Magnetic Field Reversal Effect in Inter-Landau Level Absorption,
and
Stimulated Spin-Flip Raman Scattering in n-Type InSb
by
Gervais F. Favrot, Jr.
B.A., Williams College (1970)
Submitted to the Department of Physics
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
September, 1998
© 1998 Gervais F. Favrot, Jr.
All rights reserved.
The author hereby grants to MIT permission to reproduce and to distribute
publicly paper and electronic copies of this thesis document in whole or in part.
Signature of Author
Department of Physics
September, 1998
Certified by
Don Heiman
Professor of Physics, Northeastern University
Supervisor
-Thesis
Certified by
Peter A. lif
Professor of Physics
Thesis Supery or
I
Accepted by
I
,
/
MASSACHUSETTS INSTITUTE
OCT 09108
LIBRARIES
Thom
. Greytak
Professf'r of Physics
Chairman, Departmental Committee on Graduate Students
Magnetic Field Reversal Effect in Inter-Landau Level Absorption,
and
Scattering in n-Type InSb
Raman
Stimulated Spin-Flip
by
Gervais F. Favrot, Jr.
Submitted to the Department of Physics in September 1998
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
ABSTRACT
This thesis is a theoretical and experimental study of inter-Landau level
Three main topics are considered: (i) The first
transitions in n-type InSb.
observation was made of a magnetic field reversal effect in a spin-conserving
(ii) Advancements were made in the
inter-Landau level absorption transition.
(iii) Experiments
theory of intraband absorption and scattering processes.
were carried out on the spin-flip Raman laser to examine crystal anisotropy.
An important feature of this research is the prediction and observation of a
level
inter-Landau
in
the
effect
reversal
field
magnetic
striking
magnetoabsorption in n-InSb. The magnitude of the absorption associated with
two transitions is observed to change by a factor of nearly 3 upon reversal of
The effect results from the inversion asymmetry of
magnetic field direction.
The transitions which show the effect
the tetrahedral zincblende structure.
are the double cyclotron resonance and the double cyclotron resonance combined
with spin-flip.
The reversal effect involves the interference between two matrix elements
which contribute to the absorption, one having tetrahedral symmetry, and the
The tetrahedral matrix element results from the
other spherical symmetry.
The isotropic matrix element is associated
inversion-asymmetry mechanism.
with the wave vector-dependent electric quadrupole EQ and magnetic dipole MD
The isotropic matrix element changes sign on reversal of q, and
absorption.
the one for inversion asymmetry changes sign on reversal of B, so the
interference term changes sign on reversal of either q or B.
Stimulated spin-flip Raman (SFR) scattering has been studied in applied
For 10-gm pumping, the SFR laser output is
magnetic fields up to 18 T.
dominated by structure associated with linear intraband absorption and shows
significant anisotropy with regard to crystal orientation in the applied
Calculations have been made for the transition matrix elements and
field.
intraconduction-band
anisotropic
the
for
coefficients
absorption
transitions
warping-induced
The theory includes
magneto-optical transitions.
model.
Brown
and
and the effects of band nonparabolicity in the Pidgeon
Reasonable agreement between theory and experiment is obtained.
Thesis Supervisor: D. Heiman, Professor of Physics, Northeastern University
Thesis Supervisor: Peter A. Wolff, Professor of Physics
TABLE OF CONTENTS
LIST OF TABLES
3
5
7
11
1. INTRODUCTION
13
2. THEORY OF INTER-LANDAU LEVEL TRANSITIONS
2.1 Overview
2.2 Formulation of the Effective Mass Theory for Magnetoabsorption
2.3 Calculation of Magneto-optical Absorption Coefficients
17
22
ABSTRACT
TABLE OF CONTENTS
LIST OF FIGURES
2.3.1 Magnetoabsorption in the Pidgeon and Brown Model
2.3.2 Decoupled Model: Semiquantitative Treatment of Magnetoabsorption
37
42
68
3. MAGNETOABSORPTION AND FIELD-REVERSAL EFFECT IN n-InSb
3.1 Background
3.2 Experimental Conditions
3.3 Results
3.4 Discussion
4. SPIN-FLIP RAMAN LASER
4.1 Introduction
Theory of Inter-Landau Light Scattering
4.2.1 Scattering Cross Section
4.2.2 Scattering in the Decoupled Representation
4.2.3 Scattering in the Pidgeon and Brown Model
4.3 Experimental Background
4.4 Experimental Conditions
4.5 Results and Discussion
4.2
89
90
92
98
112
133
137
139
140
150
166
168
170
5. CONCLUSION
181
6. REFERENCES
187
APPENDICES
A. Extended Pidgeon and Brown Hamiltonian
B. Irreducible Spherical Tensors
C. Rotation of the PB Hamiltonian
D. Kane and Yafet Models
E. Magneto-optical Matrix Elements for Scattering and Absorption
F. Matrix Elements of exp(iq.r) for MD and EQ Transitions
G. Decoupling of the PB Hamiltonian
H. Effective Scattering Operator to Order k2
ACKNOWLEDGEMENTS
195
221
229
263
275
279
281
293
317
LIST OF FIGURES
2.1
2.2
Energy bands and Landau levels for InSb.
Inversion-asymmetry function F (a,3,y): (a) Magnitude versus a and
1, and
2.3
2.4
(b) 18F0 versus angle in planes of high symmetry.
Inversion-asymmetry function F (a,3,y): (a) Magnitude versus a and
0, and (b) j 16F1 versus angle in planes of high symmetry.
Inversion-asymmetry function F2(a,13,y): (a) Magnitude versus a and
0, and (b) 18F 2
2.5
versus angle in planes of high symmetry.
Inversion-asymmetry function F (a,3,y): (a) Magnitude versus a and
1, and
(b) 14F 3 1 versus angle in planes of high symmetry.
Warping function W0 (a,1,y): (a) Magnitude versus a and
13,
and (b)
13,
and (b)
of high symmetry.
(a) Magnitude versus a and
13,
and (b)
of high symmetry.
(a) Magnitude versus a and
13,
and (b)
18W 3 versus angle in planes of high symmetry.
2.10 Warping function W4 (a,3,y): (a) Magnitude versus a and
13,
and (b)
2.6
SWoI versus angle in planes of high symmetry.
2.7
2.8
2.9
Warping function W (a,c,y): (a) Magnitude versus a and
18W I versus angle in planes
Warping function W2 (a,1,y):
18W 2 I versus angle in planes
Warping function W3(a,x,y):
18W 4 1 versus angle in planes of high symmetry.
2.11 oc(a ) ED cyclotron resonance, peak absorption versus magnetic
field for n-InSb. y(HWHM)=15 cm -1, and B 11[111], [110] and [001].
2.12 oc+() ED combined resonance, peak absorption versus magnetic
field for n-InSb. y(HWHM)=15 cm -1, and B 11[111], [110] and [001].
2.13 as(r) MD (with some EQ) spin resonance, peak absorption versus
magnetic field for n-InSb, with B II [111], [110] and [001].
2.14 2oc(oL) EQ resonance, peak absorption versus magnetic field for
n-InSb, with y(HWHM)=15 cm - 1. Theoretical curves are given for
the decoupled approximation and the spherical PB approximation.
2.15 2 oc+co (t) EQ/MD resonance, peak absorption versus magnetic field
cs1
for n-InSb, with y(HWHM)=15 cm - 1. The dashed curve is from the
decoupled model, neglecting g" and y0 .
2.16 2oc(it) ED absorption due to inversion asymmetry parameters C' and
G, matched to 8 =65 a.u.; peak absorption versus magnetic field
for n-InSb.
Absorption is for B II [001] and is approximately
proportional to I F 2.
2.17 2Oc(a_) ED absorption due to inversion asymmetry parameters C' and
G, matched to 68 =65 a.u.; peak absorption versus magnetic field
for n-InSb.
Absorption
proportional to jF3
is for B ll [111]
56
and is approximately
57
2.
2.18 2oc(a+) ED absorption due to inversion asymmetry parameters C' and
G, matched to 68=65 a.u.; peak absorption versus magnetic field
for n-InSb.
Absorption is
for B II [110]
and is
approximately
proportional to IF1 2.
58
2.19 2oc+o)s()
ED absorption due to inversion asymmetry parameters C'
and G, matched to 8 =65 a.u.; peak absorption versus magnetic
field
for
n-InSb.
Absorption
is
for
B0ll [110]
and
is
approximately proportional to IF 1 2
2.20 2oc+o (a ) ED absorption due to inversion asymmetry parameters C'
and G, matched to 65=65 a.u.; peak absorption versus magnetic
field for n-InSb.
Absorption is for B 11[001]
and is
approximately proportional to F2 2.
2.21
59
60
2coc+Cos(a+)
ED absorption due to inversion asymmetry parameters C'
and G, matched to 8 0=65 a.u.; peak absorption versus magnetic
field for n-InSb.
Absorption is for Boll [111] and is
approximately proportional to IF0 2.
2.22 os() ED absorption due to inversion asymmetry parameters C' and
G, matched to 8 =65 a.u.; peak absorption versus magnetic field
for n-InSb.
Absorption is for B II [110] and is approximately
proportional to IF12.
2.23 s( _) ED absorption due to inversion asymmetry parameters C' and
G, matched to 8 0=65 a.u.; peak absorption versus magnetic field
for n-InSb.
Absorption is for B ll [111] and is approximately
proportional to IF0 2
61
62
63
2.24
s(a+) ED absorption due to inversion asymmetry parameters C' and
G, matched to 8 0=65 a.u.; peak absorption versus magnetic field
for n-InSb.
Absorption is for B II [001] and is approximately
proportional to IF* 2.
64
2.25 30c(r) ED absorption due to warping parameter I = 0.55; peak
absorption versus magnetic field for n-InSb.
Absorption is for
B 11[111] and is approximately proportional to W 2.
65
2.26 3cc(_) ED absorption due to warping parameter I = 0.55; peak
absorption versus magnetic field for n-InSb.
Absorption is for
B 11[001] and is approximately proportional to W 12.
2.27 3cc(a+) ED absorption due to warping parameter i = 0.55; peak
absorption versus magnetic field for n-InSb.
Absorption is for
B II [110] and is approximately proportional to W2 2.
66
3.1
Experimental setup for inter-Landau level absorption spectroscopy.
93
3.2
Light
3.3
3.4
10.6 pm for n-InSb. B II [111], q II [T10] and T = 25 K.
Fan diagram of intra-conduction-band transition energies for n-InSb.
Light transmission (detector output) versus magnetic field at
10.6 pm for n-InSb. B II [110], q II [T10] and T = 77 K.
3.5
3.6
3.7
3.8
3.9
transmission
(detector
output)
versus
magnetic
field
at
Magnetoabsorption versus B at 10.6 pm for n-InSb with B01 [111],
q II [110] and T = 25 K. (a) it polarization; (b) a 1 polarization.
Magnetoabsorption versus B at 10.6 gm for n-InSb with B II [110],
q II [T10] and T - 77 K. (a) 7t polarization; (b) L polarization.
Magnetoabsorption versus B at 10.6 p.m for n-InSb with B 011[T10],
q II [111] and T = 25 K. (a) it polarization; (b) a 1 polarization.
Magnetoabsorption versus B at 10.6 jim for n-InSb with B II [001],
q II [T10] and T = 25 K. (a) 7t polarization; (b) _L polarization.
A
Real and imaginary parts of F 3 + F versus 0 about nlII [T10].
A
A
'A
Zero angle corresponds to
= z,
II [110].
Dashed line is
and imaginary
Zero angle corresponds to
is F + 0.85F .
1
104
105
108
109
111
parts of F + F
A3
3
99
100
124
imaginary part of F3 + 0.85F 1.
3.10 Real
67
versus 0, for il11[111].
A
Dashed line
II [110] ,
II [I12].
1
125
A
3.11 Real and imaginary parts of F versus 0 about rl II [T10].
A
A
A
angle corresponds to
= z,
II [110].
A
3.12 Real and imaginary parts of F versus 0 about l 11[111].
A
A
angle corresponds to
II [110] ,
II [112].
1
Zero
126
Zero
127
A
3.13 Absorption versus 0 about T1 II [T10] for 20c( 1,_) transition. Zero
C
A
A
A
angle corresponds to
= z,
II [110].
A
3.14 Absorption versus 0 about ri II [111] for 2 0c(o L) transition. Zero
C
A
A
angle corresponds to
II [110] ,
II [112].
A
3.15 Absorption versus E about 1 II [T10] for 2o +so (rt) tramsition.
A
A
C
A
4.2
4.3
4.4
129
S
Zero angle correspond s to 5 = z,
11[110].
A
3.16 Absorption versus ( about ] II [111] for 2c+s(x)
A
,
A
Zero angle correspond s to
II [110] , ( 11[112].
4.1
128
130
tra nsition.
131
Raman weight factor versus B for o , 2o c , and 2 c+ s transitions
at kz=0 in n-InSb, for incident photon energy E 1 =117 meV.
159
Raman weight factor versus B for c , 2 c , and 2c +o s transitions
at kz=0 in n-InSb, for incident photon energy E 1 =232 meV.
160
20
Raman weight factor versus k for o and
transitions
z
C
C
B =35 kG in n-InSb, for incident photon energy E 1 =117 meV.
Relative
Stokes
output power
versus
applied
magnetic
at
161
field for
InSb spin-flip Raman laser.
Threshold pump intensity versus applied magnetic field for InSb
spin-flip Raman laser.
Low magnetic-field cutoff for stimulated SFR scattering versus
pump photon energy for n-InSb. Also shown is the field-dependent
energy gap calculated from the PB model.
Relative Stokes output power versus applied magnetic field in InSb
spin-flip Raman laser.
170
Magnetoabsorption coefficient [uc(B o) - o(0)] versus B for n-InSb.
176
4.9 Stokes frequency shift versus applied magnetic field B II [110].
4.10 Relative Stokes output power versus B 0 11[110] for 4.867 gm pumping.
178
4.5
4.6
4.7
4.8
172
173
175
179
LIST OF TABLES
2.1
2.2
Effective velocity operators for axial model ED transitions.
Transitions due to H(a ).
74
76
2.3
Transitions due to H(yo).
Magnitudes of warping functions W (a,p,y) along directions of high
77
symmetry.
77
2.4
2.5
Operators
transitions.
causing
first-order
inversion-asymmetry-induced
79
of
inversion-asymmetry
functions
along
F (a,,)
2.6
Magnitudes
2.7
directions of high symmetry.
Effective velocity operators for axial model MD/EQ transitions.
Matrix elements and peak absorptions for Voigt transitions.
2.8
3.1
4.1
ED
Matrix elements of the velocity operator
transitions in the decoupled model.
for MD/EQ
Electronic transitions in Landau level Raman scattering
80
86
87
and ED
118
146
1. INTRODUCTION
magneto-optical
This thesis explores
single-crystal
This material
n-type InSb.
is
which
semiconductor
the
among
electrons
of conduction
effects
in
is a narrow-gap III-V compound
most
extensively
studied
narrow-gap
It is the classic example of a narrow gap semiconductor with
semiconductors.
direct energy gap at k = 0.
The energy gap is between an s-like conduction
This greatly simplifies
band (cb) and a spin-split p-like valence band (vb).
of effective
the application
mass theory to the dynamics
of electrons
and
The narrowness of the gap causes a
holes, and to magneto-optical processes.
small effective mass and large g-factor, resulting in large cyclotron and spin
resonance frequencies in comparison with wider-gap materials.
has been extremely
Magneto-optics
determining
characteristic
allow
studies
absorption
Interband
semiconductors.
of
parameters
for
useful
determination of the energy gap, and intraband absorption measurements allow
determination
of
magnetic field B
levels
associated
field.
the
effective
masses
and
g-factors.
Application
of
a
causes the conduction and valence bands to split into Landau
with
quantized
orbital
and
spin
motion in
the
magnetic
The density of states diverges at the energy minima associated with
these levels.
intraband
This
absorption.
causes easily
observable
peaks in both
interband
and
Pioneering work on semiconductor magneto-optics was
done by groups led by B. Lax at MIT's Lincoln Lab and C. Kittel at U.C.
Berkeley.
Many useful reviews of magneto-optics have been given. [Lax 60, 67,
McCombe 75, Zawadzki 79, Pidgeon 80, Weiler 81,
Seiler 92].
Kim 89a, Rashba 91
and
Work on free-carrier magneto-Raman scattering by Wolff, Yafet, Wright et
al.,
and
Makarov
identified
three
[Wolff 66,
inter-Landau-level
occur in n-type InSb at k
B.
Yafet 66,
Kelley 66, Wright 68,
Raman
= 0, where k
scattering
processes
Makarov 68]
which
could
is the momentum of the electron along
These were the double cyclotron, spin-flip double cyclotron and spin-flip
processes at frequencies 2o c , 20c+ co,
and
An = 2 processes, without and with spin flip.
orbital
quantum
frequencies.
number,
and
cc
and
os
The first two of these are
s .
The symbol n is the Landau or
are
the
cyclotron
and
spin-flip
The last scattering process is pure spin flip, with An = 0, and
shows a much narrower line width than the first two which are broadened
because
the
spin precession
is
less
affected
by collisions.
The
spin-flip
Raman (SFR) process was predicted to have the largest cross section, and SFR
scattering
was soon
experiments.
The
observed in
stimulated
both spontaneous
process
was
and
recognized
stimulated
for
applications in the form of a magnetically tunable IR laser.
scattering
having
device
The device is
known as the SFR laser.
A large portion of the experimental and theoretical work in this thesis
is devoted
to obtaining a more precise
magneto-optics.
description of free-carrier
intraband
The theoretical work attempts to unify and give some new
perspective to previous work.
[Rashba 61a, 61b, Pidgeon 68, 69, Ohmura 68,
Weiler 76, 78, Zawadzki 76, Trebin 79, Gopalan 85, Wlasak 86, La Rocca 88a,b].
The experimental work resolves a significant problem concerning the 20
2oc+Oc
absorptions.
The formalism of Lipari
c
and
and Baldereschi [Baldereschi
73, 74] has been applied to characterization of the angular dependence of the
magnetoabsorption which was first determined for general Bo by Rashba, Sheka,
and Zaslavskaya [Rashba 61a, Sheka 69]. The Pidgeon and Brown (PB) model has
been used to find the magnetic field dependence of the inversion asymmetry and
warping induced absorptions as well as isotropic absorptions which are caused
by wavevector-dependent
mechanisms.
magnetic dipole (MD) and electric quadrupole (EQ)
The decoupled theory for the conduction band has been used to
obtain the limiting behavior of the absorptions at low fields, and a better
understanding of the physical nature of the absorptions.
Chapter
2,
inter-Landau-level
which
follows
magneto-optical
perspective of the PB model
perspective.
It
is
introduction,
transitions
presents
from
the
the
theory
coupled
and also from the decoupled
of
band
conduction band
The latter is the more useful one for understanding the physics.
analogous
to
the
wavefunction is treated
operator.
this
Pauli
model
for
Dirac
electrons
in
which
the
as a two-component spinor and R is a 2x2 matrix
Here we show that the EQ matrix elements for the 2ec(a)
absorption
and the EQ/MD matrix element for the 2c+ os(rt) absorption are of sufficient
magnitude to interfere with the ED contribution due to inversion asymmetry and
produce and observable effect upon reversal of B
or q.
These effects are
analogous to the one observed in the spin resonance by Dobrowolska et al.
(1983).
This theoretical analysis suggests that the B-reversal and q-reversal
effects should be observable in the 2oc(ol) and
as in the Os(Ir) absorption.
2
oc+os(7) absorptions, as well
This is shown to be related to the way in which
the 20 c , 2oc+o s and Os scattering processes occur together at kB = 0.
Chapter
3 gives a description of the experiment
which was conducted
recently at the MIT Francis Bitter Magnet Laboratory to observe the B-reversal
effect in the 20 c
and
2 0c+s
absorptions.
compared with the theory given in Chapter 2.
The results are described
and
Chapter
4
presents
the
results
of
the
motivated the investigation of the magnetic
Chapters
cross
SFR
laser
experiment
which
field reversal effect described in
2 and 3, and includes some new theoretical calculations of Raman
sections in both the PB and decoupled models.
It demonstrates the
importance of intraband absorption in the output of the SFR laser and shows
the dependence of this effect on the crystal orientation.
My work on SFR
scattering was carried out at the Francis Bitter National Magnet Laboratory.
Previous work was conducted by Aggarwal, Weiler and Lax [Aggarwal 71a&b,
Weiler 74].
The 10.6 gm pumped SFR laser had been studied at fields up to
10.4 T using a high-power pulsed TEA CO 2 laser as a pump. The Magnet Lab had
the capability of 18 T in a 2-inch bore Bitter solenoid, and there was great
interest in studying the SFR output at fields above 10 T.
The main interest
was in determining whether electron-LO phonon coupling would: (i) increase the
threshold
for
stimulated
increase
the
Stokes
SFR
scattered
scattering
output
by
at
increasing
saturation
the
by
linewidth;
(ii)
increasing
the
relaxation of the upper level; or (iii) change the spin resonance frequency by
polaron
coupling
at
fields
near
LO-phonon energies are equal.
12.5 T
where
the
spin
splitting
and
the
Other phenomena of interest were one- and
two-photon absorption processes which were known or expected to affect the
output, and the effect of large B
at near-bandgap energies.
crystal
orientation
became
on the resonant enhancement SFR scattering
The linear magnetoabsorption and its dependence on
the
main
interest
in
my
work
when
the
electron-spin-LO-phonon coupling was not observed [Favrot 75, 76a].
The last chapter presents the conclusions and recommendations for future
work.
2. THEORY OF INTER-LANDAU LEVEL TRANSITIONS
2.1
Overview
The
starting
absorption
in
point
InSb
[Pidgeon 66,69].
is
which
the
I
use
8x8
for
the
theory
Pidgeon
and
Brown
of
magneto-optical
(PB)
Hamiltonian
This Hamiltonian is a generalization of Kane's Hamiltonian
[Kane 57,66] to the case of finite magnetic fields, and contains terms up to
order k2 in the k-p effective mass theory.
energy bands, wavefunctions,
It permits the calculation of
and optical transition matrix elements involving
the lowest conduction band and three highest valence bands.
used to calculate interband absorption.
[Weiler 78]
to
explain
It was originally
This model was extended by Weiler
experimental
results
for
intraband magneto-optical
absorption [Weiler 74, Favrot 76] arising from inversion-asymmetry and warping
effects.
The
present
theory
uses
this
model
to
calculate
the
absolute
Also, in order to obtain simple results in the limit of small
absorption.
applied magnetic fields and small electron momenta, the reduction of the PB
model to decoupled models for the conduction and valence bands is considered,
following Rashba and Sheka [Rashba 61b], and Braun and Rtissler [Braun 85].
The PB model is significantly more accurate than the decoupled model for large
magnetic
fields B0 and wavevectors
k . However,
the decoupled
model
is
simpler, and easier to interpret physically.
Figure
2.1
shows
calculated energy
bands and Landau
levels for the
conduction band (cb) and valence bands (hh, lh and sb) which are obtained by
diagonalization of the 8x8 Kane and PB Hamiltonians.
to denote the heavy hole, light hole and spin
Here I use hh, lh and sb
split-off bands, respectively.
Kane model
PB model
0.5
a,
z
wU
-0.5
-0.2
-0.2
-0.1
k/k[X]
Fig. 2.1
Kane model energy
Figure 1 of [Weiler 78].
bands and PB
levels,
after
The Kane model calculation is for an average
direction of k (or neglects warping).
10 T.
model Landau
The PB model is for B 11 [111] at
The PB level separations from the band edges have been scaled up by
a factor of 5, for better visibility.
Band parameters of [Goodwin 83].
The Kane model bands are shown for wave vectors out to 20% of the distance to
The PB
the X point of the Brillouin Zone, in a spatially averageddirection.
model
Landau
k = 0
at
levels
are
for
a field of
10 T
the
in
[111]
crystallographic direction, with the level spacings exaggerated by a factor of
5 for better visibility.
The present chapter deals only with intra-cb Landau
For the conduction band the a-set and b-set levels may be
level transitions.
regarded simply as spin up and spin down, and the optical transitions between
Landau
transitions
which
valence
the
at the aC(O) level
thesis originate
The
demonstrate
band
levels
are
The two new
and spin-flip.
levels may be both spin conserving
described
effect
field-reversal
in
this
and end at the aC(2) and bc(2) levels.
important
the
for
calculation
of
scattering
amplitudes, and will be described in Chapter 4.
A general formulation of the PB model based on irreducible spherical
tensors
has
Lipari 70,
been
developed
who
Baldereschi 73,74]
Hamiltonian [Luttinger 56].
the large
spherical
coordinate
frame
so
the
applied
lines
Lipari
of
this
technique
Baldereschi,
and
to
the
Luttinger
This provides the best way to take advantage of
component
that
along
the
of Hef and to express Heff in
quantization
direction
coincides with the direction of the magnetic field B
r
.
a rotated
of the basis
states
The usefulness of this
for the B = 0 case has been discussed by Johnson [Johnson 84].
For finite B0
this makes it easy to obtain eigenstates for the spherical (or axial) part of
the
Hamiltonian.
explanation
of the
The
spherical
terms in
tensor
the anisotropic
coordinate frame. [Rashba 61a, La Rocca 88a].
formulation
part
provides
of Hef in an
a
simple
arbitrary
The
quantization
[Luttinger 55]
direction
of the Luttinger-Kohn
can be chosen arbitrarily only in cases where
transform like irreducible
representations
crystal point group.
functions
these functions
(irreps) of the full
R(3), or like irreps of R(3) multiplied by some
and
(LK) basis
rotation group
1-dimensional
irrep of the
This occurs in the cases of the cubic point groups Td, 0
Oh , which possess a particularly high level of symmetry.
When this
condition applies, Hff usually contains a large spherical component so that
the
anisotropy
may
be regarded
as a
In
perturbation.
the case
of the
conduction band in InSb the anisotropy occurs in fourth order in momentum in
the decoupled Hamiltonian, so perturbation theory works particularly well.
In
this
development
elementary particle-like
the
perspective
of
the
theory
PB
lattice
disappears
is
useful
to
notice
and holes near k = 0.
nature of electrons
discrete
it
and
is replaced
by
the
In this
an effective
Hamiltonian, the EMA Hamiltonian, which reflects the point-group symmetry but
not the discrete translational
formulation
symmetry of the lattice.
was devised by
of Hff which
I use the invariant
[Luttinger 56]
for the case of
germanium, expanded by [Bir 74], applied to the PB model by [Trebin 79], and
further developed by [R6ssler 84] and [Braun 85].
electric
dipole
optical
[Dresselhaus 65],
perturbations,
[Suzuki 74],
I
follow
[Zawadzki 76],
In treating the effect of
[Rashba 61a],
[Grisar 78],
and
[Stickler 62],
[Trebin 79] in
replacing k in Hef(k) by k + (e/c)A l and expanding to first order in the
potential A1 , as is done in the Dirac and Pauli
perturbing
electromagnetic
theories.
This is equivalent to setting the perturbing Hamiltonian equal to
(e/c)A . V H
S
, or to (e/c)A ,v , with
k eff
v =V H
k eff
(i/h)
r]
= (ih)[H , r] .(2.1.1)
eff
The importance of this formulation for the PB model, as well as the fact that
interband and intraband transition matrix elements are identical in form when
obtained this way in the PB model, was first pointed out by [Zawadzki 76].
A
of
calculation
the
the
in
magnetoabsorption
anisotropy-induced
conduction band includes a generalization of the calculations of [Rashba 61]
transitions [Sheka 69].
and [Gopalan 85] to the case of the warping-induced
The high-field treatment using the 8x8 PB model for both the warping- and the
transitions
inversion-asymmetry-induced
improvement
a
represents
and
simplification
of the theory derived by [Weiler 78 & 82],
[Zawadzki 76] and
The selection rules which arise when B 11[001], [111] and [110]
[Wlasak 86].
are explained by the symmetry arguments used by [Suzuki 74] and [Trebin 79]
for
the valence band magneto-optical
contained
are
rules
selection
clearly
quite
These
transitions in Ge and InSb.
in
of
mathematics
the
the
irreducible spherical tensor formulation of Hef .
In this treatment of the 8x8 PB model Hamiltonian I extend the invariant
symmetry, which were
to
all
obtained by [Weiler 78].
formalism
tensor
spherical
irreducible
applied
to all twenty of the terms in Hef
t
of [Trebin 79]
formulation
of
these
terms,
allowed by
(See Appendix C.)
The
has
been
(Oh)
and
wavefunctions
it is
of Lipari
and Baldereschi
including
the
warping
inversion-asymmetry (Td) terms.
In order to calculate
necessary
to diagonalize
numerical energy
the axial
levels and
part of the PB
described in Chapter 4 and Appendix A.
Hamiltonian,
which is
2.2
Formulation of the Effective Mass Theory for Magnetoabsorption
In this section the PB Hamiltonian
is formulated
using the irreducible
tensor methods of Lipari and Baldereschi. [Lipari 70, Baldereschi 73&74]
addition
to obtaining
an
accurate
calculation
of the
In
absorption, the results
give a clear understanding of the selection rules and orientation dependence
of the intraconduction band magneto-optical transitions in InSb.
The crystal
orientation dependence is contained in nine angular functions that are shown
in polar plots.
coefficients
The present formulation will be used to obtain absorption
which
will
later
be
compared
with
previous
[Favrot 76]
and
current experimental results on InSb.
Inversion-asymmetryzincblende-type
and
semiconductors
warping-induced
have
been
magneto-optical
the
subject
transitions
of experimental
theoretical studies over the past 40 years. [Seiler 92, Rashba 91]
relative
La Rocca 88a&b]
to the crystallographic x, y and z directions.
and
A general
treatment of the problem requires use of a coordinate frame 4r1
rotated
in
which is
[Rashba 61,
The zeta direction is chosen to coincide with the direction
of the applied magnetic field B .
The LK basis states [Trebin 79] may be
quantized along zeta because each degenerate set of LK basis states transforms
with
respect
to the point
group Td like basis states of the full rotation
group R(3), and not just the group Td. [Trebin 79]
operators
of the PB model
to form sets
which
spherical
tensors
with
to
reflections.
vector
respect
The kinetic momentum operator k
potential,
irreducible
[Lipari 70]
can
be
used
in
polynomials
This causes the matrix
transform
arbitrary
R(3)
like irreducible
rotations
and
p + (e/c)A, where A is the
of different
spherical tensor operators of arbitrary rank.
orders
to
form
The PB Hamiltonian
Heft contains
products
of
involving
parts
matrix
operators
and
operators
formed from the components of k and B [Trebin 79].
The effective Hamiltonian may be divided into parts which have spherical
octahedral
R(3),
Oh
and antisymmetric
The latter terms are antisymmetric
terms
contain
irreducible
combinations
tensors
of
of
with respect to inversion.
irreducible
zero,
rank
tetrahedral Td symmetry
like
tensor
k 2,
B-J
[Trebin 79].
The spherical
operators
or
K(2)J
which
(2 )
form
[Lipari 70,
The octahedral (warping) terms in the PB model may be
Baldereschi 73,74].
made irreducible of rank 4, as shown for the valence band block of Hef by
The antisymmetric (inversion-asymmetry) terms in the
Lipari and Baldereschi.
PB model are irreducible of rank 3.
When
the direction
of B
0
the terms in Heff which were
is chosen,
spherical become axial, with symmetry group C h.
A
This part of Hef commutes
A
with the operator N1F + F , where N is the Landau level number operator for a
spinless electron,
1F
is the unit 8x8 matrix, and F
whose elements are the m
is the diagonal matrix
values of the LK basis states. [Trebin 79]
The
eigenfunctions of the axial part of Hef have a simple form and conserve the
A
total angular momentum along
Conservation
Yafet,
of this
A
(spin plus orbital) associated with N1F + F .
in the axial
quantity
approximation
was
discussed by
[Yafet 73] by Trebin et al. [Trebin 79] and recently by Rashba and
Sheka. [Rashba 91]
[Luttinger 56,
(The valence band case had been considered by Luttinger
Suzuki 74].)
eigenvalue of (N -
21F
I
use
X (and
sometimes
1) to
denote
the
+ F , so that X = 0 for the lowest energy level in the
cb, and I call this the 'axial quantum number'.
x4 +
cubic harmonic
like the irreducible
terms in H ef transform
The fourth-rank octahedral
- (3/5)r4 , and have the irreducible tensor form
4+
[Baldereschi 73]
T(4)
C
2
T+4)]
4)+T(4)
4
5
-4
(2.2.1)
0
The third-rank antisymmetric terms transform like the function xyz and have
the form
T23)-
TA( 3)
7--23)
(2.2.2)
tetrahedral
antisymmetric
The
considered
by
irreducible
of
components.
Baldereschi
and
Lipari
i = 3,
rank
invariants
can
and
be
in
were
not
Baldereschi 73]
are
which
Hef,
[Lipari 70,
expressed
in
irreducible
tensor
Using Weiler's designation of the band parameters [Weiler 78],
and with t = v3 T of Trebin et al. [Trebin 79] some of these are
+ c.p.]
H(G) = - iG[tx {k,kz
= -Y
H(N) = 2
--6
G[t(')® K (2)]A(3)
(2.2.3)
N [(k - k)o t + c.p.]
2
X
y
N [K(2) t
2
H(C) = (C/~[k({Jx,
X
Jy2
= - (2C/V3)[K'(® j(3)
zz
)(2)](3)
A
(2.2.4)
z + c.p.]
(3)
A
(2.2.5)
H(G'), H(C') and H(N3) are similar to H(G).
Rotation of the antisymmetric terms may be carried out with the Cartesian
form of the invariants by the Euler angle transformation method of Rashba,
La Rocca's
[Rashba 61] Gopalan, [Gopalan 85], and La Rocca [La Rocca 88a&b].
description is the easiest to follow, but neglects to make the connection of
the
antisymmetric
coefficients
with
(Ic;gv)
symmetric
(KXlR)
coefficients
introduced by Rashba and Sheka, [Rashba 61] but expressible in a simpler form.
These coefficients
are
expressed
in terms
angles, [La Rocca 88a] F through F3.
of four
functions
of the Euler
Polar plots of the magnitudes of these
functions are given in Figs. 2.2 - 2.5 at the end of this section.
Rotation
of the octahedral invariants is accomplished in a similar fashion, as was done
by Sheka and Zaslavskaya [Sheka 69] and recently by Obuchowicz and Wlasak.
[Obuchowicz 91]
The symmetrical coefficients (kgXv)
which are obtained for
this case may be expressed in terms of five functions of the Euler angles
which are denoted W0 through W4 .
Polar plots of the magnitudes of these
The
functions are given in Figs. 2.6 - 2.10 at the end of this section.
rotation of Hef may be simply expressed when the irreducible spherical tensor
form is used.
The spherically symmetric terms have the same form in all
rotated frames.
For the Oh case note that the f = 4 warping terms represented
(4
) contain only the m = 0 and ±4 spherical tensor components in the xyz
by TC
frame. The t = 3 antisymmetric Td terms represented by T(3) contain only the
m = ±2 components. If Heff is expressed in a rotated coordinate frame, all 9
of the t = 4 components will occur, in general,
components
will
occur.
The complete
Hamiltonian is given in Appendix C.
rotation
and all 7 of the
of all terms
The general expressions are
in
e=
3
the PB
T(C4)
m
TA( 3) =
D( 4C)((4,y) T 4)',
m
m
(2.2.6)
D(m 3A)(X, 3,y Tm( 3)
m
(2.2.7)
where
D(4c)
y)
m
2
[D 4,m
,)*
( 4 ) (a,
+ D(4) (
-4,m
,y)+
V
D( 4 )(a,,)*]
5
O,m
(2.2.8)
,
and
D ( 3 A)(ca, p' ,)
m
and Tm( 4)
frame,
- 1[D(3 ) (,,y)*
2
2,m
+ D(3 ) (a,,y)*] ,
and Tm( 3 ) ' represent the spherical tensor components in the rotated
i.e. with
components.
k+ defined
Here 4TI
as k
+ ik1 , and similarly
the
R(3).
representation
for
other vector
are the coordinates in the rotated frame, specified with
respect to the xyz frame by the Euler angles acpy.
are
(2.2.9)
-2,m
matrices
[Tinkham 64]
for
The matrices D(,)(a,P,y)
m'm
the eth representation of
The operator T(e)
raises the axial quantum number ? by m when B is
m
A
along
.
The
functions
Dm( 3A)(a,p,y)
are
proportional
to
the
angular
functions
F(xP,y) defined by La Rocca et al., and originally derived by Rashba and
Sheka
[Rashba 61] for y = 0.
The functions D(m4C)(3,P,y) are the analogous
angular functions for the warping terms in Hef , and are proportional to the
, for m # 0, and to (W - 3/5) for m = 0.
functions W
The y-dependence in
both cases is simply exp(imy).
The zeros of the angular functions which occur for specific
specified
by
and
z4 - (3/5)r4
y4+
4+
ox
J3,
with
rotations about the [001]
are
easily
understood.
Oh symmetry
is
The
symmetric
A
directions,
cubic
with respect
directions, C3 rotations about the [111]
and C2 rotations about the [110] directions.
to
C4
directions,
Coordinate rotations by an angle
y about zeta cause the mt h component of the cubic harmonic in the r1j
be multiplied by exp(imy).
harmonic
frame to
For this term to be invariant with respect to C4
rotations, m can only be 0 or ±4.
Thus all of the angular functions Wm
A
associated with warping with m # 0 or ±4 must vanish when
is along [001] or
Similarly, the functions must vanish when m # 0 or +3
a similar direction.
when zeta is along a [111] direction, and they must vanish when m # 0, +2 or
A
±4 when
is along a [110] direction.
Similarly, the cubic harmonic xyz is
antisymmetric with respect to C4 rotations about [001], symmetric with respect
to C3 rotations about [111], and antisymmetric
about [110].
when
A
with respect to C2 rotations
This implies that the angular functions Fm must vanish if m # ±2
A
is along a [100] direction, if m # 0 or ±3 when
fl1 or ±3 when
direction, and if m
A
is along a [111]
is along a [110] direction.
The vanishing of the angular functions along the high-symmetry directions
is
exactly
what
is
required
to
produce
the
selection
rules
obtained
for
magneto-optical transitions by Trebin et al. [Trebin 79] when B is along those
directions.
(a)
(b)
90.
90"
90"
120'
135'
0.8
45'
0.8
~0.8
.4 0.8
180.
0.1' 180
0.8
0
....
225'
60'
0.8
.....
18
210
33 C
315'
240'
Fig. 2.2
300'
270'
270'
270'
(001)
(110)
(111)
Inversion-asymmetry
muthal angle ax and polar angle
function
B
F (a,3,y).
on 4.50 grid (stereo pair).
angle in planes of high symmetry.
F =
0
8
(a) Magnitude
i sin(2a)sin 2 3 cosp
vs.
(b) 18Fo
azivs.
(a)
(b)
90"
90"
90"
60'
120'
135*
0.8
0.8
45'
180'
0.8
0'
180'
33
210'
225'
315'
300
240
(001)
Fig. 2.3
270
2-70
270"
(111)
(110)
Inversion-asymmetry
muthal angle cc and polar angle
function
P
Fl(a,0,y).
(a) Magnitude
on 4.50 grid (stereo pair).
angle in planes of high symmetry.
F - ' e [cos(2a)sin(2)
1
16
+ i sin(2ca)sinp(3cos 2P - 1)]
vs.
(b) 116F1
azivs.
(a)
(b)
90"
90"
90"
120'
135'
8
45'
8
[11
0.4
8
180'
0.4
O'
60*
.
0.8
150
.
0.8
180'
4 0.8
O'
180'
210
225'
\..30
.
.33
315'
240*
Fig. 2.4
300
270'
270*
270'
(001)
(110)
(111)
Inversion-asymmetry
muthal angle ca and polar angle
function
3
F 2(x,,y).
(a) Magnitude
on 4.50 grid (stereo pair).
(b)
angle in planes of high symmetry.
F = ~ e2i[2cos(2x)cos(23) + i sin(2o)cosp(3cos
2
- 1)]
vs.
azi-
8Fz
vs.
(a)
(b)
90so
90
90
120'
135
..
180
60*
0.8
0.8
45'
0.8
0'
O'180'
180'
33
210'
225i
315
240'
(111)
(110)
(001)
Fig. 2.5
300
270
27
270'
Inversion-asymmetry
function
F (a,3,y).
(a) Magnitude
muthal angle a and polar angle 0 on 4.50 grid (stereo pair).
angle in planes of high symmetry.
F
3
=3 e3iY[cos(2)sin(2)
16
+ i sin(2a)sinp(1 + cos 2 3)]
vs.
azi-
(b) 14F 3 1 vs.
(a)
(b)
90'
90'
90'
120'
135*
60
45"
0.8
0.40.4
__.4
0.4
180
0*
..
..
I
5.......
315'
225'
8
0.4
"
'""
.
"'" '"1...3....
150
11
....
0.4
210
...
240'
Fig. 2.6
40.8
30'
" ......
330
-
300
270'
270'
270'
(001)
(110)
(111)
Warping function W (c,,y). (a) Magnitude vs. azimuthal angle a and
polar angle
P
on 4.50 grid (stereo pair).
(b)
IW
vs. angle in planes of
high symmetry.
Wo
=
(COS 4
0
+ sin 4 C)sin4p
+
coS 4
=
(cos4a + 3)sin 4 p
4
+
cos 4
(a)
(b)
90"
90"
90
60'
120'
0.8
08
45'
0.8
135"
[111]
0.
-
180'
30'
0
4
0.8
O'
.
-
150*
.4
.4
0.8
0.8
0
1801
0'
180
330'
210*
315'
225.
300'
240.
Fig. 2.7
270*
270*
270
(001)
(110)
(111)
Warping function W (cc,,y).
polar angle
[3
(a) Magnitude vs. azimuthal angle cc and
on 4.50 grid (stereo pair).
(b) 18W 1
vs. angle in planes of
high symmetry.
W =
eY[(cos4a sin2p + 7sin2p - 4)sinp cosp + i sin4a sin 3]
(a)
(b)
Fig. 2.8
270'
270'
(001)
(11o0)
Warping function W2(a,,y).
polar angle
3 on 4.50 grid (stereo pa
(111)
(a) Magnitude vs. azimuthal angle a and
(b) 18W)
vs. angle in planes of
high symmetry.
w2 =
16
e,2 iy(cos 4 (cos2
+ 1) + 7cos 2 p _-
)sin2
+
i (2sin4a sin2
cos]
(a)
(b)
9C
0.8
[111]
0.4
0.8
270'
(001)
Fig. 2.9
(110)
Warping function W (a,3,y).
polar angle
3
(111)
(a) Magnitude vs. azimuthal angle ca and
on 4.50 grid (stereo pair).
(b) 18W 3 1 vs. angle in planes
high symmetry.
33
32
e3iy [(cos4o(4 - sin2p) - 7sin 2 [)sinp cos3 + i (sin4a sinp(3cos 2 3
+
1))]
(a)
(b)
90'
0.4
0.8
n*
180'
,w
330*
-
225"
315'
(3001
270*
270)
(110)
(001)
Fig. 2.10 Warping function W4(c,3,y).
polar angle
13on
270
(111)
(a) Magnitude vs. azimuthal angle cc and
4.50 grid (stereo pair).
(b) 18W 4 I vs. angle in planes of
high symmetry.
w44 =
j e4iy (cos4a(sin4
64
+ 8cos 2 ) + 7sin 4j3) + i (4sin4a cos3(cos
2
+ 1))]
1]
2.3
Calculation of Magneto-optical Absorption Coefficients
In
this
section
I
compute
using the effective
transitions
the
mass
absorption
theory presented
coefficients
of
2.2.
in section
basic formulation follows that of Bassani et al. [Bassani 88]
several
The
The key feature
is the importance of some electric quadrupole (EQ) and magnetic dipole (MD)
These processes for the conduction
processes that were overlooked previously.
are
band
of
sufficient
asymmetry-induced
q-reversal
magnitude
to
with
interfere
the
inversion-
electric dipole (ED) absorption and produce B-reversal and
effects like the one observed
in spin resonance
at lower fields.
The two new transitions which show the interference effect are the 21c().
transition, which has an EQ component that is larger than the ED one, and the
0o (r) transition which has a mixed EQ-MD component that is smaller than
20+
the ED component.
Here, 2o c refers to the second harmonic of CR, and 2c+ os
is a combined resonance transition which changes both Landau orbital and spin
state.
The
ol
polarization
is for
the perpendicular
Voigt
configuration,
while it is the parallel Voigt configuration.
The relation between the absorption coefficient and the transition matrix
element
is
electrons
and
obtained
light.
by
considering
Following
the
optical
interaction
Bassani et al. [Bassani 88]
between
the
I express
the
interaction Hamiltonian as
(2.3.1)
H' = (eA /2c)(Ve - int + h.c.) ,
where
A 10 is
the
'magnitude'
of the vector
potential
which
is
real
and
positive, and
velocity.
V is an operator
(frequently nonhermitian)
with dimensions of
The vector potential of the radiation field is
A (r,t) = '[Al ei(q-r - ot) + c.c.] ,
(2.3.2)
A
A
with complex amplitude A1 - A We , and complex unit polarization vector e
AA
2
Note that e-e* = 1 , and A -A* = A 2
10
10
to the greatest magnitude of Al(r,t).
10
,
and that A
10
is not generally equal
The complex amplitudes of the electric
and magnetic fields are
E
10
=
iA
iA
C
iA
=
10
=i A e,
C
B
10
A
10
iqxAo =iAqxe .
10
10
(2.3.3)
The electronic transition rate R due to the radiation is given by the "Golden
Rule"
(eA/2c)2 1Vfo
R = 2
2
(hc - E ) .
(2.3.4)
The absorption coefficient, in the extreme quantum limit, is found from
oc = n ehR/I ,
(2.3.5)
where ne is the carrier density, and II is the radiation intensity given by
1
-
where
c2
2
cn A 0 (cgs units),
8where
is the refractive index. If we assume a Lorentzian lineshape
nI is the refractive index.
If we assume a Lorentzian lineshape
(2.3.6)
1
y
g()
+
(2.3.7)
2
where y is the half-width at half-maximum and the peak height is given by
g(0) = /liy, we find
po mlVfol 2 g(0 - ofo),
=
t()
h
(2.3.8)
n Oc
47tn e2
e
is the squared plasma frequency for free electrons.
where C 2
pO
Then
m
finally, the peak absorption is given by
1 o
a(max)= -p
mlVfo
hec n Oy
The
next
formalism
processes.
step in
2 .
computing
(2.3.9)
the
absorption
needed to calculate the matrix elements
is
to develop
the basic
for ED, EQ,
and MD
For this I start from the most common form of the interaction
Hamiltonian
(2.3.10)
H' = 2 e(A (r,t)-v + v-Al(r,t))
where v = V H
k eff
V =
where
. v,eiqr}
The operator
V is then given by
,
{ , } denotes the anticommutator.
(2.3.11)
The usual approach expands eiq 'r to
first order in q-r and separates the ED, MD, and EQ parts as follows.
A
A
V = e-v +
e v,q.r + . . .
(2.3.12)
(eq)
e- v + -(exq)-(vxr - rxv) +
(rv2) +
-[(vr)2)
where (rv)( 2) is the 2nd rank tensor product formed from the vectors r and v
according
to
the
prescription
(rv)(2) (r1)®(v))(2) [Edmonds 73]
and
it
is
A
e.q = 0 , which is valid if the magnetic birefringence is small.
assumed that
The
first
three
terms
correspond
to
the
ED,
MD,
and
EQ
processes,
A
respectively.
One sees that e -v corresponds to the ED part, since this can be
obtained
simply
produces
both
by
neglecting
isotropic
the
spatial
and anisotropic
magneto-optical transitions.
variation
of
A (r,t).
(with respect to crystal
This
part
orientation)
A
The term proportional to exq is the MD term, and
the one proportional to (eq) (2 ) is the EQ term.
For the case of unbound carriers in Landau levels localized only in the x
direction,
it
q-dependent
is
easier
part
only
to
work
directly
after computing
with
Eq.
the matrix
(2.3.11)
elements.
and
find
When
the
q is
perpendicular to B the first-order effect of eiq-r is found to be equivalent
to that of (qxk) zeq-r
(See Appendix F).
Two additional forms for the interaction Hamiltonian may occur.
The
first is
H' = eE.R ,
(2.3.13)
where R is an operator with dimensions of length.
spin-orbit interaction, where R = K kxa, and K
S
of h2/m2
2
S
An example is the ordinary
is a constant with dimensions
In this case
.
A
V = ie-R .
(2.3.14)
EQ and MD effects are obtained by replacing this by the expression
V
=
i -e. R,eiqr .
(2.3.15)
Finally, the simplest MD interaction has the form
H' = g*. BBI.J =
-*BI-J
.
(2.3.16)
Using this we obtain
V= ig*(qxe)J
where J is replaced by
=
-ig*e -(qxJ),
2
for the spin-!2 case.
(2.3.17)
2.3.1 Magnetoabsorption in the Pidgeon and Brown Model
The method of calculating
demonstrated through examples.
magnetoabsorption in the PB model will be
I will first consider the inversion-asymmetry
induced cyclotron harmonic transition 2 c( c),
transition 2oc + cos(r),
computing
other
and then the combined resonance
both at k = 0 . Prescriptions will also be given for
transitions.
The
resulting
curves
of
absorption
versus
magnetic field are presented at the end of the section.
For the 20(0 ) transition the initial state is the aC(O) conduction band
state, with axial quantum number
e=
with
2 .
components
The
transition
t = n + m
has
J
At = +2,
of the inversion-asymmetry
--
= 0.
2
and
The final state is aC(2)
results
from the
m = 3
Hamiltonian in the rotated coordinate
A
frame with B
function F .
along
B
along the
axis.
These terms are proportional to the angular
Notice that the incident
by
one
unit
of
h
Y photon lowers the angular momentum
in
the
axial
approximation,
so
the
inversion-asymmetry part of Hef must supply the plus three units required to
bring At to +2.
The perturbed component of the velocity
v 1- = 2V k+H1
(2.3.18)
is an operator with m=2 if Vk+ operates on the m=3 part of the perturbing
Hamiltonian H , since Vk+ removes a factor of k+.
We will assume
for simplicity that the transition
results entirely from
the part of the inversion-asymmetry Hamiltonian associated with the parameter
G.
The parameters C, C', N2 and N 3 may be treated similarly.
From Appendix C
we see that the m = 3 part of H(G) in the 4rTI
frame is
,
-iGf3t+k
HI(G) I = ~f
3 +~~
where
t+ is the basis
(2.3.19)
matrix
associated
with
the '+'
component
interband velocity (or momentum), in the cv block of Hf.
of the
To convert this
into an 8x8 matrix, one must replace it+ by
0
it+
0
(it ) t
0
0
0
0
0
0
= i
-(t
0
t+
0
0
0
0
0
(2.3.20)
where the dagger denotes hermitian conjugate. For v 1- we have
v
=-
4iGf3 t+k+ .
(2.3.21)
expressions apply for H(G') in the cs block, with G' ---
Identical
t+--- t .
We will use the single-group approximation which sets
G and
G' = G.
We next evaluate the transition matrix element
Vfo = (1//2)[(v)fo + i[v _,S ]fo] ,
which results from the three processes:
(1)
a 0
1-
ao
(e=O)
v
(2)
a
(E=O)
-
a
af
(e=2)
H
ar
(t=- 1)
-
a
(f=2)
(2.3.22)
H
(e=0)
Here,
v0
v
1
a
(3)
a ,
o-
af.
(e=3)
is
the
(e=2)
velocity
operator
obtained
from
the
axial
(unperturbed)
portion of Hef, which contains parts that are both constant and linear in k.
Processes (1), (2) and (3) are evaluated as
= <aflvl_ao>
(v_)f
(1)
r
(3)
0-fr
iro
r
0
C (H )fr,(vo_)r'o/(Ef- Er) ,
where the r sum is over the a-set states with
the
a-set
states
e=
with
3.
Note
that
e =-
1 and the r' sum is over
v0 ,
v1 _
and
HI
are
all
spin-conserving operators at k = 0 .
For evaluating these three terms a program which specializes in matrix
manipulations,
such as Matlab,
extremely useful.
(Clo
C2+1
where I use 't'
from The MathWorks
Inc.,
Natick, MA,
is
The unperturbed wave functions of definite t have the form
C3 0-
1
C65
,
c5,+
to denote 'transpose',
1
C4+
2
cs
C7+l) t
(2.3.23)
so that the result is a column vector.
The labeling of the c i coefficients is that of Pidgeon and Brown, which gives
odd numbers to the a-set states, but the ordering is by decreasing m
within
the
conduction
bands.
(c I C2 ),
valence
(c 3 c6 c 5 c4 ),
and
split-off
valence
(c8 c7 )
The matrix representation of the above state is just the column vector
of the coefficients without the spatial functions, i.e.
(C
C2
C3
C6
C4
C5
c8
(2.3.24)
c 7 )t
The k+ operators, which are simply related to the Landau level raising and
lowering operators, may be converted to matrix operators which depend on the
-value of the column vector which they act upon.
(The k
operator is simply
a constant associated with the column vector state, which is taken here to be
zero.)
The k+ operator is replaced by the matrix
k+ =
VT V-TT1 A+2 V+3
diag[ VNT Vf+Z
e+T VT2 ], (2.3.25)
where s=eB /hc=ec-2, where the magnetic length ec is the smallest quantized
cyclotron radius, and 'diag'
refers to the diagonal matrix with the enclosed
values ordered from top left to bottom right. Similarly we have
k = V~ diag[
V
T
VViT
iT
The ordering of operators is important.
on
a column
vector
of
specific
/Ai T
e]-2
.
(2.3.26)
If the operator k+t+ or k+J+ operates
e, one must increase
e by
1 in
the
specification of k+ , since the matrix operators t+ and J+ increase e by one.
It is easy .to see that the matrix for the product k+k
is proportional to the
number operator
k k_ = 2s diag[
e+l e-l
e1
e+2 e e+1].
(2.3.27)
Similarly, we have
k = 2s diag[ v(e+1)(e+2)
/e(7+1) ...
¢(e+2)(e+3)
i
,
(2.3.28)
and
k = 2s diag[ e(T-TT)
(e+-l
V(-1)(-2) ...
. .
] ,
(2.3.29)
The evaluation of terms (2) and (3) may be simplified by writing each as a
product of matrices.
For (2) we have
S(v)(H1)ro/(E - E)
where af and a
- Rt H a
at v= R (E-E )
(2.3.30)
are real column vectors, and all of the matrices are real,
except for the factor if3 in H which is factored out.
3
whose columns
,
are the eigenvectors of H
for f = 3,
diagonal matrix whose elements are (E i - E )-1,
eigenvectors in R.
R is the real matrix
11
and (E-E )-1 is a
ordered the same way as the
I have used dots to denote matrix multiplication.
Care
must be taken in defining the matrices vo_ and H to account for the
-value
of the vector which they act upon.
In the present example, H 1 acts on vectors
with f = 0, and v0_ acts on vectors with e = 3.
Term (3) may be expressed as
X (H)fr,(vo )ro/(Ef-Er,) = a HI R' (Ef-E') Rt v
where R' is made from the eigenvectors with
e =-
1.
ao ,
(2.3.31)
The 2oc+ os(nt) inversion asymmetry-induced ED transition has At = 1 with
initial state aC(O) and final state bc(2). The At = 1 part of H(G) is
H(G) =
where k
k2
iGf[ -t
was dropped.
+ 2t k+k]
-
The last term is zero when H operates on states with
k = 0 , but is needed to compute v
v
(2.3.32)
, which is found to be
= - iGfI(2t k+) .
(2.3.33)
The transition matrix element is
o
(2.3.34)
1=(vS)f
+ i[V0 'SlIf 0
which results from the three processes:
V
0
(1)
(2)
f
f=l
(6=1)
(o
( =0)
ao
v
b
.
b
H
r
r
(t=O)
(e=o)
(= 1)
H
(3)
The
v
a
a
(e=o)
( =1)
o
matrix
bb
-
r"
calculation
different intermediate states.
b
f •
(e=1)
proceeds
as
before,
with
different
matrices
and
Next we consider the ED cyclotron resonance absorption ac(al) in the
Voigt geometry, and the EQ CR-harmonic absorption 2mc(al).
The EDCR
absorption is easily found from
(v+)fo = af v0+ a ,
with
a = aC(O)
replacing
v0+
af = aC(1).
and
by
(2.3.35)
The EQ 2oc(a)
absorption is found by
(-iqk+/2s)vo+, where we have chosen
A
q = qrl.
This
requires the evaluation of
(vo+k+)f = a
with
a = aC(O)
v+ k+ ao
af = aC(2).
and
(2.3.36)
The low-field dependence of the ED and EQ
matrix elements is B 0/2 and B10, respectively.
The Voigt MD spin resonance transition cos(t) and the mixed MD and EQ
cyclotron harmonic combined resonance transition 2oc+ os(t) are both found in
the PB model by a simple modification of the calculation of the ED combined
resonance
transition
O+ Oas(7).
This
last transition
requires
evaluation
of
the matrix element
(v)fo = b
with
vo0 ao
ao = 0aC(0) and
by replacing v0 ; by
(2.3.37)
,
b = bC(1).
f
{vo, - sqk4)
The MD and EQ matrix elements are obtained
and keeping only the k_ component of k
for the MDSR evaluation, and the k+ component for the 2o0 + o
the cs(x) MDSR transition one must then evaluate
evaluation.
For
-(iq/4s) {Iv ,k}
,
where the symmetrization is required because v0
is linear in k+.
contains a small part which
The transition matrix element also has a small contribution
from the magnetic dipole terms in Hef , which contain the parameters N , K,
K' and K". For the 20+
s(t) transition one must evaluate
- (iq/4 s) {v ,k }fo
B1 and
It is significant that the B -dependence of the matrix elements is Bo,
0
0
0
B20
for the
os,
c
+
0s and 20C+ co
transitions,
transition has a lower-order dependence on B
because
less cancellation
occurs
among the
respectively.
+
than the c
terms which
The
os
(s ED transition
contribute
to the
matrix element.
The result of calculations of the peak absorption for various transitions
is given in the figures.
In all of the calculations I assumed a free electron
concentration of 2 x 1016 cm - 3, and a linewidth of 15 cm - , which is the
observed half-width at half-maximum (HWHM) linewidth of the 2wc
transitions at
12 T. I also assumed quantum limit conditions and neglected the Reststrahl
absorption which occurs near photon energies of 25 meV.
The band parameters
used were those of Littler et al. [Littler 83].
Figures 2.11 and 2.12 show the ED cyclotron and ED combined resonance
absorptions
versus
B
for
the
o1 -Voigt
and
r-Voigt
geometries.
Figs. 2.13-2.15 show the MD and EQ absorptions, which result from the finite
of the incident
wavevector
These
light.
are
harmonic
resonance
and the cyclotron harmonic
7r-Voigt,
a_-Voigt
and
7t-Voigt
the
combined resonance,
The
respectively.
geometries,
cyclotron
spin-resonance,
in the
actual
spin
resonance linewidth is much less than 18 cm - 1, but this value was used so that
the
integrated
strengths
absorption for the 2
c+
of the
co
lines
can be
The
compared.
integrated
transition starts off lower than that of the spin
resonance but becomes greater than the latter at fields above 7.5 T.
Figures 2.16-2.18
show
the
absorption
for
the
cyclotron
20c
harmonic
resonance for three polarizations, assuming that only the inversion asymmetry
mechanism contributes.
The curves are matched at low fields to a value 50=65
a.u. for the cb inversion asymmetry parameter.
This value corrects the value
of 56 a.u., found by [Chen 85b], to account for energy-band nonparabolicity.
Figures 2.19-2.21 show the absorption for the
2
c+ co
combined resonance
for three polarizations, assuming that only the inversion asymmetry mechanism
contributes.
Figures 2.22-2.24 show the absorption for the os spin resonance for three
polarizations,
assuming
that
only
the
inversion
asymmetry
mechanism
contributes.
show the absorption for the 30 c transition
for three
polarizations, assuming that only the warping due to the parameters
t and pt'
Figures 2.25-2.27
contributes.
The single-group approximation p. = pt' is assumed.
7000
6000
5000
4000
3000
2000
1000
0
20
40
60
80
100
120
140
160
180
200
Magnetic Field (kG)
Fig. 2.11
oc(al) ED cyclotron resonance, peak absorption versus magnetic
field for n-InSb, with ne= 2 x1016 cm - 3 and y(HWHM)=15 cm - 1, and B II [111],
[110] and [001], from top to bottom.
10-
0L
0
20
40
60
80
100
120
140
160
180
200
Magnetic Field (kG)
Fig. 2.12
wc+os(n) ED combined resonance, peak absorption versus magnetic
field for n-InSb, with ne=2x1016 cm - 3 and y(HWHM)=15 cm - 1, and B II [111],
[110] and [001], from top to bottom.
0.03
0.025 -
0.02-
.
0.015 -
0.01
0.005 -
0
0
20
40
60
80
100
120
140
160
180
200
Magnetic Field (kG)
Fig. 2.13
o)s(7t) MD (with some EQ) spin resonance, peak absorption versus
magnetic field for n-InSb, ne =2 x016 cm - 3 , with B 11[111],
from top to bottom.
[110] and [001],
The linewidth was assumed to be 15 cm - 1.
The actual
linewidth ys is much narrower, so the peak absorption should be multiplied by
15 cm-ly .
0.7
0.6
0.5
0.4
0
0.3 -
0.2 -
0.1
0
20
40
60
80
100
120
140
160
180
200
Magnetic Field (kG)
Fig. 2.14
2%c(_l) EQ double
cyclotron resonance, peak
absorption versus
magnetic field for n-InSb, with n =2x106 cm 3 and y(HWHM)=15 cm'.
dashed line at left is the decoupled approximation.
right is the spherical PB approximation.
The dashed curve on the
The solid curves are PB model for
B II [001], [110] and [111], ordered top to bottom.
[Littler 83] are used.
The
The band parameters of
0.03
0.025
0.02
0.015
0.01 -
0.005 -
0
0
20
40
60
80
100
120
160
140
180
200
resonance,
peak
Magnetic Field (kG)
Fig. 2.15
absorption
2o c+-ws(r)
versus
EQ/MD
magnetic
double
field
for
cyclotron
n-InSb,
combined
with
ne= 2 x1016 cm
3
and
y(HWHM)=15 cm -1. The solid curves are PB model for B II [111], [110] and
[001],
ordered top to bottom.
The dashed curve at the left is from the
decoupled model, neglecting g" and y0 .
are used.
The band parameters of [Littler 83]
0.9
0.8
0.7
a
0.6 -
o
0.5
C
0.4
0.30.2
0.1
0
20
40
60
80
100
120
140
160
180
200
Magnetic Field (kG)
Fig. 2.16
2o)c(7)
ED absorption due to inversion asymmetry parameters C' and
G, matched to 8 =65 a.u.; peak absorption versus magnetic field, for n-InSb
with ne=2x10" cm - 3 , B II [001] and y(HWHM)=15 cm
inversion
asymmetry resides in the indicated parameter.
the decoupled approximation.
G, respectively.
IF2 12_
1.
Each curve assumes the
The dashed line is
The upper and lower solid curves are for C' and
The angular
dependence
is
approximately
proportional
to
0.7
0.6
0.5 0.4
a
0.3
0.2
0.1
,
20
0
40
60
80
100
120
140
160
180
200
Magnetic Field (kG)
Fig. 2.17
2coc(a)
ED absorption due to inversion asymmetry parameters C' and
G, matched to 50=65 a.u.; peak absorption due to v
versus magnetic field, for
n-InSb with n = 2 x1016 cm - 3 , B II [111] and y(HWHM)=15 cm -l .
e
0
Each curve
assumes all of the inversion asymmetry to reside in the indicated parameter.
The dashed line is the decoupled approximation.
curves
are
for
C'
and
G,
respectively.
The upper and lower solid
The
angular
dependence
approximately proportional to IF3 12 if one neglects the contribution of v+.
is
0.06
0.05
0.04
S0.03
0.02
0.01
0
0
20
40
60
80
100
120
140
160
180
200
Magnetic Field (kG)
Fig. 2.18
2)c(Y)
ED absorption due to inversion asymmetry parameters C' and
G, matched to 8 =65 a.u.; peak absorption due to v+ versus magnetic field, for
n-InSb with ne= 2 x10O 16 cm,-3
B
Ii [110] and y(HWHM)=15cm.-
E
Each
curve
assumes all of the inversion asymmetry to reside in the indicated parameter.
The dashed line is the decoupled approximation.
curves
are
for
C'
and
G,
approximately proportional to F
respectively.
2
The upper and lower solid
The
angular
dependence
if one neglects the contribution of v
is
1.2 1-
"
0.6-
C
-"
0.4-
_
0.2
0
0
20
40
60
80
100
120
140
160
180
200
Magnetic Field (kG)
2
Fig. 2.19
and
G,
c+o s(7c) ED absorption due to inversion asymmetry parameters C'
matched
to
8 =65 a.u.; peak
absorption
versus
magnetic
n-InSb with ne= 2Xl016 cm -3 , B 11[110] and y(HWHM)=15 cm .
field,
for
Each curve
assumes all of the inversion asymmetry to reside in the indicated parameter.
The dashed line is the decoupled approximation.
curves
are
for
G
and
C',
approximately proportional to IF
respectively.
2
The upper and lower solid
The
angular
dependence
is
0.9
0.8
0.7 0.6 0.5 M
0.4
,
0.3 -
0.2
0.1 -
0
20
40
60
80
100
120
140
160
180
200
Magnetic Field (kG)
Fig. 2.20
2mc+ms(al)
ED absorption due to inversion asymmetry parameters C'
and G, matched to 8 =65 a.u.; peak absorption due to v
versus magnetic field,
for n-InSb with ne=2x106 cm 3, B II [001] and y(HWHM)=15 cm - 1.
Each curve
assumes all of the inversion asymmetry to reside in the indicated parameter.
The dashed line is the decoupled approximation.
curves
are
for
G
and
C',
respectively.
The upper and lower solid
The
angular
dependence
approximately proportional to IF2 12 if one neglects the contribution
of v+.
is
0.8
0.7 0.6
E
0.5 -
.9
0.4
<
0.30.2-
C
.
-
0.1
00
40
20
80
60
100
120
140
160
180
200
Magnetic Field (kG)
2c+s(al
Fig. 2.21
)
ED absorption due to inversion asymmetry parameters C'
and G, matched to 8 =65 a.u.; peak absorption due to v+ versus magnetic field,
for n-InSb with ne= 2 xlO16 cm - 3, B 11[111] and y(HWHM)=15 cm-1.
Each curve
assumes all of the inversion asymmetry to reside in the indicated parameter.
The dashed line is the decoupled approximation.
curves
are
for
G
and
C',
approximately proportional to IF
respectively.
2
The upper and lower solid
The
angular
dependence
if one neglects the contribution of v_.
is
1.4 1.21
0
&
0.8
o0
"
0.6
0.4
0.2
0
20
40
60
80
100
120
140
160
180
200
Magnetic Field (kG)
Fig. 2.22
%s(o) ED absorption due to inversion asymmetry parameters C' and G,
matched to 8 =65 a.u.; peak absorption versus magnetic field, for n-InSb with
16
-3
ne= 2 x10 6 cm,
B
-1
11[110] and y(HWHM)=15 cm
.
Each curve assumes all of
the inversion asymmetry to reside in the indicated parameter.
is the decoupled approximation.
and C', respectively.
IF 12
The dashed line
The upper and lower solid curves are for G
The angular dependence is approximately proportional to
0.06
0.05 -
0.04
o
0.03
o0
0.02
0.01
0
0
20
40
60
80
100
120
140
160
180
200
.Magnetic Field (kG)
os(al_ ) ED absorption due to inversion asymmetry parameters C' and
Fig. 2.23
G, matched to 50=65 a.u.; peak absorption due to v_ versus magnetic field, for
n-InSb with ne= 2 x1016 cm,- 3
B0 II [111] and y(HWHM)=15 cm - 1.
Each curve
assumes all of the inversion asymmetry to reside in the indicated parameter.
The dashed line is the decoupled approximation.
curves
are
for
G
and
C',
approximately proportional to F0
respectively.
2 if
The upper and lower solid
The
angular
dependence
one neglects the contribution of v+.
is
0.2
0.18 0.16 0.14
E
a
0.12
o.
0.1 -
S0.08 0.06
0.04
0.02 0
20
40
60
80
100
120
140
160
180
200
Magnetic Field (kG)
Fig. 2.24
os(l) ED absorption due to inversion asymmetry parameters C' and
G, matched to 8 =65 a.u.; peak absorption due to v+ versus magnetic field, for
n-InSb with ne=2xl016 cm - 3 , B II [001] and y(HWHM)=15 cm
.
Each curve
assumes all of the inversion asymmetry to reside in the indicated parameter.
The dashed line is the decoupled approximation.
The upper and lower solid
curves
absorption
are
for
C'
and G,
respectively.
The
proportional to IF*2 12 if one neglects the contribution of v D
is
approximately
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
00
20
40
60
80
100
120
140
160
180
200
Magnetic Field (kG)
Fig. 2.25
absorption
3WCc(t)
ED absorption due to warping parameter gt = 0.55; peak
versus magnetic field, for n-InSb with n =2x106 cm - 3
and y(HWHM)=15 cm - '.
B II [111]
The dashed curve is the decoupled approximation.
angular dependence is approximately proportional to IW3
2
The
0.6
0.5
0.4
.
0.3
0.2
0.1
0
0
20
40
60
80
100
120
140
160
180
200
Magnetic Field (kG)
Fig. 2.26
absorption
B0 11[001]
3Oc(ol) ED absorption due to warping parameter g = 0.55; peak
due to v
and
approximation.
versus magnetic field, for n-InSb with ne=2 x1016 cm - 3
y(HWHM)=15 cm 1.
The
dashed
curve
is
the
decoupled
The angular dependence is approximately proportional to IW
if one neglects the contribution of v+.
2
0.1
0.09 0.08
0.07 -
0.06 -
.5
0.05 -
o
-
0.04-
0.03
0.02 0.01 -
0
0
20
40
60
80
100
120
140
160
180
200
Magnetic Field (kG)
Fig. 2.27
3
c(()
ED absorption due to warping parameter p = 0.55; peak
absorption due to v+ versus magnetic field, for n-InSb with ne= 2 X1016 cm -3
B II [110]
and
approximation.
y(HWHM)=15 cm-1.
The
dashed
curve
is
the
decoupled
The angular dependence is approximately proportional to
if one neglects the contribution of v .
W 22
Decoupled Model: Semiquantitative Treatment of Magnetoabsorption
2.3.2
The decoupled model for the conduction band (cb) provides greater insight
into
of the
the physics
model.
in n-InSb than
transitions
the PB
In this section we consider magnetoabsorption in the decoupled model
We consider all terms to order k4 as was done by Ogg
in which Hef is 2x2.
and
magneto-optical
McCombe,
Barticevic 87].
complete.
and
more
We
follow
However,
recently
I
the
by
latter
include
Barticevic
treatment
the
[Ogg 66,
because
effective
is quite significant
it
is
spin-orbit
leads
the
most
interaction,
to measurable
proportional
to E.kxo, which
absorption.
This term was introduced by Yafet [Yafet 63] and was used by
[Sheka 65], [McCombe 69], and [Kim 89a].
and
McCombe 69,
See also the review by Rashba and
The decoupled
Sheka [Rashba 91] and a recent application by [Jusserand 94].
Hamiltonian to fourth order is [Barticevic 87]
Hff = k/2m
eff2
*
{k
x k
+ g*gB-a
+ eKSEkxo + 2(4(
z
0 x x y - k2}
B
o({k 2 z + c.p.) + yoB(axBxkx + yz).
It would be useful to redefine parameters so that
H(g") = 2g"t(GB)(2) K (2)
H(ox)
= -
k
c.p.)
B 2
+ g'gB.TBk2 + 2g"g (-k)(B-k)
2OB
+ C0k4 +
+
+
4)
(2.3.38)
K(2 1](4)
H(y o) = YOR[(aB)2'(
In H(c%)
k4 k+
k 2 ,k2 } + c.p.= -
I used
k4 + k4 + k4
x
y
z
Appendix C).
(2.3.39)
C
) + k4
, and then made
irreducible by subtracting the part which has rank zero.
(See
The general form of the Hamiltonian in a rotated coordinate
frame is given in Appendix C.
For the purpose of computing magneto-optical
transitions
we take
the
unperturbed Hamiltonian to be
Ho = k2/2m * + g*
B0 .
(2.3.40)
The remainder of Hef is treated as a perturbation, and is denoted H.
We use
the Landau gauge
A° = B 0
1
,
(2.3.41)
A
so
A
B = B0
is along the
axes oriented relative
(See Appendix
C.)
axis in a general coordinate frame with the 4ri
to the xyz axes via the Euler
The unperturbed eigenfunctions
A
functions of N, k7,
different
gauge
k
and a .
and
different
angle transformation.
are taken to be eigen-
Here we follow Barticevic et al., but use a
set
of
eigenfunctions.
The
explicit
eigenfunctions are
n,krk,a(r) = e ik l
eik
n(4 - kllIs) X
,
where the 0n is a simple harmonic oscillator function,
(2.3.42)
On(x) = (n!)-1/2(bt)n 40(x) .
Here b! = (p /h + isx)/V2s
0 (x)
XG is
0
=(s/h)
4 e -(1)
(2.3.43)
(dimensionless, with s = eB /hc) and
2
(2.3.44)
if a is up (a-set) and
if cy is down (b-set).
is real if n is even, and imaginary if n is odd.
The phase of On(x)
The operators k+ = k + ik
are related to the Landau level raising and lowering operators at and a via
k+ = v2sa
,
k_ = V2-a.
(2.3.45)
When the fourth and higher-order corrections to H
0
are considered, one finds
A
that N and (% are no longer conserved separately, but that the combination
N +
-a
is conserved.
In particular, the fourth-order term H(g") causes
mixing of spin-up and spin-down states at finite k
First consider the electric dipole (ED) transitions.
If we consider just
the unperturbed Hamiltonian, we see that the only ED transition for a free cb
electron is cyclotron resonance (CR), which is caused by the operator
v+ = k+/m* ,
at the frequency
(2.3.46)
o c = eB lm*c = h2slm*.
For the aC(O) to aC(1 ) transition,
using the notation of [Pidgeon 66] and [Weiler 78], the matrix element is
(v )fo
=
2s-/m* .
(2.3.47)
Terms in the perturbing Hamiltonian H 1 produce new ED transitions by
lowering the symmetry of H ef relative to that of H .
isotropic
perturbations.
cyclotron
resonance,
The
but no
k4 ,
involving
and
The terms
new transitions.
produce two
however,
2g"l B(*-k)(B-k),
terms
First consider the
(B.o)k 2
produce
eKsE-kxa
and
first term is
new transitions. The
analogous to the ordinary spin-orbit Hamiltonian in the Pauli Hamiltonian for
a free electron,
which
is derived
from decoupling
proportional
to g"
does
the Dirac
not
Hamiltonian
occur in the
Pauli
[Foldy 50].
The term
Hamiltonian.
It produces mixing of a-set and b-set (spin-up and spin-down)
wave functions when k # 0.
The new ED transitions which are produced are the
combined resonance coc + os(t) which occurs for
A
polarization (denoted 7r) in
the Voigt geometry, and the ED-induced spin resonance transition ws(a_,k
which occurs for a_ polarization in the Faraday geometry, for finite k
in
the
_1
polarization
in the Voigt
geometry.
The combined
)
, and
resonance
transition (sometimes denoted KR) involves a change of both orbital and spin
states, in such a way that the sum n + m
At = 0 transition, where
t = n + m - I
remains constant.
i.e. This is a
is the integer-valued quantum number
which is conserved in the axial approximation, and which is zero for the
lowest conduction subband aC(0).
The matrix element which appears in the expression for the absorption
coefficient may be written
A
V = e-v
where
Veff is
(2.3.48)
an effective
velocity operator which is the
operator in the case where the optical perturbation
ordinary velocity
is proportional to A v.
1
For the spin-orbit-like term H(K) we may write
H(K) = eE-R ,
s
(2.3.49)
where
R - K S kxo.
(See [Rashba 91].)
v eff = i3KSkx
(2.3.50)
We then have
.
(2.3.51)
The ED transitions, and the operators which produce them, are
veff =
v eff
Occ + c s (t):
(ED spin resonance)
K ka
Ks k+ a
2
S+-
(Cyclotron resonance)
(Combined resonance) .
where we have included only the parts of the operators which give energy-
absorbing
transitions.
is
g-factor
transitions
eff
v"
-2,
I
free
in
the
Pauli
correction,
approximation
and
the
the
spin-reversing
with
os(o+,kL)
a .
electrons
the radiative
neglecting
are
= -oKsk
For
vf
- oKf k
, and
c - os(7r ) with
cannot originate from the ground
+
The latter transition cannot originate from the ground
state.
For the term
H(g") = 2g"g B(a-k)(B-k)
it is useful to consider only the
part which involves the second rank tensor components Km 2).
To obtain this
part we use
(c-k)(B-k) = (oB) (2 . K(2) +
noting that the part
-B)k 2
3
!(oxB).(kxk)
(2.3.52)
is zero since kxk is proportional to iB.
The reason for doing this is that the axial approximation requires this part
to be combined with the m = 0 part of the cubic term H(yo), as is done in
Appendix C. The ED transitions and velocity operators then are
(a_,k)
•
v~ =-
(g" I(W
+
I y(W
S=o(g-
-))k
(ED spin res.)
a
(Cyclotron res.)
-))kaa
)
Oc( OS(C)
ac
+
where W
as( t) :"
V
=
o
o(Wo
))k_
,
(Combined res.)
is the angular function for m = 0 terms in the warping (octahedral)
part of Hef,
W = (B
o
The
ox
+ B4
oy
matrix element
+
B4 )/B4
or
for the
(2.3.53)
o
os(c_,k ) transition
must be
multiplied by a
correction factor
g1 =1 0 s/((
S
This
is
S
+ (0c)
C
required
to
(2.3.54)
account
for the
A
contribution
[e-v o, iS]fo which results
Sifo
from wave function mixing, as shown in Appendix F. Table 2.1 summarizes the
results for ED transitions which occur in the axial approximation.
Table 2.1
Effective velocity operators for axial model ED Transitions.
Transition
At
s-,k )
(Oc+)
Velocity operator
-1
v_ =
+1
v+ = k+/m*
c
k
=
C
2
k(Y[s
+ os(Y+,k )
The
anisotropy
+1
v
of
the
+
- (co
(g
Sv
o(g-
W
o -
the
incident
light
)
= Order k5 and ok3
(oc + (os((t)
transition
might
account
observed anisotropy in the output of the 10.6 [pm-pumped SFR laser.
is strongest when B
-
))]
for
the
The output
is directed along a [111] crystallographic direction with
polarized
parallel
to
B.
The
oc + os(7)
absorption
contributes to the SFR scattering cross section, as pointed out by Aggarwal
[Aggarwal 71] and this is the orientation for which the o c + os(rt) absorption
is strongest.
Next consider
the ED transitions induced
terms H(o0 ) and H(y)
first
because
they
warping
We consider the warping-induced transitions
in Hef.
contribute
by the fourth-order
to
the
axial
varying degrees of anisotropy in them.
model
transitions
and
produce
For a general direction of B
the
warping terms produce many other transitions which violate the axial model
selection
rules.
Zawadzki 76,
in
all
As
Trebin 79],
polarizations,
considers
was
all orders
considers
out
previously
all inter-Landau
general
for a
of perturbation
restricted to zero, or if B
or if one
pointed
[Rashba 61a,
level transitions
direction
of B 0 at
theory.
Selection
Weiler 78,
become
finite k
rules exist
possible,
, if
if k
one
is
is chosen to be along a high symmetry direction,
only transitions
which occur in first-order
perturbation
in first-order
perturbation
theory in the anisotropic terms.
In order to compute
the
strengths
transition
theory we need the Hamiltonian and velocity matrix elements in the rotated
A
reference frame 4rli with B along
H(y0 ) are given in Appendix C.
.
The general
expressions for H(cx0) and
The velocity operators are easily computed
from these, and the results are given in Tables 2.2 - 2.4.
are
indicated
which
account
for
wavefunction mixing [Gopalan 85].
transitions on k
the
contribution
to
the
Correction factors
transitions
from
The magnetic field and k dependence of the
may be easily understood.
fourth-order terms in Heff is third-order in k.
The velocity associated with the
Thus
An = 1 or 3 transitions
may be independent of k , while An = 0 or 2 transitions must contain k once
or
three
times.
proportional
The
first
set
of
transition
matrix
to B3/2 since each factor of k+ or k
An = even matrix elements will be proportional to B k
elements
be
will
contributes B 2.
The
to lowest order in
k .
Table 2.2
Transition
Transition operators and angular dependences due to X(ao).
Operator factor
At
-
S1
C
(k -5k2 -s)k
+
0
vl+
v1
W'
-3W
2
2c
2
- ak 2k
-6W
30
3
- aok 3
-2W
Correction factor:
0
1
1
-6W
12W
24W
4W
8W
1
c
v_
2
0
c
The correction factor for the cc transition due to vl+ must be replaced by 1,
both here and in Tables 2.3 and 2.5.
Table 2.3 Transition operators and angular dependences due to f(y
o ).
Transition
At
0
- 1
S
0
C
-
S1
v+
v1
vI
cy sk
4W
2W
- W
%sk+
- W
2W 2
4W 3
y sk+
+
- W'
2W
4W
2
1
2
S1
C
S+
C
Operator
210
0S
0
0
y Ysk
-+
-W
20
Correction factor:
2
-2
0-
-1
0
2
c
0
1
W'0
- W1
1
C+
C
Magnitudes of warping functions Wm(aj,y)
along
m
directions of high symmetry.
W0 1
1.000
W|
0
W3
0
W4
8I
- 0.125
[111]
3
=
0.3333
[110]
1 = 0.5
2
0
12
0
0
-/ 0.1179
0
c(0+
C
Table 2.4
[001]
2
3
32
=
0.09375
Finally,
consider
the
inversion-asymmetry
Hamiltonian
H(50 ),
which
is
third order in k. This is the Hamiltonian which Rashba and Sheka introduced
[Rashba 61a]
and used to predict ED-induced
resonances.
In the absence of an applied magnetic
third-order
splitting
of the conduction
spin resonance and combined
field it produces
the
band which was first predicted
and
analyzed by Dresselhaus [Dresselhaus 55].
This is the Hamiltonian which is
important in the interference effect in spin resonance in which the absorption
changes upon reversal of either B
or q, where q is the optical wavevector
[Dobrowolska 83, Chen 85].
The absorptions, including
absorptions
[Rashba 61a], were treated by Gopalan et al.
not considered in
the spin conserving
[Gopalan 85].
The general expression for H(80) in a rotated coordinate frame was given
by [La Rocca 88].
operators
2.5 - 2.6.
The result is reproduced in Appendix C, and the velocity
to the different
which contribute
The correction
which must be
factors,
in Tables
are given
transitions
applied
to the matrix
elements of v+ and v_ , give the correction due to mixing of the unperturbed
wave
functions
first-order
by
the
perturbation
perturbation.
theory
have
Only
been
transitions
which
considered.
result
from
Higher-order
transitions are expected to be much weaker, but some transitions are predicted
in the PB model which have a higher-order dependence on B than the ones in
the tables.
These might be observable
at large magnetic fields.
In the
decoupled model, all of the transition matrix elements are of order k 2 , which
means order B
or Bkk.
The low-field absorption then is proportional to B
for the transitions occurring at k
finite kr.
= 0, and to k 2
for those which require
Note that now it is the An = odd transitions which depend on k
and the An = even transitions which occur at k = 0.
The magnetic field
dependence of the absorptions, obtained with the PB model, are found in the
preceding section.
Table 2.5
Operators
induced ED transitions.
Transition
o
At
causing
Operator
- 1 80(k2 - 3k)
S
0
-
first-order
v
inversion-asymmetry-
V1
4F_2
-2
V1
6F
-4F
- 8F
4F
-1
0
c - a
2
8 +k +
12F
C
1
8 0ak +k
0
8F1
16F
c + 0
0
8
k k
12F
8F
- 20F
2F
F
2
- 2F
4F
k2
- 2F
- 5F
20c - Os
3
80 k 2
20
2
8
C
2c+
C
S
1
Correction factor:
0
60
+
0 -+
2
0
0-
SC
- 6F
3
-6F
2
1o
C
Table 2.6
Magnitudes
of
inversion-asymmetry
functions
Fm((a,P,y) along directions of high symmetry.
[001]
F0 1
[111]
0
12
=
-
S0
[110]
0.1443
0
0
1 = 0.0625
0
0
16
1
IF2
0.125
2
IF
3
0
12
83
= 0.2041
3
16
- 0.1875
Now we consider magneto-optical transitions which are induced by the
[Bassani 88, La Rocca 88, Kim 89a,
finite wavevector q of the incident light.
Kim 89b
and
references
consider
We
therein.]
approximations, which are linear in q.
and
absorption.
At
high
the
MD
and
EQ
The allowed transitions are the same
ones which occur as 2-photon transitions.
scattering
only
These include the cases of both
magnetic
fields
the
transitions
are
generally mixed MD and EQ in character, except for At = 2 transitions which
must be pure EQ.
For example, the spin resonance transition is pure MD only
in the lowest-order approximation in B 0, reflecting the time-reversal symmetry
which exists in the limit B 0 = 0.
The
allowed
approximation are
3m
MD
and
s,
2
+ ms at finite k
.
[Yafet 66, 71],
absorption
transitions
20 + 0
at k
[Wright 68]
[Zawadzki 76],
These
in
-0,
the
spherical
and (
-
3
(or
+ 0,
,OC
are
and
[Makarov 68].
[Trebin 79],
all
the
possible
For
2-photon
[Weiler 82],
[Wlasak 86],
energy-absorbing
axial-model
transitions with At < 2, for positive g-factor materials.
All but the
transition
in
are
able
to
axial)
(For the analogous Raman scattering transitions see
[Kelley 66],
see
[Seiler 92].)
c,
EQ
occur
in
the
quantum
limit,
which
c-
only
s
the
ground-state aC(O) Landau subband is occupied.
The MD contribution to the optical absorption results most clearly from
terms
in
Hff which
contributions
contain
B explicitly.
Other
MD
as well
are obtained by expanding eiq-r in the optical
as
EQ
perturbation to
first order in q before, or sometimes more conveniently after, computing the
desired matrix element.
An = +1
transitions
functions.
(i/s)(qxk)
The part of q which is perpendicular to B0 causes
between
the
spatial
(i.e.
spinless)
Landau
level
wave
The effect of exp(iql.r) is found to be equivalent to that of
(See Appendix F, or take matrix elements
, to first order in ql.
of the identity [r,H] = ih2k/m.)
The effect of exp(iq () is to change k
of
the state which the exponential operates on, producing a mixture of a-set and
b-set states with the same n + ms .
choosing
k
= q/2.
the
initial
state
to
have k
The effect at k
=-q/2 and
= 0 is obtained by
the final
state to have
(See [Kim 89a], after Eq.(49).)
First, we will compute the matrix element for spin resonance which is MD
to lowest order. From Section 2.2 we have
H'g*
B '
(2.3.55)
with effective velocity operator
i
A
V =- g*e-qxo
.
(2.3.56)
Keeping only the part which produces a to b transitions we have
V = -Ig*(e q
- e+q)G_ .
(2.3.57)
This produces spin resonance for the it-Voigt and o_-Faraday geometries.
A
that
e+ = V2
when
A
A
(Note
A
e = e_ = (4 - ir)/V2.)
The appearance of q+ in the first
term implies a phase which varies with the direction of q 1 .
The matrix
A
elements
I take q = qrl for the Voigt geometry,
are given in Table 2.5.
following
[Bassani 88]
and
[La Rocca 88].
The
matrix
element
for
spin
resonance with 7t polarization, then, is
(2.3.58)
V = (i/4)qg* ,
fo
for the transition from aC(O) to bc(O).
particularly
important in the
calculation
The phase of the matrix element is
of the interference
the absorption changes upon reversal of B
effect
in which
or q, for certain geometries and
crystal orientations.
Next we consider the 2(o
transition.
This transition can only occur in
the ol-Voigt geometry since 2 units of angular momentum are transferred when
the photon is absorbed.
The transition velocity operator, then, is
V = !.{k/m*,(i/s)(qxk)
A
A
}
.
(2.3.59)
A
Taking e = ( and q = qr this becomes
V
iq/sm*)k .
(2.3.60)
Keeping only the part which produces the 2c
c
transition, we have
V = (- iq/4sm*)k+ .
For the aC(O) to aC(2), 2 oc(a)
V
(2.3.61)
transition we then have the matrix element
= - iq/V2m* .
(2.3.62)
This result has the same dependence on q, and lack of explicit dependence on
B o, as the matrix element for spin resonance.
The matrix elements have an
implicit linear dependence on B ° since q = nol/c must be evaluated at the
transition
frequency,
which depends
linearly
on B
in both cases.
It is
interesting to note that the two matrix elements are very different in terms
of their physics.
The one for spin resonance comes from a permanent magnetic
dipole moment which exists when B ° = 0, while the one for 20 c comes from an
induced electric quadrupole moment proportional to e2 which exists only in the
transition state, and which diverges like 1/B
0 as00B -
0.
The matrix element V
for the 20o transition is a factor qt smaller
B
c
fo
than the one for cyclotron resonance with the same initial state.
This
relationship was noted by Johnson and Dickey [Johnson 70] who considered the
possibility that a wavevector-dependent process could cause the 2o c absorption
The value of (qe ) 2 which
which they observed in n-InSb at 1.2 T and 2.0 T.
they calculated is small by a factor of about 2500, which is (2nn 1)2 times a
The corrected value of (qB)2 is 10-4 at 2 T
round-off factor of about four.
instead of 4 x 10- 8 , but this is still too small to account for the strength
of the 2o
The
c
transition which they observed.
wavevector-induced
matrix
element
for
the
20o
c
+ o
s
transition
involves higher-order band parameters than the previous two matrix elements
which
resulted
optical
from
interaction.
the
unperturbed
For
the
effective
n-Voigt
Hamiltonian
geometry
the
Ho ,
calculation
with
the
requires
modification of the calculation of the matrix element for the ED combined
resonance
transition,
A
oc + 0) (n).
The velocity
operator for that transition,
A
due to H(K ), with e = C, was
V = ioKs(kxo)
a ).
(2.3.63)
A
For incident wavevector q = qqr this becomes
V = io)Ks(kxo)4( - i/s)qk4 .
If we keep only the component which contributes to the 2o
(2.3.64)
c
+ 0os transition we
get
V =
q
2oKk
(2.3.65)
.
The matrix element between aC(O) and bc(2) is
(2.3.66)
Vfo = v2iqoK ,
which is proportional to B.
This is higher order by a factor of B
than the
matrix elements for spin resonance and 20o.
A
In the Faraday geometry, with the optical wavevector q parallel to
is impossible for H(Ks) to cause the 2
that the contribution of K
mechanisms.
c
+ ws(a+) transition.
, it
This implies
to the transition is a 1-to-1 mix of EQ and MD
This is understood as follows.
If the transition were pure EQ in
nature, the matrix element would involve the combination eq_ + eq , which is
the m =
-
1 component of (eq)(2).
Similarly, if the transition were pure MD in
nature, the matrix element would involve the combination eq - e_q , which is
A
i(exq) .
A 1-to-1 mixture of these two combinations is required to make the
e_q part to drop out, producing the result of the direct calculation.
Contributions to the 2c c + co
s matrix elements from the parameters g" and
a few
0 are computed in a similar fashion, except that there are
The first is that there is an additional MD part which comes
complications.
from the parts of H(g") and H(y) which depend explicitly on B1 .
This must be
For the
added to the parts which result from the explicit dependence on A .
A
parameter g" the two parts cancel for n-Voigt.
For q parallel to
the B
part of H(g") gives a finite contribution to the (a-Faraday transition.
There
is an additional contribution proportional to g" which results from mixing of
a-set and b-set wave functions due to H(g"), proportional to k . This part is
computed with the A l part of H(g"), including the eiq
discussed above.
spatial variation, as
Notice that the contribution of H(g") to 2m + m
is also a
1-to-1 mixture of MD and EQ mechanisms, but that now the two parts subtract
instead of add, so the matrix element is finite for a+-Faraday and zero for
x-Voigt.
The matrix elements,
given in Table 2.7.
including the contributions from H(yo),
are
It is of interest that the contribution of Ks, g" and yo
to the spin resonance matrix elements may be obtained in a similar fashion,
and that the result also contains a
mixture of MD and EQ components.
The
dependence of the Voigt-geometry matrix elements on B0 , based on a PB model
calculation, was given in the preceding section.
Table 2.7
Effective velocity operators for axial model MD-EQ transitions.
A
Transition
A
At
Type
q, e
V
0s(Rt)
-1
MD
1, 5
(i/8)qa_g*
os(a_)
-1
MD
, -
(-
2oc(a_)
+2
EQ
I,
( - i/4)qk+/(sm*)
2o c + )s(n)
+1
EQ+MD
1,
20 c +
+1
EQ+MD
r, +
s(o+)
Correction factors pg
c+
2 = (20
_k
2/8)qa_g*
_
(1/v )qa
[K
0co+
0 (W 0 -3)
_[g g" +
)/(o + o ) and gt3 =-(30c
-
8 3T0 (W
-5)]
o )/(o) + cO)
are
A
required to account for the contribution [e-v0 , iS]fo which results from wave
function mixing.
Here
W - (B4 + B 4 +
x
y
)/B4.
z
(2.3.67)
The transition matrix elements originating from the ground-state subband aC(O)
limit conditions)
quantum
(assuming
k
,
k+ --
simply makes the replacements
geometry with
spherical model
for the principal
A
q = qy
2sv'2 ,
Y_
and
one
for
2
-
The resulting matrix elements and
spin-flip transitions, and zero otherwise.
peak absorptions
since
compute,
are quite easy to
the Voigt
transitions in
are given in Table 2.8.
Matrix elements and peak absorptions for Voigt transitions at k =0.
Table 2.8
a(peak)
V
Type
At
Transition
2
po
v2-/2m*
ED
+1
+(l)
2
j
C
+ o0 (t)
S
0
ED
-1
MD
-
s2o0K
S
pO (4p K 2)(2g B )2
B 0
1 S
7
2
2cny
n o
Ss(t)
1
qg*
2
o
3
B
m
m4
2cy
2oC(1l)
EQ
+2
- V2iq/2m*
2cy
no
20 +
s(X)
+1
EQ+MD
V2iqwK
'1p
p
=
m/m* + g*/2
and p 2
2mm* + g*/2
mc
m*
(2
2
2cy
24 B = free electron cyclotron energy.
o
0
2
3 4g B
2
n
8g B
(4p K2
20
B B)
B 0
mc
3
3. MAGNETOABSORPTION AND FIELD-REVERSAL EFFECT IN n-InSb
In
this chapter I present
the results of my recent magnetoabsorption
experiments in n-InSb at wavelengths near 10 pm and magnetic fields up to
15 T.
The most striking feature of these new measurements
geometry
is
the
observation
of
a
significant
in the Voigt
field-reversal
effect
in
the
absorptions at 2o c and 2(oc+Ocs , in good agreement with the theory presented in
the previous chapter.
factors
of 2.5
The observed effect involves a significant change, by
to
3,
of the
magnetoabsorption
coefficient
for
these
transitions upon reversal of the direction of the applied magnetic field.
polarizations for which the effect is observed are the a_
20c
transition and the TCpolarization for the
result
is
(EQ
and/or
surprising
because
MD)
inversion-asymmetry
the
which
causes
2 0c+Os
wavevector-dependent
the
effect
by
transition.
absorption
interference
The present
mechanism
with
the
induced ED absorption was overlooked or thought to be
of the interference
the q-dependent
The
polarization for the
insignificant in all of the previous work on magnetoabsorption.
occurrence
two
The actual
effect is reasonable from the perspective
absorptions have selection
rules which are
that
similar to those
for 2-photon processes (absorption and scattering) in the isotropic model.
In
this latter case the An = 2 processes are known to accompany the An = 0
processes.
Here n is the Landau level quantum number in the conduction band.
The observation of the field-reversal effect in the 2oc(a)
transition is the
first time that the effect has been seen in a spin-conserving transition.
spin-conserving
effect
transition between
levels (PB
was
predicted by
Bassani
et al. [Bassani 88]
The
for a
the a+(0) and a+(2) heavy and light hole valence band
notation) with the same polarization, but the An = 2 conduction
band transition was said to be too weak to be of significance.
This
chapter is organized
as follows.
After
a brief review
of the
relevant background for the experiment, I will describe the conditions of the
experiment.
I will then present the experimental
results and
discuss how
these results compare with predictions of the theory presented in the previous
chapter.
3.1
Background on Magnetoabsorption and B-Reversal Experiments
Magnetoabsorption
in n-InSb has been the subject of numerous research
papers and review articles over the past 40 years.
The present work was begun
as an extension of work done at the MIT Francis Bitter National Magnet Lab by
Aggarwal et al. [Aggarwal 71a] on the SFR laser, and Weiler et al. [Weiler 74]
on magnetoabsorption
magnetoabsorption,
and
experimental
studies.
served
the
as
SFR
motivation
These included
[Huant 85].
Pidgeon 80,
affects
laser.
Reviews
which
interference
resonance
is
effect
in
early
articles
called
in
n-InSb
the
by
the
subsequent
[Weiler 78],
include
Seiler 92].
Weiler 81,
for
work
on
theoretical
[Goodwin 83],
[McCombe 75,
Zawadzki 79,
reviews
of
Dobrowolska
et
case,
al.
intraband
The magnetic field reversal
electric-dipole-magnetic-dipole
spin-resonance
and
[Grisar 78],
Earlier
magneto-optics were given by Lax [Lax 60, 61].
effect,
My
presented in Chapter 4, demonstrated the anisotropy of the
absorptions,
and
which
was
discovered
[Dobrowolska 83].
(EDMD)
in
spin
It
was
extensively developed in a series of papers by the Purdue group during the
period between
[Kim 89a].
1983
and
1988,
culminating
in the review
by Kim et al.
The effect is also covered in the recent review of magneto-optics
by Rashba and Sheka [Rashba 91].
Notable
historic
work,
in the review
discussed
articles,
includes
the
observation of CR in InSb by Dresselhaus et al. [Dresselhaus 55a]; observation
of CR
harmonic
absorption
in p-type
a semiclassical explanation;
McCombe
Ge by Lax
et al.
[Lax 60],
with
observation of combined resonance in n-InSb by
[McCombe 69]; observation of CR harmonic absorption by Johnson
[Johnson 70].
The LO phonon assisted absorptions were observed by Enck et al.
and described theoretically by Bass and Levinson [Enck 69, Bass 65].
As mentioned above, the field reversal effect was overlooked or rejected
in
several
[Grisar 78],
[Johnson 70].
effect.
studies.
These
[Chen 85b],
The
include
[Gopalan 85],
first
four
[Weiler 74],
[Favrot 75, 76],
[Lee 76],
[Bassani 88], and the earlier work of
studies
did
not
look
for
a
field-reversal
The articles by Johnson, Gopalan and Bassani imply or state that the
B-reversal effect does not occur to an observable degree for the 20 c
20 +o
C
S
conduction band transitions.
and
3.2
Experimental Conditions
This
apparatus,
section
and
provides
procedures
a description
used
to
of the InSb
study
the
samples,
magnetoabsorption
experimental
for
intra-
conduction band transitions.
SAMPLES:
Magnetoabsorption measurements were made on two InSb samples in different
orientations at temperatures T near 25 K and 77 K.
The InSb samples were
n-type, tellurium doped, with carrier concentration ne about 2 x 1016 cm - 3 and
length about 2 cm.
The samples were obtained from Cominco American, Inc.,
Spokane, WA, and their specifications are the following.
InSb #16 -- Lot W3173-B, Te doped, n-type
Dimensions
9.7 x 8.1 x 21.3 mm 3 - along [111], [112], [T10]
n = 1.5 - 2.4 x 1016 cm -3
p
-3
= 2.6 - 3.4 x 10
2 cm
(77 K)
(77 K)
g = 1.0 - 1.2 x 105 cm2/VN s (77 K)
InSb #17 -- Lot W3262-C, Te doped, n-type
Dimensions 9.5 x 7.9 x 22.6 mm 3 - along [110], [112], [111]
n = 1.7 - 2.2 x 1016 cm -3
(77 K)
p
= 2.6 - 3.1 x 10- 3 Q cm
g = 1.1 - 1.2 x 105 cm2VN s
(77 K)
(77 K)
The end faces (smallest dimensions) are ground and polished, but not etched.
EXPERIMENTAL SETUP:
Optical measurements were made in the Voigt geometry with the magnetic
field
B
wavevector
directed
q.
vertically
and
the
light
propagating
They were made in the middle infrared
horizontally
with
spectral range
at
wavelengths between 9.6 and 11.6 microns.
The experimental setup consisted of
the following apparatus shown in Fig. 3.1.
Liquid He
Cold-finger
Dewar
Bitter
Magnet
Liq. He cooled
Germanium
Bolometer
IRCoil
Heater
Chopper
Fig. 3.1
Polarizer
Filter
Experimental
IF
setup
spectra in the Voigt geometry.
in the text.
for
obtaining
magneto-optical
absorption
The individual components are described
MAGNET:
The magnet was a Bitter solenoid capable of fields of 0 to ±15 T in a
4-inch diameter bore.
The InSb samples were mechanically fastened to a copper
sample holder which was screwed against the copper end plug of the liquid
helium cold-finger of a stainless steel research dewar made by Janis Research
Co. The optical windows were 2 mm thick zinc selenide, at room temperature.
IR SOURCE:
The IR source was a Model PE-1 coilform heater from Buck Scientific,
rated at 12 to 14 amps. It was operated at 14.0 amps, at a power of 44 watts.
The effective black body temperature is estimated to be approximately 1000 0 C.
(This was not measured, nor was the information available from the supplier.
Globars of SiC supplied by Perkin Elmer are rated at temperatures of about
1100 0 C.) The active area is 0.128 inch diameter by 1 inch long, and consists
of 4 strands of nichrome wire wound tightly on a porcelain cylinder.
source
is
imaged
reflective optics.
on
the
entrance
slit
of the
monochromator
The
with
f/3.5
A long-wave-pass filter is placed immediately in front of
the slit to eliminate diffraction orders higher than the first.
MONOCHROMATOR:
A Spex model 1680B, 0.22 meter double grating monochromator was used.
The gratings were 75 lines/mm, blazed for about 9.8 microns.
A long wave pass
filter at the entrance slit was dielectric coated Ge, with 5% cut-on at 7.6
microns, transmitting 80% (average) from 7.72 microns to 15 microns.
Coating Lab, Inc., Santa Rosa, CA, No. L07600-9B)
is given
by
:D"
"
(Optical
The spectral resolution R
R(gm) = 0.5(slit width/220 mm)(13.33 gm) cosO
6 cm'
for 2.5 mm slit, at 10 gm
OPTICS:
Reflecting optics were used to image the exit slit of the monochromator
onto the sample and then onto the detector.
surface aluminized glass or pyrex.
The mirrors were standard front-
The focusing mirror was of 8-inch diameter
and 1 m radius, in a 1:1 imaging configuration.
The collecting mirror was of
6-inch diameter and 0.8 m radius and approximately 1:1 imaging.
DEWAR:
The stainless steel optical cold finger dewar was made by Janis, Inc..
The main body of the dewar was 8 inches in diameter, with a 1 1/8-inch
diameter liquid helium cold finger.
The 2-inch o.d. outer tail has two 2-mm
thick by 3/4-inch diameter ZnSe side-facing windows for IR transmission.
SAMPLE HOLDER:
The InSb sample holder was machined from 1.1 inch diameter copper.
The
sample is held against the copper with Dow High Vacuum Grease and clamped with
beryllium
copper
differential
finger
contraction
stock.
Stresses
of copper and
induced
in
the
InSb were sufficient
sample
by
the
to fracture
the
bottom of the sample after several thermal cycles. The result was that the
bottom broke free of the remaining sample to a depth of about 1 mm.
The
induced stress in the sample was nonuniform, since it was applied at only one
face, and the experimental results are in good agreement with the strain-free
theory.
Additional work should be done to determine whether there are any
stress related effects.
DETECTOR:
A Ge bolometer detector from Infrared Laboratories, Inc., Tucson, AZ was
used to detect the IR light.
It consisted of a Model HD-3 liquid helium dewar
with model LN-6 amplifier and Ge bolometer unit #325.
thick by
vertical.)
1 inch diameter KRS-5
There
is an
internal
The Detector has a 2 mm
window facing the horizontal.
filter wheel
having
six
1/2-inch
(Plane is
diameter
slots occupied primarily by far infrared filters which were not used in the
present experiment.
One slot was vacant, and this one was aligned between the
window and the light cone which led to the bolometer element.
The optimum
signal was obtained when this vacant aperture was misaligned by about two
thirds of its diameter from the axis of the light cone.
SIGNAL AND NOISE:
The light was chopped at 150 Hz and the output from the detector preamp
was measured by a Princeton Applied Research (PAR) 5101 lock-in amplifier.
With the PAR set on 250 mV scale the output was 1 V dc for 250 mV input at the
reference frequency.
With 2.5 mm slit width, the maximum dc output signal for
light transmitted through the sample at B = 0 was:
0.54, 0.31 and 0.15 V for 9.6, 10.6 and 11.6 pm light in al polarization, and
0.38, 0.15 and 0.05 V for 9.6, 10.6 and 11.6 gm light in n polarization.
The noise level in all cases was 0.002 V.
PROCEDURE:
1. Set the heater current to 14.0 amps.
2. Turn on Spex 1680B controller and 150 Hz chopper
(30 Hz motor, with 5-slot wheel).
3.Calibrate Spex 1680B (75 lines/mm) and set to 600 for 9.6 microns,
662.5 for 10.6 microns, or 725 for 11.6 microns.
4. Set wire grid polarizer (gold on AgBr) for 7t or a 1 polarization.
5. Set Spex slits to 2.5 mm.
6. Turn Ge bolometer bias and preamp on and uncover KRS-5 window.
7. Set lock-in amplifier (PAR #5101) to 250 mV scale,
with prefilter at 0.3 s and postfilter at 0.1 s.
8. Set Keithley 182 digital multimeter to 3 V scale with internal filter off.
9. Adjust optics and dewar position for optimum signal.
10. Adjust filter wheel on bolometer, and reoptimize optics.
11. Sweep B from 0 to ±15 T and back. (5-min or 2-min sweep rate from minimum
to max, or max to min.)
3.3
Results of the Magnetoabsorption Experiment
The
most
significant
feature
of
the
experimental
results presented
in
this section is the observation of a strong magnetic field reversal effect in
the 2(oc+o
the
Yl
s
absorption for the n polarization, and in the 2o
polarization, in the Voigt geometry.
investigated,
and
the
3.2,
shows
theory
predicts
c
absorption for
(The Faraday geometry was not
no
effect
for
this
geometry.
See
configuration
which
[Gopalan 85].)
Figure
displayed
the largest
B0 11[111].
the
transmission
field-reversal
data
effect.
for
the
The data is for InSb #16
The wavelength is fixed at 10.6 gm in the
the magnetic field varies from -15 T to +15 T.
with
Yl polarization, and
The curve consists of two
traces for each sign of B0, corresponding to the field being swept in the two
possible directions.
labeled
with
the
The transmission minima in the figure are identified and
aid of the fan diagram
transition energies versus B .
[Goodwin 83].)
of Fig. 3.3
which displays
the
(Small anisotropy of the energies is neglected
In Fig. 3.2 we note that the infrared transmission is maximum
when the magnetic
field is zero, and decreases,
the field is increased in either direction.
with strong oscillations,
Both positive and negative
as
field
directions are shown in order to emphasize the change in the 2m c absorption,
labelled
'2',
inter-Landau
when the field direction is reversed.
level transitions as follows:
The labels identify the
The number 1 to 4 identifies the
change in the Landau level quantum number in the transition.
The letter s
indicates that spin-flip from spin up to spin down has occurred.
The letter L
indicates that an LO phonon has been emitted in the absorption process.
of the observed transitions are considered to
All
0.3
3
3L
0 .2 .
. .. .. ..
.....
3....
25
0..12
2
0-
-0.1
-15
-10
-5
0
5
10
Magnetic Field (T)
Fig. 3.2
a
Transmission (detector output) vs. magnetic field at 10.6 pLm,
polarization, for InSb #16 with ne=2x10'6 cm 3 , B,11 [111], q II [T10]
and T = 25 K.
15
250
.
3L
..
3s
3
2sL
200
2L
2s
150
11
1sL
9. 6 mO
10, 6 urn
- - - -, -
0
"
Is
20
0
- -- -
--
40
60
80
100
140
120
160
180
200s
Magnetic Field (kG)
Fig. 3.3
Fan
diagram
following [Weiler 74]
of
intra-conduction
and [Goodwin 83].
band
transition
energies,
The dashed curves give the
energies of the LO-phonon assisted transitions, in which an LO phonon is
emitted.
The curves were computed with the PB model parameters of
Goodwin et al., using their value of h0Lo = 24.4 meV.
in
the
center
Circles give
are
photon
the approximate
energies
used
in
the
Horizontal lines
present
experiment.
magnetic field positions of the absorption
peaks for 10.6 tm.
100
occur in the extreme quantum limit and to originate from the lowest Landau
sublevel, denoted aC(O), in the conduction band.
Therefore
'2'
denotes the
double cyclotron resonance transition between the aC(O) and aC(2) conduction
band levels.
is
Note from the fan diagram that the energy of the 2%c
somewhat
transition
smaller
between
than
ac(O)
twice
and
the
ac(1),
energy
so
the
of
the
designation
energy as 2o c is more symbolic than numerically precise.
cyclotron
of
transition
resonance
the
transition
The most important
aspect of the transmission shown in Fig. 3.2 is the significant change in the
depth of the 20C minimum when the direction of B 0 is reversed.
present case we see that the 2
c
For the
transmission nearly saturates to zero for the
negative direction of field, but remains well above zero, at about 14%, for
positive B
The 'zero'
.
line which is displaced about 0.005 units below the
field axis is the detector signal obtained by sweeping the field with the IR
source blocked.
In order to convert transmission to absorption I used the formula
Ac(B0) = a(Bo) - a(O) = - Iln[I(B )/I(O)]/L .
Here I(B )
and I(0) are
the detector
signals
(3.3.1)
at
finite field
and
at zero
field, after subtracting the detector signal with the beam blocked, and L is
the length of the sample.
This formula assumes that the zero field absorption
is high enough that multiple passes inside the sample can be neglected.
corrected
formula,
taking
multiple
passes
into
account,
but
neglecting
The
any
change in the absorption which occurs when the direction of the wavevector q
of the radiation is reversed, requires use of the formula for the transmission
(1 - R)e-aL
(3.3.2)
T =
1- R
2e -2cL
101
Here
(n -
1)2
R =
- 0.36
(3.3.3)
(n + 1)2
is the single
surface reflection,
a
is the absorption coefficient,
and L
is
In the present case R2e - 2 aL is less than 0.01 in the
the sample length.
9.6 pm to 11.6 pm range at B ° = 0 if one uses the value 1.0 cm - 1 for the
absorption at 11.26 ptm [Patel 71, Weiler 74], with L = 2.1 cm.
One must use
the X- 2 dependence of the free carrier absorption to correct for the different
wavelengths.
We next use Eq. (2.3.1) to compute the absorption for the 20 c peaks.
Using I(B )/I(0) = 0.14 for positive B and 0.015 for negative B
absorptions Aat of 0.92 cm -
and 1.97 cm - .
we get
A reasonable estimate of the error
bars for the stronger absorption is obtained by assuming that the zero offset
(the 'dark' signal) could be wrong by 50%.
This would make I(B )/I(0) equal
to 0.0075 or 0.0225 instead of 0.015, since the transmitted signal averages to
These values would correspond to absorptions
about 0.00 at the peak position.
of 2.30 cm -1
and
1.78 cm - , instead
absorption near the 2o
by taking
c
the
slower magnetic field
absorption
Lorentzian lineshape.
Better results
for the
peak could be obtained by using a shorter sample, or
blocked and then unblocked.
chosen,
of 1.97 cm - 1.
peaks
scans near the peak with the light beam
Note that if the transmission zero is improperly
will
deviate
significantly
from
the
expected
If too low a zero is used, the absorption peaks will
appear to be flattened on top, and if too high a zero is used, the peaks will
be excessively pointed on top.
Care was taken in the experiments to mask the samples so that no light
102
However, there was evidence of some light
could leak around the samples.
leakage in
an early
which was the first sample
data set for sample #17,
The evidence for this came from the aL absorption at 11.6 gm at high
studied.
fields near +15 T where the absorption saturates due to the strong absorption
at the
Here the transmitted signal from
(See Fig. 3.3).
transition.
C+LO
signal with the beam
the detector at saturation was higher than the 'dark'
Correction was made
blocked by about 10% of the transmitted signal at B ° = 0.
by shifting the 'dark'
The data was used to obtain
line up by this amount.
better understanding of the absorptions, but is not presented in the figures.
in the early experiments
of the data were made
Slight field displacements
where the time constant of the electronics was excessively long.
Fig 3.4 shows typical transmission data for the i
polarization.
For this
polarization the detected signal at zero field is weaker than it was for Gl'
due to the lower reflectivity
of the gratings, but the magnetoabsorption
is
The strongest absorption peaks in this polarization cause the
also weaker.
transmission to drop to only about 20% of the zero-field value, so the error
Note that only the
due to uncertainty in the zero line is lower than before.
20 c +oos absorption, labeled '2s', shows a significant when B0 is reversed.
Fig.
shows
3.5
the
magnetoabsorption
vs.
B
for
sample
B II [111] and q II [T10], for polarizations n and oL, at T = 25 K.
significant
2o c
feature in this figure
in the
absorption
is
_l polarization
measured
from
with
#16
The most
is the strong change in the absorption
when the magnetic field is reversed.
the base of the
peak
it is
at
If the
found that
the
absorption due to the 2o c transition changes by a factor of about three upon
reversal
of the
field.
This
compares
103
with
a factor
of 2 for the
spin
resonance absorption measured by Dobrowolska et al. and Chen et al. in samples
with lower carrier concentration at 118 jIm [Dobrowolska 83, Chen 85b].
None
of the other peaks for this polarization change by significant amounts when
the field is reversed.
are those at 2o c+o
s
The peaks of greatest interest besides the 20 c peak
and 3
c,
which are normally due to inversion asymmetry and
0.15
0.1
0.05
0
-0.05 '
-15
-10
-5
0
5
10
Magnetic Field (T)
Fig. 3.4
Transmission (detector output) vs. magnetic field at 10.6 lam,
t polarization, for InSb #16 with ne =2 x06 cm-3
and T = 77 K.
104
B 11 [110],
q II [T10]
0.6
0.5
E
0
0.4
C
0.3
a
I
0.2
0.1
0
-0.1
-15
E
C.)
1.5
01
I
1
-10
-5
0
5
10
-10
-5
0
5
10
e
0.5
0
-15
Magnetic Field (T)
Fig. 3.5
Magnetoabsorption vs. magnetic field at 10.6 Lm for InSb #16 with
n e=2x1016 cm - 3 , B oII [111],
(b)
q II [110] and T = 25 K.
L- polarization.
105
(a) ix polarization;
warping
[Weiler 78].
geometry
with
and
warping,
polarization
as
was
and
are
is
in violation
of the
selection
pointed out by Zawadzki
peaks
LO-phonon-assisted
orientation,
The existence of the 30 c peak (or shoulder) in this
at nC +LO
of
are
no particular
[Zawadzki 90,
isotropic
interest
rules
with
in the
associated
92].
respect
present
The
to crystal
work,
except
that they provide a good reference against which to measure the peaks of
interest.
C+0 Lo, occurs near 18 T for 10.6 gm radiation and is
The n=l peak,
largely responsible for the increasing absorption observed at large B .
The
peaks for n=2 to about 5 or 6 are seen as a series with decreasing intensity
at fields below 9 T.
The linewidth of the absorptions in meV may be found by
multiplying the linewidth
in magnetic field by the slope
of the curve of
The absorption for the it polarization displays
transition energy in Fig. 3.3.
only three peaks out to ±15 T.
These are identified, with the aid of the fan
C +0 +LO,
diagram of Fig. 3.3, as the
and 3C transitions.
20 +
C +0 S+0LO peak is seen to be quite a bit wider than the other two.
The
Part of
the extra width is real, and part is due to the lower slope of the transition
energy curve vs. magnetic field.
Of these peaks, only the weak 20c+0 s peak,
which violates the selection rules of Table 3.1, shows any significant change
upon reversal of B .
The smaller of these two peaks is seen to have a
pronounced shoulder on the low-field side.
in agreement with the selection rules.
The 3%oc peak is quite prominent,
The it spectrum is simpler than the 0l
because the absorbed photon contributes no angular momentum, corresponding to
v -induced
transitions,
while
in the latter case
the photon
contributes both
+1 and -1 units of angular momentum,
Z_ corresponding to the operators v+
+- and v.
Figure
3.6 qshows
and
II [10],the formagnetoabsorption
polarizations
SII [110]
B0 11[110] and q II [110], for polarizations
106
vs. B
nt and
for
#16
0and
, at sample
T
77 K.
TL, at T= 77 K.
with
(The
(The
higher temperature was used because a cold leak in the dewar prevented the use
of liquid helium.)
This geometry is the optimum one for the field-reversal
effect in the 2co +os absorption for n polarization.
The effect in the 2s peak
is seen to be quite pronounced, with a change in absorption by a factor of
is reversed, where the absorption is measured relative to
about 2.6 when B
the average height of the bases of the peaks.
quite a bit larger than the
C+
+oLO
peak on the positive field side, and
side.
quite a bit smaller on the negative B
strong peaks at 2o)
and 3o c .
Notice that the 2cc+cs peak is
Notice the rather surprisingly
All of the absorption peaks are broader at the
higher temperature, but the breadth of the 3c c peak in particular causes one
to
speculate
2o + o
that
other
some
absorption
be
might
which occurs at nearly the same field.
+
is in
polarization
violation
The peak at 20
rules for
of the selection
inversion
s
polarization,
For the
effect
is
still
strong
the
for
2
For the present case the absorption changes by a factor of 2 instead of 3.
2 (o cos
c+
c
[111].
absorption, but not as strong as in the previous figure for B along
see that the
7
peak when the temperature is lowered, as
field-reversal
the
for
asymmetry
will be seen in another figure for the case where q is along [111].
S01
as
The peak shifts to
and warping, and might be somehow related to impurities.
lower fields closer to the 2C0c+o
such
contributing,
We
absorption has completely disappeared and that the 3oc
absorption is now quite resolvable above the broader 2oC+cLO peak.
Figure
B II [T10]
geometry
3.7 shows
and
the
the magnetoabsorption
q II [111],
interference
20c(
20 +O (r)
and
B 0 II [110],
q 11[T10]
)
geometry.
but
One
107
is
at
sample
with
In
this
for
both
the
in
the
observed
weaker
than
significant
#17
T = 25 K.
is
effect
(field-reversal)
for
B
polarizations,
for both
transitions,
vs.
difference
it
was
in the present
case is the reversal of the positions of the stronger of the two peaks at
2wcc(o
)
relative to those at 20c+(s(c).
This effect will be shown to be in
0.6
0.5
0.4
-E-
0.3
CO,
&0
0
0.2
0.1
0
-0.1
-
-10
-5
0
5
10
-10
-5
0
5
10
(b)
E
1.5
0
1
I-
es
0.5
0 L
-15
Magnetic Field (T)
Fig. 3.6
Magnetoabsorption vs. magnetic field at 10.6 gm for InSb #16 with
n =2x1016 cm- 3 , B 11 [110],
e
o
(b) .1 polarization.
q II [110] and T = 77 K.
108
(a) it polarization;
15
(a)
0.6
0.5
E
0.4
ov
0.3
I
el
0.2
0.1
0
-0.1
-10
-5
0
5
10
-10
-5
0
5
10
(b)
2
E
1.5
C
1
I
e~
0.5
0
-15
Magnetic Field (T)
Fig. 3.7
Magnetoabsorption vs. magnetic field at 10.6 lam for InSb #17 with
n =2 x1016 cm - 3, B II [T10],
(b)
q II [111] and T = 25 K.
_L
1 polarization.
109
(a) it polarization;
agreement with the predictions of the theory.
Notice in the 7n polarization
that the forbidden 2o c peak appears to have split into two peaks.
is not understood,
liquid
nitrogen
The reason
but the higher-field peak is found to become strong at
temperature,
temperatures near 25 K.
while
the
lower-field
peak
becomes
strong
at
The simultaneous appearance of the two peaks leads to
speculation that T was slightly higher than 25 K, possibly due to sparseness
of the silicone grease used for thermal contact.
The amount of grease was
minimized in an attempt to reduce the mechanical stress induced in the sample
upon cooling.
Figure
3.8 shows
the magnetoabsorption
B II [001]
and
q II [T10],
orientation
the
interference
for
both
effect
vs.
polarizations,
is
not
B
at
expected
polarization because the 2+os( (n) and 2oc(0)
for
sample
#16
T = 25 K.
to
occur
In
for
are both zero when
the appearance
is along [001].
of a strong 20c(O
)
direction of B.
change
transition for the B 11[001] case was
We see that both the
2 0c(YI)
and
The 2c +0 s(t) absorption is quite small, but shows a clear
The
approximately 1.0 cm - ' when B
it
and F3
are finite and show small changes on reversal of the
on reversal of B.
For the
either
The violation of the selection rules by
first pointed out by Weiler [Weiler 76, 78].
20c+os(t)absorptions
this
absorptions are both forbidden
via the inversion asymmetry mechanism, since the angular functions F
A
with
2 0c(aL)
absorption is rather
is negative and 1.1 cm -' when B
large, being
is positive.
polarization the 20 c absorption is allowed via inversion asymmetry
and has a strength of about 0.3 cm - 1.
The 20 +cos and 30
transitions are both
forbidden, but are seen to have finite magnitudes of about 0.03 to 0.05 cm- .
For the
_L polarization the 20c+ s and 30c absorptions are both allowed and
observed.
110
0.6
E
0.5
0
0.4
&
O
0.3
O
0.2
0.1
0
-0.1
-10
-5
0
5
10
-10
-5
0
5
10
(b)
2
E
1.5
CO
0
rnIs
1
0.5
0
-15
Magnetic Field (T)
Fig. 3.8
Magnetoabsorption vs. magnetic field at 10.6 gm for InSb #16 with
ne =2 x 10 16 cm- 3 , B o II [001],
q II [T10] and T= 25 K.
(b) al polarization.
111
(a)
t polarization;
3.4
Discussion of Magnetoabsorption Results
I now present a quantitative description
of the magnetic field reversal
effect in the magnetoabsorptions at 2o c and 2cc+
s .
The effect is caused by
interference between the q-dependent (MD and EQ) and inversion-asymmetryinduced (ED) components of the absorption.
I use the decoupled model because
of its simplicity, but make modifications based on the PB model to obtain
results
which
are
more
accurate.
After
the discussion
of the transitions
which show the field-reversal effect, I make a comparison with theory of the
other inversion-asymmetry and warping-induced transitions.
We begin with the
function F (oapy).
the
spin
2
Oc+s(7) transition which depends on the angular
The description
resonance
transition,
and
of this transition is similar to that of
is
simpler
than
transition which depends on both FI and F3 .
2
o c+o
s(t)
transition is mixed EQ and MD.
that
of
the
20c(1_)
The q-dependent part of the
If this mechanism is considered by
itself, the absorption computed from the PB model is 0.015 cm - 1 at a field of
10.2 T and a transition energy of 117 meV (10.6 pm wavelength) for carrier
concentration 2 x 1016 cm - 3 and FWHM linewidth 30 cm - 1.
(See Fig. 2.15.)
The
inversion-asymmetry induced electric dipole (ED) part of the transition, taken
alone, causes an absorption of 0.32 cm - 1 for B
2.19).
directed along
[110]
(Fig.
Notice from the figure that this absorption is nearly independent of
whether the parameter C' or G causes the transition.
The matrix element for
this part of the transition is proportional to F 1, which is maximum along the
[110]
directions,
and
zero
along
the
[001]
and
[111]
directions.
The
interference effect is maximized when q is along [110], since this makes both
the
q-dependent
and
inversion-asymmetry-dependent
matrix element purely imaginary,
parts
of
as in the case of electron
112
the
transition
spin resonance
[Chen 85b].
SFI , we
If we choose B ° along [110] with q II [T10], so as to maximize
would
expect
coefficients
absorption
the
for
two
the
opposite
directions of B to be
0
jiVO-0T5 + i03-212 = 0.47 cm-' or 0.20 cm- 1,
with an average value of 0.335 cm - . This agrees well, to within about 20%,
with
the
absorptions
measured
in
the
experiment,
background due to other absorption processes.
after
subtracting
the
Next consider the experimental
geometry which places B 0 along [T10] and q along [111].
In this geometry the
q-dependent part of the transition matrix element is the same as before, since
it is only weakly dependent on sample orientation, and since B
direction which is equivalent to [110].
is along a
The ED part of the matrix element,
proportional to F 1, has the same magnitude as before, but now has both real
and imaginary parts.
This causes the interference to be somewhat weaker.
Evaluation of F for this geometry gives
FI= (1/16)-J-
compared
with
i
(1/16)( - i)
,
for
the
previous
geometry.
The
absorption
coefficients for the two opposite directions of B 0 are now given by
IiV.OTT5 + V032 (0.577 ± 0.816i)1 2 = 0.45 cm
1
or 0.22 cm -1.
(Notice that only the imaginary part of F changes sign when B
is reversed.)
The interference effect is only slightly weaker for the 2 c+cs(t) transition
in this geometry, as was the case for the as(7) transition [Chen 85b].
Now we consider the interference effect for the 20c(O)
transition.
The
analysis is similar to that given by [Bassani 88] for the heavy to light hole
113
valence band transition a+(0)-- a+(2).
level even though the wave
ladder.)
The q-dependent
meaning
that
moment.
(Note that a+(0) is a heavy mass
function is the bottom rung in the light-hole
2
part of the
the transition
state
contains
c(l)
transition is EQ in nature,
an oscillating
electric
quadrupole
If this mechanism is considered alone, the absorption computed from
the PB model is 0.62 cm -
at a field of 11.6 T and a transition energy of
117 meV, for carrier concentration 2 x 1016 cm - 3 and FWHM linewidth 30 cm (See Fig. 2.14).
.
The inversion-asymmetry-induced electric dipole (ED) part of
the transition, considered alone, causes an absorption of about 0.10 cm - 1 when
B
is along [111] (Fig. 2.17).
Thus the EQ-induced absorption for
calculated to be stronger than the inversion-asymmetry-induced
This result contradicts
statements by Johnson and Dickey
2
(c 1l)
is
ED absorption.
[Johnson 70]
and
Bassani et al. [Bassani 88] claiming that the EQ component of the absorption
is negligible.
It differs from the 20c+o
(n) and
o (n) spin resonance cases
where the q-induced absorption is smaller than the ED absorption.
element for the ED part of the
2 oc((L)
The matrix
transition is proportional to F 3 + FI
(decoupled approximation), which is maximum near the [111] direction, and zero
along the [001] direction.
The interference effect is maximized when q is
along
makes
[T10],
since
asymmetry-dependent
this
both
the
q-dependent
and
inversion-
parts of the transition matrix element purely imaginary,
as in the 2o +o (n) and os(
)
cases.
Figure 3.9, at the end of this section,
A
shows the imaginary part of F 3 + F1 versus angle 8, measured from the z axis,
for the case q II [T10].
the
value
If we choose B along [111] the function F 3 + F has
i/24 = 0.2041i,
and
the
expected
directions of B are
0
ilx/076
± iVO-
= 1.22 cm-
or 0.22 cm ,
114
absorptions
for
opposing
with an average value of 0.72 cm - '.
experimental
values.
contribution
The
These are significantly smaller than the
discrepancy
is
expected
of ionized donors [Zawadzki 80].
to
result
from
the
The matrix element for this
mechanism is expected to have a random phase, so the absorption due to this
mechanism
should simply add to the absorption
inversion-asymmetry mechanisms.
of the combined
EQ and
We can estimate the contribution of the ions
from the data for case with B II [001] and q II [110], for which both F
and
F 3 are zero, implying no contribution from inversion asymmetry and no field
reversal effect.
The peak 20oc(aj) absorption for this case is 1.0 cm
suggests an absorption of 0.38 cm - ' due to ionized impurities.
1
which
The absorption
for the opposite directions of B0, for the [111] case, would then be
IiVO'O2 + i(TM-12 + 0.38
in
much
better
0.53 cm - '.
agreement
with
the
experimental
values
of
1.7 cm'
and
Increasing the inversion asymmetry contribution from 0.10 cm - 1 to
0.13 cm - 1 improves
the theoretical result
purely phenomenological
require absorptions
fit, neglecting
to 1.70 cm
1
and 0.56 cm - 1.
the contribution of the ions,
of 1.03 cm - 1 due to EQ alone, and 0.083 cm'
inversion asymmetry alone.
what follows
= 1.60 cm - 1 or 0.60 cm - 1,
A
would
due to
These values differ substantially from theory.
In
we will use the value 0.13 cm - 1 for the ED absorption for
B 11[111], which is 30% higher than the value obtained with the PB model with
the parameters of [Goodwin 83] with inversion asymmetry parameter 8 = 65 a.u.
We next consider the experimental geometry with B
along [T10].
along [110] and q
The EQ part of the transition matrix element is unchanged, being
purely imaginary, and causing an absorption of 0.62 cm - '.
The ED part of the
matrix element is proportional to F 3 + F1 , which now has the value il8.
115
This
is smaller by a factor 0.6124 = VT38
ED
absorption
then
changes
than the B 11[111] case.
0.13 cm -'
from
to
0.049 cm - 1.
The effective
The
actual
absorptions predicted by theory, for opposing directions of B 0, are
2
SiVC062 ± iV1T4
with an average
+ 0.38 = 1.40 cm - ' or 0.70 cm - 1,
of 1.05
-1 .
cm -
The experimental
values for the 2o(o
transition in this geometry are 1.3 cm -1 and 0.59 cm - ' at 77 K.
1)
The values at
25 K are expected to be about 25% larger, or about 1.63 and 0.74 cm - 1.
The
change in peak absorption with temperature was obtained from measurements made
at 77 K and 25 K with B II [111] and q II [T10].
0
We now consider the experimental geometry with B along [T10] and q along
[111].
The EQ part of the transition matrix element remains imaginary, and
unchanged.
The ED part of the matrix element, proportional to F 3 + F1, now is
and differs from its previous value both in magnitude and phase.
complex
Evaluation of F 1 and F 3 for gives
F
=
(1/16)-
i
-
,
F3 = (1/16)( 5F - i
compared with -(i/16) and 3i/16 for the previous case.
F3 + F = (1/8)
2'
,
The sum is
- i
(magnitude v2/8), compared with i/8.
The absorption coefficients for the two
opposite directions of B are now given by
ivD--62 + VOT49(1.155 ± 0.816i) 12 + 0.38
= (V07-2
. +
T.3) + 0.44 = 1.38 cm - I or 0.81 cm -
with an average of 1.09 cm - .
,
The interference effect is slightly weaker for
the 2o0(01) transition in this geometry.
116
Somewhat better agreement between experiment and theory is obtained if we
observe
that
the
FI
(o+) contribution
to
the
a1
absorption
is
smaller,
relative to the F 3 (_)
contribution, than it is in the decoupled model by a
factor of about 0.85 .
This is shown by the PB model results of Figs. 2.17
We therefore use F + 0.85F 1 in place of F 3 + F in the oa
and 2.18.
element.
matrix
If we choose B along [110], with q along [T10], the sum F3 + 0.85F
has the value 0.1344i, which is smaller than the value 0.2041i which it has
when B
is along [111].
This reduces the absorption due only to inversion
asymmetry to 0.4334 times 0.13 cm -' which equals 0.0563 cm - .
We would then
expect the absorption coefficients for the two opposite directions of B to be
Ii/-.-6 ± i/0.056-31 2 + 0.38 = 1.43 cm-1 or 0.68 cm - 1,
If B
is along [T10] and q along [111], the ED part of the matrix element is
proportional to
F 3 + 0.85F = (1/16)( 4.15
- 1.85i{-
compared with (2.15/16)i for the previous case.
The absorption coefficients
for the two opposite directions of B 0 are now given by
Si-.-.
+ /T.056
(1.1144 ± 0.7026i)12 + 0.38
= (V0-.62±V 027)2 + 0.45 = 1.36 cm-1 or 0.84 cm - ,
with an average of 1.10 cm .
The interference effect is slightly weaker for
the 2o(a_) transition in this geometry.
117
The results are summarized below.
General case:
ac = Ii62± V2(T3(F 3+.85F)
q[T10], B[111]:
a =
q[T10], B[001]:
a = 0.62 + 0.38 = 1.00 cm-1
q[T10], B[110]:
a =
iv-
2+
0.38
± i-312 + 0.38 = 1.70 or 0.56 cm - 1
_i2i56I
+ 0.38
= 10.7874i ± 0.2373il 2 + 0.38 = 1.43 or 0.68 cm - '
q[111], B[T10]:
a = 10.7874i + (0.2645 + 0.1668i)j 2 + 0.38
= 10.7874i ± 0.1668il 2 + 0.45 = 1.36 or 0.84 cm - 1
The
complete
angular
variation
of the
field-reversal
rotated in planes perpendicular to q is computed below.
matrix elements which contribute to each absorption.
effect
as
B0 is
Table 3.1 gives the
The wavevector q is
A
parallel to il.
Table 3.1
Matrix
elements
of the effective velocity operator for
q-dependent magnetic dipole/electric quadrupole, and electric dipole
transitions in the Voigt geometry, based on the decoupled model.
Transition:
V (MD/EQ):
fo
V (ED):
fo
20
Cs()
-l g*q
12F* 8 s
10
(o )
- 4V(F +F )8 s
13
(t)
-v2 icO KsIq
S
V2m *
1
2 c+(
0
-20V2 F 8 s
10
The ED contribution of the ionized impurities is not included in the table and
is taken to add incoherently to the
2
oc((l) absorption.
the other two transitions.
118
It is neglected for
The MD/EQ matrix elements in the table all have phase - i. The ED matrix
elements for the
-F 1,
and
20o
s,
and 2cc+
since
respectively,
[Cardona 86c].
a = (v-
q
s
and
86
both
are
and
real
positive
is then found from
The absorption for os()
y1
Re(16F,)) 2
0
+ Vii 110 Im(16F 11))2 + (V
Im(F *) = - Im(F1).
where I used
transitions have phases of F 1*, -(F 3 +F1 )
s
The absorption for 2Oc+os(nt) has the same
The two absorptions therefore
expression, but with different aq and a o.
show constructive or destructive interference on the same side of B0=0.
The
absorption for 2c(a ) is found from
a = (v
+V
111
Im(F)) 2 + (vrc'
Re(F)) 2 + a.
111
ions
IF,
is 3 times larger
Note that IF3
F - V24(F3+F ), with XL0.85 at 12 T.
where
than
q
16F 1 has
where the latter is maximum.
for B 11[110]
unit
magnitude along [110], and F3 has magnitude 0.204 along [111].
Figures 3.9 through 3.12, at the end of this section, show the variation
of F33 +FI 1
B0 is
and F 1 when
rotated
in
the
(T10)
perpendicular to q, with q taken along [T10] and [111].
3.15
expressions
given above.
a 111 =0.13 cm - '
For
2 Oc(a~)
a ions=0.38 cm - l .
and
calculation of Figs. 2.17 and 2.18.
but
it
causes
B II [110],
2oc+ws()
a
slight
q II [110]
I
relative
X=0.85,
used
of
used a =0.62 cm',
based
to
field-reversal
the
the
B II [T10],
I used a =0.015 cm l and a 10 =0.32 cm'.
q
from the
on
the
PB
The change from the case X=1 is small,
enhancement
case
at B 0= 12 T I
planes
Figs. 3.13 through
and 2c+%s(nr) computed
at 20c(a)
show the absorptions
and (111)
110
119
effect
q 11[111].
for
the
For
Figure 3.13 shows the field reversal effect for the 2c(a)
with q II [110].
line is for B =B
A
Zero angle corresponds to
A
absorption
A
=[001] and 4=[110].
and the dotted line is for B= -B
A
0
.
The solid
One can also consider
the dotted line as the continuation of the solid line for 0 between 1800 and
3600.
The
figure
shows
that
the
field
reversal
effect
is
finite
for
all
angles except 0 and 1800, where B II [001] and [00T] and the field reversal
effect vanishes.
For the assumed orientation we see that the absorption is
larger when B is up (along [001]) than when B is down, which is opposite to
the case observed in my Sample #16 in Figs. 3.5 and 3.6.
Rotation of my
sample by 1800 about q would achieve the orientation of Fig. 3.13.
Fig. 3.13 that the maximum field-reversal
direction towards the [001] direction.
Notice in
effect is displaced from the [111]
The magnitude of the displacement will
depend on k in the function F, and careful experimental determination of the
displacement might be used to measure k. For low fields with
=1I the maximum
should occur for sin0=2/3 or 0=41.80.
Fig. 3.14 shows the field reversal effect for the 20c(a 1 ) absorption with
A
q II [111].
A
Zero angle corresponds to C=[110] and
=[Ii2].
A
again is for B =B0
that
the
field
The solid line
A
and the dotted line if for B= -B
reversal
effect
is
maximum
when
B
~.
The figure shows
is
direction, and disappears when it is along any [112] direction.
maxima occur every 600.
than
it
was in
weaker
[110]
The zeros and
is down, which is opposite to the case
observed in my Sample #17 in Fig. 3.7.
substantially
any
For 0=0 we see that the absorption is smaller when B°
is up (along [T10]) than when B
is
along
for
B 11[110]
the previous
case
Notice that the field-reversal effect
in
with
experiment.
120
the present case
q II [T10],
with q II [111]
which agrees
with the
Figure 3.15 shows the field reversal effect for the 2oc+os(t) absorption
for the same geometry as Fig. 3.13.
Chen
et al. for the
The figure is quite similar to Fig. 6 of
spin resonance
[Chen 85b],
except that
the effect
is
stronger in the present case.
It shows smaller absorption when B
[110]
[110], which is opposite to the case of the
than when
it is along
is along
experiment shown in Fig. 3.6, as expected from the 20c(_L) result.
Figure 3.16 shows the field reversal effect for the
for the geometry of Fig. 3.14.
et al. [Chen 85b],
case.
2
oc+os((r) absorption
The figure is quite similar to Fig. 7 of Chen
except that again the effect
It shows smaller absorption when B
is stronger in
the present
is along [110] than when it is
along [1TO], which is opposite to the case of the experiment shown in Fig.
3.7, as expected from the 2%c( L) result.
Next we will compare the observed absorption strengths at 20
3c c with those predicted by the anisotropy theory alone.
et
al.
[Weiler 78]
[Favrot 75,
76],
but
Weiler
for
the
the comparison
experimental
used
20c+os and
This was done by
obtained
data
an incorrect
c,
formulation
in
1975
of the
radiation interaction and neglected the q-dependent EQ and MD mechanisms which
cause the field-reversal effect in the 2Oc(o)
and
2 cc+s(rt)
absorptions.
The transitions which are expected to result primarily from anisotropy at
k
= 0 are 2oc(t) and 2cc+os(_L) from inversion asymmetry, and 3oc(nt) from
The other
warping.
expected
to
have
three transitions,
additional
with
contributions
the
which
opposite
result
polarizations,
from
the
are
ionized
donors, and also from the electric quadrupole (EQ) and magnetic dipole (MD)
mechanisms.
121
For the observed Lorentzian linewidth 2y = 30 cm - ' (FWHM) the PB model
predicts
an
absorption
of 0.18
10.6 jtm for B II [001].
to
0.28 cm -
for
the 2 0c(t) transition
at
Here the first number is the value which results
from the parameter G in the single-group approximation (G' = G) if no other
parameters contribute to the absorption, and the second number is the value
which
results from the parameter
C' if all other parameters
are neglected.
Both G and C' are matched to a value of the decoupled third-order inversion
asymmetry parameter 68 of 65 a.u.
(I neglect C relative to C'.)
The observed
absorption
about
-1
cm 1.
2oc+os(LY)
transition
for
this
the PB
0.18 cm -1
for
transition
model
is
predicts
B II [111].
The
0.3
0.26 cm -'
observed
0.2 cm - '. The absorptions at 2oc+os(Tl)
For
absorption
for
absorptions
are
B II [001]
and
0.3 cm-1
and
are simplified by the fact that the
angular functions F 0 and F 2 associated with the
7
transition matrix element are nonoverlapping when B
[111].
the
and a+ parts
of the
is parallel to [001] and
This implies that the absorptions for these two directions of B
are
independent of the direction of E in the plane normal to B 0 .
The
3%c(7c)
transition
is
caused
by
warping
and
is
(approximately)
proportional to the squared magnitude of the angular function W3.
model
predicts
an absorption
10.6 Lm for B0 11[111].
Next
consider
the
of 0.28 cm - '
for a linewidth
The observed absorption is 0.4 cm 3mc(ol)
transition,
for
which
The PB
of 30 cm - '
at
absorption
is
.
the
proportional to IW2 -2W 4 12, in the decoupled approximation, if we neglect the
contribution
from the ions, and possibly from bound impurity atoms.
absorption should be finite for the cases of B
but zero for B0 11[111].
This
parallel to [001] and [110],
We see in the data that the absorption is in fact
finite but weak for this latter case, indicating that a mechanism other than
122
warping is present [see Zawadzki 90, 92].
W2 = 0
and
W -2W
2
IW2 -2W1
= 0.3125.
SW-
= 0.2577.
2W4
slightly
relative to
12W 4 -0.85W2
= 0.25.
4
For
For
For the B II [001] case we have
B 0 11[110]
B II [T10]
with
with
q II [T10]
q 11[111]
we
we
have
have
In the PB model the magnitude of the W2 part is reduced
that of W
so that at 7.5 T the
relevant
quantity is
and the values become 0.250, 0.294 and 0.244, respectively.
The
absorption for the B II [001] case obtained with the PB model (Fig. 2.26) is
0.17 cm -1 , so the other two absorptions would be 0.235 cm after
multiplying
absorption
absorption
by
(.294/.25) 2
(.244/.25) 2.
and
7.5 T
for
B 11[001]
subtracting the background absorption.
absorption
2 mC+oLO
peaks overlap with the broader
near
is
about
absorption at 3o
0.30 cm - '.
is about 0.25 cm -1 .
c
is
For
For
The
and 0.16 cm - 1
observed
absorption.
10.6 Lm
The peak
0.35 cm- 1
approximately
B II [110], q 11[111], the peak
B II [110],
q II [110],
the
correction
is
applied
to
account
absorption increases to 0.31 cm - .
should be larger
smaller.
than that
peak
This last value is for T = 77 K as
opposed to 25 K, so the peak is lower due to the increased linewidth.
25%
after
for
the
change
in
linewidth
If a
the
The PB model predicts that this absorption
for B II [001],
but it appears,
Finally, the peak absorption for B° 11[111],
instead to be
q II [110] is found to
be about 0.1 to 0.15 cm - ', in violation of the warping model selection rules
which indicate that this should be zero.
Additional data taken at 9.6 gm at 77 K show that the 3Oc(a_)
absorption
is larger when q II [110] than when q II [11], for B II [110], as predicted.
Absorption at 9.6 gm and 25 K shows 3c(al) peak absorption of 0.4 cm - ,
nicely
separated
from the 20c+ L absorption.
was measured to be about 0.35 cm-1
123
The absorption
at 11.6 jim
SII11 001
111
112
114
221
110
221
111
112
114
0.25
0.2
0.15
0.1
0.05
0
-0.05
30
60
90
120
150
Angle e (Degrees)
A
Fig. 3.9
A
A
F
vs. 0 about q = [110].
= [110];
the rotation is right-handed about rl.
Zero angle corresponds to
A
= z,
are
F3 + F
3
+ 0.85F.
1
F = ll(sin36 + 5sin6).
3
64
124
F = /(3sin3O
1
64
Dashed lines
- sinG).
180
AII 112
1
1I112
01il
121
110
-60
-30
0
211
101
0.2
0.15
0.1
0.05
0
-0.05
-0.1
-0.15
-
-90
30
60
Angle 0 (Degrees)
Fig. 3.10 F + F
A
A
vs. 0 about ii= [111].
Zero
angle corresponds
A
= [110] , 4 = [112]; the rotation is right-handed about r.
is F 3 + 0.85F.1
3 16
3
3
125
1
16
3
Dashed line
90
AII 001
11001
114
114
112
112
221
111
110
221
111
112
114
0.04
0.02
0
-0.02
-0.04
-0.06
-0.08
30
60
90
120
150
180
Angle 8 (Degrees)
Fig. 3.11
A
= [110];
F 1 vs. 0 about
A
Ti
= [T10].
Zero angle corresponds to
the rotation is right-handed about
126
fj.
A
A
= z
F 1 = 64-(3sin30 - sine).
AII
112
01i
121
i10
211
i01
0.06
0.04
Im(F)
0.02
0.02
(111) Plane
-0.02
-0.04
Re(F)
-0.06
-90
-60
0
-30
30
90
60
Angle 8 (Degrees)
A
F
Fig. 3.12
vs.
0
r = [111].
about
Zero
A
A
A
[110] ,
angle
= [112]; the rotation is right-handed about TI.
F
F1
=
-
16
3
127
i cos3
F
.
corresponds
to
II 001
112
114
111
221
110
221
111
112
114
2
1.5
E
0o
C
0
L.
<0
0.5
0
0
30
60
90
Angle
Fig. 3.13
2
120
150
180
e (Degrees)
absorption for B° in (110) plane, with q II Ti = [T10].
_oc(1l)
A
A
A
Zero angle corresponds to 0 = z,
= [110].
A
Solid line is for B II z;
A
dashed line is for B II- z.
Circles are experimental data normalized to
linewidth at 25 K.
128
A
11112
01i
121
iO
-60
-30
0
101
211
1.5
E
1
O
O
04-
0
Cn
0.5
-90
-90
60
30
90
Angle 0 (Degrees)
A
Fig. 3.14
Zero
2c(_1l) absorption for B0 in (111) plane, with q II T11= [111].
angle corresponds
to
A
= [110],
B II [110]; dashed line is for B II [110].
129
A
( = [112].
Solid
line
is for
II 001
114
112
111
221
110
221
111
112
114
0.5
0.4
E
0.3
0
0
Cn
0.2
0.1
0
30
60
90
120
150
180
Angle 8 (Degrees)
Fig. 3.15
20c+os(x)
Aq
q II1T1 = [T10].
absorption
for
Zero angle corresponds to
A
A
is for B 0 II z; dashed line is for B II-z.
0
130
B0
in
= z ,
(110)
= [110].
plane,
with
Solid line
S II 112
01i
121
110
-60
-30
0
101
211
0.5
0.4
E
0.3
0
O
0
ff0
0.2
i
0.1
0-90
90
60
30
Angle 0 (Degrees)
Fig. 3.16
A
q II Ti = [111].
2o +c+s(t)
absorption
for
B0
in
AZero
angle
corresponds
to
(111)
Zero angle corresponds to ( = [110] ,
line is for B II [110]; dashed line is for Bo II [110].
131
A
plane,
= [112].
with
Solid
4. SPIN-FLIP RAMAN LASER
Introduction
4.1
Spin-flip Raman (SFR) scattering in n-InSb is a process in which infrared
light
scatters inelastically
from conduction
electrons near k = 0 in InSb in
the presence of an applied dc magnetic field, causing the spins of individual
electrons
to
flip.
In
transition
from a lower energy spin-up state to a higher energy spin-down
the Stokes
scattering process
state, gaining the spin-flip energy ho s =
RB
is the Bohr magneton, and B
an electron
makes a
IBB 0 .g*
Here g* is the g-factor,
.
is the applied magnetic field.
The scattered
photon loses the amount of energy that the electron gains, so
)SR = p--s
SR
p
s
where OSR is the scattered Raman Stokes frequency and (p is the pump (or
incident)
frequency.
If
the
pump
is
sufficiently
intense
the
scattering
process can become stimulated, provided that the end faces of the crystal are
polished to form an optical cavity and the temperature is below about 40 K.
This can produce significant power conversion from the pump frequency to the
Stokes frequency with free carrier concentrations ne in the range 1x1014 cm - 3
to about 5x10
cm - 3 . The device which is based on this process is known as
the spin-flip Raman laser, or SFR laser.
Features which make spin-flip Raman scattering in n-InSb of interest are
the
large
g-factor
and
the
very
large
scattering
cross
resonant when the incident photon energy approaches
section
which
is
the band gap energy
The effective Hamiltonian for conduction electrons in InSb is
E = 0.23 eV.
g
spherical to lowest (second) order in momentum, and contains terms which are
similar
to
those
in
the
free-electron
133
Hamiltonian,
but
with
modified
coefficients.
The spherical symmetry of the lowest-order terms is a result of
the cubic symmetry (Td point group) of InSb.
The effective mass m* is about
0.0136m, where m is the free electron mass, and the g-factor at B° = 0 is
about
which is 25.5 times larger in magnitude than the free electron
-51,
g-factor
and of opposite sign.
term is
- (1/4)ag*/laE (Yafet model), which has a value 108 times larger than
The coefficient
the free electron value of - 1/(4mc2).
of the spin-orbit interaction
Here, the electron energy E may be
measured relative to the bottom of the conduction band or the top of the
valence band.
the large
energies
It is the large effective spin-orbit interaction which produces
SFR cross section
which
band-valence
are
band
The
resonance.
a
for conduction
significant
approach
resulting
fraction
must
cross
electrons.
be
of
used
section
is
E ,
to
At incident photon
a
coupled
properly
significantly
account
larger
conduction
for
the
than
the
Thomson cross section for elastic scattering from free electrons, exceeding it
by factors of the order of (m/m*) 2 , or (g*)2 , times [EgaOp/(E-2 h 2- )]2
The
2000 over the Thomson
cross
first factor
section.
gives
an enhancement of about
The second factor, which is resonant at E , is about 4/9 for 10.6 jLm
CO 2 laser radiation with photon energy equal to half the band gap, but becomes
very large for wavelengths near 5.3 gm.
In the above approximation the cross
section is independent of B 0, and is finite at B ° = 0.
For magnetic fields of
1 T or more the nonparabolicity of the bands and the variation of the energy
gap with B
cause the low-field formula to become inaccurate, so that more
detailed calculations based on the coupled-band model are required.
Enhanced electronic Raman scattering from conduction electrons in InSb in
a magnetic field was predicted by [Wolff 66].
band
models by
[Yafet 66]
and
[Kelley 66]
134
Calculations based on coupled
indicated
that
spin-flip Raman
scattering with no change of the orbital quantum number (An=0) should be the
most
important
of the
various
inter-Landau
level
and
spin-flip
scattering
processes. Spontaneous SFR scattering of 10.6 gm radiation from electrons in
n-InSb was reported by [Slusher 67], and stimulated SFR scattering was first
reported by [Patel 70].
SFR scattering from bound electrons and holes in CdS,
which is a wide-gap hexagonal material, was observed by [Thomas 68].
theory
in
this case
is
different
from that
for
InSb
due
to
the
The
lower
point-group symmetry and the need to consider both bound and excitonic states
of electrons and holes.
cw operation of an InSb SFR laser was first obtained
by [Mooradian 70], using a CO pump laser operating at 5.32 jim, and high-power
operation of an SFR laser using a TEA-CO 2 pump laser at 10.6 gm was obtained
Since 1970 SFR scattering has been observed and studied in
by [Aggarwal 71].
many semiconductors, including narrow-gap and wide-gap, cubic and hexagonal,
nonmagnetic
exist,
and
magnetic
including
ones
materials.
by
Excellent
[Dennis 72],
reviews
of SFR
[Scott 75, 76, 80],
[Wolff 76 & 77], [Smith 77], [Geschwind 84] and [Hdfele 91].
scattering
[Colles 75],
SFR scattering
in dilute magnetic semiconductors (DMS) is reviewed by [Ramdas 88], and is
also covered in [Hifele 91] and in a recent review of laser spectroscopy in
high magnetic fields by [Heiman 92].
The purpose of the experimental work was to study effects influencing the
operation of the InSb SFR laser at magnetic fields up to 18 T.
This was the
highest field which was readily attainable in a 2-inch bore Bitter magnet at
the time the work was conducted.
there
was
interest
in
Particular interest centered
processes
Previous studies had ended below 11 T, and
which
could
occur
at
on the magnetic field region near
higher
12.5 T where
h0o s becomes equal to the zone-center LO-phonon energy hOLO= 24.6 meV.
135
fields.
This
implies
the possibility of increased spin relaxation due to emission
phonons,
and
polaron
pinning effects
similar
to those
observed
cyclotron resonance energy hwc becomes equal to hCoLO.
studies a frequency-doubled
of LO
when
the
For these high-field
TEA-CO 2 laser producing radiation in the 4.8 to
5.4 gm region was chosen as the pump.
This choice took advantage of the
resonant enhancement in the scattering cross section which occurs when hCo is
P
near to Eg , and also avoided the free-carrier absorption which becomes large
at fields above 10 T for 10 im pumping.
I made a detailed study of the operation of the 10 jm-pumped SFR laser at
fields up to 13 T, in addition to the high-field operation of the 5 jm-pumped
SFR
laser.
Emphasis
magnetoabsorption
harmonics,
orientation
along
at
was
placed
on
cyclotron-resonance
with
the
similar
the
(CR)
effect
of
harmonics
LO-phonon-assisted
linear
and
free-carrier
spin-flip
transitions.
CR
The
dependence in the applied field (anisotropy) and pump-wavelength
dependence of these transitions was studied in detail.
136
4.2
Theory of Inter-Landau Level Light Scattering
In
this
I
section
discuss
theory
the
of
level
inter-Landau
light
scattering from free carriers in InSb for comparison with my experiments on
stimulated
spin-flip
(SFR)
Raman
scattering.
2o
s,
transition
at
2 0 c+s
and
finite
k .
discuss
and
origin
the
and present numerical results for
approximations for the cross section do/dI,
the
I
Raman scattering transitions at k = 0, and the o c
The
20c
and
2
c+'Os transitions,
although
not
observed in the stimulated SFR experiments, are important because of their
connection with the magnetic field-reversal effect in magnetoabsorption
The numerical calculations for these two cases
is discussed in Chapter 3.
correct results
given
which
by Wright
et al. [Wright 68].
Small effects
which
result from anisotropy will be reviewed and discussed.
Two treatments of the cross section do/dQ are considered: the decoupled
model, and the axial 8x8 Pidgeon and Brown model.
expressions which are valid at low fields.
the selection
The first employs analytic
It provides a good description of
rules in the axial approximation,
and the lifting of the sel-
ection rules, due to warping and inversion asymmetry, for arbitrary values of
The second treatment obtains numerical results for
the momentum k along B .
scattering
in the axial
electrons,
from conduction
approximation,
which are
accurate for arbitrary values of magnetic field B and incident frequency o 1.
For the purpose of scattering, the free carriers in InSb are regarded as
free
strong
elementary
spin-orbit
particles
with
interaction.
effective
small
The
spin-orbit
mass,
large
interactions
g-factor,
result
from
and
the
multiband nature of the infinite Luttinger-Kohn or 8x8 Pidgeon and Brown
137
Hamiltonian,
just
as
the
atomic
spin-orbit
two-band nature of the Dirac Hamiltonian.
understood
interaction
which
Luttinger-Kohn
are
(LK)
approximation
obtained
from
transformation
(EMA) results
the
the discrete
coupled
[Luttinger 55].
from the lowest-order
utilizes decoupled Hamiltonians to order k2 .
approximations,
kinetic energy
nature of the lattice
representation
of the
free
the
and valence band
Hamiltonian
The
via
effective
the
mass
LK transformation,
and
In this, and higher finite-order
associated with the point group symmetry remains.
the
from
These interactions are most easily
in the framework of the decoupled conduction
Hamiltonians
in
results
carriers,
disappears,
but anisotropy
The anisotropy is present
in contrast
with
the
atomic
where the anisotropy is present in the potential energy of the
lattice, V(r).
Some general properties of inter-LL Raman scattering are listed below.
The strongest scattering is electric dipole (ED) in nature, and the associated
electronic
transitions
conserve
parity
at
k = 0.
Magneto-optical
effects
resulting from cubic anisotropy tend to be small, with a few exceptions where
heavy holes
act as initial or final states.
The strongest
transitions
occur
in the axial approximation, in which the quantum number n+m is conserved,
where n is the Landau level number and m is the "effective" angular momentum
of the LK basis state along B0 .
The actual point group symmetry is the
intersection of the tetrahedral point-group Td with the axial symmetry group
Cooh, which contains only the identity element for a general direction of B0
[Rashba 61a, Trebin 79].
In spite of this, the axial approximation
good for the conduction and light-hole bands in InSb near k = 0.
dipole
nature of the inter-LL
scattering causes the intensity
is very
The electric
pattern of the
scattered radiation to be the same as that produced by an oscillating electric
138
dipole [Yafet 66, Romestain 74].
This applies to both the magnetic scattering
caused by the spin of the scatterer, and the charge scattering caused by the
charge.
light
The dependence of the scattering on the polarization of the incident
has
occurs
electric
via
virtual
dipole
character
interband
also.
electric
This
dipole
is
because
transitions
in the
the
scattering
coupled-band
approximation.
4.2.1
Scattering Cross Section
The scattering cross section is given by
d/dl
2,
= (e2/mc 2 )2(o 2 /1)IAfo
(4.1)
where e2/mc 2 is the classical electron radius, 2.82x10 the incident and scattered frequencies, and A
perturbation
theory via the
Golden-Rule.
the effects
associated with
the InSb band
involving
the refractive
indices,
3
cm, o
and
w2
are
is an amplitude computed from
This
latter quantity
structure.
contains
An additional
(n2/n 1 2, must be included
if nl n 2 .
all
factor
If
the cross-section is computed as a ratio of energy fluxes rather than photon
fluxes, one must replace (n2/n 1 )2(o2/)
In
the
infinite
by (n2/n)(2/31) 2.
coupled-band
Luttinger-Kohn
representation
at
k=O
[Luttinger 55] and the 8x8 PB model, the
scattering amplitude A
A
forOt-
in the ED approximation is given by the expression
A
f I ro
E
A
A
I
fr
hO - E fr
(4.2)
+[(e V)(e V )H
139
A
A
Here e
and e 2 are the complex unit polarization vectors of the incident and
scattered
level
radiation;
o, f
wave functions;
A
where
first-order
initial,
final
and
intermediate
in the infinite LK
It is
A
(e*.e )fo in the free electron
the intermediate
It
the
vV kHeff is the velocity operator
it only contributes
A-v.
r label
The last term results from terms of order k2 in Hff.
representation.
replaced by
and
states
has
to elastic
scattering.
The first term, involving
Ir>, results from second-order perturbation
resonances
optical
case and the infinite LK case,
transitions.
when
the
incident
In
the
large-field,
radiation
theory in
excites
allowed
low-temperature
(extreme
quantum limit) approximation, the sum over intermediate states is carried out
as
in
the
intermediate
one-electron
and final
problem,
states
valence band intermediate
virtual
with
Ir> and
initial
state
If> unoccupied
lo>
occupied
and
[Wherrett 69].
For
states, the many-electron viewpoint requires that a
transition from r to the final state f
in the conduction band
(cb)
precede the relaxation of the cb state o to the hole r created in the vb by
the first transition.
4.2.2
(See [Dennis 72].)
Scattering in the Decoupled Representation
This approximation
functions,
and an
uses decoupled
effective
conduction
scattering amplitude
between these wave functions.
and valence
operator
band wave
which acts
on or
The amplitude operator A seat is obtained as a
power series in the kinetic momentum operator k, and is useful when B and
kz
are sufficiently small.
both
the cases
description
lifting
This approach has the advantages that it works for
of finite k
of the selection
of the
selection
and finite
rules
rules,
to
anisotropy,
and provides
for the axial approximation,
lowest
140
order in
k,
for
the
a simple
and of the
anisotropic
The disadvantages are that the approximation is valid only for very low
case.
fields if the incident photon energy is close to the band-gap energy Eg, and
that the coefficients of the k-dependent terms can be quite complicated.
In the low-B
limit the scattering amplitude A
the matrix element of a matrix operator A
decoupled wave functions.
may be approximated by
between two- or four-component
Thus [see Wolff 75]
A = <f
olAs
tlo >,
0 scat 0
fo
where
Ifo>
and
o0>
are
the
component wave functions If> and
the
simplest
representation
of the
(4.3)
decoupled
Io>.
representations
of
the
infinite-
The decoupled wave functions provide
actual
multiband
wave
functions.
The
latter can be derived from the decoupled wave functions by inverting the LK
transformation.
The operator A scat is obtained as a power series in k.
lowest order is ko which gives elastic and spin-flip scattering.
The
The result
in this order is quite simple, but the result to order k 2 contains many terms,
and is rather difficult to obtain even in the Yafet approximation.
In this
order the scattering can cause a change in the orbital state as well as the
spin
of
the
scatterer.
Both
orders
show
resonances
corresponding to the interband energies Eg and E +A.
at
frequencies
One of the terms in
order k 2, associated with Faraday rotation, shows an additional resonance when
o= 0.
I
have
used
operator
techniques
to
valence, and split-off to valence band cases.
Ac , A
and A
.
obtain A scat
for the
conduction,
These operators will be denoted
The derivation to order ko (or unity) is quite simple.
result for A c is
141
The
2
2P2
A=
(1+2F) + -
1
-
3m
9 (0) +
-
I
E
(4.4)
G*
E"
g
2P
-
ih(a)
2
1
the
derivation
c0)
E
3m
where
1
.-
is
Xe 2-)
2
-
2' 2
2( )
on
the
invariant
,
1
2
g
based
expansion
of
the
PB
Hamiltonian, and makes use of some matrix algebra given in Appendix A. Notice
that the summation over intermediate states has been replaced by a sum over
bands [see Braun 85].
Here 1+2F is the contribution from the free electron
mass and from higher p-like conduction bands, P is the valence to conduction
band momentum matrix element, E is the valence to conduction band energy gap.
A
e
A
and
scattered
e2
are
the
radiation,
complex
co
is the
unit polarization
frequency
vectors
of the
of the
incident
incident and
radiation
(actually
(01),
E'
E +A ,
g
(4.5)
and
E'2
E2
E_2
E
g
-
p(o
h22
My definition of the resonant function
,2
E'2
p,2(0o)
2
h2
2
(4.6)
is larger than that of Makarov
[Makarov 68] by a factor (E'IE )2. My choice causes both (p2 and (pl to equal 1
when 0)=O.
142
The following are important points for the isotropic model:
* The first term in A c (Eq. 4.4) causes elastic scattering.
causes
spin-flip
approximation
scattering
and Faraday
is independent
of B .
rotation.
The
The second term
Notice that
spin-flip
Ac
in this
scattering is elastic
if
B = 0, and inelastic (Raman) if Bo# 0.
* The first term in A c may be regarded as scattering from the charge of the
electron (charge scattering), and the second term as scattering from the spin
(magnetic
scattering).
scattering
amplitude
In
is
this
small-B0
antisymmetric with
A
approximation,
respect
A
to
A
scattered
polarization
light
of
must have
the
a component
light.
incident
that
Notice,
magnetic
interchange
of
the
A
polarization vectors e1 and e 2, i.e. proportional to e xe*.
the
the
This means that
is perpendicular
that
the
to the
scattering
is
independent of the optical wavevectors, and is therefore electric dipole (ED)
in
nature,
for
both
the
charge
scattering
and
the
magnetic
scattering
[Yafet 66; see also Romestain 74].
* Notice
this
also the
conservation
of parity for the state of the
A one-photon
approximation.
scatterer
in
ED transition changes the parity, so a
two-photon ED transition conserves parity.
* My
expression
for
Ac
omits
resonance
effects
associated
with
higher
conduction bands, since these resonances are at energies large compared with
It also omits contributions of higher bands to the magnetic scattering,
E .
g
since the 1/E2 dependence of the coefficients make these terms small for InSb.
* My expression for the resonant coefficient of the magnetic term, obtained
from the 8x8 PB model, is an improvement on the one obtained by Wolff.
143
The
latter was obtained from a 2-band model, with inclusion of the "free-spin"
part of the 2-band Hamiltonian
[Brown 72, Wolff 75].
The more accurate
result, consistent with the PB model, is obtained by replacing
(1/2)(1/m
s
- 1/m)(Pl/E ) = - (P2/3)(I /Eg)(1/Eg
- lIE')
1
in Wolff's expression for the "Raman Hamiltonian" (where m/m s - g*/2) by
2/E'2
- (P2/3)(9 /E2
In
the
low
frequency
limit the
first
quantity
approaches
(1/E )(g*/2 - 1),
while the second approaches K = (1/E + 1/E')(g*12 - 2N - 1). For InSb the
S
g
1
three-band expression is larger than Wolff's expression by 23% at 0=-0, and by
29% when ho=E . Furthermore, the three-band expression makes it clear that
g
the scattering is associated with the effective spin-orbit term in the
decoupled cb Hamiltonian, proportional to eK s , instead of the effective Zeeman
term, proportional to gtBg*.
above
quantities,
The SFR cross section involves the squares of the
so the three-band
formula gives
cross sections
which are
larger by 50% to 67% as the incident frequency ao varies from 0 to E /h.
I
g
In the limit o
0, the coefficients in A c simplify so that
-
M AA
A C = -(e
-e* 1 + 2iht
M
1 2) G
A A 4).
KS(e 1xe*.-)
,
2
(4.7)
where
m
2P2
1
2
- 1+2F +
+
m*
3m
E
g
(4.8)
E'
is the reciprocal conduction band effective mass at k=0, and
K S
Pl
3m
3m
1
1
E
E2
E
(4.9)
g
144
is
the
coefficient
of the effective
spin-orbit
interaction
for
the
decoupled
conduction band [Yafet 63, Rashba 91, Jusserand 95].
To order k2 for conduction electrons, transitions become allowed in which
the Landau level index changes.
If we consider only order k 2 we find
occurs to order ko.)
A
C
(Only spin-flip Raman scattering, with An=0,
= (a)(e "e*)k + i(b)(e xe*)'Bo + (c)(e
1
A
A
1
2
k2
i(e)(e xe*) .k2
-
*
2
0
12
)(2 )-(
A
+ (d)(e
AA
1
2
(Bo
0
( 2)
(f(elxe*) -{k, k a} + (g)(e e*)().(
(4.10)
The coefficients (a) through (g) have been derived from the Yafet model and
are
given
in
Appendix
H.
contain
the
parameters,
and
expressions
result,
and
APi..j ,
and
effectively,
are
therefore
The expressions
resonance
from
quite
4th
are
functions
order
functions
9 ( w)
and
perturbation
complicated
[Yafet 63].
of
the band
9(2(o).
theory
The
The
in k.Pi
Raman
transitions obtained in this order by expressing A c in spherical components
are summarized in Table 4.1.
145
Table 4.1 Electronic transitions in Landau level Raman scattering from
conduction electrons for finite B in the axial approximation, in order k 2 and
Here n is the integer Landau level quantum number and s=+11/2 is the
higher.
The coefficients are from Eq. (4.10).
spin quantum number.
Transition
Operator
A
0 sS
Coefficient
A
k2
- i(elxe*)
1
2+
A
0c - a s
C
30
Here,
(z,+), (-,z)
(z,+), (-,z)
(g)
(z,+), (-,z)
Order k4 & higher
---
1
1
(+,-)
(c)
1
0
(z,-), (+,z)
A
A
c
-1
(f)
- ii(e 1xe*)
k k yZ
2-+Z
2
0
A
A
2t
Polarization
(e)
e*) ( 2)k k
S
As
- ii(e 1xe*)
k k_
2+-+
(e e(2)(Y' _B
S(e
C
An
(f)
A
kk
- i(exe*)
1 2Z Z+
(f)
Order k4 & higher
---
2
(e e*)(2)k
2 -2 +
1
C
(z,-), (+,z)
-1
1
(+,+), (-,-)
(z,z)
(c)
2
0
(+,-)
+ 03s
- i(exe*) k2
(f)
2
- 1
(z,-), (+,z)
+ o
Order k4 & higher
---
3
-1
(+,-)
S
2-
1
+-
o s is the "spin-flip frequency"
Nonparabolicity
and
oc is the
"cyclotron frequency."
effects cause both to depend on the Landau level index n.
Note that At=An+As varies from - 1 to 2, for these energy-absorbing electronic
processes, where t-n+ms
2
is the quantum number which is rigorously conserved
in the axial approximation.
(z,z) polarization,
important
in
conservation
approximation,
requires
Note that the
c+os transition also occurs in the
order k4 , but this is not
law
for
the
axial
shown
model,
that At for the electronic
in
the
transition
the total change in the photon
angular
momentum alongt:) B 0 .
Z:)
t")
146
in the table.
electric
exactly
An
dipole
opposes
Only the terms proportional to (c) and (f) produce inter-LL transitions
with An O.
(c) is associated with nonparabolicity
spin-orbit interaction (or A) is zero.
and is finite when the
It causes the 2
c
Raman transition at
k = 0, and the coc Raman transition at k 0.
(f) is finite only if A is
z
c
z
finite. It causes the 2o + s Raman transition at k = 0, and the oc+s Raman
transition at k # 0. It also contributes to the oa Raman transition at k # 0.
z
c
z
The term proportional to (g) gives a contribution to SFR scattering which is
A
symmetric
in
amplitudes
A
e
to
and
e2.
differ
for
A
A
This
B0>0,
causes
since
the
the
(z,+)
and
scattering
at
(-,z)
scattering
B0=0
is
purely
antisymmetric in e1 and e .
2
As an example of computation
of the polarization selection rules, note
that
A
A
i(elxe*)
S2+
=(e e * 2
1Z 2-
e*)
1+ 2Z
associated with the ws Raman (SFR) transition, is maximized in magnitude when
AA
the
incident
and
scattered
have
photons
polarizations
A
abbreviated (z,+) and (-,z).
(z,e )
A
or
A
(e_,z),
A
(Note that e2 = V2 when e2 = e.)
Quantitative Estimate of Anisotropic Scattering
When one goes beyond the isotropic approximation one finds three new
effects:
dependent
The
(1)
on
isotropic
crystal
model
orientation.
transitions
listed
(2)
k -induced
The
in
Table
4.1
transitions
become
of
the
isotropic approximation become allowed at k =0 due to inversion asymmetry.
(3) The axial model selection rules are lifted.
These effects are small for
the case of inter-LL scattering by conduction electrons.
147
The orientation
the isotropic model
dependence
of scattering cross sections which
is found by considering
the axial
terms (Oh symmetry) in the effective Hamiltonian.
angular function W defined in Appendix C.
occur
in
part of the warping
These terms involve the
The magnitude of the anisotropy of
SFR scattering in the low-field limit is of the order of (y70 s/oK), which
is
8% at B = 5 T and zero at B = 0.
0
0
Inversion
asymmetry
causes finite-k
transitions of the isotropic
model
to become allowed at k =0. For the An=1 scattering transition the magnitude
z
of this effect at k =0 relative to k -is of the order (8/v3)2 , which is
1% at B = 5 T.
The
axial and nonaxial
parts
associated
with inversion
asymmetry are of similar magnitudes.
The
scattering
nonaxial
part
transitions,
of inversion
and
asymmetry
new polarizations
and
for the
warping
causes
old transitions.
new
This
may be dealt with by an adaptation of the approach used by Rashba and Sheka,
Gopalan
et al.,
and La Rocca et al. for the case
[Rashba 61, Gopalan 85, La Rocca 88a,b].
of inter-LL
absorption
Details are given in Appendix H.
The new warping-induced scattering transitions have An=even at k =0, and are
weaker
than
the
axially-allowed
An=2
(aE0)2=0.004
or
( 0/E )2=0.04.
The
transition
new
by
inversion
factors
of
order
asymmetry-induced
transitions have odd An at k =0, and are weaker than the axial transitions by
a factor of order 0.01 at B =5 T.
Quantitative Estimate of Isotropic Scattering
Quantitative
estimates
of
the
cross
148
section
for
the
various
inter-LL
Raman
are
transitions
representation.
easily
obtained
in
the
completely
decoupled
The SFR scattering process is the only Raman process which
occurs to zero order in k, and is finite for Bo-- 0.
For this case there are
two enhancements over the Thompson cross section for light scattering from a
free electron.
(or
average)
effective
The first enhancement is the factor hoK which is the incident
s
photon
spin-orbit
energy
times
interaction
in
the
band parameter
effective
the
mass
associated
with
approximation.
the
For
hO) = E /2 (10.6 pm wavelength) this factor has a dimensionless value of 7.8,
g
which is squared in computing the cross section. The second factor comes from
the presence of the resonant denominators and may be expressed as Ks(o)/Ks(0),
s
s
where K(w) is an o-dependent expression which evaluates to K for co=0. The
value of this second factor for h0 = E /2 is 4.15. The overall enhancement of
g
the cross section relative to the Thompson cross section then is about 1000.
This factor becomes much larger when he approaches the resonance at E , as
will be seen in the next section.
For the An=2 scattering transition the k2 dependence near k=0 causes the
coefficient to be more complicated than the one for SFR.
(away
from
resonance)
the
relevant
factor
is
E k 2 or
When o is small
Eos ,
where
e0
is
the coefficient of the k4 term in the decoupled cb Hamiltonian, and s=-eB 0 /hc
is the
reciprocal
of the Landau
radius
squared.
The
obtained from the Yafet model, is
= (1/8)d(m/ m *)2/ dE = ( 1/4)(m/m*) 2(1/E ).
If we evaluate s at B0=10 T and convert it to an energy using
h2s/m = 2. B o
we obtain a dimensionless factor
149
expression
for Eo,
0
s
-
(1/4)(m/m*) 2 (2p BBIE )
6.0 ,
where we used m/m* = 70, 2g B° = 1.15 meV, and E = 235 meV.
This factor
becomes 36 when squared for the scattering cross section.
The coefficient
for the An=1
the one for An=2 if we set k =1/e =1/v,
z
at k 0 is close to
z
is the Landau radius. This
scattering transition
c
where t
neglects the contribution of the coefficient
c
(f) relative to (c) in Eq. 4.10.
Note that <k> is of the order of l/~ in the experiment of Slusher et al.
z
[Slusher 67].
4.2.3
Scattering in the Pidgeon and Brown Model
This approach uses the 8x8 axial coupled-band model of Pidgeon and Brown
[Pidgeon 66, 68, Weiler 78]].
This model is accurate for light scattering by
conduction electrons in InSb for arbitrary values of the magnetic field B0 and
arbitrary frequencies o) of the incident light.
It takes account of the axial
component of the warping anisotropy and can be modified to include the axial
component of the inversion asymmetry anisotropy.
anisotropy
requires
diagonalization
of much
larger
Treatment of the nonaxial
matrices,
as
has
been
discussed by Evtuhov for the case of the 4-fold degenerate valence band in
germanium, and by Trebin and by Pfeffer for the more complicated cases of InSb
and GaAs [Evtuhov 62, Bell 66, Trebin 79, Pfeffer 90.]
Wave functions in the PB model have the multicomponent form
fi(r)},
where the index i is the label associated with the eight Luttinger-Kohn (LK)
basis functions for the s and p-like conduction and valence bands of InSb at
k=0.
The Hamiltonian is an 8x8 operator matrix, where 'operator' implies that
150
the matrix elements contain the kinetic momentum operator k-p+ecA, in addition
to
perturbing
of the
Construction
invariants"
potentials
which
and fields
Hamiltonian
may
[Luttinger 56, Bir 74, Trebin 79]
tensor techniques
by use
is facilitated
of r
be functions
along
of the
with
and
t.
"method of
irreducible
spherical
[Baldereschi 73, 74] and numeric computation software, such
as Matlab [Mathworks 89], capable of matrix operations.
(See [Favrot 94] for
a brief description of the application of irreducible spherical tensor methods
to the PB model.)
Matrix algebra can be used along with the method of
invariants
simplify the
to greatly
particularly
to third
computations
of k-p perturbation
order in k and the applied
and fourth
theory,
fields.
See
Appendix A.
is
In the invariant formulation, Heff is broken into blocks, and each block
expanded via products of basis matrices times operators containing
components of k or B. We then have [Trebin 79, R6ssler 84]
Hcc
cc HcCV HsCS
Heff
(4.11)
Hvc Hvv Hvs
where hermiticity implies that
H
VC
=H ,
CV
H
SC
H
=H t
=H t
SV =
Cs'
VS
The invariant formulation of the blocks is given in Appendix A.
In the axial approximation, the Landau level wave functions have the form
W = exp(ik y)exp(ikzz)[ct
c2tq
+1
C3 (Pt
1
(4. 12)
C6 (p
151
C5 (P+
1
C4 (P+2 IC8 (
C7 (P+1 It
where the transpose symbol 't'
is used to indicate that the wave function is
actually a column vector, and the vertical bars are used to visually separate
the components
associated with basis functions for the F
band (cb), F (=3 ) valence band (vb) and F (=1)
8
2
7
2
(j=i)
conduction
split-off band.
The labels
of the coefficients refer to the PB basis functions ui(r), and the ordering is
by decreasing mj within each band.
From the form of the wave function one
finds that n+m is conserved, where n is the index of the Landau level wave
function, and mj is the component of "effective" angular momentum of the LK
(or PB) basis function
"effective"
in
quotes
angular momentum,
in the direction
because
of the magnetic
the basis
states
are
not
but transform like the corresponding
field B0 .
true
I use
eigenstates
atomic states
of
with
respect to the restricted set of rotations contained in the Td point group.
I
will refer to the quantity
= n+mJ
(4.13)
2
as the axial quantum number, which is conserved rigorously in the axial model.
It is related to Luttinger's constant of motion Q for the case of germanium,
and to Rashba's "angular quasimomentum," n+m .
Suzuki 74,
Bir 74,
A
operator (N
Trebin 79,
Rashba 91.]
[See Luttinger 56, Yafet 73,
It is
the
eigenvalue
of
the
A
-)1
+ F [Trebin 79], where N is the Landau level number operator
1
is the unit 8x8 matrix, and F is the 8x8
F
z
generalization of the angular momentum matrices s and J
WhnZ
Z
for free spinless electrons,
F
When Heff operates in an 8-dimensional subspace of fixed e it is replaced
by a purely numerical matrix [Yafet 66, Pidgeon 66, Weiler 78, Jimenez 94].
(The
basis
n+m - =
e,
functions
for
this
subspace
are
the
functions
<pn(r)ui, with
where u. is best regarded as a 8-component column vector with a
152
single unit element in the location corresponding to the basis function ui(r),
and
zeros
elsewhere.)
The
representation
of the
wave
function
in
this
subspace is the column vector of the coefficients
[cc
1cc c
1 C22 1 C3 C6 C 5
The representation
c]t.
C4 1IcC8 C7
(4.14)
of Hff is obtained by moving
all k i operators to the
right, and replacing them by their -dependent numerical representations
k = V2 diag[~T1k = v2- diag[b/
T-+ V/a
VT
VT
v
V
+2 v/m
VT'2
VTT
rt+3
1 vti 2]
it
eTl-]
(4.15)
(independent of )
k = k 1
A
2sN = k+ k- = 2s diag[
+le-+11e
e+l t+2 e t+1]
etc., where
s
eB /hc
is the magnetic wavevector
squared (k2 ), and the reciprocal
squared (t-2), and 'diag[-]'
diagonal
elements.
operations
require
Note
denotes
that k
the diagonal matrix with the enclosed
and
k
attention to the changing
raise
t by
value of e.
with the spherical tensor form Xm raise (or lower) e by m.
Diagonalizing
energies
Ei
and
the matrix
eight
sets
magnetic length
representation
of coefficients
+1,
so
Matrix
sequential
operators
(See Appendix C.)
for Hff generally
yields eight
eigenvectors).
Exceptions
(or
occur for the three lowest values of t given below.
I use the notation of
[Luttinger 56], [Pidgeon 66] and [Weiler 78]:
e=-2: b+(0)
e=-1:
(4.16)
bc(O), b+(1), a+(O), aS(O)
153
f=0: bC(1), aC(O), b-(2), b+(2), a+(1), aS(1), bs(O)
= 1: bC(2), aC(1), b-(3), a-(2), b+(3), a+(2), aS(2), bS(1)
etc.
The conduction and split-off bands (cb and sb) are both spin-- like, but their
LK basis states
have opposite
parities.
For the cb, the a-set designation
implies a large ms=+!
component near kZ=0; for the sb the a-set designation
i l2
implies a large ms=- component near k =0.
largest component of the wave function.
band edge.
The band-edge
The index n is the index of the
It is 0 for the level nearest to the
levels at finite field are aC(O) and aS(0); the
bc(0) and bs(O) levels both lie within the respective bands.
The Luttinger
notation for the light hole and heavy holes is more complicated.
The a-set
Landau levels in the decoupled (j= ) representation have two finite components
For the a-set levels the index is that of the m=--
at k =0 instead of one.
z
J 2
component function, and for the b-set the index is that of the m function.
The
light
hole
effective mass and g-factor.
a+(1) nearest
(lh)
Landau
levels
have
a
fairly
component
well-defined
The initial lh levels are a+(1) and b+(1), with
to the band edge.
The '+'
superscript
denotes
'light
hole'
except for the levels a (0) and b+(0) which are heavy hole levels, along with
the a-(n) and b-(n) levels which are defined for n>2.
The heavy hole Landau
levels have effective mass and g-factor which are well-defined only at large
n, due to interactions known as the Luttinger effect.
contained
in
the
4x4
decoupled representation.
diagonalization
of the
depends
on
the
effective
Hamiltonian
in the
The heavy hole Landau levels nearest to the band
edge are a+(0), b+(0), a-(2) and b-(2).
energy
These interactions are
values
of the
orientation of the magnetic field B0 .
0
154
The ordering of these four levels in
Luttinger
parameters,
and
on
the
The calculation of matrix elements of the velocity operator between LL
which
wave functions requires the 8x8 matrix representations of v+, v_ and v,
depend on the f value of the wave function (or subspace) which is acted upon.
The 8x8 components of the velocity operator are found from
v = 2aH/ak ,
v
v = 2aHlak ,
(These are the spherical components of v in the 471(
, with v+-v +ivl.)
directed along
(4.17)
= aH/ak .
coordinate frame with B
In the matrix representations of these
A
operators, one again moves factors of k , k and N to the right before making
the matrix substitutions given in Eqs.(4.15).
Note that v
raises the axial
quantum number e by one unit, v_ lowers e by one unit, and v
unchanged.
leaves Z
Thus the eigenvector of coefficients on the left-hand side of v
has an t value which is larger by one than the eigenvector to the right of v.
To compute Afo
for SFR scattering we must consider the processes in
Eq.(4.2) which have aC(O) and bc(O) as initial and final states, and therefore
At=- 1. These are
v
(a)
aC(0)
v
(t=0 intermed.)
-
(t=O)
bc(0)
-
(R=O)
(=- 1)
v
v
(b)
aC(0)
(t=-1 intermed.)
(e=0)
bc(0)
(e=-1)
(t=-1)
and
V VH
k+
(c)
kz
eff
aC(O)
> bc(O)
(t=o)
(=- 1)
155
.
The last process proceeds without an intermediate state in the PB model.
means that the actual intermediate
bands
treated
explicitly
"spin-conserving",
produces
only
i.e.
in
the
state is outside the
PB
a-to-a or
a-to-b
and
model.
b-to-b
b-to-a
Note
that
transitions,
transitions
at
8-dimensional
at
v
set of
produces
k =0;
k =0.
This
and
The
that
number
only
v
of
intermediate states is therefore fewer at k =0.
A
For
Eq.(4.2)
second
(z,+)
scattering,
contains
only
sum
contains
with
the
only
A
A
e =z
E=0
and
A
intermediate
the t=-1
A
e =(x+iy)/v',
states
intermediate
the
(process
states
first
sum
in
(a))
and
the
(process(b)).
The
resonances for (z,+) scattering are caused by the following absorptions:
v
v
-2
aC(0)
bC(1),
V
a+(0)~-
bc(0) ,
and
a(0) ~
The second resonance occurs at energy ho =E +Ih(O +OWs)
I
the low-field limit.
g2
C
bc(O) .
in the Yafet model, in
S
This resonance cannot be reached, however, due to the
resonant interband absorption
v
aC(0),
b (2)-
which occurs at energy
ho I=Eg2+Ih(O-Os).
C
A
For
Eq.(4.2)
second
(-,z)
scattering,
contains
only
sum
contains
with
the
only
S
A
A
e =(x-iy)/2
t=-1
intermediate
i=0
the
A
and
states
intermediate
A
e2 =z,
the
first
sum
in
(process
(b))
and
the
states
(process(a)).
resonances for (-,z) scattering are caused by the following absorptions:
V
b (2) --
v
bc()
,
b+(2) -
v
bc(O) ,
156
and
bs(O) -
bc(O)
The
The first resonance occurs at energy ho I=Eg2+h(
C
+0 ) in the Yafet model, in
S
the low-field limit. It is blocked by the resonant interband absorption
V
aC(O),
a-(2) at energy
h) =E +!h() -0).
g2
I
S
C
The resonances associated with the split-off band are blocked by strong
interband absorption that occurs when
h0lI >Eg2+h(w C-0).S
The conduction band resonance at energy h0 =h(Oc+ms) for (z,+) causes the
cross
section
section for
for
(-,z).
this
polarization
(EII B0)
to
be
larger
than
the
cross
Observation of this resonance, however, is prevented by
light, and increasing
of the scattered
CR absorption
the strong free-carrier
combined resonance absorption at h(O c+(0s) of the incident radiation.
See the
discussion of [Dennis 72].
The 2o c and
2 0oc+o)s
The 20
fashion to the spin-flip.
(+,-) polarization.
The
2 (oc+o)s
and (+,z) polarizations.
B =0,
Raman scattering amplitudes are computed in a similar
c
process has At=2, and is maximum for the
process has Ae=1,
and is maximum for the (z,-)
The cross sections for these processes are zero at
and increase in proportion to B2 at small fields [Yafet 66, Kelley 66,
Makarov 68,
Wright 68].
The cyclotron
resonance
(coc)
amplitude is zero at k =0, but is comparable to the 2
when kz=k =(tB)'.
c
Raman
scattering
scattering amplitude
This scattering process was observed by Slusher et al.
[Slusher 67] for conditions in which several Landau levels were occupied, and
was explained by Makarov [Makarov 68].
157
Numerical Results
The weight factor |Af
k =0 is presented
12
for the cs, 20c and
in Figs. 4.1
and 4.2.
2
0c+0 Raman transitions at
The first figure is for 10.6 gtm
incident radiation at approximately half the band-gap energy, and the second
is for near-resonant 5.3 gtm radiation.
band parameters of [Littler 83].
The PB model calculations use the InSb
They show the anisotropy which occurs in the
axial model due to the warping parameter i.
The dashed curve in the figures
shows the spherical average which results when g. is set equal to zero.
curve
furthest
remaining
away from
curves
the average
correspond
opposite side of the average.
corresponds to Bo [001].
to BII [110]
and [111],
moving away
The
The two
on the
Figure 4.3 shows the weight factor for the Oc
and 20 Raman transitions as a function of k at B =50 kG.
C
Z
0
158
12
10
8
6
4
2
0
0
200
160
120
80
C
E
Cu
40
0
1000
800
600
400
200
0
0
20
40
60
80
100
120
140
160
180
Magnetic Field (kG)
Fig. 4.1
Calculation of Raman weight
Afo 12 versus magnetic field for 10.6 gm
incident wavelength for three transitions at k z=0.
159
1004
1003
1002
1001
CJ
10o00
1004
0
1003
10o
E
1002
10 06
1005
10os
1004
1003
0
25
50
75
100
125
150
175
200
Magnetic Field (kG)
Fig. 4.2
Calculation of Raman weight IAf 12 versus magnetic field for 5.3 Lm
incident wavelength for three transitions at k =-0.
z
160
14-
2O(+-)
a
-
(+,z)
a
c(z,
a
12
--
2mc(+,-) b
-10
"o
8
6
(a(z,-) b
4
2
0
0
0.2
0.1
0.3
0.5
0.4
0.6
0.7
0.8
0.9
1
k z /'2
Fit. 4.3
Calculation of Raman amplitude
up transitions
'a'
and
Af'b'
versus kz for
spin-down transitions
incident radiation, where s-eB O/hc.
161
'b'
c and 2
at B =50 kG for
c
spin-
10.6 Am
Comments for Fig. 4.1, 10.6 gtm:
* The SFR weight factor for (z,+) is larger than that for (-,z) because it has
This resonance is associated with the
a resonance at high fields [Dennis 72].
bC(1) intermediate state, which is nonresonant for (-,z).
at
the
origin
intermediate
is due
to
the trail-off
The
states.
weight
The downward slope
from resonance
factor
B 11[111] than for the other orientations.
for
(z,+)
is
with
valence
slightly
band
for
larger
It has a minimum near 90 kG, and
then increases towards a resonance beyond 180 kG.
The (-,z) weight decreases
monotonically, and never reaches zero in either the PB or Yafet model.
(The
(z,+) amplitude crosses zero, in the Yafet model, when Os=o I and 02=0.)
* The weight factor for 2
c
(+,-) is zero at B =0, and increases as B2
reaches a peak near 45 kG where its square is about 1/3 that of SFR.
It
It goes
to zero at B =115 kG, which is very near to the point where 2c =0 and o 2=0.
The
curve
beyond
this
point
absorption with (+,+) polarization.
represents
the
weight
factor
for
2-photon
(Note: If the Yafet model is used, the
amplitude goes through zero at exactly the field at which 2oc =
and 02=0.
It
is not known if this condition would occur in theories which go beyond the PB
model.)
Wright's prediction of a finite amplitude for the (z,z) polarization
is an error, and violates the axial model selection rule of conservation
of
the total angular momentum component (photon plus electron) along B .
If
nonaxial
the
2 0co
scattering
is considered,
the weight factor for (z,z) scattering with
transition is much smaller than that computed by Wright.
* The weight factor for
2 0c+
s (z,-) reaches a maximum at 32 kG, and is larger
than that of (+,z) which is maximum at 21 kG.
factor is about 1/20 that of 2o c .
The peak for the (z,-) weight
This is in agreement with the prediction of
162
The amplitudes go to zero
Makarov, but differs from that of Wright et al.
near 100 kG, where 2oc+os0 =
Wright et al.
and o2=0, in disagreement with the prediction of
(This is again exact in the Yafet model.)
of finite amplitudes
Wright's predictions
for the (z,+) and (-,z) polarizations are also in error,
in violation of the axial model selection rules.
* The Yafet model calculations (not shown) differ from the dashed curves by
amounts which are similar to the amounts for the solid curves.
The present
curves differ from the Yafet result by at most 5% in the region shown, which
is about the same as the magnitude of the anisotropy.
The SFR weight factors
for the Yafet and PB models are the same at B =0.
Comments for Fig. 4.2, 5.3
m:
* The incident photon energy used in this figure is 232 meV.
This is only
3.2 meV lower than the zero field energy gap between the cb and vb.
(For
hio >E the resonant condition for the scattering amplitude occurs at some
ig
finite field.) The Raman weight factor now peaks very sharply near the origin
for all three graphs.
A logarithmic scale is used on the y-axis to show the
large variation.
* The weight factor for SFR scattering at the origin is about 2 orders of
magnitude larger than the peak values for the 2
the origin.
c
and
2 wc+os
transitions near
At 10 T the SFR weight factor is about 15 times larger than the
2o c weight, and the latter is about 5 times larger than the larger of the
20c+s
weights.
* Notice that the weights for the 20o
and 2
+o
s
scattering transitions are of
comparable size near the peak, with 2oc+o2 s now being slightly stronger than
163
2o c .
At larger fields the strength of 20c+o) s decreases more rapidly, so
2o c
becomes the stronger one, as at 10.6 gm. Notice the different scales used for
20 and 2o +o .
c
c
S
Comments for Fig. 4.3 -
k dependence at 10.6 gtm:
* The weight factors for the 0c and 20
functions of k
out to kH-(eB /hc)1 /2 , for B 0=50 kG.
the magnetic length
the Fermi level
Raman transitions are plotted as
e
kH is the reciprocal of
, and is the approximate cutoff for occupied states when
drops just below the n=l spin up Landau
attainment of the quantum limit.
level,
i.e. on
(The extreme quantum limit is attained when
the Fermi energy drops below the n=0 spin down Landau level.)
* The weight factor for cc scattering is zero at k =0.
increases
quadratically.
At
small B
and
0
k
z
the
c
c
For small k
it
scattering transitions
result from the coefficients (c) and (f) in the decoupled scattering amplitude
operator, which cause the 2o) and 20c+
s
transitions, respectively.
* Near kz=kH the weight factors for oc scattering are comparable to that for
20o
(+,-) at k =0.
The weight factor for the latter transition is smaller at
kH than it is at k =0.
The effective weight factor is the average over the
occupied states [Makarov 68].
* I have also computed the weight factors for the other scattering transitions
which are allowed by finite k
Wc
transitions,
these are
(+,-), with A=2, and 30c +
0c +o
in the axial approximation.
0c+0s (+,+),
s
(-,-) and (z,z),
(+,-), also with A1=2.
In addition to the
with Af=0,
(See [Makarov 68].)
(+,+) and (-,-) transitions at kZ=kH are weaker than co,
in strength to 20c+cs (z,-) at kz=0.
164
c -m
The
and are similar
The (z,z) amplitude is of order k 4,
compared to k2 for (+,+) and (-,-), but it is of comparable magnitude at kH
for B 0=50 kG.
The amplitude of the CocC-co
is weaker than o +o .
and is weaker still.
The 3o +o
(+,-) transition is of order k 4, and
(+,-) transition amplitude is of order k 6,
It is exactly zero in the Yafet model, but has small but
finite amplitude in the PB model.
165
4.3 Experimental Background
Stimulated
spin-flip
Raman
(SFR)
scattering
in
InSb
[Patel
71]
was
measured in magnetic fields up to 13 T for 10 gm pumping, and 18 T for 5 gm
pumping.
I have made measurements of the Stokes output power and threshold
pump intensity as a function of magnetic field, pump wavelength, and crystal
orientation.
versus B
For pumping in the 10 gm region, the curves of Stokes output
show considerable structure which varies with both pump wavelength
and crystal orientation.
This was the first observation of the anisotropy of
the output with crystal orientation
ever reported.
I also reported the first
observation of stimulated SFR scattering with pump wavelengths between 9.6 and
9.73 jim in the P branch of the 9.6 gm band of the CO 2 laser spectrum, and with
pump wavelengths between 4.75 and 4.87 gm obtained by frequency doubling these
laser lines.
The
data
indicate
that
all
of
the
observed
structure
between
the
low-field and high-field cutoffs of the stimulated output is due to intraband
This is in agreement with
absorptions at the pump and Stokes wavelengths.
previous
work by Dennis
et al.
[Dennis 72]
[Weiler 74] at magnetic fields below -
and later by Weiler
et al.
10 T, but is in disagreement with the
work of Wachernig and Grisar [Wachernig 74] at fields between 10 and 14 T.
At
magnetic fields above 10 T, the Stokes output shows two minima at - 10.4 and 12.2 T for 10.6 gm pumping.
Wachernig and Grisar [Wachernig 74] interpreted
these minima as arising from the combined effect of intraband absorptions and
resonant interaction between the electron spin and the transverse optical (TO)
and longitudinal optical (LO) phonons, respectively.
spin-phonon
interactions, if present,
would be
166
These resonant electron
expected to
occur when the
electron spin-flip frequency
phonon frequency,
WoTO
os becomes equal to the zone-center TO or LO
or COLO.
[Koteles 74]
I attribute the minima at -
10.4
and - 12.2 T to intraband absorption only.
The results with 10 tm pumping show no evidence of any resonant electron
spin-phonon
interaction
effects.
This
is
further
supported
by
the
data
obtained with pumping in the 5 gm region, for which intraband absorptions are
essentially negligible.
167
4.4
Experimental Conditions
The InSb samples used in the
10 gtm pump experiments were n-type,
single crystals with carrier concentration ne=2x10 6 cm -3 and
tellurium doped
e=lxl105 cm2 V sec - 1 at 77 K.
mobility
They were cut from large ingots and
lapped to the approximate dimensions 8 mm x 9 mm x 22 mm. The 8 mm x 9 mm end
faces were polished plane parallel to within 0.250.
The samples were held on
the cold finger of a liquid-helium cryostat in the bore of a 4 in. Bitter
The sample temperature was estimated to
solenoid providing fields up to 13 T.
be at - 20 K.
The pump laser was a transversely-excited, atmospheric-pressure (TEA) CO2
laser equipped with an intracavity NaCI Brewster window for polarized output
and
a
blazed
repetition
grating
of
rate
-
for
1 pps.
single
line
operation.
The output
It
showed
was
strong
operated
random
at
a
self-mode
locking, and the envelope of the pulse train was observed to have a width of
- 200 nsec (full-width at half-maximum).
Values for peak power were obtained by ignoring the mode locking and
simply
dividing
Raman
Spin-flip
qpll q
_B
the
average energy
scattering
and Ep II B
was
per pulse by the above
done
in
the
p
distance
configuration
with
where qp and qs are the propagation vectors of the
pump and Stokes radiation, respectively, and E
pump radiation.
collinear
pulse width.
is the electric vector of the
A 7 mm square aperture was placed in the pump beam at a
1.4 m in front of the InSb sample.
The full beam width 2w 0,
corresponding to l/e intensity points at the sample position, was measured to
be 5 mm.
The values for pump intensity, as used in this paper, were obtained
168
by dividing the peak pump power by tw 2 .
used to filter out the pump
Long-wavelength-pass filters were
radiation from the Stokes
latter was detected with a Ge:Cu detector at 4.2 K.
radiation,
and the
The output voltage pulses
from the detector were converted by a peak voltmeter to a dc signal which was
displayed on a chart recorder.
169
4.5
Results and Discussion
The SFR Stokes output and threshold pump intensity vs BII [111] for three
pump wavelengths X = .10.59, 10.15, and 9.73 gm are shown respectively in
P
Figs. 4.4 and 4.5. The output data were taken by sweeping the magnetic field
and the
at fixed pump intensity,
similar
sweeps
after
attenuating
data were
threshold
intensity
the pump
obtained
with
by making
CaF 2 crystals
of
The output data of Fig. 4.4 were taken at pump intensities
varying thickness.
of 1.4, 1.3, and 0.8 MW/cm 2, respectively.
0.8(c) Xp= 9.73psm
0.6
0.4
30.2
0
a 0.0
_jI
20
40
60
80
100
120
140
C6
D
O
W
4
0
(p
2
'.-_2
W
(p)
20
40
60
80
(s)
100
,C-LO
120
140
S12E (a)Ep 10.59
" ,
I
2C+WLO
.
20
Fig. 4.4.
with carrier
Relative
60
80
100
120
140
MAGNETIC FIELD (kOe)
Stokes output power vs applied magnetic field for InSb
concentration
and qpll [110].
40
n = 2x10' 6 cm 3
and oriented with E IB 11 [111]
oe denotes the spin-flip frequency.
170
The low-field cutoff observed in the Stokes output of Fig. 4.4, and the
corresponding sharp rise in the threshold pump intensity of Fig. 4.5, can be
understood by the results of Fig. 4.6. For pump photon energies less than half
the zero field energy gap, E (B =0), the cutoff results from the decrease in
free-carrier
the effective
concentration for SFR scattering at lower magnetic
For pump photon energies greater
fields as discussed previously. [Patel 71]
of the pump and
than !E (B =0), the cutoff is due to increased absorption
Stokes radiation
by holes
this
shown,
at
disappears
absorption
as a result of two-photon
Since the energy gap E (B ) increases with B °
absorption across the band gap.
as
59] created
[Kurnick
sufficiently
high
B
which
for
IE (B ) >hW .
2g
0
At
threshold
p
fields
magnetic
pump intensity
for
cutoff
the
above
the
Stokes
output
and
show considerable
10.15 gm pump
and
10.59
point,
I attribute the observed structure
structure, as may be seen in the figures.
to intraband absorptions of the pump and Stokes radiation by the conduction
electrons,
whether
as
the
indicated.
given
The
absorption
p
letter
is
for
or
the
s
in
pump
the
indicates
parenthesis
radiation
or
the
Stokes
radiation.
A comparison of the data for 10.59 and 10.15 pm pump shows that the
observe structure
in the Stokes
output shifts to higher magnetic
fields for
shorter pump wavelengths, in agreement with the shift of the absorption peaks
to higher magnetic
fields
[Weiler
74].
Our identification
of the observed
structure with intraband magnetoabsorption is consistent with previous work at
magnetic fields below - 10 T [Dennis 72, Weiler 74, Wachernig 74].
171
1200
(c) Xp = 9 .7 3pm
. 800
N
400
SI
>I-
0
m
20
a
.
I
I
..
60
40
II
g
i
80
100
A
I
II
120
I-
I
140
2 WC
' 400 Ia
3Wc
(p)
0
O _C
I
()
a.
o
.I
m
(b) Xp I(3.15 1
z
I
. I.
20
w400
40
I
60
.
I
80
(s)
I
.100
.
I
120
S()Xp= I0.59pm
I
140
(+
I"
200
V
Fig. 4.5.
20
Threshold
100
40
60
80
MAGNETIC FIELD (kOe)
120
140
pump intensity versus applied magnetic field for InSb
with carrier concentration n = 2x106 cm
e
and qpll [110].
172
3
and oriented with E II B II [111]
p
0
1
o 120-
I
S100-
o I [100oo]
"H II [Ill ]
H II (110]
L- 800
6040-
o
20J
115
120
125
PUMP PHOTON ENERGY, Ep (meV)
Fig. 4.6.
photon
130
Low-magnetic-field cutoff for stimulated SFR scattering versus pump
energy for InSb with carrier concentration n e = 2x106 cm - 3 .
Also
shown is the field-dependent energy gap calculated from the band parameters
given in [Weiler 74].
173
In the magnetic field region between 10 and 13 T, two minima are observed
in the Stokes output at - 10.4 and -
12.2 T for 10.59 gm pumping.
These
minima were attributed by Wachernig and Grisar [Wachernig 74] as primarily due
to
resonant
interaction
between
the
conduction
electron
zone-center TO and LO phonons, respectively.[Koteles 74]
spin
c
+ os , pump
absorption,
where
oc,
the
Instead, I attribute
these minima to intraband absorptions only: the minimum at 2c
and
is the cyclotron
10.4 T to the
frequency
of the
conduction electrons and the minimum at - 12.2 T to the strong Co + oLo Stokes
absorption.
phonon
field
The minimum at -
10.4 T cannot be due to the electron spin-TO
interaction because (i)
expected
for
this
changing pump wavelength.
it is not observed at the value of magnetic
interaction,
and
(ii)
it
changes
position
with
The minimum at 12.2 T cannot be due to the electron
spin-LO phonon interaction because it shifts completely away from 12.2 T for
9.73 gtm pumping
[Fig. 4.4(c)].
These
observations
are
supported
by
the
threshold data of Fig. 4.5.
One significant difference between the Stokes output data of Fig. 4.4(a)
for 10.59 gm pump and that of Wachernig and Grisar is the strength of the
minimum at 10.4 T.
Whereas I observe a weak minimum at that point, they
observe a complete disappearance of Stokes output.
The reason for this is
made clear in Fig. 4.7 where I show the relative Stokes output vs applied
magnetic field obtained with 10.59 gm pump intensity of samples oriented with B0 1[111],
of
the
observed
qualitatively,
structure
in
1.4 MW/cm 2 from
[110], and [100] directions.
the
Stokes
output
correlates,
with the observed anisotropy in magnetoabsorption
Fig. 4.8 for 10.59 jim radiation.
174
The anisotropy
at
least
as shown in
(c)HII [10
n-
0)(
8 F-
(s)
/
3,c, 2w+,La 4 \
(s) (s)
\
I
F-
,
ir
I/
/,
20
-(b)
0
I
,
40
HII (1101
2"
Zc+We
(p)
2
2Ic
ss)
1
I
60
80
(s)
(p)
\
-,
+
.
100
I/
,
120
140
I'
I
~
I
I
2
/
I"
m
-124
wc 4
34&c,2WC+WLc'~~jI~s
Cs
Cs),
r(S)
2
"€,
I
•]s)
L
, I ,"/rA2'
I
-
20
,
2,c+('.e
C+LO
1 (s)
,c "(P)
i , -- ;-s .,
I
--
40
e
--
100
80
60
120
1
140
-(a)H II
[IJI
cr_
2 c+L
(s)
s))
2c+ weC
4C
(s)
SI
20
Fig. 4.7.
I
,
I
,
I
40
60
80
100
MAGNETIC FIELD (kOe)
,
1
120
Relative Stokes output power versus applied magnetic field in InSb
with carrier concentration n e = 2x10
6
cm - 3, for pump intensity of 1.4 MW/cm2
at 10.59 gm, and qp 11 [110].
175
2wc
0.500.0
o 0.0
:
20
41
(b)
EpIIR
0.3 -0.2 -
[10], S
H-11[ S
0.1-
0.0
Fig. 4.8.
20
4(
Mt ,GNETIC FIELD (kOe)
Magnetoabsorption
coefficient
with n = 2x1016 cm - 3 at 10.59 gm.
[a(B ) - a(0)] versus
B 0 for
InSb
Sample No 17 with qpll [111], others with
q Ip [110].
176
For unambiguous observation of possible electron spin-phonon
interaction
effects near 11.4 and 12.5 T, it is essential to use pump wavelengths in the
5 gm region to avoid interference
by the strong intraband
absorptions just
By doubling the frequency of a TEA CO 2 laser in a crystal of
discussed.
second harmonic
tellurium oriented for phase-matched
generation, I measured
the Stokes output of the SFR laser using pump wavelengths between 4.75 and
5.3 gtm
concentrations
18 T.
fields as high as
at magnetic
InSb
samples with carrier
of ne=2 x106 and 5x1016 cm - 3 were studied at peak incident
powers of 4 to 7 kW.
No structure was observed either in the Stokes output or
the tuning behavior of oa
versus B
which could be attributed to the electron
spin-phonon interactions.
Figure 4.9 shows the Stokes frequency
shift as a function of magnetic
field up to 18 T for an InSb sample of carrier concentration ne = 2 x 1016
cm 3 .
For fields
above -10 T it was essential to use pump radiation at
wavelengths shorter than 5 gm in order to take advantage of band gap resonance
enhancement
[Mooradian 70]
since
the
effective
band
gap
increases
with
magnetic field. Note that there is no discontinuity in the tuning behavior or
"pinning" near 12.5 T where o s = cOLo.
Such pinning had been observed in the
polaron case when the cyclotron frequency coincides with (OLO [Dickey 67].
Figure 4.10 shows the relative Stokes output as a function of magnetic
field from -10.5 T to 14.5 T for the above sample with pump wavelength
X =4.867 jLm. The pump beam of -4 kW peak power was focused onto the InSb
P
The Stokes radiation was
sample with a BaF9 lens of 32 cm focal length.
separated
from the pump
radiation by passing the output beam through
a
Perkin-Elmer Model 98G monochromator and detected with a liquid nitrogen
177
cooled Ge:Au detector.
The monochromator slit width was set at 2 mm and the
drum was swept manually to track the Stokes frequency.
The Stokes output
shown in Fig. 4.10 has not been corrected for the instrument function of the
monochromator.
The dips indicated by the arrows are due to atmospheric water
vapor absorption [Plyler 60].
magnetic
field
region
Note that there is no anomalous behavior in the
around
12.5 T where
=
Lo,
indicating
that
the
resonant electron spin-LO phonon interaction is either nonexistent or too weak
to affect the Stokes output.
250-
E 200
i, 150
W 100
o X p 4.867 sm
I00P
aX
p 5.I59um
500
Fig. 4.9.
I
20
I
40
I
I
60
80
MAGNETIC
I
I
I
100
120
140
FIELD (kOe)
I
160
180
Stokes frequency shift versus applied magnetic field B II [110] for
InSb with n =2 x 10 16 cm.-3
e
178
1880
STOKES FREQUENCY (cm - )
1870
1860
1850
1840
MAGNETIC FIELD (kOe)
Fig. 4.10.
Relative
Stokes
output power
versus
B 0 II [110]
pumping from an InSb sample with n e = 2 x 1016 cm - 3.
atmospheric water vapor absorption lines.
179
for
4.867 p.m
Arrows indicate the
In conclusion,
ne=2 x0l6 cm
3,
I have
shown that for InSb with
carrier concentration
the structure in the Stokes output power as well as that in
the threshold pump intensity for the InSb SFR laser pumped with radiation in
the 10 gm region, can be attributed to anisotropic intraband magnetoabsorption
of the pump and Stokes radiation.
Furthermore, I have not observed any
effects of resonant electron spin-LO or TO phonon interactions on the SFR
scattering
in
InSb.
Wachernig
et al.
[Wachernig
75]
have
reported
the
observation of a small anomaly in the tuning behavior of the InSb SFR laser
around 12.4 T in a sample with ne=5x1015 cm -3 .
If this anomaly is indeed due
to the electron spin-LO phonon interaction, our results would imply that this
interaction is shielded in samples of higher carrier concentration.
180
5. CONCLUSION
A
effect
magnetic-field-reversal
significant
2 oc+O
s
magnetoabsorption at 20 c and
in
the
electron
conduction
in n-InSb has been predicted and observed.
The only previous observations of field reversal effects in magnetoabsorption
for
were
interband
absorption in n-InSb
absorption
in
[Hopfield 60]
CdS
Chen 85].
[Dobrowolska 83,
and
spin
resonance
previous magneto-
Three
absorption experiments done at MIT did not focus on a field-reversal effect.
These were [Weiler 74],
did not report observation
in either polarization
of 2oc+Cs
geometry, nor 2o c in the i polarization.
suggests the sample
geometry
would
may have been oriented with B II [110]
made
the
field-reversal
detect
[Chen 85].
a
field-reversal
relatively
small
for
Other experiments which did
[Grisar 78],
[Johnson 70],
are
effect
effect
and
This
(see absorption data).
compared with some other orientations.
2Oc(a_)
not
have
in the Voigt
The absence of 20c 0+ in the a 1
q II [111], as in the case of my Sample #17
orientation
The first of these
[Favrot 75,76] and [K. Lee 76].
and
It is not clear whether the effect is actually observable under
The experiment of Johnson
the conditions of these three latter experiments.
and Dickey [Johnson 70] observed the 2oc(al) transition in Te-doped n-type
InSb with n e
1.4x1015 cm - 3 at T = 20 K with B II [110].
Wavelengths
of
66.1 gm and 42.0 gm were used, on opposite sides of the Reststrahl absorption,
and the 2oc(a 1 ) transition was observed at B ° = 1.2 T and 2.0 T, respectively.
The
The B 11[110] orientation is not
sample thickness was about 2 mm.
optimum for the field reversal effect, but should produce an observable effect
for q II [110] or [111].
absorption
electric
data
The effect would not occur if q II [001].
for InSb #16
quadrupole
mechanism
and
#17.)
is too
181
weak
Johnson's
to
cause
observation
the
2o c
(See my
that the
absorption
(although his numbers are incorrect)
suggests the possibility of an unknown
absorption mechanism, since my calculations show that the EQ mechanism has a
larger
probability
interesting
than
the
inversion-asymmetry
to repeat Johnson's experiment
mechanism.
It
would
be
to see if the field-reversal effect
can be observed under his conditions.
The magnetoabsorption experiments of Chen et al. [Chen 85] were done with
n-type
InSb
samples
with
ne
lxl014 cm - 3 to 5x105 cm - 3 in
from
many
orientations, at T = 4.5 K and wavelengths of 96.5, 118.8, 163 and 251.1 gm.
These wavelengths are all longer than the Reststrahl band from 51 to 54 Lm.
The sample thicknesses varied from 1.9 to 4.5 mm.
This paper only discusses
the spin resonance absorption, but the following paper in Physical Review by
Gopalan et al. [Gopalan 85] states that the field-reversal effect was not seen
for
the
2o
or
20c+
transitions
s
in
any
of Chen's
data.
information would be required to find the reason for this.
Additional
A possibility is
that the absorptions were too weak for unambiguous observation of the effect.
The weakness relative to spin resonance results both from the larger linewidth
and from the decrease in absorption strength at low magnetic fields.
unknown
factor
field-reversal
is how
effect,
occupation
since
of additional
calculations
have
Landau
been
conditions, which would not apply for the 2%o
based
and 2o +o
Another
levels affects
on
quantum
the
limit
transitions at low
fields in Chen's higher-concentration samples.
The
magnetophotoconductivity
experiments
of Grisar
were done in InSb samples with ne= 8x1013 cm
3
wavelengths
The
B01 [001],
of 9.54,
10.26 and
[110] and [111].
10.59 gim.
et al.
[Grisar 78]
at T =12 K and CO2-laser
sample
orientations were
The main differences between their experimental
conditions and mine, then are the low conduction electron concentration, lower
182
temperature,
and use of photoconductivity
the light absorption.
instead of transmission
to detect
It would be useful to repeat these experiments to see
whether the field-reversal effect can be observed.
It has taken a considerable time to develop a general treatment for the
orientation
carrier
dependence
of the warping
magnetoabsorptions.
inversion-asymmetry-induced
The
and inversion-asymmetry-induced
complete
spin-changing
angular
dependence
transitions,
of
neglecting
free
the
the
B-reversal effect caused by MD and EQ contributions, was obtained by Rashba
and Sheka in 1961 [Rashba 61a] via the decoupled effective mass approximation
for the conduction band.
Later work on the anisotropic
magnetoabsorption
using the coupled conduction and valence band (PB) theory largely ignored the
work of Rashba and Sheka and its angular generality, focusing on the case of
B
in the (110) plane.
Pikus,
Following Suzuki and Hensel, [Suzuki 74] and Bir and
[Bir 74] Trebin et al. [Trebin 79] produced the invariant formulation
of the Pidgeon and Brown model which was needed for application of the general
rotation approach of Rashba and Sheka to the coupled-band case.
They also
gave a symmetry-based characterization of the magneto-optic selection rules in
InSb, which extended the work of Suzuki and Hensel on valence band transitions
in germanium.
A
N + F
They emphasized the importance of conservation of the quantity
in the axial approximation, and the breakdown of the axial model
selection rules when warping and inversion asymmetry are considered.
Gopalan et al. [Gopalan 85] applied the analysis of Rashba and Sheka to
the
spin-conserving
obtained
all
of
magneto-optical
asymmetry.
magneto-optical
the
transition
absorptions
in
the
transitions
matrix
in the conduction
elements
conduction
band
needed
caused
to
by
band, and
find
the
inversion
A connection was made between the matrix elements of the velocity
183
operator and the components of the second-rank irreducible tensor formed from
the components of the kinetic momentum operator k.
The theory which I have presented is the first to make the complete
connection
of the work of Rashba
and Sheka with the
formulation of Baldereschi and Lipari.
irreducible
tensor
La Rocca improved Rashba's rotation
technique by including the third Euler angle, which simplifies the treatment
of cases where the optical propagation and polarization are arbitrary.
I have
attempted to combine the different approaches to obtain a better formulation
of the magneto-optical
problem
for free
carriers,
including
the
anisotropy,
and considering both the coupled model for accuracy and the decoupled model
for simplification and for verification at low B .
4x2
off-diagonal
matrices
La Rocca's coordinate
with
model
rotation to all terms
of the Hamiltonian.
functions
in the PB
I
have pointed out
components
of the
By factoring the 2x4 and
I made
in Trebin's
it
simpler to
invariant expression
the connection
irreducible
tensor
apply
of the
operators,
angular
and
have
related this to the selection rules for the magneto-optical transitions.
After achieving this general formulation of the PB model in a rotated
coordinate frame it was discovered that the EQ or MD contributions to the
20cc(a)
and 2oc+0s(1r)
field-reversal
Voigt transitions were sufficiently large to produce a
effect in the magneto-optical absorptions.
The experiment was
set up and carried out, and the resulting data showed large changes in the
absorption which were even greater than those seen previously
in the spin
resonance absorption.
The comparison between experiment and theory indicates
agreement
50%,
to
within
with
stronger than what is predicted.
the
observed
absorptions
being
generally
Some of the uncertainty in the prediction
comes from lack of knowledge of the multiple parameters in the PB model which
184
Better theoretical and, hopefully,
are needed to compute the matrix elements.
experimental estimates of the inversion asymmetry parameters G and G', C and
and N
C', N
are needed, with consideration of deviation from the single
group approximation which sets G = G' and P = P'. Work should also be done to
find
the
influence
of exchange
effects
on
experimental determination of the parameters.
experimental
results
and
the
The linear-k parameter C in the
valence band block Hvv of the PB Hamiltonian is expected to be small compared
to the off-diagonal-block linear-k parameter
estimate
of the off-diagonal
C' in Hvc, based on Cardona's
spin-orbit interaction
parameter
A.
Additional
work should be done to determine both of these parameters more precisely.
More work could be done on measurements of the 2oc(al) and 2c+s
Voigt absorptions.
(tn)
The measurements should include the use of higher magnetic
fields and shorter wavelengths, and should use exchange gas cooling to reduce
stress in the sample.
orientation,
temperature,
They should involve greater variation of wavelength,
doping
and
concentration,
and
should
include
photoconductivity detection as well as transmission.
It would be interesting to conduct some new SFR experiments to determine
how the field-reversal effect in the 20(a)
the
output
anisotropy
of the SFR
of the
laser.
oc+os(t)
and 2c+)s()
It would be of
Voigt absorption,
which
absorptions affects
interest to measure
contributes
the
via virtual
transitions to operation of the SFRL, and the effect of this anisotropy on the
the SFR laser output.
Finally, it would be of interest to apply the theoretical formalism which
I have developed to the case of semiconductor heterostructures, where effects
of inversion asymmetry have been recently observed [Jusserand 95].
185
References
[Aggarwal 71a] R. L. Aggarwal, B. Lax, C. E. Chase, C. R. Pidgeon, D. Limbert,
and F. Brown, Appl. Phys. Lett. 18, 383 (1971).
[Aggarwal 71b] R. L. Aggarwal, B. Lax, C. E. Chase, C. R. Pidgeon, D. Limbert,
in Proc. 2nd Int. Conf on Light Scattering in Solids, Paris,
1971, ed. by M. Balkanski (Flammarion, Paris, 1971), p. 184.
[Aggarwal 72] R. L. Aggarwal, in Semiconductors and Semimetals, ed. by R. K.
Willardson and A. C. Beer (Academic, New York, 1972), Vol. 9,
p. 151.
[Anderson 89] H. L. Anderson, ed., A Physicist's Desk Reference (American
Institute of Physics, New York, 1989).
[Baldereschi 73] A. Baldereschi and N. O. Lipari, Phys. Rev. B 8, 2697 (1973).
[Baldereschi 74] A. Baldereschi and N. O. Lipari, Phys. Rev. B 9, 1525 (1974).
[Barticevic 87] Z. Barticevic, M. Dobrowolska, J. K. Furdyna, L. R. Ram Mohan,
and S. Rodriguez, Phys. Rev. B 35, 7464 (1987).
F.G. Bass and I.B. Levinson, Zh. Eksp. Teor. Fiz. 49, 914
[Bass 65]
(1965) [Sov. Phys.-JETP 22, 635 (1966)].
F. Bassani, G. C. La Rocca, and S. Rodriguez, Phys. Rev. B 37,
[Bassani 88]
6857 (1988).
R.L. Bell and K.T. Rogers, Phys. Rev. 152, 746 (1966).
[Bell 66]
G. L. Bir and G. E. Pikus, Sov. Phys.-Solid State 3, 2221
[Bir 62]
(1962).
G. L. Bir and G. E. Pikus, Symmetry and Strain-Induced Effects
[Bir 74]
in Semiconductors (Wiley-Halsted, New York, 1974).
J. D. Bjorken and S. D. Drell, Relativistic Quantum Mechanics
[Bjorken 64]
(McGraw-Hill, New York, 1964).
[Blount 62a]
E. I. Blount, in Solid State Physics, ed. by F. Seitz and D.
Turnbull (Academic, New York, 1962), Vol. 13, p. 305.
E. I. Blount, in Lectures in Theoretical Physics (University of
[Blount 62b]
Colorado, Boulder, 1962), ed. by W. E. Britten, B. W. Downs,
and J. Downs, (Interscience, New York, 1963), Vol. 5, p. 422.
R. Bowers and Y. Yafet, Phys. Rev. 115, 1165 (1959).
[Bowers 59]
M. Braun and U R6issler, J. Phys. C 18, 3365 (1985).
[Braun 85]
T. L. Brown and P. A. Wolff, Phys. Rev. Lett. 29, 362 (1972).
[Brown 72]
S. R. J. Brueck, A. Mooradian and F. A. Blum, Phys. Rev. B 7,
[Brueck 73]
5253 (1973).
M. Cardona, J. Phys. Chem. Solids 24, 1543 (1963).
[Cardona 63]
M. Cardona, F. H. Pollak, and J. G. Broerman, Phys. Lett. 19,
[Cardona 65]
276 (1965).
[Cardona 66] M. Cardona and F.H. Pollak, Phys. Rev. 142, 530 (1966).
M. Cardona, in Semiconductors and Semimetals, ed. by R. K.
[Cardona 67]
Willardson and A. C. Beer (Academic, New York, 1967), Vol. 3,
p. 125.
[Cardona 86a] M. Cardona, N. E. Christensen, and G. Fasol, Phys. Rev. Lett.
56, 2831 (1986).
[Cardona 86b] M. Cardona, Phys. Rev. B 34, 7402 (1986).
[Cardona 86c] M. Cardona, N. E. Christensen, M. Dobrowolska, J. K. Furdyna,
and S. Rodriguez, Solid State Commun. 60, 17 (1986).
[Cardona 86d] M. Cardona, N. E. Christensen, and G. Fasol, in Proc. 18th Int.
Conf Physics of Semiconductors, Stockholm, 1986, ed. by
O. Engstr6m (World Scientific, Singapore, 1987), p. 1133.
187
[Cardona 88]
M. Cardona, N. E. Christensen, and G. Fasol, Phys. Rev. B 38,
1806 (1988).
[Chen 85a]
Y.-F. Chen, M. Dobrowolska, and J. K. Furdyna, Phys. Rev. B 31,
7989 (1985).
[Chen 85b]
Y.-F. Chen, M. Dobrowolska, J. K. Furdyna, and S. Rodriguez,
Phys. Rev. B 32, 890 (1985).
[Christensen 84] N. E. Christensen and M. Cardona, Solid State Commun. 51, 491
(1984).
[Colles 75]
M. J. Colles and C. R. Pidgeon, Rep. Prog. Phys. 38, 329
(1975).
[Condon 35]
E.U. Condon and G.H. Shortley, The Theory of Atomic Spectra,
(Cambridge University Press, Cambridge, 1935).
[Dennis 72]
R. B. Dennis, C. R. Pidgeon, S. D. Smith, B. S. Wherrett and
R. A. Wood, Proc. R. Soc. Lond. A. 331, 203 (1972).
[Dickey 67]
D. H. Dickey, E. J. Johnson and D. M. Larsen, Phys. Rev.
Letters 18, 599 (1967).
[Dobrowolska 83] M. Dobrowolska, Y. Chen, J. K. Furdyna, and S. Rodriguez,
Phys. Rev. Lett. 51, 134 (1983).
[Dresselhaus 55a] G. Dresselhaus, A. F. Kip, and C. Kittel, Phys. Rev. 98, 368
(1955).
[Dresselhaus 55b] G. Dresselhaus, Phys. Rev. 100, 580 (1955).
[Dresselhaus 65] G. Dresselhaus and M. S. Dresselhaus, in Optical Properties
of Solids (Proc. Int. School of Physics "Enrico Fermi", Course
XXXIV, Varenna, 1965), ed. by J. Tauc (Academic, New York,
1966), p. 198.
[Edmonds 74] A. R. Edmonds, Angular Momentum in Quantum Mechanics, 2nd ed.
(Princeton University Press, Princeton, NJ, third printing,
with corrections, 1974).
[Enck 69]
R.C. Enck, A.S. Saleh and H.Y. Fan, Phys. Rev. 182, 790 (1969).
[Evtuhov 62] V. Evtuhov, Phys. Rev. 125, 1869 (1962).
[Favrot 75]
G. Favrot, R. L. Aggarwal and B. Lax, in Proc. 3nd Int. Conf.
on Light Scattering in Solids, Campinas, 1975, ed. by
M. Balkanski, R. C. C. Leite and S. P. S. Porto (Flammarion,
Paris, 1975), p.
286.
[Favrot 76a]
G. Favrot, R. L. Aggarwal and B. Lax, Solid State Commun. 18,
577 (1976).
[Favrot 76b]
G. Favrot, R. L. Aggarwal and B. Lax, in Proc. 13th Int. Conf.
on Physics of Semiconductors, Rome, 1976, ed. by F. G. Fumi
(North Holland, Amsterdam, 1976), p. 1035.
[Favrot 94]
G. Favrot, in Proc. 11th Int. Conf. on High Magnetic Fields in
the Physics of Semiconductors, Cambridge, 1994, ed. by
D. Heiman (World Scientific, Singapore, 1995), p. 698.
L. L. Foldy and S. A. Wouthuysen, Phys. Rev. 78, 29 (1950).
[Foldy 50]
[Geschwind 84] S. Geschwind and R. Romestain, in Light Scattering in Solids
IV, ed. by M. Cardona and G. Giintherodt (Springer, Berlin,
1984), p. 15 1 .
[Goodman 61] R. R. Goodman, Phys. Rev. 122, 397 (1961).
[Goodwin 83] M.W. Goodwin and D.G. Seiler, Phys. Rev. B 27, 3451 (1983).
[Gopalan 85]
S. Gopalan, J. K. Furdyna, and S. Rodriguez, Phys. Rev. B 32,
903 (1985).
[Gorczyca 91] I. Gorczyca, P. Pfeffer, and W. Zawadzki, Semicond. Sci.
Technol. 6, 963 (1991).
[Grisar 77]
R. Grisar and H. Wachernig, Applied Phys. 12, 1 (1977).
188
[Grisar 78]
R. Grisar, H. Wachernig, G. Bauer, J. Wlasak, J. Kowalski, and
W. Zawadzki, Phys. Rev. B 18, 4355 (1978).
[Groves 70]
S. H. Groves, C. R. Pidgeon, A. W. Ewald, and R. J. Wagner, J.
Phys. Chem. Solids 31, 2031 (1970).
[Hifele 91]
H. G. Hifele, in Landau Level Spectroscopy, ed. by G. Landwehr
and E. I. Rashba, (North-Holland, Amsterdam, 1991), p.2 0 7 .
[Hamilton 68] D.C. Hamilton and A.L. McWhorter, in Light Scattering Spectra
of Solids (Proc. Int. Conf, New York University, New York,
1968), ed. by G. B. Wright (Springer, New York, 1969), p. 3 0 9 .
[Heiman 92]
D. Heiman, in Semiconductors and Semimetals, vol. 36, ed. by
R. K. Willardson and A. C. Beer (Academic, New York, 1992), pl.
[Hensel 69]
J. C. Hensel and K. Suzuki, Phys. Rev. Lett. 22, 838 (1969).
[Herman 55]
F. Herman, J. Electron. 1, 103 (1955).
[Hermann 77] C. Hermann and C. Weisbuch, Phys. Rev. B 15, 823 (1977).
[Huant 85]
S. Huant, L.C. Brunel, M. Baj, L. Dmowski, N. Coron, and
G. Dambier, Solid State Commun. 54, 131 (1985).
[Jim6nez 94]
H.J. Jimenez, R.L. Aggarwal and G. Favrot, Phys. Rev. B 49,
4571 (1994).
[Johnson 70]
E. J. Johnson and D. H. Dickey, Phys. Rev. B 1, 2676 (1970).
[Johnson 84]
E. J. Johnson,
R. J. Seymour,
and
R. R. Alfano,
in
Semiconductors Probed by Ultrafast Laser Spectroscopy, vol. II,
ed. by
R. R. Alfano (Academic, New York, 1984), p 19 9 .
[Jusserand 95] B. Jusserand,
D. Richards,
G. Allan,
C. Priester,
and
B. Etienne, Phys. Rev. B 51, 4707 (1995).
[Kane 56]
E. O. Kane, J. Phys. Chem. Solids 1, 82 (1956).
E. O. Kane, J. Phys. Chem. Solids 1, 249 (1957).
[Kane 57]
[Kane 66]
E. O. Kane, in Semiconductors and Semimetals, ed. by R. K.
Willardson and A. C. Beer (Academic, New York, 1966), Vol. 1,
p. 75.
E. O. Kane, in Narrow Gap Semiconductors - Physics and
[Kane 79]
Applications (Proc. Int. Summer School, Nimes, 1979), ed. by
W. Zawadzki (Springer, Berlin, 1980), p. 13.
[Kelley 66]
P. L. Kelley and G. B. Wright, Bull. Am. Phys. Soc. 11, 812
(1966).
N. Kim, G.C. La Rocca, S. Rodriguez, and F. Bassani, Riv. Nuovo
[Kim 89a]
Cimento 12, No. 2 (1989), pp 1-67.
N. Kim, G. C. La Rocca, and S. Rodriguez, Phys. Rev. B 40, 3001
[Kim 89b]
(1989).
[Kittel 63]
C. Kittel, Quantum Theory of Solids (Wiley, New York, 1963.
Second revised printing, with problem solutions, 1987.)
[Kjeldaas 57] T. Kjeldaas and W. Kohn, Phys. Rev. 105, 806 (1957).
G. F. Koster and H. Statz, Phys. Rev. 113, 445 (1959).
[Koster 59]
G. F. Koster, J. O. Dimmock, R. G. Wheeler, and H. Statz,
[Koster 63]
Properties of the Thirty-Two Point Groups (MIT Press,
Cambridge, MA, 1963).
E. S. Koteles, W. R. Datars and G. Dolling, Phyw. Rev. B 9, 572
[Koteles 74]
(1974).
S. W. Kurnick and J. M. Powell, Phys. Rev. 116, 597 (1959).
[Kurnick 59]
[Landolt 82]
Landolt-Birnstein Tables, ed. by O. Madelung, M. Schulz, and
H. Weiss (Springer, Berlin, 1982), Vol. 17a.
[La Rocca 88a] G. C. La Rocca, N. Kim, and S. Rodriguez, Phys. Rev. B 38, 7595
(1988).
189
[La Rocca 88b] G. C. La Rocca, S. Rodriguez, and F. Bassani, Phys. Rev. B 38,
9819 (1988).
[Lawaetz 71]
P. Lawaetz, Phys. Rev. B 4, 3460 (1971).
[Lax 60]
B. Lax and J. G. Mavroides, in Solid State Physics, ed. by
F. Seitz and D. Turnbull (Academic, New York, 1960), Vol. 11,
p. 261.
[Lax 67]
B. Lax and J. G. Mavroides, in Semiconductors and Semimetals,
ed. by R. K. Willardson and A. C. Beer (Academic, New York,
1967), Vol. 3, p. 321.
[Lee 76]
K. F. Lee, B.S. Thesis, Massachusetts Institute of
Technology (1976).
[Lipari 70]
N. O. Lipari and A. Baldereschi, Phys. Rev. Letters 25, 1660
(1970).
[Littler 83]
C. L. Littler and D. G. Seiler, Phys. Rev. B 27, 7473 (1983).
[Liwdin 51]
P. Ltwdin, J. Chem. Phys. 19, 1396 (1951).
[Luttinger 48] J. M. Luttinger, Phys. Rev. 74, 893 (1948).
[Luttinger 55] J. M. Luttinger and W. Kohn, Phys. Rev. 97, 869 (1955).
[Luttinger 56] J. M. Luttinger, Phys. Rev. 102, 1030 (1956).
[Makarov 68] V.P. Makarov, Zh. Eksp. Teor. Fiz. 55, 704 (1968) [Sov.
Phys.-JETP 28, 366 (1969)].
[Matlab 92]
The Student Edition of Matlab (Prentice Hall, Englewood Cliffs,
NJ, 1992).
[Mayer 90]
H. Mayer and U. R6ssler, in High Magnetic Fields in
Semiconductor Physics III (Proc. Int. Conf., Wurzburg, 1990),
ed. by G. Landwehr (Springer, Berlin, 1992), p. 589.
[Mayer 91]
H. Mayer and U. R6ssler, Phys. Rev. B 44, 9048 (1991).
[McCombe 69] B. D. McCombe, Phys. Rev. 181, 1206 (1969).
[McCombe 74] B. D. McCombe, in Proc. Int. Conf. on the Application of High
Magnetic Fields in Semiconductor Physics, Wurzburg, 1974
(unpublished), p. 146.
[McCombe 75] B. D. McCombe, Advances in Electronics and Electron Phys. 37, 1
(1975)
Mooradian 70] A. Mooradian, S.- R. J. Brueck, and F. A. Blum, Appl. Phys.
Lett. 17, 481 (1970).
[Mooradian 77] A. Mooradian, in Nonlinear Optics, ed. by P. G. Harper and
B. S. Wherrett (Academic, New York, 1977), p. 2 13 .
[Obuchowicz 91] A. Obuchowicz and J. Wlasak, Acta Phys. Polonica A 79, 329
(1991).
[Ohmura 68]
Y. Ohmura, J. Phys. Soc. Japan 25, 740 (1968).
[Ogg 66]
N. R. Ogg, Proc. Phys. Soc. (London) 89, 431 (1966).
[Patel 70]
C. K. N. Patel and E. D. Shaw, Phys. Rev. Lett. 24, 451 (1970).
[Patel 71]
C. K. N. Patel and E. D. Shaw, Phys. Rev. B 3, 1279 (1971).
[Pfeffer 90]
P. Pfeffer and W. Zawadzki, Phys. Rev. B 41, 1561 (1990).
[Phillips 70]
J. C. Phillips, Rev. Mod. Phys. 42, 317 (1970).
[Pidgeon 66]
C. R. Pidgeon and R. N. Brown, Phys. Rev. 146, 575 (1966).
[Pidgeon 68]
C. R. Pidgeon and S. H. Groves, Phys. Rev. Lett. 20, 1003
(1968).
[Pidgeon 69]
C. R. Pidgeon and S. H. Groves, Phys. Rev. 186, 824 (1969).
[Pidgeon 79]
C. R. Pidgeon, in Narrow Gap Semiconductors, Physics and
Applications (Proc. Int. Summer School, Nimes, 1979), ed. by
W. Zawadzki (Springer, Berlin, 1980), p. 125.
190
C. R. Pidgeon, in Handbook on Seemiconductors, ed. by
T.S. Moss, Vol. 2, ed. by M. Balkanski (North-Holland,
Amsterdam, 1980), p. 223.
[Platzman 73] P. M. Platzman and P. A. Wolff, Waves and Interactions in Solid
State Plasmas (Solid State Physics, Suppl. 13), ed. by
H. Ehrenreich, F. Seitz and D. Turnbull (Academic, New York,
1973).
[Pertzsch 78]
B. Pertzsch and U. R6~ssler, Phys. Stat. Sol. B 89, 475 (1978).
[Plyler 60]
E. K. Plyler, A. Danti, L. R. Blaine, and E. D. Tidwell, J.
Res. Nat. Bur. Stand. - A. Phys. Chem. 64A, 7 (1960).
Semiconductors and
in
S. Rodriguez,
and
A. K. Ramdas
[Ramdas 88]
Semimetals, vol. 25, ed. by R. K. Willardson and A. C. Beer
(Academic, New York, 1988), p. 3 4 5 .
[Ranvaud 79] R. Ranvaud, H. -R. Trebin, U. R6ssler, and F. H. Pollak, Phys.
Rev. B 20, 701 (1979).
[Rashba 61a]
E. I. Rashba and V. I. Sheka, Soviet Phys.-Solid State 3, 1257
(1961).
[Rashba 61b] E. I. Rashba and V. I. Sheka, Soviet Phys.-Solid State 3, 1357
(1961).
E. I. Rashba and V. I. Sheka, in Landau Level Spectroscopy,
[Rashba 91]
ed. by G. Landwehr and E. I. Rashba (North-Holland - Elsevier,
Amsterdam, 1991), p. 131.
[Rodriguez 88] S. Rodriguez, in High Magnetic Fields in Semiconductor Physics
II (Proc. Int. Conf., Wurzburg, 1988), ed. by G. Landwehr
(Springer, Berlin, 1989), p. 550.
[Romestain 74] R. Romestain, S. Geschwind and G. E. Devlin, Phys. Rev. Lett.
33, 10 (1974).
M. E. Rose, Elementary Theory of Angular Momentum (Wiley, New
[Rose 57]
York, 1957).
[Rissler 84]
U. R6ssler, Solid State Commun. 49, 943 (1984).
[Roth 59]
L. M. Roth, B. Lax, and S. Zwerdling, Phys. Rev. 114, 90
(1959).
[Schiff 68]
L. I. Schiff, Quantum Mechanics (McGraw-Hill, New York, 3d ed.
1968).
[Scott 75]
J. F. Scott, in Laser Applications to Optics and Spectroscopy,
ed. by S. F. Jacobs, M. Sargent, J. F. Scott and M. O. Scully
(Addison-Wesley, Reading, Mass., 1975), p. 12 3 .
[Scott 76]
J. F. Scott, in Laser Photochemistry, Tunable Lasers, and Other
Topics, ed. by S. F. Jacobs, M. Sargent, M. O. Scully and
C. T. Walker (Addison-Wesley, Reading, Mass., 1976), p 32 5 .
J. F. Scott, Rep. Prog. Phys. 43, 951 (1980).
[Scott 80]
K. Seeger, Semiconductor Physics (Springer, Berlin, 1984)
[Seeger 89]
Chap. 8.
in
M. H. Weiler,
and
C. L. Littler,
[Seiler 92]
D. G. Seiler,
Semiconductors and Semimetals, ed. by R. K. Willardson and
A. C. Beer (Academic, New York, 1992), Vol. 36, p. 293.
V. I. Sheka, Soviet Phys.- Solid State 6, 2470 (1965).
[Sheka 65]
V. I. Sheka and I. G. Zaslavskaya, Ukr. Fiz. Zh. 14, 1825
[Sheka 69]
(1969).
[Shockley 50] W. Shockley, Phys. Rev. 78, 173 (1950).
[Slusher 67]
R. E. Slusher, C. K. N. Patel and P. A. Fleury, Phys. Rev.
Lett. 18, 77 (1967).
[Pidgeon 80]
191
[Smith 77]
S. D. Smith, R. B. Dennis and R. G. Harrison, Progr. in Quantum
Electron. 5, 205 (1977).
[Stickler 62]
J. J. Stickler, H. J. Zeiger, and G. S. Heller, Phys. Rev. 127,
1077 (1962).
[Suzuki 74]
K. Suzuki and J. C. Hensel, Phys Rev. B 9, 4184 (1974).
[Thomas 68]
D. G. Thomas and J. J. Hopfield, Phys. Rev. 175, 1021 (1968).
[Tinkham 64] M. Tinkham, Group Theory and Quantum Mechanics (McGraw-Hill,
New York, 1964).
[Trebin 79]
H.-R. Trebin, U. R6ssler, and R. Ranvaud, Phys. Rev. B 20, 686
(1979).
[Von der Lage 47] F. C. Von der Lage and H. A. Bethe, Phys. Rev. 71, 612
(1947).
[Wachernig 74] H. Wachernig and R. Grisar, Solid State Commun. 15, 1365 (1974)
[Wachernig 75] H. Wachernig, R. Grisar, P. Kacman, and W. Zawadzki, in Conf.
Digest of Int. Conf Infrared Physics, Zurich, 1975, ed.
by H. Wiesendanger and F. Kneubiihl, p. 250.
[Weiler 74]
M. H. Weiler, R. L. Aggarwal, and B. Lax, Solid State Commun.
14, 299 (1974).
[Weiler 76]
M. H. Weiler, R. L. Aggarwal, and B. Lax, Bull. Am. Phys. Soc.
21, 429 (1976).
[Weiler 77]
M. H. Weiler,
Ph.D.
Thesis,
Massachusetts
Institute
of
Technology (1976).
[Weiler 78]
M. H. Weiler, R. L. Aggarwal, and B. Lax, Phys. Rev. B 17, 3269
(1978).
[Weiler 79]
M. H. Weiler, J. Magn. Magn. Mater. 11, 131 (1979).
[Weiler 81]
M. H. Weiler, in Semiconductors and Semimetals, ed. by R. K.
Willardson and A. C. Beer (Academic, New York, 1981), Vol. 16, p. 119.
[Weiler 82]
M. H. Weiler, Solid State Commun. 44, 287 (1982).
[Weyl 46]
H. Weyl,
The
Classical Groups, Their Invariants and
Representations (Princeton University Press, Princeton, NJ,
1946).
[Wherrett 69] B.S. Wherrett and P.G. Harper, Phys. Rev. 183, 692 (1969).
[Wlasak 86]
J. Wlasak, J. Phys C 19, 3459 (1986).
[Wolff 66]
P. A. Wolff, Phys. Rev. Lett. 16, 225 (1966).
[Wolff 68]
P. A. Wolff, Phys. Rev. 171, 436 (1968).
[Wolff 75a]
P. A. Wolff, in Physics of Quantum Electronics, Vol. 4: Laser
Photochemistry, Tunable Lasers, and Other Topics (Proc. Summer
School, Santa Fe, NM, 1975), ed. by S. F. Jacobs, M. Sargent,
M. O. Scully, and C. T. Walker (Addison-Wesley, Reading, MA,
1976), p. 305.
[Wolff 75b]
P. A. Wolff, in Nonlinear Optics (Proc. 16th Scottish
Universities Summer School in Physics, Edinburgh, 1975), ed. by
P.G. Harper and B.S. Wherrett (Academic, New York, 1977), p.
169.
[Wolff 76]
P. A. Wolff, R. N. Nucho and R. L. Aggarwal, IEEE J. Quant.
Electron., QE-12, 500 (1976).
[Wolff 77]
P. A. Wolff, in Nonlinear Optics, ed. by P. G. Harper and
B. S. Wherrett (Academic, New York, 1977), p. 16 9 .
[Wright 68]
G.B. Wright, P.L. Kelley and S.H. Groves, in Light Scattering
Spectra of Solids (Proc. Int. Conf, New York University, New
York, 1968), ed. by G.B. Wright (Springer, New York, 1969),
p.335.
Y. Yafet, Phys. Rev. 115, 1172 (1959).
[Yafet 59]
192
[Yafet 63]
[Yafet 66]
[Yafet 73]
[Zawadzki 73]
[Zawadzki 76]
[Zawadzki 79]
[Zawadzki 90]
[Zawadzki 92]
[Ziman 72]
[Zucca 70]
Y. Yafet, in Solid State Physics, ed. by F. Seitz and D.
Turnbull (Academic, New York, 1963), Vol. 14, p. 1.
Y. Yafet, Phys. Rev. 152, 858 (1966).
Y. Yafet, in New Developments in Semiconductors, ed. by P. R.
Wallace, R. Harris, M. J. Zuckermann (Noordhoff, Leyden, 1973),
p.469.
W. Zawadzki, in New Developments in Semiconductors, ed. by
P. R. Wallace, R. Harris, M. J. Zuckermann (Noordhoff, Leyden,
1973), p. 441.
W. Zawadzki and J. Wiasak, J. Phys. C 9, L663 (1976).
W. Zawadzki, in Narrow Gap Semiconductors, Physics and
Applications (Proc. Int. Summer School, Nimes, 1979), ed.
by
W. Zawadzki (Springer, Berlin, 1980), p. 85.
W. Zawadzki, X. N. Song, C. L. Littler, and D. G. Seiler, Phys.
Rev. B 42, 5260 (1990).
W. Zawadzki, I. T. Yoon, C. L. Littler, X. N. Song, and P.
Pfeffer, Phys. Rev. B 46, 9469 (1992).
J. M. Ziman, Principles of the Theory of Solids, 2nd ed.
(Cambridge University Press, Cambridge, 1972).
R. R. L. Zucca and Y. R. Shen, Phys. Rev. B 1, 2668 (1970).
193
Appendix A
Extended Pidgeon and Brown Hamiltonian
In this appendix I will derive the Pidgeon and Brown (PB) Hamiltonian with the
extensions made by Weiler et al. using the invariant expansion approach of
Trebin
et al. [Pidgeon 66,69,
Hamiltonian
contains
20
spin-orbit splitting A.
Weiler 78,81,
parameters
besides
Trebin 79].
the
The
energy
8x8
matrix
gap
E and the
g
Of these parameters, four are associated with terms
of order k and sixteen are associated with terms of order k2 , where k - p+-A
C
is the kinetic momentum of an electron located in a vector potential A which
results
from
a
constant
electromagnetic field.
background
magnetic
field
plus
a
perturbing
The invariant expansion formalism is a generalization
of Luttinger's formulation for the valence band of germanium, based on work by
The invariant formulation allows the
Pikus and Bir [Luttinger 56, Bir 74].
separation of Heff into terms which are manifestly spherical, octahedral
and
antisymmetric tetrahedral, and simplifies the expression of the Hamiltonian in
coordinate
frames
which
are
rotated
relative
to
the
cubic
crystallographic
axes.
The PB Hamiltonian has the form [Trebin 79]
Hcc HH
Hcv
Hef
cs
Hvc Hvv Hvs
Hs
Hsv
(A.1)
ss
in which the individual matrix blocks are 2x2, 2x4, 4x2 or 4x4.
wave functions of the form
195
It acts on
fc(r)
X(r) =
(A.2)
fv(r)
fs(r)
where
f3 (r)
f
fc(r) =
f(r)(r)
f6(r)
If(r) (r)
f2 (r)
fs(r) =
fv(r) =
f 5 (r)
L 7(
J
f 4 (r)
(A.3)
The indices c, v and s refer to the
heavy
hole
valence
respectively.
band
6 conduction band (F6), Fs light and
(T), and
F7
split-off
valence
The numerical indices refer to the Luttinger-Kohn
states, as labeled by [Pidgeon 66].
band
(v),
(LK) basis
The basis states will be discussed below.
Hermiticity of Hef requires that
Hcc =
cc ,
Hvv = Ht
vv
and Hss = H ss
(A.4a)
HSC = H CS
and Hsv = Htvs
(A.4b)
and
H
VC
= Ht
CV
,
The wave function is often represented as
W(r) = . fi(r)ui(r)
where
the
sum
involves
(A.5)
the
eight
(LK)
basis
functions
ui(r) for
conduction and valence bands at the center of the Brillouin zone.
the
The spatial
character of the basis functions, including the periodicity, drops out of the
196
[Luttinger 55]
mass theory
final effective
being
effectively replaced
by
the
Equation (A.5) is the same as Eqs.(A.2) and (A.3) if the
band parameters.
functions ui(r) are replaced by 8-component unit spinors, each with seven zero
elements and a single unit element.
The LK basis functions uI through u8 are the conduction band-edge and
valence band-edge eigenfunctions of the crystal Hamiltonian at k=O, with the
The conduction band basis functions ul and
spin-orbit interaction included.
u2 form a basis for the 16 representation of the Td point group.
functions
u3
through
u6
form
a basis
the
for
I8
The basis
representation
of
T
associated with the light and heavy holes, and u7 and u8 form a basis for the
F7
representation,
associated
with
the
split-off
valence
band.
The
three
The basis functions for these three
Iv 8 and t v.
bands at k=O are denoted f,
7
6
basis functions of the full
D
and D
bands transform like the D
rotation
group
R(3),
1/2
3/2
1/2
respectively
corresponding
to
spin j=1/2 even
parity,
j=3/2 odd parity, and j=1/2 odd parity, as was pointed out by Trebin et al.
The association
of the actual basis functions with the D+1/2, D-3/2 and D-1/2
basis functions
is a mathematical
basis
functions
transformation
do
properties
not
with
convenience,
definite
possess
respect
to
become particularly clear in Appendix C.)
R(3).
or artifice,
parity
(The
or
since the actual
definite
'convenience'
group
will
In the tables of Koster et al. it
is the D +3/2 basis functions of R(3) which are used for the TF 8 basis states,
and this caused Weiler et al. to conclude that the 16 and 1F7 representations
were reversed in Koster's tables of coupling coefficients.
By reordering and
changing some signs it is possible to make the D3 2 basis functions transform
like the D_
3/2
functions with respect to the discrete Td operations, and vice
197
versa.
In the present case it is better to make the valence band functions
transform like D 3/2 and D_1/2 rather than D+ and D
because this maximizes
3/2
1/2
the part of Hef which is isotropic (spherically symmetric).
the D3
the
and D/
single-group
bands
representations result from the spin-orbit transformation of
basis functions
(representations
F 1 and
functions with respect to T.
of 'parity'
This is because
S, X, Y, Z for the conduction
F1 5) which
transform like
(See [Kane 57,66].)
and valence
atomic
s
and
p
If one particular choice
is made for all the F8 bands, all the off-diagonal matrices of the
spin-orbit Hamiltonian between F8 bands will be proportional to the 4x4 unit
matrix 1j.
the
The double-group basis states actually contain some admixtures of
functions
F ®D ,
12
1/2
F ®D+
25
and
1/2
F ®D
2
1/2
due
to
the
spin-orbit
interaction, but are principally composed of F ®D+2 and IF @D+ , where D
I
1/2
1/2
15
1/2
represents the ordinary spin of an electron.
For our representation
Weiler's
basis
functions
of the basis functions uI through u8 I will use
multiplied
by
i.
This
and
D/1/2 differs by
an overall
the
functions
Notice that this choice for
slightly, and does not change the Hamiltonian.
D2
3/2
simplifies
sign from the usual
Clebsch-Gordon
[Anderson 89??].
coefficients
The basis functions then are
6
Su
10
-
D+
iS
I
1/2
(1
2
u2= ]iSj>
rV:
U
=
u
=
(A.6a)
,
-
x+i)
(X+iY)
30
8
+
2
3
(3
[(X+iY), - 2ZT]
F60
198
32
3/2
)
[(-iY)'
Uso = -
(, --)
+ 2ZJ
2
[=
u80 = F5(x+irY)l+
tV:
7
U
)-
D 1/2:
z]
[(X-iY) - Z .
=
(A.6b)
( ,(A.6c)
These functions may be regarded as the zero-order LK basis functions with
respect to the spin-orbit interaction [Pidgeon 66],
represent
the
properties.
actual
LK
basis
functions
and may be considered to
with
respect
to
transformation
Here S,X,Y,Z are real functions which may be regarded either as
as LK basis states for the f
and Tv conduction and valence bands neglecting
spin, as mentioned above, or as atomic s and p wave functions.
and % denote the spin-up and spin-down
(
spinors
The arrows
().
and
functions are ordered so that m decreases from top to bottom.
The basis
The label on
the right is (j,m)+ of the atomic basis states, with the superscript denoting
the parity.
The numbering
[Weiler 78].
associates
This
odd
gives
numbered
is assigned
consecutive
basis
according
numbers
states
with
to
the
to
time
[Pidgeon 66,69]
reversed
a-set
so-called
pairs
and
and
('spin-up')
axial model wave functions at k =0, and even numbered basis states with the
('spin-down')
b-set
wave
functions.
(The
axial
model
will
be
discussed
later.)
An advantage of the present formulation is that the basis matrices can be
constructed purely from the Clebsch-Gordon coefficients contained in the basis
functions uio(r).
The chosen phases and ordering of these functions produce
the standard representations for total angular momentum L+I
199
when the basis
states are viewed as atomic wave functions.
Here L = rxp , and ax, y and a
are
the
standard
Pauli
spin
matrices
are the standard Pauli spin matrices
Ox=(
1 01) ,
a =
(0-i0
1-01 ,
,0
which operate on two-component spinors.
the D"
1/2
bases are just -
2
(A.7)
The matrix representations of L+-
, and the matrix representation in the D
in
basis
3/2
is J, where
3
0 O0
0 2
2 0
0 V3
Jx =
L0
0V3 0
0 -2
i0
0
0
0
J
Y
20
0 O'
O'
0
1
2
v0
0 VS
0
0
-1
0
0
0 -3
J
(A.8)
Note that
0+
00)
a=(
(A.9)
o0
and
0
0
0
0
V3
0
20
0
0
0]
0
J+=
0
0
0
0]
J
I,
(A.10)
L0
The basis matrices for the cc and ss blocks are
1,
ax ,
ay, a
or
1
and (;
)
.
(A.11)
The basis matrices for the vv block are (see Appendix B)
1
, J",j),
(A.12)
J2) (3),
200
is the irreducible spherical tensor or rank I formed from the
J()
where
[Luttinger 56].
The
Cartesian matrices obtained from j(2) and j(3 ) are given in Table A.1.
The
components
simple
of J,
was
as
demonstrated by Luttinger
numerical representations
of the J(f) matrices
makes them easier to
use than the Cartesian ones.
Terms in the invariant expansion of Hef have the form
C
a
)K(
teff -
)*
as discussed by Luttinger, and further developed by Bir and Pikus, and applied
to InSb by Trebin et al.
specifies
the
basis
function
Bir 74, Trebin 79].
numerical, and
K
Here e labels a single-group ,irrep of Td and u
of
('partner')
the
representation
[Luttinger 56,
a matrix operator which
Here X represents
is purely
represents a spatial operator formed from the components of k
or B or both, or also the electric field E.
Basis matrices for Hcv can be found from the representation of k p in the
Here k is taken to operate only
cv block, which will be abbreviated (k.P)cv
on spinor components fi(r), while p operates only between the basis functions
Ui(r), so (k-P)cv could be written as k pcv . The k-p matrix is
(k -p)
= P
2
+
R
6
(A. 13)
-
where
P - <iS pxlX> = <iSI py Y> = <iS PzIZ> .
The matrix of k-i , where
lattice
potential,
has
the
h
= p + -2
mc
same
form.
201
(A.14)
xVV(r) and V(r) is the periodic crystal
The
contribution
of
the
term
proportional to exVV to P is so small that it is generally neglected.
The
invariant representation of (k p)cv is
(k -P)cv = Pk t ,
(A.15)
where the components of t are
t
F-i
t =
t
(A.16a)
,
O
i
0
(A. 16b)
0 -
0
0
=
(A.16c)
-0
0
0.-
with
t+ =2
(A. 17a)
0
2
2
0
0
-
O
(A.17b)
0
0
0
The ti matrices are proportional to the matrices Ti defined by Trebin et al.
[Trebin 79].
couple
the
They transform like a polar vector (negative parity) since they
DI
basis
states with
the D+
basis states.
The
remaining
matrices required for a complete set for Hcv are obtained from the second-rank
tensor
(at)(2 ) formed
from
the
products
202
a.t. .
ii
Note
that
t.
J
is
a
cv
cross-space
also
operator while
a i operates in the c subspace,
in the cv cross-space as desired.
Trebin's matrices
proportional to a.t. and a.t.+ a.t. [Trebin 79].
II
t. =- 3T.;
I
Ij
=
o.t.
I
j I
-3T.:
II
T.. and T.. are
The relations are
ot + at
xy
1
so the result is
yx
=- 2ViT
xy
and c.p.
(A.18)
Basis matrices for the Hvs block of Heff are found from the matrix
representations of L, ( and L ( 2 ) in the vs cross space.
For the vv, vs and ss
blocks we find [Luttinger 56, Suzuki 74]
vv
ss
L.1
2
-J-.
31
2
3
.
2
-J.
3 1
--.3
1
1
1-
1
1
2
1
(A.19)
-- u.1
1 (o)(2)
0
J(2)
L(2)_
vs
3
3
Here
u.I = t l.
(A.20)
numerically, but u is an axial vector, like 8 and J, instead of polar, since
it connects DT
3/2
with D.
1/2
Notice that the sum
L.+ c.
1
2 1
gives
J.,
1
1
2 1
and 0 in the three blocks, as one would expect for the total angular momentum.
(See
the
last
transformation
section
of
[Luttinger 56].)
of products of L.
Suzuki and Hensel [Suzuki 74].
the entry 3ilx should be
- 3ilx .)
The
complete
spin-orbit
and L L. with a. is given in Table I of
(In the second to last column of this table,
The basis matrices for Hvs are obtained
from
u'
and
(Ua)(
2 ).
(A.21)
203
Finally the basis matrices for Hcs are found from
(A.22)
3+
3
-
where
(A.23)
t'
Here pi is numerically equal to o i but p is a polar vector instead of axial
[Trebin 79].
The remaining basis matrix is IP which is a 2x2 unit matrix
transforming like a pseudoscalar.
In summary, the complete set of basis matrices is
:
H
Hvv :
1
1,
and (o7)
) j( 2 ),
(3)
(even parity)
(A.24a)
(even parity)
(A.24b)
Hss :
1a and (Y0 )
(even parity)
(A.24c)
Hc
Cv
t( l ) and (ot)(2)
(odd parity)
(A.24d)
(odd parity)
(A.24e)
(even parity)
(A.24f)
Hcs : 1p and p(l),
Hvs
vs
or
1p and t'
1)
u(1) and (uo()(2) .
The Cartesian decomposition of these is given in Table A.1 along with the
time-reversal properties.
Note that the components of j(2) are reducible with
O h and T , so that the set {3J - J2 , V3(j 2 -_ 2)
transforms
h
d
z
x y
according to F+12 of O0 h and F12 of T d , and the set {{J ,J}, {J ,J
{Jxy,J }
yz
zx
respect
to
transforms according to F+25 of O h and F 15 of T
d
204
Reductions of the other
Table A.1. Irreducible basis matrices for PB
Table II of Trebin et al. [Trebin 79].
Block
CC
Hamiltonian.
Parity /
Td basis
R(3) basis
+/+/+
rF: J , J , J
J
25
j( 2)
j( 3)
x
F:
12
3J2Z -
r
{J
155
V3(j2 -j 2
X
J }
y z
rF :
JJJ
2
r
1
yxy
:
CV
t(l)
zxz
+/+
, , az
tx, t
r
+/-/-
t
(2 )
-/+
(ot)
rF:
25
cs
1
t (1)
u( )
vs
rF :
:
15
r2F:
25
(uo) (2)
+/-
and c.p.
1
25 :
R
and c.p.
X
-JJJ
F : JJJ
Ga
+/-
zyx
F25
: J3X _J 20
25
+/+
and c.p.
+JJJ
xyz
15
+/-
z
y
yt + ot
yz
zy
1s
-/+
1
-I+
t', t', t'
x y z
-I-
ux
X,
u o
yz
+/-
Uy, uz
)
Z
F12 : 3u
3 ZY
Z,
F :
and c.p.
V(uxx -u G
+ uo
L
zy
205
and c.p.
Time
rever sal
+1+
1
:1
vv
Adapted from
+/+
+1+
sets of components
oxt = - it ,
so
are given in the table.
these
combinations
7-t = 02x4
Note that
of components
give
no
new
and
matrices.
= 04x2 and uxo = - iu .
Similarly u
The following algebraic relations among the basis matrices are useful:
00 = 3()
,
J-J = -(1)
,
JxJ = iJ
o-t = t-J = 0 ,
oxt
t- t
txt(tt)(2)=
= 2(1)
tt t
These
xo = 2io ,
J(4 )=
,
-2tx
5
(A.25b)
(ot)(2) = 2(tJ) (2 ) ;
t)(2)
1j(
used in decoupling
for
the
2
3
(t3t)2)
Hamiltonians
(A.25c)
0 ;
4
may be
(A.25a)
0;
=it
15
relations
effective
,
G( 2 ) = 0;
(A.25d)
(A.25e)
).
the PB
conduction
and
Hamiltonian to
valence
bands.
obtain
Relations
involving u are found from Hermitian conjugates of relations involving t.
The
Cartesian decomposition of the second-rank equations may be used to obtain
t tl = t tt and
xx
yy
relations like
In the present case,
with
respect
approach
to
to
the
T
like
invariant
t t =-t tt.
xy
yx
where
irreps
the crystal
basis
of
the
full
expansion
may
be
functions
rotation
applied
connection with cubic harmonics and Cartesian tensors.
considers
and
cubic
all
possible
(octahedral
invariants
and
obtained
tetrahedral)
etc.
206
by
group,
which
a
has
different
a
direct
In this approach one
polarization
invariants like
ui(r) transform
2
r,
of
4
r,
the
spherical
xyz,
x 4+y4 +z4
Using the definitions of the band parameters given by Weiler et al., the
components of Hef are (with h = m = 1)
= Egl
H
H
VV
_=-
2
+ (F+-)k
2
k
J
21
+ [{kk
+ (N+)B a
22
+ yk
2 XX
+ k2
J
y
Ik 2
Hs = (-A)1
+ k
2
ZZ
-
k 2 2]
3
xJ
-- q [B
+ c.p.] -
}{,JJ
C[k {J , J 2 -
+
2
YY
(A.26)
2
+
ByJ + B
} + c.p.]
(A.27)
z
(A.28)
(Ic'+)Ba
Hcv = Pkt - iG[{k,k}t + c.p.]
zz +c.p.]
yy
+ ¢-t) 2 x (at
- FiN[B
H
CS
(at +
t ) + c.p.]
(A.29)
(A.30)
= P'kt' - iG'[{k,k }t'z + c.p.]
X Y
Hv = -v ' y[k2U ax + kUy a + kZuo]
V- yk[{k,k }(ua
23
x y xy
+"I
where
{a,b)}
e(c"+1)B.u +
ab + ba
+ Uy
yx
x)
+ c.p.]
iC'[kx(uy
z
+ uz
is the anticommutator.
completely general as written.
+ c.p.]
(A.31)
The above components are
The choice of the numerical coefficients makes
the primed and unprimed band parameters, except for C and C', equal in the
207
limit
of
small
numerical
spin-orbit
constant
free-electron
coupling
(single
group
approximation).
The
associated with F results from the contribution of the
-
2
kinetic energy to the effective
Hamiltonian,
and
the constants
, I and 1 associated with N, x' and K" result from the free-electron Zeeman
2' 2
term.
Notice that factors of i are required in the off-diagonal terms H(G),
H(G'),
H(N)
and
H(C') in
order
to
satisfy
the
time-reversal
symmetry
requirement.
We can use irreducible spherical tensor components to reexpress Hef
t in a
form
which
unambiguously
separates
[Lipari 70, Baldereschi 73, 74].
H
cc
the
spherical
and H
SS
and
anisotropic
parts
contain no anisotropic terms to
second order in k, and therefore remain unchanged, except that the +, -, and z
components
of all vectors should be used in place of the usual Cartesian
components.
H , Hcv , H , and H may be reexpressed as
vv
cv
cs
vs
H
VV
1k2
yK 2 ) . j(2 )
C
Hcv = Pk t
-
+q)Bj
1
-
20
j( 2 )] (4)
C
e [B(10)
.
j( 3)]( 4)
2C [k(')® j(3 )] (3)
CC
VA
e
[B'
= P'k -t' - V3G'[K(2)(
(A.32)
6N [K(2)( (at)(2)](3)
A
F
CS
2[K(2)®
V3G[K(2) t(1)] (3)-
A2 C 3
H
-
-1
2
-
Cv
+
2
(ot)(2)] (3)
A
t'(1 ] (3)
A
(A.33)
(A.34)
A
208
'+1
-~(2'.(u)(2) + ~2C ("+)B
Hvs = -V
+ 2V3 g'[K(2)® (UG)(2)](4) +V
C
C'[K(')® (uo)(2)] (3)
A
2
(A.35)
where
-
2
3
SY +
(A.36)
3'
and
r I
(3
(A.37)
2) .
Here, I use the tensor dot product [Edmonds 74]
(_l)m K(2) j(
K(2)j(2) _
m
2
)
-m
(A.38)
= (k
2
+ k2
) +
({k,kS} { J,J}
2
+ h.c.) + l3k - k2)(3
2 .
I also define the irreducible 4th-rank octahedral invariant
T(4)(ABCD) =ABCD +ABCD +ABCD
c
xxxx
yyyy
zzzz
(A
-1
.CD. + AB.CD. + AB.CD)
ij
+
[T 44 + T(+4)
-4
_
which is
4 th
4
6 T 04)
"5
(A.39)
order in the components of four vectors A, B, C and D.
I
similarly define the 3drank tetrahedral invariant
T A(3)(ABC)
V3
[A (B C + B C
X y Z
Z
+ c.p.]
(A.40)
= T ( 3) - T ( 3)
2
-2
which is 3d order in the components of A, B and C.
209
The above expressions can be regarded as the complete polarization of the
irreducible cubic harmonics
x4 + y4+
Weyl 46, Von der Lage 47].)
-
3
4
and
(See [Edmonds 74,
xyz .
The first expression has full octahedral (Oh
symmetry if the set of vectors (ABCD) contains an even number of polar and an
even
number of axial vectors.
The second expression has tetrahedral
(Td)
symmetry and is odd under inversion if the set of vectors (ABC) contains an
odd number of polar vectors and an even number of axial vectors.
expression
is
easily
generalized
to
include
the
case
of
This latter
four
vectors
replacing A, B or C by the cross product of a pair of vectors D x E.
by
This may
be used to obtain invariants of the form
A B (C D - C D) + c.p. = -V3[(AB)(2)
(CD)(2)]
ZZ
Yy
xx
(A.41)
(3)
A
and
k (JJJ
I
use
the
- J J J ) + c.p. = -[k()
standard
definitions
[Edmonds 74, Anderson 81].
k
are
and
k
factor
,
of
for
k+-
-{, k ,
-,(3k
k2)
than
the
smaller
[Baldereschi 73, 74].
(A.42)
irreducible
tensors
and
tensor
products
For example, the components of k(' ) are -l
k , where
3
J(3)(3).
k
ik.
The components of K 2 )
k ,k}),
and
definition
used
k2
-
k+ ,
[k(')® k ](2)
This definition is a
by
Baldereschi
et
al.
Care must be taken when vector components fail to
commute, as with k and J.
Expressions for K(3) and j(3) are given in Appendix
B, along with some useful tensor products and expressions.
In order to change H(y2) and H(y3) in Hvv from irreducible Cartesian form
to irreducible spherical form one uses the relations
210
2 + k 2 j 2 + k2J 2 -!kJ
H(y) = ,2[k2J
]2
2 2
yy
ZZ
3
=-2 [K(2).j(2) + [K(2)
(2)
(A.43)
4)]
and
H(y,3 )= ;Y[{k, k }{J Jy } + c.p.]
S73
[K(2) J( 2 ) -
2
K(2). j( )
Notice that
K(2). j(2) = kJ2
2
xx
[K
2
(2 )] 4)] .
(A.44)
can be written in Cartesian components as
+ k 2J
2
yy
+ kJ2
2
zz
-
3
k2
+ 2 [{kX ,kY }{JX ,Jy
+ c.p.]
(A.45)
and also as
Compare
eB
-k2j2
K(2).j(2) = (k J)2
LC
3
[Luttinger 56].
This
.
(A.46)
last expression
can
be
obtained
from
the
recoupling relation
(a-p)(b.q) = (a-b)(p.q) + (axb) (pxq)+
(ab)2).(pq 2),
which is valid if the components of b and p commute.
(A.47)
Notice finally that
2 is often expressed in the literature as -k2, i.e.
Jk2J
4
ik22
=
3
-k2,
4
(A.48)
which is true when j=32
Note that
H(g) = - 2j[K(2)®
=-
24k2j
2
xx
2
j( )] (4)
C
+ k2j 2 + k2j 2
yy
zz
k2j2 _
-
3
211
K(2) (2)
5
.
(A.49)
and
2V3jg'[K(2)0 (U)
H(3') =
= 2VL
(2)] (4)
C
Xa + k2u y + k2u
2zC
K(
(A.50)
()
Similarly
H(q) =
-
-q [BJJJ] (4) +
C
C
5C
[BJJJ]4) +
-
(3J 2
B.-
where we used
[BJJJ](4)
c
[B J3 + B J3 + B J3
y
x
1)B-J]
-1(3J25
zz
(A.51)
and
(3J2 - 1) = 41/20 ,
The
simplified
representation
by
the
for j= .
of H
(A.52)
of the PB
irreducible
spherical
model in
tensor
8x8 matrix form is
formulation.
This
is
particularly true for the axial part, which contains only the m=0 components
of the tensors T(3)
and T m( 4), but it is also true for the nonaxial terms.
m
reason
for this
is that the basis matrices
of the
elements along a single diagonal or off-diagonal line.
finite
elements
only
at positions
11,
22,
elements only at positions 13 and 24.)
33
form Xm()
The
have finite
(For example, j(02) has
and 44,
and j(22) has finite
In addition, the spherical components
of the operators k and K(2) are optimum for acting on Landau level wave
functions
of the
axial model,
since they
operators.
212
function
as raising
and
lowering
The axial part of Hef is obtained by keeping only the spherical terms,
and the m=O components of the nonspherical terms.
The inversion asymmetry
terms in Heff are usually omitted from the axial approximation, so that both
the
axial
and
nonaxial
parts
are
treated
by
perturbation
theory.
The
warping terms, which involve the parameters I, i' and q, are all of order k 2 .
The
first two contain the components of K(2), and the third contains the
components of B. The axial parts are (see Appendix C)
HU( )=
- 2(W 0
-
k2J -
k k {J+'JQ
+
1(3k - k 2 )(3J -
2)
2-2
lk
k{J_,J} + 1+-)
'k
2 +k-{
H (L') = 2
'(W
+
H=
-
)-k2ua
(
-
4(3k - k)(
4 )3
)(W
-
kk(ua + uo)
J2
B+(5J + 5J
e
-3B
k (ua + uo)
2
+
-
+ 2)J + 'B (5
2
(5J -5J-
J2 + 2)J_)
If we include axial part of H(g) with H(y) we get
H(, )
= H(7) + H(L)
= y' [(3k
+
y"'[
-J 2)] + y"[(kY + h.c.)]
k2)(3J
k k {J,J} + h.c.] ,
213
()41e
-
+
6k2U
12
G_
- 3 2 + 1)J
BJ
where
-
(= 2
0
5
y'" = 7 + 2g(W 0 -
3)
5
Notice that H (7,g) is obtained from H(y) by replacing y by y' along the 4x4
matrix diagonal, replacing
places
removed
from
j by y" along the off-diagonal
the
diagonal,
and
replacing
j
lines which are 2
by
y"'
off-diagonal lines which are 1 place removed from the diagonal.
along
the
We similarly
find
Hax
(y,')= H(T) + H (')
ax
ax
_)]
=y'[(3k2 -k2)(3u%)] + y"[J(k2u a + ku
+y"'[
1
k_k(ua+ + u,)
+ 1
kk(u_
+
_)1.
Here I avoid the use of a second level of primes by invoking the single-group
approximation,
which
sets
y'= y and
obtained from H(Y) by replacing
g'= g .
Once again, H (T',g') is
' by y' along the Am=O "diagonal," replacing
Y' by y" along the off-diagonal lines which are 2 places removed from the
"diagonal," and replacing
Y7
by y"' along the off-diagonal lines which are
place removed from the "diagonal."
Finally we have
H ax (K,q) = H(K) + H ax(q)
S-
s
(K +
q)J
+
(
)(5J-
3J 2 +
1)J
A
where I have chosen B O to be directed along
214
.,and used s-eB 0/hc.
1
[Pidgeon 66, Weiler 78,81]
Axial model Hamiltonian
and su
Terms proportional to k21 , k21 , so , sJ
in
matrix
form.
The
and
similar
[K(2)0 j(2 ) (40)
complicated.
Notice
that
proportional
terms
terms
only
H Cv
in
and
H
to
are easy to express
j(2)
k -t,
K(2).
are
slightly
more
anisotropic
axial
and
H VS
H
contain
and
components, proportional to [L, R' and q, if the inversion asymmetry terms are
omitted.
Pk-t in Hcv is given by Eq.(A.13).
P'k t' in Hcv is given by Eq.(A.22)
with P replaced by P'. For Hcc ,' HCV and Hcs we have the 2x8 representation:
A
hc(1,1) = E
+ s[(2F+1)(N + 0.5 +
2)
+ (N +0.5)]
N ->
hc(2,2) = E
+ s[(2F+1)(N + 0.5 +
2)
- (N +0.5)]
N -
A
aft
hc(1,3) = Pv2- V172 at
e+1
vq
hc(1,4) = - Pv'2~ V273
a -- >
hc(1,5) = - Pv2s VI-6 a
T
hc(1,7) = PZ VTIT3
7
hc(1,8) = PV2s vT3 a
a
hc(2,4) = PV2
at"
V'T6 at
-
/e+1
Ve+ 1
VVT-2
hc(2,5) = -Pv2s vT273
hc(2,6) = -PV2-s v2 a
a"
hc(2,7) = PV2sZ VT3 at
VV+-r
hc(2,8) = -PV2s V1T73
I
For Hvv
H
and H
(valence band blocks) it is straightforward to derive the
following
6x6 representation:
following 6x6 representation:
215
A
hv(1,1) = -s[y,(N + 0.5 + ~2) + 1.5 K + y'(N + 0.5-22)]
A
A
hv(2,2) = -s[y(N + 0.5 + 2) + 0.5
hv(3,3) = -s[y (N + 0.5 +
A
hv(4,4) = -s[y(N + 0.5 +
N ---> -1
A
- y'(N + 0.5- 22)
N--
--2)0.5 K - y'(N + 0.5- 22)]
N-
A
e+1
A
-_2)1.5 K + y'(N + 0.5-22)]
N-
+2
2
hv(1,3) = sy"V.3 a2
a --- > e(+1)TT(
hv(2,4) = sy"V3 a2
a2-
hv(1,2) = 2sy"'V3 a
a -- > II
hv(3,4) = - 2s y"'3 a
a --
hv(2,5) =- sV2[0.5(+1) + y'(N + 0.5 - 2(2)]
N -->
A
A
hv(3,6) = - sV2[0.5(K+1) - y'(N + 0.5 - 2(2)
N -t ve(+1
t2 -_(+l)
hv(1,6) = - sy"6 a2
hv(4,5) = sy"A6 (at)2
hv(1,5) = - s (y"'
2
(at)2
a
a --
hv(3,5) = 3s y"'2 at
at ____
hv(4,6) = - sy"'6 at
at -
2)
+ (+0.5)]
N -->
hv(6,6)
2)
- (K+0.5)]
N --
0.5 +
FT
__
__----
A
hv(5,5) =-A - s[y (N 4 0.5 +
(N 4
( +1)(/ +2
a
hv(2,6) = 3s y"'2 a
=-A- s[
VN+2
A
+1
The replacements indicated on the right are the ones which occur when the
Hamiltonian operates on a multicomponent wave function of the form which
diagonalizes
Haxia,
described
below.
The
unspecified
elements
hamiltonian are the hermitian conjugates of the ones above, or zero.
used 5 to represent the wavevector along B0 divided by V2s
Sk /V2s .
216
of
the
I have
Axial model wave function
fc ( r )
f(r) = fv(r)
fs(r)
with
f(r) = exp(ik g)exp(ik ) [ci
fy(r) = exp(ikl1)exp(ik )
]
6
Cs Pe+
C4(P
f (r) = exp(ik)exp(ik )
Here,
t pn
oscillator
[
8
1
+2
+i
is an abbreviation for (pn(4+(krlIs)), which is a simple
function
(hermite-gaussian)
centered
quantities, or quantum numbers, are e, k
at
and kr.
- (k Is).
The
harmonic
conserved
Notice that in the Landau
A
gauge, with A=Bo0 r , k is a raising operator for the single-component LL wave
functions with the form
exp(ikrrj)exp(ik )(p
(4+(ic Is)).
When Hial operates
on a wave function of the above form, then the energies can be found by the
diagonalization of the numerical matrix obtained by making the substitutions
indicated above.
The numerical Hamiltonian may be regarded as the eth block
217
of the representation of Haxial among basis functions of the form (pXi , where
seven
Xi is a unit spinor with
zeros and
a single
corresponding to the LK basis function ui(r),
When nonaxial
components
one for components,
remembering that
e=
of Heff are considered, the blocks
values of t become coupled.
n+mJ -2
of different
[See Evtuhov 62, Trebin 79, Pfeffer 90.]
For k =0 the 8x8 numerical Hamiltonian matrix decouples into two 4x4
blocks [Pidgeon 66, Weiler 78] corresponding to the a-set and b-set solutions.
The corresponding wavefunctions have only four nonzero components.
subspaces are coupled both for finite k
The two
and for finite inversion asymmetry.
This means that at least some of the magneto-optical
transitions which are
induced by the lack of inversion symmetry are also induced by finite k .
[See
Trebin 79.]
Band parameters
We now consider the band parameters.
The parameters which are finite in
the single-group approximation of the PB Hamiltonian are
F, P=P', G=G', " =( , 2- '3
which includes 14 parameters.
K==33
Those parameters which are finite only as a
result of corrections to the single-group approximation are
N1 , N2 , N,
These
are
spin-orbit
all
q, C and C'.
to spin-orbit
proportional
couplings between
combination of the two.
splittings
the valence band
of higher
bands,
and higher bands,
or
to
or to a
The differences between the primed and unprimed
parameters which are finite in the single-group approximation are proportional
to these same splittings or couplings, as well as to the spin splitting A of
218
the valence band.
Notice that since C and C' both result from the spin-orbit
interaction there is no reason to expect them to be equal.
The coefficient of
C' was originally chosen so it would be equal to C if both parameters resulted
exclusively from the off-diagonal spin-orbit coupling between the T v
band and higher T 2 bands above the conduction band.
valence
(Cardona has recently
stated that C results primarily from F12 levels among the core states deep in
the valence band [Cardona 86].)
The largest contribution to C' probably comes
from the off-diagonal spin-orbit coupling A
between the tv
valence band and
the Tc15 higher conduction band, which does not contribute at all to C.
[Kane 57]
and
[Bell 66].
For the estimated
location
of the
(See
single group
levels at the Brillouin zone center see [Dresselhaus 55] and [Cardona 66].)
The set of band parameters which I have used in my calculations is the
one which was determined by Goodwin et al. and by Littler et al. for intra-cb
and intra-vb processes, respectively [Goodwin 83, Littler 83].
The parameters
are listed in Table A.2.
Table A.2. Band Parameters for the PB model. [Goodwin 83, Littler 83]
E = 0.2352 eV,
g
F = - 0.2,
l
=
K=
3.25,
-1.3,
A = 0.803 eV, E = 23.2 eV
p
N = -0.55
' 2 = -0.20,
73 = 0.90
q= 0
219
Estimates of some band parameters based on k p perturbation theory in the
5-level model:
See Groves et al. and Suzuki and Hensel [Groves 70, Suzuki 74].
Also see Cardona et al. and Pfeffer and Zawadzki [Cardona 88, Pfeffer 90].
See Dresselhaus et al. for the single-group parameters [Dresselhaus 55].
G
P Q
-
-1.0
E
2
QA=
C'V3
E
0.033
1
C= 0
1 Q
N
N
3Y1/2 E 2
3
2
-
--
[P1A1 - PA-]
1
1
(small, depends on the sign of A-)
4 Q2
-
q
0.12
-
E2
9
P
N
_
3E2
[-P A + 2PA-]
(small, depends on the sign of A-)
1
Here E l is the energy of the 1 S band relative to the top of the valence band,
neglecting the spin splitting.
it involves coupling to
I have set C to be approximately zero because
2 and r
bands above '
uncertainty in the sign of A- and hence of C'.
.
Note that there is an
Cardona et al. [Cardona 86a]
assign a plus sign, but this causes too large a magnitude for the cb inversion
asymmetry parameter 50.
220
Appendix B
Irreducible Spherical Tensors
In this appendix I define some irreducible tensors and tensor products
which
are
useful
in
the
expression
following
Hamiltonian.
The
Baldereschi 73,74,
Pertzsch 78,
and
references
rotation
are
Trebin 79, Condon 35,
of
the
effective
useful:
Edmonds 74,
mass
[Lipari 70,
Anderson
89].
First consider a vector operator A with cartesian components A x , Ay and
The irreducible first-rank
A .
spherical
tensor formed from A is denoted
A(1) , with components A'm ) defined by
A (I ) =
o
A")
A
Z
IA ,
where
A+ - Ax
iA Y
Higher-rank tensors are formed by use of the tensor product and the vector
coupling, or Clebsch-Gordon coefficients.
(AB) (2)
[A(')
(ABC) (3) _ [A(1)'
B()]
(2)
We use the definitions:
;
B()® C()](3) ;
etc.
221
and
A(n)=
[A(1)
A(n-l)(n)
Here '®' denotes the tensor product.
The coefficients in the expression for
A(n) are given in Table 3.3 of Condon, reproduced in Table 5.2 of Edmonds.
The above expressions represent the highest-rank tensors which can be formed
from sets of vectors, so the result is independent of the order in which the
vectors are coupled.
Note that
A(n)n =A +)n
and
A(n)
=
(
A_)n
The tensors R (n) formed from the coordinate operator r have the components
( (2n- !11/2
•
1)!!
RRm ) = 4Tn
rr n
m(00)1
'
where Ym(0, ) is the spherical harmonic of rank n.
The components of A (n )
may be obtained by expressing rnY (0,4) in terms of x+ and z, symmetrizing
with respect to these components, then replacing them by A+ and A
symmetrization is unnecessary if the components of A commute.
The components (AB) m(2) of the tensor (AB) (2) are
m =2:
AB
1
m =-1:
m =-2:
2
j~AB+
6
1
-1
-1
B) +
+ +
= -(A
(AIB + AB I)
m = 1:
m = 0:
= A+B
1
3
A0B0
Af(A B + A_B o)
-
-
A B
Bz+ AzB+)
j(3A B - A-B)
(AB + AB)
2 2--
-1 -1
222
-
.
The
The superscripts '(1)' on A(1)
and B(1)
have been omitted for brevity.
m
m
The first rank product obtained from two first rank tensors is
B(1)1)
[A(1)(
(AB)(1)
with the components
m= 1:
m = 0:
m =-1:
(AIB o- AoB )
!(A B - AB
!'(AB-
Note that
=- (A Bz - AzB )
2
+Z
Z+
)
= - 1(A+B_-- AB+)
8
+
-AA_ 1Bo
=
B).
2(AB-A
Z- Z
(AxB) = (AB - AB), so
AB+
Z 2 +
I (AxB))
(AB)(1)_
where 'x' denotes the ordinary vector cross product.
The zero rank product of two first rank tensors is
[A(')( B(1)](0)
(AB)()
=
3
(A11B+
AB
- AoBo)
-1
-1 1
0
=-f
AB.
Here A-B is the ordinary vector dot product
A-B = 1(A+B+AB+) +AB
2
+
zz
The general tensor dot product is defined by [Edmonds 74, page 72]
A (n)B(n ) _
(1)
m
which also equals
m
(
Am(n ) B -m
n)
(- 1)nV2nT-T[A(n)® B(n)] (
223
).
Some additional tensor products which will be of use are
K(2) ] ( 3 )
[t(')(
m = 3:
m=2:
m=l:
tK
1 2
tK +
11
302
-5 1 t 1K 0+ 715 t0 K 1 + ]it15
m = 2
0
-tK -1 +
511
m =-1:
m =-2:
m =-3:
The
tK
J--
5
t K 2+
1 -2
StK +
-1 K 2
toK o + - tlK
0 0
5 -11
3
5
tK
5
0
1
+-
-
t1
-1
-1
K°
Et K
t-1 K -2
superscripts
'(1)' and
'(2)'
on t (
m
)
and K ( 2) have been
m
brevity.
[K(2)
m = 4:
(2) 1 (4)
KJ22
m= 3:
(K2 J1 + KJ 2)
m=2
(KJ+ KJ 2 ) +
14
i-714 K1J 1
m = 1:
]-(K2 J+
m = 0:
]-(K J + K J2 ) + ]s(K J + K J ) +
-70
etc.
(For m ----
K J2 ) + -(K
-2
2
70
J + KoJ 1)
1 -1
-1 1
37 K J
70
0
0
m , change signs of subscripts m and m2. )
224
omitted for
IK(2)()
M(2)]
(3)
m =3:
-(KM - K MM2)
m = 2:
'(KM o- KM 2 )
m = 1:
I2(KM
- KM2)
+
10
2
-1
-1 2)
m =- 0:
(K2 M 2- K 2M2 ) +
m =-:
J-(K1 M - K 2M )
m =-2:
J'(KoM_2- K2Mo)
m =-3:
where
- (KIM
2-
M (2 ) = ((CV)
-2(K M - K0 M )
(KIM-
KM 1 )
K1M 0 )
+ J(KM-
40 0 -1
-
KM_ )
(2 )
The tensor products given above contain tensors K(e)
up to rank 2 formed
from the components of k, and tensors J(I) up to rank 3 formed from the
components of J. We will later also need tensors K
are listed the components of K
for J.
and J()
of rank 3 and 4.
Below
for I up to 4 for K, and up to 3
I have included P(t) to demonstrate the form of the tensor components
The components of K
when the vector components commute.
obtained by symmetrizing the components of P()
components,
and
then
using
the
commutation
components.
225
)
and
) are
with respect to the vector
relations
for
the
vector
( 2)
(2)
m
K(2)
m
m
1
1
-2
2
2a+
2
J+
- k k
v"2po(P
I{J ,
V(3J
(Yj2 - J)
k 2)
-
F(3k
}
F
6
k k_
V2P0 P 1
1 2
2
2
P-1
K(
p m(3)
1
2
2 2 -
-
3)
j(
3)
m
m
13
V3p
2
3p
+
B(2p2 +
5
o0
F(5k
o(k
p_-)po
P
l1
-
k 2 + s)k+
+ P P_- )P
f(2p
5(p
kkk
)
f348
j2-
5J + 2)J+
- 3k 2)k
- 3J 2 + 1)J
2
- k - s)k
_
f
PoP-1
-
-d(5J
1)J 2
+
-
A
P-1
j2
+ 5J
+ 2)J
2
(J + 1)J
-
For noncommuting components, as with Jm and km , one must make the replacements
2
POP1I
PPo
pp
-
-
2o + pipI
6(PPop -
pP2)
+ 1op
+ P P- Po + c.p.)
etc., before replacing p by J or k.
226
4
p(m )
K(
4)
m
1k4
I+
4+
3
-
7kf k
2p
(3p +
F(2p
-- (2p
pP
p)p
+
3
F
2
(7k
- 7(7k
plp-I)Popl
+ 12pp P_l
1 + 3p2P_ )
2O1
k2 + 2s)k
3k2
-
(35k
-
+ 3s)k k+
2
30k 2k2 + 3k 4 + 3S
2
--(7 k - 3k - 3s)k k
(2p + 3p p )pp_
- k2 - 2s)k 2
(3p2 + PP_ )p_
2pop
-
3
4
k4
P-1
4 -
227
Appendix C
Rotation of the PB Hamiltonian
In order to compute the Landau levels when the field B0 points in an arbitrary
direction relative to the crystal axes it is necessary to perform a coordinate
rotation from the frame xyz of the cubic crystal axes to a frame
B
is directed along
A
.
1rl in which
This appendix shows how the rotation is accomplished,
derives the angular functions which are required for the expression of Hef in
the rotated frame, and gives the complete expressions for the components of
Hef in this frame.
generalization
where
It follows the derivation of La Rocca et al. which is a
of the original derivation by Rashba and Sheka to the case
the third
Euler
angle
y is finite.
Inclusion
the
of y simplifies
magneto-optics problem by allowing the wavevector and polarization directions
of the light to be along coordinate axes, without losing generality.
The rotation of a term in Hef is accomplished by replacing all vector
components in the xyz frame by their expressions involving spherical vector
components in the rotated (Ti)
k = RX+k+ + Rk
reference frame. One makes the replacements
+ Rk
+R Yk =R
+ Ry
+k +R
= Ry+kk
k = R +k+ + R k
,
,k
(C.1)
+ Rz k,
and similarly for the components of B0 and the matrix vector operators i, J,
t, t' and u.
The elements of R i1 are functions of the Euler angles a,
229
3, y,
and were given by La Rocca et al. [La Rocca 88b]
R
= R*
X,+
=
X,
le1 (cos
R
= R * =
y,+
y,2
R
This
= R *
Z,+
-e
Z,-
transformation
+ i sina) ,
cos
2
(sina cosP - i cosx)
sin3
2
specializes
to
the
unitary if Ri+ were replaced by V2Ri+
k+/v2 . (See [Rashba 61a].)
k
+
= cosa sin
Ry,
= sina sinP ;
Rz=
case
= 0 and y = 0 [Luttinger 56].
a = 7t/2 ,
Rx,
,
;
cosp ;
(C.2)
by
when
considered
Luttinger
The transformation matrix would be
corresponding to replacement of k+ by
This implies that the inverse transformation is
2R*
+ 2R k + 2R* k
x+x
y+y
z+z'
= 2R* k
-x- x + 2R*- ky + 2R*z- k z
k
k
= R
+ Ryk
+ Rk
A~ Y
zZ
k
xx
(C.3)
and also that
RR
-RR
RRyR
- Ry_R
= iRZ-
Rx R
- RRx+
= iR
x+ y-
y+ x-
= R
2 z
'
(C.4)
.
These last equations are also valid for cyclic permutations (c.p.) of xyz.
As
spherical
a first
example
terms in Heff,
of this
transformation
consider
such as k-t and B -J and K(2)-J (2).
230
its
application
to
The expressions
which result from the transformation are independent of the Euler angles of
the transformation, as one would expect. Thus
k.t = I(k+t_ + k t+) + k t
2+-
(C.5a)
B.J= (B+J + BJ
(C.5b)
-+
+)
+ Bo J
and
K(2)
(2)
=
kkJ2
(k
k ,k} J ,J
+
+(
+ { k_,k} {J+,J))
+ I(3k - k 2)(3J-J2 ) .
Several
key
direction of
drop out.
points
of the
transformation
are
the
(C.5c)
following.
First,
the
A
may be chosen to be along B so that the terms containing B ±
0
Second, k+ become raising and lowering operators for free-electron
Landau level wave functions, satisfying
[k ,k ] = 2s
where
(C.6)
s = eB /hc , and
(C.7)
[ki,k] = 0 .
Third, the basis functions may be requantized so that the matrix operators
have the same expressions (representations) in the rotated frame which they
had in the original xyz frame.
This is what causes the spherical terms to be
The requantization is obtained by
truly independent of the coordinate frame.
A
applying the transformation U(apy) = exp(iqn-F) to the basis functions, where
i
F. is the 8x8 matrix obtained by placing
J. and 1 . in the 2x2, 4x4 and
2x2 diagonal blocks of the 8x8 matrix filled with zeros.
Trebin et al. [Trebin 79].)
Here
A
(F was defined by
p and n may be expressed in terms of the
231
Euler angles; the transformation is found by applying successive rotations by
each
of
the
three
angles
y,
3 and
a.
The
transformation are just the rotation matrices for j-
diagonal
blocks
of
and j-2 spinors.
The rotation of the nonspherical terms in Heff is more complicated.
will
use Hvv(C) as an example.
the
Following La Rocca,
we
We
substitute the
expressions for ki and Ji into
[k {J ,J 2
H(C) = -
X
(C.8)
+ c.p.]
2
z
Xy
Defining the coefficients (rk;gv) by
RxRx (R
(KX; gv)
=(R 1R-
R
- RzR
R
- RyIyIRxR)
= 1 EijkRiRiRj
ijk
) + c.p.
+
c.p.
(C.9)
.-.R
we find
k J (J2
(K;Rv)kKJkJJ
+ c.p. =
-
z)
+ c.p.
(v
(C.10a)
)kK
and
k (J2
X Y
-
)J + c.p.
ZX
I
1c2V
(1C;g)kJgJJ
(C. 1Ob)
One then gets
C
H(C)=-
I
/3- I(X4V
(iX;gv) kK{J
JX,
232
'
(C.11)
The coefficients (C;giv) were defined by La Rocca, and may be expressed in
terms of four complex valued functions of the Euler angles,
which
F3 ,
are
a
generalization
functions
of the
of
F , F 1 , F2 ,
Rashba
and
Sheka
[Rashba 61a] to the case of ganeral y. Setting
F0
F -++;
(++;--) , F -- 4-(+;,),
2
F = (++;+) ,
4
(C.12)
direct evaluation gives [La Rocca 88a, 88b]
F = -T6 sin(2a)sin(20)sin
S=
2=
F3 = 16
,
(C.13a)
ei cos(2a)sin(2P) + i sin(2a)sino(3cos
2
-1)]
2
e2i[2cos(2a)cos(20) + i sin(2a)cosP(3cos 3 - 1)]
3i-e[cos(2a)sin(2P) + i sin(2a)sinp(1 + cos 2 )]
(C.13b)
(C.13c)
(C.13d)
The coefficients are found to have the following properties:
1. ( X;jgv) is symmetrical in the interchange of Kiand A, and of gt and v.
(ICX;g v) = (XK;Lv)
2.
(X;gxv)
second two.
= (rX;vgL) = (XK;vL) .
is antisymmetrical
on swapping
Thus
(C.14)
the first two indices with the
Thus
(C.15)
(K;gxv) = - (pyv;X)
This implies
(C.16)
(-;K&) = 0,
233
= (++;++) = (+
(+;+)
3. (1&;+ -) = - (
;
= ... = 0.
=;+)
(C.17)
)/4.
(C.18)
;)/4 = 0 .
(C.19)
Thus
(
4. (-
;+ -) =
;)
- (
(- -;)
= (+ ;Q)*,
= (++;*, etc.
(C.20)
i.e. Swapping plusses with minusses is equivalent to complex conjugation.
The finite coefficients with non-negative total index, then, are
(++;+)
= F3
(++;)
= - 4F
(++;+-) = F2
(+S;T)
= - 4F
(+5;+ -) = F
(+;-) = - 2F0
(++; - r) = 3F 1'
(++;--) = F •
'
where the total index is the sum of the spherical indices.
(C.21)
The negative-index
coefficients are determined by the symmetry and complex conjugation properties
listed above.
(+;-
Notice that
) =
(-A;+)
= -(-SA)
(C.22)
,
which implies that F 0 is pure imaginary, i.e.
(C.23)
F o = -F*.0
One can define negative-index coefficients by
F -m - -F*.m
(C.24)
234
The functions Fm are proportional to the angular functions obtained by Rashba
eim'y .
and Sheka multiplied by exponential factors
Rashba's functions Bijk stand for - +
(The indices
123 on
respectively, and 1 must be subtracted
from the sum to get the index on the corresponding function F m .)
The
magnitudes of the functions are shown in Figs. 3.1 through 3.4 as functions of
P
polar angle
and azimuthal angle a.
about the c-axis,
rotations
prominent
feature
the
of
The angle y, which gives coordinate
are
drops out when the magnitudes
is
functions
that
vanish
they
taken.
in
A
certain
high-symmetry directions. IF0 1 has maxima in the [111] directions, and nodes
along [001] and [110] directions.
IF1 I has maxima along the [110] directions
and nodes along [001] and [111] directions.
IF I has maxima along the [001]
directions and nodes along [111] and [110] directions.
IF 3 I has maxima along
the [111] directions and nodes along [001] directions.
The magnitudes in the
high-symmetry directions are summarized in Table C.1.
Table
Magnitudes
C.1.
inversion-asymmetry
of
functions
Fm(a,,y) along directions of high symmetry.
[111]
[110]
-3= 0.1443
0
[001]
IFol0
0
IF1
0
IFI
IF 33
12
0
=
2
=
235
0.2041
0.0625
0
0
- 0.125
0
1
3
16
=
0.1875
When
these
functions
are
substituted into
the expansion
for H(C) in the
rotated frame and terms associated with each function are collected, one gets
the expression for H(C) obtained by La Rocca, and given at the end of this
appendix.
The set of terms associated with F
are found to be proportional to
the mth component of the irreducible spherical tensor (see Appendix B)
[k(1 )
3
j( 3 )]( )
(C.25)
.
In the xyz coordinate frame, with Euler angles o = 3 = y = 0, the angular
functions are F = F = F
0
1
3
=
F
0,
-
2
We see that H(C) contains only the
8
m = +2 tensor components, and evaluates as
)
H(C) = - (C/I)[(k(@ j( 3 ) (33)
(k1)@
2
j( 3 ))
(3
-2
)
(C.26)
The functions Fm can be related to the rotation matrices for third-rank
irreducible spherical tensors. For any third-rank tensor we have
T
(3)
m
=
D
(3 )
m,m
T(3
)
(C.27)
m
where T (3 ) is the tensor formed from the vector spherical components in the
xyz frame, and T(3) is the tensor formed from the spherical components, such
as k+, k_, k
H(C) =
, in the 4in
-
(Cv3)
frame.
(D(3) 2,m
We then have
D (3
)
(k(1)
-2,m
3
3
j( )) ( )
m
(C.28)
By comparison with the expression for H(C) at the end of this appendix we find
that
236
F3
-
F
=
2
8
8
F1
F
0
2,3
D(3))
;
-2,2
2,2
F
(D(3)
- D (3) ) * ;F
2,1
-2,1
(D(3) - D(3)
40
-
2,0
-2
(D(3)
6
8
-3
-2,3
(D(3) -
40
-
F
(D(3) - D (3)
-
-1
2,-2
40
D (3)
-2,-3
2,-3
(D(
8
-
- D_)
;
-2,-2
D (3)
(D(3) 2,-1
)*
-2,-1
(C.29)
.
-2,0
This may be verified by direct evaluation of the D's. [Tinkham 64, p1 10.]
of the F functions for certain directions of P may be
The vanishing
A
understood very easily.
When
is along a [001] direction, one obtains a C4
rotation by keeping the angles ca and 3 constant while letting y change by R/2.
But X1 has Td symmetry
A C2 rotation is obtained by letting y change by xi.
and
is
invariant
respect
with
to
C2 rotations
about
[001]
directions
changes sign under C4 rotations, which are contained in Oh but not Td.
symmetry
axes
about
proportional to e
2i
in
its
The
direction.
This
does not lower the
is directed along ( because B
remains true when B
but
functions
F
and
F
2
are
and therefore satisfy the above requirements, while none
Therefore F
of the other functions do.
2
may be finite along [001] while the
Similarly, when
other functions must vanish.
A
is along a [111] direction R
A
is invariant
with
respect
F - 1 and F+ 3 - e
vanish when
A
3 yi
to
C3
- e 3iy.
about
This
r.
, but not by the other functions.
is along a [111] direction.
341 changes sign under C2 rotations.
F
rotations
Therefore
When
A
is
satisfied by
Therefore F+ 1 and F
is along a [110] direction
This is satisfied only by F+1 ~ e i
F and F+2 vanish.
237
Note:
2
and
F multiplies terms with
A
m=3 in R.
by (p therefore results in multiplication by a
Rotation about
phase factor e3ip and similarly for terms with other values of m.
The effective Hamiltonians H(N2 ) and H(50) may be rotated in a fashion
analogous to H(C).
The main difference is that H(N2 ) contains the tensor
product of two second rank irreducible tensors instead of first and third rank
tensors.
The results are given at the end of this appendix.
The Hamiltonian terms associated with the parameters C', G, G' and N
3
have the form of the general tetrahedral invariant
Ax(ByC + BC) + c.p.
(C.30)
as shown in Appedix A, where A, B and C are three vector operators, or one
vector and two axial vectors.
In the case of H(N) the operators are B, a and
t, where B is the magnetic field and a and t are matrix operators defined
previously.
The rotation
of this invariant
is easier than the previous
which contained four vectors, one of which was axial.
Ax(B C + BzC) + c.p. =
one
One now has
7t (C1g)AC
'Cg
(C.31)
where the symmetrical coefficients (rxjg) are defined by
(q.Lv)
Rx(Ry RLz
+ R zR ) + c.p.
(C.32)
Evaluation of the coefficients gives
(
= 8iF8)
f ;
(+5)
=-8iF
(+
)
f
;
(+ - ) =-2iF0 = -f
;
(++-) = 2iF =-
;
8iF41
238
(++)
= -4iF
f2 ;
(+++) = 2iF =3 ;
(C.33)
(-() = (+() ,
(C.34)
with
etc.
with negative
The last equation allows one to define the angular functions
coefficients by
(C.35)
fm-- fm
and implies that fo is real.
Notice that
(C.36)
(c+ - ) = - (K )/4 .
These
coefficients
analogous
are
to
the
coefficients
B(o1y)
[Rashba 61], although the labelling convention here is simpler.
of these symmetrical coefficients to the coefficients (Ic&;gLv)
defined
by
The relation
may be found
through the application of Eqs. (C.4).
The rotated expressions for H(C'), H(G), H(G'), and H(N3 ), are given at
the end of this appendix.
Notice that an invariant of this kind cannot occur
in a diagonal block because
respect to time reversal.
a factor of i is needed for invariance
with
[See Trebin 79.]
Finally we note that the squared magnitudes of the F 's can be expressed
in terms of cubic harmonics.
This was pointed out by [Rashba 91].
239
The
expressions are
F1 2
IF 1 2
F2
(yz)
=
2
r ,
2 +y 2z 2 +
1 2
2=
IF 3 2 =
(C.37a)
I [r6 - 4r2 (x 2y2 + yz
2
64
2
9[r
(x 2y
2
2
2
9x 2 2 2] r 6 ,
2X2 )
z
(C.37b)
+ z2x2) + 9xy 2z2 ] / r6
2 2z 2 ]
2
2)
/
6
(C.37c)
,
(C.37d)
,
where reducible forms of the cubic harmonics are used for simplicity.
Von der Lage 47.]
[See
Here, x, y and z may be expressed in terms of the azimuthal
and polar angles a and P via
x = r sin3 cosa ,
y = r sinp sina ,
z = r cos3 .
We
(C.38)
next consider the octahedral
(Oh) terms
in Yeff"
The general
rotation applicable to these terms was first done by [Sheka 69], but the Euler
angle y was again set to zero.
The present derivation follows the procedure
used by La Rocca for the tetrahedral terms H(C) and H(60).
We note that an octahedral invariant (cubic harmonic) of the form
ABCDX + AB CD + ABC D x x x
x
y yyy
z zzz
5
(AB.CD.
1i
+ AB.C.D. + A.B.CD.))
j
I j
j
i
ij
(C.39)
240
is irreducible.
This may be regarded as the complete polarization [Weyl 46 or
Edmonds 74] of the cubic harmonic
4 +
y4
3 4
4
(C.40)
5
Using a technique similar to that used on H(C), and first carried out by Sheka
and Zaslavskaya [Sheka 69, in Russian], we write
ABCD
xxxx
+ABCD +ABCD
yyy
y
z Z Z
=
O (&y) ABCD
(C.41)
where
(
R
R R
)R
XKCxX4L XV
RR R
+ RyyXyyVZ1Z
+
Rz
R
zRzRzv
ZV
(C.42)
These new coefficients are completely symmetrical in the four indices, and are
in terms of five complex-valued
expressible
W , with
W° , W ,...,
0
1
4
W
-m
functions
or the Euler angles:
= W* . Evaluating the coefficients we find
m
(++++) = W ;
(+++
= W3
(++5) = w ;
(+rr
(
(+++-) = -W/4
;
w,;
(++-
) = - W /4 ;
) = W;
(+ -
) = (1 - W)/4 ;
)=
(C.43)
(++--) = (1 + Wo)/16 ;
with
(-
) = (+
(C.44)
) , etc.
(Complex conjugate interchanges all +'s with -'s.
241
W is real.)
Explicit evaluation of the W's gives
= (Cos 4
W
+ sin4X)Sin 4 p
COS4
+
= (cos40 + 3)sin4P + Cos 4 P
W
W1 =8
eiY[(cos4c
(C.45a)
sin 2 3 + 7 sin 2 p - 4)sin
cosp
+ i sin4a sin 3 3
2W
16
cos
(cos
2
+
1) + 7 cos
(C.45b)
2
-
1)sin 2 P
+ i (2 sin4oa sin 2 p cosl3)
W3 =
e3iy [(cos4
(4 - sin 2p) - 7 sin 2 3)sin
(C.45c)
cos3
+ i (sin4ca sin3 (3 cos 2 3 + 1))]
W4
64
cos4
(C.45d)
sin
+ i (4 sin4ca cosp (cos 2 3 + 1))]
Polar plots
of the magnitudes
of these
functions
azimuthal angle a are shown in Fig. 3.5 through 3.9.
high-symmetry directions are given in Table C.2.
242
vs.
(C.45e)
polar
angle
P
and
The magnitudes in the
Table C.2. Magnitudes of warping functions Wm(xa,j,y) along
directions of high symmetry.
[001]
[111]
[110]
1Wo1
1.0
0.3333
0.5
W1
0
0
0
IWl
0
0
0.125
1W1
03
1W4 1
W
is just
4
+ y4 +z 4 )/r4
IWl
z = r cosp.
0
0.125
(
0
0.1179
V
12
with
3
= 0.09375
x = r sinp cosa, y = r sino sina, and
vanishes in all three high-symmetry directions.
It has a
maximum value of 0.1403 in the (110) plane for P = 25.50, and equals 1/8 in
the (100) plane for 3 = 45(n+4) degrees, where n is any integer.
The nodes of the Wm may be understood in the following manner.
Wm is
associated with terms in the Hamiltonian which are multiplied by eimy when
A
rotated by y about 5.
If
A
points along [001], 3
rotations because of the Oh point group symmetry.
is invariant under C4
The factor eim s/ 2 equals 1
for m = 0 and +4, so W4 , Wo and W 4 may be finite for
ii [111]
II [001].
For
the factor eim(2 / 3 ) equals 1 for m = 0 and ±3, so W3 , W 0 and W-3
may be finite.
2, and
A
Finally, for
II [110]
the factor eim
4, so W , W , W , W2 and W4 may be finite.
in the table above.
243
t
equals 1 for m = 0,
All of this is confirmed
Expressions for H(g), H(g'), H(q), H(%)
and H(y ) are given at the end
of this appendix.
We note that W occurs only in the combination (W- 1) when
0
0 5
the Hamiltonians are irreducible with regard to R(3).
From
the
perspective
of
irreducible
tensors,
the
rotation
of
the
irreducible cubic parts of H(o) or H(y0 ) is accomplished as follows:
-
=
H (a)
C
K(4)
0 C
-
0
-4
K4),
0
5
3
y,[B(l) 0 ((1)( 0 K(2 ) ( 3)
=eH
2C
S0
[K (4) +
+ K(4) +V
+
(4)
C
(C.46)
(C.47)
We use the convention
T(4)
T(4)
+
1 [T(4)+
C
2
4
-4
+
(C.48)
T(4)]
5
0
Notice that the irreducible tensor involving B and a is a partial polarization
of the K (4) irreducible tensor.
If we use the D matrices as before to carry
out the transformation to the coordinates of the rotated frame, we find
1 [D(4) + D(4)
4
3
4,4
8
W-
W
1 =
28
4,3
[D (4)4
128
4,2
+
0,4
+ "7
5
[D(4)
-4,1 + D(4)
4,1
-4, 1
(4)
0,3
5
-4,3
-4,2
(C.49a)
D(4)]
5
D(4) + D(4) +
16
'-
2
+
-4,4
+
D
5
(C.49b)
(C.49c)
(4)]
0,2
D(,4)
0, 1
244
(C.49d)
W
W
28
[D((4)
[D
=
+ D (4)
(4)
+
-4,-1
5
4,-2
-3
S
16
+
(4) + D(4)
-4,-3
4,-3
1 [D(4) + D(4)
-4
8
(C.49g)
0, 2
(C.49h)
D(4) ]*
5
0,-3
D(4)]*
+ V
5
-4,-4
4,-4
(C.49f)
*
0,-1
D(4)
-4,-2
28
W
D(4)
5
+
-2
(C.49e)
0,0
5
-4,0
+ D(4)
4,-1
D(4)]
D(4) +
4,0
-
-1
[D(4) +
2
3
5
0
(C.49i)
0,-4
The square magnitudes of angular functions Wm are given below in terms of
reducible cubic harmonics [Sheka 69].
(X4 + y4 + Z4)2 / r8
(C.50a)
SW 012
0w
W
2= 1 [r 2 (X 6
2
|W3
-
2 12
4
4-2[(x y
64
+
4
[r2(x
+
6
+
y 4 z 4+
4
z6)
-
(x 4
4
-
4 2
+ Z ) ]/
+
2
r
2 2 2
y
] / r
(C.50b)
r8
8
(C.50c)
+
4
6)
+
+ y6 + Z _ (x + y4 + 4)2
24r2x2yz2] /r
8
(C.50d)
S
12
[r
8
+
(X 8 + y8 + z
8)
2
- 28r 2y2
245
8
] / r
(C.50e)
In this final section of the appendix we catalog the general rotated form
of all the terms in the PB Hamiltonian to order k2 , as well as the anisotropic
terms to order k 4 in the decoupled conduction-band Hamiltonian
R .
The
rotations were obtained by applying the technique of [La Rocca 88b], which
generalizes the technique of [Rashba 61a].
component B
We assume that the uniform dc
A
of the total magnetic field B is directed along C, which allows
us to set [k+, k] to zero, and [k, k+] to 2s = 2eB0/hc.
However, we have
retained terms containing B+ since these are needed to demonstrate all of the
tensor components,
and also to describe magnetic dipole transitions when
perturbing field is present.
is
that
a unitary
An important aspect of the rotated Hamiltonians
transformation
basis functions causes
a
("rotation")
applied
to
the
Luttinger-Kohn
all of the matrix operators in the rotated frame to
have the same numerical expression which the corresponding operators had in
the original frame.
resentation
Thus, for example, {J+, J } has the same numerical rep-
with respect to the "rotated" basis functions
had with respect to the original basis functions.
subscripts on the Hamiltonians H(x) below.
which {Jx+iJ , Jz
We have omitted the band
The parameters F and NI are in
Hcc; P, G, N2 and N are in H ; P' and G' are in H ; Y ,
are in H ;
vv
Y
and i'
are in H;
ss
~1c,
',
I' and C' are in H .
expressions below, h.c. means hermitian conjugate, (+
by '-' and vice versa, and c.p. means cyclic permutations.
246
, ic, g,
vs
q
and C
In the
-) means raplace '+'
Spherical terms:
12
H(F) = (F + $)k2l
H(N 1)
.
(C.51)
;
o
=
H(Ne
-(N1+ )BGi
c
(N+ 2) (B+
+ B+
I
(C.52)
) + Beo]
H(P) = Pk-t
= P [(k+t_ + kt ) + k t%]
(C.53)
H(P') = P'k-t'
= P' [(k+t' + kt4) + k t]
(C.54)
1
H(71 )
H()
(C.55)
2
= yK( 2) . j( 2 )
[(k
=
+ k2J
2
-
+
)
k2)(3J
+ l(3k 6,
-
J2)
(C.56)
+
H (Kq)
S
H(y')
e
-- (K
C
=-k21
-q)B.
+ 41
20
o
H(K') = - e(K'+
= -
J
e(K +
(k+k{J , J } + h.c.)]
2q) [(B+
+ B_)
+ Br
]
(C.58)
;
=
)B- (
H(') = - V3 Y"K(2).(UG) (2)
(C.57)
=-
3
-
(K+
)
[(k u
(B c
+ B-
+ k2 u
+ I(kk((u a
) + B (]
; (C.59)
+) + -(3k - k2)(3u a)
+ u%)
+ (+
-))
;
(C.60)
H(K") =
2C
(K"+
1)B.u
=
e (K"+ 1) (B+u
+ Bu+) + Butj ;
(C.61)
247
Cubic (octahedral) terms:
j(2 )] (4)
H(g) = - 2[K(2)
(C.62)
H(t') = 2V3jg'[K(2)® (u)(2) (4);
H (q) =
(C.63)
j(3)]( 4 )
e q[B'(1)
C
(C.64)
C
Antisymmetric tetrahedral terms:
H(C) = - (2C/V3)[k(@
j( 3 )] (3),
(C.65)
H(C')= (3C'/2)[k( 1 )® (uo)(2)] A(3)
(C.66)
H(G) = - /3G[K(2)@ t(1)]
(C.67)
(3),
A
H(G') = - "G'[K(2)& t' ( 1) ] (3)
H(N2)
=
2
H(N) = - f
6N[K(2)
2
()]
N[B"®
3)A
2
(C.68)
(3)(C.69)
A(C.69)
(C.70)
(at)(2)A3)
(C)
In the cubic terms (point group Oh) and antisymmetric tetrahedral terms (point
group Td) we use
T ( 4)
c
T
xxxx
+ T
+ T
yyyy
zzzz
S[T(4)+
nd-4
2
(4) +
5
T(4)]
0
1
5
ij
(Tiijj.... + T.... + Tij.)
d9
(C.71)
,
and
248
and
T ( 3) =A
(T
VF
xyz
= T (3)
-
2
+ T
+ c.p.)
xZy
T ( 3)
(C.72)
-2
where Tijkl and Tijk , here, stand for polyadic quantities like A iB CkD
and
A BICk. In terms of the rotated components we have
T(4)
=
C
D 4 C a,,y) T (4 )
m
m
(C.73)
m
is the mt h component of T (4) with all of the vector components
where T ( 4)
m
replaced by the corresponding components in the rotated frame (e.g. Jx + iJy
is replaced by
J+ = J + iJ); we define
D4C = 1 [D(4) + D(4) +
m
where
2
4,m
Dmm
( 4 (ot,3,y)
D(4)]*
5
-4,m
is the e = 4 representation matrix of the proper rotation
group R+(3) given by [Tinkham 64] and [Edmonds 74].
T(4)
m
m
,
(C.74)
O,m'
Note that
T m4I) D mm((C.75)
(oM,
},y)
(C.75)
( l implies, from this, that
[Edmonds 74, Eq. (5.2.1)]; the unitarity of Dmm
( 4)
Tm
=
m
,
Dmm
(4
,y)
T4
m
)
(C.76)
.
The functions D 4mC are related to the functions Wm defined in Eqs. C.43 and
C.45 by
249
W
1
1
W
W
2
3
4C
1
1
-2
D4C ,
8
4
4
W
3
D4C
-1
-1
W
2
-
W
4C
1
-
W
1
W
4
D4C
= W*
-
1
- W*
2
-1
I
D4C
-2
= W*
-VD4
3
8
-3
-3
4
-4
4
(C.78)
(C.79)
(C.80)
1 D 4C
4
-4
(C.81)
4 -4
In a similar fashion we have
( 3)
A
D 3 A(3,)
=
T(3
m
Mn
(C.82)
m
where
Dm3A
(C.83)
D(3)
-2,m ]*
(3)
[D2,m
These functions are related to the functions f
m defined in Eqs. (C.12)
(C.13) by
f3
fo0
5
- iD3A
(C.84)
0
Sfl*
iD
=-
3A
5
f2
2
,
f
3
4
4
2
iD
3A
3
and
f
2
f
3
=f2 *=f
f
* =3
The f m functions are related to the F m functions by
250
-1
iD3A
1
-2
6
4
iD
3A
-3
(C.85)
(C.86)
(C.87)
The f m functions are related to the F m functions by
f
= 8iF ,
f = - 8iF ,
f 2 = - 4iF2 ,
f
=
2iF .
(C.88)
The functions for negative m are given by (see Eqs. C.24 and C.35)
f -m
fm*
m ,
(C.89)
=-F*.
(C.90)
=
and
F
We
give,
below,
the
rotated
terms in the PB Hamiltonian.
technique
and
octahedral
and
antisymmetric
The rotation is accomplished by La Rocca's
[La Rocca 88b], using the coefficients (1yLiv), (2g.)
the relationships
among
tetrahedral
the coefficients
and Eqs. (C.34) and (C.36).
251
described
and (X;LV),
in Eqs. (C.14)-(C.20)
Rotation of Octahedral terms in 3eff
H(g) = - 2g
xx
yy
=- 2
k k J J
(KXg)
-
3
9v
=-2
+WI-
+ (W
0
k2j2
-
j(2)
K(2).
_ 2
5
j(2)
{J+,J})
rJ+
3
14 +t +
K(2).
5
+ W (2k+k J+ +k
[W4 kJ
+W 2 (3k
2
k2j2 _ 1 k2_
zz
3
+ k22+
k2 2
k 2 )J_ + 2k+k {J,J} + k(3J'
J2)
2+
lk k rJj + (3k& - k 2 ){J ,J} + k+k<(3J - J 2 )
+
3
(W5
)
Ik2j2 -
16 -
+
-k k{JJJ} + (3k2 - k2)(3J 4
2 -+
-
+W*I(
lk 2 {J,J
-
+,( _
2 2
} + kk<(3J
-
-
J2) +
(3k
+J
2 &r2Jj k
+ W* 2kk J2 + k 2 {J ,J
3 (
-
2
-
2
1k 2
{JJ
}
2)
k k {J
l
+
,
-16
j2"
-k2
+ -)
k 2 ) {J_,J }- lk k J2)
-3k)J_)
+ W*k 22]
4
-
-
(C.91)
252
H(2)
= 24
k2 u
(y + k2 uy
(idgy)
=2/3'
+ k2u
kkXua
2
Y
K(2). (UG)(2
K(2).
-2
Rotated H(g') is the same as H(g) with the replacements
J2
{J+,J } ---
32 - J2
z
g
-
g', and
2
+_
+
(C.92)
(U)(2)
u+
3u
+ u
%
,
{ J_,J
I
uO
+ u
_.
(C.93)
.
253
The irreducible, e = 4, octahedral part of H(q) is
Hc(q)
eq [B J + B J3 + B J - I(3J - 1)B.J]
c
5
xx
yy
zz
-
-
(l4Lv)B
I
J J Jv - -(3J - 1)B-J
e W4
cq
Bq
C 4+J
+ w
3B (J
(5J
+ W2
24
J
+w4 B+(5J
+ Wl{
+ (W o -
) -
-
1)J4 + B J
- 5J - J + 2)J+ + 3B (J
2 + 1)J +
- 3J
3
B(5J
- 3J 2 + 1)J
+ WB_(5J + 5J
-
5J
-
-
B_J3
2 + 2)J+ -
- J + 2)J + 2B (5J 2 - 3
B (5J + 5J
-
1+W B (5sJ
-
1)J
J
B_(5J 2 - 5J
- J
B_(J
+ 1)J
J
+ 2)J
+ !B(5J + 5J - J2 + 2)J -
+ 2)J
- 1)J+
B (
+ 1)J-
2
+ 3B (J + 1)J - - 4B3)
+_j)
+ W* 3B_(J + 1)J2 + BJ3)
+ W]
4
-:-
(C.94)
254
Rotation of antisymmetric tetrahedral terms in Reff
H(C) = (2C/V)[k(J
J J - Jzxz
J J) + c.p.]
x yxy
= (C/V)[k x {Jx ,(Jy2 J2z)} + c.p.]
= (C/3)
= (2C/)
(&lg;4v) kK{ JJlJv}.
[F 3 (k+(J
-
1) +- k J+
+ F (- k+(5J - 5J - 2 + 2)+
+ 4k (J - 1)J - kJ+
+ FI -2k+(5J2-3J 2+ 1)J + k(5J- 5J~-J2
+ 2)J - 5k (J -1)J
+F + F
(-
k(5J+
2)J + k(5J -5J
2k_(5J - 3j2+ 1)J + k (5J~+ 5J -
+ F* -k(5
+ F* (k(J
3(C)
5J -J2+
+ 5J - J+ 2)J
+ 4k (J
-J + 2)J'
2+
2)J - 5k+(J + 1)J2)
+ 1)2 - k+)
+ 1)J2 - kJ3.
(C.95)
255
H(C') = (iC'/2)[k (u
xyz
= (iC'/2)
+ u
zy
) + c.p.]
(KiXg) k u o
= (iC'2) [f3 k+u++
+ f 2 (k+(u+%
+f
+ uo+ + ku++)
( 3 k (u
+ k(u
++ u
o(5 + u~C_) +
+
"
2*
k(ua
+f3* k u
k
-
+k (uao + uo_)
2ku
+
%
uo_) +k u_
1 ku+]+
)-
-
k (u (
+ u
)
1 k+uca
)
.
(C.96)
256
H(G) = -iG[t {k , k } + c.p.]
- iG
(01t) t kkt
S- iG[f3 t+k
+ f2 2t+k+k + t~k+J
+ f (t+(3k-
+f(
-t
+ fl* (t
2-
k2) + 2tk+k -
k
(3k2
+ f* (2tkk
+
t%(3k - k2 ) -
k 2) + 2tk k
-
-
t k+2
tk
k
t k2)
4+-
+ tk
3* t k2].
(C.97)
H(G') is obtained by replacing G by G' and t by t'.
257
(C.98)
- V2N 2[ xx
t (ky2 - k z) + c.p.]
H (N2)
-
-
V2N 2
S-V2N 2
+ F
2
(KXl; v) c t
EF
3 2+kk
(2+t+(k- 3k)
+ F, (6 tk
+ F0
k
+ F* 6t6_tk
- (o+t
+ 6
+ 2(a%t+
+t+k2 -4(o+t
k tk v
+ ot)k
tk +
ot
)(k2 - 3k ) + 12
t k k - 3(T t + a t_)k2
+o t+)kk + 4(ot + ot)k+k -
+ 2(ot +
t)(k - 3k ) + 12
t kk
k
-
tkt
-
1-)+
- 3(+t +
t+)k2'
+ F* 20 t (k2 - 3k ) + 60tk2
+F2 (Gk
-+
)k]
+ F*2(yt
k - k - (Cyt + Gtj)
3 (-(C.99)
258
i e N [B (yytz + Ot ) + c.p.
H(N)
yz
3
AcC
e N3
(;g) BKGt
Sc
+ f(B+(++
(3
+ %t+) + B G+t+)
B
+B(
f
+f+-
B(i
B () t
+f*(
B5ct
-
+f2
+
t)
+
tt)-
1B_
oat)+
Bot
t
-B(t
t +- %t)
+
+f2* B_(at +
+f* B
t
-
4
_t
+
B+tJ
I
t].
(C. 100)
259
The
fourth-order
(in k)
anisotropic
terms
in the
decoupled conduction-band
Hamiltonian are [Ogg 66, McCombe 69, Barticevic 87, Kim 89]:
H(()
o
= a[Ik 2, k2 + {k2 , k2 + {k2x , k2 ]
o y z
z x
y
4
0[(k +
-
ky4 + k)z - k4]
(C.101)
and
H(yo)
2
2 C
0
x
B k2 + oyB k2
B k2
y y y
xx
Z zz
.
(C.102)
The irreducible, e = 4, octahedral part of H(cx) is
H (%o)
c
= - a [k 4 + k4 + k4 - I(k 4 +
ox
y
Z
5
=
-
(
(Kgv) k k
WkW4
0
+
4+
kkv
4W k k3 +
3
+
S2)]
- k4 + s2)
-W2(7k
2
- k2
+ 2s)k
+
+ WI(7k2 - 3k 2 + 3s)k k+
+ (Wo-
k2
)[35k4 - 30k
+ 3(k 4
+ S2)]
+ W*(7k2 - 3k 2 - 3s)k k
+ W*(7k2 - k2 - 2s)k 2 +
2
4W*k
k -3 + W*k4
3
4
(C.103)
260
The irreducible,
e=
4, octahedral part of H(y) is
-((aB)k2 + 2((;k)(B-k)]
k2 - 51
k2 + B
7OB k2 + TB
(y
2 C~Yoxx
y yy
Z Z
C 0S
-
e
=
Yo[%
l)
[W4
Bk
2 CYO
+ W2 (3a2B
(B
+ W(-
k k
a2
7(0B
k
-
k)(B-k)
+ 2(
+ W (a+B + ;B+)k< + 2Y+B+k+kJ
-2( k
B
+(3
+ oB)k
+!(aB
+aB )(3k
2( +
B )k+
-
-B
"
W*(-
2
+
W* (oBr
+
(3
+
aB )kk
2
- ( 5Bk
-)2(a
B
B
2
-
B)(3k
B)k
BS - &B)kk
)k2 + 20aB k)k
S(C.104)
+
+
k2)
+
!B
kk
- k2)
+-BAk
)(3ka -k 2 )
a+Bkk%+)3(a B2+2B
(W+
-
B -
(3
+
B
+
- k2)
+ a B )k k + jB(3k
B)k + 2(B
-
+
2(+.1%4)
261
+
-
B
3a B~
+ W aBk2]
(cB
k2
+ asB )kJ
.B)kk2)
The third-order
antisymmetric
tetrahedral
term
in the
decoupled
conduction-
band Hamiltonian is [Rashba 61a, Ogg 66, McCombe 69, Barticevic 87]:
H(8 0)= 50[a(k k k - kzk z) + c.p.]
S12 [cxY{k ,(k' - k )} + c.p.]
Oxx
y
z
= 9 S-
=
0
(ily;Rv) (Y {kk k k I
[F3 [+k<k - ok+]
+ F2 [o+(k2 -Sk-s)k
+ F [2a (3k 2 - 5k2)k
+ 4o k k - ok+]
- a (k 2 - 5k
-
s)k+ - 5ak2k ]
(k2 - 5k -s)k+]
+ F[o (k2 - 5k2 + s)k_ -
+ F* [2a(3k - 5k )kr - a~(k 2 - 5k2 + s)k - 5a+k2k]
(k2 - 5k
+ F* [
+ 4s)kk
+ s
-
-
2-
+ F*[o k2k -
kk2k
3]
Ak
k3]
(C.105)
The
relationship
of
Eqs. (G.23) - (G.31).
a 0, yo
and
8
to
the
PB
parameters
is
given
in
Corrections which go beyond the second-order PB model
are expected to be of the order of a few percent.
262
Appendix D
Kane and Yafet Models
The Kane and Yafet models are simple analytical models for the interacting
conduction and valence bands in InSb which take account of the spin splitting
of the valence bands and of the k-p interaction between the these bands and
the conduction band, but neglect coupling to higher or lower bands.
The
models have spherical symmetry with respect to the crystal orientation, and
axial
symmetry
Hamiltonian
Yafet 59,
in
the presence
applied
magnetic
field B .
The
and omits terms of order k2 or higher [Kane 57, 66;
is 8x8
66].
of an
The
effective
equation, but has two hole bands with spin
hole band with spin
.
equation
Schridinger
3
resembles
the
Dirac
and spin I instead of a single
The Yafet model is significant for having predicted
observable spin-flip Raman scattering
in InSb.
The effective Schr6dinger equation is
g -E
Pk-t
Pk.tt
-E
Pk-t ' t
0
Pk-t'
fc
0
f
-A - E
= 0.
(D.1)
f
Here
k-t=
1
k,
and
263
(D.2)
k
k-t'
k
3 Z
3 -
(D.3)
Here k is the kinetic momentum operator in a magnetic field.
matrix equation
functions fc
, fv
(D.1)
as three equations
in three
If we regard the
multicomponent
unknown
and fs, we may use the second and third equations to obtain
Pk -t f
E
f-
(D.4)
and
Pk.t't
f
(D.5)
f .
E +A
c
Substituting these into the first equation gives
S
E+A
E
E)
Using Eq. (2.142) and the algebraic relations of Appendix D we have
(k-t)(k-tt)
2
3
k2
(k.t')(k.t' t) =
1 eB o ,
(D.7)
3C0
k2 + ' B.- .
(D.8)
Thus
E E)+P2k2
3m
2
2 +
EE
1
+ A
P2
3m 2
1
E
1
e
E + Ac o
f
c
(D.9)
which has the form of the effective mass Schrodinger equation for conduction
electrons
in
the decoupled
approximation.
264
A significant
difference
is the
presence of the energy E in the denominators.
For E =
it is useful to
write this as
(
9
- E) +
k
2m*(E)
+
2
g*(E)gB
B 0
f C(D.)
=
(D10)
where
- 2P2
m
2 +
3m
m*(E)
E
(D.11)
1
E + A
is the reciprocal conduction band effective mass, and
4P2 1
3m E
g*(E)
1
E+A)
(D.12)
is the effective g-factor.
(We have restored factors of the free mass m.)
The case of a finite uniform magnetic field was first treated by Yafet
In this case the problem has axial symmetry and the operator
[Yafet 59, 66].
A
N + F
is conserved [Luttinger 56, Yafet 73, Suzuki 74, Trebin 74, Rashba 91].
Note that N is the number operator for Landau levels in the absence of spin,
and F
is the 8x8 matrix with m along the diagonal, so the sum represents the
total orbital plus spin angular momentum in the direction of B .
S= n + m 3 the operator
2
We define
as the good integer quantum number (ef _-2) associated with
N + F
-
~.
For a given t the wave functions have the form
265
'c3Pe-1
f(r) =
,
fc(r) = C2 et+
fS(r)
,
C6 (P
C5
=
c8
,
(D.13)
C7+'C7+1
C4Pt+2
where
q n represents
Eq. (2.373).
a Landau level wave function with the form given in
Now k is an operator instead of an ordinary vector.
If we
A
choose B to be along e , then Eq. (D.9) becomes
2 +
,+P2k2
3m 2
P2s
3m 2
E +A
E
1
E
1
E + A
(D.14)
Three types of solutions exist [Yafet 66], corresponding to the cases
fc ~
The
fc ~
n
first two
cases
n
(D. 15)
c~
'
apply when E
0 and the
heavy-hole solutions which have E = 0 for all k .
last case
applies
to the
For the first two cases,
multiplication of (D.9) by E and E + A gives
E(E -
where
g
)(E + A)
) --
k 22((E
the upper sign applies
applies to the second case (b-set).
+
p 2sA
A)
3
=
3
to the first case
0 ,
(D.16)
(a-set), and the lower sign
Here
s - eB 0/hc
(D.17)
k2 =
(D.18)
and
s(2n + 1) +
is evaluated for n = e for the a-set case, and n = +1 for the b-set.
266
The
solutions of the two cubic equations give E(k) for the conduction, light-hole,
and split-off bands. The multicomponent wave functions are found from
2x2
Pk .t
E
Pk - t
f(r) =
(D.19)
E+A
with the normalization condition
<fc 1
3m
k2
2
f3m
2
2
E
I
E
P 2s( z
3m
+ E + A
=1
E + A)fc
Designating the a-set (spin-up) wave functions of band-index g., by IgnT>, with
f1 a =
c
, we have
NO
a[(
(Pn
0
VTi Pk /E
IJl'>
-
1
-~T2
PkI/E
Na
-
Pk IE
T
PVMT
1
(Pn
+
0
V
N
--VYT
a
/E
Pk
Pn/E 9,
--PVs( £+1 )/3 IE
Pn+ 1
...............................
I..............
P k / (E+A)
ViT 3Pk
VI/T Pk + /(E+A)
PV/sCT+l) /3
/ (E+A) (Pn
/ (E+A)
Pn+l,
(D.20)
267
Here
p2k2
N2a =1+
2
1
P2s
P2
1
(E + A)2
3
E2
+
E2
and g may be c, +, or s, corresponding
split-off bands, respectively.
1
(D.21)
(E + A)2
to the conduction, light-hole and
(The subscript
'a'
also applies to the energy
E, but has been omitted to simplify the notation.)
For the b-set (spin-down) wave functions with
C
1 [
Nb n
f
and
=
n-l,
we have
0
1
0
1
Ignj
Nb
Nb
Pk /E
1
- V2
Pk /E
Nb
-V-i2
Pk /E
I /6
113
-1i3s
--v2/3-
Pk
Pn-
1
/E
-PV s( n+1 ) / E Pn
+
PV'2sn3 / (E+A)
Pk /(E+A)
Pkk
PVsn73 /E
(E+A)
1
Pn- 1
-- vT7 Pk / (E+A) (pn
(D.22)
Here
P k2
N2
b
= 1 +
3
2
E +
E2
1
P2s
1
(D.23)
+
(E + A) 2
268
3
(E + A) 2
where k 2 is evaluated for n = e + 1 .
A special case exists when n=0.
In
this case the split-off band wave function IsOj> has an energy which equals -A
at k
=0 and is approximately
I2ki/(2
-A-
msI) for small k
The energy
.
denominator E + A in the expression for the wave function then is proportional
to k
for this particular state.
Normalization requires multiplication of all
of the components by a factor which is proportional to k
limit.
in the small-k
The largest components then are the m = I component in f
, which is
f,
which is
approximately (p for small k , and the m = proportional to k
for small k
.
component in
This wave function is therefore seen to be
the first member of the a-set Landau ladder rather than the first member of
the b-set.
Its
designation then
is aS(O).
Note that it differs from the
other as(n) states in that it has a finite m =-I component in the conduction
band.
This component is zero for all the other states, which are pairwise
degenerate with b-set states in the low-B 0, low-k limit.
The heavy
hole solutions have E = 0, and wave functions which
are
determined by
f=f
o
(D.24)
,
and
(D.25)
(k-t)fv = 0 .
269
The a-set heavy hole wave functions are
+ k 2 + 3s
3k
1
(4k
2V3 k k
ShnT> Na
a
1
Pn-1
V'3 k2
N
+
+ 2s(n+l)) 'Pn-1
2V96si krp
a
n
2sv3n(n+1 ) 9Pn+
1
0
0
0
0
0
(D.26)
with
N 2 = (4k2 + 6s)(3k2 + k 2 + 3s)
a
(D.27)
= [4k2 + 2s(4n + 1)][4k2 + 2s(n + 1)]
Here
k=
s(2n + 1) + k2
has been evaluated for
valid
for
n 2 1 .
(D.28)
n = e-1 .
The
state
The expression for the wave functions is
with
labelling.
270
n = 1 is
a-(2),
using
the
standard
The b-set heavy
hole wave
functions, with E = 0, are given by
0
0
0
0
0
0
v/3 k2
1
1
Ihnj>
Nb
- 2V
3k
k_ k
+ k
Nb
2sV3(e+1) pe-1
-2V6s( e+1) k Pe
(4k' + 2st) 9+l
- 3s
0
0
0
0
(D.29)
with
Nb = (4k2 -6s)(3k
+ k2 - 3s) ,
(D.30)
= [4k2 + 2s(4n + 3)][4k + 2sn].
Here
k2 =
s(2n + 1)+
k
has been evaluated for
valid for
n 2 -1
.
labelling, and equals
(D.31)
n = e + 1.
The state
The expression for the wave functions is
with n = -1
p u4 (m = - 2) for all k.
The state with n = 0 is
a+(0), which is actually in the a-set, not the b-set.
This state differs from
the other a-set heavy hole states in having a finite m = low-B , low-k
limit the u5 (m = -)
component is q
component is proportional to kpl
271
using the standard
is b+(0),
2
component.
In the
and the u4 (m=-)
The Kane model result for B0 = 0, neglecting coupling to higher bands, is
easily obtained from the finite-B0 results by making k+ and k
numbers instead
of operators, and replacing the Landau-level wave functions c(p by 1.
For small values of n and f we make the following connections between the
wave functions of the PB model [Pidgeon 66, Weiler 78, Luttinger 56] and those
of the Yafet model.
S= -2:
b+(0) =
h(-2)1> (E = 0) ;
S= -1:
a+(0) =
h(- 1)j>
bc(0) =
c(-1)> ,
b+(1)
L(-1)j> ,
(D.32a)
(E = 0) ,
aS(0) = Is(-1)j> (E =-A at k = 0) ;
e=
0:
b-(2) = Ih(0)J>
(D.32b)
(E = 0) ,
bC(1) = Ic(0)1> ,
b+(2) = IL(0)>,
bs(0) = Is(0)1> .
aC(O) =
Ic(0)'>
a+(1) =
IL(0)T>
(Lowest cb level at k = 0) ,
(Highest light hole at k = 0) ,
aS(1) = Is(0)t> ;
= 1:
b-(3) = Ih(1)j>
D.32c)
(E = 0)
bC(2) = Ic(1)1> ,
b+(3) = IL(1)> ,
bS(1) =
js(1),>
a-(2) = Ih(l)t>
.
(E = 0) ,
272
aC(1) = Ic(1)IT>
a+(2) = IL(1)T> ,
aS(1) = Is(1)t> .
(D.32d)
For small B the energies are determined by
(E2)k2
g
3m
2
2 +___
P2s
1
G G + A
g
g
3m
2
&
2
P 2 k2
3m 2
3m 2
3m
_
2
g
(D.33b)
1-
3m 2
+ A
g
(D.33a)
cb
g
S1
(E + A) -
0
1
32s
g
1=
.
(D.33c)
s-o
+ A
g
The wave functions and energies for small B and k =0 are:
Conduction band: fc = (n
aC(0) = ((P
0) for the a-set, and (0 9 ) for the b-set
0)
a
E =
I = n+1
bc(0) =(0 (0n
)
a-(n) = ( a 3Pn-2
Valence band:
where
a 3 = [3(n - 1)/(4n-3)]
3
a+(0) = ( 0
0
Po
a5 Pn
5p9
a-(2) = ( (Po
0
0
5)
mc
(D.34a)
4gco
]
Ig ho
c= co
(D.34b)
0 )
5
e =-1
)
92
c
1 /2
and a = - [n/(4n-3)]
0 )
a+(1) = ( 0
a+(2) = ( -
0
/2
0
Eb
m
E a =0
Ea
0 )
0 )
S=1
e.= 1
273
Ea =
Ea = 0
2 mL
M+
4
LCO
L]hco
(D.35)
etc., where
Valence band
where
b-(n) = ( 0
b6c n_
0
b4pn )
b6 = [(n - 1)/(4n-1)] 1 /2 and b4 = - [3n/(4n-1)] 1/2
b+(0) = ( 0
b+(1) = ( 0
b+(2) = ( 0
b-(2) = ( 0
0
0
0)
0o
-
c
0
o0
0
p
e =-2
E
e=-0
Ebm
[
b
)
m
m
'gh]
gL
CO
co
mL
e=O
262
e=0
F)2
(D.36)
etc.
Split-off band
aC(0) = ( 0
bc(O) = ( (P
n
fs= (0
(nn2
)
C)
and
e=
((pn
0 )
n+l
E
a = L2 m
a
0)
2 m
+ I
hI
CO
s] hO)C
4Sj
(D.37)
(D.37)
Here the masses are determined from Eqs. (D.33).
Note that the valence band wave functions are the same as those obtained from
Luttinger's model with
I= 272= 273 = 2K = 2P2/3
274
.
Appendix E
Magneto-optical Matrix Elements for Scattering and Absorption
If H 1 is a small perturbation, the perturbed wave function is
o
(H)io
p=
o + E -E. (Pi +
i o I
Z
(E.1)
If V is a component of the velocity or the scattering operator, and V is the
perturbation to V due to H then
0
1
<fl(V +V )Io> = < (f + E
if
(H).
-E.o
1
I(V+Vl)I (o + S 1 E>
E -E.J
E-E. (Pi
f
I
j0o o
(H) if
(Hf(V)ioo_io
)Io + : (H)
=(V
E-E.
if
f
> +...
(V ) (H)
E -E.
I
i#O 0
I
(E.2)
= (V)fo + [Vo,iS
]fo
where
(S l)jj = 0 , and
I 11
i(Hd)
E.-E
-jk
Jk
(E.3)
for jsk .
This first order perturbation theory approach is valid for the cb but not the
vb. We now consider some examples of absorption and scattering.
(Example la) Let V = k lm*, and V = v
We
+ then have1+
We then have
275
(H)f(k).
[V0 , iS]f = S(l/m*)
(k )f(H ).io
E.-E
Ef- E i
= (l/m*)
1
E
(k)fi(Hl)io
(H) (k)io
Eff (E 0 + ho)c
= (l/m*)
and
(EF- haO)
f
c
- E
o
(E.4)
[H ,k ]fO
H+fo
fo-
Use [k - ,k+ ]=2s
O
°c
[H ,k ] 1 = +12s(8H8lak - ) = sv 1
to get
[V , iSIfo =
(V l+)fo
ho)c
ifo
- hoc
CE
ho
C
fo
_
C
0-0c
(0-0)
(V)
.
(E.5)
fo
C
This implies a 'correction factor' of
(3
(0
1+
(0-0
(0-0)
(Note that (0=fo for absorption.)
fo
(Example ib) For V0 = k /m*
(E.6)
the correction factor is
0)
1-
(E.7)
C
O +
)
C
(0+0)
(Example lc)
For V = k /m*
0
z
(Example 2a)
Let VWe= A , and
then1
the correction factor is
V = A ,with
We then have
276
1 - 0 = 1.
A 0 = ih6KSave
(axb)+
(Ao)f(H) io
E.-E
1
0
(Hl)f(Ao)io
E -Ei
(E.8)
=
[Ha_]
iiKs(axb)+ [HI-f
fo
s
For A = 2ih-Ks(axb)z
(Example 2b)
we get
z
[H ,z]fo
(E.9)
2iiK (axb)z
Ofo
(Example 2c)
For A 0 = ih iK S(axb)_a +
we get
[Ho]
iOK (axb) [H +f
s
fo
(E.10)
In Example 2c consider the case of inversion asymmetry, with
H( 8) = 80[oa(kkk - kkk)k
1
x yxy
zxZ
+ c.p.]
(E.11)
8 F [oa (k2 -5k 2 + s)k
00
+
2-
-o(k
5k2 -s)k]
+
+
.
Then
[H ,]
since
= 4Fo(k
1 00Z
+
2
- 5k - s)k
[_,+] = 2i[
[o,oy] =-4
-+xy
+ . . .
(E.12)
.
(E.13)
zz
Note that
E K = - (P/3)E [1/E 2 - 1/E'2] = 15.6 in InSb.
gs
g
g
277
(E.14)
We then obtain the result
i[A ,S ]f = i6Ks(axb)0
0f
s
[H ,+]fo
s
fo
(E.15)
1
icK (axb)=
where
F
This treatment
+
-= 0.1443
parallels
s
[46 Foy(k 2 - 5k2 s)k
00
+f
when B ll[111].
0
that of Rashba and
[Rashba 61a, Gopalan 85].
278
Sheka, and of Gopalan
et al.
Appendix F
Matrix Elements of exp(iq.r) for MD and EQ Transitions
The matrix element of exp(iq.r) between Landau level wave functions is
8 (k+ q - k') e
(k + qy - k')
Z
Z
y KZ
K
<ezq-r p
-iqx(k + k')/ 2s -q
e
Y
/ 4s
M,
n
(F.1)
where
y_b
-y+b_
e
< n'j ee
MMn n =
(F.2)
n>,
with
+ = (qx
+
(F.3)
iqy)/ v'Ts .
This result corresponds to the gauge A = B0 xy and Landau Level wave functions
A
A
A
It was
which are localized in the x direction, and extended along y and z.
derived
through
the
normal
ordering
of
raising
and
obtained by application of the Baker-Hausdorf theorem.
lowering
operators,
This theorem states
that
eA+B =e -[A,B]/2 eeA
B
(F.4)
if the commutator [A,B] commutes with both A and B.
The quantity
M ,
may
be expressed in terms of Laguerre polynomials [Bass and Levinson, 1966].
To first order in q, neglecting higher and lower orders, we have
M ,
n n = < n'
y_b+ -
+
y+b_
+-
(F.5)
n >
279
If we notice that
(q x k) =
(q k-qk
+)
(F.6)
we see that interlevel effect of exp(iq.r) to first order in q is the same as
that of
(q x k)z eiq-r
(F.7)
to lowest order in q.
A
For q = qy , we have, by simple expansion,
<' leiY
=
K(k
(
+ q - k') 8 K(k
- k')
< n' -iq(b
+ b_)/ v7- I n >
(F.8)
If we neglect
q
in the first Kronecker delta function,
the result is the
same as that of
<w - iq(k + k_)/ 2s Ir> = <lI' - iqkx
280
s I>
.
(F.9)
Appendix G
Decoupling of the PB Hamiltonian
We now consider the decoupling of the PB Hamiltonian to obtain effective
Hamiltonians
for the
and
conduction
valence
Decoupling
bands.
of
the
conduction and valence bands from other bands to order k2 is the basis of the
usual
effective-mass
The resulting
theory.
effective
Heft
Hamiltonians
are
useful for computing optical and transport properties of electrons in a single
band if hco and he c and E
f
c
compared
with
g,
as
in
are small compared to 8g.
g
the
case
of interband
If hfo
transitions,
is not small
it
is
still
possible to use the decoupled wave functions to compute optical transitions as
was done by Roth et al. [Roth 59].
For narrow-gap semiconductors like InSb,
it is relatively easy to have experimental conditions in which the cyclotron
energy hOc or the Fermi energy Ef is a sizable fraction of the energy gap, in
which case the decoupled scheme becomes deficient.
This is why the coupled
models of [Kane 57, 66], [Yafet 59] and [Pidgeon 66] were developed.
cases, the
In these
decoupled Hamiltonian is still useful for determining the optical
and transport properties in the limit of small B or small E .
Rashba and Sheka [Rashba 61b] considered Hef for the conduction band of
InSb to order k3.
those
of the 8x8
They obtained the parameters of the decoupled model from
Kane model,
valence band blocks.
neglecting
the Luttinger parameters
in the
In particular, they found expressions for m/m*(E) and
g*(E) as functions of P and 8g and A, and the expression for 80 as a function
of C, C' and G (which they call K, Q and G).
They incorrectly assumed C'= C.
Ogg later obtained Hef to order k4 [Ogg 66], including warping as well as
281
inversion
asymmetry
terms,
taking
account
of
the
tensor
transformation
properties of the operators and matrices occurring in the Hamiltonian.
This
approach was further generalized by [Braun 85].
The
result
of
the
decoupling
equivalent to that obtained
process
described
from ordinary perturbation
by
[Rashba 61b]
theory
is
if one allows
k-p to be a matrix (containing the spatial operator k) and takes care to make
the result hermitian
[see Braun 85].
To fourth order the Wigner-Brillouin
perturbation theory (WBPT) expansion is
E
+ V+
= Eo + V+
n
n
nn
+
V
V
nm mn +
E - E0
n
In
V
V
V
nm ml In
0
(En - E mIn
) ( En - E )
Vnm VmlV k Vkn
E) (E - EOk
)(E
(En - Eo
m
n
I
n
k
(G.1)
summed over man, l4n and k-n.
The Rayleigh-Schr6dinger (RSPT) expansion is obtained by expanding (E -E m)Zn m
with the result
En = Eo +
n
0
V
+
nn
V
Eo n
E
V
V
V
Vnm Vml Vn
(Eon - E m
+
V
) ( E
n
-
m
Vnm Vm n Vnn
E)
(Eon1 -
Vnm VmlV 1k Vkn
o) (Eo - Eo)
(Eo - E ) (Eo
n
m
n
I
n
k
282
Em )2
Vnm Vmn Vnl V n
(Eon -
E m0 )
2
n - Eo)
(E n
1
(En -
1
V nV
Vnm
Eo n
Eo) (E - Eo
n
m
1
E
o
m
V
V
(E
Eo - Eo
l
n
o
n
(Vn n 2
-
)3
E
m
(G.2)
summed over man, ln, ken.
degenerate.
Here we have assumed that all levels are non-
When degeneracy exists the expression becomes a hermitian matrix
which is an effective Hamiltonian for the degenerate subspace.
ordering
The correct
of the matrices within the expression requires an analysis like that
of [Luttinger 55], [see also Kjeldaas 57 and Yafet
63], which uses successive
unitary transformations. The result is
Hn = Eon + Vnn +
V
V
nm
V
+
VV
V
lnmIn
9 nm &nl
V V
+VVV
nm mn nn
nn nm mn
29 2
nm
VVnmVlml V lkV kn
V
V
VV
Vnmmn
Vl In
nm nl nk
2nm nl
nn nm mil n
n Im mn nn
2nm nl
+
1
1
nm
nl
i
nm
V V
V
V
nn nm mn nn
nl
(G.3)
3
nm
summed over man, ln,
k-n, with
8nm - Eon - Eo
m
(G.4)
283
This result contains
some fourth-order
[Braun 85],
given by
terms which are
and do not contribute
additional
to those
to order k4 to the decoupled
conduction band Hamiltonian.
When
time
derivation
of
described
by
dependent
Heff requires
[Foldy 50]
Bjorken 64.]
fields
the
and
use
potentials
of
for the case
i(V
-V
V
conduction
electric
V
i( nm mn - Vnm mn
n2 2
nm
to
the
correct
transformations
as
[See also
The correction to the above result due to time dependence of the
summed over men.
respect
time-dependent
present,
of the Dirac equation.
perturbing fields, to third order in h2k2/2mg
H
Hn
are
and hOo0 g, is
)
(G.5)
(G.5)
The dots over the V's stand for partial derivative with
time.
This
electrons
proportional
term
gives
to
rise
the
to
a
transverse
spin-orbit
part
of
interaction
the
for
perturbing
field.
The largest part of the decoupled conduction band Hamiltonian Hc may be
obtained by considering the spherical Yafet model in which terms quadratic in
k are neglected in the effective
8x8 Hamiltonian [Yafet 59, 66].
The PB
Hamiltonian then reduces to [see Rashba 61b, and Kim 89]
H
=
S
g
C
C"
Ct
0
0
C.
0
-A
(G.6)
284
with
and
C = Pk.t
C' - Pk.t' .
(G.7)
The partial derivatives
where t and t' are defined in Eqs. (x.xx) and (x.xx).
of C and C' with respect to time are
C = P e At = -PeE T -t
(G.8)
C
and
C'= peC A-t' = -PeE
T
(G.9)
t'
where ET is the transverse part of the perturbing electric field.
To fourth order in perturbation theory we find
cct
H =g1
c
g 2x2
+
C'c'
+
g
g
(CC ) 2
3
(C'C't) 2
h.c
.
h
+
+2
(
+
A)
g
C-hc.
h.c.
+
+ ----
g
gg
g
+-
1
1
-
+
+
g
CC'
+
CCtC'C't
+
2
(G.10)
which we will evaluate below.
We have included fourth order time-dependent
terms not given by Eq. (G.5), and have assumed that the perturbing field is
transverse, i.e. (p(r,t) = 0.
285
The most general expression for H
for the
6
conduction and the
s
split-off bands, obtained by the method of invariants [Ogg 66, McCombe 69,
Braun 85] to fourth order in perturbation theory, is [Barticevic 87]
H
eff
= H +H + H
0
1
(G. 11)
2
with
2
k
H =2m* +
2m
tgBo.a - eo(r) ,
H1 = 8 [x (k kk - kk k)+
0 X YXY
zxz
+ e0
+
(G.12)
2
k4
+ ( [{ k2,k2
B
c.p.]
+ c.p.]
+ g jo -B k2
+ 2g".B(.k)(B .k)
+ YoL [oxBk + c.p.] ,
(G.13)
and
H2 = eKSaExk + eKD V-E + e2KE
E
2
+ eK[E {k ,k ) + c.p.] .
(G.14)
The terms in H are the kinetic energy and the electron spin contribution to
the Zeeman energy.
The term in H 1 involving 86 is anisotropic.
symmetry and is odd with respect to inversion [Rashba 61a and 61b].
in HI
proportional
symmetry.
to ao and Yo are anisotropic
These are even with respect to inversion.
286
warping
It has Td
The terms
terms with
Oh
The remaining terms in
H possess spherical symmetry.
not
considered
interaction
term
in
the
proportional
proportional
H contains terms which depend on E and were
above
references.
These
include
the
to a-Exk, the Darwin term proportional
to
the
electric
energy
density
at
the
spin-orbit
to V-E, a
position
of the
electron [Wolff 66, Eq. (5)], and an anisotropic term with Td symmetry which
is independent of spin.
Note that Exk must be symmetrized when components of
E and k fail to commute, and that a small term proportional to E (a B +B ) +
x yz zy
c.p. has been omitted.
If we consider the Yafet model in which the 1C,
6
FV and Tv7 interact only
8
via the momentum matrix element P [Yafet 59, 66], then Hc for
c is spherical
and may be expressed as [Rashba 61b]
H =
k2
g
+
+
S2m*(E)
o(E)
BBo.
,
(G.15)
with
m
2P2
3
3
m
m* ( E)
2
E
+
1
+
E + A
(G.16)
and
4P2
1
1
and with E measured from the top of the t
Eqs. (G.12)
(G. 17)
,
g 0(E) =-
and
(G.13)
may be obtained
v
valence band.
The coefficients of
by evaluating
Eq. (G.10),
or by
replacing E in Eqs. (G.16) and (G.17) by He and expanding about 9g [Kim 89].
The coefficients of Eq. (G.14) must be obtained by evaluating Eq. (G.10).
results are:
287
The
m* = m*(g) ,
o( g) '
go
1 d(m *)
E
o
(G. 18a)
(G. 18b)
-2
-
8
dE
1 dg
(G.18c)
E=9
2
dE
(G.18d)
E=E-
1 d(g / m*)
4
dE
(G.18e)
E=g
g
1 dg
P2
4 dE E=6
3
K
s
g
G2
4
P2
K
dE
(&g+ A)2
g
1 d(m / m*)
KD
D
(G.18f)
1
E=g
g
P2
2
6
G2
(G.18g)
g
(Eg+ A)2]
2
(+
)
]
(G.18h)
8o = CC=yo = g" = K A = 0 .
(G.18i)
E
3
3
The coefficient K D is evaluated by assuming that a finite perturbing potential
(p(r,t) exists so that the matrices V , V
and V
Vcc = V = - ep(r,t)12x2
CCSS2x2
satisfy
(G.19)
and
288
V=
- eqp(r,t)14
(G.20)
are considered in Eq. (G.3).
The parameter g" is zero for the conduction band
in the Yafet approximation, but is finite in the Hamiltonian for the split-off
band Tv7 .
It becomes finite for the conduction-band Hamiltonian if we include
terms proportional to k2 in the
decoupling.
inversion
'v
x Fv block of the PB Hamiltonian before
The parameter 80 becomes finite for the conduction band when
asymmetry terms in the PB model are taken into
Rashba 61b].
The parameters a
terms in the PB Hamiltonian.
account
[see
and yo become finite as a result of warping
The present approximation to Hc , obtained from
the Yafet model, may be compared with that obtained by Wolff using the
"two-band" model [Wolff 66].
The latter model may be obtained from the
present one by dropping the terms which have & by itself in the denominator,
and replacing
g + A in the remaining denominators by g.
g
g
We will now obtain expressions for 80, a(0 and yo based on the PB model.
The
expressions
anisotropic
are
important
conduction-band
c(0
(G,N) + h.c.
+
g
Hcv(P) H
parameters
determine
in the low-field
limit,
the
for
a
Starting with 8 we have [compare Rashba 61b]
o)
+
these
magnetoabsorption
model which is easy to compute.
Hc (P) H
because
Hc(P') Hs(G') + h.c.
g
HCV(P) Hv(C') Hc (P') + h.c.
(C) HVC(P)
+
g(9
gg
@2
g
+ A)
(G.21)
289
As an example of how this can be evaluated, consider
H (P) Hv(G) = -iPG[k.t][{k,k }t + c.p.]
-iPG[k
k k }t tt + k {k ,kt tt
x z
xy
x y z xx
+ k k k }ttt + c.p.].
y y z yx
This
after evaluation of
is simplified
the matrix products.
(G.22)
The complete
expression which is obtained for 60 is
8 = 8 (G,G') + 56(N0
0
0
2
)
(G.23)
+ 56(C,C'),
0
with
4
GP
G' P'
(G.24)
6 (G,G') =
3
9+
g
g
N P
= -2v"2
6o(N)
0 2
(G.25)
2
&
P
2C'P'
CP
S0(C,C') =
(G.26)
+
g
g
g
A similar analysis for o and yo begins with
290
Hc(ao, o ) =
Hcv ( P ) Hvv (,q)
H c(P)
Hsc(P') + h.c.
Hcv(P) Hvs (')
g ( g+
A)
gP
2 'P'
gg
(G.27)
The result is
2P
a
=
-
Qo(R,4')
+
g
(G.28)
g
g
and
(G.29)
o =Y(R,R') + Yo(q) ,
with
12P
7o(g4,')
=
'P'
gP
(G.30)
_
g
g
g
qP 2
(G.31)
yo(q) = 3
g
291
Appendix H
Effective Scattering Operator to Order k2
This
operator
appendix
Ascat
deals
which
with
the
derivation
describes
the
limiting
of
an
behavior
effective
scattering
of
scattering
the
amplitude Afo at small values of magnetic field B 0 and momentum k z along the
field.
The operator is obtained as a power series expansion in the kinetic
momentum k, and acts between initial and final wave functions Io> and If> in
the decoupled representation.
The approach is especially useful for scattering
from conduction band (cb) electrons because the effects of anisotropy are small
and can be treated by first-order perturbation theory.
be applied to valence band (=3
less
accurate.
(Accurate
The same approach may
vb or j=- sb) electrons, but the results are
results
require
diagonalization
of
large
matrices.
See [Evtuhov 62].)
Two methods are used.
Both make use of invariant expansions and techniques of
matrix and tensor algebra.
(1) The
first method is
[Wolff 66].
conduction
This
band,
a generalization
approach
including
starts
terms
of the
approach
used by
with the decoupled Hamiltonian
containing
the
electric
field
obtains an operator Ac which describes intra-cb light scattering
Wolff
for the
E(r,t),
and
in the limit
where the incident photon energy is small compared to the valence to conduction
band energy gap, ho1<<Eg.
(2) The second method is based on the approach used by Yafet and Makarov
[Yafet 66, Makarov 68].
It starts with the coupled-band effective Hamiltonian
of Pidgeon and Brown [Pidgeon 66, 69] as formulated by Weiler and Trebin
[Weiler 78, Trebin 79], and uses a k-p expansion of the multicomponent wave
functions
for
the
initial,
final
and
intermediate
293
states.
The
scattering
operator which results is valid for all values of the incident photon energy
below Eg, but gives useful results for Afo only if the Raman transition energy
is less than (Eg - ho).
H.1
Computation of AC from decoupled conduction band Hamiltonian H
C
Here we start with the decoupled cb Hamiltonian computed from fourth-order
k-p perturbation theory [Ogg 66, Braun 85, Barticevic 87],
with electric field-
dependent terms which result from the Luttinger-Kohn transformation [Foldy 50,
Luttinger 55, Blount 62, Yafet 63, Rashba 91]
2
H - 2m + 1g*g B - + eK E (kx)
+ E k4 + g'gk 2B ' G +
2of3ZB2
+ 2g"g K(2) (Bo)(2) + Iea 2{k 2,(E kx() } + bg E. kxB
o [kkBY]() + eK[(Exk)kk](4) + eK[Ekk(ko)
K (4)+
-
+ eK [Ekk](3) + egK [EB](3)
+ 6 [K()(3)
(H.1)
The term proportional to g" has been modified so that it has no overlap with
the term involving g', and the anisotropic
been made irreducible.
terms in the last two lines have
I have omitted terms of order E2 and higher,
which
produce terms of order 0 2 and higher in the effective scattering operator.
The second-order k-p parameters m/m* and g*, computed from the PB model, are
m - 2P2[2+ 1
+ 2F
4P2I-
-__
1,
(H.2)
+2
-
+
4N1
H3
294
K
The
The spin-orbit parameter Ks is
E'-Eg+A.
where
P2
1 dg* _
4 dE,
s
1
3
1
2
E
L C
crystal)
(isotropic
axial-model
fourth-order
(H.4)
E'2
parameters
E'
g'
and
P ,'
computed from the Yafet model, are
m
d
P4
_
2
8 dEg m*
0
1
+ 1
1
2
E'
2
4
9 E3
_ 2P 4
S1 d g*
go 4 dE, m*
1
dEg
6
P
a2
9
E'
Eg
E3
12
b 2 = 18 E144
1
9 E,
E3E'
8
Ea2E'
t,
+
(H.6)
E E'2
Er2E'
E'
+
1
1
1
4P4
)2
-d
(H.5)
E'2
E2
(H.7)
E2
7
3]
+ 3
EE'2
E'3
8 + E'12
E2E'2
3
(H.8)
(H.9)
E'14
The fourth-order axial-model parameter g" computed from the PB model is given
by
g
S-
P2
3
1
Kl
P2Y
E'2
Eg2
+
6E, Eg
The fourth-order warping parameters
E'
6
2
Eg
EE'
E'2
(H.10)
o and y o, computed from the PB model, are
given by
=
o( ,0')
2P
E
2
E,- 22Wlp'
EL+
,
(H.11)
295
o(,)
2P
=
-
(H.12)
,
(H.13)
2
y(q) = 3 P
Eg2q
The third-order inversion-asymmetry parameter 56 computed from the PB model is
the sum of terms given by
G, G')
3
,
Eg
(H.14)
E'
PN
80(N 2 ) = - 2V2
S(C, C')
-
E
,
(H.15)
+
(H. 16)
I have assumed that P'=P for simplicity, where P' and P are the momentum
parameters in the H
and H
blocks of the PB Hamiltonian, respectively.
This
may be corrected by replacing P by P' in a term wherever E' appears in place of
Eg.
Similar expressions exist for K and K4 , which are not important here.
We
next use first and second order perturbation theory to compute A c to order k4 .
First-order perturbation theory:
Replace k=p+-A 0 in H by k+eA, where A represents the perturbing fields at the
incident frequency Co
A 2 and EA.
and scattered frequency 0o2.
Then collect terms of order
(Alternatively use the "polarization" of Hc obtained by applying
the operator A-Vk once or twice [Weyl 46, Edmonds 74].)
Thus
296
----
(elc)2A 2
E -(kx)
-
(e2 /c)E -(Axo)
=
k4
-
(e/c)2 [2A2k2 + 4(A.k) 2]
= (elc) 2[LA2k2 + 4A(2) K(2)
(e/c) 2 (A-B)(A-o)
= (elc)2[A2(B-o) + A(2).(Bo)
(k.B)(k.a)
(e2/c)(ExA) -a
etcetera.
(2
)
(H.17)
Next make the replacements
A
A
2A2
(e/c)2A2
-
2(el e*)
(e/c)2A (2)
-
2(e e)(2)
1 2
(e2/c)(ExA)
2i
(e 2/c)(EA)
,
i(O
-
e e2
The bilinear expressions may also be written as
(ere )(EA)).
where 0 - (1/2)(01+02).
(e2/c)(EA.
I ]
(H.18)
(elxe*)
---
The bilinear expressions may also be written as
i( 01.2)(e .e2 +e .e ).
.A
J i)
11 2]
Notice that the factor (co -
0 2)
1J 21
in the symmetric
proportional to the Raman transition energy ef,
product of E with A is
which is of order k 2 (or Bo0)
for intraband transitions.
Results of first-order perturbation theory in A 2 and EA:
Ac(m*) = (m/m*)(e,1 e*)l
20
A (K )
Ac (
o)
2iCK (elxe*) -a
= 28[(e -e)k
2
+ 4(e e*)(2)-K(2)
(H.19)
Ac(g') = (e/c)g'(e e2*)(a.B)
Ac(g")
Ar
A
) + A )(2)
2(e/c)g"[(e
)(02 -B
I-[4k
- {k,o.k}
Ac(a 2) = ioa2(e 1xe*)
2
p (e xe*) -B
Ac(b 2) = 2iwb2B
I 2
0
297
0(2)
Anisotropic terms (using a-e and b-e*):
Ac(ao) = - 12a[abkk](4)
Ac(y o) = (e/c)[yoabBo](
4
)
Ac(K 1) = 4i0K [(axb)kk ](c) + o(k4)
(H.20)
Ac(K ) = - i@K2 [(axb)kkoa](4 + o(k4 )
A (8o) = 280[(axb - ayby)kzY
=
-(axb
+ ayb)(kxo)
+ c.p.]
238 [[(ab)(2)(ko)(2)](3' + i[(ab)(2)(kxo)](3)]
-
I have neglected the contributions of K and K which are of order k 3, due to
the
factor
computed,
(
in
1
- 02).
analogy
Additional
to
contributions
the case
due
of anisotropic
similarly for the contribution due to g".)
to
i[Aco,S 1]
magnetoabsorption.
must
be
(And
The factor of i in the last term of
Ac(8) occurs because (kxo) =(il2)(k a_ - k_ ).
Second order perturbation theory in
(e/m*c)(A-k) + eK E -(kxa):
In the second-order terms in Ascat, make the replacements
A
e1 v
A
-
A
e*-v -2
e [(k/m*) + io K (kxo) ,
(H.21)
A
e* -[(k/m*) - i) K (kxo)]
2
Neglect Ero and E
2
S
in the energy denominators, relative to hw 1 or h'c 2, since
the intra-cb energy differences are of order k 2.
With these approximations we have
1
A
^
^
Ac(2nd order p.t.) - - 1^
(e*.v)(e *v) - (e -v)(e* -v) .
298
(H.22)
Use the following expressions to simplify the results, where R-kx:
(0)
a.
= 3(1)
(1)
Cyxo = 2io
(1)
kxk = - i(ec)Bo
(0)
k-R - R-k = - 2i(elc)(B- 0)
(H.23)
k-R + R-k = 0
(1)
kxR - Rxk = {(k-&),k} - 20k2
(H.24)
kxR + Rxk = -i(e/c)(B xy)
(2)
[kR] ( 2 ) - [Rk](2)= i(e/c)[BG](2)
( 2)
[kR] (2 1 + [Rk](2) = - 26i[K(2)C]
(0)
RR = 2k 2 1
(1)
RxR = i[{(k-&),k} - (e/c)Bo0y]
(2)
[RR](2) = - K(2)1,
+ (elc)(B -
+
y)
(elc)[B0
(H.25)
](2)
The number in parentheses specifies the tensor rank of each expression.
Note
that a factor of the Bohr radius squared (a0 ) must be restored to give R
dimensions of length.
Second-order perturbation theory results, to order k 2:
A (2nd order p.t.) = (i/o)(e/c)(m/m*)2 (elxe*) Bol
+ 2(elc)(mlm*)K[(2/3)(e . e*)(Bo)
S
-
+ iK(e
A
1
A
2
0
e*) - [(e/c)B ol - {k,k-}]
S 1 2
0 G
299
-
(e e*)( 2). (B
1 2
0
)(2)]
(H.26)
Summary
of
first
and
second-order
perturbation
theory
axial
terms
(see
Table (H.1)):
C
= (m/m*)(e 1 .e*)1
2
+ (a)(e e*)k21
1
A
A
A
A
A
2
+ 2ioKS1(e xe*)
-a
2
G
+ i(b)
(e xe*) -B 1
Cl
2
0
+ (c)(e e*)( 2). K(2)l
12
+ (d) -(e .e*)(.B) - i(e)(e xe*). k2 - i(f)(e 12xe*).{k, -k}
C
+ (g)
1
2
1
0
e(ele*)(2 ). (B
0
2
(2)
(H.27)
where
(a) = (20/3)E
(b) = (1/l)(m/m*)2 + C(K2 + 2b)
(c) = 8
°
(H.28)
(d) = (4/3)(m/m*)K + g' + (2/3)g"
(e) = - 4ca
(f) =
2
(KS + a2)
2
(g) = - 2(m/m*)K s + 2g"
If ) in the zero-order term is replaced by co
then an additional term occurs to
order k2:
eA
A
(h) (exe*) -(xB) ,
(H.29)
(h) = (1/2)g*K
(H.30)
with
.
300
Table H.1 Contribution of axial terms in the decoupled cb Hamiltonian Hc to
the scattering amplitude operator Ac .
Component of A c
Terms in Hc (2nd order)
Term in He (1st order)
0
(m/m*)A2
1
0
KS ExA o
Co
0
E (A-k)2
(e 1 e )
A
A
- i(elxe*) c
1
2
^A
(a) (el1 e*)k
2
A
2
A
m
2
0
A.k), Ks(E R)
A
(d) (e .e2)(B o.)
A
A
(e) - i(e xe*)ok
A
{k,k-a}
(e
A
)(2) - (B
)(2)
eo(A k) 2
1
'A2 (Bo.)
1
and H(g")
o
Ks(E-R), Ks(E-R)
a2(A -k)(E R)
Co
-(A
k), Ks(E R)
A
R
O
a2 (A-k)(E-R)
2g"A ( 2) (B
1
(2)
1
KExA- a
0
(h) (e xe*). (Boxo)
where
b2 ExA -BB
0
A
(f) -i(elxe*)
(g)
2
/o
0
A k)
Ks(E-R), Ks(E -R)
Ae)(2). K(2)
1
0
k), m(A
-- (A k),
(b) i(exe) -B
(c) (
n
a2Okxo , and a 0 is the Bohr radius.
The anisotropic
terms in Ac
are given in
Eqs.(H.20).
There are
contributions from second-order perturbation theory to lowest order in k.
The
part of A c which contributes to [Ac, iS1], to lowest order in k, is
SA
A
-
(H.31)
A = 2ioK (e xe*)).
S 1 2
0
301
In order to evaluate [Ao,iS]fo
0 fo one may use the relations
H ]fo
[a+,iS fo = [O
E +hw
fo
s
,islfo
.J
[,iSlfo
[[ H ]fo
E - n1
fo
(H.32)
[a ,H]fo
E
Efo
These are computed from first-order perturbation theory in H 1, the anisotropic
part of the decoupled cb Hamiltonian.
302
H.2
Computation of A scat for finite hoI 1 from coupled Pidgeon and Brown
Hamiltonian
To find the effective scattering operator, we proceed by first expanding
the multiband wave function for each band in a k-p expansion applied to the
The multiband effective Hamiltonian is then used to
decoupled wave function.
If the PB model
evaluate optical matrix elements between the different bands.
is used, we have
A m
2 tv)fr(
e CV
> ro
[(Y
(f
*-v e
=4m
A
fo
2
-
Ofr
2
2
E
fr
hO)
ro
I
r
The
V
hW -E
r)
o
(e*.v)_1
(eVfr
+ (e Vk)(e2 *-V ef
1k 2
k)Heff
evaluation of A scat to zero order in k is
(H.33)
to the
simple relative
A
higher-order
The
cases.
operator
occurs
in
sums
the
over
A
A
A
which
e v,
intermediate levels with e replaced by e or e*, is given by
0
e- v = (l/m)
A
A
e'pcv e*pcs
.
0
(H.34)
.Pv
0
0
SPsc
The momentum matrices are
pc
It
is
=
Pvc= Pt
Pt
usual
approximation
to
set
Ps
is
P'=P, which
[Weiler 78].
The
P
P'
2x4
=
consistent
matrices t ,
(H.35)
P't'.
with
the
'single-group'
ty and t z are
given
t'y and t'z are given by t' = (1/V3)p
Appendix A, and the 2x2 matrices t',
x
where px
in
,
Py and pz are numerically equal to the Pauli spin matrices, but the
vector p is polar instead of axial [Trebin 79].
The
zero-order
(in
k)
initial
and
303
final
wave
functions
for
intra-cb
scattering are
FcoI
V =
c
[O
and
,
[v]
(H.36)
where
co and XCf are the decoupled initial and final state wave functions,
Oc
and Os are 2-components zero vectors (or 'spinors'), and 0v is the 4-component
zero vector.
The intermediate states in the light-hole heavy-hole valence band
(vb) and split-off valence band (sb) are
r = Xv
Os
(H.37)
Ov
r
and
,
Xsr
We substitute these wave functions into Eq. (H.33) for A
The sum over
intermediate states is accomplished by neglecting the k-dependence
(or B
and
k dependence) of the energy denominators, and using the completeness relations
for the two-component and four-component wave functions in the intermediate
A
bands.
A
We find, neglecting (e Vk)(e2 V )Hef ,
A
fo
= (1/m)
<XC
c
C()
x(1
NO - E
2I
OI -E
2
X >
-CO
I
(H.38)
where o may be w , (02 or an average, since the difference is of order k 2 .
operator part of this expression,
The
which acts between XCf and Xco, may be
written as
A
c
P
m
+
fr
hO - Eg
ro
2 fr( I
h + E,
P2 2
m
(e*-_t' fe
t')fr(
he +
')r
'
I
(e
-
t')fr(
ro
e*')
30hO - E'
304
')ro
1
(H.39)
This may be simplified by using the algebra of the t. and t' matrices (Appendix
A), along with the recoupling identity
(a.p)(b.q) = (1/3)(a-b)(p-q) + (1/2)(axb).(pxq) + (ab)(2).(pq
)
,
(H.40)
which is valid if the components of a and b are ordinary numbers, or more
generally
if
the
components
of
b
commute
with
those
of
p.
After
simplification, and inclusion of (e- V )(e* -V )H, we obtain
I k 2 k CC
A=
2F+ 3--m+
()
+
El 92(w)
el'e
1o
11
- iho-
where
and E'
(0)
Eg+A.
used by Makarov
9 ((o)=l
1
(PE(o) -
2E'22 (
Eg 2
(o)
(ho))2
-
a
xe)
.
E2and
-
2
()
(H.41)
2
(H.42)
The definition of the resonant function (p2(o) differs from that
[Makarov 68].
and (p2(o)=
when
-=0.
I have defined the functions
The constant (1+2F)
associated with free mass and higher-band terms in H
[Weiler 78, Trebin 79].
so that both
in the first term is
of the extended PB model
When co-)0 we obtain
A c = (m/m*)(e.e*) 1
+ 2ioK(e xe*)
X
as was obtained previously from the decoupled Hamiltonian He.
305
(H.43)
A
scat
to second order in k:
We now consider the second-order dependence of Ac on k. For this we must
compute the k2 dependence of the wave functions, interaction Hamiltonian and
energies, and must include conduction-band as well as valence band intermediate
states.
F wave-functions:
6
12
1 KK
cv vc
E2 2
K
1 KK
cs sc
E'2
vc
(H.44)
Eg
K
sc
E'
F8 wave-functions:
F7 wave-functions:
KKcv
K
cs
E'
K K
VC
xv
CS
E'A
K K
SC CV
EgA
1
2
306
sc cs
E "2
xss
(H.45)
Here,
K
Pt -k,
K
P't'-k,
K
Pt k,
K
-P't'-k
,
(H.46)
and we again assume P'=P.
Next use these wave functions to compute the optical matrix elements of
the A -v optical interaction and substitute into the expression for Afo
Additional dependence of A
on k comes from the energy denominators in
This dependence becomes much stronger, near resonance,
the expression for Af.
than the dependence of the wave function components on k.
In evaluating A
we
must consider terms like
AB
fr ro
hO) - E
1 ro
and
AB
fr ro
ho - E
1 fr
(H.47)
These may be expanded to order k 2 as follows.
First we write the denominators
in the more symmetrical form
hO -E ro = h
+ E + E - E
r
2f
20
(H.48)
ho -E
- (E
2f
(H.49)
So
and
fr
= h
+
2 0
Er) .
We then separate each energy on the right into a band-edge energy plus an
energy
relative to the band edge.
The latter energy
will be obtained
allowing the decoupled Hamiltonian to act on a decoupled wave function.
the case of initial and final states in the cb and intermediate
vb, we write
307
by
For
states in the
(hf 1 - Ero)
=(h
+ Eg +
EC + E
- E
(H.50)
,
where the superscripts c and v indicate that the energies
respect to the band edges.
We then expand the right hand side in powers of
(Ec + Eo Er)/(h(i
+ Eg),
(It
a term
also
contains
considered.)
noting that the numerator is of order k2 and higher.
of order
k
if the
linear-k
term
in the
vb
is
To first order in the k-dependent energies we have
(hI- E )
1
are measured with
ro
=
1
hw + Eg
2f
+
+1Ec
+Ec
2 0
fr ro
r
(H.51)
+ Eg) 2
(h
Next we multiply this by A B
- Ev
and sum over intermediate states r in the
valence band. In doing this, we make the replacements
c
= Ho
co = Hccf '
ErXcr = HvXcr
(H.52)
where the expressions on the right involve the decoupled Hamiltonians and wave
functions. Note that
SAfBro
(H.53)
r
depends on k because of the k -p expansion of the multiband wave functions, and
that
AfrBr(Ec + Eo - E )
(H.54)
r
depends on k because of the k-dependence of the energies.
To lowest order in
k, for r in the vb, and neglecting the k-dependence of the wave functions, we
have
A B
fr
(Ec + E c
ro
f
o
E) = <X
r
cf
2
HA
C
CV
B
VC
+ !A
2
B H -A
CV VC
C
CV
H B]
V VC
I
>
r
(H.55)
308
Here
A cv - P(a t) ,
A
A
A
(H.56)
P(b.tt)
B VC
an d
A
where a and b are e 1 and e*2 or e*2 and e 1
A
B
ro
'f
A similar evaluation is used for
h(o -E
fr
r
To second order in k we find
A
scat
1
2P2 2
3m Eg
- ihO
E(
3m
A
(
^^
2
) (e *•e*2
2
1
i-p2
Eg
2
A
eA
A
A2)
2 + i(b)(e xe .B + (c)(e e )(2)
K(2)
+ (a)(e -e*)k
0012
2
1
1 2C
After
the
A-
+ (d)
eA
+ (g)
£(e e*)(2).
(e e* )(
C
12
traditional
A
A
B ) - i(e)(e xe*)
2
A
A
k - i(f)(e xe*) -{k,
(H.57)
k}
(2)
0
"straightforward
but
tedious"
algebraic
manipulations,
using h=m=l and E-E', I have found the coefficients [compare Makarov 68]
309
(a)
(a)
7
20
4P4
9
+I]i+
3+
] 2
3E,2
t
(H.58)
7
5 92
+
3EE
3E E
8
4P4
(c)
9
1+
+ .0t
+-
E
14
4
(d) 2P 4
9 L .3ECa
2
3
02
2
5
E,
E,
i2 +
Eg
4
3E A
3EgE
1 2022
[ g+
+ -
+ -
E E4
2
3 (P2
E
A E2
2 4 1
E Eg4
3E,
(H.59)
2
(H.60)
+
[1
E2
(g)
=
2P4
9
(b)
-
m
(0
(m*)2
+
4
3EA
i +2+ 0
+ 1-E291
E,5
4
+ -1
E
E
1
7
3EE
2
P4 9
+6
9
EE
E2
2
+
2 (P
E
12 E,
2
+
2
E,
3E2 E
5
2
+
1
r22
3E E4
92
Eg
E
2
E2 Eg 2
E
6
4
E
E
922] (H.61)
2
(p 1
(H.62)
+ f2
E2
6
EE
2 (P2
EAI E
310
4
9221
E E3J
2P4
(e) = -
3
4
L
Eg2
9
1
EgEEgA
+
8
Eg
1 9 +
E2 JEg2
-
2 P12
-EJE 3
(H.63)
P4
(f)0=
1
9[
g
+
5
E
1 + 2 2
2
3
E92
EgE
3(
1
3
AEg
6
PP
EA
912
Eg4
42+4 4 (p22
E E3
E
1
2 3P2
- + -+
Eg
E AE 3
(H.64)
In the coefficients, o may be taken to be CoI or @, since the difference affects
terms of order k4 and higher in A c.
If 5 is replaced by Co
in the zero-order
it introduces an extra term of order k2. This term is
part of A
e
A
(h) -(e xe*) -(xB)
Cl
1
2
(H.65)
0
where
(h) = - g* 3)-
2E'2
Eg4 ( I
()2() -
E'4(P2]
(0)
(H.66)
(H.66)
with
g* =
4P2
I
I
3m
E
E'
(H.67)
.
Use of coI instead of 5 avoids the complication of the dependence of 6 on B 0, at
the expense of an extra term in A c to order k2. In the limit of small
expression for (h) approaches that of Eq.(H.30).
311
Co
the
For the anisotropic nonaxial part of A
to order k 2 in the PB model, one
must consider four contributions to the anisotropic part of Ascat
1) Anisotropic component of v in
2nd order terms.
2) Anisotropic component of energy in denominators of 2nd order terms.
3) Anisotropic component of VkVkHeff in 1st order term.
4) Anisotropic component of multicomponent wave functions.
The warping contribution to A scat for conduction electrons due to H (g) and
Hvs(g' ) to first order in g and g' has the form
c ()[abkk]
+ c (co)(e/c)[abBoa]4) + ic (o)[(axb)kkoa (4)
(4)
(H.68)
In the off-resonance, small-o limit, one finds
= 24
I
P 2-g 1 + 2
E E
E'
- = 12
,
c = 4
P2
3
Eg Eg2
(H.69)
1
EgE'
3]
E'2J
This may be compared to the result obtained from the decoupled cb Hamiltonian
H to order k2:
C
Ac(warping) = - 120o[abkk]( 4 )1
+ (elc)o[abBo](4)
+ ico(4K - K )[(axb)kko]] 4)
The parameters ao
(H.70)
and o of the decoupled model are expressed in terms of g,
and q in Appendix G.
312
'
The inversion asymmetry contribution to Aset due to H (G), H (G'), H (C')
and
the
smaller
components
HCv(N2)
and
H v(C),
to
first
order
in
the
parameters, has the form
d ()[(axbx - ayb )k
+ c.p.] + d2()[(axby
z
In the off-resonance, small-o, limit one finds
+ ab )(kxc)) + c.p.]
(H.71)
d= -d = 28 , so
A sct(asym.) = 28~[((abx - ab )kz a + c.p.)
satam
xx
y yz
- ((ab +ab )(kx)
+ c.p.)
(H.72)
The parameter 8 of the decoupled model is expressed in terms of G, G', C, C'
and N in Appendix G.
Light scattering from holes:
The scattering from free carriers in the valence bands may be approached
in the same fashion as
Auyang 72,74]
conduction band.
light
scattering
from
carriers
in
the
cb.
[See
To order ko we only need to consider intermediate states in the
For scattering within the j=3/2 valence band to order ko
obtain both At0O and A~-2 Raman transitions at kz=0, which include
we
spin-flip,
spin conserving, and heavy to light or light to heavy hole transitions.
This
contrasts with the case of scattering from electrons, in which the A-=2
Raman
processes occurred only to order k2 and higher, and n is an approximately good
quantum number.
We also find
scattering
which
produces
between the two spin-split valence bands [Auyang 72,74].
313
Raman
transitions
For scattering from holes in the
A
v
=
(e*-v)
2
(l/m)
L
*
(e v)
(e* -v)
I
cV
VCr I
vb, we find
j=3/2
VU
(e* -v)
O
2
+ (el'Vk)(e* Vk)H
I
ho + E W
hco -E
k
2
k
VV
Ecv
in
in cb
(H.73)
which simplifies to
v
A2
3
3m
E
j
0
+ ( AA)(2) J(2)
-
)
(
1
2P 2 1
+ ihO 3m g 2 P
3m Eg2
t(e e)1
J
+
2(ele2
.
A
2
J )
(AA
(E
el e 2J)
1 2
) (2)
j(2)
(H.74)
-
(2)(2 ) (2) (4)
4
Here [Littler 83]
2P2
3mE,,
32.9
g = 0.55 .
y = 0.46,
y, = 3.25 ,
The last three terms in A v , obtained from the PB model, are nonresonant.
(H.75)
In a
more exact treatment they would contain resonances at interband energies higher
than Eg and E'. For o -~O we obtain
AV
IL
(e -e*)
1
2
-
+ 2(e e) )(2)2P(2) + ii
L 1 2
4(e
12
)(2)
(2) (4)
C
2
3mEg
g
(
1
J)
2
(H.76)
,
which may be obtained from the decoupled Hamiltonian H if an effective spinorbit term (P 2/3mE2)(eE.kxJ) is included.
314
Here
YIL =(2P2 /3mEg) + YI
(H.77)
YL (P
2/3mE,)
It is significant
+
.
that
spin-orbit interaction
terms,
proportional
to E
in
the
PB Hamiltonian and the completely decoupled Hamiltonians, contribute negligibly
to
the
transition
energy
of
bound
amplitudes.
states
but
contribute
significantly
(But see [Jusserand 95] for
case of asymmetric quantum wells.)
spin-orbit
of large
Additional
theory [Evtuhov 62].
i[A , S1].
matrices,
the
Note that the contribution of the warping
scattering results
from
warping on the decoupled valence band wave functions.
diagonalization
optical
effects in
parameter g. to the scattering amplitude is of order ko, but that
smaller than y7 L and YL.
to
and
cannot
be
. is much
the effect of
This requires the
computed
by
perturbation
So the contribution cannot be expressed in the
form
Approximate results may be obtained by considering pairs of axial
model levels which interact strongly via the warping terms in H v [Pidgeon 69].
For initial
states
in
the
sb,
final states
in the vb,
and
intermediate
states in the cb we find
(
vs = (l
V)v(e
v)
v)cs
(e *-v) (e*
-v)s
ho - E-
hCo + E
1
)(eV
k 2
k)Hvs
in cb
(H.78)
which simplifies to
315
p2
AvS -3m
1
1
hC -E'
-
ho,+ EgJ
iP2
(2),.U G,(2)
2
AA
11
i2hm
1
+
1
(exe* -u )
2
hao)+ E
2v4m hl - E'
2V3( ee(
-
2
). (UTG)(2)
+
[(
4v
(H.79)
A*)(2)®(U)(2) (4)
This may also be expressed in the form (less convenient for computation)
-2
2p2
2P
As
vsmE
+ 4
~[(A
(e
e1 2 )(2)u
+ 2
3()
1
AA
^ u)
2 + VmE E -(C) e 1xe*
2
2)((U()(2)(
(H.80)
where (setting h=1),
Tp()
(H.81)
,
=
with
o2
+
1
2
)C0=
1
+
2
(H.82)
A ,
and
E=
2
(E'+Eg) = Eg + '.
2
(H.83)
Two-photon absorption operators may be found in an identical fashion. The
operator Avc for two-photon
interband transitions between
first order in k (to lowest order).
the vb and cb is
The one-photon transitions are of order ko
(= 1).
316
ACKNOWLEDGEMENTS
I am grateful to Prof. Don Heiman, now at Northeastern University, for being
the kind of advisor I wanted to deserve. He has, in fact, been much better
than anyone deserves.
Perhaps I could have finished this work without him,
but I can't quite imagine how. I appreciate his example of organization and
hard work, his appreciation of things both physics and non-physics (most
significantly art), for his caring, his time, and his determination that I
should finish this thesis.
I am grateful to Dr. Hector Jim6nez-Gonzilez for
being
a positive
influence
over the
past ten years. His
cheerfulness,
and
eloquence, and enthusiasm for life and for science have been uplifting, and
have often pulled me up from depression. He has been a great help to me in
discussions of magneto-optics, and in setting up the experiment on the
field-reversal effect.
I appreciate his sharing his family with me, many good
times, and much good advice, scientific and otherwise.
I am grateful to
Margaret O'Meara for her friendship, advice, and helping to keep things at MIT
in perspective.
Also for lunches, Fourth of July outings, apartment moves,
advice on plants and letter-writing, horseback riding, encouragement to focus
and to "work, work, work."
I am grateful to Dr. Oliver Shih for many
wonderful discussions about irreducible spherical tensors, the Luttinger
model, and magneto-optics, and for all those late-nighters in 1991-92 when he
and Hector and I were the Three Musketeers, struggling together to complete
our theses.
(Mine just took a little longer.)
I am grateful to Dr. Hide
Okamura for all the friendship and encouragement he has provided, and for
significant help in setting up the field reversal experiment.
He set a great
example of diligence for me to try to follow.
I am grateful to him for
discussions about physics, and for such finer things as music at the Boston
Symphony and dinner at Dixie Kitchen.
I appreciate the encouragement from,
and interesting discussions about magneto-optics in quantum wells with Dr.
Lushalan Liao, along with some memorable outings for horseback riding and
chinese cooking. I am grateful to Dr. Sam Sprunt for keeping things lively on
the
fourth
floor
viruses, prions,
with
discussions
about
and many other interesting
physics,
light
scattering,
topics.
I appreciated
killer
having a
friend at the lab who had shared the "Williams experience."
Sam, along with
others, like Drs. George Nounesis, Manfred Dahl, Ulf Gennser, Xichun Liu, Alex
Fung, Flavio Plentz, and doctoral students Zachary Lee and David Rovnyak,
317
helped make life at the lab much more interesting and bearable.
I thank
Zachary for discussions of physics, and for donation of some drafting
software, and David for discussions of colleges, computers and magnetic
resonance.
I am grateful to Dr. Roshan Aggarwal and Prof. Benjamin Lax for suggesting the
early experiments on magneto-optics and spin-flip Raman scattering. To Roshy
for teaching me the experimental techniques and making me focus on physical
concepts instead of mathematical techniques, and to Ben for teaching me about
quantum electronics, and for supporting my research at the Magnet Lab for five
years.
Also to Dr. Margaret Weiler for teaching me about the Pidgeon and
Brown model, and for sharing her computational techniques and her unpublished
work on magneto-optics and SFR scattering.
I am particularly grateful to
Prof. Peter Wolff for his stimulating course on semiconductors which
introduced me to the k-dot-p method for band structure calculation, following
the
approach
of Kane, Yafet, Luttinger and Blount, and for agreeing to
co-supervise my work, with Don. I am grateful to Prof. Mildred Dresselhaus
for teaching me group theory, for reading and making useful suggestions
regarding this thesis, and for taking an interest in my research and my
career.
I am grateful to Prof. Robert Griffin, who directs this Lab, for his
friendship and interest over many years, and his hospitality in providing me
with office space, particularly during the past three years.
I am also grateful:
To the staff of the Francis Bitter National Magnet Lab for significant help
over the years. To Larry Rubin for scheduling, for help with instrumentation,
for interesting discussions about the physics of levitation and other topics,
and for bringing to my attention a wonderful address about "The University"
given by A. Bart Giamatti at the 1988 MIT commencement.
To Larry Sousa for assisting with my early experiments and sample preparation.
For his enthusiasm, joie de vivre, and the positive example he set for me and
everyone around him.
To Dr. Thang Vu for discussions about physics and computers, and for helping
set up the data collection in my recent experiments.
To Jim Coffin, Jean Morrison, Nancy Tucker, Ed McNiff, John Chandnoit and Paul
Emery for help with scheduling and apparatus.
318
To Dr. Simon Foner for his interest, concern, and numerous discussions.
To Irene Ferriabough (Mother Superior) and Aldona Shumway for their help and
cheerfulness.
To Dr. Emanuel Bobrov for his Russian translations and lore, and in particular
for his translation of the article, in Russian, by Sheka and
To Dr. Georgia Papaefthymiou for her encouragement
discussions about M6issbauer spectroscopy.
To Dr. Yaacov Shapira for his interest, advice,
conversations, both in physics and non-physics, over many
Zaslavskaya.
and good advice, and
and
many
interesting
years.
To Dr. Jagadeesh Moodera for many discussions, and for a shared interest in
physics teaching.
To Dr. L. R. Ram-Mohan for a discussion about band decoupling techniques.
To John Morris of Infrared Labs in Tucson, for lending me a KRS-5 window for
the germanium bolometer.
To Peggy Berkovitz and Pat Solakoff of the Physics Department for their help
this past summer, and in previous years.
To Prof. Tom Greytak for approving my readmission to MIT.
To my friends at Tulane: Profs. Alan Goodman, John Perdew and Mel Levy who
encouraged my pursuit of physics, and to Prof. George Rosensteel, who
stimulated my interest in algebra.
To my former teachers: Dr. Roy Ellzey, Prof. Fielding Brown and Prof. David
Park, who set wonderful examples of concern, enthusiasm and accomplishment.
Of the people at home, I am most grateful to Louise Pipes Byrne for being my
It was largely her encouragement that
best friend, encourager and advisor.
nudged me back into physics. She has a heart of gold, and has been kind to
share her life, her family and her concern. She has helped me in more ways
than I can describe, or ever repay. My thanks also to Dr. Christian Byrne,
and to my wonderful friend, Gifford.
I am grateful to my sisters Bev and Joan for their encouragement, love and
support; for being so nice and for sharing so much.
Also to my cousins Michael Richardson, Sean, and especially Mary, for their
patience, moral support, perspective on life, King Cakes, and best of times
This work is dedicated to them (and also to
when we've been together.
others).
319
To my friends the Woodwards, to Barbara and Robert, who have been so much a
part of my life both in Boston and New Orleans.
I thank them for sharing
their home, where I lived for four months, their children, their advice and
support over most of the past 30 years. Special thanks to Laura for caring so
much, and for sharing her friends and many good times. She helped me maintain
some finite level of sanity. And thanks to Bo, Ann and Peter for all their
patience, good wishes, moral support, and great times too numerous to count.
I am grateful to my friend Dr. Annette Ten Elshof for caring and taking time
to help and advise; for telling me I could finish, and for many good times in
New Orleans, Covington and Denver.
I especially appreciate her remark about
the importance of friends in completing a Ph.D. thesis.
My observations at
MIT seemed to teach me the opposite, but I think she was right after all.
To Nelva Van Sickle and her family for maintaining Annette's concern, for her
good advice and encouragement, and many more fun times in Denver and N.O. She
is among the best people I know.
I am grateful to Dr. Ali Ahmadi for his friendship, encouragement and prayers,
and
for his diligence
and faithfulness
in completing
his thesis years
ago,
setting a example for me to follow.
To the Cooches, Dorans, Flowers, Montgomeries, Ruddocks, Tekippes, Ronald
Woodwards, and many others who have cared.
To Karen Seeley and Courtney; to Gail and Joe Richardson; to Renee and Marthe
Mann, and cousins too numerous to mention.
To Aunt Betty and Miggy and Chris and Paulette and their families.
To my God-children, for their patience and encouragement:
Neil,
Peter,
Richard, Michael, Sean, Audrey; and Gifford Glenny (Pipes) Byrne.
To my parents, for believing in me and supporting me in my education, my
research and my life; and for their great love.
To God for providing me with family and friends, and more; for the "assurance
of things hoped for, the conviction of things not seen."
I know this work is not worthy of them, but I am grateful, and it is dedicated
to all of them.
.,.
320