
\chapter{Scattering Theory} \label{chap:theory2}

This chapter focuses on the theoretical framework and derivations necessary to understand key scattering mechanisms that influence electron transport, spin dynamics, absorption and emission in GeSn alloys. 

In Section \ref{sec:scatrates}, we begin by using Fermi's Golden Rule to present the expressions for alloy and electron–phonon scattering rates, following established formulations from the literature, which govern transport and spin-relaxation in GeSn. We derive expressions for the alloy scattering matrix elements, which govern alloy scattering processes, and describe the frozen phonon method, which is used to determine the deformation potential associated with a particular phonon - a key parameter in determining the intervalley scattering rate by phonons. We summarise the standard derivation of the electron mobility from the Boltzmann Transport Equation in the relaxation-time approximation in Section \ref{sec:mobder}. Finally, in Section \ref{sec:abs}, we use quasi-degenerate perturbation theory to derive an expression used to calculate the absorption coefficient and photoluminescence of GeSn over a wide spectral range. Our derivations in Section \ref{sec:abs} present a simplified adaptation of previous work \cite{tiwari2024unified}.






\section{Electron Scattering} \label{sec:scatrates}

From Ashcroft and Mermin \cite{ashcroft1976solid}, in bulk cubic materials, the formal definition of the relaxation time $\tau^{(\alpha)}(\mathbf{k})$ at band valley $\alpha$ (and thus the total momentum scattering rate $R(\mathbf{k}_{\alpha})$) is given by:
\begin{equation} \label{eq:tauformal}
    \frac{1}{\tau^{(\alpha)}(\mathbf{k})}=R(\mathbf{k}_{\alpha})=\sum_\beta \int \frac{d^3\mathbf{k_\beta'}}{(2\pi)^3}\ \Gamma(\mathbf{k}_\beta',\mathbf{k}_{\alpha})
\end{equation}
where $\Gamma(\mathbf{k'}_{\beta},\mathbf{k}_{\alpha})$ is the total scattering kernel from the state $\mathbf{k}$ in valley $\alpha$ to the state $\mathbf{k'}$ in valley $\beta$. This transition rate, as we've seen in Chapter \ref{chap:spin}, is described using Fermi's Golden Rule (FGR), where the change in state is caused by a perturbation $H'$:
\begin{equation}
    \Gamma(\mathbf{k'}_{\beta},\mathbf{k}_{\alpha})= \frac{2\pi}{\hbar} \left| \left\langle \psi_{\mathbf{k}_{\beta}^{\prime}} \middle| H^{\prime} \middle| \psi_{\mathbf{k}_\alpha} \right\rangle \right|^2 \delta \left( E(\mathbf{k}_\alpha)-E(\mathbf{k}_{\beta}^\prime) \right)  
\end{equation}
By applying the formal definition of the density of states (DOS) per unit volume:
\begin{equation} \label{eq:dosdel}
    \rho_{\beta}(E)=\int \frac{1}{(2\pi)^3}\delta \left(E(\mathbf{k}_\alpha)-E(\mathbf{k}_{\beta}^\prime ) \right) d^3\mathbf{k}
\end{equation}
%where $\Omega$ is the volume of the system,
Eq. \ref{eq:tauformal} can be re-expressed as:
\begin{equation} \label{eq:FerGR}
    R(E(\mathbf{k}_\alpha))= \frac{2\pi}{\hbar} \sum_\beta  \left| \left\langle \psi_{\mathbf{k}_{\beta}^{\prime}} \middle| H^{\prime} \middle| \psi_{\mathbf{k}_\alpha} \right\rangle \right|^2 \rho_{\beta}(E) 
\end{equation}
where we define the scattering matrix element from initial state $\ket{i}$ to final state $\ket{f}$ as $M_{fi}=\left| \bra{f} H' \ket{i} \right| =  \left\langle \psi_{\mathbf{k}_{\beta}^{\prime}} \middle| H^{\prime} \middle| \psi_{\mathbf{k}_\alpha} \right\rangle$. The DOS per unit volume for parabolic bands can be described by the analytic expression:
\begin{equation} \label{eq:dos1}
    \rho(E)=\frac{\sqrt{2} \ m^{3/2} \sqrt{E-E_c}}{\pi^2\hbar^3}\Theta(E-E_c)
\end{equation}
where $E_c$ is the conduction band edge, $m$ is the effective mass at the conduction band and $\Theta(x)$ is the Heaviside function, where $\Theta(x)=1$ if $x>1$, $0$ otherwise, which stresses the fact that there are no conduction band states below $E_c$. Eq. \ref{eq:dos1} can be easily derived from Eq. \ref{eq:dosdel}. 

As electrons predominantly scatter due to phonons and alloy impurities in GeSn, we have two key goals in the remainder of this section: (i) to derive an expression for the alloy scattering matrix elements using the T-matrix formalism in order to determine the alloy scattering rate and (ii) to outline all necessary expressions for calculating the phonon scattering rates relevant to GeSn, which includes a description of how to calculate the deformation potentials using the frozen phonon method.





\subsection{Derivation of the Alloy Scattering Matrix} \label{sec:Vab}


To fully understand electron scattering in GeSn, it is critical to be able to determine the alloy scattering parameters that give the strengths of the scattering between the various conduction band valleys. In this section, we follow the work of Murphy-Armando and Fahy \cite{fma06} to derive expressions for the alloy scattering parameters. We begin by examining the T-matrix Formalism; a framework used in quantum mechanics to describe and calculate the probability amplitudes of transitions between quantum states due to a perturbation.

\subsubsection{T-matrix Formalism}

Let's consider a Hamiltonian $H=H_0+V$ with eigenstates $\ket{\phi}$, where $H_0$ is a perturbation-free Hamiltonian with eigenstates $\ket{\psi}$, and $V$ is a perturbation. We can therefore rearrange the Schrödinger equation $H\ket{\phi}=E\ket{\phi}$ to obtain the expression:
\begin{align}
    H_0\ket{\phi}+V\ket{\phi}&=E\ket{\phi} \nonumber \\
    \ket{\phi}&=(E-H_0)^{-1}\ V\ket{\phi}
\end{align}
If we consider the Hellmann–Feynman theorem \cite{feynman1939forces,guttinger1932verhalten,pauli1933principles,hellman1937einfuhrung}, which requires the energy eigenvalues of the Hamiltonian to change continuously with continuous changes in the Hamiltonian, we would therefore require $\ket{\phi}\rightarrow\ket{\psi}$ as $V\rightarrow0$. In other words, as the distance from the scattering centre $\mathbf{r}\rightarrow\infty$, $| \phi(\mathbf{r})-\psi(\mathbf{r})|^2\rightarrow 0$, meaning the effect of the perturbing potential $V$ vanishes at a sufficiently large distance from the scattering centre \cite{fma06}. This gives us what's known as the Lippmann-Schwinger equation:
\begin{equation} \label{eq:phi}
    \ket{\phi}=\ket{\psi}+(E-H_0)^{-1}\ V\ket{\phi}
\end{equation}
where the second term describes the scattering effects caused by $V$. Assuming a strong correspondence between the eigenstates $\ket{\phi}$ and $\ket{\psi}$, we can rewrite Eq. \ref{eq:phi} explicitly in terms of the wavefunction $\ket{\psi}$ by defining a linear operator $T$ such that:
\begin{equation} \label{eq:T}
    T\ket{\psi}=V\ket{\phi}
\end{equation}
where $T$ is known as the Transition Matrix. By left-multiplying Eq. \ref{eq:phi} by $V$, we can apply Eq. \ref{eq:T} to Eq. \ref{eq:phi}:
\begin{equation} \label{eq:psi}
    T\ket{\psi}=V\ket{\psi}+V(E-H_0)^{-1}\ T\ket{\psi}
\end{equation}








%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\newpage
\subsubsection{Alloy Scattering Matrix Elements}

From Fermi's Golden Rule, the scattering between these two wavefunctions will be proportional to the T-matrix element squared: $T=\bra{f} H' \ket{i}=\bra{\psi} T \ket{\phi}$. If we now consider two ideal crystal states $\ket{\psi_{\mathbf{k}}}$ and $\ket{\psi_{\mathbf{k'}}}$ of equal energy, we can equivalently express $T$ as a function of the initial and final wavefunctions at wavevectors $\mathbf{k'}$ and $\mathbf{k}$:
\begin{equation}\label{eq:Tk}
T(\mathbf{k},\mathbf{k'})=\bra{\psi_{\mathbf{k}}} T \ket{\psi_{\mathbf{k'}}} \nonumber =\bra{\psi_{\mathbf{k}}} \Delta V \ket{\phi_{\mathbf{k'}}}
\end{equation} 
where we apply the T-matrix formalism to produce Eq. \ref{eq:Tk}, with $\Delta V$ denoting the perturbing potential due to an impurity in the alloy, and $\ket{\phi_{\mathbf{k'}}}$ is the exact eigenstate of the Hamiltonian with an impurity present. 



The scattering amplitude for each site $i$ is therefore given by:
\begin{equation} \label{eq:Ti}
    T_i(\mathbf{k},\mathbf{k'})=\bra{\psi_{\mathbf{k}}} V_i \ket{\phi_{\mathbf{k'}}}
\end{equation}
where $V_i$ is the change in potential due to an impurity at site $i$. 



In any alloy composed of atoms $A$ and $B$ which are randomly distributed, the crystal potential $V$ is not periodic. In general, the crystal potential of an alloy can be represented as a sum of an average potential $\Bar{V}$, which is periodic, and a fluctuating potential $\delta V$. To obtain a first-principles expression for the total scattering amplitude due to an alloy impurity, we follow the work of Murphy-Armando \cite{fma06} by expressing $T_i(\mathbf{k},\mathbf{k'})=\Bar{T}_i(\mathbf{k},\mathbf{k'})+\delta T_i(\mathbf{k},\mathbf{k'})$, where $\Bar{T}_i(\mathbf{k},\mathbf{k'})$ is the average of over the entire system and $\delta T_i(\mathbf{k},\mathbf{k'})$ is the variation from the average. Given that we can express the total scattering amplitude as:
\begin{equation}
    T(\mathbf{k},\mathbf{k'})=\sum_{i=1}^N T_i(\mathbf{k},\mathbf{k'})
\end{equation}
where $N$ is the total number of sites, the average of the square of the scattering amplitude over the entire ensemble of configurations of the random alloy is therefore given by: 
\begin{align} \label{eq:delT}
    \left< \left| T(\mathbf{k},\mathbf{k'}) \right|^2 \right> &= \sum_{i=1}^N \sum_{j=1}^N  \left<  \left(\Bar{T}_i^*(\mathbf{k},\mathbf{k'})+\delta T_i^*(\mathbf{k},\mathbf{k'}) \right) \left(\Bar{T}_j(\mathbf{k},\mathbf{k'})+\delta T_j(\mathbf{k},\mathbf{k'}) \right) \right> \nonumber \\
    &= \sum_{i=1}^N \sum_{j=1}^N  \left< \delta T_i^*(\mathbf{k},\mathbf{k'}) \delta T_j(\mathbf{k},\mathbf{k'})\right> \nonumber \\
    &= \sum_{i=1}^N \left< \left| \delta T_i(\mathbf{k},\mathbf{k'}) \right|^2 \right> \nonumber \\
    &= N \left< \left| \delta T_i(\mathbf{k},\mathbf{k'}) \right|^2 \right>
\end{align}
where we assume that the average crystal does not scatter, implying that the mean scattering matrix satisfies $\Bar{T}_j(\mathbf{k},\mathbf{k'})=0$. This assumption aligns with the Coherent Potential Approximation (CPA) \cite{elliott1974theory,harrison1976alloy}. Additionally, we utilise the property that $\left< \delta T_i^*(\mathbf{k},\mathbf{k'}) \delta T_j(\mathbf{k},\mathbf{k'})\right>$ represents the covariance between $T_i$ and $T_j$. If we assume that Ge and Sn are weak scatterers in the alloy, allowing us to treat each site as scattering carriers independently, this covariance is zero unless $i=j$. The final step in Eq. \ref{eq:delT} is plausible based on the knowledge that $\left< \left| \delta T_i(\mathbf{k},\mathbf{k'}) \right|^2 \right>$ is the variance of $T_i(\mathbf{k},\mathbf{k'})$, which is the same for all sites.



In GeSn, these sites are occupied by either Ge (with probability $1-x$) or Sn (with probability $x$). In that sense, we can let $T_i=T_A=\bra{\psi_{\mathbf{k}}} V_A \ket{\phi_{\mathbf{k'}}}$, where $A=$ Ge or Sn. We can therefore express $T$ in terms of the scattering produced by the substitution of an atom of one type for another:
\begin{align}
    \left< \left| T(\mathbf{k},\mathbf{k'}) \right|^2 \right> &= N \left< \left| \delta T_i(\mathbf{k},\mathbf{k'}) \right|^2 \right> \nonumber \\ 
    &=Nx(1-x) \left| T_{Ge}-T_{Sn} \right|^2 \nonumber \\
    &= Nx(1-x) \left| \bra{\psi_{\mathbf{k}}} V_{Ge} \ket{\phi_{\mathbf{k'}}}-\bra{\psi_{\mathbf{k}}} V_{Sn} \ket{\phi_{\mathbf{k'}}} \right|^2
\end{align}
Finally, since the matrix elements do not depend strongly on the wave-vector $\mathbf{k}$, we remove the k-dependence from the last equation; instead expressing the wavefunctions in terms of specific BZ valleys $\alpha$ and $\beta$:
\begin{align} \label{eq:Tmat}
        \left< \left| T(\mathbf{k},\mathbf{k'}) \right|^2 \right>
    &=Nx(1-x) \left| T^{\alpha\beta}_{Ge}-T^{\alpha\beta}_{Sn} \right|^2 \nonumber \\
    &= Nx(1-x) \left| \bra{\psi_{\alpha}} V_{Ge} \ket{\phi_{\beta}}-\bra{\psi_{\alpha}} V_{Sn} \ket{\phi_{\beta}} \right|^2
\end{align}







\bigskip
%\newpage
\subsubsection{Alloy Scattering Rate}

To determine the elastic scattering rate at valley $\alpha$ due to alloy disorder, consider that site $i$ in a GeSn alloy occupies a volume of $\frac{a_0^3}{8}$ for lattice constant $a_0$, i.e. half the volume of a FCC primitive cell. As a result, the volume of the system $V=N\frac{a_0^3}{8}$. Combining Eq. \ref{eq:FerGR} with Eq. \ref{eq:Tmat} in Section \ref{sec:Vab}, and removing the k-dependence, we find:
\begin{equation} \label{eq:fgral1}
    R(E(\mathbf{k}_\alpha))= \frac{2\pi}{\hbar}x(1-x)\frac{a_0^3}{8} \sum_\beta  \left| \left\langle V_{\alpha\beta} \right\rangle \right|^2 \rho_{\beta}(E) 
\end{equation}
where we define:
\begin{align} 
\left\langle V_{\alpha\beta} \right\rangle 
&= \left\langle V_{\alpha\beta}^{Sn} \right\rangle - \left\langle V_{\alpha\beta}^{Ge} \right\rangle \nonumber \\
&=N( \bra {\psi_\alpha} {\Delta V^{Sn}} \ket{\phi_{\beta}} - \bra{ \psi_\alpha }{ \Delta V^{Ge} } \ket{ \phi_{\beta} }) \label{pot}
\end{align}








%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Electron-Phonon Scattering}

\subsubsection{Frozen Phonon Method} \label{sec:froz}

In this section, the frozen phonon method of calculating the deformation potentials (often denoted $D_\mathbf{q}$ for phonon wavevector $\mathbf{q}$) is described. This is a necessary step in determining the e-ph matrix elements $M_{fi}$, and thus the corresponding scattering rate via FGR. Results obtained using this method are used in Chapter \ref{chap:spinrel} to determine the electron spin-phonon scattering rate between the $\Gamma$ and $L$ valleys in GeSn.

The frozen phonon method is a widely used computational technique whereby atomic displacements corresponding to specific phonon modes are "frozen" into the crystal lattice, and the resulting distorted structure is analysed using first-principles methods, such as DFT. 
%Specifically, we displace each atom $i$ in order to explicitly calculate the eigen-energies and wavefunction coefficients
%Specifically, we displace each atom $i$ in order to explicitly calculate the forces between the atoms in the crystal in order to construct the force constant matrix of the crystal. This matrix allows us to calculate the normal modes of at any particular wavevector, $\mathbf{q}$. 

The displacement ($\delta \mathbf{R}_i=\mathbf{R}_i-\mathbf{R}^0_i$) of atom $i$ from $\mathbf{R}^0_i$ to $\mathbf{R}_i$ due to a phonon of momentum $\mathbf{q}$ is given by Eq. \ref{eq:phmod}:
\begin{equation} \label{eq:phmod}
   \mathbf{R}_i = \mathbf{R}^0_i + A \ \mathbf{u} \ e^{i\mathbf{q}\cdot \mathbf{R}^0_i}
\end{equation}
where $\mathbf{u}$ is the phonon eigenvector, which describes the atomic displacement pattern for a given phonon mode. This is typically calculated using density functional perturbation theory (DFPT), which can also be used to compute deformation potentials. In this work, however, we use the frozen phonon method for the latter, as it is computationally more efficient. The parameter $A$ is the displacement amplitude, typically set to some small number whereby matrix elements should converge with decreasing $A$. This convergence must be checked systematically.

%q - L-phonon - 2pi(-1/2,1/2,1/2)......

For a given wavevector $\mathbf{q}$, we choose the smallest supercell in which $\mathbf{q}$ is compatible with periodic boundary conditions. Using this supercell, $D_\mathbf{q}$ is defined as:
\begin{equation}
    D_\mathbf{q}(\mathbf{k})=\frac{\delta E_n(\mathbf{k})}{\delta R_\mathbf{q}}
\end{equation}
where $\delta E_n(\mathbf{k})$ is the band energy of electronic state $n$ at wavevector $\mathbf{k}$. We will now show how these deformation potentials can be used in determining the e-ph scattering rates.

%Forces... Dynamical matrix...



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\newpage
\subsubsection{Electron-Phonon Scattering Rate}

For momentum scattering due to phonons, the matrix element $M_{fi}$ depends primarily on deformation potentials, the phonon frequency $\omega_{\mathbf{q}}$ and the phonon occupation number $N_{\mathbf{q}}$, which is given by the Bose-Einstein distribution:
\begin{equation}
    N_{\mathbf{q}} = \frac{1}{e^{\hbar \omega_{\mathbf{q}} / k_B T} - 1}
\end{equation}
For phonon absorption, the probability scales as $N_{\mathbf{q}}$, while for emission, it scales as $N_{\mathbf{q}}+1$. We calculate the e-ph scattering in GeSn by following Fischetti and Laux \cite{Fisch} and Jacoboni and Reggiani \cite{jac} using the deformation potential method. We must consider intra-valley acoustic phonon scattering, intra-valley optical phonon scattering, and inter-valley electron-phonon scattering. The relaxation rates for the latter two processes can be described more straightforwardly compared to acoustic phonon scattering, as bulk cubic materials retain symmetry under reflections around the Cartesian planes \cite{Fisch}.

Optical (op) phonons are short wavelength (higher-energy) waves where atoms in the unit cell move out of phase. When an electron scatters as a result of optical phonons, the square of the matrix element $M_{fi}$ can be expressed as:
\begin{equation} \label{eq:phmat}
|M_{fi}|^2 = \frac{D_\mathbf{q}^2 \hbar}{2 \rho V \omega_{\mathbf{q}}} \begin{bmatrix} N_{\mathbf{q}} \\ N_{\mathbf{q}}+1\end{bmatrix}
\end{equation}
where $\rho$ is the material density and $V$ is the primitive cell volume. This results in the following expression for the e-ph scattering rate for optical phonons:
\begin{align} \label{eq:Rop}
    R_{op}=\frac{(D_{op})^2 m^{3/2}}{\sqrt{2}\pi\rho\hbar^3 \omega_{op}} \begin{bmatrix} N_{op} \\ N_{op}+1\end{bmatrix} \sqrt{E-E_c\pm\hbar\omega_{op})} \nonumber \\ \times \left[ 1+\frac{5}{2}\gamma (E-E_c\pm\hbar\omega_{op})  \right]
\end{align}
where $D_{op}$, $\omega_{op}$ and $N_{op}$ represent the deformation potential, the frequency and the distribution of optical phonons, respectively. The final term accounts for any non-parabolicity in the bands \cite{jac}, where $\gamma$ is the valley-dependent non-parabolicity factor (formula shown in Chapter \ref{chap:trans}), and the rooted term ensures the conservation of energy is obeyed. 

Acoustic (ac) phonons behave like sound waves where atoms in the unit cell move in phase. They are long-wavelength (low frequency) vibrations, such that $\hbar \omega_{\mathbf{q}} << k_B T$. For small $x$, we can use a first order Taylor series to expand $e^x$, resulting in $e^x\approx1+x$. Applying this to the Bose-Einstein distribution gives $N_{\mathbf{q}} =\frac{k_BT}{\hbar \omega_{\mathbf{q}}}$. For acoustic phonons, we also replace $D_\mathbf{q}^2$ is Eq. \ref{eq:phmat} with $\Xi^2\mathbf{q}^2$ \cite{jac}, where $\Xi$ is $\mathbf{q}$-independent. As we discussed in Chapter \ref{chap:gesn}, the deformation potential operator for a given band can be expressed in terms of two or three deformation potential constants, depending on the crystal symmetry. These are often denoted by $\Xi_{u}$, the uniaxial deformation potential constant, and $\Xi_{d}$, the dilatation deformation potential constant, which we list for Ge in Chapter \ref{chap:trans}, along with all relevant deformations in Ge. Equation \ref{eq:matac} describes the squared matrix element for acoustic phonons, which is expressed as a sum over each phonon mode $\eta$:
\begin{equation} \label{eq:matac}
    |M_{fi}|^2 = \sum_{\eta}\frac{\Xi_{ac}^2 k_BT}{\rho V \nu_{\eta}^2}
\end{equation}
where $\nu_{\eta}$ is the average sound velocity such that $\omega_{\mathbf{q}}=\nu_\eta\left| \mathbf{q}\right|$, and $k_BT$ is the thermal energy. Eq. \ref{eq:Rac} shows the relaxation rate due to intra-valley acoustic phonon emission and absorption ($R_{ac}$):  
\begin{equation} \label{eq:Rac}
    R_{ac}=\sum_{\eta}\frac{\sqrt{2}\ \Xi_{ac}^2 m^{3/2}k_BT}{\pi\rho\hbar^4 \nu_{\eta}^2}  \sqrt{E-E_c} \left[ 1+\frac{5}{2}\gamma (E-E_c)  \right]
\end{equation}
For intervalley e-ph scattering processes, electrons are scattered either by optical or acoustic phonons. The phonon wave vector $\mathbf{q}$ for such a transition is very close to $\Delta k$, the distance between the initial and final valleys \cite{jac}. Therefore, $\hbar\omega_\mathbf{q}$ is roughly constant for a given branch of phonons, since the phonon wave vector lies near the BZ edge, where the phonon dispersion is relatively flat. This is also true of intravalley optical phonon e-ph scattering. Consequently, intervalley scattering is formally treated in the same way as intravalley scattering by optical phonons, but with two distinct differences. Firstly, the energy term $E$ becomes $E-\Delta E_{fi}$ in Eq. \ref{eq:Rop}, where $\Delta E_{fi}$ is the difference in energies of the initial and final bands. Secondly, since the electrons can scatter into multiple degenerate valleys, we include the term $Z_f$ in the numerator, which represents the number of possible final conduction band valleys in the scattering process.

% should I be writing these expressions for R in terms of the dos \rho instead of analytically?





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\bigskip
\section{Mobility from the Boltzmann Transport Equation} \label{sec:mobder}



\subsection{The Boltzmann Transport Equation}

% discuss desire to plot mobility.

% define mobility

% maybe discuss that for GeSn, we have L mobility and G mobility, and they require separate deerivations. Show how they sum to total, and remove from results part. 



In order to describe the movement of carriers in a solid, we begin by defining the carrier distribution function $f_k(\mathbf{r},t)$ as the probability that a state in a region near $\mathbf{r}$ with momentum $\mathbf{k}$ at time $t$ is occupied. If we assume that the carriers remain in thermal equilibrium in the absence of an external field, $f_k$ equates to the Fermi-Dirac (FD) distribution function:
\begin{equation} \label{eq:fd} %4.46 
    f_k^0=\frac{1}{e^{\frac{E_\mathbf{k}-\mu_F}{k_B T}}+1}
\end{equation}
where $\mu_F$ is the chemical potential, which at $T=0$ is equal to the Fermi energy $E_F$. The superscript $0$ denotes the distribution $f$ is at equilibrium.


The total differential of a distribution function $f_k(\mathbf{r},t)$ is given as:
\begin{align} \label{eq:dt}
    df &= \frac{\partial f}{\partial t}dt + \nabla_{\mathbf{r}} f \cdot d\mathbf{r} + \nabla_{\mathbf{p}}f \cdot d\mathbf{p} \nonumber \\
   % &= \frac{\partial f}{\partial t}dt + \mathbf{v} \cdot \nabla_{\mathbf{r}} f \ dt +\mathbf{F} \cdot \nabla_{\mathbf{p}}f \ dt \nonumber \\
    &= \frac{\partial f}{\partial t}dt + \mathbf{v} \cdot \nabla_{\mathbf{r}} f \ dt +\frac{1}{\hbar}\mathbf{F} \cdot \nabla_{\mathbf{k}}f \ dt
\end{align}
recalling that $\mathbf{p}=\hbar \mathbf{k}$, and the rate of change of the momentum $\mathbf{k}$ is given by the external force $\mathbf{F}$ acting on the particle. Taking the derivative with respect to time gives us the Liouville equation, which governs the evolution of $f_k(\mathbf{r},t)$:
\begin{equation} \label{eq:loui}
    \frac{d f}{d t} = \frac{\partial f}{\partial t} + \mathbf{v} \cdot \nabla_{\mathbf{r}} f + \frac{1}{\hbar}\mathbf{F} \cdot \nabla_{\mathbf{k}} f
\end{equation}
where on the right hand side of Eq. \ref{eq:loui}, the first term gives the change in $f$ with time at a fixed point, the second term captures how $f$ changes as particles move through space with velocity $\mathbf{v}$ and the third term describes the change due to external forces acting on the particles. In the absence of any scattering events, the left-hand side is $0$. However, if we include scattering, we obtain the Boltzmann Transport Equation (BTE):
\begin{equation} \label{eq:bte}
    \left(\frac{\partial f_{\mathbf{k}}}{\partial t}\right)_{scatt} = \frac{\partial f_{\mathbf{k}}}{\partial t} + \mathbf{v_g} \cdot \nabla_{\mathbf{r}} f_{\mathbf{k}} + \frac{1}{\hbar}\mathbf{F} \cdot \nabla_{\mathbf{k}} f_{\mathbf{k}}
\end{equation}
where $v_g$ is the group velocity, which defines the trajectory of the particles in a crystal, and the external force  $\mathbf{F}=-e(\mathcal{E}+\mathbf{v}_g\times\mathbf{B})$ is the force felt by the electrons due to an external electromagnetic field. Eq. \ref{eq:bte} essentially explains how distributions evolve over time. Mathematically, the group velocity can be expressed as in Eq. \ref{eq:vg} below:
\begin{equation} \label{eq:vg}
    v_g\equiv\frac{\partial \omega}{\partial k}=\frac{1}{\hbar}\frac{\partial E}{\partial k}
\end{equation}
where $\omega$ is the angular frequency of the wave. For a steady-state system ($f_{\mathbf{k}}$ does not explicitly depend on time), then the first term on the right-hand side of Eq. \ref{eq:bte} becomes $0$. We can make two other assumptions of consequence: one is we assume consistent atomic distributions in our alloy, given that local variation in composition (such as clusters) are often minimised through advanced fabrication techniques. We also assume that the applied electric field is uniform and there are no thermal gradients, meaning the driving forces affecting the carriers are the same throughout the material. As a result, $f_{\mathbf{k}}$ remains spatially uniform, leading to the second term on the right-hand side of Eq. \ref{eq:bte} vanishing. This leaves us with a simplified expression of the BTE:
\begin{equation} \label{eq:bte1}
    \left(\frac{\partial f_{\mathbf{k}}}{\partial t}\right)_{scatt} = \frac{1}{\hbar}\mathbf{F} \cdot \nabla_{\mathbf{k}} f_{\mathbf{k}}
\end{equation}




%\bigskip
\subsection{The Relaxation Time Approximation}

The Relaxation Time Approximation (RTA) is a simplification used in the BTE to model how a system relaxes back to equilibrium after a weak out-of-equilibrium perturbation. In this approximation, the collision term $\left(\frac{\partial f_{\mathbf{k}}}{\partial t}\right)_{scatt}$ is assumed to cause the distribution function $f_k$ to relax towards the equilibrium distribution $f_k^0$ with a characteristic time constant $\tau$ (the relaxation time), which is representing the average time between scattering events, such that:
\begin{equation} \label{eq:bterta}
    \left(\frac{\partial f_{\mathbf{k}}}{\partial t}\right)_{scatt}\approx-\frac{f_{\mathbf{k}}-f_{\mathbf{k}}^0}{\tau_\mathbf{k}}  = \frac{1}{\hbar}\mathbf{F} \cdot \nabla_{\mathbf{k}} f_{\mathbf{k}}
\end{equation}
Thus, the distribution of carriers becomes:
\begin{align} \label{eq:delf}%4.54
   f_k &= f_k^0-\frac{1}{\hbar}\tau_k \nabla_k f_k \cdot \mathbf{F} \nonumber \\
   &\approx f_k^0-\frac{1}{\hbar}\tau_k \nabla_k f_k^0 \cdot \mathbf{F}
\end{align}
where we have considered a weak applied field, whereby $f\approx f^0$.%, and comparing to equation \ref{eqn:fkexp}, we have $h_\mathbf{k}(\mathbf{F})=-\frac{1}{\hbar}\tau_\mathbf{k}\nabla_\mathbf{k}f_k^0 \cdot \mathbf{F}$.









%\bigskip



%\newpage
\subsection{Mobility from the BTE and the RTA}


The current density (charge per unit time that flows through a unit area of a chosen cross section) can be expressed generally as:
\begin{equation} \label{eq:j}
    \mathbf{j}=-2e\int_{BZ}\frac{d^3\mathbf{k}}{(2\pi)^3}\ f_{\mathbf{k}}\ v_g(\mathbf{k})
\end{equation}
where the $2$ in the numerator accounts for the two possible spin states. By substituting Eq. \ref{eq:delf} into Eq. \ref{eq:j}, we have:
\begin{align} \label{eq:j1}
    \mathbf{j}&=-\frac{2e}{(2\pi)^3}\int_{BZ}d^3\mathbf{k}\left( f_k^0-\frac{1}{\hbar}\tau_k \nabla_k f_k^0 \cdot \mathbf{F}\right) v_g(\mathbf{k}) \\
    &\approx -\frac{2e^2}{(2\pi)^3}\int_{BZ}d^3\mathbf{k}\ \frac{1}{\hbar}\tau_k \nabla_k f_k^0 \cdot \mathcal{E} \ v_g(\mathbf{k})
\end{align}
where the external force is given by $F=-e\mathcal{E}$. The equilibrium term vanishes in Eq. \ref{eq:j1} because the Fermi–Dirac distribution is even in $\mathbf{k}$ while the group velocity is odd, so their product integrates to zero over the Brillouin zone. We now remove the $\mathbf{k}$ subscript, and instead introduce the superscript $\alpha$ to denote the band valley. For an isotropic material, considering the electric field $\mathcal{E}$ along the $\lambda$ axis, the carrier mobility at a given band $\alpha$ is given by: %cite????
\begin{equation} \label{muj}
    \mu^{(\alpha)}_{\lambda}=\frac{j^{(\alpha)}_{\lambda}}{n^{(\alpha)}e\mathcal{E}}
\end{equation}
where $n^{(\alpha)}$ is the carrier concentration at valley $\alpha$, which is calculated using Eq. \ref{eq:n}, where $N_c$ denotes the valley degeneracy:
\begin{equation}
    n^\alpha=N_c\int_0^\infty \rho^\alpha(E)f(E)dE \label{eq:n}
\end{equation}
We can therefore express the mobility at band $\alpha$ as:
\begin{equation} \label{mu1}
    \mu^{(\alpha)}_{\lambda}=-\frac{e}{\hbar \ n^{(\alpha)}}\int_{BZ}\frac{d^3\mathbf{k}}{4\pi^3}\ \frac{\partial f}{\partial \mathbf{k}} \ v_g^{(\alpha)} \tau^{(\alpha)}_{\lambda}
\end{equation}
Finally, we perform a change of variables, such that $\frac{\partial}{\partial \mathbf{k}}=\frac{\partial E}{\partial \mathbf{k}}\frac{\partial}{\partial E}$. Recalling the definition of the group velocity in Eq. \ref{eq:vg}, we have our final expression for the carrier mobility at band $\alpha$:
\begin{equation} \label{eq:mu2}
    \mu^{(\alpha)}_{\lambda}=\frac{e}{ n^{(\alpha)}}\int_{BZ}\frac{d^3\mathbf{k}}{4\pi^3}\ \frac{\left(v_g^{(\alpha)} \right)^2}{R^{(\alpha)}_{\lambda}}\left( \frac{-\partial f}{\partial E} \right)
\end{equation}
where $R$ is the total scattering rate such that $\frac{1}{\tau}=R$.





\bigskip
%\newpage

\subsection{Carrier Mobility for Parabolic Bands}

%Say something about it being easier to determine the mobility as an integral over energy as opposed to k-space, and that the group velocity can in turn be easily expressed in terms of the energy. Make sure to reference Fischetti when discussing non-parabolicity term in equation.

While Eq. \ref{eq:mu2} is applicable for the carrier mobility at any conduction band valley, it is computationally simpler to express the mobility as an integral over energy as opposed to $k$-space, in particular for parabolic bands. We can perform a change of variables using the energy-momentum relationship for a free electron ($E=\frac{\hbar^2k^2}{2m}$). We find:
\begin{enumerate}
    \item $v_g^2=\frac{2E}{m}$
    \item $\int d^3k=4\pi^3\int E \ \rho(E)\ dE$
\end{enumerate} 
However, at a given band valley $\alpha$, the transverse and longitudinal effective masses differ in materials with anisotropic band structures, where the electron's energy depends differently on the direction of its momentum. We also introduce the non-parabolicity factor $\gamma$ into the DOS to account for deviations from a purely parabolic energy-momentum relationship \cite{Fisch}. Therefore, Eq. \ref{eq:dos1} becomes:
\begin{equation}
\rho^\alpha (E)=\frac{\sqrt{2}\sqrt{m_l^\alpha}{m_t^\alpha \sqrt{(E-E_c^\alpha)(1+\gamma^\alpha(E-E_c^\alpha))}}(1+2\gamma^\alpha(E-E_c^\alpha))}{\pi^2\hbar^3}\Theta(E-E_c^\alpha)
\label{eq:dos}
\end{equation}
We also introduce a non-parabolicity term into the mobility, leading to our updated expression for the carrier mobility (as per Fischetti and Laux \cite{Fisch}):
\begin{equation} \label{eq:mob}
\mu_{\lambda}^{(\alpha)} =\frac{2e}{3n^{(\alpha)}m_{\lambda}^{(\alpha)}}\int_{0}^{\infty} dE \ \frac{ E \rho^{(\alpha)}(E) \left(\frac{-\partial f}{\partial E} \right)}{R_{\lambda}^{(\alpha)}(E)} 
 (1+\gamma^{(\alpha)} E/2)   
\end{equation}
where the $3$ in the denominator can be attributed to the equipartition theorem, describing how energy is distributed among the degrees of freedom of a system in thermal equilibrium. The total mobility is obtained by summing over the bands:
\begin{equation}
    \mu=\sum_\alpha r^{(\alpha)}\mu^{(\alpha)}
\end{equation}
where $r^{(\alpha)}=\frac{n^{(\alpha)}}{n}$ is the carrier concentration ratio, and $n=\sum_\alpha n^{(\alpha)}$ is the total carrier concentration.


In Chapter \ref{chap:trans}, where we present our results on the electron mobility of GeSn, we employ both Eq. \ref{eq:mu2} and Eq. \ref{eq:mob}, as GeSn exhibits parabolic band dispersion at the $L$ valley and non-parabolic behaviour at the $\Gamma$ point.









%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bigskip
%\newpage

\section{Optical Properties: Absorption and Emission Mechanisms} \label{sec:abs}

%\subsection{Expressions for Electron Absorption and Emission} 

The absorption coefficient $\alpha$ of a medium can be expressed using Eq. \ref{eq:abs} below: 
\begin{equation} \label{eq:abs}
    \alpha = \frac{\omega \epsilon_i}{cn}
\end{equation}
where $\omega$ represents the frequency of the photon, $\epsilon_i$ is the imaginary part of the dielectric function ($\Tilde{\epsilon}=\epsilon_r+\epsilon_ii$) and $n$ is the real part of the refractive index, such that ($\Tilde{n}=n+i\kappa$), where $\kappa$ is known as the extinction coefficient. Given that the real and imaginary parts of the dielectric function can be written as $\epsilon_r = n^2-\kappa^2$ and $\epsilon_i = 2n\kappa$ respectively, we can describe $n$ as:
\begin{equation} \label{eq:nr}
    n = \sqrt{\frac{\epsilon_r + \sqrt{\epsilon_r^2+\epsilon_i^2}}{2}}
\end{equation}
We can also express the material's photoluminescence (PL) in terms of its refractive index and dielectric function. We use the van Roosbroek-Shockley relation \cite{van1954photon}, given in Eq. \ref{eq:PL}, which computes the total rate of radiative recombination per unit volume \cite{bebb1972photoluminescence}:
\begin{equation}
    PL= \frac{2 n}{\pi c^3}\frac{\omega^3\epsilon_i}{e^{(\hbar\omega-\Delta \mu)/k_BT}-1} \label{eq:PL}
\end{equation}
where $\hbar\omega$ is the photon energy and $\Delta \mu=\mu_c-\mu_v$ is the difference in the quasi-Fermi levels for electrons ($\mu_c$) and holes ($\mu_v$), which is $0$ in thermal equilibrium. We calculate $\mu_c$ and $\mu_v$ by assuming a steady-state condition in which the electron (e) and hole (h) populations are fixed and equal ($n_e=n_h$).%, where $n_e$ is determined using Eq. \ref{eq:n}, while $n_h$ is calculated using $n_h=\int_{-\infty}^E_v(1-f_v$).


Following Section $6.2$ of Yu and Cardona \cite{peter2010fundamentals}, for an electric field $E$, $\epsilon_i$ can be described by:
\begin{equation} \label{eq:ei}
    \epsilon_i=\frac{2R\hbar}{\epsilon_0 \lvert E\rvert^2}
\end{equation}
where $R$ is the transition rate (given by Fermi's Golden Rule) of an electron transitioning from initial state $\ket{i}$, with energy $E_i$ at wavevector $\Vec{k}_i$ to final state $\ket{f}$, with energy $E_f$ at wavevector $\Vec{k}_f$. To determine $\epsilon_r$, we use the Kramer's-Kronig relations (KKRs) \cite{kronig1926theory,kramers1927diffusion}, shown in Eq. \ref{eq:er}, assuming the incident field is weak enough for polarisation to depend linearly on the electric field. Expressing these parameters as functions of the photon frequency $\omega$, $\epsilon(\omega)=\epsilon_r(\omega)+i \epsilon_i(\omega)$ represents the medium's linear response, which inherently satisfies the KKRs.
\begin{equation}
        \epsilon_r(\omega) = 1+ \frac{2}{\pi} \mathcal{P} \int_{0}^{\infty} \frac{\omega' \epsilon_i(\omega')d\omega'}{\omega'^2-\omega^2} \label{eq:er}
\end{equation}
Optical absorption and emission processes in semiconductors are governed by the energy of the photon absorbed/emitted and the material’s electronic band structure. Electrons can get excited into the conduction band (c), or recombine into the valence band (v), either directly ($\Vec{k}_v=\Vec{k}_c$), as in many III–V semiconductors, or indirectly ($\Vec{k}_v \neq \Vec{k}_c$), as is primarily the case at low photon energies for indirect-gap semiconductors. %Furthermore, the rates of these optical transitions are influenced by the presence of energy-degenerate conduction band states. 

We have seen that the band structure of GeSn enables a continuous transition from indirect to direct bandgap behaviour through compositional tuning. As a result, both direct and indirect transitions contribute significantly to their absorption and emission properties, with their contributions depending on the Sn content and temperature. Understanding and quantifying these processes is essential for optimising optoelectronic devices based on GeSn.

In the remainder of this section, we derive expressions for $\epsilon_i$ for (i) direct transitions between the valence and conduction bands, and (ii) indirect transitions from $\ket{k}$ to $\ket{k'}$ in a particular band using non-degenerate perturbation theory, as well as (iii) transitions into quasi-degenerate conduction band states using degenerate perturbation theory. This approach largely follows the unified formalism introduced by Tiwari et al. \cite{tiwari2024unified}, where a single expression is derived to govern all optical transitions. This approach enables a detailed and accurate description of the optical response across the full range of GeSn compositions. The derived expressions can then be applied to calculate the absorption coefficient and PL over a range of frequencies $\omega$ using equations \ref{eq:abs}, \ref{eq:nr} and \ref{eq:er}, results for which are given in Chapter \ref{chap:Opt}.



%\bigskip

\subsection{Direct Transitions}

The scattering rate for the direct transition of electrons can be derived using first-order perturbation theory. Consider the Hamiltonian $H=H_0+H_\gamma$, where $H_0=\frac{p^2}{2m_e}+V(\Vec{r})$ is the unperturbed one-electron Hamiltonian, where $\vec{p}$ is the momentum operator, $m_e$ is the free electron mass and $V$ represents the periodic potential of the crystal lattice, while $H_\gamma=\frac{e}{m_e c}\Vec{A} \cdot \Vec{p}$ represents the electron-radiation interaction. The vector potential $\Vec{A}$, given by Eq. \ref{eq:A}, describes an electromagnetic field:
\begin{equation} \label{eq:A}
    \Vec{A}(\Vec{r},t)=\frac{-E}{2k}e^{i(\Vec{k}\cdot\Vec{r}-\omega t)} +c.c
\end{equation}
for wavevector $\Vec{k}$ at position $\Vec{r}$. The complex conjugate term ($c.c$) corresponds to stimulated emission. We therefore apply the first term for absorption and the complex conjugate for emission. The mathematical reasoning for this can be found in Ref. \cite{peter2010fundamentals}. Without it, $\Vec{A}$ describes absorption, and its field strength must be weak enough to apply perturbation theory to calculate $R$. Using a Coulomb gauge ($\nabla \cdot \Vec{A}=0$) for simplicity and defining $\Vec{A}=A\Vec{e}$, where $\Vec{e}$ is a unit vector parallel to $\Vec{A}$, we obtain the matrix element:
\begin{align}
    \lvert \bra{c} H_{\gamma} \ket{v} \rvert^2 
    &= \left( \frac{e}{m_e c} \right)^2 \lvert A \rvert^2 \ \lvert \bra{c} \Vec{e} \cdot \Vec{p} \ket{v} \rvert^2 \nonumber \\
    &= \left( \frac{e}{m_e c} \right)^2 \left( \frac{E(\omega)}{2k}\right)^2 \lvert P_{cv} (\Vec{k}) \rvert^2 \label{eq:Hdir}
\end{align}
where $P_{cv}=\bra{c} \Vec{e} \cdot \Vec{p} \ket{v}$ is the momentum matrix element (also called the electric-dipole matrix element), which quantifies how likely an electron is to transition between states at a given point in k-space due to interaction with light (which has an electric field $\vec{E}$). Accurate computation of $P_{cv}$ requires proper evaluation of wavefunction derivatives in real space. However, as discussed in Chapter \ref{chap:theory}, DFT employs the pseudopotential method, which smooths pseudo-wavefunctions in the core region, whereas true wavefunctions exhibit strong oscillations near atomic cores. This discrepancy results in inaccuracies in momentum matrix elements. To correct for this, standard adjustments have been implemented in ABINIT post-processing, as outlined below: 
\begin{equation}
    P_{cv}=\bra{c} \Vec{e}\cdot\Vec{p}  \ket{v} + \frac{i m_e}{\hbar}\bra{c} [ U,\Vec{r} ] \ket{v}
\end{equation}
where $U$ is the non-local part of the pseudopotential and $\Vec{r}$ is the positional vector. %The positional matrix elements can be accurately determined from the pseudopotential method since the tails of the wave functions define the elements. Also, the positional matrix elements have a universal relationship with the momentum matrix elements, which results from the non-locality of the pseudopotential \cite{kageshima1997momentum,starace1971length,read1991calculation}.

Using $k=\omega/c$, we combine equations \ref{eq:FerGR}, \ref{eq:ei} and \ref{eq:Hdir} to obtain the following relation for direct transitions:
\begin{equation} 
    \epsilon_i(\omega)=\frac{\pi e^2}{\epsilon_0 m_e V} \frac{1}{N}\frac{1}{\omega^2}\sum_{\Vec{k}} \ \lvert P_{cv}(\Vec{k}) \rvert^2 \ \delta ( E_c(\Vec{k})-E_v(\Vec{k})-\hbar\omega ) \label{eq:ei1}
\end{equation}
where the summation runs over a uniform BZ grid with $N$ points, and $V$ is the primitive cell volume. In GeSn, by confining the photon energy within a $0-1eV$ range, this only encompasses transitions from around the top of the three uppermost valence bands around the $\Gamma$ valley to points around the conduction band at $\Gamma$.


%\bigskip %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Indirect Transitions} \label{sec:ndpt}

Indirect absorption occurs when an electron transitions not only  between energy bands via photon absorption/emission, but also between wavevectors within a band. This intra-band scattering can occur either through phonon absorption/emission or by alloy scattering. The resulting Hamiltonian takes the form $H=H_0+H_\gamma+H_{ep}+H_{al}$, where $H_{ep}$ is the electron-phonon (e-ph) interaction, the matrix element for which is shown in Eq. \ref{eq:phmat}, and $H_{al}$ is the alloy scattering Hamiltonian, derived in Section \ref{sec:Vab}. Second order perturbation theory is applied to obtain the transition rate for indirect absorption \cite{peter2010fundamentals}. The correction to the final eigenstate $\ket{f}$ is given generally by:
\begin{equation}
    \ket{f'}=\ket{f}+\sum_{m\neq f} \frac{\bra{m}H_{ep}\ket{f}}{E_{f}-E_{m}}\ket{m}
\end{equation}
where $\ket{m}$ is some intermediate state. The scattering rate for indirect transitions is thus given by Eq. \ref{eq:Rind}:
\begin{equation} \label{eq:Rind}
  R_{ind}  =\frac{2\pi}{\hbar}\sum_{\Vec{k}_f,\Vec{k}_i} \ \left| \sum_m  \frac{\bra{f} H_{ep} \ket{m} \bra{m} H_{\gamma} \ket{i}}{E_{m}-E_{f}\pm E_p} \right|^2 \ \delta ( E_f-E_i-\hbar\omega \pm E_p)  
\end{equation}
In Eq. \ref{eq:Rind}, $E_p=\hbar\omega_q$ is the phonon energy, whereby the $\pm$ represents phonon absorption/emission, respectively, and $\vec{k_f}=\vec{k_i}+\vec{q}$ for phonon wavevector $\vec{q}$. The matrix element is a sum over all possible intermediate states $\ket{m}$. In GeSn, within a 0-1eV range, phonon or alloy-assisted transitions only will occur between points around the $\Gamma$ valley and the four $L$ valleys. %We can therefore re-express Eq. \ref{eq:Rind} as:
%\begin{equation} \label{eq:RindGeSn}
%  R_{ind}  =  \frac{2\pi}{\hbar}\sum_{\Vec{k}_c,\Vec{k}_v} \ \left|  \frac{\bra{L} H_{ep} \ket{\Gamma} \bra{\Gamma} H_{\gamma} \ket{v}}{E_{\Gamma}-E_{L}\pm E_p} \right|^2 \ \delta ( E_c(\Vec{k}_c)-E_v(\Vec{k}_v)-\hbar\omega \pm E_p)
%\end{equation}
Applying Eq. \ref{eq:phmat} and Eq. \ref{eq:Rind} to Eq. \ref{eq:ei} gives:
\begin{align} \label{eq:eiInd}
    \epsilon_i(\omega)&=\frac{\pi e^2}{\epsilon_0 m_e V} \frac{1}{N}\frac{1}{\omega^2}
    \left(\begin{bmatrix}
        N_q \\ N_{q}+1
    \end{bmatrix} D_{\Gamma L}^2 \left( \frac{\hbar}{2\rho V \omega_q}
 \right) +\left(x(1-x)V_{L\Gamma}\right)^2\right)\nonumber \\ &\times\sum_{\Vec{k}_f,\Vec{k}_i} \left| \frac{ P_{cv}}{{E_{m}-E_{f}\pm E_p} } \right|^2  \delta ( E_f(\Vec{k}_f)-E_i(\Vec{k}_i)-\hbar\omega \pm E_p )
\end{align}

where $x$ is the alloy content and $V_{L\Gamma}$ is the alloy scattering potential between the $L$ and $\Gamma$ valleys.






\subsection{Transitions into Quasi-Degenerate Bands} \label{sec:dpt}

From the denominator of Eq. \ref{eq:Rind}, it is evident that the equation breaks down when $E_{m}=E_{f}\pm E_p$, whereby the energy of the intermediate state approaches degeneracy with the energy of the final state plus/minus phonon energy, causing the transition amplitude to diverge. Consequently, Eqs. \ref{eq:ei1} and \ref{eq:eiInd} alone are insufficient, as they fail to accurately account for such transitions.

To address this, we follow the approach of Tiwari et al. \cite{tiwari2024unified} by implementing degenerate perturbation theory to accurately describe optical transitions into quasi-degenerate conduction band states. In our GeSn calculations, shown in Chapter \ref{chap:spinrel}, we focus on the short-wave infrared light range, in which only optical transitions between the top of the valence band (at $\Gamma$) and the conduction band minima around the $\Gamma$ valley and the four $L$ valleys are considered. Consequently, in this framework, the diagonal elements of the effective Hamiltonian correspond to the energies around the $\Gamma$ and $L$ valleys, with phonon energies added or subtracted to represent phonon absorption or emission, respectively. Assuming negligible intravalley scattering and intervalley scattering between the 4 $L$ bands, the only off-diagonal elements are the interactions due to phonons and alloy disorder between the $\Gamma$ and $L$ energies (we assume a constant electron-phonon interaction term between the states rather than a $\mathbf{q}$-dependent term to reduce computational cost. Below (Fig. \ref{fig:Hdeg}) is a schematic of the Hamiltonian matrix for such as system, whereby we utilise the following notation:
\begin{itemize}
    \item $E_{\Gamma_1}$, $E_{\Gamma_2}$, etc are the energies of the states around the $\Gamma$ valley, while $E_{L_1}$, $E_{L_2}$, etc are the energies of the states around the $L$ valley. 
    \item $M_{fi}^{ab}$ and $M_{fi}^{em}$ denote the intervalley electron-phonon interaction terms for absorption and emission, respectively, as described in Eq. \ref{eq:phmat}.
    \item $V_{\Gamma L}$ represents the intervalley alloy scattering parameter between the $\Gamma$ and $L$ valleys, and expression for which is given in Eq. \ref{eq:Tmat}. This parameter is calculated in Chapter \ref{chap:trans}.
\end{itemize}

\begin{figure}[H]
    \centering
    \includegraphics[width=0.99\linewidth]{Theory/Figures/Hdeg.png}
    \caption{Schematic of the Hamiltonian matrix describing electron–phonon and alloy interactions between conduction band states near the band edge, specifically the $\Gamma$ and the $L$ valleys.}
    \label{fig:Hdeg}
\end{figure}


In this system, each mixed (eigen-)state $\ket{\psi_i}$ is a weighted linear combination of the unperturbed conduction band states, with the weights given by the components of the corresponding eigenvector $v^{(i)}$, which effectively acts as the diagonaliser of the perturbation, while the energy of each state is given by the corresponding eigenvalue of the matrix. If $N$ is the number of mixed states, we can express each new state $\ket{\psi_i}$ as:
\begin{equation}
    \ket{\psi_i}=\sum_j^{n}v_j^{(i)}\ket{\Gamma_j}+\sum_{j=n+1}^{m}v_j^{(i)}\ket{L_j+\hbar\omega_q}+\sum_{j=m+1}^Nv_j^{(i)}\ket{L_j-\hbar\omega_q} \label{eq:psisum}
\end{equation}
where $n$ and $m$ are the number of unperturbed $\Gamma$ and $L$ states, respectively, where $N=n+2m$. The states $\ket{\Gamma_j}$ and $\ket{L_j\pm\hbar\omega_q}$ represent the wavefunctions of the states around the $\Gamma$ and $L$ valleys, respectively. Given that an electron will only transition between the valence and conduction band into states around the $\Gamma$ valley, only the components of the eigenvectors corresponding to the unperturbed $\Gamma$ states contribute to the momentum matrix elements. The resulting expression for the imaginary part of the dielectric function for transitions into quasi-degenerate bands is given in Eq. \ref{eq:eideg}
\begin{equation} \label{eq:eideg}
    \epsilon_i(\omega)=\frac{\pi e^2}{\epsilon_0 m_e V} \frac{1}{N}\frac{1}{\omega^2}\sum_{\Vec{k}}^N\sum_{\Vec{k_\Gamma}}^{n} \ \lvert v^{(\vec{k})}_{\vec{k_\Gamma}}\rvert^2 \ \lvert  P_{cv}(\Vec{k}) \rvert^2 \ \delta ( E_c(\Vec{k})-E_v(\Vec{k})-\hbar\omega ) 
\end{equation}
where $\vec{k_\Gamma}$ runs over the $n$ unperturbed $\Gamma$ states.

This approach largely follows the work of Tiwari et al. \cite{tiwari2024unified}, with a few key distinctions. Our expression of $\epsilon_i$ is a simplified form of their Eq. 48, as the photon energies of interest limit the indirect transitions to conduction-band–to–conduction-band processes, excluding contributions from the top of the valence band at $L$. Their work also considers $\mathbf{q}$-dependent electron-phonon interaction terms, while we assume a constant valued term. Additionally, we include and alloy scattering interaction between the $\Gamma$ and $L$ valleys, since we are operating with an alloy. This is the first study to include alloy disorder in optical transitions.

%Interestingly, while the interaction term is the electron–phonon coupling in pure Ge, in GeSn alloys, additional contributions from alloy scattering must also be included, where the alloy scattering interaction term $V_{L\Gamma}$ can be found using Eq. \ref{eq:Tmat}. While the absorption coefficient and PL have been calculated for Ge using quasi-degenerate perturbation theory via electrons interacting with phonons, no calculation exists examining transitions via alloy scattering.



%This is especially relevant for GeSn, not only at the indirect-to-direct bandgap transition (where the $\Gamma$ and $L$ bands become degenerate), but also at $\Vec{k}$-points near the $L$ band which have the same energy as $\Vec{k}$-points near the $\Gamma$ band (as shown in Fig \ref{fig:bandLk}). We denote these states near the $L$ and $\Gamma$ bands as $L(k)$ and $\Gamma(k)$ respectively.

%\begin{figure}[H]
%    \centering
%    \includegraphics[height=10cm]{Theory/Figures/Ge_bands_EL(k).png}
%    \label{fig:bandLk}
%    \caption{Electronic band structure of Ge, highlighting the energy near the $L$ valley at which there is (quasi-)degeneracy with $\Gamma$.}
%\end{figure}

%In such a scenario, an electron absorbing a photon may transition into a superposition of states rather than a single, well-defined eigenstate. These two possible states, denoted $\ket{\psi_1}$ and $\ket{\psi_2}$, are an admixture of the two original states ($\Gamma(k)$ and $L(k)$), such that:
%\begin{equation}
%    \ket{\psi_{1,2}}=a_{1,2}\ket{L(k)}+b_{1,2}\ket{\Gamma(k)}
%\end{equation}

%for some solvable coefficients $a_1$,$a_2$, $b_1$ and $b_2$. Tiwari et al \cite{tiwari2024unified} treat this problem using degenerate perturbation theory. Since there are two near-equivalent bands for the electrons to transition into, the Hamiltonian will take the form of a 2x2 matrix:
%\begin{equation} \label{eq:ham}
%  \hat{H} = \begin{pmatrix}
%E_L(k) & H_{L\Gamma} \\
%H_{\Gamma L} & E_\Gamma(k) 
%\end{pmatrix}  
%\end{equation}

%where we denote $H_{L\Gamma}$ as the interaction term between the $L$ and $\Gamma$ bands. For Ge, this is simply the e-ph matrix element $\bra{L(k)} H_{ep} \ket{\Gamma(k)}$, which coincides with Eq. \ref{eq:phmat}. However, in GeSn, $H_{L\Gamma}$ is a sum of this e-ph interaction and the alloy scattering interaction term $V_{L\Gamma}$, which can be found using Eq. \ref{eq:Tmat}. The coefficients $a$ and $b$ are determined by solving the Schroedinger equation using this Hamiltonian, to produce the following normalised eigenvectors:
%\begin{equation}
%    v_{1,2}=     \begin{bmatrix}
%a_{1,2}  \\
%b_{1,2}
%\end{bmatrix} 
%    = \left( \frac{2\epsilon^2+2(2H_{L\Gamma})^2\pm 2\epsilon\sqrt{\epsilon^2+(2H_{L\Gamma})^2}}{(2H_{L\Gamma})^2} \right) ^{-1}
%    \begin{bmatrix}
%\frac{\epsilon \pm \sqrt{\epsilon^2 + (2H_{L\Gamma})^2}}{2H_{L\Gamma}}  \\
%1
%\end{bmatrix}
%\end{equation}

%where we denote $\epsilon=E_L(\Vec{k})-E_\Gamma$. Given that $\bra{v} H \ket{L(k)} =0$,
%\begin{align}
 %   \bra{v} H \ket{\psi_{1,2}} &= \bra{v} H \left( a_{1,2} \ket{L(k)}+b_{1,2} \ket{\Gamma(k)}\right) \nonumber \\
 %   &= b_{1,2} \bra{v} H \ket{\Gamma(k)}\nonumber\\
 %   \Rightarrow \lvert \bra{v} H \ket{\psi_{1,2}} \rvert^2 &= b_{1,2}^2 \ \lvert \bra{v} H \ket{\Gamma(k)} \rvert^2
%\end{align}

%Applying this term to FGR, the resultant imaginary part of the dielectric function takes the form:
%\begin{equation} \label{eq:eideg}
%    \epsilon_i (\omega) = \frac{1}{4\pi \epsilon_0} \left( \frac{2\pi e}{m_e \omega} \right)^2 \sum_k \left( b_1^2+b_2^2 \right) P_{cv}^2 \ \delta \left( E_c(\Vec{k}) - E_v(\Vec{k}) -\hbar \omega \right)
%\end{equation}

%where $b_1$, $b_2$ and $P_{cv}$ are all $k$-dependent, and the new admixed conduction band states have energy $E_c(\Vec{k})$, which can be obtained by calculating the eigenvalues of the matrix in Eq. \ref{eq:ham}.







%\bigskip

%\subsection{Photoluminescence} 


%Following the work of Murphy-Armando et al \cite{murphy2021light}, the PL for free carriers due to direct band-to-band electron-hole recombination is calculated using Eq. \ref{eq:photo}:
%\begin{equation} \label{eq:photo}
%    PL (\omega) \propto \sum_k \omega\frac{2 P_{cv}^2 f_c (1-f_v)}{m_e \hbar \left( E_c(\Vec{k}) - E_v(\Vec{k}) \right)^2}    \ \delta \left( E_c(\Vec{k}) - E_v(\Vec{k}) -\hbar \omega \right)
%\end{equation}

%where $f_c$ and $f_v$ represent the Fermi distribution of electrons and holes, which are functions of the electron and hole chemical potentials, respectively. This formula assumes the system is in a state of quasi-equilibrium before electron-hole recombination \cite{chow2012model}. By applying the same mathematical steps to the PL as we did to the absorption coefficient before, we can produce a generalised expression for the PL encompassing all manner of transitions, including into degenerate states. This expression is given in Eq. \ref{eq:photo1}:
%\begin{equation} \label{eq:photo1}
%    PL (\omega) \propto \sum_k \left( b_1^2+b_2^2 \right)\omega\frac{2 P_{cv}^2 f_c (1-f_v)}{m \hbar \left( E_c(\Vec{k}) - E_v(\Vec{k}) \right)^2}    \ \delta \left( E_c(\Vec{k}) - E_v(\Vec{k}) -\hbar \omega \right)
%\end{equation}








