## 2 Free Carrier-Exciton Scattering
### Differential Scattering Cross Section  
Let us begin by considering a two-body system consisting of an exciton and a free carrier (electron or hole). The reduced mass of such a system is given by  
\[M=\frac{m_{c}(m_{e}+m_{h})}{m_{c}+m_{e}+m_{h}}, \tag{1}\]  
where \(m_{c/e/h}\) is the mass of the free carrier/electron/hole. Following the derivations by [PERSON] and [PERSON] [37; 41] detailed in Appendix A, the differential scattering cross section for our free carrier-exciton system is written as  
\[I_{fi}(\theta)=\frac{M^{2}}{2\pi\hbar^{4}k_{i}}\bigg{|}\int d^{2}\mathbf{r}\, d^{2}\mathbf{R}\,e^{i\mathbf{q}\cdot\mathbf{R}}V(\mathbf{r},\mathbf{R}) \chi_{f}^{\dagger}(\mathbf{r})\chi_{i}(\mathbf{r})\bigg{|}^{2}, \tag{2}\]  
where \(\mathbf{q}=\mathbf{k}_{i}-\mathbf{k}_{f}\) is the difference between the initial and final relative momentum of the system, \(\chi_{i/f}\) represent the initial/final exciton wave functions, and \(V(\mathbf{r},\mathbf{R})\) represents the interaction potential between the free carrier and the exciton. In Equation (2), \(\mathbf{r}\) is the relative position vector of the electron and the hole in the exciton, and \(\mathbf{R}\) is the relative position vector from the free carrier to the center of mass of the exciton. The interaction potential between the free carrier and the exciton in the central field approximation [38] will be modeled by the Rytova-Keldysh potential [42; 43], usually employed to describe excitonic phenomena in mono- and few-layer materials and obtained by solving the Poisson equation for a charge embedded in a thin film of vanishing thickness. In real space, the Rytova-Keldysh potential is given by  
\[V_{RK}(\mathbf{r})=\frac{\hbar\epsilon\alpha}{\epsilon}\frac{\pi}{2r_{0}} \bigg{[}H_{0}\bigg{(}\epsilon\frac{r}{r_{0}}\bigg{)}-Y_{0}\bigg{(}\epsilon \frac{r}{r_{0}}\bigg{)}\bigg{]}, \tag{3}\]  
where \(\alpha=1/137\) is the fine-structure constant, \(\epsilon\) is the mean dielectric constant of the medium above/below the layered material, \(H_{0}(x)\) is the zeroth-order Struve function, and \(Y_{0}(x)\) is the zeroth-order Bessel function of the second kind. The parameter \(r_{0}\) corresponds to an in-plane screening length related to the 2D polarizability of the material and can be calculated from the single particle Hamiltonian of the system [44]. Additional contributions to this screening length can also originate from the 2D free carriers gas of varying density [45]. In the limit of zero screening length, the Rytova-Keldysh potential becomes the Coulomb potential. Considering again the interaction between the free carrier and the exciton, the interaction potential is \[V(\mathbf{r},\mathbf{R})=\pm[V_{RK}(\mathbf{r_{ch}})-V_{RK}(\mathbf{r_{ce}})], \tag{4}\]  
where \(\pm\) distinguishes between the free carrier being a hole (\(+\)) or an electron (\(-\)), \(\mathbf{r_{ch}}\) is the distance between the free carrier and the hole of the exciton, and \(\mathbf{r_{ce}}\) is the distance between the free carrier and the electron of the exciton. These two vectors can be written from \(\mathbf{r}\) and \(\mathbf{R}\) as in [37]  
\[\mathbf{r_{ch}}=\mathbf{R}-\frac{\sigma}{1+\sigma}\mathbf{r}, \mathbf{r_{ce}} =\mathbf{R}+\frac{1}{1+\sigma}\mathbf{r}, \tag{5}\]  
where \(\sigma=m_{e}/m_{h}\) is the ratio between the effective electron and hole masses.