## 5 Conclusions  
In this paper, we studied the effects of scattering between free carriers and excitons in monolayer materials and its contribution to the excitonic linewidth. To this end, we began by reviewing the general form of the expressions for the differential cross section between free carriers and excitons in two dimensions [37], as well as the inclusion of screening in the differential cross section. Briefly reviewing the computation of variational functions for the exciton wave functions, we discussed both elastic and inelastic scattering processes. Figure 4: Contribution to exciton linewidth from scattering with free carriers as a function of the temperature at constant free carrier area density \(n=10^{9}\,\mathrm{cm}^{-2}\) (**left**) and \(n=10^{12}\,\mathrm{cm}^{-2}\) (right). Insets show the low–temperature region of the plot for clarity. Figure 5: Contribution to exciton linewidth from scattering with free carriers as a function of the free carrier area density at constant temperature \(T=10\,\mathrm{K}\) (**top-left**), \(T=100\,\mathrm{K}\) (**top-right**), \(T=200\,\mathrm{K}\) (**bottom-left**), and \(T=300\,\mathrm{K}\) (**bottom-right**). While elastic scattering was the sole contributor to the scattering cross section for low relative momentum, inelastic \(1s\to 2p\) scattering dominated the total cross section when allowed by conservation of energy. This dominant scattering had a maximum cross section roughly 40 times larger than that of elastic scattering, as seen in Figure 3. Considering inelastic scattering processes to higher energy states, namely \(2s\) and \(3d\), their amplitude was negligible when compared to the dominant process (roughly 400 times smaller). This dominant behavior of \(1s\to 2p\) transitions was very similar to what was presented in Figures 2 and 9 of Ref. [37], although the specific ratios were highly dependent on the electron and hole masses. After discussing the relation between the total scattering cross section and the excitonic linewidth, we considered its dependence on both temperature and free carrier density. First, looking at the exciton linewidth for reasonable values of the carrier density in layered materials [63; 64; 65], namely \(10^{9}\) cm\({}^{-2}\) and \(10^{12}\) cm\({}^{-2}\), we observed that the difference in linewidth between the two regimes was clearly noticeable as the linewidth became finite at \(T=0\) K for larger carrier densities. Additionally, the contribution from exciton-hole scattering became larger than that of electron-exciton scattering slightly earlier at a higher carrier density, namely at \(T\gtrsim 150\) K for \(n=10^{9}\) cm\({}^{-2}\) and at \(T\approx 140\) K for \(n=10^{12}\) cm\({}^{-2}\). Further increasing the carrier density to values above \(10^{13}\) cm\({}^{-2}\) would produce even greater differences. These higher carrier densities would already imply an average free carrier separation \(d_{c}\) much smaller than the RMS exciton radius for the \(2p\) excitonic state, which would make more complex excitonic phenomena, such as biexcitons and trions [66; 67; 68; 69], increasingly significant. Regarding the excitonic linewidth at fixed temperatures, the free carrier scattering contribution remained in the 0.5-20 meV range for temperatures between 100 and 300 K. For temperatures in this range, the computed linewidth remained essentially constant at a temperature-dependent value as the carrier density increased until roughly \(n\approx 10^{11}\) cm\({}^{-2}\). Past \(n\approx 10^{11}\) cm\({}^{-2}\), the computed linewidth rapidly increased for all values of the temperature, although the growth occurred sooner and faster for lower temperatures, as can be seen in the top-left plot of Figure 5 for \(T=10\) K.  
From these results, it should be feasible to measure the free-carrier scattering contribution to the excitonic linewidth, as good quality samples of both hBN and transition metal dichalcogenides have been grown in the past that presented excitonic linewidths in this order of magnitude [11; 12; 13; 14; 15].