Exciton transport in strained monolayer WSe₂

Transition metal dichalcogenides (TMDs) have received special attention in the last decade due to their potential to enable optoelectronic devices with novel functionalities.^1–3 In addition, the ability to tune their band structures via strain, temperature, and pressure along with their room-temperature stable excitonic resonances has made these materials outstanding candidates for excitonic device applications.^4,5 Various devices exploiting TMD's optoelectronic potential have been demonstrated in recent years.^1,6–14 Typically, these devices employ an electrical bias to dissociate the excitons and drive the charges towards a contact. In contrast, purely excitonic devices leverage TMD's large binding energy^15–18 and highly tunable band structure^19,20 to control the transport of exciton densities at room-temperature without dissociating them to collect the charges. Hence, it is critical that the transport properties of excitons in monolayer TMDs be well characterized to enable the design of next-generation excitonic devices.

The transport of excitonic energy under strain fields in TMDs is particularly interesting due to the large tunability of their bandgap via strain and the resulting capability to manipulate their motion via spatially modulated strain fields. Several studies exploring the effect of in-plane strain on monolayer TMDs' band structures have shown that for small strain levels, their direct bandgap varies linearly with strain.^21–24 Their large sensitivity to strain can enable the design of excitonic potentials via the spatial modulation of their bandgap with non-uniform strain. Such modulation can result in the “funneling” of excitonic energy in the direction of the strain gradient.²⁵ While there have been attempts to characterize the transport of excitons in bulk and monolayer TMDs,^26–30 their combined drift and diffusion under strain gradients have not been studied in these materials. Here, we simultaneously quantify the diffusion and drift of excitons under the strain gradients created by transferring a WSe₂ monolayer over a nanostructured substrate.

Monolayer WSe₂ samples were prepared via mechanical exfoliation³¹ and were transferred over SiO₂ nano-pillars on a Si substrate. The resulting strain profiles on the WSe₂ monolayers were obtained by measuring the local PL spectrum across an area strained by a SiO₂ pillar. Figure 1(a) shows an optical image of a WSe₂ monolayer over an ∼1.5 $μ$m diameter, ∼250 nm tall SiO₂ pillar, and Fig. 1(b) shows the monolayer's integrated PL spectra from the area enclosed by the square on the optical image. The PL spectra were integrated between 622 nm and 881 nm since the peak intensity of WSe₂ occurs typically at around 750 nm. The integrated spectra were normalized with respect to their maximum value within the enclosed area to emphasize the effect of the pillar.

The direct bandgap of monolayer TMDs is expected to decrease (increase) linearly with tensile (compressive) strain²⁴ which should be evidenced by red (blue) shifts in their PL spectra. Moreover, an enhancement of their PL intensity at high tensile strain points is expected due to the funneling of excitons towards the lower bandgap energy (red-shifted) regions of the monolayer. Such an enhancement in the WSe₂ monolayer's local PL intensity is observed in Fig. 1(b) around the SiO₂ pillar, which is consistent with the funneling of excitons towards the points of the highest tensile strain. The additional high strain points on the strain map shown in Fig. 1(c) could be the result of substrate roughness³² or polymer residue from the fabrication process. AFM images of the sample can be found in the supplementary material showing the possible origin of such high-strain points.

The strain field on the WSe₂ monolayer was estimated by calculating the shift of the local PL spectra's centroids with respect to an unstrained point on the monolayer. The spectral shifts were converted to bandgap energy shifts and strain using the sensitivity of WSe₂'s direct bandgap to strain (−55 meV/%) from Ref. 24. Figure 1(c) shows the WSe₂ monolayer's bandgap shift as a result of the applied strain. The white dot indicates the location of the unstrained PL spectrum baseline. In the supplementary material, we compare the strain calculated from PL spectra's centroid and peak shifts and show that the strain estimates are nearly identical.

The transport of excitons on an unstrained point of the WSe₂ monolayer was visualized first to set a reference for the transport of excitons on the strained portion of the monolayer. A map of the time-dependent exciton density as a function of position was constructed using the technique from Refs. 33 and 34. The spatial information is obtained by scanning a detector across the PL emission spot resulting from a pulsed excitation while the temporal information is obtained via time-correlated single photon counting. Figure 2(b) shows the exciton density as a function of time and position where the excitation was fixed at an unstrained point as indicated by the cross in Fig. 2(a). The normalized exciton density was fitted with Gaussians to extract the time-dependent mean-squared displacement (MSD) of the exciton density.

In systems where excitons diffuse freely, the MSD denoted by $Δx2t$ is proportional to time according to $Δx2t≡σ2t−σ2(0)=2Dt$⁠, where $2σ(t)$ represents the width of the Gaussian exciton density, $D$ is the diffusion coefficient, and $t$ denotes the time.^35,36 However, when excitons diffuse in a disordered environment, the MSD does not increase linearly with time, and the diffusion coefficient becomes time-varying. In such cases, the MSD is better described by the power law model $Δx2t=Γtα$⁠, where $Γ$ and $α$ are known as the transport factor and anomalous coefficient, respectively.^35–39 Accordingly, the time-dependent diffusion coefficient is defined as $Dt=1/2∂Δx2t/∂t=αΓtα−1/2.$³⁵ 

In addition to the typical defects found in semiconductors, the two-dimensional nature of monolayer TMDs makes them especially susceptible to defects such as folding and wrinkling.⁴⁰ This means that the type of substrate supporting the monolayer can radically affect the amount of disorder experienced by excitons and hence their diffusion.^41,42 As a result, a sub-diffusive regime characterized by an anomalous coefficient $α<1$ and a diffusion coefficient varying between 1.8 and 0.6 cm²/s is identified on the unstrained WSe₂ monolayer as shown in Fig. 2(d). In addition, the peak's position of the exciton density remains virtually unchanged in space, which is consistent with the absence of strain gradients that are expected to generate a drift of the density's peak.

On the other hand, when the excitation was positioned at a strained point on the monolayer near the SiO₂ pillar, the exciton density's peak did not remain localized. Specifically, when the excitation was positioned just above the high tensile strain point on top of the pillar and the PL beam was monitored from top to bottom, the exciton density's peak shifted down and towards the high tensile strain point on the monolayer as shown in Fig. 3(b). Similarly, when the excitation was placed just to the left of the high tensile strain point on top of the pillar and the PL beam was monitored from left to right, the exciton density's peak shifted to the right and towards the high tensile strain point as shown in Fig. 3(c). A shift in the exciton density's peak position is expected to occur due to the strain gradient on the monolayer. Figure 3(d) shows the PL spectra measured at three positions along the x-axis near the SiO₂ pillar.

To explain our observations, we developed a model that encompasses the diffusion, drift, and relaxation of the exciton density of the WSe₂ monolayer based on Boltzmann's transport theory with the relaxation time approximation and assuming a uniform exciton mobility.^43–45 The resulting differential equation describing the exciton dynamics is given by

where the first, second, and third terms to the right-hand side of the equal sign represent the relaxation of the exciton density denoted by $nx,t$ with relaxation time $τ$⁠, the diffusion of the exciton density with time-varying diffusion coefficient $D(t)$⁠, and the drift of the exciton density with strain mobility $μϵ$ due to the strain gradient $∂ϵx/∂x, respectively$⁠. The strain mobility is simply defined as $μϵ≡μ∂Eg/∂ϵ=|vd∂ϵx/∂x−1|$⁠, where $μ$ is the traditional mobility, $∂Eg/∂ϵ$ represents the sensitivity of the monolayer's direct bandgap to strain, and $vd$ denotes the density's drift velocity due to the applied strain gradient. The units of the strain mobility are length squared per unit time and percent strain, typically expressed in cm² s⁻¹%–¹. The traditional mobility is defined as $μ≡ t¯/m*$⁠, where $t¯$ is the mean free time of the excitons, and $m*$ is their effective mass.⁴⁶ 

In our case, the strain mobility is not uniform throughout the WSe₂ monolayer because the non-uniform strain field modifies the monolayer's band structure, affecting the excitons' effective mass⁴⁷ and their mean free time. We observed such non-uniformity after extracting the drift velocity from our measurements of the excitonic density as a function of position and time. We calculated the peak position's shift of the exciton density as a function of time, which essentially represents the drift velocity of the exciton density due to the applied strain gradient. Figure 4 shows the exciton density's peak position as a function of time when the excitation was placed just to the left of the high tensile strain point on top of the SiO₂ pillar [L in Fig. 3(a)], and the mobility resulting from the local strain experienced by the density's peak. Note that the exciton density's peak position follows the direction of the strain gradient and moves towards the position of highest tensile strain as expected. With the approximate local strain profile shown in Fig. 4(a), the exciton density appears to reach drift velocities as large as 250 m/s and drift distances as large as 500 nm.

Both the local strain as a function of position and the exciton density's peak position as a function of time were fit with second-order polynomials to obtain smooth strain gradient, drift velocity, and mobility curves. As expected, the resulting strain mobility appears to be non-uniform over the range where the density's peak moves. However, assuming that such variation is negligible, the model given by Eq. (1), where the mobility is assumed to be uniform, may still be used to qualitatively explain our observations. Our simulation result of the exciton density under an experimentally obtained strain profile along one spatial dimension and time is shown in Fig. 4(e). The inputs to the numerical solution were the experimentally obtained exciton density at $t=0$ as the initial condition, and the experimentally obtained parameters $Dt$⁠, $ϵx$ and $μϵ$⁠, where the strain mobility was averaged over the spatial range where the exciton density's peak moved. As expected, the numerical exciton density's peak shifts towards the high tensile strain point on top of the pillar. While the experimental exciton density appears to become slightly asymmetric probably as the result of variations in the local strain field that our diffraction-limited imaging technique is unable to identify, the numerical and experimental peak positions are in reasonable agreement as illustrated in Fig. 4(d), indicating that the drift-diffusion-relaxation model described by Eq. (1) is able to replicate the main feature of our observations.

In summary, we have shown that the motion of excitonic energy in monolayer WSe₂ can be controlled via strain gradients. The sensitivity of WSe₂'s direct bandgap to strain enabled the creation of local excitonic potentials that resulted in the directional drift of exciton densities towards high-tensile strain points. Furthermore, the large susceptibility of WSe₂ monolayers to disorder led to anomalous diffusive transport with a time-varying diffusion coefficient. These observations are captured by the proposed drift-diffusion-relaxation model that includes the strain field on the WSe₂ monolayer. The capability to control the motion of excitonic energy on monolayer TMDs with engineered strain profiles opens up the possibility to design novel, room-temperature-stable excitonic devices^4,5 that prescind from the resistive-capacitive delays, heat generation, and scalability limitations that hinder conventional optoelectronic technologies.

See supplementary material for details of the substrate fabrication and its topography, additional transport and spectral data, and details of the model derivation and numerical solution.

This material is based upon work supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE 1256260. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. This work was also funded through AFOSR Grant No. 16RT1256 and startup grant from the University of Michigan. D.C L. would like to acknowledge support from NSF's Graduate Research Fellowship Program and University of Michigan's Rackham Merit Fellowship. The authors would like to acknowledge the Lurie Nanofabrication Facility at the University of Michigan where the substrates were fabricated, and Kanak Datta for helpful discussions.