Download Electron and nuclear spins in semiconductor

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
c
2009
- Jacob Jonathan Krich
All rights reserved.
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 semiconductor 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 spinorbit 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, studying 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
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cause the effective magnetic fields to have large differences.
Abstract
Contents
Title Page . . . . . . . . . . . . .
Abstract . . . . . . . . . . . . . .
Table of Contents . . . . . . . . .
Citations to Previously Published
Acknowledgments . . . . . . . . .
Dedication . . . . . . . . . . . . .
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Work
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1 Introduction
1.1 Spin-orbit interaction . . . . . .
1.2 Time-reversal symmetry . . . . .
1.3 Random matrix theory . . . . . .
1.4 2DES and quantum dots . . . .
1.5 Organization of thesis . . . . . .
1.6 Summary of new results obtained
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1
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2 Cubic Dresselhaus spin-orbit
coupling in 2D electron quantum dots
3 Spin polarized current generation
from quantum dots without magnetic fields
3.1 Introduction . . . . . . . . . . . . . . . . . . .
3.2 Setup and symmetry restrictions . . . . . . .
3.3 Random matrix theory . . . . . . . . . . . . .
3.4 Dephasing . . . . . . . . . . . . . . . . . . . .
3.5 Finite bias and temperature . . . . . . . . . .
3.6 Discussion . . . . . . . . . . . . . . . . . . . .
4 Fluctuations of spin transport
through chaotic quantum dots with
4.1 Introduction . . . . . . . . . . . . .
4.2 Setup . . . . . . . . . . . . . . . .
4.3 Random matrix theory . . . . . . .
4.4 Discussion . . . . . . . . . . . . . .
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spin-orbit
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coupling
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5 Inhomogeneous nuclear spin flips
5.1 Introduction . . . . . . . . . . . . . . . . . . .
5.2 Electronic states of the double dot with N=2
5.3 Nuclear spin flip location . . . . . . . . . . .
5.4 Nuclear states . . . . . . . . . . . . . . . . . .
5.5 Discussion . . . . . . . . . . . . . . . . . . . .
Contents
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6 Preparation of non-equilibrium nuclear spin states in quantum dots
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A Appendix to Chapter 2
114
A.1 Further cubic spin-orbit terms . . . . . . . . . . . . . . . . . . . . . . . . . . 114
A.2 Refitting var g data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
A.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. Krich
and 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 following 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 phenomena 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 Zumbühl 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 defense. 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 educational 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 Johnson, 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 encouragement 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 quantum 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 information. 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 interaction (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 sending (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 fluctuations 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 detail 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 exchanges 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.
4
1.1
Chapter 1: Introduction
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
~ in a magnetic field is
for spin angular momentum S
Hs = g
e ~ ~
B · S,
2mc
(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
~ experiences an effective magnetic field B
~ =E
~×
field E
~v
c
+ O(v 2 /c2 ). This magnetic field
couples to the spin, producing a Hamiltonian
Hso =
ge ~
~
E×~
p · S,
4m2 c2
(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
~ → −S
~ and E
~ → E;
~ this will be explored more formally
field. Upon time reversal ~
p → −~
p, S
Chapter 1: Introduction
5
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 = eik·~r uk,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 =
X
∞
X
µ
γnml
kxn kym kzl σµ ,
(1.3)
µ={x,y,z} n,m,l=0
µ
where γnml
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
~ → −S.
~ Since spinof the system. As mentioned above, under time reversal, ~k → −~k and S
µ
orbit coupling does not violate time-reversal symmetry (TRS), we immediately require γnml
to be zero unless n+m+l is odd. If the crystal has inversion symmetry, then H(~k) = H(−~k),
which requires that n, m, l are all even. We thus immediately see that breaking inversion
6
Chapter 1: Introduction
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 semiconductors, 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 leading terms were worked out by Dresselhaus, so the coupling is generally called Dresselhaus
coupling. It has the form [44]
Hd = γσx kx (ky2 − kz2 ) + 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
2 kz (σy ky − σx kx ) + σx kx ky2 − σy ky2 ,
(1.5)
Chapter 1: Introduction
7
where hkz i = 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
γkF3 and the linear terms have scale γ kz2 kF , so we can neglect the cubic terms when
kF2 / kz2 ≪ 1. If we model the confinement as an infinite square well, then the second
subband becomes occupied when kF2 ≥ 3kz2 ; for an infinite triangular well[42], the second
subband is occupied when kF2 & 2kz2 . 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 · ẑ,
(1.6)
where ẑ 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,
~ ~k) · S/~.
~
Hso = gµB B(
(1.7)
8
Chapter 1: Introduction
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
~2 k2
+ α(k2 σ1 − k1 σ2 ) + β(k1 σ2 + k2 σ1 ) + V (~r)
2m
~2 k2
=
+ (α + β)k2 σ1 + (β − α)k1 σ2 + V (~r),
2m
Hl =
(1.8)
(1.9)
where the components are with respect to the axes ê1 = [11̄0], ê2 = [110], and β = γ kz2 .
We can complete the squares to rewrite this as
Hl =
i2
~2 h
m
m
m
k1 + 2 (β − α)σ2 , k2 + 2 (β + α)σ1 + V (~r) − 2 (α2 + β 2 ),
2m
~
~
~
(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
ir1 σ2 ir2 σ1
−
2λ1
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-
Chapter 1: Introduction
9
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 |αi, |βi,1
hT β|T αi = hα|βi .
(1.13)
For any operator O, we can define its time-reverse OR by
hβ| O |αi = hT α| OR |T βi ,
(1.14)
OR = T O† T −1 = U CO† CU −1 = U OT U −1 .
(1.15)
and it is easy to show that
Since reversing time twice should have no physical effects, we further require that T 2 = α11
with |α| = 1. Then
U CU C = U U ∗ = α11,
so by unitarity
U = αU T = α(αU ),
(1.16)
(1.17)
so α = ±1, corresponding to integral or half-integral angular momentum.
If we change the basis of states by sending |ψi → R |ψi with R a unitary transformation, then T becomes RU CR−1 = RU RT C, that is, U → RU RT . 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
1
Note that since T is antiunitary, it can only be applied to kets, so hT β| is defined as the dual of T |βi.
10
Chapter 1: Introduction
choice, note that it is often stated that Hamiltonians of spinless time-reversal invariant systems 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 = H R = H T . 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 = −UhT , and Uh is antisymmetric, 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
0 1 repeated on the diagonal. That is,
matrix with −1
0

0


−1


Uh = 
0


0


..
.
We recognize
0 1
−1 0

0 . . .


0 0 0 . . .


0 0 1 . . .
.


0 −1 0 . . .


..
..
.. . .
.
.
.
.
1
0
(1.18)
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 = σy OT σy . 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 = RUh RT . The set of all such matrices R is called the symplectic group Sp(N ).
Chapter 1: Introduction
1.3
11
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 Hamiltonian 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 timereversal symmetry, we begin the random matrix theory (RMT) by noting that the Hamiltonian must be a real symmetric matrix. The most important energy scale for the singleparticle 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]
trH 2
P (H) dH ∝ exp −
2σ 2
dH,
(1.19)
12
Chapter 1: Introduction
a)
b)
w
N
M
y
x
Figure 1.1: a) Section of an infinite quasi-1D wire. b) scattering region (shaded) connected
to two quasi-1D wires. The wire on the left has N open modes while the wire on the right
has M open modes.
where the volume element dH is
dH =
N
Y
dHij ,
(1.20)
i≤j
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 formulation 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 Schrödinger equation is
~2 2
−
∇ + V (y) ψ(x, y) = Eψ(x, y),
2m
(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) =
r
nπy m
cos
e±ikx x ,
2π~2 kx w
w
(1.22)
Chapter 1: Introduction
13
where n is an integer, E = ~2 [kx2 + (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=
$r
2mEw2
~2 π 2
%
(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 ψn± (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
X
j=1
+
αLj ψLj
+
M
X
−
αRi ψRi
,
(1.24)
i=1
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 ψLi
+
and ψRi
with coefficients βLi , βRi . The solution to the full quantum mechanical scattering
problem for the region is then given by the matrix relation




 βL1 
 αL1 




 .. 
 .. 
 . 
 . 












 βLN 
 αLN 

=S
,




β 
α 
 R1 
 R1 




 .. 
 .. 
 . 
 . 








βRM
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
14
Chapter 1: Introduction
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 integerspin systems that Ui = 11, we must express our states with respect to the basis of cos(kx x)
and sin(kx x). Then eikx x = cos(kx x) + i sin(kx x), and under time reversal we recover the
±
∓
→ α∗ ψRn
. Assuming that the system and thus the S-matrix is
standard result that αψRn
not modified by time reversal, we find




∗
∗
 αL1 
 βL1 




 .. 
 . 
 .  = S  ..  ,








∗
α∗RM
βRM
so we can multiply by S † and take the complex conjugate to find




 αL1 
 βL1 




 .. 
T  .. 
 .  = S  . .








βRM
αRM
(1.26)
(1.27)
For this spinless time-reversal invariant system, S = S T , just as we found for the Hamiltonian 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 = q 0 σ0 + q 1 iσ1 + q 2 iσ2 + q 3 iσ3 for
Chapter 1: Introduction
15
complex numbers q j . 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 ∗ = q 0∗ σ0 + q 1∗ iσ1 + q 2∗ iσ2 + q 3∗ iσ3
(1.28)
q R = q 0 σ0 − q 1 iσ1 − q 2 iσ2 − q 3 iσ3
(1.29)
q † = q R∗ = 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 conjugate. The quaternion dual is denoted q R 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 quaternion representation of the complex matrix Uh QTc Uh−1 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,
16
Chapter 1: Introduction
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 wavefunctions, 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 spinorbit 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
Chapter 1: Introduction
17
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 discussed 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 ensemble 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
18
Chapter 1: Introduction
generally of the form w(ǫ) = |ψǫ i hψǫ |, so from the rule that
|ψout i = S |ψin i ,
(1.34)
wout (ǫ) = S(ǫ)win (ǫ)S † (ǫ),
(1.35)
we find
where the density matrix win (ǫ) represents the states at energy ǫ ingoing toward the scattering 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) = PL Θ(µL − ǫ) + PR Θ(µR − ǫ),
(1.36)
where Θ(x) is the unit step function. Thus, the net right-moving current in the left lead is
−e
IL =
h
Z
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.2 Similarly, the
R
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.
2
The choice of normalization in Eq. 1.22 is not arbitrary but rather comes from considering the 1D density
of states and velocity. That is, the current carried per unit energy by a 1D system is the velocity v(ǫ) times
the density of states ρ(ǫ). By simple arguments, the density of right-moving states in a spinless 1D system
Chapter 1: Introduction
19
We then find a total current
Z
n
o
−e
I =IL =
dǫ tr PL [win (ǫ) − Swin (ǫ)S † ]
h
Z
o
n
−e
dǫ tr PL Θ(µL − ǫ) − PL S(ǫ)[PL Θ(µL − ǫ) + PR θ(µR − ǫ)]S † .
=
h
(1.40)
(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
−e
I=
h
Z
µL
µR
dǫ tr[PL − PL S(ǫ)PL S(ǫ)† ].
(1.42)
If S(ǫ) is a constant over this range of energy (i.e., in the linear response regime [40]),
−e
(µL − µR )tr(PL − PL SPL S † )
h
e2
I = VLR tr(PL − PL SPL S † ),
h
I=
(1.43)
(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


′
r t 
S=
,
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
r
m
2π 2 ~2 ǫ
1
,
=
2π~v(ǫ)
1
ρ(ǫ) =
2
(1.38)
(1.39)
for velocity v. It is thus clear that the velocity cancels out of the current carried per unit energy. The
current per unit energy is then i = ρ(ǫ)v(ǫ) = h1 , which is the choice made in the main text.
20
Chapter 1: Introduction
transmission matrix from right to left. By unitarity of S, rr † + t′ t′† = 11N . We then see
that Eq. 1.44 is
e2
VLR tr(11N − rr † )
h
e2
e2
= VLR tr(t′ t′† ) = VLR tr(t† t),
h
h
I=
which is the usual Landauer formula, giving G =
e2
†
h tr(t t).
(1.46)
(1.47)
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/Al0.3 Ga0.7 As. 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
Chapter 1: Introduction
21
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.
22
1.4.1
Chapter 1: Introduction
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 hybridization 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


−ǫ
γ
c






 γc 0






,
Hc = 
(1.48)
−Ez







0




EZ
where Ez = |g∗ µB Bext | 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.
Chapter 1: Introduction
23
a)
gate
depleted
region
Ohmic
contact
AlGaAs
2DEG
GaAs
b)
I DO T
I QP C
I QP C
200 nm
Figure 1.2: a) Schematic of a double quantum dot device in GaAs/AlGaAs 2DES, showing
gates on the top of the chip which can deplete the electron gas beneath them. b) SEM
micrograph of a double quantum dot, showing the locations of the two dots. Figure indicates
a device with current passing through it, unlike those discussed in this work. Adapted from
[64].
(0,2)
(0,2)
S
S
(1,1)
T−
(1,1)
S
0
(1,1)
S
Energy
Energy
Ez
(1,1)
(1,1)
0
S
T0
(1,1)
S
−Ez
(1,1)
T+
(0,2)
(0,2)
S
0
ε
S
0
ε
ε̃
Figure 1.3: a) Energy levels of the lowest lying charge states in the two-electron double
quantum 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 the
two-electron double quantum dot in the presence of an external magnetic field.
24
Chapter 1: Introduction
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
E
S̃ = cos θ |(1, 1)S i + sin θ |(0, 2)S i
E
G̃ = − sin θ |(1, 1)S i + cos θ |(0, 2)S i ,
where
θ = arctan
and the energies are 12 (−ǫ ±
1.4.2
2γ
p c
ǫ − 4γc2 + ǫ2
!
,
(1.49)
(1.50)
(1.51)
p
ǫ2 + 4γc2 ), with |S̃i the lower energy state [147].
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,
and
71 Ga,
and arsenic having just one stable isotope,
75 As,
69 Ga
all three of which have spin
Chapter 1: Introduction
25
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 X ′
~ k,β )S
~ · I~k,β ,
Aβ δ(~r − R
~2
(1.52)
k,β
~ k,β is the position of the kth spin of species β, ~r is the
where β is the nuclear species, R
~ is the electron spin, I~k,β is the kth nuclear spin of species β, A′ is the
electron position, S
β
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
~
2S
~ nuc
·B
~
2
v0 X ′ ~
= ∗
Aβ ψ(Rk,β ) I~k,β .
g µB ~
Hhf = g∗ µB
~ nuc
B
(1.53)
(1.54)
k,β
~ nuc has the form of a magnetic field coupled to the electron spin, in analogy
We see that B
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)
~ kβ )|2 and normalize
is smoothly varying on the scale of the lattice. We let dβ = |u(R
R
|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 Å−3 ,
dGa = 58 Å−3 .
The number of As atoms per unit cell is xAs =1, and x69 Ga = 0.6 and x71 Ga = 0.4 [113]. We
26
Chapter 1: Introduction
~ nuc as
can rewrite B
~ nuc =
B
v0 X ′
~ 2 ~
A
x
d
(
R
)
f
Ik,β
β
β
k
β
g ∗ µB ~
(1.55)
k,β
=
X
k,β
where
~ 2 ~
bβ f (R
k ) Ik,β /~.
bβ =
v0
∗
g µ
B
A′β xβ dβ .
(1.56)
(1.57)
Using v0 = a3 /4 with a = 5.63 Å the GaAs lattice constant defined with eight atoms per
unit cell [101] gives the result [113]
b75 As = −1.84 T,
b69 Ga = −0.91 T,
b71 Ga = −0.78 T,
using g∗ = −0.44. If all of the nuclei are polarized in the ẑ direction, then
X
~ nuc = 3
B
bβ ẑ = −5.3 T ẑ,
2
(1.58)
β
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′β xβ dβ v0 = g∗ µB bβ , 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
A75 As = −47 µeV,
A69 Ga = −23 µeV,
A71 Ga = −20 µeV.
Chapter 1: Introduction
27
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 quantum dots, but has considered only a single electron. In the case of a double quantum dot
~1 and S
~2 . We consider only two orbital
with two electrons, there are two electron spins, S
wave functions, ψL/R (~r), with L/R indicating the left/right dot. We can write the Hamiltonian in a simplified form by considering the single-particle electron wavefunctions ψ(~r1 )
R
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 = Atot
i
v0 X h
~ k )S
~1 · I~k + δ(~r2 − R
~ k )S
~2 · I~k .
δ(~
r
−
R
1
~2
(1.59)
k
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

00
0 0
1
Hhf = √ 
2 2
A
A†
B


,
(1.60)
28
Chapter 1: Introduction
where

√ RL
RR
LL
RR
I− − I−
 2 I− − hR|Li I−



√ RR

 LL
A =  2 hR|Li IzRR − IzRL
2[Iz − Iz ]


√ 
RR − I RL
RR − I LL
I
2 hR|Li I+
+
+
+


√ LL
RR
LL
RR
0
 2[Iz + Iz ] I− + I−



 LL

RR
LL
RR
B =  I+ + I+
,
0
I− + I−




√
LL + I RR − 2[I LL + I RR ]
0
I+
+
z
z

where
v0
I~AB ≡ Atot
~
X
~ k )ψB (R
~ k ),
I~k ψA (R
(1.61)
k
for A, B either R(ight) or L(eft), which is correct to first order in the wavefunction overlap
hR|Li = hψR |ψL i. We see from the second column of A that transitions from the (1, 1)S to
~ = (I~LL − I~RR )/2.
the three triplet states T+ are mediated by the nuclear difference field D
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+
RR ) or flipping one in the barrier (I RL ).
either by flipping a spin in the right dot (hR|Li I−
−
These overlap terms are generally small, so the usual course is to set hR|Li = I~RL = 0,
giving the simpler Hamiltonian


0
0
0
0 
0


√


0
0
D+ − 2Dz −D− 



1 
√

Hhf = √ 0
D−
2Sz
S−
0 
,
2



√
0 − 2Dz
S+
0
S− 




√
0 −D+
0
S+
− 2Sz
(1.62)
where S~ = (I~LL + I~RR )/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 hR|Li
Chapter 1: Introduction
29
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
~ couples the singlet states to the triplet states and the sum field S~
the difference field D
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.
S̃, T0 subspace
In the limit that ǫ < 0, |ǫ/γc | ≫ 1, the states S̃ and T0 come close to degeneracy,
and the Hamiltonian for just that subspace in the basis {S̃, T0 } is approximately


−J(ǫ) −Dz 
HST0 = 
,
−Dz
0
(1.63)
where the exchange splitting J(ǫ) is the hyperfine-free energy difference between S̃ and
p
T0 . In the model of Eq. 1.48, J(ǫ) = 12 (−ǫ + ǫ2 + 4γc2 ) ≈ γc2 / |ǫ|. We see from Eq. 1.63
that the S̃ 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
~ ∗ µB | ≈ 1.3 mT [124].
B~ext . This is a good approximation as long as Bext ≫ |S/g
The {S̃, T0 } subspace has been extensively studied for quantum information applications [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 T2∗ is measured [116] by initializing
the system at large ǫ in the (0, 2)S state. Then ǫ is reduced rapidly compared to the
30
Chapter 1: Introduction
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 {S̃, 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 T2∗ [116].
~ evolves slowly on the timescale on which the electron experThe hyperfine field D
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 {S̃, T0 }
space is greater than 1 µs [116]. If, however, no spin-echo techniques are used, each element 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 T2∗ of
approximately 15 ns [124]. If Dz could be reduced, then the {S̃, 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 S̃, 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 S̃ 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 {S̃, T+ }
Chapter 1: Introduction
31
is
HST+

 −J(ǫ)
=
√θ
D− cos
2

√θ
D+ cos
2 
.
−Ez + Sz
(1.64)
We define ǫ̃ to be the value of the detuning ǫ such that J(ǫ̃) = 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 S̃ 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 T2∗ 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 ǫ̃.
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 ǫ̃ 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
32
Chapter 1: Introduction
Adiabatic nuclear polarization
Prepare Singlet
T- T0
(0,2)S
Energy
T+
(0,2)S
ε~ 0
ε
Rapid Adiabatic Passage
(0,2)S
T- T0
t=0
Energy
T+
(0,2)S
εS
ε
0
Slow Adiabatic Passage
(0,2)S
t=τS
T- T0
Energy
T+
(0,2)S
0
εF
ε
Figure 1.4: Schematic of the DNP cycle, adapted from [117]. This figure has the energy
levels rotated 45◦ from those in Figure 1.3, which corresponds to ǫ/2 added to the Hamiltonian of Eq. 1.48. At top, the system is loaded in the (0, 2)S state. At middle, the detuning
ǫ is moved rapidly (compared to the magnitude of D− ) past ǫ̃, not allowing transition into
the T+ state. At bottom, ǫ is increased slowly, flipping an electron and nuclear spin on
transitioning from singlet to T+ .
Chapter 1: Introduction
33
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 T2∗ 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 S̃ 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 electron double quantum dots. We give a brief overview of each chapter.
34
1.5.1
Chapter 1: Introduction
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 Zumbühl and collaborators on
a set of four nominally chaotic quantum dots of two different sizes [171, 172]. A random matrix 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 kF2 / kz2 , as discussed in section 1.1, but rather λ1,2 kF2 /L kz2 . 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 twodimensional billiards, to estimate the magnitude of the cubic Dresselhaus contribution to
the conductance fluctuations. We combine these results with Cremers’ RMT formulation
and refit Zumbühl’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
Chapter 1: Introduction
35
widely cited value for the Dresselhaus SOI constant, γ of Eq. 1.4, is incompatible with our
results and the conductance fluctuations measured by Zumbühl. 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 eVÅ3 , 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 eVÅ3
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 proposals 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 polarization 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
36
Chapter 1: Introduction
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
Chapter 1: Introduction
37
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 |S̃i-|T+ i 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
1.6.1
Summary of new results obtained in this research
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-
38
Chapter 1: Introduction
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 improvement 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. Understanding 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.
~ are
When all three components of the total difference nuclear magnetic field, D,
zero, the system is in a “dark state” and does not evolve. We simulate the system numer-
Chapter 1: Introduction
39
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, we
estimate the magnitude of the spin-orbit induced avoided crossings in a closed
quantum dot in a Zeeman field. Using previous analyses based on random matrix
theory, we calculate corresponding effects on the conductance through an open
quantum dot. Combining our results with an experiment on an 8 µm2 quantum
dot [D. M. Zumbühl et al., Phys. Rev. B 72, 081305 (2005)] suggests that 1) the
3
GaAs Dresselhaus coupling constant γ is approximately 9 eVÅ , significantly
3
less than the commonly cited value of 27.5 eVÅ , 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-orbit
coupling 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 Bk that couples only to the electron spin. With no
SO, each eigenstate can be written as a product of orbital state |αi and spin quantized
along Bk . Eigenstates |α ↑i and |β ↓i 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
1
This neglects other spin-flip processes, e.g., hyperfine coupling, which in GaAs is much smaller than SO
effects for quantum dots on the micron scale.
Chapter 2: Cubic Dresselhaus spin-orbit
coupling in 2D electron quantum dots
42
crossings, which can be larger than the linear terms’ contribution since the latter are suppressed for small Bk [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 Zumbühl et al.
[172, 171] and find that agreement is possible only if the CD coupling constant in GaAs γ
3
is considerably less than the frequently cited value of 27.5 eVÅ [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
γ
+ 3 (p22 − p21 )(p × σ) · ê3
2m
2~
1
+ V (r) + gµB B · σ
2
(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 = ê1 ~σ2 /2λ1 − ê2 ~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 ê1 =[110], ê2 =[11̄0], and ê3 =[001̄]. 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 → U HU † ≡ H′ where U = exp(ir · Aso ) [5].
2
We ignore other cubic terms, which are allowed by symmetry, but are estimated to be small. See
Appendix A.
Chapter 2: Cubic Dresselhaus spin-orbit
coupling in 2D electron quantum dots
43
Expanding to leading order in L/λ,
H′ =
+
1
(p − a⊥ − ak )2 + bZ + bZ⊥
2m
γ
(p2 − p21 )(p × σ) · ê3 + V (r),
2~3 2
(2.2)
where a⊥ = (~σ3 /4λ1 λ2 )[ê3 × r], ak = (~/6λ1 λ2 ) × (x1 σ1 /λ1 + x2 σ2 /λ2 )[ê3 × r], bZ =
gµB B · σ/2, and bZ⊥ = −gµB (B1 x1 /λ1 + B2 x2 /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 = |hα ↑| Hso |β ↓i|,
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 =
|(Hso )α↑,β↓ |2 δ(ǫα − ǫβ − EZ )δ(ǫα − EF ),
X
αβ
(2.3)
where the overbar indicates ensemble averaging and Oa,b ≡ ha| O |bi.
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 =
Z
∞
−∞
dt −iωZ t
so (t)Hso † (0) |αi,
e
hα| HSF
SF
∆2π~
(2.4)
where ωZ = EZ /~ is the Zeeman frequency, |αi is a typical orbital eigenstate with ǫα ≈ EF ,
so ≡ h↑| Hso |↓i is the spin-flip part of Hso . We consider CD alone, Hc = (γ/2~3 )(p2 −
and HSF
2
p21 )(p1 σ2 − p2 σ1 ), 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
Chapter 2: Cubic Dresselhaus spin-orbit
coupling in 2D electron quantum dots
0.8
0.6
0.6
c
0.8 a)
0.4
Λ
Λ
c
44
b)
0.4
[010]
[100]
[110]
[010]
0.2
0
0
20
40
E (µeV)
0.2
[100]
0
0
60
20
0.8
0.6
0.6
c
0.8 c)
0.4
0.2
0
0
40
E (µeV)
60
40
EZ (µeV)
60
Z
Λ
Λ
c
Z
d)
0.4
0.2
20
40
EZ (µeV)
60
0
0
20
Figure 2.1: Normalized rms avoided crossing Λc due to cubic Dresselhaus spin-orbit coupling
as a function of Zeeman energy EZ for four billiards with in-plane magnetic field along the
3
indicated directions, with γ = 8.5 eVÅ [31]. Insets show the billiard shapes with crystal
axes. Solid (dashed) lines indicate specular (diffuse) boundary conditions. a) has a mixed
phase space with small regions of regular trajectories, b) is a stadium billiard, c) is similar
to the dot in Ref. [172], with diffuse boundaries to ensure chaos, and d) is a square with
diffuse 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
R
′
c (t)Hc† (t′ ) for 100
or bounces does not change the results. We calculate dte−iωZ (t−t ) HSF
SF
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 corresponding 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
3
resulting Λc (EZ ) for three orientations of Bk , with γ = 8.5 eVÅ . For other choices, Λc
Chapter 2: Cubic Dresselhaus spin-orbit
coupling in 2D electron quantum dots
45
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 Bk k x̂, in the limit EZ → 0,
Λ2c EZ →0
γ2
=
∆2π~7
Z
∞
−∞
dtpy p2x (t)py p2x (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
*
+
∞
X
γ 2 mLy
2
2
n 2
Λc =
(−1) px,n ,
|py (0)| px (0)
∆2π~7
EZ →0
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 − p2x,0 )−1/2 , where P
is the probability density on [−pF , pF ]. Since the diffuse boundaries in this billiard are the
P
n
top and bottom, P(px,i6=0 ) = 1/2pF . We regularize the infinite sum by ∞
n=−∞ (−1) = 0,
and, noting that ET = ~pF /mLy , we find Λ2c = 4γ 2 p6F /(45π 2 ∆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
Bk k x̂, 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
Chapter 2: Cubic Dresselhaus spin-orbit
coupling in 2D electron quantum dots
46
dot by Zumbühl 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 hgi 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
′
, given by association to H′ in Eq. 2.2 (without CD), as
perturbation Hamiltonian, HRMT
′
HRMT
=
∆h
iA0 (x11 + a⊥ σ3 ) + iak (A1 σ1 + A2 σ2 )
2π
i
−b · σ + b⊥ Bh σ3 ) ,
(2.7)
D
E
where Ai , i = 0, 1, 2, are real antisymmetric matrices with tr Ai ATj = δij M 2 , Bh is a
real symmetric matrix with tr Bh2 = M 2 , M ≫ 1 is the number of channels in the stub,
and x, a⊥ , ak , b, and b⊥ are dimensionless parameters, with x corresponding to B⊥ , b
to the Zeeman field, and a⊥ , ak , 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 hgi and var g as functions of x, a⊥ , ak , b, b⊥ , and Neff
to leading order in 1/Neff [39]. Zumbühl et al. use these results to fit their data.
Without CD, the correspondence between Eqs. 2.7 and 2.2 gives the following
Chapter 2: Cubic Dresselhaus spin-orbit
coupling in 2D electron quantum dots
47
mapping from physical parameters to RMT parameters:
x2 = πκET /∆(4πΦ/Φ0 )2
b = πEZ /∆
a2⊥ = πκET /∆(A/λ2SO )2
(2.8)
a2k = 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 = λ1 λ2 , L1,2 are the linear dimensions of the roughly rectangular dot, oriented along
ê1,2 , 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 ak
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 HRMT
by setting a2k = a2k,l + a2k,c , where ak,l is the Rashba and LD contribution, given by Eq. 2.8,
′
contains the SO part of the Hamiltonian of
and ak,c is the CD contribution. Since HRMT
the closed quantum dot, we relate ak,c to Λc by finding the rms spin-flip matrix element
(with spins quantized along Bk ) due to ak,c , giving ak,c = 2πΛc . Including the CD term
′
in this way lifts the constraint that ak ≪ a⊥ [5], similar to spatially varying SO
in HRMT
strengths [21].
Zumbühl 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].
Zumbühl et al. measure var g as a function of Bk with time reversal symmetry
Chapter 2: Cubic Dresselhaus spin-orbit
coupling in 2D electron quantum dots
48
broken by a small B⊥ . They fit to the expression of Cremers et al. [39], with ak (and all
parameters except κ′′ ) fixed to the value determined from the hgi data. We redo the fits to
the var g data, constraining only ak ≥ ak,c , with ak,c from our simulations, and τφ fixed to
the value determined from hgi. From Fig. 2.1, a typical value of Λc in all our billiard shapes
3
is 0.4, giving ak,c = 2.5 (8.1) for γ = 8.5 (27.5) eVÅ (recalling that Λc ∝ γ). We find that
a value of ak,c ≈ 2.5 is compatible with the experimental data. However, if we require that
ak ≥ 8.1, the fits to the data become markedly worse (see Appendix A).
Zumbühl et al. also measure hgi as a function of B⊥ , which they use to determine
ak , finding ak = 3.1 [171, 172, 170]. Since a2k = a2k,l + a2k,c , we must have ak ≥ ak,c , so
3
3
we conclude that γ = 27.5 eVÅ is inconsistent with these results, while γ = 8.5 eVÅ is
consistent with the data. So both hgi and var g data indicate that γ should be closer to 9
3
3
eVÅ than 28 eVÅ 3 . Moreover, even with the smaller value of γ, the CD term gives the
dominant contribution to ak .
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
3
by Richards et al. in which they found γ = 11.0 eVÅ [126]; the same group also found
3
3
γ = 16.5 eVÅ in a different sample [77]. A recent experimental value of γ = 28 eVÅ 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
3
not probing the bulk Dresselhaus coupling, though Ref. [31] predicts γ = 8.5 eVÅ in bulk
GaAs. We include in Appendix A a table with experimental and theoretical values of γ in
3
Even using Λc = 0.2, the lowest value in Fig. 2.1, gives ak,c = 4.1 for the larger value of γ, which is still
inconsistent with the experiment.
Chapter 2: Cubic Dresselhaus spin-orbit
coupling in 2D electron quantum dots
49
GaAs.
Strictly speaking, our calculations are not directly applicable to the hgi data of
Zumbühl et al., as our calculations assume EZ ≫ ∆, and hgi is measured with Bk =0. We
do not believe, however, that ak,c changes significantly as Bk → 0; similarly, Cremers et al.
consider ak to be constant for all Bk [39]. We believe that ak,c (Bk = 0) can be estimated
by simply averaging our results from the different field directions in the limit Bk → 0.
Our reinterpretation that a2k = a2k,l + a2k,c requires, of course, that ak,l be less than
3.1 in the experiment of Zumbühl et al. This reduction of ak,l can be absorbed into the
geometric parameter κ′ (which was set to 1 without fitting in Zumbühl et al.) without
affecting any of the physical parameters, τφ , λSO , found by Zumbühl et al.. Reducing κ′ is
reasonable, as Ref. [39] predicts κ′ = 1/3 for a circular diffusive system.
Since ∆ ∝ A−1 and ET ∝ A−1/2 n1/2 , we can see that if the thickness of the 2DES
does not change with density, ak,l ∝ A7/4 n1/4 , while ak,c ∝ Λc ∝ A3/4 n5/4 . 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
3
3
heterostructures has a value near 9 eVÅ and not the frequently cited value of 27.5 eVÅ ,
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.
Chapter 2: Cubic Dresselhaus spin-orbit
coupling in 2D electron quantum dots
50
Acknowledgments
The authors acknowledge a careful reading by Charlie Marcus and helpful conversations with Dominik Zumbühl, 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
Physics Department, Harvard University, Cambridge, Massachusetts
Abstract
An unpolarized charge current passing through a chaotic quantum dot with spinorbit coupling can produce a spin polarized exit current without magnetic fields or ferromagnets. 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
Chapter 3: Spin polarized current generation
from quantum dots without magnetic fields
52
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 accumulation 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 spinorbit 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 observed 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 generation
from 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
Chapter 3: Spin polarized current generation
from quantum dots without magnetic fields
54
Figure 3.1: A special model quantum dot with N = 1 channels in the left lead and M = 2
channels in the right lead. A skew scatterer sends all ẑ spins to the top channel and all
−ẑ spins to the bottom channel. The shaded area in the bottom channel has Rashba spinorbit coupling of precisely the strength to rotate a down spin at the Fermi energy to an up
spin, thus producing a perfectly spin polarized exit current from any input current, while
respecting 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
P
these 2 × 2 matrices using quaternions, where a quaternion q = q (0) 112 + i 3µ=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 → S R , 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 (outgoing) current, wout = Swin S † . The density matrix describing the unpolarized incoherent
combination of all N incoming channels is
win =

1 11N

2N
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.
Chapter 3: Spin polarized current generation
from quantum dots without magnetic fields
55
The Landauer-Büttiker formula gives the conductance in terms of the S-matrix
′ [25]. We write the K × K quaternion S-matrix as rt rt ′ 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-Büttiker formula in units of 2e2 /h as
G = tr(tt† ),
(3.2)
= tr(PR SPL S † ),
= 2N tr(PR Swin S † ),
= 2N tr(PR wout ),
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(PR wout ) 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
~ s in units of e/2π as
by voltage) G
~ s = tr(~σ tt† ),
G
(3.3)
= 2N tr(~σ PR wout ).
Then
g~s = 2tr(~σ PR wout )
(3.4)
is the spin conductance per channel in the left lead. Hence, gµs 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~ = g~s /g, with |p| ≤ 1.
We can, of course, construct g, g~s , and p~ using only the S-matrix and not the
density matrices win and wout . The density matrix approach, however, gives the flexibility
Chapter 3: Spin polarized current generation
from quantum dots without magnetic fields
56
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 coupling. 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 downpolarized electrons, giving a conductance matrix [107]


G↑↑ G↑↓ 
G=

G↓↑ G↓↓
(3.5)
with the total charge conductance being G = 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 w↑in =
1+σ3
2N PL .
The output density matrix
3 †
is w↑out = Sw↑in S † and the portion representing the output in the right lead is t 1+σ
2N t . The
Landauer-Büttiker formula gives, in units of 2e2 /h,
out
3
G↑↑ =N tr(PR 1+σ
2 w↑ ),
3 1+σ3 †
=tr( 1+σ
2 t 2 t ).
(3.6)
3 1+σ3 †
G↓↑ =tr( 1−σ
2 t 2 t ),
(3.7)
3 1−σ3 †
G↑↓ =tr( 1+σ
2 t 2 t ),
(3.8)
3 1−σ3 †
G↓↓ =tr( 1−σ
2 t 2 t ),
(3.9)
Similarly,
from which we see that G = tr(tt† ), which is the usual Landauer-Büttiker 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,
Chapter 3: Spin polarized current generation
from quantum dots without magnetic fields
57
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=
≡
1
2

0
0
1
0
0
0
0 0 0 0 −1
0 0 1 0 0
0 0 0 0 0 
0 0 0 eiθ 0 
1 0 0 0 0
0 −eiθ 0 0 0
0
1−σz
−σx −iσy 1+σz
0
eiθ (σx −iσy ) ,
σx +iσy −eiθ (σx −iσy )
0
(3.10)
(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
1/2
in
matrix is w =
, giving
0
0
wout


0
0 
0


1


= Swin S † = 0 1 + σz
,
0 

4


0
0
1 + σz
(3.12)
so Eq. 3.4 gives g~s = p~ = ẑ, 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
Chapter 3: Spin polarized current generation
from quantum dots without magnetic fields
58
weighted incoherent current is sent into N = 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 11N

2N

 11K − PK
,
=
2N
0
(3.13)
where PK is the projection onto the K th channel. The quaternion scattering matrix satisfies
S = S R since TRS is unbroken, and
wout =
11K − SPK S †
SS † − SPK S †
=
.
2N
2N
(3.14)
Note that S = S R implies both S † = S ∗ and Sii ∈ C for i = 1 . . . K (see Eq. B.3).
out ](µ) , where [q](µ) is the µUsing Eq. 3.4, the spin conductance is gµs = 2i[wKK
out has no quaternion part, then g s = 0.
component of the quaternion q. In particular, if wKK
µ
We have
out
wKK
=
∗
1 − SKK SKK
,
2N
(3.15)
∗
out ∈ R and g~s = 0. This proof applies with channels that are
and SKK SKK
is real, so wKK
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.
Chapter 3: Spin polarized current generation
from quantum dots without magnetic fields
59
Figure 3.2: Generic device with one input channel. Unpolarized current consisting of equal
parts spin-up and -down components is incident on a chaotic quantum dot from a single
channel on the left. The irregularly shaped region in the middle is a quantum dot in the
strong spin-orbit regime. The spin-up incident current has a probability to exit into each of
the three channels of the right lead with a different spin direction in each channel, indicated
by the direction of the dark arrows. Similarly for the spin-down incident current, indicated
by 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 spin
current in the exit lead is given by summing the probabilities and directions of the six spin
polarizations shown, which is the meaning of Eq. 3.4 and is indicated by the arrow on the
right.
3.3
Random matrix theory
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
Chapter 3: Spin polarized current generation
from quantum dots without magnetic fields
60
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 symplectic 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 magnitude 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, gµs = 0 for µ = 1, 2, 3. Using Eq. 3.4, we evaluate
D
E
(gs )2 = 4 tr(σµ PR Swin S † )tr(σµ PR Swin S † ) ,
(3.16)
Chapter 3: Spin polarized current generation
from quantum dots without magnetic fields
61
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 hF1 (S)i = 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]
1
[2tr(A)tr(B) − tr(AB R )].
(3.17)
2K − 1
The second average we need is hF2 (S)i = tr(ASBS † )tr(CSDS † ) where A, B, C, D are
hF1 i =
constant K × K quaternion matrices and AB = AD = CB = CD = 0. We find[18]
hF2 i =
1
(K − 1)[4trAtrBtrCtrD + tr(AC)tr(BD)]
Λ
− [trAtrCtr(BD) + tr(AC)trBtrD] ,
(3.18)
where Λ = K(2K − 1)(2K − 3).
Using Eq. 3.18, we find
(gs )2
= 3
M (M − 1)
,
NΛ
(3.19)
where we used tr(σµ PR ) = 0, tr(PR2 ) = 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 =
s 2 −2 , we can approximate it by (gs )2 / hgi2 . This approximate form is useful for
(g ) g
analytical progress and will be compared to numerical results. Using Eq. 3.17,
hgi =
2M
,
2K − 1
(3.20)
which, combined with Equation 3.19, gives
2
p
≈
3(M − 1)(2K − 1)
.
4M N K(2K − 3)
(3.21)
Chapter 3: Spin polarized current generation
from quantum dots without magnetic fields
62
1
s 2 1/2
<g>
2 1/2
<(g ) >
<p >
0.4
0.5
0.2
N=1
N=2
N=3
N=4
0
1
3
5
M
7
0
1
3
5
M
7
1
3
5
M
7
Figure 3.3: Numerical (symbols) and analytical (lines) results for normalized mean conductance hgi, rms spin conductance gs , and rms spin polarization p of current exiting a
chaotic quantum dot with N (M ) channels in the entrance (exit) lead. An average over
60000 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 )2 g−2 ≈ (gs )2 / hgi2 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 = U U R , 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
2
p is 7%.
Chapter 3: Spin polarized current generation
from quantum dots without magnetic fields
1
63
s 2 1/2
<g>
2 1/2
<(g ) >
<p >
0.4
0.75
0.2
Nφ=1
Nφ=2
N =3
φ
0.5
1
3
5
M
7
0
1
3
5
M
7
1
3
5
M
7
Figure 3.4: Model with spin-conserving dephasing lead. Numerical (symbols) and analytical
(lines) results for normalized mean conductance hgi, rms spin conductance gs , and rms spin
polarization p of current exiting a chaotic quantum dot with N = 1 channel in the entrance
lead. 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. The
lines are from Eqs. 3.28 and 3.29.
Chapter 3: Spin polarized current generation
from quantum dots without magnetic fields
64
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(σµ Pφ Sw0in S † ), where µ = 0, 1, 2, 3, σ0 = 112 , Pφ is the projection operator
onto the dephasing lead’s channels, and w0in 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


0K

w1φ = 
 = c1µ σµ Pφ ,
c1µ σµ 11Nφ
(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(σν Pφ Sσµ Pφ S † ),
which gives the charge/spin current in the dephasing lead due to this reinjection. Defining
Chapter 3: Spin polarized current generation
from quantum dots without magnetic fields
65
win = w0in + wφin , this procedure gives
wφin
=
∞
X
wnφ
n=1
=
Pφ σµ tr(σν Pφ Sw0in S † )
∞
X
(Ξn−1 )µν
n=1
Nφn
= Pφ σµ tr(σν Pφ Sw0in S † )(Nφ δµν − 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 w0in discussed here.
We approximate wφin by replacing Ξµν with its average in Eq. 3.23, similar to Eq.
3.21. Using Eq. 3.17,
hΞµν i =
Nφ
[2(Nφ − 1)δµ0 δν0 + δµν ],
2Kφ − 1
(3.24)
where Kφ = N + M + Nφ . We further replace tr(σν Pφ Sw0in S † ) by its average,
D
E
δν0 Nφ
,
tr(σν Pφ Sw0in S † ) =
2Kφ − 1
(3.25)
which gives
win ≈ w0in + 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
wφin = Pφ
tr(Pφ Sw0in S † )
.
Nφ − tr(Pφ SPφ S † )
(3.27)
Interestingly, approximating by taking averages separately of the numerator and denominator 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
Chapter 3: Spin polarized current generation
from quantum dots without magnetic fields
66
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
3M
(gs )2 ≈
N Λφ
M − 1 + Nφ
M 2 + 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 spinconserving 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 g~s = 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 that M = 1 implies g~s =
0, but it is not obvious that dephasing will nonetheless produce polarization. Indeed, a single
Chapter 3: Spin polarized current generation
from quantum dots without magnetic fields
67
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 g~s can be produced for Nφ > 1 (results not shown).
Brouwer and Beenakker modified the third lead dephasing model to make dephasing 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 general. 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 / hgi2 , as in Sec. 3.3, where we modify hgi to include the dephasing lead. Using Eqs.
Chapter 3: Spin polarized current generation
from quantum dots without magnetic fields
68
3.17 and 3.26, we have
hgi ≈
2M Kφ
.
K(2Kφ − 1)
(3.29)
We estimate p2 , using Eqs. 3.28 and 3.29. Comparison of these approximations to numerical 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 1 11N
∂ǫ 2N
0M
(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
D
E
D
E
†
†
Sab (ǫ)Scd
(ǫ′ ) = ∆′ δ(ǫ − ǫ′ ) Sab (ǫ)Scd
(ǫ) ,
(3.31)
and
D
E
†
†
Sab (ǫ)Scd (ǫ′ )Sef
(ǫ)Sgh
(ǫ′ )
D
ED
E
†
†
= Sab (ǫ)Sef
(ǫ) Scd(ǫ′ )Sgh
(ǫ′ )
D
E
†
†
+ ∆′ δ(ǫ − ǫ′ ) Sab (ǫ)Scd (ǫ)Sef
(ǫ)Sgh
(ǫ) ,
(3.32)
Chapter 3: Spin polarized current generation
from quantum dots without magnetic fields
69
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,
Z
dǫ dǫ′ h(ǫ, ǫ′ ) =
4
trAtrBtrCtrD
(2K − 1)2
=
4
trAtrBtrCtrD
(2K − 1)2
∆′
+
Λ
+
Z
dǫ f ′ (ǫ)2 (K − 1)[4trAtrBtrCtrD + trACtrBD] − trAtrCtrBD − trACtrBtrD
∆′ (K − 1)[4trAtrBtrCtrD + trACtrBD] − trAtrCtrBD − trACtrBtrD .
6T Λ
(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, hgi 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 11N
2N Vsd
0M
,
(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 .
Chapter 3: Spin polarized current generation
from quantum dots without magnetic fields
70
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
p
(∆′ /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 (gµs )2 =
s 2
(g ) /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
Chapter 3: Spin polarized current generation
from quantum dots without magnetic fields
71
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 choosing 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
Physics Department, Harvard University, Cambridge, Massachusetts
Abstract
As devices to control spin currents using the spin-orbit interaction are proposed
and implemented, it is important to understand the fluctuations that spin-orbit
coupling can impose on transmission through a quantum dot. Using random
matrix theory, we estimate the typical scale of transmitted charge and spin
currents when a spin current is injected into a chaotic quantum dot with strong
spin-orbit coupling. These results have implications for the functioning of the
spin transistor proposed by Schliemann, Egues, and Loss. We use a density
matrix formalism appropriate for treating arbitrary input currents and indicate
its connections to the widely used spin-conductance picture. We further consider
the case of currents entangled between two leads, finding larger fluctuations.
72
Chapter 4: Fluctuations of spin transport
through chaotic quantum dots with spin-orbit coupling
4.1
73
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 heterostructures [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] spinorbit 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,
Chapter 4: Fluctuations of spin transport
through chaotic quantum dots with spin-orbit coupling
74
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
Chapter 4: Fluctuations of spin transport
through chaotic quantum dots with spin-orbit coupling
75
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 {σ 0 , iσ 1 , iσ 2 , iσ 3 }, where
the σ i are the Pauli spin matrices. In this basis, a 2 × 2 matrix q = q 0 σ 0 + i~
q · ~σ , with
q 0 , ~q ∈ C, which is also called a quaternion [105]. Then q is defined to have a complex
conjugate q ∗ = q 0∗ σ 0 + i~
q ∗ · ~σ , dual q R = q 0 σ 0 − i~
q · ~σ , and Hermitian conjugate q † = q R∗ .
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 (S R )ij = (Sji )R , and Hermitian conjugate
S † = S R∗ . This representation is convenient because for time reversal invariant systems,
P
S = S R . The quaternion representation has the standard convention that tr(S) = i Sii0 ,
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 , where m ∈ {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
76
Chapter 4: Fluctuations of spin transport
through chaotic quantum dots with spin-orbit coupling
defining this chemical potential, so we give an example. If reservoir m is spin-split along axis
x̂, then µ0m is the average chemical potential in the reservoir, 2µxm is the chemical potential
difference between spin-up and spin-down electrons quantized along x̂, and µy,z
m = 0. In
general, if the quantization axis is n̂ and the chemical potential difference along that axis
is 2µs , then µi = µs (n̂ · î). 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
w̃ nm (ǫ) = δnm [f (ǫ − µ0n ) − ~σ · ~
µsn f ′ (ǫ − µ0n )],
(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
e
jn0 (ǫ) = 2tr{Pn [w̃(ǫ) − w̃out (ǫ)]} ,
h
(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 = δan δbn ). Similarly, the spin-current in the nth channel
is
jni (ǫ) = 2tr{Pn σ i [w̃(ǫ) − w̃out (ǫ)]}
e
.
2π
(4.3)
Chapter 4: Fluctuations of spin transport
through chaotic quantum dots with spin-orbit coupling
77
We choose units in which e = h = 2π, so Eq. 4.3 can describe both charge and spin currents
if we let σ 0 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 w̃(ǫ), since w̃out = S w̃S † 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

0
0
′
0
µs1
 f ǫ − µ1 − f ǫ − µ2 − f ǫ − µ1 ~σ · ~
w(ǫ) = 

0
′
0
′
0
µs1
−f ǫ − µ δµ − f ǫ − µ1 ~σ · ~
≈
−f ′
−f ′
0
0
ǫ − µ2 ~σ

ǫ − µ2 ~σ
·µ
~ s2
·~
µs2

,



(4.4)
(4.5)
where µ0 = (µ01 + µ02 )/2 and δµ0 = µ01 − µ02 . 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


0
s
~1
δµ + ~σ · µ

w=

~σ · µ
~ s2
(4.6)
and
jnν = 2tr[Pn σ ν (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
78
Chapter 4: Fluctuations of spin transport
through chaotic quantum dots with spin-orbit coupling
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 Büttiker-type conductance equation [23]
jlν =
X
k,ρ
ρ
ν
Gνρ
lk µk − 2Ml µl ,
(4.8)
where Gνρ
lk 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
Chapter 4: Fluctuations of spin transport
through chaotic quantum dots with spin-orbit coupling
79
there is no spin-orbit coupling in the leads) implies
X
k
Gν0
lk − 2Ml = 0.
(4.9)
Further, the conservation of charge current implies that
X
G0ν
lk = 2Mk δν0 .
(4.10)
l
Specializing to the case of two leads with N and M modes in the left and right leads
with potentials µνL , µνR , respectively, we can express Gνρ
lk simply in terms of the S-matrix.
Setting µνR = 0 and µνL = δνα , Eq. 4.8 gives
ν
jR
= Gνα
RL .
Eq. 4.6 says that w =
σα 11N
0M
(4.11)
ν = 2tr(σ ν P Sσ α P S † ) =
= σ α PL , and by Eq. 4.7 we have jR
R
L
να
ν
α
†
Gνα
RL . Similarly, GRR = 2tr(σ PR Sσ PR S ). If the system is time reversal invariant, then
S = S R , which imposes some relations between the different conductance matrix elements.
Since tr(AR ) = tr(A), we have the Onsager-like relations
ν ρ ρν
Gνρ
lk = h h Gkl ,
(4.12)
for k, l = R, L where hν = (1, −1, −1, −1).
We thus see that we can express all of the Gνρ
ij 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
ν
jw
≡ 2tr[σ ν PR (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.
Chapter 4: Fluctuations of spin transport
through chaotic quantum dots with spin-orbit coupling
80
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 =
−t
2K−1 w,
or
3. w is invertible and ∃α ∈ R such that w−1 =
4.3
w−[t−2α(K−1)]11
−α[t−α(2K−1)] .
Random matrix theory
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 polarization is zero, in any particular case there will still be some residual polarization in some
Chapter 4: Fluctuations of spin transport
through chaotic quantum dots with spin-orbit coupling
81
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 spinrelaxation 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 spinorbit 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
ν , we define the outgoing current
density matrix w, in addition to jw
ν
jout
= 2tr σ ν PR SwS † ,
(4.15)
and the current due only to the input state
ν
jin
= 2tr(σ ν PR w)
(4.16)
ν = j ν − j ν . The charge current is j 0 and the spin current is ~
s =
so jw
jw . We define jw
out
w
in
~ 0 . A small number
pw = ~jw /jw
jw . The polarization of the current in the right lead is ~
of parameters of the input current are sufficient to describe the effects of any w in a two-
Chapter 4: Fluctuations of spin transport
through chaotic quantum dots with spin-orbit coupling
82
terminal configuration. In particular, we define
t = 2trw
C = 2tr(w 2 )
(4.17)
(4.18)
ν R
Dν = 2tr(σ ν PR w R ) = (jin
)
(4.19)
E ν = 2tr(σ ν PR w R w R )
(4.20)
F ν = 2tr(σ ν PR w R PR σ ν w R ),
(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 E ν and F ν are more measures of coherence. By adding a multiple of 11 to
w, we can choose D0 = 2tr(PR w) = 0, and all results below assume this choice. Note that
if current is incident only from the left lead, then Dν = E ν = F ν = 0.
We take averages over the uniform ensemble of all S-matrices in the strong spinorbit 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,
1
[2trAtrB − tr(AR B)]
2K − 1
1
= trAtrB.
K
hf1 iCSE =
(4.22)
hf1 iCUE
(4.23)
The other average we need is of f2 = tr(ASBS † )tr(ASBS † ) for A, B constant
Chapter 4: Fluctuations of spin transport
through chaotic quantum dots with spin-orbit coupling
K × K quaternion matrices. We find [18]
n
1
{K − 1} 8[trA]2 [trB]2 + 2tr[A2 ]tr[B 2 ] + 4[tr(AB R )]2
hf2 iCSE =
2ΛS
o
− 8tr[A]tr[B]tr[AB R ] − 2tr[AAB R B R ]
n
− 2[trA]2 tr[B 2 ] + tr[A2 ][trB]2 − 4tr[A]tr[AR B 2 ] − 4tr[B]tr[A2 B R ]
o
R
R
R
R R
+ 4tr[A]tr[B]tr[AB ] + tr[AB AB ] + tr[AAB B ]
hf2 iCUE =
83
(4.24)
1 4K(trA)2 (trB)2 + Ktr(A2 )tr(B 2 ) − tr(A2 )(trB)2 − (trA)2 tr(B 2 ) , (4.25)
ΛU
where ΛS = K(2K − 1)(2K − 3) and ΛU = K(4K 2 − 1).
ν . Using Eq. 4.22,
We consider the mean and fluctuations of jw
ν
ν
ν
hjw
i = hjout
i − jin
ν
hjout
i = δν0
2tM
Dν
− δS
,
2K − δS
2K − 1
(4.26)
(4.27)
where δS = 1 for averages over the CSE and δS = 0 for averages over the CUE. The relevant
ν = j ν − hj ν i, which satisfy
fluctuations to study are of ∆jw
w
w
D
E ν 2
ν 2
ν
(∆jw
) = jout
− hjout
i2
(4.28)
Using Eqs. 4.24 and 4.25, we find
1
ν 2
jout
=
M δν0 4M t2 (K − δS ) − 2M C + 4δS E 0
Λ
− M t2 + (K − δS )(2M C + 2δS Dν 2 )
(4.29)
− δS [E 0 (2K − 1) − F ν ] ,
ν 2
does not
where Λ = ΛS , ΛU for the CSE, CUE, respectively. We note that jout
CUE
depend on D, E, or F . Combining this result with Eq. 4.27,
ν 2
1n
M t2 (1 + δS )
2
ν
0
jout − hjout i =
M δν0
− 2M C + 4δS E + 2M C(K − δS )
Λ
K − δS /2
o
2K 2 − 3K + 2
− M t2 + δS [(Dν )2
− E 0 (2K − 1) − F ν ]
2K − 1
(4.30)
Chapter 4: Fluctuations of spin transport
through chaotic quantum dots with spin-orbit coupling
84
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 11N (tσ0 + ~s · ~σ )

2N
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
0
0
current is jw
= 2tM/(2K − δS ). The reduction of jw
as TRS is broken (δS → 0) is the
signature of weak antilocalization [12, 13, 29]. The rms spin current in the right lead is
M [(M − δS )t2 + (K − δS )s2 ]
s2
jw
=3
.
NΛ
(4.32)
The fluctuations in the charge current are
D
0
∆jw
2
E
=
M
[4M N − δS (4M − 1/N )]t2 + (1 − δS /N )s2
Λ
(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
Chapter 4: Fluctuations of spin transport
through chaotic quantum dots with spin-orbit coupling
85
1
<j0 >
<(js )2>1/2
w
<(ps )2>1/2
w
w
0.8
0.6
0.4
CUE
N=2
0.2
0
N=1
N=2
N=3
N=4
2
4
M
6
8 2
4
M
6
8 2
4
M
6
8
Figure 4.1: For the fully spin-polarized current represented by Eq. 4.31 with t = s = 1 and
time reversal symmetry, comparison of numerical (symbols) and analytical (lines) results
for 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. The
lines are from Eqs. 4.26, 4.32, and 4.35. The right panel shows that the expected spin
polarization in the exit lead is still appreciable, even for several open modes in each of the
leads. Also shown are the equivalent CUE results with N = 2, showing that the rms spin
polarization is nearly unchanged by breaking TRS in this case.
Chapter 4: Fluctuations of spin transport
through chaotic quantum dots with spin-orbit coupling
86
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
D
E M (N − δ )s2
S
02
∆jw
=
,
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
2
0 i
[3]. We note that hjw
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.
0 . It
We can further consider the spin-polarization of the exit current, p~w = ~jw /jw
is clear that h~
pw i = 0, just as h~jw i = 0, but there is some rms spin polarization of the exit
s 2 0 2
/ jw , we can use the above results to find
current. If we approximate p2w ≈ jw
t2 (M − δS ) + s2 (K − δS )
.
p2w ≈ 3(K − δS /2)2
Λt2 M N
(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.
Chapter 4: Fluctuations of spin transport
through chaotic quantum dots with spin-orbit coupling
87
1
0.16
0.14
0.8
<(j0 )2>1/2
w
0.12
<jz >
<(∆ js )2>1/2
w
w
0.6
0.1
0.08
0.4
0.06
0.04
CUE
N=2
0.2
N=1
N=2
N=3
N=4
0.02
0
2
4
6
M
8
0
2
4
6
M
8 2
4
6
M
8
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 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. The lines are from Eqs. 4.37–4.40. The left
panel shows that this pure spin current should still be expected to produce significant charge
currents, with a nonmonotonic dependence on the number of open channels N and M . Also
shown are the equivalent CUE results with N = 2.
Chapter 4: Fluctuations of spin transport
through chaotic quantum dots with spin-orbit coupling
88
Case 2: Pure spin current from both leads
We consider a pure spin current incident from both leads, represented by the
density matrix

σz
11N 2N
w2 = 
σz
−11M 2M


.
(4.36)
This density matrix represents a spin current of +ẑ incident from the left and a spin current
of −ẑ incident from the right, which together are an incident pure spin current from left
to right with polarization +ẑ. In this case, t = 0, C = K/2M N , Dν = (0, 0, 0, 1), E ν =
(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
D
02
jw
E
N2 + M2
K
1 − δS
.
=
Λ
MNK
(4.37)
We note that when M = N = 1, j02 CSE = 0, showing that no charge current can be produced. 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
i
jw
= (0, 0, 1 − δS
1
).
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.
Chapter 4: Fluctuations of spin transport
through chaotic quantum dots with spin-orbit coupling
89
The fluctuations around the mean are
K(K − δS ) 1 − δS MNK
=
(4.39)
NΛ
z 2
(M − 1)K 2
1
3M
N
K
2
∆jw =
K + δS
− K(M + 2 −
)+
+
(K − δS /2)N Λ
M
2M
2
K
x,y 2
∆jw
(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 +ẑ incident from the left and a pure
spin current −ẑ 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 offdiagonal 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), E ν = (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 E ν = (1/2, 0, 0, 0).
Chapter 4: Fluctuations of spin transport
through chaotic quantum dots with spin-orbit coupling
90
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
2 1/2
2 1/2
0 i
0
mean charge current, but the rms fluctuations are hjw
CSE = 0.41, hjw iCUE = 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
i
to have jw
= (0, 0, 1 − δS /3), as in case 2. We find fluctuations around the mean of
2 1/2
2 1/2
i i
i
h∆jw
CSE = (0.58, 0.58, 0.62), h∆jw iCUE = (0.63, 0.63, 0.63). With TRS, the fluctuations
are larger along the polarization axis, but not markedly so. The total spin polarization
s 2 1/2
s 2 1/2
fluctuations are ∆jw
=
1.03,
∆jw CUE = 1.10 which is larger than the mean
CSE
s , showing that coherence between the
current and equal in scale to the input current jin
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
Chapter 4: Fluctuations of spin transport
through chaotic quantum dots with spin-orbit coupling
91
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 polarization, 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, MA
2) Physics Department, Harvard University, Cambridge, Massachusetts
Abstract
We discuss a feedback mechanism between electronic states in a double quantum
dot and the underlying nuclear spin bath. We analyze two pumping cycles for
which this feedback provides a force for the Overhauser fields of the two dots to
either equilibrate or diverge. Which of these effects is favored depends on the
g-factor and Overhauser coupling constant A of the material. The strength of
the effect increases with A/Vx , where Vx is the exchange matrix element, and
also 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 systems [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 fundamental 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 (A69 Ga = −23 µeV , A71 Ga = −20 µeV
Chapter 5: Inhomogeneous nuclear spin flips
Energy E (µeV)
94
10
0
0.8
c
0.6
b
Τ+
~
ε
Ψ
a
c/b
1.3
c/b
a,b,c
-10
0.4
1.1
0.2
0.0
0.0
0.5
1.0
1.5
detuning ε (mV)
Figure 5.1: Top: electronic states near the (1,1) to (0,2) stability diagram transition. Crossing of Ψ and T + at ε̃ becomes anti-crossing in presence of transverse Overhauser field gradient. Bottom: overlap of the Ψ state with S(0, 2) (a), with |L↑ R↓ i (b) and with |L↓ R↑ i
(c). Parameters: Bext = 0.2 T , γ = 1.2 µeV , ∆ = 1000, EC = 0.6meV , Vx = 1µeV .
and A75 As = −47 µeV , 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
Chapter 5: Inhomogeneous nuclear spin flips
95
spatial dependence of nuclear spin flips induced by electronic spin “flops” is transparent.
The basis is: {ξn } ≡ {|R↑ R↓ i , |L↑ R↓ i , |L↓ R↑ i , |L↑ R↑ i}, 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↓ i and |L↑ L↓ i 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↓ i is the standard S(0, 2) state and |L↑ R↑ i is the standard T+ state. The
hyperfine Hamiltonian for two electrons is properly written:
M
Hhf =
vA X
[δ(r1 − Rm )S1 · Im ⊗ 1 + 1 ⊗ δ(r2 − Rm )S2 · Im ]
~2 m
(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 5.3 T [146] leads to an average coupling
constant A = −270 µeV . 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↓ i

EC − ε




H=




|L↑ R↓ i
IzLR
−
|L↓ R↑ i
IzRR hL|Ri
IzLL − IzRR
+γ
IzLR
|L↑ R↑ i
−
IzRR hL|Ri +
γ
RR
I+
hL|Ri −
Vx
RR
I+
IzRR − IzLL
LL
I+

LR
I+

IzRR + IzLL + EZ




,




(5.2)
where ε is the potential “detuning” [116], γ is the tunneling coefficient and EZ ≡ gµB Bext
is the Zeeman energy with µB the Bohr magneton and Bext the external magnetic field,
√
More√traditionally, the linear combinations S(1, 1) ≡ [|L↑ R↓ i − |L↓ R↑ i]/ 2 and T0 (1, 1) ≡ [|L↑ R↓ i +
|L↓ R↑ i]/ 2, are employed.
1
96
Chapter 5: Inhomogeneous nuclear spin flips
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, hL|Ri, of the non-orthogonal single particle basis. Higher order terms
(O(| hL|Ri |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
M
A X ∗
αβ
~
I ≡v
ψ (Rm )ψβ (Rm )I~m
2~ m=1 α
(5.3)
where α, β ∈ {L, R}. Note that previous researchers have typically ignored the transition
LR , which we see from Eq. 5.2 can lead to a direct transition between |R R i and
term I+
↑ ↓
|L↑ R↑ i 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↓ i and
|L↑ R↑ i anti-cross deep in the (0,2) regime, we will, in this paper, concentrate on spin flipflop 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↑ i state is coupled to |L↑ R↓ i via
RR ) and it is coupled to |L R i by a
a term which flips a nuclear spin in the right dot (I+
↓ ↑
LL ) . In the absence of flip-flop coupling to
term that flips a nuclear spin in the left dot (I+
the |L↑ R↑ i state, the upper left 3x3 matrix in Eq. 5.2 (see also yellow highlighted region of
2
Coulomb
are defined in the usual way in our two state basis, α, β, γ, δ ∈ {L, R}:
R R matrix ∗elements
Vαβγδ ≡
dr1 dr2 ψα (r1 )ψβ∗ (r2 )V (r1 , r2 )ψγ (r1 )ψδ (r2 ).
3
We have also used the identity: S · Im = Sz Imz + [S− Im+ + S+ Im− ]/2.
Chapter 5: Inhomogeneous nuclear spin flips
97
Fig. 5.2) has a ground state, which we denote:
|Ψi = a(ε) |R↑ R↓ i + b(ε) |L↑ R↓ i + c(ε) |L↓ R↑ i .
(5.4)
As shown in figure 5.1, at large (positive) ε, |Ψi → |R↑ R↓ i ≡ S(0, 2) and at large negative ε,
|Ψi becomes an unequal superposition of |L↑ R↓ i and |L↓ R↑ i. Even when Vx > |hIzRR −IzLL i|,
the inhomogeneous Overhauser effect will produce a preference for either the |L↑ R↓ i or the
|L↓ R↑ i 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 |Ψi ≈ S(0, 2) and detuning is swept
approximately adiabatically through the Ψ - |L↑ R↑ i anti-crossing. The position of this anticrossing, ε̃, is determined by the energy of |L↑ R↑ i ≡ T+ (see figure 5.1) which is determined
by Bext . Insofar as b(ε̃) 6= c(ε̃), a transition from Ψ to |L↑ R↑ i will preferentially induce a
nuclear spin flip (down) on the side with the larger Iz . This tends to equilibrate the values of
IzRR and IzLL . In the second pulse sequence the electrons are initialized into |L↑ R↑ i 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
|IzLL − IzRR | 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
98
Chapter 5: Inhomogeneous nuclear spin flips
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 Iα2 are conserved, where
P
I~α ≡ (vA/V) m∈α I~m , 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 (I~αα )2 and Iαz is the eigenvalue
of Izαα ). 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 ε̃. This
depends on ∆ and on Bext . For example, smaller Bext results in smaller (or more negative)
ε̃, 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(ε̃) as a function of Bext for various values of ∆.
The key point is that r(ε̃) increases monotonically with ∆ (cf. yellow highlighted region
of Fig. 5.2), however it also decreases monotonically with Bext (and hence ε̃). Interestingly, because the |R↑ R↓ i state is coupled equally to |L↑ R↓ i and |L↓ R↑ i, 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↑ i
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
4
Self consistent electronic structure calculations show (Stopa, unpublished) that for lateral double quantum dots, Vx can range from 250 µV to less than 1 µV . Here, we have chosen the lower value.
Chapter 5: Inhomogeneous nuclear spin flips
∆−1,s−1
EC-ε
γ*
γ*
0
0
0
0
0
γ
∆−1
Vx*
0
0
0
0
0
γ
0
Vx
0
−∆−1
0
s-1+EZ
0
0
0
IR+
0
0
0
0
0
99
∆,s
∆+1,s−1
0
0
0
0
EC-ε
γ*
γ*
0
0
0
0
IRγ
0
0
0
0
γ
∆
V x*
0
Vx
−∆
0
0
0
0
0
0
0
0
0
0
0
0
IL-
0
0
0
0
0
0
0
s+EZ
0
0
0
0
0
0
0
0
EC-ε
γ*
γ*
0
0
0
0
0
γ
∆+1
V x*
0
0
0
0
0
γ
0
0
IL+
0
0
0
0
Vx
−∆+1
0
s-1+EZ
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 manifestations 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 |Ψi ⊗ {IL , IR , ILz , IRz }
and the detuning is moved quickly to ε̃ 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
RR
|hILz IRz − 1|I−
|ILz IRz i|2
~2
A2 Ω2R− τ 2 2
=
|b|
4~2
=
(5.5)
where we have suppressed the IL , IR dependence for brevity and where the matrix elements of the ladder operators are given by the well-known formulas: Ωα± ≡ hIα , Iαz ±
p
1|I± |Iα , Iαz i = 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) =
A2 Ω2L− τ 2 2
|c| .
4~2
(5.6)
100
Chapter 5: Inhomogeneous nuclear spin flips
Ψ−component ratio c/b
1.10
∆ = 200
∆ = 600
∆ = 1000
∆ = 1400
∆ = 1800
1.08
1.06
1.04
1.02
0.2
0.4
0.6
0.8
1.0
Magnetic Field B [T]
Figure 5.3: The wave function ratio r ≡ c/b evaluated at the Ψ − T + crossing point, ε̃, as
a function of Bext for various values of the nuclear spin z-component difference ∆. r(ε̃) is
monotonically 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, ∆)
Ω2 (s, ∆)
|b(∆)|2
W (s, ∆ + 1) = W (s, ∆) 2 R−
ΩL− (s, ∆ + 1) |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 ε̃).
Recursion relation Eq. 5.7 can be solved iteratively and the influence of the narrowing 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 described 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
5
We assume Nnuc ∼ 106 nuclei in each dot leading to an average spin ∼
√
Nnuc ∼ 1000.
Chapter 5: Inhomogeneous nuclear spin flips
distribution W
6
(b) I
(a)
s
Rz
7
6
∆,s
5
ΓR
σΤ/σ
-3
8x10
101
4
ΓL
∆+1,s+1
∆+1,s-1
3
4
2
0.2
0.4
0.6
∆
Bext (T)
ILz
Bext=0.05,0.10,...,0.75 T
high temperature thermal distr.
0
0
500
1000
1500
∆=ILz-IRz
2000
Figure 5.4: (main) Reduced distribution W (∆, s), calculated from Eq. 5.7 (solid lines), for
Vx = 1 µeV as a function of ∆ for various Bext = 0.05, 0.10, ..., 0.75 T (lower fields have
narrower 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 |Ψi to |L↑ R↑ i 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
102
Chapter 5: Inhomogeneous nuclear spin flips
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, I~L and I~R , 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 ε̃, 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 Infrastructure Network Computation Project for computational support. We gratefully acknowledge
support from the Fannie and John Hertz Foundation, NSF grants PIF-0653336 and DMR05-41988 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 and
Technology, 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 dot
system undergoing adiabatic pumping. Our model, while semiclassical, allows
to explore a wide range of parameters. In different parameter regimes, we find
the system exhibits either a strong reduction or a large growth in the difference
Overhauser 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
1
Contributed 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
a
b
|T−
|T−
|T0
Energy
Energy
|T+
Dz
D+
Bext S−
|T0
D − S− S −
J | t=T
|S
|T+
|S
J | t=−T
ε
d
|T−
∆
Dz
|T−
Λ+
|T0
|T+
|T+
|S
|T−
D+
D−
|T+
|S
|T−
Λ
|T0
|T−
ΓR
|T0
S+
|T+
|S
|T0
D−
Sz
|S
|T−
Γ
|T0
Dz
∆-
105
|T0
D−
S−
|T+
|S
|S
D+
|T+
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 between
the s and T+ states as the exchange energy J(ε) is swept through the crossing, showing
the nuclear operators involved in each path. Each pathway is a term in D̃− in Eq. 6.2. c)
Annular approximation to the electron wavefunction in the double dot. d) Key processes
contributing 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
P 2 1/2
gkd = ahf v0 |ψd (rkd )|2 and the rms Overhauser energy is Ω = ( k gkd
) , 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
106
Chapter 6: Preparation of non-equilibrium nuclear spin states in quantum dots
P
operators to denote the Overhauser fields in the left (L) and right (R) dots, L = k gkl Ikl
P
and R = k gkr Ikr 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 {|si , |T+ i , |T0 i , |T− i}, where
the Tm are the (1, 1) triplet states, the Hamiltonian is [146]


√
vD+
− 2vDz
−vD− 
 −J(ε)



√

 v D−

−Bext + Sz S− / 2
0


H= √
.
√
√ 


− 2vDz
S+ / 2
0
S− / 2 




√
0
S+ / 2 Bext − Sz
−vD+
√
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 m̂ = (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, m̂
Heff
where


v(ε)D̃+ 
−J(ε) + hs
=
,
v(ε)D̃−
−Bext + hT
2v 2 †
v2
D̃z D̃z − D̃−
D̃+ ,
J
J + Bext − Sz
1
hT = Sz − (S− S+ m̂ + m̂S− S+ ),
4
1
1
2
D̃− = D− + m̂S− Dz − m̂S−
m̂D+ − m̂ S− S+ m̂D− ,
4
4
1
D̃z = Dz − S+ m̂ D− + S− m̂D+ .
2
hs = −
(6.1)
(6.2)
Chapter 6: Preparation of non-equilibrium nuclear spin states in quantum dots
107
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 |si at large negative t = −T . We identify the (nuclear spin) eigenstates of the operator
ˆ D̃
ˆ
2
D̃
+ − , labeled |D⊥ i with eigenvalues D⊥ . After the crossing, the state is unchanged or
ˆ |D i.
′ i ⊗ |T i where |D ′ i ≡ D −1 D̃
flips an electron and nuclear spin to the state |D⊥
+
−
⊥
⊥
⊥
ˆ D̃
ˆ
′ i is an eigenstate of D̃
2
We note that |D⊥
− + with eigenvalue D⊥ . The problem is now
reduced to finding Landau-Zener solutions for each independent two-level system |D⊥ i⊗|si,
′ i ⊗ |T i, where we approximate ĥ → hD | ĥ |D i, ĥ
′
′
|D⊥
+
s
T → hD⊥ | ĥT |D⊥ i, valid to
⊥ s
⊥
leading order in 1/J and 1/N . We model the actual sweep of ε by a linear sweep of J so
q
J(t) = −β 2 t + 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.
ˆ
D̃−
After one cycle, the state is |Ψi = (cS |si + cT D
|T i) |D⊥ i. For β 2 T ≫ 1, the flip
⊥
probability is pf = 1 − exp(−2πω 2 ), where ω = vD⊥ /β, and
cS =
φS ≈
φT ≈
p
Z
1 − pf exp(−iφS ),
cT =
√
pf exp(−iφT )
T
hS dt
−T
Z t0
−T
(6.3)
hS dt + (T − t0 )hT + φAD (ω),
where the crossing occurs at a time t0 ≈ Sz /β 2 . We include in φT the phase picked up
by following the adiabat, φAD . We approximate φAD by interpolating between the limits
ω = v D˜⊥ /β → 0 and ω → ∞, giving [153]
φAD
where τ = T β.
2 τ
2
= 2πω + pf ω 1 − 2π + log
− π/4 ,
ω2
2
108
Chapter 6: Preparation of non-equilibrium nuclear spin states in quantum dots
We move from the independent two-level systems to the general case by noting that the components of |Ψi depend only on the eigenvalue D⊥ and on the polarization Sz (which we approximate as commuting). Thus we can write an operator pˆf =
P
D⊥ pf (D⊥ ) |D⊥ i hD⊥ |, and similarly for φ̂S , φ̂T . 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 distribution, we choose such a spin coherent state and evolve it, where we interpret expectation
values h...i 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 , D̃± ) are only functions of L and R. A spin coherent state is entirely
described by its expectation values iid = hIid i. For the jth spin on the left dot, we expand
the discrete time difference hIjl in − hIjl in−1 after n and n − 1 cycles in the small parameter
gjl , giving an evolution equation
where
X
dijl
= gjl
Pl,µ i[∂gjl Lµ , Ijl ] = gjl Pl × ijl ,
dt
µ
Pl =
(6.4)
1
[h1 − pf i h∇l φS i + hpf i h∇l φT i − Im(γl )] ,
2T
where ∇l = (∂Lx , ∂Ly , ∂Lz ) and
γl = D̃+
pf
D̃− D̃+
∇l D̃− ,
(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
Chapter 6: Preparation of non-equilibrium nuclear spin states in quantum dots
109
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
P
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 In2 is a conserved quantity, so we can study the evolution of M ≪ N
p
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.
−1
To illustrate, to first order in m0 = Bext
, for d = l, r,
Pd =pf λ (Λ+ ẑ − Λ0 S⊥ ) + m0 Γ0
pf Dz
ẑ × D
2πω 2
(6.6)
h
β2
+ ΓR pf ∇d φAD ∓ Γ0
Im (γl − γr )
4πv 2
i
+ (1 − pf λ/2)(∆0 Dz ẑ + ∆− 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 ×
2 . It is clear from Eqs. 6.4-6.6 that all dynamics stop if D = 0. As indicated in
ẑ)pf /D⊥
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
110
Chapter 6: Preparation of non-equilibrium nuclear spin states in quantum dots
of only the Landau-Zener sweep, with instantaneous eject and reload, the parameters have
values
∆0 =
2v 2
J(t)
c
Λ+ = 1/4,
Γ0 =
2πv 2 fc
,
β2
≈ m0 ,
∆− =
v2
J(t) + Bext
c
≈ m0 /4
Λ0 = m0 /4
ΓR = f c
where fc = 1/2T is the cycle frequency and h.ic 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
v 2 ≈ 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
P
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 ≈ −In ẑ 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
Chapter 6: Preparation of non-equilibrium nuclear spin states in quantum dots
111
(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 observed. 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 m̂S− Dz term in D̃− 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− → D̃− . 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
112
Chapter 6: Preparation of non-equilibrium nuclear spin states in quantum dots
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, hSz ie is the median value of Sz at
each time step in an ensemble of 1000 trajectories. Gray shaded region shows the 84th and
16th percentiles. Above, with independent y-axis, is similar h|Dz |ie . Thin red line is a single
trajectory. For these parameters, the trajectories have |Dz | → 0 quickly, without time for
strong polarization. (inset) Mean of the maximum value of |Dz | reached on each trajectory
for the same parameters as in (b) (open circles) except M varied between 20 and 160, with
5000 trajectories per point. Closed circles show similar results with m0 = 0.05, τ = 4 and
all other parameters scaled appropriately. The physical system has M → N ≈ 106 , so we
interpret 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 presented
as 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 each
time. 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 red
line, where |Dz | is reduced initially and then goes unstable to large |Dz |.
Chapter 6: Preparation of non-equilibrium nuclear spin states in quantum dots
113
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 performed 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
In2 , 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 conduction band electrons consistent with the crystal symmetries is
HDresselhaus = γ[σx kx (ky2 − kz2 ) + 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, kz2 can be sent to hkz2 i and the 3D Dresselhaus
term produces the 2D linear and cubic Dresselhaus terms,
H001 = γhkz2 i(σy ky − σx kx ) + γ(σx kx ky2 − σy ky kx2 ).
(A.2)
Asymmetric confinement in the (001) plane also produces spin-orbit coupling of
the Rashba type [26],
HRashba = α(ky σx − kx σy ).
114
(A.3)
Appendix A: Appendix to Chapter 2
115
These three terms together give the effective Hamiltonian for 2D conduction electrons 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 (kx3 σx − ky3 σy ) while the cubic Rashba
terms are α1 (kx3 σy − ky3 σx ) + α2 (ky kx2 σx − kx ky2 σy ).
Winkler showed that γ1 (which he calls b6c6c
53 ) 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
γ2 k2 (kx σx − ky σy ) 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
hHext i along that direction. Noting that the only contribution of the confinement field is
∝ h∇z V i 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 = α3 k2 (ky σx − kx σy ), which gives the axial (C∞v ) approximation to the
cubic Rashba terms.
Using third-order perturbation theory [157, 108],
(0)
λ
P2 1
1
=
−
.
3 E02 (E0 + ∆0 )2
(A.5)
Fifth-order perturbation theory gives [50]
(1)
λ
=−
P 4 ∆0 24E03 + 41E02 ∆0 + 26E0 ∆20 + 6∆30
9E04 (E0 + ∆0 )4
2
.
(A.6)
4
In GaAs, these yield λ(0) = 5.3 Å and λ(1) = −870 Å .
Because this estimate does not include edge effects, it only provides a rough estimate 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 corresponding to the sample considered in the main text,
1
k2 λ(1)
≈ .
(0)
17
λ
(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 Dresselhaus is about twice the strength of the cubic Dresselhaus (i.e., hkz2 i/kF2 ≈ 2, using the
results in Ref. [106]), giving γ/α3 ≈ 8.
It is possible that a more careful calculation of the cubic Rashba strengths, including 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 ak 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 Zumbühl et al. var g versus Bk data discussed in the main text
[172]. Also shown are the original curvefit and our refit with ak constrained to be greater
than 8.1. This constrained fit clearly does not match the data. We thank Dominik Zumbühl
for making the data available to us.
−5
7
x 10
Data
Original Fit
a||≥8.1
6.5
var(g) (e2/h)2
6
5.5
5
4.5
4
3.5
3
0
1
2
3
4
5
6
7
B|| (T)
Figure A.1: Variance in conductance as a function of in-plane field for quantum dot of area
A = 8 µm2 and electron density n = 5.8 × 1015 m−2 , from Ref. [172]. A small perpendicular
magnetic field is applied to break time reversal symmetry. The original fit to the data is
shown along with our refit constraining only ak ≥ 8.1 and τφ = 0.39 ns (the value found
from hgi).
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
3
Table A.1: Magnitude of Dresselhaus spin-orbit coupling constant γ in GaAs (eVÅ ). All
results are in bulk GaAs unless otherwise specified.
Year
Ref
|γ|
’83
[102]
24.5
’83
’92
[8]
[45]
20.9
26.1
’93
[127]
23.5
’94
’95
’96
[76]
[77]
[126]
34.5
16.5±3
11.0
’03
[106]
28±4 ; 31±3
’09
[86]
5.0
’84
[36]
9 ; 8.5
’84
’85
’86
[128]
[16]
[99]
19 ; 30
28
27.57 ; 21.29
’88
’88
’90
’92
[120]
[27]
[118]
[154]
14.0
14.9 ; 28.2
24.12
18.3 ; 7.6 ; 36.3
’93
’94
[104]
[132]
19.8
8.9
’95
’96
’96
’03
[133]
[119]
[83]
[157]
8.5
24.21
10 ; 27.5
27.48
’05
’06
’09
[71]
[31]
[97]
23.6
8.5
5.7 ; 8.3
Technique/Comments
EXPERIMENT
Optical orientation using Dyakonov-Perel (DP) spin relaxation time. Conduction electrons in p-type GaAs.
Same as [102].
GaAs/AlGaAs heterostructure magnetoconductance
measurements. Sets kz2 → 0 instead of hkz2 i.
Raman scattering from 180Å GaAs quantum well
(QW). Does not include Rashba term.
Same as [127]. No Rashba term.
Raman scattering from GaAs/AlGaAs heterostructure.
Raman scattering from asymmetric GaAs/AlGaAs
QW. More data than earlier.
GaAs/AlGaAs heterostructure magnetoconductance
measurements.
GaAs/AlGaAs superlattice, transient spin grating measurement
THEORY
14 × 14 k · p theory ; Linear Muffin Tin Orbitals
(LMTO). Room temperature band gap.
8 × 8 ; 14 × 14 k · p models.
14 × 14 k · p.
14 × 14 k · p. Bulk GaAs ; Ga.65 Al.35 As/GaAs heterostructure.
14 × 14 k · p.
LMTO ; 16 × 16 k · p.
14 × 14 k · p.
14 × 14 k · p in bulk GaAs; bulk Al0.35 Ga0.65 As ;
GaAs/AlGaAs superlattice.
k · p.
sp3 s∗ tight binding (TB) model of a GaAs/AlAs superlattice.
TB model of 100 Å GaAs/AlAs QW.
14 × 14 k · p.
sp3 s∗ TB model ; k · p.
14 × 14 k · p theory. Notes that value is reduced to 19.6
in higher order perturbation theory (p. 74).
TB model used to refine 14 × 14 k · p.
Quasiparticle self-consistent GW method (QPscGW).
Pseudopotentials.
Appendix B
Appendix to Chapter 3:
Quaternions
A quaternion q is a 2 × 2 matrix of complex numbers
q = q (0) 112 + i
3
X
q (µ) σµ ,
µ=1
where σµ are the Pauli matrices and q (µ) ∈ C. We define three conjugates of q:
P
q ∗ = q (0)∗ 112 + i q (µ)∗ σµ ,
P
q R = q (0) 112 − i q (µ) σµ ,
q † = q R∗
complex conjugate:
quaternion dual:
and Hermitian conjugate:
119
(B.1)
120
Appendix B: Appendix to Chapter 3: Quaternions
If the 2×2 matrix
quaternion,
a b
c d
is expressed as a quaternion q, then in the 2 × 2 notation for the


q∗ = 
d∗
−c∗
−b∗



a∗


 d −b 
qR = 

−c
a


∗
∗
a c 
q† = 
.
b∗ d∗
(B.2)
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 = Qij ∗ ,
QR ij = QjiR ,
Q†
= (Q∗ )R ,
(B.3)
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 =
X
(0)
Qii ,
(B.4)
i
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 spinconserving 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 = Pφ σµ tr(σν Pφ Sw0 S † )(Nφ δµν − Ξµν )−1 ,
(C.1)
out ](µ) , where P
th channel
and gµs = 2tr(σµ PK wout ) = 2[wKK
K is the projection onto the K
(the output channel), and [q](µ) is the µ-component of the quaternion q.
The outgoing density matrix is wout = w0out + wφout where w0out = Sw0 S † and
121
122
Appendix C: Appendix to Chapter 3: Spin polarization forbidden with M = Nφ = 1
wφout = Swφ S † . For convenience of notation, the outgoing channel has index K and the
dephasing lead channel has index 1, so Pφ = P1 . Now
w0 =
1
(11 − PK − P1 ),
2N
(C.2)
1
∗
∗
(1 − SKK SKK
− SK1 S1K
).
2N
(C.3)
so
(w0out )KK =
As in Sec. 3.2, the first two terms are real, but the third term can have a quaternion
∗ is exactly canceled by (wout )
component. We show that SK1 S1K
KK in the final result for
φ
wout . We have
tr(σν P1 Sw0 S † ) =
hσ
ν
2N
i(0)
∗
∗
(1 − S1K SK1
− S11 S11
)
(C.4)
and
∗
,
Ξµν = tr(σν P1 Sσµ P1 S † ) = δµν S11 S11
(C.5)
where the last equality follows because S11 commutes with σµ , since S11 ∈ C. Together,
these give
wφ = P1 σµ
σµ
2N (1
∗ − S S ∗ ) (0)
− S1K SK1
11 11
∗
1 − S11 S11
(C.6)
with implied summation over µ. For q a quaternion, σµ [σµ q](0) = q, so
P1
wφ =
2N
∗
S1K SK1
1−
∗
1 − S11 S11
.
(C.7)
Then
wφout KK
1
=
2N
∗
SK1 S1K
∗ S∗
SK1 S1K SK1
1K
−
∗
1 − S11 S11
.
(C.8)
∗ in Eq. C.3. The second term is real, since S = S R ,
The first term in Eq. C.8 cancels SK1 S1K
R ∈ C. Thus, wout is real and g~s = 0. Since this result relies only
so SK1 S1K = SK1 SK1
KK
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 g~s 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:

0


0



1
S=

0



0


0
0
0
0
0
α
α

0 α −α


0 1 0 0 



0 0 0 0 
,

0 0 α α



α 0 0 0 


−α 0 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, g~s = ẑ/4,
and ~
p = ẑ/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, g~s = ẑ/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 g~s = 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

0 0 0


0 0 0



1 0 0



0 0 0
S=

0 α α



0 0 0



0 −α α


0 0 0
0 0 −α
1 0
0
0 0
0
0 0
α
0 0
0
0 0
0
0 0
0
0 1
0
0

α


0 0



0 0



0 α
.

0 0



−1 0 



0 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, g~s = ẑ/4,
and ~
p = ẑ/3. Using Eq. 3.23, wφ = Pφ /4, just as in the previous example.
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