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



\section{Band Theory and Bloch's Theorem}

Before examining the changing band structure of GeSn, we discuss the foundations of band theory. Band theory indicates that in solids, electrons do not possess discrete energy, but instead have values of energy within specific ranges called ``bands" separated by band energy gaps where electrons cannot exist. The behaviour of these electrons in a solid is intricately tied to the arrangement of neighbouring atoms, and the permissible energy bands in a solid stem from the discrete energy levels of individual, isolated atoms. As atoms come together to form a solid, these discrete energy levels are perturbed, leading to the occupation of a continuous band of energy levels known as ``valence bands". Similarly, the vacant energy states broaden into ``conduction bands". Electrons in a solid can become excited and move from the valence band to the conduction band by means of photon absorption, and can move down energy bands through photon emission. Valence bands are separated from the conduction bands by an energy gap, often referred to as the bandgap. The size of this gap determines a material's insulating, semiconductor, or conductive properties.


\bigskip

Reciprocal space provides a more intuitive representation of the periodicity of a crystal and how it affects electronic properties. The reciprocal lattice vectors correspond to the allowed crystal momentum values for electrons. The Brillouin zone (BZ) is a region in reciprocal space that corresponds to the periodicity of the material's primitive cell, which is the smallest repeating structure that defines the entire lattice. In essence, the BZ breaks up the reciprocal lattice in the same way the Bravais lattice is divided up into Wigner–Seitz cells. Figure \ref{fig:brill} displays the first BZ of a FCC lattice, a truncated octahedron, as is the case with Ge and Sn, showing symmetry labels for high symmetry lines and points.

\begin{figure}[h]
    \centering
    \includegraphics[width=0.5\linewidth]{Background/Figures/brill.png}
    \caption{First BZ of a FCC lattice showing critical symmetry points.}
    \label{fig:brill}
\end{figure}

\bigskip
Bloch's theorem \cite{bloch1929quantenmechanik}, which provides a framework for understanding how electrons behave in periodic crystal lattices, states that the wave function $\psi_{nk}(\Vec{r})$ for a bulk material can be expressed as the product of a plane wave term $e^{i\Vec{k}.\Vec{r}}$ and a function $u_{nk}(\Vec{r})$ with the same periodicity as the lattice:
\begin{equation}
    \psi_{nk}(\Vec{r}) = \frac{1}{\sqrt{N_{cell}}}e^{i\Vec{k}.\Vec{r}} u_{nk}(\Vec{r})
\end{equation}
where $N_{cell}$ is the number of primitive cells, $n$ is the $n^{th}$ state corresponding to wavevector $\Vec{k}$, and $\Vec{r}$ is the position vector. If $\Vec{R}$ is a Bravais lattice vector, then:
\begin{equation}
    u_{nk}(\Vec{r}+\Vec{R})=u_{nk}(\Vec{r})
\end{equation}

This implies that states $\Vec{k}$ which lie beyond the first Brillouin zone can be translated back into the BZ by adding a suitable reciprocal lattice vector $\Vec{G}$. Electron wave functions of the form $\psi_{nk}(\Vec{r})$ are called Bloch states. If we plot the energy states $E_{nk}$ in a periodic solid as a function of their corresponding wavevectors $\Vec{k}$, this gives the band structure of the solid. A proof of Bloch's theorem can be found in Ashcroft and Mermin \cite{ashcroft2022solid}. 




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

\section{Band Structure of GeSn}

The change in the band structure of GeSn with composition can be understood through a knowledge of the band structures of Ge and Sn. Germanium is an indirect-gap semiconductor, with a bandgap minimum at the $L$-point of $0.66\ eV$ and a direct gap of $0.8\ eV$ \cite{Fisch,rideau,kurdi,kao}. This makes it less suitable for many electronic applications compared to direct-gap materials, as discussed in Chapter \ref{chap:intro}. However, Ge can be considered to have a pseudo-direct bandgap, whereby a difference of around $140 \ meV$ separates the $L-$ and $\Gamma -$ conduction band minima. In contrast, Sn has a negative direct bandgap (or anti-gap) of $-0.41\ eV$ \cite{bauer2003tunable}. The conduction bands around the $L$ points show a quadratic dispersion in $\Vec{k}$-vector near the band minimum at $\Vec{k}_L$ for both Ge and Sn, i.e. 
\begin{equation*}
    E(\Vec{k}) = E(\Vec{k}_L) + \frac{\hbar^2 \Delta k_\parallel^2}{2m_l^L} + 2\frac{\hbar^2 \Delta k_\perp^2}{2m_t^L}
\end{equation*}

Carriers at $L$ can be considered as free electrons with different effective masses, i.e. the masses that an electron (or any particle) seems to have when responding to forces or interacting with other identical particles. Effective masses can be differentiated by their direction of motion: the longitudinal mass $m_l^L$ for transport along the longitudinal axis direction $\Vec{k}_L$, and the transverse mass $m_t^L$ for transport along the transverse axes perpendicular to $\Vec{k}_L$. In contrast, conduction bands in the $\Gamma$ valley only show a quadratic dispersion in $\Vec{k}$-vector in Ge. With the introduction of Sn, parabolicity is lost, with increased Sn content leading to larger deviations from a simple parabolic shape. The non-parabolic shape of the $\Gamma$-band results in electrons possessing a lower effective mass and higher group velocity, which leads to a high electron mobility, as discussed in Chapter \ref{chap:intro}. In high-Sn GeSn, there are essentially no effective masses. The effective masses of Ge at $\Gamma$ and $L$, in units of electron mass $m_e$, can be found in Table \ref{tab:GeSn}, along with the bandgaps of Ge and Sn at $L$ and $\Gamma$, and their lattice constants.


%\begin{table}[h]
%\caption{Lattice constants (in $\AA$), bandgaps (in eV) and effective masses (in $m_e$) in Ge and $\alpha$-Sn at room temperature ($300K$) taken from \cite{Chang2010}.} 
%\medskip
%\centering
%\large
%\renewcommand{\arraystretch}{1.5} % Adjust row spacing
%\begin{tabular}{ |c|c|c|c|c|c|c| } 
% \hline 
% \textbf{Element}  &  $\mathbf{a_0}$   & $\mathbf{E_\Gamma}$ &   $\mathbf{E_L}$  &  $\mathbf{m^\Gamma}$  & $\mathbf{m_l^L}$ &  $\mathbf{m_t^L}$ \\ 
% \hline
%Ge         &  5.66  &  0.8  & 0.66  & 0.05    &    1.645   &  0.074\\
%$\alpha$-Sn & 6.49  & -0.41 & 0.09  &  - & - & -\\
% \hline
%\end{tabular}
%\label{table:GeSn}
%\end{table}

\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. }
    \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   & 0.66 \cite{song2019band,precker2002experimental} & 0.09  \\
     $\mathbf{E_\Gamma}$  &   0.8  &  0.8 \cite{song2019band,el2010control}  & -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}



%\bigskip
% cbs
Figures \ref{fig:Geband} and \ref{fig:gesn12band} show the calculated band structures of bulk Ge and GeSn (at $12\%$ Sn concentration) respectively, highlighting how the energy bands can be displayed as a function of crystal momentum $\Vec{k}$. These graphs show the bands for states near the highest filled valence bands and lowest unoccupied conduction bands, calculated using density functional theory (discussed in Chapter \ref{chap:theory}). The zero of the energy is taken at the valence band maximum. The critical symmetry points of the conduction band, known as valleys, at $\Gamma$, $L$ and $X$ 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 $\frac{2\pi}{a_0}[1,0,0]$ 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 $X$ band has $6$ equivalent valleys in the $[100]$, $[\bar{1}00]$, $[010]$, $[0\bar{1}0]$, $[001]$ and $[00\bar{1}]$ directions (see Figure \ref{fig:scat}).


% vbs
The valence band for GeSn 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 with three varying effective masses: 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 also causes the SO band maximum to rise to the LH and HH maxima. 


% Sn isn't right, need to fix

%\begin{figure}[H]
%    \centering
%    \begin{subfigure}
%        \centering
 %       \includegraphics[width=0.9\linewidth]{Background/Figures/Ge_bands.PNG}
 %       \caption{Electronic band structure of Ge in eV.}
  %      \label{fig:Geband}
  %  \end{subfigure}
    
 %   \bigskip
 %   \bigskip

  %  \begin{subfigure}
   %     \centering
   %     \includegraphics[width=0.6\linewidth]{Background/Figures/snband.png}
   %     \caption{Electronic band structure for $\alpha$-Sn in eV.}
    %     \label{fig:snband}
    %\end{subfigure}
%\end{figure}

\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=0.67\linewidth]{Background/Figures/gesn12bands.png}
    \caption{Electronic band structure for GeSn for $12\%$ Sn concentration in eV.}
    \label{fig:gesn12band}
\end{figure}

% scatt
Electron scattering in indirect-gap GeSn is dominated by intravalley and intervalley scattering between the $L$ valleys, each of which are defined by a single scattering matrix element. In Figure \ref{fig:scat}, we label intervalley scattering between the $L$ valleys as $LL$. In direct-gap GeSn, scattering between the $L$ and $\Gamma$ valleys, as well as intravalley scattering at $\Gamma$, dominates. With the application of an electric field or compressive strain via a Si substrate, the $X$ bands can become the band minima. Scattering between the $X$ valleys is characterised by g-type scattering ($[100]\rightarrow [\bar{1}00]$) and f-type scattering ($[100]\rightarrow [010]$), as shown in Figure \ref{fig:scat}.


\begin{figure}[H]
    \centering
    \includegraphics[width=0.55\linewidth]{Background/Figures/Scattering.png}
    \caption{Schematics of the intervalley g-type and f-type scattering between the six equivalent $X$ points, and the intervalley $LL$ scattering between the four equivalent $L$ points.}
    \label{fig:scat}
\end{figure}


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

\section{Achieving a Direct Gap in GeSn}

Recent research endeavours aimed at achieving a direct band gap in Ge have predominantly centred around two approaches; the first of which is to incorporate a sufficient amount of Sn \cite{sau,Chang2010}. Alloying Ge with Sn shifts the conduction band towards the valence bands, whereby the energy at the $\Gamma$ valley reduces more rapidly than the $L$ valley. 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}



where $x$ is the Sn concentration. The temperature-dependent bowing parameters for GeSn are $b_\Gamma=2.24 \ eV-4\times 10^{-4}\ T$ and $b_L=0.89 \ eV+7\times 10^{-4} \ T$ \cite{mukh,von2018epitaxy}. Figure \ref{fig:GeSncrossover} shows our calculated results for change in the direct and indirect bandgaps of GeSn, revealing that GeSn has direct bandgap at a Sn concentration of $8.4\%$ and higher.

\bigskip

\begin{figure}[H]
    \centering
    \includegraphics[width=0.75\columnwidth]{Background/Figures/GeSnbandsnostrain.png}
    \caption{Energy (eV) of $\Gamma$ (red) and $L$ (black) valleys in GeSn against Sn concentration at $300K$.}
    \label{fig:GeSncrossover}
\end{figure}


The second approach in producing a direct-gap in GeSn is the application of biaxial tensile strain to the bulk alloy, whereby at $i\%$ strain, $\epsilon^i_{xx}=\epsilon^i_{yy}=\frac{a_i-a_0}{a_0}>0$, and $\epsilon^i_{zz}$ is determined by atomic relaxation. As per the addition of Sn to Ge, biaxial tensile strain reduces the energy at $\Gamma$ valley faster than at the $L$ valley, with strain $\geq 1.5\%$ \cite{kurdi,kao,clavelfma} enough to produce a direct-gap in Ge. 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}. 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. In the next chapter, we will show how the electronic band structure of group-IV materials such as Ge and Sn can be calculated theoretically.

\bigskip

\begin{figure}[H]
    \centering
    \includegraphics[width=0.75\columnwidth]{Background/Figures/straincrossover.png}
    \caption{Energy (eV) of $\Gamma$ (red) and $L$ (black) valleys in Ge against strain percentage at $300K$.}
    \label{fig:straincrossover}
\end{figure}



