Exciton-Phonon Coupling in Monolayers
of T ransition Metal Dichalcogenides
v or gele gt v on
M. Sc. Malte Selig geb . in Prenzlau
v on der F akult ¨
at II - Mathematik und Naturwissenschaften
der T echnischen Uni v ersit ¨
at Berlin
zur Erlangung des akademischen Grades
Doktor der Naturwissenschaften
- Dr . rer . nat. -
genehmigte Dissertation
Promotionsausschuss:
V orsitzender: Prof. Dr . Stephan Reitzenstein
1. Gutachter: Prof. Dr . Andreas Knorr
2. Gutachterin: Jun-Prof. Dr . Doris Reiter
T ag der wissenschaftlichen Aussprache: 18.09.2018
Berlin 2018
F ¨
ur P eter
ii
Abstract
Monolayers of transition metal dichalcogenides (TMDs) attracted much attention in
recent research due to their promising optical and electronic properties for future tech-
nological applications. In this thesis, an e xcitonic description within the Heisenberg
equation of motion formalism for the optical response of these semiconducting, ultra-
thin materials is de veloped. Hereby the main focus lies on the interaction of e xcitons
with phonons.
The de veloped model is e xploited to compute the homogeneous line width of excitons
which is found to be in the order of some tens of meV . F or instance, the line width
increases from about 10 meV at 0 K to about 25 meV at room temperature in WS 2 .
The main relaxation pathway in tungsten based materials is the relaxation to e xcitonic
states at the Λ v alley ener getically belo w the optical bright state. Since these processes
are mediated by Λ phonon emission, they contrib ute to the coherence lifetime e ven at
v ery low temperatures. Additionally , the homogeneous line width in bilayer WS 2 w as
found to e xceed the monolayer width by at least 30 meV . This dif ference was attrib uted
to a changed e xciton dispersion, making the relaxation with M phonons predominant.
Both observ ations are in excellent agreement with e xperimental results.
W ithin this model, the impact of non-Mark ovian e xciton-phonon coupling to the exci-
tonic lineshape in the absorption spectrum is in v estigated. On the one hand emission
and absorption of acoustic phonons gi ves rise to the formation of a pronounced shoul-
der of the high ener gy side of the main excitonic line. On the other hand emission
of optical phonons leads to the formation of a sideband well abo ve the e xcitonic line.
Furthermore, a pronounced polaron red shift is found.
The de veloped model is further applied to describe the phonon mediated thermalization
and photoluminescence of e xcitons after resonant optical excitation. As a highlight,
dark states in some TMD materials lead to a quenching of the luminescence intensity
at lo w temperatures and an increasing luminescence intensity as a function of temper-
ature. In molybdenum based materials, the bright state is the ground state, resulting
in a decreasing luminescence yield as a function of temperature. This is in line with
observ ations from recent experimental studies.
Additionally , the impact of these lo w lying dark states on the interv alley e xchange cou-
pling is in v estigated. In the literature, the interv alle y e xchange coupling is predicted to
be the dominant relaxation mechanism for optically injected spins. Here a quenching
of this relaxation due to thermalization in lo w lying dark states is illustrated.
Last, a heterostructure of a TMD layer and a graphene layer is considered. The F ¨
orster
induced transition rate of optically generated e xcitons from the TMD to the graphene
layer is computed microscopically and is found to be in the order of 1 ps for a closely
stacked heterostructure. This v alue coincides nicely with recent e xperimental findings.
All in all, the de veloped model leads to man y prediction related to the exciton-phonon
coupling in TMD monolayers which are in o verall in nice agreement with e xperimental
studies.
Zusammenfassung in deutsc her Sprac he
Monolagen v on ¨
Uber gangsmetall Dichalcogeniden (TMDs) haben aufgrund ihrer viel-
v ersprechenden optischen und elektronischen Eigenschaften viel Aufmerksamkeit in
den letzten acht Jahren generiert. In dieser Doktorarbeit wird eine exzitonische Theo-
rie innerhalb des Heisenber g Formalismus entwick elt, um die optische Antwort dieser
ultrad ¨
unnen Materialien zu beschreiben. Der Schwerpunkt dabei lie gt auf der Behand-
lung der Exziton-Phonon-W echsel wirkung.
Das entwickelte Modell wird ange wendet um die homogene Linienbreite v on Exzi-
tonen zu berechnen. Diese befindet sich in der Gr ¨
oßenordnung v on einigen 10 meV
f ¨
ur alle untersuchten Materialien. Beispielsweise steigt sie in WS 2 von etw a 10 meV
bei 0 K auf 25 meV bei Raumtemperatur . Interessanterweise dominiert die Relaxation
in dunkle, niederener getische Zust ¨
ande am Λ Punkt. Selbst bei ultrakalten T empera-
turen tragen diese Prozesse signifikant zur Linienbreite bei, da sie Phononemissions
getrieben sind. W eiterhin wurde die Linienbreite in der Bilage WS 2 untersucht. Diese
ist mindestens um 30 meV gr ¨
oßer als in der Monolage, was auf eine v er ¨
anderte Band-
struktur zur ¨
uckzuf ¨
uhren ist. W ¨
ahrend in der Monolage die Relaxation mit Λ Phononen
dominiert, streuen Exzitonen in der Bilage ¨
außerst ef fizient mit M Phononen. Die
pr ¨
asentierten theoretischen Er gebnisse sind in exzellenter ¨
Ubereinstimmung mit e x-
perimentellen Resultaten.
Eine nicht-Marko vsche Beschreib ung des Einflusses der Exziton-Phonon W echsel-
wirkung zeigt die F ormierung v on ausgepr ¨
agten Phononseitenbanden. Beispielsweise
f ¨
uhren Emission und Absorption v on akustischen Phononen zur Formierung einer aus-
gepr ¨
agten Schulter ¨
uberhalb der e xzitonischen Resonanz. Emission v on optischen
Phonon sor gt dar ¨
uber hinaus zur Auspr ¨
agung einer starken Seitenbande einige 10 meV
oberhalb der e xzitonischen Resonanz.
Das Modell wird weiterhin genutzt, um die Phonon-vermittelte Thermalisierung und
Photolumineszenz nach optischer Anre gung zu untersuchen. In e xzellenter ¨
Uberein-
stimmung mit e xperimentellen Studien, steigt die Lumineszenzintensit ¨
at in W olfram
basierten Materialien mit der T emperatur , was auf die niederener getischen dunklen
Zust ¨
ande zur ¨
uckzuf ¨
uhren ist. In Molybd ¨
an basierten Materialien hinge gen f ¨
allt die
Intensit ¨
at mit steigender T emperatur , da hier der Grundzustand der optisch akti ve Zu-
stand ist.
W eiterhin ist der Einfluss der dunklen Zust ¨
ande auf die Interv alley Exchange K op-
plung, dem dominanten Spin Relaxations Mechanismus, Ge genstand der Untersuchung.
Hier stellt sich heraus, dass die Spin Relaxation durch die Anwesenheit dunkler
Zust ¨
ande stark unterdr ¨
uckt wird.
Zuletzt wird eine Heterostruktur bestehend aus einem TMD und Graphen untersucht.
Mikroskopisch wird die F ¨
orstertransferrate v on TMD Exzitonen ins Graphen berech-
net. Diese betr ¨
agt etwa 1 ps f ¨
ur direkt aufeinanderlie gende Monolagen gefunden, was
in guter ¨
Ubereinstimmung mit j ¨
ungsten e xperimentellen Resultaten ist.
iv
Pub lications
Publications in P eer -Re viewed Journals
• Florian Katsch, Malte Selig , Alexander Carmele and Andreas Knorr ”Theory of
Exciton-Exciton Inter actions in Monolayer T r ansition Metal Dic halcog enides” ,
accepted for publication in physica status solidi (b), 1800185 (2018)
• Archana Raja, Malte Selig , Gunnar Ber gh ¨
auser , Jaeeun Y u, Heather M. Hill,
Albert Rigosi, Louis E. Brus, Andreas Knorr , T on y F . Heinz, Ermin Malic and
Ale xe y Chernikov ”Enhancement of Exciton-Phonon Scattering fr om Mono-
layer to Bilayer WS 2 ” , Nano Letters 18 (10), 6135 (2018)
• Jessica Lindlau, Malte Selig , Andre Neumann, Leo Colombier , Jonghw an Kim,
Gunnar Ber gh ¨
auser , Feng W ang, Ermin Malic and Ale xander H ¨
ogele ”The r ole
of momentum-dark e xcitons in the elementary optical r esponse of bilayer WSe 2 ” ,
Nature Communications 9 , 2586 (2018)
• Samuel Brem, Malte Selig , Gunnar Bergh ¨
auser and Ermin Malic ”Exciton Re-
laxation Cascade in two-dimensional T ransition Metal Dic halcog enides” , Sci-
entific Reports 8 (1), 8238 (2018)
• Malte Selig , Gunnar Berg ¨
auser , Marten Richter , Rudolf Bratschitsch, Andreas
Knorr and Ermin Malic, ”Dark and bright exciton formation, thermalization and
photoluminescence in monolayer tr ansition metal dichalcog enides” , 2D Materi-
als 5 , 035017 (2018)
• Iris Niehues, Robert Schmidt, Matthias Dr ¨
uppel, Philipp Marauhn, Dominik
Christiansen, Malte Selig , Gunnar Bergh ¨
auser , Daniel W igger , Robert Schnei-
der , Lisa Braasch, Rouven K och, Andres Castellanos-Gomez, T ilmann Kuhn,
Andreas Knorr , Ermin Malic, Michael Rohlfing, Stef fen Michaelis de V ascon-
cellos, and Rudolf Bratschitsch ”Strain Contr ol of Exciton-Phonon Coupling in
Atomically Thin Semiconductors” , Nano Letters 18 (3), 1751-1757 (2018)
• Maja Feierabend, Gunnar Bergh ¨
auser , Malte Selig , Samuel Brem, T imur She-
gai, Siegfried Eigler and Ermin Malic ”Molecule signatur es in photolumines-
cence spectr a of transition metal dic halcogenides” , Physical Re vie w Materials
2 , 014004 (2018)
• Iv an Da vid Bernal V illamil, Gunnar Ber gh ¨
auser , Malte Selig , Iris Niehues, Robert
Schmidt, Robert Schneider , Philipp T onndorf, Paul Erhart, Stef fen Michaelis de
V asconcellos, Rudolf Bratschitsch, Andreas Knorr and Ermin Malic ”Exciton
br oadening and band r enormalization due to Dexter -like intervalle y coupling” ,
2D Materials 5 (2), 025011, (2018)
v
• Philipp Steinleitner , Philipp Merkl, Alexander Graf, Philipp Nagler , K enji W atan-
abe, T akashi T aniguchi, Jonas Zipfel, Christian Sch ¨
uller , T obias K orn, Alex ey
Cherniko v , Samuel Brem, Malte Selig , Gunnar Ber gh ¨
auser , Ermin Malic, and
Rupert Huber ”Dielectric Engineering of Electr onic Corr elations in a van der
W aals Heter ostructur e” , Nano Letters 8 (2), 1402-1409 (2018)
• Ermin Malic, Malte Selig , Maja Feierabend, Samuel Brem, Dominik Chris-
tiansen, Florian W endler , Andreas Knorr and Gunnar Ber gh ¨
auser ”Dark e xci-
tons in tr ansition metal dichalcog enides” , Physical Re vie w Materials 2 , 014002
(2018)
• Dominik Christiansen, Malte Selig , Gunnar Bergh ¨
auser , Robert Schmidt, Iris
Niehues, Robert Schneider , Ashish Arora, Stef fen Michaelis de V asconcellos,
Rudolf Bratschitsch, Ermin Malic, and Andreas Knorr ”Phonon Sidebands in
Monolayer T ransition Metal Dic halcog enides” , Physical Re vie w Letters 119 ,
187402 (2017)
• Malte Selig , Gunnar Bergh ¨
auser , Archana Raja, Philipp Nagler , Christian Sch ¨
uller ,
T on y F . Heinz, T obias K orn, Alex ey Cherniko v , Ermin Malic and Andreas Knorr
”Excitonic line width and coher ence lifetime in monolayer transition metal dic halco-
genides” , Proc. SPIE 10102, Ultrafast Phenomena and Nanophotonics XXI,
101021F (2017)
• Malte Selig , Gunnar Bergh ¨
auser , Archana Raja, Philipp Nagler , Christian Sch ¨
uller ,
T on y F . Heinz, T obias K orn, Alex ey Cherniko v , Ermin Malic and Andreas Knorr
”Excitonic line width and coher ence lifetime in monolayer transition metal dic halco-
genides” , Nature Communications 7 13279,(2016)
• Robert Schmidt, Gunnar Ber gh ¨
auser , Robert Schneider , Malte Selig , Philipp
T onndorf, Ermin Malic, Andreas Knorr , Stef fen Michaelis de V asconcellos, and
Rudolf Bratschitsch ”Ultr afast Coulomb-Induced Intervalle y Coupling in Atom-
ically Thin WS2” , Nano Letters, 16 , 2945 (2016)
Submitted Manuscripts
• Zahra Khatibi, Maja Feierabend, Malte Selig , Samuel Brem, Christopher Lin-
der ¨
alv , Paul Erhart, Ermin Malic ”Impact of str ain on the excitonic line width in
tr ansition metal dichalcog enides” , arXi v:1806.07315 (2018)
• Simon Ov esen, Samuel Brem, Christopher Linder ¨
alv , Mikael K uisma, Paul Er -
hart, Malte Selig , Ermin Malic ”Interlayer exciton dynamics in van der W aals
heter ostructur es” , arXi v:1804.08412 (2018)
vi
• Malte Selig , Ermin Malic, Kwang Jun Ahn, Norbert K och and Andreas Knorr
”Theory of optically induced F ¨
orster Coupling in van der W aals coupled Het-
er ostuctur es” , (2018)
Conf erence Contrib utions
• Contrib uted T alk, ”Exciton Dynamics in van der W aals Heter ostructur es and two
dimensional T ransition Met al Dic halcogenides” , Flatlands Beyond Graphene,
Leipzig 2018
• In vited T alk, ”Impact of dark states on e xcitonic spectra of tr ansition metal
dic halcogenides” , CPQC18, Dresden 2018
• Contrib uted T alk, ”Lifetime of V alle y Excitons in Monolayer T r ansition Metal
Dic halcogenides” , Spring Meeting of the German Physical Society , Berlin 2018
• In vited T alk (in representation for Andreas Knorr), ”Exciton based description
of atomically thin materials: Optical lineshape , intervalle y coupling and lumi-
nescence dynamics” , 5th International W orkshop on the Optical Properties of
Nanostructures, M ¨
unster 2018
• Poster Presentation ”Exciton dynamics in atomically thin 2D materials” , Flat-
lands be yond Graphene, Lausanne (Switzerland) 2017
• In vited T alk, ”Influence of dark states on e xcitonic spectra of tr ansition metal
dic halcogenides” , Spring Meeting of the German Physical Society , Dresden
2017
• In vited T alk, ”Interplay of Bright and Dark Excitons in Monolayer T ransition
Metal Dic halcogenides” , Optics Seminar at the Stanford Univ ersity , Stanford
(California) 2017
• Contrib uted T alk, ”Excitonic linewidth and coher ence lifetime in transition metal
dic halcogenides” , Photonics W est, San Francisco (California) 2017
• Poster Presentation, ”Excitonic line width and coher ence lifetime in transition
metal dic halcogenides” , Nonlinear Optics and Excitation Kinetics in Semicon-
ductors NOEKS 13, Dortmund 2016
• Contrib uted T alk, ”Micr oscopic modeling of the homogeneous line width in ab-
sorption spectr a of TMDs” , Spring Meeting of the German Physical Society ,
Re gensbur g 2016
vii
Contents
1 Intr oduction 2
2 Theoretical Basics 5
2.1 T ransition Metal Dichalcogenides . . . . . . . . . . . . . . . . . . . 5
2.2 Man y Particle Hamiltonian in Second Quantization . . . . . . . . . . 8
2.2.1 Schr ¨
o d i n g e r E q u a t i o n ...................... 8
2.2.2 Canonical Quantization . . . . . . . . . . . . . . . . . . . . . 10
2 . 3 O b s e r v a b l e s ............................... 1 7
2.3.1 Absorption Coefficient . . . . . . . . . . . . . . . . . . . . . 17
2.3.2 Photoluminescence Intensity . . . . . . . . . . . . . . . . . . 19
3 System Hamiltonian 21
3.1 Man y Particle Fermion Hamiltonian . . . . . . . . . . . . . . . . . . 21
3.1.1 Free Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . 22
3.1.2 Classical Carrier-Light Coupling . . . . . . . . . . . . . . . . 24
3.1.3 Quantized Carrier-Light Coupling . . . . . . . . . . . . . . . 24
3.1.4 Carrier -Carrier Coupling . . . . . . . . . . . . . . . . . . . . 25
3.1.5 Carrier -Phonon Coupling . . . . . . . . . . . . . . . . . . . . 28
3.2 De velopment of the Excitonic Hamiltonian . . . . . . . . . . . . . . . 31
3.2.1 Exciton Operators . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2.2 W annier Equation and Exciton Bandstructure . . . . . . . . . 37
3.2.3 Numerical Evaluation of the W annier Equation . . . . . . . . 39
3 . 2 . 4 O b s e r v a b l e s ........................... 4 0
3.3 Man y Particle Exciton Hamiltonian . . . . . . . . . . . . . . . . . . . 41
3.3.1 Free Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . 41
3.3.2 Exciton-Light Coupling . . . . . . . . . . . . . . . . . . . . . 43
3.3.3 Quantized Exciton-Light Coupling . . . . . . . . . . . . . . . 44
3.3.4 Exciton-Exciton Coupling . . . . . . . . . . . . . . . . . . . 44
3.3.5 Exciton-Phonon Coupling . . . . . . . . . . . . . . . . . . . 46
4 Linear Spectr oscopy and Excitonic Line width 48
4.1 Excitonic Bloch Equation and Elliott Formula . . . . . . . . . . . . . 48
4.2 Excitonic Linewidth in the Monolayer . . . . . . . . . . . . . . . . . 51
viii
Contents
4.2.1 Radiati ve Broadening . . . . . . . . . . . . . . . . . . . . . . 51
4.2.2 Non-Radiati ve Broadening . . . . . . . . . . . . . . . . . . . 53
4.2.3 Numerical Ev aluation of the Non-Radiati ve Broadening . . . . 56
4.2.4 Excitonic Line width in WS 2 and MoSe 2 ............ 5 7
4 . 2 . 5 C o n c l u s i o n ........................... 6 1
4.3 Excitonic Linewidth in the Bilayer . . . . . . . . . . . . . . . . . . . 62
4 . 3 . 1 C o n c l u s i o n ........................... 6 4
4 . 4 P h o n o n S i d e b a n d s ............................ 6 5
4.4.1 Non-Marko vian T reatment of the Exciton-Phonon Scattering . 65
4.4.2 Excitonic Lineshape in MoSe 2 and WS 2 ............ 6 8
4 . 4 . 3 C o n c l u s i o n ........................... 7 2
5 Exciton Dynamics and Photoluminescence 73
5.1 Exciton Boltzmann Scattering Equations . . . . . . . . . . . . . . . . 73
5.1.1 Deri v ation of the Equations of Motion . . . . . . . . . . . . . 74
5.1.2 Numerical Evaluation of the Boltzmann equation . . . . . . . 80
5.2 F ormation and Thermalization of Excitons . . . . . . . . . . . . . . . 82
5.2.1 Momentum Resolv ed Dynamics in WSe 2 at 77 K . . . . . . . 84
5.2.2 Momentum Resolv ed Dynamics in MoSe 2 at 77 K . . . . . . . 86
5.2.3 V alle y Resolv ed Dynamics at 77 K and 300 K . . . . . . . . . 86
5 . 2 . 4 C o n c l u s i o n ........................... 8 9
5.3 Photoluminescence of Thermalized Excitons and Quantum Y ield . . . 90
5 . 3 . 1 C o n c l u s i o n ........................... 9 3
5.4 Intrinsic V alley Lifetime . . . . . . . . . . . . . . . . . . . . . . . . 94
5.4.1 Interv alley coupling in MoSe 2 ................. 9 6
5.4.2 Interv alley coupling in WSe 2 .................. 9 8
5.4.3 De gree of Polarization and V alle y Lifetime . . . . . . . . . . 101
5 . 4 . 4 C o n c l u s i o n ........................... 1 0 4
6 F ¨
or ster Coupling in van der W aals coupled Heterostructures 105
6 . 1 T h e o r e t i c a l M o d e l ............................ 1 0 6
6 . 2 R e s u l t s .................................. 1 0 9
6 . 3 C o n c l u s i o n ............................... 1 1 2
7 Summary and Outlook 113
A Appendix 115
A.1 Exfoliation and Spectroscopic In v estigation of Monolayer WSe 2 . . . 115
A . 2 M a t r i x E l e m e n t s ............................. 1 1 6
A.2.1 Classical Carrier-Light Coupling . . . . . . . . . . . . . . . . 116
A.2.2 Quantized Carrier-Light Coupling . . . . . . . . . . . . . . . 118
A.2.3 Carrier -Carrier Coupling . . . . . . . . . . . . . . . . . . . . 120
A.2.4 Carrier -Phonon Coupling . . . . . . . . . . . . . . . . . . . . 122
ix
Contents
A.2.5 F ¨
orster Coupling Element in a TMD Graphene Heterostructure 124
A . 3 I n t e g r a l s ................................. 1 2 8
A.3.1 Dipole-Dipole Interaction . . . . . . . . . . . . . . . . . . . . 128
A . 4 P a r a m e t e r s ................................ 1 2 9
A.4.1 Uni versal Constants . . . . . . . . . . . . . . . . . . . . . . . 129
A.4.2 Lattice Structure . . . . . . . . . . . . . . . . . . . . . . . . 129
A.4.3 Electronic Bandstructure . . . . . . . . . . . . . . . . . . . . 130
A.4.4 Excitonic Bandstructure . . . . . . . . . . . . . . . . . . . . 131
A.4.5 Phonon Dispersion . . . . . . . . . . . . . . . . . . . . . . . 132
A.4.6 Electron-Phonon Coupling Strength . . . . . . . . . . . . . . 133
Ac knowledgement 148
x
W enn ich es eilig habe , gehe ich ganz langsam.
Unbekannt
1. Intr oduction
The disco very of graphene by A. K. Geim and K. No v oselov in 2004 w as a milestone
in the research of quantum confinement, since it represents the lo wer limit for two
dimensional structures as an atomically thin layer [1, 2] and was a warded with the
nobel prize in 2010 [3]. 1 Graphene is examplarily f abricated by micromechanical
clea v age (commonly known as Scotch tape method) from b ulk graphite [1], which
is possible since graphite as a v an der W aals material consists of indi vidual layers of
graphene which are bond due to v an der W aals forces [5]. By means of its extraordinary
properties such as high mechanical stability and fle xibility [6, 7], high thermal and
electric conducti vity [8, 9, 10, 11] and lo w absorption in the visible range [12, 13], it
became a promising candidate for optoelectronic de vices [14]. From the theoretical
vie wpoint graphene was v ery interesting since its band extrema, which are located
at the corners of the hexagonal Brillouin zone, are Dirac cones e xhibiting a linear
dispersion and v anishing bandgap [15, 16]. This opens possiblities for ne w physical
phenomena such as carrier multiplication due to Auger scattering of carriers [17, 18].
In 2010 another class of atomically thin materials, namely the monolayers of transi-
tion metal dichalcogenides (TMDs), attracted much attention since the y were found
to be direct band gap semiconductors with strong light matter interaction [19, 20].
Comparable to graphite, bulk transi tion metal dichalcogenides are v an der W aals ma-
terials [21] opening the possibility to fabricate them from micromechanical clea vage
[22, 23]. Other common f abrication techniques are growth by chemical v apor deposi-
tion [24, 25, 26] and v an der W aals epitaxy [27, 28]. While being indirect semicon-
ductors in the b ulk and in the multilayer , monolayers are direct semiconductors with a
band gap in the visible range, which is at the K points at the corners of the he xagonal
Brillouin zone [29, 30, 31, 32, 33, 34, 35, 36]. Due to a pronounced spin orbit interac-
tion the v alence bands are split into two spin bands with ener getic separation of 0.2 eV
in molybdenum based and 0.4 eV in tungsten based TMDs. The conduction band is
split by some tens of meV [33, 34, 35, 36, 37, 38]. This leads to the apperance of
two optical transitions where the ener getically lo wer is referred as A transition and the
ener getically higher as B transition in the literature. F ollo wing from the time-re versal
symmetry , the K and the K 0 points are non-equi valent leading to opposite spins in the
ener getically similar bands and both high symmetry points [33, 34, 37, 39]. TMDs
possess an e xtraordinary strong light matter interaction, leading to an absorption in the
1 As a funfact, A. K. Geim also recei ved the ig nobel prize in 2000 for bringing a li ving frog to le vitate
in a magnetic field [4]. He is the only person who recei ved both, the nobel prize and the ig nobel
prize.
order of 10 % in the visible range, which is ev en more impressi ve when considering
that the monolayers are less than 1 nm thick [19, 20, 40, 41, 42, 43, 44]. TMDs e xhibit
a pronounced v alley selecti ve circular dichroism, meaning that transitions at the K v al-
le y are excited with left handed polarized light and transitions at the K 0 are addressed
with right handed polarized light [37, 45, 46, 47, 48, 49]. This results in spin selecti ve
e xcitation of electrons under resonant excitation of the lo west lying optical transition.
Due to their e xciting optical properties TMDs became promissing candidates for fu-
ture applications [50, 51]. The y are fa vorable for the production of electronic de vices
such as transistors [52, 53], optoelectronic de vices such as phototransistors [54] and
photodetectors [55], gas sensing de vices [56] and ener gy storage device [57]. Through
their ultra thin nature, the screening of the Coulomb interaction is strongly suppressed
in TMDs [58, 59, 60], which leads to the formation of strongly bound electron hole
pairs, so called excitons [61, 62, 63, 64]. Besides bright excitons, which are accessible
through optical spectroscopy , TMDs possess exciton states which are optically dark
either due to spin selection rules [65, 64, 66, 67, 68, 69] or an exciton momentum well
abo ve the light cone [65, 64, 66, 70]. Some of these dark states are located belo w
the optical bright state. Therefore they are e xpected to strongly influence the optical
and electronic properties of TMDs. As the most intuiti ve relaxation mechanism, the
interaction of e xcitons with lattice vibrations, phonons, is crucial for intrinsic proper-
ties such as the coherence lifetime [43, 71, 72] and the thermalization of e xcitons and
electrons [73, 74, 75].
Scope of this Thesis
The scope of this thesis is the de velopment of an e xcitonic frame work within the
Heisenber g equation of motion formalism for TMDs. Here, in the main focus is
the e xciton-phonon coupling. The de veloped frame work is applied to in vestig ate the
impact of e xciton-phonon coupling to the coherence lifetime and the homogeneous
line width in TMDs. Here, a particular focus is the in vestigation of the influence of
dark states on the line width. Further the phonon mediated e xciton thermalization and
the photoluminescence is studied in the de velop ed e xcitonic frame work. It will be of
special interest, ho w dark states influence the emission properties of TMDs. Finally ,
as already mentioned e xcitons can be excited spin selecti vely in TMDs. F or technical
applications the lifetime of such spins is of crucial interest. Therefore, in the last part
connected to e xciton-phonon coupling, the most common spin relaxation mechanism,
the interv alley e xchange coupling, is in vestig ated under influence of phonon mediated
e xciton relaxation in the appearance of dark exciton states.
1. Introduction
Structure of this Thesis
This thesis is structured as follo ws: In the second chapter , the material system and
the fundamental theoretical frame work in second quantization, is introduced. In the
third chapter , the electronic many particle Hamiltonian including all rele v ant coupling
elements is introduced and the transformation to an e xcitonic Hamiltonian is performed
in the follo wing. The last three chapters discuss applications of the de veloped model.
In the first, the coherence lifetime is in vestigated for monolayer as well as for bilayer
TMDs. Here, radiati ve decay and e xciton-phonon coupling are considered as most
prominent microscopic mechanisms. In the next chapter , the phonon mediated exciton
formation and thermalization is studied. Here, first a study within considering the
spin de gree of freedom is presented. Thereafter , spin resolved e xciton dynamics after
helical optical e xcitations are considered. The last chapter deals with a heterostructure
consisting of a TMD and a graphene layer . Here, the relaxation of TMD excitons to the
graphene layer under influence of interlayer F ¨
orster interaction is of particular interest.
2. Theoretical Basics
In this chapter , the basic principles which are required for the detailed understanding
of this thesis are gi ven. In the first section, the considered material system, the mono-
layers of transition metal dichalcogenides are introduced and their lattice structure is
discussed. In the second section, the fundamental many particle Schr ¨
odinger equation
including the interaction of carriers with carriers and ions in the material and with op-
tical light fields is established. In the follo wing it is transformed into a more ef ficient
description which is called the second quantization. Therefore the general procedure
of the canonical field quantization is illustrated. Last, the most important experimental
observ ables which are needed throughout this thesis are defined within the frame work
of the second quantization.
2.1. T ransition Metal Dic halcog enides
In this section, the considered material system, namely monolayers of transition metal
dichalcogenides (TMDs) are introduced. In the bulk, TMDs are layered materials,
meaning that the y consist of monolayers which are stacked on top of each other [34].
Strong co valent bonds are responsible for the bonding within the monolayers whereas
relati vely weak v an der W aals interactions lead to interlayer bondings [21]. As a result
monolayers of TMDs can be easily fabricated by e xfoliation from the b ulk crystals, for
e xample by exploiting the scotch tape method [22, 23].
As illustrated in figure 2.1, TMDs e xhibit a hexagonal hone ycomb lattice, where tran-
sition metal atoms are located at the one site and tw o chalcogen atoms are located
abo ve each other on the other site. The most common TMD materials are formed by
molybdenum or tungsten as transition metal atoms and sulfur or selenium as chalcogen
atoms. Also materials with dif ferent chalcogen atoms, which are called Janus materi-
als, ha v e been already fabricated [76]. TMDs can be described by a hexagonal lattice
with a three atomic basis. The corresponding fundamental lattice v ectors r 1 and r 2 are
gi ven by
r 1 = a 0
2 √ 3 e x + e y (2.1)
r 2 = a 0
2 √ 3 e x − e y , (2.2)
where a 0 denotes the lattice constant, which is the distance between two transition
metal atoms, cf. figure 2.1 (a). T ypically , the lattice constant is on the order of
2. Theoretical Basics
Figure 2.1.: Schematic illustration of the lattice structur e in real space. T op vie w (a) and side
vie w (b) of the lattice structure of transition metal dichalcogenide monolayers. Blue
dots denote chalcogen atoms and red atoms denote transition metal atoms. The lattice
vectors R i address the dif ferent unit cells. a 0 denotes the lattice constant and d 0 the
vertical distance between tw o chalcogen atoms.
a 0 = 0.3 nm for the most common TMD materials [21, 77, 78, 34]. A proper choice
of the unit cell is then the parallelogram which is spanned by the two principle lattice
v ectors. The area of the unit cell can be computed as the absolute value of the cross
product of the lattice v ectors
A uc = | r 1 × r 2 | = √ 3
2 a 2
0 . (2.3)
All lattice v ectors can be obtained by R i = n i
1 r 1 + n i
2 r 2 , with n i
1 and n i
2 being inte ger
numbers. The v ectors pointing from the chalcogen atoms to their next neighbors, cf.
figure 2.1 (a), are defined with respect to the fundamental lattice v ectors
b 1 = 1
3 ( r 1 + r 2 ) , (2.4)
b 2 = 1
3 ( r 1 − 2 r 2 ) , (2.5)
b 3 = 1
3 ( − 2 r 1 + r 2 ) . (2.6)
The lattice is symmetric under rotation around the angle π
3 and mirror symmetric
with respect to the x-y-plane. In contrast to other two dimensional materials such
as graphene, TMDs exhibit no in version symmetry . Monolayers of TMDs can also
be re garded as a layer of transition metal atoms, which is sandwiched by two layers
of chalcogen atoms. This is depicted in figure 2.1 (b). Here d 0 denotes the vertical
distance between the two chalcogen atoms which is typically about 0.6 nm [34].
The fundamental lattice v ectors of the reciprocal space are defined by
r i · g j =2 πδ ij . (2.7)
2.1. T ransition Metal Dichalcogenides
Figure 2.2.: Schematic illustration of the lattice structur e in recipr ocal space. The principle
lattice vectors g 1 and g 2 span the first Brillouin zone. The non-equiv alent high sym-
metry points K and K are located at the corners of the first Brillouin zone and the M
point is located between them.
All other reciprocal lattice vectors are gi ven by G i = n i
1 g 1 + n i
2 g 2 , with n i
1 and n i
2 being
inte ger numbers. As a result it holds the important relation exp ( i R i · G j )=1 ∀ i, j .
The fundamental reciprocal lattice v ectors read
g 1 = 2 π
a 0 1
√ 3 e x + e y (2.8)
g 2 = 2 π
a 0 1
√ 3 e x − e y . (2.9)
The high symmetry points of the reciprocal lattice are the K and K points at the
corners of the he xagonal Brillouin zone and the M point in the middle of them. Addi-
tionally the Λ and Λ points being located between the K / K and the Γ point will be of
certain interest.
The most rele v ant bands for the optical properties of a TMD, namely the valence band
and the conduction band, are formed from d orbitals of the transition metal atoms and
p orbitals of the chalcogen atoms. In particular the conduction band at the K point
and the v alence band at the Γ point are mainly formed by d z 2 orbitals of the transition
metal and the conduction band at the Λ point and the v alence band at the K point are
mainly formed by d x 2 − y 2 and d xy orbitals of the transition metal atoms. Additionally
at all rela v ant points, one finds a small impact of p x and p y orbitals of the chalcogen
atoms [34, 79].
2. Theoretical Basics
2.2. Man y P ar tic le Hamiltonian in Second
Quantization
In this section the fundamental man y particle Hamiltonian of electrons in a periodic
ion lattice including the interaction of the electrons with an e xternal light field is gi v en.
Thereafter the whole Hamiltonian is transformed into the second quantized v ersion,
where the quantized description of electrons, lattice vibrations (phonons) and the elec-
tromagnetic field (photons) is introduced.
2.2.1. Schr ¨
odinger Equation
Starting point for the in vestig ation is the Schr ¨
odinger equation of the electrons and
ions in a co valently bound crystal
H ( { R i } , { r i } )Ψ( { R i } , { r i } ) = E Ψ( { R i } , { r i } ) , (2.10)
with the man y particle wa ve function Ψ( { R i } , { r i } ) and the man y particle Hamilto-
nian H ( { R i } , { r i } ) . Here, the notation { r i }{ R i } indicates, that both, the Hamiltonian
and the wa vefunctions depend on the full set of all electronic r i and ionic R i coordi-
nates. The many particle Hamiltonian describing only the interaction of the solid state
system with itself is gi ven as
H ( { R i } , { r i } ) = X
i
P 2
i
2 M i
+ X
i
p 2
i
2 m
+ ( X
i,j
U ( r i − R j ) + X
i,j
V ( r j − r i ) + X
i,j
W ( R j − R i )) (2.11)
Here, the first line describes the kinetic energy of ions and electrons, P i = ~
i ∇ R i
denotes the momentum operator with the nabla operator ∇ R and ~ being the Planck
constant. m ( M i ) denote the electronic (ionic) masses. The second line describes the
Coulomb interaction between electrons and ions, electrons and electrons and ions and
ions with the respecti ve interaction potentials U( r ),V( r ) and W( r ). The mass of the
ions is typically some orders of magnitude lar ger than the electron mass. F ollo wing
the equipartition theorem the mean kinetic ener gy per degree of freedom in a macro-
scopic system is 1
2 k B T , where k B denotes the Boltzmann constant and T the absolute
temperature. As a result, the mean kinetic energies of ions and electrons are equal
which yields that the electrons mo ve much f aster than the ions. Therefore the kinetic
ener gy of the ions can be regarded as a pertubation in the lo west approximation [80]
H ( { R i } , { r i } ) = H 0 ( { R i } , { r i } ) + X
i
P 2
i
2 M . (2.12)
2.2. Many P article Hamiltonian in Second Quantization
H 0 describes the electron dynamics and only contains the ion coordinates { r i } as pa-
rameters, meaning that it describes the interaction of the electrons with fix ed ion coor-
dinates { R i } . Therefore the ion coordinates enter only as parameters in the electronic
wa vefunctions and the Schr ¨
odinger equation for the electrons in a fixed ion lattice can
be e v aluated for each configuration { R i }
H 0 ( { R i } , { r i } ) ψ α ( { R i } , { r i } ) = α ( { R i } ) ψ α ( { R i } , { r i } ) . (2.13)
Here α denotes a full set of the electronic quantum numbers. The wa vefunction of
the full system, equation 2.10, can be expanded after the electronic w av efunctions
Ψ( { R i } , { r i } ) = P α ξ α ( { R i } ) ψ α ( { R i } , { r i } ) . Inserting this e xpression in equation
2.10 yields the Schr ¨
odinger equation for the ions [80]
− X
R i
~ 2 ∇ 2
R i
2 M + α ( { R i } ) ! ξ α ( { R i } ) = E ξ α ( { R i } ) , (2.14)
where contrib utions of the order ∇ R i ( { R i } , { r i } ) and ∇ 2
R i ( { R i } , { r i } ) ha ve been ne-
glected. These can be sho wn to be at least a factor ( m
M ) 3
4 smaller than the electronic
ener gies [80]. They stem from the electron-ion interaction, and will be treated in a per -
turbation e xpansion to include interaction of the electrons with ion lattice vibrations in
the follo wing. The electronic eigenenergies α ( { R i } ) act as a ef fectiv e potential for the
ion dynamics, where all contributions to the bonding of the crystal are included. The
adiabatic elimination of the electron dynamics from the ion dynamics is commonly
kno wn as Born-Oppenheimer approximation [80]. Hence for the electronic dynamics,
only H 0 has to be considered.
The interaction of the electrons is assumed to be weak and the resulting lattice vibra-
tions are small. Hence it is suf ficient to linearize the electron ion interaction around
the equilibrium positions { R 0
i }
X
i,j
U ( r i − R j ) = X
i,j
U ( r i − R 0
j ) + X
i,j
u i · ∇ R j U ( r i − R j ) | R j = R 0
j , (2.15)
where the first term describes the interaction of the electrons with the fix ed ion lattice
and the second term describes the interaction of electrons with lattice vibrations. The
latter is re garded as a perturbation in the follo wing. u i denotes the de viations of the
ions from the equilibrium positions R 0
j .
The interaction of the electrons in the solid state system with an e xternal light field
can be introduced by considering a classical point char ge in an electro magnetic field.
Therefore the Lagrange function of a particle which is under influence of the Lorentz
force is considered
L ( r , ∂ t r ) = m
2 ( ∂ t r ) 2 + e∂ t r · A − e Φ . (2.16)
2. Theoretical Basics
Here A denotes the v ector potential of the electromagnetic field, Φ denotes the scalar
potential of the electromagnetic field and e denotes the elementary char ge. The scalar
and the v ector potential are defined from the electric field E and the magnetic field B
E = − ∂ t A − ∇ Φ (2.17)
B = ∇ × A . (2.18)
Ne xt the corresponding Hamilton function can be obtained as the Legendre transforma-
tion of the Lagrange function with respect to the generalized momentum p i = ∂ ∂ t r i L
H ( p ) = 1
2 m ( p − e A ) 2 + e Φ . (2.19)
T o e v aluated this e xpression first the e xpression for the momentum opetor p = ~
i ∇
is e xploited. The A 2 can be neglected under assumption of small incident v ector po-
tential [81]. W ithin the Coulomb gauge ∇ · A = 0 , the Φ term v anishes under the
assumption of the absence of free char ges [82]. Summing ov er all electrons yields the
final e xpression for the light matter coupling Hamiltonian
H 0 + H el − f = − X
i
~ 2
2 m ∇ 2
r i − X
i
e ~
im A · ∇ r i . (2.20)
T o summarize, the total Hamiltonian of the electron system in the semiconductor reads
H 0 ( { R 0
i } , { r i } ) = X
i
p 2
i
2 m + X
i,j
U ( R 0
j − r i ) + X
i,j
V ( r j − r i ) + X
i,j
V ( R 0
j − R 0
i )
+ X
i,j
u j · ∇ R j U ( r i − R j ) | R j = R 0
j − X
i
e ~
im A · ∇ r i . (2.21)
Again, the first term describes the kinetic ener gy of the electrons, the second term the
interaction of the electrons with the periodic Coulomb potential of the ions and the
third term describes the Coulomb interaction between the electrons. The first term in
the second line describes the interaction of the electrons with lattice vibrations and the
last term describes the interaction of electrons with an external light field. In principle
with this Hamiltonian the Schr ¨
odinger equation, eq. 2.10, can be e v aluated. Problem-
atic is that macroscopic systems contain typically particles in the order of the A v ogadro
constant N A = 6.02 · 10 23 [83], which would mean that 10 23 coupled dif ferential equa-
tions ha ve to be solv ed. This is practically impossible.
2.2.2. Canonical Quantization
The canonical quantization is a procedure to quantize a classical theory . In particular
it is applied to get a quantized description of lattice vibrations (phonons) and electro-
magnetic fields (photons). In order to get a unified description of phonons, photons
2.2. Many P article Hamiltonian in Second Quantization
and char ge carriers, it also can be applied for electrons which are already described
quantum mechanically by the Schr ¨
odinger equation, equations 2.10 and 2.21. Hence,
this approach is often referred to as second quantization. In the follo wing the general
procedure is described roughly [84]. In all considered cases first a Langrange density
of the considered field is required. If only a field equation is a v ailable, the Lagrange
density has to be guessed such that it has the field equation as an Euler Lagrange equa-
tion. W ith a Legendre transformation with respect to the generalized field momentum
this Lagrange density is transformed to a Hamilton density . According to the cor -
respondence principle no w the classical field observ able and the corresponding field
momentum and transformed to quantum operators in the Hilbert space by claiming
commutator relations. The last step is to expand these operators after a orthonormal
set of eigenfunctions of the initial field equations, by introducing annihilation and cre-
ation operators. This quantization scheme is applied to phonons, photons and electrons
in the follo wing.
Phonons
Starting point for the quantization of the lattice vibrations is the Hamiltonian for the
ions, equation 2.14,
H phon = X
i
P 2
i
2 M i
+ ( { R i } ) . (2.22)
Here the first term accounts for the kinetic ener gy of the ions. The second term stems
from the electronic motion in the ion potential with fix ed ion coordinates and accounts
for an ef fecti ve interaction between the lattice ions. The ef fecti ve interaction potential
can be e xpanded in a T aylor series around the equilibrium positions R 0
i + u i and reads
in harmonical approximation
H phon = X
i
P 2
i
2 M i
+ 1
2 X
i,j
u T
i · D ij · u j , (2.23)
The linear term v anishes since { R 0
i } are the equilibrium position of the crystal ions
( ∇ R i ( { R i } )=0 , ∀ i holds in a local minimum of ( { R i } ) ). This Hamiltonian can be
identified as the Hamiltonian of N coupled harmonic oscillators. The matrix
D ij = ∇ R i ⊗ ∇ R j ( R i , R j ) R 0
(2.24)
is the Hessian matrix of the ef fecti ve ion-ion interaction potential and is real, sym-
metric and positi ve definite. It is called the dynamical matrix. After determining the
eigenener gies ~ ω α
K and eigen vectors e α
K of the dynamical matrix, the lattice displace-
ments can be e xpanded after eigen modes
u i = 1
√ N X
α, K
e i K · R 0
i e α
K q α
K , (2.25)
2. Theoretical Basics
with coef ficients q α
K . Here K denotes the phonon momentum and α the phonon mode
inde x. Defining the corresponding momenta p α
K = ∂ t q α
K yields for the Hamiltonian
H phon = X
α, K
( p α
K ) 2
2 M + 1
2 M ( ω α
K ) 2 ( q α
K ) 2 . (2.26)
M denotes a properly defined mass of the unit cell. The step to a quantum mechanical
description can be made by assuming p α
K and q α
K as hermitian operators and claiming
the fundamental commutation relations
[ q α
K , p α 0
K 0 ] − = i ~ δ α,α 0
K , K 0 . (2.27)
As for the harmonic oscillator , creation and annihilation operators are defined
b ( † ) α
K = r M ω α
K
2 ~ q α
K + ( − ) i s 1
2 ~ M ω α
K
p α
K (2.28)
These operators fulfill the fundamental bosonic commutation relation
[ b ( † ) α
K , b ( † ) α 0
K 0 ] − = 0 (2.29)
[ b α
K , b † α 0
K 0 ] − = δ α,α 0
K , K 0 . (2.30)
Inserting the definitions for the annihilation and creation operator in the Hamiltonian,
equation 2.26, one ends up with the quantized free Hamiltonian for the lattice vibra-
tions
H phon = X
α, K
~ ω α
K b † α
K b α
K + 1
2 . (2.31)
Since it will be required for the in vestig ation of the coupling of carriers to lattice vi-
brations, the quantized e xpression for the lattice displacements is giv en
u i = X
K ,α s ~
2 ρω α
K A ( b α
K + b † α
− K ) e α
K e i K · R 0
i . (2.32)
Here, ρ denotes the mass density of the unit cell, ω α
K the frequenc y of the phonon
with two dimensional w av e vector K and mode α and A the area of the material. b ( † ) α
K
denote phonon annihilation (creation) operators and e α
K the polarization of the phonon.
More details about the quantization of lattice vibrations can be found in the standard
literature [85, 80].
Photons
Similar to the lattice vibrations, the free electromagnetic field can be quantized. Start-
ing point here is the Langrange density of the electromagenetic field in Coulomb gauge
∇ · A = 0
L = 0
2 ( ∂ t A ( r , t )) 2 − c 2 ( ∇ × A ( r , t )) 2 . (2.33)
2.2. Many P article Hamiltonian in Second Quantization
Here, 0 denotes the v acuum permitti vity and c the velocity of sound in v acuum. The
corresponding Lagrange function can be obtained by inte grating the Langrange den-
sity o ver the considered v olume V . The ne xt step is to define the corresponding field
momentum
Π ( r , t ) = ∂ ( ∂ t A ) L ( r , t ) = 0 ∂ t A ( r , t ) = − 0 E ( r , t ) , (2.34)
where in the last step the definition of the v ector potential was e xploited. The Hamilton
density of the free electromagnetic field can be obtained by Le gendre transforming the
Lagrange density with respect to the generalized momentum
H = 1
2 1
0
( Π ( r , t )) 2 + 1
µ 0
( ∇ × A ( r , t )) 2 . (2.35)
The corresponding Hamilton function is obtained by inte grating the Hamilton density
H = R V d 3 r H . As for phonons, the next step is to quantize the Hamiltonian by defining
A and Π as hermitian operators and claiming the commutator relation
h ( A ( r , t )) i , ( Π ( r 0 , t )) j i − = δ ij δ T ( r − r 0 ) . (2.36)
where δ T ( r − r 0 ) denotes the transv erse Dirac distribution [86]. The v ector potential
can be e xpanded in terms of plane wa ves assuming a finite quantization v olume
A ( r , t ) = X
K ,k z s ~
Ω σ
K ,k z 0 V e σ
K ,k z e i K · r e ik z z d σ
K ,k z ( t ) + h.c.. (2.37)
W ith the definition of the vector potential, equations 2.17 and 2.18, one obtains for the
electric and the magnetic field
E ( r , t ) = X
K ,k z s ~ Ω σ
K ,k z
0 V e σ
K ,k z e i K · r e ik z z d σ
K ,k z ( t ) + h.c. (2.38)
B ( r , t ) = X
K ,k z s ~
Ω σ
K ,k z 0 V ( K , k z ) × e σ
K ,k z e i K · r e ik z z d σ
K ,k z ( t ) + h.c. . (2.39)
Here e σ K ,k z denotes the polarization v ector of the light field. For the light matter inter -
action of transition metal dichalcogenides, it is con venient to take a circular polarized
basis e σ = σ + , σ − . ~ Ω σ
K ,k z denotes the dispersion relation of photons. V is the quanti-
zation v olume. For the in vestig ation of the interaction of the electromagentic field with
the quasi two dimensional semiconductor , it is suf ficient to split the momentum already
into an in-plane component with respect to the semiconductor K and one component
perpendicular to the semiconductor k z . Further annihilation (creation) operators d σ
K ,k z
for photons in the mode ( K , k z ) were introduced. Inserting the e xpression for the
v ector potential in the commutator relation yields for the photon operators
[ d σ
K ,k z , d σ 0
K 0 ,k 0
z ] − = 0 (2.40)
[ d σ
K ,k z , d † σ 0
K 0 ,k 0
z ] − = δ σ ,σ 0 δ k z ,k 0
z
K , K 0 . (2.41)
2. Theoretical Basics
As for the phonon operators, equation 2.30, these commutation relation constitute the
fundamental bosonic commutation relations. Inserting the plane wa ve e xpansion of the
v ector potential, equation 2.37, in the Hamiltonian yields the quantized expression
H phot = X
K ,k z ,σ
~ Ω σ
K ,k z d † σ
K ,k z d σ
K ,k z + 1
2 . (2.42)
The dispersion relation of photons reads Ω σ
K ,k z = c | ( K , k z ) | and depends linearly on
the photon momentum. F or practical computations the term proportional to 1
2 can
be left out, since it commutes with e very operator and hence ne ver appears in the
equations of motion.
Electr ons
In this subsection the quantization of the electrons is performed. As for photons and
phonons, a full basis set of orthogonal functions is required. It is con venient to take
the wa vefunctions of non-interacting electrons in the periodic lattice of solid state core
ions. The corresponding Hamiltonian, compare equation 2.21, reads
H non
0 ( { R i } , { r i } ) = X
i
( − ~ 2 ∇ 2
r i
2 m ) + X
i,j
U ( R 0
j − r i ) = X
i
h i . (2.43)
Here ef fecti ve one-particle Hamiltonians h 1 were defined. The wa vefunctions of N
non interaction electrons can be obtained by e v aluating the Schr ¨
odinger equation H non
0 Ψ =
i ~ ∂ t Ψ . Since electrons are fermions, the N -particle wa ve function has to be anti-
symmetric und e xchange of particles and can therefore be written as a Slater deter-
minant of one-particle wa ve functions Ψ 1 ( r 1 , t ) . Exploiting this for the Hamiltonian,
equation 2.43 , an ef fecti ve one particle Schr ¨
odinger equation can be written
− ~ 2 ∇ 2
r
2 m + U ( r ) Ψ( r , t ) = i ~ ∂ t Ψ( r , t ) , (2.44)
where the abbre viation U ( r ) = P i U ( R i − r ) was introduced. In order to apply the
same quantization scheme as for phonons and photons, no w a Lagrange density has to
be found, which has the one-particle Schr ¨
odinger equation as Euler -Lagrange equation.
According to Richard Fe ynman one possible procedure is to fiddle around. Doing so,
one finds
L = i ~
2 (Ψ ∗ ∂ t Ψ − ( ∂ t Ψ ∗ )Ψ) − ~ 2
2 m ( ∇ Ψ ∗ ) · ( ∇ Ψ) − U Ψ ∗ Ψ . (2.45)
The corresponding momentum to the v ector field Ψ is obtained via
Π = ∂ ( ∂ t Ψ) L = i ~
2 Ψ ∗ . (2.46)
2.2. Many P article Hamiltonian in Second Quantization
A Le gendre transformation of the Lagrange density with respect to the generalized
momentum and inte grating ov er the quantization volume yields the Hamilton function
H = Z V
d 3 r Ψ ∗ ( r , t ) − ~ 2 ∇ 2
2 m + U ( r ) Ψ( r , t ) . (2.47)
T o obtain a quantized v ersion of this Hamiltonian, the field Ψ and the momentum Ψ ∗
are redefined as hermitian operators. According to the spin statistic theorem, an anti
commutation relation is claimed
[Ψ( r , t ) , Ψ † ( r 0 , t )] + = δ ( r − r 0 ) . (2.48)
The field operators are e xpanded after a complete set of eigenstates of the stationary
single particle Schr ¨
odinger equation, equation 2.44,
Ψ( r , t ) = X
β
Ψ β ( r ) a β ( t ) . (2.49)
Here, annihilation and creation operators, a β and a †
β , of an electron with quantum
numbers β where introduced. The eigenstates and eigenenergies of the single parti-
cle Schr ¨
odinger equation in a periodic lattice are discussed in more detail in the ne xt
section. The anti-commutator of these operators is obtained by exploiting the commu-
tation relation for the field operators, equation 2.48,
[ a ( † )
β , a ( † )
β 0 ] + = 0 (2.50)
[ a β , a †
β 0 ] + = δ β ,β 0 . (2.51)
These are the fundamental commutation relations of fermions. The first ensures the
antisymmetry of the multi-particle wa vefunctions and the second ensures that a state
can not be occupied by more than one particle. W ith the expansion after eigenstates of
the Schr ¨
odinger equation, the Hamiltonian in second quantization becomes
H = X
β ,β 0 Z V
d 3 r Ψ ∗
β ( r ) − ~ 2 ∇ 2
2 m + U ( r ) Ψ β 0 ( r ) a †
β a β 0 = X
β
β a †
β a β . (2.52)
In the second step, the facts that Ψ β ( r ) is an eigenfunction of the single particle Hamil-
tonian and that the eigenfunctions of the single particle Hamiltonian are orthogonal
were e xploited.
In general, one-particle operators in first quantization A (1) ( r ) can be transformed to a
second quantized e xpression via
A (2) = X
β ,β 0 Z V
d 3 r Ψ ∗
β ( r ) A (1) ( r )Ψ β 0 ( r ) a †
β a β 0 . (2.53)
2. Theoretical Basics
This is applied to the Hamiltonian of the electron phonon coupling, fifth term in equa-
tion 2.21 together with the quantized e xpression for the lattice vibrations, equation
2.32, and to the light matter coupling, last term in equation 2.21 together with the
quantized e xpression for the vector potential, equation 2.37. T w o-particle operators
in first quantization B (1) ( r , r 0 ) , such as the Coulomb interaction Hamilton operator in
equation 2.21, can be transformed as
B (2) = X
α,α 0 ,β ,β 0 Z V
d 3 r Z V
d 3 r 0 Ψ ∗
α ( r )Ψ ∗
β ( r 0 ) B (1) ( r , r 0 )Ψ β 0 ( r 0 )Ψ α 0 ( r ) a †
α a †
β a β 0 a α 0 .
(2.54)
F or practical applications, macroscopic observ ables such as the absorption coef ficient
or the photoluminescence intensity ha ve to be e xpressed within the second quantization
in terms of annihilation and creation operators of electrons, phonons or photons. This
is done in the ne xt subsections. Equations of motion for an arbitrary operator ˆ
X can
then be obtained by e xploiting Heisenbergs equation of motion
i ~ ∂ t ˆ
X = [ ˆ
X , H (2) ] . (2.55)
2d Bloch theorem
Here, the general form of the electronic wa vefunctions in a two-dimensional material
is discussed. In general the wa vefunction depends on the threedimensional v ector r ∈
R 3 , which can be split into one component within the layer r k ∈ R 2 and one component
perpendicular to it r ⊥ ∈ R . This yields for the wa vefunction Ψ( r ) = Ψ( r k , r ⊥ ) . The
electronic wa vefunction is determined by the solution of the single particle Schr ¨
odinger
equation, compare equation 2.44. The single particle Schr ¨
odinger equation is lattice
periodic, meaning that T k
n H ( r k , r ⊥ ) = H ( r k , r ⊥ ) . T k
n denotes the translation operator ,
which shifts the function by a lattice vector . Follo wing from that, the same relation
holds for the absolute square of the electronic wa v efunction, and one finds for the
wa vefunction
T k
n Ψ( r k , r ⊥ ) = Ψ( r k + R n , r ⊥ ) = t k
n Ψ( r k , r ⊥ ) . (2.56)
with t k
n ∈ C , | t k
n | = 1 . Assuming two translation operators which are applied after
each other , one finds the relation t k
n t k
m = t k
n + m . For the commutator of the transla-
tion operator and the Hamilton operator , it holds [ T n , H ] = 0 since the Hamiltonian
is lattice periodic. Hence there e xists a simultaneous system of eigenstates for the
translation operator and the single particle Hamiltonian
H Ψ λ
k = E λ
k Ψ λ
k (2.57)
T k
n Ψ λ
k = t k λ
n k Ψ λ
k . (2.58)
Here ( λ , k ) are the quantum numbers which are associated with the translational sym-
metry of the system as the y are discussed in chapter 3. λ denotes the band index and
2.3. Observ ables
k denotes the two dimensional momentum in in-plane direction. As a result one finds
t k
n k = e i k · R n . This yields the two-dimensional Bloch theorem
Ψ λ
k ( r k + R n , r ⊥ ) = e i k · R n Ψ λ
k ( r k , r ⊥ ) . (2.59)
A common ansatz for the wa ve function reads
Ψ λ
k ( r ) = 1
√ V e i r k · k u λ
k ( r ) , (2.60)
with V denoting the quantization v olume and u λ
k ( r ) a lattice periodic function. The
v olume can be expressed as a product Al z with A denoting the area of the material,
and l z being the length of the quantization v olume in perpendicular direction to the
material. It will be mer ged into the lattice periodic function [87]. This ansatz will be
carried out throughout this thesis and is suf ficient to compute all rele v ant electronic
coupling elements, which is explicitly performed in appendix A.2 T o further simplify
this e xpression, a commonly used approximation is the en velope function approxima-
tion [87], where a product ansatz is made for the lattice periodic part of u λ
k ( r ) , leading
to one factor associated with the direction of translation symmetry u λ
k ( r k ) and one part
associated with the confinement direction ξ n ( r ⊥ )
u λ
k ( r ) ≈ u λ
k = k 0 ( r k ) ξ n ( r ⊥ ) , (2.61)
where k 0 denotes the momentum with respect to the band e xtremum and n denotes the
quantum number associated with the confinement in perpendicular direction.
2.3. Obser v ab les
In this section the most rele v ant macroscopic observ ables are defined within the frame-
work of the second quantization.
2.3.1. Absorption Coefficient
In this section important quantities to study the interaction of a tw o dimensional sheet
with an e xternal electric field in linear spectroscopy are determined.
Therefore one considers an incident electrical light field E ( r , t ) e xciting a polarization
P ( r , t ) in the material. Assuming a linear response, the most general ansatz for the
induced polarization in the linear limit reads
P ( r , t ) = 0 Z R 3
d 3 r 0 Z [ −∞ ,t ]
dt 0 χ ( r , r 0 , t, t 0 ) E ( r 0 , t 0 ) , (2.62)
with the optical susceptibility χ ∈ C 3 × 3 . The optical susceptibility is assumed as
homogeneous in space and time. Hence it just depends on the spatial and temporal
dif ferences. Further it becomes diagonal
χ = χ ( r − r 0 , t − t 0 ) ∈ C . (2.63)
2. Theoretical Basics
Considering the electrical field as a plane wa ve and assuming further that the material
is thin compared to the wa velength of the e xciting field the space dependence drops
and one can write
P ( t ) = 0 Z [ −∞ ,t ]
dt 0 χ ( t − t 0 ) E ( t 0 ) . (2.64)
Exploiting the con v olution theorem the Fourier transform of the macroscopic polariza-
tion reads
P ( ω ) = 0 χ ( ω ) E ( ω ) (2.65)
Projecting this equation on the orthogonal polarization directions of the incident light
field, yields for the optical susceptibility
χ ( ω ) = P ( ω )
0 E ( ω ) . (2.66)
As it will turn out in chapter 3, circularly polarized light modes are an excellent basis to
study the light matter coupling for TMDs. T aking no w use of j = ∂ t P and E = − ∂ t A
one ends up with
χ ( ω ) = j ( ω )
0 ω 2 A ( ω ) . (2.67)
The optical susceptibility is in general a comple x quantity . T o understand the impact of
this on the wa ve propagating through the material one writes the tele graphers equation,
which can be obtained from the Maxwell equations
∇ 2 − n 2
c 2 ∂ 2
t E ( r , t ) = µ 0 σ ∂ t E ( r , t ) . (2.68)
Here, c denotes the v acuum speed of light, n the refractiv e index of the material, µ 0
the v acuum permeability and σ the conducti vity of the material. Using a plane wa ve
ansatz for the electric field E ( r , t ) = E 0 e i ( k · r − ω t ) , one obtains the dispersion relation
k 2 = ω 2
c 2 ( n 2 + i σ
0 ω ) = ω 2
c 2 ˜ , (2.69)
where the comple x permitti vity ˜ = 0 + i 00 was introduced. A comple x permittivity
immediately yields a comple x wa ve v ector ˜
k = k 0 + ik 00 , with real and imaginary part
k 0 = 0 ( ω ) ω
c , k 00 = ω
2 p ( 0 ( ω )) c 00 ( ω ) (2.70)
Exploiting this in the plane wa ve ansatz for the electric field one obtains
E ( r , t ) = E 0 e i ( ˜
k ) r − ω t = E 0 e − k 00 r e i ( k 0 r − ω t ) . (2.71)
The intensity of the electromagnetic wa ve is gi v en by the squared absolute value, we
obtain
I ( r ) ∝ e − 2 k 00 r . (2.72)
2.3. Observ ables
Consequently the imaginary part of the wa v e vector , and thus the imaginary part of
the permitti vity 00 leads to a damping of the intensity of the light field. Writing the
permitti vity in terms of the susceptibility ( ω ) = 1 + χ ( ω ) , one ends up with a formula
of the absorption coef ficient
α ( ω ) = ω
nc 0 = ( χ ( ω )) = 1
0 c 0 nω = ( j ( ω ))
A ( ω ) . (2.73)
As a result, the absorption coef ficient is determined by the ratio of the imaginary part
of the current density of the material and of the v ectors potential of the exciting electric
field. The expression for the quantum mechanical probability current reads [88]
j ( t ) = e
2 m X
k , k 0 ,λ,λ 0 Z R 3
d 3 r Ψ λ
k ( r )( p − e A )Ψ λ 0
k 0 ( r ) h a † λ
k a λ 0
k 0 i (2.74)
with electron being in the state | λ k i with band inde x λ , momentum k and p the mo-
mentum operator for electrons. Assuming that the photon momentum is small com-
pared to the electron momentum one ne glects intraband and of f diagonal optical tran-
sitions, which yields k = k 0 . Further the second term is kno wn to be small in the
appearance of man y body Coulomb interactions and therefore is neglected in the fol-
lo wing [88]. Thus one obtains
j ( t ) = X
k
M cv
k p v c
k ( t ) + c.c. (2.75)
Here M cv
k = 1
V
i ~ e
m R R 3 Ψ ∗ c
k ( r ) ∇ Ψ v
k ( r ) denotes the optical matrix element as it appears
in the optical Hamiltonian, equation 3.3 in the next chapter 3, and p v c
k = h a † v
k a c
k i the
microscopic polarization between the v alence ( v ) and the conduction ( c ) band.
2.3.2. Photoluminescence Intensity
The intensity of the emitted light can be defined as the temporal e volution of the ener gy
of the electromagnetic field [89]
I = ∂ t Z R 3
0
2 0 ( Π ( r , t )) 2 + c 2 ( ∇ × A ( r , t )) 2 (2.76)
Inserting the quantized e xpression for the vector potential A , equation 2.37, yields for
the intensity [89, 90]
I = ∂ t X
σ , K ,k z
~ Ω σ
K ,k z h d † σ
K ,k z d σ
K ,k z i ! . (2.77)
Again K denotes the momentum parallel to the semiconductor plane and k z denotes the
momentum perpendicular to it. Writing the parallel momentum in polar coordinates
2. Theoretical Basics
K = K ( cosφ, sinφ ) one can identify φ as the emission angle within the x-y-plane.
The angle with respect to the area normal is gi ven by θ = arctan( K
k z ) . This allo ws
to re write the momentum sum and define the luminescence intensity which is emitted
under a certain angle ( φ, θ ) . This is v alid as long as the distance between the sample
and the detector compared to sample size and wa velength of the emitted light [90]
I φ,θ = V
(2 π ) 3 ∂ t X
σ Z ∞
0
dK K 2 ~ Ω σ
K ,k z h d † σ
K ,k z d σ
K ,k z i ! . (2.78)
The total intensity of the emitted light is then obtained by inte grating ov er both an-
gles I = R 2 π
0 dφ R π
0 dθ sin ( θ ) I φ,θ . Here the sum ov er the three dimensional photon
momenta was re written to an integral P K = V
(2 π ) 3 R d 3 K .
3. System Hamiltonian
In this chapter , the system Hamiltonian which is required to deri ve the fundamen-
tal equations of motion is introduced. The chapter is or ganized as follo ws: first, the
fermionic man y particle Hamiltonian in second quantization is discussed according to
the results of the pre vious chapter 2. All rele v ant electronic coupling elements are
defined. Since the scope of this thesis is a description of the excitonic properties of
TMDs, a more suitable description, namely an e xcitonic Hamiltonian, is introduced.
The deri v ation of the excitonic Hamiltonian is moti vated and illustrated in v ery detail.
Last, the complete e xcitonic Hamiltonian in the weak excitation limit is gi ven, includ-
ing all rele v ant coupling elements. W ith the results of this chapter , the fundamental
equations of motion for e xcitonic quantities can be deri ved, which will be done in the
ne xt chapters.
3.1. Man y P ar tic le Fermion Hamiltonian
The theoretical description of the electronic system in transition metal dichalcogenides
is challenging due to the v ery rich quasi particle band structure. Both, valence band
and conduction band e xhibit many v alle ys at dif ferent high symmetry points in the
first Brillouin zone. In order to increase the readability of the fermionic Hamiltonian,
a compound notation for the v alley de gree of freedom and the electron spin is intro-
duced. Starting point is the annihilation operator for an electron λ s
k with the spin s
and the momentum k , which is located within the first Brillouin zone. The band in-
de x λ denotes the conduction band ( c ) or the v alence band ( v ). Next, the momentum
k is e xpanded around a gi v en high symmetry point i , corresponding to the dif ferent
band e xtrema of conduction and v alence band. The conduction band electron i could
either be located at the K , the K 0 , the Λ or the Λ 0 point. For v alence band electrons,
i is located at the K , the K 0 or the Γ point. Thus, one ends up with λ s
i + k = λ is
k
where a compound notation seems appropriate ξ = ( is ) . F or the follo wing, we define
¯
ξ = ( ¯
i, ¯ s ) , with ¯ s being the opposite electron spin and ¯
i being the opposite v alley with
respect to the point in version at the Γ point in the Brillouin zone. Further for phonon
scattering processes as well as Coulomb scattering processes where the high symmetry
point may change b ut the spin remains unchanged the notation ξ + j = ( i + j, s ) is
settled. In this thesis in general momenta referring to electronic motion are designated
with small letters k and momenta referring to the e xcitonic center of mass motion are
designated with capital letters K .
3. System Hamiltonian
Figure 3.1.: Schematic illustration of the quasiparticle bandstructur e in WX 2 . (a) Energeti-
cally lo west lying conduction band minima in the 1. Brillouin zone in WX 2 . The
lo west lying bands hav e spin down (red) at the K points and the Λ points. The lowest
lying minima at the K points and the Λ points ha ve spin up (blue). (b) Ener getically
highest lying v alence band maxima in the first Brillouin zone in WX 2 . The highest
lying band at the K point has spin up (blue) whereas the highest lying band at the
K has spin do wn (red). The Γ valle y is spin degenerated. (c) Bandstructure in WX 2
including spin orbit coupling. Blue indicates spin up bands and red indictes spin do wn
bands.
3.1.1. Free Hamiltonian
The free Hamiltonian for the man y particle system reads [87, 88]
H c − 0 =
λ, k ,ξ
λξ
k λ † ξ
k λ ξ
k +
α,i, K
ω iα
K b † iα
K b iα
K +
σ, K ,k z
Ω σ
K ,k z d † σ
K ,k z d σ
K ,k z . (3.1)
Here, the first term describes the free ener gy of the electrons in the band λ with valle y-
spin ξ =( i, s ) and the two dimensional momentum k with respect to the high symme-
try point i . The dispersion λξ
k is treated in parabolic approximation and reads
λξ
k = E λξ + λ 2 k 2
2 m λξ . (3.2)
Here, E λξ denotes the ener getic separation of the gi ven v alle y from the fermi le vel,
and m λξ denotes the ef fecti ve mass of the band. Further the sign of the dispersion is
defined as λ = +1 , if λ = c, and λ = − 1 , if λ = v . The parabolic approximation of
the electronic bands (ef fecti ve mass approximation) is v alid for energies in the order
of hundreds of meV [34].
3.1. Many P article Fermion Hamiltonian
As depicted in figure 3.1 (a) and (c), the minima of the conduction band are located at
the corners of the he xagonal Brillouin zone at the K and K 0 points. Further arround
each K ( K 0 ) points, there are three Λ ( Λ 0 ) points which also exhibit a minimum in the
conduction band dispersion. Due to a strong spin oribit coupling, the conduction band
is split into two spin bands by some tens of meV [34]. In tungsten based TMDs, the
lo west band at the K point is a spin do wn band and at the Λ point it is a spin up band.
In contrast, in molybdenum based TMDs, the lo west lying spin band at the K point
is a spin up band, whereas the spin configuration at the Λ point is a spin up band too.
F ollowing from the time re versal symmetry , the spin configuration at the K 0 ( Λ 0 ) point
is re versed with respect to the K ( Λ ) point in all TMDs. The maxima of the v alence
band are located at the corners, at the K and K 0 points, and in the center at the Γ point
of the he xagonal Brillouin zone. In accordance to the conduction band, the v alence
band is split into two spin bands by about 0.4 eV [34], where the higher lying band at
the K ( K 0 ) point is a spin up (do wn) band. The v alence band maximum at the Γ point
is spin de generated.
The second term in equation 3.1 denotes the free ener gy of the phonons. Here, b ( † ) iα
K
denote phonon annihilation (creation) operators and ~ ω iα
K the dispersion for phonons
in the mode α at the high symmetry point i and two dimensional momentum K . Again
as for electrons the total momentum of the phonon with respect to the Γ point of the
first Brillouin zone is gi ven by i + K . Since the unit cell in TMDs consists of three
atoms, one finds 9 dif ferent phonon modes in TMDs, which are illustrated in figure 3.2.
These are three acoustic modes, namely LA, T A and ZA mode and six optical modes,
where the most prominent are the LO, T O and A 1 mode. The optical and the acoustic
modes are ener getically well separated with a phononic bandgap of about 15 meV in
the e xemplary material WS 2 [91]. F or optical phonons with short momenta, i.e. i = Γ ,
the phonon dispersion is treated in the Einstein approximation ~ ω Γ α
K = ~ ω Γ α = const.
and in Debye approximation for acoustic phonons ~ ω Γ α
K = ~ c Γ α | K | , where c Γ α denotes
the v elocity of sound for the giv en phonon mode. Zone-edge phonons, e.g. phonons at
the Λ or the K point are treated in the Einstein approximation ~ ω iα
K = ~ ω iα = const.
[73, 92, 91]. For acoustic Λ phonons this is a rather strong approximation. In an
en vironment of 2 nm − 1 around the Λ point, where the most physics tak es place, the
phonon ener gy dif fers by approximately ± 30 % from the Einstein approximation.
The last term in equation 3.1 denotes the free ener gy of photons with dispersion ~ Ω σ
K ,k z
and annihilation (creation) operators d † σ
K ,k z with light polarization σ and momentum
( K , k z ) . Here, K denotes the projection of the three dimensional photon momentum
onto the semiconductor plane and k z the component perpendicular to the material.
3. System Hamiltonian
Figure 3.2.: Phonon dispersion in WS 2 . TMDs exhibit three acoustic modes (red) and six optical
modes (blue). The data was obtained from Zahra Khatibi (Chalmers).
3.1.2. Classical Carrier -Light Coupling
The Hamiltonian describing the coupling of free carriers to a classical light field reads
[87, 63]
H c − f =
k ,i ∈{ K,K } ,λ,s
M λ ¯
λis
k · A λ † is
k ¯
λ is
k . (3.3)
Here M λ ¯
λis
k = i eN
mA u ∗ λ
i ∇ r n u ¯
λ
i denotes the optical matrix element with u λis
k being
the lattice periodic part of the Bloch wa ve and A the v ector potential of the electro-
magnetic field. The deri v ation of this e xpression can be found in the appendix A.2.1.
The optical matrix element is accessible through tight binding computations [63, 70],
where the oscillator strength can be adjusted to e xperimental measurements of the di-
electric function [41]. As it can be seen from equation 3.3 optical transitions occur
only around the K point and the K point respecti vely . This is a v alid approximation
as long as the e xcitation frequencies are not strongly detuned from the lo west e xcitonic
resonance. Additionally to the transition at the K and K v alley which are denoted by
A and B transition, also transitions at other locations in the Brillouin zone occur , such
as the C transition denoting a transition from v alence to conduction band between Λ
and Γ point [93, 94, 95]. These transitions are ne glected in the follo wing. Recent ex-
perimental and theoretical in vestig ation predicted a pronounced circular dichroism in
transition metal dichalcogenides monolayers [37, 47, 39, 48, 96, 63, 97]. This means,
that at the K point only left handed light can be absorbed, i.e. M K · σ − =0 . At the
K point it follo ws from time re versal symmetry that only right handed polarized light
can be absorbed M K · σ + =0
3.1.3. Quantiz ed Carrier -Light Coupling
The Hamiltonian for the quantized carrier -light interaction reads [87]
H c − phot =
k ,i, K ,k z ,λ,s,σ
M λ, ¯
λ,i,s,σ
k + K , k ,k z λ † is
k + K ¯
λ is
k ( d † σ
− K , − k z + d σ
K ,k z ) . (3.4)
3.1. Many P article Fermion Hamiltonian
M λ, ¯
λ,i,s,σ
k + K , k ,k z = i ~ eN
mA q ~
Ω σ
K ,k z 0 V h u ∗ λ
i ∇ u ¯
λ
i i · e σ e ik z z is the carrier -photon coupling ele-
ment with e σ
K ,k z the light polarization, V the quantization v olume K the transversal
photon momentum and k z the longitudinal photon momentum. The directions of the
photon momentum are defined with respect to the normal of the semiconductor plane.
z 0 denotes the position of the TMD monolayer along the z-axis. The deriv ation of
the carrier -photon coupling element is illustrated in appendix A.2.2. For the quan-
tized optical matrix element, one finds the same selection rules as in the case of the
interaction with a classical light field, which is ob vious since the electronic parts of
the quantized and the classical light matter coupling element coincide. T ransitions at
the i = K point in the first Brillouin zone can be e xcited with left handed polarized
photons σ = σ + and at the K 0 point with right handed polarized photons σ = σ − .
Note that this Hamiltonian also contains terms, which violate the ener gy conserv ation.
Ex emplary , an electronic transition from the conduction to the v alence band can be
assisted by the absorption of a photon. Ho we ver these processes can be consistently
remo ved from the Hamiltonian by performing a rotating wa ve approximation [98]. The
resulting Hamiltonian becomes
H c − phot = X
k ,i, K ,k z ,s,σ
M v ,c,i,s,σ
k + K , k ,k z v † is
k + K c is
k d † σ
− K , − k z + h.c. (3.5)
Here, h.c. denotes the hermitian conjugated e xpression of the former .
3.1.4. Carrier -Carrier Coupling
Ne xt the interaction between the carriers in the TMD monolayer is introduced. In
general all possible transitions of the carriers including intra- and interband scatter -
ing as well as intra- and interv alley scattering ha v e to be considered. The Coulomb
Hamiltonian takes the general form [87, 88]
H c − c = 1
2 X
k , k 0 , q ,i,i 0 ,j
s,s 0 ,λ,λ 0 ,ν,ν 0
V λλ 0 ν 0 ν ii 0 j ss 0
k , k 0 , q λ † i + j s
k + q λ 0 i 0 − j s 0
k 0 − q ν 0 i 0 s 0
k 0 ν is
k (3.6)
Here V λλ 0 ν 0 ν ii 0 ss 0 j
k , k 0 , q = P G , G 0 V j + q + G + G 0 h u λi + j s
k + q | e − i G · r | u ν is
k ih u λ 0 i 0 − j s 0
k 0 − q | e i G 0 · r | u ν 0 i 0 s 0
k 0 i de-
notes the Coulomb matrix element with V q being the F ourier transform of the Coulomb
potential. G and G 0 denote reciprocal lattice v ectors. The form of the coupling ele-
ment is deri ved in the appendix A.2.3. In general not all combination of the v alley
inde xes i , i 0 and j are allo wed, since i + j and i 0 − j alw ays hav e to match a band
e xtremum of the respectiv e carriers. In the follo wing long and short range interaction
will be discussed separately for reasons of clarity .
Long Range Interaction
Long range interaction is generally attrib uted to the transfer of small momenta [79].
Therefore the carriers can not change the v alleys during the scattering process, which
3. System Hamiltonian
Figure 3.3.: Scr eened Coulomb interaction. The Fourier transform of the Rytov a-Keldysh poten-
tial (pink) interpolates between the 3D Fourier transform of the bare Coulomb potential
(red) for lar ge momentum transfers (short range limit) and the 2D Fourier transform of
the bare Coulomb potential (blue) for small momentum (long range limit) transfers.
simplifies the sum o ver the v alleys in the general form of the Coulomb Hamiltonian,
equation 3.6. First the intraband scattering is discussed. Here the Hamiltonian reads
H LR,intr a
c − c = 1
2
k , k , q ,i,i
λ,λ ,s,s
V λλ ii ss
k , k , q λ † is
k + q λ † i s
k − q λ i s
k λ is
k . (3.7)
The matrix element appearing in the general Hamiltonian can then be simplified to
V λλ ii ss
k , k , q = V q since the o verlap of the lattice periodic functions becomes equal to 1
in the lo w wa venumber approximation u λis
k + q | u λis
k ≈ 1 [84], compare the deriv ation
in the appendix A.2.3, equation A.31. T o account for the screening of the Coulomb
interaction due to the dielectric en vironment, consisting of the monolayer with a di-
electric constant ⊥ and the surrounding materials with dielectric constants 1 and 2
the F ourier transform is treated within the Ryto va-K eldysh frame work [58, 59, 60, 61]
V q = e 2
2 0 sub A
1
| q | (1 + r 0 | q | ) , (3.8)
where e denotes the elementary char ge, 0 the vacuum permitti vity , sub the mean di-
electric constant of the substrates and r 0 the screening length. The latter is gi ven by
r 0 = d 0 ⊥
sub with d 0 being the thickness of the monolayer , cf. figure 2.1 (b). The e x-
pression, equation 3.8, can be obtained by solving the Poisson equation for the gi v en
dielectric en vironment [58, 59, 60].
Figure 3.3 illustrates the Ryto va-K eldysh potential as a function of the momentum in
units of the in verse screening length. F or small momenta q 1
r 0 the Ryto va-K eldysh
potential equals with the two dimensional F ourier transform of the Coulomb potential.
This can be attrib uted to the fact that for distances lar ger than the screening length
the potential landscape can be treated as two dimensional in good approximation. In
contrast, for lar ge momenta q 1
r 0 the Ryto va-K eldysh potential coincides up to
a prefactor with the three dimensional F ourier transform of the Coulomb potential
3.1. Many P article Fermion Hamiltonian
indicating that for small distances the three dimensional nature of the TMD layer has
to be taken into account for accurate treatment of the Coulomb interaction.
Recently T imothy Berkelbach and co-w orkers re vealed that the K eldysh potential accu-
rately describes the e xcitonic binding energies in comparison with first principle Bethe
Salpeter compuations [61]. Further , Ale xe y Cherniko v and co-work ers v alidated ex-
perimentally the use of the K eldysh potential by studying the exact ener getic positions
of e xcited excitonic states in the linear spectrum [62].
Ne xt, the interband scattering is in vestig ated. Here the intra v alle y contribution reads
H LR,inter
c − c = 1
2 X
k , k 0 , q ,i,i 0
λ,s,s 0
X λ ¯
λii 0 ss 0
k , k 0 , q ¯
λ † is
k + q λ † i 0 s 0
k 0 − q ¯
λ i 0 s 0
k 0 λ is
k . (3.9)
X λ ¯
λii 0 ss 0
k , k 0 , q = V q h u ¯
λis
k + q | u λis
k ih u λi 0 s 0
k 0 − q | u ¯
λi 0 s 0
k 0 i . Since the momentum transfers are small
which allo ws to e v aluate the scattering cross sections by using the k · p expansion
which yields for the interband Coulomb matrix element
X λ ¯
λii 0 ss 0
k , k 0 , q = V q
1
e 2
1
∆ E λ ¯
λi
k ∆ E λ ¯
λi 0
k
q · M ¯
λλis
k q · M λ ¯
λi 0 s 0
k . (3.10)
The deri v ation of this expression can be found in the appendix A.2.3, equation A.31.
Here M ¯
λλis
k denotes the classical light matter coupling element as it appears in equation
3.3. The Hamiltonian contains a number of dif ferent spin and v alley combinations.
W ith respect to the spin typically two dif ferent cases can be considered. The first
one is where all spin quantum numbers are equal. Here ex emplary a transition of
the A e xciton in the K v alley causes a transition from in the same v alley at the A
transition, cf. figure 3.4 (a). This process is referred as intra v alley e xchange coupling
in the literature [64, 66]. The same A transition could also cause a transition in the
opposite K 0 v alley b ut at the B transition (dashed arro w). Due to the lar ge energetic
miss match of the A and B transition of some hundreds of meV [34] the latter is less
fa vorable. Another interesting case is where both electronic transitions ha ve opposite
spin, cf. figure 3.4 (b). Exemplary , a transition at the A exciton in the K v alle y can
cause a transition at the A transition in the K 0 v alley . This process ef fecti vely transfers
ener gy from the K to the K 0 valle y and is often referred as the interv alley e xchange
coupling in the literature [64, 66]. Note that this process is formally equi v alent to
the F ¨
orster coupling of spatially separated structures [99, 100, 101, 100]. Again, also
the e xcitation of an electron in the K valle y at the B transition is contained in the
Hamiltonian (dashed arro w), which is less fa v orable.
Shor t Rang e Interaction
Last for reasons of completeness the Hamiltonian describing short range interaction is
gi ven. Here, in general lar ge momentum transfers are required. The Hamiltonians for
intraband and interband scattering read
3. System Hamiltonian
Figure 3.4.: Schematic illustration of the exchange interaction. (a) Intrav alley exchange cou-
pling in volv es electronic transitions with same spins. The dashed arrow indicates the
excitation of the B transition which can be e xcited off resonant. (b) Intervalle y ex-
change coupling in volv es electronic transitions with opposite spins. The dashed arro w
indicates the B transition which can be excited of f resonant.
H S R,intr a
c − c = 1
2
k , k , q ,i,i ,j
s,s ,λ,λ
V λλ ii j ss
k , k , q λ † i + js
k + q λ i − js
k − q λ i s
k λ is
k (3.11)
H S R,inter
c − c = 1
2
k , k , q ,i,i ,j
s,s ,λ,
X λ ¯
λii j ss
k , k , q λ † i + js
k + q ¯
λ i − js
k − q λ i s
k ¯
λ is
k (3.12)
where the first line accounts for the intraband interaction and the second line accounts
for the interband interaction. The coupling elements read
V λλ ii j ss
k , k , q = G , G = G V j + q + G + G u λi + js
k + q | e − i G · r | u λis
k u λ i − js
k − q | e i G · r | u λ i s
k and
X λ ¯
λii j ss
k , k , q = G , G = G V j + q + G + G u λi + js
k + q | e − i G · r | u ¯
λis
k u ¯
λi − js
k − q | e i G · r | u λ i s
k .
In contrast to the intra v alley interaction the electronic scattering cross sections appear -
ing in the Coulomb matrix element can not be e v aluated within the lo w wa ve number
approximation an ymore, which requires direct computations of these integrals, for e x-
ample with first principle methods. Further the F ourier transform of the Coulomb
potential V q can not be written as Ryto va-K eldysh potential any more. First principles
computation ha ve demonstrated, that the actual Coulomb potential in TMDs interpo-
lates between a pure 2d potential in the short range limit ( q lar ge) and the Rytov a-
K eldysh potential in the long range limit ( q → 0 )[ 64, 102, 103].
3.1.5. Carrier -Phonon Coupling
No w the Hamiltonian which describes the coupling of carriers to phonons is intro-
duced. It reads [87, 88, 85, 80, 73, 92]
H c − phon =
k ,i,i , K ,λ,α,s
g λii sα
k + K , k , K λ † is
k + K λ i s
k b αi − i
K + b † αi − i
− K . (3.13)
3.1. Many P article Fermion Hamiltonian
Figure 3.5.: P ossible carrier -phonon scattering channels. Possible scattering channels for con-
duction band electrons including intra valle y scattering with Γ phonons and interv alley
scattering with Λ , M and K phonons. (b) Possible scattering channels for valence
band electrons including intra valle y scattering with Γ phonons and interv alley scatter -
ing with K phonons.
Here g λii sα
k + K , k , K denotes the carrier -phonon matrix element for electronic transitions
from the i v alley to the i v alley in the electronic band λ and spin s . α denotes the
phonon mode and includes 3 optical ( LO , TO , A 1 ) and 2 acoustic modes ( LA , TA ).
Electrons scatter from the momentum i + k to the momentum i + k + K under ab-
sorption (emission) of a phonon with momentum i − i + K ( − i + i − K ). Figure 3.5
illustrates possible scattering channels for (a) conduction band electrons and (b) v a-
lence band electrons and illustrates the in v olv ed phonon modes. In general, two cases
can be considered: (i) the intrav alley scattering, meaning i = i and the interv alley
scattering, i = i . Interband scattering can be neglected due to the lar ge electronic
bandgap compared to the phonon ener gies [73, 92, 91].
In general the carrier -phonon matrix element takes the form
g λii sα
k + K , k , K =
2 ρω i − i α
K A G λii sα
k + K , k , K =
2 ρω i − i α
K A u λis
k + K | δV i − i α
K | u λi s
k , (3.14)
where δV i − i α
K denotes the deri v ati ve of the potential between carriers and ions and ρ
the mass density of the unit cell. The deri v ation of this expression can be found in the
appendix A.2.4. In general dif ferent coupling mechanisms contrib ute to the carrier-
phonon coupling. These are deformation potential coupling and piezoelectric coupling
for acoustic phonons and deformation potential coupling and Fr ¨
ohlich coupling for
optical phonons [85, 73, 104],
( G λiisα
k + K , k , K ) ac =( G λiisα
k + K , k , K ) DP
ac + i ( G λiisα
k + K , k , K ) PE
ac (3.15)
( G λiisα
k + K , k , K ) opt =( G λiisα
k + K , k , K ) DP
opt + i ( G λiisα
k + K , k , K ) Fr
opt , (3.16)
where the first equation accounts for acoustic phonon scattering and the second equa-
tion for optical phonon scattering. The imaginary unit i in front of the piezoelectric
3. System Hamiltonian
/ Fr ¨
ohlich coupling ensures that the dif ferent coupling mechanisms are out of phase
and do not interfere [85]. In TMDs, there are fi ve phonon modes with suf ficient high
coupling strength [73, 92, 91]. These are two acoustic modes: the longitudinal (LA)
and the transv ersal (T A) mode, which couple approximately equally in deformation
potential coupling and in piezoelectric coupling [104]. The three optical modes cou-
ple ef ficiently with electrons, which are the homopolar (A 1 ) the longitudinal optical
(LO) and the transv ersal optical (TO) mode. The A 1 and the TO mode couple mainly
in deformation potential coupling whereas the LO mode couples mainly in Fr ¨
ohlich
coupling [73].
F or the in vestig ation of intrav alley e xciton phonon coupling the relati ve signs between
v alence band and conduction band coupling elements will be of particular importance.
The sign for piezoelectric and Fr ¨
ohlich coupling elements is dictated by the char ge of
the scattering carrier . As a result the signs for the coupling elements of v alence band
and conduction band electrons coincides [85].
F or deformation potential coupling only the absolute v alue is obtained from first prin-
ciple calculations in the literature [92, 91]. Hence the sign of the coupling elements
has to be e v aluated with more care. For deformation potential coupling the sign of the
coupling elements is gi ven by the direction of the band shift under the deformation
of the unit cell associated with the respecti ve phonon mode. As a result, comparing
the absolute v alues of the deformation potential coupling constants with measurements
[105, 106] or computations [107] of the bandshift under strain fix es the relati v e sign
between conduction and v alence band coupling elements [108]. As a result in TMDs
v alence band and conduction band deformation potential coupling elements ha ve op-
posite signs, yielding for the coupling elements defined abo ve in equation 3.16
( G λiisα
k + K , k , K ) ac = λ | G λiisα
k + K , k , K | D P
ac + i | G λiisα
k + K , k , K | P E
ac (3.17)
( G λiisα
k + K , k , K ) opt = λ | G λiisα
k + K , k , K | D P
opt + i | G λiisα
k + K , k , K | F r
opt , (3.18)
with λ = 1 if λ = c and λ = − 1 if λ = v , which is v alid as long as the contributions
in equation 3.16 are real. In this thesis, the phonon coupling is treated in deforma-
tion potential approximation [92, 91], which is due to a lack of coupling-mechanism
resolv ed coupling strength for all TMD materials in the literature. In the deformation
potential approximation the scattering cross section appearing in the coupling element
(including all dif ferent coupling mechanisms) are approximated by a T aylor series in
terms of the phonon momentum h u λis
k + K | δ V i − i 0 α
K | u λi 0 s
k i = D λ Γ α
0 + D λ Γ α
1 | K | . The cou-
pling constants are then formally equi vilant to the deformation potential coupling [85].
According to reference [92, 91] acoustic phonons scattering is described by first order
ef fecti ve deformation potential coupling | G λii 0 sα
k + K , k , K | = D λ Γ α
1 | K | . In reference [104]
the authors demonstrated that acoustic phonons couple approximately equally in defor -
mation potential coupling and piezoelectric coupling for momenta which are suf ficient
to describe e xciton phonon coupling, which yields for acoustic phonon coupling ele-
ments
3.2. De velopment of the Excitonic Hamiltonian
( G λiisα
k + K , k , K ) ac = 1
√ 2 λ + 1 D λ Γ α
1 | K | . (3.19)
F or the optical phonon scattering the different coupling mechanisms are well disen-
tangled: The T O and the A 1 mode couple in deformation potential coupling whereas
the LO mode couples mainly in Fr ¨
ohlich coupling. All three coupling are described in
zeroth order deformation potential coupling | G λ Γ sα
k + K , k , K | = D λ Γ α
0 .
F or the intervalle y scattering the coupling element is treated in zeroth order deforma-
tion potential approximation G λisα
k + K , k , K = D λiα
0 . Here the signs are not important for
calculating the e xciton-phonon coupling elements.
3.2. De velopment of the Excitonic Hamiltonian
In this section, the transformation of the electronic Hamiltonian to the excitonic Hamil-
tonian is performed. Se veral shortcoming of the electronic Hamiltonian mak e it nec-
essary to introduce such a description:
• TMDs possess strongly bound e xcitons which dominate the electronic and op-
tical properties in these materials [19, 61, 62, 63]. Therefore it is con venient to
de velop a theoretical description which is also based on e xcitonic quantities.
• In the computation of the linear spectrum in the electron hole picture, a renormal-
ization of the Rabi frequenc y due to many particle interaction between electrons
and holes appears [87, 109, 63]. This renormalization couples microscopic po-
larizations between dif ferent electron momenta and makes the computation of
the linear spectrum non-tri vial. These renormalization terms ha ve been sho wn
to v anish, if the microscopic polarization is expanded after e xcitonic eigenstates,
which is called the excitonic basis [87, 109, 90, 63]. As a result, the excitonic
basis is the easiest way to access the linear spectrum of a system with strong
man y particle interaction between electrons and holes.
• Another adv antage of formulating exciton ph ysics in a fully-excitonic theory is
the fact, that in older approaches first equations of motion had to be deri ved in
the electron hole picture and then had to be transformed into the e xcitonic picture
[87, 109, 90, 110, 63]. Depending on the considered problem this procedure is
more or less complicated.
• Excitonic operators consist of electron and hole operators (conduction band and
v alence band operators creation and annihilation operators). Ex emplary the exci-
tonic density consists of two electron and tw o hole operators but in a lo w density
e xcitonic description only of two excitonic operators. Thus, the appearing hi-
erarchy problem within the Heisenber g equation of motion formalism [110] is
e xpected to reduce in complexity .
3. System Hamiltonian
• In the scope of this thesis is the computation of the luminescence dynamics of
e xcitons in the TMD after weak optical excitation. T o the photoluminescence
generally electronic and e xcitonic contributions ha ve to be considered [111]. The
electronic contrib utions appear in the third order of the exciting electromagnetic
field whereas the e xcitonic contributions appear already in the second order of
the e xciting electromagnetic field. Therefore e xcitonic photoluminescence de-
termines the total photoluminescence in the lo west order .
T o o vercome these issues, an e xcitonic Hamiltonian is introduced in this section. From
this Hamiltonian, excitonic equations of motion can be obtained by e xploiting an ex-
citonic Heisenber g equation of motion.
During the past decades man y studies focused on the deri v ation of an excitonic Hamil-
tonian from an electron hole Hamiltonian. The first approach was gi ven by T . Usui
in 1960 [112, 113, 114]. His approach required a cumbersome ordering restriction in
the pair space which stimulated ongoing research. T oshio Marumori and co-work er
introduced a transformation procedure without ordering restriction in the pair space
[115, 116, 117]. In the early 90s, A. Iv anov and H. Haug introduced a much sim-
pler method to de velop an e xcitonic Hamiltonian [118, 119]. The main idea was to
introduce a unit-operator in the pair space and expand the so obtained multioperator
quantities after pair creation and annihilation operators. This approach will be carried
out in the follo wing to de velop an e xcitonic TMD Hamiltonian.
In this section the transformation from electronic to e xcitonic operators is demon-
strated and finally the e xcitonic Hamiltonian is gi ven. The resulting e xcitonic Hamil-
tonian will be able to describe the full e xciton interaction in the low e xcitation limit
including optical e xcitation, inter v alley Coulomb coupling as well as e xciton-phonon
coupling. Most importantly the e xcitonic Hamiltonian will also include interv alley e x-
citons with momenta far abo ve the light cone where electron and hole are located at
dif ferent high symmetry points.
3.2.1. Exciton Operator s
Haug and Iv anov established the unit operator method to de velop an e xcitonic Hamil-
tonian which will be discussed in this subsection [118, 119]. F or the scope of this
thesis, this method turns out to be the simplest a v ailable method to obtain excitonic
equations of motion in the lo w excitation limit.
The first step is to define pair operators from the electronic operators
P ξ h ξ e
k h , k e = v † ξ h
k h c ξ e
k e . (3.20)
Note that pre vious studied defined the excitonic operators in terms of hole operators
h ξ
k := v † ξ
− k instead of v alence band operators [118]. In principle the creation (anni-
hilation) of a hole is formally equi v alent to the annihilation (creation) of a v alence
band electron [118]. Ho we ver , to a void confusions the introduction of hole operators
3.2. De velopment of the Excitonic Hamiltonian
is circumv ented, and the excitonic operators are directly defined from v alence band
operators. Exploiting the fundamental commutation relations for fermions, equation
2.50 and 2.51, the commutation relation for the pair operators can be obtained
[ P ( † ) ξ h ξ e
k h , k e , P ( † ) ξ 0
h ξ 0
e
k 0
h , k 0
e ] − = 0 , (3.21)
[ P ξ h ξ e
k h , k e , P † ξ 0
h ξ 0
e
k 0
h , k 0
e ] − = δ ξ h ,ξ 0
h
ξ e ,ξ 0
e δ k h , k 0
h
k e , k 0
e − δ ξ e ,ξ 0
e
k e ,k 0
e v ξ h
k h v † ξ 0
h
k 0
h − δ ξ h ,ξ 0
h
k h , k 0
h c † ξ 0
e
k 0
e c ξ e
k e . (3.22)
Up to a correction term which depends on electronic operators, the commutation rela-
tions of pair operators constitute the fundamental bosonic commutation relations. The
correction takes account to the f act that electron hole pairs consist of fermions and en-
sure the fermionic character in the commutation of pair operators. In the lo w excitation
limit ho we ver , where electron and hole densities v anish, the commutation relations are
purely bosonic. Increasing excitation densities lead to a more pronounced fermionic
character of the commutation relation.
The fermionic Hamiltonian is b uild up of electronic operators. Hence a transformation
procedure of electronic operators in terms of pair operators is required. For operators
with an equal number of electron annihilation (creation) and hole annihilation (cre-
ation) operators (v alence band annihilation (creation)), for example the carrier -light
Hamiltonian, equation 3.4 and 3.3, this can be done directly by identifying the pair op-
erators. Ho we ver , for operators with non-equal number of electron and hole operators,
for e xample the kinetic Hamiltonian, equation 3.1, or the Coulomb Hamiltonian, equa-
tion 3.6, a deriv ation scheme is required. This can be done by using a representation
of the unit operator , as first discussed by Haug and Iv anov in 1993 [118]. This will be
illustrated in the follo wing.
The completeness of electronic states in the F ock space can be expressed by the sum-
mation o ver projection operators on the electronic states which is called the complete-
ness relation
ˆ
1 = X
α | α ih α | , (3.23)
with an arbitrary state in the F ock space | α i for now . The sum can expressed in terms
of 0-particle, 1-particle, 2-particle ... states, reading
ˆ
1 = | 0 ih 0 | + X
1
c †
1 | 0 ih 0 | c 1 + X
1
v 1 | 0 ih 0 | v †
1 + X
12
c †
1 v 2 | 0 ih 0 | v †
2 c 1
+ 1
2 X
1 , 2
c †
1 c †
2 | 0 ih 0 | c 2 c 1 + 1
2 X
1 , 2
v 1 v 2 | 0 ih 0 | v †
2 v †
1
+ 1
2 X
1 , 2 , 3
c †
1 c †
2 v 3 | 0 ih 0 | v †
3 c 2 c 1 + 1
2 X
1 , 2 , 3
v 1 v 2 c †
3 | 0 ih 0 | c 3 v †
2 v †
1
+ O ( na 2
B ) 2 . (3.24)
3. System Hamiltonian
Here, | 0 ih 0 | denotes the projector on the v acuum state in Fock space, which consists
of a completely filled v alence band and an unoccupied conduction band. The sums
o ver the compound indices 1 , 2 , 3 run o ver all possible single particle quantum states.
Further , n denotes the excitation density of the sample and a B denotes the Bohr radius
of an e xciton. The number na 2
B is a measure for the number of e xcitons which are
closer than the Bohr radius to a gi ven e xciton. Note the truncation of the unit operator
is applied in terms of the excitonic density and not in terms of the free particle densities.
Under the assumption that the system is alw ays symmetric in the total electron and hole
numbers, the v acuum projector can be written as
| 0 ih 0 | = ˆ
1 − X
1 , 2
c †
1 v 2 v †
2 c 1 + O ( na 2
B ) 2 . (3.25)
The assumption of a symmetric system in the total numbers of electrons and holes im-
plies that no doping is present in the sample. Ho we ver the only considered mechanisms
which change the absolute number of electrons and holes are interactions with an e x-
ternal light field, equations 3.3 and 3.4 and interband Coulomb interaction, equation
3.9. As it is obvious from the respecti ve Hamiltonians, electrons and holes act al ways
pairwise. As a result the total electron and hole numbers stay equal for all times, which
justifies the approach.
No w , the transformation of an operator from the electron hole picture to the pair picture
is discussed. The first step is to insert the unit operator in the middle of the operator .
Hereby it is con venient to bring creation (annihilation) operators of conduction (v a-
lence) band electrons to the left and annihilation (creation) operators of conduction
(v alence) band to the right. Ne xt, only terms are taken into account which are symmet-
ric in the electron and hole numbers. Then the pair operators can be identified. W ith
this procedure the operators c †
1 c 2 and v 1 v †
2 which appear e xemplary in the commuta-
tion relation of the pair operators, equation 3.22, as well as in the kinetic Hamiltonian,
equation 3.1, can be transformed, which gi ves
c †
1 c 2 = X
3
P †
31 P 32 − 1
2 X
345
P †
31 P †
45 P 45 P 32 + O ( na 2
B ) 3 , (3.26)
v 1 v †
2 = X
3
P †
13 P 23 − 1
2 X
345
P †
13 P †
45 P 45 P 23 + O ( na 2
B ) 3 . (3.27)
Here, terms being in third order and higher in the exciton density ha ve been ne glected.
In principle, the operators in the electron hole picture are transformed to an infinite or -
der of pair operators resulting from the form of the unit operator , equation 3.24. T o treat
this problem, a truncation at a certain order is a v alid procedure [120, 121, 122]. Ha v-
ing kno wledge about these fundamental transformation, operators with higher number
of electron and hole operators can be transformed.
As a first result, the commutation relation of pair operators, equation 3.22 can be writ-
ten in terms of pair operators
3.2. De velopment of the Excitonic Hamiltonian
[ P ( † ) ξ h ξ e
k h , k e , P ( † ) ξ 0
h ξ 0
e
k 0
h , k 0
e ] = 0 , (3.28)
[ P ξ h ξ e
k h , k e , P † ξ 0
h ξ 0
e
k 0
h , k 0
e ] = δ ξ h ,ξ 0
h
ξ e ,ξ 0
e δ k h , k 0
h
k e , k 0
e
− δ ξ e ,ξ 0
e
k e , k 0
e X
k 00
e ,ξ 00
e
P † ξ h ξ 00
e
k h , k 00
e
ˆ
N P † ξ 0
h ξ 00
e
k 0
h , k 00
e
− δ ξ h ,ξ 0
h
k h , k 0
h X
k 00
h ,ξ 00
h
P † ξ 00
h ξ 0
e
k 00
h k 0
e
ˆ
N P ξ 00
h ξ e
k 00
h , k e , (3.29)
where terms being in fourth order and higher in the pair operators ha ve been ne glected.
Again, in principle the commutator in the second equation would depend on infinite
orders in the pair operators. The operator ˆ
N reads
ˆ
N = ˆ
1 − X
k e ,ξ e , k h ,ξ h
P † ξ h ξ e
k h , k e P ξ h ξ e
k h , k e + O ( na 2
B ) 2 . (3.30)
In the lo w excitation limit, the terms quadratic in the pair operators can be ne glected in
equation 3.29. This yields the fundamental bosonic commutation relation, compare ex-
emplary equation 2.41. The higher order corrections take account to the Pauli principle
stemming from the fundamental fermionic commutation relations and ensure that elec-
trons and holes constituting the electron hole pair still obe y the fermionic commutation
relations.
The ne xt step is to define e xciton operators from the pair operators. This can be done by
first performing a coordinate transformation from electron-hole-coordinates ( k e , k h ) to
relati ve q and center of mass Q coordinates. Due to the complex TMD quasiparticle
band structure it is necessary to also discuss excitons where the constituting electrons
and holes are not located at the same high symmetry point. The coordinate transfor -
mation then must be e xpanded to momenta which are defined with respect to a gi ven
v alley spin ( ξ h , ξ e ) , which includes high symmetry point ( i h , i e ) and spin configuration
( s h , s e ) ,
q = α ξ h ,ξ e k h + β ξ h ,ξ e k e (3.31)
Q = k e − k h . (3.32)
Here the relati ve electron and hole masses with respect to the v alley spins ξ e/h of
electron and hole are defined as
α ξ h ,ξ e = m ξ e
e
m ξ e
e + m ξ h
h
(3.33)
β ξ h ,ξ e = m ξ h
h
m ξ e
e + m ξ h
h
(3.34)
3. System Hamiltonian
Figure 3.6.: Schematic illustration of the coordinate transf ormation. The dif ference vector of
electron momentum and hole momentum (a) determines the position of the bound
electron hole pair in the excitonic Brillouin zone.
The global center of mass momentum of the e xciton with respect to the Γ point of the
e xcitonic Brillouin zone can then be obtained by
Q g l obal = i h − i e + Q . (3.35)
T o eliminate the coordinate of the relati ve motion q from the pair operators, it is pro-
jected to a complete set of basis functions. A con venient choice are the e xcitonic
wa vefunctions which can be obtained as solutions of the w annier equation [87, 61, 63],
which will be discussed in the ne xt section 3.2.2. Thus the definition of the e xciton op-
erators with e xciton state µ , electron and hole high v alley spins ξ e and ξ h and respecti ve
center of mass momentum Q reads
P ξ h ξ e
µ, Q =
q
ϕ ∗ ξ h ξ e
µ, q P ξ h ξ e
q − β ξ h ξ e Q , q + α ξ h ξ e Q . (3.36)
Here ϕ ∗ ξ h ξ e
µ, q denotes the e xcitonic wa vefunction with quantum number µ and relati v e
momentum q . The w av e function depends also on the hole and electron v alley spin
ξ h =( i h ,s
h ) and ξ e =( i e ,s
e ) with the high symmetry point i h/e and spin s h/e . The
total center of mass momentum of the e xciton can be obtained from Q tot = i e − i h + Q .
Figure 3.6 (a) and (b) illustrates the transformation from electron and hole coordinates
to center of mass coordinates of the e xciton, which will be called excitonic Brillouin
zone shortly in the follo wing. The location of the e xcitonic valle ys is gi ven by equation
3.35, yielding that the positions of the v alley minima of the e xcitonic v alle ys can be
computed as the dif ference of the positions of the electronic band extrema. An exciton
consisting of an electron and a hole from the K point is located at the Γ point in the
e xcitonic Brillouin zone. Excitons with non-equal electron and hole high symmetry
points are in general located far abo ve the Γ point in the excitonic Brillouin zone.
Ex emplary an exciton with an electron at the K ( Λ , Λ ) point and a hole at the K point
is located at the K ( Λ , M ) point in the e xcitonic Brillouin zone.
T ransforming the fundamental commutation relations of the pair operators, equation
3.29, to the e xcitonic picture, one obtaines
3.2. De velopment of the Excitonic Hamiltonian
[ P ( † ) ξ h ξ e
λ, Q , P ( † ) ξ 0
h ξ 0
e
λ 0 , Q 0 ] = 0 , (3.37)
[ P ξ h ξ e
λ, Q , P † ξ 0
h ξ 0
e
λ 0 , Q 0 ] = δ ξ h ξ 0
h
ξ e ξ 0
e δ λ,λ 0
Q , Q 0
− δ ξ e ,ξ 0
e X
K ,ξ µ,µ 0
Ξ eξ h ξ 0
h ξ λµλ 0 µ 0
Q , Q 0 , K P † ξ 0
h ξ
µ 0 , Q 0 + K ˆ
N P ξ h ξ
µ, Q + K
− δ ξ h ,ξ 0
h X
K ,ξ ,µ,µ 0
Ξ hξ e ξ 0
e ξ λµλ 0 µ 0
Q , Q 0 , K P † ξ ξ 0
e
µ 0 Q 0 + K ˆ
N P ξ ξ e
µ, Q + K . (3.38)
Up to correction terms in the second and third line in equation 3.38, these are the fun-
damental bosonic commutators. The corrections in equation 3.38 take account to the
fermionic constituents of the e xcitonic operators and can be interpreted as a conse-
quence of the P auli blocking. Here the operator ˆ
N reads
ˆ
N = ˆ
1 − X
Q ,ν,ξ h ,ξ e
P † ξ h ξ e
ν, Q P ξ h ξ e
ν, Q + O ( na 2
B ) 2 . (3.39)
The appearing matrices in equation 3.38 read
Ξ eξ h ξ 0
h ξ λµλ 0 µ 0
Q , Q 0 , K = X
q , q 0
δ q , q 0 + β ξ 0
h ξ e Q 0 − β ξ h ξ e Q ϕ ∗ ξ h ξ e
λ, q ϕ ξ h ξ
µ, q + β ξ h ξ K ϕ ξ 0
h ξ e
λ 0 , q 0 ϕ ∗ ξ 0
h ξ
µ 0 , q 0 + β ξ 0
h ξ K , (3.40)
Ξ hξ e ξ 0
e ξ λµλ 0 µ 0
Q , Q 0 , K = X
q , q 0
δ q , q 0 − α ξ h ξ 0
e Q 0 + α ξ h ξ e Q ϕ ∗ ξ h ξ e
λ, q ϕ ξ ξ e
µ, q + α ξξ e K ϕ ξ h ξ 0
e
λ 0 , q 0 ϕ ∗ ξ ξ 0
e
µ 0 , q 0 + α ξξ 0
e K (3.41)
F or the in vestig ation of lo w density phenomena it is suf ficient to consider pure bosonic
commutation relations only , which is to truncate the equations of motion on the third
order of the e xciting electron magnetic field [118]. The discussed transformation
scheme can be applied to the fermionic many body Hamiltonian to obtain an e xcitonic
Hamiltonian which will be introduced in the ne xt section.
In the e xcitonic picture, equations of motion for any operator can be obtained by e x-
ploiting Heisenber gs equation of motion, equation 2.55. The corresponding dynamics
of electron hole and e xciton picture are obtained by [118, 87, 88]
i ~ ∂ t ˆ
X = [ ˆ
X , H exc ] , (3.42)
with ˆ
X being an arbitrary e xcitonic operator .
3.2.2. W annier Equation and Exciton Bandstructure
The definition of the excitonic operators, equation 3.36, already required the e xcitonic
wa vefunction ϕ ξ h ξ e
ν, q with hole v alley spin ξ h , electron v alley spin ξ e , quantum number ν
and relati ve momentum with respect to the high symmetry points q . This w a vefunction
can be obtained as a eigen vector of the W annier equation which also determines the
3. System Hamiltonian
Figure 3.7.: Schematic illustration of bandstructur e transformation. An electron hole pair con-
sisting of an electron and a hole at dif ferent high symmetry points in electron hole
picture (a) is located at the zone edge in the excitonic picture (b). Ex emplary an ex-
citon consisting of a hole at the K point and a electron at the K ( Λ , Λ , K ) point is
located at the Γ ( Λ , M , K ) point in the excitonic picture. This picture was published in
a similar form in reference [70].
binding ener gies of excitons. The W annier equation will be discussed in the current
subsection. It reads [87, 63, 123, 124]
2 q 2
2 m ξ h ξ e ϕ ξ h ξ e
ν, q −
k
V v cξ h ξ e
q + k ϕ ξ h ξ e
ν, k = E ξ h ξ e
ν ϕ ξ h ξ e
ν, q . (3.43)
The first term on the left hand side denotes the kinetic ener gy of the relati ve motion of
the e xciton, with the reduced mass m ξ h ξ e = m ξ h
h m ξ e
e
m ξ h
h + m ξ e
e
. The second term on the left hand
side takes account to the attracti ve Coulomb interaction of the electron hole pair , where
V v cξ h ξ e
q + k denotes the matrix element, stemming from the density independent Hartree
F ock contribution of the intraband long range interaction in the fermionic Hamiltonian,
equation 3.7. E ξ h ξ e
ν denotes the e xciton binding energy . The W annier equation is an
eigen v alue equation, where the exciton binding ener gies can be obtained as eigen v alues
and e xciton wa v efunctions as eigen vectors. It can also be applied to excitons where
electron and hole are not located at the same high symmetry point in the first Brillouin
zone. Interestingly , the momentum dif ference of the electron and hole high symmetry
point does not enter in the W annier equation, and so the binding ener gy is independent
of it. The appearing sum o ver the momentum is an intra valle y sum, meaning that it
does not connect electron and hole states which are located in dif ferent v alleys. As
it turns out from the numerical e v aluation, higher reduced masses of the respecti ve
e xciton state lead to larger binding ener gies [70, 75].
Ha ving kno wledge about the excitonic binding ener gies for the dif ferent v alleys, equa-
3.2. De velopment of the Excitonic Hamiltonian
tion 3.43, the e xcitonic dispersion can be computed [90, 70, 123, 124]
E ξ h ξ e
µ, Q = ( E cξ e − E v ξ h ) + E ξ h ξ e
µ + ~ 2 Q 2
2 M ξ h ξ e . (3.44)
Here, the first term denotes the electronic bandgap as the dif ference of the conduction
band and v alence band energies. The second term is the exciton binding ener gy , which
is obtained from the W annier equation, equation 3.43 and the third term describes the
kinetic ener gy of the exciton with total mass M ξ h ξ e = m ξ e
e + m ξ h
h . Figure 3.7 illustrates
the transformation of the dispersion of electron hole pairs (a) to an e xcitonic dispersion
(b) for pairs with the hole being at the K v alley . The dispersions of electron and hole
mer ge into one excitonic dispersion. The resulting high symmetry point of the exci-
ton is determined according to figure 3.6. Note that the separations of the electronic
v alleys are not completely reflected by the ener getic positions of the excitonic v alleys.
This is due to the fact, that on top to the energetic separations the binding ener gy of
each e xciton state has to be tak en into account. For some applications it is suf ficient to
denote the e xcitons by their center of mass momentum. Ho wev er , in scenarios where
e xemplary the spins of electron and hole are additionally tak en into account, this no-
tation can lead to confusions. Here, it is more adv antageous to denote the e xcitons by
their hole and electron high symmetry point. In the abov e described situation, a KK
e xciton would be equi v alent to a Γ exciton.
3.2.3. Numerical Ev aluation of the W annier Equation
In this subsection, the numerical ev aluation of the W annier equation shall briefly be
discussed. For reasons of clarity the v alley and spin indices can be dropped without
loss of generality
~ 2 q 2
2 m ϕ ν, q − X
k
V v c
q + k ϕ ν, k = E ν ϕ ν, q . (3.45)
F or a radial symmetric Coulomb potential, the w av efunctions can be separated into
a radial dependent and an angular dependent part. In the case of a two dimensional
potential, the wa vefunction can be written as [87]
ϕ n,l ( r ) = f n ( r ) e ilφ . (3.46)
with n being the principle quantum number and l being the angular quantum number .
F or the lowest lying s-lik e ( l = 0 ) exciton, which is in the scope of this thesis, the
angular dependence drops. As a result, the Fourier transform of the w a v efunction does
not e xhibit a angular dependence. Therefore the W annier equation reads
~ 2 q 2
2 m ϕ ν,q − X
k
V v c
q + k ϕ ν,k = E ν ϕ ν,q . (3.47)
3. System Hamiltonian
T o include also l 6 = 0 excitons to the quantitati ve analysis, the W annier equation has to
be solv ed angular dependent, which is in principle possible and has done in reference
[124]. The next step is to re write the sums as an integral
~ 2 q 2
2 m ϕ ν,q − A
4 π 2 Z R 2
d 2 k V v c
q + k ϕ ν,k = E ν ϕ ν,q . (3.48)
Here, the area A cancels with the area in the Coulomb matrix element. Next the W an-
nier equation is a veraged o ver the angle φ q
~ 2 q 2
2 m ϕ ν,q − A
8 π 3 Z 2 π
0
dφ q Z R 2
d 2 k V v c
q + k ϕ ν,k = E ν ϕ ν,q . (3.49)
The adv antage of treating the angular dependence analytically is that only a one-
dimensional eigen v alue equation has to be solved instead of a two-dimensional one.
Last the inte grals are written as Riemann sums. Therefore the W annier equation can
be written in the form
X
k
W q ,k ϕ ν,k = E ν ϕ ν,q . (3.50)
with the matrix
W q ,k = ~ 2 q 2
2 m δ q ,k − A
8 π 3 X
φ q ,φ k ,k
∆ φ q ∆ φ k ∆ k k V v c
q + k . (3.51)
Here the step sizes for angle ∆ φ and momentum ∆ k ha ve been introduced. The dis-
cretized W annier equation can then be solv ed numerically by an eigen value solv er and
the s-like w av efunctions and the corresponding excitonic ener gies are obtained.
3.2.4. Observab les
The last thing which needs to be done for the introduction of the e xcitonic Hamiltonian
is to e xpress the macroscopic observables in terms of e xcitonic operators. First the
absorption coef ficient, equation 2.73 and 2.75, is transformed. Since the pair operators
can be identified in the e xpression for the macroscopic current density , emplo ying
equation 3.20 and 3.36 yields
j ( t ) = − X
ξ ,µ X
k
M cv
k ϕ ξ ξ
µ, k ! h P ξ ξ
µ, 0 i ( t ) + c.c., (3.52)
where the e xcitonic polarization h P ξ ξ
µ, 0 i ( t ) was introduced. The brackets are reserv ed
for the quantum mechanical expectation v alue in the Fock space. The projection of the
optical matrix element onto the excitonic w a v e function denotes the e xcitonic optical
matrix element. Then the absorption can be simply obtained as the imaginary part of
3.3. Many P article Exciton Hamiltonian
the current density normalized with the v ector potential of the incident electromagnetic
field, equation 2.73
α σ ( ω ) = 1
0 c 0 nω = 1
A σ ( ω ) X
ξ ,µ
σ · M µξ h P ξ ξ
µ, 0 i ( ω ) + c.c. ! , (3.53)
Here the v ectorpotential was restricted to a certain light polarization σ . Further the
e xcitonic optical matrix element was defined by M µξ = P k M cv
k ϕ ξ ξ
µ, k .
3.3. Man y P ar tic le Exciton Hamiltonian
In this section the lo w density exciton Hamiltonian is gi ven. It reads
H = H 0 + H x − lig ht + H x − phot + H x − x + H x − phon , (3.54)
with the free contrib ution H 0 , the interaction of e xcitons with a classical electromag-
netic field H x − lig ht , the interaction of excitons with a quantized electromagnetic field
H x − phot , the contrib ution from exciton-e xciton interaction H x − x and the coupling of
e xcitons to phonons H x − phon . The indi vidual parts will be discussed in the follo wing.
Since in this thesis, the beha vior of excitons after weak optical e xcitation is of interest,
the main emphasis lays on the lo w density exciton Hamiltonian. In this context weak
optical e xcitation means that ef fects stemming from the third order of the exciting
electromagnetic field are ne glected in the equations of motion. Therefore, contribu-
tions being of the fourth order in the e xciton operators in the excitonic Hamiltonian are
ne glected from the beginning.
3.3.1. Free Hamiltonian
The Hamiltonian describing the free ener gy of excitons, phonons and photons is ob-
tained from the free fermionic Hamiltonian, equation 3.1. It reads
H 0 = X
Q ,µ,ξ h ,ξ e
E ξ h ξ e
µ, Q P † ξ h ξ e
µ, Q P ξ h ξ e
µ, Q
+ X
K ,i,α
~ ω iα
K b † iα
K b iα
K + X
K ,k z ,σ
~ Ω σ
K ,k z d † σ
K ,k z d σ
K ,k z . (3.55)
The first term represents the free ener gy of the excitons with the dispersion E ξ h ξ e
µ, Q ,
which is defined in equation 3.44 with respect to the e xciton center of mass momentum
Q , the exciton state µ , and the electron and hole v alleys and spin ξ e = ( i e , s e ) and
ξ h = ( i h , s h ) . The Hamiltonian takes e xplicitly excitons into account, where electrons
and holes are located in the vicinity of dif ferent high symmetry points ( i e 6 = i h ) and
where electrons and holes ha ve dif ferent spins ( s e 6 = s h ).
3. System Hamiltonian
Figure 3.8.: Exciton bandstructur e in WX 2 for excitons with a hole at the K v alley . A excitons
with electron and hole ha ving spin up (blue) are formed from the energetically higher
lying hole at the K v alley wheras B e xcitons with electron and hole having spin do wn
(red) are formed from the lo wer lying hole at the K valle y . The dashed lines denote
the ener getic positions of the Γ valle y minima.
Figure 3.8 illustrates the e xciton dispersion for 1 s e xcitons with equal electron and hole
spins which ha ve a hole at the K point. Excitons where electron and hole are located
at the K v alley are located at the Γ v alley in the e xcitonic Brillouin zone. Here, the
dispersion is split into two due to the spin orbit splitting of v alence and conduction
band. The lo wer lying band can be referred to as A excitons ( s e = s h = ↑ ) whereas
the higher lying band is referred to B excitons ( s e = s h = ↓ ). Note that the interv alley
e xchange coupling is not included in th excitonic dispersion, b ut is treated separately in
the e xciton-exciton Hamiltonian, equation 3.58. This contrib ution is kno wn to split the
e xcitonic dispersion around the Γ point into a parabolic band and a Dirac cone [64, 66].
The same holds for the Λ ( K ) v alley . Here, the electron is located at the Λ ( K ) v alle y
in the electronic Brillouin zone. Interestingly the A e xcitons at the Λ and the K v alley
are located belo w the A excitons at the Γ v alley in tungsten based materials. For K
e xcitons this is due to the spin ordering of the conduction bands, cf. figure 3.1 . For
the Λ v alley the reason is the much higher ef fecti ve mass [34], which leads to larger
binding ener gies, equation 3.43. The corresponding dispersion for e xcitons with a hole
at the K v alley w ould be spin in verted which follo ws from time rev ersal symmetry .
As a result, the exciton dispersion at each Q point is de generate. Exemplary for A
e xciton dispersion with equal electron and hole spins, one band can be attributed to the
spin up e xcitons with a hole at the K point and one can be attrib uted to the spin down
e xcitons with an hole at the K point. The ener getic positions and curv atures of the
in vestig ated valle ys in this thesis can be found in table A.4 in the appendix A.4 for the
most common TMD materials.
The second term in equation 3.55 denotes the free ener gy of the phonons. Here, b ( † ) iα
K
denote annihilation (creation) operators and ω iα
K the dispersion for phonons in the
mode α at the high symmetry point i and with two dimensional momentum K . Details
for the phonon dispersion can be found in the respecti ve subsection 3.1.1 discussing
the free fermionic Hamiltonian.
The last term in equation 3.1 denotes the free ener gy of photons with dispersion Ω σ
K ,k z
and annihilation (creation) operators d † σ
K ,k z with light polarization σ and momentum
3.3. Many P article Exciton Hamiltonian
Figure 3.9.: Schematic illustration of the exciton-light interaction. KK excitons (blue) couple
to left handed polarized light σ + and K K excitons couple to right handed polarized
light σ − .
( K ,k
z ) . Here, K denotes the projection of the three dimensional photon momentum
onto the semiconductor plane and k z the component perpendicular to the material.
3.3.2. Exciton-Light Coupling
The Hamiltonian describing the e xciton-light coupling can be obtained from the elec-
tron light Hamiltonian, equation 3.3 and reads
H x − lig ht =
µξ
M ∗ µξ · A P † ξξ
µ, 0 + h.c., (3.56)
with the v ector potential of the incident electromagnetic field A and the excitonic op-
tical matrix element M ∗ µξ = i e
m k u cξ
k |∇| u vξ
k ϕ ∗ ξξ
k , which is basically the projection
of the electronic optical matrix element onto e xcitonic wa vefunctions. Thus the opti-
cal selection rules translate from the electronic matrix element to the e xcitonic matrix
element, which is illustrated in figure 3.9 .A KK e xciton at the Γ v alle y can be created
by e xcitation with left handed circular polarized light. A K K e xciton can be created
by e xcitation with right handed polarized light [37, 47, 39, 48, 96, 63, 97]. In contrast
to the electronic picture where optical e xcitations occur mainly at the K v alley , e xci-
tons are created at the Γ v alley in the center of the e xcitonic Brillouin zone, reflecting
the fact, that a classical light field can not transfer a momentum in a perpendicular
geometry [109, 90].
Here, an interesting feature can be seen if one assumes the electronic matrix element to
be independent of the electron momentum. Then the definition of the excitonic matrix
element can be simplified to M ∗ µξ = i e
m u cξ
K |∇| u vξ
K ϕ ∗ ξξ
µ ( r = 0) , where the Fourier
transform of the e xcitonic wa vefunction was identified. As a result, only excitons with
a non-v anishing wa vefunction at the position r =0 in real space couple to electro-
magnetic fields. This is to say , that electron and hole need to hav e a non-v anishing
probability to be located at the same position in real space.
3. System Hamiltonian
3.3.3. Quantiz ed Exciton-Light Coupling
The Hamiltonian for the quantized exciton-photon interaction is obtained from the
Hamiltonian of the quantized electron photon interaction, equation 3.4, and reads
H x − phot = X
K ,k z ,µ,ξ ,σ
M µξ σ
K ,k z d † σ
K ,k z P ξ ξ
µ, K + h.c., (3.57)
with the e xciton-photon coupling element M µξ σ
K ,k z = P k M v ,c,ξ ,σ
k + K , k ,k z ϕ ξ ξ
µ, k . Here M v ,c,ξ ,σ
k + K , k ,k z
denotes the electron-photon coupling element. The exciton-photon coupling element
fulfills the same selection rules as the e xcitonic optical matrix element, i.e. an K K
( K 0 K 0 ) e xciton is created by absorbing a photon with left (right) handed polarization
σ + ( σ − ). The three dimensional photon momentum is explicitly tak en into account
and its transv ersal component directly translates as exciton center of mass momentum.
Ho we ver the optically induced center of mass momenta are typically v ery small and
on the order of 10 µ m − 1 , compare equation 5.8 and the follo wing discussion.
3.3.4. Exciton-Exciton Coupling
In this subsection the Hamiltonian of the e xciton-e xciton interaction in the lo w excita-
tion limit is introduced. It is obtained by transforming the carrier carrier Hamiltonian,
equation 3.6. Again, as for the carrier -carrier interaction the long range interaction and
the short range interaction is discussed separately .
Long Range Interaction
The lo w density contrib ution of the long range interaction to the excitonic Hamiltonian
reads
H LR
x − x = X
Q ,µ,µ 0 ,
ξ ,ξ 0
X µµ 0 ξ ξ 0
Q P † ξ ξ
µ, Q P ξ 0 ξ 0
µ 0 , Q , (3.58)
with X µµ 0 ξ ξ 0
Q = P k , k 0 X v cξ ξ 0
k , k 0 , Q ϕ ∗ ξ ξ
µ, k ϕ ξ 0 ξ 0
µ 0 k 0 denoting the e xciton-exciton coupling element.
X v cξ ξ 0
k , k 0 , q is the e xchange matrix element, stemming from the interband scattering of the
fermionic Hamiltonian, equation 3.9. The electronic matrix element takes account to
the fact that only e xcitons with equal electron and hole spins contribute to the e xchange
coupling. In the lo w wa v enumber approximation, where the momentum dependence of
the electronic matrix element can be omitted, the coupling element can be further sim-
plified to X µµ 0 ξ ξ 0
Q = X v cξ ξ 0
Q ϕ ∗ ξ ξ
µ ( r = 0) ϕ ξ 0 ξ 0
µ 0 ( r = 0) . As for the electronic Hamiltonian
three dif ferent processes can be distinguished: (i) interv alley e xchange coupling for A
( B ) e xciton, where the spin up K K exciton couples to spin do wn K 0 K 0 A ( B ) excitons,
i.e. ξ 0 = ¯
ξ (ii) intra v alley e xchange coupling, where the considered exciton couples to
itselfs, i.e. ξ 0 = ξ , (iii) coupling between A and B e xcitons where either i 0 = i and
s 0 6 = s or i 0 6 = i and s 0 = s holds. Ho we ver since the coupling between A and B
3.3. Many P article Exciton Hamiltonian
e xcitons is off resonant due to the lar ge splitting of the dispersion in the v alence band,
the latter processes can be remo ved from the Hamiltonian within a rotating frame ap-
proximation. Note, that the intraband scattering does not contribute to the lo w density
e xcitonic Hamiltonian. The electron-electron and hole-hole contrib ution only account
to an ener gy renormalization of the free dispersion and the electron-hole contribution
appears as e xciton binding term sho wing up in the W annier equation, equation 3.43.
These contrib utions are implicitly included in the free Hamiltonian, equation 3.55.
Shor t Rang e Interaction
F or reasons of completeness the contribution of the short range interaction to the e xci-
tonic Hamiltonian is discussed.
The lo w density contrib ution of the intraband interaction reads
H S R,intr a
x − x = X
Q ,µ,µ 0
ξ h ,ξ e ,ξ 0
h 6 = ξ h ,ξ 0
e 6 = ξ e
V ξ h ξ e ξ 0
h ξ 0
e
µ,µ 0 , Q P † ξ h ξ e
µ, Q P ξ 0
h ξ 0
e
µ 0 , Q . (3.59)
V ξ h ξ e ξ 0
h ξ 0
e
µ,µ 0 , Q = P q , q 0 V v ci h i e i h − i 0
h s h s e
q + α ξ h ξ e Q , q − β ξ h ξ e Q , q 0 ϕ ∗ ξ h ξ e
µ q − q 0 ϕ ξ 0
h ξ 0
e
µ 0 q δ s h ,s 0
h
s e ,s 0
e δ i 0
h − i h ,i 0
e − i e denotes the e xci-
tonic coupling element. Here only the contribution of the electron hole scattering trans-
forms to the lo w density contrib ution of the Hamiltonian. The electron-electron and
the hole-hole scattering do not contrib ute. F ormally the short range interaction part
of the Hamiltonian is equi v alent to the momentum space representation of the Dexter
coupling in realspace [125, 100]. In the electronic picture it can be seen as the tran-
sition of an electron from one v alley to another which is assisted by the interv alley
transition of a hole, where the momentum dif ferences match.
The lo w density contrib ution of the interband interaction reads
H S R,inter
x − x = X
Q ,µ,µ 0 ,
ξ h ,ξ e ,ξ 0
h ,ξ 0
e
X ξ h ,ξ e ξ 0
h ξ 0
h
µµ 0 Q P † ξ h ξ e
µ, Q P ξ 0
h ξ 0
e
µ 0 , Q . (3.60)
Here, the matrix element reads X ξ h ,ξ e ξ 0
h ξ 0
h
µµ 0 Q = P k , k 0 X v ccv i h i 0
e i e − i h ,s h s e
k , k 0 , q ϕ ∗ ξ h ξ e
µ k ϕ ξ 0
h ξ 0
e
µ 0 k 0 . It
ensures the conserv ation of the global center of mass momentum and the conserv a-
tion of the spin during the interband transitions of the underlying electronic processes
X ξ h ,ξ e ξ 0
h ξ 0
h
µµ 0 Q ∝ δ s h ,s e
s 0
h ,s 0
e δ i e − i h ,i 0
e − i 0
h .
3. System Hamiltonian
3.3.5. Exciton-Phonon Coupling
Last the Hamiltonian of the e xciton-phonon coupling is discussed. The Hamiltonian
reads
H x − phon = X
Q , K ,µ,µ 0 ,ξ h ,ξ e ,j
g ξ h ξ e j α
eµµ 0 K P † ξ h ( ξ e + j )
µ, Q + K P ξ h ξ e
µ 0 , Q −
− X
Q , K ,µ,µ 0 ,ξ e ,ξ h ,j
g ξ h ξ e j α
hµµ 0 K P † ( ξ h − j ) ξ e
µ, Q + K P ξ h ξ e
µ 0 , Q ! b † α ( − j )
− K + b αj
K , (3.61)
where the first term in brackets accounts for the underlying electron scattering and the
second term accounts for scattering of holes. The transitions of excitons are assisted
by emission or absorption of phonons. The sum runs ov er all possible initial exciton
states with electron and hole v alley-spin ξ e and ξ h and center of mass momentum µ .
Further the sum runs o ver the e xciton quantum numbers of initial µ 0 and final state
µ as well as o ver the transferred phonon momentum K of the mode α from the high
symmetry point j .
The matrix elements appearing in the e xciton-phonon Hamiltonian, equation 3.61, read
[70, 75, 124]
g ξ h ξ e j α
eµµ 0 K = X
q
ϕ ∗ ξ h ( ξ e + j )
µ, q + β v h v K ϕ ξ h ξ e
µ 0 q g c v v 0 α
K δ ss 0 (3.62)
g ξ h ξ e j α
hµµ 0 K = X
q
ϕ ∗ ( ξ h − j ) ξ e
µ, q − α v v e K ϕ ξ h ξ e
µ 0 q g v v v 0 α
K δ ss 0 . (3.63)
Both, the coupling element for electron and hole scattering consist of the respecti v e
electronic matrix elements g c/v v v 0 α
q 0 and an e xcitonic form factor P q ϕ ∗ ξ h ξ
µ, q + β v h v q 0 ϕ ξ h ξ 0
µ 0 q .
The spin delta functions in the coupling elements account for the fact, that the electron
and hole spins can not be flipped during a phonon scattering e vent.
F or intrav alley scattering, i.e. j = Γ , the Hamiltonian can be simplified to
H x − phon = X
Q , K ,µ,µ 0 ,ξ h ,ξ e
g intra ξ h ξ e α
µµ 0 K P † ξ h ξ e
µ, Q + K P ξ h ξ e
µ 0 , Q b † α Γ
− K + b α Γ
K , (3.64)
with the intra v alley e xciton-phonon coupling element g intra ξ h ξ e α
µµ 0 q 0 = g ξ h ξ e Γ α
eµµ 0 q 0 − ˜ g ξ h ξ e Γ α
hµµ 0 q 0 . In
general the intra v alley coupling strength is gi ven by the dif ference of the conduction
band electron- and the v alence band electron-phonon coupling strength. Therefore the
signs of both coupling elements ha ve a crucial impact on the e xciton-phonon coupling
strength. F or coupling mechanisms which depend on the charge of the in volv ed carrier
which are Fr ¨
ohlich coupling and piezoelectric coupling, both, the v alence band and the
conduction band matrix element ha ve the same sign, equation 3.18. Hence in TMDs
where the electron hole mass are comparable, the excitonic form f actors in equation
3.64 are equal and excitons do not couple in Fr ¨
ohlich and piezoelectric coupling when
3.3. Many P article Exciton Hamiltonian
the theoretical in vestig ation is restricted to the lo west lying 1 s exciton state [124].
Thus, the LO mode which couples mainly in Fr ¨
ohlich coupling can be ne glected for
the intra v alley scattering. Further the piezoelectric contribution of the T A and LA
phonon coupling is ne glected in the following.
4. Linear Spectr oscop y and
Excitonic Line width
In this chapter the impact of e xcitons on the properties of the linear spectrum of TMDs
is discussed. Therefore, the fundamental equations of motion, the so called e xcitonic
Bloch equations, are deri ved [110, 87, 109, 90, 118]. Having kno wledge about the
Bloch equations, the absorption coef ficient can be computed quantitati vely , which
results in the famous Elliott formula [110, 63]. Afterwards, dif ferent origins of the
broadening of the e xcitonic line will be discussed. In principle dominating mechanism
determining the line shape are exciton light interaction and e xciton phonon coupling
in the lo w excitation limit [43, 70]. Interestingly , it turns out that despite a very pro-
nounced radiati ve broadening of some meV , dark states in tungsten compounds are
v ery fa vorable for the phonon assisted relaxation of e xcitons, leading to a significant
broadening of the e xcitonic lines ev en at cryogenic temperatures [70].
4.1. Excitonic Bloch Equation and Elliott Form ula
In this section, the excitonic spectrum in linear spectroscopy is discussed. F or the
computation of the e xcitonic absorption coefficient, an e xpression for the excitonic
polarization is required, equation 3.53. It can be obtained by applying Heisenber gs
equation of motion to the excitonic polarization P ξ h ξ e
µ, Q under usage of the excitonic
Hamiltonian by e xploiting the fundamental bosonic commutation relations, equations
3.37 and 3.38. The excitonic Bloch equation in the lo w excitation limit reads
i ~ ∂ t P ξ h ,ξ e
µ, Q = E ξ h ξ e
µ, Q − iγ ξ h ,ξ e
µ, Q P ξ h ,ξ e
µ, Q + M ∗ µξ h · A δ ξ h ,ξ e
Q , 0 + X
σ ,k z
M ∗ µξ h σ
Q ,k z d σ
Q ,k z δ ξ h ,ξ e
+ X
µ 0 ,ξ 0
h
X ξ h ξ 0
h
µµ 0 Q δ ξ h ,ξ e P ξ 0
h ξ 0
h
µ 0 Q + X
ξ 0
h 6 = ξ h ξ 0
e 6 = ξ e ,µ 0
V ξ h ξ e ξ 0
h ξ 0
e
µ,µ 0 , Q δ s e ,s 0
e
s h ,s 0
h P ξ 0
h ξ 0
e
µ 0 , Q
+ X
K ,α,µ 0
g intra ξ h ξ e α
µ,µ 0 , K S ξ h ξ e Γ α
µ 0 , Q − K , K + ˜
S ξ h ξ e ( − Γ) α
µ 0 , Q − K , − K
+ X
K ,α,µ 0 ,j
g ξ h ξ e j α
eµ,µ 0 , K S ξ h ( ξ e − j ) j α
µ 0 , Q − K , K + ˜
S ξ h ( ξ e − j )( − j ) α
µ 0 , Q − K , − K
+ X
K ,α,µ 0 ,j
g ξ e ξ h j α
hµ,µ 0 , K S ( ξ h + j ) ξ e j α
µ 0 , Q − K , K + ˜
S ( ξ h + j ) ξ e ( − j ) α
µ 0 , Q − K , − K . (4.1)
4.1. Excitonic Bloch Equation and Elliott Formula
The first term in the first line describes the oscillation of the excitonic polarization
with the e xcitonic ener gy . T o account for a damping of the excitonic polarization, a
for no w phenomenological dephasing constant γ ξ h ,ξ e
µ, Q was introduced. The second term
describes the e xcitation of the excitonic polarization with a classical electromagnetic
field. The Kroneck er - δ ensures the spin conserv ation and takes account to the fact, that
in a perpendicular geometry , no center of mass momentum of the e xciton is excited.
The third term describes the interaction of an exciton with a quantized electromagnetic
field. Here, the transv erse photon momentum is directly transferred to the exciton as
a consequence of the momentum conserv ation. Ho we ver for the in vestig ation of the
properties under plane wa v e laser e xcitation of the material in a perpendicular geom-
etry , the latter is ne glected in the following. The second line describes the Coulomb
coupling of the e xcitonic polarization. The first term accounts for the interv alley e x-
change coupling, which is only acti ve for e xcitons with same electron and hole spins.
Note, that the coupling element v anishes in the limit Q = 0 meaning that this term
has no direct impact on optical bright e xcitons, equation 3.10. The second term in
the second line describes the De xter-lik e interv alle y coupling. The matrix element re-
quires spin conserv ation during the scattering process [126, 127]. Assuming a bright
A 1s e xciton on the left hand side of the Bloch equation, the energetically lo west lying
state would be a B 1s e xciton in the opposite v alley which is ener getically separated
by hundreds of meV depending on the material. Therefore this coupling is strongly
of f-resonant and is neglected in the follo wing. The third, fourth and fifth line describe
the interaction of the excitonic polarization with phonons. The third line describes the
interaction with Γ phonon, i.e. intra v alley scattering of both, electron and hole. Line
four (fi ve) describe the interaction with zone edge phonons with the electrons (holes)
forming the e xciton. Here the so called phonon assisted excitonic polarizations are
defined as S ξ h ξ e j α
µ 0 , Q + K , K = h P ξ h ξ e
µ 0 , Q + K b j α
K i and ˜
S ξ h ξ e − j α
µ 0 , Q + K , K = h P ξ h ξ e
µ 0 , Q + K b †− j α
− K i .
T o in vestigate the linear spectrum all coupling processes e xcept the interaction with the
classical light field are neglected for no w . Then the resulting optical Bloch equation
may be F ourier transformed and inserted in the expression for the absorption coef fi-
cient. For simplicity the v ector potential is projected on left handed and right handed
light polarizations σ = σ + , σ − , equation 3.53. The result is the well kno wn Elliott
formula, which describes the e xcitonic absorption spectrum
α ( ω ) = 1
0 c 0 nω X
ξ ,µ,σ
| M ξ µσ | 2 γ
( ~ ω − E ξ ξ
µ, 0 ) 2 + γ 2 . (4.2)
Here, M ξ µσ denotes the excitonic matrix element which was projected on circular po-
larized light modes. T o account for radiati ve and non-radiati ve dephasing a for no w
phenomenological damping constant γ = 10 meV was chosen for the e xcitonic po-
larization being independent of the excitonic state µ . This damping constant will be
discussed in the follo wing section. As already can be seen from equation 4.2, the
absorption spectrum is gi ven by a sum of Lorentzian lines. The sum runs ov er all op-
tically bright e xciton states µ and valle y spin ξ = ( i, s ) . T o the e xciton state µ both,
4. Linear Spectroscopy and Excitonic Line width
Figure 4.1.: Excitonic absorption spectrum. T otal absorption spectrum (yello w) of MoSe 2 (a) and
WS 2 (b) as a function of the photon energy with respect to the free particle bandg ap.
The total spectrum consists of the spectrum of the A series (blue) and the B series
(red). Here for all exciton states, a phenomenological broadening of γ = 10 meV w as
assumed.
bound and unbound electron hole pairs, contrib ute which are obtained as a solution of
the W annier equation, equation 3.43. F or the excitonic spectrum close to the band gap,
only transition at the K and K point ha ve to be considered.
Figure 4.1 illustrates the absorption spectrum of monolayer MoSe 2 (a) and monolayer
WS 2 (b). In both materials, the spectrum sho ws contributions from the A transition
( s = ↑ ) and the B transition ( s = ↓ ). Both contributions are separated by the splittings of
conduction and v alence band due to spin orbit interaction of about 200 meV in MoSe 2
and 400 meV in WS 2 . Both contrib utions sho w distinct features belo w the free particle
band gap, which are ascribed as absorption of bound excitons. Here the lo west lying
e xciton of each series is the 1 s exciton, the next is the 2 s e xciton and so on. Abo ve
the free particle bandgap, the absorption is constant originating from the continuum
of unbound electron hole pairs, which are also obtained as solutions of the W annier
equation, equation 3.43, and enter in the Elliott formula, equation 4.2. In MoSe 2 , the
A 1 s and the B 1 s absorption peaks appear well belo w the free particle band gap, since
the binding ener gies of both excitons are about 450 meV e xceeding clearly the splitting
due to spin orbit coupling. Interestingly in WS 2 , the spin orbit splitting and the exciton
binding ener gies of the lo west lying excitons match almost, leading to the f act, that the
B 1 s e xciton resonance already appears in the free particle bandgap of the A transition.
F or the rest of the thesis, when it comes to the computation of the spectral width of
e xcitons and photoluminescence dynamics, only the A 1 s exciton is considered.
4.2. Excitonic Line width in the Monolayer
4.2. Excitonic Line width in the Monola yer
In this section, the excitonic line width in the absorption spectrum of monolayer of
transition metal dichalcogenides is discussed. In general, there are two qualitati ve dif-
ferent origins of the line width in the absorption spectrum which are the homogeneous
and the inhomogeneous broadening. The reason for the inhomogeneous broadening
are impurities in the monolayer sample such as lattice missmatches or adatoms [128].
This leads to local v ariations of the energies of the e xcitonic resonances resulting in
a Gaussian broadened line [43, 129]. For thicker tw o dimensional confined excitons
such as quantum wells, also v ariations in the thickness of the sample contrib ute to the
inhomogeneous broadening [130, 131]. Since the inhomogeneous broadening depends
on the quality of a gi ven sample it is not of interest of this section. In monolayers of
TMDs the inhomogeneous broadening is typically on the order of some meV [43, 129].
Recent studies demonstrated that the inhomogeneous broadening can be reduced if the
sample is encapsulated in a substrate such as boron nitride which is addressed to a
reduction of v ariation of dirt on the sample [132, 72]. The homogeneous line width,
ho we ver , is gi v en by the dephasing of the Bloch equation and can be identified as the
in verse coherence lifetime of the e xciton meaning that it is an intrinsic material prop-
erty [110, 70]. The dephasing of the Bloch equation is determined by microscopic
scattering mechanisms such as radiati ve coupling of the e xciton, exciton phonon scat-
tering or e xciton exciton scattering. The e xciton exciton scattering depends strongly
on the e xcitation density and can be neglected in the lo w e xcitation limit. The homo-
geneous width is typically about fe w meV at 4 K and about some tens of meV at room
temperature [43, 70, 72, 132]. Using two-dimensional spectroscop y Fourier transform
spectroscopy , the homogeneous width has been measured recently for cryogenic tem-
peratures, finding a linear increase of the line width as a function of temperature in
WSe 2 which was ascribed to acoustic phonon scattering [43]. Other studies report a
linear as well as a super -linear increase of the homogeneous width as a function of
temperature, where the latter is a strong indication for scattering with optical phonons
[71, 133, 134, 135].
4.2.1. Radiative Br oadening
In this subsection the radiati ve contrib ution to the homogeneous line width is discussed.
The starting point is the excit onic Bloch equation, equation 4.1, under ne glection of
e xciton-photon, exciton-e xciton and exciton-phonon interaction. It reads
i ~ ∂ t P ξ ,ξ
µ, 0 = E ξ ξ
µ, 0 P ξ ,ξ
µ, 0 + X
σ
M ∗ µσ ξ A σ . (4.3)
Excitonic polarizations which can not be e xcited optically ( ξ e 6 = ξ h and Q 6 = 0 )
ha ve been ne glected. Since the vector potential of the incident electromagnetic field is
kno wn to ha ve two linear independent polarizations, the polarization v ector of the field
4. Linear Spectroscopy and Excitonic Line width
was already projected onto the optical matrix element M ∗ µσξ = M ∗ µξ · σ . From the
Maxwell equations, the inhomogeneous Helmholtz equation can be obtained
∇ 2 − n 2
c 2 ∂ 2
t E ( r , t ) = µ 0 ∂ t j ( r , t ) , (4.4)
where E ( r , t ) denotes the electromagnetic field and j ( r , t ) the current density in the
material, c the v elocity of light and n the refrectiv e index of the surrounding mate-
rial. The electromagnetic field e xhibits two linear independent polarizations (i.e. left
handed and right handed polarized light modes). Assuming that the electromagnetic
field propagates in z -direction perpendicular to the monolayer plane, the Helmholtz
equation for each polarization component is simplified to
∂ 2
z − n 2
c 2 ∂ 2
t E σ ( z , t ) = µ 0 ∂ t j σ ( z , t ) . (4.5)
This equation can be solv ed according to the references [109, 110] and one obtains
E σ + ( t ) = E σ +
0 ( t − z n/c ) − cµ 0
2 n j σ + ( t − | z − z 0 | /c ) . (4.6)
Here, E σ +
0 denotes the initial condition, i.e. the incoming light field, and the second
term denotes the response from the material. F ourier transforming this equation, and
identifying E ( t ) = − ∂ t A ( t ) , with A ( t ) denoting the v ector potential, yields
A σ ( ω ) = A σ
0 ( ω ) + icµ 0
2 nω j σ ( ω ) . (4.7)
Expressing the macroscopic current density in terms of the microscopic e xcitonic po-
larization, equation 3.52, it follo ws
A σ ( ω ) = A σ
0 ( ω ) + cµ 0
2 nω X
µ,ξ
M ξ σ µ P ξ ξ
µ, 0 + h.c. ! . (4.8)
The last step is to insert this e xpression in the Fourier transform of the e xcitonic optical
Bloch equation, equation 4.3. The result can be written as
P ξ
µ, 0 ( ω ) = X
σ
M µξ σ A σ
0 ( ω )
~ ω − E ξ
µ, 0 − iγ ξ σ µ
r ad
, (4.9)
where the radiati ve broadening can be identified
γ ξ σ µ
r ad = cµ 0
2 nω | M µξ σ | 2 . (4.10)
In correspondence to a first order perturbation theory , one obtains a Fermi Golden rule
for the radiati ve broadening, where the radiati ve dephasing is basically gi v en by the
square of the e xcitonic optical matrix element.
4.2. Excitonic Line width in the Monolayer
4.2.2. Non-Radiative Br oadening
In this subsection the non-radiati ve contrib ution to the homogeneous line width is dis-
cussed. Starting point is the excitonic Bloch equation, equation 4.1, ne glecting the
e xciton-exciton, e xciton-photon and the exciton field coupling.
i ~ ∂ t P ξ h ,ξ e
µ, Q = E ξ h ξ e
µ, Q P ξ h ,ξ e
µ, Q
+ X
K ,α,µ 0
g intra ξ h ξ e α
µ,µ 0 , K S ξ h ξ e Γ α
µ 0 , Q − K , K + ˜
S ξ h ξ e ( − Γ) α
µ 0 , Q − K , − K
+ X
K ,α,µ 0 ,j 6 =Γ
g ξ h ξ e j α
eµ,µ 0 , K S ξ h ( ξ e − j ) j α
µ 0 , Q − K , K + ˜
S ξ h ( ξ e − j )( − j ) α
µ 0 , Q − K , − K
+ X
K ,α,µ 0 ,j
g ξ e ξ h j α
hµ,µ 0 , K S ( ξ h + j ) ξ e j 6 =Γ α
µ 0 , Q − K , K + ˜
S ( ξ h + j ) ξ e ( − j ) α
µ 0 , Q − K , − K . (4.11)
The e xcitonic polarizations couple to phonon assisted polarizations, which were de-
fined in the be ginning of this section. T o obtain an e xpression for the phonon induced
dephasing of the e xcitonic polarization, equations of motion for the phonon assisted
polarizations ha ve to be deri ved. They read
i ~ ∂ t S ξ h ξ e j α
µ Q , K = E ξ h ξ e
µ, Q + ~ ω j α
K − i ˜ γ ξ h ξ e
µ Q S ξ h ξ e j α
µ Q , K
+ X
ν
g ξ h ξ e ( − j ) α
eµν − K P ξ h ( ξ e + j )
ν, Q + K 1 + n j α
K − X
ν
g ξ e ξ h ( − j ) α
hµν − K P ( ξ h − j ) ξ e
ν Q + K 1 + n j α
K
(4.12)
i ~ ∂ t ˜
S ξ h ξ e j α
µ Q , K = E ξ h ξ e
µ, Q − ~ ω j α
K − i ˜ γ ξ h ξ e
µ Q ˜
S ξ h ξ e j α
µ Q , K
+ X
ν
g ξ h ξ e j α
eµν K P ξ h ( ξ e − j )
ν, Q − K n j α
K − X
ν
g ξ e ξ h j α
hµν K P ( ξ h + j ) ξ e
ν Q − K n j α
K . (4.13)
Here in both equation the first term on the right hand side accounts for the free oscil-
lation of the phonon assisted polarizations. The second term on the right hand side
accounts for the electron scattering with phonons whereas the third term on the right
hand side in both equations accounts for the hole scattering with phonons. T o treat the
upcoming hierarchy problem, four operator correlations ha ve been ne glected [110].
n j α
K = δ h b † j α
K b j α
K i denotes the phonon occupation and is treated in bath approximation
[88]. Contrib utions from exciton field, exciton photon and e xciton exciton interac-
tion ha ve been ne glected, since for the broadening only terms in the second order of
the matrix elements are taken into account. T o respect higher order contrib utions, a
dephasing for the phonon assisted e xcitonic polarizations ˜ γ ξ h ξ e
µ Q was introduced. As
a first approximation, this can be set to be equal with the dephasing of the excitonic
polarization [136].
4. Linear Spectroscopy and Excitonic Line width
Born Mark ov Appr o ximation
In principle the system of coupled dif ferential equations, equations 4.11 and 4.13, can
be solv ed within a Born Markov approximation [87, 88]. Therefore the general pro-
cedure shall be briefly discussed here. Starting point is the equation of motion of
the phonon assisted e xcitonic coherence S ξ h ξ e Γ α
µ Q , K . For reasons of simplicity only the
intra v alley contrib ution leading to phonon emission processes is considered in the fol-
lo wing. The treatment of the remaining contributions is equi v alent. The equation of
motion reads
∂ t S ξ h ξ e Γ α
µ Q , K = 1
i ~ E ξ h ξ e
µ, Q + ~ ω Γ α
K − i ˜ γ ξ h ξ e
µ Q S ξ h ξ e Γ α
µ Q , K
+ 1
i ~ X
ν
g intra ξ h ξ e α
µν − K 1 + n Γ α
K P ξ h ξ e
ν, Q + K (4.14)
The first line on the right hand side accounts for the homogeneous part of the dif feren-
tial equation whereas the second line accounts for the inhomogeneous part. A formal
inte gration of this equation yields
S ξ h ξ e Γ α
µ Q , K ( t ) = 1
i ~ Z ∞
0
ds e s
i ~ E ξ h ξ e
µ, Q + ~ ω Γ α
K − i ˜ γ ξ h ξ e
µ Q X
ν
g intra ξ h ξ e α
µν − K 1 + n Γ α
K P ξ h ξ e
ν, Q + K ( t − s ) .
(4.15)
This e xpression can no w be inserted into the equation of motion for the excitonic po-
larization, equation 4.11,
∂ t P ξ h ,ξ e
µ, Q ( t ) = 1
i ~ E ξ h ξ e
µ, Q P ξ h ,ξ e
µ, Q − 1
~ 2 X
K ,α,µ 0 ,ν
g intra ξ h ξ e α
µ,µ 0 , K g intra ξ h ξ e α
µ 0 ν − K 1 + n Γ α
K ×
× Z ∞
0
ds e s
i ~ E ξ h ξ e
µ 0 , Q − K + ~ ω Γ α
K − i ˜ γ ξ h ξ e
µ 0 Q P ξ h ξ e
ν, Q ( t − s ) . (4.16)
One obtains an inte gro-differential equation for the excitonic polarization. The e xci-
tonic polarization at a time t is determined by the e xcitonic polarization at past times
t − s . T o solv e the equation within a Marko v approximation, one writes the e xci-
tonic polarization into the rotating frame P ξ h ,ξ e
µ, Q ( t ) = ˜
P ξ h ,ξ e
µ, Q ( t ) exp ( t
i ~ E ξ h ξ e
µ, Q ) . Addition-
ally , to restrict to the diagonal dephasing, coupling to other excitonic states is omitted
ν = µ . One obtains
∂ t ˜
P ξ h ,ξ e
µ, Q ( t ) = − 1
~ 2 X
K ,α,µ 0 | g intra ξ h ξ e α
µ,µ 0 , K | 2 1 + n Γ α
K ×
× Z ∞
0
ds e s
i ~ E ξ h ξ e
µ 0 , Q − K − E ξ h ξ e
µ, Q + ~ ω Γ α
K − i ˜ γ ξ h ξ e
µ 0 Q ˜
P ξ h ξ e
µ, Q ( t − s ) . (4.17)
Due to the lar ge extension of the semiconductor plane, the K summation is performed
o ver a continuous v ariable (thermodynamic limit). Therefore it may be written as an
4.2. Excitonic Line width in the Monolayer
inte gral P K = A
4 π 2 R R 2 d 2 K . Here, A denotes the area of the semiconductor plane.
It cancels with the Area in the e xciton phonon coupling element. Reg arding the K
summation, the exponential factor in equation 4.17 represents a f ast oscillation term.
Under the assumption that ˜
P ξ h ξ e
µ, Q ( t − s ) v aries much slo wer in time compared to the
e xponential factor , which is v alid as long as the phonon coupling induced dynamics of
˜
P ξ h ξ e
µ, Q ( t − s ) is weak compared to the ener gy scale in the e xponent, it can be approx-
imated ˜
P ξ h ξ e
µ, Q ( t − s ) ≈ ˜
P ξ h ξ e
µ, Q ( t ) and can be taken out of the inte gral. The remaining
inte gral is ev aluated to gi ven
∂ t ˜
P ξ h ,ξ e
µ, Q ( t ) = − 1
~ X
K ,α,µ 0 | g intra ξ h ξ e α
µ,µ 0 , K | 2 1 + n Γ α
K ˜
P ξ h ξ e
µ, Q ( t ) ×
× π L ˜ γ ξ h ξ e
µ 0 Q
( E ξ h ξ e
µ 0 , Q − K − E ξ h ξ e
µ, Q + ~ ω Γ α
K ) + i E ξ h ξ e
µ 0 , Q − K − E ξ h ξ e
µ, Q + ~ ω Γ α
K
( E ξ h ξ e
µ 0 , Q − K − E ξ h ξ e
µ, Q + ~ ω Γ α
K ) 2 + ( ˜ γ ξ h ξ e
µ 0 Q ) 2 ! .
(4.18)
The real part yields a Cauchy distrib ution L ˜ γ ξ h ξ e
µ 0 Q
with the width ˜ γ ξ h ξ e
µ 0 Q . The broad-
ening stems from the dephasing of the phonon assisted polarization. It is assumed
to conincide with the dephasing of the e xcitonic polarization [136], which yields a
self consistent treatment of the phonon induced dephasing of the excitonic coherence.
The imaginary part yields a Cauchy principle v alue which is in general div er gent. It
accounts for an ener gy renormalization which is not in the interest of this work. There-
fore it is ne glected in the follo wing. Applying the scheme also to the other equations of
motion for the phonon assisted quantities including also interv alley scattering, equation
4.13, yields the e xpression for the phonon induced dephasing. It reads [70, 137, 138]
γ µξ h ξ e
Q = π X
K ,α,µ 0 , ± | g intra ξ h ξ e α
µ,µ 0 , K | 2 1
2 ± 1
2 + n α Γ
K L γ ξ h ξ e
µ 0 , Q + K E ξ h ξ e
µ, Q − E ξ h ξ e
µ 0 , Q + K ∓ ~ ω α Γ
K
+ π X
K ,α,µ 0 ,j 6 =Γ , ± | g ξ h ξ e j α
eµµ 0 , K | 2 1
2 ± 1
2 + n αj
K L γ ξ h ( ξ e + j )
µ 0 , Q + K E ξ h ξ e
µ, Q − E ξ h ( ξ e + j )
µ 0 , Q + K ∓ ~ ω αj
K
+ π X
K ,α,µ 0 ,j 6 =Γ , ± | g ξ e ξ h j α
hµµ 0 , K | 2 1
2 ± 1
2 + n αj
K L γ ( ξ h − j ) ξ e
µ 0 , Q + K E ξ h ξ e
µ, Q − E ( ξ h − j ) ξ e
µ 0 , Q + K ∓ ~ ω αj
K .
(4.19)
Here, the first term accounts for intra v alley scattering, the second term for the inter-
v alley scattering of electrons and the third term for interv alley scattering of the holes.
F or each contrib ution the sum runs ov er the phonon momentum with respect to its high
symmetry point K , the phonon mode α and the quantum number of the final exciton
state µ . Additionally , the sum runs ov er the electron v alley inde x ξ 0
e and hole v alley
ξ 0
h of the final e xciton state for the interv alley scatterings. The ± sum accounts for the
fact that both, phonon emission ( + ) and phonon absorption ( − ) e vents contrib ute to
4. Linear Spectroscopy and Excitonic Line width
the dephasing rate. The dephasing rate depends on the square of the e xciton phonon
matrix elements, on the phonon occupation n αi
K for phonon absorption processes and
on (1 + n αi
K ) for phonon emission processes. Here, the term proportional to 1 accounts
for spontaneous emission of a phonon and the term proportional to n αi
K accounts for
the stimulated phonon emission. Note, that the spontaneous emission does not depend
on the temperature and is e ven present for v anishing temperatures. The Lorentzians
L γ (∆ E ) with width γ and ener gy maximum ∆ E account for the ener gy conserv ation
during a phonon scattering e vent. From the exciton phonon matrix elements, equation
3.63 it follo ws directly that initial and final e xciton state need to hav e the same spin
configuration of the underlying electron and hole states.
4.2.3. Numerical Ev aluation of the Non-Radiative
Br oadening
Here, the numerical e v aluation of the non-radiativ e broadening, equation 4.19, shall be
discussed briefly . Therefore for reasons of simplicity the interv alley contrib ution are
ne glected in the following. The discussed method also applies for these cases.
First, the appearing K sum can be written as an inte gral
γ Q = A
4 π X
α, ± Z R 2
d 2 K | g intra α
K | 2 1
2 ± 1
2 n α Γ
K L γ Q + K E Q − E Q + K ∓ ~ ω α Γ
K . (4.20)
Here, the appearing area of the surface A cancels with the surface in the e xciton phonon
coupling element. As it is ob vious, the broadening of a gi v en state γ Q depends also on
the broadenings of all other states entering in the Lorentzian, which ensures the energy
conserv ation. A strategy to o vercome this problem is to compute the non-radiati ve
broadening self consistently . Therefore, the broadening in the first step γ 1
Q is computed
with the initial v alue of γ 0
Q = 0 . In the i ’ th step the broadening γ i
Q is e v aluated using
the broadening obtained in the ( i − 1) ’ th step γ i − 1
Q . This procedure is repeated until
a con ver gence is reached. According to experience about 20 iterations are required to
obtain a well con ver ged value for the broadening.
T o e v aluate the non-radiati ve dephasing the K inte gral has to be written as a Riemann
inte gral in polar coordinates
Z R 2
d 2 K → X
K,φ K
∆ φ ∆ K K . (4.21)
Since the center of mass coordinates are introduced with respect to the high symme-
try point where the exciton is located in the e xcitonic Brillouin zone ( Q denotes the
momentum with respect to the gi ven high symmetry point), the procedure also applies
for interv alley scattering, since no global coordinates enter in the dephasing rates. The
non-radiati ve rates con ver ge with a discretization of 80 K -points and 40 φ -points. The
4.2. Excitonic Line width in the Monolayer
Figure 4.2.: Relaxation channels determining the excitonic coher ence lifetime. A K − K ex -
citon in the Γ v alley can decay via radiati ve decay γ rad (blue arrow) or non-radiati ve
dephasing γ i , with i being the corresponding phonon momentum. The latter occurs
through exciton-phonon scattering within the K − K v alley (orange) with Γ phonons
or to dark K − Λ (red) or K − K (bro wn) excitonic states mediated by Λ (red arro w)
and K phonons (bro wn arrow). For WS 2 , the indirect K − Λ and K − K e xcitons lie
ener getically below the bright K − K e xciton allowing ef ficient scattering via emis-
sion of phonons e ven at v ery low temperatures.The dashed dispersion curv es refer to a
situation typical in MoSe 2 , where the indirect exciton states are located ener getically
abov e the bright state. This picture was published in a similar form in reference [70].
results discussed in the ne xt section are obtained with a discretization of 160 K -points
and 80 φ K -points for each high symmetry point.
4.2.4. Excitonic Line width in WS 2 and MoSe 2
In this subsection the e xcitonic linewidth in WS 2 and MoSe 2 is discussed quantita-
ti vely and compared to e xperimental results which were obtained from reflectance and
photoluminescence measurements in a collaborati ve w ork with the Stanford Uni ver -
sity (California) and the Uni versity of Re gensb urg. The results of this section where
published in Nature Communications, reference [70].
The radiati ve dephasing rate as well as the phonon induced broadening, equation 4.19
and 4.10, are e valuated numerically and inserted in the Elliott formula equation 4.2.
F or the phonon induced broadening the complex e xcitonic band structure is explicitly
taken into account including intra valle y as well as interv alley scattering channels. Due
to the lar ge binding energies of the e xcitons in TMDs, the higher lying excitons are
ener getically separated by at least half of the binding energy justifying that higher
e xcitonic states can be neglected for the computation of the homogeneous width of the
lo west lying exciton.
Figure 4.2 illustrates the microscopic origins of the homogeneous broadening, which
are radiati ve decay and e xciton phonon scattering. F or the phonon scattering all rel-
e v ant states in the Brillouin zone are taken into account. In WX 2 energetically lo w
lying states at the Λ and the K v alley are e xpected to ha ve a significant impact on the
broadening since the y are fa v orable for phonon scattering. In contrast, in MoX 2 these
dark states are located well abov e the bright state. Thus only a minor impact of these
states is e xpected.
4. Linear Spectroscopy and Excitonic Line width
Figure 4.3.: Excitonic absorption. Absorption spectrum of (a) WS 2 and (b) MoSe 2 for the ener-
getically lo west lying 1s A exciton including radiati ve coupling (blue) and radiati ve
and exciton phonon coupling (yello w to black) into account. All spectra ha ve been
normalized with respect to the maximum absorption. This picture was published in a
similar form in reference [70].
Figure 4.4.: Excitonic absorption and excitonic polarization. The en velope of the e xcitonic po-
larization in (a) WS 2 and (b) MoSe 2 after optical excitation with a 20 fs pulse at t =0
(filled curve). Color code as in figure 4.3. This picture was published in a similar form
in reference [70].
Figure 4.3 illustrates the absorption spectrum of the 1s A e xciton for the two represen-
tati ve materials WS 2 (a) and MoSe 2 (b). The spectra are broadened due to radiati ve
decay by 7 meV full width at half maximum in WS 2 and about 4.3 meV in MoSe 2 . The
reason for the higher radiati ve broadening in WS 2 compared to MoSe 2 is a lar ger oscil-
lator strength [41]. The observed radiati ve broadening can be attrib uted to the lifetime
of an optically injected coherent e xciton of 100 fs in WS 2 and 150 fs in MoSe 2 . These
numbers are in e xcellent agreement with results from pre vious theoretical [44, 139]
and e xperimental [43] studies. In both materials the line width is increasing as a func-
tion of temperature. This can be adressed to the impact of exciton phonon coupling.
While at lo w temperatures the spectra including radiati ve and phonon broadening are
only slightly broader than the spectra taking only radiati ve broadening into account,
the e xciton phonon coupling dominates the spectral width at room temperature. Here,
the broadening is significant lar ger compared to lo w temperatures.
In figure 4.4 the temporal e v olution of the en velopes of the e xcitonic coherence, equa-
tion 4.11, in WS 2 (a) and MoSe 2 (b) is depicted. In general the same temperature
trends as for the width of the e xcitonic lines can be observed: While at croygenic
4.2. Excitonic Line width in the Monolayer
Figure 4.5.: Excitonic linewidth and lifetime in WS 2 Radiati ve dephasing (blue) and non-
radiati ve dephasing contrib ute to the homogeneous line width. T o the latter contribute
scattering with Γ phonons as well as scattering K phonons (bro wn) and Λ phonons.
This picture was published in a similar form in reference [70].
temperatures the decay constant are still in the order of 100 fs in both materials, the
decay constant at room temperature drops on the order of 10 fs. This can again be
addressed to more ef ficient exciton phonon coupling at ele v ated temperatures, being a
consequence of the fact that the phonon occupation directly enters into the scattering
rate, equation 4.19.
Ne xt, the different contrib ution to the homogeneous line width are discussed quantita-
ti vely . The figures 4.5 and 4.6 illustrate the full homogeneous linewidth 2 γ in WS 2 and
MoSe 2 as a function of temperature and resolves the dif ferent origins of it. Further , the
corresponding e xcitonic coherence lifetime, which is related by 2 γτ = to the total
homogeneous line width, is depicted in the second axis. The theoretical computation
e xhibits an e xcellent agreement with the experimental results for both in vestig ated ma-
terials. The experimental measurements were carried out at the Stanford Uni versity
(California) and the Uni versity of Re gensb urg. Details on the e xperimental techniques
can be found in [70].
F or linewidth of both e xamined materials, a temperature independent of fset of the ho-
mogeneous line width can be found, which is due to the radiati ve recombination of the
e xciton. The radiati ve contrib ution to the homogeneous width are in good agreement
to recent calculations [44, 139]. In contrast to the radiati ve contrib ution, the non-
radiati ve contrib ution via exciton phonon scattering introduces a strong temperature
dependence. Interestingly , a significant dif ference in the behavior of MoSe 2 and WS 2
can be observ ed. While the homogeneous broadening in MoSe 2 is mainly determined
by intra v alley e xciton phonon coupling, the e xcitonic width is dominated by interv al-
le y exciton phonon coupling in WS 2 , namely by relaxation of the e xcitonic coherence
to e xciton states located at the K − Λ states at the Λ valle y in the exciton dispersion
and K − K states at the K v alley in the e xciton dispersion. The microscopic origin
for this dif ference lies in the qualitati vely dif ferent e xciton dispersions in both materi-
als, as demonstrated in figure 4.2. While in WS 2 the K − Λ states are located belo w
4. Linear Spectroscopy and Excitonic Line width
Figure 4.6.: Excitonic linewidth and lifetime in MoSe 2 . Color code as in figure 4.6. Radiati ve de-
phasing (blue) and non-radiati ve dephasing contrib ute to the homogeneous linewidth.
T o the latter contribute scattering with Γ phonons as well as scattering K phonons
(bro wn) and Λ phonons. This picture was published in a similar form in reference
[70].
the bright state by approximately 70 meV , they are located abo ve the bright state by
approximately 100 meV in MoSe 2 . Hence, K − Λ states turn out to be very ef ficient
for the e xciton relaxation through phonon emission which is e ven possible at 0 K. This
leads to a contrib ution of non-radiati ve broadening to the total width e ven at v ery lo w
temperatures, compare figure 4.5.
In this calculations the separations of the v alleys in the electronic bandstructures were
assumed to be independent of the temperature. Changes of the bandstructure as a
function of the temperature, which are due to lattice extensions are not included. The
parameters used in this calculations, which were obtained from ab initio calculations
[34] can be attrib uted to the room temperature situation, which w as determined by
comparing the lattice constant obtained from the ab initio computation and the lattice
constant obtained from e xperimental measurements. Ho we ver , reference [140] reports
a e ven lar ger separation between the K − K and the K − Λ transitions at cryogenic
temperatures, which would further increase the non-radiati ve contrib ution to the ho-
mogeneous width. Further , one observ es a larger coupling to K − Λ e xcitons than to
K − K e xcitons which is due to a more efficient electron phonon coupling [91].
In MoSe 2 the situation is entirely dif ferent, cf. figure 4.6. Here, the dark exciton states
at the Λ v alley are located approximately 100 meV abo ve the bright state. Since the
phonon ener gies for zone edge phonons are typically on the order of some tens of meV
(30 meV for optical and 15 meV for acoustic phonons), not e ven phonon absorption
can mediate a scattering to these states. The numerical computations re veal further that
intra v alley scattering with acoustic Γ phonons is the crucial mechanism determining
the e xcitonic coherence lifetime in MoSe 2 . This results in an almost linear increase
of the homogeneous width at cryogenic temperatures being in nice agreement with the
temperature trends reported for monolayer MoT e 2 [135] and MoS 2 [71].
The linear contrib ution of acoustic phonons to the homogeneous width can be under-
4.2. Excitonic Line width in the Monolayer
stood as follo ws: the lo w phonon energies allo w a linearization of the Bose-Einstein
distrib ution which appears as phonon occupation in equation 4.19. This results for the
intra v alley scattering with acoustic phonons in the e xpression γ K K
ac = 2 π 2 | g intra Γ
K 0 | 2 k B T
2 ~ M c 2
ac ,
which is linear as a function of temperature T . The slope is determined by the Boltz-
mann constant k B , Planck constant ~ , the excitonic mass M , the v elocity of sound c ac
in the material and the e xciton phonon matrix element at the position K 0 , where the en-
er gy conserv ation, equation 4.19 is fulfilled. Additionally to the linear increase gi v en
by the scattering with acoustic Γ phonons, the homogeneous width in MoSe 2 and WS 2
further e xhibit a superlinear increase, which can be ascribed to scattering with optical
Γ phonons or zone-edge phonons.
F or better comparison with experimental results, the total calculated homogeneous
width can be phenomenologically fitted by
γ = γ 0 + c 1 T + c 2
e ~ ω
kT − 1 . (4.22)
Here, γ 0 accounts for the temperature independent contrib ution due to radiati ve decay
and spontaneous phonon emission processes. The second term accounts for the scatter-
ing with acoustic phonons and the third term for the scattering with optical Γ phonons
and zone-edge phonons. In WS 2 , the temperature independent of fset can be deter-
mined to γ 0 = 9.2 meV , which consists of 7 meV due to radiati ve decay and 2.2 meV
due to zone-edge phonon emission. The slope of the acoustic phonon scattering is de-
termined to c 1 = 28 µ eV K − 1 . For the optical Γ and zone-edge contrib ution a v alue of
6 . 5 meV and an a verage phonon ener gy of ~ ω = 20 meV can be determined. This can
be ascribed to the scattering with lar ge momentum acoustic phonons. Here, the zone-
edge contrib ution dominates the contribution of intra valle y optical phonon scattering,
which is due to the small coupling elements of the latter appearing in equation 4.19.
F or MoSe 2 the fitted parameters read γ 0 = 4.3 meV , c 1 = 91 µ eV K − 1 , c 2 = 15.6 meV
and ~ ω = 30 meV . In general, the presented results nicely correspond to the observ a-
tions in the e xperimental studies [133, 71].
As already discussed, the relati ve position of the dif ferent exciton states in the e xci-
tonic band structure mainly depend on the transition metal of the materials. Therefore
similar beha viors can be expected for the related materials MoS 2 and WSe 2 . Ev entu-
ally , the dielectric constant enters the excitonic wa vefunctions, which determine the
e xcitonic coupling elements. Further , the dielectric constant enters directly in the ra-
diati ve rate, equation 4.10. Therefore, both, the radiativ e dephasing and the phonon
induced dephasing are e xpected to decrease as a function of the substrate dielectric
constant.
4.2.5. Conc lusion
In this section qualitati ve dif ferent origins of the homogeneous line width in molyb-
denum based and tungsten based TMDs ha ve been discussed. In molybdenum based
4. Linear Spectroscopy and Excitonic Line width
TMDs, the line width is mainly determined by radiati ve decay and intra v alley phonon
scattering whereas in tungsten based TMDs interv alley phonon scattering has a sig-
nificant impact on the homogeneous width. The results of this section shine light on
e xcitonic properties, which are crucial for technological applications of TMD mate-
rials. The theoretical approach, which was used to describe the homogeneous width
can be further generalized to other materials of the semiconducting two-dimensional
materials family be yond monolayers of transition metal dichalcogenides.
4.3. Excitonic Line width in the Bila yer
In this section the line width in bilayer WS 2 is discussed and compared to monolayer
WS 2 . The results of this section were obtained in a collaborativ e work together with the
Stanford Uni versity (California) and the Uni versity of Re gensbur g and are published
in reference [141].
First the e xcitonic bandstructure of the bilayer material has to be computed. The elec-
tronic band structure can be obtained from reference [142, 143, 33, 144, 145]. In the
ne xt step excitonic eigenener gies and wa vefunctions using the W annier equation, 3.43,
are calculated. Since the bilayer is approximately double as thick as the monolayer ,
also the screening of the Coulomb matrix element entering in the W annier equation,
equation 3.8, has to be adjusted. The exciton dispersion is then computed by e xploiting
equation 3.44 and is depicted schematically in figure 4.7 for all excitons with electrons
and holes ha ving spin up. This procedure was also e xploited for the similar mate-
rial bilayer WSe 2 in a collaborati ve w ork with the Ludwig Maximilians Uni versity in
Munich and which led to a publication in Nature Communications, reference [146].
Here se veral dif ferences compared to the situtation in monolayer WS 2 can be observed:
while in the monolayer the K − Λ v alley is the lo west lying v alley and the K − Λ 0 v alley
is located well abo v e the bright state the situation is re versed in the bilayer . The reason
for this beha vior originates in the electronic bandstructure. In the monolayer , both
spin bands of the conduction band cross between the K and the Λ point, no crossing is
found in the bilayer [142]. As a result, the K − Λ 0 exciton is the lo west lying e xciton
v alley . Additionally the Γ − K exciton is located belo w the bright state in the bilayer .
The reason is that the electronic Γ v alley lies higher in ener gy in the bilayer due to
the interaction of the indi vidual monolayers [142, 33]. The position of the K − K 0
is approximately equal in monolayer and bilayer WS 2 , since it depends mainly on
the splitting of the conduction band, which is approximately independent on the layer
number [142].
The last ingredient, which is required for the computation of the homogeneous linewidth
in bilayer WS 2 , equations 4.10 and 4.19, are the e xciton phonon coupling elements,
equation 3.63. Here, the entering wa vefunctions are a v ailable through the W annier
equation. The underlying electron phonon coupling elements are assumed to be equal
compared to the monolayer [73, 92, 91]. Now e verything is together to compute the ra-
diati ve and non-radiati ve contrib utions to the homogeneous line width, equations 4.10
4.3. Excitonic Line width in the Bilayer
Figure 4.7.: Relaxation channels determining the excitonic coher ence lifetime in bilayer WS 2 .
A K − K exciton in the Γ v alley can decay via radiati ve decay γ rad (blue arrow) or
non-radiati ve dephasing γ i , with i being the corresponding phonon momentum. The
latter occurs through exciton-phonon scattering within the K − K v alley (orange) with
Γ phonons or to dark K − Λ (red dashed), K − Λ (red), K − K (bro wn dashed) or
Γ − K (bro wn) excitonic states mediated by Λ (red dashed arro w), M (red arrow) and
K phonons (bro wn arrows). In the bilayer , the K − Λ , the K − K and the Γ − K
state lie belo w the optical bright one. The respecti ve dispersion of the monolayer is
indicated as a reference (dashed). The ener getic position of the K − K state does not
dif fer from the bilayer qualitativ ely .
and 4.19. Intrav alley scattering with lo w momentum acoustic and optical phonons gi ve
a linear increasing contrib ution to the homogeneous line width.
In figure 4.8 the homogeneous line width in bilayer WS 2 and its dif ferent contributions
are depicted. The line width sho ws a monotonous behavior as a function of tempera-
ture ranging from 40 meV at v ery lo w temperatures to 80 meV at room temperature,
being in e xcellent qualitativ e agreement with the results for the monolayer samples,
cf. figure 4.5. Radiati ve decay gi ves a temperature independent contrib ution of about
4 meV to the line width. This v alue is smaller compared to the monolayer , cf. figure
4.5, due to the stronger screening of the Coulomb interaction in the bilayer resulting in
broader e xciton wa vefunctions in realspace. Intra valle y scattering with coupling with
acoustic phonons gi ves a linear increasing contrib ution to the homogeneous line width
being consistent with the discussion in the pre vious section. The slope almost matches
the slope of the monolayer samples. Small de viations stem from the fact, that the
line width is computed self consistently . A lar ger linewidth results in a lar ger broad-
ening of the Lorentzian appearing in the formula for the homogeneous broadening,
equation 4.19, which leads to a slight reduction of the line width. In contrast to the
situation in the monolayer , where relaxation to K − Λ states manifests the dominating
relaxation channel, this contrib ution is almost zero in the bilayer , since the K − Λ are
located ener getically well abov e the bright state, cf. table A.4. The contrib ution of
scattering to K − K states is almost equi v alent to the situation in the monolayer , since
the ener getic separation of K − K states is mainly gi ven by the splitting of the conduc-
tion bands at the K point due to spin orbit coupling. This value appears not to depend
strongly on the layer number [142]. The dominating contributions to line width in the
bilayer stem from scattering with acoustic and optical M phonons to K − Λ states.
Interestingly scattering to K − Λ states can take place e ven at v anishing temperature
4. Linear Spectroscopy and Excitonic Line width
Figure 4.8.: Excitonic linewidth and lifetime in bilayer WS 2 . Radiati ve decay (blue) as well as
non-radiati ve decay contrib utes to the line width in the bilayer . T o the latter contribute
intra valle y scattering (yello w), scattering of the constituent electron with Λ phonons
(red dashed), scattering of the constituent electron with K phonons (dashed bro wn),
scattering of the constituent electron with M phonons (red) and scattering of the con-
stituent hole with K phonons (bro wn).
through phonon emission, gi ving about 20 meV to the total line width for v anishing
temperature. This scattering process is ener getically fa vorable since the K − Λ states
are the lo west lying exciton states in bilayer WS 2 accessible through phonon scatter -
ing from the bright state. Further it turns out that scattering with M phonons is very
ef ficient, which is due to the underlying electron phonon coupling elements [91]. An
additional strong contrib ution stems from the scattering with K phonons to Γ − K ex -
citon states. This process is acti v ated in the bilayer , since here the Γ − K excitons are
located belo w the bright state which is in contrast to the situation in the monolayer . The
reason is that the v alence band maximum at the Γ point in the electronic Brillouin zone
shifts up in ener gy when going from monolayer to bilayer samples. The experimental
line width were obtained from reflectance and luminescence measurements. Compar-
ing the theoretical results with the e xperimentally obtained linewidths, one finds an
e xcellent qualitativ e and quantitati ve agreement between them.
4.3.1. Conc lusion
In this section, the de veloped model for the monolayer was applied to bilayer WS 2 .
The theoretically obtained line width are in excellent agreement with e xperimentally
obtained data for the homogeneous line width. As it turned out, the model was able to
describe the phonon mediated relaxation in the bilayer gi ving nice agreement with the
e xperimentally measured data. Most interestingly relaxation in the bilayer does not
take place mediated by Λ phonons as in the monolayer , but with M phonons. This can
be ascribed to a qualitati vely dif ferent excitonic band structure in the bilayer .
4.4. Phonon Sidebands
4.4. Phonon Sidebands
So far , the impact of e xciton-phonon coupling on the optical spectrum was treated
within the Marko v approximation. As demonstrated, within this approximation the
e xciton-phonon coupling gi ves rise to a temperature dependent homogeneous broad-
ening of the Lorentzian line, where the temperature dependence is introduced by the
occupation of the contrib uting phonon modes, equation 4.19. In this limit, no changes
of the e xcitonic line shape and energy renormalizations can be observ ed. Both should
e xist as a consequence of the fluctuation-dissipation theorem [147]. In the present sec-
tion the non-Marko vian treatment of the exciton-phonon coupling shall be discussed.
As it turns out it, it will gi ve rise to the formation of pronounced phonon sidebands
enabled by light absorption into virtual states, cf. figure 4.9. This section is orga-
nized as follo ws: first the general form of the e xcitonic line is deri ved analytically . In
the follo wing subsection, the numerical results are discussed and compared to experi-
mental data. The numerical e v aluation of the phonon sideband spectrum was done by
Dominik Christiansen under supervision of the author of this thesis. The experimen-
tal data was obtained from absorption measurements in a collaborati v e work with the
Uni versity of M ¨
unster . This joint e xperiment-theory study was published in Physical
Re vie w Letters, reference [148].
4.4.1. Non-Mark ovian T reatment of the Exciton-Phonon
Scattering
In this subsection, the solution of the system of coupled dif ferential equations 4.11 and
4.13 be yond the Markov approximation is discussed. This can be obtained by a F ourier
transform of the equations, turning the system of coupled dif ferential equations to a
system of algebraic equations. The F ourier transform of the e xcitonic Bloch equation,
equation 4.11, reads
0 = E ξ h ξ e
µ, Q − ~ ω P ξ h ,ξ e
µ, Q + M ∗ µξ h · A δ ξ h ,ξ e
Q , 0
+ X
K ,α,µ 0
g intra ξ h ξ e α
µ,µ 0 , K S ξ h ξ e Γ α
µ 0 , Q − K , K + ˜
S ξ h ξ e ( − Γ) α
µ 0 , Q − K , − K
+ X
K ,α,µ 0 ,j 6 =Γ
g ξ h ξ e j α
eµ,µ 0 , K S ξ h ( ξ e − j ) j α
µ 0 , Q − K , K + ˜
S ξ h ( ξ e − j )( − j ) α
µ 0 , Q − K , − K
+ X
K ,α,µ 0 ,j
g ξ e ξ h j α
hµ,µ 0 , K S ( ξ h + j ) ξ e j 6 =Γ α
µ 0 , Q − K , K + ˜
S ( ξ h + j ) ξ e ( − j ) α
µ 0 , Q − K , − K . (4.23)
The first term in the first line describes the free energy of the e xcitonic coherence. The
second term accounts for the optical excitation of the e xcitonic coherence. The third
line describes the intra v alley scattering of e xcitons, whereas the fourth and the fifth line
describe interv alley scattering of the constituent electrons and holes. In the follo wing,
4. Linear Spectroscopy and Excitonic Line width
the interv alley scattering of holes can be ne glected due to the lar ge separations of the
respecti ve e xcitonic states, which are addressable through interv alley scattering of the
hole. The Fourier transformation of the equations of motion for the phonon assisted
quantities, equation 4.13, reads
0 = E ξ h ξ e
µ, Q + ~ ω j α
K − ~ ω − i ˜ γ ξ h ξ e
µ Q S ξ h ξ e j α
µ Q , K
+ X
ν
g ξ h ξ e ( − j ) α
eµν − K P ξ h ( ξ e + j )
ν, Q + K 1 + n j α
K − X
ν
g ξ e ξ h ( − j ) α
hµν − K P ( ξ h − j ) ξ e
ν Q + K 1 + n j α
K (4.24)
0 = E ξ h ξ e
µ, Q − ~ ω j α
K − ~ ω − i ˜ γ ξ h ξ e
µ Q ˜
S ξ h ξ e j α
µ Q , K
+ X
ν
g ξ h ξ e j α
eµν K P ξ h ( ξ e − j )
ν, Q − K n j α
K − X
ν
g ξ e ξ h j α
hµν K P ( ξ h + j ) ξ e
ν Q − K n j α
K . (4.25)
The first line of both equations accounts for the free oscillation of the phonon assisted
quantities and the second line accounts for exciton-phonon coupling. This system
of equations is solv ed by inserting the equations 4.24 and 4.25 into equation 4.23.
T ogether with the expression for the absorption coef ficient, equation 3.53, one obtains
for the non-Marko vian absorption spectrum
α ( ω ) = 1
ω 0 X
µ,ξ ,σ = | M µξ σ | 2
~ ω − E ξ ξ
µ, 0 − iγ r ad − Σ ξ ξ
µ, 0 ( ω ) ! . (4.26)
This e xpression differs from the former deri ved Elliott formula, equation 4.2, by the
appearance of a frequenc y dependent self energy Σ ξ ξ
µ, 0 ( ω ) which describes the non-
Marko vian coupling of excitons to phonons. As a result, the absorption does not hav e
a Lorentzian lineshape an ymore. The self energy is in general a comple x quantity ,
gi ving rise to lineshifts as well as to broadenings at the indi vidual frequencies. The
self ener gy has the expression
Σ ξ ξ
µ, Q = X
K ,α,µ 0 , ± | g intra ξ h ξ e α
µ,µ 0 , K | 2 1
2 ± 1
2 n α Γ
K
~ ω − E ξ h ξ e
µ 0 , Q + K ∓ ~ ω α Γ
K + iγ ξ h ξ e
µ 0 , Q + K
+ X
K ,α,µ 0 ,j 6 =Γ , ± | g ξ h ξ e j α
eµµ 0 , K | 2 1
2 ± 1
2 n αj
K
~ ω − E ξ h ( ξ e + j )
µ 0 , Q + K ∓ ~ ω αj
K + iγ ξ h ( ξ e + j )
µ 0 , Q + K
+ X
K ,α,µ 0 ,j 6 =Γ , ± | g ξ e ξ h j α
hµµ 0 , K | 2 1
2 ± 1
2 n αj
K
~ ω − E ( ξ h − j ) ξ e
µ 0 , Q + K ∓ ~ ω αj
K + iγ ( ξ h − j ) ξ e
µ 0 , Q + K
. (4.27)
The first line accounts for the scattering with Γ phonons, the second line accounts
for the scattering of the constituent electrons with zone-edge phonons and the third
line describes the scattering of the constituent holes with zone-edge phonons. The ±
sum accounts for phonon emission (+) and absorption (-) processes. In general also
4.4. Phonon Sidebands
coupling to higher e xcitonic states µ 0 has to be taken into account. Howe ver , since in
TMDs e ven the 2 s / 2 p e xcitonic states are separated by at least 200 meV from the 1 s
e xciton [63, 149], these contributions can be ne glected by restriction to the lo west lying
e xciton states. The same argumentation holds for the scattering of the constituent holes
of the e xcitons, where the coupling is suppressed by the lar ge splitting of the valence
band and the lar ge separation of Γ − K excitons from the bright state, compare table
A.4 in the appendix.
T o analytically in vestigate the impact of the self ener gy to the e xcitonic absorption
spectrum, a simplified model is considered for no w: the in vestig ation is restricted to
the ener getically lo west lying excitons. Additionally only the emission of one optical Γ
phonon with v anishing momentum K = 0 to the same e xcitonic state is assumed. For
reasons of simplicity further the v alley indices are dropped. Therefore the absorption
formula simplifies to
α ( ω ) = 1
ω 0 = | M | 2
~ ω − E 0 − iγ r ad − | g 0 | 2 (1+ n 0 )
~ ω − E 0 + ~ ω 0 + iγ 0 ! . (4.28)
As it can be seen from this e xpression, the non-Markovian treatment of the e xciton
phonon coupling gi ves rise to a peak splitting of the e xcitonic resonance. The two
peak positions can be found by setting the denominator equal to zero. T o do so the
dephasing constants are ne glected, since they ha ve only minor influence on the peak
position.
~ ω ± = E 0 + ~ ω 0
2 ± s ~ ω 0
2 2
+ | g 0 | 2 (1 + n 0 ) . (4.29)
The splitting between both peaks is then gi ven by the ener gy of the in v olved phonon
and the coupling strength to this phonon. The latter introduces a temperature depen-
dence through the phonon occupation and is later attrib uted to a polaron shift. In the
general e xpression for the self ener gy , equation 4.27, a sum ov er all phonon modes as
well as a sum o v er all momenta appears. T aking this into account, one can expect the
formation of sidebands and a shift of the main e xciton resonance. In the limit of v an-
ishing e xciton phonon coupling strength g 0 , equation 4.29 still yields tw o resonances,
where the first coincides with the excitonic ener gy E 0 and the second is giv en by the
sum of the e xcitonic energy and the phonon ener gy E 0 + ~ ω 0 . Ho wev er , exploiting
equation 4.28, one finds that the latter has v anishing oscillator strength, and therefore
the spectrum is only gi ven by the e xcitonic resonance.
Figure 4.9 illustrates the dif ferent mechanisms which contrib ute to the formation of
phonon sidebands. A sideband at the lo w energy side of the e xcitonic resonance ap-
pears if the optical transition is assisted by absorption of a Γ phonon or by emission
and absorption of a Λ phonon, if the excitonic Λ v alley is located suf ficiently lo w in en-
er gy . This is the case in tungsten based materials. Due to the low v elocities of sound, Γ
acoustic phonons do not contrib ute to phonon sidebands on the lo w energy side of the
4. Linear Spectroscopy and Excitonic Line width
Figure 4.9.: Micr oscopic processes leading to sideband f ormation. A phonon sideband occurs,
if the optical transition (blue) is assisted by scattering with Γ (yello w) or Λ phonons
(red). The latter is suppressed in mo ybdenum based materials, where the Λ valle y is
ener getically located well abov e the bright state (dashed). This picture was published
in a similar form in reference [148].
Figure 4.10.: Phonon sidebands in MoSe 2 . The picture illustrates the non-Marko vian absorption
spectrum in MoSe 2 (blue) at 300 K. For comparison the Mark ovian Lorentzian spec-
trum (yello w) and the Lorentzian shifted and normalized to the Non-Markovian spec-
trum are depicted. The inset illustrates the dif ference between the Non-Markovian
and the shifted and normalized spectrum. This picture w as published in a similar
form in reference [148].
e xcitonic resonance. A phonon sideband at the high energy side appears if the optical
transition is assisted by the emission and absorption of Γ and Λ optical and acoustic
phonons.
4.4.2. Excitonic Lineshape in MoSe 2 and WS 2
In this subsection, the numerical results for the non-Marko vian absorption spectra in
the e xemplary materials MoSe 2 and WS 2 are discussed. The numerical e v aluation was
done by Dominik Christiansen.
Figure 4.10 sho ws the non-Marko vian absorption spectrum of monolayer MoSe 2 at
room temperature. It is redshifted by about 30 meV in comparison to the Marko vian
spectrum. This is attrib uted to a polaron shift. Additionally an asymmetric broadening
due to phonon sideband formation, especially at the high energy side of the spectrum
can be observ ed. T o simplify the discussion about the appearing sidebands the shifted
4.4. Phonon Sidebands
Figure 4.11.: Phonon sidebands in WS 2 . The picture illustrates the non-Marko vian phonon spec-
trum (red) in WS 2 at 300 K. For comparison the Mark ovian Lorentzian spectrum
(yello w) and the Lorentzian shifted and normalized to the Non-Marko vian spectrum
are depicted. The inset illustrates the dif ference between the Non-Marko vian and the
shifted and normalized spectrum. This picture was published in a similar form in
reference [148].
and normalized Marko vian spectrum is subtracted from the non-Marko vian spectrum.
This is depicted in the inset of figure 4.10, where se veral sidebands are visible. At
approximately 80 meV abo ve the e xcitonic resonance, a sharp feature appears which
can be attrib uted to the emission of optical Γ phonons. The ener gy of optical phonons
is about 36 meV , compare table A.5 in the appendix. The energy dif ference between
the optical phonon ener gy and the relativ e position of the optical phonon sideband can
be e xplained by the fact, that the splitting between the main excitonic resonance and
the sideband is also af fected by the coupling strength of the exciton to the respecti ve
phonon mode, equation 4.29. At approximately 50 meV abo ve the main e xcitonic reso-
nance a broad shoulder appears. This is attributed to the emission and the absorption of
acoustic Γ phonons. This observ ation is supported by the fact that acoustic Γ phonons
ha ve a linear dispersion for small momenta and therefore can co ver a broad range of
ener gy . At approximately 10 meV belo w the main excitonic resonance, a sharp feature
appears, which is due to the absorption of optical Γ phonons. This v alue is smaller
than the optical phonon ener gy , since the excitonic main resonance is strongly red-
shifted due to the emission and absorption of acoustic phonons and the emission of
optical phonons, which reduces the energetic distance between both lines. In general
one observ es a strongly asymmetric lineshape in MoSe 2 since more phonon assisted
process occur at the high ener gy side of the main excitonic line.
Figure 4.11 illustrates the non-Marko vian spectrum of monolayer WS 2 . In line with
the observ ations for MoSe 2 , the non-Marko vian spectrum in WS 2 exhibits a redshift
of about 10 meV with respect to the Mark ovian spectrum. Also a broad shoulder and
the high ener gy side of the main exciton line due to coupling to acoustic phonons is
present. In contrast to the results for MoSe 2 no sidebands due to optical phonon emis-
sion are visible, which reflects the weak exciton-optical phonon coupling strength in
WS 2 . A major dif ference between WS 2 and MoSe 2 is the appearance of dark e xcitonic
states at the Λ v alley belo w the optical bright one. Due to the numerical e valutation,
these states are located belo w the bright state by about 50 meV . The Λ v alley represents
4. Linear Spectroscopy and Excitonic Line width
Figure 4.12.: P olaron shift. (a) Non-Marko vian spectra of MoSe 2 for fiv e selected temperatures.
(b) Polaronshift as a function of temperature for MoSe 2 (blue) and WS 2 (red). This
picture was published in a similar form in reference [148].
a broad band which is a v ailable for phonon scattering, i.e. emission and absorption
processes of optical and acoustic Λ phonons. This results in the appearance of phonon
sidebands belo w and abov e the main excitonic resonance, reducing the asymmetry of
the non-Marko vian spectrum in WS 2 .
Figure 4.12 (a) illustrates the non-Marko vian spectra of MoSe 2 for fi ve selected tem-
peratures. The spectral width of the main e xcitonic line increases as a function of
temperature being consistent with the results of the pre vious section, 4.6. Further a
pronounced redshift of the main e xcitonic resonance which increases as a function of
temperature is found. This redshift is attrib uted to a renormalized excitonic ener gy
due to the e xciton-phonon coupling and is called polaron shift. Also the formation of
sidebands can be observ ed. While at v ery low temperatures, no signatures of e xciton-
acoustic phonon coupling are visible, a pronounced shoulder on the high energy side of
the main e xcitonic resonance ev olves with increasing temperature. This is compatible
with the lar ger exciton-phonon coupling strength at ele v ated temperatures due to larger
phonon occupations in equation 4.27. Also the formation of optical phonon sidebands
can be observ ed: while at very lo w temperatures only a small kink in the absorption is
visible approximately 50 meV abo ve the e xcitonic resonance, at ele v ated temperature
it turns to a pronounced sideband.
T o study the polaron shift more quantitati vely , it is depicted in figure 4.12 (b) for
MoSe 2 and WS 2 as a function of temperature. In MoSe 2 the polaron shift increases
monotonous as a function of temperature from 11 meV at 4 K to 28 meV at room tem-
perature. Hereby the temperature independent of fset of 11 meV is addressed to the
emission of optical phonons: The respectiv e sideband appears on the high ener gy side
of the main e xcitonic resonance, leading to a redshift of the latter , pre viously discussed
together with equation 4.29. The slope of the polaron shift is ascribed to the emission
and absorption of acoustic phonons, since the respecti ve sidebands appear only on the
high ener gy side of the main excitonic resonance. Interestingly scattering with optical
phonons does not induce a strong temperature dependence of the polaron shift, which
4.4. Phonon Sidebands
Figure 4.13.: Measur ed and calculated spectra. Calculated (filled area) and measured (solid
lines) spectra for MoSe 2 (blue) and WS 2 (red). The experimental curv es were ob-
tained from a non-linear fit of the experimental measured spectrum with Pearson IV
functions. This picture was published in a similar form in reference [148].
is due to the fact that sidebands stemming from emission and absorption of optical
phonons appear on opposite sides of the main e xcitonic resonance. This leads to a can-
cellation of the induced shifts. The polaron shift in WS 2 exhibits the same qualitati ve
beha vior as in MoSe 2 . The smaller temperature independent of fset reflects the weaker
coupling of e xcitons to optical phonons in WS 2 . The smaller slope compared to the
situtation in MoSe 2 accounts for the weaker coupling of e xcitons to acoustic phonons.
Since sidebands stemming from coupling to indirect e xciton states at the Λ v alley ap-
pear on both ener gy sides of the main excitonic resonance, the induced polaron shift
due to these coupling processes is considerably small. The calculated redshifts o ver
the full temperature range from 5 K to room temperature are in the order of some fe w
10 meV . This v alue is smaller than the experimental measured v alue of about 100 meV
[148] since it only reflects the e xcitonic contribution to the total shift. The dominant
contrib ution stems from bandgap renormalization due to an e xtended lattice constant
with increasing temperature, which can be described by the phenomenological V arshni
relation [150].
Last the computed spectra are compared to e xperimentally measured spectra. T o ex-
tract the asymmetric lineshape of the 1 sA exciton from the measured spectrum, a
multi-peak-fit was performed with the e xperimental data. T o take account to the asym-
metric broadening of the e xcitonic resonances, Pearson IV functions were chosen for
the fitting procedure [151]. These allo w to subtract not ev en line width and peak po-
sition b ut also the asymmetry of the line from the experimental data. More details on
the e xperimental procedure can be found in reference [148]. Figure 4.13 illustrates the
comparison of the computed and the measured e xcitonic spectra for monolayer MoSe 2
and WS 2 . Note, that the theoretical data was shifted to match the peak position of the
e xperimental data. In general the comparison of experiment and theory e xhibits a nice
agreement re garding the approximate line width and the asymmetry of the e xcitonic
resonance for both in vestig ated materials. Due to the choice of the fitting function,
the sideband due to emission of optical phonons for MoSe 2 is not contained in the
e xperimental curve. At the lo w energy side of the e xcitonic resonance where phonon
4. Linear Spectroscopy and Excitonic Line width
sidebands due to the absorption of optical phonons occur , the theoretical curves e xceed
the e xperimental data in both materials. This could be due to an o verestimation of the
optical phonon coupling strength.
4.4.3. Conc lusion
In conclusion, in this section a model for the excitonic absorption with non-Mark ovian
e xciton-phonon coupling was presented. Non-Marko vian exciton-phonon coupling
gi ves rise to the formation of pronounced phonon sidebands in the linear spectrum.
The contrib utions of intrav alley acoustic and optical as well as interv alley phonon
scattering were disentangled. Additionally a pronounced polaron shift of the exci-
tonic resonance was found. The presented theoretical results are in good agreement
with e xperimentally obtained data.
5. Exciton Dynamics and
Photoluminescence
In this chapter the formation of e xcitons after resonant optical e xcitation, the thermal-
ization and the e xcitonic photoluminescence are discussed. Further the scope of this
chapter is the spin selecti ve e xcitation, spin resolved e xciton dynamics and polariza-
tion resolv ed photoluminescence in TMDs. Therefore, in the first section the general
momentum and spin resolv ed Boltzmann scattering equation for excitons in the lo w e x-
citation limit are deri ved [90]. These equations of motion include the e xciton photon
as well as the e xciton phonon scattering. Furthermore the lo w excitation contrib u-
tion of the Coulomb interaction, meaning terms up to the linear order in the e xciton
density , is included. In the next section the spin independen t e xciton formation and
thermalization is discussed. As a prominent result, e xcitons are formed from the e x-
citonic coherence within the coherence lifetime through non-radiati ve dephasing. The
so formed e xciton densities thermalize within a fe w picoseconds. Here, the forma-
tion and thermalization of dif ferent TMDs with and without lo w lying dark states will
be compared quantitati vely . In the next section, the photoluminescence dynamics is
in vestig ated. As a highlight it will turn out that in molybdenum based TMD materi-
als the photoluminescence intensity is decreasing as a function of temperature which
is kno wn from con ventional tw o dimensional semiconductors such as GaAs quantum
wells. In strong contrast in tungsten based TMD materials, the photoluminescence
intensity is increasing as a function of temperature which is a direct consequence of
the appearance of ener getically low lying dark states in the material. The theoretical
predictions will turn out to be in qualitati vely e xcellent agreement with e xperimental
results [133, 134, 65]. The results of the latter section were published in reference [75]
In the last section the influence of dark states on the interv alley e xchange interaction
is in vestig ated, which is the common mechanism to e xplain the short spin lifetimes in
TMD materials in the literature. Howe ver , it will be demonstrated, ho w the interv al-
le y e xchange coupling is suppressed by the appearance of energetically lo w lying dark
states leading to an increase of the spin lifetime by orders of magnitudes compared to
the situation without dark states.
5.1. Exciton Boltzmann Scattering Equations
In this section the equations of motion of e xciton densities and of the emitted light
are deri ved from the e xcitonic Hamiltonian. After this, interesting limiting properties
5. Exciton Dynamics and Photoluminescence
as the detailed balance of the e xcitonic equation of motion are discussed. Last, some
basic informations about the numerical implementation of the scattering equations are
gi ven.
5.1.1. Deriv ation of the Equations of Motion
In the current subsection, the fundamental equations of motion for the intensity of the
emitted light and the Boltzmann scattering equations for excitons in the lo w excitation
limit are deri ved.
The intensity of the emitted light is gi ven by the photon rate
I ( t ) ∝ X
K ,k z ,σ
∂ t ~ Ω σ
K ,k z n σ
K ,k z . (5.1)
with the photon density n σ
K ,k z = h d † σ
K ,k z d σ
K ,k z i and dispersion ~ Ω σ
K ,k z . Both depend
on the three dimensional photon momentum ( K , k z ) , where K denotes the projection
of the photon momentum onto the two dimensional semiconductor plane and k z the
component perpendicular to it. Further the sum runs ov er both possible light polariza-
tions. The photon density consists of two parts, namely the coherent and the incoherent
contrib ution to the photon density
n σ
K ,k z = |h d σ
K ,k z i| 2 + δ n σ
K ,k z . (5.2)
An equation of motion for the coherent and the incoherent photon occupation can be
obtained by applying Heisenber gs equation of motion to the excitonic Hamiltonian.
F or the coherent photons, the equation of motion reads
i ~ ∂ t h d σ
K ,k z i = ~ ω σ
K ,k z h d σ
K ,k z i + X
µ,ξ
M µξ σ
K ,k z P ξ ξ
µ, K . (5.3)
The first term accounts for the free oscillation with the photon frequency ω σ
K ,k z and
the second part identifies the e xcitonic coherence as a source term of the coherent
photons. Further M µξ σ
K ,k z denotes the exciton photon coupling element as it w as defined
in equation 3.57. The equation of motion for the incoherent photon density reads
i ~ ∂ t δ n σ
K ,k z = X
µ,ξ M µξ σ
K ,k z δ T ξξσ
µ, Q ,k z − M ∗ µξ σ
K ,k z δ T ∗ ξξσ
µ, K ,k z
= 2 i X
µ,ξ = M µξ σ
K ,k z δ T ξξσ
µ, Q ,k z , (5.4)
where the photon assisted polarization T ξξσ
µ, Q ,k z = h P ξξσ
µ, K d † σ
K ,k z i was defined and in the
second step the imaginary part was identified.
5.1. Exciton Boltzmann Scattering Equations
Last, the equation of motion of the photon assisted polarization is gi ven
i ~ ∂ t δ T ξξσ
µ, K ,k z = E ξ ξ
µ, K − ~ ω σ
K ,k z δ T ξξσ
µ, K ,k z
+ X
k 0
z σ 0
M ∗ µξ σ 0
K ,k z δ h d † σ
K ,k z d σ 0
K ,k 0
z i − X
ξ 0 ,µ 0
M ∗ µ 0 ξ 0 σ
K ,k z δ h P † ξ ξ
µ, K P ξ 0 ξ 0
µ 0 , K i . (5.5)
The equation for the photon assisted polarization can be solv ed within a Born Marko v
approximation, equation 4.18, which yields for the intensity of the emitted incoherent
light
I incoh = X
σ
I σ ∝ X
K ,k z ,σ ,
µ,µ 0 ,ξ ,ξ 0
M µ,ξ ,σ
K ,k z M ∗ µ 0 ξ 0 ,σ
K ,k z δ h P † ξ ξ
µ, K P ξ 0 ξ 0
µ 0 , K i δ ( E ξ 0 ξ 0
µ 0 , K − ~ ω σ
K ,k z ) . (5.6)
Here, the product of the optical matrix elements is only non v anishing if ξ = ξ 0 which
follo ws from the optical selection rules and the ener gy conservation. Further , oscilla-
tion between dif ferent exciton states µ , µ 0 can not be e xcited in a homogeneous system
in the lo w excitation limit, which yields the final e xpression
I incoh = X
σ
I σ ∝ X
K ,k z ,σ ,µ,ξ | M µ,ξ ,σ
K ,k z | 2 N ξ ξ
µ, K δ ( E ξ ξ
µ, K − ~ ω σ
K ,k z ) , (5.7)
where the incoherent e xciton density N ξ ξ
µ, K = δ h P † ξ ξ
µ, K P ξ ξ
µ, K i was identified. Since the
e xciton dispersion v aries only slo wly in the region where the e xciton photon interaction
takes place E ξ ξ
µ, K ≈ E ξ ξ
µ , the δ distribution can be further e v aluated. Writing the photon
dispersion ω K ,k z = c | ( K , k z ) | , the δ function can be re written to 1
~ c δ ( E ξξ
µ
~ c − | ( K , k z ) | )
which ef fecti vely restricts the inte gration ov er the three dimensional photon momen-
tum onto the surface of a tw o dimensional circle. Therefore, the k z sum can be e v alu-
tated, gi ving [90, 75, 152]
I incoh ∝ X
K ,σ ,µ,ξ | M µ,ξ ,σ
K , q E 2
~ 2 c 2 − K 2 | 2 N ξ ξ
µ, K 1
| K | < E ξξ
µ
~ c
, (5.8)
where 1
| K | < E ξξ
µ
~ c
= 1 if | K | < E ξ ξ
µ
~ c hold and 1
| K | < E ξξ
µ
~ c
= 0 otherwise. The K sum
thus runs only within a circle with radius E ξ ξ
µ
~ c which is called the light cone. For a
typical e xcitonic ener gy in the visible range of 2 eV , the light cone radius is only about
10 µ m − 1 . As a result the total amount of excitons which is located within the light
cone determines the intensity of the emitted light.
Similar the coherent part of the emitted light can be e v aluated exploiting equation 5.1,
5.2 and 5.3 [90, 75]
I coh ∝ X
K ,σ ,µ,ξ | M µ,ξ ,σ
K , q E 2
~ 2 c 2 − K 2 | 2 | P ξ ξ
µ, K | 2 1
| K | < E ξξ
µ
~ c
, (5.9)
5. Exciton Dynamics and Photoluminescence
Therefore, the total intensity of the emitted light is determined by the sum of the co-
herent and the incoherent emission I = I coh + I incoh .
The equation of motion for the incoherent e xciton density N ξ h ξ e
µ, Q = δ h P † ξ h ξ e
µ, Q P ξ h ξ e
µ, Q i can
be obtained by e xploiting Heisenbergs equation of motion, equation 3.42, and reads
i ~ ∂ t N ξ h ξ e
µ, Q = − 2 i X
k z ,σ = M µξ h σ
Q ,k z T ξ h ξ e σ
µ, Q ,k z δ x h ,ξ e
+ 2 i X
µ 0 ,ξ 0
h ,ξ 0
e
= X ξ h ξ e ξ 0
h ξ 0
e
µµ 0 , Q + V ξ h ξ e ξ 0
h ξ 0
e
µµ 0 , Q δ h P † ξ h ξ e
µ, Q P ξ 0
h ξ 0
e
µ 0 , Q i
+ 2 i X
K ,µ 0 ,α,j = g ξ h ( ξ e + j ) j α
eµ,µ 0 Q P ξ h ξ e − j
µ 0 , Q − K − g ( ξ h − j ) ξ e j α
hµ,µ 0 Q P ξ h + j ξ e
µ 0 , Q − K ×
× ˜
S ∗ ξ h ξ e j α
µ, Q , K + S ∗ ξ h ξ e ( − j ) α
µ, Q , − K
+ 2 i X
K ,µ 0 ,α,j = g ξ h ( ξ e + j ) j α
eµµ 0 K O ξ h ξ e ξ h ( ξ e − j ) j α
µ,µ 0 , Q , Q − K , K + ˜
O ξ h ξ e ξ h ( ξ e − j )( − j ) α
µ,µ 0 , Q , Q − K , − K −
− g ( ξ h − j ) ξ e j α
hµµ 0 K O ξ h ξ e ( ξ h + j ) ξ e j α
µ,µ 0 , Q , Q − K , K + ˜
O ξ h ξ e ( ξ h + j ) ξ e ( − j ) α
µ,µ 0 , Q , Q − K , − K (5.10)
Here, the first line accounts for exciton photon interaction, where the photon assisted
e xcitonic coherence can be identified as a source term of the incoherent exciton density .
Note the opposite sign of this term compared to the equation of motion for the photon
density , which immediately yields a detailed balance between e xcitons and photons.
The second line accounts for the lo w density contrib ution of the Coulomb interaction.
The first term accounts for interband and the second term accounts for intraband inter -
action. In general, the Coulomb interaction couples the incoherent exciton density to
coherences between e xcitonic states δ h P † ξ h ξ e
µ, Q P ξ 0
h ξ 0
e
µ 0 , Q i which mediate the transition from
one state to the other , as it will discussed later . The third line accounts for the formation
of incoherent e xciton densities from the excitonic coherence through e xciton phonon
coupling. Here the notation S ∗ ξ h ξ e ( − j ) α
µ, Q , − K = ( S ξ h ξ e ( − j ) α
µ, Q , − K ) ∗ = δ h P † ξ h ξ e
µ, Q b † ( − j ) α
− K i was intro-
duced. The last line accounts for the scattering of incoherent excitons with phonons,
where the phonon assisted excitonic transitions O ξ h ξ e ξ h ( ξ e − j ) j α
µ,µ 0 , Q , Q − K , K = δ h P † ξ h ξ e
µ, Q P ξ h ( ξ e − j )
µ 0 , Q − K b j α
K i
and ˜
O ξ h ξ e ξ h ( ξ e − j )( − j ) α
µ,µ 0 , Q , Q − K , − K = δ h P † ξ h ξ e
µ, Q P ξ h ( ξ e − j )
µ 0 , Q − K b † ( − j ) α
− K i were defined. Note that in the case
of long range phonon scattering, i.e. j = Γ , the electron and hole contributions can be
mer ged to one term.
5.1. Exciton Boltzmann Scattering Equations
The equation of motion for the intrae xcitonic coherence reads
i ~ ∂ t δ h P † ξ h ξ e
µ, Q P ξ 0
h ξ 0
e
µ 0 , Q i = E ξ 0
h ξ 0
e
µ 0 , Q − E ξ h ξ e
µ, Q δ h P † ξ h ξ e
µ, Q P ξ 0
h ξ 0
e
µ 0 , Q i
+ X
µ 00 ,ξ 00
h ,ξ 00
e X ξ 0
h ξ 0
e ξ 00
h ξ 00
e
µ 0 µ 00 , Q + V ξ 0
h ξ 0
e ξ 00
h ξ 00
e
µ 0 µ 00 , Q δ h P † ξ h ξ e
µ, Q P ξ 00
h ξ 00
e
µ 00 , Q i
− X
µ 00 ,ξ 00
h ,ξ 00
e X ξ 00
h ξ 00
e ξ h ξ e
µ 00 µ, Q + V ξ 00
h ξ 00
e ξ h ξ e
µ 00 µ, Q δ h P † ξ 00
h ξ 00
e
µ 00 , Q P ξ 0
h ξ 0
e
µ 0 , Q i (5.11)
In order to truncate the exciton dynamics at the second order in the matrix elements
[90], the impact of the exciton photon and the e xciton phonon coupling ha ve been ne-
glected for the equation of motion for the intrae xcitonic coherence. Since the exciton-
e xciton Hamiltonian, equation 3.58, has been truncated in the second order of the ex-
citonic operators, here no higher order correlations of the excitonic operators appear .
W ith the same argument, one can claim, that the intrae xcitonic coherences do not cou-
ple among each other . Thus the correlations appearing at the right hand side do only
couple back to e xcitonic densities, and so the following equation of motion reads
i ~ ∂ t δ h P † ξ h ξ e
µ, Q P ξ 0
h ξ 0
e
µ 0 , Q i = E ξ 0
h ξ 0
e
µ 0 , Q − E ξ h ξ e
µ, Q δ h P † ξ h ξ e
µ, Q P ξ 0
h ξ 0
e
µ 0 , Q i
+ X ξ 0
h ξ 0
e ξ h ξ e
µ 0 µ, Q + V ξ 0
h ξ 0
e ξ h ξ e
µ 0 µ, Q N ξ h ξ e
µ, Q − N ξ 0
h ξ 0
e
µ 0 , Q . (5.12)
Here, the first line accounts for the oscillation of the intrae xcitonic coherence with the
ener gy dif ference of the in v olved states. The second line accounts for the Coulomb
coupling. Intrae xcitonic coherences are driv en with the occupation dif ference between
the in v olved states.
The equation of motion for the phonon assisted e xcitonic transition reads
i ~ ∂ t O ξ h ξ e ξ 0
h ξ 0
e j α
µ,µ 0 , Q , Q 0 K = E ξ 0
h ξ 0
e
µ 0 , Q 0 − E ξ h ξ e
µ, Q + ~ Ω j α
K O ξ h ξ e ξ 0
h ξ 0
e j α
µ,µ 0 , Q , Q 0 K
+ X
µ 00
g ξ 0
h ( ξ 0
e + j )( − j ) α
eµ 0 ,µ 00 , − K δ h P † ξ h ξ e
µ, Q P ξ 0
h ( ξ 0
e + j )
µ 00 , Q 0 + K i 1 + n j α
K
− X
µ 00
g ( ξ 0
h − j ) ξ 0
e ( − j ) α
hµ 0 ,µ 00 , − K δ h P † ξ h ξ e
µ, Q P ( ξ 0
h − j ) ξ e
µ 00 , Q 0 + K i 1 + n j α
K
− X
µ 00
g ξ h ξ e ( − j ) α
eµ 00 ,µ, − K δ h P † ξ h ( ξ e − j )
µ 00 , Q − K P ξ 0
h ξ 0
e )
µ 0 , Q 0 i n j α
K
+ X
µ 00
g ξ h ξ e ( − j ) α
hµ 00 ,µ, − K δ h P † ( ξ h + j ) ξ e
µ 00 , Q − K P ξ 0
h ξ 0
e
µ, Q 0 i n j α
K (5.13)
Here the first line accounts for the free oscillation of the phonon assisted excitonic
transition. The other terms account for e xciton phonon scattering. T erms, leading to
dynamics in the fourth order of the matrix elements ha ve been ne glected. The equation
of motion for ˜
O ξ h ξ e ξ 0
h ξ 0
e j α
µ,µ 0 , Q , Q 0 K can be obtained by complex conjug ation of the equation of
5. Exciton Dynamics and Photoluminescence
motion of O ξ h ξ e ξ 0
h ξ 0
e j α
µ,µ 0 , Q , Q 0 K . In order to treat the appearing hierarchy problem, correlations
containing two e xciton and two phonon operators were treated within the second or -
der Born approximation, yielding h P † P b † b i≈h P † P ih b † b i . Higher order terms hav e
been ne glected. The resulting phonon occupations are treated in bath approximation.
Further terms in quadratic order of the exciton occupations ha ve been ne glected since
the in vestig ation is restricted to the lo w excitation (linear) limit. No w , the equation
of motion for the phonon assisted e xciton transition, equation 5.13, the photon as-
sisted polarization, equation 5.5 and the intra excitonic transitions, equation 5.12 can
be solv ed within a Born Marko v approximation. Inserting the result backs in equation
of motion for the e xcitonic densites, equation 5.10, yields the excitonic Boltzmann
scattering equation
∂ t N ξ h ξ e
µ, Q =
= 2 π
~ X
σ ,k z | M µ,ξ h ,σ
Q ,k z | 2 n σ
Q ,k z − N ξ h ξ e
µ, Q δ ξ h ,ξ e δ E ξ h ξ e
µ, Q − ~ ω σ
Q ,k z
+ 2 π
~ X
µ 0 ,ξ 0
h ,ξ 0
e
| X ξ h ξ e ξ 0
h ξ 0
e
µ,µ 0 , Q + V ξ h ξ e ξ 0
h ξ 0
e
µ,µ 0 , Q | 2 N ξ 0
h ξ 0
e
µ 0 , Q − N ξ h ξ e
µ, Q δ E ξ 0
h ξ 0
e
µ 0 , Q − E ξ h ξ e
µ, Q
+ 2 π
~ X
K ,α,µ 0
Γ inξ h ξ e Γ α
µ,µ 0 , Q , K N ξ h ξ e
µ 0 , Q + K + | P ξ h ξ e
µ 0 , Q + K | 2 − Γ outξ h ξ e Γ α
µ,µ 0 , Q , K N ξ h ξ e
µ, Q
+ 2 π
~ X
K ,j,α,µ 0
Γ inξ h ξ e j α
eµ,µ 0 , Q , K N ξ h ( ξ e − j )
µ 0 , Q + K + | P ξ h ( ξ e − j )
µ 0 , Q + K | 2 − Γ outξ h ξ e j α
eµ,µ 0 , Q , K N ξ h ξ e
µ, Q
+ 2 π
~ X
K ,j,α,µ 0
Γ inξ h ξ e j α
hµ,µ 0 , Q , K N ( ξ h + j ) ξ e
µ 0 , Q + K + | P ( ξ h + j ) ξ e
µ 0 , Q + K | 2 − Γ outξ h ξ e j α
hµ,µ 0 , Q , K N ξ h ξ e
µ, Q . (5.14)
The first line accounts for the e xciton photon coupling, where the first term describes
the absorption of an incoherent photon density and the second term describes the radia-
ti ve decay of the e xcitons. The Kronecker symbol δ ξ h ,ξ e tak es account to the f act, that
electron and hole ha ve to be located at the same high symmetry point and need to ha ve
the same spin to contrib ute to the radiativ e coupling. The Dirac distrib ution ensures
the conserv ation of ener gy and momentum. As a result only exciton states within the
light cone can contrib ute to the radiativ e coupling in the considered limit (truncation
at the second order of the matrix elements). The second term accounts for the lo w
density contrib ution of the Coulomb interaction. Here, the first coupling element X
accounts for interband interaction and the second coupling element V accounts for the
intraband interaction. The e xciton density is dri ven by the occupation dif ference of the
in v olved states. As a result, this term leads to an equilibration of the in volv ed exciton
states where the time constant is propotional to the in verse squared coupling elements.
The appearing Dirac distrib ution accounts for energy conserv ation during the scatter-
ing e vent. The last three lines account for the exciton phonon coupling, where the third
5.1. Exciton Boltzmann Scattering Equations
line accounts for intra v alle y scattering, the fourth line accounts for interv alley scatter -
ing with a transition of the electron and the fifth line accounts for interv alley scattering
with a transition of the hole. In each line, the first term accounts for the in scatter-
ing of incoherent e xcitons from all other exciton states, the second term accounts for
the formation of incoherent e xcitons through phonon driv en non-radiati ve decay of the
e xcitonic coherence and the last term accounts for out scattering of the gi ven e xciton
state to all other e xciton states.
The in scattering rates for intra as well as inter v alley scattering are gi ven by
Γ inξ h ξ e Γ α
µ,µ 0 , Q , K = X
± | g ξ h ξ e Γ α
µ,µ 0 , K | 2 1
2 ± 1
2 + n Γ α
K δ E ξ h ξ e
µ, Q − E ξ h ξ e
µ 0 , Q + K ± ~ Ω Γ α
K (5.15)
Γ inξ h ξ e j α
eµ,µ 0 , Q , K = X
± | g ξ h ξ e j α
eµ,µ 0 , K | 2 1
2 ± 1
2 + n j α
K δ E ξ h ξ e
µ, Q − E ξ h ( ξ e − j )
µ 0 , Q + K ± ~ Ω j α
K (5.16)
Γ inξ h ξ e j α
hµ,µ 0 , Q , K = X
± | g ξ h ξ e j α
h,µ,µ 0 , K | 2 1
2 ± 1
2 + n j α
K δ E ξ h ξ e
µ, Q − E ( ξ h + j ) ξ e
µ 0 , Q + K ± ~ Ω j α
K (5.17)
The rates can be identified as Fermi rules, since they depend on the square of the
e xciton phonon matrix element, a factor depending on the phonon occupation and
a Dirac distrib ution ensuring energy conserv ation during a phonon scattering ev ent.
In each rate, there appear two terms, denoted by ± , where the + term accounts for
phonon emission processes and the − term accounts for phonon absorption processes.
The respecti ve out scattering rates in equation 5.14 can be written in terms of the in
scattering rates
Γ outξ h ξ e Γ α
µ,µ 0 , Q , K = Γ inξ h ξ e Γ α
µ 0 ,µ, Q + K , − K , Γ outξ h ξ e j α
eµ,µ 0 , Q , K = Γ inξ h ( ξ e + j )( − j ) α
eµ 0 ,µ, Q + K , − K ,
Γ outξ h ξ e j α
hµ,µ 0 , Q , K = Γ in ( ξ h − j ) ξ e ( − j ) α
hµ 0 ,µ, Q + K , − K . (5.18)
Here, first the phonon momentum has to be transformed and then the e xciton momen-
tum. No w , some properties of the e xciton Boltzmann equation shall be discussed. First,
ne glecting the radiati ve coupling and the contrib ution from the excitonic coherence,
one finds that the total amount of e xcitons is a conserved quantity . T aking additionally
into account the dynamics of the incoherent photons one also finds a conserv ation la w
between e xcitons and photons. Therefore it holds
∂ t X
Q ,µ,ξ e ,ξ h
N ξ h ,ξ e
µ, Q + X
K ,k z ,σ
n σ
K ,k z ! = 0 . (5.19)
As already mentioned the Coulomb contrib ution leads to an equilibration of the states
with the same ener gy , which are connected through either intraband or interband con-
trib ution. The dominating contribution in a scenario where the lo west lying bright
e xciton (A exciton) is e xcited resonantly is the interband interaction leading to a cou-
pling between the A e xcitons in the K valle y and the K 0 v alley .
5. Exciton Dynamics and Photoluminescence
The phonon mediated e xciton scattering leads to a thermalization of the exciton density
and fulfills a detailed balance. This shall be demonstrated in a simplified model in
the follo wing: Ne glecting the electron and hole v alley spin as well as the e xcitonic
quantum numbers and restricting the phonon spectrum to one optical mode, the phonon
contrib ution of the Boltzmann equation reads
∂ t N Q = 2 π
~ X
K , ± | g K | 2 1
2 ± 1
2 + n K N Q + K δ ( E Q − E Q + K ± ~ Ω K )
− 2 π
~ X
K , ± | g K | 2 1
2 ± 1
2 + n K N Q δ ( E Q − E Q + K ∓ ~ Ω K ) . (5.20)
In the equilibrium where ∂ t N Q = 0 the amount of in scattering and out scattering
e xcitons are equal. The δ distribution can only be satisfied by a certain momentum | Q |
which allo ws to eliminate the sums ov er the momenta. As a result, the relation
N Q + K
N Q ∝ n K
1 + n K
(5.21)
holds, where the assumption E Q + K > E Q was made without loss of generality . In-
serting the Bose Einstein distrib ution for the phonon occupations, one obtains
N Q + K
N Q ∝ exp − E Q + K − E Q
k B T , (5.22)
with k B T being the thermal ener gy . Therefore in the lo w density limit, excitons follo w
a Boltzmann distrib ution.
5.1.2. Numerical Ev aluation of the Boltzmann equation
In this subsection the numerical e v aluation of the Boltzmann scattering equation is
discussed. T o reduce the notation for no w the simplified form in equation 5.20 is
considered, but with respect to all phonon modes. Since the exciton phonon element
carries no angular dependenc y , the formation of excitons, equation 5.14 is also angular
independent. The exciton-phonon in- and out-scattering terms, equations 5.20, 5.17
and 5.18, further conserve the isotropy of the e xciton density . This can be prov en
easily by summing the in- and out-scattering terms o ver the angle φ K on the right
hand side of equation 5.20, by exploiting the isotrop y of the exciton-phonon coupling
elements, of the exciton dispersion and of the phonon dispersion around the respecti ve
high symmetry points. As a result the e xciton density is angular independent for all
times ∂ φ N Q = 0 . Therefore the momentum dependence can be can be av eraged out
before the numerical e v aluation of the Boltzmann equation. The sum has to be written
as an inte gral, compare equation 3.48. Then the momentum space is discretized in a
5.1. Exciton Boltzmann Scattering Equations
spherical parametrization Q = Q ( cosφ Q , sinφ Q ) . The resulting equation of motion
reads
∂ t N Q = X
K
W Q,K N K , (5.23)
which is solv ed within a Runge Kutta algorithm of the fourth order . The transition
matrix reads
W Q,K = 1
4 π 2 ~
X
φ Q ,φ K ,α
∆ φ Q ∆ φ K K ∆ K Γ inα
Q , K − Q
−
− δ Q,K
1
4 π 2 ~ X
K
X
φ Q ,φ K ,α
∆ φ Q ∆ φ K K ∆ K Γ outα
Q , K − Q
. (5.24)
The appearing in scattering and out scattering rates contain Dirac distrib utions which
ensure the ener gy conserv ation during a phonon scattering e vent, equation 5.17. This
Dirac distrib ution hav e to be written as Kronecker symbols to treat them numerically .
Therefore the Dirac distrib ution, being a function of the energy dif ference, is written
as a function of the momentum
δ (∆ E ( Q, φ Q , K , φ K )) = X
i
δ ( K − K i )
| ∂ K | K = K i ∆ E ( Q, φ Q , K , φ K ) | (5.25)
where K i solv es 0=∆ E ( Q, φ Q , K , φ K ) . For the appearing form of the Dirac dis-
trib ution, 5.17 not more than one solution exists, which remo ves the sum from the
e xpression. Ne xt since the Dirac distribution is not defined on a discrete grid, it has to
be written as a Kronecker symbol
δ ( K − K i ) → δ K,K i . (5.26)
On a discrete representation of the K space the condition K = K i can ne ver be ful-
filled e xactly . Therefore the Kronecker distrib ution has to be rounded on the closest
grid point in the zeroth approximation. A better con v ergence can be obtained if the
Kronecker distrib ution is interpolated between the neighboring K points of the solu-
tion K i . It turns out that e ven the linear interpolation between the neighboring grid
points leads to a drastic increase of the con ver gence. Thus, one ends up with the final
e xpression
δ (∆ E ( Q, φ Q , K , φ K )) → X
i =( up,dow n )
α i δ K − K i
| ∂ K | K = K i ∆ E ( Q, φ Q , K , φ K ) | (5.27)
where K up = ceil ( K 0 ) and K dow n = f l oor ( K 0 ) with K 0 being the unique solution of
0 = ∆ E ( Q, φ Q , K , φ K ) . The interpolation factors can then be obtained as
α i = | ∆ E ( Q, φ Q , K ¯
i , φ K ) |
| ∆ E ( Q, φ Q , K i , φ K ) | + | ∆ E ( Q, φ Q , K ¯
i , φ K ) | . (5.28)
5. Exciton Dynamics and Photoluminescence
The nai ve implementation w ould be to discretize the K space linear . Ho wev er since
the crucial parameters here are the ener gy separations between the dif ferent v alleys, a
discretization of K 2 seems to be con venient. It turns out that both, the linear and the
quadratic discretization gi ve similar results, but the numerics with K 2 discretization
con ver ges approximately four times faster than the numerics with linear discretization.
5.2. Formation and Thermalization of Excitons
In the current section, a particular solution of the exciton Boltzmann scattering equa-
tion, equation 5.14 is discussed. T o get a general impression of the exciton formation
and the phonon dynamics, the spin degree of freedom of electron and hole and the
e xcited excitonic states are ne glected, corresponding to the excitation of the TMD
monolayer with unpolarized light. The e v aluation of the Boltzmann equation is then
restricted to the lo west exciton states which can be adressed in an optical e xperiment
and its corresponding states far abo ve the light cone including interv alley e xcitons
with the electron being at the Λ and K 0 v alley , cf. figure 5.1. Further , states where
the electron is located at the Λ 0 v alley are included in the numerical e v aluation. As
it will turn out, the occupation of these states is neglectible and there the data are not
sho wn and not discussed explicitly . Since a resonant optical e xcitation is in the focus
of this e v aluation, the non-resonant Coulomb terms are neglected for the numerical
e v aluation. F or the same reason interv alley scattering of the constituent holes can be
ne glected since the final states with holes at other high symmetry points are separated
by at least 200 meV from the bright state [34]. W ithout loss of generality the hole
forming the e xciton is considered to be located at the K v alley . The dynamics of the
e xcitons with a hole at the K 0 v alley are quantitati vely equi v alent. These states are
coupled through interv alley e xchange coupling, equation 3.58, but since the e xcitation
occurs unpolarized it has no impact on the dynamics because the occupation of both
spin configurations are equal. Therefore the whole Coulomb contribution in equation
5.14 can be neglected and the simplified equation of motion for the incoherent e xciton
density reads
∂ t N K i e
Q =
= 2 π
~ X
k z | M 1 s,K,σ +
Q ,k z | 2 n σ +
Q ,k z − N K i e
µ, Q δ K,i e δ E K i e
1 s, Q − ~ ω σ +
Q ,k z
+ 2 π
~ X
K ,α
Γ in K i e Γ α
1 s, 1 s, Q , K N K i e
Q + K + | P K i e
Q + K | 2 − Γ out K i e Γ α
1 s, 1 s, Q , K N K i e
Q
+ 2 π
~ X
K ,j,α
Γ in K i e j α
e 1 s, 1 s, Q , K N K ( i e − j )
Q + K + | P K ( i e − j )
Q + K | 2 − Γ out K i e j α
e 1 s, 1 s, Q , K N K i e
Q . (5.29)
The first line describes the coupling of the incoherent e xciton densities to the incoher-
ent photon field, where the second term accounts for photon emission and the first term
5.2. Formation and Thermalization of Excitons
Figure 5.1.: Exciton f ormation, thermalization, and luminescence. (a) The e xcitonic coherence
is created optically (blue). Phonon induced dephasing creates exciton density o ver the
whole excitonic Brillouin zone. This includes intrav alley scattering (yello w) as well as
interv alley scattering with Λ phonons (red) and K phonons (bro wn). (b) Phonon me-
diated scattering leads to the thermalization of excitons. T o the photoluminescence, all
excitons contrib ute which are located within the lightcone (blue shaded). This picture
was published in a similar form in reference [75].
accounts for photon absorption. Since the e xcitation of the system in the current model
occurs with a coherent laser pulse, the latter is neglected. The second and the third line
describe the e xciton phonon coupling where the second line accounts for intrav alley
scattering in K − K , K − Λ , K − K and K − Λ states and the third line the interv alley
scattering between these states. The definitions of the scattering rates can be obtained
from equation 5.17.
Optical e xcitation with a classical light pulse leads to the formation of an excitonic
coherence which oscillates with the e xcitonic energy . In a perpendicular geometry
this e xcitonic coherence exhibts a v anishing center of mass momentum, cf. figure 5.1
(a). Radiati ve coupling and e xciton phonon scattering leads to a decay of the excitonic
coherence as discussed in the pre vious chapter . The latter accounts for the formation
of incoherent e xciton densities within the Γ valle y ( K − K states) as wells as in the
indirect states, i.e. the Λ v alley ( K − Λ states) and the K v alley ( K − K states).
Since the e xcitonic coherence decay within some tens of femtoseconds, the formation
of e xciton densities also occurs on this timescale [70] As a direct consequence of the
Boltzmann scattering equation, in the absence of radiati ve coupling the sum of squared
e xcitonic coherence and the incoherent exciton density is a conserv ed quantity . The
results of this section and the follo wing section ha ve been published in 2D Materials,
reference [75].
5. Exciton Dynamics and Photoluminescence
Figure 5.2.: Exciton dynamics in WSe 2 at 77 K. Exciton occupation N i
Q after optical excitation
as a function of the excitonic kinetic ener gy and time for (a) Γ ( K − K ), (b) Λ ( K − Λ ),
and (c) K ( K − K ) excitonic states. Snapshots for selected times are sho wn in the
insets. Note the log scale for the occupation axis. This picture was published in a
similar form in reference [75].
5.2.1. Momentum Resolved Dynamics in WSe 2 at 77 K
First the e xciton dynamics in the ex emplary material WSe 2 at the temperature of 77 K
shall be discussed. Figure 5.2 sho ws the time- and momentum dynamics of the inco-
herent e xciton occupation N i h i e
Q after resonant optical e xcitation of the 1s A exciton.
The most rele v ant dynamics takes place within the Γ v alley corresponding to an e xci-
ton with electron and hole within the electronic K v alley , cf. figure 5.2 (a), the Λ v alley
corresponding to an e xciton with the hole at the K v alley and an electron at the Λ v al-
le y in the electronic Brillouin zone, cf. figure 5.2 (b) and the K v alley corresponding
to an e xciton with a hole at the K valle y and an electron at the K v alley , cf. figure
5.2 (c). Note, that also the exciton occupation at the M v alley consisting of an e xciton
with the hole at the K point and an electron at the Λ v alley w as included to the nu-
merical e v aluation of equation 5.14. Ho we ver due to the lar ge separation of hundreds
of meV to the bright state, the contribution of these e xciton states to the dynamics is
ne glectible small and therefore not illustrated in figure 5.2.
W ithin the first 100 fs the formation of incoherent e xciton occupations can be observ ed.
This is due to non-radiati ve decay from the e xcitonic coherence located at Q =0 . The
5.2. Formation and Thermalization of Excitons
formation of e xcitons takes mainly place at the Γ v alley ( K − K states) through emis-
sion and absorption of acoustic Γ phonons and absorption of optical Γ phonons, cf.
figure 5.2 (a), and in the Λ v alley , cf. figure 5.2 (b), due to emission and absorp-
tion of acoustic and optical Λ phonons. Also a small formation of K e xcitons can
be observ ed, cf. figure 5.2 (c), b ut weak compared to the formation of Λ excitons
because of a smaller e xciton phonon coupling element originating from a weaker car -
rier phonon coupling [92, 91]. Interestingly the formation of e xcitons at the Γ v alle y
occurs at v ery low kinetic ener gies of the e xciton since the in v olved momenta of the
formation mediating acoustic phonons is on the order of some meV . In strong contrast,
the formation of the incoherent e xcitons in the indirect Λ and K v alle ys ( K − Λ and
K − K 0 states) takes place at ele v ated kinetic energies. This is due to the fact that
the separation between the Λ v alley minimum and the bright states is approximately
-70 meV and the phonon ener gies are about 13 meV for acoustic and 30 meV for opti-
cal phonons [92, 91, 73]. According to the source terms in equation 5.14, the dif ferent
source terms of the incoherent e xciton occupation through non-radiativ e decay of the
e xcitonic coherence can be identified in figure 5.2 (b): (i) at 40 meV abo ve the v al-
le y minimum a small maximum in the exciton occupation can be observ ed, which can
be addressed the emission of optical Λ phonons, (ii) formation at 55 meV abov e the
v alley minimum can be addressed to exciton formation through emission of acoustic
Λ phonons and (iii) at approximately 80 meV the e xciton formation through absorp-
tion of acoustic Λ phonons takes place. Note that in principle also the formation of
e xcitons through optical phonon absorption takes place b ut is not clearly visible since
the coupling strength are not suf ficiently lar ge at cryogenic temperatures. According
to the scattering rates, equation 5.17, the formation of excitons is stronger for phonon
emission processes, since here the corresponding coupling element depends on 1 + n
whereas the coupling for phonon emission depends on the phonon occupation n only .
In figure 5.2 (b), the re gions of e xciton formation do not appear as sharp features in
the e xciton density and the exact ener gies do not perfectly match the energy dif ference
between v alley separation and phonon ener gy , which has two reasons: the two optical
( T O , A 1 ) and acoustic ( LA , T A ) modes hav e slightly dif ferent energies which leads to
a broadening of the formation re gion. Further already the thermalization of excitons
sets in, such that excitons relax do wn to lo wer energies which shifts the ener gies at
which the e xcitons appear in the first snapshot. In the same way , the dif ferent features
in the occupation of the K e xcitons can be explained.
The formation of incoherent e xcitons ends after approximately 100 fs which is consis-
tent with the obtained coherence lifetime under the gi ven conditions [70]. After this
time, the phonon mediated thermalization is the dominating mechanism determining
the incoherent e xciton dynamics, which leads to a down scattering of the incoherent
e xcitons to the energetically lo west states. After approximately 1 ps the steady state
distrib ution is reached, being a Boltzmann distribution in nice agreement with equa-
tion 5.22.
In vestig ating the relaxation dynamics more closely , steps in the exciton occupation can
5. Exciton Dynamics and Photoluminescence
be observ ed in the transient regime. These steps appear at approximately 30 meV and
can be identified as an optical phonon bottleneck: The intrav alley relaxation due to
optical phonon emission appears to be faster than the relaxation mediated by acoustic
phonon emission due to lar ger exciton phonon coupling elements for optical phonons.
Therefore, excitons with ener gies larger than 30 meV can relax v ery fast under emis-
sion of an optical phonon. Since excitons with ener gies less than 30 meV can only relax
under emission of acoustic phonons which appears to be slo wer , pronounced steps are
formed in the transient re gime. Note that excitons do not only relax within one v alle y
by emission of Γ phonons. Due to the very ef ficient coupling to zone-edge phonons
the most probable way of e xciton relaxation is to scatter from one v alley to another
if the e xcitonic kinetic energies are suf ficiently lar ge. In particular M phonons which
mediate the coupling of Λ e xcitons and K e xcitons turn out to be very crucial for the
e xciton thermalization, resulting in multi peak features in the e xciton occupation in the
transient re gime. Interestingly most excitons being created at the K v alley are formed
from Λ e xcitons through M phonon emission which were formed by non-radiati v e de-
cay from the e xcitonic coherence.
5.2.2. Momentum Resolved Dynamics in MoSe 2 at 77 K
Figure 5.3 illustrates the e xciton dynamics in MoSe 2 at a temperature 77 K. In line
with the results obtained for WSe 2 , within the first 100 fs the formation of incoherent
e xcitons can be observed. In contrast to the situtation in WSe 2 here most e xcitons are
formed within the Γ v alley due to absorption and emission of acoustic and optical Γ
phonons, cf. figure 5.3 (a). Also a weaker exciton formation within in the K v alley can
be observ ed, cf. figure 5.3 (c). Since the K v alley in MoSe 2 is located abo ve the bright
state by about 7 meV , only phonon absorption of acoustic and optical K phonons can
contrib ute to the exciton formation here. Ha ving a closer look in the formation dy-
namics interesting features can be observ ed which were not visible in the formation
dynamics of WSe 2 . Here, the coupling to optical phonons is much more ef ficient com-
pared to the situtation in WSe 2 [92, 91], which leads to the formation of pronounced
peaks in the occupation of Γ excitons at approximately 40 meV due to optical phonon
absorption being consistent with the respecti ve phonon ener gies. Also a very small
amount of Λ e xcitons is formed, cf. figure 5.3 (b). Here, the Λ v alley is separated
by 130 meV from the bright state such that not e v en phonon absorption processes can
fulfill the ener gy conserv ation in equation 5.14. Ho wev er , since the excitonic coher -
ence is broadened with the homogeneous width, also of f resonant phonon scattering
can take place leading to the formation of a small amount of Λ e xcitons.
5.2.3. V alle y Resolved Dynamics at 77 K and 300 K
T o further in vestigate the thermalization of e xcitons, in figure 5.4 the temporal e v olu-
tion of the total e xciton occupation for each valle y N K i e = P Q N K i e
Q for WSe 2 (a)
5.2. Formation and Thermalization of Excitons
Figure 5.3.: Exciton dynamics in MoSe 2 at 77 K. Exciton occupation N i
Q after optical excitation
as a function of the excitonic kinetic ener gy and time for (a) Γ ( K − K ), (b) Λ ( K − Λ ),
and (c) K ( K − K ) excitonic states. Snapshots for selected times are sho wn in the
insets. Note the log scale for the occupation axis. This picture was published in a
similar form in reference [75].
Figure 5.4.: Exciton densities at 77 K. the figure illustrates the coherent | P 0 | 2 and total incoherent
v alley occupations N i = Q N i
Q /A as a function of time, with i = K − K , K − Λ ,
and K − K in (a) WSe 2 and (b) MoSe 2 at 77 K. This picture was published in a similar
form in reference [75].
and MoSe 2 (b) including the optically initialized excitonic coherence is depicted. In
WSe 2 a thermalization time of about 2 ps can be re vealed after which the total e xciton
5. Exciton Dynamics and Photoluminescence
occupation for the indi vidual v alley remains approximately constant. Interestingly the
steady state occupation in the Γ v alley is v ery small. In contrast, most excitons are
located within the Λ v alley follo wed by the K v alley . Remarkably the occupation of
the Λ v alley is more than three times lar ger than the occupation of the K v alley , which
seems curious since the ener getic separation of both v alleys is only about 3 meV . This
contradiction can be traced back to the degenerac y of the Λ v alley since there e xist
three independent Λ v alleys in the electronic Brillouin zone, cf. figure 3.1 (a), and so
three independent Λ v alleys in the e xcitonic Brillouin zone.
The dynamics of the total incoherent excitons in MoSe 2 turns out to be significantly
dif ferent, cf. figure 5.4. According to the momentum resolved study most incoherent
e xcitons are formed within the Γ v alley , which is the lo west lying valle y here. After a
initial sharp increase of the occupation in the K v alley , which originates in the broad-
ened formation of e xcitons and thus enabled scattering to K states, the equilibration
between Γ and K states takes place on a pico second time scale being consistent with
the weak coupling of e xcitons with K phonons. Here, it becomes e ven more ob vious
that the fast thermalization in WSe 2 between the Γ and the K v alley , cf. figure 5.4, is
due to the presence of the Λ v alley where stepwise phonon scattering e vents via the Λ
v alley lead to ef fecti ve e xciton scattering between the Γ and the K valle y . Since the Λ
v alley is separated by 130 meV from the ground state, these states are ne ver occupied
significantly . After the thermalization a relati vely lar ge occupation of the K e xciton
states can be observ ed, cf. figure 5.4 which is due to the relati vely small separation of
the K v alley minimum to the bright state of about 7 meV .
Interestingly , in both in vestigated materials, the total amount of incoherent e xcitons is
smaller than the optically e xcited coherent exciton occupation, i.e. the individual con-
trib utions from Γ , Λ and K excitons do not add to 1. This is to the fact that the e xcitonic
coherence is damped by both, phonon mediated and radiativ e coupling. Therefore, a
certain amount of the coherent e xcitons, namely the ratio of radiati v e decayed to total
decayed coherent e xcitons γ r ad
γ r ad + γ non r ad , decayed radiati v ely and thus is not con verted to
an incoherent e xciton occupation. The indi vidual contribution can be estimated from
figure 4.5 and 4.6. As a result in the limit of vanishing non-radiati ve dephasing no in-
coherent e xcitons are formed after coherent e xcitation. Ho we ver , incoherent e xcitons
can also be formed by absorption of incoherent light, equation 5.14 line 1.
Until no w , only the e xciton formation and thermalization dynamics at 77 K hav e been
discussed. Figure 5.5 illustrates the time resolved dynamics of the total v alley occupa-
tions at 300 K in WSe 2 (a) and MoSe 2 (b). Here, se veral dif ferences to the dynamics at
77 K can be found. First one finds in both materials a much faster coherence lifetime,
leading to a faster decay of the e xcitonic coherence (Note the dif ferent timescales in
the figures 5.4 and 5.5). This is due to a more ef ficient exciton phonon interaction
at ele v ated temperatures due to a higher phonon occupation, equation 5.17. This is
consistent with the results in the pre vious chapter , cf. figure 4.5 and 4.6. The direct
consequence from the more ef ficient exciton phonon scattering is the larger ratio of
coherent e xcitons which is con verted into incoherent excitons due to e xciton phonon
5.2. Formation and Thermalization of Excitons
Figure 5.5.: Exciton densities at 300 K. the figure illustrates the coherent | P 0 | 2 and total incoher -
ent v alley occupations N i = Q N i
Q /A as a function of time, with i = K − K ,
K − Λ , and K − K in (a) WSe 2 and (b) MoSe 2 at 300 K. Note the dif ferent timescale
compared to figure 5.4. This picture was published in a similar form in reference [75].
scattering. Another consequnence of the more ef ficient exciton phonon coupling is the
faster thermalization of the incoherent e xcitons. While at 77 K the occupations of the
dif ferent v alleys were thermalized after 1 ps in WSe 2 and 2 ps in MoSe 2 the steady
state at 300 K is reached after approximately 300 fs in WSe 2 and MoSe 2 . Interestingly
also the contrib utions from the dif ferent v alleys to the total amount of e xcitons change
with increasing temperature, meaning that states which are located higher in ener gy
are occupied by more e xcitons at elev ate temperature. Exemplary the ratio between
K and Γ e xcitons is about 0.3 at 77 K and 0.8 at 300 K. This can be understood from
the detailed balance, equation 5.22. The distribution of e xcitons follo ws a Boltzmann
distrib ution where the width is gi ven by the thermal ener gy k B T . While at 77 K the
termal ener gy is only about 6.5 meV it is 25.7 meV at 300 K. Therefore also exciton
states well abo ve the bright state are occupied at ele v ated temperatures.
5.2.4. Conc lusion
In conclusion, in this section the formation and thermalization of excitons of TMDs
in the lo w excitation limit were in vestigated. The exciton formation w as found to be
phonon dri ven from the e xcitonic coherence occuring on the timescale of the coher -
ence lifetime after optical e xcitation. The phonon mediated thermalization of e xcitons
takes place on a picosecond timescale at 77 K and speeds up with increasing tempera-
ture. In molybdenum based materials where the bright state is the ground state of the
system, most excitons are located in the Γ v alley after the thermalization. In contrast
in tungsten based materials, most excitons are located in dark Λ states, which are the
ground state of the system here.
5. Exciton Dynamics and Photoluminescence
Figure 5.6.: photoluminescence after optical excitation. The photoluminescence (solid red lines)
in (a) WSe 2 and (b) MoSe 2 consists of a coherent (blue) and an incoherent part (yel-
lo w). For comparison, the bro wn curve e xhibits the total luminescence at room tem-
perature illustrating the dif ferent temperature trend in WSe 2 and MoSe 2 . This picture
was published in a similar form in reference [75].
5.3. Photoluminescence of Thermaliz ed Excitons
and Quantum Yield
Ha ving no w a detailed knowledge about the incoherent e xciton dynamics, the temporal
e v olution of the emitted light can be in vestigated. As discussed pre viously the inten-
sity of the emitted light is determined by both, incoherent as well as coherent exciton
occupation which is located within the light cone, equation 5.8 and 5.9
I ∝
K | M K
K , E 2
2 c 2 − K 2 | 2 | P KK
1 s, K | 2 + N KK
1 s, K 1
| K | < E KK
1 s
c
, (5.30)
where the first term denotes the coherent emission and the second term the incoherent
emission. Since the focus here lies on the emission after unpolarized resonant excita-
tion of the lo west lying exciton at the Γ point ( KK e xciton states), the exciton state
quantum number as well as the spin inde x were dropped compared to the pre vious
e xpression, equation 5.8 and 5.9.
Figure 5.6 illustrates the intensity of the emitted light in WSe 2 (a) and MoSe 2 (b) after
resonant optical e xcitation. In WSe 2 at 77 K the luminescence intensity is determined
by coherent emission in the first 1 ps which e xceeds the incoherent emission by some
orders of magnitude. The decay time of the coherent emission is determined by the
e xcitonic coherence lifetime being about 60 fs at 77 K, which is in agreement with the
temporal e v olution of the coherent exciton density , cf. figure 5.4. After approximately
1 ps the incoherent emission takes o ver . Since the lifetime of the coherent emission
is determined by radiati ve decay and phonon mediated scattering, it e xhibits a strong
temperature dependence gi ving only 30 fs at room temperature, again consistent with
the pre vious discussion, cf. figure 5.5. In MoSe 2 , cf. figure 5.6, in principle similar
qualitati ve trends can be observ ed. At 77 K the total intensity is determined by coherent
5.3. Photoluminescence of Thermalized Excitons and Quantum Y ield
emission up to approximately 0.5 ps. After this time the incoherent emission takes
o ver . The coherence lifetime in MoSe 2 is about 60 fs and decreases to 15 fs at room
temperature due to a more ef ficient exciton phonon scattering at higher temperatures.
The more pronounced temperature trend in MoSe 2 in comparison to WSe 2 can be
addressed to lar ger exciton phonon elements, equation 3.63 originating from lar ger
carrier phonon elements [92, 91]. In particular the strong non-radiati ve contrib ution
to the coherence lifetime in MoSe 2 can be identified as the formation of incoherent
e xcitons in the Γ v alley . There both, acoustic and optical Γ phonons contribute. This
is again in line with pre vious results discussed in the previous chapter , cf. figure 4.6
[70].
In both materials, the incoherent emission exhibits an initial f ast increase which can
be addressed to the formation of incoherent e xcitons through non-radiati ve decay from
the coherent e xcitons. Additionally , in WSe 2 the incoherent emission sho ws a fast drop
during the first 1 ps after optical e xcitation which is due to the fact that the e xcitons,
which are created within the Γ v alley , relax to ener getically lo wer states at the Λ v alley .
This drastic decay is absent in MoSe 2 since here the bright state is the ground state of
the system. In both materials the incoherent emission is smaller than the initial co-
herent emission. The reason is, that the incoherent exciton occupations is thermalized
and only a small amount of them is located within the lightcone. For the same rea-
son the incoherent emission decays on ultra long timescales compared to the coherent
emission. In equation 5.22 it w as demonstrated, that e xciton phonon scattering con-
serv es the total amount of excitons. Therefore in this model, radiati v e decay is the only
mechanism which leads to a decay of the total exciton occupation. Since the amount
of radiati ve acti ve e xcitons is small, the radiati ve decay is strongly suppressed.
Comparing the intensity of the incoherent emission at 77 K and 300 K in WSe 2 and
MoSe 2 and interesting beha vior can be found. While in WSe 2 the intensity of the
emitted light at 300 K is lar ger than at 77 K in MoSe 2 the in verse beha vior can be
found. This can be understood considering MoSe 2 as a direct material and WSe 2 as an
indirect one: in MoSe 2 the bright state is the ground state of the system. An increase
of the temperature leads to a broadening of the e xciton occupation, cf. figure 5.4 and
5.5, and so to an ef fecti ve reduction of the relati ve occupation within the light cone. In
contrast, in WSe 2 the optically dark Λ states are the ground state of the system. Here,
an increasing temperature leads again to a broadening of the e xciton occupation and
therefore to a lar ger relati v e occupation of the bright state, equation 5.30.
This beha vior is in vestigated quantitati vely in the follo wing. In figure 5.7 the e xci-
tonic occupation lifetime T 1 as a function of temperature (a) and the ratio of e xcitons
within the lightcone after thermalization as a function of temperature (b) is depicted
for both materials MoSe 2 and WSe 2 . As already discussed the luminescence intensity
as a function of temperature e xhibits an in v erse beha vior in MoSe 2 and WSe 2 : it is
increasing in WSe 2 while it is decreasing in MoSe 2 as a function of temperature. For
the occupation lifetime also an in verse beha vior can be found: While for MoSe 2 the
occupation lifetime increases from 210 ps at 50 K to 1.3 ns at room temperature, the
5. Exciton Dynamics and Photoluminescence
Figure 5.7.: Luminescence yield and decay time. (a) Radiati ve Lifetime T 1 of the incoherent
exciton population after thermalization for WSe 2 (red) and MoSe 2 (blue) and (b) pho-
toluminescence yield defined as the ratio of bright excitons N rad and all generated
excitons N all as a function of temperature. This picture w as published in a similar
form in reference [75].
occupation lifetime decreases in WSe 2 from 260 ms at 50 K to 33 ns at room tempera-
ture, cf. figure 5.7 (a). The increasing occupation lifetime in MoSe 2 can be attrib uted
to a reduction of the relati ve occupation within the light cone at ele v ated temperatures
which is connected to a suppressed radiati ve coupling of the e xcitons. In contrast, in
WSe 2 ele v ated temperatures lead again to a broadening of the e xciton occupation but
therefore to an increasing relati ve occupation of the radiati ve acti v e e xcitons within the
light cone, which e xplains the in verse beha vior .
Interestingly the computed radiati ve lifetimes of the e xcitons exceed the e xperimen-
tally measured lifetimes by some orders of magnitude [133, 134, 65], which leads
to the conclusion that the radiati ve recombination rates ha ve not been measured e x-
perimentally so far . Possible reasons are that other decay mechanisms which are be-
yond the scope of this thesis such as defect assisted recombination and higher exci-
tation processes such as Auger scattering may dominate the exciton recombination.
Ex emplary recent experimental findings suggest that non-radiati ve recombination by
e xciton-exciton annihilation may be the dominating recombination mechanism for e x-
citation densities well abo v e 10 12 cm − 2 [153, 154] Therefore, one can conclude that
the lo w excitation treatment of e xcitons is v alid up to this v alue.
The quantum yield, which is a technologically important quantity can be defined as the
ratio of emitted photons to absorbed photons. Since the, by photon absorption, created
e xcitons can either decay radiativ ely or non-radiati vely , the quantum yield can also be
e xpressed by the time integrated ratio of e xcitons which decay radiativ ely to the total
amount of decaying e xcitons. In the steady state it is described by the rates only
QY ( T )= Q Γ r ad
Q N Q ( T )
Q (Γ r ad
Q +Γ
non − r ad
Q ) N Q ( T ) , (5.31)
Unfortunately the quantum yield is not a v ailable from this computation, since the non-
radiati ve decay channels are not included in the theoretical approach so f ar . Ho we ver
5.3. Photoluminescence of Thermalized Excitons and Quantum Y ield
under some assumptions some qualitati ve statements can be made. Since in the e xper-
iment the non-radiati ve decay is dominating the radiati ve decay of e xcitons, it is fair
to state P Q (Γ r ad
Q + Γ non − r ad
Q ) N Q ( T ) ≈ P Q Γ non − r ad
Q N Q ( T ) . Assuming further that
the non-radiati ve rate Γ non − rad is equal for all exciton states and the radiati v e rate is
constant within the light cone, the quantum yield can be written in this limit as
QY ( T ) = Γ r ad
Γ non − r ad
dar k
N r ad
N tot
( T ) . (5.32)
Again note that this e xpression is only v alid in the limit of dominating non-radiati ve
decay of the e xcitons. Here, N r ad denotes the exciton occupation within the lightcone
and N tot the total amount of excitons. As a result the ratio of bright to all excitons
is a measure for the quantum yield in the limit of dominating constant and tempera-
ture independent non-radiati ve e xciton decay . T o av oid confusions, this ratio is called
luminescence yield in the follo wing.
In figure 5.7 (b) the luminescence yield is depicted as a function of temperature in
WSe 2 and MoSe 2 . For WSe 2 the luminescence yield is increasing as a function of
temperature which can be addressed to the appearance of ener getically low lying states
in this material. Again, at lo w temperature most excitons are located in the lo w lying
states and ele v ating the temperature fills the bright state. This behavior is in clear con-
trast to the situation in MoSe 2 . Here, the bright state is the ground state which results
in a decreasing yield as a function of temperature. This theoretical obtained beha vior
was recently observ ed in se veral e xperimental studies [134, 133, 65] These results can
be e xtrapolated to the whole molybdenum based and tungsten based TMD families
since within these families the e xcitonic band structures dif fer only quantitati v ely but
not qualitati vely . In particular also WS 2 exhibits lo w lying dark exciton states at the Λ
v alley in the e xcitonic Brillouin zone.
5.3.1. Conc lusion
In conclusion, in this section the luminescence properties of TMDs in the lo w e xci-
tation limit were in vestig ated. Surprisingly it turned out, that the luminescence yield
in tungsten based TMD materials is increasing as a function of temperature which is
in contrast to the beha vior in molybdenum based TMD materials. This behavior is in
e xcellent agreement with experimental observ ation [65, 133, 134] and was addressed
to the appearance of dark states ener getically belo w the bright state in tungsten based
materials.
5. Exciton Dynamics and Photoluminescence
5.4. Intrinsic V alle y Lifetime
In this section another limiting case of the general lo w density exciton Boltzmann scat-
tering equation is in vestig ated. Since optical excitation with circularly polarized light
e xcites excitons at opposite corners of the electronic Brillouin zone, energy selecti ve
e xcitation of the A exciton further selects e xcitons with opposite spins of constituting
electrons and holes. More specific the excitation of the A e xciton resonance with left
handed circular polarized light σ + e xcites an excitonic coherence at the K v alley (in
the electronic Brillouin zone) with electron and hole ha ving spin up ↑ and excitation
of the A e xciton resonance with right handed circular polarized light σ − excites an
e xcitonic coherence at the K 0 valle y with electron and hole ha ving spin do wn ↓ . Here
the lifetime of such optically injected spins, in this thesis referred to as spin lifetime,
is of crucial interest for future optoelectronic and spintronic applications.
In recent research the spin dynamics were in vestig ated experimentally [39, 155, 156,
157, 129, 158, 159] and theoretically [160, 161, 162, 163, 164]. Experimentally , the
spin lifetime is accessible through pump probe techniques [155, 165], where the mea-
sured spin lifetimes are typically on the order of 1 ps. Additionally , the spin lifetime
can be deduced from the polarization degree of the emitted light, where typical po-
larization de grees are on the order of 30-50 % [156, 157]. As possible e xplanations
for the underlying microscopic mechanisms reducing the spin lifetime the Dyakono v-
Perel [162, 163], the Elliott-Y afet [163] and the Silv a-Sham mechanism, often referred
as interv alley e xchange coupling, ha ve been considered [166, 167].
Here, in the scope of the in vestigation is the interv alley e xchange coupling which leads
to a coupling of e xcitons with dif ferent spins of the constituting electron and hole,
equation 3.58 and 5.14. As an e xperimental scenario the resonant excitation of the
1s A e xciton with left handed circular polarized light shall be considered. Thus an
e xciton occupation with spin up and hole and the K point is formed. V ia the interv alley
e xchange coupling, the e xcitons within the Γ valle y ( K − K exciton states) couple to
states in the K 0 − K 0 states resonantly , cf. figure 5.8 (a). The excitons in the K 0 − K 0
states ha ve electron and hole spin do wn. Couplings between the e xcitons in the A series
and the B series, cf. figure 3.8, are neglected due to the lar ge energetic separation. For
the same ar gument, coupling to higher excitonic states is ne glected. Since no spin flip
processes are contained in the model, it is suf ficient to take a mer ged electron and hole
spin in the follo wing. Further the coupling of excitons to indirect states f ar beyond the
light cone is considered, cf. figure 5.8 (b). Due to the lar ge v alence band splitting and
the lar ge separation of the electronic Γ from the upper K v alle y , interv alley scattering
of the holes is ne glected too.
5.4. Intrinsic V alle y Lifetime
Figure 5.8.: Schematic illustration of the excitonic band structur e in WX 2 . Blue bands denote
spin up electrons and holes whereas red bands illustrate spin do wn bands respectiv ely .
(a) The direct excitons ( K − K ↑ and K 0 − K 0 ↓ ) couple mediated by interv alley
exchange coupling, which occurs between excitons with opposite electron and hole
spins. (b) Excitons in K − K ↑ ( K 0 − K 0 ↓ , not sho wn) states can relax to indirect
K − Λ ↑ ( K 0 − Λ 0 ↓ ) and K − K 0 ↑ ( K 0 − K ↓ ) exciton states mediated by phonon
scattering.
The Boltzmann scattering equation for the incoherent e xciton densities read
∂ t N i h i e s
Q =
= 2 π
~ X
σ ,k z | M 1 s,i h ,s,σ
Q ,k z | 2 n σ
Q ,k z − N i h i e s
Q δ i h ,i e δ E i h i e s
1 s, Q − ~ ω σ
Q ,k z
+ 2 π
~ X
i 0
h ,i 0
e ,s 0 | X i h i e si 0
h i 0
e s 0
1 s, 1 s, Q | 2 N i 0
h i 0
e s 0
Q − N i h i e s
Q δ E i 0
h i 0
e s 0
1 s, Q − E i h i e s
1 s, Q
+ 2 π
~ X
K ,α
Γ ini h i e s Γ α
1 s, 1 s, Q , K N i h i e s
Q + K + | P i h i e s
Q + K | 2 − Γ outi h i e s Γ α
1 s, 1 s, Q , K N i h i e s
Q
+ 2 π
~ X
K ,j,α
Γ in i h i e sj α
e 1 s, 1 s, Q , K N i h ( i e − j ) s
Q + K + | P i h ( i e − j ) s
Q + K | 2 − Γ outi h i e sj α
e 1 s, 1 s, Q , K N i h i e
Q . (5.33)
The first line describes the radiati ve coupling of incoherent e xciton densities. The sec-
ond term describes the emission of incoherent photons and the first term describes the
absorption of incoherent photons. The latter is ne glected here, since the e xcitation of
the material is assumed to be performed with a coherent laser pulse. Due to the se-
lection rules of the optical matrix element spin up excitons ↑ only contrib ute to the
5. Exciton Dynamics and Photoluminescence
emission of left handed polarized light σ + and spin do wn excitons ↓ only contrib ute
to the emission of right handed polarized light σ − . The second line describes the
interv alley e xchange coupling between excitons with dif ferent spin. Interesting only
e xcitons located at the Γ valle y contrib ute to the interv alley e xchange coupling since
states located in indirect e xciton states violate either momentum or energy conserv a-
tion, equation 3.58. The third and fourth line describe the exciton phonon scattering,
where the third line accounts for intra val le y scattering and the fourth line accounts for
interv alley scattering assisted by electronic transitions. Note that during phonon scat-
tering e vents, neither the electron nor the hole spin is changed. The intensity of the
polarized emitted light is gi ven by
I σ ∝ X
K ,is | M 1 s,i,s,σ
K , q E 2
~ 2 c 2 − K 2 | 2 N iis
K + | P iis
K | 2 1 K < E
~ c . (5.34)
The luminescence intensity is proportional to the square of the optical matrix element
and the amount of incoherent (first term) and coherent (second term) e xcitons which
are located within the light cone. Note, that according to the selection rules of the
optical matrix element, excitons with constituting electron and hole at the i = K v alley
contrib ute to the emission of left handed polarized light σ = σ + and excitons with
constituting electron and hole located at the i = K 0 v alley contrib ute to the emission
of right handed polarized light σ = σ − .
5.4.1. Inter v alley coupling in MoSe 2
In this subsection the numerical solution of the spin resolv ed exciton dynamics, equa-
tion 5.33, and polarization resolv ed photoluminescence, equation 5.34, is discussed in
MoSe 2 at a temperature of 77 K is discussed.
In figure 5.9 (a) the temporal e v olution of the total spin up and down e xciton occu-
pation N ↑ = P i, Q N K i ↑
Q and N ↓ = P i, Q N K 0 i ↓
Q in MoSe 2 at a temperature of 77 K
is depicted. Further the excitonic coherence after resonant optical e xcitation with a
δ -shape left handed polarized light pulse is illustrated. The e xcitonic coherence decay
with a time constant of 60 fs being consistent with the results of the pre vious section,
cf. figures 5.4 (b). This decay originates in radiati ve and non-radiati ve decay where
the latter is accompanied by the formation of incoherent e xcitons in spin up states.
This can be seen in figure 5.9 since the risetime of the spin up occupation N ↑ fits the
decay time of the e xcitonic coherence. Interv alley e xchange coupling leads to the for -
mation of spin do wn excitons N ↓ from the spin up e xcitons N ↑ , which is described by
the second line in equation 5.33. The dif ferent spin occupations are equilibrized after
approximately 1 ps.
Additionally in figure 5.9 (b) the temporal dynamics of the intensity of the photolu-
minescence with respect to the polarization is depicted. The left handed polarized
emission rises on a similar timescale as the corresponding spin up exciton density N ↑
5.4. Intrinsic V alle y Lifetime
Figure 5.9.: T ime evolution of exciton density and intensity of photoemission at 77 K in MoSe 2 .
(a) sho ws the time ev oluation of the total ↑ and ↓ exciton densities. In figure (b) we
sho w the corresponding emission intensities of σ + and σ − light. Further we sho w the
contrib utions to figure (a) of Γ excitons (c) and K excitons (d).
since the corresponding e xciton states within the light cone are also driv en directly
from the phonon mediated decay of the e xcitonic coherence. After this initial rise the
emission intensity drops which is due to thermalization of the e xcitons through exciton
phonon scattering and interv alley e xchange coupling to spin do wn exciton states N ↓ .
Contrary the emission of the right handed polarized light σ − rises on a much longer
timescale of about 500 fs. The reason for the slo w rise is, that the corresponding spin
do wn exciton density is not initialized optically and therefore has first to be created due
to interv alley e xchange coupling from the spin up excitons. Further as is obvious from
comparing the risetimes of the spin do wn exciton density , cf. figure 5.9 (a), and the
right handed polarized emission a significant delay between both is visible. The rea-
son is, that the coupling element for the interv alley e xchange coupling is proprotional
to the e xcitonic center of mass momentum [64, 66], equation 3.10. As a result, the
bright states does not contrib ute to the interv alley e xchange coupling. Therefore first
an e xciton with non-v anishing center of mass momentum has to be generated through
a phonon absorption process, then has to transferred from the spin up states to the spin
do wn states and then has to scatter do wn to the light cone under emission of a phonon.
This e xplains the delay between the risetimes of spin down e xcitons and right handed
5. Exciton Dynamics and Photoluminescence
polarized light emission. Interestingly , despite the equilibration of the photolumines-
cence intensity starts delayed with respect to the equilibration of the e xciton density ,
the steady state of unpolarized emission is reached faster than the steady state of equal
spin occupations. This can be understood when considering the dif ferent contribution
to the total e xciton occupation, i.e. the occupation in Γ and in K states, cf. figure 5.9
(c) and (d). The spin up occupation in the Γ v alley is directly formed from the non-
radiati ve decay from the e xcitonic coherence which explains the f ast risetime of about
60 fs. The corresponding spin do wn occupation rises with a time constant of about
500 fs being consistent with the risetime of the total spin do wn occupation. In contrast
in the K states the risetime of the spin up states is about 200 fs. They are created di-
rectly from non-radiati ve decay of the e xcitonic coherence. Howe ver , exciton phonon
coupling to these states is weak since the y are located approximately 7 meV abov e the
bright state and thus only phonon absorption processes contrib ute to the formation of
e xcitons in these states. Further the corresponding exciton phonon elements are weak
due to weak electron phonon matrix elements [92, 91]. This is in excellent agreement
with the results in the pre vious section, cf. figure 5.4. The corresponding spin do wn
occupation rises on a much slo wer time scale of about 1 ps. The reason is that for the
formation of e xcitons in these states at least three scattering e vents are required: (i) first
an e xciton density in the Γ ↑ states has to be formed, (ii) interv alley e xchange transfer
takes place and (iii) the e xciton has to scatter from the Γ ↓ to the K ↓ states. Further if
an e xciton was created in the K ↑ v alley , it takes ag ain at least three scattering e v ents
for scattering to the K ↓ states. Therefore the K states act as a trap for exci tons with
respect to the interv alley e xchange coupling and therefore slo w do wn the equilibration
of the spin occupations.
5.4.2. Inter v alley coupling in WSe 2
After ha ving well in vestigated the spin and v alley resolv ed dynamics in monolayer
MoSe 2 no w the spin and v alley resolv ed dynamics in monolayer WSe 2 is in vestigated.
Therefore in figure 5.10 (a) the temporal e v olution of the total spin up and spin do wn
occupation at a temperature of 77 K is depicted. Again consistent with the results of the
pre vious section, cf. figure 5.4 (a), the optically injected e xcitonic coherence decays
with a time constant of about 60 fs due to radiati ve and non-radiati ve decay . The latter
accounts for the formation of incoherent excitons in spin up states N ↑ o ver the whole
e xcitonic Brillouin zone. Also a weak formation of spin do wn excitons within the first
100 fs can be observ ed. Here in contrast to the situation in MoSe 2 , cf. figure 5.9, the
magnitude is much smaller . As a result both spin occupations remain asymmetry on a
long timescale of approximately 10 ns. In the follo wing it will be discussed that this
interesting beha vior is due to the relaxation of spin up excitons N ↑ to indirect Λ and K
states which ef fecti vely blocks the interv alley e xchange coupling within the Γ states.
In figure 5.10 (b) the temporal e v olution of the photoluminescence intensity with re-
spect to the light polarization is sho wn. Again and comparable to the situation in
5.4. Intrinsic V alle y Lifetime
Figure 5.10.: T ime evolution of exciton density and intensity of photoemission at 77 K in
WSe 2 . (a) sho ws the time ev oluation of the total ↑ and ↓ exciton densities. In figure
(b) we sho w the corresponding emission intesities of σ + and σ − light. Further we
sho w the contributions to figure (a) of Γ e xcitons (c), Λ excitons (d) and K e xcitons
(e).
MoSe 2 , cf. figure 5.9 (b), an ultrafast rise of the incoherent left handed polarized
emission σ + can be observ ed which is due to the fact that the corresponding spin up
e xciton density is directly formed by non-radiati ve scattering from the optically in-
jected e xcitonic coherence. The rapid initial rise of the photoluminescence intensity
of the left handed polarized light σ + is follo wed by a decay on a timescale of some
picoseconds, which is attrib uted to the relaxation of the excitons within the light cone
to ener getically lo w lying states located at the Λ and K v alley mediated by e xciton
phonon scattering with zone-edge phonons. This is perfectly in line with the results
5. Exciton Dynamics and Photoluminescence
of the pre vious section, cf. figure 5.6 (a). Again consistent with the observ ations for
MoSe 2 the emission of right handed polarized light σ − rises on a longer time scale of
about 200 fs. Similar to the emission of left handed polarized light this rise is follo wed
by a pronounced decay within 5 ps which is again due to the relaxation to ener geti-
cally lo w zone edge exciton states. As for the corresponding e xciton densities the left
and right handed polarized emission dif fer by a factor of 10 after 5 ps. This dif ference
decays after approximately 10 ns.
T o study the blocking of the interv alley e xchange coupling in more detail, in figure 5.10
(c), (d) and (e) the temporal e v olution of the incoherent e xciton densities for Γ , Λ and
K states with respect to the spin de gree of freedom are depicted. The incoherent spin
up e xciton density at the Γ v alley , cf. figure 5.10 (c) rises within the coherence lifetime
and e xhibits a pronounced decay within 5 ps due to relaxation to lo w lying states. An
e xciton density within the Γ ↑ valle y experiences interv alley e xchange coupling which
leads to the formation of excitons in Γ ↓ states as long as there is suf ficient density in
the Γ ↑ states. Again, the excitons in Γ ↓ states relax to the corresponding dark states
being e xpressed by a 5 ps decay of this density after the initial rise.
Figure 5.10 (d) e xhibits the temporal ev olution in Λ states with respect to the spin
de gree of freedom. Excitons within the Λ ↑ states are formed through phonon mediated
decay of the e xcitonic coherence and through relaxation of Γ ↑ excitons which is also
mediated by scattering with phonons. Here, a formation rate of about 300 fs can be
identified which perfectly matches the results from the pre vious section, cf. figure 5.4
(a). Contrary to this, the formation of the Λ ↓ excitons occurs on a longer timescale,
since the y can only be formed by phonon scattering from Γ ↓ excitons. These ha ve
first to be created through interv alley e xchange coupling from Γ ↑ excitons, which
demonstrates the longer formation time. The residual occupations dif fer by a factor of
approximately 10 after 5 ps. This dif ference stays for about 10 ns.
F or the temporal e volution of K e xcitons, which is depicted in figure 5.10 (e) with
respect to the v alley de gree of freedom, in principle a similar qualitativ e beha vior as for
the Λ states can be observ ed. Ho wev er the corresponding magnitudes of the excitonic
occupations are much smaller which is due to the separation of the K states from the
Λ states by about 7 meV . Further the reader shall be reminded that there exist three Λ
v alleys in the e xcitonic Brillouin zone.
Concluding most e xcitons relax do wn to energetically lo w lying states at the Λ ↑ and
K ↑ states. Since here, no interv alley e xchange transfer occurs, the appearance of
lo w lying states ef fecti vely blocks the interv alley e xchange coupling. For bringing an
e xciton from Λ ↑ to Λ ↓ states at least three scattering e vents are required: (i) scattering
to Γ ↑ states mediated by Λ phonon absorption, (ii) interv alley e xchange transfer to the
Γ ↓ states and (iii) relaxation to the Λ ↓ states. Due to the lar ge energy separation
of the Γ and the Λ v alley of about 60 meV one phonon absorption e vent is not e ven
enough for e xcitons in the valle y minimum due to the small phonon ener gies of about
30 meV for optical phonons [91].
5.4. Intrinsic V alle y Lifetime
Figure 5.11.: Depolarization dynamics W e sho w the normalized dif ference in the occupation
∆ N = N K ↑ − N K ↓
N K ↑ + N K ↓ as a function of time after optically e xcitation at different tem-
peratures in MoSe 2 (a) and WSe 2 (b)
5.4.3. Degree of P olarization and V alley Lif etime
After ha ving a detailed kno wledge about the spin resolved dynamics in both in vesti-
gated materials at 77 K, no w the influence of the temperature on the interv alley e x-
change coupling shall be in vestig ated. Therefore in figure 5.11 the normalized occu-
pation dif ference ∆ N = N ↑ − N ↓
N ↑ + N ↓ for MoSe 2 and WSe 2 (b) is depicted. According to
the pre vious results in MoSe 2 the spin lifetime is about 100 fs at 77 K corresponding
to the time, where the occupation diff erence reaches e − 1 . As it can be observed, the
spin lifetime depends only weakly on the temperature ha ving a value of approximately
140 fs at room temperature. This beha vior can be understood from the counteracting
of two ef fects: First the interv alley e xchange coupling element is linearly increasing as
a function of the center of mass momentum of the exciton. Thus a larger temperature
results in broader e xciton distributions leading to a more ef ficient intervalle y transfer .
A counteracting ef fect can be found when considering the exciton landscape in MoSe 2 .
The dark KK states at the K v alley in the e xcitonic Brillouin zone are located abo v e
the bright state by only 7 meV . Therefore these states are a more occupied at room
temperature than at 77 K due to a broader e xciton distribution in ener gy . This is in
agreement with results from the pre vious section, 5.5. Since the KK excitons do not
contrib ute to the interv alley e xchange coupling the larger occupation of the K v alley
causes a reduction of the interv alley coupling ef ficiency . Both ef fects together explain
the almost temperature independent beha vior of the interv alley relaxation.
In WSe 2 the opposite beha vior can be observed, cf. figure 5.11 (b), where a pronounced
bie xponential decay of the normalized occupation difference is present. Here, the first
5. Exciton Dynamics and Photoluminescence
Figure 5.12.: P olarization degree and V alley Lifetime (a) V alley lifetime in both in vestigated
materials as a function of temperature. (b) Degree of polarization: the solid curv e
sho ws the degree of polarization for the incoherent emitted light. The dashed curve
sho ws the degree of polarization for the total emitted light. T o account for the non-
radiati ve relaxation of the e xcitons a dark relaxation rate of 1 ns was assumed.
decay which occurs in the first hundreds of fs after optical e xcitation can be ascribed
to the interplay of interv alley e xiton phonon coupling and phonon mediated relaxation
in the transient re gime: in contrast to the situation in the thermalized state a certain
amount of e xcitons is located at the Γ v alle y , cf. figure 5.4 (a) and figure 5.5 (a). They
can contrib ute to the interv alley e xchange coupling which results in a pronounced ini-
tial decay of the normalized occupation dif ference shortly after the optical excitation.
Interestingly this observ ation is less pronounced at ele v ated temperatures since here,
the relaxation to the lo w lying Λ states is more ef ficient and therefore the steady state
is reached faster . Since excitons in the Λ states can not contrib ute to the spin relaxation
due to interv alley e xchange coupling, the initial decay is less pronounced. The second
decay occurs on an ultralong timescale compared to the first decay . It is ascribed to
the interv alley e xchange coupling of the thermalized excitons. Here, in good agree-
ment with pre vious results, the spin lifetime at 77 K is about 10 ns. The reason is that
most e xcitons are located in Λ states under these conditions and thus the interv alley e x-
change coupling is ef fecti vely block ed. The second decay constant sho ws a decreasing
beha vior as a function of temperature being 50 ps at room temperature. The reason is
that at ele v ated temperatures also states with larger ener gies are occupied. As a result,
e xcitons in the Γ valle y are lar ger occupied leading to a larger lifetime.
After ha ving in vestigated the spin lifetime as a function of temperature qualitati vely ,
no w a quantitati ve description shall be gi ven. Therefore in figure 5.12 (a) the spin
lifetimes in MoSe 2 and WSe 2 are depicted as a function of temperature. In MoSe 2
the spin lifetime is almost constant as a function of temperature ranging from 110 fs
at 77 K to 140 fs at 300 K. This beha vior was attrib uted to the interplay of the broader
5.4. Intrinsic V alle y Lifetime
e xciton distributions which lead in general to a f aster intervalle y transfer and to the
occupation of indirect e xciton states which slow do wn the interv alley relaxation, since
the interv alley e xchange coupling only takes place within the Γ v alley . In WSe 2 the
spin lifetime is decreasing as a function of temperature from 8 ns at 77 K to 7 ps at
room temperature. This observ ation was attrib uted to lo w lying dark states at the Λ
v alley and the K v alley which suppress the interv alley e xchange coupling.
A quantity which is easily accessible through e xperiments is the de gree of polarization
of the emitted light. Therefore the whole intensity of the emitted light has to be inte-
grated o ver the measuring time with respect to its polarization n σ = R ∞
0 I σ ( t ) . Then
the polarization of the emitted light is gi ven by n σ + − n σ −
n σ + + n σ − . As already discussed no
non-radiati ve relaxation channels are included in the model. As a result the computed
lifetimes, cf. figure 5.7 (a), exceed the e xperimentally obtained lifetimes by orders of
magnitude. Therefore to match the e xperimental situation better , a ov erall dark recom-
bination rate of 1 ns − 1 is included to the numerical e v aluation. This rate is in agreement
with accessible e xperimental luminescence lifetimes [65].
The obtained light polarizations are depicted in figure 5.12 (b). The y are very small
in MoSe 2 ranging from 0.2 % at 77 K to approximately 0.02 % at room temperature.
The small polarizations is explained with the f ast interv alley transfer of e xcitons on a
subpicosecond timescale compared to a nanosecond timescale for the radiati ve decay
of the e xcitons. The decreasing beha vior as a function of temperature for the light
polarization is in contradiction to the e xpectation: at ele vated temperatures the spin
lifetime of e xcitons is comparable to that at 77 K and therefore a similar polarization
of the emitted light would be e xpected. Howe ver the decreasing character of the nu-
merical data can be understood as follo ws: As can be seen from the figure 5.9 (a) and
(b) there is a delay between the rises of the spin do wn e xciton density and the right
handed polarized light emission. The reason was that the emission of right handed
polarized light requires e xcitons within the light cone. Therefore a spin do wn exciton
has first to scatter do wn via phonon emission into the light cone. This decay decreases
drastically at room temperatures where the relaxation of e xcitons is further dri ven by
stimulated phonon emission processes. This explains the une xpected beha vior .
In WSe 2 a decreasing de gree of polarization from 80 % at 77 K to 1 % at room temper -
ature can be observ ed. This observ ation can be directly deduced from the temperature
dependence of the spin occupation dif ference: At lo w temperatures the spin lifetimes
are lar ge compared to the dark recombination rate and therefore the emitted light is
strongly polarized. In contrast at room temperature, the spin lifetimes are small com-
pared to the dark recombination rate and therefore the observ ed light polarization is
small. As already discussed in the pre vious section, also the coherent exciton occupa-
tion contrib utes to the photoluminescence intensity , cf. figure 5.6. In a pure system,
where the e xcitonic coherence is not scattered at disorder in the sample, excitonic co-
herences with non-v anishing center of mass momentum Q 6 = 0 hav e no source terms.
Since further the interv alley e xchange coupling requires a center of mass momentum
(the coupling element v anishes for Q = 0 ) the optically injected excitonic coher -
5. Exciton Dynamics and Photoluminescence
ence e xperiences no intervalle y coupling. As a result, it is interesting to consider the
coherent emission to the total polarization resolv ed emission, which is illustrated in
figure 5.12 (b) with dashed lines. One finds that for MoSe 2 the degree of polariza-
tion decreases from 52 % at 77 K to 28 % at room temperature. This beha vior can be
attrib uted to more ef ficient phonon induced dephasing of the excitonic coherence at el-
e v ated temperatures. F or v anishing phonon induced dephasing, the total emitted light
would be coherent and therefore the polarization w ould be 100 %. The more important
the non-radiati ve coupling channels are, the lar ger is the portion of incoherent emis-
sion. The incoherent emission is kno wn to be nearly unpolarized (solid lines in figure
5.12). As a result, the ratio of coherent emission determines the degree of polariza-
tion. In WSe 2 the situation is slightly dif ferent. Here, the incoherent emission exhibits
a pronounced polarization. T aking additionally coherent contributions into account,
one observ es still a decreasing behavior from nearly 100 % at 77 K to 87 % at room
temperature.
5.4.4. Conc lusion
In conclusion in this section, the ef ficiency of the interv alley e xchange coupling in
the appearance of dark e xcitonic states belo w the optical bright state was studied. In
molybdenum based materials, where the bright state is the ground state of the system,
the interv alley e xchange coupling causes a v alley lifetime belo w 1 ps at all in vestig ated
temperatures. In tungsten based materials, most excitons thermalize in dark e xcitonic
states which do not contrib ute to the interv alley e xchange coupling. This enlar ges the
v alley lifetime by orders of magnitudes for all in vestigated temperatures.
6. F ¨
or ster Coupling in v an der
W aals coupled Heter ostructures
So far only properties of monolayers of transition metal dichalcogenide monolayers
ha ve been discussed. Besides the production of monolayer samples with high quality ,
also heterostructures of v an der W aals materials hav e been produced by stacking the
e xfoliated monolayers on top of each other [27, 168]. Of particular interest were het-
erostructures of a TMD layer and graphene, which ha ve been in vestig ated theoretically
[169] and e xperimentally [170, 40, 171, 172, 173, 174, 175]. Of certain interest here is
relaxation ef ficiency of TMD e xcitons to the graphene layer and the underlying micr -
socopic mechanisms. Dif ferential reflectance measurements re vealed a transition time
of about 1 ps of e xcited carriers in tungsten disulfide to graphene [171]. Further , pho-
toluminescence measurements indicated a decreased e xciton lifetime in molybdenum
diselenide - graphene heterostructure do wn to 1 ps at room temperature [175], which is
orders of magnitude faster than for unstack ed TMD samples [65]. Reflectance spectra
of a tungsten disulfide - graphene heterostructure exhibit an additional line broadening
of 5 meV and a reduction of the e xcitonic ener gy of 23 meV compared to the unstacked
tungsten disulfide [174]. Ho we ver despite a quiet rich situation for the a vailable e x-
perimental data, no theoretical studies ha ve been focused the microscopic mechanism
leading to line broadening and e xcitation transfer from the TMD monolayer to the
graphene layer so far .
As a first indication the optical dipole moments of both materials could hint a sig-
nificant impact of F ¨
orster ener gy transfer in these heterostructures, cf. figure 6.1.
F ¨
orster mediated ener gy transfer was recently studied in heterostructures of v ariant
quantum confinement, for example between zero-dimensional quantum dots [176], be-
tween two-dimensional graphene and zero-dimensional attached molecules [101] and
between two tw o-dimensional quantum wells [177].
Therefore in this chapter a microscopic model is de veloped to describe the F ¨
orster
coupling in a TMD-graphene heterostructure. Since the scope of this chapter is to de-
v olope a simple analytic model to study the F ¨
orster coupling between the TMD layer
and the graphene layer , indirect exciton states at the zone edge in the e xcitonic Bril-
louin zone are not discussed [64, 66, 70, 75]. The results of this chapter are submitted
for publication in Physical Re view B.
6. F ¨
orster Coupling in v an der W aals coupled Heterostructures
Figure 6.1.: Schematic illustration of the F ¨
orster coupling. (a) In real space the TMD layer and
the graphene layer are distanced by z .F
¨
orster coupling mediates an ener gy transfer
between them. (b) In momentum space e xcitons in the TMD layer and in the graphene
are F ¨
orster coupled under conserv ation of energy and momentum.
6.1. Theoretical Model
First the free ener gy of the electron hole pairs in graphene is introduced. It reads
H 0 =
q , Q ,ξ
E ξ, q
Q R † ξ, q
Q R ξ, q
Q (6.1)
R ( † ) ξ , q
Q denote annihilation (creation) operators for free electron hole pairs with center
of mass momentum Q and relati ve momentum q in graphene. ξ =( i, s ) denotes a
mer ged v alle y spin inde x in graphene. They obe y the fundamental bosonic commu-
tation relations in the lo w excitation limit equations 3.37 and 3.38. The ener gy of a
unbound electron hole pair is gi ven by E ξ, q
Q =2 v F | q + Q | , where v F denotes the
Fermi v elocity . The index ξ does not enter e xplicitly in the energy , since it refers to the
equi v alent K and K points and to the degenerated spin de gree of freedom.
As deri ved in detail in the appendix A.2.5 the F ¨
orster Hamiltonian between the TMD
layer and the graphene layer reads
H F =
µ, q , Q ,ξ ,ξ
V ξξ
µ Q ( z ) P † ξ
µ, Q R ξ , q
Q + h.c. (6.2)
The F ¨
orster coupling element reads V ( Q , z , µ, ξ )= a ξξ ( Q , z ) d ξ
T d ξ
G ϕ µ ( r =0)
4 π 0 √ A . Here, z de-
notes the distance between the TMD and the graphene layer and d ξ
T /G denote the abso-
lute v alues of the dipole elements in the TMD layer (T) and graphene (G). The F ¨
orster
Hamiltonian can be interpreted as the annihilation of a bound exciton in the TMD
monolayer and the creation of a free electron hole pair in the graphene monolayer , cf.
figure 6.1 (b). The function a ξξ ( Q , z ) describes the momentum and distance depen-
denc y and reads
a ( Q , z )= d 2 s e − i Q · ( s + z )
| s + z | 5 ( | s + z | 2 e ξ
1 · e ξ
2 − 3 e ξ
1 · ( s + z ) e ξ
2 · ( s + z )) . (6.3)
6.1. Theoretical Model
e 1 denotes the direction of the dipole element in the TMD layer and e ξ 0
2 denotes the
direction of the dipole element in graphene. The concrete form of this function is
e v aluated in the appendix A.3.1. Using the Heisenber g equation and exploiting the
fundamental bosonic commutation relations for e xciton and pair operators, equations
3.37, 3.38 and 3.29, one obtains the Bloch equations for the excitonic polarization in
the TMD layer and for the electron hole pair in graphene
i ~ ∂ t P ξ
ν, Q = E ξ
ν, Q P ξ
ν, Q + X
q ,ξ 0
V ξ ξ 0
ν, Q ( z ) R ξ 0 q
Q , (6.4)
i ~ ∂ t R ξ 0 q
Q = E ξ 0 , q
Q R ξ 0 q
Q + X
ξ 00 ,µ
V ξ ξ 0
ν, Q ( z ) P ξ 00
µ, Q . (6.5)
In both equations the first term takes account for the free ener gy of the polarizations
E ξ
ν, Q and E ξ 0 , q
Q . In the follo wing the energy in graphene is assumed to be independent
of Q , which is v alid as long | q | 2 1
2 q · Q holds. This assumption is further justified
since the dispersion in the in vestiagted q re gion is linear (for optical excitons in the
TMD layer). The second terms stem from the F ¨
orster coupling between the monolay-
ers. The simplest strategy to solv e this system of coupled dif ferential equations would
be to assume the graphene layer as a bath and solv e the respectiv e equation of motion
for R q
Q within the Born Marko v approximation [178, 136, 81]. This procedure is well
justified as long as the amplitude of R q
Q v aries slo wer than the excitonic polarization in
the TMD which is equi v alent to the frequency to the e xciting laser field under resonant
e xcitation conditions. Inserting the result back into the excitonic Bloch equation in the
TMD layer yields
∂ t P ξ
ν, Q = − i
~ ( E ξ
ν, Q + ∆ ξ
ν, Q − i Γ ξ
ν, Q ) P ξ
ξ , Q . (6.6)
where a additional broadening Γ ξ
ν, Q and a lineshift ∆ ξ
ν, Q of the excitonic line were
identified. The additional broadening reads
Γ ξ
ν, Q = | V ξ ,ξ 0
ν, Q ( z ) | 2 π X
ξ 0 , q
δ ( E ξ 0 , q − E ξ
ν, Q ) (6.7)
= | d T | 2 | d G | 2 | ϕ ξ
ν ( r = 0) | 2 E ξ
ν, Q
32 π 2 2
0 2 ~ 2 v 2
F X
ξ 0 | a ξ ξ 0
Q ( z ) | 2 . (6.8)
F or the additional lineshift one obtains
∆ ξ
ν, Q = −| V ξ ,ξ 0
ν, Q ( z ) | 2 X
ξ 0 , q
1
E ξ 0 , q − E ξ
ν, Q
(6.9)
= − | d T | 2 | d G | 2 | ϕ ξ
ν ( r = 0) | 2 q max
16 π 3 2
0 2 ~ v F ×
× 1 + E ξ
ν, Q
2 ~ v F q max
l n 1 − 2 ~ v F q max
E ξ
ν, Q ! X
ξ 0 | a ξ ξ 0
Q ( z ) | 2 (6.10)
6. F ¨
orster Coupling in v an der W aals coupled Heterostructures
Here the influence of dif ferent TMD excitons via F ¨
orster coupling was ne glected, since
the remaining linebroadenings and lineshifts would be on the fourth order of the cou-
pling element. This e xpression can be further simplified by approximating the TMD
e xciton energy by the optical e xcitation frequency E ξ
ν Q ≈ ~ ω opt . The summation o ver
the spin de gree of freedom, which is contained in the multi inde x ξ , gi v es effecti vely
a factor of 2 to both e xpressions. A momentum cutof f in graphene q max had to be
introduced in the summation for the lineshift, equation 6.10, to a void di ver gence of
the principle v alue integral, equation 6.9. The momentum cutof f was determined by
angular a veraging the dispersion relation of graphene around the K point, which can
be obtained from tight binding computations. The cutof f was then chosen to fulfill
q max = E max
2 ~ v F , where E max denotes the maximal bandgap of the mean dispersion.
Since both, the dephasing rate, equation 6.8, and the lineshift, equation 6.10, depend
both linear on the function | a ξ ξ 0
Q | 2 which inserts the only Q and z dependence, both
quantities are e xpected only to differ by a constant. In the in vestig ated model, no tem-
perature dependence for the dephasing rate and the lineshift is present. As already
discussed for the phonon contrib ution to the excitonic Bloch equation, the dephasing
rate gi ves rise to a broadening of the e xcitonic line in linear spectroscopy , where the full
line width is additionally broadened by twice the dephasing rate. An e ventual lineshift
due to F ¨
orster coupling would be e xperimentally accessible by a shifted excitonic res-
onance.
T o e v aluated both, dephasing rate and lineshift, quantitativ ely the function | a ξ ξ 0
Q | 2 has
to be e v aluated. As demonstrated in appendix A.3.1, a solution of the integral a ξ ,ξ 0
Q ( z )
is obtained by e xploiting a Schwinger of the denominator [179],
1
x k = 1
Γ( k ) Z ∞
0
dtt k − 1 e − tx , (6.11)
with x, k ∈ R and Γ( k ) being the well kno wn Gamma function. After using this in
the function a ( Q , z ) the d 2 s integration can be carried out easily . The remaining dt
inte gration ov er hermitian gaussians is also easy to perform. The exact procedure is
illustrated in the appendix A.3.1. The result reads
a ξ ,ξ 0
Q ( z ) = 2 π e ξ
1 · Q e ξ 0
2 · Q
Q e − Qz . (6.12)
This results was pre viously gi ven in [177]. T o keep the model simple, this e xpression
is a veraged o ver the angle of the center of mass momentum of the TMD e xciton Q
which gi ves
a ξ ,ξ 0 ( Q, z ) = π Qe − Qz e ξ
1 · e ξ 0
2 (6.13)
Squaring this e xpression and performing the ξ 0 sum appearing in the equations 6.8 and
6.10 the final result can be obtained
| a ( Q, z ) | 2 = π 2 Q 2 e − 2 Qz . (6.14)
Here, the fact that the dipole moments at the K and the K 0 point are orthogonal in
graphene was e xploited.
6.2. Results
6.2. Results
The equations for the F ¨
orster induced dephasing rate, equation 6.8, and lineshift, equa-
tion 6.10, are exploited for the e xemplary heterostructure consisting of monolayer
tungsten disulfide and graphene on a quartz substrate. All rele v ant parameters can
be found in table 6.1.
Figure 6.2 (a) depicts a contour plot of the F ¨
orster induced dephasing rate as a func-
tion of the e xcitonic center of mass momentum Q and the separation z between the
tungsten disulfide and the graphene layer . In general the dephasing rates are in the
range from o ver 6 meV at at Q = 2 nm − 1 and z = 0.5 nm, to 0 meV for Q = 0 and
z = 0.5 nm. The distance of 0.5 nm corresponds to the situation of a closely stack ed
heterostructure [174]. Obvious from the e xponential factor in equation 6.14 the de-
phasing rates e xhibits an ov erall decreasing behavior as a function of the layer separa-
tion. The lineshift, which is not illustrated in figure 6.2 (a), follo ws exactly the same
beha vior , since it dif fers only by a prefactor from the dephasing rate, equation 6.8 and
6.10.
T o get more into detail, in figure 6.2 (b), the F ¨
orster induced dephasing rate is plotted
as a function of the e xcitonic center of mass momentum Q for three selected inter-
layer spacing of 0.5 nm, 1.0 nm and 2 nm. For all spacings an initial rise of the F ¨
orster
induced dephasing rate can be observ ed which is followed by an exponential decay .
This beha vior can be traced back to the form of the function | a Q ( z ) | 2 , equation 6.14.
Interestingly , excitons with v anishing center of mass momentum, e.g. e xcitons which
are accessible in optical experiments in perpendicular geometry [110, 87, 63], equa-
tion 3.57, experience no F ¨
orster induced dephasing. Therefore the impact of F ¨
orster
coupling between the layers should not be a v ailable through linear spectroscopy . This
observ ation is in nice agreement with pre vious theoretical in vestigations [177]. Ho w-
e ver in reference [174] the authors found an additionally broadened e xcitonic line in
a WS 2 -graphene stack compared to the unstacked WS 2 of about 5 meV . Since F ¨
orster
coupling can be ruled out as a source of broadening, ongoing in vestigations are re-
quired to clarify the source of the broadening. One possibility would be the stacking
induced changes of the e xcitonic band structure WS 2 which could lead to an enhanced
T able 6.1.: P arameters used in the computation. ∗ determined numerically by using the method
gi ven in [63, 70]
P aram. Ref.
d G 0.25 nm [180]
v F 1 nm fs − 1 [181]
d W S 2 0.4 nm [70]
| ϕ W S 2 ( r = 0) | 0.49 nm − 1 ∗
E 1 s
W S 2 2.0 eV [148]
q max 8 nm − 1
6. F ¨
orster Coupling in v an der W aals coupled Heterostructures
Figure 6.2.: F ¨
orster induced br oadening and energy renormalization of excitonic r esonances.
(a) Surface plot of the dephasing rate, eq. 6.8 as a function of the excitonic center -of-
mass momentum and the distance between the graphen and the WS 2 layer . Dephasing
rate, eq. 6.8, and ener gy renormalization, eq. 6.10, (b) as a function of the center-of-
mass momentum of the TMD exciton for fix ed layer distances and (c) as a function of
the interlayer distance for fixed e xciton momenta.
e xciton phonon coupling and thus broadened lines.
The calculations e xhibit a significant influence of the dielectric en vironment. It en-
ters in the screening of the Coulomb interaction and also in the e xplicit form of the
e xcitonic wa vefunction. Exemplary in a free standing closely stack ed WS 2 -graphene
heterostructure the F ¨
orster induced dephasing is about 10 meV at Q = 2 nm − 1 . Con-
trary , encapsulating the heterostructure in he xagonal boron nitride leads to a drastic
decrease of the F ¨
orster induced dephasing to 1.5 meV at Q = 2 nm − 1 .
No w the F ¨
orster induced dephasing as a function of the interlayer separation is studied
which is illustrated for three dif ferent center of mass momenta in figure 6.2 (c). For
all center of mass momenta, the dephasing rate e xhibits, as depicted, an e xponentially
decreasing beha vior , being a result of equation 6.14. In the special case of Q =0 , the
dephasing rate v anishes.
After ha ving understood the momentum resolved dephasing rate due to F ¨
orster cou-
pling between the layers, no w the temperature dependent transition rate for thermalized
e xcitons 1
τ T in the TMD layer to the graphene layer shall be in vestigated. Therefore a
6.2. Results
Figure 6.3.: T ransition Rate (a) T ransition rate, eq. 6.8 and 6.18, as a function of the temperature
of the exciton in the WS 2 for three dif ferent layer distances. (b) T ransition rate as a
function of the interlayer distance for three dif ferent exciton momenta.
thermal a verage for the momentum dependent F ¨
orster induced transition rate 1
τ Q , which
can be obtained by 1
τ Q = 2
Γ Q , is applied. Assuming the excitons in the TMD l ayer
to be Boltzmann distrib uted, which is a fair assumption in the lo w excitation limit, see
for e xample the results in the previous chapter 5, the thermal a v erage can be computed
via
1
τ T
= 1
τ Q T , (6.15)
= 1
Ω d 2 Qe − E Q /k T 1
τ Q
, (6.16)
where Ω= d 2 Qe − E Q /k T denotes a normalization factor . Since the only Q depen-
dence of the F ¨
orster induced dephasing rate is gi ven through the function | a ( Q ,z ) | 2 , it
is suf ficient to exploit the thermal a verage for this function. The result reads
| a ( T, z ) | 2 = | a ( Q ,z ) | 2 T (6.17)
= π 2
2
1
λ 6 2 λ 2 ( λ 2 + z 2 ) − e z 2 /λ 2 λ √ πz (3 λ 2 +2 z 2 ) er f c ( z /λ ) . (6.18)
Here, the error function er f c ( z /λ ( T )) and the thermal wa velength λ ( T )=
√ 2 Mk B T
were defined. F or the numerical implementation a high numerical accuracy is re-
quired to treat the second term accurately , since e z 2 /λ ( T ) 2 is a fast increasing and
er f c ( z /λ ( T )) is a fast decreasing function of z /λ .
6. F ¨
orster Coupling in v an der W aals coupled Heterostructures
The obtained transition rate is depicted as a function of temperature T for three se-
lected interlayer spacings z . F or a closely stacked heterostructure ( z = 0.5 nm) at room
temperature the transition rate is approximately 5 ps − 1 , which qualitati vely matches re-
cent e xperimental results from photoluminescence [175] and reflectance contrast [171]
measurements. The transition rate increases as a function of temperature for all in-
v estigated interlayer separations. T o understand this, one can consider the exciton
Boltzmann distrib ution which is broadened with the thermal energy k T . As a result
at v ery low temperatures, this distribution is v ery small, meaning that e xcitons only
occupy states with small center of mass momenta Q . Here the F ¨
orster induced transi-
tion rate is v ery small, cf. figure 6.2. An increasing temperature leads to a broadening
of the Boltzmann distrib ution. Hence e xcitons occupy states with lar ger momentum
dependent F ¨
orster rates, cf. figure 6.2 (a) and (b). Consequently the F ¨
orster induced
transition rate increases.
Figure 6.3 (b) illustrates the F ¨
orster induced transition rate as a function of the inter -
layer spacing for three selected temperatures. In the limit of small interlayer spac-
ings the transition rate e xhibits a clear e − z law for all discussed temperatures. In the
lar ge interlayer spacing limit, the transition rate follo ws a z − 4 law . This is consistent
with the calculated z dependenc y of the F ¨
orster induced transition rate in porpyrin-
functionalized graphene [101].
6.3. Conc lusion
Concluding, in this chapter a simple analytic model to describe the F ¨
orster coupling
in a TMD-graphene-stack was de veloped. The calculated F ¨
orster induced transition
rates from the TMD to the graphene are on the order of 5 ps for a closely stacked
WS 2 -graphene heterostructure at room temperature, being consistent with recent ex-
perimental observ ations [171, 175]. The model can also be applied to heterostructures
of other v an der W aals materials, such as multilayers of TMDs or stacks of dif ferent
TMD materials [182, 183, 184, 185, 186].
7. Summar y and Outlook
In this thesis an e xcitonic description of two dimensional crystals of transition metal
dichalcogenides was de veloped, which turned out to nicely describe the excitonic prop-
erties of TMDs.
T aking into account exciton light and e xciton phonon interaction the homogeneous
line width for ex emplary TMD materials were computed microscopically . The line widths
were found to range from some fe w meV at 4 K to some tens of meV at room temper -
ature, being in excellent agreement with e xperiments which were carried out at the
Stanford Uni versity (California) and the Uni versity of Re gensbur g. In particular , a
strong contrast between molybdenum and tungsten based TMDs was found: while in
molybdenum based TMDs, the line width is dominated by radiati ve decay and e xciton
phonon coupling within the e xcitonic Γ valle y , relaxation to indirect exciton states lo-
cated at the Λ v alley is the crucial mechanism determining the line width in tungsten
based TMDs.
A follo w-up study focused on the e xcitonic line width in bilayer tungsten disulfide.
Here, the line width was found to exceed the monolayer width by at least 30 meV which
was addressed to a changed e xciton dispersion in the bilayer . While in the monolayer
the e xcitonic Λ valle y is the lo west lying exciton state, in the bilayer it is the M v alle y .
The phonon scattering to these states turns out to be much more ef ficient. Further the
e xciton states with an hole at the electronic Γ valle y shift do wn in energy in the bilayer ,
leading to ef ficient scattering into these states. The results are in e xcellent agreement
with the e xperimental results.
Additionally the formation of phonon sidebands was studied. Hence a non-Markovian
treatment for the exciton-phonon scattering w as performed. Pronounced shoulders
due to acoustic emission and absorption processes slightly abo ve the e xcitonic reso-
nance and strong sidebands due to optical phonon emission well abo ve the e xcitonic
resonance ha ve been found. The theoretical results are in good agreement with e xper-
imental results.
In another study the phonon mediated e xciton formation and thermalization was in ves-
tigated. Here again dark states in tungsten based TMDs turned out to be crucial for
the relaxation, since most excitons relax do wn to these energetically f av orable states
within a fe w picoseconds. Contrary , in molybdenum based TMDs most excitons are
located in the e xcitonic Γ valle y close to the radiati ve cone after thermalization. This
beha vior has interesting consequences for the light emitting properties of both ma-
terials: while in molybdenum based materials the luminescence yield increases as a
function of temperature, tungsten based TMDs exhibit the opposite beha vior . This is
7. Summary and Outlook
in e xcellent agreement with recent experimental finding [133, 134, 65].
In a third study , the impact of dark states combined with phonon mediated relaxation
was in vestigated on the ef ficiency on the most common spin relaxation mechanism in
TMDs, the interv alley e xchange coupling. While in molybdenum based TMDs, the
spin lifetime is shorter than 1 ps, it is orders of magnitudes longer in tungsten based
materials. The reason is that interv alley e xchange coupling is forbidden for indirect
e xcitons.
In the last study , F ¨
orster ener gy transfer was analyzed for a heterostructure consisting
of a TMD layer and a graphene layer . The relaxation rate of TMD e xciton to the
graphene was found to occur on a picosecond timescale, being in e xcellent agreement
with recent e xperimental results [171, 175].
In conclusion, a robust theoretical model describing the lo w density physics of TMD
e xcitons was presented and was v erified with se veral e xperimental studies.
In the considered limit, only interactions which conserve the electron and hole spin
were in vestig ated. Therefore the theoretical model was not able to describe the for -
mation of e xcitons with opposite electron and hole spin. Ho we ver , these excitons can
dominate the optical response of the materials under certain conditions [69, 68]. A
consistent theoretical description of the formation of such excitons has not been gi ven
so far and w ould be of particular interest for the 2D community as well as for the
general field of solid state physics.
A non-Marko vian treatment of the exciton-phonon coupling re vealed the formation of
pronounced sidebands in the absorption spectrum. Such a procedure was not gi ven for
the emission spectrum of TMDs so far . Here, it would be of particular interest, ho w
dark e xcitons appear in the photolumiscence spectrum and which influence they ha ve
on the radiati ve e xciton lifetime.
The deri ved Hamiltonian describes the e xciton physics after weak optical e xcitation.
Se veral e xperimental studies ha ve re vealed the impact of e xciton exciton scattering,
specifically Auger like e xciton exciton recombination, on the e xciton dynamics in
TMDs [153, 154]. A momentum and time resolved study of such processes is miss-
ing in the current literature and would contrib ute to a more detailed understanding of
e xcitons in thin materials.
A. Appendix
A.1. Exf oliation and Spectr oscopic In vestigation
of Monola y er WSe 2
In June 2017 the author of this thesis visited the group of Rudolf Bratschitsch at the
Uni versity of M ¨
unster . There the author got the possibility to e xfoliate a flake of WSe 2
under kind assistance of Iris Niehues. This flake was brought to the T echnical Uni ver -
sity of Berlin and was e xperimentally characterized by the author in the laboratory of
the group of Ulrike W oggon under kind assistance of Sophia Helmrich. The experi-
mental procedure as well as the results shall be discussed in the follo wing briefly .
Sample Preparation
The flake w as exfoliated from b ulk crystal by micromechnical clea v age [22, 23]. First
the crystal was transferred to scotch tape. Then the b ulk crystal was remov ed from
the scotch tape carefully with a tweezer to ensure that only fe w material is located on
the scotch tape. T o further reduce the amount of material on the scotch tape, it w as
transferred from one scotch tape to another , which was repeated se veral times. After
this, the scotch tape was pressed on a polymer film. No w on the polymer film was
a v ery sparse concentration of WSe 2 . Unfortunately during this step the sample fell
on the pants of the author , which could possibly lead to a pollution of the sample. In
the last step the WSe 2 was transferred to a Si/SiO 2 substrate. Figure A.1 (a) sho ws a
microscope image of the produced flake with a magnification of 100.
Experimental Characterization
Emission spectra were measured at room temperature with a resonant continuous wa v e
e xcitation with an excitation po wer of approximately 160 µW . The emitted light of the
sample was then recorded in a spectrometer . Figure A.1 (b) illustrates the measured
emission spectrum. It e xhibits a pronounced line around 1.65 eV being consistent with
other v alues reported for the emission of the 1 s exciton in WSe 2 in the literature [148].
The homogeneous part of the excitonic line w as fitted with a Lorentzian which is de-
picted for comparison. The line width was determined to be approximately 20 meV
being in nice agreement for monolayer samples [148, 187]. The additional broad
emission on the lo w energetic side of the spectrum is attrib uted to emission of indi-
rect e xciton states [146]. The origin of the sharp feature at approximately 1.54 eV
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