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%%%% Appendix B:  %%%%
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\chapter{Interface softening in the plane wave expansion method} \label{ch:appendix2}
\lhead{\emph{Interface softening in the plane wave expansion method}}

In this Appendix, the procedure outlined in Chapter~\ref{ch:chapter4} to address the emergence of the Gibbs phenomenon in calculated superlattice (SL) envelope functions is supplemented with further detail. We present the procedure for applying interface softening to an arbitrary multi-quantum well heterostructure, which in Chapters~\ref{ch:chapter4} and \ref{ch:chapter5} is applied to InAs/GaSb SLs and strain-balanced InAs/InAs$_{1-x}$Sb$_{x}$ SLs respectively.

We consider an arbitrary multi-quantum well heterostructure comprised of a total of $N_{l}$ material layers. The spatial variation $P (z)$ of a generic material parameter $P$ along the growth direction $z$ is given by

\begin{equation}
    P(z)=\sum_{l=1}^{N_{l}} P_{l} \chi_{l}(z) \, ,
    \label{eq:appendix_2_gen_mat_param}
\end{equation}

\noindent
where $P_{l}$ is the value of $P$ in the material comprising of the $l^{\text{th}}$ layer of the heterostructure, and $\chi_{l}(z)$ is the characteristic function associated with the $l^{\text{th}}$ layer of the heterostructure (cf.~Sec.~\ref{sec:ch_4_elec_carrier_localisation}). Assuming ``hard'' -- i.e.~spatially abrupt -- interfaces between layers, each layer is taken to lie in the range $z_{l} \leq z \leq z_{l+1}$, with $\chi_{l}(z)$ then defined to have value $= 1$ within the layer and $= 0$ otherwise. We can therefore write

\begin{equation}
    \chi_{l}(z)=\theta\left(z-z_{l}\right)-\theta\left(z-z_{l+1}\right) \, ,
    \label{eq:appendix_2_spatially_abrupt}
\end{equation}

\noindent
where $\theta$ is the Heaviside step function.

We emphasise that Eq.~\eqref{eq:appendix_2_spatially_abrupt} describes spatially abrupt interfaces between layers in the heterostructure, such that $P (z)$ is a piecewise continuous function of position $z$. To ``soften'' this interface -- e.g.~to simulate non-abrupt interfaces in realistic heterostructures, by emulating intermixing between material layers -- we can convolute $\chi_{l}(z)$ with a Gaussian distribution

\begin{equation}
    g(z)=\frac{1}{\sigma \sqrt{2 \pi}} \exp \left(-\frac{z^2}{2 \sigma^2}\right) \, ,
    \label{eq:appendix_2_gaussian}
\end{equation}

\noindent
to obtain the ``softened'' layer characteristic function

\begin{equation}
    \begin{aligned}
    \left(\chi_l * g\right)(z) & =\int_{-\infty}^{+\infty} \chi_l(z) g(z-t) \mathrm{d} t=\frac{1}{\sigma \sqrt{2 \pi}} \int_{z_j}^{z_{j+1}} \exp \left(-\frac{(z-t)^2}{2 \sigma^2}\right) \mathrm{d} t \\
    & =\frac{1}{2}\left[\operatorname{erf}\left(\frac{z-z_j}{\sqrt{2} \sigma}\right)-\operatorname{erf}\left(\frac{z-z_{j+1}}{\sqrt{2} \sigma}\right)\right] \, ,
    \end{aligned}
    \label{eq:appendix_2_soft_char_func}
\end{equation}

\noindent
where $\operatorname{erf}$ is the error function.

In the plane wave expansion method, geometric information is encapsulated via the Fourier transforms $\tilde{\chi}_{l}(G)$ of the layer characteristic functions (cf.~Sec.~\ref{sec:ch_2_pwem}). To compute the Fourier transform of the ``softened'' layer characteristic function of Eq.~\eqref{eq:appendix_2_soft_char_func}, we exploit the convolution theorem

\begin{equation}
    \mathcal{F} \left\{\left(\chi_l * g\right)(z)\right\}(G)=\tilde{\chi}_l(G) \tilde{g}(G)=\tilde{\chi}_l(G) \exp \left(-\frac{G^2 \sigma^2}{2}\right) \, ,
    \label{eq:appendix_2_conv_theorem}
\end{equation}

\noindent
where $\widetilde{g}(G)=\exp \left(-\frac{\sigma^{2} G^{2}}{2}\right)$ is the Fourier transform of Eq.~\eqref{eq:appendix_2_gaussian}. This therefore allows the Fourier transform of the ``softened'' layer characteristic function to be obtained by simple multiplication of its ``hard'' equivalent $\tilde{\chi}_{l}(G)$. We now provide an example of the implementation of this technique in Fig.~\ref{fig:app_B_envelope_functions}.

In Fig.~\ref{fig:app_B_envelope_functions} we show the calculated $e1$ probability density, and associated conduction band offsets and energy of the $e1$ state, for an exemplar InAs/GaSb SL having equal InAs and GaSb layer thickness (cf.~Sec.~\ref{sec:ch_4_elec_carrier_localisation}) (a) with, and (b) without interface softening. In these calculations $M = 25$ $\left( 2M+1 = 51 \right)$. This calculation does not attempt to treat intermixing effects, but instead to introduce minimal softening that mitigates the Gibbs phenomenon. These calculations, and those of Chapters~\ref{ch:chapter4} and \ref{ch:chapter5} use $\sigma = 0.075 $ nm. Comparing Figs.~\ref{fig:app_B_envelope_functions}(a) and \ref{fig:app_B_envelope_functions}(b), demonstrates that this small standard deviation for the Gaussian distribution is sufficient to address the emergence of the Gibbs phenomenon, while producing minimal change in the energy of the bound eigenstate ($< 2$ meV at $z = 0$). Therefore, the implementation of this interface softening mitigates the oscillations in the probability density generated by the Gibbs phenomenon, while not meaninfully impacting the SL eigenstates in any other manner.

\begin{figure}[H]
\centering
	\includegraphics[width=1.0\textwidth]{./figures/appendix/figure_appendix_B1.pdf}
	\caption{(a) With no interface softening implemented -- i.e.~``hard'' interfaces -- calculated conduction band (solid red line) offsets, illustrated for two periods of a InAs/GaSb SL having equal InAs and GaSb layer thickness $t = 2.14$ nm (SL period $L = 4.28$ nm), and total number of plane waves $2M+1 = 51$ $\left( M = 25 \right)$. The dash-dotted red line denotes the energy of the lowest energy bound electron state $e1$. The $e1$ probability density is shown using a solid red line, shifted so that the zero of probability density lies at the $e1$ energy. (b) Same as in (a) with interface softening implemented -- i.e.~``softened'' interfaces.}
	\label{fig:app_B_envelope_functions}
\end{figure}
