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Quantum magnetotransport studies of semiconductor heterostructure devices
Citation
Marquardt, Ronald R.
(1995)
Quantum magnetotransport studies of semiconductor heterostructure devices.
Dissertation (Ph.D.), California Institute of Technology.
doi:10.7907/P7Z1-T478.
Abstract
NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document.

Resonant interband tunneling is perhaps the most generalized example of quantum transport in semiconductor heterosystems, and yet few studies of interband magnetotunneling have been reported. This thesis presents the results of the first in-depth study of quantum magnetotransport through valence band-like well states in broken-gap semiconductor heterostructures. The transport characteristics of a series of InAs/AlSb/GaSb/AlSb/InAs resonant interband tunneling (RIT) diodes were investigated in magnetic fields as great as 8 tesla oriented both parallel and perpendicular to the tunnel current density. The resulting observations advance the understanding of interband transport and reveal interesting and complex phenomena previously unseen in magnetotunneling investigations.

In magnetic fields perpendicular to the tunnel current density, evidence of both low-mass, negative dispersion, and high-mass, positive dispersion states in the GaSb well is shown. The magnetic field in this geometry has no discernible effect on the RIT transport characteristics at low fields, however, and the transition between this low-field regime, and the expected response at higher field strengths is extremely abrupt. Associated with this marked change in behavior is an additional, narrow peak present in the negative differential resistance (NDR) region of the device at the threshold, or critical, magnetic field. This anomalous discontinuity in device transport violates semi-classical theory and is suggestive of a dramatic and fundamental change in resonant quantum transport.

With the field applied normal to the epitaxial layers, Landau levels form and resonant tunneling through them is observed indirectly via Shubnikov-de Haas-like oscillations of the tunneling conductance. The non-conservation of Landau level index in interband magnetotunneling is first proposed theoretically, and subsequently verified experimentally. Evidence for this effect is asserted from the discontinuous changes in the Shubnikov-de Haas oscillatory phase as a function of applied bias. These phase shifts result from bias-dependent changes in the resonant current path through well hole states of differing longitudinal angular momentum. The data are the first observation of Landau level mixing in interband tunnel devices, and only the second report for magnetotunneling in general. The lack of separate Shubnikov-de Haas oscillations for coupling to spin-up and spin-down states is interpreted to be evidence that the well masses are significantly less than the electrode mass, such that the reduced mass [...] of the two is approximately equal to the well mass, [...].
Item Type:
Thesis (Dissertation (Ph.D.))
Subject Keywords:
Applied Physics
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.
Vahala, Kerry J.
Yariv, Amnon
Nicolet, Marc-Aurele
Defense Date:
20 June 1994
Record Number:
CaltechETD:etd-10172007-111750
Persistent URL:
DOI:
10.7907/P7Z1-T478
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QUANTUM MAGNETOTRANSPORT
STUDIES OF SEMICONDUCTOR
HETEROSTRUCTURE DEVICES

Thesis by
Ronald R. Marquardt

In Partial Fulfillment of the Requirements
for the Degree of

Doctor of Philosophy

California Institute of Technology

Pasadena, California

1995
(Submitted June 20, 1994)

to my family and friends

ili

Acknowledgments

When attempting to summarize the contributions of many colleagues and friends
to a work encompassing such a vast part of your life, it is difficult to know where
to begin. Certainly, this thesis is in no small part due to the unique vision and
guidance of my advisor, Tom McGill. Long ago he told me there would be some-
thing interesting to see if I'd just look at these interband devices in magnetic fields.
At times, I admit, I thought him the techno-warrior equivalent of the grizzled old
miner who swears by the gold locked deep in some favorite mountain. I was quite
gratified to find a nugget or two along the way, and I am indebted to Tom for the
experience.

If Tom is the brains behind the operations, then it is Marcia Hudson who is the
glue which keeps it from devolving into a chaotic soup of primordial ooze. If you
don’t believe me, just wait until the next time she’s off on vacation... I can honestly
say that Marcia’s friendship and encouragement have contributed greatly to this
thesis. I also must thank her son, David, for keeping me up to date on all things
Jurassic. The administrative assistance of Sandy Brooks and Carol McCollum was
also appreciated.

It really is cliché to say, after reading so many thesis acknowledgments before
this, but the real strength of Tom’s research group is the quality and character
of its members, and I can honestly claim to have enjoyed working with all that

have overlapped with my tenure. David Ting has been a mentor and good friend,

and has always been willing to provide cheerful reminders of my fallibility. Doug
Collins grew the samples that I studied and provided great insight into the resulting
data. He has also proven to be both an apt guide to the links in Scotland (“Putt,
dammit, putt!!”) and a good friend. As contemporaries, I owe Mike Wang, Johanes
Swenberg, and Rob Miles many thanks for their encouragement, support, and
good-natured practical jokes. If I only knew how to get them back... Harold Levy
has always been a constant source of witty observations and insightful ideas. His
contributions to the group’s computer system have been much appreciated. Yixin
Liu provided the theoretical support for this study, without which it would not be
complete; I have also enjoyed his stories of China and its culture.

IT owe much to Mark Phillips, Ed Yu, Mike Jackson, and David Chow, who,
as senior students and post-docs provided much-needed assistance to my research.
Chris Springfield and Per-Olov Pettersson have, between them, currently taken
on responsibility for nearly every piece of equipment the group owns, and should
clearly be thanked for their efforts. I have enjoyed working with Erik Daniel and
Alicia Alonzo, and I wish them well as they begin their scientific Odyssey. My
interactions with Andy Hunter, Gerry Picus, Odgen Marsh, Ron Grant, and Jim
McCaldin have been fruitful and rewarding.

Outside the group, many people have made life as a graduate student at Caltech
not only possible, but enjoyable. I must thank Lisa Lorden for her continuing faith
and support. James and Debbie Larkin have been great friends and roommates,
and I will miss their company. I can always count on John and Gemma Iannelli
to make me smile, and have truly appreciated their friendship. Carl and Susan
Gaines, Pat Ritto, Craig Sangster and David Fields all deserve credit for supporting
and encouraging me.

Last but not least, there is my family. Somehow my brother, Jon, managed to

put up with me as a roommate for nearly two years during my graduate tenure, and

I am glad that we have become better friends as a result. My parents deserve more

credit than all others combined. Their faith and love have been an inspiration.

Abstract

Resonant interband tunneling is perhaps the most generalized example of quan-
tum transport in semiconductor heterosystems, and yet few studies of interband
magnetotunneling have been reported. This thesis presents the results of the first
in-depth study of quantum magnetotransport through valence band-like well states
in broken-gap semiconductor heterostructures. The transport characteristics of a
series of InAs/AISb/GaSb/AISb/InAs resonant interband tunneling (RIT) diodes
were investigated in magnetic fields as great as 8 tesla oriented both parallel and
perpendicular to the tunnel current density. The resulting observations advance
the understanding of interband transport and reveal interesting and complex phe-
nomena previously unseen in magnetotunneling investigations.

In magnetic fields perpendicular to the tunnel current density, evidence of both
low-mass, negative dispersion, and high-mass, positive dispersion states in the
GaSb well is shown. The magnetic field in this geometry has no discernible effect
on the RIT transport characteristics at low fields, however, and the transition
between this low-field regime, and the expected response at higher field strengths is
extremely abrupt. Associated with this marked change in behavior is an additional,
narrow peak present in the negative differential resistance (NDR) region of the
device at the threshold, or critical, magnetic field. This anomalous discontinuity
in device transport violates semi-classical theory and is suggestive of a dramatic

and fundamental change in resonant quantum transport.

vii

to the well mass, mj.

vill

List of Publications

Work related to this thesis will be published under the following titles:

Resonant magnetotunneling spectroscopy of p-well interband tun-
neling diodes,

R. R. Marquardt, D. A. Collins, Y. X. Liu, D. Z.-Y. Ting, and T. C. McGill,
submitted to Phys. Rev. B.

Evidence of Landau level mixing in resonant interband tunneling
diodes,

R. R. Marquardt, D. A. Collins, Y. X. Liu, D. Z.-Y. Ting, and T. C. McGill,
to be submitted to Appl. Phys. Lett.

Contents

Acknowledgments iii
Abstract vi
List of Publications viii
List of Figures xii
List of Tables xiv
1 Introduction 1
1.1 Introduction to Thesis .........0.....0.00000 00000. 1
1.1.1 Motivation... 2.2... . 0.0.0.0. 000002 eee 1

1.1.2. Summary of Results ...................02, 4

1.2 Thesis Outline... . 0.0... 0.0.00 000200000. eee 6
References... ee ee 7

2 Background and Motivation 9
2.1 Introduction... 2... 0.0.0.0... 000. ee een 9
2.2 Heterostructures and Quantum Devices. ............... 10
2.2.1 Background ..................0..000-000. 10

2.2.2 UniqueIssues ...........0.0.0.0 00000002 ban 12

2.2.3 Electronic Properties... 2... 2... 0.00. eee es 19
2.3 The InAs/AISb/GaSb Material System ...............0. 35
2.3.1 Transport Properties ...........0.0.20 000008 37
2.3.2 Technological Applications ................000.4 44
2.4 Magnetic Fields in Bulk Semiconductors ............... AT
2.4.1 Landau Levels... 2... 0.0.20... ..0.0-00.0000,4 47
2.4.2 Shubnikov-de Haas Oscillations .............0.. 51
References... ee 54
Resonant Magnetotunneling Spectroscopy 57
3.1 Introduction. . 2... 2... ee 57
3.2 Technique... 2... a9
3.2.1 Basic Model... 2... 2... 2... ee 59
3.2.2 Experimental Setup... 2.2... 0.0... ...0000004 64
3.3 Experimental Results. ...............0000.0.00000, 67
3.3.1 I-V Peak and Shoulder... ........0.......0.., 67
3.3.2 Difference Spectra 2... . 2... ee 74
3.3.3 Critical Field and NDRI-V Peak .............0.. 79
3.4 Analysis... 2... ee 81
84.1 B> Beit oe 81
3.4.2 B< Beit oe e 86
3.5 Conclusions .. 1... ee 91
References . 2. 93
Interband Tunneling Through Landau Levels 95
4.1 Introduction... ... 2... ee 95
4.2 Theory... ...... ee ee 97

4.2.1 Landau Levels in Heterostructures............... 97

4.2.2 Tunneling Through Landau Levels ............2.. 104
4.3 Experiment 2... . 2. ee ee 110

4.3.1 Measurement Technique .................00. 110

4.3.2 Current-Voltage Characteristics ...........0..00.. 111

4.3.3 Oscillatory Tunneling Conductance ...........0... 113

4.3.4 Fan Diagrams... ..........0 0.00000 00 bee 119
4.4 Conclusions .. 2... eee 129
References... 0. ee 132
Magnet History and Operation 134
A.l Introduction... ... 0... 2 ee a 134
A.2 History. 2... ee 136
A.3 Operation .. 2... ee 138
References... 0. 144
Bulk &- p Theory Incorporating Magnetic Fields 145
B.1 Introduction... ..... 0.020.000.0000 0000. eee, 145
B.2 Bulkk-pTheory ... 0.200000. 00.0 eee, 146
B.3 Incorporation of the Magnetic Field... ....020002020000.. 149

xii

List of Figures

2.1
2.2
2.3
2.4
2.9
2.6
2.7
2.8
2.9
2.10
2.11

3.1
3.2
3.3
3.4
3.0
3.6

3.7

McCaldin diagram for Si, Ge, and selected III]-V compounds .... 15
Heterointerface types... 2. 18
Heterostructure device classification. ..............0.. 20
Quantum states in quantum wells and DBH’s ............ 25
Transfer matrix calculation of DBH transmission coefficient... . . 29
Calculated Type I DBH transmission coefficients. .......... 30
Type I unipolar DBH I-V curve evolution... ........002, 34
InAs/GaSb/AISb band alignments .................. 37
The p-well resonant interband tunneling (RIT) diode ........ 38
Calculated RIT transmission coefficients ..............., Al
Interband tunneling I-V curve evolution ..............., 42
Resonant magnetotunneling spectroscopy geometry ......... 61
RMTS technique in Type I unipolar intraband tunneling ...... 63
RMTS technique in interband tunneling ............02.. 65
InAs/AISb RTD I-V Characteristic ..............000., 68
RMTS results for InAs/AlISb RTD ...............0.. 69

Example RIT I-V characteristics at 0.0 T and 8.0 T, and corre-
sponding difference spectrum ....................4, 71

Peak and shoulder voltages vs. applied perpendicular magnetic field 72

3.8

3.9

3.10
3.11
3.12

3.13
3.14
3.15
3.16
3.17

4.1
4,2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10

xii

Observation of second peak develop from shoulder in p-well RIT I-V

CUIVE, 2 ee 73
Difference spectra at fields less than Bay 2... ...0000004 76
Difference spectra at fields equal to, or exceeding, Bayz ... 2... 78
Integrated AJ in NDRregion ..................004. 82

Current-voltage characteristics of device A near the critical magnetic
field. In both devices A and C, a second, narrow peak was observed

in the NDR region at B,,;:. In fields only + 300 gauss greater, the

main peak was seen to widen and encompass it entirely... ..... 83
Temperature dependence of NDR peak at By. 2... 02 84
Theoretical subband dispersions at0 T.............020. 87
Theoretical subband dispersions at8T ..............., 88

Calculated transmission curves for a RIT in various magnetic fields 89
Calculated RIT current density-voltage characteristics in 0, 2, and

6 T magnetic fields .................0.0-.00.0004 90

Landau tunneling (B L J) geometry ............200.0, 105

Representative RIT I-V characteristics in a parallel magnetic field . 112

Conductance oscillations of 7.0 nm-wide well RIT .......0.. 114
Tunneling conductance of 11.9 nm-wide well RIT .......... 115
SdH oscillation transition region of 11.9 nm-wide RIT sample. . . . 117
Fan diagram for SdH oscillations in 7.0 nm p-well RIT ....... 121
Fan diagram for SdH oscillations in 11.9 nm p-well RIT... .... 122
Slope and Landau index phase of 7.0 nm well fan lines ...... . 124
Slope and Landau-index phase of 11.9 nm well fan lines... .. . . 125

Theoretical zero-field well band structure of 7 nm and 11.9 nm p-well

xiv

List of Tables

2.1 Band structure and transport parameters for selected semiconductors 13
3.1 RIT growth parameters... .......0..... 0000000004 66

A.1 Magnet design specifications... 2... . ee ee ee ee 135

Chapter 1

Introduction

1.1 Introduction to Thesis

This thesis details research of quantum transport in semiconductor heterojunction
devices in the presence of external magnetic fields. Specifically, the devices studied
are epitaxial resonant tunneling diodes (RT'D’s) fabricated in the nAs/AISb/GaSb
material system. The purpose of this study was two-fold. First, the peculiar nature
of the InAs/GaSb band alignments makes the system attractive for basic physics
studies of tunneling phenomena. Second, it was hoped that a better understand-
ing of the underlying physics might help in the realization of practical uses and
optimizations of this technology for real-world problems. Much of this thesis will
focus on the first of these goals, both because of the breadth of the fundamen-
tal results and the lack of technological maturity which hinders the adoption of

quantum phenomena-based devices.

1.1.1 Motivation

Magnetic fields have proven to be a useful, if not indispensable, tool in the investi-

gation of low-dimensional systems. In particular, the understanding of the physics

of resonant tunneling in semiconductor heterostructure devices has been advanced
by studies of quantum transport in the presence of magnetic fields. While magnetic
fields do no work on individual charge carriers, their effect on the energy and mo-
menta spectra allow subband structure, occupation, and dispersion to be observed
experimentally [1]. With the field aligned perpendicular to the growth plane of a
semiconductor heterostructure device (parallel to the tunnel current density), the
energy spectrum of the quasi-bound states is modified by the addition of discrete
ladders of Landau levels. Studies of devices in this configuration [2, 3, 4, 5, 6],
and in tilted fields, with magnetic field components both parallel and perpendic-
ular to the epitaxial layers [7, 8], investigate transport properties in systems with
increased quantization and reduced dimensionality. The resulting changes in the
current-voltage characteristic of a device are indicative of the energies of quantized
subbands in the well, the in-plane subband dispersions, and the coupling of the well
states to those in the electrodes. In addition, resonant tunneling through Landau
levels in the well may be observed through Shubnikov-de Haas-like oscillations of
one or more transport variables.

In contrast, with a magnetic field aligned parallel to the growth plane quanti-
zation effects due to the field and the crystal potential are coaxial. Except for the
highest fields, or widest quantum wells, the magnetic field in this geometry may be
treated perturbatively. As a result, the device operates without additional quanti-
zation or significant modification to the eigenstates and energies of the system; the
only effect of the field, semi-classically, is to bend the carrier trajectories into cy-
clotron orbits about the axis of the field. This alteration of the carrier distribution
in momentum space is exploited in resonant magnetotunneling spectroscopy [9]
(RMTS) to probe the energy subband dispersion in semiconductor quantum wells.
The shift in k-space induced by the magnetic field is used to experimentally se-

lect the parallel k-vector of elastically tunneling electrons. The subsequent effect

upon the current-voltage (I-V) behavior of the device, specifically the peak voltage,
provides an indirect map of the band structure in the well.

While the aforementioned applications of magnetic fields to the study of quan-
tum transport, and specifically to RTD’s, are well-known, relatively few observa-
tions have been reported for interband devices employing the InAs/GaSb Type
II broken-gap band alignment. The unique properties of InAs/GaSb heterostruc-
tures have been known for some time [10], and recently have been exploited to
create a new class of quantum transport device [11, 12]. Because the conduc-
tion band edge of InAs lies 0.15 eV lower in energy than the valence band edge of
GaSb [13], transport through this system involves resonant tunneling between con-
duction band states in InAs and valence band states GaSb. The few observations
of magnetotunneling in devices in this system have focused on the double barrier
heterostructure with an InAs well, AlSb barriers, and GaSb electrodes [14, 15, 16],
or have investigated the barrierless InAs/GaSb/InAs system [17]. The lack of de-
tailed studies of interband transport in magnetic fields, especially for devices with
the rich valence subband structure of GaSb wells, provided the motivation for this
thesis. The transport characteristics of a series of InAs/AISb/GaSb/AISb/InAs
resonant interband tunneling (RIT) diodes were investigated in magnetic fields as
great as 8 tesla applied both parallel and perpendicular to the tunnel current den-
sity. Magnetotransport in this system combines the complexities of hole subbands,
interband tunneling, and quantum magnetic effects. The resulting data represent
the first in-depth study of interband quantum magnetotransport through valence
band-like states, and the data have revealed interesting and complex phenomena

previously unseen in magnetotunneling investigations.

1.1.2. Summary of Results

The resonant magnetotunneling spectroscopy (RMTS) technique, with the mag-
netic field aligned in the plane of the epitaxial layers, provides a probe of the
quasi-bound state in-plane dispersions in double barrier heterostructures. When
applied to GaSb-well RIT diodes, anomalous behavior is observed. There exist
two distinct regimes of differing character in the RIT data, one for fields less than
a critical magnitude, and another for fields at, or exceeding, this same B,,,. For
fields less than B.,i¢ ($5 T) little change in the current-voltage characteristics of
the RIT samples was observed. The exact magnitude of the critical field varies be-
tween samples, but does not appear to depend monotonically on well width. Above
the critical field, the RIT I-V characteristics are strongly affected by the field, and
demonstrate behavior consistent with the RMTS theory. The data indicate the
presence of both low-mass, negative dispersion, and high-mass, positive disper-
sion, states in the well. At the critical field, there is an additional, narrow peak
that occurs in the NDR region of the I-V characteristic over a ~ 300 G range of
fields. This second peak is not seen to be dependent upon external circuit param-
eters, and is therefore not a circuit instability or oscillation effect. Furthermore,
the abruptness of its appearance (less than a 50 G change in field at 5 T) is not
consistent with a simple, cyclotron orbit resonance. Similar behavior is not seen
in InAs/AJSb double barrier structures measured with the same equipment, and
thus the phenomenon seems directly related to the interband nature of transport
in the InAs/AlSb/GaSb system. The observation of such an abruptly occurring
peak has never been reported before in magnetotunneling experiments, and the
associated change in magnetotransport behavior at the critical field is suggestive
of a dramatic and fundamental change in device transport physics.

As a magnetic field parallel to the tunnel current density is applied to an RTD,

Landau levels form and fundamental changes occur in the criteria for resonant
elastic transport. As the transverse momentum is replaced by discrete Landau in-
dices, forcing new conservation laws upon the tunneling carriers, resonant Landau
level tunneling may be evident by either direct observation of individual I-V peaks
corresponding to each magnetic energy state, or Shubnikov-de Haas oscillations
of transport-related device characteristics. While there is some direct evidence for
Landau level-associated peaks in the RIT I-V’s, it is mostly within the NDR region
of the device, and suffers from concurrent magnetoresistance effects. Shubnikov-
de Haas oscillations of tunneling conductance are also observed, however, at all
biases less than the peak voltage, in fields exceeding 2 tesla. In the limit of 0D-
OD tunneling between an emitter notch state and the well, these oscillations are
attributable to the alternate alignment and misalignment of the Landau levels on
either side of the barrier layer. As such, the frequency of these oscillations, and
their bias dependence, reflects the character of both the emitter and well states.
Specifically, the band edge energy difference between the two subbands and their
reduced mass both determine the nature of the Shubnikov-de Haas conductance
oscillations. Additionally, however, interband transport introduces the possibil-
ity of Landau level index non-conservation through the coupling of states having
different angular momentum projections. With a perpendicular field, the total lon-
gitudinal angular momentum projection, including that induced by the cyclotron
orbits of the carriers, is conserved, and, consequently, tunneling carriers may mix
with states of different Landau index. In turn, this change in Landau level index
effects the phase of the resulting Shubnikov-de Haas oscillations. Experimental ob-
servations of large shifts in Shubnikov-de Haas phase in RIT conductance data are
interpreted as direct evidence for Landau level mixing, and conservation of total
longitudinal angular momentum. These data are the first observation of Landau

level non-conservation in an interband system, and only the second report for

magnetotunneling in general. This work also represents both the first theoretical
prediction, and subsequent experimental verification, of Shubnikov-de Haas phase

shifts due to Landau level mixing in semiconductor heterostructures.

1.2 Thesis Outline

The remainder of the thesis is divided into three chapters and two appendices.
Chapter 2 develops background knowledge of semiconductor heterostructures, the
InAs/AlSb/GaSb material system, and the effects of magnetic fields upon bulk
transport, with the intent to motivate the experimental studies and provide a
solid context in which to view them. Chapter 3 provides details of the RMTS
experiments on GaSb-well RIT’s, and Chapter 4 presents the results of the res-
onant Landau level tunneling experiments in the same devices. Two appendices
are provided for the reference of future students. The history and operation of
the superconducting magnet used for this thesis are detailed in Appendix A, and
Appendix B develops details of bulk k- p theory, and the means by which magnetic

fields are incorporated in it.

Bibliography

[1] Molecular Beam Epitaxy and Heterostructures, proceedings of the NATO Ad-
vanced Study Institute on Molecular Beam Epitaxy (MBE) and Heterostruc-
tures, Erice, Italy, edited by L. L. Chang and K. Ploog (Martinus Nijhoff,
Dordrecht, 1985).

[2] E. E. Mendez, L. Esaki, and W. I. Wang, Phys. Rev. B 33, 2893 (1986).

[3] V. J. Goldman, D. C. Tsui, and J. E. Cunningham, Phys. Rev. B 35, 9387
(1987).

[4] L. Eaves et al., Appl. Phys. Lett. 52, 212 (1988).
[5] C. E. T. Gongalves de Silva and E. E. Mendez, Phys. Rev. B 38, 3994 (1988).

[6] E. E. Mendez, H. Ohno, L. Esaki, and W. I. Wang, Phys. Rev. B 43, 5196
(1992).

[7] J. J. Koning et al., Phys. Rev. B 42, 2951 (1990).

[8] M. L. Leadbeater, F. W. Sheard, and L. Eaves, Semicond. Sci. Technol. 6,
1021 (1991).

[9] R.K. Hayden et al., Phys. Rev. Lett. 66, 1749 (1991).

[10] L. L. Chang and L. Esaki, Surf. Sci. 98, 70 (1980).

[11] D. H. Chow et al., in Quantum-Well and Superlattice Physics III, proceedings
of the SPIE 1283 (Bellingham, WA, 1990).

[12] D. A. Collins e¢ al., in Resonant Tunneling in Semiconductors: Physics and
Applications, edited by L. L. Chang, E. E. Mendez, and C. Tejedor (Plenum,
New York, 1991).

[13] G. J. Gualtieri, G. P. Schwartz, R. G. Nuzzo, and W. A. Sunder, Appl. Phys.
Lett. 49, 1037 (1986); G. J. Gualtieri et al., J. Appl. Phys. 61, 5337 (1987).

[14] E. E. Mendez, in Resonant Tunneling in Semiconductors: Physics and Appli-

cations, ibid.
[15] E. E. Mendez, J. Nocera, and W. I. Wang, Phys. Rev. B 45, 3910 (1992).
[16] E. E. Mendez, Surf. Sci. 267, 370 (1992).

[17] T. Takamasu et al., Surf. Sci. 263, 217 (1992).

Chapter 2

Background and Motivation

2.1 Introduction

This chapter provides a thorough background for the understanding of semicon-
ductor heterostructures (Section 2.2), the nAs/AISb/GaSb material system (Sec-
tion 2.3), and the effects of magnet fields upon transport in bulk semiconductors
(Section 2.4). The purpose of this treatment is several-fold. First, these discussions
set the framework for the research presented in later chapters. The technological
and scientific problems surrounding magnetotunneling in broken-gap heterostruc-
tures lend motivation for the work, and provide meaning to the results. Second,
too often the physics of heterostructures is reduced to simple arguments that map
to textbook problems without the added complexities of the real-life problem being
truly addressed. Interband tunneling is perhaps the most generalized example of
quantum transport in semiconductor heterosystems, and its study requires a firm
understanding of the contributing physical processes. Lastly, magnetic fields have
a profound impact upon all transport phenomena in semiconductors, and an intu-
itive discussion of the basic theory highlights the complexities they introduce and

justifies their experimental utility. All of these topics are important background

for the study of magnetotunneling in interband heterostructure devices, and as

such, provide the proper context and motivation for the work contained in this

thesis.

2.2 Heterostructures and Quantum Devices

2.2.1 Background

Semiconductor heterostructures are comprised of two or more dissimilar semicon-
ductors physically mated through an epitaxial or pseudomorphic growth process
to form a artificially-constructed single crystalline structure. While this definition
encompasses a wide variety of possible semiconductor systems, one standout excep-
tion happens also to be the most commercially viable at the present: silicon.! The
reasons for this success are many-fold, including the maturity of silicon processing
technology, the quality (near-ideal) of the Si-SiO, interface, and the relative im-
maturity of heterojunction device technology. As optical devices proliferate, the
use of non-silicon materials, and heterojunction systems in particular, will flourish,
as silicon, an indirect semiconductor, is a notably poor performer in the optical
arena. The electrical properties of heterostructures are of great interest as well,
and have been extensively studied since their technological inception [3, 4]. Het-
erostructures offer the designer the flexibility to tailor a given device for optimal
performance for a specific set of criteria. Additionally, as device sizes shrink to the

scale that quantum mechanics plays a dominant role in their operation, new device

1Estimates of the world-wide semiconductor market place the total value of shipments in
1992 at roughly $77.3 billion [1]. Of that total, $66.0 billion was spent on integrated circuits
exclusively produced in silicon. In contrast, the total semiconductor diode laser market, one of
the only commercial markets to include heterostructure devices, was a mere $287 million that

same year [2].

(and computational) paradigms must be developed to constructively exploit these
quantum effects. Heterostructure-based devices are poised to perform exactly that
role.

The feasibility of heterostructure devices must be attributed to the develop-
ment of growth techniques which allow the precise, ordered deposition of crys-
talline constituents on an atomic layer scale. The advent of molecular beam epi-
taxy (MBE) [5], and other growth techniques such as metal-organic chemical vapor
deposition (MOCVD), have made the investigation of semiconductor heterostruc-
tures possible. MBE growth takes place in a vacuum chamber under ultra-high
vacuum (UHV) conditions (P<107"* torr). High purity source material (elemental
or compound semiconductor constituents) is evaporated in several crucibles and
directed toward a single-crystal growth substrate. Growth conditions (substrate
temperature, source flux, etc.) are controlled to cause the molecular beams to
condense a monolayer at a time on the substrate surface in a lattice ordered to
the bulk substrate crystal. Typical growth rates are on the order of a few A /sec.
Because of the UHV environment, the ultra-pure source material, the high-quality
semiconductor substrates, and the controlled thermodynamic conditions, the re-
sulting crystals are typically of the highest quality attainable in the laboratory [6].
The absolute quality of MBE deposited thin films depends strongly on the relative
difference between the lattice constants of the grown layers and the underlying
substrate, the chemistry of the various compounds, and the skill of the opera-
tor. The first may be controlled via the selection of appropriately lattice-matched
compounds, the second defines areas of active research, and the last involves as
much individual artistry as technical skill. In general, the well-developed material
systems for MBE growth are all III-V semiconductors, such as AlAs/GaAs, which
has been the canonical system for heteroepitaxy research [6].

As discussed earlier, given that they can indeed be grown as requested, het-

erostructures give device designers a great deal of flexibility in determining the
characteristics, both electrical and optical, of a tailored structure. Table 2.1 lists
several of the relevant properties of some of the major semiconductor elements
and compounds. The use of heterostructures allows the band gap, mobilities, ef-
fective masses, and other material and transport parameters to be gradually or
abruptly altered. Provided that the interfacial chemistry between two abutting
semiconductors is favorable, such heterojunctions may be as sharp as a single
monolayer. While the interface plane might exhibit in-plane inhomogeneities (in-
terface roughness), in many systems it is believed that the adjoining layers consist
of relatively pure monolayers of the two junction materials [7]. In addition to pure,
binary semiconductor compounds, ternary and quaternary alloys of two or more
binary systems are commonly grown. This practice is generally used to satisfy the
structural and chemical demands of epitaxial growth, but may also be utilized to
spatially grade an alloy composition so as to smoothly vary the material properties
over many monolayers, rather than create abrupt heterojunctions. In addition to
the controlled variation of transport parameters, the differences in band gap be-
tween the various layers of a heterostructure can lead to quantum confinement of
the charge carriers; the reduced dimensionality of such a system will in turn induce

radical changes in its electrical and optical properties.

2.2.2 Unique Issues

There are many issues unique to heterostructures which are not present in tradi-
tional homojunction technology. The first conceptual problem encountered when
placing two dissimilar semiconductors adjacent is the ambiguous offset between
their respective energy scales. While the band gaps of most semiconductors are
well known, the experiments which are used to determine them, such as optical

absorption or photoluminescence, are relative energy measurements; they are not

300K Band Gap 300K Mobility Effective Mass

(eV) (cm?/V-sec) (m*/mo)
IV
Si 1.12 fle = 1500 ms = 0.98
ben = 450 mj, = 0.16
II-V
AlAs 2.14 be = 294 m* = 0.35
ms, = 0.15
AISb 1.58 bbe = 200 ms = 0.39
Ln = 420 mj, = 0.11
GaAs 1.42 be = 8,500 m* = 0.065

Lun = 400 ms, = 0.087

GaSb 0.72 Me = 5,000 m* = 0.049
Ln = 850 mi, = 0.056
InAs 0.36 [le = 33, 000 mr = 0.023
Lp, = 460 mj, = 0.025
InSb 0.17 He = 80, 000 m* = 0.014

Ln = 1,250 my, = 0.016

Table 2.1: Room temperature band gaps, mobilities, and effective masses of se-
lected III-V semiconductors, in comparison to silicon. Although not all of these
compounds may be grown epitaxially upon one another, those that can may be
used in heterostructures where the spatial variation of material parameters can
be used to tailor a device for a given application. The data was compiled from a

number of sources [8, 9, 10].

made with respect to a universal energy reference. As a result, although the band
gaps of GaAs and AlAs are known to be 1.42 eV and 2.14 eV, respectively, that in-
formation alone does not convey the difference in energy (if any) between electrons
in the valence band of AlAs, say, and that of GaAs. Band offsets are not easily
predicted by theory [11], although a general rule of thumb proposed by McCaldin,
McGill and Mead is that two semiconductors sharing the same anion should have,
to first order, no energy difference between their respective valence band edges [12].
The known exceptions to this rule are compounds with aluminum cations, such as
AlAs. This common anion rule works reasonably well, but experimental observa-
tions of band offsets are required to accurately assess the relative band alignments
between components of any given heterosystem. This experimental data, such as
compiled in Reference [13], is essential for the proper engineering of heterostructure
devices.

As differing compounds are grown together, it is especially important that the
preferential spatial ordering of the atoms of each semiconductor be compatible,
both in lattice constant and crystallographic structure. If this mechanical condi-
tion is not met, the resulting crystal will at best be highly dislocated, and may
even become polycrystalline or amorphous, and therefore generally of limited use.
The crystallographic structures of all the III-V compounds are zincblende, and
therefore compatible, but their lattice constants vary considerably. To graphically
display the two parameters of immediate significance in the design of heterostruc-
ture devices, both the lattice constant data and experimental band offsets, Mc-
Caldin introduced a useful plotting paradigm [14]. Figure 2.1 is an example of
such for silicon, germanium, and the non-phosphorous III-V semiconductors listed
in Table 2.1. Shown are the conduction (triangles) and valence (squares) band

edges for these materials plotted against their lattice constant.? Thus, in addition

7In addition, the symbols for the band edges may be open or filled, depending upon the n- and

2 pees Torres Terr Torres reer Teer

1.6 + 4 , 4

1.2 ul yy 7

> |

2 08 + ' -

B 0.4 i Mm Ge GoSb InSb |

woo si Goss i AISb ]

-0.4 + .

L ll AlAs |

-0.8 F- -
5.4 5.6 5.8 6 6.2 6.4 6.6

Lattice constant (Angstroms)

Figure 2.1: McCaldin diagram [14] showing the band alignments, lattice constants,
and dopability of silicon, germanium, and the non-phosphorous III-V semiconduc-
tors. The triangular and square symbols indicate the conduction and valence band
edges, respectively. Were any of these materials undopable either n- or p-type, the

appropriate symbol would be left unfilled as an indication.

to identifying which combinations of materials may be grown epitaxially, the band
alignments of these systems are simultaneously displayed. From this diagram, it is
evident that GaAs and AlAs make a particularly well lattice-matched system. Be-
cause of their respective band offsets, AlAs can be used to confine carriers in both
the conduction and valence bands of GaAs. Additionally, InAs, GaSb, and AlSb
form yet another combination identified as a potential heteroepitaxial system. It is

clear from the McCaldin diagram that all three are fairly well lattice-matched, and

p-type dopability of the material, which was important for the II-VI compounds the original ar-
ticle addressed. The H-VI compounds introduce this additional complication for heterostructure

device development, which thankfully is not an issue for III-V materials.

AlSb may be used as a barrier for carriers in all but the valence band of InAs. While
some strain from mismatched lattices can be tolerated in epitaxial growth, espe-
cially for thin layers (thin being less than some experimentally-determined critical
thickness), large strains (greater than a few percent difference in lattice constant)
are undesirable. Thus, while the low mass and high mobility of InSb look attrac-
tive for high-speed electronic devices, its large lattice constant precludes its use
in typical heteroepitaxy systems. In addition, strain can have a large effect upon
the band offsets, and is often used in device design to engineer appropriate band
alignments or quantum confinement.

Several types of heterojunctions are possible, as characterized by the band
gaps and band offsets of the two adjacent materials. Examples of all three types
of offsets are evident in Figure 2.1, but are more explicitly shown in Figure 2.2,
where the band gaps (shaded) of two materials are plotted as a function of position
perpendicular to the junction (along the growth direction). Figure 2.2(a) shows
a Type I heterointerface, in which the band alignments are such that both band
edges of the smaller gap material lie between the conduction and valence band
edges of the other. Such is the case for the GaAs/AlAs system. Figure 2.2(b)
shows, as mentioned earlier, how the junction may be graded so as not to be
abrupt. Here a spatially varying alloy concentration is used to gradually grade
from one material to another. Figure 2.2(c) shows the broken-gap Type II het-
erojunction, in which the band gaps of the two materials do not overlap in energy
at all. The InAs/GaSb system is a unique example of this interface configuration.
Lastly, Figure 2.2(d) shows the remaining possibility, the staggered Type II inter-
face. The InAs/AISb system is representative of this type of alignment. All of the
interfaces are shown under flat band conditions, where the charge transfer between
layers due to the energy discontinuities is ignored. Such band bending leads to a

built-in electric field at the interface, and subsequent charge accumulation or de-

pletion. Band bending can be accounted for theoretically in most systems through
a straightforward Thomas-Fermi model of the electrostatic potential [15]. The
Type II broken-gap system of InAs/GaSb is quite pathological for these codes,
however. A large amount of charge transfer occurs between these materials, as
electrons energetically prefer InAs to GaSb, and there is no barrier to their dif-
fusion, and likewise for holes in GaSb. Consequently, numerical models must be
more realistic, and solution algorithms computationally robust. As a general rule,
these requirements are necessary for all accurate models of the InAs/GaSb system.

Having said what criteria are important to the design of heterostructure de-
vices, several of the most important types may now be introduced. Figure 2.3
shows the three most prevalent classes of quantum heterostructure devices. In
choosing these to discuss, we are deliberately ignoring such devices as heterojunc-
tion bipolar transistors (HBT’s) and high electron mobility transistors (HEMT’s)
which may be viewed as heterojunction-based optimizations of normal bipolar and
field-effect transistor technology. All three heterostructure device types shown in
Figure 2.3 make use of the variation of band gap between the heterolayers to quan-
tize carriers and form two-dimensional (2D) or quasi two-dimensional carrier gases.
In the quantum well, shown in Figure 2.3(a), carriers in a central well layer having
typical dimensions on the order of the DeBroglie wavelength of the carriers, having
an energy between the band edges of the two materials, are spatially confined by
the absence of allowed energy states in the barrier layers. Consequently, one or
more quantized levels are formed within the well. Quantum wells are important
optical systems, and are also used to form channels for in-plane 2D conduction.
In contrast, in the double barrier heterostructure (DBH), shown in Figure 2.3(b),
well states are only partially confined to the well, and as such are only quasi-2D,

so long as the barriers are thin enough for the carrier wavefunctions to commu-

x x

a) Type I Interface b) Type I Graded
Junction

A A

> >

x x

c) Type II Broken-gap d) Type Il Staggered
Interface Interface

Figure 2.2: The possible types of heterojunction band alignments under flat band
conditions. The Type I and staggered Type II junctions are common interface
alignments. The InAs/GaSb system is the unique example among the common
semiconductor systems of the broken-gap Type II junction. As a reminder that all

heterojunctions need not be abrupt, the graded Type I junction is shown as well.

nicate with the electrode material.? Perpendicular transport through a double
barrier heterostructure involves quantum mechanical tunneling of carriers from
the emitter electrode, through quasi-bound well states, to the collector. Resonant
tunneling diodes (RTD’s) rely upon this heterojunction design to preferentially
allow the conduction of carriers at the quasi-bound energies. The superlattice, as
demonstrated in Figure 2.3, is a heterosystem in which layers of two materials are
symmetrically alternated, defining a long-range, artificial periodicity to the device.
This additional periodicity (superperiodicity) results in the creation of minibands,
much as the lattice periodicity leads to bulk band structure. Superlattices are
important optical systems, as the absorption characteristics may be finely tuned
to meet specific operational requirements. While Figure 2.3 shows these devices
for Type I heterojunctions, similar devices are possible with Type II offsets. The
overlap of conduction and valence band states in Type II broken-gap structures

excludes the possibility of a truly confining quantum well in this system.

2.2.3. Electronic Properties

The proper understanding of the quantum mechanical behavior of carriers in het-
erostructures is essential to the discussion of the electrical properties of these de-
vices. At the simplest level, the variation of band edges from material to ma-
terial provide classic piecewise linear quantum mechanics problems. Thus, the
physics is very accessible. However, additional factors, such as changes in effective
mass, multi-dimensional effects, and realistic band structure, provide complica-

tions which make the field far richer than any textbook model. As a result, we

3In the limit that the barriers are thick enough that essentially none of the well wavefunction
leaks out to the electrodes, we recover the simple quantum well. Depending upon the barrier
height, typically several hundreds of Angstroms of barrier material must exist, at a minimum,

for such a distinction.

a) Quantum Well

> >

x x

b) Double Barrier c) Superlattice
Heterostructure

Figure 2.3: The three major classes of heterostructure devices, showing the band
edges as a function of position. The band gap regions of the two constituent
materials are shaded. Quantized states in the quantum well, (a), are entirely
confined within the well material, while coupling to the electrode regions of the
double barrier heterostructure, (b), and between wells of a superlattice, (c), leads

to partially-confined quasi-bound states.

first discuss the basics of quantum transport in these systems for Type I unipolar
devices, where the physics is more straightforward, and then contrast Type II in-
terband device transport. We will furthermore focus our discussions on resonant
tunneling diodes. The physics of quantum wells will be presented as a motivation
for the development of RTD’s, but superlattices and other heterojunction devices

will not be discussed.

Type I Unipolar Devices

To begin, we will treat Type I unipolar devices with a simple, single-band model.
For such, the energy dispersion of the bulk materials is assumed to be quadratic,

and described by a single effective mass parameter, such that,

h2
E(ke, ky, kz) = Se (he + ko + k?), (2.1)

where m* is the effective mass. For heterostructures, it is assumed that the bulk
effective mass parameters are equally valid for thin layers of material, before the
addition of quantization. We treat a quantum well as a central layer of width L
lodged between two semi-infinite slabs of cladding barrier material with a positive
conduction band edge discontinuity of AF from the & = 0 band edge of the well.
For 0 < E, < AE, we can calculate the quantized energies in a quantum well, and
the wavefunctions, by in addition assuming that the total carrier wavefunction is
separable into a Bloch component with the periodicity of the underlying lattice,

and a smoothly varying envelope function[16, 17] such that,
y=) Kit ui (r) by(x), (2.2)

where kj is the in-plane wavevector, u(r) is the Bloch function of the mate-
rial indexed by 7, and ¢,,(z) is the envelope function for the nth quantized state

wavefunction along the growth direction, z. Due to the quantization in z, the

wavevector k, is replaced with the quantum index n. To further simplify the
problem, it is usually also assumed that the Bloch periodic functions for the ma-
terials of a heterostructure are identical. While this approximation is reasonable,
given the similar symmetry properties and material characteristics of semiconduc-
tor compounds that are typically grown together, if the problem is solved with
more complete generality, deviations from this approximation are observed [18].
Under these assumptions, the Hamiltonian acting on such a wavefunction is itself
separable into a Bloch function component dealing with the wavefunction varia-
tions over a single lattice site, and therefore containing the physics at an atomic
level, and the piecewise linear envelope function Hamiltonian incorporating the

large-scale variations due to the heterostructure,

A? Ody (a) _

where V(x) now represents the potential energy variation along the growth axis
due to the band edge discontinuities of the heterostructure,
V(x) =AE, |x| > L/2 (Barrier)
= 0, |x| < L/2 (Well),

(2.4)

for the quantum well with k; = 0. Because we are interested in the macroscopic
quantum properties of the entire heterosystem, the Bloch components and their
Hamiltonian are typically ignored. The envelope function Hamiltonian is itself
dependent upon position along the growth axis, both through V(z) and the effec-
tive masses, and thus changes at a heterointerface. Aside from this complexity,
the problem has reduced to a textbook piecewise linear potential problem, and
the solution method is identical. Because of the change in effective mass between
the different materials, the boundary conditions set by probability current density

conservation are,

b¢ (int) = be (xint); and, (2.5)

1 af

mi On (int) (2.6)

1 en
mi, Ox

where D and R refer to the left and right component materials of the interface at
position 2;,;. Thus, as in the classic textbook problem, we expect bound states
with sinusoidal wavefunction behavior in the well and exponential decay in the
barrier materials, as shown in Figure 2.4(a). The carrier described by the wave-
function is completely localized in the well; it cannot classically exist outside the
well, and since the cladding layers are semi-infinite, it cannot tunnel out quantum
mechanically. Following the textbook solution [19], the stationary state wavefunc-
tions must be of even or odd parity, having a sinusoidal period of 1/k,, where
Ak, = 2miy En; my is the bulk effective carrier mass of the well material, and,

similarly, m3 is the bulk effective mass of the barrier. The decay constant in the

barriers is given by « = (1/h) V 2m,(ALE — E,,). Even solutions exist for,

(k, /m#,) tan(k, L/2) = «/m%3, (2.7)
and odd solutions for,

(k /myy) cot(k, L/2) = «/m. (2.8)

Solutions to these boundary-imposed equations yield valid values for k, , and there-
fore the eigenvalues E,,. Unlike the textbook problem, however, quantum wells
fabricated from semiconductors are three dimensional, and the potential profile

along the x-axis is dependent upon the parallel crystal momentum, ky,

nk
V(c) =AE+ at |x| > L/2 (Barrier)
2A? B (2.9)
= may’ jz] < L/2 (Well),

and as a result, the barrier seen by carriers at arbitrary parallel momentum is
given by,

Ak? ( me
AB =Agp-—l(™5 _1). (2.10)
mz \ mt

Using this AE" in Equations 2.7 and 2.8 yields the equivalent k)-dependent so-
lutions. Because the effective masses for large band gap materials are typically
larger than those with smaller band gaps, the effect of non-zero ky is to reduce the
quantization of the carriers with respect to the ky = 0 problem. However, as the
effective mass approximation is only valid for small wavevectors, this model can
only treat small perturbations about k = 0. A further complication exists for the
quantization of holes. Because the valence band exhibits J = 3 angular momentum
symmetry, a simple one-band model is usually insufficient to adequately describe
the quantum behavior of hole states. As discussed in Appendix B, the dispersion
about k = 0 must be expanded as a function of the J,, Jy, and J, angular mo-
mentum operator matrices, following the model of Luttinger [20]. This additional
complication leads to multiple hole subbands, with mixing between the states at
finite ky. The resulting problem is not analytical, in general, and Equation 2.3
must be solved numerically as a piecewise linear differential equation.

In contrast to the quantum well, the double barrier heterostructure does not
have wholly confined states, as shown in Figure 2.4. A carrier in the well has a
wavefunction which penetrates through the narrow barrier regions into the semi-
infinite electrodes. As a consequence of these open boundary conditions, the prob-
lem is no longer a discrete eigenvalue and eigenvector problem for the quantized
energies and wavefunctions, respectively. The behavior of the DBH must be con-
sidered in the context of perpendicular quantum transport though the active region
containing the well and barriers. Energy- and momentum-dependent transmission
coefficients for carriers incident from the emitter electrode may be calculated using
standard quantum mechanics. We motivate the standard calculations to distin-
guish concepts and physical behavior that will be important in further discussions
in this thesis, but leave the details to the reference texts [19, 10]. We define our

geometry and energies in a manner similar to the discussion of quantum wells, as

YN.

a) Quantum Well

—_'

aC et LAN.

b) Double Barrier Heterostructure
(Resonant Tunneling Diode)

Figure 2.4: Comparison of bound state wave functions in a quantum well and
double barrier heterostructure. The eigenstates in a quantum well are fully con-
fined in the well region, with only partial leakage of the wave function into the
semi-infinite barrier layers. In contrast, the quasi-bound well states of a DBH
communicate quantum mechanically with the unbound electrode states through

the thin barriers.

shown in Figure 2.5, and initially consider the problem under flat band conditions.
The electrode and well band edges are assumed to be at E = 0, and the barrier
band edges AF greater. The well is taken to be of width W and the barriers
each width a. As for the quantum well, we define the oscillation wavevector and

decay constant, respectively, as Ak, = ,/2m%E, and hk = (2m%(AB —F,),

where m% is now the effective mass in both the well and the electrodes, and m}
is the effective mass in the barriers. Considering just the first heterointerface, we
see that the wavefunction on either side may be written as a sum of positive and
negative exponentials. Where the carrier energy is greater than the band edge of
the material these are complex exponentials, and therefore traveling waves incident
upon, and reflected from, the DBH, respectively. In the barrier, for carrier ener-
gies in the band gap, the exponentials are real and consist of growing and decaying
states. As the barrier is of finite thickness, the growing exponential states cannot
be neglected, as was the case for the quantum well. Consequently, we have at the

first interface,

d(x) = Ae*+* + Be~*1* (Electrode

(z) ( ) (2.11)
= Ce-** + De**, (Barrier).

Matching these at the boundary z = —a/2, using Equations 2.6, the coefficients

of the exponentials may be related as,

inms Ka/2+ik, a IRM, —Ka/Qtikya
A\ 1 (1+ Ste) « poe (1- Ee) « ™ (2.12)
B 2 1 iKm’s eha/2-ikya 1+ inme e7a/2-ik a D ,
kims kims

If this matrix is inverted, we can express the rightmost wavefunction coefficients

as a function of the leftmost,

C A
= T ; (2.13)
D B
where T is the transfer matrix linking the wavefunction across the interface. In

general, a transfer matrix is necessary to relate the wavefunction across any change

in band edge energy, as indicated by the arrows in Figure 2.5. Thus, in a realistic

device with band bending, the full transfer matrix of the active region is given by,
T=[[T:, (2.14)

where each T; is a transfer matrix linking adjacent lattice sites at position i. Typi-
cally this product must be extend over not only the active quantum heterostructure
region, but the regions of the electrodes for which there is significant band bend-
ing. Thus, even for an active region a couple hundred A’s in width, typically a
half micron or more of material must be included in the calculation. Because the
transfer matrix consists of exponentials of the product of very large (k, and «)
and very small (a) numbers, the transfer matrix method, while easy to describe,
and intuitive, is numerically unstable and seldom used for device modeling. Using
this transfer matrix method analytically, however, it can be shown [10] that the

overall transmission coefficient for carriers incident on a symmetric, flat band Type

I DBH is given by,
T2

Trot = jit [r[2e2&E +e) 2?

(2.15)

where 7: is the total transmission coefficient for the DBH, T is the transmission

coefficient for one of the barriers, |r|? = R is the complex reflection coefficient for
a single barrier, DL is the well width, and w = 7 —ka, where w is the transcendental

solution to the following:

*” *
km KM,

tan y = ; ( tanh Ka. (2.16)

KM kms
The most important and relevant fact to be gleaned from Equation 2.15 is its
behavior as a function of incident energy, shown in Figure 2.6 as calculated by the
transfer matrix method for a flat band InAs/AlSb DBH with 36 A barriers and a
90 A well. For most values of incident energy, the total transmission coefficient is

roughly the square of that for a single barrier; the incident carrier is transmitted

as if the barriers and well were absent and a single barrier of twice the width (2a)
were instead present. At certain energies, however, the transmission coefficient
is unity, and thus the barriers become completely transparent. This condition
occurs when the exponential in the denominator of the expression for T;,; is —1,
or 2(kL +w) = (2n+1)m. At these energies, the carriers resonate with quasi-
bound states in the well, and the two electrodes are completely coupled through
a resonant tunneling process. In the limit as the barrier widths become infinitely
thick, the carriers in the well are completely confined, and the problem reduces
to that of the quantum well, with the resonant quasi-bound energies approaching
those of the true eigenvalues of an equivalent quantum well.

The addition of k) to the quantum mechanical treatment of a DBH has an effect
similar to that described for quantum wells. Because it is a transport problem,
however, the additional restriction that k, be conserved must be added. Thus, for
resonant tunneling, not only must the incident energy match that of the quasi-
bound state, but the parallel wavevectors of the incident electrode state and the
final well state into which it tunnels must be identical, and similarly for tunneling
from the well to the collector. In addition, the width of the transmission resonances
are related to the time required for tunneling to occur through the uncertainty
principle, 67 6E = h, where 67 is the tunneling time, and 6F is the width of
the transmission resonance at Ti., = 0.5. Hole tunneling is again similar, albeit
more complex, and involves additional conservation criteria for orbital angular
momentum. Coupling between angular momentum states at non-zero ki, however,
alleviates this restriction through the mixing of hole angular momentum states.

The current-voltage (I-V) characteristic is the single-most important transport
property of an RTD. Applying a bias between the emitter and collector of a DBH
RTD creates a non-equilibrium carrier distribution in which emitter states are of

greater energy than those in the collector. As a result, current flows between the

-a/2 0 a/2

Figure 2.5: Details of the transfer matrix calculation for the transmission coefficient
of a double barrier RTD. Incident and reflected states are shown in the emitter
region, and exponentially growing and decaying solutions within the first barrier.
The quasi-bound states leading to enhanced resonant transport across the device
are shown in the well. At each change in potential energy seen by a tunneling
electron, a transfer matrix relates the amplitudes of the wavefunction on either
side of the interface, as shown. The transfer matrix for transport across the entire

device is the product of the individual transfer matrices.

B10 F

(o)

5 10

q 10

10° L I ry L L I i 1 ri

0.00 0.10 0.20 0.30 0.40 0.50

Incident Energy (eV)

Figure 2.6: Transmission coefficient of an InAs/AISb double barrier structure hav-
ing 36 A barriers and a 90 A well, calculated as a function of incident energy by
the transfer matrix method. The transmission coefficient of the entire device is
roughly the square of the transmission coefficient, Tsingie, of a single barrier at all
but several isolated energies where the barriers are transparent to the incident car-
rier. The results are shown for both zero and non-zero ky. The non-zero ky (= Aky)

transmission curve is essentially the k, = 0 curve shifted in energy by nh Aki /2m*,.

electrodes as allowed through the limiting active quantum region. As was evident
in the flat band, zero-bias analysis of transmission through a Type I unipolar DBH
(c.f. Figure 2.6), current transport through such a device has a strong dependence
upon the energies of the carriers relative to the quasi-bound states (QBS’s) in the
well. Consequently, as the bias is increased across the RTD, the magnitude of the
current ebbs and flows with the degree to which resonant quantum transmission
through the well states is allowed. In Type I unipolar devices, such as GaAs/ AlAs
or InAs/AlSb double barrier structures, the net current through the structure
therefore exhibits a peak at those voltages which cause the preponderance of in-
cident carriers to align in energy with a quasi-bound state in the well. At biases
greater than this peak voltage, the transmission coefficient drops precipitously, and
the current likewise reaches a minimum. The region over which the current drops
results in a negative differential resistance (NDR), which is an inherent property of
resonant tunneling devices. A typical I-V curve is shown at the top of Figure 2.7.
The ratio of peak and valley currents, or peak-to-valley ratio (PVR), is the single
figure of merit quoted to summarize the NDR behavior of a given quantum device.
Much of the current transmission in the valley region is experimentally linked to
non-resonant, inelastic transport across the structure. Consequently, the PVR is
indicative of the degree to which inelastic processes compete with resonant quan-
tum transport to yield the net current. Other parameters of the I-V characteristic
which are of importance for the technological application of RTD’s are the linearity

of the onset regime,’ the absolute magnitude of the peak current density, and the

*As indicated in Figure 2.7, the onset of most Type I RTD I-V characteristics is, in fact,
exponential, due to the Fermi distribution of emitter carriers. Such behavior should be contrasted
with the onset of a resonant interband tunneling (RIT) diode, discussed in Section 2.3 and the
subject of much of the experimental work in this thesis. Such a distinction is of great importance

in the operation of some classes of systems based upon RTD’s.

width of the NDR region.

The theoretical understanding of I-V characteristics in RTD’s may be developed
on the basis of early work by Tsu and Esaki [21]. The current density can clearly
be seen to depend upon two quantities: the transmission probability of a device
and the population and distribution of carriers in the electrodes. Each of these,
in turn, is dependent upon F,, the energy due to kinetic motion along the growth
direction. The current from emitter to collector, jzc, is given by,

jec = € >, (hk,/m’s) Trc(Ex) Na(Ex), (2.17)
ky
where fk, /mz is the semi-classical carrier velocity in the emitter, Tzc(E,) is the
emitter-to-collector transmission coefficient, and Ng(£,) is the carrier population
in the emitter at energy H,. Similarly, the current from emitter to collector, which
must in general be included, is,
Re
For a spatially symmetric device, T(E.) = Trc(Ez) = Tes(E,), and the total

current is given by,

jn = Jno ~ jon = *Y TBs) (Na(Bs) —No(Ee))- (219)

Fermi statistics yields expressions for the electrode populations functions,

kT'm* Bp- Fk
B(E;) he n|1+ exp ‘i , (2.20)
and,
kT'm* Ep — E,—eV
No( Bs) = S28 in| +-exp (“$a .
o( Ez) - In |1 + exp 7 ; (2.21)

where V is the applied bias. Given accurate transmission coefficients, Equa-
tion 2.19 generally predicts peak voltages and currents which, when adjusted for

series resistance, agree reasonably well with experiment. Equation 2.19 does not

include non-resonant processes, and thus grossly underestimates the valley cur-
rent. This theory also makes the assumption that the transmission depends only
upon the perpendicular motion, and thus all states with a given HE, are treated
identically, despite any differences in kj. While Figure 2.6 shows that the trans-
mission characteristics of carriers with identical total energies, but differing parallel
wavevectors, are quite distinct, the assumption in the Tsu and Esaki theory of I-V
characteristics is that the transmission is identical if the perpendicular energies are
equal. The validity of this approximation is evident in Figure 2.6 as the ky, = 0
curve can be translated along the z-axis by an amount equal to Ak? /2m*, to lie
roughly on top of the k; = Ak) transmission curve. This assumption is valid for

Type I unipolar conduction band tunneling only. Since additional kj increases the

coupling of m; = 5 and m; = 3 states, the simple Tsu/Esaki assumption is not
valid for hole tunneling and Type II interband tunneling involving conduction and
valence band states.

The evolution of a Type I unipolar I-V curve is best developed through an
understanding of the process via the simultaneous conservation of energy and mo-
mentum. Figure 2.7 shows a sample Type I unipolar (intraband, to contrast it
from the later discussions of InAs/AISb/GaSb interband devices) I-V characteris-
tic. Three points along it are labeled, and the emitter and well in-plane dispersions
at these three voltages are detailed in the lower panels. The degree of carrier oc-
cupation from Fermi statistics is shown via shading. It is easiest to first consider
3D-2D tunneling involving a continuum of states in the emitter, where the lowest
such state, corresponding to the band edge, is shown in the figure. In such a situ-
ation, the number of states available for resonant, elastic tunneling in which both
energy and parallel wavevector are conserved increases exponentially with applied

bias. At voltages less than the peak voltage, the well state band edge (at ky, = 0)

lies higher in energy than the emitter. As the well is brought lower in energy by

Intraband f ab oe
Tunneling

Current- I

Voltage

Characteristics .

a) V < Vex b) V = Vin
Well Well and Emitter

E J,

Cc) V > View
E Emitter
Ey
/ Well
Ye

—_—_—

Figure 2.7: The development of a Type I unipolar device I-V characteristic. Shown
in the top panel is a representative I-V curve, and beneath, the emitter and well

band alignments at each of the three indicated voltages.

the potential difference, it approaches the Fermi level, Ey, and the carrier occu-
pation in the Fermi tail changes as the exponential of the difference between the
band edge energy and the Fermi energy. Thus the exponential onset characteristic
of intraband tunneling. The peak in the I-V curve must occur when the well and
emitter dispersions overlap entirely, and the greatest number of carriers are acces-
sible for elastic transport. At voltages greater than V,-az, the well lies below the
emitter continuum at all k) and there are no states available to elastically tunnel.
As mentioned earlier, however, inelastic process contribute greatly to the current
in this regime. At very low temperatures, and also in the case of 2D-2D tunneling
where there is a notch state in the emitter accumulation region, inelastic processes
also dominate for V < Vyeax (panel (a) in the figure). However, the availability
of carriers to inelastically tunnel is also limited by the Fermi distribution, and the

onset is nonetheless exponential.

2.3 The InAs/AISb/GaSb Material System

The InAs/AlSb/GaSb material system provides an unique family of semiconductor
compounds for the study of quantum heterostructure science and the potential for
heterostructure-based technologies. Figure 2.8 shows the band gaps of all three
materials, with the conduction and valence bands indicated, and their relative en-
ergy alignments, all at room temperature. The InAs/AISb/GaSb material system
exhibits all three band offset possibilities: Type I (InAs/AISb), staggered Type II
(GaSb/AISb), and broken-gap Type II (InAs/GaSb). It is this latter alignment
which makes the system unique, and the subject of a long history of optical and
electrical studies. As indicated in the figure, the valence band edge of GaSb lies
0.15 eV above the conduction band edge of InAs [22]. This band alignment re-

sults in peculiar heterostructure transport involving conduction band-like states

in InAs and valence band-like states in GaSb. Although the unique properties
of InAs/GaSb heterostructures have been known for some time [23], only recently
were tunnel diodes based upon this material family proposed [24], and subsequently
exploited in the creation of a new class of quantum transport device [25, 26]. While
the nature of the InAs/GaSb band offset is unique and distinctive, the viability
of this system for heterostructure physics and technology is greatly increased by
the availability of a nearly lattice-matched barrier material (AlSb) for both holes
in GaSb and electrons in InAs. While there is more strain in this system (the
lattice constant of GaSb is roughly 0.5% greater than that of InAs, and the AlSb
lattice constant is about 1.0% larger still than GaSb) than the near-perfect match
of GaAs and AlAs, this trio combine to create one of only a few possible III-V
binary compound heterostructure families. One potential drawback is the lack
of confinement symmetry provided for by the AlSb barrier. Fortunately, AlSb
does provide a barrier for the majority carrier-type in each material, and provides
greater confinement for the low mass InAs electrons. However, the valence band
of InAs can couple quite strongly to that of GaSb, increasing quantum-mechanical
communication between the two across an AlSb barrier, and providing demonstra-
ble effects on the transport of even InAs/AISb Type I unipolar devices [26]. It has
been proposed [27] that strained In, Al,_,As barrier layers may be used to increase
confinement in the valence band of these structures.

Figure 2.9 details the interband tunneling equivalent of the double barrier struc-
ture discussed in Section 2.2. Shown is the p-well (so named due to the valence
band nature of the GaSb well) resonant interband tunneling (RIT) diode, with
InAs electrodes, AlSb barriers, and a GaSb well. Transport through this device
involves resonant tunneling of electrons from the InAs emitter, through unoccu-
pied electron states in the subbands of the GaSb well, and subsequently back into

the conduction band of the collector. The converse device, with an InAs well

0.15 eV

Figure 2.8: Band alignments in the InAs/GaSb/AlSb material system, with the

band gap of each material shaded for clarity.

and GaSb electrodes, has been studied as well. The availability of three mate-
rials, and the variety of band offsets between them, has produced more than 10
unique heterostructure configurations in this material system, all of which have
been fabricated and studied [26]. The unique properties of this material system
and interband tunneling devices will be presented in reference to the p-well RIT
alone, however, for simplicity. The transport properties of this device, and the ap-
plications which result, were the primary motivation behind the research contained

in this thesis.

2.3.1 Transport Properties

As indicated previously, transport in interband devices such as the RIT differs sig-

nificantly from that of Type I unipolar devices. The added complexity of transport

InAs AlSb GaSb

AlSb InAs

2K
S oL
YL
>>
oy
0 |

| VB

Figure 2.9: Band diagram of a p-well RIT, showing the conduction and valence

band edges as a function of position. Quantum transport through this device

involves electrons from the emitter conduction band tunneling into empty valence

band states in the well, and then out into the conduction band of the collector.

in these structures has both theoretical and experimental implications. Theoret-
ically, quantum transport calculations of interband systems require sophisticated
multiband treatments; 1- or 2-band models are not sufficient to explain the physics
of these devices. This criterion places further requirements on the stability and
efficiency of numerical methods for the solutions of these problems. Transfer ma-
trices are not sufficiently accurate to perform the computations correctly. Thus,
better computational techniques are a requirement for the proper simulation of
interband tunneling. Experimentally, current-voltage characteristics do not offer
a direct measure of the properties of states in the well. Typically, only one I-V
peak is observed, even at low temperatures in devices with multiple well subbands.
The opposite dispersions of the electrode and well states allow resonant tunneling
through multiple well states at a single voltage, convoluting the contributions of
the individual subbands, and leading to a single I-V peak.

Figure 2.10 shows the transmission coefficients as a function of energy for a RIT
having 36 A AlSb barriers and a 90 A GaSb well, at both zero and non-zero ky.
These calculations were made with an efficient, stable model developed by Ting et
al. to accurately model interband devices [28]. The device parameters correspond
identically to those of the InAs/AlSb DBH for which calculated transmission coef-
ficients are shown in Figure 2.6, except that the InAs well has been replaced with
an identical thickness of GaSb. Note that the RIT transmission characteristics
differ greatly from those of a simple unipolar double barrier device, and thus the
simple transfer matrix model developed earlier is insufficient to explain the inter-
band problem. At zero parallel wavevector, two peaks exist in the transmission
coefficient, one very close to the conduction band edge of InAs (the energy origin),
and another near the GaSb valence band edge (at roughly 0.15 eV). Between these
peaks, the transmission coefficient for the RIT is more than three orders of magni-

tude greater than that of the InAs DBH. At incident energies exceeding the GaSb

valence band edge (E' > 0.15 eV), the transmission coefficient drops precipitously
(again roughly three orders of magnitude from 0.15 eV to 0.20 eV). This effect
has been termed “band gap blocking” and represents a fundamental distinction
between interband tunneling and Type I unipolar transport. At these energies,
the incident carrier is in the energy gaps of both the GaSb and AlSb. Thus, the
wavefunction is attenuated greatly, and little transmission through the interband
device can occur. This band gap blocking phenomenon is responsible for the large
peak-to-valley ratios seen in interband devices (the RIT in particular). PVR’s as
high as 21:1 at room temperature and 88:1 at 77K have been recorded for p-well
RIT’s [26].

The additional complexities of heavy hole states is seen in the figure as well, for
k, #0. Here, the resonances take on a characteristic Fano-like lineshape, with a
minimum closely associated with every maximum. As is true for intraband devices
as well, heavy hole states are purely \3 + ) only at ky = 0 (where they cannot
couple to conduction band states due to the conservation of angular momentum
projection, m;). At non-zero parallel wavevector, the light (j = 3,m, = ++) and
heavy (j = 2m; = +3) hole states intermix due to their closeness in energy,
and similar symmetry. As a result, transmission through heavy hole-like states
is permissible at finite k). While these resonances are typically more narrow and
do not contribute as much to the transmitted current, band warping can lead to
conduction band-like behavior in heavy hole states, which greatly increases their
coupling to the emitter states, as shown in Chapter 3.

The development of the I-V characteristic of a RIT diode is markedly different
for an interband device as well. Figure 2.11 shows a sample I-V curve for a p-
well RIT as well as the alignment of the electrode and well subband dispersions
at the biases indicated in the top panel. Electron occupation at a given energy

is indicated by the shading of the region above the emitter curve. First consider

10 :
a) 10° HB i 71
3) \ i
(oe) ry
0 . f
iS) i
5 10 - | 4
EH
—— k,=0.0 nm
staseseeenenn k, = 0.2 nm
10° 1 i rn i 1 | rn
0.00 0.05 0.10 0.15 0.20

Incident Energy (eV)

Figure 2.10: Calculated RIT transmission coefficients for a device with 36 A bar-
riers and a 90 A well. The energy dependence of the transmission probabilities
for an incident carrier are plotted for zero and non-zero ky). The resulting curves
are significantly more complex than the results for an identical InAs/AISb double
barrier device. The results were attained through the use of a sophisticated 8-band

model attributable to Ting et al [28].

p-well Interband f 2 b c
Tunneling |
Current- I
Voltage
Characteristics .
a) V < Vix b) V = Vix

Figure 2.11: The development of an interband p-well RIT I-V characteristic.
Shown in the top panel is a representative I-V curve, and beneath, the emitter

and well band alignments at each of the three indicated voltages.

the case of 3D-2D tunneling where there is continuum of states, of which only the
lowest energy dispersion is indicated in the figure. The first difference that should
be noted is the ohmic onset to the interband I-V, as compared to the exponential
turn-on of the Type I unipolar I-V curve shown in Figure 2.7. Unlike the Type I
unipolar DBH, the RIT does not rely on the thermal excitation of carriers for low
voltage transport. As shown in Figure 2.11a, there is overlap between the emitter
and well states such that elastic tunneling, for which both energy and parallel
wavevector are conserved, is allowed at voltages less than V,eax. For some of the
well states, this condition may even hold at zero bias. As a result, conduction
through RIT diodes at low biases is not limited to the exponential thermal Fermi
distribution tail. Another difference, which becomes a significant complication
for certain experiments with RIT diodes, is that the current peak does not occur
when the k = 0 band edges of the two subbands are of equal energy. Clearly,
in that limit, the current is nearly zero as there are very few incident carriers to
contribute to elastic transport; thus this voltage is somewhere in the valley region.
The peak is instead found at some intermediate voltage, as shown in Figure 2.11b.
A simple model [29] of interband I-V curves postulates that the resonant current
at a voltage V < Vea, should be proportional to the product of a semiclassical
carrier velocity, v «x V, and the number of states available for elastic transport.
Such a simple model provides reasonable two-parameter fits to many interband
I-V curves [29]. From Figure 2.11, it is clear that these two factors have opposite
differential dependencies on the applied voltage; while the carrier velocity is linear
in applied voltage, the number of carriers that can elastically tunnel is reduced as V
is increased. Analytically, this latter quantity, in the effective mass approximation,
is « (Vo — V)~*/? for 3D emitter states, or «x (Vj — V)~!/? for 2D emitter states,
where Vo is a constant depending on the bulk effective mass. While the peak

voltage may be determined in principle from this model, it is not sophisticated

enough to bring full consideration to the quantum aspects of the problem, such
as variations in transmission coefficient. Therefore, the exact alignment of the
subband dispersions at the peak voltage is not a priori known, and can at best
only be estimated.

The technical advantages that the InAs/AlSb/GaSb system offer stem from
these unique transport properties. Due to band blocking in the valley region,
both elastic and inelastic transport through the device at these voltages is severely
curtailed, leading to PVR’s at 300 K which rival Type I unipolar device param-
eters at cryogenic temperatures. Additionally, the onset of these devices is not
exponentially dependent upon temperature. Interband devices are thus far more
likely to provide convenient room temperature operation for solutions to real-world
problems than are GaAs/AIAs or other Type I devices. This fact is of great conse-
quence; as interband resonant tunneling devices have a greater likelihood of making
positive economic contributions, the understanding of their transport properties,
and ensuing optimizations, have significance beyond academic study. The existence

of many potential applications for these devices emphasizes their importance.

2.3.2 Technological Applications

Because of their unique band alignments, and the resulting performance improve-
ment at room temperature, interband quantum effect devices fabricated in the
InAs/AlSb/GaSb material system have been proposed for use in a wide variety of
technologies. One advantage of devices in this system is their intrinsic switching
speed. An InAs/AISb double barrier device grown by the McGill group at Caltech
has been observed to oscillate at frequencies as high as 712 GHz at room tem-
perature, representing the current record for a solid-state electronic oscillator [30].
Extrapolated device performance yielded an estimated maximum oscillation fre-

quency of 1.24 THz. The measured power density of this device at 360 GHz, 90 W

cm~?, is fifty times greater than that generated by GaAs/AIAs diodes at similar
frequencies. While this diode was a purely Type I unipolar DBH, it nonethe-
less demonstrates the technological advantages enjoyed in this material system.
A Stark effect transistor, consisting of an InAs collector, a GaSb emitter, and a
GaSb base, has been demonstrated with room temperature current gains in excess
of 50 [31]. This unique device was first proposed in the AlAs/GaAs system [32], but
later only demonstrated transistor action at cryogenic temperatures (7 K) [33]; the
performance advantages of the InAs/AISb/GaSb material system therefore take the
Stark effect device from the position of scientific curiosity to that of real technical
feasibility unattainable in Type I systems.

The RIT diode has also provided large performance benefits in computational
systems based upon neural paradigms. A single, vertically-integrated RIT diode
replaces 16 silicon CMOS transistors for the generation of a resistive fuse-like I-V
characteristic for the extraction of spatial discontinuities in artificial retinas [34].
It does so with lower power, far less lithographic real estate, and without the
common mode difficulties of active devices. RIT diodes have also been used by
Levy et al. to store analog weights in feedforward artificial neural networks [35].
A three-layer prototype using RIT’s provided room temperature computation of
the fundamental logic functions of XOR, AND, OR, and NOT. Clearly, interband
tunneling diodes have demonstrated competitive solutions to current and future
computational needs with room temperature, quantum effect-based operation.

Given the success of interband devices in other arenas, it was believed very
early in the magnetotunneling investigations that RIT diodes might make highly
sensitive, tunable detectors of magnetic fields for such technological applications as
magnetic media read sensors for digital data storage. Many empirical issues con-
tributed to this presumption. First, semiconductor heterostructure devices may

be engineered for precise control of the quantized energy states, and the energies

of these states, which fundamentally depend upon the magnitude of external mag-
netic fields, in turn determine the nature of the diode current-voltage characteris-
tic. Furthermore, the interband nature of the RIT diode intuitively appears more
sensitive to changes in magnetic fields, as conduction band states are raised, and
valence band states lowered, by external fields. Since these devices are also capa-
ble of duplicating the cryogenic results of other systems at technologically-realistic
temperatures, the RIT appeared to be ideal as a room temperature B-field detec-
tor. Several realities, however, became evident that have lessened the applicability
of RIT’s as a technological solution. First, the anomalous response of RIT’s to per-
pendicular fields (c.f. Chapter 3), all but eliminates them for magnetic sensing at
technologically-realistic fields. For room temperature operation, the perpendicular
field orientation is necessary to induce any appreciable change in the I-V curve, but
the critical field phenomenon discussed in Chapter 3 eliminates this orientation as
a viable detection geometry. The peculiarities of transport in this system there-
fore may hinder, as well as improve, device performance in certain technological
applications. In addition to the intrinsic limitations of the RIT diodes, the develop-
ment of reproducible devices utilizing giant magnetoresistance (GMR) phenomena
have greatly raised the metric for device performance, and further weakened the
case for magnetic field detectors based upon current InAs/AISb/GaSb technology.
However, as nanostructures with inherent 0D-like carrier behavior are developed
and perfected, the size and potential detectivity advantages of antimonide-based

resonant tunneling magnetic field sensors may ultimately lead to their adoption.

47
2.4 Magnetic Fields in Bulk Semiconductors

2.4.1 Landau Levels

The derivation of the effect of an external magnetic field upon the quantum me-
chanics of charge carriers in semiconductor heterostructures begins with the mod-
ification of the single-particle Hamiltonian to include an arbitrary constant field.
The field enters the Hamiltonian through both potential and kinetic energy effects.
The former arise from the interaction between the field and the carrier’s intrinsic
angular momentum (spin). An additional term, gug a: B , is added to the Hamilto-
nian in the effective mass approximation, where 41g is the Bohr magneton, and g is
the gyromagnetic ratio relating the carrier spin to the intrinsic magnetic moment.
As with electric fields, a magnetic field does not enter the kinetic portion of the
Hamiltonian directly, but rather is incorporated through a related vector potential,
A, defined by B= 7 x A. This definition of A requires that it be a unique vector
field only to within the divergence of a scalar function, v f. The specification of
this function f yields a particular choice of gauge; the physics of the problem are
the same in all gauges, but the ease of solution depends upon a wise selection of
A. The use of the Landau gauge, A= —By 2, or, equivalently, A=Br y, fora
constant field B = Bz, simplifies the kinetics, reducing the field-dependence to
a one dimensional eigenproblem. In general, for all A, the kinetic portion of the
Hamiltonian transforms as H(p) —+ H(g+ eA), where @ is the momentum opera-
tor, and the negative charge of the electron is implicitly included in the expression.
Thus, for a charge carrier in the effective mass approximation, the Hamiltonian in

the presence of an external field becomes,

n= ia (B+ed)' + 92

(2.22)

m*?

where m* is the carrier effective mass, and g* = m*g/2m, is the effective Landé
factor. To determine the change in kinetic behavior, we will ignore the spin-
related term for the remainder of the calculation. This term is a constant for
carriers of a given spin state, and has little effect on the quantum mechanics of
the problem. The magnitude of this term, proportional to m*/mz,, is negligible in

many semiconductors. Considering simply the kinetic portion of the Hamiltonian,

and choosing the gauge A= —By@, the Schrodinger equation for the wavefunction
w becomes, ;
mn (0 0 6] eBy
— t+ —Y+ —2- 8 = 2.23
2m* (2+ ay! Oz th :) v= eb, (2.23)
or,

1 o{ & oP 0? a a),
7 fn (i + 5g + gga) ~ 2heBug, — (eBy) | v = ev. (2.24)

Taking the anzatz that ~ is of the form, (7) = e~***+*:4) d(y), substitution into

Equation 2.24 yields a differential equation for the unknown function ¢(y),

dp 2m* Ah? (k2 +h?) eB
€ _
dy? hi? 2m m 2m

+ + yk, - — (eB) ¢=0. (2.25)
Defining w, = eB/m* (the cyclotron frequency), y' = y — k,/s?, where s? = eB/h,
and ¢ = € — f’k?/2m*, and substituting in the differential equation for b(y), we
obtain the Schrédinger equation for a quantum harmonic oscillator (QHO) in the
primed variables:

Ww Po 1,
— ors dy? + 5™ wy! =eéd¢. (2.26)

The solutions of this equation are well-known,

d= (I+ 5) fe, 1=0,1,2..., (2.27)

and,

s? 1/4 1
dily) = (=) TEN v'/? Fh (sy) , (2.28)

where H,,(y) is the 1" Hermite polynomial, and s, as defined earlier, is now clearly
an inverse scaling length. The discrete levels indexed by / are called Landau levels,
and are characteristic of the quantum-mechanical transverse kinetic energy in the
presence of a constant magnetic field.

Since the behavior of a semiconductor carrier in a magnetic field has reduced
to a simple, well-known 1D problem, the raising operator, which acts on a state ¢y
to yield a state proportional to ¢),1, may defined in the same manner as a simple

QHO,

1 Dp
Tj = LL fog ky
as su iPy) , (2.29)
_ 1 ky Dy
= c (v =) ibe| , (2.30)
le

where Ak; = p; + eA; is the generalized k-vector in a magnetic field. The lowering
operator, a, is simply the complex conjugate of al. These operators have the

expected result when operating on the Landau level wavefunctions, ¢;(y),

alg) = Vi+1dn41, (2.32)
ag = Vigy-1. (2.33)

The functional form of a and a! emphasize the the fundamental effect of the field on
the quantum mechanics of the carriers; since A modifies the momentum-dependent
term of the Hamiltonian, and therefore gA/ his, in a very real sense, a field- and
position-dependent crystal momentum vector, the essential effect of the field is to
directly modify the allowed states in k-space. Consequently, as developed in Ap-
pendix B, magnetic field effects are best incorporated in rigorous band structure
models with k-space (k - #) methods, rather than real-space (tight binding) tech-

niques. Because the vector potential may be viewed as an entity in k-space, yet

is itself proportional to a position in real-space along a perpendicular axis, there
is an interesting duality to magnetic field problems. Despite the reduction of the
Landau level problem to a single 1D problem in one of the two coordinate vari-
ables perpendicular to the field, the problem is equally well-formulated along the
other transverse axis. This fact is easiest seen in the equivalent gauges, A= Br gy
and A = —By&. The raising and lowering operators in the latter gauge may be

expressed as rotations of those derived earlier,
al —> ial, (2.34)
a — -—ia. (2.35)

Note that under this rotation, the number operator, defined as N = aia, in terms
of which the transverse Hamiltonian may be expressed, is invariant, and thus the
energy spectra are unaltered by the difference in gauge. Thus, while although
expressed as a 1D problem, the magnetic field scrambles k-space in both directions
normal to the field; choosing the symmetric gauge A= —;Byé + 5Bxg is perhaps
the most intuitive way of expressing the general problem, but requires a more
complicated derivation to achieve the same results. Thus, while the use of the 1D
gauges yield results that appear to apply to only a single transverse direction, they
may be extended, with caution, to the entire k-space plane normal to the field.
Therefore, using the derived raising and lowering operators, we can find ex-
pressions for the position, y’, and momentum, Py operators in a magnetic field.

Rewriting Equations 2.31, y’ and p, may be reexpressed in terms of a! and a as,

,_ 1 t
y = Yas (a + a) , (2.36)
and,

p, = Szhs (a! ~a), (2.37)

respectively. It is immediately obvious that (y’) and (p,) are zero, as (¢)|¢y) = dy.

In general, every a! must be matched with an a to yield non-zero matrix elements.

The square average deviations of both variables, however, are non-zero, and given

by,
(Ay')? = (+5) (2.38)
(Api)? = (1+ 5)nPs?, (2.39)

yielding RMS expectations for the mean deviation in y’ and ky = p,/h of \/l + 5 i
and ,/i+ 5 s, respectively. Semi-classically, these values correspond to the orbital
radii in real- and k-space. As the field increases, the orbits get tighter in real space

and simultaneously move to larger values of transverse crystal momentum.

2.4.2 Shubnikov-de Haas Oscillations

The variation of orbital radius in k-space as a function of the applied field results
in periodic variations in the Fermi energy at low temperatures, and subsequent: os-
cillations in the bulk transport properties. With zero magnetic field, the transverse
k states form a periodic 2D reciprocal lattice with regular spacing between each
of the allowed k-values. When a magnetic field is applied in the z-direction, these
states begin to coalesce onto circular orbits in the k,—k, plane, forming coaxial
cylinders of radii fi+é s. The radius of each cylinder thus increases linearly with
the magnitude of the applied field. The degeneracy of each cylindrical cross-section
in the k,—-k, plane can be determined from the number of cyclotron orbits possible
in a rectilinear crystal of dimensions L, and L, transverse to the field. In the gauge
A= —ByZ, k, is a valid quantum number, and the eigenstates of the Hamiltonian
keep their plane wave-like nature along this axis. The density of states along k, is
thus L,/27 and is unaffected by the magnetic field. In the y-direction, the orbits
are centered about y) = y—y' = k,/s?. Because these must lie within the confines
of the crystal, 0 < yo < Ly, the total number of orbits allowed along the ky-axis is

L,s*; therefore the total density of states per unit area per Landau level is given by

s?/2nx = eB/h, and likewise increases linearly with field. It should be noted that
the average number of states in zero field contained in the area in k-space between
two Landau levels at field B is given by AE D(E) = (hw.)(m*/(27h”)) = eB/h,
and so the average areal density of allowed states is unchanged, although their
distribution is greatly altered.

With the carriers lying upon these cylinders in k-space, the degeneracy and
radius of which are linear functions of the field, the Fermi wavevector (and conse-
quently the Fermi energy) oscillates as a function of applied field. At an arbitrary
field chosen such that the upper most Landau level is only partially populated,
the Fermi wavevector must lie at the radius of the outermost k-space orbit at zero
temperature. The radius of this orbit increases linearly with applied field, and the
Fermi level must follow as well. As the field is increased, however, the degeneracies
of the lower-lying levels also increase linearly, and carriers from the upper Landau
level must drop into these lower states to satisfy the Fermi population criterion.
At a certain threshold field, the last carrier drops into a lower level, and the Fermi
level shifts from the / to the /—1 Landau level. The field at which this shift occurs
is such that the total degeneracy of the lowest Ima, levels equals the total volume
charge density in the crystal. Since the degeneracy per unit area of each Landau
level is given by eB/h, and the density of states along the direction of the field is
L,/2n, the Fermi level is expected to jump whenever ImaxeBL,/2th = N3p, where

N3p is the three-dimensional carrier density, or,

1 eL,

where 1/B; is the field at which the Fermi level shifts to the Landau level indexed
by !. Thus, the shift in Fermi level is periodic in 1/B, and any transport charac-

teristic of the bulk material pertaining to the behavior of carriers at Ep should be

periodic as well. Shubnikov-de Haas oscillations in the measured bulk resistivity

of a semiconductor in a magnetic field are due to the increased carrier scatter-
ing rates at fields where the Fermi energy drops down to that of a fully occupied
Landau level. In two-dimensional carrier gas systems, Shubnikov-de Haas oscilla-
tions also are observed for in-plane transport, although only the component normal
to the 2D gas, Bcos@, contributes to the periodicity. Other transport variables
are observed to oscillate as well, and Shubnikov-de Haas-like phenomena occur in
quantum transport devices such as those studied in this thesis, as discussed in

Chapter 4.

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[19] E. Merzbacher, Quantum Mechanics, (Wiley and Sons, New York, 1970).
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[27] R. R. Marquardt, unpublished.

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Chapter 3

Resonant Magnetotunneling

Spectroscopy

3.1 Introduction

The resonant magnetotunneling spectroscopy technique was developed for the
study of hole tunneling in AlAs/GaAs double barrier structures [1]. While simple
in concept, it is an experimentally powerful method for probing the in-plane energy
dispersion of quantized carrier states in semiconductor double barrier structures.
As it implicitly relies upon the tunneling of carriers into the quantizing region, it
is not a technique which can be used in the study of quantum wells. It is also
typically used to study subbands in which the effective mass approximation is not
expected to hold for all magnitudes or directions in the k)-plane. With a magnetic
field aligned parallel to the growth plane quantization effects due to the field and
the crystal potential are coaxial. Except for the highest fields, or widest quantum
wells, the magnetic field in this geometry may be treated perturbatively. As a
result, the device operates without additional quantization or significant modifi-

cation to the eigenstates and energies of the system; the only effect of the field,

semi-classically, is to bend the carrier trajectories into cyclotron orbits about the
axis of the field. This alteration of the carrier distribution in momentum space, and
the resulting effect it has upon the elastic tunneling of carriers through the device,
is exploited in resonant magnetotunneling spectroscopy (RMTS) [1] to probe the
energy subband dispersion of quantized carriers in semiconductor heterostructures.

A vast majority of work with the RMTS technique has been with traditional
Type I unipolar intraband tunneling diodes [1, 2, 3, 4, 5], where either valence
band states or in-plane band warping was studied. Few quantum magnetotrans-
port observations of any type have been reported for interband heterostructures
employing the InAs/GaSb Type II broken-gap band alignment. Previous magne-
totunneling studies in this material system have focused primarily on the n-well
RIT, with an InAs well and GaSb electrodes [9, 10, 11], or have investigated the
barrierless InAs/GaSb/InAs system with the magnetic field perpendicular to the
layers [12]. This chapter details the experimental results of quantum transport
through InAs/AlSb/GaSb/AISb/InAs heterostructures having well widths of 7.0
nm, 8.0 nm, and 11.9 nm, respectively, in magnetic fields of up to 8.0 tesla aligned
in the RMTS geometry parallel to the epitaxial growth planes. Unlike the pre-
vious magnetotunneling experiments in InAs/AISb/GaSb, this study investigates
the rich valence subband structure of GaSb wells, and reveals correspondingly in-
teresting and complex quantum transport phenomena. The results of these studies
are detailed in the remainder of this chapter. In Section 3.2 we examine the RMTS
technique, and provide a simple theory for the interpretation of experimental data.
The measurements are presented in Section 3.3, and analyzed in Section 3.4. Sec-

tion 3.5 summarizes and concludes the chapter.

59
3.2 Technique

3.2.1 Basic Model

The theory of resonant magnetotunneling spectroscopy is similar for tunneling
mechanisms involving initial and final states with either the same, or opposite,
energy dispersions. Thus, the theory for both may be developed in tandem. Refer-
ring to Figure 3.1, we take the z-axis to lie along the growth direction, while the y-
and z-axes both lie in the plane parallel to the heterointerfaces. With the B-field
along the z-axis, B=B , and using the Landau gauge vector potential A = Bz y,
the sole effect of the field on the Hamiltonian is to replace the canonical crystal
momentum operator p with p— gA. Assuming the vector potential term can be
treated as a perturbation, a valid assumption provided the cyclotron orbital radius
is significantly larger than the width of the well (1/s < Ly), and the cyclotron
energy is small relative to the quantizing material potential (hw, < AE), the
eigenstates of the Hamiltonian remain unchanged to first order, and the mechani-
cal momentum along the y-axis is increased by eB(x)/h, where (zx) is the expected
value of the position operator averaged over a given localized state under the per-
turbation. The origin along the x-axis is arbitrary; a change in origin amounts to
a change of gauge, for which the physics is invariant. Changes in (x), however,
such as tunneling from a localized emitter notch state to a quasi-bound state in
the well, result in an increased mechanical momentum along the y direction. The
conjugate momentum, p, = fky, however, remains conserved in elastic transport.
The parallel energy dispersions of the well states, dependent upon the mechanical
momentum, M, = p, + eB(Az), are shifted in k-space when plotted against the
conserved quantity, k,, as shown in Figure 3.2. As the magnetic field is increased,
and the mechanical momentum is likewise boosted, the well states are further

shifted along k,, changing the quasi-bound states available for elastic tunneling

processes conserving both # and k,. An ensemble of electrons, thermalized in an
emitter subband such that |k,| < kp and M, = p, = hk,, will have mechanical
momenta distributed from —kp + eB(Azx)/h to kp + eB(Az)/fh after tunneling
an expected distance (Az) to a localized quasi-bound state in the well. As a result,
the energy of the ensemble has been altered, and the change in energy must be
compensated by a change in the applied voltage required to meet the resonance
condition for tunneling. The magnetic field, doing no work on the charge carriers,
cannot alter their energies. As the well states are shifted, the energy dispersion of
the destination states in the well changes the energy required for resonance at a
given ky. Thus, as a function of applied magnetic field, the current-voltage (I-V)
characteristics of the device should be affected in a manner which is fundamentally
related to the energy dispersion of the subbands in the well. It is important to
note that this analysis rests upon two implicit assumptions: first, that tunneling
may be pragmatically viewed as a sequential process, in which a carrier tunnels
first into the well, and, subsequently, from it, and, secondly, that there exists a
quasi-bound state in the emitter accumulation region (a notch state) such that
the quantity (Az) is well-defined. In the absence of either of these conditions, the
semi-classical tunneling distance (Ax) becomes ill-defined, and the theory loses
validity.

For devices in which the initial and final states are of similar dispersion (Type
I unipolar intraband tunneling), peaks in the I-V curves can be associated with
the energy of a given subband dispersion at a ki value given by the increase in
mechanical momentum induced by the applied magnetic field, AM, = eB(Ac) [1].
Graphically, this criterion is shown for intraband tunneling devices in Figure 3.2.
This figure shows the energy dispersion for both the emitter and well quantized
states along the k,-direction (perpendicular to both the tunnel current and the

applied field) in the effective mass approximation for a Type I unipolar DBH

Figure 3.1: Energy-band diagram of an InAs/AISb/GaSb/AISb/InAs RIT diode
at 300K, in the emitter-barrier region of the device. Electrons from the quasi-
bound state in the emitter tunnel an expected distance (Az) through the barrier
into empty valence band states in the well. In the presence of a magnetic field
perpendicular to this tunnel current, carriers experience a semi-classical mechanical

momentum shift perpendicular to both the applied field and the growth direction.

at the peak voltage. In the absence of any magnetic field, the peak in the L
V characteristic corresponds to the condition that the ky band edges of the well
and electrode dispersions are coincident in energy, as shown. In a magnetic field,
the well state is shifted along the ky-axis, and, as the figure shows, the I-V peak
again occurs at ky = 0 (with |k,| < kp), where the electrode has the greatest
number of carriers available for tunneling. The degree to which the peak voltage
is shifted is therefore given by the well dispersion in the k,-direction at a relative
wavevector eB(Az)/h from the well state band edge. Thus, plotting the change in
peak voltage (or, equivalently, simply the peak voltage itself) as a function of the

applied magnetic field traces out the parallel band structure of the quasi-bound

state in the well. From this argument, it is clear that the well dispersion need not
obey the effective mass approximation, and the technique is capable of mapping
complicated well band structures.

The mechanics of interband tunneling are, unfortunately, far more complex, and
they obscure the direct interpretation of the experimental results. The opposite
dispersions of the electrode and well result in elastic tunneling through a broad
distribution of states in k-space at a single bias. As shown in Figure 3.3, in
the absence of a magnetic field the two dispersions have coincident energies and
momenta at two values along the k,-axis (for k,, = 0), symmetrically located about
k, = 0. With the consideration of non-zero k, as well, the overlap between the two
bands involves all k, values between these extrema. It can be shown in the effective
mass approximation that, for an arbitrary magnetic field, the set of elastically-
allowed states is a circle in the ky plane, centered along the k,-axis at k, = 0, and

along the k,-axis at,

_ mad
Ko = mi + my’ (3.1)
having a radius,
AE — (eV + (A762) /(2(m*, + m*
Key = | RBom (eV + (08) /(2(miy +m) (3.2)

fh? [2p
where A£p is the zero-field difference in the ky) = 0 band edges, m%, and m%, are
the electrode and well effective masses, respectively, 6 is the field-induced increase
in the mechanical momentum along the k, direction (= eB(Az)/h), V is the
applied bias, and y* is the reduced mass of the electrode and well dispersions.
As shown in the figure, this intersection occurs over a range of energies as well.
As the magnetic field is increased, k-space is sampled in a region centered about
a ky value that is dependent upon the masses of both dispersions and linearly
proportional to the applied field. Thus, while this shift in k, results in changes to

many of the factors determining the tunnel current, including the number of states

Well

Electrode
Overlap

B>0O

Figure 3.2: Electrode and well dispersions in the k,-direction (for k, = 0) in
the effective mass approximation for an intraband double barrier structure, with
and without a magnetic field applied perpendicular to the tunnel current. Both
cases are shown at the peak current bias, with states meeting the elastic tunneling
condition at the highlighted overlap. The well subband dispersion is shifted by a

Ak(B) linearly proportional to the magnetic field, increasing the peak voltage by

involved, and their quantum mechanical transmission probabilities, the change in
peak voltage is related, albeit in a complex manner, to the well (and electrode)

dispersions.

3.2.2 Experimental Setup

Three p-well RIT samples, grown by molecular beam epitaxy (MBE) [6, 7, 8] on
(100) n*-GaAs substrates, were investigated. The three devices, having well widths
of 7.0 nm (device A), 8.0 nm (device B), and 11.9 nm (device C), were otherwise
grown with identical layer thicknesses and doping. These device parameters are
summarized in Table 3.1. In addition, an InAs/AlSb Type I unipolar RTD with a
16.0 nm well (designed to have roughly identical confinement as that for light holes
in the 8.0 nm RIT) was investigated for comparison. For each device, mesas were
defined by standard photolithography and Au/Ge liftoff, and etched approximately
500 nm with a 1:8:80 H2SO4:H202:H2O wet etch, leaving a thick InAs electrode
beneath to minimize series resistance. Au/Ge lateral contacts were then deposited
in close proximity to the mesas. Au/Ge was used solely for its beneficial adhesion
properties; InAs has a negative Schottky barrier for electrons, and the formation of
ohmic contacts requires no special processing. Devices A and B each had octagonal
mesa cross-sections, 150 ym across, with a total area of 17,300 um?. Device C was
fabricated with a 100 zm square mesa geometry. Devices A, B, and C had zero-field
peak current densities of 126 A/cm?, 44 A/cm”, and 138 A/cm?, respectively, at
4.5 K. The higher current densities in samples A and C may be indicative of greater
non-resonant leakage current in these devices. The peak-to-valley ratios (PVRs) of
all three were low, typically 2-8:1, at 4.5 K. These PVRs are low even for devices of
this size, and are indicative of enhanced inelastic transport across the quantum-well
region. Once fabricated, the samples were mounted with silver paint on custom,

non-magnetic 8-pin TO-5 headers manufactured by Coors Ceramic Corporation.

Well

—_— —

> Ake) <— ky

Figure 3.3: Electrode and well dispersions in the k,-direction (for k, = 0) in the
effective mass approximation for the investigated p-well interband RIT structure,
with and without a magnetic field applied perpendicular to the tunnel current. The
overlap at any given bias involves a distribution of values in k,, as indicated by
the thick dashed lines. Experimental RMTS data for this device therefore cannot

be attributable to a single, field-dependent parallel wavevector.

Table 3.1: RIT growth parameters

Material Thickness Doping

nt-InAs >0.25 um 3 x 108 cm73

n-InAs 50.0 nm 1x 10'7 cm=3
InAs 5.0 nm undoped
AlSb 4.0 nm undoped

GaSb 7.0, 8.0, & 11.9 nm undoped

AlSb 4.0 nm undoped

InAs 5.0 nm undoped
n-InAs 50.0 nm 1x 10"7 cm-?
n*-InAs >1.2 um 3 x 108 cm?

These headers consisted of an AlgO3 ceramic compound base with copper pins
coated in a Cu/Ag euthectic braze alloy.

All of the measurements were performed in a cryogenic dewar housing a Nb/Ti
superconducting magnet and a variable temperature cryostat. The sample tem-
perature was monitored with a carbon glass thermometer, and was maintained at
4.5 K during all observations. The angle of the sample with respect to the applied
field was continuously adjustable from 0° to 360°. A Hall probe mounted adjacent
to the sample was used to align the sample surface parallel to the field to within
less than 0.5°. The magnet coil was rated for a maximum field of 8.0 tesla, with a
homogeneity of +0.1% over a 1 cm diameter central region. A voltmeter, monitor-

ing a calibrated shunt voltage, was used to determine the magnitude of the applied

field.

Current-voltage (I-V) curves were obtained with a Hewlett-Packard 4145A
semiconductor parameter analyzer. All I-V curves were scanned from low to high
voltage, although typically no more than 2-3 mV of hysteresis was observed when
the scan direction was reversed. Most of the mesas displayed nearly-symmetric
I-V curves, but in all cases the mesa and polarity with the highest PVR at 4.5 K
was selected for study. The header-to-4145A series resistance was measured to be
1.4 Q at 4.5 K, and the total resistance in series with the devices was estimated
not to exceed 2.5 9 from point-to-point measurements on the lower InAs electrode
layer. For each device, I-V curves from 0.0 to 0.5 V were taken in 1 mV steps at

regularly spaced (< 0.1 T) magnetic fields from 0.0 to 8.0 tesla.

3.3 Experimental Results

3.3.1 I-V Peak and Shoulder

A representative I-V curve from the InAs DB sample at zero applied magnetic
field and a temperature of 4.5 K is shown in Figure 3.4. As marked by arrows in
the figure, there are four well states identifiable from peaks or inflections in the
I-V characteristic. As the field is increased, not only do these states move as per
the RMTS theory mentioned earlier, they broaden as the well dispersion is shifted
away from ky = 0. As a result, the location of only two of these states could be
ascertained accurately for fields greater than several tenths of a tesla, and only the
resonance at the highest voltage could be tracked for B > 2.1 T. The peak voltages
of the two prevalent states are plotted as a function of magnetic field in Figure 3.5.
Both obey the effective mass approximation, as expected for conduction band
states, with effective masses of 0.044 + 0.008 m, and 0.006 + 0.002 m., assuming a

(Az) of 25.0+5.0 nm as estimated from band bending calculations and a one-band

InAs/AISb DB I-V Characteristic
B=0T; T=4.5K
1.50 1 1 :

25 7 |
629 L, =4.0nm |

1.00 + Ly, = 16.0 nm | 4

0.75 F | 7

0.50

Current (mA)

<—-

0.25 4

0.00 n i n i ‘ 1 ‘ L 4. n
0.00 0.10 0.20 0.30 0.40 0.50 0.60

Voltage (V)
Figure 3.4: Zero-field I-V characteristic of an InAs/AlSb double barrier structure
at 4.5 K. Four resonant states are obvious, two of which persisted at non-zero

magnetic fields.

InAs/AISb DB RMTS Results
4.0 nm barriers/16.0 nm well
0.80 ' , 1 :

m = 0.044 +/- 0.008 m,

OOOA ROGER CERR a FRGACMOOCCOCKOOOOD
PCO

Peak Voltage (V)
Oo
ra)

0.20 f m’ = 0,006 +/- 0.002 m,
0.00 | : : : ! : ! :
0.0 2.0 4.0 6.0 8.0

Magnetic Field (T)

Figure 3.5: Peak positions of resonant states of an InAs/AISb double barrier device
as a function of magnetic field applied perpendicular to the tunnel current. The
resulting curves are expected to follow the transverse energy dispersion of the states

in the InAs well, both of which appear to obey the effective mass approximation.

model of the emitter quasi-bound state. These values should be compared to the
bulk InAs effective mass of 0.023 m,.

In contrast, the I-V characteristics of the interband RIT devices exhibit only
a single peak, despite several quantized states in each well. Because of the oppo-
site dispersions, and small energy range over which resonant transport occurs, the
presence of a single peak is typical. These interband devices differ from the InAs
DB as well in their response to an applied perpendicular magnetic field. A typical
zero-field current-voltage characteristic from device B at 4.5 K is shown in Fig-
ure 3.6a. The I-V curve of the same device in an 8 T field is shown in Figure 3.6b.
Many of the general trends seen in all three samples are evident in this data. First,
at high fields, a shoulder formed and moved rapidly to lower voltages as the ap-
plied magnetic field was increased. In addition, the peak of the I-V curve shifts,
less dramatically, to higher bias. The peak current was observed to decrease with
increasing applied field in samples A and B, while in sample C the peak current
was constant until it was observed to increase in magnetic fields above 6 T. The
peak current was never observed to change by more than = 10% of its zero-field
value.

Figure 3.7 shows the peak and shoulder positions for each sample plotted as
a function of applied field. The shoulder position was determined from a local
minimum of the conductance curves for each sample. In cases where the broad
peak appeared to contain two or more maxima, each position is shown. For all
samples, the behavior is similar; little, if any, appreciable change at low fields
until some critical field (B.,i¢) is exceeded, at which point both the main peak
and subsidiary shoulder are observed to shift in voltage with changing applied
magnetic field. The shift of the main peak saturates at high fields and is not seen
to change in any of the samples at fields greater than 7 T. The shoulder position,

however, appears to be a linear function of magnetic field in all three samples. For

6.0 F (b)8.0T 7

2.0 F 7

Current (mA)
aN
ra)

130 p>————_—_,
0.75 F (c) I(8.0T) - 1(0.0T) |
0.00 |
-0.75 |

“1.50 #4 __1___.__
0.00 0.10 0.20 0.30

Voltage (V)

Figure 3.6: Example current-voltage curves for device B at (a) 0.0 T, and (b) 8.0 T.
The magnetic field shifts the peak to higher voltage, while pushing out a shoulder
to lower bias. In addition, the peak current has decreased. The position and
magnitude of the valley current is unchanged by the magnetic field. The difference

spectrum, shown in (c), is formed by the subtraction of the 0.0 T data from the

8.0 T data.

280
260
240

220 §

200
180
160

190

110

70
180

Peak Voltage (mV)

160 |

140
120

100 |
80 F (c) 11.9 nm well %
60 rn l rt l 1 i L

0.0 2.0 4.0 6.0 8.0

170 E OF,ed,es0ad,e
150 |
130 |

90 |

Magnetic Field (T)

Figure 3.7: Peak and shoulder voltages as a function of applied magnetic field for

devices A, B, and C, respectively. Little change is observed for fields below some

sample-dependent critical value. Above this critical field, the peak voltage appears

to saturate at high fields, whereas the location of the shoulder is a linear function

of applied field for all three devices.

InAs/AISb/GaSb/AISb/InAs RIT

4.0 nm barriers, 7.0 nm well

15.0 T q ‘ q ' 1 ’ ' q
B=7.5T
T=45K
~ 10.0 4
s)
5.0 + A
0.0 4 1 rn 1 1 ‘i ri 1 1 1 1
0.00 0.05 0.10 0.15 0.20 0.25 0.30

Voltage (V)

Figure 3.8: Observation of the shoulder in a RIT I-V form new second peak at low
voltages. Presumably the contacts had lower series resistance and there was less
inelastic leakage current in the device when this data was taken. Under less ideal
conditions, as were normally experienced, the second peak was never observed as

more than a broad, flat shoulder in the I-V curves at low voltages.

device A, a linear regression of the shoulder position yielded a slope of —18.9 + 0.3
mV/T, with a squared correlation coefficient, r?, of 0.996. Regressions of devices
B and C found slopes of —21.3 + 0.5 mV/T (r? = 0.986) and —18.8 + 0.3 mV/T
(r? = 0.992), respectively. The slopes for devices A and C are therefore statistically
identical, whereas that of device B is more than 10% greater. Recent investigations
of Sample A actually have shown a separate peak where a shoulder was previously
observed, apparently as a result of less inelastic leakage current in the device. This
data is shown in Figure 3.8. The movement of this second peak was found to be

in good agreement with that previously noted for the shoulder.

3.3.2 Difference Spectra

Unlike the data from RMTS investigations of traditional intraband tunnel devices,
much of the structure of the subbands in the GaSb well is missing from the data
in Figures 3.6a, 3.6b and 3.7. Whereas, in intraband devices, multiple peaks, each
associated with a separate subband, can be tracked with applied magnetic field to
trace the ky well dispersion, the opposite dispersions of the electrode and the well in
interband devices, as well as the narrow voltage window in which NDR can occur,
result in resonant transmission through several well states at the same bias. The
resulting I-V curve is a complicated convolution of the contributions of the various
well states. Consequently, it was useful to examine the difference spectra, such
as shown in Figure 3.6c, of the three devices over the range of investigated fields,
so as to separate the changes in conduction induced by the applied field from
the complexities of the zero-field transport. Additionally, each of these spectra
details the effect of the magnetic field more completely than the simple plot of
peak positions shown in Figure 3.7. Each difference spectrum was formed by
subtracting the zero-field current-voltage data for each device from the I-V curve at

a given magnetic field. This spectrum may be positive or negative since the shift in

carrier distribution caused by the magnetic field both adds and removes conduction
channels at a given bias relative to the zero-field equilibrium distribution. The
difference spectrum in Figure 3.6c is typical of all three samples. The NDR region
is demarked by two peaks in the spectrum, each indicating additional current at
those biases. When the peak or valley voltage shifts, as in this case with an
8.0 T field, the subtraction necessary to form the difference spectrum mimics the
calculation of the finite element approximation to the local derivative, and thus
in regions of large |dJ/dV| the difference spectrum has narrow peaks (or valleys,
depending upon the sign of the peak shift). Comparison of the behavior of the
difference spectra for the three samples highlights their differing subband structure
and the effect of the magnetic field on resonant transport.

Figure 3.9 shows the low-field (1.0 and 3.0 T) difference spectra for all three
samples. These magnetic fields are less than the critical fields of the three devices.
The difference spectra indicate that there are field-induced changes in the current-
voltage characteristics not fully evident in Figure 3.7. The peak of device B shifted
to lower voltage at both 1.0 and 3.0 T, as evidenced by the negative spikes in the
difference spectrum. The peak of device C was pushed to higher voltage, however.
The device with the narrowest well, sample A, showed little shift in the peak
voltage at either field, but a negative shift of the valley position at 3.0 T. All of
these shifts are small, no more than 10 mV. In addition, the non-resonant current
beyond the NDR region was affected in devices A and C. Device A showed a
decrease in current at voltages greater than 0.3 V in the 3.0 T field, while device
C showed a broad increase in conduction at voltages greater than 0.2 V in the 1.0
T field. In contrast, sample B showed no change in current at voltages beyond its
NDR region (> 0.25 V). Samples A and C had peak-to-valley ratios significantly
lower than sample B, and the additional transverse momentum provided by the

magnetic field appears to have affected the inelastic transport responsible for these

1.0 | T T r T T T T T T ]

0.5 - 7

0.0 SILT caaemennd
L an ee 1

-0.5 F ~~ 4

“1.07 (a) 7.0 nm well
0.5 ————

0.0 f
0.5 +
-1.0 : ,

“1.5 i (b) 8.0 nm well 7
2.9 /-——1++_1___1__.__1_.
1.0 f

0.5
ee ee
ee one ne ee

—— 1.0 T
---- 3.0T

A Current (mA)

0.5 fF (c) 11.9 nm well 7
-1.0 n l i | ‘ | A 1 L
0.00 0.10 020 030 0.40 0.50

Voltage (V)

Figure 3.9: Difference spectra for samples A, B, and C, respectively, at both 1.0
and 3.0 tesla, below the critical field of each sample. Each spectrum is obtained by
subtracting the zero-field current-voltage characteristic from the same data taken
at the given magnetic field. Below the critical field, there are slight changes in the

observed behavior of each sample not evident in the I-V peak position.

low PVRs.

The difference spectra for the three samples in high magnetic fields (at and
greater than B.,;,) are shown in Figure 3.10. It is in this regime that the behavior
of each sample is distinct. Referring to Figure 3.10a, we first examine the data
for device A. From the 4.0 T data, and the previous results shown in Figure 3.9,
it is clear that both the peak and valley of device A are shifted to lower voltage
at intermediate magnetic fields. At 5 T, this trend is continued, with additional
loss of current across the entire NDR region. In fields greater than 5 T, however,
there is a sudden shift of the peak to higher voltages, while the valley is similarly
shifted lower, shrinking the NDR region by > 50% at 8 T. The difference lineshape
at the peak and valley are both nearly symmetric, and at 5.0 T and greater, the
spectra all meet and cross at 270 mV. For voltages less than the peak, there is a
broad decrease in current, widening, and extending down to lower voltages with
increasing magnetic field. In contrast, the data in Figure 3.10b for sample B shows
significantly less change in transport across this sample due to the field. While
the I-V peak is shifted to higher voltages by the field, the width of the difference
spectra peak is roughly one third that of sample A, and less dependent upon the
magnitude of the magnetic field. While not obvious from the figure, the difference
peak height and width of the 8.0 T data are both slightly less than data taken at
7 T. In addition, the valley, while shifted to slightly higher voltages than the zero-
field data, is independent of the field in this regime. The NDR region also shows
little change in conduction, and the decrease in current at voltages below the peak
is broader than that of sample A. The data for the sample with the widest well,
device C, is shown in Figure 3.10c. Here, a peak in the difference spectrum occurs
at 5 T, with no shift in the valley and no reduction in conduction for voltages below
the peak. At greater than 5 T, the peak in the difference spectrum encompasses

the entire NDR region, growing in magnitude and width with increasing field.

2.0
1.0
0.0
-1.0
-2.0
-3.0
-4.0

4.0 7
3.0 +
2.0 |
1.0 |
0.0 |
-1.0 | “ISP

-2.0 ;
4.0 1
2.0 F ir ;
0.0 (Sse —
-1.0

-2.0
0.00 0.10 0.20 0.30

Voltage (V)

A Current (mA)

ews eeneconsagtesansaasmaeceresaecnsel

Figure 3.10: Difference spectra for samples A, B, and C, respectively, at 4.0, 5.0,
6.0, 7.0, and 8.0 tesla, at or above the critical field of each sample. Each spectrum is
obtained by subtracting the zero-field current-voltage characteristic from the same
data taken at the given magnetic field. In this regime, at or above the critical mag-
netic field, each sample displays distinct behavior. The transport characteristics
of all three samples, however, show a far greater dependence on magnetic fields

greater than B.pit.

The valley voltage remains constant throughout this range of magnetic fields, and
there is a wide region of lower conduction just below the peak voltage. This latter
behavior results in wide parabolic minima in the spectra. In contrast, the minima
in Figure 3.10a are only semi-parabolic, and the broad features in Figure 3.10b are
not at all quadratic. The lineshape of the difference spectra peak is also distinctly
different than that of devices A or B. The peak is asymmetric, with a hunchback
feature at lower voltages and a plateau-like shoulder above. There is also a slight
maximum, roughly independent of field, at the lowest voltages, not present in either
of the other devices. Attempts were made to model these difference spectra using a
simple, intuitive picture of interband tunneling [13], but they were unsuccessful. As
is often true for simulations in this system, even sophisticated, computationally-
intensive 8-band models yield results with only qualitative resemblance to the
actual data. Without a more complete understanding of all the physical processes
involved, elastic and inelastic, further quantitative details of the parallel dispersions

are unattainable from the difference spectra.

3.3.3 Critical Field and NDR I-V Peak

The unique, and unexpected, aspect of this data was the existence of a threshold,
or critical, magnetic field, below which little change was affected by the field upon
quantum transport in these devices, and above which peak shifts attributable to
the simple RMTS theory are observed. While the critical field, B,.;:, for each
sample can be estimated from Figure 3.7, it is more easily obtained by looking at
the integrated area of the difference spectra in the NDR region of the sample, as
a function of applied field. This data is shown in Figure 3.11; the three critical
fields are marked by arrows. Devices A and C show very sharp discontinuities in
the region of their critical fields, between 5.1-5.2 T and 4.5-4.6 T, respectively.

These values are in good agreement with the data in Figure 3.7. The data for

device B shows no discontinuity, but rather a smooth transition, at a critical field
of approximately 4.1-4.2 T. The critical field does not, therefore, seem to depend
monotonically on well width. Samples A and C also displayed additional behavior
at their respective critical fields not evident in sample B. Figure 3.12 shows the
NDR region of the I-V curves for sample A at 5.1, 5.2, and 5.3 T. In this regime
near the critical field, a significant second peak, located in the device’s NDR region,
was observed to appear abruptly as the field was increased above B,,;. A similar
peak was observed at the critical field of sample C. The presence of this peak was
wholly repeatable in each sample, across many mesas on each die. In both sam-
ples, this secondary peak was only evident over a narrow range of fields (~ 300G),
above which the main peak would broaden and envelope the other, as shown in
Figure 3.12. In addition, in Sample A, for which this phenomena was most greatly
studied, the critical field was observed to be dependent upon the polarity of sample
bias, being almost 1.0 T greater for substrate positive bias (Beit © 6.2T). This
polarity-dependence was also repeatable across the die, and in each polarity the
critical fields for different mesas of the same sample were identical to within the
resolution of the magnet. The addition of series resistance to the measurement
circuit merely shifted both peaks to higher voltage and did not decrease the mag-
nitude of either. Additionally, as the temperature was increased, the width of the
secondary peak increased, until around 55 K it was no longer distinguishable from
the main resonance, as shown in Figure 3.13. For these reasons, we do not believe
that the secondary peak is related to the circuit oscillations and hysteresis known
to exist in the NDR region of resonant tunneling diodes [13, 14]. Given the nature
of this peak, it seems apparent that it appears at exactly the critical field for a
given sample. As its appearance was so abrupt (the exact field at which it appears
can be determined to the resolution limit of the magnet, roughly 50 gauss), it is

the cause of the discontinuities seen in Figure 3.11. The extra current from the

new peak adds to the total integral over the NDR region discontinuously as the
peak occurs. Both samples in which this second peak was observed (A and C)
have discontinuities in the NDR integral, whereas sample B displays neither of

these behaviors.

3.4 Analysis

3.4.1 B> Bort

Because of the dichotomous nature of the data, these experiments are best analyzed
separately for fields below and above the critical field. At fields larger than Bepit,
the expected RMTS behavior was observed, and although the nature of interband
tunneling obscures direct quantitative analysis of the valence subband structure,
significant qualitative information may be inferred. Figures 3.14 and 3.15 show the
theoretical parallel band structure of the three samples investigated experimentally,
at 0.0 and 8.0 tesla, respectively. These dispersions were calculated in our group
by Yixin Liu using an efficient, numerically stable 8-band k- p method generalized
to include magnetic fields.[15] Figure 3.14 demonstrates the complexity of the well
subbands in this device structure, even in the absence of magnetic fields. Despite
the fact that several subbands are present, only one peak is evident in the current-
voltage characteristic due to resonant transmission through multiple subbands at
a single bias. As can be seen in Figure 3.15, the addition of the magnetic field
complicates the situation even further. Clearly, the field has more effect than to
simply shift the bands in k-space. There is a great deal of interaction between
the subbands, in addition to lifting of the spin degeneracy. From the basis of
the well width dependent shift of the band edges, theory predicts that (Az) is

~ 11nm + 6+ w/2, where b and w are the barrier and well widths, respectively.

0 . 1 0 L ' '
p 4

0.05 F é 7

Integrated AI in NDR Region (mA-V)
co)
, F ,
if

~% Sample A
-0.05 F ° . re
* ‘e ” ra
“e e
‘.
-0.10
0.0 2.0 4.0 6.0 8.0

Magnetic Field (T)

Figure 3.11: The change in device current integrated over the NDR region (as
defined for each sample) plotted as a function of magnetic field. The apparent
critical magnetic fields are indicated by arrows. The critical fields so obtained agree
well with values derived from Figure 3.7. The abrupt nature of this transition is

evident in samples A and C.

22.0

Sample A (7.0 nm well)

20.0

amd
9°

Current (mA)

16.0

14.0

rl 1 rn l r i 1
0.15 0.20 0.25 0.30 0.35
Voltage (V)
Figure 3.12: Current-voltage characteristics of device A near the critical magnetic

field. In both devices A and C, a second, narrow peak was observed in the NDR

region at B.,. In fields only % 300 gauss greater, the main peak was seen to widen

and encompass it entirely.

Temperature Dependence

NDR peak of 7.0 nm well RIT

LU . UJ i t . * i . 1 ‘ 1
Bh. oe. A B=B,,+200G |
3 \, j \ Ne
O Lien e nr re eee ee an os ‘ s “
5 Vr i ie
oe a oe we ee oh
os [77 WF 1 ~ 7
= an id. NN 55K:
oO — | tet eee eee
~ r - on
er po |r ---~
oO +f
a ena

0.24 0.25 0.26 0.27 0.28
Voltage (V)

0.20 0.21 0.22 0.23

Figure 3.13: Observed temperature dependence of the I-V peak seen in the NDR
region of device A at its critical field. The peak here is broader and wider than
that seen in Figure 3.12 because the magnetic field was roughly 100G greater for
these I-V traces. At a temperature of roughly 55 K, the secondary peak in the

NDR region was indistinguishable from the main I-V peak.

In addition to the complexities of the subband structure, the quantum mechan-
ical couplings of the subbands to the electrode states vary quite strongly between
bands and, to a lesser degree, as a function of k). The calculated transmission
coefficients for the narrow-well device, and their dependence upon applied field,
are shown in Figure 3.16. Due to band mixing, at non-zero magnetic field, or
equivalently, non-zero ky, the states are strongly mixed, and the LH (light hole)
and HH (heavy hole) labels are strictly valid only for the band symmetries at
k = 0. From this data, however, it is clear that the greatest coupling from emit-
ter occurs to the HH2 and LH1 bands, and not to the HH1, HH3, or HH4 bands
shown in Figures 3.14 and 3.15. Consequently, these bands provide the greatest
transmission channels for current. It is expected that the LH1 band would couple
well to the conduction band states of the emitter, due to its symmetry. The HH2
couples more strongly than the other HH bands since it has a strongly positive
dispersion for small k), and is therefore more conduction band-like. As a result, it
is suggestive that the movements of the peak and shoulder in the experimental J-V
data for fields greater than B,,,, correspond in some manner to the HH2 (for the
peak) and LH1 (for the shoulder) subbands, subject to the previously mentioned
caveats. The movement of the shoulder was in fact consistent with a lighter mass
negative dispersion, and that of the peak similarly consistent with a heavier, posi-
tive dispersion, as found in Figures 3.14 and 3.15 for the LH1 and HH2 subbands,
respectively. While the thick barriers used in the experiments make the calculation
of a theoretical I-V curve intractable, Figure 3.17 shows the results of an I-V curve
calculation for a RIT with a 7.0 nm GaSb well and significantly thinner 1.5 nm
AlSb barriers. Even at zero field, a shoulder is evident, and as the magnetic field
is increased, the theory predicts behavior similar to that experimentally observed
(compare to Figure 3.8). Additionally, the nature of the low voltage shoulder can

be theoretically traced to transmission through the LH1 state. The theory also

indicates that for the widest well, the LH1 and HH3 bands interact strongly, but it
is doubtful that the experimental results should be capable of distinguishing this
behavior from the general, low mass trend of the coupled bands. In general, for
fields larger than B.,,., the experimental data appear well-understood, and indicate
the presence of a low, positive mass dispersion (for ky values accessible at these
fields) and a separate high, negative mass dispersion at lower well energies. These
observations agree well with the theoretically predicted behavior of the HH2 and

LH1 states, respectively.

3.4.2 B< Berit

The behavior of the experimental results at fields equal to and below the criti-
cal field is incongruent with the semi-classical analysis presented. The magnetic
field has little outward effect on the experimental results at values less than Bopit.
Additionally, the transition at the critical field is hyperabrupt, experimentally dis-
tinguishable down to the resolution limit of the magnet, roughly 50 gauss. The
sharp distinction between behavior above and below the critical field, and the ex-
tra, narrow peak occurring within the NDR region of the device at the critical field
raise a number of questions. This phenomenon has not been reported in earlier
RMTS investigations of intraband devices, and was not seen with our apparatus
in the investigation of transport in an InAs/AISb double barrier heterostructure.
It would therefore seem that this behavior is attributable to the interband nature
of transport in the InAs/GaSb/AlISb system. Either the quasi-bound states in the
GaSb well have infinite effective masses for some set of wavevectors, or the basic
model itself is invalid for fields less than B.,;,. The latter possibility introduces a
number of potential explanations. As the RMTS theory assumes that a 2D state
exists in the emitter accumulation region, one failure mechanism considered was

lack of such a state at low magnetic fields. The experimental evidence disputes

B=-0T
7 nm Well Som Well 12nm Well

0.15 -— rr ee ee
GaSb
HHI HHI VB
HH1 Edge
JN ave
ry
/ ‘\
0.10 : NY ae
. / \
/ \
/ \
oo / \
> HH2 A HH3
> i \ .
sy LHI \
Pos A / \
aa ” / \
/ \ i \
/ \ i \
0.05 - /LHI\ 7 7 F :
/ \
/ \
/ \
/ \
Ul \
HH4
a em
See) ra LH2 ‘\ Edge
L4 / Ls tt J
L

0.00 ae er eee er ee a eon a A
2-101 2-2-1101 2-2-1012

k, (% n/a)

Figure 3.14: Theoretical zero-field well subband structure for RIT devices having
GaSb well widths of 7.0, 8.0, and 12.0 nm, respectively, in the energy range between
the InAs conduction band edge and GaSb valence band edge. Based upon the
calculated quantum mechanical transmission coefficients (not shown), the majority

of the resonant current tunnels through the HH2 and LH1 subbands.

B=8T

7 nm Well 8nm Well 12nm Well
0.15 red f PF fT reer y © Tf Ara
aSb
HHI HHI | ve
HHI HH2 Edge
x \
lax LH1
HH2 yw
Vy
0.10 - 5 ws x \\ 4
\\.
S HH2 Yo
co ONLI a
> ph 4 t ;
eb yA HH3
i) Vy
5 cr 4
” LHI \\
y ) \\
0.05 y “ “ ~ \\ 7 — 4
\\ Vy
\\ \
\ \\
‘ \ HH4
‘ |
‘\ HH3 jos LH InAs
\ | 7) aN Edge
0.00 -4+41 +H pt pritd 1 |g
-2-1012-2-1012-2-101 2
k, (% n/a)

Figure 3.15: Theoretical well subband structure for RIT devices having GaSb well
widths of 7.0, 8.0, and 12.0 nm, respectively, at a magnetic field of 8.0 tesla. The
predicted shift of the well subbands is shown, along with the additional features
induced by the applied field.

Transmission Coefficients in RIT
Wasp = 7-0nm, Wy, =4.0 nm, k, = 0

10 ' i qT . T Lf qT
(a) B=0.0T
10° +
10°° !
10° : 1
5 (b) B=2.0T
‘2 19° 4
5 10°° a l A i
& 10° F qT ' i
: (c) B=40T
2 10°
" 10°" a : my
10° : 1 1 :
HH1
(d) B=6.0T LHI
10° } 4
10°° . 1 i : l F l i Lao]
0.00 0.03 0.05 0.08 0.10 0.12 0.15

Incident Energy (eV)

Figure 3.16: Calculated transmission curves for a RIT having a 7.0 nm-wide GaSb
well, at zero, 2.0, 4.0, and 6.0 tesla in the RMTS geometry. At non-zero fields,
additional resonances attributable to HH1 and HH2 states are present in addition

to the LH1 resonance found at B = 0.

Calculated I-V of RIT Structure

Waoasp = 7-0 nm, Wai, = 1.5 nm
60.0

40.90 | (@ B=00T ]

20.0 F

60.0 :
400 | (b) B=2.0T

20.0 F z
0.0 ; *
60.0 , ,

40.0 | =6(c) B=6.0T ]

Current Density (10° A/cm’)

20.0 F A

0.0
0.0 100.0 200.0 300.0

Applied Voltage (mV)
Figure 3.17: Calculated current density-voltage characteristics for a 7.0 nm-wide
p-well RIT in zero, 2.0, and 6.0 tesla fields. This calculation was done for T = 77 K,

and for a device having thinner barriers than those investigated (primarily to speed

the calculation), but the results bear a strong resemblance to the experimental

data.

this possibility, however. The InAs/AISb double barrier exhibited no signs of such
an effect, even though it had identical electrodes to the RIT structures. Also, the
additional band bending induced by the transfer of charge between the GaSb well
and InAs electrodes of a RIT should actually increase the likelihood and degree of
quantization in the InAs emitter in comparison to the InAs/AISb double barrier.
Lastly, the mechanism by which such an abrupt transition may be attributed to a
change in dimensionality of the emitter electrons remains unclear. It is similarly
unlikely that this phenomenon is due to a classical cyclotron effect. For the n =
0 state and the InAs bulk mass, a 50 G change in field strength at 5 T (all that
is necessary to distinguish the onset of the additional I-V peak) results in only a
0.13 A change in cyclotron radius, and a 2.3 jeV alteration of the cyclotron en-
ergy. Furthermore, since this failure of the semi-classical RMTS model occurs over
a large range of magnetic fields, a theory evoking the field-dependent cyclotron
motion of the carriers cannot wholly explain the experimental data. The exact

cause of this behavior is still unknown.

3.5 Conclusions

The complex nature of the interband tunneling mechanism was shown to lead to
a complicated, and indirect, relationship between the observed effects of the mag-
netic field and the parallel subband dispersions of the quasi-bound states in the
well. As interband transport at a given bias involves a broad spectrum of states
in both the k, and ky directions, transport through several subbands concurrently,
and the resultant effect of the field upon the peak position involves changes in the
number of states tunneling, their respective energies, and their quantum mechani-
cal transmission probabilities, the observed peak shift cannot be uniquely assigned

to the well dispersion at a field-dependent wavevector, as it can for intraband

transport. However, multiband theoretical calculations provide strong evidence
that the peak and shoulder positions above B,,i correspond to the HH2 and LH1
subbands, respectively.

The data presented raise interesting questions as to the nature of transport
in interband tunneling devices. The abrupt, catastrophic changes observed at the
critical field are not understandable within the semi-classical theory invoked to
explain RMTS experiments. Furthermore, simple explanations for deviations from
this model fail to satisfy, or are disproved by, the experimental data. Further obser-
vations are warranted at higher magnetic fields and additional device geometries,
both to study the peak shifts further, and to also possibly lend further insight into

the nature of the observed critical magnetic field.

Bibliography

[1]
[2]

R. K. Hayden et al., Phys. Rev. Lett. 66, 1749 (1991).

J. P. Eisenstein, T. J. Gramila, L. N. Pfeiffer, and K. W. West, Phys. Rev. B
44, 6511 (1991).

S. Y. Lin et al., Appl. Phys. Lett. 60, 601 (1992).

U. Gennser et al., Phys. Rev. Lett. 67, 3828 (1991).

T. Osada, N. Miura, and L. Eaves, Solid State Commun. 81, 1019 (1992).
D. A. Collins et al., Appl. Phys. Lett. 57, 683 (1990).

D. H. Chow et al., in Quantum- Well and Superlattice Physics III, proceedings
of the SPIE 1283 (Bellingham, WA, 1990).

D. A. Collins et al., in Resonant Tunneling in Semiconductors: Physics and
Applications, edited by L. L. Chang, E. E. Mendez, and C. Tejedor (Plenum,
New York, 1991).

E. E. Mendez, in Resonant Tunneling in Semiconductors: Physics and Appli-

cations, ibid.

[10] E. E. Mendez, J. Nocera, and W. I. Wang, Phys. Rev. B 45, 3910 (1992).

[11] E. E. Mendez, Surf. Sci. 267, 370 (1992).

[12] T. Takamasu et al., Surf. Sci. 263, 217 (1992).
[13] D. A. Collins, Ph.D. Thesis, California Institute of Technology, 1993.

[13] V. J. Goldman, D. C. Tsui, and J. E. Cunningham, Phys. Rev. Lett. 58, 1256
(1987).

[14] T. C. L. G. Sollner, Phys. Rev. Lett. 59, 1622 (1987).

[15] Y. X. Liu, D. Z.-Y. Ting, and T. C. McGill, unpublished.

Chapter 4

Interband Tunneling Through

Landau Levels

4.1 Introduction

The transport properties of quantum heterostructure devices are greatly altered
by the application of magnetic fields at arbitrary angles to the epitaxial layers.
Unlike the RMTS experiments detailed in Chapter 3, in this orientation the en-
ergy spectra, wavefunctions, and density of states are dramatically altered. The
field component along the growth direction induces additional in-plane quantiza-
tion, resulting in the formation of Landau levels. Experimentally, the effect of the
field and the existence of Landau levels is observed through one of two distinct
phenomena, either from the direct observation of individual peaks (corresponding
to single Landau levels) in the I-V characteristic, or via Shubnikov-de Haas-like
oscillations in one or more of the transport variables. While much of the histori-

cal study of low dimensional transport in semiconductor devices has involved the

investigation of Shubnikov-de Haas (SdH) oscillations in 2D carrier gases,! these
techniques were characteristically in-plane transport measurements, often using Si
MOS technology for precise, adjustable control of the 2D carrier confinement and
population. Only recently have perpendicular quantum transport studies observed
Landau level tunneling and SdH oscillations. Such observation of magnetotunnel-
ing has furthered the understanding of quasi-2D states in heterostructure devices
by revealing the relative energies of well subbands, their effective masses, and the
coupling between well and electrode states.

As with the RMTS technique (c.f., Chapter 3), a vast majority of the obser-
vations of resonant tunneling through Landau levels have been in Type I unipolar
DBH diodes. While many of these studies report individual Landau-level resonance
peaks in I-V [2, 3, 12, 5) or conductance [6], still others observe SdH oscillations of a
variety of transport variables, including current [7], conductance [8, 5], d?7/dV? [10]
and dI/dB [9]. The only reports of experiments in the InAs/GaSb/AISb interband
system have been for an n-well RIT [11], where conduction band-like Landau lev-
els are formed in the well, and the barrierless InAs/GaSb/InAs quantum well-like
structure [13]. This chapter presents experimental results of resonant Landau level
tunneling in p-well RIT’s having well widths of 7.0 nm and 11.9 nm in magnetic
fields as high as 8 tesla aligned parallel to the tunnel current. Qualitatively dif-
ferent results are obtained for these samples with valence band-like Landau levels.
The remainder of this chapter details these results and their interpretation. In Sec-
tion 4.2 we examine the physics of resonant tunneling in this regime, and develop a
simple theory for the explanation of the experimental results. The measurements
are presented and analyzed in Section 4.3. A summary and conclusions are found

in Section 4.4.

' The discovery of the quantum Hall effect [1] was made during the experimental study of SdH

phenomena in Si MOS accumulation regions.

4.2 Theory

4.2.1 Landau Levels in Heterostructures

The theoretical treatment of Landau levels in heterostructures is simplified by the
orientation of the magnetic field relative to the epitaxial layers. With the magnetic
field applied solely along the growth direction, its effect is seen only in the growth
plane. Consequently, the single-carrier Hamiltonian is separable into contributions

parallel and perpendicular to the layers,
Hy = A, + A, (4.1)

where, in the envelope function approximation, Hy operates on the in-plane com-
ponent of the single-particle wavefunction, and H, acts solely on the portion of the
wavefunction dependent upon the growth direction, z, of the heterostructure. The
band structure problem Hoy = E(k) therefore simplifies through a separation of
variables. The single-particle envelope function in the z-direction, H,, is written
as,

H,= ome +V(z), (4.2)

where V (z) represents the variation in potential energy due to the heterointerfaces.
In the transverse direction, Hy incorporates the normal magnetic field through the

in-plane vector potential, A,

(a + eA)’ + gt neB (4.3)

A = 5

m*
The spin contribution is ignored as negligible for materials with small band gaps,
and subsequently small effective masses. The contributions of H, are identical to

the zero-field problem, and the solution of H, |, in the effective mass approximation,

becomes identical to the bulk problem discussed in Chapter 2. The energy structure

for carriers in a simple quantum well therefore is of the form,
E(q,l) = E,+ (1+ 5) hws (4.4)

where qg and / are the quantum well subband and Landau level indices, respectively,
HE, is the zero-field, ky = 0 energy of the gth quantum well subband, and hw, =
eB/m* is the well cyclotron energy.

Unfortunately, the effective mass approximation does not provide a good model
for the complex parallel dispersions that exist in many quantum well problems,
especially those involving hole-like states. To consider the problem in general,
deviations from the effective mass approximation, E' « k?, must be developed. An
arbitrarily complex scalar Hamiltonian may be expanded in a Taylor series in k

about k = 0 for an arbitrary magnetic field,
H(k) = SX Cak", (4.5)

a=0
where the C, are the Taylor expansion coefficients, k is the generalized crystal
momentum operator defined by k; = (p; + eA,)/h, and A is the vector potential.
With such a Hamiltonian, the Landau levels are no longer evenly spaced in en-
ergy and the eigenstates of the effective mass problem are not stationary states of
the generalized Hamiltonian. As motivated in the derivation of Landau levels in

the effective mass approximation (c.f. Chapter 2), raising (at) and lowering (a)

operators may be defined in terms of the in-plane components of k,

at = —s5 (b+ ,) (4.6)
a = — 5 (be ~ iy) (4.7)

where 1/s = ,/h/eB is the magnetic length. These operators applied to individual

wavefunctions relate a given Landau level indexed by I to its adjacent neighbors

with indices /+1 and/—1,

ayy = Vit din (4.8)
amy = Vida (4.9)

aad = Ivy (4.10)

Whereas the effective mass Hamiltonian may be rewritten in terms of the number
operator, Nz=alaz= ke +k, , thereby leaving the state vector unaltered, arbitrary
powers of k, which may be reexpressed as various powers of af and a, (a!)*(a)%,
thereby raise and lower Landau level wavefunctions to other indices, and allow for
coupling between Landau levels. Solutions of the Schrédinger Equation will thus,
in general, involve linear combinations of many Landau level basis functions. If
the effective mass Hamiltonian is only slightly perturbed with additional powers
of k, solutions result which are primarily composed of a single Landau level, but
which retain non-zero contributions from other Landau indices as well.

The simplest model for which deviations from the effective mass approximation
may be investigated is the two-band model due to Kane [15]. Calculating the
conduction and light hole bands simultaneously, while ignoring the contributions
of other states, the model is first presented in the absence of a magnetic field. The
Kane theory is, in fact, one of a number of realistic band structure models that
may be derived from the more general ke p single-particle Hamiltonian. For a given
band, indexed by m, we can express the wave function as the product of a plane
wave factor and a Bloch function, Um(k, ¥), such that Vm(k, Pr) = eu, (k, P).
Substituting this form into the general Hamiltonian in Equation 4.3 results in the
so-called k - p Hamiltonian,

P? hk po WPR?
+ +

2Me Me 2Me

+V(F)| Um(k,?) = Bin(B) Um(E, *), (4.11)

where p is the momentum operator and Em(k) is the energy eigenvalue for the

100

m* band. Abstractly, this Hamiltonian requires solutions expanded in terms of all
zone center Bloch functions indexed by m. Typically, an infinite number of such
Bloch states are implicitly included in Equation 4.11, although the Hamiltonian
may be block diagonalized to incorporate only the bands of interest. The details
of the approximations which go into the adaptation of the k - p Hamiltonian to a
finite problem are detailed in Appendix B. Following Fasolino and Altarelli [16],
the in-plane eigensystem, Hj = E\(k)2, for the 2-band system consisting of the

lowest conduction band and the light hole valence band may be written as,

BOE) Leva { YY), (aa)

where E,, Ey, %-, and w, are the conduction and valence band edges and basis

Ayy =

states, respectively. The parameter P is the Kane interband matrix element of the
momentum operator, iP = (s|p,|x), where |s) and |x) are the spherically symmet-
ric s-like and z-symmetric p-like zone-center states. The convenient consequence
of the 2D Hamiltonian in this representation is the one-to-one mapping between
the k, and k, dependence of H, | and the raising and lowering operators, a! and a.
Therefore, for non-zero magnetic field Hj) may be expressed in terms of at and a

rather than the non-commuting quantities of k, and ky. Rewriting H, \| aS,

EF. —/2Psal
Ay = ; (4.13)
~—V/2Psa Ey

we find that eigenstates of the Hamiltonian may be written in this basis as,

Cy An
v= ; (4.14)
Cg hy_y

where h,, and h,_; are the harmonic oscillator wavefunctions for the 1 = n and
! = n—1 Landau levels, and the constants c, and cy determine the fractional

contribution of the conduction band and valence band basis states, respectively.

101

Here n is a parameter of the wavefunction in which the Landau level index, J,
of each component may be expressed. Note that the constant cz must equal 0 for
n = 0 as there is no! = —1 harmonic oscillator wavefunction. What is important to
note is the unique nature of the eigenstates. In this model, the stationary states of
the Hamiltonian involve more than one Landau level, but each band has a distinct,
single Landau index. Solutions of this form are not unique to the 2-band model.
In the 8-band k - p formalism (see Appendix B for discussion), the eigenvector in

the Kramers basis set may be written as,

b= (cr hn, €3 hn-1, C5 hast, C7 hnti, C2 hn41, Cahny2, Cohn, Cahn), (4.15)

where the odd-c components correspond to spin-up states and the even-c com-
ponents to spin-down, and the vector is ordered so as to simplify the resulting
Hamiltonian. The association between the basis components, |1) through |8), and

the various physical bands is given by,

It) = ICBt) = [s4)cn

3) = |HHt) = |83)

5) = |LHt) = [3 -3)

7) = |SOt) = |§ —3)ve 4:16)
2) = |CBL) = |} -d)cep

|4) = |HH\|) = [8 -3)

J6) = |LHL) = |$4)

8) = |SOL) = [84)va,

where CB, HH, LH, and SO refer to the conduction band, heavy hole, light hole,
and split-off bands, respectively, and + and | signify spin up and down states;
in addition, the |jm,) angular momentum state definitions are shown. While
many authors [16, 17, 18] have demonstrated these interdigitated Landau ladders

in multiband models, none has proposed a physically-intuitive model for their

102

existence; all have merely used them to simplify the resulting calculation. The
underlying physics itself, however, is important to the proper understanding of
Landau-level tunneling in interband systems.

The Landau indices of components of the two- and eight-band eigenstates are
not as arbitrarily related as they might first appear. Each eigenstate in fact rep-
resents a state for which the total angular momentum projected along the z-axis
is a constant. To verify this assertion, it is important to consider the additional
angular momentum induced in the z-direction by the magnetic field through the
semi-classical cyclotron orbits in which the carriers travel. In general, the angular
momentum L is defined as,

L=Fx@. (4.17)

To calculate the component of angular momentum along the applied field, con-
sider the nature of semi-classical Landau orbits. As these orbits are circular, the
components of 7 and p projected into the plane must be perpendicular. Therefore,
ignoring the components of 7 and 7 parallel to B , the magnitude of the cross prod-
uct is the product of the vector lengths, and L is either parallel or anti-parallel to
B. As discussed in Chapter 2, the semi-classical orbital radii in real- and crystal

momentum-space are given by,

[+-, (4.18)

K=sl+5, (4.19)

where 1/s is again the magnetic length, \/A/eB. Thus, the magnitude of the

and,

cyclotron orbital angular momentum is given by |? x #] = ARK = (1+ 5)h. To

determine whether L. B = +1, one must examine the Lorentz force on a carrier,
F=

tox B=p=nhk. (4.20)

103

Thus, for an electron with charge q = —e,
. Bk
kg = —— (4.21)
. eBk

and the rotation is clockwise with respect to the field for negatively charged car-
riers. The angular momentum vector is consequently parallel to B, and the z-
component of the cross product 7 x fj is positive.

The field induces a positive angular momentum along the z-axis (for electrons),
and this additional macroscopic angular momentum component must be conserved
in addition to the projection of the orbital angular momentum of the band and the
carrier’s intrinsic spin. As the macroscopic cyclotron orbital angular momentum
is proportional to J, the conservation of all three angular momenta corresponds
to the conservation of f = 1+ m,, where m, is the projection of the sum of
orbital and spin angular momenta along the z-direction. As all carriers in the
Landau level indexed by / have the projected angular momentum (J+ 5h in the z-
direction, no additional hybridization of the states due to the quantum mechanical
addition of angular momentum must occur. Implicitly the eigenvalues of L, are
wholly specified, and the eigenvalues of L? are complicated by the rigors of angular
momentum addition. For the current problem, the total angular momentum after
the addition of the cyclotron motion is irrelevant, and will be ignored. The basis

of the Hamiltonian may be relabeled with the Landau index, in addition to j and

104

m,;, as |n:jm,), becoming,

Il) = [CBT) = |n: $3)cs

3) = |HHt) = |n—1: $3)

5) = |LHt) = |nt+1: 3 -})

7) = |SOt) = In+1: 3 —3)vo i423)
2) = |CBl) = |nt+1: 3 -4)en

|4) = |HH|) = |nt+2: 3-3)

]6) = |LHJ) = |n: $3)

8) = |SOl) = |n: d4)vs,

where n is again a parameter of the eigenstate relating the Landau level index of
each band to the state as a whole. For all components of the eigenvector, the sum
[+m, is constant and equal to n+ 5. Thus, in the absence of scattering and other
inelastic processes, tunneling may only occur between states of equal [+m,, in the
current approximation to the full single-particle Hamiltonian. Perturbations from

this idealized model will in turn couple the states to additional Landau levels in

each band.

4.2.2 Tunneling Through Landau Levels

The physics of Landau level tunneling in double barrier heterostructures depends
critically upon the dimensionality of carriers in the emitter of the device. If the
source of carriers tunneling into the well from the emitter is simply the bulk, 3D
electrode, then 3D-2D tunneling occurs in the absence of a magnetic field, and 1D-
OD tunneling with a B-field applied along the growth direction. Consequently, the
well states define a more limited subset of energies and momenta, and transport
properties therefore reflect the well states alone. In contrast, if carriers from the

emitter tunnel from a quantized notch state, either 2D-2D (B = 0) or OD-OD (|B| >

105

Figure 4.1: Emitter and well regions of a p-well RIT in a magnetic field perpendic-
ular to the heterointerfaces. In this geometry, Landau levels are formed, some of
which may deviate significantly from the evenly-spaced ladders predicted by effec-
tive mass theory, as shown in the well. Because of the intrinsic angular momenta
of the well subbands, and the additional angular momentum provided by the mag-
netic field, carriers tunneling from the emitter subband to the well are expected to

conserve the quantity | + m,;, or the total projection of angular momentum along

the field.

0,B || J) tunneling occurs, and the properties of both the well and emitter states
equally affect the transport characteristics. Unlike many experiments [6, 8, 9],
there is evidence to support the existence of a 2D accumulation region in the
interband RIT structures that have been studied, as detailed in Chapter 3. The
existence of such a notch level is consistent with the large charge transfer expected
between InAs and GaSb. Thus, only the specifics of 0D-0D Landau level tunneling
will be developed.

Figure 4.1 shows the effect a magnetic field perpendicular to the heterointerfaces

106

has upon the energy states in both the 2D emitter accumulation region and the
well of an interband RIT device. Non-ideal Landau level behavior is indicated
to highlight the potential complexity of the GaSb valence band-like well states.
Each emitter and well quasi-bound state has a ladder of Landau levels associated
with it, and consequently, the criteria for elastic resonant tunneling between such
states is modified. While energy must still be conserved, the conservation of ky
is alternatively stated as a condition on the initial and final state Landau level
indices. Typically, as there is a direct mapping from Landau levels to kj, this
criterion is expressed as the conservation of Landau level index, Al = 0 [7, 3, 11].
While this selection rule is valid for n-type Type I unipolar devices, as studied in
References. [7] and [3], it is not so in general, in light of the additional angular
momentum contributed by the cyclotron orbits of carriers in Landau levels. For
n-type Type I devices, the general tunneling criteria A(J + m,;) = 0, dictated by
the conservation of the z-projection of angular momentum, is equivalent to the
simple rule Al = 0, as the initial and final states are inherently identical angular
momentum states. In interband devices, where both conduction and valence bands
are involved as initial and final states, and for hole tunneling in Type I structures,
for which states of various m, are simultaneously involved in transport, the my, of
initial and final states may differ in the presence of a magnetic field, and the full
selection rule A(/ + m;) = 0 must be applied. This conclusion is not inconsistent
with the results of Mendez et al. [11], who, in their study of the I-V data for a
GaSb/AISb/InAs/AlSb/GaSb n-well RIT, cite the lack of additional ladders of
peaks, corresponding to inter-Landau level coupling, as evidence that Al = 0 is
valid in interband systems. The general A(/ + m;) = 0 rule allows a conduction
band state of given | to couple to only select valence band states, which should
lead to behavior such as that observed in their data.

While many studies observe Landau levels directly through a field-dependent

107

ladder of peaks [2, 3, 12, 5] in I-V characteristics, still others observe oscillatory
Shubnikov-de Haas behavior of either current or conductance [8, 7, 5]. Referring to
Figure 4.1, the origin of SdH-like oscillations in perpendicular transport through
RTD’s may be argued for zero temperature in the effective mass approximation.
With the field applied normal to the epitaxial layers, Landau levels form, and the
Fermi level in the emitter lies at the energy of the highest occupied Landau level.
As the field is increased, the spacing between the levels, iw,, and the maximum
carrier occupation of each level, eB/h, both increase. The Fermi energy thus rises
linearly with the applied field as the Landau level to which it is fixed increases in
energy. When the last carrier falls from the uppermost Landau level / into the ]—1
state, due to the increased occupation density of the lower states, the Fermi level
discontinuously jumps down to the next rung of the Landau ladder. This shift in
the Fermi level occurs whenever the total number of states in the lowest L Landau
levels identically equals the total 2D charge density in the emitter notch state,
LeB/h = N2. Thus, for fixed bias, and therefore fixed N2», the fields at which
E'p jumps to a new level are periodic in 1/B with a fundamental frequency defined
by 1/By = e/(ANZ”). In 2D transverse transport measurements, the resistivity
shows an equivalent periodic behavior as scattering increases dramatically when-
ever the Fermi energy jumps down to a fully populated level. In perpendicular
transport through an RTD, however, resonant elastic transport involves both well
and emitter states and the SdH periodicity arises from the alternate alignment
and misalignment of emitter and well states conserving energy and | + m,;. Fora
constant bias, at magnetic fields for which the emitter state at the Fermi energy
and a well state of identical /-+m, have equal energies, conductance is maximized.
At intermediate fields, the tunnel current is decreased, and at sufficiently low tem-
perature should fall to zero when no states meet the elastic criteria. However, at

more elevated temperatures, Landau level broadening results in non-zero tunnel-

108

ing conductance minima. The periodicity of Landau level tunneling as a function
of the applied field is therefore not determined from the oscillation of the Fermi
energy in the emitter alone, but rather by the structure of both the emitter and
well Landau level ladders.

The fields for which the tunneling conductance is maximized may be derived
from simple considerations of energy and longitudinal momentum conservation.
First, consider resonant tunneling in a RIT p-well device obeying the simplified
selection rule, Al = 0. In the effective mass approximation, the energies of the

emitter and well states are,
E=Eg+(l+4) fw, (Emitter) (4.24)
and,
E=Ey-(i+4) hw”, (Well) (4.25)
respectively, where Hg and Ey are the emitter and well B = 0, ky = 0 band edges,
and hw? and hw” are the emitter and well cyclotron energies, respectively. For

tunneling processes conserving both energy and Landau index, these energies are

equal, and conductance maxima occur for,

AB = By ~Ep=(lr+5)* B

; (4.26)

pr?
where Ip is the index of the emitter Landau level in which the Fermi level resides,
and y* is the reduced mass defined by 1/u* = 1/m%, + 1/m4,. Therefore, the
inverse field at which the conductance is maximized by transport through Landau

level J is given by,

1 eh 1
—= i+-— .
B, AE ( + 5) , (4.27)
and the fundamental frequency of the SdH conductance oscillations is,
1 1 1
ee — 4.28
By Bur B (4.28)
= — (4.29)

109

The fundamental field thus depends upon the mass and band edge of both emitter
and well states.

The situation is more complex when the full z-component of angular momentum
is conserved such that A(l+m,) = 0. For the RIT structures studied, the incident
carriers all lie in the conduction band (CB) of the InAs emitter, and therefore
correspond to angular momentum states of |} + $) (the split-off bands will be
ignored for the remainder of the discussion, and all ls + $) states will represent
conduction band states and not split-off valence band states). In the absence of
an applied field, at ky = 0 the |} 5) state in InAs only couples to the light hole
(LH) |2 $) state in the GaSb well. It is forbidden for a tunneling electron to end
up in a heavy hole (HH) |$ + 3) state. However, for k 4 0, the LH and HH states
mix, becoming mixed angular momentum states. The LH and HH labels then, in
general, refer to energy bands and not specific angular momentum eigenstates. The

|HH ¢) and |HH |) bands at non-zero k, may be written as linear combinations

of several of the zone center basis states, such that, for By > By, and a > By, By

for |ky| < *%,
33 3. 6d 3 1
HH t) =a|~ 5 — <= .
and, 6, > 64, and y > 5+, 6, for |ky| « 2,
3.8 31 3. 1
H =7|= —-)+6-= =~-—-), .
|HH ) = 915 — 3) +6135) +413 — 3) (4.31)

Thus, for a non-zero kj, CB states in the InAs emitter of a RIT can couple to

the HH band through the LH-like m; = +3 components. With the magnetic

field on, tunneling electrons conserve | + m,; and CB states may couple to either
LH-like (m; = +5) or HH-like (m; = +3) states with a change in Landau index
Al = —1,0,+1, +2 for the |$ 3), [2 5), |2 — 4), and |$ — 3) states, which in turn
correspond to the primary components of the HHt, LH|, LHt, and HH bands,

110

respectively. Writing the energies in the emitter and well as,
E = Ep + (1+ 5) twF, (Emitter) (4.32)
and,
B= By —(1+Al+3)fw, (Well) (4.33)

the magnetic fields corresponding to conductance maxima of Shubnikov-de Haas

oscillations are given by,

I eh 1 eh,
—= i+ — Al 4.34
B AE ( + 5) + AEm, ° ( )

and the fundamental field of the oscillations is defined by,

1 1 1 eh
Se TL 4.35

Thus, the frequency of the Shubnikov-de Haas oscillations remains the same un-
der the selection rule A(I + m,;) = 0, but the phase is altered by an amount,
ehAl/AEm*,, dependent upon the change in Landau index (determined by the
final hole subband eigenstate), the effective mass of the hole subband, and the

zero-field energy difference of the well and emitter subband edges.

4.3 Experiment

4.3.1 Measurement Technique

T'wo p-well RIT samples were investigated in magnetic fields as great as 8 tesla
aligned parallel to the tunnel current. These samples, having well widths of 7.0 nm
and 11.9 nm, respectively, are the same devices as samples A and C in Chapter 3.
The specifics of the device geometries were summarized in Table 3.1. A 16.0
nm well InAs/AlSb double barrier RTD was investigated as well, to contrast the

behavior of otherwise-identical Type I and Type II broken-gap structures. The

111

details of device fabrication and nominal zero-field characteristics are summarized
in Section 3.2.2. The alignment of the sample relative to the field was verified to
within 0.5° by an adjacent Hall probe. All measurements detailed in this chapter
were made at a temperature of 4.5 K. Several experiments of transport at % 1.6 K
were conducted, but no significant deviations were observed from the data taken at
higher temperature. All conductances were derived from the I-V data by fourth-
order nine-point Savitzky-Golay [19] numerical differentiation. The I-V data was
taken in 1 mV steps, and discrete conductance values were obtained at 5 mV

intervals.

4.3.2 Current-Voltage Characteristics

Figure 4.2 shows the I-V characteristics of the 11.9 nm-wide well sample at 2
and 6 tesla, respectively. The behavior of the 7.0 nm well device was similar.
While some change to the I-V curve is noticeable, especially at 6 tesla, the most
profound changes in behavior are noticeable in the conductance at a fixed voltage
for magnetic fields between these two values. For B < 2 T,, little effect was observed
directly in the I-V curve, except for a subtle shift of all features to higher biases as
the field strength was increased. Above 6 tesla, the NDR region narrows rapidly,
and disappears entirely by 8 T, presumably due to the effects of magnetoresistance
in the contact layers. As the back contacts of these devices were not on the opposite
face of the wafer, but instead surrounded the perimeter of each device, current
from the back contact to the mesas experiences a maximum magnetoresistance
in this geometry. This field-dependent resistance was unavoidable, but does not
appear to significantly impact the experimental results. The features seen in the
NDR region in Figure 4.2b, labeled A through D, may be the signature of Landau
level formation at this field, although their presence in the NDR region limits the

degree of confidence which may be assigned to them. The entire structure of the

112

p-well RIT I-V Characteristics (B Il J)
T=4.5K, Log, = 11.9 nm

a)B=2.0T
10.0 F a

7.5 5

5.0 F .

os)

12.5 , ! ¥ T ¥ T ¥ T

Current (mA)

10.0 F

7.5 5

5.0 F

2.9 5

0.0 ‘ i . L ry L A i n
0.00 0.05 0.10 0.15 0.20 0.25
Voltage (V)

Figure 4.2: Representative current-voltage characteristics of the 11.9 nm well RIT
device at 2 and 6 tesla. These fields represent the extremal values for which
Shubnikov-de Haas oscillations of the conductance were observed. The uniform
spacing between the features labeled A through D on the B = 6 TLV may be

signatures of Landau levels.

113

NDR region is thought to be dominated by circuit instabilities due to the negative
differential RTD resistance [20, 21]. However, the regular spacing of these small
peaks (the A-B and B-C spacing is 10 mV, while the C-D spacing is 5 mV) is
suggestive of a Landau ladder with iw, = 10 mV. In such an interpretation, peak
D would correspond to the band edge of the quasi-bound state, and peaks A, B,
and C to the | = 2, 1, and 0 Landau levels, respectively. This interpretation is
consistent with the presumed structure of the highest energy valence band state
in the GaSb well (the HH1 state), as the higher Landau levels should be accessible
at lower voltage, due to the negative dispersion. With sufficient level broadening,
the band edge might be experimentally accessible, and the assignment of peak D
to such is consistent with the abrupt onset of the valley current at slightly higher
biases. A 10 mV cyclotron energy at 6.0 tesla is consistent with an effective mass of
0.07 m,, which is a reasonable number for the HH1 state (the 8-band k-§ model of
Liu et al. [14] predicts a value of © 0.12 m,). Unfortunately, the B-dependent width
of the NDR region and the related change in effective experimental resolution do
not allow the field dependence of these peaks to be observed with enough accuracy

to unequivocally attribute them to individual Landau levels.

4.3.3 Oscillatory Tunneling Conductance

Without additional features in the experimental I-V characteristics, it was nec-
essary to look at the tunneling conductance, dJ/dV, to observe the effect of the
field on quantum interband transport. Figures 4.3 and 4.4 plot the conductance of
the 7.0 nm- and 11.9 nm-wide well samples, respectively, against inverse magnetic
field at fixed biases of 25, 50, 75, and 100 mV. For clarity, arbitrary offsets have
been added to the data. In both samples, the peak-to-peak magnitudes of the
oscillations are no greater than 10% of the local average conductance at any given

field. Superimposed on the 7.0 nm well data, however, are a strong field-dependent

114

SdH Conductance Oscillations
T =4.5K, Lo, = 7.0 nm

Conductance (arbitrary)

0.20 0.25 0.30 0.35 0.40

1/B (T )
Figure 4.3: Tunneling conductance, dI/dV, of the 7.0 nm well RIT diode as a func-
tion of inverse magnetic field, at biases of 25, 50, 75, and 100 mV, respectively. An
arbitrary offset has been added to each curve for clarity. The peak-to-peak oscil-
lation amplitude is no greater than roughly 10% of the local average conductance

at any given field.

115

SdH Conductance Oscillations
T =4.5 K, Lo, = 11.9 nm

t . J I

Conductance (arbitrary)

ane ayo

V=100mV

\ , ! R R
0.20 0.30 0.40 0.50
1/B (T")
Figure 4.4: ‘Tunneling conductance, d//dV, of the 11.9 nm well RIT diode as a
function of inverse magnetic field, at biases of 25, 50, 75, and 100 mV, respectively.
An arbitrary offset has been added to each curve for clarity. The peak-to-peak
oscillation amplitude varies considerably but is no greater than roughly 10% of the

local average conductance at any given field.

116

monotonic background variations in conductance which cause the apparent mag-
nitude of the oscillations to appear less significant. In contrast, there is some
field-dependent background conductance at 25 mV and 50 mV in the 11.9 nm
well sample, but it is of much less consequence. At higher fields (lower 1/B), the
density of states in both the well and the emitter become highly non-linear, as
the Landau levels separate and become more distinct. As this occurs, the condi-
tions for resonant, elastic tunneling become more restrictive, leading to a lower net
tunneling conductance. However, when proper Landau levels providing resonant
energy- and angular momentum-conserving current paths align at a given field, the
conductance increases, causing the observed Shubnikov-de Haas-like conductance
oscillations. Thus, the magnitude of such oscillations is expected to be greater at
larger fields, when the difference between the on- and off-resonance current should
be maximized. As seen in both figures, this behavior is observed in the experi-
mental data. At sufficiently low temperatures, the magnitude of these oscillations
should increase, as the level broadening decreases. Qualitatively little improvement
was observed for either sample at a temperature of 1.6 K, the lowest temperature
achievable in our dewar. Oscillatory behavior was observed, to varying degrees,
at all biases lower than the peak voltage in each sample. These oscillations were
evident for magnetic fields in the range of roughly 2 to 6 tesla. Below that field,
presumably the level broadening due to scattering processes [22] is of the same
order as the cyclotron energy, giving a lower limit on the energy resolution of the
experiments. Above 6 tesla, the inverse field spacing between maxima requires
substantially higher fields for the observation of additional oscillations. SdH-like
oscillations were not observed in the InAs/AISb double barrier sample at voltages
both above and below the current peak, in all magnetic fields between 0 and 8
tesla.

Of particular note in each sample are the oscillatory characteristics as a function

117

Oscillation Transition Region
Lcasp = 11.9 nm

V =60 mV

y / V=63 mV

Conductance (arbitrary)

, ,

0.2 0.3 0.4
1/B (T")

Figure 4.5: Observed conductance values of the 11.9 nm-wide device plotted
against inverse magnetic field at biases of 60 mV, 62 mV, 63 mV, and 65 mV,
respectively. In this voltage range, there is a noticeable transition in oscillation
behavior. Additional peaks, highlighted by arrows, appear and grow at higher
biases. Especially sharp are the distinctions between the data taken at 62 mV

and 63 mV bias. With the additional 1 mV of bias come many additional peaks,

observable at even relatively low magnetic fields.

118

of applied bias. While Figures 4.3 and 4.4 show the behavior at only four voltages,
many of the general trends are contained in this data. In both samples, the phase
of the SdH-like oscillations is seen to vary with applied voltage. This trend is not
surprising, as changes in voltage alter the value of AF in Equation 4.34, causing a
voltage-dependent phase shift. Also, the quality of the oscillations deteriorates at
higher biases, especially as the applied voltage nears the I-V peak. This dependence
is especially noticeable in the wider-well sample. Whereas the oscillation peaks are
discernible in the narrow well RIT at voltages up to 110 mV, they persist to even
higher voltages in the 11.9 nm well sample, although their form becomes rather
complex at biases above 90 mV. The 100 mV data in Figure 4.4 show the non-
ideal conductance oscillation behavior in the wide-well device, having two or more
distinct frequencies and associated amplitudes. The anomalous oscillations at 75
mV in the 11.9 nm well are part of a transition region in which the conductance
oscillations all but die away and then reappear at higher biases with different
frequency. The nature of the conductance oscillations in the transition region is
shown in Figure 4.5. While the amplitude of the oscillation maxima seen at 60 mV
lessons, additional peaks, as shown by the arrows, appear as the bias is increased to
65 mV. In the 1 mV bias increase from 62 mV to 63 mV, many of the peaks rapidly
become obvious. For voltages between 75 mV and 90 mV, well-discerned, single-
frequency periodicity is observed in the wide-well data. The added complexities
at high bias should be expected in the wider-well sample; the quasi-bound states
clump together deeper in the well, increasing their interaction and complicating
transport at these voltages relative to the more-diffuse subbands of the narrow

well.

119

4.3.4 Fan Diagrams

A compact, efficient method of summarizing the Shubnikov-de Haas oscillatory
phenomena is the use of fan diagrams. These diagrams are a common means
of displaying the field-dependence of numerous transport features simultaneously.
Consider maxima occurring at inverse fields predicted by Equation 4.34. At a fixed
bias, the positions of the conductance maxima in 1/B should maintain a linear
relationship when plotted against a sequential series of integers. In the absence
of a change in Landau index (A/ = 0), when these integers are chosen such that
the z-intercept of the fan line lies between 0 and —1 (ideally at —}), they in turn
represent the Landau indices corresponding to each of the maxima. With such a
calibration, the physical parameters (AF and y*) of the system may be extracted
from the fan line slope, and multiple lines may be displayed in a fan-like pattern
to show the data for a variety of applied biases. When Al + 0, a distinction must
be made between the emitter and well Landau indices. If, as in Equation 4.34, the
emitter and well states are indexed by / and / + Al, respectively, the maxima are

found at,

arr GrareE (4.36)
In general, the resulting fan lines are not strictly plotted against either Landau
level index. The Al-dependent phase term can be no larger than Al itself; in
that limit (mj, « m*) the fan diagram ordinate is the well Landau index. For
Miy > Mz (u*/ms, — 0), the diagram ordinate is the emitter Landau index.
Without a priori knowledge of the masses, the Landau indices of the conductance
maxima cannot be properly calibrated when Al # 0. However, the phase itself
provides an important clue as to the biases at which transport through the well

switches from one subband to another; an abrupt change in Landau phase should

be observed when such a change in conduction channels occurs. Furthermore, the

120

bias-dependence of the slopes may still be used to estimate AF and p*.

Figures 4.6 and 4.7 show the fan diagrams of the narrow and wide well samples,
respectively. For each bias, the set of conductance maxima are plotted in addition
to a linear least-squares fit of the resulting fan line. A representative set of biases is
shown for each sample, although space and labeling difficulties limit the displayed
fan lines to coarse bias steps. Fan lines in addition to those shown were considered,
the parameters of which will be used in later analyses. Each fan diagram is divided
into regions delineated by thick lines and labeled by Roman numerals. The fan lines
in each region have similar Landau index phase, , or z-axis intercept. There are
abrupt changes in phase between these regions, which in turn are related to changes
in Landau level index during tunneling. As resonant transport shifts between well
subbands, there occurs a corresponding change in Al, and, ultimately, the observed
y. Thus, if in one bias region the incident carriers tunnel into well states in the
LH1 band, and consequently couple primarily to the [3 + +) hole components,
they should have Al = 0 or +1, depending on the incident spin state. This
change in Landau index introduces a phase shift of (eh/AEyu*)Al into the inverse
fields predicted by Equation 4.34. When, at slightly higher biases, the incident
electrons instead couple to a well state having a different mass or Al (or both),
the subsequent change in Landau phase is observed in the fan diagram. Strictly
speaking, the differing phases for opposite incoming spin states should cause two
sets of oscillations, of the same frequency, but of different phases. Such behavior
is not seen at most biases, and it therefore is likely that the well mass, My, and
the reduced mass, u*, are nearly identical, causing the phase difference between
spin states to be an integral multiple of the overall periodicity.

In general, all of the fan lines, for both samples, are linear and of a single period,
with greater variance at higher bias. This trend is consistent with the effective

mass approximation, and Landau levels equally spaced in energy. Deviations from

121

Conductance Fan Diagram

Leash = 7-0 nm
0.50 . J . T ¥ T uy T T T
0.40
is
= 0.30
4 5 6 7 8 9 10
Assigned Landau Index

Figure 4.6: Fan diagram showing the inverse fields at which maxima occur in the 7.0
nm well RIT conductance oscillations, plotted versus assigned Landau level index.
The ordinate value is assigned such that the fan line of a given bias intersects the
z-axis between 0 and —1, calibrating the individual maxima to well Landau indices

in the limit of wy, = wy. Thick lines separate regions of differing Landau-index

phase.

122

Conductance Fan Diagram

Leasp = 11.9 nm
0.50 7
Vi, 30mv

0.40

we
= 0.30

0.20

2 4 6 8 10 12
Assigned Landau Index

Figure 4.7: Fan diagram showing the inverse fields at which maxima occur in
the 11.9 nm well RIT conductance oscillations, plotted versus assigned Landau
level index. The ordinate value is assigned such that the fan line of a given bias
intersects the z-axis between 0 and —1, calibrating the individual maxima to well
Landau indices in the limit of wj, = w%. Thick lines separate regions of differing

Landau-index phase.

123

ideal Landau level behavior are expected at higher biases and in the wider well
sample, where interactions between bands deep in the GaSb well result in strongly
nonparabolic dispersions. Such behavior is somewhat evident in the 70 mV and
80 mV fan lines in region IV of the wide well diagram. The high bias cutoff of
fan data in both samples is attributable to non-ideal, multi-periodic conductance
oscillations and decreasing oscillatory amplitude. The narrow well fan lines in
Figure 4.6 are well-ordered and evenly spaced as a function of bias, although there
is a large shift in fan position between biases of 100 mV and 110 mV. In contrast,
the data for the wider, 11.9 nm well are less ideal. There is a very abrupt shift in
slope between the 80 mV and 90 mV fan lines, causing the latter to more nearly
approximate the 30 mV data. In addition, although they could not all be shown,
the biases intermediate to 40 mV and 70 mV in region III have slightly decreasing
slopes as a function of bias, while the slopes from the 15 mV to 30 mV data in
region IT actually increase. The added complexity of transport in the wide well
sample is also evidenced by the larger number fan diagram regions and observed
shifts in Landau phase.

Figures 4.8 and 4.9 plot the fan line slope and Landau phase ¥ at 5 mV bias in-
crements for the narrow and wide well fan diagrams, respectively. The dashed verti-
cal lines, and corresponding Roman numerals, denote the various regions of similar
Landau phase. The narrow well data shows very distinct and abrupt changes in
both slope and phase between each of these regions. The bias dependence of these
slopes is useful for the determination of the well state mass and energy. As in-
dicated in Equation 4.34, the slope M of the fan lines should be related to these
parameters as M = eh/y*AE. The voltage across the device enters this equality
through the energy difference, AF = AE) — aeV, where AE) = Ew — Ep is the
zero-bias energy difference of the B = 0, k = 0 band edges of the well and emitter

states, and a is the fraction of the total voltage dropped between the the emitter

124

7.0 nm GaSb well RIT
fan slope and phase
0.050 | t | r qt ’ v ‘ ‘| . 1.00
lo 4 I | Ili Il
— | |
| ol
l Po | "|
0.045 + o| | ! 4 075
"| log se”
| I a | =<
+ | | © 6 "ao | 5
© 0.040 F | | s | 71 0.50 €
& | | 2 (0°O | sc
n | | | FS)
| | S96 }| @
| ir <——o'l"
| | |
0.035 + | | | o 7 0.25
| 7 | |
le es | lo
| I |
i a | |
| a! |
0,030 Wb ts 0.00
0 20 40 60 80 100 120
Bias (mV)

Figure 4.8: The slope and Landau index phase (y) for fan lines of the 7.0 nm
well oscillation data plotted against applied bias at 5 mV increments. The ver-
tical dashed lines separate regions between which large, discontinuous changes in

Landau phase occur.

125

11.9 nm GaSb well RIT
fan slope and phase
0.100 , } 1 1 T r 1.00
I | II | II | Vi V
| | | ,
| | | |
I | | | o
0.080 + ! ! ! | 4 0.75
| | | |
| | | | =
7 7 ] a
& °| ol ee j ">| 2
Q 0.060 F 6 | ° I | " | 7 050 €
fe) | I ao)
s Ke IOP o001 I B
| a| | | o
| | | | at
| | | |
0.040 mw | | | ml og 4 0.25
| |
| | ° or
la I | |
| | | {
ell 1 | i . | : Ll L.
0.020 0.00
0 20 40 60 80 100
Bias (mV)

Figure 4.9: The slope and Landau index phase (7y) for fan lines of the 11.9 nm
well oscillation data plotted against applied bias at 5 mV increments. The vertical
dashed lines separate regions between which large, discontinuous changes in fan
slope occur. Unlike the 7.0 nm well data, these discontinuities in slope are not
matched by simultaneous shifts in Landau index phase. The phase shifts appear
offset by up to 5 mV from the discontinuities in fan slope, perhaps because the

transitions between subbands are less abrupt in this sample.

0.15

0.10

Energy (eV)

0.05

0.00

7 om Well

126

B=0T

12 nm Well

HH1

HH2 ee ee
4 N
/ LHI
L. / N = a
/ N
/ \
/ \
f \
/ \
/ \
HH4
pHa
“ LH2 *
[ re ar 1 CA eT ee ee
-l1 0 1 2-2 -1 O 1
k, (% n/a)

GaSb

Edge

InAs
CB
Edge

Figure 4.10: Theoretical [14] zero-field well subband structure for RIT devices

having GaSb well widths of 7.0 and 11.9 nm, in the energy range between the InAs

conduction band edge and the GaSb valence band edge.

127

state and the well. The parameter a is expected to be less than 0.5 as the potential
drop from the outer emitter to the center of the well should be one half the total
voltage in a symmetric device. As the bias is increased, the emitter notch state
falls lower into the accumulation region, and while a is not strictly constant, it is
estimated from band bending calculations to be be roughly 0.35 for the voltage
range in question. This accuracy of this estimate, and the lack of a voltage depen-
dence to a, is no worse than the other assumptions and approximations that will
be made in this analysis. The overall voltage dependence of the fan line slope is
therefore inversely proportional to the applied bias. Rewriting the inverse slope,

1/M, as a function of the applied voltage yields an expected linear dependence,

1 yt
Mah [AB — 0.35 eV], (4.37)
having a slope,
6 = O(1/M)/OV = —0.35 u*/h (4.38)
and an offset,

Fitting the slopes in region I of the narrow well diagram gives estimates of 3 =
36 + 13 T/V and (1/M)y = 20.0 + 0.3 T. Likewise, in region II, fitting gives
6 = 7445 T/V and (1/M)o = 20.0+0.4 T. In both regions, the voltage-dependence
of the inverse slopes is positive, whereas Equation 4.37 predicts they should be
negative. This fundamental difference can only be explained by the existence of
positive dispersions for these valence band states. In such a case, y* is no longer
the reduced mass of the two bands, but rather the related quantity 1 [my —1/m*,.
When my < m%, this quantity is negative, and leads to the observed behavior
of the fan line slopes. Because both masses are implicitly involved in determining
the voltage dependence of the fan line slopes, the well mass cannot be analytically

determined from the values obtained in fitting the data to Equation 4.37. However,

128

if the emitter state mass is assumed to be identical to the InAs bulk mass, then
details of the well subbands may be extracted from these fits. In region I with these
assumptions, the data imply mj, = 0.008m, and AE) = —200 meV. Similarly, in
region II, these assumptions yield mj = 0.012m, and AE’ = —95 meV. Neither of
these sets of values is reasonable, however. The bulk energy difference between the
InAs conduction band and GaSb valence band edges is only 150 meV, and band
bending and quantization in the heterosystem reduced that further. Thus, the
large energy differences, while still possible, are less plausible. Additionally, since
the well mass differs substantially from the reduced mass in both of these cases,
two sets of conductance oscillations of identical frequency, but offset by a phase
factor, would be expected from the two possible spin states, but were not seen.
Thus, it appears more likely that the in-plane mass of the emitter band is rather
large, relative to the bulk value, such that the well mass is roughly equal to the
magnitude of the reduced mass. This assumption would give masses of 0.012 m,
and 0.024 m,, as well as energy offsets of -135 meV and —45 meV, respectively, for
the well states in regions I and II. For comparison, the zero-field band structure
predicted by Liu et al for these devices is shown in Figure 4.10. While theory
predicts a positive dispersion for the HH2 subband, the exact relationship between
the theory and experiment is uncertain. First, only one band is predicted to have
positive dispersion, and yet a single band cannot account for the behavior in both
regions I and II of the fan diagram, especially since these together span nearly 100
mV of bias. It is possible that the HH1 band has a positive dispersion for small k
as well, yet the cause of this deviation from the theory is unclear.

The wide well fan data display even greater complexity. The Landau phase
changes discontinuously at four different biases, although, unlike the narrow well
device, the discontinuities in slope do not, in general, occur at the same biases. In

region IV, the data for the bias range of 60 - 70 mV is omitted due to the extended

129

change in behavior shown in Figure 4.5. In all but region III, the slope increases as
a function of bias, as predicted by Equation 4.37 for negative dispersion bands. In
region III, G@ is statistically indistinguishable from 0, indicating that p* is essentially
infinite. Transport in this voltage regime is thus likely due to a positive dispersion
state of mass nearly identical to the emitter mass, such that 1/m% — 1/m}, = 0.
Least-squares fits to the inverse slope in region II give 6 = —154+ 29 T, and
(1/M)o = 21+0.7 T. If we again assume m}, < m%, as we did for the narrow well
data, these fit parameters correspond to a mass of roughly 0.051 m, and energy
difference AKy of 47 meV. In region IV, the data does not appear to behave
linearly, and the effects of the transition observed between 60 mV and 70 mV
may introduce nonidealities at higher biases. Referring to the theoretical band
structure in Figure 4.10, transport in region II is most likely through the LH1 well
state, while region III corresponds to the HH2 band. Since the HH2 band has both
positive and negative dispersion at small k, (low Landau level index), the transition
occurring at 60 mV, and the subsequent nonideal (but negative dispersion-like) fan
behavior at higher biases may be experimental evidence of the change in sign of
the local differential mass. Or, more likely, as the HH1 and HH2 bands are only 10
meV apart, transport in the 60 mV to 70 mV region may involve mixed transport
through different Landau levels of each hole subband. As the wider well data
indicates, the closer the well subbands approach each other, the more complex and

varied transport through the device becomes.

4.4 Conclusions

Shubnikov-de Haas oscillations of the tunneling conductance through RIT diodes
provide an indirect means of exploring the subband structure of the GaSb well

and observing 0D-0D quantum transport behavior in a unique hybrid system.

130

The observation of distinct and abrupt changes in the Landau phase of fan lines
as a function of applied bias provide evidence that the Landau level index | is
not conserved in quantum transport involving states of differing z components of
angular momentum. The complete selection rule for 0D-0D Landau tunneling,
A(i+m,;) = 0, reflects the conservation of total longitudinal angular momentum,
including the macroscopic contribution of cyclotron orbits induced by an external
magnetic field. Through the study of the bias dependence of fan line slopes for
regions of similar Landau phase, the mass and band edge of states in the well may
be determined, though only in the limit of m¥, < m*. In general, only the reduced
mass of the emitter and well states can be determined experimentally. The former
limit, however, may be justified by the lack of separate, out-of-phase oscillation
sequences for the spin-up and spin-down components of each subband. In both
samples, the anomalous bias-dependence of the fan line slopes is indicative of a
positive, low-mass well dispersion for the range of Landau indices experimentally
accessible. The additional complexity of the structure in the wide well was observed
in variety of nonideal behaviors, including a 5 mV-wide region where oscillations
having several phases coexist.

This study, as one of the first complete investigations of magnetotransport in
Type II broken-gap resonant tunneling diodes, demonstrates the novel nature of
transport in the InAs/AISb/GaSb system. The evidence for the non-conservation
of Landau index in Shubnikov-de Haas oscillations of resonant, elastic tunneling
conductance is unique, and has implications for transport in other quantum trans-
port systems for which states of differing longitudinal angular momentum coexist
(e.g., hole tunneling in Type I heterostructures). Only very recently have ex-
perimental observations of Landau level mixing in unipolar hole tunneling been
reported [23]. The data contained in this chapter represent the first evidence of

Landau level mixing in interband tunneling, and the first observation of the conser-

131

vation of total projected angular momentum through phase shifts of Shubnikov-de
Haas oscillations. Additional ultra-low temperature (< 4.2K) studies of this and
other systems appear warranted in light of the data presented in this chapter. Ul-
timately, if evidence of Landau level tunneling remains limited to the conductance,
additional measurements with a tunneling spectrometer, in which the conductance

is measured directly and with greater accuracy, may be appropriate.

132

Bibliography

[1] K. von Klitzing, Rev. Mod. Phys. 58, 519 (1986).
[2] C. E. T. Gongalves de Silva and E. E. Mendez, Phys. Rev. B 38, 3994 (1988).

[3] M. L. Leadbeater, F. W. Sheard, and L. Eaves, Semicond. Sci. Technol. 6,
1021 (1991).

[4] C. H. Yang, M. J. Yang, and Y. C. Kao, Phys. Rev. B 40, 6272 (1989).
[5] E. E. Mendez, Surf. Sci. 267, 370 (1992).

[6] E. E. Mendez, L. Esaki, and W. I. Wang, Phys. Rev. B 33, 2893 (1986).
[7] S. Ben Amor et al., Appl. Phys. Lett. 54, 1908 (1989).

[8] V. J. Goldman, D. C. Tsui, and J. E. Cunningham, Phys. Rev. B 35, 9387
(1987).

[9] M. L. Leadbeater et al., Solid-State Elec. 31, 707 (1988).
[10] L. Eaves et al., Appl. Phys. Lett. 52, 212 (1988).

[11] E. E. Mendez, H. Ohno, L. Esaki, and W. I. Wang, Phys. Rev. B 48, 5196
(1992).

[12] C. H. Yang, M. J. Yang, and Y. C. Kao, Phys. Rev. B 40, 6272 (1989).

133

[13] Takamasu et al., Surf. Sci. 263, 217 (1992).
[14] Y. X. Liu, D. Z.-Y. Ting, and T. C. McGill, unpublished.

[15] E. O. Kane, in Semiconductors and Semimetals 1, edited by R. K. Willardson
and A. C. Beer (Academic Press, New York, 1966).

[16] A. Fasolino and M. Altarelli, Surf. Sci. 142, 322 (1984).
[17] D. Smith, private communication.
[18] J. M. Luttinger, Phys. Rev 102, 1030 (1956).

[19] A. Savitzky and M. J. E. Golay, Anal. Chem. 36, 1627 (1964); See also W.
H. Press, 5. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical
Recipes in C, 2°4 edition, (Cambridge University Press, Cambridge, 1992), pp.
650-655.

[20] V. J. Goldman, D. C. Tsui, and J. E. Cunningham, Phys. Rev. Lett. 58, 1256
(1987).

[21] T. C. L. G. Sollner, Phys. Rev. Lett. 59, 1622 (1987).
[22] T. Ando, A. B. Fowler, and F. Stern, Rev. Mod. Phys. 54, 437 (1982).

[23] A. Zaslavsky et al., Surf. Sci. 305, 307 (1994).

134

Appendix A

Magnet History and Operation

A.1 Introduction

Well, if you are reading this appendix, you are probably planning on using the 8
tesla superconducting magnet. Since you are, and I was the individual responsible
for the original purchase, | feel obligated to provide you with a certain amount
of historical background and a few heuristic operating guidelines. For details of
the magnet design, and the “official” factory instructions, you should refer to
the documentation provided by Janis [1, 2, 3]. After reading this appendix, I
hope you understand the strengths and weaknesses of the magnet design, the
potential problems you may encounter, and the proper operation of the magnet
and associated cryogenic dewar. Let us both hope that you need not repeat the
same mistakes, and encounter the same design flaws, that I have.

The magnet itself is a split-coil Nb/Ti alloy superconducting solenoid (from
American Magnetics), and is mounted in a D2-style Janis dewar with optical access
both perpendicular and parallel to the field axis. The details of the design are
summarized in Reference [1]. The magnet specifications, taken from this document

as well as determined empirically in routine operation, are summarized below:

135

Rated Central Field at 4.5 K

Rated Current (at 4.5 K)

Field-to-Current Ratio

Homogeneity

Inductance

Max recommended charging voltage, B < 5 T
Max recommended charging voltage, B > 5 T
Total magnet resistance, 300 K

Total magnet resistance, 77 K

Total LHe chamber volume (w/out magnet)

Estimated 100% LHe volume

Inner-most windows

Outer windows

8.00 T

73.1 Amps
0.1094 T/A

0.1% (1 cm DSV)
39.6 Henries

2.5 V

1.5 V

21.7 Q

16.6 Q

rw 26 liters

ey 20 liters

Zns

ZnS or Quartz

Table A.1: Magnet design specifications

136

A.2 History

The magnet apparently has a long history in the McGill group dating back long
before it was ever purchased. It was a phantom piece of equipment that appeared
in many proposals, but which never found a student sponsor. Its true justification
came when the work with the interband InAs/AlSb/GaSb devices progressed to
a stage that magnetotransport measurements were seen as important. I believe
that you will be amused at exactly how many former students return for a visit
to be amazed that a magnet was actually bought and used. Unfortunately, there
were many pitfalls associated with the purchase of the magnet itself, the details
of which you should be aware in case the problems arise yet again. In general, I
think it fair to state that we (namely, Tom and I) are not impressed with either
Janis or American Magnetics, the two vendors which assembled the system.
While there were many of the problems which might be considered typical for a
large equipment purchase, such as shipping delays, inappropriately-wired electron-
ics, and poor quality control at the factory, the biggest single problem encountered
with the dewar was with the inner window seals around the sample tube. The
dewar was specified to have ZnS windows so that the system may be used for
magneto-optical experiments on far-IR detectors. Janis however, does not, ap-
parently, have a good track record with the installation of specialized windows.
Problems arose with a previous purchase of a dewar with ZnSe windows as well.
Basically, two flaws were found with the windows when the dewar arrived at Cal-
tech. First, they were absolutely filthy; so much so, in fact, that they were nearly
opaque in one direction. While the outer windows may be removed and cleaned, or
replaced, in a straight-forward manner, the inner-most set is not designed for easy
access, and is hermetically sealed to maintain vacuum isolation of the sample tube

(in principle, at least, see more later). Because ZnS is very soft, we were loath to

137

clean them ourselves, lest we scratch the surfaces. When the second problem with
the dewar was discovered, a leak in the sample tube, the whole magnet was sent
back to the factory for repair, and while there, the inner windows were cleaned, and
scratched, by Janis personnel. The leak in the sample tube was rather extreme;
when pumped down to a few hundred microns, it would return to atmosphere in
about 15 or 20 minutes. The problem, it turned out, was the inner window seals
themselves, and the complications introduced by ZnS. When the magnet returned,
it was cooled down and first-field was achieved on June 15, 1992. Unfortunately,
by August of that year, there was again a leak in the sample tube, and the whole
dewar was sent back to Janis yet again for repair of the inner window seals. The
entire system was not fully operational until January of 1993, fully two years after
it was first ordered. The inner window seals appear fine to date, but they may
cause problems in the future, and should be the first suspect should a leak in the
sample tube develop.

In addition to the windows, there are several other design flaws that I have
encountered of which you should be made aware. Some of these were simply due to
naiveté on my part when the system was specified, but others are true engineering
problems. First, a thermometer of some sort should have been directly mounted
onto the top of the magnet coil, so that the temperature of the magnet could be
directly monitored during the cool-down process. The danger is that if the magnet
is not completely pre-cooled to 77 K, the subsequent cooling step to 4.2 K will
require vast quantities of LHe. I have determined empirically that the resistance
of the coil when at 77 K is 16.6 Q, and this fact may be used in lieu of a thermometer
during the magnet pre-cool. The meter used to display the magnet current is only
accurate to a tenth of an Ampere, which represents an uncertainty in the field
greater than the designed magnet homogeneity. This problem is especially bad at

low fields, and is compounded by the fact that the meter also seems to have an offset

138

associated with it. As a work-around, I used a microvolt meter to directly monitor
the current shunt voltage, and get a more accurate reading of the field strength.
This procedure worked well, and is recommended in all further operation. The
computer control aspect of the system is admittedly crude, but also seems plagued
with offsets and drift. Consequently, the field could not be reliably set or read
remotely, and all attempts at full computer control were abandoned as the delays
associated with the inner windows developed. Lastly, the current reversal switch
never seems to work (although you will probably never need to use it) because of
the offsets in the built-in current ammeter. The reversing switch refuses to operate,
and chimes an annoying warning, if it thinks there is current still present in the
coils. This feature is good, but because of the offset in the meter (which never was
fully compensated), the reversal switch always “sees” current in the magnet, even
when the power supply is off. Lastly, the evacuation valve for the vacuum jacket
is positioned at such an angle that the LN2 access vent nearest it interferes with
connection to the turbo pump. As a result, I rigged together an extension tube
for the pump. It should be stored in the toolbox near the magnet. If it cannot be
found, another must be acquired before the dewar can be used. None of these are
major problems, and so long as the inner windows hold, I think you will find the
system (as I left it, at least), to be quite functional and straightforward to learn

to use.

A.3 Operation

The exact cool-down procedure recommended by Janis is found in Reference [1].
Incidentally, as these documents are often referred to during magnet operation,
a set is kept in the lab with the magnet. To satisfy proper instruction manual

archiving, however, an additional copy should be located in the manual storage file

139

cabinets.

The most important concept to keep in mind when operating the magnet is
safety. There are two main safety concerns: electrical shock, and cryogenic liquid
handling. While the magnet operates at high currents (upwards of 75 Amps),
the voltages involved are low (< 5 V), and therefore there is usually not much
of a hazard. However, if the current path is disrupted for whatever reason, the
massive inductance of the coil will cause immense voltages across (most likely) the
input terminals on top of the dewar. Thus, you should always be careful around
the magnet when it is charged, even if the power supply is off (I find that the
menacing sound of the power supply is more than enough to remind me of the
dangers involved, but in persistent current mode, where the power supply is off,
the calm and quiet belie the possible dangers). The danger from cryogenic liquids
is real as well. When filling the dewar with LHe, both the magnet dewar and the
LHe dewar may have cryogenic liquids under high pressure, and consequently, there
is a very real danger of frostbite if the flow out of either becomes uncontrolled. In
addition, there is an asphyxiation hazard, although with the proper ventilation,
this should not be a concern. Should the magnet ever quench (go normal during
operation), all 20+ liters of LHe will be dumped out the release valve on the side
of the dewar in a matter of a minute, and it is best to leave the room. The power
supply controller should drop the current to zero when a quench is detected, but
there is a danger that very large voltages will cause the current in the magnet to
discharge to a convenient nearby potential. Try to not make yourself the lightning
rod. Also, the risk of asphyxiation and frostbite is greatly increased in the presence
of that much LHe angrily venting from the side of the dewar. On the one occasion
when a quench occurred as I was using the system the room was subsequently
cooled to the low 60° F-range. The entire experience was quite impressive. It

should be noted that there are occasions in which you will find it necessary to

140

clamp off the release valve for the LHe chamber. Often this is done to increase the
He gas overpressure in the chamber if there are flow problems through the capillary
to the sample tube. If you do clamp off the release valve, which, although I have
done so, I should not recommend, be always on the ready to yank the clamp off
in the case of a quench. When the quench did occur as I was using the magnet,
the valve was clamped off, and if I had not released it, the dewar may very well
have been a twisted shard of metal at this moment. In general, it is best not to
fiddle with the safety features of the equipment. Lastly, beware of the effect the
field has on external items. Always keep loose tools away from the magnet while
in operation. The opaque plexiglass window guards were installed to keep metal
objects accelerated by the field from breaking the exterior windows, in addition to
keeping the sample free of external illumination.

The cool-down procedure, detailed in Reference [1], is summarized below.
1. Evacuate vacuum jacket (using turbo pump).

2. Evacuate helium reservoir (using mech pump attached to Tri-clamp connec-

tor off of the release valve. Open needle valve and evacuate sample tube as

well).

3. Back-fill helium reservoir and sample tube with He gas and seal. Close needle

valve to isolate the two, and maintain He pressure to both.
4, Fill nitrogen reservoir with LNo.

5. Fill helium reservoir with LN2. Use same transfer line for both LNg fills. The
copper tube at the end is designed to fit in the seal for the initial LHe fill

port.

6. As the LN2 is added to the LHe reservoir, there will at first be a rapid and

violent boil-off, and subsequent high pressure in the LHe chamber. It is

10.

11.

141

easiest to relieve this by keeping the release valve open with a spare gasket
or such jammed between the two faces of the spring-loaded valve. Also, at
this stage, continue He gas overpressure to the sample tube, but discontinue

such to the LHe chamber once the LNg creates a suitable overpressure.

. Monitor the coil resistance across the input terminals. The power supply

should be off, but the controller, and current reversal switch need to be on
due to the active circuitry in the energy absorber module. Once the magnet

reaches 77 K, fill for a little while longer, and then turn off LN flow.

At this point the dewar can sit overnight, with positive He gas pressure to
both the sample tube and the LHe reservoir. Usually, I performed all steps

to this point the night prior to the actual use of the dewar.

Siphon off the LN» from the LHe reservoir (usually the next morning) by
inserting the LHe transfer tube down the initial fill port, and applying an
overpressure of He gas. Continue roughly 5 minutes beyond the time the last

LNo2 flows from the tube.

Pump down LHe reservoir. It should reach pressures down in the hundreds
of microns. If it does not, and only goes as low as 1 torr (or greater), there is
still LN» in the chamber, and He gas should be flowed in, and then pumped

on, sequentially, until the pressure lowers well below 1 torr.

Backfill with He gas, and then begin LHe transfer. Start cooling down the
transfer tube first, while preparing to open the seal to the initial fill port.
When LHe is coming out of the transfer line, insert the tube into the fill
port. Jam open the release valve to minimize the overpressure in the LHe
chamber, and transfer the LHe until the dewar is full. Note that you may

have to pressurize the LHe dewar with He gas to continue the transfer beyond

142

the first few minutes. The LHe chamber must be a minimum of 65% full for

the magnet to be used without fear of quenching.

12. When the LHe chamber is full, apply He gas overpressure, and open needle
valve to cool sample to desired temperature. Open valve to manifold of the
vapor-cooled leads (if not opened earlier). Allow leads to cool before current
is applied. These must be kept cold by a flow of He, or they will vaporize at

high currents!

13. Let entire system sit for 15-30 minutes before charging the magnet. (Be sure

to let the power supply warm up at this time.)

14. You are ready to magnetize!!!! Be sure to keep the LN» reservoir at no less

than 80%, or else the LHe evaporation rates will be excessive.

Aside from safety, the most important thing to do during cool-down is not
allow atmospheric gases into any portion of the dewar that is cold. Usually it is
sufficient to keep a 1-3 PSI He gas overpressure to both the LHe reservoir and the
sample tube. If you do not, Nz ice can form in the LHe chamber, possibly causing
a rupture. And the capillary tube between the LHe reservoir and the sample tube
may freeze. If this happens, the whole dewar must be warmed up to clear the ice,
and allow proper sample temperature control.

There are very few maintenance issues with the system, so long as it is used
properly. If you have to change windows, you will need the big yellow monster
crane that the group owns. It takes about half a dozen strongish people to setup,
and needs to fit on either side of the optical table where the dewar is mounted.
There are two chain hoists associated with this crane. The dewar (and stand)
must first be lifted off the table and onto the floor before the cryostat can be lifted

from the vacuum jacket. Attach one hoist to either side of the dewar with the

143

eyelet fasteners on the top. Disconnect the seating screws on the underside of
the main O-ring flange, and lift the inner guts of the dewar out by evenly lifting
both sides with the chain hoists. There is only a few microns of tolerance between
the upper-most reaches of the crane, and the height to which you have to lift the
cryostat. Try and minimize the height of your connections between the eyelets
and the hoists (we had to bypass the hooks and attach the eyelets to the chain
directly). Once the dewar is out, the second layer of windows is easily accessible. It
can be harrowing to see the cryostat floating so precariously above the ground, but
I have personally gone through this procedure at least 20 times (due to the window
problems) without a major incident. The only other maintenance that I have had
to perform was to sand down the copper contacts for the power supply connection
to the magnet. These get wet and have large amounts of current going through
them and therefore corrode very quickly. Sanding both sides of the contacts down
can dramatically improve the magnet operation. The series resistance due to the
corrosion can adversely change the charging characteristics of the magnet. I had
to do this when I began to notice large offset voltages were necessary to charge the
magnet.

Well, good luck, and enjoy your new cryogenic toy!

144

Bibliography

[1] “Operating Instructions for Superconducting Magnet Systems,” Janis Re-
search Company. (This document contains specific design parameters of our
dewar in addition to generic operational guidelines for the magnet cryostat
and the Supervaritemp variable temperature insert. It is a must-read before

attempting to cool down the dewar.)

[2] “Operating Instructions for Supplied Ancillary Equipment,” Janis Research
Company. (This is basically a collection of manuals for various pieces of equip-
ment bought with the dewar and magnet. Many are from American Magnetics,

which made the superconducting coil.)

[3] M.N. Jirmanus, “Introduction to Laboratory Cryogenics,” Janis Research

Company. (A good overview of cryogenic laboratory technique.)

145

Appendix B

Bulk &.% Theory Incorporating
Magnetic Fields

B.1 Introduction

As was motivated in Chapter 2, the effect of a magnetic field upon the physics
of carriers in bulk semiconductors fundamentally occurs in k-space, where the
vector potential, A, describing the field, modifies the Hamiltonian. Aside from
spin-dependent potential energy terms, the field enters the general single-particle
Hamiltonian through the kinetic energy term. The momentum operator @ becomes
p- gA in the presence of a magnetic field, and the vector gA/ A may be viewed
as a B- and position-dependent vector in k-space. Because the field enters the
Hamiltonian naturally in k-space, solutions of the Hamiltonian are best attempted
using k- p, rather than real-space methods such as tight binding. Because this
thesis makes reference to the k - p formalism, and the specifics of 2- and 8-band
models derived from it, the goals of this chapter are to motivate this theory in bulk,
show how the zero-field Hamiltonian is changed in the presence of a magnetic field,

and lend a degree of intuition to the predicted results.

146

B.2. Bulk &-p Theory

We will first consider the zero-field, bulk Hamiltonian, and derive the formalism
on which & - p solution techniques are based. We start with the time-independent

single particle Hamiltonian,
Dp ~
HW = |}—4V(r)| V= EV, B.1
E+ ve] (B.1)
where m is the free electron mass, p and 7 are the momentum and position oper-
ators, respectively, and V(r) is the bulk crystal potential. We consider solutions

taking the form of a plane wave multiplied by a crystal-symmetric Bloch function,

u,z(7), having the periodicity of the lattice:
Uae Fy (7). (B.2)

Substitution of this form of the wavefunction into the general Hamiltonian gener-

ates the k - @ Hamiltonian for the unzl?),

& + és | ung?) + V(F)u,c(*) =Enu,c(7). — (B.3)

2m ™m 2m " " ”

In principle, this is an infinite-dimensional system, indexed (through n) by the
bands to which the wu, ;¢(7) correspond. To solve Equation B.3, we must first pick
a basis. It is traditional to chose the orthonormal zone center Bloch functions,
Uno(7), as the basis in which to express the wu, ;(7). The choice of this basis set
is convenient only when band structure near a k = 0 minimum is relevant. To

expand about a different ko, the Bloch functions at that reciprocal lattice vector

should be chosen. If the general u,,;(7) are written as,

and we note that the un/9(7) satisfy,

(= + via) uno (®) = Eno tyo(?), (B.5)

147

then substitution of the expansion into Equation B.3 yields the relationship,

A(k - 2k?
( P) Un'o(F) + om

(B.6)

yu Crn()

If we multiply both sides of this expression by u*,(7) and integrate over the unit

cell, we end up with the C-number Hamiltonian,

A? k? hk ~
n 7_>.. Onn! — Dron! nn! = En k nn!» B.
¥ |{e0+ | +p C. (k) C. (B.7)
where the parameters ,, are defined by,
Ban! = I ut oF) Puno(*) dF. (B.8)

This Hamiltonian, H,,, is still infinite in extent, but for most problems only a
few bands are of primary importance. Taking only the lowest order symmetries
of the crystal, we consider states s, x, y, and z having spherical, x-, y-, and z-like

symmetry, respectively. For these zone-center basis states, H,, in matrix form

becomes,
8 x y z
Whe AT > ho. ho.
6 E's + 3m ak ‘Psa ak ‘Psy ak * sz
kL. > h2 k? AL. > kE.p
yon 7 [mb Bes Bet oor im Poy inh Pre (B.9)
an he hes paw apie
Yojm** Pys n° Pyx 9 Sm im * Pyz
AL oo hoo hi. h2k?
2 [mb Pes mb Bex mB Dey Be + “Om

where extending in the matrix all around these entries are the contributions of other
bands. If the magnitude of the off-diagonal terms fi/m(k - Pin’) are small however
(valid for this expansion for all k near zone center), the effects of other bands may
be treated as a perturbation. If, in general, we divide the states into two sets, A

and B, such that set A contains all the sets in which we are interested (s, x, y, and

148

z), and set B contains all additional states, then the perturbation theory due to
Léwdin [1] may be used to iteratively remove the interactions between states in the
two sets. In essence, this perturbation theory takes the infinite Hamiltonian and
block-diagonalizes it in terms of the states for which there is interest. Formally,

matrix elements of the renormalized Hamiltonian are related to those of H,, by,

, - (H.0)ig (Hoo) aj
Hi, = Hoods + 2 BE, —(Ha)on (B.10)

All coupling between A and B states is removed in the procedure, and a finite
Hamiltonian results. The infinite sums from the perturbative calculation are typ-
ically treated as parameters to the theory, and fit to experimental data. The re-
sulting 4 x 4 matrix for the A € {s,z, y, z} system is the 4-band k- p Hamiltonian
for the states closest to the band gap of the crystal.

When spin-orbit coupling is added, the more complicated Kramers basis (in
which the spin-orbit terms are diagonal) must be used as a basis for H,,, and
greater complexity arises. The Kramers basis, expressed as linear combinations of

S, X, y, and z, may be written as,

wi | 19m) | Dim Band
1115.5). | Ist) CBt
31155) | dpi +iy)t) HET
5 113.3) | Jel —y)t) +2l24)] | LEY
7 |1d.—2)e | Sql -iv)t) + 2H] [SOL ,
2 || 15.-4). | és) CBY
4119,-5) | gle- i) HH
6 113.9) | gl-M@+e)L) + 2let)] | Let
8 [laa)» | agli +iv)) + let] | sor

where the corresponding bands are indicated, and ft and | refer to spin up and

149

down, respectively. There are several sources in which the full derivation of the 8-
band k- p Hamiltonian is given [2, 3, 4]. The details which remain in the calculation
are the exact specification of the infinite sums as parameters, and the reduction
of these to a minimal set through consideration of the symmetry properties of the

crystal.

B.3 Incorporation of the Magnetic Field

The classic seminal work on the incorporation of magnetic fields into k- p the-
ory is due to Luttinger [4]. This paper is a must-read for anyone considering the
calculation of realistic band structure in a magnetic field. In the zero-field, bulk
Luttinger Hamiltonian, the k- p matrix elements are dependent upon the Lut-
tinger parameters, 71,72, ¥3, *, and q, which are in turn related to the infinite sums
resulting from the perturbative renormalization of H,,. It should be noted that
these parameters are not identical to those used by Kane [2], and the differences
are the source of great consternation to those learning k- p theory. For a magnetic
field applied along one of the principle axes of the crystal, the problem reduces to
that of introducing the field dependence into the general bulk Hamiltonian. For a
field at an arbitrary angle with respect to the crystal, however, more rigor must be
introduced to properly treat the angular momentum states of the various bands in
a new basis aligned with the magnetic field (see Luttinger [4]). In either case, rais-
ing and lowering operators, defined as in Chapter 2, may be introduced in terms
k, and ky. If at and a are introduced, and the anisotropy terms proportional to
(73 — 72)a! (and similarly for a) are neglected (which is a reasonable assumption
for the [’-valleys of most III-V compounds), the eigenvector of the 8-band ke Dp

Hamiltonian takes the form,

W = (er hn, 63 Pn—1, C5 Pngi, C7 nti, Cohnsi, Cahnse, Ce hn, C8 An) , (B.11)

150

where the h; are simple quantum harmonic oscillator wavefunctions. The resulting
eigenstate contains a mixture of Landau levels indexed by /, and is, in fact, a
state of constant projected angular momentum along the axis of the magnetic
field. As shown in Chapter 4, the orbital angular momentum attributable to the
Landau orbits is proportional to the Landau index /, and the sum of / and m,,
the z-component of the total band angular momentum, is constant for the above
eigenstates of the 8-band Hamiltonian. We therefore expect a selection rule A(/ +
m,) = 0 for resonant, elastic tunneling in magnetic fields applied perpendicular to
the heterointerfaces. The incorporation of the bulk 8-band k- # Hamiltonian in

a magnetic field into heterostructure transport calculations is the work of Liu et

al. [5].

151

Bibliography

[1]
[2]

[3]
[4]
[5]

P. Léwdin, J. Chem. Phys. 19, 1396 (1951).

E. O. Kane, in Semiconductors and Semimetals 1, edited by R. K. Willardson
and A. C. Beer (Academic Press, New York, 1966).

G. Wu, Ph.D. Thesis, California Institute of Technology, 1988.
J. M. Luttinger, Phys. Rev. 102, 1030 (1956).

Y. X. Liu, D. Z.-Y. Ting, and T. C. McGill, unpublished.