## 4 Scattering Contribution to Exciton Linewidth  
To conclude our study of the scattering of excitons with free carriers in layered materials, we will now discuss the contribution from these scattering processes to the excitonic linewidth. This will provide a point of comparison against experimental studies [54]. Although other phenomena will also contribute to this linewidth, such as radiative lifetimes [27] and phonon scattering [25], the dependence of the free carrier scattering on both temperature and carrier density should provide a good distinction of the various contributing processes. As hBN is a large bandgap insulator, carrier doping in this material is usually performed chemically by diluting impurities [55; 56; 57; 58].  
The contribution to the excitonic linewidth from free carrier scattering is given by [59; 60; 24; 61]  
\[\Gamma_{\text{Total}}=\sum_{f}\frac{2\hbar^{2}}{\pi M}\int_{0}^{\infty}dk\,k^{ 2}f\bigg{(}\frac{m_{c}+m_{e}+m_{h}}{m_{e}+m_{h}}k\bigg{)}Q_{1s\to f}, \tag{29}\]  
where \(Q_{1s\to f}\) is, as described earlier, the scattering cross section associated with a specific transition from the excitonic ground state to a final state \(f\). As we are summing over final states \(f\), and only \(Q_{1s\to f}\) depends on the final state, this is equivalent to switching the sum over final states and the integral and writing  
\[\Gamma_{\text{Total}}=\frac{2\hbar^{2}}{\pi M}\int_{0}^{\infty}dk\,k^{2}n_{F} \bigg{(}\frac{m_{c}+m_{e}+m_{h}}{m_{e}+m_{h}}k\bigg{)}Q_{\text{Total}}, \tag{30}\]  
where \(Q_{\text{Total}}\) is the total scattering cross section as plotted in Figure 3. Here, \(n_{F}(k)\) is the Fermi-Dirac distribution for free carriers, given by  
\[n_{F}(k)=\frac{1}{e^{\frac{k_{2}-E_{F}}{k_{B}T}}+1} \tag{31}\]  
where the dispersion relation is given by  
\[E_{k}=\frac{\hbar^{2}}{2m_{c}}k^{2} \tag{32}\]  
and the Fermi energy is  
\[E_{F}=2\pi\frac{\hbar^{2}}{2m_{c}}n, \tag{33}\]  
with \(n\) the area density of free carriers. We consider carrier densities up to a maximum of \(10^{12}\,\text{cm}^{-2}\), where the average separation of free carriers \(d_{c}=2(\pi n)^{-1/2}\) is still larger but already of the order of the root mean square (RMS) exciton radius, given by  
\[r_{\text{RMS;}n}=\int_{0}^{\infty}\int_{0}^{2\pi}rdrd\theta\,\psi_{n}(r,\theta )^{+}r^{2}\psi_{n}(r,\theta). \tag{34}\]  
For the two excitonic states most relevant to the scattering cross section, \(1s\) and \(2p\), the RMS exciton radius is \(r_{\text{RMS;}1s}=19.5\,\text{\AA}\) and \(r_{\text{RMS;}2p}=73.2\,\text{\AA}\), respectively, while the average separation between free carriers is \(d_{c}=113\,\text{\AA}\).  
As before, we must compute the integral of Equation (30) numerically. The specific methodology for the discretization of Equation (30) is discussed in Appendix B. Choosing a Gauss-Legendre quadrature [62] of size \(N=450\), the results for the contribution of scattering with free carriers to the excitonic linewidth are presented in Figure 4 as a function of temperature for free carrier area densities of \(n=10^{9}\) cm\({}^{-2}\) and \(n=10^{12}\) cm\({}^{-2}\). In Figure 5, we present the excitonic linewidth as a function of the free carrier area density for four distinct values of the temperature \(T\) between \(10\,\mathrm{K}\) and \(300\,\mathrm{K}\). As expected, the excitonic linewidth contribution increases with both temperature and carrier density as these phenomena increase the kinetic energy of the carriers and their proximity, respectively.