Download Topics on the theory of electron spins in semiconductors

Topics on the theory of
electron spins in semiconductors
DISSERTATION
Presented in Partial Fulfillment of the Requirements for the Degree Doctor of
Philosophy in the Graduate School of The Ohio State University
By
Nicholas J. Harmon, B. A.
Graduate Program in Physics
The Ohio State University
2010
Dissertation Committee:
William O. Putikka, Advisor
John Wilkins
Ezekiel Johnston-Halperin
Richard Furnstahl
T
AF
c Copyright by
Nicholas J. Harmon
DR
2010
Abstract
As electron spin continues to be sought for exploitation in technological devices, understanding the spin’s coupling to its environment is essential. This dissertation theoretically
explores spin relaxation in two systems: the cubic zinc-blende and hexagonal wurtzite crystals.
First, bulk systems with the zinc-blende crystal structure are studied. A model that
includes both localized and itinerant spins and their interaction successfully explains the
observed phenomena. When this model is applied to certain quasi-two-dimensional structures of the same crystal type, it succeeds again after the exciton and exciton-bound-donor
spin species are introduced. For the first time a quantitatively accurate explanation of the
strange temperature dependence in intrinsic (110)-GaAs quantum wells is given.
Second, the properties of the wurtzite crystal are studied in regards to their usefulness
in spintronic devices. Research along these line in wurtzite is undeveloped in comparison
to zinc-blende. The theory of the D’yakonov-Perel’ and Elliott-Yafet spin relaxation mechanisms is developed. n-ZnO is concentrated on in bulk systems. The impurities must be
carefully considered when explaining the experimentally observed spin relaxation times; the
presence of a deep donor and a shallow donor give rise to the observed phenomena. Wurtzite
quantum wells have properties that could be especially beneficial spintronic devices. The
D’yakonov-Perel’ mechanism, which the dominant spin relaxation mechanism in the materials considered here, can be suppressed at low temperatures much more effectively than
can be done in zinc-blende due to the difference in spin-orbit fields. Suppression of spin
relaxation is also greater in wurtzite than in zinc-blende at room temperature due to the
smaller spin-orbit coupling in the examined wurtzite semiconductors.
Finally, the role an external magnetic field plays in the interacting localized-itinerant
spin picture is examined and used to explain a ‘spin beating’ phenomenon.
ii
“We usually only want to know something so that we can talk about it; in other words, we
would never travel by sea if it meant never talking about it, and for the sheer pleasure of
seeing things we could never hope to describe to others.”
-Pascal-
iii
Acknowledgments
First I would like to thank my advisor Prof. William O. Putikka, who provided guidance
and support through the entire period of my graduate research. Next I would like to thank
my advisory committee, Prof. Richard Furnstahl, Prof. Ezekiel Johnston-Halperin, and
Prof. John Wilkins who served as my committee members in my final oral exam and in my
candidacy exam. I would also like to thank Prof. Robert Joynt and Prof. Jay Kikkawa who
were invaluable sources and answered many of my questions over the past several years.
The various projects in this dissertation were supported by National Science Foundation
through Grant No. NSF-ECS-0523918 and by the Center for Emergent Materials at the
Ohio State University, an NSF MRSEC (Award No. DMR-0820414).
The list would be incomplete without thanking my friends: Sheldon Bailey, James C.
Davis, Kevin P. Driver, Ben Dundee, Michael Fellinger, Dave Gohlke, Robert Guidry, Adam
Hauser, William Parker, Patrick D. Smith, Jeffery Stevens, Rakesh Tiwari, Gregory Vieira,
and the Dr. K’s softball team for making my graduate student life more enjoyable than
it should be. I also thank my friends from back home and from Wooster: Christopher
Doherty, Joe Hall, Evan Rae, Josh Ross, Bradley Thomas, and the whole 5:15 crew. The
support from 1011H Beverage Company and it’s customers is also appreciated.
Many thanks to Dad and Mom who never taught me much physics but taught me the
more important things of life.
Last but not least, I would like to thank my beloved wife Barbara for putting up with
me working late nights and for letting me tell her more about that physics stuff.
iv
Vita
March 7, 1982 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Born - Bismarck, North Dakota
2004 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. A., The College of Wooster, Wooster,
OH
2004–2010 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Graduate Research/Teaching Assistant,
Dept. of Physics, The Ohio State University, Columbus, Ohio
Publications
[1] N.J. Harmon, W.O. Putikka, and R. Joynt, “Prediction of extremely long electron spin
relaxation times in wurtzite quantum wells”, in submission.
[2] N.J. Harmon, W.O. Putikka, and R. Joynt, “Theory of electron spin relaxation in ndoped quantum wells”, Phys. Rev. B 81, 085320 (2010).
[3] N.J. Harmon, W.O. Putikka, and R. Joynt, “Theory of electron spin relaxation in ZnO”,
Phys. Rev. B 79, 115204 (2009).
[4] J.F. Lindner, M.I. Rosenberry, D.E. Shai, N.J. Harmon, and K.D. Olaksen, “Precession
and chaos in the classical two-body problem in a spherical universe”, Internation J. of
Bifurcation and Chaos 18, 455-464 (2008).
[5] N.J. Harmon, C. Leidel, and J.F. Lindner, “Optimal exit: Solar escape as a restricted
three-body problem”, Am. J. Phys. 71, 871-877 (2003).
Fields of Study
Major Field: Physics
v
Table of Contents
Abstract . . . . . .
Dedication . . . .
Acknowledgments
Vita . . . . . . . .
List of Tables . .
List of Figures .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Page
.
ii
.
iii
.
iv
.
v
.
ix
.
x
Chapters
1 Introduction
1.1 Motivation . . . . . . . . . . . . . . . . . .
1.1.1 Spin field effect transistors . . . . . .
1.2 Organization of dissertation . . . . . . . . .
1.3 What is spin? . . . . . . . . . . . . . . . . .
1.3.1 A brief history of spin . . . . . . . .
1.4 Magnetic dipoles in a constant external field
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
1
1
4
6
7
7
9
2 Spin relaxation
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Spin density matrices and spins in a static magnetic field .
2.3 Spin relaxation due to random magnetic fields . . . . . . . .
2.3.1 The Redfield theory of relaxation . . . . . . . . . . .
2.3.2 The Redfield equation in the eigenstate formulation
2.3.3 The Bloch equations . . . . . . . . . . . . . . . . . .
2.3.4 Phenomenology . . . . . . . . . . . . . . . . . . . . .
2.4 The modified Bloch equations . . . . . . . . . . . . . . . . .
2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
11
11
13
15
15
17
20
24
24
26
.
.
.
.
.
.
28
28
28
31
33
40
41
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
3 Spin relaxation mechanisms in semiconductors
3.1 Introduction . . . . . . . . . . . . . . . . . . . . .
3.2 Conduction spin relaxation mechanisms . . . . .
3.2.1 The Elliott-Yafet mechanism . . . . . . .
3.2.2 The D’yakonov-Perel’ mechanism . . . . .
3.2.3 Finite temperatures . . . . . . . . . . . .
3.2.4 Hyperfine interaction . . . . . . . . . . . .
vi
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
3.3
.
.
.
.
.
43
43
45
48
48
4 Phenomenological approach to spin relaxation in semiconductors I; case
studies in bulk and quasi-2D zinc-blende crystals
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Bulk crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Spin relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.2 Comparison with experiments . . . . . . . . . . . . . . . . . . . . . .
4.3 Quasi-2D nanostructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Spin polarization in quantum wells . . . . . . . . . . . . . . . . . . .
4.3.2 Modified Bloch equations . . . . . . . . . . . . . . . . . . . . . . . .
4.3.3 Occupation concentrations . . . . . . . . . . . . . . . . . . . . . . . .
4.3.4 Spin relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.5 Results for GaAs/AlGaAs quantum well . . . . . . . . . . . . . . . .
4.3.6 Results for CdTe/CdMgTe quantum well . . . . . . . . . . . . . . .
4.3.7 Comparison of GaAs and CdTe quantum wells . . . . . . . . . . . .
4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
51
52
54
57
59
60
64
65
67
69
73
73
76
3.4
3.5
Localized spin relaxation mechanisms .
3.3.1 Hyperfine interaction . . . . . . .
3.3.2 Anisotropic exchange interaction
Cross-relaxation . . . . . . . . . . . . .
Summary . . . . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
5 Phenomenological approach to spin relaxation in semiconductors II;
case studies in bulk and quasi-2D wurtzite crystals
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Bulk crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.1 The Elliott-Yafet mechanism in bulk wurtzite crystals . . . . . . . .
5.2.2 The D’yakonov Perel’ mechanism in bulk wurtzite crystals . . . . . .
5.2.3 ZnO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Quasi-2D nanostructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.1 Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.2 Temperature dependence of DP mechanism in wurtzite and zincblende QWs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.3 Comparison between wurtzite and zinc-blende . . . . . . . . . . . . .
5.3.4 Tuning of spin-orbit parameters . . . . . . . . . . . . . . . . . . . . .
5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
99
102
106
108
6 Magnetic field effects
109
7 Conclusions
112
77
77
77
79
84
85
95
98
Appendices
A Material parameters
125
B An integral involving spherical harmonics
127
vii
C Important integrals
128
D The polylogarithm function
130
viii
List of Tables
Table
3.1
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
Page
The Dresselhaus spin-orbit Hamiltonians for bulk zinc-blende and wurtzite
semiconductors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Table of several semiconductor nanostructures with different growth orientations and their respective crystal axes. Also included is the parameter βD
which gives the strength of the linear Dresselhaus terms in the Dresselhaus
spin-orbit interaction (see Table 5.2). . . . . . . . . . . . . . . . . . . . . . .
The spin-orbit Hamiltonians for several semiconductor QWs with different
crystallographic orientations. The parameter βD is tabulated in Table 5.1. .
A table of the various quantities needed in determining the DP spin relaxation
rate in wurtzite and zinc-blende QWs. The quantities δν and ∆ν are located
in Table 5.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A table of the various quantities needed in determining the DP spin relaxation
rate inP
wurtzite and zinc-blende QWs. . . . . . . . . . . . . . . . . . . . . .
zz
Γzz = ∞
n=−∞ Γn as used in Eq. (5.64). Using αR = 0 for zb-(110). . . . .
1/τDP ,the DP spin relaxation rate, for several types of QWs in both the
degenerate and non-degenerate limits. ζ(F ) = 2m∗ kB T(F ) /~2 . . . . . . . . .
∗ ,the minimum DP spin relaxation rate, for several types of QWs in
1/τDP
both the degenerate and non-degenerate limits. . . . . . . . . . . . . . . . .
Parameters for several semiconductors. . . . . . . . . . . . . . . . . . . . . .
D.1 Using ν = 0. ζ(F ) = 2m∗ kB T(F ) /~2 . . . . . . . . . . . . . . . . . . . . . . .
ix
31
96
97
98
99
100
101
103
107
131
List of Figures
Figure
1.1
1.2
1.3
1.4
1.5
1.6
2.1
2.2
Page
The amount of transistors per die have approximately doubled every 24
months in the last several decades. A die is the block of material on which
the circuit is fabricated; its size is typically in the hundreds of millimeters squared. For comparison, the Pentium IV chips contains about 108
transistors/cm2 [1]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
In conventional electronics, electrons ‘see’ a potential profile (solid line denoted by E). When the potential is large (OFF), electrons cannot pass
through leading to reduced current. The opposite occurs when the potential
is low (ON). The potential is controlled by the gate voltage. Figure adapted
from [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Datta-Das spin transistor [3]. The source and drain are ferromagnets
(F). The channel is the two dimensional electron system (2DES) which has
a Rashba spin-orbit coupling induced by the gate voltage Vg . . . . . . . . .
The Stoner-Wohlfarth model of a ferromagnet’s energy dispersion. Polarization, P , is less than unity since at the Fermi energy, carriers of both spin
exist. Notice that at the Fermi level, the Fermi wave vector, kF , is different
for the two spin types. The two conduction band minima are offset by the
exchange splitting energy, ∆. . . . . . . . . . . . . . . . . . . . . . . . . . .
Schematic detailing the basic features of the spin relaxation transistor. The
source and drain are antiparallel ferromagnets (F). . . . . . . . . . . . . . .
Picture of the magnetic moment µ resultant from a current, I, flowing around
a rectangular loop of area A in an applied field B. The direction of the dipole
moment is determined from the right hand rule as shown. The torque due
to the external field is directed tangentially to the moment and causes it to
precess around the magnetic field. The energy of interaction between the
moment and the field is lowered if the moment aligned with the field. . . .
2
3
4
5
5
10
A π/2 pulse from an alternating magnetic field can rotate the magnetization
into the plane orthogonal to the static field. Relaxation processes will work
to restore the equilibrium magnetization. . . . . . . . . . . . . . . . . . . .
12
Depiction of T2 relaxation in rotating reference frame (with frequency ω0 ). In
this rotating reference frame, moments precessing at ω0 will appear stationary. 13
x
2.3
2.4
3.1
3.2
3.3
4.1
4.2
4.3
4.4
4.5
4.6
4.7
Occupations of conduction (red) and localized (blue) states for n-GaAs doped
at 2 × 1016 cm−3 (solid lines) and 1 × 1016 cm−3 (dashed lines). Parameters
used: m∗ = 0.067me , εB = −5.8 meV. Higher temperature are required to
deplete localized electrons in the higher doped system. . . . . . . . . . . .
Spin relaxation rate versus temperature in n-GaAs doped at 1016 cm−3 . Data
is from Ref. [4]. Solid lines are least squares fit using Eq. (2.45). Dasheddotted curve: (nl /nimp )(1/τl ) for B = 0 T. Dashed curve: (nl /nimp )(1/τl )
for B = 4 T. Dotted curve: (nc /nimp )(1/τc ) for B = 0 T. Inset: momentum
relaxation times for two different dopings: a) 1016 cm−3 b) 1018 cm−3 . Figure
is from Ref. [5]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Pictorial descriptions of three common conduction spin mechanisms: Elliott
Yafet (EY), D’yakonov Perel’ (DP), and Bir Aronov Pikus (BAP). For the
first two, each vertex represents a scattering event. For BAP, electrons and
holes are depicted by different colors and the arrows between them signify
the exchange interaction. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Spin polarization depends on k. For each k there are two mutually antiparallel pseudospins (only one of each pair is shown for a select few wave vectors).
Scattering that alters the momentum (from k1 to k2 ) also changes the spin
orientation. Graphic taken from [1] . . . . . . . . . . . . . . . . . . . . . . .
Spin relaxation time versus doping density. Three distinct regimes are observed, indicating three different spin relaxation mechanisms: hyperfine,
anisotropic exchange, and D’yakonov-Perel’. Figure taken from [6]. SNS
refers to non-invasive spin-noise-spectroscopy measurements while conventional probes are optical orientation experiments. Data point references are
located in [6]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Conventional cubic cell of the zinc-blende crystal structure. . . . . . . . . .
Band structure of GaAs at 300 K near the Γ-point (k = 0). Spin-splittings
of the conduction band due to the Dresselhaus interaction are too small to
be seen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Measured and theoretical spin relaxation rates below the metal-insulatortransition (nM IT ≈ 2 × 1016 cm−3 ) in n-GaAs at low temperatures (below
10 K). Symbols are various experiments referenced in [7]. The theory curves
contain no fitting parameters. The maximum spin relaxation time appears
around 0.15nM IT . A maximum spin relaxation time has been also seen in
one study on CdTe [8]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Adapted from [9] showing how the temperature dependence of the spin relaxation depends on the doping density. . . . . . . . . . . . . . . . . . . . .
Solid blue circles from Kikkawa using n-GaAs with nimp = 1 × 1016 cm−3 at
zero applied field [4]. Solid line is fit with Eq. (2.45). . . . . . . . . . . . . .
Solid blue circles from Malajovich et al. using n-ZnSe with nimp = 5 × 1016
cm−3 at zero applied field [10] Solid line is fit with Eq. (2.45). Used same
mobility as for GaAs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Solid blue circles from Sprinzl et al. using n-CdTe with nimp = 4.9 × 1016
cm−3 at zero applied field [8]. Solid line is fit with Eq. (2.45). Transport
time taken from mobility measurements of [11] . . . . . . . . . . . . . . . .
xi
27
27
30
32
50
52
53
56
57
58
59
60
4.8
Illustration of optical spin pumping in bulk semiconductor. CB and VB are
conduction and valence bands respectively. σ ± denotes the helicity of the
absorbed and emitted photons (wiggly lines). Photon promotes one electron
from VB to CB leaving a hole behind. The hole spin (thick arrows) is assumed
to relax much quicker than the electron spin (thin arrows). Sz is total electron
spin. Graphic taken from [12]. . . . . . . . . . . . . . . . . . . . . . . . . . .
4.9 Illustration of optical spin pumping in QWs when photo-excitation is resonant with the trion formation. When excited at the donor-bound-exciton
resonance instead, the picture is similar except that the exciton ‘steals’ an
electron from a neutral donor (or is actually captured by the neutral donor)
instead of a free electron. The key difference is that after hole spin relaxation
and recombination, the neutral donors are left with a net polarization instead
of the free electrons. Graphic taken from [12]. . . . . . . . . . . . . . . . . .
4.10 Illustration of optical pumping in QWs when photo-excitation is resonant
with exciton formation. The conduction band (CB) starts with no spin polarization as the exciton is created (left most panel). The hole spin bound
in the exciton rapidly relaxes (second panel). An antiparallel resident electron spin is grabbed from the electron gas (or the exciton binds to a neutral
donor). This leaves the resident electron polarized (third panel). Oppositely
oriented electron and hole spins recombine, casting the third electron back
into the electron sea adding more net spin moment (rightmost panel). Taken
from [12]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
5.1
78
The wurtzite crystal structure. . . . . . . . . . . . . . . . . . . . . . . . . .
xii
61
62
Chapter 1
Introduction
1.1 Motivation
The electronic transistor is an ubiquitous staple in early 21st century life. In the little
more than half century since John Bardeen, Walter Brattain, and William Shockley’s 1947
discovery that under certain circumstances output power could be larger than input power
for electrical contacts on germanium, the transistor has revolutionized every facet of modern
life. The world’s reliance on electronic devices stems not only from the original invention
but also from the continual improvements and extensions to electronic circuitry that has
contributed to the prominence of the transistor. The uncanny progress of the electronic
industry is best summarized by Moore’s Law: the number of transistors inexpensively
placed on an integrated circuit doubles approximately every two years [13]. As Figure 1.1
displays, this trend has continued for the last 50 years. When this exponential growth
will stop has been a matter of speculation and has often been predicted to occur in the
near future only to be proved wrong. However there are fundamental limits to how small
transistors can be manufactured and still operate satisfactorily.
To address the limits of conventional transistors that are rapidly being approached,
the chief characteristics of the transistor must first be discussed. The most rudimentary
definition of a transistor is a three terminal device where one terminal modulates the flow
of current between the other two terminals. The usefulness of the transistor comes from
the prospect that a small change in the voltage at one terminal leads to a large modulation
of current between the other two terminals; in other words there is gain. The type of
transistor to be considered here is the field effect transistor (FET). It is comprised of three
key terminal components: a source, drain, and gate. The voltage at the gate controls the
current between the source and drain by altering the potential barrier faced by electrons;
see Figure 1.2. The metal insulator field effect transistor (MISFET) is a popular type of
transistor. In the MISFET, the source and drain are metallic contacts with a semiconducting
region in between them. On top of the semiconductor is stacked, perpendicular to the path
1
Figure 1.1: The amount of transistors per die have approximately doubled every 24 months
in the last several decades. A die is the block of material on which the circuit is fabricated;
its size is typically in the hundreds of millimeters squared. For comparison, the Pentium
IV chips contains about 108 transistors/cm2 [1].
between the source and drain, an insulating layer and a metallic gate. Depending on the
applied voltage at the gate, a two dimensional electron gas may form at the surface of the
insulator and semiconductor which allows for n-type conduction [14]. As exemplified in
Figure 1.2, the most simplistic explanation of the MISFET is a resistor whose resistance is
controlled by a gate voltage.
The issues that are run into when the number of transistor per area is increased are
now addressed. A key feature of an operable transistor is the clear delineation between ON
and OFF; this means that the conductance of the ON-state should be much greater than
conductance of the OFF-state. Current ON to OFF ratios are in the range 106 [15]; channel
lengths now are as small as a few tens of nanometers whereas in the early 1970s lengths were
around ten microns. Miniaturizing transistors leads to larger leakage currents: unwanted
currents between source and drain when the transistor is in the OFF-state. This is also
known as standby or static power dissipation. The reason for this is that the potential
barrier that prohibits large conductances must be thinner and therefore tunneling through
the barrier becomes a possibility. The tunneling can be reduced by increasing the barrier
height but there is an energy cost for doing so as well as a lengthening of the time to switch
from OFF to ON [2]; the switching energy goes as 12 CVG2 where C is the gate capacitance
and VG is the gate voltage. The switching energy plays into the dynamic power dissipation
2
Figure 1.2: In conventional electronics, electrons ‘see’ a potential profile (solid line denoted
by E). When the potential is large (OFF), electrons cannot pass through leading to reduced
current. The opposite occurs when the potential is low (ON). The potential is controlled
by the gate voltage. Figure adapted from [2].
and is sought to be minimized. Additionally a high barrier must be maintained to avoid
substantial thermionic emission from the source over the barrier; the amount of electrons
thermalized into the channel goes as exp(−eVG /kB T ).
So there is a dilemma: transistors can be made smaller but to avoid standby power
dissipation barrier heights must be made larger which has the negative consequence of creating larger dynamic power dissipation [2]. The issue of power dissipation in MISFETS is
so fundamental that the International Technology Roadmap for Semiconductors has labeled
it a ‘red brick wall’ [16]. Bandyopadhyay and Cahay [15] have estimated the power dissipation on a chip in the year 2025 if Moore’s Law holds true; their calculation is sobering. A
Pentium IV chip contains about 108 transistors/cm−2 . Each transistor dissipates 1500-2500
eV when switched (ON ↔ OFF). Given a switching speed of 2.8 GHz and 5% activity level
at any one time, the power dissipation is around 5 W cm−2 . In 2025, if the density increases
to 1013 cm−2 and the clock speed increases to 10 GHz, the dissipation will be 2 MW cm−2
which is equivalent to the thermal load in the nozzle of a rocket ship.
Encoding information by charge displacement is an inherently inefficient means since
it requires an energy |∆V (Q1 − Q2 )| (charge into the channel, charge out of the channel).
Moreover, such encoding works best when Q1 and Q2 are very different in magnitude such
that the two states can be clearly distinguished. A much more energy efficient mechanism to
switch the transistor would be realizable if the charge in the channel could remain constant.
The fundamental enterprise of spin-electronics (spintronics) is to utilize carriers’ spin degrees
of freedom, which is a vector quantity, instead of its scalar charge. If altering spin states
is an energy cheap process, then different spin states, as opposed to charge states, would
be responsible for changing the conductance between source and drain and the problem of
thermal dissipation could be largely avoided.
3
1.1.1 Spin field effect transistors
In 1990, Datta and Das published the seminal article of semiconductor spintronics. The
article is benignly entitled “Electronic analog of the electro-optic modulator” [3]. In this
paper, they proposed a device very similar in structure to the traditional charge transistor
but functionally very different; current between the source and drain of the FET would
be modulated by altering the spin polarization of carriers. A simple schematic of the operation is shown in Figure 1.3. The source and drain in this spin field effect transistor
Figure 1.3: The Datta-Das spin transistor [3]. The source and drain are ferromagnets (F).
The channel is the two dimensional electron system (2DES) which has a Rashba spin-orbit
coupling induced by the gate voltage Vg .
(SPINFET) are ferromagnets. In the simple Stoner-Wohlfarth band structure of a ferromagnet (see Figure 1.4 where spin up is the majority carrier), the Fermi wave vectors of
p
√
the two spin carriers are kF,↑ = 2mEF /~ and kF,↓ = 2m(EF − ∆)/~ where ∆ is the
exchange splitting energy which is typically several electron volts. Electrons with majority
spin have a higher velocity and encounter less scattering than a minority spin. In other
words the amount of resistance experienced by a carrier will depend on that carriers spin.
If the carriers’ spins can be changed in the channel between the source and drain, it is
then possible to affect transisting action. The use of the semiconductor in the channel is to
allow the spins to be manipulated by an effective magnetic field created by the gate voltage.
The details of this effective magnetic field are discussed at length in a later chapter. This
effective field produces transistor action by rotating the spin of a carrier from majority to
minority (as shown in Figure 1.3) which in turn increases the resistance. In general, the
angle of rotation is θ = 2m∗ αR L/~2 where αR is a constant giving the strength of the effective field and L is the length of the channel [3]. Recently, the construction of this type of
transistor using InAs was reported [17] but it was questioned whether those results actually
display transistor action [18]. One main reason prohibiting the creation and utility of a
Das-Datta SPINFET is the smallness of the spin-orbit coupling and the inability to change
4
Figure 1.4: The Stoner-Wohlfarth model of a ferromagnet’s energy dispersion. Polarization,
P , is less than unity since at the Fermi energy, carriers of both spin exist. Notice that at
the Fermi level, the Fermi wave vector, kF , is different for the two spin types. The two
conduction band minima are offset by the exchange splitting energy, ∆.
the coupling with the application of small voltages [15]. To rotate the spin 180◦ with a
small spin-orbit field requires a longer channel which is detrimental to the overall aim of
the further miniaturization of transistors.
Many other ideas have been proposed and are discussed in [1]. One of these ideas the spin relaxation transistor [2] - is highlighted here. The spin relaxation transistor does
not rely on spin precession to modulate the current but instead relies on spin relaxation.
Similar to other conceptions, the ON-OFF states are determined by changing a spin-orbit
field via a gate voltage. Figure 1.5 shows how the ON and OFF states are differentiated.
Figure 1.5: Schematic detailing the basic features of the spin relaxation transistor. The
source and drain are antiparallel ferromagnets (F).
5
Rapid spin relaxation is not seen as a negative but instead defines the ON state since in
an unpolarized current, half of the carriers will be aligned with the ferromagnetic drain.
However the spin relaxation rate must be controlled since the OFF state requires very little
spin relaxation such that all spins will be anti-parallel with the drain. A successful spin
relaxation transistor entails a relaxation rate that can be tailored by a small gate voltage.
Though not implemented, one experiment suggests difficulties for the spin relaxation transistor. This study [19] found that the gate voltage needed to change by 3 V to decrease the
spin diffusion length by 2.5%.
The aim of this dissertation is to investigate materials in the hope that their spintronic
properties may be suitable for utilization in a new class semiconductor devices.
1.2 Organization of dissertation
In the remainder of this first chapter, the quantum mechanical concept of spin is introduced.
Chapters 2 and 3 can appropriately be labeled as background chapters; in chapter 2, the
subject of spin relaxation is discussed. The density matrix formalism is briefly outlined.
Finally, the Redfield theory of spin relaxation is derived and the most pertinent results from
the theory are discussed. Chapter 3 lays out the important spin relaxation mechanisms in
semiconductors and especially those mechanisms relevant to this dissertation.
Chapter 4 is divided into two sections: bulk and confined zinc-blende semiconductors.
The treatment of bulk zinc-blende is the application of Putikka and Joynt’s earlier work
[5] to more semiconductor materials (ZnSe and CdTe). Quasi-two-dimensional zinc-blende
nanostructures are also investigated. New ideas were developed to explain the experimental
phenomena. The analysis offers insight into the usefulness of certain experimental practices
to accurately measure electron spin relaxation times. The work in quasi-two-dimensional
nanostructures culminated in an article in Physical Review B [20]. Chapter 5 is set up
likewise except for wurtzite semiconductors. The bulk wurtzite semiconductor ZnO is examined carefully for the first time. In order to explain the observed spin relaxation times,
the analysis that was sufficient for zinc-blende systems is not so in ZnO. This work led to
another publication in Physical Review B [21]. As of now, spin lifetime measurements in
wurtzite quantum wells have not been conducted. The novel calculations here show that
wurtzite quantum wells may offer many benefits over their zinc-blende counterparts in the
emerging technologies that seek to utilize electron spin. This work has been submitted
to Physical Review Letters [22]. Chapter 6 includes current work exploring the effects of
magnetic fields on spin relaxation. Conclusions are drawn in chapter 7.
Several appendices have been included to aide in the coherence of the dissertation and
also offer the reader convenient reference. Appendix A contains a list of the notation used
throughout. Additionally, all relevant material parameters for the semiconductors studied
6
are compiled. Appendices B and C are guides to derivations and calculations that are too
long and would impede the flow of the dissertation if included within the main text.
1.3 What is spin?
“It appears to be one of the few places in physics where there is a rule which can be stated
very simply, but for which no one has found a simple and easy explanation. The explanation
is deep down in relativistic quantum mechanics. This probably means that we do not have
a complete understanding of the fundamental principle involved”[23]
- Richard Feynman
1.3.1 A brief history of spin
Much of the information in the following sections can be found in any standard treatment
of quantum mechanics [24, 1]. The lectures by Sin-itrio Tomonaga are especially insightful
[25].
In the mid-1910’s, Arnold Sommerfield and Peter Debye refined Niels Bohr’s atomic
model to account for observed energy splittings in magnetic fields. They did this by introducing the orbital and magnetic quantum numbers l and ml where −l ≤ ml ≤ l. The
angular momentum in a magnetic field was quantized with values m~. These ideas explained most of the possible atomic transitions but not all. In large magnetic fields energies
split once more - doubling the amount predicted in the contemporary theory of angular
momentum. This was termed the anomalous Zeeman effect.
In 1925, Ralph Kronig proposed that another angular momentum, in addition to the orbital angular momentum, must be present that couples to the magnetic field. He postulated
that this new angular momentum was due to the electron spinning on its own axis. If the
magnitude of this angular momentum was fixed at ~/2 he was able to explain the observed
atomic transitions. Kronig realized that his idea contained a serious flaw: it required the
electron to spin so rapidly that the surface velocity would exceed the speed of light by
over 60 times. He soon publicly pointed out this fallacy when Uhlenbeck and Goudsmit
published a similar idea. However Uhlenbeck and Goudsmit also realized that their results
were a factor of two off from experiment. L.H. Thomas soon corrected this discrepancy by
clarifying the electron’s rest frame. In 1927, Ronald Fraser found further confirmation of
the spin-model when he reinterpreted an experiment by Otto Stern and Walther Gerlach in
1922. In this experiment they had measured some sort of angular momentum quantization
but had ascribed it to orbital angular momentum. Fraser noted that their system of silver
atoms should have no orbital angular momentum in the ground state and hence actually
the spin angular momentum had been measured.
At about the same time the matrix mechanics and wave mechanics of Heisenberg and
7
Schrödinger, respectively, were being formulated. Pauli figured out how spin should be
incorporated. He noted that (1) the components of spin, since it is an angular momentum, should obey commutation relations akin to orbital angular momentum and (2) spin
measured along any coordinate axis yielded magnitudes ±~/2. The commutation relations
are then easy to predict: [Sx , Sy ] = i~Sz plus the cyclic permutations. The two possible
values of spin (as measured in Stern-Gerlach experiment for instance) suggest that the spin
operator, Sz , is a 2 × 2 matrix with two eigenvalues of ±~/2. Of course there is nothing
intrinsically special about the z-direction - the magnetic field could just as well be oriented
along x or y - so the operators Sx and Sy should also have the same eigenvalues as Sz .
With this constraint, in addition to the angular momentum commutation relations, Pauli
deduced that spin must be an operator of the form S = ~σ/2 where
!
!
!
0 1
0 −i
1 0
σx =
,
σy =
, and σz =
1 0
i 0
0 −1
are the Pauli spin matrices.
P.A.M. Dirac extended the Schrödinger equation to incorporate relativity. In the process, spin was introduced naturally and not post facto as Pauli has done. Dirac’s seminal
equation is
∂Ψ
qA
= (cα · (p −
) + βmc2 + qV )Ψ,
(1.1)
∂t
c
where Ψ is a four component wave function (χΦ) and Φ, the large component, and χ, the
i~
small component, are two component spinors themselves [24]. p is the momentum operator,
c is the speed of light in a vacuum, A is the magnetic vector potential, V is a scalar potential,
q is the charge of the particle, and m is its mass. α and β are the Dirac matrices which are
!
!
0 σ
I2
0
α=
and β =
.
σ 0
0 −I2
I2 is the 2 × 2 identity matrix. From now on only electrons (spin-1/2) are considered.
When considering the Dirac equation to order v 2 /c2 , the presence of the magnetic vector
potential naturally leads to the Zeeman Hamiltonian, HZ = −γe S · B = −gµB σ · B where
µB =
e~
2mc
is the electron Bohr magneton and γe = gµB is the gyromagnetic ratio [24]. In
the theory of Dirac, g = 2 - this implies that the gyromagnetic ratio for spin is twice as
large as for orbital angular momentum. If electrons really did spin in the physical sense, g
would expected to be one. When working to order v 4 /c4 , the spin-orbit effect ‘falls out’ of
the Dirac equation:
Hs.o. =
~
σ · [∇V × p].
4m2 c2
8
(1.2)
Assuming a spherically symmetric potential,
Hs.o. =
~ 1 dV (r)
σ · L.
4m2 c2 r dr
(1.3)
A certain case when the potential is not symmetric is of particular importance; taking
V = eF z where F is an electric field in the ẑ direction yields
Hs.o. =
e~F
(σy px − σx py ),
4m2 c2
(1.4)
which has the same form as the Rashba interaction to be discussed further in this dissertation.
1.4 Magnetic dipoles in a constant external field
Since electrons possess an intrinsic magnetic dipole moment, it is beneficial to recall some
results from classical physics. A current carrying loop is succinctly described by its magnetic
dipole moment. The moment, µ, is defined as IAn̂ where I is the current in the loop, A is
the area enclosed by the loop, and n̂ is a unit vector normal to the plane of the loop. It is
well known from the Lorentz force law that a current carrying wire feels a magnetic force
when the wire is in an applied field. For a loop of wire, a torque is exerted: T = IAB sin θ
where θ is the angle between B and n̂ [26]. The torque generalizes to
T = µ × B,
(1.5)
where µ = IAn̂. From picture in Fig. 1.6, it is evident (use the right hand rule) that
the torque causes the dipole to precess around the applied field. If θ = 90◦ is defined as
the zero of energy, then the potential energy of the dipole at an arbitrary angle can be
determined by calculating the work done to rotate it away from 90◦ . The potential energy
R
R
is Vint = T (θ)dθ = µB sin θdθ = −µB cos θ = −µ · B. Now instead of a loop of wire
consider a circling charged particle. The current, I, due to the charge traveling past any
point in the circle of radius r is qv/2πr. The dipole moment is found by multiplying by the
area of the orbit, giving µ = qvr/2 = (q/2m)mvr = ql/2m where l = mvr is the angular
momentum’s magnitude. In vector format, µ =
shown to be
dµ
dt
q
2m l.
The time evolution of the moment is
= −T = −µ × B.
For completeness, the quantum mechanical description of a spin in an external field
is treated now [27]. Given a field in the ẑ direction, the Zeeman Hamiltonian is HZ =
−γe Bz Sz = γe Bz ~σz /2. The eigenstates are just that of Sz : χ↑ and χ↓ . The eigenvalues
are ±γe Bz ~/2. Solutions to the time dependent Schrödinger equation (HZ χ = i~χ̇) are of
the form
χ(t) = aχ↑ e−ε↑ t/~ + bχ↓ e−ε↓ t/~ ,
9
(1.6)
Figure 1.6: Picture of the magnetic moment µ resultant from a current, I, flowing around
a rectangular loop of area A in an applied field B. The direction of the dipole moment
is determined from the right hand rule as shown. The torque due to the external field is
directed tangentially to the moment and causes it to precess around the magnetic field. The
energy of interaction between the moment and the field is lowered if the moment aligned
with the field.
where a and b are determined by initial conditions. In anticipation of the interpretation
to follow, these constants are taken to be cos(α/2) and sin(α/2) respectively. Expectation
values of the spin operators can readily be calculated by hSi i = χ(t)† Si χ(t) to obtain:
~
sin α cos(γe Bz t)
2
~
hSy i = − sin α sin(γe Bz t)
2
~
hSz i = cos α.
2
hSx i =
(1.7)
It is now clear that α marks the angle between B and hSi. By Ehrenfest’s theorem,
the expectation value of an operator should evolve like its classical analog. Therefore the
expectation value of the spin operator should evolve like the classical dipole moment µ,
dhSi
= γe hSi × B.
dt
This equation also holds for time dependent magnetic fields [28].
10
(1.8)
Chapter 2
Spin relaxation
2.1 Introduction
What is meant by ‘spin relaxation’ ? In non-magnetic systems, there is no net spin polarization under equilibrium conditions when an external magnetic field is absent. Hence
along any quantization direction, there is roughly an identical number of up and down
spins. When certain perturbations are applied to such a system (to be discussed later), a
nonequilibrium non-zero net spin polarization will form. Spin relaxation is the process by
which the spins of the system return to equilibrium (i.e. zero net spin polarization). When
an external magnetic field is present, there will be a non-zero net spin polarization under
equilibrium conditions (to be calculated in Section 2.2); this is the context in which nuclear
magnetic resonance (NMR - the spins be that of nuclei) and electron spin resonance (ESR
- the spin being that of electrons) is utilized.
In NMR experiments a second, alternating, magnetic field is applied in the plane orthogonal to the static field. The effect of this time varying field is to rotate the magnetization
away from its equilibrium position. If applied indefinitely, the magnetization continually
rotates between up and down. This phenomena is known as Rabi oscillations [24]. If applied
for a specific time period, the magnetization is rotated from the z-direction to the x-y plane
as shown in Figure 2.1. After the alternating field is shut off, the magnetization precesses
around the static field as expected from classical dynamics. Thorough discussions of NMR
can be found in many texts [28, 24]. Will the magnetic moment precess indefinitely? No,
it will relax - meaning equilibrium will be restored. ‘Relaxation’ can occur along several
pathways. Relaxation necessarily implies that the transverse moment (in x-y plane) decreases. The obvious way in which this happens is that the moment reorients along the
static field. The timescale for which this occurs is known as T1 . The more subtle way
is that some parts of the macroscopic moment may precess either quicker or slower than
the Larmor frequency, ω0 . When the microscopic moments are added vectorally, the net
moment is seen to decrease as the individual moments dephase. This is shown in Figure
11
2.2. The transverse relaxation timescale is T2 though there are other subtleties regarding
transverse relaxation that will be discussed later.
Figure 2.1: A π/2 pulse from an alternating magnetic field can rotate the magnetization
into the plane orthogonal to the static field. Relaxation processes will work to restore the
equilibrium magnetization.
The following observations can be made. T1 processes must result in a transfer of energy
since it involves magnetic dipoles reorienting in a magnetic field. Quantum mechanically,
it is a change of populations between spin-down states to spin-up states which are nondegenerate in a magnetic field. Since the energy is typically gained by the lattice, T1 is
frequently termed as the lattice or longitudinal spin relaxation time. Rarely, the term
‘spin relaxation’ is used to only describe T1 . Any longitudinal relaxation necessarily causes
transverse relaxation. Hence T2 should be constrained by T1 but later it will be shown that
it need not necessarily be less than T1 . T2 processes do not involve an energy transfer since
the moment is transverse to the static field. T2 is known as the transverse or spin-spin
relaxation time. It may also be referred to as a dephasing or decoherence time.
To describe spin relaxation, F. Bloch presented a set of equations which included T1
and T2 as phenomenological parameters in 1946 that now bear his name [29]. Bloembergen,
Purcell, and Pound devised a method by which these spin relaxation times could be derived
for nuclear systems affected my molecular motions [30]. This theory was further refined by
Wangsness and Bloch [31] and Redfield [32] in the 1950’s. It is this Wangsness, Bloch, and
Redfield theory (or often simply Redfield theory) that is described in the following sections.
Excellent reviews can be found in recent literature [33, 34, 35, 28]. It should be stated that
12
Figure 2.2: Depiction of T2 relaxation in rotating reference frame (with frequency ω0 ). In
this rotating reference frame, moments precessing at ω0 will appear stationary.
while much of this theory and nomenclature was developed for NMR and ESR, it is also
useful for the study of spin dynamics in general.
2.2 Spin density matrices and spins in a static magnetic field
In this section density matrices are introduced; the density matrix is a vital tool in the
theory of spins and their interactions. A wave function ψ describes some system such
that the expected value of some observable is hOi = hψ|O|ψi. If the wave function is
expanded in terms of basis states φn (that are complete and orthonormal), we obtain hOi =
P
∗
n,m cn cm hφn |O|φm i where the c’s are coefficients in the expansion of ψ. If the system is
a mixture of particles in different states, the coefficients will vary. So the expected value of
our observable will depend on the distribution of states. This can be expressed now as
hOi =
X
Wk hψk |O|ψk i =
k
XX
k
Wk c∗n,(k) cm,(k) hφn |O|φm i
(2.1)
n,m
where Wk denotes the probability of a specific ci,(k) occurring. The idea behind the density
matrix is that instead of describing a system in terms of a wave function and its expansion
coefficients (cn ), the system’s information is carried in the ensemble average of the product of
P
coefficients ( k Wk c∗n,(k) cm,(k) ). We set this ensemble average equal to the matrix elements
of the density matrix hφm |ρ|φn i. The density matrix can be explicitly written as
ρ=
XX
k
Wk c∗n,(k) cm,(k) |φm ihφn |.
(2.2)
nm
What information do the diagonal components of the density matrix carry? hm|ρ|mi =
P
ρmm = k Wk |cm,(k) |2 which is the probability of finding the system in the state |φm i.
The expected value of some observable is expressible in terms of the density matrix
P
as hOi = n,m hφm |ρ|φn ihφn |O|φm i = Tr(ρO). As shown in standard texts [36, 28], the
Schrödinger equation can be redrafted in terms of the density matrix which becomes what
13
is known as the von Neumann or Liouville equation:
dρ
i
= [ρ, H ].
dt
~
(2.3)
As an instructive example, spin- 12 particles with magnetic moments µi =
~
2 γe σ i
are
considered in an external magnetic field. The Hamiltonian is H0 = −µ·H. We are interested
in the expected value of the various magnetic moment components hµi i =
~
2 γe hσi i.
γe is
the electronic gyromagnetic ratio and is equal to gµB . The time dependence of a Pauli spin
matrix is found by using the properties of the density matrix:
i~
dhσi i
dt
d
dρ
(Tr(ρσi )) = i~Tr( σi ) = Tr([H0 , ρ(t)]σi ) = Tr([σi , H0 ]ρ(t))
dt
dt
1
(2.4)
= − γe ~Hz Tr([σi , σz ]ρ(t)).
2
= i~
Any 2×2 matrix can be decomposed into a sum of the identity and Pauli spin matrices as
P
ρ(t) = 21 (I2 + k mk σk ) where I2 is the 2×2 identity matrix. It is simple to show that
hσi i = mi . Substituting for the density matrix in Eq. (2.4), we ascertain
i~
X
dhσi i
1
= − γe ~Hz Tr([σi , σz ]) +
mk Tr([σi , σz ]σk ) .
dt
4
(2.5)
k
The first term in the sum is zero since the commutator of two identical operators is zero and
the commutator of two different Pauli spin matrices is traceless. The second term can be
compactly written as Tr([σi , σz ]σk ) = 4iizk , where izk is the antisymmetry (or Levi-Civita)
tensor defined as
ijk

if i,j,k are cyclical

 1
=
.
−1 if i,j,k are not cyclical


0
it two indices are identical
We therefore are left with
X
dhσi i
dmi
=
= −γe
Hz mk izk .
dt
dt
(2.6)
k
For an arbitrary magnetic field, the above equation generalizes to
X
dmi
= −γe
Hj mk ijk .
dt
(2.7)
j,k
Looking at a single component of the spin polarization mx ,
dmx
= −γe (Hy mz − Hz my ) = γe (m × H)x .
dt
14
(2.8)
This generalizes to the familiar result
dm
= γe m × H.
dt
(2.9)
Another useful example of the density matrix is from equilibrium statistical mechanics:
what is the magnetization of an ensemble of spins in a static magnetic field? The diagonal
elements of the density matrix give information on the populations of the eigenstates here spin up and down. In equilibrium, it is assumed that these populations are given
P −Em /k T
−En /k T
B
by the Boltzmann factor: e Z B where Z =
is the partition function.
me
The off-diagonal terms are zero by the ‘hypothesis of random phases’ [28]. This assumption
implies nothing more than that the magnetization transverse to the external field will vanish.
Therefore ρ =
1 −H0 /kB T
Ze
where H0 = −γe ~Hz σz /2; in matrix form
1
ρ = −γ ~H σ /2k T
e
z
z
B
e
+ eγe ~Hz σz /2kB T
!
e−γe ~Hz σz /2kB T
0
0
eγe ~Hz σz /2kB T
Using the simple results from the theory of density matrices, N µz =
γe ~
2 Tr(ρσz )
.
≈
γe2 ~2 Hz
4kB T
in
the high temperature limit. This is the familiar Curie law for paramagnetic magnetization
[37].
2.3 Spin relaxation due to random magnetic fields
In the previous section, the density matrix formalism was introduced and used in two
simple situations: spin dynamics and populations in a uniform magnetic field. While the
phenomenological times T1 and T2 have been introduced to describe spin relaxation, no
physical mechanisms have been touched upon. The goal of this section is to show in detail
how one certain environmental interaction, that of a fluctuating magnetic field, can lead to
spin relaxation. Explicit formulae will be derived for T1 and T2 in terms of the interaction
parameters.
The theory that allows this description is the aforementioned Redfield theory of spin
relaxation. It is a semi-classical theory in that spin is accounted for quantum mechanically
but the environment (or lattice) is dealt with classically. It is helpful to remember that it
is essentially a second order time dependent perturbation theory calculation. It is only the
density matrix notation that causes the development of the theory to appear foreign.
2.3.1 The Redfield theory of relaxation
Consider the following time dependent Hamiltonian:
H (t) = H0 + H1 (t)
15
(2.10)
where H0 is due to the presence of a time independent external magnetic field and H1 (t)
is a perturbative Hamiltonian that results from a much smaller magnetic field that is a
random function of time turned on at t = 0.
dρ(t)
i
= [ρ(t), H1 (t)]
dt
~
(2.11)
An operator, O(t), in the interaction representation is denoted by
Õ(t) = eiH0 t/~ O(t)e−iH0 t/~ .
(2.12)
dρ̃(t)
i
= [ρ̃(t), H˜1 (t)]
(2.13)
dt
~
Integration of the Liouville equation of motion in the interaction representation leads to
Z
i t
ρ̃(t) = ρ̃(0) +
[ρ̃(t0 ), H˜1 (t0 )]dt0 .
(2.14)
~ 0
This can then be substituted back into Eq. (2.13) which yields
dρ̃(t)
i
1
= [ρ̃(0), H˜1 (t)] − 2
dt
~
~
Z
t
[[ρ̃(t0 ), H˜1 (t0 )], H˜1 (t)]dt0
(2.15)
0
This iterative solution is exact. To proceed, we consider an ensemble of ensembles that
are identical at t = 0 but then evolve with different perturbative Hamiltonians H1 (t) [38].
We assume that the ensemble of ensemble average of H1 (t) vanishes for all times. The
averaging is denoted by a bar such that H˜1 (t) = H1 (t) = 0. If this is not the case,
whatever non-vanishing piece can be subsumed into H0 . So Eq. (2.15) averaged becomes:
dρ̃(t)
1
=− 2
dt
~
Z
t
[[ρ̃(t0 ), H˜1 (t0 )], H˜1 (t)]dt0 .
(2.16)
0
The first term of Eq. (2.15) vanishes because ρ̃(0) is independent of the perturbing Hamiltonian so this term is linear in H˜1 (t) and therefore vanishes as mentioned above. To proceed,
several assumptions must be made concerning the perturbing Hamiltonian. First a correlation time, τc is introduced. Physically this quantity is the timescale on which the random
field fluctuates. Values of the field separated by times longer than τc are therefore uncorrelated. It is assumed that this correlation time is much shorter than the timescale on which
ρ̃(t) changes. This is justifiable since ρ̃(t) would be time independent if the perturbing
Hamiltonian did not exist (Eq. (2.16)); since the fluctuating field is small, the time evolution of ρ̃(t) is slow [33]. This separation of timescales allows the following:
• ρ̃(t0 ) may be replaced by ρ̃(t). Eq. (2.16) implies that ρ̃(t) depends on its history
from times t0 = 0 to t0 = t. However we know that the system’s “memory” is erased after
a short time τc . As previously shown, ρ̃ weakly depends on time so ρ̃(t − τc ) is not an
16
appreciable change and ρ̃(t0 ) ≈ ρ̃(t). In other words the time rate of change of the density
matrix depends on the current density matrix and not on its past history. This is known
as the Markov approximation [36, 39, 35].
• Secondly, correlations between ρ̃(t) and H˜1 (t) are neglected due to the difference in
time scales.
• Thirdly, the upper limit of the integral can be taken to ∞ since the short correlation
time implies the integrand vanishes at long times. Hence this theory is limited to describing
the dynamics at times longer than τc .
As we proceed, these points will be revisited as necessary. With these assumptions in
mind, a master equation is ascertained:
dρ̃(t)
1
=− 2
dt
~
Z
∞
[[ρ̃(t), H˜1 (t0 )], H˜1 (t)]dt0 .
(2.17)
0
Alternative derivations can be found in Refs. [33, 34]. In order to obtain useful results (spin
relaxation rates) from the master equation, we will examine the master equation element by
element. In doing so, we will derive the Redfield equation. This is known as the eigenstate
formulation of Redfield theory.
2.3.2 The Redfield equation in the eigenstate formulation
In this section, the Redfield equation is derived in a basis set where the spin eigenstates
are of the unperturbed static Hamiltonian. We would like to calculate each element of the
master equation in this basis:
dρ̃(t)αα0
dhα|ρ̃(t)|α0 i
1
≡
=− 2
dt
dt
~
Z
∞
hα|[H˜1 (t), [H˜1 (t0 ), ρ̃(t)]]|α0 idt0 ,
(2.18)
0
where |αi is an eigenstate of H0 . The double commutator expands to
H˜1 (t)H˜1 (t0 )ρ̃(t) + ρ̃(t)H˜1 (t0 )H˜1 (t) − H˜1 (t0 )ρ̃(t)H˜1 (t) − H˜1 (t)ρ̃(t)H˜1 (t0 ).
(2.19)
Three complete sets of states are inserted (such that the matrix elements of all three operators can be specified) into the matrix element hα|[H˜1 (t), [H˜1 (t0 ), ρ̃(t)]]|α0 i such that
XX
β
β0
hα|
X
H˜1 (t)|γihγ|H˜1 (t0 )|βihβ|ρ̃(t)|β 0 ihβ 0 | +
γ
|βihβ|ρ̃(t)|β 0 ihβ 0 |H˜1 (t0 )|γihγ|H˜1 (t) −
H˜1 (t0 )|βihβ|ρ̃(t)|β 0 ihβ 0 |H˜1 (t) −
!
H˜1 (t)|βihβ|ρ̃(t)|β 0 ihβ 0 |H˜1 (t0 ) |α0 i.
17
(2.20)
By using the relation between the Schrödinger and Interaction representations, hm|H˜1 (t)|ni =
hm|H1 (t)|niei(Em −En )t/~ (where Em and En are the eigenvalues of H0 ), we ascertain
X
0
hα|H1 (t)|γihγ|H1 (t0 )|βihβ|ρ̃(t)|β 0 ihβ 0 |α0 iei(Eγ −Eβ )t /~ ei(Eα −Eγ )t/~ +
β,β 0 ,γ
X
0
hα|βihβ|ρ̃(t)|β 0 ihβ 0 |H1 (t0 )|γihγ|H1 (t)|α0 iei(Eβ0 −Eγ )t /~ ei(Eγ −Eα0 )t/~ −
β,β 0 ,γ
X
0
hα|H1 (t0 )|βihβ|ρ̃(t)|β 0 ihβ 0 |H1 (t)|α0 iei(Eα −Eβ )t /~ ei(Eβ0 −Eα0 )t/~ −
β,β 0
X
0
hα|H1 (t)|βihβ|ρ̃(t)|β 0 ihβ 0 |H1 (t0 )|α0 iei(Eβ0 −Eα0 )t /~ ei(Eα −Eβ )t/~ .
(2.21)
β,β 0
The time t0 can be redefined as t0 = t + τ . Now consider ensemble (or time) averaging as in the master equation. As mentioned earlier, correlations between the perturbing
Hamiltonian and the density matrix are neglected. So we are interested in quantities like
hα|H1 (t)|βihβ 0 |H1 (t + τ )|α0 i = hα|H1 (t − τ )|βihβ 0 |H1 (t)|α0 i which we define to be correlation functions Gαβα0 β 0 (τ ) which is real and an even function of τ [28]. These results allow
us to write the master equation as
Z
X
dρ̃(t)αα0
1 X ∞
=− 2
dτ δα0 β 0
Gαγβγ (τ )ei(Eγ −Eβ )τ /~ ei(Eα −Eβ )t/~ (2.22)
dt
~
0
γ
β,β 0
X
−i(Eγ −Eβ 0 )τ /~ i(Eβ 0 −Eα0 )t/~
Gβ 0 γα0 γ (τ )e
e
+δαβ
γ
−Gαβα0 β 0 (τ )ei(Eα −Eβ )τ /~ ei(Eα −Eβ +Eβ0 −Eα0 )t/~
!
−Gαβα0 β 0 (τ )e−i(Eα0 −Eβ0 )τ /~ ei(Eα −Eβ +Eβ0 −Eα0 )t/~ ρ̃(t)ββ 0 .
To proceed, we must use fact that the perturbations are stationary which implies that [28]
Gαβα0 β 0 (τ ) = hα|H1 (t)|βihβ 0 |H1 (t + τ )|α0 i = hα|H1 (t − τ )|βihβ 0 |H1 (t)|α0 i
= hβ 0 |H1 (t + τ )|α0 ihα|H1 (t)|βi = Gβ 0 α0 βα (−τ ).
(2.23)
This allows us to write
Z
X
dρ̃(t)αα0
1 X ∞
=− 2
dτ δα0 β 0
Gγβγα (−τ )e−i(Eγ −Eβ )(−τ )/~ ei(Eα −Eβ )t/~
dt
~
γ
β,β 0 0
X
−i(Eγ −Eβ 0 )τ /~ i(Eβ 0 −Eα0 )t/~
+δαβ
Gβ 0 γα0 γ (τ )e
e
−
(2.24)
γ
Gβ 0 α0 βα (−τ )e−i(α−Eβ )(−τ )/~ ei(Eα −Eβ +Eβ0 −Eα0 )t/~
!
−Gαβα0 β 0 (τ )e−i(Eα0 −Eβ0 )τ /~ ei(Eα −Eβ +Eβ0 −Eα0 )t/~ ρ̃(t)ββ 0 .
18
We now define what are known as spectral density functions:
Z ∞
0
0
0
0
dτ Gαβα0 β 0 (τ )e−i(Eα0 −Eβ0 )τ
Jαβα β (Eα − Eβ ) = 2
(2.25)
0
which results in the master equation
X
dρ̃(t)αα0
1 X
Jγβγα (Eγ − Eβ )ei(Eα −Eβ +Eβ0 −Eα0 )t/~
=− 2
δ α0 β 0
dt
2~
γ
β,β 0
X
+δαβ
Jγα0 γβ 0 (Eγ − Eβ 0 )ei(Eα −Eβ +Eβ0 −Eα0 )t/~ −
(2.26)
γ
Jαβα0 β 0 (Eα − Eβ )ei(Eα −Eβ +Eβ0 −Eα0 )t/~
!
−Jαβα0 β 0 (Eα0 − Eβ 0 )ei(Eα −Eβ +Eβ0 −Eα0 )t/~ ρ̃(t)ββ 0
after indices of the second and third terms have been rearranged as allowed. Also the
first two terms have been multiplied by different exponential terms which has no effect
(multiplying by unity) because of the Krönecker delta in each of those terms. The spectral
function is a complex function but the imaginary part which produces the dynamic frequency
shift which is in effect a small effective field which can be added to the large static field [34].
This phenomenon is ignored here. These results are summed up in the Redfield equation:
X
dρ̃(t)αα0
=
Rαα0 ββ 0 ei(ωα −ωβ +ωβ0 −ωα0 )t ρ̃(t)ββ 0
dt
0
(2.27)
β,β
where ωi = Ei /~ and
Rαα0 ββ 0 = −
X
X
1
0β0
Jγα0 γβ 0 (ωγ − ωβ 0 ) −
J
(ω
−
ω
)
+
δ
δ
γ
γβγα
β
αβ
α
2~2
γ
γ
!
Jαβα0 β 0 (ωα − ωβ ) − Jαβα0 β 0 (ωα0 − ωβ 0 )
(2.28)
is known as the Redfield relaxation matrix. Three more steps are in order:
• it is common practice to leave off the over-bars which denote ensemble averaging.
• terms with ωα − ωβ + ωβ 0 − ωα0 6= 0 oscillate rapidly and average to zero such that
ωα − ωβ + ωβ 0 − ωα0 = 0 dominates (secular approximation). The exponential terms also
will vanish exactly when the representation is switched to that of Heisenberg (see Section
2.3.3) [38].
• so far the calculation has been done at infinite temperature - physically this means
that there would be no equilibrium magnetization along the direction of the static field.
This is remedied phenomenologically by ρ̃(t)ββ 0 → ρ̃(t)ββ 0 − ρ̃eq
ββ 0 . A quantum mechanical
treatment of the lattice has been been reviewed by several authors [33, 35].
19
In conclusion we write the Redfield equation in its final simplified form
X
dρ̃(t)αα0
=
Rαα0 ββ 0 (ρ̃(t)ββ 0 − ρ̃eq
ββ 0 )
dt
0
(2.29)
β,β
where the relaxation matrix is defined as above. Redfield theory can also be derived through
the operator formulation [33, 34, 35, 38].
2.3.3 The Bloch equations
After developing the formalism above, it is now instructive to obtain some useful results from
it. The Bloch equations are derived here from a simple model perturbative Hamiltonian.
The spin relaxation rates T1 and T2 will be obtained for this model.
Recalling that H (t) = H0 + H1 (t) we set forth a general perturbation Hamiltonian
that models a fluctuating magnetic field:
1
H1 (t) = ~ ωx (t)σx + ωy (t)σy + ωz (t)σz .
2
(2.30)
H0 = 21 ~ω0 σz for a large static field in the z-direction. ω0 = γe H0 is the Larmor frequency
and γe =
gµB
~ .
Several more simplifications must be made to make the problem tractable:
• different components of the random field are independent (e.g. ωx (t) is not correlated
to ωy (t)).
• random fields in the same direction are correlated up to a correlation time τc .
• after making these assumptions, we substitute Eq. (2.30) into the correlation function
2 P
0
0
Gαβα0 β 0 (τ ) = hα|H1 (t)|βihβ 0 |H1 (t + τ )|α0 i = ~4
i,i0 ωi (t)ωi0 (t + τ )hα|σi |βihβ |σi0 |α i.
• a component of the random field will be correlated for a short time and then at longer
times be uncorrelated. A simple model of the correlation function is then ωi (t)ωi0 (t + τ ) =
δi,i0 ωi2 e−τ /τc [39, 28]. The correlation and spectral density functions then become
2 P
0
0 2 −τ /τc ,
Gαβα0 β 0 (τ ) = ~4
i hα|σi |βihβ |σi |α iωi e
Jαβα0 β 0 (α0 − β 0 ) = 2
Z
∞
dτ
0
= 2
~2
4
~2 X
hα|σi |βihβ 0 |σi |α0 iωi2 e−τ /τc cos((ωα0 − ωβ 0 )τ )
4
i
X
hα|σi |βihβ 0 |σi |α0 iωi2
i
τc
,
1 + ωα2 0 β 0 τc2
(2.31)
where we have neglected any imaginary portions of the spectral functions.
At this juncture, we transform the density matrix back into the Heisenberg representation:
dρ̃αα0
dt
d
d i(ωα −ωα0 )t
iH0 t/~ −iH0 t/~ 0
0
=
hα|e
ρe
|α i =
e
hα|ρ|α i
dt
dt
d
= ei(ωα −ωα0 )t hα|ρ|α0 i + i(ωα − ωα0 )ei(ωα −ωα0 )t hα|ρ|α0 i.
dt
20
(2.32)
The Redfield equation in the Heisenberg representation then becomes
X
dρ(t)αα0
= i(α0 − α)ρ(t)αα0 +
Rαα0 ββ 0 (ρ(t)ββ 0 − ρeq
ββ 0 )
dt
0
(2.33)
β,β
where all the exponential terms of Eq. (2.27) have conveniently vanished without the need
to invoke the secular approximation. We now write these differential equations in terms of
observables - the magnetization vector m. Recalling the definition of the trace operator in
P
the familiar expression mi = ~/2Tr(ρσi ), we write mi = ~/2 α hα|ρσi |αi. If a complete set
is inserted, then the observable is expressible in terms of the elements of the density matrix:
P
P
mi = ~/2 α,α0 hα|ρ|α0 ihα0 |σi |αi = ~/2 α,α0 ραα0 hα0 |σi |αi. After summing Eq. (2.33) over
α and α0 and multiplying by ~hα0 |σi |αi/2, the following is ascertained:
dmi
dt
=
~ X dρ(t)αα0 0
hα |σi |αi =
2
dt
0
α,α
=
~
~iX
[ρ, H0 ]αα0 hα0 |σi |αi +
2~ 0
2
α,α
=
=
~
~i
Tr([ρ, H0 ]σi ) +
2~
2
X
0
Rαα0 ββ 0 (ρ(t)ββ 0 − ρeq
ββ 0 )hα |σi |αi
β,β 0 ,α,α0
0
Rαα0 ββ 0 (ρ(t)ββ 0 − ρeq
ββ 0 )hα |σi |αi
X
β,β 0 ,α,α0
~ i ~ω0
~
Tr(ρ[σz , σi ]) +
2~ 2
2
X
0
Rαα0 ββ 0 (ρ(t)ββ 0 − ρeq
ββ 0 )hα |σi |αi.
(2.34)
β,β 0 ,α,α0
Recalling Section 2.2, the first term on the right hand side is equal to γe (m × H0 )i . The
P
second terms can be shown to be ~3 /8 α,β,j hβ|σj |αihα|[[σi , σj ], ρ]|βiωj2 1+ωτ2c τ 2 [28]. Now
αβ c
consider specific components - dmz /dt for instance. In the sum over j, j = z has no
contribution since [σz , σz ] = 0. σx and σy have no diagonal elements so β and α will differ
P
ωx2 +ωy2
eq
0
by ω0 . In the end β,β 0 ,α,α0 Rαα0 ββ 0 (ρ(t)ββ 0 − ρeq
ββ 0 )hα |σi |αi = −τc 1+ω 2 τ 2 (mz − mz ). In
0 c
totality,
ωx2 + ωy2
dmz
= γe (m × H0 )z − τc
(mz − meq
z ).
dt
1 + ω02 τc2
(2.35)
This is indeed the Bloch equation for the z-component of the magnetization. The first
term describes the precession in the external static field. The second term describes the
relaxation due to the small fluctuating magnetic field. The timescale on which the nonequilibrium magnetization is restored to its equilibrium value of meq
z is T1 - the longitudinal
or spin-lattice spin relaxation time:
ωx2 + ωy2
1
τc
= τc
= (ωx2 + ωy2 )
.
2
2
T1
1 + ω0 τc
1 + ω02 τc2
(2.36)
The evolution of mx and my can be treated similarly except that now there will be no
equilibrium magnetization for those components since the static field is oriented along z.
21
Also for j = z, t α = β since σz is diagonal. The following are obtained:
ωy2
dmx
mx − τc ωz2 mx
= γe (m × H0 )x − τc
dt
1 + ω02 τc2
dmy
ωx2
my − τc ωz2 my
= γe (m × H0 )y − τc
dt
1 + ω02 τc2
(2.37)
which is more succinctly expressed as
1
dmx
mx
= γe (m × H0 )x −
dt
T2x
dmy
1
my
= γe (m × H0 )y − −
dt
T2y
(2.38)
with
ωy2
1
= τc
+ τc ωz2
T2x
1 + ω02 τc2
1
ωx2
+ τc ωz2 .
= τc
T2y
1 + ω02 τc2
(2.39)
Note that both magnetizations decay due to fluctuating fields orthogonal to the magnetization direction. This is sensible since fields parallel to a moment cannot act on the moment.
The in-plane magnetization, m⊥ relaxes with a rate
1
T2
= 21 ( T12x +
1
T2y )
which is found by
averaging over one precessional period [28]. Combining these results, the final form of the
Bloch equations is written as
dmx
1
= γe (m × H0 )x − mx
dt
T2
dmy
1
= γe (m × H0 )y − my
dt
T2
dmz
1
= γe (m × H0 )z − (mz − meq
z ),
dt
T1
(2.40)
where
1
τc
= (ωx2 + ωy2 )
T1
1 + ω02 τc2
τc ωx2 + ωy2
1
1
1 1
1 1
=
+ τc ωz2 =
+ 0.
+ τc ωz2 =
2
2
T2
2 1 + ω0 τc
2 T1
2 T1 T2
(2.41)
We now will discuss T1 and T2 and in the process gain some physical interpretations to the
above results. The term with ωz either aides or inhibits the external field and therefore
causes variations in the precession frequency. How this causes transverse relaxation is seen
via a simple model [28]. Consider ωz jumping from +ωz to −ωz in increments of the
correlation time, τc . A spin experiencing this fluctuation will precess an additional angle
22
according to dφ = ±ωz τc . After n intervals of fluctuating (in a time t), the mean square
dephasing angle will be ∆φ2 = ndφ2 = nωz2 τc2 . The condition of relaxation is defined as
when the mean square dephasing angle is one radian; this occurs at t = T20 . So 1/T20 = ωz2 τc .
T20 is known as secular broadening in the literature. The strange relation 1/T20 ∼ τc is a
phenomenon known as motional narrowing; the quicker the field changes, the longer the
resulting relaxation time since the phase does not have time to accumulate before the field
changes (i.e. dφ is small when τc is small). If the fluctuating field is isotropic (ωx2 = ωy2 = ωz2 )
and the correlation time is short such that ω0 τc 1 (extreme motional narrowing), then
T1 = T2 . Since the correlation time is often short, T1 ∼ T2 is applicable in small magnetic
fields.
If the secular broadening is eliminated (ωz2 = 0), the result T2 = 2T1 holds. Yafet
has shown that the restraint T2 ≤ 2T1 is quite general [40]. Lastly, the regime in which
the transverse fluctuations are small will cause negligible longitudinal relaxation and all
relaxation will be dominated by T2 = T20 . It is to be expected that T2 would be limited in
some sense by T1 since any moment that leaves the transverse plane for the longitudinal
direction will necessarily cause transverse relaxation. However transverse relaxation can
occur in the absence of any longitudinal relaxation; this is pure spin dephasing. It occurs
when fluctuations perpendicular to the external magnetic field are suppressed but parallel
fluctuations are not. It will also occur when the external field is very large. In later chapters,
it will be seen that systems can be fabricated that indeed suppress certain components of
the fluctuating field. This will lead to relaxation anisotropy.
Relations between T1 and T2 are useful since both are rarely either measured or calculated together. T1 is tractable through calculations while T2 is not [41]. As mentioned
earlier, it is not quite T2 that is measured in NMR and ESR experiments. Transverse magnetization can also decay due to inhomogeneous broadening. Inhomogeneous broadening or
precessional dephasing results from spatially inhomogeneous magnetic fields which cause
different precession rates. It is to be distinguished from true decoherence or relaxation
since it is potentially reversible via spin-echo experiments. Inhomogeneous magnetic fields
will cause true decoherence if the spin’s orientation loses correlation with its position [42].
The totality of transverse magnetization loss is then summed up in the quantity T2∗ . Since
the reversible losses are added to the irreversible losses, T2∗ ≤ T2 necessarily. For localized
electrons (on quantum dots or donors), T2∗ has been observed to be much smaller than
T2 [41]. Due to the itinerant nature of conduction electrons, spatial inhomogeneities are
‘washed out’ due to motional narrowing and T2∗ ≈ T2 . Since the notation for spin relaxation
is extensive, commonly a single quantity τs is used for all types of relaxation when external
fields are small. This thesis adopts that notation except when it is necessary to delineate
between different types of relaxation.
23
2.3.4 Phenomenology
The Bloch equations that were derived in the previous section resulted from a specific Hamiltonian (small random magnetic fluctuations). For different interaction Hamiltonians, one
would need to check if Bloch equations formed at all [28]. Typically the Bloch equations are
treated exactly how they were originally developed: phenomenologically. For many systems
of interest (namely that of itinerant electrons), T1 and T2 are assumed to be comparable
(and equal to τs ) and inhomogeneous broadening is neglected. When there is no applied
field, dm/dt = −m/τs where the rate 1/τs can be calculated by various means. This relaxation rate could be a sum of different mechanisms but typically one will be dominant and
show up in experiments. Chapter 3 will be devoted to describing the arsenal from which
mechanisms are chosen. In the next section, a phenomenological modification of the Bloch
equations is presented which is necessary when spins in multiple spin species (e.g. itinerant
and localized) interact through an exchange interaction.
2.4 The modified Bloch equations
An important principle to be discussed in this thesis is the spin exchange between electrons
in different environments. Most important for the purposes here, is the exchange interaction
between localized and itinerant electrons in semiconductors. In semiconductors with sufficiently high purity, the localized electrons are situated on donor sites. Itinerant electrons
are in the conduction band. In 1981 D. Paget found striking experimental evidence that the
exchange of spins was necessary to explain the observed spin lifetimes [43]. The evidence
was that both conduction and donor spins relaxed similarly by the hyperfine interaction
(to be discussed in chapter 3) even though it was expected that only donor spins would
experience large enough nuclear fluctuations. Paget posited that the conduction electrons
depolarized not by their own interaction with nuclei but by their strong interaction with
the electrons bound to donors. In recent years this mechanism has been used to describe
phenomena in several experiments on spins in semiconductors [44, 45, 5, 46, 7].
Paget wrote a pair of coupled Bloch equations1 to phenomenologically account for the
exchange interaction [43].2
1
The equations in Ref. [5] are identical to Paget’s except that Paget writes his equations in terms of
mean magnetizations and not total magnetizations as Ref. [5] does. The notation of Ref. [5] is followed
here.
2
For the sake of simplicity, the spin relaxation from the two states τc and τl have been neglected. At the
current time, only the effects of cross relaxation want to be examined.
24
dmc (t)
1
= − (nl mc (t) − nc ml (t))
dt
γcr
dml (t)
1
= − (nc ml (t) − nl mc (t)),
dt
γcr
(2.42)
where the quantity3 nimp /γcr = nimp Γcr is the time scale for the mean magnetization of
the two populations to become equal which can simply be obtained by setting the time
derivatives equal to zero: mc /nc = ml /nl . Magnetization here is defined as mi = ni,↑ − ni,↓
for some i species of particles. The density of that species of particles is ni = ni,↑ + ni,↓ .
The mean magnetization is mav
i = mi /ni . The subscripts c and l stand for conduction and
localized states respectively. The determination of this timescale - the cross relaxation time
- will be set aside until later. The solution to Eq. (2.42) is
mc (0)
nc + nl e−nimp Γcr t
nc + nl
mc (0)nl
ml (t) =
1 − e−nimp Γcr t ,
nc + nl
mc (t) =
(2.43)
where the following initial conditions have been used: ml (0) = 0 and mc (0) 6= 0. As
expected since exchange is a spin preserving interaction, ṁ = ṁc + ṁl = 0 and m(t)
is a constant dictated by the initial conditions. At long times, Eq. (2.43) reduces to
mc (t)/nc = mc (0)/nimp and ml (t)/nl = mc (0)/nimp which confirms the expectation that
the mean magnetizations are equal.
When spin relaxation is added,
dmc (t)
1
1
= − (nl mc (t) − nc ml (t)) − mc
dt
γcr
τc
dml (t)
1
1
= − (nc ml (t) − nl mc (t)) − ml .
dt
γcr
τl
(2.44)
The solution, though more cumbersome than before, can still be found exactly. When the
cross relaxation time is assumed short, the decay constant in the solution simplifies to
1
nc 1
nl 1
=
+
.
τs
nc + nl τc nc + nl τl
(2.45)
Two decades after Paget’s initial work, the group from St. Petersburg, Russia found similar
evidence supporting the rapid cross relaxation between the two spin systems [44, 45]. At low
temperatures in a low doped n-GaAs quantum well, they were able to increase measured spin
lifetimes from a few nanoseconds to nearly 300 nanoseconds by modifying the conduction
electron concentration. As can be seen from Eq. (2.45), this will cause the rapid relaxation
3
This quantity is equal to Paget’s 1/τe [43].
25
from the localized spins (due to hyperfine interaction) to be suppressed while the slower
conduction spin relaxation becomes important.
The theory of exchanging spins between localized and conduction electrons was examined most carefully by Putikka and Joynt [5] in 2004 when they explained the anomalous
temperature dependence of spin relaxation in bulk semi-insulating n-GaAs [4]. By knowing
the approximate impurity density to be nimp = 1016 cm−3 , they were able to ascertain
the occupations of both states for all temperatures using standard statistical mechanics.
The concentration of conduction and donor-bound electrons for a given impurity density is
known to be [47]
Z
∞
dεg(ε)f0 (ε, µ),
nc =
nl =
0
1+
nimp
1 (εB −µ)/kB T
2e
(2.46)
where g(ε) = (2m∗ )3/2 ε1/2 /2π 2 ~3 is the electronic density of states in three dimensions,
f0 (ε, µ) = 1/(1 + e(ε−µ)/kB T ) is the Fermi-Dirac function, µ is the chemical potential, and
εB < 0 is the binding energy of the electron bound to the donor. nc and nl are constrained
by nc + nl = nimp . In general, the chemical potential cannot be solved in closed form in
three dimensions. It is not difficult to solve numerically. Once done, the concentrations
of both electron types are calculated as a function of temperature which is shown in Fig.
2.3. From Eq. (2.45), it is obvious that the occupations of the states will figure into the
observed relaxation times. This is indeed what was seen by Kikkawa and Awschalom (data
in Fig. 2.4) [4]. Due to the population statistics, the dominant spin lifetimes observed
at low temperatures and high temperatures should be that due to localized and itinerant
electrons respectively. What these lifetimes are, is the subject of chapter 3. It is evident
from this picture that a non-monotonic dependence of relaxation rate on temperature is
possible and indeed confirmed for B = 4 T. This behavior is also seen at zero applied field
in n-ZnO, covered in chapter 5.
2.5 Summary
In this chapter the framework of what is to come has been laid. The key concepts within
the theory of spins in semiconductors have been identified. The Bloch equations have
been introduced as phenomenological equations but have also been rigorously shown to be
valid given some model interactions. Lastly, it was shown that the exchange interaction
is an important feature in semiconductors that must be accounted for in the study of
semiconducting electron spins. No mention of the actual spin relaxation types that affect
spin polarization in real systems have been identified. This is the subject of the next chapter.
26
1.0
0.6
ncnimp
nlnimp
0.8
0.4
0.2
0.0
0
50
100
150
200
250
300
T HKL
Figure 2.3: Occupations of conduction (red) and localized (blue) states for n-GaAs doped
at 2 × 1016 cm−3 (solid lines) and 1 × 1016 cm−3 (dashed lines). Parameters used: m∗ =
0.067me , εB = −5.8 meV. Higher temperature are required to deplete localized electrons in
the higher doped system.
Figure 2.4: Spin relaxation rate versus temperature in n-GaAs doped at 1016 cm−3 . Data
is from Ref. [4]. Solid lines are least squares fit using Eq. (2.45). Dashed-dotted curve:
(nl /nimp )(1/τl ) for B = 0 T. Dashed curve: (nl /nimp )(1/τl ) for B = 4 T. Dotted curve:
(nc /nimp )(1/τc ) for B = 0 T. Inset: momentum relaxation times for two different dopings:
a) 1016 cm−3 b) 1018 cm−3 . Figure is from Ref. [5].
27
Chapter 3
Spin relaxation mechanisms in
semiconductors
3.1 Introduction
As discussed in chapter 2, magnetic field fluctuations that couple to spin can cause spin
relaxation. In semiconductors there is an arsenal of several mechanisms to choose from that
contribute to spin relaxation. Which mechanism dominates depends on system variables
such as temperature, field, and doping. The easiest delineation is between the metallic
and insulating regimes. In the metallic regime, electrons are itinerant as they populate the
conduction band. Several relaxation mechanisms are found to be especially important in
this case; they are called conduction spin relaxation mechanisms and are detailed below.
One expects these mechanisms to be predominant in highly doped systems or systems at
high temperatures. When a semiconductor is insulating, electrons are localized - typically at
donor sites in n-type materials. A different set of mechanisms is important and are entitled
localized spin relaxation mechanisms. One expects these mechanisms to be most efficient in
low doped systems or systems at low temperatures when electrons are not thermalized and
their wave functions are weakly overlapping.
3.2 Conduction spin relaxation mechanisms
The three prevalent relaxation mechanisms of conduction spin can be attributed to three
different manifestations of the spin-orbit (SO) interaction. They are briefly outlined here
and discussed further in sections 3.2.1 and 3.2.2.
In 1954 Elliott asserted that spins could relax due to spin-independent scattering [48].
However this is only possible due to the SO interaction which prohibits spin from being a
‘good’ quantum number (the operator Sz does not commute with the SO Hamiltonian). This
implies that the spin component of the wave function is not pure but is an admixture of spin
eigenstates. The amount of ‘mixing’ depends on the size of the SO interaction. Typically
28
the amount of mixing is small and a pseudospin is defined by the dominant component (up
or down). For instance, a pseudospin is up (↑) if hΨk |σz |Ψk i > 0; a pseudospin is down
(↓) if hΨk |σz |Ψk i < 0 [49]. For example, a pseudospin down state is labeled as Ψk,↓ . A
consequence of this spin admixture is that scattering induced transitions from pseudospin
up to pseudospin down states are non-zero probability events as they would be if the spinor
were pure. A cartoon that exaggerates the amount of spin flips is shown in Fig. 3.1. Yafet’s
contribution came in 1961 when he posited that since the SO interaction is caused by lattice
ions, when those ions vibrate the SO interaction is modulated [40]. The Yafet process is
also known as the short range interaction where the Elliott process is called the long range
interaction [39]. The long range interaction is most important in III-V semiconductors and
the short range interaction will not be considered further in this dissertation. Aside from
the short range interaction, Yafet also systematized the theory of Elliott by applying it to
specific band structures and electron-phonon scattering. Hence, the long range process still
goes by the name Elliott-Yafet (EY). Determining the EY relaxation rate will be discussed
in section 3.2.1.
D’yakonov Perel’ (DP) spin relaxation also originates from the SO effect but in a very
different way than EY. Besides the necessity of the SO effect, DP also requires the crystal
to be non-centrosymmetric (no inversion symmetry). The importance of lacking inversion
symmetry can be seen from quantum mechanics. First consider the time reversal operator,
K̂.4 If a system possesses time reversal symmetry (e.g. no external magnetic field), both
Ψk and K̂Ψk are orthogonal eigenstates with the same eigenvalue [49]. If the time reversal
operator is applied to a semiconducting crystal whose wave function are Bloch functions,
KΨk,↑ = Ψ−k,↓ and K̂Ψk,↓ = Ψ−k,↑ since the time reversal operator affects k → −k and
also reverses angular momentum. So
εk,↑ = ε−k,↓ ,
εk,↓ = ε−k,↑ .
(3.1)
This implies that bands are not degenerate but opposite pseudospins (just spin if spin mixing
of wave functions is ignored) are related by opposite wave vectors. If spatial inversion is a
characteristic of the crystal then k is undifferentiated from −k; hence,
εk,↑ = εk,↓ ,
(3.2)
which is the familiar result of doubly (pseudo)spin degenerate bands.
Since non-centrosymmetric crystals lack inversion symmetry, Eq. (3.2) does not hold
but Eq. (3.1) does hold when external fields are absent. Eq. (3.1) alone suggests that bands
are not doubly degenerate (they are spin-split)5 except for certain k values. Obviously there
4
The operator K̂ is represented by iσy Ĉ where Ĉ is the conjugation operator [49]
A point worth mentioning because it is rarely stated in the literature: in analytical calculations of DP
relaxation, the pseudospin nature is neglected in favor of pure spin eigenstates. Since spin admixture is
5
29
Figure 3.1: Pictorial descriptions of three common conduction spin mechanisms: Elliott
Yafet (EY), D’yakonov Perel’ (DP), and Bir Aronov Pikus (BAP). For the first two, each
vertex represents a scattering event. For BAP, electrons and holes are depicted by different
colors and the arrows between them signify the exchange interaction.
is degeneracy at k = 0.
The quantity and characteristics of spin-splitting are determined from band structure
[39]. The Dresselhaus splitting can be derived using the extended Kane model for the band
structure [50, 51, 39]. It is found from this calculation that the conduction band splits as
if there is a k-dependent magnetic field. Additionally, the form of the spin-splitting can be
found by group-theoretic techniques [52]. Consequently, this effective field is described by a
Zeeman-like Hamiltonian: Hso (k) = ~2 Ωk · σ. Table 3.1 contains the Dresselhaus spin-orbit
Hamiltonian for bulk zinc-blende and wurtzite semiconductors.
Figure 3.1 depicts the evolution of a spin in the effective k-dependent magnetic field
which is termed now as the spin-orbit field; between each scattering event, the spin precesses
around a field with a random precession axis. Over time, the spin ‘forgets’ its initial
direction. φ = Ωk τp is the angle of precession between a scattering event. The mean-square
assumed small, this approximation is valid when considering DP [42].
30
Bulk Material
zinc-blende
wurtzite
H1
H3
0
β1 (ky σx − kx σy )
β3 kx (ky2
− kz2 )σx + ky (kz2 − kx2 )σy + kz (kx2
β3 (bkz2 − kx2 − ky2 )(ky σx − kx σy )
− ky2 )σz
Table 3.1: The Dresselhaus spin-orbit Hamiltonians for bulk zinc-blende and wurtzite semiconductors.
angle increases as the number of scattering events, n, increases: hφ2 i = hΩ2k iτp2 n. The
spin is sufficiently randomized to warrant relaxation when hφ2 i = 1. This happens at time
t = τs and after n = τs /τp scattering events. The result for the relaxation rate is then
τs−1 ∼ hΩ2k iτp . This brief calculation assumes the the precession period is much smaller
than the scattering time; this is known as the motionally narrowed regime. Section 3.2.2
contains a more formal calculation of the DP mechanism.
In semiconductors where holes are plentiful, electron polarization can decay due to
exchange interactions with holes (see Figure 3.1. This is known as the Bir-Aronov-Pikus
(BAP) mechanism [53, 54]. No materials examined in this dissertation are p-type and it is
therefore safe to ignore the BAP mechanism [55, 56]. In section 4.3 an intrinsic sample is
investigated and the BAP may be important in such a case after photo-excitation when a
equivalent amount of electrons and holes are present in the form of excitons. For the time
being the relaxation of spin in excitons is left as a phenomenological parameter.
In semiconductors consisting of host nuclei with non-zero nuclear spin, the hyperfine
interaction causes conduction electron spin relaxation. This is discussed further in section
3.2.4.
3.2.1 The Elliott-Yafet mechanism
The wave functions in the presence of spin-orbit coupling are of the form
Ψk,↑ = |k, ↑ieik·r = ak | ↑i + bk | ↓i eik·r .
(3.3)
When a magnetic field is absent, we also the know that the time reversed wave function is
K̂|k, ↑ieik·r = a∗k | ↓i − b∗k | ↑i e−ik·r ,
(3.4)
which has the same energy as the prior wave function and is orthogonal to it. What is the
expected value of σz in the two states? hΨk,↑ |σz |Ψk,↑ i = |ak |2 − |bk |2 and hΨk,↓ |σz |Ψk,↓ i =
−|ak |2 + |bk |2 which says that the pseudospins are antiparallel. Typically the spin-orbit
interaction is weak and a b such that the pseudospins approximate pure spin which gives
31
±1 for the above expectation values.6
When an electron scatters, it transitions from k −→ k0 . As was just pointed out, states
with the same k and opposite pseudospin are orthogonal and a spin-independent potential
should not induce any change in the spin structure of the wave function. However when
k0 6= k, the coefficients of Ψk0 change which allows for the pseudospin to flip at scattering
events. This process is shown in Fig. 3.2 Here the framework of calculating the Elliott
Yafet relaxation rate is described. It is calculated explicitly for the bulk wurtzite crystal in
chapter 5.
Figure 3.2: Spin polarization depends on k. For each k there are two mutually antiparallel
pseudospins (only one of each pair is shown for a select few wave vectors). Scattering that
alters the momentum (from k1 to k2 ) also changes the spin orientation. Graphic taken from
[1]
In general, the scattering rate from a potential Vscatt. is found by Fermi’s Golden Rule
[24]
X 2π
~
k0
|hf |Vscatt. |ii|2 ,
(3.5)
where the state vector i and f refer to the initial and final states respectively. Using the
notation outlined above, consider the matrix element hk0 , ↓ |Vscatt. |k, ↑i which describes an
initially ‘up’ state scattering to a ‘down’ state:
Z
0
hk0 , ↓ |V |k, ↑i =
ei(k−k )·r Vscatt. dr hk0 , ↓ |k, ↑i ,
V
6
Wave functions are assumed to be normalized such that |a|2 + |b|2 = 1.
32
(3.6)
where the factorization of the integral (over the total volume V ) involving the envelope and
potential functions is possible since they are slowly varying over the unit cell which is not
necessarily true for the Bloch functions [57]. The first term in parentheses is simply the
Fourier transform of the potential [58]; the second in parentheses is an integral of Bloch
functions over a single unit cell; so Vk,k0 hk0 , ↓ |k, ↑i is left or more generally
Vk,k0 hk0 , s0 |k, si,
(3.7)
where s and s0 are the initial and final pseudospins. To proceed, the actual conduction
band Bloch functions must be known. They are known from k · p theory for both cubic
and wurtzite direct gap semiconductors [57, 58, 59]. The Elliott-Yafet mechanism has
been studied extensively in zinc-blende semiconductors [57, 58, 60, 61]. In chapter 5, it is
discussed in wurtzite crystals for the first time.
3.2.2 The D’yakonov-Perel’ mechanism
In this section the derivation of the spin relaxation rate in non-centrosymmetric bulk semiconductors proposed in 1971 by D’yakonov and Perel’ is presented. While we follow their
derivation [62, 63], other sources have clarified their work [61, 39]. Spin relaxation by this
same mechanism is examined in two-dimensional semiconductors which was taken up in 1986
by D’yakonov and Kachorovskii [64]. In both systems, the mechanism is commonly referred
to as the D’yakonov-Perel’ (DP) mechanism. Since the DP mechanism is most important
over a significant temperature range (especially at room temperature) for the zinc-blende
and wurtzite crystals studied here, the spin relaxation due to DP will be derived here fully
for bulk systems in general.
The framework in which the DP relaxation rate is solved is the semi-classical spin
Boltzmann equation. In the following, the semi-classical spin Boltzmann equation is derived
and used to calculate the DP relaxation rate in the motional narrowing regime (scattering
time shorter than spin-orbit precession period). The derivation follows that of [39]. Consider
the semi-classical evolution of a wave packet’s position and momentum in external fields:
∂εk
~∂k
~k̇ = Fk = −eE − evk × B,
ṙ = vk =
(3.8)
where F is the Lorentz force, E is the electric field, and εk is the electronic dispersion of
the conduction band. Spin is introduced quantum mechanically. By assigning each electron
wave packet a 2×2 density matrix, the electron’s spin state can be described in general. The
single electron density matrix is δk,k0 δ(r−r 0 )ρk (r, t) ≡ ρk (r, t) where we have approximated
position and momentum to be diagonal. The same restriction is not placed on spin. The
properties of the density matrix are such that the trace of the density matrix in state k at
33
r is the number of electrons in that state. The total electron density is found by summing
over all space and momenta:
1
n=
Ω
Z
d3 r
X
Trρk (r, t).
(3.9)
k
The spin density is found similarly,
m(t) =
1
Ω
Z
d3 r
X
Tr ρk (r, t)σ .
(3.10)
k
While the position and momenta evolution are evaluated semi-classically, the spin evolution
is evaluated quantum mechanically. We consider the spin-orbit Hamiltonian which was
discussed earlier:
~
Ωk · σ.
2
The time evolution of the density matrix ρ (≡ ρk (r, t)) is
Hs.o. =
∂ρ
i
= − [Hs.o. (k), ρ].
∂t
~
(3.11)
(3.12)
How does the density matrix at time ρ(t − dt) change in a small amount of time dt?
ρ(t) − ρ(t − dt)
i
= − [Hs.o. (k), ρ],
dt
~
i
ρ(t) = ρ(t − dt) − [Hs.o. (k), ρ]dt.
~
(3.13)
This is the first order in dt expansion of the exact evolution
ρ(t) = e−iHs.o. dt/~ ρ(t − dt)eiHs.o. dt/~ .
(3.14)
From this expression of the density matrix, the Boltzmann equation can be derived in the
standard way [47]. This is done by Taylor expanding Eq. (3.14) to first order in dt:
ρ = ρ(k(t), r(t), t) = e−iHs.o. dt/~ ρ(k(t − dt), r(t − dt), t − dt)eiHs.o. dt/~
∂ρ
∂ρ
∂ρ
i
≈ ρ(r, k, t) −
· F dt −
· vk dt −
dt − [Hs.o. , ρ]dt.
~∂k
∂r
∂t
~
(3.15)
Then by taking the time derivative,
∂ρ
∂ρ
∂ρ
i
+
·F +
· vk + [Hs.o. , ρ] = 0.
∂t
~∂k
∂r
~
(3.16)
The density matrix at phase space point (r, k) can also change due to electrons scattering
into and out of that point. Such collisions are not included in Eq. (3.15); Eq. (3.15) assumes
electrons travel from (r − vdt, k − F dt/~) to (r, k) in time dt [47]. In reality, collisions will
either facilitate or hinder this motion. The effect of collisions can be added to Eq. (3.15)
34
by hand and simplified to obtain
∂ρ ∂ρk
∂ρk
∂ρk
i
k
,
+
·F +
· vk + [Hs.o. , ρk ] =
∂t
~∂k
∂r
~
∂t coll.
(3.17)
where the k’s have been made explicit again. Due to the Pauli Exclusion Principle the
probability per unit time, Wk,k0 , must be modified because certain states (those occupied)
will not be allowed to be scattered into. So the modification is Wk,k0 −→ Wk,k0 ρk (1 − ρk0 )
for the probability of scattering k −→ k0 . Electrons can also scatter into state k in which
case the rate is Wk0 ,k ρk0 (1 − ρk ). By using the principle of detailed balance [47], we obtain
for the collision piece:
∂ρ k
∂t
coll.
=−
X
Wk,k0 (ρk − ρk0 ).
(3.18)
k0
In summary, we write the spin Boltzmann equation now as
X
∂ρk
∂ρk
∂ρk
i
+
·F +
· vk + [Hs.o. , ρk ] = −
Wk,k0 (ρk − ρk0 ).
∂t
~∂k
∂r
~
0
(3.19)
k
By using Eq. (3.10), the transport equations for spin density can be derived though they
will not be used here in this general form:
X
∂mk ∂mk
∂mk
+
·F +
· vk − Ωk × mk = −
Wk,k0 (mk − mk0 ).
∂t
~∂k
∂r
0
(3.20)
k
The second term on the left hand side describes spin drift due to external fields. The third
terms encompasses the diffusion present in inhomogeneous spin distributions. The fourth
term contains the spin precession due to the spin-orbit field.
Let us now go back to Eq. (3.19) and make several simplifications that allow an analytic
solution for the DP spin relaxation rate. The following approximation/simplifications are
used:
1. Elastic scattering (k = k 0 )
2. Isotropic energy (εk = ~2 k 2 /2m)
3. Homogeneous distributions (∂ρk /∂r = 0)
4. No external fields (F = 0).
These conditions simplify Eq. (3.19) to
X
i
∂ρk
+ [Hs.o. , ρk ] = −
Wk,k0 (ρk − ρk0 ).
∂t
~
0
(3.21)
k
We now separate the effects of momentum relaxation from spin relaxation; the density
matrix can be decomposed into two parts ρk = ρ + ρ0k where ρ is the isotropic portion of
the density matrix, meaning that it is ρk averaged over all directions of k and is therefore
independent of k [61, 65, 39]. ρ0k is the part of the density matrix that deviates from the
35
average value. It is called the anisotropic piece. Since ρk = ρ + ρ0k , it follows that ρ0k = 0.
ρ0k results from the spin-orbit Hamiltonian Hs.o. and is therefore much smaller than the
isotropic piece since the spin-orbit Hamiltonian is assumed to be a perturbation [65]. If Eq.
(3.21) is k-averaged, we obtain
∂ρ
i
+ [Hs.o. (k), ρ0k ] = 0,
∂t
~
(3.22)
since Hs.o. ρ = 0 because Hs.o. is an odd function of k. The right-hand side of Eq. (3.21)
vanishes exactly upon averaging over k. We also substitute the decomposed density matrix
into Eq. (3.21) to get
X
∂ρ ∂ρ0k
i
+
+ [Hs.o. , ρ + ρ0k ] = −
Wk,k0 (ρ + ρ0k − ρ − ρ0k0 ).
∂t
∂t
~
0
(3.23)
k
We use the expression for the time derivative of the isotropic piece of the density matrix,
Eq. (3.22), in Eq. (3.19) to obtain
X
∂ρ0
i
i
i
− [Hs.o. (k), ρ0k ] + k + [Hs.o. , ρ] + [Hs.o. , ρ0k ] = −
Wk,k0 (ρ0k − ρ0k0 ).
~
∂t
~
~
0
(3.24)
k
The anisotropic part of the density matrix should return to isotropy on the order of the
momentum relaxation time. So in the quasi-static limit ∂ρ0k /∂t = 0. Keeping terms to first
order in the spin-orbit coupling,
X
i
Wk,k0 (ρk − ρk0 ).
− [Hso (k), ρ] =
~
0
(3.25)
k
At this point, it is useful to consider the dimensionality of the electron gas. We look at
three and two dimensions separately in the following sections.
Three dimensions
The first step is to decompose the spin-orbit Hamiltonian into spherical harmonics. The
order (l) of the spherical harmonic match the order of the spin-splitting in k. Examples are
shown in chapters 4 and 5. For now, we write
Hso (k) =
l
X X
l
Hl,n Yln (ϑk , ϕk ).
(3.26)
n=−l
Now an ansatz is made for the solution to Eq. (3.25)[63, 61, 39],
ρk = −
l
iX X
τl Yln (ϑk , ϕk )[Hl,n , ρ].
~
l
n=−l
36
(3.27)
Commonly only one l is used in derivations of the DP mechanism since the spin-splitting is
assumed to be solely linear or cubic but not both [61, 39, 56]. In such cases,
l
X
i
i
Yln (ϑk , ϕk )[Hl,n , ρ] = − τl∗ [Hso (k), ρ].
ρk = − τl∗
~
~
(3.28)
n=−l
The scattering is assumed to be elastic (k = k 0 ) so only the spherical angles of ρk will differ
from ρk0 . Using the ansatz of Eq. (3.27), the right-hand side of Eq. (3.25) is recast as
X
Wk,k0 (ρk − ρk0 ) = −
k0
X
iX
Wk,k0
τl [Hl,n , ρ](Yln (ϑk , ϕk ) − Yln (ϑk0 , ϕk0 ))
~ 0
(3.29)
l,n
k
By converting the sum over final k-states to an integral and using an identity of spherical
harmonics (Eq. (B.5) proved in Appendix B), the following is found
X
Wk,k0 (ρk − ρk0 )
k0
Z π
Z
iX
n
0 W (θ)
=−
τl [Hl,n , ρ]Yl (ϑk , ϕk )
dΩ
−
sin θdθW (θ)Pl (cos θ)
~
2π
0
l,n
Z
π
iX
sin θdθW (θ) 1 − Pl (cos θ)
(3.30)
τl [Hl,n , ρ]Yln (ϑk , ϕk )
=−
~
0
l,n
where the trivial ϕ0 integral was computed in going from line one to two. Also the full
P P
P
summation over n is suppressed but it is understood that l,n = l ln=−l . By going
from Wk,k0 → W (θ), it is assumed that the scattering probability depends only on the
angle θ between k and k0 . The final equality in Eq. (3.30) is equal to − ~i [Hso (k), ρ] by Eq.
(3.25). If both sides are multiplied by Yn∗0 ,l0 (ϑ, ϕ) and then integrated over the solid angle
of the unprimed coordinates,
Z
δn,n0 δl,l0 [Hl,n , ρ] = δn,n0 δl,l0 τl [Hl,n , ρ]
sin θdθW (θ)(1 − Pl (cos θ)).
This allows the rate τl−1 to be expressed as
Z π
1
=
sin θdθW (θ)(1 − Pl (cos θ)).
τl
0
(3.31)
(3.32)
Now the ansatz of Eq. (3.27) is substituted into Eq. (3.22) to get a tractable differential
equation for the isotropic portion of the density matrix,
∂ρ
1 XX
=− 2
τl [Hso (k), Yln (ϑk , ϕk )[Hl,n , ρ]],
∂t
~
n
l
37
(3.33)
with the τl ’s given by Eq. (3.32). Then expanding the spin-orbit Hamiltonian in terms of
spherical harmonics yields
∂ρ
1 XX
0
=− 2
τl [Hl0 ,n0 Yln0 (ϑ, ϕ), Yln (ϑ, ϕ)[Hl,n , ρ]].
∂t
~
0
0
(3.34)
l,l n,n
Recall that the overbar is an angular average. The only angular quantities in the above
equation are the spherical harmonic functions so
∂ρ
∂t
1 XX
0
τl Yln0 (ϑ, ϕ)Yln (ϑ, ϕ)[Hl0 ,n0 , [Hl,n , ρ]]
2
~
l,l0 n,n0
X
X Z dΩ 0
1
Yln0 (ϑ, ϕ)Yln (ϑ, ϕ)[Hl0 ,n0 , [Hl,n , ρ]]
τl
= − 2
~
4π
0
0
= −
l,l n,n
1 X X (−1)n
= − 2
τl δ−n,n0 δl,l0 [Hl0 ,n0 , [Hl,n , ρ]]
~
4π
0
0
l,l n,n
=
l
1 X X
−
(−1)n τl [Hl,−n , [Hl,n , ρ]],
4π~2
l
(3.35)
n=−l
where certain properties of the spherical harmonics are used. See Appendix B for information on spherical harmonics. The coefficients are found by multiplying by Yl∗0 ,n0 (ϑ, ϕ) and
integrating over the solid angle,
Z
Z
XX
0
n0 ∗
dΩHso (k)Yl0 (ϑ, ϕ) = dΩ
Hl,n Yln (ϑ, ϕ)Yln0 ∗ (ϑ, ϕ) = Hl,n δn,n0 δl,l0 ,
(3.36)
n
l
which in the end gives
Z
Hl,n =
dΩHso (k)Yln∗ (ϑ, ϕ).
(3.37)
From our equation for the evolution of the density matrix, we determine how the components
of the spin magnetization’s expectation value evolve in time by the relations m = T r(ρk σ) =
Tr(ρσ) and ρ = 12 (I2 +mx σx +my σy +mz σz ). Due to the commutators, the identity matrix,
I2 , gives no contribution. When ρ is substituted into Eq. (3.35), Bloch-like equations result,
dmi
1
=−
mj
dt
τs,ij
(3.38)
for which the relaxation tensor is
1
τs,ij
l
X X
1
n
=
Tr
(−1)
τ
[H
,
[H
,
σ
]]σ
.
j
i
l
l,−n
l,n
8π~2
l
n=−l
38
(3.39)
Two dimensions
The calculation in two dimensions proceeds similarly to that in three dimensions [65]. The
difference in the derivation lies in the fact that the spin-orbit Hamiltonian is now expanded
in terms of plane waves instead of spherical harmonics,
Hso (k) =
∞
X
Hn einϕ ,
(3.40)
n=−∞
where k = kx x̂ + ky ŷ if we consider motion in the x − y plane only. ϕ = ϕk is the angular
measure of the k direction. The rest of the analysis proceeds as in three dimensions. One
finds that this allows the rate τl−1 to be expressed as
1
=
τl
2π
Z
dθW (θ)(1 − cos nθ).
(3.41)
0
where θ = ϕ0 − ϕ, is the angle between k and k0 . Mirroring the previous calculation, we
obtain
∂ρ
1 X
=− 2
τn [Hn0 ein0 ϕ , einϕ [Hn , ρ]].
∂t
~
0
(3.42)
n,n
The angular averaging is easier in two dimensions. We find that
∂ρ
∂t
1 X
τn ein0 ϕ einϕ [Hn0 , [Hn , ρ]]
~2 0
n,n
Z
X
1
dϕ in0 ϕ inϕ
= − 2
τn
e
e [Hn0 , [Hn , ρ]]
~
2π
0
= −
n,n
1 X
= − 2
τn δ−n,n0 [Hn0 , [Hn , ρ]]
~
0
n,n
=
−
∞
1 X
τn [H−n , [Hn , ρ]],
~2 n=−∞
where the coefficients are
Z
Hn =
0
2π
dϕ
Hso (k)einϕ .
2π
(3.43)
(3.44)
The relaxation tensor for two dimensions is found to be
1
τs,ij
∞
X
1
= 2 Tr
τn [H−n , [Hn , σj ]]σi .
2~
n=−∞
(3.45)
Examples of this spin relaxation mechanism are demonstrated in chapters 4 and 5. So far
the calculation has been done at T = 0 K. Finite temperatures are considered now.
39
3.2.3 Finite temperatures
The previous calculations assume zero temperature. In actuality temperatures are finite
and if electron spins are to be used in technology, room temperature is the most important.
Typically, one determines the density of particles by summing over all possible energies and the number of electrons occupying each of those energies; this looks like n =
R∞
0 g(ε)f0 (ε, µ)dε where g(ε) is the number of states in an interval ε → ε + dε and f0 (ε, µ)
is the Fermi-Dirac distribution where µ is the chemical potential. Since the existence
of spin polarization implies an imbalance of populations, the up and down spins, along
some axes, should have different chemical potentials. This assumes that equilibrium is
achieved much quicker than the spins can relax.
The spin along some direction ŝ is
s(ε) = ŝ(f+ (ε, µ+ ) − f− (ε, µ− )) where f+ and f− are distributions of electrons with spin
projection in ŝ [65]. The total spin density component i, mi , is determined from
Z ∞
Z ∞
mi =
g(ε)(f+ (ε, µ+ ) − f− (ε, µ− ))dε =
g(ε)si (ε)dε.
0
(3.46)
0
The time dynamics are determined by
Z ∞
Z ∞
Z ∞
dmi
dsi (ε)
1
1
=
g(ε)
dε = −
g(ε)
sj dε = −
g(ε)
(f+ (ε, µ+ )−f− (ε, µ− ))dε,
dt
dt
τ
τ
s,ij
s,ij
0
0
0
(3.47)
where Eq. (3.38) has been used. Now the last equation on the right-hand side of Eq. (3.47)
can be multiplied by 1 = mj /mj to give
R∞
1
dmi
1
0 g(ε) τs,ij (f+ (ε, µ+ ) − f− (ε, µ− ))dε
= − R∞
mj = −
mj ,
dt
τs,ij
0 g(ε)(f+ (ε, µ+ ) − f− (ε, µ− ))dε
where now the temperature dependent spin relaxation rate is defined as
R ∞ g(ε) 1 (f (ε, µ ) − f (ε, µ ))dε
+
−
−
1
τs,ij +
0
.
= R∞
τs,ij
0 g(ε)(f+ (ε, µ+ ) − f− (ε, µ− ))dε
(3.48)
(3.49)
Simplifications can be made if |µ+ −µ− | |µ+ |, |µ− |. In such a case, f+ −f− ≈ −∆µ∂f0 /∂ε
which simplifies Eq.(3.50) to
1
R∞
=
τs,ij
1
∂f0 /∂εdε
g(ε) τs,ij
R∞
.
0 g(ε)∂f0 /∂εdε
0
If the DP mechanism is considered,
R ∞ g (d) (ε) 1 ∂f0 dε
1
Iˆ(d) [εn τl (ε)]
τs,ij ∂ε
0
= R∞
=
α
.
s
τs,ij
Iˆ(d) [1]
g (d) (ε) ∂f0 dε
0
(3.50)
(3.51)
∂ε
where Iˆ(d) [h] defines an integral operator operating on an arbitrary function h in d dimen40
sions:
Iˆ(d) [h] =
R∞
0
0
g (d) (ε) ∂f
∂ε h(ε)dε
,
R∞
∂f0
(d)
0 g (ε) ∂ε dε
(3.52)
and the DP relaxation rate has been decomposed into 1/τs,ij = αs εn τl (ε) to highlight
its energy dependence. The density of states is now labeled as g (d) (ε) to highlight its
dimensional dependence. It is generally that the case the scattering mechanisms have a
power law dependence on energy, τl ∼ εν ; this allows us to write 1/τs,ij = αs εn sl εν where
sl is a proportionality constant. As we have seen, the relaxation rate may be a sum of
different power laws in which case our Iˆ operator would have to act on the relaxation rate
term by term. Since Iˆ(d) [1] = 1, we rewrite Eq. (3.51) as
1
τs,ij
= αs Iˆ(d) [εn τl (ε)]
(3.53)
From now on the angular brackets denoting the thermal averaging are dropped since whether
or not a quantity is averaged is evident from the context. The integrals in Eq. (3.51) are
important and are computed in Appendix C. The results are used frequently in what follows.
The transport time is a useful quantity to define since it can be determined from mobility
measurements. It is defined as [66, 67]
R ∞ (d)
0
g (ε)ετ1 ∂f
Iˆ(d) [εs1 εν ]
Iˆ(d) [εν+1 ]
∂ε dε
τtr = R0 ∞
=
=
s
.
1
∂f0
(d)
Iˆ(d) [ε]
Iˆ(d) [ε]
0 g (ε)ε ∂ε dε
(3.54)
The idea is to solve for the proportionality constant s1 and then express the scattering time
τ1 in terms of the experimental quantity τtr . This is now easily accomplished:
τ1 = s1 εν = τtr
Iˆ(d) [ε]
Iˆ(d) [εν+1 ]
εν = εν τtr β ν
Id+1 (βµ)
,
Id+ν+1 (βµ)
(3.55)
where the results of Appendix C are used. Other scattering times are found similarly,
τ3 = γ3 τ1 = γ3 εν τtr β ν
Id+1 (βµ)
,
Id+ν+1 (βµ)
(3.56)
where the order-unity constant γ3 can be determined from Eq. (3.32) when the scattering
mechanism is known. Now that τl (ε) is expressible in terms of the measurable quantity, τtr ,
it is appropriate to substitute Eq. (3.55) or Eq. (3.56) into Eq. (3.53). The temperature
dependence of the DP relaxation rate has been shown; examples of the calculation are given
in the proceeding chapters. Similar analysis follows for the other relaxation rates.
3.2.4 Hyperfine interaction
Both electronic and nuclear spins relax due to their mutual magnetic interaction. These
interactions are collectively called hyperfine interactions [28, 68]. The first type of magnetic
41
interaction is the dipolar coupling between the magnetic dipoles of the electron and nuclear
angular momenta. For an electron and nucleus suitably far apart, the dipolar coupling
Hamiltonian is [69]
µ0 γ e γ n 3
I
·
S
−
(I
·
r)(S
·
r)
,
(3.57)
4πr3
r2
where γn is the nuclear gyromagnetic ratio and I is the nuclear spin. For the spherically symHdip. =
metric s-state electrons, the expectation value of this Hamiltonian disappears [69]. However
the s-states must be considered carefully; the wave function is concentrated on the nucleus
(at r = 0) which calls the dipole approximation into question due to the proximity of the
two spins (divergence at r = 0) [28]. This problem is remedied by considering the finite size
of the nucleus. Assume a uniform magnetization over the spherical nucleus M = γn I/V .
The flux density is then 2µ0 M /3 after the demagnetizing field has been subtracted [68].
Thus an electron that enters the nuclear region will experience a magnetic field
BN =
2µ0 γn I
δ(r).
3V
(3.58)
Unlike the previous interaction, electrons with l > 0 are not affected by this interaction
since they have zero amplitude at r = 0. The field of Eq. (3.58) interacts with the electron
spin in what is known as the Fermi contact interaction. The energy cost associated with
this interaction is
Econtact =
2µ0
γe γn |ψ(0)|2 I · S,
3
(3.59)
where the factor V |ψ(0)|2 enters because it gives the probability of the electron to be found
at the nucleus’ location (r = 0).7 This energy can be determined from the Hamiltonian
Hcontact =
2µ0
γe γn I · Sδ(r),
3
(3.60)
by using Econtact = hψ(0)|Hcontact |ψ(0)i.8 The dipolar interaction for l > 0 is typically
considerably weaker than the contact interaction (for l = 0) and hence is neglected [70, 68].
The Fermi contact Hamiltonian can be written in a more compact way:
Hcontact = Ah.f. VN σn · σe δ(r),
(3.61)
where Ah.f. = 2µ0 γe γn ~2 /12VN is the hyperfine coupling constant and has units of energy;
VN is the nuclear volume and σe,n are the Pauli spin matrix operators for electrons and
nuclei. The spin relaxation time due to this interaction is found by the rate of mutual spin
flips between the two spin systems; that is electron state |k ↑i and nuclear state |I, µi to
|k ↓i and |I, µ + 1i respectively. A.W. Overhauser first did this for metals in 1953 [71];
7
The wave function ψ(r) is assumed to not vary much over the nucleus which justifies using ψ(0).
This quantity is commonly written in Gaussian units in which Eq. (3.60) must be multiplied by a factor
of 4π/µ0 [28, 70]
8
42
Fishman and Lampel extended the calculation to non-degenerate semiconductors in 1977
[60]. Following the work of Fishman and Lampel, the transition probability for the mutual
spin flip is [60]
w↑,µ−→↓,µ+1 =
2
2π 2
A
V
|ψ(0)|
[I(I + 1) − µ(µ + 1)]δ(ε − ε0 ),
N
h.f.
~2
(3.62)
where the term in square brackets is determined by the nuclear spin raising operator, σn+ .
If the nuclei are considered to be weakly polarized, each spin state, µ, has an identical
probability equal to 1/(2I + 1); then the transition probability for an electron spin flip from
up to down due to all identical nuclei is
w↑−→↓ =
nnuc. V X
w↑,µ−→↓,µ+1 ,
2I + 1 µ
(3.63)
where nnuc. is the density of nuclei. The total relaxation rate is found by summing over
final k states and accounting for spin-flips of the opposite variety; this yields [60]
√
∗3
1
2
2
2 8I(I + 1) 2m ε
= nnuc. (Ah.f. V ) (|ψ(0)| V )
,
τh.f.
3
2π~4
(3.64)
where the quantities Ah.f. V and |ψ(0)|2 V have either been calculated or measured for several
semiconductors [72, 43, 60]. These processes have also been recently studied in semiconductor nanostructures [73, 74, 75, 76]. Spin relaxation due to the hyperfine coupling has been
determined to be too weak (τs ≈ 103 − 104 ns [60]) to be observed in the metallic regime in
bulk GaAs [77, 7, 39]; however the hyperfine interaction is very relevant in the insulating
regime as will be looked at in the upcoming section.
3.3 Localized spin relaxation mechanisms
In the previous section, it was seen how conduction spins relax. Those types of relaxation
processes depend on the electron state being itinerant. In this section spin relaxation for
localized electrons is discussed. Localized relaxation is important at low temperatures and at
low impurity concentrations when the semiconductor is insulating. The localization centers
are considered to be positively charged donor impurities. The two relaxation mechanisms
discussed below are not too foreign to the previous discussions on itinerant spin relaxation;
they are again hyperfine coupling and another interaction induced by the spin-orbit effect
called the anisotropic spin exchange interaction.
3.3.1 Hyperfine interaction
When host nuclei possess a non-zero magnetic moment (proportional to the nucleus’ spin,
I), the nuclear spins interact with the electron spin and cause spin relaxation as if the nuclei
43
were randomly oriented magnetic fields. There are two ways this can occur for localized
electrons.
1. Frozen nuclear fields [78]. The magnetic moment of nuclei is much smaller than the
magnetic moment of electrons; this leads to the fact that nuclear spin evolves much slower
than electron spin. First, no applied magnetic field is assumed. The randomly oriented
nuclei produce a nuclear magnetic field, BN , that is felt by the electrons. The electron
spin will precess in this nuclear field whereas the nuclear spin will be nearly stationary in
the presence of the electronic magnetic field, Be . So in essence, the electron ‘sees’ a frozen
array of ∼ 105 [70] randomly oriented BN ’s. The magnitude and direction of the nuclear
field is normally distributed so
W (BN ) =
1
2
2
e−(BN ) /∆B ,
π 3/2 ∆3B
(3.65)
where ∆B is the width of the nuclear field distribution. The evolution of a spin moment in
a fixed magnetic field is [78]
m(t) = (m0 · n)n + m0 − (m0 · n)n cos ωN t + m0 − (m0 · n)n × n sin ωN t, (3.66)
where n = BN /BN is the unit vector of the nuclear magnetic field, ωN = γe BN , and m0 is
the initial electronic spin moment. When the spin ensemble is averaged over the Gaussian
spread of nuclear fields, the polarization is found to decay as
!
m0
t 2 −t2 /Tnuc.
2
hm(t)i =
1 + 2 1 − 2(
) e
3
Tnuc.
(3.67)
where the characteristic dephasing time is
Tnuc. =
1
.
γe ∆B
(3.68)
The spin evolves non-exponentially and unexpectedly; the polarization decreases (on the
timescale of Tnuc. ) to 33% of its initial value and then is maintained. This remarkable time
dependence has been observed in InAs/GaAs quantum dots [79]. Relaxation of this sort is
reversible (Tnuc. = T2∗ ); each electron spin is coherent in its nuclear environment and it is
the spin ensemble that is seen to decay. This loss of polarization can be reversed via spin
echo experiments. When this is done, it is a more difficult problem to determine the much
longer electron spin relaxation in the slowly varying nuclear fields [80, 39].
2. Motional narrowing. When strong orbital or spin correlations are present, the effect of
the nuclear fields may be motionally narrowed [41]. Examples include electrons hopping
from site to site, spin exchange between donors, and spin exchange with free electrons in
44
which the spin precession is interrupted after the correlation time. This regime is met under
2 i1/2 τ 1 where hB 2 i1/2 is the root mean square of the nuclear field.
the condition γe hBN
c
N
The relaxation time is then [7]
1
2
≈ γe2 hBN
iτc ,
τs
2 i1/2 ≈ 54 G in GaAs [7].
where hBN
(3.69)
As the metallic regime is approached, the hyperfine
interaction becomes negligible [41]. This is mainly because of strong motional narrowing;
delocalized electrons sample many nuclear fields for a short amount of time leading to no
substantial effect.
3.3.2 Anisotropic exchange interaction
The spin-orbit interaction is very important for understanding spin relaxation of itinerant
electrons. How does this interaction manifest itself in insulating systems? Since the spinorbit interaction ‘ties’ together the electron’s spin and position variables, the wave function
is expected to be affected by the spin-orbit interaction. This is indeed the case; each
component of the spinor becomes a different function of coordinates [7]. As before, the spinorbit interaction in systems lacking inversion symmetry leads to the Hamiltonian Hs.o. =
~
2 Ωk · σ
where Ωk is odd in wave vector. For localized electrons, wave vectors are obviously
small and near the localization center the effect of the spin-orbit Hamiltonian on the wave
functions is negligible in comparison to the binding potential. Interestingly though, far
from the potential, in the asymptotic region, the wave function can be altered significantly
since the spin-orbit interaction is now comparable to the binding potential. The asymptotic
form of the wave function is determined in the Wentzel-Kramers-Brillouin (WKB) or semiclassical approximation [24, 81, 7].
An electron moving in a constant one dimensional potential, V , with energy ε has
p
a wave function Ψ ∼ e±ikx where k =
2m(ε − V )/~2 is the wave vector and the ±
sign indicates the possibility of right and left moving waves. The wave vector can also
be considered the phase shift per unit length. If the potential is not quite constant but
varies slowly, the wave function over a small region in which the potential has not changed
drastically is modified such that the wave vector depends on the potential at each x such
p
that k(x) = 2m(ε − V (x))/~2 . Over a traversed distance then (x = 0 → x = x0 for
Rx
instance), the wave function acquires a phase shift 0 k(x0 )dx0 . More generally, the wave
function becomes
Ψ ∼ ei
Rr
0
k(r 0 )·dr 0
.
(3.70)
In a classically forbidden region such that ε < V (r), k becomes imaginary and the wave
function assumes a decaying exponential as expected. In the absence of spin-orbit coupling
45
~2 k02
+ V (r) = ε,
(3.71)
2m
where k0 specifies the unperturbed wave vector (with no spin-orbit effect). So k0 is simply
p
k0 = 2m(ε − V (r))/~2 . Since we want to consider the asymptotic region where r is large,
p
k0 ≈ i 2m|εB |/~2 because ε−V (r) < 0. When spin-orbit coupling is included, this changes
to
~2 k 2 ~
+ Ωk · σ = ε − V (r),
(3.72)
2m
2
where now k = k0 +∆k where ∆k is the piece of the wave vector that is due to the spin-orbit
term. Expanding k around k0 yields
~2 k02
~2 k0 ∆k ~2 ∆k 2 ~
+ V (r) +
+
+ Ωk0 +∆k · σ = ε;
2m
m
2m
2
(3.73)
the first two terms are the unperturbed energy which should be close to ε since the spin-orbit
energy is small. So the first two terms cancel with the right hand side leaving
~2 k0 ∆k ~2 ∆k 2 ~
+
+ Ωk0 +∆k · σ = 0
m
2m
2
(3.74)
Since ∆k k0 , the second term can be ignored. Also the spin-orbit term can be assumed
to depend only on k0 for similar reasons:
~2 k0 ∆k ~
+ Ωk0 · σ = 0.
m
2
(3.75)
Solving for the perturbed wave vector,
∆k =
mΩk0 · σ
mΩk · σ
= √ 0
,
2~k0
2i 2mεB
(3.76)
where ∆k is real since Ω is an odd function of k0 . The asymptotic form of the wave function
is
Ψ∼e
q
mΩk ·σ
2mεB
i √ 0 r
−
2i 2mεB
2 r
e
~
.
(3.77)
The first exponential is familiar; it is the ground state hydrogenic wave function. The second
exponential is due to the spin-orbit interaction and it has the form of a finite rotation on
the spin σ, exp(iγ(r) · σ/2) where [24, 7]
mΩk r
γ(r) = √ 0 ,
i 2mεB
(3.78)
which is real. The effect can be described as follows: near the localizing center the spin has
a projection 1/2 on some axis. At a distance r away the spin will have the same projection
on an axis rotated by the angle γ(r) [82, 7]. If an electron tunnels between two sites with
displacement Rij = |Ri − Rj |, then the spin rotates an angle γ(Rij ) ≡ γij . The presence
46
of the spin-orbit coupling changes the nature of the spinor:
e−r/aB eiγ·σ/2 χ,
(3.79)
since the exponential operator acts on the spin part of the wave function. It is helpful
p
to define a spin-orbit length Ls.o. such that hγ(Ls.o. )2 i = 1 where the angular brackets
denote an averaging over the directions of γ. This leads to hγ 2 (r)i1/2 = r/Ls.o. . If hopping
is considered, the spin of the hopping electron rotates at each hopping event. If the angle
of rotation is small the spin is considered relaxed when the mean square of the accumulated
P 2
angle is one radian:
hγi,j i = 1. For the
cubic Dresselhaus Hamiltonian9 in Table (3.1),
√
the spin-orbit length becomes Ls.o. =
35 ~2
1
2 2m |k0 |2 β3 .
For a quantum well confined in the
z-direction, the linear Dresselhaus term gives a spin-orbit length [83] Ls.o. =
~2
1
2m∗ β3 hkz2 i
where hkz2 i is due to the quasi-2D confinement and is of the form β 2 L−2 ; for infinite well
confinement β = π. When both linear and cubic terms are present, interference between
them may lead to diverging spin-orbit lengths similar to what occurs in DP spin relaxation.
It has been shown though that hopping is an inefficient mechanism for this type of spin
relaxation [7]. A stronger type of effect exists that has been shown to be more effective
in destroying spin polarization; this is the anisotropic exchange effect. In semiconductors
with the type of spin-orbit interaction discussed extensively in this chapter, there exists a
correction to the standard isotropic exchange interaction between two electrons at different
sites. The form of this correction is anisotropic in spin space and has been worked out in
the lab frame to be [82, 83]
Hex. = Hiso. +Haniso. = 2Jij cos γij Si ·Sj +sin γij γ̂ij ·Si ×Sj +(1−cos γij ) γ̂ij ·Si γ̂ij ·Si ,
(3.80)
where the first term on the right hand side is the isotropic exchange and the second two
terms make up anisotropic portions of the exchange interaction: the Dzyaloshinskii-Moriya
and the pseudo-dipole interactions. When spins are exchanged between sites, the anisotropic
correction acts to rotate the spin through the angle γ and much like in the case of hopping
the polarization can relax over a series of exchanges.
The spin relaxation due to the anisotropic exchange interaction can be considered in
the regime of motional narrowing if the spin-orbit interaction is sufficiently small; in such a
2 i1/2 /~ is the spin precession frequency in the anisotropic
case Ωτc 1 where Ω ≈ hJij ihγij
field and τc ≈ ~/hJij i is the time between exchange interactions between localized electrons
[81, 7, 39, 56]. Using the heuristic equation for motional narrowing spin relaxation, τs−1 ∼
9
In some literature [82, 83, 7] a dimensionless parameter αs.o is often used instead of β1 or β3 . In zincblende crystals where
p there is no intrinsic linear spin splitting, the relationship between these two notations
is β3 = αs.o. ~3 /m∗ 2m∗ Eg .
47
hΩ2 iτc , we obtain [7, 39]
2
τs−1 ∼ hΩ2 iτc ∼ hJij ihγij
i/~.
(3.81)
The exchange integral for hydrogen-type atoms has been determined in two and three
dimensions to be [84, 85, 86]
r 5/2
ij
e−2rij /aB
aB
r 7/4
ij
Jij = 15.2εB
e−4rij /aB
aB
Jij = 0.82εB
where εB =
m∗
Ryd
m2r
(3D)
(3.82)
(2D),
(3.83)
is the effective binding or Rydberg energy. r is the dielectric constant
and Ryd = 13.6 eV is the hydrogen atom’s Rydberg energy in three dimensions. Reduced
dimensionality reduces the Rydberg energy by a factor of four in two dimensions. These
formulae are accurate for rij > aB . The exchange constant must be averaged over the
cluster of donors (inter-donor distances vary); this is achieved in the bulk by substituting
−1/3
rij −→ αnimp /2 where α is the constant 1.73 [7].
3.4 Cross-relaxation
Chapter 2 introduces the possibility of cross relaxation between localized and itinerant
states. This phenomenon is also referred to as spin-exchange scattering; a free electron
with some spin scatters from a bound electron (to donor or hole) with opposite spin which
can lead to a mutual spin-flip [46]. This process is responsible for the Kondo Effect in metals
and is characterized by the interaction JSl · Sc where J is the exchange integral and the
subscripts denoted the l ocalized and conduction spins respectively. Mahan and Woodworth
recently calculated this rate in detail and the their results will not be repeated here [46]. It is
extremely rapid; in bulk n-GaAs with nimp = 1016 cm−3 , the cross relaxation time is ∼ 100
fs. The net effect of this process, is that moments rapidly equilibrate such that the average
magnetizations in the two states are equal (ml /nl = mc /nc ). Spin relaxation takes affect
on a longer time scale than cross relaxation. Hence the spin relaxes according to Eq. (2.45)
which is the weighted average of the two environments’ relaxation times. The presence of
an average total spin relaxation rate as oppose to two single rates was first observed by
Paget in n-GaAs [43]. This idea reoccurs in the future chapters of this dissertation.
3.5 Summary
Figure 3.3 shows various measurements of the spin relaxation time versus doping concentration in bulk n-GaAs. It is characterized by three distinct regimes: below the metal-insulator
transition (MIT), near the MIT, and above the MIT which can also be identified as the
insulating, donor band, and metallic regimes respectively. Dzhioev et al. and Müller et al.
48
point out that different mechanisms dominate in each of these regimes [77, 6]. The low doping regime consists of hyperfine induced relaxation. As the impurity content is augmented
the hyperfine relaxation gets motionally narrowed and longer times are observed. The
spin-noise-spectroscopy (SNS) measurements yield shorter times because there production
of free electrons that lead to further motional narrowing as seen in the optical orientation
experiments (conventional probes). The longest times, near the MIT, are due to anisotropic
exchange relaxation. In the metallic regime, the donor wave functions strongly overlap and
become conducting which leads to DP relaxation.
Some researchers have argued in the past that the DP mechanism is responsible for
the non-monotonic impurity and temperature dependences shown in Figures 2.4 and 3.3
[87, 88, 55].
Fully microscopic kinetic Bloch equation calculations do suggest a non-
monotonic dependence but the relaxation times are very different than those measured
[55]. Also, such calculations did not account for donor-bound spins which are now realized
to be of vital importance [7, 56]. Also the recent emergence of spin-noise-spectroscopy (SNS)
experimental technique of relaxation time measurements [9, 6] have elucidated the intrinsic
spin relaxation times in semiconductors. The method is non-invasive in that electrons and
holes are not photo-excited and therefore cannot ‘influence’ the relaxation times. Instead
SNS measures statistical fluctuations of spin polarization by Faraday rotation of linearly
polarized laser light at frequencies below the band gap. The fact that SNS measures the
same trend in relaxation times (as shown in Figure 3.3), points out that the general features
are not artifacts of photo-electrons.
The next two chapters examine spin lifetime measurements and deduce which mechanisms from the options listed in this chapter are relevant.
49
Figure 3.3: Spin relaxation time versus doping density. Three distinct regimes are observed,
indicating three different spin relaxation mechanisms: hyperfine, anisotropic exchange, and
D’yakonov-Perel’. Figure taken from [6]. SNS refers to non-invasive spin-noise-spectroscopy
measurements while conventional probes are optical orientation experiments. Data point
references are located in [6].
50
Chapter 4
Phenomenological approach to
spin relaxation in
semiconductors I; case studies in
bulk and quasi-2D zinc-blende
crystals
4.1 Introduction
In this chapter and the next, a phenomenological approach to explaining spin relaxation
rates is undertaken in zinc-blende (this chapter) and wurtzite (chapter 5) semiconductors.
Each study will be divided into bulk and quasi-2D constituents.
The approach is made up of the following steps:
1. First the question is asked: what mechanisms are responsible for a given set of
measured spin relaxation rates? This is not an easy question to answer and it is shown in
these chapters that sometimes the existing experiments are inadequate in providing a clear
answer. In many instances, an answer can be given though several parameters must be
accounted for: temperature, magnetic field, and doping density among other experimental
factors.
2. The approach is phenomenological in that it begins with the modified Bloch equations
of Eq. (2.44) which yield the spin relaxation rate, Eq. (2.45),
1
nc 1
nl 1
=
+
.
τs
nc + nl τc nc + nl τl
(4.1)
From the experimentally measured impurity density nimp , nc and nl are determined as described in chapter 2.
3. The spin relaxation times for the localized and conduction electrons must be determined. By knowing the qualitative characteristics of the mechanisms described in chapter
51
3, most candidates can be eliminated when the experimental parameter space is known.
The mechanism can be determined quantitatively by a least squares fit of Eq. (2.45) to the
data.
4. As shown in the following sections and chapter, occasionally the modified Bloch
equations must be altered to match the experimental set-up in which case Eq. (2.45) will
change accordingly. Also when the theory of certain spin relaxation mechanisms is lacking,
it must be developed as will be done for the wurtzite structure in the next chapter.
The analysis in zinc-blende (zb) bulk is similar to that done in n-GaAs by Putikka and
Joynt [5]. In addition to n-GaAs, also n-ZnSe and n-CdTe are examined in the following
section. In section 4.3, this phenomenological approach outlined above is applied to quasi2D semiconductors for the first time as presented in [20].
4.2 Bulk crystals
The actual material zinc-blende refers to ZnS (zinc sulfide) but many other semiconductor
compounds form similarly under ambient conditions; they include GaP, AlP, InP, InSb,
InAs, and GaAs to name a few. It is similar to the diamond structure (tetrahedral bonds
and cubic) except that each ion has four opposite ions as nearest neighbors as shown in
Figure 4.1. GaAs has been the archetypal zinc-blende semiconductor for spintronic study.
Figure 4.1: Conventional cubic cell of the zinc-blende crystal structure.
It is a direct band gap material with band structure shown in Figure 4.2. Zinc-blende
materials lack inversion symmetry which gives rise to Dresselhaus spin-splitting though this
splitting between the up and down spin conduction bands is too small to be seen in the
52
Figure 4.2: Band structure of GaAs at 300 K near the Γ-point (k = 0). Spin-splittings of
the conduction band due to the Dresselhaus interaction are too small to be seen.
figure. The Dresselhaus Hamiltonian for bulk zinc-blende is expressed as
HD (k) =
~ D
ω (k) · σ
2 3
(4.2)
where

ω3D (k) =
kx (ky2 − kz2 )

2β3 

 ky (kz2 − kx2 )  ,
~
kz (kx2 − ky2 )
(4.3)
where β3 is the cubic Dresselhaus parameter and gives the strength of Dresselhaus splitting.
The functional form of ω3D (k) in Eq. (4.3) is true for only zinc-blende. It is different for
semiconductors with different crystal symmetries as discussed in chapter 5.
The Dresselhaus spin-orbit splitting is determined in quasi-2D structures by spatially
averaging the Hamiltonian in Eq. (4.2) along the direction of confinement [64]. A common
quantum well orientation is the (001) plane; the Dresselhaus Hamiltonian then becomes
HD (k) =
~ D
~
ω1 (k) · σ + ω3D (k) · σ
2
2
(4.4)
where

ω1D (k) =
−kx

2β3 2 

hkz i  ky 
~
0
53
(4.5)
and

ω3D (k) =
kx ky2

2β3 

 −ky kx2  .
~
0
(4.6)
(110)-quantum wells have engendered interest because the Hamiltonian is
HD (k) =
~ D
~
ω1 (k) · σ + ω3D (k) · σ
2
2
(4.7)
where

ω1D (k) =
2β3 2 
hk i 
~ z
0

0


(4.8)
−kx /2
and

ω3D (k) =
2β3 

~
0

0

,
kx (kx2 − 2ky2 )/2
(4.9)
which is interesting because the effective magnetic field is always in the z-direction and
should not cause spin relaxation for spins parallel to it. Both (001) and (110) quantum
wells are discussed in section 4.3.
4.2.1 Spin relaxation
EY has been studied extensively in bulk zb systems [60, 58, 89, 5, 55]. Calculations of
the EY spin relaxation rate have shown it to be much too weak to produce the measured
rates in III-V zb-bulk semiconductors such as those considered here [89, 5, 55]. The BAP
mechanism is also neglected here since it is irrelevant in n-type systems where the amount
of holes is minuscule [89, 55]. This leaves the DP mechanism which is shown below to give
the correct qualitative and quantitative description of the higher temperature data.
Let us calculate the explicit temperature dependence of the DP mechanism in bulk
zinc-blende crystals. First recall section 3.2.2 that in bulk,
1
τs,ij
=
l
X X
1
n
Tr
(−1)
τ
[H
,
[H
,
σ
]]σ
.
j
i
l
l,−n
l,n
8π~2
l
(4.10)
n=−l
In bulk zinc-blende, Hs.o. ∼ k 3 so only l = 3 will yield a non-zero result. τ3 can easily be
related to τ1 by using Eqs. (3.32) for l = 1, 3 once the scattering type is discerned: τ3 = γ3 τ1
54
where γ3 is a constant that depends on the scattering type. When the sum is expanded,
3
2 X
n
(−1)
τ
Tr
[H
,
[H
,
σ
]]σ
=
3
3,−n
3,n
j
i
8π~2
n=1
τ3 −
Tr
[H
,
[H
,
σ
]]σ
−
Tr
[H
,
[H
,
σ
]]σ
3,−1
3,1 j
i
3,−3
3,3 j
i
4π~2
+Tr [H3,−2 , [H3,2 , σj ]]σi
.
(4.11)
The traces can quickly be computed; the first trace is −64πm∗3 β32 ε3 /21~6 , the second trace
is −64πm∗3 β32 ε3 /35~6 , and the third trace is 512πm∗3 β32 ε3 /105~6 . We obtain10
256
1
m∗3 ε3
=
γ3 τ1 β32
,
τs
105
~8
(4.12)
where spin relaxation can be identified as the diagonal part only of the relaxation tensor
since the relaxation is isotropic (τxx = τyy = τzz and all other components are zero).
We now proceed through the details of the thermal averaging:
256
m∗3
1
=
γ3 β32 8 Iˆ(3) [τ1 ε3 ].
τs
105
~
(4.13)
After substituting for the scattering time and using the processes developed in Section 3.2.3
and Appendix C, we ascertain the following:
1
τs
=
=
=
I7/2+ν (βµ)
s1 ˆ(3) 256m∗3 β32 3+ν 256m∗3 s1 γ3 β32
γ3 I [
ε ] =
(kB T )3+ν
2
6
8
~
105~
105~
I1/2 (βµ)
256m∗3 τtr (kB T )−ν I
I3/2 (βµ)
3/2+ν (βµ)
γ3 β32
105~8
(kB T )3+ν
I7/2+ν (βµ)
I1/2 (βµ)
I3/2 (βµ)I7/2+ν (βµ)
256m∗3 τtr γ3 β32
(kB T )3
.
8
105~
I3/2+ν (βµ)I1/2 (βµ)
(4.14)
Unfortunately in three dimensions, unlike two, the chemical potential cannot be solved
exactly in an ideal Fermi gas. The chemical potential, when Fermi-Dirac statistics are used,
R ∞ √ eβµ e−x
is intractable in the expression for the electron density, n = n0 0 dx x 1+e
βµ e−x where
n0 =
3/2
1 2m∗
k
T
,
B
4 ~2 π
T 3/2 4
n
F
√ .
=
n0
T
3 π
(4.15)
In the non-degenerate limit, when Boltzmann statistics is valid, n = n0 eβµ and the chemical potential can be determined analytically [90]. However analytic approximations exist
in the general case that allow expressions for the chemical potential to be valid even at
relatively high levels of degeneracy. The most famous of these is the Joyce-Dixon approx10
This is equivalent to
3
32
γ τ α2 ε
105 3 1 c ~2 Eg
which is a formulation often seen in the earlier literature [61].
55
Figure 4.3: Measured and theoretical spin relaxation rates below the metal-insulatortransition (nM IT ≈ 2 × 1016 cm−3 ) in n-GaAs at low temperatures (below 10 K). Symbols
are various experiments referenced in [7]. The theory curves contain no fitting parameters.
The maximum spin relaxation time appears around 0.15nM IT . A maximum spin relaxation
time has been also seen in one study on CdTe [8].
imation [91]. However even better approximations have been devised by using the L/M
Padé approximation [92, 90]:
n
n
n
µ = kB T ln
,
+ 4.897 ln(0.045 + 1) + 0.133
n0
n0
n0
(4.16)
In the non-degenerate regime, Boltzmann statistics is valid, and the integrals of Eq. (4.14)
reduce to
5
I3/2 (βµ)I7/2+ν (βµ) T TF ( 32 )!( 72 + ν)!
37
−→ 1 3
=
+ν
+ν .
I3/2+ν (βµ)I1/2 (βµ)
2 2
2
( 2 )!( 2 + ν)!
(4.17)
The final result for the DP relaxation rate at high temperatures is then
8Qm∗3 τtr β32
(kB T )3 ,
105~8
where Q =
16
35 γ3 (7/2 + ν)(5/2 + ν).
(4.18)
Q is of order unity for bulk scattering mechanisms [61].
Localized mechanisms are expected to be dominant at low temperatures when impurity sites are near full occupation. The two pertinent mechanisms are the hyperfine and
anisotropic exchange interactions detailed in chapter 3. Which of these two prevails depends
56
Figure 4.4: Adapted from [9] showing how the temperature dependence of the spin relaxation depends on the doping density.
on the impurity concentration and effective Bohr radius of the impurities [77, 7]. In GaAs
(aB = 10.4 nm), the hyperfine interaction dominates the spin relaxation for impurity concentrations below ∼ 4 × 1015 cm−3 and the anisotropic exchange interaction affects larger
rates in the range 4 × 1015 < nimp < 2 × 1016 cm−3 = nM IT = (0.25/aB )3 where nM IT is
the impurity concentration at the metal-insulator-transition (MIT). This is shown in Figs.
3.3, 4.3, and 4.4. For even higher concentrations, a metallic regime is entered. ZnSe and
CdTe have smaller Bohr radii (4.6 nm and 5.3 nm, respectively) and as a consequence the
above regimes are expected be at higher concentrations than for GaAs. Unfortunately these
two materials have not been studied experimentally nearly as much as GaAs but one study
does suggest this to be true for CdTe [8].
4.2.2 Comparison with experiments
Figures 4.5,4.6, and 4.7 present three experiments (solid circles) in which the spin relaxation
rate was measured over a temperature range in the following n-type semiconductors: GaAs,
ZnSe, and CdTe respectively. In GaAs, the theory curves in Fig. 4.3 agree with the large
amount of studies which suggest that the anisotropic exchange mechanism is operative in
the doping range of the data shown in Figure 4.5 [77, 5, 7]. The lack of data for ZnSe
57
1T2* Hns-1L
10
1
0.1
0.01
0.001
5
10
20
50
100
200
T HKL
Figure 4.5: Solid blue circles from Kikkawa using n-GaAs with nimp = 1 × 1016 cm−3 at
zero applied field [4]. Solid line is fit with Eq. (2.45).
and CdTe makes determinations of the precise mechanism difficult at low temperatures.
This is because both likely mechanisms - motionally narrowed hyperfine and anisotropic
exchange (see section 3.3) - are exponential in nature when the correlation time for the
hyperfine interaction is assumed to be due to exchange. So in GaAs we can be certain that
anisotropic exchange is the largest effect at nimp = 1 × 1016 cm−3 whereas for the impurity
concentrations in ZnSe and CdTe we are not sure which is dominant and it is possible both
could be similar in effect. Due to this uncertainty in any fit, we leave the localized relaxation
rate as a fitting parameter in Figs. 4.6 and 4.7.
The situation is much clearer at high temperatures when the DP mechanism takes
charge. Eq. (4.18) can be inserted into Eq. (2.45) for 1/τc . There is some variance in
the experimental and theoretical determinations of the Dresselhaus (cubic) spin splitting
coefficient β3 . In GaAs, by using a value near the lower end of the recently measured and
calculated spectrum (6.5 meV nm3 ) [94, 95], a least squares fit results in Q = 2.8 which
agrees well with acoustic phonon scattering (Q = 2.7) [61]. The results of the fits for the
three materials is summarized in Table 4.2.2. The fits are strikingly good in determining Q.
The fits also suggest that the localized relaxation mechanism is not strongly temperature
dependent.
The conclusion of this section is that the modified Bloch equation approach, which
58
1T2* Hns-1L
10.00
5.00
1.00
0.50
0.10
0.05
0.01
5
10
20
50
100
200
T HKL
Figure 4.6: Solid blue circles from Malajovich et al. using n-ZnSe with nimp = 5 × 1016
cm−3 at zero applied field [10] Solid line is fit with Eq. (2.45). Used same mobility as for
GaAs.
takes into account exchange interactions between localized and itinerant electrons, provides
a good description of the observed spin relaxation times in bulk zinc-blende semiconductors
in no field. In the next section this idea will be extended to structures with reduced
dimensionality.
4.3 Quasi-2D nanostructures
In recent years, uniformly doped quantum wells (QWs) have generated increasing interest
due to the long relaxation times measured therein [96, 97, 98]. The long relaxation times
are due to spins localized on donor centers. While similar relaxation times have been
measured in modulation doped systems, their duration has not been as reliable due to the
weaker binding energy of localized states and potential fluctuations from remote impurities
[99, 12, 100]. Localization is either not seen at all [99] or localization centers thermally ionize
rapidly with increasing temperature due to a small binding energy [100, 12]. QWs uniformly
doped within the well have the advantage of being characterized by well defined impurity
centers with a larger binding energy. The experimental control in the amount of doping
59
1T2* Hns-1L
100.0
50.0
10.0
5.0
1.0
0.5
0.1
5
10
20
50
100
200
T HKL
Figure 4.7: Solid blue circles from Sprinzl et al. using n-CdTe with nimp = 4.9 × 1016 cm−3
at zero applied field [8]. Solid line is fit with Eq. (2.45). Transport time taken from mobility
measurements of [11]
and well size make doped QWs particularly appealing to the study of quasi-two-dimensional
spin dynamics.
Much of the theoretical study of spin relaxation in semiconducting systems (QWs in
particular) has either focused solely on itinerant electrons [101, 102, 67] or solely on localized electrons [83, 7] without regard for either the presence of the other state or the
interaction between the two states. Recently the existence and interaction between itinerant and localized states has been dealt with in bulk systems in [5, 21, 46]. The results of
these calculations are in very good quantitative and qualitative agreement with experimental observations [4, 103] in bulk n-GaAs and n-ZnO. In the following section, the theory of
two interacting spin subsystems is applied to QWs.
4.3.1 Spin polarization in quantum wells
In QWs at low temperatures the creation of non-zero spin polarization, in the conduction
band and donor states, proceeds from the formation of trions (charged excitons, X ± ) and
exciton-bound-donor complexes (D0 X) respectively, from the absorption of circularly po60
Bulk material
GaAs
ZnSe
CdTe
γ3 (meV nm3 )
6.5
1.3
8.5
1/τl (ns−1 ) (fit)
0.010
0.017
0.43
Q (fit)
2.8
3.7
5.4
Q (th.)
2.7
2.7
2.7
Table 4.1: Results of fits in Figs. 4.5, 4.6, and 4.7. Acoustic phonon scattering is assumed
for each (give Q = 2.7). Mobilities for GaAs come from [5] and references therein. Mobilities
of ZnSe are not available so that of GaAs were used. Mobilities for CdTe come from [93].
larized light. Figure 4.8 and 4.9 illustrates the absorption and polarization process in bulk
semiconductors and QWs via the trion route.
Figure 4.8: Illustration of optical spin pumping in bulk semiconductor. CB and VB are
conduction and valence bands respectively. σ ± denotes the helicity of the absorbed and
emitted photons (wiggly lines). Photon promotes one electron from VB to CB leaving a
hole behind. The hole spin (thick arrows) is assumed to relax much quicker than the electron
spin (thin arrows). Sz is total electron spin. Graphic taken from [12].
Polarization via the trion avenue is most relevant for modulation doped QWs where
donor centers in the well are sparse [99, 98]. Due to the modulation doping outside the well,
the number of conduction electrons in the well may be plentiful. In such cases, assuming
incident σ + pump pulse, a + 32 hole and − 12 electron are created. These bind with a resident
−
electron from the electron gas in the QW to form a trion (X3/2
). The trion’s binding
energy is ∼ 2 meV [100, 12]. The ‘stolen’ electron will be + 21 to form a singlet state with
the exciton’s electron. Hence, the electron gas will be left negatively polarized since the
excitons are preferentially formed with spin up resident electrons. If the hole spin relaxes
faster than the trion decays the electron gas will remain polarized [99]. Selection rules
61
dictate + 32 (− 32 ) holes will recombine only with − 21 (+ 12 ) electrons. Therefore if the hole
spins relax rapidly, the released electrons will have no net polarization and the polarized
electron gas will remain predominantly negatively oriented.
Figure 4.9: Illustration of optical spin pumping in QWs when photo-excitation is resonant
with the trion formation. When excited at the donor-bound-exciton resonance instead, the
picture is similar except that the exciton ‘steals’ an electron from a neutral donor (or is
actually captured by the neutral donor) instead of a free electron. The key difference is
that after hole spin relaxation and recombination, the neutral donors are left with a net
polarization instead of the free electrons. Graphic taken from [12].
A very similar picture is given for the polarization of donor bound electrons in uniformly
doped QWs where the donor bound electrons play the role of the resident electrons [96, 97].
At low temperatures the donors are nearly all occupied and the density of the electron gas
is negligible. When excitations are tuned at the exciton-bound-donor resonance, instead
of photo-excitons binding with the resident electron gas, they bind with neutral donors to
form the complexes D0 X3/2 . This notation implies that a + 32 hole - − 21 electron exciton
is bound to a + 12 donor bound electron. The D0 X3/2 ’s binding energy is ∼ 4.5 meV [104].
Once again for very short hole relaxation times, the donor bound electrons can be spin
polarized.
The measured long spin relaxation times in uniformly doped QWs imply that spin
polarization remains after short time processes such as X and D0 X recombination have
completed. In other words, the translational degrees of freedom thermalize much more
quickly than the spin degrees of freedom. The occupational statistics of itinerant and
localized electrons are important and can be determined from equilibrium thermodynamics.
As the temperature is increased, the electrons bound to donors thermally ionize and become
itinerant. As the number of electrons in the conduction states increases, the spin that exists
on the donors equilibrates by cross relaxing to conduction states by the isotropic exchange
62
Figure 4.10: Illustration of optical pumping in QWs when photo-excitation is resonant
with exciton formation. The conduction band (CB) starts with no spin polarization as the
exciton is created (left most panel). The hole spin bound in the exciton rapidly relaxes
(second panel). An antiparallel resident electron spin is grabbed from the electron gas (or
the exciton binds to a neutral donor). This leaves the resident electron polarized (third
panel). Oppositely oriented electron and hole spins recombine, casting the third electron
back into the electron sea adding more net spin moment (rightmost panel). Taken from
[12].
interaction much like what occurs in bulk. If cross relaxation is rapid enough, the total
spin, which is conserved by exchange, exists in the donor and conduction states weighted
by their respective equilibrium densities [5, 46, 21]. The polarized electron moments will
then proceed to relax via different processes for the localized and itinerant states - the total
relaxation given again by Eq. (2.45).
The above description is complicated when the photo-excitation energy is at the exciton
resonance and not the exciton-bound-donor resonance. In such a case, the excitons may
recombine, the electron-in-exciton spin may relax before recombination, or the electron
spin may cross-relax to the localized or free electrons. One expects the low temperature
spin relaxation to reflect also the exciton spin dynamics instead of the donor electron spin
dynamics alone [100]; see Figure 4.10. In essence, the electrons in an exciton represent a
third spin environment with a characteristic spin relaxation time scale different from that
of the localized donor and itinerant electrons. Because of the electron’s proximity to a hole,
relaxation may result from electron-hole spin exchange or recombination.
Therefore to understand the spin dynamics in QWs, it is imperative to examine the
relaxation processes that affect the polarized spin moments of the various spin systems.
63
4.3.2 Modified Bloch equations
First photo-excitation energies near the exciton-bound-donor resonance are considered. After rapid exciton-donor-bound complex formation, recombination, and hole relaxation, we
model the zero field spin dynamics of the system in terms of modified Bloch equations as
initially discussed in chapter 2:
1
dmc
nl nc
+ cr mc + cr ml
=−
dt
τc γc,l
γc,l
dml
nl
1
nc
= cr mc −
+ cr ml
dt
γc,l
τl γc,l
(4.19)
where mc (ml ) are the conduction (localized) magnetizations, nc (nl ) are the conduction
(localized) equilibrium occupation densities, τc (τl ) are the conduction (localized) spin recr = τ
laxation times, and γc,l
exch nimp is a parameter describing the cross relaxation time
between the two spin subsystems. Mahan and Woodworth [46] have shown the cross relaxation time between impurity and conduction electron spins to be much shorter than any of
the other spin relaxation times relevant here. We shall assume below that the same is true
for the cross relaxation between electrons bound in an exciton and conduction or impurity
electron spins. The motivation of these modified Bloch equations is set forth in [5, 21].
Eqs. (4.19) is valid for photo-excitation energies that do not cause free exciton formation
(only two relevant spin systems). It is important to note that Eqs. (4.19) hold only for
time scales that are long compared with laser pulse times, energy relaxation times that
determine subsystem populations, and donor-bound-exciton formation times. Fortunately,
these time scales are the ones probed in the experiments.
Standard methods can be used to solve these differential equations with initial conditions
mc (0) and ml (0). We assume that the initial spin polarization is perpendicular to the QW’s
growth plane and that the excitation density, Nx , is small enough such that the resultant
spin relaxation time, τs , will not depend strongly on Nx [97]. The solutions yield a time
dependence of the total magnetization m(t) = mc (t)+ml (t) to be a sum of two exponentials
- one of which is exp(−t/τs ) and the other of which has a time constant proportional to the
cross relaxation time. In the case of rapid cross relaxation (faster than all spin relaxation
mechanisms), only one exponential survives and the total relaxation rate is expressed as
1
nl 1
nc 1
=
+
τs
nimp τl nimp τc
(4.20)
where nimp = nl + nc is the total impurity concentration. This model, or variations of it,
has been successfully applied to bulk n-GaAs and bulk n-ZnO [5, 21].
If the photo-excitation energy is set near the exciton energy, the Bloch equations must
be modified to take into account exciton spin relaxation and multiple cross relaxations:
64
γi,j for i, j ∈ c, l, x for conduction, localized, and exciton spins respectively. The exciton
spin relaxation is modeled as electron-in-exciton spin relaxation [105] and the hole spin
relaxation is assumed to be very rapid. Eq. (5.54) generalizes to
1
dmc
nl
nx nc + Nx − nx
nc + Nx − nx
=−
+ cr + cr mc +
ml +
mx
cr
cr
dt
τc γc,l γc,x
γc,l
γc,x
1
dml
nx nl
nl
nc + N x − nx
+
ml + cr mx
= cr mc −
+
cr
cr
dt
γc,l
τl
γc,l
γl,x
γl,x
dmx
nc + N x − nx
nx
nx
1
nl +
= cr mc + cr ml −
+
cr mx ,
cr
dt
γc,x
γl,x
τx
γc,x
γl,x
(4.21)
where τx represents spin lifetime of an electron bound to a hole. nx (mx ) is the number
(magnetization) of electrons bound in an exciton. Nx is the initial density of photo-excited
electrons and the quantity Nx − nx is the number of photo-excited electrons that do not
participate in an exciton. We assume quasi-equilibrium such that nx is determined from
thermodynamics (see Section 4.3.3). It should be stated that Eq. (4.21) is valid only for
times shorter than the recombination time; in other words, on a time scale where Nx can
be assumed to not change significantly. Recombination times have been measured [106]
in similar systems as to those studied here to be longer than the observed spin relaxation
times so this approximation seems justified. In Section 4.3.5, we find that the effects of
recombination of free carriers can be added to 1/τc to obtain excellent agreement with the
experimental data.
If the system of equations in Eq. (4.21) is solved as done for Eq. (4.19), the following
relaxation rate is obtained:
nl
1
nc + Nx − nx 1
nx
1
1
=
+
+
.
τs
nimp + Nx τl
nimp + Nx τc nimp + Nx τx
(4.22)
For both Eqs. (4.20) and (4.22), we allow τl , τc , and τx to be phenomenological paP
rameters of the form τi−1 = j 1/τj where j refers to a type of spin relaxation mechanism.
From the experimental constraints and results, the important relaxation mechanisms are
determined.
4.3.3 Occupation concentrations
As shown above, the relative occupations of localized and itinerant states play an important
role in our theory. Fortunately, in two dimensional systems, the occupation probabilities of
the two states (nl /nimp and nc /nimp ) can be determined exactly. However the densities we
are interested in are dilute enough such that the non-degenerate limit (Boltzmann statistics)
can be utilized.
The probability for a donor to be singly occupied (only the ground state needs to be
65
Figure 4.11: Occupation probabilities of localized (solid line) and conduction (dash-dotted
line) states with impurity density nimp = 4 × 1010 cm−2 determined from Eqs. (4.25, 4.26).
Other parameters for GaAs are a∗B = 10.4 nm and m∗ = 0.067m.
considered [107]) is [47]
nl
1
= 1 (ε −µ)/k T
.
B
B
nimp
+1
2e
(4.23)
The density of itinerant states is given by
nc = Nc eµ/kB T
(4.24)
where Nc = m∗ kB T /~2 π and the conduction band edge is taken to be zero energy. The
chemical potential µ can be found using the constraint
nl
nc
+
= 1.
nimp nimp
(4.25)
p
1 + Q(T, nimp ) − 1
nl
=p
,
nimp
1 + Q(T, nimp ) + 1
(4.26)
8nimp −εB /kB T
e
.
Nc
(4.27)
Using the result for µ, one obtains
where
Q(T, nimp ) =
An example of the temperature dependence of these occupation probabilities is shown
for a GaAs QW in Figure 4.11 where nimp = 4 × 1010 cm−2 . At the lowest temperatures,
66
the donors are fully occupied. As the temperature increases, nl decreases and nc increases
to where at around 50 K, the two occupation probabilities are equal. From Eqs. (4.20,
4.22), it is evident that these occupational statistics have ramifications in the measured
spin relaxation times. The results here are also applied to the excitons in quasi-equilibrium.
4.3.4 Spin relaxation
We now discuss the relevant spin relaxation mechanisms for both localized and conduction electrons. The electron-in-exciton spin relaxation, τx , is a combination of electron-hole
recombination and electron-hole exchange relaxation. Due to its complicated nature, calculation of τx is deferred to future work. Here we treat it as a phenomenological parameter.
Localized spin relaxation
First we discuss spin relaxation via the anisotropic spin exchange for donor bound electrons.
This has been treated extensively elsewhere [108, 109, 82, 83]. Most recently it has been
examined by Kavokin in [7]. It is his treatment that we detail below for semiconducting
QWs.
Kavokin argues [7] that some portion of localized relaxation results from spin diffusion
due to the exchange interaction between donors. Anisotropic corrections to the isotropic
exchange Hamiltonian cause a spin to rotate through an angle γi,j when it is transferred
between two donor centers located at positions ri and rj . The angle-averaged rotation angle
2 i1/2 = hr 2 i1/2 /L
is hγi,j
s.o. where Ls.o. is the spin orbit length [7]. The spin is relaxed when
i,j
P 2
the accumulated rotation angle Γ becomes on the order of unity such that Γ2 = hγi,j
i=
P 2
hri,j i/L2s.o. = 1. It is assumed that the spin rotates in a motionally narrowed fashion such
that Eq. (3.81) holds. The results of section 3.3.2 for the exchange integral and spin-orbit
length combine to obtain the relaxation rate of Eq. (3.81) in terms of a dimensionless
impurity separation scale, x:
1
β 2 hk 2 i2 m∗ 2 7/4 −4x
= 2 · 15.21 3 z3
hx ihx e i
τex
~
(4.28)
where x = ri,j /aB and the exponential dependence is noted. Only linear-in-k spin-splitting
is considered.How ri,j is to be determined will be discussed in Section 4.3.5.
Localized electron spins may also relax due to nuclear fields as described in section
3.2.4. A localized electron is coupled to many nuclear spins by the hyperfine interaction.
To the electrons, these nuclear spins appear as a randomly fluctuating field but these nuclear
fields can be assumed quasi-stationary since the nuclear evolution time is much longer than
electron evolution time due to the contrast in magnetic moments [7]. Merkulov et al. [78]
67
find a dephasing rate, Eq. (3.68), for quantum dots to be
s P
16 j Ij (Ij + 1)A2j
1
=
Tnuc
3~2 NL
(4.29)
where the sum over j is a sum over all nuclei in the unit cell, Ij is the nuclear spin, Aj is
the hyperfine constant, and NL is the number of nuclei in the electron’s localized volume.
If the hyperfine interaction is motionally narrowed by exchange, the relaxation time would
tend to be longer since a spin would freely precess in the nuclear field a shorter amount of
time.
Conduction spin relaxation
Conduction band states undergo ordinary impurity and phonon scattering. Each scattering
event gives a change in the wave vector k, which in turn changes the effective magnetic field
on the spin that comes from spin-orbit coupling. This fluctuating field relaxes the spin. This
is known as the D’yakonov-Perel’ (DP) spin relaxation mechanism [110, 64] and is treated
extensively in chapter 3. The effective field strength is proportional to the conduction band
splitting. We are interested in conduction spin relaxation in (001) and (110) oriented QWs.
For (001) QWs the spin relaxation rate results from a spin-orbit term in the Hamiltonian,
Hs.o = ~2 ωD (k|| ) · σ where [67]

ωD (k|| ) =
kx (ky2 − hkz2 i)

2β3 

 ky (hkz2 i − kx2 )  .
~
0
The angular brackets denote spatial averaging across the well width. β3 is a band parameter
that governs the magnitude of the spin-orbit splitting. For GaAs, β3 ∼ 17 meV nm3
[111]. The QWs have been grown symmetrically and therefore any Rashba contribution is
negligible [112].
The resulting spin relaxation has been worked out in detail by Kainz et al. in [67]. For
the experiment [113] we compare to, we find the non-degenerate limit to be applicable and
hence use the relaxation rate for spin oriented in the z-direction,
"
∗
1
4
β32 hkz2 i 2m∗ kB T 2
2 2 2 2m kB T
=
τ
(T
)
β
hk
i
−
j2 +
p
3 z
τz
~2
~2
2
~2
#
∗
3
2 1 + τ3 /τ1 2m kB T
β3
j3
16
~2
(4.30)
where j2 ≈ 2 and j3 ≈ 6 depend on the type of scattering mechanism. We assume Type
I scattering as defined in [67]. The ratio τ3 /τ1 is unity for Type I scattering. τp (T ) is the
68
momentum relaxation time which can be extracted from mobility measurements.
A more interesting case is that of (110) QWs where the spin-orbit Hamiltonian is [114]
Hs.o. = −β3 σz kx
1 2
1
hkz i − (kx2 − 2ky2 )
2
2
(4.31)
which is obtained from the (001) Hamiltonian by transforming the coordinate system such
that x||[110], y||[001], and z||[110]. As can be seen from the form of this Hamiltonian, the
effective magnetic field is in the direction of the growth plane. Hence, spins oriented along
the effective field will experience no spin relaxation.
Conduction spins also relax due to the Elliott-Yafet (EY) mechanism [48, 40] which arises
from spin mixing in the wavefunctions. Due to spin-orbit interaction, when a conduction
electron is scattered by a spin-independent potential from state k to k0 , the initial and final
states are not eigenstates of the spin projection operator Sz so the process allows for spinflips. In bulk, the relaxation rate is known to be of the form 1/τEY = αEY T 2 /τp (T ) where
αEY is a material-dependent parameter and τp is the momentum relaxation time [58].
However the EY mechanism in quasi-two dimensions will not take the same form since
k will be quantized in one direction (the direction of confinement). The treatment in bulk
[59] has been extended to QWs to obtain [115]
1
τEY
≈
∆s.o. 2 m∗ 2 Ec kB T 1
1−
,
∆s.o. + Eg
m
Eg2 τp (T )
(4.32)
where ∆s.o. is the spin-orbit splitting energy and Ec is the QW confinement energy.
Spins may also relax due to the Bir-Aronov-Pikus (BAP) mechanism [53] which arises
from the scattering of electrons and holes. This relaxation mechanism is commonly considered efficient only in p-type materials when the number of holes is large [89]. We fit the
experimental data in Section 4.3.5 without consideration of this mechanism.
We now examine how these relaxation mechanisms are manifest in two different QWs.
4.3.5 Results for GaAs/AlGaAs quantum well
We apply our method to measured spin relaxation times of two GaAs/AlGaAs QWs by
Ohno et al. [113, 116]: (100) n-doped QW with doping nimp = 4 × 1010 cm−2 , well width
L = 7.5 nm; and a (110) undoped QW with well width L = 7.5 nm. In both (pump-probe)
experiments, the pump or photo-excitation energy was tuned to the heavy hole exciton
resonance and normally incident on the sample. As mentioned in Section 4.3.1, the exciton
spin becomes important at low temperatures for such excitation energies. The experimental
spin relaxation times as a function of temperature are displayed (solid circles) in Figures
4.12 and 4.13.11
11
The experimental rates of Ohno et al. depicted here are twice that of what is reported in the actual
experiments by that group. This is due to different definitions of spin relaxation time [67].
69
Figure 4.12: Spin relaxation versus temperature in undoped (110) GaAs QW. Points are
experiment of [116]. Dash-dotted line: Using only conduction portion of Eq. (4.33) and
1/τc = 1/τEY + 1/τr . Intersubband spin relaxation is also important when DP is suppressed
[117]. Dashed line: using only excitonic portion of Eq. (4.33). Solid line: Eq. (4.33). Spin
relaxation rate of excitons decreases with temperature increase due to thermal ionization.
Conduction spin relaxation is longer in (110) QW than in other oriented QWs due to
vanishing DP mechanism.
For the undoped (110) QW, Eq. (4.22) is modified to become
1
Nx − nx 1
nx 1
=
+
.
τs
Nx τc Nx τx
(4.33)
For this sample, at low temperatures, nx = Nx so the τs = τx ≈ 0.15 ns which is seen from
viewing Figure 4.12. At higher temperatures, recombination (in time τr ) and EY act to relax
conduction spins since DP relaxation is significantly reduced for the (110) QW orientation.
To account for the quasi-two dimensional nature of the QW, we use an intermediate value
(between 2D and 3D values) for the exciton’s binding energy [118]. Eq. (4.33) (solid line)
fits the data (points) with excellent agreement in Figure 4.12 when Nx = 1.5 × 1010 cm−2
and τr = 2 ns which are near the experimentally reported values (Nx ≈ 1010 cm−2 and
τr ≈ 1.6 ns) [106]. The contributions from the excitons and conduction electrons are also
shown (dashed and dash-dotted lines respectively). The trend in the data is well described
by our theory using Eq. (4.33) - at low temperatures excitons predominate and the spin
relaxation time is τx . When the temperature increases, the excitons thermally ionize leading
to net moment in the conduction band. Since the conduction band spin relaxation time is
longer than the exciton spin relaxation time, the measured relaxation time increases with
70
temperature as described in Eq. (4.33). We expect the relaxation times to eventually
level out as the excitons disappear. Eventually, the relaxation time will decrease as the
temperature dependence of EY takes effect.
For the doped (100) QW, Eq. (4.22) should be used to describe the temperature dependence of the relaxation rate. Using the values from above and nimp = 4 × 1010 cm−2 ,
τs = 0.35 ns, we can extract the approximate value of τl . In doing so we obtain τl ≈ 0.5
ns. We stress that this value has considerable uncertainty due to the uncertainty in the
parameters (namely Nx ) that determine τl . The presence of impurities has lengthened the
observed low temperature spin relaxation time by more than a factor of two. The relaxation
time in the doped sample can be further increased by reducing the excitation density. As
the temperature is increased, donors become unoccupied and conduction electrons will play
a larger role in relaxation as expressed in Eq. (4.33). The main conduction spin relaxation
mechanism is determined by investigating its temperature dependence.
Figure 4.13: Spin relaxation versus temperature in n-doped (100) GaAs QW. Points are
from [113]. Dashed line: excitonic contribution in Eq. (4.22). Dotted line: localized
contribution in Eq. (4.22). Dash-dotted line: conduction contribution in Eq. (4.22). Solid
black line: Eq. (4.22). Both exciton and localized spin relaxation contribute to the observed
low temperature spin relaxation. Conduction spin relaxation is the most strong contributor
to the observed relaxation at higher temperatures.
We are now left with the task of determining what the localized and conduction spin
relaxation mechanisms are. We plot the relaxation rate for the n-doped GaAs QW as a
function of temperature in Figure 4.13. The dashed, dotted, and dash-dotted lines refer to
71
the three terms of Eq. (4.33) - the density weighted average of the respective relaxation
rates. The solid line is the sum of all three terms.
We begin by calculating spin relaxation due to spin exchange diffusion in Eq. (4.28).
This is difficult due to the exponential dependence on ri,j . Parameters for GaAs can be
found in Appendix A. To calculate hkz2 i = β 2 /L2 , we need to know the band offsets and
assume a finite square well. The potential depth for a AlGaAs QW is about V0 = 0.23
eV. This comes from
∆Ec
∆Eg
= 0.62 and ∆Eg = 0.37 eV in GaAs [119] . From this we can
determine β which will also depend on the well width L. For L = 7.5 nm, β = 2.19. In the
limit of V0 → ∞, β → π. What remains to be determined is ri,j which is proportional to the
−1/2
inter-donor separation ri,j = γnimp . For average inter-donor spacing in two dimensions,
γav = 0.564. When we allow γ to be fitting parameter, we obtain ri,j = 19.5 nm which
corresponds to γ = 0.4.
We now determine the relaxation rate due to the hyperfine interaction using Eq. (4.29).
Since nearly all nuclei have the same spin [120] (I = 3/2), Eq. (4.29) is expressed as
s P
5 j A2j
1
=2
,
(4.34)
Tnuc
~2 NL
P
with j A2j = 1.2 × 10−3 meV2 and NL ∼ 2.1 × 105 [78]. This yields Tnuc = 3.9 ns. Due
to the donor’s confinement in the QW, its wave function may shrink thereby reducing the
localization volume and therefore also reducing NL and Tnuc [118]. Motional narrowing
by exchange interactions would lengthen this time, taking it further from the experimental
value.
In Figure 4.13, we find find excellent agreement with experiment over a large temperature
range when τp (T ) in Eq. (4.30) is made a factor of three smaller than what is reported in
[67]. We attain approximately the same quantitative accuracy as in [67] but since we also
take into account the localized spins, we find excellent qualitative agreement as well. It
should be emphasized that the DP’s quadratic and cubic terms of Eq. (4.30) are important
in the high temperature regime. The EY rate is qualitatively and quantitatively different
from the data. For instance, 1/τEY ≈ 0.1 ns−1 at 300 K so is ruled out of contention.
We also now ignore recombination of carriers since an appreciable amount of equilibrium
carriers exist (n-doped system) leading to recombination of primarily non-polarized spins.
One would not expect these results to agree with spin relaxation measurements in modulation doped QWs. In modulation doped systems, the occupation densities nl and nc cannot
be calculated as we have done here. In such systems different spin relaxation dependencies
are seen [106, 117].
72
4.3.6 Results for CdTe/CdMgTe quantum well
The experiment by Tribollet et. al. on a n-CdTe QW offers an instructive complement to the
previous experiments on GaAs. In their experiment, Tribollet et al. measure spin relaxation
times τs ≈ 20 ns for CdTe/CdMgTe QWs with nimp = 1 × 1011 cm−2 . Importantly, they
excited with laser energies at the donor bound exciton frequency instead of the heavy hole
exciton frequency.
Parameters for CdTe can be found in Appendix A. To obtain potential well depth for
CdTe QW, Eg (xM g ) = 1.61 + 1.76xM g where xM g gives fraction of Mg in Cd1−x Mgx Te
[121]. If xM g = 0.1, V0 = 0.12eV which leads to β = 2.18.
We now determine the relaxation rate due to the hyperfine interaction. Since all nuclei
with non-zero spin have the same spin [120] (I = 1/2), Eq. (4.29) is
sP
2
1
j Aj Pj
,
=2
Tnuc
~2 NL
(4.35)
where Pj has been appended to account for isotopic abundances [96]. The natural abundances of spin-1/2 Cd and Te nuclei dictate that PCd = 0.25 and PT e = 0.08. The remaining isotopes are spin-0. NL = 1.8 × 104 , ACd = 31 µeV, and ATe = 45 µeV which yields
Tnuc = 4.4 ns [96]. The confined donor wave function in CdTe shrinks less than in GaAs
since the effective Bohr radius is half as large.
This value is within an order of magnitude of what is calculated for relaxation due to the
hyperfine interaction. We can also compare the experimental time to what we obtain for
spin exchange diffusion. When we allow γ to be a fitting parameter, we obtain ri,j = 19.3
nm which corresponds to γ = 0.61.
Unfortunately no relaxation measurements have been performed at higher temperatures
in n-doped CdTe QWs that we are aware of. We are also not aware of mobility measurements
in n-doped CdTe QWs. The prevalent mechanism (DP or EY) will depend on the mobility
so we forgo determining the more efficient rate. However, in analogy to bulk systems, we
expect the CdTe QW mobilities to be less than the GaAs QW mobilities [5, 93]. In the
next section we analyze CdTe’s spin relaxation rate for (110) grown crystal so DP can be
ignored.
4.3.7 Comparison of GaAs and CdTe quantum wells
First we discuss the low temperature spin relaxation. Interestingly, the localized relaxation
time in CdTe is about 20 times longer than in GaAs. This can be explained by the spin
exchange relaxation despite the larger spin orbit parameter in the CdTe. This is more than
offset by the smaller effective Bohr radius in CdTe (5.3 nm vs. 10.4 nm) and the exponential
behavior of the anisotropic exchange relaxation. However due to the exponential factor,
73
any discrepancy between the two QWs can be explained by adjusting their respective γs
appropriately, though the fitted γ’s do fall near γavg . The discrepancy in times is difficult to
explain by the hyperfine interaction since the two calculated relaxation times are very near
each other. Additionally, no plateau effect is seen that is indicative of frozen-field hyperfine
dephasing [96, 79]. Another possibility is that one QW is governed by relaxation from spin
exchange and the other from hyperfine interactions. The metal-insulator-transitions for the
two materials is nM IT (GaAs) ≈ 6 × 1010 cm−2 and nM IT (CdTe) ≈ 2 × 1011 cm−2 which
are just above the experimental doping densities. Without additional experimental data,
answering these questions is difficult. It is our hope that further experiments will be done
to sort out these questions. However, we can propose ways in which these answers can be
discovered.
Relaxation by anisotropic spin exchange is strongly dependent on the impurity density.
By altering the impurity doping within the well, one should see large changes in the spin
relaxation time if this mechanism is dominant. From Eq. (4.28) we see that this mechanism
will also depend on the confinement energy. Hence this mechanism should also be affected by
changing the well width. The hyperfine dephasing mechanism should be largely unaffected
by impurity concentration differences as long as they are not so extensive as to cause the
correlation time to become very short and enter a motional narrowing regime. Varying
the well width will have an effect on the donor wavefunctions, but as long as they are not
squeezed too thin as to approach the motional narrowing regime the effect should not be
dramatic.
For spin relaxation at higher temperatures, DP prevails in (100) GaAs QWs as mentioned earlier. Whether DP or EY is more efficient in CdTe depends on the momentum
relaxation time. By changes in momentum relaxation times (by changing well width or
impurity concentration), we predict the the possibility to induce a clear ‘dip’ in the temperature dependence which we see in Figure 4.14. This same non-monotonicity has been
observed bulk GaAs and ZnO [4, 103, 5, 21].
Using our results we propose that n-doped (110) QWs should optimize spin lifetimes
(when excited at exciton-bound-donor frequency) since DP is suppressed. Figure 4.15 displays our results for GaAs and CdTe (110) QWs at impurity densities nimp = 4 × 1010 cm−2
and nimp = 1 × 1011 cm−2 respectively. The decrease seen in GaAs is now due to depopulation of donor states instead of exciton thermalization. The depopulation is much slower
in CdTe since the doping is higher. The up-turn in the CdTe curve as room temperature
is reached is due to EY which is too weak to be seen in GaAs. We plot the data points
from the undoped (110) GaAs QW for comparison. By avoiding the creation of excitons
and their short lifetimes, long spin relaxation times can be achieved.
74
Figure 4.14: Spin relaxation in GaAs (100) QWs with different well widths (all other parameters, including τl and τx , do not change). Points are from Ref. [113] where L0 = 7.5
nm. Dotted: 2L0 ; dash-dotted: 3L0 /2; solid: L0 ; dashed: L0 /2.
Figure 4.15: Spin relaxation in (110) GaAs (nimp = 4 × 1010 cm−2 ): dashed-dotted line.
Spin relaxation in (110) CdTe (nimp = 1 × 1011 cm−2 ): solid line. Points from undoped
(110) GaAs QW experiment[116] are included for comparison. For both systems, τp (T )
from [67] were used. EY is too weak over the temperature range depicted to be seen in the
GaAs system. However EY is the cause of the increase in spin relaxation rate for the CdTe
system.
75
4.4 Summary
The spin relaxation times in n-doped bulk and QW semiconductors are well described
by a theory invoking the exchange interaction between spin species. In undoped (110)
QWs, where DP is absent, we find that exciton spin relaxation is important and leads to
the observed surprising temperature dependence. We predict that a similar temperature
dependence (though with longer relaxation times) should be observed in n-doped (110) QWs
when excited at the exciton-bound-donor frequency. The DP mechanism is the dominant
spin relaxation mechanism at high temperatures except for (110)-QWs when it is completely
suppressed. We have suggested future experimental work to resolve what mechanisms relax
spin localized on donors in n-doped GaAs and CdTe QWs. The theory allows us to predict
experimental conditions that should optimize the measured spin relaxation times in GaAs
and CdTe QWs.
76
Chapter 5
Phenomenological approach to
spin relaxation in
semiconductors II; case studies in
bulk and quasi-2D wurtzite
crystals
5.1 Introduction
This chapter is similar to the previous one except that wurtzite (w) semiconductors are
studied instead of zinc-blende (zb). The enterprise of the previous chapter is first closely
followed for bulk n-ZnO in which the cross-over from localized to conduction spin relaxation
is observed to be much like in the systems of chapter 4. However the phenomenological
approach requires new additions due to the new system as shown in section 5.2.3 and
reported in [21]. The experimental and theoretical situation in w-QWs is not nearly as
developed as in zb-QWs; for this reason section 5.3 develops the DP mechanism in w-QWs
for the first time as recently reported in [22].
5.2 Bulk crystals
The actual material wurtzite refers to ZnO but several other semiconductor compounds form
similar crystal structures under ambient conditions; they include GaN, AlN, and InN among
others. The wurtzite structure is similar to zinc-blende in that it is non-centrosymmetric and
its bonds are tetrahedral; however its stacking sequence is different which leads to hexagonal
as opposed to face centered cubic symmetry (see Figures 4.1 and 5.1). The hexagonal nature
of the lattice leads to two lattice constants: c is the height of the hexagonal cylinder and a
is the basal length of an edge. The four tetrahedral bonds in wurtzite are not identical and
this leads to spontaneous electric polarization, in the absence of strain, which has recently
77
Figure 5.1: The wurtzite crystal structure.
generated interest in the nitrides [59].
For bulk wurtzite, the spin-orbit Hamiltonian due to bulk inversion asymmetry (BIA)
is [122, 123]
HD (k) =
~ D
~
ω1 (k) · σ + ω3D (k) · σ,
2
2
(5.1)
where

ky

2β1 

 −kx  ,
~
0


ky
2β3


(bkz2 − k||2 )  −kx  ,
ω3D (k) =
~
0
ω1D (k) =
(5.2)
(5.3)
where k|| = kx2 + ky2 . In an asymmetric quantum well (with structural inversion asymmetry
- SIA), spin-splitting will also occur due to the Rashba effect which will present another
linear-in-k term (ω1R ) above such that β1 → β1 + αR . If the QW is confined along the
z-direction [0001], after a spatial average along the direction of confinement
H = HR + HD = αR + βD − β3 k||2 ) (ky σx − kx σy )
is obtained where
π2
,
(5.5)
L2
is the Rashba coupling and is proportional to the
βD = β1 + bβ3 hkz2 i ≈ β1 + bβ3
where σ are the Pauli matrices; αR
(5.4)
78
external electric field in the z-direction, produced either by electrodes or by asymmetry in
the structure; β3 is the cubic-in-k coupling, while βD = β3 kz2 is a Dresselhaus-type term
that is controlled by confinement. kz2 is the expectation value of the operator kz in the
QW wave function. If L denotes the well width, then kz2 ∼ (π/L)2 for the lowest electric
subband and small structural asymmetry. αR is proportional to the electric field Ez and can
thus be tuned by applying a gate voltage or producing structures with varying asymmetry;
βD depends on L; β3 depends only on the material and cannot be turned off. When elastic
scattering is assumed, k||2 ∝ ε is a conserved quantity.
5.2.1 The Elliott-Yafet mechanism in bulk wurtzite crystals
In addition to creating the Rashba and Dresselhaus interactions, the spin-orbit interaction
prevents Sz from being a good quantum number since this operator will not commute with
the spin-orbit Hamiltonian. Wave functions are then impure spin mixtures which may lead
to EY spin relaxation when scatterers are present as discussed in chapter 3. While the DP
mechanism has proved to be dominant in zinc-blende semiconductors [55], the situation is
not clear in wurtzite since EY and DP have yet to be calculated. The goal of the next two
sections is to calculate the Elliott-Yafet and D’yakonov-Perel’ spin relaxation rates in bulk
wurtzite semiconductors.
Conduction band wave functions in bulk wurtzite semiconductors
The energies and wave functions of conduction and valence band electrons can be found
using the k · p approximation. The method is described for zinc-blende semiconductors in
many texts [66] and the review by Fabian et al. [39] is especially detailed in its treatment.
It has also been extended to the lower symmetry wurtzite crystal and those results [59] are
used in the following discussion. The full calculations are not repeated here but the key
differences between the zinc-blende and wurtzite solutions are noted.
In k · p theory, the Hamiltonian equation is
where H0 =
~2 k 2
~
H0 +
+ k · p + Hs.o
2m
m
p2
2m
un (r, k ↑)
un (r, k ↓)
!
= εn
un (r, k ↑)
un (r, k ↓)
!
,
(5.6)
+ V (r) is the kinetic energy and periodic crystal potential; Hs.o. is the
spin-orbit Hamiltonian in Eq. (1.2) and is written here as Hs.o. = Hx σx + Hy σy + Hz σz ;
un (r, k) are Bloch functions with band index n. Eight bands are included in this simple
version of the theory: three degenerate valence bands and one conduction band, all doubled
for spin. The eight basis functions are
uc1 = |iS ↑i, u1 = | −
X + iY
X − iY
√
↑i, u2 = | √
↑i, u3 = |Z ↑i,
2
2
79
(5.7)
uc2 = |iS ↓i, u4 = |
X − iY
X + iY
√
↓i, u5 = | − √
↓i, u6 = |Z ↓i,
2
2
(5.8)
where S, X, Y , and Z are angular quantities expressible in terms of spherical harmonics [66].
The spin quantization axis is along the c-axis, [0001]. The wave function of a conduction
electron is a linear combination of the eight basis functions. By solving Eq. (5.6), the
conduction band wave function is
|k ↑i = auc1 + bu1 + cu2 + du3 + f u5 + gu6 ;
(5.9)
the wave function with the opposite spin can be found by applying the time reversal operator
K̂ (see section 3.2). The following is obtained:
|k ↓i = auc2 + f u2 − g ∗ u3 − b∗ u4 − c∗ u5 + du6 ,
(5.10)
where the arrows in the bra-ket refer to the pseudospin of the wave function. Due to the
lower symmetry of the wurtzite crystal (compared to zinc-blende), the spin-orbit matrix
elements are not all equal: hX|Hz |Y i = −i∆2 6= hY |Hx |Zi = hZ|Hy |Xi = −i∆3 . For the
same reasons hZ|H0 |Zi = Ev 6= hX|H0 |Xi = hY |H0 |Y i = Ev + ∆1 where ∆1 is the crystal
field splitting energy which is zero in cubic crystals. Ev is the valence band edge when
there is no spin-orbit effect. Also the momentum matrix elements are hS|pz |Zi = mP1 /~ 6=
hS|pz |Y i = hS|pz |Xi = mP2 /~. The cubic case can be retained when ∆1 = 0, ∆2 = ∆3 ,
and P1 = P2 .
The lower symmetry in wurtzite forces the coefficients of Eqs (5.9, 5.10) to be more
complicated than zinc-blende’s; the wurtzite coefficients are
a=
Ea Eb − 2∆23
−k− P2 (Ea Eb − 2∆23 )
k + P2 E a
√
,b =
,c = √
D
2Eg D
2D
√
kz P1 Eb
kz P1 2∆3
k+ P2 ∆3
d=
, e = 0, f =
,g =
,
D
D
D
(5.11)
(5.12)
where
D2 (k) = (Ea Eb − 2∆23 )2 (1 +
2 P2
2 2
k⊥
2
2
2 k⊥ P2
)
+
(E
+
2∆
)
+ (Eb2 + 2∆23 )kz2 P12 ,
a
3
2Eg2
2
(5.13)
with Ea = Eg + ∆1 + ∆2 , Eb = Eg + 2∆2 , P1 ≈ Eg ~2 (1 − m/m∗z )/2m∗z , and P2 ≈ Eg ~2 (1 −
m/m∗⊥ )/2m∗⊥ . The effective mass is not strongly anisotropic so it is reasonable to make
the simplification P1 ∼ P2 = P which will be done from now on [59]. From the defined
wave functions it is determined that hk0 ↓ |k ↑i = cf 0 + gd0 − dg 0 − f c0 where the primed
coefficients refer to coefficients of the final wave vector k0 ; this shows that states with
antiparallel pseudospins are not necessarily orthogonal.
80
EY relaxation time
The calculation starts with Fermi’s Golden Rule:
Wk,σ→k0 ,σ0 =
2π
|hf |U (r)|ii|2 δ(εk0 − εk )
~
(5.14)
where Wk,σ→k0 σ0 is the scattering rate from an initial state, |ii, with wave vector k and
spin σ to the final state, |f i, k0 and spin σ 0 . Using the approximations in section 3.2.1, the
spin-flip rate is
2π
|hexp −ik0 · r|U (r)| exp ik · ri|2 |hk0 ↓ |k ↑i|2 δ(εk0 − εk )
~
4π 2
U 0 |hk0 ↓ |k ↑i|2 δ(εk0 − εk ).
~V 2 |k −k|
Wspin−f lip = 2
=
(5.15)
(5.16)
The spin-orbit interaction is assumed to be small such that the wave function is nearly
one type of spin. Hence, the rate at which an electron scatters and does not flip spin is
approximated by the total scattering rate:
Wtotal ≈ Wno−spin−f lip =
4π 2
U 0 |hk0 ↑ |k ↑i|2 δ(εk0 − εk ).
~V 2 |k −k|
(5.17)
From the wave functions of Eqs. (5.9, 5.10), the inner products can be expressed as
hk0 ↑ |k ↑i = B(k, k0 )
(5.18)
hk0 ↓ |k ↑i = A(k, k0 ) sin θeiφ ,
(5.19)
and
where we have chosen k to be along the polar axis so θ is the angle between k0 and k and
we have assumed that the wave vector is conserved throughout such that k 0 = k. φ is the
azimuthal angle of k0 . Most generally,
A(k, k0 ) = (2Eg + ∆1 + 3∆2 )
P1 P2 k 2 ∆3
D(k)D(k0 )
(5.20)
and
B(k, k0 ) =
0 k + k k 0 )P 2 ) + 2(2k k 0 P 2 + k 0 k P 2 )∆2
2Eg4 + Eg2 (2kz kz0 P12 + (k−
+
− + 2
z z 1
− + 2
3
0
2D(k)D(k )
(5.21)
The spin relaxation time will be Wspin−f lip summed over all possible final k-states such that
X
1
=
Wspin−f lip −→
τs
0
k
81
Z
d3 k 0 V
Wspin−f lip
(2π)3
(5.22)
Making the necessary substitutions for W ,
Z
sin θdθφk 02 dk 0 V 4π 2
1
=
U 0 |A(k, k0 )|2 sin2 θδ(εk0 − εk )
τs
(2π)3
~V 2 |k −k|
By using δ(εk0 − εk ) = m∗ /(k 0 ~2 )δ(k 0 − k),
Z
1
sin θdθdφk 02 dk 0 V 4π 2
=
U 0 |A(k, k0 )|2 sin2 θδ(k 0 − k)m∗ /(k 0 ~2 ).
τs
(2π)3
~V 2 |k −k|
(5.23)
(5.24)
Now we do the trivial integrals over k 0 and φ to obtain
Z
sin θdθk 8π 2 2
1
(5.25)
=
U 0 |A(k, k0 )|2 sin2 θm∗ /~2 .
τs
(2π)3 ~V |k −k|
√
Realizing that the three dimensional density of states is g(ε) = 2m∗ εm/π 2 ~3 and using
√
k = 2m∗ ε/~ we ascertain
Z
sin θdθ 8π 4 2
1
=
U 0 |A(k, k0 )|2 sin2 θkm∗ /π 2 ~2
(5.26)
τs
(2π)3 ~V |k −k|
Z
sin θdθ 8π 4 2
U 0 |A(k, k0 )|2 sin2 θg(ε)
(5.27)
=
(2π)3 ~V |k −k|
and by simplifying we get
1
π
g(ε)
=
τs
~V
Z
sin θdθU 2 (k, θ)|A(k, k0 )|2 sin2 θ
(5.28)
The momentum relaxation time will be similar except a transport factor is added in the
standard way so it can be compared to mobility measurements [119],
Z 3 0
Z
1
d kV
π
=
Wtotal (1 − cos θ) =
g(E) sin θdθU 2 (k, θ)|B(k, k0 )|2 (1 − cos θ) (5.29)
τ̃p
(2π)3
~V
We divide the spin relaxation rate by the momentum relaxation rate to find
R
sin θdθU 2 (k, θ)|A(k, θ)|2 (1 − cos2 θ)
τ̃p
= R
τs
sin θdθU 2 (k, θ)|B(k, θ)|2 (1 − cos θ)
(5.30)
We now make a first round of approximations: the spin-orbit and crystal field energy
splittings are much less than the bandgap so we neglect those terms. This is obviously
true for the wurtzite materials with parameters listed in Appendix A. Also, as suggested
by Ridley [59], we can take P1 = P2 = P and assume isotropic effective mass. The matrix
element, P , can be expressed as [59]
P2 =
Eg ~2
m∗
(1
−
).
2m∗
m
82
(5.31)
These simplifications yield
D2 (k) = Eg4 + Eg2 P 2 k 2 = Eg4 + (1 −
m∗ 3
)E ε,
m g
(5.32)
∗
2(1 − mm )Eg2 ε∆3
2Eg P 2 k 2 ∆3
A(k, k ) ≈ 4
=
,
∗
Eg + Eg2 P 2 k 2
Eg4 + (1 − mm )Eg3 ε
0
and
(5.33)
∗
2Eg4 + 2Eg2 P 2 k 2 cos θ
Eg4 + (1 − mm )Eg3 ε cos θ
B(k, k ) ≈
=
,
∗
2(Eg4 + Eg2 P 2 k 2 )
Eg4 + (1 − mm )Eg3 ε
0
(5.34)
where the fact that kx = ky = 0 and k = k 0 have been used to greatly simplify matters. All
this allows Eq. (5.30) to be simplified to
τ̃p
=
τs
m∗ 2
2(1 −
)E ε∆3
m g
!2
sin θdθU 2 (k, θ)(1 − cos2 θ)
. (5.35)
∗
sin θdθU 2 (k, θ)(Eg4 + (1 − mm )Eg3 ε cos θ)2 (1 − cos θ)
R
R
Rearrangement of the previous result leads to
τ̃p
m∗ 2 ε
= 4(1 −
)
τs
m
Eg
!2
∆3
Eg
!2
sin θdθU 2 (k, θ)(1 − cos2 θ)
.
∗
sin θdθU 2 (k, θ)(1 + (1 − mm ) Eεg cos θ)2 (1 − cos θ)
(5.36)
R
R
The ratio of integrals is a factor that is of order unity; call it Q. Since we are mainly
interested in an order of magnitude estimate, we are not concerned with correctly evaluating
it as it depends on the scattering potential. Since ∆3 = ∆0 /3 [124, 125, 59], we obtain
1
m∗ 2 ε
4
)
≈ (1 −
τs
9
m
Eg
!2
∆0
Eg
!2
Q
,
τ̃p
(5.37)
where ∆0 is valence band spin-splitting energy (between heavy/light hole band and split-off
band; see Fig. 4.2). To obtain an order of magnitude estimate of the EY relaxation rate at
high temperatures, kB T is substituted for ε:
4
m∗ 2 kB T
1
≈ (1 −
)
τs
9
m
Eg
!2
∆0
Eg
!2
Q
.
τp
(5.38)
Eq. (5.38) is nearly identical to the expression found in zinc-blende semiconductors [56].
Using the numbers from Appendix A, τs ∼ 3 ms in GaN at 300 K if the scattering time
is taken to be 1 ps. This time is much longer than what is found in GaAs for at room
temperature: τs ∼ 60 ns. The massive discrepancy is due to the much smaller spin-orbit
splitting and the larger band gap in GaN. Appendix A shows that this is true for other
wurtzite semiconductors as well.
83
5.2.2 The D’yakonov Perel’ mechanism in bulk wurtzite crystals
The conduction band states undergo ordinary impurity and phonon scattering. Each scattering event gives a change in the wavevector k, which in turn changes the effective magnetic
field on the spin that comes from spin-orbit coupling. This fluctuating field relaxes the spin.
The effective field strength is proportional to the conduction band spin splitting. Bulk zincblende crystals have conduction band splittings cubic-in-k due to bulk inversion asymmetry
(Dresselhaus effect) [52]. In addition to cubic terms, bulk wurtzite conduction bands also
possess spin splittings proportional to linear terms in k due to the hexagonal c axis which
gives bulk wurtzite a reflection asymmetry similar to the Rashba effect [112, 126, 111, 127].
Let us calculate the explicit temperature dependence of the DP mechanism in bulk
wurtzite crystals. First recall section 3.2.2 that in bulk,
1
τs,ij
l
X X
1
n
(−1)
τ
[H
,
[H
,
σ
]]σ
.
=
Tr
j
i
l
l,−n
l,n
8π~2
l
(5.39)
n=−l
In bulk wurtzite, unlike zinc-blende, Hs.o. ∼ k and k 3 so both l = 1, 3 will yield a non-zero
result. τ3 can easily be related to τ1 by using Eqs. (3.32) as was done for zinc-blende in the
previous chapter: τ3 = γ3 τ1 . Calculation of Eq. (5.39) yields
1
τs,xx
=
1
τs,yy
=
2 ∗2
1
8τ1 m∗ 2 4β1 β3 m∗
τ3
2 4β3 m
=
(β
ε+
(b−4)ε
+
(7(4−b)2 +8(1+b)2 )ε3 ),
1
4
2
4
2τs,zz
3~
5~
175~
τ1
(5.40)
where the spin relaxation tensor contains a slight anisotropy between the in-plane and outof-plane components. The constant b has been determined [128, 129] to be near four which
simplifies the rates to
1
τs,xx
=
1
τs,yy
=
1
8τ1 m∗ 2
256γ3 τ1 β32 m∗3 3
≈
β
ε
+
ε ,
1
2τs,zz
3~4
21~8
(5.41)
which effectively eliminates the interference term (quadratic in ε) and one of the cubic terms
as demonstrated implicitly in [21] and confirmed explicitly in [129]. Due to the identical
symmetry of the Rashba and linear Dresselhaus terms, the presence of an electric field along
ẑ results in the substitution: β1 → β1 + αR .
The temperature dependence can be found by using Eq. (C.12),
Id+1 (βµ) In+ν+d (βµ)
,
Iˆ(d) [τ1 εn ] = τtr β −n
Id (βµ) Id+ν+1 (βµ)
(5.42)
to obtain
I3/2 (βµ) 256m∗3 τtr γ3 β32
I3/2 (βµ)I7/2+ν (βµ)
1
8m∗ τtr β12
=
k
T
+
(kB T )3
,
B
4
8
τs
3~
I1/2 (βµ)
21~
I3/2+ν (βµ)I1/2 (βµ)
84
(5.43)
where d = 1/2 has been used since the density of states goes as ε1/2 in three dimensions. The
temperature dependent momentum relaxation time, τtr , can be determined from electron
mobility (µe ) measurements from µe = eτtr /m∗ where e is the charge of an electron.
With the EY and DP mechanisms in the arsenal now, they are compared to experiments
to determine which is operable in n-doped ZnO in the following section.
5.2.3 ZnO
Zinc oxide has been the subject of considerable experimental and theoretical investigation
for many years [130]. Its band gap is in the near ultraviolet, making it useful as a transparent
conductor and as sunscreen. Its piezoelectricity opens up transduction applications. The
activity has intensified more recently because of the possibility that ZnO might be useful
for spintronics or spin-based quantum computation. It has been predicted to be a roomtemperature ferromagnet when doped with Mn [131]. Furthermore, its spin-orbit coupling
is generally thought to be very weak compared with GaAs.
The usual measure of the
strength of spin-orbit coupling in semiconductors is the energy splitting at the top of the
valence band. It is said that the spin-orbit coupling is negligible in ZnO because the valenceband splitting is −3.5 meV [125], as opposed to 340 meV for GaAs.
Smaller spin-orbit
coupling should lead to long spin relaxation times. Long relaxation times are required if
spin information is to be transported over appreciable distances.
The spin relaxation time τs has been measured by Ghosh et al. [103] to be about 20
ns from 0 to 20 K in optical orientation experiments. τs is sometimes called T2∗ even in
the absence of an external field. Since the data from [103] used here were taken at zero
field, the relaxation time is taken to be τs to avoid confusion with experiments conducted
at finite field.
The data show two surprising features. First, the relaxation times are
actually somewhat shorter than the longest relaxation times in GaAs, which are about 100
ns [4]. One might expect the opposite given the relative strength of spin-orbit coupling
in the two materials. Second, τs shows a non-monotonic temperature dependence, first
increasing slightly and then rapidly decreasing with increasing temperature - but increasing
temperature usually promotes spin relaxation.
We show that the theory previously developed for τs in GaAs [5] can account for these
observations. The theory must be modified to take account of the different impurity levels
and binding energies of ZnO. This is important, because, in spite of intensive investigation,
the nature of the impurities that govern the electrical properties of ZnO remains controversial, and our analysis sheds some light on this issue. Even more interestingly, it turns
out that the wurtzite crystal structure has very important consequences for the D’yakonovPerel (DP) [63, 110] scattering that dominates the relaxation at higher temperatures. Thus
the crystal structure must be taken into account fully. The final message will be that the
“weak” spin-orbit coupling of ZnO is not negligible for spin relaxation, and it does not lead
85
to long relaxation times.
In ZnO produced by the hydrothermal method, it is generally thought that there are two
sets of impurity states, one shallow and quasi-hydrogenic, one deep and very well localized
[132, 133]. Their precise physical nature is not known. In the case of the deep impurity, it
is believed that a lattice defect accompanies the chemical impurity. The binding energies
are in the range of a few 10s of meV for the shallow impurity and a few 100s of meV for
the deep impurity. We shall demonstrate below that the optical orientation data can put
bounds on these numbers.
ZnO crystallizes in the wurtzite structure rather than the zinc-blende structure familiar
from the III-V compounds. This has very important implications for the conduction band
states. The spin-orbit interaction lifts the spin degeneracy in the conduction band. In zincblende structures crystal symmetry implies that the splitting is cubic in the magnitude of
the wave vector k, but in the wurtzite structure the splitting is linear [126]. However, the
spin relaxation time of the low-lying conduction band states depends mainly on the spin
splitting near the conduction band minimum, and this is larger in ZnO than in GaAs for
small enough k.
In optical orientation experiments, electrons are excited from the valence band to the
conduction band by circularly polarized light tuned close to the bandgap energy (pump
step). The population of conduction electrons so created is spin-polarized [70]. Energy
relaxation then occurs on a short time scale (≤ 1 ns), but most of this relaxation is from
spin-conserving processes, so there is a longer time scale (or time scales) on which the spin
of the system relaxes. This longer time scale is measured using Faraday or Kerr rotation
(probe step).
The important physical point is that the fast energy relaxation leads to a thermal charge
distribution for the electrons by the time 1 ns has elapsed, but the spin distribution relaxes
on longer time scales. The thermal charge distribution means that the localized donor
impurity states are mostly full at the relatively low temperatures of the experiment. The
spins of the localized electrons must be included along with the conduction electron spins.
The spins of localized and extended states can be interchanged by the exchange coupling,
a process we call cross-relaxation. This is often a rather fast process and is particularly
important when the relaxation times of the localized and extended states are very different
in magnitude. In GaAs this process is important in all the regimes of temperature, applied
field, and impurity density that have been studied, and it is important in ZnO as well.
In the following section a set of modified Bloch equations is derived to describe the
aforementioned spin dynamics.
86
Modified Bloch equations
We consider a conduction electron in the semi-classical approximation. It moves as a wave
packet with a well-defined momentum and scatters from impurities and phonons at time
intervals of average length τp , where τp is the momentum relaxation time. Its spin operator
is sc . The spin-dependent part of its Hamiltonian in the absence of an applied external
magnetic field is:
H c = H1c + H2c = −
1 X
µB
J (r − Ri ) si · sc − g
b (t) · sc .
2}
~
(5.44)
i
The first term, H1c , is the exchange interaction with impurity spins si located at an positions
Ri . It is the same interaction that is responsible for the Kondo effect, but the temperatures
here are all much greater than the Kondo temperature. The range of the function J (r − Ri )
is roughly aB , where aB is the effective Bohr radius.
The second term, H2c , represents
other spin relaxation mechanisms that we model as a small random classical field b(t) with
a correlation time much shorter than τs . An analogous Hamiltonian H l can be written for
a localized electron.
First, we concentrate on the spin dynamics resulting from the spin-spin term and ignore
1/3
the second term. In the dilute limit (aB nimp 1), a conduction electron encounters
impurities with randomly aligned spins if no short-range order is present in the impurity
system. An effective field from the impurity spin affects the conduction electron when it
is within
∼ aB of the impurity. When |r − Ri | > aB , the conduction electron proceeds
unhindered by the effective field. This effective field is a result of the exchange potential.
An itinerant electron will spend an average time of aB /v within the range of the effective
field where v is the velocity of the electron. Thus the time between encounters is 1/nl a2B v
[11].
In a semi-classical picture the spin of the itinerant electron undergoes precession of
magnitude ∆φ = JaB /2v through a random angle during each encounter with an impurity.
The spin of the impurity electron also precesses but with angle −∆φ. Since the sum of
spins, sc + sl , commutes with H1c + H1l , the total spin in the system must be conserved.
However the spin in each subsystem may shift between one another; this is cross-relaxation.
It turns out for the parameters of the system under consideration that ∆φ ∼ 1, and we
then find that
τccr ∼
1
nl a2B v
(5.45)
which implies that the spin is essentially randomized after one impurity encounter.
If we consider an ensemble of conduction electrons with a net magnetization mc , this
magnetization is exchanged at a rate of 1/τccr . As previously mentioned, any magnetization
lost from the conduction electrons must be gained by the localized electrons and vice-versa.
87
For clarity we write 1/τccr = nl /γ cr and 1/τlcr = nc /γ cr where γ cr = 1/a2B v. Mahan and
Woodworth find a similar expression for low velocity electrons using a quantum mechanical
argument [46].
We now examine the second term of the Hamiltonian
1
H2c (t) = − gµB bx (t)σx + by (t) σy + bz (t) σz .
2
(5.46)
This Hamiltonian relaxes the conduction electron spin. To extract a relaxation rate
from this Hamiltonian, we use the equation of motion
i
dρ(t)
= [ρ(t), H2c (t)]
dt
~
(5.47)
where ρ(t) is the 2 × 2 spin density matrix for an electron of a given momentum. We assume
that the total density matrix for the conduction electron factorizes; we neglect off-diagonal
terms that come from correlations. By iteration, we can write this equation as
dρ (t)
=
dt
Z t
i
1
c 0
[ρ (0) , H2 t ] − 2
ρ t0 , H2c t0 , H2c (t) dt0
~
~ 0
(5.48)
where the angular brackets indicate averaging over all orientations of b(t). To simplify
notation, from now on angular brackets will be suppressed on the density matrix. Since
hbi (t)i = 0, the first term is zero. We assume that different directions of bi are uncorrelated
and (since the external field is zero) different direction are equivalent.
hbi (t)bj
(t0 )i
=
hb(t)b(t0 )iδi,j .
Then we have
Therefore, Eq. (5.48) reduces to
g 2 µ2B
dρ (t)
=−
dt
2~2
Z tX
0
[ρ(t0 ), σi ]σi hb(t)b(t0 )idt0 .
(5.49)
i
The correlation function is assumed to be stationary in time so hb(t)b(t0 )i = g(t0 − t) =
g(τ ).[28] If the correlation time of the b-fluctuations, τe , is short, ρ will not change on that
timescale and g(τ ) will be nearly a δ- function. Eq. (5.49) can then be written as
2g 2 µ2B 1 X
dρ (t)
=−
[ρ(t), σi ]σi
dt
~2 4
i
Z
∞
b(t)b(t0 ) dt0 .
(5.50)
0
The integral is approximated by hb2 iτe . Define the relaxation time scale τc by
gµ 2
1
B
=2
hb2 iτe
τc
~
(5.51)
dρ (t)
1 X
=−
[ρ(t), σi ]σi .
dt
4τc
(5.52)
giving
i
88
The density matrix can be expanded in Pauli spin matrices
1
1X
ρ (t) = I +
mi (t) σi .
2
2
(5.53)
i
where I is the 2 × 2 identity matrix and mi = T r(σi ρ) is the expected value of the magnetization. Inserting Eq. (5.53) in Eq. (5.52) and matching coefficients of Pauli matrices gives a
set of equations for the dynamics of m. For instance for conduction electron magnetization
mc in the x-direction, dmc /dt = T r(σx dρ/dt) = −mc /τc . As with H1c , similar expressions
for the localized magnetization ml can be found: dml /dt = T r(σx dρ/dt) = −ml /τl .
By combining the effects of H1 = H1c + H1l and H2 = H2c + H2l , the modified Bloch
equations for the magnetizations can be expressed as
1
dmc
nl nc
=−
+ cr mc + cr ml
dt
τc γ
γ
1
dml
nl
nc = cr mc −
+
ml .
dt
γ
τl γ cr
(5.54)
for two spin systems - itinerant and localized spins. τc and τl in Eq. (5.54) are now described
in terms of well known relaxation mechanisms which will be discussed in the next section.
This model was successfully applied to GaAs [5]. For ZnO, these Bloch equations are easily
extended to account for the multiple-type impurities present.
Method
We now seek to write equations like those of Eq. (5.54) with regard given to the two
types of impurities in ZnO - shallow and deep. As mentioned above, we find that the
cross-relaxation is important to understand the data. These rates come from the Kondolike Jsl · sc interaction between an impurity spin sl and a conduction band spin sc .
An
expression for J in terms of tight-binding parameters can be derived using the SchriefferWolf transformation [134]. One expects that the cross-relaxation between conduction and
shallow donor electrons to be much more rapid than the cross-relaxation between conduction
and deep donor electrons because of the greater binding energy of the deep impurity and
its larger on-site Coulomb energy. This is confirmed by the fit to the data. In fact we find
that terms involving cross-relaxation between the deep donors and either the conduction
band electrons or the shallow donor electrons can be neglected. With these simplifications,
89
for ZnO Eq. (5.54) extends to
1
nls dmc
nc
+ cr mc + cr mls
=−
dt
τc γc,s
γc,s
1
nls
nc dmls
= cr mc −
+ cr mls
dt
γc,s
τls γc,s
dmld
1
= − mld .
dt
τld
(5.55)
In this equation, mc , mls , and mld stand for the magnetizations of the conduction electrons,
the electrons on shallow impurities, and the electrons on deep impurities, respectively. The
n’s denote the corresponding volume densities.
Each of the populations has a relaxation
time τc , τls , and τld . From Eq. (5.55), we find the magnetization as a function of time.
Standard methods are used to solve these differential equations. The solutions yield a
time dependence of the total magnetization, m(t) = mc (t) + mls (t) + mld (t), to be a sum
of three exponentials, exp(−Γ+ t), exp(−Γ− t), and exp(−Γd t) where
!
1 1
1
nc + nls
1
Γ± =
+
+
± S , Γd =
cr
2 τc τls
γc,s
τld
(5.56)
with S given by
v
u
u
S=t
1
1
nc − nls
− +
cr
τls τc
γc,s
!2
+
4nc nls
.
cr 2
γc,s
(5.57)
No net moment can exist on the deep donor sites since no moment is excited into the
deep states on account of them being significantly below the conduction band, and no net
moment cross relaxes into these states. Therefore Γd can be ruled out as being the observed
cr 1/τ , 1/τ , the rate Γ simplifies to
relaxation rate. In the regime that (nls + nc )/γc,s
c
+
ls
cr and is very rapid and the rate Γ is slower,
(nc + nls )/γc,s
−
Γ− =
nc
1
nls
1
+
.
nc + nls τc nc + nls τls
(5.58)
We fit the data with this equation and associate it with τs . We see that the relaxation rate
depends on two factors: the thermodynamic occupations of the shallow donors (the deep
donors are always nearly full in the temperature range studied here) and the form of the
relaxation rates for the conduction and localized shallow states.
The densities can be computed using standard formulas from equilibrium statistical
mechanics, since we deal only with time scales long compared to the fast energy relaxation
scale.
As a function of temperature T , the ratio nc /nls naturally increases rapidly as
T → |εls |/kB , where εls is the binding energy of the shallow impurity. |εld | is so large that
these states are always occupied at the experimental temperatures, which range from 5 K
to 80 K.
90
τc is fairly complicated to calculate because there are several mechanisms that can
relax the conduction electron spins. The simplest such mechanism is the Elliot-Yafet (EY)
process [48] that arises from spin mixing in the wavefunctions. When a conduction electron
is scattered by a spin-independent potential from state k to state k0 , the initial and final
states are not eigenstates of the spin projection operator Sz so the process relaxes the spin.
The rate of relaxation due to the EY process is well known to be of the form: 1/τEY =
αEY T 2 /τp (T ) where αEY is a material dependent parameter and τp is the momentum
relaxation time [58]. We estimate αEY (th) = 4.6 × 10−15 K−2 .
The Bir-Aronov-Pikus
(BAP) mechanism [53] arises from the scattering of electron and holes. This relaxation
mechanism is commonly considered to be negligible in n-type materials like those under
consideration here since the number of holes is small [89]. The D’yakonov-Perel’ (DP)
mechanism [110] arises from the ordinary scattering of conduction-band states. Since this
has not previously been calculated in a wurtzite structure, we devote the next section to it.
This calculation yields an expression for τc as a function of temperature.
τls and τld are due to non-spin-conserving anisotropic exchange (Dzyaloshinski-Moriya)
interactions [108, 109]. The anisotropic exchange term is important. It arises from spin-orbit
coupling and produces a term proportional to d · s1 × s2 where d is related to the interspin
separation and the exchange integral between the wave function on sites 1 and 2. However,
it is not possible to calculate it in detail when the nature of the impurities is not well known.
We estimate the rate as 1/τDM = αDM (nimp,s + nimp,d ) where nimp,s and nimp,d are the
total impurity concentrations of the shallow and deep impurity respectively and αDM has
a weak temperature dependence that we neglect. The main contribution comes from the
the overlap of the shallow impurity wavefunctions, which we take to be hydrogenic, with
the deep impurity wavefunctions, which we take to be well-localized on an atomic scale.
The details of how to estimate the resulting relaxation may be found in [5, 82, 83]. The
numerical value we find from theory is αDM (th) = 1.12 × 10−20 cm3 ns−1 . When nuclei
possess nonzero magnetic moments, the hyperfine interaction between electron and nuclear
spin is a source of spin relaxation for localized electrons [41]. However, zero nuclear spin
isotopes of Zn and O are 96% and 99.5% naturally abundant respectively. Therefore we
rule out the hyperfine interaction from being an observed relaxation mechanism in [103].
DP mechanism in ZnO
There had been no calculations of the DP mechanism in wurtzite crystals until the publication of [21]. The general DP equation of Eq. (5.43) can be used to determine the DP
relaxation rate in bulk ZnO:
I3/2 (βµ) 256m∗3 τtr γ3 β32
1
8m∗ τtr β12
3 I3/2 (βµ)I7/2+ν (βµ)
=
k
T
+
(k
T
)
,
B
B
τs
3~4
I1/2 (βµ)
21~8
I3/2+ν (βµ)I1/2 (βµ)
91
(5.59)
the factors of I reduce to
I3/2 (βµ) T TF 3
−→ ,
I1/2 (βµ)
2
5
I3/2 (βµ)I7/2+ν (βµ) T TF 3 7
−→
+ν
+ν
I3/2+ν (βµ)I1/2 (βµ)
2 2
2
(5.60)
in the non-degenerate limit. In anticipation of curve fitting, we write
1
(1)
τDP
(1)
where αDP (th) =
4m∗ β12
kB
~4
(3)
= αDP τtr T + αDP τtr T 3 ,
(3)
and αDP (th) =
3 m∗3 β 2
40QkB
3
7~8
(5.61)
16
35 γ3 (7/2 + ν)(5/2 + ν)
1.1 × 10−4 eV-nm which
with Q =
being of order unity. β1 has been calculated in ZnO [126] to be
(1)
gives a theoretical value of αDP (th) = 34.6 K−1 ns−2 . β3 has been calculated in ZnO [126]
(3)
to be 3.3 × 10−4 eV-nm3 which yields αDP (th) = 9.3 × 10−4 K−3 ns−2 .
The sample from which the momentum relaxation times τtr (T ) were extracted [135] was
hydrothermally grown by the same company as the Ghosh et al. sample in [103].
Results and discussion
In Figure 5.2 we show that temperature dependence of τs as measured in a bulk ZnO sample
and our fit (using Eq. (2.45)) to the data. It is seen immediately that the temperature
dependence is not monotonic and that this is well-reproduced by the theory. The reason is
simple. At low temperatures T |εls |/kB nearly all the electrons are in localized states.
These states relax by the temperature-independent DM mechanism: 1/τls = 1/τDM . This
mechanism alone determines the T = 0 values. When T approaches |εls |/kB , the deep
impurities are all occupied but the rest of the population is shared by shallow localized and
conduction band states. Initially, the conduction band electrons have a longer spin lifetime
at low temperatures as seen from Eq. (5.61) so the DP mechanism that relaxes them is
not very effective. However, the DP mechanism increases rapidly as T increases and the τs
curve turns around. At T |εls |/kB , the shallow impurity level is empty and the relaxation
(1)
is dominated by the DP mechanism in the conduction band: 1/τc = 1/τDP (T ).
At this point it is necessary to point out why only the linear-in-T DP mechanism is
needed to explain the observed conduction spin relaxation. The other two viable candidates
(cubic DP and EY) for relaxation are much too weak to explain the observed relaxation
(1)
(3)
times in ZnO. We use the calculated values for αDP (th) and αDP (th) in the previous section
to obtain the relative relaxation efficiencies between the linear and cubic DP mechanism
terms:
(1)
1/τDP
(3)
1/τDP
(1)
=
αDP (th)
(3)
αDP (th)T 2
=
3.72 × 104 K2
T2
(5.62)
which demonstrates that the efficiency of the cubic-in-T term does not become comparable
to the linear-in-T term at temperatures below 200 K which is far above the temperature
92
range investigated here. For this reason we can ignore the cubic-in-T DP mechanism term
in our fit though it will surely be important at higher temperatures than experimentally
probed. The crystal structure of ZnO therefore makes its spin relaxation qualitatively
different from spin relaxation in bulk n-GaAs. We also compare the efficiencies of the DP
and EY mechanisms:
(1)
(1)
α (th)τp2 (T )
7.5 × 1015 τp2 (T ) K ns−2
1/τDP
= DP
=
.
1/τEY
αEY (th)T
T
(5.63)
Even if the momentum relaxation time taken to be unrealistically low, say 1 fs, the DP
mechanism is still nearly two orders of magnitude more efficient at relaxing spins than the
EY mechanism in the temperature range studied here. Due to the drastic qualitative and
quantitative differences between relaxation mechanisms, we have unequivocally determined
the relevant conduction electron spin relaxation mechanism in ZnO.
-1
1/τs (ns )
0.5
0.1
0.01
10
T (K)
50
100
Figure 5.2: Plot of 1/τs vs. temperature. Points are experiment of [103]. Dashed curve:
[nls /(nc + nls )](1/τDM ). Dotted curve: [nc /(nc + nls )](1/τDP ). Solid curve: total 1/τs .
nimp,s = 6.0 × 1014 cm−3 , nimp,d = 5.0 × 1017 cm−3 , εls = −23 meV, and εld = −360 meV.
The fit of theory to the experimental data is clearly very good.
We found that no
reasonable fit was possible using only a single impurity level, though this worked very well
for GaAs [5], so we used two levels. A good fit by this method was possible by adjusting the
(1)
coefficients αDP (exp) and αDM (exp), and the binding energies εls , εld and concentrations
93
nimp,s , nimp,d of the two donors, subject to the constraint that the room temperature carrier
density should equal the measured [103] value of 1.26 × 1015 cm−3 . Qualitatively, one finds
that nimp,d nimp,s and |εld | |εls | to get the right order of magnitude of the relaxation
at low T. Physically, the deep impurity spins are important because they relax the shallow
impurity spins by the DM mechanism, and the strength of the low T relaxation implies that
the deep impurities must be quite numerous. Quantitatively, a least squares fit to the data
(1)
of [103] yields αDP (exp) = 134.5 K−1 ns−2 , αDM (exp)nimp,d = 0.06 ns−1 , |εld | = 360 meV,
|εls | = 23 meV, and nimp,s = 6.0 × 1014 cm−3 .
(1)
(1)
αDP (exp) is about four times larger than the theoretical value of αDP (th) given above,
possibly due to strain effects. We also note that the values of τp that we used were taken
from a different sample.
If we take nimp,d to be near the highest values measured for the deep donor (see below)
then αDM (exp) = 12 × 10−20 cm3 ns−1 is about one order of magnitude larger than the
theoretical estimate αDM (th) given above. In view of the very poor understanding of the
impurity wavefunctions, and the exponential dependence of αDM on the overlaps, this is
perhaps not too disturbing.
The presence of a shallow donor and a very deep donor has been seen in hydrothermally
grown ZnO samples of the type investigated here [133, 136]. Donor concentrations up to
nearly 5.0 × 1017 cm−3 (nimp,d ) have been measured for donors 330 − 360 meV (|εld |) deep
[137, 133, 136]. Donors as shallow as 13 − 51 meV (|εls |) have been measured [132] at lower
concentrations ∼ 5.0 × 1014 cm−3 (nimp,s ). Comparison with our values indicates that the
parameters extracted from the fit are very reasonable for this material.
From this analysis, we predict that in ZnO samples with fewer deep impurities, the
relaxation time at low temperatures can be increased. As the impurities of ZnO vary
greatly between different growth techniques [138], this prediction could be tested by further
optical orientation experiments on different samples.
We have found that τs in bulk ZnO can be understood by invoking previously known
spin relaxation mechanisms.
The dominant mechanisms in the material turn out to be
the DP (scattering) relaxation of the conduction electron spins for T > 50 K and the DM
(anisotropic exchange) mechanism for the localized spins for T < 50 K. In addition, it is very
important to include the cross-relaxation between localized and conduction states previously
proposed for GaAs.
These physical ingredients explain quantitatively the relatively fast
relaxation at low temperatures as being due mainly to the DM mechanism which in turn
depends on having both deep and shallow impurity states.
At high temperatures, the
conduction states are dominant, and the DP mechanism gives an excellent fit to the data.
The combination explains the very surprising non-monotonic temperature dependence of
τs .
Finally, there are two aspects of the data in [103] that we have not addressed here: the
94
applied magnetic field dependences on the spin relaxation and the spin relaxation observed
in ZnO epilayers. We plan on addressing the former issue in a future publication. As for
the latter issue, the epilayers are doped three to four orders of magnitude higher than in the
bulk case. At such high dopings, spin glass effects become important and localized donor
states coalesce to produce donor bands; we do not expect our theory to be applicable in
such a regime.
The theory has now been sufficiently developed that optical orientation experiments
can actually serve as a characterization tool for doped semiconductors, giving information
about the binding energies and concentrations of the electrically active impurities in n-type
materials.
5.3 Quasi-2D nanostructures
Much of semiconductor research in recent years has focused on the electron spin degree of
freedom [41]. Electron spins in quantum dots can serve as a qubit - the electron spin couples
relatively weakly to the environment and provides an ideal two-level system.
For many
quantum even classical spintronics applications, however, one requires mobile electrons.
Even as a carrier of classical information the spin of mobile electrons offers advantages:
translation and rotation are in principle dissipationless, offering great potential advantages
over charge motion.
For all types of applications, long spin coherence times and ease of
spin manipulation at room temperature are of crucial importance.
Several types of devices based on mobile spins have been proposed: ballistic [3] and
non-ballistic [139, 140] spin field effect transistors, and double-barrier structures [2].
A
central insight was that tuning of the spin-orbit (SO) parameters is possible by applying a
gate voltage to a quantum well (QW) to vary the Rashba coupling. In addition, systematic
variation of the well width has made it possible to independently tune other couplings and
observe momentum-dependent relaxation times in (001)-GaAs wells [95]. Possible realization of the Datta-Das device [3] has recently been achieved in an InAs heterostructure [17].
Generally speaking, experimental studies relevant to the realization of these devices have
been carried out at rather low temperatures T or short spin lifetimes τs . The spin effects
in InAs disappear at about T = 40 K and the disappearance was attributed to additional
scattering [17]. In the GaAs work, signs of enhanced lifetime (due to the persistent spin helix [141]) decreased rapidly with temperature and at T = 300 K, τs was only slightly above
100 ps; the loss of coherence is due to cubic-in-k terms in the spin-orbit Hamiltonian that
relax the spin by the D’yakonov-Perel’ (DP) mechanism [62, 63]. Long spin lifetimes are
fundamentally limited by the strength of the SO interaction in direct band gap zinc-blende
semiconductors. Wurtzite QWs offer great advantages over the aforementioned zinc-blende
structures in that long coherence times can be maintained at room temperature due to their
95
Material
w-(0001)
zb-(001)
zb-(111)
zb-(110)
x
[100]
[110]
[112]
[110]
y
[010]
[110]
[110]
[010]
z
[001]
[001]
[111]
[110]
βD
β1 + bβ3 hkz2 i
β3 hkz2 i
cβ3 hkz2 i
β3 hkz2 i/2
Table 5.1: Table of several semiconductor nanostructures with different growth orientations
and their respective crystal axes. Also included is the parameter βD which gives the strength
of the linear Dresselhaus terms in the Dresselhaus spin-orbit interaction (see Table 5.2).
small spin-orbit couplings.
This body of work, and indeed nearly all of the theory and experiment along this research
direction, has been carried out on materials with the zinc-blende (zb) structure. However,
there has also been work on bulk wurtzite (w) materials because their SO splittings are
small [142] and there is hope of room temperature ferromagnetism in magnetically doped
w-GaN and w-ZnO [143]. Spin dynamics in bulk w-GaN has been studied by Beschoten et
al. [142, 129], and there has been experimental [103, 144] and theoretical [21, 145] work on
spin lifetimes in w-ZnO. However, it is often difficult to separate contributions coming from
localized and mobile electrons in bulk doped semiconductors. For example, the rather long
spin lifetimes measured in bulk zb-GaAs at low temperatures [4] were shown to come from
localized spins [5]. This strongly suggests that w-QWs should be studied, and indeed there
is some very recent work along these lines.
Experimentally, the SO splittings have been
measured by Lo et al. [146] in Alx Ga1−x N/GaN QWs by either Shubnikov-de Haas (SdH)
or weak antilocalization (WAL) measurements.
WAL measurements unambiguously point
to SO coupling [147] and such measurements are found to agree with theory [148]. There
are calculations for w-ZnO wells that did not consider tuning [145].
There are striking differences between the zb symmetries and the hexagonal symmetry
of wurtzite materials such as GaN, ZnO, and AlN. The Dresselhaus Hamiltonians for (001),
(110), (111)-zb and (0001)-w are tabulated in Table 5.2.
For (0001) QWs with wurtzite symmetry we have [146, 128]
(w)
w
HSO
= αR + βD − β3 kk2 (ky σx − kx σy ) ,
(w)
where βD
= β1 + bβ3 kz2 and kk = (kx , ky ) is the in-plane wavevector.
This form is
clearly very different from the (001) and (110)-zb case, and the difference has been confirmed
experimentally [147]. The (111)-zb case is similar to (0001)-w. As before, αR can be tuned
by applying a gate voltage or otherwise varying the asymmetry; however, β1 6= 0 even in
the absence of an applied electric field - the wurtzite structure does not have the mirror
96
H1
H3
2
−β3 k|| (ky σx
Material
w-(0001)
zb-(001)
αR (ky σx − kx σy ) + βD (−kx σx + ky σy )
zb-(111)
(αR + βD )(ky σx − kx σy ))
zb-(110)
(αR + βD )(ky σx − kx σy )
− kx σy )
β3 (kx ky2 σx − ky kx2 σy )
− 2β√33 k||2 (ky σx − kx σy ) +
β3 (−kx2 /2
αR (ky σx − kx σy ) + βD kx σz
+
√
β3√ 2
k (3kx2
2 3 y
2
ky )kx σz
− ky2 )σz
Table 5.2: The spin-orbit Hamiltonians for several semiconductor QWs with different crystallographic orientations. The parameter βD is tabulated in Table 5.1.
(w)
symmetry z ↔ −z. βD can be tuned by changing L.
zb ’s and H w :
There are important formal differences between the HSO
SO
(zb)
• (001)-zb : When αR = ±βD
the linear-in-k terms produce a k-independent effective
magnetic field in the [110] or [110] direction [139, 140]. This can enhance spin coherence for
spins oriented along the effective magnetic field. Aside from other possible mechanisms, the
spin relaxation time will be limited by the cubic-in-k terms of the Dresselhaus Hamiltonian.
w differs in that the linear-in-k term can be eliminated for spins oriented in any direction
HSO
(w)
when αR = −βD . Second, at the cubic-in-k level for a circular Fermi surface and elastic
scattering, |kk | is conserved and we can set kk = kF , the Fermi wavevector. This gives the
possibility of canceling the effective field all the way to third order by enforcing the condition
(w)
αR + βD − β3 kF2 = 0, eliminating what appears to be the major source of spin decoherence
in the experiments to date [146, 128]. Note that the final term can be independently tuned
by changing the electron density. The ability to ‘tune’ away the spin-orbit terms up to
cubic-in-k gives the wurtzite structure an advantage over zinc-blende.
zb is very similar to H w ; the linear terms have the exact same
• (111)-zb : This HSO
SO
form and one of the cubic terms can be canceled√ out for zinc-blende as happens for wurtzite.
However an additional cubic term remains,
β3√ 2
k (3kx2
2 3 y
− ky2 )σz ; since the effective field of
this ‘left-over’ piece is always in the z-direction, it will not relax z-oriented spins. Spin not
w .
oriented along ẑ will relax and this is different than what happens for HSO
• (110)-zb : This structure has engendered much interest [149] since in the absence
of structural inversion asymmetry (SIA), the spin-orbit Hamiltonian is Zeeman-like with
the field in a single direction, ẑ. Spins aligned in the z-direction do not relax though
other components will. In the presence of the Rashba term, no lifetime is infinite and the
relaxation rate tensor can only be diagonalized in a basis dependent on the SIA and BIA
couplings [140, 150].
As pointed out by Lo et al., these properties are what makes their Alx Ga1−x N/GaN
structure an excellent candidate for the non-ballistic spin FET [146]. In this section, we
compute the temperature-dependent spin relaxation by the DP mechanism, varying the
97
Exact
ˆ 1 E]
I[τ
ˆ 1E2]
I[τ
ˆ 1E3]
I[τ
T TF
T TF
τtr kB TF
”
“
2
1 + TT 2 δν
F
”
“
2
3
τtr kB
TF3 1 + TT 2 ∆ν
τtr kB T
T =0
0)
τtr kB T II10 (βµ
(βµ0 )
0 )Iν+2 (βµ0 )
τtr (kB T )2 II10 (βµ
(βµ0 )Iν+1 (βµ0 )
0 )Iν+3 (βµ0 )
τtr (kB T )3 II10 (βµ
(βµ0 )Iν+1 (βµ0 )
τtr kB TF
τtr (kB TF )
2
2
τtr kB
TF2
τtr (kB TF )3
(ν + 2)τtr (kB T )2
(ν + 3)(ν + 2)τtr (kB T )3
F
Table 5.3: A table of the various quantities needed in determining the DP spin relaxation
rate in wurtzite and zinc-blende QWs. The quantities δν and ∆ν are located in Table 5.4.
w .
parameters in HSO
The DP mechanism is dominant at room temperature in bulk zb-
GaAs [5] and w-ZnO [21] and this is expected to be true in modulation doped QWs as well.
We discuss other mechanisms that may take over when DP is suppressed. We also discuss
the feasibility of the needed tunings.
5.3.1 Formalism
Conduction band states undergo ordinary impurity and phonon scattering. Each scattering
event gives a change in the wave vector k, which in turn changes the effective magnetic field
on the spin that comes from spin-orbit coupling. This fluctuating field relaxes the spin. This
is known as the D’yakonov-Perel’ (DP) spin relaxation mechanism [63, 64] . The effective
field strength is proportional to the conduction band splitting. Following the treatment
in chapter 3, the dynamics of elastically scattered 2D electrons with spin polarization S is
described by Ṡi = − τ1ij Sj where i, j ∈ {x, y, z} and
∞
1
1 X
= 2
τij
2~ n=−∞
R∞
ji
0 R dE(F+ − F− )τn Γn
∞
0 dE(F+ − F− )
(5.64)
where
Γji
n = −Tr{[H−n , [Hn , σj ]]σi },
and
Hn =
Z
0
2π
dφk
H e−inφk ,
2π
1
=
τn
Z
(5.65)
π
dθWk,k0 (1 − cos nθ).
(5.66)
0
The angle φk is between the x-axis and k; Wk,k0 is the spin-conserving scattering rate
between an initial wave vector k to a final wave vector k0 . θ denotes the angle between k
ˆ 1 E n ] is needed in the
and k0 . Other symbols are defined in section 3.2.3 The quantity I[τ
determination of the DP rate’s temperature dependence and is computed in the limiting
cases in Tables 5.3 and 5.4.
98
ν
0
δν
1
2π 2
1
3 1+π 2 T 2 /3TF2
1+7T 2 /15T 2
π 2 1+π2 T 2 /T 2F
F
2
∆ν
π2
π2
3
2
2
5π 2 1+7T /25TF
3 1+π 2 T 2 /3TF2
2
2
7π 2 1+T /TF
3 1+π 2 T 2 /TF2
Table 5.4: A table of the various quantities needed in determining the DP spin relaxation
rate in wurtzite and zinc-blende QWs.
5.3.2 Temperature dependence of DP mechanism in wurtzite and zincblende QWs
Experimental situations are neither at T = 0 nor exactly non-degenerate. Since the behavior
at T = 0 and at high temperatures are dramatically different, it is important to examine
how the relaxation rate changes as the temperature and electron density vary in general.
In general, we find from Eq. (5.64):
"
∗
1
1
2τtr I1 (βµ0 )
(w) 2 2m
=
=
(α
+
β
)
kB T
R
D
(w)
2 τz(w) (T )
~2 I0 (βµ0 )
~2
τx,y (T )
1
(5.67)
4m∗2
Iν+2 (βµ0 )
8m∗3
Iν+3 (βµ0 )
(w)
− 2(αR + βD )β3 4 (kB T )2
+ β32 6 (kB T )3
~
Iν+1 (βµ0 )
~
Iν+1 (βµ0 )
#
where In (βµ) is a function related to the polylogarithm defined in Appendix C and is
tabulated in Table D.1. This simplifies at zero temperature, where the substitution E →
εF = kB TF can be made, to
1
4τtr m∗ εF
1
=
=
(w)
2 τz(w) (T )
~4
τx,y (T )
1
αR +
(w)
βD
2m∗ β3 εF
−
~2
2
.
εF is the Fermi energy. TF is the Fermi temperature and is related to the electron density
by TF = ~2 πn/kB m∗ . Clearly the T = 0 relaxation times diverge when the tunable quantity
(w)
αR + βD − 2m∗ β3 εF /~2 vanishes. This divergence is cut off by finite temperatures. All
other components of relaxation tensor vanish at all T. Preliminary expressions (before
thermal average) for zb-QWs can be determined similarly and can be found in Table 5.5.
For zb-QWs the ratio η = τ3 /τ1 is required:
(
(2−ν)(3−ν)
, ν ≤ 3/2
τ3
ν 2 −ν+6
η=
=
τ1
1/9,
ν > 3/2
which is only true in two-dimensional systems. It should be stated that additional k 6 terms
should exist due to the product of linear-in-k and quintic-in-k terms in the above formalism.
99
Γzz
Material
2
4τ1 k||
~2
w-(0001)
zb-(001)
4τ1 k2
~2
αR + βD − β3 k||2
2
zb-(110)
Material
zb-(110)
0
Γxx,yy
2
2τ1 k||
w-(0001)
zb-(111)
β 2 (1+η) 4
k
2 + β 2 − k β3 βD + 3
αR
D
2
16
β32 2 2
4τ1 k2
αR + βD − 2√3 k
~2
zb-(111)
zb-(001)
2
~2
αR + βD − β3 k||2
2
2
β 2 (1+η)
2τ1 k2
(±αR − βD )2 + k β3 (±α2 R −βD ) + 3 16 k 4
~2
2
β 2 (1+2η) 4
+β )
2τ1 k2
2 − k β3 (α
√R D + 3
(α
+
β
)
k
R
D
2
12
~
3
β32 (1+9η) 4
k2 β3 βD
2τ1 k2
2
β
−
+
k
2
D
4
64
~
Table 5.5: Γzz =
P∞
zz
n=−∞ Γn
as used in Eq. (5.64). Using αR = 0 for zb-(110).
We are not aware of expressions for the spin-orbit Hamiltonian to such a high order in k.
T TF (degenerate regime)
The degenerate regime must he dealt with carefully. The low temperature regime is determined from the content in Appendices C and D. Following the notation of the Appendices,
−z = −eβµ0 = 1 − eTF /T ≈ −eTF /T where µ0 = β −1 ln(eεF β − 1) is an exact expression
for the chemical potential in a two dimensional electron gas [67]. In the degenerate regime,
one can enforce Lin (1 − eTF /T ) ≈ Lin (−eTF /T ). At T = 0, one expands Lin (−eTF /T )
to first order which is −(TF /T )n /n!. However since the cancellation occurs in the terms
for wurtzite, we should also look at the second leading terms when the polylogarithm is
expanded. The higher order terms of the polylogarithm must be retained especially near
(w)
αR + βD − 2m∗ β3 εF /~2 = 0 where the first order terms vanish completely.
Table 5.6 shows the low temperature spin relaxation rates for wurtzite and several zincblende quantum wells. The factorization that allows the spin relaxation time to vanish
at T = 0 does not occur at a small finite temperature; unfortunately, due to the thermal
averaging we cannot get infinite spin lifetime for any ν.
What must the Rashba coefficient be tuned at to achieve a maximum relaxation time?
By
∂
1
∂αR τDP
∗ , the
= 0, in the degenerate regime, we obtain the elements of Table 5.7 for 1/τDP
∗ . The maximum
minimum DP relaxation rate, and the corresponding Rashba coefficient, αR
spin lifetimes can have different behaviors depending on the QW and its orientation. Figure
5.3 illustrates this fact.
100
Material
w-(0001)
1/τx
zb-(001)
1/τz
zb-(001)
1/τ−
zb-(111)
1/τz
zb-(111)
1/τx
zb-(110)
1/τz
zb-(110)
1/τx
Material
Rate
w-(0001)
1/τx
zb-(001)
1/τz
zb-(001)
1/τ−
zb-(111)
1/τz
zb-(111)
1/τx
zb-(110)
1/τz
zb-(110)
T TF
Rate
1/τx
2τtr ζF
~2 “
αR + βD − β3 ζF
2
+
2
2τtr ζ 2 β3
~2”
β3 ζF ∆ν − 2(αR + βD )δν
“
”
2
β (1+η)
(1+η)
4τtr ζF
2
2
αR
+ βD
− β23 βD ζF + 3 16 ζF2 + 4τtr~ζ2 β3 β3 ζF16
∆ν − β2D δν
~2
“
”
“
”
2
β (1+η) 2
2τtr ζF
(αR +βD )
β
2τtr ζ 2 β3 β3 ζF (1+η)
(αR + βD )2 − 23 (αR + βD )ζF + 3 16
ζF +
∆ν −
δν
2
2
16
2
~
4τtr ζF
~2
2τtr ζF
~2
αR + β D −
β√3
ζ
2 3 F
2
+
4τtr ζ 2 β3
~2
~
β3 ζF
12
∆ν −
(αR +βD )
√
δν
3
“
“
”
β 2 (1+2η) 2 ”
(αR +βD )
β
2τtr ζ 2 β3 β3 ζF (1+2η)
√
(αR + βD )2 − √3 (αR + βD )ζF + 3 12
ζF +
∆ν −
δν
2
12
3
2τtr ζF
~2
2 −
βD
β3 βD
4 ζF
3
~
+
β32 (1+9η) 2
ζF
64
0
+
2τtr ζ 2 β3
~2
1+9η
64 β3 ζF ∆ν
− 41 βD δν
T TF
(αR + βD
− 2(ν + 2)(αR + βD )β3 ζ 2 + (ν + 3)(ν + 2)β32 ζ 3
2
4τtr ζ
2 + β 2 − (ν + 2) β3 βD ζ 2 + (ν + 3)(ν + 2) β3 (1+η) ζ 3
α
2
R
D
2
16
~
β32 (1+η) 3
β3 (αR +βD ) 2
2τtr
2
(α
+
β
)
ζ
−
(ν
+
2)
ζ
+
(ν
+
3)(ν
+
2)
ζ
R
D
2
2
16
~
β32 3
(ν+2)β3
4τtr
2
2
√
(α
+
β
)
ζ
−
(α
+
β
)ζ
+
(ν
+
3)(ν
+
2)
ζ
R
D
R
D
2
12
~
3
2
2τtr
3
2 ζ − (ν+2)β
2 + (ν + 3)(ν + 2) β3 (1+2η) ζ 3
√
(α
+
β
)
(α
+
β
)ζ
R
D
R
D
12
~2
3
2τtr
~2
)2 ζ
0
2τtr
~2
2ζ
βD
− (ν +
2) β34βD ζ 2
+ (ν + 3)(ν + 2)
β32 (1+9η) 3
ζ
64
Table 5.6: 1/τDP ,the DP spin relaxation rate, for several types of QWs in both the degenerate and non-degenerate limits. ζ(F ) = 2m∗ kB T(F ) /~2
.
101
105
1Τ*DP
1000
10
0.1
0.001
1
2
5
10
20
T HKL
50
100
200
Figure 5.3: Maximum DP relaxation times for several QWs: w-(0001), τx solid black, zb(001) τz solid blue, zb-(001) τ− dashed blue, zb-(111) τz solid red, zb-(111) τx dashed red.
∗
These times are determined from 1/τDP
in Table 5.7. Wurtzite parameters are for GaN;
zinc-blende parameters are for GaAs (see Appendix A); TF = 20 K.
T TF (non-degenerate regime)
Since In /Im → n!/m! when T TF , we obtain the expressions in the lower half of Table 5.6
for the relaxation rates at high temperatures. The maximum times achievable for various
QWs are given in the lower half of Table 5.7.
5.3.3 Comparison between wurtzite and zinc-blende
From the preceding analysis, it is apparent that the spin relaxation times can be modified
by altering the Rashba coupling, αR , and thereby causing interference with Dresselhaus
couplings. The same could be done by changing βD though this would have to be done
by changing the QW geometry (e.g. well width) which would be impractical for devices.
The interference between spin-orbit terms is well known in zinc-blende semiconductors [65]
and can also be seen in Table 5.6. In this section we contrast the aforementioned tabulated
results for zb-QWs and w-QWs with actual material parameters. For zb we use GaAs and
for wurtzite we examine both GaN and AlN.
First, in Figure 5.4 displays DP relaxation times at zero temperature. There are a few
points to be made: both τx,z for w-(0001) and τz for zb-(111) diverge when tuned correctly.
Despite this divergence, the zb-(001) crystal has garnered more attention [139, 140] though
the cubic term prevents divergence at zero temperature. τ± for zb-(001) suppresses spin
when αR = ±βD but the τz component causes decay very rapidly in comparison.
102
Material
∗
:1/τDP
w-(0001) :1/τx
at α∗R
zb-(001) :1/τz
8τtr m∗ kB
~4
T TF 16τtr m∗3 (∆ν −2δν ) 3 2
δν2
T2
2 1−
k
T
T
τ
β
tr
F
8
2
3
B
∆ν−2δν TF
~
2m∗ kB TF
T2
−βD +
β3 1 + δν T 2
~2
F
h
“
”i
2
2 T3
∗
2
∗
m∗ kB TF
β 2 (1+η) m∗2 kB
2
F + m kB T β3 β3 m kB TF (1+η) ∆ − β δ
TF βD
− β3 βD
+ 3 4
ν
D ν
2
4
2
2
~
~
zb-(001) :1/τ−
at α∗R
zb-(111) :1/τz
at α∗R
zb-(111) :1/τx
at α∗R
τtr m∗3 3 3 2
kB TF β3 η
~8
∗
at α∗R
zb-(111) :1/τz
at α∗R
zb-(111):1/τx
at α∗R
4
F
δν2 T 4
2η TF4
F
at α∗R
zb-(001) :1/τ−
2
2
kB T
−βD + m√3~
2 β3 (ν + 2)
2
∆ν
8τtr m∗3 3 3 2
kB TF β3 η 1 + ( 2η + ∆ν − δν ) TT 2 −
3~8
F
∗
T2
B TF
−βD + m√k3~
2 β3 1 + δν T 2
∗
:1/τDP
w-(0001) :1/τx
at α∗R
1 + η1 ((1 + η)∆ν − 2δν ) TT 2 − δην TT 4
F
F
∗k T
T2
B F
−βD + m 2~
β
1
+
δ
3
ν T2
2
F
8τtr m∗3 (∆ν −δν ) 3 2
δν2
T2
2
k
T
T
τ
β
1
−
F tr 3
B
∆ν −2δν T 2
3~8
Material
zb-(001) :1/τz
4~
~
0
at α∗R
4τtr ζ
~2
T TF
16τtr m∗3 3 3 2
kB T β3 (ν + 2)
~8
∗
−βD + 2m ~k2B T β3 (ν + 2)
2 − β β (ν + 2)ζ 2 +
βD
3 D
6β32 (1+η)(ν+3)(ν+2) 3
ζ
16
0
τtr m∗3 3 3 2
kB T β3 (ν + 2)(1 + η(ν + 3))
~8
∗
−βD + m 2~kB2 T β3 (ν + 2)
8τtr m∗3 3 3 2
kB T β3 (ν + 2)
3~8
∗k T
B
−βD + m√3~
2 β3 (ν + 2)
4τtr m∗3 3 3 2
kB T β3 (ν + 2 + 2(ν + 2)(ν + 3)η)
3~8
∗k T
B
−βD + m√3~
2 β3 (ν + 2)
∗ ,the minimum DP spin relaxation rate, for several types of QWs in both
Table 5.7: 1/τDP
the degenerate and non-degenerate limits.
103
ΤDP HnsL
1000
10
0.1
0.001
-4
-3
-2
-1
ΑR HmeV nmL
0
1
Figure 5.4: DP relaxation times for several QWs at T = 0 K with TF = 25 K: w-(0001), τx
solid black, zb-(001) τz solid blue, zb-(001) τ− dashed blue, zb-(111) τz solid red, zb-(111)
τx dashed red. These times are determined from upper half of Table 5.7 by setting T = 0
K. Wurtzite parameters are for GaN; zinc-blende parameters are for GaAs (see Appendix
A).
The low temperature regime (T = 5 K) is graphed in Figure 5.5. The infinite lifetimes
seen at T = 0 K no longer exist. As seen in Table 5.7, the maximum time is limited by
the cubic parameter β3 . Thus the much longer times in wurtzite are a consequence of the
smaller SO coupling (see Table 5.8); AlN achieves the longest spin lifetime (> 1 ms).
The non-degenerate regime (T = 300 K) is shown in Figure 5.6. The elements in the
lower half of Table 5.6 are used in the figure. As was the case at low temperatures, β3 again
limits the maximum spin lifetime for all QWs (see expressions in lower half of Table 5.7).
Room temperature behavior is most important for device functions; here w-AlN yields what
would be the longest spin lifetime (∼ 0.5µs) for mobile electrons at room temperature if
other mechanisms are assumed to be even weaker. Appropriately tuned zb-GaAs reaches
a longest time of just over 100 ps - three orders of magnitude shorter than w-AlN. The
intermediate β3 (Table 5.8) of w-GaN places its spin relaxation time between w-AlN and
zb-GaAs.
A desired feature for a semiconductor suitable for spintronic applications is an easy to
manipulate spin relaxation time. As discussed in the first chapter of this dissertation, a
device’s ON/OFF state is dictated by carrier spin orientations at a ferrogmagnetic drain. It
is highly desirable to change the spin lifetime from long to short without needing to change
the SO coupling drastically. In light of Figure 5.6, it appears that the two wurtzite plots in
the left panel are ‘sharper’ than the zinc-blende plots in the right panel. This is better seen
104
7
10 (a)
6
10
5
10
τx
w-AlN
w-GaN
τz, zb-(001)
τ-, zb-(001)
τz, zb-(111)
τx, zb-(111)
4
10
3
τs (ns)
10
2
10
T=5K
T = 5 K (b) zb-GaAs
1
10
0
10
-1
10
-2
10
-3
10
-4
10 -2 -1.5 -1 -0.5 0 0.5
αR (meV nm)
-2.5 -2 -1.5 -1 -0.5 0 0.5 1
αR(meV nm)
Figure 5.5: Low temperature DP spin relaxation times versus Rashba coupling, αR for
wurtzite (left panel) and zinc-blende (right panel) semiconductors. AlN: TF = 15 K, GaN:
TF = 25 K, GaAs: TF = 80 K (all Fermi temperatures correspond to an electron density
∼ 2 × 1011 cm−2 . Entries from top half of Table 5.6 are used in the plot.
in linear-linear axes which is shown in Figure 5.7 where the relaxation rate is plotted instead
of the relaxation time. The ‘sharpness’ of the relaxation rate versus αR is governed by the
2 . For all types of QWs, the curvature goes as ∼ m∗ T . This informs
curvature or ∂ 2 τs−1 /∂αR
us that in a spin relaxation transistor, device performance due to a large ON/OFF ratio
should be optimized at high temperatures and large effective masses. AlN in the wurtzite
phase has an effective mass ∼ 6 times larger than zb-GaAs. To reduce the maximum spin
relaxation time by 105 , one needs to change the Rashba coefficient by ∆αR = ±0.7 meV
nm. To do the same in (001)-GaAs, ∆αR = ±300 meV nm.
Symmetric zb-(110) structures are unique from the others because the DP mechanism
can be completely suppressed for spin oriented along the growth axis [110]. Spin oriented
off that axis do relax. The prospect of going between extremely long and very short relaxation times is very appealing and has fostered interest in this type of QW. However other
relaxation mechanisms present themselves when DP is suppressed and inhibit the theorized
long times. In 2004 Döhrmann et al. discovered a new spin relaxation mechanism they
termed intersubband spin relaxation (ISR) [117]. Despite the suppression of DP in their
(110)-GaAs QW, they measured relaxation times on the order of only a few nanoseconds at
temperatures greater than 100 K. Much longer times are expected due to the absence of DP
105
3
10 (a)
2
10
τx
w-AlN
w-GaN
(b)
zb-GaAs
1
τs (ns)
10
T = 300 K
τz, zb-(001)
τ-, zb-(001)
τx, zb-(111)
τz, zb-(111)
T = 300 K
0
10
-1
10
-2
10
-3
10 -2
-1
0
1
αR (meV nm)
-3
-2
-1
0
αR(meV nm)
1
Figure 5.6: High temperature DP spin relaxation times versus Rashba coupling, αR for
wurtzite (left panel) and zinc-blende (right panel) semiconductors. Entries from bottom
half of Table 5.6 are used in the plot.
and the inefficiency of EY. A limiting mechanism was proposed where due to spin mixing
there is a probability for the spin of an itinerant electron to flip when scattered to a different
electronic subband [117, 153]. The amount of mixing depends quadratically on the size of
the SO coupling and the energy gap between subbands. Their estimate of this relaxation
time agreed well with experiment. This mechanism does not erode the results mentioned
in this chapter; due to the SO coupling discrepancy, ISR in AlN is 107 times weaker than
in GaAs which means ISR times around several milliseconds - longer than what is found
from DP at room temperature. Additionally, ISR is weakened by narrowing the QW and
creating larger energy gaps between different subbands. Reducing the well width increases
hkz2 i which could affect larger DP relaxation but this can be tuned away along with all the
other linear terms of the SO Hamiltonian in Tables 5.1 and 5.2.
5.3.4 Tuning of spin-orbit parameters
We now address the tunability of possible devices.
We note first that in zb-GaAs it
has been possible to achieve quite substantial variations in the appropriate parameters
(zb)
[95, 154]. For example, Koralek et al. were able to change βD
by making structures
with different well widths and to change αR by adjusting the dopant concentrations on
106
Material
w-GaN
w-AlN
w-ZnO
zb-GaAs
β3 (meV nm3 )
−0.32 [111]
−0.01 [128, 152, 151]
0.33 [111]
6.5 − 30 [95]
β1 (meV nm)
0.90 [111]
0.09 [151]
−0.11 [111]
0
m∗ /m0
0.2
0.4
0.25
0.067
Eg (eV)
3.2
6.2
3.3
1.5
Table 5.8: Parameters for several semiconductors.
5000
1ΤDP Hns-1L
4000
3000
2000
1000
0
-6
-4
-2
0
ΑR HmeV nmL
2
4
Figure 5.7: Spin relaxation rates for w-AlN τx (black) and zb-GaAs (111) τx (red). Plot is
linear-linear to demonstrate the different curvatures which go as m∗ T .
the sides of the well, corresponding to a maximum electric field of 5.4 × 10−3 V/nm. In
(zb)
this way a range of αR /βD
of about 0.25 to 1.25 was achieved, without even needing a
gate. β3 was inaccessible and remained constant for all structures, setting a upper limit
on spin lifetimes.
Significant experimental tuning of SO coupling in w-GaN has not yet
been achieved. However, calculations have been done [155] for w-GaN, which produce the
correct magnitude for the spin splittings overall (∼ 5 meV at Fermi wavevector of typical
structures). These authors do not compute αR , β1 , and β3 explicitly, but their computed
spin splittings at a typical Fermi wavevector shows that changes in spin splittings by a
factor of 4 or so can be achieved by changing the well width from 10 to 2 unit cells; to
achieve the same sort of change due to external electric fields required very strong fields of
order 1 V/nm. This suggests that changing well width and electron density rather than
electric field will be the most favorable route for tuning of wurtzite QWs.
107
5.4 Summary
An important quantitative difference between zinc-blende and wurtzite semiconductors is
the strength of the SO interaction. The wurtzites AlN, GaN, and ZnO have considerably
less nuclear charge than GaAs and therefore also a much smaller SO coupling (see Table
5.8. Of prime importance is the fact that the material w-AlN possesses a much smaller
SO coupling than its wurtzite relatives GaN and ZnO. This is understandable given the
positive charge of Al and N to be 13 and 7 respectively; Zn and O have 30 and 8; Ga and N
have 31 and 7. As has 33 protons which makes GaAs a much heavier compound though it
is the lightest of the direct band gap zinc-blende semiconductors. AlN is the lightest direct
band gap wurtzite material. By this simple analysis, it is not surprising that the wurtzites
have considerably longer spin relaxation times when SO coupling is the main reason for spin
relaxation. However wurtzite crystals offer enough flexibility such that short lifetimes can
also be achieved through the existence of linear-in-k terms (αR 6= ±βD ). Recently the idea
that strain may allow further tuning of the Dresselhaus Hamiltonian suggests the possibility
of eliminating even the cubic-in-k term at all temperatures [156]; if possible much longer
lifetimes would be achievable at room temperature.
In conclusion, the findings of this dissertation assert that semiconductors possessing the
wurtzite crystal structure offer enough promising features that could lead to it supplanting
the current zinc-blende paradigm of semiconductor spintronics.
108
Chapter 6
Magnetic field effects
The physics of spin relaxation in applied magnetic fields in the insulating regime is unclear.
The recent review of Wu et al. and the references therein consider many of the quandaries
[56]. These mysteries are not discussed here; instead the dynamics of the localized and
itinerant spin systems are considered when a field is present.
When magnetic fields are present, the different gyroscopic ratios of the localized and itinerant electrons must be included. If the magnetic moments, instead of the magnetizations,
are considered, the Bloch equations are
dµl
nc
nl
γl
= γl µl × B −
µl +
µc
dt
γcr
γcr γc
(6.1)
nl
nc γl
dµc
= γc µc × B −
µc +
µl ,
dt
γcr
γcr γc
(6.2)
where the relaxation mechanisms specific to the two environments are omitted for convenience. The difference between γc and γl is small (∼ 5%). The term γncrl µc γγcl can be written
as γncrl µc γc +(γγcl −γc ) = γncrl µc 1 + (γl − γc )/γc ≈ γncrl µc . The same is done for the last term
of Eq. (6.1), giving
dµl
nc
nl
= γl µl × B −
µl +
µc
dt
γcr
γcr
(6.3)
nl
nc
dµc
= γc µc × B −
µc +
µl ,
dt
γcr
γcr
(6.4)
where γl = gl∗ µ~B and γc = gc∗ µ~B .
When no field is present, the total magnetization is constant (in the absence of relaxation
mechanisms) since ṁl + ṁc = 0. With the two above equations,
d(µl + µc )
= γl µl × B + γc µc × B,
dt
(6.5)
which describes precessing moments as expected in an external field. However, unlike in a
109
single g ∗ case, µl + µc will depend on γcr because dµl /dt − dµc /dt depends explicitly on
γcr . In a reference frame rotating at ω,
µi × γi B → µi × (γi B − ω).
(6.6)
So if we take ω = (γ1 + γ2 )B/2 which is the average angular precession frequency,
γl µl × B →
γl − γc
µl × B
2
(6.7)
γl − γc
µc × B.
(6.8)
2
can be substituted into the above equations. In this rotating reference frame, we see that
γc µc × B → −
d(µl + µc )
γl − γc
=
(µl − µc ) × B
dt
2
(6.9)
and it is clear that the cross relaxation rate does not affect the total spin vector only if
γl = γc .
For γl 6= γc , µl and µc can both be solved for analytically in the lab frame (non-rotating).
The solutions are sums of sinusoidal exponential products with exponential dependence as
e−Γi t where Γi is a relaxation rate:
where
S=
i
1
Γ1 = (nl + nc )Γcr − (γl + γc )B + S
2
2
(6.10)
1
i
Γ2 = (nl + nc )Γcr − (γl + γc )B − S,
2
2
(6.11)
1p
((nc − nl )Γcr − i(γc − γl )B)2 + 4nl nc Γ2cr ,
2
(6.12)
−1 .
and Γcr = γcr
Let us try to simplify by going to the regime ni Γcr (γc − γl )B; this forces nl,c to be
appreciable and we assume nl ∼ nc . By expanding S to second order in B,
1
(γc − γl )2 B 2 nl nc
nc − nl
i
S ≈ nimp Γcr −
− (γc − γl )B
.
2
2
Γcr nimp nimp 2
nimp
(6.13)
We then obtain approximate relaxation rates:
Γ2 ≈
i
i
nc − nl
Γ1 ≈ nimp Γcr − (γl + γc )B − (γc − γl )B
2
2
nimp
(6.14)
i
(γc − γl )2 B 2 nl nc
i
nc − nl
− (γl + γc )B + (γc − γl )B
,
Γcr nimp n2imp 2
2
nimp
(6.15)
110
where the imaginary part gives the precession frequency of the moment. The frequency
here is ∼ (γl + γl )B/2 = ωavg. , the average frequency of the two moments. The correction
frequency is small since γl − γc is small as is nc − nl . This result is sensible because the spin
moment rapidly distributes itself evenly among the two similarly populated electron states;
thus the total moment reacts to the field like an average of the two moments. Hence, in
this regime of quick cross-relaxation and low field, we would not expect to see any beating
phenomena.
The real part of the above Γ1 is a very rapid rate which disappears quickly. The real
part of the Γ2 is much slower. It will be largest when nl = nc ; (γc − γl )B ≈ 2 ns−1 at
B = 1 T in GaAs. This rate is then R(Γ2 ) ≈
1
nimp Γcr
ns−2 . As stipulated earlier, the limit
nimp Γcr (γc − γl )B is under consideration. If the cross-relaxation time is taken to be 1
ps [5], 1/R(Γ2 ) ≈ 1000 ns which is far longer than what has been measured; hence other
mechanisms overshadow this effect. Even at B = 6 T, 1/R(Γ2 ) ≈ 25 ns which again is
too large to match experiments. If the cross-relaxation time is longer the rate due to this
mechanism will be faster and may be observable. If other mechanisms could be suppressed,
this effect offers a way to experimentally determine the cross-relaxation rate nimp Γcr which
has never been measured.
In high fields (such that precession is quicker than cross-relaxation), the two environments act independently and two Larmor frequencies are predicted: γl B and γc B. If the
cross-relaxation time is again taken to be 1 ps, then the field would have to be 100s of tesla
to have large enough precession frequencies. However beating has been observed for fields
> 3 T [157]; see Figure 6.1. If the cross-relaxation time is slower, ∼ 100 ps as suggested by
[43], then beating may be observable in fields of a few tesla. Future research is aimed at an
accurate calculation of the cross-relaxation rate.
Figure 6.1: Left: Faraday rotation data from [157] showing spin beating. Right: Fourier
transform of data depicting two distinct Larmor frequencies. T = 1.8 K, B = 6 T, nimp =
3 × 1016 cm−3 .
111
Chapter 7
Conclusions
This dissertation studies several topics regarding electron spin relaxation in semiconductors.
In chapter 4, a model that incorporates cross-relaxation between localized and itinerant
electrons in a Bloch equation approach explains the temperature dependence of the spin
relaxation in several bulk zinc-blende crystals. Due to the differing thermal occupations, localized and conduction spin relaxation mechanisms take affect at low and high temperatures
respectively. The same idea applies to zinc-blende quantum wells though some modifications are necessary. One important factor is the excitation process - whether electrons are
photo-excited into donor-bound-exciton or just exciton states. The measured spin lifetimes
reflect this difference since the two spin environments are different and hence have different
relaxation times. Also important is the impurity concentration and quantum well growth
direction as shown for i-GaAs confined in the [110]-plane and n-GaAs confined in the [001]plane. The Bloch equation model phenomenologically describes the experiments in both
systems though the operative localized and exciton spin relaxation mechanisms is not yet
a settled issue.
Chapter 5 focuses on applying some of the mechanisms described in chapter 3 to the
wurtzite crystal. Previous research has been in zinc-blende crystals only. The work in this
dissertation finds that the D’yakonov-Perel’ mechanism is qualitatively and quantitatively
different in wurtzite crystals. The phenomenological model utilized in chapter 4 is used
to describe the temperature dependent spin relaxation in bulk n-ZnO. Due to the more
complex impurity profile in ZnO, a three spin environments are required: itinerant, shallow donor, and deep donors. This work in ZnO demonstrates that this method can be
used as a characterization tool for semiconductor samples, yielding information on impurity concentrations and donor binding energies. The chapter concludes by examining the
D’yakonov-Perel’ mechanism in wurtzite quantum wells and showing much longer lifetimes
are achievable and therefore may be of use in future semiconductor spintronic devices.
Chapter 6 explores certain anomalous effects seen in the magnetic field dependence in
the spin relaxation rates. One such effect is the ‘spin beat’ phenomenon seen in some
112
optical orientation experiments. The theory of two interacting subsystems (itinerant and
localized) suggest that the two different g ∗ ’s of the electron species cause the two interfering
frequencies observed in the time-resolved experiments. Also the modified Bloch equations
result in a new relaxation mechanism when an external field is present.
All these topic together give a better understanding of the roles localized and itinerant spins play in spin relaxation processes. It is not adequate to consider only one spin
type; both are required and moreover the exchange interaction between the two is crucial.
Through the work in this dissertation, a better understanding of how electron spins may be
implemented into semiconductor spintronics is achieved.
113
Bibliography
[1] S. Bandyopadhyay and M. Cahay. Introduction to Spintronics. CRC Press, New York,
2008.
[2] K.C. Hall and M.E. Flatté. Perfomance of a spin-based insulated gate field effect
transistor. Appl. Phys. Lett., 88:162503–1–162503–3, 2006.
[3] S. Datta and B. Das. Electronic analog of the electro-optic modulator. Appl. Phys.
Lett., 56:665–667, 1990.
[4] J.M. Kikkawa and D.D. Awschalom. Resonant spin amplification in n-type GaAs.
Phys. Rev. Lett., 80:4313–4316, 1998.
[5] W.O. Putikka and R. Joynt. Theory of optical orientation in n-type semiconductors.
Phys. Rev. B, 70:113201–1–113201–4, 2004.
[6] G.M. Muller, M. Oestreich, M. Romer, and J. Hubner. Semiconductor spin noise
spectroscopy: fundamentals, accomplishments, and challenges. arXiv:1008.2191v1,
2010.
[7] K.V. Kavokin. Spin relaxation of localized electrons in n-type semiconductors. Semicond. Sci. Technol., 23:114009–1–114009–13, 2008.
[8] D. Sprinzl, P. Horodyska, E. Belas, R. Grill, P. Maly, and P. Nemec. Systematic investigation of influence of n-type doping on electron spin dephasing in CdTe.
arXiv:1001.0869v1, 2010.
[9] M. Romer, H. Bernien, G. Muller, D. Schuh, J. Hubner, and M. Oestreich. Electron spin relaxation in bulk gaas for doping densities close to the metal-to-insulator
transition. Phys. Rev. B, 81:075216–1–075216–6, 2010.
[10] I. Malajovich, J.M. Kikkawa, D.D. Awschalom, J.J. Berry, and N. Samarth. Resonant amplification of spin transferred across a GaAs/ZnSe interface. J. Appl. Phys.,
87:5073–5075, 2000.
[11] K. Seeger. Semiconductor Physics. Springer, New York, sixth edition, 1997.
[12] G.V. Astakhov, M.M. Glazov, D.R. Yakovlev, E.A. Zhukov, W. Ossau, L.W.
Molenkamp, and M. Bayer. Time-resolved and continuous-wave optical spin pumping
114
of semiconductor quantum wells. Semicond. Sci. Technol., 23:114001–1–114001–14,
2008.
[13] G.E. Moore. Cramming more components onto integrated circuits. Electronics, 38,
1965.
[14] B. Sapoval and C. Hermann. Physics of Semiconductors. Springer-Verlag, New York,
1995.
[15] S. Bandyopadhyay and M. Cahay. Electron spin for classical information processing:
a brief survey of spin-based logic device, gates, and circuits. Nanotechnology, 20:1–35,
2009.
[16] International Techonology Roadmap for Semiconductors. Semiconductor Industry
Association, San Jose, CA, 2003, http://public.itrs.net.
[17] H.C. Koo, J.H. Kwon, J. Eom, J. Chang, S.H. Han, and M. Johnson. Control of spin
precession in a spin-injected field effect transistor. Science, 325:1515–1518, 2009.
[18] P. Agnihotri and S. Bandyopadhyay. Analysis of the two dimensional Datta-Das spin
field effect transistor. Physica E, 42:1736–1740, 2010.
[19] J.H. Kwon, H.C. Koo, J. Chang, S.H. Han, and J. Eom. Gate field effect on spin
transport signals in a lateral spin-valve device. J. Korean Phys. Soc., 53:2491–2495,
2008.
[20] N.J. Harmon, W.O. Putikka, and R. Joynt. Theory of electron spin relaxation in
n-doped quantum wells. Phys. Rev. B, 81:085320–1–085320–8, 2010.
[21] N.J. Harmon, W.O. Putikka, and R. Joynt. Theory of electron spin relaxation in zno.
Phys. Rev. B, 79:115204–1–115204–6, 2009.
[22] N.J. Harmon, W.O. Putikka, and R. Joynt. Prediction of extremely long electron
spin lifetimes in wurtzite semiconductor quantum wells. arXiv:1003.0494v1, pages
1–4, 2010.
[23] R.P. Feynman, R.B. Leighton, and M. Sands. The Feynman Lectures on Physics,
volume 3. Addison-Wesley, Reading, MA, 1965.
[24] R. Shankar. Principles of Quantum Mechanics. Oxford University Press, New York,
2001.
[25] Sin itiro Tomonaga. The Story of Spin. University of Chicago Press, Chicago, 1997.
[26] D. Halliday, R. Resnick, and J. Walker. Fundamentals of Physics. John Wiley and
Sons Inc., New York, 6th edition, 2001.
[27] D.J. Griffiths. Introduction to Quantum Mechanics. Prentice Hall Inc., Upper Saddle
River, New Jersey, 1995.
[28] C.P. Slichter. Principles of Magnetic Resonance. Springer-Verlag, New York, 3rd
edition, 1990.
115
[29] F. Bloch. Nuclear induction. Phys. Rev., 70:460–477, 1946.
[30] N. Bloembergen and T.E. Purcell and R.V. Pound. Relaxation effects in nuclear
magnetic resonance absorption. Phys. Rev., 73:679–718, 1948.
[31] R.K. Wangsness and F. Bloch. The dynamical theory of nuclear induction. Phys.
Rev., 89:728–739, 1953.
[32] A.G. Redfield. On the theory of relaxation processes. IBM J. Res. Dev., 1:19–31,
1957.
[33] A. Abragam. The Principles of Nuclear Magnetism. Oxford University Press, 1961.
[34] N. Murali and V.V. Krishnan. A primer for nuclear magnetic relaxation in liquids.
Concepts Magn. Reson. Part A, 17A:86–116, 2003.
[35] M. Goldman. Formal theory of spin-lattice relaxation. J. Mag. Res., 149:160–187,
2001.
[36] K. Blum. Density Matrix Theory and Applications. Plenum Press, New York, 2nd
edition, 1981.
[37] R.K. Pathria. Statistical Mechanics. Elsevier, Oxford, 2nd edition, 2004.
[38] J. Kowalewski and L. Mäler. Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications. CRC Press, Boca Raton, FL, 2006.
[39] J. Fabian, A. Matos-Abiague, C. Ertler, P. Stano, and I. Zutic. Semiconductor spintronics. Acta Phys. Slov., 57:1–180, 2007.
[40] Y. Yafet. g factors and spin-lattice relaxation of conduction electrons. In F. Seitz and
D. Turnbull, editors, Solid State Physics, volume 14, page 2. Academic, New York,
1963.
[41] I. Źutić, J. Fabian, and S. Das Sarma. Spintronics: Fundamentals and applications.
Rev. Mod. Phys., 76:323–410, 2004.
[42] M.E. Flatte, J.M Byers, and Wayne H. Lau. Spin dynamics in semiconductors. In
D.D. Awschalom, D. Loss, and N. Samarth, editors, Semiconductor Spintronics and
Quantum Computation. Springer-Verlag, Berlin, 2002.
[43] D. Paget. Optical detection of nmr in high-purity gaas under optical pumping: Efficient spin-exchange averaging between electron states. Phys. Rev. B, 24:3776–3793,
1981.
[44] R.I. Dzhioev, B.P. Zakharchenya, V.L. Korenev, D. Gammon, and D.S. Katzer. Long
electron spin memory times in gallium arsenide. JETP Lett., 74:182–185, 2001.
[45] R.I. Dzhioev, V.L. Korenev, I.A. Merkulov, and B.P. Zakharchenya. Manipulation
of the spin memory of electrons in n-GaAs. Phys. Rev. Lett., 88:256801–1–256801–4,
2002.
116
[46] G.D. Mahan and Ryan Woodworth. Spin-exchange scattering in semiconductors.
Phys. Rev. B, 78:075205–1–075205–5, 2008.
[47] Neil W. Ashcroft and N. David Mermin. Solid State Physics. Brooks/Cole Thomson
Learning, Belmont, 1976.
[48] R.J. Elliott. Theory of the effect of spin-orbit coupling on magnetic resonance in some
semiconductors. Phys. Rev., 96:266–279, 1954.
[49] C. Kittel. Quantum Theory of Solids. John Wiley and Sons, New York, 1963.
[50] U. Rossler. Nonparabolicity and warping in the conduction band of gaas. Solid State
Communications, 49:943–947, 1984.
[51] H. Mayer U. Rossler. Spin splitting and anisotropy of cyclotron resonance in the
conduction band of gaas. Phys. Rev. B, 44:9048–9051, 1991.
[52] G. Dresselhaus. Spin-orbit coupling effects in zinc blende structures. Phys. Rev.,
100:580–586, 1955.
[53] G.L. Bir, A.G. Aronov, and G.E. Pikus. Spin relaxation of electrons scattered by
holes. Sov. Phys. JETP, 42:705–712, 1976.
[54] A.G. Aronov, G.E. Pikus, and A.N. Titkov. Spin relaxation of conduction electrons
in p-type III-V compounds. Sov. Phys. JETP, 57:680–687, 1983.
[55] J.H. Jiang and M.W. Wu. Electron-spin relaxation in bulk III-V semiconductors from
a fully microscopic kinetic spin Bloch equation approach. Phys. Rev. B, 79:125206–
1–125206–21, 2009.
[56] M.W. Wu, J.H. Jiang, and M.Q. Weng. Spin dynamics in semiconductors. Physics
Reports, 493:61–236, 2010.
[57] W. Zawadazki and W. Szymanska. Elastic electron scattering in InSb-type semiconductors. Phys. Stat. Sol. (b), 45:415–432, 1971.
[58] J.N. Chazalviel. Spin relaxation of conduction electrons in n-type indium antimonide
at low temperature. Phys. Rev. B, 11:1555–1562, 1975.
[59] B.K. Ridley. Electrons and Phonons in Semiconductor Multilayers. Cambridge University Press, Cambridge, 2nd edition, 2009.
[60] G. Fishman and G. Lampel. Spin relaxation of photoelectrons in p-type gallium
arsenide. Phys. Rev. B, 16:820–831, 1977.
[61] G.E. Pikus and A.N. Titkov. Ch. 3: Spin relaxation under optical orientation in
semiconductors. In F. Meier and B.P. Zakharchenya, editors, Optical Orientation.
North Holland, Amsterdam, 1984.
[62] M.I. D’yakonov and V.I. Perel’. Spin orientation of electrons associated with the
interband absorption of light in semiconductors. Sov. Phys. JETP, 33:1053–1059,
1971.
117
[63] M.I. D’yakonov and V.I. Perel’. Spin relaxation of conduction electrons in noncentrosymmetric semiconductors. Sov. Phys. Solid State, 13:3023–3026, 1971.
[64] M.I. D’yakonov and V.Y. Kachorovskii’. Spin relaxation of two-dimensional electrons
in noncentrosymmetric semiconductors. Sov. Phys. Semicond., 20:110–112, 1986.
[65] N.S. Averkiev and L.E. Golub. Giant spin relaxation in zinc-blende heterostructures.
Phys. Rev. B, 60:15582–15584, 1999.
[66] P.Y. Yu and M. Cardona. Fundamentals of Semiconductors. Springer, New York,
third edition, 2001.
[67] J. Kainz, U. Rossler, and R. Winkler. Temperature dependence of dyakonov perel relaxation in zinc-blende semiconductor quantum structures. Phys. Rev. B, 70:195322–
1–195322–9, 2004.
[68] S. Blundell. Magnetism in Condensed Matter. Springer, New York, 2nd edition, 1994.
[69] J.D. Jackson. Classical Electrodynamics. John Wiley and Sons, New York, 3rd edition,
1999.
[70] M.I. D’yakonov and V.I. Perel. Ch. 2: Theory of Optical Spin Orientation of Electrons
and Nuclei in Semiconductors. In F. Meier and B.P. Zakharchenya, editors, Optical
Orientation. North Holland, Amsterdam, 1984.
[71] A.W. Overhauser. Paramagnetic relaxation in metals. Phys. Rev., 89:689–700, 1953.
[72] M. Gueron. Density of the conduction electrons at the nuclei in indium antimonide.
Phys. Rev., 135:A200–A206, 1964.
[73] I. Tifrea and M.E. Flatté. Electric field tunability of nuclear and electronic spin
dynamics due to the hyperfine interaction in semiconductor nanostructures. Phys.
Rev. Lett., 90:237601–1–237601–4, 2003.
[74] I. Tifrea and M.E. Flatté. Nuclear spin dynamics in parabolic quantum wells. Phys.
Rev. B, 69:115305–1–115305–8, 2004.
[75] I. Tifrea. Nuclear Spin Dynamics in Semiconductor Nanostructures. In M.E. Flatté
and I. Tifrea, editors, Manipulating Quantum Coherence in Solid State Systems.
Springer, 2007.
[76] W.A. Coish and J. Baugh. Nuclear spins in nanostructures. Phys. Status Solidi B,
246:2203–2215, 2009.
[77] R.I. Dzhioev, K.V. Kavokin, V.L. Korenev, M.V. Lazarev, B. Ya. Meltser, M.N.
Stepanova, B.P. Zakharchenya, D. Gammon, and D.S. Katzer. Low-temperature spin
relaxation in n-type GaAs. Phys. Rev. B, 66:245204–1–245204–7, 2002.
[78] I.A. Merkulov, A.L. Efros, and M. Rosen. Electron spin relaxation by nuclei in semiconductor quantum dots. Phys. Rev. B, 65:205309–1–205309–8, 2002.
118
[79] P.F. Braun, X. Marie, L. Lombez, B. Urbaszek, T. Amand, P. Renucci, V.K. Kalevich, K.V. Kavokin, O. Krebs, P. Voisin, and Y. Masumoto. Direct observation of
the electron spin relaxation induced by nuclei in quantum dots. Phys. Rev. Lett.,
94:116601–1–116601–4, 2005.
[80] A.V. Khaetskii, D. Loss, and L. Glazman. Electron spin decoherence in quantum dots
due to interaction with nuclei. Phys. Rev. Lett., 88:186802–1–186802–4, 2002.
[81] B.I. Shklovskii. Dyakonov-Perel spin relaxation near the metal-insulator transition
and in hopping transport. Phys. Rev. B, 73:193201–1–193201–3, 2006.
[82] L.P. Gorkov and P.L. Krotkov. Spin relaxation and antisymmetric exchange in ndoped III-V semiconductors. Phys. Rev. B, 67:033203–1–033203–4, 2003.
[83] K.V. Kavokin. Symmetry of anisotropic exchange interactions in semiconductor
nanostructures. Phys. Rev. B, 69:075302–1–075302–5, 2004.
[84] L.P. Gorkov and L.P. Pitaevskii. Term splitting energy of the hydrogen molecule.
Sov. Phys.-Dokl., 8:788–790, 1964.
[85] C. Herring and M. Flicker. Asymptotic exchange coupling of two hydrogen atoms.
Phys. Rev., 134:A632–A366, 1964.
[86] I.V. Ponomarev, V.V. Flambaum, and A.L. Efros. Spin structure of impurity band
in semiconductors in two- and three-dimensional cases. Phys. Rev. B, 60:5485–5496,
1999.
[87] M.W. Wu and C.Z. Ning. Dyakonov-Perel effect on spin dephasing in n-type GaAs.
Phys. Stat. Sol. (b), 222:523–534, 2000.
[88] M.W. Wu. Kinetic theory of spin coherence of electrons in semiconductors. Journal
of Superconductivity: Incorporating Novel Magnetism, 14:245–259, 2001.
[89] Pil Hun Song and K.W. Kim. Spin relaxation of conduction electrons in bulk III-V
semiconductors.
[90] H. Haug and S.W. Koch. Quantum Theory of the Optical and Electronic Properties
of Semiconductors. World Scientific, New Jersey, 4th edition, 2004.
[91] J.B. Joyce and R.W. Dixon. Analytic approximations for the Fermi energy of an ideal
Fermi gas. Appl. Phys. Lett., 31:354–357, 1977.
[92] V.C. Aguilera-Navarro, G.A. Estevez, and A. Kostecki. A note on the Fermi-Dirac
integral function. J. Appl. Phys., 63:2848–2851, 1988.
[93] B. Segall, M.R. Lorenze, and R.E. Halsted. Electrical properties of n-type CdTe.
Phys. Rev., 129:2471–2481, 1963.
[94] A.N. Chantis, M. van Schilfgaarde, and T. Kotani. Ab initio prediction of conduction
band spin splitting in zinc blende semiconductors. Phys. Rev. Lett., 96:086405–1–
086405–4, 2006.
119
[95] J.D. Koralek, C.P. Weber, J. Orenstein, B.A. Bernevig, S.C. Zhang, S. Mack, and
D.D. Awschalom. Emergence of the persistent spin helix in semiconductor quantum
wells. Nature, 458:610–613, 2009.
[96] J. Tribollet, E. Aubry, G. Karczewski, B. Sermage, F. Bernardot, C. Testelin, and
M. Chamarro. Enhancement of the electron spin memory by localization on donors
in a CdTe quantum well. Phys. Rev. B, 75:205304–1–205304–6, 2007.
[97] B. Eble, C. Testelin, F. Bernardot, M. Chamarro, and G. Karczewski. Inversion of
spin polarization of localized electrons driven by dark excitons. arXiv:0801.1457v1,
2008.
[98] M. Chamarro, F. Bernardot, and C. Testelin. Spin decoherence and relaxation processes in zero-dimensional semiconductor nanostructures. J. Phys.: Condens. Matter,
19:445007:1–18, 2007.
[99] J. Tribollet, F. Bernardot, M. Menant, G. Karczewski, C. Testelin, and M. Chamarro.
Interplay of spin dynamics of trions and two-dimensional electron gas in a n-doped
CdTe single quantum well. Phys. Rev. B, 68:235316–1–235316–6, 2003.
[100] E.A. Zhukov, D.R. Yakovlev, M. Bayer, M.M. Glazov, E.L. Ivchenko, G. Karczewski,
T. Wojtowicz, and J. Kossut. Spin coherence of a two-dimensional electron gas induced
by resonant excitation of trions and excitons in CdTe/(Cd,Mg)Te quantum wells.
Phys. Rev. B, 76:205310–1–205310–16, 2007.
[101] M.Q. Weng and M.W. Wu. Spin dephasing in n-type GaAs quantum wells. Phys.
Rev. B, 68:075312–1–075312–11, 2003.
[102] J. Zhou, J.L. Cheng, and M.W. Wu. Spin relaxation in n-type GaAs quantum wells
from a fully microscopic approach. Phys. Rev. B, 75:045305–1–045305–9, 2007.
[103] S. Ghosh, V. Sih, W.H. Lau, and D.D. Awschalom. Room-temperature spin coherence
in ZnO. Appl. Phys. Lett., 86:232507–1–232507–3, 2005.
[104] K. Kheng, R.T. Cox, Y. Merle d’Aubigne, F. Bassani, K. Saminadayar, and
S. Tatarenko. Observation of negatively charged excitons X− in semiconductor quantum wells. Phys. Rev. Lett., 71:1752–1755, 1993.
[105] T. Adachi, T. Miyashita, S. Takeyama, Y. Takagi, and A. Tackeuchi. Exciton spin
dynamics in GaAs quantum wells. Journal of Luminescence, 72-74:307–308, 1997.
[106] T. Adachi, Y. Ohno, F. Matsukura, and H. Ohno. Spin relaxation in n-modulation
doped GaAs/AlGaAs (110) quantum wells. Physica E, 10:36–39, 2001.
[107] D.C. Look. Electrical Characterization of GaAs Materials and Devices. John Wiley
and Sons, New York, 1989.
[108] I. Dzyaloshinskii. A thermodynamic theory of “weak” ferromagnetism of antiferromagnetics. Phys. Chem. Solids, 4:241–255, 1958.
[109] T. Moriya. Anisotropic superexchange interaction and weak ferromagnetism. Phys.
Rev., 120:91–98, 1960.
120
[110] M.I. D’yakonov and V.I. Perel’. Spin orientation of electrons associated with the
interband absorption of light in semiconductors. Sov. Phys. JETP, 33:1053–1059,
1971.
[111] J.Y. Fu and M.W. Wu. Spin-orbit coupling in bulk ZnO and GaN. J. Appl. Phys.,
104:093712–1–093712–7, 2008.
[112] E.I. Rashba. Properties of semiconductors with an extremum loop. 1. cyclotron and
combinational resonance in a magnetic field perpendicular to the plane of the loop.
Sov. Phys. Solid State, 2:12241238, 1960.
[113] Y. Ohno, R. Terauchi, T. Adachi, F. Matsukura, and H. Ohno. Electron spin relaxation beyond D’yakonov-Perel’ interaction in GaAs/AlGaAs quantum wells. Physica
E, 6:817–820, 2000.
[114] T. Hassenkam, S. Pedersen, K. Baklanov, A. Kristensen, C.B. Sorensen, P.E. Lindelof, F.G. Pikus, and G.E. Pikus. Spin splitting and weak localization in (110)
GaAs/Alx Ga1−x As quantum wells. Phys. Rev. B, 55:9298–9301, 1997.
[115] A. Tackeuchi, T. Kuroda, S. Muto, Y. Nishikawa, and O. Wada. Carrier mobility
dependence of electron spin relaxation in GaAs quantum wells. Jpn. J. Appl. Phys.,
38:2546–2551, 1999.
[116] Y. Ohno, R. Terauchi, T. Adachi, F. Matsukura, and H. Ohno. Spin relaxation in
GaAs (110) quantum wells. Phys. Rev. Lett., 83:4196–4199, 1999.
[117] S. Dohrmann, D. Hagele, J. Rudulph, M. Bichler, D. Schuh, and M. Oestreich.
Anomolous spin dephasing in (110) GaAs quantum wells: anisotropy and intersubband effects. Phys. Rev. Lett., 93:147405–1–147405–4, 2004.
[118] P. Harrison. Quantum Wells, Wires and Dots. Wiley & Sons, West Sussex, 2nd
edition, 2005.
[119] J. Davies. The Physics of Low Dimensional Semiconductors. Cambridge University
Press, New York, 1998.
[120] J. Schliemann, A. Khaetskii, and D. Loss. Electron spin dynamics in quantum dots and
related nanostructures due to hyperfine interaction with nuclei. J. Phys.: Condens.
Matter, 15:R1809–R1833, 2003.
[121] S.V. Zaitsev, M.K. Welsch, A. Forchel, and G. Bacher. Excitons in artificial quantum
dots in the weak spatial confinement regime. Journal of Experimental and Theoretical
Physics, 105:1241–1258, 2007.
[122] I. Zorkani and E. Kartheuser. Resonant magneto-optical spin transitions in zincblende and wurtzite semiconductors. Phys. Rev. B, 53:1871–1890, 1996.
[123] V.I. Litvinov. Polarization-induced Rashba spin-orbit coupling in structurally symmetric III-nitride quantum wells. Appl. Phys. Lett., 89:222108–1–222108–3, 2006.
[124] S-H. Wei and A. Zunger. Valence band splittings and band offsets of aln, gan, and
inn. Appl. Phys. Lett., 69:2719–2721, 1996.
121
[125] W.J. Fan, J.B. Xia, P.A. Agus, S.T. Tan, S.F. Yu, and X.W. Sun. Band parameters
and electronic structures of wurtzite zno and zno/mgzno quantum wells. J. Appl.
Phys., 99:013702–1–013702–4, 2006.
[126] L.C. LewYan Voon, M. Willatzen, M. Cardona, and N.E. Christensen. Terms linear
in k in the band structure of wurtzite-type semiconductors. Phys. Rev. B, 53, 1996.
[127] W. Weber, S.D. Ganichev, S.N. Danilov, D. Weiss, and W. Prettl. Demonstration of
Rashba spin splitting in GaN-based heterostructures. Appl. Phys. Lett., 87:262106–
1–262106–3, 2005.
[128] W.T. Wang, C.L. Wu, S.F. Tsay, M.H. Gau, Ikai Lo, H.F. Kao, D.J. Jang, J.C.
Chiang, M.E. Lee, Y.C. Chang, C.N. Chen, and H.C. Hsueh. Dresselhaus effect in
bulk wurtzite materials. Appl. Phys. Lett., 91:082110–1–082110–3, 2007.
[129] J.H. Buß, J. Rudolph, F. Natali, F. Semond, and D. Hagele. Temperature dependence
of electron spin relaxation in bulk GaN. Phys. Rev. B, 81:155216–1–155216–5, 2010.
[130] U. Özgür, Y.I. Alivov, C. Liu, A. Teke, M.A. Reshchikov, S. Doğan, V. Avrutin, S.J.
Cho, and H. Morkoç. A comprehensive review of ZnO materials and devices. J. Appl.
Phys., 98:041301–1–041301–103, 2005.
[131] T. Dietl, H. Ohno, F. Matsukura, J. Cibert, and D. Ferrand. Zener model description
of ferromagnetism in zinc-blende magnetic semiconductors. Science, 287:1019–1022,
2000.
[132] H. von Wenckstern, H. Schmidt, M. Grundmann, M.W. Allen, P. Miller, R.J. Reeves,
and S.M. Durbin. Defects in hydrothermally grown bulk ZnO. Appl. Phys. Lett.,
91:022913–1–022913–3, 2007.
[133] U. Grossner, S. Gabrielson, T.M. Børseth, J. Grillenberger, A.Y. Kuznetsov, and B.G.
Svensson. Palladium Schottky barrier contacts to hydrothermally grown n-ZnO and
shallow electron states. Appl. Phys. Lett., 85:2259–2261, 2004.
[134] J.R. Schrieffer and P.A. Wolff. Relation between the Anderson and Kondo Hamiltonians. Phys. Rev., 149:491–492, 1966.
[135] D.C. Look. Quantitative analysis of surface donors in ZnO. Surface Science, 601:5315–
5319, 2007.
[136] C.H. Seager and S.M. Myers. Quantitative comparisons of dissolved hydrogen density
and the electrical and optical properties of ZnO. J. Appl. Phys., 94:2888–2894, 2003.
[137] G.H. Kassier, M. Hayes, and F.D. Auret. Electrical and structural characterization
of as-grown and annealed hydrothermal bulk ZnO. J. Appl. Phys., 201:014903–1–
014903–5, 2007.
[138] D.C. Look. Recent advances in ZnO materials and devices. Materials Science and
Engineering B, 80:383–386, 2001.
[139] J. Schliemann, J.C. Egues, and D. Loss. Nonballistic spin-field-effect transistor. Phys.
Rev. Lett., 90:146801–1–146801–4, 2003.
122
[140] X. Cartoixa, D.Z.Y. Ting, and Y.C. Chang. A resonant spin lifetime transistor. Appl.
Phys. Lett., 83:1462–1464, 2003.
[141] B.A. Bernevig, J.O. Orenstein, and X.C. Zhang. Exact SU(2) symmetry and persisten
spin helix in a spin-orbit coupled system. Phys. Rev. Lett., 97:236601–1–236601–4,
2006.
[142] B. Beschoten, E. Johnston-Halperin, D.K. Young, M. Poggio, J.E. Grimaldi, S. Keller,
S.P. DenBaars, U.K. Mishra, E.L. Hu, and D.D. Awschalom. Spin coherence and
dephasing in GaN. Phys. Rev. B, 63:121202(R)–1–121202(R)–4, 2001.
[143] C. Liu, F. Yun, and H. Morkoc. Ferromagnetism of ZnO and GaN: a review. J. Mater.
Sci.: Mater. Electron., 16:555–597, 2005.
[144] S. Ghosh, D.W. Steuerman, B. Maertz, K. Ohtani, H. Xu, H. Ohno, and D.D.
Awschalom. Electrical control of spin coherence in ZnO. Appl. Phys. Lett., 92:162109–
1–162109–3, 2008.
[145] C. Lü and J.L. Cheng. Spin relaxation in n-type ZnO quantum wells. Semicond. Sci.
Technol., 24:115010–1–115010–5, 2009.
[146] I. Lo, M.H. Gau, J.K. Tsai, Y.L. Chen, Z.J. Chang, W.T. Wang, J.C. Chiang,
T. Aggerstam, and S. Lourdudoss. Anomalous k-dependent spin splitting in wurtzite
Alx Ga1x NGaN heterostructures. Phys. Rev. B, 75:245307–1–245307–6, 2007.
[147] N. Thillosen, S. Caba nas, N. Kaluza, V.A. Guzenko, H. Hardtdegen, and Th.
Schäpers. Weak antilocalization in gate-controlled Alx Ga1−x N/GaN two-dimensional
electron gas. Phys. Rev. B, 73:241311(R)–1–241311(R)–4, 2006.
[148] S.B. Lisesivdin, N. Balkan, O. Makarovsky, A. Patane, A. Yildiz, M.D. Caliskan,
M. Kasap, S. Ozcelik, and E. Ozbay.
[149] K.C. Hall, W.H. Lau, K. Gundogdu, and M.E. Flatte. Nonmagnetic semiconductor
spin transistor. Appl. Phys. Lett., 83:2937–2939, 2003.
[150] X. Cartoixa, D.Z.Y. Ting, and Y.C. Chang. Suppression of the Dyakonov-Perel spinrelaxation mechanism for all spin components in [111] zincblende quantum wells. Phys.
Rev. B, 71:045313–1–045313–5, 2005.
[151] J. Nicklas. Unpublished density functional theory calculation, 2010.
[152] M.E. Lee. Private communications, 2009-2010.
[153] D. Hagele, S. Dohrmann, J. Rudolph, and M. Oestreich. Electron spin relaxation in
semiconductors. Adv. in Solid State Physics, 45:253–261, 2006.
[154] M. Studer, G. Salis, K. Ensslin, D.C. Driscoll, and A.C. Gossard. Gate-controlled
spin-orbit interaction in a parabolic GaAs/AlGaAs quantum well. Phys. Rev. Lett.,
103:027201–1–027201–4, 2009.
123
[155] I. Lo, W.T. Wang, M.H. Gau, J.K. Tsai S.F. Tsay, and J.C. Chiang. Gate-controlled
spin splitting in GaN/AlN quantum wells. Appl. Phys. Lett., 88:082108–1–082108–3,
2006.
[156] C-L. Wu, S-F. Tsay, W-T. Wang, M-H. Gau, J-C. Chiang, I. Lo, H-F. Kao, Y-C. Hsu,
D-J. Jang, M-E. Lee, and C-N. Chen. Crystal-field and strain effects on minimumspin-splitting surfaces in bulk wurtzite materials. J. of the Phys. Soc. of Japan,
79:093705, 2010.
[157] D.D. Awschalom J. Kikkawa and A. Kirby. Unpublished time resolved faraday rotation experiments.
[158] L. Lewin. Polylogarithms and Associated Functions. Elsevier North Holland, New
York, 1981.
124
Appendix A
Material parameters
zb-CdTe
aB = 5.3 nm (Iodine donor)
εB = 12.4 meV (Iodine donor)
β1 = 0 meV nm
β3 = 8.5 meV nm3
Eg = 1.606 eV
m∗ = 0.11me
zb-GaAs
aB = 10.4 nm (Silicon donor)
εB = 5.8 meV (Silicon donor)
β1 = 0 meV nm
β3 = 6.5 − 30 meV nm3
∆0 = 340 meV
Eg = 1.51 eV
m∗ = 0.067me
zb-ZnSe
aB = 4.6 nm (Chlorine donor)
εB = 14 meV (Chlorine donor)
β1 = 0 meV nm
β3 = 1.3 meV nm3
Eg = 2.82 eV
125
m∗ = 0.13me
w-AlN
β1 = 0.09 meV nm
β3 = −0.01 meV nm3
∆0 = 19 meV
Eg = 6.1 eV
m∗ = 0.4me
w-GaN
β1 = 0.9 meV nm
β3 = −0.32 meV nm3
∆0 = 13 meV
Eg = 3.2 eV
m∗ = 0.2me
w-ZnO
β1 = 0.11 meV nm
β3 = −0.33 meV nm3
∆0 = −3.5 meV
Eg = 3.3 eV
m∗ = 0.2me
126
Appendix B
An integral involving spherical
harmonics
Spherical harmonics have several different definitions depending on how authors choose to
deal with the normalization factor and orthogonality relation. We follow the definition of
Jackson [69]:
s
Yln (θ, φ) =
(2l1 )(l − n)! n
P (cos θ)einφ .
4π(l + n)! l
(B.1)
Pln (cos θ) are associated Legendre polynomials that can be determined for positive and
negative n from Rodrigues’ formula:
Pln (x) =
l+n
(−1)n
2 n/2 d
(1
−
x
)
(x2 − 1)l .
2l l!
dxl+n
(B.2)
From this recursive formula, it is simple to show
Pl−n (x) = (−1)n
(l − n)! n
P (x).
(l + n)! l
(B.3)
It is tempting to think to use the standard orthogonality relation of spherical harmonics,
R
0
dΩYln0 (θ, φ)Yln∗ (θ, φ) = 1, to compute the angular average in Eq. (3.35). However the
equation actually reads
0
Yln0 (θ, φ)Yln (θ, φ)
Z
=
dΩ n0
Y 0 (θ, φ)Yln (θ, φ).
4π l
However by using the above definitions, it can be shown that
Z
dΩ n0
(−1)n
Yl0 (θ, φ)Yln (θ, φ) =
δn0 ,−n δl0 ,l .
4π
4π
R 2π
i(n0 +n)ϕ = 1.
For the two dimensional situation, ein0 ϕ einϕ = 0 dϕ
2π e
127
(B.4)
(B.5)
Appendix C
Important integrals
Often we would like to compute the following quantity:
R∞
n ∂f0
0 g(ε)τ1 ε ∂ε dε
= Iˆ(d) [τ1 εn ].
R∞
∂f0
g(ε)
dε
∂ε
0
(C.1)
where f0 is the Fermi-Dirac function and g(ε) is the density of states. To do so, other
quantities such as Iˆ(d) [εn ] need to be determined.
First we look at the top integral in Eq. (C.1). Constant factors in the density of states
will be discarded since they will eventually cancel with the denominator. So we are left
with
∞
Z
εd εn+ν
s1
0
e−βε+βµ
dε,
(1 + e−βε+βµ )2
(C.2)
where d is 1/2 for three dimensions and 0 for two dimensions. By defining x = βε and
z = exp(βµ),
Z
s1
β n+ν+d+1
∞
xd xn+ν
0
ze−x
dx,
(1 + ze−x )2
(C.3)
which is the final integral we want to solve. The integral can also be written as [67]
R∞
s1
xd xn+ν
dx. It is not a simple integral so we look to expand the inteβ n+ν+d+1 0 4 cosh2 ((x−ln z)/2)
grand in a series of z [90]. This yields
∞
Z
s1
β n+ν+d+1
d n+ν
x x
0
∞
X
(j + 1)z j+1 e−x(j+1) dx.
(C.4)
j=0
When integrating term by term the following series is obtained:
−
s1
β
(n + ν + d)!
n+ν+d+1
∞
X
(−z)j
.
j n+ν+d
(C.5)
j=1
The series is defined as the polylogarithm function of order n + ν + d, Lin+ν+d (−z) [158].
128
We define the dimensionless integral as In+ν+d (βµ) = −(n + ν + d)!Lin+ν+d (−z) so
Z
0
∞
εd εn+ν e−βε+βµ
s1
s1
dε = − n+ν+d+1 (n + ν + d)!Lin+ν+d (−z) = n+ν+d+1 In+ν+d (βµ),
−βε+βµ
2
(1 + e
)
β
β
(C.6)
where z = exp(βµ). The bottom integral of Eq. (C.1) is now trivial to solve:
Z
0
∞
∂f0
g(ε)
dε =
∂ε
Z
0
So the quantity in Eq. (C.1) is finally
R∞
n ∂f0
0 g(ε)τ1 ε ∂ε dε
= Iˆ(d) [τ1 εn ] =
R∞
∂f0
g(ε)
dε
∂ε
0
∞
εd
e−βε+βµ
Id (βµ)
dε = d+1 .
−βε+βµ
2
(1 + e
)
β
s1
I
(βµ)
β n+ν+d+1 n+ν+d
Id (βµ)
β d+1
=
s1
n+ν
β
In+ν+d (βµ)
Id (βµ)
(C.7)
(C.8)
where the s1 is yet to be determined. As shown in Eq. (3.55) in the main text,
s1 = τtr
Iˆ(d) [ε]
Iˆ(d) [εν+1 ]
,
(C.9)
so we need to detemined the result of the integral operator acting on the energy to an
arbitrary power. This has actually been done already in the above analysis so we just show
the final result:
Id+i (βµ)
Iˆ(d) [εi ] = β −i
.
Id (βµ)
(C.10)
s1 is then simply
Id+1 (βµ)
Id (βµ)
τtr
I
(βµ)
β −(ν+1) d+ν+1
Id (βµ)
β −1
s1 =
= τtr β ν
Id+1 (βµ)
.
Id+ν+1 (βµ)
(C.11)
We are now in a position to express the temperature dependence of our desired quantity,
Id+1 (βµ) In+ν+d (βµ)
Iˆ(d) [τ1 εn ] = τtr β −n
.
Id (βµ) Id+ν+1 (βµ)
129
(C.12)
Appendix D
The polylogarithm function
The polylogarithm is a special function defined by the series
∞
X
zk
Lin (z) =
kn
(D.1)
k=1
with |z| ≤ 1. By analytic continuation, the polylogarithm can be defined over a larger range
of z. An important result, known as the inversion equation, is[158]
bnc
2
X
lnn−2k (z)
1
Li2k (−1),
Lin (−z) + (−1)n Lin (−1/z) = − lnn (z) + 2
n!
(n − 2k)!
(D.2)
k=1
whereb n2 c is the greatest integer contained in n/2. For the purposes of this article, we use
the inversion equation in the quasi-degenerate regime and obtain
n
Lin (−eTF /T ) ≈ −(TF /T )n /n! + 2
b2c
X
(TF /T )n−2k
k=1
(n − 2k)!
Li2k (−1).
(D.3)
To obtain this result we have used TF /T 1 such that Lin (1 − eTF /T ) ≈ Lin (−eTF /T ).
Also, Lin (−e−TF /T ) is small and therefore neglected. Note that the first term on the right
hand side is what we would obtain at T = 0. We ascertain These results are equivalent to
the Sommerfield expansion.
130
n
0
1
2
3
4
5
Lin (−1)
−π 2 /12
−7π 2 /720
Lin (1 − eTF /T )
−1
−TF /T
2
− 12 ( TTF )2 − π6
2
− 16 ( TTF )3 − π6 TTF
2
2
1 TF 4
( T ) − π12 ( TTF )2 − 7π
− 24
360
2
2
TF
1 TF 5
( T ) − π36 ( TTF )3 − 7π
− 120
360 T
In (βµ0 )
1
TF /T
2
( TTF )2 + π3
( TTF )3 + π 2 TTF
2
TF 4
( T ) + 2π 2 ( TTF )2 + 7π
15
2
2
TF 3
7π TF
( TTF )5 + 10π
3 ( T ) + 3 T
Table D.1: Using ν = 0. ζ(F ) = 2m∗ kB T(F ) /~2
131