## Introduction

The mechanical exfoliation of atomically thin nanomaterials has initiated a new research field1. Since then, in particular transition metal dichalcogenides (TMDs) have been intensively studied2,3,4. The family of these truly two-dimensional materials exhibits a strong Coulomb interaction giving rise to the formation of excitons, which are Coulomb-bound electron-hole pairs. Having binding energies of a few hundreds of meV, they dominate the optoelectronic response of TMDs even at room temperature3. Recent studies have shown that besides the regular bright excitons accessible in optical spectra, dark excitonic states play a significant role for understanding the optical response and non-equilibrium dynamics in TMDs5,6,7,8. Here, the Coulomb-bound electrons and holes either exhibit an opposite spin or are located at different valleys (K, K$${}^{\prime}$$, Λ) in momentum space. The required spin-flip or momentum-transfer cannot be provided by photons, making these states dark and difficult to access in optical spectra. Very recently, momentum-dark KΛ excitons (with the hole localized at the K and the electron at the Λ valley) could be visualized in angle-resolved photoemission spectroscopy9.

Being atomically thin, flexible, and exhibiting strong absorption and ultrafast non-equilibrium dynamics, TMDs have been considered as promising candidates for next-generation opto-electronic applications including light-emitting, detecting, and harvesting devices10. Optical excitation, relaxation dynamics and transport of carriers represent the crucial many-particle mechanisms governing the efficiency of such devices. The optical response8,11 and non-equilibrium dynamics12 of excitons in TMDs have been well studied. Exciton transport, however, has still remained elusive. In conventional materials, external electrical fields are used to control the transport of optically excited carriers13. However, deeply-bound excitons as neutral particles (consisting of negatively charged electrons and positively charged holes) are only weekly affected by in-plane electrical fields, e.g., via Stark shifts and exciton dissociation. Only in the case of spatially separated interlayer excitons in van der Waals heterostructures, an electrical control of exciton transport has been demonstrated14 via the energetic tuning allowed by the dipole moment15. Thus, strain engineering has been introduced to manipulate the propagation of excitons in atomically thin semiconductors and could be exploited to e.g., boost the energy conversion efficiency in photovoltaics devices16. TMDs are remarkably sensitive to strain resulting in a large tunability of their energy landscape. In particular, strain was shown to significantly shift exciton resonances17,18. As a result, strain-induced energy gradients allow us to control and manipulate exciton propagation. Excitons have been observed to funnel up to a micrometer range towards spatial regions with the strongest strain gradient, where the energy is minimal16,19,20,21,22,23,24.

Tungsten-based TMD monolayers (WS2, WSe2) are indirect semiconductors, where dark excitons are the energetically lowest states9,25. Although these excitons are not directly accessible in optical spectra, they have been demonstrated to crucially influence optics, dynamics, and even diffusion in TMDs3,4,26. However, their role on exciton funneling has still remained in the dark. In this work, we combine spatially and temporally resolved photoluminescence (PL) measurements with a microscopic many-particle theory to investigate exciton funneling in presence of spatially inhomogeneous strain gradients in the exemplary case of a WS2 monolayer. Surprisingly, we find an inverse exciton funneling (anti-funneling) toward spatial regions with low strain (Fig. 1)—opposite to the conventional funneling behavior observed so far20,21,22,23,24. Shedding light on the many-particle processes behind the funneling process, we explain this counter-intuitive behavior. In presence of strain, lattice distortions induce valley-dependent variations of energies, i.e., bright and momentum-dark excitons shift in different directions (Fig. 1c). In particular, the energy of momentum-dark KΛ excitons increases with strain, and thus this type of excitons funnels opposite to the strain gradient toward spatial regions with minimal strain (Fig. 1e). Our joint experiment-theory study represents a major advance for the microscopic understanding of exciton funneling in atomically thin semiconductors including the entire landscape of bright and dark excitons.

## Results

### Spatiotemporal exciton funneling

We investigate spatiotemporal exciton dynamics in a strongly spatially dependent strain profile. To create such a strain gradient, we use the concept of a nanopatterned surface, which has previously been exploited to create single-photon emitters in TMD monolayers27,28,29. We transfer the WS2 monolayer on an array of polymer micropillars (120 nm height and 2 μm width) on a borosilicate substrate (cf. Methods and Supplementary Information). Mechanical strain strongly modifies the electronic band structure and shifts exciton resonances (Fig. 1b). As a result, the applied strain can be mapped by measuring the energetically lowest exciton energy in spatially resolved optical absorption spectra (cf. SI for more details). Stamping the monolayer onto the micropillar array creates an inhomogeneous strain profile with the highest strain values directly between two pillars (Fig. 2a), allowing us to study exciton propagation in spatial regions of the highest strain gradients without interfering with the micropillar.

To investigate the time-resolved exciton propagation in the experiment, we spatially map the photoluminescence (PL) from the WS2 monolayer with picosecond time resolution. We pump the monolayer with ultrashort laser pulses (≈200 fs) at 2.1 eV. An objective lens focuses the laser onto a fixed sample position. By scanning a second objective lens in transmission geometry, we image the PL distribution onto a streak camera. In that way, we achieve a temporal and a two-dimensional spatial resolution of the PL (cf. Methods for details on the experimental setup and measurement routine). From this data, we obtain ultrafast PL movies tracking exciton motion within the strain profile created by micropillars (see the movies in the SI). Figure 2a–c shows cuts of the spatiotemporal dynamics highlighting three different excitation spots (A, B, C). Here, the orange color illustrates the exciton PL created by optical excitation (0 ns), while purple corresponds to the drifted exciton PL due to the funneling after 1.8 ns (cf. Supplementary Video S1 for the whole evolution). The blue colormap represents the strain field induced by the micropillars. For better visualization of the funneling process, we also show the PL and the strain profile along the x axis, cf. Fig. 2d–f. Interestingly, due to the stamping procedure a maximum strain of 0.6% is reached at the center between two micropillars, rather than directly at the position of the pillars. In these regions, the monolayer is lying flat on the substrate, which ensures a homogeneous dielectric environment. First, we optically pump at a location where the strain profile is flat (spot A). As a result, we find no observable exciton propagation (orange and purple line overlap in Fig. 2d). At locations B and C, we excite excitons in spatial regions with a significant strain gradient (dashed line in Fig. 2d–f). Here, we find a clear exciton drift, which is larger for a higher strain gradient (cf. Fig. 2e, f, and Supplementary Information). Unexpectedly, we observe exciton funneling towards low strain regions in contrast to conventional funneling20,21,22,23,24.

### Dark-exciton anti-funneling in WS2

To be able to explain this intriguing and unexpected anti-funneling behavior, we complement our experiments with microscopic calculations of the spatiotemporal dynamics of excitons. Here, we exploit a generalized drift-diffusion equation for exciton densities Nv(r, t) yielding

$${\dot{N}}_{v}({{{{{{{\bf{r}}}}}}}},t)= \;\nabla \left(\right.D({{{{{{{\bf{r}}}}}}}})\nabla {N}_{v}({{{{{{{\bf{r}}}}}}}},t)\left)\right.+\frac{1}{{k}_{B}T}\nabla \left(\right.D({{{{{{{\bf{r}}}}}}}}){N}_{v}({{{{{{{\bf{r}}}}}}}},t)\nabla {E}_{v}({{{{{{{\bf{r}}}}}}}})\left)\right.\\ -{\gamma }_{t}{N}_{v}({{{{{{{\bf{r}}}}}}}},t)+{\dot{N}}_{v}^{{{{{{{{\rm{th}}}}}}}}}({{{{{{{\bf{r}}}}}}}},t).$$
(1)

Here, we explicitly take into account the entire exciton landscape, where the valley index v includes the energetically lowest 1s states of bright and momentum-dark excitons (such as KΛ, KK’, ΓK). The total exciton density N(r, t) = ∑vNv(r, t) is initially spatially localized with a Gaussian profile reflecting the confined optical excitation pulse in the experiment. The spatial dependence of all other quantities stems from the applied strain gradient s(r), which in particular affects exciton energies Ev(r). The latter are obtained by solving the Wannier equation7 starting from the unstrained single-particle dispersion relation5 and adding strain-induced shifts30, cf. Methods and Supplementary Information. The spatial variation of energy (i) impacts the stationary diffusion coefficient D(r) by altering the occupation of single exciton states and opening/closing exciton-phonon scattering channels26 (see also Supplementary Information) and (ii) induces a drift force −Ev(r) for excitons, see Methods. The first term in Eq. (1) corresponds to the standard Fick’s equation for diffusion, i.e., it leads to a spatial broadening of the exciton occupation Nv due to the optically induced spatial gradient (Nv(r, t)). This occurs already for homogeneous (s(r) = s0) or vanishing strain (s = 0). The appearing diffusion coefficient D(r) is microscopically obtained via a full evolution of the Wigner function26, cf. the Supplementary Information. In the second term in Eq. (1), the energy Ev plays the role of a scalar potential, whose spatial variation induces a drift force driving excitons toward spatial regions with a minimal Ev. This force is present only for spatially inhomogeneous strain and leads to exciton funneling (Fig. 1b). Since with larger strain Ev decreases for bright (v = KK) and increases for dark (v = KΛ) excitons (Fig. 1c), we expect an opposite funneling behavior (cf. Supplementary Information). As a result, momentum-dark KΛ excitons could provide an explanation for the anti-funneling behavior observed in the experiment (Fig. 2).

Previous studies have shown that the interaction with high-dipole molecules31 or phonons8 can brighten dark exciton states and make them visible and even dominate the low-temperature PL of tungsten-based TMDs. Through funneling, dark KΛ excitons can reach spatial regions, where the spectral separation of dark and bright states EKK − E becomes small, significantly enhancing the phonon-driven intervalley scattering. As a direct consequence, a large amount of originally dark KΛ excitons scatters into bright KK states and becomes visible in PL spectra (cf. Figs. 1b and SI). Additional defect-induced activation mechanisms could further boost the effect. More details on the theoretical approach (including the total decay rate γt and intervalley thermalization $${\dot{N}}_{v}^{{{{{{{{\rm{th}}}}}}}}}$$ appearing in Eq. (1) as well as funneling-induced activation of dark excitons) can be found in the Supplementary Information.

To further illustrate the anti-funneling behavior, we show the spatiotemporal evolution of the experimentally measured PL signal in Fig. 3c, d. We focus on spatial cuts along the x axis (cf. the dashed white line in Fig. 2b, c) for optical excitation close to the maximum strain gradient (corresponding to spot B and C). For a better orientation with respect to the funneling direction, we also plot the spatially dependent strain profile at these spots (cf. Fig. 3a, b). Opposite to the expected behavior, we see a clear leftward propagation away from regions with maximal strain, i.e., an anti-funneling of excitons. The propagation is more pronounced in the spot C due to the presence of a higher strain gradient. A quantitative analysis shows that excitons propagate for a few hundreds of nm within the first nanoseconds. This is sensitive to the strain profile, i.e., at spots with a larger strain gradient, the drift efficiency is higher (cf. the Supplementary Material). The anti-funneling length is in the same range as the conventional exciton propagation observed so far in literature20,21.

The PL at room temperature is usually associated with bright excitons3,8. However, these states cannot explain the observed anti-funneling behavior, since they propagate toward regions of high strain, where their energy is minimal (cf. Fig. 1c). In contrast, the energy of momentum-dark KΛ excitons decreases with strain and is thus the driving force for the observed anti-funneling behavior. To illustrate this, we show in Fig. 3e, f the spatially and temporally resolved occupation N of momentum-dark KΛ excitons in direct comparison to the PL measurements in Fig. 3c, d. The theoretical data is obtained by numerically evaluating the drift-diffusion Eq. (1) exactly at the conditions of the experiment, i.e., assuming the same spatially inhomogeneous strain profile shown in Fig. 2e, f. In both experiment and theory, we find a clear exciton motion to the left of the excitation spot, i.e., towards spatial regions with a smaller strain gradient (cf. the strain profile in Fig. 3a, b). This indicates that the surprising anti-funneling observed in the PL can indeed be directly ascribed to the propagation of dark excitons. This can only occur at excitation spots, where momentum-dark KΛ excitons are energetically lower or close to the bright KK states (Fig. 3h) and thus exhibit a considerable occupation. We find a more pronounced funneling after optical excitation at location C closer to the maximum of the strain gradient, since here the spatial variation of strain is steeper, resulting in a stronger drift force −E(r). In addition, we observe both in experiment and theory that interestingly the funneling is initially faster and then it slows down approaching a stationary value for the propagation. This effect is well reproduced by the theoretically predicted evolution of dark-exciton occupation and can be understood as an interplay of two effects: First, the diffusion coefficient D is spatially-dependent and exhibits smaller values left from the excitation spots (cf. the green line in Fig. 3e). While funneling leftward, dark excitons move in regions with a smaller D, hence resulting in a decreasing funneling velocity. Second, dark excitons are subject to phonon-driven intervalley thermalization. A part of dark KΛ excitons scatters into KK states, which funnel to the right and effectively slow down the overall exciton funneling.

## Discussion

By combining spatiotemporal photoluminescence experiments with quantum-mechanic many-particle modeling we provide deep microscopic insights into the transport of excitons in atomically thin semiconductors. Creating a strain gradient via nanopatterned surfaces and investigating exciton propagation via spatiotemporal photoluminescence, we find an anti-funneling behavior in WS2 monolayers. Complementing our experiments with microscopic theory, we attribute the anti-funneling to dark KΛ excitons, which possess an opposite strain-induced energy variation compared to bright excitons and thus propagate in the opposite direction. To further prove that momentum-dark KΛ excitons are the origin of the inverse funneling observed in our experiments, we have performed similar measurements on a molybdenum diselenide (MoSe2) monolayer, cf. Fig. 4 and Supplementary Video S2. Contrary to the case of WS2, we find a conventional exciton funneling, i.e., the PL signal moves from low- toward high-strain regions (cf. orange and purple area indicating initial and final distributions, respectively). This observation is in full agreement with our theoretical predictions, since in MoSe2 dark KΛ excitons are located above the bright states7. Therefore, dark states are weakly occupied and are expected to have only a minor impact on exciton funneling (cf. Supplementary Information). Remarkably, a quantitative analysis of the funneling behavior shows an efficient propagation of excitons exceeding 1 μm within 0.8 ns (cf. Supplementary Information), which is beyond the reported values of some hundreds of nanometers in previous studies performed on WSe220,21.

Our results demonstrate that strain engineering is an efficient tool to manipulate the propagation of neutral excitons. In particular, the interplay of dark and bright states offers new possibilities for tuning the optical response, dynamics, and transport, which are key processes for the realization of novel ultrathin and flexible optoelectronic devices.

## Methods

### Theoretical methods

The strain-dependent exciton energies entering Eq. (1) are obtained starting from the unstrained single-particle dispersion relation5, adding strain-induced shifts30 and solving on microscopic footing the Wannier equation7,32

$$\frac{{\hslash }^{2}{k}^{2}}{2{m}_{v}}{{{\Psi }}}_{v}({{{{{{{\bf{k}}}}}}}})-\mathop{\sum}\limits_{{{{{{{{\bf{q}}}}}}}}}{W}_{{{{{{{{\bf{q}}}}}}}}}{{{\Psi }}}_{v}({{{{{{{\bf{k}}}}}}}}+{{{{{{{\bf{q}}}}}}}})={E}_{v}^{{{{{{{{\rm{b}}}}}}}}}{{{\Psi }}}_{v}({{{{{{{\bf{k}}}}}}}}),$$
(2)

where mv is the reduced exciton mass in exciton valley v. Here, $${E}_{v}^{{{{{{{{\rm{b}}}}}}}}}$$ corresponds to the exciton binding energy, which depends weakly on strain via mv, cf. Table in Supplementary Information for the list of all appearing parameters. Furthermore, Ψv(k) describes the excitonic wave function in momentum space, while Wq is the Coulomb interaction obtained via a modified form of the potential for charges in a thin film of thickness d surrounded by a dielectric environment33. Taking into account anisotropic dielectric tensors and solving the Poisson equation with the boundary conditions described above yields Wq = Vq/ϵscr(q) with the bare 2D-Fourier transformed Coulomb potential Vq and a non-local screening

$${\epsilon }_{scr}(q)={\kappa }_{1}\tanh \left[\frac{1}{2}\left[{\alpha }_{1}dq-{{{{{{\mathrm{ln}}}}}}}\,\left(\frac{{\kappa }_{1}-{\kappa }_{2}}{{\kappa }_{1}+{\kappa }_{2}}\right)\right]\right],$$
(3)

where $${\kappa }_{i}=\sqrt{{\epsilon }_{i}^{\parallel }{\epsilon }_{i}^{\perp }}$$ and $${\alpha }_{i}=\sqrt{{\epsilon }_{i}^{\parallel }/{\epsilon }_{i}^{\perp }}$$ account for the parallel and perpendicular component of the dielectric tensor ϵi of monolayer (i = 1)34 and environment (i = 2)35, see Table in Supplementary Information.

The exciton energy reads $${E}_{v}={E}_{v}^{0}-{E}_{v}^{{{{{{{{\rm{b}}}}}}}}}$$, where $${E}_{v}^{0}$$ is the minimal energy separation between conduction and valence band in valley v. This quantity depends crucially on strain5,30, see Table in Supplementary Information. Due to large spectral separations, we focus on the energetically lowest 1s states of the 6 most relevant exciton valleys (KK, KK$${}^{\prime}$$, KΛ, K$${{{\Lambda }}}^{\prime}$$, ΓK, ΓK$${}^{\prime}$$). These states exhibit a center-of-mass momentum Q, such that the state in valley v and momentum Q has an energy Ev,Q = Ev + 2Q2/(2Mv) with Mv being the total exciton mass. These energies depend strongly on strain mostly via Ev, hence they are space-dependent in the case of a spatially-inhomogeneous strain profile, i.e., Ev,Q ≡ Ev,Q(r). As explained in the main text, this induces (i) a spatially-dependent diffusion coefficient D(r) that has been microscopically evaluated in Ref. 26 (cf. Supplementary Information for more details) and (ii) a drift force −Ev with a generalization of the electrical mobility provided by D/kBT according to the Einstein relation36,37. It is well known that trions38,39,40,41 can also be observed in the presence of space-dependent strain profiles22, however, we focus here on the intrinsic undoped regime. The impact of spin-dark states40,42 has also been neglected, since they are expected to only lead to quantitative changes due to their similar character as KK$${}^{\prime}$$ excitons considered here.

In the presence of spatially-homogeneous strain, optically excited excitons reach an equilibrium characterized by a (local) thermalized energy distribution $${N}_{v}^{\circ }({{{{{{{\bf{r}}}}}}}},t)$$43 and a conventional diffusive regime44. This occurs on a femtosecond timescale at room temperature. Since the timescale of exciton funneling is much longer, we have assumed that the stationary conditions are always fulfilled with $${N}_{v}^{\circ }({{{{{{{\bf{r}}}}}}}},t)$$ corresponding to the Boltzmann distribution $${N}_{v}^{\circ }({{{{{{{\bf{r}}}}}}}},t)\propto {\sum }_{{{{{{{{\bf{Q}}}}}}}}}{g}_{v}{{{{{{{\rm{Exp}}}}}}}}\big[-{E}_{{{{{{{{\bf{Q}}}}}}}},v}({{{{{{{\bf{r}}}}}}}})/{k}_{B}T\big]$$ with gv as the valley-degeneracy factor (gv = 3 for v = KΛ,K$${{{\Lambda }}}^{\prime}$$, otherwise gv = 1). At every position, the relative amount $${\bar{n}}_{v}({{{{{{{\bf{r}}}}}}}})={N}_{v}^{\circ }({{{{{{{\bf{r}}}}}}}},t)/{\sum }_{v}{N}_{v}^{\circ }({{{{{{{\bf{r}}}}}}}},t)$$ of exciton occupation in valley v with respect to the total exciton population $${\sum }_{v}{N}_{v}^{\circ }({{{{{{{\bf{r}}}}}}}},t)$$ is time-independent, while it depends on space via the excitonic energies. In particular, the ratio $${\bar{n}}_{{{{{{{{\rm{K}}}}}}}}{{\Lambda }}}/{\bar{n}}_{{{{{{{{\rm{KK}}}}}}}}}$$ of KΛ and KK excitons decreases for increasing strain due to the opposite strain-induced shifts of the exciton energy Ev, cf. Fig. 1c in the main text.

We describe the spatiotemporal evolution of exciton densities Nv(r, t) by exploiting the generalized drift-diffusion equation Eq. (1). As stated in the main manuscript, the first two terms provide diffusion and valley-intrinsic drift, respectively. The third term in Eq. (1) describes the decay of exciton occupation. Bright excitons localized in the light cone with a zero center-of-mass momentum decay via the radiative recombination rate γrad = 4.5 meV/, which was previously calculated30 and shows only small variations with strain. Once depleted via direct recombination, bright excitons are then refilled via scattering from states with a non-zero center-of-mass momentum. Since only a small amount of excitons resides in the light cone, the total excitonic density N(r, t) decays via an effective rate $${\gamma }_{{{{{{{{\rm{rad}}}}}}}}}{\bar{n}}_{{{{{{{{\rm{0}}}}}}}}}({{{{{{{\bf{r}}}}}}}})$$ with $${\bar{n}}_{{{{{{{{\rm{0}}}}}}}}}$$ being the fraction of the exciton occupation in the light cone at the position r. This results in a slower decay through radiative recombination of the overall excitonic density, which takes place on a timescale of tens of nanoseconds for tungsten-based monolayers at room temperature43. However, the PL signals decay typically much faster, suggesting that the leading decay mechanism in most TMDs is given by defects and impurities (except for hBN-encapsulated samples45,46,47). Since this is an extrinsic process and varies for each sample, we assume a typical value of γt = 1/τ, with τ = 0.35 ns in accordance with recent strain experiments on WS218. The last term in Eq. (1) provides the intervalley thermalization which can be written as $${\dot{N}}_{v}^{{{{{{{{\rm{th}}}}}}}}}({{{{{{{\bf{r}}}}}}}},t)=\bar{\gamma }\left[{N}_{v}^{\circ }({{{{{{{\bf{r}}}}}}}},t)-{N}_{v}({{{{{{{\bf{r}}}}}}}},t)\right]$$. The thermalization time $$\bar{\tau }={\bar{\gamma }}^{-1}$$ depends on exciton valley and strain (and as such being in principle space-dependent), however, it occurs on a femtosecond timescale at room temperature43,44. Compared to the much slower exciton diffusion and funneling taking place on a timescale of hundreds of ps to ns, we assume the thermalization to occur instantaneously at all spatial positions. Note that we have evaluated Eq. (1) along the x direction of strain variation exploiting the symmetry of the system.

### Sample and experimental setup

To create inhomogeneous strain fields in an atomically thin semiconductor, we transfer a TMD monolayer onto a borosilicate substrate (170  μm thickness) with an array of micropillars on top28,29,48. The arrays of polymer pillars (Novolak Allresist AR-N 7520.07, 10 μm distance between pillars) are fabricated using electron beam lithography. They have a height of 140 nm and circular or elliptical shapes with a diameter of ≈2 μm. To characterize the created strain landscape, we record spectrally resolved white-light transmission images of the sample and extract the shift of the bright XKK exciton for every sample position (cf. Supplementary Information). The strongest exciton shifts appear between two pillars (X-direction). The exact strain profile depends on the varying experimental conditions (angle, pressure, shear forces) during the transfer of the TMD monolayer onto the pillars and, hence, varies from sample to sample. From these exciton shifts, we can extract the strain in the monolayer using the uniaxial gauge factor for monolayer WS2 (− 50 meV/%)18.

To measure the temporally and spatially resolved exciton motion, we use a home-built optical transmission microscope. Femtosecond pulses from a tunable Er:fiber laser (E = 2.1 eV, τ ≈ 200 fs, H = 0.9 μJ/cm2) or from a Ti:Sapphire laser (E = 1.71 eV, τ < 200 fs, H = 3.8 μJ/cm2) are focused onto the sample using an objective lens (Nikon TU Plan ELWD 50x, NA = 0.6) with a focus size <1 μm. The photoluminescence is collected by a second objective lens (Nikon TU Plan 100x, NA = 0.9) and imaged onto the entrance slit of a streak camera (Hamamatsu C10910-01) with an attached SCMOS sensor. Residual pump light is removed using an appropriate long pass filter. The streak camera provides temporal and spatial resolution along the X dimension. To obtain a spatial resolution in the Y dimension, the collection objective lens is mounted on a piezo positioner and moved in the Y direction. This configuration allows us to spatially map the photoluminescence by scanning the objective lens in the vertical direction, projecting different sample positions onto the entrance slit of the streak camera, while exciting the monolayer at a fixed position. In that way, we obtain ultrafast videos of the exciton motion in monolayer TMDs.