\chapter{Transport Properties of GeSn} \label{chap:trans}

% Interest in high-mob materials
Carrier mobility, expressions for which are derived in Chapter \ref{chap:theory2}, measures how quickly charge carriers (electrons or holes) move through a material when subjected to an electric field. High-mobility semiconductors are desirable for several reasons in the context of electronic devices and integrated circuits. In particular, the speed at which a transistor can switch between on and off states is determined in part by the mobility of charge carriers. However, a material's mobility is limited by carrier scattering, and this scattering (in particular electron-phonon scattering) produces heat, which limits the achievable switching rates of transistors \cite{sau}. High-mobility semiconductors ultimately enable faster switching speeds in transistors, leading to more responsive and efficient devices. 


% GeSn as high-mob material
As discussed in Chapter \ref{chap:gesn}, the addition of Sn to Ge induces an indirect-to-direct bandgap transition between $6-11\%$ Sn concentration. Since the $\Gamma$ valley usually has the lightest effective mass $m^*$ in diamond semiconductors, GeSn has already been proposed as a high-mobility semiconductor. We can therefore expect that when the material achieves a direct gap, the steepness of the $\Gamma$ band will cause the electron mobility of GeSn to increase. We also know that the application of biaxial tensile strain can cause an indirect-to-direct bandgap transition at a lower Sn concentration, which is desirable given the solubility issues associated with the mixing of Ge with Sn (discussed in Chapter \ref{chap:intro}).




% Alloy scattering effect
However, by introducing Sn to Ge, we are now working with an alloy (we assume an ordered, random alloy), and as a result, alloy scattering will occur in the material. This additional scattering should therefore cause the carrier mobility to decrease, thereby potentially offsetting the advantages achieved by alloying Ge with Sn \cite{Fisch}. To accurately calculate the electron mobility of GeSn, we must therefore calculate the alloy-scattering parameters of GeSn from first-principles methods using methods described in Chapter \ref{chap:theory2}. This includes all intravalley and intervalley scattering processes between the $L$ and $\Gamma$ valleys, since the band minima in Ge and Sn are at $L$ and $\Gamma$, respectively. In a previous calculation of the electron mobility of GeSn, Sau and Cohen \cite{sau} calculated a single alloy scattering parameter using a first-principles method developed by Murphy-Armando and Fahy \cite{fma06}, predicting an electron mobility of order $10^6$ for $Ge_{0.92}Sn_{0.08}$. However, given that both intra- and intervalley alloy scattering to and from the $L$ valley were not included in their calculation, this is likely to be an overestimation, even in direct-gap GeSn \cite{mukhopadhyay2021maximum}, since the $L$-mobility is much more dominant than the $\Gamma$-mobility at $8\%$ Sn due to its 33-times larger density of states. 



In this work, we concentrate specifically on electron mobility in the [$100$] direction, which has higher symmetry and serves as a standard reference direction, allowing for better comparability with results for other semiconductor materials such as Si \cite{takagi1994universality,jiang2012study} or Ge \cite{Fisch,jin2007modeling}. Since most semiconductor devices operate under ambient conditions, our primary interest is in the room-temperature electron mobility. However, low-temperature electron mobility is also of significant interest for understanding fundamental physical processes and in specialised applications such as quantum computation, which is done near $0K$. In fact, GeSn is already a candidate material in quantum computation \cite{myronov2023holes,liu2023vertical}. We wish to add to recent experimental findings cited in Chapter \ref{chap:intro} by calculating the electron mobility of (strained) GeSn at low temperatures.





% Layout of results chapter...
The remainder of this chapter is divided into two primary sections. In Section \ref{sec:alloy}, we discuss the theoretical framework behind the calculation of the alloy scattering parameters of GeSn, which was introduced in Chapter \ref{chap:theory2}. We also display the results of these first-principles calculations, and summarise the significance of our findings. In Section \ref{sec:mob}, we first detail the theoretical model of the our temperature-dependent calculation of the electron mobility of GeSn, based on our derivations in Chapter \ref{chap:theory2}. We must note at this point that in our derivation of the mobility, we implemented the energy-dependent, anisotropic relaxation time, which is suitable for anisotropic energy valleys such as those of GeSn, as well as the effects of the non-parabolicity and multiplicity of the valleys, unlike the constant relaxation time approximation. We subsequently outline in detail our findings on the room-temperature electron mobility of GeSn. We then look at the effect of biaxial tensile strain on the electron mobility. We also examine the mobility at low temperature ($15K$), before looking at a temperature-dependent calculation of the mobility at fixed Sn concentrations. We finish the chapter by summarising our main conclusions from this work. The parameters we calculate for this body of work, while first-principles, are easily transferable to models used to simulate devices, time-dependent models and other applications.





\bigskip
%\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Alloy Scattering Parameters of GeSn} \label{sec:alloy}

 % Maybe now is the time to discuss why I need to find alloy scattering parameters.

In real alloy systems, the local atomic environments play a crucial role in determining electronic and scattering properties. The VCA typically neglects these local environment effects and does not account for structural variations or local distortions that can impact the electronic structure and, consequently, scattering properties. To calculate the alloy scattering parameters in any alloy, one must understand how host atoms in a crystal respond to substitutional impurities, or how an electron's direction or velocity may change depending on which atoms it scatters between. Such transitions are not captured by the VCA, since each atom is treated as an alloy and are in effect equal to each other. In order to investigate the scattering properties of an alloy, one must analyse a supercell. Elastic momentum alloy scattering occurs due to the breaking of the translational symmetry by disorder. This effect can be determined by introducing one small change in composition in a periodic crystal \cite{fma06}.

\subsection{Theoretical Framework}





% zero potential shift
We calculate the alloy scattering parameters $\left\langle V_{\alpha\beta} \right\rangle$ using Eq. \ref{pot} in Chapter \ref{chap:theory2}. The perturbing potential $\Delta V^A$ is calculated by inserting an atom of type $A$ as a substitutional defect in a supercell of $N-1$ host atoms, and the host atoms are relaxed around the impurity using DFT. $\Delta V^A$ tends to $0$ at a large distance from the impurity site. The intervalley scattering parameters can be obtained directly from the conduction band splittings. However, a problem that occurs in finding the correct intravalley scattering matrix elements from a finite supercell is that the zero of the potential is arbitrary. The intravalley terms depend on the overall shift in the band edge energy of the supercell with $N$ host atoms and the supercell with $N-1$ host atoms and an impurity. To examine the potentials of the two supercells, we average the local DFT potential over points in the supercell far away from the impurity and compare with the same average in the pure supercell. The difference in these averages shows the necessary reference shift in the potentials. This potential difference converges to a constant, which then gets subtracted from the conduction band eigenvalues. In a supercell larger than $16$ atoms, the distance from the impurity can be sufficiently large so that the average difference in potentials is well defined \cite{fma06}. The supercell used to calculate the scattering rate must be large enough to allow for realistic structural relaxation of the host around the substitutional Ge or Sn atom. Also, the Bloch state wave vectors considered are limited by periodic boundary conditions being applied on the supercell single-particle wave functions. Supercells of sizes 64 and 128 atoms were used to find the scattering matrices between degenerate Bloch states. In the case where Sn is placed as a substitutional impurity into a Ge supercell, the reference shift was calculated to be $-270 meV$ for a 64-atom cell and $-130 meV$ for a 128-atom cell.


% mixing of nearly degenerate states.
% I don't fully understand this
Another issue arises in finding the scattering matrix elements, which is that the presence of the impurity breaks the translational symmetry of the host and therefore mixing bands which are (nearly) degenerate in the host supercell. Due to this breakage in symmetry, the boundary condition $\phi(\Vec{r})=\psi(\Vec{r})$ cannot be met. We therefore must define a state $\ket{\phi_\beta}$, which is a linear combination of the $M$ nearly degenerate energy eigenstates $\ket{\phi_i}$ with energies $E_i$, $i=1,...,M$ which has maximum overlap with the Bloch state $\ket{\psi}$ with energy $E_0 \approx E_i$. Then, we have:
\begin{equation*}
    \ket{\phi_\beta}=\sum_{i=1}^M \ket{\phi_i} \bra{\phi_i}{\psi_\beta}\rangle
\end{equation*}
and therefore the scattering matrix elements are:
\begin{equation*}
    \bra{\psi_\alpha}\Delta V \ket{\phi_\beta}=\bra{\psi_\alpha}\sum_{i=1}^M dE_i\ket{\phi_i} \bra{\phi_i}{\psi_\beta}\rangle
\end{equation*}
where $dE_i=E_i-E_0$. Using a 64-atom supercell, we have $M=17$ to include the $\Gamma$ state, the four $L$ states, the six $\frac{1}{2}X$ states and the six $X$ states.

% image like pg32 of philip's thesis could be nice

The alloy scattering parameters have not been previously calculated theoretically for GeSn, and generally they are difficult to determine from experimentally measured quantities \cite{fma06}. First-principles methods developed in Ref. \cite{fma06} are employed to determine the intravalley and intervalley electronic scattering parameters due to alloy scattering in n-type $Ge_{1-x}Sn_x$ alloys. We consider $17$ conduction band states, with only the intravalley and intervalley scattering parameters between $L$ and $\Gamma$ affecting our mobility calculation. Therefore, we seek four scattering matrix parameters: the intravalley elements ($V_L$ and $V_\Gamma$),  the intervalley term between the L valleys ($V_{LL}$) and the intervalley term between $L$ and $\Gamma$ ($V_{\Gamma L}$ or $V_{L\Gamma}$). However, we do report on the intravalley and g-type and f-type intervalley scattering in the $X$ and $\frac{1}{2}X$ Brillouin zone points, since they become important to determine the scattering of the $\Delta$ conduction band valleys in measurements where an electric field causes electrons to accelerate to higher energies, or when Ge(Sn) is grown on Si \cite{paul20088}. The alloy scattering parameters weakly depend on Sn concentration and can be considered as roughly independent of the host lattice at low Sn concentration \cite{fma06}. The first-principles calculations are performed using Hartwigsen-Goedecker-Hutter (HGH) \cite{hgh} plane-wave pseudopotentials in all of our Density Functional Theory calculations with ABINIT code \cite{romero2020abinit,gonze2020abinit}. We converge our results using an energy cut-off of 950eV and 8x8x8 Monkhorst-pack k-meshes for the first Brillouin zone integration. We apply the local density approximation for the exchange-correlation for stable and fast convergence.


% Maybe this isn't the correct place for a lot of the detail above. I'll probably go through a lot of this in the methods chapter.







\subsection{Results}

% Use Philip's thesis

% Table, or barchart?
% Importance of relaxation - show unrelaxed stuff?

In Figure \ref{fig:Vab} below, we show the alloy scattering parameters that contribute to the electron mobility of GeSn, where we denote the intravalley terms at $L$ and $\Gamma$ as $V_L$ and $V_\Gamma$, respectively, while the intervalley elements between the $L$ valleys and the $\Gamma$ and $L$ valleys are $V_{LL}$ and $V_{L\Gamma}$, respectively. In Tables \ref{tab:intraVab} and \ref{tab:interVab}, we show a more complete set of the intravalley and intervalley alloy scattering parameters of GeSn respectively. It is important to consider atomic relaxation near the impurity in the supercell, which affects the matrix elements by up to $1eV$ in magnitude, as we show in Fig. \ref{fig:Vab}. In general, the inclusion of atomic relaxation results in increased scattering, which is likely due to disorder of the atomic coordinates, enlarging the scattering area around the impurity atom. The same trend is observed in SiGe \cite{fma06}.

We find that the calculated alloy scattering parameters for GeSn are 3-4 times larger than those for SiGe \cite{fma06}, resulting in GeSn having alloy scattering 9-16 times stronger than SiGe. Overall, the inter- and intravalley matrix elements are similar in magnitude. We also note that our intravalley alloy scattering parameter at the $\Gamma$ valley is significantly larger than the $0.788 \ eV$ value calculated by Sau and Cohen \cite{sau}. This may be a consequence of atomic relaxation, where Sau and Cohen only consider atomic relaxation around the impurity up to second nearest neighbours, rather than relaxing all atoms in the supercell.



\begin{figure}[H]
    \centering
    \begin{tikzpicture}
        \begin{axis}[
            ybar,
            bar width=14pt, % Fatter bars
            width=12cm, height=8cm,
            symbolic x coords={$|V_L|$, $|V_\Gamma|$, $|V_{LL}|$, $|V_{L\Gamma}|$},
            xtick=data,
            ymin=0, ymax=4,
            ylabel={Matrix Elements $|V|$ (eV)}, 
            enlarge x limits=0.2,
            legend style={
                at={(0.79,0.99)},
                anchor=north east,
                draw=none,
                fill=none,
                font=\small
            },
            legend cell align={left},
            nodes near coords,
            nodes near coords align={vertical},
            every axis plot/.append style={area legend}, % cleaner legend icons
        ]
        % Relaxed data (shift left slightly)
        \addplot+[style={fill=blue!70}, xshift=-6pt] 
            coordinates {($|V_L|$, 2.21) ($|V_\Gamma|$, 3.56) ($|V_{LL}|$, 1.62) ($|V_{L\Gamma}|$, 2.93)};
        
        % Unrelaxed data (shift right slightly)
        \addplot+[style={fill=gray!60}, xshift=6pt] 
            coordinates {($|V_L|$, 1.27) ($|V_\Gamma|$, 2.97) ($|V_{LL}|$, 1.64) ($|V_{L\Gamma}|$, 2.32)};
        
        \legend{Relaxed, Unrelaxed}
        \end{axis}
    \end{tikzpicture}
    \caption{Calculated intravalley and intervalley scattering matrix elements for $Ge_{1-x}Sn_x$ with relaxed and unrelaxed atomic positions around the perturbing atom. All parameters are calculated using a 64-atom supercell, with Ge as the host atom and a single Sn impurity.}
    \label{fig:Vab}
\end{figure}




\bigskip

\begin{table}[h!]
\caption{Calculated intravalley scattering matrix elements (in eV) for all relevant valleys in $Ge_{1-x}Sn_x$.} 
\centering
\begin{tabular}{ ccccc } 
 \hline \hline
 Symbol  &  $V_L$       & $V_\Gamma$ &   $V_{\frac{1}{2}X}$   &    $V_X$ \\ 
 \hline
Value      &    -2.21      & -3.56            & -0.86                          &  -0.76\\ 
 \hline \hline
\end{tabular}
\label{tab:intraVab}
\end{table}

\bigskip

\begin{table}[h!]
\caption{Calculated intervalley scattering matrix elements (eV) for all relevant valleys in $Ge_{1-x}Sn_x$.} 
\centering
\begin{tabular}{ cccccccccc } 
 \hline \hline
 Symbol  &  $V_{LL}$    &  $V_{L\Gamma}$ & $V_{LX}$ & $V_{\frac{1}{2}X_f}$  &  $V_{\frac{1}{2}X_g}$    &    $V_{X_f}$   &   $V_{X_g}$   &     $V_{\Gamma \frac{1}{2}X}$   &    $V_{\Gamma X}$ \\ 
 \hline
Value      &   -1.62     &     -2.93      &    0.83      &    -0.28      &      0.34        &     0.23      &     -0.47       &       -1.34        &   -1.73      \\ 
 \hline \hline
\end{tabular}
\label{tab:interVab}
\end{table}



\bigskip

% Must show results for 128 system also

% Also need to look at difference of results once I change position of a second impurity.











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

\section{Electron Mobility} \label{sec:mob}

Given that the band minima of Ge and Sn occur at the $L$ and $\Gamma$ valleys respectively,  electrons in GeSn alloys primarily occupy these two valleys. Therefore, all necessary parameters required to determine the total mobility of GeSn are calculated at these two valleys. From Fischetti and Laux \cite{Fisch}, the total electron mobility $\mu$ in the [100] direction is given as a weighted sum of the mobilities at $L$ and $\Gamma$, which we denote $\mu^L$ and $\mu^\Gamma$ respectively:
\begin{equation}
    \mu=\frac{4}{3}r^L(\mu^L_l+2\mu^L_t)+r^\Gamma\mu^\Gamma
\end{equation}
where the parameters $r^L$ and $r^\Gamma$ are the carrier concentration ratios of the $L$ and $\Gamma$ bands respectively, while the $l$ and $t$ denote the longitudinal and transverse directions, respectively. The $4$ on the L-part comes from the 4-fold degeneracy of the L-valley, since all $4$ L valleys contribute to transport in the $[100]$-direction. The prefactor $1/3$ arises from the anisotropic nature of the $L$ valley mobility tensor and the geometric projection of each valley’s mobility onto the $[100]$ direction. Each $L$ valley has a mobility tensor consisting of a longitudinal component ($\mu^L_l$) along the valley axis ($[111]$) and two orthogonal transverse components ($\mu^L_{t,\parallel}$ and $\mu^L_{t,\perp}$). To determine the contribution of a single $L$ valley to transport along $[100]$, the tensor must be projected onto the $[100]$ axis. The longitudinal direction forms an angle $\theta$ with $[100]$, such that $cos^2\theta=1/3$, leading to a projected contribution of $\mu^L_l/3$. The transverse directions, typically taken as $[1\overline{1}0]$ and $[11\overline{2}]$, are orthogonal to $[111]$ and contribute according to the squares of their directional cosines with $[100]$, which are $cos^2\theta=1/2$ and $cos^2\theta=1/6$, respectively. Assuming the transverse mobilities are equal ($\mu^L_{t,\parallel}=\mu^L_{t,\perp}$), the total projected mobility from a single $L$ valley is: $\mu^L_{[100]}=1/3(\mu^L_l+2\mu^L_{t})$. 


% remainder of section
This Section is organised as follows: We firstly outline the theoretical framework used, reminding the reader of the detailed expressions for the electron mobility of a semiconductor and how we calculated all relevant parameters in its determination. We focus in particular on the calculation of the DOS at the non-parabolic $\Gamma$-valley. We subsequently present our results for the the electron mobility of GeSn, particularly focusing on the effect of alloy scattering, bandgap nature, strain and temperature on the mobility. 




\subsection{Theoretical Framework}

Vital to our methodology was the difference in the shape of the $L$ and $\Gamma$ bands. Recall that the $L$-bands are more-or-less parabolic, and any non-parabolicity can be accounted for by introducing a non-parabolicity factor ($\gamma^L$) to many of our mathematical expressions. Recalling formula \ref{mu1} from Chapter \ref{chap:theory2}, we can describe the intrinsic electron mobility at the L-valley in the direction $\lambda$ ($\mu^L_\lambda$) as:
\begin{equation} \label{eq:mobL}
\mu_{\lambda}^{L} =\frac{2e}{3n^{L}m_{\lambda}^{L}}\int_{0}^{\infty} dE \ \frac{ E \rho^{L}(E)\left( -\frac{\partial f}{\partial E} \right)}{R_{\lambda}^{L}(E)} 
 (1+\gamma^{L} E/2)   
\end{equation}
where $m$ represents the effective masses (given in Table \ref{tab:GeSn} in Chapter \ref{chap:gesn}), and the isotropic non-parabolicity parameters ($\gamma^L_{Ge}=0.43\ eV^{-1}$ and $\gamma^L_{Sn}=0.52\ eV^{-1}$), shown in Eq. \ref{eq:np}, are in good agreement with previous calculations \cite{fmathermo}. 
\begin{equation} \label{eq:np}
    \gamma^L=\frac{1}{E_{L_{3v}}-E_{L_{1c}}}
\end{equation}
%\gamma^L_{Ge}&=\frac{1}{2(E_{\Gamma_{3v}}-E_{\Gamma_{1c}})}\left(1-\frac{m_t}{m_e} \right)^2&=0.43\ eV^{-1}
The DOS per unit volume at the $L$ valley ($\rho^L$) is expressed analytically (Eq. \ref{eq:dos} in Chapter \ref{chap:theory2}). In contrast, the $\Gamma$-band becomes increasingly non-parabolic with increasing Sn concentration, such that the effective mass at $\Gamma$ $m_\Gamma\rightarrow 0$. As a result, the analytic expression for the DOS breaks down, meaning Eq. \ref{eq:mobL} is not applicable to the $\Gamma$-mobility. We instead express the intrinsic electron mobility at the $\Gamma$-valley using Equation \ref{eq:mu2} from Chapter \ref{chap:theory2}:
\begin{equation} \label{eq:mobg}
\mu_{\lambda}^{\Gamma} =\frac{e}{n^{\Gamma}}\int_{0}^{\infty} \frac{d^3k}{4\pi^3} \ \frac{ \left(v_g^{\Gamma}(k) \right)^2}{R_{\lambda}^{\Gamma}\left(E(k)\right)} \left( \frac{-\partial f}{\partial E} \right)
\end{equation}
In Equations \ref{eq:mobL} and \ref{eq:mobg}, $f$ describes the in-equilibrium distribution of electrons in the conduction band, given by Fermi-Dirac (FD) distribution (see Eq. \ref{eq:fd}). However, as we approach room temperature, the energy $E$ of the conduction band state is much higher than the Fermi energy ($E-E_F>>kT$). As a result, FD distribution simplifies to Boltzmann distribution. 




%where $v_g$ is the group velocity of the electrons at the $\Gamma$ valley. 
The momentum scattering rates $R^\alpha$ in Equations \ref{eq:mobL} and \ref{eq:mobg} are sums of the alloy-scattering and electron-phonon scattering at valley $\alpha$. We calculate phonon scattering following Fischetti and Laux \cite{Fisch} using the deformation potential method (shown in Eqs. \ref{eq:Rop} and \ref{eq:Rac} in Chapter \ref{chap:theory2}). Since we confine our calculations to Sn compositions below $25\%$, we used the phonon energies and deformation potentials for Ge, which have been previously calculated by Murphy-Armando and can be found in Refs. \cite{fma08,murphy2011giant}. For the $\Gamma$ valley, intravalley scattering with optical phonons vanishes around the mainly s-type wavefunctions \cite{Fisch}. The intravalley $\Gamma$ deformation potential is a linear interpolation Ge \cite{murphy2011giant} and Sn \cite{Chang2010} values. The phonon parameters used are given in Tables \ref{tab:defs} and \ref{tab:phenergies}, with the phonon energies taken from Ref. \cite{murphy2011giant}. 
  % proof that phonon terms change very little with addition of Sn?


\begin{table}[H]
\centering
\renewcommand{\arraystretch}{0.9} % reduces row height
\begin{tabular}{|c|c|c|c|c|c|}
\hline
\textbf{Scattering Type} & \textbf{Valleys} & \textbf{Symbol} & \textbf{Value} &  \textbf{Unit} & \textbf{Reference }\\
\hline \hline
intravalley & L & $(D_L)_{op}$ & 363       & $10^8eV/m$  & \cite{murphy2011giant} \\
            &   & $\Xi_u$     & $16.975$  &   $eV$    &  \cite{fma08}   \\
            &  & $\Xi_d$     & $-6.265$    &   $eV$    &  \cite{fma08}  \\
      & $\Gamma$& $(D_\Gamma)_{ac}$  & $-5.3 - 0.7 x$  &    $eV$   & \cite{mukh} \\
  \hline
intervalley  & $LL$ & $(D_{LL})$ & 78.8  & $10^8eV/m$ & \cite{murphy2011giant} \\
  & $L\Gamma$ & $(D_{L\Gamma})$    & 286      & $10^8eV/m$ & \cite{murphy2011giant} \\

\hline \hline
\end{tabular}
\caption{Deformation Potentials for Ge and GeSn.}
\label{tab:defs}
\end{table}


\begin{table}[H]
\centering
\renewcommand{\arraystretch}{0.9} % reduces row height
\begin{tabular}{|c|c|c|c|c|}
\hline
\textbf{Scattering Type} & \textbf{Valleys} & \textbf{Symbol} & \textbf{Value} &  \textbf{Unit} \\
\hline \hline
intravalley & L & $\hbar\omega_{op}$ & $37$   & $meV$  \\
  \hline
intervalley  & $LL$ & $\hbar\omega_{LL}$ & $29$  & $meV$ \\
  & $L\Gamma$ & $\hbar\omega_{L\Gamma}$    & $30$      & $meV$ \\
\hline \hline
\end{tabular}
\caption{Phonon Energies for Ge.}
\label{tab:phenergies}
\end{table}




Both the total scattering rate (via Fermi's Golden Rule) and the carrier concentration (Eq. \ref{eq:n}) at the $\Gamma$ valley depend on the DOS at $\Gamma$ ($\rho^\Gamma$). Rather than implementing the analytic expression, we instead express the DOS as a sum of delta-functions centred at the discrete energy levels $E_k$:
\begin{equation}
    \rho^{(\alpha)}(E)\equiv \sum_k\delta\left( E-E_k \right) = \int \delta\left( E-E_k \right) d^3\Vec{k}
\end{equation}
We can rewrite this expression by utilising a change of variables $d^3\Vec{k}=d\Vec{S}dh$, where $h=E-E_k$ and $d\Vec{S}$ is the surface in k-space at constant $h$, i.e. perpendicular to $dh$ \cite{murphy2023electronic}. We can then exploit the scaling properties of a delta function applied to a function, whereby $\delta \left( f(x) \right)$ can be written as a sum of $\delta$s at the zeros of $f$:
\begin{equation}
    \delta(f(x)) = \sum_i \frac{\delta(x - x_i)}{|f'(x_i)|}
\end{equation}
Applying this to the DOS gives:
\begin{equation}
    \rho^{(\alpha)}(E)= \int d\Vec{S} \ dh \frac{\delta(h)}{|\nabla_{\vec{k}} h|}
\end{equation}
Finally, integrating over $h$, we obtain our analytical expression (Eq. \ref{eq:dosg}) for the DOS using the sifting property of the delta function $\int_{-\infty}^{\infty} f(x) \delta(x - x_0) \, dx = f(x_0)$:
\begin{align}
    \Rightarrow \rho^{(\alpha)}(E)&=\frac{d\Vec{S} }{|\nabla_{\vec{k}} h|} \Big|_{\vec{k}|h=0} \nonumber \\
    \Rightarrow \rho^{\Gamma}(E)&=\frac{d\Vec{S} }{|\nabla_{\vec{k}} (E-E_{\vec{k}})|} \Big|_{\vec{k}|E=E_{\vec{k}}} \label{eq:dosg}
\end{align}
In order to accurately determine $E_k$, the conduction band energies around the $\Gamma$-point, we first use DFT to find the band dispersion around the $\Gamma$-point. We then fit the DFT eigen-energies to a $3D$-Kane model \cite{kane1957band}, as described in Section \ref{sec:Kane} of Chapter \ref{chap:theory}. The DFT eigen-energies are fitted to the eigenvalues of the $4\times4$ Hamiltonian matrix, whereby we solve for parameters $a$ and $b$. We finally correct the DFT-calculated direct-gap $E_g$ for GW and temperature. Using these energies around the $\Gamma$-point, we calculated the group velocity of the electrons, defined by $v_g^{(\alpha)}(k)\equiv \frac{1}{\hbar}\frac{\partial E}{\partial k}$. Since we are interested in calculating the electron mobility in the $[100]$-direction, we calculate the group velocity of the electrons in the $x$-direction. 




In all our calculations on unstrained GeSn, we consider Sn concentrations of up to $25\%$, keeping with the realistic amount of Sn that can be incorporated into Ge. While as much as $2\%$ strain has been successfully applied to Ge, we apply $0.5\%$ and $1\%$ biaxial tensile strain to GeSn, where we consider Sn concentrations up to $20\%$. We expect that alloy scattering will reduce the electron mobility of Ge upon the introduction of Sn. However, once the alloy transitions to a direct-gap material, we anticipate that the mobility will increase to many times that
of intrinsic Ge, especially with the application of strain.


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

\subsection{Results}


\subsubsection{Room Temperature Mobility of unstrained GeSn}

We first examine the behaviour of the room-temperature intrinsic n-type electron mobility of unstrained GeSn, where we assume a random alloy. Figure \ref{fig:mob} shows the behaviour of the mobility of unstrained GeSn as a function of Sn concentration. We calculated the mobility of Ge to be $4300 \ cm^2/(Vs)$, which is comparable to the $3900$ observed by Fischetti and Laux \cite{Fisch} and experimentally \cite{prince}. As Sn is introduced, the mobility decreases due to alloy scattering, becoming as low as $510 \ cm^2/(Vs)$  at $8.75\%$ Sn (just beyond the indirect-to-direct gap transition). Thereafter, the mobility increases rapidly as the electrons begin to populate the $\Gamma$ valley, where the mobility is much higher than at the $L$ valley. To achieve an electron mobility above that of Ge by alloying with Sn, a Sn concentration of at least $13.5\%$ must be added. Figure \ref{fig:mob} shows that at $17.5\%$ Sn, GeSn has a mobility $5$ times greater than that of Ge, while $Ge_{0.75}Sn_{0.25}$ has a mobility over $25$ times greater than that of Ge. 

Figure \ref{fig:mob} also shows in the subplot the carrier concentration ratios of the $\Gamma$ and $L$ valleys ($r_\Gamma$ and $r_L$ respectively) for intrinsic unstrained GeSn, which displays the gradual rise in the electron population at $\Gamma$ due to the high temperature (Boltzmann distribution). 

\begin{figure}[H]
    \centering
    \includegraphics[width=\linewidth]{Result1-Transport/Figures/GeSn_mob_r.png}
    \caption{Room-temperature intrinsic n-type electron mobility of unstrained GeSn as a function of Sn concentration and difference in bandgaps $E_{L\Gamma}=E_L-E_\Gamma$ (in eV). The second scale is better representative of the behaviour of the electron mobility, since the mobility only begins to rise after $E_{L\Gamma}$ becomes positive, and Sn concentration may not correspond to $E_{L\Gamma}$ directly due to clustering.}
    \label{fig:mob}
\end{figure}

Perhaps surprisingly, the mobility continues to rise when the carriers have almost entirely populated the $\Gamma$-valley, with no eventual decay as a result of alloy scattering. This can be attributed to two key reasons: one is an increase in the group velocities of the electrons with Sn concentration, and two is the lowering of the DOS at $\Gamma$ with Sn concentration. In Figures \ref{fig:vg} and \ref{fig:dos}, we show the thermal averages of the group velocity and $\Gamma$-DOS as functions of Sn concentration, whereby we integrate each quantity over all states weighted by the distribution function. 


\begin{figure}[H]
    \centering
    \includegraphics[width=0.85\columnwidth]{Result1-Transport/Figures/v_g_therm_av.png}
    \caption{Thermal average of the group velocity at $\Gamma$ of GeSn in m/s as a function of Sn concentration.}
    \label{fig:vg}
\end{figure}


%\bigskip

\begin{figure}[H]
    \centering
    \includegraphics[width=0.85\columnwidth]{Result1-Transport/Figures/dos_therm_av.png}
    \caption{Thermal average of DOS of the $\Gamma$ valley as a function of Sn concentration.}
    \label{fig:dos}
\end{figure}



Despite this continuous increase in mobility beyond $8.75\%$ Sn, it is nevertheless limited by alloy scattering. The full effect of alloy scattering in presented in Figure \ref{fig:noalloy}.
The black line corresponds to the mobility of GeSn at room temperature, as was shown in Figure \ref{fig:mob}. The red and blue lines are the hypothetical mobilities disregarding intervalley alloy scattering and all alloy scattering, respectively. Clearly, intervalley alloy scattering affects the mobility at Sn concentrations $0-11\%$. The fact that the outcomes are several orders of magnitude different highlights that intravalley alloy scattering in particular needs to be considered with precision. %rather than a single effective alloy scattering potential, as per the calculation of Sau and Cohen \cite{sau}.


\begin{figure}[H]
    \centering
    \includegraphics[width=0.99\columnwidth]{Result1-Transport/Figures/mobnoal.png}
    \caption{Room temperature mobility with alloy scattering (black), without intervalley alloy scattering (red) and without alloy scattering (blue) contributions, as a function of Sn concentration.}
    \label{fig:noalloy}
\end{figure}







\subsubsection{Room Temperature Mobility of biaxially tensile strained GeSn}


The application of biaxial tensile strain induces an indirect-to-direct bandgap transition at lower Sn concentrations (as we showed in Figure \ref{fig:straincrossover} in Chapter \ref{chap:gesn}), Consequently, electrons populate the $\Gamma$ valley at lower Sn concentrations where there is less alloy scattering, resulting in a higher electron mobility. In order to achieve mobility higher than that of unstrained Ge, a Sn concentration of at least $11.8\%$ with $0.5\%$ strain is required, or $9\%$ Sn with $1\%$ strain. In Figure \ref{fig:RTmob}, we see that under $1\%$ strain, a Sn concentration of just $13\%$ is required to achieve a mobility $5$ times higher than Ge. This makes biaxially strained GeSn experimentally more attractive than unstrained GeSn, given the segregation issues in GeSn would likely lead to a significant loss in mobility. 



\bigskip

\begin{figure}[H]
 \includegraphics[width=\columnwidth]{Result1-Transport/Figures/1strain_mob_comp.png}
    \caption{Room-temperature intrinsic n-type electron mobility of unstrained GeSn (black) and $1\%$ biaxially tensile strained GeSn (red), as a function of Sn concentration and difference in bandgaps $E_{L\Gamma}=E_L-E_\Gamma$ (in eV).}
    \label{fig:RTmob}
\end{figure}




\subsubsection{Low Temperature Mobility (15K) of (strained) GeSn}

We now examine the electron mobility of GeSn at $15K$. At such a low temperature, the electron distribution $f_0(E)$ from Equations \ref{eq:mob} and \ref{eq:mobg} describes the Fermi-Dirac (FD) distribution, which behaves like a step-function. We can therefore expect changes in the electron distribution across the $L$ and $\Gamma$ valleys (and by extension the mobility) to be less gradual compared to higher temperatures. 

In Figure \ref{fig:lowTmob}, we show the electron mobility of unstrained and $1\%$ biaxially strained GeSn, as well as the carrier concentration ratios of unstrained GeSn at $15K$. The electron mobility for unstrained Ge is $2.52\times 10^4 \ cm^2/(Vs)$, almost ten times the room-temperature value, since electron-phonon scattering is orders of magnitude lower than at room temperature. Beyond the indirect-to-direct bandgap transition ($8.8\%$ Sn in unstrained GeSn), electrons quickly populate the $\Gamma$ valley (as we see in the subplot of the carrier concentration ratios), causing the mobility to rise rapidly. A Sn concentration of $9.1\%$ is needed to achieve a mobility higher than the Ge value, where the $L$ valley is $4.6 \ meV$ above the $\Gamma$ valley and the carrier concentration ratio of electrons in $\Gamma$ is $0.022$. The mobility has an order of magnitude of $10^6$ beyond $25\%$ Sn for unstrained GeSn. 

With the addition of tensile strain, a direct gap occurs at a lower Sn concentration, causing the mobility to increase at a lower Sn concentration, with less alloy scattering as a result. At $0.5\%$ and $1\%$ strain, GeSn transitions to a direct gap material at $7.75\%$ and $5.57\%$ Sn respectively. As more Sn is introduced beyond these concentrations, the mobility is almost immediately higher than for strained Ge. Under $1\%$ strain, as we see in Figure \ref{fig:lowTmob}, the mobility of GeSn is $10^5$ at $5.95\%$ Sn, where $E_{L\Gamma} \simeq 0$ and the carrier concentration ratio of electrons in $\Gamma$ is $0.58$. At high Sn concentrations, the effect of strain in increasing the mobility of GeSn in minimal.

\begin{figure}[H]
    \centering
    \includegraphics[width=\columnwidth,keepaspectratio]{Result1-Transport/Figures/15K_1strain_mob_comp.png}
    \caption{Low-temperature (15K) intrinsic n-type electron mobility and carrier concentration ratios at $L$ and $\Gamma$ as a function of Sn concentration and difference in bandgaps.}
    \label{fig:lowTmob}
\end{figure}




\subsubsection{The Effect of Temperature on the Mobility of (strained) GeSn}

% More detail here would be good

Reducing temperature increases the electron mobility of GeSn for two reasons. Firstly, electron-phonon scattering decreases with decreasing temperatures. Secondly,  the Fermi-distribution becomes increasingly step-function-like at lower temperatures, causing the carriers to populate the $\Gamma$ valley at the indirect-to-direct bandgap transition at lower $E_{L\Gamma}$ energy difference than at higher temperatures. This results in a higher electron mobility at lower Sn concentrations. From an experimental perspective it is easier to observe transport via the direct $\Gamma$ valley by varying the temperature, since this is easier to control than the Sn concentration. Therefore in Figure \ref{fig:mobvT}, we highlight the effect of temperature on the mobility of unstrained Ge, $Ge_{0.9}Sn_{0.1}$ and $Ge_{0.85}Sn_{0.15}$, as well as $Ge_{0.93}Sn_{0.07}$ under $1\%$ biaxial strain. We can see from Figure \ref{fig:mobvT} that across the temperature range of $15-300K$, $Ge_{0.85}Sn_{0.15}$ provides the optimal mobility. However, as temperature decreases, the mobility of $Ge_{0.9}Sn_{0.1}$ becomes increasingly comparable to $Ge_{0.85}Sn_{0.15}$ and may be preferred experimentally to show the transition of $L\xrightarrow{} \Gamma$ transport. In addition, below $50K$, the mobility of $Ge_{0.93}Sn_{0.07}$ at $1\%$ biaxial strain is almost the same as unstrained $Ge_{0.9}Sn_{0.1}$, which may be even more desirable experimentally due to the lower Sn concentration.
\bigskip

\begin{figure}[H]
    \centering
    \includegraphics[width=\columnwidth,keepaspectratio]{Result1-Transport/Figures/mobvT.png}
   \caption{Intrinsic n-type electron mobility of Ge and $Ge_{0.9} Sn_{0.1}$ as a function of temperature (K).}
    \label{fig:mobvT}
\end{figure}



% Also little section on strain in 111 direction?

% Also little section on mobility with group velocity in z-direction, so low mobility, low scattering?




\section{Conclusions}


The overall aim of this chapter was to calculate the intrinsic n-type electron mobility of GeSn as a function of alloy content, temperature and biaxial tensile strain. We began by presenting a full ab initio calculation of the n-type intra- and intervalley scattering parameters of GeSn, where atomic relaxation was critical in determining accurate parameters. Our values were larger than those for SiGe, with both our intra- and intervalley scattering parameters having a substantial impact on limiting the electron mobility of GeSn.

Our alloy scattering parameters were used to determine the  mobility of GeSn using a first iteration of the Boltzmann transport equation. We find that alloy scattering causes an initial decrease in the mobility to $10\%$ of that of Ge at Sn concentrations of up to $9\%$. However, the indirect-to-direct bandgap transition initiates a steep increase in the mobility. At room temperature, the electron mobility of unstrained GeSn can rise to above $25$ times higher than Germanium's mobility at sufficient Sn concentrations. The inclusion of a biaxial tensile strain reduces the Sn content required to achieve a high mobility.  At cryogenic temperatures, the considerable increase in mobility occurs at the Sn concentration of the indirect-to-direct transition, requiring much less Sn than at room temperature. We propose that measuring the mobility against temperature in an alloy of $10\%$ Sn content will clearly show the dramatic effects of the indirect-to-direct transition. 


Our approach allows for the transfer of parameters to  multi-scale models, enabling more comprehensive and accurate modelling of carrier transport phenomena in semiconductor heterostructures. This model can be used in the future as part of a larger model that includes both n-type and p-type carriers. Given the promise of 2D hole gas mobilities in strained Ge, we propose the use of strained GeSn for n-type 2D electron gases. The broader impact of these findings could lead to significant advancements in electronic device performance. By enabling faster carrier transport and reducing energy dissipation, strained GeSn-based devices have the potential to support the development of next-generation low-power, high-speed transistors. This has direct implications for improving processor speeds, extending battery life in mobile electronics, and enhancing energy efficiency in data centres and high-performance computing applications.

Comparing our calculated mobilities to recent experimental measurements, Tien et al. \cite{tien2025extremely} reported an electron mobility of Ge$_{0.88}$Sn$_{0.12}$ to be $1500 \ \text{cm}^2/\text{Vs}$ at room temperature, which compares reasonably well with our calculated value of $2085 \ \text{cm}^2/\text{Vs}$. At 15 K, their measured mobility was $6200 \ \text{cm}^2/\text{Vs}$, significantly lower than our predicted value of $8.07 \times 10^4 \ \text{cm}^2/\text{Vs}$. The lower experimental mobility can be attributed to two main factors: (i) the mobility in GeSn alloys is highly sensitive to Sn content in the range of $8.5-12\%$, especially at lower temperatures, where even small variations in composition can lead to significant changes in mobility, and (ii) the presence of additional defects in experimental samples, such as point defects, dislocations, and other imperfections, can further reduce the mobility compared to idealised theoretical predictions.

As we will see in the next chapter, where we explore the potential of GeSn in spintronics, the alloy scattering parameters can be used as a foundation to determine the spin-dependent alloy scattering parameters, which will help us to understand the extent to which electrons change their spin polarisation as they scatter off alloy impurities.