Abstract
Atomically thin transition metal dichalcogenides can be stacked to van der Waals heterostructures enabling the design of new materials with tailored properties. The strong Coulomb interaction gives rise to interlayer excitons, where electrons and holes are spatially separated in different layers. In this work, we reveal the time and momentumdependent elementary processes behind the formation, thermalization and photoemission of interlayer excitons for the exemplary MoSe_{2}–WSe_{2} heterostructure. We identify tunneling of holes from MoSe_{2} to WSe_{2} on a ps timescale as the crucial process for interlayer exciton formation. We also predict a drastic reduction of the formation time as a function of the interlayer energy offset suggesting that interlayer excitons can be externally tuned. Finally, we explain the experimental observation of a dominant photoluminescence from interlayer excitons despite the vanishingly small oscillator strength as a consequence of huge interlayer exciton occupations at low temperatures.
Introduction
A direct band gap in the optical range, efficient electronlight coupling and a remarkably strong Coulomb interaction make transition metal dichalcogenides (TMDs) highly interesting materials for both fundamental research and technological applications^{1,2,3,4,5,6}. Tightly bound excitons, quasiparticles of Coulombbound electronhole pairs, dominate the optical response of these materials^{7,8,9,10,11,12,13,14}. They have binding energies that are one to two orders of magnitude larger than in conventional materials^{15,16}. As a result, excitonic features are stable at room temperature and dominate the optical response and nonequilibrium dynamics in TMDs. Besides regular bright excitons, TMDs also exhibit a variety of optically forbidden dark excitons^{17,18,19,20,21,22,23,24}, which cannot be addressed optically due to the required momentum transfer or spin flip. The fascinating exciton physics becomes even richer, when considering that atomically thin materials can be vertically stacked to form Van der Waals (vdW) heterostructures^{1}. In these systems the strong Coulomb interaction gives rise to interlayer excitons, where the involved electrons and holes are located in different layers (Fig. 1)^{25,26,27,28,29}. After optical excitation of a regular intralayer exciton (IaX), the hole can tunnel to the other layer forming an interlayer exciton (IeX). Due to an offset in the alignment of the monolayer band structures (type II heterostructures), these interlayer excitons lie energetically below the excitons confined within one layer^{29} (Fig. 1). Depending on spin and momentum of the states involved, interlayer excitons can be either bright or dark. VdW heterostructures present an emerging field of research, as evidenced by an increasing number of studies, in particular demonstrating the appearance of interlayer excitons in photoluminescence (PL) spectra^{25,26,28,30,31,32,33,34,35,36,37,38,39}. In the lowtemperature regime, a pronounced additional resonance is observed at an energy below the intralayer excitons of the single layers^{26,27,33,40,41}. The PL intensity of this lowenergy peak is very pronounced compared to the intralayer exciton in the weak excitation regime. This behavior can be traced back to formation of interlayer excitons that due to their spectral position are highly occupied. Furthermore, in timeresolved PL measurements a spectrally narrow resonance was observed exhibiting lifetimes of tens to hundreds of nanoseconds^{26,27,33,40,41}. Theoretical studies of vdW heterostructures have so far been primarily restricted to static observables, such as screening^{42}, excitonic binding energies^{29}, and lattice mismatch effects^{35}. Microscopic insights into the ultrafast dynamics of interlayer excitons have remained literally in the dark yet.
In this work, we present a comprehensive microscopic study of the ultrafast kinetics of excitons in vdW heterostructures, in particular revealing the elementary timeresolved and energyresolved processes behind the formation, thermalization and decay of interlayer excitons. We predict the binding energy of 150 meV for the energetically lowest interlayer excitonic state in the exemplary MoSe_{2}–WSe_{2} heterostructure, as well as a significant reduction of intralayer binding energies by about 30% due to the increased screening within a heterostructure. Moreover, we shed light on the exciton dynamics demonstrating that excited interlayer excitons are formed via hole tunneling on a picosecond timescale, followed by a phonondriven relaxation to the ground state on a femtosecond timescale. The radiative interlayer recombination is by orders of magnitude slower reflecting the small oscillator strength of spatially separated excitons. Finally, we provide a microscopic explanation for the recently performed timeresolved photoluminescence experiments demonstrating the surprisingly dominant contribution of interlayer excitons at low temperatures.
Results
Theoretical approach
We investigate the exemplary MoSe_{2}–WSe_{2} heterostructure on a typical SiO_{2} substrate. To bring the system into a nonequilibrium, we apply a laser pulse at the energy resonant to the 1s exciton of the MoSe_{2} layer. We restrict the study to 1s excitons consisting of Coulombbound electrons and holes located at the K point of the Brillouin zone.
To provide microscopic access to the dynamics of the coupled excitons, phonons and photons, we apply the density matrix formalism^{43,44,45,46} and derive the luminescence Bloch equations explicitly including excitonic effects^{9,44}. The emitted luminescence intensity I_{ω}(t) is obtained from the temporal change of the photon density and reads^{44,47}.
Here, \(M_{\mathbf{0}}^{l_{\mathrm{h}}l_{\mathrm{e}}}\) is the optical matrix element for excitons that are composed of electrons in layer l_{e} and holes in layer l_{h}. The index 0 indicates that only excitons with vanishing centerofmass momentum Q = 0 contribute to PL. While the optical matrix element can be obtained analytically using a tightbinding approach for TMD monolayers and adjusting the coupling strength to experimentally measured absorption^{9,22}, firstprinciple calculations have been performed to determine the coupling for interlayer excitons. The Lorentzian \({\cal L}_\gamma\) accounts for energy conservation, i.e., an exciton decays into a photon with the same energy. We calculate on a microscopic level the components of the dephasing rate γ stemming from excitonphonon and excitonphoton interactions^{22,48}. At lower temperatures disorderinduced dephasing becomes important and has been accounted for phenomenologically by using the full width at half maximum (FWHM) values measured in ref. ^{27}.
The PL strength is determined by the excitonic polarization \(P_{\mathbf{Q}}^{l_{\mathrm{h}}l_{\mathrm{e}}}(t)\) reflecting the optically driven coherence (often referred to as coherent excitons^{43,44}) and by the exciton occupation \(N_{\mathbf{Q}}^{l_{\mathrm{h}}l_{\mathrm{e}}}\) describing the formation and thermalization of incoherent excitons induced by a nonradiative decay of the excitonic polarization. Equations of motion for \(P_{\mathbf{Q}}^{l_{\mathrm{h}}l_{\mathrm{e}}}(t)\) and \(N_{\mathbf{Q}}^{l_{\mathrm{h}}l_{\mathrm{e}}}\) are derived in the low density regime taking into account excitonphoton coupling giving rise to a spontaneous radiative decay of excitons as well as excitonphonon coupling allowing for a relaxation of the excited hot exciton distribution into a thermal equilibrium. Furthermore, we take into account the tunneling of carriers between layers. This process is included via the tunneling Hamilton operator \(H_{\mathrm{T}} = \mathop {\sum}\nolimits_{a,b} T^{ab}a_b^\dagger a_b\), where a, b are compound indices containing layer, band and momentum of the electron. The coupling element T is given by the overlap integral 〈Ψ_{a}V_{T}Ψ_{b}〉 of Bloch waves Ψ with the interlayer potential V_{T}. The appearing spatial overlap of the wavefunctions from different layers is obtained from density functional theory calculations that have been carried out using the gpaw package^{49}. More details on the calculation of the tunneling matrix element can be found in the methods section. Finally, applying the Heisenberg equation of motion, we obtain the luminescence Bloch equations for the excitonic polarization \(P_{\mathbf{Q}}^{l_{\mathrm{h}}l_{\mathrm{e}}}(t)\) and the incoherent exciton occupation \(N_{\mathbf{Q}}^{l_{\mathrm{h}}l_{\mathrm{e}}}\), see Eqs. (2) and (3) in the methods section. Numerical solution of this coupled set of differential equations provides microscopic access to timeresolved and momentumresolved formation, relaxation, and radiative decay dynamics of intralayer and interlayer excitons. It allows us to calculate the photoluminescence of heterostructures as a function of time and energy.
Excitonic band structure
The excitonic eigenenergies and wave functions are obtained by solving the Wannier equation^{9,43,46} (Equation (1) in the methods section). To obtain the interaction potential between charge carriers in a heterostructure, we generalize the widely used Keldysh potential for the monolayer case^{9,50}. We solve the Poisson equation for two aligned homogeneous slabs (Supplementary Fig. 1 and Supplementary Note 1 for more details). The obtained excitonic band structure and the corresponding excitonic wave functions are shown in Fig. 2 for the three energetically lowest excitonic states (1s, 2s, 3s) for intralayer and interlayer excitons. Here, the electronic band alignment has been extracted from PL measurements^{26,27}. We find that the interlayer exciton is the energetically lowest state at 1.39 eV, while the intralayer excitons lie at 1.65 eV (MoSe_{2}) and 1.75 eV (WSe_{2}), Fig. 2. The resulting excitonic binding energies are displayed in Table 1. As expected, the binding energy for interlayer excitons is reduced (150 meV for 1s) compared to the value for intralayer excitons (205 meV for MoSe_{2} and 177 eV for WSe_{2}), however it is still much larger than the thermal energy. Thus, interlayer excitons are expected to be stable at room temperature and significantly contribute to the PL. Our calculations also reveal that the binding energy of intralayer excitons is considerably reduced by roughly 30% when the monolayers are stacked into a heterostructure (Table 1). This is due to an increased screening of the Coulomb potential. These results are consistent with previous theoretical studies based on DFT calculations^{29,37}.
Exciton dynamics
Solving the luminescence Bloch equations (the Methods section) we can resolve the dynamics of intralayer and interlayer excitons. Figure 3a shows the temporal evolution of exciton densities, i.e., momentumintegrated exciton occupations \(n = \mathop {\sum}\nolimits_{\mathbf{Q}} N_{\mathbf{Q}}\), at the exemplary temperature of 77 K (Supplementary Fig. 3 and Supplementary Note 3 for 4 K and room temperature). The system is excited by a 100 fs long Gaussian pulse centered at 0.5 ps and a frequency resonant to the intralayer 1s exciton of the MoSe_{2} layer. We find that the optically excited coherent excitons (dashed red line in Fig. 3) decay on a timescale of hundreds of femtoseconds due to radiative emission and excitonphonon scattering. The latter leads to the formation of incoherent intralayer excitons (IaX) through the socalled polarizationtopopulation transfer^{23,43,51,52}. After about 10 ps, these intralayer excitons are completely transferred to interlayer excitons (IeX). This occurs through tunneling between energetically resonant states of the two layers and results in a transfer of holes to the WSe_{2} layer (Fig. 1). The subsequent phononinduced relaxation of holes in the WSe_{2} layer to the valence band maximum effectively traps the holes within that layer, since tunneling back to MoSe_{2} is energetically forbidden. The resulting interlayer excitons have lifetimes orders of magnitudes longer than the intralayer excitons, since recombination mechanisms are suppressed due to the spatial separation of Coulombbound electrons and holes. We predict an interlayer exciton lifetime in the range of hundreds of microseconds at 77 K. Experimentally measured subnanosecond values^{27} suggest that the investigated radiative decay is not the dominant channel, but rather nonradiative decay e.g., induced by disorder might play the crucial role.
To provide the microscopic background of the exciton formation and thermalization process, we now show the timeresolved and energyresolved dynamics of incoherent intralayer and interlayer excitons. Figure 4a illustrates how 1s intralayer excitons (IaX) are created in the MoSe_{2} layer after optical excitation at 0.5 ps. The process is followed by a phononassisted thermalization and tunnelingdriven depletion of these excitons. The main features of these dynamics are illustrated in representative snapshots along the energy axis at fixed times (Fig. 4b). At 0.6 ps, a significant number of excitons is still located in the MoSe_{2} layer. The distribution is in a strong nonequilibrium due to the efficient polarizationtopopulation transfer, i.e., the excitonic polarization is converted into incoherent exciton occupations with nonvanishing centerofmass momentum. This is the origin of the observed peaks (green line in Fig. 4b) corresponding to the position of intersections of exciton and phonon dispersion. After the coherence has decayed (1 ps), the occupation starts to thermalize into a Boltzmann distribution (blue line). The occupation within the light cone centered around the exciton dispersion minimum E_{0} = 1.65 eV is lowered as a result of radiative recombination. Furthermore, tunneling of holes to the WSe_{2} layer causes a considerable decay of the intralayer exciton occupation (black line). Finally, the temporal evolution of the exciton occupation at a fixed energy corresponding to the 1s resonance of intralayer excitons (1.65 eV) is shown in Fig. 4c. This occupation represents the optically active exciton with vanishing centerofmass momentum. We see how intralayer excitons are created already during the optical excitation and how they decay on a ps timescale due to tunneling of holes into the neighboring WSe_{2} layer.
Figure 5a illustrates the corresponding dynamics of interlayer excitons (IeX). They first emerge at E = E_{0} corresponding to the energy of intralayer excitons in MoSe_{2}. The exciton population generated at this discrete energy consequently relaxes to the band minimum of the interlayer dispersion mostly by emitting LO and TO phonons in MoSe_{2}, which have approximately a constant energy of 36 meV. This yields a cascadelike redistribution of excitons through intermediate energy levels after a series of optical phonon emissions. The slow interaction with acoustic phonons, which provide a continuous range of energies, leads to a smoothing of the occupation distribution, but on a much larger timescale. The energy conserving nature of the tunneling interaction eventually forces the entire system into a single Boltzmannlike distribution as can be seen in Fig. 5b, spanning over both intralayer and interlayer excitonic states.
Figure 5c is analogous to Fig. 4c, and depicts the temporal evolution of interlayer excitons at fixed energies. The green curve first shows how the interlayer exciton occupation at the energy 1.65 eV resonant to the intralayer exciton rises with its intralayer counterpart. Driven by scattering with phonons, excitons then accumulate at the minimum of the dispersion at 1.39 eV (orange curve).
It should be noted that the excitonphonon scattering has an important effect on the effective tunneling rate. Since the tunneling interaction is symmetric, the effective interlayer tunneling rate is quenched by backtunneling from energyallowed states. On a short timescale, the impact is mostly determined by the rate of the phononinduced downscattering (Fig. 5). As the Boltzmann distribution is formed (Fig. 5b), the temperature starts becoming an important factor. The spectral width of the IeX distribution increases at enhanced temperatures, leading to more excitons available for backtunneling. Nevertheless, for the investigated heterostructure the change in temperature has still a relatively small effect on the IeX formation time τ. We find a reduction of τ by approximately 200 fs from 4 K to 200 K, after which it remains nearly constant up to room temperature (Fig. 3b).
The interlayer energy offset ΔE in type II heterostructures (Fig. 1) turns out to be a crucial quantity for the IeX formation time. For the investigated MoSe_{2}–WSe_{2} heterostructure, the offset reads 315 meV and results in an IeX formation time τ = 2.61 ps at room temperature. The latter is defined as the elapsed time between the pump pulse maximum and the IeX density reaching 1 − e^{−1} of its maximum value. Increasing (decreasing) the offset by 100 meV, the formation time becomes considerably enhanced to τ = 4.91 ps (reduced to τ = 0.69 ps). This behavior can be ascribed to the reduced (enhanced) tunneling rate reflecting the increased (decreased) momentum transfer, as the interlayer energy offset ΔE becomes larger (smaller). To put it in other words, the spectral overlap of the wavefunctions of the involved states in different layers is highly sensitive to ΔE. The larger the overlap (small ΔE), the more efficient is the tunneling and the faster is the formation time. The disorderassisted momentum transfer function (Methods) also decreases with increasing ΔE. For this reason tunneling to the higher excitonic states might be nonnegligible considering their lower binding energies, and will have to be addressed in a future study.
A modification of the interlayer energy offset could be e.g., achieved by introducing a doping of a single layer^{26} suggesting that the IeX formation time in heterostructures can be externally tuned. Furthermore, the interlayer wavefunction overlap and the tunneling rate could be tuned by modifying the substrateinduced screening. The smaller the dielectric constant, the stronger is the Coulomb interaction and the broader are the excitonic wavefunctions in the reciprocal space resulting in a more efficient tunneling. Finally, different layer stackings can also have a significant effect on the tunneling rate, as demonstrated by ref. ^{38} predicting formation times ranging from ps to subps timescales.
Photoluminescence
The exciton dynamics discussed above determines the light emission from the heterostructure. Figure 6a shows the timeresolved and energyresolved PL spectrum at the exemplary temperature of 77 K in (Supplementary Fig. 4 and Supplementary Note 3 for 4 and 300 K). We find that in the first few ps, the emission from intralayer excitons (IaX) at 1.65 eV clearly dominates the PL. However, after approximately 41 ps, the contribution of interlayer excitons becomes pronounced. To better understand the underlying processes, we show again snapshots at fixed times and energies in Fig. 6b, c, respectively. The emission stemming from the intralayer exciton at 1.65 eV (red line in Fig. 6c) shows a maximum PL intensity in the first hundreds of femtoseconds originating from the efficient coherent emission (radiative decay of polarization on a fs timescale). The following slower decay on a time scale of tens of ps reflects the decrease of intralayer excitons due to hole tunneling to the WSe_{2} layer. In this time, interlayer excitons are formed (solid blue line in Fig. 6c). After approximately 41 ps, the contribution of the interlayer exciton surpasses the emission from the intralayer exciton. Snapshots of the energydependent PL along these characteristic times further demonstrate by far most pronounced coherent IaX emission (orange line, 0.6 ps), the reduced emission due to incoherent IaX (red, 1 ps), equal IaX and IeX emission (blue, 41 ps), and finally the dominant IeX emission (purple, 65 ps).
After 60 ps an equilibrium situation is reached that is characterized by a constant IaX–IeX intensity ratio of approximately 100 (dashed lines in Fig. 6c). This is determined by the ratio of the square of the corresponding optical matrix elements M_{IeX}^{2}/M_{IaX}^{2} and exciton occupations \(N_0^{{\mathrm{IeX}}}/N_0^{{\mathrm{IaX}}}\) within the light cone. Since the occupations are described by a Boltzmann distribution in equilibrium, we can explicitly calculate the temperature, at which the PL ratio is 1. We find that the interlayer exciton emission dominates until approximately 200 K. Above this temperature, the thermal occupation of energetically higher intralayer excitons increases and considering the significantly larger optical matrix element for intralayer emission, the intralayer peak exceeds the emission from the interlayer exciton.
A recent DFT study^{37} on MoSe_{2}/WSe_{2} heterostructures suggests that the interlayer coupling might shift the Λ valley below the K valley in both layers, giving rise to a momentum mismatch between the lowest conduction and the highest valence band in the heterostructure. The resulting enhanced occupation of momentumforbidden dark interlayer excitons and their impact on luminescence spectra (e.g., via phononassisted recombination) is of particular interest for future studies. In particular, for tungstenbased heterostructures the inclusion of other highsymmetry points of the Brillouin zone is expected to be important to capture the entire interlayer exciton dynamics^{41,53,54}
Discussion
In conclusion, we have presented a microscopic view on the dynamics of inter and intralayer excitons in van der Waals heterostructures. Solving the Wannier equation, we report a binding energy of 150 meV for the interlayer exciton as well as a clear 30% reduction of the binding energy of the intralayer excitons due to the enhanced screening within a heterostructure. Solving the luminescence Bloch equations, we reveal the timeresolved and energyresolved processes behind the formation, thermalization, and decay of interlayer excitons. We predict that tunneling of holes from the optically excited into the neighboring layer is the dominant formation channel occurring on a picosecond timescale. We demonstrate the crucial role of phonons in both their formation and subsequent relaxation dynamics. Furthermore, we suggest a possibility to externally tune the formation time of interlayer excitons by changing the interlayer energy offset in the type II heterostructure. Finally, we show that although the light emission from interlayer excitons is strongly quenched due to the spatial separation of charge carriers, they actually dominate the photoluminescence spectrum of heterostructures at temperatures below 200 K. The gained insights will trigger new experimental studies on van der Waals heterostructures. In particular, the predicted formation dynamics of interlayer excitons can be experimentally addressed by pumping and probing the intralayer exciton transition in different layers, where a clear bleaching is expected due to the efficient interlayer tunneling.
Methods
Binding energies
To model the interlayer exciton dynamics in heterostructures, we derive the luminescence Bloch equations for the excitonic polarization \(P_{\mathbf{Q}}^{l_{\mathrm{h}}l_{\mathrm{e}}}(t) = \mathop {\sum}\nolimits_{\mathbf{q}} \varphi _{\mathbf{q}}^{ \ast l_{\mathrm{h}}l_{\mathrm{e}}}\left\langle {v_{{\mathbf{q}} + \beta ^{l_{\mathrm{h}}l_{\mathrm{e}}}{\mathbf{Q}}}^{\dagger l_{\mathrm{h}}}c_{{\mathbf{q}}  \alpha ^{l_{\mathrm{h}}l_{\mathrm{e}}}{\mathbf{Q}}}^{l_{\mathrm{e}}}} \right\rangle\) and the incoherent exciton occupation \(N_{\mathbf{Q}}^{l_{\mathrm{h}}l_{\mathrm{e}}}(t) = \mathop {\sum}\nolimits_{{\mathbf{q,q}}\prime } \varphi _{{\mathbf{q}}\prime }^{ \ast l_{\mathrm{h}}l_{\mathrm{e}}}\phi _{\mathbf{q}}^{l_{\mathrm{h}}l_{\mathrm{e}}}\delta \left\langle {c_{{\mathbf{q}}  \alpha ^{l_{\mathrm{h}}l_{\mathrm{e}}}{\mathbf{Q}}}^{\dagger l_{\mathrm{e}}}v_{{\mathbf{q}} + \beta ^{l_{\mathrm{h}}l_{\mathrm{e}}}{\mathbf{Q}}}^{l_{\mathrm{h}}}v_{{\mathbf{q}}\prime + \beta ^{l_{\mathrm{h}}l_{\mathrm{e}}}{\mathbf{Q}}}^{\dagger l_{\mathrm{h}}}c_{{\mathbf{q}}\prime  \alpha ^{l_{\mathrm{h}}l_{\mathrm{e}}}{\mathbf{Q}}}^{l_{\mathrm{e}}}} \right\rangle (t)\) corresponding to electronhole pair correlations^{44}. Here, we have introduced φ_{q} as excitonic wavefunctions in momentum space, \(v^{(\dagger )},c^{(\dagger )}\) as operators for annihilation (creation) of valence and conduction band electrons as well as relative q and centerofmass momenta Q with the coefficients α = m_{e}/(m_{h} + m_{e}) and β = m_{h}/(m_{h} + m_{e}) describing the relative electron and hole masses.
To obtain the excitonic binding energies and wave functions, we solve the Wannier equation^{9,43,46}
where μ is the reduced mass, E_{b} the excitonic binding energy, and \(V_{\mathbf{k}}^{l_{\mathrm{h}}l_{\mathrm{e}}}\) the Coulomb matrix element for the electronhole interaction. Solving the Poisson equation for two aligned homogeneous slabs, we obtain an effective 2D Coulomb potential \(V_{\mathbf{k}}^{l_{\mathrm{h}}l_{\mathrm{e}}} = \frac{{e_0^2}}{{k\varepsilon _0\varepsilon ^{l_{\mathrm{h}}l_{\mathrm{e}}}({\mathbf{k}})}}\) with a dielectric function \(\varepsilon ^{l_{\mathrm{h}}l_{\mathrm{e}}}({\mathbf{k}})\), depending on the momentum transfer k and the overall composition of the heterostructure, Supplementary Note 1 for more details.
Dynamics
Equations of motion for the exciton polarization and the exciton occupations are derived taking into account all relevant interaction mechanisms in the low density regime. The carrierphonon matrix elements are taken from DFT calculations^{55} and are treated in analogy to refs. ^{22,23,52}. The interaction mechanism that distinguishes the exciton dynamics in a vdW heterostructure from the dynamics in a bare TMD monolayer is the tunneling of carriers between layers. This process is included via the tunneling Hamilton operator \(H_T = \mathop {\sum}\nolimits_{a,b} T^{ab}a_b^\dagger a_b\), where a, b are compound indices containing layer, band and momentum of the electron. The coupling element T is given by the overlap integral 〈Ψ_{a}V_{T}Ψ_{b}〉 of Bloch waves Ψ with the interlayer potential V_{T}. The latter can be separated into an outofplane component V_{z} and an inplane disorder potential V_{ρ}^{56}. The first is given by a step function, which is only nonzero within the region between the two layers and its value was fixed to 5 eV corresponding to the ionization energy of TMD monolayers^{57,58}. The tunneling matrix element can then be expressed as T^{ab} = V_{ρ}(k_{b} − k_{a})V_{z} 〈u^{a}u^{b}〉_{uc}, where u are the latticeperiodic parts of the Bloch waves, which are integrated over one unit cell (uc). This integral was obtained from density functional theory calculations, yielding an overlap of approximately 1 × 10^{−2}. The calculations were carried out using the gpaw package^{49}. The wave function was expanded on a grid and exchangecorrelation effects were represented using the PBE exchangecorrelation functional^{59}. The details behind this process are written in Supplementary Note 4. Finally, the Fourier transform of the inplane component of the disorder potential reads^{56} \(V_\rho ({\mathbf{k}}_b  {\mathbf{k}}_a) = \sqrt \pi L_{\mathrm{C}}/\left( {1 + \frac{{{\mathbf{k}}_b  {\mathbf{k}}_a^2L_{\mathrm{C}}^2}}{2}} \right)^{\frac{3}{4}}\) with L_{C} as the correlation length. It has been set to 1 nm in accordance with the excitonic Bohr radius^{60}, an approximation that applies for shortrange disorder^{61,62}.
Applying the Heisenberg equation of motion, we obtain the luminescence Bloch equations for vdW heterostructures
where the excitonic compound indices α, β contain center of mass momentum and electron/hole layers. The dynamics of the exciton polarization P_{α}(t) is determined by the Rabi frequency Ω_{α} containing the driving optical pump pulse and the decay processes stemming from radiative damping \(\left( {\gamma _{\mathrm{r}}^\alpha } \right)\) and excitonphonon interaction \(\left( {\Gamma _{\mathrm{P}}^{\alpha \beta }} \right)\). The latter has been obtained by truncating the electronphonon interaction by a second order BornMarkov approximation^{44,46}, giving rise to energydependent scattering rates. Those incorporate all excitonphonon scattering channels, i.e., all energy allowed exciton transitions, weighted by the excitonic scattering crosssections, electronphonon couplings and the occupations of involved phonon modes. The dynamics of the incoherent exciton occupation N_{α}(t) is determined by formation processes driven by phononassisted decay of the excitonic polarization, the radiative decay, and excitonphonon scattering \(\left( {N_\alpha ^{{\mathrm{scatt}}}} \right)\) driving the excited system towards an equilibrium Boltzmann distribution^{23,52}. Note that that radiative decay scales with N_{α,}_{0} = N_{α}δ_{Q,}_{0}, where only states within the light cone with a nearly zero centerofmass momentum Q = 0 can contribute. Finally, \(\Gamma _{\mathrm{T}}^{\alpha \beta }\) describes resonant tunneling between different layers. It depends on the occupation difference (ΔN_{αβ} = N_{β} − N_{α}) of the involved excitonic states and causes the formation of interlayer excitons. It was obtained by applying the same methodology as for \(\Gamma _{\mathrm{P}}^{\alpha \beta }\). The explicit form of the scattering rates is given in Supplementary Note 5.
Code availability
The code used to calculate the findings of this study is available from the corresponding author upon request.
Data availability
The data that supports the findings of this study is available from the corresponding author upon request.
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Acknowledgements
This project has received funding from the European Union’s Horizon 2020 research and innovation program under grant agreement No 696656. Furthermore, we acknowledge financial support from the Swedish Research Council (VR), Stiftelsen Olle Engkvist, and the Deutsche Forschungsgemeinschaft (DFG) through SFB 951 and the School of Nanophotonics (SFB 787).
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E.M. conceived and supervised the project. S.O. derived the theory and developed the code, based on previous work by S.B. and M.S. C.L., M.K., and P.E. performed the DFT calculations. S.O. wrote the main bulk of the manuscript, with the help of multiple revisions by mainly S.B. and E.M. T.K. provided fruitful discussions and experimental data to compare to.
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Ovesen, S., Brem, S., Linderälv, C. et al. Interlayer exciton dynamics in van der Waals heterostructures. Commun Phys 2, 23 (2019). https://doi.org/10.1038/s420050190122z
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