Abstract
Bound electron–hole pairs called excitons govern the electronic and optical response of many organic and inorganic semiconductors. Excitons with spatially displaced wave functions of electrons and holes (interlayer excitons) are important for Bose–Einstein condensation, superfluidity, dissipationless current flow, and the lightinduced exciton spin Hall effect. Here we report on the discovery of interlayer excitons in a bulk van der Waals semiconductor. They form due to strong localization and spinvalley coupling of charge carriers. By combining highfield magnetoreflectance experiments and ab initio calculations for 2HMoTe_{2}, we explain their salient features: the positive sign of the gfactor and the large diamagnetic shift. Our investigations solve the longstanding puzzle of positive gfactors in transition metal dichalcogenides, and pave the way for studying collective phenomena in these materials at elevated temperatures.
Introduction
Monolayers of transition metal dichalcogenides (TMDCs) such as MoS_{2} are direct band gap semiconductors compared with their indirect gap multilayer counterparts. The hexagonal crystal structure gives rise to six energetically equivalent extrema in momentum space, which are alternatingly denoted as K^{+} and K^{−} valleys. Their electronic and optical properties are governed by strongly bound excitons, which exist at room temperature^{1}. In monolayers, excitons are naturally confined to two dimensions (2D), i.e. a single layer. In a bilayer or a bulk system, a strong spinlayer locking exists^{2,3,4,5}. Owing to this effect together with a weak van der Waals interlayer interaction, strongly confined electrons and holes might exist within different layers of a multilayer TMDC (interlayer excitons^{6,7,8,9}), even without the requirement of a spacer layer^{10, 11}. Interlayer excitons are potential candidates for unraveling a wealth of interesting physical phenomena, such as BoseEinstein condensation, high temperature superfluidity, dissipationless current flow, and the lightinduced exciton spin Hall effect^{10,11,12,13,14,15}. Interlayer excitons have been successfully created in coupled GaAs quantum wells at cryogenic temperatures^{16}. In contrast, TMDCs hold the promise for observing aforementioned phenomena at much higher temperatures^{10, 11}. In TMDCs, interlayer excitons have been created in artificial heterostructures by stacking monolayers of two different TMDC materials on top of each other^{6,7,8,9}. However, the present fabrication procedures are nontrivial. The rotation angle between the individual layers has to be precisely set^{8}. Furthermore, coupling of the layers critically depends on the quality of the formed interfaces, which is difficult to control^{17}.
Here, we show that interlayer excitons exist in an inversionsymmetric bulk TMDC semiconductor, 2HMoTe_{2}, which are accompanied by an inherent spinlayer locking. MoTe_{2} is a promising TMDC for novel electronic devices, because it has a band gap similar to silicon^{18,19,20} and extends the spectral range of TMDCs to the infrared region. Only recently, excitons in monolayer MoTe_{2} have been investigated using magnetooptical spectroscopy^{21}.
Results
Optical spectroscopy of excitonic resonances
The microreflectance contrast (μRC) spectrum of a mechanically exfoliated 40 nmthick bulklike 2HMoTe_{2} crystal on a SiO_{2}(80 nm)/Si substrate is presented in Fig. 1b (see Methods section for details). Four prominent resonances are identified: \({\rm X}_{\rm{A}}^0\), \({\rm X}_{\rm{A}}^{\rm{*}}\), X _{IL}, and \({\rm X}_{\rm{B}}^0\) with corresponding transition energies \(E_{\rm{A}}^0 = 1.131 \pm 0.001\) eV, \(E_{\rm{A}}^* = 1.155 \pm 0.002\) eV, \({E_{{\rm{IL}}}} = 1.183 \pm 0.003\) eV, and \(E_{\rm{B}}^0 = 1.42 \pm 0.01\) eV, respectively. \({\rm X}_{\rm{A}}^0\) and \({\rm X}_{\rm{B}}^0\) correspond to the wellknown A and B intralayer excitons with electron and hole confined to the same layer of bulk MoTe_{2} (Fig. 1a)^{18}. As explained in the next sections, we assign \({\rm X}_{\rm{A}}^{\rm{*}}\) and X _{IL} to the ‘2slike’ excited state of the intralayer A exciton and the ground state interlayer exciton, respectively.
Figure 1c depicts helicityresolved magnetoμRC spectra of MoTe_{2} under magnetic fields of up to B = 29 T. The σ^{+} and σ^{−} components of all excitonic resonances exhibit a Zeeman splitting ΔE, which increases linearly with rising magnetic field (Figs. 2a–d). \({{\Delta}}E = {E_{{\sigma ^ + }}}\!\!{E_{{\sigma ^  }}} = {g_{\rm{X}}}{\mu _{\rm{B}}}B\), where \({E_{{\sigma ^ + }}}\) and \({E_{{\sigma ^  }}}\) are the excitonic transition energies for the two circular polarizations, g _{X} is the excitonic gfactor, and μ _{B} is the Bohr’s magneton. The excitonic resonances \({\rm X}_{\rm{A}}^0\) and \({\rm X}_{\rm{A}}^{\rm{*}}\) have negative gfactors of similar magnitude: \(g_{\rm{A}}^0 =  2.4 \pm 0.1\) and \(g_{\rm{A}}^{2{{s}}} =  3 \pm 0.6\), respectively. The similarity of these values suggests a common origin. Strikingly, the effective gfactor of X _{IL} is of opposite sign, with g _{IL} = +4 ± 0.5, indicating a different physical mechanism. Each excitonic resonance exhibits a diamagnetic shift depending on B ^{2}: \({{\Delta}}E_{{\rm{dia}}}^{\rm{X}} = \left( {{e^2}{{\left\langle {{r_{\rm{X}}}} \right\rangle }^2}{B^2}} \right)/8\mu _{\rm{X}}^{\rm{*}}\), with e the electronic charge, \(\left\langle {{r_{\rm{X}}}} \right\rangle \) the excitonic rootmeansquare (RMS) radius, and \(\mu _{\rm{X}}^*\) the reduced mass^{22}. \({{\Delta}}E_{{\rm{dia}}}^{\rm{X}}\) is the deviation of the exciton’s mean transition energy \(\left( {{E_{{\sigma ^ + }}} + {E_{{\sigma ^  }}}} \right)/2\) for \(B \,\ne\, 0\) from the zerofield value. Figure 2e reveals that \({\rm X}_{\rm{A}}^0\) shows a significantly smaller diamagnetic shift (\({{\Delta}}E_{{\rm{dia}}}^0 = 0.2 \pm 0.1{{\rm{T}}^{  1}}\)) compared with those for \({\rm X}_{\rm{A}}^{\rm{*}}\) (\({{\Delta}}E_{{\rm{dia}}}^* = 1.6 \pm 0.1\) μeVT^{−1}) and \({X_{{\rm{IL}}}}\left( {{{\Delta}}E_{{\rm{dia}}}^{{\rm{IL}}} = 2.3 \pm 0.1\,{\rm{\mu e}}V\,{{\rm{T}}^{  1}}} \right)\). This observation points towards relatively large RMS radii and smaller excitonic binding energies for \({\rm X}_{\rm{A}}^{\rm{*}}\) and X _{IL} compared with \({\rm X}_{\rm{A}}^0\).
Identification of interlayer excitons
To confirm the identification of the observed resonances, we calculate the quasiparticle band structure and the optical absorption spectrum for bulk 2HMoTe_{2} using the GW–Bethe–Salpether equation (BSE) method within the LDA + GdW approximation^{23} (see Methods section for details). The quasiparticle band structure is represented by solid lines in Fig. 3a. Our calculated valence bands (VBs) are in good agreement with the results of angleresolved photoemission spectroscopy measurements performed for 2HMoTe_{2}.^{24} The calculated direct and indirect band gaps are 1.39 and 1.09 eV. A flat dispersion along the k _{ z } (or K–H) direction around the K point implies a large outofplane effective mass and strong inplane carrier confinement, which demonstrates the 2D character of the charge carriers even for bulk material. In the calculated absorption spectrum (Fig. 3b), we find that the first three lowenergy optical transitions originate from the K point of the Brillouin zone. The first resonance at 1.24 eV corresponds to the intralayer exciton \({\rm X}_{\rm{A}}^0\) (binding energy \(E_{\rm{b}}^0 = 150\) meV), which is in good agreement with the experimental transition energy of 1.131 eV. Figure 3c shows the spatial distribution for \({\rm X}_{\rm{A}}^0\), obtained by fixing the positive hole in the vicinity of a Mo atom and calculating the probability distribution of the electron around it. Strikingly, this exciton with a calculated RMS radius \(\left\langle {{r_0}} \right\rangle \) = 1.2 nm is strongly confined (>90%) within a single layer, demonstrating its 2D excitonic character. From the experimentally observed value for the diamagnetic shift (see methods), we determine \(\left\langle {{r_0}} \right\rangle \) = 1.7 ± 0.4 nm, which is in good agreement with the calculated value. The next two resonances at higher energies, 1.3 and 1.35 eV (Fig. 3b), are identified as the \({\rm X}_{\rm{A}}^{\rm{*}}\) and X_{IL} features observed in Fig. 1b. The calculated relative oscillator strengths of these three transitions in increasing order of energy are 1, 0.07, and 0.51, respectively. Experimentally, we obtain 1, 0.01, and 0.19, which reasonably agrees with the calculated trend. The spatial distribution of the resonance at 1.34 eV has a ‘2slike’ excitonic character (top view in Fig. 3d), verifying its assignment as \({\rm X}_{\rm{A}}^{\rm{*}}\). The RMS radius obtained from the experimentally derived diamagnetic shift is \(\left\langle {{r_*}} \right\rangle \) = 4.8 ± 0.4 nm. \({\rm X}_{\rm{A}}^{\rm{*}}\) is also spatially extended to the third layer (Fig. 3d). This is in qualitative agreement with the radial probability distribution of the 2 s orbital of the H atom, where one expects a large extent of the wave function (see Supplementary Fig. 3). As we find that both \({\rm X}_{\rm{A}}^0\) and \({\rm X}_{\rm{A}}^{\rm{*}}\) are related to the same interband transitions, we expect their gfactors to be similar, which is indeed observed in the experiment. It must be noted that in the GWBSE approach it is difficult to converge the details of the absorption spectrum near the direct band gap at 1.386 eV. Both the dense Rydberg series just below the band gap, as well as the onset (at the gap) of a Coulombenhanced continuous spectrum, require many more kpoints than used in our study and could better be described in an appropriate Wannier exciton model with appropriate parameters. Therefore, we concentrate on the wellconverged discrete exciton states below 1.37 eV.
Strikingly, for the 1.35 eV resonance (X _{IL}), with the hole fixed in one layer, the probability of finding an electron within the same layer is much smaller ( ~ 19%) than in the neighboring layers ( ~ 57%) (Fig. 3e), demonstrating its interlayer nature. It is weakly bound (calculated \(E_{\rm{b}}^{{\rm{IL}}} = 60\) meV) compared with the intralayer exciton. In addition, the RMS radii obtained for X _{IL} from our ab initio calculations and the one using the diamagnetic shift are \(\left\langle {{r_{{\rm{IL}}}}} \right\rangle \) ~ 2.0 nm and \(\left\langle {{r_{{\rm{IL}}}}} \right\rangle = 5.0 \pm 0.4\,{\rm{nm}}\), respectively. Although these values differ significantly, possibly due to errors in the computed exciton reduced mass and in the exciton wave function with respect to the used kmesh (see Methods), they overall point towards a larger spread of this state in realspace. This is consistent with the weakening of the Coulomb binding for a large spatial separation of electron and hole for X_{IL} ^{10}. Nevertheless, the binding energy of the bulk TMDC interlayer exciton \(E_{\rm{b}}^{{\rm{IL}}}\) is almost an order of magnitude larger compared with those in coupled quantum wells^{25}, which renders the existence of bulk TMDC interlayer excitons possible at elevated temperatures (Supplementary Fig. 6) or even at room temperature.
Discussion
To understand the magnetooptical transitions of the intralayer and interlayer excitons, we derive the interband transition selection rules for circularly polarized light by calculating the circular optical dipole matrix elements between the bands^{2} (see Supplementary Fig. 4). The wellestablished case of monolayer 2HMoTe_{2} ^{21} is presented in the left panel of Fig. 4a. The spin of the electron or hole is locked to the valley and is conserved during an optical transition^{26}. For B = 0, the σ^{+} and σ^{−} transitions occur in the K^{+} and K^{−} valleys, respectively, which are degenerate in energy. For B > 0 (right panel of Fig. 4a) the valley degeneracy is lifted^{21}. The energy of the σ^{+} (σ^{−}) transition decreases (increases), giving rise to a negative excitonic gfactor.
For a bilayer or bulk TMDC, every band is doubly degenerate with opposite spin configurations due to inversion and timereversal symmetry. For the 2H polytype, adjacent layers are rotated by 180° (Fig. 1a), which results in an inversion of the spin orientation when going from one layer to the next (“spinlayer locking”)^{2,3,4,5} in all bands. For instance, for the K ^{+} valley, the VB and the CB1 for layer 1 (layer 2) have spin down (up) character (top row of Fig. 4b). The 2D confinement of excitons within the individual neighboring layers of bulk material results in spatially direct (intralayer) transitions (vertical arrows in Fig. 4b). These transitions are similar to the monolayer case, except that for a given valley, the σ^{+} and σ^{−} transitions take place in adjacent layers (see Supplementary Fig. 5a, e for calculated transition selection rules). The gfactor of the intralayer transitions in bulk has a negative sign and similar magnitude (see Fig. 4c for \({\rm X}_{\rm{A}}^0\)), reaffirming their monolayerlike 2D character. In contrast, the highenergy spinorbitsplit conduction band CB2 participates in the formation of interlayer transitions (diagonal arrows in Fig. 4b). The carrier spins for the σ^{+} and σ^{−} interlayer transitions are reversed when compared with the corresponding intralayer transitions. The calculated transition selection rules for this case are presented in Supplementary Fig. 5d, h. For B > 0, the interlayer σ^{+} transitions increase in energy compared with the B = 0 case, whereas the σ^{−} transitions decrease in energy (Fig. 4b, right panel) giving rise to a positive gfactor (see Fig. 4c for X_{IL}). The contribution from the orbital magnetic moment to interlayer excitons is expected to be similar to intralayer excitons, but with opposite sign, i.e., + 4, which is consistent with our measured value of + 4 ± 0.5.
In summary, we have performed magnetoreflectance contrast spectroscopy under high magnetic fields and GWBSE ab initio calculations to unveil interlayer excitons in the bulk van der Waals semiconductor 2HMoTe_{2}. The interlayer excitons form due to strong localization and spinvalley coupling of the charge carriers and are characterized by gfactors with opposite sign and large diamagnetic shift compared with intralayer excitons. Interestingly, unexplained positive gfactors have been reported also for bulk MoSe_{2} and MoS_{2} using magnetic circular dichroism spectroscopy^{27}, which suggests that interlayer excitons might also exist in these bulk TMDCs. Indeed, our GWBSE calculations indicate that bulklike MoS_{2} and MoSe_{2} also exhibit interlayer excitons similar to MoTe_{2} (Supplementary Fig. 7).
Methods
Experiment
For micromagnetoreflectance (μRC) measurements, an optical fiberbased lowtemperature probe is placed inside a 50 nmdiameter bore of a resistive magnet, which generates magnetic fields up to 29 T at the center of the magnet. A 50 μmdiameter optical fiber carries unpolarized light to the sample, which is placed on x–y–z piezonanopositioners. The light used for excitation is focused on the sample using an aspheric lens of focal length 3.1 mm and a numerical aperture of 0.68. Light reflected from the sample is circularly polarized in situ using a combination of a quarter wave plate and a polarizer. The polarized light is collected using a 200 μm diameter optical fiber and detected with a monochromator and an InGaAs array. The measurements are performed using a fixed circular polarization configuration of the quarter wave platepolarizer assembly, whereas the magnetic field is reversed to acquire spectral information corresponding to the other direction of circular polarization due to timereversal symmetry^{28}. The reflectance contrast spectrum C(λ) is defined as \(C\left( \lambda \right) = \left[ {R\left( \lambda \right)  {R_0}\left( \lambda \right)} \right]/\left[ {R\left( \lambda \right) + {R_0}\left( \lambda \right)} \right]\), where R(λ) and R _{0}(λ) denote the reflectance of the MoTe_{2} crystal on the substrate and the bare substrate, respectively, for a given wavelength λ. μRC spectral lineshapes are modeled using a transfer matrix approach for obtaining the transition energies (solid lines in Fig. 1c, d). The excitonic contribution to the dielectric response function is calculated by a Lorentz oscillator model^{29}
where n _{b}+ik _{b} is the background complex refractive index of MoTe_{2} in the absence of excitonic effects, and is assumed to be equal to that of bulk material^{30}. E _{0} is the transition energy, A is the oscillator strength parameter, and γ is the full width at half maximum line width. The oscillatory features in the data of Fig. 2 are due to Faraday rotation effects of a small linearly polarized component (<10%) in detection^{21}.
Calculations
In the density functional theory calculations, we employ a basis of three shells of localized Gaussian orbitals with s, p, d and s* symmetry and decay constants (in \(a_B^{  2}\)) of [0.14, 0.4, 1.1] for Te and [0.16, 0.51, 1.42] for Mo along with a kmesh of 12 × 12 × 3 points. The GW calculations are conducted in the GdW approximation to the selfenergy, where an atomresolved dielectric model function based on the randomphase approximation is used for representing the dielectric screening properties^{23}. We use the experimentally obtained lattice structure of 2HMoTe_{2} from ref. ^{24}. The energy cutoff of the auxiliary plane wave basis is set to 2.5 Ry (see Supplementary Fig. 1 for energy gap convergence as a function of cutoff energy and the kmesh). Figure 3b depicts the calculated absorption spectrum resulting from the solution of the BSE equation using a 30 × 30 × 4 kgrid in the first Brillouin zone and four valence and eight conduction bands (see Supplementary Fig. 2 for convergence of transition energies as a function of kmesh). The used meshes are identical for the electronhole interaction and the quasiparticle band structure so no interpolation is needed. Spin–orbit interaction is fully accounted for during all computational steps.
The energetic separation of the two resonances \({\rm X}_{\rm{A}}^{\rm{*}}\) and X _{IL} from \({\rm X}_{\rm{A}}^0\) are 24 and 52 meV in the experiment (Fig. 1b), whereas we find 100 and 110 meV in the calculation. The two main sources of discrepancy in the theoretical description (apart from the methodological error) are the limits of convergence of our calculation for such small energies, and experimental uncertainties in the determination of the lattice structure we used as input for our calculation, which is discussed as follows: (1) the numerical convergence limit, which depends mostly on the used kmesh density and the number of bands included in the BSE Hamiltonian, as represented by the Supplementary Figs. 1 and 2. The kmesh convergence for a 2slike state \({\rm X}_{\rm{A}}^{\rm{*}}\) is especially demanding due to a large spatial extent and internal structure of the wave function that needs to be well captured^{31}. (2) We use the experimental lattice structure^{24} for the covalent bonds as well as the interlayer separation, which is difficult to calculate within density functional theory (DFT) due to the van der Waals interaction. As an alternative, we have relaxed the covalent bonds within each layer and only use the experimental lattice constant for the layer separation. We found that the energetic separation between \(X_{\rm{A}}^0\) and \(X_{\rm{A}}^{\rm{*}}\) critically depends on the details of the lattice structure. For instance, the relaxed inplane lattice constant is merely 0.1% larger than the experimental one, but the energetic difference between \({\rm X}_{\rm{A}}^0\) and \({\rm X}_{\rm{A}}^{\rm{*}}\) reduces from 100 to 80 meV.
The RMS radius of the exciton is calculated using \(\sqrt {\left\langle {r_{\rm{X}}^2} \right\rangle } = {\left[ {\frac{{\left\langle {{\Phi ^{\rm{X}}}\left {{{\left( {{{\bf{r}}_{\bf{e}}}  {{\bf{r}}_{\bf{h}}}} \right)}^2}} \right{\Phi ^{\rm{X}}}} \right\rangle }}{{\left\langle {{\Phi ^{\rm{X}}}{\Phi ^{\rm{X}}}} \right\rangle }}} \right]^{\frac{1}{2}}}\), where \({\Phi ^{\rm{X}}}\) denotes the threedimensional electron distribution in real space (i.e., the exciton wave function with a fixed hole), and r _{ e }−r _{ h } is the distance between the electron and the hole. When evaluating the RMS radius on a grid, special care has to be taken since the electron distribution repeats itself, e.g., after 30 × 30 × 4 lattice vectors for a kmesh of 30 × 30 × 4. To avoid double counting, a cutoff radius for the evaluation in real space is introduced. The chargecarrier effective masses used for determining the Bohr radius from experimentally obtained diamagnetic shifts are calculated using the LDA + GdW band structure^{23}. This is done by fitting parabolas in an area of \(0.03\frac{\pi }{a}\) at the band extrema and calculating the band curvature. The values of the effective masses are \(m_{{\rm{CB}}1}^* = 0.83;m_{{\rm{CB}}2}^* = 0.44; m_{{\rm{VB}}}^* = 0.56\) (in units of electron mass), where CB1 and CB2 are the lower and higher energy spin–orbit–split conduction bands in Fig. 3, and VB is the topmost valence band at the K point. Although CB1 participates in the creation of intralayer excitons \({\rm X}_{\rm{A}}^0\) and \({\rm X}_{\rm{A}}^{\rm{*}}\), CB2 is involved in the formation of interlayer excitons X_{IL}. It must be noted that the used effective masses are an average over the effective mass in KГ and KM direction. The GdW approximation in the quasiparticle band structure, the fitting procedure and the mass averaging increase the uncertainty of the calculated effective masses and hence of the RMS radii derived from experiment.
In Fig. 4 we schematically present the circularpolarizationresolved optically allowed transitions after calculating the optical dipole matrix elements between the conduction and VBs. The calculations are performed for monolayer and bulk MoTe_{2}. The spin orientations of the bands are determined from the expectation values of the spin polarizations of the DFT wave functions. The solution of the BSE Hamilton operator provides the absorption spectrum with exciton energies, along with the wave function of every exciton. The real space wave functions for \({\rm X}_{\rm{A}}^0\), \({\rm X}_{\rm{A}}^{\rm{*}}\), and X _{IL} are depicted in Figs. 3c–e. The wave functions in kspace are used to determine which bands and kpoints contribute the most to the excitation at hand. For example the interlayer exciton X_{IL} involves a transition mainly at the K point between the (degenerate) topmost VB and the degenerate CB1 (see Fig. 3a).
The polarization of excitations between a valence and a conduction band can be obtained using the optical dipole matrix elements. We calculate the degree of circular polarization, η(k), which is defined by \(\eta \left( {\boldsymbol{k}} \right) = \frac{{{{\left {{\cal P}_{{\rm{cv}}}^ + } \right}^2}  {{\left {{\cal P}_{{\rm{cv}}}^  } \right}^2}}}{{{{\left {{\cal P}_{{\rm{cv}}}^ + } \right}^2} + {{\left {{\cal P}_{{\rm{cv}}}^  } \right}^2}}}\), where \({\cal P}_{{\rm{cv}}}^ \pm = \frac{1}{{\sqrt 2 }}\left[ {{\cal P}_{{\rm{cv}}}^x \pm i{\cal P}_{{\rm{cv}}}^y} \right]\), with \({\cal P}_{cv}^{x/y}\) as the interband matrix elements of the dipole operator evaluated with the LDA–DFT wave functions. The results of the calculations are shown in Supplementary Fig. 4 for the monolayer, and Supplementary Fig. 5 for the bulk system.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
We acknowledge financial support from Alexander von Humboldt foundation, European Research Council (MOMB project number 320590) and the EC Graphene Flagship project (number 604391). The computing times granted by the John von Neumann Institute for Computing (NIC) and the JURECA supercomputing facility at Jülich Supercomputing Centre (JSC) are also acknowledged. We thank Peter Krüger for helpful discussions on the calculated MoTe_{2} band structure. We gratefully acknowledge support by the Center for Multiscale Theory and Computation and the Open Access Publication Fund of the University of Münster.
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A.A., R.B. and M.P. conceived the project. A.A. and R. Schmidt performed the magnetooptical experiments. A.A. analyzed data and wrote the manuscript with significant inputs from M.D. and R.B. R. Schneider prepared the samples. M.D., T.D. and P.M. performed theoretical calculations. M.R.M. participated in the measurements. S.M.d.V., M.P., M.R. and R.B. supervised the project. All authors discussed the results and commented on the manuscript.
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Correspondence to Rudolf Bratschitsch.
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Arora, A., Drüppel, M., Schmidt, R. et al. Interlayer excitons in a bulk van der Waals semiconductor. Nat Commun 8, 639 (2017). https://doi.org/10.1038/s41467017006915
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