\chapter{Optimisation of optical absorption in strain-balanced InAs/InAs$_{1-x}$Sb$_{x}$ superlattices} \label{ch:chapter5}
\lhead{\emph{Optimisation of optical absorption in strain-balanced InAs/InAs$_{1-x}$Sb$_{x}$ superlattices}}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% Section 1: Overview %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Overview}
\label{sec:ch_5_overview}

In this Chapter we present a systematic optimisation of InAs/InAs$_{1-x}$Sb$_{x}$ strain-balanced superlattices (SBSLs) for 3 - 5 $\mu$m photodetection, based on rigorous calculations of the optoelectronic properties using the plane wave \textbf{k}$\cdot$\textbf{p} computational framework outlined in Chapter~\ref{ch:chapter2}. There exists significant demand to develop inexpensive and efficient emitters and photodetectors in the mid-infrared (mid-IR) spectral range for sensing applications in environmental, medical, agricultural, industrial and defence domains \cite{Wang_OSA_2016,Wang_Small_2019,Alshahrani_ADPR_2022}. Existing technologies for these applications are largely based on II-VI Hg$_{1-x}$Cd$_{x}$Te materials which possess several limitations including high fabrication cost, required cooling to cryogenic temperatures, high non-radiative recombination rates, and large dark current due to band-to-band tunneling at high temperature \cite{Alshahrani_ADPR_2022,Delli_ACSP_2019,Downs_Sens_2013,Razeghi_RPP_2014}. Beyond these limitations Hg$_{1-x}$Cd$_{x}$Te materials also pose significant problems for the environment as their constituent materials are toxic, posing challenges for safe disposal \cite{Alshahrani_ADPR_2022}. Type-II III-V SLs are an emerging platform for applications as mid-IR detectors and emitters since they offer, via their highly tunable electronic structure, the opportunity to deliver high-performance devices with simplified fabrication and lower cost \cite{Delli_ACSP_2019,Delli_APL_2020,Alshahrani_ADPR_2022,Bouschet_AS_2022,Downs_Sens_2013,Razeghi_RPP_2014}. SBSLs are desirable for these applications, since strain compensation facilitates growth of thick (several $\mu$m) SL absorber layers possessing low defect densities due to the suppression of dislocations associated with strain relaxation \cite{Lackner_JAP_2012}. InAs/InAs$_{1-x}$Sb$_{x}$ SBSLs have attracted significant interest for several reasons. Firstly, a Ga-free design simplifies growth \cite{Alshahrani_ADPR_2022} and mitigates the deleterious impact of chemically inhomogeneous interfaces between material layers, which can impede performance via the formation of localised electronic defect states \cite{Steenbergen_APL_2011,Steenbergen_JVSTB_2012,Olson_APL_2012,Bader_APL_2022,Razeghi_RPP_2014}. Secondly, in addition to having a highly tunable band gap, InAs/InAs$_{1-x}$Sb$_{x}$ SLs possess type-II band offsets which reduces the non-radiative (Auger-Meitner) recombination rate \cite{Zegrya_APL_1995,Aytac_JAP_2016}. Additionally, integration of InAs/InAs$_{1-x}$Sb$_{x}$ SLs on Si has been achieved via growth on GaSb-on-Si buffer layers \cite{Delli_ACSP_2019,Delli_APL_2020}, and high performance has been demonstrated in prototype photodetectors \cite{Wu_AIPA_2020,Wu_IPT_2020}.

Previous analysis of InAs/InAs$_{1-x}$Sb$_{x}$ SBSLs has often focused on the zone-centre electron-hole spatial overlap associated with the SL band gap as a figure of merit \cite{Lackner_JAP_2012,Webster_APL_2015,Manyk_RP_2018,Ting_JEM_2020,Petticrew_SST_2023}. While such analysis has provided valuable guidance, the associated consideration of the electronic and optical properties is limited. Considering a single transition at a single energy neglects that photodetection generally encompasses a range of wavelengths, within which the in-plane ($\textbf{k}_{\parallel}$) band dispersion, miniband ($q$) dispersion and $( \textbf{k}_{\parallel}, q )$-dependent optical matrix elements all play important roles in determining the optical absorption \cite{Livneh_PRB_2012,Klipstein_JEM_2014}. For a defined cut-off wavelength we uniquely identify a SBSL at each Sb composition $x$ by imposing strain balancing to constrain the InAs and InAs$_{1-x}$Sb$_{x}$ layer thicknesses. Following this procedure there are two primary design regimes: (i) thicker layers at lower Sb composition, and (ii) thinner layers at higher Sb composition. We extend the formalism described in Chapter~\ref{ch:chapter4}, including the renormalisation of the Kane parameter to mitigate spurious solutions, and compute the optical absorption spectrum. The SL absorption spectra are used in a model calculation of the internal quantum efficiency (IQE), which is employed as a figure of merit to quantify photodetector performance. This constitutes a detailed interrogation of the InAs/InAs$_{1-x}$Sb$_{x}$ SBSL design space, quantitatively comparing the endpoints (i) thicker layers at lower Sb composition, and (ii) thinner layers at higher Sb composition for SBSL design. We demonstrate that increasing the Sb composition to $x \approx 30$\% can increase the (relative) IQE at fixed cut-off wavelength by $\approx 15$\%, with minimal additional improvement in performance expected at higher Sb compositions.

\begin{sloppypar}\tolerance 900
The rest of this Chapter is organised as follows. In Sec.~\ref{sec:ch_5_elec_structure} we identify and analyse InAs/InAs$_{1-x}$Sb$_{x}$ SBSLs at fixed cut-off wavelength and describe their associated electronic properties. In Sec.~\ref{sec:ch_5_elec_structure_identification} we describe the procedure we applied to identify InAs/InAs$_{1-x}$Sb$_{x}$ SBSLs having fixed target cut-off wavelength. We also quantify properties relevant to the epitaxial growth of InAs/InAs$_{1-x}$Sb$_{x}$ SBSLS, including the in-plane strain, strain-balanced layer thicknesses, and critical thickness. In Sec.~\ref{sec:ch_5_elec_structure_carriers_bands} we describe the electronic properties of InAs/InAs$_{1-x}$Sb$_{x}$ SBSLs and outline the impact of carrier localisation and the electronic band structure on the optical (momentum) matrix elements, and their implictions for the optical absorption. In Sec.~\ref{sec:ch_5_optical_abs} we describe the trends in optical properties of InAs/InAs$_{1-x}$Sb$_{x}$ SBSLs pertinent to mid-IR photodetection. In Sec.~\ref{sec:ch_5_optical_abs_spectra_vs_x} we investigate the trends in the optical absorption spectra of InAs/InAs$_{1-x}$Sb$_{x}$ SBSLs as a function of Sb composition $x$. In Sec.~\ref{sec:ch_5_optical_abs_opt_iqe} we quantify the potential to optimise the photodetector performance in the 3 - 5 $\mu$m wavelength range by computing the IQE, and offer guidelines for epitaxial growth based on our calculations. In Sec.~\ref{sec:ch_5_optical_abs_opt_jdos_ome} we analyse contributions to the optical absorption at the level of the electronic structure to quantify the origin of the observed trends in the calculated IQE. Finally, in Sec.~\ref{sec:ch_5_conclusions} summarise our analysis, and guidelines for material growth, and conclude.
\end{sloppypar}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% Section 2: Electronic structure of SBSLs %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Structural and electronic properties}
\label{sec:ch_5_elec_structure}

In this section we present our framework for the identification of InAs/InAs$_{1-x}$Sb$_{x}$ SBSLs having fixed target cut-off wavelength $\lambda_{\text{cut}}$. Having established this framework we analyse the trends in the identified SBSLs and consider structural and elastic properties pertinent to the viability of epitaxial growth. Then, we investigate the electronic band structure and trends in carrier localisation and outline their expected impact on optical properties.

\subsection{Fixed cut-off wavelength strain-balanced superlattice identification}
\label{sec:ch_5_elec_structure_identification}

% Structure identification

We consider InAs/InAs$_{1-x}$Sb$_{x}$ SLs to be grown on a GaSb substrate. When grown pseudomorphically on a [001]-oriented GaSb substrate, InAs is under 0.62\% tensile strain in the (001) plane. We employ linear interpolation to compute the InAs$_{1-x}$Sb$_{x}$ lattice parameter, meaning we calculate that pseudomorphic InAs$_{1-x}$Sb$_{x}$ is lattice-matched to GaSb for $x = 8.9$\%, and is under tensile (compressive) strain for $x < 8.9$\% ($x > 8.9$\%). To obtain strain-balancing we therefore consider compressively strained InAs$_{1-x}$Sb$_{x}$ layers with Sb composition $x > 8.9$\% chosen to balance the tensile strain in the InAs layers. Formally, strain-balancing between two adjacent pseudomorphic layers occurs when they admit zero net stress in the plane perpendicular to the growth direction \cite{Ekins-Daukes_CGD_2002}. We denote via subscripts ``$0$'' and ``$x$'' properties associated respectively with the electron-confining InAs ($x = 0$) and hole-confining InAs$_{1-x}$Sb$_{x}$ layers. Using elastic continuum theory we construct strain-balanced SLs by enforcing strain-balancing between a tensile strained InAs layer thickness $t_{0}$ and a compressively strained InAs$_{1-x}$Sb$_{x}$ layer of thickness $t_{x}$, such that each two-layer SL period is strain-balanced.

The in-plane biaxial strain is defined as $\epsilon_{\parallel} = ( a( \text{GaSb} ) - a( \text{InAs}_{1-x}\text{Sb}_{x} ) )/a( \text{InAs}_{1-x}\text{Sb}_{x} )$, where $a$ is the unstrained lattice parameter. For a pseudomorphic strained layer under biaxial stress in the (001)-plane this then gives the stress along the (001) direction as $\sigma_{zz} = C_{12} ( \epsilon_{xx} + \epsilon_{yy} ) + C_{11} \epsilon_{zz} = 2 C_{12} \epsilon_{\parallel} + C_{11} \epsilon_{\perp}$. Setting $\sigma_{zz} = 0$ yields $\epsilon_{\perp} \equiv \epsilon_{zz} = -2C_{12} \epsilon_{\parallel} / C_{11}$, where $C_{ij}$ denote the material elastic constants. We can then express the elastic energy per unit volume $U$ as \cite{Landau_book_1986,Ekins-Daukes_CGD_2002,ODonnell_thesis_2021}

\begin{equation}
    U = \frac{ C_{11}^{2} + C_{11} C_{12} - 2 C_{12}^{2} }{ C_{11} } \, \epsilon_{\parallel}^{2} \, .
    \label{eq:ch_5_elastic_energy_per_unit_V}
\end{equation}

For two strained layers, the average strain energy density -- for a pair of strained layers of thickness $t_{0}$ and $t_{x}$ with SL period $L = t_{0} + t_{x}$ -- is given by \cite{Ekins-Daukes_CGD_2002}

\begin{equation}
    U_{\text{av}} = \frac{ U_{0} t_{0} + U_{x} t_{x} }{ t_{0} + t _{x} } = \frac{ A_{0} \epsilon_{\parallel,0}^{2} t_{0} + A_{x} \epsilon_{\parallel,x}^{2} t_{x} }{ t_{0} + t_{x} } \, ,
    \label{eq:ch_5_average_strain_density}
\end{equation}

%\begin{eqnarray}
%    U_{\text{av}} &=& \frac{ U_{0} t_{0} + U_{x} t_{x} }{ t_{0} + t _{x} } \nonumber \\
%    &=& \frac{ A_{0} \epsilon_{\parallel,0}^{2} t_{0} + A_{x} \epsilon_{\parallel,x}^{2} t_{x} }{ t_{0} + t_{x} } \, , \label{eq:ch_5_average_strain_density}
%\end{eqnarray}

\noindent
where $A = C_{11} + C_{12} - 2 C_{11}^{-1} C_{12}^{2}$. We can derive the average in-plane stress from the derivative of the average energy density with respect to the strain in a given layer. The strain-balancing condition mandates that this quantity vanishes -- i.e. $ \frac{ \partial U_{\text{av}} }{ \partial \epsilon_{\parallel , x} } = \frac{ \partial U_{\text{av}} }{ \partial \epsilon_{\parallel , 0} } = 0 $. Applying this condition to Eq.~\eqref{eq:ch_5_average_strain_density} gives, for each individual SL period, the strain-balancing condition such that there is zero net stress in the (001) plane as \cite{Ekins-Daukes_CGD_2002,Broderick_SR_2017}

\begin{equation}
    \frac{ t_{x} }{ t_{0} } = - \frac{ A_{0} \, \epsilon_{\parallel,0} \, a_{x} }{ A_{x} \, \epsilon_{\parallel,x} \, a_{0} } \, .
    \label{eq:ch_5_strain_balancing_condition}
\end{equation}

Given a target optical absorption cut-off wavelength $\lambda_{\text{cut}}$, which we treat as being equal to the SBSL zone-centre ($\textbf{k}_{\parallel} = q = 0$) band gap \cite{Petticrew_SST_2023}, we identify candidate SBSLs as follows. For a given Sb composition $x$ we choose an InAs layer thickness $t_{0}$ and solve Eq.~\eqref{eq:ch_5_strain_balancing_condition} to obtain the InAs$_{1-x}$Sb$_{x}$ layer thickness $t_{x}$ that imposes strain balancing. This combination of $x$, $t_{0}$ and $t_{x}$ then fully specifies a candidate SBSL, for which we calculate the zone-centre band gap between the lowest-energy bound electron state $e1$ and highest-energy bound hole state, the latter possessing purely heavy-hole (HH) Bloch character at $\textbf{k}_{\parallel} = q = 0$ and which we therefore label as $hh1$. To determine the Bloch band character of a given SL eigenstate we note that the calculated SL eigenstates posess eight envelope function components, each associated with one of the zone-centre Bloch basis states employed in the underlying bulk $\textbf{k} \cdot \textbf{p}$ Hamiltonian (cf.~Chapter~\ref{ch:chapter2}). The Bloch character can then be evaluated by projecting the SL eigenstate $ \vert \psi_{n \textbf{k}} \left( z \right) \rangle $ onto the bulk basis states $ \vert u_{b} \rangle $ (cf.~Eq.~\eqref{eq:ch_2_bloch_band_character}).

Repeating this procedure as a function of $t_{0}$ at fixed $x$ yields the zone-centre $e1$-$hh1$ band gap vs.~$t_{0}$, from which we identify the SBSL having band gap $E_{g}$ corresponding to the desired $\lambda_{\text{cut}}$ -- i.e.~$E_{g} = hc / \lambda_{\text{cut}}$. As an example of this procedure, we present in Fig.~\ref{fig:ch_5_structure_identification_exemplar} the results of these calculations for a single Sb composition $x = 18$\% -- i.e.~the full set of candidate InAs/InAs$_{0.82}$Sb$_{0.18}$ SBSLs. In Fig.~\ref{fig:ch_5_structure_identification_exemplar}(a) we present the identified InAs$_{0.82}$Sb$_{0.18}$ thicknesses $t_{x}$ as a function of InAs thickness $t_{0}$, calculated using Eq.~\eqref{eq:ch_5_strain_balancing_condition}. In Fig.~\ref{fig:ch_5_structure_identification_exemplar}(b) we show the calculated $e1$-$hh1$ energy gap $E_{g}$ as a function of SL period $L$ for the candidate SBSLs detailed in Fig.~\ref{fig:ch_5_structure_identification_exemplar}(a).

\begin{figure}[t!]
	\centering
   \includegraphics[width=0.95\textwidth]{./figures/chapter5/figure_0.pdf}
	\caption{(a) Computed InAs$_{0.82}$Sb$_{0.18}$ layer thicknesses $t_{x}$ as a function of InAs layer thicknesses $t_{0}$, specifying identified InAs/InAs$_{0.82}$Sb$_{0.18}$ ($x = 18$\%) SBSLs computed via Eq.~\eqref{eq:ch_5_strain_balancing_condition}. (b) Calculated $e1$-$hh1$ energy gap $E_{g}$ as a function of SL period $L$ of the SBSLs detailed in (a).}
	\label{fig:ch_5_structure_identification_exemplar}
\end{figure}

The design space for InAs/InAs$_{1-x}$Sb$_{x}$ SLs is expansive, consisting of structures having varying InAs and InAs$_{1-x}$Sb$_{x}$ layer thicknesses $t_{0}$ and $t_{x}$ and Sb composition $x$. Imposing (i) fixed $\lambda_{\text{cut}}$, and (ii) strain balancing, significantly reduces the extent of this space by reducing a 3D optimisation problem in ($x, t_{0}, t_{x}$) to a 1D optimisation problem in $x$. Applying constraints (i) and (ii) allows us to identify a unique SBSL -- defined by $x$, $t_{0} (x)$ and $t_{x} (x)$ -- at each Sb composition $x$. In the first instance, we carry out analysis of SBSLs having $\lambda_{\text{cut}} = 5$ $\mu$m, calculated at a representative detector operating temperature $T_{\scalebox{0.7}{\text{D}}} = 150$ K. This choice of $\lambda_{\text{cut}}$ reflects the upper limit of the mid-IR atmospheric transmission window \cite{Henini_chapter_2002,Bouschet_AS_2022}. Moving $\lambda_{\text{cut}}$ to longer wavelengths beyond 5 $\mu$m is one approach employed to enhance photodetection in the mid-IR spectral range \cite{Zhou_APL_2019_2}. For this reason we also carry out analysis of InAs/InAs$_{1-x}$Sb$_{x}$ SBSLs having $\lambda_{\text{cut}} = 5.5$ and 6 $\mu$m. For operation at $T_{\scalebox{0.7}{\text{D}}} = 150$ K, test calculations indicate that it is not possible to achieve $\lambda_{\text{cut}} = 5$ $\mu$m for Sb composition $x < 18$\%, $\lambda_{\text{cut}} = 5.5$ $\mu$m for Sb composition $x < 20$\%, or $\lambda_{\text{cut}} = 6$ $\mu$m for Sb composition $x < 22$\%. We therefore restrict our attention to SBSLs having $x = 18$ - 40\%, $x = 20$ - 40\% and $x = 22$ - 40\% for $\lambda_{\text{cut}} = 5$, 5.5 and 6 $\mu$m respectively.

\begin{figure}[t!]
	\centering
   \includegraphics[width=0.90\textwidth]{./figures/chapter5/figure_1.pdf}
	\caption{(a) Calculated InAs (closed blue circles) and InAs$_{1-x}$Sb$_{x}$ (closed red squares) layer thicknesses for SBSLs having fixed cut-off wavelength $\lambda_{\text{cut}} = 5$ $\mu$m as a function of Sb composition $x$. (b) Equivalent to (a) for SBSLs having $\lambda_{\text{cut}} = 5.5$ $\mu$m. (c) Equivalent to (a) for SBSLs having $\lambda_{\text{cut}} = 6$ $\mu$m. (d) Calculated critical thickness $t_{c}$ (closed black circles and left-hand vertical axis) and compressive in-plane strain $\epsilon_{\parallel,x}$ (closed red squares and right-hand vertical axis) as a function of Sb composition $x$.}
	\label{fig:ch_5_layer_thickness_critical_thickness_strain}
\end{figure}

Following this procedure we systematically identify layer thicknesses for InAs/InAs$_{1-x}$Sb$_{x}$ SBSLs having $\lambda_{\text{cut}} = 5$, 5.5 and 6 $\mu$m ($e1$-$hh1$ band gap = 0.248, 0.225 and 0.207 eV). The resulting structures are summarised in Figs.~\ref{fig:ch_5_layer_thickness_critical_thickness_strain}(a), \ref{fig:ch_5_layer_thickness_critical_thickness_strain}(b) and \ref{fig:ch_5_layer_thickness_critical_thickness_strain}(c) respectively. The upper limit of the investigated Sb composition range, $x = 40$\%, was selected based on the fact that the (001) in-plane compressive strain $\vert \epsilon_{\parallel,x} \vert$ reaches a high value of 2.1\% at $x = 40$\% (cf.~Fig.~\ref{fig:ch_5_layer_thickness_critical_thickness_strain}(d)). As the Sb composition increases the layer thicknesses $t_{0}$ and $t_{x}$ required to maintain $\lambda_{\text{cut}}$ decrease, with the higher $e1$ and $hh1$ confinement energies in the thinner layers compensating for the reduced InAs$_{1-x}$Sb$_{x}$ bulk band gap. These results highlight the two primary design regimes: requiring thicker layers to reach the target $\lambda_{\text{cut}}$ at lower $x$, changing to thinner layers with increasing $x$ to retain both fixed $\lambda_{\text{cut}}$ and strain-balancing. Higher Sb compositions can pose challenges for epitaxial growth. For larger Sb composition $x$ strain-balancing mandates thin highly-strained layers, which make the optoelectronic properties susceptible to factors including minor variations in layer thickness and Sb intermixing at the interfaces \cite{Milosavljevic_JAP_2023}. For $\lambda_{\text{cut}} = 5$ $\mu$m, at $x = 40$\% we find SBSLs requiring InAs$_{1-x}$Sb$_{x}$ layers having $t_{x} < 1$ nm ($\lesssim 4$ monolayers). These concerns are lessened moving to longer $\lambda_{\text{cut}}$ since, at fixed Sb composition, thicker InAs and InAs$_{1-x}$Sb$_{x}$ layers are sufficient to maintain fixed $\lambda_{\text{cut}}$. At $x = 40$\% we find that InAs$_{1-x}$Sb$_{x}$ layer thicknesses $t_{x} = 1.32$ and 1.48 nm are respectively required to achieve $\lambda_{\text{cut}} = 5.5$ and 6 $\mu$m. These low layer thicknesses still likely pose challenges for expitaxial growth since the InAs$_{1-x}$Sb$_{x}$ layer thickness $t_{x}$ is less than several monolayers. Nonetheless, the work of Ref.~\cite{Bouschet_AS_2022} has demonstrated that the types of structures which we identify in this work are compatible with recently demonstrated epitaxial growth. The challenge for epitaxial growth will now become the reproducible growth of many SL periods with uniform low layer thickness and high Sb composition.

% Critical thickness

When considering epitaxial layer design to guide materials growth and inform device fabrication, it is important to consider the strain-thickness limits of the constituent materials. To ensure that these limits are not exceeded during epitaxial growth of InAs/InAs$_{1-x}$Sb$_{x}$ SBSLs, we calculate the critical thickness $t_{c}$ associated with our InAs and InAs$_{1-x}$Sb$_{x}$ layers. This is motivated by noting that our identified SBSLs include highly strained InAs$_{1-x}$Sb$_{x}$ layers, it is therefore pertinent to assess their compatibility with strain-thickness limits for growth on [001]-orientated GaSb. If an epitaxial layer is grown with thickness exceeding its critical thickness, it becomes energetically favourable to relieve strain via the formation of a dislocation (plastic lattice relaxation), rendering the strained layer thermodynamically unstable \cite{OReilly_SST_1989}. To obtain an expression for this critical thickness we begin by considering the elastic energy density $E_{\text{st}}$ of a dislocation-free pseudomorphically strained layer. We define the shear modulus as $G = \frac{1}{2} ( C_{11} - C_{12} )$ and Poisson's ratio as $\sigma$ $= \frac{ C_{12} }{ C_{11} + C_{12} }$. We then multiply Eq.~\eqref{eq:ch_5_elastic_energy_per_unit_V}, the bulk elastic energy density, by the layer thickness $t$ to obtain the elastic energy per unit area stored in a [001]-orientated pseudomorphically strained layer of thickness $t$, which gives \cite{OReilly_SST_1989,vanderMerwe_SS_1972,ODonnell_thesis_2021}

% J per m^2 - areal energy density
\begin{equation}
    E_{\text{st}} = 2 G \left(\frac{1+\sigma}{1-\sigma}\right) \epsilon_{\parallel , x}^{2} \, t \, .
    \label{eq:ch_5_elastic_strained_layer_energy}
\end{equation}

Following the approach of Voisin \cite{Voisin_PS_1988}, the energy per unit length associated with a plastic dislocation $E_{\text{dis}}$ can be evaluated as

%J m^-1 - per unit length
\begin{equation}
    E_{\text{dis}} = \frac{ G }{ 16 \pi } \left( \frac{ a }{ \sqrt{2} } \right) ^{2} \left( \frac{ 4 - \sigma }{ 1 - \sigma } \right) \left(1+\ln \left(\frac{ \sqrt{2} \, t }{ a }\right)\right) \, ,
    \label{eq:ch_5_elastic_dislocation_energy}
\end{equation}

\noindent
where $a$ is the substrate lattice constant. The critical thickness occurs when Eqs.~\eqref{eq:ch_5_elastic_strained_layer_energy} and \eqref{eq:ch_5_elastic_dislocation_energy} are equal, with scaling of the former with the average spacing between dislocations $s = a / \left( \sqrt{2} ~ \vert \epsilon_{\parallel,x} \vert \right)$ \cite{Fitzgerald_MSR_1991}. The critical thickness of a strained layer can then be determined as \cite{Voisin_PS_1988,OReilly_SST_1989,Tomic_IEEE_2003,Broderick_SR_2017,ODonnell_thesis_2021}

\begin{equation}
    t_{c} = \frac{ a }{8 \sqrt{2} \pi \left|\epsilon_{\parallel,x}\right|}\left(\frac{4-\sigma}{1+\sigma}\right)\left(1+\ln \left(\frac{ \sqrt{2} \, t_{c} }{ a }\right)\right) \, .
    \label{eq:ch_5_critical_thickness}
\end{equation}

% $C_{12} \approx C_{44} \approx C_{11} / 2$, and so

For tetrahedral semiconductors $C_{12} \approx C_{11} / 2$, and so the Poisson ratio is $\sigma \approx 1/3$, this is the value we use in our calculations of $t_{c}$ \cite{OReilly_SST_1989}. A number of approaches exist for computing $t_{c}$, however the results from the method that we have outlined above are in good agreement with experimental measurements for a number of semiconductor materials \cite{OReilly_SST_1989,Voisin_PS_1988,Tomic_IEEE_2003,Broderick_SR_2017,ODonnell_thesis_2021}. Equation \eqref{eq:ch_5_critical_thickness} is an implicit equation in $t_{c}$, from which $t_{c}$ can be computed numerically via any standard root-finding algorithm. To assess the practical viability of our identified InAs/InAs$_{1-x}$Sb$_{x}$ SBSLs we calculate the InAs$_{1-x}$Sb$_{x}$/GaSb critical thickness by solving Eq.~\eqref{eq:ch_5_critical_thickness}. The calculated InAs/GaSb critical thickness is 41.94 nm, corresponding to a strain-thickness limit $t_{c} \vert \epsilon_{\parallel} \vert = 41.94$ nm $\times \, 0.62\% = 26$ nm \%. For the InAs$_{1-x}$Sb$_{x}$ layers the critical thickness is a function of the Sb composition $x$, since the composition determines the values of the lattice parameters, in-plane strain and elastic moduli that enter Eq.~\eqref{eq:ch_5_critical_thickness}. The calculated $t_{c} ( x )$ for InAs$_{1-x}$Sb$_{x}$/GaSb is shown in Fig.~\ref{fig:ch_5_layer_thickness_critical_thickness_strain}(d), alongside the composition dependent in-plane compressive strain $\epsilon_{\parallel , x}$. The identified InAs SBSL layer thicknesses $t_{0}$ for $\lambda_{\text{cut}} = 5$, 5.5 and 6 $\mu$m are $\leq 25$\% of the InAs critical thickness, a result of the low tensile strain in InAs/GaSb. The identified InAs$_{1-x}$Sb$_{x}$ SBSL layer thicknesses $t_{x}$ for $\lambda_{\text{cut}} = 5$, 5.5 and 6 $\mu$m at $x = 40$\% are $\leq 38$, 44 and 50\% of the corresponding critical thickness respectively. We therefore expect the identified SBSLs to lie safely within the expected strain-thickness limits across the investigated composition range, and hence to be compatible with established epitaxial growth \cite{Bouschet_OR_2023}.


\subsection{Carrier localisation and band structure}
\label{sec:ch_5_elec_structure_carriers_bands}

In Fig.~\ref{fig:ch_5_offsets_bands}(a) we show the calculated conduction band (CB) and HH valence band (VB) offsets, and the $e1$ and $hh1$ probability densities at $\textbf{k}_{\parallel} = q = 0$, for the identified InAs/InAs$_{0.70}$Sb$_{0.30}$ ($x = 30$\%) SBSL having $\lambda_{\text{cut}} = 5$ $\mu$m. The zero of energy is set at the VB maximum of unstrained InAs. Our calculations employ the unstrained InSb/InAs VB offset recommended by Vurgaftman et al.~\cite{Vurgaftman_JAP_2001} of 0.59 eV. Our calculations predict a smaller strained, with respect to an [001]-orientated GaSb substrate, InSb/InAs VB offset of 0.49 eV. We note, consistent with literature reports \cite{Bouschet_AS_2022,Bouschet_OR_2023,Delli_ACSP_2019}, that our model solid theory calculation predicts type-II band alignment, which generates spatially indirect confinement of the electron and hole envelope functions. As $x$ increases we see the type-II band alignment begin to approach type-III (broken-gap) alignment, however this transition takes place beyond the Sb composition range investigated in this work. Our model solid theory calculations predict a type-II to type-III band alignment transition at $x \approx 43$\%. While we observe the $hh1$ state to be strongly localised in the InAs$_{1-x}$Sb$_{x}$ layers, we note that the weakly bound $e1$ state is significantly delocalised. This delocalisation of the $e1$ state is a result of the shallow CB offset, low InAs CB edge effective mass, and the low InAs$_{1-x}$Sb$_{x}$ layer thickness. This behaviour holds for all the identified SBSLs across the investigated range of Sb composition and cut-off wavelengths. We note that as $x$ increases, and the corresponding layer thicknesses decrease (cf.~Fig.~\ref{fig:ch_5_layer_thickness_critical_thickness_strain}), we expect increasing spatial overlap of the zone-centre $e1$ and $hh1$ envelope functions. This increasing spatial overlap will act to enhance the inter-band momentum matrix elements, and hence increase the optical absorption close in energy to the band edge.

\begin{figure}[t!]
	\centering
   \includegraphics[width=0.95\textwidth]{./figures/chapter5/figure_2.pdf}
	\caption{(a) Calculated CB (red) and VB (blue) offsets, illustrated for three periods of an InAs/InAs$_{0.70}$Sb$_{0.30}$ ($x = 30 \%$) SBSL with $\lambda_{\text{cut}} = 5$ $\mu$m having respective InAs and InAs$_{1-x}$Sb$_{x}$ layer thicknesses $t_{0} = 3.57$ nm and $t_{x} = 1.67$ nm (SL period $L = t_{0} + t_{x} = 5.24$ nm). The dash-dotted red (blue) line denotes the energy of the lowest energy bound electron state $e1$ (highest bound hole state $hh1$). The corresponding $e1$ ($hh1$) probability density at $\textbf{k}_{\parallel} = q = 0$ is shown using a solid red (blue) line, shifted so that the zero of probability density lies at the $e1$ ($hh1$) energy. (b) Calculated band structure and density of states (DOS) at temperature $T_{\scalebox{0.7}{\text{D}}} = 150$ K for the SBSL of (a). The left-hand, centre and right-hand panels respectively show the calculated miniband ($q$) dispersion, in-plane ($\textbf{k}_{\parallel}$) band dispersion and DOS.}
	\label{fig:ch_5_offsets_bands}
\end{figure}

In Fig.~\ref{fig:ch_5_offsets_bands}(b) we present the calculated SL electronic structure, consisting of the miniband dispersion ($q$) at $\textbf{k} = 0$ and the in-plane dispersion ($\textbf{k}_{\parallel}$) at $q = 0$, in addition to the density of states (DOS) for the SBSL of Fig.~\ref{fig:ch_5_offsets_bands}(a). The zero of energy is as in Fig.~\ref{fig:ch_5_offsets_bands}(a). In the miniband dispersion we note a highly dispersive $e1$ and nearly dispersionless $hh1$ miniband. The nature of the strongly delocalised $e1$ and strongly localised $hh1$ states are responsible for this behaviour with, e.g., the weakly bound $e1$ state possessing strong spatial overlap with its periodic image in other SL periods, leading to strong $e1$ miniband dispersion \cite{Murphy_JPDAP_2023}. For the full suite of identified SBSLs (cf.~Fig.~\ref{fig:ch_5_layer_thickness_critical_thickness_strain}), our calculations show that for increasing $\lambda_{\text{cut}}$ CB (VB) states move closer in energy to the CB (VB) edge. This is a result of the increasing layer thicknesses, and hence SL period, which act to confine a greater number of states (cf.~Sec.~\ref{sec:ch_5_optical_abs_opt_jdos_ome}) while also reducing the confinement energy of each bound state. Our calculated in-plane and miniband dispersion are in quantitative agreement with similar calculations performed for comparable InAs/InAs$_{1-x}$Sb$_{x}$ SLs \cite{Krizman_JAP_2021,Bouschet_OR_2023}.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% Section 3: Trends in optical absorption %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Analysis of optical properties for photodetection}
\label{sec:ch_5_optical_abs}

In this section we present our calculated optical absorption spectra for the InAs/InAs$_{1-x}$Sb$_{x}$ SBSLs described in Sec.~\ref{sec:ch_5_elec_structure}. In analysing the trends in optical absorption we identify the potential for optimisation of photodetector performance as a function of Sb composition $x$. Then, for the 3 - 5 $\mu$m mid-IR window, we compute the IQE to determine to what extent photodetection can be optimised, and in which Sb composition range it is then predicted to be favourable to target growth. Finally, we analyse the SL electronic structure to elucidate the factors that to and dominate the calculated trends in IQE.

\subsection{Impact of varying Sb composition on optical absorption}
\label{sec:ch_5_optical_abs_spectra_vs_x}

Following the same procedure previously outlined to calculate the spontaneous emission spectrum including the contribution of the miniband dispersion (cf.~Chapter~\ref{ch:chapter4}), we calculate the band-to-band optical absorption spectrum including $q$ dependence in the same manner. Using our calculated SL eigenstates we compute the band-to-band optical absorption spectrum as \cite{Chuang_book_2009,Li_PRB_2010}

%\begin{eqnarray}
%
%    \alpha^{ ( \widehat{e} ) } ( \hbar \omega ) &=& \frac{ \pi e^{2} }{ \epsilon_{0} m_{0}^{2} c n_{r} \omega L } \sum_{ n_{c}, n_{v} } \left( \frac{ 2 \pi }{ L } \right)^{-1} \int_{ - \frac{ \pi }{ L } }^{ + \frac{ \pi }{ L } } \text{d} q \int \frac{ \text{d} \textbf{k}_{\parallel} }{ ( 2 \pi )^{2} } \nonumber \\
%
%    &\times& \vert \widehat{e} \cdot \textbf{p}_{ n_{c} n_{v} \scalebox{0.7}{\textbf{k}} } \vert^{2} \, \delta ( E_{ n_{c} \scalebox{0.7}{\textbf{k}} } - E_{ n_{v} \scalebox{0.7}{\textbf{k}} } - \hbar \omega ) \, , \label{eq:ch_5_optical_absorption} 
%
%\end{eqnarray}

\begin{equation}
    \alpha^{ ( \widehat{e} ) } ( \hbar \omega ) = \frac{ \pi e^{2} }{ \epsilon_{0} m_{0}^{2} c n_{r} \omega L } \sum_{ n_{c}, n_{v} } \left( \frac{ 2 \pi }{ L } \right)^{-1} \int_{ - \frac{ \pi }{ L } }^{ + \frac{ \pi }{ L } } \text{d} q \int \frac{ \text{d} \textbf{k}_{\parallel} }{ ( 2 \pi )^{2} } \vert \widehat{e} \cdot \textbf{p}_{ n_{c} n_{v} \scalebox{0.7}{\textbf{k}} } \vert^{2} \, \delta ( E_{ n_{c} \scalebox{0.7}{\textbf{k}} } - E_{ n_{v} \scalebox{0.7}{\textbf{k}} } - \hbar \omega ) \, ,
    \label{eq:ch_5_optical_absorption}
\end{equation}

% This choice is made as existing SL photodetectors and recent device development has focused on photodetection via edge absorption in device structures grown on [001]-orientated substrates \cite{Alshahrani_ADPR_2022,Bouschet_AS_2022}.

\noindent
where $\hbar \omega$ is the photon energy, $L$ is the SL period, $E_{ n_{c} (n_{v}) \scalebox{0.7}{\textbf{k}} }$ is the energy of conduction subband $n_{c}$ (valence subband $n_{v}$) at wave vector $\textbf{k} = ( \textbf{k}_{\parallel}, q )$, and $\textbf{p}_{ n_{c} n_{v} \scalebox{0.7}{\textbf{k}} }$ is the inter-band momentum matrix element between subbands $n_{c}$ and $n_{v}$ at wave vector \textbf{k} (cf.~Eq.~\eqref{eq:ch_2_momentum_matrix_element}). $\textbf{k}_{\parallel}$ is the wave vector in the (001) plane, while $q$ describes the miniband dispersion along [001] (cf.~Sec.~\ref{sec:ch_4_model_energy_temp_dependence}) \cite{Murphy_JPDAP_2023}. The unit vector $\widehat{e}$ denotes the photon polarisation, which we consider to be transverse electric (TE; $\widehat{e} = \widehat{x}$), i.e.~incident normal to the growth plane of the SBSL. The calculation of the inter-band momentum matrix elements $\textbf{p}_{ n_{c} n_{v} \scalebox{0.7}{\textbf{k}} }$ follows the procedure outlined in Secs.~\ref{sec:ch_4_model_energy_temp_dependence} and \ref{sec:ch_2_pwem_8-band_inclusion}. As done in Sec.~\ref{sec:ch_4_model_energy_temp_dependence} (cf.~Eq.~\ref{eq:ch_4_se_rate}), for numerical evaluation we replace the Dirac delta distribution in Eq.~\eqref{eq:ch_5_optical_absorption} by a hyperbolic secant lineshape \cite{Marko_SR_2016} of width $\delta =$ 4 meV \cite{Murphy_JPDAP_2023}.

In Fig.~\ref{fig:ch_5_absorption_spectra} we show the calculated TE-polarised ($\widehat{e} = \widehat{x}$) optical absorption in the 3 - 6.5 $\mu$m range for a selection of the $\lambda_{\text{cut}} = 5$, 5.5 and 6 $\mu$m SBSLs of Figs.~\ref{fig:ch_5_layer_thickness_critical_thickness_strain}(a), \ref{fig:ch_5_layer_thickness_critical_thickness_strain}(b) and \ref{fig:ch_5_layer_thickness_critical_thickness_strain}(c). We note excellent quantitative agreement with the measured absorption of Ref.~\cite{Bouschet_OR_2023} for an InAs/InAs$_{0.65}$Sb$_{0.35}$ SL having equivalent layer thicknesses and $\lambda_{\text{cut}} = 5$ $\mu$m. Our calculations across the investigated range of $\lambda_{\text{cut}}$ predict an increase in the optical absorption with increasing $x$. For $\lambda_{\text{cut}} = 5$ $\mu$m, beginning at $x = 18$\% we see an increase in optical absorption up to $x \approx 30$\%, beyond which there is minimal additional increase up to $x = 40$\%. In the 3 - 5 $\mu$m wavelength range, for SBSLs designed to have longer $\lambda_{\text{cut}}$ ($= 5.5, 6$ $\mu$m) our calculations indicate enhanced scope to the increase the optical absorption for Sb compositions $x > 30$\%. However, for these longer wavelength SBSLs we note that the rate at which the magnitude of the optical absorption increases with increasing $x$ still begins to reduce for $x > 30$\%. These results suggest the possibility to optimise photodetector performance in a given target wavelength range. It is to that optimisation that we now turn our attention.

\begin{figure}[H]
	\centering
   \includegraphics[width=0.46\textwidth]{./figures/chapter5/figure_3.pdf}
	\caption{(a) Calculated optical absorption spectra $\alpha ( \lambda )$ for InAs/InAs$_{1-x}$Sb$_{x}$ SBSLs with $\lambda_{\text{cut}} = 5$ $\mu$m having Sb compositions $x = 18$, 22, 28 and 38\% (darkening red lines), at temperature $T_{\scalebox{0.7}{\text{D}}} = 150$ K. (b) and (c) are as in (a), but for respective cut-off wavelengths $\lambda_{\text{cut}} = 5.5$ and 6 $\mu$m, and for Sb compositions $x = 20$, 24, 30 and 40\% and $x = 22$, 26, 30 and 40\%.}
	\label{fig:ch_5_absorption_spectra}
\end{figure}

\subsection{Optimising optical absorption and internal quantum efficiency}
\label{sec:ch_5_optical_abs_opt_iqe}

When assessing photodetector performance there are several figures of merit that can be employed. These metrics are, in general, device-based and include, e.g. responsivity, normalised or specific detectivity, and noise-equivalent power \cite{Datskos_EOPE_2016}. Such figures of merit typically assess full device performance, with their calculation requiring definition of a full device structure and knowledge of parameters including the root mean square detector noise and signal bandwidth \cite{Datskos_EOPE_2016}. Our aim in this theoretical analysis is to provide general guidance by quantifying the performance of an InAs/InAs$_{1-x}$Sb$_{x}$ SBSL photodetector in a generic manner that is independent of the surrounding device structure. As such, we choose a figure of merit that quantifies the performance of the SL absorber region in isolation. For this purpose, we compute and analyse the IQE. Our evaluation of the IQE is based on using the calculated absorption spectrum for a given SL structure to compute the photocurrent generated by the absorption of photons in an an energy range $[ \hbar \omega_{\text{min}}, \hbar \omega_{\text{max}} ]$. This model calculation does not consider transport of photogenerated carriers within a device structure and, as such, corresponds to the idealised limit of long carrier diffusion lengths.

The spectral photocurrent density generated by absorption of photons of energy $\hbar \omega$, incident normal to the growth plane of the SBSL absorber from a radiating blackbody source, is \cite{Nelson_Imperial_2003}

\begin{equation}
    \frac{ \text{d} J }{ \text{d} ( \hbar \omega ) } = e \, [ 1 - R( \hbar \omega ) ] \, a ( \hbar \omega, T_{\scalebox{0.7}{\text{D}}} ) \, b ( \hbar \omega, T_{\scalebox{0.7}{\text{S}}} )  \, ,
    \label{eq:ch_5_spectral_photocurrent_density}
\end{equation}

\noindent
where $R$ is the surface reflectivity, $a$ is the SBSL (photodetector) absorptivity, and $b$ is the spectral photon flux of a radiating blackbody source at temperature $T_{\scalebox{0.7}{\text{S}}}$, with

\begin{eqnarray}
%
    a ( \hbar \omega, T_{\scalebox{0.7}{\text{D}}} ) &=& 1 - \exp \left( - \alpha ( \hbar \omega ) d \right) \, , \label{eq:ch_5_absorptivity} \\
%
    b ( \hbar \omega, T_{\scalebox{0.7}{\text{S}}} ) &=& \frac{ 2 \pi }{ h^{3} c^{2} } \, \frac{ ( \hbar \omega )^{2} }{ \exp \left( \hbar \omega / k_{\scalebox{0.7}{\text{B}}} T_{\scalebox{0.7}{\text{S}}} \right) - 1 } \, , \label{eq:ch_5_blackbody_spectral_photon_flux}
%
\end{eqnarray}

\noindent
where $\alpha ( \hbar \omega )$ is the calculated SBSL absorption at photon energy $\hbar \omega$ and detector operating temperature $T_{\scalebox{0.7}{\text{D}}}$ (cf.~Eq.~\eqref{eq:ch_5_optical_absorption}), and $d$ is the total thickness of the SBSL absorber (which, in practice, typically consists of several hundred SL periods). Assuming zero reflection of photons impinging on the SBSL, $R ( \hbar \omega ) = 0$, integrating Eq.~\eqref{eq:ch_5_spectral_photocurrent_density} yields the photocurrent density $J$ generated by absorbing photons having energies between $\hbar \omega_{\text{min}}$ and $\hbar \omega_{\text{max}}$ in an absorber of thickness $d$ as

\begin{equation}
    J (d) = \frac{ 2 \pi e }{ h^{3} c^{2} } \int_{ \hbar \omega_{\text{min}} }^{  \hbar \omega_{\text{max}} } \frac{ ( \hbar \omega )^{2} \, \left[ 1 - \exp \left( - \alpha ( \hbar \omega ) d \right) \right] }{ \exp \left( \hbar \omega / k_{\scalebox{0.7}{\text{B}}} T_{\scalebox{0.7}{\text{S}}} \right) - 1 } \, \text{d} ( \hbar \omega ) \, .
    \label{eq:ch_5_photocurrent_density}
\end{equation}

\noindent
We assume a typical photodetector operating temperature $T_{\scalebox{0.7}{\text{D}}} = 150$ K -- employed in the electronic structure calculations whose eigenstates are used to evaluate Eq.~\eqref{eq:ch_5_optical_absorption} -- for the detection of photons emitted by a blackbody at ambient temperature $T_{\scalebox{0.7}{\text{S}}} = 300$ K. Using our calculated SBSL absorption spectra $\alpha ( \hbar \omega )$ as input to Eq.~\eqref{eq:ch_5_photocurrent_density}, we compute $J(d)$ for $\lambda_{\text{cut}} = 5$, 5.5 and 6 $\mu$m and absorber thickness up to $d = 5$ $\mu$m (corresponding to SBSLs containing $\approx$ 170 - 1,100 periods), for absorption between 3 and 5 $\mu$m. As we are targeting photodetection in the 3 - 5 $\mu$m window, we apply this as the integration range of Eq.~\eqref{eq:ch_5_photocurrent_density}, i.e. $\hbar \omega_{\text{min}} = 0.248$ eV and $\hbar \omega_{\text{max}} = 0.413$ eV. We note that the 3 - 5 $\mu$m range encompasses 0.32\% of the photon flux obtained by integrating Eq.~\eqref{eq:ch_5_blackbody_spectral_photon_flux}.

Finally, we compute the IQE as

\begin{equation}
    \eta (d) = \frac{ J (d) }{ J ( \infty ) } \, ,
    \label{eq:ch_5_internal_quantum_efficiency}
\end{equation}

\noindent
where $J ( \infty )$ is the photocurrent density at unit absorptivity, corresponding to absorption of all incident photons (cf.~Eq.~\eqref{eq:ch_5_absorptivity}). We note that the 3 - 5 $\mu$m wavelength (0.248 - 0.413 eV energy) integration range is applied to all calculations of IQE that follow, independent of $\lambda_{\text{cut}}$.

In Fig.~\ref{fig:ch_5_iqe}(a) we present the calculated IQE vs.~absorber thickness $d$ for $\lambda_{\text{cut}} = 5$ $\mu$m SBSLs having Sb compositions $x = 18$, 28 and 38\%. At fixed $d$ we note a relative increase in IQE of up to 17.5\% with increasing Sb composition. The majority of this enhancement is achieved between $x = 18$ and 28\%, with minimal further increase between $x = 28$ and 38\%. This mirrors the behaviour observed for the optical absorption in Fig.~\ref{fig:ch_5_absorption_spectra}(a), as described in Sec.~\ref{sec:ch_5_optical_abs_spectra_vs_x}. Equivalent calculations for SBSLs with $\lambda_{\text{cut}} = 5.5$ and 6 $\mu$m reveal similar trends, where we note that the rate of enhancement of the IQE with increasing $x$ is reduced beyond $x \approx 30$\%. However, as in the optical absorption (cf.~Figs.~\ref{fig:ch_5_absorption_spectra}(b) and \ref{fig:ch_5_absorption_spectra}(c)), at longer $\lambda_{\text{cut}}$ we do not observe the same saturation of the IQE as the cut-off wavelength moves further beyond the upper bound (5 $\mu$m) of the wavelength range investigated in our IQE calculations. This is consistent with our calculated absorption spectra since, for $\lambda_{\text{cut}} > 5$ $\mu$m, the absorption does not tend towards zero in the 3 - 5 $\mu$m range. For $x = 30$\% and $d = 5$ $\mu$m, we see in Fig.~\ref{fig:ch_5_iqe} a 23\% and 24\% increase in IQE moving from $\lambda_{\text{cut}} = 5$ $\mu$m to $\lambda_{\text{cut}} = 5.5$ and 6 $\mu$m, respectively. As detailed in Sec.~\ref{sec:ch_5_elec_structure_identification}, for increasing $\lambda_{\text{cut}}$ the required layer thicknesses remain safely within our calculated critical thickness limits. Therefore, we conclude that increasing $\lambda_{\text{cut}}$ to 5.5 $\mu$m for the SBSLs investigated in this work is a viable approach to increase IQE in the 3 - 5 $\mu$m wavelength range, but little further benefit is expected when increasing $\lambda_{\text{cut}}$ to 6 $\mu$m. As expected, based on Eq.~\eqref{eq:ch_5_photocurrent_density}, at fixed $x$, we also note an overall increase of the IQE with increasing $d$. This is expected based on the absorptivity, Eq.~\eqref{eq:ch_5_absorptivity}, which increases with increasing $d$.

\begin{figure}[t!]
	\centering
   \includegraphics[width=0.90\textwidth]{./figures/chapter5/figure_4.pdf}
	\caption{(a) Calculated IQE $\eta$ vs.~total SBSL absorber thickness $d$ for InAs/InAs$_{1-x}$Sb$_{x}$ SBSLs having $\lambda_{\text{cut}} = 5$ $\mu$m, for Sb compositions $x = 18$\% (closed orange circles), 28\% (closed red squares) and 38\% (closed dark red triangles), calculated assuming detector operating temperature $T_{\scalebox{0.7}{\text{D}}} = 150$ K for the absorption of 3 - 5 $\mu$m wavelength radiation emitted by a blackbody source at temperature $T_{\scalebox{0.7}{\text{S}}} = 300$ K. (b) Calculated IQE $\eta$ vs.~Sb composition $x$ for SBSLs having $\lambda_{\text{cut}} = 5$ $\mu$m for absorber thicknesses $d = 3$ $\mu$m (closed light blue circles), 4 $\mu$m (closed blue squares) and 5 $\mu$m (closed dark blue triangles), with $T_{\scalebox{0.7}{\text{D}}} = 150$ K and $T_{\scalebox{0.7}{\text{S}}} = 300$ K. In each case the IQE is shown relative to its calculated value at $x = 18$\%. (c) Calculated IQE $\eta$ vs.~Sb composition $x$ for SBSLs having $\lambda_{\text{cut}} = 5.5$ $\mu$m for absorber thicknesses $d = 3$ $\mu$m (closed light blue circles), 4 $\mu$m (closed blue squares) and 5 $\mu$m (closed dark blue triangles), with $T_{\scalebox{0.7}{\text{D}}} = 150$ K and $T_{\scalebox{0.7}{\text{S}}} = 300$ K. (d) Equivalent to (c) for SBSLs having $\lambda_{\text{cut}} = 6$ $\mu$m. In each case the IQE is calculated with $\hbar \omega_{\text{min}} = 0.248$ eV and $\hbar \omega_{\text{max}} = 0.413$ eV}
	\label{fig:ch_5_iqe}
\end{figure}

In Figs.~\ref{fig:ch_5_iqe}(b), \ref{fig:ch_5_iqe}(c) and \ref{fig:ch_5_iqe}(d) we show the calculated IQE as a function of Sb composition $x$ for SBSL absorber thicknesses $d = 3$, 4 and 5 $\mu$m, at cut-off wavelengths $\lambda_{\text{cut}} = 5$, 5.5 and 6 $\mu$m respectively. In Fig.~\ref{fig:ch_5_iqe}(b), to quantify trends across different absorber thicknesses for $\lambda_{\text{cut}} = 5$ $\mu$m, for each value of $d$ we normalise the IQE relative to the minimum IQE calculated for a given cut-off wavelength, $\eta_{\text{min}} ( \lambda_{\text{cut}} )$, which for $\lambda_{\text{cut}} = 5$ $\mu$m is at $x = 18$\%. As detailed in Sec.~\ref{sec:ch_5_elec_structure_identification}, increasing $\lambda_{\text{cut}}$ increases the lowest Sb composition for which that cut-off wavelength can be reached in a SBSL. In Figs.~\ref{fig:ch_5_iqe}(c) and \ref{fig:ch_5_iqe}(d), for $\lambda_{\text{cut}} = 5.5$ and 6 $\mu$m we show the IQE for absorption between 3 and 5 $\mu$m on an absolute scale. Firstly, we examine the predicted trends for $\lambda_{\text{cut}} = 5$ $\mu$m. Independent of $d$ we observe a consistent trend: the IQE initially increases rapidly up to $x = 30$\%, demonstrating a relative increase of up to $\approx 15$\%. Beyond $x = 30$\% we see minimal additional enhancement of the IQE up to $x = 40$\%. This trend is strongly mirrored for $\lambda_{\text{cut}} = 5.5$ $\mu$m, with a larger rate of increase of the IQE with increasing $x$ for $x < 30$\%. We compute a relative increase of up to $\approx 30$\% at $x = 30$\%. For SBSLs having $\lambda_{\text{cut}} = 6$ $\mu$m, for $x < 26$\% we see IQE lower than that of SBSLs having $\lambda_{\text{cut}} = 5.5$ $\mu$m. For $\lambda_{\text{cut}} = 6$ $\mu$m we predict a larger rate at which the IQE increases with increasing Sb composition for $x < 30$\%. However, this larger rate of increase only allows for minimal further enhancement in the IQE compared to SBSLs having $\lambda_{\text{cut}} = 5.5$ $\mu$m for $x > 30$\%.

These results, in conjunction with the results of Sec.~\ref{sec:ch_5_elec_structure}, suggest that growth of InAs/InAs$_{1-x}$Sb$_{x}$ SBSLs for mid-IR photodetectors should not exclusively target structures having either lower Sb composition with thicker layers, or higher Sb composition with thinner layers. Instead, our calculations predict that device development should target intermediate Sb compositions close to $x = 30$\%. We have reached this general conclusion based on consideration of three key factors: (i) for $x > 30$\% we generally observe a rapid saturation of the calculated IQE, particularly for shorter $\lambda_{\text{cut}}$, (ii) approaching $x = 40$\% we predict large in-plane compressive strains in excess of 2\% which, while within estimated strain-thickness limits, nonetheless increase the chance that deleterious strain-related defects form in the lattice, and (iii) approaching $x = 40$\% we also observe layer thicknesses corresponding to a small number of material monolayers where, in such thin layers, the SBSL electronic and optical properties will be strongly sensitive to sub-nm variations in layer thicknesses, in addition to Sb composition profiles due to unintentional intermixing \cite{Magri_PRB_2002,Osbourn_JVST_1983,Ihm_APL_1987}. The latter two factors represent key challenges for epitaxial growth and device development.

\subsection{Origin of calculated trends in the internal quantum efficiency}
\label{sec:ch_5_optical_abs_opt_jdos_ome}

Having described trends in the calculated SBSL IQE vs.~both cut-off wavelength and Sb composition, we now turn our attention to elucidating and quantifying the origin of these trends. In practice, there are two key contributions to the optical absorption described by Eq.~\eqref{eq:ch_5_optical_absorption}: the joint density of states (JDOS) and the ($\textbf{k}_{\parallel} , q$)-dependent momentum matrix elements, with the latter weighting the individual ($\textbf{k}_{\parallel} , q$)-resolved pairwise subband contributions to the JDOS. We therefore analyse in detail the interplay of these two quantities, which allows to identify the consequences of their evolution and competition with increasing Sb composition as the underlying driver of the calculated trends in IQE discussed above.

Using the calculated electronic structure for each SBSL, we evaluate the JDOS per unit volume as \cite{Chuang_book_2009}

\begin{equation}
    \text{JDOS}(E) = \frac{1}{L} \, \sum_{n_c, n_v}\left(\frac{2 \pi}{L}\right)^{-1} \int_{-\pi / L}^{+\pi / L} \mathrm{~d} q \int \frac{\mathrm{d} \mathbf{k}_{\|}}{(2 \pi)^2} \delta\left(E_{n_c \mathbf{k}}-E_{n_v \mathbf{k}}-E\right) \, .
    \label{eq:ch_5_joint_density_of_states}
\end{equation}

The leading factor of $1 / L$ matches that in the optical absorption, thereby removing the explicit dependence on the SL period and allowing comparison between structures having different $L$ on equal footing. For consistency with our calculated optical absorption spectra (cf.~Eq.~\eqref{eq:ch_5_optical_absorption}) in our evaluation of the JDOS we replace the Dirac delta distribution in Eq.~\eqref{eq:ch_5_joint_density_of_states} by a hyperbolic secant lineshape \cite{Marko_SR_2016} of width $\delta =$ 4 meV \cite{Murphy_JPDAP_2023}. In Figs.~\ref{fig:ch_5_jdos_transitions}(a) and \ref{fig:ch_5_jdos_transitions}(b) we show the calculated JDOS for InAs/InAs$_{1-x}$Sb$_{x}$ SBSLs with $\lambda_{\text{cut}} = 5$ $\mu$m having $x = 18$ and 30\% respectively, for the 0.248 - 0.413 eV energy (3 - 5 $\mu$m wavelength) range considered in the IQE calculations. We show the calculated full JDOS which includes contributions from all conduction and valence subband pairs $n_{c}$ and $n_{v}$. Firstly, at $x = 18$\% we note the presence of a larger number of individual subband pairs in the investigated energy range, with the thicker SBSL structures at lower Sb composition supporting a larger number of bound states. For example, the $x = 38$\% SBSL supports only two bound hole states: the HH-like $hh1$ and a predominantly light-hole-like $lh1$, the latter identified via its calculated Bloch band character \cite{Krizman_JAP_2021,Bouschet_OR_2023}. Secondly, we note that the overall JDOS is decreased by a factor of approximately two at $x = 38$\%, but the $e1$-$hh1$ contribution to the JDOS is little changed between $x = 18$ and 38\%. This reduction in JDOS between $x = 18$ and 38\% is out of step with the calculated increase in optical absorption in this energy range (cf.~Fig.~\ref{fig:ch_5_absorption_spectra}(a)). This indicates that an increase in optical matrix elements must dominate over the reduction in JDOS and account for the enhanced optical absorption.

\begin{figure}[b!]
	\centering
   \includegraphics[width=0.84\textwidth]{./figures/chapter5/figure_5.pdf}
	\caption{(a) Calculated JDOS of an InAs/InAs$_{0.82}$Sb$_{0.18}$ ($x = 18$\%) SBSL with $\lambda_{\text{cut}} = 5$ $\mu$m, including contributions from all conduction and valence subband pairs (solid blue line). Dotted, dashed and dash-dotted lines show contributions to the JDOS from individual subband pairs, which in this SBSL corresponds to one bound electron state $e1$ and four bound hole states $hh1$, $lh1$, $hh2$ and $hh3$ (in order of decreasing energy at $\textbf{k}_{\parallel} = q = 0$). (b) As in (a), but for an InAs/InAs$_{0.62}$Sb$_{0.38}$ SBSL supporting two bound hole states $hh1$ and $lh1$. (c) Calculated $\textbf{k}_{\parallel} = q = 0$ squared optical (momentum) matrix elements $\vert \hat{x} \cdot \textbf{p}_{0} \vert ^{2}$, scaled as the oscillator strength $f$ vs.~inter-band transition energy $E$ for a selection of SBSLs having $\lambda_{\text{cut}} = 5$ $\mu$m and Sb compositions $x = 18$, 22, 26, 30, 34 and 38\% (darkening red). Shaded region highlights the $\hbar \omega = 0.248$ - 0.413 eV photon energy ($\lambda = 3$ - 5 $\mu$m wavelength) range used in evaluation of the IQE. Inset: $\textbf{k}_{\parallel} = q = 0$ $e1$-$hh1$ squared optical matrix elements vs.~$x$ for $\lambda_{\text{cut}} = 5$ $\mu$m. The squared optical matrix elements are shown relative to their calculated value at $x = 18$\%.}
	\label{fig:ch_5_jdos_transitions}
\end{figure}

The full ($\textbf{k}_{\parallel} , q$)-dependent momentum matrix elements between subband pairs are not readily visualised. To corroborate that increasing Sb composition at fixed $\lambda_{\text{cut}}$ can strongly increase individual optical matrix elements, we show in Fig.~\ref{fig:ch_5_jdos_transitions}(c) the calculated $\textbf{k}_{\parallel} = q = 0$ squared momentum matrix elements $\vert \hat{x} \cdot \textbf{p}_{0} \vert ^{2}$ for SBSLs having $\lambda_{\text{cut}} = 5$ $\mu$m and Sb compositions $x = 18$ - 38\% in steps of 4\%. These results are presented as the dimensionless oscillator strength $f$, calculated via the Kane parameter $E_{P}$ as \cite{Peeters_PRB_1993}

\begin{equation}
    f = \frac{ E_{P} }{ E_{g} } = \frac{ 2 \vert \hat{x} \cdot \textbf{p}_{0} \vert ^{2} }{ m_{0} E_{g} } \, ,
    \label{eq:ch_5_oscillator_strength}
\end{equation}

\noindent
where $E_{g}$ is the transition energy between a subband pair. Using Eq.~\eqref{eq:ch_5_oscillator_strength} to calculate the dimensionless oscillator strength $f$ allows comparison of the strength of transitions with different energies. The inset of Fig.~\ref{fig:ch_5_jdos_transitions}(c) displays the calculated $e1$-$hh1$ momentum matrix elements vs.~Sb composition $x$, shown relative to the calculated value at $x = 18$\%. The 3 - 5 $\mu$m range employed in our IQE calculations is shown as a grey shaded region. We note the additional dipole-allowed $e1$-$hh3$ optical transitions seen for $x = 18$ and 22\%, which respectively lie within and outside the IQE calculation range. As with the additional bound subband contributions to the JDOS for lower Sb compositions, the thicker InAs and InAs$_{1-x}$Sb$_{x}$ layers required to maintain $\lambda_{\text{cut}}$ for lower Sb composition $x$ drive this behaviour. As the $e1$-$hh3$ inter-band transitions do not lie within the 3 - 5 $\mu$m range for the majority of the $x = 18$ - 40\% composition range, we restrict our analysis to the $e1$-$hh1$ and $e1$-$lh1$ transitions. For $x = 18$\% we note that the $e1$-$lh1$ (squared) optical matrix element is 55\% of that associated with the $e1$-$hh1$ transition. The $e1$-$hh1$ optical matrix element increases by 21\% up to $x = 30$\%, beyond which it increases by a maximum of 25\% at $x = 38$\%. This increase is significantly larger than that associated with $e1$-$lh1$, which increases by only 4\% between $x = 18$ and 38\%. When examined independently, the trend in the $e1$-$hh1$ optical matrix elements strongly mirror that of the IQE (cf.~Figs.~\ref{fig:ch_5_iqe}(b) and \ref{fig:ch_5_jdos_transitions}(c)). This explicitly validates the $e1$-$hh1$ momentum matrix element -- which is largely determined by the $e1$-$hh1$ spatial overlap -- as a useful heuristic for high-level analysis. As $x$ increases the layer thicknesses decrease to maintain $\lambda_{\text{cut}} = 5$ $\mu$m, resulting in a larger $e1$-$hh1$ spatial overlap that in turn increases the optical matrix element. Our analysis therefore indicates that the increase in IQE with increasing $x$ is driven and dominated by a strong increase in the strength of the zone-centre $e1$-$hh1$ transitions with increasing Sb composition.


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%%%% Section 4: Conclusions %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Conclusions}
\label{sec:ch_5_conclusions}

In summary, the methodology developed to investigate radiative recombination in unstrained InAs/GaSb SLs in Chapter~\ref{ch:chapter4} has been extended here to treat optical absorption in strained, and strain-balanced, InAs/InAs$_{1-x}$Sb$_{x}$ SLs for applications in mid-IR photodetectors operating in the 3 - 5 $\mu$m wavelength range. By enforcing fixed cut-off wavelength $\lambda_{\text{cut}}$, in addition to mandating strain balancing with respect to a GaSb substrate, we systematically identified mid-IR SBSLs having Sb compositions $x = 18$ - 40\%. For $\lambda_{\text{cut}} = 5$, 5.5 and 6 $\mu$m we identified that SBSLs can reach the desired $\lambda_{\text{cut}}$ for Sb compositions $x \geq 18$, 20 and 22\% respectively. At $x = 40$\% the InAs$_{1-x}$Sb$_{x}$ layers have in-plane compressive strain $> 2$\%, with strain balancing mandating small layer thickness approaching $\lesssim$ five monolayers. The high Sb composition regime thus poses challenges for epitaxial growth. Based on rigorous SL electronic structure calculations we computed ($\textbf{k}_{\parallel} , q$)-dependent optical matrix elements and absorption spectra. These calculations predict an increase in optical absorption in the 3 - 5 $\mu$m range with increasing $x$. The calculated SBSL absorption spectra were employed in a model calculation of the IQE for 3 - 5 $\mu$m photodetection of radiation from an ambient blackbody source. At fixed Sb composition $x$ and SBSL absorber thickness $d$, we demonstrated the potential to increase the IQE in the 3 - 5 $\mu$m window by targeting longer cut-off wavelength $\lambda_{\text{cut}}$. For $x = 30$\% and $d = 5$ $\mu$m, we see a 23\% and 24\% increase in IQE moving from $\lambda_{\text{cut}} = 5$ $\mu$m to $\lambda_{\text{cut}} = 5.5$ and 6 $\mu$m, respectively. For $\lambda_{\text{cut}} = 5$ $\mu$m, the calculated IQE vs.~$x$ demonstrated a (relative) enhancement of $\approx 15$\% between $x = 18$ and 30\%, with minimal subsequent increase at higher Sb composition. For $\lambda_{\text{cut}} = 5.5$ and 6 $\mu$m, we predict a greater rate of increase in IQE up to $x = 30$\% and increased enhancement of the IQE beyond $x = 30$\%. Moving from $\lambda_{\text{cut}} = 5.5$ to 6 $\mu$m sees a reduction in the IQE for $x < 26$\%. Also, while the rate of increase of IQE is larger for $\lambda_{\text{cut}} = 6$ $\mu$m, minimal additional enhancement of the IQE is observed for $x > 30$\%. We also note that interrogation of the electronic structure vs.~$x$ elucidated that this trend derives from a competition between the JDOS and optical matrix elements, which respectively decrease and increase in SBSLs with increasing $x$. The increase of the latter being sufficient to drive a net increase of the IQE.

The results of our analysis suggest targeting growth of InAs/InAs$_{1-x}$Sb$_{x}$ SBSLs having Sb compositions $x \approx 25$ - 30\%. SBSLs in this composition range are predicted to deliver appreciable enhancement of the IQE, while avoiding the large in-plane strain and narrow layer thicknesses required to maintain strain-balancing at higher Sb composition ($x \approx 40$\%). The optimum SBSLs identified by our analysis are compatible with recently demonstrated epitaxial growth, suggesting a promising route for device design to guide the development of III-V SL photodetectors for real-world applications.