Journal accepted version of the manuscript. Published version at https://doi.org/10.1063/5.0042683

Magnetooptics of layered two-dimensional semiconductors and heterostructures: progress and prospects

Ashish Arora †Institute of Physics and Center for Nanotechnology, University of Münster, Wilhelm-Klemm-Strasse 10, 48149 Münster, Germany Email: [email protected] Fax: +492518333641

Beginning with the ‘conventional’ two-dimensional (2D) quantum wells (QWs) based on III-V and II-VI semiconductors in the 1970s, to the recent atomically-thin sheets of van der Waals materials such as 2D semiconducting transition metal dichalcogenides (TMDCs) and 2D magnets, the research in 2D materials is continuously evolving and providing new challenges. Magnetooptical spectroscopy has played a significant role in this area of research, both from fundamental physics and technological perspectives. A major challenge in 2D semiconductors such as TMDCs is to understand their spin-valley-resolved physics, and their implications in quantum computation and information research. Since the discovery of valley Zeeman effects, deep insights into the spin-valley physics of TMDCs and their heterostructures has emerged through magnetooptical spectroscopy. In this perspective, we highlight the role of magnetooptics in many milestones such as the discovery of interlayer excitons, phase control between coherently-excited valleys, determination of exciton-reduced masses, Bohr radii and binding energies, physics of the optically-bright and dark excitons, trions, other many-body species such as biexcitons and their phonon replicas in TMDC monolayers. The discussion accompanies open questions, challenges and future prospects in the field including comments on the magnetooptics of van der Waals heterostructures involving TMDCs and 2D magnets.

1. INTRODUCTION magnets. In the current section, we begin with a brief overview of magnetooptical effects from a historical The field of magnetooptics broadly deals with the perspective along with providing a glimpse into recent interaction of electromagnetic waves with magnetism. developments in the area of 2D materials. In section 2, we Since many decades, it has played a central role in providing describe new spin-valley phenomena discovered in layered insights into the band-structure of solids 1–4. On the one semiconductors under magnetic fields in the last few years, hand, magnetooptics is important for fundamental as outlined in Figure 1. For instance, valley Zeeman research 2, on the other hand, it is vital to many technological splitting and valley polarization under magnetic fields have applications 1. In this perspective , we highlight the crucial contributed in various ways in identification and role played by magnetooptics in many new discoveries in investigation of bright and dark excitons, trions, interlayer the emerging area of layered two-dimensional (2D) excitons, and four- and five-particle Coulomb-bound materials involving van der Waals semiconductors and

Figure 1. This perspective focuses on recent progress and future directions in the area of optical spectroscopy of 2-dimensional layered semiconductors under magnetic fields. Magnetooptical spectroscopy of Coulomb-bound quasiparticles such as excitons, trions, biexciton s and charged biexcitons is discussed in 2-dimensional systems. Titles of the gray panels describe the effects broadly covered in this perspective. Text in blue are some of the deduced quasiparticle parameters by studying these effects.

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Journal accepted version of the manuscript. Published version at https://doi.org/10.1063/5.0042683 complexes in 2D. While a large part of the perspective the dimensionality of the system by two. 24,25 An observation covers Zeeman effects in 2D semiconductors (i.e. section of cyclotron resonance of electrons and holes 26 and 2.1 which is further divided into sections 2.1.1 till 2.1.6), measurement of magnetoabsorption between quantized other phenomena such as Landau quantization and Landau levels27,28 led to important experimental diamagnetic shifts are discussed as well in appropriate advancements to study quantum mechanical phenomena in details in sections 2.2 till 2.5, which have revealed solids. This was accompanied with many theoretical important band-structure information such as exciton developments related to magnetoabsorption 29,30 . Roth et al. reduced masses, Bohr radii, and carrier magnetic moments. showed that the spin-orbit interaction can strongly modify We also recapitulate the recent results involving the effective g factors of the electrons in a semiconductor magnetooptics of the emerging area of van der Waals leading to deviations from its free-space value of .30 2 heterostructures. Such heterostructures provide a rich Interband Faraday rotation was also extensively studied platform for studying new physics in moiré potential theoretically, and experimentally 31–36 . In the 1970s, landscape. In each subsection, we point out open questions molecular beam epitaxy began producing high quality and possible future directions of the field. In section 3, we quantum wells (QWs) based on GaAs opening a new conclude by highlighting the role and immense future paradigm to investigate 2D physics using optical and potential of magnetooptics in the area of layered magnetic transport methods 37 . By the 1980s, magnetooptical materials, and their heterostructures made with 2D spectroscopy had become a powerful tool to study the semiconductors. electronic band structure of 2D quantum wells 2. Rich low- dimensional physics of many-body quasiparticles such as A brief history of magnetooptics. Magnetooptics was born excitons, trions and biexcitons was revealed 2,4,46–55,38,56–58,39– after Michael Faraday’s seminal works on the effects of a 45 . magnetic field on the polarization of light in the 1840s 5. He noticed a rotation in the plane of polarization of light when The discovery of the giant magnetoresistance effect it passed through a piece of lead-borosilicate glass placed (GMR) around 1988 triggered the area of spintronics under a magnetic field. The material under a magnetic field promising a new generation of energy-efficient and high- offers different refractive indices to the left- and right- speed device applications. This brought the electron spin, a circularly polarized components of the incident linear purely quantum-mechanical entity discovered in 1920s, on polarization, leading to the ‘Faraday effect’. In 1876, John the forefront of modern and future technology and research. Kerr discovered a similar effect (magnetooptical Kerr effect With the development of ultrafast-spectroscopy (sub or MOKE) while the light reflected off from a magnetized picosecond) techniques around the same time 59–62 , time- surface (an iron pole piece of a magnet) 6,7. Augusto Righi resolved magnetooptical spectroscopy opened up a new found that in addition to the rotation of the state of regime to study spin-relaxation processes and coherent spin polarization, the light becomes elliptically polarized in manipulation in semiconductors and their QWs 63,64,73–82,65– MOKE 8. The resulting ellipticity is known as ‘Kerr 72 . The area of research envisions spin based quantum- ellipticity’ (or ‘Faraday ellipticity’ in transmission information processing, and quantum computation. The geometry). The effect was quantitatively measured by Pieter observation of the spin-Hall effect in thin films of GaAs and Zeeman 9. In subsequent years, Zeeman discovered the much InGaAs established MOKE as an important tool for celebrated “Zeeman effect” by observing a splitting of Na spintronics research 83 . lines in a magnetic field 10 . Since then, a variety of other Similar to the possibilities of spin manipulation in solids, magnetooptical effects have been discovered. Woldemar a new area of research involving the manipulation of local Voigt observed that a magnetic field applied transverse to minima in the momentum space i.e. valleys the direction of propagation of light through Na vapor (“valleytronics”) has recently emerged 84 . Although, ideas of creates linear birefringence, known as ‘Voigt effect’ 11 . A valley manipulation date back to 1970s 85,86 , atomically-thin similar effect was observed by Quirino Majorana in sheets of layered semiconductors in the family of transition colloidal metal particles 12 , and Aimé Cotton and Henri metal dichalcogenides (TMDCs) provide an ideal platform Mouton in paramagnetic liquids (‘Cotton-Mouton effect’) 13 . for studying the valley physics 87,88 . Monolayers of TMDCs In the first half of the twentieth century, the Hanle effect of the type MX 2 (M=Mo, W; X=S, Se, Te) have energy- was discovered which dealt with the reduction of degree of degenerate valleys at ± points of the Brillouin zone, which K 87–91 polarization of light emission from atoms under a transverse host the ground-state optical transitions . The and K K magnetic field, when they are excited using linearly valleys are connected by time-reversal symmetry, and 14,15 polarized light . Later on, this effect was used to couple to and circularly polarized light (Figure σ σ determine spin relaxation times, nuclear fields and spin- 2(b))89–91 . This is because an absence of an inversion center 16–20 orbit effects . In the 1950s, the capability of combined with a strong spin-orbit interaction leads to a MOKE/Faraday effect to observe magnetic domains 21 , and strong coupling (‘locking’) of the carrier spin to the to readout the magnetically-stored information (i.e. valleys 89–91 . An external magnetic field applied magnetooptic recording) 22,23 raised a lot of interest in perpendicular to the plane of the crystal breaks time- magnetooptics. It was realized that an application of a reversal symmetry, and lifts the valley degeneracy by magnetic field leads to Landau quantization, and reduces shifting the two valleys in opposite directions as shown in

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Journal accepted version of the manuscript. Published version at https://doi.org/10.1063/5.0042683

Figure 2. (a) Schematic representing intralayer excitons in a monolayer (top and side views), and a few-layer and bulk (side view) semiconducting transition metal dichalcogenide. (b) Interband A and B exciton ‘bright’ (optically allowed) and transitions for the σ σ intralayer excitons in monolayer and few-layer bright (left) and dark (right) materials are shown in the absence and presence of magnetic field. These transitions occur between the conduction and valence bands of the same spin. Horizontal dashed lines are the energies of the band extrema at . The energetic movement of the bands in the presence of a magnetic field is represented following the conventions 0 in Ref. 146. The expected exciton g factor is about from this atomic picture (more details in text). – 4 Figure 2(b) (“valley Zeeman splitting”) 87,92 . This opens Boltzmann constant, being the temperature). In such a doors to control phase,93–98 and population between the case, excitons appear as a strong resonance at an energy valleys 87,99–106 . More recently, intrinsic spin ordering below the quasiparticle band gap by an amount equal to (ferromagnetism and antiferromagnetism) has been i.e. . may also be called as the optical discovered in new families of layered materials down to the band gap of the material. atomically thin limit 107–110 . Some of the examples of these 2D magnetic materials (2DMMs) include thiophosphates of The exciton binding energy in a 2D semiconductor the form MPX3 (M = Mn, Fe, V, Zn, Co, Ni, Cd, Mg; X = S, is theoretically expected to be larger than the 3D case by a 111,112 125 Se) , chromium trihalides of the form Cr X3 (X = Cr, Br, factor of in Wannier approximation i.e. . 113,114 115 116 117 4 4 I) , and others e.g. Cr 2Ge 2Te 6 , VSe 2 , Fe 3GeTe 2 . Typically, the binding energies of excitons in conventional Heterostructures made from TMDCs and 2DMMs are quantum wells ranges from a few meVs to a few 10s of promising to open new directions for valley manipulation meVs 46,47,122,123,125 . Mostly cryogenic temperatures are using smaller magnetic fields through proximity effects 114 . required to study the exciton ground and excited states in these systems. On the other hand, is much larger i.e. from Although, this perspective focuses on TMDCs and few 10s of meVs to 100s of meVs in semiconducting 2DMMs, research in the area of layered materials is TMDCs 87,88,126–131 . Therefore, TMDCs are an excellent continuously evolving. Nicolosi et al. explore the potential platform to study these 2D hydrogen-like quasiparticles, as of liquid exfoliation techniques in many families of layered well as other many-body species such as charged excitons, materials such as metal halides, oxides, III-VI layered Fermi polarons, biexcitons and charged biexcitons at semiconductors, layered and zirconium phosphates and elevated temperatures. phosphonates, layered silicates (clays), layered double hydroxides, and ternary transition metal carbides and Some of the most important parameters which determine nitrides 118 . Offering future directions in the field, a Coulomb-bound composite’s contribution to the optical computational techniques have sufficiently matured to response of a material are its binding energy, reduced mass predict hundreds of exfoliable and stable layered materials ∗ and Bohr’s radius. Magnetooptical methods are powerful with desired characteristics to choose from 119,120 . tools to determine these parameters 2,4 . An externally applied magnetic field can affect the optical signatures in many 2. 2D QUASIPARTICLES IN MAGNETIC FIELDS ways: a) Zeeman splitting, b) Valley polarization and Coulomb-bound electron-hole quasiparticles such as neutral coherence effects, c) Landau quantization and oscillator and charged excitons, Fermi polarons, and biexcitons are strength enhancement, and d) Diamagnetic shift, e) important for the optical response of semiconductors 87,88,121– Binding-energy enhancement. We discuss these effects one 124 . Consider the case of an undoped semiconductor where by one as follows. neutral excitons dominate the optical spectra, especially 2.1 Zeeman splitting. In this section, we discuss the when the binding energy of the exciton is comparable to Zeeman effects in ‘bright’ and ‘dark’ neutral and charged or larger than the thermal energy scale given by ( is excitons, biexcitons, and interlayer excitons. Conduction

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Journal accepted version of the manuscript. Published version at https://doi.org/10.1063/5.0042683 and valence bands (CB and VB respectively) of a semiconductor undergo a Zeeman splitting under a magnetic field characterized by their effective g factors. The effective g factor of a carrier in a band is strongly influenced by the spin-orbit interaction-induced band mixing 68,132–135 . If the conduction and the valence bands which are involved in the exciton creation undergo Zeeman shifts of and Δ under a magnetic field, then the Zeeman splitting of Δ the exciton line can be approximated by 87

(1) Δ 2Δ Δ ≅ where and are the effective CB, VB and exciton , g factors, is the Bohr’s magneton, 0.05788 meV/T and is the applied magnetic field 2. The Zeeman-split exciton lines for bright excitons are circularly polarized and are detected separately by helicity-resolved photoluminescence (PL)96,99,139–141,100–103,105,136–138 , reflectance 87,104,142–146 , transmittance 56,101,147 or photoconductivity spectroscopy 2,148 . Zeeman splitting in terms of circularly-polarized components is given as

(2) Δ Here and are the transition energies for the left and Figure 3. (a) and (b) False color maps of the helicity-resolved right circularly polarized light in the helicity-resolved photoluminescence of monolayer MoSe 2 on Si/(300 nm)SiO 2 spectra. Alternatively, polarization-modulation- substrate under out-of-plane magnetic field from to . 10 T 10 T spectroscopy measurements are performed such as Faraday Panels below the color maps are PL spectra for both circular rotation 35 , MOKE spectroscopy 149,150 , and magnetic circular polarizations at . Magnetic-field-induced variation of ±10 T dichroism (MCD)151,152 to determine the Zeeman splitting. PL intensity signifies the valley polarization effect discussed in the Other techniques such as quantum beats in time-resolved text. (c) and (d) Neutral ( ) and charged ( ) excitons shift in X X pump-probe spectroscopy 64 , quantum beats in time- energy with magnetic field corresponding to a g factor of about 66,68–71,73–76,79 . See Ref. 99 for further details. Reproduced with permission resolved PL in Voigt geometry , time-resolved 4 67,72,78 from Li et al., Phys. Rev. Lett. 113, 266804 (2014). Copyright Faraday spectroscopy in Faraday geometry , time- resolved Kerr spectroscopy 77,81 , four-wave mixing 153 , Spin- 2014 American Physical Society. flip Raman scattering 154 and hole burning 65 have also been the increasing g factor of electrons in GaAs with rising used to measure the carrier and exciton g factors in quantum temperature 135,161 . wells. In the case of TMDCs, strong spin-orbit interaction leads We briefly revise the well-established Zeeman effects in to a splitting of the conduction ( ) and valence ( ) bands QWs before moving on to the layered semiconductors. The Δ Δ as shown in Figure 2(b). is of the order of few meVs to a g factors of excitons in QWs have been found to be strongly Δ few 10s of meVs, while ranges from about 150 meV to dependent upon the well thickness. For instance, the heavy- Δ 500 meV 162–165 . Due to the specific sign of the conduction- hole exciton g factor in GaAs QW is negative for small well band splitting, the ground-state exciton in WS , WSe , and widths (about for a well width of nm), while it changes 2 2 2 2 MoS is found to be optically “spin forbidden” or sign for a well width of around 6 nm, and approaches a value 2 “dark” 87,162,172–181,164,182,165–171 (see Figure 2(b)). In MoSe of about +1 for large well widths ( nm)136,150 . The light- 2 25 and MoTe 2, however, the ground-state exciton is an hole exciton g factor stays between 6 to 8 for this range of optically “bright” state due to a reversed conduction band well widths 150 . The g factor of electrons has been studied splitting compared to the other three materials 165,182–185 extensively as a function of well width and it increases (Figure 2(b)). The brightness or the darkness of the ground- monotonically from the bulk GaAs value of to about 0.44 state excitons have a strong influence on the temperature- with reducing well thickness to nm 133,155,156 . 0.4 3 and carrier-density-dependent optical and magnetooptical Theoretical attempts have been made to calculate the g properties (absorption and PL) of TMDCs 166,167,183,186,187 . factors using perturbation theory 133,134,157 . While a ⋅ We first discuss the Zeeman effects of bright excitons, good agreement with the experimentally obtained electron followed by a discussion on dark excitons. and heavy-hole g factors was obtained 133,134 , a satisfactory explanation of the trends for the light-hole g factors requires 2.1.1. Neutral ‘bright’ excitons in TMDCs. For more work 158,159 . The monotonically increasing electron’s monolayer TMDCs, most of the works report the ‘bright’ A and heavy-hole g factors in CdTe/Cd (Mg )Te QWs with and B exciton g factors close to using 1-x x 4 decreasing well-width were also successfully modeled by magnetoreflectance, transmittance and PL at liquid He theory 160 . This method has also been used to explain temperatures, for out-of-plane applied magnetic ⋅

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Journal accepted version of the manuscript. Published version at https://doi.org/10.1063/5.0042683

The fact that this atomic picture works for the A and B excitons could be a complete coincidence, since it does not explain the findings of the recent experiments 146,189 and ab initio theory 190–193 concerning g factors of conduction and valence bands in TMDCs. The authors of Ref. 146 determine the g factor for the lower conduction band equal to and for WS and MoSe , respectively. A recent 1.08 1.84 2 2 work reports the g factors of the bottom (top) conduction

Figure 4. Effective g factors of the ground-state A exciton of a monolayer to the bulk limit in WSe 2, MoSe 2 and MoTe 2. Green spheres are the experimental data and lines are the fit using ⋅ perturbation theory 145 . Reproduced from Arora et al., 2D Mater. 6, 015010 (2018). Copyright 2018 Institute of Physics Publishing. All rights reserved. fields 87,96,145–147,188,100–105,139,141 . As an example, Figure 3 shows the circular-polarization-resolved magneto PL of 99 monolayer MoSe 2 as measured by Li et al. Contrary to the case of III-V and II-VI QWs, a simplistic ‘atomic picture’ incidentally provides a qualitatively correct value of g factors in certain cases involving excitons in TMDCs. An example is the proposed explanation of the g factors of A and B excitons in monolayer TMDCs100,146 . Here the g factors of the bands participating in the exciton formation are assumed to be equal to the sum of spin, orbital and valley moments. In this picture, the transition metal’s ± hybridized orbitals at the top of the valence band at the and valleys in the Brillouin zone are mainly K K Figure 5. (a) Single-particle band structure of a monolayer WSe 2 supposed to contribute to the exciton’s g for the ± valleys by choosing a complete set of electron K factor 99,100,103,104,139,146 . The magnetic moment associated eigenstates in the irreproducible representations of . (b) Band with these orbitals is . Then the Zeeman splitting structure in the exciton picture at the point of the exciton ±2 Γ would be obtained as yielding an effective excitonic Brillouin zone. Exciton states belonging to the and 4 Γ,Γ Γ g factor of . It has been argued that the contributions due representations are in-plane-dipole allowed (“bright”), out-of- 4 to valley magnetic moment, which depends on the effective plane-dipole allowed (“gray”) and dipole forbidden (“dark”), masses of the charge carriers forming an exciton, would respectively. The arrows represent the electric spin components. (c) Magneto-PL of the gray and dark excitons in the Faraday lead to deviations from this value in a geometry. At , only gray exciton is visible. In the presence of a 99,100,103,139,142,146 0 T monolayer . An attempt was made to find the magnetic field, the gray and dark excitons may couple to each spin, orbital and valley contributions to the exciton g factor other and the dark exciton acquires a small oscillator strength 146 separately, in monolayers of WS 2 and MoSe 2 . The valley making it possible to measure the exchange splitting 0.6 meV term was found to significantly contribute to the energies of and a dark exciton g factor of . More details are found in Ref. 9.4 the conduction and valence band states with a g factor of 170. Reproduced with permission from Robert et al., Phys. Rev. B for the two materials 146 . 96, 155423 (2017). Copyright 2017 American Physical Society. 1.5 ± 0.5

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Figure 6. Schematic depicting dispersion of bright (orange color ) and dark (grey color) excitons and an in-plane magnetic-field- Figure 7. Energies of bright, dark and grey excitons deduced from induced mixing between them (Voigt geometry). Filled and magneto-PL spectra of an hBN encapsulated MoSe 2 monolayer empty arrows represent electrons and hole spins, respectively. In- under a tilted magnetic field ( ∘ with respect to the sample plane) plane magnetic field mixes the conduction band spins between 45 up to . The in-plane component of the magnetic field bright and dark excitons. Therefore, the otherwise optically- 30 T brightens the dark excitons, while the out-of-plane component disallowed dark excitons acquire some oscillator strength. The separates them in energy (valley Zeeman splitting). Figure adapted splitting between bright and dark excitons in this geometry is from Robert et al., Nat. Commun. 11, 4037 (2020) ( Creative used in determining the electronic g factor of about .182,185 2 Commons Attribution 4.0 International License) bands as ( ) and for the top valence 0.86 ± 0.1 3.84 ± 0.1 band as 189 in an hBN-encapsulated WSe an based on ab initio methods. Furthermore, the thickness 6.1 ± 0.1 2 monolayer. These values differ from the atomic picture dependence of the B exciton g factors is not reported yet, involving simple additions of the spin, valley and orbital and could provide more surprises 136,150 . magnetic moments highlighting the requirement of a more One puzzle which is a topic of discussion in the complete theory. Recently, ab initio theory has matured to community is that the g factor of A excitons in MoS be able to calculate the g factors of the bands and excitons 2 monolayer encapsulated with hBN has been found to in TMDCs 190–193 . The values of the band g factors measured deviate significantly from the value of ,182,201,202 where in 146 and 189 are in a good agreement with these ab initio 4 the values as low as are obtained 201 . Authors of Ref. calculations. Experiments on other TMDCs are required to 1.7 202 find that the g factor in MoS monolayers shows a be performed before their CB and VB g factors could be 2 systematic trend where the poor quality samples (larger compared with the ab initio theory. exciton linewidths) show larger magnitudes of the g factor When the thickness of the TMDC is increased, it is found when compared to those exhibiting narrow linewidths. that the magnitude of the A exciton g factor reduces while What needs to be checked is if this behavior is typical only 145 staying negative (See Figure 4) . This reduction is stronger to MoS 2, or to the other TMDCs as well. Another question for Mo-based TMDCs (MoSe 2 and MoTe 2 in Ref. 145) one might ask is how the exciton and band g factors vary as compared to WSe 2. The physics of thickness-dependent g one raises the temperature. The g factor of electrons in GaAs factors of K-point excitons in TMDCs cannot be directly quantum wells is known to change from its low temperature compared with their well-width dependence in quantum value of to about at room temperature 68,135,161 . 0.44 0.33 wells. It is because in TMDCs thicker than a monolayer, the Interestingly, initial calculations predicted the ⋅ neighboring layers are only weakly coupled to each other opposite trend 68 . However, later on it was realized that one through van der Waals interactions 141,143,145,194–198 . needs to consider an integration over many Landau levels Therefore, a multilayer TMDC acts more like a weakly and over in the theory to explain the experimental coupled QW system for the states at the K point of the results 135 . Such temperature-dependent measurements and Brillouin zone, with every individual layer acting as a QW theories have not been reported for TMDCs so far. on its own. The K-point electron and hole wavefunctions It is noteworthy that in the PL spectra of WS and WSe are largely localized within a single layer of a multilayer 2 2 monolayers, many features lower in energy to the bright TMDC 143,199 . Therefore, A and B excitons in a multilayer trions are observed, which were initially attributed to and a bulk TMDC largely possess two dimensionality 143,200 . localized exciton states 203 . The origin of these peaks has When the thickness of the TMDC increases, theory is ⋅ been a topic of debate. Koperski et al. found that many of able to model the reduction of the g factor of A excitons by these PL lines have anomalous g factors ranging from -4 to incorporating interlayer-interaction-induced mixing of the -13 146 . This raised speculations that these lines could be conduction and valence states 145 . However, there is a scope related to the intravalley dark excitons with predicted g for a more detailed theory for calculating the g factors such

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Journal accepted version of the manuscript. Published version at https://doi.org/10.1063/5.0042683

Figure 8. Examples of some (a) momentum-forbidden and (b) Figure 9. Possibility of a (a) spin-forbidden and (b) momentum - spin-forbidden transitions in dark TMDCs such as WSe 2 in the forbidden transition assisted by a chiral phonon. Transitions in (a) absence and presence of a magnetic field, as described in Ref. 146 . transitio ns show similar g factors in Faraday geometry as that of dark excitons and trions i.e. . A transition depicted in (b) These transitions are expected to exhibit much larger g factors ~ 9 than the bright exciton value of . For instance, the expected g shows a larger g factor . (a) is reproduced from Liu et al., 4 ~ 12 factors of momentum and spin-forbidden transitions in this work Phys. Rev. Res. 1, 032007 (2019), Creative Commons Attribution are and respectively. Reproduced fr om Koperski et al., 4.0 International license . (b) is reproduced with permission from 10 8 2D Mater. 6, 015001 (2018). CC BY 3.0 license. Li et al., ACS Nano 13, 14107 (2019). Copyright 2019 American Chemical Society. factors of , other shake-up processes involving a spin flip waveguide coupled monolayer WSe device, providing 8 2 of excess electrons from low to upper conduction bands 146 . access to the out-of-plane polarization 210 . Alternatively, Recent magneto-PL experiments and theoretical these darkish states can be brightened by applying a developments have confirmed some of these speculations to magnetic field in the plane of the sample (Voigt ∥ be true. For instance, emission lines corresponding to the geometry, see Figure 6)181,185,204–206,209,211 . The in-plane dark excitons 170,182,185,204–206 , dark trions 176,178,181,205,207 , magnetic field mixes the bright and dark excitons through a phonon replicas of spin- and momentum-forbidden dark Zeeman coupling of where is the transverse (in- ∥∥ ∥ excitons 179,180 , and phonon replicas of positive and negative plane) g factor of the conduction electrons 204,205 . As a result, dark trions 177,179,189,207,208 have been identified in monolayer the dark excitons acquire some oscillator strength from the WSe 2 magneto PL, on the bases of their observed g factors bright excitons leading to their emission in the out-of-plane (ranging from -6 to -13). Another recent work also identifies direction 204,205 . has been determined to be about for 209 ∥ 182,185 2 these lines in hBN-encapsulated monolayer WS 2 . We MoS 2, MoSe 2 and WSe 2 . In another work, Yang et al. discuss some of these features in the following. use time-resolved Kerr rotation under in-plane magnetic 2.1.2. Neutral ‘dark’ excitons in TMDCs. As shown in fields to determine the electronic g factor of in 212 ||~1.86 Figure 5, the dark excitons in TMDC monolayers exhibit a CVD-grown monolayer MoS 2 . Strikingly, this is very fine structure doublet due to an exchange interaction which close to the free electron g factor and requires a theoretical 30 mixes the valleys and leads to a splitting understanding . 170,181,206 . The low-energy state is dipole forbidden ( Magnetic-field-brightened dark and gray excitons are 1 meV symmetry), while the higher state (“gray” exciton with polarized along and perpendicular to the direction of the in- symmetry) has an out-of-plane dipole moment 170,206 . The plane magnetic field, respectively 181,204 . The lifetime of dark gray excitons couple to the out-of-plane polarization of the excitons is longer than the bright excitons by about two to light and their emission can be studied by collecting a PL three orders of magnitude 170 . Therefore, they hold potential component along the sample plane 168,170 . In a recent work, in investigating Bose-Einstein condensates and future spin-forbidden dark excitons are resonantly controlled in a

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peaks previously under debate 87,166,203 . It needs to be understood if the emission lines exhibiting single-photon emission in TMDCs have their origin related to dark excitons/trions or their phonon replicas as well 213–217 . The author suspects this because the g factors of these emission lines are also large ( or approximately) 214–217 . 9 13 2.1.3. Bright and dark trions. Trions are three-particle bound states with either two positive charges and a negative charge (positive trion), or two negative charges and a positive charge (negative trion). These quasiparticles have been extensively studied in doped quantum wells 57,58,218 and TMDCs 87,88,223–226,146,165,186,209,219–222 using optical spectroscopy and from theoretical perspectives. The binding energies of the trions in TMDCs are of the order of Figure 10. (a) Schematic showing conduction and valence bands a few 10s of meVs, which is one to two orders of magnitude involved in the creation of the bright ( ) and the dark ( ) trions larger than the traditional quantum wells (typically 1 meV X X in a doped monolayer TMDC. (b) In monolayer WSe 2 and WS 2, to few meVs) 218 . Therefore, it is possible to study trions in dark trions are the lower in energy than the two bright trions. In TMDCs at elevated temperatures. Similar to the neutral MoSe 2, the bright trion is the lowest in energy, while in MoS 2, the exciton, trions could be bright or dark depending upon the dark trion is almost coincident with the bright trion . Fi gure 165 spin-forbidden optical selection rules . We discuss bright adapted from Arora et al., Phys. Rev. B 101 , 241413 (2020). Copyright 2020 American Physical Society. trions first. The initial state in the absorption process (or trion technological applications involving quantum information creation) is a free electron (hole) for the negative (positive) and computation using TMDCs. trion, while the final state is a trion 186,218 . In PL, the trion in To measure the magnetooptical response of dark its initial state relaxes to a free electron with the emission of excitons, two methods can be used. First, a large numerical a photon 186,218 . Depending upon the presence of the excess electron in the or valleys, three types of bright trions aperture (NA) objective lens to collect PL emitted parallel K K to the sample plane can be used and magneto PL is are possible 165 . Two of the bright trions and one dark trion performed in Faraday geometry 170 . Alternatively, if no PL are shown in Figure 10. Normally, these two bright trions from dark excitons using high NA objective is observed, are observed in the PL and absorption spectra of n-doped 186,227 105,226,228,229 170,186 one can apply tilted magnetic fields to the sample 181,182 . In MoS 2 , WS 2 and WSe 2 , while one has 185,186 this case, a component of the magnetic field within the plane been seen in MoSe 2 so far . This is an ongoing matter of the sample brightens the dark excitons, and the out-of- of debate in the community where three close-lying trion plane component splits them. An example is shown in resonances could be expected in MoSe 2 monolayer as Figure 7. well 165 . Furthermore, a discussion concerning the identification of trions (intravalley or intervalley) in the Dark excitons are observed to exhibit larger g factors than absorption and PL spectra of TMDCs has not been settled 170,172–174,181,206 the bright excitons . The dark exciton g factors yet 187,228,230 . for the case of WSe 2, MoSe 2 and MoS 2 are found to be 170,172,174,206,208 (approximately), 182 , and 182 . Both magnetoabsorption (reflectance) 146 and PL 99– 9 8.63 6.5 105,139,187 From the atomic picture considering contributions from the have been used to determine the Zeeman splitting magnetic moments of the d-orbitals of the transition metal, of the trions. In principle, in the low-density limit, the g the g factor of dark excitons is expected to be which is factor of the trion is expected to be similar to that of the 8 close to the observed values (see Figure 8) 146 . For the case neutral exciton, which has been the case in most of the 99,101–105,146,187 of MoS 2, a strong deviation of the dark exciton g factor from works . This is because the magnetic moment other materials is puzzling 182 , and requires further of the extra electron contributes equally to the initial and theoretical and experimental work. This is similar to the final states in the absorption and PL processes. However, at larger carrier densities ( ), many-body effects bright exciton g factor of MoS 2 (discussed earlier) where a 10 cm range of g factors in MoS are found (between and (such as excitons in a Fermi sea of electrons, or ‘Fermi 2 1.7 about ) depending upon the exciton linewidth in the polarons’) could change the magnetooptical response of the 4 monolayer 182,201,202 . It needs to be checked if it is the case trions considerably, leading to a significant deviation from a value of 124,187,227,231 . A theoretical treatment (ab initio for the dark exciton in MoS 2 as well. Finally, phonon 4 replicas of dark excitons, for instance in Figure 9 are or parameter-based theories) explaining the variation of the observed energetically below the dark excitons in g factor of trions due to these many-body effects is required monolayer WS 2 and WSe 2 on the basis of their observed g to completely understand these phenomena. As such, the 146 189 factors (either about or ), energetic locations, and magnetoabsorption and magneto PL of bright trions 9 12 valley polarization 177,179,180,209 . Therefore, magnetooptical have proven extremely useful to provide information on the spectroscopy has been crucial in the identification of PL g factors of the conduction and the valence bands forming

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Journal accepted version of the manuscript. Published version at https://doi.org/10.1063/5.0042683

146 189 the A excitons (and trions) of WS 2 , WSe 2 and 146 MoSe 2 . This is an important step towards developing a complete understanding of the spin-valley band structure of the TMDCs. However, many questions remain open. For instance, the g factors of the higher energy conduction bands were not determined in Ref. 146. Furthermore, the spin-orbit-split valence band which is involved in the formation of B exciton has not been investigated. In the future, the bright trions corresponding to the B excitons could be magnetooptically investigated to deduce this information. Similar to dark excitons, magnetooptical spectroscopy has played a central role to identify and to investigate dark trions and their phonon replicas in monolayers of WS 2 and 176–178,181,189,205,207,208 WSe 2 . An in-plane magnetic field can be used to brighten the dark trions (see Figure 11)181,205 . Alternatively, a high NA objective lens may be used to capture a part of the in-plane emission from the dark trions 176–178,208 . The g factor of the dark trions are found to be around or 176–178,181,189,205,207,208 , on the similar 9 10 order as that of the dark excitons discussed previously. This is close to the expected value of from the atomic picture. 8 On the other hand, a few works identify dark trion phonon replicas on the basis of their specific g factors (lying Figure 11. (a) Magnetic brightening of dark exciton and dark trion between and ), their emission energy positions with 9 13 in monolayer WSe 2 demonstrated using magneto-PL in the respect to the dark trions and their chirality using magneto presence of an in-plane magnetic field for various field strengths 177,179,207,208 PL spectroscopy (see Figure 10) . The author is at a temperature of . (b) PL intensity of the dark exciton 30 K aware of one report where effective mass of the lower and trion with an in-plane magnetic field. Both features rise conduction band in hBN-encapsulated WSe 2 monolayer is quadratically in intensity. (c) Time-resolved PL of the bright and found to be ∗ using magneto PL of intra- and the dark trions under an in-plane magnetic field at ∥ 17.5 T 0.46 207 . The dark trion decays much slowly than the bright trion. intervalley trion emission lines . 4 K Reproduced with permission from from Zhang et al., Nat. 2.1.4. Biexcitons and charged biexcitons. Magneto-PL Nanotechnol. 12, 883 (20 17). Copyright 2017 Springer Nature. spectroscopy and the Zeeman effects have been useful in identifying the multiparticle complexes such as biexcitons intravalley and intervalley scattering processes 237 . More and charged biexcitons in TMDC monolayers 172–174,232–234 . theoretical inputs would be required to satisfactorily answer These investigations point out that a biexciton could these questions. possibly be considered as a composite of a bright and a dark exction 172–175,211,232,233 . Similarly, a charged biexciton is 2.1.5. Interlayer excitons in homolayers. ‘Interlayer reported to form from a bright trion and a dark exction 172– excitons’ with spatially-separated electron-hole 175,233 . These quasiparticles are expected to have a g factor wavefunctions have long lifetimes, and are promising for close to that of the neutral exciton and trion (i.e. about their role in investigating exciton Bose-Einstein 4 238 for TMDCs)211,235 , which has been observed 172–175,232,233 . condensates (BEC) . They have a unique magnetooptical 143,144,239 However, a magnetooptical signature of these many-body response in few-layer and bulk TMDCs . The species is still lacking in Mo-based TMDCs and could be adjacent layers of these ‘homolayers’ with 2H stacking are 236 oriented ∘ with respect to each other as shown in Fig. observed in future experiments . 180 12(a). This results in a reversal of the magnetic moments of The identification of dark excitons and trions, their the corresponding bands (conduction and valence bands) in phonon replicas, and other multiparticle complexes has the neighboring layers, within a valley (Figure 12(d)). settled some debate in the identification of the low-lying PL While the g factor of the A exciton in a monolayer or features in monolayer WS 2 and WSe 2. The question remains ‘intralayer’ exciton (exciton wavefunction largely located why is it different in the case of Mo-based TMDCs where within a layer 143 ) in few-layer or bulk TMDCs is of a no such low-energy peaks are observed in the PL spectra. A negative sign, the g factor of interlayer exciton has a possible answer could emerge by comparing the dark- positive sign as shown in Figure 12. This was used as a exciton occupation (expected to be low in monolayer MoSe 2 fingerprint in the discovery of interlayer excitons in the and MoTe 2 at low temperatures), exciton-phonon coupling reflectance spectrum of a 2H-stacked bulk TMDC (g factor 143 strengths, and two energy scales in various TMDCs: i) the of in bulk MoTe 2, see Figure 12(b)) , and later 4 ± 0.5 239 magnitude and sign of the conduction-band splitting, and ii) on in a bilayer and trilayer 2H-MoS 2 . Such positive g the optical-phonon energies which participate in the factors in Mo-based bulk TMDCs had been observed in

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Journal accepted version of the manuscript. Published version at https://doi.org/10.1063/5.0042683

Figure 12. (a) A representation of intralayer and interlayer excitons in a bilayer TMDC (homolayer) with adjacent layers denoted as L1 and L2. (b) Helicity-resolved magneto-reflectance-contrast spectroscopy of intralayer and interlayer A excitons in a bulk MoTe 2 at 143 . Effective exciton g factors of intralayer and interlayer excitons are negative and positive, respectively. (c) Magneto-reflectance 4 K spectroscopy of a bilayer MoS 2 with an interlayer exciton g factor of . (d) Transition selection rules in the (upper panel) and 7.8 ± 0.3 K (lower panel) valleys of a bilayer or a bulk TMDC. Layer 1 and Layer 2 denote adjacent layers as shown in (a). Left and right sides of K the figure represent the movement of bands in the absence and presence of an out-of-pl ane applied magnetic field. Interlayer excitons have a positive g factor ( ) compared to the negative g factors of intralayer excitons, in accordance with this simplistic picture. (b) and (d) ~ 8 are adapted from Arora et al., Nat. Commun. 8, 639 (2017) (Creative Commons Attribution 4.0 International License) 1970s 240 , but their origin remained unexplained until the interlayer exciton with the intralayer A exciton in W- 144 recently. Magneto-reflectance spectra of a bilayer MoS 2 based TMDCs also prohibits its detection . It has been measured by us are presented in Figure 12(c) where an recently found that the oscillator strength of interlayer interlayer exciton g factor of is obtained. We get excitons in bilayer MoS 2 can be increased using an external 7.8 ± 0.3 244,245 a similar g factor for bulk MoS 2 as well (data not shown). electric field . The question remains if the interlayer This is close to an expected value of about from a simple excitons in W-based TMDCs could be magnetooptically 8 atomic picture in Figure 12(d). While resonances with detected by enhancing their oscillator strength (e.g. using an positive g factors are found in MoS 2, MoSe 2, and MoTe 2, electric field). Such long-lived dark interlayer excitons, if they are absent in W-based TMDCs 144 . The dissimilarity in created as a lowest-energy state in a bilayer system, such as the absolute value of the interlayer exciton g factor in bulk by applying a strain, could hold potential in studying MoTe 2 and MoS 2 is a matter of debate. It is worth noting collective physics such as BEC. that in 3R stacked MoS bilayers with a relative twist angle 2 2.1.6. Interlayer excitons in heterobilayers. So far, we of ∘ (compared to ∘ in 2H bilayers), the interlayer hole 0 180 have discussed magnetooptics of monolayer and few-layer tunneling is symmetry disallowed and interlayer excitons TMDC homostructures. However, following the traditional are not observed 241 . heritage of III-V and II-VI heterostructures, research is Ab initio theory-based calculations showed that a progressing at a rapid pace in the area of heterostructures of relatively low spin-orbit coupling in Mo-based materials layered materials 106,246–252 . Stacks of layered crystals can be was responsible for a mixing between the interlayer readily created by stamping monolayers or few layers of excitons and the B excitons, providing a large oscillator TMDCs on the top of each other 246 . In a heterobilayer, strength to the interlayer excitons 144,242,243 . This mixing is hexagonal lattice units of the adjacent layers can be twisted negligible in W-based materials, leading to an with respect to each other at an angle ∘ ∘ and the 0 60 unmeasurable interlayer exciton resonance in their electronic properties could be tweaked. This emerging area reflectance spectra. Furthermore, an expected proximity of of research is termed ‘twistronics’ 253 . For a slightly different

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Journal accepted version of the manuscript. Published version at https://doi.org/10.1063/5.0042683

Figure 13. (a) Alignment of ± valleys in MoSe 2-WSe 2 heterostructure with AB stacking (From Ref. 106). (b) Type-II band align ment K and creation of the interlayer exciton in this heterostructure (From Ref. 247). (c) Sketch of circularly-polarized interlayer exciton transitions in AB stacked heterostructure in (a) in the absence (dashed lines) and presence (solid lines) of an out-of-plane magnetic field. Characteristic alignment of ± valleys between two layers lead to a large interlayer exciton g factor (From Ref. 106 ). (d) Moiré superlattice K formed in a heterobilayer with a small twist angle , and (e) exciton in a moiré potential (From Ref. 248 ). (a) and (c) are reproduced with permission from Nagler et al., Nat. Commun. 8, 1551 (2017), Creative Commons Attribution 4.0 International License. (b) is reproduced with permission from Ciarrocchi et al., Nat. Photonics 13, 131 (2019), C opyright 2019 Springer Nature. (d) and (e) are reproduced with permission from Seyler et al., Nature 567, 66 (2019), Copyright 2019 Springer Nature. lattice constant and alignment angle between the stacked been highlighted 251 . The area of research is still in its layers, a moiré pattern is formed as shown in Figure infancy, and in the future, heterobilayers of other 254 13(d) . For ∘ and ∘, AA and AB stacked semiconducting TMDCs are expected to be investigated 0 60 configurations are created, respectively 106,247–249 . An magnetooptically. Many parameters such as twist angle, example of an AB stacked MoSe 2-WSe 2 heterobilayer is layer thickness, doping, electric fields etc. could be tuned to shown in Figure 13(a). One notices that the ± valleys of create heterostructures exhibiting interesting physics and K MoSe align with ∓ valleys of WSe . The interlayer potential for future valleytronics and twistronics device- 2 K 2 excitons created between the type-II band alignment (Figure based applications. 13(b)) in this structure is lower in energy than the A excitons 2.2 Valley polarization and coherence in magnetic in the individual layers. Due to the specific valley fields. alignment, the valley magnetic moment contributions enhance the g factor ( which corresponds to a When the valley degeneracy is lifted, a magnetic field ~ 15 Zeeman splitting of about at ) of interlayer leads to a redistribution of the carriers between the valleys. 26 meV 30 T excitons significantly compared to the case of monolayers This leads to an uneven intensity of the emission of the two (Figure 14 (c)) 106 . In the case of AA stacked layers, a circular polarizations, and is termed as magnetic-field- doublet with a spin-conserving and spin-flipping transition induced ‘valley polarization’ 99–106 . An example is shown in is observed, with g factors of and Figure 3(a) and (b). In Ref. 146, the authors assumed a 8.5 ± 1.5 7.1 ± 1.6 respectively 247 . It has been shown in multiple theoretical Boltzmann distribution of carriers (low-doping-density works that such spin-flipping transitions may be brightened regime) in the conduction bands of doped monolayers of in heterobilayers with matrix elements similar to those of WS 2 and MoSe 2, and studied the valley polarization of spin-conserving transitions, due to the presence of a trions in the magnetic field. It was used to determine the 255,256 lower conduction band g factor equal to and for periodic moiré potential (Figure 13(d) and (e)) . A few 1.08 1.84 other recent works have also observed interlayer excitons in WS 2 and MoSe 2, respectively. The magneto-reflectance moiré potential with large negative ( ) and positive g contrast of trions in monolayer MoSe 2 is shown in Figure ~ 16 factor ( )248,250–252 . The role of chiral-phonon-induced 14 where the trions appear to be completely polarized ~ 7 around . Magnetic-field-induced valley polarization of exciton-phonon interactions in the WSe 2/MoSe 2 10 T heterostructures in the determination of the optical selection trions in WS 2 is much more complicated. Two bright trions rules and brightening of spin-flip (triplet) transitions has (singlet and triplet, shown in Figure 10(a)) are observed in

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Journal accepted version of the manuscript. Published version at https://doi.org/10.1063/5.0042683

Figure 15. Schematic depicting the proposed explanation of observed valley polarization of trions under an out-of-plane 227 magnetic field in the case of a doped MoS 2 monolayer . (a) Figure 14. (a) Magneto-reflectance contrast spectra of trions in an represents the valley Zeeman effect in the conduction bands unintentionally doped monolayer of MoSe 2 under magnetic fields under magnetic field . The first electrons (green s pheres) B of up to for the two circular polarizations. Trions polarize enter the lowest energy band. (b) As the electron density is 30 T strongly in the presence of magnetic field due to a redistribution raised, the intra- and inter-valley exchange interactions pull the of carriers between the energetically non-degenerate valleys. The same-spin conduction bands to low energy, resulting in a spin- oscillator strengths of the trions are shown in (b) while (c) depicts polarized 2D electron gas. (c) and (d) depict the low- and high- the degree of polarization. The data has been fitted with a model energy exciton polarons and respectively . The X X based on Boltzmann distribution of carriers and conduct ion band corresponding trions in both cases are spin singlets. (e) The spin g factor is determined equal to . Reproduced with permission triplet trion is unbound which results in a high-energy tail of the 1.84 from Koperski et al., 2D Mater. 6, 015001 (2018). CC BY 3.0 neutral exciton of polarization. Reproduced with permission σ license. from Roch et al., Nat. Nanotechnol. 14, 432 (2019). Copyright reflectance and PL and show different polarization response 2019 Springer Nature. under magnetic fields 146,230 . In reflectance, the polarization is of opposite sign and similar magnitude 146,230 . This is a microscopic theoretical description of this effect in MoS 2 requires to be developed for a complete picture. This is a explained by an uneven occupation of states in the ± K developing area of research, where alternative views are valleys by free carriers due to valley Zeeman splitting, emerging 187 . From an experimental perspective, the which affects the oscillator strengths in this peculiar way. possible future directions could be to test the spin- However, in magneto PL, only the singlet trion is polarization effects by placing monolayer MoS on significantly polarized 230 . This has not been explained 2 magnetic substrates, or on 2D layered magnetic materials. It satisfactorily and could involve an interplay between many might be possible to achieve significant spin polarization processes such as intervalley relaxation of carriers, spin-flip under low magnetic fields (compared to the high fields processes, and laser-induced photodoping effects. Magneto ) in Ref. 257. PL as a function of gate voltage (carrier density) and 9 T excitation intensity is expected to shine light on this Linearly-polarized light can be written as a superposition problem in the future. of two circular polarizations of opposite orientations. Therefore, both and valleys can be excited It is worth discussing the peculiar case of doped MoS 2 K K monolayer under magnetic fields. Roch et al. performed simultaneously using linearly-polarized light, and a coherent superposition of states can be created in the two helicity-resolved magneto-reflectance on this system as a 93–98 function of carrier density 227,257 . The unusual trion valleys (‘valley coherence’) . TMDCs such as polarization under magnetic fields in the low-density monolayer WS 2 and WSe 2 when excited using linearly- polarized light also emit a linear polarization, while the regime ( ) seems to suggest that the 5 10 cm ground-state is spin polarized. At a certain critical density polarization angle of the emission follows that of the excitation 93–98 . An external magnetic field leads to a ( in Ref. 257), the system is reported 3.0 10 cm to undergo a ferromagnetic-to-paramagnetic first-order Zeeman splitting between the two valleys, lifting the valley phase transition 257 . The mechanism behind the spontaneous degeneracy. This leads to a phase difference between the spin polarization is assumed to be the strong intra- and inter- two circularly-polarized components. As a result, the emission polarization rotates with respect to the excitation valley exchange interaction shown in Figure 15. However,

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Journal accepted version of the manuscript. Published version at https://doi.org/10.1063/5.0042683 polarization as shown in Figure 16. The coherent state decays with a lifetime (or ∗) which is of the order of a few 100 fs at liquid He temperatures 96–98,258 . This decay leads to a depolarization of the emission polarization. The extent of this depolarization increases as the external magnetic field rises 96–98 . Therefore, external magnetic fields have helped in understanding valley-coherence effects, deducing the coherence lifetimes, and controlling the intervalley phase. These developments mark important steps towards the realization of an excitonic valleytronic gate for future quantum technologies such as in quantum computation. However, at the moment, many roadblocks persist. The coherence life times are small, and the magnetic fields required for such a phase control are much higher (up to 25 T external fields were used 96,97 ) than suitable for practical device purposes. Additionally, the experiments were done at cryogenic temperatures, while investigations remain to be performed at room temperature. On the contrary, a long spin coherence lifetime of resident carriers exceeding many nanoseconds to a few microseconds is observed in monolayer TMDCs using time- resolved Kerr rotation spectroscopy 212,259–266 . This is many orders of magnitude larger than the PL decay time and the Figure 16. Linear polarization of luminescence from coherently excitonic valley coherence decay time discussed above. In emitting valleys can be manipulated using an out-of-plane externally applied magnetic field. An example of the magnetic- fact, while the electron polarization decay (nanosecond field-induced rotation and depolarization of emission polarization regime) was seen to accelerate in the presence of an in-plane from a monolayer WS 2 under a linearly-polarized excitation along 212,259,262,263 magnetic field (Hanle effect) , the hole ∘ (red double-headed arrow) is show n. (a), (c) and (d) are the 0 polarization in WSe 2 monolayer persisted for a couple of emission-polarization intensities as a function of angle under microseconds, and did not depend upon the magnetic magnetic fields of , and , respectively . (e) and 25 T 12.5 T 25 T field 262 . This verified a strong spin-valley locking of holes (f) are the rotation angle and the polarization degree as a function in the monolayers. Therefore, the application of a robust of magnetic field. The data is used to determine the decay time of the coherent light emission i.e. ∗ . Reproduced from spin coherence in the future monolayer TMDC-based 260 fs spintronic devices is promising. However, so far, this long Schmidt et al., Phys. Rev. Lett. 117, 077402 ( 2016). Copyright 2016 American Physical Society. spin coherence persists only at cryogenic temperatures which might limit usage in device applications, and requires of a magnetic field (Landau quantization of the cyclotron further work. orbits), and the effective dimensionality of the system is [2] reduced by two. Energy of the Landau level of the Recently, there are proposals to optically control the entanglement of electron and nuclear spins in TMDC conduction (c) or the valence (v) band is given by monolayers via intervalley terms of the hyperfine interaction 267–269 . It has been argued that similar to the case , , 1 , (3) ℏ of electron and hole hyperfine interactions in III-V 2 semiconductors 270,271 , they are of short range nature in where , ∗ is the cyclotron frequency of the /, TMDCs as well. Such a short-range hyperfine interaction is conduction or valence band, and ∗ is the effective mass , expected to lead to a coupling between electronic states in of the band. The inter-Landau-level transition follows 4 the ± valleys providing a spin relaxation mechanism. On the selection rule . In the presence of K Δ 0 the other hand, within a valley of TMDC, the electron- the Coulomb interaction, the th inter-Landau transition nuclear spin flip is suppressed due to a large (4 orders of corresponds to optically probing the Ns exciton magnitude larger than hyperfine splitting) spin-orbit (magnetoexciton) state 38,52,53,55,56,129–131,202 . Apart from splitting 267 . Further understanding of the role of hyperfine separating the excited exciton states from each other, a interaction in spin coherence dephasing is expected from magnetic field also enhances their oscillator strengths linear future experimental and theoretical works. with , by squeezing the exciton wave functions 38,51,272 . 2.3 Landau quantization and Rydberg spectroscopy: This is especially advantageous for performing Rydberg spectroscopy where Rydberg states with very large could exciton binding energy, reduced masses, and oscillator 45,47,55,56,129–131,202 strength enhancement. The problem of a 3D and a 2D be observed under magnetic fields . Recently, Rydberg states up to have been observed in electron-hole system under strong magnetic fields has been 11s extensively studied theoretically 38,40–44,50–52,272 . The hBN-encapsulated monolayer WSe 2 in moderately large magnetic fields of 131 (see Figure 17 (e) and (f)). conduction and valence bands are quantized in the presence 17 T

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Journal accepted version of the manuscript. Published version at https://doi.org/10.1063/5.0042683

Figure 17. (a) Rydberg states of excitons up to in monolayer WS 2 measured using reflectance contrast spectroscopy in the absence 5 of magnetic field. (b) Circular-polarization-resolved magnetotransmittance of Rydberg excitons up to in hBN-encapsulated 4 monolayer WSe 2 under pulsed high magnetic fields of up to . 1s and 2s show a quadratic diamagnetic shift throughout the field range, 65 T suggesting a low-field limit. 4s energy blue shifts approximately linear with at large fields, representing a crossover to the high-field limit. (c) Experimental and theoretical ns exciton transition energies from (a), along with a 2D hydrogen model (red solid line) for comparison. The inset shows the corresponding effective dielectric constant. (d) Screened 2D interaction potential (black lin e, Eq. 4) and 2D hydrogen potential (red) along with the corresponding energy levels and radial wave functions up to for data in (c). The inset 3 shows the potentials in a semilogarithmic plot. (e) Valley polarization (magnitudes marked by color scale in the legend) and (f) energy positions (Landau-fan diagram) of Rydberg excitons up to deduced from the magnetophotocurrent spectroscopy of hBN- 11 encapsulated WSe 2 monolayer, with solid-line fits using a numerical model based on Rytova-Keldysh potential. Valley polarization is defined as . A linear blue shift of 9s to 11s excitons for points towards a high-field limit of / 10 T these states. The A exciton binding energy determined from the fan diagram is . (a), (c) and (d) Reproduced with permis sion 168.6 meV from Chernikov et al., Phys. Rev. Lett., 113, 076802 (2014). Copyright 2014 American Physical Society. (b) Reproduced with pe rmission from Stier et al., Phys. Rev. Lett. 120, 057405 (2018). Copyright 2018 American Physical Society. (e) and (f) Reprodu ced with permission from Wang et al., Nano Lett. 10, 7635 (2020). Copyright 2020 American Chemical Society. Due to the excitonic effects, one does not observe a Under low magnetic fields, i.e. the Coulomb ≪ 1 simple linear -dependent blue shift of the exciton energy dominates. Often, in the case for the ground-state transition energy, especially for low inter-Landau exciton, until under very high fields. In this limit, ≪ 1 transitions. An example is shown in Figure 17(b) for exciton only a -dependent diamagnetic shift of the exciton is 129 states up to in hBN-encapsulated WSe 2 . We clearly observed. This low-field limit is discussed in the next 4 observe a non-linear behavior of 1s and 2s exciton energies section. In high magnetic fields, when , the system is ≫ 1 even under high magnetic fields of up to . However, 3s purely Landau like. This high-field limit is readily 65 T and 4s states begin to crossover to a linear blue shift regime approached under moderate fields for magnetoexcitons for higher fields (above about ). To understand this, we with a large , while the exciton energy blue shifts linearly 50 T 2,129–131 compare the Coulomb energy scale given by with magnetic field approximated by . For Δ ∗ (where ∗ is the effective Rydberg) with instance, the binding energy of the exciton in Figure / 1/2 11 the ground-state cyclotron energy , . We define a 17(f) is , which is much smaller than the Landau ℏ /2 1.2 meV , 2 quantization energy of at . Therefore, we are dimensionless parameter ℏ . The parameter 8.69 meV 15 T / in the high-field limit. Another example of an hBN- 130 is also written in terms of a magnetic length ℏ encapsulated monolayer WSe2 is shown in Figure 18. Here, the 5s exciton nearly lies in moderately high-field which depends only on magnetic field, and the Bohr radius limit ( ) above . The energies of the ground and ∗ ~1 9 T ∗ as . excited exciton states are commonly plotted as a function of

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in a Landau-fan diagram with exciton energy and standing monolayers still requires to be determined magnetic field as the x- and y-axis respectively such as in magnetooptically. One would have to wait for high-quality Figures 17(f) and 18(d) 2,4,45,47,55,56,129–131,202 . An analysis of free-standing samples where strain-free monolayers would this data has been readily used in the literature for a be exfoliated or CVD grown on substrates with holes (at determination of the exciton-reduced masses ∗ and their least diameter to avoid diffraction effects in 10 μm binding energies in III-V and II-VI QWs 2,47,55 as well as transmission mode) drilled through them. The ab initio in TMDCs 129–131,202 . We now discuss the parameters derived theory calculations predict that the binding energies in this for TMDCs. case could be of the order of for monolayers 126,221,224 , 1 eV and 0.1 eV for bulk TMDCs 143,199 . Furthermore, the Fitting a nearly linear shift of 3s and 4s excitons at high Rydberg spectroscopy of a few-layer and bulk TMDCs has magnetic fields in Figure 17(b) (moderately high-field limit not been performed so far for . To understand the i.e. or slightly larger), the authors determine an upper 2 ~1 variation of exciton-reduced masses with layer thickness, limit on the exciton reduced mass equal to 129 ( 0.23 magneto-reflectance or transmittance measurements would is the free electron mass). Molas et al. use magneto PL of be preferable over magneto PL. It is because thicker layers the state and find this value 130 . In Figure 5 0.25 being indirect gap semiconductors are inefficient emitters of 16(d), the high-field limit ( ) of , and ≫ 1 9s 10s 11s light around their direct band gaps166,169,183,280–285 . Rydberg states in hBN-encapsulated WSe 2 monolayer is used to determine a reduced mass of 131 . Theory Table 1. Reduced mass and binding energy of A excitons in 0.2 predicts a value of , deviations from which are a TMDC monolayers determined from a combined analysis of the 0.17 matter of debate 163,221 . Wang et al. measure inter-Landau ground- and excited-exciton-state energies in the low- and high- field limits, along with the references. Level transitions in a gated (for tuning the carrier density) hBN-encapsulated WSe monolayer using 2 Material Reduced A A exciton magnetoreflectance spectroscopy when the electron-hole 273 exciton mass ∗ binding energy interaction could be neglected . They find the reduced ( ) (meV) mass equal to . A few more recent works also 142 0.268 1L WS 2 on SiO 2 - 410 reported inter-Landau-level transitions in doped hBN- 202 130 202 130 274–276 277 1L hBN/MoS 2/hBN 0.275 , 0.26 221 , 217 encapsulated monolayer WSe 2 and MoSe 2 . 202 130 202 130 Notably, the reduced masses are also determined in a more 1L hBN/MoSe 2/hBN 0.35 , 0.44 231 ,216 consistent fashion by analyzing the energy shifts of the 202 130 202 130 excitons in the low-field limit (diamagnetic shift). We 1L hBN/WS 2/hBN 0.175 , 0.15 180 , 174 discuss this separately in the next section. 202 130 202 129 1L hBN/WSe 2/hBN 0.20 , 0.21 , 167 , 167 , 131 131 130 The Landau-fan diagram provides an estimate of the 0.20 168.6 , 167 exciton binding energy. The extrapolation of the high 1L hBN/MoTe /hBN 0.36 202 177 202 2 Rydberg states to 0 magnetic field asymptotically approaches the free-particle band gap as shown in Figure 17(d). Subtracting the 1s exciton energy from this provides 2.4 Diamagnetic shift: exciton masses and Bohr radii. an estimate of the exciton binding energy. Popularly, a Magnetooptical measurements on excitons in the low-field 278,279 limit ( ) are useful in determining the exciton masses nonhydrogenic screened Rytova-Keldysh potential ≪ 1 2,48 and Bohr radii . The dependent diamagnetic shift of has been used to explain the observed energy shifts of the 2,38,129–131,142,202,286,287 excited exciton states in hBN-encapulated monolayers of exciton energy in this limit is given by 129,130,202 TMDCs (5) Δ ∗ 〈 〉 (4) 8 8 where is the root-mean-square (RMS) radius 〈 〉 where and are Struve and Bessel functions of the exciton, and ∗ is the reduced mass. This is a general respectively, is the average dielectric constant of the top result, applicable equally to 3D and 2D ground and excited- and the bottom layer, is the permittivity of the vacuum, state excitons. For the exciton ground ( ) state, the 1 is the screening length and is the 2D diamagnetic shift is given in 3D and 2D as 2,49 2 polarizability. Solid lines in Fig. 17(f) are the numerical fits ∗ to the data using this model where the best fit parameters , , ℏ (6) 131 Δ 1 ∗ ∗ are and ∗ . The A 4.3, 4.5 nm 0.2 exciton binding energies determined using this treatment in 3 (7) various works are mentioned in Table 1. Δ 1 16 Therefore, magnetooptical spectroscopy has provided is the dielectric constant of the material, and ∗ , answers to some of the most fundamental questions ∗ is the 3D exciton Bohr radius. The 4ℏ / concerning exciton band structure. Indeed, the binding exciton Bohr radius is defined here as the radius at which energies and reduced masses of the excitons in the free- the wavefunction has decayed to of its maximum 288 . It 1/

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Figure 18. (a) and (b) Circular-polarization-resolved magneto-PL spectra of an hBN-encapsulated WSe 2 monolayer under out-of-plane magnetic fields of up to . (c) Magnetic-field-dependent energies of the exciton states up to for the two polarizations showing 14 T 5 valley Zeeman splitting of the Rydberg states. (d) and (e) Mean exciton energies versus and respectively. 1s and 2s states show quadratic diamagnetic shifts throughout the field range while 3s to 5s follow a quadratic behavior in low magnetic fields. T his represents the low-field limit. 5s state shows a linear blue shift above about suggesting a crossover to the high-field limit. Reproduced with 9 T permission from Molas et al., Phys. Rev. Lett. 123, 136801 (2019). Copyright 2019 American Physical Society. is worth noting that the 3D Bohr radius is twice as large than dependent diamagnetic shifts in hBN-encapsulated WSe 2 129 130 the 2D case [ ∗ ∗ ] and the 3D are shown in Figures 17(b) and 18(d,e) . For bulk , 2ℏ / exciton diamagnetic shift is larger by a factor of than TMDCs, is large (e.g. 14.5 for bulk MoS 2, and expected 16/3 in 2D. Also, the exciton RMS radius in 3D is to be of the same order for other TMDCs 289 ) resulting in , larger diamagnetic shifts 142,143,290 . The exciton-reduced ∗ and in 2D is ∗ . The diamagnetic masses determined for A excitons in TMDCs following a √3, , , shifts of the excited states of the excitons are much larger combined analysis of the exciton energies in the low-field than the ground state (large Bohr radii), and are discussed and high-field limits are compiled from various works in in Refs. 2,38,48,50. For the 3D exciton, the diamagnetic Table 1. The RMS radii of the excitons are noted in Table shifts of the excited states are given by 2. A change of the dielectric environment around the monolayer affects the exciton binding energy and RMS 286 (8) radius, resulting in a different diamagnetic shift . Δ Table 2. RMS radii of the ground and excited states of A and B However, for 2D , and excitons, the diamagnetic excitons and ) in various TMDC crystals determined using 2 3 4 X X shifts are expected to be larger by a factor of 39, 275, and magnetooptics by measuring diamagnetic shifts in the low-field 1029 respectively, compared to the 2D exciton 50 . The limit ( ). The references are mentioned along with the 1 ≪ 1 measured diamagnetic shifts are used to determine the numbers as well. In addition to these, Wang et al. have recently measured the RMS radii of Rydberg states up to where exciton-reduced mass, binding energy, and/or radius. The 11 they get the RMS radius of state equal to 131 . reduced mass determined this way can be compared with 11 214 nm the high-field limit case described in the previous section. (nm) (nm) For a 2D exciton in a GaAs quantum well, is Material X X Δ 1 about at , and at 45 . 1s 2s 3s 1s 2 meV 10 T 7 meV 20 T However, in a layered semiconductor such as a monolayer 142 142 1L WS 2 on SiO 2 1.53 - - 1.16 WS 2, this is extremely small for the A excitons ( 1.5 meV 202 at )142 . This is because the exciton reduced masses in 1L hBN/MoS 2/hBN 1.2 - - - 65 T TMDCs are much larger than in III-V semiconductors ( 202 221 0.15 1L hBN/MoSe 2/hBN 1.1 - - - in monolayer WS 2 compared to 0.04 – 0.08 for heavy hole excitons in GaAs QWs 46,47 ), while their out-of-plane 1.8 202 , 1L hBN/WS /hBN 5 – 8287 - - dielectric constants are smaller (4.13 in monolayer WS 289 2 2287 2 versus 12.5 in GaAs 46 ). Therefore, one needs very large 1.7 202 , magnetic fields (still staying within the low-field limit) to 6.6 129 , 1.7 129 , 14.3 129 , measure the diamagnetic shifts of ground-state excitons in 1L hBN/WSe /hBN 6.8 131 , - 2 1.75 131 , 15.45 131 TMDCs with a reasonable accuracy 142 . For this reason, the 7.7 291 2.7 291 diamagnetic shifts were not measured in monolayer 142 TMDCs until 2016 . Examples of the measured -

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Figure 19. (a) Hysteresis loops of monolayer (1L), 2L and 3L thick layered magnet CrI 3, measured using MOKE spectroscopy. is the measured Kerr rotation. (b) Heterostructure made from 10 nm CrI 3 and monolayer WSe 2 sandwiched between thin layers of hBN. An exchange field of about has been detected at the heterojunction. (c) Magneto PL spectra of trions in monolayer WSe 2 in (b) in the 13 T two excitation/detection configurations RR and LL, where R and L denote right and left circularly polarized light. Strong valley polarization is observed under relatively small magnetic fields, arising from the exchange interaction at the heterojunction. (d) Valley Zeeman splitting between ± valleys of the monolayer WSe 2 derived from the magneto PL measurements. (e) Spin-orientation-dependent K charge hopping between monolayer WSe 2 and thin CrI 3, resulting in polarized PL response as a function of magnetic field. (a) is reproduced with permission from Huang et al., Nature 546 , 270 (2017) (copyright 2017 Springer Nature), (b) – (e) are reproduced with permission from Zhong et al., Sci. Adv. 3, e1603113 (2017) (copyright 2017 American Association for the Advancement of Science)

202 1L hBN/MoTe 2/hBN 1.3 - - - understand if it could be neglected. This could be important

144 144 to check if it contributes to the discrepancies between the Bulk WSe 2 on Al 2O3 2.0 4.3 experimentally deduced exciton reduced masses and the 143 143 130,163,202,221 Bulk MoTe 2 on SiO 2 1.2 4.8 - - theoretical calculations . 3. LAYERED MAGNETIC MATERIALS AND OUTLOOK The experimentally obtained reduced masses are a matter of debate since discrepancies are found compared to Research in the area of layered materials got another boost ab initio calculations 130,163,202,221 . The disagreement is the in 2017 with the discovery of a ferromagnetic ordered state strongest for MoSe 2 (up to deviation) where the in monolayers of Cr-trihalide CrI 3 (Curie Temperature 70% 113 115 calculations predict a value of 163 . For other ) , and bilayers of Cr 2Ge 2Te 6 ( ) . Here, 0.27 45 K 30 K materials, the disagreement is of the order of polarization modulation and Sagnac interferometry based 10 15 % which might be considered reasonable. MOKE spectroscopy was used, respectively, for detecting ferromagnetic ordering as a function of temperature113 . 2.5 Binding-energy enhancement. The binding energy Since then, many other 2D ferromagnets such as other 109 of the exciton (ground and excited states) also increases chromium trihalides i.e. CrCl 3 and CrBr 3 , and more e.g. with magnetic field as due to an enhanced overlap Cr Ge Te 115 , VSe 116 , and Fe GeTe 117 have been √ 2 2 6 2 3 2 between electron and hole states under a magnetic discovered. While chromium trihalides are semiconductors field 38,51,272 . However, this effect is normally not noticed in with optical band gaps in the visible electromagnetic the data since the other effects such as the -dependent 110,292 spectral region , VSe 2 and Fe 3GeTe 2 are diamagnetic shift and -dependent Landau quantization are metallic 107,116,117 . Another emerging family of 2D magnetic more dominant. Therefore, it has been neglected in all semiconductors are antiferromagnets of the type MPX3 (M works on TMDCs known to the author so far. It might be = Mn, Fe, V, Zn, Co, Ni, Cd, Mg; X = S, Se) 111,112 . Many useful in the near future, to quantify and understand the surprises are unraveled when a layer-thickness-dependent extent of this effect theoretically for TMDCs so as to ordering is investigated. For instance, in the case of CrI 3, a

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Journal accepted version of the manuscript. Published version at https://doi.org/10.1063/5.0042683 monolayer and a trilayer are ferromagnetic, but a bilayer that fiber-based setups which are normally used for shows antiferromagnetic ordering (see Figure 19(a))109 . The magnetoreflectance/transmittance or PL measurements are 87,104,148 ferromagnetic order is restored in bulk CrI 3. CrBr 3 exhibits not very suitable for modulation spectroscopy . In ferromagnetism irrespective of its layer thickness 109 . future works, these methods are expected to pave a way to Recently, magneto-Raman spectroscopy has been used to perform Zeeman spectroscopy on quasiparticle resonances 293 identify magnons in the layered antiferromagnets FePS 3 , in TMDCs under tiny magnetic fields. It would be important 294 and MnPS 3 . This approach could be extended to other to check if the magnetic-field-dependent linearity of the layered magnetic materials to study magnetic excitations. valley Zeeman splittings of quasiparticles in TMDCs persist For a summary of developments in this area of the research, to very low fields ( ) where no reliable data is available 2 T the reader is referred to other recent reviews 107–110,295,296 . so far. Furthermore, modulation Zeeman spectroscopy under relatively large magnetic fields ( and more) It was also shown in 2017 that an exchange interaction 10 T could be vital to shine light on the physics of highly-excited between CrI thin film and a WSe monolayer in a 3 2 Rydberg states of excitons whose signatures in non- heterostructure could be used to control the valley Zeeman modulated spectra remain feeble such as in Figure 17(b and splitting and valley polarization in WSe using small 2 c). external magnetic fields (see Figure 19 (b-e)) 114 . The authors reported an exchange-induced equivalent out-of- In summary, this perspective highlights the progress and plane magnetic field of at the heterojunction. This future prospects of many scientific questions of 13 T provides unprecedented opportunities for exploring spin- fundamental importance in the area of 2D semiconductors valley physics of quasiparticles in TMDCs and device using magnetooptical spectroscopy. Although the area of applications under external magnetic fields of practical research is relatively recent (5-6 years), a rapid progress has importance (< ). Many future directions could be been witnessed. Many puzzles, sometimes decades old, 1 T envisioned. For instance, the intervalley phase could be have been solved, and there is an enormous scope for future controlled by substantial amounts under small magnetic research from both theoretical and experimental fields in such a heterostructure. Previously, such perspectives. Overall, the author firmly believes that non- experiments required large magnetic fields up to 25 T 96,97 . destructive magnetooptical techniques have a strong Room-temperature control of intervalley phase and potential to be useful in future fundamental explorations of population under low-magnetic fields would be an essential 2D semiconductors and magnetic materials, as well as requirement for a valleytronic gate. As another example, a device applications in many emerging fields such as suitable heterostructure between a 2D magnet and a TMDC spintronics, valleytronics, straintronics, twistronics, and could be used to create an in-plane exchange field and quantum information and computation. brighten the dark excitons using small external fields. Dark- Acknowledgements. The author gratefully acknowledges exciton brightening has been proposed to be useful in countless discussions on magnetooptics of semiconductors chemical sensing applications 297 . Uniaxial strain on the with Marek Potemski over the course of many years and has layered magnets could be used to tweak the magnetic benefitted immensely from discussions with Rudolf properties such as changing the orientation of the easy axis Bratschitsch, Xavier Marie, Thorsten Deilmann, Alexey or the Curie/Néel temperature for future ‘straintronic’ Chernikov, Tobias Korn, Clément Faugeras, Sandip Ghosh, devices 298,299. Overall, the author believes that the Andreas Stier, Maciej Molas, and Cedric Robert while magnetooptical methods would play an important role in the writing this paper. The work was supported by the DFG 2D-materials-based future technologies. projects AR 1128/1-1 and AR 1128/1-2. Help from Shreya For unlocking the full potential of magnetooptics in Agrawal during manuscript preparation is gratefully layered materials, it would be essential to perform acknowledged. modulation magnetospectroscopy (e.g. magnetocircular Data availability statement. The data that support the dichroism, MOKE, Faraday rotation etc.). Although findings of this study are available from the corresponding monochromatic MOKE has been used for studying authors of the original manuscripts discussed. magnetism in 2D materials, the use of modulation techniques to detect valley-Zeeman splitting effects on References. quasiparticle resonances such as excitons is scarce152 . These 1 A.K. Zvezdin and V.A. Kotov, Modern Magnetooptics and methods are capable of measuring very low Zeeman Magnetooptical Materials (Institute of Physics Publishing, Bristol splittings (few )150,152,300,301 . Furthermore, they provide and Philadelphia, 1997). μeV 2 a non-destructive way to study magnetic ordering in 2D N. Miura, Physics of Semiconductors in High Magnetic Fields (Oxford University Press Inc., New York, 2008). magnetic materials. However, one requires to perform 3 B. Lax and J.G. Mavroides, in Semicond. Semimetals Ser. 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