Abstract
Large spin–orbit coupling in combination with circular dichroism allows access to spinpolarized and valleypolarized states in a controlled way in transition metal dichalcogenides. The promising application in spinvalleytronics devices requires a thorough understanding of intervalley coupling mechanisms, which determine the lifetime of spin and valley polarizations. Here we present a joint theory–experiment study shedding light on the Dexterlike intervalley coupling. We reveal that this mechanism couples A and B excitonic states in different valleys, giving rise to an efficient intervalley transfer of coherent exciton populations. We demonstrate that the valley polarization vanishes and is even inverted for A excitons, when the B exciton is resonantly excited and vice versa. Our theoretical findings are supported by energyresolved and valleyresolved pumpprobe experiments and also provide an explanation for the recently measured upconversion in photoluminescence. The gained insights might help to develop strategies to overcome the intrinsic limit for spin and valley polarizations.
Introduction
The promising implementation of monolayer transition metal dichalcogenides (TMDs) in spinvalleytronics is based on the spinselective and valleyselective excitation of excitonic states. An important prerequisite for these applications is a profound understanding of the dynamics of optically excited spin and valley polarization. The challenge is to be able to control manyparticle mechanisms by coupling and mixing excitonic states of different spin and valley. In previous theoretical and experimental studies, it has already been shown that in addition to relatively slow intervalley spinflip scattering processes^{1,2,3,4,5} there is an efficient intervalley coupling mechanism via Coulomb exchange processes^{6,7,8,9,10,11,12,13,14,15}. This Coulombinduced dipole–dipole interaction couples resonant excitonic states in K and K′ valleys, giving rise to a decay of valley polarization on a picosecond time scale. However, the coupling is relatively small and only occurs as a secondorder process, since it requires a nonzero centerofmass momentum that first has to be generated by excitondisorder coupling^{14}, biexcitonic excitations,^{15} or excitonphonon coupling^{16}.
In this work, we reveal a new intrinsic and direct Coulombdriven intervalley coupling mechanism that is much more pronounced than the Coulomb exchange term and determines the optically accessible spinvalley polarizations in TMDs. We show that this interaction resembles the Dexter coupling between two spatially separated systems^{17,18}; however, now coupling different valleys in momentum space. It gives rise to an efficient intervalley transfer of coherent exciton populations between excitonic states with the same spin in different valleys. The theoretical predictions are supported by measurements of the valley polarization in energyresolved and valleyresolved femtosecond twocolor pumpprobe experiments, and also provide a microscopic explanation for recently published photoluminescence experiments measuring an upconversion from A to B excitons in the opposite valley^{19}.
Results
Theoretical model
The theoretical approach is based on the density matrix formalism, providing microscopic access to the timeresolved and energyresolved dynamics of microscopic quantities, such as the microscopic polarization \(p^{ij} = \left\langle {a_j^\dagger a_i} \right\rangle \), which is a measure for the transition probability between the states i and j^{20}. We use the picture of second quantization introducing electron creation and annihilation operators \(a_j^\dagger \) and a_{ i }. The states are described by the compound indices i, j including the electronic momentum k, the spinvalley index ξ_{s} and \(\bar \xi _s\) denoting opposite valleys with the same spin, and the electronic band index λ = v, c standing for the valence and the conduction band. First, we define the manyparticle Hamilton operator consisting of the interactionfree carrier contribution H_{0}, the light–matter interaction H_{l−m}, and the Coulomb interaction H_{C}. Then, we derive the TMD Bloch equation for the microscopic polarization p^{ij} by exploiting the Heisenberg equation of motion^{21}. This quantity couples to incoherent carrier densities \(\rho ^{ii} = \mathop {\sum}\nolimits_j {\kern 1pt} \left( {\left {p^{ij}} \right^2 + {\mathrm{\Delta }}N_x^{ij}} \right)\) including coherent \(\left( {\left {p^{ij}} \right^2} \right)\) and incoherent (\({\mathrm{\Delta }}N_x^{ij}\)) excitons^{16,20,21}. The microscopic polarization is directly excited via an external light source that introduces the coherent excitons, which then decay via phonons into incoherent excitons^{16}. In this work, we focus on the coherent exciton dynamics, where we include dephasing and the formation of incoherent excitons via phononscattering constants calculated on a microscopic footing^{22}. A microscopic description of the dissipation dynamics of incoherent excitons into dark states away from the light cone is beyond the scope of this analysis^{22}. Here we have included the incoherent exciton dynamics within a Lindblad formalism^{23} cf. Methods section.
The full TMD Bloch equations for the microscopic polarization and the incoherent excitons can be found in Eq. (3) in the Methods section. The equation includes a term describing the intrinsic Coulombinduced intervalley coupling of excitonic states in different valleys (last line in Eq. (3)). This interaction is formally analog to the wellknown Dexter interaction^{17}; however, now coupling states are in the same material but in different valleys. In contrast to the exchange term, no centerofmass momentum transfer between the valleys is necessary. Furthermore, since carriers in valence and conduction band carry the same geometric phase, the Coulombinduced intraband exchange of carriers between different valleys conserves the total angular momentum. Therefore, the process does not violate the threefold rotational symmetry of the system, leading to valleyselective circular dichroism.
The corresponding Dexterlike matrix element reads
It is inversely proportional to the valley distance \( {{\bf{\Delta }}_{\xi {\bar \xi}}}\)= K − K′ = 4π/3a_{0}. Although the valley separation in the momentum space is large, the Dexter coupling is still very important due to the remarkably strong Coulomb interaction in TMDs^{24,25,26,27,28,29,30,31,32,33}. Note that the distance between the valleys \( {{\bf{\Delta }}_{\xi {\bar \xi}}}\) enters the Coulomb potential, while it does not effect the excitonic wave functions \(\theta _{\bf{k}}^{n\xi _s}\) and \(\theta _{{\bf{k{\prime}}}}^{m\bar \xi _s}\). Furthermore, in the vicinity of the K valleys the coefficients \(C_{{\bf{k}},{\bf{k{\prime}}}}^{cc,\xi {\bar \xi }}C_{{\bf{k{\prime}}},{\bf{k}}}^{vv{\bar \xi }\xi }\) are given by a phase, which cancels the geometric phase of the excitonic wave functions^{34}, cf. Methods section for details on the applied theory. We can estimate the coupling analytically by exploiting \(\left {{\bf{\Delta }}_{\xi {\bar \xi }}} \right \gg \left {{\bf{k}}  {\bf{k{\prime}}}} \right\), which allows us to evaluate the sums over the excitonic wave functions using twodimensional 1 s excitonic functions. Assuming an excitonic Bohr radius of 1 nm we find \(\mathop {\sum}\nolimits_{\bf{k}} {\kern 1pt} \theta _{\bf{k}}^{1s\xi _s} = \theta ^{1s\xi _s}(r = 0)\) = \(\sqrt {\frac{2}{\pi }} \frac{1}{{{{a}}_{\mathrm{B}}}} \approx 0.6{\kern 1pt} {\mathrm{nm}}^{  1}\). Finally, assuming the TMD monolayer on a borosilicate substrate that is characterized via a highfrequency dielectric constant of ε = 4.6 and exploiting that for large q the internal manyparticle screening is negligibly small^{35}, we can estimate the coupling strength to be in the range of 50 meV reflecting the high importance of the Dexterlike mechanism.
Before evaluating the full TMD Bloch equations (Eq. (3) in the Methods section), we first consider a simplyfied model to illustrate the efficiency of the initially offresonant coupling. The Dexter interaction couples states with the same spin and in different valleys, i.e., A (B) excitons in the K valley couple to B′ (A′) excitons in the K′ valley, cf. Fig. 1. The nature of the Dexter coupling can be understood from a simple example: we assume two states with the eigenenergies E_{1} and E_{2} in different valleys. Valley 1 is optically excited with the Rabi frequency Ω_{1}(t) giving rise to the microscopic polarization p_{1}(t), which then interacts with the other valley via a coupling constant C_{12}, inducing a polarization p_{2}(t) in the unpumped valley 2. In Fourier space, the corresponding simplified Bloch equation can be solved analytically resulting in \(p_1(\omega ) = \frac{{{{\Omega }}_1(\omega )}}{{E_1  \hbar \omega  \left {C_{12}} \right^2{\mathrm{/}}\left( {E_2  \hbar \omega } \right)}}\) for the optically excited valley and \(p_2(\omega ) = \frac{{C_{12}{\kern 1pt} p_1(\omega )}}{{E_2  \hbar \omega }}\) for the unpumped but indirectly driven valley. We can now express the relative energydependent polarization difference as \(r_v^1(\omega ) = \frac{{p_1(\omega )  p_2(\omega )}}{{p_1(\omega ) + p_2(\omega )}} = \frac{{1  v_{21}(\omega )}}{{1 + v_{21}(\omega )}}\), where we have introduced the ratio between the indirectly and directly driven valley as \(v_{21}(\omega ) = \frac{{p_2(\omega )}}{{p_1(\omega )}} = \frac{{C_{12}}}{{E_2  \hbar \omega }}\). For resonant excitation to E_{1} (ħω = 0), we find that the reduction of the valley polarization is determined by \(\tilde v_{21} = v_{21}(\omega = 0) = \frac{{C_{12}}}{{E_2  E_1}}\), i.e., the ratio of the coupling constant and the energy separation of the states, e.g., analog to the spin–orbit coupling in TMDs. In contrast, for resonant excitation to E_{2}, we find that the valley polarization is completely inverted \(r_v^1(\omega ) \to  1\). This is a surprising result as the state E_{2} is only indirectly driven. The intervalley coupling leads to an energydependent dominance of the indirectly driven polarization. For the investigated exemplary material tungsten disulfide (WS_{2}), A_{1s} and B_{1s} excitonic states are separated due to the spin–orbit coupling by E_{1} − E_{2} ≈ 400 meV. Using the approximated value of 50 meV for the Dexter coupling, we obtain a scaling factor of \(\tilde v_{21} \approx 0.125\). This estimation suggests that when resonantly pumping the A_{1s} state, approximately \(1{\mathrm{/}}\left( {1{\mathrm{/}}\tilde v_{21} + 1} \right)\) ≈ 11% of the light field is indirectly absorbed by the B_{1s} state in the unpumped valley due to the Dexter coupling. This means that the induced valley polarization at the A exciton is decreased by 11% and inverted at the energy of the B exciton, where the Dexter coupling leads to an intervalley upconversion. In the case of MoS_{2}, where the energy separation of A_{1s} and B_{1s} is in the range of 150 meV, the same estimation predicts an indirect absorption of 30% at the energy of A_{1s} and B_{1s} excitons in the unpumped valley suggesting a crucial role of Dexter coupling, c.f. Supplementary Note 4 for details on the estimation for different TMDs and substrates, portrayed in Supplementary Fig. 6. Note also that the discussed simple scaling model only takes into account the 1s states and therefore presents a lower limit for the Dexter coupling. Taking into account all possible coupling excitonic states as well as nonlinear effects, Dexter coupling turns out to be even more efficient, as discussed below.
Now, we model ultrafast pumpprobe experiments by evaluating the Bloch equation (Eq. (3) in the methods section) for a nonlinear excitation. We calculate the temporal evolution of the coherent and incoherent exciton densities in both valleys and for both spin systems, and evaluate the corresponding exciton population \(N_s^\xi \) consisting of coherent and incoherent excitons for different delay times. Here \(N_n^\xi \) describes the total exciton population in the valley ξ and the excitonic state n. This allows us to compare the spin and valley polarization as a function of excitation energy, pump fluence, and delay time.
Experimental methods
To address the same quantities in experiments, we measure transient differential transmission of monolayer WS_{2} with energyresolved and valleyresolved femtosecond pumpprobe experiments. A WS_{2} monolayer on a borosilicate substrate is pumped by left circularly polarized laser pulses (200 fs), creating a population of excitons in a specific valley (K or K′). To probe the temporal and spectral dynamics of excitons we use linearly polarized pulses with a spectral bandwidth of 250 meV covering either the spectral range of A or B excitons. By analyzing the linearly polarized probe pulses transmitted through the monolayer for their circularly polarized components, we can simultaneously measure the dynamics in the pumped (“same circular polarization”, SCP) and unpumped valley (“opposite circular polarization”, OCP), cf. Supplementary Fig. 1. Observed changes of the spectral shapes at positive delay times are due to shifts of the exciton due to Coulomb renormalization, in particular biexcitons^{15}, and bleaching of the exciton absorption due to Pauli blocking^{14}. Integrating the spectra in the range of the exciton resonances (A or B) yields the dynamics proportional to the respective exciton densities in two valleys^{14} and provides the valley polarization degree at different delay times and energies. Additional details of the experimental setup are given in Supplementary Note 1.
Timedependent valley polarization
We first focus on the dynamics of the valley polarization (VP) \(P_v^n\) of A (n = A) and B excitons (n = B). The latter is defined as \(P_v^n\) = \(\left( {N_n^K  N_n^{K{\prime}}} \right){\mathrm{/}}\left( {N_n^K + N_n^{K{\prime}}} \right)\). Figure 2 illustrates the timedependent exciton populations in A and B states after resonant excitation of the A_{1s} exciton in the K valley with a temporally broad (200 fs) and spectrally narrow (20 meV) pump pulse. Simultaneously to the excitation of the A exciton (Fig. 2c), we find an increase in the B′ exciton population (Fig. 2a) resulting in an inversion of the VP (Fig. 2b). This VP follows the pump pulse in time, and it decays as soon as the pump pulse has vanished. The ultrafast dynamics of the VP clearly underlines the coherent nature of the observed process. This is consistently described via the coherent Dexterlike oscillation transfer between the valleys (Fig. 2b, d). The resonantly excited A exciton coherence drives the B′ exciton populations via the Dexterlike coupling. This induces an immediate formation of coherent B excitons in the unpumped valley, followed by a dephasing of the coherence via phonons. This dephasing induces incoherent excitons during the first 200 fs in both A and B′ states. The latter dissipate over the whole Brillouin zone, which leads to a decay of the valley polarization for B and A excitons, which has been characterized by a typical scattering constant γ_{K−K′} = 1 meV^{36}. Furthermore, we have extracted the lifetime of excitons from the experiments resulting in γ_{ x } = 50 μeV and γ_{ x } = 200 μeV for A and B excitons, respectively.
We find in both experiment and theory a strong inversion of the VP of B excitons in the range of −80% when the A_{1s} exciton is resonantly excited (Fig. 2b). However, the theoretical calculation predicts a VP of A excitons of 90%. This is qualitatively in line with the experimentally extracted A exciton time series for nearresonant pump pulses shown in Supplementary Note 2 (Supplementary Fig. 3 for A excitons and Supplementary Fig. 4 for B excitons). For A and B excitons, we find that the VP grows during the excitation process and decays rapidly on a subpicosecond time scale due to dissipation of the excitonic system. This instantaneous response to the pumping field strongly supports the Dexterlike transfer mechanism, which couples the excitonic coherences of A and B excitons in opposite valleys.
Figure 2e, f show the pumpprobe spectra for zero delay time, when the A_{1s} exciton is pumped close to the resonance. In Fig. 2e, the A_{1s} excitons in both valleys (K and K′) are probed. While the SCP curve shows a mainly positive signal, the OCP curve has a dispersive shape and is smaller which indicates the creation of a positive valley polarization^{14}. This situation is reversed when probing B_{1s} (Fig. 2f), far away from the initial pump resonance at A_{1s}. This shows that the VP for the offresonant B exciton is inverted with respect to the resonantly pumped A exciton. In addition, in Supplementary Fig. 5 (cf. Supplementary Note 3) we have portrayed the timedependent polarization of pumped and unpumped valleys. To underline these findings we analyze the population dynamics of the A_{1s} and B_{1s} excitons.
Pumpdependent valley polarization
After having revealed the timeresolved exciton population and valley polarization, we now present the impact of Dexterlike coupling when varying the pump energy and the pump fluence at time t = 0 fs. First, we analyze the energy dependence of the valley polarization shown in Fig. 3a, b. Here we calculate the generated exciton populations for different pump energies, from 20 meV below A_{1s} up to 50 meV above B_{1s}. In the experiments, we tune the excitation energy from 2.10 to 2.48 eV while the pump fluence is fixed at 6 μJ cm^{−2}. Figure 3a, b show a direct comparison between the experimental estimated (dots) and theoretical (lines) VP for different pump energies of A and B excitons, respectively. The numerical calculations are in line with analytically estimated intervalley transfer of ~10% for WS_{2}, e.g., the VP is close to +90% for resonant excitation and inverted in the offresonant case.
Though the theoretical model can reproduce well the trend of the experiment, it does not capture all aspects of the VP dependence for different pump energies. While the theory shows a minor inversion of the A exciton VP, the experiment only indicates a clear reduction, it does not reach the inversion.
In Fig. 3a, both experiment and theory exhibit a pronounced decrease in the VP of A excitons for pump energies in the vicinity of the B_{1s} exciton. We find the inversion only in the case shown in Fig. 3b. Here experiment and theory yield an inverted valley polarization for the B_{1s} excitons for pump energies resonant and close to A_{1s}, confirming the crucial role of the Dexterinduced intervalley exciton transfer. Though the theoretical model can reproduce well the trend of the experiment, it does not capture all aspects of the VP dependence for different pump energies. The quantitative discrepancy to the experiment might be due to the fact that the B exciton is in close proximity to a number of higher excitonic A states. This allows additional excitations and coupling processes including phononinduced scattering and Coulomb exchange coupling. Another difference lies in the experimentally broader absorption spectrum, which can be ascribed to inhomogeneous dephasing and defectinduced absorption, which can reduce the valley polarization.
To reduce the role of defects, we study the resonant excitation of the A exciton while probing the B exciton states for different pump fluences. Under these conditions the major part of the coherence is generated in resonance to the intrinsic states, which reduces the relative amount of defectassisted absorption. Furthermore, we can control the excitation density in experiment and theory in a comparable way. In Fig. 3c, d, we show the enhanced excitation of the B exciton as the pump fluence of the excitation pulse (resonant to the A exciton) is increased. For larger excitation densities, we observe in both experiment (Fig. 3c) and in theory (Fig. 3d) an increase in the excitation of B excitons in both valleys. Here the offresonantly excited B excitons in the pumped valley (purple line) and the B excitons induced via Dexter coupling in the unpumped valley (blue line) are compared. The latter becomes more efficient for stronger excitation, where the phasespace filling induces an asymmetry between the directly excited A exciton and the indirectly (weaker) excited B′ exciton. As a result, the oscillation transfer from the A exciton to B′ is more efficient than the backcoupling process. Therefore, stronger excitation leads to a higher relative increase of indirectly excited B excitons. As a result, we find a higher increase of the population of B excitons in the unpumped valley. Note that this result is in analogy to the findings of ref. ^{19}, where an increased signal of the B excitons was observed for larger CW pump fluences. The inversion of the VP is also evident in the experiment, when pumping the A exciton and probing the B exciton. Here theory and experiment show an inversion of up to −80%.
Photoluminescence
To further demonstrate the importance of Dexterlike intervalley coupling, we calculate the valley polarization in the case of continuously pumped WSe_{2} aiming at the explanation of the recently observed upconversion in ref. ^{19}. Here we selectively excite A_{1s} excitons with cw σ_{+} polarized light. We expect the Dexter coupling to be even more pronounced due to the static excitation leading to ultralong lifetimes of the coupling microscopic polarizations. Evaluating the Bloch equation, we have access to coherent exciton populations \(\left {p_n^{\xi _s}} \right^2\) that determine the coherent contributions to the photoluminescence^{37}
Here we assume a small dephasing of γ = 2 meV, which is in agreement to the measured excitonic linewidth at low temperatures in ref. ^{19}. We observe a strong photoluminescence from higher energetic states despite the initial excitation at the A_{1s} exciton corresponding to the experimentally measured upconversion^{19}. In particular, we find a considerable PL signal from the A_{2s} and the B_{1s} state (Fig. 4), which arises from the nonlinear coupling terms in the TMD Bloch equation (second line in Eq. (3) in the Methods section). They enable coupling of excitonic states with different quantum numbers n,m giving rise to an increasing upconversion to higher excitonic states at higher pump intensities (cf. orange arrows in Fig. 1a). As a result, we find a two orders of magnitude stronger upconversion from the A_{1s} to the A_{2s} state compared to the case without coupling (cf. dashed lines in Fig. 4). Furthermore, we calculate the PL signal for σ_{+} and σ_{−} polarized light. Without the Dexter coupling (dashed lines), only σ_{+} is emitted. Switching on the Dexter coupling, we also observe a considerable contribution of σ_{−} light. The VP for the A_{1s} is reduced to 28%, which is in good agreement with the experimentally estimated value of ~20%^{38}. Similarly to the pumpprobe experiments, we observe a Dexterinduced inversion of the VP of the B exciton. Here the contribution of the σ_{−} light is even higher and exceeds the value for σ_{+} light—again in excellent agreement with the experiment^{19}.
The Dexterlike intervalley coupling can explain the surprising valley inversion during and shortly after the optical excitation on the subpicosecond time scale, as well as for CW excitation on longer time scales. For the latter case, it may be an additional process, since one cannot rule out higherorder relaxation mechanisms occurring on a picosecond or a nanosecond time scale. An alternative explanation for the observed upconversion and inversion of the VP is given in ref. ^{19}, where the inversion is attributed to twophoton absorption followed by a relaxation mechanism under the influence of attractive boson interactions. While this process may be relevant in the CW experiment situation, it can be ruled out in the presented ultrafast pumpprobe experiment, where the formation of excitons and the formation of the inverted VP take place on the same time scale (<200 fs)^{22,39,40} (cf. Fig. 2).
Discussion
In summary, we have presented a joint theory–experiment study on the decay of optically accessible valley polarization in TMD. On the basis of a microscopic approach and highresolution twocolor pumpprobe experiments, we reveal the crucial importance of the Coulombinduced Dexterlike intervalley coupling mechanism. We find an efficient coupling of A and B excitonic states in different valleys giving rise to an intervalley transfer of coherent exciton populations. As a result, the valley polarization breaks down and is quasiinstantaneously inverted for B excitons when the A excitonic state is resonantly excited, and vice versa. We show that the upper limit for the achievable valley polarization is given by the ratio of the Dexter coupling strength and the energy difference between A and B excitons that is determined by the spin–orbit coupling. The theoretical prediction is confirmed in spectrally and valleyresolved pumpprobe experiments.
Methods
Theoretical formalism
On the basis of the density matrix formalism, we derive the TMD Bloch equation for the microscopic polarization p^{ij} within the coherent limit^{20,21,41}. To account for the crucial importance of excitons, we project the solution for the microscopic polarization into an excitonic basis using the transformation \(p_{\bf{k}}^{vc\xi _s} = \mathop {\sum}\nolimits_n {\kern 1pt} p_n^{\xi _s}\theta _{\bf{k}}^{n\xi _s}\). The appearing excitonic wave functions \(\theta _{\bf{k}}^{n{\kern 1pt} \xi _s}\) are calculated by solving the Wannier equation, an eigenvalue equation for excitons that also provides the excitonic energies \(E^{n\xi _s}\) of the state n^{20,21,31}. The TMD Bloch equation reads in the excitonic basis for the microscopic polarization:
inducing incoherent exciton densities via phonon scattering,
The dynamics of the excitonic polarization \(p_n^{\xi _s}\) is driven by an external optical field denoted by the Rabi frequency \({\mathrm{\Omega }}^{n\xi _s}(t) = \mathop {\sum}\nolimits_{\bf{k}} {\kern 1pt} \theta _{\bf{k}}^{n\xi _{s^ \ast }}{\bf{M}}_{\bf{k}}^{vc\xi _s} \cdot {\bf{A}}(t)\) including the vector potential A(t) and the optical matrix element \(M_{\bf{k}}^{{\bf{vc}}\xi _{\bf{s}}}\) weighted by the excitonic wave function \(\theta _{\bf{k}}^{n\xi _{s^ \ast }}\). The optical matrix element is analytically obtained by using nearestneighbor tightbinding wave functions^{31}. Nonlinear phasespace filling effects have also been taken into account in \({\tilde{\mathrm \Omega }}_{ij}^{n\xi _s}(t) = \mathop {\sum}\nolimits_{\bf{k}} {\kern 1pt} 2\theta _{\bf{k}}^{i\xi _s}\theta _{\bf{k}}^{j\xi _{s^ \ast }}{\mathrm{\Omega }}_{\bf{k}}^{n\xi _s}(t)\). Furthermore, the excitonic polarization oscillates with \(\tilde E_n^{\xi _s} = \left( {E_n^{\xi _s} + i\gamma _{{\mathrm{hom}}}} \right)\), where γ_{hom} corresponds to the radiative decay rate determining the homogeneous dephasing of the polarization^{16,42} (cf. Supplementary Fig. 2).
The second and the third line in Eq. (3) describe the Coulombinduced interexcitonic coupling (Γ term) and intervalley bandgap renormalization (Yterm). The first couples states of the same spin within one valley (Fig. 1), while the latter leads to a redshift of excitonic resonances in one valley due to the optically excited carrier occupations in the other valley^{14}. The corresponding coupling elements read
with the screened Coulomb matrix elements \(V_{\lambda \bar \xi _s\lambda {\prime}\xi _s{\bf{k{\prime}k}}}^{\lambda \xi _s\lambda {\prime}\bar \xi _s{\bf{kk{\prime}}}}\).
Finally, the last line in Eq. (3) describes the intrinsic Coulombinduced intervalley coupling of excitonic states in different valleys and has been discussed in the main text. Similarly to the Rabi frequency we also consider nonlinear phasespace filling effects scaling with \(\tilde D_{ij}^{nm\xi _s\bar \xi _s}\) = \(\mathop {\sum}\nolimits_{{\bf{kk{\prime}}}} {\kern 1pt} 2\theta _{\bf{k}}^{i\xi _s}\theta _{\bf{k}}^{j\xi _{s^ \ast }}D^{nm\xi _s\bar \xi _s}\).
The appearing Coulomb matrix elements are calculated within the nearestneighbor tightbinding approach and by exploiting an effective Keldysh screening \(\left. {V_q = V_q^{{\mathrm{2D}}}\varepsilon _q^{  1}} \right)\) which has been demonstrated to describe excitonic properties well in atomically thin twodimensional materials on a specific substrate^{28,31,43}. Note, however, that the regular Keldysh screening with ε_{ q } = 1 + r_{0}q can overestimate the screenings for processes with a large momentum transfer. Since Dexter coupling bridges different valleys, we have defined an effective Keldysh screening ε_{ q } = 1 + r_{0}q/((qa_{0})^{5/2} + 1) that has been adjusted to recent DFT calculations^{35}. Here a_{0} ≈ 0.3 nm denotes the lattice constant, r_{0} = ε_{⊥}d/(ε_{s}) is the screening length with the dielectric tensor of the monolayer ε_{⊥} ≈ 11.7^{28} and the surrounding substrate ε_{s} = ε_{1} + ε_{2}, and the thickness of the material is approximated to 0.7 nm, matching the experimentally measured spectral separation between A_{1s} and A_{2s} states. The general form of the screened Coulomb matrix elements reads:
where \(C_{{\bf{k}},{\bf{k{\prime}}}}^{ii,lj}\) are the valleydependent phase factors determined by lattice symmetry^{34}. Furthermore, the transferred momentum \(q = \left {{\bf{k}}  {\bf{k{\prime}}} + {\bf{\Delta }}_{\xi {\bar\xi }}} \right\) corresponds to the distance of the involved states located within one valley (Δ_{ ξξ } = 0) or in different valleys (\({\bf{\Delta }}_{\xi \bar \xi } = {\bf{K}}  {\bf{K{\prime}}}\)).
The values of the effective masses \(m_s^\lambda \) of the spindegenerated parabolic bands around the K points are taken from PBEbased DFT calculations^{44}. In WS_{2} they correspond to \(m_ \uparrow ^c = 0.36\), \(m_ \downarrow ^c = 0.27\) for the conduction band, and \(m_ \uparrow ^v = 0.36\), \(m_ \downarrow ^v = 0.50\) for the valence band.
Data availability
The data that support the findings of this study are available from the corresponding author upon request.
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Acknowledgements
This project has received funding from the European Unions Horizon 2020 research and innovation program under grant agreement No 696656—Graphene Flagship (E.M.). Furthermore, we acknowledge support by Stiftelsen Olle Engkvist (I.B.V.), Swedish Research Council (G.B. and E.M.) as well Deutsche Forschungsgemeinschaft through the SONAR EU Project (A.K.) and SFB 787 (G.B. and A.K.).
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Affiliations
Department of Physics, Chalmers University of Technology, Gothenburg, 41296, Sweden
 Gunnar Berghäuser
 , Ivan BernalVillamil
 , Paul Erhart
 & Ermin Malic
Institute of Physics and Center for Nanotechnology, University of Münster, Münster, 48149, Germany
 Robert Schmidt
 , Robert Schneider
 , Iris Niehues
 , Steffen Michaelis de Vasconcellos
 & Rudolf Bratschitsch
Institut für Theoretische Physik, Technische Universität Berlin, Berlin, 10623, Germany
 Andreas Knorr
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Contributions
G.B. and I.B.V. have performed the theoretical calculations of the valley polarization. The experimental studies have been carried out by the coauthors from the University of Münster. All coauthors have contributed to the writing of the manuscript and the interpretation of results.
Competing interests
The authors declare no competing interests.
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Correspondence to Gunnar Berghäuser.
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1.
npj 2D Materials and Applications (2018)
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