level in nanoscale semiconductor systems and

in simulations [24, 25, 26, 27, 28] of molecular transport. The collective be-

havior captured by self-consistency can sometimes change the current-voltage

relationship in a qualitative way, leading to an I-V curve that is not smooth

or even to circumstances in which there is no steady state response to a time-

independent bias voltage. Some of the most useful and interesting examples

of this type of effect occur in ferromagnetic metal spintronics. The quasipar-

ticle Hamiltonian is a ferromagnetic metal that has a large spin-splitting term

which lowers the energy of quasiparticles whose spins are aligned with the

magnetization (majority-spin quasiparticles) relative to those quasiparticles

whose spins are aligned opposite to the magnetization (minority-spin quasi-

particles). When current flows in a ferromagnetic metal the magnetization di-

rection is altered [29]. The resulting change in the quasiparticle Hamiltonian

[29] is responsible for the rich variety of so-called spin-transfer torque collec-

tive transport effects which occur in magnetic metals and semiconductors.

These spin-transfer torques [30, 31] are often understood macroscopically1 as

the reaction counterpart of the torques which act on the quasiparticle spins

that carry current through a non-collinear ferromagnet. Spin-transfer torques

[30] can lead to discontinuous I-V curves and to oscillatory [32] or chaotic

response to a time-independent bias voltage. Similar phenomena [33] occur

in semiconductor bilayers for certain ranges of external magnetic field over

which the ground state has spontaneous [34, 10, 9] interlayer phase coher-

ence (see Section 1.1). Indeed when bilayer exciton condensate ordered states

are viewed as pseudospin ferromagnets, the spectacular transport anomalies

[9, 11, 35, 36, 12] they exhibit have much in common with those of ferromag-

netic metals.

1The microscopic and macroscopic pictures are essentially equivalent in ferromagnetsbecause the spin-splitting term is much larger than other spin-dependent terms in thequasiparticle Hamiltonian.

3

1.3 Interacting Bilayer System with Zero Field

In this thesis we address interlayer transport in separately contacted nanome-

ter length scale semiconductor bilayers, with a view toward the identification

of possible interaction-induced collective transport effects. In particular, we

consider semiconductor bilayers at zero magnetic field, using a pseudospin

language [37, 38] in which top layer electrons are said to have pseudospin up

(| ↑〉) and bottom layer electrons are said to have pseudospin down (| ↓〉).Although interlayer transport in semiconductor bilayers has been studied

extensively in the strong field quantum Hall regime, work on the zero field

limit has been relatively sparse and has focused on studies of interlayer drag

[39, 40, 41, 42], counterflow [43], and on speculations about possible broken

symmetry states [44, 45]. We concur with the consensus view that pseu-

dospin ferromagnetism is not expected in conduction band two-dimensional

electron systems2 [46] except possibly [47] at extremely low carrier densi-

ties. There are nevertheless pseudospin-dependent interaction contributions

to the quasiparticle Hamiltonian. When a current flows, the pseudospin ori-

entation of transport electrons is altered, just as in metal spintronics, and

some of the same phenomena can occur. The resulting change in the quasi-

particle Hamiltonian is responsible for a current-induced pseudospin-torque

which alters the state of non-transport electrons well away from the Fermi

energy. Indeed although the spin-splitting field appears spontaneously in

ferromagnetic metals, spintronics phenomena usually depend on an inter-

play between the spontaneous exchange field and effective magnetic fields

due to magnetic-dipole interactions, spin-orbit coupling, and external mag-

netic fields. In semiconductor bilayers the pseudospin external field is due

to single-particle interlayer tunneling amplitude and is typically of the same

order [48] as the interaction contribution to the pseudospin-splitting field.

While theoretically possible and potentially useful as a logic device [49], to

be experimentally observed, these pseudospin exchange field signatures must

be robust against material disorder, which these semiconductor bilayer sys-

tems are well-known to contain. Therefore, understanding the observable

deviations in interlayer transport caused by material disorder is also studied.

2Pseudospin ferromagnetism (or equivalently exciton condensation) is, however, pre-dicted in systems with equal densities of conduction band electrons and valence band holesin separate quantum wells.

4

1.4 Figure

Figure 1.1: A plot of longitudinal (ρxx) and Hall (ρxy) resistivity as afunction of magnetic field. The low-disorder 2DEG in a high magnetic fieldreveals three distinct electronic features identified through temperaturedependence (left inset). The right inset shows measurement geometry andthe vertical bar inside the plot indicates Landau level filling factor ν [6].

5

CHAPTER 2

NON-EQUILIBRIUM GREEN’S FUNCTION

2.1 Introduction

In order to study transport characteristic under non-equilibrium conditions,

we adopt the non-equilibrium Green’s function (NEGF) formalism [50, 51].

As a typical system of consideration is illustrated in Figure 2.1, the electron

dynamics of the system (or channel) are described by the system Hamilto-

nian, H, connected to the two contacts. The coupling between contacts and

channel is described by self-energy matrices, Σ1 and Σ2, which are defined

in Section 2.2. The chemical potential of each leads, µ1 and µ2, are adjusted

to make non-equilibrium current flow and a chemical potential inside the

channel is determined by the Poisson equation,

O2φ(x) =e2

ε[ND(r)− n(r)] , (2.1)

where φ(r) is the potential profile, ND(r) is the charge density from doners

in the material and n(r) is the electron density which is evaluated by NEGF

formalism (see Eq. (2.10)).

In this work, we define a GaAs quantum well Hamiltonian as

H0 =∑<i,j>

−τ | i〉〈j | +(4τ + Vi) | i〉〈i |, (2.2)

where lattice points i and j are nearest neighbors. τ = ~2/2m∗a2 is the near-

est neighbor hopping energy, m∗ is the electron effective mass of GaAs, and

a is the lattice constant of our simulation. Vi = φ(ri) is the on-site potential

for GaAs from the normal tight-binding description and is calculated via a

Poisson solver. With the proper Hamiltonian of Eq. (2.2), the following

sections describe how to calculate observables such as electron density and

6

current via NEGF formalism.

2.2 Retarded Green’s Function

Within the NEGF formalism, we define the retarded Green’s function as

below [50, 51]

G(E) =[(E + i0+) I−H0 −Σ1(E)−Σ2(E)

]−1, (2.3)

where 0+ is an infinitesimal positive number, I is the identity matrix and

H0 is the system Hamiltonian in Eq. (2.2). In order to complete the above

definition, we need to define Σ1(2)(E), which is a self-energy function of the

semi-infinite contact 1 (2). Although we assume contacts are semi-infinite,

we can devide the contact system by blocks with a contact Hamiltonian, Hc,

and a hopping matrix between blocks, τc. Then, the surface Green’s function,

gs, is defined as

gs =[(E + i0+) I−Hc − τcHcτ

†c

]−1. (2.4)

As a result, the self-energy of the contact is defined as,

Σ(E) = τcpgsτ†cp, (2.5)

where τcp is a coupling matrix between channel and contact. The broadening

matrix due to the contact is obtained from the imaginary part of the self-

energy,

Γ = i(Σ−Σ†) (2.6)

whose meaning is the rate at which carriers are injected into the channel.

In case we have two contacts as in Figure 2.1, we define the self-energy, Σ1

and Σ2, and the broadening matrix, Γ1 and Γ2, for the contact 1 and 2,

respectively.

7

2.3 Observables

With the definitions in Section 2.1 and 2.2, the transmission is defined as

T12(E) = Trace[Γ1GΓ2G

†] = Trace[Γ2GΓ1G

†] . (2.7)

When we consider the coherent transport inside the channel, the terminal

current is calculated by the Landauer formula:

I =e

h

∫dE T12(E) (f1(E)− f2(E)) , (2.8)

where f1, f2 are Fermi-Dirac distribution of each contacts. As long as we

only consider the interaction between the system and each lead, the inscat-

tering and outscattering function is defined solely by contact self-energy as

Σin(E) =∑

p fp(E)Σp and Σout(E) =∑

p(1− fp(E))Σp, where p runs over

contacts which are connected to the system [50]. As a result, we can define

electron (hole) correlation function, Gn (Gp), as

Gn = GΣinG†, Gp = GΣoutG†. (2.9)

Using the electron and hole correlation function in Eq. (2.9), the electron

and hole density are defined as below,

n =

∫dE

2πGn(E), p =

∫dE

2πGp(E). (2.10)

For example, the matrix expression of the one-dimensional system described

by Eq. (2.2) is

H0 =

4τ + V1 −τ 0 · · · 0

−τ 4τ + V2 −τ · · · 0...

......

......

0 0 · · · −τ 4τ + VN

, (2.11)

8

and the corresponding density matrix from Eq. (2.10) is

n =

ρ11 ρ12 ρ13 · · · ρ1N

ρ21 ρ22 ρ23 · · · ρ2N

......

......

...

ρN1 ρN2 ρN3 · · · ρNN

. (2.12)

As the n (or p) in Eq. (2.12) is a matrix, it is possible to obtain on-site density

from diagonal elements as well as off-diagonal terms which will be used to

evaluate interlayer interaction in the case of the semiconductor bilayer system

(for detail, see Eq. (3.4) and the following discussion in Chapter 3.1.1). Then,

the electron (or hole) density at a certain bias condition is fed back to the

Poisson solver to obtain Vi in Eq. (2.2) and observables are calculated after

this self-consistent loop is converged within a defined tolerance.

9

2.4 Figure

Figure 2.1: A schematic of the system [51]. The system consists of achannel with Hamiltonian H and two contacts placed with correspondingself-energy Σ1,2.

10

CHAPTER 3

PSEUDOSPIN TORQUE THEORY IN THESEMICONDUCTOR BILAYER SYSTEM

We consider an Al0.9Ga0.1As/GaAs bilayer heterostructure with top and bot-

tom Al0.9Ga0.1As barriers which act to isolate the coupled quantum wells from

electrostatic gates, as illustrated in Figure 3.1a. The bilayer consists of two

15 nm deep GaAs quantum wells with an assumed 2DEG electron density

of 2.0× 1010 cm−2 separated by a 1 nm Al0.9Ga0.1As barrier. The quantum

wells are 1.2 µm long and 7.5 µm wide with the splitting between the sym-

metric and antisymmetric states set to a small value ∆SAS = 2t = 2 µeV.

We define the z-axis as the growth direction, the x-axis as the longitudinal

(transport) direction, and the y-axis as the direction across the transport

channel as shown schematically in Figure 3.1a. We connect ideal contacts

attached to the inputs and outputs of both layers. The contacts inject and

extract current and enter into the Hamiltonian via appropriate self-energy

terms [15].

3.1 System Hamiltonian

In the following sections, we begin this chapter by formulating the system

Hamiltonian considering the interlayer exchange interaction with pseudospin

language.

3.1.1 Interacting System Hamiltonian

We construct the system Hamiltonian from a model with a single-band effec-

tive mass Hamiltonian for top and bottom layers with a phenomenological

11

single-particle inter-layer tunneling term:

H =

[HTL 0

0 HBL

]+∑

µ=x,y,z

µ ·∆⊗ σµ. (3.1)

The first term on the right-hand side of Eq. (3.1) is the single-particle non-

interacting term while the second term is a mean-field interaction term. In

the first term in the right-hand side, HTL and HBL are an individual quan-

tum well Hamiltonian, H0, which is described in Eq. (2.2). In the second

term in the right-hand side, σµ represents the Pauli spin matrices in each

of the three spatial directions µ = x, y, z, ⊗ represents the Kronecker prod-

uct, and ∆ is a pseudospin effective magnetic field which will be discussed

in more detail later in this section. To explore interaction physics in bilayer

transport qualitatively, we use a local density approximation in which the in-

teraction contribution to the quasiparticle Hamiltonian is proportional to the

pseudospin-magnetization at each point in space. If we take the top layer as

the pseudospin up state (| ↑〉) and the bottom layer as the pseudospin down

state (| ↓〉), the single-particle interlayer tunneling term contributes a pseu-

dospin effective field with magnitude t = ∆SAS/2 and direction x. In real spin

ferromagnetic systems, interactions between spin-polarized electrons lead to

an effective magnetic field in the direction of spin-polarization. Bilayers with

pseudospin-polarization due to tunneling have a similar interaction contri-

bution to the quasiparticle Hamiltonian. Including both single-particle and

many-body interaction contributions, the pseudospin effective field ∆ term

in the quasiparticle Hamiltonian is [52, 37],

∆ = (t+ Umxps) x+ Umy

ps y (3.2)

where the pseudospin-magnetization mps is defined by

mps =1

2Tr[ρpsτ ]. (3.3)

In Eq. (3.3), τ = σx, σy, σz is the vector of Pauli spin matrices, and ρps is

the 2× 2 Hermitian pseudospin density matrix which we define as,

ρps =

[ρ↑↑ ρ↑↓

ρ↓↑ ρ↓↓

]. (3.4)

12

The diagonal terms of the pseudospin density matrix (ρ↑↑, ρ↓↓) are the elec-

tron densities of the top and bottom layers. In Eq. (3.2), we have dropped the

exchange potential associated with the z component of pseudospin because

it is dominated by the electric potential difference between layers induced

by the interlayer bias voltage (see below). From the definition of Eq. (3.3),

the pseudospin-magnetization of x, y, z directions are defined in terms of the

density matrix as

〈mxps〉 =

1

2(ρ↑↓ + ρ↓↑),

〈myps〉 =

1

2(−iρ↑↓ + iρ↓↑),

〈mzps〉 =

1

2(ρ↑↑ − ρ↓↓).

(3.5)

Using Eqs. (3.2) and (3.5), we may express the system Hamiltonian in terms

of pseudospin field contributions,

H =

[HTL + ∆z ∆x − i∆y

∆x + i∆y HBL −∆z

]. (3.6)

Here ∆z is the electric potential difference between the two layers which we

evaluate in a Hartree approximation, disregarding its exchange contribution.

The planar pseudospin angle which figures prominently in the discussion

below is defined by

φps = tan−1

(〈myps〉

〈mxps〉

). (3.7)

This angle corresponds physically to the phase difference between electrons

in the two layers.

3.1.2 Enhanced Interlayer Tunneling

With the quasiparticle Hamiltonian for our semiconductor bilayer defined, we

now address the strength of the interlayer interactions present in the system.

The interaction parameter U in Eq. (3.2) is chosen so that the local density

approximation for interlayer exchange reproduces a prescribed value for the

interaction enhancement of the interlayer tunneling amplitude. In equilib-

rium the pseudospin magnetization will be oriented in the x direction and

13

the quasiparticle will have either symmetric or antisymmetric bilayer states

with pseudospins in the x and −x-directions, respectively. The majority and

minority pseudospin states differ in energy at a given momentum by 2teff

where

teff = t+ UNs −Na

2. (3.8)

The population difference between symmetric and antisymmetric differences

may be evaluated from the differences in their Fermi radii illustrated in Figure

3.2:Ns −Na

2= ν0 teff , (3.9)

where ν0 is the density-of-states of a single layer. Combining Eq. (3.8) and

Eq. (3.9) we can relate U to S, the interaction enhancement factor for the

interlayer tunneling amplitude:

teff =t

1− Uν0

≡ S t. (3.10)

The physics of S is similar to that responsible for the interaction enhancement

of the Pauli susceptibility in metals. According to microscopic theory [48]

a typical value for S is around 2. We choose to use S rather than U as a

parameter in our calculations and therefore set

U =1− S−1

ν0

' 1

2ν0

. (3.11)

3.2 Pseudospin Transfer Torque Theory

In the following section we report on simulations in which we drive an in-

terlayer current by keeping the top left and top right contacts grounded and

applying identical interlayer voltages, VINT , at the bottom left and bottom

right contacts. We choose this bias configuration so as to focus on interlayer

currents that are relatively uniform. The transport properties depend only

on the quasiparticle Hamiltonian and on the chemical potentials in the leads.

Because the pseudospin effective field is the only term in the quasiparticle

Hamiltonian which does not conserve the z component of the pseudospin, it

14

follows that every quasiparticle wavefunction in the system must satisfy [53]

∂tmzps = −∇ · jz − 2

~(mps ×∆)z = 0 (3.12)

where jz is the z component of the pseudospin current contribution from that

orbital, i.e. the difference between bottom and top layer number currents,

and mps is the pseudospin magnetization of that orbital. For steady state

transport, the quasiparticles satisfy time-independent Schrodinger equations

so that, summing over all quasiparticle orbitals, we find,

2|mps||∆| sin(φps − φ∆) = 2tmyps = ~∇ · jz. (3.13)

In Eq. (3.13), φ∆ is the planar orientation of ∆. The first equality in Eq.

(3.13) follows from Eq. (3.2). The pseudospin orbitals do not align with the

effective field they experience because they must precess between layers as

they transverse the sample. The realignment of transport orbital pseudospin

orientations alters the total pseudospin and therefore the interaction contri-

bution to ∆. The change in mps×∆ due to transport currents is referred to

here as the pseudospin transfer torque, in analogy with the terminology com-

monly found in metal spintronics. Integrating Eq. (3.13) across the sample

from left to right and accounting for spin degeneracy, we find that

4etA〈myps〉

~= IL + IR = I (3.14)

where A is the 2D layer area, the angle brackets denote a spatial average, and

IL and IR are the currents flowing from top to bottom at the left and right

contacts. (The pseudospin current jz flows to the right on the right and is

positive on the right side of the sample, but flows to the left and is negative

on the left side of the sample.) If the bias voltage can drive an interlayer

current I larger than 4etA〈myps〉/~, it will no longer be possible to achieve

a transport steady state. Under these circumstances the interlayer current

will oscillate in sign and the time-averaged current will be strongly reduced.

In the next section we use a numerical simulation to assess the possibility of

achieving currents of this size.

A similar conclusion can be reached following a different line of argument.

The microscopic operator J describing net current flowing top layer (↑) to

15

bottom layer (↓) given by [42, 54] is

J =−iet~∑k,σ

(c†k,σ,↑ck,σ,↓ − c

†k,σ,↓ck,σ,↑

)=−2et

~i∑k

(c†k,↑ck,↓ − c

†k,↓ck,↑

)=

4et

~myps,

(3.15)

where k, σ are the momentum and spin indices, respectively. Here we have

introduced a common notation ∆SAS = 2t for the pseudospin splitting be-

tween symmetric and antisymmetric states in the absence of interactions and

added a factor of 2 to account for spin-degeneracy. The pseudospin density

operator myps in Eq. (3.15) is given by

myps =

−i2

∑k

(c†k,↑ck,↓ − c

†k,↓ck,↑

). (3.16)

As a result, current density flowing in the given system may be written as⟨J⟩

=4et

~⟨myps

⟩=

4et

~|mxy| sinφ

≤ 4et

~|mxy| = Jc,

(3.17)

where |mxy| =√〈mx〉2 + 〈my〉2, φps is defined in Eq. (3.7) and Jc is defined

as a critical current density. Assuming the current flow is uniform across the

device, it is possible to obtain the critical current simply by multiplying Eq.

(3.17) by the system area A to obtain

Ic = Jc × A =4etA

~|mxy|. (3.18)

3.3 Simulation Details

As we neglect disorder in this simulation, we may write the Hamiltonian in

the form of decoupled 1D longitudinal channels in the transport (x) direction,

taking proper account of the eigenenergy of transverse (y) direction motion

16

[55]. The calculation strategy follows a standard self-consistent field proce-

dure as is described in a flow chart of Figure 3.1b. Given the density matrix

of the 1D system, we can evaluate the mean-field quasiparticle Hamiltonian.

For the interlayer tunneling part of the Hamiltonian, we use the local-density

approximation outlined in Section 3.1.2. The electrostatic potential in each

layer is calculated from the charge density in each of the layers by solving a

2D Poisson equation using an alternating direction implicit method [56] with

appropriate boundary conditions. The boundary conditions employed in this

situation were hard wall boundaries on the top and sides of the simulation

domain and Neumann boundaries at the points where current is injected to

insure charge neutrality [57]. Given the mean-field quasiparticle Hamiltonian

and voltages in the leads, we can solve for the steady state density matrix

of the two-dimensional bilayer using the NEGF method with a real-space

basis. The density-matrix obtained from the quantum transport calculation

is updated at each state in the iteration process. The update density is then

fed back into the Poisson solver and on-site potential is updated using the

Broyden method [58] to accelerate self-consistency. The effective interlayer

tunneling amplitudes are also updated and the loop proceeds until a desired

level of self-consistency is achieved. The transport properties are calculated

after self-consistency is achieved by applying the Landauer formula, i.e. by

using

I(Vsd) =2e

h

∫T (E)[fs(E)− fd(E)]. (3.19)

3.4 Results and Discussions

3.4.1 Linear Response

In Figure 3.3, we plot the interlayer conductance at temperature T = 0 as a

function of the interlayer exchange enhancement, S by setting VBL = VBR =

VINT and VTL = VTR = 0 in Figure 3.1a. The S2 dependence demonstrates

that the conductance in our nanodevice is proportional to the square of the

quasiparticle tunneling amplitude, as in bulk samples [39, 40, 41, 42, 59, 60],

and that the quasiparticle tunneling amplitude is approximately uniformly

enhanced even though the finite-size system is not perfectly uniform. The

range of S used in this figure corresponds to the relatively modest enhance-

17

ment factors that we expect in bilayer systems in which the individual layers

have the same sign of mass. For systems in which the quasiparticle masses

are opposite in the two layers, we expect values of S that are significantly

larger. Spontaneous interlayer coherence, which occurs in a magnetic field

[11, 61, 62, 63, 64] but is not expected in the absence of a field, would be

signaled by a divergence in S.

The physics of the results illustrated in Figure 3.3 can be understood qual-

itatively by ignoring the finite-size-related spatial inhomogeneities present in

our simulations and considering the simpler case in which there is a single

bottom-layer source contact and a single top-layer drain contact on opposite

ends of the transport channels. The interlayer tunneling is then diagonal

in a transverse channel, and the transmission probability from top layer to

bottom layer in channel k is

Tinterlayer = sin2

(δkL

2

). (3.20)

In Eq. (3.20), δk is the difference between the current direction wavevectors

of the symmetric and antisymmetric states and L is the system length in

the transport direction. We may simplify the expression in Eq. (3.20) by

rewriting it in terms of the Fermi velocity, vf , in the transport-direction and

making use of the small angle expansion of the sin function to obtain

Tinterlayer =

(teffL

~vf

)2

, (3.21)

where teff is the quasiparticle tunneling amplitude proportional to the inter-

layer exchange enhancement S, which results in the power law dependence

seen in Figure 3.3. As this approximate expression suggests, we find that

transport at low interlayer bias voltages is dominated by the highest energy

transverse channel, which has the smallest transport direction Fermi velocity.

3.4.2 Transport Beyond Linear Response

We have seen that at small bias voltages the inter-layer current is enhanced

by interlayer exchange interactions. In Figure 3.4 , we compare our interlayer

transport result with the S = 1 case. We see that all of the curves are sharply

18

peaked near zero bias, as in bulk 2D to 2D tunneling [39, 40, 41, 42, 59, 60].

In all cases the decrease and change in sign of the differential conductance at

higher bias voltages is due to the build-up of a Hartree potential difference

(∆z) between the layers, which moves the bilayer away from its resonance

condition. The peak near zero bias is sharper for the enhanced interlayer

tunneling (filled triangle and circle in Figure 3.4) cases. To get a clearer

picture of S > 1 transport properties in the non-linear regime, we examine

the influence of bias voltage on steady-state pseudospin configurations. Ini-

tially, all orbitals are aligned with the external pseudospin field (the single-

particle interlayer tunneling term, t), and it follows from Eq. (3.14) that

φps = 0. The enhanced magnitude of this equilibrium pseudospin polariza-

tion is m0 = ν0St = ν0teff . When current flows between the top and bottom

layer, there must be a y-component pseudospin, as we have explained previ-

ously, whose spatially averaged value is proportional to the current. If it is

possible to drive a current that is larger than allowed by Eq. (3.18), it will

no longer be possible to sustain a time-independent steady state.

In Figures 3.5 and 3.6 we plot the magnitudes of the x, y, and z directed

pseudospin fields, along with |mxy| =√m2x +m2

y, evaluated at the center

of the device, as a function of current and bias voltage, respectively. The

pseudospin densities are plotted in units of their equilibrium values. The

fifteen data points for each of these curves correspond to the 15 data points

in the current vs. bias potential plot in Figure 3.4, so that the maximum

current corresponds to a bias voltage of ∼20 µV and the last data point

corresponds to a bias voltage of ∼100 µV. The first thing to notice in this

plot is that the z pseudospin component increases monotonically with bias

voltage as a potential difference between layers builds up. This is the effect

which eventually causes the current to begin to decrease. The x component

of pseudospin increases very slowly with bias voltage in the regime where the

current is increasing, but drops rapidly when current is decreasing. At the

same time the y component of pseudospin evaluated at the device center rises

steadily with current until the maximum current is reached and then remains

approximately constant. When the two effects are combined |mxy| first in-

creases slowly with bias voltage and then decreases slightly more rapidly.

The relatively weak dependence of |mxy| on current can be understood as a

competition between two effects. Because of the Coulomb potential which

builds up and lifts the degeneracy between states localized in opposite layers,

19

the direction of the pseudospin tilts toward the z direction, decreasing the

x− y plane component of each state. At the same time the total magnitude

of the pseudospin field increases, increasing the pseudospin polarization. In

a uniform system the two effects cancel when a z-direction pseudospin field

is added to a uniform system.

Our simulations suggest the following scenario for how the critical current

might be reached in bilayers. The width of the linear response regime is lim-

ited by the lifetime of Bloch states, which is set by disorder in bulk systems,

and in our finite-size system by the time for escape into the contacts. In the

linear-response regime the current is enhanced by a factor of S2 by inter-layer

exchange interactions. The maximum current which can be supported in the

steady state is however proportional to the bare inter-layer tunneling and to

the x− y pseudospin polarization and is therefore enhanced only by a factor

of S. From this comparison we can conclude that more strongly enhanced

inter-layer tunneling quasiparticle tunneling amplitudes (larger S) increases

the chances of reaching a critical current beyond which the system current

response is dynamic. For the parameters of our simulation, the maximum

current of around 4 nA is reached at a bias voltage of around 10 µV. At this

bias voltage the average x− y plane angle of the pseudospin field is still less

than 90◦, and steady state response occurs. As the bias voltage increases

further, the total current decreases, but the pseudospins become strongly

polarized in the z direction. We illustrate this behavior in Figure 3.7 by

plotting

σ = ∆z/√

∆2z + ∆2

c (3.22)

vs. bias voltage, where ∆z is z directional pseudospin field and ∆c is a con-

stant beyond which charge imbalance of the system becomes significant [65].

The magnitude of the x − y direction pseudospin polarization consequently

decreases. In our simulations the critical current decreases more rapidly

than the current in this regime. As shown in Figure 3.5, φps approaches π/2

(mx → 0) at the largest bias voltages (∼100 µV) for which we are able to ob-

tain a steady state solution of the non-equilibrium self-consistent equations.

20

3.5 Figures

Figure 3.1: (a) Device cross section in x− z direction. For the clean system(Chapter 3), interlayer bias configuration of VTR = VTL = 0,VBR = VBL = VINT is used, whereas VTL = VTR = VBR = 0, VBR = VINTconfiguration is used for the disordered system (Chapter 4). (b) A flowchart of the simulation procedure.

Figure 3.2: Schematic of the Fermi surfaces of the bilayer system containingpopulations of symmetric and antisymmetric states separated in energy bya single-particle tunneling term, ∆SAS = 2t.

21

1 2 3 4 50

1

2

3

4

5

Co

nduct

ance

(2e2 /h)

S ( u n i t l e s s )

1 2 3 4 5 6048

1 21 62 0

S ( U n i t l e s s )

Cond

uctan

ce (N

orm.)

Figure 3.3: Plot of the height of the interlayer conductance as a function ofthe interlayer exchange enhancement S at interlayer bias VINT = 10 nV. Inthe inset, we plot of the height of the interlayer conductance normalized tothe non-interacting conductance (S = 1) as a function of S with the samex-axis. We clearly see that the height of the interlayer conductance followsan S2 dependence. The single-particle tunneling amplitude t = ∆SAS/2 =1 µeV.

22

- 1 0 0 - 7 5 - 5 0 - 2 5 0 2 5 5 0 7 5 1 0 0- 1 0 00

1 0 02 0 03 0 04 0 05 0 06 0 0

S = 1 S = 2 S = 4

dI/dV

(mS)

V o l t a g e ( m V )

- 1 0 0 - 5 0 0 5 0 1 0 0- 6- 4- 20246

I (nA

)

V o l t a g e ( m V )

Figure 3.4: Plot of the differential conductance of the bilayer system atS = 4 (orange triangle), S = 2 (blue circle), and S = 1 (openedrectangular) as a function of interlayer bias VINT at 0 K. In the inset, thesame plot in negative interlayer bias with same y-axis is plotted with x-axisin log scale. In a small bias window, almost constant interlayertransconductance is observed. After the interlayer bias reachesVINT ≈ 10 µV, an abrupt drop in transconductance can be seen in theinteracting electron cases of S = 2, 4. We plot the current vs. bias voltagefor each enhancement factor in the inset.

23

0 . 0 0 . 4 0 . 8 1 . 2 1 . 6 2 . 00 . 0

0 . 2

0 . 4

0 . 6

0 . 8

1 . 0

m/m 0 (U

nitles

s)

I n t e r l a y e r c u r r e n t ( n A )

< m x > / m 0 < m y > / m 0 < m z > / m 0 | m x y | / m 0

Figure 3.5: Pseudospin polarization vs. current. The quantities areexpressed in units of the value of mx at zero bias (m0). The points in thisplot correspond to the same points that appear in the inset of current vs.bias voltage in Figure 3.4. For zero-current, the pseudospin is in thex-direction and has a value close to m0 = ν0St. mx decreasesmonotonically, mz increases monotonically, and my increasesnearly-monotonically before saturating at the largest field values. Thearrow in the plot indicates the direction in which a bias voltage increases.

24

0 2 0 4 0 6 0 8 0 1 0 00 . 0

0 . 2

0 . 4

0 . 6

0 . 8

1 . 0

m/m 0 (U

nitles

s)

V o l t a g e ( m V )

< m x > / m 0 < m y > / m 0 < m z > / m 0 | m x y | / m 0

0 2 5 5 0 7 5 1 0 0048

1 21 62 02 4

m/m 0(U

nitles

s)

V o l t a g e ( m V )

Figure 3.6: Pseudospin polarization vs. bias voltage. The quantities areexpressed in units of the value of mx at zero bias (m0). The inset of thisplot clearly shows that mz increases monotonically with bias voltage whicheventually causes the current to decrease. The |mxy| decreases only slowlywith bias voltage because of the compensating effects of pseudospinrotation toward the z-direction and an increase in the overall pseudospinpolarization. There is no steady state solution to the self-consistenttransport equations beyond the largest bias voltage plotted here, becausethe pseudospin orientation φps has reached π/2.

25

0 2 0 4 0 6 0 8 0 1 0 00 . 0

0 . 2

0 . 4

0 . 6

0 . 8

1 . 0

s (Un

itless)

V o l t a g e ( m V )

1 E - 3 0 . 0 1 0 . 1 1 1 0 1 0 00 . 00 . 20 . 40 . 60 . 81 . 0

V o l t a g e ( m V )

s (Un

itless)

Figure 3.7: Quasiparticle layer-polarization parameter σ as a function ofinterlayer bias. When the source-drain bias exceeds around ∼10 µV, themismatch match between subband energy levels exceeds the width of theconductance peak, σ increases rapidly, and the x-y plane pseudospinpolarization and critical current decrease. The inset shows the same data inlog scale to emphasize the low bias behavior.

26

CHAPTER 4

THE EFFECT OF DISORDER IN THESEMICONDUCTOR BILAYER SYSTEM

In this chapter, the effects of disorder on the interlayer transport properties

are evaluated by performing self-consistent quantum transport calculations.

4.1 Modeling of Disorder

We are interested in electron transport of GaAs/AlGaAs heterojunction sys-

tem with impurities. In this thesis, we are particularly interested in the

effects of ionized impurity and roughness at the interface of heterojunction

in GaAs/AlGaAs system [66, 67, 68, 69]. The impurities are assumed to

be highly screened in 2DEG and, as a result, the impurity potential can be

described as a δ-function

U(r) = u∑l

δ(r − rl), (4.1)

where rl is position of lth impurity with amplitude u [66, 67]. If the impurity

is a major scattering source, a mean free path Λ is expressed within the Born

approximation as [67]

Λ =~3vF

ni〈U2〉m∗(4.2)

where vF is Fermi velocity, ni is the impurity number per unit area and

m∗ is an effective mass. Within a lattice model with a lattice constant a,

random disorders can be inserted into the system Hamiltonian by adding

a uniform distribution having a window of W to the diagonal term in the

Hamiltonian. Then, the on-site energy in the disordered system satisfies the

inequality −W/2 ≤ εi − ε0 ≤ W/2, where εi(0) is an on-site energy of the

system with (without) disorder [69]. As a result, the lattice model containing

27

high concentration of δ-function impurities satisfies the relation

ni〈U2〉 = a2〈(εi − ε0)2〉 =a2W

12. (4.3)

From Eqs. (4.2) and (4.3), we can get a width of uniform distribution W as

a function of a mean free path Λ [68, 69]

W

EF=

(6λ3

F

π3a2Λ

)1/2

, (4.4)

where EF , λF are Fermi energy and Fermi wavelength, respectively. Con-

sequently, the impurity level is now characterized by a mean free path, Λ.

At a given MFP, the random disorder potential changes the on-site potential

landscape of the device according to Eq. (4.4), and variations in the poten-

tial landscape act as rigid scatterers. The effect of those scatterers on the

electron propagation is evaluated in the limit of coherent transport, as there

is no phase-breaking mechanism included (e.g. phonon scattering) [51, 50].

Therefore, the current may be calculated by the Landauer formula as has

been done in previous studies on doping and surface roughness in Si [70] and

bulk and edge disorder in graphene nanoribbons [71].

4.2 Simulation Details

Now we consider the system which is comprised of two 16 nm deep GaAs

quantum wells separated by a 2 nm Al0.9Ga0.1As barrier. The gates on the

top and bottom are isolated from their respective quantum well with 60 nm

thick top and bottom Al0.9Ga0.1As barriers. The structure of the system

is identical to the system we considered in Chapter 3 which is illustrated

in Figure 3.1a, but now we have two-dimensional channels instead of one-

dimensional decoupled channels in Chapter 3. We use the tunneling bias

configuration (VTL = VTR = VBL = 0, VBR = VINT ) in which electrons are

injected into the bottom right contact and are extracted from the top left

contact. The length and width of the system are each 1.2 µm and we assume

that each 2DEG contains an electron concentration of 3 × 1010 cm−2. The

system temperature for all of our simulations is set to the zero temperature

limit or T = 0 K.

28

The calculation strategy follows a standard self-consistent field procedure

as shown in Figure 3.1b. We can evaluate the mean-field quasiparticle Hamil-

tonian with the NEGF formalism coupled with a 3D Poisson equation. The

effective interlayer interaction, ∆ in Eq. (3.2), is also updated during the

loop and the loop proceeds until a desired level of self-consistency is achieved.

When the MPF is larger than the system length, there is no significant modi-

fication to the interlayer transport. Thus, we only focus on the MFP compa-

rable to or smaller than the device size. When the MFP is comparable to the

device size, however, the many-body contribution to ∆ shows local fluctua-

tions around the region where the on-site potential is significantly modified.

As these numerical fluctuations are not critical for global observables and

depend on the particular disorder configurations, the data obtained from the

simulation are averaged over six different disorder configurations in order to

mitigate the numerical fluctuations.

4.3 Results and Discussions

In Figure 4.1, we analyze the I − V characteristics of the disordered system

sweeping the interlayer bias up to VINT = 100 µV. We now introduce impu-

rities into the system corresponding to three different MFPs of Λ = 1.2, 0.6,

and 0.3 µm. The range of MFPs we consider here adds a significant per-

turbation to the on-site energy of 10 ∼ 20% of the self-consistent Hartree

potential. In Figure 4.1a, we plot the current flowing from the bottom right

contact (BR) to the top left contact (TL) and immediately see the effects of

disorder. By comparing the interacting (S = 2) and non-interacting (S = 1)

systems, we notice that the interacting system has a uniform current en-

hancement over the non-interacting case over all interlayer voltages. How-

ever, when disorder corresponding to a MFP commensurate with the system

size is introduced, we see that the interlayer current is increased at low VINT .

This is a consequence of local regions of stronger interlayer interactions set

up by disorder-induced localization. As VINT is increased, the interlayer en-

hancement quickly disappears and leaves only a small exchange enhancement

over the non-interacting case. As the MFP is further decreased, we see that

the trend of decreased interlayer exchange enhancement continues as the per-

turbations to the carriers via the disorder potential are now large enough to

29

energetically separate the two layers, leaving only weak single-particle inter-

layer interactions. A separate way of visualizing the effects of disorder is

via Figure 4.1b where we plot the current injected from the bottom right

(BR) contact and extracted from the top right contact (TR). Here, we would

expect the interlayer current to be close to zero as we see for the S = 1 and

S = 2 cases without disorder. However, when disorder localizes the electron

states in the bottom layer, interlayer tunneling is locally enhanced, leading to

increased interlayer transmission near the injection contact. Beyond locally

enhanced transmission, the presence of disorder in the top layer perturbs the

electron states leading to backscattering of injected states. The overall effect

is an increasing current from BR to TR as disorder increases.

A crucial component to understanding the interlayer dynamics in the pres-

ence of materials disorder is the critical current (Ic), or amount of interlayer

current the system can sustain by simply altering the interlayer phase. As

we have not passed a phase boundary, we do not expect large differences in

interlayer currents between before and after the critical current. The critical

current of the clean system from the simulation is Icleanc = 0.89 nA, which

is smaller than the predicted critical current [72] of Ianalc = 1.44 nA. This

discrepancy can be understood by considering the charge imbalance of two

layers, which is addressed in Chapter 3.4. As the bottom layer is biased, the

charge imbalance of two layers builds up the electrostatic potential difference

between two layers. When the charge imbalance between two layers is suffi-

ciently large, two nested Fermi surfaces are separated and the system is in an

off-resonant condition. As a result, the interlayer current is decreased before

the system reaches the critical current. We calculate a critical current for

each disorder configuration and MFP and find that I1.2µmc = 0.67± 0.02 nA,

I0.6µmc = 0.72±0.09 nA, and I0.3µm

c = 0.74±0.07 nA, all of which are smaller

than the disorder-free case. We find the critical current saturates with in-

creasing disorder but this is simply because the exchange enhancement has

been lost and the system responds as if it is non-interacting for MFPs shorter

than the system dimensions.

The effect of disorder can be quantitatively analyzed by probing the inter-

layer conductance. With the presence of the scattering source in the system,

the spectral function of 2DEG has Lorentzian form within the Born approxi-

mation [42] and, therefore, the tunneling conductance also follows Lorentzian

30

shape as [59]

G =IINTVINT

= G0Γ2

Γ2 + (∆EF + eVINT )2, (4.5)

where G0 is a conductance constant and VINT is an interlayer bias. ∆EF is

the electrostatic potential difference between the top and bottom layer and

∆EF = 0 at VINT = 0 when two layers are well balanced. The tunneling

linewidth Γ is the half width at half maximum in a G vs. VINT plot and

satisfies the following relation

Γ =~τ, (4.6)

where τ is the average electron scattering time in the system. Figures 4.2a -

4.2c depict tunneling conductance which is defined as G = 〈ITL+ITR〉/VINT .

We find that the width of the tunneling conductance peak is broadened

with the increasing material disorder, which agrees with Eq. (4.5). With a

given MFP, the linewidth, Γ, is calculated from Eq. (4.6). The predicted

broadening (Γ/2) for Λ = 1.2, 0.6, 0.3 µm are∼ 20, 41, 82 µeV, respectively.

Our results show energy broadening of 22.1±2.0, 36.7±5.8, 74.5±9.1 µeV,

respectively, in a good agreement with the predicted values. For a disordered

system which has a well-balanced electron density and satisfies Ef � Γ, the

conductance constant, G0, in Eq. (4.5) is described as [42]

G0

A=

2e2t2ν0τ

~2=

2e2t2ν0

~2vfΛ, (4.7)

where A is system area, t is a tunneling amplitude, and ν0 = m∗/π~2 is the

two-dimensional density of states. Consequently, the conductance near the

zero bias reveals a linear dependence on MFP. In Figure 4.2d, we plot the

interlayer conductance as a function of MFP at a small bias (VINT = 0.1 µV).

We observe a linear dependence in the calculated interlayer conductance,

especially at the MFP shorter than Λ = 0.6 µm. The broadening in tunneling

peak and linear dependence of interlayer conductance in MFP indicate that

the interlayer transport is no longer controlled by the intrinsic nature of the

system and is instead controlled by the material disorder when the mean free

path is shorter than the device length.

As important as it is to understand the variations in the magnitude of the

interlayer currents, we must also understand how disorder shifts the local

interlayer current flow. In Figure 4.3, we plot the spatially resolved inter-

31

layer current profile below and above critical current for a clean system (left

column) and one with a MFP of Λ = 0.3 µm (right column). In the linear

response regime, the interlayer transmission probability of the channel k is

proportional to v−2k , where vk is a quasiparticle Fermi velocity of the channel

k in a transport direction [72]. The quasiparticle subband with the high-

est transverse velocity, or the highest quantized energy level, has the lowest

transport directional velocity and, consequently, makes the largest contribu-

tion to the interlayer transport. As a result, the quasiparticle wavefunction

of the highest quantized level contributes most significantly to the tunneling

current. The effect is manifest as a transverse directional oscillation in the

current profile in Figure 4.3 due to the small degree of confinement in our

1.2 µm wide structure. In the clean case, we see in Figure 4.3a that the

interlayer current density is peaked near the contacts at a small interlayer

bias. This is clearly not the case in Figure 4.3b, where we see that above

critical current the clean system exhibits an even spatial distribution of the

interlayer current.

In the presence of strong disorder, however, the interlayer current flows

randomly across the system. The current profile of the disordered system

has been broken up into regions of strong and weak interlayer current flow

controlled by the disorder pattern both in the small and large bias case. Even

though the current profile is arbitrarily distributed, the regions where the

tunneling is strongly suppressed are consistent independent of the interlayer

bias as is shown in right column of Figure 4.3 (dark regions). Meanwhile,

there is a significant difference in current flow profile from different disorder

configurations, which further indicates the strong dependence of the current

flow on the disorder configuration. The net current averages out the locally

enhanced and suppressed current flow and averages to a smaller current than

that of the clean system. This gives further evidence of how the disorder

controls the interlayer transport properties, when the corresponding MFP is

smaller than the system dimension.

32

4.4 Figures

Figure 4.1: A plot of current as a function of interlayer bias for electronsinjected into the bottom right contact and extracted from (a) the top leftcontact and (b) the top right contact.

0 2 0 4 0 6 0 8 0 1 0 001 02 03 04 05 06 0

I n t e r l a y e r b i a s ( m V )

G (m

S)

S = 2 S = 1 0 . 3 µm0

1 02 03 04 05 06 0

G (m

S)

S = 2 S = 1 0 . 6 µm0

1 02 03 04 05 06 0

G (m

S)

S = 2 S = 1 1 . 2 µm

( c )

( b )

0 . 0 0 . 3 0 . 6 0 . 9 1 . 21 0

2 0

3 0

4 0

5 0

6 0

7 0

8 0

M e a n f r e e p a t h ( m m )

G (m

S)

( d )( a )

Figure 4.2: Plots of an interlayer conductance, G, as a function of interlayerbias with mean free path of (a) 1.2 µm, (b) 0.6 µm, and (c) 0.3 µm. (d)Plot of max height of the interlayer conductance as a function of mean freepath at VINT = 0.1 µV.

33

Widt

h (mm

)

0 . 0

0 . 4

0 . 8

1 . 2 0 . 0 0 . 4 0 . 8 1 . 20 . 0 00 . 0 40 . 0 80 . 1 2

0 . 0 0 . 4 0 . 8 1 . 2

- 0 . 1

0 . 0

0 . 1

0 . 2

0 . 30 . 00 . 10 . 2

0 . 0

0 . 4

0 . 8

1 . 2 0 . 0 0 . 4 0 . 8 1 . 20 . 00 . 10 . 20 . 3

0 . 0 0 . 4 0 . 8 1 . 2

- 0 . 5- 0 . 20 . 10 . 40 . 71 . 0

0 . 00 . 30 . 6

L e n g t h ( m m )L e n g t h ( m m )

Widt

h (mm

)L = 0 . 3 m mC l e a n

( b )

( a )I(p

A)I(p

A)W

idth (

mm)

Figure 4.3: A plot of spatially resolved current as a function of device widthand length at (a) VINT = 10 µV and (b) VINT = 80 µV in units of pA foran assumed enhancement factor of S = 2. The current profile of the cleansystem (disordered system of Λ = 0.3 µm) is presented in top and bottomleft (top and bottom right) panels of this figure. Above eachtwo-dimensional plot, there is a current profile taken in the middle of thesystem indicated as a white dashed line.

34

CHAPTER 5

CONCLUSION

For a balanced double quantum well system, the equilibrium electronic eigen-

states have symmetric or antisymmetric bilayer wavefunctions. When the

interlayer exchange interaction of the bilayer system is described using a

pseudospin language, the difference between symmetric and antisymmetric

populations corresponds to a x-direction pseudospin polarization. When the

bilayer system is connected to reservoirs that drive current between layers,

it is easy to show that the non-equilibrium pseudospin polarization must tilt

toward the y-direction. The total current that flows between layers is in

fact simply related to the total y-direction pseudospin polarization. These

properties suggest the possibility that electron-electron interactions can qual-

itatively alter interlayer transport under some circumstances. In equilibrium

interactions enhance the quasiparticle inter-layer tunneling amplitude, but

not the total current that can be carried between layers for a given y-direction

pseudospin polarization. If the inter-layer quasiparticle current can be driven

to a value that is larger than can be supported by the inter-layer tunneling

amplitude, the self-consistent equations for the transport steady state have

no solution and time-dependent current response is expected. This effect

has not yet been observed, but is partially analogous to spin-transfer-torque

oscillators in circuits containing magnetic metals. We have demonstrated by

explicit calculation for a model bilayer system that it is in principle possible

to induce this dynamic instability in semiconductor bilayers.

The current-voltage relationship in semiconductor bilayers is character-

ized by a sharp peak in dI/dV at small bias voltages, followed by a regime of

negative differential conductance at larger bias voltages. In our simulations

the pseudospin instability occurs in the regime of negative differential con-

ductance where dynamic responses might also occur simply due to normal

electrical instabilities. It should be possible to distinguish these two effects

experimentally, by varying the circuit resistance that is in series with the

35

bilayer system.

In addition, we have shown that the exchange-enhanced interlayer trans-

port properties in semiconductor bilayers are very sensitive to the strength

and location of material disorder. We find that as soon as the MFP of the

electrons becomes commensurate with the spatial system dimensions, the

disorder controls the interlayer transport properties and reduces the inter-

acting electron system to a non-interacting one. These findings demonstrate

that to observe pseudospin instability and interaction driven interlayer en-

hancement experimentally, one requires systems with MFPs greater than the

system spatial dimensions.

36

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