Abstract
The optical properties of monolayer and bilayer transition metal dichalcogenide semiconductors are governed by excitons in different spin and valley configurations, providing versatile aspects for van der Waals heterostructures and devices. Here, we present experimental and theoretical studies of exciton energy splittings in external magnetic field in neutral and charged WSe_{2} monolayer and bilayer crystals embedded in a field effect device for active doping control. We develop theoretical methods to calculate the exciton gfactors from first principles for all possible spinvalley configurations of excitons in monolayer and bilayer WSe_{2} including valleyindirect excitons. Our theoretical and experimental findings shed light on some of the characteristic photoluminescence peaks observed for monolayer and bilayer WSe_{2}. In more general terms, the theoretical aspects of our work provide additional means for the characterization of single and fewlayer transition metal dichalcogenides, as well as their heterostructures, in the presence of external magnetic fields.
Introduction
Monolayer (ML) and bilayer (BL) transition metal dichalcogenides (TMDs) such as WSe_{2} represent semiconductor building blocks for novel van der Waals heterostructures. By virtue of sizable light–matter coupling governed by excitons^{1}, they exhibit versatile potential for applications in photonics and optoelectronics^{2,3}, optovalleytronics^{4,5}, and polaritonics^{6}. Most recently, the optical interface to TMDs has been instrumental for the observation of strongly correlated electron phenomena in twisted homobilayer and heterobilayer moiré systems^{7,8,9}.
The key to further developments of van der Waals heterostructures for fundamental studies and practical devices using TMD MLs and BLs is the detailed understanding of their optical properties. While substantial understanding of zeromomentum excitons in ML and BL WSe_{2} has been established^{1}, some important aspects remain subject of debate^{10}. This holds, in particular, for valleydark excitons with finite centerofmass momentum that escape direct optical probes by virtue of momentum mismatch with photons. In MLs, they complement the notion of intravalley spinbright and spindark excitons^{1}, and they entirely dominate the photoluminescence (PL) from the lowestenergy states in native homobilayers of WSe_{2} (ref. ^{11}).
Within the realm of optical spectroscopy techniques, magnetospectroscopy provides means for studying the exciton spin and valley degrees of freedom. Magnetoluminescence experiments on ML WSe_{2} in the presence of outofplane and inplane magnetic fields, for instance, have been used to quantify the valley Zeeman splitting of bright excitons^{12,13,14,15,16,17} or to brighten spindark excitons^{18,19,20}, respectively. To date, however, a rigorous assignment of exciton gfactors to intervalley excitons with finite momentum falls short mainly due to the lack of theoretical predictions^{10}.
In this work, we develop theoretical methods to evaluate gfactors for excitons in different spin and valley configurations, and provide explicit values for WSe_{2} ML and BL excitons composed from electron and hole states away from high symmetry points of the first Brillouin zone. Our calculations go beyond the existing tightbinding models by employing the density functional theory (DFT). We compare our theoretical results with experimentally determined gfactors of intravalley excitons and use them to interpret ambiguous peaks in the PL spectra of ML and BL WSe_{2} attributed to intervalley excitons. The technique can be expanded to other materials like WS_{2} with similarly complex spectra^{21} and large gfactors of ML^{17} and BL^{22} excitons.
Results
Magnetoluminescence spectroscopy of chargecontrolled ML and BL WSe_{2}
In our experiments, we used a fieldeffect heterostructure based on an exfoliated WSe_{2} crystal with extended ML and BL regions encapsulated in hexagonal boron nitride (hBN). The device layout is shown schematically in Fig. 1a (see the Methods section for details) and the first Brillouin zone of ML and BL WSe_{2} with most relevant points in Fig. 1b. The sample was cooled down to 3.2 K, and the PL was probed as a function of voltagecontrolled doping with laser excitation at 1.85 eV and powers below the regimes of neutral and charged biexcitons^{23,24,25,26}. Magnetoluminescence experiments were performed in Faraday configuration with a bidirectional solenoid at magnetic fields of up to 9 T (see the Methods section for experimental details).
The evolution of the PL with the gate voltage is shown in Fig. 1c, d for representative spots of ML and BL regions, respectively. In Fig. 1c, the ML reaches the intrinsic limit at gate voltages <−5 V consistent with residual ndoping of the exfoliated crystal^{27,28}. The neutral regime is characterized by the bright exciton PL (X^{0}) at 1.72 eV and a series of redshifted peaks that we label as \({M}_{1}^{0}\), \({M}_{2}^{0}\), and \({M}_{3}^{0}\). None of these peaks with respective redshifts of 35, 60 and 75 meV from the bright exciton peak is to be attributed to the PL of dark excitons (D^{0}) with 42 meV redshift^{20,29,30}. In our sample, this feature is a rather weak shoulder at the lowenergy side of \({M}_{1}^{0}\). At positive gate voltages, the ML is charged with electrons and thus exhibits the characteristic signatures of a bright trion doublet (\({X}_{1}^{}\) and \({X}_{2}^{}\)) split by the exchange energy of ~6 meV (ref. ^{28}), the dark trion (D^{−}) at 28 meV redshift from \({X}_{1}^{}\) (refs. ^{31,32,33,34}), and a series of lowenergy peaks dominated by the peak \({M}_{1}^{}\) at 44 meV redshift^{33,34}.
The PL from the BL region in Fig. 1d is characterized by a multipeak structure, >100 meV below X^{0}. It exhibits the same limits of charge neutrality and electron doping as a function of the gate voltage, consistent with the charging behavior of the ML in Fig. 1c. The BL peaks, labeled by an increasing subscript number with decreasing peak energy as \({B}_{1}^{0}\) through \({B}_{3}^{0}\) and \({B}_{1}^{}\) through \({B}_{3}^{}\) in the neutral and negative regime, respectively, correspond to phonon sidebands of neutral and charged momentumindirect excitons with a global redshift of 22 meV at about −7 V (ref. ^{11}) in Fig. 1d. According to the singleparticle band structure of BL WSe_{2} (refs. ^{35,36}), the fieldinduced electron concentration is accommodated at the conduction band edge by the six inequivalent Qvalleys. However, the nature of the hole states that constitute the lowestenergy momentumdark excitons as longlived reservoirs of phononassisted PL remains ambiguous. The energetic proximity of the valance band edge states at K and Γ in BL WSe_{2} (ref. ^{37}) renders QK and QΓ excitons and trions (composed from electrons at Q and holes at K or Γ) nearly degenerate, which in turn complicates their energetic ordering^{11}.
To examine the origin of the BL peaks and to shed light on the nature of ML peaks with ambiguous or partly controversial interpretation, we performed magnetospectroscopy in the two welldefined limits of charge neutrality and negative doping. The external magnetic field B was applied along the zaxis perpendicular to the sample. It removes the valley degeneracy and splits the exciton reservoirs by their characteristic Zeeman energies proportional to the exciton gfactor in WSe_{2} (refs. ^{12,13,14,15,16,17}). The respective polarizationcontrasting spectra recorded at −8 T under linearly polarized excitation (π) and circularly polarized detection (σ^{+} and σ^{−}) for the neutral (negatively charged) ML and BL are shown in the top (bottom) panel of Fig. 2a, b.
At each magnetic field, we quantified the experimental Zeeman splitting for every PL peak as the energy difference Δ = E^{+} − E^{−} between the peak energies E^{+} and E^{−} recorded under σ^{+} and σ^{−} polarized detection. The left and right panels of Fig. 3a, b show Δ as a function of the magnetic field for all peaks of the neutral and negatively charged ML and BL, respectively. The set of data derived from magnetoPL measurements was complemented for X^{0}, \({X}_{1}^{}\), and \({X}_{2}^{}\) by performing magnetoreflectivity under circular excitation and detection. The corresponding experimental exciton gfactors, obtained from Δ = gμ_{B}B as the slopes of best linear fits to the data in Fig. 3 scaled by the Bohr magneton μ_{B}, are summarized in Table 1. The negative sign of the gfactors reflects the energy ordering of exciton states that exhibit higher (lower) energy for σ^{−} (σ^{+}) polarized PL peaks at positive magnetic fields.
In ML WSe_{2}, the gfactors of both neutral and negatively charged excitons with the corresponding PL peaks X^{0}, D^{0}, \({X}_{1}^{}\), \({X}_{2}^{}\), and D^{−} have been established in previous experiments on a wide range of different samples^{12,13,14,15,16,17,18,19,20,31,32,33,34}. Our results for the bright exciton and the trion doublet in Table 1 agree well with these reports if we discard the magnetoluminescence result for \({X}_{1}^{}\) that is compromised by both a vanishingly small PL intensity at high magnetic fields and the relatively broad linewidth of 6 meV in our sample. Due to this inhomogeneous broadening, we are unable to track the dispersion of the relatively weak spindark exciton peak D^{0}, with gfactors ranging between 9.1 and 9.9 in previous reports^{20,31,33,34} nor its chiralphonon replicum with the same gfactor at 65 meV redshift from X^{0} (refs. ^{33,34,38}). The signature of the latter is overwhelmed in our spectra by the peak \({M}_{2}^{0}\) with 60 meV redshift and a gfactor of −12.9 ± 0.7 in agreement with values reported from samples with spectrally narrow PL^{33,34}. The redmost peak \({M}_{3}^{0}\) features the same gfactor within the experimental error bars as \({M}_{1}^{0}\), suggesting a joint reservoir as their origin. The negatively charged trion D^{−} was reported to have the same gfactor as its neutral counterpart^{31,32,33,34}, whereas we determine −12.2 ± 0.1. The agreement with previous reports is better for the peak \({M}_{1}^{}\) with a gfactor of −9.0 ± 0.1 that is supposed to be a phonon sideband of D^{−} (refs. ^{33,34}). The latter studies also reported an intense PL peak between \({M}_{1}^{}\) and D^{−} with a remarkably small gfactor of −4.1 (ref. ^{33}) and −3.4 (ref. ^{34}). This peak of unidentified origin is missing in our spectra from the negative doping regime.
There are other peaks in ML WSe_{2} without conclusive assignment, and in particular \({M}_{1}^{0}\) has received controversial interpretation as phononassisted PL from virtual trions^{39}, phonon sidebands of momentumdark Qexcitons^{21}, or zerophonon PL of finitemomentum excitons in spinlike configuration^{34} that we denote as \({K}_{L}^{\prime}\). Due to the lack of theory for the gfactors of excitons with finite centerofmass momentum, the task of confronting the competing hypotheses with the characteristic valley Zeeman splittings of controversial ML peaks has remained elusive. The same shortcoming holds for both neutral and charged BL excitons with finite centerofmass momentum. To shed additional light on the nature of PL peaks in both ML and BL WSe_{2}, we calculate in the following the gfactors for excitons in different spin and valley configurations from DFT.
Ab initio calculations of exciton gfactors
We consider a crystal electron in a Bloch state \({\psi }_{n{\bf{k}}}({\bf{r}})={S}^{1/2}\exp ({\rm{i}}{\bf{kr}}){u}_{n{\bf{k}}}({\bf{r}})\) with energy E_{nk}, where n is the band number, k is the wave vector, u_{nk}(r) is the periodic Bloch amplitude, and S is the normalization area. In the presence of a weak perturbation by a static magnetic field B, the firstorder correction to the electron energy is proportional to B and given by^{40}:
where μ_{B} = ∣e∣ℏ/(2m_{0}c) is the Bohr magneton, e and m_{0} are the charge and mass of the free electron, ℏ is the Planck constant, and c is the speed of light. The expression in square brackets is usually called the effective magnetic moment^{41,42}, which contains both spin and orbital contributions. In particular, the first term is proportional to the free electron Landé factor g_{0} ≃ 2 and the spin angular momentum s = σ/2, where σ denotes the Pauli matrix.
The second term, \({{\bf{L}}}_{n}({\bf{k}})=\left\langle {\psi }_{n{\bf{k}}}({\bf{r}})\right{\bf{L}}\left{\psi }_{n{\bf{k}}}({\bf{r}})\right\rangle\), is the orbital angular momentum with the operator L = ℏ^{−1}[r × p]. To obtain its matrix elements, one can reduce the calculation to the interband matrix elements of the space coordinate operator r^{14,41,42,43}:
where m is the sum over all bands with energy E_{nk} but the band of interest, and \({{\boldsymbol{\xi }}}_{nm}({\bf{k}})={\rm{i}}\left\langle {u}_{n{\bf{k}}}({\bf{r}})\right\partial /\partial {\bf{k}}\left{u}_{m{\bf{k}}}({\bf{r}})\right\rangle\) is the interband matrix element of the crystal coordinate operator.
In the following, we restrict our analysis to the orientation of the magnetic field along the zaxis and define the electron Zeeman splitting as the difference between the energy of the electron state with wave vector +k and spin projection +s along the zaxis and the state with −k and −s as:
Thus the electron gfactor of Bloch electrons in the nth band can be written as:
with +(−) for s = +1/2 (−1/2) corresponding to spin up (down) projections along z denoted as ↑ (↓), and the explicit expression for the zcomponent of the orbital angular momentum:
where \({\xi }_{mn}^{(\pm )}=({\xi }_{mn}^{(x)}\pm {\rm{i}}{\xi }_{mn}^{(y)})/\sqrt{2}\).
To calculate the contributions of the conduction (c) band electron with k_{c}, s_{c} and the hole (h) with k_{h}, s_{h} to the exciton gfactor, we neglect electron–hole Coulomb interactions^{14,44}. In this case, the exciton Zeeman splitting simplifies to the sum of the Zeeman splittings of the electron and the hole. Using time reversal symmetry that relates the spin and wave vector of the hole to the corresponding spin and wave vector of the empty electron state in the valence (v) band (s_{h} = −s_{v} and k_{h} =−k_{v}), we obtain the exciton gfactor as
Finally, by reference to the valence band electron with k_{v} = K or Γ with spinup projection s_{v} = +1/2, we discriminate spinlike (L) excitons (with s_{c} = s_{v}) from spinunlike (U) excitons (with s_{c} = −s_{v}). Their respective exciton gfactors are given by:
Using Eqs. (7) and (8), we calculate in the following the exciton gfactors from the orbital angular momenta L_{c}(k_{c}) and L_{v}(k_{v}) of conduction and valence bands obtained from Eq. (5) within DFT calculations on the Γcentered \(\overrightarrow{k}\) grid of 12 × 12 divisions with 300 (600) bands (see the Methods section for details of DFT calculations). In Fig. 4a, b, we show the convergence of the orbital angular momenta L_{n}(k) within our ML and BL calculations as a function of the number of bands taken into account in the sum of Eq. (5). For the ML, Fig. 4a shows the results for the topmost valence band state v at K (blue solid line) and the highest valence band state v at Γ (gray solid line), as well as the two lowest conduction band states c and c + 1 at K and Q (red and black solid and dashed lines). As the BL bands are doubly degenerate, each kpoint of the Brillouin zone has at least two bands with L_{n}(k) = L_{n+1}(k) or L_{n}(k) = L_{n−1}(k). For the BL in Fig. 4b, we consider the same kpoints as for the ML and show the corresponding bands where the orbital angular momenta have the same sign as in the ML case of Fig. 4a.
For the orbital angular momenta of these states, convergence is observed above 275 and 550 bands in the case of ML and BL in Fig. 4a, b, respectively, with the factor of two difference related to the doubled number of atoms in BL calculations. We note that the values for the valence band states at Γ must vanish by symmetry arguments, whereas our numerical calculations yield ±0.01 for both ML and BL. This marginal discrepancy is due to a finite number of bands taken into account and can be used to estimate the precision of our numerical calculations. The corresponding bound on the absolute error of the exciton gfactors from DFT, given explicitly in Table 2 for selected exciton configurations, is thus in the order of ±0.05.
As evident from Fig. 4a, particular bands make decisive contributions to the gfactor. To discuss this behavior for the ML case in more detail, we consider the 24th and 26th bands that correspond to the highest valence band (v) and the second conduction band (c + 1), respectively, and give rise to largest mutual contributions in the gfactors. This is expected according to Eq. (5), where the orbital angular momentum is proportional to the product of the interband matrix elements, which in turn are largest for the fundamental Aexciton transition X^{0} between the 24th and 26th bands. Similar arguments apply for the mutual contributions of the 23rd and 25th bands to the gfactor of Bexcitons. It is also instructive to note the different dependencies of the orbital momenta for the two lowest conduction bands (L_{c} and L_{c+1}) and the top valence band (L_{v}) on the number of bands included. In Fig. 4a, L_{c} and L_{c+1} exhibit jumps at 23rd and 24th bands, respectively, and then increase only marginally. In contrast, L_{v} in Fig. 4a increases nearly monotonously beginning from m ~ 30 on. A closer inspection shows that for m > 26, the sign of the square bracket in Eq. (5) alternates with increasing m, and the terms of comparable absolute values therefore cancel each other for both L_{c} and L_{c+1}. For L_{v}, on the other hand, the absolute values of the positive terms systematically exceed the negative terms and thus L_{v} continues to grow with increasing band number. Further analysis will be required to understand this behavior in more detail.
The DFT results for L_{n}(k) within the first Brillouin zone are shown in Fig. 5. Since spin–orbit effects were included at the DFT level, it is instructive to show both spin–orbit split highest valence bands (v and v − 1) and lowest conduction bands (c and c + 1). With the matrix elements of the orbital angular momenta of the valence and conduction bands in Fig. 5, it is straight forward to calculate the gfactors of the lowestenergy ML excitons in various configurations. In Table 2, we list the gfactors obtained from our DFT results for excitons in different configurations of valleys (k_{c}, k_{v}) and spins (s_{c}, s_{v}, with ↑ or ↓ projection along z).
In the top block of Table 2, we list excitons with the hole at K and the electron at K or \(K^{\prime}\) in spinlike and spinunlike configurations with short exciton notation for zeromomentum bright and dark neutral excitons X^{0} and D^{0} and their finitemomentum counterparts \({K}_{L}^{\prime}\) and \({K}_{U}^{\prime}\). The block below shows the results for the spinlike and spinunlike Qexcitons with the electron in Q and the hole in K, followed by two blocks without short exciton notation for momentumindirect excitons composed from electrons in K or Q and holes in Γ. Note that the sign of the gfactor can be determined without further assumptions only for X^{0} with established valleycontrasting dipolar selection rules. For Zeemansplit momentumindirect excitons, on the other hand, the sign will depend on the symmetry of the actual phonons involved in phononassisted PL^{34}. This is analogous to the case of spindark excitons D^{0} with linearly polarized inplane zerophonon emission^{30} contrasted by circularly polarized PL sidebands of the same reservoir mediated by chiral phonons^{33,45}. In principle, not only the gfactor signs but also the absolute values of the respective PL peaks should be distinct due to different exciton–phonon coupling not accounted for in our model. However, we expect such higherorder corrections to be well below the resolution of our measurements.
Discussion
First, we discuss the results of our calculations for excitons in ML WSe_{2}. The gfactor of −4.0 from our DFT model is in excellent agreement with the experimental value of −4.1 for X^{0} (refs. ^{12,13,14,15,16,17}). The good agreement in the gfactor of spindark excitons with g ≃ 9.4 in experiment^{20} and 10.1 in DFT provides further confidence in our model. According to our calculations, the states \({K}_{L}^{\prime}\) and \({K}_{U}^{\prime}\), which are the momentumindirect counterparts of X^{0} and D^{0}, exhibit different gfactors with large values of 13.6 and 19.6, respectively. The gfactors of Qmomentum excitons (9.2–14.5) are similar to those of D^{0} and \({K}_{L}^{\prime}\), whereas excitons with the hole at Γ are predicted to have rather small gfactors (<5.8) except for the spinunlike configuration with \(K^{\prime}\) electron (9.8). As expected, the gfactors of intralayer excitons in BL WSe_{2} are close to the values of the corresponding ML excitons^{46}. In addition to intralayer excitons, the BL hosts interlayer counterparts (e.g., intralayer Q_{L} and interlayer \({Q}_{L}^{\prime}\), intralayer Q_{U} and interlayer \({Q}_{U}^{\prime}\), so on) that exhibit the same gfactors within our model, which neglects Coulomb corrections for intralayer and interlayer excitons.
By providing explicit gfactor values for momentumindirect excitons, our DFT results complement the experimental observations in ML and BL WSe_{2}. In the framework of neutral MLs, however, they do not resolve the ambiguity between the two competing explanations of the peak \({M}_{1}^{0}\). The assignment of the peak as a phonon sideband of Qmomentum excitons^{21}, on the one hand, is consistent with the gfactors of 9.2 and 14.5 for Q_{L} and \({Q}_{U}^{\prime}\) states in Table 2 (note that Q_{U} and \({Q}_{L}^{\prime}\) excitons, 250 meV above degenerate Q_{L} and \({Q}_{U}^{\prime}\) states^{47}, are irrelevant in this context) and the structured peak \({M}_{1}^{0}\) in Fig. 2 with a gfactor of 11.5. On the other hand, the interpretation of the peak as direct PL emission by momentumdark \({K}_{L}^{\prime}\) excitons^{34} is also consistent with the theoretical gfactor of 13.6 from DFT. Our DFT results also identify KΓ and QΓ with small gfactors as potential candidates to explain the bright PL peak between \({M}_{1}^{}\) and D^{−} in the negatively charged regime of highquality samples with narrow spectra^{33,34}.
For the neutral BL, our results help to rule out QΓ excitons and suggest spinunlike interlayer QK and intralayer \(Q^{\prime} K\) exciton reservoirs rather than \(K^{\prime} \Gamma\) as a joint origin of phonon sidebands \({B}_{1}^{0}\), \({B}_{2}^{0}\), and \({B}_{3}^{0}\) (ref. ^{11}). Whereas a detailed assignment of the neutral BL peaks to the specific reservoirs and phonon sidebands is yet to be developed, the values of the exciton gfactors in the charged regime can be understood, as in the ML case, by regarding the additional electron in the charged complex simply as a spectator to the Zeeman effect of the neutral finitemomentum exciton reservoir.
In summary, our work provides exciton gfactors for neutral and charged ML and BL WSe_{2} from both experiment and DFT. For ML WSe_{2}, the gfactors obtained from firstprinciples calculations are in excellent quantitative agreement with previous reports and complement these studies by providing theoretical gfactors for momentumindirect excitons in different configurations of spins and valleys. For BL WSe_{2}, our work adds insight into the origin of PL peaks on the basis of theoretical gfactor values. In the broad context of research on layered semiconductors and their applications, the theoretical aspects of our work provide guidelines for magnetooptical studies of singlelayer TMDs, homobilayer or heterobilayer systems, and other realizations of TMDbased van der Waals heterostructures.
Note: During the submission of our manuscript, we became aware of three related works on the theory of exciton gfactors in TMD MLs and heterostructures from first principles^{48,49,50}.
Methods
Experimental methods
The fieldeffect heterostructure consisted of an exfoliated WSe_{2} crystal (HQ Graphene) with extended ML and BL regions encapsulated in hBN (NIMS). To control the charge doping, the crystal was contacted by a gold electrode deposited on a 50nmthick thermal silicon oxide layer of a pdoped silicon substrate. With the electrode grounded, a gate voltage applied to the highly doped silicon was used to control the doping level in ML and BL WSe_{2}. The sample was mounted in a cryogenic confocal microscope and cooled down in a closedcycle magnetocryostat (attocube systems, attoDRY1000) with a base temperature of 3.2 K. The PL was excited at 1.85 eV with a few μW power of a continuouswave laser diode focused to the diffractionlimited confocal excitation and detection spot of a lowtemperature apochromatic objective (attocube systems, LTAPO/VISIR/0.82), dispersed with a monochromator (Roper Scientific, Acton SP2500), and detected with a nitrogencooled CCD (Roper Scientific, Spec 10:100BR/LN). Magnetoluminescence experiments were performed in Faraday configuration with a bidirectional solenoid at magnetic fields of up to 9 T.
DFT calculations
DFT calculations were performed within the generalized gradient approximation with the PBEsol exchangecorrelation functional^{51} as implemented in the Vienna ab initio simulation package. The van der Waals interactions were considered with the DFTD3 method with Becke–Johnson damping^{52,53}; the spin–orbit interaction was included at all stages. Elementary cells with a vacuum thickness of 30 Å were used in order to minimize interactions between periodic images. The atomic positions were relaxed with a cutoff energy of 400 eV until the change in the total energy was <10^{−6} eV. The band structure of ML (BL) was calculated on the Γcentered \(\overrightarrow{k}\) grid of 12 × 12 divisions with 300 (600) bands.
Data availability
The data that support the findings of this study are available from the corresponding authors upon reasonable request.
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Acknowledgements
The authors thank M. M. Glazov, T. Deilmann, and P. Hawrylak for fruitful discussions. This research was funded by the European Research Council (ERC) under the Grant Agreement No. 772195 and the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy EXC2111390814868. N.V.T. acknowledges support from the Federal Target Program for Research and Development of the Ministry of Science and Higher Education of the Russian Federation (No. 14.587.21.0047, identifier RFMEFI58718X0047). S.Y.K. acknowledges support from the Austrian Science Fund (FWF) within the Lise Meitner Project No. M 2198N30. A.S.B. has received funding from the European Union’s Framework Programme for Research and Innovation Horizon 2020 (2014–2020) under the Marie SkłodowskaCurie Grant Agreement No. 754388 and from LMU Munich’s Institutional Strategy LMUexcellent within the framework of the German Excellence Initiative (No. ZUK22). A.H. acknowledges support from the Center for NanoScience (CeNS) and the LMUinnovativ project Functional Nanosystems (FuNS). K.W. and T.T. acknowledge support from the Elemental Strategy Initiative conducted by the MEXT, Japan, Grant Number JPMXP0112101001, JSPS KAKENHI Grant Numbers JP20H00354 and the CREST (JPMJCR15F3), JST. Open access funding provided by Projekt DEAL.
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J.F., J.L., V.F., and M.F. performed experiments on samples with highquality hBN provided by K.W. and T.T.; J.F., J.L., A.S.B., and A.H. analyzed the data; A.S.B. developed the theory of exciton gfactors from first principles; N.V.T., S.Y.K., and A.S.B. performed numerical calculations; J.F., A.S.B., and A.H. prepared the figures and wrote the manuscript. All authors commented on the manuscript.
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Förste, J., Tepliakov, N.V., Kruchinin, S.Y. et al. Exciton gfactors in monolayer and bilayer WSe_{2} from experiment and theory. Nat Commun 11, 4539 (2020). https://doi.org/10.1038/s41467020180191
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DOI: https://doi.org/10.1038/s41467020180191
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