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Charles University in Prague Université Louis Pasteur Strasbourg Faculty of Mathematics and Physics Faculté de Physique

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Niveau: Supérieur, Doctorat, Bac+8
Charles University in Prague Université Louis Pasteur Strasbourg Faculty of Mathematics and Physics Faculté de Physique The Czech Republic France Model Cal ulation of Four{Wave Mixing Polarization and Dynami s in Bulk and Confined Semi ondu tors Doctoral Thesis (These en co–tutelle) Tomá? Ostatnick? Prague, August 2005

  • semiconductor bloch equations

  • semiconductor nanostructures

  • exciton spin

  • particle–particle interactions

  • petr n?- mec

  • physics faculté de physique

  • optical bloch

  • wave mixing

  • model cal

  • modified optical


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Charles University in Prague Université Louis Pasteur Strasbourg
Faculty of Mathematics and Physics Faculté de Physique
The Czech Republic France
Doctoral Thesis
(Th`ese en co–tutelle)
Tomáš Ostatnický
Prague, August 2005
Model
Calcula
and
and
Polariza
tion

of
in
F
tion
our{W
D
a
Confined
ve
Bulk
Mixing
ynamics2The work presented in this thesis was done at Charles University in Prague, the Czech
Republic, and Université Louis Pasteur in Strasbourg, France, in the framework of the
project “The`ese en co–tutelle” (joint supervision of thesis) of the French government. I
spent 18 months at the university in Strasbourg where I’ve gained a great experience in
physics,cultureandsociallife.Iwouldliketoexpressmythankstobothuniversitieswhich
made these studies possible and I thank the French government which supported my stay
in France financially.
I thank both my advisors, prof. Petr Malý from Charles University and prof. Bernd
Hönerlage from Université Louis Pasteur, for their leadership, support and inspiring dis-
cussions. I could always discuss my ideas with other people in the lab which helped me
in this way to organize my thoughts and to formulate the theory — I thank Petr Ně-
mec, Pierre Gilliot, Mathieu Gallart and Jean–Pierre Likforman for their comments and
interest.
I would like to thank in particular my wife Katka for her best personal effort to make
my stays in Prague comfortable. She helped me to cope with all my personal problems
and she was the source of the spiritual power necessary to finish my studies.
Prague, August 2005
3
wledgments

A4Acknowledgments 3
Contents 5
1 Introduction 9
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2 Subject of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4 Symbol convention . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2 Systems of identical particles 15
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Systems of identical particles . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 Density matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4 Atomic systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.5 Optical Bloch Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3 Semiconductors 31
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2 Band structure and excitons . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2.1 Electron bands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2.2 Excitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2.3 The most frequent crystal structures . . . . . . . . . . . . . . . . . . 35
3.3 Exciton spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.3.1 Exciton spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.3.2 Exciton Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.3.3 Spin precession and relaxation . . . . . . . . . . . . . . . . . . . . . 40
3.4 Particle–particle interactions . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.4.1 Exciton–exciton interactions. . . . . . . . . . . . . . . . . . . . . . . 41
3.4.2 Biexcitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.4.3 Polaritons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.5 Semiconductor nanostructures . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4 Description of four–wave mixing experiments 51
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.2 Modified Optical Bloch Equations . . . . . . . . . . . . . . . . . . . . . . . 53
4.3 Microscopic theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.3.1 Semiconductor Bloch equations . . . . . . . . . . . . . . . . . . . . . 55
4.3.2 Four–particle correlation theories . . . . . . . . . . . . . . . . . . . . 58
4.3.3 Weakly interacting boson model . . . . . . . . . . . . . . . . . . . . 58
4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5
Contents6 CONTENTS
5 FWM on low–dimensional structures 61
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.2 Optical Bloch equations on quantum dots . . . . . . . . . . . . . . . . . . . 62
5.2.1 “V” system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.2.2 “E” system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.2.3 “O” system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.2.4 “II” system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.2.5 Summary of polarization selection rules . . . . . . . . . . . . . . . . 69
5.3 Spin structure of excitons in quantum wells . . . . . . . . . . . . . . . . . . 70
5.3.1 Zinc–blende in [001] direction . . . . . . . . . . . . . . . . . . . . . . 71
5.3.2 Zinc–blende in [011] direction . . . . . . . . . . . . . . . . . . . . . . 74
5.3.3 Zinc–blende in [111] direction . . . . . . . . . . . . . . . . . . . . . . 76
5.3.4 Wurtzite in [001] direction . . . . . . . . . . . . . . . . . . . . . . . . 77
6 FWM on bulk materials: The model 79
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.2 Linear coupling of photons to crystal . . . . . . . . . . . . . . . . . . . . . . 81
6.3 Nonlinear coupling of photons to crystal . . . . . . . . . . . . . . . . . . . . 87
6.4 Polariton–polariton interaction . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.5 Bipolaritons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.6 Relaxation and dephasing rates . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.7 Derivation of simplified equations of motion . . . . . . . . . . . . . . . . . . 99
6.7.1 Truncation of the scheme . . . . . . . . . . . . . . . . . . . . . . . . 100
6.7.2 Coulomb interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.7.3 Reduction of the number of states . . . . . . . . . . . . . . . . . . . 101
6.7.4 Equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
6.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
7 FWM on bulk materials: Results and discussion 107
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
7.2 Principle of wave mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
7.2.1 Scenario of the wave mixing process . . . . . . . . . . . . . . . . . . 108
7.2.2 Perturbative solution of equations of motion: t <t <t . . . . . . 1091 3 2
7.2.3 Perturbative solution of equations of motion: t <t <t . . . . . . 1101 2 3
7.3 FWM polarization selection rules . . . . . . . . . . . . . . . . . . . . . . . . 110
7.4 Time–resolved experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
7.5 Spectrally–resolved experiments . . . . . . . . . . . . . . . . . . . . . . . . . 119
7.6 Time–integrated experiments . . . . . . . . . . . . . . . . . . . . . . . . . . 120
7.7 Discussion of the features of the model . . . . . . . . . . . . . . . . . . . . . 120
7.8 Extension of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
7.9 Simulations of experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
7.10 Measurements of wave vector dependent interactions . . . . . . . . . . . . . 124
7.10.1 Overview of the method . . . . . . . . . . . . . . . . . . . . . . . . . 124
7.10.2 Possible polarization setups . . . . . . . . . . . . . . . . . . . . . . . 127
7.10.3 Proposal of experimental setup . . . . . . . . . . . . . . . . . . . . . 131
8 Conclusions 135CONTENTS 7
A Biexciton Hamiltonian in quantum wells 137
A.1 Zinc–blende in [001] direction . . . . . . . . . . . . . . . . . . . . . . . . . . 137
A.2 Zinc–blende in [110] direction . . . . . . . . . . . . . . . . . . . . . . . . . . 138
A.3 Zinc–blende in [111] direction . . . . . . . . . . . . . . . . . . . . . . . . . . 139
A.4 Wurtzite in [001] direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
B Equations of motion of the model 141
B.1 General equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
B.2 Solution of equations of motion for t <t <t . . . . . . . . . . . . . . . . 1451 3 2
B.3 Solution of equations of motion for t <t <t . . . . . . . . . . . . . . . . 1461 2 3
Bibliography 149
Summary 155
Model Calculation ofFour–Wave Mixing Polarization and Dynamicsin Bulk and
Confined Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
Modelování polarizace a dynamiky čtyřvlnového směšování v objemových polo-
vodičích a polovodičových nanostrukturách . . . . . . . . . . . . . . . . . . 159
Calcul mod`ele de polarisation et dynamique de mélange de quatre ondes sur des
semi–conducteurs massifs et confinés. . . . . . . . . . . . . . . . . . . . . . . 1628 CONTENTS1.1 Motivation
Solid state physics involves many subjects of interest like transport phenomena, cooling
of carrier plasma, nanostructures, etc. Research of electron spin has been one of the most
popular fields in solid state physics in the past since it provides a possibility for direct
applications. Scaling down dimensions of devices to the sizes of the order of nanometers,
one might use the electron spin and its coherence as the carrier of either a classical or
a quantum information. Such devices then could be used in new spintronic or quantum
computers, opening the new possibilities in future technologies.
Solid state research is connected with various types of materials. Semiconductors are
oneofthemostpopulargroupsofthesolidsbecauseoftheirbandstructure.Theyremainin
the ground state even at nonzero thermodynamic temperatures since thermal fluctuations
are too weak to provide enough energy for any excitation. It is then possible to excite a
defined number of electrons in well defined states in a crystal. These electrons then may
stay excited for long times because of the bottleneck effect what makes them very good
candidates as the carriers of informations.
The recent progress in technology allows preparation of semiconductor devices with
welldefinedparametersonthesizescalesofnanometers.Suchobjectsreveal manyinteres-
ting phenomena connected with the quantum coherence, therefore we call them quantum
objects or nanostructures. Properties of the nanostructures strongly differ from proper-
ties of macroscopic crystals even they were made from a same material. Research in the
nanoscale then opens new areas of interest.
Laser physics provides a very powerful tool for experimental investigation of the
electron spin as well as for investigation of semiconductors and their nanostructures. La-
ser spectroscopy is then very well suitable for the experimental research of the spin in
semiconductor nanodevices. Since many phenomena take effect on a time scale of several
hundredsof femtoseconds (fs), it is necessary to use ultrafast nonlinear spectroscopy besi-
des the usual linear optical spectroscopy. Four–wave mixing (FWM) is one of the methods
suitable for observation of the dynamics of spins inside the investigated crystal due to its
sensitivity to spin of excited carriers.
Semiconductor devices are also widely investigated theoretically. There exist many
models for description of various types of semiconductor structures valid under different
conditions. Theoretical works, however, are mostly developed in order to describe and
simulate selected phenomena with high precision showing the accuracy of the theoreti-
cal approach. One usually describes dynamics of electrons and holes in a crystal under
assumption of some initial conditions (initial populations of the electron states) and the
9
oduction
Intr
1
Chapter10 CHAPTER 1. INTRODUCTION
optical field, used for excitation and probe of the system’s dynamics, is considered to be
only some source term in equations. No attention is often paid to transmission of optical
field through the crystal boundaries, to strong exciton–photon coupling (resulting in the
polariton effect) and the role of optical fields is underestimated.
Dynamical models formulated on the basis of microscopic electron–hole Hamiltonian
are usually very precise, however derived equations of motion must be solved numerically.
Needofnumericalsolutionmightbeveryinconvenientforexperimenters—theyoftenneed
a tool simple in use in order to evaluate polarization selection rules for FWM response
from a system under particular experimental setup. In many cases the FWM response
hasn’t a definite polarization and this polarization may evolve in time. In such cases, one
usually wants only an estimate of the polarization dynamics when building an experiment
putting emphasis on fast calculations.
Theaim of the theory presented in this thesis isto providea model usablein nonlinear
ultrafastlaser spectroscopyofsemiconductorsandtheirnanostructures.Uniqueproperties
ofthird–ordernonlinearopticalprocessesinsystemsofagivendimensionalityarediscussed
inordertoshowwhichtypeofmodelshouldbechosenfordescription.Theworkisfocused
on polarization of the outgoing signal since it becomes very important in many types of
experiments. As an application of a model for bulk materials, a method for measurement
of a weak wave vector dependent electron–hole exchange interaction is proposed.
1.2 Subject of the thesis
Four–wave mixing isan experimental technique used for observation of nonlinear coherent
processes in materials. Three photons which come from laser sources excite the sample
and the fourth photon is radiated and detected, see the sketch in Fig. 1.1. Separation of
the photons from the excitation beams transmitted through the sample from the FWM
responseisdonebyspatialfiltering:ifthewavevectorsoftheexcitation beamsareK ,K1 2
andK ,theFWMresponsecanbedetectedinthedirections−K +K +K ,K −K +K3 1 2 3 1 2 3
and K +K −K in transmission. In reflection, the diffraction direction are given by1 2 3
the reflection of the three aforementioned directions by the plane of the sample surface.
DirectionK +K −K is chosen to be the detection direction throughout this thesis. The1 2 3
FWM response in other FWM directions can be calculated simply by permutation of the
indices.
The third–order optical response of semiconductors, in particular FWM response, is
under experimental and theoretical investigation for a long time. The aim of the four–
wave mixing experiments is mainly to detect various channels of the coherence decay or
coherent changes in the system. Because of a long history of research in this field, there
already exists a big number of models which describe the dynamics of semiconductors
under various experimental conditions. Models for bulk semiconductors can be divided
into two main groups: on one side, there are phenomenological models based on Optical
Bloch Equations (OBE) [1, 2], on the other side, one finds microscopic theories [3, 4, 5].
A detailed discussion of the two theoretical approaches can be found in chapter 4.
Theaforementioned modelsincludingphenomenological OBEweresuccessfulinexpla-
nation andmodellingofthebasicbehaviouroftheFWMresponsewithoutspin.Therefore
most of theories are formulated using two bands, one valence and one conduction band.
Currentresearch on semiconductors and their nanostructuresis, however, focused on spin.
Itistheninterestingtoexaminewhichofthemodelsarealsoabletoreproducepolarization
properties of the FWM response.