Enhancing and quantifying spatial homogeneity in monolayer WS2

Controlling the radiative properties of monolayer transition metal dichalcogenides is key to the development of atomically thin optoelectronic devices applicable to a wide range of industries. A common problem for exfoliated materials is the inherent disorder causing spatially varying nonradiative losses and therefore inhomogeneity. Here we demonstrate a five-fold reduction in the spatial inhomogeneity in monolayer WS2, resulting in enhanced overall photoluminescence emission and quality of WS2 flakes, by using an ambient-compatible laser illumination process. We propose a method to quantify spatial uniformity using statistics of spectral photoluminescence mapping. Analysis of the dynamic spectral changes shows that the enhancement is due to a spatially sensitive reduction of the charged exciton spectral weighting. The methods presented here are based on widely adopted instrumentation. They can be easily automated, making them ideal candidates for quality assessment of transition metal dichalcogenide materials, both in the laboratory and industrial environments.


Three-Voigt Model
We used the three-Voigt Model to describe the spectral shape of single layer WS2 at ambient conditions. We justify this here.
The spectral line shape of bound excitons results from optical transitions that decay exponentially in time. In the frequency space, this is the Lorentzian function in the case of a truly isolated emitter. For well-isolated systems often, the Lorentzian approximation is good enough for the intrinsic linewidth, but thermal effects bring with it a Gaussian component. The Voigt profile. This is given by, ϕ(x, y) = 1 α G � ln 2 π K(x, y), Here, x and y are given by, This contains the Gaussian and Lorentzian half-widths α G and α L , respectively, as well υ − υ 0 that defines the centre of the Voigt line profile. This function is approximated to compute the Voigt function quickly and rationally [1]. The fitting algorithm is written in MATLAB as a nonlinear least-squares minimisation problem, with trust-region. Four starting parameters are specified for each emission state, the exciton (X0), charged exciton (CX) and the localised exciton (LX): The peak position, intensity, Gaussian and Lorentzian widths. These starting values are presented in the table below. Peak position for the exciton state is known to be around 2 eV for WS2, the values of the charged and localised exciton states are taken based on experimentally obtained and theoretically predicted binding energies for the CX and LX states, which are approximately 30 meV and 75 meV respectively. As the spectral intensity varies across the map, due to photoluminescence inhomogeneity, the maximum numerical value of measured PL intensity across all positions on the map (M) was used as the reference for the intensity starting values. Lastly, the starting values of the width parameters were chosen to be the middle of the constraints shown above. Before applying the fitting, The fitting procedure is performed by MATLAB, using the nonlinear least-squared method, it was found that within 35 iterations, convergence was reached for all twelve parameters.
The above procedure is applied to all points on recorded PL maps, from which parameter maps, e.g. were extracted. In order to reduce computation time, pixels that do not correspond to possible WS2 spectra were ignored by the fitting algorithm, this was achieved by appropriately thresholding the integrated intensity across the PL map prior to applying fitting procedures.
Next it is demonstrated that the above fitting procedure yields optimal results and is highly adaptable in spite of spectral inhomogeneity. The PL map from Figure 3 in the main manuscript will be used as the example. Due to the large number of spectral samples in the map, three co-ordinates will be chosen from the map with different brightness levels, to represent the range of possible spectral shapes and signal-to-noise ratios with the corresponding measured and fitted PL, as well as the residuals. Finally a histogram showing the residuals for all pixels will be shown. All other spectral fits on this flake had similar residuals with the largest being shown in SM- Figure 3. The autocorrelation plots in each case show that the fit had accounted for most of the PL emission signal, with the remainder caused by white noise or shot noise. In SM- Figure 2 (top panel), where we plotted the residuals for each fit on the same graph, clearly the residuals are shot-noise limited and the fit cannot be improved further. The relative errors of all shot-noise limited spectral fits are quantified by dividing the residual by the measured intensity at each energy value. According to the histogram shown in SM- Figure 5 (Lower panel), 92 % of all shotnoise limited data have relative error 10 % or less. Shot noise can also be eliminated from the residual analysis, by normalising the residual from each spectral fit by the mean residual of that fit, after which the relative error are reduced to below 2%. We compared this with a three-Gaussian model and plotted the analysed residuals in SM- Figure 4. Residuals from the three-Gauss model is not entirely random with oscillatory trends, with a visibly larger range compared to the three-Voigt model. The relative errors for the shot-noise corrected and uncorrected cases also follow the same contrast.  Figure 4 Comparing the residuals from the entire PL map shown in SM- Figure 1,2 and 3 after fitting, with a 3-Voigt, against a 3-Gauss model.

Steady state model of the spectral weighting
The steady state model defined in Equations (1.1-1.5) returns the scattering parameters and , which are scattering rates describing the formation of charged excitons and localised exitons, respectively. The scattering rates obtained from the curve fitting are significantly slower compared to the values reported elsewhere [4], however studies that report much faster scattering rates, such as in MoS2 also reported higher CX spectral weightings than X0. Here, the PL spectral characteristics imply 0 > , , this could be the result of a shift in the dielectric constant of the PDMS substrate compared to the SiO2/Si substrate.  4. Changes to spectral homogeneity after 5 months.
After leaving the large flake in ambient conditions for 5 months after the final measurement, the spectral homogeneity was measured again. This data is presented here, serving to compare the spectral homogeneity of this flake before the laser illumination, immediately after laser illumination ( Figure 5 from our manuscript), and then after several months with the monolayer/PDMS sample stored at ambient conditions. SM- Figure 6 Illumination-time dependence for a small monolayer region studied using time-stepped PL maps. (a) PL intensity and (b) values per − position extracted at different laser illumination times. From top to bottom: = 0 (before laser illumination) and after repeated laser illumination time steps = 2.5 , 5 , 7.5 and 10 . Histograms of values from each map are plotted in column (c). In (d), the fitted PL intensity histograms are plotted. In (e) histograms of and 0 values at each stage is plotted. Correlation statistics between the PL intensity and each of the spectral weighting maps and 0 are extracted and plotted against the laser illumination time in (f), with the absolute value taken for .