ABSTRACT
Title of Dissertation: EXCITED STATES IN MONOLAYER TRANSITION
METAL DICHALCOGENIDES
Julia C. Sell
Doctor of Philosophy, 2022
Dissertation Directed by: Professor Mohammad Hafezi
Department of Physics
Monolayer two-dimensional transition metal dichalcogenides (2D TMDs) represent
a class of atomically thin semiconductors with unique optical properties. Similar to
graphene, but with a three-layer (staggered) honeycomb lattice, TMDs host direct-gap
transitions at their ?K valleys that exhibit circular-dichroism due to their finite Berry
curvature. The reduced dimensionality of materials in this system, combined with large
effective carrier masses, leads to enhanced Coulomb interaction and extremely tightly
bound excitons (EB ? 150 ? 300 meV). Here, we seek to exploit the unusually tight
binding of the excitons to probe two different types of higher energy exciton species in
TMDs.
First, we experimentally probe the magneto-optical properties of 2s Rydberg exciton
species in WSe2. The magnetic response of excitons gives information on their spin
and valley configurations, nuanced carrier interactions, and insight into the underlying
band structure. Recently, there have been several reports of 2s/3s charged excitons in
TMDs, but very little is still known about their response to external magnetic fields.
Using photoluminescence excitation spectroscopy, we verify the 2s charged exciton and
report for the first time its response to an applied magnetic field. We benchmark this
response against the neutral exciton and find that both the 2s neutral and charged excitons
exhibit similar behavior with g-factors of gX2s=-5.20?0.11 ?B and gX2s=-4.98?0.11 ?B,0 ?
respectively.
Second, via theoretical calculations, we investigate the exciton spectrum generated
in 2D semiconductors under illumination by twisted light. Twisted light carries orbital
angular momentum (OAM) which can act as an additional tunable degree of freedom in
the system. We demonstrate that twisted light does not have the ability to modify the
exciton spectrum and induce dipole-forbidden excitons, in contrast to atoms. This result
stems from the fact that the additional OAM is transferred preferentially to the center-of-
mass (COM) of the exciton, without modifying the relative coordinate which would allow
dipole-forbidden, higher energy excitons to form.
EXCITED STATES IN MONOLAYER TRANSITION METAL
DICHALCOGENIDES
by
Julia C. Sell
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
2022
Advisory Committee:
Professor Mohammad Hafezi, Chair/Advisor
Professor Sunil Mittal
Professor Glenn Solomon
Professor Ichiro Takeuchi
Professor You Zhou
? Copyright by
Julia Sell
2022
Dedication
To my grandfather, who inspired me to pursue my Ph.D. from the time I was a little
girl. To my parents, for their unwavering support through all of my years of school. To
Garrett and Layla for all of the love, encouragement, and tail-wags that fueled me through
the finish line.
ii
Acknowledgments
I would like to thank my advisor, Dr. Mohammad Hafezi, for his kind support
throughout graduate school. He has built a group unlike any other, where experimentalists
and theorists coexist and collaborate within the group; it fosters a wonderful sense of
scientific community and curiosity. This model allowed to step out of my comfort zone
and try new projects, like some theoretical modeling. I found this work to be extremely
rewarding and believe it helped me grow as a scientist in the later years of my doctorate.
Mohammad also ?talks physics? better than just about anyone I have met, with an uncanny
ability to break down extremely difficult problems into simple terms. I strive to one day
be able to communicate my thoughts so succinctly; I?m extremely grateful that he took a
chance on me at the end of my second year and let me join such a wonderful group.
I was lucky enough to have quite a few mentors in graduate school who were not my
advisor. My first was Rodney Snyder, who was the senior graduate student who helped
me through my first semester. He turned out to be one my best friends in graduate school
as well and made the journey much more enjoyable. I also received wonderful council on
both science and the business of being a scientist from Dr. Utso Bhattarcharya, Dr. Tobias
Gra?, Dr. Sunil Mittal, Dr. Daniel Sua?rez-Forero, and Dr. Glenn Solomon. They taught
me so much, and I?m a better scientist for having known them; thank you all. I would
also like to thank Dr. Ichiro Takeuchi and Dr. You Zhou for taking the time to serve as
iii
committee members, I greatly appreciate it!
I would also like to thank the other group members: Jon Vannucci (who also put a
huge amount of work in to making the contents of this thesis a reality), Deric Session, and
Bin Cao. In no particular order: Sandy, Zach, Natalia, Christie, Liz, Phil, Harry, Kristi,
and Brittany - I?m lucky to have met all of you through graduate school. Outside of the
university, I also received wonderful support over the years from Irene, Caroline, Aurora,
Jackie, and all my NOVA friends. Shelby, who was my roommate for three years of
graduate school, is the best Ph.D. soul-sister I could?ve ever asked for and I?m so grateful
to have her in my life.
I would like to thank Berry Jonker?s group at the Naval Research Laboratory who
we collaborated with for samples and sample material. On campus, I would like to
sincerely thank the clean room staff - Tom, Mark, and John - who taught me so much
about fabrication and working in a clean room environment. Additionally, I would like
to thank Dr. Karen Gaskell, who is the caretaker of the AFM on campus and was always
game to try whatever crazy AFM technique I read about but wasn?t quite sure how to
actually execute.
I would like to thank Dr. Tom O?Haver for his assistance in understanding and
implementing signal processing techniques. His book was extremely helpful, and he
was kind enough to correspond with me and provide insight on some fitting issues I
encountered while analyzing magnetic field data. Though the project I originally contacted
him about was scrapped, his expert advice proved to be invaluable and helped me hone
my processing techniques for other data that did make the cut for this thesis.
I was fortunate to be supported by the National Science Foundation Graduate Research
iv
Fellowship Program (NSF GRFP) from 2015-2020 and the ARCS MWC Scholar program
from 2020-2022, for which I am eternally grateful.
Last, but not least I would like to thank my parents, my wonderful partner, and our
dog. My parents have provided so much unwavering support to me since I announced
that I wanted a Ph.D. when I was 6. Though the field has changed (paleobotany turned
out to not be my thing), their support never has and I?m so grateful to have a family
that understands this journey that I have been on. As for Garrett, I strongly feel that it
takes a special person to be willing to walk the path of a Ph.D. with someone in their
relationship. It was a commitment that I made before we met, but he has really taken it on
as his burden to share. Garrett has been such a wonderful partner while I work towards
accomplishing my goals, doing everything from taking on extra household chores when
I?m busy to answering random programming syntax questions. He has been my emotional
rock when research has been hard, and I appreciate everything he has done so much. Our
dog, Layla, was my pandemic work-from-home buddy while the university was closed.
I really missed her when I returned to working on campus, but she?s always at the door
wagging and happy to see me after a long day.
v
Table of Contents
Dedication ii
Acknowledgements iii
Table of Contents vi
List of Tables x
List of Figures x
List of Abbreviations xii
Chapter 1: Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 The Rise of Two-Dimensional Materials . . . . . . . . . . . . . . . . . . 2
1.3 Goals of this Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Outline of this Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Chapter 2: Review of Transition Metal Dichalcogenide Fundamental Properties
and Exciton Species 9
2.1 Band Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Selection Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.1 Massive Dirac Hamiltonian . . . . . . . . . . . . . . . . . . . . . 14
2.2.1.1 Light-Matter Interaction . . . . . . . . . . . . . . . . . 15
2.2.1.2 Polarization . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2.1.3 Valley Hall Effect . . . . . . . . . . . . . . . . . . . . 18
2.2.2 Spin-Orbit Coupling . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3 Exciton Species . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3.1 Single-Particle Models . . . . . . . . . . . . . . . . . . . . . . . 23
2.3.1.1 Bright Excitons . . . . . . . . . . . . . . . . . . . . . . 24
2.3.1.2 Dark Excitons . . . . . . . . . . . . . . . . . . . . . . 28
2.3.1.3 Spin-Dark Excitons . . . . . . . . . . . . . . . . . . . 29
2.3.1.4 Momentum-Dark Excitons . . . . . . . . . . . . . . . . 34
2.3.1.5 More Exotic Pairings: Biexcitons and Trions . . . . . . 36
2.3.1.5.1 Biexcitons . . . . . . . . . . . . . . . . . . . 36
2.3.1.5.2 Trions . . . . . . . . . . . . . . . . . . . . . 39
2.3.2 Many-Body States . . . . . . . . . . . . . . . . . . . . . . . . . 43
vi
2.3.2.1 Break Down of the Single-Particle Model . . . . . . . . 43
2.3.2.2 Historical Context of Polarons . . . . . . . . . . . . . . 44
2.3.2.3 2D Fermi Polarons in TMDs . . . . . . . . . . . . . . . 45
Chapter 3: Device Fabrication Methodology 48
3.1 Materials Synthesis and Extraction . . . . . . . . . . . . . . . . . . . . . 48
3.1.1 Mechanical Exfoliation or the ?Scotch-Tape Method? . . . . . . . 48
3.1.2 Chemical Vapor Deposition Growth . . . . . . . . . . . . . . . . 51
3.2 Hexagonal Boron Nitride Encapsulation . . . . . . . . . . . . . . . . . . 52
3.2.1 Dry Encapsulation Techniques . . . . . . . . . . . . . . . . . . . 55
3.2.1.1 Visoelastic Stamping . . . . . . . . . . . . . . . . . . . 55
3.2.1.2 Sacrificial Polymers A.K.A the ?Drop-Down? Method . 56
3.2.1.3 van der Waals Pick-Up with Heat-Expansive Polymers . 57
3.2.2 Capillary Action Assisted Encapsulation . . . . . . . . . . . . . . 59
3.2.3 Comparison of Encapsulation Methods . . . . . . . . . . . . . . . 61
3.3 Sample Flattening via Atomic Force Microscopy A.K.A ?Nano - Squeegeeing? 63
3.4 Lithographic Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.4.1 Electron-Beam Lithography . . . . . . . . . . . . . . . . . . . . 65
Chapter 4: Measuring Light-Matter Interaction 67
4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.2 Optical Measurement Techniques . . . . . . . . . . . . . . . . . . . . . . 67
4.2.1 Photoluminescence Measurements . . . . . . . . . . . . . . . . . 68
4.2.2 Photoluminescence Excitation Measurements . . . . . . . . . . . 69
4.3 Experimental ?Knobs? . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.3.1 Light Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.3.2 Electrostatic Gating . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.3.3 Magnetic Field (Zeeman Effect) . . . . . . . . . . . . . . . . . . 73
4.3.4 A Note on Measurement Energies . . . . . . . . . . . . . . . . . 75
4.4 Measurement Specifications . . . . . . . . . . . . . . . . . . . . . . . . 75
Chapter 5: Understanding Exciton Dynamics Under Applied Magnetic Field in
Transition Metal Dichalcogenides 77
5.1 The Hydrogen Atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.1.1 The (Normal) Zeeman Effect . . . . . . . . . . . . . . . . . . . . 77
5.1.2 Lande? g-factor . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.2 The Zeeman Effect in Transition Metal Dichalcogenides . . . . . . . . . 81
5.2.1 Single Particle Model . . . . . . . . . . . . . . . . . . . . . . . . 82
5.2.1.1 Intercellular Component Correction . . . . . . . . . . . 84
5.2.1.2 Application of the Single-Particle Model to Other Exciton
Species and Break Down of the ?Atomic? Picture of
Interaction . . . . . . . . . . . . . . . . . . . . . . . . 84
5.2.2 Many-Body Regime Enhancement of g-factor . . . . . . . . . . . 86
5.2.3 High Field Considerations: Paschen-Back Effect, Diamagnetic
Shift, and the Quantum Hall Effect . . . . . . . . . . . . . . . . . 92
vii
Chapter 6: Magneto-Optical Measurements of the Negatively Charged 2s Exciton
in WSe2 95
6.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
6.2 Manuscript . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.2.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.2.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.2.3 Identification of the Excited Charged State via Gate-Dependent PLE100
6.2.4 Valley Zeeman Effect in the Excited Charged State . . . . . . . . 103
6.2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.3 Supplemental Material . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6.3.1 Fabrication Details . . . . . . . . . . . . . . . . . . . . . . . . . 106
6.3.1.1 CVD growth of WSe2 . . . . . . . . . . . . . . . . . . 107
6.3.1.2 Heterostructure construction . . . . . . . . . . . . . . . 107
6.3.1.3 Electrical contact fabrication . . . . . . . . . . . . . . . 108
6.3.2 Optical Configuration . . . . . . . . . . . . . . . . . . . . . . . . 108
6.3.3 Electrostatic Gate Mapping of the Ground State . . . . . . . . . . 109
6.3.4 Extraction of Carrier Concentration . . . . . . . . . . . . . . . . 111
6.3.5 2s Photoluminescence . . . . . . . . . . . . . . . . . . . . . . . 113
6.3.6 X2s? Photoluminescence . . . . . . . . . . . . . . . . . . . . . . . 114
6.3.7 X2s? Emission Channel Evolution with Gate . . . . . . . . . . . . 115
6.3.8 X2s? Vertical Cross-Section g-factor . . . . . . . . . . . . . . . . . 116
6.3.9 Fitting Procedure for Extracting g-factor . . . . . . . . . . . . . . 118
6.3.10 Ground State PL and g-factor . . . . . . . . . . . . . . . . . . . . 121
6.3.11 Estimating the Intercellular Contribution to g for Charged Excitons
in Single-Particle Formalism . . . . . . . . . . . . . . . . . . . . 122
6.3.12 Raman Phonon Lines in X0 PLE Emission Response . . . . . . . 124
Chapter 7: Fundamentals of Optical Orbital Angular Momentum 126
7.1 Optical Orbital Angular Momentum . . . . . . . . . . . . . . . . . . . . 126
7.2 Types of Angular Momentum in Light . . . . . . . . . . . . . . . . . . . 127
7.2.1 Linear Momentum . . . . . . . . . . . . . . . . . . . . . . . . . 127
7.3 Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
7.3.1 Spin Angular Momentum . . . . . . . . . . . . . . . . . . . . . . 130
7.3.2 Orbital Angular Momentum . . . . . . . . . . . . . . . . . . . . 131
Chapter 8: Two-dimensional excitons from twisted light and the fate of the
photon?s orbital angular momentum 134
8.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
8.2 Manuscript . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
8.2.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
8.2.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
8.2.3 General Analytical Model . . . . . . . . . . . . . . . . . . . . . 138
8.2.4 Light-matter coupling . . . . . . . . . . . . . . . . . . . . . . . . 138
8.2.5 Exciton transitions . . . . . . . . . . . . . . . . . . . . . . . . . 141
8.2.6 Example: Excitons from twisted light . . . . . . . . . . . . . . . 144
viii
8.2.7 Band-to-band transitions in cylindrical basis . . . . . . . . . . . . 145
8.2.8 Exciton transitions . . . . . . . . . . . . . . . . . . . . . . . . . 148
8.2.9 Numerical evaluation . . . . . . . . . . . . . . . . . . . . . . . . 150
8.2.10 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . 153
8.3 Supplemental Material . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
8.3.1 Band-to-band transition matrix for a Bessel beam . . . . . . . . . 154
8.3.2 Hankel transform of the exciton . . . . . . . . . . . . . . . . . . 155
Chapter 9: Conclusions and Future Avenues of Work 162
Appendix A: Fabrication Recipes 164
A.1 Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
Appendix B: Expanded Derivations for Magnetic Field Effects 166
B.1 The Weak-Field Zeeman Effect . . . . . . . . . . . . . . . . . . . . . . . 166
B.2 The Bohr Magneton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
B.3 Modified Landau Level Spectrum in TMDs . . . . . . . . . . . . . . . . 172
Appendix C: Elionix Standard Operating Procedure 176
Bibliography 194
ix
List of Figures
1.1 Periodic table of transition metal dichalcogenides . . . . . . . . . . . . . 4
1.2 A brief history of transitional metal dichalcogenide research . . . . . . . 5
2.1 Transition metal dichalcogenide monolayer and polytypes . . . . . . . . . 10
2.2 Band structure schematic of the band edges in TMDs . . . . . . . . . . . 12
2.3 Illustration of the valley polarization and Berry curvature in TMDs . . . . 20
2.4 General exciton energy structure . . . . . . . . . . . . . . . . . . . . . . 23
2.5 Illustration of dielectric screening bulk vs. monolayer transition metal
dichalcogenides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.6 Illustration of the formation of bright vs. dark states in TMDs . . . . . . . 29
2.7 Dark Excitons: Effects of Numerical Aperture on Light Collected . . . . . 31
2.8 Illustration of biexciton formation . . . . . . . . . . . . . . . . . . . . . 36
2.9 Illustration of singlet and triplet exciton band configurations . . . . . . . 40
2.10 Illustration of the difference in configuration for bright vs. dark trions . . 43
2.11 Illustration of single-particle vs. many-body models . . . . . . . . . . . . 45
2.12 Illustration of repulsive and attractive polarons . . . . . . . . . . . . . . . 47
3.1 Schematic mechanical exfoliation process . . . . . . . . . . . . . . . . . 50
3.2 Schematic of Chemical Vapor Deposition Process and Resulting Material 51
3.3 Schematic showing the steps for visoelastic stamping . . . . . . . . . . . 55
3.4 Schematic illustrating the Elvacite?-based encapsulation process . . . . . 56
3.5 Schematic for using heat-expansive polymers . . . . . . . . . . . . . . . 58
3.6 Schematic for capillary-action assisted pickup . . . . . . . . . . . . . . . 60
3.7 Schematic illustrating the nano-squeegee process . . . . . . . . . . . . . 63
3.8 Schematic for EBL Processing . . . . . . . . . . . . . . . . . . . . . . . 65
4.1 Photoluminescence process and model emission spectra . . . . . . . . . . 68
4.2 Photoluminescence Excitation process and model emission spectra . . . . 70
4.3 Gate Response for Representative Sample . . . . . . . . . . . . . . . . . 72
4.4 Illustration of the basic Zeeman effect . . . . . . . . . . . . . . . . . . . 74
4.5 Schematic of the Measurement Setup . . . . . . . . . . . . . . . . . . . . 76
5.1 Detailed illustration of the Zeeman effect in hydrogen . . . . . . . . . . . 80
5.2 Zeeman splitting for excitons in TMDs in the single-particle model . . . . 83
5.3 Illustration of the relationship between carrier density, Fermi energy, Wigner-
Seitz radius, and exchange interaction strength in TMDs . . . . . . . . . 88
x
5.4 Stages of carrier polarization under applied magnetic field . . . . . . . . . 89
6.1 Devices schematic and images, PLE process, and base PLE map . . . . . 98
6.2 PLE as a function of gate voltage for the X0 emission channel . . . . . . . 100
6.3 PLE as a function of gate voltage for the Xt? emission channel . . . . . . 101
6.4 Valley Zeeman effect for 2s charged and neutral excitons . . . . . . . . . 102
6.5 More detailed images from the fabrication images . . . . . . . . . . . . . 106
6.6 PL gate charactierzation map . . . . . . . . . . . . . . . . . . . . . . . . 109
6.7 Extracted carrier concentration and Fermi energy as a function of applied
gate voltage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.8 Characterization PL measurements of the 1s and 2s exciton lines . . . . . 114
6.9 Comparison of the PL signal for X , X2s0 ? , and X
2s
0 for PL measurements
with gate dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.10 Gate dependent PLE data monitoring the Xs? recombination channel in
the -K-valley (????) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.11 Valley Zeeman measurements of the excited neutral and charged states in
WSe2 through the Xs? emission channel . . . . . . . . . . . . . . . . . . 117
6.12 Data fitting procedure for valley Zeeman measurements via PLE . . . . . 118
6.13 1s PL spectra at Vg = 0.6 V for X0, Xt?/X
s
? and their extracted g-factors . 121
6.14 Valley orbital magnetic moment contribution for a charged exciton . . . . 123
7.1 Illustration of the Nichols radiometer. . . . . . . . . . . . . . . . . . . . 128
7.2 Angular moment of optical beams . . . . . . . . . . . . . . . . . . . . . 129
7.3 Intensity and phase of LG modes with different ? . . . . . . . . . . . . . 133
8.1 Depiction of a 2DEG illuminated with a spherical Bessel beam and resulting
selection rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
8.2 Transition strength into the center-of-mass modes after illumination with
Bessel beams with different ? values . . . . . . . . . . . . . . . . . . . . 152
B.1 Illustration of Bohr magneton derivation . . . . . . . . . . . . . . . . . . 170
B.2 Schematic of valley-resolved LLs in TMDs . . . . . . . . . . . . . . . . 172
xi
List of Abbreviations
2D two-dimensional
2DEG two-dimensional electron gas
AFM atomic force microscopy
BZ Brillouin zone
CPL circularly polarized light
CVD chemical vapor deposition
DI deionized
DOF degree of freedom
DOS density of states
EBL electron beam lithography
EEA exciton-exciton annihilation
hBN hexagonal boron nitride
HOPG highly oriented pyrolitic graphite
IPA isopropanol
LCPL left circularly polarized light
LG Laguerre-Gauss
NA numerical aperture
OAM orbital angular momentum
PC poly(Bisphenol A carbonate)
PDMS polydimethylsiloxane
PL photoluminescence
PLE photoluminescence excitation
PMMA poly(methyl methacrylate)
PPC poly(propylene carbonate)
RCPL right circularly polarized light
SEM scanning electron microscopy
TMD transition metal dichalcogenide
QE quantum emitters
QHE quantum Hall effect
FQHE fractional quantum Hall effect
vdW van der Waals
xii
Chapter 1: Introduction
1.1 Motivation
Traditional electronics are at a crossroads; while our thirst for increased computational
power to tackle larger problems is ever growing, the semiconductor industry is struggling
to keep pace. In recent years, the rate of computational power increases has diverged from
the expectations of Moore?s Law as development of new chips is hindered by minimum
component (transistor) size constraints and electronic Joule heating arising from the increasing
density of transistors on functional computer chips. Photonics ? which relies on photons
to transmit information instead of electrons ? can circumvent these issues and has the
potential to bridge the gap between conventional electronics and our computational needs.
Currently, most commercially available optoelectronic components are constructed
from Si or GaAs because there is well established large-scale manufacturing for both
materials. However, both materials have their drawbacks. Si is inefficient at emitting
light due to its indirect band gap, while GaAs is toxic to manufacture and confined
to the infrared regime. There exists a need for other materials that circumvent these
issues, especially ones that emit in the visible spectrum. Within this thesis, we discuss a
family of two-dimensional (2D) semiconductors that have the potential to fill this hole:
the transition metal dichalcogenides (TMDs).
1
1.2 The Rise of Two-Dimensional Materials
The study of 2D materials dates back nearly 100 years. Many experimentalists tried
to synthesize these materials to no avail. Theory backed up their lack of success; both
Landau and Peierls published work arguing that the thermal fluctuations in the lattice of
a 2D crystal would be so large that atomic displacement could take place on the scale of
the lattice rendering the whole system unstable [157, 158, 228]. Over the years, clever
scientists managed to grow monolayers of material on top of lattice-matched, single-
crystal bulk materials. No one thought that it would be experimentally feasible for 2D
materials to exist in a free-standing form until Geim and Novosolev surprised the world
with their isolation of graphene in 2004 from larger highly oriented pyrolytic graphite
(HOPG) crystals [217]. This discovery earned them the Nobel prize in 2010.
The isolation of graphene sparked a revolution in science, where more than 15
years later the number of papers per year on graphene is still on the rise and new, ground-
breaking discoveries come nearly yearly. In the early days of graphene research, the
physics accessible was hampered by the quality of the material available and the size
of the sheets that researchers were able to extract. However, improvements in bulk
crystal synthesis, exfoliation, and fabrication techniques have opened up a huge range
of possible physics to study. Due to the special Dirac nature of graphene, the presence of
massless Dirac fermions in the system allows for carriers densities researchers previously
couldn?t even dream of (200,000 cm2/Vs), and with it astonishing discoveries [27, 41,
219]. Not only could researchers see the quantum Hall effect (QHE), they could see
it at room temperature [220, 351]. Along with this came the measurement of the more
2
exotic fractional quantum Hall effect (FQHE) [28], and the chance to probe Hofstadter?s
butterfly in a condensed matter system [60, 116, 234]. The measurement of Hofstadter?s
butterfly sparked further interest in moire? lattices formed with graphene, which led to the
discovery of unconventional superconductivity in twisted bilayer graphene [34].
However, for all of its many interesting properties, one of graphene?s strengths is
also one of its weaknesses: graphene doesn?t have a band gap. Practically speaking,
our understanding of how to make electronics hinges upon using the band gap present
in semiconductors (currently predominantly Si) to act as a switch. Researchers have
successfully been able to produce semimetallic graphene, through artificially inducing
very small band gaps. However, as the name suggests, these materials are still more
similar in their electronic properties to a metal than a semiconductor, which makes control
in both the electronic and optoelectronic regimes difficult [4, 258, 328, 336].
In the mid-2000s, as researchers were trying to make semimetallic graphene, interest
arose in another group of 2D materials: the TMDs. Similar in structure to graphene ?
both in terms of the (staggered) honeycomb lattice and the fact that the layers are held
together by weak van der Waals attraction ? it offered a simple way to extend the 2D
materials family. Like HOPG, studies of these materials in bulk had existed for many
years [322]. Interestingly, the first report of monolayer TMDs is actually from the 1980s,
with few layer thick studies dating even earlier, but these works were lost to time until
the renewed interest in the 2000s [83, 124]. Materials in this family are of the chemical
form MX2, where M is the transition metal atom and X is the chalcogen atom. Despite
having this similar chemical structure, different degrees of filling in the non-bonding d
bands of the transition metal result in a family of ? 60 different materials that range from
3
Figure 1.1: Table of elements capable of forming TMDs. TMDs are of the chemical form
MX2, where M is the transition metal atom and X is the chalcogen atom. The chalcogen
atoms are highlighted in orange and the transition metals are highlighted in teal. Figure
from Ref. [90].
insulators and semiconductors to superconductors with competing charge density waves.
This variety has opened a new field that allows researchers to study nearly every electronic
system in the extreme quantum limit of 2D. Figure 1.1 shows all of the transition metals
and chalcogens that can form TMDs.
Over the last decade, the semiconducting TMDs have become an extremely promising
platform for photonics research as it allows us to study light-matter interaction in truly the
quantum limit. In bulk, these materials possess indirect band gaps, but in the monolayer
limit they undergo an electronic band structure transition that leads to the formation
of a direct band gap [197, 287]. Applying strain allows researchers to tune this gap
deterministically, introducing a simple means for band gap engineering to suit different
4
Figure 1.2: A brief, and ever-evolving history of a discoveries in semiconducting TMDs.
The references included: A: [83, 124], B: [218], C: [197], D: [241], E: [195, 344], F1:
[257], F2: [89], F3: [357], G1: [188, 235, 256], G2: [246], G3: [46], H1: [109, 288, 300],
H2: [8], I1: [163], I2: [334], J1: [208, 350], J2: [42], J3: [319], J4: [15], K1: [48], K3:
[358], L1: [303], L2: [304], M: [243, 283, 356], N: [45, 265].
device needs [14, 252, 357].
Along with the direct band gap, the broken inversion symmetry and strong spin-
orbit interaction gives rise to spin-valley polarized excitons in the system that are selectively
addressable with circularly polarized light. This is effectively a naturally occurring on-
off switch and makes these materials of great interest for both optoelectronics [199,
315] and valleytronics. The strong 2D confinement in the monolayer sheets and large
carrier mass leads to exceptionally tightly bound excitons (bound electron-hole pairs
with EB ? 100 ? 300meV) which makes it possible to observe them even at room
temperature [19, 190, 196, 257]. This alluring combination means that many desirable
5
and interesting quantum effects can be observed without expensive and cumbersome
cryogenic equipment, which is a boon for industrial scaling of cutting edge technology.
It also results in a semiconductor platform where many exciton species, including multi-
particle states like trions [191, 289], biexcitons (both neutral and charged) [44, 172, 181,
290], and most recently exciton-polarons [15, 69, 91, 275] can all coexist with distinct
energetic signatures. Further state tuning has recently become experimentally accessible
with the advent of moire? lattices [303], leaving us with seemingly endless possibilities to
tune these systems to fit many needs.
TMDs have also attracted interest in the field of quantum computation. The valleys
in the material have the capacity to form a qubit and the planar nature of the materials has
the potential to lend itself to large-scale integration of qubits in a way that trapped ions
currently do not [307]. Beyond that, dark exciton states have been found to exist in TMDs
[19]. Due to the fact that these states require a second-order process to recombine, and
therefore have long lifetimes, they are a promising candidate in the field quantum photonic
memory research [110, 120, 262]. A brief, and ever-evolving timeline of discoveries in
TMDs is shown in Fig. 1.2. This figure highlights breakthroughs most relevant to the
semiconducting TMDs and this work.
1.3 Goals of this Work
The focus of this thesis is on higher energy exciton species in TMDs, which we
define here as those with principle quantum number n > 1. As n increases, the binding
energy of the exciton species decreases. This makes observing higher n exciton species
6
difficult in most traditional semiconductor systems, since the binding energy of the n = 1
exciton is small to start with e.g. EB ? 10 meV in GaAs. However, the large binding
energy in TMDs gives us a unique opportunity to probe higher energy exciton dynamics.
In this work, we study two types of higher energy species:
(EI) In analogy to the hydrogen atom, excitons are also known to form higher
energy Rydberg series [128]. However, until its recent discovery in TMDs, a similar series
for trion-like particles was thought to be energetically impossible. We, for the first time,
explore the magneto-optical properties of the 2s charged state in WSe2 via observation of
the valley Zeeman effect and extract a g-factor for this 2s charged exciton. Origins of the
observed g-factor are discussed.
(EII) The excitons in (EI) are all s exciton species with orbital quantum number
l = 0. However, also in analogy to the hydrogen atom, it is possible for excitons to
form higher energy states with non-zero orbital quantum number l. A typical example of
this are 2p excitons, which have been observed in many different semiconductor systems
including TMDs. However, because of the additional angular momentum, these states
are dipole-forbidden and therefore require a nonlinear optical process ? like two-photon
spectroscopy ? to access them. Here, we theoretically examine if light that carries additional
orbital angular momentum (OAM) ? can modify the exciton spectrum and, for example,
allow transitions beyond the dipole approximation. We explicitly determine if the OAM
carried by the light is transferred to the internal degree of freedom of the exciton where
is could create a p exciton or if it would instead transfer to the center-of-mass (COM) of
the exciton where it could induce vortex-like motion or dispersion.
7
1.4 Outline of this Work
We begin this thesis in Ch. 2 with an overview of the properties of the semiconducting
TMDs. Here, the physical crystal structure is discussed, as well as the various types of
exciton species that can be optically generated along with their selection rules; this also
includes a comparison of single-particle and many-body interaction. Next, in Ch. 3 we
explore different experimental methods for TMD sample fabrication. Following sample
preparation techniques, in Ch. 4, we discuss measuring light-matter interaction and the
experimental ?knobs? that are at our disposal. As a primer for understanding our experimental
findings regarding (EI), in Ch. 5 we examine the different effects that a magnetic field can
have on excitons with a specific emphasis on the Zeeman effect. Based on this context,
we then discuss our work on magneto-optical characterization of the 2s charged exciton
in WSe2 (EI) in Ch. 6. Following this, we review the fundamentals of OAM in Ch. 7 as a
primer for our theoretical findings regarding (EII); those findings are highlighted in Ch. 8.
Finally, a summary of this work and prospective direction of future study is presented in
Ch. 9.
8
Chapter 2: Review of Transition Metal Dichalcogenide Fundamental Properties
and Exciton Species
This section is meant to discuss the physical and electronic properties of TMDs.
In bulk form, TMDs have been studied extensively. However, in monolayer form they
are truly in the atomic limit and their properties are distinctly different than their bulk
counterparts. Their reduced dimensionality, paired with strong Coulomb interaction,
supports a wide variety of stable, optically accessible exciton species that remain difficult
to traditional semiconductors systems like Si and GaAs [15, 191, 198, 290, 293].
Before we dive into the formalism, it is instructive to look at the crystal structure of
TMDs. TMD monolayers, unlike their graphene counterparts, are actually three atomic
layers thick and comprised of a plane of transition metal atoms between two sheets
of chalcogen atoms. These monolayer sheets come in two phases: trigonal prismatic
and octahedral. In both cases, the transition metal atom is six-fold coordinated, but
the positions of the chalcogen atoms are different: in trigonal prismatic monolayers
the chalcogens are directly over one another, while in octahredal monolayers they are
staggered. Which structure a given TMD monolayer takes depends mostly on the d
orbital filling, with the naturally occurring Mo- and W- based (semiconducting) TMDs
falling into trigonal prismatic category [137]. Modification of the structure is possible,
9
Figure 2.1: (A) An illustration of stacking periodicity and coordination of the different
TMD polytypes. C is the unit repeat in the z? direction. In this labeling scheme, the
number denotes the monolayers in the unit cell and T, H, and R correspond to trigonal,
hexagonal, and rhombohedral respectively [145]. (B) An illustration of a monolayer
TMD lattice extracted from a 2H crystal from a top-down view. In both panels, the black
atoms correspond to the transition metal atoms and the yellow atoms correspond to the
chalcogen atoms. This figure is adapted with some modification from Ref. [137]
but usually require significant engineering to do so, and usually results in some lattice
distortion [284, 340].
Fig. 2.1(A) shows the three common polytypes ? referred to as 1T, 2H and 3R
based on their respective unit cell geometries. 1T crystals are comprised of octahedral
monolayers with the top-down view showing effect of the staggered chalcogens. Both
2H and 3R crystals are comprised of trigonal prismatic monolayers, and are the type used
within our work that result in the honeycomb-like lattice. Though beyond the scope of
this thesis, it is worth noting that though the underlying monolayers are the same, the
difference in stacking between 2H and 3R can lead to different electronic and optical
properties of homobilayer systems [169, 206]. This highlights the important role that the
underlying structure plays in the physics observed. Fig. 2.1(B) shows a larger, top-down
view of a trigonal prismatic monolayer.
10
2.1 Band Structure
One of the very interesting properties that the semiconducting TMDs exhibit is an
indirect-to-direct band gap transition as the material is thinned from bulk to monolayer
form. This was originally observed in 2010 by Mak et al. in MoS2 and has since been
observed experimentally countless times in the the group VIB TMDs [146,197,287,315]
and the origins of this phenomenon confirmed with first principle density function theory
(DFT) calculations [337]. An important consequence of this property is that, compared
to its bulk form, the monolayer TMDs are optically very bright since this direct transition
enhances our ability to access states without having to provide additional momentum.
Referring to Fig. 2.2 for an illustration, when the TMD is in monolayer form the
direct band gap is formed at the two inequivalent, high symmetry points (?K) which
correspond to the corners of the BZ. Though they are inequivalent, ?K do constitute a
time-reversed pair, which is important to a variety of phenomena in the system. Similar to
graphene, we call these points valleys, and since they are inequivalent they act as another
degree of freedom within the system. Manipulation of this degree of freedom forms the
basis for valleytronics in the system and allows for manipulation of the population of
each valley independently [261, 307]. We will discuss this more explicitly in relation to
selection rules in the next section and also look at the origin of the valley Hall effect in
the system.
In 2H TMDs, the unit cell is comprised of two monolayers that are rotated in-plane
by 180?with respect to the other. This means if the crystal is thinned to a monolayer
unit, the resulting 2D piece lacks inversion symmetry. This leads to spin-splitting of
11
Figure 2.2: Transition metal dichalcogenide band structure schematic of the band edges
showing the spin splitting of the valence band/conduction band at the ?K valleys. The
schematic is not to scale. The figure is derived from Reference [327].
the bands that is a manifestation of strong spin-orbit interaction in the system; thus, it
is present even with no applied magnetic field. Both the conduction band and valence
band at the ?K points are spin-split, though the there is roughly an order of magnitude
difference between the magnitudes with the valence band splitting in the 100s of meV and
the conduction band splitting is generally 10s of the meVs [141, 142, 186]. As a general
rule of thumb, the magnitude of the spin-splitting in both the valence and conduction band
increases with the mass of the transition metal (e.g. ?EMo < ?EW), and then for a given
transition metal the splitting increases further with increasing mass of the chalcogen (e.g.
?ES < ?ESe < ?ETe). This can be understood in the context spin-splitting resulting
from the spin-orbit coupling is a relativistic effect. It is therefore felt more strongly with
increasing nuclear mass; as is the case with the heavy transition metals with their valence
d orbitals [201].
Because ?K constitute a time-reversed pair, the spin-splitting between the two
12
valleys is opposite, which implies that the spin and valley degrees of freedom are coupled,
or rather the system is spin-valley polarized [33, 195, 198, 199, 326, 327, 339, 344]. In
fact, the coupling is so robust that it has even been observed to translate to the Landau
level (LL) structure in the quantum Hall regime [98, 166, 185, 231, 319]. In the next
section, we will discuss the implications of the band structure on optical selection in more
quantitative detail. Later in the chapter, we will explore the important role of the spin-
valley polarization in the formation of different exciton species under optical excitation.
2.2 Selection Rules
To describe the 2H-TMDs we will use two terms,
H = HMD +HSOC. (2.1)
The first term, HMD is the massive Dirac Hamiltonian which serves as the most basic
model for low-energy electronic properties and yields the well-known valley coupled
selection rules. Recalling from the previous section that the spin-orbit coupling plays
an important role in this system, the second term HSOC corrects the energies from HMD
to account for spin-splitting. Additional terms that correct this Hamiltonian become
relevant in specific cases ? for example in the case of magnetic impurities one must add an
exchange term [38] ? but these two terms suffice to understand our elementary selection
rules. Here, we also ignore Coulomb interaction, but we later include it when explicitly
looking at exciton formation.
13
2.2.1 Massive Dirac Hamiltonian
Graphene, the first of the aptly named ?Dirac material? is gapless (i.e. metallic)
and described with well known massless Dirac equation. The TMDs, as we noted earlier,
have a similar honeycomb lattice and a gapped band structure. The gap implies that the
particles in the system are massive, so it is not a stretch to think that the massive Dirac
equation would describe our system well [327]. The massive Dirac Hamiltonian is given
as,
H? ?MD(k) = ?vF(?zkx??x + ky??y) + ??z. (2.2)
2
Here vF is the Fermi velocity, ?z is the valley index and can take the values ?1, k =
(kx, ky) are the electron wave-vectors, ?x,y,z are the Pauli matrix elements, and ? is the
direct band gap energy [327]. Note that the fermi velocity v = atF ? where a is the lattice
constant and t is the hopping parameter. We can rewrite Eqn. (2.2) in matrix form using
the sub-lattice sigma basis as,
???? ? ?vF(?kx ? ik2 y)?
?
?? (2.3a)
?vF(?kx + iky) ??? 2 ?
??? ? ?v ?i??k2 Fke= ??? (2.3b)
?v kei??k ??F .2
14
Here, tan(?k) = ky/kx. Diagonalizing Eqn. (2.3b) to solve for the eigenvalues, we get
? ?1
?
? = ?2 + 4(?vFk)2 = Ec/v(k). (2.4)
2
We can solve for the corresponding eigenvectors and find that,
?? ( ) ?
| k? ?c, ? = ??
c(os )?k2 ???? (2.5a)
sin ?k ei??k
2
?? ( ) ???sin ?k(e?)i??k2 ?|v,k?? = ? ??? . (2.5b)
?cos ?k
2
Here, cos(? ) ? ?k [93, 348]. We now have the eigenvectors corresponding to the2?
conduction and valence bands, and in the next section we will use them to find the
selection rules for forming a bound pair (exciton) between the bands.
2.2.1.1 Light-Matter Interaction
Suppose now that we illuminate the TMD with circularly polarized light (CPL);
applying minimal coupling (p ? p + eA) and assuming that the frequency of the light
?? ? ? so that we can make the rotating wave approximation, the light-matter interaction
Hamiltonian can be written as,
H?LM = e?vFA(r)[? ??xAx + ??yAy]. (2.6)
15
( ) ( )
Where in the circularly polarized basis A = 1 A+ ?x +A and A = ? i A+?A?y . We2 2
can rewrite Eqn. (2.6) as
?? ?0 (? + 1)A+ + (? ? 1)A??
H? = e?vFA(r)?? ??
?(? ? 1)A+?+ (? + 1)A? 0? (2.7)0 A??
= ??evFA(r)?? ?? .
A?? 0
With this, we can write the amplitude P(ke,kh) of creating an electron-hole pair ? or
rather an exciton, which we will discuss more extensively in the next section ? in the
valley with associated index ? ,
P? (ke,kh) =? ?c,ke| r[? ?r|HLM(|r?)?r| v,kh?? ( ) ] ?
? ? 2 ?k ?k 1= ? (evF) cos A? + e?2i??ksin2 A?? ? d2rA(r)eir?(ke?kh) .2 2 S ?? ?
=constant
(2.8)
We note that the portion of Eqn. (2.8) with the underbrace is equal to a constant because
CPL has no spatial profile. Here S is the area of the system. From this we can find our
selection rules: s excitons couple to A? light and d excitons couple to A?? light. However,
the coupling strength is vastly different. Recall that cos(?k) ? ? .(Nea)r the ?K p(oints,2? )
where the band gap is direct, 2? ? ? which means that the cos2 ?k ? sin2 ?k .
2 2
This implies that the s exciton formation process with A? light is near 1 and the d exciton
amplitude with A?? is roughly zero. In the dipole approximation, the d excitons should be
16
completely dark ? i.e. there should not be a non-zero transition amplitude associated with
them. This amendment to traditionally expected results for selection rules is due to the
winding number w of the Bloch bands, which is a topological quantity; here w =1 [348].
There are is another additional correction due to reducing the symmetry of the system
from C? to C3 to reflect the true symmetry of the crystal lattice which is referred to as
trigonal warping. Though we will not explore it here, it also brightens dipole forbidden
transitions. The curious reader can find an excellent overview in the article by Gong et
al. [93].
2.2.1.2 Polarization
The selection rules we have derived above let us finally show a property that we
have already alluded to: valley polarization. We define the polarizability as
|P? (ke,kh)|2 ? |P 2k ?? (ke,kh)|?( ) =
|P (k ,k )|2? e h + |P?? (k 2e,kh)|
| ? 1|2 ? | ? 0|2
= (2.9)
| ? 1|2 + | ? 0|2
? 1.
Thus, we see that from theory we would expect that there is near unity coupling of
the +K valley to right CPL and similarly between the -K valley and left CPL. Sadly,
in experimental practice, researchers do not observe near unity polarization in the TMDs
without taking great pains [212]. A typical laboratory experiment see somewhere between
20-70% polarization [102,150,195,259,325,344]. According to research done by McCreary
17
et al. this is largely attributable to the fact that is it desirable to have samples with
strong and sharp photoluminescence (PL) response for performing experiments, but this
correlates with long radiative lifetimes. The longer the radiative lifetime, the higher
the chance the exciton will experience a decoherence event (i.e. intervalley phonon
scattering) and lose polarization leading to an overall lower degree of polarization in
the system. Conversely, systems with short non-radiative lifetimes experience fewer
decoherence events and have higher overall polarization [205].
2.2.1.3 Valley Hall Effect
While we are on the topic of valley properties related to the massive Dirac Hamiltonian,
it is worth taking a brief look at the Berry curvature in each valley. The Berry curvature
can be generically written as,
?[? ???? ? ?? ? ? ??Hu ?n,k ?? ?u ? ?H ?? ]j,k uj,k ? ?un,k?kx ?ky
?(k) = i ? c.c. . (2.10)
(Ej ? En)2
j=? n
Here, En and un are the energy and wavefunction of the nth band, respectively [40,
289, 327]. In our case, we have a two-band model and can apply our massive Dirac
Hamiltonian and associated energy eigenvalues/eigenvectors and find that the Berry curvature
in the conduction band is,
2 2
?c(k) = ? ( 2? vF?? ) .? (2.11)3/2
?2 + 4?2v2 2Fk
Note that we can do a similar calculation for the valence band and we find that
18
?v(k) = ??c(k). Additionally, we see that Eqn. (2.11) has a ? dependence indicating
that the magnitude of the Berry phase in each valley is the same, but the signs are different
between the ?K valleys. The Berry phase is commonly described as a pseudomagnetic
field occurring in k-space and, as such, can be used to drive Hall effect physics. Specifically,
if there is an electronic bias in the system the difference in Berry curvature between the
two valleys can drive the valley Hall effect (VHE). Similar to the classical Hall effect, in
the VHE electrons and holes from the same valley will move in opposite directions, but it
requires no applied magnetic field; the motion is instead induced by the Berry curvature.
It is worth noting that because the resulting current would be equal in magnitude, but
opposite in sign for the two valleys in equilibrium, in order to observe the VHE the
system needs to be driven our of equilibrium. This is easy to achieve experimentally by
preferentially pumping on one valley with circularly polarized light [198]. Researchers
are very interested in this phenomenon for its applications in valleytronics [198, 307]
2.2.2 Spin-Orbit Coupling
As we can see in the above section, the massive Dirac Hamiltonian described there
is a simple two-band model that does not take into account the spin-splitting that occurs
in the conduction and valence band through spin-orbit coupling. As we discussed in the
previously in Section 2.1, this interaction is far from negligible in TMDs and as such
should be included in order to understand the full selection in this system. We have,
H? ?c(1+ ??z) + ?v(1? ??z)SOC = ? sz. (2.12)
2
19
Figure 2.3: Valley- and spin-selection rules in TMDs the polarization to excite each valley
is denoted (??) and the dipole allowed transitions are denoted with blue (higher energy,
?B? excitons) and red (lower energy,?A? excitons) transitions. Spin up is denoted with
the dashed line and spin down is denoted with the solid line. On the outside the Berry
curvature in each valley is depicted. This figure is derived with modification from Ref.
[198]
20
Here, similar to the definition in the massive Dirac equation, 2?c/v are the respective
conduction and valence band splittings and sz = ?1 is the spin index labeling spin-up
(+1)/spin-down (-1) [38]. Then, the whole Hamiltonian thus far is equal to,
H?tot?MD/SOC = ?I? H?MD + H?SOC? ??? ? + ? ?i??k? 2 c
? ?vFke 0 0 ??
?? ???vFkei??k ?? + ? ? 0 0 ???? (2.13)? 2 v= ? .?? 0 0 ? ? ?c? ?v ke?i??k2 F ????
0 0 ?vFkei??k ?? ? ?v?2
We can see in Eqn. (2.13) that the additional of spin-orbit interaction lifts the spin-
degeneracy of the valence and conduction band and transforms our two-band model into
a four-band model. We are not overly concerned here with the specifics of solving this
Hamiltonian, the more interesting correction here is to the energy of the system. Noting
that Eqn. (2.12) bares a striking resemblance to Eqn. (2.2), we can add a correction to the
band gap that with spin-orbit coupling included is
E?,sz ?gap = ?+ ?sZ(?c ? ?v) = ? . (2.14)
Calling this new quantity ?? we can solve but analogy the eigenvalues and we see that
here,
?
? 1? ?2 2MD+SOC = ? ? + 4(?vFk)
2 (2.15)
?1
?
= (? + ?sz(? ? ? ))2 + 4(?v k)2c v F .
2
21
From this we can see that between the outer ? and inner ? and sz we now have four
distinct energy eigenvalues. Additionally, because we can see that the splitting depends
on ? we expect that the ordering of the levels with spin will be reversed between the
?K valleys. An illustration of this, along with the Berry curvature we discussed in the
previous section, is shown in Fig. 2.3. In the diagram we see that we can select the
valley with the polarization and by tuning the energy of the light we can access excitation
associated with each spin state separately. This is commonly referred to as spin-valley
polarization. Note that the energy of the blue and red transition are degenerate, so this
is technically two degenerate Kramer doublets. This degeneracy can be broken with an
applied magnetic field and lead to an extremely large degree of polarization, which will
be discussed further in Ch. 5.
2.3 Exciton Species
Now that we understand the band structure and resulting selection rules in TMDs,
we will look at the different species of optically generated excitons that result from
them. This section is broken down into two models for describing these species: single-
particle and many-body states. The single-particle interpretation, which describes exciton
species as a finite number of electrons and holes bound together, is the simplest picture
to understand and we capture most of the basic underlying physics. The many-body
interpretation, on the other hand, treats all of the electrons and holes in the system as an
interacting ensemble which makes the description more complete. It has been successfully
used to describe optical excitations in TMDs when there are a large number of free carriers
22
Figure 2.4: (A) An illustration of the general energy configuration for excitons as mid-
gap states in semiconductors. (B) An illustration of the different kinds of exciton species
possible within the TMDs. Here, we ignore the lower valence band due the energy
discrepancy between the two bands (>100 meV). Excitations originating with with a hole
in the lower valence band make up the B series of excitons in TMDs. Panel (A) is adapted
with modification from Ref. [132].
present in the system.
2.3.1 Single-Particle Models
Excitons are bound electron-hole pairs that are created in semiconductors when a
photon excites an electron from the valence band to a mid-gap ?defect?-like state below
the conduction band. They have been studied in semiconductors for many years, but
the TMD system has allowed researchers to push the boundaries of exciton physics. As
discussed earlier in this chapter in the band structure section (Section 2.1), the TMDs
undergo an indirect-to-direct band gap transition when the crystal is thinned from bulk to
monolayer form [197, 287]. The direct gap strongly enhances the oscillator strength of
the bright exciton, which makes the study of the exciton states more readily accessible
23
from optical measurements.
Since that discovery, researchers have found quite a number of interesting and
sometimes exotic excitonic states in the TMDs. Not only are traditional bright excitons
visible, but many-particle phenomenon like biexcitons [16, 290, 342] and trions [18, 181,
190, 196, 257] have also been observed. Owing to band inversion in some TMDs, spin-
forbidden dark excitons have been observed in W-based TMDs [19, 120, 181, 208, 350,
355], and intervalley momentum-forbidden dark excitons are also possible under the right
conditions [229,304]. Localized excitons within the gap of the material have been host to
quantum emitters (QE) [39, 109, 139, 288, 300]. A visual representation of these excitons
can be found in Fig. 2.4(B). This section will serve to provide further explanation of these
phenomenon.
Though beyond the scope of this work, it is worth mentioning that the wealth of
excitons is even richer in the realm of traditional vdW heterostructures and Moire? physics.
It is not only possible to get interlayer excitons [74, 246], but more recently so-called
?Moire? excitons? have been found to exist in Moire? heterostructures and exhibit properties
different than their more ?traditional? (if one can truly use that word in a field still so new)
interlayer excitonic states [268, 303].
2.3.1.1 Bright Excitons
Excitons are typically discussed in analogy to the hydrogen atom since they are also
a bound pair consisting of a negatively and positively charged particle. Thus, the distance
between the electron and hole is called the exciton Bohr radius, aB. The energy of the
24
state is sometimes called the optical gap, and is defined as:
EX = E(optical)gap = Ebandgap ? Ebinding (2.16)
The selection rules discussed in the previous section apply to ?bright? excitons
found in TMDs (see Fig. 2.4(B)). Bright excitons are a bound e?/h+ pair where the e?
and h+ both are in bands with the same electron spin. They are thus a dipole-allowed
transition which recombine and emit light readily, leading to the moniker that they are
?bright.? In this case, the dipole is in-plane and thus the bright exciton emits in the out-
of-plane direction. The ease with which they recombine lends itself to a short lifetime (<
10ps).
The exciton spectrum of a material depends strongly on the dimensionality of the
system. Because many materials are well described with a hydrogenic approach, a good
jumping off point for describing excitons is the Rydberg model.In general, for a ?-
dimensional space, the Rydberg series is given as,
En
E0
binding = ?( )2 . (2.17)
n+ ??3
2
Here, Enbinding is the binding energy of the n-th level, n is the principal quantum number,
? is the dimensionality of the system, and E0 is the Rydberg energy. If we for a moment
ignore any dielectric effects, we see from Equation 2.17 that the energy of the lowest
energy state will be four times larger in 2D than in 3D. An illustration of excitons,
Eqn. 2.16, and the Rydberg states is shown in Fig. 2.4. The Rydberg series is particularly
25
Figure 2.5: (A) exciton formed in a bulk TMD. The entire exciton is contained within
the same dielectric environment, ?3D. (B) exciton formed in a monolayer TMD. Much
of the field associated with the exciton extends out of the TMD dielectric environment,
?2D, into the surrounding atmosphere, ?0. This leads to reduced dielectric screening on
2D excitons. Figure is adapted from Ref. [46].
prolific in TMDs because of the strong 2D confinement, and have been observed through
a variety of different optical techniques up to n =11 [44, 46, 94, 292, 293, 317].
However, there are other effects to take into account when discussing the structure
of the exciton series in TMDs. In the atomically thin limit, i.e. in the monolayer regime,
the dielectric screening is reduced substantially as the electric field between the bound
e?/h+ pair forming the exciton extends outside of the monolayer and into the surrounding
atmosphere (generally air or vacuum, which has a lower dielectric constant than the
semiconductor). This is unlike bulk excitons, in which the field emanating from the bound
e?/h+ pair is all contained within the same, bulk dielectric environment. An illustration of
this effect can seen in Fig. 2.5. This also means that the excitons themselves are strongly
confined within the plane of the monolayer sheet.
These two factors ? increased binding energy and reduced dielectric screening ?
26
have major implications for the behavior of excitons within TMDs. First, the optical gap
is expected to increase in monolayer TMDs. Second, the binding energy of the excitons
is also expected to increase (beyond the 4x factor from reduced dimensionality) due to
enhanced electron-hole interaction from the confinement [46]. In fact, this phenomenon
results in such a strong confinement of the exciton, that the binding energy of the exciton
is on the order of 100-300 meV, which is much larger than the thermal energy provided
at room temperature (? 25 meV) [292, 293]. This means that researchers observe robust
bright excitons in the in TMDs without ever having to cool their samples to cryogenic
temperatures, which is typically required to study this kind of physics [36,187,189,323].
To dive in to this a little more, let?s rewrite Eqn. 2.17,
?e4
En = . (2.18)
2?2?2(n? 1/2)2
Here ? is the reduced exciton mass, ? is Planck?s constant/2?, e is the charge of the
electron, and ? is the effective dielectric constant. In practice, when experimentalists
measure the Rydberg series they are measuring difference in energy that is actually Eqpg -
En where Eqpg is the quasiparticle gap. Thus, experimentally both the binding energy for
the nth state and the quasiparticle gap can be obtained if the series behaves in a hydrogenic
fashion.
As mentioned briefly, another quantity used to describe excitons are their Bohr
radius. For hydrogen, aHB = 0.529 A? but the radius scales with n like the energy does.
As with the binding energy, if we again account for the mass and dielectric environment
difference of the exciton, we can define the exciton Bohr radius as
27
n m
H 2
e?a
a = B
n
B . (2.19)?
Here, all previously defined symbols retain their meanings.
To throw a wrench in this formalism, researchers mapping the Rydberg series in
TMDs have found that they behave in a non-hydrogenic fashion with respect to the the
energy spacing between the levels in the series [46, 107]. In essence, researchers found
that the energy scaling is weaker with respect to n; in particular when n=1 and n=2. For
n=3-5, the spacing is much closer to what would be expected for a hydrogenic series
and this allows researchers to still extract physical information about the exciton binding
energy, etc. However, this still does not explain the diversion from the hydrogenic model.
If we recall Fig. 2.5, we see that the field from the exciton extends outside of the dielectric
environment of the monolayer itself. As the radius of the exciton extends with increasing
n, the charge separation leads to a larger portion of the electric field exists in this low-
dielectric region outside of the TMD. This gives rise to what researchers refer to as ?anti-
screening,? whereby the dielectric screening is reduced and leads to non-hydrogenic-like
behavior [46].
2.3.1.2 Dark Excitons
An exciton, like those discussed above, is said to be ?bright? if it has a one photon,
dipole-allowed transition (i.e. has even parity). If we refer back to Fig. 2.4(B) from
earlier in this section, we see that there are two ?dark? kinds of excitons: spin-forbidden
(intra-valley) and momentum-forbidden (inter-valley).
28
Figure 2.6: Comparison of the bright/dark states in (A) W-based TMDs and (B) Mo-
based TMDs. Due to d-orbital chemistry in the W-based TMDs the conduction band
orbitals are inverted compared with the conduction band in Mo-based TMDs.
2.3.1.3 Spin-Dark Excitons
As discussed in the previous section, bright excitons are formed from a e?/h+
pair that comes from bands with the same spin. Spin-forbidden dark excitons, on the
other hand, are formed from an e?/h+ pair in bands with the opposite spin. This means
that in order to recombine a second-order process (spin-flip) must occur. This leads to
significantly longer lifetimes than for bright excitons, with lifetimes >100 ps vs. fs scale
in bright states [249].
Not all TMDs form dark excitons readily. In particular, TMDs with W as the
transition metal have highly populated dark states. Turning our attention to Fig. 2.6,
let us examine why. In W-based TMDs resulting d-orbital hybridization in the heavier
system (6d in W vs. 5d in Mo) leads to a band inversion in the conduction band [141,
29
142, 144, 186]. Thus, when we look at this in Fig. 2.6 (A), we see that the bright state
(same spin) is higher in energy than the dark state (opposite spin) in this system. The
electrons can undergo an intervalley exchange interaction via phonon and relax down in
the lower energy state, thus populating the dark state and leaving a large reservoir of
dark excitons trapped there due to the long lifetimes. On the other hand, we can see in
Fig. 2.6 (B) that in Mo-based TMDs the bright state is lower in energy than in the dark
state. Therefore, while it is possible to form a dark exciton, the bright state is much more
energetically favorable and will populate in the same manner that the dark state does in
W-based TMDs [19].
The dark state was first observed indirectly in temperature resolved PL measurements
[11, 313, 349]. Researchers found that as the temperature was increased, so did the PL
intensity of the 1s state. Since the dark state is lower energy than the bright state, as
the system is cooled the electrons are more likely to non-radiatively thermalize into the
dark state, thereby quenching the emission and lowering the intensity. After this initial
work, researchers set about trying to determine how to ?brighten? the dark states and
interact with it in a controlled fashion. In contrast to the bright state, the dark state
has an out-of-plane dipole transition which means that if it emits light it does so in the
plane of the monolayer which makes detecting signatures difficult in common laboratory
setups [67, 249, 299, 349]. Naturally, upon experimental evidence of it?s existence the
research community set about determine how to best probe it.
The most obvious method is simply to collect light from the in-plane direction by
rotating the objective used to collect the light by 90?in order to have it aligned with the
edge of the monolayer plane. Wang et. al did just this and successfully observed the dark
30
Figure 2.7: Illustration of low NA and high NA configuration showing the difference in
the light collected. Additionally, there are some different NA values calculated on the
right side of the illustration to show the correlation between angle and NA more clearly.
In our experiments, we need NA > 0.6 to see dark excitons. Figure is adapted with some
modification from Ref. [59]
state directly in-plane [312]. Though rotating the setup like this may seem trivial, for
many groups performing their measurements at low temperatures in fixed cryostats it?s
simply not achievable depending on their optical access. Remarkably though, Wang et
al. also showed that with the correct common-place lenses it was possible to observe the
dark state emission (albeit weakly) without having to flip the setup.
Referring to Fig. 2.7, we see that the numerical aperture (NA) of a lens is defined
as
NA = nmedsin(?). (2.19)
Here, nmed in the refractive index of the medium through which the light travels and ?
is the collection angle of the light. Thus, for low NA lenses, almost all of the light we
collect is emitted in the out-of-plane direction. However, with increasing NA we collected
progressively more and more light that is emitted in-plane. In W-based TMDs, especially
at low temperatures, the dark state population becomes so large that even though the
31
lifetime of the dark state is long and the amount of in-plane light collected is small, we
can capture enough in-plane emission with a high NA lens to directly observe the dark
exciton even when the optics is aligned perpendicular to the sample. In fact, this is how
we observe dark state within our experimental work.
Though the high-NA lens method is popular, it does have the drawback of producing
a relatively weak signal. For those who still want to measure the state directly, a slightly
more refined method has become popular as it is easier to implement in existing measurement
setups. This method relies on high magnetic fields that are generally already accessible in
many measurement systems. Because of its reliance on a magnetic field, this technique is
referred to as ?magnetic brightening.?
The premise of magnetic brightening is simple: applying an in-plane (parallel)
magnetic field B = (Bx, By) in the plane of the monolayer TMD results in a Zeeman
interaction that mixes the spin states in both the valence and conduction band which
results in a small, but non-zero, tunable emission in the out-of-plane direction of the dark
exciton. Diving into the math, we see how this mixing occurs,
H 1Z = gc?B(?xBx + ?yBy), (2.20)
2
?B is the Bohr magneton, ?x,y are the Pauli matrices, and here gc is the in-plane Lande?
g-factor for the conduction band. For the remainder of this section we will be working in
the basis of {|?, b? , |?, d?}, where ? is still our valley index and b/d represent the bright
and dark states within the material.
In this formalisn, the Hamiltonian for the unperturbed bright and dark states is,
32
?? ???Eb 0 ?H ?0 = ? . (2.21)
0 Ed
Here, Eb and Ed are the energies of the bright and dark states, respectively, and Eb - Ed =
?EcSO. ?E
c
SO is our chosen form for the difference in energy between the bright and dark
states in the conduction band that are split due to spin-orbit coupling.
We know that the combined Hamiltonian accounting for interaction of the applied
B
parallel field is H ||int = H0+HZ , so combining Equations 2.20 and 2.21 and applying the
matrix form of the Pauli matrices we get,
?? ??? E gc?BB b (Bx ? iB2 y)?H || ?int = ? (2.22a)
gc?B (B + iB ) E
?2
x y d
? ?E 1b gc?2 BB???
= ?? ?? . (2.22b)
1gc?BB+? E2 d
To simplify matters, B? = Bx?iBy is introduced in Equation 2.22b. With our
Hamiltonian in place, we can solve for our eigenvectors through the usual process and
that,
| B?, b? || 1int = |
?
?, b? ? ?B+? |?, d? (2.23a)
1 + ?/2
and
|?, d?B|| 1 ?int = |?, d?+ ?B?? |?, b? (2.23b)1 + ?/2
33
where, ? = g2?2c BB
2/(4?E2SO). Since ? ? 1, the perturbed states in Equations 2.23a and
2.23b are very close to those of the original bright and dark states in the system. We can
see easily from this that the bright state now has a non-negligible mixture of the dark
state attached, making it possible to detect signatures of the dark state in out-of-plane PL
measurements. From Equation 2.23a we can easily read off some information about the
intensity of the PL signal from such a dark state,
Id ? ndIb? ? ndI B2b . (2.24)
Here, nd is the is the population of the dark exciton state and Ib is the intensity of the
bright exciton emission without the applied external field.
Even with the magnetic brightening method, which is limited by the field strengths
readily available in a lab setting, the intensity of this dark state is still very small. However,
it has been successfully measured and the preceding derivation is based on the work
presented by these successful experiments [29, 208, 350]. Other groups have since found
more creative methods as well to measure and interact with the dark states, such as chiral
phonons and through use of waveguides [184, 299, 355].
2.3.1.4 Momentum-Dark Excitons
Momentum-dark excitons, as the same suggests, occur when an exciton pair is
formed between and electron and a hole that are at distinctly different points in momentum
space. Most commonly, this phrase references an exciton formed between a hole in
the ?K valley and an electron in the ?K valley (see Fig. 2.4(B)). Researchers have
34
found that there is a prominent phonon-assisted decay path that allows for electron in
the upper conduction band in one valley to undergo intervalley scattering and end up in
the lower conduction band of the other valley (spin/valley degree of freedom are retained
here) [131,170]. This is then considered a dark transition since another phonon is needed
to allow the electron to move back to the correct momentum in order to radiatively
recombine.
Similar to the spin-dark exciton, the momentum-dark energy is lower energetically
than the bright exciton. However, it lies in between the spin-dark and bright excitons. In
a na??ve model, one would expect that the spin-dark and momentum-dark excitons would
be energetically degenerate. After all, they are both formed because a hole in the upper
valence band and an electron in the lower conduction band. However, in this na??ve model,
there is no accounting for the exchange effect on an exciton species formed between two
valleys. This intervalley exchange effect is rather substantial and raises the energy of the
momentum-dark exciton by 7 ? 10 meV ? depending on the TMD ? from the spin-dark
exciton energy [180]. The recombination can be observed directly, as well as additional
signature of a chiral phonon replica ? which comes from the chiral phonon required for
the exciton to recombine ? in PL measurements [170, 180, 358].
Ongoing research has found evidence of many different kinds of momentum-dark
excitons betwen the ?K valleys and different symmetry points within the band structure.
This is rapidly developing area of research and outside of the scope of this work. For the
curious reader, please see References 30, 76, 179, 194, 229, 304.
35
Figure 2.8: (A) shows two excitons, which are higher densities will interact and form a
biexciton compound (B)
2.3.1.5 More Exotic Pairings: Biexcitons and Trions
2.3.1.5.1 Biexcitons So far we have only discussed a single exciton in a TMD, but
what happens when there are many excitons in a TMD sample? One possibility is that
as the incident excitation power density increases, two excitons will undergo a 4-body
interaction where their energy/momentum is transferred and results in the creation of
energetic free carriers (so-called hot carriers) through the destruction of the exciton pairs.
This is known as exciton-exciton annihilation (EEA). This is a well-known process in
many semiconducting systems and several groups have reported EEA in TMDs in the
literature [55, 147, 211, 341]. The interest in this particular mechanism is two-fold: first,
the destruction of excitons results in loss of optical signal which is undesirable in many
experiments [8, 316] and second, on the flip side, research has shown that the efficient
of generation of hot-carriers from this project could be an excellent means of energy
generation in photovoltaics [24, 100, 178, 200].
In a process quite opposite of EEA, as the exciton density increases in the monolayer,
36
excitons in close proximity can bind together into a larger complex referred to as a
biexciton (Fig. 2.8). Biexcitons have been studied extensively in bulk semiconductors
as well as in other quantum systems ? like quantums wells, nanowires, and quantum
dots [?, 210]. Since the biexciton consists of two holes and two electrons, it can be
modeled similarly to positronium. In bulk,
2 2
Eb
? k
XX(k) = 2EX ? EXX + . (2.25)2?X
Here, EbXX(k) is the biexciton binding energy as a function of momentum, EX is the energy
of the exciton, EXX is the energy of the biexciton, and ?X is the reduced mass of the
exciton (2?X = ?XX) [135]. In systems with reduced dimensionality, research has shown
that this manifest as increasing biexciton binding energy with increasing confinement
[134, 226, 280].
Biexcitons have been studied extensively in the semiconducting TMD systems [16,
104, 233, 276, 291, 306, 342]. As an illustration, we will discuss results in WSe2 since
that is our system of interest. The biexciton was the subject of controversy in the TMD
community for several years. Based on theoretical numerical simulations, the expected
binding energy of the biexciton in WSe2 is ? 20 meV [149, 204, 297, 345]. However, in
early experimental measurements, researchers have found that it is actually closer to ?
50 meV [342]. The theoretical and experimental results were later reconciled when the
researchers determined that there were two biexciton complexes that had formed in the
material: a neutral biexciton with a binding energy of ? 17 meV (in good agreement with
the numerical simulations) as well as the a negatively charged biexciton residing at the
37
first reported position with a binding energy of ? 50 meV [172]. Since then, further work
has revealed biexciton fine structure that is the result of different configurations for the
electrons and holes between the valleys where the fine structure splitting in energy results
from different exchange term values for each configuration [290]. Additionally, there
have been predictions of an excited biexciton in the literature that at this time remains
illusive [345].
Aside from the predictive modeling for binding energies, there is another signature
that can determine whether a features is a biexciton: the relationship between the intensity
of the exciton peak and the suspected biexciton peak. Following simple kinetic theory laid
out in Reference 230,
dn 2X
= ?? nX nXX ? nX 1 nXX+ 2 + 2 (2.26a)
dt ?X ?XX n? ?C ?C
and
dn n n2XX ? XX X 1= + ? ?
nXX (2.26b)
dt ?XX n ?C ?C
Here, ? is the continuous-generation rate (? the laser pump power), nX is the
number of excitons, nXX is the number of biexcitons, ?X is the excitons lifetime, ?XX
is the biexciton lifetime, ?C is the interconversion time between excitons and biexcitons
(and vice versa) when n ?X = nXX = n where n? is the equilibrium constant.
When one solves these equations, the following solution is obtained,
[[ ] ]1/2
nX ?
?
1 + ? 1 (2.27a)
?0
38
and [[ ] ]1/2 2
nXX ?
?
1 + ? 1 . (2.27b)
?0
Here, ?0 = (n?? /4? 2XX X)(1+?C/?X) and is the characteristic generation rate that determines
the separation of the exciton/biexciton-dominant regions [95]. Thus, from this we can
predict from this at low generation rates (? < ?0), the exciton population will grow
linearly, while the biexciton population will grow quadratically [135]. We expect this to
manifest as,
I ? I?XX X. (2.28)
Here, IXX and IX are the PL intensities of the biexciton and excitons states, respectively.
As we can see from Equations 2.27a and 2.27b, ? = 2 when the system is in equilibrium.
Returning to our experimental system, in the WSe2 the researcher found for the
lower binding energy peak ? = 1.94, and for the higher binding energy peak ? = 1.82.
These are nearly 2 and in reasonably good agreement with the expected power relation
between the exciton and biexcitons emission intensities. In other quantum-well type
systems where biexcitons have been previously studied, a typical ? value falls in the
? = 1.2-1.9 range. Researchers have generally attributed these lower values to systems
that are varying degrees out of equilibrium [25, 286]. In light of this, researchers felt
comfortable taking this as additional confirmation of the biexciton state.
2.3.1.5.2 Trions As the name suggests, trions are quantum systems consisting of three
charged particles. Positively charge trions consist of two holes and an electron, while
39
Figure 2.9: Focusing just on K-valley trions (A) shows the intravalley singlet trion and
(B) shows the intervalley triplet trion and its associated exchange interaction. It should
also be noted the bands are structured in this diagram for WX2 systems. All trions here
are negative and a similar configuration to both (A) and (B) can be produced in the -K
valley.
negatively charge trions consist of two electrons and a hole. They were originally predicted
and observed in bulk semiconductor systems, but they have also been widely observed in
TMD systems [10, 53, 190, 196, 288].
Much like biexcitons, the formation of trions via the reversible process X0+e?(h+) ?
X?(+) is a function of the electrons (or holes) density in the system. Moving forward, we
focus solely on negative trions, but a similar formalism can be used to discuss positive
trions. We show this in the following, using a dynamical equilibrium model,
nbackground = ne + nX? . (2.29)
Here, nbackground is the initial background charge density controlled by gate voltage.
ne, nX, nX? are the density of free electrons, neutral excitons, and trions and are considered
to be steady state variables [257]. Our relationship between these quantities is defined
40
through the reaction rate from above.
Next, we apply the law of mass action onto our trion state:
T
nXn ? Ee B
= AkBTe kBT = ?X? (2.30a)nX?
and
4MXme
A = . (2.30b)
??2MX?
Here, kB it the Boltzmann constant, T is the temperature, ETB is the trion binding energy,
and MX/MX? are the composite exciton and trion masses, respectively, and the right-hand
side becomes ?X? which is the trion formation coefficient [257, 281].
If we rewrite Eqn. (2.30a) as nX? = ?X?nXne and substitute Eqn. (2.29) into it,
we get a final expression for the trion density as a function of initial background charge
in the system,
?X
n ?
nX
X? = nbackground. (2.31)1 + ?X?nX
From this, we can see when the exciton density is low in the system (?X?nX ? 1) then
nX? ? nX and the exciton and trion densities are roughly proportional. However, when
the exciton density in the system is high (?X?nX ? 1), the number of trions in the system
nX? ? nbackground approaches the background charge concentration [175]. As indicated
earlier, this is a reversible process X + e?0 ? X? and population in X(X?) will feed
into X?(X) as the system evolves in time. In fact, researchers have been able to show this
using two-color pump probe spectroscopy [36, 103, 154, 278].
Now that we know how trions are formed, let?s look a little more closely at what
41
kinds of trions we can form. Like excitons, trions can exist in both bright and dark states.
In order for a trion to be bright, the selection rules mandate one of the electrons must
have the same valley and spin index as the hole in the trion [2,279]. This amounts to four
total bright ground states ? two intervalley and two-intravalley ? which are illustrated in
Fig. 2.9 [191]. If we examine the spin configurations of the bands of the different bright
excitons in Fig. 2.9, we can see in the intravalley trions exist in a singlet state (Xs?), while
the intervalley trions exist in a triplet state (Xt?).
In the case of the intervalley trions, there is an intervalley exchange interaction
between the e?/h+ pair and the extra electrons, which researchers expect to be on the
order of about 6 meV in WSe2 [53,191,338] and reflects the energy preference for aligning
spins in the system. This results in a difference in energy between the singlet and triplet
exciton states, essentially splitting them and manifesting as trion fine structure. The trion
fine structure is well documented in the literature in the WX2 system [125, 172, 191, 232,
279, 305].
Just like bright excitons have dark exciton counterparts, bright trions have dark trion
counterparts due to spin forbidden transitions. Dark trion form in a triplet configuration,
but with the electron in the same valley as the hole in the opposite spin state. Like in
excitons, this suppresses the radiative recombination. An illustration of the difference
between a bright and dark triplet trion can be found in Fig. 2.10. The existence of dark
trions is well documented in the literature in PL studies of dark excitons [184, 350, 355].
However, there has been recent interest in the specific study and control of these dark
states due to their extremely long lifetimes. Typically, the lifetime of bright excitons is
< 1ps, triplet trions have lifetimes of a few ps and singlet trions have lifetimes of >
42
Figure 2.10: (A) the formation of a bright trion in a TMD, denoting the positions in the
bands of the carriers. (B) the formation of a dark trion in a TMD, denoting the positions in
the bands of the carriers. It should also be noted the bands are structured in this diagram
for WX2 systems.
25ps [279]. Recent work has shown that the lifetime of dark trions is > 1ns, which is
an order of magnitude higher than even dark excitons, and because of the charge of the
species this lifetime can be controlled by means of electrostatic gating [181]. The tunable
nature is extremely exciting for those interested in TMDs for quantum computation, and
these findings will open more serious study into manipulating the dark trion state.
2.3.2 Many-Body States
2.3.2.1 Break Down of the Single-Particle Model
As mentioned briefly earlier, all of the excitations we have discussed in this section
(bright excitons, dark excitons, etc.) are exciton species that fit within the single-particle
framework. Because of the finite number of particles in the system, these kind of excitations
are much simpler to model mathematically which makes them a convenient tool. However,
the realistic use of this type of model is limited to situations in which there are very few
43
free carriers present in the system and these single-particle type species can remain largely
isolated from one another.
In real experiments, there are usually many carriers present in the system (O ?
1011 ? 1012cm?2) if we do not artificially hold the system at neutrality by means of
electrostatic gating (a technique we will discuss more in Ch. 4). In this many-body regime,
the properties predicted by the single-particle model begin to diverge from observed
physics and a new model is needed to describe the scenario playing out where excitons
interact with and are dressed by the Fermi sea of carriers that now exist.
2.3.2.2 Historical Context of Polarons
If we cannot use single-particle physics anymore, what do we replace it with? It
turns out this question comes up frequently in condensed matter systems where we are
faced with impurity problems ? that is, what happens when a quantum impurity interacts
with a reservoir of bosons or fermions. A very common technique is to introduce the
concept of polarons as the many-body quasiparticle resulting from that interaction. In
traditional polaron theory, a polaron is an electron or hole (impurity) that is dressed by a
cloud of virtual phonons (boson reservoir). As the impurity travels through the lattice, the
virtual phonons distort the lattice and other excess charges around the impurity creating a
traveling polarization cloud ? hence the name polaron.
This concept of the polaron dates back to the early days of quantum theory in
the early-to-mid 20th century, with the pioneering work performed by Lev Landau and
Solomon Pekar in the 1930s-1940s [156, 159]. Later Holstein and Fro?hlich built on this
44
Figure 2.11: (A) Neutral (exciton) and charged (trion) single-particle species and their
(B) many-body interpretations, the repulsive and attractive polarons.
to further develop the two regimes of large and small polarons [84, 85, 112, 113]. This
idea has become a very useful one to describe many-body induced quasiparticles and
today the framework has been extended to include ideas like bipolarons [5, 215, 353],
polaron excitons [115,302], and 2D Fermi-polarons ? which are the kind found in TMDs
[15, 69, 275]. For the curious reader, Franchini et al. have an excellent and recent review
on the large variety of polarons currently known [81].
2.3.2.3 2D Fermi Polarons in TMDs
In the presence of excess carriers, we look to replace the concept of excitons and
trions with 2D Fermi exciton-polarons. Fermi polarons were widely explored in cold-
atom systems in the early 2000s [136, 143]. During all of this work, two species of
polaron were identified: the repuslive Fermi polaron and the attractive Fermi polaron. As
the name would suggest, the attractive (stable) Fermi polaron interacts attractively with
the Fermi sea by pulling in the charges to screen itself, which the repulsive (metastable)
Fermi polaron repels the Fermi sea and carves out a neutral environment for itself. See
Fig. 2.11 for an illustration of the transition from single-particle to polaron states.
45
The discovery of trion-like resonances became a catalyst for applying polaron theory
in TMDs. As discussed earlier in this chapter, significant work has been done to characterize
and model the trion state in TMDs. The largest snag in that effort lies in the interpretation
of the observed resonances themselves. The single-particle model for trions is only
theoretically-applicable when the Fermi energy in the system EF ? EX? , the binding
energy of the trion. However, experimental results for the trion are limited to the regime
in which EF ? EX? .
To resolve this issue, Efimkin et. al, amongst others, suggested that much like in
cold-atom systems, the exciton could form a Fermi polaron in the presence of excess
charges in the system [68?70]. In TMDs, this would mean that the extension of the
exciton state under doping becomes the repulsive polaron branch and the state originally
thought to be trions are actually attractive polaron states. It can be helpful to think of
these polarons as continuum excitons, where the excited bound electron now lies in the
actual conduction band in contrast to the below band states typical of a single-particle like
exciton. In this context, we can illustrate the repulsive and attractive polarons in a similar
manner to the excitons and trions, which can be seen in Fig. 2.12. The experimental
confirmation of these states was performed shortly after their prediction by Sidler and
Back et al., who also contributed to the theoretical interpretation [15, 275].
One important question that remains is determining when the single-particle and
many-body interpretations are valid. We can think of the carrier concentration as the
knob that lets us tune the strength of the many-body state (we will discuss this more in
Ch. 4.3.2). Glazov showed that in the extremely low carrier limit, the physical properties
predicted by both the trion and polaron interpretations converge [91]. This indicates that
46
Figure 2.12: (A) Repulsive polaron, (B) intervalley attractive polaron, and (C) intravalley
attractive polaron in WSe2. Figure is adapted with some modification from Ref. [91].
the many-body interaction involved in polaron formation is more of a sliding scale than
an on-off switch.
In this thesis, we discuss briefly the influence of many-body interaction our experiments.
Our chosen metric to measuring this interaction strength is the g-factor extracted from
Zeeman splitting. In many way, the g-factor can act as a barometer for correlated magnetism
and indeed researchers have found atypically large g-factors [15, 133, 174, 320] in the
polaron regime. We will discuss more about interpreting this g-factor metric in Ch. 5. In
Ch. 6, our work studying the g-factor of the excited trion-like state is highlighted.
47
Chapter 3: Device Fabrication Methodology
3.1 Materials Synthesis and Extraction
In order to experimentally probe the properties of TMDs, we must first produce
samples or devices from them. Generally speaking, there are several methods for producing
the pieces needed to do this. In this work, two methods were used: mechanical exfoliation
and chemical vapor deposition (CVD) growth. The details of each method are outlined in
the following subsections and serve to help explain this type of fabrication from the very
building blocks up. As a rule of thumb, mechanically exfoliated materials tend to have
higher mobilities and therefore better transport properties, while CVD grown materials
tend to produce fewer optical defects but have lower mobilities.
3.1.1 Mechanical Exfoliation or the ?Scotch-Tape Method?
Mechanical exfoliation was the technique originally used to isolate graphene [217].
Since the original method used by Geim and Novosolev utilized everyday Scotch tape,
the method has since been colloquially dubbed the ?Scotch-tape method.? The individual
layers of graphene, hexagonal boron nitride (hBN), and TMDs are all held together in the
bulk by van der Waals (vdW) bonds. These bonds are extremely weak compared to the
48
intralayer covalent bonds that hold the atoms in the plane of each sheet together, and the
shearing force applied mechanically by the tape is enough to cleave the bonds and thin
the material down to monolayer thickness when it is repeated several times.
The process for mechanical exfoliation is illustrated in Figure 3.1. First, the tape
is brought into contact with the bulk crystal and used to remove a chunk of materials
(Figure 3.1 (A)/(B)). Then the tape is repeatedly stuck together and peeled apart to thin
the material removed from the bulk and to cover the entirety of the tape (not pictured).
After the crystal has been sufficiently thinned through mechanical exfoliation, the tape is
applied to a substrate on which to deposit the material (Figure 3.1 (C)). For the work
outlined in this thesis, the substrate used was degenerately doped Si, with 285nm of
SiO2 on top. While the Si/SiO2 can be utilized for electrostatic gating or transport
measurements, the SiO2 thickness turns out to be the key to actually being able to optically
identify graphene and other 2D materials. At this thickness, the contrast between monolayer
materials and the background SiO2 is optimal under magnification to the human eye due
to thin-film interference effects [26]. Even a 15 nm difference in thickness of the oxide
is enough to almost completely obscure the monolayer material on the surface of the
sample. Uniform pressure is applied gently to the surface of the tape now in contact with
the SiO2 and heat is applied from the bottom using a hot plate. Both the pressure and
elevated temperature help facilitate the fracturing and adhesion of the exfoliated crystal
from the tape to the substrate. Finally, the tape is gently and slowly peeled back from the
surface of the SiO2 (Figure 3.1 (D)). Images taken of a piece of exfoliated WSe2 on SiO2
and cleaved hBN on Si/SiO2 can be seen in Figure 3.1 (E)/(F), respectively.
Though the process is simple conceptually, it turns out to be incredibly nuanced to
49
Figure 3.1: Panels (A)-(D) shows the mechanical exfoliation or ?Scotch-tape method? for
extracting monolayers crystals and panel (E)/(F) shows a microscope image of resulting
monolayer WSe2 and hBN respectively, obtained using mechanical exfoliation. Figure
adapted with some modification from Ref. [216].
achieve consistent results from the process in terms of both uniform thickness and sheet
size. Each material has its own quirks. For example, it is necessary to plasma clean the
surface of the SiO2 with oxygen for graphene to achieve large sample sizes of monolayer
material, but it?s nearly impossible to get monolayer material of any size without extensive
plasma cleaning for both hBN and TMDs. The type of adhesive, the pressure applied to
the tape, and the temperature at which the Si/SiO2 substrates are heated turn out to play
crucial roles in the success of exfoliation as well.
After exfoliation, monolayer flakes are optically identified under a microscope. As
mentioned earlier, the oxide thickness used (285nm) has been experimentally determined
to be optimum for identifying monolayer materials [26]. If there was ever a question of
if the sample was in fact monolayer, atomic force microscopy (AFM) was used to verify
the optical signature.
50
Figure 3.2: (A) Setup of the tube furnace for depositing monolayers of TMDs using
the chemical vapor deposition (CVD) process. The examples materials here are raw
selenium (Se) for the chalcogen and tungsten trioxide (WO3) for the transition metal
precursor, since our experiments focus on WSe2. (B) CVD grown WSe2 monolayers on
SiO2. The natural termination of the monolayer into a triangle is due to the underlying
crystal structure and a typical size is ?30-40 ?m per side of the triangle.
Generally speaking, these crystals are sourced from various growers, as we do not
grow them in house. During the course of this work, we purchased HOPG, natural
graphite, hBN, and several kinds of bulk TMDs from HQ+ graphene, natural graphite
from Manchester Nanomterials, and bulk TMDs from 2D Semiconductor. Through academic
collaboration, we also received hBN from the Watanabe group at the University of Tokyo.
3.1.2 Chemical Vapor Deposition Growth
One of the drawbacks to mechanical exfoliation is that extracted monolayer flakes
tend to be small in their lateral dimensions and the yield of even these small pieces is
very low. It requires significant time, effort, and skill to produce just one monolayer
sheet which makes the process of fabricating devices even more difficult. CVD growth of
monolayers is one method that overcomes this issue. In this method, instead of thinning to
one atomic unit from a bulk crystal, you grow just the monolayer sheets themselves. This
method, though requiring its own set of skills, produces numerous large-area monolayer
51
sheets at a time and does so consistently once a recipe for growth is developed.
Through our collaboration with researchers in Dr. Berry Jonker?s group at the
Naval Research Laboratory (NRL) in Washington, D.C., we had access to CVD grown
TMDs that their group specialized in producing. An illustration of the CVD process for
producing TMDs is shown in Figure 3.2(A). Referring to Figure 3.2(A), we see that a
chalcogen source (S or Se) is placed upstream in a tube furnace from the metallic source
(typically MoO3 or WO3 [164, 352]) with the desired substrates placed very close to
the metallic source. During this process, an inert gas like N2 or Ar is flushed through
the tube to prevent atmospheric contamination and the furnace is heated in the range
of 600-1000 ?C to promote reaction between the precursors. In the gas of Se-based
TMDs, hydrogen gas is used to help enhance the reactivity of Se since it is reductive
by nature. The final result of this deposition is shown in Figure 3.2(B). This is simply a
brief overview of the techniques involved, but for a more in-depth look at the growth of
TMDs and other 2D materials, please refer to Reference [250] which is a review of CVD
techniques coauthored by one of our NRL collaborators. A recipe provided by our NRL
collaborators specifically for growing WSe2 can be found in Ch. 6.3.1.1
3.2 Hexagonal Boron Nitride Encapsulation
Following the optical identification of material on the surface of the Si/SiO2chips,
the next step in fabricating a device is to ?encapsulate? the 2D material between two layers
of hBN. This process was originally pioneered in graphene by Philip Kim?s research
group in 2010 [61]. There, Dean et al. showed that they could achieve mobilities of
52
10,000 cm2/Vs [217]. Though low by today?s standards, it was extremely groundbreaking
at the time and opened up the possibility of further improvement to 2D samples. In fact,
within a year there were theoretical works providing predictions that graphene should
be able to achieve mobilities of 1,000,000 cm2/Vs, provided the sample is clean enough
[1, 41]. The allure of achieving these high mobilities was not simply intellectual, but
practical: with high mobilities comes richer physics to probe.
To understand why this encapsulation process is so revolutionary, let us consider the
2D system further. By their nature 2D materials are composed almost entirely of exposed
surfaces, which means that they are strongly susceptible to environmental disorder. Early
graphene devices made simply from graphene mechanically exfoliated on Si/SiO2 had
low mobilities and high disorder due to surface phonons, charge impurities native to the
oxide substrate, and strain from lattice mismatch between the crystal and substrate [40,
64, 203]. The first attempt to fix this issue was to fabricate suspended graphene samples
where a portion of the SiO2 was selectively etched [27, 66]. However, these devices
were extremely fragile, difficult to fabricate, and had substantial issues with strain. The
advance by Dean et al. solved most of these problems and allowed more researchers the
chance to enter the field as the work became more accessible.
While the improvement seems simple at first glance, significant work has gone
into understanding the subtleties that led to improved material properties. To begin with,
because graphene and TMDs are atomically thin, they tend to conform to the substrate
they are placed on. On SiO2, this can lead to a surface roughness of ?0.5nm due to
the amorphous structure of the SiO2 [54]. However, results of microprobe studies have
found that the roughness on hBN is < 50pm, which will lead to reduced potential traps
53
[62, 331]. Additionally, since hBN also has a hexagonal lattice structure with similar
lattice parameters to both graphene and the TMDs this greatly reduces strain induced
from a lattice mismatch between the substrate and the material of interest when compared
to traditional SiO2 substrates.
The technique that evolved from hBN encapsulation can be utilized to stack just
about any combination of 2D materials and led to even further developments in the field.
The next step came when researchers sandwiched the hBN encapsulated structure with
few-layer graphite sheets and used the graphite to electrostatically gate ? i.e. control the
number of free carriers ? in the TMD layer [360]. Research has shown that this technique
also effectively eliminated charge inhomogenity in the graphene or TMDs which further
enhances the mobility [331]. We utilize this type of electrostatic gating in our devices
as well, as it allows us to tune the Fermi level within our sample and probe different
regions of charge interaction within the system. From there, 2D materials have become
a playground for interesting physics with the addition of using twist angle between the
layers to create moire? lattices with many astonishing properties from superconductivity
to Mott states [34, 243, 268, 303].
In the next section, we will discuss the different techniques that can be used to
perform the kind of stacking described above. There are a significant number of variations
in methods for stacking with nearly every research group having their own recipes, so this
section is not meant to be all-encompassing. Instead, outlined are the techniques used
by this author through her doctoral research. The first several years of work performed
for this thesis are nearly all encompassed by different transfer techniques. Since we
were building our experiment from scratch, many different attempts were made with
54
Figure 3.3: Panels (A)-(D) show chronologically the steps for creating one layer of a stack
using visoelastic stamping.
different techniques before a successful method was found through expertise from and
collaboration with NRL. Note that all techniques discussed below focus only on encapsulating
a monolayer sheet with hBN to serve as a standard example between techniques. All
techniques could be used to create more complex structures like moire? lattices or electrostatically
gated samples.
3.2.1 Dry Encapsulation Techniques
Dry encapsulation techniques, as the name might suggest, refer to all techniques
that do not utilize a liquid in the process. Ultimately, the polymer-based techniques with
Poly(Bisphenol A carbonate) (PC) and poly(propylene carbonate) (PPC) proved to be the
most fruitful in terms of dry techniques. In addition to these methods, we will discuss
some earlier methods as they represent significant effort and energy by this author.
3.2.1.1 Visoelastic Stamping
Visoelastic stamping was the first method that was used in an attempt to pickup
and ultimately stack 2D materials. The method was based off of work by Castellanos-
Gomez et al. [37]. In this method, the top hBN and TMD are both directly mechanically
55
Figure 3.4: Panels (A)-(E) illustrate, in order, the process for producing one layer of an
encapsulated stack using the ?drop-down? method. The method must be repeated with
each layer of the stack.
exfoliated onto a base polydimethylsiloxane (PDMS) stamp that is attached to a transparent
glass slide. PDMS itself is also transparent, so it is possible to see straight through both
the slide and stamp with a microscope. In our own adaption, to make it easier to keep
track of the exfoliated material we pre-patterned the PDMS with embossed alignment
marks using soft lithography ? a technique commonly used in microfluidics [209, 347].
The TMD is optically aligned with the bottom piece of hBN and gently pressed down onto
it and the stamp is slowly peeled back, leaving the TMD on the surface of the bottom hBN.
An illustration of this can be seen in Figure 3.3. This process is repeated to place the top
hBN. Because the pieces of the stack are dropped down, this is considered a bottom up
method for encapsulation.
3.2.1.2 Sacrificial Polymers A.K.A the ?Drop-Down? Method
Following the visoelastic stamping method, we attempted a similar method that
relied on the release of a sacrificial polymer layer to drop the 2D material onto the
surface of the sample. The method was originally pioneered by Zomer et al. and relies
on an industrial resin known as Elvacite? 2550 [362]. The preparation procedure for
this polymer can be found in Appendix A. Similar to the visoelastic stamping method,
56
the 2D material is exfoliated directly onto the stamp that has been pre-embossed with
alignment marks using soft lithography. However, this time the PDMS stamp has a thin
layer of Elvacite? (Figure 3.4 (A)) spun onto the surface prior to exfoliation. The stamp
is then loaded into the encapsulation system and aligned over the Si/SiO2 (Figure 3.4
(B)). Because Elvacite? has a low glass temperature, it melts around 75-100?C. In order
to drop the 2D layers, the Si/SiO2 substrate is heated to about 85?C and the stamp in
then gently touched to the surface and then retracted, leaving behind the 2D material
with the Elvacite? melted on top (Figure 3.4 (C)). The Elvacite? is then removed in a
bath of chloroform or acetone, leaving behind a clean layer of the 2D material (Figure
3.4 (D)/(E)). Between steps (D) and (E) the sample is annealed to remove residue. This
process must be repeated for each piece of the stack, though Figure 3.4 shows only the
process for the bottom piece of hBN. Since this method ?drops? the sample using the
weight of the polymer, this is also a bottom-up encapsulation process.
3.2.1.3 van der Waals Pick-Up with Heat-Expansive Polymers
The final, and most reliable dry-method used for stacking 2D materials is based
on heat-expansive polymers. Both PC and PPC based polymers work in this manner
and recipes for these polymers can be found in Appendix A. Cleanliness of samples
is an ongoing issue in all areas of research. In order to try to expose the ?functional?
components of a stack (say the graphene or TMD) to as few contaminants as possible,
Wang et al. pioneered a method that relied on vdW attraction and heat-expansive polymers
[314]. This is an incredibly popular method since it reduces the introduction of contaminants
57
Figure 3.5: Dry pickup method for sequentially picking up 2D materials to form
heterostructures with PPC.
between the constituent layers. As a result, a large of number of variations are seen in the
literature with PPC as the heat-expansive polymer. This method also works with PC as
the polymer as well [363].
Figure 3.5 illustrates how this dry pickup technique works with heating specifications
for PPC. A PDMS stamp is adhered to a glass slide and wrapped in clear tape. A thin layer
of 6-15% PPC is spin-coated on the surface of the PDMS stamp. The PDMS stamp is
brought just barely into contact with the top piece of hBN and then heat is used to slowly
expand the PPC polymer to pickup the top layer and then it is cooled to retract it from the
surface. The target temperature for this step is 60?C, the heating rate is 2?C/min, and the
cooling rate is simply based on radiative cooling after the heater is turned off. The hBN
is then used to pickup the TMD via vdWs attraction using the same heating procedure
(Figure 3.5 (A)). The bottom piece of hBN is picked up in the same fashion (Figure 3.5
(B)). Finally, the whole stack is dropped down onto a pre-patterned piece of Si/SiO2 with
gold alignment marks. This is done by heating the PPC to a higher temperature than to
pickup, so that it is heated beyond its elastic limit and drops to the surface (about 105?C,
58
Figure 3.5 (C)). The PPC is cleaned in a chloroform bath leaving an encapsulate stack
on the surface of the Si/SiO2 (Figure 3.5 (D)). The procedure for picking up with PC is
largely the same, except the pickup stages are done at 80?C and the final drop is done in
the 160-175?C range.
3.2.2 Capillary Action Assisted Encapsulation
A problem that plagued our pursuit of high-quality samples was that CVD grown
TMDs are incredibly difficult to remove from the surface they are grown on. Significant
efforts were made to try to utilize the vdW method with heat-expansive polymer, but the
TMD would always remain stubbornly stuck to the surface it originated on. Researchers
had found many years ago with graphene that it?s difficult to remove CVD grown material
in general, not just for stacking. They could grow large area sheets of graphene, but were
unable to remove them from the growth substrate without causing significant damage to
the material.
Eventually, scientists determined that they could use a polymer spin-coated onto the
surface of the CVD grown material to support it and then float the CVD grown material
off after disrupting the bond between the sheet and the substrate [346]. This method
became the basis for what is referred to as the ?wedging transfer method? for stacking 2D
materials [264]. In this method, the transfer relies on different affinities for water of the
materials in the system. A thin layer of water is inserted between the hydrophilic substrate
and the hydrophobic polymer that is spun onto the 2D material, allowing them to separate.
However, the group found that because the material ends up free-floating on the water, it
59
Figure 3.6: Capillary-action assisted pickup of a TMD stack using a combination of liquid
injection and dry transfer.
leads to significant wrinkles and water trapped between the layers of the stack [31].
Several years later, another group showed that the same process could be achieved
using heated water vapor [192]. This led to fewer wrinkles, but is very tedious to perform.
However, the concept of capillary action also inspired one of our group?s collaborators at
NRL and led to the development of a more user-friendly fast pick-up method that only
relies on a very small amount of liquid [101]. Dr. H.-J. Chuang was kind enough to give
us a tutorial on his method, the process of which is outlined below.
On this capillary-action assisted pickup method, we start with a bare piece of PDMS.
Since the hBN is in direct contact with the PDMS it is important that you use a very clean
piece of PDMS. It is recommended that you use homemade PDMS made with Sylgard
184 (182 can also be used, a recipe can be found in Appendix A) instead of purchased,
pre-made PDMS. The PDMS is brought gently into contact with the to top piece of hBN
and approximately 1dL of a mixture of deionized (DI) water/isopropanol (IPA) is injected
between the PDMS and the substrate using a syringe. The PDMS is then retracted with
the hBN attached and the sample is gently dried with N2 (Figure 3.6 (A)). Similar to the
60
vdW method, the process is then repeated to pickup the TMD and the bottom hBN (Figure
3.6 (B)/(C)). Finally, the whole stack is brought into contact with a Si/SiO2 substrate with
alignment markers and then gently peeled back without the DI water/IPA, leaving the final
stack on the substrate (Figure 3.6 (D)).
3.2.3 Comparison of Encapsulation Methods
We worked with the visoelastic stamping method for a short period of time. It
proved to be very difficult in our setup to get TMDs and hBN to detach from the PDMS
stamp like Castellanos-Gomez, et al. were able to, which may have been influenced by
the soft lithography we had perform as it could leave residues on the surface of the PDMS.
The yield of larger 2D pieces from exfoliating directly onto PDMS is also very low, which
was a problem with the ?drop down? method as well.
In terms of cleanliness, there were a variety of issues. One of the main reasons
that we abandoned the ?drop down? method with Elvacite? was the residue that was left
behind. No matter the cleaning process, there always seemed to be white, chalky residue
that was left on the surface of the sample. We believe this residue was likely a result
of the temperatures needed to melt the Elvacite that cause it to completely break down.
Moreover, since the process needed to be repeated for each layer, with the sacrificial
polymer melted on top of the 2D material each time, there is a large amount of contamination
introduced at each step. Large spots of residue that could not be removed in the solvent
bath or through high-temperature annealing remained and caused bubbles/wrinkles at
each step.
61
While the procedure for performing pickup is roughly the same for both PC and
PPC, there are differences in their functionality. Because the elastic limits of PPC is
lower than that of PC, it tends to come off more cleanly during the chloroform cleaning
and not leave as much residue on the sample. The residue is an important factor (which
we will discuss further in a later subsection of this chapter) and can greatly influence the
quality of the sample. However, PC has a desirable quality that PPC does not: it picks
up TMD or graphene/graphite not directly underneath the hBN and leave edges exposed.
Making 1D contact is notoriously difficult in graphene and nearly impossible with TMDs
due to the barrier potential on these edges. Therefore, especially in TMDs, it can be
desirable to leave enough of the functional material to make 2D contact. PPC only allows
for pickup of material that is directly underneath of the top layer so it is largely useful for
making samples that only require optical measurement. Recipes for both polymers can
be found in Appendix A.
Ultimately, the samples used in our experiments were made by our collaborators
in Berry Jonker?s group at NRL. Though countless samples were produced in-house
using the techniques above, they were not showing the same consistent quality as those
produced at NRL. Those samples were produced using the capillary-action assisted method.
Generally speaking, this is a very clean process since it does not rely on any sort of
polymer and there is no heating. However, additional care must be taken to make sure that
the PDMS is sufficiently clean and thin, as this can lead to poor pick-up and contamination.
A sample specific overview of this process can be found in Ch. 6.3.1.2.
62
Figure 3.7: Panels (A) and (B) show the nano-squeegee process whereby the bubbles
that can appear during encapsulation can be flattened out using the force of the AFM tip.
Figure is adapted with some modification from Ref. [255].
3.3 Sample Flattening via Atomic Force Microscopy A.K.A ?Nano -
Squeegeeing?
As we recall from Section 3.2, the purpose of encapsulating graphene, and later
TMDs, was to increase the quality of the sample through reducing charge inhomogeneities,
strain, etc. Encapsulating a sample goes a long way to improving sample quality, but it can
also introduce new contaminants during the process. Though the vdW pickup methods
don?t have the active layer in direct contact with the polymer, there is often an extremely
thin, uniform layer of contaminants between each layer of the stack and wrinkles and
bubbles with additional contaminants can form as well [31, 99, 253]. Some groups will
annealing to try remove contaminants through evaporation, but the technique is not well
controlled or predictable [8, 329]. Sometimes it can even lead to the destruction of the
sample, not an improvement.
Enter the concept of the nano-?squeegee.? Through our group?s collaboration with
scientists at NRL, we became aware of a technique that they pioneered to flatten samples
and remove contaminants [255]. An illustration of this method can be seen in Figure
63
3.7. In essence, the method is very simple: one simply uses an AFM in contact mode to
line-by-line push the material out from between the layers using the force of the tip.
There are a few experimental parameters that are important to success. Generally
speaking, ?soft? AFM tips with a low spring constant are generally used for contact mode
scans. Since the tip is deflected from the surface in this mode, springier tips help protect
the sample and keep the tip from cracking under the pressure. Rosenberger et. al found
that it takes about 100 nN of force to flatten a monolayer of a TMD, while flattening an
encapsulated stack required more than 1000 nN of force. Since flattening a stack requires
a significant amount of force non-traditional, stiffer tips (spring constant (k) of about
40 N/m) proved to be neccessary in order to perform adequate flattening. Additionally,
the radius of an AFM tip is nominally 10 nm. In order to ensure the overlap between
subsequent passes with the AFM tip, the resolution of the scan needs to be about the
same. Otherwise, you will leave space in between each pass and have pocketed lines of
residue instead of removing it out of the edges of the sample altogether.
Since the development of this technique by the group at NRL, other researchers
have adopted this technique as well. A recent study suggests that the process can introduce
charge traps that lead to dominant trion features even at zero bias [114]. They also found
that the addition of a few-layer graphite cap (gate) on the surface of the hBN during the
squeegee process allows these charge traps to disperse and the dominant trion feature at
zero bias disappears. It should be noted that all samples made in our collaboration with
NRL featured a graphite gate due to experimental design needs that allow us to control
the strength of charged features.
64
Figure 3.8: EBL process for defining electrical contacts on a sample.
3.4 Lithographic Processing
Once the 2D material has been encapsulated in hBN and/or gated, it is common to
perform some sort of lithographic processing like etching or making electrical contact.
Much like a snowflake, each stack is unique due to the random nature of the constituent
flakes. This makes a more sophisticated technique, electron-beam lithography (EBL), a
more practical choice for this process over, say, photolithography since it allows us to
tweak each pattern to fit the sample without having to send out for a new mask.
3.4.1 Electron-Beam Lithography
The premise of EBL is similar to photolithography, but due to the fact that electrons
have a much smaller wavelength than light, the user is able to achieve much finer control
over the features they pattern. Like photolithography, the patterning process hinges on the
use of an energy sensitive polymer (in our case, we use polymethyl methacrylate (PMMA)
950A4) that is spin-coated in a uniform layer onto the sample. After the polymer is
65
cured to remove an adequate amount of solvent (per the instructions provided by the
manufacturer), the sample is loaded into an EBL machine. These machines operate
similarly to a scanning electron microscope (SEM), except they are also outfitted with
an electron gun capable of delivering a certain ?dose? of current (?C/cm2) to the sample
selectively through use of a finely calibrated stage that can move in x and y. This patterning
is controlled through use of a CAD file. The electrons delivered to the polymer break up
the bonds, but only where they interact. After the bonds are broken, these areas will
preferentially wash away in a chemical referred to as a developer. Resists that work
this way leave a positive image of what you have ?written? with EBL down to the bare
substrate, with the rest of the sample remaining covered in polymer until such time as it is
removed with a different solvent; they are aptly named positive resists. It is also possible
to purchase a negative resist, where the area written remains after development instead.
Figure 3.8 shows the total EBL process for defining electrical contacts. The example
illustrated is for direct contact to a TMD that has the edges exposed after encapsulation.
First, you begin with a stack of 2D materials and then spin on a thin coat of PMMA and
cure the PMMA per manufacturer instructions. Next, the correct e-beam dose is delivered
in the pattern of the contacts, and then developed to remove the exposed PMMA all the
way down to the substrate. A thin metallic film ? typically either Ti/Au or Cr/Au ? is then
deposited onto the sample. Post-deposition, the sample is placed in a solvent to ?liftoff?
the Ti/Au after deposition, leaving behind metal only in the areas that were exposed. This
is the basic process for creating all devices. More details and recipe information can be
found in Ch. 6.3.1.3.
66
Chapter 4: Measuring Light-Matter Interaction
4.1 Overview
The goal of this section is to briefly introduce the experimental measurement techniques
used within this work.
4.2 Optical Measurement Techniques
Throughout this work, we utilized optical measurements. Though there are merits
to performing transport measurements, in our system there are distinct advantages to
using optical techniques. TMDs tend to have poor mobility (? 100-1000 cm2V?1s?1,
in contrast to graphene which has an upward limit of ? 200,000 cm2V?1s?1 when free-
standing), which makes transport difficult. Poor mobility does not prevent light from
creating optical excitations though, which circumvents that signal issue in transport. Optical
measurements inherently give us the ability to selectively interact with the two valleys by
choosing the polarization of the light used for excitation. Comparing the response from
the two valleys gives us information about carrier dynamics in state formation, leaving
a rich area of physics to explore in this manner. In the following subsections, a we?ll
provide brief overview of the measurement techniques used in this work as a primer for
67
Figure 4.1: Panel (A) shows a diagram representing the energetics of the
photoluminescence (PL) process and panel (B) shows a typical PL emission spectra for
WSe2 with labels denoting the different exciton species emission origins.
understanding later experimental results in Ch. 6.
4.2.1 Photoluminescence Measurements
The most common measurement technique we use is photoluminescence (PL). The
PL process is outlined in Figure 4.1 (A). A photon is used to excite an electron from the
valence band and create a quasiparticle (e.g. exciton or polaron). The electron then has
a chance to non-radiatively decay into other, lower energy configurations. Eventually,
the electron drops back into the valence band and radiatively recombines with the hole it
left behind. These emitted photons can be counted as a function of their energy with
a spectrometer and produce a spectrum like that in Figure 4.1 (B). Figure 4.1 (B) is
real data from one of our WSe2 samples under selective excitation and measurements
of the ?K valleys. We see here the neutral exciton-like state, X0, and the singlet and
68
triplet trion-like states, Xs /Xt? ? , which are prominent since we have electrostatically
electron doped the system (Vg = 0.4 V). A nice feature of this technique is the non-
radiative relaxation inherently involved makes it easier to observe exciton species with
lower oscillator strengths that can be obscured in techniques like reflectivity.
4.2.2 Photoluminescence Excitation Measurements
As the name would suggest, photoluminescence excitation (PLE) is very similar to
PL. Sometimes referred to as resonant PL, the main difference between the techniques is
instead of using just one excitation energy (PL), we collect many spectra as a function
of a range of excitation energies (PLE). In this scheme, the photon is used to excite an
electron from the valence band to a higher energy state. There, it undergoes non-radiative
relaxation from an higher energy state (e.g. X2s0 ) to a n=1 state (e.g. X0, X
t s
?, or X?)
and then radiatively recombines with the n = 1 emission channels monitored. Fig. 4.2(A)
provides an illustration of this process. We can indirectly monitor higher energy states that
have a lower radiative oscillator strength, which is what makes this technique particularly
useful. Unlike reflectivity, because it involves relaxation it can also give information on
decay branching ratio from higher n states.
Since these plots can be less intuitive to read, an illustration of how this works is
shown in Figure 4.2 (B). Here, the cooler color indicates low intensity and red indicates
the highest intensity of emission. Generally speaking, when the excitation energy is
resonant with the optical gap of an exciton species, this state becomes very highly populated.
If a non-radiative decay channel exists to a lower, monitored state we can expect that
69
Figure 4.2: Panel (A) shows the diagram representing the energetics of the
photoluminescence excitation (PLE) process. It is similar to PL, but the excitation
wavelength is varied in this case. Panel (B) shows a diagram of the expected PLE plot,
indicating emission to the lowest state from a resonantly excited state, and panel (C)
shows a typical PLE emission spectra for WSe2 in the 2s energy region.
70
the increase in population in the resonantly excited state will lead to an increase in the
population of the monitored state. Thus, when we resonantly excite a state, the intensity
from the monitored emission state increases dramatically (see the red spot in Figure
4.2 (B)). With an understanding of how this process works in theory, we can see a real
example of this process in WSe2 in Figure 4.2 (C). Here, we are exciting in the 2s energy
region and monitoring 1s states. The resonantly excited states (X2s,X3s0 0 ), monitored decay
channels (X , Xt , or Xs0 ? ?), and Raman modes are all marked. This spectra is related to
the work in Ch. 6 and will be discussed in greater detail there. It serves as an excellent
illustration of the power of this technique for studying higher n states.
4.3 Experimental ?Knobs?
Below is an outline of the experimental ?knobs? in the system used in this work.
That is, what can we change easily in an experiment to investigate the phenomena of
interest and what effect does that ?knob? have. Note there are other ?knobs?, such as
temperature dependence and spatial resolution, but they are not used in this body of work.
4.3.1 Light Polarization
This is the simplest parameter in our system. If we recall Ch. 2.2, one of the
interesting properties of TMDs is that the populations of the two valleys are preferentially
accessed with LCPL and RCPL. This allows us to access the population of one valley at
a time and contrast their behavior under other applied conditions. Polarization selective
excitation is easily achievable in the lab by passing linearly polarized laser light through a
71
Figure 4.3: PL measurement of the gate response in a representative sample of WSe2.
quarter-wave plate; theoretically the degree of polarization should be 100%, but typically
lab experiments realize ?20-70%. This, is generally enough to extract valley dependent
behavior. .
4.3.2 Electrostatic Gating
Electrostatic gating allows us to tune the Fermi level in our sample and investigate
different doping regimes. Note that frequently the term ?doping? is used in the semiconductor
field to refer to atomic substitution in the lattice to increase the number of donor/acceptor
state (depending on the dopant). However, here we will use it to refer to excess electrons
and hole introduced through electrostatic gating. In Ch. 2.3 we discussed the formation
of both single-particle (trion) and many-body (attractive/repulsive polaron) states that are
72
the result of excess charge in the system. With the use of an electrostatic gate, we can
tune the sample to the ?neutral? region (intrinsic doping of electrons/holes only) and
observe the exciton strongly. Or, we can pull the Fermi level into the conduction (valence)
band and introduce excess electrons (holes) into the system and promote the formation
of trions/polarons. This type of gating allows us to focus our study on either neutral or
charged exciton species by promoting their respective formation by varying the Fermi
level.
In order to make sure that the sample is functioning correctly and that this electrostatic
gating process is working, one of the first measurements is a PL sweep as a function of
the back gate voltage. A representative spectra of what we are looking to see in a high-
quality, working WSe2 sample is shown in Figure 4.3. In this spectra, all the data has
been normalized by the 1s exciton peak at Vg = 0 V, and the white dashed lines divide
the spectra into three regions: neutral, n-doped, and p-doped. We will discuss this type of
electrostatic characterization in much greater detail in Ch. 6.
4.3.3 Magnetic Field (Zeeman Effect)
The Zeeman effect, discovered by Pieter Zeeman in the late 1890s, describes the
splitting of atomic transitions based on their angular momentum when a magnetic field
is applied [343]. When we performs spectroscopy on a material in the presence of no
magnetic field, all of the fine structure is degenerate and hidden (Figure 4.4(A)). However,
Zeeman found that when a magnetic field is applied, there were more optically bright
transitions observed. It would take another couple of decades and the advent of quantum
73
Figure 4.4: Visual representation of the effect of applying a magnetic field and observing
the splitting of different angular momentum states (the Zeeman effect).
mechanics to understand the origin of the splitting. This splitting occurs because when
electrons move around their nuclei in an atom they generate a magnetic field with a
magnetic moment. The magnitude of this magnetic moment is determined by which
orbitals the electrons originate from. When an external magnetic field is applied, the field
does work on this moment to align it. This work changes the energy of the affected state
as a function of the field and the change in energy is directly proportional to how much
work it takes to perform the alignment.
Importantly for us, the application of a magnetic field is a friend to those performing
spectroscopy. The splitting of states by magnetic field allows one to observe energetically
bunched states more easily, and the evolution with the amount of field applied can tell us
something of the character of the state itself and even quantify the degree of many-body
interaction in the system. In the next chapter, we will dived heavily into the specifics of
this and look at a comparison of the effect in 3D with the unusual case of valley dependent
74
Zeeman splitting in TMDs. The phenomenon is discussed further in Ch. 5.
4.3.4 A Note on Measurement Energies
Many of the ?knobs? discussed can result in a change in the emission energy of
different excitonic states in TMDs. In the literature, it is common to see the absolute
energy of any particular state (ex. exciton, dark exciton, etc.) has a wide variety of
reported values. This is simply a function of different sample parameters, like dielectric
environment and stress. The most reliable metric is the relative spacing in energy between
features. For example, the dark exciton and bright exciton are separated in energy by the
spin-splitting in the conduction band, which is a more fundamental quantity. Thus, in our
own work, we rely on the relative position of features to help us determine their origin
more than the absolute value for the energy of any given signal.
4.4 Measurement Specifications
A full diagram of the dilution unit with the optical setup is provided in Fig. 4.5.
There were two different excitation sources used throughout the main manuscript and
the supplemental information. For PLE measurements in the main text, a dye laser
with 4-(Dicyanomethylene)-2-methyl-6-(4-dimethylaminostyryl)-4H-pyran (DCM) dye
and pumped with a 2.33 eV (532 nm) green laser was used to access a dynamic range
from 1.92-1.75 eV. The dye laser was also utilized for PL measurements of the 1s state,
while for PL measurements accessing both the n=2 and ground states a 2.33 eV green
diode laser was used. As with the PLE measurements in the main text, all supplemental
75
Figure 4.5: Optical diagram depicting the excitation and emission collection scheme used
for photoluminescence and photoluminescence excitation measurements. Figure is taken
from our own work [266].
measurements were also performed in the confocal configuration and Faraday geometry.
The system is equipped with a 12 T superconducting magnet and the PL/PLE
measurements were performed in Faraday geometry. We estimate that with residual
heating from the laser and magnet, the ambient temperature of the sample is <300 mK.
76
Chapter 5: Understanding Exciton Dynamics Under Applied Magnetic
Field in Transition Metal Dichalcogenides
This chapter serves as a reference for understanding the effect of magnetic field on
exciton fine structure in TMDs. We will focus mainly on the Zeeman effect, which is
important for understanding the results of Ch. 6, but will also touch on some other high
field effects in TMDs. First we will look at the Zeeman effect in the hydrogen atom,
since there are many similarities between excitons and the hydrogen atom. Next we will
review the single-particle model for predicting g-factor in TMDs. Then we will look at
the effects of many-body correlation on the g-factor of polaronic particles. Finally we
will briefly touch on some other magnetic field effects that can compete with the Zeeman
effect at higher fields and discuss whether we need to consider them in our work.
5.1 The Hydrogen Atom
5.1.1 The (Normal) Zeeman Effect
The Zeeman effect is conceptually simple: energy bands in both atoms and crystals
have magnetic moments associated with them that arise from the magnetic field produced
by electrons moving around their respective nuclei at high speeds. When an external
77
magnetic field is applied, that field will do work on the atom to try to align its magnetic
moment with the field. This work results in an energy shift of the underlying bands as
the magnetic field is increased. It follows that this energy shift scales with the magnetic
moment since larger magnetic moments will require more work to align with the applied
field. As we recall from Ch. 2.3, an exciton is similar in many ways to the hydrogen atom.
Thus, understanding the Zeeman effect in hydrogen is an excellent starting point for our
discussion.
We begin by constructing the Hamiltonian that describes a single hydrogen atom
subject to a perpendicular magnetic field. The magnetic dipole moment for the electron
spin and orbital motion are, respectively,
e
??s = ? S? (5.1a)
me
and
??l = ?
e
L?. (5.1b)
2me
The Zeeman Hamiltonian is then,
H?Z = ?(??l + ??s) ? B?ext
(5.2)
e
= (L?+ 2S?) ? B?ext.
2m
Recall that the general Hamiltonian for the hydrogen atom is,
H?H = H?Coulomb + ?H?fine?structure. (5.3)
78
Here, the term fine structure refers to interactions such as spin-orbit coupling and relativistic
corrections. The spin-orbit coupling is of particular interest because it arises from the
effective internal magnetic field that the electron sees from the nucleus. This implies that
we actually have two magnetic fields at play: the internal magnetic field from spin-orbit
interaction and the external applied field. How we proceed in our treatment depends
on which field is dominant. Here, we will treat the weak-field Zeeman effect where
Bext ?Bint, since that is most relevant to the observed Zeeman effect in TMDs. At the
end of this section, we will briefly discuss some general high field effects.
Since we assume that we are in the weak-field regime, we can evaluate the Zeeman
Hamiltonian using first order perturbation theory. Recall, that under first order perturbation
theory the energy correction is given as,
E(1) = ?? |Hperturb |?? , (5.4)
where E(1) is the first-order energy correction to the eigenstate |?? resulting from the
perturbation to the system. Assuming that our applied field is along z?, for hydrogen we
can rewrite Eqn. (5.4) as,
? ?
(1) eB ? ?
En,l,j,m = n, l, j,m
?
j Lz + 2S ?z n, l?, j,m?j . (5.5)j 2m
Here n is the principle quantum number, l is the azimuthal (angular momentum) quantum
number, j is the total angular momentum in the system, and mj is the projection of j along
our axis of choices (z?). These are the ?good? quantum numbers in our system. Note that
79
Figure 5.1: Detailed illustration of the Zeeman effect in hydrogen for the ground state.
because of the assumed presence of spin-orbit interaction, the magnetic quantum number
ml and the spin quantum number ms are not conserved, but their combined total mj is.
After some algebraic massaging we find the solution,
(1) e?
En,l,j,m = Bgj J(l)mJ2me (5.6)
= ?BBgJ(l)mJ.
Here, ?B = 0.05788 meV/T is the Bohr magneton and gJ(l) is the Lande? g-factor. The
Lande? g-factor will be discussed in detail later in this chapter, but it essentially represents
how much work the magnetic field must do in order to align the magnetic moment of
the energy level with the direction of the applied field. Full derivations of the energy
correction due to the weak field Zeeman effect in hydrogen and of the Bohr magneton can
be found in Appendix B. Figure 5.1 shows a visual representation of the consequences of
80
Eqn. (5.6) for the ground state of hydrogen. In panel (A) we see the transitions between
the 1s and 2p states. If no field is applied, the three energy levels with different ml (and
hence different mj) for the 2p are degenerate with energy E = E?10, E00, E10. If we
apply a magnetic field, the Zeeman effect breaks the energetic degeneracy and we can
observe three distinct levels each with different ml, which is shown in panel (B).
5.1.2 Lande? g-factor
The Lande? g-factor (hereafter referred to just as g-factor) is an important quantity
of interest in this chapter, as it can provide a fingerprint for different exciton species and
be used as a barometer for many-body interaction. In an atomic system, like the hydrogen
atom, the g-factor arises from the internal configuration of the angular momentum and is
written as,
j(j + 1)? l(l + 1) + 3
gJ = 1 +
4 . (5.7)
2j(j + 1)
A full derivation for the g-factor can be found in Appendix B.
5.2 The Zeeman Effect in Transition Metal Dichalcogenides
Now that we have a basic understanding of the Zeeman effect, we need to look at
how it manifests in our 2D system. In this section, we will explore the single-particle
expectation, g-factor enhancement in the many-body regime, and the high field limit.
81
5.2.1 Single Particle Model
We begin our discussion of the Zeeman effect in TMDs in the single-particle regime,
focusing on the exciton. In contrast to the hydrogen atom, excitons exist in a crystal lattice
and the resulting Zeeman splitting is from the magnetic moment contributions from the
underlying band structure. Again, we can treat the Zeeman effect as a perturbation in our
system and look at the change in energy due to applied field in each valley,
?E?K = ?(??c ? ??v) ? B? (5.8)
?c/v are the magnetic moments of the conduction and valence bands the electron and hole
reside in, respectively. In TMDs, the total Zeeman splitting is defined as the difference in
splitting between the two valleys (we assume a field along z?):,
?EZ = E+K ? E?K = ?[(?+K,c ? ?+K,v)? (??K,c ? ??K,v)]Bz. (5.9)
?c/v each have underlying contributions from ?s and ?l so it is helpful to evaluate Eqn. 5.9
in this context. A visual representation of the contributions in each valley is shown in
Fig. 5.2. To evaluate these components, we will first use an ?atomic? perspective that
relies on basic spin/orbital band configurations.
Beginning with the spin component ?s, we see in Fig. 5.2 that for a bright transition
in each valley, both the valence and conduction bands have the same spin associated with
them. This means that ?s will shift the energy of each band by the same amount, resulting
in ?s,c ? ?s,v = 0 for the bright exciton in each valley.
82
Figure 5.2: Zeeman splitting for excitons in TMDs in the single-particle model. Here,
only the bands for the A series bright exciton are shown. Note that if the applied field was
negative, we would expect the K-valley to blue shift and the -K-valley to red shift instead.
Next we look at the orbital component ?l. The value ?l for the valence and conduction
bands can be estimated based on which bonding orbitals contribute to the formation of the
band. For WSe2, Liu et al. showed that the K-valley conduction band is made primarily
of dz2 orbitals which have ml = 0 ?B, while the valence band is dominated by dxy + dx2+y2
orbitals, which have ml = ?2 ?B [186]. Using this as an estimate, they find that for the
K-valley, ml = -2 ?B. K/-K are a time-reversed pair (?K,c/v = ???K,c/v), which implies
that ml = +2 ?B in the -K valley. Putting all the contributions together, we get
?EZ = ?((?l,+K,c ? ?l,+K,v)? (?l,?K,c ? ?l,?K,v))Bz
= ?((0 ? (5.10)B ??2 ?B)? (0 ?B ? 2 ?B))
= ?4 ?BBz.
Thus, the expected g-factor in the single-particle model for the bright exciton is g=-4.
83
5.2.1.1 Intercellular Component Correction
Experimentally, many groups observed small deviations from the expected g=-4
value within their work [2, 12, 140, 172, 193, 289, 292] in both 1s and higher Rydberg
states. It was proposed throughout many of these works that the discrepancy could be
attributed to orbital magnetic moment that arises due to the finite Berry curvature in the
system [2, 44, 140, 172, 193, 289, 292]. This term is generally called the intercellular or
valley contribution. We can calculate the valley orbital magnetic moment using a multi-
band (>2) tight-binding Hamiltonian [186, 289] using the following relation:
m ?[?u ?Hn,k| |u ? ? ?u | ?H |u ? ]?k j,k j,k ?k n,k
Ln(k) = i
x y ? c.c. . (5.11)
? E ? E
j=? j nn
This generally leads to ?10% correction (? 0.4 ?B) resulting in a g ? ?4.4 ?B estimate
for 1s excitons, which accounted for the small deviations observed [289].
5.2.1.2 Application of the Single-Particle Model to Other Exciton Species
and Break Down of the ?Atomic? Picture of Interaction
Due to its apparent success, several groups attempted to increase the accuracy of
the additive model by explicitly determining the spin, orbital, and valley contribution.
Attempts were made via both experiment [140,248] or ab initio calculations [63,80,324,
330] for bright excitons and extend its treatment to other exciton species. These works
has been rather successful in predicting g-factor values for neutral exciton species. For
example, the predictions for the spin-dark excitons are |gXD| ? 8?10?B and momentum-0
84
dark excitons would have |g D(inter)| ? 10 ? 13.6 ?B [80, 140]. This correlates very wellX0
with experimental results which have found that |gXD| ? 9.1 ? 9.9 ?B [108, 170, 184,0
247] and |g D(inter)| ? 12.1 ? 12.6 ?B [80, 108, 170], respectively. For charged particlesX0
like trions, |gX?| ? 4 ?B is predicted since the excess electron contributes equally to
both the initial and final state resulting a similar g to the exciton. A similar correction
can be made to account for the intercellular contribution, although the formulation is
slightly different (see Ch. 6. This prediction has proven to be fairly accurate at low carrier
densities with |gX? | ? 3.8 ? 4.4 ?B observed experimentally [140, 170, 184]. We will
discuss the influence of many-body interaction at higher carrier densities on g-factor in
the next section.
Though this additive method has been rather successful for many exciton species,
both the experimental work and ab initio calculation revealed something startling: the
early success of the ?atomic? model ? where the majority orbital contributions are used to
estimate ?l ? turned out to be correct purely by coincidence. For example, if we were to
explicitly sum ?s, ?l, and ?v in the ?atomic? picture, we would find that ?+K,c = +3.5?B
and ?+K,v = +5.5?B. However when these contributions were measured experimentally,
Robert et al. found that ?+K,c = +3.84 ?B and ?+K,v = +6.1 ?B [248]. This is in
excellent agreement with the ab initio calculations that find ?+K,c = +3.81 ? 3.97 ?B
and ?+K,v = +5.81 ? 5.91 ?B [63, 80, 324, 330] and still preserves that prediction of
gX0 ? 4.4 ?B that is seen experimentally throughout the literature.
85
5.2.2 Many-Body Regime Enhancement of g-factor
Despite the significant work done to explain the results of valley-dependent Zeeman
measurements within the single-particle model, there were still numerous reports of g-
factor values that diverged from these expectations. In particular, Mo-based TMDs [15,
94, 118, 133, 282] and charged exciton species [2, 289, 319] reported the most variation
with values from |g| = 1.57 ? 18 ?B reported for bright excitons and trions (polarons).
In early TMD work, electrostatic control of the carrier concentration through a top- or
back-gate was not a standard technique, and most of the experiments in the literature
were performed using either bare monolayer on SiO2 or a monolayer with simple hBN
encapsulation. In those cases, the carrier concentration in the sample was fixed and non-
deterministic: the as-grown TMD monolayers can possess an intrinsic hole- or electron-
heavy concentration. Other parts of the fabrication process ? such as squeegeeing without
a graphite contact, or simply resting on bare SiO2 (see Ch. 3) ? can also result in additional,
uncontrolled electrostatic doping. In other semiconductor systems, it is well known that
the g-factor observed can be manipulated through control of the carrier concentration
[73, 238, 244, 270, 271, 274]. Thus it was not a stretch to attribute the range of observed
g-factors to variation in the carrier concentration from experiment-to-experiment.
The advent of controlled electrostatic gating in TMDs has allowed researchers to
explicitly investigate the relationship between carrier concentration and g-factor. Based
on these studies it has become clear that the preference for these systems to minimize
their energy by inducing carrier polarization as a result of strong many-body interaction
is responsible for the variety of g-factors observed. To fully understand why many-body
86
interaction leads to this observed variation in g-factor, we will first discuss how we might
quantify the strength of many-body interaction and then show why this interaction can
lead to carrier polarization.
The first step to quantifying many-body interaction is to understand how the number
of carriers controls the Fermi level in a system. Throughout this discussion, we will focus
on a specific illustration in WSe2 since this is where we do our own measurements, but it
can be similarly applied to other TMDs. We begin by estimating the Fermi level via the
following relation,
n n
EF = = . (5.12)
?(E) m?/(??2)
Here, ?(E) = m?/(??2) is the density of available states when the Fermi level is below
the upper conduction band and we take the effective mass of the electron m?e = 0.4 me
[186], and the effective mass of the hole to be m?h = 0.36 me [141]. When we pass
into the upper conduction band, we go from a degenerate to non-degenerate state and
the density of states becomes ?(E) = m?/(2??2). Using these physical parameters, we
generate Fig. 5.3(A) and indicate the splitting of the upper and lower conduction band for
our system as ?EWSe2CB = 38 meV [186]. The density required to make this crossing is
indicated with a vertical line marked n ? 6.4 ? 1012 cm?20 . Note that in all cases, the
Fermi energy is measured from the bottom of the lower conduction band at B=0T.
Next, we calculate the Wigner-Seitz radius rs, which is a dimensionless quantity
that is commonly used to gauge the strength of many-body effects in a system [58, 251,
319]. Defined as
1 me2
rs = ? , (5.13)
?n ??2
87
Figure 5.3: Illustration of the relationship between carrier density, Fermi energy, Wigner-
Seitz radius, and exchange interaction strength in TMDs. (A) Calculated Fermi energy
from carrier dependence with gate voltage. The Fermi energy needed to cross from the
lower to upper conduction band is marked, as is the associated density for that transition.
(B) Extracted Wigner-Seitz radius for the sample as a function of carrier density. The
threshold for crossing into the upper conduction band is marked. (C) The calculated
exchange parameter, highlighting the effect of doubling the spin-valley degeneracy upon
crossing into the upper conduction band.
the Wigner-Seitz radius can be interpreted as the ratio between average Coulomb potential
energy of the carriers and their Fermi energy [58]. It is used to separate carriers into two
regimes: (I) rs > 1 (the dilute and strongly interacting regime) and (II) rs < 1 (the dense
and weakly interacting regime). Fig. 5.3(B) shows the corresponding Wigner-Seitz radius
as a function of carrier density. As we can see, the Wigner-Seitz radius is greater than one
even at carrier densities approaching 1013 cm?2, indicating that many-body interaction is
strongly favored even in the presence of strong carrier doping.
The exchange parameter, rx, has been proposed as a related metric to the Wigner-
Seitz radius as it characterizes the ratio of the average exchange energy to the average
kinetic energy (EF) at a given carrier density [58]. The exchange parameter is an important
quantity because the price that the system has to pay to become fully valley polarized
must be less than the exchange energy gained in order to make the polarization to be
energetically favorable [15]. Thus when rx > 1, full polarization of the carrier population
88
Figure 5.4: Stages of carrier polarization under applied magnetic field (A) B = 0T and
there are an equal number of carriers in each valley, (B) 0<B<BC and there is a carrier
population imbalance between the two valleys, but both valleys have excess carriers, (C)
B>BC and all of the excess carriers now resides in just one of the two valleys. If we
reversed the field, the occupation sequence would flip valleys as |B| increased. In all
panels, the dashed horizontal line indicates where the Fermi energy EF is in the sample.
is favorable to minimize the energy of the collective ensemble of carriers. We write rx as
?
rx = lslvrs. (5.1)
Here, lslv characterizes the spin-valley degeneracy. When we cross into the upper conduction
band at n0, we have lslv = 2 ? lslv = 4, which results in an abrupt change in the
exchange parameter. We illustrate where this transition would occur in our model system
in Fig. 5.3(C) and see the resulting discontinuity as rx increases again when we begin to
dope into the upper conduction band. We also see that rx > 1 even at extremely high
doping, indicating that valley polarization is highly favorable in the system.
To visualize how this interaction induced valley polarization occurs in the system,
we recall the picture of polaron formation from Ch. 2.3.2.3. At B = 0 T, Fig. 5.4(A) shows
89
that the population of excess carriers in each valley is the same ? as indicated with the
horizontal dashed line representing EF. The spin-polarization is defined as
N ?N
?s =
? ? ,
N?+N?
and we have ?s = 0 when B = 0 T. Once we turn on a positive field, we know from the
previous section that our bands offset with the field. In our previous discussion though,
we had no excess carriers to worry about. Here, in 5.4(B), we see that the red(blue) shift
of the +K(-K) valley results in a population imbalance and a net spin polarization. If
we keep the Fermi level fixed, as we have in Fig. 5.4, increasing the field strength will
result in progressively more spin-polarization until we reach the critical field BC. At this
point, we have shifted the bands so much the system is fully spin polarized, ?s = +1 as
in Fig. 5.4(C). Intuitively, we can see that BC depends on the carrier concentration and
the g-factor in the system since the polarization condition requires that EZ > EF. If we
reverse the field, the valleys will shift in the opposite direction and at BC we will end up
with ?s = ?1 instead.
Since it is strongly energetically favorable to fully polarize the excess carriers into
one valley, it can be reasoned that an enhanced, carrier density dependent g-factor would
be the most efficient way for the system to force the lower energy configuration at a small
applied field. Thus, what is observed experimentally is a g ? rs. Experimentally, the
earliest signature of this effect in TMDs was demonstrated by Back et al. in 2017 where
they showed a strongly enhanced g-factor of ? 18 ?B in MoSe2 when the system was
fully valley polarized [15]. Following this discovery, both Wang et al. [319] and Lin et al.
[176] showed independently that they could correlate g-factor strength with the Wigner-
Seitz radius in WSe2 and MoS2, respectively. More recently, Klein et al. performed an
incredibly thorough study of all exciton species in MoS2 and showed that each species
90
exhibited smoothly varying g-factor with carrier concentration that was correlated with
the degree of polarization in the system. This implies that all species of excitons display
many-body physics [133].
In this scenario described above, instead of seeing a linear Zeeman effect, we would
see something that looks much more like a net magnetization curve for a paramagnet
where we asymptotically approach full polarization. Thus, the net spin imbalance in this
case this give by the,
SZ(B) = ?sz[?BJ(x) ( ) ( )] (5.14)
? 2sz + 1 2sz + 1 ? 1 1= sz coth x coth x ,
2sz 2sz 2sz 2sz
where sz = 1/2 and x = g?BszB and the total magnetization is M ? (N ?Nk T ? ?) ?SZ(B). xB
is explicitly the ratio of the Zeeman energy to the thermal energy, but the thermal energy
is inherently related to the Fermi energy of the excess carriers, so this quantity can be
used to extrapolate the critical field for complete polarization. Within this formulation,
the carrier dependent g-factor is
g?(ne,h, B) = gX0 + gm(ne,h) ? Sz(B), (5.15)
where gX0 is the g-factor of the exciton in the intrinsic regime where the linear Zeeman
effect dominates and gm(ne,h) is the carrier dependent g-factor that is tuned by the strength
of the many-body interaction. In this model, the g-factor is only tuned by the strength of
the many-body interaction and degree of magnetization at a given applied field, so it is
the same for any exciton species that is present at that carrier concentration [133].
91
5.2.3 High Field Considerations: Paschen-Back Effect, Diamagnetic Shift,
and the Quantum Hall Effect
Besides the Zeeman effect, which is dominant at low fields, there are several other
effects that are routinely observed in semiconductors under the application of magnetic
field. Here, we discuss the conditions needed to observe the anomalous Zeeman effect
(the so-called Paschen-Back effect [227]), diamagnetic shift [207], and the quantum Hall
effect (QHE) [155, 308]. We will also discuss whether we would expect those conditions
to be met in our experimental work.
The Paschen-Back effect occurs when the applied magnetic field becomes large
enough that the ?normal? Zeeman splitting would be on the order of the splitting due
to spin-orbit coupling in the system. At this point, the energetic splitting due to the
external magnetic field becomes dominant and the spin-orbit interaction is treated as the
perturbation in the system. Since this results in the reordering of the fine structure in the
system, it is treated as a distinct phenomenon from the Zeeman effect and is sometimes
also referred to as the ?anomalous? Zeeman effect. We can recall from Ch. 2.2.2 that the
spin-orbit interaction in the conduction band in our system is on the order of 30 meV. For
a typical lab experiment, where we achieve ? |10| T field at most, this would equate to
a g-factor of 3 meV/T? 50 ?B in order for the spin-orbit interaction to be comparable
to splitting due to the Zeeman effect. Since g ? 50 ?B is outlandish, even with the
enhancement due to many-body effects, we can safely rule out the Paschen-Back effect
from being relevant in our system.
When a magnetic field is applied, we recall that charged particles will undergo
92
?
cyclotron motion. We extract a quantity called the magnetic length lB = ? thateB
describes the scale of these cyclotron orbits. When the magnetic field is small-to-intermediate,
t(hese)orbits are large and span many lattice sites/exciton Bohr radii in a solid, i.e. ? =2
aB ? 1. In this limit, the diamagnetic shift ? which is an additional, higher-order
lB
correction to the Zeeman shift ? can be observed. Generally, the diamagnetic shift is only
significant in its effect in larger radius excitons and high fields. We can see why this is
true in the following,
e2
?Edia = ?r2 ?B2 = ?B2. (5.16)
8m ?r
Here, m?r is the reduce mass of the exciton and ?r
2
?? is the expected radial component and
?
rrms = ?r2?? = 8mr?/e [207, 292]. In WSe2, the parabolic bending of the Zeeman
splitting resulting from the diamagnetic shift is only strongly observed at B> 10 T for
n = 1, 2 exciton species, which is beyond our experimental range. Additionally, since it
affects both valleys equally, when we take the difference of the energy splitting between
the valleys to calculate the Zeeman shift the influence of the diamagnetic shift is removed
as well [293].
As the field increases, the magnetic length decreases and can become comparable
to the Bohr radius of the material, i.e. ? ? 1. When this is true, the QHE occurs and
overtakes typical Zeeman splitting/diamagnetic correction. In the quantum Hall regime,
carriers are forced into a level scheme similar to that of a quantum harmonic oscillator and
which we call Landau levels (LLs). The field regime at which this becomes the dominant
phenomenon can be approximated by looking at the relevant length scales in the system.
93
If we take l ? 25?nmB and aB ? 1.5 nm:B
( )2
25?nm 25nm= 1.5 nm ? B = = 277 T (5.15)
B 1.5nm
Based on this comparison, we would expect the QHE to only be relevant in TMDs at
extremely high fields (unachievable in a lab). However, the QHE has been observed
optically and through transport in several TMD systems [98,166,176,185,231,282,319].
The strong many-body interaction and resulting valley polarization is thought to play
a large role in observing this effect at much lower fields than would traditionally be
expected and has also allowed researchers to experimentally determine the LL ordering
through carrier-dependent measurements [166,176]. However, at fields less than 10 T, the
spacing of the LLs is sufficiently small that the Zeeman effect dominates as the observable
effect [166]. There have been a couple of reports near B ? 10 T, but only in systems
with large numbers of excess carriers n ? 7 ? 1012 cm?2e . In our system the carrier
concentration is more than an order of magnitude smaller (see Ch. 6), so we can safely
rule out the QHE as a complicating factor in our g-factor measurements.
94
Chapter 6: Magneto-Optical Measurements of the Negatively Charged
2s Exciton in WSe2
6.1 Notes
This chapter is largely taken verbatim from our paper ?Magneto-Optical Measurements
of the Negatively Charged 2s Exciton in WSe2? J.C.Sell, J.R.Vannucci, D. Suarez, B. Cao,
D. Session, H.-J. Chuang, K.M. McCreary, M.R. Rosenberger, B.T. Jonker, S. Mittal, M.
Hafezi, arXiv:2202.06415 (2022) [266]. It is currently under review for peer-reviewed
publication. Small alterations have been made to fit references from earlier sections to
this chapter.
As discussed in Chs. 2 and 5, the trion peaks are sometimes interpreted as polarons
depending on the carrier concentration in the system [15, 275]. This remains an active
area of research at the time of this thesis, with work showing that polaron and trion
interpretations converge in their predictions at low-to-intermediate carrier density [91,
361]. These results highlight the difficulty in drawing a dividing line between the two
interpretations, and for simplicity we refer to them as charged states throughout this
chapter.
95
6.2 Manuscript
6.2.1 Abstract
Monolayer transition metal dichalcogenides host a variety of optically excited quasiparticles
species that stem from two-dimensional confinement combined with relatively large carrier
effective masses and reduced dielectric screening. The magnetic response of these quasiparticles
gives information on their spin and valley configurations, nuanced carrier interactions,
and insight into the underlying band structure. Recently, there have been several reports
of 2s/3s charged excitons in TMDs, but very little is still known about their response to
external magnetic fields. Using photoluminescence excitation spectroscopy, we observe
the presence of the 2s charged exciton and report for the first time its response to an
applied magnetic field. We benchmark this response against the neutral exciton and find
that both the 2s neutral and charged excitons exhibit similar behavior with g-factors of
gX2s=-5.20?0.11 ?B and gX2s=-4.98?0.11 ?B, respectively.0 ?
6.2.2 Introduction
Monolayer semiconductor TMDs have attracted significant attention in the last
decade due to their unique optical properties. Similar to graphene, but with a three-layer
(staggered) honeycomb lattice, TMDs host direct-gap transitions at their ?K valleys and
exhibit circular-dichroism due to their finite Berry curvature [197,198,327]. The reduced
dimensionality of materials in this system, coupled with techniques like hexagonal boron
nitride (hBN) encapsulation, lead to enhanced Coulomb interaction and excitons with
96
large binding energies (EB ? 150? 500 meV) [46, 107, 293].
When there is excess charge present in the system during exciton formation, the
exciton may lower its energy by capturing an electron or hole and form a bound, charged
three-body state referred to as a charged exciton [153, 257]. Charged excitons are a
ubiquitous feature of semiconductors, but are difficult to observe in traditional systems
? like GaAs/AlGaAs quantum wells ? due to their small binding energies (1-2 meV)
[79, 129, 285]. In TMDs, however, both singlet and triplet charged species have been
discovered with EB ? 20 ? 40 meV [53, 191, 196, 257, 289]. In the high carrier density
regime, these resonances have been alternatively interpreted as many-body polaron states
[15, 69, 91, 275].
In analogy to the hydrogen atom, excitons are known to form a Rydberg series
of higher energy states [128]. In TMDs, they have been observed through a variety of
different optical techniques up to principal quantum number n =11 [44, 46, 94, 292, 293,
317]. However, even in the presence of excess charge, a corresponding series for the
charged exciton has remained elusive. Lack of experimental observations of these states
has been thought of analogously to the H? ion, for which there exists no bound excited
state [111]. Recently, there have been a series of compelling reports of metastable 2s/3s
charged excitons in TMDs [10, 92, 182, 309] coupled with theoretical work [77, 242, 273,
332] showing their existence is possible.
The difficulty in observing these higher n states is two-fold: (I) the weak radiative
decay rate of excitons with higher n makes them increasingly dim in typical PL measurements
[105] and (II) even once the state is observed optically, further carrier-density and magnetic
field dependent measurements are needed to correctly identify the exciton species [108,
97
Figure 6.1: (A) Schematic of the hBN encapsulated WSe2 with graphite (Gr) back gate
and electrodes. (B) Optical image of device after full fabrication. The compressed area
in the center indicates the region of the sample that underwent nanosqueegeeing. (C)
Schematic of the PLE process highlighting the higher n states (e.g. X2s0 ) and emission
monitored channels (e.g. X0, Xt s ? ??, or X?). (D) ? ? PLE spectra taken at Vg = 0 V
and sample temperature of <300 mK. The monitored emission channels are marked with
arrows (X0, Xt?, and X
s
?) [16, 181]. Two Raman modes are identified as diagonal dashed
lines (ZO(hBN) and ZO(hBN)+A1g(WSe2)), see SM [119, 121]. The resonances of the
excited states are marked with horizontal dashed lines (X2s, X3s0 0 ) [292].
168, 171, 184, 191, 193, 289].
In this work, we confirm the presence of negatively charged 2s-exciton (X2s? ) in
WSe2 via PLE. In PLE, we monitor the emission from the 1s (lowest energy) exciton
species while the excitation laser?s energy was swept in the energy regime needed to
resonantly probe higher n states. This provides a superior signal-to-noise ratio compared
to PL. Additionally, we report on the response of X2s? to an applied magnetic field. To
the best of our knowledge, these represent the first magneto-optical measurements of a 2s
charged exciton in any TMD system. We measure the valley dependent Zeeman splitting
for both the 2s neutral (X2s0 ) and charged (X
2s
? ) excitons in the carrier density regime in
which they coexist. From this, we extract similar g-factors for X2s/X2s0 ? , gX2s=-5.20?0.110
?B and gX2s=-4.98?0.11 ?B, and discuss the possible physical origins of this result.?
In our experiment, a monolayer of CVD grown WSe2 is encapsulated in hBN along
98
with few-layer graphite (Gr) contacts and bottom gate electrode. Encapsulation was
performed via the wet capillary action method and interlayer contamination was removed
via the nano-squeegee method [255] (see Fig. 6.1(A) for a schematic of the sample and
Fig. 6.1(B) for an image of the final device). The full fabrication details are reported in
the supplemental material (SM). The joint hBN and Gr encapsulation allows for a high-
quality device with electrostatic control over the carriers in the system via the applied gate
voltage Vg [61].
Throughout our work, we utilize PLE to resonantly probe the 2s exciton states.
In PLE, the energy of the input photons is varied and when their energy resonantly
matches a 2s exciton state, electrons are excited from the valence band to form these
excitons (e.g. X2s0 ). There, the 2s excitons undergo non-radiative relaxation to a 1s state
(e.g. the neutral exciton X0) where they radiatively recombine and emit photons. An
illustration of this process is shown in Fig. 6.1(C). For these measurements, the excitation
beam is generated using a dye laser with a dynamic excitation range of 1.77-1.99 eV.
We use a confocal configuration with circular polarization resolution in both excitation
and detection. Throughout the text, we denote the excitation/emission polarization in the
format ?excitation?emission. The sample was placed in a dilution refrigerator equipped with
a 12 T superconducting magnet in a Faraday geometry. We estimate that with residual
heating from the laser and magnet, the ambient temperature of the sample is <300 mK.
Fig. 6.1(D) shows a baseline PLE spectrum taken with ???? (-K-valley selective)
at Vg = 0V and B = 0T. We identify the 2s and 3s neutral Rydberg excitons (X2s0 , X
3s
0 )
by their binding energies [44, 183] and labeled them with white dashed lines at 1.859 eV
and 1.887 eV, respectively. The 1s neutral (X0) exciton?s emission channel and the triplet
99
Figure 6.2: (A) PLE data with increasing ne-doping while monitoring the X0
recombination channel in the -K-valley (????). (B) Waterfall plot of vertical cross-
sections from Vg = 0 - 0.9 V. The integration region is annotated in panel (A). The counts
were summed over the emission width for each excitation energy.
(Xt )/singlet (Xs? ?) charged excitons? emission channels were identified by their binding
energies [16, 173] and PL gate voltage dependence [321] (see SM).
6.2.3 Identification of the Excited Charged State via Gate-Dependent
PLE
Next, we tune Vg to ne-dope the system and look for signs of an emerging charged
2s exciton in our PLE spectra. Fig. 6.2 highlights the results of this while monitoring
the X0 emission channel; Fig. 6.2(A) shows the full PLE spectra at selected Vg, while
Fig. 6.2(B) is the integrated vertical cross-section of the emission spectrum around the X0
signal. The integration region used for all gate voltages is denoted in the Vg = 0.6 V panel
of Fig. 6.2(A) by the vertical dashed lines. As in Fig. 6.1, we identify the resonance at
1.859 eV as X2s0 .
At Vg = 0.3 V, a lower energy resonance begins to emerge at 1.838 eV. We label
this state as the 2s charged exciton X2s? and base this identification on two observations:
100
Figure 6.3: (A) PLE data with increasing ne-doping while monitoring the Xt?
recombination channel (????). (B) As in Fig. 6.2, the waterfall plot corresponds to
vertical cross-sections from Vg = 0 - 0.9 V.
(I) Vg = 0.3 V corresponds to the transition of the sample from charge neutrality to ne-
doped and the emergence of the negatively charged 1s excitons Xt /Xs? ? (see SM for 1s
PL data). The X2s? resonance displays a similar onset at Vg = 0.3 V indicating a similar
negative charge character. (II) When the X2s? resonance first appears at Vg = 0.3 V, we
find that ?E(X2s?X2s) = 21 meV while ?E(X ?Xt = 29 meV and ?E s = 35 meV.0 ? 0 ?) (X0?X?)
This reduction indicates that the 2s charged exciton is less tightly bound than its 1s state
counterpart. This is in accordance with other observations in the literature [?,92,182,309]
and consistent with the fact that Rydberg states display a reduction in relative binding
energy with each increasing n.
Since the 2s charged exciton is expected to be a doublet, as observed for the 1s
charged excitons, the extracted position of X2s? is an average. X
t
? and X
s
? have a narrow
linewidth and a strong intervalley exchange interaction that splits them (? 6 meV [53,
338]) which allows us to spectrally resolve them. However, the broadness of the 2s states
combined with a reduced intervalley exchange energy (theoretically predicted to be ? 1
meV [10, 338]), prevents us from resolving the doublet of the 2s charged exciton. There
101
Figure 6.4: Vertical cross-sections from the Xt? emission channel as a function of field for
(A) (????) and (B) (?+?+) marked with the corresponding peak positions (black dots)
for the X2s and X2s0 ? states from fitting with the dashed line serving as a guide to the eye.
(C) Extracted g-factors for X2s0 and X
2s
? states. The thickness of the fit line in panel (C)
corresponds to the error in the fit.
is, however, indication of the two states in the asymmetric lineshape of the X2s? peak (see
SM).
In Fig. 6.2(B), we see the spectral dependence of X2s 2s0 and X? with carrier density.
As the ne-doping increases with increasing gate voltage, the X2s0 resonance broadens,
decreases in intensity, and spectrally blueshifts. The broadening and loss of spectral
intensity are consistent with more rapid decoherence from interaction with the Fermi
sea. The blueshift results from the competing effects of band gap and binding energy
renormalization due to decreased e??e? and e??h+ interaction from screening by the
Fermi sea [47, 251, 309].
In contrast, X2s? peak grows in intensity and experiences minimal spectral drift
with increased carrier density. In the case of a three-body quasiparticle, one expects a
redshift that is linearly dependent on the charge concentration in the system resulting from
momentum conservation [47, 69, 196]. This competes with the effects of band gap and
binding energy renormalization previously discussed for the neutral excitons that favor
a blueshift [47], and leads to the minimal spectral drift observed. Both the increase in
102
intensity and small spectral shift are consistent with the behavior of 1s and 2s charged
excitons previously observed [47, 251, 309].
Since X2s? emerges in the ne-doped regime, we expect X
t
? and X
s
? to be the most
prominent emission channels for 2s exciton species (see SM). To verify this, we monitor
the Xt? emission channel in a similar manner to X0 and show the results as a function of
Vg in Fig. 6.3 (the results for Xs? can be found in the SM). We confirm that the behavior
(spectral position, shift with gate, etc.) of X2s0 and X
2s
? is independent of the monitored
decay channel.
6.2.4 Valley Zeeman Effect in the Excited Charged State
We turn our attention to extracting the behavior of the X2s0 and X
2s
? with applied
magnetic field. We chose to take the data at Vg = 0.6 V because both the neutral
and charged exciton have similar intensity. Integrated vertical cross-sections of the Xt?
emission channel presented in Fig. 6.4 (A)/(B) show the response of the -K(????) /
+K(?+?+) valleys, respectively, with magnetic field. The extracted peak centers from
fitting are marked with black dots. Applying a magnetic field breaks the time-reversal
symmetry in the system, and results in a red(blue) shift with positive field for the +K(-K)
valley and vice versa with applied negative field [15, 289].
Using the definition for the Zeeman splitting in terms of polarization components,
?E = E?
+?+ ? ? ?Z E? ? = g?BB, we fit a linear model to our data and extract a g-factor
of -5.20?0.11 ?B and -4.98?0.11 ?B for X2s0 and X2s? , respectively. This fit and extracted
difference is shown in Fig. 6.4(C). Results that agreed within experimental error were
103
found for both X2s0 and X
2s
? for a similar analysis of the X
s
? emission channel (see SM).
Frequently, a single-particle model is used to interpret the g-factor for 1s excitons.
In this model, the contributions to the Zeeman splitting are defined as ?EZ = ????B?. The
magnetic moment ?? is composed of additive terms for the orbital and spin contributions
(intracellular components ?O, ?S) along with a correction for the effects of the finite
Berry curvature in the system (intercellular component ?V) [80, 140, 168, 193, 338] in
each relevant band. Within this interpretation, we expect gX0 ? ?4.4 ?B and ?11
?Bg t/s ? 4 ?B (depending on the method used to calculate ?V, and whether the doubletX?
is resolved [181, 191, 289]).
To serve as a reference point between the literature and our 2s results, we also
extracted the g-factors for X0 and
t/s
X? . These values are gX0 = -4.22?0.04 ?B, gXt =?
-4.12?0.04 ?B, and gXs = -3.86?0.05 ?B in our system at Vg=0.6 V. They are consistent?
with the results from the single particle interpretation, but highlight a distinct increase
in our 2s g-factors with respect to the corresponding 1s states. We discuss two possible
contributions to this enhancement.
(I) Enhancement of the g-factor for the 2s neutral exciton has been observed in
magnetic Rydberg measurements in both intrinsic and electrostatically neutral samples
[44, 94, 317]. Since the observation in neutral samples rules out doping effects, the
divergence from gX0 ? ?4.4 ?B has been attributed to enhanced intercellular contributions
arising from the increased k-space localization of the wavefunctions with each subsequent
n [44]. Extending this technique to charged excitons gives an intercellular component
that decreases as the Bohr radius increases. This is compounded by an increased k-space
localization of the charged exciton (see SM). While this model could explain the results
104
for X2s it would underestimate the g-factor for X2s0 ? .
(II) A second possibility is the onset of many-body interaction (polaron picture)
between the excitons and the emerging Fermi sea from electrostatic gating. Many-body
interactions are expected to be very favorable in WSe2 which has a Wigner-Seitz radius
greater than 1 even at extremely high densities [58,319]. The interaction strength will vary
with the Fermi sea?s population and the Bohr radius, and induce Fermi sea polarization.
Carrier dependent enhancement of the g-factor in TMDs has been documented for many
materials/quasiparticles, with the strength of enhancement dictated by the degree of the
induced Fermi sea polarization [15, 133, 171, 319].
In the many-body picture, it has been observed that as doping levels are varied there
is a convergence of the g-factor between competing quasiparticles (e.g. X0 and X?) in
regions in which they coexist. In analogy to the Kondo effect, the impurity (exciton) is
dressed with either an attractive or repulsive interaction with the Fermi sea. As carrier
density increases, the state dressing will become more similar for all exciton species ?
regardless of the type of interaction ? resulting in a convergence of the g-factors [133] for
X0-like and X?-like excitons. Such behavior is not expected to be limited to the 1s state
excitons and can explain the convergence of our extracted values of g for the X2s0 and X
2s
?
within experimental error.
6.2.5 Conclusions
Our results serve as the first marker in mapping the behavior of the 2s charged state,
X2s? , with magnetic field in TMDs. The stability of the X
2s
? state offers a possible medium
105
for studying the cross-over from exciton Rydberg physics to the quantum Hall regime for
charged species at high magnetic fields. Recent work by Klein et al. used carrier density
dependent g-factor measurements to demonstrate tunable many-body physics through all
1s exciton species in MoS2 [133]. Our initial results indicate that it would be possible
to produce this type of map for 2s species with access to higher magnetic fields and
devices with larger dynamic carrier density range. This opens up a unique opportunity
to study many-body interactions in higher energy exciton species that is generally limited
in traditional semiconductors systems with smaller exciton binding ? like GaAs quantum
wells.
6.3 Supplemental Material
6.3.1 Fabrication Details
Figure 6.5: More detailed images from the fabrication process showing (A) the as-grown
WSe2, (B)/(C)/(D) graphite leads/gate, (E) full hBN encapsulation, (F) fully encapsulated
stack after nano-squeegee process (highlighted with dashed box), (G) an AFM image
of the nano-squeegee region highlighting the removal of interlayer impurities from the
stacking process, and (H) a PL image of the WSe2 monolayer and graphite leads, with the
quenching of emission at the contact points indicating good physical contact.
106
6.3.1.1 CVD growth of WSe2
Chemical vapor deposition (CVD) synthesis of WSe2 was performed in a two inch
quartz tube furnace on SiO2/Si (275 nm oxide) substrates. Prior to use, all SiO2/Si
substrates were cleaned in acetone, isopropanol (IPA), and Piranha etch (H2SO4 + H2O2)
then thoroughly rinsed in deionized water. At the center of the furnace was positioned
a quartz boat containing approximately 1g of WO3 powder (Alfa Aesar 99.999%). Two
SiO2/Si wafers were positioned face-down, directly above the oxide precursor. A separate
quartz boat containing approximately 500 mg selenium powder (Alfa Aesar 99.999%)
was placed upstream, outside the furnace-heating zone. The upstream SiO2/Si wafer
contained perylene- 3,4,9,10-tetracarboxylic acid tetrapotassium salt (PTAS) seeding molecules
to promote lateral growth, while the downstream substrate was untreated. Pure argon
(65 sccm) was used while the furnace ramped to the target temperature. Upon reaching
the target temperature of 825 ?C, 10 sccm H2 was added to the Ar flow and maintained
throughout the 10 minutes soak and subsequent cooling to room temperature.
6.3.1.2 Heterostructure construction
Mechanical transfer from the deposition substrate and van der Waals heterostructure
construction was performed using a wet capillary technique outlined in the experimental
methods section of Ref. [255]. The polydimethylsiloxane (PDMS) used in this process
was made from a commercially available SYLGARD 184 silicone elastomer kit. To
prepare the PDMS mixture, we thoroughly mix Silicone Elastomer and curing agent with
a weight ratio of 10:1 followed by a debubbling process under rough vacuum for 30
107
minutes. This mixture was spin coated on a silicon wafer with a spin rate of 350 rpm for
30 s, then cured at 80 ?C for 20 minutes on a hot plate. The resultant PDMS is easily
peeled off the silicon wafer for use.
The top and bottom hBN in the heterostructure are 12 nm and 15 nm thick, respectively,
as measured by AFM. After the heterostructure was complete, interlayer interfacial contamination
was removed via the nano-squeegee method [255]. Fig. 6.5 shows more detailed images
from the encapsulation process.
6.3.1.3 Electrical contact fabrication
After the full heterostructure was placed onto the SiO2/Si substrate, electron beam
lithography (EBL) was used to define leads to the graphite contacts and back gate. For
this, a bilayer of poly (methyl methacrylate) (PMMA) 950 A4 was spun onto the sample
and baked at 185 ?C for 5 and 10 minutes, respectively. The patterning was performed
on an 100kV system. Post patterning and development, 3 nm Cr/70 nm Au contacts were
deposited using thermal evaporation. Excess metal was removed using a standard solvent
liftoff procedure in a bath of room temperature acetone for 1 hour.
6.3.2 Optical Configuration
See Ch. 4.4 for optical setup and experimental details regarding light sources and
magnetic field configuration.
108
Figure 6.6: PL of the 1s exciton species as a function of gate voltage. Here, there are three
regions marked based on their charge character: n-doped, neutral, and p-doped. Exciton
species are noted. Data collected while the sample was illuminated with an excitation
energy of 1.95 eV.
6.3.3 Electrostatic Gate Mapping of the Ground State
As discussed in Ch. 4.3.2, an initial gate map was produced to identify different
doping regions. For this measurement, polarization-resolved PL was collected as a function
of gate voltage from -2 V to 2 V. The results presented in Fig. 6.6 are from a mapping
using ????, but ?+?+ was also collected and found to produce identical results. The
spectra is normalized to the 1s exciton intensity at Vg = 0 V.
In this measurement, we identify three charge regimes: neutral, n-doped, and p-
doped. In the neutral regime we identify the 1s exciton X0 [181,289], the neutral biexciton
cluster XX0 [290], the intervalley momentum-dark exciton X
D(inter)
0 [180], and the intravalley
109
spin-dark exciton XD0 [181, 208, 350].
In the charge neutral regime, the Fermi level remains in the band gap and only
localized disorder states increase in occupation. These band gap states have little influence
on the overall electronic properties of the sample outside of allowing for a small probability
that charged exciton states will form through coupling to these localized electrons. In an
ideal device, the neutral point in the sample would be just one gate voltage because we
have perfect carrier injection efficiency. However, in a real sample we can face issues like
Schottky barriers and Fermi level pinning that prevents us from injecting carriers without
overcoming a barrier first [354]. This results in a neutral regime with a finite voltage
width.
The sample enters the n-doped regime when carriers start to populate the lower
conduction bands at Vg ? 0.3 V. This region is host to the triplet Xt? and singlet Xs?
charged excitons [16, 181, 191]. Their appearance has a significant impact on the exciton
emission spectrum from the sample. As the Fermi level crosses the lowest conduction
band, both the Xs and Xt? ? quickly increase in brightness while the neutral exciton starts
to rapidly blue-shift and dim.
On the other side of charge neutrality, the sample enters the p-doped regime at
Vg ? ?0.3 V. Here, the dominant excitation is the positively charged exciton X+ [181].
Exciton species labeled in Fig. 6.6 were identified based on the doping regime and
their spacing from the fundamental X0 excitation at onset gate voltage.
110
Figure 6.7: (A) Extracted carrier density as a function of applied gate voltage in a
capacitive model (B) Calculated Fermi energy from carrier dependence with gate voltage.
The energy corresponding to the conduction band spin-splitting is noted.
6.3.4 Extraction of Carrier Concentration
The capacitive model is the simplest model for estimating the carrier density in a
sample, and it is usually accurate within 5-10%. Reduction in accuracy is highest near the
onset of carrier injection due to nonlinearities that can occur just as carriers overcome the
Schottky contact barrier [15]. Generally we reduce the problem to simply the geometric
component, but there are technically two components that determine the capacitance of
the system [282, 355]:
Ctot = (C
?1 + C?1 ?1quantum geometric) (6.1)
Where,
Cquantum = e
2?(EF) (6.2)
and
?0?hBN
Cgeometric = (6.3)
thBN
111
We can see that Cquantum only plays an important role is the density of states (DOS)
?(EF) is negligible. In the case that we are in either the conduction or valence band, it
is generally assumed that the DOS is sufficiently large enough that Cquantum becomes a
negligible contributing factor to the overall capacitance and that the geometric contribution
is sufficient to accurately extract carrier density information.
In this case, to extract the number of free carriers in the system we can use the
relation:
?V Cgeometric ?V ?0?hBN
?ne/h = = (6.4)
e ethBN
Here, ?V is defined as difference between the current gate voltage and the onset or either
valence band (VB) or conduction band (CB) filling. ?0 is the permittivity of free space,
?hBN is the dielectric constant for hBN, e is the fundamental charge, and thBN is the
thickness of the dieletric spacer. In our case, the dieletric spacer is the hBN between the
graphite back gate and our sample, which is estimated to be 15 nm thick. The thickness
was measured using AFM and is expected to have a margin of error of approximately 5%.
The average dielectric background for the hBN is given as ?? ? 2.5 [320]. We estimate
the charge neutrality region based on the extension of neutral states in Fig. 6.6, which
extends from Vg = -0.3 to 0.3 V. Fig. 6.7(A) shows the results of applying Eqn. 6.4 with
our system parameters.
We can use the carrier density to extract an approximate associated Fermi level in
the sample. Here, we estimate the Fermi energy based on the carrier density as:
ne/h ne/h
EF = = , (6.5)
?(E) m?/(??2)
112
with ?(E) = m?/(??2) as the density of available states when the the Fermi level is below
the upper conduction band. We take the effective mass of the electron m?e = 0.4 me [186]
and the effective mass of the hole to be m?h = 0.36 me [141]. WSe2 transitions from a
degenerate to a non-degenerate state as the Fermi level enters the upper conduction band.
This results in the density of states becoming ?(E) = m?/(2??2). The Fermi energy is
measured from the bottom of the lower conduction band at B = 0 T. Using these physical
parameters, we generate 6.7(B) and indicate that the splitting of the upper and lower
conduction bands for our system is ?EWSe2CB = 38 meV [186].
We estimate that the carrier concentration in the system is ne ? 2.7?1011 cm?2
at Vg = 0.6 V based on the above capacitive model of our device. Based on extracted
g-factors, magnetic field range, and carrier concentration we expect the carriers in the
system to only be partially valley polarized.
6.3.5 2s Photoluminescence
To collect PL from both the 1s and 2s excitons, we used a green laser (Eex = 2.33
eV) and hold the sample at Vg = 0 V. The results of this measurement are shown in
Fig. 6.9. We find that EPLX = 1.728 eV and E
PL
2s = 1.859 eV. This is comparable to the0 X0
values extracted from PLE, EPLEX = 1.727 eV and E
PLE
2s = 1.859 eV. In either case, we find0 X0
the ?E1s?2s ? 132 meV. This is comparable to a spacing of ?E1s?2s = 130 meV found
for WSe2 in the literature [200, 293].
We note that while the excitation energy is high enough to probe the 3s exciton as
well, we do not observe this state in our PL measurement. We attribute this to increased
113
Figure 6.8: Comparison of the PL signal for X0 and X2s0 for PL measurements with Eex =
2.33 eV. Both spectra are normalized to the emission maximum of their respective exciton
lines and the spectral position of each is marked with a dashed line.
noise (evident in Fig. 6.9 around the X2s0 region) and extremely low oscillator strength for
the state due to Kasha?s rule [43, 127].
In Fig. 1(D) of the main text, we are able to measure a resonance attributed to X3s0
at EPLE3s = 1.884 eV. Its offset of ?E2s?3s ? 25 meV is comparable to the ?EX 2s?3s = 220
meV measured in previous reports [293].
6.3.6 X2s? Photoluminescence
To collect PL from both the 1s and 2s excitons, we used a green laser (Eex = 2.33
eV) and hold the sample at Vg = 0V. The results of this measurement are shown in Fig. 6.9.
We find that EPLX = 1.728 eV and E
PL
2s = 1.859 eV. This is comparable to the values0 X0
extracted from PLE, EPLEX = 1.727 eV and E
PLE
2s = 1.859 eV. In either case, we find the0 X0
?E1s?2s ? 132 meV. This is comparable to a spacing of ?E1s?2s = 130 meV found for
WSe2 in the literature [200, 292].
114
Figure 6.9: Comparison of the PL signal for X 2s0 and X0 for PL measurements with Eex =
2.33 eV. Both spectra are normalized to the emission maximum of their respective exciton
lines and the spectral position of each is marked with a dashed line.
We note that while the excitation energy is high enough to probe the 3s exciton as
well, we do not observe this state in our PL measurement. We attribute this to increased
noise (evident in Fig. 6.9 around the X2s0 region) and increasingly low oscillator strength
for the higher n states [105].
In Fig. 1(D) of the main text, we are able to measure a resonance attributed to X3s0
at EPLE3s = 1.884 eV. Its offset of ?E2s?3s ? 25 meV is comparable to the ?EX 2s?3s = 220
meV measured in previous reports [292].
6.3.7 X2s? Emission Channel Evolution with Gate
A similar measurement technique to what was outlined in the previous section was
used to collect the PL response of the sample as the gate voltage was swept into the n-
doped regime. In addition to collecting the PL recombination energies from the 1s and 2s
neutral excitons, we can also see the emergence of the 2s charged exciton at Vg ? +0.3V.
115
Figure 6.10: Comparison of the PL signal for X0, X2s 2s? , and X0 for PL measurements
with gate dependence. All spectra have been normalized to the same color bar with the
scale factor highlighted in the top left of each plot. (A) The 1s exciton species were
collected while the sample was illuminated with an excitation energy of 1.95 eV (633
nm). (B/C) The 2s exciton species were collected while the sample was illuminated with
an excitation energy of 2.33 eV. The sample was illuminated with linearly polarized light
and the collection was polarization resolved for ??.
The data presented in Fig. 6.10 shows the emergence of the X2s resonance at EPL? 2s = 1.841X?
eV. This is in close agreement with our reported value of EPLE2s = 1.838 eV from the PLE.X?
Although we are able to spectrally resolve the X2s0 and X
2s
? through PL in Fig. 6.10
(B/C), the number of counts from the radiative recombination of X2s0 is over two-orders of
magnitude smaller than for X0. The PL contrast is even higher between the X2s? and X
t/s
? .
Hence, in order to reliably measure the spectral profile of the higher energy excitons, we
moved to PLE.
6.3.8 X2s? Vertical Cross-Section g-factor
In the main text, we analyzed the magneto-optical dependence of the 2s neutral and
charged exciton resonances while monitoring the Xt? emission channel. From their valley
116
Figure 6.11: Vertical cross-sections in excitation energy as a function of field through
the Xs? emission channel for (A) (?
???) and (B) (?+?+) marked with the corresponding
peak positions for the X2s0 and X
2s
? excitons from fitting.(C) Extracted g-factor for X
2s
0 and
X2s? . The shaded regions in panel (C) on the fit line include the error in the extracted slope.
dependent Zeeman splittings, we extracted corresponding g-factors:
gtriplet2s = ?5.21? 0.11 ? and gtripletB 2s = ?4.98? 0.11 ?B. (6.6)X0 X?
In this notation, the triplet superscript denotes the monitored emission channel
while the subscript denotes the exciton resonance for each g-factor.
We expect that changing the monitored emission channel would not change the
overall results of the extracted g-factors and verify this hypothesis by conducting the same
analysis on the Xs? emission channel. The results of this analysis are shown in Fig. 6.11,
which is counterpart to Fig. 4 of the main text. We find that g-factors extracted from the
Xs? emission channel for the excited exciton species are:
gsinglet2s = ?5.16? 0.16 ?B and gsingletX X2s = ?4.90? 0.09 ?B. (6.7)0 ?
The g-factors extracted from the Xs t? and X? emission channels agree within experimental
error. This allows us to make the important conclusion that the results from our PLE
117
Figure 6.12: (A) Raw Vertical cross-sections of the Xt? emission channel at -10T and
0.6V. The data has been normalized to the maxima of the extracted spectra. (B) The same
data after the lowest value in the spectra has been subtracted from all points and data has
been renormalized to the maxima. (C) Power sharpening of the spectra in panel (B). (D)
Peak fitting of the X2s0 and X
2s
? . The denoted input/output polarizations, magnetic field,
and gate voltage in panel (A) are the same for all panels.
measurement are independent of the analyzed emission line.
6.3.9 Fitting Procedure for Extracting g-factor
The Zeeman interaction induced splitting is quite small in TMDs, so it is important
to take the utmost care to extract accurate peak center information to produce reliable
g-factor measurements. In the literature, common techniques for doing this include using
the peak maxima or using a weighted-average fitting scheme [319]. Both techniques work
well for reasonably separated peaks. However, if there is overlap between peak envelopes
in a multipeak spectra, both can give erroneous results [191]. In our work, to combat
118
this issue, we introduce use of digital signal processing (DSP) techniques to enhance a
peak-fitting approach. Specifically we introduce the ?power law method? [56, 221, 310].
In this technique, the spectra is sharpened by raising each point in the data to a
power greater than 1. The result is that peaks ?sharpen? i.e. become narrower and the
background/overlap areas between the peaks becomes minimized, effectively giving us a
better signal contrast between overlapping peaks. This is namely due to a narrowing of
the peaks, so it must be noted that this technique not advisable for extracting information
such as full width at half max as a function of a varied parameter. Additionally, it is worth
pointing out that while the peaks become narrower the overall shape of the envelope will
not change ? i.e. if a peak is symmetric this method will preserve that symmetry just
as it would preserve asymmetry in a peak that has an asymmetric envelope. Some work
has indicated that excessive peak sharpening in this method can reduce asymmetry but it
cannot create or destroy that style of line shape; that is dictated by the underlying physics
of the system [310]. Crucially though for our purposes, this method preserves the central
signal of the peak which is our desired quantity to extract. To walk through the power-law
method used in this manuscript, we turn to Fig. 6.12.
Fig. 6.12 (A) contains an example of a raw, vertically integrated cut through the
Xt? emission line with ?
??? excitation/collection. Since this technique works best with
a minimal background, we first subtract the lowest value of the spectra from the entirety
of the signal ? Fig. 6.12 (B). Next, we apply a power filter and renormalize the data to the
new maxima in Fig. 6.12 (C).
Once the sharpening is performed, the data can be fit with the model of choice.
Here, we use a Voigt function to model X2s0 and an asymmetric Voigt function to model
119
X2s? . While the intrinsic lineshape of an exciton is generally assumed to be Lorentzian,
there are many sources of potential inhomogeneous broadening (such as lattice defects,
exciton-carrier scattering, or temperature induced thermal broadening) that are generally
modelled as a Gaussian envelope. Thus, the Voigt lineshape seems to be the best choice
since it is a convolution of these two possible sources of signal and it makes no assumptions
about the present sources of broadening [3]. Both in the raw and power-sharpened spectra
it is clear that the X2s? state is asymmetric. An asymmetric tail on the lower energy side of a
peak is relatively common and is usually indicative of a phonon side band. Higher energy
asymmetry is less commonly observed, and generally is attributed to inhomogeneity in
the dielectric environment, usually from contamination during the encapsulation process
[177, 359]. However, in that case, since optical measurements are local in nature one
would expect to see this blue tail on all peaks in a spectra collected at a given location
on the sample. Since it is obvious in Fig. 6.12 (C)/(D) that the asymmetry is rather
limited to the X2s? signal, we attribute this asymmetry to the presence of two broad,
closely spaced peaks that cannot be spectrally resolved. This picture is consistent with
the expected presence of a negatively charged doublet as in the ground state, combined
with the reduced intervalley exchange splitting in the excited state discussed in the main
text as well as in the literature [10].
The fitting was performed using the LMfit library in python, which includes Voigt
and asymmetric (skewed) Voigt models as built-in functions [213]. The library builds
on the Levenberg?Marquardt algorithm ? sometimes also referred to as a damped least-
squares optimization approach ? for optimizing the fit of a input function to the data
provided [165,202]. The repeated fitting during the optimization process by the algorithm
120
Figure 6.13: 1s PL spectra at Vg=0.6V for the (A) X0, (B) Xt?/X
s
?. (C) Extracted g-factors
from fitting PL as a function of field for X t0, X?, and X
s
?. For all measurements Eex=1.92
eV
allows a standard error to be extracted for all input parameters. Error bars used in the main
and supplementary texts for extracted points correspond to the standard error of that point.
An example of the resulting fit using this method is shown in Fig. 6.12 (D).
6.3.10 Ground State PL and g-factor
The vast majority of g-factor measurements reported for TMDs in the literature are
for 1s states with a limited number of results for X0 Rydberg states. For the purpose of
comparison to both the literature and between differing n states, we measured the g-factor
of X0, Xt s?, and X? in our system at Vg = 0.6 V. This is same carrier environment as our
main text measurements of the 2s.
The supplemental measurements of these states were performed using PL with Eex
= 1.92 eV. The extracted plots for X0 and Xs /Xt? ? are shown in Fig. 6.13(A) and (B),
respectively. We note that the relative intensity of Xt?, and X
s
? changes from -10 T to 10
121
T indicating that there is a degree of valley polarization that is induced by the Zeeman
splitting. However, as neither state is fully suppressed, the system is never fully valley
polarized.
To extract the peak centers, we use a similar peak-fitting technique as described in
6.3.9, but with the exception of using symmetric Voigt functions to fit each relevant peak.
Using this method we find that ground state g-factors of these states are all ? ?4?B:
gX0 = ?4.22?0.04?B, gXt = ?4.12?0.04?? B, and gXs = ?3.86?0.05?B. (6.8)?
These results are plotted in Fig. 6.13(C). Our findings are consistent with the relevant
literature on WSe2 [171, 289].
6.3.11 Estimating the Intercellular Contribution to g for Charged Excitons
in Single-Particle Formalism
It is common practice in the literature to use the orbital magnetic moment (OMM)
to calculate the intercellular contribution to g for excitons arising from the Berry curvature
[44,63,80,289]. Charged excitons also experience a contribution from the Berry curvature
that is induced by the exchange gap near the ?K points [338]. The resulting OMM is
written as,
122
Figure 6.14: Valley orbital magnetic moment for a charged exciton as a function of the
center of mass k from the valley center K for different carrier concentrations (expressed
in kTF) for (A) 1s charged exciton and (B) 2s charged exciton. Note the difference in
scale for the resulting magnetic moment in each panel.
m
Ln(k) = E(g(k)?? X?(k) )1/2 ( )2 4 2 2 ?3/24J k 2J k (k + 2kTF) 4J2k4
= ??ex 1 + ?? ? ? 1 + .K2?2ex(k + k 2TF) K2?2ex (k + k 3TF) ?? K2?2ex(k + k 2TF) ?
Eg(k) ?X? (k)
(6.9)
Thus, the charged exciton valley magnetic moment resulting from this exchange interaction
is written as,
( )?1
e eJ2 k2(k + 2kTF) 4J
2k4
?X?(k) = L(k) = 1 + . (6.10)2m ?K2?ex (k + k 3 2 2TF) K ?ex(k + k 2TF)
123
Here, J is the e?/h+ exchange coupling strength, kTF is the Thomas-Fermi wavevector
for the carrier screening, ?ex is the exchange-induced gap discussed previously, and K is
the position of the valleys in k-space [338].
Fig. 6.14 shows the results of evaluating Eqn. 6.10 to extract the magnetic moment
as a function of the center of mass (COM) momentum k with respect to its distance from
the valley center K and Thomas-Fermi wave vector. The Thomas-Fermi wave vector is
used as a proxy in the system the carrier concentration present with 10 ? /c ? 1011cm?20
[289,338]. Panel (A) shows the results for a 1s charged exciton while panel (B) shows the
results for a 2s charged exciton. Though the plots look very similar, note the difference in
scale for ?X?: the resulting valley OMM for the 2s charged exciton is much smaller than
for the 1s charged exciton. This difference largely results from an increase in the Bohr
radius, while both J and ?ex decrease.
6.3.12 Raman Phonon Lines in X0 PLE Emission Response
Previous experiments exploring the X2s 3s0 - X0 energy regime via PLE have demonstrated
two prominent Raman modes resulting from electron-phonon coupling between WSe2
and hBN that become bright when their emission energies match X0 [49, 121, 267];
both are labeled in Fig. 1(D) and 2(A) of the main text. The first is an optical phonon,
ZO(hBN), that is silent in pure hBN and becomes prominent only when in close proximity
to WSe2 [49, 121, 267]. The second is a combined mode of ZO(hBN) and an out-of-
plane optical vibrational mode in WSe2, A1g(WSe2) [121]. We measure the spectral
displacement of these two phonon line to be 31.1 meV, which is in good agreement with
124
prior reports [123].
Since the ZO(hBN) + A1g(WSe2) phonon has the same energy as the gap between
X2s0 and X0, when the laser is tuned to the energy of X
2s
0 , the Raman signal overlaps the
X2s0 resonance and becomes orders of magnitude brighter than any other signal from the
sample. This degeneracy can be broken through gating as the X2s0 resonance blue-shifts
with increasing carrier density and the Raman modes are unaffected. Fig. 2(A) depicts
this shift, showing the spectral isolation of X2s0 from the ZO(hBN)+A1g(WSe2).
125
Chapter 7: Fundamentals of Optical Orbital Angular Momentum
As we discussed in the introduction, the second set of work detailed in this thesis is a
theoretical study looking at the effects of twisted light on the spectrum in 2D semiconductors
(including TMDs). An overview on twisted light, or light that carries additional orbital
angular momentum (OAM), is given here as a primer for understanding Ch. 8. First we
discuss the history of twisted light, and then take a detailed look at the types of momentum
present in light.
7.1 Optical Orbital Angular Momentum
In the late 1980s, Coulett et al. discovered vortex solutions to the optical Bloch
equations and proposed the concept of optical vortices or ?twisted light? [52]. However,
it was not until a few years later that it became clear that the twisted phase of these light
beams was associated with non-zero OAM. In 1992, Allen et al. published a landmark
paper in which they proposed that Laguerre-Gaussian (LG) laser modes had well defined
OAM due to their intrinsic azimuthal phase dependence ei?? [7]. This phase dependence
results in tunable quantized momentum given as ?? per photon, where ? is the azimuthal
mode index. Scientists had known for a long time that multipole transitions could produce
radiation that possessed additional angular momentum beyond spin [57]. However, study
126
of these ?forbidden? transitions remained difficult experimentally, and the left the origins
of this additional angular momentum as a topic of debate through the 1990s. He et al.
verified Allen?s proposal in 1995 [106].
The work by Allen et al., which now has more than 5,000 citations, spawned
an entirely new field in the study of light-matter interaction. Research utilizing OAM
spans the fields of microscopy [245], large-scale telecommunications [167,269,333], and
astronomy [296, 298] amongst many others. Some excellent review on the topic can be
found in Refs. [17, 82, 222, 223, 225, 272].
7.2 Types of Angular Momentum in Light
7.2.1 Linear Momentum
The concept of radiation pressure significantly predates our modern understanding
of electromagnetism that derives from Maxwell?s equations. Nearly 200 years before
Maxwell?s work, Kepler utilized the concept to explain why the tails of comets always
point away from the sun [222]. Radiation pressure stems from the linear momentum
of light. In 1884, while exploring Maxwell?s work, Poynting found that the directional
energy flux of the electromagnetic field is,
1
S? = E? ? B?. (7.1)
?0
Here S? is the Poynting vector, E? is the electric field vector, B? is the magnetic field vector
and ?0 is the vacuum permeability [236].
127
Figure 7.1: Illustration of the Nichols radiometer.
From this, one can find that the linear momentum of the electromagnetic field is,
S?
p? = , (7.2)
c2
where c is the speed of light. The radiation pressure exerted by that field is,
???? ?S?
?? , for a perfect absorbercprad = ??? (7.3)2?S? , for a prefect reflector
c
Here, ?S? = I , the intensity of the incident light.
In 1901, Nichols and Hull devised an experiment to measure radiation pressure
using two silvered mirrors forming a torsion balance hanging from a fiber; a simple
illustration of this is shown in Fig. 7.1. In the real experiment, the torsion pendulum was
inside a bell jar where the ambient pressure could be controlled. They were able to show
that the mirror did indeed deflect under illumination and that the deflection depended on
the intensityof the light as expected from Eqn. 7.3 [214].
Adapting this work to the concept of wave-particle duality, in 1905 Einstein showed
128
Figure 7.2: (A) Spin angular momentum with illustration of the electromagetic field
of circularly polarized light at some instant, t, along with the direction of spin angular
momentum (S). (B) Intrinsic OAM arising in a vortex beam with illustration of the phase
front at some instant, t, along with the direction of momentum (Lint). In both panels P is
the direction of propagation.
that photoelectric effect (and blackbody radiation) could be explained if the linear momentum
of a single was given by,
p? = ?k?, (7.4)
where is the wave number associated with the wavelength of the photon [71]. Today, the
relationship for linear momentum is directly utilized by those working the field of atomic
and molecular optical physics for trapping and cooling schemes [222].
7.3 Angular Momentum
In addition to linear momentum, light also possesses angular momentum. Here, we
will discuss spin angular momentum (SAM), which is associated with the polarization of
the light beam and OAM, which is associated with the phase evolution of the light.
129
7.3.1 Spin Angular Momentum
Beyond his work with linear momentum, Poynting made an important contribution
to our understanding of angular momentum in light in 1909 when he inferred that CPL
should carry angular momentum [237]. Making an analogy to a rotating cylinder, in this
work Poynting reasoned that CPL should posses non-zero angular momentum in the same
direction as it propagated. To test this, he suggested that CPL should exert a torque on a
wave plate as it passes through. Poynting?s theory was experimentally verified by Beth in
1935 [21, 22].
Today, we generally referred to this kind of angular momentum as SAM, due to its
quantum mechanical origins. Photons are spin-1 particles and RCPL and LCPL are the
manifestations of the two eigenstates of the spin operator. In this formalism, we find that
SAM for a single photon is given by,
Sz = ?z?. (7.5)
Here, ?z = ?1 and corresponds to the handedness of the light. Linearly polarized light,
which is equal parts LCPL and RCPL that are in phase, has no spin angular momentum.
In his classical theory, Poynting found that the ratio of the angular to linear momentum
with respect to the axis of the beam was ?/2?; the photon picture yields the same result.
An illustration of SAM can be found in Fig. 7.2(A).
130
7.3.2 Orbital Angular Momentum
Finally we come to intrinsic orbital angular momentum (IOAM) or OAM, the type
of angular momentum that is more relevant to this work. As mentioned in the introduction,
OAM is intrinsically linked to vortex beams. Vortex beams have a helical phases structure
with an azimuthal (?) angular dependence ? ei?? ? which causes them to carry OAM that
is independent of the polarization of the light [7]. An illustration of the twisted phase of
a beam with OAM is shown in Fig. 7.2(B).
We can use a ray optics picture to understand how this phase dependence results in
OAM. The phase rotation is correlated with twisting of the Poynting vector itself, which
results in rays that are skewed at some angle ? at every point on the wavefront [224]. If
we pick a point along the phase front at fixed radius r, we can think of the beam like a
phase ramp with constant gradient in the azimuthal direction. At point r this means that
the base of the phase ramp has a length of 2?r and a height of ??. This allows us to
calculate the skew of a ray at any point along the phase front,
??
? =
2?r (7.6)
?
=
kr
in terms of both the wavelength and the wavevector k = 2? . Each of these rays carriers
?
131
linear momentum, so the OAM carrier by the beam is given by,
|L?| = |r? ? P? |
= |r?ksin(?)| (7.7)
= ??.
Unlike SAM, the quantized values of ? are not limited to ?1 and can take on any integer
value [162]. In fact, researchers have been able to generate beams up to ? = 10, 000? in
the lab [78].
The phase rotation results in a singularity at the center beam which causes a ?donut?-
like beam spatial profile where the intensity at the center of the beam is zero. As an
illustration for what this looks like we turn to LG modes which take the from,
? ( ? )|?| ( )
2p! 1 r 2 2r2 ?r2 ikr2
LG?p(r, z, ?) = L
|?| e w2 e 2Rp e
?i(2p+|?|+1)?e?i??.
?(p+ |?|)!w w w2
(7.8)
Here, p is the radial mode index, |?|Lp (x) is a Laguerre polynomial. w, R, and ? have an
implicit z-dependence and take the form,
?
w(z) = w0 1 + (z/zR)2) (7.9a)
R(z) = z(1 + (zR/z)
2) (7.9b)
?(z) = arctan(z/zR) (7.9c)
1
zR = kw
2
2 0
, (7.9d)
132
Figure 7.3: Intensity (top panels) and phase (bottom panels) of LG modes with different
?. The intensity if normalized to the maximum. From left-to-right the modes are:
LG01, LG11, and LG21. The figure is from Ref. [335].
w(z) is the laser spot size (defined by the radius for which the intensity of the Gaussian
beam is 1/e2 its central value), R(z) is the radius of curvature of the wavefront, ?(z) is the
Gouy phase, zR is the Rayleigh range, and w0 is the waist radius of the beam [13, 318].
These definitions are the same as those found for a traditional Gaussian beam. In the
event that ? = 0, we get light with no azimuthal phase dependence and therefore no OAM.
However, for light with ? ?= 0, the phase dependence produces OAM and the beam profile
consists of p + 1 concentric rings [13, 225]. An illustration of the spatial profile and
corresponding phase distribution for different LG modes can be found in Figure 7.3.
Though we do not discuss it further here, the curious reader can refer to Ref. [335]
for a nice overview on different labs methods for generating beams with OAM, since LG
beams are by no means the only option.
133
Chapter 8: Two-dimensional excitons from twisted light and the fate of
the photon?s orbital angular momentum
8.1 Notes
This chapter is largely taken verbatim from our paper ?Two-dimensional excitons
from twisted light and the fate of the photon?s orbital angular momentum? T. Gra?, U.
Bhattacharya, J.C. Sell, and M. Hafezi, arXiv:2201.13058 (2022) [96]. It is currently
under review for peer-reviewed publication. Small alterations have been made to fit
references from earlier sections to this chapter. It should be noted that in comparison
to the other sections of this thesis these results apply not only to TMDs, but also other 2D
semiconductor systems.
8.2 Manuscript
8.2.1 Abstract
As the bound state of two oppositely charged particles, excitons emerge from optically
excited semiconductors as the electronic analogue of a hydrogen atom. In the two-
dimensional (2D) case, realized either in quantum well systems or truly 2D materials
such as transition metal dichalcogenides, the relative motion of an exciton is described by
134
two quantum numbers: the principal quantum number n, and a quantum number j for the
angular momentum along the perpendicular axis. Conservation of angular momentum
demands that only the j = 0 states of the excitons are optically active in a system
illuminated by plane waves. Here we consider the case for spatially structured light
sources, specifically for twisted light beams with non-zero orbital angular momentum
per photon. Under the so-called dipole approximation where the spatial variations of the
light source occur on length scales much larger than the size of the semiconductor?s unit
cell, we show that the photon (linear and/or angular) momentum is coupled to the center-
of-mass (linear and/or angular) momentum of the exciton. As a result, our study shows
explicitly that the additional orbital angular momentum imparted by spatially structured
light sources cannot be used modify the 2D exciton spectrum by facilitating dipole forbidden
transition as it does in atomic systems.
8.2.2 Introduction
Excitons are the bound states formed by an electron-hole pair in a semiconductor
crystal [105, 260], and as such, they are close analogues of the hydrogen atom. Excitons
manifest themselves as optical absorption or emission lines within the band gap of the
material. In the theoretical treatment of exciton formation, light-matter interaction is
most often described within the dipole approximation. This approximation disregards the
spatial structure of the light field, which is justified by the tiny length scale on which the
de Broglie waves vary, as compared to the wavelength of the light [51].
The dipole approximation gives rise to optical selection rules related to the conservation
135
of angular momentum during an optical transition. Within the dipole approximation,
light may carry only one quantum of angular momentum per photon, realized through
the circular polarization of the light; therefore, optical transitions can change angular
momentum quantum numbers of the matter only by one unit. In atoms, this rule selects
the s-to-p or p-to-d transitions; in semiconductors, this affects the orbitals of the bands
in an analogous way. As a consequence of the dipole approximation, excitons created
from such a dipole transition do not carry angular momentum; that is, the respective
quantum number j is zero. Moreover, as established by the Elliott formula [72], the
transition amplitude quickly decays with the principal quantum n (as (n + 1/2)?3 in
2D), such that the exciton spectrum is strongly dominated by transitions into the 1s state,
i.e. the state corresponding to the hydrogenic ground state. A common technique to
allow optical access to the p exciton series in semiconductors is nonlinear, two-photon
spectroscopy [20, 311].
It is clear that effects beyond the dipole approximation can modify the exciton
spectrum. In particular, the dipole approximation disregards the possible spatial structure
of the light beam, which in the case of twisted light results in a well-defined orbital
angular moment (OAM) per photon [7, 301, 335]. The photon OAM essentially adds
another tunable degree of freedom for tailoring light-matter interaction. It has been
proposed to use this new degree of freedom for generating a topological band structure by
breaking time-reversal symmetry [23], for pumping electrons in a magnetic field through
the Landau level [32, 88, 97], or for producing topological defects such as vortices or
skyrmions [50, 86, 87, 130]. A striking demonstration of how the optical spectrum can be
modified by photon OAM has been achieved in an experiment with trapped ions [263],
136
showing a dipole-forbidden atomic s-to-d transition in the presence of a twisted light
field that is the result of additive interaction between SAM and OAM. Given the analogy
between an exciton and a hydrogen atom, it might be expected that twisted light can
generate transitions into dipole-forbidden excitonic levels in a similar manner. This
indeed has theoretically been suggested for the case of Rydberg excitons [138]. On the
other hand, there have also been experiments with atomic and polaritonic condensates
in which twisted light has led to the formation of vortices [9, 148]. This indeed would
suggest that the orbital angular momentum of the photon is absorbed by the center-of-
mass (COM) degree of freedom of the exciton, rather than by the relative motion of
electron and hole. Also, in the strong drive limit and absence of Coulomb binding, OAM
of light can lead to Floquet vortex creation, but it is an open question how strong drive
and exciton formation compete with each other.
To better understand the fate of the photonic OAM in excitonic transitions, the
present paper studies the case of a single exciton in a two-band semiconductor model
in 2D in the presence of a twisted light source. We extend theoretical studies of band-
to-band transitions in semiconductors or graphene with twisted light, presented in Refs.
[75,239,240], to the case where Coulomb interactions give rise to exciton formation. Our
analysis demonstrates that, under the assumption that the spatial variation of the light
occurs on a length scale much larger than the size of the unit cell of the semiconductor
crystal, transitions into excitonic levels j ?= 0 remain completely forbidden even in the
presence of twisted light. Instead, the structure of the light field selects the COM degree
of freedom of the excitons. Since it is the relative motion that essentially determines the
energy of an exciton, it follows that the twist of the light source does not modify the
137
excitonic spectrum. In this context, we also note that small shifts of the spectrum are
possible if the COM dispersion of the exciton is taken into account. This indeed has been
observed in a recent experiment with excitons in a Dirac material, which found a blueshift
of the exciton lines for sufficiently large values of photon OAM [277].
Our paper is organized in the following way: In Sec. II, we develop the general
analytical formalism to describe exciton transitions in structured light beam. In Sec. III,
we specifically address the case of a Bessel beam. To evaluate this case, we make use
of the rotational symmetry of the beam which makes an explicit numerical treatment
feasible. With this we are able to show, for a finite system size, that the s states are
optically bright, in quantitative good agreement with the 2D Elliot formula, independent
from the choice of the photon OAM. Our numerical calculation also confirms that the
COM momentum of the exciton is peaked at the linear momentum of the photon.
8.2.3 General Analytical Model
8.2.4 Light-matter coupling
We consider a 2D semiconductor with Bloch bands ? and wave vector k, described
by Bloch functions ??,k(r) = ?1 eik?ru?,k(r), where u?,k(r + Ri?) = u?,k(r) with RS i a
lattice vector. In this basis, the crystal Hamiltonian reads H0 = ? c
?
?,k ?,k ?,kc?,k, with
c (c??,k ?,k) being the annihilation (creation) operators, and ??,k the dispersion. We assume
a light field given by a vector potential A(r) = A(r) ? e in the Coulomb gauge, such that
138
the light-matter Hamiltonian is given by:
?? ie?
H = ???,k?LM |A(r) ? ?r|?,k?c???,k?c?,k. (8.1)M
??,? k?,k
We are only interested in the matrix element
?
??h ,?k?,k = ??
?,k?|A(r) ? ? 2 ?r|?,k? = d r???,k?(r)A(r)e ? ?r??,k(r), (8.2)
with ? = c and ?? = v, i.e. transitions amplitudes between conduction and valence band.
Taking into account the orthonormality of the bands, the derivative operator has to act
onto the lattice-periodic function u?,k to yield non-zero contributions. Explicitly, we have
?
hv,c
1 2 ?
k?,k = d rA(r)e
i(k?k )?ru?
S v,k
?(r)e ? ?ruc,k(r). (8.3)
Here, S is the size of the system.
At this stage, we make the approximation (A1): the vector potential and the exponential
do not vary within a unit cell. By keeping variations beyond the scale of a unit cell, this
approximation is less restrictive than the dipole approximation which would fully ignore
the spatial structure of the light. Yet without considering a particular choice of vector
potential, the spatial variations of the beam is generally limited by a length scale on the
order of the wavelength of the light. The same length scale also determines the variation
of the exponential ei(k?k?)?r, since it will turn out a posteriori that k ? k? is determined
through the photon momentum. Since the optical wavelength is usually several orders of
139
magnitude larger than the size of unit cell, the approximation is totally valid in the usual
cases, but might not hold in some special cases, e.g. of Moire? lattices with enlarged unit
cells [6, 122, 268, 303]. Applying (A1) to Eq. (8.3), we write:
?
hv,c
1
= A(R )ei(k?k
?)?Ri
k?,k iN sites
R[?i ]
? 1 e ? d2ru? (r)? u (8.4)v,k? r c,k(r)Scell cell
? A? ? e ? pvck?,k,
?
where ? = k ? k? and A = 1? R A(Ri)ei??Ri the Fourier transform of the vectorN sites i
potential. The dipole moment between the k? state in the valence band and the k state in
the conduction band is denoted by pvck?,k.
We proceed by making a second approximation (A2): the dipole moment depends
only weakly on the wave vectors k and k?. Indeed, the most radical implementation of
this approximation in which the dipole moment is set to a constant pvc0 is commonly used
in the literature, cf. Ref. [105]. To be less restrictive, we argue that pvck?,k may depend on
k+k? (which can take relatively large values), whereas the dependence on k?k? (which
remains small since it is equivalent to the photon momentum) is neglible. To this end we
introduce the quantity K = 1(k+ k?) and assume a linear (or linearized) dependence on
2
K:
pvc = pvcK 0 + (? ?K)pvc1 . (8.4)
For notational convenience, we write e ? pvc = pvcK K = pvc0 + (? ?K)pvc1 . The light-matter
140
matrix element is finally written as:
hv,ck?,k = h
vc
?,K = A p
vc
? K. (8.5)
This expression makes it immediately clear that the wave vector ? is exclusively selected
by properties of the light field, whereas the wave vector K is exclusively determined by
material properties. In the following, we will find that, in the case of exciton transitions,
? (K) is related to the COM (relative) momentum of the exciton.
8.2.5 Exciton transitions
We are now interested in the transition amplitude for exciton formation TX ?
?X|HLM|vac?. Here, |X? denotes an excitonic state, which in 2D is characterized through
four quantum numbers for relative and COM motion. We choose |X? = |kcom, n, j?, i.e.
we describe the excitonic state by its linear COM momentum kcom, and its hydrogenic
quantum numbers n and j, representing the relative degrees of freedom. The vacuum
state |vac? corresponds to a filled valence band and an empty conduction band.
The excitonic wave function can be written as
?R, r|X? ? ?kcom ? (com) ? (rel)n,j (R, r) ?k (R) ?n,j (r), (8.6)com
where r = re ? rh are relative coordinates of an electron-hole pair, and R = 1(re + rh)2
are the COM coordinates. The relative motion of electron and hole is described by the
141
solutions to the 2D hydrogen atom, which are given by [105]
(rel) N? ij?r N? |j| ?
|?| 2|j|
?n,j (r) = n,jfnj(r)e =
ij?r
n,j? e 2 Ln?|j|(?)e , (8.7)
where ? = r?n, with the inverse length scale given by ? 2n = . The material-(n+1/2)a0
specific length scale a 20 = ? ?/(e2M) is the effective Bohr radius depending on effective
mass M and dielectric constant ?. The normalization of the relative wave function is given
by
?
N? (n? |j|)!
(? )2n 1 1
n,j = . (8.8)
(n+ |j|)! 2 ? n+ 1/2
For the COM part, we simply assume plane waves, (com)? (R) = eikcom?Rk .com
Without making use of the explicit solution for the excitonic wave functions, we
write for the transition amplitude:
? ? ?
TX = d2R d2r ?X|R, r??R, r|?,K?
?,K (8.9)
? ??,K|HLM|vac?.
The last term corresponds to the band-to-band transition amplitude evaluated above,
? | | ? ? ? ??,K HLM vac = K |HLM|
?
K+ ?
2 2 (8.10)
= hv,c = A pvcK?? ,K+? ? K,
2 2
142
which is Fourier transformed to spatial coordinates by the second term,
? ?
? 1 1R, r|?,K? = ei(K+ )?re?i(K? )?rh = ei(K?r+i??R)2 2 . (8.11)
S2 S2
This expression explicitly shows that the wave vector ? (K) is conjugate to the COM
(relative) variable.
Plugging all expressions into Eq. (8.9), the R integral is immediately evaluated
into a Kronecker-Delta ??,kcom , so photon momentum ? and COM momentum kcom must
match. We obtain:
?
T (2?)
2N?nj 2 ij?r
X = Akcom d rf? 2 nj
(r)e
S
(8.12)
? eiK?r(pvc0 + ? ?Kpvc1 ).
K
?
The K-sum is a Fourier transform into the relative variable r, and we can write eiK?rK (p
vc
0 +
? ? Kpvc1 ) = (2?)2[pvc0 ?(r) ? ipvc1 ? ? ?r?(r)]. From this expression it can immediately
be seen that a constant dipole moment leads to non-zero transition amplitudes only if the
relative exciton wavefunction fnj(r) is non-zero at r = 0. This is the case only for s-
excitons. The linear dependence of the dipole momentum on K, expressed by the second
term, gives rise to non-vanishing transition amplitudes if the first derivative of fnj(r) is
non-zero at r = 0.
This second term enables the formation of excitons in higher momentum states than
the s series. However, we emphasize that this term is independent from the light source,
and with respect to the relative degrees of freedom (n, j), we get the same transitions,
143
no matter what the spatial structure of the light might be (as long as approximation (A1)
holds). Our analysis shows that the Elliott formula is unchanged by spatial structure of
light beyond the scale of the unit cell.
Our result agrees with a recent experiment in 2D transition metal dichalcogenides
(TMDs) where twisted light has been used to reveal light-like exciton dispersion [277].
Using non-resonant Laguerre-Gaussian beams, they observed a blue shift of the exciton
energy that increased with ?; this indicates that the OAM was transferred preferentially
to the COM of the exciton during its creation. However, we stress that our results apply
much more generally since no particular 2D semiconductor or spatial light profile was
specified during the analysis. This implies that the dispersion of all 2D excitons could
be probed in a similar manner, presenting an alternative method to the traditional angle-
resolved photoluminescence measurements [148].
8.2.6 Example: Excitons from twisted light
Thus far, we have been very general in our treatment with respect to the profile
of the optical excitation. In the following, we are going to treat the specific case of a
Bessel beam and, besides the analytical treatment along the lines presented in the previous
sections, we will also present the result of numerical evaluations. With this choice of the
vector potential, our system exhibits a cylindrical symmetry, since the light field has an
azimuthal phase dependence exp(i??), where ? defines the OAM per photon (in units
?). To match this symmetry, we consider a cylindrical sample, noting that the sample
geometry becomes irrelevant in the thermodynamic limit. Accordingly, we adopt our
144
- R
+
Figure 8.1: (a) A Bessel light beam A with photon momentum/angular momentum
(q?, ?) creates an electron-hole pair in a 2D electron gas (2DEG). The electron [hole] is
characterized by quantum numbers (m, ?) [(m?, ? ?)] for angular momentum/ momentum.
The pair can form a bound state, and the degrees of freedom of such an exciton are
the center-of-mass motion, characterized by angular momentum/ momentum quantum
numbers (J,N), and the relative motion, characterized by quantum numbers (j, n) for
angular momentum and energy. (b) The selection rules for optical transitions and exciton
formation reflect (i) conservation of angular momentum, and (ii) conservation of linear
momentum, reflected by the illustrated triangle conditions.
theoretical description to this symmetry, and express the light-matter coupling in terms of
a cylindrical wave functions, see also Refs. [75, 240]. We illustrate this case in Fig. 8.1,
where we also sketch the resulting selection rules in terms of the good quantum numbers
for the cylindrical symmetry.
8.2.7 Band-to-band transitions in cylindrical basis
Instead of plane waves with linear momentum quantum number, the electronic basis
in a cylindrical sample is best described by wave functions ?m,?(r) = Nm,?Jm(km,?r) exp(im?),
which solve the Schro?dinger equation for free electrons with cylindrical boundary conditions.
145
Here, Jm(x) denotes the mth Bessel function, and the momenta km,? must be chosen such
that the wave function vanishes at the system boundary (i.e. for |r| = R0). Therefore,
we have km,? = xm,?/R0, with xm,? the ?th zero of the mth Bessel function. The
?
normalization is given by N ?1m,? = (R0 ?|J|m?1|(xm,?)|) . As in the previous section,
the crystal lattice is taken into account by multiplying the wave functions ?m,? with
lattice-periodic Bloch functions u?(r) for the bands ?. For simplicity, the Bloch functions
are assumed to be independent from the quantum numbers m and ?. With this, the
electronic basis is given
??;m,?(r) = Nm,?Jm(km,?r) exp(im?)u?(r). (8.13)
In this basis, the light-matter transition amplitudes in the Coulomb gauge (within
the weak field limit) are given by
?
v,c ?i?ehm,?;m?,?? = dr??+,m?,??(r) [A(r) ? ?r]??,m,?(r), (8.14)SM
with ?? denoting the complex conjugate of ?.
Before we proceed, let us first fix the vector potential. We consider a vector potential
which in the sample plane reads A(R) = A i??0a(R)e e?, where e? is the polarization in
the plane. For a circularly polarized Bessel beam with OAM ?, we have a(R) = J?(q?R),
with q? the in-plane photon momentum. Note that a vertical contribution to the vector
potential, needed to fulfill Maxwell?s equation, is neglected here since it is not relevant
for light-matter interaction with a two-dimensional medium. However, it is important
146
to keep in mind that the frequen?cy ? of the photon depends also on the perpendicular
momentum component qz, ? = c? q
2
? + q
2
z .
Invoking the approximations (A1) and (A2), both A(r) and Jm(km,?r) exp(im?)
can be considered constant on the scale of the lattice constant a. Thus, the evaluation of
hv,cm,?;m?,?? can be split into an integral I restricted to the unit cell and a sum over units cells
Sm,?;m?,?? , that is, we can write hv,c ?i?m,?;m?,?? = SSM m,?;m?,?? ? I. As before, the sum over
the whole system takes into account the variation of the light field, occurring on larger
scales, whereas the unit cell integral determines the material?s dipole moment taken to be
a constant.
Explicitly, the two contributions are given by
?
I = e dru?+(r)(e? ? ?r)u?(r), (8.15)
c
and
?
Sm,?;m?,?? =Nm,?Nm?,??A0 Jm(km,?Ri)Jm?(km?,??Ri)
i (8.16)
? J?(q?Ri) exp[i(m?m? + ?)?i],
where (Ri, ?i) denote the lattice vectors. In analogy to Eq. (8.4), we can read off the
results of the cell integral I as the interband dipole moment pvc ? e? ? d = I, which
depends only on the material. Since we take it to be constant here, it will enter the
transition amplitudes only as a prefactor. To evaluate Sm,?;m?,?? , we replace the summation
over cells by an integral. With this, we immediately arrive at a first selection rule from
147
the angular part of the integral:
Sm,?;m?,?? ? ??+m?m? . (8.17)
?The radial integral in Sm,?;m?,?? is over a product of three Bessel functions, Sm,?;m?,?? ?R0 dr rJm(km,?r)J ?m(km?,??r)J?(q?r). Its analytic solution (in the limit R0 ? ?) has0
been derived in Ref. [117], and can also be found in the Supplemental Material (SM).
Here, we only consider that, from this solution, the integral takes non-zero values only if
a triangle condition is fulfilled: The three scalars km,? , km?,?? , and q? must be such that
they can form a triangle (including the limit in which the triangle is squeezed to a line).
Therefore, this condition yields a second selection rule: the change of electron momentum
upon a band-to-band transition is bounded by the in-plane momentum of the photon.
8.2.8 Exciton transitions
The amplitudes hv,cm,?;m?,?? quantify the band-to-band transition which generates an
electron-hole pair characterized by m ? m? = ? and |km,? ? km?,?? | ? q?. Next, we
have to ask which excitonic states can be formed from these pairs. In accordance with
the presumed cylindrical symmetry of the system, we now also describe the excitonic
states in terms of cylindrical-symmetric quantum numbers, |X? = |N, J, n, j?, where
n, j account for the state of relative motion (as before), and N, J for the COM degrees of
freedom (instead of kcom used in the previous section). Again, the excitonic wave function
is a product of the relative (r) and COM (R) contributions: ?R, r|X? ? ?J,Nn,j (R, r) ?
(com) (rel)
?N,J (R)??n,j (r). The relative part is unchanged, given by Eq. (8.7). Since the COM
148
of the exciton is subject to the same boundary conditions as electron and hole individually,
its wave function is given by:
(com)
?J,N (R) = NJ,NJJ(kJ,NR) exp(iJ?com), (8.18)
where the quantum number J denotes the angular momentum of the COM, and both J
and N together define the total COM momentum Qcom = xJ,N/R0.
Projecting the excitonic wave function onto the rotationally symmetric basis for
electron and hole wave functions is equivalent to a Hankel transform. This projection
yields a quantity Bn,j;J,Nm,?;m?,??:
? ?
Bn,j;J,Nm,?;m?,?? =N
J,N
m,?Nm?,?? dre drh??n,j (R, r)
? Jm(km,?re)Jm?(k ?m?,??rh) exp[i(m?e ?m ?h)]. (8.19)
An explicit analytic expression which solves this integral is provided in the SM. As before
for Sm,?;m?,?? , we also encounter a triangle condition in the evaluation of Bn,j;J,Nm,?;m?,??: it is
non-zero, only if the lengths km,? , km?,?? , and kJ,N form a triangle, i.e. |km,? ? km?,?? | ?
kJ,N . More importantly, as shown in the SM, one of the integrals in Eq. (8.19) yields a
Kronecker-?:
Bn,j;J,Nm,?;m??? ? ?j+J,m?m? . (8.20)
Together with the selection rule for band-to-band transitions, Eq. (8.17), Eq. (8.20) reflects
149
conservation of angular momentum.
We are now in the position to calculate the exciton transition amplitude T J,Nn,j ?
?XJ,Nn,l |HLM|vac?:
?
T J,N v,c n,j;J,Nn,j = hm,?;m?,??Bm,?;m??? . (8.21)
m,?;m?,??
These sums should go over all occupied (empty) levels m?, ? ? (m, ?), but a more practical
limitation of these sums is due to the fact that Bn,j;J,Nm,?;m??? ? 0 when either km,? or km?,??
become much larger than the inverse of the Bohr radius, a?10 .
8.2.9 Numerical evaluation
The last observation allows us to introduce a cutoff momentum k ? a?1cut 0 at
which the sums can be truncated. With this, the numerical evaluation of Eq. (8.21)
become feasible. For concreteness, by comparison of T J,Nn,j obtained from different cutoff
momenta kcut, we estimate that the relative error remains below 0.1 for kcuta0 ? 3.75.
In the numerical evaluation of T J,Nn,j presented below, we have included 47,100 Bessel
functions. With that, kcuta0 > 3.75 for system sizes up to R0/a0 = 12, 500. We note
that kcut also restricts the sums in m and ? in the following way: |m| < ?kcutR and/or
? < kcutR.
The numerical evaluation confirms the analytical result from Sec. II that the Elliott
formula remains unchanged by the spatial structure of the light source. To this end, we
obtained the height of the spectral lines, T?n,j , by summing the contributions from all COM
150
1s 2s 3s
OAM 0 0.998 0.037 0.0079
OAM 1 0.995 0.037 0.0079
2D Elliott 0.9993 0.037 0.0080
Table 8.1: Relative transition strength, |T? |2n,j , for a system of size R0 = 104a0 in a Bessel
beam with OAM ? = 0 and ? = 1 and in-plane photon momentum q? = 10?3a?10 . For
comparison, we also provide the results from 2D Elliott formula for an infinite system in
a Gaussian beam.
momentum modes at a given n and j:
T? 1
?
J,N
n,j = Tn,j . (8.22)NT
N,J
?To?make th? is qua?ntity independent from the intensity of the light, we normalize by N =??
T
2
T J,N ?n,j,N n,j ? . The results are shown in Table 8.1 for a system of size R0 = 104a0 in a
Bessel beam with OAM ? = 0 and ? = 1 and in-plane photon momentum q ?3 ?1? = 10 a0 .
For comparison, we also provide the results from the 2D Elliott formula for an infinite
system. All values agree very well with each other.
We note that the numerical evaluation also yields small but finite values for transitions
into p-states. However, in contrast to the values for the transitions into s-states, these
values show a strong and non-monotonic dependence on the system size and/or photon
momentum. This suggests that, in accordance with our general arguments presented in
Sec. II, the finite transition amplitudes into p-states are numerical artifacts, and the only
bright transitions occur into the s states.
Our numerical evaluation also confirms that the COM momentum of the exciton is
determined by the linear in-plane momentum of the photon. To this end, we focus on
the 1s transition and evaluate the transition strengths T ?,N0,0 into the different COM modes
151
1
0.8
0.6
0.4
0.2
0
0 0.5 1 1.5 2 2.5 3 3.5 4
Figure 8.2: For the 1s transition, we plot the transition strength T ?,N0,0 into the different
COM modes k?,N , normalized by the peak value max (T ?,NN 0,0 ), for illumination with ? =
0 and ? = 1 Bessel beams. The peak is obtained for the best match between COM
momentum k?,N and in-plane photon momentum, q ?3 ?1? = 10 a0 .
k?,N . The results, normalized by the peak value maxN(T ?,N0,0 ), are shown in Fig. 8.2 for
different values of photon OAM ? and photon momentum. The transition strength is
clearly peaked for the COM momenta which match with the momentum of the photon,
but barely depends on the OAM.
Let us finally discuss the different length scales which appear in the calculation,
that is, the effective Bohr radius a0, the sample size R0, and the inverse of the photon
momentum q?1? . In the calculation, we have taken the effective Bohr radius a0 as the
unit of length. With typical values of the dielectric constant being much greater than
1 (e.g. ? 13 in GaAs [294] and 7 in semiconducting TMDs [161]), and the effective
mass being much smaller than the electron mass (e.g. 0.067 and 0.39 electron masses
for the conduction band in GaAs [35] and model TMDs [141], respectively), the effective
Bohr radius can significantly exceed the Bohr radius of the hydrogen atom (? 0.05nm).
152
Typical values range between 0.1 to 1 nm. Taking the numerical constraints into account
(i.e. truncation errors), our study examines sample sizes R0 on the order of 104 effective
Bohr radii which corresponds to sample sizes on the order of 1-10 microns. Importantly,
this size is significantly larger than the optical vortex. Regarding the in-plane photon
momentum q?, an upper limit is given by the inverse of the wavelength, 2?/?0, assuming
vertical incidence on the sample. The wave length ?0 is determined by the band gap of
the material. As an estimate for this limit, we obtain 100nm?1. Thus, our choice of
q = 10?3a?1? 0 corresponds to the upper limit if a0 = 0.1 nm, while this choice remains
below that limit if a0 is larger.
8.2.10 Summary and Conclusions
We have shown that the vector potential selects the COM quantum numbers (absolute
value of COM momentum + COM angular momentum), but has no effect on the transition
amplitudes into states with different relative quantum numbers n and j. This implies that
Elliott?s formula is unchanged by the structure in the light field. The approximation which
gives rise to these conclusion is the separation of length scales: unit cell vs. wavelength.
This assumption implies that A(x) and eiqx are constant on the the level of a unit cell, and
for the dipole moment, pvc vck,k+q ? pk,k. We have confirmed our general analytical result
by performing numerical evaluations for the concrete case of Bessel beams in a circularly
symmetric sample. Qualitatively, we have shown that, for a transition to be optically
bright, the sum of both relative and COM angular momenta, j + J , must be equal to the
OAM value ? of the light. Quantitatively, we have evaluated that the transition amplitudes
153
are given by the Elliott formula.
While our results rule out twisted light for the generation of dark excitons, the
predicted transfer of OAM to the COM degree of freedom can be useful from the perspective
of quantum simulation, and especially from the point of view of artificial gauge fields. In
Ref. [126], it has been shown that artificial flux is generated in a photonic system when
OAM light is injected into a waveguide lattice. Excitons in tunable lattices have recently
be shown to form strongly correlated many-body phases, such as Mott insulating phases
[152] or checkerboard phases [151]. If, in the future, twisted light provided excitonic
lattices with artificial magnetic fluxes, this could give rise to chiral Mott insulators [65]
or extended supersolid regimes [295].
8.3 Supplemental Material
8.3.1 Band-to-band transition matrix for a Bessel beam
The band wave functions are expressed in terms of Bessel functions Jm(km,?r),
and the spatial profile of the light is given by a Bessel beam J?(q?r). Thus, the matrix
elements for band-to-band transitions are proportional to the integral over a product of
three Bessel functions:
? R0
S ?m,?;m?,?? ? dr rJm(km,?r)Jm(km?,??r)J?(q?r). (8.23)
0
To evaluate this integral, we take the limit R0 ? ?, and follow the procedure as described
in Ref. [117]; that is, we perform a plane-wave expansion of the Bessel functions. With
154
this, the integral is found to take the following value:
? ?
J ? ? dr rJm(k ?m,?r)Jm(km?,??r)J?(q?r)?0??? 1? cos (m? ?m
?
2 ?1) , if km,? , km?,?? , q?can form a triangle , (8.24)
?2?A?= ??0 otherwise.
If a triangle with lengths km,? , km?,?? , and q? can be formed, the quantity A? denotes
the area of this triangle, and ?1 and ?2 are exterior angles of this triangle. Defining
? ? (k1 + k2 + k3)/2, we have
?
A? = ?(?? k1)(?? k2)(?? k3). (8.25)
The angles are given by:
( )
k2 ? k2 2
? = arccos 1 2
? k3
1 ( , (8.26)2k2k3 )
k2 ? k2 2
? = arccos 2 3
? k1
2 . (8.27)
2k1k3
8.3.2 Hankel transform of the exciton
In a circularly-symmetric system, the electronic bands are conveniently given by
Bessel functions. Decomposing the excitonic wave function in terms of these single-
particle states is equivalent to a Hankel transform of the exciton. Let the relative motion
of electron and hole of the exciton be described by a principal quantum number n, and an
155
angular momentum quantum number j, and the center-of-mass (COM)motion be described
by quantum numbers J for angular momentum and momentum and N for momentum/energy.
The overlap of such exciton with an electron in a state described by quantum numbers
m, ?, and a hole described by m?, ? ? reads:
? ?
Bn,j;J,N (com) (rel)m,?;m?,?? = Nm,?Nm?,?? dre drh??J,N (R)??n,j (r)
(8.28)
? Jm(km,?re)Jm?(km?,??rh) exp [i(m?e ?m??h)] .
Here, R = (re + rh)/2 describes the COM motion, and r = re ? rh describes the relative
motion, and re,h are expressed in polar coordinates (re,h, ?e,h). As a first step to evaluate
B, we will expand the Bessel functions (including the one contained in the definition of
(com)
??J,N , see main text) in terms of plane waves. To this end, we note that
n?=?
eiz cos(?) = inein?Jn(z). (8.29)
n=??
Therefore,
n?=?
eik??r?e = eikre cos(?e??k) = inein?ee?in?kJn(kre). (8.30)
? n=??
Integrating both sides over 2? d? eim?kk gives0
? 2?
J (kr )eim?
1
e m ik??r?e im?k
m e = (?i) e e d?k. (8.31)
2? 0
Applying this expansion, below we associate k ? (k, ?k) with electron momentum, and
k? ? (k?, ?k?) with hole momentum, where k and k? are introduced as short-hand notations
156
for km,? and km;,?? . Similarly, the center-of-mass momentum will be associated with a
vector Q = Q, ?Q. With this, and re-expressing electron and hole coordinates in terms of
relative and center-of-mass coordinates, we get
m?m?+J
Bn,j;J,N ?im,?;m?,?? = ?N?n,jNJ,N?Nm,?Nm??,?? (2?)32? 2? 2? ? ?
? ?d? d? d? dr dReiR?(k?k ?Q)? (8.32)k k Q
0 0 0
|?|
? eir?(k+k?)/2eim? ?k ?im ? 2|j|e k?e?iJ?Q?|j|e? 2 Ln?|j|(?)e
?ij?rel ,
where ? ? r?n with ?n ? 2r/[a0(n+ 1)]. The integration in R yields a ?-function, which2
imposes a triangle condition for the momenta:
?
?
dReiR?(k?k ?Q) = (2?)2?(2)(k? k? ?Q). (8.33)
Let us next carry out the angular part of the integral in r.:
?
d? e?ij?releirq cos(?rel??q) = 2?ije?ij?qrel Jj(qr). (8.34)
157
Here, as a short-hand notation, we have introduced q = (q, ? ) ? k+k?q . For the radial2
part of the relative position integral we write:
? ( )
? q
f(q, n, j) ? d??j+1e? 2j2Ln?j(?)Jj ?
? ?? ? nn j ( 2(?1)= )s( q )j?n ?(n+ j + 1)?(2j + s+ 2)j+s+ 3
2q2 1 2 ?(s+ 1)?(j + 1)?(n? j ? s+ 1)?(2j + s+ 1)s=0 (?2 +n 2 )
2
? ?s ?1? s 4q2F1 , , 1 + j,? ,
2 2 ?2n
(8.35)
with 2F1 being the hypergeometric function Note that here we have changed the integration
variable from r to ?, which yields a factor 1/?2n. For the analytic solution of the integral,
we have taken the integration boundary to be at infinity.
Putting all together, we arrive at the following intermediate result:
m?m?+J+j
Bn,j;J,N im,?;m?,?? = ?N?n,jNJ,N2? ?Nm,?Nm
??,?? f(q, n, j)??2n2? 2?
? d? d? (2) ? i(m? ?m?? ?j? ?J? )k k? d? ? (k? k ?Q)e k k? q QQ .
0 0 0
(8.36)
Let us for a moment assume ?Q to be fixed. Then, in the remaining two integrals, ?k and
?k? will be fixed such that the triangle co(ndition expr)essed by the ?-function is met. To
proceed, we write the delta function ?(2) k? ? k?? ? Q? explicitly in terms of ?k, ?k? , and
158
?Q:
( )
?(2) k? ? k?? ? Q? = ? (k cos? ? k?k cos?k? ?Q cos?Q)
(8.37)
? ? (k sin? ? k?k sin?k? ?Q sin?Q) .
To perform the integral over these ? functions, we first note the identity
? ? ?g(?w) ??
d?g(?)?(h(?)) = ??? ?dh(?) ?w ? ??? , (8.38)?=?w
where ? = ?w denote the solutions to h(?) = 0. We define h1(?k) = k cos?k ?
k? cos?k? ? Q cos?Q and h2(?k?) = k sin?k ? k? sin?k? ? Q sin?Q. The zeros of h1
and h2 are given by
k? cos?k? +Q cos?Q
cos?k = , (8.39)
k
k sin?k ?Q sin?Q
sin?k? = ? . (8.40)k
If there is a triangle with lengths given by k, k?, and Q, the angles ?k and ?k? of such
triangle solve these equations. Note that a second solution can then be obtained from a
mirror transformation of the triangle, but since the solutions are equivalent, considering
only one of them is sufficient. On the other hand, if such triangle does not exist, the
equations (8.39) have no solution, and the integral is zero. Explicitly, the solutions can be
159
written as
( )
k2 +Q2 ? k?2
?k = ??k + ?Q with ??k = arccos ( ), (8.41)2kQ
k2 ?Q2 ? k?2
?k? = ??k? + ?Q with ??k? = arccos . (8.42)
2k?Q
The angle ?q in Eq. (8.36) is given by:
( )
?1 qy?q = tan (qx )
? ky + k
?
y
= tan 1(k + k?x x )
k sin(?? ??1 k ? ?Q) + k sin(??k? ? ?Q)= tan ( ) (8.43)k cos(??k ? ? ) + k?Q cos(??k? ? ?Q)
?
?1 k sin(??k) + k sin(??k?)= tan + ?
? Qk cos(??k) + k cos(??k?)
? ??q + ?Q.
We can also express q in terms of k, k?, ??k, and ??k?:
?
1
q = k2 + k?2 + kk? cos(??k ? ??k?). (8.44)
2
??? ????1 ??? ??1The term dh1(?k) ? dh2(?k? )?? , evaluated at ?k = ??k+?Q and ?k? = ??k? +?Q, whichd?k d?k?
we get according to Eq. (8.38) from the two integrals in ?k and ?k? , is given by:
|h? 11(??k + ? )|?1|h?Q 2(??k + ?Q)|?1 = . (8.45)
kk?| sin(??k ? ??k?)|
160
Putting all together, we arrive at
m?m?+J+j
Bn,j;J,Nm,?;m?,?? = N?n,jNJ,NN
i 1
m,?Nm??,?? f(q, n, j)?2n kk?| sin(??k ? ??k?)| (8.46)2?
? ei(m??k?m???k??j??q) d? ei(m?m??j?J)?QQ .
0
From this, we finally obtain the selection rule ?m?m?,j+J which expresses the conservation
of angular momentum. Recalling that k = k , k?m,? = km?,?? , Q = kJ,N , we write as the
final result
Bn,j;J,N 2?m,?;m?,?? = ?m?m?,j+JN?n,jNJ,NNm,?Nm?,?? f(q, n, j)?2n (8.47)
? 1 ei(m?? ?k?m ??k??j??q).
km,?km?,??| sin(??k ? ??k?)|
The angles ?k, ?k? , ??q are defined in Eqs. (8.41) and (8.43). The value of q is given by
Eq. (8.44).
161
Chapter 9: Conclusions and Future Avenues of Work
As we discussed in Ch. 1, the fields of photonics/optoelectronics are in need of
new optically controllable materials that operate efficiently in the visible regime to help
overcome the computational issues that we face today. From a practical perspective,
three things need to be achieved in order for a material to becomes useful: (I) the system
needs to be well characterized, meaning that all particle interactions are documented, (II)
selective control of optical states, and (III) reliable control mechanisms. The study of
higher energy exciton species in this thesis addresses a mixture of points (I) and (II).
We have demonstrated the first magnetic field measurements of the novel charged
2s exciton and utilized the valley Zeeman effect to extract an anomalously large g-factor.
Recent reports in the literature show that 1s exciton species in TMDs exist along a
sliding scale of carrier density induced many-body states. Our measurements place the
first marker in mapping this interaction for 2s charged excitons, and indicate that the
behavior observed for 1s exciton species likely continues into the 2s regime. From a
fundamental physics perspective, this opens up a unique opportunity to study many-
body interactions in higher energy exciton species. This would be energetically very
difficult in traditional semiconductors systems with smaller exciton binding energies ?
like GaAs quantum wells ? showcasing another area in which TMDs outperform their
162
industry standard counterparts.
From an applications perspective, many-body interaction effects could be particularly
useful for controlling excitons selectively. When a system is fully valley-polarized ?
a possible result of many-body interaction that we discussed in Ch. 5 ? it can induce
complete, preferential quenching of an exciton species which depends on the direction of
the applied magnetic field (?z?). Having a larger family of documented exciton species
over an an increased energy range allows for more options when engineering optical
devices, making the 2s species a valuable addition.
In the later part of the thesis, we showed that twisted light does not modify the
exciton spectrum in 2D semiconductors (including TMDs). This work was initially motivated
by the idea that we could utilize twisted light to brighten dipole-forbidden transitions in
TMDs (and other semiconductor systems), which would have been an novel, selective
optical technique. However, this work yielded the interesting result that the momentum
is instead transferred to the COM which has its own uses. As discussed in the Ch. 8,
this could be used to provide excitonic lattices with artificial magnetic flux to aid in the
production of chiral Mott insulators [65].
Another particularly interesting implication is that twisted light could be used as an
alternative for performing in-plane momentum resolved PL ? a technique that is particularly
useful for studying band dispersion ? but can be tricky to achieve. Our theoretical results,
coupled with some early experimental results by another group indicate that the transfer
of OAM to the COM can be used to reconstruct the dispersion of exciton bands [277].
Application of light with increasing ? pushes the electron to higher energies/momentum
within the band and creating another way to measure the dispersion through PL.
163
Appendix A: Fabrication Recipes
A.1 Polymers
? Elvacite
1. Per online instructions, the polymer should prepared in a 20% concentration
by weight solution. The 80% volume of methyl isobutyl ketone (MIBK)
should be agitated by the magnetic stirrer prior to and during the inclusion
on the powdered Elvacite 2550 to reduce clumping.
2. Agitate the polymer for 4 hours.
3. Every hour after the first 4 hours, a seed sampling test should perform. This is
done by spreading a small dot of the mixture on a glass slide and evaporating
the solvent to see if seeds remain. Repeat until no seeds are left during the
seed test.
4. Spin at 750 rpm for 1 minute, bake at 120 ?C for 10 minutes to cure.
? Polycarbonate (PC) - 15% Solution by Weight
1. Carefully measure and pour 15g of PC and 57mL of chloroform into an amber
bottle and replace the lid.
164
2. Place the sealed bottle into a sonicator filled with just enough water that the
bottle isn?t floating and let sonicate for about 6 hours. Note: There will still
be a solid in the middle of the sample, but as long as it has formed a ball in
the middle that?s okay.
3. Spin at 750 rpm for 1 minute, bake at 100 ?C for 5 minutes to cure.
? Polypropolene Carbonate (PPC) - 15% Solution by Weight
1. Carefully measure and pour 15g of PPC and 57mL of chloroform into an
amber bottle and replace the lid.
2. Place the sealed bottle into a sonicator filled with just enough water that the
bottle isn?t floating and let sonicate for about 6 hours. It should completely
dissolved.
3. Spin at 750 rpm for 1 minute, bake at 70 ?C for 4 minutes to cure.
165
Appendix B: Expanded Derivations for Magnetic Field Effects
B.1 The Weak-Field Zeeman Effect
Beginning by restating Eqn. 5.9
? ? ? ?
(1) eB ? ?
En,l,j,m = n, l, j,mj ? L?z + 2S?z ?n, l?, j,m? . (B.1)j 2m j
To start, we can look to see if Eqn. B.1 commutes with any of the operators in the system
to try to reduce our task at hand. We start with the commutation relation,
[L2, L?z + 2S?z] = 0. (B.2)
Recalling our elementary quantum mechanics, we know this to be true because L2 commutes
with any L operator and L and S are compatible observables. This implies that ?l,l? is true,
which allows us to reduce to,
?
(1) eB
En,l,j,m = n, l, j,mj ??? ?? ?L?z + 2S?z ?n, l, j,m?j . (B.3)j 2m
Similarly,
[J?z, L?z + 2S?z] = [L?z + S?z, L?z + 2S?z] = 0. (B.4)
166
Which implies that ?mj ,m? is true, and thatj
(1) eB
? ?? ?? ?
En,l,j,m = n, l, j,mj ? L?z + 2S?z ?n, l, j,mj . (B.5)j 2m
This allows us to use non-degenerate perturbation theory and tells us that the matrix for
our Zeeman Hamiltonian is perfectly diagonal. Now the task at hand is to solve for the
energy correction. We begin by noting that if we apply the definition of J?z we can turn
L?z + 2S?z = J?z + S?z, and we get,
? ? ? ?
(1) eB ? ?
En,l,j,m = [?n, l, j,mj j ? J?? z +? S?z ?n, l, j2meB ? ?
,mj? ? ? ? ]
= ?n, l, j,mj ? J?2m[ ??
?
z ? ? ?n, l, j,mj ?+ ?n, l, j,mj ? S???z ?n, l, j,mj
I ] ? (B.6)II
eB
= ?
2m ???? ?? ?
? ??
mj + n, l, j,mj ? S???z ?
?
n, l, j,mj ? .
I
II
The result of term I in Eqn. B.6 is a simple consequence of the J?z operator, but term II is
a little trickier; we will now focus just on this to get our full result.
To begin with, we start with the statement that S? is a vector operator under J? ,
[J?i, S?j] = i??ijkS?k. (B.7)
In quantum mechanics, vector operators have an unusual property that is,
[J2[J2, S?]] ? (S? ? ? 1J?)J? (J2S? + S?J2). (B.8)
2
167
Supposed we take,
? ???? ? ? ? ? ? ?1 ?jm [J2[J2, S?]] ?? ? 1 ?j jm ?j = jmj ? (S? ? J?)J? ? (J2S? + S?J2) ?? jmj , (B.9)? 2
where |jmj? is some generic eigenstate of J? . Since there will always be a J2 term near
both the bra and ket in the expectation value on the left-hand side, we can see that the
terms they produce will cancel and we?ll be left with,
? ??
0 = ?jmj ?
??
? (S? ? J?)J? ?
? ? ??
jm + jm ? ?? ?j j ??1(J2S? + S?J2) ?? jmj
? 2
= jmj ? ??? ? ? ??? ??? ?
(B.10)
(S? ? J?)J? jmj ? ?2j(j + 1) jmj S? jmj .
This implies that,
? ? ?(S? ? J?)J??jS? j = , (B.11)
?J?2?j
which is a projection formula, and we can use this result to finish computing Eqn. B.6.
Applying this formula to
[
eB ? ?? ?? ? ]
II = n, l, j,m[j ??S?z ?n, l, j,?mj2meB ?m ?? ? ? ] (B.12)j ?= n, l, j,mj S? ? J? ?n, l, j,m?2 j ,2m j(j + 1)
we can use the fact that J? = L?+ S? ? S? ? J? = 1(J2 + S2 + L2) and assume a spin =1/2
2
168
system,
[
eB ?mj 1 ? ? ? ? ]
II = n, l, j,m ? J2 + S2 + L2 ?n, l, j,m
2m ?2 j jj(j + 1) 2
? ?2 ? ?? ? (B.13)eB mj= (j(j + 1) + s(s+ 1)+l(l + 1)).2m ?2j(j + 1) 2
=3/4
Plugging into Eqn. (B.6) and reducing, we find that
[ ( )]
(1) eB j(j + 1)? l(l + 1) + 3
En,l,j,m = ?mj 1? + 4 . (B.14)j 2m 2j?(?j + 1) ?
gJ (l)
gJ(l) is the Lande? g-factor which takes into account all of the contributions to the angular
momentum of the multiplet. It was first derived in 1921 by Alred Lande? in 1921 [160].
We can further reduce and find that,
(1)
En,l,j,m = ?BBgJ(l)mj. (B.15)j
Here, ?B is the Bohr magneton, which is derived in the next section. Overall, we can take
away that the magnetic field will split the energy linearly.
B.2 The Bohr Magneton
The magnetic dipole moment is the amount of torque caused by an external field on
the dipole on a magnetic object and the Bohr magneton ?B is the magnetic dipole moment
of the hydrogen atom. Since most of our understanding of quantum mechanics begins
with a fundamental understanding of the hydrogen atom, ?B is generally considered the
169
Figure B.1: (A) Current carrying loop with magnetic moment ?? and (B) the Bohr
magneton (?B) arising from the Bohr model of the hydrogen atom.
natural unit of the magnetic dipole moment. Where does this unit come from though?
As the name might suggest, ?B arises when we envision the hydrogen atom in the Bohr
picture (Figure B.1(B)). The single electron orbiting around the proton core of the hydrogen
atom is very similar to a current carrying loop (Figure B.1(A)), since current is fundamentally
moving electrons (or holes in a semiconductor).
Recall from elementary physics class that the magnetic moment (derived through
the torque on a current carrying loop) is,
?? = IA?. (B.16)
Where ?? is the magnetic moment, I is the current in the loop, and A? is the area vector of
the loop. Extending this to the Bohr picture, the current in that system is
I = ? e . (B.17)
?
Where -e is the single electron charge, and ? is the time that it takes to orbit the nucleus.
If we assume that the electron travels in uniform circular motion around the nucleus, we
170
can define the orbital period as,
2?r
? = . (B.18)
v
Here, r is the radius of the orbit and v is the velocity the electron. The assumption that the
electron is experiencing uniform circular motion also allows us to define the area vector
in the system and rewrite Eqn. (B.16) as,
e evr
?? = IA? = ? 22?r ?r = . (B.19)2
v
Recall, again from elementary physics, that the orbital angular momentum L? is
defined as,
L? = r? ? p? (B.20)
Here, p? is the linear momentum. Since r? and p? as always at a right-angle from one
another in uniform circular motion, we can show that the magnitude of the orbital angular
momentum as,
L = |L?| = |r? ? p?| = rp sin(?) = rmv. (B.21)
Substituting this result into Eqn. (B.19) and applying the full vector form,
( )
e
?? = ? L?. (B.22)
2me
If we consider a field applied along z?, the momentum eigenstates are all multiples of ?,
so we can take this to be the fundamental unit of momentum in this system, which gives
171
Figure B.2: Valley-resolved LLs in TMDs resulting from a real magnetic field. Note that
the energy axis is split so as to showcase both the valence and conduction bands at the
same time.
us the true definition of the Bohr magneton:
? e??B = . (B.23)
2me
B.3 Modified Landau Level Spectrum in TMDs
We start with the 2-band, basic k ? p Hamiltonitan for TMDs [327]:
H? ?0 = vF(?px??x + py??y) + ??z. (B.24)
2
Here, vF is the Fermi velcoty, ? = ?1 is the valley index in the ?K valley, px/py are
the momentum in the respective directions, ? is the optical gap, and ??i are the Pauli spin
172
matrices.
We want to study the application of magnetic field, so we want to replace the
momentum with a gauge-invariant form (i.e. use minimal substitution)
p? ? ?? = p?+ eA?(r?). (B.25)
Here, A?(r?) is the vector potential describing the magnetic field. We can use this substitution
in Eqn. B.24 to obtain
?? ??? ? vF(??x ? i?y)H? 2= ?B ?? . (B.26)
vF(??x + i?
?
y) ? 2
Since, in many way the QHE is similar to the quantum harmonic oscillator, is serves to
take advantage of this and write the gauge invariant momentum operators in terms of the
classic ladder operators, ( )
1 x p
a? = ? ? i (B.27a)
2 x0 p0
and ( )
a?? = ?1 x p+ i . (B.27b)
2 x0 p0 ? ?
Here, x and p are the position and momentum, x = ?0 , and p0 = ?m?. Workingm?
through some algebra, it can be shown that
??? = (a??x + a?) (B.28a)
2lB
173
and
??? ?y = (a? ? a?). (B.28b)
? i 2lB
Here, l ?b = is the magnetic length associated with the cylcotron motion of carriers ineB
the system. Substituting this into Eqn. B.26 and making the substitution that ? = ?vF
2 2?lB
we get,
?? ??? ? 1??(a?(1 + ?) + (? ? 1)a??)H? 2 2 ?= ?B ? . (B.29)
1??(a?(? ? 1) + (1 + ?)a??) ??
2 2
We now want to solve for the new eigenvalues in the system with the applied field,
so we take |H?B ? ?n?| = 0 and solve for ?n. This gives us,
1?
? 2 2n = ? ? + a (?1 + ? 2)?2?2 + 2a(1 + ? 2)?2?2a? + (?1 + ? 2)?2?2a?2. (B.30)
2
We want to simplify things, and also know what the behavior is in both of the
valleys, so we can substitute ? = 1 to get the +K-valley behavior
1? 1?
??=+1n = ? ?2 + 4?2?2aa? = ? ?2 + 4?2?2n, (B.31)2 2
and for ? = ?1 to get the -K-valley behavior
? ?
??=?1
1 1
n = ? ?2 + 4?2?2aa? = ? ?2 + 4?2?2n. (B.32)2 2
174
If we take the cyclotron frequency to be ? = e?Bc and do a first order expansion in bothm
valleys we get,
? ???n ? ??cn. (B.33)
2
A peculiarity in TMDs arising from inequivalent Berry curvature at each valley
(?+K = ???K) is that the n index labeling for the Landau levels is different in each
valley given by +Kc = n = 0,1,2,3... and -Kc = n = 1,2,3... while +Kv =-Kc and -Kv
=+Kc [254, 320]. An illustration can be found in Fig. B.3.
175
Appendix C: Elionix Standard Operating Procedure
Between the end of the first and second year of my doctoral study I was heavily
involved with the commissioning of the Elionix GS-100 system on campus. During that
time, it was my job to train and clear new users and as part of that work I wrote a detailed
standard operating procedure as a reference for users. That reference is still in use, and has
assisted dozens of users during their training and work. It details the standard operating
procedure for many common tasks that users perform. The rest of this section holds a
copy of the procedure.
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
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