THEORETICAL STUDIES OF THE ELECTRONIC, MAGNETO-OPTICALAND TRANSPORT PROPERTIES OF DILUTED MAGNETIC

SEMICONDUCTORS

By

YONGKE SUN

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOLOF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREE OFDOCTOR 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 Christopher

J. Stanton, for giving me an invaluable opportunity to work on challenging and

extremely interesting projects over the past four years. He has always 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 Abernathy 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 always 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.

iii

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 always 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.

iv

TABLE OF CONTENTSpage

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv

CHAPTER

1 INTRODUCTION AND OVERVIEW . . . . . . . . . . . . . . . . . . . 1

1.1 Spintronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 The II-VI Diluted Magnetic Semiconductors . . . . . . . . . . . . 4

1.2.1 Basic Properties of II-VI Diluted Magnetic Semiconductors 41.2.2 Exchange Interaction between 3d5 Electrons and Band Elec-

trons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3 The III-V Diluted Magnetic Semiconductors . . . . . . . . . . . . 13

1.3.1 Ferromagnetic Semiconductor . . . . . . . . . . . . . . . . 131.3.2 Effective Mean Field . . . . . . . . . . . . . . . . . . . . . 21

1.4 Open Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231.4.1 Nature of Ferromagnetism and Band Electrons . . . . . . . 231.4.2 DMS Devices . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2 ELECTRONIC PROPERTIES OF DILUTED MAGNETIC SEMICON-DUCTORS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.1 Ferromagnetic Semiconductor Band Structure . . . . . . . . . . . 292.2 The k · p Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.2.1 Introduction to k · p Method . . . . . . . . . . . . . . . . . 302.2.2 Kane’s Model . . . . . . . . . . . . . . . . . . . . . . . . . 342.2.3 Coupling with Distant Bands-Luttinger Parameters . . . . 382.2.4 Envelope Function . . . . . . . . . . . . . . . . . . . . . . . 42

2.3 Landau Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432.3.1 Electronic State in a Magnetic Field . . . . . . . . . . . . . 432.3.2 Generalized Pidgeon-Brown Model . . . . . . . . . . . . . . 442.3.3 Wave Functions and Landau Levels . . . . . . . . . . . . . 49

2.4 Conduction Band g-factors . . . . . . . . . . . . . . . . . . . . . . 53

v

3 CYCLOTRON RESONANCE . . . . . . . . . . . . . . . . . . . . . . . . 56

3.1 General Theory of Cyclotron Resonance . . . . . . . . . . . . . . . 563.1.1 Optical Absorption . . . . . . . . . . . . . . . . . . . . . . 563.1.2 Cyclotron Resonance . . . . . . . . . . . . . . . . . . . . . 60

3.2 Ultrahigh Magnetic Field Techniques . . . . . . . . . . . . . . . . 633.3 Electron Cyclotron Resonance . . . . . . . . . . . . . . . . . . . . 64

3.3.1 Electron Cyclotron Resonance . . . . . . . . . . . . . . . . 643.3.2 Electron Cyclotron Mass . . . . . . . . . . . . . . . . . . . 72

3.4 Hole Cyclotron Resonance . . . . . . . . . . . . . . . . . . . . . . 743.4.1 Hole Active Cyclotron Resonance . . . . . . . . . . . . . . 743.4.2 Hole Density Dependence of Hole Cyclotron Resonance . . 833.4.3 Cyclotron Resonance in InMnAs/GaSb Heterostructures . . 833.4.4 Electron Active Hole Cyclotron Resonance . . . . . . . . . 90

4 MAGNETO-OPTICAL KERR EFFECT . . . . . . . . . . . . . . . . . . 96

4.1 Relations of Optical Constants . . . . . . . . . . . . . . . . . . . . 964.2 Kerr Rotation and Faraday Rotation . . . . . . . . . . . . . . . . 1014.3 Magneto-optical Kerr Effect of Bulk InMnAs and GaMnAs . . . . 1044.4 Magneto-optical Kerr Effect of Multilayer Structures . . . . . . . 107

5 HOLE SPIN RELAXATION . . . . . . . . . . . . . . . . . . . . . . . . . 112

5.1 Spin Relaxation Mechanisms . . . . . . . . . . . . . . . . . . . . . 1135.2 Lattice Scattering in III-V Semiconductors . . . . . . . . . . . . . 115

5.2.1 Screening in Bulk Semiconductors . . . . . . . . . . . . . . 1175.2.2 Spin Relaxation in Bulk GaAs . . . . . . . . . . . . . . . . 118

5.3 Spin Relaxation in GaMnAs . . . . . . . . . . . . . . . . . . . . . 1225.3.1 Exchange Scattering . . . . . . . . . . . . . . . . . . . . . . 1225.3.2 Impurity Scattering . . . . . . . . . . . . . . . . . . . . . . 124

6 CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

vi

LIST OF TABLESTable page

1–1 Some important II-VI DMS . . . . . . . . . . . . . . . . . . . . . . . . 4

2–1 Summary of Hamiltonian matrices with different n . . . . . . . . . . . 50

2–2 InAs band parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3–1 Parameters for samples used in e-active CR experiments . . . . . . . . 67

3–2 Characteristics of two InMnAs/GaSb heterostructure samples . . . . . 85

5–1 Parameters for GaAs phonon scattering . . . . . . . . . . . . . . . . . 121

vii

LIST OF FIGURESFigure page

1–1 The band gap dependence of Hg1−kMnkTe on Mn concentration k. . . 5

1–2 The band structures of Hg1−xMnxTe with different x. . . . . . . . . . 6

1–3 Cd1−xMnxTe x-T phase diagram. . . . . . . . . . . . . . . . . . . . . 7

1–4 Average local spin as a function of magnetic field at 4 temperaturesin paramagnetic phase. . . . . . . . . . . . . . . . . . . . . . . . . . 10

1–5 Magnetic-field dependence of Hall resistivity ρHall and resistivity ρ ofGaMnAs with temperature as a parameter. . . . . . . . . . . . . . 14

1–6 Mn composition dependence of the magnetic transition temperatureTc, as determined from transport data. . . . . . . . . . . . . . . . . 16

1–7 Variation of the RKKY coupling constant, J , of a free electron gas inthe neighborhood of a point magnetic moment at the origin r = 0. . 17

1–8 Curie temperatures for different DMS systems. Calculated by Dietlusing Zener’s model. . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1–9 Schematic diagram of two cases of BMPs. . . . . . . . . . . . . . . . . 20

1–10 Average local spin as a function of magnetic field at 4 temperatures. . 22

1–11 The photo-induced ferromagnetism in InMnAs/GaSb heterostructure. 25

1–12 Spin light emitting diode. . . . . . . . . . . . . . . . . . . . . . . . . . 27

1–13 GaMnAs-based spin device. . . . . . . . . . . . . . . . . . . . . . . . 28

2–1 Valence band structure of GaAs and ferromagnetic Ga0.94Mn0.06Aswith no external magnetic field, calculated by generalized Kane’smodel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2–2 Band structure of a typical III-V semiconductor near the Γ point. . . 35

2–3 Calculated Landau levels for InAs (left) and In0.88Mn0.12As (right) asa function of magnetic field at 30 K. . . . . . . . . . . . . . . . . . 52

2–4 The conduction and valence band Landau levels along kz in a mag-netic field of B = 20 T at T = 30 K. . . . . . . . . . . . . . . . . . 53

viii

2–5 Conduction band g-factors of In1−xMnxAs as functions of magneticfield with different Mn composition x. . . . . . . . . . . . . . . . . 54

2–6 g-factors of ferromagnetic In0.9Mn0.1As. . . . . . . . . . . . . . . . . . 55

3–1 Quasi-classical pictures of e-active and h-active photon absorption. . . 62

3–2 The core part of the device based on single-coil method. . . . . . . . . 64

3–3 A standard coil before and after a shot. . . . . . . . . . . . . . . . . . 65

3–4 Waveforms of the magnetic field B and the current I in a typical shotin single-turn coil device. . . . . . . . . . . . . . . . . . . . . . . . . 65

3–5 Waveforms of the magnetic field B and the current I in a typical fluxcompression device. . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3–6 Experimental electron CR spectra for different Mn concentrations xtaken at (a) 30 K and (b) 290 K. . . . . . . . . . . . . . . . . . . . 68

3–7 Zone-center Landau conduction-subband energies at T = 30 K asfunctions of magnetic field in n-doped In1−xMnxAs for = 0 andx = 12%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3–8 Electron CR and the corresponding transitions. . . . . . . . . . . . . 70

3–9 Calculated electron CR absorption as a function of magnetic field at30 K and 290 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3–10 Calculated electron cyclotron masses for the lowest-lying spin-up andspin-down Landau transitions in n-type In1−xMnxAs with photonenergy 0.117 eV as a function of Mn concentration for T = 30 Kand T = 290 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3–11 Hole cyclotron absorption as a function of magnetic field in p-typeInAs for h-active circularly polarized light with photon energy 0.117 eV. 75

3–12 Calculated cyclotron absorption only from the H−1,1−H0,2 and L0,3−L1,4 transitions broadened with 40 meV (a), and zone center Lan-dau levels responsible for the transitions (b). . . . . . . . . . . . . . 76

3–13 Experimental hole CR and corresponding theoretical simulations. . . 77

3–14 Observed hole CR peak positions for four samples with different Mnconcentrations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

3–15 The dependence of cyclotron energies on several parameters. . . . . . 79

3–16 Hole CR spectra of InAs using different sets of Luttinger parameters. 80

3–17 Calculated Landau levels and hole CR in magnetic fields up to 500 T. 81

ix

3–18 k-dependent Landau subband structure at B = 350 T. . . . . . . . . . 82

3–19 Band structure near the Γ point for InAs calculated by eight-bandmodel and full zone thirty-band model. . . . . . . . . . . . . . . . . 82

3–20 The hole density dependence of hole CR. . . . . . . . . . . . . . . . . 84

3–21 Cyclotron resonance spectra for two ferromagnetic InMnAs/ GaSbsamples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

3–22 Theoretical CR spectra showing the shift of peak A with temperature. 87

3–23 Average localized spin as a function of temperature at B = 0, 20, 40,60 and 100 Tesla. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

3–24 Relative change of CR energy (with respect to that of high tempera-ture limit) as a function of temperature. . . . . . . . . . . . . . . . 89

3–25 Band diagram of InMnAs/GaSb heterostructure. . . . . . . . . . . . . 90

3–26 Schematic diagram of Landau levels and cyclotron resonance transi-tions in conduction and valence bands. . . . . . . . . . . . . . . . . 91

3–27 The valence band Landau levels and e-active hole CR. . . . . . . . . . 92

3–28 Experimental and theoretical hole CR absorption. . . . . . . . . . . . 93

3–29 Valence band structure at T = 30 K and B = 100 T for In1−xMnxAsalloys having x = 0% and x = 5% . . . . . . . . . . . . . . . . . . . 94

3–30 The primary transition in the e-active hole CR under different Mndoping. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4–1 Diagram for light reflection from the interface between medium 1 withrefractive index N1 and medium 2 with refractive index N2. . . . . 100

4–2 Schematic diagram for magnetic circular dichroism. . . . . . . . . . . 102

4–3 Diagrams for Kerr and Faraday rotation. . . . . . . . . . . . . . . . . 103

4–4 Kerr rotation of InMnAs. . . . . . . . . . . . . . . . . . . . . . . . . . 105

4–5 The band diagram for InAs. . . . . . . . . . . . . . . . . . . . . . . . 106

4–6 Kerr rotation of GaMnAs. . . . . . . . . . . . . . . . . . . . . . . . . 107

4–7 The band diagram for GaAs. . . . . . . . . . . . . . . . . . . . . . . . 108

4–8 The absorption coefficients both in InMnAs and GaSb layers (a) andthe reflectivity of InMnAs/GaSb heterostructure(b). . . . . . . . . 109

x

4–9 Reflectivity of In0.88Mn0.12As(9 nm)/GaSb(600 nm) heterostructureat T = 5.5 K measured by P. Fumagalli and H. Munekata. . . . . . 110

heterostructure under a magnetic field of 3 T atT= 5.5 K. 111

5–1 Light-induced MOKE. Signal decays in less than 2 ps. . . . . . . . . . 113

5–2 Light-induced magnetization rotation. . . . . . . . . . . . . . . . . . . 114

5–3 The heavy hole spin relaxation time as a function of wave vector (a),and temperature at the Γ point (b). . . . . . . . . . . . . . . . . . . 123

5–4 Spin relaxation time for a heavy hole as a function of k along (0,0,1)direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

5–5 Spin relaxation time of a heavy hole as a function of hole density atdirection (a) (0,0,1) and (b) (1,1,1). . . . . . . . . . . . . . . . . . . 127

xi

4–10 Measured (a) and calculated (b) Kerr rotation of InMnAs(19 nm)/AlSb(145 nm)

Abstract of Dissertation Presented to the Graduate Schoolof the University of Florida in Partial Fulfillment of theRequirements for the Degree of Doctor of Philosophy

THEORETICAL STUDIES OF THE ELECTRONIC, MAGNETO-OPTICALAND TRANSPORT PROPERTIES OF DILUTED MAGNETIC

SEMICONDUCTORS

By

Yongke Sun

December 2005

Chair: Christopher J. StantonMajor 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 Pidgeon-Brown model, and the magneto-optical 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

play 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 III-V DMS

xii

from paramagnetic to ferromagnetic; ii) g-factors 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 p-doped DMS arise from

the optical transitions of heavy-hole to heavy-hole and light-hole to light-hole

Landau levels, in lower and higher magnetic fields, respectively; iv) Electron-active

cyclotron resonance takes place in p-doped DMS samples. This is unusual since a

simple quasi-classical argument would suggest 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 magneto-optical Kerr rotation up to

several tenths of a degree occurs in DMS systems. The Kerr rotation in multilayer

structures depends on quantum confinement and multi-reflections 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.

xiii

CHAPTER 1INTRODUCTION 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

semi-magnetic semiconductors (SMSC).

Since the initial discovery of DMS in II-VI semiconductor compounds [1], more

than two decades have passed. The recent discovery of ferromagnetic DMS based

on III-V 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 played 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 says

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

1

2

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 ever-tinier 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 electron-spin

effects in magnetic materials composed of ultra-thin multilayers, 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 layers which let through more electrons when the spin orientations in

the two layers 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 1’s (magnetizations of the layers parallel

or antiparallel) as in a conventional transistor memory device. A straightforward

application could be in the magnetic random access memory (MRAM) device which

is non-volatile. These devices would be smaller, faster, cheaper, use less power and

would be much more robust in extreme conditions such as high temperature, or

high-level radiation environments.

3

Currently, besides continuing to improve the existing GMR-based technology,

people are now focusing on finding novel ways of both generating and utilizing spin-

polarized currents. This includes investigation of spin transport in semiconductors

and looking for ways 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 semiconductor-based devices

to be integrated with traditional semiconductor technology, and the semiconductor

based spintronic devices could in principle provide amplification, in contrast with

existing metal-based devices, and can serve as multi-functional 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 well-understood. Fortunately, there is another approach

to investigating spin control and transport in all-semiconductor devices. This

approach has become possible since the discovery of DMS.

The most common DMS studied in the early 1990s were II-VI 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 IV-VI

(e.g., PbS, SnTe) and most importantly, III-V (e.g., GaAs, InSb) crystals. Most

commonly, Mn ions are used as magnetic dopants.

4

1.2 The II-VI Diluted Magnetic Semiconductors

1.2.1 Basic Properties of II-VI Diluted Magnetic Semiconductors

The first II-VI DMS was grown in 1979 [1], and has been given a great deal of

attention ever since [6]. The most studied II-VI DMS materials are listed in Table

1–1.

Table 1–1: Some important II-VI DMS

Material Crystal Structure x range1

Hg1−xMnxTe Zinc-blende x ≤ 0.30Hg1−xMnxSe Zinc-blende x ≤ 0.30Cd1−xMnxTe Zinc-blende x ≤ 0.75Cd1−xMnxSe Wurtzite x ≤ 0.50Zn1−xMnxTe Zinc-blende x ≤ 0.75

1 x refers the range of x for which the crystals are usually studied. Whenx’s become relatively large, phases like MnTe or MnTe2 occur, and thecrystal qualities are poor [7].

The II-VI DMS have attracted so much attention since their discovery because

of the following important properties.

• Unique electronic properties: The wide variety of both host crystalsand magnetic atoms provides materials which range from wide gap to zerogap semiconductors, and reveal many different types of magnetic interaction.Several of the properties of these materials may be tuned by changing theconcentration of the magnetic ions. The bandgap, Eg, of Hg1−xMnxTe caneven change from negative to positive. This property becomes favorableas far as designing infrared devices is concerned. The dependence of Eg ofHg1−xMnxTe on x is given in Fig. 1–1 [8].

With the definition of the band gap as Eg = EΓ6 − EΓ8 , the band structuresof Hg1−xMnxTe with different x are given in Fig. 1–2 [8]. With x ≤ 0.075,Eg < 0, and with x > 0.075, Eg > 0. Without spin-orbital coupling, weshould have a six-fold degenerate valence band at the Γ point. Consideringspin-orbital coupling, the valence band splits into two bands-Γ7 and Γ8

(split-off band), with an energy difference of ∆.

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 ofelectrons can be very high. For instance, µ ∼= 106 cm2/V · s for Hg1−xMnxTewhen x ∼= 0.07 at 4.2 K.

5

Figure 1–1: The band gap dependence of Hg1−kMnkTe on Mn concentration k.Reprinted with permission from Bastard et al. Phys. Rev. B 24: 1961-1970, 1981.Figure 10, Page 1967.

• Broad phase behavior: With different Mn concentration x and temper-ature T, each II-VI DMS presents a different (phase) property, but their x-Tphase diagrams are very similar. Shown in Fig. 1–3 is the phase diagram ofCd1−xMnxTe obtained from specific heat and magnetic susceptibility measure-ments [9]. The DMS system may be considered as containing two interactingsubsystems. The first of these is the system of delocalized conduction andvalence band electrons/holes. The second is the random, diluted system oflocalized magnetic moments associated with the magnetic atoms. Thesetwo subsystems interact with each other by the spin exchange interaction.The fact that both the structure and the electronic properties of the hostcrystals are well known means that they are perfect for studying the basicmechanisms of the magnetic interactions coupling the spins of the bandcarriers and the localized spins of magnetic ions. The coupling between the

6

Figure 1–2: The band structures of Hg1−xMnxTe with different x. ∆ 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 24:1961-1970, 1981. Figure 1, Page 1961.

localized moments results in the existence of different magnetic phases such asparamagnets, spin glasses and antiferromagnets.

• Important magnetic phenomena: As described above, if we don’tconsider the spin exchange interaction between the band electrons andlocalized magnetic moments, DMS materials are just the same as the othersemiconductors. When we consider the spin exchange interaction, however,DMS materials present many important properties, such as very big Landeg-factors, extremely large Zeeman splitting of the electronic bands, giantFaraday rotation, and huge negative magnetoresistance. Therefore, to studyDMS, one has to first understand the spin exchange interaction between thelocalized magnetic ions and band electrons.

7

Figure 1–3: Cd1−xMnxTe x-T phase diagram. P: Paramagnet; A: Antiferromagnet;s: spin-glass, mixed crystal when x > 0.7. Reprinted with permission from GaÃlazkaet al. Phys. Rev. B 22: 3344-3355, 1980. Figure 12, Page 3352.

8

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 magneto-optical 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 II-VI

DMS.

Suppose the state of Mn ions in DMS material is Mn2+. The electronic struc-

ture of Mn2+ is 1s22s22p63s23p63d5, in which 3d5 is a half-filled 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 l = 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

Lande g-factor is

g = 1 +J(J + 1) + S(S + 1)− L(L + 1)

2J(J + 1)= 2. (1–1)

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

Hex = −Jσ · S, (1–2)

where σ 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 non-interacting paramagnetic phase, a very simplified model will be

described in the following. Since L = 0 for Mn2+, the magnetic momentum for

9

Mn2+ is µ = (−ge/2m0)J = (−ge/2m0)S. Assuming a magnetic field B along the

z direction, the additional energy in this field of a Mn2+ ion is −µ · B = gµBmsB,

where ms = 5/2, 3/2, 1/2, -1/2, -3/2, -5/2. Assuming non-interactive spins, and

using a classic Boltzman distribution function egµBmsB/kBT , the average magnetic

moment in the z direction is then

〈µz〉 =

∑5/2ms=−5/2(gµBms)e

−gµBmsB/kBT

∑5/2ms=−5/2 egµBmsB/kBT

. (1–3)

This can be written as

〈µz〉 = −gµBSBs(y), (1–4)

where Bs(y) is the Brillouin function

Bs(y) =2S + 1

2Scoth

(2S + 1

2Sy

)− 1

2Scoth

( y

2S

),

S = 5/2, y = gµBSB/kBT. (1–5)

The average spin of one Mn2+ ion then is

〈Sz〉 = −SBs(y). (1–6)

The antiparallel orientation of B and 〈Sz〉 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 (paramagnetic) dependence of 〈Sz〉on magnetic field and temperature is shown in Fig. 1–4.

From Eq. 1–2, the exchange Hamiltonian of one band electron with spin σ

interacting with the 3d5 electrons from all Mn2+ ions is,

Hex =∑

i

J(r−Ri)Si · σ, (1–7)

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

10

Figure 1–4: Average local spin as a function of magnetic field at 4 temperatures inparamagnetic phase.

with the 3d5 electrons in the ith Mn2+ ion. Si is the total angular momentum of

the 3d5 electrons in the ith Mn2+ ion.

Next we will use a virtual crystal approximation to deal with Hamiltonian

1–7. Due to the fact the the wave function of a band electron actually extends

over the whole crystal, 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〉 = 〈Sz〉, and Si · σ = 〈Sz〉mch2. mc = ±1/2 here indicates the spin quantum

number of the band electrons. The 〈Sz〉 is given by Eq. 1–6. The exchange

11

Hamiltonian then can be written as

Hex = 〈Sz〉mc

∑i

J(r−Ri). (1–8)

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〈Sz〉at each cation site. So, Eq. 1–8 becomes

Hex = x〈Sz〉mc

∑R

J(r−R). (1–9)

Here R becomes the position vector of each cation site. In Eq. 1–9 the exchange

Hamiltonian now has the same periodicity as the crystal.

From the Hamiltonian 1–9, the exchange energy can then be obtained by

Ecex = 〈ψck|Hex|ψck〉 = x〈Sz〉mc〈ψck|

∑R

J(r−R)|ψck〉. (1–10)

For the electrons at the conduction band edge, the wave function is ψck∼= uc0(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. 1–10. Considering the periodicity of

J(r−R), the Eq. 1–10 can be rewritten as,

Ecex = Nx〈Sz〉mc

∫

Ω

u∗c0J(r)uc0dr = αNx〈Sz〉mc, (1–11)

α =

∫

Ω

u∗c0J(r)uc0dr. (1–12)

where N is the number of unit cells in the crystal.

For zinc-blende semiconductors (most II-VI and III-V semiconductors), the

states for conduction band-edge (k = 0) electrons are s-like, and those for valence

12

band-edge holes are p-like. So the use of mc = ±1/2 is justified. Then in a

magnetic field B, the conduction band energy is,

Enc = (n +1

2)hωc + mcgcµBB + mcαNx〈Sz〉, (1–13)

where ωc = eB/m∗c is the cyclotron frequency, and gc is the conduction band

g-factor. In Eq. 1–13, 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 ωc by ωv = eB/m∗v, gc by gv, mc by mv, and importantly, α by β, where

β =

∫

Ω

u∗v0J(r)uv0dr. (1–14)

α and β are called exchange constants for s − d and p − d exchange interactions

between band electrons and localized Mn2+ ions.

We can introduce an effective Lande g-factor in the conduction band

geff = gc +αNx〈Sz〉

µBB, (1–15)

which indicates the strength of the spin splitting of the first Landau level in

the conduction band. In the low field approximation, Eq. 1–6 becomes 〈Sz〉 =

−gµBS(S + 1)B/3kBT , so in this limit

geff = gc − αNxgS2(S + 1)

3kBT. (1–16)

At low temperature, the effective g-factor can reach very large values. The

g-factor depends on temperature through 〈Sz〉 in Eq. 1–15. We will have a more

detailed discussion of g-factors in Chapter 2.

The above discussion is a very simplified qualitative model, and only appropri-

ate for II-VI DMS in a paramagnetic phase, where the Mn concentration is not so

13

high that they don’t have a direct exchange interaction. This discussion can also

be applied to paramagnetic III-V DMS, in which commonly the Mn solubility are

very low. As a matter of fact, although Eq. 1–13 can give a qualitative discription

of the conduction band structure, it does not work in real cases. Chapter 2 gives a

quantitative model.

Since the discovery of ferromagnetism in III-V DMS, much research now

focuses on exploring ferromagnetism mechanisms, looking for new materials and

obtaining higher Curie temperatures. Recently, ferromagnetism in II-VI DMS was

also reported by several groups [11, 12, 13].

1.3 The III-V Diluted Magnetic Semiconductors

1.3.1 Ferromagnetic Semiconductor

Although II-VI DMS combine both semiconducting and magnetic properties

and manifest spectacular properties, other characteristics such as ferromagnetism

are also desirable. From Eq. 1–15 and Eq. 1–16, we can see that at low tempera-

tures, the g-factor can be very large, but it is strongly temperature dependent. As

we mentioned above, the g-factor 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 II-VI 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 II-VI DMS to III-V DMS should have been very natural. But

unlike II-VI semiconductors, Mn is not very soluble in III-V semiconductors.

It can be incorporated only by non-equilibrium growth techniques and it was

not until 1992 that the first III-V DMS, InMnAs was grown and investigated.

Ferromagnetism was soon discovered in this system [14]. Higher ferromagnetic

14

Figure 1–5: Magnetic-field dependence of Hall resistivity ρHall and resistivity ρ ofGaMnAs with temperature as a parameter. Mn composition is x = 0.053. Theinset shows the temperature dependence of the spontaneous magnetization Ms

determined from magnetotransport measurements; the solid line is from mean-field theory. Reprinted with permission from Matsukura et al. Phys. Rev. B 57:R2037-R2040, 1998. Figure 1, Page R2037.

15

transition temperatures were also achieved in GaMnAs [15]. Shown in Fig. 1–5 is

the magnetic-field 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 III-V

DMS led to an explosion of interest [14, 15, 17, 18]. Many new matetials 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 III-V 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 Ga1−xMnxAs films with 8% nominal Mn doping after annealing [19].

The dependence of the Curie temperature of Ga1−xMnxAs on Mn concentration x

is shown in Fig. 1–6 [16]. The Curie temperature reaches the highest value when

x ∼= 5.3% 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 III-V DMS is still controversial, however,

there is consensus that it is mediated by the itinerant holes. Unlike the case in

II-VI DMS in which Mn ions have the same number of valence electrons as the

16

Figure 1–6: Mn composition dependence of the magnetic transition temperature Tc,as determined from transport data. Reprinted with permission from Matsukura etal. Phys. Rev. B 57: R2037-R2040, 1998. Figure 2, Page R2038.

cations, Mn ions in III-V DMS are not only providers of magnetic moments, they

are also acceptors. Due to compensating defects like As-antisites 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 relativelylarge distances. It is the dominant exchange interaction in metals where thereis little or no direct overlap between neighboring magnetic impurities. Ittherefore acts through an intermediary which in metals are the conductionelectrons (itinerant electrons) or holes. This type of exchange was firstproposed by Ruderman and Kittel [30] and later extended by Kasuya [31] andYosida [32] to give the theory now generally know as the RKKY interaction.Ohno et al. explained the ferromagnetism in GaMnAs for Mn concentrationx = 0.013 using the RKKY mechanism [14]. In the interaction Hamiltonian,

H = −JRKKYi,j Si · Sj, (1–17)

the coupling coefficient JRKKYi,j assumes the form[33],

JRKKY (r) ∼ [sin(2kF r)− 2kF rcos(2kF r)]/(2kF r)4 (1–18)

17

where kF is the radius of the conduction electron/hole Fermi surface, r isthe distance away from the origin where a local moment is placed. TheRKKY exchange coefficient, J , oscillates from positive to negative as theseparation of the ions changes with the period determined by the Fermiwavevector k−1

F and has the damped oscillatory nature shown in Fig. 1–7. Therefore, depending upon the separation between a pair of ions theirmagnetic coupling can be ferromagnetic or antiferromagnetic. A magnetic ioninduces a spin polarization in the conduction electrons in its neighborhood.This spin polarization in the itinerant electrons is felt by the moments ofother magnetic ions within range, leading to an indirect coupling.

Figure 1–7: Variation of the RKKY coupling constant, J , of a free electron gas inthe neighborhood of a point magnetic moment at the origin r = 0.

In the case of DMS, the average distance between the carriers rc =

(4πp

3

)− 13

is usually much greater than that between the spins rS =

(4πxN

3

)− 13

. A

simple calculation show that the first zero of the RKKY function occurs atr ≈ 1.17rc. This means that the carrier-mediated interaction is ferromagneticand effectively long range for most of the spins.

The RKKY interaction as the main mechanism for the ferromagnetism inIII-V DMS is questionable in some cases such as in the insulating phase(x < 3% for GaMnAs), in which carriers are not itinerant. When the holedensity is low, and there is no Fermi surface (Fermi level in the gap), RKKYtheory cannot predict ferromagnetism. The other problem, maybe fatal, isthat in the RKKY approximation the exchange energy is much smaller than

18

the Fermi energy, which is not commonly the case in DMS. As a matter offact, these two energies are comparable in most cases.

2. Zener’s model: Zener’s model is a continuous-medium limit of the RKKYmodel. Zener’s model was first proposed by C. Zener in 1950 [34] to interpretthe ferromagnetic coupling in transition metals. Similar to the RKKY model,it describes an exchange interaction between carriers and localized spins. TheHamiltonian of Zener’s model in a transition metal is [34]

Hspin =1

2αS2

d − βSsSc +1

2γS2

c , (1–19)

where Sd and Sc are the mean magnetization of the d-shell electron andthe conduction electron, respectively, and α, β, and γ are three couplingconstants. The main assumption here is that the exchange constant β isalways positive, which under certain circumstances leads to ferromagneticcoupling. Comparing Hamiltonian 1–17 and 1–19, we can see that β inEq. 1–19 plays the similar role of J in Eq. 1–17. One big difference is thatZener’s model neglects the itinerant character and the Friedel oscillations ofthe electron spin polarization around the localized spins.

Dietl [35] applied Zener’s model to ferromagnetic semiconductors andpredicted the Curie temperature TC for several Mn doped DMS systems.The results are shown in Fig. 1–8. This quite accurately predicts the 110 Ktransition temperature in GaMnAs, but certainly this is still a quite coarsemodel. Even so, the trend shown in Fig. 1–8 has stimulated the enthusiasmof people investigating GaN based materials looking for higher transitiontemperatures.

Some of the problems in the RKKY model remain in Zener’s model. Forinstance, Zener’s model still has limited application when carriers are mostlylocalized because it still requires itinerant carriers to mediate the interactionsbetween localized spins. Besides, when the carrier density is higher thanthe Mn concentration, important changes in the hole response functionoccur at the length scale of the mean distance between the localized spins.Accordingly, the description of spin magnetization by the continuous-mediumapproximation, which constitutes the basis of the Zener model, ceases to bevalid. 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 undercertain conditions. In DMS, localized moments can also be aligned in thevicinity of carriers to form what are known as “magnetic polarons”. Thecarrier spin creates an effective exchange field for the magnetic ions due tothe exchange interaction which is similar in form to Eq. 1–7, and this fieldcauses ferromagnetic coupling of these local spins. The net spin alignment

19

Figure 1–8: Curie temperatures for different DMS systems. Calculated by Dietlusing Zener’s model.

again creates a self-consistent exchange field for the carriers. In this process,the carrier spin creates a magnetic potential well resulting in formation ofa “spin cloud”, a magnetic polaron. Due to the localized character of thesemagnetic polarons in DMS, they are called bound magnetic polarons (BMP).

There have been extensive studies of BMP in II-VI DMS [6], in which BMPare accountable for many optical and phase transition properties. Recently,Bhatt et al. [36] and Das Sarma et al. [37] generalized BMP theory for III-VDMS. They studied the coupling between two adjacent BMPs, and concludedthat the exchange coupling is ferromagnetic. There are two different cases.In one case two polarons overlap and the overlap integral accounts for theferromagnetic coupling. The ferromagnetic transition can be regarded as apercolation occurring through the whole system when the temperature dropsbelow the Curie temperature. In the other case one does not need overlappingpolarons, their effect on the magnetic moment being taken into accountthrough a local magnetic field. Ferromagnetic coupling has been shown toresult when the carrier is allowed to hop between the ground state of onemagnetic atom and excited states of the other. A diagram of these two casesare shown in Fig. 1–9.

The BMP model quite naturally and successfully explains the magnetism ofthe DMS in the insulating phase. With a much higher carrier density, most

20

Figure 1–9: 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 carriersare localized and have exchange interaction with the localized spins, mostcarriers have extended wave functions, which tend to interact with the othercarriers and spins in the whole band. The condition for the BMP model doesnot exist any more. In such a case, the RKKY mechanism should dominate.

4. Double exchange theory: Double exchange can be considered as chargetransfer exchange which leads to ferromagnetism in ferromagnetic perovskitesSuch as LaMnO3. Akai et al. [38] performed first principle DFT calculationswhich show that the majority of the carriers comes from Mn d states. Thehopping of the carriers between the impurity bands and valence bands causesthe ferromagnetic ordering. Later, Inoue et al. [39] also discussed a similarmechanism. They calculated the electronic states of III-V DMS and foundthat resonant states were formed at the top of the down spin valence banddue to magnetic impurities and the resonant states gave rise to a stronglong-ranged ferromagnetic coupling between Mn moments. They proposedthat coupling of the resonant states, in addition to the intra-atomic exchangeinteraction between the resonant and nonbonding states was the origin of theferromagnetism of GaMnAs. We can classify this kind of mechanism causedby the hopping of carriers between impurity states and valence states as adouble exchange mechanism. Double-exchange-like interactions in GaMnAswere reported by Hirakawa et al. [40].

In the four models of ferromagnetism in III-V DMS, the first three are mean-

field based theories, and the last is based on d-electrons. Though each of them is

21

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 Heisenberg-like Hamiltonian

H = −∑

i6=j

Ji,j(Si · Sj), (1–20)

where i, j specify atomic sites, say, of the magnetic moments in the crystal, and

Ji,j is the interatomic exchange interaction constant. The molecular field (effective

mean field) is simply given by

Bex =1

gµB

∑j

Ji,j〈Sj〉, (1–21)

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

〈Sz〉 = −SBs(y), (1–22)

with Bs, the Brillouin function, given by Eq. 1–5, and where

y = [gµBS(B + Bex)]/kBT. (1–23)

After substitution of Eq. 1–22 to above equation, we get

y = [gµBSB + J0S〈Sz〉]/kBT, (1–24)

J0 =∑

j

Ji,j. (1–25)

22

Figure 1–10: Average local spin as a function of magnetic field at 4 temperatures.The Curie temperature is 110K.

Equation 1–22 can be solved by standard root finding programs to find 〈Sz〉.The solution for 〈Sz〉 6= 0 exists even when B = 0 due to the internal exchange

field. When |〈Sz〉| ¿ 1,

Bs(y) ∼ 1

3(S + 1)y. (1–26)

When J0 > 0, the condition for 〈Sz〉 6= 0 then is

T < TC = J0S(S + 1)/3kB. (1–27)

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. 1–27 to find the exchange interaction constant J0, and thus

the spontaneous magnetization for T < TC .

23

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 Chapter 5.

1.4 Open Questions

Although the research of III-V 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 III-V 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 III-V DMS

will introduce host-like-hole states that will interact via RKKY-type 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 host-like, which undermines

the basis of applying RKKY theory to DMS. The ensuing ferromagnetism by

the holes induced by Mn ions is then not RKKY-like, but “has a characteristic

dependence on the lattice-orientation of the Mn-Mn interactions in the crystal

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

24

hopping. The nature of the ferromagnetism then is closely related to the nature

of the band electrons. Photo-induced 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

host-like 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

predicable.

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 employed

designing novel devices. Semiconductor optical isolators based on II-VI DMS,

CdMnTe, which has a low absorption and large Faraday rotation for light with

0.98 µm wavelength, have been developed. This is the first commercial semiconduc-

tor spintronic device [45]. Since II-VI DMS is paramagnetic at room temperature, a

magnetic field is needed to obtain Faraday rotation. Ferromagnetic semiconductor

based on III-V 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.

Photo-induced ferromagnetism has been demonstrated by Koshihara et al.

[44] and Kono et al. [46]. In Koshihara’s experiment, ferromagnetism is induced

by photo-generated carriers in InMnAs/GaSb heterostructures. The effect is

illustrated in Fig. 1–11. Due to the special band alignment of this heterostructure,

electrons and holes are spacially separated, and holes accumulate in the InMnAs

layer. The photo-generated holes then cause a transition of the InMnAs layer to

a ferromagnetic state. This opens a possibility to realize optically controllable

magneto-optical devices. In Kono’s experiment, ultrafast demagnetization takes

place after a laser pulse shines on InMnAs/GaSb heterostructure and produces

25

ferromagnetism. The time scale is typically of several ps. They propose a new and

very fast scheme for magneto-optical recording.

Figure 1–11: The photo-induced ferromagnetism in InMnAs/GaSb heterostructure.Reprinted with permission from Koshihara et al. Phys. Rev. Lett. 78: 4617-4620,1997. Figure 3, Page 4619.

Recently, Ohno et al. [2] achieved control of ferromagnetism with an electric

field. They used field-effect transistor structures to vary the hole concentrations in

DMS layers and thus turn the carrier-induced 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

26

electric control of ferromagnetism or spin states makes possible a unification of

magnetism and conventional electronics, and thus has a profound meaning.

Low-dimensional 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 suggested for use in quantum

computers where quantum dots can be used as quantum bits, since they offer a

two-level 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, low-dimensional structures are very easy to construct, so unique

low-dimensional properties can be employed. 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 spin-polarized electrons is injected from the diluted

magnetic semiconductor BexMnyZn1−x−ySe into a GaAs/GaAlAs light-emitting

diode. Circularly polarized light is emitted from the recombination of the spin

polarized electrons with non-polarized holes. An injection efficiency of 90% spin

polarized current has been demonstrated. As BexMnyZn1−x−ySe is paramagnetic,

the spin polarization is obtained only in an applied field and at low temperature.

27

Figure 1–12: Spin light emitting diode.

A ferromagnetic III-V DMS based spin injector does not need an applied

field. Shown in the left panel of Fig. 1–13 is a GaMnAs-based spin injection and

detection structure [55], in which spin-polarized holes are injected from GaMnAs

to a GaAs quantum well. The emitter and analyzer are both made of layers 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 DMS-based 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.

28

Figure 1–13: GaMnAs-based spin device. Left: GaMnAs-based spin injector andanalyzer structure. Right: Temperature dependence of spin life in the GaAs quan-tum well in the structure shown in left panel.

CHAPTER 2ELECTRONIC 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 Γ

point. Specifically, a generalized Pidgeon-Brown 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

29

30

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 splittings. Shown in Fig. 2–1 are

calculated valence band structures for Bext = 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.

Figure 2–1: Valence band structure of GaAs and ferromagnetic Ga0.94Mn0.06Aswith no external magnetic field, calculated by generalized Kane’s model. The spinsplitting 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 k0, which

31

is an extremum of the band structure. In the case of the band structure near the Γ

point in a direct bandgap semiconductor, k0 = 0.

The Hamiltonian for an electron in a semiconductor can be written as

H =p2

2m0

+ V (r), (2–1)

here p = −ih∇ is the momentum operator, m0 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), (2–2)

where R is an arbitrary lattice vector, the solution of the Schrodinger equation

Hψk(r) = Eψk(r) (2–3)

satisfies the condition

ψk(r) = eik·ruk(r) (2–4)

where

uk(r + R) = uk(r), (2–5)

and k is the wave vector. Equations 2–4 and 2–5 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. 2–3 split into a series of bands [64]. Consider the

Schrodinger equation in the nth band with a wave vector k,

[p2

2m0

+ V (r)

]ψnk(r) = En(k)ψnk(r). (2–6)

Inserting the Bloch function Eq. 2–4 into Eq. 2–6, we obtain

[p2

2m0

+h

m0

k · p +h2k2

2m0

+ V (r)

]unk(r) = En(k)unk(r). (2–7)

32

In most cases, spin-orbit coupling must also be considered and added into the

Hamiltonian. The spin-orbit interaction term is

h

4m20c

2(σ ×∇V ) · p. (2–8)

Including the spin-orbit interaction, Eq. 2–7 becomes

[p2

2m0

+hk

m0

·(p +

h

4m0c2(σ ×∇V )

)+

h

4m20c

2(σ ×∇V ) · p +

h2k2

2m0

+ V (r)

]unk(r)

= En(k)unk(r).

(2–9)

The Hamiltonian in Eq. 2–9 can be divided into two parts

[H0 + W (k)]unk = Enkunk, (2–10)

where

H0 =p2

2m0

+h

4m20c

2(σ ×∇V ) · p + V (r) (2–11)

and

W (k) =hk

m0

·(p +

h

4m0c2(σ ×∇V )

)+

h2k2

2m0

. (2–12)

Only W (k) depends on wave vector k.

If the Hamiltonian H0 has a complete set of orthonormal eigenfunctions at

k = 0, un0, i.e.,

H0un0 = En0un0, (2–13)

then theoretically any lattice periodic function can be expanded using eigenfunc-

tions un0. Substituting the expression

unk =∑m

cnm(k)um0 (2–14)

33

into Eq. 2–9, and multiplying from the left by u∗n0, and integrating and using the

orthonormality of the basis functions, we obtain

∑m

[(En0 − Enk +

h2k2

2m0

)δnm +

hk

m0

· 〈un0|(p +

h

4m0c2(σ ×∇V )

)|um0〉

]cnm(k) = 0.

(2–15)

Solving this matrix equation gives us both the exact eigenstates and eigenenergies.

Usually, people only consider the energetically adjacent 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 non-diagonal terms in Eq. 2–15, the

eigenfunction is unk = un0, and the corresponding eigenvalue is given by Enk =

En0 + h2k2

2m0. This solution can be improved by second order perturbation theory, i.e.

Enk = En0 +h2k2

2m0

+∑

m6=n

〈un0|H ′|um0〉〈um0|H ′|un0〉En0 − Em0

, (2–16)

where

H ′ =hk

m0

·(p +

h

4m0c2(σ ×∇V )

). (2–17)

In the calculation shown above, we used the property 〈un0|(p + h

4m0c2(σ×∇V )

)|un0〉 =

0, which holds for a cubic lattice periodic Hamiltonian due to the crystal symmetry.

If we write

π = p +h

4m0c2(σ ×∇V ) (2–18)

then the second order eigenenergies can be written as

Enk = En0 +h2k2

2m0

+h2

m20

∑

m6=n

|πnm · k|2En0 − Em0

. (2–19)

34

Equation 2–19 is often written as

Enk = En0 +h2

2

∑

α,β

(1

m∗

)

αβ

kαkβ, (2–20)

where

1

m∗ =1

m0

δαβ +2

m20

∑

m6=n

παmnπβ

nm

En0 − Em0

(2–21)

is the inverse effective mass tensor, and α, β = x, y, z. The effective mass generally

is not isotropic, but we can see it is not k-dependent, this is because at this level

of approximation, the eigenenergies in the vicinity of the Γ 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. 2–15 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 split-off 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. 2–2 is the band structure of a typical III-V direct band gap

semiconductor. Due to crystal symmetry, the conduction band bottom belongs to

the Γ6 group, the valence band top belongs to the Γ8 group, and the split-off band

belongs to the Γ7 group. The spatial part of the wave functions at the conduction

band edge are s-like and those at the valence band top are p-like. Symbols of

|S〉, |X〉, |Y 〉, and |Z〉 are used to represent the one conduction band edge and

35

Figure 2–2: Band structure of a typical III-V semiconductor near the Γ point.Kane’s model considers the doubly spin degenerate conduction, heavy hole, lighthole and split-off 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 |S ↑〉, |S ↓〉, |X ↑〉, |X ↓〉,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

36

and mJ , as the basis states for the eight-band Kane’s model,

u1 = |12,1

2〉 = |S ↑〉 = |S ↑〉,

u2 = |32,3

2〉 = |HH ↑〉 =

1√2|(X + iY ) ↑〉,

u3 = |32,−1

2〉 = |LH ↓〉 =

1√6|(X − iY ) ↑ +2Z ↓〉,

u4 = |12,−1

2〉 = |SO ↓〉 =

i√3| − (X − iY ) ↑ +Z ↓〉,

u5 = |12,−1

2〉 = |S ↓〉 = |S ↓〉,

u6 = |32,−3

2〉 = |HH ↓〉 =

i√2|(X − iY ) ↓〉,

u7 = |32,1

2〉 = |LH ↑〉 =

i√6|(X + iY ) ↓ −2Z ↑〉,

u8 = |12,−1

2〉 = |SO ↓〉 =

i√3| − (X − iY ) ↑ +Z ↓〉. (2–22)

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 2–11. Because of spin degeneracy at k = 0, the eigenenergies for |S〉,|HH〉, |LH〉 and |SO〉 are Eg, 0, 0, −∆, respectively, with the selection of energy

zero at the top of Γ8 band, where Eg is the band gap, and

∆ =3ih

4m20c

2〈X|∂V

∂xpy − ∂V

∂ypx|Y 〉, (2–23)

is the split-off band energy.

At this level of approximation, the bands are still flat because the Hamiltonian

3–10 is k-independent. Including W (k) in Eq. 2–12 into the Hamiltonian, and

defining Kane’s parameter as

P =−ih

m0

〈S|πz|Z〉, (2–24)

37

we obtain a matrix expression for the Hamiltonian H = H0 + W (k), i.e.,

Eg + h2k2

2m0Pk+ − 1√

3Pk− −

√23Pk− 0 0 −

√23Pkz

1√3Pkz

Pk− h2k2

2m00 0 0 0 0 0

− 1√3Pk+ 0 h2k2

2m00 −

√23Pkz 0 0 0

−√

23Pk+ 0 0 −∆ + h2k2

2m0

1√3Pkz 0 0 0

0 0 −√

23Pkz

1√3Pkz Eg + h2k2

2m0Pk− 1√

3Pk+

√23Pk+

0 0 0 0 Pk+h2k2

2m00 0

−√

23Pkz 0 0 0 1√

3Pk− 0 h2k2

2m00

1√3Pkz 0 0 0

√23Pk− 0 0 −∆ + h2k2

2m0

(2–25)

where k+ = kx + iky, k− = kx − iky, and kx, ky, kz are the cartesian components

of k. The Hamiltonian 2–25 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,

Ec = Eg +h2k2

2mc

,1

mc

=1

m0

+4P 2

3h2Eg

+2P 2

3h2(Eg + ∆). (2–26)

For the light hole and split-off bands,

Elh = − h2k2

2mlh

,1

mlh

= − 1

m0

+4P 2

3h2Eg

; (2–27)

Eso = −∆− h2k2

2mso

,1

ms0

= − 1

m0

+2P 2

3h2(Eg + ∆). (2–28)

For the heavy hole band we have

Ehh =h2k2

2mhh

,1

mhh

=1

m0

. (2–29)

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

38

effect of the distant band coupling will make the heavy hole band curve downward

rather than upward.

2.2.3 Coupling with Distant Bands-Luttinger Parameters

The coupling with distant bands can be parameterized by Lowdin’s pertur-

bation method [65], in which the bands are classified as A and B. In our case, we

select the basis states 2–22 as class A and label them with subscript n and all the

other (energetically distant) states as class B which we label with subscript α.

Suppose all states are orthonormal, the Schrodinger equation then takes the

form∑

l

(Hlm − Eδlm)am = 0, (2–30)

where l and m run over all states. Rewrite this equation using class A and B, and

we obtain

(E −Hmm)am =A∑

n6=m

Hmnan +B∑

α 6=m

Hmαaα (2–31)

or

am =A∑

n 6=m

Hmn

E −Hmm

an +B∑

n 6=α

Hmα

E −Hmm

aα, (2–32)

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,

am =A∑n

UAmn −Hmnδmn

E −Hmm

an (2–33)

and

UAmn = Hmn +

B∑

α 6=m

HmαHαn

E −Hαα

+B∑

α,β 6=m,nα 6=β

HmαHαβHβn

(E −Hαα)(E −Hββ)+ · · · (2–34)

A little algebra shows that Eq. 2–33 is equivalent to

A∑n

(UAmn − Eδmn)an = 0. (2–35)

39

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. 2–35. The effect from class B is

treated as a perturbation using Eq. 2–34 to second order.

Truncating UAmn to the second term, and using Hamiltonian in Eq. 2–9, it can

be rewritten as

UAmn = Hmn +

B∑

α 6=m,n

HmαHαn

E0 −Hα

= Hmn +B∑

α 6=m,n

H ′mαH ′

αn

E0 −Hα

(2–36)

where

Hmn = 〈um0|H|un0〉 =

[Em(0) +

h2k2

2m0

]δmn (2–37)

and

H ′mα = 〈um0| h

m0

k · π|uα0〉 ∼=∑

a

hka

m0

pamα, (2–38)

where a = x, y, z and πamα ' pa

mα for m ∈ A and α ∈ B. Thus

UAmn =

[Em(0) +

h2k2

2m0

]δmn +

h2

m20

B∑

α 6=m,n

∑

a,b

kakbpamαpb

αn

E0 − Eα

. (2–39)

Applying basis set 2–22, we can define parameters A, B, C and F as follow,

A =h2

2m0

+h2

m20

B∑α

pxXαpx

αX

Ev − Eα

,

B =h2

2m0

+h2

m20

B∑α

pyXαpy

αX

Ev − Eα

,

C =h2

m20

B∑α

pxXαpy

αY + pyXαpx

αY

Ev − Eα

,

F =1

m0

B∑α

pxSαpx

αS

Ec − Eα

(2–40)

Rewriting these parameters in terms of “Luttinger” parameters γ1, γ2, γ3 and γ4

defined as

40

− h2

2m0

γ1 =1

3(A + 2B), (2–41)

− h2

2m0

γ2 =1

6(A−B), (2–42)

− h2

2m0

γ3 =C

6, (2–43)

γ4 = 1 + 2F, (2–44)

we can obtain the the Hamiltonian Hmn = Umn including the distant band coupling

under the basis set listed in Eq. 2–22 as

Eg + h2k2

2m0γ4

i√2V k+

i√6V k− 1√

3V k− 0 0

√23V kz

i√3V kz

− i√2V k− −P −Q −M i

√2M 0 0 −L − i√

2L

− i√6V k+ −M+ −P + Q i

√2Q −i

√23V kz L 0 i

√32L+

1√3V k+ −i

√2M+ −i

√2Q −P −∆ − 1√

3V kz − i√

2L i

√32L+ 0

0 0 i√

23V kz − 1√

3V kz Eg + h2k2

2m0γ4 − 1√

2V k− − 1√

6V k+

i√3V k+

0 0 L+ i√2L+ − 1√

2V k+ −P −Q −M+ i

√2M+

√23V kz −L+ 0 −i

√32L − 1√

6V k− −M −P + Q i

√2Q

− i√3V kz

i√2L+ −i

√32L 0 − i√

3V k− −i

√2M −i

√2Q −P −∆

,

(2–45)

where

k2 = k2x + k2

y + k2z ,

P =h2

2m0

γ1k2,

Q =h2

2m0

γ2(k2x + k2

y − 2k2z),

L = −ih2

m0

√3γ3(kx − iky)kz,

M =h2

2m0

√3(γ2(k

2x − k2

y)− 2iγ3kxky

),

V =

√h2

m0

Ep

2, (2–46)

41

and

Ep =2m0

h2 P 2 (2–47)

related to the Kane’s parameter P defined in Eq. 2–24. We can see that if k = 0 or

kz = 0, the Hamiltonian is block diagonalized.

In practice, one important thing needs to be noted that the Luttinger parame-

ters defined in Eq. 2–44 are not the “usual Luttinger” parameters which are based

on a six-band model since this is an eight-band model, but instead are related to

the usual Luttinger parameters γL1 , γL

2 , and γL3 through the relations [66]

γ1 = γL1 −

Ep

3Eg

,

γ2 = γL2 −

Ep

6Eg

,

γ3 = γL3 −

Ep

3Eg

. (2–48)

This takes into account the additional coupling of the valence bands to the

conduction band not present in the six-band Luttinger model. We refer to γ1

etc. as the renormalized Luttinger parameters.

The Hamiltonian 2–45 is based on an eight-band 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

1

mc

=1

m0

(γ4 +

Ep

3

[2

Eg

+1

Eg + ∆

]). (2–49)

In DMS materials without magnetic fields, the Hamiltonian 2–45 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 magneto-optical Kerr

effect, which is to be studied in chapter 4.

42

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 six-band Luttinger model), all the bands

in class A are considered as degenerate at the Γ point. Away from the Γ 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

Hmn = Umn =

[Em(0) +

h2k2

2m0

]δmn +

h2

m20

B∑

α 6=m,n

∑

a,b

kakbpamαpb

αn

E0 − Eα

= Em(0)δmn +∑

a,b

Dabmnkakb,

(2–50)

where

Dabmn =

h2

2m0

[δmnδab +

B∑α

pamαpb

αn + pbmαpa

αn

m0(E0 − Eα)

], (2–51)

the eigenequation is given by

8∑n=1

Hmnan(k) =8∑

n=1

[Em(0)δmn +

∑

a,b

Dabmnkakb

]an(k) = E(k)am(k) (2–52)

where am is the superposition coefficients defined as

ψnk(r) =∑

n

an(k)un0. (2–53)

Now we consider a spatial perturbation U(r) added to the Hamiltonian Hmn.

The eigenequation now becomes

[H + U(r)]ψ(r) = Eψ(r). (2–54)

43

If we write the solution to the equation as

ψ(r) =8∑

m=1

Fm(r)um0(r), (2–55)

Luttinger and Kohn [67] have shown that we need only solve the following equa-

tion,

8∑n=1

[Em(0)δmn +

∑mn

Dabmn

(−i

∂

∂xa

)(−i

∂

∂xb

)+ U(r)δmn

]Fn(r) = EFm(r)

(2–56)

This means that we only need to replace the wave vector in the Hamiltonian ka 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

2m(−ih∇+ eA)2ψ(x) = εψ(x), (2–57)

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. (2–58)

and assuming a solution like

ψ(x, y, z) =1√

LxLz

ei(kxx+kzz)φ(y), (2–59)

44

where Lx, Ly, and Lz are lengths for the bulk crystal in three dimensions. After

substituting into the effective mass equation, we have an equation for φ(y),

1

2m

[(kx − eBy

h)2 − h2 ∂2

∂y2+ h2k2

z

]φ(y) = εφ(y). (2–60)

Defining ε′ ≡ ε − h2k2z

2m, the equation above is a simple harmonic oscillator equation

with

ε′ = (n +1

2)hωc, (2–61)

where ωc = eB/m is the cyclotron frequency. Thus the total energy is

ε = (n +1

2)hωc +

h2k2z

2m. (2–62)

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 z-direction

is still continuous. The original states in one band now split into a series of Landau

levels whose eigenfunctions are

ψnkxkz =1√

LxLz

φn

(y − hkx

eB

)ei(kxx+kzz). (2–63)

The electronic energies in Eq. 2–62 is only related to n and kz. They are

degenerate for different kx. In Eq. 2–63 the center of y0 = hkx

eBcan only be from 0

to Ly. Using the periodic boundary condition, the interval for kx is 2π/Lx, thus the

interval for y0 is h/eBLx, the corresponding number of values for y0 is eBLxLy/h.

Therefore, for given n and kz, the degeneracy is eBLxLy/h.

2.3.2 Generalized Pidgeon-Brown 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

45

valence bands is strong, 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 (kz) 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. 2–22. 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 =1

h

(p +

e

cA

), (2–64)

where p = −ih∇ is the momentum operator. For the vector potential, we still use

the Landau gauge as in Eq. 2–58, thus B =∇×A = Bz.

Now we introduce two operators

a† =λ√2(kx + iky) (2–65a)

and

a =λ√2(kx − iky) (2–65b)

where λ is the magnetic length which is defined as

λ =

√hc

eB=

√h2

2m

1

µBB. (2–66)

The operators defined in Eqs. 2–65 obey the commutation rules of creation and

annihilation operators. The states they create and annihilate are simple harmonic

oscillator functions, and a†a = N are the order of the harmonic functions. Using

these two operators to eliminate kx and ky in Hamiltonian 2–45, we arrive at the

Landau Hamiltonian

HL =

La Lc

L†c Lb

, (2–67)

46

with the submatrices La, Lb and Lc given by

La =

Eg + A iVλa i

√13

Vλa†

√23

Vλa†

−iVλa† −P −Q −M i

√2M

−i√

13

Vλa −M † −P + Q i

√2Q

√23

Vλa −i

√2M † −i

√2Q −P −∆

(2–68)

Lb =

Eg + A −Vλa† −

√13

Vλa i

√23

Vλa

−Vλa −P −Q −M † i

√2M †

−√

13

Vλa −M −P + Q i

√2Q

−i√

23

Vλa† −i

√2M −i

√2Q −P −∆

(2–69)

Lc =

0 0√

23V kz i

√13V kz

0 0 −L −i√

12L

−i√

23V kz L 0 i

√32L†

−√

13V kz −i

√12L i

√23L† 0

(2–70)

The operators A, P , Q, L, and M in Eq. 2–67 now are

A =h2

m0

γ4

2

(2N + 1

λ2+ k2

z

), (2–71a)

P =h2

m0

γ1

2

(2N + 1

λ2+ k2

z

), (2–71b)

Q =h2

m0

γ2

2

(2N + 1

λ2− 2k2

z

), (2–71c)

L =h2

m0

γ3

(−i√

6 kza

λ

), (2–71d)

and

M =h2

m0

(γ2 + γ3

2

) (√3

λ2a2

). (2–71e)

47

The parameters γ1, γ2, γ3 and γ4 are defined in Eq. 2–48 and 2–44. Usually,

the Luttinger parameters γ2 and γ3 are approximately equal (spherical approxima-

tion), so we have neglected a term in M proportional to (γ2 − γ3)(a†)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 non-zero angular momentum (thus a non-zero magnetic

moment µ) in a magnetic field, the energy due to the interaction between the

magnetic moment and the magnetic field is −µ · B, which is called Zeeman energy

which we discussed in Section 1.2.2. The electrons in III-V DMS conduction or

valence bands possess both orbital angular momenta and spin, so there is one extra

Zeeman term proportional to (K0L · B + K1σ · B), where L and σ 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 κ as

κ = κL − Ep

6Eg

(2–72)

where

κL = γL3 +

2

3γL

2 −1

3γL

1 −2

3(2–73)

is the Luttinger κ parameter, and we obtain the Zeeman Hamiltonian

HZ =h2

m0

1

λ2

Za 0

0 −Z∗a

(2–74)

where the 4× 4 submatrix Za is given by

Za =

−12

0 0 0

0 −32κ 0 0

0 0 12κ −i

√12κ

0 0 i√

12κ κ

. (2–75)

48

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 (∑

I J(r−RI)SI ·σ). Under a mean field and virtual crystal

approximation (see Section 1.2.2), and defining the two exchange constants

α =1

Ω〈S|J |S〉 (2–76a)

and

β =1

Ω〈X|J |X〉, (2–76b)

we can arrive at an exchange Hamiltonian

HMn = x N0 〈Sz〉

Da 0

0 −D∗a

(2–77)

where x is the Mn concentration, N0 is the number of cation sites in the sample,

and 〈Sz〉 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× 4 submatrix Da is

Da =

12α 0 0 0

0 12β 0 0

0 0 −16β −i

√2

3β

0 0 i√

23

β −12β

. (2–78)

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

49

similar to that discussion, the total Hamiltonian here can be written as

H = HL + HZ + HMn. (2–79)

We note that at kz = 0, the effective mass Hamiltonian is also block diagonal like

the Hamiltonian 2–45.

2.3.3 Wave Functions and Landau Levels

With the choice of Gauge 2–58, translational symmetry in the x direction is

broken while translational symmetry along the y and z directions is maintained.

Thus ky and kz are good quantum numbers and the envelope of the effective mass

Hamiltonian 2–79 can be written as

Fn,ν =ei(kyy+kzz)

√A

a1,n,ν φn−1

a2,n,ν φn−2

a3,n,ν φn

a4,n,ν φn

a5,n,ν φn

a6,n,ν φn+1

a7,n,ν φn−1

a8,n,ν φn−1

(2–80)

In Eq. 2–80, n is the Landau quantum number associated with the Hamilto-

nian matrix, ν labels the eigenvectors, A = LxLy is the cross sectional area of the

sample in the x− y plane, φn(ξ) are harmonic oscillator eigenfunctions evaluated at

ξ = x− λ2ky, and ai,ν(kz) are complex expansion coefficients for the ν-th eigenstate

which depend explicitly on n and kz. Note that the wave functions themselves will

be given by the envelope functions in Eq. 2–80 with each component multiplied by

the corresponding kz = 0 Bloch basis states given in Eq. 2–22.

50

Substituting Fn,ν from Eq. 2–80 into the effective mass Schrodinger equation

with H given by Eq. 2–79, we obtain a matrix eigenvalue equation

Hn Fn,ν = En,ν(kz) Fn,ν , (2–81)

that can be solved for each allowed value of the Landau quantum number, n, to

obtain the Landau levels En,ν(kz). The components of the normalized eigenvectors,

Fn,ν , are the expansion coefficients, ai.

Since the harmonic oscillator functions, φn′(ξ), are only defined for n′ ≥ 0,

it follows from Eq. 2–80 that Fn,ν is defined for n ≥ −1. The energy levels are

denoted En,ν(kz) where n labels the Landau level and ν labels the eigenenergies

belonging to the same Landau level in ascending order.

Table 2–1: Summary of Hamiltonian matrices with different n

n Dimension of Hamiltonian Eigenenergy No. Label as

-1 1× 1 1 (−1, 1)0 4× 4 4 (0, ν), ν = 1 · · · 41 7× 7 7 (1, ν), ν = 1 · · · 7≥ 2 8× 8 8 (n, ν), ν = 1 · · · 8

For n = −1, we set all coefficients ai to zero except for a6 in order to prevent

harmonic oscillator eigenfunctions φn′(ξ) with n′ < 0 from appearing in the

wavefunction. The eigenfunction in this case is a pure heavy hole spin-down state

and the Hamiltonian is now a 1 × 1 matrix whose eigenvalue corresponds to the a

heavy hole spin-down 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 a1 = a2 = a7 = a8 = 0 and the Landau levels

and envelope functions are then obtained by diagonalizing a 4 × 4 Hamiltonian

matrix obtained by striking out the appropriate rows and columns. For n = 1, the

51

Hamiltonian matrix is 7 × 7 and for n ≥ 2 the Hamiltonian matrix is 8 × 8. The

summary of Hamiltonian matrices for different n is given in Table 2–1.

The matrix Hn in Eq. 2–81 is the sum of Landau, Zeeman, and exchange

contributions. The explicit forms for the Zeeman and exchange Hamiltonian

matrices are given in Eq. 2–74 and 2–77 and are independent of n.

Table 2–2: InAs band parameters

Energy gap (eV)1

Eg (T = 30 K) 0.415Eg (T = 77 K) 0.407Eg (T = 290 K) 0.356Electron effective mass (m0)m∗

e 0.022Luttinger parameters 1

γL1 20.0

γL2 8.5

γL3 9.2

κL 7.53Spin-orbit splitting (eV) 1

∆ 0.39Mn s-d and p-d exchange energies (eV)N0 α -0.5N0 β 1.0Optical matrix parameter (eV) 1

Ep 21.5Refractive index 2

nr 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 2–2. Shown in Fig. 2–3 are the conduction band Laudau levels for InAs

and In0.88Mn0.12As as a function of magnetic field at k = 0 for a temperature of

30 K. The dashed lines represent spin-up levels, and the solid lines represent the

spin-down levels. This illustrates the energy splitting of the conduction band at

the Γ point. The right panel for InMnAs is only different from the left panel for

52

Figure 2–3: Calculated Landau levels for InAs (left) and In0.88Mn0.12As (right) as afunction 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. 1–13 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 α (exchange constant defined in Eq. 2–76) dependence of

the cyclotron energy, which is the energy difference between two adjacent 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. 2–4, 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. 2–3 and Fig. 2–4, 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

Figure 2–4: 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 andIn0.88Mn0.12As, respectively.

2.4 Conduction Band g-factors

In practice, spin-splitting is represented the g-factor. For a free electron, the

g-factor is the ratio between the magnetic moment due to spin in units of µB and

the angular momentum in units of h. The g-factor 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 spin-orbital interaction (and other interactions, for

example in DMS, the exchange interaction), the g-factor for an electron is not 2.

Usually the g-factor in the solid state is defined as

g =hωspin

µBB, (2–82)

where hωspin is the spin-splitting. Roth et al. [70] have calculated the g-factor in

semiconductors based on Kane’s model, and have shown that the g-factor in the

54

conduction band is

gc = 2

[1 +

(1− m0

mc

)∆

3Eg + 2∆

]. (2–83)

Using this equation, the g-factor for bulk InAs is about −15.1, which is close to the

experimental value −15 [71].

Figure 2–5: Conduction band g-factors of In1−xMnxAs as functions of magneticfield with different Mn composition x. For the left figure, T = 30 K and for theright, T = 290 K. Note at high temperatures we lose the spin splitting.

Due to the exchange interaction, the spin-splitting 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. 1–15 for a few percent of Mn

doping. In that case, if we take x = 0.1, Nα = −0.5 eV, and T = 30 K, then

geff ∼ 256. If we only consider the exchange interaction, from Eq. 2–78, the

spin-splitting in the conduction band is exactly that in Eq. 1–13. However, this

is not correct because the first conduction band spin-down level comes from the

n = 0 manifold, while the first conduction band spin-up level comes from the n = 1

manifold. Different manifold numbers result in different matrix elements, which

will cause different state coupling, and thus spin-splitting 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. 2–5. This

clearly demonstrates how Mn doping affects the g-factors. At 290 K, the g-

55

Figure 2–6: g-factors of ferromagnetic In0.9Mn0.1As. 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 employed, due to the internal

exchange field, a strong alignment can be expected even at high temperatures. Now

we suppose a high-TC In0.9Mn0.1As system in which a Curie temperature of 110 K

is achieved. The g-factor for this system is shown in Fig. 2–6. Even at relatively

high temperature (still below the transition temperature though), big g-factors are

still obtained. The g-factor reaches infinity at zero field when temperatures are

below TC because there is still spin-splitting even though there is no external field.

CHAPTER 3CYCLOTRON 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. 2–3

and 2–5. Apart from the theoretical calculation, optical experiments are always

good ways 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 inter-subband optical properties and effective masses of

carriers. Cyclotron resonance is a high-frequency 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 adjacent 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, α, can be determined by calculating the absorption

rate T of incident light with angular frequency ω in a unit volume. Suppose the

56

57

energy flux of the incident light is S, then the photon flux density is S/hω, and we

have Tdx = Sαdx/hω, i.e.

α(ω) =hωT (ω)

S. (3–1)

T (ω) is the sum of the transition probabilities Wif under the illumination of light

with angular frequency ω divided by the volume, namely

T =1

V

∑

i,f

Wif (3–2)

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,

Wabs =2π

h|H ′

if |2δ(Ef − Ei − hω), (3–3)

and for emission

Wems =2π

h|H ′

fi|2δ(Ei − Ef + hω), (3–4)

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 δ 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/λ is much smaller than the typical electron momentum, we

generally consider the optical transition to be “vertical”, 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

58

equilibrium is described by a Fermi-Dirac distribution function

f =1

1 + eE−EF /kBT, (3–5)

and so the rate of absorption in the whole crystal can be written as

Ti→f =1

V

∑

i,f

2π

h|H ′

if |2δ(Ef − Ei − hω)fi(1− ff ) (3–6)

and the emission rate

Tf→i =1

V

∑

i,f

2π

h|H ′

fi|2δ(Ei − Ef + hω)ff (1− fi) (3–7)

Due to the hermitian property of H ′, |H ′if | = |H ′

fi|. The net absorption rate per

unit volume then is

T = Ti→f − Tf→i =1

V

∑

i,f

2π

h|H ′

if |2δ(Ef − Ei − hω)(fi − ff ). (3–8)

When a semiconductor is illuminated by light, the interaction between

the photons and the electrons in the semiconductor can be described by the

Hamiltonian,

H =1

2m0

(p + eA)2 + V (r) (3–9)

where m0 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 crystal periodic potential (in

DMS, including the virtual crystal exchange potential). Thus the one-electron

Hamiltonian without the optical field is

H0 =p2

2m0

+ V (r) (3–10)

and the optical perturbation terms are

H ′ =e

m0

A · p +e2A2

2m0

. (3–11)

59

Optical fields are generally very weak and usually only the term linear in A is

considered, i.e., we treat the electron-photon interaction in a linear response regime

and neglect two-photon absorption. The transition due to the optical perturbation

in Eq. 3–11 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 = eA0cos(K · r− ωt) = eA0

2eiK·re−iωt + e

A0

2e−iK·reiωt (3–12)

where K is the electromagnetic wave vector, ω is the optical angular frequancy, 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×H. Using the relations E = −∂A/∂t, H = ∇× S/µ, and ω/K = c/nr, the

averaged energy flux then is

S =nrω

2A20

2µc. (3–13)

Using this relation and Eq. 3–8, the absorption coefficient then is

α(ω) =hωT

S=

hω

(nrω2A20/2µc)

1

V

∑

i,f

2π

h|H ′

if |2δ(Ef − Ei − hω)(fi − ff ). (3–14)

According to Eq. 3–12, the interaction Hamiltonian can be written as

H ′i,f =

eA0

2m0

〈f |e · p|i〉 =eA0

2m0

e · pfi, (3–15)

so the absorption coefficient 3–16 becomes

α(ω) =πe2

nrcε0m20ω

1

V

∑

i,f

|e · pif |2δ(Ef − Ei − hω)(fi − ff ). (3–16)

60

Note that the interaction 3–15 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 Γ through the replacement of the δ function by a Lorentzian function [72]

as

δ(Ef − Ei − hω) → Γ/2π

(Ef − Ei − hω)2 + (Γ/2)2(3–17)

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

ωc =eB

m0

(3–18)

where m0 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. 2–62, the energy of the electron splits into a series of

Landau levels. If the energy quanta hω of the applied electromagnetic wave are

exactly the same as the energy difference hωc between two adjacent 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 3–10, in DMS system,

is replaced by the one in Eq. 2–79. We already have the eigenstates for this

61

Hamiltonian. For convenience, we rewrite them here as

ψn,ν =ei(kyy+kzz)

√A

a1,n,ν φn−1u1

a2,n,ν φn−2u2

a3,n,ν φnu3

a4,n,ν φnu4

a5,n,ν φnu5

a6,n,ν φn+1u6

a7,n,ν φn−1u7

a8,n,ν φn−1u8

. (3–19)

The eigenfunction above can be considered as the linear superposition of eight

basis states, each of which is composed of two parts. φn 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, ν 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′ν′nν =

∑

i,i′a∗i,n,νai′,n′,ν′ × (〈ui|p|ui′〉〈φn

i |φn′i′ 〉+ 〈ui|ui′〉〈φn

i |p|φn′i′ 〉). (3–20)

Since the Bloch functions ui are quickly varying functions, their gradients are much

larger than those of the envelopes φi. 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.

62

We can factorize e ·p to e ·p = e+p−+ e−p+ + ezpz where e± = (x± iy)/√

2, and

p± = (px± ipy)/√

2. In the Faraday 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, ν|p+|n′, ν ′〉 ∝ δn−1,n′ (3–21)

and

〈n, ν|p−|n′, ν ′〉 ∝ δn+1,n′ . (3–22)

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-active” (e-active). 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-active” (h-active). The quasi-classical picture for

the two types of absorption is shown in Fig. 3–1. To comply with conservation of

Figure 3–1: Quasi-classical pictures of e-active and h-active photon absorption.

both energy and angular momentum, in a quasi-classical picture, electrons can only

absorb photons with e-active polarization, and holes can only absorb photons with

h-active polarization. In a quantum mechanical treatment, we will see that the

63

true situation is more complicated than this. In particular, we find that e-active

absorption can also take place in p-type materials.

When the temperature is not zero, EF in Eq. 3–5 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 NC = 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

NC =1

(2π)2λ2

∑n,ν

∫ ∞

−∞dkz[fn,ν(kz)− δv

n,ν ], (3–23)

where δvn,ν = 1 if the subband (n, ν) is a valence band and vanishes if (n, ν) is a

conduction band. Given the net donor concentration and the temperature, the

Fermi energy can be found from Eq. 3–23 using a root finding routine.

3.2 Ultrahigh Magnetic Field Techniques

Since the mobility of a ferromagnetic III-V 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 ωcτ > 1, where ωc is the cyclotron

frequency and τ 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 single-turn coil technique [76, 77] and the

electromagnetic flux compression method [77, 78]. The single-turn 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

µs) to the single-turn coil to generate an ultrahigh magnetic field. The core part of

a real single-turn coil device is demonstrated in Fig. 3–2 [76]. Although the sample

is intact, the coil is damaged after each shot. A standard coil is shown in Fig. 3–3

64

Figure 3–2: The core part of the device based on single-coil method. The coil isplaced 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 theconnection 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 µs. The time dependence of the pulsed

magnetic field and of the current flowing through the coil is shown in Fig. 3–4 [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. 3–5 [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 e-active cyclotron resonance,

the light polarization vector is e− = (x − iy)/√

2, corresponding to momentum

operator p+ = (px + ipy)/√

2. This operator will result in an n → n + 1 transition.

65

Figure 3–3: A standard coil before and after a shot.

Figure 3–4: Waveforms of the magnetic field B and the current I in a typical shotin single-turn coil device.

66

Figure 3–5: Waveforms of the magnetic field B and the current I in a typical fluxcompression 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 e-active transition, both

angular momentum and energy for an electron-photon 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 semi-insulating GaAs substrates

at 200 C. 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 3–1, together with the electron cyclotron

masses obtained at a photon energy of 117 meV (or a wavelength of 10.6 µm).

Typical measured CR spectra at 30 K and 290 K are shown in the left and

right panel of Fig. 3–6, 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 12%. All the samples show pronounced absorption

67

Table 3–1: Parameters for samples used in e-active CR experiments

Mn content x 0 0.025 0.050 0.120

Density (4.2 K) 1.0× 1017 1.0× 1016 0.9× 1016 1.0× 1016

Density (290 K) 1.0× 1017 2.1× 1017 1.8× 1017 7.0× 1016

Mobility (4.2 K) 4000 1300 1200 450Mobility (290 K) 4000 400 375 450m/m0 (30 K) 0.0342 0.0303 0.0274 0.0263m/m0 (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 12% results in a 25% decrease in cyclotron mass (see

Table 3–1). At high temperatures [e.g., Fig. 3–6(b)] the x = 0 sample clearly

shows nonparabolicity-induced CR spin splitting with the weaker (stronger) peak

originating from the lowest spin-down (spin-up) Landau level, while the other three

samples do not show such splitting. The absence of splitting in the Mn-doped

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 spin-down level is substantially thermally populated (see

Fig. 2–5).

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 × 1016/cm3 and 1 × 1018/cm3) are calculated. The

conduction band Landau levels and the Fermi energies are shown in Fig. 3–7 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 3–1. 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 density-dependent mass due

68

Figure 3–6: Experimental electron CR spectra for different Mn concentrations xtaken at (a) 30 K and (b) 290 K. The wavelength of the laser was fixed at 10.6µmwith e-active 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. 3–8, we simulate cyclotron resonance experiments in n-type InAs for e-

active circularly polarized light with photon energy hω = 0.117eV . We assume a

temperature T = 30 K and a carrier concentration n = 1016/ cm3. The lower panel

of Fig. 3–8 shows the four lowest zone-center Landau conduction-subband energies

and the Fermi energy as functions of the applied magnetic field. The transition at

the resonance energy hω = 0.117eV is a spin-up ∆n = 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. 3–8 shows the resulting cyclotron

69

Figure 3–7: Zone-center Landau conduction-subband energies at T = 30 K as func-tions of magnetic field in n-doped In1−xMnxAs for = 0 and x = 12%. Solid linesare spin-up and dashed lines are spin-down levels. The Fermi energies are shown asdotted 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 ground-state Landau

level is occupied at low electron densities. For higher electron densities, more

Landau levels are occupied. For example, if both spin-up and spin-down states of

the first Landau level are occupied, one obtains multiple resonance peaks.

Our simulation of the experimental e-active cyclotron resonance in the

conduction band shown in Fig. 3–6 is shown in Fig. 3–9. The left and right panel

demonstrate the calculated cyclotron resonance absorption coefficient for e-active

70

Figure 3–8: Electron CR and the corresponding transitions. The upper panel showsthe resonance peak and the lower panel shows the lowest four Landau levels withspin-up states indicated by solid lines and spin-down states indicated by dashedlines. Vertical solid line in the lower panel indicates the transition accountable forthe resonance.

circularly polarized 10.6µm 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 0%, 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. 3–6(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 spin-down Landau level. The peak does

71

Figure 3–9: Calculated electron CR absorption as a function of magnetic field at30 K and 290 K. The curves are calculated based on generalized Pidgeon-Brownmodel and Fermi’s golden rule for absorption. They are broadened based on themobilities reported in Table 3–1.

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 Mn-Mn interactions such as pairing

and clustering.

The e-active 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 α because the exchange interaction will shift all levels by the same amount.

This shift comes from the complicated conduction-valence band mixing, and

depends on the value of (α − β) [79]. We can qualitatively explain this shift using

the cyclotron mass, which will be discussed in the following subsection.

72

The CR peaks shown in Fig. 3–9 are highly asymmetric. 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 asymmetry.

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 hω by the definition

mCR

m0

=2µBB∗

hω. (3–24)

This equation can be derived from Eq. 3–18 if we set magnetic field B so that

hωc = hω, which is the cyclotron resonance condition.

The calculated cyclotron masses for the lowest spin-down and spin-up tran-

sitions are plotted in Fig. 3–10 as a function of Mn concentration x at a photon

energy of hω = 0.117 eV. Cyclotron masses are computed for several sets of α and

β values. The cyclotron masses in Fig. 3–10(a) and (b) correspond to the computed

cyclotron absorption spectra shown in Fig. 3–9 (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 3–10 clearly shows that the cyclotron mass depends on both exchange

constants and x. With increasing x, spin-down (spin-up) 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 (α − β) [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 conduction-valence band mixing, the exchange

73

0 2 4 6 8 10 12Mn concentration, x (%)

3.0

3.5

4.0

4.5

5.0

MC

R (

10−

2 m0)

Ephoton

= 0.117 eV

T = 30 K

up

down

α = −0.5, β = 1.3

α = −0.5, β = 1.0

α = −0.3, β = 1.0

0 2 4 6 8 10 12Mn concentration, x (%)

3.2

3.4

3.6

3.8

4.0

MC

R (

10−

2 m0)

Ephoton

= 0.117 eV

T = 290 K

down

up

α = −0.5, β = 1.3α = −0.5, β = 1.0α = −0.3, β = 1.0

Figure 3–10: Calculated electron cyclotron masses for the lowest-lying spin-up andspin-down Landau transitions in n-type In1−xMnxAs with photon energy 0.117 eVas a function of Mn concentration for T = 30 K and T = 290 K. Electron cyclotronmasses are shown for three sets of α and β values.

74

interaction constants α and β, 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 spin-down cyclotron mass. Due to the smaller downward slope in

the spin-down 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. 2–4, 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 Γ point. The

split-off band also contributes strongly to the valence band-edge 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

conduction-valence band mixing to produce strong enough oscillator strength.

Interband mixing across the band gap is small in wide-gap 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 narrow-gap semiconductors. Our collaborators [80, 81,

82] have performed cyclotron resonance experiments on p-doped InAs and InMnAs

at ultrahigh magnetic fields up to 500 T. The typical h-active CR absorption of

InAs below 150 T is shown in Fig. 3–11, in which the incident light is h-active

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× 1019/cm3 and a broadening factor of 40 meV is also displayed in

Fig. 3–11 for comparison.

75

Figure 3–11: Hole cyclotron absorption as a function of magnetic field in p-typeInAs for h-active circularly polarized light with photon energy 0.117 eV. The up-per curve is experimentally observed result and the lower one is from theoreticalcalculation.

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 heavy-hole to heavy-hole transition, and the peak at higher fields is from

the light-hole to light-hole transition. We now use Hn,ν to specify the heavy hole

level, and Ln,ν to specify the light hole level, where (n, ν) are the quantum numbers

defined in Eq. 3–19. 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 two-state absorption in Fig. 3–12 along with the Landau level structure as a

function of magnetic field.

It is seen from Fig. 3–12 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. 3–11. The cyclotron absorption peak around 40 T

is due to a transition between the spin-down ground state heavy hole Landau level

H−1,1, and heavy hole Landau level H0,2, which near the zone center is primarily

76

Figure 3–12: Calculated cyclotron absorption only from the H−1,1 − H0,2 andL0,3 − L1,4 transitions broadened with 40 meV (a), and zone center Landau levelsresponsible for the transitions (b).

spin-down. The other absorption peak around 140 T, is a spin-down 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 In1−xMnxAs with x = 2.5%

have also been performed. They are shown in Fig. 3–13 along with our theoretical

simulation. The CR measurements were made at temperatures of 17, 46, and

70 K in h-active circularly polarized light with photon energy hω = 0.224 eV.

In our simulation, the hole density is taken as 5 × 1018/ cm3, and the curves are

77

α

Figure 3–13: Experimental hole CR and corresponding theoretical simulations.The low temperature CR has an abrupt cutoff at low fields due to the fermi levelsharpening 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. 3–11 and Fig. 3–12.

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 asymmetric 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

78

the absorption. This can be attributed to the sharpness of the Fermi distribution

at low temperatures.

Figure 3–14 shows the observed CR peaks as a function of magnetic field. The

y-axis 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 (L0,3)and

H0,2 (L1,4). The theoretical calculation shows an overall consistency with the

experiments.

Figure 3–14: Observed hole CR peak positions for four samples with different Mnconcentrations. 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 eight-band

effective mass theory itself. In Fig. 3–11, 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 γ1, γ2, γ3, 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 γ1 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. 3–15.

Figure 3–15: The dependence of cyclotron energies on several parameters. Leftpanel shows the heavy hole CR energy dependence, and the right panel shows thelight hole CR energy dependence.

Figure 3–16 illustrates how the CR absorption depends on three Luttinger

parameters while keeping all the other parameters unchanged. It can be seen that

80

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.

Figure 3–16: Hole CR spectra of InAs using different sets of Luttinger parame-ters. Light hole transition is more significantly affected by change of the Luttingerparameters.

In Fig. 3–14, there is one peak around 450 T labeled as ‘C’ when the light

energy is hω = 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. 3–17, 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 eight-band Pidgeon-Brown model may

break down; the other is that transitions contributing to the peak take place away

from the zone center where eight-band k · p theory is not adequate to describe the

energy dispersion. The band structure along kz is plotted at Fig. 3–18, where we

see that the Landau levels H2,6 and H3,6 both have camel back structures. At a

hole density p = 1 × 1019/ cm3, the zone center part of H2,6 is not occupied. The

81

Figure 3–17: Calculated Landau levels and hole CR in magnetic fields up to 500T. The upper panel shows the k = 0 valence band Landau levels as a function ofmagnetic field and the Fermi level for p = 1019 cm−3 (dashed line). The hole CRabsorption in p-type InAs is shown in the lower panel for h-actively polarized lightwith hω = 0.117 eV at T = 20 K and p = 1019 cm−3. A FWHM linewidth of 4 meVis assumed.

lowest energy for this heavy hole Landau level resides at about kz = 0.75(1/nm).

Checking the transition element along kz, it is also found that this transition indeed

takes place away from the zone center. Displayed in Fig. 3–19 is the comparison of

the eight-band model versus a full-zone thirty-band model. At the zone center, the

eight-band model fits well with the thirty-band 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.

82

Figure 3–18: k-dependent Landau subband structure at B = 350 T.

Figure 3–19: Band structure near the Γ point for InAs calculated by eight-bandmodel and full zone thirty-band model.

83

3.4.2 Hole Density Dependence of Hole Cyclotron Resonance

Cyclotron resonance depends on Fermi energy through Eq. 3–16, thus CR

spectra depend strongly on carrier densities. In Fig. 3–12 the hole density is

1 × 1019 cm−3. At such a hole density, the Fermi energy is below the H−1,1 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. 3–20(a) are

the CR spectra for four different hole densities. The Landau levels along with the

corresponding Fermi energies are plotted in Fig. 3–20(b). Resonant transitions at

0.117 eV are indicated by vertical lines. We can see that the CR peak 2 is almost

always present, because at low magnetic fields, the heavy hole state H−1,1 is almost

always occupied. The CR peak 1 changes dramatically with hole density, and

nearly vanishes at p = 5 × 1018 cm−3. The relative strengths of the heavy and

light-hole 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. 3–20(a), we see that the itinerant hole concentration is around 2 × 1019 cm−3.

From Fig. 3–20(a), we can rule out p < 1019 cm−3 and n > 4 × 1019 cm−3. We

estimate that an error in the hole density of around 25% should be achievable at

these densities. Because of the existence in III-V 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 3–2.

84

Figure 3–20: The hole density dependence of hole CR. (a) Theoretical hole CRcurves in InAs from bottom to top with hole densities of 5× 1018, 1019, 2× 1019 and4 × 1019 cm−3; (b) Landau levels involved in observed CR along with Fermi levelscorresponding to theoretical curves in (a).

85

Table 3–2: Characteristics of two InMnAs/GaSb heterostructure sam-ples

Sample No. TC(K) Mn content x Thickness (nm) Density(cm−3)

1 55 0.09 25 1.1× 1019

2 30 0.12 9 4.8× 1019

The experimentally observed CR transmission of a 10.6 µm 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 hole-active circularly polarized. In the left panel of Fig. 3–21, 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. 3–21. Using different wavelengths of the

incident light, similar CR spectrum behavior has also been observed.

For zinc-blende semiconductors, the CR peaks A and B are due to the tran-

sitions of H−1,1 −→ H0,2 and L0,3 −→ L1,4, respectively, which we have already

pointed out. We attribute the temperature-dependent peak shift to the increase

in the carrier-Mn ion exchange interaction resulting from the increase of magnetic

ordering at low temperatures. The theoretically calculated results are shown in

Fig. 3–22 for bulk In0.91Mn0.09As. The CR spectra was broadened using a FWHM

linewidth of 4 meV. The theoretical results clearly show a shift of peak A to lower

86

Figure 3–21: Cyclotron resonance spectra for two ferromagnetic InMnAs/ GaSbsamples. The transmission of hole-active circular polarized 10.6 µm radiation isplotted 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

∼ 50 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× 1, and when n = 0,

the Hamiltonian factorizes into two 2 × 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

87

α

Figure 3–22: Theoretical CR spectra showing the shift of peak A with temperature.

for the H−1,1 −→ H0,2 transition is

ECR = −Eg

2+

1

4x〈Sz〉(α− β)−

√[Eg

2− 1

4x〈Sz〉(α− β)

]2

+ EpµBB. (3–25)

In the field range of interest (∼ 40 T),√

EpµBB is the same order as Eg/2,

while the exchange interaction is much smaller even in the saturation limit.

Expanding the square root in Eq. 3–25, we obtain an expression of the form

ECR =Eg

2

(1

δ− 1

)+

1

4x〈Sz〉(α− β)(1− δ) (3–26)

where

δ =Eg√

E2g + 4EpµBB

. (3–27)

If we assume the temperature dependence of Eg and Ep is small, it follows

from Eq. 3–26 that the CR peak shift should follow the temperature dependence

of the magnetization 〈Sz〉, which in a mean field theory framework is given by

88

Figure 3–23: Average localized spin as a function of temperature at B = 0, 20, 40,60 and 100 Tesla. The Curie temperature is assumed to be 55 K.

Eq. 1–22. The temperature dependence of 〈Sz〉 is shown in Fig. 3–23 at several

magnetic fields. The magnetic field dependence of 〈Sz〉 has already been shown in

Fig. 1–10.

The relative change of the CR energy, calculated using Eq. 3–25 and 3–26,

as a function of temperature is presented in Fig. 3–24. It shows that from room

temperature to 30 K the cyclotron energy increases about 20%, which corresponds

to an approximately 20% decrease in the resonant magnetic field, approximately

the result observed in the experiment. In addition, we found that the shift is

nonlinear in temperature and the main shift occurs at temperatures well above TC .

These features are also consistent with experiment.

Along with the CR peak shift, experiment indicates a significant narrowing of

the linewidth. We speculate that this effect may be associated with the suppression

of localized spin fluctuations at low temperatures. A similar effect has been

observed in II-VI dilute magnetic semiconductors (see Ref. [84] and references

therein). Spin fluctuations become important when a carrier in the band interacts

simultaneously with a limited number of localized spins. This takes place, for

89

Figure 3–24: Relative change of CR energy (with respect to that of high tempera-ture limit) as a function of temperature. Vertical dashed line indicates TC .

example, in magnetic polarons and for electrons in dilute magnetic semiconductor

quantum dots. The strong in-plane localization by the magnetic field may also

result in a reduction of the number of spins which a carrier in the band feels, thus

increasing the role of spin fluctuations. However, it is possible that the CR peak

narrowing is the result of the increased carrier mobilities. Although the InMnAs

layer is heavily doped and thus the hole mobility is very low, holes in the GaSb

layer, if they exist, will have much higher mobilities. So we can speculate that near

the transition temperature, the band structure of InMnAs changes in such a way

that a fraction of the holes move into the GaSb layer or/and the InMnAs/GaSb

interface, where the hole CR has a much narrower linewidth. Shown in Fig. 3–25 is

the band diagram of the InMnAs/GaSb heterostructure [46]. The interface states

of InMnAs/GaSb are very complicated, and we have not carried out calculations

incorporating them in our CR simulations.

90

Figure 3–25: Band diagram of InMnAs/GaSb heterostructure.

3.4.4 Electron Active Hole Cyclotron Resonance

We discussed the CR selection rules in Section 3.1.2 where we defined the term

“h-active”, which is associated with e+ light and the p− operator, and “e-active”,

which is associated with e− light and the p+ operator. Selection rules for CR are

a direct result of the conservation of energy and angular momentum. For a free

gas of particles, CR can only be observed for a specific circular polarization. For

instance, for an electron gas, CR transitions occur in e− polarization while for a gas

of positively charged particles, CR will occur only for e+ polarization.

The situation in real semiconductors, however, differs from that in a classical

free electron gas. Matsuda et al. [82] have experimentally observed e-active CR

in p-doped InAs and InMnAs. The temperature was quite low (12K) and the hole

concentration was high enough (1019 cm−3) to safely eliminate the possibility that

the e-active CR comes from the thermally excited electrons in the conduction band.

The possibility of the existence of electrons in the interface or surface inversion

layers has been also excluded. Thus, the results suggest that e-active CR comes

from the valence band holes, in contradiction with the simple picture of a free hole

gas.

We find that e-active cyclotron resonance in the valence bands is an intrinsic

property of cubic semiconductors and results from the degeneracy of the valence

bands. As we discussed before, heavy hole and light hole bands will both split into

91

a series of Landau levels. This complexity allows one to satisfy conservation of

angular momentum in CR absorption for both e+ and e− polarization, provided one

switches band type.

In the conduction band, increasing the manifold quantum number always

increases the energy. As a result, only transitions with increasing n may take place

in absorption, that is, only e-active (e−) CR can be observed in the conduction

band.

Figure 3–26: Schematic diagram of Landau levels and cyclotron resonance transi-tions in conduction and valence bands. Both h-active and e-active transitions areallowed in the valence band because of the degenerate valence band structure. Onlye-active transitions are allowed in the conduction band.

The valence band, however, consists of two types of carriers: heavy holes

(J = 3/2,Mj = ±3/2) and light holes (J = 3/2,Mj = ±1/2). Each of them

has their own Landau ladder in the magnetic field. An increase of n always

decreases the energy only within each ladder. Similar to the conduction band

case, transitions within a ladder (HH → HH or LH → LH) can take place only in

h-active (e+) polarization. However, the relative position of the two ladders can be

such that interladder transitions (LH → HH) in e-active polarization are allowed.

92

This process is schematically shown in Fig. 3–26. Note that this figure is extremely

simplified and should be used only as a qualitative explanation of the effect.

Figure 3–27: The valence band Landau levels and e-active hole CR. (a) The low-est three pairs of Landau levels in the e-active transition; (b) The separate CRabsorption contributing to the e-active CR.

We’ve examined e-active CR in p-type InAs at T = 12 K with a free hole

density of 1 × 1019/ cm3. The computed k = 0 valence band energies as a

function of magnetic field, the e-active optical transitions and the corresponding

CR absorption spectra are shown in Fig. 3–27. The most pronounced e-active

transitions take place between the HH state H0,2 and LH state L1,5 and between

the H1,3 and L2,6 states. There are some other less pronounced transitions, which

contribute to the absorption spectra at lower fields.

93

The calculated and experimental CR are shown in Fig. 3–28 for both e-

active and h-active polarizations. There is good agreement between theory and

experiment. As discussed above, the electron active absorption is determined by

the HH → LH transitions. The main contribution to the h-active absorption (left

panel in Fig. 3–28) comes from the transitions within the heavy hole ladder, which

we have already discussed in detail in the last section.

The calculation and observation of e-active CR in p-type DMS can aid in

understanding the valence band structure of DMS systems. Using both h and

e-active CR one can explore the whole picture of the valence bands.

Figure 3–28: Experimental and theoretical hole CR absorption. Solid lines are ex-perimental hole CR spectra as a function of magnetic field for h-active and e-activepolarizations. Corresponding theoretical calculations are shown in dashed lines.

94

In the last chapter, it was seen that the Mn doping greatly affects the valence

subband alignment and the cyclotron resonance. On the one hand, doping with

Mn impurities will greatly enhance the scattering of carriers thus increasing the

linewidth and reducing the strength of the CR spectra. On the other hand, it will

also shift the CR peak positions due to the changes in the valence band structure.

Figure 3–29: Valence band structure at T = 30 K and B = 100 T for In1−xMnxAsalloys having x = 0% and x = 5% . For x = 0%, the first HH state,H−1,1, lies belowthe light hole state L1,5. For x = 5%, the order of these two states is reversed. Twopossible CR transitions are shown using upward arrows, namely an h-active(σ−)transition between H0,2 and H−1,1, and an e-active(σ+) transition from H0,2 toLH1,5. The dashed lines are the Fermi energies for a hole density of 1019cm−3.

We illustrate how Mn doping affects the optical transitions in Fig. 3–29. In

this case, we assume a carrier density of 1019 cm−3, without and with Mn doping at

30 K, in a magnetic filed of 100 T. The primary h-active and e-active transitions

are both indicated in these figures. Only three levels are involved in both h- and

e-active transitions. The h-active transition is from the H−1,1 to H0,2 state as we

mentioned above. The e-active transition is from the L1,5 to H0,2 state. We see

that without Mn doping, the L1,5 state sits on the top, but with doping, H−1,1 state

is shifted to the top while L1,5 is shifted to a lower position. Thus both h-active

and e-active absorption will be affected.

95

Line shapes and peak positions are very sensitive to Mn doping. Note that

absorption takes place not only at the Γ point, but also in regions away from

the zone center. Even though we broaden the CR lines with the same line width

(4 meV), the doped sample has a much broader line shape due to the energy

dispersion change along kz and the energy position change relative to the Fermi

level brought about by the the exchange interaction. This is shown in Fig. 3–30.

Furthermore, the height of the peak of the CR spectrum of the Mn doped sample

is reduced by about 30 times compared to the undoped one. This may come from

the Fermi filling effect, since the L1,5 state become less occupied when the sample is

doped with Mn ions at a carrier density of 1019cm−3.

Figure 3–30: The primary transition in the e-active hole CR under different Mndoping. Mn doping changes the Landau level alignment and the transition strengthas well. It also shifts the CR peak position to a lower field.

CHAPTER 4MAGNETO-OPTICAL KERR EFFECT

In magneto-optical experiments, transmission of the sample is usually mea-

sured, since transmission measures absorption inside the sample, and in most

cases, absorption probes the intrinsic electronic and transport properties. However,

when a sample is too thick, direct measurement of transmission is impossible.

In this case, one can measure the reflection of the sample, and use the relations

between the optical constants to derive the absorption coefficient, thus obtaining

the intrinsic properties through quantitative analysis.

The magneto-optical Kerr effect (MOKE) is related to light reflection. When

linearly polarized light is reflected by the surface of a ferromagnetic sample, the

polarization plane will undergo a rotation. Similarly, there is also the Faraday

effect, which is related to the rotation of the polarization plane of the transmitted

light. Magneto-optical effects may be observed in non-magnetic media such as glass

when a magnetic field is applied. However, the intrinsic effects are usually small in

such cases. In magnetic media (ferromagnetic or ferrimagnetic) the effects are much

larger. For cubic crystals, when there is no ferromagnetism, no MOKE signal will

be present, so when studying the dynamical magnetic properties of DMS, MOKE

can serve as a powerful tool for detection and measurement of magnetic moments

or time dependent magnetic moments, with time resolved optics [46].

4.1 Relations of Optical Constants

In this section, we introduce the relations between optical variables. In the

last chapter, we described how the absorption coefficient can be used to explore

electronic properties, so in the following, we will find the relations between all

96

97

the other optical variables and the absorption coefficient, especially the relation

between the reflection and absorption coefficients.

When an electromagnetic wave is propagating in a medium with a magnetic

relative permittivity µr, and electronic relative permittivity εr, it satisfies Maxwell

equations,

∇× E = −µ0µr∂H

∂t, (4–1)

∇×H = σE + ε0εr∂E

∂t, (4–2)

∇ ·H = 0, (4–3)

∇ · E = − ρ

ε0εr

. (4–4)

When there are no free charges, Eq. 4–4 becomes

∇ · E = 0. (4–5)

Taking the curl of Eq. 4–1, and using the relation

∇×∇× E = ∇(∇ · E)−∇2E (4–6)

Eq. 4–5 becomes

∇2E− σµ0µr∂E

∂t− µ0µrε0εr

∂2E

∂t2= 0. (4–7)

For a plane wave propagating in z-direction,

E = E0e−i(ωt−kz), (4–8)

and

k2 =µrεrω

2

c2+

iσµrω

ε0c2. (4–9)

In general, the wave vector can be written as a complex number

k =ω

cN, (4–10)

98

where N is the complex refractive index give by

N2 = µrεr +iσµr

ε0ω. (4–11)

If we write N = n + iκ, where n and κ are the real and imaginary part of N ,

then k = nωc

+ iκωc. Substituting this into Eq. 4–8, we obtain

E = E0e−i(ωt−kz) = E0e

−κk0ze−i(ωt−nk0z) (4–12)

where k0 = k/N is the wave vector in vacuum. Thus we can see that the imaginary

part of N , κ, is related to light absorption, i.e. the extinction coefficient. The light

intensity is proportional to |E|2, so we can write the intensity as

I = I0e−2κk0z = I0e

− 2ωκc

z. (4–13)

Thus the absorption coefficient is

α =2ωκ

c. (4–14)

From Eq. 4–9, if there is no energy loss, then

k =ω

c

√µrεr (4–15)

is real. But since there are losses, we write

k =ω

c

√µr εr (4–16)

where the complex dielectric constant εr is defined as

εr = εr +iσ

ε0ω= ε1 + iε2. (4–17)

where ε1 and ε2 are the real and imaginary parts of εr.

Comparing Eq. 4–10 and 4–16, we have

N =√

µrεr. (4–18)

99

Generally, µr is very close to 1, in such a case, the relationships between the real

and imaginary parts of N and εr are

ε1 = n2 − κ2, (4–19a)

ε2 = 2nκ, (4–19b)

and

n =1√2

(ε1 +

√ε21 + ε2

2

)1/2

, (4–20a)

κ =1√2

(−ε1 +

√ε21 + ε2

2

)1/2

. (4–20b)

Thus, we can calculate n and κ from ε1 and ε2, and vice versa. In the weakly

absorbing case, i.e. κ ¿ n, Eq. (4–20) can be simplified to

n =√

ε1, (4–21)

κ =ε2

2n. (4–22)

From Eq. 4–14, the relation between the absorption coefficient and dielectric

constant is

α =ωε2

nc(4–23)

Many measurements of optical properties in solids involve normally incident

reflectivity. Inside the solid, the wave will be attenuated. We assume for the

present discussion that the solid is thick enough so that reflections from the back

surface can be neglected.

Consider the reflection of a plane wave moving in the z direction. The

interface between a half-infinite medium 1 with refractive index N1 and a half-

infinite medium 2 with a refractive index N2 is taken to be z = 0. This situation is

illustrated in Fig. 4–1. Assuming E//x, we have an incoming and reflected wave in

100

Figure 4–1: Diagram for light reflection from the interface between medium 1 withrefractive index N1 and medium 2 with refractive index N2.

medium 1

Ex = E1eiω(N1z/c−t) + E2e

−iω(N1z/c+t). (4–24)

In medium 2, the transmitted wave is

Ex = E0eiω(N2z/c−t). (4–25)

Continuity of electric field at the interface requires E0 = E1 + E2. With E in the x

direction, the second relation between E0, E1 and E2 follows from the continuity of

the tangential magnetic field Hy across the interface. From Eq. 4–1, we have

∂Ex

∂z= iµ0µrωHy. (4–26)

The continuity condition on Hy thus yields a continuity relation for ∂Ex/∂z so that

from Eq. 4–26 we obtain

N1(E1 − E2) = N2E0 (4–27)

101

The normally incidenct reflectivity R is

R =

∣∣∣∣E2

E1

∣∣∣∣2

=

∣∣∣∣N1 −N2

N1 + N2

∣∣∣∣2

(4–28)

and the reflection coefficient r is given by

r =E2

E1

=N1 −N2

N1 + N2

(4–29)

where N1, N2 and r are all complex variables. According to Eq. 4–14, absorption

measurements can be used to determine the reflection coefficient.

Usually, the real and the imaginary parts of the optical constants are not

independent. They are related by the Kramers-Kronig relation [85]. For example,

for N = n + iκ,

n(ω′) = 1 +

2

π℘

∫ ∞

0

ωκ(ω)

ω2 − ω′2dω, (4–30)

or using Eq. 4–14,

n(ω′) = 1 +

c

π℘

∫ ∞

0

α(ω)

ω2 − ω′2dω, (4–31)

where ℘

∫ ∞

0

is the principal value of the integral.

4.2 Kerr Rotation and Faraday Rotation

In non-magnetic cubic crystals, the complex refractive index (and thus the

absorption coefficient, the dielectric constant, and the reflection coefficient) does

not depend on the polarization of the incident light. This can be understood from

symmetry considerations. However, in a ferromagnetic cubic crystal, there is a

spontaneous magnetization direction, and the cubic symmetry is broken. Suppose

the magnetization direction is along z, then for σ+ (right-circularly polarized,

e = (x − iy)/√

2) and σ− (left-circularly polarized, e = (x + iy)/√

2) light, the

complex refractive index will have different values. This can also be understood

from the splitting of bands by magnetization. Without spontaneous magnetization,

the conduction band edge is two-fold degenerate, and the valence band edge is

102

Figure 4–2: Schematic diagram for magnetic circular dichroism.

four-fold degenerate. With spontaneous magnetization which produces a self-

consistent effective magnetic field, the conduction and valence band states will

split (see also, Fig. 2–1). Thus the absorption for σ+ and σ− polarization will be

different. A very simple schematic diagram for this effect is illustrated in Fig. 4–2,

in which we only show the two heavy hole valence bands. Corresponding to the two

circular polarizations, we define two complex refractive indices N+ and N−, where

N+ = n+ + iκ+, and N− = n− + iκ−. The real parts of the refractive indices do not

have a strong polarization dependence, so we set n+ = n− = n.

At the interface, the reflection coefficient for σ+ polarization, following Eq. 4–

29, is

r+ =1−N+

1 + N+

=1− n− iκ+

1 + n + iκ+

, (4–32)

and for σ−,

r− =1−N−1 + N−

=1− n− iκ−1 + n + iκ−

. (4–33)

Due to difference between κ+ and κ−, r+ and r− have different phase factors. That

means after reflection, σ+ and σ− light will have different phases.

Now consider the case where linearly polarized light propagates normal to the

surface of a ferromagnetic crystal and is reflected by the surface, as illustrated in

103

Fig 4–3(a). Linearly polarized light can be decomposed into σ+ and σ− compo-

nents. From the discussion above, these two circularly polarized beams will have

different phase changes after reflection, so the reflected light, if it is still linearly

polarized, will not stay in the same polarization plane. The polarization plane will

be rotated and this is called Kerr rotation. Due to the differences in absorption of

the two circularly polarized beams, the reflected light will in general be elliptically

polarized. Defining a complex rotation

ΦK = θK + iηK (4–34)

we have

ΦK =i(r+ − r−)

r+ + r−=

i(N+ −N−)

N+N− − 1. (4–35)

The quantity θK specifies the rotation of the major axis of the reflected elliptically

Figure 4–3: Diagrams for Kerr and Faraday rotation. (a) Kerr rotation; (b) Fara-day rotation.

polarized light, and ηK is the ellipticity, which is defined as the ratio of the minor

to the major axes of an ellipsoid.

104

Similar to the Kerr effect, after being transmitted through a ferromagnetic

crystal, linearly polarized light will also have a rotation and in general be ellipti-

cally polarized. This is called the Faraday effect, which is illustrated in Fig. 4–3(b).

It is easy to show that the rotation angle per unit length is

θF =ω

2c<(N+ −N−), (4–36)

and the elliplicity

ηF = − ω

2c=(N+ −N−). (4–37)

The ellipticity is related to the magnetic circular dichroism (MCD), which is

defined by the difference ∆α(ω) between the absorption coefficient of the right and

left circularly polarized light. From the relations between the optical constants, we

have

∆α(ω) = α+(ω)− α−(ω) = −4ηF (ω)

l, (4–38)

where l stands for the light transmitted length.

Note that the incident light is along the magnetization direction, which

we defined as the z-direction. In the longitudinal case where the magnetization

vector is in the plane of the surface and parallel to the plane of incidence or in

the transverse case where the magnetization vector is in the plane of the surface

and transverse to the plane of incidence, no Kerr rotation is observed at normal

incidence.

4.3 Magneto-optical Kerr Effect of Bulk InMnAs and GaMnAs

For bulk InAs with no magnetic field, there is no Kerr rotation. When

doped with Mn, and if the doped sample is ferromagnetic, then below the Curie

temperature, there is a spontaneous magnetization, and the absorption for σ+

and σ− light will be different. In our model, e-active (σ+) absorption and h-active

(σ−) absorption coefficients are calculated, then following Eq. 4–14, the extinction

105

Figure 4–4: Kerr rotation of InMnAs. (a) Absorption coefficient of In0.94Mn0.06Asas a function of photon energy for e- and h-active light at T = 30 K; (b) Thecorresponding Kerr rotation.

coefficients are computed. Then from Eq. 4–35, we obtain

θK = =(

N+ −N−1−N+N−

)= =

(κ+ − κ−

1− (n + κ+)(n + κ−)

). (4–39)

Suppose we have a In1−xMnxAs sample with x = 6%, a Curie temperature

TC = 55 K, at T = 30 K. The computed e- and h-active absorption coefficients are

shown in Fig. 4–4(a). It can be seen that due to the ferromagnetism, the sample

has different absorption coefficients for e- and h-active polarization. This gives rise

to a non-zero Kerr rotation, which is shown in Fig. 4–4(b). The rotation is about

several tenths of a degree.

Actually, the eight-band k · p theory is not capable of calculating the light

absorption for a very wide range of photon energies, because not only Γ-Γ valley,

but also Γ-L valley, and even Γ-X valley absorption need to be considered. This

task requires a full zone band structure. The L-valley lies about 1.08 eV above the

valence band edge, and the X-valley about 1.37 eV above the valence band edge.

A schematic band diagram for InAs is shown in Fig. 4–5 [86]. Even so, we can still

106

Figure 4–5: The band diagram for InAs.

get a qualitative picture of the Kerr rotation for InMnAs for photon energies below

1 eV.

The results for Ga0.94Mn0.06As are shown in Fig. 4–6(a) and (b), in which we

suppose the Curie temperature is TC = 110 K, at T = 30 K. Comparing Fig. 4–

4(b) and Fig. 4–6(b), we see that the Kerr rotations are of the same order, about a

tenth of a degree.

The schematic diagram for the GaAs band structure is shown in Fig. 4–7 [87].

The L-valley lies about 1.71 eV above the valence band edge, and the X-valley

about 1.90 eV above the valence band edge. So use of the eight-band k · p theory

in GaAs is even worse. However, we expect the transition from Γ-L valley is not as

effective as Γ-Γ valley transition because the former is an indirect process.

107

Figure 4–6: Kerr rotation of GaMnAs. (a) Absorption coefficient of Ga0.94Mn0.06Asas a function of photon energy for e- and h-active light at T = 30 K; (b) Thecorresponding Kerr rotation.

4.4 Magneto-optical Kerr Effect of Multilayer Structures

According to the definition of Kerr rotation, we have

θK =1

2arg(

r−r+

), (4–40)

where r− and r+ are the two complex reflection coefficients for σ− and σ+ circularly

propagating light beams in the medium, and arg(x) represents the phase of the

complex number x. It is easy to prove that this θK is exactly the θK in Eq. 4–35.

According to Ref. [88], for a multilayer structure, the coefficients r± depend on

the amplitude of the reflection coefficients ri,i+1± at the interfaces of successive layers

i and i + 1. If we approximate a single quantum well by a three-layer structure, in

the case of normal reflection, r± takes the form

r± =r01± + r123

± e2iβ±1

1 + r01± r123± e2iβ±1, (4–41)

where

r123± =

r12± + r23

± e2iβ±2

1 + r12± r23± e2iβ±2. (4–42)

108

Figure 4–7: The band diagram for GaAs.

The reflection coefficients ri,i+1± , can be obtained from Eq. 4–32 and 4–33. β±i =

(w/c)liNi± denotes the dephazing of the electric field radiation after crossing the

layer i of thickness li .

At the interface of i, i + 1, the complex reflection coefficient becomes

ri,i+1± =

N i± −N i+1

±N i± + N i+1

±, (4–43)

which can be easily obtained from the same procedure we described in Section

4.1. The complex refractive indices N± are obtained by calculating the absorption

coefficients in each layer.

Now consider a In0.88Mn0.12As/GaSb heterostructure with a InMnAs layer

thickness of 9 nm in a paramagnetic phase (thus r+ = r−) at temperature

T = 5.5 K. To compare with experiment, the reflectivity of this structure is

calculated and shown in Fig. 4–8(b). Along with the reflectivity, the absorption

coefficients in the InMnAs and GaSb layer are also shown in Fig. 4–8. In the

109

Figure 4–8: The absorption coefficients both in InMnAs and GaSb layers (a) andthe reflectivity of InMnAs/GaSb heterostructure(b).

calculation, we assume a carrier density of p = 1019 cm−3 in the InMnAs layer, and

no carriers in the GaSb layer.

The experimental result [89] is illustrated in Fig. 4–9. Up to 1.5 eV, we can

see that the calculation successfully reproduces the oscillating structure of the

reflectivity, and the calculated mean reflectivity is very close the the experimental

one. In our calculation, there are several energies where the reflectivity is very

close to zero, while in the experiments, the lowest reflectivity is still around 40%.

We suppose this is because in our calculation, we have not considered interface

roughness, which can significantly contribute to the reflection.

Now we consider a ferromagnetic In0.88Mn0.12As/AlSb heterostructure with

a 9 nm thick InMnAs layer and a 136 nm thick AlSb layer grown on a 400 nm

thick GaSb layer. This structure has a Curie temperature of 35 K, and the

MOKE signal has been measured at a magnetic field of 3 T. The experimental

result is shown in Fig. 4–10(a). Following the same procedure, the absorption

coefficients in the InMnAs layer are calculated including quantum confinement

effects using a finite difference method [90]. The calculation has been performed

in the 0.3 − 1.5 eV photon energy range, in which the AlSb layer is transparent.

110

Figure 4–9: Reflectivity of In0.88Mn0.12As(9 nm)/GaSb(600 nm) heterostructure atT = 5.5 K measured by P. Fumagalli and H. Munekata. Reprinted with permissionfrom P. Fumagalli and H. Munekata. Phys. Rev. B 53: 15045-15053, 1996. Figure3, Page 15047.

The light absorption in the GaSb layer is also calculated, and the light reflections

from the InMnAs/AlSb and AlSb/GaSb interface are both taken into account.

The Kerr rotation is obtained using Eq. 4–40 and the calculated result is shown in

Fig. 4–10(b).

111

Figure 4–10: Measured (a) and calculated (b) Kerr rotation ofInMnAs(19 nm)/AlSb(145 nm) heterostructure under a magnetic field of 3 T atT = 5.5 K. Panel (a) is reprinted with permission from P. Fumagalli and H.Munekata. Phys. Rev. B 53: 15045-15053, 1996. Figure 8, Page 15049.

CHAPTER 5HOLE SPIN RELAXATION

Holes in III-V DMS play an important role in mediating the ferromagnetism

and participating in magneto-optical transport processes. Unlike in II-VI DMS

where Mn2+ ions are isoelectronic with the cations, Mn ions in III-V DMS are

acceptors. Due to the As anti-site defects and interstitial Mn, both of which act

as double donors, the hole concentration is usually much lower than the Mn con-

centration. However, the mediation of the exchange interaction between localized

magnetic moments by holes is the cornerstone of most theories of ferromagnetism

in III-V DMS (please refer to Section 1.3.1). Light induced ferromagnetism in

p-InMnAs/GaSb has been observed by Koshihara et al. [44], and Kono et al. [46],

where hole-electron pairs have been excited and the hole density greatly enhanced

by the incident light. In the latter experiment, ultrafast lasers have been employed

and the time-dependent MOKE signal has been measured. The ultrashort laser

pulses create a large density of transient carriers in the InMnAs layer and the

MOKE signal decays less than 2 ps after laser pumping, as shown in Fig. 5–1.

Recently, Mitsumori et al. [91] has studied the photo-induced magnetization

rotation in ferromagnetic p-GaMnAs. They found that when shining circularly

polarized light normal to the sample surface, which is parallel to the magnetiza-

tion, and probing with linearly polarized light, a non-zero Kerr rotation is seen

which has a decay time of ∼ 25 ps. As we discussed in the last chapter, there is

no longitudinal Kerr rotation at normal incidence, so this Kerr rotation was due to

the rotation of the magnetization due to the light-induced non-equilibrium carrier

spins. This light-induced rotation of magnetization is illustrated schematically in

Fig. 5–2.

112

113

Figure 5–1: Light-induced MOKE. Signal decays in less than 2 ps.

In these pump-probe optical experiments, the behavior of the non-equilibrium

carrier spins which induce the exchange interaction is a key factor that deserves

to be studied. The electron spin relaxation in the conduction band has been

thoroughly investigated by many authors [4, 92], however, theoretical studies on

hole spins, especially on the non-equilibrium hole spins, are sparse. In this chapter,

we will focus on the hole spin relaxation in bulk III-V DMS, and discuss the

mechanisms which induce the relaxation.

5.1 Spin Relaxation Mechanisms

Spin relaxation is a process that leads to spin equilibration. Through a long

history of studies, four main spin relaxation mechanisms have been proposed for

the conduction band electrons: the Elliot-Yafet (EY) [93, 94], Dyakonov-Perel (DP)

[95], Bir-Aronov-Pikus (BAP) [96], and hyperfine interaction processes. In the

EY mechanism electron spins relax because the electron wave functions normally

associated with a given spin have an admixture of the opposite-spin states, due to

spin-orbit coupling induced by the ions, thus spin-flipping processes accompany

momentum relaxation. The DP mechanism explains spin dephasing in solids with

114

Figure 5–2: Light-induced magnetization rotation. Reprinted with permission fromMitsumori et al. Phys. Rev. B 69: 33203-33206, 2004. Figure 1, Page 33203.

inversion asymmetry, which causes spin splitting. Spin dephasing occurs because

electrons feel an effective magnetic field, due to the spin-splitting and spin-orbit

interactions, which changes in random directions every time the electron scatters

to a different momentum state. The BAP mechanism is important for p-doped

semiconductors, in which the electron-hole exchange interaction gives rise to

fluctuating local magnetic fields which flip electron spins. In semiconductors with a

nuclear magnetic moment, there is also a hyperfine interaction between the electron

spins and nuclear moments which will cause spin-flipping of electron spins.

In the valence band, the Luttinger Hamiltonian describing the 4 heavy hole

and light hole bands is [97]

HL =h2

2m0

[(γ1 +

5

2γ2

)k2 − 2γ2(k · J)2

](5–1)

where Jx, Jy and Jz are 4 × 4 matrices corresponding to angular momentum

J = 3/2. Due to the spin-orbit interaction, and the k · J term in the Hamiltonian,

the spin states in the valence band mix very strongly. The light hole and split-off

hole basis states are not spin eigenstates. Even for a pure heavy state at k = 0, it

becomes strongly mixed not far away from the Γ point. Unlike in the conduction

band, where the overlap integral between two electron spin states is very small

115

(¿ 1) because of the weak mixing, the overlap integral between two strongly mixed

valence band states is usually big. The chances are high that a hole scattered from

one electronic state (say, mainly spin-up) to another (say, mainly spin-down) may

totally reverse its spin. So we conclude that for hole spin relaxation in the valence

band, the EY mechanism is most effective.

Suppose a hole in state |k〉 has spin Sk. After one scattering event, this hole

transits to another state |k′〉, in which the spin is Sk′ . In this process, the spin

change is ∆S = Sk′ − Sk. The spin rate of change is

dS

dt=

∫W (k,k′)∆Sdk′ =

∫W (k,k′)(Sk′ − Sk)dk

′ (5–2)

where W (k,k′) is the scattering rate between state k and k′.

Assume we can use one time constant to describe this spin relaxation process,

then we can write down an equation like

dS

dt= −Sk

τ, (5–3)

where τ is the spin relaxation time, which states how long a non-equilibrium hole

spin will take to completely lose its previous orientation. For calculating τ , we need

to know the scattering rate W (k,k′).

5.2 Lattice Scattering in III-V Semiconductors

In bulk intrinsic III-V semiconductors, momentum relaxation is realized by

phonon scattering. A unit cell in III-V semiconductors contains two atoms, so

there are three acoustic phonon modes and three optical phonon modes. Normally,

III-V semiconductors are polar materials, and unlike Si, they have no inversion

symmetry. In such a situation, the strain caused by acoustic vibration will cause

macroscopic electric fields. The vibration of oppositely charged atoms will also

give rise to long-range macroscopic electric fields. Both kinds of electric field will

provide additional scattering channels. The former is called the piezoelectric effect,

116

and the latter is polar optical phonon scattering. These two mechanisms dominate

phonon scattering in bulk III-V semiconductors.

Using Fermi’s golden rule, a general 3-D phonon scattering rate can be written

as [98]

W (k) =V

8π2NM ′

∫C2

q,bI2(k,k′)

ωq,b

δk±q−k′,K

× (n(ωq,b) +1

2∓ 1

2)δ(Ek′ − Ek ∓ hωq,b)dk

′(5–4)

where V is the crystal volume, N is the number of unit cells in the crystal, M ′ is

the reduced mass of the unit cell, hωq,b is energy of a phonon with wave vector

q in mode b, and n(ωq,b) is the phonon density for such a mode. δk±q−k′,K takes

into account momentum conservation in a scattering event, where K is a reciprocal

lattice vector. For normal processes, K = 0, and processes with K 6= 0 are umklapp

processes. Usually we only consider long wavelength phonons, where scattering

mainly takes place near the Brillouin zone center. For this kind of situation, K = 0.

In Eq. 5–4, the upper sign is for phonon absorption and the lower sign for emission.

C2q,b is the electron-phonon interaction coupling constant from the interaction

Hamiltonian, and

I(k,k′) =

∫

cell

ψ∗n′k′(r)ψnk(r)dr (5–5)

is the overlap integral between initial and final electronic states.

In DMS materials, generally the hole density is rather high, so screening must

be taken into account. When considering optical phonon scattering, in which the

vibration frequency is very high, the plasma mode may be mixed, and therefore a

dynamic screening model is desirable. But here, we just assume a static screening,

which works well for acoustic phonon and impurity scattering, and is a good

approximation for optical phonon scattering.

117

5.2.1 Screening in Bulk Semiconductors

The Coulomb potential of carriers or charged impurities is actually screened in

an electron gas or in semiconductors. If there is a semiconductor crystal without

any additional potential, the carriers should be uniformly distributed, and the

density of electrons is

n(r) =∑

i

N(Ei)f(Ei) (5–6)

where N(Ei) is the density of states at energy Ei, and f(Ei) is the Fermi-Dirac

distribution function representing the occupation probability. The sum runs over

all energy levels. When an electric potential V (r) is present, it will change the

electron density to

n′(r) =∑

i

N(Ei)f(Ei − eV (r)). (5–7)

Here we suppose the perturbation potential is small and does not affect the density

of states. Because of this potential, there is a perturbation in the space charge

density

δn(r) = n(r)− n′(r) = −∑

i

eN(Ei)(f(Ei − eV (r))− f(Ei)) (5–8)

The charge density is related to the potential by Poisson’s equation,

∇2V (r) = −δn(r)

ε=

e

ε

∑i

N(Ei)[f(Ei − eV (r))− f(Ei)] (5–9)

Consider V (r) to be a small perturbation, and expand the right-hand side of the

above equation. The leading term gives

∇2V (r) = −e2

εV (r)

∑i

N(Ei)df(Ei)

dEi

. (5–10)

Defining the reciprocal Debye screening length q0 by

q20 = −e2

ε

∑i

N(Ei)df(Ei)

dEi

, (5–11)

118

the solution for Eq. 5–10 is

V (r) =A

re−q0r (5–12)

for spherical symmetry. The value of A can be obtained using boundary conditions.

For example, for a point charge, V (r) → 0, r → ∞ and V (r) → Ze2/4πεr, r → 0,

so A = Ze2/4πε. Equation 5–12 is known as the Yukawa potential.

The derivative of Fermi’s function with respect to E in Eq. 5–10 is

df(Ei)

dEi

= −f(Ei)(1− f(Ei))

kBT, (5–13)

and thus Eq. 5–11 becomes

q20 =

e2

εkBT

∑i

N(Ei)f(Ei)(1− f(Ei)). (5–14)

We will use this equation to calculate the reciprocal screening length later when

dealing with phonon and impurity scattering.

5.2.2 Spin Relaxation in Bulk GaAs

Recently, Hilton et al. [99] measured the hole spin relaxation time in the GaAs

valence band using a pump-probe technique. They generated oriented holes using

an 800 nm laser in heavy and light hole bands, and probed using a laser pulse

(3 µm) with energy corresponding to the split-off hole to heavy hole or spilt-off

hole to light hole transitions, then measured the circular polarization change of the

transmitted light. Within an error of 10%, they obtained a hole spin relaxation

time of 110 fs. In pure intrinsic III-V semiconductors, polar optical phonon and

piezoelectric scattering are responsible for this hole spin relaxation.

Polar optical phonon scattering dominates in both II-VI and III-V intrinsic

semiconductors when temperature is not too low. At very low temperature, due to

the high energy of the optical phonons, their density is very low, too. Furthermore,

the emission of an optical phonon requires a large energy difference, which is also

not favorable at low temperatures.

119

Polar scattering occurs in connection with the contrary motion of the two

atoms in each unit cell and only takes place in the longitudinal optical mode, as

described by Frohlich [100] and Callen [101].

We can write the polar interaction Hamiltonian as

Hpopep =

∫ρ(R)φ(R) (5–15)

where ρ(R) is the charge density of the electrons and φ(R) is the electric potential

associated with polarization in the unit cell centered at R.

Following the discussion in [98], if we take into account the screening effect,

then the Hamiltonian is

Hpopep =

1√N

ee∗

V0ε0

∑q

q

q2 + q20

(iQqeiq·r + c.c.) (5–16)

where e∗ is the effective charge on the atoms in a unit cell and V0 is the volume of

a unit cell. Qq are the normal coordinates of this longitudinal optical mode. The

coupling coefficient in Eq. 5–4 is the given by

C2q =

(ee∗

V0ε0

)2q2

(q2 + q20)

2, (5–17)

and the polar optical phonon scattering rate is

W (k) =V0

8π2M ′ω0

(ee∗

V0ε0

)2 ∫q2I2(k,k′)(q2 + q2

0)2δk±q−k′,0

× (n(ω0) +1

2∓ 1

2)δ(Ek′ − Ek ∓ hω0)dk

′,

(5–18)

where hω0 is the optical phonon energy, M ′ is the reduced mass in a unit cell, and

q0 is the reciprocal Debye screening length we derived in the last section.

The “ω − q” dispersion curve for optical phonons is flat at the Γ point, very

flat in the whole Brillouin zone, and perpendicular to the Brillouin zone boundary.

So in a long-wave approximation, which means scattering near the Brillouin zone

120

center, we adopt the Einstein approximation and use the LO-phonon energy at the

Γ point, hω0, for all q in Eq. 5–18.

The effective charge in Eq. 5–4 is related to the difference between the

permittivities at low and high frequencies, and is given by [102, 103, 104]

e∗2 = M ′V0ω20ε

20

(1

ε∞r− 1

ε r

). (5–19)

Substituting Eq. 5–19 into Eq. 5–18, we get

W (k) =e2ω0

8π2ε0

(1

ε∞r− 1

ε r

) ∫q2I2(k,k′)(q2 + q2

0)2δk±q−k′,0

× (n(ω0) +1

2∓ 1

2)δ(Ek′ − Ek ∓ hω0)dk

′.

(5–20)

We can see that the polar scattering rate does not depend on the details of the unit

cell such as the volume and reduced mass, etc.

The acoustic phonon energy in the long wavelength limit can be expressed

as hω = hvsq, where vs is the sound velocity in the crystal. With a wave vector

q = 107 cm−1, the acoustic phonon energy is below 1 meV. Piezoelectric scattering

is an acoustic phonon effect, so for piezoelectric scattering, the phonon density in

Eq. 5–4 is the acoustic phonon density. Due to the very low energy, the density is

still appreciable even at low temperatures. The piezoelectric effect is due to the

acoustic strain, which is in contrast with the polar optical effect due to the optical

polarization.

The electron-phonon interaction Hamiltonian in the piezoelectric case can be

written as

Hpiezoep = − 1

ε0

∫dRD(R)P(R) (5–21)

where D(R) is the electric displacement vector related to the electric field and

acoustic strain, P(R) is the electric polarization caused by the acoustic vibration,

and R is the position of the unit cell.

121

Following the discussion in [98], in a plane wave approximation, we obtain

Hpiezoep =

ee14

ε√

N

∑q

q2

q2 + q20

(2i(a1βγ + a2γα + a3αβ)Qqeiq·r + c.c.) (5–22)

where α, β, γ are the direction cosines with respect to the crystal axis of the

direction of propagation of the waves, a1, a2 and a3 are the components of the unit

polarization vector a, and e14 is the only nonvanishing piezoelectric constant in

zinc-blende crystals. Thus the coupling coefficient of Eq. 5–4 is

C2q =

(ee14

ε

) q4

(q2 + q20)

24(a1βγ + a2γα + a3αβ)2. (5–23)

For acoustic phonon scattering, we can assume that the rates for absorption

and emission are the same due to the fact that at temperatures above several

Kelvin, n(ω) À 1 in the long wavelength limit. Combining the longitudinal and

transverse modes together, averaging over all the directional dependence, and using

the equipartition approximation, we reach the following expression

W (k) =e2K2

avkBT

8π2εh

∫q2I2(k′,k)

(q2 + q20)

2δk±q−k′,0δ(Ek′ − Ek ∓ hωq)dk

′(5–24)

where K2av is an average electromechanical coupling coefficient related to the

spherical elastic constants. As a further approximation, we can assume the acoustic

phonon scattering is an elastic process, and take the δ−function in the above

equation to be q-independent.

Table 5–1: Parameters for GaAs phonon scattering

LO phonon wavelength (Γ)1 285.0( cm−1)K2

av2 0.00252

ε∞r 10.6εr 12.4

1 Reference [105].2 Reference [98].

122

Considering both polar optical phonon and piezoelectric scattering, we have

calculated the heavy hole spin relaxation time in intrinsic GaAs. The parameters

used are listed in Table 5–1.

The calculated spin relaxation time for a spin-up heavy hole near the Γ point

as a function of the electronic wave vector at T = 300 K is shown in Fig. 5–3

(a). We can see that away from the Brillouin zone center, the spin relaxation

time decreases for a hole density of 1019/cm3, and increases for a hole density of

1018/cm3. This is believed to be connected with changes in the available density

of states at different Fermi energies. In Fig. 5–3 (b), the spin relaxation time of

a heavy hole at the Γ point is plotted as a function of temperature. We can see

that at T = 300 K with a hole density of 1018/cm3, which we think should be close

to the experimental case, the hole spin relaxation time is around 110 fs, which is

very close to the experimental value. This reveals that phonon scattering is indeed

dominant in pure GaAs.

5.3 Spin Relaxation in GaMnAs

In DMS, apart from the phonon scattering, which is pretty much the same

as we have discussed in the last section, scattering from magnetic moments

and charged impurities also occurs. The exchange interaction between localized

moments and itinerant carriers can also cause spin-flipping. The screened Coulomb

potential of an impurity atom will also couple different electronic states. In DMS,

usually the Mn doping is quite high (several percent), so exchange and impurity

scattering are much stronger than phonon scattering, so we only consider the

former two mechanisms.

5.3.1 Exchange Scattering

As discussed in chapter 1 and chapter 2, the exchange interaction between

localized moments and band carriers can be described by the exchange Hamiltonian

123

Figure 5–3: The heavy hole spin relaxation time as a function of wave vector (a),and temperature at the Γ point (b).

124

Hex = Jpd

∑I

SI · s(r)δ(r−RI). (5–25)

In a mean field approximation, this Hamiltonian can be written as [106]

Hex = JpdNMnSΩ · s, (5–26)

where Ω is the orientation of the substitutional Mn local moments, s is the carrier

spin, and Jpd is the exchange constant. Then the scattering rate in the Fermi’s

golden rule approximation is

W (k) =2π

hNMn

∫dk′

(2π)3|Mk,k′|2 × δ(Ek − Ek′) (5–27)

where

Mk,k′ = JpdS〈k|Ω · s|k′〉. (5–28)

Thus

W (k) =h

16π2NMnJ

2pd

∫|〈k|S · σ|k′〉|2 × δ(Ek − Ek′). (5–29)

Since S · σ = Sxσx + Syσy + Szσz, we have 〈k|S · σ|k′〉 = (Sxσx)kk′ + (Syσy)kk′ +

(Szσz)kk′ . The squared term will have (Sxσx)2kk′ -like terms in it. In the absence

of spontaneous magnetization, (Sxσx)2kk′ = 〈S2

x〉(σx)2kk′ , and 〈S2

x〉 = 〈S2y〉 = 〈S2

z 〉.For spontaneous magnetization (ferromagnetic case), assuming the magnetization

direction is along z, 〈Sz〉 can be found using Eq. 1–22, which results from the

self-consistent effective field approximation. From the relation S2 = S2x + S2

y + S2z ,

〈S2x〉 and 〈S2

x〉 are also found. The 〈Sx〉〈Sz〉(σ∗x)kk′(σz)kk′ -like cross terms are all

averaged to zero in first order approximation.

5.3.2 Impurity Scattering

The Mn ions in Mn doped DMS are acceptors, each of which contributes one

hole to the system and serves as a positively charged impurity site. The reason why

the carrier density is usually lower than the Mn concentration is that there exist

125

in III-V DMS materials many anti-site defects and Mn interstitials [27, 28, 29].

They both serve as Z = 2 compensating defects. The Coulomb potential of these

impurity sites can couple different electronic states.

We will use Brooks-Herring’s approach [107] in the following to deal with the

impurity scattering in GaMnAs crystals.

We can write down the equation for a screened Coulomb potential as

V (r) =Ze

4πε|r−R|e−q0|r−R| (5–30)

where q0 is the reciprocal Dybye screening length. Using Fermi’s golden rule, the

scattering rate due to this screened Coulomb coupling is given by

W (k) =2π

h

∫V

8π3|〈k′|eV (r)|k〉|2δ(Ek′ − Ek)dk

′. (5–31)

In this equation, if we assume the incident carriers can not penetrate very close to

the impurity site, we can factorize the matrix element 〈k′|eV (r)|k〉 into two parts.

One part is the rapidly varying Bloch part, the other is the slowly varying plane

wave part times the exponentially decaying Coulomb potential. Thus we have

〈k′|eV (r)|k〉 =Ze2

εV

1

q2 + q20

〈zk′|zk〉 (5–32)

where zk is the eight-component envelope spinor. So if the impurity density is NI ,

then

W (k) =Z2e4NI

4π2ε2h

∫1

(q2 + q20)

2|〈z′k|zk〉|2δ(Ek′ − Ek)dk

′. (5–33)

In GaMnAs, Mn is an impurity, its concentration is x, and its density NMn

is proportional to x. Suppose the hole density is p, then the density for the

compensating defects is (NMn − p)/2. Including both exchange interaction and

impurity scattering, the spin relaxation time as a function of the wave vector

along the Γ − X direction is plotted in Fig. 5–4. In this case, the sample is

ferromagnetic with a Curie temperature TC = 55 K at T = 30K. The hole

126

Figure 5–4: Spin relaxation time for a heavy hole as a function of k along (0,0,1)direction.

density is assumed to be p = 1019/cm3. Fig. 5–4 reveals that normally in DMS,

τimpurity ¿ τexchange ¿ τphonon. In fact, the impurity scattering is 1000 times

stronger than phonon scattering. This is natural, because for 6% Mn doping, the

Mn impurity density itself can reach 1021/cm3. The other point we can see is that

the phonon scattering in GaMnAs is weaker than in GaAs. This is because the

valence band splitting in the ferromagnetic phase makes some states energetically

unavailable for scattering due to energy conservation.

The spin relaxation time at the hole Fermi surface is illustrate in Fig. 5–5 as

a function of hole density. Fig. 5–5(a) shows the hole spin relaxation time in the

Γ − X direction and Fig. 5–5(b) shows the hole spin relaxation time in the Γ − L

direction. Here the Fermi surface can be considered as that of the unperturbed

system. The different behavior along different k direction is due to the GaAs

valence band anisotropy, which is enhanced in GaMnAs. The spin relaxation time

at the Fermi surface helps one understand the properties of the holes mediating

127

Figure 5–5: Spin relaxation time of a heavy hole as a function of hole density atdirection (a) (0,0,1) and (b) (1,1,1).

128

the ferromagnetism in DMS. In the RKKY model (or its low density limit, Zener’s

model), it is the holes at the Fermi surface that mediate the exchange interaction

which results into a ferromagnetic phase change.

Here we have only talked about the spin transport properties of a single hole.

However, the collective behavior of holes determines the properties of the hole

system. In DMS, usually the magnetization due to holes themselves is negligible

compared to that due to the localized magnetic moments. In the pump-probe

experiments we mentioned in the beginning of this chapter, it is the change of

the magnetization due to the localized moments that gives an observable result.

The change of the magnetization of localized moments, i.e., the rotation of the

magnetization direction, is induced by the spin alignment of the itinerant holes

through the exchange interaction. The change of the magnetization in return will

also have a feedback to these holes. Thus the hole-localized moments system is a

complicated system and must be treated in a self-consistent manner. This collective

self-consistent system should have a much longer life time than that of a single

hole.

CHAPTER 6CONCLUSION

In this thesis, the development and current research situation of diluted

magnetic semiconductors, including both II-VI and III-VI semiconductor-based, has

been introduced and discussed. In calculating the band structure, an eight-band

k · p theory has been employed together with the sp − d exchange interaction. In

the absence of an external magnetic field, a generalized Kane’s model is appropriate

for calculating the band structure , while in the case of an applied magnetic field,

one band will split into a series of Landau levels. In order to deal with this, we

developed a generalized Pidgeon-Brown model which incorporates the exchange

interaction and also takes into account finite kz effects. Calculations have shown

that in a diluted magnetic semiconductor, the band structure is very different from

that in a pure semiconductor. For example, the g-factors in InMnAs can be above

100 in contrast with a computed g-factor of −15 in InAs.

Cyclotron resonance in ultrahigh magnetic fields (up to 500 T) has been

simulated and compared with experiments. The method for calculating optical

transitions has been introduced and Fermi’s golden rule has been utilized. We have

successfully reproduced the cyclotron absorption in both conduction and valence

bands in InMnAs. We pointed out that the shift of cyclotron resonance peaks in

the conduction band had a dependence on the exchange constants (α − β), and the

peak line shape depended on the nonparabolicity. The h-active CR resonance in

valence bands has been decomposed into heavy-hole to heavy-hole and light-hole

to light-hole transitions in a field up to 150 T. The selection rules for optical

transitions have been discussed in a dipole approximation and we have pointed

out that due to the degeneracy in the valence band, not only h-active cyclotron

129

130

resonance, but also e-active cyclotron resonance can take place in semiconductor

valence bands. Generally the h-active transitions take place between heavy

hole or light hole Landau levels themselves, but e-active transitions take place

between heavy hole and light hole levels. The CR strength and line shape strongly

depend on carrier density, which provide an alternative way to measure the carrier

concentration. We have given an analytical expression which explains the CR

peak shifts with temperature in InMnAs/GaSb heterostructure. The pronounced

narrowing may be due to the suppression of spin fluctuation or transfer of the holes

to the InMnAs/GaSb interface at low temperatures.

We have discussed the relations between the optical constants, and from

the calculation of absorption coefficients, the reflection coefficients and magneto-

optical Kerr rotation has been calculated in bulk InMnAs and InGaAs or their

heterostructures. Because of ferromagnetism, the e-active and h-active cross-band

absorption have different dependence on photon energies. This magnetic circular

dichroism results in the polarization plane rotation of linearly polarized light.

We have simulated this magnetic circular dichroism and compared our results to

experiments.

Due to the importance of holes in diluted magnetic semiconductor systems,

we have carried out calculations for hole spin relaxation times in bulk GaAs and

GaMnAs valence bands. In GaAs, phonon scattering dominates and gives a hole

spin relaxation time around 100 fs at room temperature. In GaMnAs, due to the

strong exchange interaction and heavy impurity doping, exchange and impurity

scattering dominate. We have briefly introduced the theory of phonon scattering,

exchange scattering and impurity scattering, and shown in calculations that

in Mn-doped DMS systems, the phonon scattering is no longer important and

only impurity scattering dominates. Assuming the external disturbance a small

131

perturbation, the hole spin relaxation time at the Fermi surface is only a few

femtoseconds.

There is still much work to be done in the future. The mean field theory

has its own drawbacks in treating ferromagnetic transitions. To obtain better

results when calculating the band structure and optical absorption in ferromagnetic

samples, a better theoretical framework is highly desirable. Our current model is

not adequate to calculate the CR absorption in a InMnAs/GaSb heterostructure,

which is a type-II heterostructure (the conduction band of InAs lies below the

valence bands of GaSb). A model that can account for the interface states needs to

be developed in the future. At the current level, we have only calculated the static

MOKE of DMS, while the time-resolved MOKE is of more importance for studying

the dynamical properties of DMS. Currently we are trying to develop a method to

study time-dependent magnetic phenomena in DMS.

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BIOGRAPHICAL SKETCH

Yongke Sun was born in a small village in Zhumadian, Henan Province,

People’s Republic of China, on March 6, 1974. He stayed there for 15 years until

he finished middle school studies. After that, he went 10 miles away from home to

study in a high school called Yangzhuang High School. In 1992, he was exempted

from the national exam and admitted to Peking University in Beijing, China. From

1992 to 1997 he studied in Peking University and received his B.S. degree in 1997.

He subsequently participated in the master’s program and obtained the M.S. degree

in 2000. In the same year, he was married to his beautiful wife, Yuan Zhang, who

was his schoolmate. In the fall of 2000 he came to the United States and became

a Gator. In the summer of 2001, he entered Prof. Stanton’s group and has been

studying the properties of diluted magnetic semiconductors, pursuing a Ph.D.

degree.

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