\chapter{Theoretical background: the $\textbf{k} \cdot \textbf{p}$ and plane wave expansion methods} \label{ch:chapter2}
\lhead{\emph{Theoretical background: the $\textbf{k} \cdot \textbf{p}$ and plane wave expansion methods}}

In this Chapter we outline the theoretical methods used in our analysis of radiative recombination in InAs/GaSb superlattices (SLs) in Chapter~\ref{ch:chapter4}, and of optical absorption in InAs/InAs$_{1-x}$Sb$_{x}$ strain-balanced SLs in Chapter~\ref{ch:chapter5}. We begin in Sec.~\ref{sec:ch_2_k.p_theory} by providing general background on the $\textbf{k} \cdot \textbf{p}$ method, and describe the 8-band $\textbf{k} \cdot \textbf{p}$ Hamiltonian employed in this Thesis. In Sec.~\ref{sec:ch_2_k.p_theory_outline} we introduce the $\textbf{k} \cdot \textbf{p}$ method for an electron in a periodic solid using time-independent (Rayleigh-Schr\"{o}dinger) perturbation theory, which allows us to treat the $\textbf{k} \cdot \textbf{p}$ coupling as a small perturbation. In Sec.~\ref{sec:ch_2_k.p_theory_outline} we also present the form of this technique for the description of degenerate energy bands in a semiconductor, using the L\"{o}wdin renormalisation approach to time-independent perturbation theory. This approach partitions the material eigenstates into two sets: (A) the bands we wish to incorporate in our $\textbf{k} \cdot \textbf{p}$ Hamiltonian, and (B) other ``energetically remote'' bands. L\"{o}wdin renormalisation block diagonalises the resulting Hamiltonian, incorporating the impact of states B in a perturbative Hamiltonian that explicitly treats states A. In Sec.~\ref{sec:ch_2_k.p_theory_8-band_Hamiltonian} we present the 8-band $\textbf{k} \cdot \textbf{p}$ Hamiltonian that underpins our SL calculations in Chapters~\ref{ch:chapter4} and \ref{ch:chapter5}. In Sec.~\ref{sec:ch_2_pwem}, we introduce the techniques employed in our implementation of the 8-band $\textbf{k} \cdot \textbf{p}$ Hamiltonian described in Sec.~\ref{sec:ch_2_k.p_theory_8-band_Hamiltonian}. In Sec.~\ref{sec:ch_2_pwem_outline} we outline the plane wave expansion method (PWEM) in the envelope function approximation (EFA) for the calculation of heterostructure electronic properties. This includes treatment of} the SL miniband dispersion. In Sec.~\ref{sec:ch_2_pwem_1-band_example}, we provide an example of the application of this approach, by considering a 1-band (Schr\"{o}dinger) model of the Kronig-Penney model of a periodic SL potential. Next, in Sec.~\ref{sec:ch_2_pwem_8-band_inclusion} we outline of the application of the PWEM in conjunction with the 8-band $\textbf{k} \cdot \textbf{p}$ Hamiltonian, and detail our procedure for the calculation of inter-band (optical) momentum matrix elements, which enable the calculation of optical properties, including the spontaneous emission and optical absorption spectra. Finally, in Sec.~\ref{sec:ch_2_conclusions} we summarise and conclude.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% Section 1: $\textbf{k} \cdot \textbf{p}$ theory %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{The $\textbf{k} \cdot \textbf{p}$ method}
\label{sec:ch_2_k.p_theory}

In this Section we will outline the $\textbf{k} \cdot \textbf{p}$ method. This method employs the eigenstates of the crystal Hamiltonian at a given wave vector $\textbf{k}_{0}$ as a basis, and allows to calculate the eigenstates at nearby wave vectors \textbf{k} via time-independent perturbation theory. In practice, the $\textbf{k} \cdot \textbf{p}$ interaction is treated as a small perturbation that couples different band states at a reference wave vector $\textbf{k}_{0}$. With this general form outlined, we then present the 8-band $\textbf{k} \cdot \textbf{p}$ Hamiltonian employed in the calculations described in Chapters~\ref{ch:chapter4} and \ref{ch:chapter5}.

\subsection{Outline of the $\textbf{k} \cdot \textbf{p}$ method}
\label{sec:ch_2_k.p_theory_outline}

% Introduce perturbation theory and rough outline of the $\textbf{k} \cdot \textbf{p}$ method?

The time-independent Schr{\"o}dinger equation for an electron in a periodic crystal, and associated Hamiltonian $\widehat{H}_{0}$, at wave vector $\textbf{k}$, is \cite{OReilly_book_2002,Chuang_book_2009,Bogusevschi_thesis_2018}

\begin{eqnarray}
%
    \widehat{H}_{0} \psi_{n} (\textbf{k},\textbf{r}) &=& E_{n} (\textbf{k}) \psi_{n} (\textbf{k},\textbf{r}) \, , \label{eq:ch_2_free_elec_schrodinger_eqn} \\
%
    \widehat{H}_{0} &=& - \frac{ \hbar^{2} }{ 2 m_{0} } \nabla^{2} +V_{0} (\textbf{r}) \, , \label{eq:ch_2_free_elec_hamiltonian}
%
\end{eqnarray}

\noindent
where $V_{0}$ is the periodic potential of the crystal with periodicity $V_{0} (\textbf{r}) = V_{0} (\textbf{r} + \textbf{R})$, with $\textbf{R} = n_{1} \textbf{a}_{1} + n_{2} \textbf{a}_{2} + n_{3} \textbf{a}_{3}$ a Bravais lattice vector described by the lattice vectors $\textbf{a}_{1}$, $\textbf{a}_{2}$ and $\textbf{a}_{3}$, with $n_{i} \in \mathbb{Z}$. Using Bloch's theorem the electron wave function, having energy dispersion $E_{n} (\textbf{k})$ described by the Bloch wave vector \textbf{k}, can be written in the form \cite{Yu_book_2010}

\begin{equation}
    \psi_{n} (\textbf{k}) = e^{i \textbf{k} \cdot \textbf{r}} u_{n} (\textbf{k},\textbf{r}) \, , \label{eq:ch_2_elec_wavefunction_bloch}
\end{equation}

\noindent
where $u_{n} ( \textbf{k}, \textbf{r} )$ is a cell-periodic function, $u_{n} ( \textbf{k}, \textbf{r} ) = u_{n} ( \textbf{k}, \textbf{r} + \textbf{R} )$. Wanting to compute the band energies $E_{n} ( \textbf{k} )$ at a given wave vector $\textbf{k}$, close to the wave vector $\textbf{k}_{0}$ at which we know the solutions of Eq.~\eqref{eq:ch_2_free_elec_schrodinger_eqn}, we can use Eqs.~\eqref{eq:ch_2_free_elec_schrodinger_eqn} and \eqref{eq:ch_2_elec_wavefunction_bloch}. Substituting $\textbf{k} \rightarrow \textbf{k} - \textbf{k}_{0}$, which shifts the origin of the wave vector expansion to $\textbf{k}_{0}$, we can write 

\begin{equation}
    \left[ -\frac{\hbar^{2}}{2m_{0}} \nabla^{2} + V_{0} (\textbf{r}) \right] \left[ e^{i (\textbf{k} - \textbf{k}_{0}) \cdot \textbf{r}} u_{n} (\textbf{k} - \textbf{k}_{0}, \textbf{r}) \right] = E_{n} (\textbf{k} - \textbf{k}_{0}) e^{i (\textbf{k} - \textbf{k}_{0}) \cdot \textbf{r}} u_{n} (\textbf{k} - \textbf{k}_{0}, \textbf{r}) \, . \label{eq:ch_2_rewritten_schrodinger}
\end{equation}

Applying the Laplacian, we can write the $\textbf{k}$-dependent Schr{\"o}dinger equation, and \textbf{k}-dependent contribution to the Hamiltonian $\widehat{H}'$, for the periodic function $u_{n} (\textbf{k}-\textbf{k}_{0},\textbf{r})$ as

\begin{eqnarray}
%
    \left( \widehat{H}_{0} + \widehat{H}' \right) u_{n} (\textbf{k}-\textbf{k}_{0},\textbf{r}) &=& E_{n} (\textbf{k} - \textbf{k}_{0}) u_{n} (\textbf{k}-\textbf{k}_{0},\textbf{r}) \, , \label{eq:ch_2_k-dep_schrodinger} \\
%
    \widehat{H}' &=& \frac{\hbar^{2}}{2m_{0}} \vert \textbf{k} - \textbf{k}_{0} \vert^{2} + \frac{\hbar}{m_{0}} (\textbf{k}-\textbf{k}_{0}) \cdot \textbf{p} \, , \label{eq:ch_2_perturbation_hamiltonian}
%
\end{eqnarray}

\noindent
where $\textbf{p} = -i\hbar\nabla$ is the momentum operator. The $\textbf{k} \cdot \textbf{p}$ method seeks to solve Eq.~\eqref{eq:ch_2_k-dep_schrodinger}, which includes the $\textbf{k}$-dependent Hamiltonian $\widehat{H}'$ of Eq.~\eqref{eq:ch_2_perturbation_hamiltonian}, otherwise referred to as the $\textbf{k} \cdot \textbf{p}$ Hamiltonian. For a known value of $E_{n} ( \textbf{k}_{0} )$, and assuming a known periodic function $u_{n} ( 0, \textbf{r} )$ at $\textbf{k} = \textbf{k}_{0}$, the band structure $E_{n} (\textbf{k}-\textbf{k}_{0})$ can then be evaluated in the vicinity of $\textbf{k}_{0}$ by applying time-independent perturbation theory. To second-order in $\textbf{k} - \textbf{k}_{0}$, and writing $u_{n} ( \textbf{r} , \textbf{k}_{0} ) = \langle \textbf{r} \vert u_{n} (\textbf{k}_{0}) \rangle$, $E_{n} (\textbf{k}-\textbf{k}_{0})$ can be written as

\begin{eqnarray}
    E_{n} (\textbf{k}-\textbf{k}_{0}) &=& \bra{ u_{n}(\textbf{k}_{0}) } \widehat{H}_{0} + \widehat{H}' \ket{u_{n}(\textbf{k}_{0})} \nonumber \\
    &=& E_{n} (\textbf{k}_{0}) + \frac{\hbar^{2} \vert \textbf{k} - \textbf{k}_{0} \vert^{2}}{2 m_{0}} \braket{ u_{n}(\textbf{k}_{0}) \vert u_{n}(\textbf{k}_{0}) } + \frac{\hbar ( \textbf{k} - \textbf{k}_{0} )}{m_{0}} \bra{u_{n}(\textbf{k}_{0})} \textbf{p} \ket{u_{n}(\textbf{k}_{0})} \, . \label{eq:ch_2_2nd_order_dispersion}
\end{eqnarray}

Operating with the momentum $\textbf{k} = -i \nabla$ introduces a change in parity of the periodic functions, $u_{n} ( \textbf{r} , \textbf{k}_{0} )$ such that the term linear in $\textbf{k}$ in Eq.~\eqref{eq:ch_2_2nd_order_dispersion} vanishes, leaving terms of second-order in $\textbf{k}$, i.e.~$\sim \mathcal{O}(\textbf{k}^{2})$. Due to the orthonormality of the periodic functions $\ket{u_{n}(\textbf{k}_{0})}$, we can write $\langle u_{m}(\textbf{k}_{0}) \vert u_{n}(\textbf{k}_{0}) \rangle = \delta_{mn}$. We can therefore write Eq.~\eqref{eq:ch_2_2nd_order_dispersion} as

\begin{equation}
    E_{n} (\textbf{k}-\textbf{k}_{0}) = E_{n} (\textbf{k}_{0}) + \frac{\hbar^{2} \vert \textbf{k} - \textbf{k}_{0} \vert^{2}}{2 m_{0}} \, . \label{eq:ch_2_2nd_order_dispersion_reduced}
\end{equation}

This description of the band structure has not taken into account spin-orbit coupling or the interaction between multiple bands, which are required to produce an accurate description of the band structure of a real semiconductor using the $\textbf{k} \cdot \textbf{p}$ method. Typical multi-band $\textbf{k} \cdot \textbf{p}$ methods aim to describe the band structure within a narrow range of wave vector $\textbf{k}$. The perturbative nature of the method allows for an accurate quantitative description in a small range of wave vectors close to $\textbf{k}_{0}$ \cite{Bogusevschi_thesis_2018}.

To account for the interaction between bands we must now introduce an inter-band coupling correction to Eq.~\eqref{eq:ch_2_2nd_order_dispersion_reduced}. Using L{\"o}wdin renormalisation \cite{Chuang_book_2009,Lowdin_JCP_1951}, we can divide the eigenstates at wave vector $\textbf{k}_{0}$ into two classes, A and B. Our chosen basis states are those in class A, the interaction between which will be included explicitly in the Hamiltonian. Class B consists of all other bands outside of class A, which are typically energetically remote from class A. The $\textbf{k} \cdot \textbf{p}$ Hamiltonian, using second-order perturbation theory for degenerate bands, is then given as \cite{Chuang_book_2009,OReilly_book_2002}

\begin{equation}
    H_{mn} ( \textbf{k} ) = E_{n} (\textbf{k}_{0}) \delta_{mn} + H_{mn}' (\textbf{k}) + \sum_{l \in \text{A} \neq n} \frac{ H_{ml}'(\textbf{k}) H_{ln}'(\textbf{k})  }{ E_{n}(\textbf{k}_{0}) - E_{l}(\textbf{k}_{0}) } + \sum_{\gamma \in \text{B}} \frac{ H_{m \gamma}'(\textbf{k}) H_{\gamma n}'(\textbf{k})  }{ E_{\text{A}}(\textbf{k}_{0}) - E_{\gamma}(\textbf{k}_{0}) } \, , \label{eq:ch_2_2nd_order_k.p_hamiltonian}
\end{equation}

% Now, in practice, the states "B" are assumed to be energetically remote from states "A". So, the index "m" in the denominator of the final term on the right-hand side is often replaced by some EA(k) to denote some "average" energy of "A" states (cf. the expressions for the Luttinger parameters, Eqs. (2.11) - (2.13) in Cónal's chapter draft). So, for consistency, I would probably suggest changing "m" --> "A" in the above, but only in the denominator of the final term on the right-hand side (and not the numerator, which must remain "m" to make sense).

\noindent
where interactions between states in class A, and the impact of coupling to class B states, are described by the first and second sum respectively. In practice, class B states are assumed to be energetically remote from class A states. This means the quantity $E_{\text{A}} (\textbf{k}_{0})$ in the second sum denotes some appropriate average energy of class A states. Further detail regarding the derivation of Eq.~\eqref{eq:ch_2_2nd_order_k.p_hamiltonian} can be found in Appendix B of Ref.~\cite{Bogusevschi_thesis_2018}.

% The matrix elements of the Hamiltonian $\widehat{H}_{nl}'$ ...

Using Eq.~\eqref{eq:ch_2_2nd_order_k.p_hamiltonian}, the band structure $E_{n} ( \textbf{k} )$ at wave vector $\textbf{k}$ can be calculated knowing (i) the periodic basis functions $u_{n}(\textbf{k}_{0})$ for the bands in class A, (ii) the band edge energies of the basis states (class A) at the reference wave vector $\textbf{k}_{0}$, and (iii) the perturbation sums describing electronic coupling between the class A and B states. This approach is typically applied for electronic structure calculations near high symmetry points in the Brillouin zone (BZ), close in energy to the conduction and/or valence band edges. These symmetry-distinct perturbation sums are then assigned values, that we treat as empirical parameters to parametrise the band structure, the latter being computed by diagonalising the Hamiltonian matrix derived in the basis of class A states. An example of such a parametrisation comes from the so-called effective mass approximation. In the effective mass approximation the energy dispersion $E_{n} (\textbf{k})$ can be written in terms of the (inverse) effective mass tensor, which, in units of free electron mass $m_{0}$, is expressed as \cite{Willardson_book_1977,Bogusevschi_thesis_2018}

\begin{equation}
    \frac{1}{m_{ij}^{\ast}} = \delta_{ij} + \frac{1}{m_{0}} \sum_{m \in \text{A} \neq n} \frac{ p_{nm}^{(i)} p_{mn}^{(j)} + p_{nm}^{(j)} p_{mn}^{(i)} }{ E_{n} (0) - E_{m} (0) } + \frac{1}{m_{0}} \sum_{m \in \text{B}} \frac{ p_{nm}^{(i)} p_{mn}^{(j)} + p_{nm}^{(j)} p_{mn}^{(i)} }{ E_{\text{A}} (0) - E_{\text{B}} (0) } \, . \label{eq:ch_2_electron_effective_mass}
\end{equation}

where $p_{mn}^{(i)}$ is the momentum matrix element describing the coupling between states $m$ and $n$ (cf.~Sec.~\ref{sec:ch_2_pwem_8-band_inclusion}).

In choosing basis functions for the $\textbf{k}\cdot\textbf{p}$ Hamiltonian, we note that in III-V semiconductors eigenstates close to the valence band (VB) and conduction band (CB) edges can be well described as $sp^{3}$ hybridised linear combinations of atomic orbitals \cite{Bogusevschi_thesis_2018}. Therefore, we work with a basis of Bloch states that are linear combinations of atomic $s$-, $p_{x}$-, $p_{y}$- and $p_{z}$-like orbitals. The subscript $m$ ($n$), and superscript $i$ ($j$), for $p_{mn}^{(i)}$ in Eq.~\eqref{eq:ch_2_electron_effective_mass} correspond to the $m$ ($n$) functions and the $i$ ($j$) component of the momentum matrix, respectively.

The zone-centre effective mass tensor for the VBs is defined using the Luttinger parameters $\gamma_{i}^{\, L}$ for cubic crystals \cite{Vurgaftman_JAP_2001}. The assumption of $p_{x}$-, $p_{y}$- and $p_{z}$-like basis states produces three symmetry-distinct perturbation sums (cf.~Eq.~\eqref{eq:ch_2_electron_effective_mass}), which define the parameters $K$, $M$ and $N$ \cite{Chuang_book_2009,Bogusevschi_thesis_2018}

\begin{eqnarray}
%    
    K &=& \frac{\hbar^{2}}{2m_{0}} \left( 1 + \frac{2}{m_{0}} \sum_{m \in B} \frac{p_{xm}^{(x)} p_{mx}^{(x)}}{E_{A}(0) - E_{m}(0)} \right) \, , \label{eq:ch_2_L} \\
%
    M &=& \frac{\hbar^{2}}{2m_{0}} \left( 1 + \frac{2}{m_{0}} \sum_{m \in B} \frac{p_{xm}^{(y)} p_{mx}^{(y)}}{E_{A}(0) - E_{m}(0)} \right) \, , \label{eq:ch_2_M} \\
%
    N &=& \frac{\hbar^{2}}{m_{0}^{2}} \sum_{m \in B} \frac{p_{xm}^{(x)} p_{my}^{(y)} + p_{xm}^{(y)} p_{my}^{(x)}}{E_{A}(0) - E_{m}(0)} \, . \label{eq:ch_2_N}
%
\end{eqnarray}

The parameters $K$, $M$ and $N$ define the Luttinger parameters as \cite{Chuang_book_2009,Bogusevschi_thesis_2018}

\begin{eqnarray}
%
    \gamma_{1}^{\, L} &=& - \frac{2m_{0}}{3 \hbar^{2}} \left( K + 2 M \right) \, , \label{eq:ch_2_luttinger_1_generalised} \\
%
    \gamma_{2}^{\, L} &=& - \frac{m_{0}}{3 \hbar^{2}} \left( K - M \right) \, , \label{eq:ch_2_luttinger_2_generalised} \\
%
    \gamma_{3}^{\, L} &=& - \frac{m_{0}}{3 \hbar^{2}} N \, . \label{eq:ch_2_luttinger_3_generalised}
%
\end{eqnarray}

These quantities are typically parametrised based on data obtained from experimental measurements or first-principles calculations (cf.~Chapter~\ref{ch:chapter6}). In particular, the Luttinger parameters, which are mathematically defined based on symmetry constraints, are treated as empirical parameters that determine the VB edge effective masses.

\subsection{8-band $\textbf{k} \cdot \textbf{p}$ Hamiltonian}
\label{sec:ch_2_k.p_theory_8-band_Hamiltonian}

Having outlined the procedure for constructing a general $N$-band $\textbf{k} \cdot \textbf{p}$ Hamiltonian (cf.~Sec.~\ref{sec:ch_2_k.p_theory_outline}), we now present the 8-band $\textbf{k} \cdot \textbf{p}$ Hamiltonian for zinc-blende, including spin-orbit coupling \cite{Cohen_PRB_1990,Bahder_PRB_1990,Bogusevschi_thesis_2018}, that underpins our analysis of the optoelectronic structure of semiconductor SLs in Chapters~\ref{ch:chapter4} and \ref{ch:chapter5}.

We work in a Cartesian coordinate system in which the $x$, $y$ and $z$ axes respectively align with the [100], [010] and [001] cubic crystal principal axes. The 8-band $\textbf{k} \cdot \textbf{p}$ Hamiltonian employed in this work is constructed using a basis of spin-degenerate $\Gamma$-point Bloch functions $\vert u_{b} \rangle$ ($b = 1, 2, \dots, 8$) associated with the lowest energy CB, as well as the heavy-hole (HH), light-hole (LH) and spin-split-off (SO) VBs \cite{Tomic_IEEEJSTQE_2003}. These basis states, constructed as a linear combination of $s$- and $p$-like atomic orbitals, are written as \cite{Bogusevschi_thesis_2018,Tomic_IEEEJSTQE_2003}

\begin{eqnarray}
%
    \left|u_{1}\right\rangle &=& |s ; \uparrow\rangle \, , \label{eq:ch_2_u_1} \\
%
    \left|u_{2}\right\rangle &=& \frac{i}{\sqrt{2}}[|x ; \uparrow\rangle+i|y ; \uparrow\rangle] \, , \label{eq:ch_2_u_2} \\
%
    \left|u_{3}\right\rangle &=& \frac{i}{\sqrt{6}}[|x ; \downarrow\rangle+i|y ; \downarrow\rangle-2|z ; \uparrow\rangle] \, , \label{eq:ch_2_u_3} \\
%
    \left|u_{4}\right\rangle &=& \frac{i}{\sqrt{3}}[|x ; \downarrow\rangle+i|y ; \downarrow\rangle+|z ; \uparrow\rangle] \, , \label{eq:ch_2_u_4} \\
%
   \left|u_{5}\right\rangle &=& -|s ; \downarrow\rangle \, , \label{eq:ch_2_u_5} \\
%
    \left|u_{6}\right\rangle &=& -\frac{i}{\sqrt{2}}[|x ; \downarrow\rangle-i|y ; \downarrow\rangle] \, , \label{eq:ch_2_u_6} \\
%
    \left|u_{7}\right\rangle &=& \frac{i}{\sqrt{6}}[|x ; \uparrow\rangle-i|y ; \uparrow\rangle+2|z ; \downarrow\rangle] \, , \label{eq:ch_2_u_7} \\
%
    \left|u_{8}\right\rangle &=& \frac{i}{\sqrt{3}}[|x ; \uparrow\rangle-i|y ; \uparrow\rangle-|z ; \downarrow\rangle] \, . \label{eq:ch_2_u_8}
%
\end{eqnarray}

These basis states $\ket{u_{b}}$ diagonalise the spin-orbit Hamiltonian, i.e.~at $\textbf{k} = 0$ the Hamiltonian is diagonal, with the diagonals then being the $\textbf{k} = 0$ band energies \cite{Tomic_IEEEJSTQE_2003}. These basis states come as Kramer's pairs $\vert u_{b + 4} \rangle = \widehat{T} \vert u_{b} \rangle$, where $\widehat{T}$ is the time-reversal operator, which relates states of opposite spin \cite{Tomic_IEEEJSTQE_2003}. Using these basis functions we can construct the 8-band $\textbf{k} \cdot \textbf{p}$ Hamiltonian as \cite{Bogusevschi_thesis_2018,Tomic_IEEEJSTQE_2003}

\begin{equation}
    \widehat{H} = \begin{pmatrix}
    E_{\text{CB}} & -\sqrt{3} T_{+} & \sqrt{2} U & -U & 0 & 0 & -T_{-} & -\sqrt{2} T_{-} \\
    & E_{\text{HH}} & \sqrt{2} S & -S & 0 & 0 & -R & -\sqrt{2} R \\
    & & E_{\text{LH}} & Q & T_{+}^{*} & R & 0 & \sqrt{3} S \\
    & & & E_{\text{SO}} & \sqrt{2} T_{+}^{*} & \sqrt{2} R & -\sqrt{3} S & 0 \\
    & & & & E_{\text{CB}} & -\sqrt{3} T_{-} & \sqrt{2} U & -U \\
    & & & & & E_{\text{HH}} & \sqrt{2} S^{*} & -S^{*} \\
    & & & & & & E_{\text{LH}} & Q \\
    & & & & & & & E_{\text{SO}} \\
    \end{pmatrix} \, ,
    \label{eq:ch_2_bulk_8_band_hamiltonian}
\end{equation}

\noindent
where Eq.~\eqref{eq:ch_2_bulk_8_band_hamiltonian} is a Hermitian matrix so the below-diagonal terms are the complex conjugates of the corresponding above-diagonal terms: $H_{nm} ( \textbf{k} ) = H_{mn}^{\ast} ( \textbf{k} )$. The functions that define the matrix elements of Eq.~\eqref{eq:ch_2_bulk_8_band_hamiltonian} are given by

\begin{eqnarray}
%
    E_{\mathrm{CB}} (\textbf{k}) &=& E_{\text{CB}_{0}}+\frac{\hbar^{2}}{2 m_{0}} s_{c}\left(k_{\parallel}^{2}+k_{z}^{2}\right) \, , \label{eq:ch_2_E_CB_k} \\
%
    E_{\mathrm{HH}} (\textbf{k}) &=& E_{\text{VB}_{0}}-\frac{\hbar^{2}}{2 m_{0}}\left(\gamma_{1}+\gamma_{2}\right) k_{\parallel}^{2}-\frac{\hbar^{2}}{2 m_{0}}\left(\gamma_{1}-2 \gamma_{2}\right) k_{z}^{2} \, , \label{eq:ch_2_E_HH_k} \\
%
    E_{\mathrm{LH}} (\textbf{k}) &=& E_{\text{VB}_{0}}-\frac{\hbar^{2}}{2 m_{0}}\left(\gamma_{1}-\gamma_{2}\right) k_{\parallel}^{2}-\frac{\hbar^{2}}{2 m_{0}}\left(\gamma_{1}+2 \gamma_{2}\right) k_{z}^{2} \, , \label{eq:ch_2_E_LH_k} \\
%
     E_{\mathrm{SO}} (\textbf{k}) &=& E_{\text{VB}_{0}}-\Delta_{\scalebox{0.6}{\text{SO}}}-\frac{\hbar^{2}}{2 m_{0}} \gamma_{1}\left(k_{\parallel}^{2}+k_{z}^{2}\right) \, , \label{eq:ch_2_E_SO_k} \\
%
    T_{\pm} (\textbf{k}) &=& \frac{1}{\sqrt{6}} P\left(k_{x} \pm i k_{y}\right) \, , \label{eq:ch_2_T_k} \\
%
    U (\textbf{k}) &=& \frac{1}{\sqrt{3}} P k_{z} \, , \label{eq:ch_2_U_k} \\
%
    S (\textbf{k}) &=& \sqrt{\frac{3}{2}} \frac{\hbar^{2}}{m_{0}} \gamma_{3} k_{z}\left(k_{x}-i k_{y}\right) \, , \label{eq:ch_2_S_k} \\
%
    R (\textbf{k}) &=& \frac{\sqrt{3}}{2} \frac{\hbar^{2}}{2 m_{0}}\left[\left(\gamma_{2}+\gamma_{3}\right)\left(k_{x}-i k_{y}\right)^{2}\right. \left.-\left(\gamma_{3}-\gamma_{2}\right)\left(k_{x}+i k_{y}\right)^{2}\right] \, , \label{eq:ch_2_R_k} \\
%
     Q (\textbf{k}) &=& -\frac{1}{\sqrt{2}} \frac{\hbar^{2}}{m_{0}} \gamma_{2} k_{\parallel}^{2}+\sqrt{2} \frac{\hbar^{2}}{m_{0}} \gamma_{2} k_{z}^{2} \, , \label{eq:ch_2_Q_k}
%
\end{eqnarray}

\noindent
where $\Delta_{\scalebox{0.6}{\text{SO}}}$ is the VB spin-orbit splitting energy, which describes the separation between the SO VB and the LH and HH VBs at $\textbf{k} = 0$. $s_{c}$ is the renormalised inverse zone-centre CB effective mass accounting for remote band contributions. Precise definition for $s_{c}$ is provided in Chapter~\ref{ch:chapter4}. $\gamma_{i}$ are the modified VB Luttinger parameters, defined as $\gamma_{1} = \gamma_{1}^{\, L} - \frac{E_{P}}{3E_{g}}$ and $\gamma_{2,3} = \gamma_{2,3}^{\, L} - \frac{E_{P}}{6E_{g}}$, where $\gamma_{i}^{\, L}$ are the bare VB Luttinger parameters (cf.~Eqs.\eqref{eq:ch_2_luttinger_1_generalised} - \eqref{eq:ch_2_luttinger_3_generalised}), $E_{g}$ is the $\Gamma$-point band gap, $E_{P}$ is the Kane parameter (cf.~Secs.~\ref{sec:ch_4_model_spurious_reparam} and ~\ref{sec:ch_6_unstrained_elec_band_parameters_kane_ellipticity}) and $P$ is the Kane matrix element (cf.~Eq.~\eqref{eq:ch_2_momentum_matrix_element}). As shown in Sec.~\ref{sec:ch_2_k.p_theory_outline}, the Luttinger parameters $\gamma_{i}$ can be used to define the VB edge effective masses \cite{Chuang_book_2009,Bogusevschi_thesis_2018}. The Luttinger parameters specify the zone-centre HH and LH VB (inverse) effective masses along different crystal directions as \cite{Vurgaftman_JAP_2001}

\begin{eqnarray}
%
    \left( \frac{1}{m^{\ast}_{\scalebox{0.7}{\text{HH}}}} \right)^{[001]} &=& \gamma_{1} - 2 \gamma_{2} \, , \label{eq:ch_2_vb_eff_mass_hh_001} \\
%
    \left( \frac{1}{m^{\ast}_{\scalebox{0.7}{\text{HH}}}} \right)^{[110]} &=& \frac{1}{2} \left( 2 \gamma_{1} - \gamma_{2} - 3 \gamma_{3} \right) \, , \label{eq:ch_2_vb_eff_mass_hh_110} \\
%
    \left( \frac{1}{m^{\ast}_{\scalebox{0.7}{\text{HH}}}} \right)^{[111]} &=& \gamma_{1} - 2 \gamma_{3} \, , \label{eq:ch_2_vb_eff_mass_hh_111} \\
%
    \left( \frac{1}{m^{\ast}_{\scalebox{0.7}{\text{LH}}}} \right)^{[001]} &=& \gamma_{1} + 2 \gamma_{2} \, , \label{eq:ch_2_vb_eff_mass_lh_001} \\
%
    \left( \frac{1}{m^{\ast}_{\scalebox{0.7}{\text{LH}}}} \right)^{[110]} &=& \frac{1}{2} \left( 2 \gamma_{1} + \gamma_{2} + 3 \gamma_{3} \right) \, , \label{eq:ch_2_vb_eff_mass_lh_110} \\
%
    \left( \frac{1}{m^{\ast}_{\scalebox{0.7}{\text{LH}}}} \right)^{[111]} &=& \gamma_{1} + 2 \gamma_{3} \, , \label{eq:ch_2_vb_eff_mass_lh_111}
%
\end{eqnarray}

\noindent
and the SO VB edge (inverse) effective mass as \cite{Vurgaftman_JAP_2001}

\begin{equation}
    \frac{1}{m^{\ast}_{\scalebox{0.7}{\text{SO}}}} = \gamma_{1} - \frac{E_{P} \, \Delta_{\scalebox{0.6}{\text{SO}}}}{3 E_{g} \left( E_{g} + \Delta_{\scalebox{0.6}{\text{SO}}} \right)} \, . \label{eq:ch_2_vb_eff_mass_so}
\end{equation}

We implement Eq.~\eqref{eq:ch_2_bulk_8_band_hamiltonian} using the axial approximation, which neglects the warping of the valence bands in the $x$-$y$ plane \cite{Altarelli_PRB_1985,Bogusevschi_thesis_2018}. This approximation eliminates the anisotropic nonparabolicity of the band structure obtained by diagonalising Eq.~\eqref{eq:ch_2_bulk_8_band_hamiltonian}, and imposes axial symmetry about $k_{z}$. In the axial approximation, the band dispersion in the (001)-plane depends only on the magnitude $k_{\parallel} = \vert \textbf{k}_{\parallel} \vert$ of the in-plane wave vector. This is achieved by neglecting the $( \gamma_{3} - \gamma_{2} )$ term in the function $R ( \textbf{k} )$ (cf.~Eq.~\eqref{eq:ch_2_R_k}).


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% Section 2: Plane wave expansion method %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Envelope function approximation and plane wave expansion method}
\label{sec:ch_2_pwem}

In this Section we outline the techniques employed in applying the 8-band $\textbf{k}\cdot\textbf{p}$ Hamiltonian presented in Sec.~\ref{sec:ch_2_k.p_theory_8-band_Hamiltonian} to compute the electronic properties of SL heterostructures. We begin by outlining the EFA and PWEM, through which we are able to express a general envelope function Hamiltonian in a plane wave basis. Next, in Sec.~\ref{sec:ch_2_pwem_1-band_example} we provide a demonstration of the technique outlined in Sec.~\ref{sec:ch_2_pwem_outline} for a 1-band PWEM using a computational supercell of a single SL period, explicitly including the miniband dispersion. We demonstrate the computational efficiency of this approach, owing largely to the relatively small size of the plane wave basis set required to achieve convergence of the calculated electronic properties. In Sec.~\ref{sec:ch_2_pwem_8-band_inclusion}, we outline the application of this approach to the 8-band $\textbf{k}\cdot\textbf{p}$ Hamiltonian used in this Thesis, including the calculation of properties such as the inter-band matrix elements. We also describe the inclusion of the miniband dispersion in the calculation of spectral properties, which will be employed in our calculation of spontaneous emission in Chapter~\ref{ch:chapter4} and optical absorption in Chapter \ref{ch:chapter5}.

\subsection{Outline of the methods}
\label{sec:ch_2_pwem_outline}

% EFA

We begin with a brief outline of the EFA, which begins using Bloch's theorem, as in Eq.~\eqref{eq:ch_2_elec_wavefunction_bloch}. Using Bloch's theorem, it is convenient to write the eigenstates $\ket{\psi_{n} (\textbf{k},\textbf{r})}$ as linear combinations of the Bloch states of the associated Hamiltonian \cite{Broderick_thesis_2015}

%\begin{equation}
%    \ket{\psi_{n} (\textbf{k},\textbf{r})} = \sum_{\textbf{k}} a_{n} (\textbf{k}) \, e^{i \textbf{k} \cdot \textbf{r}} \, \ket{u_{n} (\textbf{k} , \textbf{r})} \, . \label{eq:ch_2_efa_eigenstates}
%\end{equation}

\begin{equation}
    \ket{\psi_{n} (\textbf{k},\textbf{r})} = \sum_{\textbf{G}} a_{n\textbf{G}} (\textbf{k}) e^{i (\textbf{q}+\textbf{G}) \cdot\textbf{r}} \ket{u (\textbf{k})} \, , \label{eq:ch_2_efa_eigenstates}
\end{equation}

\noindent
where $\textbf{G}$ are the reciprocal lattice vectors. In the EFA and expanding the wave function at wave vector $\textbf{k}$, it is assumed that the periodic Bloch function $\ket{u (\textbf{\textbf{k}})}$ weakly depend on the wave vector $\textbf{q} = \textbf{k} - \textbf{k}_{0}$ in the vicinity of $\textbf{k}_{0}$. As such, Eq.~\eqref{eq:ch_2_efa_eigenstates} can be approximated as

%\begin{eqnarray}
%    \ket{\psi_{n} (\textbf{k}, \textbf{r})} &\approx& \ket{u_{n} (\textbf{k}_{0} , \textbf{r})} \, \sum_{\textbf{k}} a_{n} (\textbf{k}) \, e^{i \textbf{k} \cdot \textbf{r}} \, , \nonumber \\
%    
%    &\equiv& F_{n} (\textbf{k},\textbf{r}) \ket{u_{n} (\textbf{k}_{0}, \textbf{r}) } \, , \label{eq:ch_2_efa_eigenstates_approx}
%\end{eqnarray}

\begin{eqnarray}
    \ket{\psi_{n} (\textbf{k}, \textbf{r})} &\approx& \left( \sum_{\textbf{G}} a_{n\textbf{G}} (\textbf{k}) \, e^{i (\textbf{q} + \textbf{G} ) \cdot\textbf{r}} \right) \ket{u (\textbf{k}_{0})} \, , \nonumber \\
%    
    &\equiv& F_{n} (\textbf{k},\textbf{r}) \, \ket{u (\textbf{k}_{0})} \, , \label{eq:ch_2_efa_eigenstates_approx}
\end{eqnarray}

\noindent
where $\textbf{k}_{0}$ is the band edge wave vector, and $F_{n} (\textbf{k},\textbf{r})$ is a slowly-varying envelope function defined as a linear combination of the Bloch components of the eigenstates $\ket{\psi_{n} (\textbf{k},\textbf{r})}$. Equation \eqref{eq:ch_2_efa_eigenstates_approx} defines the EFA, where the probability densities associated with $\ket{u (\textbf{k}_{0})}$ and $F_{n} (\textbf{k},\textbf{r})$ are assumed to be normalised over the unit cell and crystal volume, respectively. In this Thesis we consider SL heterostructures in which carrier eigenstates are localised along the growth direction $z$ via quantum confinement arising from the periodic SL potential, which for an $N$-band $\textbf{k} \cdot \textbf{p}$ Hamiltonian means we can write Eq.~\eqref{eq:ch_2_efa_eigenstates_approx} as

\begin{equation}
    \ket{\psi_{n} (\textbf{k},z)} = \sum_{b=1}^{N} \, F_{nb} (\textbf{k},z) \, \ket{u_{b}} \, , \label{eq:ch_2_efa_eigenstates_N_band}
\end{equation}

\noindent
where $F_{nb} (\textbf{k},z)$ is the component of the carrier envelope function in subband $n$ associated with bulk band $b$, and $\ket{u_{b}}$ is the zone-centre Bloch state associated with bulk band $b$. For the 8-band $\textbf{k} \cdot \textbf{p}$ Hamiltonian employed in this Thesis, the corresponding Bloch states are given in Eqs.~\eqref{eq:ch_2_u_1} - \eqref{eq:ch_2_u_8}. In our analysis of SL heterostructures we also consider wave vector $\textbf{k}$ to not only consist of the in-plane wave vector $\textbf{k}_{\parallel}$, but also the wave vector associated with the miniband dispersion due to periodicity along the growth ($z$) direction. We denote the miniband wave vector along the growth direction as $q$, which gives $\textbf{k} = \left( \textbf{k}_{\parallel}, q \right)$. As described in Chapter~\ref{ch:chapter1}, the inclusion of the miniband dispersion is crucial in accurate description of the SL optoelectronic properties, and is typically neglected in $\textbf{k} \cdot \textbf{p}$-based methods. The choice of neglecting the miniband dispersion is typically made since (i) thick surrounding barrier layers in QW structures make the SL period $L$ large, and hence $q$ small, and (ii) electronic coupling between QW eigenstates localised in neighbouring SL periods can generally be neglected. As we will describe in Chapters \ref{ch:chapter4} and \ref{ch:chapter5}, the SLs considered in this Thesis possess narrow barrier layers. This means that the contributions associated with $q$ cannot be neglected. The multi-band envelope function equation can be expressed as

\begin{equation}
    \sum_{b=1}^{N} \widehat{H}_{b'b} (\textbf{k}) \, F_{nb}(\textbf{k},z) = E_{n} (\textbf{k}) \, F_{nb'} (\textbf{k},z) \, , \label{eq:ch_2_efa_env_func_eqn}
\end{equation}

\noindent
where $E_{n} (\textbf{k})$ is the SL band structure, and Eq.~\eqref{eq:ch_2_efa_env_func_eqn} is obtained using the orthonormality of the Bloch basis states, which gives $\braket{u_{b'} \vert u_{b}} = \delta_{b'b}$. We note, the operator $\widehat{H}_{b'b} (\textbf{k}) = \bra{u_{b'}} \widehat{H} (\textbf{k}) \ket{u_{b}}$ is obtained by substituting $k_{z} \rightarrow -i\frac{\text{d}}{\text{d}z}$ in the $\textbf{k} \cdot \textbf{p}$ matrix element $H_{b'b}(\textbf{k})$, and the resulting differential operators are ordered with respect to the position-dependent material parameters to ensure that the envelope functions $F_{nb} (\textbf{k},z)$ are continuous across the SL interfaces. Equation \eqref{eq:ch_2_efa_env_func_eqn} then represents a set of coupled differential equations, which can be solved to obtain the SL band structure $E_{n} ( \textbf{k} ) = E_{n} ( \textbf{k}_{\parallel}, q )$ and envelope functions $F_{nb} ( \textbf{k}, z )$. We will now outline the PWEM, which can be used to solve Eq.~\eqref{eq:ch_2_efa_env_func_eqn}.

% PWEM

The ``semi-analytical'' PWEM approach we employ in this work allows to write the quantised matrix elements of a general $N$-band $\textbf{k} \cdot \textbf{p}$ Hamiltonian as analytical expressions. The evaluation of these matrix elements does not rely on numerical differentiation or integration unlike, e.g., finite difference methods \cite{Broderick_thesis_2015}. The resulting matrix representation of the heterostructure Hamiltonian can then be diagonalised to solve Eq.~\eqref{eq:ch_2_efa_env_func_eqn}. We will begin our description of the PWEM with the example of a 1-band (Schr\"{o}dinger) model for the Kronig-Penney SL.

The 1-band Schr{\"o}dinger equation in the EFA is

\begin{equation}
    \widehat{H} \, F_{n} (z) = E_{n} \, F_{n} (z) \, , \label{eq:ch_2_pwem_1_band_schrodinger}
\end{equation}

\noindent
with the envelope function Hamiltonian $\widehat{H}$ is given as

\begin{equation}
    \widehat{H} = - \frac{\hbar^{2}}{2m_{0}} \frac{\text{d}}{\text{d}z} \left( \frac{1}{m^{\ast}(z)} \frac{\text{d}}{\text{d}z} \right) + V(z) \, , \label{eq:ch_2_pwem_1_band_env_func_hamiltonian}
\end{equation}

\noindent
where $m^{\ast}(z)$ and $V(z)$ are the position-dependent relative effective mass and potential, respectively. The PWEM consists of expanding the envelope function in a Fourier series as

\begin{equation}
    F_{n} (z) = \frac{1}{\sqrt{L}} \sum_{m=-M}^{M} a_{nm} \, e^{i ( G_{m} + q ) z} \, , \label{eq:ch_2_pwem_1_band_env_func_expansion}
\end{equation}

\noindent
where $G_{m} = \frac{2m\pi}{L}$, and $m \in \left[ -M, \dots , M \right] \in \mathbb{Z}$. As in our outline of the EFA, here we have included the miniband wave vector $q$. Equation \eqref{eq:ch_2_pwem_1_band_env_func_expansion} indicates that each eigenstate of Eq.~\eqref{eq:ch_2_pwem_1_band_env_func_hamiltonian} is described as a linear combination of a set of $2M+1$ plane wave basis functions, given as $\frac{1}{\sqrt{L}} e^{i (G_{m}+q) z}$. These basis functions are periodic along the growth direction $z$ such that, the envelope functions $F_{n}(z)$, which represent the eigenstates of $\widehat{H}$, are periodic on the closed interval $z \in \left[ -\frac{L}{2} , \frac{L}{2} \right]$. Here $L$ is the period of this interval, or the SL period. describe this interval as our calculational ``supercell'', the miniband dispersion associated with which is described by the wave vector $ -\frac{\pi}{L} \leq q \leq +\frac{\pi}{L} $. Substituting Eq.~\eqref{eq:ch_2_pwem_1_band_env_func_expansion} in Eq.~\eqref{eq:ch_2_pwem_1_band_schrodinger} leads to a reciprocal space eigenvalue equation, which is the plane wave representation of the Schr\"{o}dinger equation. In this equation, the eigenvectors are comprised of the Fourier coefficients $a_{nm}$ of the envelope function $F_{n} (z)$.

We now conclude this section by outlining the derivation of the general form of the envelope function equation in the plane wave representation. The reciprocal space equivalent of Eq.~\eqref{eq:ch_2_pwem_1_band_schrodinger}, substituting in Eq.~\eqref{eq:ch_2_pwem_1_band_env_func_hamiltonian}, is expressed as \cite{Broderick_thesis_2015}

\begin{equation}
    \sum_{m=-M}^{M} \left( \frac{1}{L} \int_{-\frac{L}{2}}^{\frac{L}{2}} e^{-i(G_{m'}+q)z} \, \widehat{H} \, e^{i(G_{m}+q)z} \, \text{d}z \right) \, a_{nm} = E_{n} \, a_{nm'} \, , \label{eq:ch_2_pwem_env_func_eqn_general_form}
\end{equation}

\noindent
where we have also multiplied across by $e^{-i(G_{m'}+q)z}$ and integrated over the supercell. We note, the orthonormality of the plane wave basis functions over the supercell gives

\begin{equation}
    \frac{1}{L} \int_{-L/2}^{L/2} e^{i(G_{m}-G_{m'})z} \text{d}z = \delta_{mm'} \, . \label{eq:ch_2_orthonormality_pwem}
\end{equation}

Rewriting the left side of Eq.~\eqref{eq:ch_2_pwem_env_func_eqn_general_form} by defining the matrix elements $H_{m'm}$ as

\begin{equation}
    H_{m'm} = \frac{1}{L} \int_{-\frac{L}{2}}^{\frac{L}{2}} e^{-i(G_{m'}+q)z} \, \widehat{H} \, e^{i(G_{m}+q)z} \, \text{d}z \, . \label{eq:ch_2_pwem_matrix_elements_general_form}
\end{equation}

so that Eq.~\eqref{eq:ch_2_pwem_env_func_eqn_general_form} becomes

\begin{equation}
    \sum_{m=-M}^{M} H_{m'm} a_{nm} = E_{n} a_{nm'} \, , \label{eq:ch_2_pwem_eigenvalue_eqn_general_form}
\end{equation}

which is the plane wave representation of Eq.~\eqref{eq:ch_2_pwem_1_band_schrodinger}. The plane wave basis states of Eq.~\eqref{eq:ch_2_pwem_1_band_env_func_expansion} have therefore been used to generate the matrix representation presented in Eq.~\eqref{eq:ch_2_pwem_matrix_elements_general_form} for the envelope function Hamiltonian $\widehat{H}$, for which $H_{m'm}$ has dimensions $\left( 2M+1 \right) \times \left( 2M+1 \right)$ and is Hermitian. The PWEM can be summarised in Eqs.~\eqref{eq:ch_2_pwem_matrix_elements_general_form} and \eqref{eq:ch_2_pwem_1_band_env_func_expansion}, which can be respectively used to calculate the matrix representation of a given envelope function Hamiltonian, and to construct the corresponding eigenstates of the system in real space. We note that Eq.~\eqref{eq:ch_2_pwem_1_band_env_func_expansion} becomes exact in the limit $M \rightarrow \infty$. The PWEM possesses significant advantages over real space techniques, e.g.~the finite difference method, since the number of components in Fourier space required to represent $F_{n}(z)$ is typically significantly less than the number of grid points required to obtain the equivalent accuracy in a real space calculation \cite{Broderick_thesis_2015}.


\subsection{Example: application of the plane wave expansion method to the Kronig-Penney model}
\label{sec:ch_2_pwem_1-band_example}

% 1-band PWEM

To demonstrate the accuracy and efficiency of the PWEM, we now apply the method to the Kronig-Penney (K-P) model. The K-P superlattice corresponds to a 1-band model of an idealised infinite SL \cite{OReilly_book_2002}. As in Sec.~\ref{sec:ch_2_pwem_outline} we assume that the calculational supercell, having period $L$, lies along the $z$-direction in the range $z \in \left[ -\frac{L}{2} , \frac{L}{2} \right]$. In this example we choose our QW to have thickness $a$ and lie in the range $z \in \left[ -\frac{a}{2} , \frac{a}{2} \right]$, with barrier thickness $b$, such that the SL has period $L = a + b$. The potential profile is given as

\begin{equation}
    V(z)=\left\{\begin{array}{lr}
0, & \vert z \vert<\frac{a}{2} \\
V_{0}, & \text{otherwise}
\end{array}\right. \label{eq:ch_2_1-band_potential_profile}
\end{equation}

\noindent
and the effective mass in these regions is described piecewise continuously as

\begin{equation}
    m^{\ast}(z)=\left\{\begin{array}{lr}
m_{w}^{\ast}, & |z|<\frac{a}{2} \\
m_{b}^{\ast}, & \text { otherwise }
\end{array}\right. \label{eq:ch_2_1-band_eff_mass_profile}
\end{equation}

\noindent
where $V_{0}$ is the QW potential depth, and $m_{w}^{\ast}$ $\left( m_{b}^{\ast} \right)$ is the effective mass in the well (barrier) region.

To solve the K-P superlattice described by Eqs.~\eqref{eq:ch_2_1-band_potential_profile} and \eqref{eq:ch_2_1-band_eff_mass_profile}, we begin by outlining the derivation of the 1-band PWEM. Using Eqs.~\eqref{eq:ch_2_pwem_1_band_env_func_hamiltonian} and \eqref{eq:ch_2_pwem_matrix_elements_general_form} we obtain

\begin{equation}
    H_{m'm} = -\frac{i \hbar^{2} (G_{m} + q)}{2 m_{0} L} \int_{-\frac{L}{2}}^{\frac{L}{2}} e^{-i (G_{m'}+q) z} \, \frac{\text{d}}{\text{d}z} \left( \frac{e^{i(G_{m}+q)z}}{m^{\ast}(z)} \right) \text{d}z + \frac{1}{L} \int_{-\frac{L}{2}}^{\frac{L}{2}} V(z) \, e^{i (G_{m}-G_{m'}) z} \, \text{d}z \, . \label{eq:ch_2_1-band_matrix_elements}
\end{equation}

The Fourier transform of a supercell-periodic function $f(z)$ in terms of the wave vector $G$ is defined as \cite{Broderick_thesis_2015}

\begin{equation}
    \widetilde{f}(G) = \frac{1}{L} \int_{-\frac{L}{2}}^{\frac{L}{2}} f(z) \, e^{-iGz} \, \text{d}z \, \label{eq:ch_2_fourier_transform}
\end{equation}

\noindent
and we define the ``characteristic function'' $\chi(z)$ for a QW of thickness $a$ lying in the range $z \in \left[ -\frac{a}{2} , \frac{a}{2} \right]$ by \cite{Broderick_thesis_2015}

\begin{equation}
    \chi(z)=\left\{\begin{array}{lr}
1, & |z|<\frac{a}{2} \\
0, & \text { otherwise }
\end{array}\right. \label{eq:ch_2_1-band_characteristic_function}
\end{equation}

\noindent
which takes a value of one within the QW only. Using Eqs.~\eqref{eq:ch_2_1-band_potential_profile} and \eqref{eq:ch_2_1-band_eff_mass_profile} we obtain

\begin{eqnarray}
    H_{m'm} &=& \left( \frac{\hbar^{2}}{2m_{0}} \frac{(G_{m'}+q) (G_{m}+q)}{m_{b}^{\ast}} + V_{0} \right) \delta_{m'm} \nonumber \\ &+& \widetilde{\chi}(G_{m}-G_{m'}) \left( \frac{\hbar^{2}}{2m_{0}} \, (G_{m'}+q) (G_{m'}+q) \left( \frac{1}{m_{w}^{\ast}} - \frac{1}{m_{b}^{\ast}} \right) - V_{0} \right) \, , \label{eq:ch_2_1-band_matrix_elements_specific}
\end{eqnarray}

\noindent
where $\widetilde{\chi} (G_{m} - G_{m'}) = \frac{a}{L} \text{sinc} \left( \frac{a}{2} (G_{m}-G_{m'}) \right)$ is the Fourier transform of the characteristic function $\chi(z)$, with $\delta_{m'm} = \text{sinc}\left( \frac{L}{2} (G_{m} - G_{m'}) \right)$. Equation \eqref{eq:ch_2_1-band_matrix_elements_specific} is an explicit expression for the matrix representation of the 1-band envelope function Hamiltonian described in Eq.~\eqref{eq:ch_2_pwem_1_band_env_func_hamiltonian}, in the plane wave basis defined in Eq.~\eqref{eq:ch_2_pwem_1_band_env_func_expansion}. We obtain bound state energies $E_{n}$ by diagonalising Eq.~\eqref{eq:ch_2_1-band_matrix_elements_specific}.

% Kronig-Penney

In order to benchmark the solutions for the 1-band PWEM we obtain via Eq.~\eqref{eq:ch_2_1-band_matrix_elements_specific}, we now present the solutions of the K-P model. The K-P model can be used to obtain the solutions, via Bloch's theorem, for a periodic SL potential \cite{OReilly_book_2002}. For a SL with well thickness $a$ and barrier thickness $b$, we apply boundary conditions requiring continuity of the envelope function and its derivative with respect to $z$ across the well/barrier interfaces, combined with the use of Bloch's theorem to enforce periodicity. Under the application of these boundary conditions the energy dispersion as a function of wave vector $q$ can be found as the solutions of the equation \cite{OReilly_book_2002}

\begin{equation}
    \text{cos} \left( q(a+b) \right) = \text{cos}(ka) \, \text{cosh}(\kappa b) + \frac{1}{2} \left( \frac{\kappa}{k} - \frac{k}{\kappa} \right) \, \text{sin}(ka) \, \text{sinh}(\kappa b) \, , \label{eq:ch_2_kronig_penney_sln}
\end{equation}

% where $k = \sqrt{\frac{2m_{0}m_{w}^{\ast}}{\hbar^{2}} E_{n}}$ and $\kappa = \sqrt{\frac{2m_{0}m_{b}^{\ast}}{\hbar^{2}} (V_{0} - E_{n})}$.

\noindent
where $k = \sqrt{\frac{2m_{w}^{\ast}}{\hbar^{2}} E_{n}}$ and $\kappa = \sqrt{\frac{2m_{b}^{\ast}}{\hbar^{2}} (V_{0} - E_{n})}$. Equation \eqref{eq:ch_2_kronig_penney_sln} is a transcendental equation in wave vector $q$, for which the solutions can be found via any standard root finding algorithm to obtain the miniband dispersion $E_{n} (q)$.

% Comparison - accuracy and efficiency

We now compare the 1-band PWEM, including the miniband wave vector $q$, and the K-P model for an exemplar periodic SL potential, with $V_{0} = 0.955$ eV and equal well and barrier thicknesses, such that $a = b = 2.3$ nm ($L = a + b = 4.6$ nm). We also take the effective mass in the well and the barrier to be equal $m_{w}^{\ast} = m_{b}^{\ast} = 0.025$ $m_{0}$. This periodic potential is shown in Fig.~\ref{fig:ch_2_K-P_PWEM_comparison}(a) for three supercell periods. The miniband dispersion $E (q)$ associated with the lowest energy bound state of the periodic potential in Fig.~\ref{fig:ch_2_K-P_PWEM_comparison}(a) is shown in Fig.~\ref{fig:ch_2_K-P_PWEM_comparison}(b). The solutions of Eq.~\eqref{eq:ch_2_kronig_penney_sln} for the Kronig-Penney model are shown using filled orange circles, and the solutions for the 1-band PWEM, obtained via the diagonalisation of the Hamiltonian matrix constructed using Eq.~\eqref{eq:ch_2_1-band_matrix_elements_specific}, are shown using a solid blue line. We note the strong agreement between the results obtained via the K-P model and the 1-band PWEM, demonstrating the ability of the PWEM to accurately describe SL bound states numerically and reproduce the K-P model miniband dispersion. In our calculations using the 1-band PWEM, the results were found to be sufficiently converged by $M = 9$, corresponding to a Hamiltonian matrix of size $19 \times 19 \left( = (2M+1) \times (2M+1) \right)$. This framework therefore provides a computationally efficient method with which to compute SL electronic structure. As we will see in Sec.~\ref{sec:ch_2_pwem_8-band_inclusion}, the PWEM allows for semi-analytical computation of SL properties having computed the energy dispersion and corresponding eigenstates, and here using a calculational supercell of a single SL period.

\begin{figure}[t!]
	\centering
   \includegraphics[width=1.0\textwidth]{./figures/chapter2/figure_1.pdf}
	\caption{(a) Potential profile for a Kronig-Penney superlattice having equal well and barrier thicknesses ($a = b = 2.3$ nm), and barrier potential energy depth $V_{0} = 0.955$ eV. (b) Solution for the energy dispersion (band structure) of the lowest energy bound states in the superlattice of (a) as a function of wave vector $q$ -- obtained using the Kronig-Penney model (filled orange circles) and the 1-band plane wave expansion method (solid blue line).}
	\label{fig:ch_2_K-P_PWEM_comparison}
\end{figure}

\subsection{Application to the 8-band $\textbf{k} \cdot \textbf{p}$ Hamiltonian}
\label{sec:ch_2_pwem_8-band_inclusion}

Having, in Sec.~\ref{sec:ch_2_pwem_1-band_example}, demonstrated the PWEM for a 1-band model, we now describe the application of this approach to the 8-band $\textbf{k} \cdot \textbf{p}$ Hamiltonian presented in Sec.~\ref{sec:ch_2_k.p_theory_8-band_Hamiltonian}. We also outline the calculation of key SL properties, such as the inter-band matrix elements, and describe the calculation of SL spectral properties with explicit inclusion of the miniband dispersion.

\textit{Application to the 8-band $\textbf{k}\cdot\textbf{p}$ Hamiltonian for SL heterostructure calculations:}

We now apply the PWEM to the 8-band $\textbf{k} \cdot \textbf{p}$ Hamiltonian (cf.~Eq.~\eqref{eq:ch_2_bulk_8_band_hamiltonian}) to compute the electronic structure of [001]-orientated quantum-confined SL heterostructures. We consider a SL in which the growth direction aligns with $z$ axis of a Cartesian coordinate system. As demonstrated in Sec.~\ref{sec:ch_2_pwem_outline}, the PWEM consists of expanding the envelope functions $F_{nb\mathbf{k}} (z)$ as a truncated Fourier series, such that a general SL eigenstate $\vert \psi_{n\mathbf{k}} (z) \rangle$ having energy $E_{n\mathbf{k}}$ at wave vector $\mathbf{k} = (\textbf{k}_{\parallel},q)$ is given by \cite{Healy_IEEE_2006,Murphy_NUSOD_2021,Murphy_JPDAP_2023}

\begin{equation}
    \left| \psi_{n \mathbf{k}} ( z ) \right. \rangle = \sum_{b = 1}^{ N_{b} } \left( \frac{1}{ \sqrt{L} } \, \sum_{m = -M}^{ +M } a_{nbm \mathbf{k} } \, e^{ i \left( G_{m} + q \right) z } \right) \left| u_{b} \right. \rangle \, ,
    \label{eq:ch_2_8_band_plane_wave_expansion}
\end{equation}

\noindent
where $N_{b} = 8$ is the number of Bloch basis states of the underlying bulk $\textbf{k} \cdot \textbf{p}$ Hamiltonian (cf.~Sec.~\ref{sec:ch_2_k.p_theory_8-band_Hamiltonian}), $\textbf{k}_{\parallel}$ is the component of the wave vector in the plane perpendicular to the growth direction, and $q$ is the component of the wave vector along the growth direction describing the miniband dispersion. We now have one envelope function component for each Bloch basis state, for which we employ a plane wave expansion akin to that in Eq.~\eqref{eq:ch_2_pwem_1_band_env_func_expansion}. Generating the plane wave representation of the associated multi-band EFA Schr\"{o}dinger equation $\widehat{H}_{\mathbf{k}} \vert \psi_{n\mathbf{k}} (z) \rangle = E_{n\mathbf{k}} \vert \psi_{n\mathbf{k}} (z) \rangle$ proceeds via the same procedure that led to Eq.~\eqref{eq:ch_2_pwem_eigenvalue_eqn_general_form}. This yields the reciprocal space matrix representation in the plane wave basis of the element $\widehat{H}_{b'b,\mathbf{k}}$ of the heterostructure Hamiltonian $\widehat{H}_{\mathbf{k}}$ \cite{Erhardt_Springer_2014,Broderick_thesis_2015}

\begin{equation}
     \left( \widetilde{H}_{b'b,\mathbf{k}} \right)_{m'm} = \frac{1}{L} \int_{ - \frac{L}{2} }^{ + \frac{L}{2} } e^{ - i \left( G_{m'} + q \right) z } \, \widehat{H}_{b'b,\mathbf{k}} (z) \, e^{ i \left( G_{m} + q \right) z } \,\text{d} z \, ,
     \label{eq:ch_2_8_band_plane_wave_hamiltonian}
\end{equation}

\noindent
which is a complex-valued $( 2M + 1 ) \times ( 2M + 1 )$ matrix. In Eq.~\eqref{eq:ch_2_8_band_plane_wave_hamiltonian}, the operators $\widehat{H}_{b'b,\mathbf{k}} (z)$ must be derived from the bulk $\mathbf{k} \cdot \mathbf{p}$ matrix elements (cf.~Eq.~\eqref{eq:ch_2_bulk_8_band_hamiltonian}). 

In the 1-band model, the Hamiltonian gets quantised to become an operator, which generates a $(2M + 1) \times (2M + 1)$ matrix representation using a set of plane wave basis states. For an $N$-band model, the same procedure is applied, but is applied to each element of the bulk $N \times N$ Hamiltonian matrix. Each bulk Hamiltonian matrix element is then assigned a $(2M + 1) \times (2M + 1)$ matrix representation, such that the matrix representation of the $N$-band Hamiltonian is of size $N(2M + 1) \times N(2M + 1)$. Previous comparative analysis between these boundary conditions and the ``exact'' Burt-Foreman EFA boundary conditions, demonstrated that asymmetric contributions in the latter play a minimal role in determining the character of eigenstates computed using an 8-band $\mathbf{k} \cdot \mathbf{p}$ Hamiltonian \cite{Meney_PRB_1994}. Symmetrising, quantising and evaluating Eq.~\eqref{eq:ch_2_8_band_plane_wave_hamiltonian} for all matrix elements of the underlying bulk $\mathbf{k} \cdot \mathbf{p}$ Hamiltonian yields the reciprocal space representation $\widetilde{H}_{\mathbf{k}}$ of $\widehat{H}_{\mathbf{k}}$, which is a $N_{b} ( 2M + 1 ) \times N_{b} ( 2M + 1 )$ Hermitian matrix. The eigenvalues of $\widetilde{H}_{\mathbf{k}}$ are the band energies $E_{n\mathbf{k}}$, while its eigenvectors are given by the Fourier coefficients $a_{nbm \mathbf{k} }$ of the envelope functions $F_{nb \mathbf{k} } (z)$. Having included the miniband dispersion in Eq.~\eqref{eq:ch_2_8_band_plane_wave_hamiltonian}, we note that the operators $\widehat{H}_{b'b,\mathbf{k}} (z)$ are in general differential operators, with their matrix representation via Eq.~\eqref{eq:ch_2_8_band_plane_wave_hamiltonian} being $q$-dependent. A detailed derivation of the matrix elements Eq.~\eqref{eq:ch_2_8_band_plane_wave_hamiltonian} for the 8-band $\textbf{k} \cdot \textbf{p}$ Hamiltonian employed in this work can be found in Ref.~\cite{Broderick_thesis_2015}.

% Bloch character

\textit{Bloch character:}

Having computed our 8-band $\textbf{k} \cdot \textbf{p}$ eigenstates using Eqs.~\eqref{eq:ch_2_8_band_plane_wave_expansion} and \eqref{eq:ch_2_8_band_plane_wave_hamiltonian}, we note that we can compute the ``Bloch character'' of these eigenstates. Having eigenstates that are linear combinations of the basis states $\ket{u_{b}}$ ($b = 1, 2, \dots, 8$), the Bloch character describes the hybridisation between the states in the 8-band description of the electronic structure, and the degree to which each heterostructure eigenstate can be associated with each basis state $\ket{u_{b}}$. For our choice of basis functions (cf.~Sec.~\ref{sec:ch_2_k.p_theory_8-band_Hamiltonian}), this would be CB, HH, LH, or SO character which they possess. The calculation of the Bloch character faciliates analysis of SL electronic structure in Chapters~\ref{ch:chapter4} and \ref{ch:chapter5}, e.g.~the identification of whether a given bound hole state is predominately HH- or LH-like. For the heterostructure eigenstate Eq.~\eqref{eq:ch_2_8_band_plane_wave_expansion}, the Bloch character associated with the bulk basis state $\vert u_{b} \rangle$ is given by

\begin{equation}
    c_{nb \textbf{k} } \equiv \int_{- L/2}^{+ L/2} \langle F_{nb \textbf{k} } (z) \vert \psi_{nb \textbf{k} } (z) \rangle \, \text{d} z = \sum_{m} \vert a_{nbm} \vert^{2} \,
    \label{eq:ch_2_bloch_band_character}
\end{equation}

\noindent
where Eq.~\eqref{eq:ch_2_bloch_band_character} can be evaluated directly using the Fourier coefficients of the envelope function $F_{nb \textbf{k} } (z)$, which we obtain via the eigenvectors of the plane wave matrix representation of the heterostructure Hamiltonian.

% OMEs

\textit{Inter-band optical (momentum) matrix elements:}

To analyse and optimise SL structures for optical emission and absorption (cf.~Chapters~\ref{ch:chapter4} and \ref{ch:chapter5}), we respectively compute the spontaneous emission (cf.~Sec.~\ref{sec:ch_4_model_energy_temp_dependence}) and optical absorption (cf.~Sec.~\ref{sec:ch_5_optical_abs_spectra_vs_x}). This requires accurate evaluation of the inter-band optical (momentum) matrix elements $\mathbf{p}_{n_{c}n_{v}\mathbf{k}}$ between an electron eigenstate $\left| \psi_{n_{c} \mathbf{k}} ( z ) \right. \rangle$ in conduction subband $n_{c}$ and a hole eigenstate $\left| \psi_{n_{v} \mathbf{k}} ( z ) \right\rangle$ in valence subband $n_{v}$, at wave vector $\mathbf{k}$. We recall from Chapter~\ref{ch:chapter1}, that optical emission or absorption follows strict $\mathbf{k}$-selection, $\Delta \mathbf{k} = 0$ for optical transitions, imposing that the electron and hole states have equal wave vector. We compute $\mathbf{p}_{n_{c}n_{v}\mathbf{k}}$ via the Hellmann-Feynman theorem \cite{Szmulowicz_PRB_1995}, using the SL Hamiltonian and eigenstates explicitly at each wave vector $\mathbf{k}$

\begin{equation} % Units = kg m s^{-1}
    p_{ n_{v} n_{c} \mathbf{k} }^{ ( e ) } \equiv \widehat{e} \cdot \mathbf{p}_{n_{v}n_{c}\mathbf{k}} = \frac{ m_{0} }{ \hbar } \langle \psi_{ n_{v} \mathbf{k} } \vert \, \widehat{e} \cdot \nabla_{\mathbf{k}} \widehat{H}_{\mathbf{k}} \, \vert \psi_{ n_{c} \mathbf{k} } \rangle \, ,
    \label{eq:ch_2_momentum_matrix_element}
\end{equation}

\noindent
where $\widehat{e}$ is a unit vector denoting the polarisation of the emitted or absorbed photon, with $\widehat{e} \cdot \nabla_{\mathbf{k}} \widehat{H}$ then denoting the directional derivative of the supercell Hamiltonian with respect to the component of $\mathbf{k}$ parallel to the photon polarisation. Evaluation of Eq.~\eqref{eq:ch_2_momentum_matrix_element} requires calculation of the reciprocal space matrix representation of $\widehat{e} \cdot \nabla_{\mathbf{k}} \widehat{H}_{\mathbf{k}}$, which can be evaluated analytically by replacing $\widehat{H}_{b'b,\mathbf{k}} (z)$ by $\widehat{e} \cdot \nabla_{\mathbf{k}} \widehat{H}_{b'b,\mathbf{k}} (z)$ in Eq.~\eqref{eq:ch_2_8_band_plane_wave_hamiltonian}. The simple analytical form of the plane wave basis states of complex exponentials allows the matrix representation of $\widehat{e} \cdot \nabla_{\textbf{k} } \widehat{H}_{ \textbf{k} }$ to be computed analytically in the same manner as for the Hamiltonian matrix elements (cf.~Eq.~\eqref{eq:ch_2_8_band_plane_wave_hamiltonian}). This analytical evaluation is possible as, the solution of the multi-band envelope function expansion is in reciprocal space (cf.~Sec.~\ref{sec:ch_2_pwem_outline}) and the plane wave basis functions possess a simple form in real space, leading to the spatial integrations associated with the calculation of matrix elements between states being analytical. As a result, the PWEM has the additional advantage of removing the need to perform the costly numerical quadrature associated with real space implementations of such calculations. By explicitly employing the SL eigenstates and Hamiltonian, we take all effects such as carrier localisation, partial reflection of wave functions at layer interfaces, and band hybridisation, explicitly into account.

% Chris thesis: Additionally, as we will demonstrate in Section 7.5, the multi-band equivalent of the expansion of F n (z) in Eq. (7.9) enables signiﬁcant simpliﬁcation of the calculation of matrix elements describing transitions between QW states. The reason for this is that because the solution of the multi-band equivalent of Eq. (7.7) proceeds in reciprocal space, the spatial integrations associated with the calculation of matrix elements between states can be performed analytically due to the simple form of the plane wave basis functions in real space. This then leads to analytical expressions for several classes of matrix elements in terms of (i) the Fourier coefﬁcients describing the eigenstates, (ii) position dependent bulk material parameters, and (iii) quantised matrix elements of the underlying bulk k·p Hamiltonian. As a result, the plane wave expansion method has the additional advantage of removing the need to perform the costly numerical quadrature associated with real space implementations of such calculations.

% Accounting for miniband dispersion in the calculation of spectral properties

\textit{Accounting for the miniband dispersion in the calculation of spectral properties:}

To compute energy-dependent properties in a QW, e.g., density of states, spontaneous emission and optical absorption, it is necessary to integrate over the in-plane degrees of freedom -- i.e.~the in-plane wave vector $\mathbf{k}_{\parallel}$. For a SL this is modified to incorporate $q$-dependent miniband dispersion and eigenstates as

\begin{equation}
    \int \frac{\mathrm{d} \mathbf{k}_{\|}}{(2 \pi)^{2}} \longrightarrow\left(\frac{2 \pi}{L}\right)^{-1} \int_{-\frac{\pi}{L}}^{+\frac{\pi}{L}} \mathrm{~d} q \int \frac{\mathrm{d} \mathbf{k}_{\|}}{(2 \pi)^{2}} \, ,
    \label{eq:ch_2_integration_conversion}
\end{equation}

\noindent
where $\frac{ 2 \pi }{L}$ is the SL BZ extent along $z$. As our calculations employ the axial approximation (cf.~Sec.~\ref{sec:ch_2_k.p_theory_8-band_Hamiltonian}), the in-plane integration is reduced to a one-dimensional integral, i.e.~$\int d \textbf{k}_{\parallel} \rightarrow 2 \pi \int k_{\parallel} d k_{\parallel}$. Thus, instead of calculating the SL electronic structure on a grid with order of $N^{2}$ points $\textbf{k}_{\parallel} = ( k_{x}, k_{y} )$ this allows for calculation of the electronic structure on a grid of $N$ points along $k_{\parallel} = \vert \textbf{k}_{\parallel} \vert$, substantially reducing computational expense. Therefore, in the axial approximation, we perform two-dimensional integrals over $k_{\parallel}$ and $q$ to compute spectral properties.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% Section 3: Conclusions %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Conclusions}
\label{sec:ch_2_conclusions}

In this Chapter we have presented the theoretical methods employed in our calculations of SL electronic and optical properties. This theoretical framework is employed in the analysis of radiative recombination in InAs/GaSb SLs in Chapter~\ref{ch:chapter4}, and of optical absorption in strain-balanced InAs/InAs$_{1-x}$Sb$_{x}$ SLs in Chapter~\ref{ch:chapter5}. In Sec.~\ref{sec:ch_2_k.p_theory}, we began by outlining the $\textbf{k} \cdot \textbf{p}$ method for an electron in a periodic solid, using time-independent (Rayleigh-Schr\"{o}dinger) perturbation theory. We then presented the form of the multi-band $\textbf{k} \cdot \textbf{p}$ Hamiltonian employing L{\"o}wdin renormalisation, which partitions the eigenstates of a periodic potential into two classes, class A and B. Class A states are those that we choose as the basis within which to generate the $\textbf{k} \cdot \textbf{p}$ Hamiltonian. Class B states are the remaining, energetically remote states, whose impact on the states in class A is important in determining the semiconductor electronic structure, notably the effective masses. The use of L{\"o}wdin renormalisation to paritition these eigenstates allows us to focus on a subset (class A), while treating their coupling to other bands (class B) perturbatively. We then introduced the 8-band $\textbf{k} \cdot \textbf{p}$ Hamiltonian, including spin-orbit splitting, for zinc blende semiconductors that will underpin our work on SL structures in Chapters \ref{ch:chapter4} and \ref{ch:chapter5}. This Hamiltonian is constructed using zone-centre ($\Gamma$-point) Bloch basis functions $\ket{u_{b}}$, consisting of one conduction and three valence spin degenerate states. This 8-band $\textbf{k} \cdot \textbf{p}$ Hamiltonian allows to compute the band structure in the vicinity of the $\Gamma$-point. Following this, we described aspects of our implementation of this multi-band $\textbf{k} \cdot \textbf{p}$ Hamiltonian for quantum-confined heterostructures. In Sec.~\ref{sec:ch_2_pwem} we presented our implementation of the PWEM in the EFA, and the application of this approach in conjunction with the 8-band $\textbf{k} \cdot \textbf{p}$ Hamiltonian. In Sec.~\ref{sec:ch_2_pwem_outline}, we provided an outline of this approach, including the miniband dispersion via the wave vector $q$ and, in Sec.~\ref{sec:ch_2_pwem_1-band_example}, demonstrated how this approach allows for efficient computation of SL electronic properties using analytical expressions for the matrix elements of the $\textbf{k} \cdot \textbf{p}$ Hamiltonian, using a 1-band PWEM to recover the K-P miniband dispersion. This demonstrated the accuracy and efficiency of the description of the miniband dispersion for SL heterostructure calculations prescribed by inclusion of $q$, which can be obtained using a small number of plane waves in a 1-band PWEM calculation. Next, in Sec.~\ref{sec:ch_2_pwem_8-band_inclusion} we described the application of the PWEM to the 8-band $\textbf{k} \cdot \textbf{p}$ Hamiltonian, which includes the description of the miniband dispersion along the growth direction $z$. As discussed in Chapter~\ref{ch:chapter1}, typical $\textbf{k} \cdot \textbf{p}$-based calculations neglect the miniband dispersion, which is required in accurate predictive calculations. Finally, in Sec.~\ref{sec:ch_2_pwem_8-band_inclusion} we also described the analytical calculation of momentum matrix elements in the PWEM using our computed SL eigenstates. The theoretical methods we have detailed in this Chapter underpin the accurate and computationally efficient calculations of SL heterostructures in this Thesis. These methods are employed in the analysis of radiative recombination in InAs/GaSb SLs in Chapter~\ref{ch:chapter4}, and of optical absorption in InAs/InAs$_{1-x}$Sb$_{x}$ strain-balanced SLs in Chapter~\ref{ch:chapter5}.