Strain-dependent exciton diffusion in transition metal dichalcogenides

Monolayers of transition metal dichalcogenides (TMDs) have a remarkable excitonic landscape with deeply bound bright and dark exciton states. Their properties are strongly affected by lattice distortions that can be created in a controlled way via strain. Here, we perform a joint theory-experiment study investigating exciton diffusion in strained tungsten disulfide (WS$_2$) monolayers. We reveal a non-trivial and non-monotonic influence of strain. Lattice deformations give rise to different energy shifts for bright and dark excitons changing the excitonic landscape, the efficiency of intervalley scattering channels, and the weight of single exciton species to the overall exciton diffusion. We predict a minimal diffusion coefficient in unstrained WS$_2$ followed by a steep speed-up by a factor of 3 for tensile biaxial strain at about 0.6\% strain - in excellent agreement with our experiments. The obtained microscopic insights on the impact of strain on exciton diffusion are applicable to a broad class of multi-valley 2D materials.

Strain is expected to have an important impact also on transport in TMDs, e.g. spatially inhomogenous strain acts as a driving force for exciton/carrier funneling [22,24,29], similarly to bias fields for charged particles [30,31] or gauge potentials for interlayer excitons in van der Waals heterostructures [32]. Even in the absence of additional driving forces, an initial spatially-localized excitonic occupation spreads in space due to the occupation gradient resulting eventually in a conventional exciton diffusion [33], as sketched in Fig. 1c. While in the presence of one single valley the diffusion is expected to be only quantitatively altered by strain through e.g. changes in the effective masses, the multi-valley exciton landscape in TMDs promises interesting strain effects. The diffusion is expected to be dominated by the most populated, energeti- * roberto.rosati@chalmers.se FIG. 1. Strain-dependent exciton diffusion. Biaxial strain implies (a) lattice deformations, which modify (b) the excitonic landscape by different spectral shifts of the bright KK and momentum-dark KΛ and KK excitons. This can introduce strain-dependent phonon bottlenecks for intervalley scattering. (c) As a result, an initial spatially-localized exciton occupation diffuses slower or faster depending on the applied strain.
cally lowest exciton valley [33][34][35], which might vary as a function of strain due to the strongly valleydependent energy shifts [ Fig. 1b]. Since different valleys have different valley-intrinsic diffusion coefficients [33], this leads to strain-induced changes of the overall diffusion, see Fig. 1c. Furthermore, straininduced energy shifts also change the possibility for phonon-induced intervalley scattering channels and may even result in phonon bottlenecks at specific strain values [ Fig. 1b]. This has also a direct impact on the efficiency of exciton diffusion.
Based on a fully quantum-mechanical approach and supported by experimental measurements, our work provides microscopic insights into the interplay of exciton diffusion and strain in the WS 2 monolayer as an exemplary TMD material. We microscopically address the evolution of optically excited, spatially localized excitons resolved in time, momentum, and space. We take into account bright and momentumdark excitonic states obtained by solving the Wannier equation under strain [27,28]. We predict nontrivial dependence of the diffusion on strain, showing a non-monotonic behaviour, where the overall diffusion is either dominated by specific dark excitons or determined by intervalley scattering. This leads to a steep speed-up of the diffusion upon small tensile strain values -in excellent agreement with our spatiotemporal photoluminescence experiments.

I. RESULTS
Theoretical approach: Starting with the unstrained single-particle dispersion [36], we implement the strain-induced variations of effective masses and band extrema E v [27]. For each strain value we then solve the Wannier equation [37][38][39][40] with a non-local Coulomb screening [41] to obtain a set of excitonic states |α ≡ |Q, v labelled by the excitonic valley v and the center-of-mass momentum Q. These states have the energy ε α = E v + 2 |Q| 2 /(2M v ), which depends on strain largely via changes in E v [27] [see also Fig. 1b], but also via changes in the total mass M v . Due to considerable energetic separations to higher excitonic states, we restrict our investigations to the ground 1s exciton, however taking into account all relevant electronic valleys and the resulting bright KK as well as momentum-dark KK , KΛ, KΛ , ΓK and ΓK excitons [3]. Here, the first and the second letter denote the location of the Coulomb-bound hole and electron, respectively. Now, we introduce the excitonic intravalley Wigner function N v Q (r, t), which summed over Q provides the intravalley spatial density [33]. At the spatial and temporal scales considered here, N v Q (r, t) can be directly interpreted as probablity of finding excitons with momentum Q in position r and valley v. An equation of motion for the spatiotemporal dynamics of excitons can be introduced by exploiting the Heisenberg equation [33]. The derived equation can then be transformed into Wigner representation [42,43] and reads in the low excitation regimė (1) The first term indicates the free evolution of excitons, while the second term takes into account the losses due to the radiative recombination γ within the light cone (δ Q,0 δ v,KK ) [38][39][40]44]. The first contribution in the second line of Eq. (1) describes the formation of incoherent excitons due to phonon-driven transfer from the excitonic polarization p Q≈0 (r, t). The process is driven by excitonphonon scattering rates Γ vv QQ describing scattering from the state |Q v to |Qv via interaction with phonons [39,40]. The last term in Eq. (1) de- , which is dominated by exciton-phonon scattering in the considered lowexcitation regime. It gives rise to a redistribution of the Wigner function in momentum toward a local equilibrium distribution, cf. the supplementary material for more details. Importantly, exciton-phonon scattering depends crucially on strain, mostly via the energies of the involved initial and final exciton states, while the variations of phonon energies or electron-phonon scattering are typically less relevant and are thus not considered here [27,45]. In particular, strain-induced energy shifts can lead to drastic changes in intervalley scattering via opening or closing of scattering channels [see phonon bottleneck in Fig. 1b]. In contrast, intravalley scattering is only slightly influenced by strain via changes in the effective mass M v [27,46,47].
In the steady-state regime the interplay between scattering-free propagation and scattering mechanisms implies that the spatial distribution evolves at first approximation according to the Fick's laẇ where decaying mechanisms have been omitted [48,49]. Here, Under the assumption of constant relaxation times τ v Q ≈ τ v the well-known steady-state relation D v = τ v k B T /M v can be recovered. The overall diffusion of the total excitonic spatial density N = v N v can be affected by strain in two major ways: (i) Strain changes the relative occupation weight N v /N of each valley. The overall Strain-dependent spatial evolution of the exciton density N in a WS2 monolayer on SiO2 after a pulsed spatiallylocalized optical excitation. The spatial evolution is shown along the x axis at the fixed time of (a) 2 ps soon after the optical excitation and after (b) 40 ps. For given strain values, we show the evolution of the corresponding (c) squared spatial width w 2 and the (d) associated effective diffusion coefficient D. diffusion will be dominated by the most populated valley and its own valley-intrinsic diffusion [33]. (ii) Strain changes the efficiency of the scattering channels through shifting the energies of initial and final scattering states, which crucially determine the velocity of exciton diffusion.
Strain-dependent exciton diffusion: Exploiting Eq. (1), we have a microscopic access to the spatially and temporally resolved dynamics of excitons in strained TMDs. In the main manuscript, we focus on the exemplary case of WS 2 monolayers on a SiO 2 substrate. In supplementary material, we also discus the exciton diffusion in strained WSe 2 , MoSe 2 , and MoS 2 monolayers. Figures 2(a)-(b) illustrate the strain dependent spatial evolution of the exciton density N (r, t) soon after the optical excitation [t=2 ps in (a)] and at a later time [t=40 ps in (b)]. We consider a pulsed optical excitation around t = 0 resonant to the bright exciton X 0 , and with a Gaussian spatial confinement corresponding to a full-widths-half-maximum of 800 nm in amplitude and a temporal duration of 200 fs. This generates an initial strain-independent excitonic distribution, cf. Fig. 2a. After few tens of picoseconds, the spatial distribution becomes broader and the width is strongly strain dependent, cf. Figs. 2(b). We find in particular a faster spatial spreading for given strain values s, e.g. s ≈ 0.6%. The differences in exciton diffusion are not monotonic in strain: Increasing the strain from negative (compressive) to positive (ten-sile) values, the diffusion initially becomes slower from −1% to 0% strain, then it speeds up steeply as the strain increases to 0.6% and finally it slows down again for larger strain values.
The spatial broadening of the exciton density N can be quantified introducing a width w whose squared modulus is proportional to the variance w 2 = r 2 N (r, t)dr/N . According to Fick's law, confined spatial distributions behave as N (r, t) ∝ exp −r 2 /w 2 (t) with w 2 (t) = w 2 0 + 4Dt [50,51], where w 0 is the initial width. It follows that also in the microscopic case when evaluating Eq. (1) one can define an effective diffusion coefficient (also called diffusivity) D = 1 4 ∂ t w 2 , i.e. as slope of the temporal evolution of the squared width w 2 . In Figs. 2(c) and (d) we plot the temporal evolution of squared width w 2 and the associated effective diffusion coefficient for three different values of strain. The squared width w 2 shows quickly a linear evolution for all strain values, although the slope varies crucially with strain indicating a strong strain-dependence of the exciton diffusion. We find that after an initial steep increase, the diffusion coefficient D(t) reaches a stationary value after a few ps. This corresponds to the transition from a ballistic regime, i.e. scattering-free evolution with a quadratic w 2 and linear D, to the conventional diffusive regime with a linear w 2 and stationary D(t) ≡ D [33]. Both strain-induced quicker ballisticto-diffusive transition and smaller D values are signatures of more efficient scattering channels, as seen e.g. at 0% strain. Due to the very short ballisticto-diffusive transition, it is the stationary diffusion coefficient D that determines the exciton diffusion in Fig. 2b. Now, we investigate the strain-dependence of the stationary diffusion coefficient covering a larger range of compressive to tensile biaxial strain values, cf. Fig. 3a. We find saturation values for the diffusion coefficient of 2.2 and 0.4 cm 2 /s for compressive strain around -1% and tensile strain above +1.6%, respectively. Furthermore, we predict a strongly non-monotonic strain-dependence including a relative minimum in the unstrained material (0% strain) and a maxium at about 0.6 % tensile strain. This can be understood by decomposing the valley-dependent contribution to the overall exciton diffusion. Thin lines in Fig. 3a show the diffusion coefficient obtained considering a reduced excitonic valley landscape. In all cases we have taken into account the optically excited KK excitons together with only KK (purple line) or KΛ and KΛ (i.e. KΛ ( ) , orange line) or ΓK ( ) (green line) excitons, Biaxial strain s (%) respectively. These thin lines reflect the scenario of valley-intrinsic diffusion, where the overall diffusion is dominated by the considered specific valley and the corresponding valley-specific diffusion determined by its total mass and intravalley scattering [33]. We find immediately that the saturation value of the diffusion at compressive (tensile) strain is determined by KΛ (ΓK) excitons, while the maximum in diffusion is governed by the contribution of KK exciton.
To better understand the valley-specific contribution to the overall diffusion, we show the strain dependent energy of the involved exciton valley minima in Fig. 3b [27]. We find that at larger compressive (tensile) strain values, KΛ (ΓK ( ) ) excitons are the energetically lowest and thus most occupied states. The same applies to KK excitons for strain values around 0.5%. As shown in our previous work [33], efficient intervalley scattering gives rise to one joint diffusion coefficient, however the weight of each valley is determined by its relative occupation. The latter depends crucially on the position of the bottom of the valley E v as well as on degeneracy (3 times larger for KΛ ( ) ) or total mass (smaller total mass leads to smaller population): This explains why KΛ or the very massive ΓK ( ) excitons still contribute even when E KK < E KΛ , E ΓK [see e.g. vertical solid line in Fig. 3a]. At very strong compressive strain it is KΛ which dominates the diffusion. Since at these strain values, the KΛ valley is already by far the energetically lowest state, the diffusion coefficient is not affected anymore by further straininduced energetic changes resulting in a stationary value. The same also applies to the situation at high tensile strain, where the ΓK valley when low enough becomes dominant also due to its large effective mass and high exciton occupation.
The predicted maximum in the exciton diffusion at about 0.6% is formed when moving from the strain regime, where KK excitons are energetically lowest states toward the regime governed by ΓK ( ) states. The latter exhibit a considerably slower diffusion due to a much larger effective mass. The dip toward the unstrained case cannot be explained by just considering valley-intrinsic diffusion and is due to intervalley scattering, which will be discussed separately in the next section. Note that the abrupt increase in the diffusion coefficient in the considered hypothetical two-valley system dominated by ΓK ( ) (thin green line) and KΛ (thin orange line) excitons for strain values around 0.5% is due to the increasing impact of KK excitons exhibiting a much smaller effective mass and thus a much larger diffusion coefficient [33]. The thin purple line does not show this steep increase, since KK and KK excitons have a very similar dependence on strain modifying their separation by only approximately 5 meV per percentage of biaxial strain (cf. solid and dashed purple line in Fig. 3b).
We compare our microscopic results with measurements of the exciton diffusion in an uniaxially strained WS 2 monolayer. To this end, the monolayer is placed on a flexible PMMA substrate and a homogeneous, uniaxial, tensile strain from 0 % to 1 % is applied via the bending method [9,14]. The similar optical properties between PMMA and SiO 2 make a comparison of experiment and theory possible. Since we apply uniaxial strain, the experimentally determined strain values are related to the biaxial ones in the theory by comparing the energy shifts of the KK ( ) . The exciton diffusion is measured by spaceand time-resolved photoluminescence after applying a spatially localized optical excitation at 2.10 eV (see supplementary information for details). We find an full no ac. no op. excellent agreement between theoretically predicted and experimentally measured diffusion coefficients in the experimentally accessible strain region, cf. Fig. 3a. We see an increase in the diffusion coefficient from approximately 0.6 to 2.2 cm 2 /s when varying the strain from 0 to 0.5%. The slope of this increase strongly depends on the formation of the minimum in the diffusion coefficient D appearing approximately for the unstrained case.
Impact of intervalley scattering: While the overall diffusion discussed so far was dominated by one exciton valley with the lowest energy and largest occupation (valley-intrinsic diffusion), there are strain regions where multiple valleys have similar energies, cf. Fig. 3b. This occurs e.g. at small compressive strain values, where KΛ (solid orange) and KK excitons (dashed purple) cross, cf. the vertical dashed line. Here, intervalley scattering turns out to play a crucial role resulting in a minimum in the exciton diffusion, cf. Fig. 3a. Now, we investigate the origin of the predicted dip in the diffusion coefficient for the unstrained WS 2 monolayer, cf. Fig. 3a. In particular, we study the role of intervalley scattering between KK and KΛ excitons, whose energies are the lowest and cross in the strain region, where the minimum appears, cf. Fig. 3b. The intervalley scattering between KK and KΛ excitons is driven by absorption or emission of high-momentum M phonons. These phonons can be approximated as nearly dispersion-free including two acoustic modes with the energies of 16.5 and 22.7 meV. The corresponding optical M phonons have larger energies between 40 and 50 meV [52]. When the strain-induced separation between valleys becomes larger than these phonon energies, excitons from the energetically lowest states cannot scatter any longer out of these states, cf. Fig. 1b. To better visualize the condition for this phonon bottleneck, we show the Wigner function for the KK excitons for three different strain values, where KK are the energetically lowest states, cf. Figs. 4(a)-(c). Here, the thin horizontal lines show the minimum momentum |Q x | required for the absorption of acoustic M phonons. In the unstrained case [ Fig. 4a], the two momentum-dark exciton valleys are aligned [ Fig. 3b], thus all intervalley channels into KΛ states are possible. The situation changes at 0.25 % strain [ Fig. 4b], where the energetic misalignment is already large enough that only a small portion of the most occupied excitons is able to scatter into KΛ states. At 0.5% strain [ Fig. 4c], there are almost no occupied states fulfilling the condition for intervalley scattering resulting in a phonon bottleneck. Now we address the impact of this phonon bottleneck for the strain-dependent evolution of the exciton diffusion. In Fig. 4d, we compare the full calculation (also shown in Fig. 3a) with the case, where the KK -KΛ intervalley scattering via acoustic or optical phonons is switched off, respectively. We see that optical phonons have in general a minor effect, since their energy is relatively high resulting in a small phonon occupation and thus a negligbly small phonon absorption. We also find that scattering with acoustic phonons does not play a role for strain values |s| 0.5%. The reason is the phonon bottleneck illustrated in Fig. 4c. However, in the strain region |s| 0.5%, switching off scattering with acoustic phonons suppresses the appearance of the dip in the exciton diffusion. This illustrates the crucial role of intervalley scattering with acoustic phonons for the formation of the predicted minimal exciton diffusion in unstrained WS 2 monolayers.
Note that the scattering with M phonons connecting KK and KΛ excitons is particularly strong according to DFT calculations [52]. In contrast, the corresponding phonon-induced intervalley scattering connecting ΓK ( ) and KK ( ) excitons is much weaker. This together with the much larger effective mass and smaller diffusion coefficient in the ΓK ( ) valley does not lead to a similar dip in the exciton diffusion around 1.5% strain, where KK -and ΓK ( ) states are energetically lowest and cross [ Fig. 3b]. Note that the efficiency of the KK -KΛ intervalley scattering together with the small energy of acoustic M phonon leads to a persistence of the dip also at smaller temperatures, as shown in the supplementary material.
Finally, we briefly discuss the strain-dependence of exciton diffusion in other TMD materials. While the physical mechanisms are the same, in Mo-based monolayers one needs a larger tensile or compressive strain to reach the multi-valley features discussed for WS 2 . In particular, the situation in MoS 2 is more involved due to the initial excitonic landscape in the unstrained case, where the position of E ΓK ( ) strongly depends on the relative distance ∆E KΓ of the valence-band maxima located in K and Γ. The latter is still being controversially discussed in literature [27,36,53,54]. We predict a qualitatively different strain-dependent exciton diffusion in MoS 2 depending on ∆E KΓ , cf. the supplementary material. An experimental study of exciton diffusion in MoS 2 could thus provide a better understanding of the relative position of the bright KK and the dark ΓK ( ) excitons.

II. DISCUSSION
In conclusion, the presented joint theoryexperiment study provides new microscopic insights into strain-dependent exciton diffusion in TMD monolayers. We find that the diffusion becomes faster or slower with strain in a non-trivial and nonmonotonic way. This is a result of the interplay between lattice-distortions and the remarkable multi-valley excitonic landscape in TMDs. Strain-induced shifts of exciton energies change the relative energy separations in the excitonic landscape of bright and momentum-dark excitons. This has an immediate impact on which state is dominant and governs the overall diffusion coefficient. In particular, we predict a dip in the diffusion for unstrained WS 2 monolayers that we microscopically ascribe to intervalley scattering with acoustic phonons. This dip is followed by a large increase of exciton diffusion by a factor of 3 for tensile biaxial strain of up to 0.6%. The theoretical prediction is found to be in excellent agreement with spatiotemporal photoluminescence experiments. Overall, our study provides microscopic insights into the impact of strain on exciton diffusion in technologically promising 2D materials and uncovers the underlying fundamental intraand intervalley scattering processes involving bright and momentum-dark excitons.
Supporting Information Additonal details on microscopic modeling and measurements on strain-dependent exciton diffusion are included. Furthermore, temperature dependence as well as exciton diffusion in different strained TMD monolayers are discussed.
Acknowledgements: This project has received funding from the Swedish Research Council (VR, project number 2018-00734) and the European Unions Horizon 2020 research and innovation programme under grant agreement no. 881603 (Graphene Flagship). The authors thank Zahra Khatibi and Maja Feierabend for fruitful discussions.

Supporting Information: Strain-dependent exciton diffusion in transition metal dichalcogenides 1 Wannier equation and exciton-phonon scattering
The energetic position of the excitonic energies E v is given by where m v is the reduced exciton mass in valley v, i.e. 1/m v = 1/m e v + 1/m h v with m e,h denoting the effective mass of electrons and holes in the excitonic valley v. Here, m v and E 0 v crucially depend on strain. We take the starting parameters for the unstrained single-particle dispersion from Ref. [1] and use the strain-dependent variation of single-particle parameters provided in Ref. [2]. Furthermore, Ψ v (k) appearing in Eq. (1) is the excitonic wave function in momentum space, while W q is the Coulomb interaction, for which we have used a modified form of the potential for charges in a thin film of thickness d surrounded by a dielectric environment [3]. Taking into account anisotropic dielectric tensors and solving the Poisson equation with the boundary conditions described above yields W q = V q / scr (q) with the bare 2D-Fourier transformed Coulomb potential V q and a non-local screening scr (q) = κ 1 tanh 1 i account for the parallel and perpendicular component of the dielectric tensor i of monolayer (i = 1) and environment (i = 2), whose parameters have been taken from Refs. [4] and [5], respectively. In this work, we consider the intrinsic undoped regime, where the effect of trions [6][7][8][9] is negligible. Moreover, we focus on intrinsic excitonic properties, hence we disregard defects/disorders, which are expected to lead to a quantitative slow-down of the overall propagation without affecting qualitatively the strain dependence. Finally, we also neglect the impact of spindark states [8,10], which are known to appear in PL spectra under certain conditions, but since they have similar properties as KK excitons (in terms of dispersion relation), they are expected to only lead to quantitative changes.
The scattering-induced dynamics for the Wigner function can be written asṄ v Q (r, t) The scattering rates Γ vv QQ of the exciton-phonon coupling read in second-order Born-Markov approximation [11,12] Q v addresses the position in the Brillouin zone of the bottom of valley v, ± refers to emission/absorption of phonons, ξ labels the phonon modes, ω ξq provides the energy of phonon ξ with momentum q and n ξ,q describes the equilibrium phonon distribution. We take into account longitudinal and transverse acoustic (LA, TA) and optical (LO, TO) modes as well as the out-of-plane A 1 optical mode, which provide the most efficient scattering channels [13]. The exciton-phonon scattering coefficients G can be written as [3] G vv , where p distinguishes between electrons/holes, F is the form fac- for p = e, h, respectively, while g are the carrier-phonon scattering coefficients which are approximated with the generic form of a deformation potential g vv pξq = 2ρAω ξq D pξq . Here, ρ denotes the surface mass density of the monolayer (whose small variations with strain have been omitted) and A the area of the system. For the coupling constant D pξq we adopt the approximations deduced from DFPT calculations in Ref. [13], where long range acoustic phonons couple linear in momentum D ξq ≡ D (1) ξ q, while optical phonons and short range acoustic modes couple with a constant strength D ξq ≡ D (0) ξ in the vicinity of high symmetry points. The phonon energies as well as deformation constants D (0) and D (1) for all possible intra and intervalley scattering channels are listed in Ref. [13], including longitudinal and transverse acoustic (LA, TA) and optical (LO, TO) modes as well as the out-ofplane A 1 optical mode. Finally, for describing the radiative decay [3,14] we use γ = 4.5 meV in accordance to a recombination-induced broadening of the linewidth of 4.5 meV in FWHM [2]. We find, however, that the radiative recombination rate has a negligible role on the qualitative aspects of the investigated effective diffusion in h-BN encapsulated WS 2 monolayers [15]. For the same reason we have also neglected strain-induced changes of γ, which are typically weak [2,16].

Exciton diffusion in WS at lower temperatures
We study the impact of temperature on the predicted dip in the exciton diffusion for unstrained WS 2 , cf. Fig. 3(a) in the main text. A decrease in temperature is accompanied by a spectral shift of excitonic resonances, often described via the Varshni equation [17]. The shift originates from an interplay of temperature-induced changes in the electron-phonon interaction (polaron shift) and a the lattice contraction at smaller temperature. We denote the latter effect as temperature-induced strain s T . To which extent the shift is due to one or the other contribution is still under debate in literature, see e.g. [18] and references therein. This goes beyond the scope of the present work. For the moment, we neglect this effect, i.e. s T = 0, and we show the exciton diffusion as a function of temperature in Fig. 1(a). We find that the predicted dip in the diffusion becomes narrower for decreasing temperatures. This can be easily explained by studying Figs. 4(a)-(c) in the main manuscript: While the position of the thin red line is assumed independent of temperature, the width of the exciton occupation in momentum space decreases with temperature. As a result, at smaller temperatures already at s =0.25 % the intervalley scattering via absorption of TA modes is suppressed due to a narrower distribution in momentum, and at this strain value the corresponding dip already saturates, as shown in Fig. 1(a).
Interestingly, the minimum of the dip at s ≈ 0% seems independent of temperature. To understand this we study the diffusion coefficient from the Fick's law by considering only one phonon mode and assuming τ Q ≡ τ ph . The resulting diffusion D ph ≡ τ ph k B T /M behaves with temperature as D ph ∝ Exp ω ph k B T T ≡ f ph (T ). In the range of temperatures between 150 and 300 K f TA varies weakly. This is not the case anymore when considering more energetic phonon modes. Here, f TO would increase at smaller temperatures, as expected from the naive consideration that weaker scattering leads to a faster diffusion. Since acoustic phonons are crucial for the generation of the dip (as discussed in the main manuscript), we have an almost temperature-independent dip minimum in the considered temperature range.
Finally, in Fig. 1(b) we consider a non-zero background strain s T such that it gives rise to the half of the observed temperature-induced shift [19]. This results in a shift of the dip in the exciton diffusion to higher strain values. This means that the non-monotonicity appears in the strain range that is experimentally accessible at room temperature. Although with state-of-the-art techniques it is complicated to induce strain at smaller temperatures, Fig. 1(b) predicts that the interplay between temperature-induced lattice shrinking and additional strain could in principle lead to observations of a pronounced dip in the exciton diffusion also at experimentally accessible tensile strain values.

Exciton diffusion in other TMDs
We now briefly discuss the impact of strain on exciton diffusion in other TMD materials than discussed in the main manuscript. In MoS 2 , the separation between the valence-band maxima located at the Γ and the K valley is still being debated in literature, cf. [1,2,20,21]. Assuming a small separation of 46 meV [1], we find that already at s ≈ 0 the diffusion is saturated, cf. the solid line in Fig. 2(a). This can be led back to the vicinity of the Γ and the K valley in the singleparticle dispersion for the unstrained case, reflecting the close position of the excitonic energies shown in Fig. 2(b). As a consequence, for tensile strain almost all excitons are located in the ΓK ( ) state. This means that the exciton diffusion is almost unchanged for experimentally accessible strain values between 0 and 0.5%. The behaviour is more involved for compressive strain, where we observe the maximum in the region where KK excitons are clearly energetically lowest states. Exciton diffusion further decreases for higher values of compressive strain, i.e. moving toward the regime, where KΛ ( ) excitons are dominant. The situation for tensile strain 1% values changes, if a larger Γ-K separation is assumed. A value of 145 meV [2] leads to a decrease of the diffusion coefficient upon increase of strain, cf. the dashed red line in Fig. 2(a). This occurs in view of the larger separation between E ΓK ( ) and E KK ( ) with the former being now able to affect the overall diffusion only if larger values of strain are applied. The increased separation also affects the maximum value of the diffusion as obtained for small compressive strain, which is now approximately two times larger than in the case of the smaller separation, again reflecting a smaller contribution stemming from the dark ΓK and ΓK excitons.
For MoSe 2 , exciton diffusion also changes only weakly for small tensile strain (Fig. 2(c)), however the microscopic origin is very different w.r.t. the solid line in Fig. 2(a), as can be seen in the excitonic energies shown in Fig. 2(d). At around 0%, KK excitons are energetically lowest states and dominate the diffusion. Only at larger tensile strain values (s ≈ 2%) ΓK ( ) excitons become dominant giving rise to a decrease of the diffusion coefficient -similar to the behaviour at slightly smaller tensile strain discussed for WS 2 in the main manuscript. Note that in the strain regimes where the overall diffusion is dominated by KK ( ) the absolute value of the diffusion coefficient is much smaller than in the analogous situation in WS 2 [see maximum at 0.6% of Fig.  3(a) of the main manuscript], reflecting in particular a much stronger effectiveness of the excitonphonon scattering with intravalley modes in MoSe 2 w.r.t. WS 2 [2].
Finally, we find that WSe 2 has a similar strain behaviour as WS 2 for compressive strain and  small tensile strain values, cf. Fig. 2(e). However, the decrease of the diffusion coefficient at larger tensile strain does not take place due to different energy positions of E ΓK ( ) and E KK ( ) , Fig. 2 To investigate the exciton diffusion in strained WS 2 , a monolayer is mechanically exfoliated onto a Polymethyl methacrylate (PMMA) substrate (Röhm GmbH Plexiglas 0F058) of 0.5 mm thickness. By bending the substrate, the monolayer on top is uniaxially strained by up to 0.96 % [22]. Excellent strain transfer is verified by the observed red-shift of the photoluminescence (Fig. 3) with a gauge factor of 50 meV/%. To directly compare the experimental and theoretical data, the experimental uniaxial strain values have to be related to the biaxial values from the theory. Both uniaxial and biaxial tensile strain cause similar energetic shifts of the excitonic transitions, and only their gauge factors differ. For exciton diffusion, the relative energetic shift of excitonic transitions is the relevant parameter. Therefore, those experimental uniaxial strain values correspond to the theoretical biaxial ones, which yield the same exciton shift. To this end, the experimental strain values are multiplied by the ratio of the measured uniaxial gauge factor and the calculated biaxial gauge factor [cf. Fig. 3
In this way, the experimental and theoretical gauge factors match, i.e. the experiment spans a biaxial strain range from 0 % to 0.42 %.

Exciton diffusion measurement and analysis
To measure exciton diffusion, the monolayer is optically excited using 250 fs pulses at 2.10 eV from a 40 MHz femtosecond fiber laser system [23]. The monolayer PL is measured in a transmission microscope using a 50x (NA = 0.55) objective lens for excitation, resulting in a focus size of 700 nm at the sample. For detection, we use a 100x (NA = 0.9) objective lens. A streak camera (Hamamatsu C10910-05) detects the photoluminescence with a temporal resolution in the picosecond range. The detection spot of the PL can be scanned across the sample by moving the detection objective lens.
The residual pump light is blocked using a long-pass filter at 2.07 eV. From the measurement we reconstruct an image sequence of the spatial distribution of the monolayer photoluminescence with a temporal resolution of ≈ 5 ps and a temporal range of up to 2 ns (Fig. 4).
Kulig et al. [24] have demonstrated that exciton diffusion in monolayer WS 2 depends on the excitation density, except for very low powers in the nanowatt regime. For higher powers, so-called "halos" are observable in the exciton photoluminescence. To be able to observe the impact of strain on diffusion coefficients predicted by the theory, exciton diffusion has to be measured in the regime, where it is independent of the excitation density.
We use two different approaches to measure exciton diffusion. In the first, we are in the excitation regime, where exciton diffusion shows the formation of halos (excitation power 2 µW). In this case, excitons are quickly pushed away from the irradiated into a pristine area of the substrate, where they can diffuse freely.
The experimental data is fitted using a donut-like function (Fig. 4): I PL (x, y) = A 0 · e −((x−x 0 ) 2 +(y−y 0 ) 2 −d/2) 2 /2σ 2 + e −((x−x 0 ) 2 +(y−y 0 ) 2 +d/2) 2 /2σ 2 Here, I PL is the intensity of the photoluminescence at the specific coordinate, x 0 and y 0 denote the center of the donut, A 0 the amplitude, d the diameter and σ the width of the ring. In this excitation regime, exciton transport consists of an excitation density-dependent part, creating the halo and the density-independent exciton diffusion. The first part dominates the temporal evolution of the parameter d, while the latter one can be extracted from the increasing width of the halo ring w, especially at long delay times, when the exciton density has already decreased by 1 to 2 orders of magnitude. The time evolution of w 2 = 2σ 2 is analyzed for delay times > 700 ps and the diffusion coefficient D can be calculated as follows: In the second approach, we use the low excitation power regime, where exciton diffusion is independent of the excitation density (excitation power 5 nW) and no halo formation is visible. To decrease the measurement time we measure the diffusion only in one dimension instead of the entire PL image, i.e. we measure a horizontal cut through the center of the excitation spot, which has a Gaussian shape (no halo). Therefore, the experimental data can be modeled by a Gaussian function with the width w and the diffusion coefficient D can be calculated by Figure 5 presents the results of the two measurement approaches and compares them to the theoretical prediction. Both approaches yield the same qualitative trend and show a very good agreement with the theory.