Electron and nuclear spins in semiconductor quantum...

Electron and nuclear spins in semiconductor quantum dots

A dissertation presented

by

Jacob Jonathan Krich

to

The Department of Physics

in partial fulfillment of the requirements

for the degree of

Doctor of Philosophy

in the subject of

Physics

Harvard University

Cambridge, Massachusetts

July 2009

Thesis advisor Author

Bertrand I. Halperin Jacob Jonathan Krich

Electron and nuclear spins in semiconductor quantum dots

Abstract

The electron and nuclear spin degrees of freedom in two-dimensional semicon-

ductor quantum dots are studied as important resources for such fields as spintronics and

quantum information. The coupling of electron spins to their orbital motion, via the spin-

orbit interaction, and to nuclear spins, via the hyperfine interaction, are important for

understanding spin-dynamics in quantum dot systems. This work is concerned with both

of these interactions as they relate to two-dimensional semiconductor quantum dots.

We first consider the spin-orbit interaction in many-electron quantum dots, study-

ing its role in conductance fluctuations. We further explore the creation and destruction

of spin-polarized currents by chaotic quantum dots in the strong spin-orbit limit, finding

that even without magnetic fields or ferromagnets (i.e., with time reversal symmetry) such

systems can produce large spin-polarizations in currents passing through a small number

of open channels. We use a density matrix formalism for transport through quantum dots,

allowing consideration of currents entangled between different leads, which we show can

have larger fluctuations than currents which are not so entangled.

Second, we consider the hyperfine interaction between electrons and approximately

106 nuclei in two-electron double quantum dots. The nuclei in each dot collectively form

an effective magnetic field interacting with the electron spins. We show that a procedure

originally explored with the intent to polarize the nuclei can also equalize the effective

magnetic fields of the nuclei in the two quantum dots or, in other parameter regimes, can

iii

iv Abstract

cause the effective magnetic fields to have large differences.

Contents

Title Page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Citations to Previously Published Work . . . . . . . . . . . . . . . . . . . . . . . vii

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

1 Introduction 1

1.1 Spin-orbit interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Time-reversal symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3 Random matrix theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.4 2DES and quantum dots . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.5 Organization of thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

1.6 Summary of new results obtained in this research . . . . . . . . . . . . . . . 37

2 Cubic Dresselhaus spin-orbit

coupling in 2D electron quantum dots 40

3 Spin polarized current generation

from quantum dots without magnetic fields 51

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.2 Setup and symmetry restrictions . . . . . . . . . . . . . . . . . . . . . . . . 53


3.4 Dephasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.5 Finite bias and temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4 Fluctuations of spin transport

through chaotic quantum dots with spin-orbit coupling 72

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.2 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74


4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

v

vi Contents

5 Inhomogeneous nuclear spin flips 92

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 925.2 Electronic states of the double dot with N=2 . . . . . . . . . . . . . . . . . 945.3 Nuclear spin flip location . . . . . . . . . . . . . . . . . . . . . . . . . . . . 965.4 Nuclear states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 975.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

6 Preparation of non-equilibrium nuclear spin states in quantum dots 103

A Appendix to Chapter 2 114

A.1 Further cubic spin-orbit terms . . . . . . . . . . . . . . . . . . . . . . . . . . 114A.2 Refitting var g data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117A.3 Value of γ in GaAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

B Appendix to Chapter 3: Quaternions 119

C Appendix to Chapter 3: Spin polarization forbidden with M = Nφ = 1 121

D Appendix to Chapter 3: Spin polarization from dephasing 124

Bibliography 127

Citations to Previously Published Work

The material of chapters 2, 3, and 5 have appeared in the following forms:

“Cubic Dresselhaus spin-orbit coupling in 2D electron quantum dots,” Jacob J. Krichand Bertrand I. Halperin, Phys. Rev. Lett. 98, 226802 (2007), cond-mat/0702667;

“Spin-polarized current generation from quantum dots without magnetic fields,”Jacob J. Krich and Bertrand I. Halperin, Phys. Rev. B 78, 035338 (2008),arXiv:0801.2592;

“Inhomogeneous nuclear spin flips,” M. Stopa, J. J. Krich, and A. Yacoby,arXiv:0905.4520.

Chapters 4 and 6 are based on manuscripts in preparation.

Electronic preprints (shown in typewriter font) are available on the Internet at the fol-

lowing URL:

http://arXiv.org

vii

Acknowledgments

This thesis has been the influenced by many people who supported, advised, and

entertained me these last six years. I cannot thank them all, but I’ll try.

Foremost, I cannot imagine a better choice for my thesis supervisor than Bert

Halperin. Though he initially played hard to get, he has been dedicated to guiding me both

in understanding physics and also in how to be a physicist. I can only hope to have picked

up a fraction of his unerring physical intuition and conscientious attention to precision and

detail. Bert will always serve for me as the model scientist, with deep physical insights and

great respect for those around him.

My road to mesoscopic physics began in the spring of 2004, when I took Charlie

Marcus’ mesoscopic physics course. Bert, while getting his caffeine fixes, often expounded

on and explained subtleties of the quantum Hall effect. Amir Yacoby made regular guest

appearances, challenging us to be more creative in imagining the implications of the phe-

nomena we discussed. Charlie so exuded enthusiasm for the subject that it was inevitable

I found myself drawn to mesoscopics for my thesis work. Though he failed to make me an

experimentalist, he gave me my field.

Charlie and Rick Heller, both on my orals committee, have provided me with much

useful advice on physics, grad school, and life. Rick was also a pleasure to work with as a

research and teaching supervisor, and I greatly benefited from conversations with his group,

including Rob Parrott, Jay Vaishnav, Jamie Walls, Tobias Kramer, Florian Mintert, and

Jamal Sakhr.

In the last eighteen months I have enjoyed a fruitful collaboration dedicated to

solving the mysteries of the so-called Zamboni effect. Conversations with Misha Lukin and

Amir Yacoby were always enlightening and provided useful direction. Mike Stopa started

me on this work and helped out in several different places in my time at Harvard; he remains

a font of knowledge about all things GaAs. Jake Taylor’s clarity of thought and expression

viii

Acknowledgments ix

are extraordinary. Michael Gullans has taught me a great deal about physics and how to

fearlessly pursue it, and I look forward to more fruitful collaboration.

I additionally thank Bert, Charlie, and Misha for serving on my thesis committee.

I have had the great privilege and pleasure of sharing an office with Emmanuel

Rashba, whose comments and advice have become an essential part of my week.

The hallways of the condensed matter theory group have provided a number of

people whose insights and company I have enjoyed, including Hansres Engel, Ilya Finkler,

Ari Turner, Jiang Qian, Naomi Chang, Caio Lewenkopf, Adrian del Maestro, David Pekker,

Mark Rudner, Izhar Neder, Michael Levin, and Wesley Wong.

Hakan Tureci helped me out with the numerics of semiclassical simulations at a

crucial point.

Able experimentalists have worked with me throughout my time at Harvard, and

I would particularly like to thank Dominik Zumbuhl and Jeff Miller, for setting me straight

about spin-orbit coupling in quantum dots, and Sandra Foletti, Hendrik Bluhm, David

Reilly, and Christian Barthel for doing the same on all matters hyperfine.

My classmates have been great friends and colleagues, from puppet show to de-

fense. Thanks go to Subhy Lahiri, Abram Falk, Mark Romanowsky, Rodrigo Guerra, Megha

Padi, Jeremy Munday, Josh Boehm, Jon Gillen, Esteban Real, Phil Larochelle, Jihye Seo,

Yi-Chia Lin, and Alex Wissner-Gross.

The life of a physics graduate student is made immeasurably easier by Sheila

Ferguson, who provides advice for navigating the whole system and an abiding concern

about each of us.

I have had the honor of being supported by the Fannie and John Hertz Foundation,

which gave me the freedom and flexibility to move at my own pace. For the last year, I

thank the support from Bert (and the NSF) and from teaching a great course on energy

x Acknowledgments

technology with Mike Aziz.

Though its influence is not apparent in this thesis, the most rewarding and ed-

ucational part of my experience at Harvard has been with the Harvard Energy Journal

Club, and I particularly thank Kurt House, Mark Winkler, Ernst van Nierop, Alex John-

son, Suni Shah, David Romps, Jason Rugolo, Kate Dennis, and Josh Goldman for their

gleeful engagement with all things energy.

For always keeping me entertained, informed, distracted, and wanting more, I must

thank Kevin Drum, without whom this thesis might have been completed a year earlier.

Through means hidden and overt, my parents fostered both my tendency to ask

the great scientific question, “why?” and the self-conscious awareness that all things can

and should be optimized. They have inspired, supported, and encouraged me during the

formation of this thesis and throughout my life.

Abby has been the best sister possible, even selflessly moving to Somerville to

make the last year of my PhD better. She has cared deeply about my work, always wanting

to know what I’m studying, even going so far as to read the first unintelligible draft of my

first paper. Her dedication to making the world a better place has stimulated my own shift

of interests in the direction of energy science.

I formally joined Harriet and John Lenard’s family in my third year at Harvard,

but I felt they had already welcomed me before I ever arrived in Cambridge. Their encour-

agement and enthusiasm have been incomparable.

It is a counterfactual the truth of which we can never be sure we have determined

correctly, but I do not believe I could have completed this thesis without the unconditional

support, love, and inspiration of Patti.

To Patti

xi

Chapter 1

Introduction

This thesis is concerned with objects small enough that quantum mechanics is

essential to understand their behavior, but large enough to contain many particles, so

statistics are important. Such systems are called mesoscopic, and they combine many of

the features of both atomic and bulk systems. Mesoscopic physics has been a rich subject

for over twenty years and continues to bring out fundamental science and an ever-expanding

set of proposed applications.

We will consider some of the smallest and largest mesoscopic systems. This may

not sound like a large range, but the differences between quantum dots on the scale of

100 nm and those on the scale of 5 µm can be profound. Quantum dots are carved out from

a two-dimensional electron system, and the larger variety retain many of the properties of

that two-dimensional system, including the relative unimportance of Coulomb interactions.

As experimentalists learn to cool electrons to lower temperatures, ever larger quantum dots

can be considered mesoscopic, the requirement being that the electrons maintain quantum

coherence across the dot. In the smallest case, single electrons can be trapped inside a quan-

tum dot, producing an artificial atom. This system has received attention in part because

1

2 Chapter 1: Introduction

it was hoped that it would avoid many of the stochastic issues of larger mesoscale devices,

making relatively easily manipulable artificial atoms for applications such as quantum infor-

mation. At least in the GaAs system, however, such single-electron quantum dots are still

firmly mesoscopic, due to the hyperfine interaction between the electron spin and millions

of nuclear spins. While this may not be the ideal situation for technological purposes, it

provides a fascinating system for probing fundamental interactions between a small number

of electrons and a large number of nuclei.

This thesis is divided into two parts. Chapters 2-4 concern the spin-orbit interac-

tion (SOI) in large quantum dots while chapters 5 and 6 concern the hyperfine coupling of

electron and nuclear spins in two-electron double quantum dots. In the case of the large

quantum dots, we begin in chapter 2 by discussing the effects of the oft-neglected cubic

Dresselhaus SOI on transport through quantum dots, showing that the cubic Dresselhaus

term is important for understanding the conductance fluctuations in the devices studied;

combined with experiments, these results imply that the Dresselhaus coupling in GaAs is

about a third of its most commonly cited value. We move in chapters 3 and 4 to discussing

the use of the SOI to manipulate the spin-polarization of currents. We use random matrix

theory to determine the typical amounts of spin-polarization that can be expected by send-

ing (charge or spin) currents through chaotic quantum dots with strong spin-orbit coupling.

We use a density matrix formalism for the currents, which allows consideration of currents

entangled between the channels of two leads; such entangled currents produce larger fluc-

tuations than currents without such entanglement. To lay the groundwork for these three

chapters, we begin in section 1.1 by providing an introduction to the SOI, especially in the

conduction band of common zincblende semiconductors. We continue in section 1.2 to de-

tail some relevant effects of time-reversal symmetry and the formal properties of operators

in time-reversal invariant systems. With this foundation constructed, we discuss in section

Chapter 1: Introduction 3

1.3 the random matrix theory of scattering, which is the primary tool used for exploring

the effects of the SOI in mesoscopic transport in chapters 2-4.

For the second part of the thesis, we put the SOI aside and focus on another

interaction of electron spins, this time with nuclear spins. Chapters 5 and 6 form the

beginning of a theoretical picture of how to control aspects of the nuclear spin configuration

by manipulation of the electron spin states in a double quantum dot with two electrons. We

show that, for some parameter regimes in the model we construct, the effective magnetic

fields produced by the nuclei in the two quantum dots can be made to equalize with each

other, while in other parameter regimes these effective magnetic fields can be made to

diverge. In section 1.4, we introduce this material by describing double quantum dots and

the hyperfine interaction with nuclear spins.

A leitmotif in this thesis is quantum coherence and its loss. As an electron ex-

changes energy with phonons, other electrons, or stray fields, its wavefunction necessarily

becomes a superposition with the states of all these other systems, which can effectively

destroy its quantum coherence. All of the systems considered in this thesis interact with a

robust environment, which causes the systems to eventually behave more classically. The

dephasing time is the time required for these decohering interactions to occur.When the

characteristic time a particle spends in the system, the dwell time, is on the order of the

dephasing time, or shorter, quantum effects on the transport properties through the device

become important. One of the appeals of technology using the spin degree of freedom is

that it does not interact as strongly with its environment as does the electron charge, so it

possibly can retain coherence for useful periods of time [160]. And even in cases where the

electron is fully dephased, coherence of the nuclear system can be important, as discussed

in chapters 5 and 6.


1.1 Spin-orbit interaction

One of the interesting and useful features of electron spin is that it is inert when

compared with electrical charge. It is relatively easy to control charged particles with

small electrical fields, and these are now easily produced in nanoscale environments. Spin,

however, does not couple directly to electrical fields, so it is both harder to manipulate than

electrical charge and resistant to various forms of environmental decoherence. The latter

property makes the spin degree of freedom useful for such things as quantum information.

For manipulating spins, then, we must concern ourselves with relatively weak couplings.

The most direct way to interact with spin is via a magnetic field. The Hamiltonian

for spin angular momentum ~S in a magnetic field is

Hs = ge

2mc~B · ~S, (1.1)

where g ≈ 2 is the electron g-factor, −e is the electron charge, m is the mass of the electron,

and c is the speed of light. We define the Bohr magneton µB = e~/2mc, where ~ is Planck’s

constant.

Special relativity requires that a particle moving with velocity v ≪ c in an electric

field ~E experiences an effective magnetic field ~B = ~E × ~vc + O(v2/c2). This magnetic field

couples to the spin, producing a Hamiltonian

Hso =ge

4m2c2~E × ~p · ~S, (1.2)

where we have put an extra factor of 2 in the denominator, to account for the relativistic

effect of Thomas precession [70]. The same result can be derived from expansion of the

Dirac equation in the nonrelativistic limit [129]. This Hamiltonian shows the coupling of

the momentum (or orbit) of the electron with its spin, when in the presence of an electric

field. Upon time reversal ~p→ −~p, ~S → −~S and ~E → ~E; this will be explored more formally


in Section 1.2. We thus see that the spin-orbit Hamiltonian is time-reversal invariant, unlike

a Hamiltonian containing an ordinary magnetic field.

In solid state systems, we are mostly interested in electrons in the conduction band

(or holes in the valence band), which move with respect to the strong electric fields of the

ionic cores of the crystal lattice. In this case, the electric field is a function which oscillates

on the length scale of the lattice constant, while electronic states of long wavelength overlap

a large number of ionic cores. In a crystal, Bloch’s theorem says that the single-particle

wavefunctions can be written ψ = ei~k·~ruk,n(~r), where ~r is the position, uk,n(~r) is a highly

oscillatory function periodic on the unit cell, n is a band index, and ~k is the wavevector

confined to the first Brillouin zone.

In the zincblende semiconductors which are the focus of this work, the conduction

band has a twofold degeneracy at k = 0. This double degeneracy behaves like the electron

spin degree of freedom, except with renormalized mass and g-factor. It is convenient to

approximate the full Hamiltonian of the solid state system by an effective Hamiltonian for

the conduction band quasiparticles, commonly called electrons. We can then write a generic

Hamiltonian for the spin-orbit coupling in the conduction band of the form

Hso,2 =∑

µ=x,y,z

∞∑

n,m,l=0

γµnmlk

nxk

my k

lzσµ, (1.3)

where γµnml is a constant, and σµ are the standard Pauli spin matrices, acting on the double

degeneracy of the band (that is, equivalent to the spin degree of freedom). This expression

does not immediately look helpful, but it is usefully simplified by considering the symmetries

of the system. As mentioned above, under time reversal, ~k → −~k and ~S → −~S. Since spin-

orbit coupling does not violate time-reversal symmetry (TRS), we immediately require γµnml

to be zero unless n+m+l is odd. If the crystal has inversion symmetry, thenH(~k) = H(−~k),

which requires that n,m, l are all even. We thus immediately see that breaking inversion


symmetry is required for spin-orbit coupling in the conduction band of such a crystal, so bulk

crystals with the diamond lattice, such as silicon, do not have conduction band spin-orbit

coupling in the absence of external fields.

A large class of experimentally relevant semiconductor alloys, most notably GaAs,

has the zincblende structure, which breaks inversion symmetry. In most such semiconduc-

tors, we are interested in the properties of states with wavevector k ≪ π/a, where a is the

lattice constant, so it is natural to consider the terms which are of lowest order in k in

the spin-orbit interaction of Eq. 1.3. Formally, the zincblende structure has point group Td

[157], and there are no terms linear in k in Eq. 1.3 consistent with the point group. The lead-

ing terms were worked out by Dresselhaus, so the coupling is generally called Dresselhaus

coupling. It has the form [44]

Hd = γσxkx(k2y − k2

z) + cyclic permutations. (1.4)

In the two-dimensional structures that are the subject of this thesis, the electrons

are confined to an interface between GaAs and AlGaAs. For a sufficiently small electron

density and temperature, the conduction band electrons occupy only one quantum state in

the confinement direction, and thus form a quasi-2D electron system (2DES). The AlGaAs

acts as a barrier, so the electrons mostly occupy the GaAs crystal. For heterostructures

grown in the (001) direction, the structure has its symmetry group broken down to C2v

or D2d, for symmetric and asymmetric confinement, respectively [157], which allows more

spin-orbit terms in Eq. 1.3. In particular, it allows spin-orbit terms linear in k, which will

be the dominant contributions for low electron density. The most important term comes

from Eq. 1.4 by simply replacing all of the powers of kz by their expectation values across

the confinement wavefunction. This gives

Hd,2 = γ[⟨

k2z

⟩

(σyky − σxkx) + σxkxk2y − σyk

2y

]

, (1.5)


where 〈kz〉 = 0, since the wavefunction is not moving in the z-direction. The linear term

in Eq. 1.5 is called the linear Dresselhaus SOI. We can roughly estimate the magnitude of

the cubic terms relative to the linear ones by considering kx ≈ ky ≈ kF , where kF =√

2πn

is the 2D Fermi momentum, with n the electron density. Then the cubic terms have scale

γk3F and the linear terms have scale γ

⟨

k2z

⟩

kF , so we can neglect the cubic terms when

k2F /⟨

k2z

⟩

≪ 1. If we model the confinement as an infinite square well, then the second

subband becomes occupied when k2F ≥ 3k2

z ; for an infinite triangular well[42], the second

subband is occupied when k2F & 2k2

z . Thus, if the quantum well is doped close to filling the

second subband, the cubic Dresselhaus interaction can be as important as the linear terms.

The cubic terms are, nonetheless, often neglected; some of their effects are considered in

chapter 2.

If the confinement in the growth direction lacks inversion symmetry, either from

the heterostructure growth process or from the application of an electric field, there is a

further contribution to the SOI from this structure inversion asymmetry (SIA). This gives

a spin-orbit interaction which is linear in k, commonly referred to as the Rashba coupling,

of the form

Hr = α(

~σ × ~k)

· z, (1.6)

where z is the confinement direction.

The groups C2v and D2d actually permit additional cubic spin-orbit terms than

the cubic terms in Eq. 1.5, but they are generally weaker (See Appendix A and Ref. [157]).

There are many ways to think about the spin-orbit interaction. One of the most

convenient is to think of it as a velocity-dependent magnetic field coupled to the electron

spin. That is,

Hso = gµB~B(~k) · ~S/~. (1.7)


In the case with only the Rashba and linear Dresselhaus SOI terms, it is convenient to

consider the SOI in a related fashion as a spin-dependent vector potential. In this case, the

Hamiltonian can be written as

Hl =~

2k2

2m+ α(k2σ1 − k1σ2) + β(k1σ2 + k2σ1) + V (~r) (1.8)

=~

2k2

2m+ (α+ β)k2σ1 + (β − α)k1σ2 + V (~r), (1.9)

where the components are with respect to the axes e1 = [110], e2 = [110], and β = γ⟨

k2z

⟩

.

We can complete the squares to rewrite this as

Hl =~

2

2m

[

k1 +m

~2(β − α)σ2, k2 +

m

~2(β + α)σ1

]2+ V (~r) − m

~2(α2 + β2), (1.10)

where the linear spin-orbit terms look just like a vector potential, except dependent on

spin, as described in Ref. [5]. Since the vector potential does not depend on position, it

would at first appear to have no effect on the dynamics of the electrons. It is, however,

non-abelian. That is, if we try to gauge away this constant vector potential with the unitary

transformation [5]

U = exp

(

ir1σ2

2λ1− ir2σ1

2λ2

)

, (1.11)

where λ1,2 = ~2/2m(β ∓ α) is the distance required for the spin to precess around the

effective magnetic field, we can successfully cancel the effects of the spin-orbit coupling

to first order in 1/λ1,2, but we will introduce effects at higher order in 1/λ1,2 [5]. Some

implications of this theory, and its modifications when the cubic Dresselhaus SOI is included,

are the subject of chapter 2.

1.2 Time-reversal symmetry

Time-reversal symmetry plays such an important role in this work that it is worth

defining its properties carefully. This discussion follows that in Mehta [105]. The time-


reversal operation T is antiunitary [130] and for any basis can be written in the form

T = UC (1.12)

where U is a unitary operator and C is the complex conjugation operator. We require that

for any states |α〉, |β〉,1

〈Tβ|Tα〉 = 〈α|β〉 . (1.13)

For any operator O, we can define its time-reverse OR by

〈β|O |α〉 = 〈Tα|OR |Tβ〉 , (1.14)

and it is easy to show that

OR = TO†T−1 = UCO†CU−1 = UOTU−1. (1.15)

Since reversing time twice should have no physical effects, we further require that T 2 = α11

with |α| = 1. Then

UCUC = UU∗ = α11, so by unitarity (1.16)

U = αUT = α(αU), (1.17)

so α = ±1, corresponding to integral or half-integral angular momentum.

If we change the basis of states by sending |ψ〉 → R |ψ〉 with R a unitary transfor-

mation, then T becomes RUCR−1 = RURTC, that is, U → RURT . In all that follows, we

choose a standard form for the time-reversal operator T and then limit ourselves to basis

transformations that leave T invariant. In the case that α = 1 in Eq. 1.17, one solution is to

have Ui = 11, where the subscript i refers to integer spin. To illustrate the importance of this

1Note that since T is antiunitary, it can only be applied to kets, so 〈Tβ| is defined as the dual of T |β〉.


choice, note that it is often stated that Hamiltonians of spinless time-reversal invariant sys-

tems are symmetric (i.e., real) [105], but all Hamiltonians are Hermitian and thus unitarily

similar to diagonal matrices with purely real components. The important claim is that if

we limit ourselves to bases in which the time-reversal operator is T = C (i.e., Ui = 11), then

all spinless time-reversal invariant Hamiltonians are simultaneously symmetric, as Eq. 1.15

implies that H = HR = HT . We see that this choice of representation of T restricts the

permissible basis transformations R to be orthogonal transformations; if we use a general

unitary (not orthogonal) basis transformation, then H will no longer be symmetric, though

it will still of course be time-reversal invariant.

In the half-integer spin case, where α = −1, then Uh = −UTh , and Uh is antisym-

metric, where the subscript h indicates half-integer spin. An antisymmetric unitary matrix

must have even dimension[105], and one can choose Uh to be the 2N × 2N block diagonal

matrix with(

0 1−1 0

)

repeated on the diagonal. That is,

Uh =

0 1 0 0 . . .

−1 0 0 0 . . .

0 0 0 1 . . .

0 0 −1 0 . . .

......

......

. . .

. (1.18)

We recognize(

0 1−1 0

)

as iσy where σy is the standard Pauli matrix, so the time reversal of

the operator O for a spin-1/2 system takes the usual form OR = σyOTσy. Once we fix Uh in

the form of Eq. 1.18, we are allowed unitary basis transformations R that preserve Uh, i.e.,

where Uh = RUhRT . The set of all such matrices R is called the symplectic group Sp(N).


1.3 Random matrix theory

When presented with a relatively simple quantum system, such as a hydrogenlike

atom or a parabolic potential, it is useful to solve for the eigenvalues and eigenfunctions

of the Hamiltonian to determine the dynamics of the system. For many more complicated

systems, such as electrons in a quantum dot in a 2DES (described in Sec. 1.4), the spectrum

and eigenfunctions depend sensitively on the exact shape of the quantum dot (e.g., on

the particular voltages applied to all of the gates defining the dot) and rapidly change as

that shape is modified. It is thus not only impractical to solve for the spectrum of such

Hamiltonians but also unhelpful, as the important, reproducible properties of the system

will be those common to many particular quantum dot shapes.

It has proven helpful in just this circumstance to throw in the towel and not

attempt to solve for the system’s properties in detail, but rather to assume that the Hamil-

tonian has effectively random entries, subject to the overall symmetries of the system. For

spinless noninteracting electrons confined to a so-called chaotic quantum dot with time-

reversal symmetry, we begin the random matrix theory (RMT) by noting that the Hamil-

tonian must be a real symmetric matrix. The most important energy scale for the single-

particle system is the mean level spacing (i.e., the inverse of the density of states), which in

the effective mass approximation is ∆ = 2π~2/2m∗A, where A is the area of the quantum

dot and m∗ is the effective mass. We can then consider the properties of Hamiltonians H

drawn from the Gaussian Orthogonal Ensemble in which the matrix elements Hij are drawn

independently from the distribution invariant under orthogonal basis transformation, which

gives the unique probability distribution for the symmetric Hermitian matrix H [105]

P (H) dH ∝ exp

(

−trH2

2σ2

)

dH, (1.19)


w

x

y

N M

a) b)

Figure 1.1: a) Section of an infinite quasi-1D wire. b) scattering region (shaded) connectedto two quasi-1D wires. The wire on the left has N open modes while the wire on the righthas M open modes.

where the volume element dH is

dH =N∏

i≤j

dHij, (1.20)

and the standard deviation is σ =√

2N∆/π, where N is the dimension of H. This method

was first introduced by Wigner [156] for studying heavy nuclei, and many interesting results

are reviewed in Refs. [105, 12]. In this work, we shall be interested in an equivalent formu-

lation for open quantum systems, most easily described in terms of scattering matrices.

1.3.1 Random scattering matrix theory

Consider an ideal quasi-one dimensional wire in a 2DES. It is oriented in the x-

direction and has a width w, as illustrated in Figure 1.1a. For spinless electrons propagating

in the wire, the Schrodinger equation is

[

− ~2

2m∇2 + V (y)

]

ψ(x, y) = Eψ(x, y), (1.21)

where V (y) is zero for |y| < w/2 and infinite otherwise. This is a free particle in the

x-direction and the classic particle in a box in the y-direction, with solutions

ψ(x, y) =

√

m

2π~2kxwcos(nπy

w

)

e±ikxx, (1.22)


where n is an integer, E = ~2[k2

x + (nπ/w)2]/2m, and the state has been normalized to

carry a probability current per unit energy of 1/h (in particular, the current is independent

of kx). For fixed energy E, there are

N =

⌊√

2mEw2

~2π2

⌋

(1.23)

propagating states (i.e., states with kx real), where ⌊.⌋ is the integer part. These independent

states are called modes or channels, and we denote them by ψ±n (E), where the ± indicates

the left- or right-moving state.

Consider two such ideal wires with N , M open channels at energy E connected

adiabatically to a scattering region such as a quantum dot, as shown in Fig. 1.1b, and let

K = N + M . The incoming states can be written as superpositions of all propagating

wavefunctions with momenta directed toward the scattering region, as

ψin =N∑

j=1

αLjψ+Lj +

M∑

i=1

αRiψ−Ri, (1.24)

where L/R indicates the wavefunctions in the wire on the left, right respectively, and αLi,

αRi are constants. The outgoing states can similarly be written as superpositions of ψ−Li

and ψ+Ri with coefficients βLi, βRi. The solution to the full quantum mechanical scattering

problem for the region is then given by the matrix relation

βL1

...

βLN

βR1

...

βRM

= S

αL1

...

αLN

αR1

...

αRM

, (1.25)

where S is a K×K complex matrix. By conservation of flux, S must be unitary. The space

of all unitary matrices of dimension K is compact [66](p. 69); it is thus mathematically


possible to take the uniform distribution over all unitary matrices of dimension K, which

is called the circular unitary ensemble (CUE).

We now consider the time-reversal operation. To use the convention for integer-

spin systems that Ui = 11, we must express our states with respect to the basis of cos(kxx)

and sin(kxx). Then eikxx = cos(kxx) + i sin(kxx), and under time reversal we recover the

standard result that αψ±Rn → α∗ψ∓

Rn. Assuming that the system and thus the S-matrix is

not modified by time reversal, we find

α∗L1

...

α∗RM

= S

β∗L1

...

β∗RM

, (1.26)

so we can multiply by S† and take the complex conjugate to find

βL1

...

βRM

= ST

αL1

...

αRM

. (1.27)

For this spinless time-reversal invariant system, S = ST , just as we found for the Hamilto-

nian in Section 1.2. The space of symmetric unitary matrices is also compact, and we can

define the uniform distribution of all such matrices, called the circular orthogonal ensemble

(COE). It is called the circular orthogonal ensemble because the distribution is invariant on

multiplication by any orthogonal matrix, which is precisely the set of basis transformations

that preserve the representation of the time-reversal operator, as shown in Section 1.2.

If we consider the spin degree of freedom, then each of the wavefunctions of Eq. 1.22

becomes a two-component spinor, and the S-matrix becomes a 2K × 2K complex unitary

matrix. Each 2× 2 matrix in S can be expressed as a linear combination of 112, σ1, σ2, and

σ3, where the σi are the standard Pauli spin matrices, and we will let σ0 ≡ 112. The most

convenient choice is to express each 2 × 2 matrix as q = q0σ0 + q1iσ1 + q2iσ2 + q3iσ3 for


complex numbers qj . Such 2 × 2 matrices are called quaternions. The 2K × 2K complex

matrix S can then be written as a K × K quaternion matrix. A quaternion has three

conjugates,

q∗ = q0∗σ0 + q1∗iσ1 + q2∗iσ2 + q3∗iσ3 (1.28)

qR = q0σ0 − q1iσ1 − q2iσ2 − q3iσ3 (1.29)

q† = qR∗ = q∗R, (1.30)

respectively called the complex conjugate, quaternion dual, and Hermitian conjugate. For

examples, see Appendix B. The Hermitian conjugate gives the same result as the Hermitian

conjugate of the equivalent complex matrix, but the same is not true of the complex conju-

gate. The quaternion dual is denoted qR because it is precisely the time-reversal operation

for a spin-1/2 particle, sending each of the spin components to its opposite. For a matrix

of quaternions Q, we define

(Qij)∗ = Q∗

ij (1.31)

(Qij)R = QR

ji (1.32)

Q† = QR∗ = Q∗R. (1.33)

If Q is the quaternion representation of the complex matrix Qc, then QR is the quater-

nion representation of the complex matrix UhQTc U

−1h with Uh from Eq. 1.18. That is, the

quaternion dual is precisely the time-reversal operation for spin-1/2 systems, which is the

reason quaternions are useful to introduce. The uniform distribution over all quaternion

unitary self-dual matrices is called the circular symplectic ensemble (CSE), which gets the

name because the distribution is invariant under symplectic transformations, precisely those

shown in Section 1.2 to preserve the form of the time-reversal operator.

These circular ensembles of scattering matrices not only exist but are also quite

useful in describing physical systems. Large quantum dots, if you’ll pardon the oxymoron,


are those on the micron scale, generally containing hundreds to thousands of electrons. For

such quantum dots with irregular boundaries, a plunger gate applied to the side of the dot

can sufficiently change the shape of the dot so as to effectively scramble the wavefunctions

inside the dot [30], thus making an easily obtained ensemble of quantum dots as a function

of plunger-gate voltage. The essential physics is that for sufficiently complicated wavefunc-

tions, the transport properties through the quantum dot are essentially random functions,

subject only to the symmetries of the system, such as time reversal. The transport will

be sufficiently chaotic to be described by RMT when the dwell time of particles inside the

dots is long compared to the time required to interact with the boundary, also known as the

bounce time or the Thouless time ET = L/v, where L is the typical linear size of the dot and

v is the particle velocity, generally taken to be the Fermi velocity. If we want an ensemble

of physical S-matrices to correspond to the circular ensembles (CUE, COE, or CSE), all the

K modes connected to the scattering region must be coupled with ideal contacts (otherwise

the reflection coefficients would generally be larger than the transmission coefficients; this

case corresponds to what is called the Poisson kernel rather than the CUE [12]).

For an ensemble of such chaotic quantum dots without time-reversal symmetry,

such as those with an external magnetic field, the CUE is a natural guess. For systems with

TRS, if there is no spin-orbit coupling, then the transport through the dot will not mix

spin directions, and the transport will be effectively two copies of the COE results. When

the different spin channels are strongly mixed with each other, for example by the spin-

orbit interaction, the CSE can be appropriate. Using the CSE requires that the spin-orbit

coupling be strong enough that by the time an electron exits the dot its spin has rotated

sufficiently that it is uncorrelated with its original orientation. The spin-orbit interaction

can be characterized by a spin-orbit time, which is the time required for the spin-orbit

interaction to rotate a spin by π. The CSE is a good ensemble to use if the dots are chaotic


and the mean dwell time is much larger than the spin-orbit time in the material. This limit

is, in practice, difficult to meet in GaAs quantum dots without making the dwell time an

appreciable fraction of the dephasing time, though technologies for ever lower temperatures

(increasing the dephasing time) make this regime increasingly accessible.

The spectral and transport properties of ensembles of quantum dots have been

shown to be well-modeled by suitable versions of random matrix theory (modified to include

dephasing, which will be discussed further in chapter 3) [30, 56, 68, 171, 172].

A microscopic justification for random matrix theory for disordered metals is dis-

cussed in the review by Efetov [48]. There is a connection between the statistics of the

closed-system Hamiltonians and the open-system scattering matrices [152]. Namely, if an

ensemble of closed quantum dots with Hamiltonians taken from the Gaussian unitary en-

semble is connected to 1D wires by ideal contacts, then the scattering matrices will be

distributed according to the CUE [57].

1.3.2 Landauer formula

Having set up the scattering matrix, it is now straightforward to present one of

the key results of transport theory in one-dimension (the wires are quasi-one dimensional),

often referred to as the Landauer formula [91, 69, 23, 24]. Consider the case of a scattering

region connected to two quasi-1D wires, as in Figure 1.1b, with the wires each connected by

reflectionless contacts to large reservoirs with fixed chemical potentials µL,R in the left, right

respectively. For convenience, we consider the case of spinless electrons at zero temperature.

All of the right-moving states in the left lead with energy less than µL are occupied,

and similarly for all the left-moving states in the right lead. In the noninteracting system,

the many-particle wavefunctions are Slater determinants, and it is natural to describe them

using a density matrix formalism on the single-particle wavefunctions. A density matrix is


generally of the form w(ǫ) = |ψǫ〉〈ψǫ|, so from the rule that

|ψout〉 = S |ψin〉 , (1.34)

we find

wout(ǫ) = S(ǫ)win(ǫ)S†(ǫ), (1.35)

where the density matrix win(ǫ) represents the states at energy ǫ ingoing toward the scat-

tering region and wout(ǫ) represents the outgoing states at energy ǫ from the scattering

region.

All the ingoing states in the left lead of energy less than µL are fully occupied,

and similarly for the right lead, so

win(ǫ) = PLΘ(µL − ǫ) + PRΘ(µR − ǫ), (1.36)

where Θ(x) is the unit step function. Thus, the net right-moving current in the left lead is

IL =−eh

∫

dǫ tr

PL[win(ǫ) − wout(ǫ)]PL

, (1.37)

where PL is the projection matrix onto the N modes of the left lead. The factor of e comes

from considering the electric (rather than probability) current, and the 1/h arises because

we normalized the states to carry a probability flux of 1/h per unit energy.2Similarly, the

net right-moving current in the right lead is IR =∫

dǫ tr[PR(win − wout)PR]e/h. Since

current is conserved, we must have IL = IR, which follows immediately from the unitarity

of S.

2The choice of normalization in Eq. 1.22 is not arbitrary but rather comes from considering the 1D densityof states and velocity. That is, the current carried per unit energy by a 1D system is the velocity v(ǫ) timesthe density of states ρ(ǫ). By simple arguments, the density of right-moving states in a spinless 1D system


We then find a total current

I =IL =−eh

∫

dǫ tr

PL[win(ǫ) − Swin(ǫ)S†]

(1.40)

=−eh

∫

dǫ tr

PLΘ(µL − ǫ) − PLS(ǫ)[PLΘ(µL − ǫ) + PRθ(µR − ǫ)]S†

. (1.41)

For ǫ > µL, µR, the integrand is clearly zero. For ǫ < µL, µR, the integrand is zero by

unitarity of S, since PL + PR = 11K . Without loss of generality, let µL ≥ µR, so

I =−eh

∫ µL

µR

dǫ tr[PL − PLS(ǫ)PLS(ǫ)†]. (1.42)

If S(ǫ) is a constant over this range of energy (i.e., in the linear response regime [40]),

I =−eh

(µL − µR)tr(PL − PLSPLS†) (1.43)

I =e2

hVLRtr(PL − PLSPLS

†), (1.44)

where VLR is the voltage between the left and right reservoirs. We then have an expression

for the conductance G = I/VLR, which is the Landauer formula, though not in the way

it is usually written down. For that, we need to express S in terms of transmission and

reflection matrices

S =

r t′

t r′

, (1.45)

where r is the N ×N reflection matrix of the left lead, r′ is the M ×M reflection matrix of

the right lead, t is the M ×N transmission matrix from left to right, and t′ is the N ×M

of free fermions is

ρ(ǫ) =1

2

√

m

2π2~2ǫ(1.38)

=1

2π~v(ǫ), (1.39)

for velocity v. It is thus clear that the velocity cancels out of the current carried per unit energy. Thecurrent per unit energy is then i = ρ(ǫ)v(ǫ) = 1

h, which is the choice made in the main text.


transmission matrix from right to left. By unitarity of S, rr† + t′t′† = 11N . We then see

that Eq. 1.44 is

I =e2

hVLRtr(11N − rr†) (1.46)

=e2

hVLRtr(t′t′†) =

e2

hVLRtr(t†t), (1.47)

which is the usual Landauer formula, giving G = e2

h tr(t†t). The key insight of the Landauer

formula is that the conductance through a device is proportional to the probability that an

electron incident from the left will exit to the right, which is the meaning of tr(t†t).

In the theory of mesoscopic transport, theorists like to consider ideal 1D wires

coupled adiabatically by reflectionless contacts into the scattering region, as used in the

discussion here. In practice, experiments are performed with quantum dots separated from

the large 2DES by a quantum point contact (QPC), which is not a long 1D wire. The

central constriction of the QPC is the closest the experiments generally come to a 1D wire.

But the conductance through that QPC is quantized [149], just as for the ideal 1D wire,

so we apply the theory, with generally good agreement. Differences in predictions for some

conduction features were studied in Refs. [60, 162, 6], among others.

1.4 2DES and quantum dots

The work in this thesis concerns electrons confined into a quasi-two dimensional

world at the interface of two semiconductors, such as GaAs/Al0.3Ga0.7As. Though more

expensive than the standard silicon heterostructures that are the workhorses of modern

computers, the GaAs/AlGaAs system provides a number of advantages. The similarity

of the lattice constants of GaAs and AlGaAs allow the interface to be grown with nearly

atomic perfection. Modulation doping, where the dopants are displaced by tens or hundreds

of nanometers from the interface, allow the creation of low density electron systems at the


interface without strong scattering from the ionized dopants[65]. Such a system with all of

the electrons occupying the ground state wavefunction in the confinement direction is called

a two dimension electron system (2DES). These systems can have long Fermi wavelengths,

on the order of 50 nm, depending on the choice of electron density, and mean free paths on

the order of tens of microns, allowing devices to be built firmly in the ballistic limit, i.e.,

where the device size is smaller than the mean free path.

A metal gate placed on the top of the semiconductor wafer can modulate the

electron density beneath, entirely depleting the electrons with a gate voltage of only a

fraction of a volt. This allows the sculpting of the lateral confinement potential of the

2DES, limiting the electrons to move in large 2D regions, small 1D wires, or confined

regions called quantum dots. Quantum dots containing as few as one electron are made

routinely, and larger quantum dots containing hundreds to thousands of electrons are also

studied. The important feature of a quantum dot is that its length be less than both the

electron mean free path and dephasing length, so the coherent properties of the system

confined to the quantum dot (and not interacting with impurities) can be probed.

Within the tiny world of quantum dots, this thesis considers the largest and the

smallest specimens. In the large, micron-scale quantum dots, the electrons are well-modeled

as non-interacting in the framework of Landau’s Fermi liquid theory [109]. It is in this regime

that we can treat the transport properties of noninteracting electrons using the random

scattering matrices of Section 1.3. In the other limit of small, few-electron quantum dots,

the Coulomb repulsion energy is much larger than the kinetic energy, so we can study single

electrons in their orbital ground states. This situation is ideal for gaining access to the spin

degree of freedom, as we will now elaborate.


1.4.1 Double quantum dots

Here we are interested in a double quantum dot system, that is, two immediately

adjacent quantum dots, as illustrated in Fig. 1.2. We will consider the situation in which

there is one orbital quantum state energetically accessible in each dot. By controlling the

voltages on gates around the edges of the dots, the two-particle ground state can be shifted

from having one electron in each dot to having both electrons in one dot. In the case that

both electrons occupy the single orbital on the right dot, they must form a spin-singlet to

satisfy the Pauli exclusion principle, where we neglect any spin-orbit splitting for simplicity.

The gate-controlled energy splitting between the (1,1) and (0,2) states (where (NL,NR)

indicates the number of electrons on the left, right dots) is ǫ, called the detuning. There are

four allowed (1,1) states, and we use the singlet/triplet basis to describe them. Combined

with the single energetically accessible (0, 2)S state, these five states form the relevant

electronic space for all that follows.

As shown in Fig 1.3a, finite tunnel coupling between the two dots allows hybridiza-

tion of the (0, 2)S and (1, 1)S states. An external magnetic field Bext will split off the triplet

states, giving the energy diagrams shown in Fig. 1.3b. For typical GaAs quantum dots,

the energy splitting to the next orbital states is ∼ 1 meV, and the tunnel coupling γc is

∼ 10 µeV [147]. With the basis (0, 2)S , (1, 1)S , T+, T0, T−, the Hamiltonian can be written

Hc =

−ǫ γc

γc 0

−Ez

0

EZ

, (1.48)

where Ez = |g∗µBBext| is the Zeeman energy, with g∗ the effective g-factor for conduction

band electrons, g∗ = −0.44 in GaAs [155], and µB the Bohr magneton.


2DEG

gate

Ohmiccontact

depletedregion

GaAs

AlGaAs

200 nm

IQP CIQP C

IDOT

a)

b)

Figure 1.2: a) Schematic of a double quantum dot device in GaAs/AlGaAs 2DES, showinggates on the top of the chip which can deplete the electron gas beneath them. b) SEMmicrograph of a double quantum dot, showing the locations of the two dots. Figure indicatesa device with current passing through it, unlike those discussed in this work. Adapted from[64].

0ε

Ene

rgy

(0,2)S

(0,2)S

(1,1)S

(1,1)S

0

0

−Ez

0

Ez

ε

Ene

rgy

(0,2)S

(0,2)S

(1,1)S

(1,1)S

(1,1)T−

(1,1)T0

(1,1)T+

ε

Figure 1.3: a) Energy levels of the lowest lying charge states in the two-electron doublequantum dot, as a function of the detuning ǫ between the (1, 1) and (0, 2) charge states,including tunnel coupling γc between the dots. b) Energy levels, including spin, in thetwo-electron double quantum dot in the presence of an external magnetic field.


The tunneling process conserves spin, so only the (1, 1)S state can tunnel over to

the (0, 2) charge state, since the (0, 2) triplet state requires one of the electrons to occupy

an energetically inaccessible orbital state. This produces the phenomenon of Pauli blockade

[111], in which the system gets stuck in one of the (1, 1) triplet states even in a situation

where (0, 2)S is energetically favorable. Pauli blockade is very useful experimentally, as

it converts the spin information (i.e., whether the state is singlet or triplet) into charge

information (i.e., whether both electrons are in the right dot or one in each). The charge

state of the dots can be measured using an adjacent quantum point contact charge sensor

[53, 75].

It is useful to diagonalize the upper 2 × 2 matrix in Eq 1.48 to find the adiabatic

singlet states. Following Taylor[146], we write them as

∣

∣

∣S⟩

= cos θ |(1, 1)S〉 + sin θ |(0, 2)S〉 (1.49)

∣

∣

∣G⟩

= − sin θ |(1, 1)S〉 + cos θ |(0, 2)S〉 , (1.50)

where

θ = arctan

(

2γc

ǫ−√

4γ2c + ǫ2

)

, (1.51)

and the energies are 12 (−ǫ±

√

ǫ2 + 4γ2c ), with |S〉 the lower energy state [147].

1.4.2 Hyperfine coupling

There are, of course, processes that can provide further off-diagonal matrix ele-

ments in Eq. 1.48. These can include spin-orbit coupling and cotunneling processes, but

the ones considered in this work are from the hyperfine interaction between the electron

and nuclear spins. Not all materials, of course, have nuclear spins, but GaAs is blessed (or

cursed) with an abundance of them, with gallium consisting of two stable isotopes, 69Ga

and 71Ga, and arsenic having just one stable isotope, 75As, all three of which have spin


3/2 nuclei. The conduction band in GaAs is formed from atomic s-orbitals, which have an

overlap with the atomic nuclei. Thus, the electron spin dipole overlaps with the nuclear

dipole, forming the Fermi contact hyperfine interaction between the electron and nuclei.

For a single electron interacting with many nuclear spins, the hyperfine Hamiltonian is [1]

Hhf =v0~2

∑

k,β

A′βδ(~r − ~Rk,β)~S · ~Ik,β, (1.52)

where β is the nuclear species, ~Rk,β is the position of the kth spin of species β, ~r is the

electron position, ~S is the electron spin, ~Ik,β is the kth nuclear spin of species β, A′β is the

hyperfine coupling constant, which depends on the type of nucleus, and v0 is the unit cell

volume (which contains two nuclei in GaAs).

Writing Eq. 1.52 as an effective spin Hamiltonian, taking matrix elements with

the electron spatial wavefunction, gives

Hhf = g∗µB2~S

~· ~Bnuc (1.53)

~Bnuc =v0

g∗µB~

∑

k,β

A′β

∣

∣

∣ψ(~Rk,β)

∣

∣

∣

2~Ik,β. (1.54)

We see that ~Bnuc has the form of a magnetic field coupled to the electron spin, in analogy

with Eq. 1.1. For conduction band electrons, the wavefunction can be written ψ(~r) =

u(~r)f(~r) where u(~r) is a highly oscillatory periodic function on the crystal lattice and f(~r)

is smoothly varying on the scale of the lattice. We let dβ = |u(~Rkβ)|2 and normalize

∫

|f(~r)|2 = v0. The dβ vary because the electron generally has greater weight on the As

sites than the Ga sites. From Ref. [113]

dAs = 98 A−3,

dGa = 58 A−3.

The number of As atoms per unit cell is xAs=1, and x69Ga = 0.6 and x71Ga = 0.4 [113]. We


can rewrite ~Bnuc as

~Bnuc =v0

g∗µB~

∑

k,β

A′βxβdβ

∣

∣

∣f(~Rk)∣

∣

∣

2~Ik,β (1.55)

=∑

k,β

bβ

∣

∣

∣f(~Rk)

∣

∣

∣

2~Ik,β/~. (1.56)

where

bβ =v0g∗µB

A′βxβdβ. (1.57)

Using v0 = a3/4 with a = 5.63 A the GaAs lattice constant defined with eight atoms per

unit cell [101] gives the result [113]

b75As = −1.84 T,

b69Ga = −0.91 T,

b71Ga = −0.78 T,

using g∗ = −0.44. If all of the nuclei are polarized in the z direction, then

~Bnuc =3

2

∑

β

bβ z = −5.3 T z, (1.58)

so the fully polarized nuclear system exerts a 5.3 Tesla magnetic field on the conduction

electrons in GaAs. We can also express the couplings as Aβ ≡ A′βxβdβv0 = g∗µBbβ, which

incorporates the rapidly oscillating part of the wavefunction and abundance of the nuclei, so

it gives the energy scale that couples only to the smooth envelope function of the conduction

electrons. The GaAs energy scales are

A75As = −47 µeV,

A69Ga = −23 µeV,

A71Ga = −20 µeV.


Thus, the typical energy scale for the GaAs hyperfine interaction is Atot = −90 µeV.

The negative sign indicates an anti-ferromagnetic interaction between electron and nuclear

magnetic moments.

This discussion has been general for conduction band electrons, in or out of quan-

tum dots, but has considered only a single electron. In the case of a double quantum dot

with two electrons, there are two electron spins, ~S1 and ~S2. We consider only two orbital

wave functions, ψL/R(~r), with L/R indicating the left/right dot. We can write the Hamil-

tonian in a simplified form by considering the single-particle electron wavefunctions ψ(~r1)

to consist only of the envelope-function (normalized so∫

|ψ(~r)|2 = 1), without the highly

oscillatory Bloch function. In that case, we can use the A coupling constant, and if we

consider only one species of nuclei, we can write the hyperfine Hamiltonian as a sum of

terms like Eq. 1.52.

Hhf = Atotv0~2

∑

k

[

δ(~r1 − ~Rk)~S1 · ~Ik + δ(~r2 − ~Rk)~S2 · ~Ik]

. (1.59)

Since the orbital wave functions in the quantum dots are non-degenerate ground states

of a confining potential, we can take them to be real. The spatial wavefunctions ψR(~r),

ψL(~r) are not necessarily orthogonal. Taking matrix elements of Hhf with the double dot

electronic basis states as in Eq. 1.48, we find

Hhf =1

2√

2

0 00 0 A†

A B

, (1.60)


where

A =

√2[

IRL− − 〈R|L〉 IRR

−]

ILL− − IRR

−

2[

〈R|L〉 IRRz − IRL

z

] √2[IRR

z − ILLz ]

√2[

〈R|L〉 IRR+ − IRL

+

]

IRR+ − ILL

+

B =

√2[ILL

z + IRRz ] ILL

− + IRR− 0

ILL+ + IRR

+ 0 ILL− + IRR

−

0 ILL+ + IRR

+ −√

2[ILLz + IRR

z ]

,

where

~IAB ≡ Atotv0~

∑

k

~IkψA(~Rk)ψB(~Rk), (1.61)

for A, B either R(ight) or L(eft), which is correct to first order in the wavefunction overlap

〈R|L〉 = 〈ψR|ψL〉. We see from the second column of A that transitions from the (1, 1)S to

the three triplet states T+ are mediated by the nuclear difference field ~D = (~ILL − ~IRR)/2.

There are direct transitions from (0, 2)S to the triplet states as long as there is finite overlap

between the wavefunctions. The first entry of A shows that (0, 2)S can transition to T+

either by flipping a spin in the right dot (〈R|L〉 IRR− ) or flipping one in the barrier (IRL

− ).

These overlap terms are generally small, so the usual course is to set 〈R|L〉 = ~IRL = 0,

giving the simpler Hamiltonian

Hhf =1√2

0 0 0 0 0

0 0 D+ −√

2Dz −D−

0 D−√

2Sz S− 0

0 −√

2Dz S+ 0 S−

0 −D+ 0 S+ −√

2Sz

, (1.62)

where ~S = (~ILL + ~IRR)/2. Eq. 1.62 shows that in this limit there is no direct hyperfine

coupling from (0, 2)S to any of the triplets, as the absence of wavefunction overlap 〈R|L〉


means there is no way to move an electron from one dot to the other by a hyperfine

transition. If we include the tunnel coupling γc between left and right, then there are

hyperfine processes for the (0, 2)S state which are second-order in γc/ǫ. We also see that

the difference field ~D couples the singlet states to the triplet states and the sum field ~S

couples the triplet states to each other. The total Hamiltonian in this subspace of 5 states

is the sum of Eqs. 1.48 and 1.62.

S, T0 subspace

In the limit that ǫ < 0, |ǫ/γc| ≫ 1, the states S and T0 come close to degeneracy,

and the Hamiltonian for just that subspace in the basis S, T0 is approximately

HST0=

−J(ǫ) −Dz

−Dz 0

, (1.63)

where the exchange splitting J(ǫ) is the hyperfine-free energy difference between S and

T0. In the model of Eq. 1.48, J(ǫ) = 12(−ǫ +

√

ǫ2 + 4γ2c ) ≈ γ2

c/ |ǫ|. We see from Eq. 1.63

that the S and T0 states are coupled by the difference in the z-components of the effective

magnetic field induced by the nuclei in the left and right dots. Eq. 1.63 has been written

down assuming that the quantization axis for the electron spin is along the external field

~Bext. This is a good approximation as long as Bext ≫ | ~S/g∗µB| ≈ 1.3 mT [124].

The S, T0 subspace has been extensively studied for quantum information ap-

plications [64, 116, 85]. Since both electronic states have zero angular momentum along

the external field, the states are unaffected to leading order by fluctuations of the external

field, which can give them long coherence times. The nuclear field is the dominant source

of dephasing.

Experimentally, the ensemble coherence time T ∗2 is measured [116] by initializing

the system at large ǫ in the (0, 2)S state. Then ǫ is reduced rapidly compared to the


hyperfine field scales but slowly compared to γc, which separates the two electrons, one in

each dot. The goal is to reduce ǫ far enough that J(ǫ) . Dz, so the S, T0 superposition

evolves due to Dz, as indicated by Eq. 1.63. The detuning ǫ is held at this value for a

time τs and then ǫ is ramped quickly back to its original value. An adjacent charge sensor

[53] can detect whether the system made a transition to the T0 state, thus staying in the

(1, 1) charge state due to Pauli blockade. This cycle is repeated many times at each τs,

accumulating a singlet-return probability PS(τs), which shows a Gaussian decay with τs,

with decay constant T ∗2 [116].

The hyperfine field ~D evolves slowly on the timescale on which the electron exper-

iments are performed [146], so the effects of Dz in Eq. 1.63 can be removed by standard

spin-echo techniques [116]. These techniques show that the coherence time T2 of the S, T0

space is greater than 1 µs [116]. If, however, no spin-echo techniques are used, each ele-

ment of the ensemble of measurements is performed with a different value of Dz, so the

Rabi oscillations are lost in ensemble averaging, giving an ensemble coherence time T ∗2 of

approximately 15 ns [124]. If Dz could be reduced, then the S, T0 space would increase

in value as a quantum information resource, as it would not require spin-echo procedures

to maintain coherence, simplifying the necessary pulse sequences.

Dynamic nuclear polarization - the S, T+ subspace

There is a second important cycle in these double quantum dots, which we call

the dynamic nuclear polarization (DNP) cycle. It uses the avoided crossing between S and

T+, i.e., near where EZ ≈ J(ǫ). With EZ large enough that the other states are separated

by a large energy gap, the effective Hamiltonian near this degeneracy in the basis S, T+


is

HST+=

−J(ǫ) D+cos θ√

2

D−cos θ√

2−Ez + Sz

. (1.64)

We define ǫ to be the value of the detuning ǫ such that J(ǫ) = EZ −Sz, as marked in Figure

1.3b.

The contact hyperfine interaction is rotationally invariant and thus conserves total

angular momentum, so flipping an electron spin up on transitioning from S to T+ must

involve lowering a nuclear spin, which is clear from HST+. This controlled flip of the nuclear

spin makes it possible to polarize the nuclei using a pulse sequence similar to the T ∗2 cycle

described above.

There are a few variants on the DNP sequence [117, 124, 55], one of which is

illustrated in Figure 1.4. If the slow sweep were truly adiabatic, this cycle should flip

one nuclear spin each time it is repeated. As the nuclear system polarizes, Sz decreases,

increasing the total effective magnetic field felt by the electrons and thus also increasing ǫ.

For typical single-electron quantum dots, the wavefunction strongly interacts with ≈ 106

nuclei in each dot [146]. Repeating this DNP cycle at 4 MHz for a second or two should

then be able to flip all of the nuclei down. In practice, polarizations of only about 1% are

observed [117, 123, 124]. That is, the shift of ǫ after running the DNP cycle is consistent

with a nuclear field strength of approximately 80 mT, while the fully polarized system would

have a magnetic field of 5.3 T, as in Eq. 1.58 [116]. Similar processes using transport through

the vertical quantum dots rather than a gate-controlled pulse sequence have succeeded in

producing nuclear polarizations of approximately 40% [11]. These nuclear polarizations are,

of course, not static. If the electrons are not in a singlet configuration, the dominant decay

mode is believed to be the electron-mediated nuclear-nuclear coupling [1, 38, 165, 123];

otherwise, the dominant decay mode is the nuclear dipole-dipole coupling, which causes the


Adiabatic nuclear polarization

Rapid Adiabatic Passage

Slow Adiabatic Passage

Prepare Singlet

t=τS

t=0

En

erg

y

ε(0,2)S

0

(0,2)S T0

T+

T-

En

erg

y

ε(0,2)S

0

(0,2)S T0

T+

T-

εS

En

erg

y

ε(0,2)S

0

(0,2)S T0

T+

T-

εF

ε~

Figure 1.4: Schematic of the DNP cycle, adapted from [117]. This figure has the energylevels rotated 45 from those in Figure 1.3, which corresponds to ǫ/2 added to the Hamilto-nian of Eq. 1.48. At top, the system is loaded in the (0, 2)S state. At middle, the detuningǫ is moved rapidly (compared to the magnitude of D−) past ǫ, not allowing transition intothe T+ state. At bottom, ǫ is increased slowly, flipping an electron and nuclear spin ontransitioning from singlet to T+.


polarization to diffuse out from the quantum dots to the surrounding material [114]. The

decay time for the dipole-dipole diffusion is on the order of 10 seconds [123], so it should not

be limiting the polarization in the lateral double dot devices to only 1%. It is possible that

something is sending D− to zero, so the avoided crossing itself closes; that is, the probability

of transitioning to the T+ state decreases as the DNP cycle is repeated, but we do not yet

know. In chapter 6 we present a small piece of the puzzle indicating that such a suppression

of D− may occur in some cases.

The subject of dynamic nuclear polarization became even more interesting when

it was reported that performing the DNP cycle increased the measured T ∗2 by a factor of

70 to about 1 µs, which implies a drastic reduction of Dz [124]. Mike Stopa had an idea

for a force which could cause this, which is detailed in chapter 5. An implication of this

work is that loading in the T+ state and transferring to the S state should produce the

opposite effect, causing a large increase in |Dz|. When Sandra Foletti and Hendrik Bluhm

in Amir Yacoby’s group tried this experiment, they indeed found results consistent with a

large |Dz| [55]. However, they also found a large |Dz| even when executing the usual DNP

cycle. There are more experiments on the subject ongoing as this thesis is being written,

but the interface between the DNP cycle and the nuclear spins is quite rich, as will be

discussed further in chapter 6.

1.5 Organization of thesis

Chapters 2-4 concern spin-orbit coupling in many-electron quantum dots while

chapters 5-6 concern hyperfine interactions between electron and nuclear spins in two elec-

tron double quantum dots. We give a brief overview of each chapter.


1.5.1 Chapter 2

We consider the effects of the cubic Dresselhaus spin-orbit interaction on the con-

ductance fluctuations in transport through a quantum dot with a magnetic field. This topic

follows experiments of just this type performed by Dominik Zumbuhl and collaborators on

a set of four nominally chaotic quantum dots of two different sizes [171, 172]. A random ma-

trix theory explanation of the results was provided by Jan-Hein Cremers, Piet Brouwer, and

Vladimir Fal’ko, which was in good agreement with the experimental results [39]. Cremers’

theory, however, started from the assumption that only the linear Dresselhaus and Rashba

SOI were important. As mentioned in section 1.1, this is generally a good approximation

for sufficiently low density electron systems.

There is, however, reason to believe that the cubic Dresselhaus term should be

anomalously important in quantum dot systems. As noted in Eq. 1.10, the linear spin-orbit

terms appear as a constant non-abelian vector potential, and so can be gauged away to first

order in the inverse spin-orbit lengths, L/λ1,2, where L is the linear size of the quantum

dot [5]. This implies that the relevant comparison for the cubic and linear spin-orbit terms

is not of the order of k2F /⟨

k2z

⟩

, as discussed in section 1.1, but rather λ1,2k2F /L

⟨

k2z

⟩

. For

sufficiently small quantum dots, then, the cubic SOI effects should be more important than

the linear SOI effects.

We perform semiclassical simulations of chaotic quantum dots, modeled as two-

dimensional billiards, to estimate the magnitude of the cubic Dresselhaus contribution to

the conductance fluctuations. We combine these results with Cremers’ RMT formulation

and refit Zumbuhl’s data. The functional form of the fitting functions is unchanged by the

inclusion of the cubic Dresselhaus contribution, but the interpretation of the parameters is

modified to include both the cubic and linear SOI. As a result, the refits cannot improve

on those in the original papers, but an interesting conclusion results nonetheless: the most


widely cited value for the Dresselhaus SOI constant, γ of Eq. 1.4, is incompatible with our

results and the conductance fluctuations measured by Zumbuhl. It turns out that the value

of γ in GaAs is disputed within a factor of three. The most commonly cited value in recent

years, 27.5 eVA3, comes from Ref. [157], but there have been an array of theoretical and

experimental results over the last thirty years. These results are summarized in Table A.1

in appendix A, which has been updated since its original publication [88] to include more

recent results [86, 97]. Our results are consistent with |γ| approximately equal to 10 eVA3

or less, which is supported by a number of experiments, though by no means all. Our

results imply both that γ should be on the lower end of theoretical and experimental results

and that the cubic Dresselhaus SOI can have an anomalously important role for spin-flip

processes in quantum dots.

1.5.2 Chapter 3

The generation of spin-polarized currents without magnetic fields or ferromagnets

is an important goal for the field of spintronics [160], and there have been a number of pro-

posals for specific devices that produce such spin-polarized currents using SOI. We consider

the spin polarization produced by sending a charge current through a chaotic quantum dot

with strong SOI in a two-terminal geometry. The nonintuitive production of spin polariza-

tion from a charge current violates no requirements of TRS, except in one special case. We

rederive a theorem which stipulates that with TRS, no spin polarization can be produced

in the exit lead if it has only one open channel. Using random scattering matrix theory, we

find that having just two open channels in the right lead and one in the left produces an

rms spin polarization of 45%.

We include the effects of dephasing, finite temperature, and finite bias to quantify

how much the spin polarization is reduced. Interestingly, we find that dephasing removes


the restriction that having only one channel forbids spin polarization in the right lead.

Since such chaotic quantum dots have been studied in laboratories around the

world for decades, we believe that the only reason these spin polarizations have not been

observed is that there is, as of now, no good way to measure spin polarizations in small

currents. When such measurement techniques improve, these currents will likely be found

to be generic features of mesoscopic devices with strong spin-orbit interaction.

1.5.3 Chapter 4

In the converse case, it is interesting to consider the effects of sending a spin-

polarized current or a pure spin current through a quantum dot with SOI, as might occur

in a spin field effect transistor [136]. We use a similar random matrix theory and density

matrix formulation as in chapter 3, which allows us to consider all the usual methods for

spin-conductance studies. This formalism is flexible enough, however, to also allow us to

study currents with arbitrary entanglement between the orbital channels, which has not

previously been considered. Currents entangled between the channels of the two leads show

larger fluctuations than currents without such entanglement.

1.5.4 Chapters 5 and 6

We move to double quantum dots with two electrons to study the hyperfine inter-

action with the nuclei. Motivated by experiments whose implications are, as of this writing,

still uncertain, we consider the forces that a dynamic nuclear polarization cycle imposes on

the distribution of Overhauser fields in the double quantum dot. In chapter 5, we study

a model in which the electron wavefunction is taken to be uniform inside a spherical box,

so we can consider each quantum dot as effectively having a single large nuclear magnetic

moment. In that model, we explore feedback between the Overhauser difference field Dz


and the probability that a nuclear spin will flip on the left or right dot during the DNP

cycle. We find that for the standard DNP cycle in GaAs, there is a feedback force in the

direction of reducing |Dz|.

In chapter 6, we consider a more sophisticated semiclassical model for the nuclear

dynamics during the DNP cycle, including a treatment of the Landau-Zener-like sweep

through the |S〉-|T+〉 avoided crossing and an inhomogeneous electron wavefunction. We

find a rich parameter space which can send |Dz| strongly to zero or, alternatively, can send

|Dz| to large values. Exactly how these results relate to the experimental findings is an area

of active research.

1.6 Summary of new results obtained in this research

1.6.1 Chapter 2

We have introduced the cubic Dresselhaus spin-orbit interaction (SOI) for the first

time into calculations of the conductance fluctuations of transport through a quantum dot.

We find that the cubic Dresselhaus effects should be large enough to be inconsistent with

experimental results [171, 172] unless the GaAs Dresselhaus coupling constant γ is about

a third of its most commonly cited value. These results provide evidence, added to that of

several experimental and theoretical works, in favor of a smaller value of γ. Understanding

the value of γ is essential for designing spintronic devices using the SOI in GaAs.

1.6.2 Chapter 3

We show that unpolarized currents incident on quantum dots with spin-orbit cou-

pling will produce spin-polarized currents in most geometries. We quantify the expected

values of spin-polarization using random matrix theory, and found expected spin polariza-


tions up to 45%. We include the effects of dephasing, finite temperature, and bias, each of

which reduces the expected spin polarizations. Such devices could provide useful sources of

spin-polarized currents for spintronic devices without magnetic fields or ferromagnets.

1.6.3 Chapter 4

Here, we extend the results of chapter 3 to arbitrary charge or spin currents inci-

dent on the quantum dot, finding the average and fluctuations of charge and spin currents

produced, with and without time reversal symmetry. This work also details a useful im-

provement in formalism from the standard method of S-matrix based transport calculations.

By considering the input currents using a density matrix, it is possible to consider entangled

currents, which are shown to produce larger fluctuations than incoherent currents. Under-

standing such fluctuations is important for designing spintronic devices using the SOI.

1.6.4 Chapters 5 and 6

Here, we try to understand how nuclear spins can be manipulated by dynamic

nuclear polarization (DNP) with the hyperfine interaction in experiments with two-electron

double quantum dots. In particular, we wish to understand circumstances where Dz, the

difference in z-components of the Overhauser fields produced by nuclei in the two dots,

can increase or decrease after many electronic cycles. Chapter 5 details one effect of the

electronic and nuclear structure which could cause the DNP cycle to reduce Dz. In chapter

6, we develop a more sophisticated model for the evolution of the nuclear spins. We break

the nuclei into groups, each with constant hyperfine coupling to the electrons. This allows

treating each group as having a single large nuclear spin, which we model semiclassically.

When all three components of the total difference nuclear magnetic field, ~D, are

zero, the system is in a “dark state” and does not evolve. We simulate the system numer-


ically and find that for some parameter regimes, the system is attracted to the vicinity of

such fixed points; for other parameters these dark states are only metastable. In the latter

case, the system tends to produce a large Dz. This is the first theoretical work indicating

that large Dz can be produced in this system. The features of the equations of motion that

cause these behaviors are still poorly understood. This work is important for evaluating

whether a quantum computer using GaAs quantum dots can ever be built.

Chapter 2

Cubic Dresselhaus spin-orbit

coupling in 2D electron quantum

dots

Jacob J. Krich and Bertrand I. Halperin

Physics Department, Harvard University, Cambridge, Massachusetts

Abstract

We study effects of the oft-neglected cubic Dresselhaus spin-orbit coupling (i.e.∝ p3) in GaAs/AlGaAs quantum dots. Using a semiclassical billiard model, weestimate the magnitude of the spin-orbit induced avoided crossings in a closedquantum dot in a Zeeman field. Using previous analyses based on random matrixtheory, we calculate corresponding effects on the conductance through an openquantum dot. Combining our results with an experiment on an 8 µm2 quantumdot [D. M. Zumbuhl et al., Phys. Rev. B 72, 081305 (2005)] suggests that 1) the

GaAs Dresselhaus coupling constant γ is approximately 9 eVA3, significantly

less than the commonly cited value of 27.5 eVA3, and 2) the majority of the

spin-flip effects can come from the cubic Dresselhaus term.

Control over electron spin in semiconductors has promise for quantum computing

40

Chapter 2: Cubic Dresselhaus spin-orbitcoupling in 2D electron quantum dots 41

and spintronics. In such applications, it is essential to understand how the transport of

an electron through a circuit affects its spin; i.e., we must understand spin-orbit coupling

(SO). In technologically important III/V semiconductor heterostructures, SO originates in

the asymmetry of the confining potential (called the Rashba term), which can be controlled

by gates, and in bulk inversion asymmetry of the crystal lattice (called the Dresselhaus

term). In quasi-2D systems, the Dresselhaus term has two components, one linear in the

electron momentum and the other cubic. The cubic Dresselhaus term (CD) is usually

neglected, as it is generally smaller than the linear contribution. Datta and Das proposed

a spin-field-effect transistor (SFET) for quasi-1D ballistic wires with Rashba coupling [41].

Schliemann, Egues, and Loss proposed an SFET that can operate in diffusive quasi-2D

systems based on tuning the Rashba and linear Dresselhaus (LD) terms to be equal in

strength, which produces long spin lifetimes, neglecting CD [136]. The strengths of the SO

terms are difficult to measure independently, but a full understanding of their strengths is

crucial to making such devices. Additionally, in confined systems such as quantum dots,

some effects of the linear SO terms are suppressed [63], and it is important to know the

magnitude of the CD contribution, which could limit or even prevent the functioning of

spintronic devices.

We characterize the strength of CD in a confined system by its effect on avoided

crossings in an in-plane magnetic field B‖ that couples only to the electron spin. With no

SO, each eigenstate can be written as a product of orbital state |α〉 and spin quantized

along B‖. Eigenstates |α ↑〉 and |β ↓〉 become degenerate when ǫα − ǫβ = EZ, where ǫα,β

are the orbital energies and EZ is the Zeeman energy,1 but SO leads to avoided crossings.

In the first half of this Letter, we estimate the CD contribution to the avoided

1This neglects other spin-flip processes, e.g., hyperfine coupling, which in GaAs is much smaller than SOeffects for quantum dots on the micron scale.

42Chapter 2: Cubic Dresselhaus spin-orbit

coupling in 2D electron quantum dots

crossings, which can be larger than the linear terms’ contribution since the latter are sup-

pressed for small B‖ [63]. In the second half of this Letter, we relate these avoided crossings

in closed quantum dots to the mean and variance of the conductance when the quantum dot

is connected to ideal leads. We compare these predictions to the results of Zumbuhl et al.

[172, 171] and find that agreement is possible only if the CD coupling constant in GaAs γ

is considerably less than the frequently cited value of 27.5 eVA3

[83, 157] from k ·p theory.

A smaller value of γ has also been suggested by experiments [126, 77] and band structure

studies [132, 133, 31]. Even with this smaller value of γ, we find that CD is the dominant

spin-flip mechanism in the sample considered.

We consider conduction electrons in a 2D electron system (2DES) grown on a (001)

surface of a III/V semiconductor confined to a small area by a potential V (r). We use an

effective Hamiltonian

H =(p− Aso)

2

2m+

γ

2~3(p2

2 − p21)(p × σ) · e3

+ V (r) +1

2gµBB · σ

(2.1)

where p = P − eA/c, P is the canonical momentum, A is the vector potential from the

perpendicular magnetic field, σ is the vector of Pauli matrices, m is the effective mass,

Aso = e1~σ2/2λ1 − e2~σ1/2λ2 is the effective SO vector potential, which contains both

the LD and Rashba SO terms, and λ1,2 are the (linear) SO lengths [5, 39] We choose a

coordinate system with axes e1=[110], e2=[110], and e3=[001]. The second term is the

CD.2

In a system of linear size L, the linear SO terms can be gauged away to first

order in L/λ by the unitary transformation H → UHU † ≡ H′ where U = exp(ir ·Aso) [5].

2We ignore other cubic terms, which are allowed by symmetry, but are estimated to be small. SeeAppendix A.


Expanding to leading order in L/λ,

H′ =1

2m(p − a⊥ − a‖)

2 + bZ + bZ⊥

+γ

2~3(p2

2 − p21)(p × σ) · e3 + V (r), (2.2)

where a⊥ = (~σ3/4λ1λ2)[e3 × r], a‖ = (~/6λ1λ2) × (x1σ1/λ1 + x2σ2/λ2)[e3 × r], bZ =

gµBB · σ/2, and bZ⊥ = −gµB(B1x1/λ1 +B2x2/λ2)σ3/4.

When we apply a Zeeman field, we can treat each induced degeneracy as a two-level

system, assuming the SO matrix elements ǫso are much less than the single spin mean level

spacing, ∆ = 2π~2/mA, with m the conduction band effective mass and A the dot area. The

magnitude of the avoided crossings at the Fermi energy is given by ǫso = |〈α ↑|Hso |β ↓〉|,

where ǫα− ǫβ = EZ and ǫα = EF . We want to find the rms value of ǫso. Following Ref. [63],

for a closed chaotic dot we may write Λ2 ≡ (ǫso/∆)2 as

Λ2 =∑

αβ

|(Hso)α↑,β↓|2 δ(ǫα − ǫβ − EZ)δ(ǫα −EF ), (2.3)

where the overbar indicates ensemble averaging and Oa,b ≡ 〈a|O |b〉.

We rewrite Eq. 2.3 as in Ref. [63] using the t-dependent representation of the delta

function and interaction picture operators, and, after summing over β, find

Λ2 =

∫ ∞

−∞

dt

∆2π~e−iωZ t〈α| Hso

SF(t)HsoSF

†(0) |α〉, (2.4)

where ωZ = EZ/~ is the Zeeman frequency, |α〉 is a typical orbital eigenstate with ǫα ≈ EF,

and HsoSF ≡ 〈↑|Hso |↓〉 is the spin-flip part of Hso. We consider CD alone, Hc = (γ/2~

3)(p22−

p21)(p1σ2 − p2σ1), and estimate its contribution to Λ, which we call Λc.

We estimate Λc semiclassically using a billiard model for the quantum dot, where

the matrix element in Eq. 2.4 is replaced by the corresponding expectation value for a

classical particle moving at the Fermi velocity vF starting at a random point in phase space.

Semiclassical methods using SO have been rigorously justified [14] and used for studying 2D



0 20 40 600

0.2

0.4

0.6

0.8

EZ (µeV)

Λc

[100][110][010]

0 20 40 600

0.2

0.4

0.6

0.8

EZ (µeV)

Λc

[100]

[010]

0 20 40 600

0.2

0.4

0.6

0.8

EZ (µeV)

Λc

0 20 40 600

0.2

0.4

0.6

0.8

EZ (µeV)

Λc

a) b)

c) d)

Figure 2.1: Normalized rms avoided crossing Λc due to cubic Dresselhaus spin-orbit couplingas a function of Zeeman energy EZ for four billiards with in-plane magnetic field along the

indicated directions, with γ = 8.5 eVA3[31]. Insets show the billiard shapes with crystal

axes. Solid (dashed) lines indicate specular (diffuse) boundary conditions. a) has a mixedphase space with small regions of regular trajectories, b) is a stadium billiard, c) is similarto the dot in Ref. [172], with diffuse boundaries to ensure chaos, and d) is a square withdiffuse scattering from the top and bottom and specular scattering from the sides (see text).

electron SO effects [167]. We consider B⊥ = 0 for these simulations. Each of (2 − 3) × 105

such trajectories is followed for an equal amount of time, which is generally about 300

bounces total in the forward and backward directions. Increasing the number of trajectories

or bounces does not change the results. We calculate∫

dte−iωZ (t−t′)HcSF(t)Hc†

SF(t′) for 100

random initial times t′ on each trajectory as a function of ωZ , and their average gives Λ2c

when multiplied by the appropriate prefactors. We add a damping function to the integrand

that sends it smoothly to zero as t approaches the simulation cutoff.

We consider four billiard shapes and, for specificity, choose parameters correspond-

ing to the largest, highest density dot in Ref. [172], with A = 8 µm2 and n = 5.8×1015 m−2.

We use g = 0.44 and m = 0.067me where me is the electron mass. Fig. 2.1(a-c) shows the

resulting Λc(EZ) for three orientations of B‖, with γ = 8.5 eVA3. For other choices, Λc


scales linearly with γ. For our method to be valid, we must have ∆ ≪ EZ ≪ ET , where

ET = ~vF /√A is the Thouless energy. For the case discussed here, ∆ = 0.9 µeV and

ET ≈ 80 µeV.

We can understand the approximate scale of Λc by using a simpler, unphysical

billiard. Consider an Lx ×Ly rectangle with specular reflections from the sides and diffuse

scattering from the top and bottom. At each collision with a diffuse wall, we choose the

tangential momentum from a uniform distribution on [-pF , pF ] [54]. This choice gives a

correct weighting for diffuse scattering and maintains detailed balance. In such a billiard,

for B‖ ‖ x, in the limit EZ → 0,

Λ2c

∣

∣

∣

EZ→0=

γ2

∆2π~7

⟨∫ ∞

−∞dtpyp

2x(t)pyp

2x(0)

⟩

, (2.5)

and we can break each trajectory into segments between collisions with the top/bottom

walls. Along each segment, p2x and py are constant, and the particle takes time t = mLy/ |py|

to move from one end of the segment to the other, so we can rewrite Eq. 2.5 as

Λ2c

∣

∣

∣

EZ→0=γ2mLy

∆2π~7

⟨

|py(0)| p2x(0)

∞∑

n=−∞(−1)np2

x,n

⟩

, (2.6)

which we can evaluate explicitly, since the p2x,i are uncorrelated between segments. The

particle begins moving in a random direction with P(px,0) = π−1(p2F − p2

x,0)−1/2, where P

is the probability density on [−pF , pF ]. Since the diffuse boundaries in this billiard are the

top and bottom, P(px,i6=0) = 1/2pF . We regularize the infinite sum by∑∞

n=−∞(−1)n = 0,

and, noting that ET = ~pF/mLy, we find Λ2c = 4γ2p6

F/(45π2∆ET ~

6). For the parameters

in Fig. 2.1, this gives Λc(EZ → 0) = 0.678. Finite values of EZ are not amenable to such

simple treatment, but simulations of this billiard appear in Fig. 2.1d, where the results for

B‖ ‖ x, shown by the solid trace, approach the analytic prediction for EZ → 0.

Avoided crossings have not yet been directly measured in chaotic dots, but our

calculations can be related to experiments measuring the conductance g through a quantum



dot by Zumbuhl et al. [171, 172]. To make this comparison, we need a connection between

avoided crossings in a closed dot and properties of the dot with leads attached. Cremers et

al., using random matrix theory (RMT), worked out a similar connection for dots with only

LD and Rashba SO [39]. We point out that CD can be added easily into the predictions of

Cremers et al. without changing their expressions for 〈g〉 and var g by reinterpreting one of

their RMT energy scales to include both linear and cubic SO terms. We now elaborate.

In Ref. [39], the chaotic quantum dot is connected to two ideal leads with N ≫ 1

open channels, giving a scattering matrix from the circular orthogonal ensemble. They

treat the magnetic field and SO with a stub model [20] in which the stub has the M ×M

perturbation Hamiltonian, H′RMT, given by association to H′ in Eq. 2.2 (without CD), as

H′RMT =

∆

2π

[

iA0(x11 + a⊥σ3) + ia‖ (A1σ1 + A2σ2)

−b · σ + b⊥Bhσ3)]

, (2.7)

where Ai, i = 0, 1, 2, are real antisymmetric matrices with⟨

trAiATj

⟩

= δijM2, Bh is a

real symmetric matrix with⟨

trB2h

⟩

= M2, M ≫ 1 is the number of channels in the stub,

and x, a⊥, a‖, b, and b⊥ are dimensionless parameters, with x corresponding to B⊥, b

to the Zeeman field, and a⊥, a‖, and b⊥ to the similarly named terms in Eq. 2.2 (without

CD). Dephasing is included by setting Neff = N + 2π~/τφ∆, with τφ the dephasing time.

Expressions are then obtained for 〈g〉 and var g as functions of x, a⊥, a‖, b, b⊥, and Neff

to leading order in 1/Neff [39]. Zumbuhl et al. use these results to fit their data.

Without CD, the correspondence between Eqs. 2.7 and 2.2 gives the following


mapping from physical parameters to RMT parameters:

x2 = πκET /∆(4πΦ/Φ0)2

b = πEZ/∆

a2⊥ = πκET /∆(A/λ2

SO)2 (2.8)

a2‖ = a2

⊥κ′[(L1/λ1)

2 + (L2/λ2)2]

b2⊥ = πκ′′(EZ)2/ET ∆(A/λ2SO)

where Φ is the magnetic flux through the quantum dot, Φ0 = h/e is the flux quantum,

λSO =√λ1λ2, L1,2 are the linear dimensions of the roughly rectangular dot, oriented along

e1,2, and κ, κ′, and κ′′ are geometric factors of order unity [39, 5]. We add CD to this theory

by noting that, as a random matrix, the CD in Eq. 2.2 has the same symmetry as the a‖

term in Eq. 2.7, i.e., it contains only σ1 and σ2 Pauli matrices. By making the simplest

assumption of no correlation between the cubic and linear terms, we include CD in H′RMT

by setting a2‖ = a2

‖,l + a2‖,c, where a‖,l is the Rashba and LD contribution, given by Eq. 2.8,

and a‖,c is the CD contribution. Since H′RMT contains the SO part of the Hamiltonian of

the closed quantum dot, we relate a‖,c to Λc by finding the rms spin-flip matrix element

(with spins quantized along B‖) due to a‖,c, giving a‖,c = 2πΛc. Including the CD term

in H′RMT in this way lifts the constraint that a‖ ≪ a⊥ [5], similar to spatially varying SO

strengths [21].

Zumbuhl et al. observe weak anti-localization (WAL) in only one of the GaAs/AlGaAs

heterostructure quantum dots they study [171, 172], and that dot gives the best defined

values of the RMT parameters; we use it for the discussion of our results. The other dots

do not contradict this discussion. The dot that displays WAL has area A = 8 µm2 and

electron density n = 5.8 × 1015 m−2. The dot has N = 2 and Neff = 13.9 [172].

Zumbuhl et al. measure var g as a function of B‖ with time reversal symmetry



broken by a small B⊥. They fit to the expression of Cremers et al. [39], with a‖ (and all

parameters except κ′′) fixed to the value determined from the 〈g〉 data. We redo the fits to

the var g data, constraining only a‖ ≥ a‖,c, with a‖,c from our simulations, and τφ fixed to

the value determined from 〈g〉. From Fig. 2.1, a typical value of Λc in all our billiard shapes

is 0.4, giving a‖,c = 2.5 (8.1) for γ = 8.5 (27.5) eVA3(recalling that Λc ∝ γ). We find that

a value of a‖,c ≈ 2.5 is compatible with the experimental data. However, if we require that

a‖ ≥ 8.1, the fits to the data become markedly worse (see Appendix A).

Zumbuhl et al. also measure 〈g〉 as a function of B⊥, which they use to determine

a‖, finding a‖ = 3.1 [171, 172, 170]. Since a2‖ = a2

‖,l + a2‖,c, we must have a‖ ≥ a‖,c, so

we conclude that γ = 27.5 eVA3

is inconsistent with these results, while γ = 8.5 eVA3

is

consistent with the data. So both 〈g〉 and var g data indicate that γ should be closer to 9

eVA3than 28 eVA

33. Moreover, even with the smaller value of γ, the CD term gives the

dominant contribution to a‖.

There are only a few other experiments pertaining to the value of γ in GaAs.

The best, most direct study is the Raman scattering in a GaAs/AlGaAs quantum well

by Richards et al. in which they found γ = 11.0 eVA3

[126]; the same group also found

γ = 16.5 eVA3

in a different sample [77]. A recent experimental value of γ = 28 eVA3

from

transport measurements [106] is less direct, includes the CD only as a density-dependent

renormalization of the LD, and assumes the Rashba coupling is independent of gate voltage.

Theoretical work has indicated that γ is smaller in AlGaAs/GaAs heterostructures and

superlattices than it is in bulk GaAs [132, 133, 99], so it is possible that experiments are

not probing the bulk Dresselhaus coupling, though Ref. [31] predicts γ = 8.5 eVA3

in bulk

GaAs. We include in Appendix A a table with experimental and theoretical values of γ in

3Even using Λc = 0.2, the lowest value in Fig. 2.1, gives a‖,c = 4.1 for the larger value of γ, which is still

inconsistent with the experiment.


GaAs.

Strictly speaking, our calculations are not directly applicable to the 〈g〉 data of

Zumbuhl et al., as our calculations assume EZ ≫ ∆, and 〈g〉 is measured with B‖=0. We

do not believe, however, that a‖,c changes significantly as B‖ → 0; similarly, Cremers et al.

consider a‖ to be constant for all B‖ [39]. We believe that a‖,c(B‖ = 0) can be estimated

by simply averaging our results from the different field directions in the limit B‖ → 0.

Our reinterpretation that a2‖ = a2

‖,l + a2‖,c requires, of course, that a‖,l be less than

3.1 in the experiment of Zumbuhl et al. This reduction of a‖,l can be absorbed into the

geometric parameter κ′ (which was set to 1 without fitting in Zumbuhl et al.) without

affecting any of the physical parameters, τφ, λSO, found by Zumbuhl et al.. Reducing κ′ is

reasonable, as Ref. [39] predicts κ′ = 1/3 for a circular diffusive system.

Since ∆ ∝ A−1and ET ∝ A−1/2n1/2, we can see that if the thickness of the 2DES

does not change with density, a‖,l ∝ A7/4n1/4, while a‖,c ∝ Λc ∝ A3/4n5/4. We therefore

expect that CD should be relatively more important in small, high density dots, precisely

the ones likely to be useful for producing an SFET.

In summary, we have used billiard simulations to estimate the effect of the cubic

Dresselhaus term on avoided crossings in a closed chaotic quantum dot. These results are

related to the conductance through a dot with ideal leads attached. The CD plays a strong

and previously ignored role in observed transport properties through quantum dots. Our

calculations suggest that 1) the Dresselhaus SO coupling constant, γ, in GaAs/AlGaAs

heterostructures has a value near 9 eVA3and not the frequently cited value of 27.5 eVA

3,

and 2) even with this smaller value of γ, in the experiments considered the cubic Dresselhaus

term provided the bulk of the spin-flip portion of the SO Hamiltonian, which had previously

been assigned to the effects of linear SO terms. The value of γ in this technologically

important system deserves further study.



Acknowledgments

The authors acknowledge a careful reading by Charlie Marcus and helpful conver-

sations with Dominik Zumbuhl, Hakan Tureci, Mike Stopa, Emmanuel Rashba, Jeff Miller,

Subhaneil Lahiri, Eric Heller, and Hans-Andreas Engel. The work was supported in part by

the Fannie and John Hertz Foundation and NSF grants PHY01-17795 and DMR05-41988.

Chapter 3

Spin polarized current generation

from quantum dots without

magnetic fields

Jacob J. Krich and Bertrand I. Halperin


Abstract

An unpolarized charge current passing through a chaotic quantum dot with spin-

orbit coupling can produce a spin polarized exit current without magnetic fields or ferromag-

nets. We use random matrix theory to estimate the typical spin polarization as a function

of the number of channels in each lead in the limit of large spin-orbit coupling. We find

rms spin polarizations up to 45% with one input channel and two output channels. Finite

temperature and dephasing both suppress the effect, and we include dephasing effects using

a new variation of the third lead model. If there is only one channel in the output lead, no

51

52Chapter 3: Spin polarized current generation

from quantum dots without magnetic fields

spin polarization can be produced, but we show that dephasing lifts this restriction.

3.1 Introduction

The generation and control of spin polarized currents, in particular without mag-

netic fields or ferromagnets, is a major focus of recent experimental and theoretical work.

This includes the spin Hall effect, which produces spin currents transverse to an electric

field in a two-dimensional electron system (2DES) with spin-orbit coupling, with spin ac-

cumulation at the edges [49]. Similarly, the magnetoelectric effect [93, 47, 7] produces a

steady state spin accumulation when an electric field is applied to a 2DES with spin-orbit

coupling. The accumulation can be uniform [148, 67] in the case of uniform Rashba spin-

orbit coupling [26] or at the edges of a channel in either the Rashba model [34, 32, 94] or

with spin-orbit coupling induced by lateral confinement [72, 161]. Experiments have ob-

served current induced spin polarization in n-type 3D samples [78] and in 2D hole systems

[59, 139] with spin polarization estimated to be up to 10% [139]. Further work suggests a

spin polarized current can be produced in quantum wire junctions [80, 61], by a quantum

point contact (QPC) [140, 51], in a carbon nanotube [74], in a ballistic ratchet [135], in a

torsional oscillator [87], in vertical transport through a quantum well [100], or in disordered

mesoscopic systems [46].

Here we show that generating a polarized current from an unpolarized current is

a generic property of scattering through a mesoscopic system with spin-orbit coupling. We

propose using many-electron quantum dots (outside the Coulomb blockade regime) with

spin-orbit coupling to produce partially spin polarized currents without magnetic fields or

ferromagnets. Due to the complicated boundary conditions of the quantum dot, we do

not solve for the spin polarization in terms of any particular spin-orbit coupling model,

Chapter 3: Spin polarized current generationfrom quantum dots without magnetic fields 53

geometry, and contact configuration. We estimate the effect for a ballistic system in the

limit of strong spin-orbit coupling by performing a random matrix theory (RMT) calculation

for the spin polarization, allowing consideration of realistic quantum dot devices robust to

details of shape and contact placement. Finely tuned systems should be able to exceed these

polarizations, but these results provide a useful benchmark for whether a particular tuned

system is better than a generic chaotic one. We use a density matrix formalism throughout,

which allows us to develop straightforwardly a spin-conserving dephasing probe, using a

new variant of the third lead technique for accounting for dephasing. Dephasing and finite

temperature both reduce the expected polarization. Without dephasing, we find that if

there is only one outgoing channel then no spin polarization is possible, which was first

shown simultaneously by Zhai and Xu [169], and Kiselev and Kim [81]. Interestingly, with

dephasing, spin polarization can be produced with only one outgoing channel. The case of

polarized input currents will be discussed elsewhere (see chapter 4).

Analogous calculations have been performed by Bardarson, Adagideli, and Jacquod

in a four-terminal geometry, to study the transverse spin current produced by an applied

charge current [10].

3.2 Setup and symmetry restrictions

We consider non-interacting electrons in a quantum dot with two attached leads

connected to large reservoirs. For any electron current entering from the leads, we can

describe the output state in the leads in terms of the S-matrix of the dot, including any

tunnel barriers between the leads and the dot. We assume negligible spin-orbit coupling in

the leads and consider the lead on the left (right) to have N (M) spin-degenerate channels

at least partially open at the Fermi energy, and let K = N + M . As usual, the channel



Figure 3.1: A special model quantum dot with N = 1 channels in the left lead and M = 2channels in the right lead. A skew scatterer sends all z spins to the top channel and all−z spins to the bottom channel. The shaded area in the bottom channel has Rashba spin-orbit coupling of precisely the strength to rotate a down spin at the Fermi energy to an upspin, thus producing a perfectly spin polarized exit current from any input current, whilerespecting time reversal symmetry.

wavefunctions are normalized so all channels have the same flux. The S-matrix S is a

2K×2K unitary matrix of complex numbers. For spin 1/2 particles with spin-orbit coupling,

however, it is convenient to consider S to be a K×K matrix of 2×2 matrices. We represent

these 2 × 2 matrices using quaternions, where a quaternion q = q(0)112 + i∑3

µ=1 q(µ)σµ,

where σµ are the Pauli matrices and q(µ) ∈ C. We give a brief introduction to quaternions

in Appendix B. The quaternion representation is convenient, as the time reversal operation

for a scattering matrix can simply be written as S → SR, where R gives the quaternion

dual (which takes the transpose and sends q(1,2,3) → −q(1,2,3), see Appendix B) [105]. The

S-matrix of a system with time reversal symmetry (TRS) is self-dual.

If win (wout) is the K × K quaternion density matrix of the incoming (outgo-

ing) current, wout = SwinS†. The density matrix describing the unpolarized incoherent

combination of all N incoming channels is

win =1

2N

11N

0M

(3.1)

That is, win = PL/2N where PL is the projection onto the channels of the left lead. We

choose trwin = 1/2, due to the quaternion trace convention (see Appendix B), so win

represents one incident electron.


The Landauer-Buttiker formula gives the conductance in terms of the S-matrix

[25]. We write the K × K quaternion S-matrix as(

r t′

t r′

)

with r (r′) being the N × N

(M ×M) reflection matrix and t (t′) the M ×N (N ×M) transmission matrix. Then we

write the Landauer-Buttiker formula in units of 2e2/h as

G = tr(tt†), (3.2)

= tr(PRSPLS†),

= 2Ntr(PRSwinS†),

= 2Ntr(PRwout),

where PR is the projection onto the channels of the right lead. Since win is normalized to

represent one input particle entering the system, g = 2tr(PRwout) is the probability for that

particle to exit through the right lead. The conductance is N times this probability, so we

call g the conductance per channel in the left lead.

Similarly, we define a vector spin conductance [169] (i.e., exit spin current divided

by voltage) ~Gs in units of e/2π as

~Gs = tr(~σtt†), (3.3)

= 2Ntr(~σPRwout).

Then

~gs = 2tr(~σPRwout) (3.4)

is the spin conductance per channel in the left lead. Hence, gsµ is the µ-component of the

spin polarization of the exit current times the probability of exiting into the right lead.

Thus, the spin polarization of the current in the right lead is ~p = ~gs/g, with |p| ≤ 1.

We can, of course, construct g, ~gs, and ~p using only the S-matrix and not the

density matrices win and wout. The density matrix approach, however, gives the flexibility



to consider arbitrarily correlated states of incoming current and also to look for arbitrary

correlations in the outgoing current (see chapter 4). We will also use it to straightforwardly

derive a method of accounting for non-magnetic dephasing in a device with spin-orbit cou-

pling. To complete the translation to the standard notation of conductances, we consider

sending up- or down-polarized electrons into a sample and collecting either up- or down-

polarized electrons, giving a conductance matrix [107]

G =

G↑↑ G↑↓

G↓↑ G↓↓

(3.5)

with the total charge conductance beingG = G↑↑+G↑↓+G↓↑+G↓↓. Gσ,σ′ is the conductance

for an input current of spin σ′ and an exit current of spin σ, for σ, σ′ =↑, ↓. We translate

the quaternion representation into the standard notation by noting that the up-polarized

incoherent input current has input density matrix win↑ = 1+σ3

2N PL. The output density matrix

is wout↑ = Swin

↑ S† and the portion representing the output in the right lead is t1+σ3

2N t†. The

Landauer-Buttiker formula gives, in units of 2e2/h,

G↑↑ =Ntr(PR1+σ3

2 wout↑ ),

=tr(1+σ3

2 t1+σ3

2 t†). (3.6)

Similarly,

G↓↑ =tr(1−σ3

2 t1+σ3

2 t†), (3.7)

G↑↓ =tr(1+σ3

2 t1−σ3

2 t†), (3.8)

G↓↓ =tr(1−σ3

2 t1−σ3

2 t†), (3.9)

from which we see that G = tr(tt†), which is the usual Landauer-Buttiker formula [25].

Though there are several proposed spin-orbit coupled systems that demonstrate

spin polarization from unpolarized input, in many cases the effect is subtle [140, 51, 73,


135, 115]. Here we give a simple, idealized thought experiment to show that time reversal

symmetry does not forbid generating a spin polarized current from an unpolarized input

current. Consider a system with N = 1 and M = 2, as illustrated in Fig. 3.1. All input

electrons are incident on a perfect skew scatterer [49], which sends spins quantized in the

+z direction into exit channel 1 and spins quantized in the −z direction into exit channel

2. Exit channel 2 has a region with Rashba spin-orbit coupling [26] which is precisely of

the strength and length necessary to rotate −z spins to +z. Thus, all spins incident from

the left lead exit with their spins up, and the system respects TRS, since skew scattering

and Rashba spin-orbit interaction are each time reversal symmetric.

We illustrate by constructing S explicitly. We can express the scattering matrix

for this thought experiment (up to an overall phase) in the 6× 6 and 3× 3 representations

as

S =

0 0 0 0 0 −10 0 0 1 0 01 0 0 0 0 00 0 0 0 eiθ 00 1 0 0 0 00 0 −eiθ 0 0 0

(3.10)

≡ 1

2

( 0 1−σz −σx−iσy

1+σz 0 eiθ(σx−iσy)

σx+iσy −eiθ(σx−iσy) 0

)

, (3.11)

where θ ∈ [0, 2π) and r and t have been determined by the above description, while the

rest of the matrix is given by TRS and unitarity. The unpolarized input quaternion density

matrix is win =(

1/20

0

)

, giving

wout = SwinS† =1

4

0 0 0

0 1 + σz 0

0 0 1 + σz

, (3.12)

so Eq. 3.4 gives ~gs = ~p = z, as stated above.

We now prove that having at least two channels in the outgoing lead is essential.

That is, for a dot with TRS and K channels in attached leads, if an unpolarized equally



weighted incoherent current is sent intoN = K−1 of the channels, then the spin polarization

in the remaining channel must be zero. This result has been shown before [169, 81], but

the quaternion formalism with density matrices makes it particularly transparent, so we

include the proof here.

We start with

win =1

2N

11N

0

=

11K − PK

2N, (3.13)

where PK is the projection onto the Kth channel. The quaternion scattering matrix satisfies

S = SR since TRS is unbroken, and

wout =SS† − SPKS

†

2N=

11K − SPKS†

2N. (3.14)

Note that S = SR implies both S† = S∗ and Sii ∈ C for i = 1 . . . K (see Eq. B.3).

Using Eq. 3.4, the spin conductance is gsµ = 2i[wout

KK ](µ), where [q](µ) is the µ-

component of the quaternion q. In particular, if woutKK has no quaternion part, then gs

µ = 0.

We have

woutKK =

1 − SKKS∗KK

2N, (3.15)

and SKKS∗KK is real, so wout

KK ∈ R and ~gs = 0. This proof applies with channels that are

fully open or have tunnel barriers, as it requires only that the S-matrix satisfy TRS and

unitarity, which are unchanged by tunnel barriers.

We note further that if K > 2 then 1) the reflected current in any of the K − 1

input channels can be spin polarized, and 2) if the input current goes through less than

K − 1 channels, then the remaining channels can have a spin polarization, as shown in the

example of Fig. 3.1.


Figure 3.2: Generic device with one input channel. Unpolarized current consisting of equalparts spin-up and -down components is incident on a chaotic quantum dot from a singlechannel on the left. The irregularly shaped region in the middle is a quantum dot in thestrong spin-orbit regime. The spin-up incident current has a probability to exit into each ofthe three channels of the right lead with a different spin direction in each channel, indicatedby the direction of the dark arrows. Similarly for the spin-down incident current, indicatedby the light arrows. The exit channels are shown spatially separated, for convenience.There is also a probability to scatter back into the left lead (not shown). The total spincurrent in the exit lead is given by summing the probabilities and directions of the six spinpolarizations shown, which is the meaning of Eq. 3.4 and is indicated by the arrow on theright.


The device illustrated in Fig. 3.1 cannot easily be made, but realistic devices with

spin-orbit coupling will show a similar (albeit weaker) spin current generation. An intuitive

picture for the generation of spin currents from an applied voltage in a realistic device is

to consider the case with N = 1 and M > 1. A current of spin-up electrons incident

from the left has some probability to exit in each of the M channels of the right lead,

each associated with a spin polarization direction; in the strong spin-orbit limit, these spin

polarizations can have arbitrary directions. The same is true for a current of spin-down

electrons incident from the left. As illustrated in Fig. 3.2, a charge current entering from

the left is a combination of these spin-up and spin-down currents. The spin polarization

of the current in the right lead is given by the sum of the spin polarizations of each of the

2M currents in the channels of the right lead, weighted by the probability of the particle to



enter that channel. Despite the presence of time reversal symmetry, there is no requirement

that this polarization sum to zero, as seen explicitly in the example of Figure 3.1. In the

chaotic strong spin-orbit limit, these 2M vectors are (almost) uncorrelated, and their sum

will generically be nonzero. As M or N are increased, however, there are more independent

vectors contributing to the sum, which generally brings the sum closer to zero. We formalize

this intuition using random matrix theory.

We estimate the expected spin polarization in realistic situations by using random

scattering matrix theory. We assume that the mean dwell time τd = 2π~/K∆ of particles

in the dot is much greater than the Thouless time τTh = Ld/vF , where K is the number

of fully open orbital channels attached to the dot, ∆ = 2π~2/mA is the mean orbital level

spacing, m is the effective mass, A is the area of the dot, vF is the Fermi velocity, and Ld

is a typical length scale of the quantum dot. We further assume the strong spin-orbit limit,

where the spin-orbit time τso is much less than τd. Since τd = mA/~K, for a sufficiently

large A, even a material with “weak” spin-orbit coupling will be in the strong spin-orbit

limit. The crossover from weak to strong spin-orbit coupling in chaotic quantum dots has

been studied in the K ≫ 1 limit in the context of adiabatic spin pumping [138].

For dots with strong spin-orbit coupling, we assume that the S-matrix is chosen

from the uniform distribution of unitary matrices subject to TRS, called the circular sym-

plectic ensemble (CSE) [105, 12]. We find the root mean square (rms) magnitude of the

spin conductance on averaging over the CSE, which gives the typical spin conductance mag-

nitude to be expected from chaotic devices. Such an averaging can be realized in practice

by small alterations of the dot shape [171, 172].

By symmetry,⟨

gsµ

⟩

= 0 for µ = 1, 2, 3. Using Eq. 3.4, we evaluate

⟨

(gs)2⟩

= 4⟨

tr(σµPRSwinS†)tr(σµPRSw

inS†)⟩

, (3.16)


where we sum over µ.

We use the technique for averaging over the CSE described by Brouwer and

Beenakker in Section V of Ref. [18]. We need just two generic averages, which we will

use repeatedly. The first is of the form 〈F1(S)〉 =⟨

tr(ASBS†)⟩

, where A and B are con-

stant K ×K quaternion matrices and the average is taken over S chosen from the CSE of

K ×K quaternion self-dual matrices. Then[18]

〈F1〉 =1

2K − 1[2tr(A)tr(B) − tr(ABR)]. (3.17)

The second average we need is 〈F2(S)〉 =⟨

tr(ASBS†)tr(CSDS†)⟩

where A, B, C, D are

constant K ×K quaternion matrices and AB = AD = CB = CD = 0. We find[18]

〈F2〉 =1

Λ

(K − 1)[4trAtrBtrCtrD + tr(AC)tr(BD)]

− [trAtrCtr(BD) + tr(AC)trBtrD]

, (3.18)

where Λ = K(2K − 1)(2K − 3).

Using Eq. 3.18, we find

⟨

(gs)2⟩

= 3M(M − 1)

NΛ, (3.19)

where we used tr(σµPR) = 0, tr(P 2R) = M , trwin = 1/2, and tr

[

(win)2]

= 1/4N . Note that

when M = 1,⟨

(gs)2⟩

= 0, consistent with the general symmetry.

If we are interested in the mean square polarization of the exit current,⟨

p2⟩

=

⟨

(gs)2g−2⟩

, we can approximate it by⟨

(gs)2⟩

/ 〈g〉2. This approximate form is useful for

analytical progress and will be compared to numerical results. Using Eq. 3.17,

〈g〉 =2M

2K − 1, (3.20)

which, combined with Equation 3.19, gives

⟨

p2⟩

≈ 3(M − 1)(2K − 1)

4MNK(2K − 3). (3.21)



1 3 5 70

0.2

0.4

M

<(gs)2>1/2

1 3 5 7M

<p2>1/2

1 3 5 70

0.5

1

M

<g>

N=1

N=2

N=3

N=4

Figure 3.3: Numerical (symbols) and analytical (lines) results for normalized mean con-ductance 〈g〉, rms spin conductance gs, and rms spin polarization p of current exiting achaotic quantum dot with N (M) channels in the entrance (exit) lead. An average over60000 S-matrices from the CSE was performed for each data point. The lines are from Eqs.3.19–3.21.

We study the approximation⟨

(gs)2g−2⟩

≈⟨

(gs)2⟩

/ 〈g〉2 numerically. We choose

a 2K × 2K complex Hermitian matrix from the Gaussian unitary ensemble [105] and find

the unitary matrix U which diagonalizes it. We multiply columns of U by random phases,

map U into a K ×K matrix of quaternions, and construct unitary self-dual S by setting

S = UUR, giving S chosen from the CSE [57].

Figure 3.3 shows the numerical and analytical results, which agree quantitatively

for⟨

g2⟩

and⟨

(gs)2⟩

and qualitatively for⟨

p2⟩

. The largest percentage disagreement for

⟨

p2⟩

is 7%.


1 3 5 70

0.2

0.4

M

<(gs)2>1/2

1 3 5 7M

<p2>1/2

1 3 5 70.5

0.75

1

M

<g>

Nφ=1

Nφ=2

Nφ=3

Figure 3.4: Model with spin-conserving dephasing lead. Numerical (symbols) and analytical(lines) results for normalized mean conductance 〈g〉, rms spin conductance gs, and rms spinpolarization p of current exiting a chaotic quantum dot with N = 1 channel in the entrancelead. M and Nφ are the numbers of channels in the exit and dephasing leads, respectively.An average over 60000 S-matrices from the CSE was performed for each data point. Thelines are from Eqs. 3.28 and 3.29.



3.4 Dephasing

We add dephasing to this setup using the dephasing voltage probe technique [24,

17, 9]. We add a fictitious voltage probe drawing no current with Nφ = 2π~/∆τφ fully open

orbital channels, so the time for escape into the dephasing lead is the dephasing time τφ.

If the dephasing process is spin independent, then it is appropriate to conserve the spin

of the reinjected electrons, and we extend the third lead model to allow this. If, however,

dephasing processes relax the spin, then it is appropriate for the dephasing lead to reinject

unpolarized current, preserving only electron number. We consider both of these models,

with emphasis on the first, as it is new in this work.

In our formulation, we explicitly model reinjection of electrons from the fictitious

voltage lead by modifying win to include incoherent reinjection from the dephasing lead. In

either model of dephasing, the reinjection matches the total charge current absorbed by the

dephasing lead, but distributes the charge current evenly between the channels and removes

the correlations. In the spin-conserving case, the reinjection also preserves the spin current.

Consider ηµ = tr(σµPφSwin0 S

†), where µ = 0, 1, 2, 3, σ0 = 112, Pφ is the projection operator

onto the dephasing lead’s channels, and win0 is the input density matrix. Then 2η0 is the

probability for a particle to enter the dephasing lead, and 2~η is the spin conductance into

the dephasing lead, which is proportional to the spin current into the dephasing lead.

We reinject from the dephasing lead with

wφ1 =

0K

c1µσµ11Nφ

= c1µσµPφ, (3.22)

where we sum over repeated index µ. To preserve both spin and charge currents, we set

c1µ = ηµ/Nφ. Some of this reinjected current reflects back into the dephasing lead, so it

must be reinjected again. We define a 4 × 4 complex matrix Ξµν = tr(σνPφSσµPφS†),

which gives the charge/spin current in the dephasing lead due to this reinjection. Defining


win = win0 + win

φ , this procedure gives

winφ =

∞∑

n=1

wφn

= Pφσµtr(σνPφSwin0 S

†)∞∑

n=1

(Ξn−1)µν

Nnφ

= Pφσµtr(σνPφSwin0 S

†)(Nφδµν − Ξµν)−1, (3.23)

where we sum over repeated indices µ, ν = 0, 1, 2, 3. This result holds for any input current,

not just the unpolarized incoherent win0 discussed here.

We approximate winφ by replacing Ξµν with its average in Eq. 3.23, similar to Eq.

3.21. Using Eq. 3.17,

〈Ξµν〉 =Nφ

2Kφ − 1[2(Nφ − 1)δµ0δν0 + δµν ], (3.24)

where Kφ = N +M +Nφ. We further replace tr(σνPφSwin0 S

†) by its average,

⟨

tr(σνPφSwin0 S

†)⟩

=δν0Nφ

2Kφ − 1, (3.25)

which gives

win ≈ win0 + Pφ/2K. (3.26)

Note that Eq. 3.26 satisfies unitarity only on average; the total probability of exiting either

through the right or left lead equals 1 only on average.

In the case of a spin-relaxing dephasing lead, Eq. 3.23 becomes

winφ = Pφ

tr(PφSwin0 S

†)

Nφ − tr(PφSPφS†). (3.27)

Interestingly, approximating by taking averages separately of the numerator and denomi-

nator gives the same answer as in Eq. 3.26.

As is clear from Eq. 3.26, the approximations of Eqs. 3.24 and 3.25 result in

unpolarized reinjection from the the dephasing lead. Averaging these terms separately



removes the coherence between the injected spin and the reinjected spin, so they separately

average to spinless quantities. As a result, this approximation reproduces the effect of a

spin-relaxing dephasing lead. In fact, we find that Eq. 3.26 matches the exact numerical

results for a spin-relaxing lead better than it matches the exact results for a spin-conserving

lead, which is consistent with this understanding of the approximations in Eqs. 3.24 and

3.25.

Using Eqs. 3.26 and 3.18, we find

⟨

(gs)2⟩

≈ 3M

NΛφ

(

M − 1 +NφM2 +N(M − 1)

K2

)

, (3.28)

where Λφ = Kφ(2Kφ − 1)(2Kφ − 3).

Note that Eq. 3.28 predicts an output spin polarization in all cases with Nφ > 0,

including when there is only one outgoing channel, M = 1. Numerical results for the spin-

conserving dephasing lead, using Eq. 3.23, are shown in Fig. 3.4, and they show an rms spin

conductance in agreement with Eq. 3.28 except for Nφ = M = 1. In the special case of Nφ =

M = 1, an exact treatment shows that ~gs = 0, contrary to Eq. 3.28, even with arbitrary

tunnel barriers between the leads and the sample, as shown in Appendix C. Appendix D

contains explicit examples of dephasing-induced spin polarization in the case M = N = 1,

in both the spin-conserving and spin-relaxing cases. Numerical results for the spin-relaxing

dephasing lead are not shown in Fig. 3.4, but fall very close to the corresponding lines for

the analytic results, even for the case Nφ = M = 1. (A finite spin-polarization is allowed

in this case, with a dephasing lead that relaxes spin.)

It may seem surprising that dephasing can produce spin polarization in the case

M = 1, where none would be produced in the absence of dephasing. Clearly, the presence of

the dephasing lead violates the conditions of the proof in Section 3.2 thatM = 1 implies ~gs =

0, but it is not obvious that dephasing will nonetheless produce polarization. Indeed, a single


channel spin-conserving dephasing lead does not, as shown in Appendix C. Nevertheless,

we may consider whether the production of spin polarization with M = 1 is an artifact of

the particular third lead dephasing models studied here or a real consequence of inelastic

dephasing processes. To further explore this question, we have considered two further

variants of the third lead dephasing model.

One could modify the spin-conserving dephasing model to have Nφ dephasing leads

each with one channel, each separately reinjecting the same charge/spin that it absorbs. In

this model, too, we find that a nonzero ~gs can be produced for Nφ > 1 (results not shown).

Brouwer and Beenakker modified the third lead dephasing model to make dephas-

ing uniform in phase space by placing a tunnel barrier with transparency Γ between the

dephasing lead and the dot, with Γ → 0 and Nφ → ∞ while maintaining ΓNφ = 2π~/∆τφ

[19]. The S-matrix is then not drawn from the CSE, and simple analytical results in the

spin-orbit coupled system are challenging. We have studied the spin-conserving variant of

this model numerically and find that for fixed τφ, it gives qualitatively similar results to

the simpler model described above; in particular it also gives a nonzero spin current when

M = 1 (results not shown).

All variants of third lead dephasing models considered here show dephasing-induced

spin currents with M = 1. Without a microscopic model of dephasing, however, we cannot

rule out the possibility that this effect is an artifact of third lead dephasing models in gen-

eral. This dephasing-induced polarization is worthy of further study and will be important

to consider when designing devices based on the lack of spin polarization with M = 1 [168].

Finally, returning to the single dephasing lead with Γ = 1, we estimate⟨

p2⟩

≈⟨

(gs)2⟩

/ 〈g〉2, as in Sec. 3.3, where we modify 〈g〉 to include the dephasing lead. Using Eqs.



3.17 and 3.26, we have

〈g〉 ≈ 2MKφ

K(2Kφ − 1). (3.29)

We estimate⟨

p2⟩

, using Eqs. 3.28 and 3.29. Comparison of these approximations to nu-

merical evaluations is shown in Fig. 3.4. Again we find that the numerical and analytical

results agree qualitatively, except when Nφ = M = 1.

3.5 Finite bias and temperature

If the source-drain bias V or the temperature T is large enough, the polarization

will be further suppressed by electrons of different energy feeling uncorrelated scattering

matrices. This effectively increases the number of orbital channels, which decreases the

residual polarization. First consider infinitesimal bias at temperature T . Adapting Datta

[40], we take

win(ǫ) = −∂f∂ǫ

1

2N

(

11N0M

)

(3.30)

where f(ǫ) is the Fermi distribution. If the scattering matrix for particles of energy ǫ is

S(ǫ), then wout(ǫ) = S(ǫ)win(ǫ)S†(ǫ). We approximate S(ǫ) as correlated only within energy

intervals of scale given by the level broadening due to escape into the leads ∆′ = ∆K/2

(see Ref. [68] for an equivalent treatment). That is, we take

⟨

Sab(ǫ)S†cd(ǫ

′)⟩

= ∆′δ(ǫ− ǫ′)⟨

Sab(ǫ)S†cd(ǫ)

⟩

, (3.31)

and

⟨

Sab(ǫ)Scd(ǫ′)S†

ef (ǫ)S†gh(ǫ′)

⟩

=⟨

Sab(ǫ)S†ef (ǫ)

⟩⟨

Scd(ǫ′)S†

gh(ǫ′)⟩

(3.32)

+ ∆′δ(ǫ− ǫ′)⟨

Sab(ǫ)Scd(ǫ)S†ef (ǫ)S†

gh(ǫ)⟩

,


which are valid only for T ≫ ∆′, which is often true for chaotic quantum dots. For T ≈ ∆′,

⟨

S(ǫ)S†(ǫ)⟩

can be calculated using the random Hamiltonian method [152].

We need an average over a new function,

h(ǫ, ǫ′) = f ′(ǫ)f ′(ǫ′)tr[AS(ǫ)BS†(ǫ)]tr[CS(ǫ′)DS†(ǫ′)],

where AB = AD = CB = CD = 0 and f ′ = ∂f/∂ǫ. We evaluate the average of h with the

K ×K quaternion matrix S(ǫ) chosen from the CSE along with Eq. 3.32, giving,

∫

dǫdǫ′⟨

h(ǫ, ǫ′)⟩

=4

(2K − 1)2trAtrBtrCtrD

+∆′

Λ

∫

dǫ f ′(ǫ)2

(K − 1)[4trAtrBtrCtrD + trACtrBD] − trAtrCtrBD − trACtrBtrD

=4

(2K − 1)2trAtrBtrCtrD

+∆′

6TΛ

(K − 1)[4trAtrBtrCtrD + trACtrBD] − trAtrCtrBD − trACtrBtrD

. (3.33)

Using Eq. 3.33 in place of Eq. 3.18, we evaluate⟨

(gs)2⟩

as above, which simply

multiplies Eq. 3.19 by ∆′

6T . Also, 〈g〉 is unaffected by temperature, so Eq. 3.21 is also

multiplied by ∆′/6T .

When dephasing and temperature are both included, the level broadening has a

component due to dephasing (effectively due to escape into the dephasing lead) so ∆′ =

∆(K/2 +Nφ/2) [68]. Eq. 3.28 is then multiplied by ∆′/6T , and Eq. 3.29 is unchanged.

If the temperature is small but the source-drain bias Vsd is large compared to ∆′,

then we can repeat this calculation with

win(ǫ) = [Θ(ǫ) − Θ(ǫ− Vsd)]1

2NVsd

(

11N0M

)

, (3.34)

where Θ is the unit step function. Using the equivalent of Eq. 3.33, this multiplies Eqs. 3.19,

3.21, and 3.28 by ∆′/Vsd.



3.6 Discussion

This spin polarization should be able to be produced and detected experimentally.

Even quantum dots in n-type GaAs/AlGaAs heterostructures have been observed to have

sufficiently strong spin-orbit coupling to approach the RMT symplectic limit [171, 172].

If the spin-orbit coupling is not strong enough for the S-matrices of the dot to be drawn

from the CSE, the spin polarization predicted here will be reduced but should still be

present. In a given material with fixed spin-orbit coupling strength, a sufficiently large

quantum dot will be well described by the CSE, with a possible increase in dephasing rate

as the dot size increases. Section 3.5 shows that the rms spin polarization goes down as

√

(∆′/Vsd) ∝ (AVsd)−1/2, so as the dot is made larger to enter the strong spin-orbit limit,

the bias range where the results are observable decreases. Thus, the effects predicted in

this paper are most likely to be observable in a material with inherently strong spin-orbit

coupling, such as p-type III/V heterostructures.

If a measurement technique or application is only sensitive to spin polarization

along a particular axis, then the rms predictions for the µ-component of the polarization and

spin conductance are only√

3 times smaller than the results stated above, since⟨

(gsµ)2⟩

=

⟨

(gs)2⟩

/3 and⟨

p2µ

⟩

=⟨

p2⟩

/3.

We have shown that quantum dots with spin-orbit coupling can generate spin

polarized currents without magnetic fields or ferromagnets, except in the case of only one

outgoing channel, when such a device can only produce a spin current if dephasing is present.

These mesoscopic fluctuations can be large enough to give appreciable spin currents in

devices with a small number of propagating channels. Even if the spin-orbit coupling is

weak, a sufficiently large device will show these effects. Dephasing generally decreases the

spin polarization, except in the case of one outgoing channel, where spin polarization cannot


be produced in the absence of dephasing.

Acknowledgments

We acknowledge helpful discussions with Caio Lewenkopf, Emmanuel Rashba, Ilya

Finkler, Philippe Jacquod, and particularly Ari Turner, who suggested the method of choos-

ing matrices from the CSE. We note the use of the Quaternion Toolbox for MATLAB,

created by S. J. Sangwine and N. Le Bihan. This work was supported in part by the Fannie

and John Hertz Foundation and NSF grants PHY-0646094 and DMR-0541988.

Chapter 4

Fluctuations of spin transport

through chaotic quantum dots with

spin-orbit coupling

Jacob J. Krich


Abstract

As devices to control spin currents using the spin-orbit interaction are proposedand implemented, it is important to understand the fluctuations that spin-orbitcoupling can impose on transmission through a quantum dot. Using randommatrix theory, we estimate the typical scale of transmitted charge and spincurrents when a spin current is injected into a chaotic quantum dot with strongspin-orbit coupling. These results have implications for the functioning of thespin transistor proposed by Schliemann, Egues, and Loss. We use a densitymatrix formalism appropriate for treating arbitrary input currents and indicateits connections to the widely used spin-conductance picture. We further considerthe case of currents entangled between two leads, finding larger fluctuations.

72

Chapter 4: Fluctuations of spin transportthrough chaotic quantum dots with spin-orbit coupling 73

4.1 Introduction

There has been much recent progress in the creation and control of spin currents.

There have been demonstrations and proposals for producing spin-polarized currents both

with [79, 141, 51, 10, 89] and without time reversal symmetry (TRS) [96, 58]. Recent

progress in measuring and controlling the spin-orbit coupling in semiconductor heterostruc-

tures [86, 92, 144] promises to enable a range of spintronic applications relying on the

spin-orbit interaction. As such devices are considered and developed, it is important to

understand the role of coherent mesoscopic fluctuations in these systems. In this paper, we

consider the effects of injecting either a spin-polarized current or a pure spin current into

a two-dimensional ballistic region with strong spin-orbit coupling and consider the scale of

the fluctuations of charge and spin currents transmitted through such a device.

For example, these effects could be important for the Schliemann-Egues-Loss spin

field effect transistor (SFET) proposal [136]. In such a SFET, spin-polarized electrons are

injected into a region (e.g., a diffusive wire or a quantum dot) with spin-orbit coupling.

In the “on” state of the device, the Rashba [26] and k-linear Dresselhaus [44, 157] spin-

orbit couplings are tuned to be equal, and the spin polarization does not decay as the

electrons cross the region, but instead undergoes a controlled rotation [136]. In the “off”

state, the Rashba and k-linear Dresselhaus strengths are tuned to be different, and the spin

polarization is lost while traversing the region due to the random spin rotations experienced

by electrons traversing different trajectories through the dot. Ideally, the on state has a fully

spin-polarized current exiting the device and the off state has no spin polarization in the

exit current. For coherent 2D quantum systems, however, the decay of the spin current in

the off state relies on having a sufficiently large number of channels to average together. In

the 1D limit, with two ideal one-dimensional wires, each having only one propagating mode,

74Chapter 4: Fluctuations of spin transport

through chaotic quantum dots with spin-orbit coupling

a fully spin-polarized current injected into the first wire is a pure state, so the transmitted

current must have a spin pointing in some direction; this fact implies that no reduction

in spin-polarization is possible in the coherent 1D limit. Other limitations to the SFET

proposal have been simulated by Shafir et al. [137].

In this paper we discuss the general problem of coherent propagation of currents

through quantum dots, focusing on the relationship of incident to exit spin-polarization of

the currents. For the case of 2D ballistic chaotic scattering regions with strong spin-orbit

interaction, we use random matrix theory to give analytic results for the expected values of

spin-polarization in the exit currents. Once we can describe the ingoing current in terms of a

density matrix, all of the conclusions will follow. Thus, the problem is generally broken into

two parts: first, find the relevant input density matrix for the system of interest; second,

propagate that density matrix to find the output currents and polarizations. We choose

the density matrix formalism to describe the input currents to the quantum dots, as it

is flexible enough to describe any current in the noninteracting system. As an important

example, we describe how to construct the density matrices representing currents produced

from potentials applied to (possibly spin-split) reservoirs. We go beyond this model and

also consider injection of spin currents entangled between the two leads, finding larger

fluctuations in this case. Similar work in a three-terminal geometry was considered in

Ref. [3]. The case of unpolarized input currents was considered in chapter 3.

4.2 Setup

We consider a quantum dot attached to two ideal leads through quantum point

contacts (QPCs). There are N , M open spin-degenerate channels in the left, right QPCs,

respectively, and we let K = N+M . We take a basis for the propagating states in the ideal


leads normalized to unit flux in each channel, as usual. We consider noninteracting spin

1/2 particles which are coherently scattered by the quantum dot, which we describe using

an S-matrix. Given a density matrix w representing the current into the dot from the K

channels, the output current is described by density matrix wout = SwS†.

With K open channels, the S-matrix S can be represented by a 2K×2K matrix of

complex numbers. In systems with time reversal symmetry, it is convenient to consider S to

be a K×K matrix of 2×2 matrices. Any 2×2 matrix can be written as a linear combination

of the four Pauli matrices, but it is convenient to consider the basis σ0, iσ1, iσ2, iσ3, where

the σi are the Pauli spin matrices. In this basis, a 2 × 2 matrix q = q0σ0 + i~q · ~σ, with

q0, ~q ∈ C, which is also called a quaternion [105]. Then q is defined to have a complex

conjugate q∗ = q0∗σ0 + i~q∗ · ~σ, dual qR = q0σ0 − i~q · ~σ, and Hermitian conjugate q† = qR∗.

The Hermitian conjugate is the same as the standard Hermitian conjugate of a complex

matrix, but the complex conjugate is not the same. For an S-matrix of quaternions, we

define complex conjugate (S∗)ij = (Sij)∗, dual (SR)ij = (Sji)

R, and Hermitian conjugate

S† = SR∗. This representation is convenient because for time reversal invariant systems,

S = SR. The quaternion representation has the standard convention that tr(S) =∑

i S0ii,

which is half of the trace of the equivalent complex matrix.

4.2.1 Constructing w from chemical potentials

Consider for the moment not two leads attached to the dot but K leads, each with

one open channel and connected to its own reservoir with adiabatic, reflectionless contacts.

Modeling the reservoirs as paramagnetic, each reservoir can be spin-split along its own

quantization axis with each spin band separately in equilibrium, having its own chemical

potential µνm, wherem ∈ 1 . . . K labels the channel and ν ∈ 0, x, y, z indicates the charge

and spin potentials [151, 145]. There has been some confusion [52] on the consistency of



defining this chemical potential, so we give an example. If reservoir m is spin-split along axis

x, then µ0m is the average chemical potential in the reservoir, 2µx

m is the chemical potential

difference between spin-up and spin-down electrons quantized along x, and µy,zm = 0. In

general, if the quantization axis is n and the chemical potential difference along that axis

is 2µs, then µi = µs(n · i). Such spin-split chemical potentials can be realized, for example,

by optical excitation in heterostructures, in an environment with inelastic relaxation much

faster than spin relaxation [151, 98, 79].

We assume the leads have negligible spin-orbit coupling and spin relaxation, so

there is a well-defined spin current in the leads. In the absence of inelastic processes, we

can consider the current carried by particles with energy ǫ. For simplicity, we assume the

number of open channels does not vary over the range of ǫ considered here. Then the

particle-currents flowing in from each channel are represented by the quaternion density

matrix

wnm(ǫ) = δnm[f(ǫ− µ0n) − ~σ · ~µs

nf′(ǫ− µ0

n)], (4.1)

where f(ǫ) is the Fermi function at temperature T , and we assume that µsn < max(T,∆),

where ∆ is the mean orbital level spacing in the quantum dot without leads attached, and

the prime indicates the derivative with respect to ǫ.

The charge current in the nth channel of particles with energy ǫ is

j0n(ǫ) = 2trPn[w(ǫ) − wout(ǫ)] eh, (4.2)

where −e is the electron charge, h is Planck’s constant, and Pn is the projection matrix

onto the nth channel (i.e., (Pn)ab = δanδbn). Similarly, the spin-current in the nth channel

is

jin(ǫ) = 2trPnσi[w(ǫ) − wout(ǫ)] e

2π. (4.3)


We choose units in which e = h = 2π, so Eq. 4.3 can describe both charge and spin currents

if we let σ0 be the identity.

The currents are the physical objects in the system, and we note that the currents

are unaffected by adding any multiple of the identity to w(ǫ), since wout = SwS† and S is

unitary. We can thus use the density matrix to represent the currents, but we do not need

to maintain trw = 1 or even that w has positive eigenvalues. In the case where there are

only two leads, we can subtract f(ǫ− µ02) from w, giving

w(ǫ) =

[

f(

ǫ− µ01

)

− f(

ǫ− µ02

)]

− f ′(

ǫ− µ01

)

~σ · ~µs1

−f ′(

ǫ− µ02

)

~σ · ~µs2

(4.4)

≈

−f ′(

ǫ− µ0)

δµ0 − f ′(

ǫ− µ01

)

~σ · ~µs1

−f ′(

ǫ− µ02

)

~σ · ~µs2

, (4.5)

where µ0 = (µ01 +µ0

2)/2 and δµ0 = µ01 −µ0

2. Note that if δµ0 = 0 then the average chemical

potential in both leads is the same, so no net charge flows and w is traceless.

If we consider an energy range in which the S-matrix does not vary (i.e., the

linear response regime [40], where δµν < T,∆), then we can represent the currents by

integrating over energy in the density matrix, giving

w =

δµ0 + ~σ · ~µs1

~σ · ~µs2

(4.6)

and

jνn = 2tr[Pnσν(w − wout)]. (4.7)

Spin-polarized injection from ferromagnetic contacts does not immediately map

onto the chemical potential formalism. It is clear that if a ferromagnet is in equilibrium

with a wire, connected by adiabatic contacts, it will not produce a spin current in the wire,

since adiabaticity requires that the lowest energy levels remain filled. For practical injection



of spin-polarized currents from a ferromagnet to a normal metal system, a tunnel barrier

at the contact is the most common form of non-adiabaticity [122, 96].

We can consider a situation where the ferromagnet injects into a semiconductor,

which serves as the reservoir for a wire connected to our quantum dot. If we consider the

case where the semiconductor has an energy relaxation time τe much shorter than the spin

relaxation time τs, then the spin-polarized current injected from the ferromagnet into the

reservoir can relax to two independent distributions with a spin-split chemical potential.

This is the same assumption used for optical excitation of spin-split chemical potentials.

We can then use the formulation in terms of potentials as described above.

The tunnel barrier at the ferromagnet introduces a second complication, as it

implies that the ingoing current in the wire contains particles injected directly from the

reservoir and also particles reflected from the scattering region and reflected back from the

barrier. The input density matrix thus needs to be determined self-consistently, including

the effects of both reflections. Such effects can be included systematically, by using the

Poisson kernel [12] rather than the circular ensemble described below and also including the

TRS-breaking effects of the ferromagnetic scattering. For a sufficiently large reservoir in

the semiconductor, this reflection can represent a small perturbation to the input currents,

and the procedure described below will be a good approximation.

4.2.2 Connection to spin conductances

We can write a generalized Buttiker-type conductance equation [23]

jνl =∑

k,ρ

Gνρlk µ

ρk − 2Mlµ

νl , (4.8)

where Gνρlk is the conductance from lead k to lead l and spin ρ to ν and 2Ml is the number of

modes, including spin, in lead l. The absence of equilibrium charge or spin currents (since


there is no spin-orbit coupling in the leads) implies

∑

k

Gν0lk − 2Ml = 0. (4.9)

Further, the conservation of charge current implies that

∑

l

G0νlk = 2Mkδν0. (4.10)

Specializing to the case of two leads withN andM modes in the left and right leads

with potentials µνL, µν

R, respectively, we can express Gνρlk simply in terms of the S-matrix.

Setting µνR = 0 and µν

L = δνα, Eq. 4.8 gives

jνR = GναRL. (4.11)

Eq. 4.6 says that w =(

σα11N0M

)

= σαPL, and by Eq. 4.7 we have jνR = 2tr(σνPRSσαPLS

†) =

GναRL. Similarly, Gνα

RR = 2tr(σνPRSσαPRS

†). If the system is time reversal invariant, then

S = SR, which imposes some relations between the different conductance matrix elements.

Since tr(AR) = tr(A), we have the Onsager-like relations

Gνρlk = hνhρGρν

kl , (4.12)

for k, l = R,L where hν = (1,−1,−1,−1).

We thus see that we can express all of the Gνρij in terms of traces over appropriate

density matrices multiplying S-matrices. We will consider the current in the right lead

associated with the input density matrix w, defined as

jνw ≡ 2tr[σνPR(SwS† − w)], (4.13)

which is proportional to the outgoing current in the right lead after injection represented

by w, where the sign is chosen so that outgoing currents to the right are positive.



4.2.3 Purity of w

We will see that the coherence properties of the currents are important, so it is

interesting to consider when w represents a pure state. Ordinarily, density matrices are

defined (with quaternion trace convention) so 2trρ = 1, and ρ is pure if ρ2 = ρ. In our

open system, normalization is a choice, and we set 2trw = t, where t gives the total current

incident on the dot. We can also add any multiple of the identity to w without affecting the

physical currents. Taking both these factors into account, w represents a pure state only if

there is a real number α such that

(

w − α11

2tr(w − α11)

)2

=w − α11

2tr(w − α11). (4.14)

This condition implies that the K ×K quaternion matrix w represents a pure state only if

1. w2 = tw,

2. w2 = −t2K−1w, or

3. w is invertible and ∃α ∈ R such that w−1 = w−[t−2α(K−1)]11−α[t−α(2K−1)] .


Though for any particular quantum dot it is difficult to determine the full scat-

tering matrix exactly, if there is a small number of open channels in the leads connected

to the dot, mesoscopic fluctuations should produce an appreciable spin polarization in the

exit current. We can understand this by considering that the current from one of the input

channels has some probability to exit into each of the M exit channels after undergoing

some spin rotation. In the chaotic strong spin-orbit limit, there is no correlation between

the entry and transmitted spin polarizations. Though on average the transmitted spin po-

larization is zero, in any particular case there will still be some residual polarization in some


direction in the exit lead. When there is only a small number of channels in the entrance

and exit, these residual polarizations can be large. We will find the root mean square spin

currents in the right lead by averaging over the ensemble of coherent cavities with strong

spin-orbit coupling. These fluctuations are due to mesoscopic interference effects inside the

quantum dots.We are primarily interested in the time-reversal invariant case, but we will

present results valid with and without TRS.

We consider coherent elastic scattering of noninteracting electrons with no spin-

relaxation in the leads. We consider the chaotic limit for the quantum dot, in which the

electron dwell time τd = 2π~/K∆ is much longer than the Thouless time τTh = Ld/vF ,

where Ld is a typical linear distance across the dot, vF is the Fermi velocity, and ∆ =

2π~2/mA is the mean orbital level spacing in the quantum dot, with m the effective mass

and A the area of the dot. We further assume the strong spin-orbit limit, where the spin-

orbit time τso is much less than τd. We assume that all of the channels have perfect coupling

into the quantum dot.

We are interested in the properties of the current in the right lead. For an input

density matrix w, in addition to jνw, we define the outgoing current

jνout = 2tr(

σνPRSwS†)

, (4.15)

and the current due only to the input state

jνin = 2tr(σνPRw) (4.16)

so jνw = jνout − jνin. The charge current is j0w and the spin current is ~jw. We define jsw =∣

∣

∣

~jw

∣

∣

∣. The polarization of the current in the right lead is ~pw = ~jw/j0w. A small number

of parameters of the input current are sufficient to describe the effects of any w in a two-



terminal configuration. In particular, we define

t = 2trw (4.17)

C = 2tr(w2) (4.18)

Dν = 2tr(σνPRwR) = (jνin)R (4.19)

Eν = 2tr(σνPRwRwR) (4.20)

F ν = 2tr(σνPRwRPRσ

νwR), (4.21)

where superscript R is the quaternion dual, t is the total flux incident on the dot, C is a

measure of the coherence of the current, Dν gives the incident charge/spin current from the

right lead, and Eν and F ν are more measures of coherence. By adding a multiple of 11 to

w, we can choose D0 = 2tr(PRw) = 0, and all results below assume this choice. Note that

if current is incident only from the left lead, then Dν = Eν = F ν = 0.

We take averages over the uniform ensemble of all S-matrices in the strong spin-

orbit limit, either with TRS (called the circular symplectic ensemble – CSE) or without

TRS (called the circular unitary ensemble – CUE) [105, 12]. Such averaging is readily

performed experimentally by small changes of the shape of a quantum dot [171]; the root

mean square (rms) fluctuations also give a typical value to be expected for any one chaotic

dot. An external magnetic field can easily break TRS, moving between these ensembles.

A convenient formalism for performing such averages was worked out by Brouwer and

Beenakker [18]. From that work, we need two averages. In the quaternion representation,

for f1 = tr(ASBS†) for A,B constant K ×K quaternion matrices,

〈f1〉CSE =1

2K − 1[2trAtrB − tr(ARB)] (4.22)

〈f1〉CUE =1

KtrAtrB. (4.23)

The other average we need is of f2 = tr(ASBS†)tr(ASBS†) for A, B constant


K ×K quaternion matrices. We find [18]

〈f2〉CSE =1

2ΛS

(

K − 1

8[trA]2[trB]2 + 2tr[A2]tr[B2] + 4[tr(ABR)]2

− 8tr[A]tr[B]tr[ABR] − 2tr[AABRBR]

−

2[trA]2tr[B2] + tr[A2][trB]2 − 4tr[A]tr[ARB2] − 4tr[B]tr[A2BR]

+ 4tr[A]tr[B]tr[ABR] + tr[ABRABR] + tr[AABRBR]

)

(4.24)

〈f2〉CUE =1

ΛU

[

4K(trA)2(trB)2 +Ktr(A2)tr(B2) − tr(A2)(trB)2 − (trA)2tr(B2)]

, (4.25)

where ΛS = K(2K − 1)(2K − 3) and ΛU = K(4K2 − 1).

We consider the mean and fluctuations of jνw. Using Eq. 4.22,

〈jνw〉 = 〈jνout〉 − jνin (4.26)

〈jνout〉 = δν02tM

2K − δS− δS

Dν

2K − 1, (4.27)

where δS = 1 for averages over the CSE and δS = 0 for averages over the CUE. The relevant

fluctuations to study are of ∆jνw = jνw − 〈jνw〉, which satisfy

⟨

(∆jνw)2⟩

=⟨

jνout2⟩− 〈jνout〉2 (4.28)

Using Eqs. 4.24 and 4.25, we find

⟨

jνout2⟩

=1

Λ

Mδν0

[

4Mt2(K − δS) − 2MC + 4δSE0]

−Mt2 + (K − δS)(2MC + 2δSDν2) (4.29)

− δS [E0(2K − 1) − F ν ]

,

where Λ = ΛS , ΛU for the CSE, CUE, respectively. We note that⟨

jνout2⟩

CUEdoes not

depend on D, E, or F . Combining this result with Eq. 4.27,

⟨

jνout2⟩

− 〈jνout〉2 =1

Λ

Mδν0

[

Mt2(1 + δS)

K − δS/2− 2MC + 4δSE

0

]

+ 2MC(K − δS) (4.30)

−Mt2 + δS [(Dν)22K2 − 3K + 2

2K − 1− E0(2K − 1) − F ν ]



Eq. 4.30 is the main result of this work, and we will now look at its implications in

some special cases. First, an arbitrarily polarized current incident from the left lead, as can

be readily created by optical methods. Second, a pure spin current uniformly distributed

between the leads. Third, a pure state pure spin current, with entanglement between the

currents incident from each lead.

Case 1: Spin-polarized current

For any current incident exclusively from the left, the total current t and the

parameter C are sufficient to describe mean and rms currents in the right lead. We consider

the input current represented by

w1 =1

2N

11N (tσ0 + ~s · ~σ)

0M

(4.31)

where ~s is the polarization magnitude and direction of the input spin current. Note that t

can be positive, negative, or zero, depending on the direction of the charge current through

the device. For |~s| = |t|, the current is fully polarized.

For the density matrix of Eq. 4.31, C = (t2 + s2)/2N , and D = E = F = 0.

Applying Eq. 4.27, the mean spin current in the right lead is zero and the average charge

current is⟨

j0w⟩

= 2tM/(2K − δS). The reduction of⟨

j0w⟩

as TRS is broken (δS → 0) is the

signature of weak antilocalization [12, 13, 29]. The rms spin current in the right lead is

⟨

jsw2⟩

=3M [(M − δS)t2 + (K − δS)s2]

NΛ. (4.32)

The fluctuations in the charge current are

⟨

∆j0w2⟩

=M

Λ

[4MN − δS(4M − 1/N)]t2 + (1 − δS/N)s2

(4.33)

In the case of an unpolarized charge current, s = 0, with TRS, spin current in the

exit lead is forbidden when M = 1 due to the combined effects of time reversal symmetry


2 4 6 8

<(jws )2>1/2

M2 4 6 8

<(pws )2>1/2

M2 4 6 8

0

0.2

0.4

0.6

0.8

1

<jw0 >

M

CUEN=2

N=1N=2N=3N=4

Figure 4.1: For the fully spin-polarized current represented by Eq. 4.31 with t = s = 1 andtime reversal symmetry, comparison of numerical (symbols) and analytical (lines) resultsfor the mean charge current (left), rms spin current (middle) and rms spin polarization(right) in the exit lead, where M (N) is the number of channels in the exit (entrance) lead.An average over 50 000 S-matrices from the CSE was performed for each data point. Thelines are from Eqs. 4.26, 4.32, and 4.35. The right panel shows that the expected spinpolarization in the exit lead is still appreciable, even for several open modes in each of theleads. Also shown are the equivalent CUE results with N = 2, showing that the rms spinpolarization is nearly unchanged by breaking TRS in this case.



and unitarity [89, 81, 169], as can be seen in Eq. 4.32. We can consider a pure spin current

incident from the left by setting t = 0. In that case, we see that

⟨

∆j0w2⟩

=M(N − δS)s2

NΛ, (4.34)

showing the scale of charge currents produced from the pure spin current. Similar effects

have recently been proposed to measure the spin conductance in a three-terminal geometry

[3]. We note that 〈j0w2〉CSE = 0 if N = 1, showing that a pure spin-current incident from

a single channel cannot produce a net charge current in the other channels. This is the

time reversed statement of the theorem that with TRS a charge current cannot produce a

spin-polarized current when M = 1.

We can further consider the spin-polarization of the exit current, ~pw = ~jw/j0w. It

is clear that 〈~pw〉 = 0, just as 〈~jw〉 = 0, but there is some rms spin polarization of the exit

current. If we approximate⟨

p2w

⟩

≈⟨

jsw2⟩

/⟨

j0w⟩2

, we can use the above results to find

⟨

p2w

⟩

≈ 3(K − δS/2)2 t

2(M − δS) + s2(K − δS)

Λt2MN. (4.35)

To test this approximation, we found⟨

p2w

⟩

by numerically averaging over the CSE. Matrices

drawn from the CSE were chosen by diagonalizing matrices from the Gaussian Unitary

Ensemble, as described in Ref. [89]. Results are shown in Fig. 4.1 for the case t = s = 1,

and it is clear that Eq. 4.35 agrees very well with the numerical results (right panel). The

case shown in the figure is the relevant one for the Schliemann-Egues-Loss SFET, in which

a fully polarized spin current is incident from one lead. In the off state, which relies on large

spin-orbit coupling, the spin polarization in the exit lead is supposed to be zero. We see in

Fig. 4.1 that even for several open channels in each lead, we expect to find an appreciable

spin polarization in the output, limiting off-state function.


2 4 6 80

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

<(jw0 )2>1/2

M2 4 6 8

<(∆ jws )2>1/2

M2 4 6 8

0

0.2

0.4

0.6

0.8

1

<jwz >

M

CUEN=2

N=1N=2N=3N=4

Figure 4.2: For the pure spin current represented by Eq. 4.36, comparison of numerical(symbols) and analytical (lines) results for the rms charge current (left), mean spin current(middle) and rms spin current fluctuations (right) in the exit lead, where M (N) is thenumber of channels in the exit (entrance) lead. An average over 50 000 S-matrices fromthe CSE was performed for each data point. The lines are from Eqs. 4.37–4.40. The leftpanel shows that this pure spin current should still be expected to produce significant chargecurrents, with a nonmonotonic dependence on the number of open channels N and M . Alsoshown are the equivalent CUE results with N = 2.



Case 2: Pure spin current from both leads

We consider a pure spin current incident from both leads, represented by the

density matrix

w2 =

11Nσz

2N

−11Mσz

2M

. (4.36)

This density matrix represents a spin current of +z incident from the left and a spin current

of −z incident from the right, which together are an incident pure spin current from left

to right with polarization +z. In this case, t = 0, C = K/2MN , Dν = (0, 0, 0, 1), Eν =

(1/2M, 0, 0, 0), and F ν = (1,−1,−1, 1)/2M . Though the mean value of the charge current

is zero, since it is as likely for the charge current to flow in as out, the spin current can

produce a mean square charge current

⟨

j0w2⟩

=K

Λ

[

1 − δSN2 +M2

MNK

]

. (4.37)

We note that when M = N = 1,⟨

j20⟩

CSE= 0, showing that no charge current can be pro-

duced. This result is another implication of the theorem that, with time reversal symmetry,

a spin current incident in one channel cannot produce a charge current, combined with a

simpler result that coherence and time reversal symmetry forbid spin-to-charge reflection

in a single channel.

The spin current in the right lead is a combination of the incident spin current,

the reflected spin current from the right and the transmitted spin current from the left.

Together, these give a mean spin current of

⟨

jiw⟩

= (0, 0, 1 − δS1

2K − 1). (4.38)

Thus, with TRS, the spin current in the right lead is, on average, reduced from 1. In the

case M = N = 1, this reduction removes 1/3 of the spin current that began in the lead.


The fluctuations around the mean are

⟨

∆jx,yw

2⟩

=K(K − δS)

(

1 − δSN

MK

)

NΛ(4.39)

⟨

∆jzw2⟩ =

K

(K − δS/2)NΛ

K2 + δS

[

(M − 1)K2

M−K(M + 2 − 1

2M) +

3M

2+N

K

]

(4.40)

These results, along with confirming numerical simulations, are shown in Fig. 4.2.

Case 3: Pure state pure spin current

We consider entanglement between the currents in the two leads, which is beyond

the standard chemical potential formulation of transport. In particular, consider a pure

state spin current entangled between both leads, rather than the mixed state spin current

of case 2. With M = N = 1, we consider

w3 =1

2

σz σx + iσy

σx − iσy −σz

(4.41)

This state has, as in case 2, a pure spin current +z incident from the left and a pure

spin current −z incident from the right, but the off-diagonal terms of w3 indicate that the

currents are entangled. The density matrix formalism easily allows consideration of such off-

diagonal correlations between the channel currents. The entanglement could be produced

by passing a current through a beamsplitter produced from quantum dots[110, 134, 84],

feeding into the two channels or from spin injection by optical orientation using entangled

photons. The density matrix w3 represents a pure state by condition 3 of section 4.2.3 with

α = −1/2.

In this scenario, t = 0, C = 3, Dν = (0, 0, 0, 1), Eν = (3/2, 0, 0, 1), and F ν =

(1,−1,−1, 1)/2. This should be compared with case 2 in the M = N = 1 limit, which is

the same except C = 1 and Eν = (1/2, 0, 0, 0).



The most significant difference from case 2 is that coherence between the channels

allows a charge current to be produced, even when M = N = 1 with TRS. There is still no

mean charge current, but the rms fluctuations are 〈j0w2〉1/2

CSE = 0.41, 〈j0w2〉1/2

CUE = 0.45. This

rms charge current is much larger than the results of case 2, even away from M = N = 1,

(see Fig. 4.2, left, and Eq. 4.37) indicating that the entangled spin current is better able

to couple into charge current than is the incoherent spin current. We have normalized w3

to have⟨

jiw⟩

= (0, 0, 1 − δS/3), as in case 2. We find fluctuations around the mean of

〈∆jiw2〉1/2

CSE = (0.58, 0.58, 0.62), 〈∆jiw2〉1/2

CUE = (0.63, 0.63, 0.63). With TRS, the fluctuations

are larger along the polarization axis, but not markedly so. The total spin polarization

fluctuations are⟨

∆jsw2⟩1/2

CSE= 1.03,

⟨

∆jsw2⟩1/2

CUE= 1.10 which is larger than the mean

current and equal in scale to the input current jsin, showing that coherence between the

channels significantly enhances the mesoscopic fluctuations; this should be compared with

Fig. 4.2 (right panel). Such large fluctuations entail a significant loss of knowledge of the

quantization axis of the spin current, so the initially z-polarized current can exit polarized

in many directions.

4.4 Discussion

Mesoscopic fluctuations of spin current on passing through a chaotic ballistic quan-

tum dot can produce large fluctuations in spin-polarization, charge currents from pure spin

currents, and spin currents from charge currents [89]. These predictions for mean and

rms currents will be modified by dephasing and effects of the energy dependence of the

S-matrix. Dephasing processes can be readily added to this model using the third lead

method [24, 17, 9], as detailed in chapter 3. Dephasing generally reduces the fluctuations

in charge and spin currents and also removes the symmetry that forbids charge or spin


currents at certain values of M and N with TRS.

If the ingoing current contains particles with energies varying over a large enough

range, the energy dependence of the S-matrix must be considered as well. The S-matrix

is generally correlated on the energy scale of the level broadening of the quantum dot

eigenstates, approximately ∆′ = ∆K/2 + γφ/2, where γφ is the dephasing rate [68, 89].

If the incident particles have energies that differ by a large amount compared to the level

broadening ∆′, as can happen at sufficiently large temperatures or δµν , then the mesoscopic

fluctuations are suppressed, as there are effectively more open channels for particles passing

through the dot.

Mesoscopic fluctuations producing spin polarized exit currents could be important

for operation of a Schliemann-Egues-Loss SFET. To avoid this impact on the off-state polar-

ization, such a device should have many scattering regions in parallel or operate in a regime

with sufficiently large temperature, bias, or dephasing so as to reduce these mesoscopic

effects.

Acknowledgments

We acknowledge useful discussions and a careful reading of this manuscript by Bert

Halperin. We acknowledge helpful conversations with Caio Lewenkopf, Emmanuel Rashba,

Ilya Finkler, and Ari Turner. This work was supported in part by the Fannie and John

Hertz Foundation and NSF grant PHY-0646094.

Chapter 5

Inhomogeneous nuclear spin flips

M. Stopa,1 J. J. Krich,2 and A. Yacoby2

1) Center for Nanoscale Systems, Harvard University, Cambridge, MA2) Physics Department, Harvard University, Cambridge, Massachusetts

Abstract

We discuss a feedback mechanism between electronic states in a double quantumdot and the underlying nuclear spin bath. We analyze two pumping cycles forwhich this feedback provides a force for the Overhauser fields of the two dots toeither equilibrate or diverge. Which of these effects is favored depends on theg-factor and Overhauser coupling constant A of the material. The strength ofthe effect increases with A/Vx, where Vx is the exchange matrix element, andalso increases as the external magnetic field Bext decreases.

5.1 Introduction

Hyperfine interaction with the host nuclei in nanoscale GaAs systems, while rel-

atively weak, can nevertheless limit the electron coherence time and thereby complicate

strategies to implement quantum information and quantum computing schemes in these sys-

tems [75, 112, 158, 95]. Conversely, ever-increasing control of angular momentum transfer

between electrons and nuclei in a range of materials enables numerous applications precisely

92

Chapter 5: Inhomogeneous nuclear spin flips 93

because of the environmental isolation of the nuclear system. These include applications to

quantum information processing employing NMR [150]. From the perspective of fundamen-

tal physics, experiments on few-electron systems with controllable coupling to the nuclear

many-body system uncover a fascinating arena of new phenomena with ramifications for

theoretical physics and engineering [64].

Experiments on double quantum dots with electron number N = 2 have uncovered

and exploited an intriguing phenomenon called the “Pauli blockade” [111] in which two

electrons with parallel spins are forbidden from combining in one dot by the exclusion

principle. In transport or in gate pulsing, even when such a transition becomes energetically

favorable, it can only proceed via a spin “flip-flop” process in which angular momentum

is exchanged with the local nuclei. Repeating the spin transfer, however, modifies the

character of the nuclear spin distribution. One metric for the nuclear state is the difference

between the total Overhauser fields of each dot. These are the effective Zeeman fields

which the electrons experience due to the hyperfine interaction. Several recent experiments

addressed transfer of angular momentum from the electron system to the nuclear bath

through various pumping cycles. One experiment claims that under a specific, repeated

pulsing sequence (see below) [124] the polarizations in the two dots tend to equilibrate; a

phenomenon which has been numerically reproduced [121, 125]. However, another similar

experiment claims to find a large difference induced between the Overhauser fields of the two

dots [55]. The theory which we describe here does not claim to explain either experiment.

Here we describe a force toward either equalizing or inducing differences between

the Overhauser fields in the two dots. The direction of this force depends on the spin of

the initial electron state (i.e. the direction of the electron-nuclear “flip-flop” process) as

well as on the sign of the product of the g-factor g and the Overhauser coupling constants

Aβ, where β is the nuclear species. Assuming GaAs (A69Ga = −23µeV , A71Ga = −20µeV

94 Chapter 5: Inhomogeneous nuclear spin flips

En

erg

y E

(µ

eV

)

0.8

0.6

0.4

0.2

0.0

a,b

,c

0.0 0.5 1.5

detuning ε (mV)

1.3

1.1

c/bΨ

Τ+

ac

b c/b

0

-10

10

ε~

1.0

Figure 5.1: Top: electronic states near the (1,1) to (0,2) stability diagram transition. Cross-ing of Ψ and T+ at ε becomes anti-crossing in presence of transverse Overhauser field gra-dient. Bottom: overlap of the Ψ state with S(0, 2) (a), with |L↑R↓〉 (b) and with |L↓R↑〉(c). Parameters: Bext = 0.2T , γ = 1.2µeV , ∆ = 1000, EC = 0.6meV , Vx = 1µeV .

and A75As = −47µeV , g = −0.44 [113]), we describe two pulse sequences which differ in

the choice of the initial electron state, which consequently have a force tending to cause the

Overhauser fields in the two dots to equilibrate or to diverge.

5.2 Electronic states of the double dot with N=2

We calculate the electronic states of the two electron (N = 2) double dot within

the Hund-Mulliken formalism [22, 103] developed for the hydrogen molecule. We focus on

the regime in the charge stability diagram [116] where the charge states (NL, NR) = (1, 1)

and (0, 2) are close to degeneracy, with NL, NR the numbers of electrons on the left, right

dots. Typically, in this method, eigenstates of total spin, singlets and triplets, are employed

as basis states. However, since we wish to study the inhomogeneous Overhauser effect due

to different effective magnetic fields in the two dots, we choose a basis which diagonalizes,

at the single particle level, the z-component of this inhomogeneous field and in which the


spatial dependence of nuclear spin flips induced by electronic spin “flops” is transparent.

The basis is: ξn ≡ |R↑R↓〉 , |L↑R↓〉 , |L↓R↑〉 , |L↑R↑〉, where L and R indicate the orbital

states of the left and right dot, the arrows denote spin direction.1 Two remaining states of

the Hund-Mulliken model, |L↓R↓〉 and |L↑L↓〉 are not relevant to our analysis, the former

due to high Zeeman energy and the latter being far away in the charge stability diagram.

Note that |R↑R↓〉 is the standard S(0, 2) state and |L↑R↑〉 is the standard T+ state. The

hyperfine Hamiltonian for two electrons is properly written:

Hhf =vA

~2

M∑

m

[δ(r1 − Rm)S1 · Im ⊗ 1 + 1⊗ δ(r2 − Rm)S2 · Im] (5.1)

where ri and Si are operators in the subspace of electron i (first quantized representation)

and m is summed over a total of M nuclei (typically M ∼ 106); and where v is the volume

per nucleus. We assume, for simplicity, a single nuclear species with spin 1/2. Then,

constraining the maximum Overhauser field to be 5.3T [146] leads to an average coupling

constant A = −270µeV . We incorporate the matrix elements of Hhf from Eq. 5.1 in our

basis ξn into the Hund-Mulliken Hamiltonian which gives (upper triangle of Hermitian

matrix shown):

|R↑R↓〉 |L↑R↓〉 |L↓R↑〉 |L↑R↑〉

H =

EC − ε ILRz − IRR

z 〈L|R〉 + γ ILRz − IRR

z 〈L|R〉 + γ IRR+ 〈L|R〉 − ILR

+

ILLz − IRR

z Vx IRR+

IRRz − ILL

z ILL+

IRRz + ILL

z + EZ

,

(5.2)

where ε is the potential “detuning” [116], γ is the tunneling coefficient and EZ ≡ gµBBext

is the Zeeman energy with µB the Bohr magneton and Bext the external magnetic field,

1More traditionally, the linear combinations S(1, 1) ≡ [|L↑R↓〉 − |L↓R↑〉]/√

2 and T0(1, 1) ≡ [|L↑R↓〉 +|L↓R↑〉]/

√2, are employed.


defining the z-direction. We have taken the orbital energies of L and R to be zero for

simplicity. We include only two Coulomb terms: the charging energy EC ≡ VRRRR −

VRLRL and the exchange matrix element Vx ≡ VLRRL.2 Equation 5.2 is written to leading

order in the overlap, 〈L|R〉, of the non-orthogonal single particle basis. Higher order terms

(O(| 〈L|R〉 |2)) occur due to the normalization of the basis states [22]. Note that the matrix

elements of H in this electronic basis remain operators in the Hilbert space of the nuclear

coordinates:3

~Iαβ ≡ vA

2~

M∑

m=1

ψ∗α(Rm)ψβ(Rm)~Im (5.3)

where α, β ∈ L,R. Note that previous researchers have typically ignored the transition

term ILR+ , which we see from Eq. 5.2 can lead to a direct transition between |R↑R↓〉 and

|L↑R↑〉 and causes a spin flip where the two wavefunctions overlap, in this case in the barrier.

While such terms could be experimentally important for large Bext, i.e. where |R↑R↓〉 and

|L↑R↑〉 anti-cross deep in the (0,2) regime, we will, in this paper, concentrate on spin flip-

flop processes occurring entirely in the left or right dot. We therefore consider the simpler

Hamiltonian with the terms proportional to the L−R overlap omitted.

5.3 Nuclear spin flip location

The crucial feature of Eq. 5.2 is that the |L↑R↑〉 state is coupled to |L↑R↓〉 via

a term which flips a nuclear spin in the right dot (IRR+ ) and it is coupled to |L↓R↑〉 by a

term that flips a nuclear spin in the left dot (ILL+ ) . In the absence of flip-flop coupling to

the |L↑R↑〉 state, the upper left 3x3 matrix in Eq. 5.2 (see also yellow highlighted region of

2Coulomb matrix elements are defined in the usual way in our two state basis, α, β, γ, δ ∈ L,R:Vαβγδ ≡

∫ ∫

dr1dr2ψ∗α(r1)ψ

∗β(r2)V (r1, r2)ψγ(r1)ψδ(r2).

3We have also used the identity: S · Im = SzImz + [S−Im+ + S+Im−]/2.


Fig. 5.2) has a ground state, which we denote:

|Ψ〉 = a(ε) |R↑R↓〉 + b(ε) |L↑R↓〉 + c(ε) |L↓R↑〉 . (5.4)

As shown in figure 5.1, at large (positive) ε, |Ψ〉 → |R↑R↓〉 ≡ S(0, 2) and at large negative ε,

|Ψ〉 becomes an unequal superposition of |L↑R↓〉 and |L↓R↑〉. Even when Vx > |〈IRRz −ILL

z 〉|,

the inhomogeneous Overhauser effect will produce a preference for either the |L↑R↓〉 or the

|L↓R↑〉 component of Ψ (see figure 5.1), with the electron down spin preferentially located

on the dot with smaller Iz. In the first electron pulsing sequence which we describe, the

electron state is initialized at large positive ε into |Ψ〉 ≈ S(0, 2) and detuning is swept

approximately adiabatically through the Ψ - |L↑R↑〉 anti-crossing. The position of this anti-

crossing, ε, is determined by the energy of |L↑R↑〉 ≡ T+ (see figure 5.1) which is determined

by Bext. Insofar as b(ε) 6= c(ε), a transition from Ψ to |L↑R↑〉 will preferentially induce a

nuclear spin flip (down) on the side with the larger Iz. This tends to equilibrate the values of

IRRz and ILL

z . In the second pulse sequence the electrons are initialized into |L↑R↑〉 and the

transition occurs into Ψ. The same feedback mechanism now preferentially causes nuclear

spins to flip up, but still on the side with the larger Iz, thus leading to a tendency for

|ILLz − IRR

z | to grow. Both of these sequences can be experimentally implemented [116, 55].

Our further analysis focuses mainly on the first pulse sequence.

Note that the preceding argument depends on the sign of A which in turn depends

on the sign of g and the sign of the effective Overhauser field which, for Ga and As, are

anti-parallel to the nuclear spins [113].

5.4 Nuclear states

To further analyze the Hamiltonian, Eq. 5.2, it is helpful to introduce a simplified

basis for the nuclear states in which all of the nuclei are either in the left or right dot and


all within a given dot interact equally with the electron. In other words, |ψL(r)|2 is taken

as a constant within a spherical “box” of some volume, V. In this model, which we refer

to as the “box model,” the squares of the total angular momenta I2α are conserved, where

~Iα ≡ (vA/V)∑

m∈α~Im, and where α ∈ L,R. Thus, the electrons essentially interact with

two composite nuclear spins, one on the left and one on the right. The nuclear state basis

is IL, IR, ILz, IRz (where Iα(Iα + 1) is the eigenvalue of (~Iαα)2 and Iαz is the eigenvalue

of Iααz ). Finally, for given IL, IR, it is convenient to transform to the basis of ∆ ≡ ILz − IRz

and s ≡ ILz + IRz. In this basis the z-components of the nuclear operators have non-zero

matrix elements on the diagonal blocks, but the raising and lowering operators connect

different (∆, s) subspaces (see Fig. 5.2).

The strength of the narrowing force depends on the ratio r ≡ c/b at ε. This

depends on ∆ and on Bext. For example, smaller Bext results in smaller (or more negative)

ε, where, as shown in Fig. 5.1, the ratio c/b increases (for ∆ > 0). Exactly how large c/b

can get depends on Vx which, in the example of Fig. 5.1, we have set to 1 µeV.4

In Fig. 5.3 we plot the value of r(ε) as a function of Bext for various values of ∆.

The key point is that r(ε) increases monotonically with ∆ (cf. yellow highlighted region

of Fig. 5.2), however it also decreases monotonically with Bext (and hence ε). Interest-

ingly, because the |R↑R↓〉 state is coupled equally to |L↑R↓〉 and |L↓R↑〉, the value of b/c is

independent of γ.

We note that the flip-flop process naturally also depends on the rate at which ε

is swept since, in order to be adiabatic and remain on the lower branch of the Ψ - |L↑R↑〉

anti-crossing the ε variation must be sufficiently slow. More generally, the character of the

state evolution can be examined as a Landau-Zener tunneling problem, as in chapter 6 or

4Self consistent electronic structure calculations show (Stopa, unpublished) that for lateral double quan-tum dots, Vx can range from 250µV to less than 1µV . Here, we have chosen the lower value.


EC-ε

∆

−∆

s+EZ

γ γ

γ*

γ*

Vx

Vx*

EC-ε

∆+1

−∆+1

s-1+EZ

γ γ

γ*

γ*

Vx

Vx*

EC-ε

∆−1

−∆−1

s-1+EZ

γ γ

γ*

γ*

Vx

Vx*

IR-

IL+

IL-

IR+

∆−1,s−1 ∆,s ∆+1,s−10 000

0 000

0 000

0 000

0 000

0 000

0 000

0 000

0 00

0 00

0 00

0 00

0 00

0 00

0 00

0 0

0 0

0 0

0 0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Figure 5.2: Hamiltonian for three sectors of the nuclear difference quantum number (∆ −1, s−1), (∆, s), (∆+1, s−1). In the above, ±∆±1 and s±1 are shorthand for (vA/V)(±∆±1)and (vA/V)(s ± 1) respectively.

else evaluated numerically [142].

The evolution of the full nuclear state is complex and the experimental manifesta-

tions of that evolution are ambiguous. Nevertheless, as a possible baseline for more detailed

studies of the nuclear evolution, we describe a simple, incoherent model which results in

narrowing of the distribution of ∆.

If we assume that the system is in the well-defined state |Ψ〉 ⊗ IL, IR, ILz, IRz

and the detuning is moved quickly to ε and held there for time τ , we can compute, by time

dependent perturbation theory, the probability for a nuclear spin to flip in the right dot as:

ΓR(ILz, IRz → ILz, IRz − 1) ≡ ΓR(s,∆ → s− 1,∆ − 1)

=τ2

~2|〈ILzIRz − 1|IRR

− |ILzIRz〉|2

=A2Ω2

R−τ2

4~2|b|2

(5.5)

where we have suppressed the IL, IR dependence for brevity and where the matrix ele-

ments of the ladder operators are given by the well-known formulas: Ωα± ≡ 〈Iα, Iαz ±

1|I±|Iα, Iαz〉 =√

Iα(Iα + 1) − Iαz(Iαz ± 1). Similarly, the flip probability in the left dot is

proportional to the c component of Ψ

ΓL(s,∆ → s− 1,∆ + 1) =A2Ω2

L−τ2

4~2|c|2. (5.6)


1.10

1.08

1.06

1.04

1.02Ψ−

com

po

ne

nt

rati

o c

/b

1.00.80.60.40.2

Magnetic Field B [T]

∆ = 200

∆ = 600

∆ = 1000

∆ = 1400

∆ = 1800

Figure 5.3: The wave function ratio r ≡ c/b evaluated at the Ψ − T+ crossing point, ε, asa function of Bext for various values of the nuclear spin z-component difference ∆. r(ε) ismonotonically increasing with ∆ and decreasing with Bext.

If we denote the probability distribution for the nuclear state (at fixed IL, IR) as W (s,∆),

then the condition for W to be stable in its dependence on ∆ can be written (cf. Fig. 5.4a):

W (s+ 1,∆ + 1)ΓL(s + 1,∆ + 1) = W (s,∆)ΓR(s,∆)

W (s,∆ + 1) = W (s,∆)Ω2

R−(s,∆)

Ω2L−(s,∆ + 1)

|b(∆)|2|c(∆ + 1)|2

(5.7)

where we have assumed that W (s) ≈ W (s + 1) and we have used the fact that b and c

depend very weakly on s (only through the s-dependence of ε).

Recursion relation Eq. 5.7 can be solved iteratively and the influence of the narrow-

ing force evaluated. In Fig. 5.4 we have plotted W (∆) computed with the ratio ΩR−/ΩL−

set to unity to show only the narrowing from the inhomogeneous Overhauser effect de-

scribed here with the same electronic parameters as in Fig. 5.1, and with IL = IR = 1000;5

including the Ω’s induces more narrowing. For comparison we show the T → ∞ thermal

distribution of ∆, averaged over s, also for IL = IR = 1000. Inset (a) shows the ratio

of the root-mean-square (rms) ∆ in the thermal distribution, σT , to the rms ∆ with the

5We assume Nnuc ∼ 106 nuclei in each dot leading to an average spin ∼√Nnuc ∼ 1000.


8x10-3

6

4

2

0

dist

ribu

tion

W

2000150010005000

7

6

5

4

3

σ Τ/σ

0.60.40.2Bext (T)

∆,s ∆+1,s+1

IRz

ILz

s

∆

∆+1,s-1

ΓL

ΓR

∆=ILz-IRz

Bext=0.05,0.10,...,0.75 T

(a) (b)

high temperature thermal distr.

Figure 5.4: (main) Reduced distribution W (∆, s), calculated from Eq. 5.7 (solid lines), forVx = 1 µeV as a function of ∆ for various Bext = 0.05, 0.10, ..., 0.75 T (lower fields havenarrower W ); and thermal W (∆) (dashed), averaged over s, all with IL = IR = 103. Inset(a) narrowing factor σT /σ(Bext) versus Bext. Inset (b) Illustration of ILz − IRz plane. ∆and s are the diagonal coordinates, with ∆ ≡ ILz − IRz.

narrowing force at varying Bext, σ(Bext). A substantial narrowing of W (∆) results from

the inhomogeneous Overhauser effect.

5.5 Discussion

Experimentally, the |Ψ〉 to |L↑R↑〉 pulse sequence polarizes only about 1% of the

nuclei, even when running sufficient cycles to flip all of the nuclei [116]. This saturation of

the nuclear polarization is still an open problem. A recent article by Yao [164] discusses

a model similar to that described herein. In that paper, no mechanism for stopping the

flip-flop process is proposed when the pumping continues (as it does in experiments) beyond

∼ 105 cycles. In our model, polarization will saturate when both ILz = −IL and IRz = −IR,

implying that s = −IL − IR. However, the resulting distribution of ∆ will then mirror the

difference in the initial distributions of IL and IR, and hence will show no narrowing of


W (∆). Thus our box model can qualitatively explain the narrowing effect or the saturation,

but not both.

We believe that a full understanding of these phenomena depends on the variable

coupling of the electron wave function to different groups of nuclei, so that conservation

of the magnitudes of two spins, ~IL and ~IR, is not required. Such a model with multiple

interacting composite nuclear spins, incorporating the narrowing effect described here as

well as the Landau-Zener tunneling behavior near ε, in some parameter regimes shows the

potential to send |∆| → 0 while reducing the spin flip probability, slowing the growth of total

polarization; for other parameters, |∆| grows large despite the narrowing force described

here, as described in chapter 6.

Acknowledgments

We thank B. I. Halperin, M. Gullans, J. Taylor, M. Lukin, S. Foletti, H. Bluhm,

Y. Tokura, D. Reilly and M. Rudner for valuable conversations. We thank E. Rashba for

a critical reading of the manuscript. We thank the National Nanotechnology Infrastruc-

ture Network Computation Project for computational support. We gratefully acknowledge

support from the Fannie and John Hertz Foundation, NSF grants PIF-0653336 and DMR-

05-41988 and the ARO.

Chapter 6

Preparation of non-equilibrium

nuclear spin states in quantum dots

M. Gullans,∗1 J. J. Krich,∗1 J. M. Taylor,† B. I. Halperin,∗ A. Yacoby,∗

M. Stopa,‡ and M. D. Lukin∗

∗ Department of Physics, Harvard University, Cambridge, MA† Department of Physics, Massachusetts Institute of Technology, Cambridge, MA

† Joint Quantum Institute, University of Maryland and National Institute of Standards andTechnology, College Park, MD

‡ Center for Nanoscale Systems, Harvard University, Cambridge, MA

Abstract

We develop a technique to model nuclear spin dynamics in a double quantum dotsystem undergoing adiabatic pumping. Our model, while semiclassical, allowsto explore a wide range of parameters. In different parameter regimes, we findthe system exhibits either a strong reduction or a large growth in the differenceOverhauser field produced by the nuclei in the two dots.

The study of non-equilibrium dynamics of nuclei in solids has a long history [2]

and has become particularly relevant as nanoscale engineering and improvements in control

1Contributed equally to this work.

103

104 Chapter 6: Preparation of non-equilibrium nuclear spin states in quantum dots

allow to probe mesoscopic collections of nuclear spins [166, 43, 131, 112, 85, 15, 90]. This

control has direct applicability to quantum information science, where nuclear spins are

often a main source of dephasing [64]. The hope and goal of developing an understanding

of electronic control of nuclei is to usher in an era of mesoscopic state engineering, in

which nuclear spins can be programmed to be useful resources [82], as indicated in recent

experiments [124, 55].

Many groups have developed specific theoretical techniques to solve subsets of

this intractably hard problem, including semiclassical solutions for the central spin [33,

4], cluster and diagramatic expansion techniques for short time non-equilibrium behavior

[159, 165, 37], and exact solutions for small systems, relevant in the case of homogenous

coupling to the electron [121, 125, 164, 143]. In this paper, we present a model for nuclear

spin dynamics in a double quantum dot undergoing dynamic nuclear polarization (DNP).

We include the inhomogeneous coupling of nuclear spins to the electrons [35], incorporate

the full electron spin dynamics, and determine the long time dynamics for the nuclear

spins while containing parameters motivated by experiments. Our treatment of nuclear

spin evolution is semiclassical, which gives results for the distribution of sum and difference

Overhauser fields of the nuclei in the two quantum dots, but does not include the quantum

corrections required to understand T2 decoherence processes. Numerical simulations show

that, for some parameters, the distribution of the Overhauser difference field between the

dots is suppressed. For other parameters, including those closer to experimental situations,

simulations show apparent instabilities leading to the growth of large Overhauser difference

fields.

An electron spin confined in a quantum dot interacts with the lattice nuclear spins

through the contact hyperfine interaction. DNP experiments operate near the crossing of the

electronic singlet s with the spin-polarized triplet T+ state. A typical example (Fig. 6.1a) is

Chapter 6: Preparation of non-equilibrium nuclear spin states in quantum dots 105

|S|S |T+

|T+|T0

|T0|T

−

|T−

DzD+

D−

S−

S−

S−

J |t=−T

J |t=T

Bext

Energy

Energy

ba

ε

|S|T+

|T0

|T−

Dz

Dz

∆

|S|T+

|T0

|T−

D+

D−

∆-

|S|T+

|T0

|T−

Sz

Λ+

|S|T+

|T0

|T−

S−

S+

Λ

|S|T+

|T0

|T−

D−

Γ

D+|S|T+

|T0

|T−

D−

ΓR

d

Figure 6.1: a) Schematic of two-electron energy levels as a function of detuning ε between(1,1) and (0,2) charge states. Arrows indicate adiabatic sweep through avoided crossing(pink) and rapid sweep back to (0,2) with reload (green). b) Spin-flip pathways betweenthe s and T+ states as the exchange energy J(ε) is swept through the crossing, showingthe nuclear operators involved in each path. Each pathway is a term in D− in Eq. 6.2. c)Annular approximation to the electron wavefunction in the double dot. d) Key processescontributing to Eq. 6.6.

an adiabatic sweep through the s-T+ degeneracy, followed by a non-adiabatic return to (0,2)

and reset of the electronic state via coupling to leads. To model this type of experiment,

we first derive an effective two-level Hamiltonian to describe the system near the crossing

of the singlet and lowest energy triplet state, T+, then solve the time dynamics.

For a double quantum dot with two electrons, we can write the Hamiltonian for

the lowest energy (1, 1) and (0, 2) electron states, where (n,m) indicates n (m) electrons

in the left (right) dot. If ψd(r) is the single-particle envelope wave function on dot d = l,r

(for the left, right dot), the hyperfine coupling constant for the nuclear spin at rkd is

gkd = ahfv0|ψd(rkd)|2 and the rms Overhauser energy is Ω = (∑

k g2kd)

1/2, where ahf is the

hyperfine coupling constant, and v0 is the volume per nuclear spin. We assume that Ω is

the same in each dot and choose units such that Ω = −~g∗µB = 1, where g∗ is the electron

effective g-factor and µB is the Bohr magneton. We introduce two collective nuclear spin


operators to denote the Overhauser fields in the left (L) and right (R) dots, L =∑

k gklIkl

and R =∑

k gkrIkr and further define S = (L + R)/2, D = (L − R)/2, where Ikd is the

angular momentum of the kth nucleus on dot d. In the basis |s〉 , |T+〉 , |T0〉 , |T−〉, where

the Tm are the (1, 1) triplet states, the Hamiltonian is [146]

H =

−J(ε) vD+ −√

2vDz −vD−

v D− −Bext + Sz S−/√

2 0

−√

2vDz S+/√

2 0 S−/√

2

−vD+ 0 S+/√

2 Bext − Sz

.

where Bext is an external magnetic field, and v = v(ε) = cos θ(ε)/√

2. The parameters

cos θ(ε), the overlap of the adiabatic singlet state with the (1, 1) singlet state, and J(ε), the

splitting between s and T0, are both functions of the energy difference ε between the (1, 1)

and (0, 2) charge states. We consider the nuclei to be spin-3/2 of a single species in a frame

rotating at the nuclear Larmor frequency.

Assuming that J,Bext ≫ 1, we perform a formal expansion in the inverse electron

Zeeman energy operator m = (Bext−Sz)−1. We apply a unitary transformation that rotates

the quantization axis of the triplet states to align with Bext − S and find, to second order

in J , m

Heff =

−J(ε) + hs v(ε)D+

v(ε)D− −Bext + hT

, (6.1)

where

hs = −2v2

JD†

zDz − D−v2

J +Bext − SzD+, (6.2)

hT = Sz −1

4(S−S+m+ mS−S+),

D− = D− + mS−Dz −1

4mS2

−mD+ − 1

4m S−S+mD−,

Dz = Dz −1

2

(

S+mD− + S−mD+

)

.


We develop a model for the evolution of the nuclear spin density matrix after one

pair of electrons has passed through the system. By coarse graining this evolution over a

cycle we can derive a master equation for the nuclear spins. The electron system is prepared

in |s〉 at large negative t = −T . We identify the (nuclear spin) eigenstates of the operator

ˆD+ˆD−, labeled |D⊥〉 with eigenvalues D2

⊥. After the crossing, the state is unchanged or

flips an electron and nuclear spin to the state |D′⊥〉 ⊗ |T+〉 where |D′

⊥〉 ≡ D−1⊥

ˆD− |D⊥〉.

We note that |D′⊥〉 is an eigenstate of ˆD−

ˆD+ with eigenvalue D2⊥. The problem is now

reduced to finding Landau-Zener solutions for each independent two-level system |D⊥〉⊗|s〉,

|D′⊥〉 ⊗ |T+〉, where we approximate hs → 〈D⊥| hs |D⊥〉, hT → 〈D′

⊥| hT |D′⊥〉, valid to

leading order in 1/J and 1/N . We model the actual sweep of ε by a linear sweep of J so

J(t) = −β2t+ Bext, where β =√

12 |dJ(ε)/dt| |t=0. We approximate that v(ε) is constant,

valid in the limit of large tunnel coupling and assume β ≪ Bext to ensure the applicability

of Eq. 6.1.

After one cycle, the state is |Ψ〉 = (cS |s〉+ cTˆD−

D⊥|T 〉) |D⊥〉. For β2T ≫ 1, the flip

probability is pf = 1 − exp(−2πω2), where ω = vD⊥/β, and

cS =√

1 − pf exp(−iφS), cT =√pf exp(−iφT )

φS ≈∫ T

−ThSdt (6.3)

φT ≈∫ t0

−ThSdt+ (T − t0)hT + φAD(ω),

where the crossing occurs at a time t0 ≈ Sz/β2. We include in φT the phase picked up

by following the adiabat, φAD. We approximate φAD by interpolating between the limits

ω = vD⊥/β → 0 and ω → ∞, giving [153]

φAD = 2πω2 + pf

ω2

[

1 − 2π + log

(

τ2

ω2

)]

− π/4

,

where τ = Tβ.


We move from the independent two-level systems to the general case by not-

ing that the components of |Ψ〉 depend only on the eigenvalue D⊥ and on the polariza-

tion Sz (which we approximate as commuting). Thus we can write an operator pf =

∑

D⊥pf (D⊥) |D⊥〉〈D⊥|, and similarly for φS , φT . The nuclear spin density matrix after

each cycle is given by tracing over the electronic states.

Rather than solve for the exact dynamics of the nuclear density matrix–still an

intractably hard computational problem for any reasonable number of nuclear spins–we

instead adopt an approximate solution to the problem using the P-representation for the

density matrix as a integral over products of spin coherent states. From the thermal distri-

bution, we choose such a spin coherent state and evolve it, where we interpret expectation

values 〈...〉 as being taken in that state. The ensemble of such trajectories represents the

physical system [4].

We organize this calculation by noting that the components of the Landau-Zener

model (φS , φT , pf , D±) are only functions of L and R. A spin coherent state is entirely

described by its expectation values iid = 〈Iid〉. For the jth spin on the left dot, we expand

the discrete time difference 〈Ijl〉n −〈Ijl〉n−1 after n and n− 1 cycles in the small parameter

gjl, giving an evolution equation

dijldt

= gjl

∑

µ

Pl,µ

⟨

i[∂gjlLµ, Ijl]

⟩

= gjlPl × ijl, (6.4)

where

Pl =1

2T[〈1 − pf 〉〈∇lφS〉 + 〈pf 〉〈∇lφT 〉 − Im(γl)] ,

where ∇l = (∂Lx , ∂Ly , ∂Lz ) and

γl =

⟨

D+pf

D−D+

∇lD−

⟩

, (6.5)

and similarly for ijr, Pr, and γr, with L replaced by R. The factorization of expectation

values is a natural consequence of our spin-coherent state approximation, as it explicitly


prevents entanglement between spins. Thus we have an effective, semi-classical picture of

nuclear spins precessing and being polarized by their interaction with the electron spin,

integrated over one cycle.

We approximate the electron wavefunction as a piecewise-flat function with M

levels, which we refer to as the annular approximation, as illustrated in Fig. 6.1c. We define

In =∑

j∈n ij where the sum is over all nuclei with the same hyperfine coupling to the

electron. Since gj is identical for all j ∈ n, we can simply replace i with I in Eq. 6.4. This

is convenient, because I2n is a conserved quantity, so we can study the evolution of M ≪ N

spins. Additionally, the typical size of In is now ∼√

N/M ≫ 1. This allows us to replace

the spin-coherent states used above with semi-classical spins, and makes taking expectation

values straightforward: all quantum operators can be replaced by their expectation values

directly.

To illustrate, to first order in m0 = B−1ext, for d = l, r,

Pd =pfλ (Λ+z − Λ0S⊥) +m0Γ0pfDz

2πω2z ×D (6.6)

+ ΓRpf∇dφAD ∓[

Γ0β2

4πv2Im (γl − γr)

+ (1 − pfλ/2)(∆0Dz z + ∆−D⊥)]

where the top sign applies for d = l, D⊥ = (Dx,Dy, 0), S⊥ = (Sx, Sy, 0), λ = 1− t0/T gives

the shift in the location of the crossing, and ∆0, ∆−, Λ+, Λ0, ΓR, and Γ0 are constants

depending on the details of the pulse cycle. To leading order in m0, Im(γl − γr) = 2(D ×

z)pf/D2⊥. It is clear from Eqs. 6.4-6.6 that all dynamics stop if D = 0. As indicated in

Fig. 6.1d, the Γ0 term originates in the hyperfine flip-flop, the ∆0 and ∆− terms are the

off-resonant effects of coupling from the singlet state to the T0 and T− states, respectively,

Λ0 comes from coupling between the T+ and T0 states, and Λ+ comes from Knight shifts

due to occupation of the T+ state. To leading order in m0, for a pulse sequence consisting


of only the Landau-Zener sweep, with instantaneous eject and reload, the parameters have

values

∆0 =

⟨

2v2

J(t)

⟩

c

≈ m0, ∆− =

⟨

v2

J(t) +Bext

⟩

c

≈ m0/4

Λ+ = 1/4, Λ0 = m0/4

Γ0 =2πv2fc

β2, ΓR = fc

where fc = 1/2T is the cycle frequency and 〈.〉c indicates an average taken over a full cycle;

these values can be modified readily by changing the details of the pulse cycle, while leaving

the Landau-Zener portion unchanged. The simulations shown below were performed with

the equations of motion correct to second order in m0 with ψd(r) a 2D Gaussian. Taking

v2 ≈ 1/2, we estimate that for experiments performed with Bext = 10 mT with T = 25 ns

[124], m0 ≈ 0.13, Γ0 ≈ 0.20, but the ∆ and Λ terms depend on the rest of the cycle. In

each of the simulations, we choose initial magnitudes and directions of the spins In by a

procedure equivalent to choosing initial directions for each of the Nn spin-3/2 nuclei in the

nth annulus and evaluating In =∑

j∈n ij explicitly. The relationship between simulation

time and laboratory time depends on the details of the pulse cycle, including pauses and

reloads not considered explicitly here, but simulation time is roughly in units of (Ωgmax)−1,

where gmax ≈ 2Ω/ahf is the largest value of gk, so t = 400 is approximately 10 ms.

When the number of annuli M is chosen to be small, the system moves rapidly to

its maximally polarized state, with In ≈ −Inz for all n. (This does not correspond to all of

the nuclei being polarized, which also requires In = 3Nn/2.) In the limit that M = 1, this

polarization is the only result, showing that inhomogeneity of the electron wavefunction

is essential for interesting phenomena involving Dz. The annular approximation should

correctly describe the nuclear dynamics for a time scale given by the inverse of the difference

between the gj of adjacent annuli. For any M , when ∆0 = 0, the system rapidly saturates


(i.e., pf → 0) without any statistical change in the distribution of Dz; coupling from the

singlet to the T0 state is an essential ingredient in all of the effects discussed below.

By tuning the parameters of this model, two features of particular note are ob-

served. First, as illustrated in Fig. 6.2a, we show an ensemble of trajectories in which D

rapidly reduces toward zero. For the parameters of Fig. 6.2a, the standard deviation of Dz

was reduced by a factor of 28. We remark that as D → 0, the singlet state does not mix

with the triplets and nuclear spin dynamics stop. Until something (outside this model, such

as nuclear dipole-dipole coupling) restores D, the DNP process is shut off, limiting the total

nuclear polarization that can build up. While not shown in Fig. 6.2a, we observe a dramatic

reduction of the total |D|, not just Dz, consistent with this qualitative observation.

There have been a number of attempts to explain [125, 164, 143] the reported

reduction in |Dz| [124], none of which has taken into account both the coupling to the T0

state and the inhomogeneity of the electron wavefunction. We only observe a narrowing of

the Dz distribution in these simulations with both of these factors. Refs. [164, 143] claim

that the force to reduce |Dz| originates in the hybridization of the singlet and T0 states before

anti-crossing with T+, which can be seen in the mS−Dz term in D− in Eq. 6.2. While we

find that this effect can help reduce |Dz|, the strongest narrowing occurs in simulations with

sufficiently large ∆0 and Λ0, regardless of the change D− → D−. We find that the most

significant factors for strong narrowing of the Dz-distribution are ∆0/Γ0 ≈ Λ0/Γ0 ≈ 1 − 2

and Λ+/Γ0 . 1. A similar phenomenon can be seem from modeling the dynamics as a

diffusion in Dz [62].

We now consider two prototypical pulse sequences motivated by experiments, one

with small m0 = 0.01 and τ = 8, the other with larger m0 = 0.05 and smaller τ = 4, with β

held constant. In both cases, over 90% of the trajectories display a growth in |Dz| as shown

in Fig. 6.2b. This increase in |Dz| indicates that the spin flips are occurring predominantly


Figure 6.2: (a) Simulations with M = 100, m0 = 0.05, ∆0/Γ0 = 1, ∆−/Γ0 = 0.25, Λ+/Γ0 =0.5, Λ0/Γ0 = 1, ΓR/Γ0 = 1, β =

√2π, τ = 4. At bottom, 〈Sz〉e is the median value of Sz at

each time step in an ensemble of 1000 trajectories. Gray shaded region shows the 84th and16th percentiles. Above, with independent y-axis, is similar 〈|Dz |〉e. Thin red line is a singletrajectory. For these parameters, the trajectories have |Dz| → 0 quickly, without time forstrong polarization. (inset) Mean of the maximum value of |Dz| reached on each trajectoryfor the same parameters as in (b) (open circles) except M varied between 20 and 160, with5000 trajectories per point. Closed circles show similar results with m0 = 0.05, τ = 4 andall other parameters scaled appropriately. The physical system has M → N ≈ 106, so weinterpret this as an instability to large |Dz|. (b) Simulations as in (a) except m0 = 0.01,∆0/Γ0 = 0.78, ∆−/Γ0 = 0.19, Λ+/Γ0 = 5.8, Λ0/Γ0 = 0.08, τ = 7.8. Results are presentedas in (a), except each trajectory was shifted so that its maximum |Dz| occurs at time zero.The thin line at top shows the fraction of trajectories contributing to the ensemble at eachtime. 4.5% of the trajectories, which do not show this peak in |Dz|, are not included.Approximately 10% of the trajectories show behavior similar to that shown in the thin redline, where |Dz| is reduced initially and then goes unstable to large |Dz|.


in one dot. We interpret these results as showing a continuing increase of |Dz|, where the

peak is an artifact of the annular approximation. Near the peak, many of the annular spins

artificially reach their maximal polarization, at which point they should be broken into more

annuli. Similar trajectories with different M show the maximum value of |Dz| increasing

with M (Fig. 6.2 inset). The physical cause of this increase in |Dz| is not clear, but it is

associated with both ∆0/Γ0 and Λ+/Γ0 being sufficiently large. This could be the same

phenomenon as seen in Ref. [55], though some other effect is required to produce the large

|Dz| ≈ Bext of that work. Future work will discuss the relationship between this effect and

the shape of the electron wavefunction, in particular its inhomogeneity.

There are a variety of outcomes possible in this model, including crossover regions

between the Dz distribution narrowing and spreading described here. Simulations per-

formed with parameters intended to approximate experiments are in this crossover regime.

These simulations do not include such effects as the nuclear dipole-dipole coupling,

so we expect them to describe accurately the effects of pumping and coherence for times

shorter than the dipole-dipole diffusion time scale. Spin diffusion processes should oppose

the buildup of polarization, return Dz to its thermal values, and, since they do not conserve

I2n, can also break up the annulus spins, possibly allowing larger |Dz| to be produced than

seen in these simulations.

Acknowledgements

We thank S. Foletti, H. Bluhm, C. Barthel, C. M. Marcus, and M. Rudner for

valuable conversations. We gratefully acknowledge support from the Fannie and John Hertz

Foundation, Pappalardo, NSF grants PIF-0653336 and DMR-05-41988, and the ARO.

Appendix A

Appendix to Chapter 2

A.1 Further cubic spin-orbit terms

The following discussion was prepared with major input from Hans-Andreas Engel,

and we are grateful for his contribution.

In 3D zincblende crystals, the lowest order in k spin-orbit coupling term for con-

duction band electrons consistent with the crystal symmetries is

HDresselhaus = γ[σxkx(k2y − k2

z) + c.p.] (A.1)

where c.p. indicates cyclic permutations and x, y, z are the cubic crystalline directions [44].

When electrons are confined on a (001) plane, k2z can be sent to 〈k2

z〉 and the 3D Dresselhaus

term produces the 2D linear and cubic Dresselhaus terms,

H001 = γ〈k2z〉(σyky − σxkx) + γ(σxkxk

2y − σykyk

2x). (A.2)

Asymmetric confinement in the (001) plane also produces spin-orbit coupling of

the Rashba type [26],

HRashba = α(kyσx − kxσy). (A.3)

114

Appendix A: Appendix to Chapter 2 115

These three terms together give the effective Hamiltonian for 2D conduction elec-

trons in Eq. 2.1 of the main text.

In a 2D zincblende system on a (001) surface, there are further cubic spin-orbit

terms originating from both bulk and structure inversion asymmetries [28]. The extra

contribution from bulk inversion asymmetry is γ1(k3xσx − k3

yσy) while the cubic Rashba

terms are α1(k3xσy − k3

yσx) + α2(kyk2xσx − kxk

2yσy).

Winkler showed that γ1 (which he calls b6c6c53 ) is much less than γ [157]. Specifically,

using his formulae and parameters for GaAs gives γ/γ1 ≈ 103 for kF = 0.19 nm−1. In a

similar calculation, von Allmen considered an axial cubic Dresselhaus term of the form

γ2k2(kxσx − kyσy) and showed that γ/γ2 ≈ 20–400, depending on the model used [154].

We are unaware of estimates of the magnitudes of α1 and α2 in GaAs. Yang and

Chang recently showed that in a 15 nm-wide GaAs/AlGaAs quantum well, the nonlinear

Rashba terms do not become significant until considerably higher electron densities than

considered in this paper [163].

An approximation of the cubic Rashba terms can also be given using the 8 × 8

Kane model, which considers the s-type conduction band and the p-type valence bands.

The model is parameterized by the band gap E0, the energy of the split-off holes ∆0, and

the matrix element P of the momentum (times ~/m0, where m0 is the free electron mass)

between the s- and p-type states. When an external electric potential is applied across the

sample, the conduction band Hamiltonian contains a term

Hext = λ σ · (k×∇V ) . (A.4)

For a system confined along the z-direction, one can take the expectation value

〈Hext〉 along that direction. Noting that the only contribution of the confinement field is

∝ 〈∇zV 〉 and for λ a constant, one obtains the k-linear Rashba Hamiltonian HRashba. If we

116 Appendix A: Appendix to Chapter 2

consider higher order corrections to λ, we can write λ = λ(0) + λ(1)k2. The λ(1) term gives

a term HRashba,c = α3k2(kyσx − kxσy), which gives the axial (C∞v) approximation to the

cubic Rashba terms.

Using third-order perturbation theory [157, 108],

λ(0) =P 2

3

[

1

E20

− 1

(E0 + ∆0)2

]

. (A.5)

Fifth-order perturbation theory gives [50]

λ(1) = −P4∆0

(

24E30 + 41E2

0∆0 + 26E0∆20 + 6∆3

0

)

9E40 (E0 + ∆0)

4 . (A.6)

In GaAs, these yield λ(0) = 5.3 A2

and λ(1) = −870 A4.

Because this estimate does not include edge effects, it only provides a rough esti-

mate for the coupling constants α, α3. We can, however, use Eqs. A.5 and A.6 to estimate

the ratio of the strengths of the linear and cubic Rashba terms. For k = 0.19 nm−1 corre-

sponding to the sample considered in the main text,

k2λ(1)

λ(0)≈ 1

17. (A.7)

We are interested, however, in the relative sizes of the cubic Dresselhaus and the

cubic Rashba. In the sample considered, the linear Dresselhaus and the linear Rashba

strengths appear to be almost the same [106], and we can estimate that the linear Dres-

selhaus is about twice the strength of the cubic Dresselhaus (i.e., 〈k2z〉/k2

F ≈ 2, using the

results in Ref. [106]), giving γ/α3 ≈ 8.

It is possible that a more careful calculation of the cubic Rashba strengths, includ-

ing band offsets and the full crystal symmetry, would modify these results substantially,

but these results indicate that the cubic Dresselhaus term considered in the main text is

the dominant cubic contribution to the spin-orbit coupling of GaAs quantum dots.

Appendix A: Appendix to Chapter 2 117

Regardless of their strengths, all of these cubic terms contribute to a‖ similarly

to the cubic Dresselhaus term discussed in the main text. We neglect the other cubic

contributions, but their inclusion would only increase the constraint on γ.

A.2 Refitting var g data

Fig. A.1 shows the Zumbuhl et al. var g versus B‖ data discussed in the main text

[172]. Also shown are the original curvefit and our refit with a‖ constrained to be greater

than 8.1. This constrained fit clearly does not match the data. We thank Dominik Zumbuhl

for making the data available to us.

0 1 2 3 4 5 6 73

3.5

4

4.5

5

5.5

6

6.5

7x 10

−5

B|| (T)

va

r(g

) (e

2/h

)2

Data

Original Fit

a||≥8.1

Figure A.1: Variance in conductance as a function of in-plane field for quantum dot of areaA = 8 µm2 and electron density n = 5.8×1015 m−2, from Ref. [172]. A small perpendicularmagnetic field is applied to break time reversal symmetry. The original fit to the data isshown along with our refit constraining only a‖ ≥ 8.1 and τφ = 0.39 ns (the value found

from 〈g〉).

A.3 Value of γ in GaAs

We include here a table of values of γ from both experiment and theory.

118 Appendix A: Appendix to Chapter 2

Table A.1: Magnitude of Dresselhaus spin-orbit coupling constant γ in GaAs (eVA3). All

results are in bulk GaAs unless otherwise specified.

Year Ref |γ| Technique/Comments

EXPERIMENT

’83 [102] 24.5 Optical orientation using Dyakonov-Perel (DP) spin re-laxation time. Conduction electrons in p-type GaAs.

’83 [8] 20.9 Same as [102].’92 [45] 26.1 GaAs/AlGaAs heterostructure magnetoconductance

measurements. Sets k2z → 0 instead of 〈k2

z〉.’93 [127] 23.5 Raman scattering from 180A GaAs quantum well

(QW). Does not include Rashba term.’94 [76] 34.5 Same as [127]. No Rashba term.’95 [77] 16.5±3 Raman scattering from GaAs/AlGaAs heterostructure.’96 [126] 11.0 Raman scattering from asymmetric GaAs/AlGaAs

QW. More data than earlier.’03 [106] 28±4 ; 31±3 GaAs/AlGaAs heterostructure magnetoconductance

measurements.’09 [86] 5.0 GaAs/AlGaAs superlattice, transient spin grating mea-

surement

THEORY

’84 [36] 9 ; 8.5 14 × 14 k · p theory ; Linear Muffin Tin Orbitals(LMTO). Room temperature band gap.

’84 [128] 19 ; 30 8 × 8 ; 14 × 14 k · p models.’85 [16] 28 14 × 14 k · p.’86 [99] 27.57 ; 21.29 14 × 14 k · p. Bulk GaAs ; Ga.65Al.35As/GaAs het-

erostructure.’88 [120] 14.0 14 × 14 k · p.’88 [27] 14.9 ; 28.2 LMTO ; 16 × 16 k · p.’90 [118] 24.12 14 × 14 k · p.’92 [154] 18.3 ; 7.6 ; 36.3 14 × 14 k · p in bulk GaAs; bulk Al0.35Ga0.65As ;

GaAs/AlGaAs superlattice.’93 [104] 19.8 k · p.’94 [132] 8.9 sp3s

∗ tight binding (TB) model of a GaAs/AlAs super-lattice.

’95 [133] 8.5 TB model of 100 A GaAs/AlAs QW.’96 [119] 24.21 14 × 14 k · p.’96 [83] 10 ; 27.5 sp3s

∗ TB model ; k · p.’03 [157] 27.48 14×14 k · p theory. Notes that value is reduced to 19.6

in higher order perturbation theory (p. 74).’05 [71] 23.6 TB model used to refine 14 × 14 k · p.’06 [31] 8.5 Quasiparticle self-consistent GW method (QPscGW).’09 [97] 5.7 ; 8.3 Pseudopotentials.

Appendix B

Appendix to Chapter 3:

Quaternions

A quaternion q is a 2 × 2 matrix of complex numbers

q = q(0)112 + i3∑

µ=1

q(µ)σµ, (B.1)

where σµ are the Pauli matrices and q(µ) ∈ C. We define three conjugates of q:

complex conjugate: q∗ = q(0)∗112 + i∑

q(µ)∗σµ,

quaternion dual: qR = q(0)112 − i∑

q(µ)σµ,and Hermitian conjugate: q† = qR∗

119

120 Appendix B: Appendix to Chapter 3: Quaternions

If the 2×2 matrix(

a bc d

)

is expressed as a quaternion q, then in the 2 × 2 notation for the

quaternion,

q∗ =

d∗ −c∗

−b∗ a∗

qR =

d −b

−c a

(B.2)

q† =

a∗ c∗

b∗ d∗

.

It is clear that q† corresponds to the usual definition of Hermitian conjugation, but complex

conjugation is not equivalent.

For a K ×K matrix of quaternions Q we similarly define

(Q∗)ij = Q ∗ij ,

(

QR)

ij= Q R

ji , (B.3)

(

Q†)

= (Q∗)R ,

where again Hermitian conjugation of a K×K quaternion matrix corresponds to the usual

Hermitian conjugation of the equivalent 2K×2K complex matrix, but complex conjugation

is not equivalent. By convention, the trace of the quaternion matrix Q is

trQ =∑

i

Q(0)ii , (B.4)

which accords with the usual definition of the trace of the equivalent complex matrix (since

the Pauli matrices have zero trace), except that the quaternion trace is a factor of 2 smaller.

Appendix C

Appendix to Chapter 3: Spin

polarization forbidden with

M = Nφ = 1

Consider a quantum dot with N input channels, 1 output channel and 1 spin-

conserving dephasing channel. We show here that if we send an unpolarized incoherent

current in the N channels and measure the spin polarization in the output channel, then

as long as TRS is unbroken, there can be no spin polarization in the measured channel,

independent of the transparency of the contacts.

From Eq. 3.23, we have w = w0 + wφ where

wφ = Pφσµtr(σνPφSw0S†)(Nφδµν − Ξµν)

−1, (C.1)

and gsµ = 2tr(σµPKw

out) = 2[woutKK ](µ), where PK is the projection onto the Kth channel

(the output channel), and [q](µ) is the µ-component of the quaternion q.

The outgoing density matrix is wout = wout0 + wout

φ where wout0 = Sw0S

† and

121

122 Appendix C: Appendix to Chapter 3: Spin polarization forbidden with M = Nφ = 1

woutφ = SwφS

†. For convenience of notation, the outgoing channel has index K and the

dephasing lead channel has index 1, so Pφ = P1. Now

w0 =1

2N(11 − PK − P1), (C.2)

so

(wout0 )KK =

1

2N(1 − SKKS

∗KK − SK1S

∗1K). (C.3)

As in Sec. 3.2, the first two terms are real, but the third term can have a quaternion

component. We show that SK1S∗1K is exactly canceled by (wout

φ )KK in the final result for

wout. We have

tr(σνP1Sw0S†) =

[ σν

2N(1 − S1KS

∗K1 − S11S

∗11)](0)

(C.4)

and

Ξµν = tr(σνP1SσµP1S†) = δµνS11S

∗11, (C.5)

where the last equality follows because S11 commutes with σµ, since S11 ∈ C. Together,

these give

wφ = P1σµ

[ σµ

2N (1 − S1KS∗K1 − S11S

∗11)](0)

1 − S11S∗11

(C.6)

with implied summation over µ. For q a quaternion, σµ[σµq](0) = q, so

wφ =P1

2N

(

1 − S1KS∗K1

1 − S11S∗11

)

. (C.7)

Then

woutφ KK

=1

2N

(

SK1S∗1K − SK1S1KS

∗K1S

∗1K

1 − S11S∗11

)

. (C.8)

The first term in Eq. C.8 cancels SK1S∗1K in Eq. C.3. The second term is real, since S = SR,

so SK1S1K = SK1SRK1 ∈ C. Thus, wout

KK is real and ~gs = 0. Since this result relies only

Appendix C: Appendix to Chapter 3: Spin polarization forbidden with M = Nφ = 1 123

on the unitarity and self-duality of the S-matrix, it is true independent of the transparency

of the dephasing contact. Note, however, that if Nφ > 1, then ~gs can be nonzero in this

theory, even when M = 1.

Appendix D

Appendix to Chapter 3: Spin

polarization from dephasing

Here we give explicit examples of dephasing-induced spin polarization in the case

M = N = 1 for both the spin-conserving and spin-relaxing dephasing leads. The key to the

operation of the dephasing leads is that they break coherence between different spin states.

The spin-relaxing dephasing lead breaks the coherence between the different spins within a

single channel and can thus produce a spin conductance with Nφ = 1. The spin-conserving

dephasing lead does not break coherence between modes within a single channel and thus

requires Nφ = 2 to produce a spin polarization.

In the case of a spin-relaxing dephasing lead, consider an example of an S-matrix

124

Appendix D: Appendix to Chapter 3: Spin polarization from dephasing 125

in the complex representation with M = N = Nφ = 1:

S =

0 0 0 0 α −α

0 0 0 1 0 0

1 0 0 0 0 0

0 0 0 0 α α

0 α α 0 0 0

0 α −α 0 0 0

, (D.1)

where α = 1/√

2. An up spin incident from the left exits as an up spin to the right. A down

spin incident from the left enters the dephasing lead as an x-polarized spin. It is reinjected

as a dephased equal combination of up- and down-polarized spins, which then exit 50% as

up spins to the left and 50% as down spins to the right. This results in g = 3/4, ~gs = z/4,

and ~p = z/3.

We can obtain this result formally, applying Eq. 3.27 to find wφ = Pφ/4. Note that

we divide by 2 because we are using the complex rather than quaternion representation.

Applying Eq. 3.4, ~gs = z/4. Note that if the dephasing lead had preserved the spin of its

absorbed current, it would have reinjected x-polarized spins, which would have all exited

as spin down to the right, giving g = 1 and ~gs = 0, as Appendix C proves must happen.

For the spin-conserving dephasing lead, consider a quantum dot with M = 1 = N

126 Appendix D: Appendix to Chapter 3: Spin polarization from dephasing

and Nφ = 2 with complex S-matrix

S =

0 0 0 0 0 −α 0 α

0 0 0 1 0 0 0 0

1 0 0 0 0 0 0 0

0 0 0 0 0 α 0 α

0 α α 0 0 0 0 0

0 0 0 0 0 0 −1 0

0 −α α 0 0 0 0 0

0 0 0 0 1 0 0 0

. (D.2)

Again, an up spin incident from the left exits as an up spin to the right. A down spin

incident from the left enters the dephasing lead as a superposition of spin up in mode 1

and mode 2. This spin is conserved but the phase between the modes is broken. After

reinjection, the current reflects back into the dephasing lead, this time as a superposition of

spin down in both modes. Again the spin is conserved but dephased, and it exits as equal

parts spin up to the left and spin down to the right, which again gives g = 3/4, ~gs = z/4,

and ~p = z/3. Using Eq. 3.23, wφ = Pφ/4, just as in the previous example.

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