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%%%% Appendix A:  %%%%
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\chapter{Considerations for 8-band $\textbf{k} \cdot \textbf{p}$ plane wave calculations} \label{ch:appendix1}
\lhead{\emph{Considerations for 8-band $\textbf{k} \cdot \textbf{p}$ plane wave calculations}}

In this Appendix, the results presented in Chapters~\ref{ch:chapter4} and \ref{ch:chapter5} are supplemented with further detailed analysis of key considerations for plane wave multi-band $\textbf{k} \cdot \textbf{p}$ calculations. In Sec.~\ref{sec:appendix1_spurious_soln}, we begin by discussing the presence and mitigation of spurious solutions. We provide additional detail on the renormalisation of the Kane parameter $E_{P}$ (cf.~Sec.~\ref{sec:ch_4_model_spurious_reparam}), the impact of spurious solutions on calculated electronic band structure, and the extent to which spurious solutions can be mitigated. In Sec.~\ref{sec:appendix1_env_func_evol}, we analyse the evolution of the envelope functions in InAs/GaSb superlattices (SLs) with miniband wave vector $q$ in our 8-band $\textbf{k} \cdot \textbf{p}$ plane wave calculations.

\section{Presence and mitigation of spurious solutions}
\label{sec:appendix1_spurious_soln}

Spurious solutions are non-physical solutions of the multi-band $\textbf{k} \cdot \textbf{p}$ envelope function (Schr\"{o}dinger) equations, which can affect the validity of results and provide an inaccurate physical description of heterostructure electronic properties \cite{Godfrey_PRB_1996,Broderick_thesis_2015,Foreman_PRB_1997}. When applying $\textbf{k} \cdot \textbf{p}$ methods and the plane wave expansion method (PWEM) care must be taken to minimise, or if possible, remove the deleterious effect of spurious solutions. As described in Sec.~\ref{sec:ch_4_model_spurious_reparam}, initial test calculations for InAs/GaSb SLs produced band structures displaying spurious solutions. The emergence of these spurious solutions was observed for basis sets containing relatively low numbers of plane waves, indicating that careful consideration is required in the choice of plane wave basis set size. For a choice of $M$, $2M+1$ is the total number of plane waves in the basis set (cf.~Sec.~\ref{sec:ch_2_pwem}). When comparing the $e1$-$h1$ peak emission wavelength to experimental measurements it was noted that, despite careful selection of $M$, further efforts were mandated to ensure confidence in the accuracy of our calculations. In the majority of well established parameter sets, for several III-V semiconductors, for the 8-band $\textbf{k} \cdot \textbf{p}$ Hamiltonian we see

\begin{equation}
    \frac{ 1 }{ m_{c}^{\ast} } < \frac{ E_{P} }{ 3 } \left( \frac{ 2 }{ E_{g} } + \frac{ 1 }{ E_{g} + \Delta_{\text{SO}} } \right) \, . \label{eq:app_A_sc_inequality}
\end{equation}

This gives $s_{c} < 0$ (cf.~Eq.~\eqref{eq:ch_4_sc}), which is the case due to the large magnitude of the Kane parameter $E_{P}$. This causes downward bending of the lowest-energy conduction band (CB) at large real wave vector $k = \vert \textbf{k} \vert$, which can produce non-physical states lying energetically within the band gap. In the PWEM $k_{z}$ is sampled via the discrete wave vectors $G_{m}$, with $\vert m \vert \leq M$, which for large $M$ can sample non-physical large-$k_{z}$ bulk states and pollute the results of heterostructure calculations with spurious solutions \cite{Godfrey_PRB_1996,Broderick_thesis_2015,Foreman_PRB_1997}. As discussed in Sec.~\ref{sec:ch_4_model_spurious_reparam}, we found that following Foreman's prescription of imposing $s_{c} = 0$ by renormalising the Kane parameter $E_{P}$ rectified this issue \cite{Foreman_PRB_1997}. However, while this established approach worked to mitigate the impact of spurious solutions in our SL calculations, further analysis revealed that the origin of the spurious solutions was not related to states at large real $k_{z}$ lying energetically within the band gap. Rather it was the presence of a spurious band having complex-valued wave vector in the bulk band structure. As such, we undertook an analytical analysis of a representative 2-band (Kane) Hamiltonian to identify the origin of the spurious band ``SB'' identified in Fig.~\ref{fig:ch_4_complex_bs}(a), and to demonstrate why imposing $s_{c} = 0$ mitigates its impact in the numerical calculations.

Following this renormalisation, we note that it is not possible to conclude that the impact of spurious solutions on the SL electronic structure calculated using the 8-band $\textbf{k} \cdot \textbf{p}$ Hamiltonian have been removed entirely. We therefore present an analytical analysis to demonstrate explicitly that imposing $s_{c} = 0$ mitigates the influence of these spurious solutions in our numerical calculations. Firstly, we note that setting $s_{c} = 0$ allows for a discontinuity to occur at an InAs/GaSb interface in amplitude of the CB Bloch component of the envelope function \cite{Foreman_PRB_1997}. Omitting the spin-split-off (SO) band by setting the valence band (VB) spin-orbit splitting $\Delta_{\scalebox{0.7}{\text{SO}}}$ equal to zero allows us to reduce $H_{3 \times 3}$ of Eq.~\eqref{eq:ch_4_3x3_hamiltonian} to a 2-band (Kane) Hamiltonian for the CB and light-hole (LH) states

\begin{equation}
H_{2 \times 2} ( k_{z} ) = \left( \begin{array}{cc}
        E_{c} + ak_{z}^{2} & Pk_{z} \\
        Pk_{z} & E_{v} - bk_{z}^{2} \\
    \end{array} \right) \, \begin{array}{l} \vert \frac{1}{2} ; + \frac{1}{2} \rangle ~ \vspace{0.0cm} \\ \vert \frac{3}{2} ; + \frac{1}{2} \rangle \\ \end{array} \, ,
	  \label{eq:app_A_hamiltonian_2x2}
\end{equation}

%\begin{equation}
%H_{2 \times 2} ( k_{z} ) = \left( \begin{array}{cc}
%        E_{c} + ak_{z}^{2} & Pk_{z} \\
%        Pk_{z} & E_{v} - bk_{z}^{2} \\
%    \end{array} \right) \, \begin{array}{l} \vert \frac{1}{2} ; + \frac{1}{2} \rangle \\ ~ \vspace{-0.15cm} \\ \vert \frac{3}{2} ; + \frac{1}{2} \rangle \\ \end{array} \, ,
%	  \label{eq:app_A_hamiltonian_2x2}
%\end{equation}

\noindent
where $a = \frac{\hbar^{2} s_{c}}{2 m_{0}}$ and $b = \frac{ \hbar^{2} }{ 2m_{0} } \left( \gamma_{\, 1} + 4\gamma_{\, 2} \right)$. We can use Eq.~\eqref{eq:app_A_hamiltonian_2x2} to understand the origin of the aforementioned discontinuity in the envelope function, which gives rise to the step-like behaviour of the $e1$ probability density observed in Fig.~\ref{fig:ch_4_exemplar_offsets}. We do so by diagonalising Eq.~\eqref{eq:app_A_hamiltonian_2x2} for small values of $a$ (equivalent to small values of $s_{c}$) and then analyse the evolution of the eigenstates as $a \to 0$. Evaluating the determinant $\vert H_{2 \times 2} - E \, I \vert = 0$, where $I$ is the $2 \times 2$ identity matrix, produces a quartic characteristic equation in $k_{z}$, for which the solutions at small $a$ are

\begin{equation}
    k_{z,\pm}^{2} (E) \approx \frac{ - \left( P^{2} + b \left( E_{c} - E \right) \right) \pm \sqrt{ \left( P^{2} + b \left( E_{c} - E \right) \right)^{2} -4ab \left( E_{c} - E \right) \left( E - E_{v} \right) } }{ 2ab } \, .
    \label{eq:app_A_k_solutions_small_a}
\end{equation}

In addition to the usual Kane solution $k_{z,+}^{2}$, which describes the evanescent band linking the VB and CB edges \cite{Das_PRA_2022}, we also obtain a second solution $k_{z,-}^{2}$ whose value at small $a$ can be approximated as

\begin{equation}
    \kappa_{-}^{2} (E) \equiv - k_{z,-}^{2} (E) \approx \frac{ P^{2} + b \left( E_{c} - E \right) }{ ab } \, ,
    \label{eq:app_A_k_minus_small_a}
\end{equation}

\noindent
with $\kappa_{-}^{2} \to \infty$ as $a \to 0$. The corresponding eigenstates of $H_{2 \times 2}$ approach $\vert \psi_{\pm} \rangle = e^{\pm \kappa_{+}z} \vert \frac{1}{2}; + \frac{1}{2} \rangle$ as $a \to 0$, where $\vert \frac{1}{2}; + \frac{1}{2} \rangle$ is the $k_{z} = 0$ CB Bloch (basis) state \cite{Bahder_PRB_1990}. To calculate allowed bound eigenstates in a heterostructure grown along the $z$ direction, we replace $k_{z} \to - i \frac{ \text{d} }{ \text{d} z }$ in Eq.~\eqref{eq:app_A_hamiltonian_2x2} \cite{Chen_PRB_1992}. Writing this quantised Hamiltonian as

\begin{equation}
    \widehat{H} \psi = A \, \frac{ \text{d}^{2} \psi}{ \text{d} z^{2}} + B \, \frac{ \text{d} \psi}{\text{d} z} + C \psi \, ,
    \label{eq:app_A_quantised_hamiltonian}
\end{equation}

\noindent
then the allowed solutions must satisfy the boundary conditions that both $\psi$ and $A \, \frac{ \text{d} \psi}{ \text{d} z } + \frac{1}{2} B \psi$ are continuous \cite{Chen_PRB_1992}, where

\begin{equation}
	A = \left( \begin{array}{cc}
            -a & 0 \\
            0 & b \\
        \end{array} \right) , \; B = \left( \begin{array}{cc}
            0 & iP \\
            -iP & 0 \\
        \end{array} \right) ,~\text{and} \; \; C = \left( \begin{array}{cc}
            E_{c} & 0 \\
            0 & E_{v} \\
        \end{array} \right) ,
	   \label{eq:app_A_A_matrix}
\end{equation}

\noindent
are the respective coefficient matrices of Eq.~\eqref{eq:app_A_hamiltonian_2x2} for terms that are quadratic in $k_{z}$, linear in $k_{z}$, and independent of $k_{z}$.

These boundary conditions appear to mandate that the CB component of $\psi$ must always be continuous. However, the term $A \, \frac{ \text{d} \psi}{ \text{d} z}$ can allow for a jump discontinuity in the amplitude of $\psi$ as $a \to 0$. For $E \approx E_{c}$ we have $\frac{ \text{d} \psi_{-} }{ \text{d} z } \approx \pm \frac{ P }{ \sqrt{ ab } } \psi_{-}$, so that the CB component of $A \, \frac{ \text{d} \psi}{ \text{d} z }$ can then vary as $- a \, \frac{ \text{d} \psi_{-}}{ \text{d} z } \approx \mp P \sqrt{ \frac{a}{b} } \, \psi_{-}$, which vanishes as $a \to 0$. This behaviour, discussed by Foreman in the $a = 0$ limit \cite{Foreman_PRB_1997}, accounts for the observed discontinuities in the $e1$ envelope function at InAs/GaSb interfaces in the full 8-band calculations (cf.~Fig.~\ref{fig:ch_4_exemplar_offsets}).

We note that our calculated InAs and GaSb bulk complex band structures, using renormalised parameters with the Kane parameter $E_{P}$ having been reduced such that $s_{c} = 0$ (cf.~Fig.~\ref{fig:ch_4_complex_bs}(b)), are in good quantitative agreement with the results of atomistic tight-binding calculations over the same ranges of energy and wave vector \cite{Schulman_PRB_1985}. States from outside of these ranges of energy and wave vector also contribute, albeit less strongly, to the determination of the variation of the electron envelope function with position in a full atomistic calculation. However, as described in Chapter~\ref{ch:chapter4}, our renormalised 8-band $\textbf{k} \cdot \textbf{p}$ calculations capture the key features of the envelope functions and inter-band optical matrix elements observed in the full atomistic SL calculations of, e.g., Refs.~\cite{Kato_SM_2018} and~\cite{Chang_PRB_1985}. This demonstrates that the impact of spurious solutions is sufficiently mitigated to validate our $\textbf{k} \cdot \textbf{p}$ calculations of the InAs/GaSb SL electronic structure. We then use these calculations as a platform to predict and analyse radiative recombination in broken-gap InAs/GaSb SLs (cf.~Chapter~\ref{ch:chapter4}) and, following this, optical absorption in InAs/InAs$_{1-x}$Sb$_{x}$ SBSLs (cf.~Chapter~\ref{ch:chapter5}).

\section{Envelope function evolution along minibands in InAs/GaSb superlattices}
\label{sec:appendix1_env_func_evol}

In this Section we analyse the evolution vs.~miniband wave vector $q$ of the envelope functions of InAs/GaSb SLs having equal InAs and GaSb layer thicknesses. In Sec.~\ref{sec:ch_4_trends_rad_recomb_omes_se} we noted that the optical (momentum) matrix element between the $\textbf{k}_{\parallel} = 0$ $e1$ and $h1$ SL eigenstates goes to zero as $q$ is varied from zero to $\frac{\pi}{L}$ (cf.~Fig.~\ref{fig:ch_4_exemplar_ome_se}(a)). In Chapter~\ref{ch:chapter4} we noted that this behaviour was a consequence of the introduction of a relative phase shift between the $e1$ and $h1$ envelope functions as both minibands are traversed in $q$. To demonstrate this explicitly, we present here the calculated Bloch components of the $e1$ and $h1$ envelope functions at both $q = 0$ and $\frac{\pi}{L}$. We recall that the heavy-hole (HH) band is decoupled at $\textbf{k}_{\parallel} = 0$, which can be seen in the 8-band Hamiltonian matrix of Eq.~\eqref{eq:ch_2_bulk_8_band_hamiltonian} by setting $k_{x} = k_{y} = 0$. As such, the $e1$ eigenstate at $\textbf{k}_{\parallel} = 0$ consists of an admixture of CB, LH and SO Bloch character only, while the $h1$ eigenstate is purely HH-like.

The results of these calculations are summarised in Fig.~\ref{fig:app_A_envelope_functions}, which shows the aforementioned non-zero envelope function Bloch components at $q = 0$ (top row) and $q = \frac{\pi}{L}$ (bottom row) for a single SL period. Panels (a), (b) and (c) in the top row, and panels (e), (f) and (g) in the bottom row, respectively show the CB, LH and SO Bloch components of the $e1$ eigenstate. Panels (d) and (h) show the HH component of the $h1$ eigenstate at $q = 0$ and $\frac{\pi}{L}$, respectively.

\begin{figure}[ht!]
\centering
	\includegraphics[width=0.98\textwidth]{./figures/appendix/figure_appendix_A1.pdf}
	\caption{Top row: real (solid blue lines) and imaginary (dashed red lines) parts of the $q = 0$ envelope function Bloch components $F_{nb\textbf{k}} (z)$ for the $\textbf{k}_{\parallel} = 0$ $e1$ and $h1$ eigenstates of the exemplar equal layer thickness InAs/GaSb SL considered in Sec.~\ref{sec:ch_4_elec} (having SL period $L = 2t = 4.28$ nm). (a) CB component of $e1$, (b) LH component of $e1$, (c) SO component of $e1$, and (d) HH component of $h1$. Bottom row: as in the top row, but for $q = \frac{\pi}{L}$. Shaded (unshaded) regions denote the electron-confining InAs (hole-confining GaSb) layers.}
	\label{fig:app_A_envelope_functions}
\end{figure}

Considering first the Bloch components of the $e1$ eigenstate, at $q = 0$ the CB (LH and SO) component is purely real and even (imaginary and odd) about the centre of the InAs layer -- shaded region -- at $z = 0$, and remains so as $q$ changes from zero to $\frac{\pi}{L}$. For the $h1$ state, we observe changing phase and parity between $q = 0$ and $\frac{\pi}{L}$, with the HH Bloch component changing from being purely real and even about the centre of the InAs layer at $q = 0$ to purely real and odd about the centre of the InAs layer at $q = \frac{\pi}{L}$. This directly demonstrates that varying $q$ from zero to $\frac{\pi}{L}$ introduces a difference in parity between the $e1$ and $h1$ SL eigenstates at $\textbf{k}_{\parallel} = 0$. This results in the $e1$ and $h1$ eigenstates having equal (opposite) parity at $q = 0$ ($q = \frac{\pi}{L}$) which, combined with the respective $s$- and $p$-like symmetry of the CB and HH Bloch basis states $\vert \frac{1}{2}; \pm \frac{1}{2} \rangle$ and $\vert \frac{3}{2}; \pm \frac{3}{2} \rangle$, gives rise to the calculated reduction of the $\textbf{k}_{\parallel} = 0$ $e1$-$h1$ inter-band optical matrix element to zero at the SL Brillouin zone edge (cf.~Fig.~\ref{fig:ch_4_exemplar_ome_se}(a)).

We note that the fact that this change in parity appears restricted to the $h1$ eigenstate in Fig.~\ref{fig:app_A_envelope_functions} is a consequence of the specific choice of calculational supercell employed in our analysis. Here, we have utilised a supercell $z \in [ - \frac{L}{2}, \frac{L}{2} ]$ which is centred about the centre of the electron-confining InAs layer. Changing to, for example, a supercell with $z = 0$ set at the centre of the hole-confining GaSb layer instead produces a change in the phase and parity of the $e1$ eigenstate as $q$ varies from zero to $\frac{\pi}{L}$, with the parity of the $h1$ eigenstate then remaining fixed. This has been confirmed via test calculations and emphasises that it is the \textit{difference} in parity between the $e1$ and $h1$ eigenstates that determines their inter-band optical matrix element, irrespective of a given choice of calculational supercell.