Contribution to Excitonic Linewidth from Free Carrier–Exciton Scattering in Layered Materials: The Example of hBN

Abstract

Scattering of excitons by free carriers is a phenomenon, which is especially important when considering moderately to heavily doped semiconductors in low-temperature experiments, where the interaction of excitons with acoustic and optical phonons is reduced. In this paper, we consider the scattering of excitons by free carriers in monolayer hexagonal boron nitride encapsulated by a dielectric medium. We describe the excitonic states by variational wave functions, modeling the electrostatic interaction via the Rytova–Keldysh potential. Making the distinction between elastic and inelastic scattering, the relevance of each transition between excitonic states is also considered. Finally, we discuss the contribution of free carrier scattering to the excitonic linewidth, analyzing both its temperature and carrier density dependence.

1. Introduction

Scattering between excitons and free carriers has been observed experimentally since the 1960s in highly excited bulk semiconductors [1,2,3,4,5,6,7,8]. In these bulk semiconductor systems, the scattering cross sections have been studied previously [9,10], 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 [11,12,13,14,15], with advances in techniques such as molecular beam epitaxy [16], chemical vapor deposition [17], or solution methods [18]. The improved quality of these samples allows 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 [19,20,21,22]. This then creates the necessity of calculating and measuring the excitonic linewidth.

Several mechanisms contribute to the excitonic linewidth, such as acoustic [23] and optical [24] phonon scattering [25], radiative recombination [26,27], as well as scattering in semiconductor alloys [28,29]. In addition to 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 a high density of free carriers and excitons [30], high pump fluences [31], in tunneling experiments [32,33], or when an electric field is applied to the semiconductor [34,35]. Additionally, electron exciton scattering processes also play an important role when studying exciton–polaron systems [36].

This paper is structured as follows. In Section 2, we briefly review the approach outlined by Feng and Spector in Ref. [37] to the scattering between free carriers and excitons in semiconducting quantum wells. This derivation was performed in the central field [38] and Born approximations [39]. In Section 3, 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 Section 4, 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.

2. Free Carrier–Exciton Scattering

In this section, we follow the expressions derived by Feng and Spector in Ref. [37] 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 [40]. The cross sections due to collisions between the carriers and excitons are then calculated using the central field [38] and Born approximations [39].

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

$$\begin{array}{cc}\hfill M& =\frac{m_c\left(m_e+m_h\right)}{m_c+m_e+m_h},\hfill \end{array}$$

(1)

where $m_{c/e/h}$ is the mass of the free carrier/electron/hole.

Following the derivations by Feng and Spector [37,41] detailed in Appendix A, the differential scattering cross section for our free carrier–exciton system is written as

$$I_{fi}\left(\theta \right)=\frac{M^2}{2\pi {\hslash }^4{k}_i}{\left|\int d^2\mathbf{r}{d}^2\mathbf{R}{e}^{i\mathbf{q}·\mathbf{R}}V\left(\mathbf{r},\mathbf{R}\right){\chi }_f^{†}\left(\mathbf{r}\right){\chi }_i\left(\mathbf{r}\right)\right|}^2,$$

(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 $V\left(\mathbf{r},\mathbf{R}\right)$ 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

$$V_{RK}\left(\mathbf{r}\right)=\frac{\hslash c\alpha }{ϵ}\frac{\pi }{2r_0}\left[H_0\left(ϵ\frac{r}{r_0}\right)-Y_0\left(ϵ\frac{r}{r_0}\right)\right],$$

(3)

where $\alpha =1/137$ is the fine-structure constant, $ϵ$ is 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 [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

$$V\left(\mathbf{r},\mathbf{R}\right)=\pm \left[V_{RK}\left({\mathbf{r}}_{\mathbf{ch}}\right)-V_{RK}\left({\mathbf{r}}_{\mathbf{ce}}\right)\right],$$

(4)

where ± distinguishes between the free carrier being a hole (+) or an electron (−), ${\mathbf{r}}_{\mathbf{ch}}$ is the distance between the free carrier and the hole of the exciton, and ${\mathbf{r}}_{\mathbf{ce}}$ 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]

$$\begin{array}{cccc}\hfill {\mathbf{r}}_{\mathbf{ch}}& =\mathbf{R}-\frac{\sigma }{1+\sigma }\mathbf{r},\hfill & \hfill {\mathbf{r}}_{\mathbf{ce}}& =\mathbf{R}+\frac{1}{1+\sigma }\mathbf{r},\hfill \end{array}$$

(5)

where $\sigma =m_e/m_h$ is the ratio between the effective electron and hole masses. Returning to the discussion on the scattering cross section from Equation (2), the integration over R reads

$$\int d^2\mathbf{R}{e}^{i\mathbf{q}·\mathbf{R}}V\left(\mathbf{r},\mathbf{R}\right)=\pm \left[\int d^2\mathbf{R}{e}^{i\mathbf{q}·\mathbf{R}}{V}_{RK}\left({\mathbf{r}}_{\mathbf{ch}}\right)-\int d^2\mathbf{R}{e}^{i\mathbf{q}·\mathbf{R}}{V}_{RK}\left({\mathbf{r}}_{\mathbf{ce}}\right)\right],$$

(6)

which can be computed directly [41] by performing a change in integration variables back to ${\mathbf{r}}_{\mathbf{ch}},{\mathbf{r}}_{\mathbf{ce}}$ and reads

$$\pm \left[e^{iq\frac{\sigma }{1+\sigma }rcos\left({\varphi }_r\right)}-e^{-iq\frac{1}{1+\sigma }rcos\left({\varphi }_r\right)}\right]2\pi \frac{\hslash c\alpha }{ϵ}\frac{1}{q\left(1+r_{0}q\right)},$$

(7)

with ${\varphi }_r$ as the angle between $\mathbf{r}$ and $\mathbf{q}$, and

$$V_{RK}\left(\mathbf{q}\right)=2\pi \frac{\hslash c\alpha }{ϵ}\frac{1}{q\left(1+r_{0}q\right)}$$

(8)

the Fourier transform of the Rytova–Keldysh potential. Finally, the differential scattering cross section can be written as

$$\begin{array}{c}\hfill I_{fi}\left(\theta \right)=\frac{M^2}{2\pi {\hslash }^4{k}_i}{\left|2\pi \frac{\hslash c\alpha }{ϵ}\frac{1}{q\left(1+r_{0}q\right)}J\left(i\to f\right)\right|}^2.\end{array}$$

(9)

with the dependence on the initial and final exciton states included in $J\left(i\to f\right)$, defined as

$$J\left(i\to f\right)={\int }_0^{+\infty }rdr{\int }_0^{2\pi }d{\varphi }_r\left[e^{i\frac{\sigma }{1+\sigma }qrcos{\varphi }_r}-e^{-i\frac{1}{1+\sigma }qrcos{\varphi }_r}\right]{\chi }_f^{†}\left(\mathbf{r}\right){\chi }_i\left(\mathbf{r}\right).$$

(10)

3. Total Cross Section

Knowing the differential cross section given by Equation (9), we can now compute the full scattering cross section. To this effect, we must simply perform an angular integration in $\theta$ as

$$\begin{array}{cc}\hfill Q_{i\to f}& ={\int }_{-\pi }^{\pi }d\theta I_{i\to f}\left(\theta \right)\hfill \end{array}$$

(11)

and, explicitly substituting Equation (9), the full cross section is given by

$$\begin{array}{cc}\hfill Q_{i\to f}\left(k_i\right)& =\frac{2\pi M^2{\left(\hslash c\alpha \right)}^2}{{\hslash }^4{ϵ}^2{k}_i}{\int }_{-\pi }^{\pi }d\theta {\left|\frac{J\left(i\to f\right)}{q(1+r_{0}q)}\right|}^2.\hfill \end{array}$$

(12)

To compute this integral, however, we must first define the exciton wave functions, which we will consider when computing Equation (10). 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.

3.1. 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

$$q=2\left|sin\left(\frac{\theta }{2}\right)\right|k_i.$$

(13)

Additionally, we also have ${\chi }_{f,i}\left(\mathbf{r}\right)={\chi }_{1s}\left(r\right)$, where we consider a simple variational ansatz [46,47] based on the eigenfunctions of the two-dimensional hydrogen atom [48,49] and given by

$${\chi }_{1s}\left(r\right)={\mathcal{N}}_{1s}{e}^{-r{\gamma }_{1s}/2},$$

(14)

with ${\mathcal{N}}_{1s}$ a normalization constant given by

$${\mathcal{N}}_{1s}={\left[\int rdrd\theta {\left(e^{-r{\gamma }_{1s}/2}\right)}^2\right]}^{-1/2}=\frac{{\gamma }_{1s}}{\sqrt{2\pi }}$$

(15)

and ${\gamma }_{1s}$ a variational parameter. This variational parameter is computed by minimization of the energy expectation value of the Wannier Equation [50]

$$H=-\frac{{\hslash }^2{c}^2}{2\mu }{\nabla }^2+V_{RK}\left(\mathbf{r}\right),$$

(16)

with $V_{RK}\left(\mathbf{r}\right)$ the Rytova–Keldysh potential.

With this ansatz, we can directly substitute the wave function into $J\left(i\to f\right)$, given by Equation (10), and obtain

$$\begin{array}{c}\hfill J_{\mathrm{elast}}\left(q\right)=J\left(1s\to 1s\right)={\gamma }_{1s}^3\left[\frac{1}{{\left[{\left(\frac{\sigma }{1+\sigma }q\right)}^2+{\gamma }_{1s}^2\right]}^{3/2}}-\frac{1}{{\left[{\left(\frac{1}{1+\sigma }q\right)}^2+{\gamma }_{1s}^2\right]}^{3/2}}\right]\end{array}$$

(17)

after integration. This is then substituted into Equation (12), reading

$$\begin{array}{cc}\hfill Q_{\mathrm{elast}}\left(k_i\right)& =\frac{2\pi M^2{\left(\hslash c\alpha \right)}^2}{{\hslash }^4{ϵ}^2{k}_i}{\int }_{-\pi }^{\pi }d\theta {\left|\frac{J_{\mathrm{elast}}\left(q\right)}{q(1+r_{0}q)}\right|}^2,\hfill \end{array}$$

(18)

where q is given by Equation (13). This integral has no analytical solution and must be computed numerically.

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 $ϵ=3.8$ [51]. The electron and hole masses in this material are $m_e=0.83m_0$, $m_h=0.63m_0$ [52], with $m_0$ the electron rest mass, and screening length $r_0=10\AA$ [53]. This material was chosen, as both the screening length and the electron/hole masses are known accurately. 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 Equation (12) in Figure 1. 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 [41] for elastic scattering.

3.2. Inelastic Scattering

Let us now consider inelastic scattering between free carriers and the exciton. The relative momenta is now given by

$$q^2=k_f^2+k_i^2-2k_i{k}_{f}cos\left(\theta \right),$$

(19)

with $k_f$ obtained from the conservation of energy as

$$k_f^2=k_i^2-\frac{2M}{{\hslash }^2}(E_f-E_i).$$

(20)

Here, $E_{f/i}$ is the energy of the final/initial state of the exciton, respectively. Substituting this relation into Equation (19), we obtain

$$q^2=2k_i^2-\frac{2M}{{\hslash }^2}{\Delta }_{f,i}-2k_i^2\sqrt{1-\frac{2M}{{\hslash }^2}\frac{{\Delta }_{f,i}}{k_i^2}}cos\left(\theta \right),$$

(21)

with ${\Delta }_{f,i}=E_f-E_i$. A threshold in $k_i$ below which no scattering is allowed, is immediately evident, obtained from Equation (21) as

$$k_{\min }=\sqrt{\frac{2M}{{\hslash }^2}(E_f-E_i)}.$$

(22)

Below this threshold, there is not enough energy in the scattering process to allow the jump between excitonic states $i\to f$.

Besides knowing the energies of the final states, we must also know their wave functions. These are obtained [46,47,48] in a similar form to Equation (14) and are given by

$$\begin{array}{cc}\hfill {\chi }_{2s}\left(r\right)={\mathcal{N}}_{2s}\left(1-\frac{r}{d}\right)e^{-r{\gamma }_{2s}/2}, & \hfill {\chi }_{2p_{\pm }}\left(r\right)={\mathcal{N}}_{2p}r{e}^{\pm i\theta }{e}^{-r{\gamma }_{2p}/2},\end{array}$$

(23)

where ${\gamma }_{2s},{\gamma }_{2p}$ are variational parameters, ${\mathcal{N}}_{2s},{\mathcal{N}}_{2p}$ are normalization constants given by

$$\begin{array}{cccc}\hfill {\mathcal{N}}_{2s}& =\frac{2{\gamma }_{2s}^2}{\sqrt{\pi }\sqrt{3{\gamma }_{1s}^2-2{\gamma }_{1s}{\gamma }_{2s}+3{\gamma }_{2s}^2}}, \hfill & \hfill {\mathcal{N}}_{2p}& =\frac{{\gamma }_{2p}^2}{2\sqrt{3\pi }},\hfill \end{array}$$

(24)

and d is a parameter obtained by imposing orthogonality between ${\chi }_{1s}$ and ${\chi }_{2s}$, given by

$$d=\frac{4}{{\gamma }_{1s}+{\gamma }_{2s}}.$$

(25)

These wave functions, together with the $1s$ wave function, are plotted in Figure 2 for monolayer $\mathrm{hBN}$ encapsulated in fused quartz.

3.2.1. $1s\to 2s$ Transitions

We will first consider $1s\to 2s$ transitions. To compute $J_{2s}=J\left(1s\to 2s\right)$, we recall Equation (10) and, after integration, obtain

$$\begin{array}{c}\hfill J_{2s}=\frac{3{\gamma }_{1s}\left({\gamma }_{1s}+{\gamma }_{2s}\right){\gamma }_{2s}^2}{\sqrt{6{\gamma }_{1s}^2-4{\gamma }_{1s}{\gamma }_{2s}+6{\gamma }_{2s}^2}}\left[\frac{q^2{\left(\frac{\sigma }{1+\sigma }\right)}^2}{{\left[q^2{\left(\frac{\sigma }{1+\sigma }\right)}^2+{\left(\frac{{\gamma }_{1s}+{\gamma }_{2s}}{2}\right)}^2\right]}^{5/2}}-\frac{q^2{\left(\frac{1}{1+\sigma }\right)}^2}{{\left[q^2{\left(\frac{1}{1+\sigma }\right)}^2+{\left(\frac{{\gamma }_{1s}+{\gamma }_{2s}}{2}\right)}^2\right]}^{5/2}}\right],\end{array}$$

where q is obtained from Equation (21) as

$$q^2=2k_i^2-\frac{2M}{{\hslash }^2}{\Delta }_{2s,1s}-2k_i^2\sqrt{1-\frac{2M}{{\hslash }^2}\frac{{\Delta }_{2s,1s}}{k_i^2}}cos\left(\theta \right).$$

(26)

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 Equation (22), we obtain $k_{\min }=0.0833{\AA }^{-1}$ for electron–exciton scattering and $k_{\min }=0.0760{\AA }^{-1}$ for hole–exciton scattering.

3.2.2. $1s\to 2p$ Transitions

For computing $J\left(1s\to 2p_{\pm }\right)$ following Equation (10), we must consider the distinction between $p_{\pm }$ states. This is, however, not important, as the two states are degenerate, and the two integrals $J\left(1s\to 2p_{+}\right)$, $J\left(1s\to 2p_{-}\right)$ 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, $J_{2p}=J\left(1s\to 2p\right)$ is given by

$$\begin{array}{c}\hfill J_{2p}=\frac{3{\gamma }_{1s}{\gamma }_{2p}^2\left({\gamma }_{1s}+{\gamma }_{2p}\right)}{2\sqrt{6}}\left[\frac{iq\frac{\sigma }{1+\sigma }}{{\left[q^2{\left(\frac{\sigma }{1+\sigma }\right)}^2+{\left(\frac{{\gamma }_{1s}+{\gamma }_{2p}}{2}\right)}^2\right]}^{5/2}}+\frac{iq\frac{1}{1+\sigma }}{{\left[q^2{\left(\frac{1}{1+\sigma }\right)}^2+{\left(\frac{{\gamma }_{1s}+{\gamma }_{2p}}{2}\right)}^2\right]}^{5/2}}\right].\end{array}$$

The relative momentum q is defined analogously to Equation (26), 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 Equation (22) are now $k_{\min }=0.0822{\AA }^{-1}$ for electron–exciton scattering and $k_{\min }=0.0749{\AA }^{-1}$ for hole–exciton scattering.

3.3. 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 $1s\to 1s$ 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

$$Q_{\mathrm{Total}}=\sum _f{Q}_{1s\to f}.$$

(27)

For the three transitions discussed previously, the sum in Equation (27) is restricted and is explicitly written as

$$Q_{\mathrm{Total}}=Q_{1s\to 1s}+Q_{1s\to 2s}+Q_{1s\to 2p}.$$

(28)

This total scattering cross section is plotted in Figure 3 for both types of free carriers, together with the dashed lines representing the thresholds for the inelastic scattering processes considered.

Analyzing Figure 3, 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 $1s\to 2p$ scattering, and the threshold momentum for electron–exciton $1s\to 2p$, i.e., between $k_i=0.0749{\AA }^{-1}$ and $k_i=0.0822{\AA }^{-1}$. In this momentum range, the dominant $1s\to 2p$ process is already allowed for hole–exciton scattering, leading to a vastly superior cross section relative to that for electron–exciton scattering.

The final threshold included, visible in Figure 3 as the dashed purple lines, originates from $1s\to 3d$ scattering, as $E_{3d}=-3.74\mathrm{meV}$ is the lowest energy state after $2s$. These take place at $k_{\min }=0.0875{\AA }^{-1}$ for electron–exciton scattering and $k_{\min }=0.0798{\AA }^{-1}$ for hole–exciton scattering. These are, however, much smaller than the peaks in Figure 3, similar to what was observed in Ref. [37], and are invisible in Figure 3.

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 [24,59,60,61]

$${\Gamma }_{\mathrm{Total}}=\sum _f\frac{2{\hslash }^2}{\pi M}{\int }_0^{\infty }dk{k}^{2}f\left(\frac{m_c+m_e+m_h}{m_e+m_h}k\right)Q_{1s\to f},$$

(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

$${\Gamma }_{\mathrm{Total}}=\frac{2{\hslash }^2}{\pi M}{\int }_0^{\infty }dk{k}^2{n}_F\left(\frac{m_c+m_e+m_h}{m_e+m_h}k\right)Q_{\mathrm{Total}},$$

(30)

where $Q_{\mathrm{Total}}$ is the total scattering cross section as plotted in Figure 3. Here, $n_F\left(k\right)$ is the Fermi–Dirac distribution for free carriers, given by

$$n_F\left(k\right)=\frac{1}{e^{\frac{E_k-E_F}{k_{B}T}}+1}$$

(31)

where the dispersion relation is given by

$$E_k=\frac{{\hslash }^2}{2m_c}{k}^2$$

(32)

and the Fermi energy is

$$E_F=2\pi \frac{{\hslash }^2}{2m_c}n,$$

(33)

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 $d_c=2{\left(\pi n\right)}^{-1/2}$ is still larger but already of the order of the root mean square (RMS) exciton radius, given by

$$r_{\mathrm{RMS};n}={\int }_0^{\infty }{\int }_0^{2\pi }rdrd\theta {\psi }_n{(r,\theta )}^{†}{r}^2{\psi }_n(r,\theta ).$$

(34)

For the two excitonic states most relevant to the scattering cross section, $1s$ and $2p$, the RMS exciton radius is $r_{\mathrm{RMS};1s}=19.5\AA$ and $r_{\mathrm{RMS};2p}=73.2\AA$, respectively, while the average separation between free carriers is $d_c=113\AA$.

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 $n={10}^9{\mathrm{cm}}^{-2}$ and $n={10}^{12}{\mathrm{cm}}^{-2}$. 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.

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.

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 ${10}^9{\mathrm{cm}}^{-2}$ and ${10}^{12}{\mathrm{cm}}^{-2}$, we observed that the difference in linewidth between the two regimes was clearly noticeable as the linewidth became finite at $T=0\mathrm{K}$ 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 $T\gtrsim 150\mathrm{K}$ for $n={10}^9{\mathrm{cm}}^{-2}$ and at $T\approx 140\mathrm{K}$ for $n={10}^{12}{\mathrm{cm}}^{-2}$. Further increasing the carrier density to values above ${10}^{13}{\mathrm{cm}}^{-2}$ 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 $n\approx {10}^{11}{\mathrm{cm}}^{-2}$. Past $n\approx {10}^{11}{\mathrm{cm}}^{-2}$, 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 $T=10\mathrm{K}$.

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].