Theoretical Methods for Spintronics in S

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Theoretical Methods for Spintronics in Semiconductors with Applications
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Cartoixà Soler, Xavier
(2003)
Theoretical Methods for Spintronics in Semiconductors with Applications.
Dissertation (Ph.D.), California Institute of Technology.
doi:10.7907/0YTD-VC11.
Abstract
Theoretical studies of the role of structural inversion symmetry (SIA) and bulk inversion symmetry (BIA) in the band structure and tunneling properties of zincblende heterostructures have been carried out.
The effective bond orbital model (EBOM) method is used to examine the spin splitting due to SIA in AlSb/InAs/GaSb asymmetric heterostructures. It is found for the resulting two-dimensional electron gas (2DEG) that large theoretical values of the Rashba coefficient in the range of 50E-10 eV.cm can be achieved for optimized structures. Structures presenting anticrossing of the conduction and valence bands show an appreciable reduction in the value of the Rashba coefficient. The possibility of extracting the Rashba coefficient from magnetization measurements is explored. An expression is derived, valid in the diffusive limit, for the spin polarization of the current resulting from a bias parallel to the plane of the quantum well.
The EBOM method is expanded to include BIA effects. The resulting formalism is then used to compute the band structure of an AlSb/GaSb superlattice, where the BIA-induced splitting is observed. The results agree with k.p calculations.
The first implementation of an 8-band Envelope Function Approximation method faithful to the T
symmetry of bulk zincblendes has been made. It has been used to compute the bands for quantum wells with and without BIA effects included, and demonstrates that the BIA effects can be of the same order of magnitude as SIA (i.e., Rashba) effects. A 2-band Hamiltonian describing BIA effects is proposed. The origin of spurious solutions for certain values of the input parameters is determined and a condition for its absence is derived. Modest modifications to the superlattice method allow the computation of spin-dependent transmission coefficients with the multiband quantum transmitting boundary method (MQTBM). The effect of BIA on the transmitted states and the spin filtering action of an asymmetric resonant interband tunneling diode are investigated.
Finally, a Monte Carlo single photon generation algorithm is devised. The photons generated are satisfactory for simulation of light emitted from band-to-band spontaneous transitions in crystals. The polarization is determined taking into account the electron spin, making the algorithm suitable for the analysis of optical detection of spin injection experiments.
Item Type:
Thesis (Dissertation (Ph.D.))
Subject Keywords:
effective mass approximation; quantum transport; resonant tunneling; spin injection
Degree Grantor:
California Institute of Technology
Division:
Engineering and Applied Science
Major Option:
Applied Physics
Thesis Availability:
Public (worldwide access)
Research Advisor(s):
McGill, Thomas C.
Thesis Committee:
McGill, Thomas C. (chair)
Hunter, A.
Yariv, Amnon
Ting, David Z.
Vahala, Kerry J.
Bockrath, Marc William
Defense Date:
18 September 2002
Record Number:
CaltechETD:etd-05232003-104331
Persistent URL:
DOI:
10.7907/0YTD-VC11
ORCID:
Author
ORCID
Cartoixà Soler, Xavier
0000-0003-1905-5979
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No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:
1967
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CaltechTHESIS
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Deposited On:
23 May 2003
Last Modified:
03 May 2021 21:29
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Theoretical Methods for Spintronics in
Semiconductors with Applications

Thesis by

Xavier Cartoixà Soler
In Partial Fulfillment of the Requirements
for the Degree of
Doctor of Philosophy

California Institute of Technology
Pasadena, California

2003
(Submitted September 18, 2002)

c 2003
Xavier Cartoixà Soler

iii

Als meus pares

List of Publications
Work related to this thesis has been or will be presented in the following publications:
[1] Theoretical investigations of spin splittings in asymmetric AlSb/InAs/
GaSb heterostructures and the possibility of electric field induced magnetization,
X. Cartoixà, D. Z.-Y. Ting, E. S. Daniel, and T. C. McGill, Superlatt. Microstruct., 30 (6), pp. 309–319, 2001.
[2] Spin injection from ferromagnetic metals into gallium nitride,
C. J. Hill, X. Cartoixà, R. A. Beach, D. L. Smith, and T. C. McGill, condmat/0010058, 2001.
[3] Theoretical investigations of spin splittings and optimization of the
Rashba coefficient in asymmetric AlSb/InAs/GaSb heterostructures,
X. Cartoixà, D. Z.-Y. Ting, and T. C. McGill, J. Comp. Elec., 2002, in press.
[4] Modeling spin-dependent transport in InAs/GaSb/AlSb resonant tunneling structures,
D. Z.-Y. Ting, X. Cartoixà, T. C. McGill, D. L. Smith, and J. N. Schulman, J.
Comp. Elec., 2002, in press.
[5] Rashba effect resonant tunneling spin filters,
David Z.-Y. Ting, Xavier Cartoixà, David H. Chow, Jeong S. Moon, Darryl L.
Smith, Thomas C. McGill, and Joel N. Schulman, IEEE Trans. Magn., 2002, to
be published.
[6] Spin filtering in asymmetric resonant interband tunneling diodes,
D. Z.-Y. Ting, X. Cartoixà, D. H. Chow, D. L. Smith, T. C. McGill, and J. N.
Schulman, in Proceedings of the 26th International Conference on Physics of
Semiconductors (ICPS-26), 2002, to be published.

[7] Resonant interband tunneling spin filter,
David Z.-Y. Ting and Xavier Cartoixà, Appl. Phys. Lett., 2002, to be published.
[8] Refraction effects in the emitted light of a spin LED,
S. R. Ichiriu, X. Cartoixà, and T. C. McGill, in preparation.
[9] Bulk inversion asymmetry effects in zincblende heterostructures,
X. Cartoixà, D. Z.-Y. Ting, and T. C. McGill, 2002, to be submitted.
[10] Description of bulk inversion asymmetry in the effective bond orbital
model,
X. Cartoixà, D. Z.-Y. Ting, and T. C. McGill, 2002, in preparation.
[11] Elimination of spurious solutions in finite difference method implementations of band structure calculation methods,
X. Cartoixà and T. C. McGill, 2002, in preparation.
Work in which the author has been involved, but not related to this thesis, has been
presented in the following publications:
[1] Coupled drift-diffusion/quantum transmitting boundary method simulations of thin oxide devices with specific application to a silicon based
tunnel switch diode,
E. S. Daniel, X. Cartoixà, W. R. Frensley, D. Z.-Y. Ting, and T. C. McGill, IEEE
Trans. Elec. Dev., 47 (5), pp. 1052–1060, 2000.
[2] Tunnel switch diode based an AlSb/GaSb heterojunctions,
X. C. Cheng, X. Cartoixà, M. A. Barton, C. J. Hill, and T. C. McGill, J. Appl.
Phys., 88 (11), pp. 6948–6950, 2000.

Acknowledgements
During my years pursuing the Ph.D. degree at Caltech, I have had the pleasure of
meeting many wonderful people. Without them, my stay here would have been far
less enjoyable. So, I want to write a few lines to thank the people who helped me
each in their own way all this time.
In Professor Thomas Conley McGill can be found a rare combination of scientific
insight, political skills and wit. This combination of talents has helped Tom attract
a group of extraordinary people around him and obtain enough capital resources to
satisfy their expensive tastes–in lab equipment, of course. Working with Tom not only
opened new avenues of thought in my scientific endeavors, but also provided me–and
the members of the group–with valuable lessons about the intricacies of running a
research operation as well as about life at large. He always protected his students from
the occasional storm, and understood that science should not be our only passion.
For this and more, I am thankful to him.
Also, I could not let the opportunity to thank David Z.-Y. Ting slip. It is largely
because of his guidance that I am able to write these lines today instead of a year
from now. Through regular collaboration and thought sharing with him, I have come
to appreciate his creativity and insight. He has always been there to listen to the
questions I posed him and, when he could not provide answers, I could count on him
for some sympathy. The development of some of his ideas constitute a good part of
this thesis.
I enjoyed lengthy lunch talks with Gerry Picus and his good humor. His wish to
keep up to date with my work was a motivation to express my ideas in a clear fashion,
and I sure appreciate his patience while trying to teach me some of his methodology
in my adventures as an experimentalist. I also enjoyed speaking with Ogden Marsh
and the late Jim McCaldin, who showed me that one can stay curious a whole life.
Joel Schulman and Darryl Smith stopped me from trying to revolutionize semiconduc-

vii
tor physics based on some considerations to which I had not given enough thought.
I am also grateful to Andy Hunter for letting me show unpublished Shubnikov-de
Haas measurements. This work would not have been possible without the support
from Larry Cooper, of the Office of Naval Research, and Stuart Wolf, of the Defense
Advanced Research Projects Agency.
The students in Tom’s group have been wonderful to work with. I will hardly
find anywhere else such companions, with whom I can discuss at length not only
science, but almost every imaginable topic. The older students: Erik Daniel, XiaoChang Cheng, Eric Piquette, Alicia Alonzo, Paul Bridger, Zvonimir Bandic and Joel
Jones always found time to guide me. Bob Beach, with his carefree, but not careless, approach to life infused us all with high spirits. Cory Hill was a loyal friend
and his relentless pursuit of knowledge was a good counterpart to my excessive inclination to handwaving. Matt Barton and Rob Goettler shared the beginnings with
me. Ed Preisler is one of the better hearted people I know. His knowledge of sports
probably can only be matched by Neal Oldham’s, who seems to know almost everything there is to know about materials science. The outgoing personality of Robert
Strittmatter served as a catalyst for a number of social events within the group. The
generosity of Stephan Ichiriu has showed up in many occasions, and we spent many
hours together–my role being mainly watch-and-learn–doing the system administration tasks. Although in the more rewarding universe of postdoctoral research, Justin
Brooke has become one more of us. The group’s administrative assistant, Tim Harris,
has been most helpful in shielding us from the bureaucracy that one must inevitably
face. To all of them I wish the best in their future endeavors.
Of course, there is life outside of the lab; and over my years at Caltech many
have been the friends with whom I shared experiences: Federico Spedalieri, Diego
Dugatkin, Γιώργoς Panatopoulos (don’t quote me on the spelling), Bjarne Bergheim,
Fok-Yan Leung, José Mumbru, Mariu Hernández, Alfredo Martı́nez, Javier González,
Mario Múnich, Pili Gainza, Seong-Min Kim, Gabriela Surpi, Álvaro González, Pedro
González, Martı́n Basch, Michela Muñoz, Enric Claverol, Maria Teresa Monserrat,
Rodrigo Quian, Guifré Vidal, Anna Fontcuberta, my friends at home. . . Thank you

viii
for all the good times!
Apart from their major role in my existing, none of this would be possible without
my parents, Albert and Mercè. They have always stood behind me, and supported
me in all my undertakings. Throughout the changing times I know their love stays
the same.
Finally, I must thank my girlfriend, Virginia. Her good spirits always lift me up
when I need it. I am really fortunate to be able to enjoy her love and companionship.
I will always be grateful for the many things she has sacrificed during my stay here.
I hope I will be able to make it up to her.
Gràcies.

Abstract
Theoretical studies of the role of structural inversion symmetry (SIA) and bulk inversion symmetry (BIA) in the band structure and tunneling properties of zincblende
heterostructures have been carried out.
The effective bond orbital model (EBOM) method is used to examine the spin
splitting due to SIA in AlSb/InAs/GaSb asymmetric heterostructures. It is found
for the resulting two-dimensional electron gas (2DEG) that large theoretical values of
the Rashba coefficient in the range of 50×10−10 eV·cm can be achieved for optimized
structures. Structures presenting anticrossing of the conduction and valence bands
show an appreciable reduction in the value of the Rashba coefficient. The possibility
of extracting the Rashba coefficient from magnetization measurements is explored.
An expression is derived, valid in the diffusive limit, for the spin polarization of the
current resulting from a bias parallel to the plane of the quantum well.
The EBOM method is expanded to include BIA effects. The resulting formalism
is then used to compute the band structure of an AlSb/GaSb superlattice, where the
BIA-induced splitting is observed. The results agree with k · p calculations.
The first implementation of an 8-band Envelope Function Approximation method
faithful to the Td symmetry of bulk zincblendes has been made. It has been used to
compute the bands for quantum wells with and without BIA effects included, and
demonstrates that the BIA effects can be of the same order of magnitude as SIA
(i.e., Rashba) effects. A 2-band Hamiltonian describing BIA effects is proposed. The
origin of spurious solutions for certain values of the input parameters is determined
and a condition for its absence is derived. Modest modifications to the superlattice
method allow the computation of spin-dependent transmission coefficients with the
multiband quantum transmitting boundary method (MQTBM). The effect of BIA
on the transmitted states and the spin filtering action of an asymmetric resonant
interband tunneling diode are investigated.

Finally, a Monte Carlo single photon generation algorithm is devised. The photons
generated are satisfactory for simulation of light emitted from band-to-band spontaneous transitions in crystals. The polarization is determined taking into account the
electron spin, making the algorithm suitable for the analysis of optical detection of
spin injection experiments.

Contents
List of Publications

Acknowledgements

Abstract

1 Introduction

1.1

Thesis overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.2

Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.2.1

Theoretical methods . . . . . . . . . . . . . . . . . . . . . .

1.2.2

Spin injectors/filters . . . . . . . . . . . . . . . . . . . . . .

1.2.3

Detection of spin injection . . . . . . . . . . . . . . . . . . .

Summary of results . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.3.1

Theoretical methods . . . . . . . . . . . . . . . . . . . . . .

1.3.2

Applications . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.3.3

Collaborative work . . . . . . . . . . . . . . . . . . . . . . .

Chapter overview . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.3

1.4

Bibliography

2 Spin splittings in asymmetric AlSb/InAs/GaSb heterostructures
and electric field induced magnetization

2.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2

Theoretical methods . . . . . . . . . . . . . . . . . . . . . . . . . .

2.3

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.3.1

Band structure and spatial part of the wavefunction . . . . .

2.3.2

Spin part of the wavefunction . . . . . . . . . . . . . . . . .

xii
2.3.3

Rashba coefficient for an asymmetric heterostructure made of
AlSb/InAs/GaSb/AlSb . . . . . . . . . . . . . . . . . . . . .

Spatial variation of the wavefunction . . . . . . . . . . . . .

2.4

Physical explanation for the results . . . . . . . . . . . . . . . . . .

2.5

Electric field induced magnetic moment . . . . . . . . . . . . . . . .

2.6

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.3.4

Bibliography
3 Optimization

27
of

the

Rashba

coefficient

asymmetric

AlSb/InAs/GaSb heterostructures

3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2

Rashba coefficient as a function of the layer thicknesses . . . . . . .

3.3

Rashba coefficient for wide wells . . . . . . . . . . . . . . . . . . . .

3.4

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Bibliography

4 Description of bulk inversion asymmetry in the effective bond orbital model

4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.2

Review of the EBOM method . . . . . . . . . . . . . . . . . . . . .

4.3

Inclusion of bulk inversion asymmetry effects in EBOM . . . . . . .

4.4

Bulk GaSb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.5

Bulk inversion asymmetry effects in symmetric superlattices . . . .

4.6

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Bibliography

5 Bulk inversion asymmetry effects on the bands of zincblende heterostructures

5.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xiii
5.2

Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.3

k · p method applied to bulk III-Vs . . . . . . . . . . . . . . . . . .

5.3.1

Invariant expansion of the Hamiltonian . . . . . . . . . . . .

5.3.2

Parameters of the model . . . . . . . . . . . . . . . . . . . .

5.3.3

Analytic expressions of the energy values near the zone center

5.3.4

Numerical calculation of the energy bands . . . . . . . . . .

Eight-band effective mass theory for superlattices and quantum wells

5.4.1

EMA Hamiltonian . . . . . . . . . . . . . . . . . . . . . . .

5.4.2

The finite difference method . . . . . . . . . . . . . . . . . .

5.4.3

Interface conditions and hermiticity in the FDM . . . . . . .

Bulk inversion asymmetry effects on symmetric quantum wells . . .

5.5.1

Symmetry group of the discretized Hamiltonian . . . . . . .

5.5.2

SQWs without BIA terms . . . . . . . . . . . . . . . . . . .

5.5.3

SQWs with BIA terms . . . . . . . . . . . . . . . . . . . . .

Bulk inversion asymmetry effects on asymmetric quantum wells . .

5.6.1

AQWs without BIA terms . . . . . . . . . . . . . . . . . . .

5.6.2

AQWs with BIA terms . . . . . . . . . . . . . . . . . . . . .

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.4

5.5

5.6

5.7

Bibliography

6 Spurious numerical solutions in the effective mass approximation

100

6.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

100

6.2

Spurious solutions in an InP/In0.53 Ga0.47 As superlattice . . . . . . .

101

6.3

Method for the study of the spurious solutions . . . . . . . . . . . .

105

6.4

Spurious solutions in the k · p method . . . . . . . . . . . . . . . .

106

6.4.1

The InP/In0.53 Ga0.47 As SL revisited . . . . . . . . . . . . . .

111

6.5

Potential for spurious solutions in sets of Luttinger parameters . . .

113

6.6

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

115

Bibliography

117

xiv
7 Spin filters based on resonant tunneling

120

7.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

120

7.2

Theoretical method . . . . . . . . . . . . . . . . . . . . . . . . . . .

121

7.2.1

The MQTBM . . . . . . . . . . . . . . . . . . . . . . . . . .

121

7.2.2

Computation of the velocity of the states . . . . . . . . . . .

126

7.2.3

Preparation of the incoming states . . . . . . . . . . . . . .

126

7.2.4

Transmission coefficients and transmitted spin for an ensemble of electrons . . . . . . . . . . . . . . . . . . . . . . . . .

131

7.3

Resonant tunneling in asymmetric double barriers . . . . . . . . . .

134

7.4

Asymmetric resonant tunneling diode (aRTD) . . . . . . . . . . . .

135

7.5

Asymmetric resonant interband tunneling diode (aRITD) . . . . . .

139

7.6

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

143

Bibliography

145

8 Photon generation for a Monte Carlo ray tracing LED simulator

147

8.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

147

8.2

The interaction Hamiltonian and Fermi’s golden rule . . . . . . . .

149

8.2.1

The Wigner-Eckart theorem for point groups applied to momentum matrix elements . . . . . . . . . . . . . . . . . . . .

151

8.2.2

Application to III-V zincblendes . . . . . . . . . . . . . . . .

152

8.2.3

Complications following the path of Fermi’s golden rule . . .

153

8.3

Polarization of the emitted photon . . . . . . . . . . . . . . . . . .

154

8.4

Application to semiconductor structures . . . . . . . . . . . . . . .

160

8.4.1

Bulk zincblende luminescence . . . . . . . . . . . . . . . . .

160

8.4.2

Quantum well luminescence . . . . . . . . . . . . . . . . . .

163

8.4.3

Bulk wurtzite luminescence . . . . . . . . . . . . . . . . . .

169

Monte Carlo photon generation . . . . . . . . . . . . . . . . . . . .

171

8.5.1

Single event generation scheme . . . . . . . . . . . . . . . .

172

8.5.2

Equivalence of the Monte Carlo and the time-dependent per-

8.5

turbation pictures . . . . . . . . . . . . . . . . . . . . . . . .

175

xv
8.6

Application to a bulk zincblende . . . . . . . . . . . . . . . . . . . .

176

8.7

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

179

Bibliography

180

A Explicit form of the 8-band k · p Hamiltonian

183

Bibliography

185

B Group theory for band structures

186

B.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

186

B.2 Degeneracies, splittings and eigenstates . . . . . . . . . . . . . . . .

187

B.3 Matrix elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

189

Bibliography

191

C Derivation of the formula for the transmission coefficients

192

C.1 Expansion to the Hellmann-Feynman theorem . . . . . . . . . . . .

192

C.2 Transmission coefficient . . . . . . . . . . . . . . . . . . . . . . . . .

193

Bibliography

200

D Velocity operator in the k · p formalism

201

Bibliography

204

xvi

List of Figures
1.1

Intel’s CPUs and Moore’s law . . . . . . . . . . . . . . . . . . . . .

1.2

Shubnikov-de Haas oscillations for a structure showing spin splitting

2.1

Structure under consideration and wavefunction. . . . . . . . . . . .

2.2

Band structure for an AlSb/InAs/GaSb/AlSb quantum well . . . .

2.3

Spin directions for a 2DEG in an AlSb/InAs/GaSb/AlSb quantum
well . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.4

Spin splitting for a 10/6/10 AlSb/InAs/GaSb superlattice . . . . .

2.5

Schematic of the origin of the Rashba splitting . . . . . . . . . . . .

2.6

Integration region for the 2DEG magnetization

. . . . . . . . . . .

2.7

Origin of the electric field-induced magnetization . . . . . . . . . .

3.1

Rashba coefficient for an AlSb/InAs/GaSb/AlSb QW and fixed InAs
thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2

Rashba coefficient for an AlSb/InAs/GaSb/AlSb QW and fixed AlSb
thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.3

Band structure of a broken gap QW with thick layers . . . . . . . .

4.1

Band structure of GaSb calculated with EBOM . . . . . . . . . . .

4.2

Bands close to the zone center showing the spin splitting . . . . . .

4.3

Comparison of EBOM and k · p superlattice bands . . . . . . . . .

5.1

Effect of BIA on HH and LH bands along [100ec near the zone center

5.2

Effect of BIA on HH and LH bands along [110ec near the zone center

5.3

Band structure for GaSb near Γ along [100ec. . . . . . . . . . . . . .

5.4

Band structure for GaSb near Γ along [110ec. . . . . . . . . . . . . .

5.5

Spin directions for a conduction subband of GaSb. . . . . . . . . . .

5.6

Polar plot of the spin splitting for the conduction band of GaSb. . .

xvii
5.7

Mesh used in the solution of the EMA. . . . . . . . . . . . . . . . .

5.8

Mesh used in the study of interface boundary conditions. . . . . . .

5.9

Effect of an extra atomic layer in a non-common atom QW. . . . .

5.10 Layer arrangement for a no-common-atom quantum well. . . . . . .

5.11 Bands for a symmetric quantum well w/o BIA terms. . . . . . . . .

5.12 Bands for a symmetric quantum well with BIA terms. . . . . . . . .

5.13 Linear and angular spin splitting dependence for a SQW. . . . . . .

5.14 Spin directions for the lower CB of an AlSb/GaSb/AlSb QW. . . .

5.15 Bands for an asymmetric quantum well w/o BIA terms. . . . . . . .

5.16 Bands for an asymmetric quantum well with BIA terms. . . . . . .

5.17 Angular dependence of the spin splitting for an AQW. . . . . . . .

5.18 Spin splitting vs. k for an AQW.

. . . . . . . . . . . . . . . . . . .

5.19 Spin directions corresponding to the lower CB of an AlSb/InAs/GaSb/AlSb
QW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.1

Bands of an InP/In0.53 Ga0.47 As SL showing spurious solutions . . .

102

6.2

Comparison of spurious and physical envelope functions . . . . . . .

103

6.3

Energy of the spurious states vs. number of mesh points . . . . . .

105

6.4

Energy vs. wavenumber of a hypothetical spurious solution . . . . .

108

6.5

Predicted and actual spurious solution energy . . . . . . . . . . . .

109

6.6

Bands of an InP/In0.53 Ga0.47 As SL showing no spurious solutions . .

112

6.7

Same as Fig. 6.4 but for GaAs and GaSb . . . . . . . . . . . . . . .

115

7.1

Band diagram of an RTD . . . . . . . . . . . . . . . . . . . . . . .

122

7.2

Bands of a bulk zincblende for finite kk . . . . . . . . . . . . . . . .

130

7.3

aRTD structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

135

7.4

Spin directions for a 2DEG in an AlSb/InAs/GaSb/AlSb quantum
well . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

136

7.5

Resonant spin of an electron incident into an aRTD structure . . . .

137

7.6

Transmission coefficient for an aRTD structure . . . . . . . . . . . .

137

7.7

Transmission coefficient for an aRTD structure with BIA inclusion .

138

xviii
7.8

kk dependence of the spin of the transmitted states in an aRTD . .

139

7.9

aRITD structure . . . . . . . . . . . . . . . . . . . . . . . . . . . .

140

7.10 Transmission coefficient for an aRITD structure . . . . . . . . . . .

141

7.11 Bands of the aRITD structure and operation with a lateral electric
field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

142

7.12 Spin polarization as a function of lateral field for several emitter dopings

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

143

7.13 Transmission coefficient for an aRITD structure w/BIA effects . . .

144

8.1

Photon emission process in a crystal

. . . . . . . . . . . . . . . . .

149

8.2

Band structure for a zincblende near the zone center . . . . . . . .

163

8.3

Band structure near the zone center for a QW with D2d symmetry .

165

8.4

Linear and circular polarization for CB-HH1 emission from a D2d QW 167

8.5

Linear and circular polarization for CB-LH1 emission from a D2d QW 168

8.6

Band structure near the zone center for a QW with C2v symmetry .

169

8.7

Band structure near the zone center for GaN . . . . . . . . . . . . .

171

8.8

Flowchart for the Monte Carlo photon generation process . . . . . .

173

8.9

Monte Carlo calculation of photon polarization for a zincblende . .

178

of integration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

194

C.2 Integration region for Jz . . . . . . . . . . . . . . . . . . . . . . . . .

196

C.1 Illustration of the independence of

Jz dx dy with respect to the plane

xix

List of Tables
4.1

Relationship between the EBOM parameters and the k · p parameters

5.1

Band gap and γc for selected III-Vs . . . . . . . . . . . . . . . . . .

5.2

Matrix basis set for the H 66 block of the Hamiltonian . . . . . . . .

5.3

Γl
of K . . . . . . . . . . . . . . . . . . . .
Irreducible components Km

5.4

Material parameters used in the k · p model . . . . . . . . . . . . .

5.5

Parameter values for some materials . . . . . . . . . . . . . . . . . .

5.6

Quantization axis for the basis functions of the Hamiltonian . . . .

5.7

Symmetry requirements on degeneracy and spin splitting for a D2d
structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.8

Symmetry requirements on degeneracy and spin splitting for a C2v
structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.1

Band structure parameters for InP and In0.53 Ga0.47 As . . . . . . . .

101

6.2

Luttinger parameters for In0.53 Ga0.47 As . . . . . . . . . . . . . . . .

112

6.3

Check for possibility of spurious solutions in the Luttinger parameters
from Lawaetz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.4

113

Check for possibility of spurious solutions in the Luttinger parameters
from Vurgaftman et al. . . . . . . . . . . . . . . . . . . . . . . . . .

114

8.1

Equivalence table of the irreducible representations of Td and SO(3)

152

8.2

Character and basis functions table for Td . . . . . . . . . . . . . .

161

8.3

Equivalence of basis functions for O and Td . . . . . . . . . . . . . .

161

8.4

Clebsch-Gordan coefficients for Td . . . . . . . . . . . . . . . . . . .

162

8.5

Character and basis functions table for D2d . . . . . . . . . . . . . .

164

8.6

Clebsch-Gordan coefficients for D2d . . . . . . . . . . . . . . . . . .

165

8.7

Character and basis functions table for C2v . . . . . . . . . . . . . .

169

xx
8.8

Character and basis functions table for C6v . . . . . . . . . . . . . .

170

8.9

Transition probability from the CB to the VB for a bulk zincblende

177

A.1 Matrix elements of the 8-band k · p Hamiltonian . . . . . . . . . . .

184

Chapter 1
1.1

Introduction

Thesis overview

The bulk of this thesis describes a theoretical study of the spin splitting that takes
place in the electronic band structure of semiconductor heterostructures when a
source of inversion asymmetry is present due to the breakdown of the conditions
for Kramers degeneracy [1]. In particular, this work focuses on the effect in the
bands of heterostructures of two different sources of asymmetry; namely, structural
inversion asymmetry (SIA) and bulk inversion asymmetry (BIA). In particular, these
effects are studied in structures built from materials belonging to the so-called 6.1 Å
system: AlSb, GaSb and InAs.
The splitting of the conduction subbands when the sequence of the layers making
a heterostructure is not symmetric (SIA) is called the Rashba effect. This is the
subject of the first two chapters. The subsequent chapters describe methods developed
to include the effect of a different source of asymmetry: bulk inversion asymmetry
(BIA), which arises from the different character of cations and anions in the unit
cell of zincblende materials [2]. Then, these methods are applied to see how our
understanding of the situation when SIA is present is affected by the inclusion of BIA
for both bands and tunneling properties of heterostructures.
Finally, the last chapter, while having in common with the rest the study of spindependent electronic properties, dwells on how these affect the optical processes in
semiconductors, and provides the backbone for Monte Carlo simulations where the
effect of the electron spin on light production in band-to-band recombination is fully
taken into account.
The major results obtained in this thesis are
• Obtention of the largest reported theoretical value of the Rashba coefficient.

• Development of efficient methods for the computation of spin-resolved band
structures and transmission coefficients in heterostructures. BIA effects need to
be included in a complete description of the behavior of the spins.
• Characterization of a new class of spurious solutions for the finite difference
method in the context of the effective mass approximation and obtention of a
test to predict their presence.
• Development of a Monte Carlo algorithm to generate photons with polarizations
in accordance to quantum mechanical selection rules.
Chronologically, Chapters 2 and 3 were developed first. The analysis of results
from spin injection using scanning tunneling microscopy [3] and the interpretation [4]
of experiments with spin polarized LEDs led to Chapter 8. The need for the understanding and quantification of BIA effects resulted in Chapter 5. The adaptation
of that k · p formalism to tunneling processes described in Chapter 7 was then implemented. Efforts to reproduce experimental data obtained at HRL gave birth to
Chapter 6. Finally, the extension of code to calculate spin polarization in spin filter
devices motivated Chapter 4.

1.2

Motivation

Since the invention of the transistor in December 1947, its uses have grown into
a multibillion dollar market. Today’s way of life relies heavily on the ability of a
few million transistors to process data and a few million more to store it, whether
temporarily or permanently. As the capabilities of modern computers grow larger,
so do the demands they must satisfy. Applications such as weather forecasting, drug
design, crash simulations, gene sequencing, nuclear test simulations. . . are sure to
exhaust all the computing power available in today’s and a near future’s machines.
The increase in performance of computers depends critically on the ability to
reduce the feature size of transistors. So far, scientists and engineers have managed
to make Moore’s “law” good. This “law,” enunciated in 1965 [5], originally stated

486™ DX

1000000

386™
286

100000

8086
10000

8080
4004
1970

8008
1975

Transistors

1E8
Pentium 4
Pentium III
1E7
Pentium II
Pentium®

1980

1985

1990

1995

1000
2000

Year
Figure 1.1: Intel’s CPUs and Moore’s law. Data taken from Intel’s web site.
that the number of components in an integrated circuit would double every year, and
in its present form that time constant is usually quoted as 18 months although, at
least for Intel’s CPUs, the number of transistors in a CPU doubles roughly every two
years (see Fig. 1.1).
Eventually, sizes will be reached where quantum effects will affect the normal
operation of the transistors. For example, tunneling leakage current in metal/oxide/
semiconductor (MOS) structures was believed in 2001 [6] to limit the gate oxide
thickness to 12 Å1 . A new approach to keep up with this continuing miniaturization
is to come up with devices based precisely on the quantum properties of the electron.
The resonant tunneling diode (RTD) [7] and the single electron transistor (SET) [8]
are good examples of these devices.

This is a typical figure, which is really application dependent and is given by how much power
density one can afford to consume.

1.2.1

Theoretical methods

Another quantum property of the electron that only recently has received attention
for its potential for information storage and processing is its spin. The starting point
for the field of SPIN elecTRONIC (spintronic) devices was the discovery of giant
magnetoresistance (GMR) in 1988 by Baibich et al. [9]. So far, the only commercially
available spintronic devices—the GMR read heads present in hard drives—or in the
development phase—magnetic RAMs (MRAMs)—are based on the behavior of spin
in metals. Nevertheless, the marriage of spin and semiconductors is a hot research
topic. In 1990 Datta and Das [10] proposed a spin transistor where a gate voltage
controlled the amount of precession of a collection of spins while traveling in a semiconductor between two ferromagnetic contacts, thus modulating the resistance seen
by the electrons in transit. Typical precession frequencies of spins in semiconductor
structures are of the order of a few THz. The prospect of logic at such frequencies
has caused great interest among the funding agencies.
For applications such and Datta and Das’ spin transistor or even quantum computation with spins in semiconductors to become a reality, a hurdle that must be
overcome is the achievement of reliable injection of spins into semiconductors. The
control over the direction of the spins forced into a material is what is understood
as spin injection. Once a nonequilibrium spin population has been established by
means of spin injection, it is necessary that the time it takes for it to come back to
equilibrium is long enough so that useful manipulations can be made.
On one hand, the proper description of transport, time evolution and interface
phenomena of spin ensembles will require having at disposal electron band structures
where all spin effects are included; in much the same way as thorough knowledge
of the “regular” band structure is required to fully understand electronic devices.
On the other hand, semiconductor heterostructures have been extensively used to
create new classes of materials with engineered electronic properties, and they show
promise to serve the same purpose for the spin properties. The methods developed
in Chapters 4 and 5 provide an efficient and easy to implement way to obtain spin

resolved band structures in heterostructures. In particular, the extension of the effective bond orbital model (EBOM) method in Chapter 4 allows for the inclusion of
BIA effects in existing code with negligible extra computational cost. Since EBOM
is a full zone method, results from EBOM calculations will be valid further away
from the zone center than k · p results. However, the EBOM method does poorly
when strain and magnetic field effects need to be accounted for. These two external
fields are straightforwardly included into the k · p formalism. This is the driving force
behind Chapter 5, where an 8-band effective mass approximation (EMA) code that
reproduces the zincblende Td symmetry is implemented.

1.2.2

Spin injectors/filters

The Rashba effect is the building block on which several proposed spintronic devices
are built. When referring to the spin splitting of the conduction band (CB) levels of
a two-dimensional electron gas (2DEG) in an asymmetric potential, it constitutes the
basis for Datta and Das’ spin transistor [10]. Clearly, a greater value of the splitting,
given by a larger value of the Rashba coefficient, would increase the frequency at
which these spins are switched. A large Rashba coefficient will also improve the
performance of the asymmetric resonant tunneling diode (aRTD) spin filter proposed
by Voskoboynikov et al. [11] in 2000. In that device, electrons transmitted through
an asymmetric double barrier heterostructure emerge with their spin aligned with
the spin of the quasi-bound state through which they have tunneled. Note that spin
filtering would be achieved without the presence of any external magnetic field or any
ferromagnetic material.
Clearly, a method for experimental determination of the Rashba coefficient in heterostructures and theoretical understanding of its origin are required for the achievement of a spintronic device based on the Rashba effect. Since the Rashba effect is
closely related to structural asymmetry, the 6.1 Å materials system, with its variety
of band alignments, will be a candidate for the construction of structures with large
Rashba splitting. This motivates the studies in Chapters 2 and 3.

Ting et al. [12] have proposed a modification of the aRTD idea that would have a
higher performance as a spin filter by letting the electrons tunnel through the strongly
spin split valence band states of an AlSb/InAs/GaSb/AlSb well region. Their original
calculation of the current polarization did not include the effects of bulk inversion
asymmetry. The formalism and structures studied in Chapter 7 address how the
tunneling properties of an asymmetric barrier change when BIA is accounted for.

1.2.3

Detection of spin injection

Once a structure is chosen as a candidate for spin injection, the question of the measurement of the spin polarization of the injected electron population must be faced.
For this, there are mainly two kinds of measurements: electrical and optical. In
the electrical measurements, one looks for small changes in the voltage drop when a
current is forced through a semiconductor between two aligned or antialigned ferromagnets. However, these kind of measurements are not free from controversy as the
presence of ferromagnets is thought to produce spurious voltages due to local Hall
effect.
The optical measurement method is preferred because of its direct relationship
to the electron spin density. It basically consists in the analysis of the polarization
of light coming from radiative recombination between an electron and a hole. The
polarization of an emitted photon will be correlated with the spin of the electron
that originated it, and this relationship is well understood. However, the polarization
of photons emitted from a device will be affected by refractive effects at boundaries
and interfaces, and these effects have not received sufficient attention. Monte Carlo
ray tracing techniques are ideal for that purpose because they can easily be adapted
to different characters of the emitting medium—bulk with varying crystal structure,
quantum well, quantum dot—or a variation in the geometrical shape of the extracting
medium. They might find an application too in the study of the optical transmission
by polarized light of information stored in spins. These techniques will require a
method of generating single photons in agreement with the quantum mechanical

selection rules. This and the understanding of the light emission process for arbitrary
situations led to the study in Chapter 8.

1.3

Summary of results

1.3.1

Theoretical methods

The methods developed in this thesis can be divided into four parts:
1. An extension to the EBOM method capable of accounting for bulk inversion
asymmetry (BIA) effects in zincblendes has been derived and implemented.
This extension correctly reproduces the space group symmetry of the zincblende
structure. By construction, it accurately describes the band structure near
the zone center, including the spin splitting in the conduction band of bulk
zincblendes. Calculations for superlattices using this new method agree with
k · p results.
2. Starting with a bulk k · p Hamiltonian having the right Td symmetry for
zincblendes, an 8-band effective mass approximation (EMA) code has been implemented for [001] heterostructures. Contrary to most of the existing EMA
implementations, the one presented here correctly describes the microscopic
symmetry of most superlattices, making it an appropriate tool for phenomena
such as the optical anisotropy [13], and the mixing of heavy hole and light hole
states at the top of the valence band [14], whose existence is a consequence of
the reduced symmetry due to the BIA.
Also, a source of spurious solutions for the finite difference method has been
characterized. The application to the k · p method allows the derivation of a
condition that the Luttinger parameters must meet for the spurious solutions
not to appear.
3. A previous implementation of the multiband quantum transmitting boundary
method (MQTBM) in the k·p framework [15] for the determination of tunneling

transmission coefficients has been cast in a form consistent with the finite difference method, eliminating some small spurious contributions. An algorithm for
generating an incoming state with a predetermined spin has also been created.
4. A Monte Carlo algorithm for single polarized-photon generation in radiative
recombination processes has been developed. This algorithm makes extensive
use of the host crystal symmetry properties to reduce to a minimum the number
of external inputs needed to determine the photon polarization. It allows the
determination of the Stokes parameters for light emitted in arbitrary directions
by a given electron spin population.

1.3.2

Applications

The 6.1 Å material system comprised of AlSb, GaSb and InAs is shown to offer great
promise for spintronic devices based on the Rashba effect. The Rashba coefficient for
AlSb/InAs/GaSb/AlSb 2DEGs is calculated theoretically to be able to achieve values
of the order of 50×10−10 eV·cm. This is the highest value reported in the literature.
The dependence of the Rashba coefficient on the thicknesses of the different layers is
studied, finding that the structures with InAs thickness between 5 and 15 monolayers
(MLs) and GaSb thickness of 8 or more MLs present optimal values of the Rashba
coefficient. It is also observed that the presence of anticrossing between the conduction
band of InAs and the valence band of GaSb in the 2DEG reduces the value of the
coefficient substantially. The application of an in-plane bias in the 2DEG is shown to
produce an amount of spin polarized current and some magnetization accompanying
it. The degree of polarization of the current is of the order of 5×10−5 for typical values
of the in-plane bias. Therefore, much larger Rashba coefficients would be needed to
use this configuration as a source of spin polarized current.
The presence of spin splitting even in structurally symmetric quantum wells (QWs)
is investigated. A 2-band Hamiltonian analogous to the Rashba Hamiltonian, but describing the splitting for these symmetric QWs due to the BIA and its interaction with
the Rashba splitting is proposed, and its validity tested with numerical 8-band calcu-

Figure 1.2: Shubnikov-de Haas oscillations for a (InAs/AlSb SL)/InAs/(InAs/AlSb
SL) structure showing spin splitting. Data taken by Andy Hunter et al. [16]
lations. BIA effects are shown to be of the same order of magnitude as Rashba effects
for the conduction band of the studied 6.1 Å heterostructures, hence its inclusion is
necessary in the description of the bands in QWs.
Experiments support the theoretical predictions made in this thesis regarding the
magnitudes of the Rashba coefficients. Figure 1.2 shows a node in Shubnikov-de Haas
oscillations of a quantum Hall measurement, taken by Andy Hunter et al. [16] at HRL
Laboratories. The node and the change of phase of the oscillations are signatures of
the presence of the Rashba effect. The measured value (6.4×10−10 eV·cm) and a calculation using the methods described in Chapter 5 (7.6×10−10 eV·cm) show reasonable
agreement.
The implementation of the MQTBM is used to study the spin-dependent tunneling
properties of an asymmetric interband resonant tunneling diode (aRITD). It is shown
that the predicted [12] spin filtering properties of the aRITD should not be largely
affected when BIA is included into the calculation.

1.3.3

Collaborative work

Unless noted otherwise, all the material presented in this thesis is the work of the
author. On the other hand, I must acknowledge David Ting for putting to my disposal
his EBOM code, used to do the calculations in Chapters 2 and 3 and modified to
obtain the numerical results in Chapter 4. Also, the code used to generate Fig. 8.9
was written by Steve Ichiriu.

1.4

Chapter overview

Chapter 2 presents the use of a previously established method for the calculation of
band structures [the effective bond orbital model (EBOM) method] to calculate the
magnitude of the Rashba effect in AlSb/InAs/GaSb/AlSb quantum wells. Then, the
consequences of the Rashba splitting on the magnetization and spin polarization current of a two-dimensional electron gas (2DEG) are explored. Chapter 3 presents the
search for a structure that will yield an optimized Rashba effect. Chapter 4 presents
an extension to the EBOM method that reproduces the reduction in the symmetry
due to the bulk inversion asymmetry (BIA) present in zincblendes, and describes the
spin splitting resulting from that. In Chapter 5, the description of BIA in the k · p
framework is discussed, and results for band structure calculations in symmetric and
asymmetric quantum wells, with and without BIA effects are presented; and the role
of BIA is delineated. Chapter 6 describes a new class of spurious solutions that can
appear when using the finite difference method to solve the Envelope Function Approximation (EFA) equations and how to make sure they don’t show up. Chapter 7
calculates the spin-dependent transmission coefficients in AlSb/InAs/GaSb/AlSb aRITDs with and without the BIA contribution and discusses the influence of BIA in
existing current polarization calculations. Finally, Chapter 8 presents an algorithm
for Monte Carlo single polarized-photon generation from electron-hole radiative recombination.

Bibliography
[1] J. J. Sakurai, Modern Quantum Mechanics, 1st ed. (Addison-Wesley, Redwood
City, CA, USA, 1985).
[2] G. Dresselhaus, Phys. Rev. 100, 580 (1955).
[3] C. J. Hill, Ph.D. thesis, California Institute of Technology, Pasadena, Ca, 2001.
[4] S. R. Ichiriu, X. Cartoixà, and T. C. McGill, , in preparation.
[5] G. E. Moore, Electronics 38, (1965).
[6] D. Frank, R. Dennard, E. Nowak, P. Solomon, Y. Taur, and H. Wong, Proc.
IEEE 89, 259 (2001).
[7] L. Esaki and R. Tsu, IBM J. Res. Develop. 14, 61 (1970).
[8] K. K. Likharev, Proc. IEEE 87, 606 (1999).
[9] M. N. Baibich, J. M. Broto, A. Fert, F. N. Vandau, F. Petroff, P. Eitenne, G.
Creuzet, A. Friederich, and J. Chazelas, Phys. Rev. Lett. 61, 2472 (1988).
[10] S. Datta and B. Das, Appl. Phys. Lett. 56, 665 (1990).
[11] A. Voskoboynikov, S. S. Lin, C. P. Lee, and O. Tretyak, J. Appl. Phys. 87, 387
(2000).
[12] D. Z.-Y. Ting, X. Cartoixà, D. H. Chow, J. S. Moon, D. L. Smith, T. C. McGill,
and J. N. Schulman, IEEE Trans. Magn. (2002), to be published.
[13] O. Krebs and P. Voisin, Phys. Rev. Lett. 77, 1829 (1996).
[14] E. L. Ivchenko, A. Y. Kaminski, and U. Rossler, Phys. Rev. B 54, 5852 (1996).
[15] Y. X. Liu, D. Z.-Y. Ting, and T. C. McGill, Phys. Rev. B 54, 5675 (1996).

12
[16] A. Hunter, P. Deelman, D. Chow, L. Warren, J. N. Schulman, J. Zinck, and S.
Skeith, , unpublished.

Chapter 2

Spin splittings in asymmetric

AlSb/InAs/GaSb heterostructures and
electric field induced magnetization
2.1

Introduction

In recent years, interest in developing spin-sensitive devices (spintronics) [1–4] has
fueled renewed investigations into spin phenomena in semiconductors. The aim is to
control not only the spatial degrees of freedom of the electron, but also the spin degree
of freedom. Useful spintronic devices can be devised if such control is achieved. A
number of such devices have already been proposed [5–8], and the search for phenomena which can lead to spin based devices is widespread [9–11]. Studies of asymmetric
quantum wells have been carried out both theoretically and experimentally in other
material systems [12–15]. One of the systems showing promise is the nearly lattice
matched system comprised of InAs, GaSb and AlSb [16, 17].
In particular, one of the phenomena that might be of importance in the
InAs/GaSb/AlSb heterojunction system is the Rashba effect [12]. The precise contributions to the magnitude of the effect are a subject of some recent studies [18, 19].
Previously, a number of studies have examined InAs quantum wells confined by
InGaAs layers [20–23] or Alx Ga1−x As/GaAs heterostructures [24, 25]. In this study,
the focus is on the effects of the InAs/GaSb unique band offsets.
In Sec. 2.2 of this chapter, the results of a study of the band structure for an
AlSb/InAs/GaSb/AlSb asymmetric well using the effective bond orbital model are
presented. These results can also be described in terms of a Bychkov-Rashba Hamiltonian [12]. In Sec. 2.3 the dependence of the spin splitting on the electron momentum
for the conduction band for several structures is shown and the orientation of the

14
electron spins is described. In Sec. 2.4 an intuitive picture of the physics behind
the Rashba effect is presented. In Sec. 2.5, a derivation of an expression for the
surface magnetization and the degree of spin polarization of the current when the
nonmagnetic two-dimensional electron gas (2DEG) is under an in-plane electric field
is shown.

2.2

Theoretical methods

Spin splitting can result from the lifting of Kramers degeneracy through the removal
of inversion symmetry. Mechanisms for inversion symmetry removal include
I. The specific spatial arrangement of the composing layers with their interfaces. [18]
II. Bulk inversion symmetry as in the zincblende structure. [19]
III. Asymmetry created by external fields and dopant-induced band bending. [18]
In this chapter, the primary interest is in examining effects in the band structure induced by an asymmetric spatial arrangement of the heterostructure. Hence, a
band structure calculation technique was selected that allows treatment of layer arrangement contributions with the highest accuracy. Bulk inversion asymmetry (BIA)
contributions are considered in Chapter 5.
The calculations shown are based on the EBOM method developed by Y.-C.
Chang [26, 27]. It consists of a tight-binding [28] model taking bond orbitals [29]
in an fcc lattice as a basis set, expanding the resulting matrix elements to second
order in k and identifying the EBOM parameters with the k · p [30, 31] parameters.
It accurately describes the lowest conduction band and the heavy hole, light hole and
splitoff bands near the zone center, making it appropriate for the calculation of the
band structure of the 2DEG, taking into account the strong coupling between the
InAs conduction band and the GaSb valence band states due to the broken gap band
alignment, as illustrated in Fig. 2.1.

15
The calculations are performed on [001] structures using superlattice boundary
conditions, and it is found that the superlattice cells are effectively decoupled when
the AlSb thickness is 8 monolayers (ML) or more, with a monolayer having a thickness
of 3.048 Å.
The EBOM method accounts for the very different nature of the various interfaces, although it cannot describe properly the lack of inversion symmetry inherent
intrinsic to the zincblende structure. An extension to the method to deal with BIA
effects is presented in Chapter 4. In the bulk single crystal, that lack of inversion
symmetry leads to the k 3 splitting [32]. This effect could lead to some corrections.
In Chapter 5 they are found to be on the order of a 20% at most. There has been
some work on BIA effects beginning with the work of Lommer et al. [33] and Cardona
et al. [34]. A discussion of the relative tradeoff is included in the papers by Silva et
al. [18, 19]. Space charge and gate generated electric fields are not included in the
calculation either. These fields do make some contribution to the Rashba splitting
[see Eq. (2.1)] [18, 35], and they are determined by the details of the doping and
applied bias.

2.3

Results

2.3.1

Band structure and spatial part of the wavefunction

The structure under consideration and a typical result for the probability distribution
of the electron wavefunction are illustrated in Fig. 2.1. The values of the band gaps
and lattice constants are taken to be the widely accepted values [16, 17] for the plots
in Fig. 2.1. The band offsets are taken from the review by Yu et al. [36] and reflect
the Type II staggered alignment of the bands for the AlSb/InAs interface and the
Type II broken gap alignment for the InAs/GaSb interface [37]. This configuration
provides a strong asymmetry of the confining potential. As a result, the Rashba
constant [see Eq. (2.1)] will have an enhanced value compared to previously studied
systems [14, 15, 35].

Distance [Å]
20

80 100 120 140 160
2.0

Probability density (a.u.)

1.5

1.0

Lower Subband
Upper Subband

AlSb

InAs

AlSb

GaSb

0.5

Energy [eV]

0.0
-0.5

10
15
Layer index

Figure 2.1: Flat band diagram (right scale) of the heterostructure under consideration.
The structure consists of a superlattice constructed by repeating 16 layers of AlSb
followed by 6 layers of InAs followed by 6 layers of GaSb. The energies reflect the
accepted values of the band gaps and band offsets. Further, the calculated probability
density (left scale) for conduction band states in the well for k = (0.02, 0, 0) 2π/a,
where a is the unit cell size. The solid line shows the lower energy spin split state.
The dashed line shows the higher energy spin split state. The axes show the choice
of coordinates.
Figure 2.2 shows the band structure near the Γ point for a superlattice composed of
16 layers of AlSb, 6 layers of InAs and 6 layers of GaSb. The bands become spin split
due to the lack of inversion symmetry in the growth direction. Note that the energy
minimum of the lower conduction subband is no longer at the point k = 0. Control
calculations with symmetrical AlSb/InAs/AlSb and GaSb/InAs/GaSb structures and
these showed no splitting.
The probability density of the spin split states is shown in Fig. 2.1. We see that
the electron is mainly localized in the InAs layer, but it displays significant leakage
into the AlSb and GaSb layers. Careful examination of this figure shows that the state

17
0.8

Energy [eV]

0.6

0.4

0.2

0.0

-0.2
-0.05

0.00

0.05

kx/(2π/a)

Figure 2.2: The calculated band structure of an 16ML (monolayers) AlSb/6ML
InAs/6ML GaSb superlattice near the Γ point for light holes, heavy holes and conduction electrons. Note the splitting in the states of the 2DEG, corresponding to the
conduction band.
corresponding to the lower (upper) energy band is displaced towards the AlSb/InAs
(InAs/GaSb) interface. This point is further discussed in Sec. 2.3.4.

2.3.2

Spin part of the wavefunction

One of the most important features of the results is the spin character of the eigenstates. The result is shown in Fig. 2.3. The spin expectation value for the different
states always lies on a circle in the kx − ky plane. The x − y plane is the quantum
well plane (see Fig. 2.1). It is seen that for the lower (upper) conduction subband the
spins point tangentially to the circle in a counterclockwise (clockwise) fashion. The z
component of the expectation value of the spin is found to be zero within numerical

18
0.04

0.04

0.02

0.00
ky/(2π/a)

Lower Subband

Upper subband

0.00

ky/(2π/a)

-0.02

-0.04

-0.04
-0.04 -0.02 0.00 0.02 0.04
kx/(2π/a)

-0.04 -0.02 0.00 0.02 0.04
kx/(2π/a)

Figure 2.3: Calculated expectation value of the spin for states of a 2DEG lying on
circles of constant k for the two conduction spin split subbands.
fluctuations. This is indeed the behavior predicted by the Rashba Hamiltonian [12]
HSO = αR (σ × k) · ν

(2.1)

and found analytically by Schäpers et al. [35]. αR is the Rashba constant (dependent
on the details of the heterostructure), σ are the Pauli spin matrices, k is the crystal
momentum of the electron in the 2DEG and ν is a unit vector parallel to the growth
direction. The specific handedness of the rotation of each subband depends on the
details of the well, and it would have been reversed had the order of the InAs and
GaSb layers been reversed.
Note that, although there is spin splitting without the presence of an external magnetic field or magnetic constituents, Kramers time reversal degeneracy is preserved,
for
E (k, | ↑in̂ ) = E (−k, | ↓in̂ ) ,

(2.2)

where the left- (right-) hand side term is the energy of a state with wavevector k
(−k) and spin up (down) in the direction of a unit vector n̂.
Another important consequence of this unique spin configuration is that, though
there is a preferred spin direction for a single k state, when averaging over the whole

19
0.030

(10/6/10) - AlSb/InAs/GaSb Superlattice

Splitting [eV]

0.025
0.020
0.015
0.010
0.005
0.000
0.00

0.02

0.04

0.06

0.08

0.10

kx [2π/a]
Figure 2.4: Spin splitting for a 10/6/10 AlSb/InAs/GaSb superlattice in the conduction band.
subband the net spin vanishes, so there is no electronic magnetism for this system in
equilibrium.

2.3.3

Rashba coefficient for an asymmetric heterostructure
made of AlSb/InAs/GaSb/AlSb

From Eq. (2.1) it can be seen that, close to the zone center, the splitting ∆R due to
the Rashba Hamiltonian is linear with the electron wavevector k, and is given by
∆R = 2αR k.

(2.3)

Figure 2.4 shows the spin splitting for a 10/6/10 AlSb/InAs/GaSb superlattice
in the conduction band along the [100] direction. As indicated by Eq. (2.3), the
calculated splitting is linear close to Γ. Further away from the zone center, the terms
of higher order in k coming from the numerical diagonalization of the 8 × 8 EBOM
Hamiltonian start taking over.

20
A coefficient αR = 38 × 10−10 eV·cm is found for the (AlSb=16 ML, InAs=6 ML,
GaSb=6 ML) heterostructure. In Chapter 3 optimization procedures are carried out
and coefficients as large as 50 × 10−10 eV·cm have been found. This is one of the
largest reported values for the Rashba coefficient.
As a validation of the method of calculation, the Rashba coefficient for an
In0.53 Ga0.47 As/In0.77 Ga0.23 As/InP heterostructure studied by Schäpers et al. [35] was
calculated. They reported an experimental value αR = (5.0 ± 0.1) × 10−10 eV·cm
for that structure. They also compute a Type I interface contribution to α R of
6.7 × 10−10 eV·cm. Making the calculation with the EBOM method one finds an in-

terface contribution of 7.8 × 10−10 eV·cm. The reasonable agreement in the computed
values for the interface contribution to the Rashba coefficient validates the procedure
followed here and lends support to the claim for a large interface contribution to α BR
in the AlSb/InAs/GaSb/AlSb heterojunction studied here.

2.3.4

Spatial variation of the wavefunction

Figure 2.1 shows that the two eigenstates of the conduction subband have slightly
different spatial behavior. This coupling of the spatial part with the spin part of the
wavefunction is not modeled by the Bychkov-Rashba Hamiltonian in Eq. (2.1). The
attraction towards one or the other interface comes from the preference of the electron
to be located in regions with lower potential energy. The interface electric fields
transform in the rest frame of the electron into magnetic fields to which the electron
spin couples. Since the two interfaces have electric fields with opposite signs, hence,
magnetic fields with opposite signs, the sign of the spin will dictate the preference
for one of the interfaces. The lowest subband state tends towards the interface with
larger band offsets, where the electric fields are larger.

2.4

Physical explanation for the results

The unusual configuration of the spins in the conduction band shown in Fig. 2.3 can
be intuitively understood using the following argument. Classically, if the electrons in

21
To the rest frame of the electron

E' B

Figure 2.5: Schematic of the process by which the Rashba spin splitting arises. A
moving electron in an electric field sees a magnetic field in its rest frame and its spin
couples to it in a Zeeman-like manner.
the 2DEG are moving with a velocity v under the presence of an electric field E, this
transforms relativistically in the rest frame of the electron into an effective magnetic
field Beff given (in SI units) by (see, for example, Ref. [38])
Beff = − 2 v × E,

(2.4)

where c is the speed of light.
The magnetic moment of the electron will then couple to Beff producing the splitting. The quantum version of this reasoning is nothing but the spin-orbit interaction,
which includes a factor of 1/2 to account for Thomas precession. Since v ∝ k and
the only electric field that is not averaged to zero by the spatial extent of the wavefunction is E = Eν where ν is a unit vector in the direction of the growth, one has
for the spin-orbit Hamiltonian
H = −µ · Beff

∝ −σ · (k × E) ,

(2.5)

where µ ∝ −σ is the magnetic moment of the electron.
As shown in Fig. 2.5, this readily explains why the spins are pointing in-plane and
perpendicular to the direction of propagation of the electron. Finally, if one makes
the substitution E = hEiν into Eq. (2.5) with hEi being the expectation value of the
electrical field for the conduction band states, the Hamiltonian (2.1) is recovered with

22
αR just being the proportionality constant. In this description, the Bychkov-Rashba
constant is proportional to the expectation value of the electrical field, agreeing with
published experimental results [35, 39]. More sophisticated calculations [35] also find
interface contributions.
It can be argued [40] that the expectation value of the electric field should vanish
because the expectation value of the force that the electrons are subject to must
definitely vanish. Hence, it would seem that the value of the Rashba coefficient must
not depend on the expectation value of the electric field, in contradiction to the
argument shown in this section. However, Malcher et al. [14] show that one can have
a nonzero expectation value of the electrical field perpendicular to the plane of the
quantum well and still have a vanishing expectation value for the force.

2.5

Electric field induced magnetic moment

A novel observation is that the presence of a DC electric field parallel to the plane of
the 2DEG Ek can produce a net magnetic moment. Consider the application of an
in-plane bias. It is a well-known result that, in the linear approximation and in the diffusive limit where the electrons are thermalized before reaching the collector, the equilibrium distribution function is displaced by a constant amount ∆k = eτ Ek /~, [41]
where e is the electron charge, τ is the relaxation time and Ek is the electric field
applied in-plane. Consider, for example, the displacement of the distribution function for the electrons in the lower conduction subband of Fig. 2.3. Figure 2.7 (a)
shows a schematic of the effect of the applied electric field on a 2DEG at zero temperature. The gray region corresponds to states k whose spins are compensated by
opposite spins at −k, thus making a zero contribution to any magnetization or spin
polarization of the current. The white regions correspond to states that contribute
to the total spin. This can be accomplished by states newly occupied (unoccupied),
represented by the black (light gray) arrows. In the figure the displacement has been

m*µ E

2

cos θ

θ

S

Figure 2.6: The hatched portion of k space S denotes the integration region of uncompensated states where the magnetic moment of each electron will be added.
magnified for clarity and, using the relationship eτ = µm∗ , it is found to be
m∗ µEk
∆k
= √
∼ 10−4 ,
kF
~ 2πn

(2.6)

where µ is the mobility of the electrons in the 2DEG and n is the surface density of
electrons. The numerical estimate has been obtained using typical values for a 2DEG;
m∗ ∼ 0.02me , µ ∼ 80000 cm2 /V·s, Ek ∼ 1 V/cm and n ∼ 1012 cm−2 . The sum of the
spins coming from the white regions will have a non zero component perpendicular
to the applied electric field. This yields a net magnetic moment and spin polarization
of the current due to electrons in that subband.
The net magnetic moment M coming from one subband can be obtained by integrating the spins over the uncompensated states in the spherical band approximation
(see Fig. 2.6 for a schematic of the integration region):
M=

λ(k) µB hσi dSk ,

(2.7)

24
where λ(k) =

¡ A ¢2
2π

is the surface density of states in k space for a single subband;

i.e., no spin degeneracy factor of 2, A is the area of the sample, g is the free electron
g—factor , µB is the Bohr magneton and hσi is the unitless spin expectation value.
For small displacements of the Fermi sphere, as it is the case here, the differential of
surface in k space dSk can be written as
dSk = 2

m∗ µ
Ek cos θ kF dθ,

(2.8)

where kF is the Fermi wavevector for the subband under consideration.
Then, performing the integral for θ between −π/2 and π/2, one gets
gµm∗ Ek
M↑ = −A
µB k F ↑ ,
8π~

(2.9)

where M↑ is the resulting net magnetic moment coming from that subband and kF ↑
is the Fermi radius for the subband with spins pointing counterclockwise. Figure 2.7
(a) shows that the magnetization will be perpendicular to the direction of the electric
field.
The combined effect of the two conduction subbands is shown in Fig. 2.7 (b). Each
subband contributes with a surface magnetization pointing in a direction opposite to
the other subband, but they do not cancel out due to the different size of the circles.
Adding the contribution from both subbands, using
(kF ↑ − kF ↓ ) =

2m∗
αR
~2

(2.10)

and assuming Ek pointing in a general direction in the plane, the resulting magnetic
moment per unit area will be
gµm∗ 2
4π~3

(2.11)

where ν is a unit vector pointing perpendicular to the 2DEG plane.
If one substitutes into this expression the values representative of the structure

(a)

(b)

Figure 2.7: Illustration of the influence of a DC electric field on Fermi surfaces in
k-space. Part (a) shows a schematic of the displacement of the Fermi circle for one
of the conduction subbands of a 2DEG under the application of an in-plane external
electric field. The amount of displacement is greatly exaggerated for clarity. The
black arrows represent the spins of the newly occupied states, while the light gray
arrows represent the contribution from the states that have become unoccupied, thus
yielding the net current polarization and magnetization in the direction perpendicular
to the displacement of the circle. In part (b), the net effect for the two conduction
subbands is shown. The difference in size of the circles has been exaggerated for
clarity. Since the size of the regions contributing to the magnetization is different,
the “up” and “down” magnetizations will not compensate and there will be a net
surface magnetization in a direction perpendicular to the applied electric field.
µ ∼ 8 m2 /V·s, m∗ ∼ 0.02me , αR ∼ 38 × 10−10 eV·cm and Ek ∼ 1 V/cm one gets a

surface magnetization of the order of 10−12 J/T·m2 . This magnetization corresponds
to ∼ 105 Bohr magnetons for a 1 mm2 sample. Measurement of this small effect
might be possible, for example, using Wernsdorfer’s µ-SQUID technique [42]. This
magnetic field would be distinguishable from a field produced by the current à la
Ampère because of the different directions that the Ampère field and the one proposed
here would have.
Also, the spin polarization of the current can be estimated by assuming all the

26
magnetization is supplied by a difference
n↑ − n ↓ =

2M
AgµB

(2.12)

to obtain the spin polarization of the 2DEG under an applied bias
P =

j↑ − j ↓
n↑ − n ↓
∼ 5 × 10−5 .
j↑ + j ↓
n↑ + n ↓

(2.13)

This small value shows that this scheme is not a good candidate as a source of spin
polarized current, unless structures with a much higher Bychkov-Rashba coefficient
can be found.

2.6

Summary

In conclusion, it has been shown that there is a large spin splitting in the conduction band of the two-dimensional electron gas (2DEG) formed in an asymmetric
AlSb/InAs/GaSb/AlSb quantum well. This theoretical value is the largest such splitting reported for nonmagnetic 2DEGs confined by asymmetric walls. This splitting
and the behavior of the spins are well described, at low k’s, by a Rashba Hamiltonian.
The Rashba coefficients obtained with the effective bond orbital model (EBOM) have
been shown to be in agreement with those obtained with other methods. However,
EBOM has the added benefits that it takes into account the coupling of the space
and spin parts of the wavefunction and the interaction between the conduction and
valence bands in situations where it cannot be treated perturbatively. Also, it is
predicted that when an electric field is applied in the plane of the 2DEG, it should
display a surface magnetization estimated to be of the order of 10−12 J/T·m2 for
values typical of the structures in this chapter. The degree of spin polarization of
the resulting current is found to be ∼ 0.01%. These results are not encouraging for
practical applications, but other approaches using the Rashba effect [43, 44] show
more promise towards the realization of a spin injector.

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Chapter 3

Optimization of the Rashba

coefficient in asymmetric
AlSb/InAs/GaSb heterostructures
3.1

Introduction

In Chapter 2, it has been shown that, in order for any device exploiting the Rashba
splitting to be useful, large Rashba coefficients are required. In particular, large
Rashba splittings would help achieve a bigger spin polarization of the current in the
proposed [1] asymmetric Resonant Tunneling Device (a-RTD) realization of a spin
injector.
It has also been shown in Chapter 2 that predictions of the magnitude of the
Rashba coefficient in the 6.1 Å system composed by the nearly lattice matched compounds AlSb, GaSb and InAs, with its unique band offsets [2], are among the largest
in the literature.
In this chapter, a systematic exploration of different AlSb/InAs/GaSb heterostructures is undertaken leading to a set of design rules for a structure having an optimized
Rashba coefficient. In Sec. 3.2 the Rashba coefficient is found for structures having a
varying number of monolayers. In Sec. 3.3 it is shown how the Rashba coefficient is
significantly reduced in wide wells and finally a summary is presented.

3.2

Rashba coefficient as a function of the layer
thicknesses

In this chapter, following the line of Chapter 2, the band structures are computed
using the effective bond orbital model (EBOM) method [3, 4]. For computational

GaSb thickness (ML)

25
20
15
50

10
45

40
30 35
15 20

AlSb thickness (ML)
Figure 3.1: The Rashba coefficient as a function of AlSb and GaSb thicknesses for
InAs thickness fixed to 9 ML. The numbers in the plot are contour lines of the Rashba
coefficient in units of 10−10 eV·cm.
convenience purposes, the calculations will be made using superlattice boundary conditions. The superlattice results will be valid for the quantum well case as long as the
AlSb layers are thick enough that all the instances of the superlattice are in effect isolated from one another. One finds that the superlattice cells are effectively decoupled
when the AlSb thickness is 8 monolayers (ML) or more, with a monolayer having a
thickness of 3.048 Å. A flat band plot of the structure with its corresponding band
alignments can be found in Fig. 2.1.
The computations of the Rashba coefficient (see Chapter 2) are made in the same
conditions as in Chapter 2. Therefore, structural symmetry breaking mechanisms
for the well other than the layer sequencing [5, 6] will not be dealt with. The bulk
inversion asymmetry inherent to bulk zincblendes and the spin splitting originating
from it [7] will not be taken into account, either.

30
25

InAs thickness (ML)

17
5.5

20
15

10
50

39
17 28
11

GaSb thickness (ML)
Figure 3.2: The Rashba coefficient as a function of InAs and GaSb thicknesses for
AlSb thickness fixed to 16 ML.
A systematic exploration of structures with different layer thicknesses has been
undertaken with the purpose of looking for the configuration optimizing the Rashba
coefficient and delineating the role of each of the composing layers. The thickness
of each of the layers has been swept from 1 to 30 ML. The Rashba coefficient αR is
calculated from the splitting at 0.2% of the zone edge along the [100] direction and
then applying
∆R = 2αR k.

(2.3)

Figure 3.1 shows the Rashba coefficient as a function of the AlSb and GaSb thicknesses while keeping the InAs thickness fixed to 9 ML. It is seen that the Rashba
coefficient is independent of the AlSb and GaSb thicknesses in a wide range. It only
changes its value when the AlSb layer is very thin—the quantum well approximation ceases to be valid—and when the GaSb layer is less than 8 ML. In this case,
a thinner GaSb layer reduces the amount of asymmetry in the quantum well and,

33
therefore, reduces the Rashba coefficient. This behavior can be understood looking
at the wavefunction in Fig. 2.1. The tail of the wavefunction vanishes into the AlSb
on one side and into the GaSb on the other. Once the point has been reached where
the tail of the wavefunction vanishes inside one layer, it is of little consequence that
more monolayers of material are added, since that would affect a region where the
electron is barely present. This explains the independence of the AlSb and GaSb
thicknesses for most of the range.
Figure 3.2 shows the Rashba coefficient as a function of the InAs and GaSb thicknesses while keeping the AlSb thickness fixed to 16 ML. It is seen that, in order to
achieve a high Rashba coefficient, the GaSb thickness must be bigger than approximately 8 ML and the InAs layer must be between 5 and 15 ML. But in the previous
paragraph it has been shown that the AlSb thickness is of little importance as long
as it is thick enough; therefore, these design rules apply for quantum wells with AlSb
barriers thicker than 15 Å.

3.3

Rashba coefficient for wide wells

Another interesting feature in Fig. 3.2 that requires attention is the region of thick
GaSb and InAs where the Rashba coefficient is appreciably diminished, i.e., the dark
region. This sudden reduction is due to the anticrossing of the InAs electron states
with the GaSb hole states.
Figure 3.3 shows an example of such anticrossing. It displays the band structure
of a 24 ML GaSb / 24 ML InAs quantum well. In bulk, the InAs conduction band
electron states lie below the GaSb valence band hole states [2]. For a thin quantum
well, the InAs electron states are pushed up in energy, while the GaSb hole states
are pushed down; thus making the well behave as if it were made of a direct band
gap material. However, for the case shown in Fig. 3.3, the GaSb and InAs layers are
not thin enough for the valence and the conduction bands to be separated, and the
anticrossing takes place.
The inset in Fig. 3.3 shows that the amount of splitting only recovers its linear

0.35

Splitting in the CB

0.30

Energy [meV]

Energy [eV]

0.25

0.00

0.02

0.04

kx/(2π/a)

0.20

0.15

-0.04

-0.02

0.00

0.02

0.04

kx/(2π/a)
Figure 3.3: Band structure of a 12ML AlSb/24ML InAs/24ML GaSb/12ML AlSb
quantum well, showing the anticrossing of the conduction and valence bands. The
inset shows the splitting in the conduction band.
k dependence once the electron states don’t couple with the hole states. In this
situation, the Rashba coefficient defined as one half of the slope of the splitting at the
Γ point loses much of its meaning, and it is more appropriate instead to look at the
total amount of splitting. For example, the splitting at k = (0.03, 0, 0) Å−1 is reduced
to about one third of the value for thin wells [cf. plot (c) in Fig. 2.2]. This illustrates
the reduction in the magnitude of splitting that takes place whenever anticrossing of
the states occurs.
It is important to note that an eight-band method is needed to fully consider the

35
interplay of the conduction and valence bands leading to the reduction of the Rashba
coefficient. This would have been impossible if, instead, a two-band method had been
employed.

3.4

Summary

In summary, it has been shown that the AlSb/GaSb/InAs system shows promise
to be the material system of choice to obtain large Rashba effect splittings. The
Rashba coefficient has been seen to be critically dependent on the InAs thickness,
while showing almost no dependence on the AlSb and GaSb thicknesses once these
are above some threshold value. The optimal thickness of the InAs layer has been
determined. Finally, the importance of designing the sample in such a way that there
is no anticrossing between the InAs electron states and the GaSb hole states in order
to achieve large values of the Rashba coefficient has been highlighted.

Bibliography
[1] A. Voskoboynikov, S. S. Lin, C. P. Lee, and O. Tretyak, J. Appl. Phys. 87, 387
(2000).
[2] E. T. Yu, J. O. McCaldin, and T. C. McGill, Solid State Phys. 46, 1 (1992).
[3] Y. C. Chang, Phys. Rev. B 37, 8215 (1988).
[4] G. T. Einevoll and Y. C. Chang, Phys. Rev. B 40, 9683 (1989).
[5] J. Luo, H. Munekata, F. F. Fang, and P. J. Stiles, Phys. Rev. B 38, 10142 (1988).
[6] J. Nitta, T. Akazaki, H. Takayanagi, and T. Enoki, Phys. Rev. Lett. 78, 1335
(1997).
[7] G. Dresselhaus, Phys. Rev. 100, 580 (1955).

Chapter 4

Description of bulk inversion

asymmetry in the effective bond orbital
model
4.1

Introduction

Since its inception in 1954 [1], the empirical tight binding method (TB) has been
extensively used to compute band structures of bulk metals [1–3], semiconductors [1,
4], and heterostructures [5, 6], yielding a good compromise between accuracy and
ease of implementation. Tight binding is a full zone method and, as such, it has been
used to describe situations where states corresponding to more than one extremum
are needed, such as short period superlattices [7] or X-point tunneling influence on
the escape time of electrons inside leaky quantum wells [8].
One of the difficulties tight binding users encounter is the choice of parameters.
Usually, tight binding parameters bear only an indirect relation with measurable
quantities, and their determination usually requires a tedious fitting procedure. The
effective bond orbital model (EBOM) method by Chang [9, 10] reviewed in Sec. 4.2
provides a way of matching the TB parameters using a reduced bond orbital basis set [11] with the measurable k · p parameters. As originally developed, the
EBOM method does not account for the bulk inversion asymmetry (BIA) present in
zincblendes, predicting no spin splitting in structures where BIA does cause a nonzero
splitting (cf. Sec. 5.5). In Sec. 4.3, an extension to EBOM capable of describing BIA
effects in the conduction band is presented. In Sec. 4.4 the method is applied to bulk
GaSb. Sec. 4.5 shows the application of the method to a symmetric AlSb/GaSb superlattice, predicting the appearance of spin splitting in the conduction band. This is
a distinct feature of this method, while standard k · p implementations or the original

38
EBOM formulation would predict the absence of splitting. These results are shown
to agree with k · p calculations that do account for BIA (cf. Chapter 5). Finally, a
summary of the results is presented.

4.2

Review of the EBOM method

The basic idea of the EBOM method is to take the TB Hamiltonian expressed in
a bond orbital model basis set, series expand it for small k’s and then compare the
matrix elements with the k · p matrix elements [12] to obtain the TB parameters in
terms of the k · p parameters.
Following Chang [9], the orbitals are located at points of an fcc lattice. A state
at a site R with character α = s, x, y or z is labeled |R, αi. The bond orbitals are
taken to be the Löwdin symmetrized orbitals [13] that most closely resemble the top
of the valence and bottom of the conduction band states. The requirement that they
originate from linear combination of atomic orbitals in a unit cell does not really
need to be made. The success of the method (and of the k · p method as well) comes
precisely from the fact that detailed knowledge of the wavefunctions isn’t needed
because the matrix elements are only determined on symmetry grounds plus a few
parameters fitted empirically [1, 14].
The interaction between the p-type orbitals |R, βi and |R0 , β 0 i (β = x, y, z) is, for
the fcc lattice, given by [10]:
hR, β |H| R0 , β 0 i =
Ep δR,R0 δβ,β 0 +
δR0 −R,τ Exy τβ τβ 0 (1 − δβ,β 0 ) + Exx τβ2 + Ezz (1 − τβ2 ) δβ,β 0 , (4.1)

where Ep is the on-site energy and Exy , Exx and Ezz are different nearest-neighbors
interaction parameters. The vectors τ join the twelve nearest neighbors, and they
can have values
τ=

([±1, ±1, 0], [±1, 0, ±1], [0, ±1, ±1]) ,

(4.2)

39
with a being the lattice constant.
The interaction of s orbitals is quite simple
hR, s |H| R0 , si = Es δR,R0 +

Ess δR0 −R,τ ,

(4.3)

with Es and Ess having analogous meanings as Ep and Exx . The remaining interaction
is between the s- and p-like orbitals at nearest neighbor sites:
hR, s |H| R0 , βi = Esx δR0 −R,τ τβ .

(4.4)

At this point, a clarification must be made. The inadequacy of the ordinary EBOM
method to describe the reduced Td symmetry of zincblendes does not originate from
the basis set being located on an fcc lattice, but rather on the fact that a basis with
a definite parity has been used to obtain Eqs. (4.1)-(4.4). In the next section this
assumption is relaxed, yielding the correction necessary to describe spin splitting in
the conduction band.
From the Löwdin functions |R, αi, Bloch sums can be written in the form
1 X ik·R
|R, αi,
|k, αi = √
N R

(4.5)

where N is the number of unit cells in the sample. Each eigenstate with a wavevector
k is written as a linear combination of Bloch sums [15]:
|Ψk i =

uα |k, αi.

(4.6)

The coefficients uα are found by seeking stationary values of hΨk |H| Ψk i /hΨk |Ψk i,
which leads to the diagonalization of a Hamiltonian with matrix elements
hk, α |H| k, α0 i =

1 X ik·(R0 −R)
hR, α |H| R0 , α0 i =
eik·R hR = 0, α |H| R0 , α0 i .
N R,R0
R0

(4.7)

40
Parameters

Es =Ec + 12 A +~a2/2m0
Ess =− A +~a2/2m0
0 +4M
Ep =Ev + 2 3~ /2m0a+2L
0 +2L
Exx =− ~ /2m
2a
−2L0 +4M
Ezz =− ~ /2m02a
Exy =Exy (110) = − Na2
a) Esx = 4a
or b) (Ezz − Exx ) = Xhl /8 = 0.5 eV
Table 4.1: Relationship between the EBOM parameters and the k · p parameters.
For the p subblock, plugging Eq. (4.1) into Eq. (4.7) yields
Hβ,β 0 (k) = Ep δβ,β 0 +

eik·τ Exy τβ τβ 0 (1 − δβ,β 0 ) + Exx τβ2 + Ezz (1 − τβ2 ) δβ,β 0 .

(4.8)

Similarly, it is easy to see that
Hs,s (k) = Es +

eik·τ Ess

(4.9)

eik·τ Esx τβ .

(4.10)

and
Hs,β (k) =

In order to find values for the EBOM parameters, the sums over first neighbors
in Eqs. (4.8)-(4.10) are evaluated. For example, it is easy to see that
Hs,x = 4iEsx sin ξ(cos η + cos ζ),

(4.11)

where ξ = kx a/2, η = ky a/2 and ζ = kz a/2. This agrees with the value in Table II
of Ref. [1], provided that terms occupied in the sc but not in the fcc lattices are
disregarded.
Then, the matrix elements are series expanded up to second order in k and compared to k · p matrix elements [14] to obtain the relations listed in Table 4.1. The

values of the k · p parameters L0 , N 0 in terms of the more common L, N are available,

41
for example, Eq. (13) in Ref. [16]. Note that the last entry in that table is not totally
determined. Taking Esx = P/4a might seem the sensible thing to do, but it produces
spurious solutions [9]. Instead, the auxiliary constraint (Ezz − Exx ) = Xhl /8, where
Xhl is the heavy hole–light hole separation at the X point, is used.
Spin-orbit effects are simply introduced by adding spin to the basis states, performing a change of basis on the Hamiltonian into a |j, mi basis and then modifying
the diagonal components of the energies to include the spin-orbit splitting.

4.3

Inclusion of bulk inversion asymmetry effects
in EBOM

As previously indicated, the EBOM Hamiltonian in zincblendes reproduces an Oh
point group symmetry rather than the reduced Td because the basis states (specifically, the p states) are implicitly assumed to be parity eigenstates. The simplest
way to introduce an inversion symmetry breaking component consistent with the Γ 5 1
symmetry of the valence states is to add some d character to the p states. Thus, the
substitutions
|R, xi → cp |R, xi + cd |R, yzi
|R, yi → cp |R, yi + cd |R, zxi
|R, zi → cp |R, zi + cd |R, xyi

(4.12)

are made, where cp and cd are taken to be real and weight the importance of the odd
and even (under inversion) components, respectively, in the new state.
Keeping with the matrix element Hs,x as an example, the change in the states will
transform it to
Hs,x = cp hk, s |H| k, xi + cd hk, s |H| k, yzi .

The KDWS notation is being used (see Ref. [17]).

(4.13)

42
Looking up again in Table II of Ref. [1], one can see that
Hs,x = 4iEsx sin ξ(cos η + cos ζ) − 4Es,xy sin η sin ζ,

(4.14)

where the coefficients ci have been absorbed into the adjustable parameters Ei . Now,
comparing Eq. (4.14) with the corresponding element in the k · p Hamiltonian in
Ref. [14], one sees that the parameter B describing the BIA in the k · p formalism
(see Sec. 5.3) can be introduced in EBOM by taking
Es,xy = −B/a2 .

(4.15)

Therefore, the inclusion of BIA is made at a negligible computational cost and
its implementation is straightforward because only a supplemental matrix element
is being added instead of extending the basis set to include, say, anion and cation
orbitals. Another computational advantage is that the number of neighbors included
in the calculation is not increased.
It remains to be seen how the remaining matrix elements are affected by the
substitutions (4.12). Hs,s is left unchanged, while the other diagonal elements become
Hx,x = |cp |2 [4Exx (cos ξ cos η + cos ξ cos ζ) + 4Ezz cos η cos ζ]+
|cd |2 [4Exy,xy (110) cos η cos ζ + 4Exy,xy (011)(cos ξ cos η + cos ξ cos ζ)] =
[|cp |2 4Exx + |cd |2 4Exy,xy (011)](cos ξ cos η + cos ξ cos ζ)+
[|cp |2 4Ezz + |cd |2 4Exy,xy (110)] cos η cos ζ =
4Exx (cos ξ cos η + cos ξ cos ζ) + 4Ezz cos η cos ζ, (4.16)
where in the last step Exx and Ezz have been redefined so that Table 4.1 still holds.
The other diagonal elements can be obtained by the appropriate cyclic permutations.

43
The nondiagonal elements between Γ5 states also change:
Hx,y = |cp |2 [−4Exy sin ξ sin η] + |cd |2 [−4Exy,xz (011) sin ξ sin η]+
cp cd [4iEx,xy (110) cos ξ sin ζ + 4iEx,xy (011) cos η sin ζ]+
cd cp [−4iEx,xy (110) cos η sin ζ − 4iEx,xy (011) cos ξ sin ζ] =
− 4Exy (110) sin ξ sin η − 4iExy (011) sin ζ[cos ξ − cos η], (4.17)
with the usual redefinition of parameters in the last step. Hy,z and Hz,x are obtained
by cyclic permutations. The results here obtained for the tight binding zincblende
matrix elements agree with those of Hass et al. [18], which correct the misprints in
Table V of Ref. [1].
Comparison with the k · p Hamiltonian doesn’t provide the value of the Exy (011)

parameter because it only introduces terms of order k 3 or higher when the corresponding matrix element is expanded. This should not be a concern when properties
are sought involving only states near Γ. A look at the matrix elements reveals that
the contribution of Exy (011) reaches its peak near the K point. Since only properties
near the zone center are of interest here, its value will be set to zero for the following
calculations.

4.4

Bulk GaSb

The considerations above are illustrated with an example calculation of bulk GaSb.
Figure 4.1 shows the bands of bulk GaSb for a few special directions. The dashed
and dotted lines in plot (a) correspond to the EBOM model without the zincblende
symmetry corrections. The dotted line is obtained under the original requirement [9]
that the separation Xhl = 4.0 eV is used to obtain Esx . This will be called X model.
, with P obtained from the value of the
The dashed line is obtained taking Esx = 4a

effective mass given by Eq. (5.17). This will be called P model. With the parameters
used, looking at the X and L points, in the P model the conduction (CB) and split
off (SO) bands are pushed further apart than in the X model. In the P model, the

Energy [eV]

-2
-4
-6

-8

-10

-12

-12
-14

-14

U,K

Figure 4.1: Band structure of GaSb calculated with EBOM under different assumptions for the parameters. The dotted line in plot (a) is obtained under the original
requirement [9] that the separation Xhl = 4.0 eV is used to obtain Esx . The dashed
. A term describing BIA has been included in plot (b),
line is obtained taking Esx = 4a
which otherwise uses the same set of parameters as in the calculations represented by
the dashed line.
positions of the CB and SO bands are very sensitive to the value of the CB effective
mass. For example, changing m∗c in InAs from 0.025 to 0.024 changes the position
of the SO band at the X point from about -10 eV to about -6.5 eV. Going one step
further and setting m∗c = 0.023 makes the SO band anticross with the heavy and
light holes, and the spurious heavy hole (HH) - light hole (LH) crossing described in
Ref. [9] appears. Therefore, it is reasonable to assume that very small changes in the
value of the m∗c parameter can not only get rid of spurious solutions present in the P
model but also tune the position of the CB at the X point.
Plot (b) in Fig. 4.1 is generated under the same conditions as model P , but with
BIA effects turned on by letting Es,xy = −B/a2 . This will be called P B model. In
agreement with predictions from the character tables for the Td group [17], the bands
become spin split in the Σ direction because of the breakdown of Kramers degeneracy.
However, the correct description of the zincblende symmetry is made at the cost of

45
1.2

0.015

0.8
Energy [eV]

CB Splitting
Fit to Splitting=γck

0.010

0.005

Energy [eV]

0.4
0.000
-0.05

-0.04

-0.03

-0.02

-0.01

0.00

k [2π/a]

0.0

-0.4

-0.8
-0.04

[110]

-0.02

0.00

k [2π/a]

0.02

0.04

[100]

Figure 4.2: Bands close to the zone center showing the spin splitting, calculated with
EBOM. The inset shows the amount of CB splitting and its k 3 dependence at low
values of k.
the loss of accuracy for the CB and SO bands, specially along the Λ line, where they
take values quite far from pseudopotential calculations [19]. The preference of having
a correct description of the bands near the Γ point or the ∆ line including spin—with
its ability to describe short period 100 superlattices—or a more accurate full zone
description will determine the model to be used. The inclusion of second nearest
neighbors matrix elements [20] might reconcile the energy values at the L point in

46
the P B model with the pseudopotential calculations and experimental findings [21].
Figure 4.2 shows the bands in more detail close to the zone center, with the spin
splitting in the bands along the [110] direction. The inset shows the splitting in the
conduction band along the Σ line for the CB, and a fit using
Splitting = γc k 3 ,

(4.18)

where γc is the k 3 splitting proportionality constant. The value used for γc is 186
eV ·Å3 , in good agreement with the measured value of 187 eV ·Å3 [22]. This shows
that the parameter B determines the CB splitting near the zone center in the P B
model in the same way as it does in the k · p method, as expected from the derivation
in Sec. 4.3. A look at the inset reveals that, for GaSb, expression (4.18) is good
until about 2% of the zone edge. The only qualitative aspect of the bulk bands
that the extension in Sec. 4.3 cannot incorporate is the linear spin splitting in the
valence bands close to the zone center [23] (cf. Fig. 5.2). In k · p, this is described
by a parameter C coming from second-order mixed k · p and spin-orbit terms in the
perturbation expansion [14, 24]. It is this relation of C to the spin orbit interaction
that makes it impossible to include its effects in the EBOM method. This is because
the starting tight binding formulation includes spin orbit effects only in a limited
and ad hoc fashion. Boykin [25] has extended the tight binding method to include
the linear k splittings in the valence band, but the fact that four extra parameters
are needed in his treatment while a single one does the job in k · p suggests that a
tight binding formulation with spin orbit effects included from the beginning should
yield the linear splitting naturally and reveal constraints due to symmetry between
Boykin’s parameters. In any case, its effects are normally small, and its importance
for heterostructures is studied, for a particular case, in the following section.

47
1.8

1.6

k.p
EBOM

1.4
18
16

1.0

0.8

14
Splitting E1 [meV]

Energy [eV]

1.2

12
10

αk.p=22x10

αEBOM=20x10

-12

eV.m
-12

eV.m

0.00

0.01

0.02

0.03

0.04

0.05

-1

k || [100] [Å ]

0.6

HH1
0.4

LH1
0.2

HH2
-0.04

-0.02

0.00

0.02

0.04

-1

k || [100] [Å ]
Figure 4.3: Comparison of EBOM and k · p superlattice bands. The structure is an
8/8/8 AlSb/GaSb/AlSb SL. The solid (dotted) lines are the k · p (EBOM) results.
The bands are spin split away from Γ due to the bulk inversion asymmetry. The inset
shows, for both methods, the amount of splitting in the E1 band and the values for
the splitting coefficients as defined in Eq. (5.69).

4.5

Bulk inversion asymmetry effects in symmetric superlattices

The extension of the EBOM method in Sec. 4.3 is tested with the calculation of the
band structure of an AlSb/GaSb/AlSb symmetric superlattice (SL). The reduction
of the symmetry due to the confinement causes the states in the CB to become spin
split even along the [100] direction [26], contrary to the predictions of most of the
“oversymmetrized” k · p implementations.

48
Figure 4.3 shows the comparison of the bands of an 16/8 AlSb/GaSb SL calculated
by both the k · p and the EBOM methods. In the k · p calculation, the parameters B
and C describing BIA are both set to finite values for GaSb. In the EBOM calculation,
as stated previously, only B can be set. Control calculations have been performed
with C = 0 and C 6= 0 for this structure and for a 16/8/8 AlSb/GaSb/InAs SL,
always finding that the inclusion of C modified the splittings only by a few tenths
of meV. Thus, at least for this system, the inability of the P B model to describe
the linear splitting in the valence bands of bulk zincblendes does not constitute a
serious drawback when studying splittings in heterostructures. The solid (dotted)
lines correspond to the k · p (EBOM) results. It can be seen that the value of the
gap is similar, although the absolute position of the levels is slightly different. In the
inset, the amount of splitting between the E1 subbands is shown, with both methods
yielding similar results.

4.6

Summary

Summarizing, an extension to Chang’s EBOM method [9] for calculating full zone
band structures has been presented. This extension can describe the most important
effects of bulk inversion asymmetry in zincblendes [23]. Also, the problem of spurious
solutions in the original formulation has been shown to be solvable with small changes
in the parameters. Finally, the new method has been applied to the calculation of
bulk GaSb and an AlSb/GaSb/AlSb superlattice, and shown to have good agreement
with k · p results close to the zone center.

Bibliography
[1] J. C. Slater and G. F. Koster, Phys. Rev. 94, 1498 (1954).
[2] M. Asdente and J. Friedel, Phys. Rev. 124, 384 (1961).
[3] M. J. Mehl and D. A. Papaconstantopoulos, Phys. Rev. B 54, 4519 (1996).
[4] J. M. Jancu, R. Scholz, F. Beltram, and F. Bassani, Phys. Rev. B 57, 6493
(1998).
[5] J. N. Schulman and Y. C. Chang, Phys. Rev. B 24, 4445 (1981).
[6] J. N. Schulman and Y. C. Chang, Phys. Rev. B 27, 2346 (1983).
[7] T. Matsuoka, T. Nakazawa, T. Ohya, K. Taniguchi, C. Hamaguchi, H. Kato, and
Y. Watanabe, Phys. Rev. B 43, 11798 (1991).
[8] M. K. Jackson, D. Ting, D. H. Chow, D. A. Collins, J. R. Soderstrom, and T. C.
McGill, Phys. Rev. B 43, 4856 (1991).
[9] Y. C. Chang, Phys. Rev. B 37, 8215 (1988).
[10] G. T. Einevoll and Y. C. Chang, Phys. Rev. B 40, 9683 (1989).
[11] W. A. Harrison, Phys. Rev. B 8, 4487 (1973).
[12] E. O. Kane, J. Phys. Chem. Solids 1, 82 (1956).
[13] P.-O. Löwdin, J. Chem. Phys. 18, 365 (1950).
[14] E. O. Kane, in Semiconductors and Semimetals, edited by R. K. Willardson and
A. C. Beer (Academic, New York, 1966), Vol. 1, pp. 75–100.
[15] W. A. Harrison, Electronic Structure and the Properties of Solids, 1st ed. (Dover,
Mineola, USA, 1989).

50
[16] R. Enderlein, G. M. Sipahi, L. M. R. Scolfaro, and J. R. Leite, Phys. Stat. Sol.
(B) 206, 623 (1998).
[17] G. F. Koster, J. O. Dimmock, R. G. Wheeler, and H. Statz, Properties of the
Thirty-Two Point Groups, 1st ed. (M.I.T. Press, Cambridge, MA, USA, 1963).
[18] K. C. Hass, H. Ehrenreich, and B. Velicky, Phys. Rev. B 27, 1088 (1983).
[19] J. R. Chelikowski and M. L. Cohen, Phys. Rev. B 14, 556 (1976).
[20] J. P. Loehr, Phys. Rev. B 50, 5429 (1994).
[21] T. C. Chiang and D. E. Eastman, Phys. Rev. B 22, 2940 (1980).
[22] G. E. Pikus, V. A. Marushchak, and A. N. Titkov, Soviet Physics
Semiconductors-USSR 22, 115 (1988).
[23] G. Dresselhaus, Phys. Rev. 100, 580 (1955).
[24] M. Cardona, N. E. Christensen, and G. Fasol, Phys. Rev. B 38, 1806 (1988).
[25] T. B. Boykin, Phys. Rev. B 57, 1620 (1998).
[26] R. Eppenga and M. F. H. Schuurmans, Phys. Rev. B 37, 10923 (1988).

Chapter 5

Bulk inversion asymmetry

effects on the bands of zincblende
heterostructures
5.1

Introduction

In recent years, interest in developing spin-sensitive devices (spintronics) [1–4] has
fueled renewed investigations into spin phenomena in semiconductors. The aim is
to control not only the spatial degrees of freedom of the electron, but also the spin
degree of freedom. Useful spintronic devices can be devised if such control is achieved.
A number of such devices have already been proposed [5–7]. If a full understanding
of the operation of spintronic devices is wanted, a thorough knowledge of the band
structure including all spin details will be needed; in much the same way as thorough
knowledge of the “regular” band structure is required to fully understand electronic
devices.
In this chapter, the band structure of AlSb/GaSb/AlSb and AlSb/InAs/GaSb/
AlSb quantum wells is calculated using the k · p method taking into account spinorbit effects. Among others, the coupling through the spin-orbit interaction between
electron and hole states is taken into account, leading to the appearance of a spin
splitting in the conduction states. This splitting can be viewed as a consequence of
the removal of inversion symmetry in these heterostructures. The contribution to the
splitting of the different sources of asymmetry will be explored quantitatively. Thus,
the apparently contradictory statements of Lommer et al. [8] and Cardona et al. [9]
regarding this matter can be clarified.
Following the literature, throughout this chapter the term spin splitting will be
used to refer to the splitting of levels otherwise degenerate due to Kramers degeneracy,

52
even though in the valence band the total angular momentum is the correct quantum
number to use.
Section 5.2 provides introductory remarks about how the breakdown of Kramers’
degeneracy translates into spin splitting, and also quotes the apparently contradictory
positions on the magnitude of the contributions to the splitting. In Sec. 5.3 the
method of the invariants for the construction of a Hamiltonian with the correct point
group symmetry is described. The validity of the constructed Hamiltonian is tested
by obtaining analytical expressions for the dispersion relation and then comparing
them to known expressions. Section 5.4 describes how the bulk Hamiltonian yields
the effective mass approximation equations and how they are solved. In Secs. 5.5 and
5.6 the effects of bulk inversion asymmetry on symmetric and asymmetric quantum
wells are explored and the results are finally summarized in Sec. 5.7.

5.2

Background

The appearance of spin splitting in the electronic band structure can be viewed as a
consequence of the removal of inversion symmetry. In systems possessing inversion
symmetry, the argument for the existence of spin degeneracy, most widely known as
Kramers’ degeneracy, goes as follows:
I. Time reversal is always a property of the system1 . Since the Hamiltonian is
invariant under time reversal, the following pair of eigenstates linked by the
time reversal operator will be degenerate:
|k, ↑i −→ |−k, ↓i,

(5.1)

where Θ is the time reversal operator.
II. If the system possesses inversion symmetry, the following pair of states will also

In ferromagnetic materials, time reversal must be applied to the whole system, including the
ions responsible for the magnetic moment. This makes the following analysis invalid for that case.

53
be degenerate:
|k, ↑i −→ |−k, ↑i,

(5.2)

with I being the inversion operator.
III. And finally, the sequential action of these two operators would yield a pair of
degenerate eigenstates at the same k point in the Brillouin zone:
IΘ
|k, ↑i −→ |k, ↓i.

(5.3)

In the case of bulk zincblendes, the splitting rising from the lack of inversion
symmetry is commonly called the k 3 splitting due to its dependence near the zone
center [10]. Along the [110] direction, this splitting can be written for the conduction
band as [11]
∆BIA [110] = γc k 3 ,

(5.4)

where ∆BIA [110] is the splitting due to the bulk induced asymmetry (BIA), k is the
modulus of the wavevector and γc is the proportionality constant. Traditionally, the
term in the Hamiltonian leading to this splitting has been omitted in k · p calculations [12, 13].
However, in the case of quantum wells, other sources of inversion asymmetry can
be present. These sources include the different composition of the confining layers, an
asymmetric doping profile, an applied external electric field. All these mechanisms
are grouped into the so-called structure induced asymmetry (SIA) [14].
There have been some contradictory statements in the literature regarding the
relative contribution of SIA versus that of BIA in asymmetric heterostructures. In
Ref. [8] Lommer, Malcher and Rössler say, “Spin splitting of subband states . . . is
ascribed to the inversion-asymmetry-induced bulk k 3 term, which dominates in large
gap materials, and to the interface spin-orbit or Rashba term, which becomes important in narrow-gap systems.” On the other hand, Cardona, Christensen and Faso̧l [9]
compile the measured coefficients γc shown in Table 5.1. From there it can be seen
that, for the compounds shown, the k 3 splitting is bigger in narrow gap than in wide

54
Material
GaAs
InP
GaSb
InAs
InSb

Band Gap (eV ) γc (eV ·Å3 )
1.52
25.5
1.42
8.5
0.81
186.3
0.418
130
0.235
226.8

Table 5.1: Band gap and γc for selected III-Vs, adapted from [9]. The value for InAs
has been taken from [11].
gap materials. The numerical calculations performed in the following sections will
clarify these statements.

5.3

k · p method applied to bulk III-Vs

In order to calculate the band structure of the heterostructures under study, the
effective mass approximation (EMA) [13] based on an 8-band k · p formalism is used.
There are several published 8-band k · p Hamiltonians [13], each including a more or
less detailed set of effects.
For the calculations shown, the Hamiltonian constructed by Trebin et al. [15] has
been implemented. This 8-band k · p Hamiltonian is constructed solely on group
theory arguments and, when applied correctly, guarantees the inclusion of all matrix
elements compatible with the Td symmetry group of the zincblendes up to the desired
order in the electron wavevector k.

5.3.1

Invariant expansion of the Hamiltonian

This method to construct k · p Hamiltonians was first outlined by Luttinger [16], and
developed generally by Bir and Pikus [17–19]. It is based solely on symmetry arguments. For the case of zincblendes, it basically goes as follows. The bulk Hamiltonian
can be written in the following way:
H=

i,j∈Td irreps

H ij ,

(5.5)

55
where the H ij operator corresponds to the block in the Hamiltonian matrix coupling
the states of the Γj irreducible representation (irrep) to the states of Γi . Taking |Γi , ki
defined as the k-th basis state of the Γi irrep of the Td point group, these operators
can be written explicitly as:
H ij =

X X

k∈Γi m∈Γj

Hkm (k) |Γi , kihΓj , m|,

(5.6)

where Hkm (k) are the matrix elements, and they are a function of the electron
wavevector k only.
Now, each of the matrices |Γi , kihΓj , m| in H ij can be thought of as a vector that

will transform according to the product group representation Γi ⊗ Γ∗j of Td . It is more
convenient to work with linear combinations of the |Γi , kihΓj , m|’s that transform

according to the irreps contained in Γi ⊗ Γ∗j .

Since it is being required that H is invariant under the point group operations,
each of the H ij must transform according to the trivial irreducible representation,
where all the group elements are represented by the unity. To achieve this, the
Hamiltonian blocks must have the following form [19]:
H ij (K) =

Γl ∈Γi ⊗Γ∗j

m∈Γl

¡ Γ ¢∗
Γl
Xm
Kml ,

(5.7)

where the first sum is carried over the irreps contained in Γi ⊗ Γ∗j , the second one
over the elements of Γl , K is a general tensor from the components of k, the strain
¡ Γ ¢∗
² and the magnetic field H; Km
is the complex conjugate of the m-th irreducible

Γl
component of K, Xm
is the m-th basis matrix coming transforming as Γi ⊗ Γ∗j , and

the al ’s are parameters that later can be mapped into standard k · p parameters.
In the case of interest here, the irreps building the Hamiltonian blocks are the
Γ6 , Γ8 and Γ7 of Td , corresponding to the conduction, valence and split off bands
respectively.
To exemplify the method, consider the H 66 block of the Hamiltonian, containing
all the possible coupling of the conduction bands states among themselves. The first

56
Block
H 66

Representations
Γ6 ⊗ Γ∗6 = Γ1 + Γ4

Matrices
Γ1 : 1σ
Γ4 : σx , σy , σz

Table 5.2: Matrix basis set for the block of the Hamiltonian coupling the conduction
band states to themselves (adapted from [15]). 1σ is the 2 × 2 identity matrix, and
the σi ’s are the Pauli matrices.
Terms involving k and H
Γ1 : 1,
√ k ¡ 2 1 2¢ 2
Γ3 : 3 kz − 3 k , kx − ky2
Γ4 : [ky , kz ] = eHx /i~c, [kz , kx ] = eHy /i~c, [kx , ky ] = eHz /i~c
Γ5 : kx , ky , kz ; {ky kz }, {kz kx }, {kx ky }
{ky kz } ≡ 12 (ky kz + kz ky )

Γl
of K (adapted from [15]).
Table 5.3: Irreducible components Km

entry of Table II in Ref. [15] shows that the basis matrices of that block can only
transform as Γ1 and Γ4 , and lists the explicit matrices (see Table 5.2). Therefore, only
the combinations of the components of k up to the desired (second) order transforming
as Γ1 and Γ4 will play a role. These are shown in Table 5.3.
Now, by application of Eq. (5.7), the most general H 66 compatible with the symmetry requirements can be constructed:
H 66 = a1 1σ + a01 k 2 1σ + a4

(σx Hx + σy Hy + σz Hz ) ,
i~c

(5.8)

where 1σ is the 2 × 2 identity matrix, and the σi ’s are the Pauli matrices. For clarity,
the strain and mixed k-strain terms are not shown. In order to find the value of the
ai parameters, one just needs to compare with the k · p perturbation results, and it

can be seen that a1 corresponds to the energy of the conduction band edge, a01 must
i~
be 2m
and a4 must be −gs 4m
, with gs being the effective g factor.

For completeness, Appendix A reproduces the full 8-band k · p Hamiltonian from
Ref. [15], which has been implemented to perform the bulk and heterostructure calculations. Note that, due to the way that it has been constructed, this Hamiltonian
takes into account all the effects of the spin-orbit interaction in the matrix elements
up to k 2 , and in particular the s − p coupling responsible for the existence of spin

57
splitting in the conduction band. Strain and coupled strain/spin-orbit effects are also
properly described by this method.
To minimize the probability of introducing hard-to-detect typos in the coded definition of the matrix elements, the Hamiltonian was first entered in Mathematica [20].
From there, the appropriate C code for each matrix element was generated automatically with the instruction CForm. This method also has the advantage that it allows
the algebraic operation of the Hamiltonian to find analytical forms for the dispersion
relation very near the zone center (cf. Sec. 5.3.3).
When implementing the Hamiltonian described in [15], one must be careful to
note the following typos in appearing in that article. The matrix Txx in Table I there
must read

√ 
−1 0
Txx = √  √
3 2 − 3 0 1 0

(5.9)

and the last equation in the group (A3) must also be corrected:
(2)
(2)
X12 = −i X2 − X−2 /2.

(5.10)

There is another remark about a point that can lead to confusion. Koster et
al. [21] have developed a set of tables for the Clebsch-Gordan coefficients for point
groups that are very helpful when constructing explicit subspace-invariant matrices
or when checking the symmetry properties of the Hamiltonian. However, in their
Table 83 the values they show can be used as displayed for the O point group, but
for Td the values should be taken according to the lookup table shown in Table 8.3.

5.3.2

Parameters of the model

When implementing a k · p Hamiltonian, one of the most crucial tasks is obtaining
a good set of parameters for the model. Unfortunately, this task is made harder
because there are in the literature a number of models that vary in the number of
bands considered and, even within the same number of basis states, with a different
set of effects included [22, 23].

Parameter
Eg
Ev
∆SO
A0
γ1
γ2
γ3

C1
Dd
Du
Du0
C2
C4
C50
C11 , C12 , C44

Description
Lattice constant.
Energy gap.
Position of the valence band edge.
Spin orbit splitting.
Correction to the conduction effective mass due to the interaction with farther bands.
First Kane parameter. Related to the effective mass of heavy
and light holes.
Second Kane parameter. Related to the effective mass of heavy
and light holes.
Third Kane parameter. Measures the warping—directional dependence of the effective mass—of the heavy and light hole
bands.
Related to the hole g-factor.
Related to the anisotropy of the hole g-factor respect to the
direction of an applied uniaxial stress [24].
Linear splitting coefficient of the HH-LH bands due to the bulk
inversion asymmetry. Measures the intra-valence band coupling
due to the atomic electric field.
Momentum irreducible matrix element.
Measures momentum coupling of conduction and valence band
via farther Γ5 bands. Contributes to most of the k 3 splitting
in the conduction band. Vanishes when the crystal possesses
inversion symmetry.
Change in the conduction band edge due to hydrostatic strain.
Change in the valence band edge due to hydrostatic strain.
Related to the Bir-Pikus b deformation potential [19].
Related to the Bir-Pikus d deformation potential [19].
Measures coupling of the conduction band to the valence band
under shear stress.
Related to uniaxial strain-k coupling.
Related to shear strain-k coupling.
Elastic moduli.

Table 5.4: Material parameters used in the k · p model and their description.

59
When taking a parameter set from the literature, one must make sure that parameters with the same name have the same meaning. The Luttinger parameters
γL [16] are one example of this. The values of the 8-band Luttinger parameters, also
known as Kane parameters, differ greatly from the more widely used 6-band Luttinger
parameters. The γL shown in Table II in Lawaetz [22] are related to the γ’s in the
current implementation (cf. Appendix A) by
1 EP
3 Eg
1 EP
γ2 = γ2L −
6 Eg
1 EP
γ3 = γ3L −
6 Eg
1 EP
κ = κL −
3 Eg

γ1 = γ1L −

(5.11)

where Eg is the energy gap of the compound, and EP has been defined as
EP ≡

2me P 2
~2

(5.12)

where me is the free electron mass and P is the irreducible momentum matrix element.
A list of the parameters employed in the implementation of the Hamiltonian in
Appendix A [15] and their meaning are shown in Table 5.4. Table 5.5 shows the
numerical values used in the calculations for the parameters of a number of materials.
The effects of an applied magnetic field will not be considered in the calculations
in this chapter. Therefore, the parameters κ and q will not be needed. Similarly,
C2 , C4 and C50 will not be needed because they only appear when the material is
under some sort of shear stress. When finding energy values for heterostructures in
Secs. 5.5 and 5.6, they will be supposed to be grown along the [001] direction, hence
introducing no shear stress. Finally, the effect of remote bands on the conduction
effective mass will be neglected. This amounts to setting the A0 parameter to zero.

InSb
6.4794
0.235
0.803
2.59
-0.6
0.67
-9.32×10−3
9.35
10.3
-6.17
0.36
-3.1
-4.3
69.18
37.88
31.32

GaSb
6.096
0.813
0.56
0.8
2.58
-0.58
0.65
7.00×10−4
9.21
49.9
-6.85
0.79
-3
-4.2
88.34
40.23
43.22

AlSb
6.136
2.219
0.11
0.75
1.44
-0.35
0.39
8.41
-6.97
1.38
-2.1
-3.7
87.69
43.41
40.76

InAs
6.058
0.356
0.41
2.05
-0.44
0.48
-1.12×10−2
9.17
13.7f
-5.08
-2.7
-3.1
83.29
45.26
39.59

GaAs
5.653
1.52
0.341
2.01
-0.41
0.46
-3.40×10−3
9.86
30.4
-7.17
1.16
-2.6
-3.9
112.6
57.1
60

AlAs
5.66
3.002
-0.55
0.279
1.74
-0.37
0.42
2.00×10−3
8.94
21.3g
-5.64
2.47
-2.6
-3.9
120.2
57
58.9

Al.08 Ga.92 As
5.6539
1.639
-0.044
0.336
-0.5
0.45
-2.97×10−3
9.79
29.7
-7.05
1.26
-2.6
-3.9
113.2
57.02
59.91

Al.12 Ga.88 As
5.6541
1.633
-0.055
0.335
1.99
-0.53
0.44
-2.86×10−3
9.77
29.5
-7.02
1.29
-2.6
-3.9
113.4
57.09
59.89

Ref. [25].
Ref. [22].
The valence band offsets are consistent within the systems comprised of (InSb), (GaSb, AlSb, InAs) and (Al x Ga1−x As).
Ref. [9].
From γc obtained in Ref. [26].
From γc obtained in Ref. [11].
From γc obtained in Ref. [27].
Adapted from the Bir-Pikus deformation potentials a, b, d in Ref. [28].

Table 5.5: Parameter values for some materials.

Parameter
a (Å)a
Eg (eV )b
Ev (eV )c
∆SO (eV )b
γ1b
γ2b
γ3b
C (eV ·Å)d
P (eV ·Å)b
B (eV ·Å2 )e
C1 (eV )h
Dd (eV )h
Du (eV )h
Du0 (eV )h
C11 (GPa)a
C12 (GPa)a
C44 (GPa)a

5.3.3

Analytic expressions of the energy values near the zone
center

Starting from the full Hamiltonian shown in Appendix A, analytical expressions for
the bands near the zone center can be found. These are useful to find measurable
quantities such as effective masses and intraband splittings as a function of the model
parameters. These expressions can also be useful when relating the parameters of
the model to the parameters used in other families of k · p Hamiltonians. Unstrained
bulk material will be assumed in this section.
In order to obtain an analytic approximation to the energy dispersion relation, the
8-band Hamiltonian is first divided into two 4 × 4 blocks for the direction required.
To achieve this decoupling, the Hamiltonian in Appendix A is expressed in a basis
whose quantization axis has been selected according to Table 5.6. The matrix that
transforms the Hamiltonian from a z quantization axis into an arbitrary θ, ϕ axis is

D1/2 (θ, ϕ)

R(θ, ϕ) = 

D1/2 (θ, ϕ) = 

D3/2 (θ, ϕ) =
θ 3
cos( 2 )

3/2

where

and

(θ, ϕ)

D1/2 (θ, ϕ)

e− 2 ϕ cos( 2θ ) −e− 2 ϕ sin( 2θ )

3 cos( θ2 ) sin( θ2 )

sin( 2θ )

cos( 2θ )

3 cos( θ2 ) sin( θ2 )

(5.13)

(5.14)

sin( θ2 )

3i
3i
3i
 √ e 32i 2ϕ
e2 ϕ
e2 ϕ
e2 ϕ
3θ
3θ
θ 2 
 3 cos( θ ) sin( θ )
cos(
)+3
cos(
sin(
)−3
sin(
sin(
cos(
e2 ϕ
4e2 ϕ
4e2 ϕ
e2 ϕ
√
i ϕ
3θ
3θ
−e
3 cos( 2 ) sin( 2 ) 
 3 cos( 2 ) sin ( 2 )
(sin( 2 )−3 sin( 2 ))
(cos( 2 )+3 cos( 2 ))
e− 2 ϕ
e− 2 ϕ
3i
3 e 2 ϕ cos( 2θ ) sin( 2θ )
3 e 2 ϕ cos( 2θ ) sin( 2θ )
e 2 ϕ cos( 2θ )
e 2 ϕ sin( 2θ )

(5.15)

62
Direction in k space
[001]
[010]
[100]
[110]
[111]

Notation for the line Quantization axis
ẑ
ŷ
x̂
ŷ − x̂
ŷ − x̂

Table 5.6: Quantization axis for the basis functions of the Hamiltonian.
After the Hamiltonian has been expressed in the basis with the new quantization
axis, another change of basis consisting of a rearranging of the vectors is applied in order to leave it in its final block diagonal form. The basis states are reordered from the
set |Γ6 , + 21 i, |Γ6 , − 12 i, |Γ8 , + 23 i, |Γ8 , + 21 i, |Γ8 , − 21 i, |Γ8 , − 23 i, |Γ7 , + 21 i, |Γ7 , − 21 i into
|Γ6 , + 21 i, |Γ8 , − 23 i, |Γ8 , + 21 i, |Γ7 , + 21 i, |Γ6 , − 21 i, |Γ8 , + 23 i, |Γ8 , − 12 i, |Γ7 , − 21 i
for the
∆ directions, or |Γ6 , + 12 i, |Γ8 , + 23 i, |Γ8 , − 21 i, |Γ7 , − 21 i, |Γ6 , − 21 i, |Γ8 , − 23 i, |Γ8 , + 21 i,
|Γ7 , + 12 i for the Σ and Λ directions.
Taking one of the 4 × 4 blocks, the secular equation to obtain the eigenvalues

would be a fourth degree polynomial. Instead, the standard procedure of taking the
unperturbed energy for the bands that are not under consideration will require the
solving of two first-degree equations—for the conduction and the split off bands—
and one second-degree polynomial for the heavy and light hole bands to yield the
approximate energies near the zone center.
Bands along the [100] direction
The energy dispersion relation for the conduction band (CB) along the [100] direction
up to second order in k is given by
ECB (k) = Eg +

~2 kx2
2me m∗CB

(5.16)

where me is the free electron mass, m∗CB is the conduction band effective mass, with
a value of
1 EP 3Eg + 2∆SO
=1+
mCB
3 Eg (Eg + ∆SO )

(5.17)

Light Holes
Heavy Holes

0.60

Energy [eV]

0.55

0.50

0.45

C=0
C≠0

0.40
-0.05

0.00

0.05
-1

k || [100] [Å ]
Figure 5.1: Linear splitting of the HH and LH bands along [100] near the zone center
due to bulk induced asymmetry (BIA) effects. The inset compares the bands with
BIA included C 6= 0 to the bands with BIA not included C = 0. The value of C has
been artificially augmented 1,000-fold for C 6= 0.
where the symbols have the meaning listed in Table 5.4. This effective mass is
isotropic—the same along any direction on k space.
For the heavy hole (HH) band, the dispersion relation is
EHH (k) = Ckx −

(3Eg γ1 ~2 /me + 2P 2 ) kx2
6Eg

(5.18)

(3Eg γ1 ~2 /me + 2P 2 ) kx2
6Eg

(5.19)

while for the light hole (LH) band
ELH (k) = −Ckx −

These results indicate that, very close to the Γ point, the valence bands have a
linear behavior. Also, the effective mass seems to be quite different from the usual

64
expressions [29, 30]. This can be reconciled with the conventional wisdom. The plot
in Fig. 5.1 shows the numerical diagonalization of the HH and LH bands close to
the Γ point for GaSb, except that the valence band splitting parameter C has been
increased by a factor or 1,000 to show more clearly its effects. It is seen that very
close to the zone center the HH and LH bands split linearly instead of quadratically,
following the behavior described in Eqs. (5.18)-(5.19); and it is not until farther into
the Brillouin zone that the bands recover the usual quadratic behavior. The inset in
Fig. 5.1 compares these bands to the case where the splitting has been set to zero.
It is seen that even with the inclusion of the bulk inversion asymmetry effects, the
bands recover soon the shape of the C = 0 case. The only difference then is that the
HH-LH separation is slightly bigger than predicted in the models not taking C into
account.
Setting C = 0 in the analytical Hamiltonian and expanding it to second order
yields the more usual expressions for the HH and LH effective masses
= γ1 − 2γ2
m∗HH
4 EP
= γ1 + 2γ2 +
mLH
3 Eg

(5.20)

which have been expressed in terms of the Kane parameters. In terms of the Luttinger
parameters (see Eq. (5.11) ), they adopt the form
= γ1L − 2γ2L
m∗HH
= γ1L + 2γ2L .
m∗LH

(5.21)

Finally, the spin-orbit split off (SO) band has the following dispersion relation
ESO (k) = −∆SO −

~2 kx2
2me m∗SO

(5.22)

65
where m∗SO is the split off band effective mass, with a value of
EP
= γ1 +
mSO
3 (Eg + ∆SO )

(5.23)

The effective mass for the SO band is also isotropic.
Note that the expressions of the effective masses of the CB and SO bands are not
affected by the inclusion of BIA effects.
Bands along the [110] direction
Along the [110] direction, the energy of the electrons in the CB as a function of the
wavevector k is, up to third order,
ECB (k) = Eg +

~2 k 2
± γc k 3 ,
2me mCB 2

(5.24)

with m∗CB given by Eq. (5.17), and the k 3 splitting coefficient (see Eq. (5.4) ) in terms
of the model parameters given by
P 2BEg ∆SO − 3CP (Eg + ∆SO )
γc =
Eg2 (Eg + ∆SO )

(5.25)

It is easy to show that the contribution to γc of the part containing C is only about
4% for InSb and InAs. That contribution goes down to about 0.3% for GaAs and
AlAs, and it drops to a mere 0.03% for GaSb. Therefore, it is a good approximation
to consider that all the splitting in the conduction band is due to the nonvanishing
bulk inversion parameter B, which has its source in the momentum coupling of the
conduction and valence bands via remote Γ5 states [15].
Note that, in order to turn off BIA effects, both parameters B and C need to be
set to zero.
As in the [100] case, the inclusion of C 6= 0 changes the characteristics of the
bands very close to the Γ point respect to the more common assumption of C = 0.
In particular, it provides them with a small linear component. But, in opposition
to the [100] case, here the HH and LH bands are not doubly degenerate. The LH1–

0.570
0.565
0.560
Energy [eV]

∆HH

∆LH

0.555
0.550

LH2
HH2
LH1
HH1

0.545
0.540
-0.010

-0.005

0.000

0.005

0.010

-1

kx=ky [Å ]
Figure 5.2: Linear splitting of the HH and LH bands along [110] near the zone center.
The value of the parameter C has been artificially augmented 1,000-fold.
LH2 and HH1–HH2—the number indicating subbands—linear splittings (∆LH [110]
and ∆HH [110] respectively) turn out to be the same for both HH and LH bands (see
Fig. 5.2), and are given by
∆HH [110] = ∆LH [110] =

3Ck.

(5.26)

This result is slightly different from the one indicated in Ref. [9] in their Eq. (7.5).
A numerical diagonalization of the Hamiltonian has been performed to check the
validity of Eq. (5.26). The discrepancy arises because the splittings in Ref. [9] are
valid in the region where the quadratic (effective mass) splitting predominates, while
the result obtained here is valid in the region where the linear splitting dominates.
In the materials studied, one needs not go far from Γ to enter a regime where the
bands basically behave according to the standard behavior, described by the effective

67
masses
= γ1L − γ2L
+ 3γ3L
m∗HH
γ2L
+ 3γ3L
1L
mLH

(5.27)
(5.28)

which agree with the expressions in Ref. [30]. In this regime, there is some LH and
HH splitting proportional to k 3 due to the effect of B:
+ 3γ3L
BP γ2L − 3γ3L + γ2L
δHH [110] =
k3
3Eg
γ2L
+ 3γ3L
+ 3γ3L
BP 3γ3L − γ2L + γ2L
k3.
δLH [110] =
3Eg
γ2L + 3γ3L

(5.29)
(5.30)

Note that the k 3 splitting in the heavy hole band is a good indicator of the
anisotropy of the hole effective masses, because it should vanish for a material with
isotropic hole effective masses (γ2L = γ3L ).
The SO band also presents k 3 splitting ∆SO [110] , proportional to the B parameter
only:
∆SO [110] =

2BP
k3.
3 (Eg + ∆SO )

(5.31)

Bands along the [111] direction
Along this direction, in the region where the linear splitting dominates, the heavy
hole (HH) band has the dispersion relation
~2 k 2
(γ1L − 2γ3L ) + O(k 4 ),
EHH (k) = ± 2Ckx −
2me

(5.32)

while for the light holes
ELH (k) = −

~2 k 2
(γ1L + 2γ3L ) + O(k 4 ),
2me

(5.33)

68
1.5

Energy [eV]

1.0

HH
0.5

LH
0.0

SO
-0.04

-0.02

0.00

0.02

0.04

-1

k || [100] [Å ]

Figure 5.3: Band structure for GaSb near Γ along [100].
from which
= γ1L − 2γ3L
m∗HH
= γ1L + 2γ3L .
m∗LH

(5.34)
(5.35)

The light hole, conduction and split-off bands are degenerate along the [111] direction, as can also be deduced by group theory arguments [10]. The heavy hole band
shows a linear splitting near the zone center.

5.3.4

Numerical calculation of the energy bands

In the preceding section, it was assumed that the k point under consideration was
close enough to the zone center that the analytic expressions derived were valid. In
this section, k will still be supposed to be within the range of validity of the k · p
theory, but farther out into the Brillouin zone, so that the analytical expressions lose
their validity. In this section the k · p Hamiltonian will be diagonalized numerically,
giving special attention to the spin behavior of the conduction band states.

a)
CB

1.0

0.5
LH

Energy [meV]

Energy [eV]

1.5

0.1

0.01

0.0
SO

-0.04

-0.02

0.00

0.02

0.04

1E-3
-3

-1

k || [110] [Å ]

-2

-1

10
-1

k [Å ]

Figure 5.4: Band structure for GaSb near Γ along [110] and spin splitting of the
conduction band.
Bands along the [100] direction
Figure 5.3 shows the band structure of bulk GaSb along the [100] direction. In
accordance with the group symmetry requirements, the bands are spin degenerate for
finite k’s.
Bands along the [110] direction and k 3 splitting
Figure 5.4.a) shows the band structure of bulk GaSb along the [110] direction. For
this direction, all bands are spin split except at the Γ point due to the bulk inversion
asymmetry (BIA) effects. Plot (b) shows the energy splitting of the CB states as
a function of k in a loglog graph. Close to the zone center, the splitting in the
conduction band follows a power law, with exponent 3. This is in agreement with the
results derived in Eq. (5.24).
This so-called k 3 splitting [10] can also be predicted by the methods described
in Sec. 5.3.1. Going up to order 3 in the combinations of components of k and
constructing an invariant 2-band Hamiltonian for the conduction band, it is found

70
0.010

-1

ky [Å ]

0.005

Spin of lowest
subband

0.000

-0.005

-0.010
-0.010

-0.005

0.000

0.005

0.010

-1

kx [Å ]

Figure 5.5: Direction for the spin of the spin-split states of the lowest conduction
subband of GaSb. This plot sweeps a circular path in k space with kz = 0. No spin
direction is specified for the h100i family because the states are spin degenerate.
that the Hamiltonian will include the following term breaking the spin degeneracy [11]:
Hk3 = γc σx kx ky2 − kz2 + cyclic permutations .

(5.36)

The fact that effects of order k 3 are being studied with a Hamiltonian with terms
of order up to k 2 might lead to some inaccuracies. A future study might deal with the
effect of these higher-order terms in the CB spin splitting. Since the main contribution
to γc comes from the parameter B, describing the coupling of the CB and the LH
band via remote states, one might expect that the third order terms in the 8-band
Hamiltonian block Hcc would have a contribution of about

EΓ8c −Γ6c
Eg

smaller than B,

where EΓ8c −Γ6c is the distance in energy from the CB to the closest, non valence band,
Γ8 band, which normally lies above the conduction band. For GaAs,

EΓ8c −Γ6c
is about
Eg

2. Basically, this would not have a big effect on the band structure calculations, but
it would affect substantially the values of the calculated matrix elements.
The two-band Hamiltonian in Eq. (5.36) also predicts the direction where the split
spins will be pointing. For example, it is easily seen from the previous equation that if

71
k = (1, δ, 0), with δ a positive infinitesimal, the spin will point along the ±y direction.
Similarly, symmetry requires the spin to point along (−1, 1, 0) or (1, −1, 0) for k along
the [110] direction. This is indeed obtained in the numerical diagonalization of the
Hamiltonian including the BIA effects, as seen in Fig. 5.5. That figure shows a
circular sweep in k space with kz = 0. The arrows represent the direction towards
which the spin of the lowest conduction subband states is pointing. The horizontal
axis represents kx , while the vertical axis indicates the ky component of the state.
The states belonging to the h100i directions are spin degenerate; therefore, no spin
direction is given for them in Fig. 5.5.
The expectation value of the spin is calculated using the following spin oper©
ator, given in the |Γ6 , + 12 i, |Γ6 , − 21 i, |Γ8 , + 32 i, |Γ8 , + 21 i, |Γ8 , − 21 i, |Γ8 , − 23 i, |Γ7 , + 21 i,
|Γ7 , − 12 i basis:
0 1 0 0 0
0 
1 0 0
0 
 0 0 0 √13 0
2 
 0 0 √3 0 3
~
− 2
√1
0 0 0
0 
Sx = 
(5.37)
√2 
2
− 3
 0 0 √0 0 √√3 0
0 0 2 0 − 2 0
−( 13 ) 
0 0 0
0 − 32 −( 31 )
 0 −1 0
0 
1 0
³0 ´
 0 0 0 − √13
− 23
0 
− 2 
 0 0 √13
−( 3 )
− 2
√1
0 0
(5.38)
Sy = 
2
0 0 0
2 
− 3
 0 0 √2 0
0 0
−( 31 )
1 0 0 0
0 
0 −1 0

0√
 0 0 1 01
0 −23 2
0 
0 0 0
√ 
~
Sz =  0 0 0 0 −( 13 ) 0 0 −23 2  .
2  0 0 0 0√ 0 −1 0
0 
 0 0 0 −2 2 0 0 − 1
(3) 0
0 0 0

−2 2

(5.39)

The amount of spin splitting in bulk materials is highly anisotropic. The polar
plot in Fig. 5.6 shows the splitting as a function of angle for a circular sweep in k
space with kz = 0 and kk = 0.01Å−1 . In that plot, an angle of 0 ◦ corresponds to an

72
CB Splitting

Splitting [µeV]

90
200
180
160
140
120
100
80
60
40
20
20
40
60
80
100
120
140
160
180
200

120

150

180

210

330

-1

240

300

k=0.01 Å

270

Figure 5.6: Spin splitting in the CB for an electron with kk = 0.01Å−1 and traveling
in the plane kx − ky .
electron traveling with positive kx only, an angle of 90 ◦ corresponds to an electron
traveling with positive ky only, and the rest taking the usual meaning. The amount
of splitting is given by the axis on the left or, equivalently, by the distance of the plot
line to the center.

5.4

Eight-band effective mass theory for superlattices and quantum wells

In this section the method used to find the band structure and eigenstates for superlattices and quantum wells is shown. Following the effective mass approximation
(EMA) theory [13], the 8 band k · p Hamiltonian of Eq. (A.1) is transformed into
a set of eight linear, second-order, ordinary differential equations. The appropriate
boundary conditions are enforced and the equations are solved by means of a finite
difference scheme.

5.4.1

EMA Hamiltonian

In the effective mass approximation (EMA) [12], the bulk Hamiltonian H0 is modified
by some gentle—changing little in a lattice constant—potential U (r). Then, the
wavefunction of the structure under study is written as
Ψkk (r) =

eikk ·r Fn (z) un0 (r) ,

(5.40)

where kk = kx x̂ + ky ŷ is the electron wavevector in the kx − ky plane, un0 is the Bloch
wavefunction at the zone center, the Fn (z)’s are the envelope function components,
a set of supposedly slow varying functions that take all the effect of U (z); and it has
been assumed that the translational bulk symmetry is broken along the z axis only
(i.e., the superlattice or quantum well has been grown along the [001] direction). Burt
has developed a rigorous effective mass theory [31], which has the Luttinger theory
as a limiting case and explains the success of the latter in treating structures with
abrupt interfaces. The simpler Luttinger approximation will be used throughout this
study.
It has been shown [12, 13] that the equation

H0 kk , kz ; z + U (z) Ψkk (r) = E Ψkk (r) ,

(5.41)

with H0 kk , kz ; z being the bulk Hamiltonian with an allowance for a change of
material as a function of z, can be written in what is the key equation of the effective
mass approximation:

H0 kk , −i∂z ; z + U (z) F(z) = E F(z),

(5.42)

where F(z) is a multicomponent vector constructed from the different Fn (z)’s. In
an 8-band theory, F would have 8 components, each one multiplying the conduction
band (CB), heavy hole (HH), light hole (LH) and split off (SO) basis states.
On the other hand, the bulk k·p Hamiltonian can be expanded into its polynomial

74
form for kz in the following manner:
¡ ¢
¡ ¢
H0 (k) = H(2) kz2 + H(1) kk kz + H(0) kk .

(5.43)

Putting together Eq. (5.42) and Eq. (5.43), a system of eight coupled differential
equations results:

¡ ¢
¡ ¢
−H(2) ∂z2 − iH(1) kk ∂z + H(0) kk + U (z) F(z) = E F(z).

(5.44)

This is the system of equations that must be solved to obtain the energies and eigenstates of the system.

5.4.2

The finite difference method

There are several methods to solve numerically the system of coupled ordinary differential equations given by Eq. (5.44), such as the transfer-matrix method [32, 33],
the finite element method [34], the basis expansion method [35]. In this study, the
finite difference method (FDM) has been employed because of its conceptual simplicity, its ability to describe tunneling phenomena with only a few changes (see Sec. 7)
and its numerical stability respect to the transfer-matrix method, which requires the
truncation of growing exponential states [36].
In the finite difference method, the differential operators are first written in a
Hermitian form and then substituted by their finite difference approximations over a
discrete mesh (see Fig. 5.7) with N points. Following Chuang and Chang [37], the
following discretization scheme is used:
¢¯
H(2) (zi+1 ) + H(2) (zi )
H(2) (z) ∂z2 f ¯zi → ∂z H(2) (z) ∂z f ¯zi ≈
f (zi+1 )−
2(∆z)2
H(2) (zi+1 ) + 2H(2) (zi ) + H(2) (zi−1 )
H(2) (zi−1 ) + H(2) (zi )
(z
f (zi−1 ) (5.45)
2(∆z)2
2(∆z)2

N1/N2/N3 Quantum Well or Superlattice

Monolayer
Boundary
Points

Mesh
Points

1 ... N1-1 N1

1 ... N2-1 N2

...

i-1 i i+1

1 ... N3-1 N3

...

N-1 Nº0

Figure 5.7: Schematic of the structure under study, with points separating the monolayers, and mesh used when solving the effective mass approximation equations. The
mesh points need not coincide with the monolayer boundaries.
¤¯
−i £ (1)
−iH(1) (z) ∂z f ¯zi →
H (z) ∂z f + ∂z H(1) (z)f ¯zi ≈
H(1) (zi−1 ) + H(1) (zi )
H(1) (zi+1 ) + H(1) (zi )
f (zi+1 ) + i
f (zi−1 ), (5.46)
−i
4∆z
4∆z
where ∆z is the separation between the mesh points, and zi is the position of the i-th
mesh point.
Now, the application of the above equations to Eq. (5.44) yields the following
system of N algebraic equations:
Hi,i−1 Fi−1 + Hi,i Fi + Hi,i+1 Fi+1 = EFi

(5.47)

76
where Fi is the eight-vector containing the envelope function components corresponding to the i-th mesh point. This eigenproblem can be written in matrix form to better
appreciate its sparse structure:

H0,0
H0,1
...
...
...
H0,−1
 H1,0
H1,1 H1,2
...
...
 0
H2,1 H2,2 H2,3
...
 F = E F, (5.48)
..
..
..
...
...
...
...
 0
...
...
0 HN −2,N −3 HN −2,N −2 HN −2,N −1 
HN −1,N
...
...
HN −1,N −2 HN −1,N −1
where F is a column vector composed of the different Fi ’s.
The discretized Hamiltonian matrices when inside only one material are given by:
2H(2)
Hi,i =
+ H(0) + Ui
(∆z)
H(1)
H(2)
Hi,i+1 = −
(∆z)2
2∆z
(2)
H(1)
Hi,i−1 = −
= H†i,i+1 .
(∆z)2
2∆z

(5.49)
(5.50)
(5.51)

Note that the above discretization scheme treats with equal footing the inner and
interface mesh points. Therefore, the general expression for the discretized 8 × 8
Hamiltonian matrices can be used without modification when dealing with interface
mesh points:
(2)

(2)

H + 2Hi + Hi−1
(0)
Hi,i = i+1
+ Hi + U i
2(∆z)
(2)

(2)

H + Hi
Hi,i+1 = − i+1
2(∆z)2
(2)

(2)

H + Hi
Hi,i−1 = − i−1
2(∆z)2

(1)

H + Hi
− i i+1
4∆z
(1)

(5.52)
(5.53)

(1)

H + Hi
+ i i−1
4∆z

(5.54)

H0,−1 and HN −1,N in Eq. (5.48) express the boundary conditions (BCs) of the
problem. When studying a quantum well, the BCs are that the wavefunction must

77
vanish far from the well region. This is accomplished by setting the barrier region
wide enough, and requesting
F−1 = FN = 0,

(5.55)

H0,−1 = HN −1,N = 0.

(5.56)

which translates into

When finding the energies and states of a superlattice, the Bloch BCs apply, and
the envelope function is requested to have the supercell periodicity d, modulated by
a phase:
FN = eiqd F0
F−1 = e−iqd FN −1

HN −1,N = eiqd HN −2,N −1

(5.57)

H0,−1 = e−iqd H1,0 ,

(5.58)

where q is the electron wavevector along the z direction, and it has been assumed
that the same material is at mesh points 0 and N − 1.

5.4.3

Interface conditions and hermiticity in the FDM

The hermiticity of the discretized Hamiltonian operator in Eq. (5.48) is ensured
if H†i,i+1 = Hi+1,i . Since the H(j) ’s are themselves Hermitian, an inspection of
Eqs. (5.52)-(5.54) shows that this is indeed the case. It is also clearly seen that
the introduction of the BCs as defined in the previous section doesn’t affect the hermiticity of the Hamiltonian.
There exist in the literature several proposals on what are the correct quantities
to match at the interface between two materials [29, 37–39]. Most of them require
the continuity of the envelope function and a quantity that has the general form:
[A∂z + B] F,

(5.59)

where A and B take different values depending on the author. Using the finite
difference formulae Eqs. (5.45)-(5.46), the continuity of Eq. (5.59) can be written in

Figure 5.8: Mesh used in the study of interface boundary conditions.
a form similar to Eq. (5.47):
H2,1 F1 + H2,2 F2 + H2,3 F3 = 0,

(5.60)

where the Hi,j take the appropriate values and the subindexes i, j are referred to
the mesh points in Fig. 5.8. Isolating F2 from Eq. (5.60) and plugging it into the
corresponding equations for the Fi ’s, one obtains
−1
H1,0 F0 + H1,1 − H1,2 H−1
2,2 H2,1 F1 − H1,2 H2,2 H2,3 F3 = E F1
−1
−H3,2 H−1
2,2 H2,1 F1 + H3,3 − H3,2 H2,2 H2,3 F3 + H3,4 F4 = E F4 .

(5.61)
(5.62)

Now, in order to preserve the hermiticity of the discretized Hamiltonian, one
should have

or, equivalently

H1,2 H−1
2,2 H2,3

H†2,3 = H3,2

¢†

= H3,2 H−1
2,2 H2,1

(5.63)

H†2,1 = H1,2 .

(5.64)

However, the requirements in Eq. (5.64) are not satisfied by the discretized version

79
of the interface conditions. From this it must be concluded that the enforcement of
interface boundary conditions of the form (5.59) is not possible if the hermiticity of
the Hamiltonian is to be preserved.

5.5

Bulk inversion asymmetry effects on symmetric quantum wells

In this section, the methods outlined in Sec. 5.4 will be used to calculate the electronic properties of a symmetric quantum well. In particular, focus will fall on
AlSb/GaSb/AlSb quantum wells. However, some of the results derived are a consequence of the underlying symmetry of the structure rather than the constituents
themselves. Therefore, these particular results will illustrate general considerations
about symmetric quantum wells.
In Sec. 5.5.1, the underlying symmetry of the discretized Hamiltonian is identified,
and its requirements on the energies and states of the well are listed. In Sec. 5.5.2,
the band structure and eigenstates of symmetric quantum wells (SQWs) without bulk
inversion asymmetry (BIA) terms will be studied, taking an AlSb/GaSb/AlSb QW as
a paradigm. Finally, in Sec. 5.5.3, the assumption of negligible BIA effects is relaxed,
the resulting energies and states are computed and the differences respect to the case
with higher symmetry are highlighted.

5.5.1

Symmetry group of the discretized Hamiltonian

This section deals with symmetric quantum wells, but the word “symmetric” needs a
more precise definition. Here, “symmetric” will be taken to mean that the sequence
of materials and their respective thicknesses are left unchanged under the inversion
operation (i.e., they are macroscopically symmetric). Thus, an AlSb/InAs/AlSb
quantum well (QW) is called symmetric, while an AlSb/InAs/GaSb/AlSb QW is
called asymmetric. This definition is made in order to avoid confusion with the
microscopic symmetry, that is, the symmetry group, of the QW. All asymmetric

a)
$O6E$O6E$O6E,Q$V,Q$V,Q$V$O6E$O6E$O6E

$O6E$O6E$O6E,Q$V,Q$V,Q$V,Q6E$O6E$O6E$O6E

Figure 5.9: Effect of an extra atomic layer in a non-common atom AlSb/InAs/AlSb
QW. (a) shows the QW having complete monolayers, belonging to the C2v point
group. (b) shows the effect of adding one extra In atomic layer in the InAs. New
symmetry operations appear and among them, the C2 symmetry axis shown. The
point group is now D2v although it is a non-common-atom QW [41].
[001] heterostructures made from zincblendes are described by the C2v point group.
On the other hand, symmetric [001] QWs can belong to either the C2v or the D2d
symmetry groups depending on an interplay of characteristics such as the parity of
the number of monolayers, the existence of a common atom in the constituents [40],
or the existence of an extra atomic layer [41] (see Fig. 5.9).
The symmetry group of the discretized Hamiltonian in Eq. (5.48) can be found
by the brute force method consisting on verifying for all the operations g of the T d
point group whether the relationship
D−1 (g)H(kk )D(g) = H(g −1 kk )

(5.65)

is sustained [19], where D(g) is the representation of the g operator in the basis of
the discretized Hamiltonian H(kk ), and kk = (kx , ky ). This tedious procedure can
be done with the help of computer software, such as Mathematica [20], which auto-

81
Point/Line
∆ [100]
Σ [110]
Other points

Point Group Symmetry Spin Splitting Spin directions
D2d
No
C2
Yes
[100],[1̄00]
Cs
Yes
[1̄10],[11̄0]
C1
Yes
Undetermined

Table 5.7: Symmetry requirements on spin splitting and directions for points in the
kx − ky plane in a D2d structure.
mates algebraic manipulations. It is seen that the EMA Hamiltonian corresponding
to structures possessing macroscopic symmetry under inversion transforms according to D2d , while for macroscopically asymmetric structures it transforms according
to C2v . This is in opposition to the majority of EMA implementations, which lack
the inclusion of bulk inversion asymmetry effects and reproduce an approximate D4h
symmetry [42] for symmetric structures. There have been reports in the literature
of other 2-band [27], 14-band [43], and 16-band [9] EMA models describing the spin
splitting effects due to the reduced symmetry. Zhu and Chang [44] have started from
an 8-band model to generate perturbatively a 2-band Hamiltonian for electrons and
a 4-band Hamiltonian for holes, and they performed their calculations of inversion
asymmetry effects in that reduced basis set. No 8-band model, combining the more
accurate description of interband couplings respect to 2-band models and the simplicity and numerical performance advantages over the 14- and 16-band models, had
been previously used in numerical studies of the BIA effects.
Tables 5.7 and 5.8 show the requirements that the underlying symmetry of the
atom arrangement imposes on the spin degeneracy of the energy levels and the direction where the spins are pointing in case the levels are not degenerate.
Although the k·p method is not designed taking into account the interface characPoint/Line
∆ [100]
Σ [110]
Other points

Point Group Symmetry Spin Splitting Spin directions
C2v
No
C1
Yes
Undetermined
Cs
Yes
[1̄10],[11̄0]
C1
Yes
Undetermined

Table 5.8: Symmetry requirements on spin splitting and directions for points in the
kx − ky plane in a C2v structure.

82
$O6E,Q$V$O6E6/

$O6E$O6E$O6E,Q$V,Q$V,Q$V$O6E$O6E$O6E

$O6E,Q6E,Q$V$O$V$O6E6/

$O6E$O6E$O6E,Q$V,Q$V,Q$V$O6E$O6E$O6E

Figure 5.10: Layer arrangements for a no-common-atom quantum well. An arrangement as in (a) in the EMA would yield D2d symmetry. The alternative arrangement
(b) yields the correct C2v symmetry of the heterostructure.
teristics of no-common-atom (NCA) heterostructures, it is possible in some situations
to modify the simulated structure to obtain at least the right symmetry effects. Figure 5.10 a) shows a NCA quantum well. The way that the boundaries of the layers
are set up, the well would be symmetric and, therefore, the Hamiltonian would have
D2d symmetry instead of the C2v corresponding to the asymmetric interface bonds.
However, the material boundaries in the k · p method are arbitrary to half a monolayer. As seen in Fig. 5.10 b), a simple rearrangement of the material boundaries
reproduces the asymmetry in the bonds and allows to take into account, at least
qualitatively, the effects of the lower symmetry. The case where a NCA QW is added
an extra atomic layer to make it symmetric, as seen in Fig. 5.9 b), does not require
any rearrangement of the layers in order to make the calculated structure have the
correct D2d symmetry.

83
The only case that cannot be modeled through these rearrangements is when a
common atom structure, such as an AlAs/GaAs/AlAs QW, has C2v symmetry due to
the well having an odd number of monolayers. In that case, even though the species
participating at the bond at the interface are the same, there is an asymmetry in the
bond orientation, which the EMA method cannot take into account.
With the control that the proposed model allows over the symmetry of the calculated heterostructures, it is possible to use the EMA in studies of the spin splitting appearing in heterostructures aiming at delineating the role of bulk inversion asymmetry
vs. structure inversion asymmetry, layer asymmetry vs. interface asymmetry. . . This
model also provides a straightforward and easy to implement tool to study effects in
QWs and superlattices derived from the possession of the reduced symmetry, such
as the presence of optical anisotropy [45, 46], and the mixing of heavy hole and light
hole states on top of the valence band [47, 48].
There is room for future improvement if the interface equations Eqs. (5.52)-(5.54)
are considered. One possible way to expand the model would be to treat the weighted
average of bulk parameters appearing in those equations as adjustable parameters.
This would improve the accuracy at the expense of simplicity. The addition of interface parameters to the EMA theory had been previously proposed in the optical
anisotropy [45, 48] and the hole spin splitting and relaxation [49] contexts.

5.5.2

SQWs without BIA terms

Figure 5.11 shows the band structures along the [100] and the [110] directions of a
common atom AlSb/GaSb/AlSb symmetric quantum well (SQW) grown along the
[001] direction and with a well thickness of 8 monolayers (24.4 Å). Since no inversion
asymmetry affects are included, the bands show Kramers degeneracy throughout the
Brillouin zone and the quantization axes of the spins are not univocally defined.
The labels E1, HH1, LH1 and HH2 shown in the plots correspond to the first
electron, first heavy hole, first light hole and second heavy hole states in the QW
respectively. They refer to the main bulk state contribution at k = 0. For a well

84
1.8

1.8

1.6

1.4

1.2

Energy [eV]

1.0

0.8

1.0

0.8

0.6

HH1

HH1
0.4

0.4

LH1

LH1
0.2

HH2
-0.04

0.2

-0.02

0.00

0.02

0.04

-1

k || [100] [Å ]

HH2
-0.04

-0.02

0.00

0.02

0.04

-1

k || [110] [Å ]

Figure 5.11: Bands along [100] and [110] for an AlSb/GaSb/AlSb SQW 8 monolayers
thick without BIA terms.
without BIA terms and in the zone center, the heavy holes decouple from the rest
of the bands, and the HHn states have only bulk heavy hole components. This is
in opposition to the En (LHn) bands, which have small bulk light hole (electron)
and split off contributions even at the zone center due to the loss of translational
symmetry along [001] caused by the well potential.

5.5.3

SQWs with BIA terms

Figure 5.12 shows the same band structures as in Fig. 5.11, but with the BIA terms
included. As predicted by the group theory (cf. Table 5.7), the bands are split along
both directions except at the zone center. This is the major difference with most
of the EMA models in the literature. Another point that must be noticed is that,
even though there is no spin splitting in bulk bands along [100], in the SQW a finite
splitting appears along that direction [27].
With the inclusion of BIA terms, the heavy hole states couple with the light holes
by means of remote states through a perturbative mixed spin orbit and k·p interaction
parametrized by C [9]. Thus, the HHn states lose their pure bulk heavy hole character

85
1.8

1.8

1.6

1.4

1.2

Energy [eV]

1.0

0.8

1.0

0.8

0.6

HH1

HH1
0.4

0.4

LH1
0.2

HH2
-0.04

LH1
0.2

-0.02

0.00

0.02

0.04

HH2
-0.04

-0.02

0.00

0.02

0.04

-1

k || [110] [Å ]

k || [100] [Å ]

Figure 5.12: Bands along [100] and [110] for an AlSb/GaSb/AlSb SQW 8 monolayers
thick with BIA terms.
and, in particular, the HHeven (HHodd) mix with the LHodd (LHeven). However,
looking at the wavefunction for the HH1 state, it can be seen that the contribution
from bulk components other than the HH to the probability density is about 8 orders
of magnitude less than the heavy hole contribution.
The linear behavior and the isotropicity close to the zone center of the spin splitting between the conduction subbands is manifest in Fig. 5.13. Plot (a) shows the
dependence of the spin splitting along the [100] line. It is seen that the splitting is
linear close to the Γ point, with a “Rashba” coefficient of αR = 22×10−10 eV·cm. Bychkov and Rashba introduced that coefficient in the context of asymmetric quantum
wells [50]. In their article, the splitting in the conduction subbands is given by
∆R = 2αR k.

(5.66)

However, this splitting is derived from a model Hamiltonian that describes only structural inversion asymmetry (SIA) effects, but not bulk inversion asymmetry. As a
consequence, the spin directions that they predict don’t apply to the SQW situation

0.018

0.5
0.4

0.014

0.3

CB Splitting [meV]

0.016

CB Splitting [eV]

0.012

0.010

0.008

0.006

AlSb/GaSb/AlSb QW
0.004

αR=22×10

-10

120

150

0.2
0.1
0.0

180

0.1
0.2
0.3

210

330

eV·cm
0.4

0.002

0.5
0.000
0.00

240

300
270

0.02

-1

k=0.001 Å

0.04
-1

k || [100] [Å ]

Figure 5.13: a) Spin splitting dependence for an AlSb/GaSb/AlSb QW along the [100]
line. The calculated “Rashba” coefficient is αR = 22 × 10−10 eV·cm. (b) Splitting
dependence along a circle in the kx − ky plane, with k = 0.001 Å.
(cf. Sec. 5.6).
A model Hamiltonian for spins in the conduction subbands of SQWs in the same
spirit as the Rashba Hamiltonian can be derived in the following fashion. From
Eq. (5.36) the operator nature of kz in the effective mass approximation can be made
explicit [27], and keeping only up to second-order terms in kx and ky the following
perturbation to the 2-band EMA Hamiltonian can be written:
¢¤
HSplit Sym = γc −σx kx ∂z2 + σy ky ∂z2 + σz ∂z kx2 − ky2 .

(5.67)

87
Lower Conduction Subband

0.010

-1

ky [Å ]

0.005

0.000

AlSb/GaSb/AlSb QW
|k|=0.01 Å

-1

-0.005

-0.010
-0.010 -0.005

0.000

0.005

0.010

-1

kx [Å ]

Figure 5.14: Spin directions for the lower conduction subband of an AlSb/GaSb/AlSb
SQW. The thickness of the well is 8 ML (24.4 Å). The spins are plot at 15 ◦ intervals,
and correspond to states lying on a circle in the kx − ky plane with k = 0.01 Å.
with the “Rashba” coefficient for symmetric heterostructures αR given by αR =
− hFs ; kx , ky |γc ∂z2 | Fs ; kx , ky i. The term depending on ∂z vanishes because the envelope function has a definite parity. So, it is readily seen that a phenomenological
Hamiltonian
H = αR (σx kx − σy ky )

(5.69)

will have the desired effect. From the definition of αR it is easy to see that in the
conduction band BIA effects will be bigger in narrow quantum wells because states
there are made from larger perpendicular wavevectors.
The behavior of the spins when BIA terms are included is very interesting. Figure 5.14 shows the direction towards where the spins of the eigenstates of the lowest
conduction subband point. The spin directions are shown for a circular sweep in the
kx − ky plane keeping k = 0.01 Å. The directions of the spins agree with what would
be predicted from Eq. (5.69). The spins at a given point in the plane point opposite
for the two subbands. Note that in a given subband, although the x and y axes are

88
equivalent, in one of the axes the spin points outward while in the other it points inward. The explanation lies in the way that the x and y axes are connected and in the
fact that spinors don’t change sign under inversion. For a QW with D2d symmetry,
the x and y axes are equivalent through a reflection by the plane containing the [110]
and [001] directions. The reflection by this plane can be thought of as a rotation of
180 ◦ along the [1̄10] direction followed by an inversion. Starting with a state |ky , ↑x̂ i
(spin pointing outward), the rotation will send it to | − kx , ↑−x̂ i (still outward). Then,
the inversion will flip k, but not the spin, sending the state to |kx , ↓−x̂ i (spin pointing
inward).

5.6

Bulk inversion asymmetry effects on asymmetric quantum wells

In this section the structure under study will be an AlSb/InAs/GaSb/AlSb asymmetric quantum well (AQW) grown along the [001] direction compliant with a GaSb
substrate. The thickness of the InAs and GaSb layers is 8 monolayers (ML) each one,
with a monolayer having 3.048 Å.

5.6.1

AQWs without BIA terms

Figure 5.15 shows the band structure along the [100] and the [110] directions of the
AQW without the inclusion of BIA terms. The structural inversion asymmetry (SIA)
reduces the symmetry group from D4h to C2v , and a spin splitting appears between
the conduction subbands. The splitting due to SIA effects is usually modeled using
a Hamiltonian first introduced by Bychkov and Rashba [50]:
HR = αR (σ × k) · ν,

(5.70)

where αR is the so-called Rashba constant, σ is a vector composed of the Pauli
matrices, k is the electron wavevector and ν is the axis of symmetry of the structure.

89
1.0

0.9

0.8

Energy [eV]

0.7

0.6

0.5

HH1
0.4

0.3

0.2

LH1
HH2

0.1
-0.04

-0.02

[110]

0.00
-1

k [Å ]

0.02

0.04

[100]

Figure 5.15: Bands without BIA effects for an AlSb/InAs/GaSb/AlSb AQW grown
along the [001] direction compliant with a GaSb substrate. The bands are along [100]
and [110]. The thickness of the InAs and GaSb layers is 8 ML each.
This Hamiltonian is valid to describe the SIA contributions close to the zone center.
It predicts a linear and isotropic splitting
∆R = 2αR k,

(5.71)

where k is the magnitude of the electron wavevector. It also predicts that the spins
will point tangentially to the circles of constant k in the kx − ky plane, which is
verified in the numerical calculations (see Fig. 2.3). The Rashba effect in asymmetric
heterostructures is studied in detail in Chapter 2.

90
1.0

0.9

0.8

Energy [eV]

0.7

0.6

0.5

HH1
0.4

0.3

0.2

LH1
HH2

0.1
-0.04

-0.02

[110]

0.00
-1

k [Å ]

0.02

0.04

[100]

Figure 5.16: Bands along [100] and [110] for an AlSb/InAs/GaSb/AlSb AQW with
BIA effects.

5.6.2

AQWs with BIA terms

The band structure for the AQW under study with BIA effects is shown in Fig. 5.16.
The effects of inversion asymmetry are highly anisotropic in bulk [10], and this reflects
on the directional dependence of the bands. Comparing with Fig. 5.15, it is seen that
the BIA effects are necessary to obtain accurate bands in the [110] direction.
The interplay of SIA and BIA effects in asymmetric quantum wells (AQWs) adds
a level of variety to the analysis of the behavior of the spins in the conduction band.
However, the inclusion of the Hamiltonian (5.69) keeps the analysis quite simple. For
a [001] structure, the SIA and BIA contributions to the splitting can be described by

91
90
12
60

120
10
150

Splitting [meV]

180

210

330

10
240

300

k=0.01 Å

-1

270

Figure 5.17: Angular dependence of the spin splitting for an AlSb/InAs/GaSb/AlSb
AQW. The solid line is the 8-band model numerical result. The dashed line is a fit
using Eq. (5.73) with αSIA = 40.3 × 10−10 eV·cm and αBIA = 15.0 × 10−10 eV·cm.
Hamiltonian HIA made from the addition of Eq. (5.69) and Eq. (5.70):
HIA = αBIA (σx kx − σy ky ) + αSIA (σx ky − σy kx ) =
σx (αSIA ky + αBIA kx ) − σy (αBIA ky + αSIA kx ) , (5.72)
where αBIA (αSIA ) is the coefficient describing BIA (SIA) effects. From here, making
an analogy with the Zeeman splitting, it is easy to find that the splitting in the
conduction band (CB) close to the zone center will be
∆IA = 2k

αSIA
+ 2αSIA αBIA sin 2θ + αBIA

(5.73)

where θ is the in-plane polar angle.
A full 8-band numerical calculation of the splitting along a circle in the k x − ky

92
plane and the 2-band prediction from expression (5.73) are shown in Fig. 5.17. The
values from the analytic expression show very good agreement with the numerical
results. The numerical results are fitted with αSIA = 40.3 × 10−10 eV·cm and αBIA =

15.0×10−10 eV·cm. This way the BIA effects are quantified, and it must be concluded
that they must be taken into account for an accurate description of the bands. This
is clearly so in the [110] direction, where the contributions are added linearly, but it
is also true in a lesser degree in the [100] direction, where the contributions are added
quadratically.
For a quantum well where the 2-band model is valid, the BIA splitting coefficient
for the CB can be estimated with
αBIA ≈

γcW
L2W

(5.74)

where γcW and L2W are the k 3 splitting coefficient of the CB and the thickness respectively of the layer where the electrons are confined. This estimate will become
more accurate as the well becomes thicker. From this expression and Eq. (5.25) it is
readily seen that BIA effects will be considerable when the material in the well layer
has a low band gap and high spin-orbit interaction, such as InAs, GaSb and InSb
(cf. Table 5.1). So, it has been seen that, a priori, it is not possible to consider only
SIA effects for an asymmetric structure even if the constituents are low band gap
materials.
Figure 5.18 compares the splitting in the CB along the [100] and [110] directions
for the same AQW with and without the BIA terms. The values of half the slope,
i.e., the “Rashba” coefficient including bulk and inversion asymmetry, are listed for
each curve under the symbol α. With no BIA terms, the slope near the origin is
the same for both directions, and the values only depart when higher order O(k 3 )
contributions start to take over. For the curves with BIA effects, the linear splitting
behavior predicted by Eq. (5.72) holds only until about 1.5% of the zone boundary.
So, that equation can be applied to wells with electron concentrations up to 10 11
cm−2 . It would be interesting to study the effects that higher order contributions to

α=40×10

-12

eV·m

α=43×10

-12

eV·m

α=40×10

-12

eV·m

α=25×10

-12

eV·m

Energy [meV]

W/o BIA [100]
W/ BIA [100]
W/o BIA [110]
W/ BIA [110]

0.00

0.02
-1
k || [110] [Å ]

0.04

Figure 5.18: Spin splitting vs. k for an AlSb/InAs/GaSb/AlSb quantum well along
the [100] and the [110] direction with and without BIA terms. The α’s are half the
slope at k = 0.
the splitting and BIA-induced anisotropy can have on the interpretation of the wide
body of Shubnikov-de Haas measurements of αR (see, for example, Refs. [51–53]) that
have so far assumed Eq. (5.70) to describe the splitting for all electrons in the well.
Finally, the electron spins are also affected by the inclusion of BIA terms. In
Fig. 5.19 the spins of the lowest conduction subband are shown for states lying on
a circle in the kx − ky plane of radius k = 0.01 Å. The direction of the spins has
changed respect to the case without BIA terms (see Fig. 2.3). As it can be deduced
from Eq. (5.72), it corresponds to the vector sum of the spins in Fig. 2.3 and the spins

94
0.010

-1

ky [Å ]

0.005

AlSb/InAs/GaSb/AlSb QW

0.000

with BIA and |k|=0.01 Å

-1

-0.005

-0.010

-0.005

0.000

0.005

0.010

-1

kx [Å ]

Figure 5.19:
Spin directions for the lower conduction subband of an
AlSb/InAs/GaSb/AlSb AQW. The spins are plot at 15 ◦ intervals, and correspond
to the lowest conduction subband states lying on a circle in the kx − ky plane with
k = 0.01 Å.
in Fig. 5.19, each one weighted by their corresponding splitting coefficients α’s.

5.7

Summary

In summary, an implementation of an 8-band effective mass approximation (EMA)
method for calculating band structures has been obtained. This implementation is
faithful to the Td microscopic symmetry of bulk zincblendes. As a consequence, all
symmetry effects close to the zone center, including the spin splitting of the bands,
are correctly described. When the method is applied to symmetric heterostructures,
linear splittings in k are predicted as a consequence of the reduced symmetry. This
is not found for standard EMA implementations. A 2-band Hamiltonian describing
this splitting due to the bulk inversion asymmetry (BIA) is derived. The bands of
asymmetric heterostructures are also studied and described in the context of the BIA
Hamiltonian obtained for heterostructures. It is seen that, in the case studied, the
SIA and BIA contributions to the spin splitting are of the same order of magnitude

95
even though the well is composed by narrow gap materials. Therefore, an accurate
description of the bands will require the inclusion of both effects.

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100

Chapter 6

Spurious numerical solutions

in the effective mass approximation
6.1

Introduction

The effective mass approximation (EMA) [1, 2] has long been a favorite among researchers as a method that is fast and easy to implement for calculating the energy levels in quantum wells and superlattices [3–7], quantum wires [8–10], quantum
dots [8, 11, 12]. . . It is also the tool of choice for engineers to model devices such as
lasers and photodetectors [13–18]. Also, its use is widespread in the calculation of
tunneling coefficients [19–21] and times [22, 23], with application to the modeling of
resonant tunneling devices (RTDs).
Some of the implementations have the undesirable characteristic that they produce spurious solutions [3, 24–26]. Their origin is traced to the statement of a secular
equation having too high a polynomial degree in the electron wavevector k. In some
cases, the presence of the spurious solutions is required for consistent boundary conditions at the interface to be satisfied [3], thus raising doubts about the validity of the
results [24]. In some other cases, interface or surface states are predicted to lie in the
gap [27], but the physical meaning of these solutions is a point still in discussion [28].
There exist several proposals to solve the k·p spurious solution problem ranging from
methods to eliminate them [29, 30] to pointing out the necessity of keeping them for
a complete description [31].
In this chapter a new class of spurious solutions (SSs) particular to the finite
difference method (FDM) [32, 33] is studied. In Sec. 6.2 these SSs are presented,
they are characterized and different failed attempts to suppress them are presented.
Then, in Sec. 6.3, a general method for the study of this class of SSs is presented.
In Sec. 6.4 a condition that the Luttinger parameters must satisfy for the SSs not to

101
Quantity
InPa
a (Å)
5.8693
Eg (eV )
1.35
Ev (eV )
-0.351
∆SO (eV )
0.108
γ1
1.49
γ2
-0.31
γ3
0.37
C (eV ·Å) -1.44×10−2
P (eV ·Å)
8.79
B (eV ·Å )
-27.5
C1 (eV )
-5.04
Dd (eV )
1.27
Du (eV )
-2.4
-3.6
Du0 (eV )
C11 (GPa)
101.1
C12 (GPa)
56.1
C44 (GPa)
45.6

In0.53 Ga0.47 Asb
5.8693
0.839
0.362
1.84c
-0.87c
0.25c
7.53×10−3
9.49
21.5
-6.06
1.08
-2.7
-3.5
97.07
50.82
49.18

Same source as in Table 5.5.
Obtained by linear interpolation unless otherwise noted.
Obtained by harmonic averaging [34].

Table 6.1: Band structure parameters for InP and In0.53 Ga0.47 As.
exist is derived. Finally, in Sec. 6.5 this condition is applied to popular compilations
of Luttinger parameters to identify sets of parameters leading to SSs, and then the
results are summarized.

6.2

Spurious solutions in an InP/In0.53 Ga0.47 As superlattice

The structure that will be used to illustrate the spurious solutions (SSs) is an
InP/In0.53 Ga0.47 As symmetric superlattice (SL) grown along the [001] direction with
an In0.53 Ga0.47 As width of 65 Å and an InP width of 88Å. The method of calculation
is described in Sec. 5.4. Table 6.1 lists the numerical parameters employed in the calculations1 . When there is no explicit source for the parameters of In0.53 Ga0.47 As, they

The meaning of the parameters is shown in Table 5.4.

102

Energy [eV]

b)
1.1

1.1

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0

-0.1

-0.1
InP

-0.2

65Å In0.53Ga0.47As

InP

-0.2

-0.3

-0.4

100

Distance [Å]

150

-0.04

-0.02

0.00

0.02

0.04

-1

kx [Å ]

Figure 6.1: Bands of an 88 Å/65 Å InP/In0.53 Ga0.47 As SL showing spurious solutions.
(a) shows the spatial profile of the bands. b) shows the bands in k space in the same
energy scale as in (a). It is seen that the first three hole bands lie in the forbidden
gap.
have been obtained as a linear interpolation between InAs and GaAs, except for the
8-band Luttinger parameters γi , where a harmonic average has been performed [34].
Figure 6.1.a) shows the band edge spatial profile for that structure. Figure 6.1.b)
shows a calculation of the first electron band and first three hole bands near the Γ
point for the SL. The energy scales in plots (a) and (b) are matched. Note that the
energy of the hole states lies inside the forbidden gap of the superlattice, showing
that there is something wrong with the computational procedure employed.
In order to obtain more insight into the nature of these spurious solutions, the
norm and the relevant envelope function components of the wavefunction corresponding to the first SS (counting from the conduction band edge) were plotted and compared to a nonspurious wavefunction. This is shown in Fig. 6.2. Plot (a) depicts

103

b)
0.08

0.08

0.07

0.06

0.05

|Ψ| [a.u.]

0.04
0.03

|Ψ| [a.u.]

0.04
0.03

0.02

0.01

0.00

-0.01

-0.01
10

0.08

0.30

0.06

0.25

0.04

0.20

0.02
0.00
-0.02
-0.04
-0.06

Re[ |HH,+3/2> ]
Im[ |HH,+3/2> ]

0.15
0.10
0.05
0.00

Re[ |CB,↑> ]
Im[ |CB,↑> ]

ML number

Amplitude [a.u.]

ML number

-0.05
20

ML number

0.3
0.00

Amplitude [a.u.]

0.2
0.1
0.0
-0.1
-0.2

-0.01

-0.02
Re[ |HH,-3/2> ]
Im[ |HH,-3/2> ]

Re[ |LH,+1/2> ]
Im[ |LH,+1/2> ]

-0.3

-0.03
10

ML number

Figure 6.2: Comparison of spurious and physical envelope functions. (a) corresponds
to a spurious state. b) corresponds to the first heavy hole state. The top plots corresponds to the probability density of finding the electron in a given monolayer, while
the bottom two show the real and imaginary parts of selected envelope components.

104
the spurious wavefunction, while plot (b) refers to the non spurious one. It can be
seen that the probability density F (z) = 8i=1 |Fi (z)|2 of finding the electron in a
given monolayer is similar in both cases and doesn’t supply any information. How-

ever, a look at the individual components of the envelope function reveals that they
are highly oscillatory in nature for the spurious case, while they are smooth for the
physical state. The period for these spurious oscillations is twice the mesh spacing,
thus indicating that they might be somewhat related to the chosen discretization
mesh (see Sec. 5.4.2). The fact that the states are located mainly in the center of
the In0.53 Ga0.47 As layer discards the possibility that the SSs might originate from
boundary condition induced interface states [28, 35], because the latter should decay
exponentially away from the interface.
The relationship of the SSs and the discretization grid is further investigated in
Fig. 6.3. There, the band edge energies are plotted vs. the number of mesh points to
look for any dependencies. Four SSs appear in the range of mesh points and energies
under study. It is seen that while, as expected, the energies of the physical states
don’t depend strongly on the number of mesh points2 , the energies of the SSs do
depend on the number of mesh points. After this, it must be concluded that the
SSs presented here are not only nonphysical, but also that they are not intrinsically
attached to the system of coupled differential equations (5.44) that must be solved
in the EMA model. Instead, they appear only due to the procedure followed to solve
these equations.
A heuristic approach was first tried to remove the spurious solutions. First, a
number of different boundary conditions for the interfaces was tried, without affecting
the SSs. The highly oscillatory behavior of the SSs might induce to think that there
was something wrong with the discretized version of the kinetic energy operator [36],
but they also remained there when different discretization schemes were tried, 2nd
neighbor difference formulae used or unevenly spaced grids employed.
Finally, it was realized that, since the role of the boundary conditions is merely to
Except for a very coarse grid, where, as per the Nyquist theorem, there are not enough mesh
points to properly describe the oscillation of the envelope function in the confinement region.

105
1.1
1.0
0.9
0.8

Spurious energies
Physical energies

0.7

Energy [eV]

0.6
0.5
0.4
0.3
0.2
0.1
0.0

-0.1
-0.2
10

# mesh points
Figure 6.3: Energy of the spurious states vs. number of mesh points for a 88 Å/65 Å
InP/In0.53 Ga0.47 As SL. The state labeled with the square (round) dot corresponds to
plot (a) [(b)] in Fig. 6.2.
connect the bulk solutions, the SSs should be present also in bulk In0.53 Ga0.47 As, because that is where the envelope functions are oscillating. After this was understood,
it was possible to study the origin of the SSs and predict when they would appear.

6.3

Method for the study of the spurious solutions

Several practical implementations of calculations of band structures of heterostructures, including the multiband k · p [32, 33] and EBOM [37] methods, require the

106
solution of a set of equations
Hσ,σ−1 Fσ−1 + Hσ,σ Fσ + Hσ,σ+1 Fσ+1 = EFσ

(6.1)

for each of the points {σ} in the mesh (cf., for example, Sec. 5.4), where E is the
sought energy, Fσ are the envelope function coefficients at the mesh point σ and the
Hσ,ν are transfer Hamiltonians.
In bulk, the envelope functions of the physical solutions at the Γ point should be
flat. On the other hand, it has been shown in the previous section that SSs oscillate
with a period of double the mesh spacing. Both cases can be studied if it is assumed
that the envelope must follow the Bloch behavior, relating the value of the envelope
at a point σ + 1 to the value at σ,
Fσ+1 = ei∆z kz Fσ ,

(6.2)

where ∆z is the mesh spacing and kz is the z component of the electron wavevector
in bulk.
Now, plugging Eq. (6.2) into Eq. (6.1), a single eigenvalue equation
£ −i∆z kz
Hσ,σ−1 + Hσ,σ + ei∆z kz Hσ,σ+1 Fσ = EFσ

(6.3)

is obtained that must be solved in order to know the effect of an oscillating envelope function on the energies. Any further advance requires an explicit form for the
Hamiltonian.

6.4

Spurious solutions in the k · p method

In the 8-band EMA method, the starting point is a model that treats the coupling
between the conduction band (CB), heavy hole (HH), light hole (LH) and split-off
(SO) bands exactly, and the interactions with the rest of the bands perturbatively.
Bulk inversion asymmetry effects [38] will be ignored to keep the results simple and

107
obtain analytical expressions. This will cause the bands to be degenerate. The diagonalization of Eq. (6.3) can be carried out numerically if some of the assumptions need
to be relaxed. Also, since only spurious solutions at the zone center are sought, the
kx and ky components of the wavevector are set to zero. Then a simple change in the
ordering of the basis states described in Sec. 5.3.3 diagonalizes the k · p Hamiltonian
into two 4×4 blocks, and each one of them is block diagonalized again into a 1×1
block describing the HH band and a 3×3 block describing the CB, LH and SO bands:

|Γ6 , + 21 i

Eg + ~2mkz
 q
H3×3 = hΓ8 , + 1 |
P kz
2 
− P√k3z
hΓ7 , + 21 |
hΓ6 , + 21 |

2 2

|Γ8 , + 21 i
P kz

2 2
− (γ1 + 2γ2 ) ~2mkz
√ ~2 k 2
2γ2 2mz

|Γ7 , + 21 i
− P√k3z
√ ~2 k 2
2γ2 2mz
~2 kz2

−∆SO − γ1 2m

,

(6.4)

where m is the free electron mass and the rest of the parameters is defined in Table 5.4.
After the application of the discretization procedure described in Sec. 5.4.2 to
Eq. (6.4) and plugging the result into Eq. (6.3) one obtains that the finite difference
algorithm is effectively solving the Hamiltonian

H3×3 FDM =
√2
P sin(∆z kz )
P sin(∆z kz )
kz )
Eg + ~m 1−cos(∆z
∆z
∆z 2
3 ∆z
 √2
2 2 2 γ2 sin( ∆z kz )
2 1−cos(∆z kz )
sin(∆z
 . (6.5)
− (γ1 + 2 γ2 ) m
∆z
∆z
∆z
∆z
P sin(∆z kz )
~2 1−cos(∆z kz )
~2 2 2 γ2 sin( 2 )
−∆SO − γ1 m
∆z 2
∆z 2
3 ∆z
Expanding this Hamiltonian about kz = 0 up to second order it is easily seen that
Eq. (6.4) is recovered, which ensures the correct description of the bands when
∆z kz ¿ 1.
Now, the eigenvalues of Eq. (6.5) can be plotted as a function of kz for both
the InP and the In0.53 Ga0.47 As parameters in Table 6.1. This is shown in Fig. 6.4.
Plot (a) shows the results for In0.53 Ga0.47 As. Consider, say, six mesh points in bulk
In0.53 Ga0.47 As with cyclic boundary conditions (see the insets in Fig. 6.4). The dif-

108

In0.53Ga0.47As

Energy [eV]

InP

-1

-2

-3

-4
-5
0.0

0.2

0.4

0.6

-4

0.8

1.0

-5
0.0

kz [π/(∆z)]

0.2

0.4

0.6

0.8

1.0

kz [π/(∆z)]

Figure 6.4: Energy vs. wavenumber of a hypothetical spurious solution. (a) [(b)]
shows the dispersion relation for In0.53 Ga0.47 As (InP). The insets show two kinds
of light hole envelope functions (kz = 0 and kz = ∆z
) satisfying cyclic boundary
conditions. ∆z is taken to be half the unit cell constant.
ferential equations from the EMA Hamiltonian (6.4) can be solved analytically to
obtain flat envelopes. The FDM should give the same results. From Fig. 6.4.a), it
is seen that the point kz = 0 reproduces the expected results. However, an envelope
with kz = ∆z
will also satisfy the boundary conditions. The existence of this kind

of solutions is unavoidable in the FDM but, at least, one can demand that they lie
far from the energy range of interest. This is indeed the case with InP in Fig. 6.4.b)
but, on the other hand, the LH band of In0.53 Ga0.47 As enters the gap, thus giving
opportunity to the presence of SSs in the midgap.
Also, one finds that, for kz = ∆z
, the energy of the three branches (with Ev set

109

1.2
1.0

Numerical Spurious Energy
Analytic Formula

Energy [eV]

0.8
0.6
0.4
0.2
0.0
-0.2

Number of mesh points
Figure 6.5: Prediction from Eq. (6.6) and actual spurious solution energy. The actual
spurious energy corresponds to the highest spurious energy in Fig. 6.3.
to zero) is

Eg +

4(γ1 + γ2 )~2 /m + ∆SO (∆z)2
2~2
,−
m∆z
2(∆z)2
48γ22 ~4 /m2 − 8(∆z)2 γ2 ∆SO ~2 /m + ∆2SO (∆z)4
. (6.6)
2(∆z)2

One can argue that, since the well region in the SL is quite wide, the first electron
and hole levels will be close to the corresponding edges. Then, the analytic expression
corresponding to the plus sign in front of the square root in Eq. (6.6), describing the
light hole (LH) states in bulk In0.53 Ga0.47 As can be used to
energy of the kz = ∆z

make a rough approximation of the energy of the SS in the superlattice (SL) studied
in Fig. 6.3. Figure 6.5 shows a comparison of the energy of the SS estimated this way

110
and the highest energy SS from the calculation in Fig. 6.3. The agreement is quite
good, and it supports the claim that the mechanism presented in the previous and
the present sections is responsible for the apparition of this class of mesh-dependent
SSs. It is expected that a similar study to the one in Sec. 6.3, but extended this way
Hσ,σ−1 Fσ−1 + Hσ,σ Fσ + Hσ,σ+1 Fσ+1 = EFσ
Hσ+1,σ Fσ + Hσ+1,σ+1 Fσ+1 + Hσ+1,σ+2 Fσ+2 = EFσ+2

(6.7)
(6.8)

and keeping Eq. (6.2) could describe lower energy SSs. The condition that will be
derived is expected at least to delay the apparition of these lower energy SSs.
The SSs will first originate from the eigenvalue in Eq. (6.6) with the plus sign
in front of the square root (i.e., the LH band). A reasonable requirement to avoid
be less than the valence band
solutions in the gap is that the LH energy for kz = ∆z

edge (which is set to zero). Therefore, it is wanted that

48γ22 ~4 /m2 − 8(∆z)2 γ2 ∆SO ~2 /m + ∆2SO (∆z)4 < 4(γ1 +γ2 )~2 /m+∆SO (∆z)2 (6.9)

or, equivalently, taking squares on both sides of the inequality,
¢ ~2
+ ∆SO (γ1 + 2γ2 )(∆z)2 .
0 < −2 2γ22 − 2γ1 γ2 − γ12

(6.10)

Assuming γ1 + 2γ2 > 0, the condition that the mesh spacing must satisfy in order
to ensure that there are no solutions in the gap is obtained:
2 (2γ22 − 2γ1 γ2 − γ12 ) ~m
(∆z) >
∆SO (γ1 + 2γ2 )

(6.11)

This condition will always be satisfied if the right hand side of that inequality is
less than zero. So this yields the condition that the modified Luttinger parameters
must satisfy in order to avoid SSs in the FDM method for any mesh spacing:
1− 3
1+ 3
γ1 < γ 2 <
γ1 ⇒ −0.3666025γ1 < γ2 < 1.366025γ1 .

(6.12)

111
If the above condition is not satisfied for a material, Eq. (6.11) can be used to find a
safe mesh spacing. Of course, for some choice of parameters the safe mesh spacing will
be too big for the solutions to be accurate. In that case, a different set of parameters
or a different solution method for the EMA equations should be employed.
On the other hand, if γ1 + 2γ2 < 0, it is easy to see from Eq. (6.10) that ∆z must
satisfy

2 (2γ22 − 2γ1 γ2 − γ12 ) ~m
(∆z) <
∆SO (γ1 + 2γ2 )

(6.13)

which can only be satisfied if the right hand side is positive, leading to the conditions
(6.12) again.
Finally, imposing that the branch with the minus sign in front of the square root
in Eq. (6.6) (the spin-orbit band) also has a negative energy at kz = ∆z
results in the

requirement
− 48γ22 ~4 /m2 − 8(∆z)2 γ2 ∆SO ~2 /m + ∆2SO (∆z)4 < 4(γ1 + γ2 )~2 /m + ∆SO (∆z)2 .

(6.14)

In principle, this requirement can be satisfied in two ways. The first way is that the
absolute value of the right hand side (RHS) is smaller than the absolute value of the
left hand side (LHS). However, this leads to conditions that are not compatible with
demanding that the light hole band has an energy below the band edge at kz = ∆z

The other way that the inequality can be satisfied is requiring that absolute value of
the RHS is positive and larger than the absolute value of the LHS. It is not hard to
see that this leads to Eq. (6.9) and, therefore, will result in the same set of conditions
(6.12).

6.4.1

The InP/In0.53 Ga0.47 As SL revisited

Looking back at the parameters for In0.53 Ga0.47 As in Table 6.1, it is seen that
−0.3666025γ1 = −0.67 > −0.87 = γ2 . Therefore, the conditions for the absence
of spurious solutions in the gap were not satisfied. For that case, any mesh spacing
smaller than 22.4 Å would have triggered the appearance of a spurious solution close

112
20/34/20 InP/In0.53Ga0.47As/InP SL
1.1

1.0

Energy [eV]

0.9

0.0

-0.1

-0.04

-0.02

0.00

0.02

0.04

-1

k || [100] [Å ]

Figure 6.6: Bands of an 88 Å/65 Å InP/In0.53 Ga0.47 As SL showing no spurious solutions.
to or in the gap.
A different set of Luttinger parameters for In0.53 Ga0.47 As was found [39] and the
band structure of the InP/In0.53 Ga0.47 As superlattice recalculated. These bands are
plotted in Fig. 6.6, and they show no spurious solutions. The Luttinger parameters
used are listed in Table 6.2. They are calculated from the ones shown in Ref. [39]
using Eq. (5.11).
γ1
In0.53 Ga0.47 As 1.63

γ2
-0.27

γ3
-1.37

Table 6.2: Luttinger parameters for In0.53 Ga0.47 As satisfying the condition for the
absence of SSs (adapted from Ref. [39]).

113
AlP
AlAs
AlSb
GaP
GaAs
GaSb
InP
InAs
InSb

γ1
2.32
1.74
1.44
1.62
2.01
2.58
1.49
2.05
2.59

γ2
-0.52
-0.37
-0.35
-0.31
-0.41
-0.58
-0.31
-0.44
-0.60

γ3
0.57
0.42
0.39
0.37
0.46
0.65
0.37
0.48
0.67

1− 3
γ1

1+ 3
γ1

-0.85
-0.64
-0.53
-0.59
-0.74
-0.95
-0.55
-0.75
-0.95

3.17
2.38
1.97
2.22
2.75
3.53
2.04
2.80
3.54

Table 6.3: Check for possibility of spurious solutions in the Luttinger parameters
from Lawaetz [41].

6.5

Potential for spurious solutions in sets of Luttinger parameters

A great amount of literature can be found about calculations and measurement of
Luttinger parameters for compounds and alloys. A comprehensive review was published by Vurgaftman et al. [40]. Another commonly used list of parameters was
tabulated by Lawaetz [41]. In this section the condition derived in Eq. (6.12) will be
applied to the listed parameters in the above two references for selected materials to
detect the potential presence of SSs when using those data.
Table 6.3 lists the 8-band Luttinger parameters adapted from Lawaetz [41] for a
set of III-Vs and the limits of the interval where γ2 must lie in order to avoid SSs. It is
seen that none of those sets of parameters present a potential for SSs. An interesting
feature of writing the modified Luttinger parameters is that they depend strongly on
the anion but weakly on the cation (except when the cation is Al).
However, the situation for alloys requires a more careful consideration. A possible
approach to obtain the modified Luttinger parameters for alloys such as Inx Ga1−x As
or Alx Ga1−x As is to take averages of the parameters of the base compounds. If this is
the approach followed, the calculated parameters are not at risk of producing SSs. On
the other hand, another plausible approach would be to take the direct [42] or reciprocal [34] averages of the true Luttinger parameters instead, and then use Eq. (5.11) to

114
AlP
AlAs
AlSb
GaP
GaAs
GaSb
InP
InAs
InSb

γ1
1.72
1.49
2.57
0.42
0.66
2.32
0.23
2.81
1.75

γ2
γ3
-0.10 0.42
-0.31 0.29
-0.12 0.66
-1.32 1.12
-1.10 -0.23
-0.84 0.46
-0.82 -0.32
-0.09 0.61
-1.02 -0.02

1− 3
γ1

1+ 3
γ1

-0.63
-0.55
-0.94
-0.15
-0.24
-0.85
-0.09
-1.03
-0.64

2.36
2.04
3.51
0.58
0.90
3.16
0.32
3.84
2.39

Table 6.4: Check for possibility of spurious solutions in the Luttinger parameters
from Vurgaftman et al. [40].
find the modified parameters. Since the k·p Hamiltonian is mostly expressed in terms
of the true Luttinger parameters, the averages of these are mainly used in the literature. Nevertheless, a linear interpolation for an alloy of all the terms in Eq. (5.11) can
introduce considerable bowing in the modified (aka. Kane [43]) parameters, so one
should be careful when calculating Kane parameters using the latter approach and
make sure to check that the condition (6.12) is satisfied. An experimental study of
the hole effective masses as a function, say, of the Ga composition, and then finding
from there the Kane parameters using Eqs. (5.20) should be able to discern which
one of the two approaches is more accurate or whether a more complex interpolation
formula should be used.
The bulk Luttinger parameters tabulated for III-Vs in Vurgaftman et al. [40] are
more dangerous to use in a FDM implementation of the EMA. Table 6.4 shows the
Kane parameters calculated from the Luttinger parameters in Ref. [40]. Any trace of
independence of the parameters respect to the cation is lost. The independence seen
in Table 6.3 might come from underlying assumptions in the way Lawaetz calculates
the parameters, but a test of this conjecture is out of the scope of this Chapter. It is
easy to check that GaP, GaAs, InP and InSb don’t satisfy Eq. (6.12), and that GaSb
is close to the lower limit. Figure 6.7 shows the energy of the states given by Eq. (6.2)
for GaAs and GaSb computed with Eq. (6.5) using parameters from Ref. [40]. Again,
the light hole and split off bands start bending downward reproducing the physical

115

GaAs

GaSb

Energy [eV]

-1

-2

-3

-4

-5

-6
0.0

0.2

0.4

0.6

kz [π/(∆z)]

0.8

1.0

-6
0.0

0.2

0.4

0.6

0.8

1.0

kz [π/(∆z)]

Figure 6.7: Plot (a) [(b)] is the same as Fig. 6.4 but for GaAs (GaSb) with Luttinger
parameters from Ref. [40].
effective mass, but at about kz = 2∆z
the LH band bends up and enters the gap

region for GaAs and finishes close to it for GaSb. In the case of GaAs the spurious
solution would be out of the energy range of interest, but still there might be problems
due to other lower light hole folded bands that could invade the gap region. The set
of GaSb parameters isn’t ideal for the FDM method, either, because the energy of
the LH solution at kz = 2∆z
is too close to the valence band edge [see Fig. 6.7.b)].

This state might interfere with states mainly in the GaSb layer in a quantum well or
superlattice.

6.6

Summary

In summary, a new class of spurious solutions for the effective mass approximation, which appears when trying to solve the EMA equations with the finite difference method, has been presented. A general approach to the study of this class of

116
SSs in systems requiring the solution of a set of equations Hσ,σ−1 Fσ−1 + Hσ,σ Fσ +
Hσ,σ+1 Fσ+1 = EFσ has been formulated, and has been applied to the 8-band EMA.
A set of conditions has been derived to predict the appearance of SSs. The proposed
theory shows excellent agreement with the numerical values of the spurious energies.
Finally, popular tabulations of Luttinger parameters have been examined with the
derived conditions to identify those that might be problematic when carrying out
FDM 8-band EMA calculations. The table by Lawaetz [41] is free from danger, but
the parameters in the review article by Vurgaftman et al. [40] can lead to SSs for
some of the compounds listed.

117

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120

Chapter 7

Spin filters based on resonant

tunneling
7.1

Introduction

A major challenge for the development of spintronics [1–4] is the fabrication of a
spin injector. In the context of this chapter, spin injection refers to injection into a
semiconductor, as opposed to injection into a paramagnetic metal, which has been
demonstrated [5]. Up to date, there have been demonstrations of spin injection from
a ferromagnet [6] and dilute magnetic semiconductors (DMS) [7, 8] into a semiconductor. Recent studies show the possibility of filtering the spin of electrons traveling
through an appropriately designed resonant tunneling structure [9] in order to obtain
a spin injector [10] with a high degree of spin polarization using only nonmagnetic
materials. The operating principle of this device would be the Rashba effect (see
Ref. [11] and Chapter 2) appearing in asymmetric structures.
In this chapter, a variation on the resonant tunneling spin filter described in
Ref. [10] proposed by Ting et al. [12] is considered. In this device, the electrons tunnel
through the valence band of the active region rather than the conduction band. This
allows the electron spins to interact with the strongly spin orbit split valence band
states. The focus will fall on how the inclusion of bulk inversion asymmetry (BIA)
affects the tunneling properties on which the device is based.
Section 7.2 describes the method used to find the transmission coefficients for
the resonant tunneling structures. In Sec. 7.4, the principles of operation of the
asymmetric resonant tunneling diode (aRTD) is reviewed and the effects of BIA on the
aRTD are studied. Section 7.5 does the same for the asymmetric resonant interband
tunneling diode (aRITD). Finally, a summary of the results is presented.

121

7.2

Theoretical method

The method used to find the transmission coefficients is the multiband quantum
transmitting boundary method (MQTBM) described in Ref. [13] adapted to the k · p
formalism. This adaptation differs from the one in Ref. [14] in that the method used
here to obtain the bulk imaginary band structure is less accurate, but is consistent
with the rest of the procedure and thus it avoids the introduction of small spurious
solutions. The MQTBM is based on the single band formalism by Frensley [15]. It
does not have the stability problems of the transfer matrix method [16, 17] while
keeping the efficiency that the S-matrix method [18] loses when solving the stability
issues. Also, the k · p implementation allows for the use of a wide body of parameters
available in the literature and the easy inclusion of strain and magnetic field effects.

7.2.1

The MQTBM

The MQTBM is a method for specifying the boundary conditions appropriate for
tunneling in heterostructures. The calculation of the transmission coefficients starts
with the construction of the Hamiltonian of the system. For that purpose, the effective
mass approximation (EMA) Hamiltonian is implemented as described in Sec. 5.4. In
that section, it was shown that the solution of the EMA equations in the finite
difference method (FDM) required the solution of a set of equations
Hσ,σ−1 Fσ−1 + Hσ,σ Fσ + Hσ,σ+1 Fσ+1 = EFσ ,

(5.47)

where the symbols were defined in Sec. 5.4. For the case of tunneling, Eq. (5.47) is
more conveniently written as
Hσ,σ−1 Fσ−1 + H̄σ,σ Fσ + Hσ,σ+1 Fσ+1 = 0,

(7.1)

where H̄σ,σ Fσ = Hσ,σ Fσ − EFσ .
Figure 7.1 shows a schematic of a general structure under an applied bias V for
which the transmission coefficients are sought. The arrows represent the incoming (I),

122

(L)

(R)

Figure 7.1: Band diagram of a conduction band RTD under an applied bias V . The
arrows represent the incoming (I), reflected (r) and transmitted (t) components. The
left (right) electrode is labeled by (L) [(R)].
reflected (r) and transmitted (t) components. As plotted, the figure would represent
an intraband device—tunneling through the same band as in the incoming state
(conduction band in this case), but the development in this section is general to any
tunneling structure. The electrodes at the left (L) and right (R) are assumed to have
bulk properties and to be in the flat band condition. Under these conditions, the
envelope function components of a Bloch state will have the form
Fk, flat band, Bloch = Bk = eik·r Ck ,

(7.2)

where the Ck are 8-component vectors whose components weight the contribution of
each of the bulk zone center states. In particular, the Bk will satisfy
Bσ = eikz d Bσ−1 ,

(7.3)

where σ is the mesh point index in the discretized structure, d is the mesh point
separation and kz is the electron wavevector in the z direction. The Bloch states Bk

123
of course satisfy Eq. (5.47), which can be rewritten in a transfer matrix form [13]

−H−1
σ,σ−1 H̄σ,σ

−H−1
σ,σ−1 Hσ,σ+1




Bσ
Bσ+1

 = e−ikz d 

Bσ
Bσ+1

.

(7.4)

The above 16×16 eigensystem yields the Bloch states and their corresponding
wavevectors—which can be complex, describing in that case evanescent states—for
a given energy of the state E and an in-plane wavevector kk . On the other hand,
a general state in the left electrode with a definite energy and kk will be a linear
combination of the Bloch states found in the solution of Eq. (7.4)

|Li = |Ii + |ri =

Ij |Bkk ,kz,j , Li +

rj |Bkk ,kz,j , Li,

(7.5)

|Bkk ,kz,j , Li = eikk ·r eikz,j z |ukk ,kz,j i = eikk ·r eikz,j z Ckl k ,kz,j |ulΓ i

(7.2’)

j=1

j=9

where the kz ’s and the Bloch states

have been ordered so that first 8 eight kz,j correspond to propagation towards the
right (i.e., kz,j is either positive real or has positive imaginary part for electron states
and viceversa for holes) and the last eight to propagation towards the left. Similarly,
for the transmitted state
|Ri = |ti =

j=1

tj |Bkk ,kz,j , Ri.

(7.6)

Following Ting et al. [13], equations for the envelopes at the left and right boundaries can be written by letting I, r and t be the column 8-vectors obtained from
putting together the coefficients {Ij }, {rj } and {tj }, respectively:
 
  
D12
 
 0  = DL   =  11
DL21 DL22
F1

(7.7)

124
and
 
  
D12
 ,
 N −2  = DR   =  11
DR
FN −1
21 D22

(7.8)

where DL and DR are 16×16 matrices whose column vectors are the normalized
eigenvectors, arranged in the same order as the eigenvalues, from the solution of
Eq. (7.4) in the left or right electrodes. For degenerate kz ’s, the eigenvectors from
Eq. (7.4) are orthogonalized. Each D matrix is divided into submatrices Dij for
convenience.
The incident state I is specified in the statement of the problem. Therefore,
Eq. (7.7) can be viewed as a system of two equations for the three 8-vector unknowns
r, F0 and F1 . It can be transformed in a single equation with two unknowns
−1
−1
F0 − DL12 DL22 F1 = DL11 − DL12 DL22 DL21 I.

(7.9)

Similarly, t can be eliminated from Eq. (7.8) to yield the other boundary condition
−1

−DR
21 D11 FN −2 + FN −1 = 0.

(7.10)

Including the boundary conditions into Eq. (5.48), the wavefunction is found by

125
solving the sparse linear system

H1,0
 0
 ..
 .
 0

−1
−DL12 DL22

...

H̄1,1

H1,2

H2,1
...

H̄2,2 H2,3
...
...

...



 0 

  F1 
...
...


  F2 
...

  .. 
..
..
...
 . 


HN −2,N −3 H̄N −2,N −2 HN −2,N −1  FN −2 

R −1
−D21 D11
FN −1
³
´ 
−1
DL11 − DL12 DL22 DL21 I
 . (7.11)
=
..

Once the envelope function components for the entire wavefunction have been
found, the coefficients of the transmitted part in terms of the Bloch states for the
right electrode can be found from Eq. (7.8):
−1

t = DR
21 FN −1 .

(7.12)

And from there, the transmission coefficient can be expressed as the sum of transmission coefficients into each Bloch state channel1 :

T E, kk

¯ ¡
¢¯
¯ ¡
¢¯2 ¯vj E, kk ; R ¯
¯tj E, kk ¯ ¯ ¡
¢¯
¯vI E, kk ; L ¯ ,
j=1

(7.13)

where vI E, kk ; L and vj E, kk ; R are the velocities along the z direction of the

incident and the transmitted bulk plane wave states respectively.

The sum should only include terms due to transmitting, as opposed to evanescent, components.

126

7.2.2

Computation of the velocity of the states

In appendices C and D it has been shown that
¯ ¯¯
¯ ¯
¢ 1 ∂E(kk , kz ; R) ¯
∂H(k)
¯ ¯ Bk ,k , R .
vj E, kk ; R =
Bkk ,kz,j , R ¯
¯ ¯ k z,j
∂kz
∂k
kz,j
kz,j
(7.14)

The expansion in Eq. (5.43) can be used to evaluate ∂H(k)/∂kz and obtain
¢ 1D
¯ (2)
(1) ¡ ¢¯
Bkk ,kz,j , R ¯2HR kz,j + HR kk ¯ Bkk ,kz,j , R ,
vj E, kk ; L =

(7.15)

which can be written in matrix notation because, apart from an overall phase factor,
the components of |Bkk ,kz,j , Ri in the zone center basis are given by the j-th column
of DR
11 , D11,j :

¢ 1
(2)
(1)
vj E, kk ; L = DR
DR
2H
11,j .
R z,j
~ 11,j

(7.16)

Incidentally, the operator

v̂ =

1 ¡ (2)
2H kz + H(1)

(7.17)

coincides with the current density operator found by Wu et al. [19].

7.2.3

Preparation of the incoming states

Since the transmission properties of the spin filter structure are assumed to be spindependent, one has to carefully prepare the incoming state and have control over
its spin (for electrons) or total angular momentum (for holes). The method used to
achieve this is described below.
Electrons
When electrons are injected into the resonant structure, the incoming state can only
be a linear combination of right propagating Bloch states in the conduction band
(CB) for the left electrode. Since the electrons are assumed to have a given energy E

127
and parallel component of the wavevector kk , the intervening Bloch states will have
that E and kk .
The incoming state is given by the first summand in Eq. (7.5):
|Ii =

j∈CB

Ij |Bkk ,kz,j , Li,

(7.18)

where the sum over Bloch states only includes the two in the CB satisfying the E
and kk requirements.
If the electron spin is required to point in the θ, ϕ direction, first one must
construct the dimensionless spin operator in an arbitrary direction σθ,ϕ :
σθ,ϕ = σx sin θ cos ϕ + σy sin θ sin ϕ + σz cos θ,

(7.19)

where the σi ’s are derived from Eqs. (5.37)-(5.39).
At first it would seem logical to look for eigenstates of σθ,ϕ , but states away from
the zone center will have a finite hole component. This will cause electron states to
cease being spin eigenstates because holes don’t have a definite spin. Therefore, one
must satisfy oneself with requiring that the expectation value of σθ,ϕ is maximized.
That is, one seeks a set of {Ij }’s such that
hσθ,ϕ iI =

j,l∈CB

Il∗ Ij

Bkk ,kz,l , L |σθ,ϕ | Bkk ,kz,j , L

(7.20)

is maximized. These matrix elements are taken with respect to the primitive cell in
the left electrode right before entering the active region of the device. This is so to
make sure the electron is coming into the structure with the desired spin. If the matrix
elements were taken all over the crystal, the elements relating states with k z,l 6= kz,j
would vanish by the space group selection rules. Physically, this translates into the
fact that, when the Kramers degeneracy is broken, it is impossible to construct a
state that will maximize hσθ,ϕ iI at all points of the space at the same time because
the difference in kz ’s will affect the relative phases of the Bloch states. This is seen

128
in the following example. Consider a free electron described by the wavefunction
 √ 
1/ 2
 + ei[(k+∆k)x−ωt]  √  ,
Ψ(x, t) = ei[kx−ωt] 
1/ 2

(7.21)

where the same ω indicates that even if the two components have different wavevector, there is some (spin-dependent) mechanism putting their energies together. It is
easily seen that the expectation value of the, say, x component of the spin is space
dependent:
hσx i(x) = Ψ† σx Ψ = cos ∆k x,

(7.22)

and that is why hσθ,ϕ iI is maximized as close to the active region as possible.
In structures presenting Rashba splitting (cf. Chapter 2), the separation between
the kz ’s with the same energy is of the order of ∆k ≈ 10−3 Å−1 . At typical values of the
Fermi radius kF , it takes about 3000Å for the relative phase of the two Bloch states
to change by π. Of course, real electrons in a crystal will have some wavepacket size.
As long as this wavepacket size is substantially smaller than these 3000Å, one can
say that the spin of the electron is the same throughout its spatial extent. Another
way to think about it is that the spin of the electron will be independent of space
if the spread in k space corresponding to its localization is bigger than ∆k. If this
condition is not fulfilled, taking a snapshot in time and looking along the z direction
one would see the spin precessing in a helix-like fashion.
Therefore, instead of maximizing the expression in Eq. (7.20), one should construct
|Ii =

Ij |ukk ,kz,j i

(7.23)

hσθ,ϕ iI = hI |σθ,ϕ | Ii

(7.20’)

hI|Ii = 1.

(7.24)

j∈CB

and maximize

subject to the constraint

129
This is an optimization problem that can be solved with the standard Lagrange
multiplier technique. Since the |ukk ,kz,j i’s in |Ii need not be orthogonal with one
another, it easily seen that the optimization procedure leads to the solution of a
generalized eigenvalue problem
σ̂I = λĝI,

(7.25)

where, for electrons, σ̂ is a 2 × 2 matrix whose matrix elements are
σjl = ukk ,kz,j |σθ,ϕ | ukk ,kz,l ,

(7.26)

and ĝ is a metric tensor with elements
gjl = hukk ,kz,j |ukk ,kz,l i.

(7.27)

Holes
The determination of the coefficients I for holes is quite similar to the electron case.
The main difference is that one is presented with the dilemma of choosing the incoming
state to have a given spin or making it have a given total angular momentum. Of
course, the same discussion as for the electron case applies, and, for a fixed E and kk ,
only states that maximize the expectation value of the chosen measurable quantity
can be constructed. Physically, holes in the valence bands of bulk zincblendes are
better characterized by their total angular momentum, so the incoming states will be
constructed maximizing the expectation value of
hJθ,ϕ iI =

j,l∈CB

Il∗ Ij Bkk ,kz,l , L |Jθ,ϕ | Bkk ,kz,j , L ,

(7.28)

where Jθ,ϕ is the (dimensionless) total angular momentum along an arbitrary axis,
and is defined analogously to Eq. (7.19). The angular momentum components along

130

0.56
0.54

Energy [eV]

0.52
0.50
0.48

0.46
0.44
0.42

kx=0.01 Å

0.40

-1

0.38
-0.04

-0.02

0.00

0.02

0.04

-1

kz [Å ]
Figure 7.2: Bands of InAs for kk = (0.01, 0)Å−1 as a function of kz . The horizontal
lines indicate possible energies of the incoming particle, and the circles signal the
states whose linear combination creates an incoming state with heavy or light hole
character.
the x, y and z axes are given by

1/2
 i

Ji =  0
with

Jx1/2 = 

0 1
1 0

Jy1/2 = 

3/2
Ji

0 ,

(7.29)

1/2

0 −i

Jz1/2 = 

0 −1

(7.30)

131
and

√
 3
3/2
Jx = 
 0

0 − 3i 0
√
 3i
−2i
0 
3/2
Jy = 
√ 
 0
2i
0 − 3i
3i
0 0
1 0
0
,
(7.31)
0 −1 0 
0 0 −3

0 
√ 
3
3 0

0
Jz3/2 = 
0

and the rest of the procedure is analogous to the electron case.

7.2.4

Transmission coefficients and transmitted spin for an
ensemble of electrons

In normal operation of the device, the incoming spins will be randomly oriented. If
the approximations that a) they come from degenerate bands and b) that spin is a
good quantum number are used to simplify the calculations, they can be described
by the following density matrix
ρ̂I =
4π

| ↑Ω ih↑Ω |dΩ,

(7.32)

where the integral is over spins pointing in any direction with equal probability and
| ↑Ω i is a spin pointing in the direction given by the solid angle Ω. This integral can
be easily performed to obtain
ρ̂I =

(| ↑θ,ϕ ih↑θ,ϕ | + | ↓θ,ϕ ih↓θ,ϕ |) ,

(7.33)

where | ↑θ,ϕ i (| ↓θ,ϕ i) labels a spin up (down) along the direction given by the polar
angles θ and ϕ. Note that Eq. (7.33) will hold no matter what the choice for θ and

132
ϕ is, therefore the angle labels are dropped.
If tij denotes the component of the transmitted state for spin j when the incident
state is a spin i, the density matrix for the transmitted ensemble will be
ρ̂t =

1 X
tij t∗ik |jihk|.
2 i,j,k∈↑,↓

(7.34)

The transmission coefficient will be given by the ratio of currents of the transmitted respect to the incident state. From Appendix C it is easily seen that ratio of
probability currents can be given as
tr(ρ̂t v̂)
T E, kk =
tr(ρ̂I v̂)

(7.35)

where v̂ is the velocity operator as defined in Appendix D. Evaluating the traces in
this equation using the | ↑i, | ↓i basis yields
¢ (|t↑↑ |2 + |t↓↑ |2 )v⊥↑,t + (|t↓↓ |2 + |t↑↓ |2 )v⊥↓,t
T E, kk =
v⊥↑,I + v⊥↓,I

(7.36)

where the dependencies of tij and vi on E and kk are not explicitly shown. With the
assumptions stated at the beginning of this section, one has v⊥↑ = v⊥↓ , and Eq. (7.36)
can be rewritten as
¢ 1 (|t↑↑ |2 + |t↑↓ |2 )v⊥,t + (|t↓↓ |2 + |t↓↑ |2 )v⊥,t
T E, kk =
v⊥,I
= [(T↑↑ + T↑↓ ) + (T↓↓ + T↓↑ )],

(7.37)

where
Tij ≡

|tij |2 v⊥,t
v⊥,I

(7.38)

Thus it has been shown that, in the approximation where the bands are degenerate, the transmission coefficient for a random population of spins will simply be
the average of the coefficients for spin up and down in an arbitrary direction even
when the properties of the barrier are spin-dependent and it might introduce “channel

133
mixing.”
A good measure of the filtering efficiency of a spin filter can be given by the ratio
of transmitted spin polarized current to the incoming current
¢ | tr(ρ̂t σv̂)|
η E, kk =
tr(ρ̂I v̂)

(7.39)

where η E, kk measures the filtering efficiency for a given energy and kk of the

incoming particles. This measure gives the maximum possible efficiency, which is

achieved when the transmitted spins are analyzed in the same direction as tr(ρ̂t σv̂).
To evaluate the trace in the numerator of Eq. (7.39), it is convenient to define
raising and lowering operators

0 1

,
(σx + iσy ) = 
0 0
0 0
,
σ− = (σx − iσy ) = 
1 0

σ+ =

(7.40)

(7.41)

where the z axis is chosen along to be parallel with the | ↑i spin.
Then
tr(ρ̂t σ+ v̂) = (t↑↓ t∗↑↑ + t↓↓ t∗↓↑ )v⊥,t
tr(ρ̂t σ− v̂) = (t∗↑↓ t↑↑ + t∗↓↓ t↓↑ )v⊥,t
tr(ρ̂t σz v̂) = (|t↑↑ |2 + |t↓↑ |2 )v⊥,t − (|t↓↓ |2 + |t↑↓ |2 )v⊥,t ,

(7.42)
(7.43)
(7.44)

or, in cartesian components
tr(ρ̂t σx v̂) = < (t↑↓ t∗↑↑ + t↓↓ t∗↓↑ )v⊥,t
tr(ρ̂t σy v̂) = = (t↑↓ t∗↑↑ + t↓↓ t∗↓↑ )v⊥,t

tr(ρ̂t σz v̂) = (|t↑↑ |2 + |t↓↑ |2 )v⊥,t − (|t↓↓ |2 + |t↑↓ |2 )v⊥,t

(7.45)
(7.46)
(7.47)

134
and, from here, the efficiency will be

η E, kk =

[ tr(ρ̂t σx v̂)]2 + [ tr(ρ̂t σy v̂)]2 + [ tr(ρ̂t σz v̂)]2
v⊥,I /2

(7.48)

The spin polarization p, defined as the magnitude of the spin polarized current
divided by the transmitted current, is related to the filter efficiency by
¢ ¡
η E, kk = p E, kk T E, kk .

(7.49)

These expressions adopt simpler forms when the incoming spins are chosen to be
resonant with the barrier (i.e., there is no up-down channel mixing). This translates
into t↑↓ = t↓↑ = 0, which simplifies the above expressions considerably to
η = T↑↑ − T↓↓

(7.50)

T↑↑ − T↓↓
T↑↑ + T↓↓

(7.51)

and
p=

7.3

Resonant tunneling in asymmetric double barriers

In this section the basic operating principle of the Rashba spin filters is explained.
To do that, a barrier structure comprised of AlSb/GaSb/InAs/AlSb clad by InAs
electrodes is considered (see Fig. 7.3), with the GaSb and InAs layer thicknesses
chosen to yield a large Rashba splitting (cf. Chapter. 3).
The quasi-bound states for the structure in Fig. 7.3 without the inclusion of bulk
inversion asymmetry (BIA) effects [20] are described in Sec. 5.6.1. The spin of these
quasi-bound states has a kk dependence shown in Fig. 7.4.
Consider now an incoming electron whose spin is pointing according to the directions in Fig. 7.4, belonging to either the lower or the highest conduction subband.

135

Distance [Å]
-20

100

2.0

Energy [eV]

1.5
1.0
0.5

(L)

AlSb

InAs GaSb AlSb

(R)

0.0
-0.5
-5

10 15 20 25 30 35
Layer Index

Figure 7.3: aRTD structure at zero bias. The AlSb, GaSb and InAs layers are 8
monolayers (MLs) each (3.048 Å/ML), and the electrodes are InAs.
A schematic of this situation is shown in Fig. 7.5. The transmission coefficient of
such an electron, with kk = (0.02, 0)Å−1 and the spin pointing in the ±y direction,
is shown in Fig. 7.6. It is seen that the transmission curves appear decoupled due to
the incident electron interacting with only one of the quasi-bound states. The transmission peaks are split by the Rashba effect. Looking at the spin of the transmitted
electron it is seen that it does not change after going through the barrier. Therefore,
Fig. 7.4 also indicates the spin of the transmitted electrons as a function of their k k .

7.4

Asymmetric resonant tunneling diode (aRTD)

The aRTD spin filter is based in the intraband tunneling phenomena through an
asymmetric structure as described in the previous section. The origin of that asymmetry can be different barrier thicknesses [10], different constituent materials of the
barrier [10, 12] or the inclusion of spatially dependent doping profiles [21].

136
0.04

0.04

0.02

0.00
ky/(2π/a)

Lower Subband

0.00

Upper subband

ky/(2π/a)

-0.02

-0.04

-0.04
-0.04 -0.02 0.00 0.02 0.04
kx/(2π/a)

-0.04 -0.02 0.00 0.02 0.04
kx/(2π/a)

Figure 7.4: Spin directions for the quasi-bound states in the aRTD (same as Fig. 2.3).
Looking at Fig. 7.4 it must be noted that, for an electron gas with thermal distribution, the spin contribution for each incoming electron with some kk will be
compensated by an electron with −kk . Therefore, in order to obtain spin injection an
anisotropy in the distribution of kk must be created. This can be done, for example,
with the addition of a lateral electric field that gives the incoming electrons a nonvanishing average kk component [10]. Then, if the lateral electric field points, say, in the
x direction, the transmitted current should be analyzed along the y axis to obtain a
maximum effect. Calculations for this family of devices show current polarizations p
of the order of 20% [12].
The inclusion of BIA effects does not change the qualitative picture for the aRTD
very much. The biggest change is due to the spins of the quasi-bound states being
tilted respect to when no BIA was considered (see, for example, Fig. 5.19). As shown
in Fig. 7.7, this introduces supplementary peaks in the transmission curves if the
incoming spins are left unchanged. This is just due to the fact that spins pointing
along ±y are no longer resonant when BIA is taken into account. However, the peak
strength is very little affected by the inclusion of the BIA terms and one expects the
current polarization prediction to be similar, with the caveat that the spin analysis
must be made along the new resonant direction.
There are mainly two effects that limit the performance of the aRTD as a spin

137

k||
kz

Figure 7.5: Resonant spin of an electron incident into an aRTD structure.

Transmission Coefficient

0.1
+y
-y
0.01

1E-3

1E-4

0.87

0.88

0.89

0.90

0.91

0.92

Energy [eV]

Figure 7.6: Transmission coefficient for the aRTD structure described in Fig. 7.3.
The incoming electrons are assumed to have kk = (0.02, 0)Å−1 and the spin pointing
in the ±y direction. This direction is chosen for the electron to couple only to one of
the two quasi-bound states.

138

Transmission Coefficient

0.1
+y
-y

0.01

1E-3

1E-4

0.87

0.88

0.89

0.90

0.91

0.92

Energy [eV]

Figure 7.7: As in Fig. 7.6, but with BIA effects included.
filter (see Fig. 7.8):
I. Cancelation of net spin between spin split subbands:
The two conduction subbands have spins pointing in different directions. Since
an electron has to tunnel through a barrier state, the amount of current polarization will go as the ratio in the number of quasi-bound states in the barrier
below the emitter Fermi level (i.e., the area enclosed by the Fermi circles for
each subband). Therefore, the amount of polarization will only be appreciable
when the Fermi level of the emitter lines close to the conduction band edge of
the barrier.
II. Cancelation in the same band:
Even within a subband there is cancelation because of the spins in the barrier
states pointing in a circular fashion (cf. Fig. 7.4). This will also decrease the
filter efficiency.

139
Lower Conduction
Subband

Higher Conduction
Subband

Figure 7.8: kk dependence of the spin of the transmitted states in an aRTD and
illustration of the difficulties in the aRTD. Not only do electrons tunneling through
different subbands have opposite spin (the net contribution is represented by the big
arrow), but also the spin contribution within a subband cancels out for states where
both kk and −kk participate in the tunneling process. The big arrows indicate the
total spin contribution for each subband.

7.5

Asymmetric resonant interband tunneling
diode (aRITD)

The aRITD [12] attempts to deal with some of the problems of the aRTD mentioned
above by making the transmitted electrons tunnel through the barrier valence band
states rather than the conduction band. A typical aRITD structure is shown in
Fig. 7.9. The aRITD would be a low voltage device because of the little voltage
(always less than ∼200 mV) necessary to align the barrier quasi-bound valence band
states with the Fermi level of the emitter.
A typical tunneling transmission curve through an aRITD structure is shown in
Fig. 7.10. The resonant peaks where the energy and kk of the incident electron
match those of the quasi-bound barrier states are clearly shown. As for the aRTD,
choosing the incoming spins to be ±y for kk along [100] decouples the two peaks.
It can be seen that having the spins point in-plane and perpendicular to kk (e.g.,
clock and counterclockwise) also decouples the peaks exactly when kk is along the
[110] direction and to a very good approximation for a general kk ’s intervening in

140

Distance [Å]
-20

80 100 120 140

2.0
(L)

AlSb InAs

GaSb

AlSb (R)

Energy [eV]

1.5
1.0
0.5
0.0
-0.5
-5 0

5 10 15 20 25 30 35 40 45
Layer Index

Figure 7.9: aRITD structure at zero bias. The AlSb and InAs layers are 8 monolayers
(MLs) each (3.048 Å/ML), the GaSb is 18 MLs thick and the electrodes are InAs.
the tunneling process. A surprising feature of this curve is that the peak strength is
heavily dependent on the incident spin orientation. This is also found in calculations
using the effective bond orbital model method. This effect can be used to minimize
the contribution of the upper subband to the transmitted current. Thus, the impact
of one of the difficulties mentioned earlier for the aRTD can be lessened.
Figure 7.11 (a) shows how the emitter (strained to a GaSb substrate) conduction
band and the barrier valence states match up for the aRITD. The structure can be
designed in such a way that, for the operating bias, tunneling takes place through the
HH1 bands only. The fact that the bands have opposite curvatures makes that only
states with big kk can tunnel. This allows the device to work far from the zone center,
where the splitting vanishes. Plots (b) and (c) show the effect of the application of a
lateral electric field on the states participating in the tunneling process. The Fermi sea
tilts [22] and some of the channels used at zero lateral field no longer contribute due to
the lack of incident states at that kk . In the structure studied here, the transmitted

141

0.748

Transmission Coefficient

0.1
0.015

0.01
1E-3

+y
-y

1E-4
1E-5
1E-6
1E-7
1E-8
0.495

0.500

0.505

0.510

0.515

0.520

Energy [eV]

Figure 7.10: Transmission coefficient for the aRITD structure described in Fig. 7.9.
The incoming electrons are assumed to have kk = (0.02, 0)Å−1 and the spin pointing
in the ±y direction. This direction is chosen for the electron to couple only to one of
the two quasi-bound states. Note the asymmetry in the peak strengths.
current will mainly originate from tunneling through the lower HH1 subband (cf.
Fig. 7.10). As the applied lateral electric field gets larger, the Fermi sea will tilt more
and the number of unwanted kk channels within a band reducing the average spin
will decrease. Thus, the aRITD is less sensible to within band cancelation effects.
The reduction of the two aforementioned adverse effects allows the increase of the
spin polarization levels up to ∼ 60% [12]. The curves in that reference, reproduced
in Fig. 7.12, show that the amount of filtering is strongly dependent on the electrode
doping. Alternatively, the regime of high filtering can be achieved with the application
of an appropriate bias across the barrier.
Figure 7.13 shows the effect in the transmission curves of the inclusion of BIA,
with the incoming spin directions left unchanged. The spin preferred directions of
the holes in the barrier are more sensitive to the BIA terms than its conduction band
(CB) counterparts. This makes new peaks appear in the transmission curves. The
asymmetry in peak strengths for the lowest and highest energy peaks (considering
the sum of both spin contributions at a given energy) isn’t affected very much by the

142

b)
a)
InAs/8ML AlSb/18ML GaSb/8ML InAs/8ML AlSb/InAs

1.0
0.9

Energy [eV]

0.8
0.7
0.6

Emitter
Fermi Level

Bulk InAs CB

c)
HH1

0.5

LH1
0.4
HH2

0.3
-0.04

-0.02

0.00

0.02

0.04

kx [100]

Figure 7.11: a) Bands of the aRITD structure and operation with a lateral electric
field. The short dashed line shows the conduction band of bulk InAs strained to a
GaSb substrate. The desired position of the Fermi level of the emitter respect to the
barrier bands is indicated. (b) [c)] shows the participating channels and their spin
contribution of the upper (lower) HH1 subband. The light grey arrow signifies the
weaker peak strength for transmission through the upper HH1 subband (cf. Fig. 7.10).
inclusion of BIA. So, one of the key aspects of the operation of the device is still there
with the BIA terms present. The origin of the asymmetry in the peak strength is a
subject still under investigation. The direction where the spins should be analyzed
for optimum performance will change, and depends on the interplay of BIA and SIA
(structural inversion asymmetry) effects for a particular structure, but that is not
expected to influence heavily on the magnitude of the polarization.

143

Figure 7.12: Spin polarization of the current as a function of lateral field for several
emitter electron densities (reproduced from Ref. [12]).

7.6

Summary

In summary, a method for calculating spin-dependent transmission properties of tunnel structures has been presented. The multiband quantum transmitting boundary
method, originally formulated in the tight binding context [13], has been adapted to
the k · p formalism. This method has been employed to calculate spin-dependent
transmission curves for candidate structures for a spin filter. Calculations without
BIA effects show the same features as equivalent calculations using the effective bond
orbital model method. Two devices, the aRTD and the aRITD, have been studied.
For the aRTD, it has been shown that the inclusion of bulk inversion asymmetry
(BIA) effects changes only slightly the transmission properties of the resonant structure. Finally, for the aRITD, it has been seen that BIA plays a role on the choice of

144

Transmission Coefficient

0.1

0.4386
0.0660

0.01
1E-3
1E-4
1E-5
1E-6
1E-7
1E-8
0.495

+y
-y

0.500

0.505

0.510

0.515

0.520

Energy [eV]

Figure 7.13: As Fig. 7.10, with BIA effects included. The numbers indicate the sum
of the transmission coefficients for both peaks.
the transmitted spin direction, but doesn’t affect appreciably the main working principle of the device, namely, the highly asymmetric peak strengths of the transmission
resonances through the HH1 subbands.

145

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K. H. Ploog, Phys. Rev. Lett. 87, art. no. (2001).
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[10] A. Voskoboynikov, S. S. Lin, C. P. Lee, and O. Tretyak, J. Appl. Phys. 87, 387
(2000).
[11] Y. A. Bychkov and E. I. Rashba, J. Phys. C 17, 6039 (1984).
[12] D. Z.-Y. Ting, X. Cartoixà, D. H. Chow, J. S. Moon, D. L. Smith, T. C. McGill,
and J. N. Schulman, IEEE Trans. Magn. (2002), to be published.
[13] D. Z.-Y. Ting, E. T. Yu, and T. C. McGill, Phys. Rev. B 45, 3583 (1992).
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146
[15] W. R. Frensley, Reviews Of Modern Physics 62, 745 (1990).
[16] E. O. Kane, in Tunneling Phenomena in Solids, edited by E. Burstein and S.
Lundqvist (Plenum Press, New York, 1969), p. 1.
[17] D. L. Smith and C. Mailhiot, Phys. Rev. B 33, 8345 (1986).
[18] D. Y. K. Ko and J. C. Inkson, Phys. Rev. B 38, 9945 (1988).
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Press, Cambridge, Uk, 1972), pp. 216–217.

147

Chapter 8

Photon generation for a

Monte Carlo ray tracing LED simulator
8.1

Introduction

A requirement for the development of spintronic devices is the achievement of reliable
spin injection. Once a candidate for a spin injector has been identified, there must
be a means of quantifying its performance. This leads to mainly two families of experiments for measuring spin injection; namely, electrical experiments [1] and optical
experiments [2–4].
In electrical experiments the spin polarized electrons inside the material of interest,
which will be considered to be a semiconductor but could be a metal as well, are
collected by a ferromagnetic contact. Then, depending on the relative orientation
of the majority electron spins and the magnetization of the collector, a higher or
lower resistance is measured. From that data, the existence of spin injection can be
established. However, in recent years electrical measurement methods have fallen into
disgrace due to the possibility of the presence of spurious local Hall voltages due to
the ferromagnetic contact fringing fields [5]. Another inconvenience is that it has also
been shown [6, 7] that one should expect only a very small effect in the standard
ferromagnet-semiconductor-ferromagnet configuration.
Optical detection of spin injection is based on the polarization state of the photon
emitted after a direct interband electron-hole recombination. If we consider the light
coming from direct transitions, say in a III-V zincblende semiconductor, the polarization state of the emitted photons will depend on the direction of emission and the
initial and final states of the electron making the transition. By analyzing the emitted light, one can obtain statistical information about the initial electron states, and
from there the spin polarization of the electrons before making the transition. This

148
method is considered to be more direct and less subject to criticism than the electrical
detection method. It has been successfully employed to detect significant amounts of
electrical spin injection both from a semiconductor into another semiconductor [3, 4],
and from a ferromagnet into a semiconductor [8].
However, the effects of refraction on the polarization of the emitted light have
been mostly overlooked. Ichiriu et al. [9] have investigated these effects using a
Monte Carlo ray tracing algorithm that follows the photons from their generation at
the p-n junction until they hit the detector. They conclude that, for emissions that
aren’t parallel or perpendicular to the junction plane, the effect of refraction will be
to increase the observed degree of polarization with respect to that actually emitted
in the bulk of the semiconductor.
In order to perform this kind of calculations, a method that generates single photons with the adequate polarization is needed. This article describes in detail such
a method. In the discussion in Sec. 8.2, it is shown how Fermi’s golden rule does
not specify the polarization of the emitted photon. The time-dependent perturbation
theory is used instead to derive the polarization of a photon emitted in an arbitrary
direction for transitions between two bands of a crystal in Sec. 8.3. In Sec. 8.4, the
formalism is applied to zincblende and wurtzite bulk and quantum well semiconductors. Section 8.5 describes the use of a Monte Carlo scheme to generate single photons
ready for use in a ray tracing algorithm and yielding the ensemble results derived in
Sec. 8.3. Finally, Sec. 8.6 shows results from the application of the method to a bulk
zincblende.
A reader not familiar with group theoretical arguments might find it useful to
refer to Appendix B for a utilitarian introduction to group theory.

149

|i>, Ei

Ps
|g>

|f>, Ef

Figure 8.1: Schematic of the interband photon emission process.

8.2

The interaction Hamiltonian and Fermi’s
golden rule

Figure 8.1 describes schematically the photon emission process under study. Πσ (Πλ )
is the label of the initial (final) irreducible representation. These labels indicate the
starting and final bands between which the interband transition takes place. |ii and
|f i denote the specific initial and final states, respectively, inside each band. Their
corresponding energies are Ei and Ef . Finally, |γi represents the state of the emitted
photon.
To study this process, consider a plane wave interacting with a crystal. The plane
wave can be described by just its vector potential in the radiation gauge [10]:
A = A0 a cos (q · r − ωt) ,

(8.1)

where A is the vector potential, A0 is the amplitude of the wave, a is a unit vector
in the direction of A, q is the wavevector of the wave and ω is its frequency. The
electric and magnetic fields can be found using the usual relations:
∂A
∂t
B = ∇ × A,
E=−

(8.2)
(8.3)

150
where the radiation gauge condition ∇ · A = 0 has been used. From Eq. (8.2) it is
seen that a determines the polarization of the plane wave. a can take complex values
to describe an elliptical polarization state.
The Hamiltonian of a crystal coupled to an applied electromagnetic plane wave
can be written, in the low-intensity limit and dipole approximation (i.e., the photon
wavelength much bigger than the primitive cell size), as [11]:
H = H0 −

P · A,
mo

(8.4)

where the dipole magnetic term has been omitted because it is much weaker than
the dipole electric term, e is the absolute value of the electron charge, mo is the free
electron mass and P is the momentum operator.
Following Ridley [12], the cosine in Eq. (8.1) can be expanded into a sum of complex exponentials and introduced into Eq. (8.4) to yield, by selecting the appropriate
terms, the space part of the Hamiltonian describing the photon emission process:
Hν em = −
mo

~ (nν + 1)
2V ²ν ων

¾1/2

a · P,

(8.5)

where ~ is the reduced Planck constant, nν is the number of photons present corresponding to the mode ν and having a frequency ων , ²ν is the optical permittivity of
that mode and V is the volume of the crystal. The time part of the Hamiltonian
leads to the appearance of the Dirac delta factor in energy in Eq. (8.6).
On the other hand, Fermi’s golden rule can be used to find Wi→f , the probability
per unit time that an initial state |ii will make a transition to any |fi belonging to
a family with dSf degenerate states due to the action of a perturbation Hν em . It
states [12]:
Wi→f =

2π
|hf |Hν em | ii|2 δ (Ei − Ef ) dSf .

(8.6)

Thus, the probability of spontaneous emission into a solid angle dΩ per unit time
Wem can be obtained by taking nν = 0 in Eq. (8.5) and plugging this equation into

151
Eq. (8.6):
e2 4 |a · Pf i |2 ηr ων2 dΩ
Wem =
4π²0 ~ων 2m2
c3 4π

(8.7)

where ²0 is the vacuum permittivity, c is the speed of light, ηr is the crystal refractive
index and Pf i is the momentum matrix element between the initial and the final
states.
The important point about Eq. (8.7) is that the probability of emission is proportional to the scalar product of the polarization vector of the plane wave and the
matrix element of the momentum operator between the initial and final states. If
only transitions between two bands are of interest, as is commonly the case in semiconductors, all the photons emitted will have the same frequency and the only thing
that will change between two emission events will be the |a · Pf i |2 factor. Therefore, the knowledge of that factor is the only thing needed to determine the relative
transition rates between some initial and final states belonging to two given bands.
Mathematically, this can be written as:
Wi→f,a
|a · Pf i |2
Wi0 →f 0 ,a0
|a0 · Pf 0 i0 |2

8.2.1

(8.8)

The Wigner-Eckart theorem for point groups applied
to momentum matrix elements

The expression in Eq. (8.8) can be further simplified by making use of symmetry
considerations. The Wigner-Eckart theorem for point groups (see Ref. [13] and appendix B.3) isolates the symmetry effects from other contributions in the matrix
elements of tensor operators. In particular, the momentum matrix elements can be
written as follows:
¯ ¯ ®
¯ ¯ ® XX
Pµmf i = f ¯Pµm ¯ i =
hf |κ, ki κ, k ¯Pµm ¯ λ, l hλ, l|ii =
k∈κ l∈λ

hκ kPµ k λi

XX
k∈κ l∈λ

hf |κ, kihµ, m; λ, l|κ, kihλ, l|ii,

(8.9)

152
Td
Γ5
Γ6
Γ7
Γ8

SO(3)
|l = 1i
|l = 0i ⊗ |s = 1/2i
|l = 1i ⊗ |s = 1/2i such that j = 1/2
|j = 3/2i

Table 8.1: Table of selected irreducible representations (irreps) of the zincblende point
group Td and their equivalent full rotation group SO(3) irreps.
where the initial and final states have been expanded into the basis states corresponding to their irreducible representation (irrep). Greek indices label irreps and
latin indices label specific basis states inside an irrep. hκ kPµ k λi is the so-called reduced matrix element [14], and will not depend on |ii nor |f i as long as these are taken
as linear combinations of basis states belonging to the Πλ and Πκ bands, respectively.
The symbol hµ, m; λ, l|κ, ki represents the complex conjugate of the Clebsch-Gordan
coefficient for the point group under consideration.

8.2.2

Application to III-V zincblendes

To exemplify this abstract formalism, consider a transition between the conduction
band and the valence band of a III-V zincblende at the zone center. For most of the
III-Vs, the conduction band edge is described by the Γ6 irrep of the Td point group;
while the valence band edge is described by the Γ8 irrep, where the Koster-DimmockWheeler-Statz (KDWS) [15] notation is being used. The treatment of Td is simplified
by the fact that some of their irreps can be identified with full rotation group irreps
according to Table 8.1. Assume also that the electron in the conduction band has
spin up in the z axis and it can go to either the heavy hole state |3/2, +3/2i or the
light hole |3/2, −1/2i, and the photon is being emitted along the z direction. The

(8.10)

The reduced matrix elements in Eq. (8.10) cancel out and, using the orthonormality of the basis kets, one is left with the simple expression:
Ws↑→|3/2,+3/2i
|h1, +1; 1/2, +1/2|3/2, +3/2i|2
= 3,
2 =
Ws↑→|3/2,−1/2i
1/3
|h1, −1; 1/2, +1/2|3/2, −1/2i|

(8.11)

where the Clebsch-Gordan coefficients are for the full rotation group, and can be
found in any standard quantum mechanics book.

8.2.3

Complications following the path of Fermi’s golden rule

In the previous two subsections it has been shown how to find ratios between the
probabilities that a photon be emitted from a transition into a given band using
Fermi’s golden rule. This might seem good enough to achieve the goal of generating single events using a random number generator. However, in the derivation of
Eq. (8.11), the formalism has left discretionary choices that make it unsuitable for
event generation. Some of the questions that arise are
• What set of final states must be chosen?
In Sec. 8.2.2, it was assumed that the initial state transitioned to either
|3/2, +3/2i or |3/2, −1/2i, where the quantization axis for the final sates was
chosen to point along the z axis. However, had another quantization axis been
chosen, the matrix element Pf i would have taken another value and the re-

154
sults for the probability of emission and photon polarization would have been
different.
• What polarization of the emitted photon must be chosen?
Also in Sec. 8.2.2, it was implicitly assumed that the polarization vector a was
parallel to Pf i . The polarization of the emitted photon is given by the vector a
[cf. Eq. (8.1)]. In principle, one might think that the polarization of the emitted
photon should be independent of the choice of polarization basis vectors. But
in this formalism, falling back to the example in Sec. 8.2.2 and considering
emission along the z axis; the following choice of polarization modes: a1 = x̂,
a2 = ŷ with x̂ (ŷ) a unit vector along the x (y) axis, would have generated
photons with linear polarizations only. This is clearly unacceptable because it
is known that when the electron spin points along the z axis and light is emitted
along that axis too, a circular polarization of 50 % is expected [16].
In principle, one could overcome these difficulties by adopting the prescription
that any basis set spanning the final state subspace can be chosen, and then pick a
polarization that is in the same direction as P⊥ f i , where P⊥ is the component of P
perpendicular to the direction of emission. Alas, this is an ad hoc prescription, which
does not shed any insight on the nature of the radiative process. But this prescription
yields, indeed, the correct results. In the next section a more natural and univocal
way of obtaining a prescription is shown.

8.3

Polarization of the emitted photon

Spontaneous emission is a purely quantum-field-theoretical effect. Consider, for the
sake of argument, an atom in an excited state. Quantum mechanics states that if the
electron is in an eigenstate of the system, it will remain there forever. The difference
between classical and quantum fields manifests itself in the definition of system. In
the classical viewpoint the electromagnetic bath surrounding that atom can totally
vanish, therefore the system is the electron and its interaction with the nucleus, and

155
the electron will remain in the excited state. In the quantum field theory (QFT)
viewpoint, the electromagnetic bath has a finite amplitude even at its ground state,
so it must be included in the system. These zero point fluctuations will interact with
the electron and cause the emission process.
In a sense, spontaneous emission is nothing more than stimulated emission caused
by the ground state of the electromagnetic environment.
The Hamiltonian in Eq. (8.4) can be rewritten [10] in terms of the photon creation
and annihilation operators a†s and as , where the s is an index including the wavevector
k and the polarization mode:
e X
H = H0 + HDE (t) = H0 −
m s

2π £
as (P · as ) e−iωt + a†s (P · a∗s ) eiωt , (8.12)
kV

where HDE (t) is the perturbation to the crystal Hamiltonian.
The two possible polarization modes are conveniently chosen to be the polar and
azimuthal unit vectors. This way, any polarization of the photon will be described in
terms of these two direction dependent linear polarization modes:
aθ̂ϕ = êθ = cos θ cos ϕ x̂ + cos θ sin ϕ x̂ − sin θ ẑ

(8.13)

aθϕ̂ = êϕ = − sin ϕ x̂ + cos ϕ ŷ.

(8.14)

All this said, now the Hamiltonian (8.12) can be applied to an initial state
|ii ⊗ | . . . 0θ̂ϕ 0θϕ̂ . . .i to generate the time evolution of that state. The symbol |ii
represents the initial state of solely the electron, and the symbol | . . . 0θ̂ϕ 0θϕ̂ . . . i says
that there are no photons propagating in the θ, ϕ direction with either ê θ or êϕ polarization. The formalism of first-order time-dependent perturbation theory will be
applied to find the quantum state after a time t has elapsed, and the question of the
probability that a photon has been emitted can be answered.
Note that this method shows the physics underlying the transition process in a
much clearer way. Once the initial state and the Hamiltonian of the system are given,
the state at any time, including the photon polarization, is fully determined. The only

156
arbitrary choice in this approach is the basis with which the states are represented;
but, of course, that does not alter the physics of the problem.
Following Ref. [14], the final state can be written as follows:

|f (t)ielectron ⊗ |f (t)iphoton =
X Z dΩ
e−iEκ t/~ |κ, ki ⊗ bkκ,i,θ̂ϕ (t)| . . . 1θ̂ϕ 0θϕ̂ . . .i + bkκ,i,θϕ̂ (t)| . . . 0θ̂ϕ 1θϕ̂ . . .i +
4π
k∈κ
X Z dΩ
e−iEλ t/~ |λ, li ⊗ 1 − blλ,i,θ̂ϕ (t) − blλ,i,θϕ̂ (t) | . . . 0θ̂ϕ 0θϕ̂ . . .i, (8.15)
4π
l∈λ
where λ represents the initial band, κ the final band, Eλ and Eκ are their respective
energies, and

bkκ,i,θ̂ϕ (t) =
hκ, k| ⊗ h1θ̂ϕ 0θϕ̂ |HDE (0)|ii ⊗ |0θ̂ϕ 0θϕ̂ i 1 − ei(ωκi +ω)t 1 − ei(ωκi −ω)t
2i~
ωκi + ω
ωκi − ω
¯P ¡
¢¯
† ∗ ¯
hκ,
|P|
ii
s s
s s
θ̂ϕ θϕ̂
θ̂ϕ θϕ̂
2π
m kV
2i~
1 − ei(ωκi +ω)t 1 − ei(ωκi −ω)t
ωκi + ω
ωκi − ω

bkκ,i,θϕ̂ (t) =
hκ, k| ⊗ h0θ̂ϕ 1θϕ̂ |HDE (0)|ii ⊗ |0θ̂ϕ 0θϕ̂ i 1 − ei(ωκi +ω)t 1 − ei(ωκi −ω)t
2i~
ωκi + ω
ωκi − ω
¯P ¡
¢¯
† ∗ ¯
0θ̂ϕ 0θϕ̂
s as a s + a s a s
2π hκ, k |P| ii · 0θ̂ϕ 1θϕ̂
m kV
2i~
1 − ei(ωκi +ω)t 1 − ei(ωκi −ω)t
(8.16)
ωκi + ω
ωκi − ω
are the time-dependent coefficients of the expansion in terms of the chosen basis
set. These coefficients can be simplified by the proper action of the creation and

157
annihilation operators:
2π hκ, k |P| ii · aθ̂ϕ
g (t; ω)
bkκ,i,θ̂ϕ (t) = −
m kV
2i~
2π hκ, k |P| ii · a∗θϕ̂
g (t; ω) ,
bkκ,i,θϕ̂ (t) = −
m kV
2i~

(8.17)

where
1 − ei(ωκi +ω)t 1 − ei(ωκi −ω)t
g (t; ω) ≡
ωκi + ω
ωκi − ω

(8.18)

has been defined.
Once the final state is known, expectation values for the polarization of a photon
emitted along a given direction can be found. For this purpose, the following photon
polarization operators are first defined following the Stokes parameters’ notation for
plane waves (see, for example, Ref. [17] for an introduction to the Stokes parameters)
Q|1θ̂ϕ i = +1|1θ̂ϕ i

Q|1θ̂ϕ i = +1|1θ̂ϕ i

Q|1θϕ̂ i = −1|1θϕ̂ i
1 ³
1 ³
U √ |1θ̂ϕ i + |1θϕ̂ i = +1 √ |1θ̂ϕ i + |1θϕ̂ i
1 ³
U √ |1θ̂ϕ i − |1θϕ̂ i = −1 √ |1θ̂ϕ i − |1θϕ̂ i
1 ³
1 ³
V √ |1θ̂ϕ i + i|1θϕ̂ i = +1 √ |1θ̂ϕ i + i|1θϕ̂ i
1 ³
1 ³
V √ |1θ̂ϕ i − i|1θϕ̂ i = −1 √ |1θ̂ϕ i − i|1θϕ̂ i

Q|1θϕ̂ i = −1|1θϕ̂ i

(8.19)

U |1θ̂ϕ i = +1|1θϕ̂ i
U |1θϕ̂ i = +1|1θ̂ϕ i

(8.20)

V |1θ̂ϕ i = +i|1θϕ̂ i
V |1θϕ̂ i = −i|1θ̂ϕ i,

(8.21)

and the expectation values for the measurement of polarization for a photon emitted
in the θ, ϕ direction can be evaluated, for example, for V:

hf (t) |V | f (t)i = hf (t)|f (t)ielectron hf (t) |V | f (t)iphoton =

XX
hκ, k 0 |κ, ki×

k0 ∈κ k∈κ

b∗k0 κ,i,θ̂ϕ (t)h1θ̂ϕ 0θϕ̂ | + b∗k0 κ,i,θϕ̂ (t)h0θ̂ϕ 1θϕ̂ | V bkκ,i,θ̂ϕ (t)|1θ̂ϕ 0θϕ̂ i + bkκ,i,θϕ̂ (t)|0θ̂ϕ 1θϕ̂ i =
ibkκ,i,θ̂ϕ (t)b∗kκ,i,θϕ̂ (t) − ib∗kκ,i,θ̂ϕ (t)bkκ,i,θϕ̂ (t). (8.22)
k∈κ

158
Equations (8.17) can be plugged into the previous expression to obtain:
e2 2π |g (t; ω)|2
hf (t) |V | f (t)i = 2
m kV
4~2
hκ, k |P| ii · aθ̂ϕ hκ, k |P| ii · aθϕ̂ − hκ, k |P| ii · aθ̂ϕ hκ, k |P| ii · aθϕ̂ . (8.23)
k∈κ

At this point, a comment about this result is required. The question that the
above equation is answering is: “Given an initial state |ii, what is the probability that
a photon is detected along a certain direction and has a given circular polarization
after a time t?”, rather than the more adequate question: “Given an initial state
|ii and that a photon has been detected along a certain direction after a time t, what
is the probability that it has a given circular polarization?” In order to answer this
last question, Eq. (8.23) must be divided by the probability that a photon has been
emitted along the required direction. That yields the expectation value of the circular
polarization for a single photon or, equivalently, the measured circular polarization
for an ensemble of photons Vmeas :
Vmeas =

hκ,
|P|
ii
hκ,
|P|
ii
hκ,
|P|
ii
hκ,
|P|
ii
ϕ̂
θϕ̂
k∈κ
θ̂ϕ
θ̂ϕ
µ¯
∗ ¯
∗ ¯2
k∈κ ¯hκ, k |P| ii · aθ̂ϕ ¯ + hκ, k |P| ii · aθϕ̂

(8.24)

The two other measured Stokes parameters can be found by an analogous calculation, yielding
µ¯
¯2 ¯
∗ ¯
∗ ¯2
k∈κ ¯hκ, k |P| ii · aθ̂ϕ ¯ − hκ, k |P| ii · aθϕ̂
µ¯
(8.25)
Qmeas =
¯2 ¯
∗ ¯
∗ ¯2
k∈κ ¯hκ, k |P| ii · aθ̂ϕ ¯ + hκ, k |P| ii · aθϕ̂
P ³
hκ,
|P|
ii
hκ,
|P|
ii
hκ,
|P|
ii
hκ,
|P|
ii
θϕ̂
θϕ̂
k∈κ
θ̂ϕ
θ̂ϕ
µ¯
Umeas =
∗ ¯
∗ ¯2
k∈κ ¯hκ, k |P| ii · aθ̂ϕ ¯ + hκ, k |P| ii · aθϕ̂

(8.26)

where the common denominator in Eqs. (8.24)-(8.26) yields the angular distribution

159
of light intensity. It can also be thought of as the Stokes parameter S:
¯2 ¯
X µ¯¯
¯2
∗ ¯
Smeas =
¯hκ, k |P| ii · aθ̂ϕ ¯ + ¯hκ, k |P| ii · aθϕ̂ ¯ .

(8.27)

k∈κ

Finally, all reference to the momentum operator can be eliminated by means of
the Wigner-Eckart theorem as in Eq. (8.9). Symmetry considerations will govern
completely the polarization of the emitted photons as long as:
• Only two bands are involved in the light emission process.
• The direction towards which the light is being emitted is such that all the
relevant momentum components belong to the same irrep of the point group of
the crystal.
In that case,

Qmeas =
¯2 ¯
X X X µ¯¯
¯2
∗ ¯
¯hµ, m; λ, l|κ, kihλ, l|iiêm · aθ̂ϕ ¯ − ¯hµ, m; λ, l|κ, kihλ, l|iiêm · aθϕ̂ ¯
k∈κ l∈λ m∈µ

¯2 ¯
X X X µ¯¯
∗ ¯2
∗ ¯
¯hµ, m; λ, l|κ, kihλ, l|iiêm · aθ̂ϕ ¯ + hµ, m; λ, l|κ, kihλ, l|iiêm · aθϕ̂
k∈κ l∈λ m∈µ

(8.28)

Umeas =
XX X
2<
hµ, m; λ, l|κ, kihκ, k|µ, m ; λ, l ihλ, l|iihλ, l |ii êm · aθ̂ϕ êm0 · aθϕ̂
k∈κ l,l0 ∈λ m,m0 ∈µ

¯2 ¯
X X X µ¯¯
∗ ¯2
∗ ¯
¯hµ, m; λ, l|κ, kihλ, l|iiêm · aθ̂ϕ ¯ + hµ, m; λ, l|κ, kihλ, l|iiêm · aθϕ̂
k∈κ l∈λ m∈µ

(8.29)

160
Vmeas =
XX X
−2=
hµ, m; λ, l|κ, kihκ, k|µ, m0 ; λ, l0 ihλ, l|iihλ, l0 |ii êm · a∗θ̂ϕ ê∗m0 · aθϕ̂
k∈κ l,l0 ∈λ m,m0 ∈µ

¯2 ¯
X X X µ¯¯
∗ ¯
∗ ¯2
¯hµ, m; λ, l|κ, kihλ, l|iiêm · aθ̂ϕ ¯ + hµ, m; λ, l|κ, kihλ, l|iiêm · aθϕ̂
k∈κ l∈λ m∈µ

(8.30)

where < and = mean real and imaginary parts respectively, and êm is a unit vector
pointing in the same direction as the basis function of the irrep µ to which it is related.

8.4

Application to semiconductor structures

The abstract formalism developed in the previous section can be employed to predict
measured polarizations for several materials of interest. In particular, it can be used
to design experiments for optical spin injection detection. The optimal location of the
light detector with respect to the predominant spin can be found this way. Several
cases of interest are explicitly worked out in the following subsections. In these
examples, it is assumed that the electrons that are spread close to a symmetry point
can be well approximated as being in that symmetry point.

8.4.1

Bulk zincblende luminescence

Bulk zincblendes possess a Td point group symmetry. The character table and basis
functions for Td , adapted from Ref. [15], are shown in Table 8.2.
In a number of direct band gap zincblendes, the conduction band minimum is
situated at the Brillouin zone center, and it is described by the Γ6 irrep. The valence
band maximum is also at the zone center, and it is described by the Γ8 irrep. This
irrep splits away from the zone center into the heavy and light hole bands, as seen in
Fig. 8.2.
Consider, for example, an electron that has been injected into the conduction
band (CB) with its spin pointing up in the z direction. The initial electron state will

161
6S4

6S̄4

-1
-1
-1

3C2
3C̄2
-1
-1

-1
-1
√2
- 2

-1
-1
-√ 2

6σd
6σ̄d
-1
-1

Ē

8C3

8C̄3

Γ1
Γ2
Γ3
Γ4
Γ5
Γ6
Γ7

-2
-2

-1

Γ8

-4

-1

Basis Functions
R or xyz
Sx Sy S√
(2z − x − y ), 3 (x2 − y 2 )
S x , Sy , Sz
x, y, z
|s = 1/2, −1/2i, |s = 1/2, +1/2i
Γ6 × Γ2
|j = 3/2, −3/2i,|j = 3/2, −1/2i,
|j = 3/2, +1/2i,|j = 3/2, +3/2i

Table 8.2: Character and basis functions table for Td
be |ii = |s = 1/2, −1/2i. If the luminescence due to recombination with the valence
band (VB) Γ8 is under study, the expressions in Eqs. (8.28)—(8.30) can be used to
calculate the light polarization in an arbitrary direction.
We know that the initial state belongs to the Γ6 irrep, the final state to the Γ8
irrep and Table 8.2 shows that the momentum operator transforms according to the
Γ5 irrep. The Clebsch-Gordan coefficients for Td (aka. coupling coefficients) can be
looked up, for example, in Table 83 of Ref. [15]. In this table the notation is confusing,
in the sense that the basis functions for O and Td are reshuffled, and the table can be
employed as is for O only. The equivalence between the O and the Td basis functions
is shown in Table 8.3. Table 8.4 shows the coefficients in a ready-to-use way for T d .
|Γ8 , −3/2i
|Γ8 , −1/2i
|Γ8 , +1/2i
|Γ8 , +3/2i

Td
|Γ8 , +1/2i
|Γ8 , +3/2i
|Γ8 , −3/2i
|Γ8 , −1/2i

Table 8.3: Equivalence table of basis functions for O and Td .

162
|xi5 | −1
2 6

|xi5 | +1
2 6

|yi5 | −1
2 6

|yi5 | +1
2 6

|zi5 | −1
2 6

|zi5 | +1
2 6

7 h+1/2|

−i
−i

√1
−1

−i

8 h−1/2|

√i

√1

−i

√1

7 h−1/2|
8 h−3/2|

8 h+1/2|

8 h+3/2|

−i

√i

−1

√i

Table 8.4: The Clebsch-Gordan coefficients for Γ5 ⊗ Γ6 belonging to Td , with |jii a
shorthand for |Γi , ji.
Therefore, for the degree of circular polarization:

Vmeas =
−2=

k∈Γ8 m,m0 ∈Γ5

hΓ5 , m; Γ6 , +1/2|Γ8 , kihΓ8 , k|Γ5 , m0 ; Γ6 , +1/2i êm · a∗θ̂ϕ ê∗m0 · aθϕ̂

¯2 ¯
X X µ¯¯
∗ ¯
∗ ¯2
¯hΓ5 , m; Γ6 , +1/2|Γ8 , kiêm · aθ̂ϕ ¯ + hΓ5 , m; Γ6 , +1/2|Γ8 , kiêm · aθϕ̂

k∈Γ8 m∈Γ5

(8.31)

and plugging in the numbers, one obtains
Vmeas = −

cos θ

= − cos θ,
4/3

(8.32)

where θ is the angle between the injected electron spin and the direction of emission.
The denominator in Eq. (8.31) yields the angular distribution of the emitted radiation,
which is isotropic for this case.
Similarly, for the two modes of linear polarization:
Qmeas = 0

Umeas = 0.

(8.33)

163

Γ6
Eg

Γ8
Γ7
LH

Figure 8.2: Band structure for a zincblende near the zone center. The irreps at k = 0
are labeled, and their respective bands too. A transition between Γ6 and Γ7 has been
depicted, yielding a photon having the gap energy Eg .

8.4.2

Quantum well luminescence

In this section, only quantum wells (QWs) constructed with zincblende semiconductors will be considered.
Most of the times, the confining and the active layers will not have the same
lattice parameter. This will produce stress in the active layer which, in turn, will
cause the lifting of the degeneracy between heavy holes and light holes at the zone
center of zincblendes. Another source of splitting will be the reduction of symmetry
due to the quantum confinement of the electrons. Stress and confinement will reduce
the symmetry point group of the active layer from Td to D2d or C2v [18], depending
respectively on whether the [110] and [1̄10] directions are equivalent or not. The
interplay of confinement and stress effects will determine whether the heavy hole or
light hole bands will remain at the top of the valence band.

164
D2d

Ē

2S4

2S̄4

Γ1
Γ2
Γ3
Γ4
Γ5
Γ6
Γ7

-2
-2

-1
-1
√0
√2
- 2

-1
-1
-√ 2

C2
C̄2
-2

2C20
2C̄20
-1
-1

2σd
2σ̄d
-1
-1

Basis Functions
Sz
(x2 − y 2 )
xy or z
S x , Sy
|s = 1/2, −1/2i, |s = 1/2, +1/2i
Γ 6 × Γ3

Table 8.5: Character and basis functions table for D2d
QWs with D2d symmetry
QWs with symmetric walls, common atom and an even number of monolayers in the
active layer (eg. AlSb/GaSb/AlSb) have D2d symmetry. The character table of D2d
is shown in Table 8.5.
By checking the Clebsch-Gordan coefficients in Ref. [15] or the compatibility relations for the basis states of C4v [19], it can be deduced that the conduction band edge
states and the top1 heavy hole-like (HH1) state will transform according to Γ6 , while
the top light hole-like (LH1) and split-off-like (SO) states will transform according to
Γ7 .
The polarization of the emitted light will depend on whether the valence band edge
is described by states with predominantly HH or LH character (see Fig. 8.3). The
other valence states are assumed to lie deep enough in energy that no transitions are
made. Table 8.6 shows the appropriate Clebsch-Gordan coefficients for the transitions
depicted in Fig. 8.3. When studying case a), it can be seen from Eq. (8.30) that, for
vertical emission and for a single spin injected,
Vmeas =

−2= {hΓ5 , x; Γ6 , +1/2|Γ6 , −1/2ihΓ6 , −1/2|Γ5 , y; Γ6 , +1/2i}
= −1,
|hΓ5 , x; Γ6 , +1/2|Γ6 , −1/2i|2 + |hΓ5 , y; Γ6 , +1/2|Γ6 , −1/2i|2

(8.34)

while no net linear polarization would be measured.

HHodd and LHeven (HHeven and LHodd) states transform according to Γ 6 (Γ7 ) [18]. Even
and odd also label the parity of the total wavefunction. The even-odd mixing is due to the fact
that zincblendes don’t possess inversion symmetry, explaining the observation of parity forbidden
transitions [20].

165
|xi5 | − 12 i6

|xi5 | + 12 i6

|yi5 | − 12 i6

|yi5 | + 12 i6

6 h+1/2|

√i

√1

7 h+1/2|

√i

6 h−1/2|
7 h−1/2|

√i
√i

−1
√1

−1

Table 8.6: The Clebsch-Gordan coefficients for Γ5 ⊗ Γ6 belonging to D2d , with |jii a
shorthand for |Γi , ji.

Γ6

Γ6
Eg

Γ6
Γ7

HH1

Γ7

Γ6

LH1

LH1
HH1

Figure 8.3: Band structure near the zone center for a QW with D2d symmetry. (a)
shows the case where the heavy hole band has higher energy, while (b) shows the case
where the light hole band has higher energy.
When the LH band is the one having higher energy [see Fig. 8.3, (b) ], a similar
analysis yields
Qmeas = 0

Umeas = 0

Vmeas = +1.

(8.35)

The fact that the z basis function—therefore Pz —belongs to a different irrep than
x and y will make an exact statement about the polarization for a direction other
than along z impossible. However, some good approximations can be made in the
case where a few assumptions are valid.
When the symmetry of a system is reduced, levels that were previously degenerate
split in energy. Another consequence of the symmetry reduction is the mixing of
states. But the mixing of states can only take place between original irreps that will

166
transform into the same irrep of the reduced symmetry group. To exemplify this,
consider the CB, HH, LH and SO bands of bulk GaSb as a starting point. Inside a
symmetric quantum well, the electron in a GaSb active layer will see an environment
with D2d symmetry. Unless the well is very narrow, the well states will keep most of
the character of the bulk states they come from [21].
So, when the HH1 band is on the VB edge, the approximation can be made that
the action of Pz is negligible because these states will have only a small pz component.
In that case, and assuming that the electron in the CB is the initial state |Γ6 , +1/2i,
the different polarizations will be given by plugging the coefficients in Table 8.6 into
Eqs. (8.28)—(8.30):

Qmeas =

1 − cos2 θ
1 + cos2 θ

Smeas ∝

1 + cos2 θ

Umeas = 0

Vmeas = −

2 cos θ
1 + cos2 θ

(8.36)

These results are plotted for both input spins in Fig. 8.4. Note that, although
some linear spin polarization can be measured, reaching a maximum for emission in
the plane of the QW, it will not be spin-dependent. So, only optical measurements
of circular polarization can be used for electron spin detection. Clearly, the direction
perpendicular to the well is favored because of its higher amount of signal for a totally
spin polarized electron population.
In the case of Fig. 8.3 (b), when the approximation that the top of the VB has
LH character holds, a similar analysis can be performed. The polarizations that one
obtains for all initial spins in the |Γ6 , +1/2i state are
Smeas ∝
Qmeas = 1 −

5 − 3 cos2 θ

Umeas = 0

Vmeas =

2 cos θ
5 − 3 cos2 θ

(8.37)

These results are plotted for both input spins in Fig. 8.5. Again, the only means
to detect spin injection is through measurements of circular polarization. The amount
of polarization expected for emission perpendicular to the QW plane is the same as

167
CB-HH Intensity
1.2

330

1.0
0.8
0.6

300

0.4
0.2
0.0 270

0.2
0.4
0.6

120

240

0.8
1.0
210

1.2

150
180

Spin Up
Spin Down

1.0

Circular polarization

Linear polarization

0.8

0.6

0.4
Spin Up
Spin Down

0.2

0.0
-1.0

-0.5

0.0

cos θ

0.5

1.0

0.5

0.0

-0.5

-1.0
-1.0

-0.5

0.0

0.5

1.0

cos θ

Figure 8.4: Approximate intensity pattern, linear and circular polarization for radiation coming from band-to-band recombination for a QW with D2d symmetry. These
results are valid only when the top of the valence band is HH-like.
in CB-HH1 transitions. However, if one is forced to perform measurements off-axis,
it is clearly more convenient to resort to a CB-HH1 structure because of the greater
maximum attainable signal.
QWs with C2v symmetry
All of the QWs having asymmetric confinement layers (eg. an AlSb/InAs/GaSb/AlSb
QW), most of the QWs having symmetric walls but noncommon atom (eg.
AlSb/InAs/AlSb) and QWs with common atoms and an odd number of monolayers [18] (eg. AlSb/GaSb/AlSb) possess C2v point group symmetry. In this class of

168
CB-LH Intensity
1.2

330

1.0
0.8
0.6

300

0.4
0.2
0.0 270

0.2
0.4
0.6

120

240

0.8
1.0
210

1.2

150
180

Spin Up
Spin Down

Circular polarization

0.8

Linear polarization

Spin Up
Spin Down

1.0

0.6

0.4

0.2

0.0
-1.0

-0.5

0.0

cos θ

0.5

1.0

0.5

0.0

-0.5

-1.0
-1.0

-0.5

0.0

0.5

1.0

cos θ

Figure 8.5: As in Fig. 8.4, but the top of the valence band is LH-like.
QWs, as opposed to the D2d QW’s, a rotation of 180 ◦ about the x or y axes will not
result in the same structure. Its character table, adapted from Ref. [15], is shown in
Table 8.7.
The x and y axis are not equivalent for this family of quantum wells. Of course,
the z axis, being the direction of growth, will not be equivalent to any of those,
either. Therefore, in a strict sense, symmetry arguments cannot help in determining
the average polarization of the emitted photons, and Eqs. (8.24)—(8.26) should be
employed.
However, in some cases the reduction of the symmetry from D2d to C2v is due
to small effects. For example, in an AlSb/InAs/AlSb QW, the symmetry is reduced

169
C2v

Ē

Γ1
Γ2
Γ3
Γ4
Γ5

-2

σv
σ̄v
-1
-1

C2
C̄2
-1
-1

σd0
σ̄d0
-1
-1

Basis Functions
Sy or x
Sz or xy
Sx or y
|s = 1/2, −1/2i, |s = 1/2, +1/2i

Table 8.7: Character and basis functions table for C2v

Γ5 ~Γ6

Γ5 ~Γ7

Γ5 ~Γ6
Γ5 ~ Γ7

HH1

Γ5 ~ Γ6

LH1

LH1
HH1

Figure 8.6: Band structure near the zone center for a QW with C2v symmetry. Γ5
is the exact irrep for any band, and their approximate counterpart irreps in D 2d are
also shown.
because of the different nature of the interface bonds; Sb-In at one interface and
As-Al at the other. This is only an interface contribution, which should cause a
very small mixing between the D2d Γ6 and Γ7 states. Following the k · p spirit, these
interface effects can be neglected, and it can be assumed that all the C2v states behave
according to their original D2d irreps (see Fig. 8.6). When these assumptions hold,
the analysis performed for D2d QWs will also be valid for asymmetric QWs.

8.4.3

Bulk wurtzite luminescence

Bulk wurtzites, such as the stable phase at room temperature of GaN, AlN, InN [22,
23], possess a C6v point group symmetry. The character table for this point group
is given in Table 8.8. In that table the notation from Koster et al. [15] has been

170
C6v

Ē

Γ1
Γ2
Γ3
Γ4
Γ5
Γ6
Γ7
Γ8
Γ9

-2
-2
-2

C2
C̄2
-1
-1
-2

2C3

2C̄3

2C6

2C̄6

-1
-1
-2

-1
-1
-1
-1

-1
-1
-1
-1
-1
-1
√3 -√ 3
- 3

3σd
3σ̄d
-1
-1

3σv
3σ̄v
-1
-1

Basis Functions
R or z
Sz
x3 − 3xy 2
y 3 − 3yx2
(Sx − iSy ), −(Sx + iSy )
Γ 3 × Γ5
|1/2, −1/2i, |1/2, +1/2i
Γ 7 × Γ3
|3/2, −3/2i, |3/2, +3/2i

Table 8.8: Character and basis functions table for C6v .
used. The reader must be careful because often in literature Herring’s scheme [24]
is used. In this scheme, the Γ5 and Γ6 irreps are interchanged respect to Koster’s
sequence [25].
Figure 8.7 (a) shows the band structure of GaN near the zone center. As opposed
to zincblendes, the top of the valence band in wurtzites is split into two spin degenerate bands. The uppermost—heavy hole (HH)—transforms according to the Γ9 irrep
of C6v , while the light hole (LH) and the crystal split band (CR) transform according
to Γ7 . The crystal split band receives this name because, even when the spin orbit
interaction is not considered, that band is split from the others due to the crystal field
being different along z than along x and y. Panel (b) shows the splittings between
the different subbands. The numeric values are for GaN, but the behavior shown is
also applicable to InN. That panel also shows the allowed dipole transitions and their
polarization for vertical emission depending on the incoming spin (quantized along
the z axis).
At low temperatures, the thermal occupancy factor will make transitions to bands
other than the HH highly improbable. In that case (or, for that matter, whenever
transitions to the HH can be resolved spectroscopically), the polarization of the emitted light in an arbitrary direction can be determined. The Clebsch-Gordan coefficients
for transitions from the CB into the HH band are basically the same than in the case
of D2d QWs, except for an overall factor. This is understandable, because in D2d

171

a)
50

b)
G9v

G7c

Energy [meV]

G7v

s- s+ s+

G7v

s+ s- sEg=3.5 eV

-50

G9v(HH)
G7v(LH)
-100

-150

G7v

0.10
kx

0.05
0.05
wavevector [Å-1]

E1=11 meV
E2=26 meV

(CR)

0.10
kz

Figure 8.7: Band structure near the zone center for GaN. (a) has been adapted from
Ref. [26]. The band structure for InN is the same except for the numerical values of
the splittings and the band gap. (b) shows the allowed dipole transitions and their
polarization for vertical emission as a function of the electron spin (taken along the
z direction).
heterostructures the x and y are also equivalent, while the z axis is singled out.
Therefore, the results shown in Fig. 8.4 will also be valid for transitions from the CB
into the Γ9 band in wurtzites.

8.5

Monte Carlo photon generation

At this point a single event generation scheme can be devised in order to reproduce
the above results and, once the photons are generated, study phenomena that might
alter their behavior depending on their polarization, such as interface refraction,
magneto-optical Kerr effect, passage through polarizers...
After that, it will be shown that the time-dependent perturbation picture and
the Monte Carlo scheme yield the same results when a large number of photons is

172
considered.

8.5.1

Single event generation scheme

Figure 8.8 displays the general steps to generate a single photon using random numbers in such a way that their electric fields have the adequate polarization to reproduce the results predicted by the time-dependent perturbation theory derivation.
This paragraph will discuss the steps inside the dashed rectangle in Fig. 8.8, namely
how to assign an electric field to a generated photon.
The recipe and its application to photon generation for a bulk zincblende are
described in the following steps:
I. A basis set of states {|κ, ki} spanning the arrival band must be chosen.
II. A direction of propagation for the photon is generated according to the intensity distribution given by Eq. (8.27). In the cases with axial symmetry, the
probability of emission towards a solid angle dΩ will depend only on the polar
angle θ:
p (θ, ϕ) dΩ =

f (cos θ)
d (cos θ) dϕ,
2π

(8.38)

with f (cos θ) properly normalized to 1 and, if x is the generated random number
between 0 and 1, the appropriate cos θ can be found by solving the following
equation:
x=

Z cos θ

f (cos θ0 ) d (cos θ 0 ) .

(8.39)

−1

III. A quantization axis for the electron spin must be chosen. The direction of the
electron spin is determined according to whether it is required to be random,
all or predominantly pointing in one direction...
IV. The momentum matrix element between the initial state and all the possible
arrival basis states hκ, k |P| ii is found.
V. The transition from |ii to the final state is partitioned to transitions to basis

173

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Figure 8.8: Flowchart for the Monte Carlo photon generation process.

(8.40)

VI. Generate a random number between 0 and 1, and choose the state to which
the transition has taken place according to the probabilities calculated in the
previous step.
VII. The electric field of the photon is set to
E = hκ, k |P| ii · a∗θ̂ϕ aθ̂ϕ + hκ, k |P| ii · a∗θϕ̂ aθϕ̂ ,

(8.41)

where the units here are of no importance because in the end the electric field
will be renormalized. The components of the electric field can also be thought
of as the coefficients of the photon state.
VIII. Propagate the photon through the desired structure.
IX. Normalize the electric field of the photon to one. The amplitude of the electric
field given before carries information about the probability that the photon is
emitted along that direction. Since, by construction, the photon has made it to
the present point, the probability that it is there is one. With the normalization,
all photons will have the same weight when averaging electric fields.
X. Compute the Stokes parameter for that photon.
XI. Perform the average of the Stokes parameters for all photons hitting the desired
location.

175

8.5.2

Equivalence of the Monte Carlo and the timedependent perturbation pictures

The fact that the prescription given in Sec. 8.5.1 yields the same results as
Eqs. (8.24)—(8.26) is not obvious at first sight, and requires some thought. Here the
equivalence will be shown for the light emitted in the semiconductor structure, when
its polarization has not been affected by any extraneous element.
In Sec. 8.5.1 different quantum states are being generated according to a classical
probability. Denstity matrices [27] are the ideal tool to describe this situation. In
particular, the density matrix describing a generated photon will be

ρ=

k∈κ

p (κ, k) |κ, kihκ, k|⊗

´³
hκ, k |P| ii · a∗θ̂ϕ |1θ̂ϕ i + a∗θϕ̂ |1θϕ̂ i aθ̂ϕ h1θ̂ϕ | + aθϕ̂ h1θϕ̂ | · hκ, k |P| ii∗
, (8.42)
¯2 ¯
∗ ¯
∗ ¯2
¯hκ, k |P| ii · aθ̂ϕ ¯ + hκ, k |P| ii · aθϕ̂

where p (κ, k) is given by Eq. 8.40 and the denominator comes from the normalization
of the electric field or, in other words, the normalization of the photon state. The
expectation value of an operator, say the linear polarization parameter Q, will be

given by the trace over the photon and electron states of that operator times the
density matrix:

Qmeas = Tr (ρQ) =
X X
X X
(hκ, k 0 | ⊗ h1s0 |) ρQ (|κ, k 0 i ⊗ |1s0 i) =
p (κ, k) ×
k0 ∈κ s0 ∈θ̂,ϕ̂

k∈κ s0 ∈θ̂,ϕ̂

´³
hκ, k |P| ii · h1s0 | |1θ̂ϕ iaθ̂ϕ + |1θϕ̂ iaθϕ̂ aθ̂ϕ h1θ̂ϕ | + aθϕ̂ h1θϕ̂ | Q|1s0 i · hκ, k |P| ii∗
¯2 ¯
∗ ¯
∗ ¯2
hκ,
|P|
ii
hκ,
|P|
ii
θϕ̂
θ̂ϕ

(8.43)

and we can use the action of operator Q as defined in Eq. (8.19) to arrive at the

176
previously given expression:
µ¯
¯2 ¯
¯2
∗ ¯
k∈κ ¯hκ, k |P| ii · aθ̂ϕ ¯ − hκ, k |P| ii · aθϕ̂
µ¯
Qmeas =
¯2 ¯
¯2 .
∗ ¯
k∈κ ¯hκ, k |P| ii · aθ̂ϕ ¯ + hκ, k |P| ii · aθϕ̂

(8.25)

The proof of equivalence for the other polarization operators U and V goes in a
very similar manner.

8.6

Application to a bulk zincblende

The general steps outlined in Sec. 8.5.1 can be made more explicit for the case of bulk
zincblendes due to the high symmetry they show. The following list is the equivalent
of the one previously shown, but adapted to bulk zincblendes:
I. The obvious choice for the basis for the arrival space are the {|3/2, mj i} states
quantized along the z axis, due to the fact that tables of Clebsch-Gordan coefficients are readily available for them.
II. The emission process—without taking polarization into account—is isotropic.
Therefore, two random numbers must be generated to obtain the polar angles
θ and ϕ of the unit vector along the direction of propagation of the photon. A
random number between -1 and 1 will yield cos θ; and a number between 0 and
2π will yield ϕ.
III. The quantization axis for the electron spin is chosen to be the z axis. Again, this
is because it is the convention used in the existing tables. Once the direction
where the spin is pointing has been set, the initial state is set to [27]
µ ¶
µ ¶
θ iϕ
|Γ6 , +1/2i + sin
e |Γ6 , −1/2i ≡
|ii = cos
b+ |Γ6 , +1/2i + b− |Γ6 , −1/2i. (8.44)

177
State
|3/2, −3/2i
|3/2, −1/2i
|3/2, +1/2i
|3/2, +3/2i

p (κ, k)
|b− |

(3 + cos(2θ))

|b+ |2
(1 + cos2 θ) + 2|b3− | sin2 θ
|b− |2
(1 + cos2 θ) + 2|b3+ | sin2 θ
|b+ |2
(3 + cos(2θ))

Table 8.9: Transition probability for a state in the conduction band and arbitrary
spin into each of the states forming a basis in the valence band edge space.
IV. By virtue of the Wigner-Eckart theorem, the only component of the momentum
matrix element between the initial state and all the possible arrival basis states
h3/2, k |P| ii will be given by the Clebsch-Gordan coefficients in Table 8.4.
V. The probability of transition to each of the states will be given by the application
of Eq. (8.40), and the results are listed in Table 8.9.
VI. Generate a random number between 0 and 1, and choose the state to which
the transition has taken place according to the probabilities calculated in the
previous step.
VII. The electric field of the photon is set to

j=x,y,z

(b+ hΓ5 , j; Γ6 , +1/2|κ, ki + b− hΓ5 , j; Γ6 , −1/2|κ, ki) ̂ · a∗θ̂ϕ aθ̂ϕ +

(b+ hΓ5 , j; Γ6 , +1/2|κ, ki + b− hΓ5 , j; Γ6 , −1/2|κ, ki) ̂ · a∗θϕ̂ aθϕ̂ , (8.45)

where ̂ can be x̂, ŷ or ẑ.
VIII. Propagate the photon through the desired structure.
IX. Normalize the electric field of the photon to one.
X. Compute the Stokes parameter for that photon.
XI. Perform the average of the Stokes parameters for all photons hitting the desired
location.

Linear Polarization (Q)

178

0.4
0.2
0.0
-0.2
-0.4

Circular Polarization (V) Linear Polarization (U)

-1.0 -0.8 -0.6 -0.4 -0.2

0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

0.4
0.2
0.0
-0.2
-0.4
-1.0 -0.8 -0.6 -0.4 -0.2
0.4
0.2
0.0
-0.2
-0.4
-1.0 -0.8 -0.6 -0.4 -0.2

cos θ
Figure 8.9: Monte Carlo calculation of photon polarization for a zincblende for spin
down electrons. The data points and their standard deviation are shown. The lines
correspond to the theoretical values.

179
Figure 8.9 shows the results for a Monte Carlo (MC) simulation of the light emitted
by a GaAs substrate [9]. It shows the different Stokes parameters for the generated
photons as a function of the cosine of the polar angle. The representation in terms of
the cosine of the polar angle is preferable because an interval in solid angle maps into
the same interval in cos θ no matter what the θ is. The points are the values obtained
following the steps just mentioned; while the error bars correspond to one standard
deviation. It is clearly seen that the photons generated using the MC method follow
the theoretical predictions of Eqs. (8.32)-(8.33). Of course, the MC values show the
fluctuations inherent to the statistical nature of that method.

8.7

Summary

In summary, a method for determining the polarization of light emitted in arbitrary
directions for an arbitrary initial electron spin population has been developed. This
method allows the generation of single photons having a polarization such that, when
averaged over a large number of events, reproduce the results from first-order timedependent perturbation theory. The limitations of Fermi’s golden rule for single
photon generation have been shown.

180

Bibliography
[1] M. Johnson and R. H. Silsbee, J. Appl. Phys. 63, 3934 (1988).
[2] S. F. Alvarado and P. Renaud, Phys. Rev. Lett. 68, 1387 (1992).
[3] R. Fiederling, M. Keim, G. Reuscher, W. Ossau, G. Schmidt, A. Waag, and
L. W. Molenkamp, Nature 402, 787 (1999).
[4] B. T. Jonker, A. T. Hanbicki, Y. D. Park, G. Itskos, M. Furis, G. Kioseoglou, A.
Petrou, and X. Wei, Appl. Phys. Lett. 79, 3098 (2001).
[5] F. G. Monzon and M. L. Roukes, J. Magn. Magn. Mater. 199, 632 (1999).
[6] G. Schmidt, D. Ferrand, L. W. Molenkamp, A. T. Filip, and B. J. van Wees,
Phys. Rev. B 62, R4790 (2000).
[7] D. L. Smith and R. N. Silver, Phys. Rev. B 64, art. no. (2001).
[8] H. J. Zhu, M. Ramsteiner, H. Kostial, M. Wassermeier, H. P. Sch onherr, and
K. H. Ploog, Phys. Rev. Lett. 87, art. no. (2001).
[9] S. R. Ichiriu, X. Cartoixà, and T. C. McGill, , in preparation.
[10] A. Messiah, in Quantum Mechanics, 1st ed. (Dover, New York, 1999), pp. 1016–
1017.
[11] C. Cohen-Tannoudji, B. Diu, and F. Laloë, in Quantum Mechanics, 2nd ed.
(Wiley, New York, 1977), pp. 1306–1307.
[12] B. K. Ridley, in Quantum Mechanics, 1st ed. (Dover, New York, 1999), pp.
184–187.
[13] W.-K. Tung, in Group Theory in Physics, 1st ed. (World Scientific, Singapore,
1985), pp. 60–61.

181
[14] C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, 2nd ed. (Wiley,
New York, 1977).
[15] G. F. Koster, J. O. Dimmock, R. G. Wheeler, and H. Statz, Properties of the
Thirty-Two Point Groups, 1st ed. (M.I.T. Press, Cambridge, MA, USA, 1963).
[16] M. Wöhlecke and G. Borstel, Phys. Stat. Sol. B 107, 653 (1981).
[17] D. Kliger, J. Lewis, and C. Randall, Polarized Light in Optics and Spectroscopy,
1st ed. (Academic Press, Inc., New York, NY, USA, 1990).
[18] R. Magri and A. Zunger, Phys. Rev. B 62, 10364 (2000).
[19] Y. Onodera and M. Okazaki, J. Phys. Soc. Jpn. 21, 2400 (1966).
[20] R. C. Miller, A. C. Gossard, G. D. Sanders, Y. C. Chang, and J. N. Schulman,
Phys. Rev. B 32, 8452 (1985).
[21] D. Z.-Y. Ting, E. T. Yu, and T. C. McGill, Phys. Rev. B 45, 3576 (1992).
[22] O. Lagerstedt and B. Monemar, Phys. Rev. B 19, 3064 (1979).
[23] S. Iwama, K. Hayakawa, and T. Arizumi, Journal Of Crystal Growth 56, 265
(1982).
[24] C. Herring, J. Franklin Inst. 233, 525 (1942).
[25] M. J. Lax, Symmetry principles in solid state and molecular physics, 1st ed.
(Wiley-Interscience, New York, USA, 1974).
[26] Y. C. Yeo, T. C. Chong, and M. F. Li, J. Appl. Phys. 83, 1429 (1998).
[27] J. J. Sakurai, Modern Quantum Mechanics, 1st ed. (Addison-Wesley, Redwood
City, CA, USA, 1985).

182

Appendices

183

Appendix A

Explicit form of the 8-band

k · p Hamiltonian
The calculations in Chapters 5, 6 and 7 are based in the 8-band k · p Hamiltonian
derived by Trebin et al. [1]. This Hamiltonian has the nice property that it has been
constructed using the theory of invariants [2], and thus it correctly describes the T d
symmetry of bulk zincblendes. In particular, it includes terms breaking the spin
degeneracy of the bands at a general point in the Brillouin zone, making it ideal for
the study of inversion asymmetry effects. It also accounts for the effects of strain and
an external magnetic field.
Although the Hamiltonian is explicitly shown in Ref. [1], it will be rewritten here
to provide a higher degree of self containment to this thesis. Some typos present in
the original work by Trebin et al. are removed and, hopefully, no new ones will be
introduced.
The 8-band k · p Hamiltonian will be expressed in the basis |Γ6 , + 12 i , |Γ6 , − 12 i,
|Γ8 , + 32 i, |Γ8 , + 21 i, |Γ8 , − 21 i, |Γ8 , − 23 i, |Γ7 , + 21 i, |Γ7 , − 21 i . It can be written in a block
diagonal form

H cc H cv H cs

H = H vc H vv H vs  ,
H sc H sv H ss

(A.1)

¢†
where, of course, H αβ = H βα , c refers to the two conduction band (CB) states, v

to the four heavy hole (HH) and light hole (LH) states and s to the two spin-orbit

split off (SO) states.
The constituent blocks of the Hamiltonian are shown in Table A.1. The meaning
of the different parameters is listed in Table 5.4 in the main text. The phases of
the wavefunctions and the prefactors in Table A.1 are chosen in a way that all the
parameters are real. The σ and the ρ matrices are the Pauli matrices; the T matrices

184
e~
H cc =Ev + Eg + ~2mk + A0 k 2 − gs 4mc
σ · H + C1 tr²
2 2

vv
vv
+ H²k
H vv =Hkvv + H²vv + Hkl
£¡
Hkvv =− ~m 12 γ1 k 2 − γ2 Jx2 − 13 J 2 kx2 + cp − 2γ3 [{Jx Jy } {kx ky } + cp] −
e~
{(κJx + qJx3 ) Hx + cp}
mc
£¡
H²vv =Dd tr² + 32 Du Jx2 − 13 J 2 ²xx + cp + 23 Du0 [2 {Jx Jy } ²xy + cp]
£© ¡
¢ª
vv √2
Hkl
= 3 C Jx Jy2 − Jz2 kx + cp
vv
H²k
=[C4 (²yy − ²zz ) kx + C50 (²xy ky − ²xz kz )] Jx + cp

e~
γ1 k 2 − 2κ 2mc
σ · H + Dd tr²
H ss =−∆so + 2m
cv
H = 3 [P (kx Tx + cp) + iB (Tx {ky kz } + cp) + iC2 (Tx ²yz + cp)]

H cs =− √13 [P (kx ρx + cp) + iB (ρx {ky kz } + cp) + iC2 (ρx ²yz + cp)]

vs
H vs =Hkvs + H²vs + H²k

e~ 3
[−3γ2 (Uxx kx2 + cp) − 6γ3 (Uxy {kx ky } + cp)]− mc
(Ux Hx + cp)
Hkvs =− 2m

H²vs =2Du (Uxx ²xx + cp) + 2Du0 (2Uxy ²xy + cp)

vs 3
H²k
= 2 [C4 (²yy − ²zz ) kx + C50 (²xy ky − ²xz kz )] Ux + cp

cp means cyclic permutation, {AB} = 12 (AB + BA), tr² = ²xx + ²yy + ²zz
Table A.1: Matrix elements of the 8-band k · p Hamiltonian.
are given by
− 3 0 1 √0
0 −1 0 3
√ ´
0 −1 0 3
Txx = 3√1 2 −√
3 0 1 0
−1 √0 − 3 0
Tyz = 2 6 0 3 0 1
Tx = 3√1 2

Ty = 3−i

³√

3 0 1 √0
0 1 0 3

√ ´
√0 −1 0 − 3
3 0 1 0
3 0
−1 √0
Tzx = 2 6 0 3 0 −1

Tyy = 3√1 2

The matrices U are simply given by Ui = Ti† .

Tz = 32 ( 00 10 01 00 )

0 0
Tzz = 32 ( 00 01 −1
0)
¡ 0 0 0 −1 ¢
Txy = √i6 −1
0 0 0

(A.2)

185

Bibliography
[1] H.-R. Trebin, U. Rössler, and R. Ranvaud, Phys. Rev. B 20, 686 (1979).
[2] G. L. Bir and G. E. Pikus, Symmetry and Strain-Induced Effects in Semiconductors, 1st ed. (Wiley, New York, USA, 1974).

186

Appendix B

Group theory for band

structures
In this thesis an extensive use of group theoretical arguments is made. The following
lines are meant to be a utilitarian crash course on group theory for the reader not
familiar with the group theory (GT) concepts and language applied to semiconductor
band structure theory. This appendix is not by any means rigorous in the derivations,
proofs or definitions, but hopefully it is with the results. What will be presented here
is mainly a repetition, rephrasing or elaboration on the notes from the Ph129 course
by Kerry Vahala and the textbook used there [1]. Falicov [2] wrote another excellent
textbook with more emphasis on the applications of GT to solid-state physics.

B.1

Definitions

First, a group G consists of a set of elements and an operation that sends a pair
of elements back into another1 element of the set. The operation must satisfy the
following three properties for the set to be a group:
• The operation must be associative: a(bc) = (ab)c for all a, b, c ∈ G
• There must be an identity element e such that ae = a for all a ∈ G.
• For each a ∈ G there must be an inverse element a−1 such that aa−1 = e.
Then, the symmetry group of a crystal will be the set of operations that send a
crystal to itself, which can be shown to form a group. This group is called the space
group of the crystal, and there are 230 of them [3]. For each space group, there will be
a subset of operations that leave a certain point in the crystal invariant. This subset

It does not need to be different from the original two.

187
also forms a group and is called the point group of a crystal. There are 32 of these
crystallographic point groups [4] (i.e., point groups compatible with the existence of
a crystal). An example of a point group not being a crystallographic point group
would be the symmetry group of a molecule with pentagonal symmetry.
Now, focusing in the electronic band structure of a solid, one must consider that
the points in the Brillouin zone will have symmetry properties associated with the
point group of the parent crystal. In particular, it can be shown [2] that a state in
the zone center has the same symmetry properties as the underlying lattice, and that
the symmetry of a general state at k inside the Brillouin zone will be governed by the
subset of operations of the point group that leave k invariant. This set of operations
forms the “small group of k.”
At this point, a mathematical object satisfying the group multiplication table is
needed. If the group is commutative (aka. Abelian), plain numbers will do the job.
If the group is not, one has to resort to matrices. A set of numbers or matrices
that satisfies the group multiplication table, each one associated to a group member
and acting on a vector space V , constitutes a representation. If the elements of a
representation can (cannot) be put in a block diagonal form all at the same time,
the representation is called reducible (irreducible). The trace of a matrix belonging
to a representation is called the character. Obviously, the character of the matrix
representing the identity element will yield the dimensionality of the representation.
There exist published tables of characters for irreducible representations of the most
important groups.

B.2

Degeneracies, splittings and eigenstates

Finally, a point has been reached where one of the most important results of GT with
regard to crystals can be enunciated [1]:
Lemma 1 (Schur’s 1st) Let U (G) be an irreducible representation of a group G on
the vector space V , and H be an arbitrary operator on V . If H commutes with all

188
the operators {U (g), g ∈ G}, then H must be a multiple of the identity operator I, i.e.
H = λI, where λ is a number.
On the other hand, the crystal Hamiltonian at a point k will commute with all
the symmetry operations S belonging to the small group of k:
SH(k) = H(k)S.

(B.1)

Putting Eq. (B.1) together with Schur’s first lemma, one obtains that the subspace upon which an irreducible representation of the small group of k acts will be
degenerate. From this conclusion the power of GT to predict degeneracies at points
of high symmetry in the Brillouin zone is drawn.
Once this has been said, the recipe to find degeneracies at a given k point inside
the Brillouin zone2 can be stated:
I. Find the small group G(k) corresponding to k.
II. Look at the character table of G(k). Koster et al. [4] provide a good compilation
of tables for the crystallographic point groups.
III. The character of the identity element yields the degeneracy of the level transforming according to the selected irreducible representation. If basis functions
are listed, they indicate the symmetry of the states.
An example the procedure described above can be the degeneracy of the levels
at the zone center in GaN. GaN has wurtzite structure. The point group at the
zone center (Γ point) is C6v . From Table 65 in page 67 in Koster et al. [4] it is
seen that (spin resolved) states at the zone center can transform according to three
different irreducible representations, each one of them twofold degenerate. A single
electron can transform only according to the irreducible representations that change
sign under a 2π rotation3 . Looking at the tables of bases, and in the cases where the

The treatment for points on the Brillouin zone boundary is more complicated.
The irreducible representations above the line are useful for the study of systems with an even
number of electrons.

189
states come from s or p states (that is, an sp model), one can see that the Γ9 states
will be proportional to |x + iyi| ↑i and |x − iyi| ↓i. Less can be said about the Γ7 and
Γ8 . d type orbitals are needed to generate Γ8 states, so they will not appear in an
sp description, but that is not an inconvenient when describing the lower conduction
band and the upper valence bands of wurtzites.

B.3

Matrix elements

The consistency of the matrix elements of an operator O with the crystal symmetry
imposes certain restrictions on these matrix elements. The Wigner-Eckart theorem
exploits these restrictions to show that the matrix elements between states belonging to given irreducible representations factor into a very small number of operatordependent constants and the rest is determined only by symmetry.
Theorem 1 (Wigner-Eckart) If {Piα } is a set of components of the operator P
transforming according to the irreducible representation (irrep) α, then the matrix
element between states |vjβ i where β denotes the irrep and j labels the specific state
inside the β irrep is given by

E X
αβ,γ ∗
cβ,γ
vnγ |Piα | vjβ =
k Uij,n ,

(B.2)

where k runs over the number of times that the irrep Γγ is contained in Γα ⊗ Γβ , cβ,γ

is the irreducible matrix element, and does not depend on the indices i, j or n 4 , and
αβ,γ
Uij,n
is the Clebsch-Gordan coefficient for the appropriate point group connecting

states transforming according to the product group Γα ⊗ Γβ to states that transform

under the irrep Γγ .

The importance of this theorem is that it provides with selection rules and strength
ratios between matrix elements. As an example, one can evaluate the ratio of the

It does, of course, also depend on the operator P .

190
matrix elements

Γ8 , −3/2 |Px5 | Γ6 , − 21
−ic6,8
1 / 2
®=
6,8
Γ8 , +1/2 |Px5 | Γ6 , − 2
−ic1 / 6

(B.3)

just by the application of the Wigner-Eckart theorem and looking up the coefficients
in Table 8.4.

191

Bibliography
[1] W.-K. Tung, Group Theory in Physics, 1st ed. (World Scientific, Singapore, 1985).
[2] L. M. Falicov, Group Theory and Its Physical Applications, Chicago Lectures in
Physics, 1st ed. (The University of Chicago Press, Chicago, 1966).
[3] G. F. Koster, in Solid State Physics, edited by F. Seitz and D. Turnbull (Academic
Press, New York, 1957), pp. 173–256.
[4] G. F. Koster, J. O. Dimmock, R. G. Wheeler, and H. Statz, Properties of the
Thirty-Two Point Groups, 1st ed. (M.I.T. Press, Cambridge, MA, USA, 1963).

192

Appendix C

Derivation of the formula

for the transmission coefficients
The aim of this appendix is to prove the formula for the transmission coefficients due
to tunneling

T E, kk

C.1

¯ ¡
¢¯
¯ ¡
¢¯2 ¯vj E, kk ; R ¯
¯tj E, kk ¯ ¯ ¡
¢¯
¯vI E, kk ; L ¯ .
j=1

(7.13)

Expansion to the Hellmann-Feynman theorem

In order to prove the formula above, the following generalization to the HellmannFeynman theorem will be useful.
Theorem 2 Let A(λ) be a Hermitian operator, and let |u(λ)i and |v(λ)i be two
eigenvectors of A(λ) having the same eigenvalue a(λ). If hv(λ)|u(λ)i does not depend
on λ, then
¯ ∂A(λ) ¯
¯ u(λ) = ∂a(λ) hv(λ)|u(λ)i.
v(λ) ¯¯
∂λ ¯
∂λ

(C.1)

Proof: Following the original proof of the Hellmann-Feynman theorem (see, for
example, Ref. [1]), the left hand side of Eq. (C.1) can be written as

½·
¸¾
¯ ∂A(λ) ¯
¯ u(λ) + a(λ)
v(λ) ¯¯
hv(λ)| |u(λ)i + hv(λ)|
|u(λ)i ,
∂λ ¯
∂λ
∂λ

(C.2)

193
which proves Eq. (C.1) because of the assumption that hv(λ)|u(λ)i is independent of
λ.
The Hellmann-Feynman theorem is recovered taking A → H and |u(λ)i = |v(λ)i
(properly normalized) into Eq. (C.1):
¯ ∂H(λ) ¯
∂E(λ)
v(λ) =
v(λ) ¯
∂λ
∂λ

(C.4)

Another useful case is when hv(λ)|u(λ)i = 0. Then, a straightforward application
of Eq. (C.1) shows

¯ ∂A(λ) ¯
¯ u(λ) = 0.
v(λ) ¯¯
∂λ ¯

(C.5)

In particular, this is useful when evaluating matrix elements of operators that are
a derivative of the Hamiltonian respect to some parameter. The above equation is
saying that ∂H(λ)
is diagonal within a degenerate subspace of states.
∂λ

C.2

Transmission coefficient

The transmission coefficient for the tunneling process in a heterostructure is given by
the ratio of probability currents flowing across a plane perpendicular to the growth
direction (chosen to be z) of the transmitted state respect to the incident state 1 :
¯ E
R D ¯¯
(r)
¯ t dx dy
¯ E
T E, kk = R D ¯¯
dx
dy
(r)
¯ z ¯

(C.6)

where L and R refer to the left and right electrode, respectively, and the current
operator is given by [2]
Ĵ(r0 ) =

[Pδ(r − r0 ) + δ(r − r0 )P] .
2m

(C.7)

The ratio should only include the current due to transmitting, as opposed to evanescent,
components.

194

Figure C.1: Illustration of the independence of Jz dx dy with respect to the plane
of integration. σ and σ 0 are equivalent through the cyclic boundary conditions. The
thick lines represent current lines, and the hexagons are primitive zone boundaries.
The number of current lines cutting through the plane z0 is independent its z position.
Also, an appropriate control volume V for the mathematical proof and its boundary
S are shown.
It is easily seen that the usual value for the probability current is recovered by taking
the expectation value of Ĵ(r):
¯ E
D ¯
J(r) = ψ ¯Ĵ(r)¯ ψ =
[ψ ∗ (r) (∇ψ(r)) − (∇ψ(r))∗ ψ(r)] .
2mi

(C.8)

However, in quantum mechanics one is normally more comfortable taking volume
integrals (i.e., matrix elements) than integrals over a plane. To progress towards that
direction, first it must be shown that the integrals over the plane in Eq. (C.6) are
independent of the position z0 of the plane. To do this, only the bulk properties of the

195
crystal need be considered. Cyclic boundary conditions for a bulk crystal are chosen
to facilitate the argument. A control volume V is defined in such a way that encloses
all the crystal along the x and y directions, but has an arbitrary thickness along the
z direction. Figure C.1 shows V limited by the closed surface S.
= 0, therefore
Since the current is being sought for a stationary state, ∂ρ
∂t
∇ · J(r) = 0.

(C.9)

This equation can be integrated over V to obtain
0=

∇ · J(r)dx dy dz =

J · dS =

Jz dx dy −

Jz dx dy,

(C.10)

where in the last step the fact that the planes σ and σ 0 (and the set perpendicular to
y) are equivalent under the cyclic boundary conditions has been used. Since z1 and
z0 were arbitrary, the independence of z Jz dx dy on z has been proved.
Now, each of the integrals appearing in Eq. (C.6) can be trivially integrated respect

to z, with the limits chosen as described in Fig. C.2. Therefore,
R zR +dhkl,R ¡R
J (r)dx dy dz/dhkl,R
zR
R z,t
T E, kk = R zL +dhkl,L ¡R
J (r)dx dy dz/dhkl,L
L z,I
zL

(C.11)

where dhkl,L (dhkl,R ) denotes the distance between planes with Miller indices [khl] in
¯ E
D ¯
the left (right) electrode and Jz,t = t ¯Jˆz (r)¯ t . Note that the indices don’t need to
be the same in both electrodes.

The integrals in Eq. (C.11) can be transformed into integrals over the electrode

volume as follows
Z zR +dhkl,R µZ
zR

Jz,t (r)dx dy dz = nhkl S

Jz,t (r)dV,

(C.12)

where nhkl is the number of lattice points per unit surface in the [hkl] plane, S is the
transverse area of the crystal and the P C symbol means that the integral is to be
performed over a primitive cell. Following Feynman [3], the integral of the current

196

z0+dhkl

Figure C.2: Integration region for Jz . dhkl is the distance between [hkl] planes,
perpendicular to which the heterostructure is grown.
density over the primitive cell becomes

¿ ¯ ¯ À
¯ Pz ¯
Jz,t (r)dV = t ¯¯ ¯¯ t ,
PC

(C.13)

but Pz /m is simply vz , the z component of the velocity operator. Therefore

Jz,t (r)dV = ht |vz | ti = vz (E, kk ; t)ht|ti,

(C.14)

where vz (E, kk ; t) is the group velocity of the transmitted state. An analogous expression is obtained for the current of the incoming state.

197
Now, using Eqs. (C.14) and (C.12), the average current through a crystal plane
can be obtained

J (r)dx dy
ht|tivz (E, kk ; t)
VP C,R
dx dy

RR z

(C.15)

where VP C,R is the primitive cell volume of the right electrode and z is the direction
perpendicular to the plane under study. This is the same result shown in Shockley [4], but generalized to states that are not pure Bloch states. From Eq. (C.15) and
Eq. (C.6)
ht|tivz (E, kk ; t)/VP C,R
T E, kk =
hI|Iivz E, kk ; I /VP C,L

(C.16)

More progress can be made substituting the expression for |ti in terms of the
Bloch states
|ti = tj |Bkk ,kz,j , Ri,

(C.17)

with |Bkk ,kz,j , Ri given by Eq. (7.2’) and being orthogonal to each other, into
Eq. (C.13):
¿ ¯ ¯ À
¯ Pz ¯
t ¯¯ ¯¯ t = t∗l tj Bkk ,kz,l , R |Pz | Bkk ,kz,j , R .

(C.18)

The Einstein summation convention is used. The matrix element on the right hand
side will only be different from zero when kz,l = kz,j because of the space group
selection rules for a position independent operator2 . The matrix element between
Bloch states can be further developed considering the following two cases:
• kz,l = kz,j and l = j
The matrix element yields the velocity of the state |Bkk ,kz,j , Ri
1 D
Bkk ,kz,j , R |Pz | Bkk ,kz,j , R = vj E, kk ; R .

(C.19)

Since

¯ ∂H(kk , kz ) ¯
1 ∂E(kk , kz ) ¯¯
vj E, kk ; R =
ukk ,kz,j ¯
¯ ukk ,kz,j , (C.20)
∂kz
∂kz
kz,j

Implicitly, the assumption is made that the tunneling process does not involve states in the edge
of the Brillouin zone.

198
where the Hellmann-Feynman theorem has been used in the last step, a velocity
operator appropriate for application to the periodic part of the Bloch states can
be defined [5]:
v̂ =

∇k H(kk , kz ).

(C.21)

Remember that now k in H(kk , kz ) plays the role of three parameters in the
k · p Hamiltonian.
• kz,l = kz,j and l 6= j
In this case, it is easily seen that
1 D
1 D
Bkk ,kz,l , R |Pz | Bkk ,kz,j , R =
ukk ,kz,l |Pz | ukk ,kz,j ,

(C.22)

and, using the form (C.21) for the velocity operator,
E 1
1 D
Bkk ,kz,l , R |Pz | Bkk ,kz,j , R =

¯ ∂H(kk , kz ) ¯
¯ uk ,k
ukk ,kz,l ¯¯
¯ k z,j = 0, (C.23)
∂kz

where the result in Eq. (C.5) has been used in the last step.

Therefore, one is only left with the diagonal elements of the velocity operator.
Collecting Eqs. (C.23), (C.19), (C.18) and (C.15) into Eq. (C.16) one obtains

T E, kk =

¯ ¡
¢¯2
¯ vj (E, kk ; R)/VP C,R
j=1 tj E, kk
¢¯
P8 ¯
¯2 vl (E, kk ; L)/VP C,L
E,
l=1

(C.24)

The volume factors appear because |Ii and |ti are subject to normalization over the
whole structure, as can be seen if one considers a case where an electron propagates
in free space and partitions one region with cells of volume VP C,L and another with
cells of volume VP C,R . But if one solves for coefficients of the envelope function, as
usually done, then each Bloch state is normalized to one within a primitive cell and
the volume factors can be dropped off, resulting in
¢¯2
P8 ¯ ¡
¯ vj (E, kk ; R)
j=1 tj E, kk
T E, kk = P8 ¯ ¡
¢¯
¯Il E, kk ¯2 vl (E, kk ; L)
l=1

(C.25)

199
which reduces to Eq. (7.13) when the incident state is supposed to be a pure Bloch
state.

200

Bibliography
[1] C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, 2nd ed. (Wiley,
New York, 1977).
[2] A. Messiah, Quantum Mechanics, 1st ed. (Dover, New York, 1999).
[3] R. P. Feynman, Statistical Mechanics, 1st ed. (Addison-Wesley, Reading, Ma,
1998).
[4] W. Shockley, Electrons and Holes in Semiconductors, 1st ed. (D. Van Nostrand,
New York, 1950).
[5] E. L. Ivchenko, A. Y. Kaminski, and U. Rossler, Phys. Rev. B 54, 5852 (1996).

201

Appendix D

Velocity operator in the

k · p formalism
Probably the following development or something very similar to it is already somewhere else, but since I came up with it and it helped me clarify some aspects of the
k · p formalism and understand better the origin of
vg =

∂E(k)
∂k

(D.1)

I decided to include it in this thesis.
The eigenstates of the Hamiltonian of a crystal can also be chosen to be eigenstates
of the translation operations corresponding to that crystal. By Bloch’s theorem [1],
these states will have the form
Ψk = eik·r unk ,

(D.2)

where n labels the band and k is the electron wavevector.
The Schrödinger equation with spin-orbit interaction states
HΨk =

P2
+ V (r) +
[∇V (r) × P] · σ Ψk = EΨk .
2m
4m2 c2

(D.3)

Plugging Eq. (D.2) into the Schrödinger equation, and equation that the unk ’s must
satisfy is obtained [2]:
Hunk =

~2 k 2
H + k·P+
[∇V (r) × k] · σ unk = Eunk .
2m
4m2 c2

(D.4)

Note that, although H is commonly called the Hamiltonian of the crystal, it is really
not a true Hamiltonian because of the presence of the extra terms. However, to stick
with the convention, the term Hamiltonian is used throughout this thesis for H or

202
H indistinctly. From a more mathematical point of view, H is just an operator that
happens to have the same spectrum as the true Hamiltonian of the system.
A velocity operator in quantum mechanics consistent with the Ehrenfest theorem
can be defined by
−i
dr
[r, H]
dt

dr
[σ × ∇V (r)] .
dt
m 4m2 c2

(D.5)

Now, keeping with the philosophy of using the unk ’s instead of the Ψk ’s, one looks
for a velocity operator v̂ such that
¯ ¯ À D
¯ ¯
n0 ¯ dr ¯ n
Ψk0 ¯ ¯ Ψk = unk0 |v̂| unk .
dt

(D.6)

Plugging Eq. (D.5) into Eq. (D.6) one obtains

ukn0 |v̂| unk

and from Eq. (D.2)

unk0 |v̂| unk

¯ À
¯ n
n0 ¯ P
¯ Ψk ,
= Ψ k0 ¯ +
[σ
∇V
(r)]
m 4m2 c2

(D.7)

¯ À
¿ ¯
~k D n0 ¯¯ i(k−k0 )·r ¯¯ n E
n0 ¯ i(k−k0 )·r P ¯ n
u k0 ¯ e
u +
¯ uk + uk0 ¯e
m¯ k
¯ À
¿ ¯
¯ n
n0 ¯ i(k−k0 )·r
¯ uk . (D.8)
u k0 ¯ e
[σ
∇V
(r)]
4m2 c2

Comparing the equation above with Eq. (D.4), one sees that the matrix elements
of the tentative velocity operator must satisfy1
D 0
E ¿ 0 ¯¯ 1 ∂H ¯¯ À
¯ u n δ k0 k ,
uk0 |v̂| uk = unk ¯¯
~ ∂k ¯ k

(D.9)

and from here the definition of the velocity operator [3] in the k · p formalism
v̂ =

1 ∂H
~ ∂k

(D.10)

Note that, since this offers a prescription for the evaluation of matrix elements between any
Bloch state, it is also valid for arbitrary states.

203
follows. However, it must be emphasized that, as shown in the above derivation, this
definition only applies when looking at matrix elements between states with the same
k.
At this point, one can find the expectation value of the velocity of a Bloch state
quite easily:
vkn =

¯ À
¿ ¯ ¯ À ¿ ¯
¯ ¯
1 ∂E n (k)
n ¯ 1 ∂H ¯ n
n ¯ dr ¯ n
Ψ k ¯ ¯ Ψ k = uk ¯
dt
~ ∂k ¯ k
~ ∂k

where, in the last step, the theorem proved in Sec. C.1 has been used.

(D.11)

204

Bibliography
[1] N. W. Ashcroft and N. D. Mermin, Solid State Physics, 1st ed. (Saunders College,
Philadelphia, PA, USA, 1976).
[2] E. O. Kane, in Tunneling Phenomena in Solids, edited by E. Burstein and S.
Lundqvist (Plenum Press, New York, 1969), p. 1.
[3] E. L. Ivchenko, A. Y. Kaminski, and U. Rossler, Phys. Rev. B 54, 5852 (1996).