\chapter{Spin Relaxation in GeSn} \label{chap:spinrel}

The ultimate goal of this chapter is to calculate the intrinsic spin relaxation time and the spin-diffusion length of GeSn as a function of temperature and Sn concentration. Adding Sn to Ge introduces a novel degree of freedom to tailor its spin properties. As a consequence of the large SOC of Sn, we expect there to be ``spin-flips" as a result of alloy scattering in GeSn; that is, a certain proportion of electrons will change their spin-orientation as they scatter due to alloy impurities. In this chapter, we calculate the spin-dependent alloy scattering parameters from first-principles. In addition, we calculate all necessary phonon parameters which define spin-dependent scattering to and from the $\Gamma$ valley, which plays an important role at higher Sn concentrations. This is the first theoretical calculation of the spin relaxation time of GeSn, and our findings could be vital in the development of future design of spintronic devices, such as those mentioned in Chapter \ref{chap:intro}. 


This chapter is divided into three sections. In Section \ref{sec:spintheory}, we present the theoretical framework for determining the spin relaxation time for GeSn, with an emphasis on our first-principles methods used to calculate the spin-dependence of the alloy and electron-phonon scattering parameters. We show our results in Section \ref{sec:spinresults} and our conclusions in Section \ref{sec:spinconcl}.






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

\section{Theoretical Framework} \label{sec:spintheory}

To calculate the spin relaxation time $\tau^\alpha$ at the valley minimum $\alpha$, we use the fact that $\tau^\alpha=\frac{1}{\langle R^\alpha\rangle}$ \cite{yafet1963g,song2012analysis}, where $\langle R^\alpha\rangle$ is the thermal average of the scattering rate $R^\alpha(E)$, such that:
\begin{equation} \label{eq:thav}
    \langle R^\alpha \rangle = \frac{\int R^\alpha(E)  \ \frac{\partial f}{\partial E} \ \rho(E) \ dE}{\int \frac{\partial f}{\partial E} \ \rho(E) \ dE}
\end{equation}
Each $R^\alpha(E)$ is a sum of the electron spin-phonon scattering rate and spin-alloy scattering rate. Given the nature of the band structure of GeSn, the total spin relaxation time $\tau$ is a weighted sum of the relaxation times at the $L$ and $\Gamma$ valleys:
\begin{equation}
    \tau=r_L\tau_L+r_\Gamma\tau_\Gamma
\end{equation}
where both $\tau_L$ and $\tau_\Gamma$ include intravalley and intervalley spin relaxation as a result of e-ph and alloy scattering, and the carrier concentration ratio $r_L$ is for the four $L$ valleys.  In this section, we discuss our methods of calculating the spin-scattering parameters due to alloy scattering and e-ph scattering. 




%\bigskip

\subsection{Spin-Alloy Scattering}

Calculating the alloy scattering parameters in GeSn in Chapter \ref{chap:trans} was a stepping stone in determining the spin-alloy scattering parameters in GeSn, i.e the potentials $V_{\alpha\beta}^{\sigma'\sigma}$ that measure the likelihood of an electron scattering from BZ valley $\alpha$ with spin orientation $\sigma$ to valley $\beta$ with spin orientation $\sigma'$. %One should expect a Pythagorean relationship between the alloy scattering parameters in Chapter \ref{chap:trans}, the spin-flips and the non-flips (scattering parameters in which $\sigma$ remains unchanged).




Our methodology largely follows that found in Section \ref{sec:alloy}. However, the presence of SOC results in a more complicated process for determining the spin-dependence of the alloy scattering parameters. The scattering rate for spin-flips caused by alloy impurities is given by Eq. \ref{pot} in Chapter \ref{chap:theory2}, with the matrix element now dependent on a change in spin orientation. We will denote this parameter as $V_{\alpha\beta}^{\uparrow\downarrow}$. Consequently, we modify Eq. \ref{pot} to instead read:
\begin{equation} 
\left\langle V_{\alpha\beta}^{\uparrow\downarrow} \right\rangle 
= N( \bra {\psi_\alpha^{\uparrow}} {\Delta V_{Sn}} \ket{\phi_{\beta}^{\downarrow}} - \bra{ \psi_\alpha^{\uparrow} }{ \Delta V_{Ge} } \ket{ \phi_{\beta}^{\downarrow} }) \label{potspin}
\end{equation}
whereby the wavefunction $\ket{\psi}=(\psi_\Uparrow,\psi_\Downarrow)^T$ can be decomposed into effective up and down spin-states $\psi_\Uparrow$ and $\psi_\Downarrow$, which diagonalise the spin operator $S^\alpha=\frac{\hbar}{2}\sigma^\alpha$ in the Kramers degenerate subspace \cite{park2020spin}, where $\sigma^\alpha$ denotes the Pauli matrices corresponding to the Cartesian components $\alpha=x,y,z$:
\begin{align} \nonumber
    \bra{\psi_\Uparrow} S_\alpha \ket{\psi_\Uparrow}&= -\bra{\psi_\Downarrow} S_\alpha \ket{\psi_\Downarrow} \\
    \bra{\psi_\Uparrow} S_\alpha \ket{\psi_\Downarrow}&= 0
\end{align}
The effective spin states are calculated from the spin matrix $S(k)$, which represents the spin operator in the wave function basis. The matrix elements of the spin operator in the Bloch state basis are given by 
\begin{equation}
    \sigma_{m,s ; n,s'}(k) = \bra{\psi_{m,k,s}} \sigma^\alpha \ket{\psi_{n,k,s'}}
\end{equation}
Here, $\ket{\psi_{n,k,s}}$ denotes the Bloch state at wavevector $k$, band index $n$, and spin $s$. We take the spin quantisation axis to be aligned with the z-axis.  We separately diagonalise each degenerate subspace in S at each k-point, obtaining the unitary matrices $D_k$ that make each of the subspaces in $D_kS(k)D^\dag_k$ diagonal, with eigenvalues equal to the effective spin \cite{park2020spin}. The spin-flip alloy matrix elements $V_{\alpha\beta}^{\uparrow\downarrow}$ are then computed using Eq. \ref{potspin} for all pairs of states with opposite effective spin. A 64-atom supercell (one Sn substitutional impurity in a VCA host) is used to find the scattering matrices between degenerate Bloch states.


% need to say somewhere that since we're working with spinors, the overlaps are a sum of spinor 1 overlaps and spinor 2 overlaps.

We calculate all intravalley and intervalley alloy-spin scattering parameters between 20 high-symmetry points in the Brillouin Zone: the $\Gamma$ point, the four $L$ points, the six $X$ points (denoted $X$, $-X$, $Y$, $-Y$, $Z$, $-Z$) and the six $\frac{1}{2}X$ points. We expect that both intravalley scattering and g-type (intra-axis, i.e. $X\rightarrow-X$) intervalley spin-flip alloy scattering will be forbidden as a consequence of rotational symmetry about each Cartesian axis. Consider a rotation of $\pi$ radians about any Cartesian axis (we arbitrarily choose the z-axis), which is denoted as $C_2(z)$ in group theory. On average in an alloy, $C_2(z)$ is an element of the alloy substitutional defect symmetry group $T_d$ \cite{koster1957solid}. Therefore, it commutes with the alloy scattering Hamiltonian perturbation. %We now take $R_z(\pi)=e^{\frac{-i\pi S_z}{\hbar}}$ as the representation of a rotation acting on spinors. 
If we examine the transformation of spin-up $\psi_{z,\uparrow}$ and spin-down state $\psi_{z,\downarrow}$ at the $Z$ valley in the BZ zone under $C_2(z)$, we find:
\begin{align} \label{eq:c2z}
    C_2(z)\ket{\psi_{z,\uparrow}}&=-i\ket{\psi_{z,\uparrow}} \nonumber\\
    C_2(z)\ket{\psi_{z,\downarrow}}&=i\ket{\psi_{z,\downarrow}} 
\end{align}
Since the up and down states transform differently under $C_2(z)$, the matrix element of the Hamiltonian $H$ between them is zero. Thus:
\begin{equation} \label{eq:noia}
    \bra{\psi_{z,\uparrow}}H\ket{\psi_{z,\downarrow}}=0
\end{equation}
Eq. \ref{eq:noia} confirms that no intravalley spin-flip alloy scattering processes are allowed in GeSn. Since we can apply Eqs. \ref{eq:c2z} to the $-Z$ valley, this yields:
\begin{equation} \label{eq:nog}
    \bra{\psi_{z,\uparrow}}H\ket{\psi_{-z,\downarrow}}=0
\end{equation}
which confirms that g-type alloy scattering where spin-orientation changes is forbidden also. 

Our Hamiltonian is also time reversal symmetric ($H=THT^{-1}$ for the time symmetry operator T), which for any two wavefunctions $\ket{\psi}$ and $\ket{\phi}$ means that $\bra{\psi}H\ket{\phi}=\bra{T\phi}H\ket{T\psi}^*$. This ensures that:
\begin{equation} \label{eq:inversion}
    \bra{\psi_{z,\uparrow}}H\ket{\psi_{z,\uparrow}}=\bra{\psi_{-z,\downarrow}}H\ket{\psi_{-z,\downarrow}}^*
\end{equation}
However, since the substitutional defect in our supercell removes the inversion symmetry of the diamond structure, we cannot necessarily guarantee that $\bra{\psi_{z,\uparrow}}H\ket{\psi_{z,\uparrow}}=\bra{\psi_{z,\downarrow}}H\ket{\psi_{z,\downarrow}}$. Eqs. \ref{eq:c2z}, \ref{eq:noia}, \ref{eq:nog} and \ref{eq:inversion} apply to valleys along the $x$ and $y$ axes also.




%\bigskip

\subsection{Electron Spin-Phonon Scattering}

To accurately model spin transport in GeSn, it is essential to evaluate the full spectrum of electron–phonon scattering mechanisms that contribute to spin relaxation. In this section, we calculate all relevant scattering parameters in the $L$ and $\Gamma$ valleys of the Brillouin zone. Intravalley and intervalley scattering within the $L$ valley have been previously reported in the literature, particularly for Ge, and serve as a reference point for GeSn. However, additional mechanisms become important in GeSn due to its evolving band structure, specifically the intervalley scattering between the $L$ and $\Gamma$ valleys and intravalley scattering at the $\Gamma$ point. Each of these processes is treated individually.



We follow the work of Tang et al \cite{tang2012electron} to calculate the spin relaxation time due to intra- and intervalley electron spin-phonon scattering between the $L$-valleys. We implement the phonon parameters for Ge, since we are confining our calculations to low Sn compositions, and in doing so, we assume that the phonon parameters depend weakly on Sn concentration. The phonon frequencies and deformation potentials used are obtained directly from Ref. \cite{tang2012electron}.% are listed in Table \ref{tab:defspin}.


% intravalley Gamma bit
To calculate the spin relaxation time for intravalley scattering at the $\Gamma$ valley, we follow the work of Tamborenea et al \cite{tamborenea2003spin}, who calculated the spin relaxation time in GaAs due to electron-electron (e-e) EY scattering at the $\Gamma$ valley. They describe the spin-flip transition rate using Fermi's Golden Rule, expressing the matrix element as a product of the spin-independent matrix element and the overlap integral, such that the scattering rate $R_{\psi_{\mathbf{k}\uparrow,\mathbf{k'}\downarrow}}$ is given by:
\begin{align}
    R_{\psi{\mathbf{k}\uparrow,\mathbf{k'}\downarrow}}&=\frac{2\pi}{\hbar} \left|\bra{\psi_{\mathbf{k'}\downarrow}}V\ket{\psi_{\mathbf{k}\uparrow}}\right|^2 \delta\left(E_{\mathbf{k'}}-E_{\mathbf{k}}\right) \label{eq:Rg1}\\ 
    &\approx\frac{2\pi}{\hbar} \ V(\mathbf{k}-\mathbf{k'})^2 \left|\left<\psi_{\mathbf{k'}\downarrow}|\psi_{\mathbf{k}\uparrow}\right>\right|^2 \delta\left(E_{\mathbf{k'}}-E_{\mathbf{k}}\right) \label{eq:Rg2}
\end{align}
We can factorise the matrix element, as shown in Eq. \ref{eq:Rg2}, by assuming that the potential $V$ is slowly varying on the scale of a unit cell \cite{chazalviel1975spin}. We replace their e-e matrix element with the equivalent e-ph matrix element for GeSn, found in Table \ref{tab:defs} in Chapter \ref{chap:trans}. % How we calculate the overlap matrix accurately
The scalar product is given by: 
\begin{equation} \label{eq:kapp}
    \left<\psi_{\mathbf{k'}\downarrow}|\psi_{\mathbf{k}\uparrow}\right>= C (k_zk_+'-k_z'k_+)
\end{equation}
where $k_+=k_x+ik_y$. We calculate the proportionality constant $C$ by calculating the wavefunction overlaps of a dense set of k-points near the $\Gamma$ valley using DFT, and comparing to the right hand side of Eq. \ref{eq:kapp} for the same set of k-points. Simple integration over $\mathbf{k}$ and $\mathbf{k'}$ obtains an expression of the spin relaxation time due to intravalley scattering at the $\Gamma$ point.



% L\Gamma 
We use the frozen phonon supercell approach (described in Section \ref{sec:froz}) to calculate the deformation potential ($D_\mathbf{q}$) for intervalley scattering between the $L$ and $\Gamma$ bands. We use long $4$-atom supercells (see Fig. \ref{fig:4at} below) to obtain the energy splittings and wavefunctions due to a phonon wave-vector $\mathbf{q}$ in the $[111]$-direction, since this is the smallest cell in which phonons with the momentum transfer needed to simulate $L-\Gamma$ transitions can be modelled. The phonon frequencies ($\hbar\omega_\mathbf{q}$) and phonon eigenvectors $\mathbf{u}$ were calculated using DFPT with ABINIT for all six phonon modes: three acoustic (one longitudinal and two transverse) and three optical (one longitudinal and two transverse). We set the scattering amplitude $A=0.0025$ and $A=0.005$, where we ensure our findings show convergence. We use a k-point grid of 6x6x6 and an energy cut-off of $950eV$.
% see philip's appendix for more


\begin{figure}
    \centering
    \includegraphics[width=0.6\linewidth]{Result2-Spin/Figures/4atom.png}
    \caption{Schematic of the long 4-atom supercell used to calculate the intervalley electron spin-phonon scattering parameters of GeSn between the $L$ and $\Gamma$ valleys, showing the Cartesian coordinates of each atom.}
    \label{fig:4at}
\end{figure}

\bigskip






%Firstly, since they define the thermal averaging of a function $g(E)$ using the Boltzmann distribution in such a manner that $\tau$ is independent of the carrier density:
%\begin{equation} \label{eq:thav1}
%   \langle g(E) \rangle_T = \frac{4}{3\sqrt{\pi}T^{5/2}}\int_0^\infty g(E) E^{3/2} e^{-E/T}dE
%\end{equation}

%Eq. \ref{eq:thav1} is applicable for non-degenerate systems. Intravalley scattering at the $L$ valley is determined using Eq. \ref{eq:Lph}:
%\begin{equation} \label{eq:Lph}
 %   \frac{1}{\tau_L}=\frac{\sqrt{2}(D_A^{\uparrow\downarrow})^2 \ m^{3/2}}{\pi \ \rho c^2\hbar^4}k_BT\langle\sqrt{E}\rangle_T
%\end{equation}
%where $D_A^{\uparrow\downarrow}=0.032$ eV for Ge. Using an effective phonon frequency $\omega$, the $L-L$ intervalley scattering rate is a sum of zeroth-order and first-order scattering rates:
%\begin{align}
%    \frac{1}{\tau^{(0)}}&=  \frac{D_0^2m^{3/2}}{\sqrt{2}\pi \rho \omega \hbar^3} [n_{\omega}\langle p_+^{(0)}\rangle_T+(n_{\omega}+1)\langle p_-^{(0)}\rangle_T]  \nonumber \\ 
%    \frac{1}{\tau^{(1)}}&=  \frac{\sqrt{2}D_1^2m^{5/2}}{\pi \rho \omega \hbar^5} [n_{\omega}\langle p_+^{(1)}\rangle_T+(n_{\omega}+1)\langle p_-^{(1)}\rangle_T]
%\end{align}

%where $n_\omega$ is the phonon occupation number, and we define:
%\begin{align}
 %   p_{\pm}^{(0)}&=\sqrt{E\pm \hbar\omega}\ \theta(E\pm \hbar\omega) \nonumber \\
 %    p_{\pm}^{(1)}&=(2E\pm \hbar\omega)\sqrt{E\pm \hbar\omega}\ \theta(E\pm \hbar\omega)
%\end{align}


%\bigskip



%intravalley gamma

% intervalley (gamma-L)





%\bigskip

\section{Results} \label{sec:spinresults}


\subsection{Spin-Alloy Scattering Parameters}

%In all of our results in this section, we were able to confirm that the sum of the squares of the flip and non-flip matrix elements equal the square of the equivalent non-spin-dependent matrix element seen in Chapter \ref{chap:trans}, that is, $(V_{\alpha\beta})^2\approx(V_{\alpha\beta}^{\uparrow\uparrow})^2+(V_{\alpha\beta}^{\uparrow\downarrow})^2$. Consider also that electron spins are aligned in the z-direction in all results presented in this section.


Our calculations confirm that intravalley and intra-axis (g-type) spin-flips are negligible, which includes interactions from $\Gamma$ to the six $X$ and $\frac{1}{2}X$ valleys. This indicates that intravalley and g-type scattering preserves the spin orientation of the electrons. Consequently, the intravalley and g-type scattering parameters that describe electron dynamics calculated in Chapter \ref{chap:trans} are not affected by the inclusion of SOC. 

%In contrast, changes in spin orientation occur via intervalley scattering, although spin-conserving scattering dominates momentum relaxation, in agreement with Li et al \cite{li2012intrinsic}. 

We first analyse the spin-flip scattering parameters relevant to the spin relaxation time of GeSn, that is, the scattering parameters between the $\Gamma$ and $L$ valleys, calculated using Eq. \ref{potspin}. Figure \ref{fig:Lflip} shows $V_{LL}^{\uparrow\downarrow}$ calculated using a supercell with a VCA host at $0\%$ Sn content (effectively representing a pure Ge supercell with a single substitutional Sn impurity). Scattering between $L$ valleys separated along the $z$ direction (denoted with blue arrows) is stronger at $5.82$ meV than scattering between L valleys in the $x$ or $y$ directions (shown with red arrows), which is $4.33$ meV. From this point, $L-L$ intervalley spin-alloy scattering parameters will be denoted as $V_{LL}^{x}$ and $V_{LL}^z$, where the superscripts indicate that the initial and final valleys are separated along the $x$ (or equivalently $y$) direction and the $z$ direction, respectively. Although it may not be immediately apparent that $L_1$ and $L_2$ in the figure are separated along the z-axis, $L_1$ corresponds to the L-point at $(\frac{1}{2},\frac{-1}{2},\frac{3}{2})$, which lies along the z-axis relative to $L_2$. There is a similar correspondence between $L_3$ and $L_4$.

\begin{figure}[H]
    \centering
    \includegraphics[width=0.8\linewidth]{Result2-Spin/Figures/Lflips.png}
    \caption{Intervalley spin-alloy scattering parameters between the $L$ valleys of GeSn (in meV) with spin-polarisation in the z-direction.}
    \label{fig:Lflip}
\end{figure}


In Fig. \ref{fig:Vflip} we compare the $L-L$ spin-flip scattering  parameters to  spin-flips between $L$ and $\Gamma$, which is over one full order of magnitude weaker, revealing that spin-flips due to alloy scattering are more prominent in high-Ge GeSn compared to direct-gap GeSn.



\begin{figure}[H]
    \centering
    \begin{tikzpicture}
        \begin{axis}[
            ybar,
            bar width=18pt,
            width=10cm, height=9cm,
            symbolic x coords={{$|V_{LL}^z|$}, {$|V_{LL}^{xy}|$}, {$|V_{L\Gamma}|$}},
            xtick=data,
            nodes near coords,
            every node near coord/.append style={font=\large},
            ymin=0, ymax=6.5,
            ylabel={Matrix Elements $|V|$ (meV)}, 
            enlarge x limits=0.2,
            xtick align=center
        ]
        \addplot coordinates {({$|V_{LL}^z|$}, 5.82) ({$|V_{LL}^{xy}|$}, 4.33) ({$|V_{L\Gamma}|$}, 0.14)};
        \end{axis}
    \end{tikzpicture}
    \caption{Calculated intervalley scattering matrix elements for GeSn in meV, emphasising the dominance of L-L intervalley alloy spin scattering.}
    \label{fig:Vflip}
\end{figure}



To capture the Sn-dependence of the spin-alloy scattering parameters, we calculate the spin-alloy scattering matrix using supercells with host atoms of varying VCA compositions ($6\%$, $12\%$ and $18\%$), with Tables \ref{tab:ieLflips} and \ref{tab:ieXfflips} showing our findings for complete set of intervalley spin-flip scattering matrix elements (in meV) for $Ge_{1-x}Sn_x$. We include interactions involving the six $X$ and $\frac{1}{2}X$ points, as they would be prominent in spin-scattering calculations for SiGeSn. The tables are divided such that Table \ref{tab:ieLflips} contains all interactions involving the L valley, while Table \ref{tab:ieXfflips} includes f-type scattering (inter-axis scattering between the various X and $\frac{1}{2}$X points, where the direction of alignment of the spins had a profound bearing on the scattering parameters). 



In general, $V_{\alpha\beta}^{\uparrow\downarrow}$ largely exhibits a weak quadratic dependence on the Sn concentration of the VCA of the host atom in the supercell. Our findings show that the intervalley spin-alloy scattering parameters can vary by almost three orders of magnitude, depending on the initial and final BZ valleys involved in the process. For $f$-type scattering between the $X$ and $Y$ valleys, the scattering magnitude is of the order of meV, and $V_{XY}^{\uparrow\downarrow}=V_{XY}^{\downarrow\uparrow}$. However, for $f$-type scattering involving transitions to or from the $Z$ or $\frac{1}{2}Z$ valleys, not only does the magnitude increase significantly, ranging from $10-100$ meV, but the up-to-down and down-to-up spin-flips differ slightly from one another. Additionally, these values swap when scattering occurs into the $-Z$ valley as opposed to the $+Z$ valley. %As we show in Table \ref{tab:ieXfflips}, a spin-up electron in the $X$ valley will scatter more into the $-Z$ valley than the $Z$ valley. This is also true for spin-conserving scattering, as we will see later in this section. %To summarise, the following spin-flips are equivalent:





\begin{table}[H]
\centering
\renewcommand{\arraystretch}{1.5} % Adjusts row height
\setlength{\tabcolsep}{12pt} % Adjusts column width
\begin{tabular}{|c|c|}
\hline
\textbf{Symbol} & \textbf{Value (meV)} \\ \hline \hline
$V_{\Gamma L}$  & $0.14+ 0.799x + 0.625 x^2$  \\ \hline
$V_{LL}^z$   & $5.818 + 15.625x -7.639x^2$  \\ \hline
$V_{LL}^{xy}$  & $4.334+12.15x -4.17x^2$  \\ \hline
$V_{XL}$  & $11.585 + 21.417 x - 6.944 x^2$ \\ \hline
$V_{ZL}$  & $16.295 + 31.583 x - 20.833 x^2$ \\ \hline
$V_{\frac{1}{2}X L}$  & $22.6 + 25 x - 27.778 x^2$ \\ \hline
$V_{\frac{1}{2}Z L}$  & $31.89 + 32.333 x - 27.778 x^2$ \\ \hline \hline
\end{tabular}
\caption{Calculated intervalley spin-scattering matrix elements $V_{\alpha\beta}^{\uparrow\downarrow}$
(in meV) which involve the L valley in $Ge_{1-x}Sn_x$, with spin polarisation in the z-direction. The Sn concentration is determined by the VCA applied to the host atoms within the supercell.}
\label{tab:ieLflips}
\end{table}

%Our results also show that scattering from $Y\rightarrow Z$ is equivalent to scattering from $X-\rightarrow -Z$, while scattering from $Y\rightarrow -Z$ is the same as $X\rightarrow Z$. To summarise, the following spin-flips are equivalent:


Our results for f-type intervalley spin-alloy scattering in GeSn respect time-reversal symmetry and obey Kramers theorem, as expected, ensuring that each state $\psi_{\vec{k},\sigma}$ has a degenerate partner $\psi_{-\vec{k},-\sigma}$, and spin-dependent alloy transitions are symmetric under time-reversal symmetry. However, inversion symmetry is broken due to the explicit alloy disorder in the supercell model. The presence of substitutional Sn atoms locally disrupts the centro-symmetry of the lattice, leading to asymmetries in the spin-dependent scattering rates that are not constrained by inversion/parity. This explains the difference in scattering in the $+Z$ direction as opposed to the $-Z$ direction. However, our substitutional impurity was arbitrarily placed at one end of a [111] bond. In the random alloy, there would be equal numbers at either end of the [111] bond. A true random alloy model would result in a single spin-flip parameter for each f-type transition (i.e. $V_{XZ}^{\uparrow\downarrow}=V_{XZ}^{\downarrow\uparrow}$, etc). We therefore recommend taking the average of our calculated values for use in future calculations. 

%\bigskip

\begin{table}[H]
\centering
\renewcommand{\arraystretch}{1.5} % Adjusts row height
\setlength{\tabcolsep}{12pt} % Adjusts column width
\begin{tabular}{|c|c|c|}
\hline 
& \textbf{Symbol} & \textbf{Value (meV)} \\ \hline \hline
\textbf{X-Y}& $V_{XY}$  & $10.403- 9.708x + 9.028x^2$  \\ \hline 
&$V_{\frac{1}{2}X\frac{1}{2}Y}$  & $3.048 - 7.117x + 15.278x^2$      \\ \hline
&$V_{\frac{1}{2}X Y}$  & $12 + 21x - 27.778 x^2$ \\ \hline %\hline


\textbf{X-Z}&$V_{XZ}^{\uparrow\downarrow}$  & $27.595 + 40.75 x - 34.722x^2$ \\ \hline
&$V_{XZ}^{\downarrow\uparrow}$  & $42.38 + 25.5 x - 13.889 x^2$ \\ \hline
&$V_{\frac{1}{2}X\frac{1}{2}Z}^{\uparrow\downarrow}$  & $87.895 + 24.083x - 6.944 x^2$      \\ \hline
&$V_{\frac{1}{2}X\frac{1}{2}Z}^{\downarrow\uparrow}$  & $92.215 + 12.75 x + 20.8333 x^2$      \\ \hline
&$V_{\frac{1}{2}X Z}^{\uparrow\downarrow}$  & $51.625+42.08x-34.72x^2 $ \\ \hline
&$V_{\frac{1}{2}X Z}^{\downarrow\uparrow}$  & $66.61+24.33x-27.78x^2$ \\ \hline
&$V_{ X\frac{1}{2}Z}^{\uparrow\downarrow}$  & $68.525+64.59x-76.39x^2$ \\ \hline
&$V_{X\frac{1}{2}Z}^{\downarrow\uparrow}$  & $49.71+1.83x+13.89x^2$ \\ \hline \hline
\end{tabular}
\caption{Calculated intervalley spin-scattering matrix elements $V_{\alpha\beta}^{\uparrow\downarrow}$
(in meV) for f-type scattering and scattering between the X and $\frac{1}{2}$X valleys in $Ge_{1-x}Sn_x$, with spin-polarisation in the z-direction. The following spin-flips are equivalent: $V_{XZ}^{\uparrow\downarrow}=V_{X-Z}^{\downarrow\uparrow}=V_{YZ}^{\downarrow\uparrow}=V_{Y-Z}^{\uparrow\downarrow}$, and $V_{XZ}^{\downarrow\uparrow}=V_{X-Z}^{\uparrow\downarrow}=V_{YZ}^{\uparrow\downarrow}=V_{Y-Z}^{\downarrow\uparrow}$.}
\label{tab:ieXfflips}
\end{table}


\bigskip



For completion, we include the non-flip intravalley and intervalley scattering parameters ($V_{\alpha\beta}^{\uparrow\uparrow}$ or $V_{\alpha\beta}^{\downarrow\downarrow}$) in Tables \ref{tab:nonflipsia}, \ref{tab:nonflipsie} and \ref{tab:fnonflipsie} respectively, where the values at $x=0$ closely correspond to our scattering parameters in Tables \ref{tab:intraVab} and \ref{tab:interVab} from Chapter \ref{chap:trans}. Table \ref{tab:nonflipsia} displays the intravalley spin-conserving scattering parameters, while the intervalley terms are divided into two tables: f-type (Table \ref{tab:fnonflipsie}) and non-f-type (Table \ref{tab:nonflipsie}). In our f-type scattering parameters, we observe the same breakage in symmetry as the corresponding spin-flips, where $Z$ and $-Z$ are non-degenerate, and $V_{\alpha\beta}^{\uparrow\uparrow}\neq V_{\alpha\beta}^{\downarrow\downarrow}$ for f-type scattering involving the Z-valley. %For full disclosure, the following parameters are equivalent:





\bigskip

\begin{table}[H]
\centering
\renewcommand{\arraystretch}{1.5} % Adjusts row height
\setlength{\tabcolsep}{12pt} % Adjusts column width
\begin{tabular}{|c|c|}
\hline
\textbf{Symbol} & \textbf{Value (eV)} \\ \hline \hline
$V_{\Gamma}$  & $-3.48 - 2.277 x + 4.984x^2$  \\ \hline
$V_{L}$  & $-2.157+ 0.433 x - 0.463 x^2$  \\ \hline
$V_{X}$  & $-0.75+ 0.814 x - 3.92 x^2$  \\ \hline
$V_{\frac{1}{2}X}$  & $-0.83 + 0.817 x - 2.863 x^2$  \\ \hline \hline
\end{tabular}
\caption{Calculated intravalley spin-conserving alloy scattering matrix elements $V_{\alpha\beta}^{\uparrow\uparrow}$ (in eV) for all relevant valleys in $Ge_{1-x}Sn_x$.}
\label{tab:nonflipsia}
\end{table}

\bigskip


\begin{table}[H]
\centering
\renewcommand{\arraystretch}{1.5} % Adjusts row height
\setlength{\tabcolsep}{13pt} % Adjusts column width
\begin{tabular}{|c|c|}
\hline
\textbf{Symbol} & \textbf{Value (eV)} \\ \hline \hline
$V_{\Gamma L}$  & $2.927 + 0.072 x - 1.15 x^2$  \\ \hline
$V_{\Gamma X}$  & $1.73- 0.361x + 0.031x^2$  \\ \hline
$V_{\Gamma \frac{1}{2}X}$  & $1.348+ 0.219x - 0.377x^2$  \\ \hline
$V_{LL}$  & $1.629- 0.433x$  \\ \hline
$V_{LX}$  & $0.838- 0.214x + 0.069x^2$  \\ \hline
$V_{LZ}$  & $0.827- 0.223x$  \\ \hline
$V_{X_g}$  & $0.471- 0.141x + 0.07 x^2$  \\ \hline
$V_{\frac{1}{2}X_g}$  & $0.328- 0.44 x + 0.278x^2$  \\ \hline
$V_{\frac{1}{2}X  X}$  & $1.94- 1.154x + 1.2x^2$ \\ \hline
$V_{\frac{1}{2}X -X}$  & $0.043+ 0.1128x- 2.33x^2 +8.951x^3$ \\ \hline \hline
\end{tabular}
\caption{Calculated intervalley spin-conserving alloy scattering matrix elements $V_{\alpha\beta}^{\uparrow\uparrow}$ (in eV) for all relevant valleys in $Ge_{1-x}Sn_x$ (excluding f-type scattering), with spin-polarisation in the z-direction. The subscript ``g" represents g-type scattering.}
\label{tab:nonflipsie}
\end{table}


% more to be added in the table below

\begin{table}[H]
\centering
\renewcommand{\arraystretch}{1.5} % Adjusts row height
\setlength{\tabcolsep}{13pt} % Adjusts column width
\begin{tabular}{|c|c|}
\hline 
\textbf{Symbol} & \textbf{Value (eV)} \\ \hline \hline
%$V_{\Gamma L}$  & $2.927 + 0.072 x - 1.15 x^2$  \\ \hline
%$V_{\Gamma X}$  & $1.73- 0.361x + 0.031x^2$  \\ \hline
%$V_{\Gamma \frac{1}{2}X}$  & $1.348+ 0.219x - 0.377x^2$  \\ \hline
%$V_{LL}$  & $1.629- 0.433x$  \\ \hline
%$V_{X_g}$  & $0.471- 0.141x + 0.07 x^2$  \\ \hline
%$V_{LX}$  & $0.838- 0.214x + 0.069x^2$  \\ \hline
%$V_{LZ}$  & $0.827- 0.223x$  \\ \hline
%$V_{\frac{1}{2}X  X}$  & $1.94- 1.154x + 1.2x^2$ \\ \hline
%$V_{\frac{1}{2}X -X}$  & $0.043+ 0.1128x- 2.33x^2 +8.951x^3$ \\ \hline
$V_{XY}$  & $0.2347 - 0.113x$  \\ \hline
$V_{\frac{1}{2}X \frac{1}{2}Y}$  & $0.296-0.224x+0.347x^2$ \\ \hline
$V_{\frac{1}{2}X  Y}$  & $0.13 + 0.04 x - 0.278x^2$ \\ \hline
$V_{XZ}^{\uparrow\uparrow}$  & $0.2247 - 0.113 x$  \\ \hline
$V_{XZ}^{\downarrow\downarrow}$  & $0.2397 - 0.13 x$  \\ \hline
$V_{\frac{1}{2}X \frac{1}{2}Z}^{\uparrow\uparrow}$  & $0.285-0.258x +0.417x^2$ \\ \hline
$V_{\frac{1}{2}X \frac{1}{2}Z}^{\downarrow\downarrow}$  & $0.28-0.234x+ 0.347x^2$ \\ \hline

$V_{\frac{1}{2}X Z}^{\uparrow\uparrow}$  & $0.124+0.175x-2.22x^2+6.94x^3$ \\ \hline
$V_{\frac{1}{2}X Z}^{\downarrow\downarrow}$  & $0.105+0.129x-2.33x^2+7.56x^3$ \\ \hline
$V_{X\frac{1}{2} Z}^{\uparrow\uparrow}$  & $0.107+0.158x-2.22x^2+6.94x^3$ \\ \hline
$V_{X\frac{1}{2} Z}^{\downarrow\downarrow}$  & $0.122+0.142x-2.22x^2+6.94x^3$ \\ \hline \hline

\end{tabular}
\caption{Calculated f-type intervalley spin-conserving alloy scattering matrix elements $V_{\alpha\beta}^{\uparrow\uparrow}$ (in eV) in $Ge_{1-x}Sn_x$, with spin-polarisation in the z-direction. The following parameters are equivalent: $V_{XZ}^{\uparrow\uparrow}=V_{X-Z}^{\downarrow\downarrow}=V_{YZ}^{\downarrow\downarrow}=V_{Y-Z}^{\uparrow\uparrow}$, and $V_{XZ}^{\downarrow\downarrow}=V_{X-Z}^{\uparrow\uparrow}=V_{YZ}^{\uparrow\uparrow}=V_{Y-Z}^{\downarrow\downarrow}$.}
\label{tab:fnonflipsie}
\end{table}



\subsection{Electron Spin-Phonon Scattering Parameters}

For our frozen phonon calculation, our non-flip intervalley scattering parameters were found to be in excellent agreement with Ref. \cite{murphy2023electronic}. Our phonon energies (in meV), their corresponding eigenvectors and calculated deformation potentials (in $eV/\AA$) for each of the six LO phonon modes are shown in Table \ref{tab:phie}. These will be used to calculate the scattering rate and spin relaxation time due to intervalley scattering between the $L$ and $\Gamma$ valleys.





\begin{table}[H]
    \centering
    \renewcommand{\arraystretch}{0.9} % Slightly more compact row spacing
    \setlength{\tabcolsep}{3pt} % Narrower columns
    \begin{tabular}{c c c c}
        \toprule
        \textbf{Mode} & $\mathbf{\hbar\omega_q}$ \textbf{(meV)} & $\mathbf{\Vec{u}}$ & 
        $\mathbf{D_{L\Gamma}^{\uparrow\downarrow}}$ $\mathbf{(eV/\AA)}$ \\
        \midrule
        1 & 5.83  & \centering \parbox{4.2cm}{\footnotesize$\begin{bmatrix} 0.58 \\ 0.26-0.006i \\ 0.26+0.006i \end{bmatrix}$} & 0.0102 \\
        2 & 5.83  & \centering\parbox{4.2cm}{\footnotesize$\begin{bmatrix} -0.005i \\ 0.5 \\ -0.5-0.005i \end{bmatrix}$} & 0.0204 \\
        3 & 21.56 & \centering\parbox{4.2cm}{\footnotesize$\begin{bmatrix} 0.41 \\ -0.41 \\ -0.41 \end{bmatrix}$} & $9.805 \times 10^{-5}$ \\
        4 & 23.23 & \centering\parbox{4.2cm}{\footnotesize$\begin{bmatrix} 0.41 \\ -0.41 \\ -0.41 \end{bmatrix}$} & $9.804 \times 10^{-5}$ \\        
        5 & 27.69 & \centering\parbox{4.2cm}{\footnotesize$\begin{bmatrix} 0.01+0.11i \\ -0.48+0.11i \\ 0.49 \end{bmatrix}$} & 0.02163 \\
        6 & 27.69 & \centering \parbox{4.2cm}{\footnotesize$\begin{bmatrix} -0.57 \\ -0.28+0.09i \\ -0.28-0.09i \end{bmatrix}$} & 0.02145 \\
        \bottomrule
    \end{tabular}
    \caption{Phonon energies $\mathbf{\hbar\omega_q}$ (in meV), displacements $\Vec{u}$ and calculated spin-flip deformation potentials $D_{L\Gamma}^{\uparrow\downarrow}$ (in $eV/\AA$) for all phonon modes relevant for scattering between the $\Gamma$ and $L$ bands in $Ge_{1-x}Sn_x$.}
    \label{tab:phie}
\end{table}







\subsection{Spin Relaxation of unstrained GeSn} \label{sec:tau}

Figure \ref{fig:tauSn} shows the calculated intrinsic spin relaxation time $\tau$ (in seconds) at different fixed temperatures (T=$30$,$100$,$200$,$300$K) as a function of Sn concentration. Our results confirm that at room temperature (black), the spin relaxation of Ge lies in the nanosecond range. At any fixed temperature, the relaxation time initially decreases upon the introduction of Sn to Ge due to alloy scattering, especially at lower temperatures, where alloy scattering is more dominant than e-ph scattering. The effect of spin-alloy scattering is very small compared to electron spin-phonon scattering at room temperature. 

However, beyond the indirect-to-direct bandgap transition, $\tau$ increases by several orders of magnitude, primarily as a result of low levels of alloy scattering from the $\Gamma$ valley. Clearly, the behaviour of the spin-relaxation time of GeSn as a function of Sn concentration is similar to its electron mobility seen in Chapter \ref{chap:trans}. At room temperature, alloying Ge with Sn results in an increase in the spin relaxation time from below the nanosecond range at $10\%$ Sn to the microsecond range above $17\%$ Sn. At lower temperatures, this increase at the indirect-to-direct bandgap transition ($8.4\%$ Sn concentration) is much more rapid as a result of reduced electron spin-phonon scattering. This results in a relaxation time which approaches $0.1$ seconds at $25\%$ Sn concentration at $30$K.

In terms of applications, our figure tells us that for applications requiring a long spin lifetime, direct-gap GeSn is desirable, while Ge is more applicable for short spin lifetime applications, irrespective of temperature.


\bigskip

\begin{figure}[H]
    \centering
    \includegraphics[width=\linewidth]{Result2-Spin/Figures/tauvsSninset1.png}
    \caption{Intrinsic spin-relaxation time of GeSn at temperatures T=300K (black), T=200K (red), T=100K (blue) and T=30K (green), as a function of Sn concentration. We include an inset of the spin relaxation time at 300K at low Sn concentration to highlight its decrease with alloy content.}
    \label{fig:tauSn}
\end{figure}

Experimentally, since it is easier to vary temperature as opposed to Sn concentration, we plot the spin relaxation time against temperature at Sn concentrations of $0\%$, $5\%$, $10\%$, $15\%$, $20\%$ and $25\%$, shown in Figure \ref{fig:tauT}. In general, it is clear that across a $15-300$K temperature range, spin relaxation decreases as temperature increases, which is a consequence of increased electron spin-phonon scattering. It is also clear from this graph that to achieve a high spin relaxation time in GeSn, it must have a direct bandgap. Our Ge plot (light blue) is in good agreement with calculations by Tang et al \cite{tang2012electron}, where $\tau\approx0.5$ ns at room temperature, and approaches the microsecond range as $T\rightarrow0$. 

Above $100$K, at least $15\%$ Sn is required to increase the spin relaxation time by several orders of magnitude. At low temperatures however, the spin relaxation time can be increased by up to six orders of magnitude with the introduction of $10\%$ Sn. Due to the change in the behaviour of the distribution function with temperature, the spin relaxation time for Ge$_{0.9}$Sn$_{0.1}$ changes by almost five orders of magnitude with temperature. Beyond $15\%$ Sn concentration, there is little advantage in adding further Sn to increase the relaxation time, especially at lower temperatures.

\bigskip

\begin{figure}[H]
    \centering
    \includegraphics[width=0.95\linewidth]{Result2-Spin/Figures/tauvsT.png}
    \caption{Intrinsic spin-relaxation time of Ge (light blue) and GeSn at Sn concentrations of $5\%$ (purple), $10\%$ (green), $15\%$ (dark blue), $20\%$ (red) and $25\%$ (black), as a function of temperature (in Kelvin).}
    \label{fig:tauT}
\end{figure}




\subsection{The Effect of Strain}

The behaviour of the spin relaxation time of GeSn under biaxial tensile strain is analogous to that of unstrained GeSn. The
difference is that strain induces an indirect-to-direct bandgap transition at lower Sn concentrations, resulting in electrons populating the $\Gamma$ valley at lower Sn concentrations; the consequence of which is a higher spin relaxation time at any given temperature. Figure \ref{fig:taustrain} below shows the spin relaxation time of $1\%$ biaxially tensile strained GeSn (at $T=300K$ and $30K$) as a function of Sn concentration. Our graph shows that at room temperature, beyond $5\%$ Sn concentration, $1\%$ biaxial tensile strain can increase the spin relaxation time by almost two full orders of magnitude. At $30K$, adding $6.7\%$ Sn to $1\%$ strained GeSn is equivalent to introducing $10-15\%$ Sn to unstrained GeSn to increase the spin relaxation time. At $30$K, $1\%$ strain increases $\tau$ by almost six orders of magnitude between $6-9\%$ Sn.

\begin{figure}[H]
    \centering
    \includegraphics[width=\linewidth]{Result2-Spin/Figures/taustrain.png}
    \caption{Spin relaxation time of unstrained (solid) and $1\%$ biaxially tensile strained (dashed) GeSn at $300$K (black) and $30$K (green), as a function of Sn concentration.}
    \label{fig:taustrain}
\end{figure}




\subsection{Spin Diffusion Length}

The spin diffusion length ($L_s$) is the average distance a spin-polarised electron travels before its relaxes, and is a key parameter in spintronics. It can be expressed as the product of the drift velocity $v_d$ and the spin relaxation time $\tau$. Since the drift velocity is proportional to the electron mobility ($v_d=\mu E$), where $E$ is the electric field), the spin diffusion length is given by:
\begin{equation}
    L_s=\mu\tau E \label{eq:ld}
\end{equation}
If we assume a voltage of $1$V and a piece of material of length $1\mu m$, we have $E=1\ Vm^{-1}$, and Eq. \ref{eq:ld} reduces to $L_s=\mu\tau$. We combine our findings in Section \ref{sec:tau} with those in Chapter \ref{chap:trans} to calculate the spin diffusion length as a function of Sn concentration. We calculate the spin diffusion length at 300K, 200K, 100K, and 30K, with our results given in Figure \ref{fig:difflength}, where $L_d$ is given in micrometers. In general, the behaviour of the spin diffusion length mirrors that of the spin relaxation time and the mobility of GeSn, in that it decreases with Sn concentration until the indirect-to-direct bandgap transition, before increasing by several orders of magnitude. We calculate the room-temperature spin-diffusion length of Ge to be 275$\mu$m. At least $11\%$ Sn concentration is required to increase the spin diffusion length at room temperature, with a Sn concentration of $25\%$ increasing $L_d$ to $100$m. As temperature decreases, we achieve a higher spin-diffusion length, with Ge$_{0.75}$Sn$_{0.25}$ possessing a spin-diffusion length of the order of $1000$km at 30K. In reality, over such large distances, other factors such as impurities, doping and crystal defects would take effect, likely reducing the spin diffusion length by several orders of magnitude.




\begin{figure}[H]
    \centering
    \includegraphics[width=\linewidth]{Result2-Spin/Figures/difflength.png}
    \caption{Spin diffusion length of GeSn (in $\mu$m) at temperatures T=300K (black), T=200K (red), T=100K (blue) and T=30K (green), as a function of Sn concentration.}
    \label{fig:difflength}
\end{figure}








\subsection{Spin-Alloy Scattering in SiGe}

In order to gain a more complete understanding of spin-alloy scattering in group-IV alloys, we also calculated the spin-alloy scattering parameters of SiGe. We calculated the parameters using two supercell systems: a Ge supercell with a Si impurity, and a Si supercell with a Ge impurity. In our calculations, we considered 17 conduction band states: the $\Gamma$ valley, the four $L$ valleys, the six $X$ valleys and the six $\frac{1}{2}X$ valleys. However, for a Si supercell with a Ge impurity, we exclude the $\Gamma$ valley as it plays a negligible role in scattering effects due to the band structure of Si. 

Our findings for the spin-flips are divided into two tables: f-type scattering (Table \ref{tab:SiGefflips}) and all other relevant scattering parameters (Table \ref{tab:SiGeieflips}), which includes scattering from $\Gamma$ to $L$, scattering from one of the $L$ valleys to another $L$ valley, one of the $X$ valleys or one of the $\frac{1}{2}X$ valleys. We list two values per each scattering parameter since our two supercells produced two different values. In future work, these two parameters should be used as part of a concentration-dependent expression that could be obtained by calculating the spin-alloy scattering parameters at different VCA concentrations. There are no intravalley or intra-axis spin-flips in SiGe, consistent with GeSn. The scattering parameters calculated using a Ge supercell are all larger than the parameters calculated using a Si supercell, all within the same order of magnitude. 


We compare both sets of parameters in Tables \ref{tab:SiGefflips} and \ref{tab:SiGeieflips} with their corresponding GeSn parameters, which were calculated using a Ge supercell with a Sn impurity.  The scattering parameters in a Ge supercell with a Sn impurity are generally higher than those with a Si impurity, due to the stronger SOC associated with Sn. For f-type scattering, we observe the same pattern as in GeSn, whereby the degeneracy between the $Z$ and $-Z$ valleys is broken, and the up-to-down and down-to-up flips differ from one another.




\begin{table}[H]
\centering
\renewcommand{\arraystretch}{1.4} % Adjusts row height
\setlength{\tabcolsep}{12pt} % Adjusts column width
\begin{tabular}{|c|c|c|c|}
\hline
\textbf{Symbol} & \multicolumn{3}{c|}{\textbf{Value (meV)}} \\ \hline \hline
 & $\mathbf{Si_{63}Ge_1}$ & $\mathbf{Ge_{63}Si_{1}}$ & $\mathbf{Ge_{63}Sn_{1}}$ \\ 
$V_{XY}$  &  4.11 & 5.17 & 10.4 \\  \hline
$V_{\frac{1}{2}X\frac{1}{2}Y}$  & 2.07  & 4.07 & 3.05  \\ \hline
$V_{\frac{1}{2}X Y}$  & 3.18 & 6.61 & 12 \\ \hline
$V_{XZ}^{\uparrow\downarrow}$  & 5.85 & 22.1 & 27.6\\ \hline
$V_{XZ}^{\downarrow\uparrow}$  & 11.7 & 29.4 & 42.38 \\ \hline
$V_{\frac{1}{2}X\frac{1}{2}Z}^{\uparrow\downarrow}$  & 27.7   &  47.1 & 87.9 \\ \hline
$V_{\frac{1}{2}X\frac{1}{2}Z}^{\downarrow\uparrow}$  & 30.6  & 41.3   & 92.21 \\ \hline
$V_{\frac{1}{2}X Z}^{\uparrow\downarrow}$  & 14.2 & 30.4 & 51.63 \\ \hline
$V_{\frac{1}{2}X Z}^{\downarrow\uparrow}$  & 19.8 & 37.7 & 66.61 \\ \hline
$V_{\frac{1}{2}Z X}^{\uparrow\downarrow}$  & 18.6 & 28.6 & 49.71 \\ \hline
$V_{\frac{1}{2}Z X}^{\downarrow\uparrow}$  & 15.4 & 39.5 & 68.53 \\ \hline \hline
\end{tabular}
\caption{Calculated f-type intervalley spin-scattering matrix elements $V_{\alpha\beta}^{\uparrow\downarrow}$
(in meV) in SiGe using a Si supercell with a Ge substitutional impurity (denoted Si$_{63}$Ge$_1$), and a Ge supercell with a Si substitutional impurity (denoted Ge$_{63}$Si$_1$), with spin-polarisation in the z-direction. We compare these findings to our GeSn parameters which were calculated using a Ge supercell with a Sn substitutional impurity (Ge$_{63}$Sn$_1$).}
\label{tab:SiGefflips}
\end{table}




\begin{table}[H]
\centering
\renewcommand{\arraystretch}{1.4} % Adjusts row height
\setlength{\tabcolsep}{12pt} % Adjusts column width
\begin{tabular}{|c|c|c|c|}
\hline
\textbf{Symbol} & \multicolumn{3}{c|}{\textbf{Value (meV)}} \\ \hline \hline
 & $\mathbf{Si_{63}Ge_1}$ & $\mathbf{Ge_{63}Si_{1}}$ & $\mathbf{Ge_{63}Sn_{1}}$ \\ \hline
$V_{\Gamma L}$  & - & 0.039 & 0.14  \\ \hline
$V_{LL}^z$   & 2.13 & 6.1 & 5.82 \\ \hline
$V_{LL}^{xy}$  & 1.5 & 4.2 & 4.33 \\ \hline
$V_{XL}$  & 3.26 & 9.3 & 11.59 \\ \hline
$V_{ZL}$  & 4.61 & 13.1 & 16.3 \\ \hline
$V_{\frac{1}{2}X L}$  & 5.67 & 14.1 & 22.6\\ \hline
$V_{\frac{1}{2}Z L}$  & 8.01 & 19.9 & 31.89 \\ \hline \hline
\end{tabular}
\caption{Calculated intervalley spin-scattering matrix elements $V_{\alpha\beta}^{\uparrow\downarrow}$
(in meV) for all relevant valleys (excluding f-type) in SiGe using a Si supercell with a Ge substitutional impurity (denoted Si$_{63}$Ge$_1$), and a Ge supercell with a Si substitutional impurity (denoted Ge$_{63}$Si$_1$), with spin-polarisation in the z-direction. Our results are compared to our GeSn parameters calculated using a Ge supercell with a Sn substitutional impurity (Ge$_{63}$Sn$_1$).}
\label{tab:SiGeieflips}
\end{table}


% spin-conserving parameters?




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


\section{Conclusions} \label{sec:spinconcl}

In this chapter, we used first-principles methods to calculate the spin-dependent alloy and e-ph scattering parameters of GeSn in order to determine the spin-relaxation time and diffusion length of GeSn as a function of Sn concentration, temperature and biaxial tensile strain. Our results show that changes in spin orientation only occur by means of intervalley alloy scattering, and not by intravalley alloy scattering. We find that intervalley alloy-spin scattering is an order of magnitude stronger when electrons scatter to/from the direction of spin-alignment. We also implemented the frozen-phonon method to calculate the electron spin-phonon scattering parameters between the $L$ and $\Gamma$ valleys. By using these parameters along with electron spin-phonon parameters from literature, we were able to accurately calculate the spin relaxation time of GeSn. 

Our calculations show that at low Sn concentrations, spin-alloy scattering reduces the spin-relaxation time, especially at lower temperatures. However, beyond the indirect-to-direct bandgap transition, the spin relaxation time increases by several orders of magnitude. At room temperature, the spin relaxation time of GeSn are up to 1000 times that of Ge at Sn concentrations above $25\%$. At cryogenic temperatures, less Sn is required for such an increase in the spin-lifetime compared to room-temperature GeSn, with $11\%$ Sn enough to achieve a spin-lifetime $10^5$ times higher than Ge at $30K$. Including biaxial tensile strain further reduces the Sn content required to achieve such a rise. For applications which require a long spin relaxation time, we propose an alloy of $15\%$ Sn above 50K, or $10\%$ below 50K. Compared to other direct-gap materials such as III-Vs (see Chapter \ref{chap:spin}), GeSn possesses a much longer spin relaxation time. 

Our calculations also reveal a high spin-diffusion length in GeSn beyond the indirect-to-direct bandgap transition. These findings highlight the potential GeSn has as a promising material for spintronic applications, especially those which require a long spin-lifetime such as spin buses, spin interconnects and quantum computing. Our model facilitates parameter transfer for multi-scale models, enabling accurate spin-relaxation and spin-diffusion length calculations for materials under arbitrary material compositions, temperatures, and strain conditions.


% Can conclude that if we introduced strain to split up the Ls, there would be no intervalley L spin-alloy scattering and fuck all L-Gamma spni-alloy since it's weak anyway, we could achieve a higher spin relaxation time.

