Magnetization Dynamics in Diluted Magnetic Semiconductor Heterostructures

Contents

1  II VI diluted magnetic  NA

1.1  Crystal structure of Te and Se  11

1.2  Band structure of Te and Se  14

1.2.1  Band structure of zincblende semiconductors  14

1.2.2  Band structure of zincblende semiconductors containing manganese  19

1.3  Magnetic properties  24

1.3.1  Basic principles of magnetism  24

1.3.1.1  Larmor Diamagnetism  25

1.3.1.2  Paramagnetism  26

1.3.1.3  Heisenberg model  27

1.3.1.4  Ferromagnetism  28

1.3.1.5  Ferrimagnetism  28

1.3.1.6  Antiferromagnetism  28

1.3.2  Magnetic effects of free electrons  29

1.3.3  Magnetic properties of Te and Se without  NA

interaction   31

1.3.4  Exchange Interactions  33

1.3.4.1  sp d exchange interaction  34

1.3.4.2  d exchange interaction  35

1.3.4.3  Magnetic properties of Te and Se with  NA

Mn  interactions  39

1.3.4.4  Giant Zeeman splitting  42

1.4  Quantum well heterostructures  46

CONTENTS

1.4.1  Single particle states in quantum wells  48

1.4.2  Spin orbit splitting in quantum wells  50

1.4.3  Heterostructures in magnetic field  50

1.4.4  Density of states in quantum wells  52

1.4.5  Selection rules and polarization degree in quantum wells  55

1.4.6  Parabolic and half parabolic quantum wells  56

1.5  Excitons  60

1.5.1  Free exciton  61

1.5.2  Interaction of excitons with Mn ions  63

1.5.3  Quasi two dimensional excitons in quantum wells  63

1.5.4  Quasi two dimensional excitons in magnetic field  64

1.5.5  Trions  66

2  Magnetization  NA

2.1  Spin and energy transfer  68

2.1.1  Coupled systems in diluted magnetic semiconductors  68

2.1.2  Theoretical formulation of spin and energy transfer  71

2.1.3  Manganese spin temperature in stationary condition  75

2.2  Mechanisms for spin relaxation  76

2.2.1  yakonov Perel mechanism  77

2.2.2  Elliott Yafet mechanism  78

2.2.3  Bir Aronov Pikus mechanism  78

2.2.4  Hyperfine interaction mechanism  79

2.2.5  Spin relaxation in excitons  79

2.3  Spin lattice relaxation  80

2.4  Spin diffusion  82

3  Experimental  NA

3.1  Optical detection of Mn spin temperature  88

3.2  Heating of the Mn spin system  92

3.2.1  Heating by laser light  92

3.2.2  Heating by electric current  94

CONTENTS

3.2.3  Heating by phonons  95

3.3  Time resolved measurements  95

3.4  Experimental setup  98

4  Interaction between carriers and Mn spin  NA

4.1  Twofold dynamic impact for Mn heating  102

4.2  Direct energy and spin transfer  104

4.3  Competition between direct and indirect energy and spin transfer  106

4.4  Influence of excitation density  110

4.5  Distinction between direct and indirect heating of the Mn system  112

4.5.1  Steady state optical excitation  112

4.5.2  Long pulses with low and moderate excitation densities  112

4.5.3  Short pulses with high excitation densities  114

5  Spin lattice  NA

5.1  Dependence of the spin lattice relaxation on the Mn content  115

5.2  Effect of free carriers in doped structures  121

6  Control of spin lattice  NA

6.1  Electric field control of DEG  130

6.2  Engineering of spin lattice relaxation by digital growth  134

6.3  Spin lattice relaxation in parabolic and half parabolic quantum wells  139

6.4  Acceleration of spin lattice relaxation by spin diffusion  145

A  Samples  NA

A.1  Preparation of the samples  159

A.1.1  Molecular beam epitaxy  159

A.1.2  Quantum well heterostructures  160

A.1.3  Structure with electric contacts  161

A.1.4  Digital growth technique  161

A.2  Tables of samples  162

A.3  Lattice and electronic properties  164

CONTENTS

B  Measurement and treatment of the experimental  NA

B.1  Giant Zeeman shift  167

B.2  Spin lattice relaxation time  176

Symbols  and  NA

List  of  NA

List  of  NA

255

List of Figures

1.1  Unit cell of zincblende structure  12

1.2  First Brillouin zone of zincblende lattice  13

1.3  Schematic representation of the band structure of zincblende semiconductors  16

1.4  Calculated band structure of ZnSe and CdTe  18

1.5  Relation between lattice constant and fundamental band gap in II  NA

ductors   20

1.6  Variation of the energy gap in Cd x Mn Te with Mn concentration  21

1.7  Variation of the energy gap in Zn x Mn Se with Mn concentration  22

1.8  Lowest energy states of the Mn shell  24

1.9  Change of the density of states in magnetic field  30

1.10  Brillouin function  32

1.11  Energy level scheme of an interacting Mn ion pair as function of magnetic field  38

1.12  Magnetic phase diagram of Cd x Mn Te  40

1.13  The dependencies of the phenomenological parameters ef and on the  NA

concentration   42

1.14  Competition between Landau level and giant Zeeman splitting term for  NA

ent  Mn concentrations  44

1.15  Schematic picture of the giant Zeeman splitting of conduction band  NA

valence  band for wide band gap A  NA

1−x  Mn alloy in magnetic field  NA

the  center of the Brillouin zone at the point  45

1.16  Temperature and magnetic field dependence of the giant Zeeman splitting  45

1.17  Band edge devolution of type I and type II quantum wells  47

1.18  Schematical illustration of type I quantum well  47

1.19  Potential change of type I quantum well in magnetic field  51

1.20  Density of states without or with magnetic field in quantum well  53

LIST OF FIGURES

1.21  Filling factors of Landau levels  56

1.22  Potential of the conduction and valence band of parabolic quantum well  NA

or  without magnetic field  57

1.23  Devolution of the conduction band edge wavefunctions and energy levels  NA

parabolic  and half parabolic quantum wells  59

1.24  Theoretical Zeeman splitting of parabolic quantum well for different  NA

larized  optical transitions in magnetic field  60

1.25  Schematical picture of exciton creation  62

2.1  Interacting systems of DMS and channels for energy transfer  69

2.2  Energy diagram of electron exchange scattering on Mn ion in external  NA

netic  field  73

2.3  Scheme of yakonov Perel mechanism  77

2.4  Schematical picture of the three possible spin phonon transition  NA

with  phonon absorption  81

2.5  Times for spin spin interaction depending on Mn content  83

2.6  Time evolution of the relative changes in magnetization in type II heterostructure  83

2.7  Band scheme of heteromagnetic nanostructure  84

3.1  PL PLE and reflectivity spectra of Se and Te QWs  89

3.2  Comparison of circular polarization degree and giant Zeeman shift of  NA

PL  line  90

3.3  Giant Zeeman shift of excitons for different excitation densities  91

3.4  Interacting systems of undoped DMS under heating by laser light  93

3.5  Mn spin temperature dependency on excitation density for different Mn  NA

centrations   94

3.6  Energy scheme for photoexcitation with different photons  95

3.7  PL spectra in different time regime  96

3.8  Temporal variation of the PL spectral line and circular polarization degree  97

3.9  Experimental setup  99

3.10  Schematical assembly of the intensifier of an ICCD camera  100

4.1  Temporal evolution of Nd YAG laser pulse and of the PL signal of  NA

(Zn,Mn)Se-based  QW  102

LIST OF FIGURES

4.2  Schematical picture of the two impacts for Mn heating  103

4.3  Schematic presentation of the dynamical response of the Mn system on the  NA

pact  pulses under various experimental conditions  104

4.4  Normalized energy shifts of PL lines induced laser pluses in magnetic field  NA

(Zn,Mn)Se-based  QWs  105

4.5  Rise in energy in based QWs with different Mn concentrations  NA

comparison  with the laser pulse integral  105

4.6  Energy scheme for photoexcitation with different photons  106

4.7  Dynamics of the Mn temperature for two different laser excitation energies  107

4.8  Spin lattice and nonequilibrium phonon relaxation times measured for  NA

powers  of nm laser excitation  108

4.9  Dynamics of the Mn temperature for based QW with low  NA

concentration  under two different laser excitation energies  109

4.10  Temporal behavior of the PL line energy shift in Te  109

4.11  Mn spin temperature versus time measured at different excitation densities  111

4.12  Maximal Mn spin temperatures achieved by direct carrier heating and  NA

nonequilibrium  phonons as function of excitation density  111

5.1  Temporal evolution of PL spectral line shift for different Mn content  116

5.2  Spin lattice relaxation time as function of Mn content for nominally  NA

(Zn,Mn)Se/(Zn,Be)Se  structures  117

5.3  Energy scheme for manganese pair cluster  118

5.4  Dependence of the spin lattice relaxation time on the concentration of free  NA

trons   122

5.5  SLR time as function of the carrier density  123

5.6  Illustration of the bypass channel for energy transfer from the Mn system to  NA

lattice  through the DEG  123

5.7  Model calculations of the SLR time as function of the magnetic field  125

5.8  Fermi energy of the DEG in CdTe and ZnSe for different electron  126

6.1  PL spectra at different magnetic fields for two gate voltages  131

6.2  Giant Zeeman shift of photoluminescence line for two different gate voltages  132

6.3  Temporal evolution of PL line shift corresponding to the cooling of the Mn  NA

system  heated by pulsed laser excitation  133

LIST OF FIGURES

6.4  SLR time dependence on gate voltage for type  NA

(Zn,Mn)Se-based  QW  133

6.5  Gate voltage dependence of the PL line maxima energy and current  134

6.6  Schematic diagram of the conduction and valence band profile and Mn ion  NA

file  in Te digital alloy structures  135

6.7  PL spectra for digital alloy samples  136

6.8  Giant Zeeman shift of the photoluminescence line for the three different  NA

samples   137

6.9  Dynamical shift of the PL lines in digital alloys showing the cooling of the  NA

spin  system heated by pulsed laser excitation  137

6.10  SLR times versus Mn content in disordered alloys and digital alloys  138

6.11  Diagram linking the static and the dynamic magnetic characteristics of  NA

dered  and digital alloys  138

6.12  Scheme of digital growth profile for parabolic and half parabolic QW  140

6.13  Giant Zeeman shift of the PL line for one HPQW and two PQW samples  141

6.14  Dynamical shift of the PL lines in PQWs showing the cooling of the  NA

spin  system heated by pulsed laser excitation  142

6.15  Temporal evolution of the PL line maximum position after laser pulse for  NA

HPQW  samples  143

6.16  Diagram linking the static and the dynamic magnetic characteristics of  NA

and  HPQWs  144

6.17  Power dependence of the magnetization dynamics in the HPQW sample  145

6.18  Comparison of the relation between effective Mn contents and for  NA

and  PQWs  146

6.19  Analytical representation of the dependence of the SLR time on the Mn  NA

centration   148

6.20  Calculated spin lattice relaxation times for different spin spin diffusion  NA

cients  for Se type II QW  148

6.21  Calculated kinetics of the Mn temperature for different diffusion coefficients  NA

the  center of Se type II QW  149

6.22  Profiles of Mn spin temperature during spin lattice relaxation  150

6.23  Temporal profile of the Mn temperature in QW  150

6.24  Comparison of model calculations including spin diffusion with  NA

results  in based digital alloys  151

LIST OF FIGURES

6.25  Comparison of model calculations including spin diffusion with  NA

results  for PQWs  152

A.1  Schematical picture of MBE chamber  160

B.1  Spectrally resolved PL line of ZnMnSe based QW for different magnetic  NA

strengths   168

B.2  Giant Zeeman splitting of the PL line in magnetic field  171

B.3  Time scheme of GCCD measurement  176

B.4  Dynamics of PL line  178

List of Tables

1.1  Crystal structures and ranges of composition of DMS ternary materials  12

6.1  Experimentally determined values for the effective Mn contents of PQW  NA

HPQW  samples  142

A.1  Technological parameters of PQW and HPQW samples  162

A.2  Technological parameters and experimentally measured values for SLR for  NA

(Zn,Mn)Se/(Zn,Be)Se  samples and after the double line for the  NA

(Cd,Mg)Te  samples  163

A.3  Structure stable at room temperature lattice constants and energy  NA

(for  zincblende semiconductors vb cb transition of binary II  NA

ductors.  For more detailed data see Mad  164

A.4  Electronic properties of CdTe and ZnSe For more detailed data see  NA

It  should be mentioned that in literature exist partly drastically different  NA

band  parameters see e Fri  164

A.5  Structure stable at room temperature and lattice constants of MnSe  NA

MnTe.  For more detailed data see Mad  165

A.6  Band gap of ternary II VI semiconductors at liquid helium temperature  NA

4.2   165

A.7  Valence band offset VBO of ternary II VI semiconductors  165

A.8  Mn electron exchange constants of Te and  165

List of Listings

B.1  Origin worksheet script for nm to eV conversion  168

B.2  OriginPro function gfit  170

B.3  Origin function definition file Brillouin fdf  173

B.4  OriginPro function partdata  175

Introduction

Already in ancient times the Greek had the knowledge about electrostatic charging of amber, ν ηλεκτρoν). The first the resin of conifers, which was denoted by the Greek word electron ( realization of this effect is accredited to the great Greek philosopher Thales of Milet 1 . Never- theless, the effect was not used for centuries until beginning of modern times in 18 th century. Since then our life was revolutionized by applications and devices based on the electric charge, so that our contemporary life is unimaginable without this technology.

Especially the rapid development in the last hundred years has its reason in the compre- hension of the underlying mechanisms. The cognition of the particle electron is of particular importance in this regard. The name electron for the unit of the electric charge was introduced by George Johnstone Stoney together with Hermann Ludwig Ferdinand von Helmholtz in 1894 [Sto94, Sto95], closely followed by the experimental discovery of the electron by Joseph John Thomson [Tho97] and Emil Wiechert [Wie97] in 1897. Motivated by the discovery of the electron, Thomson developed the famous “Plum pudding model” of the atom [Tho04], which was later proved incorrect by Ernest Rutherford and substituted by the “Rutherford model” [Rut11]. However, this model could not explain origin and principle of the observed spectral lines of different gases like e.g. hydrogen, for which already several empirical correlations had been discovered [Bal85, Lym06, Pas08]. Therefore, Niels Bohr has advanced the “Rutherford model” to the “Bohr model” in 1913 [Boh13]. In the “Bohr model” the electrons have dis- crete orbits around the nucleus. Although this model achieved success, it could not explain the abnormal Zeeman-effect and the fine structure of atomic spectra.

These phenomena could be explained by an eigen angular momentum of electrons, the so- called spin. The half-integer electron spin was postulated by George Eugene Uhlenbeck and Samuel Abraham Goudsmit in 1925 [Uhl25, Uhl26] because of spectroscopic investigations. They have interpreted the spin as the fourth quantum number, which was proposed by Wolf- gang Pauli [Pau25] beside the energy E, the orbital angular momentum L and its projection L z

. This concept of an intrinsic angular momentum was very successful and could simultane- ously explain earlier experiments by Albert Einstein and Wander Johannes de Haas [Ein15], as well as Otto Stern and Walter Gerlach [Ger22c, Ger22b, Ger22a, Ste88]. Thus, the detected

1 * ∼624 v. Chr.; † ∼546 v. Chr.

INTRODUCTION 2

twofold splitting of a silver atom ray in an inhomogeneous magnetic field on the “Stern-Gerlach experiment” in 1922 is regarded as the first direct observation of the electron spin.

Because many fundamental effects in solid state physics, like e.g. ferromagnetism, are spin- related, the discovery of spin had one of the biggest impacts on modern physics. The new degree of freedom, which is offered by the spin, is technologically used in magneto-electronics, which is based on spin-polarized currents in metallic structures [Aki02, Aws02, Wol01, ˇ Zut04].

Spin-dependent transport structures based on the famous giant magnetoresistance (GMR) [Bai88, Bin89, Gr¨ u86, Pat07, Pri98] or tunneling magnetoresistance (TMR) effect [Jul75, Miy95, Moo96, Moo95], like spin valves or magnetic tunnel junctions (MTJ) [Hir02], are used e.g. in read heads of hard disks [Tsa94], magnetic field sensors [Dau94, Ton98] or magnetic memory modules (magnetoresistive random access memory (MRAM)) [Kat00, Par99, Teh00] and challenge the conventional semiconductor electronics [Dau99, Har00, Hir02, Wol01]. MRAMs for instance have, compared to common semiconductor memory modules, the big advantage, that they do not loose the stored information without being refreshed [Dre04]. Hence, electronic devices (e.g. computers) can be realized, which are immediately operable after switching-on, without loading operationally necessary data from a permanent storage into the random access memory (RAM). Because MRAMs combine the advantages of exist- ing memory technologies (high integration density and low costs of dynamic random access memory (DRAM), high speed of static random access memory (SRAM) and nonvolatility of flash-memory) [Ino02], they have the potential for “universal memory”, replacing current mem- ory technology.

Analogue to conventional charge-based electronics, where the contemporary information and communication technology was only enabled by semiconductor electronics, crucial disad- vantages of metallic components limit further progress. In metals the carrier density cannot be changed continuously and metals have no band gap, so that they are unsuitable for many electronic and all opto-electronic components. Adding the spin degree of freedom for main- stream charge-based electronic devices has the potential advantages of nonvolatility, increased data processing speed, decreased electric power consumption, and increased integration densi- ties compared with conventional semiconductor devices [Wol01]. Furthermore, the proceeding reduction of structural sizes in electronic components will soon result in a dominance of quan- tum mechanical effects (especially the spin-dependent exchange interaction among carriers). Compared with the spatial coherence of carriers, the spin is a relatively stable value and, thus, potentially very suitable for future electric components. From these aspects arises the recent big attention for the spin in semiconductor electronics [Ohn98, Wol01]. This new research area is called “spintronics” and may denote the next evolution-step for electronics and can be the be- gin of the age of spin-electronics [Win04]. It is envisioned that in particular merging photonics with spintronics will lead to a multitude of spin-based multifunctional devices such as spin-light emitting diodes (LED), spin-field-effect transistors (FET), spin-vertical cavity surface emitting

INTRODUCTION 3

lasers (VCSEL), spin-resonant tunneling devices (RTD), optical switches working at terahertz frequency, modulators, encoders, decoders, and quantum bits for quantum computation and communication [Wol01, ˇ Zut04]. Especially in the field of quantum computing [DiV95, Ste98], where the required quantum mechanical two-level-system (the so-called qubit) can be realized by the two possible spin-states of a particle with spin

1 [Ima99, Los98, Sas01], a big impact by

Successful incorporation of spins into the existing semiconductor technology inevitably implies experimental possibilities to create spin-polarized electrons and currents respectively (spin-injection), to controllably switch their spin-state (spin manipulation), to transport and store orientation and/or phase of the spin-state (spin-transport, spin memory, spin coherence), and finally to reliably read out the spin-state (spin detection). Good progress is already achieved on spin-injection [Fie99, Oes99, Ohn99b] and spin-transport [H¨ ag98, Kik99]. Spins can be in- jected 2 in a semiconductor by several methods: Traditionally optical injection [Mei84, Oes02] has been used, where the angular momentum of absorbed circularly polarized photons is trans- ferred to electrons through spin-orbit interaction. As a light source is needed, this method is unfavorable for electronic components. More desirable is electrical spin-injection, where trans- fer from a magnetic layer to a nonmagnetic semiconductor layer was used at the beginning [Gar99, Ham99, Ham00, Joh98, Sch00b]. To avoid mismatch between the layers, alternatively doping of semiconductors with magnetic impurities was established [Fie99, Jon00, Oes99]. Semiconductors doped with magnetic ions are nowadays the most promising materials for spintronic devices [Die94, Fur88a, Fur88b, Jai92]. Examples are well-known A III B V and A II B V I semiconductors like GaAs, InAs, GaN, CdTe, ZnSe, ZnO etc., which have metal ions in the cation sublattice isoelectronically substituted by magnetic ions, like Mn, Fe, Cr, V etc. This allows growing ternary alloys with a wide range of magnetic ion concentrations up to 100 %. These materials are called diluted magnetic semiconductors (DMS) as opposed to “concentrated” magnetic semiconductors. In the latter the magnetic ion is a part of its reg- ular lattice, whereas in the former the magnetic ions partially substitute the nonmagnetic host atoms [Aki02]. The ternary nature gives good possibility of tuning lattice constant and band parameters by varying the amount of substituted cations. As first investigations were done on macroscopic mono-crystals, predominate thin films [Deb81, Kol84b] and heterostructures [Bic84, Kol84a] since the 80 th . Nowadays it is possible to create stoichiometric compositions, which are not stable as volume crystals [Dur89, Kol86b]. This progress in fabrication leads to a rich spectrum of entirely new physical phenomena not accessible by bulk growth [Fur96].

DMS combine optical and transport properties typical to a semiconductor (narrow optical resonances, high electron mobility) with features specific for magnetic materials (antiferro- magnetism, ferromagnetism or paramagnetism) [Aki06a]. Most popular as magnetic ions are manganese Mn 2+ -ions because of several advantages: Mn 2+

-ions can be incorporated in siz- 2 Spin-injection usually means creation of nonequilibrium spin population.

INTRODUCTION 4

able amounts in A II B V I semiconductors without considerably change of the crystallographic quality of the host material, Mn 2+ -ions are electrically neutral in A II B V I semiconductors and Mn 2+ -ions possess a relatively large magnetic moment due to the half-filled d-shell [Fur88b]. Comprehensive information about the properties of DMS with Mn-ion is collected in several reviews [Die94, Fur82, Fur86, Fur88a, Fur88b, Fur96, Goe88, Jai92].

If Mn-ions are incorporated in A II B V I semiconductors, like (Cd,Mn)Te and (Zn,Mn)Se, which are investigated in this thesis, a strong paramagnetism appears. Thus, the conduction band electrons can be completely aligned already by weak external magnetic fields. But be- cause of the needed magnetic field and additionally because of the covering of paramagnetism by the thermic motion of the electrons at room temperature, DMS are presently unsuitable for integrated components. Nevertheless A II B V I DMS have also several advantages. On the one hand they have good structural quality, sharp absorption edge formed by excitons, and show strong band edge photoluminescence (PL), which allows one to apply the broad spectrum of optical experimental techniques, in particular in magnetic field [Die94, Fur88b, Fur88a]. By contrast, luminescence is poor in A III B V DMS materials, such as (Ga,Mn)As. On the other hand they have the unique possibility to tailor electronic and magnetic properties independently. Thus, A II B V I DMS are very suitable for testing novel design concepts for spintronics applica- tions and are nowadays widely used in this respect [Aws02, Fre99].

In DMS the carrier spin manipulation arises from the spin-flip exchange scattering of free carriers on the localized magnetic moments of the magnetic ions. Formation of magnetic polarons [Die83, Naw81] and a variety of exceptional strong magneto-optical and magneto- transport effects originate from the strong exchange interaction of the localized Mn 2+ magnetic moments with spins of the conduction band electrons (s-d-exchange interaction) and/or va- lence band holes (p-d-exchange interaction). Among the effects are giant Zeeman-splitting of the band states, which may exceed 100 meV at low temperatures, giant Faraday rotation [Gaj78] and Kerr rotation effects [Die94]. The magnetic properties of the Mn-ion system, which depend strongly on the Mn concentration, play a key role in these effects. For example, neighboring Mn-ions interact antiferromagnetically, which leads to the formation of high ordered clusters and spin-glass phases at higher Mn concentration [Fur88a]. An overview about the basic prop- erties of A II 1−x Mn x B V I DMS, like crystal and band structure and magnetic and optic properties, is presented in the first chapter of this thesis. Comprehensive collection of the data for para- magnetic ions can be found in [Abr70] and references therein, and for conduction electrons and holes in semiconductors in [Mei84].

The prerequisite of spintronic implementations in DMS is detailed knowledge about the cou- pled systems of magnetic ions, lattice (the phonon system) and free carriers which determine transport, magneto-optical and magnetic properties of DMS. The spin and energy transfer be- tween the coupled systems, where spins are hold by free carriers (electrons and holes) and by Mn-ions, controls the magnetization dynamics in DMS. Present state of knowledge on mag-

INTRODUCTION 5

netization dynamics in A II B V I semiconductors with Mn 2+ -ions is given in the second chapter. The focus lies thereby on the description of the coupled systems. Theoretical basis for spin and energy transfer between the systems will be considered, which often can not be separated from each other.

Access to the magnetization dynamics is offered by the impact of the Mn-spin system tem- perature on the magneto-optical properties. As already mentioned, the giant magneto-optical effects originate from the strong sp-d-exchange interaction between the localized electrons in the 3d-shell of the Mn-ions and the delocalized s-type electron states in the conduction band and the p-type hole states in the valence band. These effects are based on polarization of the carrier spins interacting with the localized magnetic Mn-ions, which in turn are polarized by an external magnetic field. As a result the magnitudes of the spectroscopic responses are propor- tional to the magnetization of the Mn-spin system. Besides the strength of external magnetic fields, the magnetization is determined by the temperature of the Mn-spin system, which can differ from the bath temperature (i.e. lattice temperature). Therefore, heating of the Mn-system can strongly influence magneto-optical and magnetotransport properties.

As internal thermometer of the Mn-spin temperature the giant Zeeman-splitting of exci- tons (band states) is exploited, which is highly sensitive to the magnetic Mn-ions due to the sp-d-exchange interaction. This allows studies in a wide temporal range from picoseconds to hundreds of microseconds. To drive the Mn-spin system out of equilibrium with the lattice, several methods are employed [Yak09]. The experimental approach via heating by means of photogenerated (by pulsed laser excitation) or electrically accelerated carriers, which is used in thesis, and the associated experimental setup for time-resolved measurements are described in chapter three.

Concerning the magnetization dynamics, detailed understanding of dynamical properties of the localized Mn-spins, namely spin dephasing and particularly spin relaxation is one emphasis of current research. As the electric charge is a conserved quantity, which cannot be affected by scattering processes for instance, the spin can loose its information - the direction of the spin-vector, i.e. the spin polarization - in scattering processes [Win04] or due to spin-orbit and hyperfine coupling [ ˇ Zut04]. While the occupation of electronic spin states in undistorted systems is given by an equilibrium distribution, is the spin polarization a nonequilibrium state. The relaxation of the spin leads therefore to equilibration of spin polarization and, thus, is of great importance for spintronics. The time within the spin polarization gets lost, is the spin relaxation time. Typical values for spin relaxation times vary from less than one picosecond for free carriers to years for some nuclears [Aki06a]. The fact that nonequilibrium electronic spin in semiconductors lives relatively long (typically a nanosecond), allows for spin-encoded information to travel macroscopic distances [ ˇ Zut04].

First findings on spin relaxation in Mn 2+ -ions were already achieved in the 60 th [Blu62, Lam60], but on materials with very low Mn 2+ content (x 0.01), where magnetic ions may be

INTRODUCTION 6

considered as isolated. In these materials different mechanisms of spin relaxation are important in comparison to the magnetic semiconductors with much higher Mn content, which are actual for spintronics. In bulk A II 1−x Mn x B V I DMS spin relaxation of Mn 2+ -ions was investigated during the last decade [Bin91, Far96, Sca88, Sca96a, Str90, Str92]. A strong dependance of the spin-lattice and spin-spin relaxation times on the Mn content x and lattice temperature T L was shown.

In this thesis spin dynamics in (Zn,Mn)Se/(Zn,Be)Se and (Cd,Mn)Te/(Cd,Mg)Te DMS quantum well (QW) heterostructures with a type-I band alignment are studied, where the carriers are quantum confined. It is already well-known that the spin relaxation time of free carriers (∼10 -12 -10 -11 s [Cam01, Cro97]) is much faster than for Mn-ions (10 -8 -10 -3 s [Far96, Kel01, Kel04, Sca96a, Sch00a]). Thus, spin relaxation of the magnetic ion becomes a bottleneck for fast spin switching in magnetic semiconductors [Aki06a]. Nevertheless, the free carriers cannot be neglected, because the presence of free carriers (their concentration, temperature and spin polarization) modifies strongly the efficiency of energy and spin transfer between the systems of DMS and, thus, is of great importance for static characteristics and dynamical properties of DMS materials. Especially the important role of free carriers in heat- ing of the Mn-system, by its interaction with photoexcited carriers with excess kinetic energy, and in the cooling of the Mn-system in the presence of cold background carriers, provided by modulation doping, is established.

The studies are separated in three chapters. In the fourth chapter of this thesis, new results on energy and spin transfer between free carriers and Mn-ion system are presented. Contribu- tions of direct heating of the Mn-system by photocarriers and indirect heating via nonequi- librium phonons are distinguished and their competition is discussed. In the fifth chapter dynamics of spin-lattice relaxation (SLR) of magnetic Mn-ions in (Zn,Mn)Se/(Zn,Be)Se and (Cd,Mn)Te/(Cd,Mg)Te DMS QW heterostructures is investigated and new experimental studies on (Zn,Mn)Se/(Zn,Be)Se heterostructures are shown. As the manganese SLR is of key impor- tance for the dynamical processes in DMS, a number of studies have already been performed for (Cd,Mn)Te-based heterostructures [Sch00a, Sch01b, Sch01a] with Mn concentrations not exceeding x = 0.05.

Crucial for spintronic devices is the ability to tune the spin relaxation time precisely, as the spin relaxation time is important in double respects. On the one hand spin polarization must be conserved over long times and distances, if the spin shall be processed or stored in a region, which is spatial separated from the spin-injector [Wol01]. Especially for the possibility of utilizing spins as quantum bits for quantum information processing, long spin polarization is needed. On the other hand short spin relaxation time is needed for fast switching between different spin-states. For instance semiconductor lasers can be switched off extremely fast by reorientation of spins [Oes02]. This very relevant topic is devoted the sixth chapter, before the thesis is summarized in the last chapter. Especially for one of the biggest drawbacks for

INTRODUCTION 7

precise tuning, that the magnetization dynamics in DMS cannot be controlled separately from the static magnetization, solutions via electric field control of the magnetization dynamics or via the technological concept of “digital alloying” are presented.

Chapter 1

II-VI  diluted magnetic  NA

1.1  Crystal structure of Te and Se  11

1.2  Band structure of Te and Se  14

1.2.1  Band structure of zincblende semiconductors  14

1.2.2  Band structure of zincblende semiconductors containing manganese  19

1.3  Magnetic properties  24

1.3.1  Basic principles of magnetism  24

1.3.2  Magnetic effects of free electrons  29

1.3.3  Magnetic properties of Te and Se without  NA

interaction   31

1.3.4  Exchange Interactions  33

1.4  Quantum well heterostructures  46

1.4.1  Single particle states in quantum wells  48

1.4.2  Spin orbit splitting in quantum wells  50

1.4.3  Heterostructures in magnetic field  50

1.4.4  Density of states in quantum wells  52

1.4.5  Selection rules and polarization degree in quantum wells  55

1.4.6  Parabolic and half parabolic quantum wells  56

1.5  Excitons  60

1.5.1  Free exciton  61

1.5.2  Interaction of excitons with Mn ions  63

1.5.3  Quasi two dimensional excitons in quantum wells  63

1.5.4  Quasi two dimensional excitons in magnetic field  64

9

CHAPTER 1. II-VI DILUTED MAGNETIC SEMICONDUCTORS 10

1.5.5 Trions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 In this thesis QW heterostructures containing (Cd,Mn)Te and (Zn,Mn)Se are investigated. To facilitate the presentation and understanding of the results, discussed in the following chapters, it is necessary to give an overview about the structural, electronic and magnetic properties of these materials in the first sections of this chapter. The crystal structure is described in section 1.1 on the basis of the binary II-VI semiconductors CdTe and ZnSe, which is due to the close

resemblance of the properties between the binary compounds and their ternary DMS derivatives. This applies analogue for (Cd,Mg)Te and (Zn,Be)Se, which are used in some of the investigated heterostructures.

In the second section 1.2 the band structure of wide-band-gap A II 1−x Mn x B V I bulk alloys 1 is presented, starting in subsection 1.2.1 with the host semiconductors CdTe and ZnSe in absence of an external magnetic field. Also here resembles the band structure of the A II 1−x Mn x B V I alloys qualitatively that of the nonmagnetic A II B V I “parent” material having the same crystal struc- ture. Because all investigated compounds have the same crystal structure, the band structures are very similar, especially at the Γ-point. Calculation of the electronic states can be done by the envelope-function-approximation [Alt83a], which describes the particular layers of a QW structure by volume bands. The latter are described in the section for CdTe and ZnSe accord- ing to the Kohn-Luttinger-theory [Lut56]. Subsection 1.2.2 discusses effects of incorporated manganese ions, especially of the Mn d-electrons, on the electronic states. The impact on the optical properties, due to intra-ion transitions in the half-filled Mn 3d 5 -shells, is considered in particular.

The subsequent section 1.3 outlines the magnetic properties of (Cd,Mn)Te and (Zn,Mn)Se. Firstly an short overview about possible types of magnetic orders in solid states is given, which is needed for the further understanding. In the second subsection 1.3.2 the case of free carriers in the host semiconductor is considered. Pauli paramagnetism as well as Landau diamagnetism occur if an magnetic field is applied. For the remaining section the focus is shifted to the pecu- liarity of DMS, namely that an amount of the cations in the lattice is exchanged by the magnetic ions. Starting point are the magnetic properties of isolated Mn-ions outlined in subsection 1.3.3.

Because of the insertion of Mn-ions, simultaneously localized spins and magnetic moments are integrated in the crystal. The localized spins of the half-filled d-shell of the Mn-ions incur two types of exchange interactions. The strong Kondo-like sp-d-exchange interaction between the band electron spins and the localized moments of the magnetic ions influences strongly the electronic properties. The Heisenberg inter-ion d-d-exchange interaction is weaker and underlies the static and dynamic magnetic properties of the A II 1−x Mn x B V I

DMS. Both exchange interactions are introduced briefly in subsection 1.3.4. Furthermore, the giant Zeeman-splitting 1 A = Cd,Zn; B = S,Se,Te

1.1. CRYSTAL STRUCTURE OF (CD,MN)TE AND (ZN,MN)SE 11

is introduced as consequence of the significant influence of the sp-d-exchange interaction on the band structure.

The approximation of non interacting Mn-spins, made in subsection 1.3.3, is only valid for very strong diluted systems (x < 0.01), so that the influence of the d-d-exchange interaction be- tween Mn-spins is introduced in the discussion of the magnetic properties in subsection 1.3.4.3. The magnetic dipole-dipole interaction can continued to be neglected. Furthermore, the mag- netic phases in A II 1−x Mn x B V I DMSs are addressed briefly in this subsection. Big importance arises from precise knowledge about the magnetic properties, because they bear on the optical and electrical properties of a DMS through the sp-d-exchange [Fur88a].

In section 1.4 the peculiarities of DMS QW heterostructures are described. A II 1−x Mn x B V I alloys are excellent candidates for these structures because of the tunability of their lattice pa- rameters and band gaps, especially with regard to flexibly band structure engineering [Dat85]. In the section the effects of the spatial confinement on the energy bands, the density of states, the spin-orbit splitting and the resulting selection rules are reviewed.

In the last section introduction into excitons in semiconductors is given. Both the free exciton and the situation of excitons in QWs is described. Finally also the situation of quantized motion of excitons in magnetic field will be considered.

1.1 Crystal structure of (Cd,Mn)Te and (Zn,Mn)Se

Solid states in crystalline phase have a spatial symmetry, arising from the continuous arrange- ment of the atoms in the crystal lattice. The materials, investigated in this thesis, are based on the II-VI binary compound-semiconductors CdTe and ZnSe. The latter have the propensity to crystallize in a variety of polymorphic modifications. In general, structures for II-VI semi- conductors are hexagonal wurtzite and cubic zincblende (sphalerite) [Ave67]. CdTe and ZnSe possess under normal conditions the zincblende structure [Yeh92].

In the ternary materials (Zn,Mn)Se, (Zn,Be)Se and (Cd,Mn)Te and (Cd,Mg)Te a small amount of the Zn 2+ -ions and Cd 2+ -ions, respectively, is exchanged by likewise bivalent Be 2+ -, Mg 2+ - and Mn 2+ -ions, respectively. Bulk crystals of BeSe exhibit the zincblende structure [Wyc63], of MgTe the wurtzite structure [Kle51, Kuh71, Par71, Zac27], of MnSe the rock- salt (NaCl) structure [Dur89] and of MnTe the hexagonal NiAs structure [Oft27]. Therefore, the crystal structure of the ternary materials is dependant on the exchanged amount of cations. The corresponding composition ranges and the upper limits for successful incorporation of the magnetic Mn-ions are given in table 1.1. It is remarkable that such high values of x

for the Mn-ions in ternary alloys can be reached, although the crystal structures of MnSe and MnTe are 2 Special epitaxial growth (e.g. molecular beam epitaxy (MBE)) enables the zincblende structure for Cd 1−x Mn x Te with x > 0.77 up to pure MnTe [Zak95] and for Zn 1−x Mn x Se with x > 0.3 up to pure MnSe [Kol86b].

CHAPTER 1. II-VI DILUTED MAGNETIC SEMICONDUCTORS 12

Table 1.1 – Crystal structures and ranges of composition of DMS ternary materials [Fur88a, Fur96]. The upper limit on x

is imposed by the fact that neither MnSe nor MnTe crystallize in the zincblende or wurtzite structure.

neither zincblende nor wurtzite. For higher concentrations a mixture of wurtzite and rocksalt structure was observed in Zn 1−x Mn x Se [Juz56]. In Cd 1−x Mn x Te even a mixture of zincblende, NiAs and pyrite-typestructure was observed [Paj78]. In this thesis only samples with low Mn concentration in zincblende structure were investigated.

The zincblende structure, which is schematically represented in figure 1.1, is based on the d -F ¯ cubic space group T 2 43m [Ave67]. 3

It has the same arrangement as diamond and consists of two face-centered-cubic (fcc) lattices. Contrary to diamond one lattice is occupied with cations and the other with anions. The two lattices are displaced against each other by a quarter of the space diagonal, so that every ion is tetraedrically surrounded by four ions of the other kind. Each cation has four next neighbors of anions at the corners of a regular tetrahedron at a

by the lattice constant a 0 . In ZnSe the lattice constant constitutes 5.668 ˚ A [Ave67] and in CdTe 6.487 ˚

A [Bot81]. The lattice constant for the ternary materials a 0,ternary

can be calculated by Vegard’s law

as linear interpolation between values of the corresponding binary semiconductors a

0,binary1;2 [Den91, Fur83, Veg21, Fur96]. For a general characterization it is more convenient to calculate

Figure 1.1 – Unit cell of zincblende struc-

ture. The open spheres represent the metal- lic cations and the others the semi-metallic anions. The two sublattices are displaced against each other by a quarter of the space diagonal.

d is the Sch¨ onflies notation and F ¯

3 T 2 43m is the crystallographic description according to Hermann-Mauguin.

1.1. CRYSTAL STRUCTURE OF (CD,MN)TE AND (ZN,MN)SE 13

Fur96]:

˚

= 4.009 − 0.366y [Mad99, Gal97].

(1.5) Even the structure transition in Zn 1−x Mn x Se does not affect the linear relation of d [Yod85]. An important aspect for zincblende structures arises from symmetry. The T d point group, which is a subgroup of the full spherical group O(3), contains 24 proper and improper

symmetry-conserving rotations. In contrast to monatomic semiconductors in diamond structure, like silicon or germanium, possess zincblende structures no center of symmetry or inversion. Hence, zincblende crystals may have different physical and chemical properties in different directions. As a consequence zincblende crystals are piezoelectric [Ave67]. The inversion as symmetry operation leaves spinors invariant and causes, in combination

with the Kramers-degeneration against time reversal, that all states in absence of magnetic field with Ψ ↑ ( k) = Ψ ↓ ( k) are at least twofold spin-degenerated [D’y71, Ell54a, Ell54b]. In contrast

holds in zincblende the general Kramers-degeneration Ψ ↑ ( k) = ˆ τ Ψ ↑ ( k) = Ψ ↓ (− k) with the

CHAPTER 1. II-VI DILUTED MAGNETIC SEMICONDUCTORS 14

The chemical bond between the metallic and semi-metallic ion in II-VI semiconductors is a tetrahedral s-p 3 -bond based on the two s-valence electrons of the metal and the six p-valence

electrons of the semi-metal. The additional Be-, Mg- and Mn-ions, respectively, substitute the group II elements on the cation places. Also these contribute with two s-valence electrons to the bond. Because of the differences in electronegativity of the involved ions, has the bonding of the sp 3 -hybridized atomic orbitals partly ionic and partly covalent character, depending on

the concrete material system.

Contrary to the lattice constants remain the bondlengths for CdTe, ZnSe, MnTe and MnSe over the hole composition range of Zn 1−x Mn x Se and Cd 1−x Mn x Te constant [Bal84, Bun87b, Bun87a, Pon90]. This can only mean that the real crystal lattice is locally highly distorted [Kos93]. Balzarotti stated that the cation-sublattice is undistorted in first order and the structural necessary alignment takes only place in the anion-sublattice, which abandons fcc symmetry [Bal84]. Contrary to that shows X-ray data that both sublattices are microscopically distorted [Abr85].

1.2 Band structure of (Cd,Mn)Te and (Zn,Mn)Se

1.2.1 Band structure of zincblende semiconductors

The special properties of the band structure of semiconductors become obvious, paying regard to the differences to metals and isolators. In metals the electronic bands are partly filled, so that many electrons can contribute to high conductivity. Isolators have either empty or completely filled bands, so that electrons cannot contribute to the conductivity due to the Pauli principle. Because of the huge energy gaps (E g 4 eV) between the lowest empty band (conduction band)

and the uppermost filled band (valence band) of isolators, conductivity cannot be achieved by thermal excitation of electrons from the valence band to the conduction band. At T = 0 K this situation is similar for semiconductors. But, in contrast to isolators, semiconductors have only small energy gaps in the region of a few eV. Therefore, electrons can be excited from the valence band to the conduction band via thermal energy for T = 0 K [Kit05].

For many optical properties of semiconductors only the energy dispersion near the funda- mental band gap is relevant. This holds also for the optical transitions utilized in the experi- mental part of this thesis, which occur near k = 0 in the middle of the Brillouin-zone at the Γ-point. Therefore, this part of the band structure is specified in this section. For a discussion of the general band structure one can refer e.g. to [Lar88b]. Starting point is the carrier wave function. This has to fulfill the Schr¨ odinger-equation ([Ash76])

1.2. BAND STRUCTURE OF (CD,MN)TE AND (ZN,MN)SE 15

in the crystalline potential

(1.7)

with the periodicity of the underlaying Bravais lattice for all Bravais lattice vectors R. Thereby is m 0 the free electron mass and Ψ n k ( r)

the electron wave function in crystal lattice with in-

Ψ n k ( r) = exp i kk r u n k ( r) (1.8) with the lattice-periodic potential u n k . The eigenvalues are characterized by eigenstates E n k . Eigenstates for fixed n in solid states are called energy bands with band index n and wave vector k. The possible energy states E n k are filled with electrons according to the Pauli principle. The energetically highest filled energy bands are called valence bands and the proximate band

conduction band. The energy difference between the energetically highest valence band and the minimum of the conduction band is denoted as energy gap E g . Using the tight-binding-model in absence of spin-orbit splitting, the valence band is p-like with orbital angular momentum of L = 1, which leads to a threefold degeneration. The con- duction band is s-like with orbital angular momentum L = 0 at the Γ-point. Using the, in solid state physics usual, nomenclature for irreducible representations by Koster [Kos69], the valence band has Γ 15 -symmetry and the conduction band Γ 1 -symmetry. The dispersion relation of the band structure is approximately parabolic in the vicinity of the Γ-point. As in zincblende the minimum of conduction band and the maximum of valence band is at the Γ-point in the center of the Brillouin-zone, the investigated materials are direct semiconductors. In materials with an

indirect band gap, like silicon or germanium, the maximum and minimum occur at different

k values, so that only phonon assisted optical excitations are possible.

p ×

∇V , (1.9)

CHAPTER 1. II-VI DILUTED MAGNETIC SEMICONDUCTORS 16

E

fold spin-degenerated Γ 6 -band. As the conduction band is mainly formed by the s-like wave functions of the atomic orbitals with L = 0 and S = 1 /2, it has total angular momentum J = 1 /2. The valence band is based on p-like wave functions with L = 1 and S = 1 /2. For k = 0 the Γ 8 states with J z = ± 3 /2 split into two bands with different bending. Because of their effective masses they are called heavy-hole (hh) band (states with J z = ± 3 /2) and light-hole (lh) band (states with J z = ± 1 /2). The spin-orbit splitting arises due to the interaction of the

intrinsic magnetic moment of the electron spin with the magnetic field, generated by electron motion. The magnitude of the spin-orbit splitting is known to affect the location of the lowest hole levels in the valence band. Thus, any change in spin-orbit splitting affects the luminescent properties of semiconductors [Lip98]. As the Γ 7 -band lies energetically about 950 meV and 430 meV, respectively, below the Γ 8 -band in CdTe and ZnSe, respectively, [Al-03, W¨ or97], it is nearly always full occupied at low temperature. Therefore, the Γ 7 -band has no effect on the optical properties in the region of the fundamental band gap. Because of the weak coupling between conduction and valence band, due to the wide fundamental band gap, the dispersion of the conduction band can be assumed as isotropic and parabolic. It can be described similar to the dispersion of free electrons E(k) = 2 k

2 . To take into account the influence of the periodic

2 k

2

In contrast the dispersion of the Γ 8 -valence band is not isotropic and mostly described by means of the Kohn-Luttinger-theory [Lut55, Lut56]. The valence band can be described by a 4×4 matrix containing the wave vector k in the second power [Kuh95]. By means of a Hamilton-

1.2. BAND STRUCTURE OF (CD,MN)TE AND (ZN,MN)SE 17

Operator with the same symmetry, the energy dispersion can be derived from the condition

, (1.11)

∂k

2 i

k [100], (1.13)

= γ 1 ± 2γ 2 m 0

m ef f,hh/lh = γ 1 ± 2γ 3 k

[111]. (1.14)

6 and for ZnSe m ef f,lh = 0.27m 0 and m ef f,hh = 0.81m 0 .

As already stressed in the previous section, the zincblende lattice possess no center of in- version because of the two-atomic basis. But spin-degeneration of electron and hole states splitting of the bands occurs already for k = 0 without external magnetic field B. The resulting spin splitting of the Γ 6 valence band can be described in third order in k by the Dresselhaus

Hamiltonian [Dre55, D’y84, Kai03]

CHAPTER 1. II-VI DILUTED MAGNETIC SEMICONDUCTORS 18

Figure 1.4 – Calculated band structure of ZnSe and CdTe according to Chelikowsky and Cohen [Che76]

with the spin operator σ

and

y ⎞ ⎟

⎠

,

1 − sin 2 ϑ

1 + 2 sin 2 (2ϕ)

1.2. BAND STRUCTURE OF (CD,MN)TE AND (ZN,MN)SE 19

1.2.2 Band structure of zincblende semiconductors containing man-

ganese

The qualitative devolution of the band structure, pictured in figures 1.4 and 1.3, remains un- changed for the ternary compounds (Cd,Mg)Te, (Cd,Mn)Te, (Zn,Be)Se and (Zn,Mn)Se, if only a small amount of cations is exchanged with respect to the binary semiconductors CdTe and ZnSe [Bec88]. Theoretically can the effect of the manganese atom on the band structure of the host semiconductor be described in terms of an average medium theory, such as the virtual crys- tal approximation, the coherent potential approximation or another appropriate theoretical pro- cedure [Agg87, Gun89, Has83, Has88, Has90, Hui89, Lar85, Lar88b, Maˇ s87, Wei86, Wei87]. In first order the changes of the band structure can be quantified as continuous transition from the band structure of the host A II B V I material to a hypothetic tetrahedrally bonded MnB V I material [Fur96, Liu04, Wei86, Wei87].

Hence, the fundamental band gap at the Γ-point becomes influenced by the concentration of the substituted cations 7 . The variation of the fundamental band gap with the manganese content can be described by a linear dependence in first approximation [Fur88a, Kos93]. The relation between band gap and lattice constant for several ternary II-VI semiconductors is shown in figure 1.5. The small deviation of the linear dependence which can be seen in the figure is known as “bowing” [Bec88, Tho67]. The effect of bowing is rather small in wide-band-gap DMS 8

, so that the band gap energy for a given composition can be estimated by the virtual crystal approximation (VCA) [Fur88a]:

with ΔE g = E g (x = 1) − E g (x = 0). Using the energy gaps of the binary compounds for liquid helium temperature (T = 4.2 K), results for the ternary compounds

Cd 1−x Mn x Te : E g (x) = 1.606 + 1.592x [eV] (x < 0.77) [Bot81, Hei86, Yod85] (1.20)

and

Zn 1−x Mn x Se : E g (x) = 2.82 + 0.48x [eV] [Fur88a].

(1.21)

As in the A II B V I

“parent” material, the temperature affects the energy gap in DMS by a linear relationship. The energy gap tends to open wider as the temperature decreases [Bec88, Fur96]. 7 This influence can be dramatic at higher concentrations. For example, in (Zn,Be)Se changes the fundamental band gap to an indirect band gap (Γ → X) for Be concentrations over 0.46. This leads to rapid degeneration of the optical properties [Cha00, Slo06].

8 In (Cd,Mn)Te bowing is even absent [Bec88].

CHAPTER 1. II-VI DILUTED MAGNETIC SEMICONDUCTORS 20

Figure 1.5 – Relation between lattice constant a 0 and fundamental band gap in several II-VI semi- conductors [Mac96]. The ternary compounds lie on the dashed lines between the binary compounds.

Apart from the bowing, an additional anomalous temperature dependence of the energy gap at low temperatures can be observed in several A II 1−x Mn x B V I alloys, especially strong in (Zn,Mn)Se [Byl86, Byl87, Dio85, Gaj87, Ike68, Kol86c, Lee84]. This might be due to second-order corrections to the energy of the band edge states, originating from the p-d- and s-d-exchange interaction with the localized magnetic moments [Byl86]. This correction can be a possible explanation for the peculiar temperature dependence of bowing in DMS [Kos93, Gaj87].

An isolated manganese atom has the electronic configuration (Ar) 4s 2 3d 5 . As already men- tioned, the 4s 2 -electrons substitute the s- valence electrons of the cations, if manganese is incor- porated in an A II B V I lattice, and contribute therefore to the formation of valence and conduc- tion band. Contrary the electrons of the half filled d-shell cannot contribute to the band structure, but their effects superimpose on the band structure of the host semiconductor. The reason for this is that the energy of the narrow band, originating from the ground state of the 3d 5 electrons, lies approximately 3.5 eV below the valence band edge (E vb ) [Fra85, Oel82, Tan86, Web81]. The ground state e

+σ is an orbital singlet. This cannot be further split by the crystal field

the wave functions increases slightly because of hybridization with the p-bands of the host semiconductor [Lee84, Lee86, Mor82, Ngu83, Tao82]. The position of the ground state is rather insensitive to the host material [Fur96]. The corresponding excited state e

1.2. BAND STRUCTURE OF (CD,MN)TE AND (ZN,MN)SE 21

Figure 1.6 – Experimental results for the variation of the energy gap in Cd 1−x Mn x Te with Mn concentration for three temperatures according to [Lee84]. The measured peak (A) corresponds to the free exciton. The concentration independent feature at higher manganese concentrations is related to manganese intra-ion transitions. [Fur88b]

CHAPTER 1. II-VI DILUTED MAGNETIC SEMICONDUCTORS 22

Figure 1.7 – Experimental results for the variation of the energy gap in Zn 1−x Mn x Se with Mn con- centration for two temperatures according to [Bec88]. Cross-hatched points originate from [Twa83] and dots from [Kol86c]. At low x appears noticeable bowing. [Fur88b]

1.2. BAND STRUCTURE OF (CD,MN)TE AND (ZN,MN)SE 23

electrons in the d-shell for a transition. Because of the parallel alignment of all five spins in the orbital ground state, this would require an exceptional energy-consuming add of an electron with an opposite spin [Has90]. In this respect the 3d 5 orbit conducts similar to a complete shell. Nevertheless, these states are very important in formulating p-d type hybridization mechanism [Bha83, Has86, Lar85, Lar88b, Lar88a] (see 1.3.4.1, especially equation 1.44).

Dissociated from this transition are spin-flip intra-ion transitions. These do not change the number of electrons occupying the 3d-shell, but flip the direction of some electron spins. While the excited spin of a Mn-ion relaxes to the ground state, the neighboring spin becomes excited. Such transitions are possible because the 3d-shell of the Mn-ion is only half-filled, which is a peculiarity of A II 1−x Mn x B V I compounds compared to binary A II B V I or other II-VI ternary compounds.

According to Hund, the 3d 5 manganese electrons of a free Mn atom constitute in the ground state a 6 S multiplet with L = 0, S = 5 /2 and J = 5 /2. The higher states require flipping of at least one of the d-electron spins. Dominant for optical properties are transitions involving the lowest energy. Such states emanate from the ground state by flipping the spin of one electron (S = 3 /2). They are labeled 4 P (L = 1), 4 D (L = 2), 4 F (L = 3) and 4 G (L = 4), corresponding to S = 3 /2 and L = 1, 2, 3, 4. According to figure 1.8 the transition 6 S → 4 G has the lowest energy.

The crystal field of the A II B V I host lattice is mainly induced by the Coulomb field of the four tetrahedrally surrounding next anions. Thus, it is not strong enough to break Hund’s rule. The ground state remains the 6 S state. It is labeled as 6 A 1 in group theoretical notation, as it changes from the spherical symmetry of the free Mn 2+ -ion to the tetrahedral symmetry of the crystal field.

The ninefold 9 degenerated excited state 4 G is lifted in a cubic crystal field and splits in four levels: 4 T 1 , 4 T 2 , 4 E and 4 A 1 [Abr70, Sug70]. As shown in figure 1.8, the 4 T 1 and 4 T 2 states are lowered by the crystal field while the 4 E and 4 A 1 states are energetically almost coincident with the 4 G state and practically unaffected by the crystal field [Gri61, Tan54].

In the free ion case transitions between the 6 S and any excited state are forbidden by the ΔS = 0 selection rule and the parity selection rule. By placing Mn 2+ -ions in a zincblende A II B V I semiconductor, the first rule is relaxed due to spin-orbit interaction and the second rule due to the absent inversion symmetry in the zincblende crystal lattice. The transitions become allowed. Energetically favored is the 6 A 1 → 4 T 1 transition, which occurs at energies around 2.2 eV [Fur96]. This transition often governs the optical properties at high values of x. At higher

manganese contents also transitions with higher states can be observed due to broadening and overlap of the states [Lan65, Mor84, Mor82, Tao82]. 9 The states of the free atom have a degeneracy of 2L + 1.

CHAPTER 1. II-VI DILUTED MAGNETIC SEMICONDUCTORS 24

Figure 1.8 – Scheme of the lowest energy states of the Mn 3d-shell. Shown are three cases: on the left the isolated Mn-ion, in the middle the splitting of the 4 T 1 level in crystal field and on the right the splitting of the ground state 6 A in external magnetic field. The internal Mn transition ( 6 A 1 → 4 T 1 ) amounts to 2.2 eV. [Kel04]

The temperature dependence of the intra-ion transitions is qualitatively similar to the depen- dence of the energy gap of the A II B V I host material [Fur96]. More detailed information on these transitions is included in [Abr70, Bal62, Gri61, Sug70].

1.3 Magnetic properties

1.3.1 Basic principles of magnetism

Magnetic properties emerge from the occurrence of a magnetic ordering in a solid state. Magnetic ordering was theoretically postulated by Weiss in 1907 [Wei07] and quantum- mechanically explained by Heisenberg in 1928 [Hei28]. Classically is magnetism not explain- able, being a direct consequence of the Bohr-van-Leeuwen theorem. This indicates that mag- netic phenomena can only have quantum-mechanical origin, since the magnetism of classical systems vanishes in thermic equilibrium.

1.3. MAGNETIC PROPERTIES 25

Source of the magnetism in matter is the magnetic moment of electrons, which is induced by spin and orbital angular momentum [Kit05]. Although nucleons also have a magnetic mo- ment, it can be neglected due to the large difference in masses between nucleons and electrons. Magnetic ordering is based upon a collective alignment of the magnetic moments of the atoms of a crystal lattice. The moments are either permanent or induced. An alignment can occur either spontaneous or enforced by an external magnetic field. Spontaneous occurrence of long- range ordering arises from the competition of energetically favorable ideal alignment of the spins (minimum of inner energy U ) and entropically favorable complete disorder (minimum of entropy S). The relation between both values is given by the free energy F = U − T S. While at

low temperatures the inner energy dominates (aligned spins), rises with increasing temperature the impact of entropy in form of fluctuations around the ground state. At the so-called order- ing temperature prevails the entropy-contribution and a phase transition to an unordered phase occurs.

Several types of magnetic ordering can be distinguished, which can be divided into two groups. The dia- and paramagnetism shows no long-range magnetic ordering in absence of external magnetic fields. In contrast occurs for ferro-, ferri- and antiferromagnetism a sponta- neous long-range magnetic ordering, manifesting in a macroscopic magnetization for ferro- and ferrimagnetism. As at zero magnetic field in dia- and paramagnetic systems the magnetic sym- metry and the crystal symmetry coincide, causes the spontaneous long-range magnetic ordering a reduction of symmetry. Due to the fact that time-inversion causes spin reversal, the symmetry is broken.

The relation between magnetization

M and magnetic field strength H

establishes via the

ˆ χ m μ 0 ·

(1.22)

B = μ 0 μ r

H and the magnetic permeability μ 0 . The magnetic susceptibility is defined as the difference of relative permeability of a material μ r and relative permeability of vacuum (μ r = 1). The relative permeability is the relation between magnetic flux density B

in mater and magnetic flux density B 0 in vacuum at the same magnetic field strength H: μ r = B