
\section{Introduction}

Scattering between excitons and free carriers has been observed experimentally since the 1960s in highly excited bulk semiconductors\cite{PhysRev.177.567,10.1063/1.1653814,10.1143/JPSJ.34.693,10.1002/pssb.2220580237,10.1002/pssa.2210220102,10.1002/pssb.2220710216,10.1002/pssb.2220780219,Levy1975ExperimentalIO}. In these bulk semiconductor systems, the scattering cross sections have been studied previously\cite{ELKOMOSS1977557,ELKOMOSS1979431}, with the distinction between elastic and inelastic scattering fundamental for the interpretation of the experimental data.

Recently, the quality of monolayer semiconductor samples has drastically increased\cite{Reynolds1986,Museur2008,Cadiz2017,Elias2019,Li2021a}, with advances in techniques such as molecular beam epitaxy\cite{Vuong2017}, chemical vapor deposition\cite{Kim2015} or solution methods\cite{Yano2000}. The improved quality of these samples allowed for a much more detailed study of excitons in mono- and few--layer materials, where well--defined resonance peaks are experimentally identifiable even at room temperature\cite{PhysRevLett.33.827,10.1063/1.93648,Poellmann2015,Merkl2019}. This then creates the necessity of calculating and measuring the excitonic linewidth. 

Several mechanisms contribute to the excitonic linewidth, such as acoustic\cite{PhysRevB.33.5512} and optical\cite{PhysRevB.34.2554} phonon scattering \cite{PhysRevB.103.235402}, radiative recombination\cite{10.1080/00018739100101522,10.1021/nl503799t}, as well as scattering in semiconductor alloys\cite{10.1063/1.94649,10.1063/1.102583}. Besides these mechanisms, others can play a part in determining the exciton linewidth. In this paper, we focus our attention on one of these mechanisms, namely scattering with free carriers. This scattering mechanism can play an important part in determining the excitonic linewidth, especially in systems with high density of free carriers and excitons\cite{10.1063/1.360578}, high pump fluences\cite{Djurii2005}, in tunneling experiments\cite{Young1990,Teran2009} or when an electric field is applied to the semiconductor\cite{10.1063/1.363972,10.1002/adom.202102132}.  Additionally, electron exciton scattering  processes also play an important role when studying exciton - polaron systems\cite{PhysRevB.103.075417}.

This paper is structured as follows. In Sec. \ref{sec:sec2}, we quickly review the approach outlined by Feng and Spector in Ref. \cite{FENG1987593} to the scattering between free carriers and excitons in semiconducting quantum wells. This derivation was performed in the central field \cite{Massey1951} and Born approximations \cite{Bates1953}.
In Sec. \ref{sec:sec3}, we turn our discussion to the total scattering cross section. We discuss the distinction between elastic and inelastic scattering, briefly reviewing the variational exciton wave functions for various states. We then explicitly compute the total cross section for a few select transitions, discussing the thresholds present in inelastic scattering processes. Finally, in Sec. \ref{sec:sec4}, we compute the contribution to excitonic linewidth from the scattering cross section with free carriers, analyzing its dependence on both the temperature of the system and the free carrier density in the monolayer.
\section{Free Carrier - Exciton Scattering\label{sec:sec2}}

In this section, we will follow the expressions derived by Feng and Spector in Ref. \cite{FENG1987593} for free carrier - exciton scattering, based on assuming two--dimensional (2D) gases of free carriers (electrons or holes) and excitons interacting with one another. The obtained expressions are derived following the same approach as those discussed by Mott and Massey for the general theory for tridimensional (3D) two--body collisions \cite{Mott1965}. The cross sections due to collisions between the carriers and excitons are then calculated using the central field \cite{Massey1951} and Born approximations \cite{Bates1953}.

\subsection{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 
\begin{align}
	M &=\frac{m_c\left(m_e + m_h\right)}{m_c + m_e + m_h},
\end{align}
where $ m_{c/e/h} $ is the mass of the free carrier/ electron/hole. 

Following the derivations by Feng and Spector\cite{FENG1987593,FENG7096} in--depth in Appendix \ref{app:derivation}, the differential scattering cross section for our free carrier - exciton system is written as
\begin{equation}
	I_{fi}\left(\theta\right)=\frac{M^{2}}{2\pi\hbar^{4}k_{i}}\left|\int d^{2}\mathbf{r}\,d^{2}\mathbf{R}\,e^{i\mathbf{q}\cdot\mathbf{R}} V\left(\mathbf{r},\mathbf{R}\right)\chi_{f}^{\dagger}\left(\mathbf{r}\right)\chi_{i}\left(\mathbf{r}\right)\right|^{2},\label{eq:generic_crossection}
\end{equation}
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\left(\mathbf{r},\mathbf{R}\right) $ the interaction potential between the free carrier and the exciton. In Eq. (\ref{eq:generic_crossection}), $ \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\cite{Massey1951} will be modeled by the Rytova--Keldysh potential\cite{rytova1967,keldysh1979coulomb}, 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
\begin{equation}
	V_{RK}(\mathbf{r})=\frac{\hbar c \alpha}{\epsilon}\frac{\pi}{2 r_{0}}\left[H_{0}\left(\epsilon\frac{r}{r_{0}}\right)-Y_{0}\left(\epsilon\frac{r}{r_{0}}\right)\right], \label{eq:RK}
\end{equation}
where $\alpha=1/137$ is the fine--structure constant, $\epsilon$ the mean dielectric constant of the medium above/below the layered material, $H_{0}\left(x\right)$ is the zeroth-order Struve function and $Y_{0}\left(x\right)$ 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 \cite{PhysRevB.99.035429}. 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 will be 
\begin{equation}\label{eq:pot_old}
	V\left(\mathbf{r},\mathbf{R}\right)=\pm \left[V_{RK}(\mathbf{r_{ch}})-V_{RK}(\mathbf{r_{ce}})\right],
\end{equation}
where $ \pm $ distinguishes between the free carrier being a hole ($ + $) or an electron ($ - $), $ r_{ch} $ is the distance between the free carrier and the hole of the exciton, and $ 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 \cite{FENG1987593}
\begin{align}\label{eq:vectors_from_masses}
	\mathbf{r}_{ch} &= \mathbf{R}-\frac{\sigma}{1+\sigma}\mathbf{r}, 
	&\mathbf{r}_{ce} &= \mathbf{R}+\frac{1}{1+\sigma}\mathbf{r},
\end{align}
where $ \sigma=m_e / m_h $ is the ratio between effective electron and hole masses. 
Returning to the discussion on the scattering cross section from Eq. (\ref{eq:generic_crossection}), the integration over $ R $ reads
\begin{equation}\label{eq:R_integration}
	\int d^{2}\mathbf{R}\,e^{i\mathbf{q}\cdot \mathbf{R}} V\left(\mathbf{r},\mathbf{R}\right)=\pm \left[\int d^{2}\mathbf{R}\,e^{i\mathbf{q}\cdot \mathbf{R}}V_{RK}(\mathbf{r_{ch}})-\int d^{2}\mathbf{R}\,e^{i\mathbf{q}\cdot \mathbf{R}}V_{RK}(\mathbf{r_{ce}})\right],
\end{equation}
which can be computed directly\cite{FENG7096} by performing a change of integration variables back to $\mathbf{r}_{ch},\,\mathbf{r}_{ce} $ and reads 
\begin{equation}\label{eq:change_variables_int}
	\pm\left[e^{iq\frac{\sigma}{1+\sigma}r\cos\left(\phi_{r}\right)}-e^{-iq\frac{1}{1+\sigma}r\cos\left(\phi_{r}\right)}\right]2\pi\frac{\hbar c\alpha}{\epsilon}\frac{1}{q\left(1+r_{0}q\right)},
\end{equation}
with $ \phi_{r} $ the angle between $ \mathbf{r} $ and $ \mathbf{q}$, and
\begin{equation}
	V_{RK}\left(\mathbf{q}\right)=2\pi\frac{\hbar c\alpha}{\epsilon}\frac{1}{q\left(1+r_{0}q\right)}
\end{equation}
the Fourier transform of the Rytova--Keldysh potential. Finally, the differential scattering cross section can be written as 

\begin{align}
	I_{fi}\left(\theta\right)=\frac{M^{2}}{2\pi\hbar^{4}k_{i}}\left|2\pi \frac{\hbar c\alpha}{\epsilon}\frac{1}{q\left(1+r_{0}q\right)}J\left(i\rightarrow f\right)\right|^{2}.\label{eq:cross-section_integrated_RK}


\end{align}
with the dependence on the initial and final exciton states included in $ J\left(i\rightarrow f\right) $, defined as
\begin{equation}\label{eq:J}
	J\left(i\rightarrow f\right)=\int_{0}^{+\infty}r\,dr\,\int_{0}^{2\pi} d\phi_{r}\,\left[e^{i\frac{\sigma}{1+\sigma}qr\cos \phi_{r}}-e^{-i\frac{1}{1+\sigma}qr\cos \phi_{r}}\right]\chi_{f}^{\dagger}\left(\mathbf{r}\right)\chi_{i}\left(\mathbf{r}\right).
\end{equation}

\section{Total Cross Section\label{sec:sec3}}

Knowing the differential cross section given by Eq. (\ref{eq:cross-section_integrated_RK}), we can now compute the full scattering cross section. To this effect, we must simply perform an angular integration in $ \theta $ as
\begin{align}\label{eq:full_scatter}
	Q_{\mathrm{i\rightarrow f}}&=\int_{-\pi}^{\pi}d\theta\,I_{\mathrm{i\rightarrow f}}\left(\theta\right).
\end{align}
Explicitly substituting Eq. (\ref{eq:cross-section_integrated_RK}), the full cross section is given by 
\begin{align}\label{eq:scatter_theta_RK_tot}
	Q_{\mathrm{i\rightarrow f}}\left(k_{i}\right)&=\frac{2\pi M^{2}\left(\hbar c\alpha\right)^{2}}{\hbar^{4}\epsilon^{2}k_{i}}\int_{-\pi}^{\pi}d\theta\,\left|\frac{J\left(i\rightarrow f\right)}{q(1+r_{0}q)}\right|^{2}.
\end{align}
To compute this integral, however, we must first define the exciton wave functions which we will consider when computing Eq. (\ref{eq:J}). We must also define the type of scattering in question, as it will introduce both the specific $ \theta $ dependence in $ \mathbf{q} $ as well as specific thresholds for the relative momentum of the free carrier - exciton system from conservation of energy.


\subsection{Elastic Scattering}

In an elastic scattering process, the exciton remains in its ground state after the collision, meaning $ \left|\mathbf{k}_i\right|=\left|\mathbf{k}_f\right| $. As such, we can write
\begin{equation}
	q= 2 \left|\sin\left(\frac{\theta}{2}\right)\right| k_i.\label{eq:elast_q}
\end{equation}
Additionally, we also have $ \chi_{f,i}\left(\mathbf{r}\right)=\chi_{1s}\left(r\right) $, where we consider a simple variational \emph{ansatz} \cite{Quintela,Gomes2021} based on the eigenfunctions of the two--dimensional Hydrogen atom\cite{Lee1979,PhysRevA.43.1186} and given by 
\begin{equation}\label{eq:ansatz}
	\chi_{1s}\left(r\right)=\mathcal{N}_{1s}e^{-r\gamma_{1s}/2},
\end{equation}
with $ \mathcal{N}_{1s} $ a normalization constant given by 
\begin{equation}\label{eq:normalization}
	\mathcal{N}_{1s}=\left[\int rdrd\theta\,\left(e^{-r\gamma_{1s}/2}\right)^{2}\right]^{-1/2}=\frac{\gamma_{1s}}{\sqrt{2\pi}}
\end{equation}
and $ \gamma_{1s} $ a variational parameter. This variational parameter is computed by minimization of the energy expectation value of the Wannier equation\cite{Wannier1937}
\begin{equation}\label{eq:wannier}
	H=-\frac{\hbar^{2} c^{2}}{2\mu}\nabla^2+V_{RK}(\mathbf{r}),
\end{equation}
with $ 	V_{RK}(\mathbf{r}) $ the Rytova--Keldysh potential.

With this \emph{ansatz}, we can directly substitute the wave function into $ J\left(i\rightarrow f\right) $, given by Eq. (\ref{eq:J}), and obtain
\begin{align}
	J_{\mathrm{elast}}\left(q\right)=J\left(\mathrm{1s}\rightarrow \mathrm{1s}\right)
	&=\gamma_{1s}^{3}\left[\frac{1}{\left[\left(\frac{\sigma}{1+\sigma}q\right)^{2}+\gamma_{1s}^{2}\right]^{3/2}}-\frac{1}{\left[\left(\frac{1}{1+\sigma}q\right)^{2}+\gamma_{1s}^{2}\right]^{3/2}}\right]\label{eq:J_transition_elastic_exponential}
\end{align}
after integration. This is then substituted into Eq. (\ref{eq:scatter_theta_RK_tot}), reading 
\begin{align}\label{eq:scatter_theta_RK_tot_elast}
	Q_{\mathrm{elast}}\left(k_{i}\right)&=\frac{2\pi M^{2}\left(\hbar c\alpha\right)^{2}}{\hbar^{4}\epsilon^{2}k_{i}}\int_{-\pi}^{\pi}d\theta\,\left|\frac{J_{\mathrm{elast}}\left(q\right)}{q(1+r_{0}q)}\right|^{2},
\end{align}
where $ q $ is given by Eq. (\ref{eq:elast_q}). This integral has no analytical solution and must be computed numerically.

\begin{figure}
	\centering{}\includegraphics{Fig1}
	\caption{Total elastic cross section for electron--exciton (black) and hole--exciton (red) scattering in $ \mathrm{hBN} $ encapsulated in fused quartz as a function of the initial relative wave vector. \label{fig:cross-section_coulomb_GaAs}}
\end{figure}

To finalize the computation of the elastic cross section, we must choose a set of material specific parameters. We consider those corresponding to monolayer hexagonal Boron--Nitride ($ \mathrm{hBN} $) encapsulated in fused quartz, with dielectric constant $ \epsilon=3.8 $ \cite{Serway2003}. The electron and hole masses in this material are $ m_e = 0.83\, m_0$, $ m_h = 0.63\, m_0$ \cite{Ferreira19}, with $ m_0 $ the electron rest mass, and $ r_0 = 10\,\text{\AA}$ \cite{henriques_optical_2020}. The obtained variational energy for the $ 1s $ excitonic state is $ E_{1s}=-58.9\,\mathrm{meV} $.

Varying the initial relative wave vector, we obtain the plot of the total elastic cross section from Eq. (\ref{eq:scatter_theta_RK_tot}) in Fig. (\ref{fig:cross-section_coulomb_GaAs}). We can see that the cross section for electron scattering is always larger than that for hole scattering, as expected from the fact that the reduced mass of the system is larger when the free carrier considered is an electron. A very quick increase from zero relative momentum up to a global maximum is also observed, consistent with the results of Feng and Spector \cite{FENG7096} for elastic scattering. 


\subsection{Inelastic Scattering}

Let us now consider inelastic scattering between free carriers and the exciton. The relative momenta is now given by
\begin{equation}\label{eq:momenta_1}
	q^{2} = k_{f}^{2} + k_{i}^{2}-2k_{i}k_{f}\cos(\theta), 
\end{equation}
with $ k_{f} $ obtained from conservation of energy as 
\begin{equation}\label{eq:conservation_energy}
	k_{f}^{2}=k_{i}^{2}-\frac{2M}{\hbar^{2}}(E_{f}-E_{i}).
\end{equation}
Here, $ E_{f/i} $ is the energy of the final/initial state of the exciton, respectively. Substituting this relation into Eq. (\ref{eq:momenta_1}), we obtain
\begin{equation}\label{eq:momenta_2}
	q^{2} = 2k_{i}^{2}-\frac{2M}{\hbar^{2}}\Delta_{f,i}-2k_{i}^{2}\sqrt{1-\frac{2M}{\hbar^{2}}\frac{\Delta_{f,i}}{k_{i}^{2}}}\cos(\theta),
\end{equation}
with $ \Delta_{f,i}=E_{f}-E_{i} $. A threshold in $ k_i $ below which no scattering is allowed is immediately evident, obtained from Eq. (\ref{eq:momenta_2}) as 
\begin{equation}\label{eq:threshold}
	k_{\mathrm{min}}=\sqrt{\frac{2M}{\hbar^{2}}(E_{f}-E_{i})}.
\end{equation}
Below this threshold, there is not enough energy in the scattering process to allow the jump between excitonic states $ i\rightarrow f $. 

Besides knowing the energies of the final states, we must also know their wave functions. These are obtained\cite{Lee1979,Quintela,Gomes2021} in a similar form to Eq. (\ref{eq:ansatz}) and are given by 
\begin{align}\label{eq:2s_2p_ansatz}
	\chi_{2s}\left(r\right)&=\mathcal{N}_{2s}\left(1-\frac{r}{d}\right)e^{-r\gamma_{2s}/2}, &
	\chi_{2p_{\pm}}\left(r\right)&=\mathcal{N}_{2p}r e^{\pm i \theta}e^{-r\gamma_{2p}/2},
\end{align}
where $ \gamma_{2s},\gamma_{2p} $ are variational parameters, $ \mathcal{N}_{2s},\mathcal{N}_{2p} $ are normalization constant given by 
\begin{align}
	\mathcal{N}_{2s}&=\frac{2\gamma_{2s}^{2}}{\sqrt{\pi}\sqrt{3\gamma_{1s}^{2}-2\gamma_{1s}\gamma_{2s}+3\gamma_{2s}^{2}}},&
	\mathcal{N}_{2p}&=\frac{\gamma_{2p}^{2}}{2\sqrt{3\pi}},
\end{align}
and $ d $ is a parameter obtained by imposing orthogonality between $ \chi_{1s} $ and $ \chi_{2s} $, given by 
\begin{equation}
	d=\frac{4}{\gamma_{1s}+\gamma_{2s}}.
\end{equation}
These wave functions, together with the $ 1s $ wave function, are plotted in Fig. (\ref{fig:wave_functions}) for monolayer $ \mathrm{hBN} $ encapsulated in fused quartz.
\begin{figure}
	\centering{}\includegraphics{Fig2}
	\caption{Radial probability density for the $ 1s $ (blue), $ 2s $ (orange) and $ 2p $ (green) states as described by the variational wave functions of Eq. (\ref{eq:ansatz}) and Eq. (\ref{eq:2s_2p_ansatz}) in a $ \mathrm{hBN} $ monolayer encapsulated in fused quartz. \label{fig:wave_functions}}
\end{figure}

\subsubsection{$ 1s \rightarrow 2s $ Transitions}
We will first consider $ 1s \rightarrow 2s $ transitions. To compute $ J_{2s}=J\left(1s\rightarrow 2s\right) $, we recall Eq. (\ref{eq:J}) and, after integration, obtain
\begin{align}\label{eq:J_1s2s}
	J_{2s}
	&= \frac{3\gamma_{1s}\left(\gamma_{1s}+\gamma_{2s}\right)\gamma_{2s}^{2}}{\sqrt{6\gamma_{1s}^{2}-4\gamma_{1s}\gamma_{2s}+6\gamma_{2s}^{2}}} \left[\frac{q^{2}\left(\frac{\sigma}{1+\sigma}\right)^2}{\left[q^2\left(\frac{\sigma}{1+\sigma}\right)^2+\left(\frac{\gamma_{1s}+\gamma_{2s}}{2}\right)^{2}\right]^{5/2}}-\frac{q^{2}\left(\frac{1}{1+\sigma}\right)^2}{\left[q^2\left(\frac{1}{1+\sigma}\right)^2+\left(\frac{\gamma_{1s}+\gamma_{2s}}{2}\right)^{2}\right]^{5/2}}\right],\nonumber
\end{align}
where $ q $ is obtained from Eq. (\ref{eq:momenta_2}) as 
\begin{equation}\label{eq:q_1s2s}
	q^{2}=2k_{i}^{2}-\frac{2M}{\hbar^{2}}\Delta_{2s,1s}-2k_{i}^{2}\sqrt{1-\frac{2M}{\hbar^{2}}\frac{\Delta_{2s,1s}}{k_{i}^{2}}}\cos(\theta).
\end{equation}

As discussed above, the energy of the $ 2s $ state is obtained by minimization of the Wannier equation with the variational wave functions and its value is $ E_{2s}=-8.83\,\mathrm{meV} $ for our system. Explicitly computing the thresholds from Eq. (\ref{eq:threshold}), we obtain $ k_{\mathrm{min}}= 0.0833\,\text{\AA}^{-1}$ for electron--exciton scattering and $ k_{\mathrm{min}}= 0.0760\,\text{\AA}^{-1}$ for hole--exciton scattering.

\subsubsection{$ 1s \rightarrow 2p $ Transitions}

For computing $ J\left(1s\rightarrow 2p_{\pm}\right) $ following Eq. (\ref{eq:J}), we must quickly consider the distinction between $ p_{\pm} $ states. This is, however, not important, as the two states are degenerate and the two integrals $ J\left(1s\rightarrow 2p_{+}\right) $, $ J\left(1s\rightarrow 2p_{-}\right) $ are, in fact, equal. As such, we take into account the two $ 2p $ states by multiplying the total cross section by an angular momentum degeneracy factor $ g_{\ell}=2 $.

Explicitly, $ J_{2p}=J\left(1s\rightarrow 2p\right) $ is given by 
\begin{align}\label{eq:J_1s2p}
	J_{2p}
	&=\frac{3\gamma_{1s}\gamma_{2p}^{2}\left(\gamma_{1s}+\gamma_{2p}\right)}{2\sqrt{6}}\left[\frac{iq\frac{\sigma}{1+\sigma}}{\left[q^2\left(\frac{\sigma}{1+\sigma}\right)^2+\left(\frac{\gamma_{1s}+\gamma_{2p}}{2}\right)^{2}\right]^{5/2}}+\frac{iq\frac{1}{1+\sigma}}{\left[q^2\left(\frac{1}{1+\sigma}\right)^2+\left(\frac{\gamma_{1s}+\gamma_{2p}}{2}\right)^{2}\right]^{5/2}}\right].\nonumber
\end{align}
The relative momentum $ q $ is defined analogously to Eq. (\ref{eq:q_1s2s}), with the only factor missing being the energy of the $ 2p_{\pm} $ states.

For our system, this energy is $ E_{2p}=-10.2\,\mathrm{meV} $. As such, the thresholds from Eq. (\ref{eq:threshold}) are now $ k_{\mathrm{min}}= 0.0822\,\text{\AA}^{-1}$ for electron--exciton scattering and $ k_{\mathrm{min}}=0.0749\,\text{\AA}^{-1}$ for hole--exciton scattering. 

\subsection{Joint Elastic and Inelastic Scattering}


Finally, we consider the joint contribution to the scattering cross section from both elastic and inelastic scattering. This cross section will, therefore, involve a sum over final states, where only the $ 1s\rightarrow 1s $ contribution originates from elastic scattering processes. Explicitly, and for an arbitrary set of final exciton states $ f $, the total scattering cross section is given by
\begin{equation}\label{eq:scatt_total_1}
	Q_{\mathrm{Total}}=\sum_{f}Q_{1s\rightarrow f}.
\end{equation}
For the three transitions discussed previously, the sum in Eq. (\ref{eq:scatt_total_1}) is restricted and is explicitly written as 
\begin{equation}\label{eq:scatt_total}
	Q_{\mathrm{Total}}=Q_{1s\rightarrow 1s}+Q_{1s\rightarrow 2s}+Q_{1s\rightarrow 2p}.
\end{equation}
This total scattering cross section is plotted in Fig. (\ref{fig:cross-section_total}) for both types of free carriers, together with the dashed lines representing the thresholds for the inelastic scattering processes considered.

Analyzing Fig. (\ref{fig:cross-section_total}), we can see that, when the same scattering process is allowed for both types of free carriers, electron--exciton scattering has a cross section roughly $ 1.5\times $ larger. This trend is, however, inverted between the threshold momentum for hole--exciton $ 1s \rightarrow 2p $ scattering and the threshold momentum for electron--exciton $ 1s \rightarrow 2p $, \emph{i.e.}, between $ k_i = 0.0749\,\text{\AA}^{-1} $ and $ k_i = 0.0822\,\text{\AA}^{-1}$. In this momentum range, the dominant $ 1s \rightarrow 2p $ process is already allowed for hole--exciton scattering, leading to a vastly superior cross section relative to that for electron--exciton scattering.

\begin{figure}
	\centering{}\includegraphics{Fig3}
	\caption{Total cross section for electron--exciton (black) and hole--exciton (red) scattering in $ \mathrm{hBN} $ encapsulated in fused quartz as a function of the initial relative wave vector. Vertical dashed lines represent the electron--exciton and hole--exciton scattering thresholds for the $ 1s \rightarrow 2p $ transition, located at $ k_{\mathrm{min},e^{-}}= 0.0822\,\text{\AA}^{-1}$ and $ k_{\mathrm{min},h}= 0.0749\,\text{\AA}^{-1}$.
	
		\label{fig:cross-section_total}}
\end{figure}

The final threshold included, visible in Fig. (\ref{fig:cross-section_total}) as the dashed purple lines, originates from $ 1s \rightarrow 3d $ scattering, as $ E_{3d}=-3.74\,\mathrm{meV} $ is the lowest energy state after $ 2s $. These take place at $ k_{\mathrm{min}}=0.0875\,\text{\AA}^{-1} $ for electron--exciton scattering and $ k_{\mathrm{min}}=0.0798\,\text{\AA}^{-1} $ for hole--exciton scattering. These are, however, much smaller than the peaks in Fig. (\ref{fig:cross-section_total}), similarly to what is observed Ref. \cite{FENG1987593}, and are invisible in Fig. (\ref{fig:cross-section_total}). 

\section{Scattering Contribution to Exciton Linewidth\label{sec:sec4}}

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\cite{Selig2016}. Although other phenomena will also contribute to this linewidth, such as radiative lifetimes\cite{Palummo2015} and phonon scattering\cite{Henriques2021}, the dependence of the free carrier scattering on both temperature and carrier density should provide a good distinction of the various contributing processes. 

The contribution to the excitonic linewidth from free carrier scattering is given by\cite{PhysRevB.34.2554,FENG1987459,feng1989excitons,10.1063/1.364274}
\begin{equation}\label{eq:linewidth}
	\Gamma_{\mathrm{Total}}=\sum_{f}\frac{2\hbar^{2}}{\pi M}\int_{0}^{\infty}dk\,k^{2}f\left(\frac{m_c+m_e+m_h}{m_e+m_h}k\right)Q_{1s\rightarrow f},
\end{equation}
where $ Q_{1s\rightarrow 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\rightarrow f} $ depends on the final state, this is equivalent to switching the sum over final states and the integral and writing 
\begin{equation}\label{eq:linewidth_tot}
	\Gamma_{\mathrm{Total}}=\frac{2\hbar^{2}}{\pi M}\int_{0}^{\infty}dk\,k^{2}n_{F}\left(\frac{m_c+m_e+m_h}{m_e+m_h}k\right)Q_{\mathrm{Total}},
\end{equation}
where $ Q_{\mathrm{Total}} $ is the total scattering cross section as plotted in Fig. (\ref{fig:cross-section_total}).
Here, $ n_{F}(k) $ is the Fermi--Dirac distribution for free carriers, given by 
\begin{equation}\label{eq:fermi_dirac-distr}
	n_{F}(k)=\frac{1}{e^{\frac{E_k-E_F}{k_B T}}+1}
\end{equation}
where the dispersion relation is given by
\begin{equation}\label{eq:free-carrier_dispersion}
	E_k=\frac{\hbar^{2}}{2 m_{c}}k^{2}
\end{equation}
and the Fermi energy is
\begin{equation}\label{eq:fermi_energy_free}
	E_F=2\pi \frac{\hbar^{2}}{2 m_{c}}n,
\end{equation}
with $ n $ the area density of free carriers. We consider carrier densities up to a maximum of $ 10^{12}\,\mathrm{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 \begin{equation}\label{eq_rms}
	r_{\mathrm{RMS};n}=\int_{0}^{\infty}\int_{0}^{2\pi} rdrd\theta\,\psi_{n}(r,\theta)^{\dagger} r^{2} \psi_{n}(r,\theta).
\end{equation}
For the two excitonic states most relevant to the scattering cross section, $ 1s $ and $ 2p $, the RMS exciton radius is $ r_{\mathrm{RMS};1s}=19.5\,\text{\AA} $ and $ r_{\mathrm{RMS};2p}=73.2\,\text{\AA} $, respectively, while the average separation between free carriers is $ d_{c}=113\,\text{\AA} $.

\begin{figure}
	\includegraphics[scale=0.78]{Fig4_a}\includegraphics[scale=0.78]{Fig4_b}
	\caption{Contribution to exciton linewidth from scattering with free carriers as a function of the temperature at constant free carrier area density $n=10^{9}\,\text{cm}^{-2}$ (left) and $n=10^{12}\,\text{cm}^{-2}$ (right). Insets show the low--temperature region of the plot for clarity.\label{fig:linewidth_const_density}}
\end{figure}

\begin{figure}
	\centering{}\includegraphics[scale=0.9]{Fig5_a}\includegraphics[scale=0.9]{Fig5_b}
	\includegraphics[scale=0.9]{Fig5_c}\includegraphics[scale=0.9]{Fig5_d}
	\caption{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). \label{fig:linewidth_const_temp}}
\end{figure}

As before, we must compute the integral of Eq. (\ref{eq:linewidth_tot}) numerically. The specific methodology for the discretization of Eq. (\ref{eq:linewidth_tot}) is discussed in Sec. \ref{app:numeric_integral}. Choosing a Gauss--Legendre quadrature\cite{Kythe2002} of size $ N=450 $, the results for the contribution of scattering with free carriers to the excitonic linewidth are presented in Fig. (\ref{fig:linewidth_const_density}) as a function of temperature for free carrier area densities of $ n = 10^{9}\,\,\text{cm}^{-2} $ and $ n = 10^{12}\,\,\text{cm}^{-2} $. In Fig. (\ref{fig:linewidth_const_temp}), 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}  $. 
\section{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\cite{FENG1987593}, as well as the inclusion of screening in the differential cross section. Quickly reviewing the computation of variational functions for the exciton wave functions, we discussed both elastic and inelastic scattering processes. 

With the differential cross section known for both elastic and inelastic scattering, we proceeded to the computation of the total scattering cross section. 
While elastic scattering is the sole contributor for low relative momentum, inelastic $ 1s\rightarrow 2p $ scattering dominates the total cross section after it becomes allowed. This dominant scattering has a maximum cross section roughly $ 40 $ times larger than that of elastic scattering, and at least $ 400 $ times larger than both $ 1s\rightarrow 2s $ and $ 1s\rightarrow 3d $ inelastic scatterings. This dominant behavior of $ 1s\rightarrow 2p $ transitions is very similar to what is is presented in Figs. (2), (9) of Ref. \cite{FENG1987593}. This behavior is, of course, dependent on the ratio between electron and hole masses, although we only focus our attention on the masses characteristic of single layer hBN. 

After discussing the relation between the total scattering cross section and the excitonic linewidth, we considered both its dependence on both temperature and free carrier density. First looking at the exciton linewidth for fixed values of the carrier density, we observe that the difference of linewidth between the two densities studied ($ 10^{9}\,\,\text{cm}^{-2} $ and $ 10^{12}\,\,\text{cm}^{-2} $) is noticeable in both the finite value of the linewidth as $ T$ approaches $0\,\mathrm{K} $ and in the crossing of the two curves happening slightly earlier at higher carrier density (at $ T\gtrsim150\,\mathrm{K} $ for $ n=10^{9}\,\,\text{cm}^{-2} $ and at $ T\approx140\,\mathrm{K} $ for $ n=10^{12}\,\,\text{cm}^{-2} $). An even greater noticeable change in behavior would be present at densities above $ 10^{13}\,\,\text{cm}^{-2} $, reasonably larger than those usually obtained in experimental works. 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\cite{Lampert1958,Kheng1993,Chen2002,Li2003}, increasingly significant.

Regarding the excitonic linewidth at fixed temperatures, the free carrier scattering contribution remained in the $ 0.5-20\,\mathrm{meV} $ range for temperatures between $ 100-300\,\mathrm{K} $, although a much faster increase for higher densities is observed at $ T=10\,\mathrm{K} $. For temperatures in the range $ 100-300\,\mathrm{K} $, the computed linewidth remained essentially constant as carrier density increases until roughly $ n\approx10^{11}\,\,\text{cm}^{-2} $, although the actual value of the linewidth is strongly dependent on the temperature. Past $ n\approx10^{11}\,\,\text{cm}^{-2} $, the computed linewidth rapidly increases for all values of the temperature, although the growth occurs sooner for lower temperatures. It should, therefore, be feasible to measure this contribution to the excitonic linewidth, as this is both an attainable carrier density \cite{Du2016,Kuiri2021,Li2021}, and good quality samples of both hBN and transition metal dichalcogenides have been grown in the past which presented excitonic linewidths in this order of magnitude\cite{Reynolds1986,Museur2008,Cadiz2017,Elias2019,Li2021a}.


\funding{M. F. C. M. Q. acknowledges the International Nanotechnology Laboratory (INL) and the Portuguese Foundation for Science and Technology (FCT) for the Quantum Portugal Initiative (QPI) grant SFRH/BD/151114/2021. 
N. M. R. P. acknowledges support by the Portuguese Foundation for Science and Technology (FCT) in the framework of the Strategic Funding UIDB/04650/2020, COMPETE 2020, PORTUGAL 2020, FEDER, and  FCT through projects PTDC/FIS-MAC/2045/2021, EXPL/FIS-MAC/0953/ 2021, and from the European Commission through the project Graphene Driven Revolutions in ICT and Beyond (Ref. No. 881603, CORE 3).}

\dataavailability{Not applicable.} 

\conflictsofinterest{The authors declare no conflict of interest.} 


\abbreviations{Abbreviations}{
The following abbreviations are used in this manuscript:\\

\noindent 
\begin{tabular}{@{}ll}
hBN & hexagonal Boron Nitride\\
RMS & root mean squared
\end{tabular}
}

