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

Charles University

Description

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

Sujets

• Louis Pasteur University
• Strasbourg
• University

Informations

• Publié par: profil-zyak-2012
• Nombre de lectures: 12
• Langue: English
• Poids de l'ouvrage: 10 Mo

Extrait

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 objemo