

Phillip Springer
Mainzer Gasse 33, Büro 02 001
35037 Marburg
AG�eoretische Halbleiterphysik
Fachbereich Physik der Philipps-Universität Marburg

Vom Fachbereich Physik der Philipps-Universität Marburg
als Dissertation angenommen am

Erstgutachter: Prof. Dr. Stephan W. Koch
Zweitgutachter: PD Dr. Sangam Chatterjee

Tag der mündlichen Prüfung: 11. Mai 2016
Hochschulkennzi�er 1180


Für meine Eltern, Joachim und Cornelia.
Für meine Liebe, Patricia.


Zusammenfassung

Kaum ein Forschungsfeld prägt unser heutiges Leben so sehr wie die Halbleiterphysik. Wir intera-

gieren ständig mit unseren Smartphones, unseren Tabletcomputern oder Heimrechnern. Kurzum:

mit Geräten, die ohne Grundlagenforschung in der Festkörperphysik nicht möglich wären. Neben

den alltäglichen Erfahrungen werdenHalbleiter auch in derMedizintechnik,Materialbearbeitung,

Umweltanalytik, Energieerzeugung oder der optischen Datenübertragung eingesetzt. Das Streben

nach Energiee�zienz bei gleichzeitig höherer Rechenleistung und kleineren Bauelementen treibt

die Wissenscha� bis heute an.

So habenHalbleiter Einzug in unser Leben gefunden und stellen einen wesentlichen Bestandteil

unsererWelt dar.Die rasante Entwicklungsumgebunghat dabei diverseAnwendungsmöglichkeiten

hervorgebracht. Diese reichen von Prozessoren (englisch: central-processing unit, CPU) bis hin zu

Datenspeichern in Form von solid-state drives (SSDs). Die Vielfältigkeit von Halbleitern ist ihrer

Kristallstruktur geschuldet, welche der Grund für die Bildung der Bandlücke ist. Diese trennt be-

setzte von unbesetzten elektronischenZuständen und bestimmt hauptsächlich, welche Energie von

derHalbleiterlegierung absorbiert, verstärkt oder emittiert werden kann.Mittels Verspannung und

Materialkomposition kann diese Bandlücke verändert werden und ist die Grundlage ihrer Flexi-

bilität. So kann man die Farbe von Leuchtdioden (englisch: light-emitting-diode, LED) über das

gesamte sichtbare Spektrum anpassen, indem man die beteiligten Materialien sorgfältig auswählt.

Daneben zeichnen sichHalbeiter durch hoheE�zienz, eine lange Lebensdauer und kompakte Bau-

form aus.

Ein bemerkenswerter Fortschritt in derKristallzüchtungunddenDotierungstechnikenwährend

den letzten Jahrzehnten hat dazu geführt, dass man mit nahezu atomarer Präzision Halbleiter-

schichten aufeinander abscheiden kann. Hierbei de�nieren die verschiedenen Schichten in der

Wachstumsrichtung des Kristalls ein Potential für Ladungsträger im Leitungs- und Valenzband.

In solchen e�ektiv niederdimensionalen Systemen ist die Bewegung der Elektronen eingeschränkt,

wodurch ihre Energie quantisiert wird. Somit ist es notwendig, die Teilchen quantenmechanisch

auf einer mikroskopischen Ebene zu beschreiben und messbare Größen von ihnen abzuleiten. In

der vorliegenden Dissertation wird eine mikroskopische Vielteilchentheorie zur Beschreibung op-

tischer Eigenscha�en von Halbleitern angewandt und erweitert. Das theoretische Verständnis und

der hohe Grad an Reinheit bei der Herstellung haben bereits zu außerordentlich hilfreichen und

leistungsstarken Applikation geführt wie etwa integrierten Schaltkreisen oder Halbleiterlasern.

Innerhalb der theoretischen Physik ist die einmikroskopischer Ansatz aus zwei Gründen beson-

ders interessant: Erstens bildenHalbleitersystemenahezu ideale quantenmechanische Systeme und

I


II Zusammenfassung

eignen sich somit ausgezeichnet als Modellsystem um fundamentale physikalische Eigenscha�en

zu untersuchen. Daneben werden die künstlich hergestellten Schichtstrukturen bereits seit gerau-

mer Zeit für praktische Zwecke eingesetzt. Ihre Weiterentwicklung und Verbesserung erfordert

ein detailliertes Wissen über die elektronischen und optischen Vorgänge der zugrundeliegenden

mikroskopischen Prozesse.

Aber auch von experimenteller Seite ist die Untersuchung von Halbleitern faszinierend. Die Be-

herrschung der Ultrakurzzeitspektroskopie erlaubt die Untersuchung von mikroskopischen Pro-

zessen durch die Erzeugung von Lichtpulsen mit einer Länge von unter einer Pikosekunde (1 ps =
10−12 s). Hierzu sindMethoden der kohärenten Spektroskopie wie etwa Vier-Wellen-Mischen [1–3]

oder Anregungs-Abfrage-Experimente [4, 5] entwickelt. Bei Halbleiterprobenmit Bandlücken von

etwa einem Elektronvolt (1 eV=̂1240nm=̂242THz) kann Licht im sichtbaren und nah-infraroten

Teil des elektromagnetischen Spektrums elektronischeAnregungen erzeugen.HierbeiwerdenElek-

tronen vomValenz- ins Leitungsband gehobenund erzeugendarau�in eineVielzahl von kohärenten

und inkohärenten Korrelationen. Umfassende Experiment-�eorie-Vergleiche machen es somit

möglich, solche Korrelationen über Dephasierungs-, Relaxations- oder Streuprozesse zu studieren,

da sie sich direkt in linearer oder nicht-linearer optischer Spektroskopie- und Lumineszenzmes-

sungen widerspiegeln. In dieser Arbeit werden sowohl nicht-linear optische Absorption als auch

Photolumineszenzspektren berechnet, um experimentelle Daten zu interpretieren.

Wohl eine der bekannteste Korrelationen ist das Exziton. Als gebundenes Elektron-Loch-Paar

ist es ein Quasiteilchen, welches viele Gemeinsamkeitenmit demWassersto� Atom aufweist. Seine

Bindungsenergie allerdings ermöglicht intraexzitonische Übergänge, die im Bereich von einigen

Terahertz (THz) liegen (1THz=̂4.1meV).Möchteman also exzitonische Vorgänge und Eigenschaf-

ten untersuchen, stellt die Terahertzspektroskopie das idealeWerkzeug dar.Neben rein spektrosko-

pischen Einsatzmöglichkeiten kann starke THz Strahlung auch genutzt werden, um exzitonische

Zustände kohärent zu manipulieren [6]. Auch intersubband Übergänge können mit elektroma-

gnetischer Strahlung im THz Bereich induziert werden [7]. Mittels des THz Feldes können dann

sogar selektiv Ladungsträger, Exzitonen oder Zweiteilchen-Korrelationen zwischen verschiedenen

Halbleiterschichten transportiert werden. ImRahmen dieser Dissertationwird die etablierte�eo-

rie zur Berechnung von Exziton-Wellenfunktionen und THz Absorptionsspektren auf Halbleiter

erweitert, welche eine Anisotropie der e�ektiven Elektron- und Lochmassen aufweisen.

Die theoretischen Grundlagen für die Erörterung aller weiteren Untersuchungen werden in Ka-

pitel 2 diskutiert. Neben dem System-Hamiltonian werden wichtige Gleichungen zur Berechnung

von Bandstrukturen, Photolumineszenz und THzAbsorption besprochen. Im anschließenden Ka-

pitel 5 wird dieWannier Gleichung aufHalbleitermit anisotropen e�ektiven Ladungsträgermassen

erweitert. Die Wannier Gleichung ist eine Eigenwertgleichung für die Wellenfunktion und Bin-

dungsenergien von Exzitonen, die bisher erfolgreich für direkte Halbleiter wie Galliumarsenid

(GaAs) angewendet wurde, in denen die e�ektiven Elektron- und Lochmassen in alle Raumrich-

tungen identisch sind. Indirekte Halbleiter wie Germanium (Ge) oder Silizium (Si) aber weisen


III

unterschiedliche Massen bezüglich der Raumrichtungen auf. Am Beispiel von Ge zeigt sich als

Konsequenz der Massenanisotropie, dass die Exziton-Wellenfunktionen ihre Form im Vergleich

zu isotropen Massen stark ändern. Da die Verallgemeinerung der �eorie auf einer Entwicklung

in Kugelächenfunktionen basiert, ist die Änderung der Form eine Folge von Kopplungen ver-

schiedener Quantenzahlen. Für Ge und Si wird außerdem das THz Absorptionsspektrum berech-

net, wobei auch die Entartung der indirekten Bandkante Beachtung �ndet. Aufgrund dieser Ent-

artung werden zwei energetisch unterscheidbare Resonanzen beobachtet, die der parallelen und

senkrechten Komponente der THz Polarisation zugeordnet werden können. Den Abschluss des

Kapitels bildet die Berechnung der optischen Absorption in Rutil nahe der Bandkante. Zu diesem

Zweck wurden ab-initio Berechnungen mit den vorher beschriebenen Exziton-Wellenfunktionen

zu einer neuen Methode kombiniert.

In Kapitel 4 werden experimentelle Anregungs-Abfrage-Experimente von Ge and Galliumin-

diumarsenid ((Ga,In)As) Quanten�lmen analysiert. Im Experiment zeigt die Ge Probe au�ällige

Seitenbanden für starke Anregungsdichten. Die (Ga,In)AsHeterostruktur zeigt unter äquivalenten

Bedingungen jedoch nur den erwarteten E�ekt der anregungsinduzierten Dephasierung. Mittels

der mikroskopischen Vielteilchentheorie kann der Unterschied auf verschiedene Dephasierungs-

mechanismen der beiden beiden Proben zurückgeführt werden.

Kapitel 5 adressiert wieder experimentelleDaten einer speziellenHalbleiterheterostruktur.Diese

wurde so hergestellt, dass sie eine Typ-II Photolumineszenz zeigt. Eine dichteabhängige Studie der

Photolumineszenz o�enbart eine ungewöhnlich Breite Bande, was mit einem Vergleich zu theo-

retischen Spektren auf die beteiligten Übergänge und Wellenfunktionen zurückgeführt werden

kann. Bedingt durch den speziellen Probenau au ist es möglich, die energetische Bandanpassung

von Galliumnitridarsenid (Ga(N,As)) zwischen GaAs Barrieren zu bestimmen, wofür ein leichter

Typ-I Übergang gefunden wird.


Acknowledgements

First and foremost, I want to thank my parents and my sister Nora who supported me during all

stages of my thesis and without whom this work would not have been possible. I especially would

like to thank Prof. Dr. Stephan W. Koch for his guidance and supervision, and for giving me the

opportunity to study theoretical physics on extremely interesting topics. His guidancewas essential

in all the small and large problems I faced. Just as much I want to thank Prof. Dr. Mackillo Kira,

especially for teaching me the basic and vital things in science. His everlasting patience and always

competent and motivating feedback on my work helped me to push forward whenever I was stuck

which really means a lot to me. �is work would also not have been possible without the experi-

ments, which I had the opportunity and pleasure to analyze. In this regard, I would like to thank

Dr. Sangam Chatterjee, Prof. Dr. Wolfram Heimbrodt, Prof. Dr. Kerstin Volz, Dr. Wolfgang Stolz,

and their coworkers. Special thanks are entitled to Dr. Sangam Chatterjee and Prof. Dr. Reinhard

Noack for agreeing to evaluate my thesis. Especially Sangam also helped me to understand all the

complex problems from an experimental viewpoint inmany inspiring discussions. I truly value the

help from Prof. Dr. John Sipe, who guided me through the �rst year in many helpful conversations

about physics and pudding. I also thank Martin, Christian, Uli, Markus, Osmo, Christoph, Lukas,

Dominik, Benjamin, Tineke, Prof. Dr. Peter �omas, and the rest of the group for a pleasant and

relaxed atmosphere, for a lot of tasty group dinners, formany interesting discussions about physics,

running, the Braveheart Battle, funny excuses from students, and other stu�. WithMartin, I shared

a great and fun time together in the o�ce and I will never forget some epic moments we shared.

Both Martin and Christian o�en helped me to �ght o� depressing times with extensive exercise

sessions whenever things did not work as expected. I also got a lot of support from old friends and

members from my former group. �ank you David, Exi, Alex, Marc, Ajanth, Norman, Tobi, and

Daniel. I want to especially emphasize my acknowledgement for Tobi and Christian for extensive

proof reading of my manuscript.

I gratefully acknowledge �nancial support by the DFG in the framework of the GRK 1782.

Last but de�nitely not least, I want to thank Patricia. You keep on being themost signi�cant part

of my life. You always protected me frommyself, always supported and encouraged my work. I am

excited to �nd out where our way leads us next. I love you

V


Author’s Contributions

Publications in Peer-Reviewed Journals

[I] T. B. Norris, P. Springer, and M. Kira, in Front. opt. 2014 (2014), LTu1I.1.

[II] K. Jandieri, P. Ludewig, T. Wegele, A. Beyer, B. Kunert, P. Springer, S. D. Baranovskii, S. W.

Koch, K. Volz, and W. Stolz, J. Appl. Phys. 118, 065701 (2015).

[III] P. Springer, S. Gies, P. Hens, C. Fuchs, H. Han, J. Hader, J. Moloney, W. Stolz, K. Volz, S.

Koch, and W. Heimbrodt, J. Lumin. 175, 255–259 (2016).

[IV] P. Springer, S.W. Koch, andM. Kira, J. Opt. Soc. Am. B 33, C30 (2016), arXiv:1602.02972.

[V] O. Vänskä, M. Ljungberg, P. Springer, D. Sánchez-Portal, M. Kira, and S. W. Koch, J. Opt.

Soc. Am. B 33, C123 (2016), arXiv:1512.05156.

Publications in Preparation

[VI] P. Springer, P. Rosenow, T. Stroucken, J. Hader, J. V. Moloney, R. Tonner, and S. W. Koch,

“An Ab-Initio Modeling Approach for the Optical Properties of Dilute Nitride Semicon-

ductors”, in preparation.

[VII] P. Springer, S. W. Koch, M. Kira, N. S. Köster, K. Kolata, S. Chatterjee, J. Cheng, and

J. E. Sipe, “Enhancement of Phonon Assisted Optical Transients in GermaniumQuantum

Wells”, in preparation.

[VIII] F. Dobener, P. Springer, P. Ludewig, S. Reinhard, K. Volz,W. Stolz, S.W. Koch, and S. Chat-

terjee, “Room-temperature optical gain in Ga(N,As,P)/(B,Ga)(As,P) heterostructures”, in

preparation.

Posters and Talks

[IX] P. Springer, S.W. Koch, M. Kira, J. Cheng, J. E. Sipe, N. S. Köster, K. Kolata, and S. Chatter-

jee, Enhancing Optically Induced Transients with Phonon Relaxation, Poster presented at

the Materialforschungstag (Material Sciences Day) Mittelhessen 2014, Gießen, Germany.

VII

http://dx.doi.org/10.1364/LS.2014.LTu1I.1
http://dx.doi.org/10.1063/1.4928331
http://dx.doi.org/10.1016/j.jlumin.2016.03.010
http://dx.doi.org/10.1364/JOSAB.33.000C30
http://arxiv.org/abs/1602.02972
http://dx.doi.org/10.1364/JOSAB.33.00C123
http://dx.doi.org/10.1364/JOSAB.33.00C123
http://arxiv.org/abs/1512.05156


VIII Posters and Talks

[X] P. Springer, S. W. Koch, and M. Kira, E�ect of Anisotropic Mass on Exciton Wave Func-

tions, Talk presented at the Seminar of the Research Training Group Functionalization of

Semiconductors (GRK 1782) 2014, Donostia-San Sebastián, Spain.

[XI] P. Springer, S. W. Koch, M. Kira, J. Cheng, J. E. Sipe, N. S. Köster, K. Kolata, and S. Chat-

terjee, Enhancing optically induced transients with phonon relaxation, Poster presented at

the Nonlinear Optics and Excitation Kinetics in Semiconductors (NOEKS) 2014, Bremen,

Germany.

[XII] P. Springer, T. Stroucken, S. W. Koch, P. Rosenow, R. Tonner, J. Hader, and J. V. Moloney,

Microscopic Calculations of the Optical Properties of Novel Bismide and Nitride Containing

III-V Semiconductors, Talk presented at the Spring Meeting 2015 of the German Physical

Society (DPG), Berlin, Germany.

[XIII] P. Springer, P. Rosenow, K. Jandieri, P. Ludewig, T. Wegele, A. Beyer, B. Kunert, R. Tonner,

S.D. Baranovskii, K.Volz,W. Stolz, and S.W.Koch,Photoluminescence inGa(NxAs1−x−yPy)

Systems, Poster presented at the Materialforschungstag (Material Sciences Day) Mittel-

hessen 2015, Marburg, Germany.

[XIV] P. Springer, S. Gies,W.Heimbrodt, and S.W. Koch,Optical Properties of Nitride Containing

QuantumWells, Poster presented at the GRK Seminar 2015, Ho�eim, Germany.

[XV] P. Springer, S. W. Koch, and M. Kira, Consequences of Anisotropic Electron Masses on THz

Absorption, Talk presented at the GRK Seminar 2015, Ho�eim, Germany.

Original Contributions

Publication [II] provides a simple model developed by Stolz et al. to extract nitrogen composi-

tions in Ga(N,As,P)/GaP quantum well heterostructures. In this regard, I conducted multi-band

electronic structure calculations based on the k ⋅ p theory with a band anticrossing extension. To
validate the simple model, I calculated the corresponding photoluminescence spectra and com-

pared them with experiments.

�e photoluminescence spectrum of amulti quantumwell heterostructure is the subject of pub-

lication [III]. Experiments have been conducted by W. Heimbrodt et al. while I applied an estab-

lishedmicroscopic many-body theory to obtain the multi-band photoluminescence of this system.

My calculations form the basis for interpretation of the experimental spectra.

In publication [IV], I developed a microscopic approach to compute the excitonic properties

and the corresponding terahertz response for semiconductors which exhibit anisotropic e�ective

masses. I applied the approach to germanium and quantitatively predicted the outcome and inter-

pretation of a future experiment.

In publication [V], the cluster-expansion method is combined with the density-functional the-

ory into a hybrid approach. It is applied to the optical absorption of rutile TiO2. For this purpose,


Posters and Talks IX

exciton wave functions and binding energies are relevant, which I provided. Optical-matrix ele-

ments were computed via density-functional theory byM. Ljungberg. �emain author, O. Vänskä,

composed the results to obtain the optical absorption and is responsible for the theoretical model.


Contents

Zusammenfassung I

Acknowledgements V

Author’s Contributions VII

1 Introduction 1

2 Theoretical Framework 5

2.1 System Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Band Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3.1 Bulk Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3.2 Semiconductor Heterostructures . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.4 Semiconductor Luminescence Equations . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.5 Semiconductor Bloch Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.5.1 Bulk Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.6 Inhomogeneous Broadening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.7 Optical Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.8 Wannier Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.9 Linear THz Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3 E�ects of Mass Anisotropy on theOptical Properties of Semiconductors 19

3.1 THz Spectroscopy of Bulk Germanium . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.1.1 Generalized Wannier Equation with Mass Anisotropy . . . . . . . . . . . . 20

3.1.2 Excitons in Germanium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.1.3 Single-Valley Response to Classical THz Excitation . . . . . . . . . . . . . . 28

3.1.4 Muli-Valley Response to Classical THz Excitation . . . . . . . . . . . . . . . 32

3.2 THz Spectroscopy of Bulk Silicon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.3 Absorption of Bulk Rutile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.3.1 Density Functional �eory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.3.2 Band Structure and Matrix Elements . . . . . . . . . . . . . . . . . . . . . . 40

3.3.3 Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

XI


XII Contents

4 Enhancement ofOptical Transients in GermaniumQuantumWells 47

4.1 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.2 Two-Level System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.3 Quantum-Well System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.3.1 Dephasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.4 Experiment-�eory Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5 Optical Properties of HighlyMismatched Alloys 61

5.1 Band Anticrossing Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5.2 Valence Band O�set Determination of Ga(N,As)/GaAs QuantumWells . . . . . . . 64

5.2.1 Sample Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.2.2 Valence Band O�set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.2.3 Dominant Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

6 Conclusion andOutlook 73

Bibliography 75

Wissenschaftlicher Werdegang 85


1 Introduction

�e good thing about science is that it’s true whether or not you believe in it.

(Neil deGrasse Tyson, American astrophysicist, cosmologist, and author)

Today, only few scienti�c �elds have such an enormous impact on our daily lives as semicon-

ductor physics. We constantly interact with smartphones, tablets, and computers; devices realized

on the basis of fundamental solid-state research. Beside our daily experiences with semiconduc-

tors they are utilized in medical engineering, material machining, environmental technology, and

optical communication. �e quest for ever increasing computing power, energy e�ciency, and

miniaturization drives the fundamental research until today.

Hence, semiconductors have become an integral part of our world. A rapid development en-

vironment has created diverse applications ranging from central processing units (CPUs) to data

storage solutions in the form of the solid-state drive (SSD).�is mutability is owed to the inherent

crystal structure of semiconductors, giving rise to the band gap, which separates occupied from

unoccupied electronic states. �e band gap of a semiconductor alloy mainly determines which

energies can be absorbed, ampli�ed, or emitted. Using strain and alloy composition allows to tune

the band gap, permitting the fabrication of light-emitting diodes (LEDs) over the entire visible

spectrum. Besides the exibility of these materials, they are featured with a high e�ciency, long

life time, and compact design.

A remarkable progress in crystal growth and doping techniques during the past few decades

has allowed the deposition of semiconductor heterostructure layers with almost atomic precision.

�e di�erent layers de�ne a potential landscape in the growth direction for charge carriers in the

conduction and valence bands. By e�ectively reducing the system’s dimensionality, the motion of

electrons is then con�ned. As a consequence, their energy becomes quantized. �erefore, a quan-

tum mechanical description on a microscopic level is required from which measurable quantities

can be derived. In this work, a microscopic many-body theory to describe optical properties of

semiconductors is used and further extended. �eoretical comprehension and a high degree of

purity during the manufacturing process has led to some extremely helpful and powerful applica-

tions, including integrated circuits and semiconductor lasers.

Concerning theoretical physics, a microscopic approach is interesting for two reasons: First,

semiconductor structures constitute almost ideal quantum mechanical systems thus qualifying as

model systems to study fundamental physical properties. In addition, arti�cially produced layered

1


2 1 Introduction

structures have been introduced for a considerable time. To improve and optimize these devices,

it is indispensable to gain detailed knowledge about the electrical and optical mechanisms of the

underlying microscopic processes.

�e experimental prospect of semiconductors is equally fascinating. Mastering ultrafast op-

tical pulses with a duration of less than a picosecond (1 ps = 10−12 s) allowed to experimentally
study microscopic processes via methods such as four-wave mixing [1–3] and pump–probe spec-

troscopy [4, 5]. Applied to semiconductors whose band gap is of the order of one electronvolt

(1 eV=̂1240nm=̂242THz), light frequencies in the visible and near-infrared regime of the electro-
magnetic spectrum can induce excitations, e.g. li� an electron from the valence into the conduc-

tion band while coherent and incoherent correlations are generated subsequently. Comprehensive

experiment–theory comparison have allowed to study such correlations via relaxation, dephasing,

and scattering processes of carriers a�er optical excitation, which in turn can be monitored by

linear and nonlinear optical spectroscopy or photoluminescence (PL) measurements. Both non-

linear absorption and PL spectra are analyzed in this thesis to interpret experimental data.

One of the most prominent examples of correlations is the exciton. �is quasiparticle con-

sists of a truly bound electron–hole pair and shows major similarities with the hydrogen atom.

However, its binding energy allows for interexcitonic transitions in the range of a few terahertz

(1 THz=̂4.1meV) in common semiconductors. �at makes THz spectroscopy an excellent tool to
investigate these transitions. Apart from purely spectroscopical applications, strong THz �elds can

coherently manipulate excitonic states [6]. Intersubband transitions can also be investigated and

manipulated via THz radiation [7]. Resonantly applied, they can selectively transport charge carri-

ers, excitons, or two-particle correlations across semiconductor layers. In the context of this thesis,

the established theory to compute exciton wavefunctions and THz-absorption spectra is extended

to semiconductors that have anisotropic e�ective electron and hole masses.

�e theoretical basis is revised inChapter 2. It introduces the systemHamiltonian and covers the

relevant equations to compute a band structure, PL, and THz-absorption spectra. In the adjacent

Chapter 3, theWannier equation is extended to semiconductorswith anisotropicmasses. �eWan-

nier equation de�nes an eigenvalue problem for the exciton wavefunctions and binding energies.

It was successfully applied tomaterials like gallium arsenide (GaAs), which is a direct semiconduc-

tor with isotropic masses. Indirect substances like germanium (Ge) or silicon (Si) however exhibit

di�erent masses regarding the spatial directions. As an example, Ge is used to show that mass

anisotropy changes the shape of the exciton wavefunctions. Since the generalization of the theory

is based on an expansion into spherical harmonics, the altered appearance of the wavefunctions

is the outcome of the coupling between di�erent quantum numbers. �e THz spectra for Ge and

Si are calculated explicitly considering the degeneracy of the band edge. Owing to this degener-

acy, two energetically close but separated resonances are observed which can be assigned to the

parallel and perpendicular component of the THz �eld. �e closing of this Chapter describes the

calculations of the near-bandgap optical absorption of rutile. For this purpose, the generalized the-


3

ory of exciton wavefunctions is combined with ab-initio calculations which provide optical-matrix

elements.

In Chapter 4, pump–probe absorption experiments of (Ga,In)As and Ge quantum wells (QWs)

are analyzed. Experimentally, prominent optical transients are observed only in the Ge sample,

while the (Ga,In)As sample only shows the usual excitation-induced dephasing (EID) signatures.

Using a fully microscopic many-body theory, the di�erence can be ascribed to di�erent charge

carrier dephasing mechanisms.

Chapter 5 addresses the PL spectrum of a speci�c heterostructure, which was designed to show

a type-II PL. An unusually broad PL line shape was observed in an excitation-density dependent

experiment but could be explained by a thorough investigation of the participating electronic tran-

sitions. �e special design of structure also allows to determine the fundamental nature of the

valence band o�set (VBO) of gallium nitride arsenide (Ga(N,As)) QWs between GaAs barriers,

which are found to have a weak type-II alignment.


2 Theoretical Framework

�is Chapter provides a short revision of the essential properties and correlations necessary in

the context of this thesis. �e derivations of the relevant equations have been covered in detail

throughout numerous publications [8–12] and shall not be repeated here. �e purpose of this

Chapter is rather the introduction of a concise notation to be used throughout the thesis.

2.1 SystemHamiltonian

�e starting point of relevant theoretical investigations is the many-body Hamiltonian

H = H0 +HC +HD +HP , (2.1)

describing non-interacting particles (including electrons, photons, and phonons) viaH0, Coulomb

or electron–electron interaction via HC, dipole or light–matter interaction via HD, and carrier–

phonon interactions via HP. It is obtained from the minimal-substitution Hamiltonian in �rst

quantization for N particles which is determined by the gauge invariance of electrodynamics [13–

15]. In the Coulomb gauge, it reads [11]

H = N∑
j=1

[ 1

2m0
(p j + ∣e∣A)2 + VCr(r j)] + 12

N∑
j,l≠ j

V(r j − r l) +Hem +Hph , (2.2)

where p and r are the canonical momentum and position operators of particle jmoving within the

periodic crystal potentialVCr and having charge e = −∣e∣ andmassm0. Carrier–carrier interactions
are described via the Coulomb potential [16]

V(r) = e2

4πє0
√
є1є2є3

1√
r21
є1
+ r22

є2
+ r23

є3

, (2.3)

containing єl , the materials background dielectric permittivity in principal directions l = {1, 2, 3},
which in general can be anisotropic, e.g. є1 ≠ є2 ≠ є3 ≠ є1. However, it is worth noting that the

Coulomb potential can always be modi�ed into a symmetric function via a coordinate transfor-

5


6 2 Theoretical Framework

mation r̃ = (r1,√ є1
є2
r2,
√

є1
є3
r3), so that

V(r̃) = V(∣r̃∣ = r̃) = e2

4πє0
√
є2є3

1

r̃
. (2.4)

Whenever the system is isotropic, є1 = є2 = є3 ≡ єBG and r̃ = r maps trivially.

In Eq. (2.2), the transverse light �eld is given by A and it has been assumed that all external

longitudinal �elds vanish resulting in no additional potential term. �e free-�eld part in Eq. (2.2)

is given byHem and phonon interactions are covered byHph, whose explicit forms can be found in

Refs. [10] and [12].

To obtain the Hamiltonian (2.1) in second quantization from Eq. (2.2), it is usually advantageous

to expand the �eld operators in the Bloch basis

Ψ̂(r) =∑
λ,k

âλ,k ϕλ,k(r) , Ψ̂†(r) =∑
λ,k

â†λ,k ϕ
⋆

λ,k(r) , (2.5)

containing the creation (annihilation) operator â†λ,k (âλ,k) of an electron characterized by a full set

of quantum numbers λ and having momentum ħk. �e single-particle wave functions ϕλ,k obey

the time-independent Schrödinger equation [17]

⎡⎢⎢⎢⎣
p2

2m0
+ VCr(r)⎤⎥⎥⎥⎦ϕλ,k(r) = єλ,k ϕλ,k(r) , (2.6)

VCr(r) = VIon(r) + 1

2m2
0c

2
0r

∂VIon(r)
∂r

S ⋅ L , (2.7)

containing the particle’s angular momentum and spin L and S, respectively, and the ionic crystal

potential VIon. �us, spin-orbit interactions have been explicitly included.

Several approaches have been developed to solve Eq. (2.6), e.g the k ⋅ p method [18–20], the

tight-binding model [21, 22], or density functional theory (DFT) [23, 24]. In the scope of this

thesis, two di�erent strategies will be used. For the modeling of semiconductor heterostructures,

realistic single-particle energies are computed using an 8-band k ⋅ p Hamiltonian [25] containing

non-parabolicities which emerge due to coupling e�ects between individual bands. Whenever

fundamental physical properties have been investigated, such subsidiary e�ects can be neglected

to identify the essential physics. �us, the e�ective-mass approximation (see Section 2.3.1) has been

used in these cases.

�e wave functions ϕλ,k contain the e�ective dimensionality of the system. In bulk semicon-

ductors, they can be separated into a lattice periodic and a free-particle wave part via Bloch’s theo-

rem [26]. In quantumwell systems, the envelope-function approximation [10, 12, 27] can be applied

to reduce the e�ective dimensionality. �e same approach can be used to obtain quantum-wire and

even quantum-dot properties.


2.2 Equations of Motion 7

2.2 Equations of Motion

�e most interesting properties of excited semiconducting materials include the spectral distri-

bution of emitted, transmitted, reected, or absorbed light. Once the temporal evolution of the

relevant expectation values such as the microscopic polarization or the photon-number correla-

tion is known, they can be obtained via a Fourier transformation into the frequency domain. �e

Heisenberg equation of motion

iħ
∂

∂t
⟨Ô⟩ = ⟨[Ô , Ĥ]

−

⟩ , (2.8)

can be used to calculate the temporal evolution of the expectation value of an operator Ô. How-

ever, Eq. (2.8) couples an N to an (N + 1)-particle operator, which leads to the well known in�nite

hierarchy of equations. A systematic way to deal with this problem is the cluster-expansion ap-

proach (CE). Here, an N-particle quantity is factorized into clusters of single particles, correlated

pairs, correlated three particle clusters, and so forth. �e truncation at one speci�c level allows to

include only relevant physical e�ects, while simultaneously reducing the numerical e�ort.

If the truncation is performed at the single-particle level, the most basic Hartree–Fock approxi-

mation is obtained. Many quantum-optical phenomena such as exciton formation or EID require

a truncation at higher cluster levels. Neglecting these contributions may yield an incomplete de-

scription of the system [28].

2.3 Band Structure

2.3.1 Bulk Semiconductors

Awave function in a periodic potential ful�lls Bloch’s theorem [26]. �us, it must necessarily have

the form

ϕλ,k(r) = 1√
L3

eik⋅r uλ,k(r) , (2.9)

where uλ,k is the lattice-periodic part of the wave function and L3 is the quantization area. �is

can be used in Eq. (2.6) to obtain

⎡⎢⎢⎢⎣
p2

2m0
+ VCr(r) + ħ

m0
k ⋅ p⎤⎥⎥⎥⎦uλ,k(r) = ⎛⎝єλ,k − ħ2k2

2m0

⎞⎠uλ,k(r) . (2.10)

Considering the term proportional to k ⋅p being a small disturbance, second-order non-degenerate

perturbation theory can be applied, once the eigenvalue problem (2.10) has been solved for one

speci�c k = k0. For momenta k within the vicinity of k0, the lattice periodic wave function is then


8 2 Theoretical Framework

given by [29]

uλ,k(r) =∑
ν

cλ,ν,k uν,k0(r) , (2.11)

where

cλ,ν,k = δλ,ν + (1 − δλ,ν) ħ

m0

(k − k0) ⋅ pν,λ(k0)
єλ,k0 − єν,k0 , (2.12)

are the expansion coe�cients containing the momentum matrix element [11]

pν,λ(k) = ⟨ν, k∣e−ik⋅rp eik⋅r∣λ, k⟩ . (2.13)

Here, Dirac’s notation ⟨r∣λ, k⟩ = uλ,k(r) is used.
�e single-particle energy of electrons in band λ and in the vicinity of k0 reads [11]

єλ,k = єλ,k0 + 3∑
j=1

ħ2

2mλ, j(k)[(k − k0) ⋅ e j]
2
, (2.14)

containing the e�ective mass

1

mλ, j(k) =
1

m0
+ 2

ħm0

pλ,λ(k0) ⋅ e j(k − k0) ⋅ e j
+ 2

m2
0

∑
ν≠λ,l

(k − k0) ⋅ e l(k − k0) ⋅ e j
pλ,ν(k0) ⋅ e j pν,λ(k0) ⋅ e l

єλ,k0 − єν,k0 . (2.15)

Equation (2.14) is essentially the e�ective-mass equation. �e pairwise orthogonal unit vectors e j

of the principal directions j are pointing away from k0 [30]. If the wave functions uλ,k0(r) have a
well de�ned parity, then pλ,λ(k) vanishes [12]. Equation (2.14) is also valid in QW systems with

the replacement k → k∥.

�e ansatz (2.11) used in Eq. (2.10) constitutes a coupled eigenvalue problem that must be solved

simultaneously for both the lattice-periodic wave functions uλ,k and the energies єλ,k . �is can

be done by diagonalizing the k ⋅ p Hamiltonian de�ned by Eq. (2.10) for a �nite set of quantum

numbers λ, resulting in realistic band structures and wave functions. Usually, k0 is chosen to be a

point of high symmetry, since the wave functions uλ,k0 and energies єλ,k0 are known here. In gen-

eral, єλ,k0 refers to a group of energetic minima of band λ. For example, in direct-gap zincblende

and wurtzite III-V and II-VI semiconductors, k0 = 0 and uλ,k0 have the symmetry of angular mo-

mentum eigenstates [25, 29], while the band-gap energies єλ,k0 must be provided by experiments.

In this thesis, the lowest conduction (e), heavy- (hh), light-hole (lh), and spin-orbit split-o� (so)

band are considered explicitly. Each band is twice spin degenerate, resulting in a set of eight bands.

�e explicit representation of the resulting fully-coupled Luttinger Hamiltonian can be found in


2.3 Band Structure 9

Ref. [25].

Although the Löwdin renormalization [31] provides perturbation theory with a framework to

include implications from remote bands, without taking them into account explicitly, an eight-band

model is insu�cient to describe the band structure of indirect semiconductors around the global

energeticminima. �is is due to the limitedmomentum range as a consequence of the perturbation

around a single point. Much more than the eight bands used in this thesis are required to calculate

a realistic band structure extending over the whole Brioullin zone [32].

Sometimes e�ects from band non-parabolicities are not of interest. It is then o�en useful to

restrict the discussion to two bands and Eq. (2.10) does not have to be solved explicitly. Instead,

it is su�cient to use Eq. (2.14) regarding a limited momentum range around the band minimum.

Although the e�ective mass (2.15) depends on the momentum, it can be assumed constant around

the band minima and is ultimately provided by experiments, which is fairly accurate concerning

optics in semiconductor systems that are excited closely to the band gap. �is approach is called

e�ective-mass approximation. In this case, Eq. (2.15) decouples concerning λ so that k0 is a free

parameter. While for direct semiconductors, k0 is identical for all λ, it di�ers in indirect materials

since electrons and holes accumulate at di�erent valleys. �en, k0 is e�ectively replaced by k0,λ,

the group of global minima for band λ.

For direct semiconductors with cubic symmetry, all masses are isotropic [29], e.g. mλ,1 = mλ,2 =
mλ,3 ≡ mλ. In indirect semiconductors however, allmasses can be di�erent although the anisotropy

is usually constrained to the conduction band. Yet, if the group of k0 contains at least a threefold

axis, two of the three conduction band masses have the same value [30], a result obtained using

group theory [20].

2.3.2 Semiconductor Heterostructures

�e envelope-function approach [27, 33] is an excellent tool to �nd the band structure of semi-

conductor heterostructures. �ese structures can be considered mesoscopic since the QW’s lateral

extension is considerably larger than the lattice constant of the utilizedmaterials, but small in com-

parison to the full sample. As a consequence, the electrons are con�ned within these regions and

quantized along the growth direction of the crystal.

�erefore, the considerations of the previous Section can be repeated with the adjusted Bloch

theorem

ϕλ,ν,k(r) = 1√
L2

eik∥⋅r∥ ξλ,ν(z)uλ,ν,k∥(r) , (2.16)

containing the con�nement function ξλ,ν characterized by a bulk band index λ and a subband

index ν. Both the momentum and spatial coordinates have been decomposed into a component in

growth direction of the crystal (kz and z) and one perpendicular to it, e.g. within the QWplane (k∥
and r∥). �e additional subband quantum number ν occurs due to the quantization in the growth


10 2 Theoretical Framework

direction z.

Similar steps as in Section 2.3.1 yield the perturbative wave function using bulk states via

ξλ,ν(z)uλ,ν,k∥(r) =∑
µ

ξ
(µ)
λ,ν,k∥
(z)uµ,k0(r) . (2.17)

Formally, ξ
(µ)
λ,ν,k∥

are the expansion coe�cients in analogy to Eq. (2.12). However, they are some-

times called con�nement functions as well.

A heterostructure system constitutes a spatial potential landscape Vλ for electrons and holes in

the conduction (λ = e) and valence (λ ∈ h ≡ {hh, lh, so}) bands [34]
Vλ∈e(h)(z) = єλ,k0(z) + Vo�set(z) −(+) ∣e∣Φ(z) , (2.18)

with contributions Vo�set to account for the relative energetic alignment of adjacent layers. �e

screening potential Φ obeys the Poisson equation [35]

d2

dz2
Φ(z) = − ∣e∣

є0єBG(z)(ρh(z) − ρe(z)) , (2.19)

where the charge density distribution ρ(z) = ρh(z) − ρe(z) has been decomposed into hole (ρh)

and electron (ρe) contributions. Using the expansion (2.17), they read [36]

ρh(z) = 1√
L2
∑

νh ,k∥ ,µ

∣ξ(µ)νh ,k∥
(z)∣2(1 − f νh

k∥
) , ρe(z) = 1√

L2
∑

νe ,k∥ ,µ

∣ξ(µ)νe ,k∥
(z)∣2 f νe

k∥
, (2.20)

containing the carrier distribution

f νλ
k∥
= ⟨â†νλ ,k∥ âνλ ,k∥⟩ . (2.21)

Here, the notation (λ, ν) ≡ νλ is introduced to label subbands directly according to their bulk

a�liation. �is is useful to emphasize a restriction of a speci�c subband to be either an occupied

(electron) or unoccupied (hole) state.

2.4 Semiconductor Luminescence Equations

Once an excited system has reached a quasi-equilibrium state, the energy of the exciting light �eld

is completely stored in its quantum uctuations. Under these conditions, it is necessary to quantize

the light �eld [12]. As a result, spontaneous emission of a photon from an excited semiconductor,

known as luminescence, is a purely quantum-optical e�ect. One of the several forms of lumines-

cence is PL, the light emission from any form of matter a�er the preceding absorption of electro-

magnetic radiation. In semiconductors, its origin is the radiative recombination of electron–hole


2.5 Semiconductor Bloch Equations 11

pairs. PL signals emerge from electronic transitions de�ned by the band structure of the system.

Hence, it is an excellent tool to investigate the fundamental band gap of semiconductors [37, 38].

But device applications of PL are numerous. In phosphor thermometry, its temperature depen-

dence is exploited to measure heat [39].

�e most relevant quantity in the description of PL is the photon-number-like correlation

Nq = ∆⟨b̂†qb̂q⟩ , (2.22)

since it is proportional to the temporal change of the amount of photons. Here, b̂†q (b̂q) is the
creation (annihilation) operator of a photon with optical frequency ωq = c0∣q∣. Via Eq. (2.8), the
steady-state luminescence in the rotating-frame approximation can be found to be [40]

PL = ∂

∂t
Nq = 2 1

L2
Re

⎡⎢⎢⎢⎢⎢⎣ ∑λe ,νh ,k∥

(F λe ,νh
q )⋆Πλe ,νh

k∥ ,q

⎤⎥⎥⎥⎥⎥⎦
, (2.23)

containing the coupling-matrix element F λe ,νh
q and the photon-assisted polarization

Πλe ,νh
k∥ ,q
= ∆ ⟨b̂†q â†λe ,k∥ âνh ,k∥⟩ . (2.24)

Equation (2.24) characterizes the emission of a photon under simultaneous recombination of an

electron–hole pair. Explicit forms for F λe ,νh
q can be found in Refs. [10] and [40]. �e dynamics of

Eq. (2.24) can again be obtained using the Heisenberg equation of motion (2.8), yielding

iħ
∂

∂t
Πλe ,νh

k∥ ,q
= (єλe ,k∥ − єνh ,k∥ − ħωq)Πλe ,νh

k∥ ,q
+W λe ,νh

q,k∥
− (1 − f λe

k∥
− f νh

k∥
)U λe ,νh

q,k∥
. (2.25)

Both the spontaneous-emission sourceW λe ,νh
q,k∥

and the stimulated-emission source U λe ,νh
q,k∥

depend

on the level of approximation made concerning contributions from scattering. In this thesis, we

treat these contributions on the second-Born level and in the Markov limit. �e Markov limit

explicitly neglects quantum-memory e�ects for the phonon interactions [41]. �e second-Born

approximation includesCoulomb e�ects up to quadratic order in the screened interaction potential

for carrier many-body e�ects [42]. Resulting explicit representations ofW λe ,νh
q,k∥

and U λe ,νh
q,k∥

can be

found in Refs. [29] and [40].

2.5 Semiconductor Bloch Equations

Ideally, a coherent laser generates a light �eld which is very close to classical light [10]. Hence, it

is not required to quantize the light �eld while investigating the absorption of a semiconductor

system. Instead, it is su�cient to assume a coherent classical light source which is described via

E(t) ≡ E(t)eP. It contains the polarization eP and acts at the position of the QW.


12 2 Theoretical Framework

In order to determine the optical response of a many-body system to this classical light �eld, the

microscopic interband polarization is introduced via pνh ,λe
k∥
= ⟨â†νh ,k∥ âλe ,k∥⟩ , which corresponds

to the transition amplitude of an excitation process. Again, Eq. (2.8) can be employed to obtain the

relevant dynamics, which read

iħ
∂

∂t
pνh ,λe
k∥
= ∑

µe ,ηh

[є̃eλe ,µe δνh ,ηh + є̃hνh ,ηh δλe ,µe]pηh ,µek∥

−(1 − f λe
k∥
− f νh

k∥
)Ωνe ,λh

k∥
+ iħΓνh ,λe

k∥
, (2.26)

∂

∂t
f λe
k∥
= − 2

ħ
Im

⎡⎢⎢⎢⎢⎣∑νh Ωλe ,νh
k∥
(pνh ,λe

k∥
)⋆⎤⎥⎥⎥⎥⎦ + Γ

λe ,λe
k∥

, (2.27)

∂

∂t
f νh
k∥
= − 2

ħ
Im

⎡⎢⎢⎢⎢⎣∑λe Ω
λe ,νh
k∥
(pνh ,λe

k∥
)⋆⎤⎥⎥⎥⎥⎦ + Γ

νh ,νh
k∥

, (2.28)

with the renormalized band energies

є̃eλe ,νe = єλe ,k∥ δλe ,νe − ∑
µe ,k

′
∥

V
λe ,µe ,νe ,µe
k∥−k′∥

f
µe
k∥

, (2.29)

є̃hλh ,νh = єλh ,k∥ δλh ,νh − ∑
µh ,k

′
∥

V
νh ,µh ,λh ,µh
k∥−k′∥

f
µh
k∥

, (2.30)

and the renormalized Rabi energy

Ωνe ,λh
k∥
= dνe ,λh

k∥
E(t) + ∑

µe ,ηh ,k
′
∥

V
λe ,ηh ,νh ,µe
k∥−k′∥

p
ηh ,µe
k∥

. (2.31)

�e scattering terms Γλ,ν
k∥

in Eqs. (2.26)–(2.28) which constitute the semiconductor Bloch equations

(SBEs), are again treated in second Born approximation and the Markov limit. Explicit versions

can be found in Refs. [29, 43]. Equations (2.29)-(2.31) also contain the dipole-matrix element

dλe ,νh
k∥
= ie

m0(єλe ,k∥ − єνh ,k∥) ∑µ,κ
∞

∫
−∞

dz [ξ(µ)λe ,k∥
(z)]⋆[δµ,κ ħk∥ + pµ,κ] ⋅ eP ξ(κ)νh ,k∥

(z) , (2.32)

and the Coulomb-matrix element

V
λ j ,ν l ,µm ,ηn
k∥

= e2

2є0єBG∣k∥∣L2
×∑

µ,κ

∞

∫∫
−∞

dz dz′ [ξ(µ)λ j ,k∥
(z)ξ(κ)ν l ,k∥

(z′)]⋆e−∣k∥(z−z′)∣ξ(κ)µm ,k∥
(z′)ξ(µ)ηn ,k∥

(z) , (2.33)

both using the expansion (2.17).


2.6 Inhomogeneous Broadening 13

2.5.1 Bulk Semiconductors

In some parts of this thesis, the SBEs for bulk semiconductors are required. Although the di�er-

ences are marginal, they are given here for completeness. Within the scope of this thesis, a two

band model is considered.

A restriction to the bulk e and hh bands yields the dynamics of the macroscopic polarization

pk ≡ ⟨a†hh,kae,k⟩ and carrier distributions f
e(hh)
k

= ⟨a†
e(hh),kae(hh),k⟩, which read

iħ
∂

∂t
pk = є̃k + (1 − f ek − f hhk )Ωk + iħΓhh,ek

, (2.34)

∂

∂t
f ek = − 2ħ Im[Ωk(pk)⋆] + Γe,ek

, (2.35)

∂

∂t
f hhk = − 2ħ Im[Ωk(pk)⋆] + Γhh,hhk

, (2.36)

containing the renormalized band dispersion є̃k = єe,k−єhh,k−∑k′ Vk−k′( f ek′ + f hh
k′
) and scattering

terms Γλ,ν
k

. �e renormalized Rabi energy is given by

Ωk = dkE(t) +∑
k′
Vk−k′ pk′ , (2.37)

containing the Coulomb potential

Vq = e2

єBG є0 L3
1

q2
, (2.38)

which is the Fourier transform of Eq. (2.4), and the interband dipole matrix element dk = de,hh
k

with

dλ,ν
k
= ⟨λk∣(−∣e∣r)∣νk⟩ = iħ∣e∣

m0

pλ,ν(k)
єλ,k − єν,k . (2.39)

2.6 Inhomogeneous Broadening

If the hierarchy problem is truncated at the level of the second-Born approximation, only ho-

mogeneous broadening e�ects are taken into account explicitly. However, realistic semiconduc-

tors are subject to other mechanisms that may contribute to the absorption or PL line shape.

Most commonly, these are local uctuations of the band gap due to small composition inhomo-

geneities, layer thickness variations, or disorder. To account for these e�ects, a homogeneous re-

sponse S(E = ħω, 0) can be convoluted with a Gaussian to obtain the inhomogeneously broadened


14 2 Theoretical Framework

one by

S(E , ∆) = 2
√
ln(2)

∆
√
π

∞

∫
0

dE′ S(E′, 0) exp
⎡⎢⎢⎢⎢⎢⎣
−4 ln(2)(E − E′)2

∆2

⎤⎥⎥⎥⎥⎥⎦
, (2.40)

which is characterized by the full width at half maximum (FWHM) ∆.

2.7 Optical Absorption

Computing the dynamics of speci�c quantities outlined previously is motivated by their suitabil-

ity to calculate the optical response of semiconductor systems. �e used microscopic many-body

approach simultaneously allows to examine interactions on the scale of fundamental particles. Op-

tically accessible quantities can be modeled via macroscopic properties derived from their micro-

scopic counterparts. While PL has already been discussed in Section 2.4, this Section focuses on

linear and nonlinear absorption.

�e response of an excited semiconductor is determined by the macroscopic polarization

P(t) = 1

L2
1

dQW
∑

λe ,νh ,k∥

pνh ,λe
k∥
[dλe ,νh

k∥
]⋆ + c.c. , (2.41)

P(t) = 1

L3
∑
k

pk d
⋆
k + c.c. , (2.42)

for QWs and bulk materials, respectively. Here, dQW = ∑ j l j is the lateral extension of a multi QW

structure consisting of active layers with lengths l j.

When a transverse electric �eld is applied to a sample, the optical response is also determined

by the transverse dielectric function [44–46],

єT(ω) = єBG + χ(ω) . (2.43)

It contains the electric susceptibility χ, which relates the macroscopic polarization P to the semi-

classical electric �eld E via

P(ω) = є0χ(ω)E(ω) . (2.44)

In more complicated QW structures such as the “W”-Laser [47–50], єBG is determined by aver-

aging over the active regions, e.g.

єBG = 1

dQW
∑
j

єBG, j l j , (2.45)

with єBG, j being the background relative permittivity of the j-th optically active layer.


2.8 Wannier Equation 15

�e absorption of a QW heterostructure as an experimentally accessible property is related to

the electric susceptibility via [10]

α(ω) = 1 − ∣R(ω)∣2 − ∣T(ω)∣2 = 2 Im [ξ(ω)]
1 + ∣ξ(ω)∣2 + 2 Im [ξ(ω)] , (2.46)

where ξ is the scaled susceptibility

ξ(ω) = ω

2
√
єBGc0

χ(ω) . (2.47)

�e reectance and transmission coe�cients are given by [10]

R(ω) = iξ(ω)
1 − iξ(ω) , (2.48)

T(ω) = 1

1 − iξ(ω) . (2.49)

If we assume that both ∣ξ(ω)∣2 and Im [ξ(ω)] are small, Eq. (2.46) reduces to

α(ω) ≈ 2 Im [ξ(ω)] = ω√
єBGc0

Im [χ(ω)] . (2.50)

2.8 Wannier Equation

�e incoherent limit describes a quasi-equilibrium con�guration in which carriers have been ex-

cited by a preceding light pulse and decayed into a thermal state. �is means that this state is

characterized by a temperature T and su�ciently described by a Fermi–Dirac distribution,

f λk = 1

exp[β(єλ,k − µλ)] + 1 , β−1 = kBT , (2.51)

containing the chemical potential µλ.

Having eliminated the time dependence for the carrier distributions in the SBEs, the di�erential

equation for the polarization (2.34) becomes homogenous, since the driving �eld has already been

decayed in the incoherent limit. �us, the homogeneous solution ψλ constitutes an eigenvalue

problem for the exciton wave function and binding energies Eλ known as the generalizedWannier

equation [51]

Eλ ψ
R
λ(k) = є̃k ψR

λ(k) − (1 − f ek − f hk )∑
k′
Vk−k′ ψ

R
λ(k′) . (2.52)

For non-vanishing carrier distributions, Eq. (2.52) is non-Hermitian thus having both le�- (ψL
λ)


16 2 Theoretical Framework

and right-handed (ψR
λ ) solutions. �e corresponding le�-handed equation reads [52]

[ψL
λ(k)]⋆Eλ = [ψL

λ(k)]⋆є̃k −∑
k′
[ψL

λ(k′)]⋆(1 − f ek′ − f hk′)Vk−k′ , (2.53)

and both solutions are related via [10]

ψL
λ(k) = ψR

λ(k)
1 − f e

k
− f h

k

. (2.54)

Although Eqs. (2.52) and (2.53) assume Γhh,e
k
= 0, dephasing mechanisms can be included mi-

croscopically. As an example, Coulomb correlated e�ects such as EID [28, 51] may be included via

complex scatteringmatrices [53]. �en, Equation (2.54) is no longer valid and Eqs. (2.52) and (2.53)

must be solved simultaneously.

�e wave functions obey the orthogonalization and completeness relation [52]

∑
k

[ψL
λ(k)]⋆ψR

ν (k) = δλ,ν , (2.55)

∑
λ

[ψL
λ(k)]⋆ψR

λ(k′) = δk,k′ . (2.56)

For vanishing carrier densities, Eq. (2.52) resembles the Schrödinger equation for the relative

motion of a hydrogen-like atom [12]. However, di�erences in the particle masses and dielectric

constants yield binding energies which di�er by about 3 orders of magnitude. While the hydrogen

binding energy depends on the bare electron mass m0 and has a well known absolute value of

13.6 eV [54], the exciton counterpart in common semiconductors is usually in the order of 1 −
20meV [55]. Covering the energy range of a fewmeV iswhatmakesTHz spectroscopy an especially

quali�ed tool to identify and study excitonic e�ects in semiconductors [56].

2.9 Linear THz Spectroscopy

�at part of the system Hamiltonian (2.1) that governs the interaction of carriers and a weak clas-

sical THz �eld ATHz ≡ ATHz(t)eA reads [10]

HTHz = −∑
λ,k

jλ,k ⋅ ATHz(t) a†λ,kaλ,k . (2.57)

�e �eld’s polarization is labeled eA while the current-matrix element is given by [6]

jλ,k = − eħ∇kєλ,k . (2.58)


2.9 Linear THz Spectroscopy 17

As a measure of the overlap of the participating exciton states, transitions are mediated via the

transition-matrix element [57]

Jλ,ν =∑
k

[ψL
λ(k)]⋆ j(k)ψR

ν (k) , (2.59)

containing the reduced current

j(k) =∑
λ

jλ,k ⋅ ep . (2.60)

Finally, the absorption follows from the THz-Elliott formula [10]

αTHz(ω) = Im⎡⎢⎢⎢⎢⎣∑λ,ν
Sλ,ν(ω)∆nλ,ν − [Sλ,ν(−ω)∆nλ,ν]⋆

є0
√
єBG c0ω(ħω + iγJ)

⎤⎥⎥⎥⎥⎦ , (2.61)

with the response function

Sλ,ν(ω) =∑
β

(Eβ − Eλ)Jλ,β Jβ,ν
Eβ − Eλ − ħω − iγ , (2.62)

and the total density of exciton correlations ∆nλ,ν, which generally includes both exciton and free

carrier contributions [58]. �e phenomenological damping constant of the exciton correlations

(THz current) is labeled γ (γJ).


3 E�ects of Mass Anisotropy on theOptical

Properties of Semiconductors

One of the most fascinating organizing principles in quantum many-body physics is the forma-

tion of quasiparticles. As such, an electron in a periodic potential kinetically behaves as if moved

through free space but with a modi�ed mass. �e recently discovered dropleton [59] consists of

collectively bound electron–hole complexes and is characterized by a pair-correlation function

identical to that of a liquid [60]. In turn, quasiparticles can modify a semiconductor’s optical re-

sponse making them an excellent phenomenon to test our understanding of quantum mechanics

with spectroscopical techniques.

�e formation of a bound state consisting of an electron and a hole via the attractive Coulomb

interaction is known as an exciton and quali�es as one of the most fundamental excitations in

semiconductors [61]. �is energetically favorable state in excited semiconductors has been stud-

ied extensively in material systems such as GaAs [62–65], where the electrons’ e�ective mass is

isotropic [66], i.e. does not depend on the direction of motion. In this Chapter, the description of

the exciton wavefunction is extended to systems where the electrons’ e�ective mass is anisotropic,

e.g. varies concerning di�erent crystal momentum paths. To con�rm its validity, it is applied to

calculate the THz absorption in Ge and Si. It will also be used to obtain the near-bandgap excitonic

optical absorption of titanium dioxide (TiO2) in rutile con�guration.

3.1 THz Spectroscopy of Bulk Germanium

�e energetic width of THz radiation roughly covers the range of 1 − 15meV, making THz spec-

troscopy a uniquely quali�edmethod to investigate inter-molecular vibrations [67], high-harmonic

generation [68], transient photoconductivity [69], or transition energies of exciton eigenstates in

quantum many-body systems [70–72]. Especially excitons take an important role in both opti-

cal and THz spectroscopy of semiconductors. As an experimentally directly observable quantity,

transition energies between exciton states critically depend on e�ective electron and hole masses.

THz excitation dynamics and spectra of many direct-gap semiconductors have been investigated

by both theoretical [73–76] and experimental [77–80] means. In these systems, exciton properties

can be obtained even analytically [73, 74].

E�ective carrier masses in indirect semiconductors are o�en anisotropic. Although excitons

have been observed in such systems, e.g. in Ge [81] or Si [82], this has been done by means of THz

19


3.1 THz Spectroscopy of Bulk Germanium 21

particles having di�erent momenta and reads

Eλ ψ
R
λ(k, k′) = є̃k,k′ ψR

λ(k, k′) − (1 − f ek′ − f hk )∑
q

Vq ψ
R
λ(k − q, k′ − q) . (3.1)

�e appropriate renormalized band dispersion is given by

є̃k,k′ = єe,k′ − єhh,k −∑
q

Vq( f ek′−q + f hk−q) . (3.2)

�e discussion will be restricted on bulk semiconductors so that the Fourier transform of the

Coulomb potential is given by Eq. (2.38).

In quasi-equilibrium, electrons accumulate around k0,e ≡ k0 while holes usually are aggregated
around the Γ valley, e.g. k0,hh = 0. Hence, k0 is the separation between the band extrema. In the

following, only diagonal excitons where k′ = k+k0 are considered. With this assumption, Eq. (3.1)

reduces to the usual Wannier equation (2.52), while the renormalized band structure is given by

є̃k = єe,k+k0 − єhh,k −∑
k′
Vk−k′( f ek′ + f hk′)

= Eg + 3∑
j=1

ħ2

2µ j
[k ⋅ e j]2 −∑

k′
Vk−k′( f ek′ + f hk′) ,

(3.3)

introducing the reduced e�ective masses µ−1j = m−1e, j +m−1hh, j and the band gap Eg = єe,k0 − єhh,0.
Next, the numerical e�ort can be reduced by explicitly accounting for the angle dependence

within the generalizedWannier equation. An e�cient approach is to project Eq. (3.1) onto spherical

harmonics,

∫ dΩ [Ym
l (Ω)]⋆ Eλ ψR

λ(k) = ∫ dΩ [Ym
l (Ω)]⋆є̃k ψR

λ(k)
−(1 − f ek − f hk ) ∫ dΩ [Ym

l (Ω)]⋆∑
k′
Vk−k′ ψ

R
λ(k′)

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶
≡Ik

, (3.4)

where radially symmetric carrier distributions fk = fk have been assumed. In terms of angles, it is

Ω = (θ , φ) and ∫ dΩ = ∫ 2π0 dφ ∫
π
0 sin θ dθ. �e appearing integrals can be solved if the wavefunc-

tion is expanded into a linear combination of spherical harmonics Ym
l ,

ψR
λ(k) = ∞∑

l=0

l∑
m=−l

Rλ,l ,m(k)Ym
l (θ , φ) , (3.5)

which is valid since they constitute an orthonormal basis. If the carrier distributions cannot be as-

sumed to be radially symmetric, they also need to be expanded into a series of spherical harmonics

similar to Eq. (3.5). However, this issue goes beyond the scope of this thesis.


22 3 E�ects of Mass Anisotropy on the Optical Properties of Semiconductors

A wavefunction’s dominant character may be de�ned via that pair {l ,m} which produces the

largest value for the expression

Cλ,l ,m ≡ ∑
k

∣Rλ,l ,m(k)Ym
l (θ , φ)∣2 . (3.6)

Equation (3.6) is a measure for the individual contributions from di�erent angular momentum

eigenstates. Any wavefunction can then be uniquely labeled by {ν, l ,m} according to its dominant

character {l ,m} and a principal quantum number ν = {1, 2, 3, . . . }. Instead of a single label λ ={1, 2, 3, . . . }, this notation also reects symmetry information and allows to use the usual notation

for complex-valued atomic orbitals {1, 0, 0} ≡ 1s, {2, 1,−1} ≡ 2p−1, {2, 1, 0} ≡ 2p0, {2, 1, 1} ≡ 2p1,{2, 0, 0} ≡ 2s, and so on.

One representation of the spherical harmonics is given by [85]

Ym
l (θ , φ) = (−1)mαml Pml (cos θ) eimφ , αml =

√
(2l+1)(l−m)!
4π(l+m)! , (3.7)

where the phase factor (−1)m is chosen not to be included in the associated Legendre polynomials

Pml (x) =√(1 − x2)m dm

dxm
Pl(x) , (3.8)

which contain the Legendre polynomials Pl . �is choice is also known as the Condon–Shortley

sign convention [86].

In a system with isotropic masses, all integrals in Eq. (3.4) except Ik are easily solved due to the

orthogonality of the spherical harmonics [87],

∫ dΩ [Ym
l (Ω)]⋆Ym′

l ′ (Ω) = δl ,l ′ δm,m′ . (3.9)

A�er inserting the expansion (3.5), Ik reads

Ik = ∞∑
l=0

l∑
m=−l

∞

∫
0

dk′ (k′)2Rλ,l ,m(k′) ∫ dΩ′ ( L
2π
)3Vk−k′Y

m
l (Ω′)

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶
≡Ik ,k′

, (3.10)

if the vectorial sum is replaced by an integral [8],∑k → ( L2π)3 ∫ d3k, which is then implemented in

spherical coordinates. In attempt to solve Ik,k′ , the Fourier transformed Coulomb potential needs

to be expressed via

Vk−k′ = 1

L3

∞

∫
0

d3r V(r) e−ik⋅reik′⋅r . (3.11)


3.1 THz Spectroscopy of Bulk Germanium 23

�e plane-wave expansion [88, 89],

e−ik⋅r = 4π ∞∑
l=0

l∑
m=−l
(−i)l j⋆l (kr)[Ym

l (Ω)]⋆Ym
l (Θ) , (3.12)

contains the spherical Bessel functions jl while r = (r,Θ) with Θ = (ϑ , ϕ) is the position-vector
coordinate in spherical coordinates. Equations (3.11) and (3.12) can be used to obtain

Ik,k′ = Ym
l (Ω)V lk,k′ , (3.13)

V lk,k′ = 2

π

∞

∫
0

dr r2 V(r) j⋆l (kr) jl(k′r) . (3.14)

Equation (3.14) only depends on the absolute values of thewave vectors and the azimuthal quantum

number l .

On the other hand, Eq. (3.13) yields

V lk,k′ = ∫ dΩ′ ( L
2π
)3Vk−k′

Ym
l (Ω′)
Ym
l (Ω) , (3.15)

where the right-hand side is known to be independent of m and Ω, see Eq. (3.14). �erefore, their

values can be chosen to be most convenient, e.g. m = 0 and θ = φ = 0. With the help of Eq. (2.3),

it follows that

V lk,k′ = e2

4π2є0єBG

π

∫
0

dθ′
sin θ′ Pl(cos θ′)

k2 + (k′)2 − 2kk′ cos θ′ . (3.16)

�is representation is bene�cial mostly due to numerical reasons, since no additional grid for the

spatial coordinate is required, contrary to Eq. (3.14).

Without radial symmetry, the projection of the kinetic part of theWannier equation in Eq. (3.4)

produces integrals of the type

I1/2 =
π

∫
0

dθ sin3θ Pml ′ (cos θ)Pml (cos θ) , I3 =
π

∫
0

dθ sin θ cos2θ Pml ′ (cos θ)Pml (cos θ) ,
(3.17)

resulting from the components of the wave vector in spherical coordinates. �ey can be solved

using the recurrence formula for the associated Legendre polynomials [90].

�e �nal radial eigenvalue equation reads


24 3 E�ects of Mass Anisotropy on the Optical Properties of Semiconductors

Anisotropic Radial Wannier Equation

ẼλRλ,l ,m(k) =
⎡⎢⎢⎢⎢⎢⎣
ε(1)l ,m,k −

∞

∫
0

dk′ (k′)2 V lk,k′( f ek′ + f hk′)
⎤⎥⎥⎥⎥⎥⎦
Rλ,l ,m(k) + ε(2)l−2,m,kRλ,l−2,m(k)

+ε(2)l ,m,kRλ,l+2,m(k) + cλ,l ,m(k) − (1 − f ek − f hk )
∞

∫
0

dk′ (k′)2 V lk,k′Rλ,l ,m(k′) .
(3.18)

�e binding energy Ẽλ = Eλ − Eg has been de�ned with regard to the band gap while the function

cλ,l ,m collects all radial parts with a magnetic quantum number di�erent from m

cλ,l ,m(k) = ε(3)l ,m,kRλ,l ,m+2(k) + ε(3)l ,m−2,kRλ,l ,m−2(k) + ε(4)l−2,m−2,kRλ,l−2,m−2(k)+ε(4)l−2,−(m+2),kRλ,l−2,m+2(k) + ε(4)l ,−m,kRλ,l+2,m−2(k) , (3.19)

containing the coupling energies

ε(1)l ,m,k = ħ
2k2

2

⎡⎢⎢⎢⎣
Km
l

µ+
+ Nm

l

µ3

⎤⎥⎥⎥⎦ , ε(2)l ,m,k = ħ
2k2

2
[ 2
µ3
− 1

µ+
]Lml ,

ε(3)l ,m,k = −ħ2k22

Xm
l

µ−
, ε(4)l ,m,k = ħ

2k2

2

Wm
l

µ−
,

(3.20)

where µ−1± = µ−11 ± µ−12 . �e appearing weights are given by

Km
l = l

2 + l +m2 − 1
4l2 + 4l − 3 ,

Nm
l = 2l2 + 2l − 2m2 − 1

4l2 + 4l − 3 ,

Lml = 1

2(2l + 3)
¿ÁÁÁÀ(l −m + 2)(l −m + 1)(l +m + 1)(l +m + 2)(2l + 1)(2l + 5) ,

Xm
l =
√(l +m + 2)(l +m + 1)(l −m)(l −m − 1)

2(4l2 + 4l − 3) ,

Wm
l = 1

4(2l + 3)
¿ÁÁÁÀ(l +m + 4)(l +m + 3)(l +m + 2)(l +m + 1)(2l + 1)(2l + 5) .

(3.21)

It can be observed in Eq. (3.18) that a coupling between di�erent quantum numbers is the con-

sequence of mass anisotropy which breaks the symmetry. More speci�cally, a pair {l ,m} couples
to itself as well as {l ± 2,m ± 2}, which may be formulated by the notation ∆l = ∆m = {0,±2}.


26 3 E�ects of Mass Anisotropy on the Optical Properties of Semiconductors

Ẽ
λ
[m

e
V
]

AnisotropymΩ/m· AnisotropymΩ/m·

E
�
p

�
→
�
p
±
�
[m

e
V
]

(a) (b)

 
�s

�p�

�s
�p±�−15

−10

−5

0

0.0

0.4

0.8

1 10 20 1 10 20

Figure 3.3— (a) Binding energies of di�erent states as function of anisotropy are

presented. (b) The energetic di�erence E2p0→2p±1
between the �rst

stateswith p0 and p±1 symmetry (solid red and dashed black lines from

Frame (a)).

carrier masses [93]

m∥ = 1.59m0 , (3.22)

m⊥ = 0.0815m0 , (3.23)

mhh = 0.33m0 . (3.24)

�roughout this Section, Ge is used as a prototype system andweak excitations ( f ek = f hk = 0) are
assumed to demonstrate the major e�ects of mass anisotropy. A numerical solution of Eq. (3.18)

requires a truncation of the maximal included l quantum number because of the expansion (3.5).

As a criterion for convergence, we can evaluate the transition energy Eλ→λ′ = ∣Eλ − Eλ′ ∣ between
two states λ and λ′. For λ = 1s and λ′ = 2p0, fast convergence is provided by Eq. (3.5), which is

shown in Fig. 3.2(a). Stable results are obtained by including seven l states producing E1s→2p0 ≡E∥ = 3.38meV (dotted line). Using the e�ective masses in Ge, the ground state has dominant s-like

symmetry while the �rst excited state possesses a dominant p0 character.

Next, it is useful to study the energetic arrangement of exciton states in Ge. Figure 3.2(b) shows

Ẽλ for di�erent dominant symmetries. Each column represents a magnetic quantum number m

while orbital quantum numbers are color coded. It can be observed that the 2s, 2p±1, and 2p0 states

are energetically non-degenerate.

To provide more insights, the gradual change of Ẽλ for these states from isotropic to anisotropic

masses is studied in Fig. 3.3(a). It depicts the binding energies of the four �rst bound excitons as

a function of anisotropy m∥/m⊥ while m∥ is kept constant. For isotropic masses (m∥/m⊥ = 1), all
the considered excited states are energetically degenerate, which is the expected result of purely

2s and 2p-like hydrogen wavefunctions. �is initial degeneracy is li�ed with decreasing m⊥ (or


3.1 THz Spectroscopy of Bulk Germanium 27

Wave Vector ka�

(a) (b)
R
λ
(k
)[
sc

a
le

d
]

Wave Vector ka�

�f
−��f
−��s�s

s

d�

f−�
h−�
p−�

 
0.0

0.5

1.0

0.0 0.5 1.0 0.0 0.15 0.3

Figure 3.4—Decomposition of excitonwavefunctions into radialpart Rλ(k). (a) The

1s ground state wavefunction is clearly dominated by the s-like radial

part. The only other non-vanishing radial part has d0 symmetry. (b)

The decomposition of the 3f−1 wavefunction shows dominant {3,−1}

character and a coupling to p−1 and h−1 radial parts. This is in agree-

ment with the scheme presented at the end of Section 3.1. All other

radial parts vanish due to nonexistent coupling.

increasing m∥/m⊥) and reveals this to be a direct consequence of mass anisotropy. �e 1s ground

state emphasizes the general trend of a fast change close to m∥/m⊥ = 1 while Ẽλ saturates with

increasing anisotropy. Increasing m∥/m⊥ freezes the motion in the parallel direction, which is

why this saturation is observed. Measuring the energetic di�erence E2p0→2p±1 between the �rst

excited states with p symmetry visualizes the initial degeneracy in Fig. 3.3(b). For Ge masses,

E2p0→2p±1 converges against 0.78meV. As a result, initially degenerated states are li�ed due to the

mass anisotropy, producing a E2s→2p0 = 0.415meV (E2s→2p±1 = 0.362meV) separation between the

2s and the 2p0 (2p±1) states. It is noted that E1s→2p±1 ≡ E⊥ = 4.16meV, since this transition energy

will be of interest in Section 3.1.3.

�e previous Section discussed that a wavefunction will contain many {l ,m} components due

to the ansatz (3.5). Figure 3.4 visualizes the resulting coupling rules of Eq. (3.1) for the 1s and 3 f−1

states in Frames (a) and (b), respectively, only showing non-vanishing contributions. As the ground

state, the 1s wavefunction possesses dominantly {0, 0} symmetry. �e only other non-vanishing

component is d0, which conforms to the coupling rules ∆l = {0, 2} and ∆m = 0 (∆l = −2 is not
possible for ν < 2). No coupling to ∆m = ±2 is provided due to identical masses within the plane

perpendicular to k0. To verify the full coupling rules, the 3 f−1 wavefunction, which is characterized

by the {3,−1} radial part, shows non-vanishing {1,−1} (or p−1) and {5,−1} (or h−1) components.

To gain more insight, modi�cations to the wavefunctions a�er the transition from isotropic to

anisotropic masses are investigated. �e e�ects are illustrated in Fig. 3.5, which shows a selection

of exciton wavefunctions ∣ψR
λ (k)∣ for isotropic (top row) and anisotropic (bottom row) electron


30 3 E�ects of Mass Anisotropy on the Optical Properties of Semiconductors

which follows from Eq. (2.60) using the e�ective-mass approximation (2.14). Accordingly, the

transition-matrix elements (2.59) become

J∥λ,ν = − eħL38π3µ3

∞

∫
0

dk k3
∞∑
l=0

l∑
m=−l

R⋆λ,l ,m[gml Rν,l+1,m + gml−1 Rν,l−1,m] , (3.28)

and

J⊥λ,ν = −i eħL38π3µ2

∞

∫
0

dk k3
∞∑
l=0

l∑
m=−l

R⋆λ,l ,m [ f ml Rν,l+1,m+1 − f −ml Rν,l+1,m−1

− f −m−1l−1 Rν,l−1,m+1 + f m−1l−1 Rν,l−1,m−1] ,
(3.29)

containing the weight functions

gml =
¿ÁÁÁÀ(l −m + 1)(l +m + 1)(2l + 3)(2l + 1) , (3.30)

f ml = − 12
¿ÁÁÁÀ(l +m + 1)(l +m + 2)(2l + 1)(2l + 3) . (3.31)

Equations (3.28) and (3.29) de�ne selection rules incorporating e�ects due to themass anisotropy.

In particular, J∥λ,ν vanishes for all transitions except those with equal m components while the l

quantum number di�ers by ±1. In the usual notation, this is ∆m = 0 and ∆l = ±1. Regarding e⊥

polarized excitons, the transition rules translate according to ∆m = ∆l = ±1. Consequently, the
energetically lowest allowed THz transition is E∥ and E⊥ for parallel and perpendicular polarized

THz �elds, respectively, assuming exclusive occupation of 1s ground-state excitons. �us, 1s and

2p0 (2p±1) states are involved for eA = e∥ (eA = e⊥) and both transitions are illustrated in Fig. 3.2

by arrows. Conclusively, these two transitions di�er by E⊥−E∥ = 0.78meV as a consequence of the

mass anisotropy and produce direction-dependent THz resonances.

�e wavefunctions describing the sample’s many-body state are determined for arbitrary carrier

masses by Eq. (3.18) as discussed in the previous Section. Once they are known, the linear THz

absorption spectrum follows from the THz Elliott Formula (2.61). Although weak carrier densities

are still assumed throughout this Section, three di�erent excitation conditions will be investigated:

(i) Coherent excitons generated at the Γ valley,

(ii) incoherent excitons formed across the L valley addressed by parallel polarization,

(iii) incoherent excitons formed across the L valley addressed by perpendicular polarization.


3.1 THz Spectroscopy of Bulk Germanium 31

A
b
so

rp
ti

o
n
[a

rb
.u

.]

Photon Energy ħω [meV]

 
αΓ
αΩ
α·

×�.�

Ek E⊥0

2

4

1 2 3 4 5

Figure 3.7— THz absorption spectra of Ge. Assuming carrier masses of excitons

forming at the Γ valley (solid line) leads to a signi�cantly stronger sig-

nal and an energetically lower fundamental transition compared to

excitons forming across the indirect L valley. The spectra for parallel

(dashed line) and perpendicular (dotted line) polarization are magni-

�ed for clarity. Thin vertical lines indicate the fundamental transition

energies for indirectly bound excitons.

While incoherent excitons exhibit the full mass anisotropy in the L valley, coherent excitons form

at the direct Γ point. �us, the carrier masses are isotropic with me = 0.041m0 [93] and no de-

pendency on the THz polarization exists. Experimentally, these two situations can be achieved via

the delay between optical-pump and THz-probe beam: Directly a�er the optical excitation at the

direct Γ point, a THz probe pulse would detect the coherent polarization as depicted in Fig 3.1(b)

as a red-shaded area. Due to the long temporal extension of a THz pulse compared to the equilib-

rium scattering times of conduction band electrons, the experimental realization of such a situation

poses an extreme challenge. In Ge, conduction band electrons eventually scatter into the indirect

L valley, which takes about 400 fs [96, 97]. Since quantum-optical many-body correlations build

up sequentially [98], the radiative lifetime must be su�ciently long to allow carriers to scatter into

the indirect valley and provide abundant exciton formation (orange-shaded area in Fig. 3.1(a)).

Typically, this presumption is ful�lled in many indirect semiconductors at low temperatures and

long a�er a weak optical excitation [91, 99–101]. �en, a THz pulse detects the indirectly bound

excitons. �e binding energies are computed to be ẼΓ1s = −1.910meV and ẼL1s = −5.228meV for

direct and indirect excitons, respectively. �ese values are in reasonable agreement with previous

theoretical and experimental �ndings [102–109].

Figure 3.7 compares the THz absorptions αΓ , α∥, and α⊥ corresponding to situations (i), (ii),

and (iii), respectively. It is assumed that only the 1s ground state exciton is occupied with a density

n1s, e.g. ∆nλ,ν = δλ,νδλ,1sn1s. �e dephasing has been chosen to be γ = γJ = 0.3meV. Contributions

due to a correlated electron–hole plasma have been neglected here since they only add a Drude-


32 3 E�ects of Mass Anisotropy on the Optical Properties of Semiconductors

∂J
�s
,λ
∂[
sc

a
le

d
]

AnisotropymΩ/m· AnisotropymΩ/m·

(a) (b)

··ΩΩ
�p�

�p�

�p�

�f�

�p�

�p�

�p�

0.0

0.5

1.0

1 10 20 1 10 20

Figure 3.8— Transition-matrix elements from the ground state for parallel and per-

pendicular THz polarization. (a) For eA = e∥, the �rst three allowed

transitions from the 1s state have p0 symmetry. (b) The �rst four end

states for eA = e⊥ have dominantly p1 and f1 symmetry.

type free carrier response to the spectrum [58]. �e coherent exciton response αΓ (solid line) is

shown besides the incoherent exciton THz absorption α∥ (dashed line) and α⊥ (dotted line) for

parallel and perpendicular polarization, respectively. Both incoherent responses have been mag-

ni�ed by a factor 2.7 for enhanced visibility. �e symmetry-based arguments about the selection

rules are corroborated by the two distinct resonance energies (thin vertical lines), exhibited by

these responses.

To study general trends of THz-transition oscillator strengths, Fig. 3.8 shows ∣J1s,λ∣ as a function
of anisotropym∥/m⊥ for selected end states λ. Frames (a) and (b) depict parallel and perpendicu-

lar polarizations according to Eqs. (3.28) and (3.29), respectively. �e selection rules determine the

symmetry of the �rst three end states to be p0 for parallel polarized excitons. Conversely, the �rst

four non-vanishing matrix elements for perpendicularly polarized excitons are of p1 and f1 sym-

metry. From the THz-Elliott formula (2.61) it can be deduced that J1s,λ determines the strength of

absorption for 1s-to-λ transitions as it is a measure of the overlap of the participating wavefunc-

tions. Since it monotonically decreases for parallel polarization while it increases in the perpen-

dicular case, the corresponding THz absorption appears stronger in perpendicular polarization as

observed in Fig. 3.7.

3.1.4 Muli-Valley Response to Classical THz Excitation

Until now, the THz absorption for purely parallel and perpendicular directions at a single L valley

has been calculated. A generalization, which takes the fourfold degeneracy of the L point into

account, is the subject of this Section.

�e issue is visualized in Fig. 3.9. A polarization eA may address the particular valley L0 in an


34 3 E�ects of Mass Anisotropy on the Optical Properties of Semiconductors

A
b
so
rp
ti

o
n
[a
rb

.u
.]

Photon Energy ħω [meV]

 
αTHz × �/�

αΩ
α·

0

1

2

1 2 3 4 5

Figure 3.10— Full THz absorption αTHz(ω) (shaded area) according to Eq. (3.37). The

individual contributions from the parallel (solid line) and perpendic-

ular (dashed line) constituents are also shown. A scaling factor of 3/8

has been applied to the full spectrum for visibility.

same response. Hence, Eq. (3.33) becomes

αTHz(ω) = α⊥(ω) 4∑
j=1

b2j + α∥(ω) 4∑
j=1

c2j . (3.34)

Next, b j and c j are calculated for all relevant L points. �e in-plane projection is found via

c j = L j ⋅ eA = b L j ⋅ e⊥ + c L j ⋅ e∥ ≡ b A j + c B j . (3.35)

�e coe�cients A j and B j are summarized in Table 3.1 and it follows that

4∑
j=1

b2j = 8

3
,

4∑
j=1

c2j = 4

3
, (3.36)

resulting in the total THz absorption

αTHz(ω) = 4

3
[ 2α⊥(ω) + α∥(ω)] , (3.37)

j 1 2 3 4

A j 0.0 −0.4714 −0.9428 −0.4714
B j 1 1/3 −1/3 1/3

Table 3.1—Coe�cients found from Eq. (3.35).


3.1 THz Spectroscopy of Bulk Germanium 35

where b2j = 1− c2j and b2 + c2 = 1 is used. Hence, the THz absorption for any arbitrary polarization

always yields the same mixture of parallel and perpendicular constituents. A simple interpretation

of this result is that the total response is determined by the directional mean value of all four rele-

vant L valleys. �e full absorption is shown along its two components in Fig. 3.10. Although α(ω)
contains a mixture composed of all directions, it still exhibits a residual double-peak structure as a

result of the mass anisotropy. �is result suggests that anisotropic masses can be detected directly

via available THz experiments [68, 72, 110]. As a prerequisite, the dephasing γ and γJ must not

greatly exceed the anisotropic splitting of 0.78meV. To the best of the authors knowledge, such

experiments have not yet been conducted using Ge samples.

Alternative Derivation

An alternative view on the THz current provides a more concise derivation of Eq. (3.37) that does

not rely on geometrical arguments. �e THz interaction Hamiltonian (2.57) may be written as

HTHz = −J ⋅ ATHz , (3.38)

containing the carrier-density dependent part of the current

J =∑
n
Jn =∑

n
∑
λ,kn

jλkn â
†

λ,kn
âλ,kn , (3.39)

where kn is in the vicinity of Ln.

It is bene�cial to introduce the vector en,m, which is obtained from the unit vector em via a

rotation so that en,3 and Ln are aligned. �is guarantees that a set en,m forms a coordinate system

where en,z points out-of and en,1/en,2 are within the n-th Brillouin zone edge plane. Hence, the

absorption follows from the single-valley response as described in Section 3.1.3.

�e current at the n-th L point is given by

Jn =∑
m
Jn,m(ω)en,m , Jn,m(ω) = Jn ⋅ en,m . (3.40)

At the same time, the current at Ln in direction m is [10]

Jn,m(ω) = 1

2
є0 ω

2χm(ω)ALn ,m(ω) , (3.41)

where

An,m(ω) = ATHz(ω) ⋅ en,m , (3.42)

is the projected THz �eld. Equation (3.41) also contains the single-valley susceptibility χm. �is ap-

proximation is valid when no crosstalk of the current is expected, e.g. in the linear regime. Hence,


36 3 E�ects of Mass Anisotropy on the Optical Properties of Semiconductors

we can compute χm at a reference valley as done in Section 3.1.3. An additional factor 1/2 accounts
for the sharing of each L point with the adjacent Brillouin zone. �en, the total response is deter-

mined by

χ(ω) ≡ J ⋅ eA
є0 ω2A(ω) = 1

2

1

A(ω) ∑n,m χmAn,men,m ⋅ eA
= eA ⋅ ∑

n,m

χm
2
(en,m ⊗ en,m) ⋅ eA ≡ eA ⋅ ∑

n
χ
n
⋅ eA , (3.43)

de�ning both the total susceptibility χ and the susceptibility tensor χ. �e Fourier transform of

A(t) is denoted by A(ω). For example, it is

χ
1
=
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

χ⊥ 0 0

0 χ⊥ 0

0 0 χ∥

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠
, (3.44)

since χ1 = χ2 ≡ χ⊥ and χ3 = χ∥. Considering all eight L valleys, it is

∑
n
χ
n
= 4

3
(2χ⊥ + χ∥)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

1 0 0

0 1 0

0 0 1

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠
. (3.45)

Applied to Eq. (3.43), the total susceptibility reads

χ(ω) = 4

3
(2χ⊥ + χ∥) , (3.46)

which yields Eq. (3.37) via the general expression (2.50) for the linear absorption.

3.2 THz Spectroscopy of Bulk Silicon

To the best of the authors knowledge, an experiment as described in the previous Section has not

yet been conducted. However, similar measurements in Si have been performed by Suzuki and

Shimano [83]. �erefore, the calculations from the previous Section shall be repeated adapted to

Si, in which the electron masses are [95]

m∥ = 0.98m0 , m⊥ = 0.19m0 , mhh = 0.49m0 . (3.47)


3.2 THz Spectroscopy of Bulk Silicon 37

Ref. [83]

Frequency [THz]

A
b
so
rp
ti
o
n
[n
o
rm

.]

αTHz

Ek E⊥
0.0

0.5

1.0

8 10 12 14 16 18

Figure 3.11— THz absorption of bulk Si. Experiments from Ref. [83] (blue-shaded

area) are compared to the full absorption obtained from Eq. (3.48)

(black solid line).

In Si, the energetic minima are situated somewhere between the Γ and X points [111] and threefold

degenerate. It is straightforward to show that the total absorption is then given by

αTHz(ω) = 6

2

1

3
[ 2α⊥(ω) + α∥(ω)] , (3.48)

which can be interpreted as the spatial average at a single valley again, multiplied by the degeneracy

of the energetic minima accounting for the sharing between adjacent Brillouin zones.

�e results are summarized in Fig. 3.11. A blue-shaded area represents the experimental data

from Ref. [83], which depicts a situation 600ps a�er a weak optical excitation. �e authors of

that study assign the double-peak characteristic at 10.1 THz and 11.3THz to the �ne structure of

an isotropic 2p excitons, arising from the simultaneous presence of both parallel and anti-parallel

spin alignments of the participating carriers. However, no e�ort was made to actually prove this

statement, although several experiments are suitable for this purpose. For example, excitons can be

spin-polarized using circularly polarized radiation [112]. An exciton population with exclusively

parallel or anti-parallel spin-polarization would not give rise to a �ne structure. Furthermore,

applying an external magnetic �eld would allow the manipulation of the �ne structure splitting

via the �eld strength and orientation [113]. Besides, the separation of the �ne structure splitted

states is usually in the range of 0.002−0.03THz [114] due to small exchange interactions. Even big

splittings of 0.5THz, which occur only in rare cases such as cadmium selenide (CdSe) quantum

dots (QDs) [114], are smaller than the observed splitting of 1.2THz in Fig. 3.11.

�e theoretically obtained spectrum (solid black line) shows a certain agreement with the ex-

periment. Especially the transition energies of parallel and perpendicular responses indicated by

thin vertical lines coincide with the measured resonances. Even the asymmetric high frequency

tail is covered by the theoretical data, while the original study does not explain this at all. Also, the


38 3 E�ects of Mass Anisotropy on the Optical Properties of Semiconductors

oscillator strengths of the relevant transitions arematching those that are observed experimentally.

�e good agreement between experiment and theory suggests that the observed features originate

from themass anisotropy rather then �ne structure e�ects. However, thorough studies are required

to sustain such assumptions.

3.3 Absorption of Bulk Rutile

For many optically interesting systems it is impossible to calculate fundamental properties such as

energy dispersions or single-particle wavefunctions using the k ⋅ p theory due to a lack of knowl-

edge of the relevant band parameters. Concerning TiO2, not a single Luttinger parameter can be

found in the literature today. �ese nontrivial systems such as complex compounds and interface-

dominated or organic/inorganic heterostructures impose a challenging task. However, with DFT

it is possible to calculate band structures and optical-matrix elements of the ground state in any

arbitrary system. On the other side, the CE outlined in Section 2.2 has been proven extremely

suitable to obtain dynamics of relevant microscopic operator expectation values.

In this Section, the CE–DFT method is presented that combines the strength of DFT and the

CE. At a glance, DFT determines ground-state optical-matrix elements while the dynamics of the

excited many-body system are computed using the CE. Under certain conditions, it is possible to

obtain the optical absorption via the wavefunctions described in Section 3.1.1. Rutile serves as a

prototype system in this Section to verify the method.

3.3.1 Density Functional Theory

Chapter 2 outlines the main results of the semiconductor many-body physics relevant for this

thesis. Knowledge of the complete system wavefunction Ψ was not necessary at any step of the

derivations. �e introduction of �eld operators reduced the needed information to single-particle

wavefunctions that obey a much simpler Schrödinger equation (2.10) for which the k ⋅ p theory

provides a useful framework with low computational e�ort. Although the expansion (2.5) can be

done using any complete set of basis functions, the natural integration of e�ective dimensionalities

via the con�nement function makes the use of single-particle wavefunctions advantageous.

Nevertheless, the complete system wavefunction Ψ obeys the Schrödinger equation

HΨ(r1, r2, . . . , rN) = E Ψ(r1, r2, . . . , rN) , (3.49)

whereH is themany-bodyHamiltonian Eq. (2.1) for N electronsmoving in the crystal lattice while

E is the electronic eigenenergy. �e attempt to solve Eq. (3.49) would obviously fail due to the huge

amount of 3N variables in a realistic many-body system and limited computing resources. Over

time, many powerful methods have been developed to solve the Schrödinger equation. �e dia-

grammatic perturbation theory based on Feynman diagrams andGreen’s functions have been used


3.3 Absorption of Bulk Rutile 39

in physics. In chemistry, a systematic expansion in terms of Slater determinants has been applied

widely but it has hardly been practicable to calculate properties of systems with more than 100

molecules with this method [24]. DFT provides a remedy concerning this limitations. Although

it might be less accurate, it presents itself as a versatile and viable alternative. Many structural,

electronic, or magnetic properties of molecules, semiconductors, and other materials have been

calculated using DFT, granting it a signi�cant role in fundamental natural sciences [115]. Award-

ing theNobel Prize in 1998 to JohnA. Pople [116] andWalter Kohn [117], who is one of the founding

fathers of this method, emphasizes its importance.

�e central quantity in DFT is the electron density

n(r) = N ∫∫∫ dr2 dr3 . . . drN ∣Ψ(r, r2, . . . , rN)∣2 . (3.50)

In their famous paper [23], Hohenberg and Kohn showed that it is in principle possible to �nd the

wavefunction Ψ0 and energy E0 of a systems ground state, given that its density is known. In other

words, the ground state density n0 and wavefunction Ψ0 are connected via a functional Ψ0[n0] and
as a consequence, all other system observables will be, too. �is e�ectively reduces the number of

variables from 3N to only 3, making a numerical solution possible.

For an arbitrary density n, the energy functional is given by

E[n] = min
Ψ→n
⟨Ψ ∣H∣Ψ⟩ , (3.51)

where Ψ → n denotes the requirement that Ψ must reproduce n via Eq. (3.50) while a ground-

state wavefunction has to minimize the system’s energy. Since the energy will be at least as large

as the ground state energy for any n ≠ n0, e.g. E[n0] ≤ E[n], a variational procedure can be ap-

plied to obtain an approximative solution for n0 and Ψ0. Usually, Eq. (3.51) is divided according

to the Kohn–Sham equations, which is the most popular formulation of DFT. Its foundation is the

representation of the interacting electrons to a system of non-interacting carriers, introducing the

Kohn–Sham orbitals from which single-particle properties (such as the band structure) can be de-

rived. �is allows the separation into an uncorrelated kinetic part—which can be solved exactly—

and the exchange correlation energy. �e latter is not exactly solvable because the exchange corre-

lation potential is unknown. Although DFT is formally exact, this embodies its greatest weakness,

since only boundary conditions for the exchange potential can be formulated. �is lack of knowl-

edge restricts DFT from being truly universal and forces one to �nd the most suitable functional

to reproduce physical properties like band gaps.

Di�erent functionals have been proposed to cope with this issue. It is well known that some

reproduce only certain physical aspects. Possibly the most famous example is the underestimation

of the band gap using the Perdew–Burke–Ernzerhof (PBE) [118, 119] functional. Nevertheless, the

PBE functional successfully describes the valence-band e�ective masses or the spatial relaxation of

atoms [120].


3.3 Absorption of Bulk Rutile 41

�
λ
,k
−

�
λ
,�
[m

e
V
]

kx a�kz a�

λ = e

λ = h

0

5

10

15

20

6 4 2 0 2 4 6

Figure 3.13—Magni�cation of the band structure into the region of interest shown

as a shaded area in Fig. 3.12(b). The dots resemble results obtained

by density functional theory calculations, while the solid lines are

parabolic �ts, allowing to extract e�ective masses.

is applied to the conduction band.

Carriers in the lowest conduction band exhibit a strong mass anisotropy due to di�erent cur-

vatures into di�erent momentum directions. Figure 3.13 presents a magni�cation of the region of

interest (shaded area in Fig. 3.12(b)). Besides the DFT single-particle energies (dots), parabolic

approximations (solid lines) according to Eq. (2.14) are shown. �ey reproduce the band struc-

ture almost perfectly, with only minor deviations at larger momenta. �is degree of agreement

is obtained using the e�ective masses me,1 = me,2 ≡ me,⊥ = 1.03m0, me,3 ≡ me,∥ = 0.519m0,

mh,1 = mh,2 ≡ mh,⊥ = 3.8m0, and mh,3 ≡ mh,∥ = 5.3m0. Here, the notation of Section 3.1.2 has

been applied with a minor modi�cation: Since the highest valence band is non-degenerate and

hence no classi�cation into hh and lh is feasible, it is labeled h instead of hh. As opposed to Ge,

carriers in the valence band also exhibit anisotropy. �e replacement of the true band structure

by the e�ective-mass approximation later allows the usage of exciton wavefunctions obtained by

Eq. (3.18).

To explain the optical-absorption spectra of TiO2, the transversal electric and magnetic �elds in

Eq. (2.2) are expanded in terms of multipoles. For this purpose, they are assumed to have the form

E(r, t) = − ∂
∂tA(r, t) ≡ E(t)E(r) and B(r, t) = ∇× A(r, t) ≡ E(t)/ω B(r). A�er the quantization

procedure, the light–matter coupling term in Eq. (2.1) reads

ĤD = −E(t)∑
λ,k
∑
ν,k′

Fν,k
′

λ,k â†λ,k âν,k′ , (3.52)


42 3 E�ects of Mass Anisotropy on the Optical Properties of Semiconductors

where its strength is given by

Fν,k
′

λ,k = ∫ d3r ϕ⋆λ,k(r) F(p, r)ϕν,k′(r) , (3.53)

containing the light–matter coupling operator

F(p, r) = −er ⋅ ∞∑
n=0

1(n + 1)! [(r ⋅∇r′)n E(r′)]∣
r′=0

− e

2m0

∞∑
l=0

l + 1
(l + 2)! {p ⋅ [r × [(r ⋅∇r′)l B(r′)]] + [r × [(r ⋅∇r′)l B(r′)]] ⋅ p}RRRRRRRRRRRr′=0 .

(3.54)

In its most popular form, the light–matter interaction in Eq. (2.1) is given including only electric

dipoles, e.g. F(p, r) = −er ⋅ E(0). Once the dipole-matrix element vanishes, such as is the case

for rutile, it is necessary to include higher multipole contributions, such as electric- and magnetic-

quadrupole interactions. In contrast to the band structure, thesematrix elements are taken directly

from the DFT calculations and depend on the �eld polarization. �e DFT calculation reveals that

magnetic dipole-matrix elements are vanishingly small and are therefore neglected.

3.3.3 Absorption

Apart from the mass anisotropy, the low-frequency dielectric constants of TiO2 also show an ex-

traordinary direction dependence with є1 = є2 ≡ є⊥ = 111 and є3 ≡ є∥ = 257 [121]. Since in the

derivation of the exciton wavefunctions (3.18), isotropic dielectric constants have been assumed

for the Coulomb potential, Eq. (3.15) does not hold any more. However, a coordinate transforma-

tion as discussed in Section 2.1 can be applied so that the Coulomb potential remains isotropic. As

a consequence, єBG and µ∥ have to be replaced according to єBG → √є∥є⊥ and µ∥ → µ∥є∥/є⊥ in
Eq. (3.15), which then can be used without further restrictions.

Since they constitute an orthonormal basis, it is possible to expand the microscopic polariza-

tion (2.42) into the exciton wavefunctions (3.1) via [10]

pk =∑
λ

pλ ψ
R
λ (k) , pλ =∑

λ

pk[ψL
λ(k)]⋆ . (3.55)

Here, diagonal excitonswith k′ = k (which follows from k0 = 0) have been assumed, see Section 3.1.

�e change of basis (3.55) can be used to transform the dynamics of pk . A subsequent Fourier

transform yields

pλ(ω) = Fλ
Eλ − ħω − iγλ(ω)E(ω) , (3.56)


3.3 Absorption of Bulk Rutile 43

A
b
so
rp
ti
o
n
[c
m

−
� ]

ħω − Eg [meV]

(a) (�, �)

(�, π/�)
(π/�, π/�)
(π/�, π/�)

1s

2p±1

0

2

4

6

−4 −2 0 2

�s
-P

e
a

k
A

b
so

rp
ti

o
n
[c

m
−
� ]

ħω − Eg [meV]

(b)
(θ , π/�)
(π/�, φ)
 

� π/� π

0.0

0.2

0.4

0.6

Figure 3.14— (a) Near band gap optical Absorption of rutile TiO2 for a light �eld

polarized in the x–y plane and propagation angles (θ , φ). When

light propagates along the z-axis (shaded area and solid line), the

absorption is mainly dominated by electric dipole interactions. A

propagationdirectionperpendicular to the z-axis (dashed anddotted

lines) yieldsweak resonanceswhich canbeassigned tonon-vanishing

quadrupole interactions. (b) Resonance strength from the 1s exciton

(spectral position is indicated in (a)) as a function of θ (solid) and φ

(dashed), when the respective other angle is �xed.

with the generalized oscillator strength

Fλ ≡ ∑
k

[ψR
λ (k)]⋆Fh,ke,k

, (3.57)

and E(ω) being the Fourier transform of E(t).
�e susceptibility follows from the macroscopic polarization (2.34),

P(ω) =∑
λ

Fλpλ(ω) , (3.58)

which yields the linear absorption

α(ω) = ω

є0
√
єBGc0

∑
λ

∣Fν ∣2 γλ(ω)(Eλ − ħω)2 + γ2λ(ω) , (3.59)

using Eq. (2.50).

Next, the near-band gap optical absorption of TiO2 is calculated via Eq. (3.59). �e electric and

magnetic �elds are assumed to be polarized within the x–y plane and their propagation direction


44 3 E�ects of Mass Anisotropy on the Optical Properties of Semiconductors

is given by the polar and azimuthal angles θ and φ, respectively. More speci�cally, it is

E(r) = eiqE ⋅reP , B(r) = eiqE ⋅r(qE × eP) , (3.60)

with the polarization

eP = xex + ye y , x2 + y2 = 1 , (3.61)

and the propagation vector

qE = ∣qE ∣(sin(θ) cos(φ)ex + sin(θ) sin(φ)e y + cos(θ)ez) . (3.62)

For the generalized oscillator strength (3.57), the wavefunctions are obtained by Eq. (3.18). Fig-

ure 3.14(a) shows the absorption for di�erent qE , uniquely identi�ed by the touple (θ , φ). If the
light propagates parallel to the z-axis (shaded area and solid line), the corresponding quadrupole

matrix elements have a vanishingly impact on the spectrum, since they are small for the applied po-

larization in the considered direction. Hence, the absorption follows almost entirely from electric

dipole interactions. However, these couple predominantly excitons with p±1 symmetry, which are

degenerate due to the symmetry of the system. �ey do not show a signi�cant angle dependency.

Figure 3.14(a) also shows angles for which the light propagates away from the x, y, or z planes

(dashed and dotted lines). Overall, the spectra are not drastically inuenced changing (θ , φ), only
a weak feature at the 1s ground state spectral position appears, a resonance which is dipole forbid-

den but quadrupole allowed.

To visualize that no propagation direction leads to a strong excitonic signal, Fig. 3.14(b) shows

the 1s-peak absorption as a function of θ or φ while the respective other angle is �xed. As a general

trend, whenever θ (φ) is varied, φ = π/4 (θ = π/2) yields the strongest 1s-related absorption. Even
using the optimal choice (π/2, π/4) only results in a weak resonance. �is is a consequence of the

extraordinary small binding energy of Ẽ1s = −0.5meV (Ẽ2p±1 = −0.1meV) for the 1s (2p±1) exciton
due to the unusually large dielectric constants.

�e experimental spectra presented in Refs. [126–128] show a distinct resonance 4meV below

the band gap, which is not reproduced in Fig. 3.14. Di�erent conclusions have been drawn from

the various authors about the origin of this resonance. While Pascual et al. claim that the excitonic

signatures possess dipole forbidden but quadrupole allowed 1s symmetry, Amtout and Leonelli

assign it to a weakly dipole allowed isotropic 2p state. To resolve this issue, a closer look at the

dielectric constant is required.

When the band structure is obtained from ab-initio calculations, it is not straightforward to de-

�ne wether the low or high frequency, or even some phenomenological dielectric constant between

these should be used when the excitonic response is modeled [130]. �is is particularly important

if the high and low frequency components of the dielectric constant di�er on a huge scale like in


3.3 Absorption of Bulk Rutile 45

Ref. [128]

ħω − Eg [meV]

A
b

so
rp

ti
o

n
[c

m
−

� ] ��.�, (�, �)
��.�, (�, �)

(a)

��.�, (π/�, π/�)

×�/��

0

10

20

30

−6 −4 −2 0 2

Ref. [126]

Ref. [127]

ħω − Eg [meV]

A
b

so
rp

ti
o

n
[c

m
−

� ]

(b)

��.�, (π/�, π/�)

0

25

50

−6 −4 −2 0 2

Figure 3.15— Experimental and theoretical absorption spectra. The computed

spectra are characterizedby a set of є and (θ , φ) containing the dielec-
tric constantє(v)according toEq. (3.62)andpropagationangles. To re-
produce the experimental data, propagation according to (π/9, π/4)
and a dielectric constant of 47.8 is necessary (blue line). Other com-

binations do not reproduce the resonance at −4meV ormisevaluate

its strength. No experimental data for a small region around −2meV

is provided as a consequence of the detection method.

TiO2. Here, the high-frequency components are є∞⊥ = 6.8 and є∞∥ = 8.4, while the low-frequency
constants are є⊥ = 111 and є∥ = 257. A pragmatic approach to cope with this issue is to obtain the

dielectric constant as a linear interpolation between the low- and high-frequency limit according

to

є̃∥/⊥(v) = є∥/⊥ − v(є∥/⊥ − є∞∥/⊥) , v ∈ [0, 1] , (3.63)

introducing the directionally averaged dielectric constant via є(v) = [є̃2∥(v)є̃⊥(v)]
1
3 . Tuning є(v)

in turn allows the adjustment of the exciton binding energies. Selecting either є(v) = 24.3 or

є(v) = 47.8, the 2p±1 or 1s binding energy coincides with the experimentally observed signature

4meV below Eg.

To study the implications of Eq. (3.62) on the absorption, Fig. 3.15(a) shows a comparison of

α(ω) for di�erent sets of dielectric constants є(v) and propagation angles (solid lines) to the ex-

perimental data from Ref. [128] (shaded area). Although for є(v) = 24.3 and propagation along

the z-axis (black line), a resonance 4meV below the band gap is clearly visible, the total scale and

the continuum absorption is not successfully reproduced. In general, only changing the dielectric

constants does not generate a signature at the appropriate spectra position while simultaneously


46 3 E�ects of Mass Anisotropy on the Optical Properties of Semiconductors

yielding a satisfactory agreement to the high-energy absorption tail. As an example, the absorp-

tion for є(v) = 47.8 and (θ = 0, φ = 0) is shown as a red line in Fig. 3.15. Moreover, it is di�cult to

reproduce the experimentally observed energetic position using values di�erent from є(v) = 47.8.
At the same time, the angles determine the strength of the resonance and in general, several com-

binations (θ , φ) can be found to reproduce the overall shape of the absorption, e.g. the choice

є(v) = 47.8, θ = π/9, and φ = π/4 (blue line) provides excellent agreement with the experiment.

Next, the theoretically obtained spectra are compared the experiments fromRefs. [126] and [127].

Again, the best agreement between experiments and theory can be obtained with є(v) = 47.8 and(π/9, π/4) (blue line). Additionally, є(v) = 47.8 and (π/2, π/4) (orange line) is shown, which
overestimates the 1s resonance strength. �e continuum absorption settles in between the di�er-

ent experiments in both cases. Generally, it is only possible to reconstruct the excitonic resonance

under the assumption that its origin is the 1s exciton. In other words, є(v) = 47.8 is a prerequisite.
To conclude this Section, a combination of DFT and the CE approach is presented and applied

to calculate the near-band gap optical absorption of rutile. For this purpose, it is necessary to in-

clude electric dipole and quadrupole interactions. �e corresponding matrix elements and single-

particle energies are provided by DFT calculations. Since the band structure is highly parabolic

with strong mass anisotropy, it has been replaced using the e�ective-mass approximation, allow-

ing to obtain the wavefunctions as described in Section 3.1.1. �ose can be used to calculate the

optical absorption, which reveals that the quadrupole interaction in rutile is highly polarization

and propagation-direction dependent. Experimentally available excitonic signatures cannot be re-

produced by omitting quadrupole interactions. In contrast, a good agreement between experiment

and theory can be obtained including quadrupole interactions via a tuning of the dielectric con-

stant and propagation angles. �e origin of