Many-Body Effects in Optical Excitations of Transition Metal Dichalcogenides

Research Collection

Doctoral Thesis

Author(s): Sidler, Meinrad

Publication Date: 2018-02

Permanent Link: https://doi.org/10.3929/ethz-b-000290909

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ETH Library DISS. ETH NO. 24988

Many-Body Eﬀects in Optical Excitations

of Transition Metal Dichalcogenides

A thesis submitted to attain the degree of

DOCTOR OF SCIENCES of ETH ZURICH

(Dr. sc. ETH Zurich)

presented by

MEINRAD SIDLER

MSc ETH Physics, ETH Zurich

born on 21.06.1989

citizen of Grosswangen LU

accepted on the recommendation of:

Prof. Ata¸cImamo˘glu,examiner Prof. Bernhard Urbaszek, co-examiner

2018

Summary

This dissertation treats a quantum impurity problem in a semiconductor system. Quantum impurity problems describe the interaction between a single quantum object and a complex environment. They are ubiquitous in physical systems and represent a fundamental field of research in many-body physics. Prominent examples are the Anderson orthogonality catastrophe and the Kondo effect. In both cases, the impurity is much heavier than the constituents of the interacting environment. If the mass of the impurity is comparable to the surrounding particles, we have a mobile impurity. These systems are usually harder to solve, as evident in the case of lattice polarons, which were first proposed in 1933 by Lev Landau. A complex but accurate description was found years later in 1955 by Richard Feynman. In recent years, strong coupling between single, mobile quantum impurities and a fermionic bath was realized in cold atoms. The interaction results in the formation of new quasi- particles called Fermi polarons. In contrast to other mobile quantum impurities such as lattice polarons, Fermi polarons can be described with a simple and quantitatively accurate model, which renders them an especially attractive field of research of many-body physics. In this work, we report the observation of Fermi polarons in a solid state environment, namely a new class of semiconductors called transition metal dichalcogenides (TMDs). TMDs consisting of the transition metal Tungsten or Molybdenum and the chalcogenide Sulphur or Selenium are semiconductors. In the monolayer limit, they feature a direct bandgap, a large Coulomb interaction and a large effective electron and hole mass as compared to GaAs. In combination with the two-dimensional confinement, these result in a large binding energy of the exciton. As a consequence, the exciton remains a rigid particle even when it is surrounded by a two-dimensional electron system (2DES) with a large electron density. When the exciton is surrounded by a 2DES, a second resonance emerges in the optical spectrum. Previously, this resonance was attributed to the trion, a bound state of two electrons and a hole. In this dissertation, we demonstrate that this emerging red-shifted resonance has to be described as a Fermi polaron. Thanks to the large binding energy of excitons in TMDs, we can test the predictions of our model qualitatively and quantitatively for a large range of electron densities. For our experimental investigations, we employ cavity quantum electrodynamics in a zero-dimensional, tunable micro-cavity to investigate the optical spectrum of the TMD monolayer for different electron densities. The possibility to reduce the cavity length to a few wavelengths allows the formation of polaron-polariton modes. The strong light-matter coupling cannot be explained with the trion model, and provides solid evidence for the validity of the Fermi polaron model to describe optical resonances in a 2DES.

Zusammenfassung

Diese Dissertation behandelt eine Form eines Quantenstörstellenproblemsin einem Halbleitersystem. Quantenstörstellenprobleme beschreiben eine Wechselwirkung zwischen einem einzelnen Quantenobjekt und einer komplexen Umgebung. Sie sind in vielfältigenphysikalischen System anzutreffen und stellen ein fundamentales Ge- biet der Vielteilchenphysik dar. Bekannte Vertreter sind beispielsweise die Anderson Orthogonalitätskatastrophe oder der Kondoeffekt. In beiden Fällenist die Störstelle viel schwerer als die Bestandteile der wechselwirkenden Umgebung. Wenn die Masse der Störstellevergleichbar mit den umgebenden Teilchen ist, spricht man von einer mobilen Störstelle. Diese Systeme sind normalerweise schwieriger zu lösen,wie man am Beispiel der Gitterpolaronen sehen kann, die schon 1933 von Lev Lan- dau vorgeschlagen wurden, fürdie aber erst 1955 Richard Feynman eine komplexe, aber genaue, Beschreibung fand. In den letzten Jahren konnte die starke Koplung zwischen mobilen Quantenstörstelln und einem Fermisee in kalten Atomen realisiert werden. Die Wechselwirkung manifestiert sich in der Formation von neuen Qua- siteilchen namens Fermipolaronen. Im Gegensatz zu anderen mobilen Störstellen, wie zum Beispiel Gitterpolaronen, könnenFermipolaronen mit einem einfachen und quantitativ präzisenModel beschrieben werden, was sie besonders als attraktives Feld der Vielteilchenphysik auszeichnet. In dieser Arbeit, berichten wir die Beobachtung von Fermipolaronen in einem Festkörpersystem, nämlich in einer neuen Klasse von Halbleitern: Den Ubergangsmetalldichalkogeniden¨ (TMD). TMDs bestehend aus den Ubergangsmet-¨ allen Wolfram oder Molybdänund den Chalkogeniden Schwefel oder Selen sind Halbleiter. Als Monolage haben sie eine direkte Bandlücke, eine starke Coulomb- Wechselwirkung und grosse effektive Elektronen- und Lochmasse. In der Kombina- tion mit der zweidimensionalen Geometrie führenletztere zu einer grossen Exziton- bindungsenergie, welche bewirkt, dass das Exziton strukturell intakt ist, wenn es von einem dichten, zweidimensionalen Elektronensystem (2DES) umgeben ist. Wenn das Exziton sich in einem 2DES befindet, bildet sich eine zweite Resonanz im optischen Spektrum. Bisher wird diese Resonanz den Trionen, einem gebundenen Dreikörperzustand, zugeordnet. In dieser Dissertation argumentieren wir, dass jene optische Resonanz als Fermipolaron beschriben werden sollte. Dank der grossen Bindungsenergie der Exzitonen in TMDs, könnenwir qualitative und quantitative Prognosen unseres Modells in einem grossen Bereich von Elektronendichten testen. Für unsere experimentellen Untersuchungen benutzen wir Quantenelektrody- namik in einem null-dimensionalen optischen Resonator, um das optische Spek- trum der TMD Monolage fürverschiedene Elektronendichten zu untersuchen. Die Möglichkeit, die Resonatorlängeauf ein paar Wellenlängenzu reduzieren, erlaubt es uns, die Bildung von Polaron-Polaritonen zu zeigen. Diese starke Kopplung von Matterie und Licht kann vom Trionenmodell nicht erklärtwerden und liefert somit f stichhaltige Hinweise, dass das Fermipolaron eine gutes Modell von optischen Res- onanzen in einem 2DES darstellt. Contents g

Contents

Title b

Summaryd

Zusammenfassungf

Contentsh

List of symbols and abbreviations1

1 Introduction4

2 Transition metal dichalcogenides6 2.1 Crystal lattice...... 6 2.2 Tight binding model...... 7 2.3 Selection rules...... 7 2.4 Exciton...... 8 2.5 Exchange interaction...... 9

3 Theory of Fermi polarons 12 3.1 Quantum impurity problems...... 12 3.2 Theoretical model...... 13 3.2.1 Model Hamiltonian...... 13 3.2.2 Chevy Ansatz...... 13 3.2.3 Contact interaction...... 14 3.2.4 Spectral function...... 16 3.2.5 Quasiparticle weight...... 17 3.2.6 Charge distribution...... 18 3.2.7 Validity of our approximations...... 20 3.3 Fermi polarons in TMDs...... 20 3.3.1 Sample structure...... 20 3.3.2 Capacitive model for the Fermi Energy...... 22 3.3.3 Spectroscopy of an exciton in a 2DES...... 22

4 Fermi polaron polaritons 26 4.1 Cavity spectroscopy...... 26 4.1.1 Fiber cavity...... 26 4.1.2 Sample structure...... 28 4.1.3 Cavity modes...... 30 4.1.4 Resonant Transmission...... 31 4.1.5 Photoluminescence...... 33 h Contents

4.2 Strong light-matter coupling...... 34 4.3 Fermi Polaron-Polaritons...... 40 4.4 Theoretical modeling of Fermi polaron polaritons...... 42

5 Hole injection 46

6 Excitons, trions and polarons in magnetic fields 50 6.1 Introduction...... 50 6.2 Exciton magnetic moment...... 50 6.3 Measurement of the exciton magnetic moment...... 50 6.3.1 Sample and Setup...... 50 6.3.2 Polarization-resolved photoluminescence and reflection spectroscopy...... 52 6.4 Magnetic moment of localized trions...... 56 6.5 Polarons in a magnetic field...... 56 6.5.1 Correcting the phase shift in the reflection spectrum..... 60 6.6 Conclusion...... 60

7 Single photon emitters in WSe2 64 7.1 Introduction...... 64 7.2 Power dependence...... 64 7.3 Spatial map...... 66 7.4 Photon autocorrelation...... 69 7.5 Lifetime, spectral ﬂuctuation and blinking...... 69 7.6 Magnetic ﬁeld...... 72 7.7 Photoluminescence excitation...... 74 7.8 Conclusion...... 76

8 Conclusion and Outlook 78

Distributed Bragg reﬂector 80

Samples 82 1 WSe2 ...... 82 2 MoSe2 ...... 82 2.1 M1...... 82 2.2 M2...... 82

BibliographyII

List of publications and conference presentations XIV

Acknowledgment XVI

List of Figures XVIII List of symbols and abbreviations i

List of symbols and abbreviations

k Wave vector m Mass n Refractive index λ Light wavelength ω Light (angular) frequency

1 eV = 1.602 · 10−19 J 1 electronvolt e = 1.602 · 10−19 As Electron charge c = 299 458 792 m s−1 Speed of light ~ = 1.0546 · 10−34 Js Reduced Planck’s constant −31 me = 9.109 · 10 kg Free electron mass −12 −1 0 = 8.854 · 10 Fm Vacuum permittivity

BS Beam splitter CCD Charge-coupled device (camera) c Cavity cw Continuous wave DBR Distributed Bragg reﬂector FSR Free spectral range FWHM Full width at half maximum GaAs Gallium arsenide LED Light-emitting diode MoS2 Molybdenum disulphide MoSe2 Molybdenum diselenide MoX2 MoS2 and/or MoSe2 MX2 MoX2 and/or WX2 PBS Polarizing beam splitter PL Photoluminescence Si Silicon SiO2 Silicon dioxide Ti:sapphire Titanium-sapphire (laser) WS2 Tungsten Disulphide WSe2 Tungsten Diselenide WX2 WS2 and/or WSe2 j List of symbols and abbreviations Introduction 1

1 Introduction

The discovery of graphene launched a novel field, rich in interesting physical phenomena and promising applications [1]–[8]. However, the absence of a bandgap is a major drawback for experiments in optics and transport [9], [10]. As a result, mechanical exfoliation was applied to other van-der-Waals force coupled layered materials [2], [11], [12]. Amongst them are transition metal dichalcogenides (TMDs), a group of materials containing several semiconducting crystals [13], [14]. The direct bandgap in the monolayer limit make these semiconductor interesting for optics [15]–[17]. Stacking of layered materials allows to assemble heterostructures forming complex devices, using hexagonal boron nitride (h-BN) as an insulator, graphene as a semimetal and TMD monolayers as semiconductors [18]–[24]. Most notably, stacking a graphene layer onto a TMD monolayer embedded in h-BN, allows to change the electron density in the TMD monolayer when a voltage is applied between the TMD monolayer and the graphene. Electron and hole-doping, can be achieved in such field effect transistors. Due to the large Coulomb interaction, light-matter coupling is strong in TMD monolayers, which manifests itself in sub-picosecond spontaneous emission rates [25]–[30]. Encapsulation in h-BN shields TMD monolayers efficiently from their environment such that exciton linewidths as small as 2 meV [27], [31], [32] were observed, which is close to the radiative decay limited lifetime. In this work, we exploit another manifestation of the extraordinarily strong Coulomb interaction: The exciton in monolayer TMDs has a binding energy which is two orders of magnitude larger than that of GaAs [21], [33]–[36]. As a consequence, the exciton can be regarded as a rigid particle, which allows us to study in detail the interaction of excitons with a surrounding Fermi sea of electrons [29]. In other semiconductors such as GaAs, the exciton resonance rapidly broadens and the trion resonance merges with the electron-hole continuum as the electron density is increased which render it much more difficult to study the interaction of excitons with Fermi sea electrons in these systems. The exciton immersed in a Fermi sea of electrons constitutes a quantum impurity problem. The latter describes the interaction of a single quantum object with a complex environment, which is a fundamental problem of many-body physics. In this thesis, we argue that we can map the theoretical model used to describe Fermi polarons in ultracold atoms to the exciton in a Fermi sea of electrons. We find a new, qualitatively and quantitatively accurate description of optical excitations of semiconductors in the presence of a two-dimensional electron system (2DES). Our model deviates strongly from the established description which attributes the optical resonances to bound states in the form of charged excitons (trions). Our findings therefore shed new light on fundamental semiconductor physics. Beyond its implications for semiconductor physics, the observation of Fermi polarons in a solid state environment also provides a testing platform for unexplored 2 Introduction frontiers of quantum impurity problems. We demonstrate coupling of polarons to cavity photons which gives rise to polaron-polaritons. For small energies, these quasi- particles have an ultra-low mass yet they interact with the surrounding electrons. The combination of these two features are often exclusive [37] and thus highlights the promising new frontier in quantum impurity problems found in polaron-polaritons.

Scope of this thesis

This thesis is organized as follows: In chapter2, we review the main physical properties of TMDs relevant for optics. We discuss the crystal lattice and the band structure. Furthermore, we derive the selection rules and discuss the properties of excitons. We discuss the different forms of exchange interaction and its role in TMD excitons. In chapter3, we present the Fermi polaron model [37]–[41] and the Chevy ansatz [42] as a simple yet accurate, analytical solution of the Fermi polaron Hamil- tonian. We use the Fermi polaron model to describe the physics of an exciton interacting with a Fermi sea of electrons. We predict two sharp resonances in the optical spectrum which we identify as the attractive and repulsive polaron. The attractive polaron has previously been interpreted as the trion, a bound three-body state of two electrons and a hole. We demonstrate that experimental results of the TMD absorption spectrum as a function of electron density can be fitted with just one fitting parameter by the polaron model. Our strongest argument in favor of the polaron model is that a trion as a molecular state should barely couple to light whereas the attractive polaron is expected to interact strongly with light. In chapter4, we therefore use cavity quantum electrodynamics to investigate the coupling strength between polarons and light. The clear anticrossing of the polaron resonance with the cavity mode of a fiber micro- cavity [43] therefore demonstrates the superior accuracy of the Fermi polaron model to describe optical excitations of a 2DES. The band structure of TMDs features two bandgap minima labeled as the ±K valleys. In chapter6, we first demonstrate lifting of the degeneracy between the two valleys by a magnetic field and investigate its signature in the exciton energy. Next, we demonstrate how the polaron model can qualitatively reproduce the spectrum of a 2DES in a TMD monolayer in a magnetic field. Finally, we demonstrate the presence of quantum dots in WSe2 in chapter7. Using photon autocorrelation measurements, we show that the quantum dots we find in WSe2 are indeed single photon emitters. We demonstrate that these quantum dots have favorable optical characteristics, such as a long lifetime and a good spectral stability. Transition metal dichalcogenides 3

2 Transition metal dichalcogenides

2.1 Crystal lattice

The semiconducting TMD monolayers feature a honeycomb crystal lattice, similar to that of graphene. Figure 2.1a shows a few lattice sites of a monolayer MX2 crystal (M ∈ {Mo,W},X ∈ {S,Se}) where the unit cell is highlighted in green. In contrast to graphene, TMDs do not feature two identical atoms within the unit cell but rather a transition metal atom on the A site and two out-of-plane separated chalcogenide atoms on the B cite of the unit cell [44]. As compared to graphene, this breaks the inversion symmetry. The breaking of inversion symmetry allows for a finite Berry curvature at the ±K points [45]. The Berry curvature at the bandgap minimum unlocks novel physical phenomena such as the observation of the valley Hall effect (VHE). In the VHE, the Berry curvature acts like a magnetic field in k-space and therefore allows to observe a Hall effect in the absence of a real magnetic field [46]– [48]. Two unit lattice vectors d1 and d2 are indicated in the figure. The point group of the TMD monolayer is D3h. The first Brillouin zone of TMD monolayers is shown in figure 2.1b. The high symmetry points K, K’ (= −K), M and Γ are indicated. The monolayers can be stacked to bulk crystals in two different ways: In the 2H phase, the crystal’s unit cell in z-direction contains two monolayers. The top monolayer hereby is obtained from the first one by inversion symmetry and aligning its metal atoms on top of the bottom layer’s chalcogen atoms (figure 2.1c). The bulk crystals, which we exfoliate monolayers from, are in the 2H phase. In the 3R phase, the unit cell contains three monolayers which simply correspond to translations of the same monolayer.

a b c a y

x d2 K d1 Γ M -K S/Se W/Mo

Figure 2.1: a Crystal structure of a monolayer of MX2. Highlighted in green is the unit cell. d1 and d2 are two lattice vectors. b First Brillouin Zone of the bandstructure. c 2H stacking of monolayer TMDs to a bulk crystal. 4 Transition metal dichalcogenides

2.2 Tight binding model

Ab initio calculations of the bandstructure of TMD monolayers show a direct bandgap at the ±K-point for MX2. The wavefunctions at the valence band maximum (VBM) and conduction band minimum (CBM) are predominantly consisting of atomic orbitals of the transition metal. The VBM wavefunctions are mostly formed from the metal’s dx2−y2 and dxy orbitals while the wavefunction of the CBM is mostly due to the metal’s dz2 orbitals. The basis for a tight binding model simply aiming at describing the bandstructure around the bandgap minimum at the K- point therefore can be made from this relatively small basis [49]. More complicated models using 5, 7, 11 or 27 bands can more accurately describe the band structure away from the K-point [50]. In the three band tight-binding Hamiltonian, nearest-neighbor hopping will mix the dx2−y2 and dxy orbitals such that orthogonal linear combinations of these two orbitals form the valence band and a band at an energy above the conduction band minimum in the K-valley. In the opposite valley, the two linear combinations exchange their roles. The c + 2 band is more than 1 eV above the conduction band at the K-point and therefore is not signiﬁcant for experiments described in this work. The conduction band is described by the dz2 orbitals that does not mix with dx2−y2 and dxy at the K-point for symmetry reasons. The simplest k · p model we can formulate at ±K point contains only two bands [46] and can be written as:

∆/2 vD (±kx − iky) H±K = ~ (2.1) vD (±kx + iky) −∆/2

Using τ = ±1 as a pseudospin to describe which valley ±K the electron is in, we can reformulate equation 2.1 as:

HτK = ~∆/2σ ˆz + ~vDτzkxσˆx + ~vDkyσˆy, (2.2)

where vD is the Dirac velocity and ~∆ = Eg the gap energy at k = 0. This model does not allow to quantitatively capture the TMD bandstructure but serves as a qualitative model. The valence band is split by a large spin-orbit interaction of ∼ 200 meV. The spin-orbit induced splitting for the conduction band is of the order of magnitude of ∼ 20 meV [51]. Interestingly, the spin orbit interaction in the conduction band has opposite sign in Mo and W compounds which yields the lowest energy transition to be dipole forbidden in WS2 and WSe2 as we will see in the next paragraph.

2.3 Selection rules ~ Starting from equation 2.1 we can√ include optical√ transitions by replacing ~k with ~ ~ ~k − eA. If we choose Ax = A0/ 2 and Ay = iA0/ 2, corresponding to right hand circularly polarized light (σ+), we obtain for k = 0: 2.4. Exciton 5

MoX2 WX2

K -K K -K

Figure 2.2: Sketch of the band structure at the ±K-points. The red bands are are spin up, the blue ones spin down. The Optical selection rules allow for optical transitions with σ+ polarized light in the K-valley and with σ− polarized light in the -K valley. All allowed transitions are indicated with a vertical line in red or blue for σ+ respectively σ− allowed transitions. The spin-orbit interaction in the conduction band of MoX2 and WX2 have opposite sign. As a consequence the transition from the highest valence band to the lowest conduction band is allowed in MoX2 but not in WX2.

√ ~∆/2 − 2vDeA0 H+K = √ (2.3) − 2vDeA0 −~∆/2 ~∆/2 0 H−K = . (2.4) 0 −~∆/2 In the rotating wave approximation (RWA), this means that we can couple the valence and conduction band at +K with σ+ polarized light but not in −K. If we do the same calculations for σ−, we will find that σ− light only excites electrons from the −K valence band to the conduction band but does not have any impact on the K valley. We would obtain the same conclusion from a group theory approach. All allowed optical transitions are summarized in figure 2.2. The correlation between spin, valley pseudospin and photon helicity opens up the possibility to measure valley selective effects such as the valley Hall effect [46], [47].

2.4 Exciton

The energy of electron-hole pairs is lower than the bare bandgap. This is due to the direct Coulomb interaction between the electron-hole pair which reduces the exciton energy by the binding energy Eb with respect to the bandgap Eg. If the exciton can recombine under the emission of a photon, it is called a bright exciton. Dark excitons consist of an electron-hole pair whose optical recombination is dipole forbidden, e.g. due to their spin conﬁguration or because the dipole moment of the exciton recombination is parallel to the optical axis [52]. The Coulomb interaction in TMD monolayers is especially strong due to their two dimensional nature. On the one hand, the binding energy Eb is increased by a factor of four just from the conﬁnement of the charge carrier within the same plane [53]. In 6 Transition metal dichalcogenides

addition, TMDs have relatively large effective electron mass mc and hole mass mh in the order of mc ≈ mh ≈ 0.5me [49], [54], where me is the electron mass. This yields a reduced mass of ∼ 0.25 me which is significantly larger than the reduced mass of ∼ 0.06 me found in GaAs [55]. Furthermore, the electromagnetic field outside the monolayer is subject to a potentially much lower screening than the field penetrating out of a quantum well embedded within a semiconductor crystal. This effect is most prominent in monolayers in vacuum but even encapsulated monolayers benefit from a lower permittivity of h-BN as compared to TMDs. Theoretical simulations and experiments suggest that the Bohr radius in TMDs is of the order of aB ≈ 0.5−1.7 nm and the binding energy is around Eb ≈ 500 meV [34], [56]–[62]. This is approximately 50 times larger than the binding energy in a GaAs quantum well [63]. It is this strong exciton binding energy that yields TMDs an excellent platform to study many-body effects of an exciton interacting with a Fermi sea of electrons. [56], [61], [64], [65] As a consequence of the small Bohr radius, we can also expect strong interaction with light [56], [60]. The small Bohr radius further implies a large spread of the exciton wavefunction in k-space, including k-vectors for which the band structure deviates strongly from a paraboloid. [56], [58], [61], [66]

2.5 Exchange interaction

In addition to the direct Coulomb interaction, we should also consider the effects of electron-hole exchange interaction. Exchange interaction increases the energy of a bound state of an electron-hole pair. This means that bright excitons will shift to higher energies due to electron-hole exchange. Dark excitons are not affected by this shift. The exchange interaction which causes a splitting between the bright and the dark exciton is called short-range exchange. Figure 2.3a shows different excitons in MoX2 materials. In this figure, as well as for the rest of this work, ESO is the spin-orbit interaction, Eg is the gap energy, Ee-h is the electron-hole exchange energy and Eb is the exciton binding energy. We notice that the lowest energy intra- valley exciton is bright because the splitting between the bright and dark intra-valley exciton due to spin-orbit interaction is larger and of opposite sign than the electron- hole exchange splitting. In addition to the the intra-valley excitons, there is a dark inter-valley exciton which might be a possible reason why in photoluminescence (PL) the trion (or attractive polaron resonance, as we will show later) is usually brighter than the exciton, even in MoX2 materials where the lowest energy intra-valley is bright [31].

Figure 2.4a shows the exciton configurations in WX2. Since the sign of the spin- orbit interaction is opposite to that of MoX2, the lowest energy exciton is dark. Long range exchange will mix the electron and holes of opposite valleys. Most likely, this inter-valley exchange is largely responsible for a poor apparent conserva- tion of the selection rules in TMDs: Upon excitation by circularly polarized light, the photoluminescence (PL) is only partly circularly polarized [30], [67]–[70]. The degree of polarization, called circular dichroism, is defined as η = (Ico − Icross)/(Ico + Icross), where Ico defines the intensity of PL emitted with the same polarization as the excitation laser and Icross describes the PL intensity in the opposite polarization. Typical values are around 30%, which means that within its lifetime, the exciton undergoes a 2.5. Exchange interaction 7

MoX2

K -K

ESO Eb Dintra E -E KK SO e-h B E intra KK e-h D inter K K'

Figure 2.3: The energy of diﬀerent excitons in MoX2. Excitons labeled ’D’ are dark, those labeled with ’B’ are bright. Due to the opposite contribution of the spin-orbit interaction and the electron-hole exchange interaction to the splitting of the bright and the dark intra-valley exciton, the lowest energy intra-valley exciton is bright. In addition, a dark inter-valley exciton has even less energy since it is not subject to electron-hole exchange. sizable part of the oscillation from one valley to the other due to long-range exchange interaction. 8 Transition metal dichalcogenides

WX2

K -K

ESO

Eb Bintra E +E KK SO e-h D D intra KK inter K K'

Figure 2.4: The energy of diﬀerent excitons in WX2. Excitons labeled ’D’ are dark, those labeled with ’B’ are bright. In WX2, both, spin-orbit interaction and the electron-hole exchange interaction, lower the dark intra-valley exciton energy with respect to the bright one. As a result, the lowest energy intra-valley exciton is dark. In addition, there exists a dark inter-valley exciton with approximately the same energy as its intra-valley counterpart. Theory of Fermi polarons 9

3 Theory of Fermi polarons

3.1 Quantum impurity problems

The non-perturbative interaction of a single quantum impurity with a reservoir of bosons or fermions constitutes an interesting field of research of many-body physics: On the one hand, the interaction is strong enough to substantially change the properties of the impurity as well as the reservoir which can best be described with the introduction of new quasi particles. On the other hand, the interaction is often simple enough such that the resulting quasi particles can be described either with an analytical model or using modest numerical resources. We can classify quantum impurity problems in two ways: by the statistics of the reservoir (Bose/Fermi) and by the mass of the impurity as compared to the reservoir constituents. Some well known representatives are the Fermi edge singularity [71] and the Kondo effect [72] where we have an infinite mass impurity interacting with a Fermi gas. The Kondo effect in turn is a nice example where the quantum impurity problem led to the development of a new numerical technique, namely the numerical renormalization group [73]. Another example of a quantum impurity is the formation of polarons. In this case, the impurity has a finite mass. A representative of the mobile impurity is the lattice polaron which is an electron dressed with lattice phonons. In this case, the reservoir is formed out of bosons. Polarons are studied for a long time and were first introduced by Lev Landau in 1933. More recently, proposals suggest the formation of polarons in a BEC. In this way, a new strong coupling regime between the impurity and the Bose-Einstein condensate could be realized [74], [75]. In 2009, the interest in polarons reignited with the first observation of a Fermi-polaron, an impurity coupling to a Fermi reservoir, in ultracold atoms [37]–[41]. In cold atoms, the impurity is represented by a few spin down 40K atoms interacting with a Fermi sea of spin up 40K atoms. Most notably, in these Fermi-polarons a metastable repulsive polaron is formed concurrently with the attractive polaron. In this chapter, I would like to argue that TMDs represent a promising platform to study Fermi-polaron physics in solid state. Moreover, a prominent absorption resonance that was so far interpreted as trions [76] can be explained qualitatively and quantitatively by a polaron model. Furthermore, we present a new frontier of quantum impurity problems: Coupling the polaron to a micro-cavity [77], [78], we can provide a scenario where the mass of the quantum impurity is not just comparable to the mass of the reservoir but rather up to four orders of magnitude lighter than the mass of the reservoir [79] within a small fraction of the polarons dispersion. This completes the scheme of infinite mass, mobile and ultra-low mass impurities. The results presented in chapters3 and4 were published in [29]. The theoretical work was done by Ovidiu Cotlet in collaboration with Eugene Demler and Atac 10 Theory of Fermi polarons

Imamoglu. The samples were fabricated by Patrick Back. The dimple was fabricated by Thomas Fink. Furthermore, Ajit Srivastava and Martin Kroner contributed to the paper.

3.2 Theoretical model

3.2.1 Model Hamiltonian What renders TMDs a good platform to study many body physics is the large exciton binding energy Eb ≈ 500 meV (see section 2.4) which is larger than any other relevant energy scale such as the optical linewidth, the trion binding energy ET ≈ 25 meV [65] and the Fermi energy EF induced in typical Field eﬀect transistors of TMD monolayers. As a consequence, an optically generated exciton in a TMD monolayer in the presence of electrons can be considered as a robust mobile bosonic impurity embedded in a fermionic reservoir. The Hamiltonian describing the system is

X † X † H = ωX (k)xkxk + kekek k k X † † + Vq(xk+qek0−qek0 xk + h.c.) , (3.1) k,k0,q

where the first line describes the energy of 2D excitons and the Fermi sea electrons, described by the exciton annihilation operator xk and electron annihilation operator ek respectively. The second line of the Hamiltonian describes the interaction of the Fermi sea electrons with the exciton. This simplified description in which we treat the exciton as a rigid body is valid if the electron density ne is low such that the average inter-electron separation is much larger than the exciton Bohr radius aB i.e. aBkF 1. We assume a simple parabolic dispersion for the Fermi sea electrons and a parabolic dispersion with an offset δ(EF) for the exciton:

~k2 ~k2 ωX (k) = −Eexc + + δ(EF), k = (3.2) 2mexc 2mc δ(EF) = βEF, (3.3)

where mexc and mc are the masses of the exciton and the electron. The proportionality constant β is a ﬁtting parameter which accounts for band gap renormalization as well as for a reduction of the exciton binding energy with increasing electron density.

3.2.2 Chevy Ansatz In order to solve the Hamiltonian of equation 3.1, we will truncate the Hilbert space. An ansatz that has proved to be remarkably accurate was proposed by F. Chevy [42] and describes an exciton which can be dressed by maximum one electron-hole pair: 3.2. Theoretical model 11

! (p) † X † † |Ψ i = φ0cp + φk,qxp+q−kekeq |0i, (3.4) k,q where we deﬁne the Fermi surface to be our vacuum state |0i which means that a hole describes the annihilation of a conduction band electron below the Fermi energy and not the usual valence band hole. Here, and for the rest of this chapter, 0 summation over q or q implies a summation from q = 0 up to q = kF , whereas a 0 summation over k or k means a summation from k = kF up to inﬁnity. We want to solve the Hamiltonian of eq. 3.1 using a variational approach. This means that we minimize hΨ(p)|E − H|Ψ(p)i. Inserting the Chevy ansatz into eq. 3.1 yields:

! (p) (p) 2 X 2 (p) hΨ |E − H|Ψ i = E |φ0| + |φk,q| − Hvar (3.5) k,q (p) (p) (p) 2 X 2 Hvar = hΨ |H|Ψ i = ωX (p)|φ0| + EX (p, k, q)|φk,q| (3.6) k,q 2 X X ∗ + |φ0| V0 + [φ0φk,qVk−q + c.c.] q k,q X ∗ + φk,qφk0,qVk−k0 + c.c. k,q,k0 X ∗ − φk,qφk,q0 Vq−q0 + c.c. k,q,q0 ∗ ∗ We take the derivative of eq. 3.6 with respect to φ0 and φkq to obtain:

X X Eφ0 = ωX (p)φ0 + φ0V0 + Vk−qφk,q (3.7) q k,q X X Eφk,q = EX (p, k, q)φk,q + Vk−qφ0 + Vk0−kφk0,q − Vq0−qφk,q0 , (3.8) k0 q0 where we obtained eq. 3.7 and eq. 3.8 by taking the derivative of eq. 3.6 with ∗ ∗ respect to φ0 and φkq respectively and EX (p, k, q) ≡ ωX (p + q − k) + (k) − (q) is the energy of an exciton dressed with an electron-hole pair. We notice that the trion in the sense of a bound three body state can be described by a subspace of the truncated Hilbert space captured by the Chevy ansatz.

3.2.3 Contact interaction

In a next step, we can approximate the interaction potential Vk between electrons and the exciton by a contact interaction. This means that we assume Vk be constant up to quasi momentum k = Ω, where Ω represents the ultra violet cutoﬀ [80]: ( V if k < Ω, Vk = (3.9) 0 otherwise. 12 Theory of Fermi polarons

Our model should work independent of the ultraviolet cutoff Ω. Therefore, we need both sides of equations 3.7 and 3.8 to converge for Ω → ∞. For the moment, we will assume that V will grow as 1/V = O (ln Ω) as Ω → ∞. In equation 3.14, we will a posteriori show that our assumption for V was justified. We notice that in the limit of Ω → ∞ and therefore V → 0, the last term of equation 3.8 tends to zero since the sum stops at kF and not at Ω. The same holds true for the second term of eq. 3.8 and eq. 3.7 which we are however going to keep for simplicity. Next, we introduce the function χ defined as:

X χq = φ0 + φk,q. (3.10) k

Using χ, we can rewrite φ0 and φk,q as:

P V q χq φ0 = (3.11) E − ωX (p) V χq φk,q = . (3.12) E − EX (p, k, q) Inserting equations 3.12 and 3.12 into equation 3.10 yields:

P V q0 χq0 X V χq χq = + . (3.13) E − ωX (p) E − EX (p, k, q) k

In the limit of kF = 0 and for p = 0, we expect to obtain a solution at the trion binding energy and therefore, we obtain:

X V χq χq = −ET − EX (p = 0, k, q = 0) k 1 X 1 ⇒ = V −ET − EX (0, k, q) k Ω X 1 = 2 2 −ET − k /(2µ) k ~ 2 2 −2 µ 1 ~ Ω = − (2π) 2 log 1 + , (3.14) ~ ET 2µ

where µ = (memX)/(me +mX) is the reduced mass of the exciton-electron system. We see that our assumption of a logarithmic growth for the cutoff tending to infinity was justified. For a finite electron density, we find:

" Ω #−1 X 1 X 1 E − ωX (p) = − ≡ ΣX (p). (3.15) V E − EX (p, k, q) q k=kF 3.2. Theoretical model 13

a b 101

0.12 E F −E C = 5 meV E F −E C = 5 meV 0 E F −E C = 20 meV 10 E F −E C = 20 meV 0.10 AP E F −E C = 35 meV E F −E C = 35 meV 10-1 0.08

10-2 0.06

RP 10-3 0.04 Spectral function (a.u.) Spectral function (a.u.) -4 0.02 10

0.00 10-5 20 0 20 40 60 20 0 20 40 60 Photon energy - exciton energy (meV) Photon energy - exciton energy (meV)

Figure 3.1: a) The spectral function A(ω) for diﬀerent values of the Fermi energy EF . The two prominent resonances correspond to the attractive and repulsive polaron. b) The same spectral function plotted on a logarithmic plot reveals a weak resonance in between the attractive and the repulsive polaron: the molecular states (i.e. trion states in the context of semiconductors).

This equation describes the self energy ΣX (p) of an exciton with momentum p and energy E dressed with a electron-hole pair introduced into the Fermi sea. Values of E for which equation 3.15 holds true correspond to the energy of eigenstates of the perturbed exciton - Fermi sea system.

3.2.4 Spectral function Instead of solving this equation, we can directly look at the spectral function, deﬁned as:

−iHt † A(t) = h0|x0e x0|0i (3.16)

In the truncated basis of the Chevy Ansatz, we ﬁnd for the Fourier transform of the spectral function:

1 1 A(ω) = Im . (3.17) π ω + iη − ωX (0) − ΣX (ω, 0)

Figure 3.1 shows the spectral function A(ω) for diﬀerent values of EF . At an energy Eb = 20 meV (binding energy) below the exciton energy, we see the resonance attributed to the attractive polaron: An exciton accompanied by a coherent displacement of the surrounding electrons of the Fermi gas towards the exciton. With increasing Fermi energy, the attractive polaron slowly shifts to higher energy and its total weight increases. The resonance that emerges out of the exciton rapidly shifts to the blue and broadens with increasing Fermi energy. We identify this resonance as the repulsive polaron: An exciton dressed with a displacement of the surrounding Fermi gas electrons away from the exciton. The weight of this resonance decreases with increasing Fermi energy. 14 Theory of Fermi polarons

5 (meV)

C 10 − E F E 15

Molecular Repulsive polaron state Attractive polaron 20 30 20 10 0 10 20 30 Photon energy - exciton energy (meV)

Figure 3.2: Middle The spectral function calculated using the Chevy ansatz showing the attractive and repulsive polaron branches as a function of the Fermi energy EF . In the ﬁgure to right, the attractive polaron is depicted as an exciton surrounded by an electron screening cloud which lowers the energy of the exciton due to exciton-electron interactions. In the left panel, the repulsive polaron is depicted as an exciton surrounded by cloud of electron-hole pairs that correspond to Fermi sea electrons that are pushed away from the exciton leading to a higher energy metastable excitation. Also depicted is the molecular state which is a bound state of one electron and one exciton.

In between the attractive and the repulsive polaron, we ﬁnd a third resonance. Its total weight is much smaller than the one of either polaron. Notably, this resonance broadens much faster than the polaron resonances. This resonance can be attributed to molecular states: A tightly bound state between the exciton and one of the electrons of the Fermi gas. The resonance increases drastically in width because the energy of the molecular state can vary depending on whether the electron included in this bound state stems from the bottom or the top of the Fermi sea. Figure 3.2 shows an illustration of the attractive and repulsive polarons as well as the molecular state. In the case of the attractive polaron, the exciton can lower its energy due to exciton-electron interactions by shifting the surrounding Fermi sea electrons closer to the exciton. Since we deﬁned our vacuum to be the Fermi sea, the displacement of the Fermi electrons is understood as a screening cloud composed of electron-hole pairs for which the electrons (depicted in dark green) is closer to the exciton than the corresponding hole (bright green). In contrast, for the meta stable repulsive polaron, the electrons of the Fermi sea are pushed away from the exciton which is equivalent to electron-hole pairs in which the electron is closer to the exciton than the corresponding hole. In this picture, the molecular state is represented by an exciton with one, closely bound electron, and a completely localized hole.

3.2.5 Quasiparticle weight The molecular states and the attractive polaron most signiﬁcantly manifest their diﬀerence in the oscillator strength: Only excitons whose momentum is within the light cone can recombine under emission of a photon. Therefore, the only term of the Chevy ansatz which contributes to the oscillator strength is φ0. In order to compare the oscillator strength of the attractive and repulsive polaron and the 3.2. Theoretical model 15

1.0 Attractive polaron Repulsive polaron 0.8 Molecule

0.6

0.4

Quasiparticle weight Z 0.2

0.0 0 5 10 15 20 25 30 Fermi energy E F −E C (meV)

Figure 3.3: The quasi particle weight Z as a function of the Fermi energy EF for the attractive (blue) and repulsive (green) polaron as well as for the molecular states.

molecular states, we therefore integrate φ0 over the energy spectrum corresponding to each of these resonances to obtain their respective quasi particle weights Z. The oscillator strength of the different resonances is therefore equal to Z times the oscillator strength of the bare exciton. In figure 3.3, the quasi particle weight Z is plotted as a function of the Fermi energy EF. We notice that the attractive polaron rapidly gains quasi particle weight and therefore oscillator strength as the Fermi energy is increased. At the same time, the repulsive polaron, which corresponds to the exciton in the absence of electrons, loses its quasi particle weight with increasing Fermi energy. We notice that the attractive polaron has a larger oscillator strength than the molecular states for every Fermi energy. In the absorption spectrum of a TMD monolayer filled with a 2DES, we therefore should see two sharp prominent resonances: the repulsive polaron which emerges from the exciton resonance and the attractive polaron below the exciton energy. The trion described in semiconductor literature as an electron tightly bound to the exciton we identify in our model with the broad resonance that corresponds to molecular states. Its lack of oscillator strength as compared to the attractive polaron supports our understanding that the quasiparticle which we see in absorption at a finite 2DES electron density should be described as an attractive polaron rather than as a trion.

3.2.6 Charge distribution

The charge distribution of a polaron differs from that of a trion. In models describing the trion as a molecular state, two different Bohr radii describe the close separation of the hole and one of the electrons and a larger Bohr radius to the other electron. As shown in figure 3.4, the charge distribution will be a double exponential with the two Bohr radii as decay scales. If we disregard the delocalized hole, the net charge of a trion is the electron charge e. The screening due to the electron gas will result in an enlargement of either Bohr radius. 16 Theory of Fermi polarons

Trion Attractive polaron n ( r ) n ( r )

r r Charge density Charge density ~aB ~1/kF

Figure 3.4: Left Sketch of the charge density distribution around a trion shows an exponential decay of the electron charge over a length scale corresponding to the outer trion Bohr radius aB,trion. Right Sketch of the charge density of an attractive polaron showing oscillations with a period ∝ 1/kF. The net charge is zero.

In contrast to the trion, the attractive polaron clearly has to be neutral because it consists of a neutral exciton dressed with electron-hole pairs, which average out to a neutral particle. The diﬀerence in net charge of the trion and the polaron could serve as an important experimental observable to distinguish between trions and attractive polarons e.g. in transport measurements [81] or in quantum Hall measurements [82], [83]. For attractive polarons, where surrounding Fermi sea electrons are shifting towards the exciton to screen it, a surplus of negative electron density is expected around the exciton. With increasing distance from the exciton, the charge density will oscillate with a period proportional to 1/kF. This can be understood by realizing that the weight of holes contributing to the attractive polaron is in good approximation equal for all hole momenta up to kF. In real space, this will lead to said oscillation as predicted by the Fourier-Bessel transform of a box distribution. Together with the slowly varying charge distribution due to the electron, the attractive polaron is charge neutral.

The difference of the trion and attractive polaron net charge at first glance seems to allow for transport measurements to confirm our hypothesis that the optically excited quasi particles in a 2DES should be describes as attractive polarons rather than as trions. However, it is important to realize that both, the trion and the attractive polaron will move into the same direction in an electric field. This is because the hole of the electron-hole pair dressing the exciton in an attractive polaron is not a valence band hole. It describes a hole in the Fermi sea in the conduction band. Therefore, this hole has the opposite mass of a valence band hole and will be accelerated in the same direction as the conduction band electron. We see that even though the attractive polaron is neutral, on a microscopic level, it is composed of charged particles which, due to the sign of their mass will move into the same direction. While the direction of the movement is the same for trions and attractive polarons, the drift speed is different and depends on the electron density. As a consequence, transport measurements could still corroborate the polaron model but the sought-for effect is of quantitative and not qualitative nature [81]. 3.3. Fermi polarons in TMDs 17

3.2.7 Validity of our approximations

In our model, we assumed a small Fermi energy EF such that we can neglect exciton- hole scattering. When the Fermi energy approaches the trion binding energy ET , this assumption is clearly not valid anymore. In this limit, the Chevy-ansatz will still allow for qualitative predictions but the results will not be accurate. When the electron density is increased, we should not just include hole-scattering but also an ansatz that considers interactions with more than one electron-hole pair. In our model, we did not include screening of Coulomb interaction electron-hole attraction due to the surrounding 2DES. This is justified since the timescale on which the 2DES can react to electrostatic changes is given by the plasma frequency, which is in turn much smaller than the exciton binding energy. Recently, two groups have analyzed the effect of dynamic screening of excitons by the 2DES by going beyond random phase approximation (RPA). While the group in ref. [84] finds that the 12 exciton binding remains large for the ne ≤ 1x10 that is of interest in my work, in ref. [85] the authors find strong screening and intervalley plasmon scattering leading to exciton sidebands. Previous work by Suris [86] and Rapaport [87] model the modification of the trion energy due to the effect of a 2DES by considering a trion dressed with a hole. The appeal of the polaron model lies in its simplicity, its accuracy and explaining both, repulsive and attractive polaron resonances with one model.

3.3 Fermi polarons in TMDs

3.3.1 Sample structure To investigate exciton-electron interactions, we used TMD samples embedded in between hexagonal boron nitride (h-BN) layers (figure 3.5). It has been shown that MoSe2 monolayers feature much narrower spectral lines when embedded in h-BN which has a similar honeycomb lattice. h-BN furthermore serves as an insulator onto which a monolayer of graphene is deposited. This heterostructure is transfered with the graphene side up onto a substrate. Depending on the experiment we use SiO2 on Si, SiO2 or a distributed Bragg reflector (DBR)evaporated onto fused silica as a substrate. The MoSe2 and the graphene flake are contacted with gold leads. Applying a positive (negative) gate voltage Vg between the TMD flake and the graphene gate will attract (repel) electrons into the MoSe2 flake. In the following we well look at data for two different samples:

1. DBR, few layers of h-BN, MoSe2 monolayer, 105 nm h-BN, graphene (M1)

2. Si, 285 nm of SiO2, few layers of h-BN, MoSe2 monolayer, few layers of h-BN (M2) Sample 1 was never measured outside the cavity which motivated a control measurement with sample M2. In the following we will focus mainly on sample M1. A more detailed discussion of the samples, can be found in appendix8. Figure 3.6 shows a micrograph of the sample structure of sample M1, where the contours of the diﬀerent ﬂakes forming the heterostructure are highlighted in a dotted line. 18 Theory of Fermi polarons

Au White light MoSe 2 graphene h-BN

SiO 2

Figure 3.5: A sketch of a typical sample structure: A MoSe2 monolayer is embedded in between h-BN ﬂakes and covered with a graphene ﬂake. The heterostructure is transfered onto a substrate (e.g. SiO2 on Si, SiO2 or a DBR). The MoSe2 monolayer and the graphene are contacted separately with gold in order to apply a voltage in between the graphene top-gate and the TMD monolayer.

h-BN

AuAu

MoSe2

graphene

Figure 3.6: An optical microscope image of the heterostructure where the contours of the MoSe2 monolayer, the 105 nm thick h-BN and the top graphene layer is iden- tiﬁed. The area where the MoSe2 and the graphene monolayers overlap represents the region of interest where we can change the electron density by applying a gate voltage Vg. 3.3. Fermi polarons in TMDs 19

3.3.2 Capacitive model for the Fermi Energy

By applying a top gate voltage Vg the electron density in the sample and therefore the Fermi energy EF is changed. We denote the smallest Vg for which the attractive polaron is observed as Vg = Vc which we interpret as the gate voltage for which we start populating the conduction band (EF > 0). In order to estimate the electron density at a given gate voltage, we can approximate the TMD - h-BN - graphene heterostructure as a planar capacitor. The capacitance per unit area C/A between top gate and sample is given by:

C t 1 −1 = + 2 , (3.18) A 0 e D(E)

where D(E) is the density of states and t ≈ 105 nm, ≈ 3 [88] are the thickness respectively the permittivity of the h-BN flake. The two terms are the geometric respectively quantum capacitance of the sample. For EF > 0, the quantum capacitance can be neglected since its effect is much smaller and within the uncertainty of the permittivity of the h-BN flake. For Vg > Vc i.e. EF > 0 and an effective electron mass of the conduction band mc = 0.49me this yields:

2 π~ 0 meV EF = (Vg − Vc) ≈ 0.77 (Vg − Vc). (3.19) temc V

3.3.3 Spectroscopy of an exciton in a 2DES In order to study the interaction of excitons with a two dimensional electron system (2DES), we want to investigate the optical response of a TMD flake as we increase the electron density. In Figure 3.7, the differential reflection (dR) of a MoSe2 flake M1 is shown as a function of gate voltage Vg. At small gate voltages, corresponding to the absence of electrons, we only observe the exciton at around Eexc ≈ 1646 meV. At low but finite electron densities, at ET ≈ 25 meV below the exciton energy, a second, weak resonance is observed. We attribute this peak to excitons that form a bound state with localized charges. As the gate voltage Vg and therefore the electron density ne and the Fermi energy EF increase, the repulsive polaron emerging from the exciton shifts to higher energy, loses in oscillator strength and broadens. At the same time, the attractive polaron emerging from the localized trions increases in oscillator strength and also shifts to higher energies, although at a lower rate than the repulsive polaron. The sizable energy shifts of the two resonances is an evidence that at high gate voltages, there is indeed a degenerate Fermi sea of electrons in the TMD monolayer, rather than just a collection of localized states. The simple Chevy ansatz does not just qualitatively describe the optical resonances of an exciton in a 2DES but even quantitatively predicts the energy of the attractive and repulsive polaron as a function of the Fermi energy with just one fitting parameter. The one fitting parameter we need describes the blue shift of the bare exciton energy as a function of the Fermi energy EF . Several effects, such as phase space filling and screening will cause a the exciton energy to increase. On 20 Theory of Fermi polarons

Photon energy (meV) Photon energy (meV) a 1680 1660 1640 1620 b 1680 1660 1640 1620 -50 -50

0 0 0 0 10 10 (meV) 50 20 50 20 (meV) C C − E 30 30 − E F F E E Gate voltage (V) 100 40 Gate voltage (V) 100 40

50 50 150 150 730 735 740 745 750 755 760 765 770 730 735 740 745 750 755 760 765 770 Wavelength (nm) Wavelength (nm)

-4 0 4 8 12 2016 -0.8 0 0.8 1.6 2.4 43.2 Differential reflection Differential reflection

Figure 3.7: a Differential reflection (dR/R) as a function of gate voltage Vg. At -50 V, the MoSe2 monolayer is charge neutral. A strong exciton peak is visible. As the electron density is increased, the resonance, which we identify as repulsive polaron, emerging from the exciton loses its oscillator strength, shifts to the blue and broadens. Simultaneous to the decrease of the exciton oscillator strength, a second resonance that we attribute to the attractive polaron emerges. Subfigure b shows the same data with increased contrast. the other hand, bandgap renormalization will reduce the exciton energy. While the increasing interaction with the 2DES with increasing electron density will move the repulsive polaron to higher and the attractive polaron to lower energies, phase space filling will cause a blue shift of the exciton that is of the order of the Fermi energy EF . Since estimations of the band gap renormalization and the reduction of the exciton binding energy as a function of EF are imprecise, we will simply assume a linear blue shift of the bare exciton energy with a proportionality constant β based on experimental observations. Figure 4.1 shows as green dots the attractive and repulsive polaron energy as a function of Fermi energy. The values obtain correspond to the maxima of the data shown in figure 3.7. The spectral function calculated using the Chevy ansatz discussed in chapter3 where we assumed the bare exciton dispersion with mexc = me to be:

2 2 EX (k) = −Eexc + βEF + ~ k /2mexc, (3.20) where β was chosen to be 0.8 to best ﬁt the experimental data. The agreement between experiment and theory is surprisingly good. The accuracy of the Chevy ansatz in describing interactions with an inﬁnite number of electrons with simply one electron-hole pair has been discussed before and was found to stem from destructive interference terms in the two electron-hole pair term. Our proposal of modeling excitons in a 2DES as polarons rather than trions and our calculations employing the Chevy ansatz does not just allow to qualitatively and quantitatively describe the attractive and repulsive polaron resonances but is also making the connection 3.3. Fermi polarons in TMDs 21

15 (meV)

C 20

− E 25 F E 30

40 60 40 20 0 20 Photon energy - exciton energy (meV)

Figure 3.8: The spectral function calculated using the Chevy ansatz showing the attractive and repulsive polaron branches as a function of EF . The center frequencies of the attractive and repulsive polaron resonances determined using differential reflection on a MoSe2 monolayer are shown as green dots. The only fitting parameter for the calculations is β = 0.8. between the two resonances. While previous models described just either the exciton or the trion resonance, employing many fitting parameters, our model recognizes the link between the two resonances and describes them accurately with just one fitting parameter.

Fermi polaron polaritons 23

4 Fermi polaron polaritons

4.1 Cavity spectroscopy

The main difference between trions and attractive polarons is the much larger oscillator strength of attractive polarons. As we will discuss in more detail in the next chapter, the strong-coupling regime of light-matter interaction allows to quantify the oscillator strength of the observed optical resonances. Having confirmed that the optical excitation spectrum in the presence of a 2DES is well described by many-body excitations termed exciton-polarons, we next turn to studying coupling of these excitations to cavity modes. While previous results had demonstrated strong-coupling of the exciton [28], [89], they could not change the electron density significantly. We perform our measurements with an open cavity setup that allows to form a tightly confined zero-dimensional cavity around any point on a flat sample. In the following paragraphs we will describe our fiber cavity setup as a well suited measurement scheme for TMDs to study the strong coupling regime of light-matter interaction, but also to independently measure the real and imaginary part of the monolayer’s susceptibility.

4.1.1 Fiber cavity A sketch of the fiber setup. The fiber cavity consisting of quartz substrate and a fiber facet. The dimple, which has a radius of curvature of 30 µm, is ablated from the cleaved fiber facet by a focused CO2 pulse [43]. Both, the quartz substrate and the fiber facet are coated with the same distributed Bragg reflector (DBR). The DBR layer facing the cavity has a low refractive index such that the intensity of the standing wave pattern of the cavity mode (depicted in red) is largest on the surface of the DBR. Details about the DBR can be found in appendix8. The sample is mounted on a titanium sample holder with a hole to allow for transmission measurements. The sample holder is attached on top of two piezoelectric actuators (orange) that allow the sample to be moved in x and y direction. The single-mode fiber that acts as the bottom mirror of the cavity is fixed by a titanium holder which can be moved along the optical axis by another slip stick actuator moving in z direction. Moving the fiber in z direction will change the cavity length while keeping the z position of the sample fixed. In order to scan the cavity length, we can change the piezo voltage up applied rather than using the normal step mode of the slip-stick actuator. The transmitted light is collimated by an aspheric lens whose position can be changed in all three directions but only at room temperature. The cavity is located at the bottom of a 2” stainless steel tube filled with 20 mbar of exchange gas (helium) at room temperature which is immersed in liquid helium (LHe) to cool the sample to 4 K. The fiber connected to the cavity as well as several electrical cables are fed through the vacuum. The collimated transmitted light exits 24 Fermi polaron polaritons

Figure 4.1: A sketch of the fiber setup. Left: The fiber cavity consisting of quartz substrate and a fiber facet with a dimple which both are coated with the same DBR. Middle: The sample can be moved in x and y direction by two piezoelectric actuators (orange). The single-mode fiber, which acts as the bottom mirror of the cavity, can be moved along the optical axis by another slip stick actuator moving in z direction. The transmitted light is collimated by an aspheric lens which can be aligned at room temperature. Right: The cavity is located at the bottom of a tube filled with 20 mbar of exchange gas (helium) which is immersed in liquid helium (LHe). The collimated light transmitted through the cavity is analyzed in an optical setup on top of the tube, coupling two opposite polarizations of the transmitted light into separate single-mode fibers. Two pellicle beam splitters (pel.) allow to couple illuminate the monolayers through the cavity and image the sample onto a CCD. 4.1. Cavity spectroscopy 25 00 TEM 00 TEM

FSR

02 02 10 / TEM / TEM 10 11 11 / TEM 01 / TEM / TEM / TEM 01 20 20 TEM TEM TEM TEM

Figure 4.2: The transmission spectrum of an empty fiber cavity. We see two different fundamental modes (TEM00) which are separated by the free spectral range (FSR). At higher energy with respect to the fundamental mode, we see a series of transverse modes. This fiber coated with a DBR was provided by the group of Prof. Jakob Reichel. the tube through a window and is analyzed in a optical setup directly mounted on top of the tube. As a first element, a quarter wave plate (QWP) or a liquid crystal retarder (LCR) transform the polarization of the transmitted light. A polarizing beam splitter cube separates the light into two different beams which each pass a pellicle beam splitter before they are coupled into fibers. Attached to the pellicle in the top arm is a achromatic lens with a focal length of 10 cm focused on a CCD chip to allow for imaging of the sample. The pellicle in the side arm reflects the collimated light from a multimode fiber down to the sample. Since the DBR is almost transparent for green light, we use a 530 nm LED to image the sample onto the CCD in order to move the monolayer into the focus of the cavity.

4.1.2 Sample structure

The device under investigation in the cavity is a h-BN/MoSe2/h-BN/graphene heterostructure transfered onto a DBR evaporated onto a quartz substrate (Figure 4.3).

The thickness t = 105 nm of the h-BN layer corresponds to t ≈ λexc/4nh-BN, where λexc ≈ 1646 meV is the wavelength of the exciton emission and nh-BN ≈ 2.4 which ensures that MoSe2 is located at an anti-node of the cavity, while the graphene monolayer is at a node where the intra-cavity field vanishes. In this configuration, the absorption by the graphene layer does not lead to a deterioration of the cavity finesse F. In order to move the monolayer into the focus of the cavity, we need to be able to image the substrate surface. Two methods have been used for this, each with their own advantages and disadvantages. The first method is to shine light at a 26 Fermi polaron polaritons

graphene

MoSe2 h-BN Lc

AuAu

Figure 4.3: A sketch of the heterostructure inside the ﬁber cavity. The thickness of the h-BN layers is chosen such that the TMD layer is in the anti-node and the graphene is in the node of the cavity mode.

wavelength within the stopband when the fiber is far enough such that there is no stable cavity mode. In this case, the sample can be seen as shadows absorbing the transmitted light. The position of the dimple can be seen once the cavity is closed and the transmitted light is observed on the CCD. Next, we can send the same light source from the top and align the spot with the position on the CCD where we observed the cavity mode. This will yield a good starting point for coupling the light transmitted from the fiber side into the cavity into the single-mode fibers of the optical setup on top of the cryostat. Once the transmitted light is coupled into the cavity, we can observe the cavity length and close it further without the risk of crashing the cavity into the monolayer. The advantage of this method is that we do not need to send light from the top. However, the disadvantage is that only a fraction of the light we use to illuminate the sample will reach the CCD since two DBRs will each reflect most of the light.

As an alternative, we can send light at 530 nm which passes the DBR. On the CCD, we will see the reflection from the exfoliated TMD layers. Figure 4.4a shows the reflection from a sample which is a few layers thick. In figure 4.4b we see the same sample but with a shorter cavity length. This already illustrates the advantage of the green light: Due to the remaining reflectivity of the DBR for green light, we observe interference between the light reflected by the substrate surface and the light reflected by the fiber facet. Due to a slight angle between the fiber facet and the substrate surface, this shows up as interference fringes in the form of stripes. The density of the fringes directly serves as a measure of the cavity length. Furthermore, the dimple is very well visible in this way and we can precisely align the cavity with the sample even at small cavity lengths. Figure 4.4c shows sample M2 inside the cavity imaged with green light. The disadvantage of this method is that the observed position of the dimple with green illumination does not agree very well with the position of the cavity mode observed with light within the stop band due to the large chromatic aberration of the aspheric lens. 4.1. Cavity spectroscopy 27

a cb

Figure 4.4: a The reflection of a green LED from the sample surface. The reflection from a multilayer TMD flake is clearly visible. b The same sample with a shorter cavity length. Fringes due to the interference between LED light reflected by the sample surface with light reflected by the fiber facet. c A micrograph of sample M2 inside the cavity.

4.1.3 Cavity modes The eigenmodes of our zero-dimensional cavity are described by the number of nodes q along the cavity axis (longitudinal mode index) as well as the number of nodes m and n in the Hermite-Gaussian mode-proﬁle in the transverse direction. An asymmetry of the mirror curvature would mean that we would describe the radius of curvature by its minimum and maximum value Rx and Ry. For the resonance frequencies of the cavity ωqmn, we ﬁnd [90]:

πc 2LDBR ωqmn = q + Leff λ0 1 r L + m + π−1 arccos 1 − eff 2 Rxneff # 1 r L + n + π−1 arccos 1 − eff , (4.1) 2 Rxneff

where LDBR is the penetration depth into the dielectric mirrors and λ0 is the center wavelength of the DBR. For the penetration depth of the of a DBR made of dielectrics with refractive indices n1 and n2, we ﬁnd:

λ0 n1n2 LDBR = (4.2) 2 n1 − n2

The cavity of length L that forms between the curved ﬁber with radius of curvature R and the ﬂat substrate mirror has beam waist w0 equal to:

r λ 1 w ≈ (LR) 4 . (4.3) 0 π

where L is the cavity length. 28 Fermi polaron polaritons

Photon energy (meV) 1680 1660 1640 1620 1600

45 40 (V)

p 35 Vg = -3V 30 25 20 15 Piezo voltage u 10 5 740 745 750 755 760 765 770 775 Wavelength (nm)

Figure 4.5: The white light transmission spectrum of the ﬁber cavity incorporating the MoSe2/h-BN/graphene heterostructure, as a function of the piezo voltage up. The insert shows the cavity transmission at Vg = −3 V and up = 20 V ﬁtted with a Lorentzian curve.

4.1.4 Resonant Transmission Figure 4.5 shows the transmitted spectrum through the cavity as a function of the piezo voltage up corresponding to a range of cavity lengths around 9.5 µm. As light source, we use a white light LED. We can identify three fundamental modes that shift to shorter wavelengths for increasing piezo voltage or decreasing cavity length. Since the bare cavity linewidth of 0.3 meV is much smaller than all other energy scales, cavity transmission allows for identifying the linear optical response of the heterostructure: Whenever the cavity mode is at a frequency absorbed by the MoSe2 flake, its linewidth increases. Consequently, the MoSe2 absorption spectrum can be measured as a frequency dependent broadening of the cavity. Figure 4.7 shows the absorption spectrum, which is measured as an increase in cavity linewidth, as a function of gate voltage. Indeed we see the same features as in the dR/R measurements without the cavity described in section 3.3.3: In the absence of a Fermi sea of electrons, we see a strong absorption from a sharp resonance: The unperturbed exciton. As we introduce electrons, the exciton resonance should be described as a repulsive polaron that broadens and rapidly shifts to higher energy. At a finite electron density, we also see the attractive polaron emerge roughly ET ≈ 25 meV below the exciton energy. As the repulsive polaron decreases in intensity, the attractive polaron increases. With increasing electron density, the attractive polaron shifts to higher energy. In section 4.5, we saw that the Chevy ansatz can not just qualitatively but even quantitatively explain the attractive and repulsive polaron energy as a function of Fermi energy EF . We should therefore check whether the resonances have the same Fermi energy dependence in this sample as in the one measured outside the cavity. Figure 4.6a shows the energies corresponding to the maximum absorption as a function of Fermi energy for the attractive and the repulsive polaron for the two different samples. We notice a constant offset between the two samples for both polaron resonances. We attribute this offset to the different dielectric environment of the two samples. 4.1. Cavity spectroscopy 29

a b 90 1720 Acvtt. polaron (dR) Cavity spectroscopy Rep. polaron (dR) Diff. reflection Rep. polaron (cav.) 80 Att. polaron (cav.) 1700 70

1680 60

50 1660 Photon energy (meV) Polaron splitting (meV) 40

1640 30

1620 20 0 10 20 30 40 50 5 0 5 10 15 20 25 30 35 40 Fermi energy EF (meV) Fermi energy EF (meV)

Figure 4.6: The attractive and repulsive polaron energy as a function of the Fermi energy measured on sample M1 in reﬂection and M2 inside the cavity. b Splitting between attractive and repulsive polaron as a function of Fermi energy.

Figure 4.6b shows the diﬀerence between the attractive and repulsive polaron peak as a function of Fermi energy. We see that in the limit of EF = 0, where we expect the splitting between the two polaron resonances to approach the trion binding energy ET, the two samples show a good agreement. The slope of the splitting as a function of Fermi energy agrees well considering the large uncertainty of the repulsive polaron energy for large Fermi energies. The uncertainty in the repulsive polaron energy is due to the large linewidth and small peak area of the repulsive polaron resonance at large Fermi energies.

In the presence of MoSe2, the index of refraction seen by the photons transmitted through the cavity is modified due to the real part of the MoSe2 susceptibility, thereby modifying the effective cavity length and leading to a shift of the cavity resonance wavelength as compared to what we would have obtained in the absence of MoSe2. For measuring the absorption in Figure 4.5, we followed the fundamental mode with 23 nodes as its wavelength changes from ∼ 740 nm to ∼ 775 nm while the cavity length is tuned. At a wavelength approximately 30 nm longer, still within the stop band of the DBR, we observe the fundamental mode with 22 nodes. Within the tuning range covered in Figure 4.5, this red detuned cavity mode does not cross any resonances of the heterostructure and therefore serves as an indicator for the cavity length. Using this information, we can measure the energy shift of the cavity resonance crossing the MoSe2 as the difference between the measured resonance energy of the cavity mode with 23 nodes and the expected resonance energy of the unperturbed cavity mode with 23 nodes as calculated from the measured cavity resonance of the mode with 22 nodes. The result is shown in Figure 4.5b: The dispersive spectrum is plotted in terms of energy shift as a function of gate voltage. For the gate voltage range where the spectrum is dominated by the exciton, the cavity mode can not be fitted with a Lorentzian which is why this gate voltage 30 Fermi polaron polaritons

a Photon energy (meV) b Photon energy (meV) 1680 1660 1640 1620 1600 1680 1660 1640 1620 1600 -10 -30 -20 0

(V) -10 g 10 0 10 20

20 30 30 Gate voltage V 40 40 50 50 740 745 750 755 760 765 770 775 740 745 750 755 760 765 770 775 Wavelength (nm) Wavelength (nm)

0.5 1.0 2.0 4.0 -1 0 +1 Linewidth (meV) Energy shift (meV)

Figure 4.7: a, The MoSe2 absorption spectrum determined by cavity spectroscopy as a function of Fermi energy. The resonances match well except for a energy offset. b The dispersion spectrum as measured through the cavity. range is not shown in the figure. In this case the reason simply is that the cavity is still too short such that the cavity is still in the strong coupling regime. However, recent experiments have shown that the non-radiative decay of TMD excitons could be so small that the monolayer will act as a mirror [91], [92]. In that case we would see a dip within the transmission spectrum when the cavity is in resonance with the flake even when the system is not in the strong coupling regime. We can look at this from a classical picture where we model this problem as changing the detuning of two coupled cavities. It could therefore in principle be impossible to measure the spectrum of TMD monolayers with this technique once the non-radiative decay rate is much smaller than the radiative decay rate. A similar measurement technique was used by ref. [93]. The data presented in Figure 4.5b is connected to the absorption data of Fig- ure 4.5a via Kramers-Kronig relations. It is a great advantage of measuring the spectrum using a weakly coupled cavity that the real and imaginary part of the susceptibility can be measured independently. Figure 4.8 shows horizontal line cuts of the data in Figure 4.5, depicting the absorption (blue) as measured through the linewidth of the cavity mode and the dispersion (black) as the energy shift of the cavity mode. The red curve is the expected dispersion as calculated from the measured absorption using Kramers-Kronig relations. The vertical displacement of the measured dispersion and the calculated dispersion is most likely a result of the simplified model of a cavity with an effective length given by the distance between the mirrors plus a wavelength independent penetration depth into the DBR. Correcting for the wavelength dependence of the penetration depth should get rid of this offset.

4.1.5 Photoluminescence In addition to resonant spectroscopy, we can also measure the PL through the cavity. For this purpose, we use the ﬁnite width of the DBR stop band to eﬃciently couple a PL laser into the cavity for any cavity length. The DBR transmission for a 532 nm 4.2. Strong light-matter coupling 31

Photon energy (meV) Photon energy (meV) Photon energy (meV) 1700 1675 1650 1625 1600 1575 1700 1675 1650 1625 1600 1575 1700 1675 1650 1625 1600 1575 0.5 1.4 a Vg =-10V b Vg = 0V c Vg =+40V 1.2 0.0 1.0

0.8 0.5 Linewidth (meV)

0.6 Energy shift (meV) 1.0 0.4 740 745 750 755 760 765 770 775 740 745 750 755 760 765 770 775 740 745 750 755 760 765 770 775 Wavelength (nm) Wavelength (nm) Wavelength (nm)

Figure 4.8: Measured absorption (blue) and measured dispersion (black) as well as the dispersion calculated from the absorption via Kramers-Kronig relations (red) for three different electron densities. laser is close to unity. For long cavity lengths, the cavity mode acts just as a filter. We can therefore tune a fundamental mode corresponding to a long cavity mode across the MoSe2 resonances and measure the PL spectrum at each cavity length. The data obtained through this procedure is shown in Figure 4.9. We can see that we observe light emitted through the cavity whenever any fundamental or transverse cavity mode is in resonance with the PL emission. In a next step, the photon counts obtained within a 3 nm wide window around the fundamental mode are binned. Plotting the collected counts versus the resonance energy of the fundamental mode for each cavity length yields the PL spectrum. Figure 4.10 shows the absorption spectrum (cross sections through Figure 4.7a) together with the corresponding PL spectrum for three different gate voltages. Fig- ure 4.10a shows the absorption (blue) and PL spectrum (green) for Vg = −10 V corresponding to EF = 5 meV: In absorption, both, attractive and repulsive polaron are visible. In PL, we only see one peak (with a small shoulder to the red). The wavelength of PL emission and attractive polaron absorption are only marginally shifted. In Figure 4.10b, the absorption spectrum shows a large blue shift and broadening of the repulsive polaron and a slight small blue shift of the attractive polaron at Vg = 0 V (EF = 13 meV). In Figure 4.10c, the absorption spectrum at Vg = 40 V shows only the attractive polaron which is shifted further to the blue and is broadened. The PL at this gate voltage is also broadened and significantly weaker. Furthermore, a sizable energy difference of ∼ 40 meV is observed between the absorption and the PL peak. Recent theoretical calculations [94] suggest that PL comes from the polaron. At a finite Fermi energy, the polaron dispersion has a minimum at finite momentum. In absorption, this minimum cannot be observed because it is outside the light cone. However, in PL, the polarons will relax to finite k and, eventually, recombine from there with the help of a third particle (e.g. a phonon) that takes care of the extra momentum.

4.2 Strong light-matter coupling

As described in chapter introduction, the strong Coulomb interaction in monolayer TMDs, leads to a large exciton binding energy Eb and, therefore, a small Bohr radius 32 Fermi polaron polaritons

Figure 4.9: Photoluminescence of the MoSe2 monolayer inside a weakly coupled cavity as a function of cavity length.

Photon energy (meV) Photon energy (meV) Photon energy (meV) 1675 1650 1625 1600 1675 1650 1625 1600 1675 1650 1625 1600

1.4 V =-10V V = 0V V =+40V a g b g c g 250

1.2 200

1.0 150

0.8 100 Linewidth (meV) PL intensity (a.u.) 0.6 50

0.4 0 740 750 760 770 780 740 750 760 770 780 740 750 760 770 780 Wavelength (nm) Wavelength (nm) Wavelength (nm)

Figure 4.10: a, Line-cut through the cavity line broadening data (blue curve) for Vg = −10 V (EF = 5 meV): both repulsive and attractive polaron features are visible. The photoluminescence (PL) data is shown in green. b, Line-cut through the cavity line broadening data for Vg = 0 V (EF = 13 meV): the absorption data is dominated by the attractive polaron which is now blue-shifted with respect to PL. c, Line-cut through the cavity line broadening data for Vg = 40 V: the PL and absorption peaks are separated by 40 meV. 4.2. Strong light-matter coupling 33

Figure 4.11: Photoluminescence spectra as a function of gate voltage. The spectrum at each gate voltage was obtained by tuning the cavity mode of a long cavity mode of length ∼ 18 µm across the spectral range of interest. The integrated PL intensity emitted through this cavity mode corresponds to the spectrum at the cavity mode resonance.

aB. As we will show in the following, the small exciton radius is responsible for a very strong light-matter interaction in TMDs. The most striking manifestation of this strong interaction is the avoided crossing between a cavity mode photon and the exciton resonance, mixing into polariton modes. Moreover, we show a new type of polaronic excitation, the polaron-polariton, that is resilient against disorder. In this section, we want to study the strong coupling regime of light-matter interaction. We will see that the normal mode splitting observed in the strong coupling regime allows for a quantitative determination of the oscillator strength of a resonance. In Section 2.4, we saw that we can write matter excitations as excitons which essentially behave as bosons. Therefore, in second quantization, we can write the non-interacting Hamiltonian of a cavity and of a TMD ﬂake as:

ˆ X † † H0 = ~ων,kcˆν,kcˆν,k + ~ωcaˆ a,ˆ (4.4) ν,k

† where ~ων,k, is the energy andc ˆν,k is the annihilation operator of an exciton with ^ quantum numbers ν (e.g. 1s ) with quasi momentum k and energy ~ωc respectively aˆ is the energy respectively the annihilation operator of cavity mode c. ˆ For the interaction Hamiltonian Hint, we ﬁnd for an electron located at rX in an electric ﬁeld Eˆ:

ˆ ˆ Hint = −eˆr · E (rX) . (4.5) 34 Fermi polaron polaritons

When we quantize the electric ﬁeld, we can write E in terms of annihilation operators a of the electric ﬁeld:

r ˆ X ~ωq iqr −iqr † E (r) = i 2 e aˆq − e aˆq , (4.6) 20n Veﬀ q | {z } Eq

where ~ωq is the energy of the mode with wavevector q, n is the effective refractive index and Veff is the effective mode volume of of mode q. Inserting eq. 4.6 into eq. 4.5 yields:

Z X † † † 1 3 ∗ iqr Hˆ = −i eˆ hˆ aˆ E d r ϕ (r)ˆε · e~rϕ 0 (r)e +h.c., (4.7) int k −k q q V c,k q v,k k,k0,q | {z } I

ˆ wheree ˆk and hk are the annihilation operators of an electron respectively hole with quasi momentum k andε ˆq is a unit vector denoting the polarization of mode q. We can simplify the integral I to:

Z 1 X i(k0+q−k) 3 ∗ iqr I = e d r u (r)ˆε · e~ru 0 (r)e = δ 0 µ , (4.8) V c,k q v,k k,k +q cv R∈Λ FBZ

where µcv is the transition matrix element from the conduction band to the valance band at the bandgap minimum. We can therefore rewrite the interaction Hamilto- nian as:

ˆ X † ˆ† † Hint = −i~ gkeˆk+qh−kaˆq + h.c. (4.9) k,q

r Ek ωk gk = µcv = µcv 2 , (4.10) ~ 20n Veﬀ~

where gk is the coupling coeﬃcient of mode k. Since we have already established excitons , we want to rewrite the interaction Hamiltonian in terms of exciton annihilation operators rather than in the basis of electrons and holes. For the basis † transformation, we recall the deﬁnition ofc ˆν,k:

† 1 X † ˆ† cˆν,k = √ ϕν(q)ˆe k h k , (4.11) 2 +q 2 −q V q

Multiplying both sides withϕ ˜ν(r = 0), the Fourier transform of ϕν(q) evaluated at r = 0 and summing over ν yields: 4.2. Strong light-matter coupling 35

X 1 X X X k ϕ˜ (r = 0)ˆc† = √ eˆ† hˆ† ϕ + q ϕ∗(k0) ν ν k +q k −q ν ν 2 2 2 ν V q k0 ν 1 X X X k = √ eˆ† hˆ† hk0|νihν| + qi k +q k −q 2 2 2 V q k0 ν | {z } δ 0 k k , 2 +q 1 X † ˆ† = √ eˆk h k . (4.12) 2 +q 2 −q V q

Therefore, we can write the interaction Hamiltonian in the exciton basis as: ˆ X † † † † Hint = i~ gexc-phot,ν(q) cˆν,qaˆq − aˆqcˆν,q (4.13) ν,q r Ek ωk gexc-phot,ν(q) = µcv =ϕ ˜ν(r = 0)µcv 2 . (4.14) ~ 20n Veﬀ~ Approximating the 1s state using a 2D hydrogen wave function, we ﬁnd:

r 8A 1 2D ϕ˜1s(r) = 2D exp −r/aB , (4.15) π aB where A is the quantization area of the 2D exciton wave function. We can therefore rewrite g as:

r 2D 2µcv ωk g1s (k) = 2 . (4.16) aB 20n Leﬀ~

To estimate µcv, we express it in terms of the Dirac velocity vD of the 2-band Hamiltonian:

ev µ = D , (4.17) cv ω

5 where Eg = ~ω. For MoSe2, we find that ~ω ≈ 1.65 eV, vD ≈ 4 · 10 , ≈ 4, aB ≈ 1 nm and Leff ≈ 1,µm, we estimate g ≈ 10 meV, which is much larger than typical inhomogeneously broadened linewidths of encapsulated MoSe2 in the order of γX ≈ 1 − 5 meV. If the surrounding cavity has a high enough Q-factor, it is possible to fulfill the conditions for the strong coupling regime:

4~g > Γc, ΓX , (4.18)

where γc is the cavity linewidth. In this case, we ﬁnd for the eigenenergy of an exciton with energy ~ωX coupled to a cavity mode a with energy ~ωa = ~ωX + δ: 36 Fermi polaron polaritons

C (V) E

p 60 = 1645.9 meV X E 40 g = 8.7 meV

Piezo voltage V 20

1630 1640 1650 1660 Photon energy (meV)

Figure 4.12: Avoided crossing of the exciton and cavity photon resonances. Scan- ning the cavity length by means of applying a voltage to a piezoelectric scanner, the cavity mode (orange line) is tuned across the exciton resonance (blue line). We assume a linear change of the cavity length as a function of piezo voltage and obtain an excellent ﬁt of the predicted upper and lower polariton energy (green crosses) with the measured transmission spectrum (density plot). In the ﬁtting procedure we determined the exciton energy ~ωX = 1645.9 meV and a normal mode splitting of 2~g = 19.4 meV.

r ! δ δ2 E = ω + ± g2 + , (4.19) ± ~ X 2 4

where E± are the energies of the upper (+) and lower (−) polariton modes, which themselves are a superposition of exciton and cavity photon. We can write the upper/lower polariton annihilation operatorq, ˆ pˆ as a result of the following basis transformation:

qˆ = cos θcˆ + sin θaˆ pˆ = sin θcˆ − cos θa,ˆ (4.20)

wherec ˆ,a ˆ are creation operators for the exciton and photon respectively and tan 2θ = 2g/δ. The hyperbolic function of equation 4.19 shows the smallest energy difference, called normal mode splitting or vacuum Rabi splitting ~ΩR, between upper and lower polariton mode at δ = 0. Figure 4.12 shows a fit of the upper and lower polariton energy as described by equation 4.19 to the transmission spectrum of a cavity, whose energy is assumed to increase linearly as a function of piezo voltage (depicted in orange), crossing the exciton resonance (blue line). The coupling coefficient ~g = 8.7 meV leads to an avoided crossing of the exciton and cavity resonance. The crosses in Figure 4.12, as derived from just four fitting parameters fits the transmission spectrum in great detail. 4.3. Fermi Polaron-Polaritons 37

Figure 4.13: a, The white light transmission spectrum as a function of the piezo voltage for an average cavity length of 1.9µm. Due to enhanced cavity electric ﬁeld, the interaction between the cavity mode and MoSe2 resonances is directly observed in cavity transmission spectra as anticrossings associated with polariton formation. For gate voltages where the MoSe2 monolayer is devoid of electrons (Vg1) the spectrum shows a prominent anticrossing with a normal mode splitting of 16 meV. The elementary optical excitations in this regime are bare exciton-polaritons without any polaron eﬀect. b, White light transmission spectrum for Vg = Vg2 = −5 V, showing two anticrossings associated with the formation of repulsive- and attractive-polaron-polaritons. The observation of anticrossings for both lower and higher energy resonances proves that these originate from Fermi-polarons with a large quasiparticle weight. c, White light spectrum for a higher gate voltage (Vg3), where only the attractive-polaron exhibits non-perturbative coupling to the cavity mode.

4.3 Fermi Polaron-Polaritons

The most convincing argument bolstering the attractive polaron model is observed in the strong coupling regime of the coupled cavity exciton system. In the following, I will discuss our results we obtained by reducing the cavity length to ∼ 1.9 µm. Figure 4.13 shows the transmission spectrum through the cavity. In Figure 4.13a, the gate voltage Vg = −30 V rendering the TMD monolayer free of electrons. Indeed, the cavity mode and the exciton resonance show a clear anti-crossing: the cavity mode and the exciton hybridize into lower and upper polariton modes. Figure 4.14a shows a cross section through Figure 4.13a for the cavity length at which the cavity mode is in resonance with the exciton, revealing a normal mode splitting of ∼ 16 meV.

When the gate voltage is increased to Vg = −5 V (Figure 4.13b), due to the ﬁnite electron density, we expect a partial transfer of oscillator strength from the exciton to the attractive polaron. The exciton itself, due to the surrounding Fermi sea electrons, should be described as a repulsive polaron for this gate voltage. Indeed, scanning the cavity length at Vg = −5 V, we see a slight reduction of the normal mode splitting around the exciton resonance and we see a broadening of the cavity mode and a typical dispersive shift of the cavity mode energy as the cavity mode crosses the attractive polaron resonance. Figure 4.14b shows two cross sections through Figure 4.13b for the cavity length for the cavity lengths at which we observe a resonance of the cavity mode with the repulsive respectively the attractive polaron.

At the even higher Fermi energy associated with a gate voltage of Vg = −2 V, we 38 Fermi polaron polaritons

a Photon energy (meV) b Photon energy (meV) Photon energy (meV) 1660 1640 1620 1600 1660 1640 1620 1600 c 1660 1640 1620 1600

1.0 1.0

0.8 0.8

0.6 0.6

0.4 0.4 Intensity (a.u.) PL intensity (a.u.) 0.2 0.2

0.0 0.0 750 760 770 780 750 760 770 780 750 760 770 780 Wavelength (nm) Wavelength (nm) Wavelength (nm)

Figure 4.14: a Line-cut through the data in Fig. 3a for the piezo voltage up = 57 V shows the transmission spectrum (red curve) at the resonance of the cavity with the exciton. b, Line-cut through the data in Fig. 3b for the piezo voltage up = 10 V respectively up = 69 V. c Line-cut through the data in Fig. 3e for up = 25 V, corresponding to the case where the cavity mode is resonant with the attractive polaron resonance. The photoluminescence spectrum in the strong coupling regime is also plotted (green shaded curve). do not observe an avoided crossing of the cavity mode with the repulsive polaron resonance. At the same time, we see a clear anti crossing of the cavity mode with the attractive polaron mode. Figure 4.14c shows in red a cross section through Fig- ure 4.13c for the resonance of the cavity mode with the attractive polaron, revealing a normal mode splitting of ∼ 8.3 meV which is in the same order of magnitude as the oscillator strength of the exciton in a TMD flake devoid of any Fermi sea electrons. The observation of the large anticrossing between the cavity and the lower energy resonance indicates the latter’s large and spectrally narrow oscillator strength. This can only occur for attractive polarons but not for trions. This is because the initial and final state of the absorption of a photon to create a trion have vanishing overlap: The initial state comprises an electron, part of the Fermi sea, which is completely delocalized. However, as part of the trion, that same electron has to be tightly bound to an exciton. This marginal overlap is part of the expression for the transition rate according to Fermi’s golden rule and will therefore ensure that the trion will only weakly couple to light. In this argumentation, we assume that the Fermi energy is larger than typical ionization energies of localized electrons. We know that we are in this regime in our experiments since we observe a sizable energy shift of the observed optical resonances as a function of electron density. Figure 4.15 shows newer data of the attractive polaron-polariton with a clearer anti-crossing. In contrast to the trion, the projection of the attractive polaron onto a an undressed exciton is finite. The undressed exciton in turn strongly couples to light ensuring a large oscillator strength of the attractive polaron. Figure 4.14 shows in green the PL spectrum obtained by integrating the PL observed through the cavity as the cavity is scanned. This is the same method as described before when we looked at the transmission spectrum in the weak coupling regime, however, now, at the short cavity length around 1.9 µm. The emission observed in PL is significantly shifted to lower energies as compared to the center of 4.4. Theoretical modeling of Fermi polaron polaritons 39

1400 Vp = 21.6 V 35 Vp = 26.1 V 1200 ) V (

1000

p 30 V

g 800 a t l o v 25 600 o z e i

P 400 Intensity (counts/0.1s)

20 200

0 762 763 764 765 766 767 768 769 770 760 762 764 766 768 770 772 Wavelength (nm) Wavelength (nm)

Figure 4.15: a Transmission spectrum of a cavity mode forming upper and lower polariton modes with the attractive polaron resonance. b, Line-cut through the data in a for the piezo voltage up = 21.6 V respectively up = 26.1 V.

the polariton avoided crossing. At this wavelength, we neither see any anti crossing nor even a broadening of the cavity mode, even at this short cavity length. In order to study the oscillator strength transfer in more detail, we measure the transmission spectrum of the MoSe2 monolayer as a function of the gate voltage Vg when the bare cavity mode is in resonance with the repulsive polaron (left hand side) or the attractive polaron (right hand side). An increase of the electron density ne and therefore of the Fermi energy EF results in a decrease respectively increase of the normal mode splitting for the repulsive respectively attractive polaron. It is worth mentioning, that the maximum normal mode splitting of the attractive polaron, which is observed at Vg ≈ 5V, is less than half of that of the exciton, measured in a ﬂake devoid of a 2DES. If the electron density ne is further increased, the normal mode splitting of the attractive polaron decreases again. This is to be expected because, in the limits of large ne, the oscillator strength is distributed over a broad energy range of the order of EF . As a result, the optical response of the MoSe2 ﬂake at the narrow cavity mode is strongly suppressed.

4.4 Theoretical modeling of Fermi polaron polaritons

We expand the model that we covered in 3.2.1 to include cavity modes with in-plane momentum k and energy ωc(k) described by annihilation operator ck: 40 Fermi polaron polaritons

Figure 4.16: a, The white light transmission spectrum of the ﬁber cavity incorporating the MoSe2/h-BN/graphene heterostructure, as a function of the gate voltage (vertical scale) for two diﬀerent settings of the cavity length: the left (right) part shows the transmission when the cavity is tuned on resonance with the repulsive (attractive) polaron. For each horizontal line, the cavity frequency is tuned so as to yield two polariton modes with equal peak amplitude. As Vg is increased, the oscillator strength transfer from the repulsive to attractive branch is clearly visible. While the normal mode splitting for the repulsive branch disappears for Vg ' −10 V, the collapse of the splitting takes place at Vg = 25 V for the attractive branch. b, The spectral function calculated using the Chevy ansatz in the strong coupling regime captures the oscillator strength transfer from the repulsive to attractive polaron but fails to predict the collapse of the normal mode splitting with increasing electron density. 4.4. Theoretical modeling of Fermi polaron polaritons 41

X † X † H = ωC (k)ckck + g(ckxk + h.c.) k k X † X † X † † + (k)ekek + ωX (k)xkxk + Vqxk+qek0−qek0 xk (4.21) k k k,k0,q ~k2 ~k2 ωX (k) = −Eexc + + δ(EF ), k = , 2mexc 2me ~k2 ωC (k) = ωc + , (4.22) 2mc

where g describes the exciton-photon coupling strength and mc is the eﬀective mass of the cavity photon. We added the ﬁrst line of equation 4.21 and the last line of equations 4.22 to model described in section 3.2.1. We extend our truncated Hilbert space and use the following extended Chevy ansatz:

! (p) † † X † † X † † |Ψ i = φ0xp + ϕ0cp + φk,qxp+q−kekeq + ϕk,qcp+q−kekeq |0i. (4.23) k,q k,q

In contrast to section 3.1, where we evaluated the spectral function of the zero- momentum exciton, here, we evaluate the corresponding expression for the cavity photon:

−iHt † A(t) = h0|c0e c0|0i, (4.24)

since we are probing the system by coupling to its photonic part. Figure 4.16b shows the spectral function in dependence of the Fermi energy EF. Our model captures the oscillator strength transfer from the repulsive to the attractive polaron, as EF is increased. Furthermore, it also captures the repulsion of the attractive and repulsive polaron-polariton branches with increasing EF. One shortcoming of the model concerns the collapse of the normal mode splitting at high Fermi energies. This lack of accuracy can be attributed either to an increase of the Bohr radius due to screening or to the breakdown of the Chevy ansatz at very high electron densities. We expect that the introduction of more electron-hole pairs dressing the exciton would show the measured spread of the oscillator strength over a larger spectral range which causes the collapse of the normal mode splitting.

Hole injection 43

5 Hole injection

A slow non-linear process that can be measured in MoSe2 is the optical injection of a population of holes. In sample M2, we observe the n-doped regime but we usually cannot achieve hole doping even at very negative gate voltages. The reason for this imbalance seems to be a much larger Schottky barrier for holes than for electrons. We can overcome this barrier by creating electron-hole pairs by non-resonant optical excitation and rely on the electron tunneling out of the monolayer at negative gate voltages. As a result, only the hole stays in the monolayer. This is especially interesting as a technique of a creating valley-polarized hole population: If we expose the sample to a circularly polarized PL laser, we will only create excitons in one valley. Under the assumption that the hole does not scatter into the other valley when the electron tunnels out of the flake, a valley polarized hole population will build up. The most visible effect of the hole injection is observed in the fiber cavity setup when the cavity mode is in resonance with the positive attractive polaron. We measure the transmission of a single-mode fiber coupled white light LED with a Gaussian spectrum centered around 750 nm and a bandwidth of 20 nm. We measure the transmission for different excitation powers which can be tuned via a control voltage vl. The conversion from Vl to output power is listed in table5. In order to inject holes, we use a diode pumped solid state laser (DPSS) at a wavelength of λ = 532 nm for which the cavity is close to transparent. The power of the PL laser is controlled by a stepper motor which turns a ND filter wheel to attenuate the laser. The conversion between the position of the stepper motor xm and the laser power is shown in table5. Figure 5.1 shows the transmission spectrum through the cavity for different LED and laser powers at a fixed cavity length around ∼ 2.3 µm. Each graph is measured at a different LED power and shows the transmission spectrum as a function of the laser power. If we consider the measurement performed with a dark LED (Vl = 0), we simply see a monotonically increasing single PL peak. If we look at the spectrum at Vl = 0.4 V, we see the effect of the above mentioned hole injection: At a low laser power, we see the transmission through a single cavity mode. As we increase the laser power, the cavity mode splits into two polariton modes which is an indicator for the creation of holes by the PL laser. However, considering the measurements for which the LED power is large, we see that the white light itself can also create holes. The same measurements are plotted in a different way in figure 5.2: Each subfigure is measured at a different laser power and shows the transmission spectrum as a function of the LED power. What is surprising is that for very low LED powers, the cavity mode splits into polariton modes, indicating the presence of holes. At inter- mediate LED and laser powers, the cavity mode does not split until further increase of the LED power splits the cavity mode into two polariton peaks independently 44 Hole injection

Motor position xm (steps) 10 k 20 k 30 k 40 k 50 k Laser power 7.1 nW 70.5 nW 770 nW 11.3 µW 206 µW

Table 5.1: PL laser power as a function of the position xm of the motor turning the ND ﬁlter wheel.

LED voltage Vl (V) 0.2 0.4 0.6 0.8 1.0 LED power 53 nW 296 nW 1084 nW 3.20 µW 8.05 µW LED voltage Vl (V) 1.2 1.4 1.6 1.8 2.0 LED power 17.7 µW 34.0 µW 57.5 µW 85.4 µW 116 µW

Table 5.2: Output power of the white light LED as a function of control voltage Vl. of the laser power. Currently, we lack a convincing explanation for the underlying dynamics. 45

Figure 5.1: Transmission spectrum as a function of the laser power for diﬀerent LED intensities. The exposure time te was adjusted to the LED power in order not to saturate the CCD of the spectrometer. 46 Hole injection

Figure 5.2: Transmission spectrum as a function of the LED power for diﬀerent laser intensities. The spectrum for each LED and laser power is normalized to its peak power. Excitons, trions and polarons in magnetic ﬁelds 47

6 Excitons, trions and polarons in magnetic ﬁelds

6.1 Introduction

As we have seen in section 2.2, for low energy excitations, charge carriers in TMDs have a twofold degree of freedom described by the valley pseudospin. In the first part of this chapter, we will demonstrate that we can lift the degeneracy between excitons in the two valleys using a magnetic field. Thanks to the correlation between valley pseudospin and the circular polarization of the exciton, we can detect this splitting in polarization resolved spectroscopy. This is an important ingredient for valleytronics, which aims at using the valley pseudospin for information processing [47], [48]) and applications of valleytronics [44]–[46]. In the second part, we extend our experiments to polarons. As we will see, polarization resolved spectroscopy of the polaron can serve as a valley selective measurement of the electron population. This confirms that we can achieve a sizable electron gas of valley polarized electrons by a magnetic field alone.

6.2 Exciton magnetic moment

Optically we cannot measure the electron or hole Landé g-factor but we can measure the magnetic field induced splitting of the exciton resonance which we can describe by an effective exciton g-factor. The magnetic field induced change in the exciton energy is the result of the difference of the conduction and the valence band electron magnetic moment.

6.3 Measurement of the exciton magnetic moment

6.3.1 Sample and Setup The work presented in this and the following section is the result of a collaboration with Ajit Srivastava, Adrien Allain, Dominik Lembke, Andras Kis and Atac Imamoglu and is published in [95]. Similar results were published in ref. [96]–[99]. The sample for this study was fabricated by our collaborators in Prof. Andras Kis’s group. We used a WSe2 monolayer that was obtained by mechanical exfoliation of synthetic WSe2 crystals onto heavily doped silicon substrates with a 285 nm thick SiO2 layer on top. The sample was not connected with any electrode and therefore its electron density could not be controlled. All the measurements mentioned in this chapter are performed in a bath cryostat (see ﬁgure 6.1). In more detail, this means that the sample is placed on top of a stack of attocube actuators, allowing it 48 Excitons, trions and polarons in magnetic ﬁelds

Figure 6.1: A sketch of the confocal setup. a The WSe2 monolayer transferred onto a SiO2 on Si substrate. b The sample can be moved in x, y and z direction by piezoelectric actuators (orange). The collimated excitation laser is focused onto the sample by an aspheric lens. The same lens collects the photoluminescence emitted by the sample which is transmitted as a collimated beam. c The sample is located at the bottom of a tube filled with 20 mbar of exchange gas (helium) which is immersed in liquid helium (LHe). A superconducting solenoid provides magnetic fields along the tube’s axis of up to 8.4 T. A window in the top of the tube and an optical setup mounted on top allow optical access to the sample. A pellicle beam splitter (Pel) allows to image the sample onto a CCD. d The sample mounted on an adapter to allow for Voigt geometry. The sample can be moved in x, y and z direction. An adapter holds an aspheric lens which collimates the PL and a mirror that deflects the beam.

to be moved in all three directions to place it in the focus of an aspheric lens. The sample and the lens are at the bottom of a stainless-steel tube filled with 25 mbar of helium. The excitation light is coupled out of a single-mode fiber in the side arm of an optical setup on top of the insert. A linear polarizer (pol) followed by a liquid crystal retarder (LCR) prepares the polarization of the laser. The laser passes a beam splitter cube (BS) and a window to reach the aspheric lens (AL) focusing the collimated beam onto the sample. The collected PL passes the beam splitter to the top arm of the optical setup, where a LCR followed by a polarizer allow to analyze the PL polarization before it is coupled into a single-mode fiber. A pellicle beam splitter (Pel) sends 8% of the light from the sample through an achromatic lens with 15 cm focal length which allows to image the sample onto a CCD. A superconducting solenoid surrounding the insert tube provides a magnetic field along the optical axis (Faraday geometry). When we want to measure our sample in Voigt geometry, which means that the magnetic field is perpendicular to the optical axis, we use an adapter consisting of a gold mirror, which deflects the collimated beam to be perpendicular to the solenoids axis, followed by an aspheric lens. The sample itself is also placed on top of an adapter which rotates the substrate to face the aspheric lense. 6.3. Measurement of the exciton magnetic moment 49

1200 0 X 1000

800

- 600 X 400

PL Intensity (a.u.) PL Intensity 200

0 700 710 720 730 740 Wavelength (nm)

Figure 6.2: A typical PL spectrum of a WSe2 monolayer at 4.2 K. The resonance at 708 nm is identiﬁed as the neutral exciton (X0). A second resonance at 722 nm stems from localized, negatively charged trions (X−). The excitation laser is linearly polarized. The PL polarized parallel respectively perpendicular to the laser is shown in orange respectively purple.

6.3.2 Polarization-resolved photoluminescence and reflection spectroscopy Since the optical polarization is linked to the valley degree of freedom, we expect that the energy difference between the bandgap at the different valleys could directly be measured in polarization-resolved PL and absorption measurements. Figure 6.2 shows a typical photoluminescence (PL) of our sample at 4.2 K. The sample was illuminated with a linearly polarized helium-neon (HeNe) laser (λHeNe = 632.8 nm). In orange (purple), figure 6.2 shows the PL that is co- (cross-) polarized to the excitation laser. We can clearly identify emission from the exciton as the peak at ∼ 708 nm. The high degree of valley coherence (linear dichroism) of ∼ 20 % confirms that our sample is indeed a monolayer [64]. At ∼ 722 nm we see the resonance of the attractive polaron. At energies below the attractive polaron, we detect broad resonances that are associated with impurities (which we will discuss in more detail in chapter7). Since we do observe the attractive polaron, we should be consequent and use the term repulsive polaron instead of exciton. However, we think that the electron density in this sample is so low that we model the repulsive polaron in very good approximation as an exciton. Two arguments support our claim that the electron density is small: A) PL from the repulsive polaron is significantly larger than from the attractive polaron and B) while we can identify the repulsive polaron in white light reflection measurements, we do not see any response from the attractive polaron. Figure 6.3a shows the PL from the exciton resonance at different magnetic fields B up to 8.4 T in Faraday configuration. We excite the sample with a linearly polarized HeNe laser. The PL is analyzed in a circularly polarized basis (corresponding to the curves in red and blue). With increasing B-field, we can clearly see a splitting between the peaks observed in PL in the two circular polarizations. Figure 6.3b shows the splitting as function of the magnetic field obtained by fitting a Lorentzian curve to either of the two circularly polarized PL peaks. We observe a linear increase 50 Excitons, trions and polarons in magnetic fields

of the splitting with a slope µ = (4.37 ± 0.15)µB, where µB = e~/2me is the Bohr magneton. Measurements of five different samples were within ±10 % of this value. The fits in Figure 6.3a correspond to a single Lorentzian lineshape on top of a cubic polynomial to account for any broad background contribution (e.g. the tail of the attractive polaron peak). In addition to photoluminescence measurements, we also measured the white light reflection from the sample. We are interested in the change in reflection from the substrate due to the presence of the WSe2 flake. Therefore, we measure the reflected light from a broadband LED on the substrate next to the sample (Roff) and on the sample (Ron). The difference between the two divided by the spectrum from the bare substrate is then plotted as ∆R/R = (Roff − Ron)/Roff. Again, we analyze the reflected spectra in the two circular polarizations. We can clearly see a difference of the peak position in the two polarizations, as shown in Figure 6.3c for a magnetic field of 8.4 T. However, the splitting that we observe in reflection corresponds to µ = 3.67µB, which is slightly less than the splitting observed in PL. We attribute the difference in the magnetic moment measured in PL as compared to reflection to the larger uncertainty of the splitting measured in reflection due to the larger number of fitting parameters. The fits in Figure 6.3c correspond to an admixture of absorptive and dispersive Lorentzian lineshapes on top of a linear background. The linear background should compensate for artifacts such as potential shifts of the focus during the translation of the sample which, due to the chromatic aberration of our aspheric lense would lead to a different filtering of the white light. We check whether reversing the direction of the magnetic field exchanges the polarizations of the red and the blue detuned exciton peaks. As expected, as we change the magnetic field from -7 T to +8.4 T, we observe that the helicity of the split peaks are reversed, as shown in Figure 6.5b. When PL is analyzed in a linear basis rather than in the circular basis, we observe no significant splitting between the two polarizations. This is expected, since we will just see an admixture of the two split lorentzians of the circular basis. Because the splitting observed in the circular basis is, even at 8.4 T, much smaller than the exciton linewidth, the two Lorentzians are practically merged into one broadened peak. Figure 6.4a shows PL at 8.4 T analyzed in the circular (top) and a linear basis (bottom). Varying the both the excitation and the collection polarization allows to confirm that the PL at finite magnetic field is indeed circularly polarized. When we excite the sample with circularly polarized light and analyze it in the circular basis, we measure the circular dichroism of our WSe2 sample. Figure 6.4b shows co- and cross-polarized PL following excitation with circularly polarized light at 0 T and at 8.4 T. The circular dichroism we measure is approximately 20%. As discussed in 2.5, this modest value is most likely due to electron–hole exchange- induced mixing of the two valleys [30], [70]. We observe that at higher magnetic fields, the degree of circular dichroism increases by an approximate factor of two which can be explained by the decrease in exchange-interaction due to the lifted degeneracy. Figure 6.5a shows the PL in one circular polarization for 0 T and 8.4 T for the case that the direction of the magnetic field lies within the plane of the sample (Voigt geometry). In this configuration, we can observe neither a shift nor a broadening of the peak as the magnetic field is increased. This extreme anisotropy in the magnetic 6.3. Measurement of the exciton magnetic moment 51

a b 2 σ� μ ��μB σ� 1

8.4 T (meV) 0

K,-K

Δ -1

-2 6 T -8 -4 0 4 8

Bz(T)

c σ1 Normalized PL Intensity Normalized μ ��μB σ2

4 T

ΔR/ R

0 T Bz = 8.4 T

704 708 712 705 710 715 Wavelength (nm) Wavelength (nm)

d K -K ΔEinter,c B Figure 6.3: a Polarization-resolvedΔEs PL of the WSe2 exciton at different magnetic σ+ fields perpendicular to the sample plane (Faraday geometry)ΔEinter,c shows an increasing σ+σ- ΔEs splitting between the two polarizations asB the= 0 magnetic field isσ- increased. b Mag- ΔEinter,v ΔEintra netic field dependence of theΔEs exciton splitting fitted with a linear function reveals a magnetic moment µ = (4.37 ± 0.15)µ . c The exciton spectrumΔEinter,v in a polarization B ΔE B ΔEs intra resolved reflection measurement at an out of plane magnetic field of Bz = 8.4 T shows a splitting corresponding to a magnetic moment µ = 3.67µB.

1.0 a b σ� 0.8 σ σ � 2 σ� 0.6

0.4

0.2 +8.4 T 0.0

1.0 π� 0.8 PL Intensity (a.u.) PL Intensity π� 0.6 0.4 0.2 0 T 0.0 705 710 715 705 710 715 Wavelength (nm) Wavelength (nm)

Figure 6.4: a PL from the exciton at an out-of-plane magnetic field of 8.4 T showing a splitting when analyzed in a circular basis but no splitting in any linear basis. b Polarization resolved exciton PL at out-of-plane magnetic fields of 0 and 8.4 T with circular (σ2) excitation. The suppression of PL which is cross polarized to the excitation laser (circular dichroism), is clearly increased for large magnetic fields. 52 Excitons, trions and polarons in magnetic fields

a b

σ� σ�

+8.4 T Bz = -7 T

Normalized PL Intensity Normalized

0 T

Bz = +8.4 T

704 708 712 700 705 710 715 Wavelength (nm) Wavelength (nm)

Figure 6.5: a The exciton PL in Voigt geometry (magnetic field parallel to the sample plane) without and with a magnetic field of 8.4 T. The exciton PL peak neither shifts nor broadens as a result of the magnetic field. b The polarization of the lower and higher energy split exciton peaks are exchanged as the sign of the magnetic field is inversed. response of the monolayer TMD sample follows directly from the fact that the orbital magnetic moment of a strictly 2d material points out of the plane and therefore can only couple to a magnetic field perpendicular to the sample surface. In addition, our experimental conditions also exclude any contribution from the spin-Zeeman effect to the B-induced splitting since the strong spin-orbit coupling in both, the valence and the conduction band of WSe2, ensures that the spin degree of freedom is frozen out [51], [100]. Recent calculations based on four-band [99] and eleven-band [101] k · p Hamil- tonians fitted to density functional theory (DFT) models improved by the GW approximation (GWA) yield for g of WSe2:

µv(K) = 4.08 µB (DFT+GWA)

µc(K) = 0.24 µB (DFT+GWA). (6.1)

The resulting exciton magnetic moment amounts to gX = gc − gv ≈ −3.84 and is therefore, besides the sign which depends on the deﬁnition of the circular basis, in good agreement with our experimental results. An important aspect in the calculation of the exciton g-factor is the large spread of the exciton wavefunction in k-space resulting from the small exciton Bohr radius. As a consequence, the g-factor of an exciton is not just the diﬀerence in the valence band and conduction band g-factor at the K-point but should rather be a weighted average of the g-factors of all k-vectors contributing to the exciton wavefunction. 6.4. Magnetic moment of localized trions 53

a b σ � 3 μ ��μB σ�

8.4 T 2

(meV)

Δ 1

0 T

Normalized PL Intensity Normalized 0 715 720 725 20 864 Wavelength (nm) Bz(T)

Figure 6.6: a Circular polarization resolved PL from the trion peak at different magnetic fields perpendicular to the sample plane. b The splitting obtained from the data shown in a as a function of magnetic field. By fitting a linear dependence we measure a magnetic moment of µ = 6.28 ± 0.32µB.

6.4 Magnetic moment of localized trions

In PL, we observe the resonance of the attractive polaron ∼ 30 meV below the repulsive polaron (exciton) resonance. In reflection, we cannot observe the attractive polaron at all. Even though there are no electrical contacts to the sample and therefore we cannot check the polarity of the charge carriers, we assume that our sample is n-doped since this is the most common natural doping of WSe2 samples. However, the doping density most likely is so low that the attractive polaron we observe are very similar to localized trions since the few extra electrons in the system most likely are trapped to a shallow bound state just below the conduction band. This electron will form a bound state with an exciton. The latter can lower its energy by the trion binding energy ET but will be localized around the trapped charge. This quasi particle is close to the model of a trion as a bound three body state. The density of localized charges however seems to be low such that we cannot observe the localized trion in resonant experiments such as reflection. In PL however, we clearly see the trion peak. Figure 6.6a shows the polarization resolved PL from the trion peak at different magnetic fields perpendicular to the sample plane. We see a clear increase of the splitting between σ1 and σ2 polarized PL. The splitting as a function of magnetic field is plotted in figure 6.6b and shows a linear dependence corresponding to a magnetic moment of µ = 6.28 ± 0.32µB.

6.5 Polarons in a magnetic ﬁeld

While the charge density in the WSe2 sample cannot be controlled, we have working gates in MoSe2 samples. Reflectivity data of this sample in the absence of a magnetic field was presented in chapter 3.3 and showed that we can reach Fermi energies as large as 50 meV in this sample which is a regime where the trion model breaks down and we can use the attractive polaron model instead. The data discussed in this section was measured by Patrick Back on the gate controlled MoSe2 sample M1. Ovidiu Cotlet developed the theoretical model. This section is based on the re- 54 Excitons, trions and polarons in magnetic fields

a b EF EF VX-e VX-e

K -K K -K

Figure 6.7: Formation of polarons between the exciton at K and an electron-hole pair at -K. sults of a collaboration with Patrick Back, Ovidiu Cotlet, Ajit Srivastava, Naotomo Takemura, Martin Kroner and Atac Imamoglu that were published in ref. [102]. In order to explain the behavior of polarons in a magnetic field, we are going to employ a model which treats polarons as interactions between electrons and excitons whose energies have been modified due to the presence of the magnetic field. More specifically, in a magnetic field we assume that the K-valley conduction band is +K shifted to higher energies by ∆Ec = gc · µB · B/2, where gc is the conduction band Landéfactor, while the −K-valley will shift to lower energies by the same amount. A priori, we assume that the magnetic field induced conduction band splitting will remain unchanged as we fill the electron states with a 2DES. At a finite out-of-plane magnetic field B, we expect the conduction and valence bands to shift according to the respective free electron and hole g-factors. As a consequence, increasing the electron density ne in the TMD monolayer starting from a neutral sample, we will first fill electrons into the −K-valley while the K-valley conduction band still remains empty. Only once the Fermi energy EF is larger than K-valley conduction band minimum, we start filling electrons into both valleys. We refer to the gate voltage at which we start filling the ±K-valley as Von,±K . As we have seen in chapter3, in the presence of a 2DES, we expect the K-valley exciton to interact exclusively with the −K-valley electrons to form the attractive and repulsive polaron resonances. This is because in MoSe2 the singlet trion, consisting of an exciton in one and an electron in the opposite valley, is the only bound trion state. The Feshbach-like increase of attractive interaction between the exciton and electrons just below the trion energy and the corresponding repulsive interaction for electrons just above the trion energy therefore only exists for electrons in the opposite valley of the exciton.

The most interesting regime is when Von,−K < Vg < Von,+K . In this case, the electrons in the monolayer are all in the −K valley (see figure 6.7). As a consequence, only +K-excitons can interact with electrons to form attractive and repulsive polarons. As in the case of WSe2, we use a circularly polarized detection scheme to probe the ±K-exciton by σ±-polarized light. The setup used for this experiment is the same as the one described in 6.3. Figure 6.8a,b show the differential reflection spectra of σ+ (a) and σ− (b) polarized light. In contrast to the measurements presented in 6.3 we define differential reflection for this measurement as (∆R/R)(Vg) = (R(60 V) − R(Vg))/R(60 V). This choice of normalization proves to be less problematic than one where we move the 6.5. Polarons in a magnetic field 55 sample out of the focus of our collection lens. Moving the sample potentially changes the distance between lens and sample and therefore exposes our normalization to aberrations of the aspheric lens. The normalization with the spectrum obtained at a large electron density is motivated by the observation in chapter 3.3 which shows that for large electron densities, the oscillator strength is distributed over a large spectral range. As expected, the gate voltage at which the attractive polaron emerges is different for the two circular polarizations: For σ+ polarized light, we observe the attractive polaron already at a gate voltage of Von,−K = 100 V while the in σ− the attractive polaron only emerges at Von,+K = 70 V. Furthermore, between Von,−K and Von,+K , the σ+ polarized attractive polaron drifts to lower energies with increasing electron densities. As discussed in chapter3, the reason why the attractive polaron in MoSe 2 increases in energy as the electron density is increased, is because the increasing splitting between attractive and repulsive polaron is combined with a global blue shift of the two resonances due to a phase space filling induced blue shift of the exciton energy. For Fermi energies such that only the −K-valley is occupied with electrons, the K valley in which the exciton part of the σ+ polaron wavefunction is generated, is not subject to phase space filling such that we only observe the attractive polaron red-shift due to the increasing electron density. At the same time, we observe a blue-shift of the −K-exciton. This blue shift is due to the phase space filling due to the −K-electrons.

For gate voltages Vg < Von,+K , both valleys are populated by 2DES electrons (see figure 6.7). As a consequence, excitons of either valley interact with the electrons of the opposite valley to form attractive and repulsive polarons. Furthermore, both excitons are suspect to phase space filling and therefore all polaron resonances are shifting to higher energies with increasing electron densities. Figure 6.7c shows the reflection spectrum at a gate voltage of 127 V, for which the sample is neutral. We observe a splitting between the two excitons corresponding to an exciton g-factor of -4.4. Whether it is justified to talk of a g-factor for the polaron resonances is debatable. For example, if look at the differential reflection at Vg = 69 V (figure 6.7d), we observe an energy difference between the σ+ and σ− polarized attractive polaron that corresponds to a g-factor of 18. However, since Vg = 69 V is just below Von,+K , increasing the magnetic field by a small amount will increase the splitting between the conduction bands in the two valleys beyond the Fermi energy. Therefore the attractive polaron will no longer be observable in σ− polarized differential reflection, yielding the g-factor ill-defined. Expressing energy splitting in units of Bohr magnetons however serves the purpose of comparing the sensitivity of resonances to magnetic fields. In that spirit, at the same Vg = 69 V, we observe a repulsive polaron Landéfactor of +7.2. The opposite sign of this energy shift comes from the fact that blue shift due to phase space filling of the σ− repulsive polaron acts in the opposite direction of the contribution to the g-factor due to polaron formation. Figure 6.7e shows the reflection spectrum at Vg = −13 V While our model worked well to qualitatively describe polarons in a magnetic field, it fails to do so quantitatively, shedding light on strong many-body interactions. The limits of this model appear when we calculate the density of electrons introduced into the monolayer while raising the gate voltage from Von,−K , corresponding to an electron depleted sample, to Von,+K . Using the capacitive model introduced 56 Excitons, trions and polarons in magnetic fields

Figure 6.8: a Differential reflection of σ+-polarized light as a function of gate voltage Vg. The attractive polaron appears at Vg = Von,K. b Differential reflection of − σ -polarized light as a function of gate voltage Vg. The attractive polaron appears at Vg = Von,-K. c Differential reflection at Vg = 127 V showing a valley splitting for the repulsive polaron corresponding to a g-factor of +7.2 and -18 for the attractive polaron. d Differential reflection at Vg = −13 V showing a valley splitting corresponding to -14.4. In addition, polarization-resolved PL is also shown in the same graph.

in chapter 3.3 and the constant density of states corresponding to one (!) parabolic band, we arrive at an electron density of 1.6 × 1012 cm−2. Since we do not observe an attractive polaron peak in σ−, we conclude that all of these electrons are valley- polarized, a major milestone for the investigation of valley-dependent effects such as the valley hall effect and potential applications in valleytronics. Furthermore, the polarization-resolve attractive polaron spectroscopy proves to be a viable alternative to Kerr-reflection measurements for probing spin population differences. The most interesting aspect of this large spin-polarized electron population appears is that the band offset equal to EF = 15 meV, which is the Fermi energy of an electron density 12 − of 1.6 × 10 cm 2, would correspond to an electron Landéfactor of gc = 38. The real conduction band g factor is of course much lower than that as the largest MoSe2 conduction band g factors obtained in simulations is gc = 5.12. The reason for the delayed filling of the K conduction band lies in many-body effects favoring single band occupancy. 6.6. Conclusion 57

6.5.1 Correcting the phase shift in the reﬂection spectrum

Figure 6.9a/b shows the differential reflection dR/R at Vg = 60 V, 31 V and 13 V. At Vg = 60 V, in the absence of free electrons, the absorption spectrum is dominated by the exciton. The spectra at Vg = 31 V and Vg = 13 V correspond to the ones depicted in Figure 6.8a. In order to pinpoint the voltage at which the attractive polaron emerges, we need to measure the oscillator strength of the exciton as well as the attractive and repulsive polaron resonances. Therefore, it is necessary to integrate the area under the peaks in the absorption spectrum. However, we notice that the differential reflection imag(Ψ) clearly shows negative values. This is because what we measure in our setup is not the absorption spectrum imag (χ(ω)) but rather imag(Ψ(ω)) = imag (exp (iα(ω)) χ(ω)) where χ(ω) is the susceptibility and α(ω) is a wavelength dependent phase shift. This highlights once more the advantage of the cavity setup where we get independent measurements of imag(χ) as well as real(χ). The mixing of real and imaginary part of the interference of χ in differential reflection is due to interference effects between reflections at different surfaces (h-BN, SiO2, Si). To first order, we assume α(ω) = α0 to be wavelength independent within the spectral range of interest. At this point, we can use another tool whose usability we tested in the context of the fiber cavity: Kramers-Kronig relations. We find:

χ = χ0 + iχ00 (6.2) Ψ = Ψ0 + iΨ00 ≈ Ψ0 + i · dR/R (6.3) imag(χ) = imag(e−iα0 Ψ) (6.4) Z ∞ 0 00 0 00 2 ω Ψ (ω ) ∝ cos(α0)Ψ (ω) + sin(α0) P 02 2 dω. (6.5) π 0 ω − ω In the absence of any Fermi sea electrons, the exciton absorption is expected 00 to feature a Lorentzian lineshape. Consequently, we choose α0 such that χ for Vg = 60 V can be accurately described by a Lorentzian. The absorption spectra 00 χ after correcting for the phase shift α0 are plotted in Figure 6.9e/f for σ+/σ−. The corrected spectra are predominantly positive which is an indication that the correction with a constant phase is a good approximation. Figure 6.9g/h shows line cuts through Figure 6.9e/f at Vg = 60 V, 31 V and 13 V indicated in the latter with dashed lines. The black dashed lines represent a ﬁt to the absorption spectrum at Vg = 60 V by a Lorentzian lineshape. The ﬁt is in excellent agreement with the experimental data after the correction of the phase shift α0.

6.6 Conclusion

We measured the Landé g-factor for excitons, localized trions as well as for attractive and repulsive polarons in a 2DES. Our experimental findings are qualitatively in good agreement with the predictions of our polaron model. Due to the large electron- electron exchange interaction, we find that the single-particle band alignment breaks down. As a result, even for moderate magnetic fields of 7 T, we observe a large, highly valley polarized, electron population. Further improvement in sample quality 58 Excitons, trions and polarons in magnetic fields

a 5 b 5

σ+Vg = 60 V σ−V g = 60 V

σ+Vg = 31 V σ−V g = 31 V 4 4 σ+Vg = 13 V σ−V g = 13 V

3 3 Ψ '' Ψ '' 2 2

1 1

0 0

730 735 740 745 750 755 760 765 770 730 735 740 745 750 755 760 765 770 Wavelength (nm) Wavelength (nm) c 6 d 6

σ+Vg = 60 V σ−V g = 60 V

5 σ+Vg = 31 V 5 σ−V g = 31 V

σ+Vg = 13 V σ−V g = 13 V 4 Fit: Lorentz 4 Fit: Lorentz

3 3 χ '' χ ''

2 2

1 1

0 0

730 735 740 745 750 755 760 765 770 730 735 740 745 750 755 760 765 770 Wavelength (nm) Wavelength (nm)

Figure 6.9: a Differential reflection for B = 7 T and σ+ polarized light. We observe the absorption from the exciton which turns into a repulsive polaron shifting to higher energies and broadening as the gate voltage Vg is decreased. In the spectra for Vg = 31 V and Vg = 13 V , we observe the attractive polaron at an energy lower than the exciton energy. b Differential reflection for B = 7 T and σ− polarized light. The attractive polaron only becomes observable at Vg = 13 V . In a,b, all observed peaks are partly negative which makes it unfeasible to determine the area under the peaks. c,d show the measurement data of a,b after correction of the phase shift α0. 6.6. Conclusion 59 in combination with measurements at lower temperatures could potentially even reveal ferromagnetism in such a system. To our knowledge, previous experiments did not show comparable valley polarization. While other experiments in magnetic fields [82], [97] have demonstrated a finite, partially polarized electron population due to the magnetic field, they most likely did not measure in the regime where exactly one of the two valleys is filled with a 2DES. Other experiments have demonstrated the creation of a surplus of electrons by optical excitation [103], [104], or through ferroelectric contacts [105], however, with a small degree of valley polarization. We also demonstrated how observables, such as the oscillator strength of the K versus the −K attractive polaron, predicted by the polaron model allow to determine the valley polarized population. Future experiments such as Kerr-reflection could be used to corroborate our findings.

Single photon emitters in WSe2 61

7 Single photon emitters in WSe2

7.1 Introduction

So far, we have studied two-dimensional properties of TMDs. From other semiconductor systems, we know that we can further reduce the dimensionality which can completely change the physics of the confined quasi particles. Solid state single photon emitters, such as color centers or self-assembled quantum dots, are promising candidates for quantum information processing [106]–[108]. All optical manipulation of single spins bound to quantum dots has been demonstrated in GaAs [109], [110]. While for excitations deeply below the bandgap, the physics of the bound states is completely disconnected from the band structure of the host crystal (as is the case e.g. in rare-earth-doped crystals [111]) , the bound states due to an adia- batically varying band profile (such as in in self-assembled quantum dots) inherits many properties from the bulk crystal. Quantum dots in WSe2 could therefore benefit from the unique band structure of TMDs such as the degenerate valleys and the non-zero Berry curvature [46] . In this chapter, we are going to discuss quantum in WSe2. Furthermore, we will investigate the stability of these quantum emitters. We will look at their behavior in a magnetic field and perform photoluminescence excitation to find similarities to the bulk crystal properties of WSe2. The results presented in this chapter are the result of a collaboration with Ajit Srivastava, Adrien Allain, Dominik Lembke, Andras Kis and Atac Imamoglu and were published in [112]. Adrien Allain and Dominik Lembke in the group of Andras Kis fabricated the sample. Similar work was conducted at the same time by other research groups [113]–[115].

7.2 Power dependence

We investigate monolayers of WSe2 obtained by mechanical exfoliation of synthetic WSe2 crystals onto heavily doped silicon substrates covered with an insulation layer of 285 nm of SiO2. The samples, which we label, W1 and W2, are discussed in8. In this sample, we do not have any control on the doping level. The sample was mounted in the same bath cryostat setup as explained in chapter6. The left hand side of figure 7.2 shows photoluminescence spectra of a monolayer flake (flake 0) of WSe2 at two different excitation powers Pexc = P0, respectively Pexc = P0/80. In the spectrum at Pexc = P0, we can clearly observe the attractive and repulsive polaron, which, as discussed in chapter6 are most likely behaving similar to a trion bound to a localized electron and an exciton at the low doping level and high defect density expected in our monolayer. In addition, we notice a third feature in the emission spectrum which is at lower energy as compared to the attractive and repulsive polarons. This broad feature has previously been attributed to defect- 62 Single photon emitters in WSe2

30 0 4 X - X P increasing 2x 20

2 10

Normalized PL

0 0 700 710 720 740 760 780 800 Wavelength (nm) Wavelength (nm)

Figure 7.1: LHS Scaled photoluminescence spectra of flake 0 for different excitation laser powers (Pexc). The curves corresponding to the attractive and repulsive polaron PL spectra for different excitation powers overlap with each other after the division with the excitation power implying a linear dependence of PL on excitation power. Even for the highest excitation power of ∼ 320 µW, no broadening of the polaron peaks is observed. RHS The impurity band emission, scaled with the excitation laser power, exhibits a sub-linear emission power and drastic changes in the spectral features with increasing laser power. trapped excitons in TMDs due to its relative increase when a sample is exposed to α particles [116]. Whether the quantum dots are indeed a result of the defects is unclear. More recent experiments have shown that it is possible to create quantum dots by locally introducing strain into the TMD monolayer [117]. We assume that the broad band stems from an ensemble of defects of different emission energy which all saturate at slightly different excitation powers. As a result, this broad peak is expected to demonstrate a sub-linear power dependence. Indeed, reducing the excitation power shows a linear power dependence of the attractive and repulsive polaron peaks and a clearly saturating defect peak. Figure 7.1 shows the PL spectrum for an extended series of excitations powers. For readability, we scale all spectra by the respective excitation power. We notice that the curves of the scaled PL spectra for the repulsive and attractive polarons overlap perfectly, confirming the linear excitation power dependence of the polarons’ PL intensity. On the right hand side of figure 7.1, we see the scaled PL emission from the impurity peak for the same excitation powers as for the left hand side. We can clearly see that the PL from the impurity peak is increasing sub-linearly as a function of excitation power. Figure 7.2 summarizes the power dependence of the integrated counts within the attractive and repulsive polaron resonances as well as in the broad impurity band peak. The right hand side of figure 7.2 summarizes the power dependence of the repulsive 7.3. Spatial map 63 a P0 600 5 P0/ 80 10

400 0 4 X 10 - X 200 3 10

PL Intensity (a.u.) PL Intensity 2 0 (a.u.) Intensity Integ. 10 700 720 740 760 780 800 0.1 1 10 100 Wavelength (nm) Excitation Power (µW)

b QD1F1 P -3 0 10 P0

)

4 3 Figure 7.2: Integrated photoluminescence3 for the attractive (green diamonds) and repulsive (blue circles) polaron, as well as the impurity peak (red squares), showing

2 2 linear power dependence for2 the polarons and sub-linear power dependence for the QD2F1 impurity peak. x10 0 1 1 0 Intensity (x10 Intensity X X 0 and attractive polaron, along0 with the impurity peak on a log-log plot. For this 700 720 740 760 780 700 720 740 760 780 purpose,Wavelength the (nm) area under the correspondingWavelength parts(nm) of the spectrum is summed up for each of the three peaks. Indeed, the attractive and repulsive polaron follow a linear c 30 γ = 118powerµeV law inQD1F1 an excitation power range larger than two orders of magnitude up to a power as high as Pexc ≈ 320 µW. The impurity peak on the other hand is sub-linear 20 over a power range larger than four orders of magnitude, even at powers as low as 10 Pexc ≈ 100 nW.

PL Intensity (a.u.) PL Intensity The sub-linear power dependence even at very low excitation powers agrees well with the saturation behavior of the red-shifted photoluminescence that stems from γ = 110µeV QD2F1 40 defects and impurities in samples of III-V semiconductor quantum wells [118]. Figure 7.3 shows the PL spectrum of another monolayer ﬂake (ﬂake 1). We believe 20 that the density of impurities in this sample is lower such that we can indeed resolve 0 1µm

PL Intensity (a.u.) PL Intensity single quantum dots. Indeed, this ﬂake shows a collection of sharp peaks within -1.0 the-0.5 impurity0.0 0.5 band1.0 that dominate the spectrum at lower excitation powers (compare Energy (meV) LHS to RHS). For excitation powers Pexc ≤ 1 µW, the emission from the repulsive polaron is so weak as compared to the emission from the quantum dots that we can hardly see the repulsive polaron in the RHS of ﬁgure 7.3. We labeled two quantum dot peaks in the PL spectrum with ”QD1F1”, respectively ”QD2F1”. We will investigate these two resonances in more detail in the following.

7.3 Spatial map

In order to conﬁrm that the sharp lines stem from diﬀerent, spatially separated localized states, we measure spatially resolved PL spectra. For this purpose, we move the sample by piezoelectric positioners. Figure 7.4 shows in black the outline of the exciton PL emission (outside the black line, the PL intensity of the exciton has dropped below 50% of the maximum PL intensity measured within the black line). Furthermore, the color map indicates the localization of three of the quantum dot emission lines. The green saturation indicates, as a function of spatial coordi- nates, the PL emission collected within a narrow window around the peak labeled a P0 600 5 P0/ 80 10

400 0 4 X 10 - X 200 3 10

PL Intensity (a.u.) PL Intensity 2 0 (a.u.) Intensity Integ. 10 700 720 740 760 780 800 0.1 1 10 100 64 Single photon emitters in WSe2 Wavelength (nm) Excitation Power (µW)

b QD1F1 P -3 0 10 P0

)

4 3 3

2 2 2 QD2F1

x10

0 1 1 0 Intensity (x10 Intensity X X

0 0 700 720 740 760 780 700 720 740 760 780 Wavelength (nm) Wavelength (nm) c 30 γ = 118µeV QD1F1 Figure 7.3: LHS Low-temperature PL spectrum of ﬂake 1 showing emission from the20 repulsive polaron, the impurity peak as well as a collection of very narrow peaks within the impurity band that we attribute to single quantum dots. Two distinct10 peaks are labeled and will be analyzed in more detail in the following. RHS Reducing the excitation power by a factor of 1000 reduces the emission from

PL Intensity (a.u.) PL Intensity the repulsive polaron so much more than the emission from the quantum dots that the PL fromγ = 110 theµeV repulsive polaronQD2F1 is barely visible. 40

0 1µm

”QD1F1”. (a.u.) PL Intensity In blue, we see the spatial distribution of the emission from the peak labeled-1.0 ”QD1F2”.-0.5 We0.0 notice,0.5 that1.0 the two peaks are spatially separated by more than a micrometer.Energy Both (meV) peaks are still within the area from which we also see sizable PL from the repulsive polaron resonance. The spectra of figure 7.3 are collected from a spot in the turquoise area in between the green and the blue speckle such that both, QD1F1 and QD2F1 show up in the spectrum. The spatial map shows the location of yet another peak depicted in red which, however, is not visible in figure 7.3. The red region seems to be outside the are in which PL from the exciton is observed. Indeed, the emission from the exciton in the red region is more than an order of magnitude weaker than the maximum exciton emission measured on this flake. This can either mean that it just is close to the edge of the flake or it could be due to an extremely efficient relaxation of any delocalized excitons, generated in the part of the flake corresponding to the red region, into localized states. Within the the area of ∼ 8 µm, we observed more than 20 sharp emission peaks, corresponding to a density of more than 2.5 defects per µm2.

On the left of the spatial map in ﬁgure 7.4, we show a high-resolution PL spectrum of the peaks QD1F1 and QD2F1 that were obtained with an excitation power Pexc < 1 µm. For both peaks we measure extremely narrow linewidths of 118 µeV, respectively 110 µeV. Changing the excitation energy does not result in a corresponding energy shift of the sharp peaks, which allows us to rule out Raman scattering as a possible cause of the sharp peaks. a P0 600 5 P0/ 80 10

400 0 4 X 10 - X 200 3 10

PL Intensity (a.u.) PL Intensity 2 0 (a.u.) Intensity Integ. 10 700 720 740 760 780 800 0.1 1 10 100 Wavelength (nm) Excitation Power (µW) 7.3. Spatial map 65 b QD1F1 P -3 0 10 P0

)

4 3 3

2 2 2 QD2F1

x10

0 1 1 0 Intensity (x10 Intensity X X

0 0 700 720 740 760 780 700 720 740 760 780 Wavelength (nm) Wavelength (nm) c 30 γ = 118µeV QD1F1

PL Intensity (a.u.) PL Intensity

γ = 110µeV QD2F1 40

0 1µm

PL Intensity (a.u.) PL Intensity -1.0 -0.5 0.0 0.5 1.0 Energy (meV)

Figure 7.4: RHS Spatial map of selected quantum dot peaks within the impurity band of flake 1. The saturation of the green (blue) color channel of each pixel indicates the intensity stemming from a narrow spectral window around the quantum dot PL from QD1F1 (QD2F1) when the respective part of the flake is in the focus of our microscope. The red region stems from yet another sharp peak which is not included in figure 7.3. The black line indicates the outline of the area within which the emission from the repulsive polaron is more than half of its maximum value. LHS High-resolution spectrum showing the extremely narrow linewidths of QD1F1 and QD2F2 fitted with a Lorentzian curve. 66 Single photon emitters in WSe2

a 1.2 1.0 1.0 0.8

) 0.8

τ 0.6 (

(2) 0.6 0.4 g QD1F1 0.4 QD4F1

0.2 (2) 0.2 (2) g (0) = 0.20 ± 0.02 g (0) = 0.18 ± 0.02 0.0 0.0 -10 -5 0 5 10 -20 -10 0 10 20 τ (ns) τ (ns)

3 2 b Figure10 7.5: a Second-order photon autocorrelation3 function g (τ) of photolumines- τ = 1.51 ns 10 τ = 2.5 ns cence stemming from the sharp peaks QD1F1 and QD4F1. The pronounced dip

below2 0.5 (antibunching) at zero time delay unambiguously conﬁrms that the origin 10 2 of the sharp peaks are zero-dimensional emitters,10 generally called quantum dots.

1 1 7.410 Photon autocorrelation10

PL Intensity (a.u.) PL Intensity QD1F1 QD3F1

In order0 to prove that the sharp peak originate0 from single photon emitters with 10 10 an anharmonic0 spectrum,1 2 we3 perform4 a5 Hanbury0 Brown2 and Twiss4 measurement6 8 to 2 2 measure the photonTime autocorrelation delay (ns) function g (τ). WeTime know delay that (ns) g (0) must be larger than one for classical light and must be larger than 0.5 for a light source that can emit two photons at the same time. A value of g2(0) < 0.5 unequivocally proves that the measured light originates from a single, anharmonic emitter, in the context of semiconductor physics, called a quantum dot. For time resolved single photon measurements, we use avalanche photo diodes (APDs). To overcome the sizable dead time of the detectors, we use a fiber beam splitter to send our PL signal to two different photo detectors. One of the detectors, upon the arrival of a photon, sends a start signal, the other one sends a stop signal to a correlation unit that measures the waiting time distribution between the two signals. We can then convert the waiting time distribution to a autocorrelation curve. Figure 7.5 shows g2(τ) for QD1F1 (LHS) and QD4F1, another quantum dot located on the same flake (RHS). For zero delay, g2(τ) is clearly smaller than 0.5 with g2(0) = 0.20 ± 0.02 for QD1F1 and g2(0) = 0.18 ± 0.02 for QD4F1. We therefore proved that the sharp peaks in the spectrum presented in figure 7.3 indeed stem from quantum dot photoluminescence.

7.5 Lifetime, spectral ﬂuctuation and blinking

We measure the lifetime of the conﬁned exciton using a pulsed Ti:sapphire laser, tuned into resonance with the delocalized repulsive polaron, with a pulse width of ∼ 5 ps for excitation. As in section 7.4, we send the collected PL through a a 1.2 1.0 1.0 0.8

) 0.8

τ 0.6 (

(2) 0.6 0.4 g QD1F1 0.4 QD4F1

0.2 (2) 0.2 (2) g (0) = 0.20 ± 0.02 g (0) = 0.18 ± 0.02 0.0 0.0 7.5. Lifetime, spectral ﬂuctuation and blinking 67 -10 -5 0 5 10 -20 -10 0 10 20 τ (ns) τ (ns)

3 b 10 3 τ = 1.51 ns 10 τ = 2.5 ns

2 10 2 10

1 1 10 10

PL Intensity (a.u.) PL Intensity QD1F1 QD3F1

0 0 10 10 0 1 2 3 4 5 0 2 4 6 8 Time delay (ns) Time delay (ns)

Figure 7.6: Time-resolved PL following an excitation of the delocalized repulsive polaron by a 5 ps long laser pulse. We observe an exponential decay of the PL with a lifetime of τ = 1.51 ns for QD1F1 (LHS) and τ = 2.5 ns for QD3F1 (RHS).

transmission grating to spectrally filter out the quantum dot PL. The filtered signal is sent to a single-photon-counting APD for detection. Figure 7.6 shows the time- resolved PL from QD1F1 (LHS) and QD3F1, yet another quantum dot located on the same flake (RHS). The measured lifetime of ∼ 2 ns is consistent with other semiconductor quantum dots and orders of magnitude longer than the lifetime of the delocalized 2D exciton, which has a lifetime of ∼ 2 ps [119]. Changes, such as charge fluctuations, in the local environment of quantum dots in solid state systems are known to cause effects such as spectral fluctuation, blinking and even photo bleaching. Spectral fluctuation is a change of the PL spectrum over time as a result of a changing environment such as electrostatic fluctuations due to shallowly trapped charges in the vicinity of the quantum dot. Blinking describes temporal fluctuations of the PL intensity. The causes for blinking are similar to spectral fluctuation when a fluctuating environment causes change of the emission intensity on timescales much longer than the exciton lifetime. As an example we consider at the NV−-centers in diamond. While all states used in quantum information schemes are states of the negatively charged NV-center, the excitation laser will eventually remove the extra electron from the color center which will render the NV-center dark until eventually another electron will find its way into the NV-center. Photo bleaching describes an irreversible reduction of PL intensity due to the laser light itself. All three effects limit the use of solid state quantum dots for applications. There- fore, we would like to quantify the extend to which we observe said effects in WSe2 quantum dots. First, we study blinking by sending the PL from QD1F1 to a single- photon-counting APD. Given the quantum dot’s lifetime and our collection efficiency, we observe photons at a rate of ∼ 13 kHz. Figure 7.7 shows the waiting time distribution W (τ)[120], [121] for τ ≥ 10 µs between two consecutive photons. We can fit W (τ) very well with just a single exponential decay with a decay −1 time τdet = (Γη) ≈ 77 µs, where Γ is the spontaneous emission rate and η is the 68 Single photon emitters in WSe2

4 a 10 τ = 77µs 3 det 10 2 10

τ) W( 1 10 0 10 0.0 0.2 0.4 0.6 0.8 τ (ms) b 100 120

Figure 7.7: Waiting time distribution W100(τ) as a function of time τ in between two consecutive80 PL photons emitted by QD1F1. The black trace shows a single exponential decay with a decay constant τdet80≈ 77 µs. The excellent agreement with an exponential decay60 tentatively indicates the absence of intensity intermittency on timescales larger than τdet. 60

Time (s) Time 40 40 detection efficiency. The fitting parameter τdet is in excellent agreement with the 20 measured photon rate. The fact that we can20 fit the W (τ) with just one exponential decay shows that there is no blinking for timescales longer than τdet. If the dot was to exhibit blinking, we would expect the system to switch between bright and dark 1.67 1.68 1.69 states. If we assume that this switching can1.645 be modeled1.650 as a Poissonian1.655 process with Energy (eV) Energy (eV) a typical timescale τblink > τdet, then W (τ) should show to characteristic timescales: τdet and τblink [122]. However, at least on the timescale from 10 µs to 0.5 ms, this does not seem to be the case. If we had blinking on timescales ≤ 1 µs, we had observed bunching in g2(τ). Our assumption of a Poissonian process causing blinking has been shown to be inaccurate in most quantum dots [123]. In reality, blinking exhibits a power-law behavior in the waiting dime distribution W (τ). This should mean that we could not fit our measured W (τ) with a exponential decay. There- fore, the excellent agreement of our fit with the measured data lets us tentatively conclude that quantum dots in WSe2 monolayers that we studied in this chapter show no significant blinking. However, we should emphasize that we focused our attention to the brightest localized emitters in the two flakes within which we found sharp emission peaks.

Spectral wandering indeed seems to be a present in WSe2 quantum dots. Fig- ure 7.8 shows the time evolution of the emission spectrum. On the left hand side, we see the spectral evolution of a dot with a small spectral fluctuation of approximately ±200 µeV which is hardly visible in this low resolution spectrum. The majority of the dots we observed showed spectral wandering comparable in magnitude. How- ever, some dots were more prone to changes in wavelength as shown on the right hand side of figure 7.8. Due to the increased resolution, we can see that the peak is composed of two close peaks separated by approximately 0.7 meV. We observe that the two peaks are changing their wavelength with a fixed splitting which confirms that they belong to the same quantum dot. As we will see later the two peaks are the electron-hole exchange split exciton transitions. This quantum dot is one of few that show as much spectral wandering as ±500 µeV.

WSe2 quantum dots have also proven to be stable against warm-up and cool- 4 a 10 τ = 77µs 3 det 10 2 10

τ) W( 1 10 7.6. Magnetic ﬁeld0 69 10 0.0 0.2 0.4 0.6 0.8 τ (ms) b 100 120

100 80

80 60

Time (s) Time 40 40

20 20

1.67 1.68 1.69 1.645 1.650 1.655 Energy (eV) Energy (eV)

Figure 7.8: LHS: Typical time evolution of the low-resolution PL spectrum of a quantum dot that shows a rather stable peak position. RHS: Time evolution of the high-resolution PL spectrum of another quantum dot showing spectral wandering with a typical fluctuation energy of ∼ 1 meV. Since both peaks visible are wandering with a fixed splitting in between suggests that they originate from the same quantum dot and are the electron-hole exchange split exciton transitions. down cycles and exposure to excitation laser (HeNe) powers as high as hundreds of microwatts which is much larger than the power needed to saturate the quantum dots. The quantum dots we observed did not suffer from bleaching.

7.6 Magnetic ﬁeld

As we mentioned in the beginning of this chapter, there are solid state single photon emitters that are completely unrelated to the host crystal, such as NV centers in diamond for example. In the following we want to show that the quantum dots that we observe in WSe2 are not of that sort but rather exhibit many similarities with the monolayer WSe2 crystal. One such indicator of inheritance from the host crystal could be a similarly high and anisotropic g-factor. We therefore measure the polarization resolved PL of the quantum dots in Faraday as well as Voigt geometry. The left hand side of figure 7.9 shows PL from QD1F2. The excitation laser is linearly polarized whereas we only collect σ1 circularly polarized light. Even without a magnetic field, we see that the quantum dot emits two split peaks. With increasing magnetic field, the two peaks separate further. At high magnetic fields (≥ 3 T), we see that the peak at higher energy is strongly suppressed in intensity, even though it is mostly σ1 polarized and therefore passes our polarization filter in the collection arm whereas the bright peak is mostly σ2 polarized and therefore strongly suppressed (as we will discuss in more detail in the paragraph describing figure 7.10). We attribute this strong domination of the low energy peak to an efficient thermalization of excitons into the lowest energy state within their lifetime. 70 Single photon emitters in WSe2

a Energy (meV) Energy (meV) 1680 1678 1676 1684 1682 1680

150 0 T 0 T 100 100 50 50

736 737 738 738 739 740 1680 1678 100 1 T 2 T 100 50 50

738 739 740 150 738 739 100 3 T 3 T 100 50 50 0 738 739 740 738 739 Wavelength (nm) Wavelength (nm) b Energy (meV) Energy (meV) Figure 7.9:1715LHS 1714The split1713 PL peaks of QD1F2 for1715 three1714 different1713 magnetic fields 300 600 in Faraday1 T geometry. The finiteσ splitting1 at 0 T1 T further split as the magneticσ2 field increases.200 An efficient thermalization leads400 to a domination of the spectrum by the lower100 energy peak, even though the circular200 polarization in which we collect the PL agrees with the polarization along which the higher energy peak emits. RHS The split PL peaks723.0 of a quantum723.5 dot724.0 in Voigt geometry723.0 shows no723.5 magnetic724.0 field induced -1 T σ -1 T σ increase200 of the splitting. 2 400 1 100 200 On the right hand side of figure 7.9, the PL from another quantum dot in Voigt geometry shows723.0 no significant723.5 724.0 change in splitting as723.0 the magnetic723.5 field724.0 is increased. This confirmsWavelength that the quantum (nm) dots inherit the strongWavelength anisotropy (nm) of the g-factor from the host crystal, for which the underlying reason for the anisotropy is the two dimensional geometry. The fact that the quantum dot inherits this anisotropy from c 2.5 2.0 the2.0 crystal strongly suggests that the quantum dots in WS22 that we observe indeed 1.5 stem1.5 from a local change in the bandgap that stretches over a few lattice sites rather 1.0 (meV) than1.0 from a single point defect as is the case e.g. in color centers. B Δ0 = 768 µeV 0.5 Δ0 = 670µeV Δ 0.5As mentioned above, at a finite magnetic field, the quantum dot emission is cir- 9.3 µ QD2F1 10.3 µ QD1F2 cularly0.0 polarizedB to a large degree. Figure0.0 7.10 depictsB the PL spectrum of QD1F2 at ±-41 T measured-2 in0 the two2 circular4 polarizations-3 -2σ1,2.-1 The0 higher1 energy2 peak3 is twice as strong in σB1 (T)than in σ2 whereas the lower energy, whileB (T) always dominating over the high energy peak, is half as bright in σ1 than in σ2. As expected, reversing the direction of the magnetic field exchanges the polarization of the two peaks. When we quantify the quantum dots g-factor, we have to take into account that a Energy (meV) Energy (meV) 1680 1678 1676 1684 1682 1680

150 0 T 0 T 100 100 50 50

736 737 738 738 739 740 1680 1678 100 1 T 2 T 100 50 50

738 739 740 150 738 739 100 3 T 3 T 100 50 50 0 7.7. Photoluminescence738 739 excitation740 738 739 71 Wavelength (nm) Wavelength (nm) b Energy (meV) Energy (meV) 1715 1714 1713 1715 1714 1713 300 600 1 T σ1 1 T σ2 200 400 100 200

723.0 723.5 724.0 723.0 723.5 724.0 200 -1 T σ2 400 -1 T σ1

100 200

723.0 723.5 724.0 723.0 723.5 724.0 Wavelength (nm) Wavelength (nm)

2.5 c Figure 7.10: PL from QD1F2 at ±1 T analyzed2.0 in the circular basis (σ , plotted 2.0 1,2 in red and blue respectively), revealing that1.5 inverting the direction of the magnetic 1.5 field swaps the polarization of the two split1.0 peaks. (meV) 1.0 B Δ0 = 768 µeV 0.5 Δ0 = 670µeV Δ 0.5 QD2F1 QD1F2 9.3 µB 10.3 µB the0.0 splitting in the absence of a magnetic0.0 field will diminish the magnetic field -4 -2 0 2 4 -3 -2 -1 0 1 2 3 dependence at small fields. More concretely, we expect the splitting to follow the B (T) B (T) following hyperbolic curve:

q 2 2 2 ∆E(B) = µ B + ∆0. (7.1)

Figure 7.11 shows the splitting between the two PL peaks as a function of the magnetic field fitted with a hyperbolic curve for two different quantum dots. We observe high g-factors in the order of ∼ 10, which is significantly larger than the g-factor of both, the delocalized repulsive and attractive polaron.

7.7 Photoluminescence excitation

Since the quantum dots we observe have their origin in a local variation of the band structure we expect that excitons created in the unperturbed part of the monolayer will, as a result of their diffusion, with some probability relax into the trap formed by the quantum dots. As a consequence, we should see an increase of PL emitted by the quantum dots if we excite the flake at the unbound exciton resonance. To confirm this, we measure the quantum dot PL intensity as a function of the excitation laser wavelength. This measurement technique is generally called photoluminescence excitation (PLE). The excitation laser, a Ti:Sapphire laser in CW mode, is kept at constant power and tuned in wavelength. Figure 7.12 shows the result of the PLE measurement of two different quantum dots, both of which show a drastic increase of PL when we excite at the exciton energy, confirming that there is an efficient transfer of excitation from the unperturbed monolayer to the quantum dot. a Energy (meV) Energy (meV) 1680 1678 1676 1684 1682 1680

150 0 T 0 T 100 100 50 50

736 737 738 738 739 740 1680 1678 100 1 T 2 T 100 50 50

738 739 740 150 738 739 100 3 T 3 T 100 50 50 0 738 739 740 738 739 Wavelength (nm) Wavelength (nm) b Energy (meV) Energy (meV) 1715 1714 1713 1715 1714 1713 300 600 1 T σ1 1 T σ2 200 400 100 200

723.0 723.5 724.0 723.0 723.5 724.0 -1 T σ -1 T σ 72200 2 400 Single photon emitters1 in WSe2

100 200

723.0 723.5 724.0 723.0 723.5 724.0 Wavelength (nm) Wavelength (nm) c 2.5 2.0 2.0 1.5 1.5 1.0 (meV) 1.0 B Δ0 = 768 µeV 0.5 Δ0 = 670µeV Δ 0.5 9.3 µ QD2F1 10.3 µ QD1F2 0.0 B 0.0 B -4 -2 0 2 4 -3 -2 -1 0 1 2 3 B (T) B (T)

Figure 7.11: Magnetic field dependence of the splitting between the two orthogo- nally polarized peaks emitted by QD2F1 (LHS) and QD1F2 (RHS). The measurement data is fitted by a hyperbole with a asymptotic g-factor of ∼ 10 and a zero field splitting of ∼ 700 µeV.

QD3F1 1.5 QD1F2 1.5

1.0 1.0

0.5 0.5

PL Intensity (a.u.) PL Intensity

700 710 720 730 710 720 730 Wavelength (nm) Wavelength (nm)

Figure 7.12: Photoluminescence excitation measurement of QD3FS (LHS) and QD1F2 (RHS) show the PL intensity emitted by the quantum dot as a result of excitation with a laser wavelength that is scanned over a wavelength range covering the attractive and repulsive polaron energy. 7.8. Conclusion 73

7.8 Conclusion

We have established that we quantum dots with a long lifetime and clear antibunching can be found in TMD monolayers. The correlation of spin and valley (which is expected to persist from monolayer to the quantum dot [100], [124]) make TMD quantum dots especially interesting for quantum information processing because the spin and valley pseudospin are therefore strongly protected. For applications it is important that quantum dots can be placed reproducibly. Recent experiments successfully demonstrated that a monolayer flake of WSe2 transferred onto an array of micro-pillars will create at least one quantum dot at each pillar. However, there are still obstacles to overcome, such as a sizable spectral distribution of the quantum dots’ emission energy. Another promising direction is the design of electrically pumped single-photon sources, which was recently demonstrated [125]. A bright single-photon source is a key ingredient for quantum information processing protocols that are already in use such as quantum random number generators or quantum cryptography. Furthermore, it would be interesting to demonstrate gate defined quantum dots [126] that confine an electron and probe the presence or absence of this electron by measuring the spectrum inside the gate confined region. While such experiments are difficult to achieve in GaAs due to its weakly bound trion resonance, TMDs might open new possibilities for combining transport and gate defined quantum dots with optics.

Conclusion and Outlook 75

8 Conclusion and Outlook

We have demonstrated that the strong Coulomb interaction and the resulting large binding energy make the exciton in TMDs an excellent probe to investigate many- body physics. In particular, we studied how the presence of a Fermi sea of electrons affects the optical spectrum of the exciton. Since the electron mass and the exciton mass are comparable, the exciton is a mobile impurity interacting with a Fermi bath. We use the Chevy ansatz, which approximates the exciton interacting with a Fermi sea of electrons by a superposition of a bare exciton and an exciton dressed with an electron-hole pair. The interaction leads to two new resonances: The attractive polaron, a quasiparticle describing a shift of the Fermi sea electrons towards the quantum impurity, and the repulsive polaron, describing an exciton surrounded by electrons displaced away from the exciton. In the limit of a vanishing electron density, the repulsive polaron resonance overlaps with the bare exciton resonance and the attractive polaron coincides with the trion resonance. When the electron density is increased, the attractive polaron is fundamentally different from the trion, especially in terms of the interaction with light. We find that our simulation based on the simple Chevy ansatz can qualitatively and quantitatively reproduce the attractive and repulsive polaron energy as a function of electron density in a TMD monolayer. For this simulation, only one fitting parameter had to be introduced to reproduce the gate dependent energy of two resonances. While the trion, as a bound three-body state, interacts only weakly with light, attractive polarons can interact strongly with it due to the bare exciton part of their wavefunction. We therefore chose to probe the coupling strength of the optical transitions of the exciton - Fermi sea system. Optical cavities are the method of choice to manipulate as well as characterize the optical coupling strength of a resonance. We used an open fiber cavity setup to measure the spectrum of the TMD monolayers in weak and in strong coupling. In the weak coupling regime, we could measure the real and imaginary part of the susceptibility independently which highlights the advantage of cavity spectroscopy for van-der-Waals heterostructures, where the interference of reflections from the various surfaces complicate the measurement of the absorption spectrum. Shrinking the cavity volume unveils an avoided crossing of the cavity mode and the attractive polaron mode. This is the most striking manifestation of the strong interaction with light of the attractive polaron and is in stark contrast to the weak interaction expected from a bound three-body state in the form of a trion. At the anticrossing of the attractive polaron with the cavity mode, we find new quasi particles that we call polaron-polaritons. Due to their light mass, the latter could allow for polarons that are resilient to disorder [83]. For the observation of polaron-polariton transport related phenomena, we envision measurements with a two-dimensional cavity. TMD monolayers in such cavity structures have already been demonstrated [89]. 76 Conclusion and Outlook

Measurements in magnetic field confirmed our assumption that the K-exciton will only form polaron states with −K-electron-hole pairs. We observed that this valley sensitivity is strong enough for us to measure the valley selective strength of the attractive polaron peak as an indicator of the spin valley imbalance. A promising prospective experiment is the investigation of polaron-polaron interaction. We expect that polarons, due to the dressing by electron-hole pairs, occupy a larger space and, therefore, interact at a lower density as compared to bare excitons. This effect could be exploited to increase the non-linearity of polaritons in a zero- dimensional cavity and thereby provide another step towards polariton-blockade. While our measurements of Fermi-polarons were performed in MoSe2, for which the inter-valley and intra-valley trions are split by the conduction band spin-orbit coupling, a promising endeavor is to extend the research of polarons to WSe2. In WSe2, the intra- and inter-valley trions are split only due to exchange interaction. Theoretical calculations predict a large Berry curvature of opposite sign for the two trion states, which could allow for the observation of a large valley Hall effect [83]. Another potential direction is to combine optics and transport: While trions have a net charge, polarons are neutral. Recent theoretical calculations have demonstrated that polarons will respond to an in-plane electric field with an acceleration that depends on the electron density. Trions on the other hand, will be accelerated according to their charge, which is a multiple of an electron charge [81]. Furthermore, it would be insightful to investigate the states from which attractive polarons recombine under the emission of PL. Recent theoretical models suggest that the attractive polaron PL originates from a minimum in the attractive polaron dispersion at a finite wave vector k.[94] The recombination therefore has to be assisted by a third particle such as a phonon. The polaron-polariton dispersion is dominated by the polaron within the whole k-space except for a small range of k around 0, where the low mass of the cavity mode describes the dispersion. As a consequence, for low electron densities, the smallest energy state of a polaron- polariton is at k = 0. For increasing electron densities, the minimum at finite K is going to be decreased further until it eventually is in resonance with the polaron dispersion at k = 0. The investigation of the effect of this crossing onto the reflection and PL spectrum could reveal more details about the polaron dispersion. Moreover, promising experiments investigating the attractive polaron could be found in the combination of gate defined quantum dots with optics. Since the trion binding energy is much larger than the linewidth of both the exciton and the attractive polaron, we can envision a scenario where the charge state of a gate defined quantum dot could be measured optically: While in the absence of electrons inside the quantum dot we expect not to see any signature of the attractive polaron in the reflection spectrum, even one electron would give rise to an attractive polaron peak which would be close to the trion peak. With every additional electron inside the quantum dot, we would expect a stepwise increase of the attractive polaron energy. This would allow to measure with optical means the absolute number of charges inside the dot [83]. Furthermore, the excellent qualities of TMD excitons to investigate many-body physics are not limited to Fermi polarons: Experiments in which a K-valley exciton interacts with Bogoliubov excitations out of a polariton condensate in the −K-valley provides an interesting system for the research of Bose-polarons [127]. Distributed Bragg reflector A

Distributed Bragg reﬂector

The DBR was grown by Laseroptik using ion beam sputtering (IBS). Three 4” fused silica wafers and an adapter with the same footprint, designed to hold 40 fibers, were coated simultaneously. The coating consists alternately of ten layers of Nb2O5 and ten layers of SiO2. The first layer grown on the substrate consists of Nb2O5, has refractive index nN ≈ 2.27 and a thickness of 1.23 × xN, where xN = 736 nm/4nN ≈ 81 nm is the thickness corresponding to a quarter wavelength. The next layer consists of SiO2, has a refractive index nS ≈ 1.48 and a thickness of 1.88 × xS, where xS = 736 nm/4nS ≈ 124 nm. The remaining 18 layers each have a thickness of xN respectively xS. The stack ends with a low index SiO2 layer which ensures a intensity maximum at the surface of the DBR. The reason for the first two layers is to decrease the group velocity dispersion. Figure1 shows the calculated transmission spectrum of the DBR structure using the transmission matrix method. The python module named ”tmm” written by Dr. Steven Byrnes was used for this purpose [128]. Figure2 shows the refractive index profile and the calculated field intensity of a plane wave reflected at the DBR. The field intensity shows a maximum at the surface of the DBR where the TMD flake is placed. B Distributed Bragg reflector

100

10 1 Transmission

10 2

500 550 600 650 700 750 800 850 900 Wavelength (nm)

Figure 1: Calculated transmission spectrum of the DBR using TMM.

2.5 4.5 vacuum SiO2 4.0 Nb2O5 2.0 3.5

3.0 n

x 2 | e 1.5 E d | n 2.5 i

y t e i s v i n t c 2.0 e t a r n I f 1.0 e

R 1.5

1.0 0.5 Normalized to incoming plain wave

0.5

0.0 0.0 500 0 500 1000 1500 2000 2500 3000 Distance from DBR surface (nm)

Figure 2: Refractive index proﬁle and ﬁeld intensity distribution of the DBR. Samples C

Samples

1 WSe2

The WSe2 samples used for the studies of quantum dots and the magnetic moment were fabricated by the group of Adrien Allain and Dominik Lembke in the group of Prof. Andras Kis at EPFL. WSe2 was exfoliated onto a substrate of SiO2 on heavily doped silicon. Some of the samples measured during the course of our collaboration were contacted by gold electrodes. Figure1 and2 show the gate dependent spectra of a WSe2 monolayer showing resonances of quantum dots, the exciton and two trion states.

2 MoSe2 2.1 M1 The heterostructure was fabricated by Patrick Back using the pickup technique [129] and consists of a MoSe2 monolayer embedded in between two h-BN layers. The sample was transferred onto a substrate of SiO2 on highly doped silicon.

2.2 M2 The sample was fabricated by Patrick Back using the same pickup technique as the one used for sample M1. The sample is transfered onto a quartz substrate coated with a DBR. Figure3 shows the sample after many cool-downs. We do not know what exactly caused the degradation and over what time period it happened. Possibly, the ﬁber facet, which regularly touches the sample, pulled part of the sample from the substrate after contact. D Samples

Figure 1: Spectrum of a WSe2 monolayer as a function of gate voltage.

Figure 2: Spectrum of a WSe2 monolayer as a function of gate voltage. 2. MoSe2 E

Figure 3: Microscope photograph of sample M2 after it broke.

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List of publications and conference presentations XIII

List of publications and conference presentations

Peer Reviewed Publications

P. Back, M. Sidler, O. Cotlet, A. Srivastava, N. Takemura, M. Kroner, and A. Imamoglu, “ Giant Paramagnetism-Induced Valley Polarization of Electrons in Charge-Tunable Monolayer MoS2 ”, Physical Review. Letters, 118, 237404 (2017)

M. Sidler, P. Back, O. Cotlet, A. Srivastava, T. Fink, M. Kroner, E. Demler, A. Imamoglu, “ Fermi polaron-polaritons in charge-tunable atomically thin semiconductors”, Nature Physics, 13, 225-261 (2017) (incl. cover)

A. Srivastava*, M. Sidler*, A.V. Allain, D.S. Lembke, A. Kis, A. Imamoglu, “Optically active quantum dots in monolayer WSe2”, Nature Nanotechnology, 10, 1038 (2015) * Equally contributing authors

A. Srivastava, M. Sidler, A.V. Allain, D.S. Lembke, A. Kis, A. Imamoglu, “Valley Zeeman eﬀect in elementary optical excitations of monolayer WSe2”, Nature Physics, 11, 141-147 (2015)

M. Sidler, P. Rauter, R. Blanchard, P. M´etivier, T.S. Mansuripur, C. Wang, Y. Huang, J.H. Ryou, R.D. Dupuis, J. Faist, F. Capasso, “Mode switching in a multi-wavelength distributed feedback quantum cascade laser using an external micro-cavity”, Applied Physics Letters 104, 5, 51102 (2014)

Invited Talks

M. Sidler, P. Back, O. Cotlet, A. Imamoglu, “Linear and non-linear optics of Fermi polarons in TMDs”, Optics of Excitons in Conﬁned Systems (OECS), Bath UK, 2017

M. Sidler, P. Back, A. Srivastava, T. Fink, M. Kroner, A. Imamoglu, “Trion Polaritons in MoSe2”, New Developments in Solid State Physics, 19th Inter- national Winterschool, Mauterndorf Austria, 2016 XIV List of publications and conference presentations

Contributed Talks

M. Sidler, P. Back, O. Cotlet, A. Imamoglu, “Fermi polaron-polaritons in MoSe2”, American Physical Society March Meeting, New Orleans LA, USA, 2017

M. Sidler, P. Back, O. Cotlet, A. Srivastava, T. Fink, M. Kroner, E. Demler, A. Imamoglu, “Fermi polaron-polaritons in MoSe2”, Flatlands beyond graphene, Bled, Slovenia, 2016

M. Sidler, P. Back, O. Cotlet, A. Srivastava, T. Fink, M. Kroner, E. Demler, A. Imamoglu, “Fermi polaron-polaritons in MoSe2”, Quantum Simulations and Many-Body Physics with Light, Chania, Greece, 2016

M. Sidler, R. Blanchard, T.S. Mansuripur, P. Rauter, S. Menzel, Y. Huang, J.H. Ryou, R.D. Durpuis, J. Faist, F. Capasso, “Discrete tuning between the modes of a multiple-wavelength quantum cascade laser using a micro-scale external cavity”, Photonics West, San Francisco, USA, 2013 Acknowledgment XV

Acknowledgment

I would like to thank Atac Imamoglu for giving me the opportunity to work as a PhD student in his group. I am thankful that he provided me with all the resources a young researcher can wish for: A research project in an emerging, diverse and incredibly fast moving field, collaborations with leading experts in that field, state of the art lab equipment and most of all his dedication to interesting discussions about physics. It is always fascinating to see how quickly Atac can find an intuitive picture of a physical effect and support it with a solid theoretical framework. His passion for fundamental physics is highly contagious. I am thankful for his trust in me, which he showed not only by supporting my work but also by giving me plenty of opportunities to present my findings at conferences. Special thanks go to Bernhard Urbaszek who agreed to be the co-referee for my PhD thesis. I am thankful for him squeezing in a trip to Zurich into his schedule, for reading and correcting my PhD thesis and for his helpful comments. I would also like to thank Ajit Srivastava who started the research on TMDs in our group and provided me with valuable know-how when I joined the project as a PhD student. I am thankful for his support and I cherished our time together in and outside the lab. Ajit’s passion for physics and mathematics is contagious and I always enjoyed to listen to him explaining his latest insights in geometry. I am grateful to Ovidiu Cotlet for his support. He did all the theoretical calculations concerning the Fermi polaron that I used in my work. What I especially cherished is his commitment to discussions about physics with me. He not only was always ready to explain mathematical details about his calculations but also man- aged to come up with intuitive models to provide me with a better understanding. My experiments would not have been possible without excellent sample quality. I would like to thank Patrick Back who brought a lot of know-how with him when he joined our group and who worked long hours in order to improve his fabrication process to provide the whole group with outstanding samples. I am deeply grateful to Andras Kis and all the people in his group that provided us with samples when our group expanded into the field of TMDs. I would like to thank Adrien Allain, Dominik Lembke and Dumitru Dumcenco for their dedication to provide us with samples and for Adrien to help us with transport measurements. I want to thank Martin Kroner for being a great colleague and an even better friend. I enjoyed many interesting discussions with Martin. I always found helpful advice when I shared a concern with him. I will never forget how he taught me how to drive in his brand-new car and never even once raised any concern about his car’s well-being. I would like to thank Eugene Demler and Richard Schmidt for our fruitful collaboration. XVI Acknowledgment

Before we obtained the fibers designed for our samples, Jakob Reichel kindly gave us test fibers, which were sent to us by Konstantin Ott. I am grateful for this generous support. A special thank-you also goes to Aymeric Delteil who I could always ask for technical but also personal advice with the benefit of getting an answer in French. I would like to thank Thomas Fink for providing me with the fiber dimples and for the great time we spent together on lake Zurich. I am grateful for Emre Togan, Thomas, Patrick and Yuya Shimazaki for helping me in the clean room. It was great to work with Li Bing Tan, Alex Popert and Naotomo Takemura: I enjoyed the many thought-provoking discussions we had inside and outside the lab. I am glad to see that Li Bing continues our project and wish her all the best. I am grateful to Sajedeh Manzeli, Kolyo Marinov, Yen-Cheng Kung and Patrick. Together we had a lot of fun organizing a summer school. I would like to thank the whole team for the valuable experience we gained and the nice time we had together. I am thankful for the great colleagues that I had the chance to share a lab with. We always had a very collaborative environment in which discussions about python, physics or popular science were an important part of otherwise boring routines such as filling helium. In addition to Ajit, Martin, Patrick, Alex, Li Bing and Naotomo, I also had the pleasure of having Priska Studer, Andres Vargas Lugo Cantu, Wolf Wüster,Yves Delley, Ido Schwarz and Yuya Shimazaki as my lab neighbors. I would also like to thank my office mates, Ajit, Alex, Hadis Abbaspour, Aroosa Ijaz and Thibault Chervy for many interesting discussions. It was great to have the opportunity to get to know first hand information about cultural and linguistic specialties of so many different countries. Many thanks go to Adrian Maier and Sina Zeytinoglu, two invaluable members of the lunch at one group, Patrick Knüppel and Sun Zhe and all QPG members, who I had overlap with, for many nice discussions. A special thanks goes to Manuela Weber-Semler and Katharina Rodharth for all the administrative support. I am deeply grateful to Julia Meier, Nadine Minder and Hubert Sidler for founding a shared flat together with me. Because of them, my apartment felt like a home. I would like to thank for all their moral support, all their help and all the interesting discussions we have had. Many thanks go to Dimitri Rettig and Michael Locher. Both of them worked on their master thesis in our group at the same time as I started my PhD. With that started a deep friendship. We have had many funny moments, relaxing days off and long discussions together. I am deeply grateful to my parents in law for welcoming me so warmly into their family. I hope that I will progress quickly with learning Farsi such that our conversations can become longer and we will get to know each other better. A big thank you also goes to my family who support me whenever they can. Above all, I would like to thank the love of my life, my wife Sajedeh Manzeli. I am so grateful that our paths have crossed, at which point I immediately fell in love with you. I am thankful that ever since, we shared a wonderful life together. Thank you for all the love and support I receive every day from you. Each day, you motivate me to be the best person I can be. List of Figures XVII

List of Figures

2.1 Crystal structure of a monolayer of MX2 ...... 6 2.2 Selectrion rules...... 8

2.3 Excitons in MoSe2 ...... 10 2.4 Excitons in WSe2 ...... 11

3.1 Spectral of an exciton interacting with a Fermi sea of electrons in the Chevy ansatz...... 16 3.2 Simpliﬁed depiction of the attractive and repulsive polaron...... 17 3.3 Quasi particle weight of the attractive and repulsive polaron as a function of Fermi energy...... 18 3.4 Charge distribution around a trion and attractive polaron...... 19 3.5 Sketch of a typical Van der Waals heterostructure...... 21

3.6 Microscope photograph of the h-BN embedded MoSe2 sample..... 21 3.7 Diﬀerential reﬂection of a MoSe2 monolayer as a function of gate voltage...... 23 3.8 Comparison of the simulated spectral function and the measured polaron energy as a function of Fermi energy...... 24

4.1 Sketch of the fiber setup...... 27 4.2 Transmission spectrum of an empty fiber cavity...... 28 4.3 A sketch of the heterostructure inside the fiber cavity...... 29 4.4 Image of the samples inside the cavity...... 30 4.5 White light transmission through a cavity weakly coupled to the MoSe2 polaron as a function of cavity length...... 31 4.6 Comparison of the Fermi energy dependence of the attractive and repulsive polaron energy for a sample measured inside the cavity and a different sample measured in differential reflection...... 32 4.7 Real and imaginary part of the susceptibility...... 33 4.8 Measured real and imaginary part of the susceptibility for three different gate voltages as well as the Kramers-Kronig transformed absorption...... 34

4.9 Photoluminescence of the MoSe2 monolayer inside a weakly coupled cavity as a function of cavity length...... 35 4.10 Comparison of absorption and photoluminescence spectra for three diﬀerent gate voltages...... 35 4.11 Photoluminescence spectra as a function of gate voltage...... 36 4.12 Avoided crossing of the exciton and cavity photon resonances..... 39

4.13 Transmission spectrum of a MoSe2 monolayer strongly coupled to a cavity as a function of cavity length for three diﬀerent gate voltages. 40 XVIII List of Figures

4.14 Transmission spectrum through the cavity which is at resonance with and strongly coupling to the repulsive and attractive polaron..... 41 4.15 Transmission spectrum through the cavity which is at resonance with and strongly coupling to the attractive polaron...... 42 4.16 Measured and simulated transmission spectra for a cavity strongly coupled to the attractive or repulsive polaron as a function of Fermi energy...... 43

5.1 Transmission spectrum as a function of the laser power for diﬀerent LED intensities...... 48 5.2 Transmission spectrum as a function of the LED power for diﬀerent laser intensities...... 49

6.1 Sketch of the confocal setup...... 51 6.2 Photoluminescence of a WSe2 monolayer...... 52 6.3 Magnetic field dependence of the exciton spectrum in polarization resolved photoluminescence and reflection spectra...... 54 6.4 Circular vs Linear analyzation of the exciton PL in a magnetic field; Magnetic field dependent circular dichroism...... 54 6.5 Exciton PL in Voigt geometry and polarization inversion for opposite magnetic fields...... 55 6.6 Trion splitting in magnetic fields...... 56 6.7 Sketch of polarons in a magnetic field...... 57 6.8 Polarons in a magnetic field...... 59 6.9 Correction of the phase shift in differential reflection...... 61

7.1 Power dependence of the impurity band spectrum of WSe2 ...... 65 7.2 Power dependence of the WSe2 photoluminescence...... 66 7.3 Photoluminescence spectrum from the WSe2 monolayer featuring several sharp quantum dot peaks...... 67 7.4 Spatial map of the quantum dots...... 68 7.5 Second-order photon autocorrelation of quantum dot PL...... 69 7.6 Time-resolved PL from a quantum dot after a pulsed excitation... 70 7.7 Waiting time distribution of photons stemming from a quantum dot. 71 7.8 Spectral wandering of quantum dot PL...... 72 7.9 Quantum dot PL for different magnetic field strengths in Faraday and Voigt geometry...... 73 7.10 Measurement of the degree of circular polarization of the split PL peak of WSe2 quantum dots...... 74 7.11 Energy splitting of the quantum dot PL as a function of magnetic field...... 75 7.12 Photoluminescence excitation measurement of WSe2 quantum dots.. 75 1 Calculated transmission spectrum of the DBR...... 81 2 Refractive index profile and field intensity distribution of the DBR.. 81

1 Spectrum of a WSe2 monolayer as a function of gate voltage..... 83 2 Spectrum of a WSe2 monolayer as a function of gate voltage..... 83 3 Sample M2 after it broke...... 84