\chapter{Band Structure of GeSn} \label{chap:gesn}


% introduction connecting DFT to this? BZ discussion? 

The electronic structure methods discussed in the previous chapter provide the theoretical foundation for analysing and predicting the band structures of semiconducting materials. In this chapter, we apply these electronic structure techniques to calculate the band structure in GeSn alloys. Special attention is given to how Sn incorporation and strain modify the conduction bands, leading to the transition from an indirect to a direct band gap. The insights gained from these models enable the understanding of key properties such as effective masses and the density of states; parameters that are essential for evaluating the potential of GeSn in electronic, spintronic and optoelectronic applications.





\section{Band structures of Ge and Sn}

Changes in the band structure of GeSn with composition can be understood by examining the electronic band structures of its constituent elements, Ge and Sn. Germanium is an indirect-gap semiconductor, with a conduction band minimum of $0.66\ eV$ at the $L$ point and a higher-energy direct gap of $0.80\ eV$ at the $\Gamma$ point \cite{Fisch,rideau,kurdi,kao} at room temperature. The relatively small energy difference of just $140\ meV$ between the $L$ and $\Gamma$ valleys gives Ge a pseudo-direct band gap character. In contrast, $\alpha$-Sn exhibits a negative direct band gap (or anti-gap) of $-0.41\ eV$ at the $\Gamma$ point \cite{bauer2003tunable}, resulting in a semimetallic nature.

% cbs
Figures \ref{fig:Geband} and \ref{fig:snband} show the band structures of bulk Ge and Sn respectively, calculated using DFT and corrected with GW and a 100-band $\mathbf{k.p}$ model. These figures show the bands for states near the highest filled valence bands and lowest unoccupied conduction bands, with the zero of the energy taken at the valence band maximum. The critical symmetry points of the conduction band, known as valleys, at $\Gamma$, $L$ and $\Delta$ represent $\Vec{k}=\frac{2\pi}{a_0}[0,0,0]$, $\frac{2\pi}{a_0}[\frac{1}{2},\frac{1}{2},\frac{1}{2}]$ and $\frac{2\pi}{a_0}[0.83,0,0]$ respectively. The $L$ band has $4$ equivalent valleys, located in the $[111]$, $[\bar{1}11]$, $[1\bar{1}1]$ and $[11\bar{1}]$ directions, while the $\Delta$ band has $6$ equivalent valleys in the $[100]$, $[\bar{1}00]$, $[010]$, $[0\bar{1}0]$, $[001]$ and $[00\bar{1}]$ directions. 


% vbs
The valence band for GeSn (at any concentration) reaches its maximum at $\Vec{k}=0$ (the $\Gamma$ point). In Ge, the constant energy surfaces around this point are made up of three warped spheres: the heavy-hole (HH), light hole (LH) and split-off (SO) bands. Only in the presence of spin-orbit coupling do we see the breakage in degeneracy of the LH and HH bands, with the SO band being slightly lower again. %They display mostly parabolic structure, however, like the conduction bands at $\Gamma$, this parabolicity is disrupted with the introduction of Sn. The introduction of Sn causes the SO band maximum to rise to the LH and HH maxima. 


\begin{figure}[H]
    \centering
    \includegraphics[width=0.78\linewidth]{Background/Figures/Ge_bands.png}
    \caption{Electronic band structure of Ge in eV.}
    \label{fig:Geband}
\end{figure}
    
    \bigskip

\begin{figure}[H]
    \centering
    \includegraphics[width=0.78\linewidth]{Background/Figures/Snband.png}
    \caption{Electronic band structure for Sn in eV.}
    \label{fig:snband}
\end{figure}

The bandgaps at critical points $\alpha$ of the BZ can be calculated using an interpolation of the two elements along with the respective bowing parameter $b_\alpha$, which describes the deviation from linearity in the interpolation:
\begin{equation} \label{eq:gapalloy}
    E_\alpha^{GeSn}=(1-x)E_\alpha^{Ge}+xE_\alpha^{Sn}+b_\alpha^{GeSn}x(1-x)
\end{equation}






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\section{Achieving a Direct Gap in GeSn}

In Chapter \ref{chap:intro}, we discussed the advantages of direct-gap semiconductors for electronic transport and optoelectronic applications. We also noted that recent research efforts to engineer a direct band gap in GeSn have primarily focused on two strategies: incorporating a sufficient amount of Sn \cite{sau,Chang2010} and applying biaxial tensile strain to the bulk alloy.


\subsection{Direct Gap through Addition of Sn}

Alloying Ge with Sn causes the conduction band to shift downward relative to the valence band, with the energy at the $\Gamma$ valley decreasing more rapidly than at the $L$ valley. Figures \ref{fig:GeSncrossover} and \ref{fig:GeSncrossover0} shows our calculated changes in the direct and indirect band gaps of GeSn at room-temperature and 0K respectively, as functions of Sn concentration. The results indicate that GeSn becomes a direct band gap semiconductor at a Sn content of approximately $8.4\%$ or higher. We calculated the bandgaps using DFT and firstly corrected using the GW approximation, which is also done at $0K$. The gaps were then corrected empirically for temperature using Varshni's empirical expression \cite{varshni1967temperature}, shown in Equation \ref{eq:varsh}, where the material-specific constants $\alpha$ and $\beta$, which quantify the rate at which the energy band gap decreases with temperature. These constants are also taken from Ref. \cite{varshni1967temperature}, and are displayed in Table \ref{tab:materialconst}. %Whilst the bandgaps can be calculated as a function of temperature using DFPT, this is computationally expensive for an exploration of alloy composition, strain and temperature. On the other hand, analytic expressions used in this work have been shown to be an excellent approximation for calculating the transport in group IV materials \cite{fmathermo}. The effect of alloy content on the band edge is calculated using the VCA, where the bandgap at each composition is given by Eq. \ref{eq:gapalloy} in Chapter \ref{chap:gesn}. 
\begin{equation}
    E_g(T) = E_g(0) - \frac{\alpha T^2}{T + \beta} \label{eq:varsh}
\end{equation}


\renewcommand{\arraystretch}{1.7}
\begin{table}[h!]
    \caption{Material constants $\alpha$ (in eV/K$^2$)and $\beta$ (in K) at band valleys L and $\Gamma$ of Ge. }
    \centering
    \begin{tabular}{ |c|c|c| }
     \hline 
      & \textbf{L} & $\mathbf{\Gamma}$ \\ \hline
     $\mathbf{\alpha}$ & $4.8\times10^{-4}$ & $5.82\times10^{-4}$  \\ \hline
     $\mathbf{\beta}$  & 235   & 296  \\   \hline
    \end{tabular}
    \label{tab:materialconst}
\end{table}


\bigskip

\begin{figure}[H]
    \centering
    \includegraphics[width=0.7\columnwidth]{Background/Figures/GeSnbandsnostrain.png}
    \caption{Calculated energy (eV) of $\Gamma$ (red) and $L$ (black) valleys in GeSn against Sn concentration at $300K$.}
    \label{fig:GeSncrossover}
\end{figure}

\begin{figure}[H]
    \centering
    \includegraphics[width=0.7\columnwidth]{Background/Figures/GeSncrossover0.png}
    \caption{Calculated energy (eV) of $\Gamma$ (red) and $L$ (black) valleys in GeSn against Sn concentration at $0K$.}
    \label{fig:GeSncrossover0}
\end{figure}



\subsection{Direct Gap through Strain}

When strain is applied to a semiconductor, it alters the interatomic spacings and symmetry of the lattice, thereby modifying the band structure. Mathematically, strain is described by the strain tensor $\epsilon_{ij}$, a second-rank symmetric tensor that quantifies the relative displacement between atoms in different crystallographic directions. For small deformations, the strain tensor components are defined as:
\begin{equation}
    \epsilon_{ij}=\frac{1}{2} \left( \frac{\partial (\delta R_i)}{\partial R_j} +\frac{\partial (\delta R_j)}{\partial R_i} \right)
\end{equation}

where $\delta R_i$ is the displacement vector field and $R_i$ are the spatial coordinates. The parameters which describe the changes in the electronic structure with respect to atomic displacements are the deformation potentials. Within the framework of deformation potential theory, which was developed by Bardeen and Shockley \cite{bardeen1950deformation} and generalised by Herring and Vogt \cite{herring1956transport}, the energy shift of a band extremum $\alpha$ is expanded in terms of the components of the strain tensor:

\begin{equation}
    \Delta E^{(\alpha)}=\sum_{ij}D_{ij}^{(\alpha)}\epsilon_{ij}
\end{equation}

where the tensor quantity $D_{ij}$ is the deformation potential. The deformation potential tensor obeys the same symmetry as the strain tensor, giving
$D_{ij}^{(\alpha)} = D_{ji}^{(\alpha)}$, limiting the number of independent components of the deformation tensor is six. In a cubic lattice, the edges of the conduction band are located on symmetry lines. Therefore, the deformation potential operator of a particular band can be expressed via two or three deformation potential constants. We list these values for GeSn in Chapter \ref{chap:trans}.  

Similar to the addition of Sn to Ge, biaxial tensile strain reduces the energy at the $\Gamma$ valley faster than at the $L$ valley. In this case, we have $\epsilon^i_{xx}=\epsilon^i_{yy}=\frac{a_i-a_0}{a_0}>0$ at $i\%$ strain, where $a_i$ and $a_0$ are the lattice constants of the strained and unstrained material, respectively. In DFT calculations, these values are fixed, while $\epsilon^i_{zz}$ is determined by atomic relaxation, yielding the Poisson ratio of the material or the Young's modulus. 

In Figure \ref{fig:straincrossover} we show the variation of the $L$ and $\Gamma$ valley conduction band energies in Ge with biaxial tensile strain. We find a direct-gap occurs at around $1.5\%$ strain, which is in excellent agreement with literature \cite{kurdi,kao,clavelfma}. Menendez et al \cite{soref2006advances} have predicted that tensile-strained Ge exceeding $2\%$ is achievable before breakage, which has been achieved in Refs \cite{saladukha2018direct,pavarelli2013optical}. 


\bigskip

\begin{figure}[H]
    \centering
    \includegraphics[width=0.68\columnwidth]{Background/Figures/straincrossover.png}
    \caption{Calculated energy (eV) of $\Gamma$ (red) and $L$ (black) valleys in Ge against strain percentage at $300K$.}
    \label{fig:straincrossover}
\end{figure}





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\section{Effective Masses}

An important consequence of the band structure is the curvature of the energy bands near their extrema, which directly influences the behaviour of charge carriers in a semiconductor. This curvature is quantitatively described by the concept of effective mass, which is the mass that an electron (or any particle) appears to have when responding to forces or interacting with other identical particles. In a three-dimensional system, the effective mass tensor $m^*_{ij}$ is calculated using the second derivative of the calculated energy bands with respect to k-space, as per Eq. \ref{eq:mass}:
\begin{equation}
    \frac{1}{m^*_{ij}}=\frac{1}{\hbar^2}\frac{\partial^2E(\Vec{k})}{\partial k_i\partial k_j} \label{eq:mass}
\end{equation}


where $i$ and $j$ are the components of the wavevector $\Vec{k}$ in different directions. Effective masses at band minimum $\alpha$ can be differentiated by their direction of motion: the longitudinal mass $m_l^{\alpha}$ for transport along the longitudinal axis direction $\Vec{k}_{\alpha}$, and the transverse mass $m_t^{\alpha}$ for transport along the transverse axes perpendicular to $\Vec{k}_{\alpha}$. This description is particularly relevant for conduction band minima that form anisotropic energy surfaces. Since all relevant effective masses in this thesis are calculated in the longitudinal and transverse directions, we can utilise a simplified expression given in Eq. \ref{eq:mass1}. 
\begin{equation} 
     \frac{1}{m^*_i}=\frac{1}{\hbar^2}\frac{\partial^2E(\Vec{k})}{\partial k_i^2} \label{eq:mass1}
\end{equation}

The surfaces of constant energy in GeSn at the $L$ and $\Delta$ valleys represent eight half-ellipsoids and six full ellipsoids of revolution, respectively. The axes of revolution are parallel to the direction of the valley $\Vec{k}_{\alpha}$, as we show in Fig. \ref{fig:scat}. The parabolicity of the $L$ valley allows its carriers to be considered as free electrons with effective masses $m_l^L$ and $m_t^L$. Expressions for the effective masses of GeSn at the $L$ valley are given in Equations \ref{eq:ml} and \ref{eq:mt} as functions of Sn concentration $x$. The masses were calculated at different concentrations of Sn (in increments of $3\%$ Sn) and fitted to a quadratic model.
\begin{align} 
    m^L_l&= (1.636- 0.117 x + 0.313x^2)m_e \label{eq:ml} \\
    m^L_t&=  (0.074 -0.036x -0.029 x^2) m_e \label{eq:mt}
\end{align}

In contrast, the conduction band at the $\Gamma$ point in Ge exhibits strong non-parabolicity, especially beyond the immediate vicinity of the band edge. Although the dispersion near the conduction band minimum at $\Gamma$ is approximately parabolic, this holds only within a narrow energy range. As the energy increases, the band rapidly deviates from a simple quadratic form, transitioning toward a more linear-like dispersion due to strong coupling with higher conduction bands. This results in electrons in Ge possessing a low effective mass and higher group velocity, which as we will show in Chapter \ref{chap:trans}, leads to a high electron mobility. The addition of Sn leads to even larger deviations from a simple parabolic shape. Consequently, in GeSn, the band dispersion is such that the effective masses tends to $0$, with an increased group velocity and electron mobility. 


The effective masses of Ge at the $\Gamma$ and $L$ valleys can be found in Table \ref{tab:GeSn} (in units of electron mass $m_e$), along with the bandgaps of Ge and Sn at $L$ and $\Gamma$, and their lattice constants. Our calculated values for Ge are compared with experiments.

\bigskip

\renewcommand{\arraystretch}{1.4}

\begin{table}[H]
    \caption{Theoretically (th) calculated bandgaps (in eV), effective masses (in $m_e$) and lattice constants (in $\AA$) of Ge and $\alpha$-Sn at room temperature ($300\,\mathrm{K}$) along with experimental (exp) values. The bandgaps are calculated using GW and corrected empirically for temperature.}
    \centering
    \begin{tabular}{ |c|c|c|c| }
     \hline 
    \textbf{Parameter}  & \textbf{Ge}$_{\mathbf{th}}$   &  \textbf{Ge}$_{\mathbf{exp}}$  &  $\mathbf{\alpha}$\textbf{-Sn} \\  
    \hline
     $\mathbf{E_L} $   &   0.66   & 0.67$\pm$0.015 \cite{song2019band,precker2002experimental,collings1980simple} & 0.09  \\
     $\mathbf{E_\Gamma}$  &   0.8  &  0.805$\pm$0.01 \cite{song2019band,el2010control,goi1989direct}  & -0.41   \\
     $\mathbf{m^L_l}$ & 1.636 & 1.64 \cite{dexter1956cyclotron}  & 1.83  \\
     $\mathbf{m^L_t}$ & 0.074 & 0.08 \cite{dexter1956cyclotron} & 0.008  \\
     $\mathbf{m^\Gamma}$ & 0.04 & 0.049 $\pm$ 0.07 \cite{clavelfma} & - \\
     $\mathbf{a_0}$ & 5.58  & 5.65  \cite{johnson1954some,Chang2010} & 6.41 \\
 \hline
    \end{tabular}
    \label{tab:GeSn}
\end{table}






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\section{Scattering in GeSn} % this is seque into Ch5

The transport of electrons and holes in a semiconductor is not determined solely by the band structure and effective masses; it is also significantly influenced by the frequency and nature of scattering events that disrupt carrier motion. Electrons in the conduction band undergo frequent collisions with phonons, impurities, and other charge carriers. These scattering processes randomise the carriers’ momentum and, in some cases, their energy, thereby limiting their ability to contribute to electrical conduction.


These scattering processes are closely tied to the band structure of the host material, particularly the positions and degeneracies of the conduction band minima. In indirect-gap GeSn, electron scattering is primarily governed by intravalley and intervalley transitions between the equivalent $L$ valleys. In Figure \ref{fig:scat}, we label intervalley scattering between the $L$ valleys as $LL$. With increased Sn concentration, scattering between the $L$ and $\Gamma$ valleys, as well as intravalley scattering at $\Gamma$, become dominant. Scattering between the $\Delta$ valleys can be of critical importance in GeSn with the application of an electric field or compressive strain via a Si substrate, whereby $\Delta$ bands can become the band minima. Scattering between the $\Delta$ valleys is characterised by g-type scattering, which is intra-axis scattering ($[100]\rightarrow [\bar{1}00]$) and f-type scattering, where an electron scattering from one Cartesian axis to another ($[100]\rightarrow [010]$), as shown in Figure \ref{fig:scat}. We often denote the six $X$ points as $\pm X$, $\pm Y$ and $\pm Z$.


\begin{figure}[H]
    \centering
    \includegraphics[width=0.6\linewidth]{Background/Figures/Scattering.png}
    \caption{Schematics of the intervalley g-type and f-type scattering between the six equivalent $\Delta$ points, and the intervalley $LL$ scattering between the four equivalent $L$ points.}
    \label{fig:scat}
\end{figure}




In the next chapter, we will discuss in depth the dominant scattering processes in GeSn, which include derivations of expressions for the various scattering rates. We will also examine photon absorption processes, where electrons become excited and move from the valence band to the conduction band, and photon emission processes, where electrons move down energy bands, emitting a photon.