\chapter{Optical Properties of GeSn} \label{chap:Opt}



In Chapter \ref{chap:intro}, we discussed some of the past experimental research demonstrating the potential of GeSn to replace III-V materials in optoelectronics in the future, primarily due to its direct bandgap (at sufficient Sn concentration and/or strain) and compatability with current CMOS processes. In this chapter, we present our findings on two key properties that define the optoelectronic behaviour of GeSn: the absorption coefficient and photoluminescence spectrum.

Electron absorption is a fundamental process in semiconductors and optoelectronic devices, where electrons gain energy by absorbing photons. This mechanism is crucial for applications such as solar cells, photodetectors and absorption modulators, as it governs the efficiency of light-matter interactions. The absorption coefficient, an expression for which is given in Eq. \ref{eq:abs} in Chapter \ref{chap:theory2}, quantifies how far photons of a given wavelength can penetrate the material before being absorbed. A high absorption coefficient is desirable in photovoltaic materials to maximise light harvesting, while a low value is beneficial in transparent conductive films such as waveguides \cite{johnson1973measurement}. Understanding electron absorption and its coefficient is essential for designing efficient electronic and photonic devices. The absorption coefficient of Ge has been previously calculated \cite{tiwari2024unified,bychto2018determination} and measured \cite{philipp1959optical,mosleh2014material}. Hsieh et al \cite{hsieh2012enhanced} have calculated the absorption coefficient of GeSn for direct transitions at Sn concentrations of up to $10\%$ in a photon energy range of $0.66-1$eV, showing a general increase in the absorption coefficient with Sn concentration, owing to the decreasing bandgap. There are several well established methods of calculating the absorption coefficient over a wide spectral range, such as DFT \cite{tiwari2024unified} and tight-binding \cite{avdeev2020absorption}. However, the spectral region near the bandgap energy is of particular importance for optoelectronic applications, as it critically influences the efficiency of photon absorption processes that govern device performance.


Photoluminescence (PL), shown in Eq. \ref{eq:PL} in Chapter \ref{chap:theory2}, is the emission of a photon when an electron that has been excited into the conduction band recombines with a hole in the valence band. Like absorption, photoluminescence is stronger in direct bandgap materials. Efficient PL emission is critical for LEDs and laser diodes, where radiative recombination directly impacts performance. The PL of Ge is well established \cite{murphy2021light,arguirov2011room}, whereby a peak occurs at the direct gap of Ge. Measurements \cite{ghetmiri2014direct,pezzoli2016temperature,schlykow2018photoluminescence} of the PL of GeSn is also largely defined by a singular peak at the direct gap. However, theoretical calculations of the absorption and emission of GeSn which include indirect transitions have not been previously determined over a wider spectral range. As we discussed in Chapter \ref{chap:theory2}, Tiwari et al \cite{tiwari2024unified} provided us with the first unified theory of optical absorption and emission which includes direct and indirect processes in Ge (as well as Si and GaAs), finding that at photon energies below the direct gap, indirect absorption is dominant in Ge. However, no such calculations currently exist for GeSn, where indirect transitions are also prevalent due to alloy scattering processes. 

Building on the theoretical framework established in Section \ref{sec:abs} of Chapter \ref{chap:theory2}, we apply degenerate and non-degenerate perturbation theory to calculate absorption and emission spectra across a range of Sn compositions (up to $12\%$ Sn). By incorporating both direct and indirect transitions, as well as transitions into energy-degenerate bands, our findings provide a comprehensive picture of the optical properties of GeSn. We focus on a photon energy range of $0.4-1$ eV ($800-3000$nm), which encompasses short-wave infrared (SWIR) light ($1400-3000$nm). This is the first work to introduce the effect of alloy disorder on the optoelectronic properties of an alloy in this range of wavelengths. Light with these wavelengths are essential in a range of biomedical applications \cite{hudait2021design} (particularly laser-based devices), such as optical coherence tomography - a non-invasive cross-sectional imaging method using an optical interferometer \cite{golovynskyi2018optical}. Additionally, SWIR imaging is used in a variety of applications including solar cell inspection surveillance, anti-counterfeiting and process quality control.

%We expect that since the $\Gamma$ valley reduces quicker than the L valley as Sn is added to Ge, the direct absorption should increase with Sn concentration, especially when GeSn transitions to a direct-gap material. We also expect a rise in the indirect absorption upon the introduction of Sn as a consequence of alloy scattering. % what about PL?


The remainder of this chapter is laid out as follows: in Section \ref{sec:theoryabs}, we summarise the theoretical framework used in extension with the formulae derived in Chapter \ref{chap:theory2}. Our
results are discussed in Section \ref{sec:absres}. Finally, Section \ref{sec:absconc} presents our conclusions.












%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Theoretical Framework} \label{sec:theoryabs}


Following Section \ref{sec:abs} in Chapter \ref{chap:theory2}, we calculate the absorption coefficient and photoluminescence of GeSn at fixed Sn concentrations up to $12\%$. To accurately capture the these properties in the SWIR region, we consider a dense k-point system centred at the $\Gamma$ point
%, with a k-point spacing of $0.005$($\frac{2\pi}{a}$)
. We need only consider transitions from the three uppermost valence bands (HH, LH, SO) to the lowest conduction band. All necessary band energies and momenta are computed at these k-points using DFT, with the energies corrected using a scissors shift. All DFT calculations are performed using ABINIT under the same criteria as Chapter \ref{chap:trans}. We use a Gaussian function to approximate the $\delta$-function in Fermi's Golden Rule.

To consider indirect transitions, we use Eq. \ref{eq:ELk} to calculate the energies $E_L(k)$ of k-points around the L valley, since Ge shows an ellipsoidal energy dispersion in this region:
\begin{equation} \label{eq:ELk}
    E_L(k) = \frac{\hbar^2}{2} \left( \frac{k_x^2}{m_{L,l}} + \frac{k_y^2}{m_{L,t}} + \frac{k_z^2}{m_{L,t}} \right) + E_L
\end{equation}



Each state is treated using degenerate and non-degenerate perturbation theory, following Sections \ref{sec:ndpt} and \ref{sec:dpt} in Chapter \ref{chap:theory2}. To calculate the eigenstates of the mixed states, we incorporate the unperturbed band energies into the diagonal elements of our Hamiltonian matrix, while the alloy and phonon interactions are included in the off-diagonal elements, as shown in Fig. \ref{fig:Hdeg} in Chapter \ref{chap:theory2}. However, applying degenerate perturbation theory to such dense k-point systems is very computationally expensive. Instead, we follow Tiwari et al \cite{tiwari2024unified} in partitioning the photon energy range into a series of intervals of energy width $\Delta E$, as shown in Fig. \ref{fig:DPT} below. The choice of $\Delta E$ is important as it must be checked for convergence in the limit $\Delta E\rightarrow\infty$. Within each interval, each state is treated using degenerate perturbation theory, while alloy and phonon-assisted transitions from states outside each interval are treated using non-degenerate perturbation theory.

\medskip

\begin{figure}[H]
    \centering
    \includegraphics[width=0.65\linewidth]{Result3-Abs/Figures/DPT2.png}
    \caption{Schematic of the $L$ and $\Gamma$ valleys divided into energy partitions of energy width $\Delta E$ in order to be treated using perturbation theory.}
    \label{fig:DPT}
\end{figure}




% NDPT, DPT
% collapsing energies to Ebar
% the full unified expression, or a sum of each bit that I have?
% point out in chapter 5 that it's all about calculating ei for getting PL and alpha 
% what we used for Delta E.... will need to add this to outlook





%\medskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Results} \label{sec:absres}


\subsection{Absorption Coefficient of GeSn}

We first examine the absorption coefficient of Ge in the SWIR range, previously calculated in Ref. \cite{tiwari2024unified}, providing a basis for validating our model against established results. Figure \ref{fig:aGe} presents the room temperature absorption coefficient of Ge, highlighting the role of indirect transitions by contrasting the complete calculation with one that includes only direct transitions. Including indirect transitions slightly enhances the absorption coefficient above the direct gap, but more importantly, introduces pronounced indirect-gap absorption arising from electron–phonon scattering. In the direct-transition absorption calculation, a sharp drop is expected at the onset of the direct bandgap. However, the application of Gaussian broadening, used to produce a smooth and continuous spectrum, artificially softens this edge, resulting in a more gradual decline in the absorption coefficient. We find an absorption coefficient of $4.5\times10^4\ $cm$^{-1}$ at 1eV (1250nm). Overall, our findings are in good agreement with Ref. \cite{tiwari2024unified}.

\begin{figure}[H]
    \centering
    \includegraphics[width=0.95\linewidth]{Result3-Abs/Figures/aGe.png}
    \caption{Room-temperature absorption coefficient ($\alpha$)  of Ge (black), given in $cm^{-1}$, compared to the absorption coefficient with only direct transitions considered (red), plotted as a function of photon energy $\hbar\omega$ and wavelength $\lambda$. }
    \label{fig:aGe}
\end{figure}

% GeSn
The room-temperature absorption coefficients of Ge and GeSn alloys with Sn concentrations of $3\%$, $6\%$, $9\%$, and $12\%$ are shown in Fig. \ref{fig:absgesn}. As expected, increasing Sn content leads to a higher absorption coefficient and extends the absorption edge to longer wavelengths, consistent with the reduction in bandgap energy and in agreement with Refs. \cite{bychto2018determination,philipp1959optical,mosleh2014material}. In indirect-gap GeSn ($3\%$ and $6\%$ Sn), both electron-phonon scattering and alloy scattering enable notable absorption below the direct gap, making the absorption coefficient non-zero even in that regime, especially at increased Sn concentration. Additionally, in these indirect-gap compositions, the absorption coefficient rises more sharply with photon energy once the photon energy exceeds the direct-gap threshold. In contrast, for direct-gap GeSn ($9\%$ and $12\%$ Sn), the absorption behaviour closely resembles that of Ge above the band edge, but with the onset occurring at lower photon energies. In direct-gap GeSn, alloy scattering plays a minimal role, and the absorption coefficient increases rapidly just above the bandgap. Notably, in the SWIR range (0.4–1eV), direct-gap GeSn ($9-12\%$) achieves absorption coefficients approaching $10^5\ $cm$^{-1}$, making it highly suitable for photonic and optoelectronic applications in this spectral range.

\begin{figure}[H]
    \centering
    \includegraphics[width=0.99\linewidth]{Result3-Abs/Figures/aGeSn.png}
    \caption{Room-temperature absorption coefficient (in $cm^{-1}$) of Ge (black) GeSn at Sn concentrations of $3\%$ (blue), $6\%$ (green), $9\%$ (red) and $12\%$ (pink) against photon energy $\hbar\omega$ and wavelength $\lambda$. For $9\%$ and $12\%$ Sn, the absorption curves are truncated at higher photon energies due to the limited energy range of the calculated conduction band states, which was constrained by the chosen k-point sampling.}
    \label{fig:absgesn}
\end{figure}

% effect of alloy -  contrast difference to Ge difference
We highlight the importance of alloy disorder in Fig. \ref{fig:absal}, where we compare the absorption coefficient of Ge$_{0.94}$Sn$_{0.06}$ with the corresponding absorption coefficient where alloy scattering is ignored. Below the direct gap, omitting alloy scattering can result in an absorption coefficient a full order of magnitude below its true value. However, beyond the direct gap, our plots converge, with absorption dominated by direct transitions into the conduction band.

\begin{figure}[H]
    \centering
    \includegraphics[width=0.95\linewidth]{Result3-Abs/Figures/aGeSnnoal.png}
    \caption{Comparison of the absorption coefficient of Ge$_0.94$Sn$_{0.06}$ with and without the inclusion of alloy scattering, plotted as functions of photon energy and wavelength.}
    \label{fig:absal}
\end{figure}

Finally, we examine the effect of temperature on the absorption coefficient by comparing room-temperature and low-temperature calculations. In Fig. \ref{fig:absT75}, we compare the absorption coefficients of Ge and Ge$_{0.88}$Sn$_{0.12}$ at 300K and 75K. As expected, the absorption coefficient decreases at a given photon energy with decreasing temperature, owing to the increase in the bandgap at lower temperatures, shifting the absorption edge to higher photon energies. 

\begin{figure}[H]
    \centering
    \includegraphics[width=0.95\linewidth]{Result3-Abs/Figures/aT75.png}
    \caption{Absorption coefficient of Ge (black) and Ge$_{0.88}$Sn$_{0.12}$ at $300K$ (solid) and $75K$ (dashed) as functions of photon energy and wavelength.}
    \label{fig:absT75}
\end{figure}








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

\subsection{Photoluminescence of GeSn}

%\subsubsection{Room-Temperature Calculations}

We begin by analysing the room-temperature emission spectrum of Ge, shown in Fig. \ref{fig:PLGe}, where we compare results that include phonon-assisted transitions with those that account only for direct transitions. We can see that the most significant contribution to the PL is from phonon-assisted light emission, whereby electrons scattering from $L$ to $\Gamma$ via optical phonons before recombining in the valence band at $\Gamma$ by photon assistance. This underscores the importance of using non-degenerate perturbation theory to capture these processes. Additionally, applying degenerate perturbation theory for quasi-degenerate states results in a noticeable enhancement near the direct bandgap. Our findings are in good agreement with Ref. \cite{tiwari2024unified}. Clearly, indirect transitions play a much more pivotal role in light emission compared to absorption in Ge.

\medskip

\begin{figure}[H]
    \centering
    \includegraphics[width=0.99\linewidth]{Result3-Abs/Figures/PLGe.png}
    \caption{Room-temperature PL spectra of Ge including both phonon assisted and direct transitions (black), compared to the PL spectrum with direct transitions only (red), plotted as a function of photon energy $\hbar\omega$. Intensities are given in arbitrary units (a.u.).
}
    \label{fig:PLGe}
\end{figure}


% GeSn
Figure \ref{fig:PLGeSnRT} shows the room-temperature PL spectra of Ge and GeSn alloys with Sn concentrations of $3\%$, $6\%$, $9\%$ and $12\%$, plotted as functions of photon energy. Overall, we observe an increase in the PL intensity with Sn content. Like in pure Ge, in indirect-gap GeSn, two distinct peaks are observed: a smaller one at the direct bandgap and a more prominent one at the indirect bandgap. When the material transitions into a direct-gap material, a single stronger peak emerges at the direct gap, which is broadened by phonon-assisted transitions. Notably, the PL intensity of Ge$_{0.88}$Sn$_{0.12}$ is up to $250$ times higher than that of Ge, underscoring the significant enhancement in radiative efficiency achieved by Sn incorporation. We compare our findings to experimental data at $9\%$ Sn, measured by Ref. \cite{ghetmiri2014direct} (also at room-temperature).



\begin{figure}[H]
    \centering
    \includegraphics[width=0.99\linewidth]{Result3-Abs/Figures/PLGeSnRTexp.png}
    \caption{Room-temperature PL spectrum (in arbitrary units) of Ge (black) GeSn at Sn concentrations of $3\%$ (blue), $6\%$ (green), $9\%$ (red) and $12\%$ (pink), as well as experimental data at $9\%$ (red, dotted), plotted against photon energy $\hbar\omega$.}
    \label{fig:PLGeSnRT}
\end{figure}


% alloy vs no alloy - 6%
To highlight the effect of alloy scattering on the emission spectrum of GeSn, we compare the calculated PL of GeSn at $6\%$ Sn concentration with the corresponding PL where alloy scattering is neglected. This comparison is shown in Fig. \ref{fig:PLnoal}, where a clear difference appears at the indirect-gap peak, which is significantly enhanced when alloy scattering is included. This contrast is a consequence of alloy scattering being $2.5$ times stronger than e-ph scattering in Ge$_{0.94}$Sn$_{0.06}$ at room temperature, and is even more prevalent at lower temperatures. %Our findings are in excellent agreement with experiment \cite{ghetmiri2014direct}. 

\begin{figure}[H]
    \centering
    \includegraphics[width=0.95\linewidth]{Result3-Abs/Figures/PLnoal.png}
    \caption{Comparison of the PL spectra of Ge$_{0.94}$Sn$_{0.06}$ with and without alloy scattering included, plotted as a function of photon energy $\hbar\omega$. Intensities are given in arbitrary units (a.u.).}
    \label{fig:PLnoal}
\end{figure}


%Figure \ref{fig:PL} shows the room-temperature photoluminescence of GeSn at Sn concentrations up to $12\%$ when only direct transitions into the conduction band are considered. Our figure shows and increase in the PL with Sn concentration in this energy region, with the peaks occurring at the direct gap, which decreases in energy with Sn concentration.


%\begin{figure}[H]
%    \centering
%    \includegraphics[width=0.9\linewidth]{Result3-Abs/Figures/PLdirect.png}
%    \caption{Photoluminescence of Ge (black), Ge$_{0.97}$Sn$_{0.03}$ (blue), Ge$_{0.94}$Sn$_{0.06}$ (green), Ge$_{0.91}$Sn$_{0.09}$ (red) and Ge$_{0.88}$Sn$_{0.12}$ (pink) at 300K in arbitrary units (a.u) for direct transitions in the NIR photon energy range as functions of photon energy $\hbar\omega$ in eV}
%    \label{fig:PL}
%\end{figure}


%\subsubsection{Low-Temperature Calculations}
We also calculate the PL of Ge, Ge$_{0.94}$Sn$_{0.06}$, and Ge$_{0.88}$Sn$_{0.12}$ at low temperature (75K), where phonon absorption is minimal. At such low temperatures, the photon distribution function $\omega^3/[exp\{(\hbar\omega-\Delta\mu)/k_BT\}-1]$ in Eq. \ref{eq:PL} becomes sharply peaked near the band edge, significantly suppressing higher-energy transitions. As a result, PL features associated with the direct bandgap in both Ge and indirect-gap GeSn are strongly diminished. The emission in GeSn becomes dominated by a single peak corresponding to its absolute bandgap. Notably, at 75K, the PL of Ge$_{0.88}$Sn$_{0.12}$ is up to $20$ times stronger than that of pure Ge. These results are in excellent agreement with the calculations on Ge provided by Tiwari et al \cite{tiwari2024unified} and experimental observations reported in Ref. \cite{ghetmiri2014direct}.


\begin{figure}[H]
    \centering
    \includegraphics[width=0.95\linewidth]{Result3-Abs/Figures/PL75.png}
    \caption{PL of Ge (black), Ge$_{0.96}$Sn$_{0.06}$ (blue) and Ge$_{0.88}$Sn$_{0.12}$ (pink) in arbitrary units at $75K$, plotted against photon energy.}
    \label{fig:PL75}
\end{figure}



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

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


In conclusion, we calculate the absorption coefficient and photoluminescence of GeSn as a function of photon energy $\hbar\omega$ (and light wavelength $\lambda$), employing both degenerate and non-degenerate perturbation theory to include all direct and indirect transitions. This study presents the first quantitative treatment of the optoelectronic properties of an alloy that explicitly incorporates indirect transitions enabled by alloy scattering. Our calculations focus on the short-wavelength infrared spectral range, considering only direct transitions between the conduction band minimum and the three uppermost valence bands around the $\Gamma$ point. All relevant band energies and momentum matrix elements were obtained using DFT, with the band energies corrected by GW and for temperature.

% abs
Our results demonstrate a significant increase in absorption with increasing Sn concentration in GeSn. The room-temperature absorption coefficient increases by up to an order of magnitude relative to pure Ge, reaching values exceeding $10^5\ $cm$^{-1}$ in the SWIR light range for direct-gap GeSn alloys. Reducing temperature reduces the absorption coefficient. In this light range, we find that optical absorption is primarily governed by direct transitions around the $\Gamma$ point, with alloy scattering contributing only modestly, particularly in Ge and indirect-gap GeSn where electrons can be absorbed into the four $L$ bands. 

% PL
Adding Sn to Ge also results in an increase in the emission spectra in the SWIR light range. In contrast to absorption however, alloy scattering plays a much more significant role in light emission. In both pure Ge and indirect-gap GeSn, the PL spectrum exhibits a peak at the direct band edge and a stronger peak at the indirect gap. In GeSn, the intensity of the indirect peak is largely driven by alloy-assisted recombination, which dominates over phonon-mediated processes. For direct-gap GeSn, the PL spectrum shows a single, pronounced peak at the band edge. At room temperature, the PL intensity of Ge$_{0.88}$Sn$_{0.12}$ exceeds that of Ge by a factor of $250$ in the SWIR range. At lower temperatures, the PL peaks become narrower, with higher energy transitions suppressed by the Bose-Einstein function. At 75K, the peak of the PL of Ge$_{0.88}$Sn$_{0.12}$ is over $20$ times the Ge value. Our findings are in excellent agreement with experiment.

We predict that applying biaxial tensile strain could further enhance both the absorption and emission in GeSn by modifying the band structure in favour of direct transitions. 

Additionally, we expect electron–hole (e-h) excitonic interactions to boost optical response by slightly lowering transition energies. However, they should have only a minor influence on the optical response, as such effects are relatively weak ($\approx meV$) in bulk materials and become significant mainly under strong confinement. 

Given our findings, coupled with its ability to be grown directly on Si substrates, makes GeSn a possible candidate to replace III-V materials in optoelectronic applications, paving the way for the integration of photonic components into established CMOS technology.