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 Theoretical Studies of the Electronic, MagnetoOptical, and Transport Properties of Diluted Magnetic Semiconductors
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 Cyclotron resonance ( jstor )
Electrons ( jstor ) Ferromagnetism ( jstor ) Ions ( jstor ) Landau levels ( jstor ) Magnetic fields ( jstor ) Magnetism ( jstor ) Magnets ( jstor ) Phonons ( jstor ) Semiconductors ( jstor )
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THEORETICAL STUDIES OF THE ELECTRONIC, MAGNETOOPTICAL
AND TRANSPORT PROPERTIES OF DILUTED MAGNETIC
SEMICONDUCTORS
By
YONGKE SUN
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
2005
To my dear wife Yuan, and my parents.
ACKNOWLEDGMENTS
I owe my gratitude to all the people who made this thesis possible and because
of whom my graduate experience has been one that I will cherish forever.
First and foremost I would like to thank my advisor, Professor ('!Ch I .lh. r
J. Stanton, for giving me an invaluable opportunity to work on challenging and
extremely interesting projects over the past four years. He has ahv ~ made himself
available for help and advice and there has never been an occasion when I have
knocked on his door and he has not given me time. His physics intuition impressed
me a lot. He taught me how to solve a problem starting from a simple model, and
how to develop it. It has been a pleasure to work with and learn from such an
extraordinary individual.
I would also like to thank Professor David H. Reitze, Professor Selman P.
Hershfield, Professor Dmitrii Maslov and Professor Cammy Ab, i il,:,I for agreeing
to serve on my thesis committee and for sparing their invaluable time reviewing the
manuscript.
My colleagues have given me lots of help in the course of my Ph.D. studies.
Gary Sanders helped me greatly to develop the program code, and we ah,ii had
fruitful discussions. Professor Stanton's former postdoc Fedir Kyrychenko also gave
me good advice and some insightful ideas. I would also like to thank Rongliang Liu
and Haidong Zhang, who made my life here more interesting.
I want to thank our research collaborators. Dr. Kono's group from Rice
University provided most of the experimental data. Collaboration with Dr. Kono
was a wonderful experience in the past four years. I also had fruitful discussion
with Prof. Miura and Dr. Matsuda from University of Tokyo.
I would also like to acknowledge help and support from some of the staff
members, in particular, Darlene Latimer and Donna Balcom, who gave me much
indispensable assistance.
I owe my deepest thanks to my family. I thank my mother and father, and
my wife, Yuan, who have ahlv stood by me. I thank them for all their love and
support. Words cannot express the gratitude I owe them.
It is impossible to remember all, and I apologize to those I have inadvertently
left out.
TABLE OF CONTENTS
page
ACKNOWLEDGMENTS ................... ...... iii
LIST OF TABLES ...................... ......... vii
LIST OF FIGURES ................... ......... viii
ABSTRACT ...................... ............ xiv
CHAPTER
1 INTRODUCTION AND OVERVIEW ................... 1
1.1 Spintronics ................... ........... 1
1.2 The IIVI Diluted Magnetic Semiconductors ............ 4
1.2.1 Basic Properties of IIVI Diluted Magnetic Semiconductors 4
1.2.2 Exchange Interaction between 3d5 Electrons and Band Elec
trons ..................... ....... 8
1.3 The IIIV Diluted Magnetic Semiconductors . . 13
1.3.1 Ferromagnetic Semiconductor ............... .. 13
1.3.2 Effective Mean Field ..... ........... .... 21
1.4 Open Questions . . ..... .......... 23
1.4.1 Nature of Ferromagnetism and Band Electrons ...... ..23
1.4.2 DMS Devices. .................. ... .. .. 24
2 ELECTRONIC PROPERTIES OF DILUTED MAGNETIC SEMICON
DUCTORS ................... ..... ....... 29
2.1 Ferromagnetic Semiconductor Band Structure . ... 29
2.2 The k p Method. ..... ...... ........ 30
2.2.1 Introduction to k p Method ... . . 30
2.2.2 Kane's Model .... . . ...... 34
2.2.3 Coupling with Distant BandsLuttinger Parameters . 38
2.2.4 Envelope Function .................. .. 42
2.3 Landau Levels .......... . . . ... 43
2.3.1 Electronic State in a Magnetic Field . . 43
2.3.2 Generalized PidgeonBrown Model . . ..... 44
2.3.3 Wave Functions and Landau Levels . . ..... 49
2.4 Conduction Band gfactors .................. ..... 53
3 CYCLOTRON RESONANCE ................... .... 56
3.1 General Theory of Cyclotron Resonance . . ..... 56
3.1.1 Optical Absorption .................. ..... 56
3.1.2 Cyclotron Resonance ................ .. .. 60
3.2 Ultrahigh Magnetic Field Techniques ............... .. 63
3.3 Electron Cyclotron Resonance ............ .. .. .. 64
3.3.1 Electron Cyclotron Resonance ............... .. 64
3.3.2 Electron Cyclotron Mass .................. .. 72
3.4 Hole Cyclotron Resonance .................. ..... 74
3.4.1 Hole Active Cyclotron Resonance . . ..... 74
3.4.2 Hole Density Dependence of Hole Cyclotron Resonance 83
3.4.3 Cyclotron Resonance in InMnAs/GaSb Heterostructures .83
3.4.4 Electron Active Hole Cyclotron Resonance . ... 90
4 MAGNETOOPTICAL KERR EFFECT ....... . . 96
4.1 Relations of Optical Constants ............... .. 96
4.2 Kerr Rotation and Farid1,i Rotation . . . .... 101
4.3 Magnetooptical Kerr Effect of Bulk InMnAs and GaMnAs . 104
4.4 Magnetooptical Kerr Effect of Multilayer Structures ...... 107
5 HOLE SPIN RELAXATION. ................... .. .. 112
5.1 Spin Relaxation Mechanisms ............... .. .. .. 113
5.2 Lattice Scattering in IIIV Semiconductors . . 115
5.2.1 Screening in Bulk Semiconductors . . ..... 117
5.2.2 Spin Relaxation in Bulk GaAs . . ..... 118
5.3 Spin Relaxation in GaMnAs .................. .. 122
5.3.1 Exchange Scattering ................. . 122
5.3.2 Impurity Scattering ............. .... . 124
6 CONCLUSION .................. ........... .. 129
REFERENCES .................. ................ .. 132
BIOGRAPHICAL SKETCH ............. . . .... 138
LIST OF TABLES
Table page
11 Some important IIVI DMS .................. ...... 4
21 Summary of Hamiltonian matrices with different n .......... ..50
22 InAs band parameters .................. ........ .. 51
31 Parameters for samples used in eactive CR experiments . ... 67
32 C'!i i i''teristics of two InMnAs/GaSb heterostructure samples . 85
51 Parameters for GaAs phonon scattering ................ .121
LIST OF FIGURES
Figure page
11 The band gap dependence of Hgl_, Mi:, Te on Mn concentration k. .. 5
12 The band structures of Hgi__,.i..Te with different x. . . 6
13 Cdl__?!. i,.Te xT phase diagram. ............... . 7
14 Average local spin as a function of magnetic field at 4 temperatures
in paramagnetic phase. .................. ..... 10
15 Magneticfield dependence of Hall resistivity pHall and resistivity p of
GaMnAs with temperature as a parameter. ........... ..14
16 Mn composition dependence of the magnetic transition temperature
T,, as determined from transport data. ............ 16
17 Variation of the RKKY coupling constant, J, of a free electron gas in
the neighborhood of a point magnetic moment at the origin r = 0. 17
18 Curie temperatures for different DMS systems. Calculated by Dietl
using Zener's model. .................. .... 19
19 Schematic diagram of two cases of BMPs. .............. ..20
110 Average local spin as a function of magnetic field at 4 temperatures.. 22
111 The photoinduced ferromagnetism in InMnAs/GaSb heterostructure. 25
112 Spin light emitting diode. .................. ..... 27
113 GaMnAsbased spin device. ............. .... 28
21 Valence band structure of GaAs and ferromagnetic Ga,,,,Mi_,,,,,,As
with no external magnetic field, calculated by generalized Kane's
model .. ......................... .... 30
22 Band structure of a typical IIIV semiconductor near the F point. 35
23 Calculated Landau levels for InAs (left) and In,, ..i,,, ,.As (right) as
a function of magnetic field at 30 K. ............. .. .. 52
24 The conduction and valence band Landau levels along kz in a mag
netic field of B = 20 T at T = 30 K. ............. 53
25 Conduction band gfactors of Inl_,i1,.As as functions of magnetic
field with different Mn composition x. .............. 54
26 gfactors of ferromagnetic In,,.,M\li,,,As. ............... 55
31 Quasiclassical pictures of eactive and hactive photon absorption. .. 62
32 The core part of the device based on singlecoil method . .... 64
33 A standard coil before and after a shot. ................ 65
34 Waveforms of the magnetic field B and the current I in a typical shot
in singleturn coil device. ............... .... 65
35 Waveforms of the magnetic field B and the current I in a typical flux
compression device. ............... ...... 66
36 Experimental electron CR spectra for different Mn concentrations x
taken at (a) 30 K and (b) 290 K. ............... . 68
37 Zonecenter Landau conductionsubband energies at T = 30 K as
functions of magnetic field in ndoped Inl_,71i..As for = 0 and
x = 12" .................. ............. .. 69
38 Electron CR and the corresponding transitions. . . 70
39 Calculated electron CR absorption as a function of magnetic field at
30 K and 290 K. ............... ....... 71
310 Calculated electron cyclotron masses for the lowestlying spinup and
spindown Landau transitions in ntype Inl_MnxAs with photon
energy 0.117 eV as a function of Mn concentration for T = 30 K
and T = 290 K. .................. .. ..... 73
311 Hole cyclotron absorption as a function of magnetic field in ptype
InAs for hactive circularly polarized light with photon energy 0.117 eV. 75
312 Calculated cyclotron absorption only from the H_1,1 H,2 and L0,3 
L1,4 transitions broadened with 40 meV (a), and zone center Lan
dau levels responsible for the transitions (b). ............ ..76
313 Experimental hole CR and corresponding theoretical simulations. 77
314 Observed hole CR peak positions for four samples with different Mn
concentrations. ............... .......... 78
315 The dependence of cyclotron energies on several parameters. . 79
316 Hole CR spectra of InAs using different sets of Luttinger parameters. 80
317 Calculated Landau levels and hole CR in magnetic fields up to 500 T. 81
318 kdependent Landau subband structure at B = 350 T . .... 82
319 Band structure near the F point for InAs calculated by eightband
model and full zone thirtyband model. ............ 82
320 The hole density dependence of hole CR. .............. 84
321 Cyclotron resonance spectra for two ferromagnetic InMnAs/ GaSb
samples ........ .......... ......... .... 86
322 Theoretical CR spectra showing the shift of peak A with temperature. 87
323 Average localized spin as a function of temperature at B = 0, 20, 40,
60 and 100 Tesla. ............... ...... 88
324 Relative change of CR energy (with respect to that of high tempera
ture limit) as a function of temperature. ............. 89
325 Band diagram of InMnAs/GaSb heterostructure. . . 90
326 Schematic diagram of Landau levels and cyclotron resonance transi
tions in conduction and valence bands. ............ 91
327 The valence band Landau levels and eactive hole CR. . ... 92
328 Experimental and theoretical hole CR absorption. ......... ..93
329 Valence band structure at T = 30 K and B = 100 T for Inl_Mi,..As
alloys having x = and x = 5'. ................. 94
330 The primary transition in the eactive hole CR under different Mn
doping. .................. .............. ..95
41 Diagram for light reflection from the interface between medium 1 with
refractive index N1 and medium 2 with refractive index N2. .... ..100
42 Schematic diagram for magnetic circular dichroism. . ... 102
43 Diagrams for Kerr and Far,idi rotation. .. . . .... 103
44 Kerr rotation of InMnAs. ................ .... 105
45 The band diagram for InAs. ............. ... 106
46 Kerr rotation of GaMnAs. .................. ..... 107
47 The band diagram for GaAs. .................. .... 108
48 The absorption coefficients both in InMnAs and GaSb lV. i (a) and
the reflectivity of InMnAs/GaSb heterostructure(b). . ... 109
49 Reflectivity of In,, ..it,, ,,As(9 nm)/GaSb(600 nm) heterostructure
at T = 5.5 K measured by P. Fumagalli and H. Munekata. ..... ..110
410 Measured (a) and calculated (b) Kerr rotation of InMnAs(19 nm)/
AlSb(145 nm) heterostructure under a magnetic field of 3 T at T 5.5 K. 111
51 Lightinduced MOKE. Signal decays in less than 2 ps. . ... 113
52 Lightinduced magnetization rotation. ................ 114
53 The heavy hole spin relaxation time as a function of wave vector (a),
and temperature at the F point (b). ...... . . ...... 123
54 Spin relaxation time for a heavy hole as a function of k along (0,0,1)
direction. .................. ............ 126
55 Spin relaxation time of a heavy hole as a function of hole density at
direction (a) (0,0,1) and (b) (1,1,1). ................. 127
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
THEORETICAL STUDIES OF THE ELECTRONIC, MAGNETOOPTICAL
AND TRANSPORT PROPERTIES OF DILUTED MAGNETIC
SEMICONDUCTORS
By
Yongke Sun
December 2005
CI, ,ir: Chi iI.l !h. r J. Stanton
Major Department: Physics
Spintronics has recently become one of the key research areas in the magnetic
recording and semiconductor industries. A key goal of spintronics is to utilize
magnetic materials in electronic components and circuits. A hope is to use the
spins of single electrons, rather than their charge, for storing, transmitting and
processing quantum information. This has invoked a great deal of interest in spin
effects and magnetism in semiconductors. In my work, the electronic and optical
properties of diluted magnetic semiconductors(DMS), especially (In,Mn)As and
its heterostructures, are theoretically studied and characterized. The electronic
structures in ultrahigh magnetic fields are carefully studied using a modified eight
band PidgeonBrown model, and the magnetooptical phenomena are successfully
modeled and calculated within the approximation of Fermi's golden rule. We
have found the following important results: i) Magnetic ions doped in DMS
pli, a critical role in affecting the band structures and spin states. The sp d
interaction between the itinerant carriers and the Mn d electrons results in a
shift of the cyclotron resonance peak and a phase transition of the IIIV DMS
from paramagnetic to ferromagnetic; ii) gfactors of the electrons in DMS can be
enhanced to above 100 by large spin splitting due to strong sp d interaction. Also
the effective masses of DMS systems strongly depend on interaction parameters;
iii) Two strong cyclotron resonance peaks present in pdoped DMS arise from
the optical transitions of heavyhole to heavyhole and lighthole to lighthole
Landau levels, in lower and higher magnetic fields, respectively; iv) Electronactive
cyclotron resonance takes place in pdoped DMS samples. This is unusual since a
simple quasiclassical argument would sil.: 1 that one could not simultaneously
increase angular momentum and energy for this type of polarized light in a hole
system. This occurs because of the degeneracy in the valence bands; v) Due to
the magnetic circular dichroism, nonvanishing magnetooptical Kerr rotation up to
several tenths of a degree occurs in DMS systems. The Kerr rotation in muiltiliv. r
structures depends on quantum confinement and multireflections from the surfaces;
vi) Quantitative calculations show that in intrinsic bulk GaAs, the hole spin life
time is around 110 femtoseconds, which is due to phonon scattering. However, in
DMS, the p d exchange interaction and the high density of impurities give rise
to other spin flip scattering channels. The nonequilibrium spin life time is only
a few femtoseconds. These research results should be helpful for gaining more
understanding of the properties of DMS systems and should be useful in designing
novel devices based on DMS.
CHAPTER 1
INTRODUCTION AND OVERVIEW
There is a wide class of semiconducting materials which are characterized by
the random substitution of a fraction of the original atoms by magnetic atoms.
The materials are commonly known as diluted magnetic semiconductors (DMS) or
semimagnetic semiconductors (SMSC).
Since the initial discovery of DMS in IIVI semiconductor compounds [1], more
than two decades have passed. The recent discovery of ferromagnetic DMS based
on IIIV semiconductors [2] has lead to a surge of interest in DMS for possible
spintronics applications. Many papers have been published investigating their
electronic, magnetic, optical, thermal, statistical and transport properties, in many
journals, and even in popular magazines [3]. This interest not only comes from
the DMS themselves as good theoretical and experimental subjects, but also can
be better understood from a broader view from the relation of DMS research with
spintronics [4].
1.1 Spintronics
Spintronics, or spin electronics, refers to the study of the role 1p i, d by elec
tron (and nuclear) spin in solid state physics, and possible devices that specifically
exploit spin properties instead of or in addition to the charge degrees of freedom.
Spin relaxation and spin transport in metals and semiconductors are of fundamen
tal research interest not only for being basic solid state physics issues, but also for
the already demonstrated potential these phenomena have in electronic technology.
There is a famous Moore's Law in the conventional electronics industry, that ,i
the number of transistors that fit on a computer chip will double every 18 months.
This may soon face some fundamental roadblocks. Most researchers think there
will eventually be a limit to how many transistors they can cram on a chip. But
even if Moore's Law could continue to spawn evertinier chips, small electronic de
vices are plagued by a big problem: energy loss, or dissipation, as signals pass from
one transistor to the next. Line up all the tiny wires that connect the transistors in
a Pentium chip, and the total length would stretch almost a mile. A lot of useful
energy is lost as heat as electrons travel that distance. Spintronics, which uses spin
as the information carriers, in contrast with conventional electronics, consumes less
energy and may be capable of higher speed.
Spintronics emerged on the stage in scientific field in 1988 when Baibich et al.
discovered giant magnetoresistance (GMR) [5], which results from the electronspin
effects in magnetic materials composed of ultrathin rmuiltil iv rs, in which huge
changes could take place in their electrical resistance when a magnetic field is
applied. GMR is hundreds of times stronger than ordinary magnetoresistance.
Basing on GMR materials, IBM produced in 1997 new read heads which are able
to sense much smaller magnetic fields, allowing the storage capacity of a hard
disk to increase from the order of 1 to tens of gigabytes. Another valuable use of
GMR material is in the operation of the spin filter, or spin valve, which consists
of 2 spin lV. i which let through more electrons when the spin orientations in
the two Iivr s are the same and fewer when the spins are oppositely aligned. The
electrical resistance of the device can therefore be changed dramatically. This
allows information to be stored as 0's and l's (magnetizations of the lv ir parallel
or antiparallel) as in a conventional transistor memory device. A straightforward
application could be in the magnetic random access memory (!I RAM) device which
is nonvolatile. These devices would be smaller, faster, cheaper, use less power and
would be much more robust in extreme conditions such as high temperature, or
highlevel radiation environments.
Currently, besides continuing to improve the existing GMRbased technology,
people are now focusing on finding novel vi, of both generating and utilizing spin
polarized currents. This includes investigation of spin transport in semiconductors
and looking for v,v in which semiconductors can function as spin polarizers and
spin valves. We can call this semiconductor based spintronics, the importance of
which lies in the fact that it would be much easier for semiconductorbased devices
to be integrated with traditional semiconductor technology, and the semiconductor
based spintronic devices could in principle provide amplification, in contrast with
existing metalbased devices, and can serve as multifunctional devices. Due to
the excellent optical controllability of semiconductors, the realization of optical
manipulation of spin states is also possible.
Although there are clear merits for introducing semiconductors into spintronic
applications, there are fundamental problems in incorporating magnetism into
semiconductors. For example, semiconductors are generally nonmagnetic. It is hard
to generate and manipulate spins in them. People can overcome these problems
by contacting the semiconductors with other (spintronic) materials. However, the
control and transport of spins across the interface and inside the semiconductor is
still difficult and far from wellunderstood. Fortunately, there is another approach
to investigating spin control and transport in allsemiconductor devices. This
approach has become possible since the discovery of DMS.
The most common DMS studied in the early 1990s were IIVI compounds
(like CdTe, ZnSe, CdSe, CdS, etc.), with transition metal ions (e.g., Mn, Fe or Co)
substituting for their original cations. There are also materials based on IVVI
(e.g., PbS, SnTe) and most importantly, IIIV (e.g., GaAs, InSb) crystals. Most
commonly, Mn ions are used as magnetic dopants.
1.2 The IIVI Diluted Magnetic Semiconductors
1.2.1 Basic Properties of IIVI Diluted Magnetic Semiconductors
The first IIVI DMS was grown in 1979 [1], and has been given a great deal of
attention ever since [6]. The most studied IIVI DMS materials are listed in Table
11.
Table 11: Some important IIVI DMS
Material Crystal Structure x range1
Hg1ixM!i, Te Zincblende x < 0.30
Hg1i__Mi, Se Zincblende x < 0.30
Cd_1 MnTe Zincblende x < 0.75
Cdl_M,?i. ,Se Wurtzite x < 0.50
Znl_?!Mi., Te Zincblende x < 0.75
1 x refers the range of x for which the crystals are usually studied. When
z's become relatively large, phases like MnTe or MnTe2 occur, and the
crystal qualities are poor [7].
The IIVI DMS have attracted so much attention since their discovery because
of the following important properties.
Unique electronic properties: The wide variety of both host < i ,1
and magnetic atoms provides materials which range from wide gap to zero
gap semiconductors, and reveal many different types of magnetic interaction.
Several of the properties of these materials may be tuned by changing the
concentration of the magnetic ions. The bandgap, E,, of Hgi__? ,.Te can
even change from negative to positive. This property becomes favorable
as far as designing infrared devices is concerned. The dependence of E, of
Hgl_?j.Mi,Te on x is given in Fig. 11 [8].
With the definition of the band gap as E, E6r Ers, the band structures
of Hgi_ _i,..Te with different x are given in Fig. 12 [8]. With x < 0.075,
Eg < 0, and with x > 0.075, E, > 0. Without spinorbital coupling, we
should have a sixfold degenerate valence band at the F point. Considering
spinorbital coupling, the valence band splits into two bandsF7 and Fs
(splitoff band), with an energy difference of A.
The electron effective mass, i.e., the band curvature, will also change with x.
At some x values, the effective mass becomes so small that the mobility of
electrons can be very high. For instance, p 106 cm2/V s for HgIi_.li,.Te
when x 0.07 at 4.2 K.
5
300 
*
200
100
oo
0 0
100
o0 0
00 x Hgl Mn1 Te
Ox T= 4.2K
0, M.O
200 ox M0
x SdH
A FarlR
3001
0 5 10 15
k (ot.%)
Figure 11: The band gap dependence of Hgi_, Mi,, Te on Mn concentration k.
Reprinted with permission from Bastard et al. Phys. Rev. B 24: 19611970, 1981.
Figure 10, Page 1967.
Broad phase behavior: With different Mn concentration x and temper
ature T, each IIVI DMS presents a different (phase) property, but their xT
phase diagrams are very similar. Shown in Fig. 13 is the phase diagram of
Cdl_,i ,.Te obtained from specific heat and magnetic susceptibility measure
ments [9]. The DMS system may be considered as containing two interacting
subsystems. The first of these is the system of delocalized conduction and
valence band electrons/holes. The second is the random, diluted system of
localized magnetic moments associated with the magnetic atoms. These
two subsystems interact with each other by the spin exchange interaction.
The fact that both the structure and the electronic properties of the host
< i I Ji; are well known means that they are perfect for studying the basic
mechanisms of the magnetic interactions coupling the spins of the band
carriers and the localized spins of magnetic ions. The coupling between the
Sh.
L h.
0.02 0.04 OD06
Mn c
"positive gap"
I r
0.08 0.10
ontent (k)
Figure 12: The band structures of Hgl_, ,.iTe with different x. A is the spin
orbital splitting, HH indicates the heavy hole band, and LH the light hole band,
respectively. Reprinted with permission from Bastard et al. Phys. Rev. B 24:
19611970, 1981. Figure 1, Page 1961.
localized moments results in the existence of different magnetic phases such as
paramagnets, spin glasses and antiferromagnets.
Important magnetic phenomena: As described above, if we don't
consider the spin exchange interaction between the band electrons and
localized magnetic moments, DMS materials are just the same as the other
semiconductors. When we consider the spin exchange interaction, however,
DMS materials present many important properties, such as very big Land6
gfactors, extremely large Zeeman splitting of the electronic bands, giant
Fa,idiv rotation, and huge negative magnetoresistance. Therefore, to study
DMS, one has to first understand the spin exchange interaction between the
localized magnetic ions and band electrons.
8
0
1 1
0.12 0.14
h.h.7
,
Vp mixed
P crystal
20 phases
IA
10 I S
S I
0I
0 0.2 0.4 0.6 0.8 1.0
X
Figure 13: Cdl_ i:,.Te xT phase diagram. P: Paramagnet; A: Antiferromagnet;
s: spinglass, mixed i< I I1 when x > 0.7. Reprinted with permission from Galgzka
et al. Phys. Rev. B 22: 33443355, 1980. Figure 12, Page 3352.
1.2.2 Exchange Interaction between 3d5 Electrons and Band Electrons
Many features of DMS, such as the special electronic properties, unique
phase diagrams, and important magnetic and magnetooptical characteristics,
are induced by the exchange interaction between the localized d shell electrons of
the magnetic ions and the delocalized band states (of s or p origin). The s d,
p d exchange, and its consequences and origin have been pointed out from the
very beginning of the history of DMS and the Heisenberg form of the exchange
interaction Hamiltonian was successfully used for this interaction [10]. In the
following, I will briefly introduce a simple qualitative theoretical approach to IIVI
DMS.
Suppose the state of Mn ions in DMS material is Mn2+. The electronic struc
ture of Mn2+ is ls22s22p63s23p63d5, in which 3d5 is a halffilled shell. According to
Hund's rule, the spin of these five 3d5 electrons will be parallel to each other, so the
total spin is S = 5/2. These five electrons are in states in which the orbital angular
momentum quantum number 1 = 0, 1, 2. Thus the total orbital angular momen
tum L = 0. The total angular momentum for a Mn2+ ion then is J = S = 5/2. The
Land6 gfactor is
+ J(J + 1) + S(S +1) L(L +1)
g = 1 + =2. (11)
2J(J + 1)
Analogous to the exchange interaction in the Hydrogen molecule, the exchange
interaction between a 3d5 electron and a band electron can be written in the
Heisenberg form
H, = J S, (12)
where a is the spin of a band electron/hole, J is the exchange constant, and S is
the total angular momentum of all 3d5 electrons in a Mn2+ ion.
In the noninteracting paramagnetic phase, a very simplified model will be
described in the following. Since L = 0 for Mn2+, the magnetic momentum for
Mn2+ is p = (ge/2mo)J = (ge/2mo)S. Assuming a magnetic field B along the
z direction, the additional energy in this field of a Mn2+ ion is p B = gpBmsB,
where ms = 5/2, 3/2, 1/2, 1/2, 3/2, 5/2. Assuming noninteractive spins, and
using a classic Boltzman distribution function egpB"5B/kBT, the average magnetic
moment in the z direction is then
Y5/2 (B5 2 s)eg"pgB/kn
Y5/2 egpSmB/kBT
m s 5/2
This can be written as
(p) =gpBSB,(y), (14)
where B,(y) is the Brillouin function
2S + t 2S 2S 2S
S = 5/2, y = gpBSB/kBT. (15)
The average spin of one Mn2+ ion then is
(S) SB,(y). (16)
The antiparallel orientation of B and (S,) is due to the difference in sign of the
magnetic moment and the electron spin. Since B is directed along the z axis, the
average Mn spin saturates at (Sz) = 5/2. The paramagneticc) dependence of (S,)
on magnetic field and temperature is shown in Fig. 14.
From Eq. 12, the exchange Hamiltonian of one band electron with spin a
interacting with the 3d5 electrons from all Mn2+ ions is,
Hex = J(r Rj)Si a, (17)
where r is the position vector of the band electron, and Ri is the position vector of
the ith Mn2+ ion, J(r Ri) is the exchange coupling coefficient of the band electron
2.5
2.0 30K
A 1.5 K
N
C)
V
1.0 150K
0.5 290K
0.0
0 20 40 60 80
B (T)
Figure 14: Average local spin as a function of magnetic field at 4 temperatures in
paramagnetic phase.
with the 3d5 electrons in the 1th Mn2+ ion. Si is the total angular momentum of
the 3d5 electrons in the 1th Mn2+ ion.
Next we will use a virtual < i I 1 approximation to deal with Hamiltonian
17. Due to the fact the the wave function of a band electron actually extends
over the whole i  I 1 it interacts with all the Mn2+ ions simultaneously. In the
mean field framework, we can replace the angular momentum of each Mn2+ ion
by the average value. Still assuming a magnetic field along z direction, we have
(S) = (S,), and Si a = (Sz)mch2. m, = 1/2 here indicates the spin quantum
number of the band electrons. The (S,) is given by Eq. 16. The exchange
Hamiltonian then can be written as
H,, (S )m, J(r Ri). (18)
Because of the extended nature of the band electron states, which interact
with the 3d5 electrons in all Mn2+ ions, the positions of these Mn2+ ions are not
important. We can distribute approximately these Mn2+ ions uniformly at cation
sites. This amounts to assuming we have an equivalent magnetic moment of x(S,)
at each cation site. So, Eq. 18 becomes
Hex = x(S,)mc J(r R). (19)
R
Here R becomes the position vector of each cation site. In Eq. 19 the exchange
Hamiltonian now has the same periodicity as the (i I I1
From the Hamiltonian 19, the exchange energy can then be obtained by
E = (. kL "'ck) x(S,)m Tc(ck J R) Ik). (110)
R
For the electrons at the conduction band edge, the wave function is Qck uco(k
0). J(r R) is the coupling coefficient as we have said above, which is the exchange
integral between the band electrons and 3d5 electrons. Due to the fact that the 3d5
electrons are strongly localized, we can assume the integral is only nonvanishing
in a unit cell range for a specific R in Eq. 110. Considering the periodicity of
J(r R), the Eq. 110 can be rewritten as,
E4 = Nx(S,)mc ua J(r)ucodr = aNx(S,)mr, (111)
a J uoJ(r)ucodr. (112)
where N is the number of unit cells in the crystal.
For zincblende semiconductors (most IIVI and IIIV semiconductors), the
states for conduction bandedge (k = 0) electrons are slike, and those for valence
bandedge holes are plike. So the use of m, = 1/2 is justified. Then in a
magnetic field B, the conduction band energy is,
Enc = (n + )khW + nmc9cpBB + mcnaNx(Sz), (113)
2
where wc eB/mu is the cyclotron frequency, and gc is the conduction band
gfactor. In Eq. 113, the first term is the Laudau splitting, the second term is the
Zeeman splitting, and the third term is the exchange splitting, which is unique for
the DMS.
Similarly, the energy structure of the valence band can also be obtained, if we
replace w by wv = eB/m*, gc by g, me by m,, and importantly, a by 3, where
3 uoJ(r)uodr. (114)
a and 3 are called exchange constants for s d and p d exchange interactions
between band electrons and localized Mn2+ ions.
We can introduce an effective Land6 gfactor in the conduction band
SaNx (Sz)
geff = 9cB+ (115)
pBB
which indicates the strength of the spin splitting of the first Landau level in
the conduction band. In the low field approximation, Eq. 16 becomes (Sz) =
gpBS(S + 1)B/3kBT, so in this limit
ff aNxgS2(S +1)
geff = gc (116)
3kBT
At low temperature, the effective gfactor can reach very large values. The
gfactor depends on temperature through (S,) in Eq. 115. We will have a more
detailed discussion of gfactors in C'! lpter 2.
The above discussion is a very simplified qualitative model, and only appropri
ate for IIVI DMS in a paramagnetic phase, where the Mn concentration is not so
high that they don't have a direct exchange interaction. This discussion can also
be applied to paramagnetic IIIV DMS, in which commonly the Mn solubility are
very low. As a matter of fact, although Eq. 113 can give a qualitative description
of the conduction band structure, it does not work in real cases. C'! lpter 2 gives a
quantitative model.
Since the discovery of ferromagnetism in IIIV DMS, much research now
focuses on exploring ferromagnetism mechanisms, looking for new materials and
obtaining higher Curie temperatures. Recently, ferromagnetism in IIVI DMS was
also reported by several groups [11, 12, 13].
1.3 The IIIV Diluted Magnetic Semiconductors
1.3.1 Ferromagnetic Semiconductor
Although IIVI DMS combine both semiconducting and magnetic properties
and manifest spectacular properties, other characteristics such as ferromagnetism
are also desirable. From Eq. 115 and Eq. 116, we can see that at low tempera
tures, the gfactor can be very large, but it is strongly temperature dependent. As
we mentioned above, the gfactor actually indicates the spin splitting. To employ
spin as a subject in research and device design, a large spin splitting is essen
tial. While most IIVI DMS are paramagnetic, the spin splitting becomes small
at high temperatures, so the realization of room temperature spintronic devices
becomes difficult. The answer for this problem is ferromagnetic semiconductors.
We can expect a large spin splitting even at high temperatures for ferromagnetic
semiconductors.
The leap from IIVI DMS to IIIV DMS should have been very natural. But
unlike IIVI semiconductors, Mn is not very soluble in IIIV semiconductors.
It can be incorporated only by nonequilibrium growth techniques and it was
not until 1992 that the first IIIV DMS, InMnAs was grown and investigated.
Ferromagnetism was soon discovered in this system [14]. Higher ferromagnetic
14
6.0
2K
IOK 112
25 K .125K
4.0 40K
55 K / 150K
E 70K .
2.0T
l ... ....85K /
'0 100 M
0.8 a
0 K
0.0
8 4 0 4
B (T)
Figure 15: Magneticfield dependence of Hall resistivity pHall and resistivity p of
GaMnAs with temperature as a parameter. Mn composition is x = 0.053. The
inset shows the temperature dependence of the spontaneous magnetization 1.,
determined from magnetotransport measurements; the solid line is from mean
field theory. Reprinted with permission from Matsukura et al. Phys. Rev. B 57:
R2037R2040, 1998. Figure 1, Page R2037.
R2037R2040, 1998. Figure 1, Page R2037.
transition temperatures were also achieved in GaMnAs [15]. Shown in Fig. 15 is
the magneticfield dependence of the Hall resistivity and the normal resistivity of
GaMnAs with temperature as a parameter [16]. In this case, the ferromagnetic
transition temperature is about 110 K. The discovery of ferromagnetism in IIIV
DMS led to an explosion of interest [14, 15, 17, 18]. Many new materials were
investigated, theories explaining the ferromagnetism mechanisms were brought
forward, and experiments aimed at increasing the Curie temperatures were carried
out.
Although InMnAs was the first MBE grown IIIV DMS, its Curie temperature
was relatively low at about 7.5 K. In 1993, a higher Curie temperature of 35 K was
realized in a p type InMnAs/GaSb heterostructure [17]. Since 1996, a number of
groups are working on the MBE growth of GaMnAs and related heterostructures,
in which the highest Curie temperature (173 K) has been achieved recently for
25 nm thick Gal) iM..As films with 's. nominal Mn doping after annealing [19].
The dependence of the Curie temperature of Gal_,Mi,.AAs on Mn concentration x
is shown in Fig. 16 [16]. The Curie temperature reaches the highest value when
x 5.;;' in this case.
GaMnN and GaMnP are also candidates for high Curie temperature III
V DMS materials. Ferromagnetism in GaMnN is elusive. While some groups
found it paramagnetic when doped with percent levels of Mn [20], some groups
have reported a ferromagnetic transition temperature above 900 K [21]. Room
temperature ferromagnetism was also reported in GaMnP [22, 23]. Besides III
V DMS, Mn doped IV semiconductors like GeMn [24, 25], SiMn [26], were also
reported ferromagnetic.
The theory for ferromagnetism in IIIV DMS is still controversial, however,
there is consensus that it is mediated by the itinerant holes. Unlike the case in
IIVI DMS in which Mn ions have the same number of valence electrons as the
120
(C) /(
80 /
40
0
0.00 0.04 0.08
X
Figure 16: Mn composition dependence of the magnetic transition temperature Tc,
as determined from transport data. Reprinted with permission from Matsukura et
al. Phys. Rev. B 57: R2037R2040, 1998. Figure 2, Page R2038.
cations, Mn ions in IIIV DMS are not only providers of magnetic moments, they
are also acceptors. Due to compensating defects like Asantisites or/and Mn
interstitials [27, 28, 29], hole concentrations are generally much lower than the Mn
concentration.
The theories of carrier induced ferromagnetism fall into four categories.
1. RKKY mechanism: Indirect exchange couples moments over relatively
large distances. It is the dominant exchange interaction in metals where there
is little or no direct overlap between neighboring magnetic impurities. It
therefore acts through an intermediary which in metals are the conduction
electrons (itinerant electrons) or holes. This type of exchange was first
proposed by Ruderman and Kittel [30] and later extended by Kasuya [31] and
Yosida [32] to give the theory now generally know as the RKKY interaction.
Ohno et al. explained the ferromagnetism in GaMnAs for Mn concentration
x = 0.013 using the RKKY mechanism [14]. In the interaction Hamiltonian,
H = JRKKYSi .Sj (117)
the coupling coefficient JyKKY assumes the form[33],
JRKKY(r) [sin(2kFr) 2kFrcos(2kFr)]/(2kFr)4
(118)
where kF is the radius of the conduction electron/hole Fermi surface, r is
the distance away from the origin where a local moment is placed. The
RKKY exchange coefficient, J, oscillates from positive to negative as the
separation of the ions changes with the period determined by the Fermi
wavevector kF1 and has the damped oscillatory nature shown in Fig. 1
7. Therefore, depending upon the separation between a pair of ions their
magnetic coupling can be ferromagnetic or antiferromagnetic. A magnetic ion
induces a spin polarization in the conduction electrons in its neighborhood.
This spin polarization in the itinerant electrons is felt by the moments of
other magnetic ions within range, leading to an indirect coupling.
JMC7 1
0 r
Figure 17: Variation of the RKKY coupling constant, J, of a free electron gas in
the neighborhood of a point magnetic moment at the origin r = 0.
In the case of DMS, the average distance between the carriers rc (4 p 3
is usually much greater than that between the spins rs ) A
simple calculation show that the first zero of the RKKY function occurs at
r t 1.17rc. This means that the carriermediated interaction is ferromagnetic
and effectively long range for most of the spins.
The RKKY interaction as the main mechanism for the ferromagnetism in
IIIV DMS is questionable in some cases such as in the insulating phase
(x < :;'. for GaMnAs), in which carriers are not itinerant. When the hole
density is low, and there is no Fermi surface (Fermi level in the gap), RKKY
theory cannot predict ferromagnetism. The other problem, maybe fatal, is
that in the RKKY approximation the exchange energy is much smaller than
the Fermi energy, which is not commonly the case in DMS. As a matter of
fact, these two energies are comparable in most cases.
2. Zener's model: Zener's model is a continuousmedium limit of the RKKY
model. Zener's model was first proposed by C. Zener in 1950 [34] to interpret
the ferromagnetic coupling in transition metals. Similar to the RKKY model,
it describes an exchange interaction between carriers and localized spins. The
Hamiltonian of Zener's model in a transition metal is [34]
1 1
Hs = aS, /3SSc + 7YS (119)
2 2
where Sd and Sc are the mean magnetization of the dshell electron and
the conduction electron, respectively, and a, 3, and 7 are three coupling
constants. The main assumption here is that the exchange constant 3 is
alvb, positive, which under certain circumstances leads to ferromagnetic
coupling. Comparing Hamiltonian 117 and 119, we can see that 3 in
Eq. 119 p]1 il the similar role of J in Eq. 117. One big difference is that
Zener's model neglects the itinerant character and the Friedel oscillations of
the electron spin polarization around the localized spins.
Dietl [35] applied Zener's model to ferromagnetic semiconductors and
predicted the Curie temperature Tc for several Mn doped DMS systems.
The results are shown in Fig. 18. This quite accurately predicts the 110 K
transition temperature in GaMnAs, but certainly this is still a quite coarse
model. Even so, the trend shown in Fig. 18 has stimulated the enthusiasm
of people investigating GaN based materials looking for higher transition
temperatures.
Some of the problems in the RKKY model remain in Zener's model. For
instance, Zener's model still has limited application when carriers are mostly
localized because it still requires itinerant carriers to mediate the interactions
between localized spins. Besides, when the carrier density is higher than
the Mn concentration, important changes in the hole response function
occur at the length scale of the mean distance between the localized spins.
Accordingly, the description of spin magnetization by the continuousmedium
approximation, which constitutes the basis of the Zener model, ceases to be
valid. In contrast, the RKKY model is a good starting point in this regime.
3. Bound polaron model: Paramagnetic spins can be aligned to form ferro
magnetic domains even in the absence of an external magnetic field under
certain conditions. In DMS, localized moments can also be aligned in the
vicinity of carriers to form what are known as ii, ,iw,.tic 1p .! iiii The
carrier spin creates an effective exchange field for the magnetic ions due to
the exchange interaction which is similar in form to Eq. 17, and this field
causes ferromagnetic coupling of these local spins. The net spin alignment
Si
Ge
AlP
AlAs
GaN
GaP
GaAs
GaSb
InP
InAs
ZnO
ZnSe
ZnTe
0 100 200 300 400
Curie Temperature (K)
Figure 18: Curie temperatures for different DMS systems. Calculated by Dietl
using Zener's model.
again creates a selfconsistent exchange field for the carriers. In this process,
the carrier spin creates a magnetic potential well resulting in formation of
a I"p' cloud", a magnetic polaron. Due to the localized character of these
magnetic polarons in DMS, they are called bound magnetic polarons (BMP).
There have been extensive studies of BMP in IIVI DMS [6], in which BMP
are accountable for many optical and phase transition properties. Recently,
Bhatt et al. [36] and Das Sarma et al. [37] generalized BMP theory for IIIV
DMS. They studied the coupling between two .,1i i,'ent BMPs, and concluded
that the exchange coupling is ferromagnetic. There are two different cases.
In one case two polarons overlap and the overlap integral accounts for the
ferromagnetic coupling. The ferromagnetic transition can be regarded as a
percolation occurring through the whole system when the temperature drops
below the Curie temperature. In the other case one does not need overlapping
polarons, their effect on the magnetic moment being taken into account
through a local magnetic field. Ferromagnetic coupling has been shown to
result when the carrier is allowed to hop between the ground state of one
magnetic atom and excited states of the other. A diagram of these two cases
are shown in Fig. 19.
The BMP model quite naturally and successfully explains the magnetism of
the DMS in the insulating phase. With a much higher carrier density, most
a \ L 4 1 N
a. Two overlapping BMPs b. Two nonoverlapping BMPs.
Electrons can hop between two
localized moments.
Figure 19: Schematic diagram of two cases of BMPs.
carriers are conducting. They are more like free band carriers. In such a case,
the BMP model may not be appropriate. Although some part of the carriers
are localized and have exchange interaction with the localized spins, most
carriers have extended wave functions, which tend to interact with the other
carriers and spins in the whole band. The condition for the BMP model does
not exist any more. In such a case, the RKKY mechanism should dominate.
4. Double exchange theory: Double exchange can be considered as charge
transfer exchange which leads to ferromagnetism in ferromagnetic perovskites
Such as LaMnO3. Akai et al. [38] performed first principle DFT calculations
which show that the i1i i ,i i ly of the carriers comes from Mn d states. The
hopping of the carriers between the impurity bands and valence bands causes
the ferromagnetic ordering. Later, Inoue et al. [39] also discussed a similar
mechanism. They calculated the electronic states of IIIV DMS and found
that resonant states were formed at the top of the down spin valence band
due to magnetic impurities and the resonant states gave rise to a strong
longranged ferromagnetic coupling between Mn moments. They proposed
that coupling of the resonant states, in addition to the intraatomic exchange
interaction between the resonant and nonbonding states was the origin of the
ferromagnetism of GaMnAs. We can classify this kind of mechanism caused
by the hopping of carriers between impurity states and valence states as a
double exchange mechanism. Doubleexchangelike interactions in GaMnAs
were reported by Hirakawa et al. [40].
In the four models of ferromagnetism in IIIV DMS, the first three are mean
field based theories, and the last is based on delectrons. Though each of them is
capable of explaining some specific aspects of ferromagnetism, none of them can be
applied universally.
1.3.2 Effective Mean Field
Each of the models we discussed above utilized one type of interaction, namely,
the interaction between two spins. In the following, we discuss how to solve this
kind of interaction inside a mean field framework.
Suppose a Heisenberglike Hamiltonian
H = J,,(Si Sj), (1 20)
where i, j specify atomic sites, iw, of the magnetic moments in the crystal, and
Jj is the interatomic exchange interaction constant. The molecular field (effective
mean field) is simply given by
B,, = j(S), (121)
91B
where g is the g factor. Using the results we got in the discussion in Section 1.2.2,
the average spin along a magnetic field B (suppose it is directed along z) will be
(S,) SB(y), (122)
with Bs, the Brillouin function, given by Eq. 15, and where
Y = [gpS(B + Be.)]/kBT. (123)
After substitution of Eq. 122 to above equation, we get
y = [gpSSB + JoS(S,)]/kBT, (124)
Jo Z= J,. (125)
J
2.5 ____r_
2.0 80K
A 1.5 T=110K 150K
V
1.0
290K
0.5
0.0 1 1
0 20 40 60 80
B (T)
Figure 110: Average local spin as a function of magnetic field at 4 temperatures.
The Curie temperature is 110K.
Equation 122 can be solved by standard root finding programs to find (S,).
The solution for (Sz) / 0 exists even when B = 0 due to the internal exchange
field. When I(S,) < 1,
B,(y) ( (S + )y. (126)
3
When Jo > 0, the condition for (Sz) / 0 then is
T < Tc JoS(S + 1)/3kB. (127)
This is consistent with the fact that J > 0 in Heisenberg Hamiltonian leads to
ferromagnetic interaction.
In a realistic calculation, Tc as a measurable parameter is easy to obtain,
hence we can use Eq. 127 to find the exchange interaction constant Jo, and thus
the spontaneous magnetization for T < Tc.
The spontaneous magnetization has fundamental effects on carrier scattering
and spin scattering, and thus affects the transport properties of both carriers and
spins. We will talk about this in C'! Ilpter 5.
1.4 Open Questions
Although the research of IIIV DMS has been carried on for more than one
decade, and people have gained lots of understanding of their properties, there
are still a lot of open questions which deserve a deep and thorough investigation.
Among these outstanding problems, the nature and origin of the ferromagnetism,
the nature of the band electrons, and the possible device applications are most
fundamental and crucial.
1.4.1 Nature of Ferromagnetism and Band Electrons
As we discussed in the last section, people have proposed a variety of theories
to explain ferromagnetism in IIIV DMS, each of which has its drawbacks. The
importance of the mechanism of ferromagnetism lies in the fact that it can predict
trends and lead people to search for suitable materials to achieve applications.
The first and widely publicized RKKY (Zener) model made predictions of above
room temperature ferromagnetism and prompted a worldwide search for materials
satisfying the conditions. The model asserts that localized spins in the IIIV DMS
will introduce hostlikehole states that will interact via RKKYtype coupling with
the Mn local moments to produce the observed ferromagnetism. Recently, Zunger
et al. [41, 42, 43] performed first principle calculations showing that contrary to
the RKKY model, the hole induced by Mn is not hostlike, which undermines
the basis of applying RKKY theory to DMS. The ensuing ferromagnetism by
the holes induced by Mn ions is then not RKKYlike, but !h 1 a characteristic
dependence on the latticeorientation of the MnMn interactions in the i i1I I
which is unexpected by RKKY". They claim that the dominant contribution
to stabilizing the ferromagnetic state was the energy lowering due to the p d
hopping. The nature of the ferromagnetism then is closely related to the nature
of the band electrons. Photoinduced ferromagnetism [44] clearly reveals the role
of holes in mediating the ferromagnetic coupling. There is no doubt carriers are
crucial in all the mechanisms accounting for the ferromagnetism, but are they really
hostlike holes, or do they have strong d component mixing? How do they behave
in the process of mediating the ferromagnetism? Only after we know the right
answer, will the manipulation of charge carriers and also the spins become more
predictable.
1.4.2 DMS Devices
The attraction of DMS mostly comes from their promising application
prospects. The special optical and magnetic properties can both be employ, ,1
designing novel devices. Semiconductor optical isolators based on IIVI DMS,
CdMnTe, which has a low absorption and large Faraday rotation for light with
0.98 pm wavelength, have been developed. This is the first commercial semiconduc
tor spintronic device [45]. Since IIVI DMS is paramagnetic at room temperature, a
magnetic field is needed to obtain Fard1vi rotation. Ferromagnetic semiconductor
based on IIIV DMS, which does not need an external magnetic field to sustain the
big Faraday rotation, should have a good potential for use in optical isolators.
Photoinduced ferromagnetism has been demonstrated by Koshihara et al.
[44] and Kono et al. [46]. In Koshihara's experiment, ferromagnetism is induced
by photogenerated carriers in InMnAs/GaSb heterostructures. The effect is
illustrated in Fig. 111. Due to the special band alignment of this heterostructure,
electrons and holes are specially separated, and holes accumulate in the InMnAs
1 vr. The photogenerated holes then cause a transition of the InMnAs ?1,r to
a ferromagnetic state. This opens a possibility to realize optically controllable
magnetooptical devices. In Kono's experiment, ultrafast demagnetization takes
place after a laser pulse shines on InMnAs/GaSb heterostructure and produces
ferromagnetism. The time scale is typically of several ps. They propose a new and
very fast scheme for magnetooptical recording.
8
(a) (*) *i*"*"
S 4 *
O0
2 /*
0
N
I 8 
w
z 4 (
8
H((T)
2
0.30 0.15 0.0 0.15 0.30
H (
Figure 111: The photoinduced ferromagnetism in InMnAs/GaSb heterostructure.
Reprinted with permission from Koshihara et al. Phys. Rev. Lett. 78: 46174620,
1997. Figure 3, Page 4619.
Recently, Ohno et al. [2] achieved control of ferromagnetism with an electric
field. They used fieldeffect transistor structures to vary the hole concentrations in
DMS 1 ,i. i and thus turn the carrierinduced ferromagnetism on and off by varying
the electric field. Rashba et al. [47] also proposed the electron spin operation by
electric fields. They also discussed the spin injection into semiconductors. The
electric control of ferromagnetism or spin states makes possible a unification of
magnetism and conventional electronics, and thus has a profound meaning.
Lowdimensional structures usually have dramatically different properties from
bulk materials. Much longer spin coherent times have been reported by several
groups in quantum dots [48, 49], which have been ,.. I 1. for use in quantum
computers where quantum dots can be used as quantum bits, since they offer a
twolevel system close to the ideal case. One ultimate goal of DMS spintronics is to
implement quantum computing. The use of semiconductors in quantum computing
has various benefits. They can be incorporated in the conventional semiconductor
industry, and also, lowdimensional structures are very easy to construct, so unique
lowdimensional properties can be employ, 1 Several proposals have been made for
quantum computing using quantum dots [50, 51, 52].
Spin manipulation needs injection, transport and detection of spins. The
most direct way for spin injection would seem to be injection from a classical
ferromagnetic metal in a metal/semiconductor heterostructure but this raises
difficult problems related to the difference in conductivity and spin relaxation
time in metals and semiconductors [53]. Although these problems are now better
understood, this has slowed down the progress for spin injection from metals. On
the other hand, this has boosted the research of connecting DMS with nonmagnetic
semiconductors for spin injection. Many experiments pursuing hign efficiency spin
injection have been carried out. Shown in Fig. 1 12 is a spin light emitting diode
[54], in which a current of spinpolarized electrons is injected from the diluted
magnetic semiconductor BetMi Zi:1_,_ySe into a GaAs/GaAlAs lightemitting
diode. Circularly polarized light is emitted from the recombination of the spin
polarized electrons with nonpolarized holes. An injection efficiency of 911' spin
polarized current has been demonstrated. As BeA Mi Zii,_,_ySe is paramagnetic,
the spin polarization is obtained only in an applied field and at low temperature.
aiGaAs
ncontact
dNM dSM
pcontact
100 nm 300 nm 100 nm 15 nm 500nm 300mm
Figure 112: Spin light emitting diode.
A ferromagnetic IIIV DMS based spin injector does not need an applied
field. Shown in the left panel of Fig. 113 is a GaMnAsbased spin injection and
detection structure [55], in which spinpolarized holes are injected from GaMnAs
to a GaAs quantum well. The emitter and analyzer are both made of l. 'ri of
ferromagnetic semiconductor GaMnAs. The temperature dependence of the spin
life time in the GaAs quantum well from magnetoresistance measurements is shown
in the right panel.
To obtain the information which a spin carries, one needs to detect an electron
spin state. Many methods for doing this have been brought forth and structures
or devices have been designed such as spin filters using magnetic tunnel junctions
[56, 57], spin filters [58], and one device involving a single electron transistor to
read out the spatial distribution of an electron wave function depending on the spin
state [59].
The development of DMSbased spintronics is now receiving a great attention,
and may become a key area in research and industry in the future. Although
enormous effort has been made, there is still a long way to go for DMS to be
extensively used in real life.
1 .0
08 0 O08
Ga1,JMnAs o0 aMagelazatim
06 O
S 0 0.6
"W life
0o.4 t0 04
GaNnAs 02 o02
0 10 20 30 40 50 60 70
Teerature (K)
Figure 113: GaMnAsbased spin device. Left: GaMnAsbased spin injector and
analyzer structure. Right: Temperature dependence of spin life in the GaAs quan
tum well in the structure shown in left panel.
CHAPTER 2
ELECTRONIC PROPERTIES OF DILUTED MAGNETIC SEMICONDUCTORS
To understand the optical and transport properties of DMS in the presence
of an applied magnetic field, we have to know the electronic band structure and
the electronic wave functions. For optical transitions, with the knowledge of the
interaction Hamiltonian, we may use Fermi's golden rule to calculate the transition
rate. In an external magnetic field, one energy level will split into a series of
Landau levels. Optical transitions can take place inside one series of Landau levels
or between different series according to the light configuration. So the knowledge of
the parities of these Landau levels need to be investigated. In this chapter, we will
use the k p method to study the band structure of DMS materials around the F
point. Specifically, a generalized PidgeonBrown model [60] will be used to study
the Landau level structures.
2.1 Ferromagnetic Semiconductor Band Structure
Ferromagnetic DMS's are different from normal semiconductors in that they
are doped with magnetic ions. These magnetic ions usually have indirect exchange
interaction resulting in an internal effective magnetic field. The electrons experienc
ing this effective field will have an extra energy gain. For a paramagnetic DMS, or
for a ferromagnetic DMS with Tc much smaller than the typical temperature, when
there is no applied magnetic field, there is no internal exchange field, so it is just
like the host semiconductor. The special properties are present only in an applied
field.
The extra energy gain in a ferromagnetic DMS can be treated in a mean
field approximation (see Section 1.3.2). The localized magnetic moments line up
along the effective field, so for each magnetic ion, it has a nonvanishing average
spin along the field direction. According to the discussion in Section 1.2.2, an
extra energy term proportional to the exchange constant will be added to the
Hamiltonian. This term is related to spin quantum numbers, thus different spin
states will gain different energies, leading to spin splitting. Shown in Fig. 21 are
calculated valence band structures for B,,t = 0 of bulk GaAs and ferromagnetic
GaMnAs, which has a Curie temperature Tc = 55 K. The calculation is actually
based on a generalized Kane's model [61], and the effective field is assumed to be
directed in z direction. Kane's model was developed from k p theory, which we
will introduce in the following section.
0.0 0.0
0 0.2 0.2
Pure GaAs, GaMnAs T=30 K
>, T=30 K E x=6% T=55 K
S0.4 0.4
0.6 0.6
1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0
(1,1,1) k (1/nm) (o,o,1) (1,1,1) k (1/nm) (0,0o,1)
Figure 21: Valence band structure of GaAs and ferromagnetic Ga,,t ., ,,, As
with no external magnetic field, calculated by generalized Kane's model. The spin
splitting of the bands is shown.
2.2 The k p Method
2.2.1 Introduction to k p Method
The k p method was introduced by Bardeen [62] and Seitz [63]. It is a
perturbation theory based method, often called effective mass theory in the
literature, useful for analyzing the band structure near a particular point ko, which
is an extremum of the band structure. In the case of the band structure near the F
point in a direct bandgap semiconductor, ko = 0.
The Hamiltonian for an electron in a semiconductor can be written as
2
H=
2mo
+ V(r),
(21)
here p = ihV is the momentum operator, mo refers to the free electron mass, and
V(r) is the potential including the effective lattice periodic potential caused by the
ions and core electrons or the potential due to the exchange interaction, impurities,
etc. If we consider V(r) to be periodic, i.e.,
V(r) V(r + R),
where R is an arbitrary lattice vector, the solution of the Schr6dinger equation
H k(r) = E k(r)
satisfies the condition
k (r) = eikrUk(r)
where
Uk(r + R) = Uk(r),
and k is the wave vector. Equations 24 and 25 is the Bloch theorem, which gives
the properties of the wave function of an electron in a periodic potential V(r).
The eigenvalues for Eq. 23 split into a series of bands [64]. Consider the
Schr6dinger equation in the nth band with a wave vector k,
(26)
p + V(r)j Qnk(r) E (k)Qnk(r).
2mo I
Inserting the Bloch function Eq. 24 into Eq. 26, we obtain
r p2 hk h2k2 ]
+ k p + + V(r) Unk(r) E,(k)Unk(r).
2mo mo 2mo
(27)
(22)
(23)
(24)
(25)
In most cases, spinorbit coupling must also be considered and added into the
Hamiltonian. The spinorbit interaction term is
S x. ( X VV) p. (28)
Including the spinorbit interaction, Eq. 27 becomes
P2 Vk V \2k2 V)
S+  p + 2 +x V) +  p + ))p + V (r) k (r)
2mo mo ii. *,,. 2mo
= E,(k)Uk (r).
(29)
The Hamiltonian in Eq. 29 can be divided into two parts
[Ho + W(k)]uk = EnkUnk, (210)
where
p2 h
Ho + a x VV) p + V(r) (211)
2m o 1',,. 
and
W k ) hk ( h $2) h2k2
W(k) p + h(x VV) + k (2 12)
mo 1 ,,:2 2mo
Only W(k) depends on wave vector k.
If the Hamiltonian Ho has a complete set of orthonormal eigenfunctions at
k = 0, uno, i.e.,
HoU.n = EnOUno, (2 13)
then theoretically any lattice periodic function can be expanded using eigenfunc
tions Uno. Substituting the expression
Unk Z Cn(k)Umo (214)
m
into Eq. 29, and multiplying from the left by u*0, and integrating and using the
orthonormality of the basis functions, we obtain
z[ (Eo En+ 2 k nm U n0 (p+ 2 (x VV) U) mO) c$(k) 0.
n2mo mo 2m .o,,,:
(215)
Solving this matrix equation gives us both the exact eigenstates and eigenenergies.
Usually, people only consider the energetically .,i.i i:ent bands when studying the k
expansion of one specific band. It actually becomes very complicated if one wants
to pursue acceptable solutions when k increases. One has to increase the number of
the basis states, go to higher order perturbations, or both.
When k is small and we neglect the nondiagonal terms in Eq. 215, the
eigenfunction is Unk = UnO, and the corresponding eigenvalue is given by Ek =
Eno + This solution can be improved by second order perturbation theory, i.e.
hE Z i 2k2 + ((Uno H' umTo) (umo H' uno) (26)
Enk = En0 + 2 + o (216)
2mo Eno Emo
where
H' . (p+ 2 (x VV). (217)
In the calculation shown above, we used the property (uo (p + 4mToc2( xVV)) u 0)
0, which holds for a cubic lattice periodic Hamiltonian due to the
If we write
7 p + 21,( (x VV) (2 18)
then the second order eigenenergies can be written as
Sh2k2 h2 Inmk2
Enk = Eno + 2+ 2 nmk (219)
2mo m Eo Enmo
rn7nn
Equation 219 is often written as
Ek= E0 + h ( kk,, (220)
a,i3 a 3
where
1 1 2 7a( _)3
S6 E + m (221)
m* nmo ai 7 m Eo Emo
is the inverse effective mass tensor, and a, 3 x, y, z. The effective mass generally
is not isotropic, but we can see it is not kdependent, this is because at this level
of approximation, the eigenenergies in the vicinity of the F point only depend
quadratically on k.
2.2.2 Kane's Model
As we mentioned in the last section, expanding in a complete set of orthonor
mal basis states in Eq. 215 gives exact solutions for both the eigenfunctions and
eigenenergies. Practically, it is not feasible to include a complete set of basis states,
so usually only strongly coupled bands are included in usual k p formalism, and
the influence of the energetically distant bands is treated perturbatively.
In Kane's model, electronic bands are divided into two groups. In the first
group, there is a strong interband coupling. Usually the number of bands in this
group is eight, including two conduction bands (one for each electron spin) and six
valence bands(two heavy hole, two light hole and two splitoff hole bands). The
second group of bands is only weakly interacting with the first group, so the effect
can be treated by second order perturbation theory.
Shown in Fig. 22 is the band structure of a typical IIIV direct band gap
semiconductor. Due to crystal symmetry, the conduction band bottom belongs to
the F6 group, the valence band top belongs to the Fs group, and the splitoff band
belongs to the F7 group. The spatial part of the wave functions at the conduction
band edge are slike and those at the valence band top are plike. Symbols of
IS), IX), Y), and Z) are used to represent the one conduction band edge and
Distant bands
Distant bands
Figure 22: Band structure of a typical IIIV semiconductor near the F point.
Kane's model considers the doubly spin degenerate conduction, heavy hole, light
hole and splitoff bands, and treats the distant bands perturbatively.
three valence band edge orbital functions. With spin degeneracy included, the
total number of states is eight. These eight states IS T), IS 1), IX ), IX 1),
etc, can serve as a set of basis states in treating these eight bands. A unitary
transformation of this basis set is still a basis set. So in practice, people use the
following expressions, which are the eigenstates of angular momentum operators J
and mj, as the basis states for the eightband Kane's model,
1
2
3
U2 
2
3
2
1
4 =  ,
2
1
2
3
U6 = ,
2
3
U7 =  2
2
1
8 2'
1
2
3
2
1
)
2
1
2
3
)
2
1
1
ST) IsT),
HH T) 1 (X + iY)
1
2
ILH T) l(X iY) 1 2Z T),
V6
ILH 1) = (X + iY) 2 ),
V3
(222)
This set of basis states is a unitary transformation of the basis which we have
mentioned above, and it can be proven that they are the eigenfunctions of the
Hamiltonian 211. Because of spin degeneracy at k = 0, the eigenenergies for IS),
HH), LH) and ISO) are Eg, 0, 0, A, respectively, with the selection of energy
zero at the top of Fs band, where E, is the band gap, and
3ikh 0V
A= 2 (XI p
4m2C2 Ox
OV
PxY),
iiy
(223)
is the splitoff band energy.
At this level of approximation, the bands are still flat because the Hamiltonian
310 is kindependent. Including W(k) in Eq. 212 into the Hamiltonian, and
defining Kane's parameter as
ih
P= (SIrZ), (224)
Tmo
we obtain a matrix expression for the Hamiltonian H = Ho + W(k), i.e.,
E, + Pk^ PkP PkI_ 0 0 Pk, Pk,
Pk_ hk 0 0 0 0 0 0
2mo
Pk 0 0 Pk 0 0 0
Pk+ 0 0 A+ 2Pk, 0 0 0
0 0 /jPPk Pk + Pk_ Pk+ Pk+
Pk 0 20 0 PI k 0 0+
+ S2o 0
 Pkz 0 0 0 Pk_ 0 2tko 0
Pk 0 0 0 Pk_ 0o o A + h 2
(225)
where k+ = k + iky, k_ = kx iky, and k1, ky, kz are the cartesian components
of k. The Hamiltonian 225 is easy to diagonalize to find the eigenenergies
and eigenstates as functions of k. We have eight eigenenergies, but due to spin
degeneracy, there are only four different eigenenergies listed below. For the
conduction band,
S2k2 1 1 4P2 2P2
E= E,+ ++ (226)
2m,' m mno 32E, 3h2(E, + A)
For the light hole and splitoff bands,
h2k2 1 1 4P2
Emlh = + 42 (227)
2mih T mh mo 3h2E
h2k2 1 1 2P2
Eo A 2k+ 2= (228)
2mso mso mo 3h2(E, + A)
For the heavy hole band we have
h2k2 1 1
Ehh 2 1 (229)
2mhh mhh mo
The effective mass of the heavy hole band is still equal to the bare electron mass,
since we have not included the distant band coupling in the Hamiltonian. The
effect of the distant band coupling will make the heavy hole band curve downward
rather than upward.
2.2.3 Coupling with Distant BandsLuttinger Parameters
The coupling with distant bands can be parameterized by L6wdin's pertur
bation method [65], in which the bands are classified as A and B. In our case, we
select the basis states 222 as class A and label them with subscript n and all the
other (energetically distant) states as class B which we label with subscript a.
Suppose all states are orthonormal, the Schr6dinger equation then takes the
form
Y(Hi Ezi)a, 0, (230)
where I and m run over all states. Rewrite this equation using class A and B, and
we obtain
A B
(E H,,m)am = Hmnna, + H,,a, (231)
n4m a4m
or
A B
amn= "E + "EH a,, (232)
/:E i + E Hnm
n4m n4a
where the first sum on the right hand side is over the states in class A only, while
the second sum is over the states in class B. We can eliminate those coefficients in
class B by an iteration procedure and obtain the coefficients in class A only,
A UA HI_ nn
am m H m na (233)
and
B B
Ur HT + H ,1m mH,,lan + YH, + (234)
z E Haa (E Haa)(E H)..
a#3
A little algebra shows that Eq. 2 33 is equivalent to
A
5(U[/ E6m,)a, = 0. (235)
n
This means that we can find the eigenenergies with the basis in class A but still
include the remote effects from class B using Eq. 235. The effect from class B is
treated as a perturbation using Eq. 234 to second order.
Truncating U,, to the second term, and using Hamiltonian in Eq. 29, it can
be rewritten as
B B H H
UTn Hmn + HmH Hmn+ Y a an (236)
a7m,n a4m,n
where
Hn = (umo H uno) = E,(0) + h2k 62 n (237)
and
H (oh = o k o)= hka, (238)
mo mo
where a = x, y, z and r, a_ pa for m E A and a E B. Thus
F h2k21 h 2 B kk rPa
n Ew(0) + 22mo 6 + 2+ ka P (239)
L OJ 7m,n a,b
Applying basis set 222, we can define parameters A, B, C and F as follow,
h2 h2 B p x
S2r Xa aX
2mo m02 z E, Ea'
B= +
B 2mo + zm0 E, Ea'
h2 pB Y 7 p7
Co
m E2 Ea
B
F = 1 PaPs (240)
Tmo Ec Ea
Rewriting these parameters in terms of "L il i ,, parameters 71, 72, 73 and 74
defined as
h2
O71
h2
2nnO 72
h2
<73
2mo
(A + 2B),
3( )
(A B),
6
74 1 + 2F,
we can obtain the the Hamiltonian Hmn
under the basis set listed in Eq. 222 as
Umn including the distant band coupling
Vk_ Vk_
M iz2M
P+Q i 2Q
i2MM+ iv2Q
0 i /lVk,
PA
0
0
i Vk,
Vk,
v3^
SVk E, 2 7 2
73 1 E+2 0<74
0 VVkk
0
L
SVL
SVk_
72
Vk,
L L
0 16L+
1/i L+ 0
 Vk+ Vk+
k2 k2 + k2 + k2,
K y ~
h2
P = o 7k2,
2mo
2T
L ih v373(kx 
mo
h2
M 3 
2mo
h2
0 = 2
V o 2 '
1Vk_
'Vk_
73'
 Q M+ i2M
M P+Q iz2Q
2M i2Q P A
(245)
 2k2),
iky)kz,
 2) 21 I. ) ,
y~ Ij
(246)
(241)
(242)
(243)
(244)
Eh2 k2
vg + 2To 74
"Vk_
72
PQ
V~Vk
3 Vk,
0
i L
P
i\
where
and
E = 2 P2 (247)
related to the Kane's parameter P defined in Eq. 224. We can see that if k = 0 or
k, = 0, the Hamiltonian is block diagonalized.
In practice, one important thing needs to be noted that the Luttinger parame
ters defined in Eq. 244 are not the "usual L iIIli:, i parameters which are based
on a sixband model since this is an eightband model, but instead are related to
the usual Luttinger parameters 7
L Ep
L Ep
72 72 6E'
6 Eg
73 3 EL p (248)
3E,
This takes into account the additional coupling of the valence bands to the
conduction band not present in the sixband Luttinger model. We refer to 71
etc. as the renormalized Luttinger parameters.
The Hamiltonian 245 is based on an eightband Kane's Hamiltonian including
the contributions of the remote bands. With the remote band coupling, the
electron effective mass at the conduction band minimum now becomes
S 4 + + (249)
m, mo 3 E9 E9 +A
In DMS materials without magnetic fields, the Hamiltonian 245 plus the
exchange interaction can be used to calculate the band structure which will be
applied to the calculation of the optical properties such as magnetooptical Kerr
effect, which is to be studied in chapter 4.
In a magnetic field, a single energy level splits into a series of Landau levels.
Optical transitions take place between two levels in one series or two in different
series.
2.2.4 Envelope Function
In the treatment of Kane's model (or sixband Luttinger model), all the bands
in class A are considered as degenerate at the F point. Away from the F point
or/and taking the remote band coupling into account, the electronic wave functions
become linear superposition of the basis states.
In last section, if we write
2k2 2 B kkba b
Hrmn mn Em(O) + 2mo 6mn + hn 2 k Tman
2Tmo Mc Er Ea
aam,n a,b (250)
= Em (0)6mn + D jb kakb,
the eigenequation is given by
H a.(k) Em(0)6 + Dab kakb a.(k)= E(k)am(k) (252)
n 1 n 1 a,b
where am is the superposition coefficients defined as
fnk(r) a,(k)uno. (2 53)
Now we consider a spatial perturbation U(r) added to the Hamiltonian Hm.
The eigenequation now becomes
[H + U(r)]b(r) = E(r).
(254)
If we write the solution to the equation as
8
(r) Fm(r)umo(r), (255)
nm=l
Luttinger and Kohn [67] have shown that we need only solve the following equa
tion,
SEm(0)6mn + a D + U(r)6+ m F,(r) = EFm(r)
(256)
This means that we only need to replace the wave vector in the Hamiltonian k" by
the operator pa/h, and solve an equation for F(r). The function F(r) is called the
effective mass envelope function.
2.3 Landau Levels
2.3.1 Electronic State in a Magnetic Field
Using the simple effective mass theory, the motion of an electron in semicon
ductors is like that of free electrons. In the simplest case, we consider a parabolic
band, and assume the effective mass to be m. The wave equation under the
effective mass approximation is
1 (ihV + eA)2(x) c~(x), (257)
2m
where A is the vector potential, and e is the electron charge. Assume the magnetic
field is directed along z. Using Landau's gauge,
A = Byx. (258)
and assuming a solution like
(x, y, z) = i +kz) )Q(y), (259)
V&xyLL
where L,, Ly, and L, are lengths for the bulk crystal in three dimensions. After
substituting into the effective mass equation, we have an equation for ((y),
2m (k e )2 h2 2+ 2k2 O(y) = e(). (260)
k2 72
Defining c' e 2, the equation above is a simple harmonic oscillator equation
with
= (n + n :. (261)
2
where wc = eB/m is the cyclotron frequency. Thus the total energy is
1 h2k2
S(n + )h + (262)
2 2m
This means that in a magnetic field, the motion of an electron in a semiconductor
now has quantized energies in the x y plane, though its motion in the zdirection
is still continuous. The original states in one band now split into a series of Landau
levels whose eigenfunctions are
S( hk )i(kx+k (263)
VLLX eB
The electronic energies in Eq. 262 is only related to n and kz. They are
degenerate for different kx. In Eq. 263 the center of yo = can only be from 0
to Ly. Using the periodic boundary condition, the interval for kx is 27/Lx, thus the
interval for yo is h/eBLx, the corresponding number of values for yo is eBLxL,/h.
Therefore, for given n and k,, the degeneracy is eBLxL,/h.
2.3.2 Generalized PidgeonBrown Model
In a realistic calculation for electronic states in a magnetic field, the simple
qualitative theory is not adequate. From the discussion of the k p theory, we
know that the band structure is very complicated. So in this section, we will
use the k p based Hamiltonian to calculate Landau levels in DMS. For narrow
gap semiconductors such as InAs, the coupling between the conduction and
valence bands is I ri,. so it is necessary to use the eight band model to calculate
the Landau levels. Pidgeon and Brown [60] developed a model to calculate the
magnetic field dependent Landau levels at k = 0. We will generalize this model to
include the wave vector (k,) dependence of the electronics states as well as the s d
and p d exchange interactions with localized Mn d electrons.
We will still utilize the basis set defined in Eq. 222. In the presence of a
uniform magnetic field B oriented along the z axis, the wave vector k in the
effective mass Hamiltonian is replaced by the operator
k (p + A), (264)
where p = ihV is the momentum operator. For the vector potential, we still use
the Landau gauge as in Eq. 258, thus B = x A = Bz.
Now we introduce two operators
at A (k + ik) (265a)
v2
and
a (k iky) (265b)
2
where A is the magnetic length which is defined as
hc h2 1
rh h2 (266)
eB 2m pBB
The operators defined in Eqs. 265 obey the commutation rules of creation and
annihilation operators. The states they create and annihilate are simple harmonic
oscillator functions, and ata = N are the order of the harmonic functions. Using
these two operators to eliminate kx and ky in Hamiltonian 245, we arrive at the
Landau Hamiltonian
HL La L (267)
Lt Lb
with the submatrices La, Lb and Lc given by
i at
v
Eg+A
a
V
Va
0
0
i 3Vk,
 1Vk,
iza
PQ
Mt
 at
PQ
M P + Q i2Q
iM i2Q P P A
0 Vk, i Vk
L 0 6i Lt
 2 L Lt 0
The operators A, P, Q, L, and M in Eq. 267 now are
A 2 Y4 (2N + 1
mo 2 A2
p 2 71
P
mo 2
h2 72
Qmo 2
mo 2
+ k2)
12) 1
2N +k
A 2 ) 1
2N +1
A 2
S h2
L = 73
mo
2k 2
iW6
\
M  + )3(
mo 2
A 2
Mat
M
VMat
i 2
iA/at
P +Q iz2Q
iz 2Mt i 2Q
PA
I a
iV2Mt
1 a
Mt
(268)
(269)
(270)
(271a)
and
(271b)
(271c)
(271d)
(271e)
The parameters 71, 72, 73 and 74 are defined in Eq. 248 and 244. Usually,
the Luttinger parameters 72 and 73 are approximately equal (spherical approxima
tion), so we have neglected a term in M proportional to (72 73)(at)2. This term
will couple different Landau manifolds making it more difficult to diagonalize the
Hamiltonian. The effect of this term can be accounted for later by perturbation
theory.
For a particle with nonzero angular momentum (thus a nonzero magnetic
moment p) in a magnetic field, the energy due to the interaction between the
magnetic moment and the magnetic field is p B, which is called Zeeman energy
which we discussed in Section 1.2.2. The electrons in IIIV DMS conduction or
valence bands possess both orbital angular moment and spin, so there is one extra
Zeeman term proportional to (KoL B + Klra B), where L and a are the orbital
angular momentum and spin operators, both of which are in matrix form. K0
and K1 are the magnetic field dependent coefficients. Following Luttinger [66], we
define the parameter K as
KL = (272)
6E,
where
L L 2 ,L 1 L 2
S= 73 + 72 371 (273)
is the Luttinger K parameter, and we obtain the Zeeman Hamiltonian
h2 1 Za 0
Hz  (274)
mo A2 0 Z*
where the 4 x 4 submatrix Za is given by
1 0 0 0
2
0 0 0
Za 2 (275)
0 0 i
0 0 i,/ K i
Due to existence of the Mn impurity ions, the exchange interactions between
the band electrons and localized moments also needs to be accounted for. This
term is proportional to (, J(r RI)SI o a). Under a mean field and virtual i I I1
approximation (see Section 1.2.2), and defining the two exchange constants
a= (SJIS) (276a)
and
P 3 (XJX), (276b)
we can arrive at an exchange Hamiltonian
HM x No(S) D (277)
0 D*
where x is the Mn concentration, No is the number of cation sites in the sample,
and (S,) is the average spin on a Mn site which is exactly the one we derived at
Section 1.2.2 for paramagnetic DMS or that in Section 1.3.2 for ferromagnetic
DMS. The 4 x 4 submatrix Da is
Si 0 0 0
0 >0 0 0
Da= 2 (278)
0 0 $3 (i2p
0 0 i 3
Here we just treat the effect of magnetic ions as an additional interaction. We
don't consider the possible effect of these magnetic ions on the band gap, etc. The
band gap changes as a result.
The discussion here is very similar to that in Section 1.2.2 where only a
qualitative model is introduced, but here we used a realistic band structure. Also
similar to that discussion, the total Hamiltonian here can be written as
H = HL + Hz + HMn. (279)
We note that at k, = 0, the effective mass Hamiltonian is also block diagonal like
the Hamiltonian 245.
2.3.3 Wave Functions and Landau Levels
With the choice of Gauge 258, translational symmetry in the x direction is
broken while translational symmetry along the y and z directions is maintained.
Thus ky and k, are good quantum numbers and the envelope of the effective mass
Hamiltonian 279 can be written as
al,n,v (n1
a2,n,v .' 2
a3,n,v
Ci(kyy+kzz) a4,n,v n
,,' A = (280)
a6,n,,v n+1
a7,n,,v n1
a8,n,v 1
In Eq. 280, n is the Landau quantum number associated with the Hamilto
nian matrix, v labels the eigenvectors, A = LxLy is the cross sectional area of the
sample in the x y plane, ,(0) are harmonic oscillator eigenfunctions evaluated at
( = x A2ky, and ai,,(kz) are complex expansion coefficients for the vth eigenstate
which depend explicitly on n and k,. Note that the wave functions themselves will
be given by the envelope functions in Eq. 280 with each component multiplied by
the corresponding k, = 0 Bloch basis states given in Eq. 222.
Substituting T,, from Eq. 280 into the effective mass Schrodinger equation
with H given by Eq. 279, we obtain a matrix eigenvalue equation
H. F.,, = E,, (k,) F,,,, (281)
that can be solved for each allowed value of the Landau quantum number, n, to
obtain the Landau levels E,,,(k,). The components of the normalized eigenvectors,
F,,,, are the expansion coefficients, ai.
Since the harmonic oscillator functions, n,'((), are only defined for n' > 0,
it follows from Eq. 280 that F,,, is defined for n > 1. The energy levels are
denoted E,,,(k,) where n labels the Landau level and v labels the eigenenergies
belonging to the same Landau level in ascending order.
Table 21: Summary of Hamiltonian matrices with different n
n Dimension of Hamiltonian Eigenenergy No. Label as
1 1 x 1 1 (1,1)
0 4x4 4 (0,v), v 1. 4
1 7x7 7 (1,, v), 1
>2 8x8 8 (n,v), v 18
For n = 1, we set all coefficients ai to zero except for a6 in order to prevent
harmonic oscillator eigenfunctions 0,, (0) with n' < 0 from appearing in the
wavefunction. The eigenfunction in this case is a pure heavy hole spindown state
and the Hamiltonian is now a 1 x 1 matrix whose eigenvalue corresponds to the a
heavy hole spindown Landau level. Please note that when we speak about a heavy
(light) hole state, it generally means that the electronic wave function is composed
mainly of the heavy (light) hole Bloch basis state near the k = 0 point.
For n = 0, we must set al = a2 = a7 = a8 = 0 and the Landau levels
and envelope functions are then obtained by diagonalizing a 4 x 4 Hamiltonian
matrix obtained by striking out the appropriate rows and columns. For n = 1, the
Hamiltonian matrix is 7 x 7 and for n > 2 the Hamiltonian matrix is 8 x 8. The
summary of Hamiltonian matrices for different n is given in Table 21.
The matrix Hn in Eq. 281 is the sum of Landau, Zeeman, and exchange
contributions. The explicit forms for the Zeeman and exchange Hamiltonian
matrices are given in Eq. 274 and 277 and are independent of n.
Table 22: InAs band parameters
Energy gap (eV)1
E, (T = 30 K) 0.415
Eg (T = 77 K) 0.407
E, (T = 290 K) 0.356
Electron effective mass (mo)
m 0.022
Luttinger parameters 1
71L 20.0
72 8.5
73 9.2
KL 7.53
Spinorbit splitting (eV) 1
A 0.39
Mn sd and pd exchange energies (eV)
No a 0.5
No f 1.0
Optical matrix parameter (eV) 1
Ep 21.5
Refractive index 2
n, 3.42
1 Reference [68].
2 Reference [69].
Now we study the Landau level of InAs and InMnAs, which in the following
we assume paramagnetic. The parameters used in the calculation are listed in
Table 22. Shown in Fig. 23 are the conduction band Laudau levels for InAs
and In,,.. i,,, As as a function of magnetic field at k = 0 for a temperature of
30 K. The dashed lines represent spinup levels, and the solid lines represent the
spindown levels. This illustrates the energy splitting of the conduction band at
the P point. The right panel for InMnAs is only different from the left panel for
0.9 0.9 
0.6 o.6
0.5 0.5
0.4 0.4
0 20 40 60 80 0 20 40 60 80
B (Tes a) B (Tesca)
Figure 23: Calculated Landau levels for InAs (left) and In, ..Mi!,, ,As (right) as a
function of magnetic field at 30 K.
InAs in that it has the exchange contributions due to the interaction between the
band electrons and the localized Mn moments. The ordering of these Landau levels
can be qualitatively explained by the simple model in Eq. 113 where we have an
analytical expression for the Landau level energy. Note that they are not linear
functions of the magnetic field. In the next chapter we will see that this simple
model cannot predict an a (exchange constant defined in Eq. 276) dependence of
the cyclotron energy, which is the energy difference between two .,i1] ient Landau
levels with the same spin. The exchange constant dependence is a consequence of
k p mixing between conduction and valence bands.
The wave vector kz dependence of Landau levels in both conduction band and
valence bands is shown in Fig. 24, where only the five lowest order Landau levels
are shown. Because of the strong state mixing, the spin states in valence bands are
not indicated. Comparing the left and right panels of Fig. 23 and Fig. 24, we can
see that Mn doping drastically changes the electronic structure. Spin splitting is
greatly enhanced in both conduction and valence bands. As a matter of fact, the
spin state ordering in the conduction band is reversed with Mn doping.
53
0.8 0.8
S0.7 0.7
> 0.6 0.6
0.4 0.2 0.0 0.2 0.4 0.4 0.2 0.0 0.2 0.4
Kz (1/nm) Kz (1/nm)
Figure 24: The conduction and valence band Landau levels along kz in a mag
netic field of B = 20 T at T = 30 K. The left and right figures are for InAs and
0In, 2, As, respectively.
0.2 0.2
0.3 ,0.3
04 0 .2 0 0.2 0.4 0.4 0.2 0.0 0.2 0.4
Figure 24: The conduction and valence band Landau levels along k, in a mag
netic field of B 20 T at T 30 K. The left and right figures are for InAs and
In,, .. 11_,,,, 1,A s, respectively.
2.4 Conduction Band gfactors
In practice, spinsplitting is represented the gfactor. For a free electron, the
gfactor is the ratio between the magnetic moment due to spin in units of pB and
the angular momentum in units of h. The gfactor for a free electron is 2 (if the
influence of the black body radiation in the universe is accounted for, it is 2.0023).
In the solid state, due to the spinorbital interaction (and other interactions, for
example in DMS, the exchange interaction), the gfactor for an electron is not 2.
Usually the gfactor in the solid state is defined as
g o=pi, (282)
P/BB
where hw8spi, is the spinsplitting. Roth et al. [70] have calculated the gfactor in
semiconductors based on Kane's model, and have shown that the gfactor in the
54
conduction band is
2r1+ 1 < j. (283)
m, 3E,+ 2A
Using this equation, the gfactor for bulk InAs is about 15.1, which is close to the
experimental value 15 [71].
150 I I 0' O
.......... 12%
0 290 K ......... 12%
 .. ..  2 .% :
S50 20 40 60 0 20 40
.15 .
C ` ~ "  5
.. .. .. .. ... ...................................... 2 .5 %
0%
50 I I I I I I I I I 25 F
0 20 40 60 0 20 40 60
B (Tesl) 8 (Tesl)
Figure 25: Conduction band gfactors of InlMn1As as functions of magnetic
field with different Mn composition x. For the left figure, T = 30 K and for the
right, T = 290 K. Note at high temperatures we lose the spin splitting.
Due to the exchange interaction, the spinsplitting is greatly enhanced.
Usually in DMS, the exchange energy is much bigger than the Zeeman energy,
which can be seen from the simple theory in Eq. 115 for a few percent of Mn
doping. In that case, if we take x = 0.1, Na = 0.5 eV, and T = 30 K, then
gff ~ 256. If we only consider the exchange interaction, from Eq. 278, the
spinsplitting in the conduction band is exactly that in Eq. 113. However, this
is not correct because the first conduction band spindown level comes from the
n = 0 manifold, while the first conduction band spinup level comes from the n = 1
manifold. Different manifold numbers result in different matrix elements, which
will cause different state coupling, and thus spinsplitting due to the exchange
interaction is not what the simple model predicts. The conduction band g
factors for InAs and InMnAs at 30 K and 290 K are shown in Fig. 25. This
clearly demonstrates how Mn doping affects the gfactors. At 290 K, the g
200 .
S30K
150 \\ In MnAs 80K
\' To=110K .... 150K
0\\ c  290 K
o \
o 100 \\
50 
0
0 20 40 60 80
B (Tesla)
Figure 26: ,jf Ii.. 1 of ferromagnetic In, ., i,,, As. Tc = 110 K.
factors are drastically reduced. This is because at high temperatures, thermal
fluctuations become so large that the alignment of the magnetic spins is less
favorable. However, if ferromagnetic DMS are employ, .1 due to the internal
exchange field, a strong alignment can be expected even at high temperatures. Now
we suppose a highTc In,, ..,Mi As system in which a Curie temperature of 110 K
is achieved. The gfactor for this system is shown in Fig. 26. Even at relatively
high temperature (still below the transition temperature though), big gfactors are
still obtained. The gfactor reaches infinity at zero field when temperatures are
below Tc because there is still spinsplitting even though there is no external field.
CHAPTER 3
CYCLOTRON RESONANCE
In chapter 2, a systematic method of calculating the electronic structure
of DMS was developed and described in detail and applied to the narrow gap
InMnAs. It has been seen that the band structure of DMS depends strongly
on Mn doping which induces the exchange interaction. The band structure also
depends on the strength of the applied magnetic field, as can be seen from Fig. 23
and 25. Apart from the theoretical calculation, optical experiments are ahliv
good v,V to detect the electronic properties of semiconductors. Among these
methods, cyclotron resonance (CR) is an extensively used and a powerful diagnostic
tool for studying the intersubband optical properties and effective masses of
carriers. Cyclotron resonance is a highfrequency transport experiment with all
the complications which characterize transport measurements. Through cyclotron
resonance, one can get the effective masses, which are determined by the peak of a
resonance line, while scattering information is obtained from the line broadening.
Cyclotron resonance occurs when electrons absorb photons and make a transition
between two .i,.] ient Landau levels. From cyclotron resonance measurements one
can infer the magnetic field dependent band structure of the material. Since the
band structure of a DMS is so sensitive to magnetic fields, this is a useful means to
study and obtain band information from a comparison between the experimental
results and theoretical calculations.
3.1 General Theory of Cyclotron Resonance
3.1.1 Optical Absorption
The absorption coefficient, a, can be determined by calculating the absorption
rate T of incident light with angular frequency w in a unit volume. Suppose the
energy flux of the incident light is S, then the photon flux density is S/hw, and we
have Tdx = Sadx/hw, i.e.
a(WU) = T() (31)
S
T(w) is the sum of the transition probabilities Wif under the illumination of light
with angular frequency w divided by the volume, namely
T = W (32)
i,f
where i, f are the labels for the initial and final states. The summation runs over
all states. For absorption between state i and f, the transition probability from
Fermi's golden rule [72] is,
2r
Wabs = Hf 26(Ef Ej hw), (33)
and for emission
27 1
Wems 2= H 26(E Ef+ hw), (34)
where Ei and Ef are the energies of the initial and final states (here we only want
the final expression for absorption, so in emission, even the electrons transit from
state f to state i, we still call state i is the initial state, and state f the final state),
respectively, and the 6 function ensures the conservation of energy in the optical
transition. H' is the electron photon interaction Hamiltonian. Essentially, in
optical transitions, momentum should also be conserved. However, since the photon
momentum p = h/A is much smaller than the typical electron momentum, we
generally consider the optical transition to be v,i I ical", which means an electron
can only transit to states with the same k, i.e., wee ignore the photon momentum.
In semiconductors when dealing with the realistic case of absorption, we need
to take into account the state occupation probability by electrons, which in thermal
equilibrium is described by a FermiDirac distribution function
1
f = t (35)
1 + EEFkBT' (5)
and so the rate of absorption in the whole
1 2 1 1
Tif= H h 26(Ef E, hw)fi( ff) (36)
iv
and the emission rate
1 2x
Tf = 2 IH, 2I(E, Ef + hw)ff(1 fi) (37)
i,f
Due to the hermitian property of H', IHi, = IH' I. The net absorption rate per
unit volume then is
T = Tf T E hu)(f, ff). (38)
i,f
When a semiconductor is illuminated by light, the interaction between
the photons and the electrons in the semiconductor can be described by the
Hamiltonian,
1
H (p + eA)2 + V(r) (39)
2mo
where mo is the free electron mass, e is the electron charge, A is the vector
potential due to the optical field, and V(r) is the < iil periodic potential (in
DMS, including the virtual crystal exchange potential). Thus the oneelectron
Hamiltonian without the optical field is
Ho + V(r) (310)
2mo
and the optical perturbation terms are
e e2A2
H' A .p+ (311)
mo 2mo
Optical fields are generally very weak and usually only the term linear in A is
considered, i.e., we treat the electronphoton interaction in a linear response regime
and neglect twophoton absorption. The transition due to the optical perturbation
in Eq. 311 can take place either across the band gap or inside a single band
(conduction or valence band) depending on the photon energy. In this chapter, we
only consider cyclotron resonance, which takes place between the Landau levels
within conduction or valence bands.
For monochromatic light the vector potential is
A = eA,,, .(K r uj) eirewt + e iKr eit (312)
2 2
where K is the electromagnetic wave vector, w is the optical angular frequency, p
is the momentum operator, and e is the unit polarization vector in the direction of
the optical field, representing the light configuration.
The energy flux of the optical field can be expressed by the Poynting vector,
S E x H. Using the relations E = OA/Ot, H V x S/p, and u/K c/n,, the
averaged energy flux then is
n, 2A2
S = rnA (313)
2/ic
Using this relation and Eq. 38, the absorption coefficient then is
hwT hw 1 2r
a() (nw2A/2pc) v IH I26 (EF E , hw)(f, f). (3 14)
i,f
According to Eq. 312, the interaction Hamiltonian can be written as
eAo eAo
H "f 0(f le pi) Pfi, (315)
2mo 2mo
so the absorption coefficient 316 becomes
a(w) pif 26 (Ef E hw)(f, ff). (3 16)
,icf
Note that the interaction 315 is based on the dipole approximation. So in the
following when we talk about selection rules, etc, they are electric dipole selection
rules.
The scattering broadening (as well as disorder) can be parameterized by the
linewidth F through the replacement of the 6 function by a Lorentzian function [72]
as
6(Ef E w) 2 (317)
(Ef E, h)2 + (F/2)2
3.1.2 Cyclotron Resonance
From a classical mechanical point of view, in the presence of a magnetic field,
an electron moves along the field direction in a spiral, whose projection in the
perpendicular plane is a circle. The angular frequency for this circular motion is
eB
c = e (318)
mo
where mo is the free electron mass (effective mass when in a semiconductor). If
an electromagnetic wave is applied with the same frequency, the electron will
resonantly absorb this electromagnetic wave.
Quantum mechanically, an electron in a magnetic field will have a quantized
motion. Referring to Eq. 262, the energy of the electron splits into a series of
Landau levels. If the energy quanta hk of the applied electromagnetic wave are
exactly the same as the energy difference hkc between two .il1i ient Landau levels,
the electron will absorb one photon to transit from the lower Landau level to the
higher one. This is called cyclotron resonance.
In the presence of a magnetic field, the Hamiltonian 310, in DMS system,
is replaced by the one in Eq. 279. We already have the eigenstates for this
Hamiltonian. For convenience, we rewrite them here as
al,n,v Onl l
a2,n,v ,' 2U2
a3,n,v QnU3
ei(kyy+kzz) a4,n,v OnU4
nv = (319)
VIA a,, .
a6,n,v Qn+lU6
a7,n,v in1U7
a8,n,v 18
The eigenfunction above can be considered as the linear superposition of eight
basis states, each of which is composed of two parts. '. is the harmonic oscillator
envelope function, which is slowly varying over the lattice, and can be considered
constant over a unit cell length scale. ui is the Bloch part of the wave function,
which varies rapidly over a unit cell and has the periodicity of the lattice.
Now let us inspect the properties of the momentum matrix element in Eq. 3
16. Using n, v as the new set of quantum numbers, and utilizing the spatial
properties of the wave functions, we can factorize the integral into two parts and
write the matrix element as
Pn"' ax ((upu)( . + (uju,)( p ')). (320)
i,i'
Since the Bloch functions ui are quickly varying functions, their gradients are much
larger than those of the envelopes 0i. As shown in Ref. [73], the first term on the
right hand side dominates both in narrow gap and wide gap semiconductors, so we
have neglected the second term in our calculation. However, it is easy to check that
these two terms obey the same selection rules.
We can factorize ep to ep = +ep_ + e_p+ + ep where e (x iy)/V2, and
p = (p, ipy)/ /2. In the Fa ,d'v configuration (light incident along the magnetic
field B), the circularly polarized light can be represented by unit polarization
vector e. In this case, we only need to consider the matrix elements of p. It is
easy to check that
(n, v p+ n', v') oc 6,_1,n/ (321)
and
(n, vlp_ n', v') oc 6,i,/. (322)
This means that p+ and p_ are raising and lowering operators for the eigenstates.
For p+, an electron will absorb an e_ photon to have an n  n+1 transition, which
usually happens in the conduction band for electrons, so we call this transition
"electron I ., (eactive). For p_, an electron will absorb an e+ photon to have
an n n 1 transition, which usually happens in the valence bands for holes,
so we call this transition "hole .. I ,ii (hactive). The quasiclassical picture for
the two types of absorption is shown in Fig. 31. To comply with conservation of
BB
electron orbit hole orbit
Figure 31: Quasiclassical pictures of eactive and hactive photon absorption.
both energy and angular momentum, in a quasiclassical picture, electrons can only
absorb photons with eactive polarization, and holes can only absorb photons with
hactive polarization. In a quantum mechanical treatment, we will see that the
true situation is more complicated than this. In particular, we find that eactive
absorption can also take place in ptype materials.
When the temperature is not zero, EF in Eq. 35 should be understood
as the chemical potential, which we still call the Fermi energy, and depends on
temperature and doping. If ND is the donor concentration andNA the acceptor
concentration, then the net donor concentration No = ND NA can be either
positive or negative depending on whether the sample is n or p type. For a fixed
temperature and Fermi level, the net donor concentration is
1 of
Nc = ( 2Y dk, [f,,(k,) (323)
where 6,, = 1 if the subband (n, v) is a valence band and vanishes if (n, v) is a
conduction band. Given the net donor concentration and the temperature, the
Fermi energy can be found from Eq. 323 using a root finding routine.
3.2 Ultrahigh Magnetic Field Techniques
Since the mobility of a ferromagnetic IIIV DMS is generally low, using ultra
high magnetic fields exceeding 100 T (megagauss field) is essential for the present
study in order to satisfy the CR condition wjr > 1, where uc is the cyclotron
frequency and 7 is the scattering time [74, 75]. The megagauss experiments have
been done at the university of Tokyo where high magnetic fields can be generated
using two kinds of pulsed magnets: the singleturn coil technique [76, 77] and the
electromagnetic flux compression method [77, 78]. The singleturn coil method
can generate 250 T without any sample damage and thus measurements can be
repeated on the same sample under the same experimental conditions. The idea be
hind this method is to release a big current in a very short period of time (several
pis) to the singleturn coil to generate an ultrahigh magnetic field. The core part of
a real singleturn coil device is demonstrated in Fig. 32 [76]. Although the sample
is intact, the coil is damaged after each shot. A standard coil is shown in Fig. 33
Figure 32: The core part of the device based on singlecoil method. The coil is
placed in the clamping mechanism as seen in the figure. The domed steel cylin
ders on each side of the coil are supports for the sample holders which protect the
connection to the sample(e.g., thin wires, helium pipes) against the lateral blast.
before and after a shot. Depending on the coil dimension, each shot generates a
pulsed magnetic field up to 250 T in several ps. The time dependence of the pulsed
magnetic field and of the current flowing through the coil is shown in Fig. 34 [76].
For higher field experiments an electromagnetic flux compression method is
used. It uses the implosive method to compress the electromagnetic flux so as to
generate ultrahigh magnetic fields up to 600 T. The time dependence of the pulsed
magnetic field and current is shown in Fig. 35 [77]. This is a destructive method
and the sample as well as the magnet is destroyed in each shot.
3.3 Electron Cyclotron Resonance
3.3.1 Electron Cyclotron Resonance
According to the discussion in Section 3.1.2, for eactive cyclotron resonance,
the light polarization vector is e_ = (x iy)/ V2, corresponding to momentum
operator p+ = (p, + ipy)//2. This operator will result in an n  n + 1 transition.
Figure 33: A standard coil before and after a shot.
300
200
100
0
0 5 10 15
Time(gs)
Figure 34: Waveforms of the
in singleturn coil device.
magnetic field B and the current I in a typical shot
4
500 I
3
400
300 2
OD 2 :
200
100
0 0
0 10 20 30 40 50 60
Time (gs)
Figure 35: Waveforms of the magnetic field B and the current I in a typical flux
compression device.
In the conduction band, the Landau subbands are usually aligned in such a way
that energy ascends with quantum number n. So for an eactive transition, both
angular momentum and energy for an electronphoton system can be conserved.
Our collaborators Kono et al. [74] measured the electron active cyclotron
resonance in InMnAs films with different Mn concentrations. The films were grown
by low temperature molecular beam epitaxy on semiinsulating GaAs substrates
at 200 oC. All the samples were n type and did not show ferromagnetism for
temperatures as low as 1.5 K. The electron densities and mobilities deduced from
Hall measurements are listed in Table 31, together with the electron cyclotron
masses obtained at a photon energy of 117 meV (or a wavelength of 10.6 pm).
Typical measured CR spectra at 30 K and 290 K are shown in the left and
right panel of Fig. 36, respectively. Note that to compare the transmission with
absorption calculations, the transmission increases in the negative y direction.
Each figure shows spectra for all four samples labeled by the corresponding
Mn compositions from 0 to 1"' All the samples show pronounced absorption
Table 31: Parameters for samples used in eactive CR experiments
Mn content x 0 0.025 0.050 0.120
Density (4.2 K) 1.0 x 1017 1.0 x 1016 0.9 x 1016 1.0 x 1016
Density (290 K) 1.0 x 1017 2.1 x 1017 1.8 x 1017 7.0 x 1016
Mobility (4.2 K) 4000 1300 1200 450
Mobility (290 K) 4000 400 375 450
m/mo (30 K) 0.0342 0.0303 0.0274 0.0263
m/mo (290 K) 0.0341 0.0334 0.0325 0.0272
peaks (or transmission dips) and the resonance field decreases with increasing
x. Increasing x from 0 to 1'". results in a '",. decrease in cyclotron mass (see
Table 31). At high temperatures [e.g., Fig. 36(b)] the x = 0 sample clearly
shows nonparabolicityinduced CR spin splitting with the weaker (stronger) peak
originating from the lowest spindown (spinup) Landau level, while the other three
samples do not show such splitting. The absence of splitting in the Mndoped
samples can be accounted for by their low mobilities (which lead to substantial
broadening) and large effective g factors induced by the Mn ions. In samples
with large x, only the spindown level is substantially thermally populated (see
Fig. 25).
Using the Hamiltonian described in Section 2.3.2, the wave functions in Section
2.3.3, and the techniques for calculating Fermi energy, the several lowest Landau
levels in the conduction band at two Mn concentrations and the Fermi energy
for two electron densities (1 x 1016/cm3 and 1 x 1018/cm3) are calculated. The
conduction band Landau levels and the Fermi energies are shown in Fig. 37 as
a function of magnetic field at T = 30 K. From these figures, we can see that at
resonance, the densities and fields are such that only the lowest Landau level for
each spin type is occupied for typical densities listed in Table 31. Thus, all the
electrons were in the lowest Landau level for a given spin even at room temperature
due to the large Landau splitting, precluding any densitydependent mass due
0.99 a) 12% 0.982 J) 12%
:0 11.00
0.96 0.92
S0.98 5%0.96 5%
0.92 0.88
0.96 y2.5% 0.94 %
1.00
0.20 0.20 
0.60 0% 0.60 2 0%
30 K 290 K
1.00 1_ 31_ 1.002
0 40 80 0 40 80
B (T) B (T)
Figure 36: Experimental electron CR spectra for different Mn concentrations x
taken at (a) 30 K and (b) 290 K. The wavelength of the laser was fixed at 10.6pm
with eactive circular polarization while the magnetic field B was swept.
to nonparabolicity (expected at zero or low magnetic fields) as the cause of the
observed trend.
The cyclotron resonance takes place when the energy difference between
two Landau levels with the same spin is identical to the incident photon energy.
In Fig. 38, we simulate cyclotron resonance experiments in ntype InAs for e
active circularly polarized light with photon energy h = 0.117eV. We assume a
temperature T = 30 K and a carrier concentration n = 1016/ cm3. The lower panel
of Fig. 38 shows the four lowest zonecenter Landau conductionsubband energies
and the Fermi energy as functions of the applied magnetic field. The transition at
the resonance energy hu = 0.117eV is a spinup An = 1 transition and is indicated
by the vertical line. From the Landau level diagram the resonance magnetic field is
found to be B = 34 T. The upper panel of Fig. 38 shows the resulting cyclotron
0.8
(0) x = 0%
0.7
> T = 30 K
8 0.6
S10'8
Lii
0.5
0.4
0 20 40 60
B (Teslo)
0 .8 1' I I '
(b) x = 12% s :
0.7
> T = 30 K
o 0.6 101'
0.4
0 20 40 60
B (Tesla)
Figure 37: Zonecenter Landau conductionsubband energies at T =30 K as func
tions of magnetic field in ndoped In i_ l,.,As for = 0 and x = 1'. Solid lines
are spinup and dashed lines are spindown levels. The Fermi energies are shown as
dotted lines for n 1016/ cm3 and n 1018/ cm3.
resonance absorption assuming a FWHM linewidth of 4 meV. There is only one
resonance line in the cyclotron absorption because only the groundstate Landau
level is occupied at low electron densities. For higher electron densities, more
Landau levels are occupied. For example, if both spinup and spindown states of
the first Landau level are occupied, one obtains multiple resonance peaks.
Our simulation of the experimental eactive cyclotron resonance in the
conduction band shown in Fig. 36 is shown in Fig. 39. The left and right panel
demonstrate the calculated cyclotron resonance absorption coefficient for eactive
4 .. II "IIII.IIIII.I.IIII 111111111
1016 cm3 x = 0%
3 30 K
E = eactive
2 0.117 eV
0 2

r 1 
t 0.6
0 .4 ... .. .. .. .. .. .... .
c 0.5
0 .4 _,, ,, ,1. ..,, ,, ,, ,, ,I ,, ,, ,
20 30 40 50 60
B (Teslo)
Figure 38: Electron CR and the corresponding transitions. The upper panel shows
the resonance peak and the lower panel shows the lowest four Landau levels with
spinup states indicated by solid lines and spindown states indicated by dashed
lines. Vertical solid line in the lower panel indicates the transition accountable for
the resonance.
circularly polarized 10.6pm light in the Faraday configuration as a function of
magnetic field at 30 K and 290 K, respectively. In the calculation, the curves
were broadened based on the mobilities of the samples. The broadening used for
T =30 K was 4 meV for ( 40 meV for 2.5' 40 meV for 5' and 80 meV for
12'. For T = 290 K, the broadening used was 4 meV for 0' 80 meV for 2.5' ,
80 meV for 5' and 80 meV for 12'" At T = 30 K, we see a shift in the CR peak
as a function of doping in agreement with Fig. 36(a). For T = 290 K, we see
the presence of two peaks in the pure InAs sample. The second peak originates
from the thermal population of the lowest spindown Landau level. The peak does
"30K 8 290
6 12% 6 12%
= 5%
4 4 5%
SI
2 .5
2 25 2
2.5%
xlOo 5 x5"
0% 0 0% 0o 0%
0 20 40 60 80 100 0 20 40 60 80 100
B(Tesla) B ( Tesla)
Figure 39: Calculated electron CR absorption as a function of magnetic field at
30 K and 290 K. The curves are calculated based on generalized PidgeonBrown
model and Fermi's golden rule for absorption. They are broadened based on the
mobilities reported in Table 31.
not shift as much with doping as it did at low temperature. This results from the
temperature dependence of the average Mn spin. We believe that the Brillouin
function used for calculating the average Mn spin becomes inadequate at large x
and/or high temperature due to its neglect of MnMn interactions such as pairing
and clustering.
The eactive CR shows a shift with increasing Mn concentration. From the
simple theory in Section 1.2.2, the cyclotron resonance field does not depend on x
and a because the exchange interaction will shift all levels by the same amount.
This shift comes from the complicated conductionvalence band mixing, and
depends on the value of (a 3) [79]. We can qualitatively explain this shift using
the cyclotron mass, which will be discussed in the following subsection.
The CR peaks shown in Fig. 39 are highly ..immetric. This is because
we have taken into account the finite kz effect in our calculation, and the energy
dispersion along kz shows high nonparabolicity. Also, the carrier filling effect due to
the Fermi energy sharpening will also contribute to the CR peak ..iiin.i. I ry.
3.3.2 Electron Cyclotron Mass
The electron cyclotron mass mcR for a given cyclotron absorption transition is
related to the resonance field B* and photon energy he by the definition
mcR 2PBB*
(324)
mo hw
This equation can be derived from Eq. 318 if we set magnetic field B so that
hkw = hw, which is the cyclotron resonance condition.
The calculated cyclotron masses for the lowest spindown and spinup tran
sitions are plotted in Fig. 310 as a function of Mn concentration x at a photon
energy of hw = 0.117 eV. Cyclotron masses are computed for several sets of a and
3 values. The cyclotron masses in Fig. 310(a) and (b) correspond to the computed
cyclotron absorption spectra shown in Fig. 39 (a) and (b), respectively. In our
model, the electron cyclotron masses depend on the Landau subband energies and
photon energies and are independent of electron concentration.
Figure 310 clearly shows that the cyclotron mass depends on both exchange
constants and x. With increasing x, spindown (spinup) cyclotron mass show
almost a linear decrease (increase). The cyclotron mass does not depend on one
single exchange constant, it depends on both exchange constants. Investigation
of the mass dependence on these two constants reveals the mass shift has a close
relation with the absolute value of (a f) [79]. This shift allows use to measure
the exchange interaction.
The calculated cyclotron mass has taken into account all the energy depen
dence on nonparabolicity due to the conductionvalence band mixing, the exchange
5.0
4.5
4.0
0 2 4 6 8
Mn concentration, x (%)
down
3 .6.. ....... .
3.4 ......
...... ....l U P. .
0 2 4 6 8
Mn concentration, x (%)
10 12
10 12
Figure 310: Calculated electron cyclotron masses for the lowestlying spinup and
spindown Landau transitions in ntype In, MnAs with photon energy 0.117 eV
as a function of Mn concentration for T = 30 K and T = 290 K. Electron cyclotron
masses are shown for three sets of a and 03 values.
I I I I I I I I I
0.5, = 1.3 E = 0.117 eV
photon
0.5, = 1.0
0.3, = 1.0 T = 290 K
'
..a=
a=
a=
interaction constants a and 3, and the Mn content x. The shift of the resonance
peaks to lower fields with increasing Mn content x is naturally explained by the
decrease of the spindown cyclotron mass. Due to the smaller downward slope in
the spindown cyclotron mass at 290 K as compared to 30 K, the resonance peak
shift at 290 K is seen to be less pronounced than at 30 K.
3.4 Hole Cyclotron Resonance
3.4.1 Hole Active Cyclotron Resonance
As shown in Fig. 24, the DMS valence band structure is much more com
plicated than the conduction band structure. Due to their energetic proximity,
heavy hole and light hole bands are strongly mixed even near the F point. The
splitoff band also contributes strongly to the valence bandedge wave functions. In
a magnetic field, these hole bands split into their own Landau levels, but optical
transitions can happen between any two levels if both angular momentum and
energy are conserved. As in the conduction band, cyclotron resonance requires
conductionvalence band mixing to produce strong enough oscillator strength.
Interband mixing across the band gap is small in widegap semiconductors, so it is
more difficult to observe cyclotron resonance in these semiconductors. As a matter
of fact, no cyclotron resonance has been reported to date in GaMnAs.
InAs and InMnAs are narrowgap semiconductors. Our collaborators [80, 81,
82] have performed cyclotron resonance experiments on pdoped InAs and InMnAs
at ultrahigh magnetic fields up to 500 T. The typical hactive CR absorption of
InAs below 150 T is shown in Fig. 311, in which the incident light is hactive
circularly polarized with photon energy 0.117 eV. Two peaks are present in the
experimental observation, one around 40 T, and another around 125 T. At even
lower fields, there is a background absorption. The theoretical simulation using a
hole density of 1 x 1019/cm3 and a broadening factor of 40 meV is also dipl i' 1 in
Fig. 311 for comparison.
Expt. 27.5KI
4 L \X=M
u 3 hactive
b E0.117eV
S2 Theory 30K 
0 50 100 150
B Tesla)
Figure 311: Hole cyclotron absorption as a function of magnetic field in ptype
InAs for hactive circularly polarized light with photon energy 0.117 eV. The up
per curve is experimentally observed result and the lower one is from theoretical
calculation.
In our model, we are capable of calculating the absorption between any two
Landau levels. Detailed calculation reveals that the peak at lower fields is due
to the heavyhole to heavyhole transition, and the peak at higher fields is from
the lighthole to lighthole transition. We now use H,,, to specify the heavy hole
level, and L,,,, to specify the light hole level, where (n, v) are the quantum numbers
defined in Eq. 319. Because of strong wave mixing H or L only labels the zone
center (k = 0) character of a Landau level. Using these labels, we illustrate
the twostate absorption in Fig. 312 along with the Landau level structure as a
function of magnetic field.
It is seen from Fig. 312 that the holes optically excited from the heavy hole
subband H_1,1 and light hole subband L0,3 give rise to the two strong cyclotron
absorption peaks shown in Fig. 311. The cyclotron absorption peak around 40 T
is due to a transition between the spindown ground state heavy hole Landau level
H_1,1, and heavy hole Landau level Ho,2, which near the zone center is primarily
101
Total (a)
C 6 I
c6
(0 4
0 .1 1 1I I I 
(b)
H_c Ho .
CM 0H
1.1 0 L L 
0.2
0.2 .. .... 14
0 50 100 150 200
B (Teslo)
Figure 312: Calculated cyclotron absorption only from the H1,1 H,2 and
Lo,3 L1,4 transitions broadened with 40 meV (a), and zone center Landau levels
responsible for the transitions (b).
spindown. The other absorption peak around 140 T, is a spindown light hole
transition between L0,3 and L1,4 Landau levels. The background absorption at
B < 30 T is due to the absorption between higher Landau levels which also become
occupied by holes at lower fields.
Cyclotron resonance absorption measurements on Inl_, li,.As with x = 2.5'.
have also been performed. They are shown in Fig. 313 along with our theoretical
simulation. The CR measurements were made at temperatures of 17, 46, and
70 K in hactive circularly polarized light with photon energy he = 0.224 eV.
In our simulation, the hole density is taken as 5 x 1018/ cm3, and the curves are
1.0 I
SExperiment
0.8
x=2.5%
0.6 70 K
z 0.4 46 K
0.2 0.224 eV
17 K hactive
0.0
10
Theory
8 p=5x10M cm3
x = 2.5%
_6 6
ad 70 K
t 4
46 K
2 0.224 eV
17 K hactive
0 I I .
50 0 50 100 150
Magnetic Field (T)
Figure 313: Experimental hole CR and corresponding theoretical simulations.
The low temperature CR has an abrupt cutoff at low fields due to the fermi level
sharpening effect.
broadened using a FWHM linewidth of 120 meV. Clearly the absorption peak is
due to the heavy hole transition which we have seen in Fig. 311 and Fig. 312.
Due to the higher photon energy, this peak shifts from around 40 T to around
85 T. The resonance field is insensitive to temperature and the line shape is
strongly .,viii i I ic with a broad tail at low fields. This broad tail again comes
from the higher order transitions resonant at low fields. We see that in both
experiment and theory at low temperature and low field, there is a sharp cutoff of
the absorption. This can be attributed to the sharpness of the Fermi distribution
at low temperatures.
Figure 314 shows the observed CR peaks as a function of magnetic field. The
yaxis indicates the photon energies used when observing the cyclotron resonance.
The solid curves show the calculated resonance positions. The curve labeled
'HH' ('LH') is just the resonance energy between Landau levels H_1,1 (Lo,3)and
Ho,2 (L1,4). The theoretical calculation shows an overall consistency with the
experiments.
0.4
o InAs
SHH / x = 0.0015
0.3 o x = 0.006
>. v x = 0.025
>, 0.2
2D LH
) 00
,c
0.1
0.0
0 100 200 300 400 500
Magnetic Field (T)
Figure 314: Observed hole CR peak positions for four samples with different Mn
concentrations. The solid curves are theoretical calculations.
There are two factors in our calculation that affect the results. One is the
selection of Luttinger parameters, the other is the limitation of the eightband
effective mass theory itself. In Fig. 311, the theoretically computed peak at
higher fields does not fit the experimental peak exactly. Due to the fact that this
transition takes place at the zone center, where the k p theory should be very
accurate, this deviation may be the result of unoptimized Luttinger parameters.
79
The empirical parameters used in the effective mass Hamiltonian can drastically
change the valence band structure and the resulting CR absorption spectra. Fig. 3
15 shows the dependence of the CR energies on several parameters such as the
Luttinger parameters 71, 72, 73, Kane's parameter Ep and the effective electron
mass m*. This figure reveals that the 'LH' transitions are affected more by small
variations in these parameters than the 'HH' transitions. For instance, a 10''.
change in 71 will result in a ~ 0.025 eV change at B = 140 T in the LH CR energy,
which in turn will result in about a 50 T CR position shift in the resonance field
when the photon energy is 0.117 eV. The Mn doping on the other hand generally
enhances the CR energy dependence on these parameters, which can be seen from
comparing the two graphs in Fig. 315.
HH
0.2 T = 30 K LH
HH ~LH
0.2 T=30K 0.2 T=30K K
=5% x=5%
.. ...... ..... ..>.x
. Y 3.
0.0 ._0.0_
U E .. P
0.2 (a) 0.2 (b) ..
HH LH
0 5= 30 100 150 0 50 100 150
x=5% x=5% 71.
0.0 (b)....2.(b)........,..,,....
Magnetic Field (T) Magnetic Field (T)
Figure 315: The dependence of cyclotron energies on several parameters. Left
panel shows the heavy hole CR energy dependence, and the right panel shows the
light hole CR energy dependence.
Figure 316 illustrates how the CR absorption depends on three Luttinger
parameters while keeping all the other parameters unchanged. It can be seen that
the CR spectra quite sensitively depends on the values of the Luttinger parameters,
providing an effective way to measure these parameters through comparison with
experiments.
10 T = 30K
10 =21, y2=8.5, y39.2 1x101cmK
8
6
2 ,=20, y2=8.5, y3=9.2
0
0 50 100 150 200
Magnetic Field (T)
Figure 316: Hole CR spectra of InAs using different sets of Luttinger parame
ters. Light hole transition is more significantly affected by change of the Luttinger
parameters.
In Fig. 314, there is one peak around 450 T labeled as 'C' when the light
energy is h = 0.117 eV. To account for this peak, CR absorption spectra up to
500 T have been computed. The k = 0 Landau levels as a function of magnetic
field, along with the CR spectra are plotted in Fig. 317, in which we can see
that this peak is due to the superposition of two transitions: L1,5 L2,5 and
H2,6 H3,6. However, the calculated peak position is around 360 T, different
from the experiment. There are two possible reasons for this big deviation. One
is that at very high magnetic fields, the eightband PidgeonBrown model may
break down; the other is that transitions contributing to the peak take place away
from the zone center where eightband k p theory is not adequate to describe the
energy dispersion. The band structure along k, is plotted at Fig. 318, where we
see that the Landau levels H2,6 and H3,6 both have camel back structures. At a
hole density p = 1 x 1019/ cm3, the zone center part of H2,6 is not occupied. The
0.1
(a) T=20 K p=1x1019 cm
0.0 hactive ,3) E =0.117 eV
00 ^ (0,3) photon
(1, 5)
.? 0.1
0) 0.2
0.2 (2,5) 
W (3,6
InAs
0.3
10
(b)
HH LH
8 HH LH T=20 K
: p=1x109 cm3
c 6 Ephoton=0.117eV
c hactive
5. InAs
0 c
n 2
0 100 200 300 400 500
Magnetic Field (T)
Figure 317: Calculated Landau levels and hole CR in magnetic fields up to 500
T. The upper panel shows the k = 0 valence band Landau levels as a function of
magnetic field and the Fermi level for p = 1019 cm3 (dashed line). The hole CR
absorption in ptype InAs is shown in the lower panel for hactively polarized light
with u = 0.117 eV at T 20 K and p 1019 cm3. A FWHM linewidth of 4 meV
is assumed.
lowest energy for this heavy hole Landau level resides at about k, = 0.75(1/nm).
C('!. L:.ig the transition element along k, it is also found that this transition indeed
takes place away from the zone center. Di ,' 't1 in Fig. 319 is the comparison of
the eightband model versus a fullzone thirtyband model. At the zone center, the
eightband model fits well with the thirtyband model. Not far away from the zone
center, a big deviation occurs. We think this deviation of the energy dispersion
is possibly responsible for the large deviation of the calculated resonance peak
position.
0.1
0.2
0.3
1.0
0.5 0.0 0.5
K (1/nm)
Figure 318: kdependent Landau subband structure at B
0.5
350 T.
L A K A X
Figure 319: Band structure near the F point for InAs calculated by eightband
model and full zone thirtyband model.
3.4.2 Hole Density Dependence of Hole Cyclotron Resonance
Cyclotron resonance depends on Fermi energy through Eq. 316, thus CR
spectra depend strongly on carrier densities. In Fig. 312 the hole density is
1 x 1019 cm3. At such a hole density, the Fermi energy is below the H1,1 and
L0,3 states so that we have two strong transitions. If the hole density is lower, the
Fermi energy will shift upward, thus these two states will become less occupied by
holes, and we can expect a decrease in the CR strength. However, the decrease
in strength for the two resonance peaks is different. Shown in Fig. 320(a) are
the CR spectra for four different hole densities. The Landau levels along with the
corresponding Fermi energies are plotted in Fig. 320(b). Resonant transitions at
0.117 eV are indicated by vertical lines. We can see that the CR peak 2 is almost
ah,i present, because at low magnetic fields, the heavy hole state H1,1 is almost
ah,v occupied. The CR peak 1 changes dramatically with hole density, and
nearly vanishes at p = 5 x 1018 cm3. The relative strengths of the heavy and
lighthole CR peaks is sensitive to the itinerant hole density and can be used to
determine the hole density. By comparing theoretical and experimental curves in
Fig. 320(a), we see that the itinerant hole concentration is around 2 x 1019 cm3
From Fig. 320(a), we can rule out p < 1019 cm3 and n > 4 x 1019 cm3. We
estimate that an error in the hole density of around 25'. should be achievable at
these densities. Because of the existence in IIIV DMS of the anomalous Hall effect,
which can often make the determination of carrier density difficult, determining
carrier density by cyclotron resonance can serve as a possible alternative.
3.4.3 Cyclotron Resonance in InMnAs/GaSb Heterostructures
Hole CR in InMnAs/GaSb heterostructures has also been experimentally
studied by Kono et al. [83]. These samples are ferromagnetic with Tc ranging from
30 to 55 K and whose characteristics are summarized in Table 32.
Expt. hactive (a)
4 27.5 K 0.117 eV
2' 2(\X=O%
Si 9 0%
S '1 0 I \ *
1 ETheory
..3 0 K ".. .............. ................. ,
0.1
B (Tesl(b))
1,1 0,2 L
0.0curves 03to top with hole densities of 5 x 9, 2 x
S *.. ..........
C0.1 E
0. w \ ~I Fermi
0 50 100 150 200
B (Tesla)
Figure 3 20: The hole density dependence of hole CR. (a) Theoretical hole CR
curves in InAs from bottom to top with hole densities of 5 x 1018s, 1019, 2 x 1019 and
4 x 1019 cm3; (b) Landau levels involved in observed CR along with Fermi levels
corresponding to theoretical curves in (a).
85
Table 32: Characteristics of two InMnAs/GaSb heterostructure sam
ples
Sample No. Tc(K) Mn content x Thickness (nm) Density(cm3)
1 55 0.09 25 1.1 x 1019
2 30 0.12 9 4.8 x 1019
The experimentally observed CR transmission of a 10.6 pm laser beam
through these two samples (Tc = 55 and 30 K, respectively) are shown in Fig. 3
21(a) and (b), at various temperatures as a function of magnetic field. The laser
beam was holeactive circularly polarized. In the left panel of Fig. 321, from
room temperature down to slightly above Tc, a broad resonance feature (labeled
'A') is observed with almost no change in intensity, position, and width with
decreasing temperature. Close to Tc, quite abrupt and dramatic changes take place
in the spectra. First, a significant reduction in line width and a sudden shift to a
lower magnetic field occur simultaneously. Also, the resonance rapidly increases
in intensity with decreasing temperature. In addition, a second feature (labeled
'B') suddenly appears around 125 T, which also rapidly grows in intensity with
decreasing temperature and saturates, similar to feature A. At low temperatures,
both features A and B do not show any shift in position. Essentially, the same
behavior is seen in the right panel in Fig. 321. Using different wavelengths of the
incident light, similar CR spectrum behavior has also been observed.
For zincblende semiconductors, the CR peaks A and B are due to the tran
sitions of H1,1 Ho,2 and L0,3 L1,4, respectively, which we have already
pointed out. We attribute the temperaturedependent peak shift to the increase
in the carrierMn ion exchange interaction resulting from the increase of magnetic
ordering at low temperatures. The theoretically calculated results are shown in
Fig. 322 for bulk Ino.90 g!i,,.,As. The CR spectra was broadened using a FWHM
linewidth of 4 meV. The theoretical results clearly show a shift of peak A to lower
Sample 1 Sample 2
RT RT
152 K 140 K
107 K 107 K
85 K " 80 K
68 K 60 K
,42 K 30 K
25 K ,21 K
13K
15K 13K
A B
0 50 100 150 0 50 100 150
Figure 321: Cyclotron resonance spectra for two ferromagnetic InMnAs/ GaSb
samples. The transmission of holeactive circular polarized 10.6 pm radiation is
plotted vs. magnetic field at different temperatures.
fields with decreasing temperature, although in bulk InAs, the transition occurs
at about 40 T, as opposed to the heterostructure where the resonance occurs at
S50 T.
The CR peak A only involves the lowest two Landau manifolds. As was
discussed in Section 2.3.3, when n = 1, the Hamiltonian is 1 x 1, and when n = 0,
the Hamiltonian factorizes into two 2 x 2 matrices, so it is easy to obtain an exact
analytical expression for the temperature dependent cyclotron energy. With neglect
of the small terms arising from the remote band contributions, the cyclotron energy
10 1
p=1xl019 cm3 In1xMnxAs
8 Ephoton=0.117 eVx=9%
hactive TC=55 K
6 6
=3 290 K
42 K
2 30 K A=
'15 K
0 1 *
0 20 40 60
B (T)
Figure 322: Theoretical CR spectra showing the shift of peak A with temperature.
for the H_1,1 HO,2 transition is
ECR = + x(S)(a ) E/ x(S) a 4) 2 + EppBB. (325)
In the field range of interest (~ 40 T), IEp pBB is the same order as E,/2,
while the exchange interaction is much smaller even in the saturation limit.
Expanding the square root in Eq. 325, we obtain an expression of the form
ECR =E ( I) + x(S,)(a 3)(1 ) (326)
where
6 = E (327)
/E + 4EpPBB
If we assume the temperature dependence of E, and Ep is small, it follows
from Eq. 326 that the CR peak shift should follow the temperature dependence
of the magnetization (S,), which in a mean field theory framework is given by

Full Text 
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IowemygratitudetoallthepeoplewhomadethisthesispossibleandbecauseofwhommygraduateexperiencehasbeenonethatIwillcherishforever.FirstandforemostIwouldliketothankmyadvisor,ProfessorChristopherJ.Stanton,forgivingmeaninvaluableopportunitytoworkonchallengingandextremelyinterestingprojectsoverthepastfouryears.HehasalwaysmadehimselfavailableforhelpandadviceandtherehasneverbeenanoccasionwhenIhaveknockedonhisdoorandhehasnotgivenmetime.Hisphysicsintuitionimpressedmealot.Hetaughtmehowtosolveaproblemstartingfromasimplemodel,andhowtodevelopit.Ithasbeenapleasuretoworkwithandlearnfromsuchanextraordinaryindividual.IwouldalsoliketothankProfessorDavidH.Reitze,ProfessorSelmanP.Hersheld,ProfessorDmitriiMaslovandProfessorCammyAbernathyforagreeingtoserveonmythesiscommitteeandforsparingtheirinvaluabletimereviewingthemanuscript.MycolleagueshavegivenmelotsofhelpinthecourseofmyPh.D.studies.GarySandershelpedmegreatlytodeveloptheprogramcode,andwealwayshadfruitfuldiscussions.ProfessorStanton'sformerpostdocFedirKyrychenkoalsogavemegoodadviceandsomeinsightfulideas.IwouldalsoliketothankRongliangLiuandHaidongZhang,whomademylifeheremoreinteresting.Iwanttothankourresearchcollaborators.Dr.Kono'sgroupfromRiceUniversityprovidedmostoftheexperimentaldata.CollaborationwithDr.Konowasawonderfulexperienceinthepastfouryears.IalsohadfruitfuldiscussionwithProf.MiuraandDr.MatsudafromUniversityofTokyo. iii
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iv
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page ACKNOWLEDGMENTS ............................. iii LISTOFTABLES ................................. vii LISTOFFIGURES ................................ viii ABSTRACT .................................... xiv CHAPTER 1INTRODUCTIONANDOVERVIEW ................... 1 1.1Spintronics ............................... 1 1.2TheIIVIDilutedMagneticSemiconductors ............ 4 1.2.1BasicPropertiesofIIVIDilutedMagneticSemiconductors 4 1.2.2ExchangeInteractionbetween3d5ElectronsandBandElectrons ............................. 8 1.3TheIIIVDilutedMagneticSemiconductors ............ 13 1.3.1FerromagneticSemiconductor ................ 13 1.3.2EectiveMeanField ..................... 21 1.4OpenQuestions ............................ 23 1.4.1NatureofFerromagnetismandBandElectrons ....... 23 1.4.2DMSDevices .......................... 24 2ELECTRONICPROPERTIESOFDILUTEDMAGNETICSEMICONDUCTORS ................................. 29 2.1FerromagneticSemiconductorBandStructure ........... 29 2.2ThekpMethod ........................... 30 2.2.1IntroductiontokpMethod ................. 30 2.2.2Kane'sModel ......................... 34 2.2.3CouplingwithDistantBandsLuttingerParameters .... 38 2.2.4EnvelopeFunction ....................... 42 2.3LandauLevels ............................. 43 2.3.1ElectronicStateinaMagneticField ............. 43 2.3.2GeneralizedPidgeonBrownModel .............. 44 2.3.3WaveFunctionsandLandauLevels ............. 49 2.4ConductionBandgfactors ...................... 53 v
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........................ 56 3.1GeneralTheoryofCyclotronResonance ............... 56 3.1.1OpticalAbsorption ...................... 56 3.1.2CyclotronResonance ..................... 60 3.2UltrahighMagneticFieldTechniques ................ 63 3.3ElectronCyclotronResonance .................... 64 3.3.1ElectronCyclotronResonance ................ 64 3.3.2ElectronCyclotronMass ................... 72 3.4HoleCyclotronResonance ...................... 74 3.4.1HoleActiveCyclotronResonance .............. 74 3.4.2HoleDensityDependenceofHoleCyclotronResonance .. 83 3.4.3CyclotronResonanceinInMnAs/GaSbHeterostructures .. 83 3.4.4ElectronActiveHoleCyclotronResonance ......... 90 4MAGNETOOPTICALKERREFFECT .................. 96 4.1RelationsofOpticalConstants .................... 96 4.2KerrRotationandFaradayRotation ................ 101 4.3MagnetoopticalKerrEectofBulkInMnAsandGaMnAs .... 104 4.4MagnetoopticalKerrEectofMultilayerStructures ....... 107 5HOLESPINRELAXATION ......................... 112 5.1SpinRelaxationMechanisms ..................... 113 5.2LatticeScatteringinIIIVSemiconductors ............. 115 5.2.1ScreeninginBulkSemiconductors .............. 117 5.2.2SpinRelaxationinBulkGaAs ................ 118 5.3SpinRelaxationinGaMnAs ..................... 122 5.3.1ExchangeScattering ...................... 122 5.3.2ImpurityScattering ...................... 124 6CONCLUSION ................................ 129 REFERENCES ................................... 132 BIOGRAPHICALSKETCH ............................ 138 vi
PAGE 7
Table page 1{1SomeimportantIIVIDMS ........................ 4 2{1SummaryofHamiltonianmatriceswithdierentn 50 2{2InAsbandparameters ........................... 51 3{1ParametersforsamplesusedineactiveCRexperiments ........ 67 3{2CharacteristicsoftwoInMnAs/GaSbheterostructuresamples ..... 85 5{1ParametersforGaAsphononscattering ................. 121 vii
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Figure page 1{1ThebandgapdependenceofHg1kMnkTeonMnconcentrationk. .. 5 1{2ThebandstructuresofHg1xMnxTewithdierentx. ......... 6 1{3Cd1xMnxTexTphasediagram. .................... 7 1{4Averagelocalspinasafunctionofmagneticeldat4temperaturesinparamagneticphase. ......................... 10 1{5MagneticelddependenceofHallresistivityHallandresistivityofGaMnAswithtemperatureasaparameter. ............. 14 1{6MncompositiondependenceofthemagnetictransitiontemperatureTc,asdeterminedfromtransportdata. ................ 16 1{7VariationoftheRKKYcouplingconstant,J,ofafreeelectrongasintheneighborhoodofapointmagneticmomentattheoriginr=0. 17 1{8CurietemperaturesfordierentDMSsystems.CalculatedbyDietlusingZener'smodel. .......................... 19 1{9SchematicdiagramoftwocasesofBMPs. ................ 20 1{10Averagelocalspinasafunctionofmagneticeldat4temperatures. 22 1{11ThephotoinducedferromagnetisminInMnAs/GaSbheterostructure. 25 1{12Spinlightemittingdiode. ......................... 27 1{13GaMnAsbasedspindevice. ....................... 28 2{1ValencebandstructureofGaAsandferromagneticGa0:94Mn0:06Aswithnoexternalmagneticeld,calculatedbygeneralizedKane'smodel. .................................. 30 2{2BandstructureofatypicalIIIVsemiconductornearthepoint. .. 35 2{3CalculatedLandaulevelsforInAs(left)andIn0:88Mn0:12As(right)asafunctionofmagneticeldat30K. ................. 52 2{4TheconductionandvalencebandLandaulevelsalongkzinamagneticeldofB=20TatT=30K. ................. 53 viii
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................ 54 2{6gfactorsofferromagneticIn0:9Mn0:1As. ................. 55 3{1Quasiclassicalpicturesofeactiveandhactivephotonabsorption. .. 62 3{2Thecorepartofthedevicebasedonsinglecoilmethod. ........ 64 3{3Astandardcoilbeforeandafterashot. ................. 65 3{4WaveformsofthemagneticeldBandthecurrentIinatypicalshotinsingleturncoildevice. ........................ 65 3{5WaveformsofthemagneticeldBandthecurrentIinatypicaluxcompressiondevice. ........................... 66 3{6ExperimentalelectronCRspectrafordierentMnconcentrationsxtakenat(a)30Kand(b)290K. ................... 68 3{7ZonecenterLandauconductionsubbandenergiesatT=30KasfunctionsofmagneticeldinndopedIn1xMnxAsfor=0andx=12%. ................................ 69 3{8ElectronCRandthecorrespondingtransitions. ............ 70 3{9CalculatedelectronCRabsorptionasafunctionofmagneticeldat30Kand290K. ............................ 71 3{10CalculatedelectroncyclotronmassesforthelowestlyingspinupandspindownLandautransitionsinntypeIn1xMnxAswithphotonenergy0:117eVasafunctionofMnconcentrationforT=30KandT=290K. ............................. 73 3{11HolecyclotronabsorptionasafunctionofmagneticeldinptypeInAsforhactivecircularlypolarizedlightwithphotonenergy0:117eV. 75 3{12CalculatedcyclotronabsorptiononlyfromtheH1;1H0;2andL0;3L1;4transitionsbroadenedwith40meV(a),andzonecenterLandaulevelsresponsibleforthetransitions(b). ............. 76 3{13ExperimentalholeCRandcorrespondingtheoreticalsimulations. .. 77 3{14ObservedholeCRpeakpositionsforfoursampleswithdierentMnconcentrations. ............................. 78 3{15Thedependenceofcyclotronenergiesonseveralparameters. ..... 79 3{16HoleCRspectraofInAsusingdierentsetsofLuttingerparameters. 80 3{17CalculatedLandaulevelsandholeCRinmagneticeldsupto500T. 81 ix
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......... 82 3{19BandstructurenearthepointforInAscalculatedbyeightbandmodelandfullzonethirtybandmodel. ................ 82 3{20TheholedensitydependenceofholeCR. ................ 84 3{21CyclotronresonancespectrafortwoferromagneticInMnAs/GaSbsamples. ................................. 86 3{22TheoreticalCRspectrashowingtheshiftofpeakAwithtemperature. 87 3{23AveragelocalizedspinasafunctionoftemperatureatB=0,20,40,60and100Tesla. ............................ 88 3{24RelativechangeofCRenergy(withrespecttothatofhightemperaturelimit)asafunctionoftemperature. ............... 89 3{25BanddiagramofInMnAs/GaSbheterostructure. ............ 90 3{26SchematicdiagramofLandaulevelsandcyclotronresonancetransitionsinconductionandvalencebands. ................ 91 3{27ThevalencebandLandaulevelsandeactiveholeCR. ......... 92 3{28ExperimentalandtheoreticalholeCRabsorption. ........... 93 3{29ValencebandstructureatT=30KandB=100TforIn1xMnxAsalloyshavingx=0%andx=5%. .................. 94 3{30TheprimarytransitionintheeactiveholeCRunderdierentMndoping. ................................. 95 4{1Diagramforlightreectionfromtheinterfacebetweenmedium1withrefractiveindexN1andmedium2withrefractiveindexN2. .... 100 4{2Schematicdiagramformagneticcirculardichroism. .......... 102 4{3DiagramsforKerrandFaradayrotation. ................ 103 4{4KerrrotationofInMnAs. ......................... 105 4{5ThebanddiagramforInAs. ....................... 106 4{6KerrrotationofGaMnAs. ........................ 107 4{7ThebanddiagramforGaAs. ....................... 108 4{8TheabsorptioncoecientsbothinInMnAsandGaSblayers(a)andthereectivityofInMnAs/GaSbheterostructure(b). ........ 109 x
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4{9ReectivityofIn 0 : 88 Mn 0 : 12 Asnm/GaSbnmheterostructure at T =5 : 5KmeasuredbyP.FumagalliandH.Munekata. ..... 110 heterostructureunderamagneticeldof3Tat T =5 : 5K. 111 5{1LightinducedMOKE.Signaldecaysinlessthan2ps. ......... 113 5{2Lightinducedmagnetizationrotation. .................. 114 5{3Theheavyholespinrelaxationtimeasafunctionofwavevectora, andtemperatureatthepointb. .................. 123 5{4Spinrelaxationtimeforaheavyholeasafunctionof k along,0,1 direction. ................................ 126 5{5Spinrelaxationtimeofaheavyholeasafunctionofholedensityat directiona,0,1andb,1,1. .................. 127 xi 4{10MeasuredaandcalculatedbKerrrotationofInMnAsnm/ AlSbnm
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Spintronicshasrecentlybecomeoneofthekeyresearchareasinthemagneticrecordingandsemiconductorindustries.Akeygoalofspintronicsistoutilizemagneticmaterialsinelectroniccomponentsandcircuits.Ahopeistousethespinsofsingleelectrons,ratherthantheircharge,forstoring,transmittingandprocessingquantuminformation.Thishasinvokedagreatdealofinterestinspineectsandmagnetisminsemiconductors.Inmywork,theelectronicandopticalpropertiesofdilutedmagneticsemiconductors(DMS),especially(In,Mn)Asanditsheterostructures,aretheoreticallystudiedandcharacterized.TheelectronicstructuresinultrahighmagneticeldsarecarefullystudiedusingamodiedeightbandPidgeonBrownmodel,andthemagnetoopticalphenomenaaresuccessfullymodeledandcalculatedwithintheapproximationofFermi'sgoldenrule.Wehavefoundthefollowingimportantresults:i)MagneticionsdopedinDMSplayacriticalroleinaectingthebandstructuresandspinstates.ThespdinteractionbetweentheitinerantcarriersandtheMndelectronsresultsinashiftofthecyclotronresonancepeakandaphasetransitionoftheIIIVDMS xii
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xiii
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Thereisawideclassofsemiconductingmaterialswhicharecharacterizedbytherandomsubstitutionofafractionoftheoriginalatomsbymagneticatoms.Thematerialsarecommonlyknownasdilutedmagneticsemiconductors(DMS)orsemimagneticsemiconductors(SMSC). SincetheinitialdiscoveryofDMSinIIVIsemiconductorcompounds[ 1 ],morethantwodecadeshavepassed.TherecentdiscoveryofferromagneticDMSbasedonIIIVsemiconductors[ 2 ]hasleadtoasurgeofinterestinDMSforpossiblespintronicsapplications.Manypapershavebeenpublishedinvestigatingtheirelectronic,magnetic,optical,thermal,statisticalandtransportproperties,inmanyjournals,andeveninpopularmagazines[ 3 ].ThisinterestnotonlycomesfromtheDMSthemselvesasgoodtheoreticalandexperimentalsubjects,butalsocanbebetterunderstoodfromabroaderviewfromtherelationofDMSresearchwithspintronics[ 4 ]. 1
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willeventuallybealimittohowmanytransistorstheycancramonachip.ButevenifMoore'sLawcouldcontinuetospawnevertinierchips,smallelectronicdevicesareplaguedbyabigproblem:energyloss,ordissipation,assignalspassfromonetransistortothenext.LineupallthetinywiresthatconnectthetransistorsinaPentiumchip,andthetotallengthwouldstretchalmostamile.Alotofusefulenergyislostasheataselectronstravelthatdistance.Spintronics,whichusesspinastheinformationcarriers,incontrastwithconventionalelectronics,consumeslessenergyandmaybecapableofhigherspeed. Spintronicsemergedonthestageinscienticeldin1988whenBaibichetal.discoveredgiantmagnetoresistance(GMR)[ 5 ],whichresultsfromtheelectronspineectsinmagneticmaterialscomposedofultrathinmultilayers,inwhichhugechangescouldtakeplaceintheirelectricalresistancewhenamagneticeldisapplied.GMRishundredsoftimesstrongerthanordinarymagnetoresistance.BasingonGMRmaterials,IBMproducedin1997newreadheadswhichareabletosensemuchsmallermagneticelds,allowingthestoragecapacityofaharddisktoincreasefromtheorderof1totensofgigabytes.AnothervaluableuseofGMRmaterialisintheoperationofthespinlter,orspinvalve,whichconsistsof2spinlayerswhichletthroughmoreelectronswhenthespinorientationsinthetwolayersarethesameandfewerwhenthespinsareoppositelyaligned.Theelectricalresistanceofthedevicecanthereforebechangeddramatically.Thisallowsinformationtobestoredas0'sand1's(magnetizationsofthelayersparallelorantiparallel)asinaconventionaltransistormemorydevice.Astraightforwardapplicationcouldbeinthemagneticrandomaccessmemory(MRAM)devicewhichisnonvolatile.Thesedeviceswouldbesmaller,faster,cheaper,uselesspowerandwouldbemuchmorerobustinextremeconditionssuchashightemperature,orhighlevelradiationenvironments.
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Currently,besidescontinuingtoimprovetheexistingGMRbasedtechnology,peoplearenowfocusingonndingnovelwaysofbothgeneratingandutilizingspinpolarizedcurrents.Thisincludesinvestigationofspintransportinsemiconductorsandlookingforwaysinwhichsemiconductorscanfunctionasspinpolarizersandspinvalves.Wecancallthissemiconductorbasedspintronics,theimportanceofwhichliesinthefactthatitwouldbemucheasierforsemiconductorbaseddevicestobeintegratedwithtraditionalsemiconductortechnology,andthesemiconductorbasedspintronicdevicescouldinprincipleprovideamplication,incontrastwithexistingmetalbaseddevices,andcanserveasmultifunctionaldevices.Duetotheexcellentopticalcontrollabilityofsemiconductors,therealizationofopticalmanipulationofspinstatesisalsopossible. Althoughthereareclearmeritsforintroducingsemiconductorsintospintronicapplications,therearefundamentalproblemsinincorporatingmagnetismintosemiconductors.Forexample,semiconductorsaregenerallynonmagnetic.Itishardtogenerateandmanipulatespinsinthem.Peoplecanovercometheseproblemsbycontactingthesemiconductorswithother(spintronic)materials.However,thecontrolandtransportofspinsacrosstheinterfaceandinsidethesemiconductorisstilldicultandfarfromwellunderstood.Fortunately,thereisanotherapproachtoinvestigatingspincontrolandtransportinallsemiconductordevices.ThisapproachhasbecomepossiblesincethediscoveryofDMS. ThemostcommonDMSstudiedintheearly1990swereIIVIcompounds(likeCdTe,ZnSe,CdSe,CdS,etc.),withtransitionmetalions(e.g.,Mn,FeorCo)substitutingfortheiroriginalcations.TherearealsomaterialsbasedonIVVI(e.g.,PbS,SnTe)andmostimportantly,IIIV(e.g.,GaAs,InSb)crystals.Mostcommonly,Mnionsareusedasmagneticdopants.
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1.2.1BasicPropertiesofIIVIDilutedMagneticSemiconductors 1 ],andhasbeengivenagreatdealofattentioneversince[ 6 ].ThemoststudiedIIVIDMSmaterialsarelistedinTable 1{1 Table1{1: SomeimportantIIVIDMS MaterialCrystalStructurexrange1 7 ]. 1{1 [ 8 ]. WiththedenitionofthebandgapasEg=E6E8,thebandstructuresofHg1xMnxTewithdierentxaregiveninFig. 1{2 [ 8 ].Withx0:075,Eg<0,andwithx>0:075,Eg>0.Withoutspinorbitalcoupling,weshouldhaveasixfolddegeneratevalencebandatthepoint.Consideringspinorbitalcoupling,thevalencebandsplitsintotwobands7and8(splitoband),withanenergydierenceof. Theelectroneectivemass,i.e.,thebandcurvature,willalsochangewithx.Atsomexvalues,theeectivemassbecomessosmallthatthemobilityofelectronscanbeveryhigh.Forinstance,=106cm2=VsforHg1xMnxTewhenx=0:07at4:2K.
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Figure1{1: ThebandgapdependenceofHg1kMnkTeonMnconcentrationk.ReprintedwithpermissionfromBastardetal.Phys.Rev.B24:19611970,1981.Figure10,Page1967. 1{3 isthephasediagramofCd1xMnxTeobtainedfromspecicheatandmagneticsusceptibilitymeasurements[ 9 ].TheDMSsystemmaybeconsideredascontainingtwointeractingsubsystems.Therstoftheseisthesystemofdelocalizedconductionandvalencebandelectrons/holes.Thesecondistherandom,dilutedsystemoflocalizedmagneticmomentsassociatedwiththemagneticatoms.Thesetwosubsystemsinteractwitheachotherbythespinexchangeinteraction.Thefactthatboththestructureandtheelectronicpropertiesofthehostcrystalsarewellknownmeansthattheyareperfectforstudyingthebasicmechanismsofthemagneticinteractionscouplingthespinsofthebandcarriersandthelocalizedspinsofmagneticions.Thecouplingbetweenthe
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Figure1{2: ThebandstructuresofHg1xMnxTewithdierentx.isthespinorbitalsplitting,HHindicatestheheavyholeband,andLHthelightholeband,respectively.ReprintedwithpermissionfromBastardetal.Phys.Rev.B24:19611970,1981.Figure1,Page1961. localizedmomentsresultsintheexistenceofdierentmagneticphasessuchasparamagnets,spinglassesandantiferromagnets.
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Figure1{3: Cd1xMnxTexTphasediagram.P:Paramagnet;A:Antiferromagnet;s:spinglass,mixedcrystalwhenx>0:7.ReprintedwithpermissionfromGalazkaetal.Phys.Rev.B22:33443355,1980.Figure12,Page3352.
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10 ].Inthefollowing,IwillbrieyintroduceasimplequalitativetheoreticalapproachtoIIVIDMS. SupposethestateofMnionsinDMSmaterialisMn2+.TheelectronicstructureofMn2+is1s22s22p63s23p63d5,inwhich3d5isahalflledshell.AccordingtoHund'srule,thespinoftheseve3d5electronswillbeparalleltoeachother,sothetotalspinisS=5=2.Theseveelectronsareinstatesinwhichtheorbitalangularmomentumquantumnumberl=0;1;2.ThusthetotalorbitalangularmomentumL=0.ThetotalangularmomentumforaMn2+ionthenisJ=S=5=2.TheLandegfactoris 2J(J+1)=2:(1{1) AnalogoustotheexchangeinteractionintheHydrogenmolecule,theexchangeinteractionbetweena3d5electronandabandelectroncanbewrittenintheHeisenbergform whereisthespinofabandelectron/hole,Jistheexchangeconstant,andSisthetotalangularmomentumofall3d5electronsinaMn2+ion. Inthenoninteractingparamagneticphase,averysimpliedmodelwillbedescribedinthefollowing.SinceL=0forMn2+,themagneticmomentumfor
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Mn2+is=(ge=2m0)J=(ge=2m0)S.AssumingamagneticeldBalongthezdirection,theadditionalenergyinthiseldofaMn2+ionisB=gBmsB,wherems=5=2,3/2,1/2,1/2,3/2,5/2.Assumingnoninteractivespins,andusingaclassicBoltzmandistributionfunctionegBmsB=kBT,theaveragemagneticmomentinthezdirectionisthen Thiscanbewrittenas whereBs(y)istheBrillouinfunction 2Scoth2S+1 2Sy1 2Scothy TheaveragespinofoneMn2+ionthenis TheantiparallelorientationofBandhSziisduetothedierenceinsignofthemagneticmomentandtheelectronspin.SinceBisdirectedalongthezaxis,theaverageMnspinsaturatesathSzi=5=2.The(paramagnetic)dependenceofhSzionmagneticeldandtemperatureisshowninFig. 1{4 FromEq. 1{2 ,theexchangeHamiltonianofonebandelectronwithspininteractingwiththe3d5electronsfromallMn2+ionsis, whereristhepositionvectorofthebandelectron,andRiisthepositionvectoroftheithMn2+ion,J(rRi)istheexchangecouplingcoecientofthebandelectron
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Figure1{4: Averagelocalspinasafunctionofmagneticeldat4temperaturesinparamagneticphase. withthe3d5electronsintheithMn2+ion.Siisthetotalangularmomentumofthe3d5electronsintheithMn2+ion. NextwewilluseavirtualcrystalapproximationtodealwithHamiltonian 1{7 .Duetothefactthethewavefunctionofabandelectronactuallyextendsoverthewholecrystal,itinteractswithalltheMn2+ionssimultaneously.Inthemeaneldframework,wecanreplacetheangularmomentumofeachMn2+ionbytheaveragevalue.Stillassumingamagneticeldalongzdirection,wehavehSi=hSzi,andSi=hSzimch2.mc=1=2hereindicatesthespinquantumnumberofthebandelectrons.ThehSziisgivenbyEq. 1{6 .Theexchange
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Hamiltonianthencanbewrittenas Becauseoftheextendednatureofthebandelectronstates,whichinteractwiththe3d5electronsinallMn2+ions,thepositionsoftheseMn2+ionsarenotimportant.WecandistributeapproximatelytheseMn2+ionsuniformlyatcationsites.ThisamountstoassumingwehaveanequivalentmagneticmomentofxhSziateachcationsite.So,Eq. 1{8 becomes HereRbecomesthepositionvectorofeachcationsite.InEq. 1{9 theexchangeHamiltoniannowhasthesameperiodicityasthecrystal. FromtheHamiltonian 1{9 ,theexchangeenergycanthenbeobtainedby Fortheelectronsattheconductionbandedge,thewavefunctionisck=uc0(k=0).J(rR)isthecouplingcoecientaswehavesaidabove,whichistheexchangeintegralbetweenthebandelectronsand3d5electrons.Duetothefactthatthe3d5electronsarestronglylocalized,wecanassumetheintegralisonlynonvanishinginaunitcellrangeforaspecicRinEq. 1{10 .ConsideringtheperiodicityofJ(rR),theEq. 1{10 canberewrittenas, whereNisthenumberofunitcellsinthecrystal. Forzincblendesemiconductors(mostIIVIandIIIVsemiconductors),thestatesforconductionbandedge(k=0)electronsareslike,andthoseforvalence
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bandedgeholesareplike.Sotheuseofmc=1=2isjustied.TheninamagneticeldB,theconductionbandenergyis, 2)h!c+mcgcBB+mcNxhSzi;(1{13) where!c=eB=mcisthecyclotronfrequency,andgcistheconductionbandgfactor.InEq. 1{13 ,thersttermistheLaudausplitting,thesecondtermistheZeemansplitting,andthethirdtermistheexchangesplitting,whichisuniquefortheDMS. Similarly,theenergystructureofthevalencebandcanalsobeobtained,ifwereplace!cby!v=eB=mv,gcbygv,mcbymv,andimportantly,by,where WecanintroduceaneectiveLandegfactorintheconductionband whichindicatesthestrengthofthespinsplittingoftherstLandaulevelintheconductionband.Intheloweldapproximation,Eq. 1{6 becomeshSzi=gBS(S+1)B=3kBT,sointhislimit 3kBT:(1{16) Atlowtemperature,theeectivegfactorcanreachverylargevalues.ThegfactordependsontemperaturethroughhSziinEq. 1{15 .WewillhaveamoredetaileddiscussionofgfactorsinChapter2. Theabovediscussionisaverysimpliedqualitativemodel,andonlyappropriateforIIVIDMSinaparamagneticphase,wheretheMnconcentrationisnotso
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highthattheydon'thaveadirectexchangeinteraction.ThisdiscussioncanalsobeappliedtoparamagneticIIIVDMS,inwhichcommonlytheMnsolubilityareverylow.Asamatteroffact,althoughEq. 1{13 cangiveaqualitativediscriptionoftheconductionbandstructure,itdoesnotworkinrealcases.Chapter2givesaquantitativemodel. SincethediscoveryofferromagnetisminIIIVDMS,muchresearchnowfocusesonexploringferromagnetismmechanisms,lookingfornewmaterialsandobtaininghigherCurietemperatures.Recently,ferromagnetisminIIVIDMSwasalsoreportedbyseveralgroups[ 11 12 13 ]. 1.3.1FerromagneticSemiconductor 1{15 andEq. 1{16 ,wecanseethatatlowtemperatures,thegfactorcanbeverylarge,butitisstronglytemperaturedependent.Aswementionedabove,thegfactoractuallyindicatesthespinsplitting.Toemployspinasasubjectinresearchanddevicedesign,alargespinsplittingisessential.WhilemostIIVIDMSareparamagnetic,thespinsplittingbecomessmallathightemperatures,sotherealizationofroomtemperaturespintronicdevicesbecomesdicult.Theanswerforthisproblemisferromagneticsemiconductors.Wecanexpectalargespinsplittingevenathightemperaturesforferromagneticsemiconductors. TheleapfromIIVIDMStoIIIVDMSshouldhavebeenverynatural.ButunlikeIIVIsemiconductors,MnisnotverysolubleinIIIVsemiconductors.Itcanbeincorporatedonlybynonequilibriumgrowthtechniquesanditwasnotuntil1992thattherstIIIVDMS,InMnAswasgrownandinvestigated.Ferromagnetismwassoondiscoveredinthissystem[ 14 ].Higherferromagnetic
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Figure1{5: MagneticelddependenceofHallresistivityHallandresistivityofGaMnAswithtemperatureasaparameter.Mncompositionisx=0:053.TheinsetshowsthetemperaturedependenceofthespontaneousmagnetizationMsdeterminedfrommagnetotransportmeasurements;thesolidlineisfrommeaneldtheory.ReprintedwithpermissionfromMatsukuraetal.Phys.Rev.B57:R2037R2040,1998.Figure1,PageR2037.
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transitiontemperatureswerealsoachievedinGaMnAs[ 15 ].ShowninFig. 1{5 isthemagneticelddependenceoftheHallresistivityandthenormalresistivityofGaMnAswithtemperatureasaparameter[ 16 ].Inthiscase,theferromagnetictransitiontemperatureisabout110K.ThediscoveryofferromagnetisminIIIVDMSledtoanexplosionofinterest[ 14 15 17 18 ].Manynewmatetialswereinvestigated,theoriesexplainingtheferromagnetismmechanismswerebroughtforward,andexperimentsaimedatincreasingtheCurietemperatureswerecarriedout. AlthoughInMnAswastherstMBEgrownIIIVDMS,itsCurietemperaturewasrelativelylowatabout7:5K.In1993,ahigherCurietemperatureof35KwasrealizedinaptypeInMnAs/GaSbheterostructure[ 17 ].Since1996,anumberofgroupsareworkingontheMBEgrowthofGaMnAsandrelatedheterostructures,inwhichthehighestCurietemperature(173K)hasbeenachievedrecentlyfor25nmthickGa1xMnxAslmswith8%nominalMndopingafterannealing[ 19 ].ThedependenceoftheCurietemperatureofGa1xMnxAsonMnconcentrationxisshowninFig. 1{6 [ 16 ].TheCurietemperaturereachesthehighestvaluewhenx=5:3%inthiscase. GaMnNandGaMnParealsocandidatesforhighCurietemperatureIIIVDMSmaterials.FerromagnetisminGaMnNiselusive.WhilesomegroupsfounditparamagneticwhendopedwithpercentlevelsofMn[ 20 ],somegroupshavereportedaferromagnetictransitiontemperatureabove900K[ 21 ].RoomtemperatureferromagnetismwasalsoreportedinGaMnP[ 22 23 ].BesidesIIIVDMS,MndopedIVsemiconductorslikeGeMn[ 24 25 ],SiMn[ 26 ],werealsoreportedferromagnetic. ThetheoryforferromagnetisminIIIVDMSisstillcontroversial,however,thereisconsensusthatitismediatedbytheitinerantholes.UnlikethecaseinIIVIDMSinwhichMnionshavethesamenumberofvalenceelectronsasthe
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Figure1{6: MncompositiondependenceofthemagnetictransitiontemperatureTc,asdeterminedfromtransportdata.ReprintedwithpermissionfromMatsukuraetal.Phys.Rev.B57:R2037R2040,1998.Figure2,PageR2038. cations,MnionsinIIIVDMSarenotonlyprovidersofmagneticmoments,theyarealsoacceptors.DuetocompensatingdefectslikeAsantisitesor/andMninterstitials[ 27 28 29 ],holeconcentrationsaregenerallymuchlowerthantheMnconcentration. Thetheoriesofcarrierinducedferromagnetismfallintofourcategories. 1. 30 ]andlaterextendedbyKasuya[ 31 ]andYosida[ 32 ]togivethetheorynowgenerallyknowastheRKKYinteraction.Ohnoetal.explainedtheferromagnetisminGaMnAsforMnconcentrationx=0:013usingtheRKKYmechanism[ 14 ].IntheinteractionHamiltonian, thecouplingcoecientJRKKYi;jassumestheform[ 33 ],
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wherekFistheradiusoftheconductionelectron/holeFermisurface,risthedistanceawayfromtheoriginwherealocalmomentisplaced.TheRKKYexchangecoecient,J,oscillatesfrompositivetonegativeastheseparationoftheionschangeswiththeperioddeterminedbytheFermiwavevectork1FandhasthedampedoscillatorynatureshowninFig. 1{7 .Therefore,dependingupontheseparationbetweenapairofionstheirmagneticcouplingcanbeferromagneticorantiferromagnetic.Amagneticioninducesaspinpolarizationintheconductionelectronsinitsneighborhood.Thisspinpolarizationintheitinerantelectronsisfeltbythemomentsofothermagneticionswithinrange,leadingtoanindirectcoupling. Figure1{7: VariationoftheRKKYcouplingconstant,J,ofafreeelectrongasintheneighborhoodofapointmagneticmomentattheoriginr=0. InthecaseofDMS,theaveragedistancebetweenthecarriersrc=4p 3isusuallymuchgreaterthanthatbetweenthespinsrS=4xN 3.AsimplecalculationshowthattherstzerooftheRKKYfunctionoccursatr1:17rc.Thismeansthatthecarriermediatedinteractionisferromagneticandeectivelylongrangeformostofthespins. TheRKKYinteractionasthemainmechanismfortheferromagnetisminIIIVDMSisquestionableinsomecasessuchasintheinsulatingphase(x<3%forGaMnAs),inwhichcarriersarenotitinerant.Whentheholedensityislow,andthereisnoFermisurface(Fermilevelinthegap),RKKYtheorycannotpredictferromagnetism.Theotherproblem,maybefatal,isthatintheRKKYapproximationtheexchangeenergyismuchsmallerthan
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theFermienergy,whichisnotcommonlythecaseinDMS.Asamatteroffact,thesetwoenergiesarecomparableinmostcases. 2. 34 ]tointerprettheferromagneticcouplingintransitionmetals.SimilartotheRKKYmodel,itdescribesanexchangeinteractionbetweencarriersandlocalizedspins.TheHamiltonianofZener'smodelinatransitionmetalis[ 34 ] 2S2dSsSc+1 2S2c;(1{19) whereSdandScarethemeanmagnetizationofthedshellelectronandtheconductionelectron,respectively,and,,andarethreecouplingconstants.Themainassumptionhereisthattheexchangeconstantisalwayspositive,whichundercertaincircumstancesleadstoferromagneticcoupling.ComparingHamiltonian 1{17 and 1{19 ,wecanseethatinEq. 1{19 playsthesimilarroleofJinEq. 1{17 .OnebigdierenceisthatZener'smodelneglectstheitinerantcharacterandtheFriedeloscillationsoftheelectronspinpolarizationaroundthelocalizedspins. Dietl[ 35 ]appliedZener'smodeltoferromagneticsemiconductorsandpredictedtheCurietemperatureTCforseveralMndopedDMSsystems.TheresultsareshowninFig. 1{8 .Thisquiteaccuratelypredictsthe110KtransitiontemperatureinGaMnAs,butcertainlythisisstillaquitecoarsemodel.Evenso,thetrendshowninFig. 1{8 hasstimulatedtheenthusiasmofpeopleinvestigatingGaNbasedmaterialslookingforhighertransitiontemperatures. SomeoftheproblemsintheRKKYmodelremaininZener'smodel.Forinstance,Zener'smodelstillhaslimitedapplicationwhencarriersaremostlylocalizedbecauseitstillrequiresitinerantcarrierstomediatetheinteractionsbetweenlocalizedspins.Besides,whenthecarrierdensityishigherthantheMnconcentration,importantchangesintheholeresponsefunctionoccuratthelengthscaleofthemeandistancebetweenthelocalizedspins.Accordingly,thedescriptionofspinmagnetizationbythecontinuousmediumapproximation,whichconstitutesthebasisoftheZenermodel,ceasestobevalid.Incontrast,theRKKYmodelisagoodstartingpointinthisregime. 3. 1{7 ,andthiseldcausesferromagneticcouplingoftheselocalspins.Thenetspinalignment
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Figure1{8: CurietemperaturesfordierentDMSsystems.CalculatedbyDietlusingZener'smodel. againcreatesaselfconsistentexchangeeldforthecarriers.Inthisprocess,thecarrierspincreatesamagneticpotentialwellresultinginformationofa\spincloud",amagneticpolaron.DuetothelocalizedcharacterofthesemagneticpolaronsinDMS,theyarecalledboundmagneticpolarons(BMP). TherehavebeenextensivestudiesofBMPinIIVIDMS[ 6 ],inwhichBMPareaccountableformanyopticalandphasetransitionproperties.Recently,Bhattetal.[ 36 ]andDasSarmaetal.[ 37 ]generalizedBMPtheoryforIIIVDMS.TheystudiedthecouplingbetweentwoadjacentBMPs,andconcludedthattheexchangecouplingisferromagnetic.Therearetwodierentcases.Inonecasetwopolaronsoverlapandtheoverlapintegralaccountsfortheferromagneticcoupling.TheferromagnetictransitioncanberegardedasapercolationoccurringthroughthewholesystemwhenthetemperaturedropsbelowtheCurietemperature.Intheothercaseonedoesnotneedoverlappingpolarons,theireectonthemagneticmomentbeingtakenintoaccountthroughalocalmagneticeld.Ferromagneticcouplinghasbeenshowntoresultwhenthecarrierisallowedtohopbetweenthegroundstateofonemagneticatomandexcitedstatesoftheother.AdiagramofthesetwocasesareshowninFig. 1{9 TheBMPmodelquitenaturallyandsuccessfullyexplainsthemagnetismoftheDMSintheinsulatingphase.Withamuchhighercarrierdensity,most
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Figure1{9: SchematicdiagramoftwocasesofBMPs. carriersareconducting.Theyaremorelikefreebandcarriers.Insuchacase,theBMPmodelmaynotbeappropriate.Althoughsomepartofthecarriersarelocalizedandhaveexchangeinteractionwiththelocalizedspins,mostcarriershaveextendedwavefunctions,whichtendtointeractwiththeothercarriersandspinsinthewholeband.TheconditionfortheBMPmodeldoesnotexistanymore.Insuchacase,theRKKYmechanismshoulddominate. 4. 38 ]performedrstprincipleDFTcalculationswhichshowthatthemajorityofthecarrierscomesfromMndstates.Thehoppingofthecarriersbetweentheimpuritybandsandvalencebandscausestheferromagneticordering.Later,Inoueetal.[ 39 ]alsodiscussedasimilarmechanism.TheycalculatedtheelectronicstatesofIIIVDMSandfoundthatresonantstateswereformedatthetopofthedownspinvalencebandduetomagneticimpuritiesandtheresonantstatesgaverisetoastronglongrangedferromagneticcouplingbetweenMnmoments.Theyproposedthatcouplingoftheresonantstates,inadditiontotheintraatomicexchangeinteractionbetweentheresonantandnonbondingstateswastheoriginoftheferromagnetismofGaMnAs.Wecanclassifythiskindofmechanismcausedbythehoppingofcarriersbetweenimpuritystatesandvalencestatesasadoubleexchangemechanism.DoubleexchangelikeinteractionsinGaMnAswerereportedbyHirakawaetal.[ 40 ]. InthefourmodelsofferromagnetisminIIIVDMS,therstthreearemeaneldbasedtheories,andthelastisbasedondelectrons.Thougheachofthemis
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capableofexplainingsomespecicaspectsofferromagnetism,noneofthemcanbeapplieduniversally. SupposeaHeisenberglikeHamiltonian wherei,jspecifyatomicsites,say,ofthemagneticmomentsinthecrystal,andJi;jistheinteratomicexchangeinteractionconstant.Themoleculareld(eectivemeaneld)issimplygivenby wheregisthegfactor.UsingtheresultswegotinthediscussioninSection1.2.2,theaveragespinalongamagneticeldB(supposeitisdirectedalongz)willbe withBs,theBrillouinfunction,givenbyEq. 1{5 ,andwhere AftersubstitutionofEq. 1{22 toaboveequation,weget
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Figure1{10: Averagelocalspinasafunctionofmagneticeldat4temperatures.TheCurietemperatureis110K. Equation 1{22 canbesolvedbystandardrootndingprogramstondhSzi.ThesolutionforhSzi6=0existsevenwhenB=0duetotheinternalexchangeeld.WhenjhSzij1, 3(S+1)y:(1{26) WhenJ0>0,theconditionforhSzi6=0thenis ThisisconsistentwiththefactthatJ>0inHeisenbergHamiltonianleadstoferromagneticinteraction. Inarealisticcalculation,TCasameasurableparameteriseasytoobtain,hencewecanuseEq. 1{27 tondtheexchangeinteractionconstantJ0,andthusthespontaneousmagnetizationforT
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Thespontaneousmagnetizationhasfundamentaleectsoncarrierscatteringandspinscattering,andthusaectsthetransportpropertiesofbothcarriersandspins.WewilltalkaboutthisinChapter5. 41 42 43 ]performedrstprinciplecalculationsshowingthatcontrarytotheRKKYmodel,theholeinducedbyMnisnothostlike,whichunderminesthebasisofapplyingRKKYtheorytoDMS.TheensuingferromagnetismbytheholesinducedbyMnionsisthennotRKKYlike,but\hasacharacteristicdependenceonthelatticeorientationoftheMnMninteractionsinthecrystalwhichisunexpectedbyRKKY".Theyclaimthatthedominantcontributiontostabilizingtheferromagneticstatewastheenergyloweringduetothepd
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hopping.Thenatureoftheferromagnetismtheniscloselyrelatedtothenatureofthebandelectrons.Photoinducedferromagnetism[ 44 ]clearlyrevealstheroleofholesinmediatingtheferromagneticcoupling.Thereisnodoubtcarriersarecrucialinallthemechanismsaccountingfortheferromagnetism,butaretheyreallyhostlikeholes,ordotheyhavestrongdcomponentmixing?Howdotheybehaveintheprocessofmediatingtheferromagnetism?Onlyafterweknowtherightanswer,willthemanipulationofchargecarriersandalsothespinsbecomemorepredicable. 45 ].SinceIIVIDMSisparamagneticatroomtemperature,amagneticeldisneededtoobtainFaradayrotation.FerromagneticsemiconductorbasedonIIIVDMS,whichdoesnotneedanexternalmagneticeldtosustainthebigFaradayrotation,shouldhaveagoodpotentialforuseinopticalisolators. PhotoinducedferromagnetismhasbeendemonstratedbyKoshiharaetal.[ 44 ]andKonoetal.[ 46 ].InKoshihara'sexperiment,ferromagnetismisinducedbyphotogeneratedcarriersinInMnAs/GaSbheterostructures.TheeectisillustratedinFig. 1{11 .Duetothespecialbandalignmentofthisheterostructure,electronsandholesarespaciallyseparated,andholesaccumulateintheInMnAslayer.ThephotogeneratedholesthencauseatransitionoftheInMnAslayertoaferromagneticstate.Thisopensapossibilitytorealizeopticallycontrollablemagnetoopticaldevices.InKono'sexperiment,ultrafastdemagnetizationtakesplaceafteralaserpulseshinesonInMnAs/GaSbheterostructureandproduces
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ferromagnetism.Thetimescaleistypicallyofseveralps.Theyproposeanewandveryfastschemeformagnetoopticalrecording. Figure1{11: ThephotoinducedferromagnetisminInMnAs/GaSbheterostructure.ReprintedwithpermissionfromKoshiharaetal.Phys.Rev.Lett.78:46174620,1997.Figure3,Page4619. Recently,Ohnoetal.[ 2 ]achievedcontrolofferromagnetismwithanelectriceld.TheyusedeldeecttransistorstructurestovarytheholeconcentrationsinDMSlayersandthusturnthecarrierinducedferromagnetismonandobyvaryingtheelectriceld.Rashbaetal.[ 47 ]alsoproposedtheelectronspinoperationbyelectricelds.Theyalsodiscussedthespininjectionintosemiconductors.The
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electriccontrolofferromagnetismorspinstatesmakespossibleaunicationofmagnetismandconventionalelectronics,andthushasaprofoundmeaning. Lowdimensionalstructuresusuallyhavedramaticallydierentpropertiesfrombulkmaterials.Muchlongerspincoherenttimeshavebeenreportedbyseveralgroupsinquantumdots[ 48 49 ],whichhavebeensuggestedforuseinquantumcomputerswherequantumdotscanbeusedasquantumbits,sincetheyoeratwolevelsystemclosetotheidealcase.OneultimategoalofDMSspintronicsistoimplementquantumcomputing.Theuseofsemiconductorsinquantumcomputinghasvariousbenets.Theycanbeincorporatedintheconventionalsemiconductorindustry,andalso,lowdimensionalstructuresareveryeasytoconstruct,souniquelowdimensionalpropertiescanbeemployed.Severalproposalshavebeenmadeforquantumcomputingusingquantumdots[ 50 51 52 ]. Spinmanipulationneedsinjection,transportanddetectionofspins.Themostdirectwayforspininjectionwouldseemtobeinjectionfromaclassicalferromagneticmetalinametal/semiconductorheterostructurebutthisraisesdicultproblemsrelatedtothedierenceinconductivityandspinrelaxationtimeinmetalsandsemiconductors[ 53 ].Althoughtheseproblemsarenowbetterunderstood,thishassloweddowntheprogressforspininjectionfrommetals.Ontheotherhand,thishasboostedtheresearchofconnectingDMSwithnonmagneticsemiconductorsforspininjection.Manyexperimentspursuinghigneciencyspininjectionhavebeencarriedout.ShowninFig. 1{12 isaspinlightemittingdiode[ 54 ],inwhichacurrentofspinpolarizedelectronsisinjectedfromthedilutedmagneticsemiconductorBexMnyZn1xySeintoaGaAs/GaAlAslightemittingdiode.Circularlypolarizedlightisemittedfromtherecombinationofthespinpolarizedelectronswithnonpolarizedholes.Aninjectioneciencyof90%spinpolarizedcurrenthasbeendemonstrated.AsBexMnyZn1xySeisparamagnetic,thespinpolarizationisobtainedonlyinanappliedeldandatlowtemperature.
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Figure1{12: Spinlightemittingdiode. AferromagneticIIIVDMSbasedspininjectordoesnotneedanappliedeld.ShownintheleftpanelofFig. 1{13 isaGaMnAsbasedspininjectionanddetectionstructure[ 55 ],inwhichspinpolarizedholesareinjectedfromGaMnAstoaGaAsquantumwell.TheemitterandanalyzerarebothmadeoflayersofferromagneticsemiconductorGaMnAs.ThetemperaturedependenceofthespinlifetimeintheGaAsquantumwellfrommagnetoresistancemeasurementsisshownintherightpanel. Toobtaintheinformationwhichaspincarries,oneneedstodetectanelectronspinstate.Manymethodsfordoingthishavebeenbroughtforthandstructuresordeviceshavebeendesignedsuchasspinltersusingmagnetictunneljunctions[ 56 57 ],spinlters[ 58 ],andonedeviceinvolvingasingleelectrontransistortoreadoutthespatialdistributionofanelectronwavefunctiondependingonthespinstate[ 59 ]. ThedevelopmentofDMSbasedspintronicsisnowreceivingagreatattention,andmaybecomeakeyareainresearchandindustryinthefuture.Althoughenormouseorthasbeenmade,thereisstillalongwaytogoforDMStobeextensivelyusedinreallife.
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Figure1{13: GaMnAsbasedspindevice.Left:GaMnAsbasedspininjectorandanalyzerstructure.Right:TemperaturedependenceofspinlifeintheGaAsquantumwellinthestructureshowninleftpanel.
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TounderstandtheopticalandtransportpropertiesofDMSinthepresenceofanappliedmagneticeld,wehavetoknowtheelectronicbandstructureandtheelectronicwavefunctions.Foropticaltransitions,withtheknowledgeoftheinteractionHamiltonian,wemayuseFermi'sgoldenruletocalculatethetransitionrate.Inanexternalmagneticeld,oneenergylevelwillsplitintoaseriesofLandaulevels.OpticaltransitionscantakeplaceinsideoneseriesofLandaulevelsorbetweendierentseriesaccordingtothelightconguration.SotheknowledgeoftheparitiesoftheseLandaulevelsneedtobeinvestigated.Inthischapter,wewillusethekpmethodtostudythebandstructureofDMSmaterialsaroundthepoint.Specically,ageneralizedPidgeonBrownmodel[ 60 ]willbeusedtostudytheLandaulevelstructures. TheextraenergygaininaferromagneticDMScanbetreatedinameaneldapproximation(seeSection1.3.2).Thelocalizedmagneticmomentslineupalongtheeectiveeld,soforeachmagneticion,ithasanonvanishingaverage 29
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spinalongtheelddirection.AccordingtothediscussioninSection1.2.2,anextraenergytermproportionaltotheexchangeconstantwillbeaddedtotheHamiltonian.Thistermisrelatedtospinquantumnumbers,thusdierentspinstateswillgaindierentenergies,leadingtospinsplittings.ShowninFig. 2{1 arecalculatedvalencebandstructuresforBext=0ofbulkGaAsandferromagneticGaMnAs,whichhasaCurietemperatureTC=55K.ThecalculationisactuallybasedonageneralizedKane'smodel[ 61 ],andtheeectiveeldisassumedtobedirectedinzdirection.Kane'smodelwasdevelopedfromkptheory,whichwewillintroduceinthefollowingsection. Figure2{1: ValencebandstructureofGaAsandferromagneticGa0:94Mn0:06Aswithnoexternalmagneticeld,calculatedbygeneralizedKane'smodel.Thespinsplittingofthebandsisshown. 2.2.1IntroductiontokpMethod 62 ]andSeitz[ 63 ].Itisaperturbationtheorybasedmethod,oftencalledeectivemasstheoryintheliterature,usefulforanalyzingthebandstructurenearaparticularpointk0,which
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isanextremumofthebandstructure.Inthecaseofthebandstructurenearthepointinadirectbandgapsemiconductor,k0=0. TheHamiltonianforanelectroninasemiconductorcanbewrittenas herep=ihristhemomentumoperator,m0referstothefreeelectronmass,andV(r)isthepotentialincludingtheeectivelatticeperiodicpotentialcausedbytheionsandcoreelectronsorthepotentialduetotheexchangeinteraction,impurities,etc.IfweconsiderV(r)tobeperiodic,i.e., whereRisanarbitrarylatticevector,thesolutionoftheSchrodingerequation satisesthecondition where andkisthewavevector.Equations 2{4 and 2{5 istheBlochtheorem,whichgivesthepropertiesofthewavefunctionofanelectroninaperiodicpotentialV(r). TheeigenvaluesforEq. 2{3 splitintoaseriesofbands[ 64 ].ConsidertheSchrodingerequationinthenthbandwithawavevectork, InsertingtheBlochfunctionEq. 2{4 intoEq. 2{6 ,weobtain m0kp+h2k2
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Inmostcases,spinorbitcouplingmustalsobeconsideredandaddedintotheHamiltonian.Thespinorbitinteractiontermis h Includingthespinorbitinteraction,Eq. 2{7 becomes TheHamiltonianinEq. 2{9 canbedividedintotwoparts [H0+W(k)]unk=Enkunk;(2{10) where and OnlyW(k)dependsonwavevectork. IftheHamiltonianH0hasacompletesetoforthonormaleigenfunctionsatk=0,un0,i.e., thentheoreticallyanylatticeperiodicfunctioncanbeexpandedusingeigenfunctionsun0.Substitutingtheexpression
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intoEq. 2{9 ,andmultiplyingfromtheleftbyun0,andintegratingandusingtheorthonormalityofthebasisfunctions,weobtain Solvingthismatrixequationgivesusboththeexacteigenstatesandeigenenergies.Usually,peopleonlyconsidertheenergeticallyadjacentbandswhenstudyingthekexpansionofonespecicband.Itactuallybecomesverycomplicatedifonewantstopursueacceptablesolutionswhenkincreases.Onehastoincreasethenumberofthebasisstates,gotohigherorderperturbations,orboth. WhenkissmallandweneglectthenondiagonaltermsinEq. 2{15 ,theeigenfunctionisunk=un0,andthecorrespondingeigenvalueisgivenbyEnk=En0+h2k2 where Inthecalculationshownabove,weusedthepropertyhun0jp+h thenthesecondordereigenenergiescanbewrittenas
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Equation 2{19 isoftenwrittenas where 1 istheinverseeectivemasstensor,and;=x;y;z.Theeectivemassgenerallyisnotisotropic,butwecanseeitisnotkdependent,thisisbecauseatthislevelofapproximation,theeigenenergiesinthevicinityofthepointonlydependquadraticallyonk. 2{15 givesexactsolutionsforboththeeigenfunctionsandeigenenergies.Practically,itisnotfeasibletoincludeacompletesetofbasisstates,sousuallyonlystronglycoupledbandsareincludedinusualkpformalism,andtheinuenceoftheenergeticallydistantbandsistreatedperturbatively. InKane'smodel,electronicbandsaredividedintotwogroups.Intherstgroup,thereisastronginterbandcoupling.Usuallythenumberofbandsinthisgroupiseight,includingtwoconductionbands(oneforeachelectronspin)andsixvalencebands(twoheavyhole,twolightholeandtwosplitoholebands).Thesecondgroupofbandsisonlyweaklyinteractingwiththerstgroup,sotheeectcanbetreatedbysecondorderperturbationtheory. ShowninFig. 2{2 isthebandstructureofatypicalIIIVdirectbandgapsemiconductor.Duetocrystalsymmetry,theconductionbandbottombelongstothe6group,thevalencebandtopbelongstothe8group,andthesplitobandbelongstothe7group.Thespatialpartofthewavefunctionsattheconductionbandedgeareslikeandthoseatthevalencebandtopareplike.SymbolsofjSi,jXi,jYi,andjZiareusedtorepresenttheoneconductionbandedgeand
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Figure2{2: BandstructureofatypicalIIIVsemiconductornearthepoint.Kane'smodelconsidersthedoublyspindegenerateconduction,heavyhole,lightholeandsplitobands,andtreatsthedistantbandsperturbatively. threevalencebandedgeorbitalfunctions.Withspindegeneracyincluded,thetotalnumberofstatesiseight.TheseeightstatesjS"i,jS#i,jX"i,jX#i,etc,canserveasasetofbasisstatesintreatingtheseeightbands.Aunitarytransformationofthisbasissetisstillabasisset.Soinpractice,peopleusethefollowingexpressions,whicharetheeigenstatesofangularmomentumoperatorsJ
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andmJ,asthebasisstatesfortheeightbandKane'smodel, 2;1 2i=jS"i=jS"i;u2=j3 2;3 2i=jHH"i=1 2;1 2i=jLH#i=1 2;1 2i=jSO#i=i 2;1 2i=jS#i=jS#i;u6=j3 2;3 2i=jHH#i=i 2;1 2i=jLH"i=i 2;1 2i=jSO#i=i Thissetofbasisstatesisaunitarytransformationofthebasiswhichwehavementionedabove,anditcanbeproventhattheyaretheeigenfunctionsoftheHamiltonian 2{11 .Becauseofspindegeneracyatk=0,theeigenenergiesforjSi,jHHi,jLHiandjSOiareEg,0,0,,respectively,withtheselectionofenergyzeroatthetopof8band,whereEgisthebandgap,and =3ih @xpy@V @ypxjYi;(2{23) isthesplitobandenergy. Atthislevelofapproximation,thebandsarestillatbecausetheHamiltonian 3{10 iskindependent.IncludingW(k)inEq. 2{12 intotheHamiltonian,anddeningKane'sparameteras m0hSjzjZi;(2{24)
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weobtainamatrixexpressionfortheHamiltonianH=H0+W(k),i.e., 3Pk00q 3Pkz1 3Pkz000q 3Pk+00+h2k2 3Pkz1 3Pk+0000Pk+h2k2 3Pkz0001 3Pk00+h2k2 wherek+=kx+iky,k=kxiky,andkx,ky,kzarethecartesiancomponentsofk.TheHamiltonian 2{25 iseasytodiagonalizetondtheeigenenergiesandeigenstatesasfunctionsofk.Wehaveeighteigenenergies,butduetospindegeneracy,thereareonlyfourdierenteigenenergieslistedbelow.Fortheconductionband, Forthelightholeandsplitobands, Fortheheavyholebandwehave Theeectivemassoftheheavyholebandisstillequaltothebareelectronmass,sincewehavenotincludedthedistantbandcouplingintheHamiltonian.The
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eectofthedistantbandcouplingwillmaketheheavyholebandcurvedownwardratherthanupward. 65 ],inwhichthebandsareclassiedasAandB.Inourcase,weselectthebasisstates 2{22 asclassAandlabelthemwithsubscriptnandalltheother(energeticallydistant)statesasclassBwhichwelabelwithsubscript. Supposeallstatesareorthonormal,theSchrodingerequationthentakestheform wherelandmrunoverallstates.RewritethisequationusingclassAandB,andweobtain (EHmm)am=AXn6=mHmnan+BX6=mHma(2{31) or wheretherstsumontherighthandsideisoverthestatesinclassAonly,whilethesecondsumisoverthestatesinclassB.WecaneliminatethosecoecientsinclassBbyaniterationprocedureandobtainthecoecientsinclassAonly, and AlittlealgebrashowsthatEq. 2{33 isequivalentto
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ThismeansthatwecanndtheeigenenergieswiththebasisinclassAbutstillincludetheremoteeectsfromclassBusingEq. 2{35 .TheeectfromclassBistreatedasaperturbationusingEq. 2{34 tosecondorder. TruncatingUAmntothesecondterm,andusingHamiltonianinEq. 2{9 ,itcanberewrittenas where and m0kju0i=Xahka wherea=x;y;zandam'pamform2Aand2B.Thus Applyingbasisset 2{22 ,wecandeneparametersA,B,CandFasfollow, Rewritingtheseparametersintermsof\Luttinger"parameters1,2,3and4denedas
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3(A+2B); 6(AB); wecanobtainthetheHamiltonianHmn=UmnincludingthedistantbandcouplingunderthebasissetlistedinEq. 2{22 as 3Vkzi 3VkzL0iq 2L+1 2L+000iq 3Vkz1 3VkzL+0iq 2L1 2L0i where
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and relatedtotheKane'sparameterPdenedinEq. 2{24 .Wecanseethatifk=0orkz=0,theHamiltonianisblockdiagonalized. Inpractice,oneimportantthingneedstobenotedthattheLuttingerparametersdenedinEq. 2{44 arenotthe\usualLuttinger"parameterswhicharebasedonasixbandmodelsincethisisaneightbandmodel,butinsteadarerelatedtotheusualLuttingerparametersL1,L2,andL3throughtherelations[ 66 ] ThistakesintoaccounttheadditionalcouplingofthevalencebandstotheconductionbandnotpresentinthesixbandLuttingermodel.Wereferto1etc.astherenormalizedLuttingerparameters. TheHamiltonian 2{45 isbasedonaneightbandKane'sHamiltonianincludingthecontributionsoftheremotebands.Withtheremotebandcoupling,theelectroneectivemassattheconductionbandminimumnowbecomes 1 InDMSmaterialswithoutmagneticelds,theHamiltonian 2{45 plustheexchangeinteractioncanbeusedtocalculatethebandstructurewhichwillbeappliedtothecalculationoftheopticalpropertiessuchasmagnetoopticalKerreect,whichistobestudiedinchapter4.
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Inamagneticeld,asingleenergylevelsplitsintoaseriesofLandaulevels.Opticaltransitionstakeplacebetweentwolevelsinoneseriesortwoindierentseries. Inlastsection,ifwewrite where theeigenequationisgivenby whereamisthesuperpositioncoecientsdenedas NowweconsideraspatialperturbationU(r)addedtotheHamiltonianHmn.Theeigenequationnowbecomes [H+U(r)](r)=E(r):(2{54)
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Ifwewritethesolutiontotheequationas LuttingerandKohn[ 67 ]haveshownthatweneedonlysolvethefollowingequation, @xai@ @xb+U(r)mn#Fn(r)=EFm(r)(2{56) ThismeansthatweonlyneedtoreplacethewavevectorintheHamiltoniankabytheoperatorpa=h,andsolveanequationforF(r).ThefunctionF(r)iscalledtheeectivemassenvelopefunction. 2.3.1ElectronicStateinaMagneticField 1 2m(ihr+eA)2(x)=(x);(2{57) whereAisthevectorpotential,andeistheelectroncharge.Assumethemagneticeldisdirectedalongz.UsingLandau'sgauge, andassumingasolutionlike
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whereLx,Ly,andLzarelengthsforthebulkcrystalinthreedimensions.Aftersubstitutingintotheeectivemassequation,wehaveanequationfor(y), 1 2m(kxeBy Dening0h2k2z 2)h!c;(2{61) where!c=eB=misthecyclotronfrequency.Thusthetotalenergyis 2)h!c+h2k2z Thismeansthatinamagneticeld,themotionofanelectroninasemiconductornowhasquantizedenergiesinthexyplane,thoughitsmotioninthezdirectionisstillcontinuous.TheoriginalstatesinonebandnowsplitintoaseriesofLandaulevelswhoseeigenfunctionsare TheelectronicenergiesinEq. 2{62 isonlyrelatedtonandkz.Theyaredegeneratefordierentkx.InEq. 2{63 thecenterofy0=hkx
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valencebandsisstrong,soitisnecessarytousetheeightbandmodeltocalculatetheLandaulevels.PidgeonandBrown[ 60 ]developedamodeltocalculatethemagneticelddependentLandaulevelsatk=0.Wewillgeneralizethismodeltoincludethewavevector(kz)dependenceoftheelectronicsstatesaswellasthesdandpdexchangeinteractionswithlocalizedMndelectrons. WewillstillutilizethebasissetdenedinEq. 2{22 .InthepresenceofauniformmagneticeldBorientedalongthezaxis,thewavevectorkintheeectivemassHamiltonianisreplacedbytheoperator hp+e cA;(2{64) wherep=ihristhemomentumoperator.Forthevectorpotential,westillusetheLandaugaugeasinEq. 2{58 ,thusB=rA=B^z. Nowweintroducetwooperators and whereisthemagneticlengthwhichisdenedas eB=s TheoperatorsdenedinEqs. 2{65 obeythecommutationrulesofcreationandannihilationoperators.Thestatestheycreateandannihilatearesimpleharmonicoscillatorfunctions,andaya=Naretheorderoftheharmonicfunctions.UsingthesetwooperatorstoeliminatekxandkyinHamiltonian 2{45 ,wearriveattheLandauHamiltonian
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withthesubmatricesLa,LbandLcgivenby aiq 3V ayq 3V ayiV ayPQMip 3V aMyP+Qip 3V aip ayq 3V aiq 3V aV aPQMyip 3V aMP+Qip 3V ayip 3Vkziq 3Vkz00Liq 2Liq 3VkzL0iq 2Lyq 3Vkziq 2Liq 3Ly0377777775(2{70) TheoperatorsA,P,Q,L,andMinEq. 2{67 noware !;(2{71d) and
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Theparameters1,2,3and4aredenedinEq. 2{48 and 2{44 .Usually,theLuttingerparameters2and3areapproximatelyequal(sphericalapproximation),sowehaveneglectedaterminMproportionalto(23)(ay)2.ThistermwillcoupledierentLandaumanifoldsmakingitmorediculttodiagonalizetheHamiltonian.Theeectofthistermcanbeaccountedforlaterbyperturbationtheory. Foraparticlewithnonzeroangularmomentum(thusanonzeromagneticmoment)inamagneticeld,theenergyduetotheinteractionbetweenthemagneticmomentandthemagneticeldisB,whichiscalledZeemanenergywhichwediscussedinSection1.2.2.TheelectronsinIIIVDMSconductionorvalencebandspossessbothorbitalangularmomentaandspin,sothereisoneextraZeemantermproportionalto(K0LB+K1B),whereLandaretheorbitalangularmomentumandspinoperators,bothofwhichareinmatrixform.K0andK1arethemagneticelddependentcoecients.FollowingLuttinger[ 66 ],wedenetheparameteras where 3L21 3L12 3(2{73) istheLuttingerparameter,andweobtaintheZeemanHamiltonian wherethe44submatrixZaisgivenby 200003 200001 2iq 200iq 2377777775:(2{75)
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DuetoexistenceoftheMnimpurityions,theexchangeinteractionsbetweenthebandelectronsandlocalizedmomentsalsoneedstobeaccountedfor.Thistermisproportionalto(PIJ(rRI)SI).Underameaneldandvirtualcrystalapproximation(seeSection1.2.2),anddeningthetwoexchangeconstants hSjJjSi(2{76a) and hXjJjXi;(2{76b)wecanarriveatanexchangeHamiltonian wherexistheMnconcentration,N0isthenumberofcationsitesinthesample,andhSziistheaveragespinonaMnsitewhichisexactlytheonewederivedatSection1.2.2forparamagneticDMSorthatinSection1.3.2forferromagneticDMS.The44submatrixDais 200001 200001 6ip 300ip 31 2377777775:(2{78) Herewejusttreattheeectofmagneticionsasanadditionalinteraction.Wedon'tconsiderthepossibleeectofthesemagneticionsonthebandgap,etc.Thebandgapchangesasaresult. ThediscussionhereisverysimilartothatinSection1.2.2whereonlyaqualitativemodelisintroduced,buthereweusedarealisticbandstructure.Also
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similartothatdiscussion,thetotalHamiltonianherecanbewrittenas Wenotethatatkz=0,theeectivemassHamiltonianisalsoblockdiagonalliketheHamiltonian 2{45 2{58 ,translationalsymmetryinthexdirectionisbrokenwhiletranslationalsymmetryalongtheyandzdirectionsismaintained.ThuskyandkzaregoodquantumnumbersandtheenvelopeoftheeectivemassHamiltonian 2{79 canbewrittenas A2666666666666666666664a1;n;n1a2;n;n2a3;n;na4;n;na5;n;na6;n;n+1a7;n;n1a8;n;n13777777777777777777775(2{80) InEq. 2{80 ,nistheLandauquantumnumberassociatedwiththeHamiltonianmatrix,labelstheeigenvectors,A=LxLyisthecrosssectionalareaofthesampleinthexyplane,n()areharmonicoscillatoreigenfunctionsevaluatedat=x2ky,andai;(kz)arecomplexexpansioncoecientsforthetheigenstatewhichdependexplicitlyonnandkz.NotethatthewavefunctionsthemselveswillbegivenbytheenvelopefunctionsinEq. 2{80 witheachcomponentmultipliedbythecorrespondingkz=0BlochbasisstatesgiveninEq. 2{22
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SubstitutingFn;fromEq. 2{80 intotheeectivemassSchrodingerequationwithHgivenbyEq. 2{79 ,weobtainamatrixeigenvalueequation thatcanbesolvedforeachallowedvalueoftheLandauquantumnumber,n,toobtaintheLandaulevelsEn;(kz).Thecomponentsofthenormalizedeigenvectors,Fn;,aretheexpansioncoecients,ai. Sincetheharmonicoscillatorfunctions,n0(),areonlydenedforn00,itfollowsfromEq. 2{80 thatFn;isdenedforn1.TheenergylevelsaredenotedEn;(kz)wherenlabelstheLandaulevelandlabelstheeigenenergiesbelongingtothesameLandaulevelinascendingorder. Table2{1: SummaryofHamiltonianmatriceswithdierentn 1111(1;1)0444(0;),=141777(1;),=172888(n;),=18 Forn=1,wesetallcoecientsaitozeroexceptfora6inordertopreventharmonicoscillatoreigenfunctionsn0()withn0<0fromappearinginthewavefunction.TheeigenfunctioninthiscaseisapureheavyholespindownstateandtheHamiltonianisnowa11matrixwhoseeigenvaluecorrespondstotheaheavyholespindownLandaulevel.Pleasenotethatwhenwespeakaboutaheavy(light)holestate,itgenerallymeansthattheelectronicwavefunctioniscomposedmainlyoftheheavy(light)holeBlochbasisstatenearthek=0point. Forn=0,wemustseta1=a2=a7=a8=0andtheLandaulevelsandenvelopefunctionsarethenobtainedbydiagonalizinga44Hamiltonianmatrixobtainedbystrikingouttheappropriaterowsandcolumns.Forn=1,the
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Hamiltonianmatrixis77andforn2theHamiltonianmatrixis88.ThesummaryofHamiltonianmatricesfordierentnisgiveninTable 2{1 ThematrixHninEq. 2{81 isthesumofLandau,Zeeman,andexchangecontributions.TheexplicitformsfortheZeemanandexchangeHamiltonianmatricesaregiveninEq. 2{74 and 2{77 andareindependentofn. Table2{2: InAsbandparameters Energygap(eV)1Eg(T=30K)0.415Eg(T=77K)0.407Eg(T=290K)0.356Electroneectivemass(m0)me0.022Luttingerparameters1L120.0L28.5L39.2L7.53Spinorbitsplitting(eV)10.39Mnsdandpdexchangeenergies(eV)N00.5N01.0Opticalmatrixparameter(eV)1Ep21.5Refractiveindex2nr3.42 68 ]. 69 ]. 2{2 .ShowninFig. 2{3 aretheconductionbandLaudaulevelsforInAsandIn0:88Mn0:12Asasafunctionofmagneticeldatk=0foratemperatureof30K.Thedashedlinesrepresentspinuplevels,andthesolidlinesrepresentthespindownlevels.Thisillustratestheenergysplittingoftheconductionbandatthepoint.TherightpanelforInMnAsisonlydierentfromtheleftpanelfor
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Figure2{3: CalculatedLandaulevelsforInAs(left)andIn0:88Mn0:12As(right)asafunctionofmagneticeldat30K. InAsinthatithastheexchangecontributionsduetotheinteractionbetweenthebandelectronsandthelocalizedMnmoments.TheorderingoftheseLandaulevelscanbequalitativelyexplainedbythesimplemodelinEq. 1{13 wherewehaveananalyticalexpressionfortheLandaulevelenergy.Notethattheyarenotlinearfunctionsofthemagneticeld.Inthenextchapterwewillseethatthissimplemodelcannotpredictan(exchangeconstantdenedinEq. 2{76 )dependenceofthecyclotronenergy,whichistheenergydierencebetweentwoadjacentLandaulevelswiththesamespin.Theexchangeconstantdependenceisaconsequenceofkpmixingbetweenconductionandvalencebands. ThewavevectorkzdependenceofLandaulevelsinbothconductionbandandvalencebandsisshowninFig. 2{4 ,whereonlythevelowestorderLandaulevelsareshown.Becauseofthestrongstatemixing,thespinstatesinvalencebandsarenotindicated.ComparingtheleftandrightpanelsofFig. 2{3 andFig. 2{4 ,wecanseethatMndopingdrasticallychangestheelectronicstructure.Spinsplittingisgreatlyenhancedinbothconductionandvalencebands.Asamatteroffact,thespinstateorderingintheconductionbandisreversedwithMndoping.
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Figure2{4: TheconductionandvalencebandLandaulevelsalongkzinamagneticeldofB=20TatT=30K.TheleftandrightguresareforInAsandIn0:88Mn0:12As,respectively. whereh!spinisthespinsplitting.Rothetal.[ 70 ]havecalculatedthegfactorinsemiconductorsbasedonKane'smodel,andhaveshownthatthegfactorinthe
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conductionbandis 3Eg+2:(2{83) Usingthisequation,thegfactorforbulkInAsisabout15:1,whichisclosetotheexperimentalvalue15[ 71 ]. Figure2{5: ConductionbandgfactorsofIn1xMnxAsasfunctionsofmagneticeldwithdierentMncompositionx.Fortheleftgure,T=30Kandfortheright,T=290K.Noteathightemperatureswelosethespinsplitting. Duetotheexchangeinteraction,thespinsplittingisgreatlyenhanced.UsuallyinDMS,theexchangeenergyismuchbiggerthantheZeemanenergy,whichcanbeseenfromthesimpletheoryinEq. 1{15 forafewpercentofMndoping.Inthatcase,ifwetakex=0:1,N=0:5eV,andT=30K,thengeff256.Ifweonlyconsidertheexchangeinteraction,fromEq. 2{78 ,thespinsplittingintheconductionbandisexactlythatinEq. 1{13 .However,thisisnotcorrectbecausetherstconductionbandspindownlevelcomesfromthen=0manifold,whiletherstconductionbandspinuplevelcomesfromthen=1manifold.Dierentmanifoldnumbersresultindierentmatrixelements,whichwillcausedierentstatecoupling,andthusspinsplittingduetotheexchangeinteractionisnotwhatthesimplemodelpredicts.TheconductionbandgfactorsforInAsandInMnAsat30Kand290KareshowninFig. 2{5 .ThisclearlydemonstrateshowMndopingaectsthegfactors.At290K,theg
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Figure2{6: factorsaredrasticallyreduced.Thisisbecauseathightemperatures,thermaluctuationsbecomesolargethatthealignmentofthemagneticspinsislessfavorable.However,ifferromagneticDMSareemployed,duetotheinternalexchangeeld,astrongalignmentcanbeexpectedevenathightemperatures.NowwesupposeahighTCIn0:9Mn0:1AssysteminwhichaCurietemperatureof110Kisachieved.ThegfactorforthissystemisshowninFig. 2{6 .Evenatrelativelyhightemperature(stillbelowthetransitiontemperaturethough),biggfactorsarestillobtained.ThegfactorreachesinnityatzeroeldwhentemperaturesarebelowTCbecausethereisstillspinsplittingeventhoughthereisnoexternaleld.
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Inchapter2,asystematicmethodofcalculatingtheelectronicstructureofDMSwasdevelopedanddescribedindetailandappliedtothenarrowgapInMnAs.IthasbeenseenthatthebandstructureofDMSdependsstronglyonMndopingwhichinducestheexchangeinteraction.Thebandstructurealsodependsonthestrengthoftheappliedmagneticeld,ascanbeseenfromFig. 2{3 and 2{5 .Apartfromthetheoreticalcalculation,opticalexperimentsarealwaysgoodwaystodetecttheelectronicpropertiesofsemiconductors.Amongthesemethods,cyclotronresonance(CR)isanextensivelyusedandapowerfuldiagnostictoolforstudyingtheintersubbandopticalpropertiesandeectivemassesofcarriers.Cyclotronresonanceisahighfrequencytransportexperimentwithallthecomplicationswhichcharacterizetransportmeasurements.Throughcyclotronresonance,onecangettheeectivemasses,whicharedeterminedbythepeakofaresonanceline,whilescatteringinformationisobtainedfromthelinebroadening.CyclotronresonanceoccurswhenelectronsabsorbphotonsandmakeatransitionbetweentwoadjacentLandaulevels.Fromcyclotronresonancemeasurementsonecaninferthemagneticelddependentbandstructureofthematerial.SincethebandstructureofaDMSissosensitivetomagneticelds,thisisausefulmeanstostudyandobtainbandinformationfromacomparisonbetweentheexperimentalresultsandtheoreticalcalculations. 3.1.1OpticalAbsorption 56
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energyuxoftheincidentlightisS,thenthephotonuxdensityisS=h!,andwehaveTdx=Sdx=h!,i.e. wherei;farethelabelsfortheinitialandnalstates.Thesummationrunsoverallstates.Forabsorptionbetweenstateiandf,thetransitionprobabilityfromFermi'sgoldenrule[ 72 ]is, andforemission whereEiandEfaretheenergiesoftheinitialandnalstates(hereweonlywantthenalexpressionforabsorption,soinemission,eventheelectronstransitfromstateftostatei,westillcallstateiistheinitialstate,andstatefthenalstate),respectively,andthefunctionensurestheconservationofenergyintheopticaltransition.H0istheelectronphotoninteractionHamiltonian.Essentially,inopticaltransitions,momentumshouldalsobeconserved.However,sincethephotonmomentump=h=ismuchsmallerthanthetypicalelectronmomentum,wegenerallyconsidertheopticaltransitiontobe\vertical",whichmeansanelectroncanonlytransittostateswiththesamek,i.e.,weeignorethephotonmomentum. Insemiconductorswhendealingwiththerealisticcaseofabsorption,weneedtotakeintoaccountthestateoccupationprobabilitybyelectrons,whichinthermal
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equilibriumisdescribedbyaFermiDiracdistributionfunction 1+eEEF=kBT;(3{5) andsotherateofabsorptioninthewholecrystalcanbewrittenas andtheemissionrate DuetothehermitianpropertyofH0,jH0ifj=jH0fij.Thenetabsorptionrateperunitvolumethenis Whenasemiconductorisilluminatedbylight,theinteractionbetweenthephotonsandtheelectronsinthesemiconductorcanbedescribedbytheHamiltonian, 2m0(p+eA)2+V(r)(3{9) wherem0isthefreeelectronmass,eistheelectroncharge,Aisthevectorpotentialduetotheopticaleld,andV(r)isthecrystalperiodicpotential(inDMS,includingthevirtualcrystalexchangepotential).ThustheoneelectronHamiltonianwithouttheopticaleldis andtheopticalperturbationtermsare m0Ap+e2A2
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OpticaleldsaregenerallyveryweakandusuallyonlythetermlinearinAisconsidered,i.e.,wetreattheelectronphotoninteractioninalinearresponseregimeandneglecttwophotonabsorption.ThetransitionduetotheopticalperturbationinEq. 3{11 cantakeplaceeitheracrossthebandgaporinsideasingleband(conductionorvalenceband)dependingonthephotonenergy.Inthischapter,weonlyconsidercyclotronresonance,whichtakesplacebetweentheLandaulevelswithinconductionorvalencebands. Formonochromaticlightthevectorpotentialis whereKistheelectromagneticwavevector,!istheopticalangularfrequancy,pisthemomentumoperator,and^eistheunitpolarizationvectorinthedirectionoftheopticaleld,representingthelightconguration. TheenergyuxoftheopticaleldcanbeexpressedbythePoyntingvector,S=EH.UsingtherelationsE=@A=@t,H=rS=,and!=K=c=nr,theaveragedenergyuxthenis UsingthisrelationandEq. 3{8 ,theabsorptioncoecientthenis S=h! AccordingtoEq. 3{12 ,theinteractionHamiltoniancanbewrittenas sotheabsorptioncoecient 3{16 becomes
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Notethattheinteraction 3{15 isbasedonthedipoleapproximation.Sointhefollowingwhenwetalkaboutselectionrules,etc,theyareelectricdipoleselectionrules. Thescatteringbroadening(aswellasdisorder)canbeparameterizedbythelinewidththroughthereplacementofthefunctionbyaLorentzianfunction[ 72 ]as m0(3{18) wherem0isthefreeelectronmass(eectivemasswheninasemiconductor).Ifanelectromagneticwaveisappliedwiththesamefrequency,theelectronwillresonantlyabsorbthiselectromagneticwave. Quantummechanically,anelectroninamagneticeldwillhaveaquantizedmotion.ReferringtoEq. 2{62 ,theenergyoftheelectronsplitsintoaseriesofLandaulevels.Iftheenergyquantah!oftheappliedelectromagneticwaveareexactlythesameastheenergydierenceh!cbetweentwoadjacentLandaulevels,theelectronwillabsorbonephotontotransitfromthelowerLandauleveltothehigherone.Thisiscalledcyclotronresonance. Inthepresenceofamagneticeld,theHamiltonian 3{10 ,inDMSsystem,isreplacedbytheoneinEq. 2{79 .Wealreadyhavetheeigenstatesforthis
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Hamiltonian.Forconvenience,werewritethemhereas A2666666666666666666664a1;n;n1u1a2;n;n2u2a3;n;nu3a4;n;nu4a5;n;nu5a6;n;n+1u6a7;n;n1u7a8;n;n1u83777777777777777777775:(3{19) Theeigenfunctionabovecanbeconsideredasthelinearsuperpositionofeightbasisstates,eachofwhichiscomposedoftwoparts.nistheharmonicoscillatorenvelopefunction,whichisslowlyvaryingoverthelattice,andcanbeconsideredconstantoveraunitcelllengthscale.uiistheBlochpartofthewavefunction,whichvariesrapidlyoveraunitcellandhastheperiodicityofthelattice. NowletusinspectthepropertiesofthemomentummatrixelementinEq. 3{16 .Usingn;asthenewsetofquantumnumbers,andutilizingthespatialpropertiesofthewavefunctions,wecanfactorizetheintegralintotwopartsandwritethematrixelementas SincetheBlochfunctionsuiarequicklyvaryingfunctions,theirgradientsaremuchlargerthanthoseoftheenvelopesi.AsshowninRef.[ 73 ],thersttermontherighthandsidedominatesbothinnarrowgapandwidegapsemiconductors,sowehaveneglectedthesecondterminourcalculation.However,itiseasytocheckthatthesetwotermsobeythesameselectionrules.
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Wecanfactorize^epto^ep=^e+p+^ep++^ezpzwhere^e=(^xi^y)=p and Thismeansthatp+andpareraisingandloweringoperatorsfortheeigenstates.Forp+,anelectronwillabsorban^ephotontohaveann!n+1transition,whichusuallyhappensintheconductionbandforelectrons,sowecallthistransition\electronactive"(eactive).Forp,anelectronwillabsorban^e+photontohaveann!n1transition,whichusuallyhappensinthevalencebandsforholes,sowecallthistransition\holeactive"(hactive).ThequasiclassicalpictureforthetwotypesofabsorptionisshowninFig. 3{1 .Tocomplywithconservationof Figure3{1: Quasiclassicalpicturesofeactiveandhactivephotonabsorption. bothenergyandangularmomentum,inaquasiclassicalpicture,electronscanonlyabsorbphotonswitheactivepolarization,andholescanonlyabsorbphotonswithhactivepolarization.Inaquantummechanicaltreatment,wewillseethatthe
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truesituationismorecomplicatedthanthis.Inparticular,wendthateactiveabsorptioncanalsotakeplaceinptypematerials. Whenthetemperatureisnotzero,EFinEq. 3{5 shouldbeunderstoodasthechemicalpotential,whichwestillcalltheFermienergy,anddependsontemperatureanddoping.IfNDisthedonorconcentrationandNAtheacceptorconcentration,thenthenetdonorconcentrationNC=NDNAcanbeeitherpositiveornegativedependingonwhetherthesampleisnorptype.ForaxedtemperatureandFermilevel,thenetdonorconcentrationis (2)22Xn;Z1dkz[fn;(kz)vn;];(3{23) wherevn;=1ifthesubband(n;)isavalencebandandvanishesif(n;)isaconductionband.Giventhenetdonorconcentrationandthetemperature,theFermienergycanbefoundfromEq. 3{23 usingarootndingroutine. 74 75 ].ThemegagaussexperimentshavebeendoneattheuniversityofTokyowherehighmagneticeldscanbegeneratedusingtwokindsofpulsedmagnets:thesingleturncoiltechnique[ 76 77 ]andtheelectromagneticuxcompressionmethod[ 77 78 ].Thesingleturncoilmethodcangenerate250Twithoutanysampledamageandthusmeasurementscanberepeatedonthesamesampleunderthesameexperimentalconditions.Theideabehindthismethodistoreleaseabigcurrentinaveryshortperiodoftime(severals)tothesingleturncoiltogenerateanultrahighmagneticeld.ThecorepartofarealsingleturncoildeviceisdemonstratedinFig. 3{2 [ 76 ].Althoughthesampleisintact,thecoilisdamagedaftereachshot.AstandardcoilisshowninFig. 3{3
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Figure3{2: Thecorepartofthedevicebasedonsinglecoilmethod.Thecoilisplacedintheclampingmechanismasseeninthegure.Thedomedsteelcylindersoneachsideofthecoilaresupportsforthesampleholderswhichprotecttheconnectiontothesample(e.g.,thinwires,heliumpipes)againstthelateralblast. beforeandafterashot.Dependingonthecoildimension,eachshotgeneratesapulsedmagneticeldupto250Tinseverals.ThetimedependenceofthepulsedmagneticeldandofthecurrentowingthroughthecoilisshowninFig. 3{4 [ 76 ]. Forhighereldexperimentsanelectromagneticuxcompressionmethodisused.Itusestheimplosivemethodtocompresstheelectromagneticuxsoastogenerateultrahighmagneticeldsupto600T.ThetimedependenceofthepulsedmagneticeldandcurrentisshowninFig. 3{5 [ 77 ].Thisisadestructivemethodandthesampleaswellasthemagnetisdestroyedineachshot. 3.3.1ElectronCyclotronResonance
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Figure3{3: Astandardcoilbeforeandafterashot. Figure3{4: WaveformsofthemagneticeldBandthecurrentIinatypicalshotinsingleturncoildevice.
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Figure3{5: WaveformsofthemagneticeldBandthecurrentIinatypicaluxcompressiondevice. Intheconductionband,theLandausubbandsareusuallyalignedinsuchawaythatenergyascendswithquantumnumbern.Soforaneactivetransition,bothangularmomentumandenergyforanelectronphotonsystemcanbeconserved. OurcollaboratorsKonoetal.[ 74 ]measuredtheelectronactivecyclotronresonanceinInMnAslmswithdierentMnconcentrations.ThelmsweregrownbylowtemperaturemolecularbeamepitaxyonsemiinsulatingGaAssubstratesat200C.Allthesampleswerentypeanddidnotshowferromagnetismfortemperaturesaslowas1:5K.TheelectrondensitiesandmobilitiesdeducedfromHallmeasurementsarelistedinTable 3{1 ,togetherwiththeelectroncyclotronmassesobtainedataphotonenergyof117meV(orawavelengthof10:6m). TypicalmeasuredCRspectraat30Kand290KareshownintheleftandrightpanelofFig. 3{6 ,respectively.Notethattocomparethetransmissionwithabsorptioncalculations,thetransmissionincreasesinthenegativeydirection.EachgureshowsspectraforallfoursampleslabeledbythecorrespondingMncompositionsfrom0to12%.Allthesamplesshowpronouncedabsorption
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Table3{1: ParametersforsamplesusedineactiveCRexperiments Mncontentx00.0250.0500.120 Density(4.2K)1:010171:010160:910161:01016Density(290K)1:010172:110171:810177:01016Mobility(4.2K)400013001200450Mobility(290K)4000400375450m=m0(30K)0.03420:03030:02740:0263m=m0(290K)0.03410:03340:03250:0272 peaks(ortransmissiondips)andtheresonanceelddecreaseswithincreasingx.Increasingxfrom0to12%resultsina25%decreaseincyclotronmass(seeTable 3{1 ).Athightemperatures[e.g.,Fig. 3{6 (b)]thex=0sampleclearlyshowsnonparabolicityinducedCRspinsplittingwiththeweaker(stronger)peakoriginatingfromthelowestspindown(spinup)Landaulevel,whiletheotherthreesamplesdonotshowsuchsplitting.TheabsenceofsplittingintheMndopedsamplescanbeaccountedforbytheirlowmobilities(whichleadtosubstantialbroadening)andlargeeectivegfactorsinducedbytheMnions.Insampleswithlargex,onlythespindownlevelissubstantiallythermallypopulated(seeFig. 2{5 ). UsingtheHamiltoniandescribedinSection2.3.2,thewavefunctionsinSection2.3.3,andthetechniquesforcalculatingFermienergy,theseverallowestLandaulevelsintheconductionbandattwoMnconcentrationsandtheFermienergyfortwoelectrondensities(11016=cm3and11018=cm3)arecalculated.TheconductionbandLandaulevelsandtheFermienergiesareshowninFig. 3{7 asafunctionofmagneticeldatT=30K.Fromthesegures,wecanseethatatresonance,thedensitiesandeldsaresuchthatonlythelowestLandaulevelforeachspintypeisoccupiedfortypicaldensitieslistedinTable 3{1 .Thus,alltheelectronswereinthelowestLandaulevelforagivenspinevenatroomtemperatureduetothelargeLandausplitting,precludinganydensitydependentmassdue
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Figure3{6: ExperimentalelectronCRspectrafordierentMnconcentrationsxtakenat(a)30Kand(b)290K.Thewavelengthofthelaserwasxedat10:6mwitheactivecircularpolarizationwhilethemagneticeldBwasswept. tononparabolicity(expectedatzeroorlowmagneticelds)asthecauseoftheobservedtrend. ThecyclotronresonancetakesplacewhentheenergydierencebetweentwoLandaulevelswiththesamespinisidenticaltotheincidentphotonenergy.InFig. 3{8 ,wesimulatecyclotronresonanceexperimentsinntypeInAsforeactivecircularlypolarizedlightwithphotonenergyh!=0:117eV.WeassumeatemperatureT=30Kandacarrierconcentrationn=1016=cm3.ThelowerpanelofFig. 3{8 showsthefourlowestzonecenterLandauconductionsubbandenergiesandtheFermienergyasfunctionsoftheappliedmagneticeld.Thetransitionattheresonanceenergyh!=0:117eVisaspinupn=1transitionandisindicatedbytheverticalline.FromtheLandauleveldiagramtheresonancemagneticeldisfoundtobeB=34T.TheupperpanelofFig. 3{8 showstheresultingcyclotron
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Figure3{7: ZonecenterLandauconductionsubbandenergiesatT=30KasfunctionsofmagneticeldinndopedIn1xMnxAsfor=0andx=12%.Solidlinesarespinupanddashedlinesarespindownlevels.TheFermienergiesareshownasdottedlinesforn=1016=cm3andn=1018=cm3. resonanceabsorptionassumingaFWHMlinewidthof4meV.ThereisonlyoneresonancelineinthecyclotronabsorptionbecauseonlythegroundstateLandaulevelisoccupiedatlowelectrondensities.Forhigherelectrondensities,moreLandaulevelsareoccupied.Forexample,ifbothspinupandspindownstatesoftherstLandaulevelareoccupied,oneobtainsmultipleresonancepeaks. OursimulationoftheexperimentaleactivecyclotronresonanceintheconductionbandshowninFig. 3{6 isshowninFig. 3{9 .Theleftandrightpaneldemonstratethecalculatedcyclotronresonanceabsorptioncoecientforeactive
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Figure3{8: ElectronCRandthecorrespondingtransitions.TheupperpanelshowstheresonancepeakandthelowerpanelshowsthelowestfourLandaulevelswithspinupstatesindicatedbysolidlinesandspindownstatesindicatedbydashedlines.Verticalsolidlineinthelowerpanelindicatesthetransitionaccountablefortheresonance. circularlypolarized10:6mlightintheFaradaycongurationasafunctionofmagneticeldat30Kand290K,respectively.Inthecalculation,thecurveswerebroadenedbasedonthemobilitiesofthesamples.ThebroadeningusedforT=30Kwas4meVfor0%,40meVfor2:5%,40meVfor5%,and80meVfor12%.ForT=290K,thebroadeningusedwas4meVfor0%,80meVfor2:5%,80meVfor5%,and80meVfor12%.AtT=30K,weseeashiftintheCRpeakasafunctionofdopinginagreementwithFig. 3{6 (a).ForT=290K,weseethepresenceoftwopeaksinthepureInAssample.ThesecondpeakoriginatesfromthethermalpopulationofthelowestspindownLandaulevel.Thepeakdoes
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Figure3{9: CalculatedelectronCRabsorptionasafunctionofmagneticeldat30Kand290K.ThecurvesarecalculatedbasedongeneralizedPidgeonBrownmodelandFermi'sgoldenruleforabsorption.TheyarebroadenedbasedonthemobilitiesreportedinTable 3{1 notshiftasmuchwithdopingasitdidatlowtemperature.ThisresultsfromthetemperaturedependenceoftheaverageMnspin.WebelievethattheBrillouinfunctionusedforcalculatingtheaverageMnspinbecomesinadequateatlargexand/orhightemperatureduetoitsneglectofMnMninteractionssuchaspairingandclustering. TheeactiveCRshowsashiftwithincreasingMnconcentration.FromthesimpletheoryinSection1.2.2,thecyclotronresonanceelddoesnotdependonxandbecausetheexchangeinteractionwillshiftalllevelsbythesameamount.Thisshiftcomesfromthecomplicatedconductionvalencebandmixing,anddependsonthevalueof()[ 79 ].Wecanqualitativelyexplainthisshiftusingthecyclotronmass,whichwillbediscussedinthefollowingsubsection.
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TheCRpeaksshowninFig. 3{9 arehighlyasymmetric.Thisisbecausewehavetakenintoaccountthenitekzeectinourcalculation,andtheenergydispersionalongkzshowshighnonparabolicity.Also,thecarrierllingeectduetotheFermienergysharpeningwillalsocontributetotheCRpeakasymmetry. ThisequationcanbederivedfromEq. 3{18 ifwesetmagneticeldBsothath!c=h!,whichisthecyclotronresonancecondition. ThecalculatedcyclotronmassesforthelowestspindownandspinuptransitionsareplottedinFig. 3{10 asafunctionofMnconcentrationxataphotonenergyofh!=0:117eV.Cyclotronmassesarecomputedforseveralsetsofandvalues.ThecyclotronmassesinFig. 3{10 (a)and(b)correspondtothecomputedcyclotronabsorptionspectrashowninFig. 3{9 (a)and(b),respectively.Inourmodel,theelectroncyclotronmassesdependontheLandausubbandenergiesandphotonenergiesandareindependentofelectronconcentration. Figure 3{10 clearlyshowsthatthecyclotronmassdependsonbothexchangeconstantsandx.Withincreasingx,spindown(spinup)cyclotronmassshowalmostalineardecrease(increase).Thecyclotronmassdoesnotdependononesingleexchangeconstant,itdependsonbothexchangeconstants.Investigationofthemassdependenceonthesetwoconstantsrevealsthemassshifthasacloserelationwiththeabsolutevalueof()[ 79 ].Thisshiftallowsusetomeasuretheexchangeinteraction. Thecalculatedcyclotronmasshastakenintoaccountalltheenergydependenceonnonparabolicityduetotheconductionvalencebandmixing,theexchange
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Figure3{10: CalculatedelectroncyclotronmassesforthelowestlyingspinupandspindownLandautransitionsinntypeIn1xMnxAswithphotonenergy0:117eVasafunctionofMnconcentrationforT=30KandT=290K.Electroncyclotronmassesareshownforthreesetsofandvalues.
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interactionconstantsand,andtheMncontentx.TheshiftoftheresonancepeakstolowereldswithincreasingMncontentxisnaturallyexplainedbythedecreaseofthespindowncyclotronmass.Duetothesmallerdownwardslopeinthespindowncyclotronmassat290Kascomparedto30K,theresonancepeakshiftat290Kisseentobelesspronouncedthanat30K. 3.4.1HoleActiveCyclotronResonance 2{4 ,theDMSvalencebandstructureismuchmorecomplicatedthantheconductionbandstructure.Duetotheirenergeticproximity,heavyholeandlightholebandsarestronglymixedevennearthepoint.Thesplitobandalsocontributesstronglytothevalencebandedgewavefunctions.Inamagneticeld,theseholebandssplitintotheirownLandaulevels,butopticaltransitionscanhappenbetweenanytwolevelsifbothangularmomentumandenergyareconserved.Asintheconductionband,cyclotronresonancerequiresconductionvalencebandmixingtoproducestrongenoughoscillatorstrength.Interbandmixingacrossthebandgapissmallinwidegapsemiconductors,soitismorediculttoobservecyclotronresonanceinthesesemiconductors.Asamatteroffact,nocyclotronresonancehasbeenreportedtodateinGaMnAs. InAsandInMnAsarenarrowgapsemiconductors.Ourcollaborators[ 80 81 82 ]haveperformedcyclotronresonanceexperimentsonpdopedInAsandInMnAsatultrahighmagneticeldsupto500T.ThetypicalhactiveCRabsorptionofInAsbelow150TisshowninFig. 3{11 ,inwhichtheincidentlightishactivecircularlypolarizedwithphotonenergy0:117eV.Twopeaksarepresentintheexperimentalobservation,onearound40T,andanotheraround125T.Atevenlowerelds,thereisabackgroundabsorption.Thetheoreticalsimulationusingaholedensityof11019=cm3andabroadeningfactorof40meVisalsodisplayedinFig. 3{11 forcomparison.
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Figure3{11: HolecyclotronabsorptionasafunctionofmagneticeldinptypeInAsforhactivecircularlypolarizedlightwithphotonenergy0:117eV.Theuppercurveisexperimentallyobservedresultandtheloweroneisfromtheoreticalcalculation. Inourmodel,wearecapableofcalculatingtheabsorptionbetweenanytwoLandaulevels.Detailedcalculationrevealsthatthepeakatlowereldsisduetotheheavyholetoheavyholetransition,andthepeakathighereldsisfromthelightholetolightholetransition.WenowuseHn;tospecifytheheavyholelevel,andLn;tospecifythelightholelevel,where(n;)arethequantumnumbersdenedinEq. 3{19 .BecauseofstrongwavemixingHorLonlylabelsthezonecenter(k=0)characterofaLandaulevel.Usingtheselabels,weillustratethetwostateabsorptioninFig. 3{12 alongwiththeLandaulevelstructureasafunctionofmagneticeld. ItisseenfromFig. 3{12 thattheholesopticallyexcitedfromtheheavyholesubbandH1;1andlightholesubbandL0;3giverisetothetwostrongcyclotronabsorptionpeaksshowninFig. 3{11 .Thecyclotronabsorptionpeakaround40TisduetoatransitionbetweenthespindowngroundstateheavyholeLandaulevelH1;1,andheavyholeLandaulevelH0;2,whichnearthezonecenterisprimarily
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Figure3{12: CalculatedcyclotronabsorptiononlyfromtheH1;1H0;2andL0;3L1;4transitionsbroadenedwith40meV(a),andzonecenterLandaulevelsresponsibleforthetransitions(b). spindown.Theotherabsorptionpeakaround140T,isaspindownlightholetransitionbetweenL0;3andL1;4Landaulevels.ThebackgroundabsorptionatB<30TisduetotheabsorptionbetweenhigherLandaulevelswhichalsobecomeoccupiedbyholesatlowerelds. CyclotronresonanceabsorptionmeasurementsonIn1xMnxAswithx=2:5%havealsobeenperformed.TheyareshowninFig. 3{13 alongwithourtheoreticalsimulation.TheCRmeasurementsweremadeattemperaturesof17,46,and70Kinhactivecircularlypolarizedlightwithphotonenergyh!=0:224eV.Inoursimulation,theholedensityistakenas51018=cm3,andthecurvesare
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Figure3{13: ExperimentalholeCRandcorrespondingtheoreticalsimulations.ThelowtemperatureCRhasanabruptcutoatloweldsduetothefermilevelsharpeningeect. broadenedusingaFWHMlinewidthof120meV.ClearlytheabsorptionpeakisduetotheheavyholetransitionwhichwehaveseeninFig. 3{11 andFig. 3{12 .Duetothehigherphotonenergy,thispeakshiftsfromaround40Ttoaround85T.Theresonanceeldisinsensitivetotemperatureandthelineshapeisstronglyasymmetricwithabroadtailatlowelds.Thisbroadtailagaincomesfromthehigherordertransitionsresonantatlowelds.Weseethatinbothexperimentandtheoryatlowtemperatureandloweld,thereisasharpcutoof
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theabsorption.ThiscanbeattributedtothesharpnessoftheFermidistributionatlowtemperatures. Figure 3{14 showstheobservedCRpeaksasafunctionofmagneticeld.Theyaxisindicatesthephotonenergiesusedwhenobservingthecyclotronresonance.Thesolidcurvesshowthecalculatedresonancepositions.Thecurvelabeled`HH'(`LH')isjusttheresonanceenergybetweenLandaulevelsH1;1(L0;3)andH0;2(L1;4).Thetheoreticalcalculationshowsanoverallconsistencywiththeexperiments. Figure3{14: ObservedholeCRpeakpositionsforfoursampleswithdierentMnconcentrations.Thesolidcurvesaretheoreticalcalculations. Therearetwofactorsinourcalculationthataecttheresults.OneistheselectionofLuttingerparameters,theotheristhelimitationoftheeightbandeectivemasstheoryitself.InFig. 3{11 ,thetheoreticallycomputedpeakathighereldsdoesnotttheexperimentalpeakexactly.Duetothefactthatthistransitiontakesplaceatthezonecenter,wherethekptheoryshouldbeveryaccurate,thisdeviationmaybetheresultofunoptimizedLuttingerparameters.
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TheempiricalparametersusedintheeectivemassHamiltoniancandrasticallychangethevalencebandstructureandtheresultingCRabsorptionspectra.Fig. 3{15 showsthedependenceoftheCRenergiesonseveralparameterssuchastheLuttingerparameters1,2,3,Kane'sparameterEpandtheeectiveelectronmassm.Thisgurerevealsthatthe`LH'transitionsareaectedmorebysmallvariationsintheseparametersthanthe`HH'transitions.Forinstance,a10%changein1willresultina0:025eVchangeatB=140TintheLHCRenergy,whichinturnwillresultinabouta50TCRpositionshiftintheresonanceeldwhenthephotonenergyis0:117eV.TheMndopingontheotherhandgenerallyenhancestheCRenergydependenceontheseparameters,whichcanbeseenfromcomparingthetwographsinFig. 3{15 Figure3{15: Thedependenceofcyclotronenergiesonseveralparameters.LeftpanelshowstheheavyholeCRenergydependence,andtherightpanelshowsthelightholeCRenergydependence. Figure 3{16 illustrateshowtheCRabsorptiondependsonthreeLuttingerparameterswhilekeepingalltheotherparametersunchanged.Itcanbeseenthat
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theCRspectraquitesensitivelydependsonthevaluesoftheLuttingerparameters,providinganeectivewaytomeasuretheseparametersthroughcomparisonwithexperiments. Figure3{16: HoleCRspectraofInAsusingdierentsetsofLuttingerparameters.LightholetransitionismoresignicantlyaectedbychangeoftheLuttingerparameters. InFig. 3{14 ,thereisonepeakaround450Tlabeledas`C'whenthelightenergyish!=0:117eV.Toaccountforthispeak,CRabsorptionspectraupto500Thavebeencomputed.Thek=0Landaulevelsasafunctionofmagneticeld,alongwiththeCRspectraareplottedinFig. 3{17 ,inwhichwecanseethatthispeakisduetothesuperpositionoftwotransitions:L1;5L2;5andH2;6H3;6.However,thecalculatedpeakpositionisaround360T,dierentfromtheexperiment.Therearetwopossiblereasonsforthisbigdeviation.Oneisthatatveryhighmagneticelds,theeightbandPidgeonBrownmodelmaybreakdown;theotheristhattransitionscontributingtothepeaktakeplaceawayfromthezonecenterwhereeightbandkptheoryisnotadequatetodescribetheenergydispersion.ThebandstructurealongkzisplottedatFig. 3{18 ,whereweseethattheLandaulevelsH2;6andH3;6bothhavecamelbackstructures.Ataholedensityp=11019=cm3,thezonecenterpartofH2;6isnotoccupied.The
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Figure3{17: CalculatedLandaulevelsandholeCRinmagneticeldsupto500T.Theupperpanelshowsthek=0valencebandLandaulevelsasafunctionofmagneticeldandtheFermilevelforp=1019cm3(dashedline).TheholeCRabsorptioninptypeInAsisshowninthelowerpanelforhactivelypolarizedlightwithh!=0:117eVatT=20Kandp=1019cm3.AFWHMlinewidthof4meVisassumed. lowestenergyforthisheavyholeLandaulevelresidesataboutkz=0:75(1=nm).Checkingthetransitionelementalongkz,itisalsofoundthatthistransitionindeedtakesplaceawayfromthezonecenter.DisplayedinFig. 3{19 isthecomparisonoftheeightbandmodelversusafullzonethirtybandmodel.Atthezonecenter,theeightbandmodeltswellwiththethirtybandmodel.Notfarawayfromthezonecenter,abigdeviationoccurs.Wethinkthisdeviationoftheenergydispersionispossiblyresponsibleforthelargedeviationofthecalculatedresonancepeakposition.
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Figure3{18: kdependentLandausubbandstructureatB=350T. Figure3{19: BandstructurenearthepointforInAscalculatedbyeightbandmodelandfullzonethirtybandmodel.
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3{16 ,thusCRspectradependstronglyoncarrierdensities.InFig. 3{12 theholedensityis11019cm3.Atsuchaholedensity,theFermienergyisbelowtheH1;1andL0;3statessothatwehavetwostrongtransitions.Iftheholedensityislower,theFermienergywillshiftupward,thusthesetwostateswillbecomelessoccupiedbyholes,andwecanexpectadecreaseintheCRstrength.However,thedecreaseinstrengthforthetworesonancepeaksisdierent.ShowninFig. 3{20 (a)aretheCRspectraforfourdierentholedensities.TheLandaulevelsalongwiththecorrespondingFermienergiesareplottedinFig. 3{20 (b).Resonanttransitionsat0:117eVareindicatedbyverticallines.WecanseethattheCRpeak2isalmostalwayspresent,becauseatlowmagneticelds,theheavyholestateH1;1isalmostalwaysoccupied.TheCRpeak1changesdramaticallywithholedensity,andnearlyvanishesatp=51018cm3.TherelativestrengthsoftheheavyandlightholeCRpeaksissensitivetotheitinerantholedensityandcanbeusedtodeterminetheholedensity.BycomparingtheoreticalandexperimentalcurvesinFig. 3{20 (a),weseethattheitinerantholeconcentrationisaround21019cm3.FromFig. 3{20 (a),wecanruleoutp<1019cm3andn>41019cm3.Weestimatethatanerrorintheholedensityofaround25%shouldbeachievableatthesedensities.BecauseoftheexistenceinIIIVDMSoftheanomalousHalleect,whichcanoftenmakethedeterminationofcarrierdensitydicult,determiningcarrierdensitybycyclotronresonancecanserveasapossiblealternative. 83 ].ThesesamplesareferromagneticwithTCrangingfrom30to55KandwhosecharacteristicsaresummarizedinTable 3{2
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Figure3{20: TheholedensitydependenceofholeCR.(a)TheoreticalholeCRcurvesinInAsfrombottomtotopwithholedensitiesof51018,1019,21019and41019cm3;(b)LandaulevelsinvolvedinobservedCRalongwithFermilevelscorrespondingtotheoreticalcurvesin(a).
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Table3{2: CharacteristicsoftwoInMnAs/GaSbheterostructuresamples SampleNo.TC(K)MncontentxThickness(nm)Density(cm3) 1550.09251:110192300.1294:81019 3{21 (a)and(b),atvarioustemperaturesasafunctionofmagneticeld.Thelaserbeamwasholeactivecircularlypolarized.IntheleftpanelofFig. 3{21 ,fromroomtemperaturedowntoslightlyaboveTC,abroadresonancefeature(labeled`A')isobservedwithalmostnochangeinintensity,position,andwidthwithdecreasingtemperature.ClosetoTC,quiteabruptanddramaticchangestakeplaceinthespectra.First,asignicantreductioninlinewidthandasuddenshifttoalowermagneticeldoccursimultaneously.Also,theresonancerapidlyincreasesinintensitywithdecreasingtemperature.Inaddition,asecondfeature(labeled`B')suddenlyappearsaround125T,whichalsorapidlygrowsinintensitywithdecreasingtemperatureandsaturates,similartofeatureA.Atlowtemperatures,bothfeaturesAandBdonotshowanyshiftinposition.Essentially,thesamebehaviorisseenintherightpanelinFig. 3{21 .Usingdierentwavelengthsoftheincidentlight,similarCRspectrumbehaviorhasalsobeenobserved. Forzincblendesemiconductors,theCRpeaksAandBareduetothetransitionsofH1;1!H0;2andL0;3!L1;4,respectively,whichwehavealreadypointedout.WeattributethetemperaturedependentpeakshifttotheincreaseinthecarrierMnionexchangeinteractionresultingfromtheincreaseofmagneticorderingatlowtemperatures.ThetheoreticallycalculatedresultsareshowninFig. 3{22 forbulkIn0:91Mn0:09As.TheCRspectrawasbroadenedusingaFWHMlinewidthof4meV.ThetheoreticalresultsclearlyshowashiftofpeakAtolower
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Figure3{21: CyclotronresonancespectrafortwoferromagneticInMnAs/GaSbsamples.Thetransmissionofholeactivecircularpolarized10:6mradiationisplottedvs.magneticeldatdierenttemperatures. eldswithdecreasingtemperature,althoughinbulkInAs,thetransitionoccursatabout40T,asopposedtotheheterostructurewheretheresonanceoccursat50T. TheCRpeakAonlyinvolvesthelowesttwoLandaumanifolds.AswasdiscussedinSection2.3.3,whenn=1,theHamiltonianis11,andwhenn=0,theHamiltonianfactorizesintotwo22matrices,soitiseasytoobtainanexactanalyticalexpressionforthetemperaturedependentcyclotronenergy.Withneglectofthesmalltermsarisingfromtheremotebandcontributions,thecyclotronenergy
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Figure3{22: TheoreticalCRspectrashowingtheshiftofpeakAwithtemperature. fortheH1;1!H0;2transitionis 4xhSzi()s Eg 4xhSzi()2+EpBB:(3{25) Intheeldrangeofinterest(40T),p 3{25 ,weobtainanexpressionoftheform 4xhSzi()(1)(3{26) where IfweassumethetemperaturedependenceofEgandEpissmall,itfollowsfromEq. 3{26 thattheCRpeakshiftshouldfollowthetemperaturedependenceofthemagnetizationhSzi,whichinameaneldtheoryframeworkisgivenby
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Figure3{23: AveragelocalizedspinasafunctionoftemperatureatB=0,20,40,60and100Tesla.TheCurietemperatureisassumedtobe55K. Eq. 1{22 .ThetemperaturedependenceofhSziisshowninFig. 3{23 atseveralmagneticelds.ThemagneticelddependenceofhSzihasalreadybeenshowninFig. 1{10 TherelativechangeoftheCRenergy,calculatedusingEq. 3{25 and 3{26 ,asafunctionoftemperatureispresentedinFig. 3{24 .Itshowsthatfromroomtemperatureto30Kthecyclotronenergyincreasesabout20%,whichcorrespondstoanapproximately20%decreaseintheresonantmagneticeld,approximatelytheresultobservedintheexperiment.Inaddition,wefoundthattheshiftisnonlinearintemperatureandthemainshiftoccursattemperatureswellaboveTC.Thesefeaturesarealsoconsistentwithexperiment. AlongwiththeCRpeakshift,experimentindicatesasignicantnarrowingofthelinewidth.Wespeculatethatthiseectmaybeassociatedwiththesuppressionoflocalizedspinuctuationsatlowtemperatures.AsimilareecthasbeenobservedinIIVIdilutemagneticsemiconductors(seeRef.[ 84 ]andreferencestherein).Spinuctuationsbecomeimportantwhenacarrierinthebandinteractssimultaneouslywithalimitednumberoflocalizedspins.Thistakesplace,for
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Figure3{24: RelativechangeofCRenergy(withrespecttothatofhightemperaturelimit)asafunctionoftemperature.VerticaldashedlineindicatesTC. example,inmagneticpolaronsandforelectronsindilutemagneticsemiconductorquantumdots.Thestronginplanelocalizationbythemagneticeldmayalsoresultinareductionofthenumberofspinswhichacarrierinthebandfeels,thusincreasingtheroleofspinuctuations.However,itispossiblethattheCRpeaknarrowingistheresultoftheincreasedcarriermobilities.AlthoughtheInMnAslayerisheavilydopedandthustheholemobilityisverylow,holesintheGaSblayer,iftheyexist,willhavemuchhighermobilities.Sowecanspeculatethatnearthetransitiontemperature,thebandstructureofInMnAschangesinsuchawaythatafractionoftheholesmoveintotheGaSblayeror/andtheInMnAs/GaSbinterface,wheretheholeCRhasamuchnarrowerlinewidth.ShowninFig. 3{25 isthebanddiagramoftheInMnAs/GaSbheterostructure[ 46 ].TheinterfacestatesofInMnAs/GaSbareverycomplicated,andwehavenotcarriedoutcalculationsincorporatingtheminourCRsimulations.
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Figure3{25: BanddiagramofInMnAs/GaSbheterostructure. Thesituationinrealsemiconductors,however,diersfromthatinaclassicalfreeelectrongas.Matsudaetal.[ 82 ]haveexperimentallyobservedeactiveCRinpdopedInAsandInMnAs.Thetemperaturewasquitelow(12K)andtheholeconcentrationwashighenough(1019cm3)tosafelyeliminatethepossibilitythattheeactiveCRcomesfromthethermallyexcitedelectronsintheconductionband.Thepossibilityoftheexistenceofelectronsintheinterfaceorsurfaceinversionlayershasbeenalsoexcluded.Thus,theresultssuggestthateactiveCRcomesfromthevalencebandholes,incontradictionwiththesimplepictureofafreeholegas. Wendthateactivecyclotronresonanceinthevalencebandsisanintrinsicpropertyofcubicsemiconductorsandresultsfromthedegeneracyofthevalencebands.Aswediscussedbefore,heavyholeandlightholebandswillbothsplitinto
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aseriesofLandaulevels.ThiscomplexityallowsonetosatisfyconservationofangularmomentuminCRabsorptionforbothe+andepolarization,providedoneswitchesbandtype. Intheconductionband,increasingthemanifoldquantumnumberalwaysincreasestheenergy.Asaresult,onlytransitionswithincreasingnmaytakeplaceinabsorption,thatis,onlyeactive(e)CRcanbeobservedintheconductionband. Figure3{26: SchematicdiagramofLandaulevelsandcyclotronresonancetransitionsinconductionandvalencebands.Bothhactiveandeactivetransitionsareallowedinthevalencebandbecauseofthedegeneratevalencebandstructure.Onlyeactivetransitionsareallowedintheconductionband. Thevalenceband,however,consistsoftwotypesofcarriers:heavyholes(J=3=2;Mj=3=2)andlightholes(J=3=2;Mj=1=2).EachofthemhastheirownLandauladderinthemagneticeld.Anincreaseofnalwaysdecreasestheenergyonlywithineachladder.Similartotheconductionbandcase,transitionswithinaladder(HH!HHorLH!LH)cantakeplaceonlyinhactive(e+)polarization.However,therelativepositionofthetwoladderscanbesuchthatinterladdertransitions(LH!HH)ineactivepolarizationareallowed.
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ThisprocessisschematicallyshowninFig. 3{26 .Notethatthisgureisextremelysimpliedandshouldbeusedonlyasaqualitativeexplanationoftheeect. Figure3{27: ThevalencebandLandaulevelsandeactiveholeCR.(a)ThelowestthreepairsofLandaulevelsintheeactivetransition;(b)TheseparateCRabsorptioncontributingtotheeactiveCR. We'veexaminedeactiveCRinptypeInAsatT=12Kwithafreeholedensityof11019=cm3.Thecomputedk=0valencebandenergiesasafunctionofmagneticeld,theeactiveopticaltransitionsandthecorrespondingCRabsorptionspectraareshowninFig. 3{27 .ThemostpronouncedeactivetransitionstakeplacebetweentheHHstateH0;2andLHstateL1;5andbetweentheH1;3andL2;6states.Therearesomeotherlesspronouncedtransitions,whichcontributetotheabsorptionspectraatlowerelds.
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ThecalculatedandexperimentalCRareshowninFig. 3{28 forbotheactiveandhactivepolarizations.Thereisgoodagreementbetweentheoryandexperiment.Asdiscussedabove,theelectronactiveabsorptionisdeterminedbytheHH!LHtransitions.Themaincontributiontothehactiveabsorption(leftpanelinFig. 3{28 )comesfromthetransitionswithintheheavyholeladder,whichwehavealreadydiscussedindetailinthelastsection. ThecalculationandobservationofeactiveCRinptypeDMScanaidinunderstandingthevalencebandstructureofDMSsystems.UsingbothhandeactiveCRonecanexplorethewholepictureofthevalencebands. Figure3{28: ExperimentalandtheoreticalholeCRabsorption.SolidlinesareexperimentalholeCRspectraasafunctionofmagneticeldforhactiveandeactivepolarizations.Correspondingtheoreticalcalculationsareshownindashedlines.
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Inthelastchapter,itwasseenthattheMndopinggreatlyaectsthevalencesubbandalignmentandthecyclotronresonance.Ontheonehand,dopingwithMnimpuritieswillgreatlyenhancethescatteringofcarriersthusincreasingthelinewidthandreducingthestrengthoftheCRspectra.Ontheotherhand,itwillalsoshifttheCRpeakpositionsduetothechangesinthevalencebandstructure. Figure3{29: ValencebandstructureatT=30KandB=100TforIn1xMnxAsalloyshavingx=0%andx=5%.Forx=0%,therstHHstate,H1;1,liesbelowthelightholestateL1;5.Forx=5%,theorderofthesetwostatesisreversed.TwopossibleCRtransitionsareshownusingupwardarrows,namelyanhactive()transitionbetweenH0;2andH1;1,andaneactive(+)transitionfromH0;2toLH1;5.ThedashedlinesaretheFermienergiesforaholedensityof1019cm3. WeillustratehowMndopingaectstheopticaltransitionsinFig. 3{29 .Inthiscase,weassumeacarrierdensityof1019cm3,withoutandwithMndopingat30K,inamagneticledof100T.Theprimaryhactiveandeactivetransitionsarebothindicatedinthesegures.Onlythreelevelsareinvolvedinbothhandeactivetransitions.ThehactivetransitionisfromtheH1;1toH0;2stateaswementionedabove.TheeactivetransitionisfromtheL1;5toH0;2state.WeseethatwithoutMndoping,theL1;5statesitsonthetop,butwithdoping,H1;1stateisshiftedtothetopwhileL1;5isshiftedtoalowerposition.Thusbothhactiveandeactiveabsorptionwillbeaected.
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LineshapesandpeakpositionsareverysensitivetoMndoping.Notethatabsorptiontakesplacenotonlyatthepoint,butalsoinregionsawayfromthezonecenter.EventhoughwebroadentheCRlineswiththesamelinewidth(4meV),thedopedsamplehasamuchbroaderlineshapeduetotheenergydispersionchangealongkzandtheenergypositionchangerelativetotheFermilevelbroughtaboutbythetheexchangeinteraction.ThisisshowninFig. 3{30 .Furthermore,theheightofthepeakoftheCRspectrumoftheMndopedsampleisreducedbyabout30timescomparedtotheundopedone.ThismaycomefromtheFermillingeect,sincetheL1;5statebecomelessoccupiedwhenthesampleisdopedwithMnionsatacarrierdensityof1019cm3. Figure3{30: TheprimarytransitionintheeactiveholeCRunderdierentMndoping.MndopingchangestheLandaulevelalignmentandthetransitionstrengthaswell.ItalsoshiftstheCRpeakpositiontoalowereld.
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Inmagnetoopticalexperiments,transmissionofthesampleisusuallymeasured,sincetransmissionmeasuresabsorptioninsidethesample,andinmostcases,absorptionprobestheintrinsicelectronicandtransportproperties.However,whenasampleistoothick,directmeasurementoftransmissionisimpossible.Inthiscase,onecanmeasurethereectionofthesample,andusetherelationsbetweentheopticalconstantstoderivetheabsorptioncoecient,thusobtainingtheintrinsicpropertiesthroughquantitativeanalysis. ThemagnetoopticalKerreect(MOKE)isrelatedtolightreection.Whenlinearlypolarizedlightisreectedbythesurfaceofaferromagneticsample,thepolarizationplanewillundergoarotation.Similarly,thereisalsotheFaradayeect,whichisrelatedtotherotationofthepolarizationplaneofthetransmittedlight.Magnetoopticaleectsmaybeobservedinnonmagneticmediasuchasglasswhenamagneticeldisapplied.However,theintrinsiceectsareusuallysmallinsuchcases.Inmagneticmedia(ferromagneticorferrimagnetic)theeectsaremuchlarger.Forcubiccrystals,whenthereisnoferromagnetism,noMOKEsignalwillbepresent,sowhenstudyingthedynamicalmagneticpropertiesofDMS,MOKEcanserveasapowerfultoolfordetectionandmeasurementofmagneticmomentsortimedependentmagneticmoments,withtimeresolvedoptics[ 46 ]. 96
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theotheropticalvariablesandtheabsorptioncoecient,especiallytherelationbetweenthereectionandabsorptioncoecients. Whenanelectromagneticwaveispropagatinginamediumwithamagneticrelativepermittivityr,andelectronicrelativepermittivityr,itsatisesMaxwellequations, @t; 0r: Whentherearenofreecharges,Eq. 4{4 becomes TakingthecurlofEq. 4{1 ,andusingtherelation Eq. 4{5 becomes Foraplanewavepropagatinginzdirection, and 0c2:(4{9) Ingeneral,thewavevectorcanbewrittenasacomplexnumber cN;(4{10)
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whereNisthecomplexrefractiveindexgiveby IfwewriteN=n+i,wherenandaretherealandimaginarypartofN,thenk=n! c+i! c.SubstitutingthisintoEq. 4{8 ,weobtain wherek0=k=Nisthewavevectorinvacuum.ThuswecanseethattheimaginarypartofN,,isrelatedtolightabsorption,i.e.theextinctioncoecient.ThelightintensityisproportionaltojEj2,sowecanwritetheintensityas cz:(4{13) Thustheabsorptioncoecientis c:(4{14) FromEq. 4{9 ,ifthereisnoenergyloss,then cp isreal.Butsincetherearelosses,wewrite cp wherethecomplexdielectricconstanterisdenedas 0!=1+i2:(4{17) where1and2aretherealandimaginarypartsofer. ComparingEq. 4{10 and 4{16 ,wehave
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Generally,risverycloseto1,insuchacase,therelationshipsbetweentherealandimaginarypartsofNand~rare and Thus,wecancalculatenandfrom1and2,andviceversa.Intheweaklyabsorbingcase,i.e.n,Eq.( 4{20 )canbesimpliedto FromEq. 4{14 ,therelationbetweentheabsorptioncoecientanddielectricconstantis Manymeasurementsofopticalpropertiesinsolidsinvolvenormallyincidentreectivity.Insidethesolid,thewavewillbeattenuated.Weassumeforthepresentdiscussionthatthesolidisthickenoughsothatreectionsfromthebacksurfacecanbeneglected. Considerthereectionofaplanewavemovinginthezdirection.Theinterfacebetweenahalfinnitemedium1withrefractiveindexN1andahalfinnitemedium2witharefractiveindexN2istakentobez=0.ThissituationisillustratedinFig. 4{1 .AssumingE==x,wehaveanincomingandreectedwavein
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Figure4{1: Diagramforlightreectionfromtheinterfacebetweenmedium1withrefractiveindexN1andmedium2withrefractiveindexN2. medium1 Inmedium2,thetransmittedwaveis ContinuityofelectriceldattheinterfacerequiresE0=E1+E2.WithEinthexdirection,thesecondrelationbetweenE0,E1andE2followsfromthecontinuityofthetangentialmagneticeldHyacrosstheinterface.FromEq. 4{1 ,wehave ThecontinuityconditiononHythusyieldsacontinuityrelationfor@Ex=@zsothatfromEq. 4{26 weobtain
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ThenormallyincidenctreectivityRis andthereectioncoecientrisgivenby whereN1,N2andrareallcomplexvariables.AccordingtoEq. 4{14 ,absorptionmeasurementscanbeusedtodeterminethereectioncoecient. Usually,therealandtheimaginarypartsoftheopticalconstantsarenotindependent.TheyarerelatedbytheKramersKronigrelation[ 85 ].Forexample,forN=n+i, orusingEq. 4{14 }Z10(!) where}Z10istheprincipalvalueoftheintegral.
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Figure4{2: Schematicdiagramformagneticcirculardichroism. fourfolddegenerate.Withspontaneousmagnetizationwhichproducesaselfconsistenteectivemagneticeld,theconductionandvalencebandstateswillsplit(seealso,Fig. 2{1 ).Thustheabsorptionfor+andpolarizationwillbedierent.AverysimpleschematicdiagramforthiseectisillustratedinFig. 4{2 ,inwhichweonlyshowthetwoheavyholevalencebands.Correspondingtothetwocircularpolarizations,wedenetwocomplexrefractiveindicesN+andN,whereN+=n++i+,andN=n+i.Therealpartsoftherefractiveindicesdonothaveastrongpolarizationdependence,sowesetn+=n=n. Attheinterface,thereectioncoecientfor+polarization,followingEq. 4{29 ,is andfor, Duetodierencebetween+and,r+andrhavedierentphasefactors.Thatmeansafterreection,+andlightwillhavedierentphases. Nowconsiderthecasewherelinearlypolarizedlightpropagatesnormaltothesurfaceofaferromagneticcrystalandisreectedbythesurface,asillustratedin
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Fig 4{3 (a).Linearlypolarizedlightcanbedecomposedinto+andcomponents.Fromthediscussionabove,thesetwocircularlypolarizedbeamswillhavedierentphasechangesafterreection,sothereectedlight,ifitisstilllinearlypolarized,willnotstayinthesamepolarizationplane.ThepolarizationplanewillberotatedandthisiscalledKerrrotation.Duetothedierencesinabsorptionofthetwocircularlypolarizedbeams,thereectedlightwillingeneralbeellipticallypolarized.Deningacomplexrotation K=K+iK(4{34) wehave K=i(r+r) Figure4{3: DiagramsforKerrandFaradayrotation.(a)Kerrrotation;(b)Faradayrotation. polarizedlight,andKistheellipticity,whichisdenedastheratiooftheminortothemajoraxesofanellipsoid.
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SimilartotheKerreect,afterbeingtransmittedthroughaferromagneticcrystal,linearlypolarizedlightwillalsohavearotationandingeneralbeellipticallypolarized.ThisiscalledtheFaradayeect,whichisillustratedinFig. 4{3 (b).Itiseasytoshowthattherotationangleperunitlengthis andtheelliplicity Theellipticityisrelatedtothemagneticcirculardichroism(MCD),whichisdenedbythedierence(!)betweentheabsorptioncoecientoftherightandleftcircularlypolarizedlight.Fromtherelationsbetweentheopticalconstants,wehave (!)=+(!)(!)=4F(!) wherelstandsforthelighttransmittedlength. Notethattheincidentlightisalongthemagnetizationdirection,whichwedenedasthezdirection.Inthelongitudinalcasewherethemagnetizationvectorisintheplaneofthesurfaceandparalleltotheplaneofincidenceorinthetransversecasewherethemagnetizationvectorisintheplaneofthesurfaceandtransversetotheplaneofincidence,noKerrrotationisobservedatnormalincidence. 4{14 ,theextinction
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Figure4{4: KerrrotationofInMnAs.(a)AbsorptioncoecientofIn0:94Mn0:06AsasafunctionofphotonenergyforeandhactivelightatT=30K;(b)ThecorrespondingKerrrotation. coecientsarecomputed.ThenfromEq. 4{35 ,weobtain SupposewehaveaIn1xMnxAssamplewithx=6%,aCurietemperatureTC=55K,atT=30K.ThecomputedeandhactiveabsorptioncoecientsareshowninFig. 4{4 (a).Itcanbeseenthatduetotheferromagnetism,thesamplehasdierentabsorptioncoecientsforeandhactivepolarization.ThisgivesrisetoanonzeroKerrrotation,whichisshowninFig. 4{4 (b).Therotationisaboutseveraltenthsofadegree. Actually,theeightbandkptheoryisnotcapableofcalculatingthelightabsorptionforaverywiderangeofphotonenergies,becausenotonlyvalley,butalsoLvalley,andevenXvalleyabsorptionneedtobeconsidered.Thistaskrequiresafullzonebandstructure.TheLvalleyliesabout1:08eVabovethevalencebandedge,andtheXvalleyabout1:37eVabovethevalencebandedge.AschematicbanddiagramforInAsisshowninFig. 4{5 [ 86 ].Evenso,wecanstill
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Figure4{5: ThebanddiagramforInAs. getaqualitativepictureoftheKerrrotationforInMnAsforphotonenergiesbelow1eV. TheresultsforGa0:94Mn0:06AsareshowninFig. 4{6 (a)and(b),inwhichwesupposetheCurietemperatureisTC=110K,atT=30K.ComparingFig. 4{4 (b)andFig. 4{6 (b),weseethattheKerrrotationsareofthesameorder,aboutatenthofadegree. TheschematicdiagramfortheGaAsbandstructureisshowninFig. 4{7 [ 87 ].TheLvalleyliesabout1:71eVabovethevalencebandedge,andtheXvalleyabout1:90eVabovethevalencebandedge.SouseoftheeightbandkptheoryinGaAsisevenworse.However,weexpectthetransitionfromLvalleyisnotaseectiveasvalleytransitionbecausetheformerisanindirectprocess.
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Figure4{6: KerrrotationofGaMnAs.(a)AbsorptioncoecientofGa0:94Mn0:06AsasafunctionofphotonenergyforeandhactivelightatT=30K;(b)ThecorrespondingKerrrotation. 2arg(r whererandr+arethetwocomplexreectioncoecientsforand+circularlypropagatinglightbeamsinthemedium,andarg(x)representsthephaseofthecomplexnumberx.ItiseasytoprovethatthisKisexactlytheKinEq. 4{35 AccordingtoRef.[ 88 ],foramultilayerstructure,thecoecientsrdependontheamplitudeofthereectioncoecientsri;i+1attheinterfacesofsuccessivelayersiandi+1.Ifweapproximateasinglequantumwellbyathreelayerstructure,inthecaseofnormalreection,rtakestheform where
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Figure4{7: ThebanddiagramforGaAs. Thereectioncoecientsri;i+1,canbeobtainedfromEq. 4{32 and 4{33 .i=(w=c)liNidenotesthedephazingoftheelectriceldradiationaftercrossingthelayeriofthicknessli. Attheinterfaceofi;i+1,thecomplexreectioncoecientbecomes whichcanbeeasilyobtainedfromthesameprocedurewedescribedinSection4.1.ThecomplexrefractiveindicesNareobtainedbycalculatingtheabsorptioncoecientsineachlayer. NowconsideraIn0:88Mn0:12As/GaSbheterostructurewithaInMnAslayerthicknessof9nminaparamagneticphase(thusr+=r)attemperatureT=5:5K.Tocomparewithexperiment,thereectivityofthisstructureiscalculatedandshowninFig. 4{8 (b).Alongwiththereectivity,theabsorptioncoecientsintheInMnAsandGaSblayerarealsoshowninFig. 4{8 .Inthe
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Figure4{8: TheabsorptioncoecientsbothinInMnAsandGaSblayers(a)andthereectivityofInMnAs/GaSbheterostructure(b). calculation,weassumeacarrierdensityofp=1019cm3intheInMnAslayer,andnocarriersintheGaSblayer. Theexperimentalresult[ 89 ]isillustratedinFig. 4{9 .Upto1:5eV,wecanseethatthecalculationsuccessfullyreproducestheoscillatingstructureofthereectivity,andthecalculatedmeanreectivityisveryclosethetheexperimentalone.Inourcalculation,thereareseveralenergieswherethereectivityisveryclosetozero,whileintheexperiments,thelowestreectivityisstillaround40%.Wesupposethisisbecauseinourcalculation,wehavenotconsideredinterfaceroughness,whichcansignicantlycontributetothereection. NowweconsideraferromagneticIn0:88Mn0:12As/AlSbheterostructurewitha9nmthickInMnAslayeranda136nmthickAlSblayergrownona400nmthickGaSblayer.ThisstructurehasaCurietemperatureof35K,andtheMOKEsignalhasbeenmeasuredatamagneticeldof3T.TheexperimentalresultisshowninFig. 4{10 (a).Followingthesameprocedure,theabsorptioncoecientsintheInMnAslayerarecalculatedincludingquantumconnementeectsusinganitedierencemethod[ 90 ].Thecalculationhasbeenperformedinthe0:31:5eVphotonenergyrange,inwhichtheAlSblayeristransparent.
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Figure4{9: ReectivityofIn0:88Mn0:12As(9nm)/GaSb(600nm)heterostructureatT=5:5KmeasuredbyP.FumagalliandH.Munekata.ReprintedwithpermissionfromP.FumagalliandH.Munekata.Phys.Rev.B53:1504515053,1996.Figure3,Page15047. ThelightabsorptionintheGaSblayerisalsocalculated,andthelightreectionsfromtheInMnAs/AlSbandAlSb/GaSbinterfacearebothtakenintoaccount.TheKerrrotationisobtainedusingEq. 4{40 andthecalculatedresultisshowninFig. 4{10 (b).
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Figure4{10: Measured(a)andcalculated(b)KerrrotationofInMnAs(19nm)/AlSb(145nm)heterostructureunderamagneticeldof3TatT=5:5K.Panel(a)isreprintedwithpermissionfromP.FumagalliandH.Munekata.Phys.Rev.B53:1504515053,1996.Figure8,Page15049.
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HolesinIIIVDMSplayanimportantroleinmediatingtheferromagnetismandparticipatinginmagnetoopticaltransportprocesses.UnlikeinIIVIDMSwhereMn2+ionsareisoelectronicwiththecations,MnionsinIIIVDMSareacceptors.DuetotheAsantisitedefectsandinterstitialMn,bothofwhichactasdoubledonors,theholeconcentrationisusuallymuchlowerthantheMnconcentration.However,themediationoftheexchangeinteractionbetweenlocalizedmagneticmomentsbyholesisthecornerstoneofmosttheoriesofferromagnetisminIIIVDMS(pleaserefertoSection1.3.1).LightinducedferromagnetisminpInMnAs/GaSbhasbeenobservedbyKoshiharaetal.[ 44 ],andKonoetal.[ 46 ],whereholeelectronpairshavebeenexcitedandtheholedensitygreatlyenhancedbytheincidentlight.Inthelatterexperiment,ultrafastlasershavebeenemployedandthetimedependentMOKEsignalhasbeenmeasured.TheultrashortlaserpulsescreatealargedensityoftransientcarriersintheInMnAslayerandtheMOKEsignaldecayslessthan2psafterlaserpumping,asshowninFig. 5{1 .Recently,Mitsumorietal.[ 91 ]hasstudiedthephotoinducedmagnetizationrotationinferromagneticpGaMnAs.Theyfoundthatwhenshiningcircularlypolarizedlightnormaltothesamplesurface,whichisparalleltothemagnetization,andprobingwithlinearlypolarizedlight,anonzeroKerrrotationisseenwhichhasadecaytimeof25ps.Aswediscussedinthelastchapter,thereisnolongitudinalKerrrotationatnormalincidence,sothisKerrrotationwasduetotherotationofthemagnetizationduetothelightinducednonequilibriumcarrierspins.ThislightinducedrotationofmagnetizationisillustratedschematicallyinFig. 5{2 112
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Figure5{1: LightinducedMOKE.Signaldecaysinlessthan2ps. Inthesepumpprobeopticalexperiments,thebehaviorofthenonequilibriumcarrierspinswhichinducetheexchangeinteractionisakeyfactorthatdeservestobestudied.Theelectronspinrelaxationintheconductionbandhasbeenthoroughlyinvestigatedbymanyauthors[ 4 92 ],however,theoreticalstudiesonholespins,especiallyonthenonequilibriumholespins,aresparse.Inthischapter,wewillfocusontheholespinrelaxationinbulkIIIVDMS,anddiscussthemechanismswhichinducetherelaxation. 93 94 ],DyakonovPerel(DP)[ 95 ],BirAronovPikus(BAP)[ 96 ],andhyperneinteractionprocesses.IntheEYmechanismelectronspinsrelaxbecausetheelectronwavefunctionsnormallyassociatedwithagivenspinhaveanadmixtureoftheoppositespinstates,duetospinorbitcouplinginducedbytheions,thusspinippingprocessesaccompanymomentumrelaxation.TheDPmechanismexplainsspindephasinginsolidswith
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Figure5{2: Lightinducedmagnetizationrotation.ReprintedwithpermissionfromMitsumorietal.Phys.Rev.B69:3320333206,2004.Figure1,Page33203. inversionasymmetry,whichcausesspinsplitting.Spindephasingoccursbecauseelectronsfeelaneectivemagneticeld,duetothespinsplittingandspinorbitinteractions,whichchangesinrandomdirectionseverytimetheelectronscatterstoadierentmomentumstate.TheBAPmechanismisimportantforpdopedsemiconductors,inwhichtheelectronholeexchangeinteractiongivesrisetouctuatinglocalmagneticeldswhichipelectronspins.Insemiconductorswithanuclearmagneticmoment,thereisalsoahyperneinteractionbetweentheelectronspinsandnuclearmomentswhichwillcausespinippingofelectronspins. Inthevalenceband,theLuttingerHamiltoniandescribingthe4heavyholeandlightholebandsis[ 97 ] 22k222(kJ)2(5{1) whereJx,JyandJzare44matricescorrespondingtoangularmomentumJ=3=2.Duetothespinorbitinteraction,andthekJtermintheHamiltonian,thespinstatesinthevalencebandmixverystrongly.Thelightholeandsplitoholebasisstatesarenotspineigenstates.Evenforapureheavystateatk=0,itbecomesstronglymixednotfarawayfromthepoint.Unlikeintheconductionband,wheretheoverlapintegralbetweentwoelectronspinstatesisverysmall
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(1)becauseoftheweakmixing,theoverlapintegralbetweentwostronglymixedvalencebandstatesisusuallybig.Thechancesarehighthataholescatteredfromoneelectronicstate(say,mainlyspinup)toanother(say,mainlyspindown)maytotallyreverseitsspin.Soweconcludethatforholespinrelaxationinthevalenceband,theEYmechanismismosteective. SupposeaholeinstatejkihasspinSk.Afteronescatteringevent,thisholetransitstoanotherstatejk0i,inwhichthespinisSk0.Inthisprocess,thespinchangeisS=Sk0Sk.Thespinrateofchangeis dt=ZW(k;k0)Sdk0=ZW(k;k0)(Sk0Sk)dk0(5{2) whereW(k;k0)isthescatteringratebetweenstatekandk0. Assumewecanuseonetimeconstanttodescribethisspinrelaxationprocess,thenwecanwritedownanequationlike dt=Sk whereisthespinrelaxationtime,whichstateshowlonganonequilibriumholespinwilltaketocompletelyloseitspreviousorientation.Forcalculating,weneedtoknowthescatteringrateW(k;k0).
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andthelatterispolaropticalphononscattering.ThesetwomechanismsdominatephononscatteringinbulkIIIVsemiconductors. UsingFermi'sgoldenrule,ageneral3Dphononscatteringratecanbewrittenas[ 98 ] 21 2)(Ek0Ekh!q;b)dk0(5{4) whereVisthecrystalvolume,Nisthenumberofunitcellsinthecrystal,M0isthereducedmassoftheunitcell,h!q;bisenergyofaphononwithwavevectorqinmodeb,andn(!q;b)isthephonondensityforsuchamode.kqk0;Ktakesintoaccountmomentumconservationinascatteringevent,whereKisareciprocallatticevector.Fornormalprocesses,K=0,andprocesseswithK6=0areumklappprocesses.Usuallyweonlyconsiderlongwavelengthphonons,wherescatteringmainlytakesplaceneartheBrillouinzonecenter.Forthiskindofsituation,K=0.InEq. 5{4 ,theuppersignisforphononabsorptionandthelowersignforemission.C2q;bistheelectronphononinteractioncouplingconstantfromtheinteractionHamiltonian,and istheoverlapintegralbetweeninitialandnalelectronicstates. InDMSmaterials,generallytheholedensityisratherhigh,soscreeningmustbetakenintoaccount.Whenconsideringopticalphononscattering,inwhichthevibrationfrequencyisveryhigh,theplasmamodemaybemixed,andthereforeadynamicscreeningmodelisdesirable.Buthere,wejustassumeastaticscreening,whichworkswellforacousticphononandimpurityscattering,andisagoodapproximationforopticalphononscattering.
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whereN(Ei)isthedensityofstatesatenergyEi,andf(Ei)istheFermiDiracdistributionfunctionrepresentingtheoccupationprobability.Thesumrunsoverallenergylevels.WhenanelectricpotentialV(r)ispresent,itwillchangetheelectrondensityto Herewesupposetheperturbationpotentialissmallanddoesnotaectthedensityofstates.Becauseofthispotential,thereisaperturbationinthespacechargedensity ThechargedensityisrelatedtothepotentialbyPoisson'sequation, XiN(Ei)[f(EieV(r))f(Ei)](5{9) ConsiderV(r)tobeasmallperturbation,andexpandtherighthandsideoftheaboveequation.Theleadingtermgives DeningthereciprocalDebyescreeninglengthq0by
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thesolutionforEq. 5{10 is req0r(5{12) forsphericalsymmetry.ThevalueofAcanbeobtainedusingboundaryconditions.Forexample,forapointcharge,V(r)!0,r!1andV(r)!Ze2=4r,r!0,soA=Ze2=4.Equation 5{12 isknownastheYukawapotential. ThederivativeofFermi'sfunctionwithrespecttoEinEq. 5{10 is andthusEq. 5{11 becomes Wewillusethisequationtocalculatethereciprocalscreeninglengthlaterwhendealingwithphononandimpurityscattering. 99 ]measuredtheholespinrelaxationtimeintheGaAsvalencebandusingapumpprobetechnique.Theygeneratedorientedholesusingan800nmlaserinheavyandlightholebands,andprobedusingalaserpulse(3m)withenergycorrespondingtothesplitoholetoheavyholeorspiltoholetolightholetransitions,thenmeasuredthecircularpolarizationchangeofthetransmittedlight.Withinanerrorof10%,theyobtainedaholespinrelaxationtimeof110fs.InpureintrinsicIIIVsemiconductors,polaropticalphononandpiezoelectricscatteringareresponsibleforthisholespinrelaxation. PolaropticalphononscatteringdominatesinbothIIVIandIIIVintrinsicsemiconductorswhentemperatureisnottoolow.Atverylowtemperature,duetothehighenergyoftheopticalphonons,theirdensityisverylow,too.Furthermore,theemissionofanopticalphononrequiresalargeenergydierence,whichisalsonotfavorableatlowtemperatures.
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Polarscatteringoccursinconnectionwiththecontrarymotionofthetwoatomsineachunitcellandonlytakesplaceinthelongitudinalopticalmode,asdescribedbyFrohlich[ 100 ]andCallen[ 101 ]. WecanwritethepolarinteractionHamiltonianas where(R)isthechargedensityoftheelectronsand(R)istheelectricpotentialassociatedwithpolarizationintheunitcellcenteredatR. Followingthediscussionin[ 98 ],ifwetakeintoaccountthescreeningeect,thentheHamiltonianis q2+q20(iQqeiqr+c:c:)(5{16) whereeistheeectivechargeontheatomsinaunitcellandV0isthevolumeofaunitcell.Qqarethenormalcoordinatesofthislongitudinalopticalmode.ThecouplingcoecientinEq. 5{4 isthegivenby andthepolaropticalphononscatteringrateis (q2+q20)2kqk0;0(n(!0)+1 21 2)(Ek0Ekh!0)dk0;(5{18) whereh!0istheopticalphononenergy,M0isthereducedmassinaunitcell,andq0isthereciprocalDebyescreeninglengthwederivedinthelastsection. The\!q"dispersioncurveforopticalphononsisatatthepoint,veryatinthewholeBrillouinzone,andperpendiculartotheBrillouinzoneboundary.Soinalongwaveapproximation,whichmeansscatteringneartheBrillouinzone
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center,weadopttheEinsteinapproximationandusetheLOphononenergyatthepoint,h!0,forallqinEq. 5{18 TheeectivechargeinEq. 5{4 isrelatedtothedierencebetweenthepermittivitiesatlowandhighfrequencies,andisgivenby[ 102 103 104 ] SubstitutingEq. 5{19 intoEq. 5{18 ,weget (q2+q20)2kqk0;0(n(!0)+1 21 2)(Ek0Ekh!0)dk0:(5{20) Wecanseethatthepolarscatteringratedoesnotdependonthedetailsoftheunitcellsuchasthevolumeandreducedmass,etc. Theacousticphononenergyinthelongwavelengthlimitcanbeexpressedash!=hvsq,wherevsisthesoundvelocityinthecrystal.Withawavevectorq=107cm1,theacousticphononenergyisbelow1meV.Piezoelectricscatteringisanacousticphononeect,soforpiezoelectricscattering,thephonondensityinEq. 5{4 istheacousticphonondensity.Duetotheverylowenergy,thedensityisstillappreciableevenatlowtemperatures.Thepiezoelectriceectisduetotheacousticstrain,whichisincontrastwiththepolaropticaleectduetotheopticalpolarization. TheelectronphononinteractionHamiltonianinthepiezoelectriccasecanbewrittenas whereD(R)istheelectricdisplacementvectorrelatedtotheelectriceldandacousticstrain,P(R)istheelectricpolarizationcausedbytheacousticvibration,andRisthepositionoftheunitcell.
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Followingthediscussionin[ 98 ],inaplanewaveapproximation,weobtain where,,arethedirectioncosineswithrespecttothecrystalaxisofthedirectionofpropagationofthewaves,a1,a2anda3arethecomponentsoftheunitpolarizationvector^a,ande14istheonlynonvanishingpiezoelectricconstantinzincblendecrystals.ThusthecouplingcoecientofEq. 5{4 is Foracousticphononscattering,wecanassumethattheratesforabsorptionandemissionarethesameduetothefactthatattemperaturesaboveseveralKelvin,n(!)1inthelongwavelengthlimit.Combiningthelongitudinalandtransversemodestogether,averagingoverallthedirectionaldependence,andusingtheequipartitionapproximation,wereachthefollowingexpression (q2+q20)2kqk0;0(Ek0Ekh!q)dk0(5{24) whereK2avisanaverageelectromechanicalcouplingcoecientrelatedtothesphericalelasticconstants.Asafurtherapproximation,wecanassumetheacousticphononscatteringisanelasticprocess,andtakethefunctionintheaboveequationtobeqindependent. Table5{1: ParametersforGaAsphononscattering LOphononwavelength()1285:0(cm1)K2av20:002521r10:6r12.4 105 ]. 98 ].
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Consideringbothpolaropticalphononandpiezoelectricscattering,wehavecalculatedtheheavyholespinrelaxationtimeinintrinsicGaAs.TheparametersusedarelistedinTable 5{1 ThecalculatedspinrelaxationtimeforaspinupheavyholenearthepointasafunctionoftheelectronicwavevectoratT=300KisshowninFig. 5{3 (a).WecanseethatawayfromtheBrillouinzonecenter,thespinrelaxationtimedecreasesforaholedensityof1019=cm3,andincreasesforaholedensityof1018=cm3.ThisisbelievedtobeconnectedwithchangesintheavailabledensityofstatesatdierentFermienergies.InFig. 5{3 (b),thespinrelaxationtimeofaheavyholeatthepointisplottedasafunctionoftemperature.WecanseethatatT=300Kwithaholedensityof1018=cm3,whichwethinkshouldbeclosetotheexperimentalcase,theholespinrelaxationtimeisaround110fs,whichisveryclosetotheexperimentalvalue.ThisrevealsthatphononscatteringisindeeddominantinpureGaAs.
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Figure5{3: Theheavyholespinrelaxationtimeasafunctionofwavevector(a),andtemperatureatthepoint(b).
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Inameaneldapproximation,thisHamiltoniancanbewrittenas[ 106 ] where^istheorientationofthesubstitutionalMnlocalmoments,sisthecarrierspin,andJpdistheexchangeconstant.ThenthescatteringrateintheFermi'sgoldenruleapproximationis where Thus SinceS=Sxx+Syy+Szz,wehavehkjSjk0i=(Sxx)kk0+(Syy)kk0+(Szz)kk0.Thesquaredtermwillhave(Sxx)2kk0liketermsinit.Intheabsenceofspontaneousmagnetization,(Sxx)2kk0=hS2xi(x)2kk0,andhS2xi=hS2yi=hS2zi.Forspontaneousmagnetization(ferromagneticcase),assumingthemagnetizationdirectionisalongz,hSzicanbefoundusingEq. 1{22 ,whichresultsfromtheselfconsistenteectiveeldapproximation.FromtherelationS2=S2x+S2y+S2z,hS2xiandhS2xiarealsofound.ThehSxihSzi(x)kk0(z)kk0likecrosstermsareallaveragedtozeroinrstorderapproximation.
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inIIIVDMSmaterialsmanyantisitedefectsandMninterstitials[ 27 28 29 ].TheybothserveasZ=2compensatingdefects.TheCoulombpotentialoftheseimpuritysitescancoupledierentelectronicstates. WewilluseBrooksHerring'sapproach[ 107 ]inthefollowingtodealwiththeimpurityscatteringinGaMnAscrystals. WecanwritedowntheequationforascreenedCoulombpotentialas whereq0isthereciprocalDybyescreeninglength.UsingFermi'sgoldenrule,thescatteringrateduetothisscreenedCoulombcouplingisgivenby Inthisequation,ifweassumetheincidentcarrierscannotpenetrateveryclosetotheimpuritysite,wecanfactorizethematrixelementhk0jeV(r)jkiintotwoparts.OnepartistherapidlyvaryingBlochpart,theotheristheslowlyvaryingplanewaveparttimestheexponentiallydecayingCoulombpotential.Thuswehave wherezkistheeightcomponentenvelopespinor.SoiftheimpuritydensityisNI,then (q2+q20)2jhz0kjzkij2(Ek0Ek)dk0:(5{33) InGaMnAs,Mnisanimpurity,itsconcentrationisx,anditsdensityNMnisproportionaltox.Supposetheholedensityisp,thenthedensityforthecompensatingdefectsis(NMnp)=2.Includingbothexchangeinteractionandimpurityscattering,thespinrelaxationtimeasafunctionofthewavevectoralongtheXdirectionisplottedinFig. 5{4 .Inthiscase,thesampleisferromagneticwithaCurietemperatureTC=55KatT=30K.Thehole
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Figure5{4: Spinrelaxationtimeforaheavyholeasafunctionofkalong(0,0,1)direction. densityisassumedtobep=1019=cm3.Fig. 5{4 revealsthatnormallyinDMS,impurityexchangephonon.Infact,theimpurityscatteringis1000timesstrongerthanphononscattering.Thisisnatural,becausefor6%Mndoping,theMnimpuritydensityitselfcanreach1021=cm3.TheotherpointwecanseeisthatthephononscatteringinGaMnAsisweakerthaninGaAs.Thisisbecausethevalencebandsplittingintheferromagneticphasemakessomestatesenergeticallyunavailableforscatteringduetoenergyconservation. ThespinrelaxationtimeattheholeFermisurfaceisillustrateinFig. 5{5 asafunctionofholedensity.Fig. 5{5 (a)showstheholespinrelaxationtimeintheXdirectionandFig. 5{5 (b)showstheholespinrelaxationtimeintheLdirection.HeretheFermisurfacecanbeconsideredasthatoftheunperturbedsystem.ThedierentbehavioralongdierentkdirectionisduetotheGaAsvalencebandanisotropy,whichisenhancedinGaMnAs.ThespinrelaxationtimeattheFermisurfacehelpsoneunderstandthepropertiesoftheholesmediating
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Figure5{5: Spinrelaxationtimeofaheavyholeasafunctionofholedensityatdirection(a)(0,0,1)and(b)(1,1,1).
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theferromagnetisminDMS.IntheRKKYmodel(oritslowdensitylimit,Zener'smodel),itistheholesattheFermisurfacethatmediatetheexchangeinteractionwhichresultsintoaferromagneticphasechange. Herewehaveonlytalkedaboutthespintransportpropertiesofasinglehole.However,thecollectivebehaviorofholesdeterminesthepropertiesoftheholesystem.InDMS,usuallythemagnetizationduetoholesthemselvesisnegligiblecomparedtothatduetothelocalizedmagneticmoments.Inthepumpprobeexperimentswementionedinthebeginningofthischapter,itisthechangeofthemagnetizationduetothelocalizedmomentsthatgivesanobservableresult.Thechangeofthemagnetizationoflocalizedmoments,i.e.,therotationofthemagnetizationdirection,isinducedbythespinalignmentoftheitinerantholesthroughtheexchangeinteraction.Thechangeofthemagnetizationinreturnwillalsohaveafeedbacktotheseholes.Thustheholelocalizedmomentssystemisacomplicatedsystemandmustbetreatedinaselfconsistentmanner.Thiscollectiveselfconsistentsystemshouldhaveamuchlongerlifetimethanthatofasinglehole.
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Inthisthesis,thedevelopmentandcurrentresearchsituationofdilutedmagneticsemiconductors,includingbothIIVIandIIIVIsemiconductorbased,hasbeenintroducedanddiscussed.Incalculatingthebandstructure,aneightbandkptheoryhasbeenemployedtogetherwiththespdexchangeinteraction.Intheabsenceofanexternalmagneticeld,ageneralizedKane'smodelisappropriateforcalculatingthebandstructure,whileinthecaseofanappliedmagneticeld,onebandwillsplitintoaseriesofLandaulevels.Inordertodealwiththis,wedevelopedageneralizedPidgeonBrownmodelwhichincorporatestheexchangeinteractionandalsotakesintoaccountnitekzeects.Calculationshaveshownthatinadilutedmagneticsemiconductor,thebandstructureisverydierentfromthatinapuresemiconductor.Forexample,thegfactorsinInMnAscanbeabove100incontrastwithacomputedgfactorof15inInAs. Cyclotronresonanceinultrahighmagneticelds(upto500T)hasbeensimulatedandcomparedwithexperiments.ThemethodforcalculatingopticaltransitionshasbeenintroducedandFermi'sgoldenrulehasbeenutilized.WehavesuccessfullyreproducedthecyclotronabsorptioninbothconductionandvalencebandsinInMnAs.Wepointedoutthattheshiftofcyclotronresonancepeaksintheconductionbandhadadependenceontheexchangeconstants(),andthepeaklineshapedependedonthenonparabolicity.ThehactiveCRresonanceinvalencebandshasbeendecomposedintoheavyholetoheavyholeandlightholetolightholetransitionsinaeldupto150T.Theselectionrulesforopticaltransitionshavebeendiscussedinadipoleapproximationandwehavepointedoutthatduetothedegeneracyinthevalenceband,notonlyhactivecyclotron 129
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resonance,butalsoeactivecyclotronresonancecantakeplaceinsemiconductorvalencebands.GenerallythehactivetransitionstakeplacebetweenheavyholeorlightholeLandaulevelsthemselves,buteactivetransitionstakeplacebetweenheavyholeandlightholelevels.TheCRstrengthandlineshapestronglydependoncarrierdensity,whichprovideanalternativewaytomeasurethecarrierconcentration.WehavegivenananalyticalexpressionwhichexplainstheCRpeakshiftswithtemperatureinInMnAs/GaSbheterostructure.ThepronouncednarrowingmaybeduetothesuppressionofspinuctuationortransferoftheholestotheInMnAs/GaSbinterfaceatlowtemperatures. Wehavediscussedtherelationsbetweentheopticalconstants,andfromthecalculationofabsorptioncoecients,thereectioncoecientsandmagnetoopticalKerrrotationhasbeencalculatedinbulkInMnAsandInGaAsortheirheterostructures.Becauseofferromagnetism,theeactiveandhactivecrossbandabsorptionhavedierentdependenceonphotonenergies.Thismagneticcirculardichroismresultsinthepolarizationplanerotationoflinearlypolarizedlight.Wehavesimulatedthismagneticcirculardichroismandcomparedourresultstoexperiments. Duetotheimportanceofholesindilutedmagneticsemiconductorsystems,wehavecarriedoutcalculationsforholespinrelaxationtimesinbulkGaAsandGaMnAsvalencebands.InGaAs,phononscatteringdominatesandgivesaholespinrelaxationtimearound100fsatroomtemperature.InGaMnAs,duetothestrongexchangeinteractionandheavyimpuritydoping,exchangeandimpurityscatteringdominate.Wehavebrieyintroducedthetheoryofphononscattering,exchangescatteringandimpurityscattering,andshownincalculationsthatinMndopedDMSsystems,thephononscatteringisnolongerimportantandonlyimpurityscatteringdominates.Assumingtheexternaldisturbanceasmall
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perturbation,theholespinrelaxationtimeattheFermisurfaceisonlyafewfemtoseconds. Thereisstillmuchworktobedoneinthefuture.Themeaneldtheoryhasitsowndrawbacksintreatingferromagnetictransitions.Toobtainbetterresultswhencalculatingthebandstructureandopticalabsorptioninferromagneticsamples,abettertheoreticalframeworkishighlydesirable.OurcurrentmodelisnotadequatetocalculatetheCRabsorptioninaInMnAs/GaSbheterostructure,whichisatypeIIheterostructure(theconductionbandofInAsliesbelowthevalencebandsofGaSb).Amodelthatcanaccountfortheinterfacestatesneedstobedevelopedinthefuture.Atthecurrentlevel,wehaveonlycalculatedthestaticMOKEofDMS,whilethetimeresolvedMOKEisofmoreimportanceforstudyingthedynamicalpropertiesofDMS.CurrentlywearetryingtodevelopamethodtostudytimedependentmagneticphenomenainDMS.
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YongkeSunwasborninasmallvillageinZhumadian,HenanProvince,People'sRepublicofChina,onMarch6,1974.Hestayedtherefor15yearsuntilhenishedmiddleschoolstudies.Afterthat,hewent10milesawayfromhometostudyinahighschoolcalledYangzhuangHighSchool.In1992,hewasexemptedfromthenationalexamandadmittedtoPekingUniversityinBeijing,China.From1992to1997hestudiedinPekingUniversityandreceivedhisB.S.degreein1997.Hesubsequentlyparticipatedinthemaster'sprogramandobtainedtheM.S.degreein2000.Inthesameyear,hewasmarriedtohisbeautifulwife,YuanZhang,whowashisschoolmate.Inthefallof2000hecametotheUnitedStatesandbecameaGator.Inthesummerof2001,heenteredProf.Stanton'sgroupandhasbeenstudyingthepropertiesofdilutedmagneticsemiconductors,pursuingaPh.D.degree. 138

