Letter | Published:

Momentum-space indirect interlayer excitons in transition-metal dichalcogenide van der Waals heterostructures

Nature Physicsvolume 14pages801805 (2018) | Download Citation


Monolayers of transition-metal dichalcogenides feature exceptional optical properties that are dominated by tightly bound electron–hole pairs, called excitons. Creating van der Waals heterostructures by deterministically stacking individual monolayers can tune various properties via the choice of materials1 and the relative orientation of the layers2,3. In these structures, a new type of exciton emerges where the electron and hole are spatially separated into different layers. These interlayer excitons4,5,6 allow exploration of many-body quantum phenomena7,8 and are ideally suited for valleytronic applications9. A basic model of a fully spatially separated electron and hole stemming from the K valleys of the monolayer Brillouin zones is usually applied to describe such excitons. Here, we combine photoluminescence spectroscopy and first-principles calculations to expand the concept of interlayer excitons. We identify a partially charge-separated electron–hole pair in MoS2/WSe2 heterostructures where the hole resides at the Γ point and the electron is located in a K valley. We control the emission energy of this new type of momentum-space indirect, yet strongly bound exciton by variation of the relative orientation of the layers. These findings represent a crucial step towards the understanding and control of excitonic effects in van der Waals heterostructures and devices.


An optical micrograph of a representative MoS2/WSe2 heterobilayer, which was fabricated by deterministic transfer and stacking10 followed by an annealing procedure, is shown in Fig. 1a. All heterobilayer and isolated regions of the constituent monolayers were thoroughly studied by micro-photoluminescence spectroscopy, and typical spectra are shown in Fig. 1b. The monolayer regions display the well-known A exciton and trion peaks11,12,13,14 near 1.65 and 1.9 eV for WSe2 (green) and MoS2 (blue), respectively. In the heterobilayer region the same two peaks are discernible, but they are slightly shifted in energy due to the modified dielectric environment15,16. However, in addition, a new peak near 1.6 eV is observed, which is absent in the monolayer regions. We assign this peak to the interlayer exciton (ILE)4,17.

Fig. 1: Interlayer excitons in MoS2/WSe2 heterobilayers.
Fig. 1

a, An optical micrograph of a sample with a twist angle of 58.7°, fabricated by deterministic transfer and stacking. Monolayer (ML) and heterobilayer (HB) regions are indicated. b, Photoluminescence (PL) spectra of the heterobilayer and monolayer regions. The occurrence of an interlayer exciton near 1.6 eV is discernible in the heterobilayer.

Now, we control the relative orientation of the transition-metal dichalcogenide (TMD) layers to reveal the k-space indirect nature of this ILE in MoS2/WSe2 heterobilayers. The twist angle is measured with respect to the zigzag direction of each layer (green and blue arrows in Fig. 2a), varying between 0° (aligned) and 60° (anti-aligned). A total of 15 heterobilayers with twist angles covering this range were fabricated, and the ILE emission was observable as a high-intensity photoluminescence peak in all samples. The twist angle was determined by second-harmonic generation (SHG) measurements and the samples were further characterized by Raman spectroscopy (for details, see Supplementary Information). The presence of the ILE becomes more obvious in Fig. 2b, which displays photoluminescence spectra from the heterobilayer region of two samples with twist angles of 33.0° and 58.7° and their decomposition into three Gaussian peaks. The comparison of the two panels shows that as the twist angle is varied, the ILE peak displays a greater energy shift than the A exciton or trion peaks. As is clear from Fig. 2c, the latter do not exhibit a distinct dependence on the twist angle. Figure 2d shows that the ILE energy (red circles) shifts as a continuous function of the twist angle over a range of 50 meV. The maximum of the curve is near 30° and it exhibits a slight asymmetry (that is, the minimum near 0° has a smaller energy than the minimum near 60°). Similar twist-angle-dependent, slightly asymmetric shifts of photoluminescence peaks have been observed in twisted bilayer MoS2 (refs 2,3,18).

Fig. 2: Tuning the interlayer exciton energy via interlayer twist.
Fig. 2

a, An atomic structure illustration of a MoS2/WSe2 heterobilayer. The twist angle is the relative lattice orientation of the two layers. Inset: side view of the heterobilayer; the layer separation d is the distance between the Mo and W planes. b, Photoluminescence spectra and their decomposition into Gaussian peaks for two heterobilayers with twist angles of 33.0° and 58.7°. Besides the A exciton or trion peaks, arising from WSe2 (green) and MoS2 (blue), an ILE near 1.6 eV has emerged (red). The dashed vertical lines allow comparison of the peak positions; the black lines are the sum of the Gaussians. The extracted peak energies are used in c and d. c, A exciton or trion energies for monolayer and heterobilayer regions for varying twist angles. The dashed horizontal lines indicate the mean values. We observe no clear dependence on the twist angle but a redshift from the monolayer to the heterobilayer. d, ILE energies and calculated transition energies for heterobilayers with different twist angles. The error bars indicate the standard deviation of the ILE energy determined from spatial averaging of the ILE photoluminescence emission (see Supplementary Information). The Γ−K and K−K values are calculated with DFT and they are rigidly upshifted by 0.445 eV (see the text). Only the trend of Γ−K is in quantitative agreement with the experiment. e, The mean layer separation (indicated graphically in the inset of a) as a function of twist angle, as calculated with dispersion-corrected DFT. Steric repulsion of chalcogen atoms, due to lattice mismatch and incommensurability, creates a twist angle dependence and leads to bigger layer separations than the mean of the layer spacings of bulk MoS2 and WSe2 samples (dashed horizontal line). The red dots correspond to ‘ILE (experiment)’ in d. A strong correlation is discernible; the linear proportionality factor is 0.44 eV Å−1.

We are able to explain this effect quantitatively via density functional theory (DFT) calculations. Details can be found in the Supplementary Information. An analysis of the geometries revealed that the mean layer separation of a heterobilayer changes as a continuous function of the twist angle over a range of 0.07 Å, as shown in Fig. 2e. This result can be ascribed to steric effects since the surface of a TMD monolayer is not atomically smooth but corrugated due to protrusion of the chalcogen atoms out of the metal-atom plane. For angles near 0° or 60°, Fig. 2e indicates a reduction of the mean layer separation by 1%. In these systems, long-wavelength moiré patterns are formed and the individual layers maximize their adhesion by adopting static spatial fluctuations. To study the consequences of these observations, we now analyse the electronic structure of the MoS2/WSe2 heterobilayer (for details, see Supplementary Information). A level-alignment diagram is shown in Fig. 3a. It illustrates the staggered band alignment of the heterobilayer, the optical transitions in the two monolayers that give rise to the A excitons (vertical arrows) and the K−K and Γ−K interlayer transitions. Due to the generally weak coupling between the monolayers in TMD heterostructures, the Bloch wavevectors defining the K valleys of the monolayers are also approximately good quantum numbers of the heterobilayer. In Fig. 2d, K−K and Γ−K interlayer transition energies of twisted heterobilayers, as obtained from DFT calculations, are plotted as a function of twist angle. To allow a visual comparison of the DFT transition energies with the photoluminescence ILE energies, the DFT values in Fig. 2d are rigidly shifted by 0.445 eV, which implies that (relative) energy differences and not absolute energies are compared. The comparison reveals a remarkable quantitative correspondence with the Γ−K transition (red) but not with K−K (yellow). This suggests that the ILE is related to the Γ−K transition. For the monolayer A type transitions, the DFT results exhibit no change with twist angle (see Supplementary Information). The behaviour exposed by the DFT calculations is in full agreement with the photoluminescence results of Fig. 2d, because the change of the Γ−K transition energy is essentially a shift of the Γ point valence band energy (white arrow in Fig. 3a), an effect that should be well captured by DFT (for details, see Supplementary Section 3.2.4)18,19.

Fig. 3: Electronic structure of MoS2/WSe2 heterobilayers.
Fig. 3

a, Band alignment diagram. The valence band maximum at the Γ valley |+Γ〉 is a hybrid state of both layers and it moves up as hybridization increases. The coloured vertical arrows indicate monolayer transitions. Γ−K and K−K are possible interlayer transitions. b, The two-dimensional band structure of the heterobilayer near the band edges. In twisted heterobilayers, the Brillouin zones of MoS2 (blue) and WSe2 (green) are misaligned. Therefore, both heterobilayer transitions, K−K and Γ−K, are k-space indirect (the wavevectors of the electron and holes states differ). However, for twist angles near 0° (aligned) or 60° (anti-aligned), K−K is k-space direct (no wavevector difference). c, Temperature-dependent photoluminescence spectra measured on isolated WSe2 (left panel) and MoS2 (right panel) monolayers and on a heterobilayer region (centre panel). Every spectrum is individually normalized to the peak of highest intensity. In the heterobilayer region, spatial averaging of photoluminescence spectra is performed due to the spatially inhomogeneous ILE emission (see Supplementary Information for details). The dotted lines trace the spectral evolution of the WSe2 and MoS2 A exciton and trion, as well as the ILE as a function of temperature. As the temperature is decreased, the ILE photoluminescence, which is the most prominent emission at room temperature, is suppressed compared to the intralayer MoS2 emission in the heterobilayer region. This finding supports our assertion that the ILE is related to a k-space indirect, phonon-assisted transition.

To better understand the impact of the layer separation on the electronic structure of the MoS2/WSe2 heterobilayer, we studied by DFT an artificial model system that is anti-aligned and lattice-commensurate by applying strain. We considered 28 different transitions between valence and conduction band extrema and calculated their energies as a function of layer separation (see Supplementary Information). Most transitions exhibit either no dependence on the layer separation or a linear dependence with a negative slope (including K−K). There is only a single transition, Γ−K, that lies within a reasonable energy range, and has the correct trend and a positive slope of 0.47 eV Å−1, in excellent agreement with 0.44 eV Å−1 found for realistic systems in Fig. 2e. These results uniquely identify the observed ILE to be related to the Γ−K transition and not to K−K that is usually assumed when studying ILEs. Additional evidence supporting this key finding of our work is provided by analysing the twist angle and temperature dependencies of the ILE photoluminescence intensity and by an exciton model.

We note that if the ILE was related to a K−K transition, its photoluminescence emission should be observable only for nearly (anti-)aligned structures because the transition probability of k-space direct transitions is higher (Fig. 3b) 20. However, the analysis of the photoluminescence intensity as a function of twist angle in the heterobilayers shows no pronounced angle-dependence (see Supplementary Information).

For indirect optical transitions, the difference between the wavevectors of the electron and holes states is compensated by coupling to a phonon and the efficiency of this process can be partially tuned by varying the temperature, which controls the phonon population. Temperature-dependent photoluminescence measurements of a heterobilayer and isolated WSe2 and MoS2 monolayers are shown in Fig. 3c. We observe a systematic blueshift of all exciton peaks with decreasing temperature. We also observe a complex behaviour of the relative photoluminescence intensities in the heterobilayer region. The ILE photoluminescence, which is the most prominent emission peak at room temperature, decreases relative to the MoS2 intralayer emission as the temperature decreases. This further supports the identification of the transition as being indirect in k-space and is in stark contrast to the supposed K−K transition observed in WSe2/MoSe2 heterobilayers5,20,21, for which the ILE photoluminescence yield monotonically increases with decreasing temperature22,23. We also observe that the WSe2 intralayer emission is quenched with decreasing temperature, as reported previously for WSe2 monolayers24,25. Additional measurements and discussion of the temperature-dependent photoluminescence are presented in the Supplementary Information.

We now analyse the localization of electron and hole wavefunctions in the MoS2/WSe2 heterobilayer. Figure 4a,b shows partial charge densities of electron and hole states for the K−K and Γ−K transitions in the model system calculated with DFT. Three unique states are involved, the K-valley electron state |−〉, the K-valley hole state |+K〉 and the Γ-valley hole state |+Γ〉. The electron–hole wavefunction overlap of an ILE can be quantified by projecting the hole state |+k〉 (k = Γ or K) onto the MoS2 layer ok = |〈MoS2|+ k〉|2. The overlap of the K−K transition (see Fig. 4a) is nearly zero (oK ≈ 0%) because electrons and holes involve only transition-metal atom d-states and reside 6.6 Å apart (see Fig. 2e). The photoluminescence intensity scales with the square of the transition matrix element, which suggests that radiative recombination of K−K ILE is suppressed and is thus not seen in our photoluminescence measurements26. This is very different for the Γ−K transition (Fig. 4b): |+Γ〉 is strongly affected by interlayer hybridization and therefore extends over both layers with Mo, S, W and Se atoms all participating. It has a large component that resides in the MoS2 monolayer (oΓ = 24%) where |−〉 is localized. Therefore, the matrix element is much larger for Γ−K transitions than it is for K−K ones.

Fig. 4: The nature of interlayer excitons.
Fig. 4

a, The hole |+K〉 and electron |−〉 states of the K−K ILE are localized in the individual layers (the pink and magenta contours are partial charge densities). b, Hole and electron states of the Γ−K ILE. While the electron state |−〉 is localized only in the MoS2 layer, the hybrid hole |+Γ〉 state is delocalized over both layers. The percentages correspond to the fraction of the wavefunction that is localized in each layer. c, Comparison between experimental (circles) and theoretical (crosses) photoluminescence peak energies for different monolayer A excitons and ILE. The arrows indicate exciton binding energies ΔEX, as calculated with our model. The theoretical bandgaps (asterisks) are G0W0 results from the literature31. The theoretical results closely match the experimental data. ΔEX for the Γ−K ILE is comparable to values in the monolayer, and the resulting emission energy is in good agreement with the experimentally observed ILE peak.

Our observations strongly imply the picture of an ILE with high photoluminescence intensity that does not represent the thermodynamically lowest-energy states (that is, the K−K excitonic transition), and is fully consistent with transitions of the Γ−K type. We note that our system is pumped with a sufficiently high energy to create carriers across a wide range of momenta with a hot, non-thermal distribution. Thus, the observed response will depend intimately on the non-equilibrium kinetics of exciton formation and recombination, as well as charge transfer27 and a host of non-radiative relaxation channels28,29. However, it should be noted that non-equilibrium effects alone are insufficient to explain why the Γ−K ILE seems to be so strongly favoured. One possibility is that the large, real-space overlap of electron and holes in the respective layers kinetically favours the formation and recombination of partially charge-separated Γ−K excitons despite the fact that such states are not formed from band-edge carriers.

The large, real-space overlap of these k-space indirect ILEs suggests a binding energy that is increased as compared to their K−K counterparts that are fully charge-separated. Therefore, we calculate the exciton binding energies ΔEX of the A excitons as well as K−K and Γ−K ILEs using the quantum electrostatic heterostructure model30 and a variational wavefunction ansatz. Excitonic interlayer interactions are described within a tight-binding approach (see Supplementary Information for details). Experimentally, the exciton binding energy is defined as \(\Delta {E}_{X}={E}_{{\rm{gap}}}^{{\rm{qp}}}-{E}_{{\rm{gap}}}^{{\rm{opt}}}\), where \({E}_{{\rm{gap}}}^{{\rm{qp}}}\) is the quasiparticle bandgap and \({E}_{{\rm{gap}}}^{{\rm{opt}}}\) is the optical gap, measured as the photoluminescence peak energy. The results of these calculations are given in Fig. 4c, where ΔEX is indicated by arrows. For the A excitons, the theoretical and experimental energies agree well. ΔEX values are of the order of 0.5 eV, in good agreement with previous results31. For the K−K ILE, ΔEX = 0.29 eV, which is also in agreement with earlier results15,32. For Γ−K, we obtain a much bigger value of 0.55 eV, comparable to those of A excitons. The main reason for this large number is the delocalization of the hole state over both layers that enhances the electron–hole Coulomb attraction and gives the Γ−K ILE also a strong monolayer character in MoS2. We note that the ILE emission energy calculated using this large binding energy is in good agreement with the experimentally observed value. It should also be noted that charge separation creates excitons with an interlayer dipole moment of μIL = (1 − ok)ed ≈ (1 − ok) × 1.4 Debye (e is the elementary charge and d is the layer separation). Its magnitude is reduced by interlayer hybridization since oK < oΓ. Thus K−K and Γ−K excitons can potentially be distinguished by measuring μIL.


Sample fabrication

Heterobilayer samples were fabricated by means of a deterministic transfer process10. For this, we initially exfoliated MoS2 and WSe2 flakes from bulk crystals (here, we utilized a natural MoS2 crystal and a synthetic WSe2 crystal bought at HQgraphene.com) onto polydimethylsiloxane substrates. Monolayer regions of these flakes were identified via optical microscopy. Then, we first transferred a MoS2 flake onto the target substrate, a silicon wafer piece covered with a SiO2 layer and pre-defined metal markers. Subsequently, the WSe2 flake was transferred on top of the MoS2. For each of the heterobilayers fabricated in this way, the relative orientation of the individual layers was chosen to optimize the overlap region of the monolayer parts of the flakes. Subsequent to the transfer, the heterobilayers were annealed. For this, they were mounted in a furnace, which was initially flushed with a H2/Ar gas mixture and then pumped to high vacuum. In vacuum, the samples were annealed at a temperature of 150 °C for 5 h.

Optical spectroscopy

Photoluminescence and Raman measurements were performed in a self-built microscope set-up (details are published elsewhere33). A continuous-wave laser (wavelength 532 nm) was coupled to a ×100 microscope objective and focused to a submicrometre spot on the sample surface. Photoluminescence and scattered light were collected with the same objective, passed through long-pass filters, coupled into a grating spectrometer and detected with a Peltier-cooled CCD (charge-coupled device). For temperature-dependent photoluminescence measurements, the sample was mounted on the cold finger of a small He-flow cryostat. For photoluminescence mapping, the sample was moved beneath the fixed microscope objective using a motorized xy stage, and photoluminescence spectra were collected for sample positions defined on a square lattice. To extract information from these spectra, an automated fitting routine was employed, which yields the integrated intensity, spectral position and full-width at half-maximum for each spectral feature extracted using a Gaussian fit function. SHG measurements were performed in a similar, self-built microscope set-up, which was optimized to yield high SHG throughput. Here, a Ti:sapphire laser oscillator (pulse length 100 fs, central wavelength 815 nm) was used as an excitation source. The laser light coupled into the microscope objective was linearly polarized, and the same polarizer was used to analyse the reflected light, so that only the signal polarized parallel to the excitation was detected. To separate the SHG signal from the reflected fundamental laser wavelength, a dichroic mirror and short-pass filters were employed before the SHG signal was either coupled into a grating spectrometer to be detected by a CCD, or focused onto an avalanche photodiode. In the measurements using the avalanche photodiode, a lock-in scheme was employed to further increase the signal-to-noise ratio. For SHG mapping, the sample was moved beneath the fixed microscope objective using a motorized xy stage. For polarization-dependent measurements, the combined polarizer/analyser was rotated using a motorized stage.

Experimental data analysis

For each heterobilayer structure, a photoluminescence map was measured at room temperature. To compensate for the spatial inhomogeneity of the ILE emission, spatial averaging was employed. For this, the average photoluminescence emission energy of the ILE, and its standard deviation, were calculated from the values extracted from an automated fitting routine applied to individual photoluminescence spectra collected in the heterobilayer region where sufficiently intense ILE photoluminescence was observed. The size of these regions varied from sample to sample, but, on average, more than 60 spectra were evaluated for an individual heterobilayer.

DFT calculations

The DFT calculations were carried out with the Perdew–Burke–Ernzerhof functional34 and the Tkatchenko–Scheffler dispersion-interaction-correction scheme35 using the projector augmented wave method36 and a plane-wave basis set with a cutoff energy of 259 eV, as implemented in VASP (Vienna Ab initio Simulation Package)37,38. For the k-point sampling, an in-plane sampling density of 0.1 Å2 was used. It was carefully checked that this density leads to converged total energies (energy differences are smaller than 1 meV per atom). The k-space integration was carried out with a Gaussian smearing method using an energy width of 0.05 eV for all calculations. All unit cells were built with at least 10 Å separation between replicates in the perpendicular direction to achieve negligible interaction. All systems were fully structurally optimized until all interatomic forces and stresses on the unit cell were below 0.01 eV Å−1 and 10 kbar, respectively. Spin–orbit interactions were generally not taken into account and the inclusion of these interactions does not alter any of our conclusions as spin–orbit-dependent interlayer interactions in TMDs have not been reported before2,18. The wavefunction overlap ok = |〈MoS2|+k〉|2 is calculated by integrating the partial charge density of the state |+k〉 (k = Γ or K) over the half of the volume of the unit cell that contains the MoS2 layer. The cutting plane between the two halves is the minimum of the plane-averaged line charge density in the van der Waals gap between the layers.

Data availability

The data that support the plots within this paper and other findings of this study are available from the corresponding authors upon reasonable request.

Additional information

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The work is financially supported by the German Research Foundation (DFG) under grant numbers SE 651/45-1, GRK 1570, KO 3612/1-1 and KO 3612/3-1. G.S. gratefully acknowledges financial support by the Ministry of Education and Science of the Russian Federation (grant no. K3-2017-064). Computational resources for this project were provided by ZIH Dresden.

Author information

Author notes

  1. These authors contributed equally: Jens Kunstmann, Fabian Mooshammer.


  1. Theoretical Chemistry, Department of Chemistry and Food Chemistry, TU Dresden, Dresden, Germany

    • Jens Kunstmann
    • , Frederick Stein
    •  & Gotthard Seifert
  2. Institut für Experimentelle und Angewandte Physik, Universität Regensburg, Regensburg, Germany

    • Fabian Mooshammer
    • , Philipp Nagler
    • , Nicola Paradiso
    • , Gerd Plechinger
    • , Christoph Strunk
    • , Christian Schüller
    •  & Tobias Korn
  3. Departamento de Fisica, Universidade Federal do Ceara, Fortaleza, Ceara, Brazil

    • Andrey Chaves
  4. Department of Chemistry, Columbia University, New York, NY, USA

    • Andrey Chaves
    •  & David R. Reichman
  5. National University of Science and Technology, MISIS, Moscow, Russia

    • Gotthard Seifert


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F.M., P.N., C. Schüller and T.K. conceived the experiments. F.M. fabricated the samples and performed the optical spectroscopy and data analysis together with P.N., G.P. and T.K. N.P. and C. Strunk annealed samples and performed AFM measurements. J.K. performed the DFT calculations together with F.S. and G.S., interpreted the results and supervised the theoretical analysis. A.C. carried out the exciton modelling under the supervision of D.R.R. using parameters provided by J.K. J.K. D.R.R. and T.K. wrote the paper together with F.M. and P.N. All authors discussed the results.

Competing interests

The authors declare no competing interests.

Corresponding authors

Correspondence to Jens Kunstmann or Tobias Korn.

Supplementary information

  1. Supplementary Information

    Supplementary Figures 1–13, Supplementary Tables 1 and 2, Supplementary References

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