Dark exciton anti-funneling in atomically thin semiconductors

Transport of charge carriers is at the heart of current nanoelectronics. In conventional materials, electronic transport can be controlled by applying electric fields. Atomically thin semiconductors, however, are governed by excitons, which are neutral electron-hole pairs and as such cannot be controlled by electrical fields. Recently, strain engineering has been introduced to manipulate exciton propagation. Strain-induced energy gradients give rise to exciton funneling up to a micrometer range. Here, we combine spatiotemporal photoluminescence measurements with microscopic theory to track the way of excitons in time, space and energy. We find that excitons surprisingly move away from high-strain regions. This anti-funneling behavior can be ascribed to dark excitons which possess an opposite strain-induced energy variation compared to bright excitons. Our findings open new possibilities to control transport in exciton-dominated materials. Overall, our work represents a major advance in understanding exciton transport that is crucial for technological applications of atomically thin materials.

Here, the Wigner distribution N v Q (r, t) summed over excitonic momenta Q provides the spatially dependent exciton density N v (r, t). It undergoes a free propagation (first term), which is counteracted by exciton-phonon coupling (last term). The latter is evaluated microscopically [5,6], where the strength of the coupling is obtained from first-principle calculations [7]. The competition between these two terms results in an initial transient phase with time-varying effective diffusion coefficients D ≡ D(t) that evolves into a stationary conventional diffusion D(t) ≡ D [8]. The Diffusion coefficient plotted for a homogeneously strained WS2 monolayer including a direct comparison between theoretically predicted (dark thick line) and experimentally measured (thin light line with error bars) values as a function of uniaxial strain [4].
duration of the transient phase crucially depends on temperature, ranging from tens of ps at very low temperatures down to hundreds of fs at room temperature [8]. The results shown in Fig. S1 are taken once the conventional diffusion is reached for different homogeneous strain values. Note that the second and the third term in Eq. (1) describe recombination of bright excitons and the phonon-assisted formation of incoherent excitons, respectively [4]. Although crucial for the optical response and formation dynamics of excitons, these processes either involve only a few states in the light-cone or occur on a much faster timescale and have thus a negligible impact on the stationary exciton diffusion. While strain has only a minor effect on excitonic masses or radiative recombination [3,9,10], it strongly affects excitonic energies, whose variations can drastically change the efficiency of exciton-phonon scattering by opening/closing intervalley scattering channels. This results in an increase of the diffusion coefficient observed in Fig. S1, which is in excellent agreement with the values obtained experimentally upon bending of a WS 2 monolayer deposited on PMMA substrate (cf. the thin line with error bars in Fig. S1). Depending on strain, the measured values of the diffusion coefficient range from tens of cm 2 /s up to a few cm 2 /s -in agreement with previous experiments for tungsten-based TMDs on SiO 2 [11][12][13]. Note that SiO 2 has similar dielectric characteristics as the PMMA substrate employed in the experiment shown in Fig. S1 [4], thus allowing a direct quantitative comparison theory and experiment. The reported values are smaller than those obtained in the presence of hBN-encapsulation [14,15]. This is expected due to the reduced dielectric disorder [16][17][18] and the increased contribution from electron-hole plasma playing a larger role due to the decreased exciton binding energy in hBN-encapsulated TMD samples [15].  Here, m0 is the free-electron mass. Strain variations are adjusted by the factor stemming from the Poisson effect [4]. Further required parameters include dielectric constants of the TMD monolayer [19] and of the surrounding environment (taking an average of SiO2 [20] and air) as well as radiative γ rad [3] and total decay rates γt [21] (the latter dominated by non-radiative mechanisms).

LINEAR STRAIN PROFILE
The spatiotemporal dynamics of excitons under spatially-inhomogeneous strain profiles depends on strain both via its absolute value and its spatial gradient, cf. Eq. (1) of the main manuscript. The former affects the value of the diffusion coefficient D, see Sec. 1, or the relative population of a given exciton valley v, while the latter induces the force −∇E v (x). Here, we assume a linear strain profile s =s + x∂ x s with boths and ∂ x s being space independent. In this way the derivative ∂ x s of the strain s is constant at all spatial points. Hence, the driving force −∇E v (x) does not depend on the position, remaining in particular identical while excitons move in different regions. Note that linear strain profiles can typically be found further away from the maximum strain gradient produced by a pillar. Here, we choose the values ofs=0.5% and ∂ x s i =0.5%/µm, cf. the black line in Fig. S2(a).
We start with an initial total exciton occupation N (x, 0) corresponding to a Gaussian centered at x 0 = 0 with a FWHM of 1µm. The strain profile gives rise to a spatial variation of the diffusion coefficient D(x) (green line in Fig.  S2(a)) and to spatial shifts of excitonic energy E v ( Fig. S2(b)). The latter are obtained via the Wannier equation, while D depends on position via strain s(x), i.e. D(x) ≡ D(s(x)), cf. Sec. 1. We find that E KK decreases with strain, while E KΛ shows the opposite behaviour. This results in different drift forces -∂ x E v of approximately 280 eV/cm for KK and -180 eV/cm for KΛ excitons. In Fig. S2(c), we show the PL profile I(x, t) ∝ γ radn0 (x)N (x, t), which depends on the total exciton population N (x, t) weighted by the radiative decay rate γ rad and the fraction of  the exciton occupationn 0 (x) populating the light cone at the position x. At t=0, the spatial PL profile is centered slightly on the right-hand side of x = x 0 ≡ 0, where the initial excitonic distribution (as generated by a laser pulse) is centered, cf. also the PL time cuts in Fig.S2(d). This can be traced back to a higher exciton occupation in the light conen 0 (x) for x > x 0 due to the lower energetic position of KK vs. KΛ excitons in this region (cf. Fig. S2(b)). The time evolution of the PL illustrates the expected exciton funneling toward spatial regions with higher strain values following the drift force −∂ x E KK for the bright KK excitons. During the funneling, the PL spatial profiles become asymmetric, displaying a more pronounced right flank, cf. Fig. S2(d). This stems from the interplay of the spatial variation of the diffusion coefficient D and the occupation of bright excitonsn 0 . The overall behaviour of the PL follows the evolution of the distribution of bright KK excitons, cf. Fig. S2(e). In contrast, momentum-dark KΛ excitons show a qualitatively different behaviour demonstrating an anti-funneling toward regions of smaller strain values ( Fig. S2(e)) -in clear contrast to N KK excitons and the behaviour observed in the PL. This reflects the opposite spectral shift in presence of strain and thus opposite drift forces −∂ x E KK and −∂ x E KΛ for KK and KΛ excitons. However, we do not see any signatures of anti-funneling in the PL, since no activation mechanism for dark excitons have been considered here.

ACTIVATION OF DARK EXCITONS
We now study a scenario, where momentum-dark KΛ excitons funneling away from regions of maximum strain can be visualized in spatiotemporal PL. To this end, we consider a Gaussian strain profile s(x) = s 0 + s i exp −x 2 /(2σ 2 ) with s 0 = 0.6%, s i = −1.2% and a FWHM of 4µm (black line in Fig. S3a). Such a strain profile induces a repulsive potential for KK and an attractive one for KΛ excitons, cf. Fig. S3b. The choice of the strain profile induces a peculiar behaviour of the diffusion coefficient D, which shows minima at approximately x = ±1µm corresponding to a strain of 0%. We start with an initial optically-excited exciton density that is spatially centered at x 0 = 1µm close to the maximum of the strain gradient. The initial stage of the PL evolution is as expected: The PL moves toward the regions of maximal strain magnitude s (x > x 0 ) reflecting the funneling of KK excitons, cf. Fig. S3c. However,  Fig. S2, but now using a Gaussian strain profile. We find a time-delayed formation of a new peak for negative x values that can be ascribed to momentum-dark excitons. The inset demonstrates the crucial role of intervalley exciton thermalization for the activation of dark states. at later times a new peak appears on the left side x < x 0 , as demonstrated in Fig. S3d showing time-cuts of the spatiotemporal PL. The origin of this peak can be traced back to the presence of KΛ excitons, whose distribution N KΛ moves towards the region of smallest strain at x = 0. Then, a fraction of KΛ excitons keeps on diffusing further to the left (due to the first term in Eq. (1) of the main manuscript). When they reach the spatial region x −1µm, where E KK < E KΛ (Fig. S3b), KΛ excitons scatter into energetically lower KK states, from where they can emit light. Thus, the interplay of exciton funneling and intervalley thermalization activates originally dark excitons and makes them visible in spatiotemporal PL profiles.
The crucial role of the intervalley thermalization for the activation of dark excitons is demonstrated in the inset of Fig. S3d, where we show the PL profile without intervalley scattering (dashed line). The additional peak at negative x vanishes, since the amount of bright excitons funneling in this spatial region from the initial excitation at x 0 = 1µm is negligibly small. Given the large spatial extent of the investigated strain profile, dark excitons have to funnel a relatively long distance (from +1 to -1 µm) before being activated. Thus, the time delay for the formation of the second peak is relatively long with a few nanoseconds. The delay time can be in principle reduced by considering spatially narrower strain gradients. Furthermore, additional activation mechanisms for momentum-dark KΛ excitons beyond the considered phonon-driven intervalley scattering could further boost the effect. Here, in particular defect-induced activation could play an important role and needs to be further investigated [22,23].

MEASURED EXCITON PROPAGATION: EXCITON FUNNELING VS ANTI-FUNNELING
To obtain a better understanding of the observed anti-funneling behaviour, in Fig. S4 we quantitatively analyze the propagation length and its direction with respect to the maximum of the strain profile in WS 2 and MoSe 2 monolayers. In the case of the conventional funneling towards spatial regions with maximal strain, we define the propagation as positive. In contrast, the anti-funneling behaviour induces a negative drift in WS2 (cf. the blue arrows in (b)). The latter considerably increases when moving from spot A to C reflecting the enhanced strain gradient.
To quantify the exciton propagation, we evaluate the temporal evolution of the central position of the PL as extracted using a Gaussian fit of the experimental results. For the case of MoSe 2 , we find a positive propagation, i.e. directed towards maximum strain (cf. Figs. S4a,c) -in agreement with with previous experiments [24,25]. We observe a very efficient exciton propagation exceeding 1 µm already within the first nanosecond. In the case of WS 2 , we first observe the crucial change of the sign in the direction of the propagation (cf. the blue lines in Fig. S4a), indicating an anti-funneling behaviour toward spatial region of minimal strain (cf. Fig. S4b). The propagation length is in the range of a few hundreds of nm within the first ns. This is less efficient compared to the regular funneling in MoSe 2 . However, we find a pronounced increase of the exciton propagation, when moving the excitation from spot A to C, i.e. closer to the maximum of the strain profile. Here, the strain gradient is higher resulting in a larger exciton drift, cf. Fig. S4b. The exciton funneling direction in MoSe 2 is opposite to WS 2 , which can be explained by the energetic alignment of bright KK and momentum-dark KΛ excitons, see Figs. S5a-d. In MoSe 2 , KΛ excitons lies energetically much higher than the bright KK states. As a result, their occupation is small and their impact on the diffusion behaviour in MoSe 2 negligible. Here, bright excitons play the dominant role, where the reduction in energy with strain creates a spatial energy landscape favoring exciton propagation in direction of spatial regions with maximal strain, cf. also Fig.  1d in the main manuscript. The situation is drastically different in WS 2 , where the momentum-dark KΛ excitons are the energetically lower states carrying the majority of the occupation. As a result, dark excitons govern the diffusion behavior. Here, the energy is enhanced with increasing strain creating a drift force in the opposite direction toward spatial regions with low strain, cf. also Fig. 1e in the main manuscript.

COMPARISON OF SAMPLE TOPOGRAPHY AND STRAIN
The strain is characterized by recording spectrally resolved optical transmission images of the sample (Fig. S6a) and extracting the shift of the bright X KK exciton for every sample position in Fig. S6b. In Fig. S7, we compare the 3D profile of the WS 2 monolayer stamped onto the micropillars with the resulting strain profile. Figure S7a depicts an atomic force microscopy (AFM) image of the relevant sample area. The height image shows that the monolayer conforms to the micropillars and the flat substrate in-between, except for a few nanoscopic folds. The strain (Fig. 7b) has its maximum in the middle between two pillars, which is due to the experimental conditions during the transfer of the monolayer. This sample allows us to measure the exciton motion in a strain gradient, where the monolayer is in contact with the substrate. Therefore, we can rule out that other effects, such as changes of the dielectric environment have a significant influence on the spatial change of exciton energies.
Finally, we illustrate the strain profiles created for WS 2 and MoSe 2 monolayers, cf. Fig. 8. The crosses indicate the excitation spots. The strain profile in WS 2 is essentially straight and orthogonal to the (leftward) anti-funneling direction of excitons. In the case of MoSe 2 , the strain profile has a more complicated shape exhibiting a maximum close to the pillar followed by a drawn-out tail.