\chapter{Revised and improved band parameters for III-P semiconductors} \label{ch:chapter6}
\lhead{\emph{Revised and improved band parameters for III-P semiconductors}}


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

\section{Overview}
\label{sec:ch_6_overview}

% III-P applications and and high-level motivation

In this Chapter we examine the band parameters of III-P semiconductors.\footnote{The calculations in this Chapter were performed in collaboration with my Co-Supervisor, Dr.~Christopher A.~Broderick, who performed fitting to spectroscopic ellipsometry data in Sec.~\ref{sec:ch_6_elec_structure_validation}, calculation of effective masses and Luttinger parameters in Sec.~\ref{sec:ch_6_unstrained_elec_band_parameters_eff_mass_luttinger}, and the valence band offset calculations in Sec.~\ref{sec:ch_6_valence_band_offsets}.} III-P semiconductors are well established as the basis of red light emitting diodes \cite{Kish_SS_1997} and diode lasers \cite{Kobayashi_EL_1985,Geels_EL_1992,Walter_APL_2001,Ryou_APL_2001,Huang_IEEEJQE_2019}, and there remains significant interest in III-P heterostructures for the development of visible-wavelength light emitters for applications in solid-state lighting and in display technologies for virtual and augmented reality \cite{Oh_OE_2018,Han_OM_2021,Jung_SR_2021}, in addition to applications of III-P quantum wells and dots in integrated Si photonics \cite{Dhingra_Optica_2021,Dhingra_APL_2020,Chang_ACSAMI_2022}, and as hosts of Wigner crystallisation for quantum computing \cite{Mintairov_PRB_2018,Mintairov_APL_2019}. Accurate knowledge of the electronic band parameters is essential to facilitate predictive theory and design of the physical properties of these nanostructures. Despite its relevance to practical applications, limited detailed investigation of the electronic structure of AlGaInP alloys and, in particular, the properties of AlP has been undertaken.

Notably, there remains large uncertainty surrounding several key band structure parameters for AlP. Several sets of published parameters for AlP have been inferred based on extrapolation of parameters for Al$_{x}$(Ga,In)$_{1-x}$P alloys \cite{Vurgaftman_JAP_2001,Vurgaftman_book_2020,Onton_JAP_1970}. Key examples include the relative energy and ordering of the $\Gamma$- and L-point conduction band (CB) valley minima above the indirect X-point CB minimum, and the magnitude of the direct $\Gamma_{6c}$-$\Gamma_{8v}$ band gap. Literature reports differ on the presence of X-L-$\Gamma$ vs.~X-$\Gamma$-L CB valley energy ordering \cite{Huang_JPCS_1985,Williams_JEM_1978,Hwang_JAC_2014}, and reported values of the direct $\Gamma_{6c}$-$\Gamma_{8v}$ band gap vary by $> 1$ eV \cite{Stukel_PR_1969,Wing_PRM_2019,Onton_JAP_1970,Huang_JPCS_1985,Vurgaftman_JAP_2001,Bastos_JAP_2018,Vurgaftman_book_2020}. While less uncertainty exists for (Ga,In)P, there are key uncertainties that, while smaller than in AlP, can still significantly impact the results of electronic structure calculations for quantum-confined heterostructures \cite{Bogusevschi_thesis_2018}. This includes a number of CB minima energy splittings between the $\Gamma$, X and L CB valleys in InP \cite{Vurgaftman_JAP_2001}. For GaP, there exists a large spread in reported effective masses \cite{Vurgaftman_JAP_2001}. Additionally, the band edge deformation potentials for III-P compounds have not been subject to rigorous examination and, in many cases, have yet to be explicitly investigated \cite{Vurgaftman_JAP_2001,Bogusevschi_thesis_2018}. A number of commonly employed parameter values have been taken from the extrapolation of results measured from Ga$_{1-x}$In$_{x}$P lattice-matched to GaAs \cite{Vurgaftman_JAP_2001,Meney_IEEE_1995,Mowbray_APL_1994,Bogusevschi_thesis_2018}.

% In GaP, the magnitudes of the $\Gamma_{6c}$-$\Gamma_{8v}$ and X$_{6c}$-$\Gamma_{8v}$ band gaps lack certainty \cite{Auvergne_PRB_1975,Dean_JAP_1967,Zallen_PR_1964,Hakki_JAP_1970}, and a number of the band gaps presented in literature are taken from the interpolation of results measured from Ga$_{1-x}$In$_{x}$P lattice-matched to GaAs \cite{Meney_IEEE_1995,Mowbray_APL_1994}.

Systematic review of literature parameters for III-P compounds highlights the challenges disparate parameter values present for predictive simulations \cite{Bogusevschi_thesis_2018}. The spread in reported values of key material parameters for III-P compounds -- and specifically for AlP \cite{Vurgaftman_JAP_2001} -- are sufficiently large to qualitatively change the nature (direct or indirect) of the predicted band gap in III-P heterostructures \cite{Bogusevschi_thesis_2018}. Improved knowledge of the band parameters and band edge deformation potentials is thus critical to underpin accurate and predictive analysis of III-P heterostructures.

% What we do

We present a detailed analysis of the band structure of AlP, GaP and InP, which hereafter we refer to as (Al,Ga,In)P, using first-principles calculations based on density functional theory (DFT) employing a hybrid exchange-correlation (XC) functionals. Our calculations follow the method outlined in Chapter~\ref{ch:chapter3}, meaning DFT calculations were performed using the projector augmented-wave (PAW) method as implemented in VASP. We use a combination of XC functionals to compute structural, elastic and electronic band parameters. For the calculation of structural parameters and elastic moduli we use the local density approximation (LDA) and revised Heyd-Scuseria-Ernzerhof for solids (HSEsol) XC functionals \cite{Schimka_JCP_2011}. Bulk electronic band structure calculations are carried out using the HSEsol XC functional, and also in the meta-generalised gradient approximation (meta-GGA) by combining the Tran-Blaha modified Becke-Johnson (TB-mBJ) exchange potential with the LDA to the electronic correlation \cite{Tran_PRL_2009,Kim_PRB_2010}. The potential-only TB-mBJ functional possesses no associated exchange-correlation energy, meaning that it is not possible to compute forces, thereby precluding TB-mBJ-based structural relaxation. We use LDA-calculated structural parameters in our TB-mBJ calculations. The TB-mBJ exchange potential, combined with the LDA to the electronic correlation, has become a popular approach to analyse semiconductor band structure, due to its resolution of the band gap underestimation problem at low (near-LDA) computational expense. This work does not perform a direct comparison between meta-GGA and hybrid functional band structure calculations. Such benchmark calculations have been performed by several other groups \cite{Laurien_PRB_2022,Borlido_CM_2020}. Therefore we employ two independent frameworks: (i) computationally expensive HSEsol hybrid functional calculations, and (ii) comparatively computationally inexpensive LDA + TB-mBJ calculations. In the context of DFT, approach (i) provides state-of-the-art accuracy, while approach (ii) has gained popularity for high-throughput calculations (e.g.~composition-dependent alloy supercells \cite{Polak_JPDAP_2017,Halloran_OQE_2019,Pearce_PRM_2022}). The HSEsol (TB-mBJ) calculations employed a plane wave cutoff energy of at least 400 eV (500 eV), and $\Gamma$-centred grids containing at least $11 \times 11 \times 11$ $\textbf{k}$-points for Brillouin zone (BZ) integration. A key initial observation is an apparent inconsistency between reported values of the direct band gap in AlP. Here, detailed analysis of literature data highlights the inadequacy of the interpretation of experimental measurements, motivating us to re-evaluate literature ellipsometry data. Careful re-evaluation of experimental data in light of our DFT calculations allows to revise the magnitude of the direct $\Gamma_{6c}$-$\Gamma_{8v}$ band gap upwards by $\approx$ 1 eV vs.~the previously accepted value. This reconciles theory and experiment for this critical material parameter. Then, with a revised set of parameters in place for AlP, we also compute parameters for GaP and InP to reduce the systematic uncertainties present in the literature. Specifically, we compute revised values of the band edge effective masses, band edge deformation potentials and direct and indirect band gap pressure coefficients. While previous III-P parameter sets have generally been assembled piecewise from the experimental and theoretical literature \cite{Vurgaftman_JAP_2001}, our calculations provide a consistent set of material parameters for III-P semiconductors. This resolves a significant and decades-old inconsistency for the direct band gap in AlP. This provides an improved basis from which to establish predictive calculations of the optoelectronic properties of III-P heterostructures for classical and quantum photonic applications.

% Chapter layout

The remainder of this chapter is organised as follows. In Sec.~\ref{sec:ch_6_elastic_structural} we present our calculated structural parameters for zinc-blende III-P materials. In Sec.~\ref{sec:ch_6_elastic_structural_lattice_param} we calculate and analyse the equilibrium lattice parameters for (Al,Ga,In)P. In Sec.~\ref{sec:ch_6_elastic_structural_elastic_param} we present our analysis of the elastic moduli for (Al,Ga,In)P. In Sec.~\ref{sec:ch_6_elec_structure} we present our analysis of the electronic structure of (Al,Ga,In)P. In Sec.~\ref{sec:ch_6_elec_structure_fitting} we describe our approach to fitting the exact exchange (Becke-Roussel) mixing in the HSEsol (TB-mBJ) calculations, to reproduce the experimentally known fundamental band gap. With this procedure outlined, in Sec.~\ref{sec:ch_6_elec_structure_validation} we validate this approach via detailed comparison to experimental results for AlP. Having validated our electronic structure calculations, in Sec.~\ref{sec:ch_6_unstrained_elec_band_parameters} we compute unstrained electronic band parameters, with comparison to existing literature. In Sec.~\ref{sec:ch_6_unstrained_elec_band_parameters_energy_gaps} we analyse the calculated direct and indirect band gaps, and key band edge energy splittings. In Sec.~\ref{sec:ch_6_unstrained_elec_band_parameters_eff_mass_luttinger} we discuss the calculated effective masses and Luttinger parameters. In Sec.~\ref{sec:ch_6_unstrained_elec_band_parameters_kane_ellipticity} we detail calculated Kane parameters $E_{P}$, and demonstrate of the ellipticity of the extracted parameters set for application in 8-band $\textbf{k} \cdot \textbf{p}$ Hamiltonians. In Sec.~\ref{sec:ch_6_strained_elec_band_parameters} we analyse the computed deformation potentials and hydrostatic pressure coefficients. In Sec.~\ref{sec:ch_6_valence_band_offsets} we analyse the predicted natural (unstrained) valence band offsets (VBOs). Finally, in Sec.~\ref{sec:ch_6_conclusions} we summarise and conclude.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% Section 2: Elastic moduli and structural parameters %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Structural parameters}
\label{sec:ch_6_elastic_structural}

In this Section we describe and present our calculations of the equilibrium lattice parameters and elastic moduli of (Al,Ga,In)P. We begin with a brief recapitulation of linear elasticity theory, which we then apply to obtain equilibrium lattice parameters $a$ for zinc-blende (Al,Ga,In)P. Our calculation of the lattice parameters is required for the calculation of the unstrained and strained electronic band parameters, analysis of these parameters follows this Section, beginning from Sec.~\ref{sec:ch_6_elec_structure}. We then describe our calculation of and present our results for the elastic moduli $C_{ij}$, bulk modulus $B_{0}$ and internal strain (Kleinman) parameter $\zeta$.

\subsection{Lattice parameter}
\label{sec:ch_6_elastic_structural_lattice_param}

% Linear elasticity theory

We consider materials with zinc-blende crystal structure, this being the equilibrium crystal structure of (Al,Ga,In)P. The face-centred cubic lattice vectors of the zinc-blende primitive unit cell are given in Cartesian coordinates by

%\begin{eqnarray}
%
%    \textbf{a}_{1} &=& \frac{a}{2} \, \widehat{y} + \frac{a}{2} \, \widehat{z} \, , \label{eq:ch_6_lattice_vector_1} \\
%
%    \textbf{a}_{2} &=& \frac{a}{2} \, \widehat{x} + \frac{a}{2} \, \widehat{z} \, , \label{eq:ch_6_lattice_vector_2} \\
%
%    \textbf{a}_{3} &=& \frac{a}{2} \, \widehat{x} + \frac{a}{2} \, \widehat{y} \, , \label{eq:ch_6_lattice_vector_3}
%
%\end{eqnarray}

\begin{eqnarray}
%
    \textbf{a}_{1} &=& \left( 0, \frac{a}{2}, \frac{a}{2} \right) \, , \label{eq:ch_6_lattice_vector_1} \\
%
    \textbf{a}_{2} &=& \left( \frac{a}{2}, 0, \frac{a}{2} \right) \, , \label{eq:ch_6_lattice_vector_2} \\
%
    \textbf{a}_{3} &=& \left( \frac{a}{2}, \frac{a}{2}, 0 \right) \, , \label{eq:ch_6_lattice_vector_3}
%
\end{eqnarray}

\noindent
where $a$ is the equilibrium lattice parameter, and the $x$, $y$ and $z$ axes are respectively chosen to align with the [100], [010] and [001] principal crystal axes, having corresponding Cartesian unit vectors $\widehat{x}$, $\widehat{y}$ and $\widehat{z}$. The equilibrium positions of the atoms in the zinc-blende primitive unit cell, a basis of two atoms, are therefore given by

%\begin{eqnarray}
%
%    \textbf{r}_{1} &=& \textbf{0} \, , \label{eq:ch_6_zb_equil_atomic_posn_III} \\
%
%    \textbf{r}_{2} &=& \frac{a}{4} \, \widehat{x} + \frac{a}{4} \, \widehat{y} + \frac{a}{4} \, \widehat{z} \, . \label{eq:ch_6_zb_equil_prim_atomic_posn_V}
%
%\end{eqnarray}

\begin{eqnarray}
%
    \textbf{r}_{1} &=& \textbf{0} \, , \label{eq:ch_6_zb_equil_atomic_posn_III} \\
%
    \textbf{r}_{2} &=& \left( \frac{a}{4} , \frac{a}{4} , \frac{a}{4} \right) \, . \label{eq:ch_6_zb_equil_prim_atomic_posn_V}
%
\end{eqnarray}

%\begin{eqnarray}
%
%    \textbf{r}_{1} &=& \left[ 0, 0, 0 \right] \, , \label{eq:ch_6_zb_equil_atomic_posn_III} \\
%
%    \textbf{r}_{2} &=& \left[ \frac{a}{4}, \frac{a}{4}, \frac{a}{4} \right] \, . \label{eq:ch_6_zb_equil_prim_atomic_posn_V}
%
%\end{eqnarray}

We then apply an arbitrary deformation to this primitive unit cell, described by the macroscopic strain tensor $\epsilon$. The positions of the atoms in the unit cell then become 

%\begin{eqnarray}
%
%    \textbf{r}_{1} &=& \textbf{0} \, , \label{eq:ch_6_zb_strained_atomic_posn_III} \\
%
%    \textbf{r}_{2} &=& \left( I + \epsilon \right) \left( \frac{a}{4} \, \widehat{x} + \frac{a}{4} \, \widehat{y} + \frac{a}{4} \, \widehat{z} \right) + \textbf{t} \, , \label{eq:ch_6_zb_strained_atomic_posn_V}
%
%\end{eqnarray}

\begin{eqnarray}
%
    \textbf{r}_{1} &=& \textbf{0} \, , \label{eq:ch_6_zb_strained_atomic_posn_III} \\
%
    \textbf{r}_{2} &=& \left( I + \epsilon \right) \left( \frac{a}{4} , \frac{a}{4} , \frac{a}{4} \right) + \textbf{t} \, , \label{eq:ch_6_zb_strained_atomic_posn_V}
%
\end{eqnarray}

%\begin{eqnarray}
%
%    \textbf{r}_{1} &=& \left[ 0, 0, 0 \right] \, , \label{eq:ch_6_zb_strained_atomic_posn_III} \\
%
%    \textbf{r}_{2} &=& \left( I + \epsilon \right) \left[ \frac{a}{4}, \frac{a}{4}, \frac{a}{4} \right] + \textbf{t} \, , \label{eq:ch_6_zb_strained_atomic_posn_V}
%
%\end{eqnarray}

\noindent
where $I$ is the $3 \times 3$ identity matrix, and $\textbf{t}$ is the internal strain vector. The internal strain vector is given by \cite{Keating_PR_1966}

%\begin{equation}
%    \textbf{t} =  -\frac{a}{2} \zeta \epsilon_{yz} \, \widehat{x} - \frac{a}{2} \zeta \epsilon_{xz} \, \widehat{y} - \frac{a}{2} \zeta \epsilon_{xy} \, \widehat{z} \, ,
%    \label{eq:ch_6_int_strain_vector}
%\end{equation}

\begin{equation}
    \textbf{t} = \left( -\frac{a}{2} \zeta \epsilon_{yz}, -\frac{a}{2} \zeta \epsilon_{xz}, -\frac{a}{2} \zeta \epsilon_{xy} \right) \, ,
    \label{eq:ch_6_int_strain_vector}
\end{equation}

\noindent
where $\zeta$ is the internal strain, or Kleinman, parameter \cite{Kleinman_PR_1962}. When the crystal is in equilibrium when the internal strain is minimised, i.e.~$\textbf{t} = 0$. When a strain having non-zero shear components is applied, the crystal develops a non-zero internal strain -- i.e.~a relative displacement of the two interpenetrating face-centred cubic lattices that make up the zinc-blende structure -- in order to minimise the lattice energy.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% III-P Table 1: lattice and elastic parameters %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{table}[b!]
\centering
	\caption{DFT-calculated equilibrium lattice parameter $a$, elastic moduli $C_{ij}$, bulk modulus $B_{0}$ and Kleinman parameter $\zeta$ for AlP, GaP and InP. HSEsol- and LDA-calculated results are respectively denoted by a dagger ($\dagger$) and an asterisk ($\ast$). The HSEsol calculations used $\alpha = 0.25$.}
    \begin{tabular}{l|cc|cc|cc} \hline
                                                                                                    & \multicolumn{2}{c}{AlP} \vline            & \multicolumn{2}{c}{GaP} \vline           & \multicolumn{2}{c}{InP}                 \\
        Parameter                                                                                   & This work & Reference                     & This work & Reference                    & This work & Reference                   \\
        \hline
        $a$ (\AA)                                                                                   & 5.449$^{\dagger}$    & 5.467$^{a}$ & 5.427$^{\dagger}$     & 5.451$^{a}$            & 5.879$^{\dagger}$    & 5.870$^{a}$            \\
                                                                                                    & 5.435$^{\ast}$ & 5.471$^{b}$ & 5.426$^{\ast}$  & 5.456$^{f}$                       & 5.878$^{\ast}$ & 5.886$^{f}$                       \\
                                                                                                    &     & 5.417$^{c}$  &      & 5.451$^{h}$            &     & 5.86$^{h}$            \\
                                                                                                    &     & 5.460$^{d}$  &      & 5.443$^{i}$            &     & 5.861$^{i}$            \\
        \hline
        $C_{11}$ (GPa)                                                                              & 139.7$^{\dagger}$    & 133.0$^{a}$ & 152.29$^{\dagger}$    & 140.5$^{a}$            & 107.74$^{\dagger}$   & 101.1$^{a}$            \\
                                                                                                    & 132.5$^{\ast}$ & 138.3$^{b}$  & 141.29$^{\ast}$ & 150.0$^{f}$                       & 99.60$^{\ast}$ & 105.9$^{f}$                       \\
                                                                                                    &     & 150.0$^{e}$  &      & 143.9$^{i}$            &     & 104.2$^{i}$            \\
                                                                                                    &     & 133.6$^{f}$  &      &                       &     &             \\
        \hline
        $C_{12}$ (GPa)                                                                              & 70.8$^{\dagger}$     & 63.0$^{a}$  & 70.15$^{\dagger}$     & 62.03$^{a}$            & 61.75$^{\dagger}$    & 56.1$^{a}$             \\
                                                                                                    & 68.9$^{\ast}$  & 67.7$^{b}$   & 66.97$^{\ast}$  & 64.3$^{f}$                       & 58.72$^{\ast}$ & 56.4$^{f}$                       \\
                                                                                                    &     & 64.2$^{e}$  &      & 65.2$^{i}$            &     & 60.5$^{i}$            \\
                                                                                                    &     & 65.1$^{f}$  &      &             &     &            \\
        \hline
        $C_{44}$ (GPa)                                                                              & 65.8$^{\dagger}$     & 61.5$^{a}$  & 73.51$^{\dagger}$     & 70.33$^{a}$            & 46.99$^{\dagger}$    & 45.6$^{a}$             \\
                                                                                                    & 61.7$^{\ast}$  & 66.5$^{b}$   & 67.80$^{\ast}$           & 78.4$^{f}$                      & 42.65$^{\ast}$ & 49.3$^{f}$                       \\        
                                                                                                    &     & 61.1$^{e}$  &      &            &     &            \\
                                                                                                    &     & 64.1$^{e}$  &      &            &     &            \\        
        \hline
        $B_{0}$ (GPa)                                                                               & 93.91$^{\dagger}$    & 86.3$^{a}$   & 97.72$^{\dagger}$     & 88.2$^{a}$ & 77.25$^{\dagger}$    & 71.1$^{a}$ \\
                                                                                                    & 90.25$^{\ast}$ & 92.8$^{e}$   & 92.15$^{\ast}$  & 91.4$^{i}$             & 72.69$^{\ast}$ & 75.1$^{i}$ \\
                                                                                                    &     & 86.0$^{g}$  &      & 89.6$^{g}$            &     & 71.0$^{g}$            \\
                                                                                                    &     & 90.3$^{g}$  &      & 88.2$^{g}$            &     & 71.6$^{g}$            \\        
        \hline
        $\zeta$                                                                               & 0.631$^{\dagger}$    & 0.576$^{j}$   & 0.568$^{\dagger}$  & 0.533$^{j}$ & 0.686$^{\dagger}$    & 0.652$^{j}$  \\
                                                                                                    & 0.619$^{\ast}$ &    & 0.583$^{\ast}$  &           & 0.703$^{\ast}$ & 
    \\ \hline
    \end{tabular}
    %\hrule
	\label{tab:III_P_structural}
         \begin{flushleft}
            $^{a}$Recommended by Vurgaftman et al., see Ref.~\cite{Vurgaftman_JAP_2001} for details \\
            Theory: $^{b}$Ref.~\cite{Tanner_PRM_2019}, $^{c}$Ref.~\cite{Wang_PRB_2002}, $^{f}$Ref.~\cite{Bastos_JAP_2018}, $^{g}$Ref.~\cite{Wei_PRB_1999}, $^{h}$Ref.~\cite{Huang_JPCS_1985}, $^{j}$Ref.~\cite{Tanner_thesis_2017} \\
            Experiment: $^{d}$Ref.~\cite{Bessolov_FTT_1982}, $^{e}$Ref.~\cite{Adachi_book_2009}, $^{i}$Ref.~\cite{Ritter_pss_1998}
        \end{flushleft}
\end{table}

At equilibrium the total energy, and the magnitude of the components of the macroscopic stress experienced by the unit cell, will be minimised, indicating that the structure has reached its most stable geometric configuration. To determine the equilibrium lattice parameter we vary the lattice parameter, performing an internal relaxation of the ionic positions $\textbf{r}_{1}$ and $\textbf{r}_{2}$, and calculate the stress on the unit cell. In a small range near equilibrium, the lattice parameter vs.~stress relationship will behave approximately linearly. In this approximately linear regime, in which we seek zero stress to compute the equilbrium lattice parameter, we extract the lattice parameter as the zero stress intercept of a linear fit of the lattice parameter and stress. This means we can then determine the equilibrium lattice parameter $a$ by computing the value of $a$ that produces zero stress on the unit cell. Our HSEsol calculations of $a$ employed exact exchange mixing $\alpha = 0.25$. Our HSEsol- and LDA-calculated lattice parameters are summarised in Table~\ref{tab:III_P_structural}. We observe good overall agreement between our calculated lattice parameters and literature values, with agreement to within 1\% in all cases. We note, in general, a slight underestimation of the calculated lattice parameters in comparison to experimental literature values. This is a consequence of lattice expansion associated either with measurements having been performed at non-zero temperature, due to the contribution of the zero-point anharmonic lattice energy \cite{Schimka_JCP_2011}. Our DFT calculations are carried out at zero temperature and do not account for these contributions. The errors in calculated lattice parameters and the contribution of the zero-point anharmonic lattice energy has been studied by Schimka et al.~\cite{Schimka_JCP_2011}, who found that the HSEsol XC functional yields the smallest deviation from experiment amongst conventionally employed DFT XC functionals. Our HSEsol-calculated $a$ deviate by $\leq 1$\% from experimental measurements, and we note that a zero-point anharmonic energy correction could further improve the accuracy of these results \cite{Schimka_JCP_2011}. Our HSEsol-calculated lattice parameters being larger than those calculated in the LDA is consistent with previous calculations and benchmarking of DFT XC functionals \cite{Rasander_JCP_2015}.

\subsection{Elastic parameters}
\label{sec:ch_6_elastic_structural_elastic_param}

Considering harmonic elasticity, the elastic energy of a material with arbitrary crystal structure is given, using Voigt notation, by

\begin{equation}
    E = \frac{\Omega}{2} \sum_{i=1}^{6} \sum_{j=1}^{6} C_{ij} \epsilon_{i} \epsilon_{j} \, ,
    \label{eq:ch_6_elastic_energy_voigt}
\end{equation}

% $\Omega = \textbf{a}_{1} \cdot ( \textbf{a}_{2} \times \textbf{a}_{3} ) = a^{3} / 4$

\noindent
where $\Omega = \vert \textbf{a}_{1} \cdot ( \textbf{a}_{2} \times \textbf{a}_{3} ) \vert$ is the unit cell volume, $C_{ij}$ are the elastic moduli, and $\epsilon_{i}$ are the components of the strain tensor. We note that, in Voigt notation $\epsilon_{1} = \epsilon_{xx}$, $\epsilon_{2} = \epsilon_{yy}$, $\epsilon_{3} = \epsilon_{zz}$, $\epsilon_{4} = 2 \epsilon_{yz}$, $\epsilon_{5} = 2 \epsilon_{xz}$ and $\epsilon_{6} = 2 \epsilon_{xy}$. The elastic energy can be described in terms of components $\sigma_{i}$ of the stress tensor produced by an applied strain $\epsilon_{j}$

\begin{equation}
    \sigma_{i} = \frac{1}{\Omega} \frac{\partial E}{\partial \epsilon_{i}} = \sum_{j=1}^{6} C_{i j} \epsilon_{j} \, ,
    \label{eq:ch_6_stress_tensor_components}
\end{equation}

\noindent
which is the generalised form of Hooke's law. Combining Eqs.~\eqref{eq:ch_6_elastic_energy_voigt} and \eqref{eq:ch_6_stress_tensor_components}, we can write the elastic crystal energy as

\begin{equation}
    E = \frac{\Omega}{2} \sum_{i} \sigma_{i} \epsilon_{i} \, .
    \label{eq:ch_6_elastic_energy_stress_strain}
\end{equation}

The elastic moduli can then be computed either by taking second derivatives of the elastic energy with respect to strain, or by taking first derivatives of the lattice stress with respect to strain, yielding

\begin{equation}
    C_{i j} = \frac{1}{\Omega} \left( \frac{\partial^{2} E}{\partial \epsilon_{i} \partial \epsilon_{j}} \right) \bigg\vert_{ \epsilon_{i} = 0 } = \left( \frac{ \partial \sigma_{i} }{ \partial \epsilon_{j} } \right) \bigg\vert_{ \epsilon_{i} = 0 } \, .
    \label{eq:ch_6_elastic_constants_hookes_law}
\end{equation}

By applying strains of selected symmetry -- so-called strain branches that impose lattice deformations with specific symmetries -- to the unit cell and calculating the resultant stress, we can use Eq.~\eqref{eq:ch_6_elastic_constants_hookes_law} to calculate the elastic moduli $C_{ij}$. For the zinc-blende compounds of interest in this work, this includes the elastic moduli $C_{11}$, $C_{12}$ and $C_{44}$ in addition to the bulk modulus $B_{0}$. Eq.~\eqref{eq:ch_6_elastic_constants_hookes_law} shows two approaches to compute the elastic moduli, via the first derivative of the stress, or second derivative of the energy. We compute the elastic moduli via computation of the first derivative of the stress. This approach has previously been shown to be more robust than a determination based on the second derivative of the energy with respect to strain, in terms of the plane-wave cut-off energies and \textbf{k}-point grid densities required to achieve convergence \cite{Caro_JPCM_2013}. $C_{11}$ and $C_{12}$ are calculated via the application of a strain branch $( \epsilon_{1}, \epsilon_{2}, \epsilon_{3}, \epsilon_{4}, \epsilon_{5}, \epsilon_{6} ) = \left( \epsilon , 0, 0, 0, 0, 0\right)$. The bulk modulus $B_{0}$ is calculated via the application of a $ \left( \epsilon, \epsilon, \epsilon, 0, 0, 0 \right) $ strain branch, while $C_{44}$ is calculated via the application of a $ \left( 0, 0, 0, \epsilon, \epsilon, \epsilon \right) $ strain branch. The Kleinman parameter $\zeta$ is also calculated from the application of the same strain branch as for $C_{44}$. This represents application of a homogeneous shear from which we can extract $\zeta$ by using Eqs.~\eqref{eq:ch_6_zb_strained_atomic_posn_V} and \eqref{eq:ch_6_int_strain_vector} in conjunction with the calculated displacement of the cation (group-III) and anion (group-V) sublattices in the unit cell, after allowing internal relaxation in response to the applied shear. For all strain branches, calculations are performed for $\vert \epsilon \vert \leq 2$\%, for which the relationship between elastic moduli and stress is linear. For consistency with the calculation of the equilbirum lattice parameter $a$, the HSEsol calculations of $C_{ij}$ employed $\alpha = 0.25$. Our explicitly calculated bulk modulus $B_{0}$ validates the relationship $B_{0} = ( C_{11} + 2 C_{12} )/3$ for the zinc-blence structure, without reference to the calculated $C_{11}$ and $C_{12}$. Our calculated $C_{ij}$ in Table~\ref{tab:III_P_structural} again display typical trends, with the HSEsol values being larger than those calculated in the LDA. We note good overall agreement with literature data. The tendency of the DFT-calculated $C_{ij}$ to slightly overestimate experiment is, in part, attributable to our calculated equilibrium lattice parameters $a$ being slightly lower than experimental values. Previous benchmarking of local, semi-local and hybrid XC functionals vs.~experiment for the elastic moduli of a range of semiconductors has shown that the HSE hybrid functional provides higher accuracy than the LDA \cite{Rasander_JCP_2015}. This, in addition to the previously discussed work of Schimka et al.~\cite{Schimka_JCP_2011}, provides confidence in our calculated elastic moduli. We therefore recommend the use of the HSEsol-calculated values in further calculations.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% Section 3: Electronic structure %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Electronic band structure}
\label{sec:ch_6_elec_structure}

In this Section we present our analysis of the electronic band structure of (Al,Ga,In)P. We detail our procedure for calculating the electronic band structure, and validate our calculations in the context of existing experimental measurements.

\subsection{Fitting to the fundamental energy gap}
\label{sec:ch_6_elec_structure_fitting}

To calculate electronic band structure for (Al,Ga,In)P we treat the exact exchange mixing $\alpha$ (Becke-Roussel mixing $c$) as an empirical parameter in the HSEsol (TB-mBJ) calculations. Rather than tuning the XC functional to fit the direct band gap \cite{Bastos_JAP_2018}, we instead fit $\alpha$ and $c$ to the fundamental (lowest-energy) band gap. This is motivated by the fact that the direct band gap can be challenging to identify in an indirect-gap material while the fundamental band gap is, whether direct or indirect, readily accessible to several classes of spectroscopic measurement. In practice, this means that the fundamental band gap can generally be determined with a high degree of accuracy, while higher energy band gaps have larger associated uncertainties. This indirect-gap fitting has successfully been applied to analyse electronic structure evolution in (Si,Ge)$_{1-x}$Sn$_{x}$ alloys \cite{Halloran_OQE_2019,Donnell_JPDAP_2021,Pearce_PRM_2022}, and it was also employed by Ziembicki et al.~\cite{Ziembicki_JAP_2022} in their recent analysis of wurtzite-structured III-V compounds. We note that fitting to one part of the band structure can have negative implications for the description of other parts. As we will describe below, this fitting procedure can effectively describe band edge parameters and offers improved description of the band edge electronic structure compared to fitting to the direct band gap in indirect band gap materials \cite{Bastos_JAP_2018}. Our HSEsol (TB-mBJ) band structure calculations use the HSEsol-calculated (LDA-calculated) equilibrium lattice parameter $a$. For AlP and GaP we fit $\alpha$ and $c$ to the well-known, low-temperature fundamental (indirect) X$_{6c}$-$\Gamma_{8v}$ band gaps of 2.52 and 2.35 eV respectively \cite{Vurgaftman_JAP_2001}. This yields $\alpha = 0.363$ and $c = 1.218$ for AlP, and $\alpha = 0.314$ and $c = 1.193$ for GaP. For InP we fit $\alpha$ and $c$ to the direct $\Gamma_{6c}$-$\Gamma_{8v}$ low-temperature band gap of 1.424 eV \cite{Vurgaftman_JAP_2001}, yielding $\alpha = 0.273$ and $c = 1.166$.

\begin{figure}[h!]
	\centering
   \includegraphics[width=1.0\textwidth]{./figures/chapter6/figure_1.pdf}
	\caption{DFT-calculated band structure (left panel) and density of states (DOS; right panel) of AlP, calculated using the HSEsol (solid lines) and TB-mBJ (dashed lines) XC functionals. The exact exchange mixing in the HSEsol calculation ($\alpha = 0.363$), and the Becke-Roussel mixing in the TB-mBJ calculation ($c = 1.218$), were adjusted to fit to the experimental zero-temperature fundamental (indirect) X$_{6c}$-$\Gamma_{8v}$ band gap of 2.52 eV \cite{Vurgaftman_JAP_2001}. The zero of energy is set at the valence band maximum.}
	\label{fig:ch_6_AlP_bands_dos}
\end{figure}

\begin{figure}[t!]
	\centering
   \includegraphics[width=1.0\textwidth]{./figures/chapter6/figure_2.pdf}
	\caption{DFT-calculated band structure (left panel) and density of states (DOS; right panel) of GaP, calculated using the HSEsol (solid lines) and TB-mBJ (dashed lines) XC functionals. The exact exchange mixing in the HSEsol calculation ($\alpha = 0.314$), and the Becke-Roussel mixing in the TB-mBJ calculation ($c = 1.193$), were adjusted to fit to the experimental zero-temperature fundamental (indirect) X$_{6c}$-$\Gamma_{8v}$ band gap of 2.35 eV \cite{Vurgaftman_JAP_2001}. The zero of energy is set at the valence band maximum.}
	\label{fig:ch_6_GaP_bands_dos}
\end{figure}

The resulting HSEsol (TB-mBJ) band structures and density of states (DOS) are respectively shown in Figs.~\ref{fig:ch_6_AlP_bands_dos}, \ref{fig:ch_6_GaP_bands_dos} and \ref{fig:ch_6_InP_bands_dos} using solid (dashed) lines. The TB-mBJ bands demonstrate systematic underestimation of energetic band widths vs.~HSEsol. This reflects the known tendency of the TB-mBJ exchange potential to overestimate effective masses \cite{Kim_PRB_2010}. For AlP, the HSEsol and TB-mBJ calculations indicate X-L-$\Gamma$ ordering of the CB valley minima, in agreement with GW \cite{Wing_PRM_2019} and HSE \cite{Bastos_JAP_2018} calculations, but contrary to the X-$\Gamma$-L ordering identified in other studies \cite{Hwang_JAC_2014}. For GaP, our calculations indicate X-L-$\Gamma$ ordering of the CB valley minima, consistent with previous literature \cite{Vurgaftman_JAP_2001,Bastos_JAP_2018,Wing_PRM_2019}. For InP, our calculations indicate $\Gamma$-L-X ordering, consistent with GW \cite{Wing_PRM_2019} and HSE \cite{Bastos_JAP_2018} calculations. Our calculations provide clarity for the disputed CB valley minima ordering in AlP (cf.~Sec.~\ref{sec:ch_6_overview}), consistent with recent theoretical investigations \cite{Wing_PRM_2019,Bastos_JAP_2018}.

\begin{figure}[t!]
	\centering
   \includegraphics[width=1.0\textwidth]{./figures/chapter6/figure_3.pdf}
	\caption{DFT-calculated band structure (left panel) and density of states (DOS; right panel) of InP, calculated using the HSEsol (solid lines) and TB-mBJ (dashed lines) XC functionals. The exact exchange mixing in the HSEsol calculation ($\alpha = 0.273$), and the Becke-Roussel mixing in the TB-mBJ calculation ($c = 1.166$), were adjusted to fit to the experimental zero-temperature fundamental (direct) $\Gamma_{6c}$-$\Gamma_{8v}$ band gap of 1.424 eV \cite{Vurgaftman_JAP_2001}. The zero of energy is set at the valence band maximum.}
	\label{fig:ch_6_InP_bands_dos}
\end{figure}

\subsection{The direct band gap of AlP: reconciling theory and experiment}
\label{sec:ch_6_elec_structure_validation}

Having examined the CB valley minima ordering and energetic band widths of our calculated HSEsol and TB-mBJ electronic band structures for (Al,Ga,In)P, before further investigation of the electronic band parameters we first examine our AlP band structure via direct comparison to experimental measurements. As discussed in Sec.~\ref{sec:ch_6_overview} the band structure, and band parameters, of (Ga,In)P are generally better understood and agreed upon than those of AlP \cite{Vurgaftman_JAP_2001}. For AlP, there exist significant spreads in literature values of several key band parameters including, critically, large discrepancies between reported values of the direct $\Gamma_{6c}$-$\Gamma_{8v}$ band gap. Given these uncertainties, we now perform rigorous benchmarking of our HSEsol-calculated AlP band structure via comparison with literature experimental data.

% AlP "issues"

In AlP, there exist significant discrepancies between the 3.63 eV low-temperature direct $\Gamma_{6c}$-$\Gamma_{8v}$ band gap recommended by Vurgaftman et al.~\cite{Vurgaftman_JAP_2001} and those predicted via state-of-the-art GW \cite{Wing_PRM_2019} or HSE-DFT \cite{Bastos_JAP_2018} calculations. GW and HSE-DFT calculations suggest that the $\Gamma_{6c}$\text{-}$\Gamma_{8v}$ band gap is $\gtrsim 0.5$ eV larger than that recommended in Ref.~\cite{Vurgaftman_JAP_2001}. The original source of the 3.63 eV value recommended by Vurgaftman et al.~is the photoluminescence excitation (PLE) measurement of Monemar \cite{Monemar_PRB_1973}, in which the measured spectrum displayed a weak ``shoulder'' at 3.63 eV that was attributed to the $\Gamma_{6c}$\text{-}$\Gamma_{8v}$ band gap. When presenting their recommended parameters in Ref.~\cite{Vurgaftman_JAP_2001}, Vurgaftman et al.~issued a caveat that little was known about the AlP band structure, and that Ref.~\cite{Monemar_PRB_1973} was the sole experimental report of the $\Gamma_{6c}$\text{-}$\Gamma_{8v}$ band gap. Despite this clear caveat, the value 3.63 eV has proliferated in subsequent literature, likely driven by the reliability of the recommended parameters for better-studied III-V compounds that have established Ref.~\cite{Vurgaftman_JAP_2001} as a canonical reference.

% AlP vs SE experimental data

We compared critical points in the HSEsol-calculated band structure to the spectroscopic ellipsometry (SE) data of Ref.~\cite{Hwang_JAC_2014}. Measurement of the dielectric function $\epsilon ( \hbar \omega )$ via SE constitutes a valuable technique to analyse critical points in the band structure, particularly at high energies where it is challenging to produce optical excitation in wide-gap semiconductors. Taking the second derivative of the measured $\epsilon ( \hbar \omega )$ with respect to photon energy $\hbar \omega$ yields a differential spectrum with high sensitivity to critical points in the band structure \cite{Costa_SSC_2006,Pearce_arXiv_2022}. This approach was applied by Hwang et al.~in Ref.~\cite{Hwang_JAC_2014}, who assigned inter-band optical transition energies at the $\Gamma$-, L- and X-points via fitting to the $\text{d}^{2} \epsilon ( \hbar \omega ) / \text{d} ( \hbar \omega )^{2}$ spectrum obtained by numerically differentiating the measured $\epsilon ( \hbar \omega )$ data. To validate our HSEsol band structure we also fit to the experimental $\text{d}^{2} \epsilon ( \hbar \omega ) / \text{d} ( \hbar \omega )^{2}$ data of Ref.~\cite{Hwang_JAC_2014} employing, for consistency, the same excitonic ($n = -1$) lineshape \cite{Hwang_JAC_2014,Aspnes_SS_1973,Lee_JAP_1993}

\begin{equation}
    \frac{ \text{d}^{2} \epsilon }{ \text{d} ( \hbar \omega )^{2} } = \sum_{j} \frac{ n ( n - 1 ) \, A_{j} \, e^{ i \phi_{j} } }{ ( \hbar \omega - E_{j} + i \Gamma_{j} )^{2 - n} } \, ,
    \label{eq:ch_6_ellipsometry_lineshape}
\end{equation}

\noindent
where $E_{j}$, $A_{j}$, $\phi_{j}$ and $\Gamma_{j}$ are, respectively, the energy, amplitude, phase and linewidth associated with the $j$th critical point \cite{Hwang_JAC_2014}. We write the complex-valued dielectric function as $\epsilon ( \hbar \omega ) = \epsilon_{1} ( \hbar \omega ) + i \, \epsilon_{2} ( \hbar \omega )$, such that $\epsilon_{1} ( \hbar \omega ) = \text{Re} \lbrace \epsilon ( \hbar \omega ) \rbrace$. In Eq.~\eqref{eq:ch_6_ellipsometry_lineshape},  $n = \left( D - 2 \right) / 2$ where $D$ is the dimensionality of the critical point \cite{Lee_JAP_1993}. The choice $n = -1$ corresponds to a discrete excitonic transition $\left( D = 0 \right)$, which was employed for AlP in Ref.~\cite{Hwang_JAC_2014} based on recommendations from earlier analyses \cite{Lee_JAP_1993,Lautenschlager_PRB_1987}.

\begin{sloppypar}\tolerance 900
To assess the accuracy of the HSEsol band structure we employ Eq.~\eqref{eq:ch_6_ellipsometry_lineshape} to perform a constrained fit to the real part of $\text{d}^{2} \epsilon / \text{d} ( \hbar \omega )^{2}$ computed from the experimental SE data of Ref.~\cite{Hwang_JAC_2014}. The fit is constrained by extracting the critical point energies $E_{j}$ from the HSEsol-calculated band structure, which are then kept fixed during a non-linear least squares (Levenberg-Marquardt) fit to Eq.~\eqref{eq:ch_6_ellipsometry_lineshape}. With the critical point energies kept fixed, during the fit the amplitude $A_{j}$, phase $\phi_{j}$ and linewidth $\Gamma_{j}$ associated with each critical point are allowed to vary. In the energy range for which experimental $\text{d}^{2} \epsilon_{1} / \text{d} ( \hbar \omega )^{2}$ data are provided in Ref.~\cite{Hwang_JAC_2014}, we identify five critical points ($j = 1, \dots, 5$) in the HSEsol band structure. These are, in order of increasing energy: $E_{0} = E ( \Gamma_{6c} ) - E ( \Gamma_{8v} )$, $E_{1} = E ( \text{L}_{6c} ) - E ( \text{L}_{5v} )$, $E_{2} = E ( \text{X}_{6c} ) - E ( \text{X}_{7v} )$, $E_{0}^{\prime} = E ( \Gamma_{7c} ) - E ( \Gamma_{8v} )$, and $E_{2}^{\prime} = E ( \text{X}_{7c} ) - E ( \text{X}_{7v} )$. The optical activity of these transitions was confirmed by computing their associated inter-band optical matrix elements, all of which were found to be non-zero.
\end{sloppypar}

\begin{figure}[b!]
	\centering
   \includegraphics[width=0.65\textwidth]{./figures/chapter6/figure_4.pdf}
	\caption{Second derivative with respect to photon energy $\hbar \omega$ of the real part $\epsilon_{1}$ of the dielectric function of AlP, computed from the measured spectroscopic ellipsometry data of Ref.~\cite{Hwang_JAC_2014} (open circles), and calculated via a constrained fit (solid line) employing Eq.~\eqref{eq:ch_6_ellipsometry_lineshape} with fixed critical point energies $E_{j}$ ($j = 1, \dots, 5$; vertical arrows) obtained from the HSEsol-calculated AlP band structure (cf.~Fig.~\ref{fig:ch_6_AlP_bands_dos}).}
	\label{fig:ch_6_AlP_spectral_ellipsometry}
\end{figure}

Before performing the fit, the HSEsol-calculated values of the critical point energies $E_{j}$ must first be adjusted to account for the SE measurements of Ref.~\cite{Hwang_JAC_2014} having been performed at room temperature. We employ the Varshni parameters of Ref.~\cite{Vurgaftman_JAP_2001} to redshift our HSEsol-calculated critical point energies, yielding suitable $E_{j}$ values for use in Eq.~\eqref{eq:ch_6_ellipsometry_lineshape}. At temperature $T = 300$ K this gives $E_{0} = 4.561$ eV, $E_{1} = 4.868$ eV, $E_{2} = 4.940$ eV, $E_{0}^{\prime} = 5.824$ eV, and $E_{2}^{\prime} = 5.952$ eV. These critical point energies at the $\Gamma$-point ($E_{0}$ and $E_{0}^{\prime}$), L-point ($E_{1}$), and X-point ($E_{2}$ and $E_{2}^{\prime}$) have respectively been redshifted by 77, 32 and 32 meV vs.~their calculated low-temperature values. Figure~\ref{fig:ch_6_AlP_spectral_ellipsometry} compares the resulting constrained fit to Eq.~\eqref{eq:ch_6_ellipsometry_lineshape} (solid line) to the experimental $\text{d}^{2} \epsilon_{1} / \text{d} ( \hbar \omega )^{2}$ SE data of Ref.~\cite{Hwang_JAC_2014} (open circles). This constrained fit provides an accurate description of the experimental SE data. As described above, Hwang et al.~assigned $E_{0} = 3.681$ eV for the $\Gamma_{6c}$\text{-}$\Gamma_{8v}$ band gap, despite the absence of a corresponding feature in the SE data \cite{Hwang_JAC_2014}. Our assignment of critical point energies in Fig.~\ref{fig:ch_6_AlP_spectral_ellipsometry} revises that of Ref.~\cite{Hwang_JAC_2014} as: $E_{1} \to E_{0}$, $E_{2} \to E_{1}$/$E_{2}$, and $E_{0}^{\prime} \to E_{0}^{\prime}$/$E_{2}^{\prime}$. Consistent with Ref.~\cite{Hwang_JAC_2014}, we find that the weak features close to 4.2 and 5.3 eV in the experimental data do not correspond to clear optical transitions in the bulk band structure, and are likely attributable to native defects and/or unintentional dopants.

Notably, our calculated $E_{0}$ corresponds to a sharp, prominent feature in the SE data, which is consistent with (i) the high optical matrix element associated with the $\Gamma_{6c}$-$\Gamma_{8v}$ transition (cf.~Kane parameter $E_{P}$ in Table~\ref{tab:III_P_effective_masses}), and (ii) the appearance of the $E_{0}$ transition in SE measurements for other semiconductors \cite{Pearce_arXiv_2022}. We assert that this constitutes sufficient evidence to ascribe a revised value of 4.639 eV to the zero-temperature $\Gamma_{6c}$-$\Gamma_{8v}$ band gap, an increase of 1.009 eV compared to the value originally proposed by Monemar \cite{Monemar_PRB_1973} and subsequently recommended by Vurgaftman et al.~\cite{Vurgaftman_JAP_2001}. This comparison reconciles theory and experiment, reducing the uncertainty caused by the significant spread in reported values of this key band parameter over the past several decades. Our revision of the direct band gap corresponds to an $\approx 1$ eV increase in the energy difference between the direct $\Gamma_{6c}$ and indirect X$_{6c}$ CB minima suggesting, e.g., a more rapid direct- to indirect-gap transition with increasing Al composition $x$ in Al$_{x}$In$_{1-x}$P alloys. It is also likely to impact on the value of other parameters that depend on the direct energy gap.

Literature quasiparticle \cite{Zhu_PRB_1991}, GW \cite{Wing_PRM_2019} and HSE-DFT \cite{Bastos_JAP_2018} calculations corroborate the likelihood of a larger $\Gamma_{6c}$\text{-}$\Gamma_{8v}$ band gap. The quasiparticle calculations of Zhu and Louie \cite{Zhu_PRB_1991}, and GW calculations of Wing et al.~\cite{Wing_PRM_2019} respectively predicted $E ( \Gamma_{6c} ) \text{-} E ( \Gamma_{8v} ) = 4.38$ eV and 4.25 eV. These are lower than the observed feature at $\approx 4.6$ eV in the SE data of Ref.~\cite{Hwang_JAC_2014}, an underestimation that can be expected since those calculations underestimate the $\Gamma_{6c}$-$\Gamma_{8v}$ band gap of some III-V semiconductors (e.g.~underestimating the well-known GaAs direct band gap by 0.30 eV and 0.57 eV in Refs.~\cite{Zhu_PRB_1991} and \cite{Wing_PRM_2019} respectively). In their HSE-DFT analysis Bastos et al.~\cite{Bastos_JAP_2018} employed the exact exchange mixing $\alpha$ as an empirical parameter, which they fit to the direct band gap of 3.63 eV recommended by Vurgaftman et al.~\cite{Vurgaftman_JAP_2001}. The resulting best-fit $\alpha = 0.127$ is anomalously lower than that obtained for all other III-(P,As,Sb) compounds, which had $\alpha \gtrsim 0.22$ \cite{Bastos_JAP_2018}. Using the HSE06 XC functional ($\alpha = 0.25$) Bastos et al.~predicted $E ( \Gamma_{6c} ) \text{-} E ( \Gamma_{8v} ) = 4.153$ eV.

To summarise, the SE measurements of Hwang et al.~\cite{Hwang_JAC_2014} displayed no evidence of an optical transition close to 3.6 eV. A clear feature was visible close to 4.6 eV in the SE data, having an appearance qualitatively consistent with the $\Gamma_{6c}$\text{-}$\Gamma_{8v}$ ($E_{0}$) transition in other semiconductors \cite{Pearce_arXiv_2022}. The authors of Ref.~\cite{Hwang_JAC_2014} ascribed a value of 3.681 eV to the $\Gamma_{6c}$-$\Gamma_{8v}$ band gap, despite their measurements not indicating the presence of an optical transition at that energy. The excitation energies employed in the PLE measurements of Ref.~\cite{Monemar_PRB_1973} were restricted to $\leq 4.1$ eV, which precluded probing optical transitions having energies $> 4.1$ eV. As such, we expect that the direct $\Gamma_{6c}$-$\Gamma_{8v}$ band gap was not visible in the PLE measurements reported by Monemar due to the range of excitation energies employed. Most recently Vurgaftman et al.~revised their recommended $\Gamma_{6c}$\text{-}$\Gamma_{8v}$ band gap from 3.63 eV to 3.99 eV \cite{Vurgaftman_book_2020}. Taken together, these observations indicated the possibility that the $\Gamma_{6c}$\text{-}$\Gamma_{8v}$ band gap in AlP is significantly larger than previously assumed. We have demonstrated that reinterpretation of the SE data of Ref.~\cite{Hwang_JAC_2014} is consistent with the zero-temperature direct band gap $E ( \Gamma_{6c} ) \text{-} E ( \Gamma_{8v} ) = 4.639$ eV obtained from our HSEsol-calculated band structure. With our calculated AlP band structure validated via comparison to experimental SE measurements, we now undertake a full re-evaluation of the (Al,Ga,In)P electronic band parameters.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% Section 4: Unstrained electronic band parameters %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Unstrained electronic band parameters}
\label{sec:ch_6_unstrained_elec_band_parameters}

In this Section we present our calculated unstrained electronic band parameters for (Al,Ga,In)P. We provide analysis of the direct and indirect band gaps, effective masses, and the inter-band Kane parameter. We also investigate the robustness of our calculated parameters for use in 8-band $\textbf{k} \cdot \textbf{p}$ calculations. To do this, we demonstrate the ellipticity of our parameter set and the renormalised inverse CB effective mass $s_{c}$ (cf.~Sec.~\ref{sec:ch_4_model_spurious_reparam}).

\subsection{Direct and indirect band gaps}
\label{sec:ch_6_unstrained_elec_band_parameters_energy_gaps}

% AlP Recap

As demonstrated in Sec.~\ref{sec:ch_6_elec_structure_validation}, our calculations suggest an increase of $\approx$ 1 eV to the AlP direct $\Gamma_{6c}$\text{-}$\Gamma_{8v}$ band gap recommended by Vurgaftman et al.~\cite{Vurgaftman_JAP_2001}. Having been validated via comparison to experimental SE measurements, our calculations predict, at zero temperature, $E ( \Gamma_{6c} ) \text{-} E ( \Gamma_{8v} ) = 4.639$ eV. We now examine the direct band gaps in GaP and InP, and the indirect band gaps in (Al,Ga,In)P. Our calculated direct and indirect band gaps, and valence band (VB) spin-orbit (SO) splitting energies, are shown in Table~\ref{tab:III_P_energies}.

% Direct gaps

% GaP
For GaP, our calculation of the direct $\Gamma_{6c}$-$\Gamma_{8v}$ band gap revises the value recommended by Vurgaftman et al.~\cite{Vurgaftman_JAP_2001} upwards by $\approx$ 190 meV. This is consistent with recent HSE-DFT calculations by Bastos et al.~\cite{Bastos_JAP_2018}, which suggest an upward revision of $\approx 50$ meV to the value recommended by Vurgaftman et al. The HSE-DFT calculations of Bastos et al.~fit $\alpha$ to the direct gap, rather than following our approach of fitting to the fundamental band gap. As a result, Bastos et al.~employed $\alpha = 0.283$, which is 10\% smaller than our calculated $\alpha$. Since the calculated band gap in a HSE(sol) calculation increases monotonically with increasing $\alpha$ the increase in the direct $\Gamma_{6c}$\text{-}$\Gamma_{8v}$ band gap predicted by Bastos et al.~is underestimated in comparison with our calculations. We note that recent GW calculations suggest a reduction to the value recommended by Vurgaftman et al.~by $\approx$ 300 meV \cite{Wing_PRM_2019}. Earlier theoretical \cite{Huang_JPCS_1985,Chelikowsky_PRB_1976} and experimental work \cite{Dean_JAP_1967,Zallen_PR_1964} suggests values consistent with that recommended by Vurgaftman et al.

% InP
InP is a direct band gap material. As such, the direct $\Gamma_{6c}$-$\Gamma_{8v}$ band gap is readily measurable and its value is therefore well established in the literature. As described in Sec.~\ref{sec:ch_6_elec_structure_fitting}, in our calculations we fit to the direct band gap of 1.424 eV recommended by Vurgaftman et al.~\cite{Vurgaftman_JAP_2001}. This recommended value is in good agreement with GW calculations \cite{Wing_PRM_2019} and early empirical band structure \cite{Huang_JPCS_1985} and empirical pseudopotential \cite{Chelikowsky_PRB_1976} calculations.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% III-P Table 2: energies %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{table}[t!]
\centering
	\caption{DFT-calculated direct $(\Gamma_{6c}\text{-}\Gamma_{8v})$ and indirect ($\text{X}_{6c}\text{-}\Gamma_{8v}$ and $\text{L}_{6c}\text{-}\Gamma_{8v}$) band gaps, and valence band spin-orbit splitting energy $\Delta_{\scalebox{0.7}{\text{SO}}}$ for AlP, GaP and InP. HSEsol- and TB-mBJ-calculated results are respectively denoted by a dagger ($\dagger$) and an asterisk ($\ast$). The exact exchange mixing $\alpha$ (Becke-Roussel mixing $c$) is $\alpha = 0.363$ ($c = 1.218$) for AlP, $\alpha = 0.314$ ($c = 1.193$) for GaP, and $\alpha = 0.274$ ($c = 1.165$) for InP.}
    \begin{tabular}{l|cc|cc|cc} \hline
                                                                                                    & \multicolumn{2}{c}{AlP} \vline          & \multicolumn{2}{c}{GaP} \vline      & \multicolumn{2}{c}{InP}      \\
        Parameter                                                                                   & This work & Reference                   & This work & Reference               & This work & Reference        \\
        \hline
        $E_{g} (\Gamma_{6c}\text{-}\Gamma_{8v})$ (eV)                                               & 4.639$^{\dagger}$    & 3.63$^{a}$ & 3.075$^{\dagger}$    & 2.886$^{a}$         & 1.424$^{\dagger}$    & 1.424$^{a}$ \\
                                                                                                    & 4.582$^{\ast}$ & 4.25$^{b}$             & 3.010$^{\ast}$ & 2.895$^{d}$                   & 1.423$^{\ast}$ & 1.422$^{e}$            \\
                                                                                                    &  & 4.38$^{c}$             &  & 2.915$^{e}$                   &  &             \\
        \hline
        $E_{g} (\text{X}_{6c}\text{-}\Gamma_{8v})$ (eV)                                             & 2.520$^{\dagger}$    & 2.52$^{a}$ & 2.350$^{\dagger}$    & 2.35$^{a}$         & 2.330$^{\dagger}$    & 2.384$^{a}$ \\
                                                                                                    & 2.520$^{\ast}$ & 2.50$^{b}$             & 2.350$^{\ast}$ & 2.36$^{d}$                   & 2.381$^{\ast}$ & 2.39$^{d}$            \\
                                                                                                    &  & 2.59$^{c}$             &  & 2.38$^{f}$                   &  & 2.35$^{f}$            \\
                                                                                                    &  & 2.51$^{d}$             &  &                    &  &             \\                                                                                                    
        \hline
        $E_{g} (\text{L}_{6c}\text{-}\Gamma_{8v})$ (eV)                                             & 4.015$^{\dagger}$    & 3.57$^{a}$ & 2.690$^{\dagger}$    & 2.72$^{a}$         & 2.218$^{\dagger}$    & 2.014$^{a}$ \\
                                                                                                    & 3.780$^{\ast}$ & 3.90$^{c}$             & 2.559$^{\ast}$ &                    & 2.098$^{\ast}$ &             \\
                                                                                                    &  & 4.00$^{g}$             &  &                    &  &             \\        
        \hline
        $\Delta_{\scalebox{0.6}{\text{SO}}}$ (eV)                                                   & 0.067$^{\dagger}$    & 0.07$^{a}$             & 0.104$^{\dagger}$    & 0.08$^{a}$         & 0.128$^{\dagger}$    & 0.108$^{a}$ \\
                                                                                                    & 0.055$^{\ast}$  & 0.06$^{d}$             & 0.084$^{\ast}$ & 0.093$^{e}$                   & 0.104$^{\ast}$ & 0.111$^{e}$            	
    \\ \hline
    \end{tabular}
    %\hrule
	\label{tab:III_P_energies}
         \begin{flushleft}
            $^{a}$Recommended by Vurgaftman et al., see Ref.~\cite{Vurgaftman_JAP_2001} for details \\
            Theory: $^{b}$Ref.~\cite{Wing_PRM_2019}, $^{c}$Ref.~\cite{Zhu_PRB_1991}, $^{e}$Ref.~\cite{Bastos_JAP_2018}, $^{f}$Ref.~\cite{Huang_JPCS_1985} \\
            Experiment: $^{d}$Ref.~\cite{Ritter_pss_1998}
        \end{flushleft}
\end{table}

% Indirect gaps

Considering the indirect band gaps, we recall that for AlP and GaP our DFT calculations were fit to the fundamental $E_{g} (\text{X}_{6c}\text{-}\Gamma_{8v})$ band gap known from experiment. We now examine the $E_{g} (\text{L}_{6c}\text{-}\Gamma_{8v})$ band gap in comparison to available literature data. Beginning with AlP, the value of 3.57 eV recommended in Ref.~\cite{Vurgaftman_JAP_2001} is from an early empirical band structure calculation \cite{Huang_JPCS_1985}, while the higher 3.90 eV is from the quasiparticle GW calculation by Zhu and Louie \cite{Zhu_PRB_1991}. Our calculated values straddle the latter, with our HSEsol (TB-mBJ) calculated L$_{6c}$-X$_{6c}$ valley splitting being $\approx 180$ meV higher (50 meV lower) than that predicted by Zhu and Louie. For GaP, our HSEsol calculated $E_{g} (\text{L}_{6c}\text{-}\Gamma_{8v})$ is in good agreement with the recommendation by Vurgaftman et al.~\cite{Vurgaftman_JAP_2001}. We note that these values are in good quantitative agreement -- to within $\approx 10$ meV -- with previous empirical linear combination of atomic orbitals \cite{Huang_JPCS_1985} and empirical pseudopotential \cite{Chelikowsky_PRB_1976} calculations.

For InP we fit to the direct $E_{g} (\Gamma_{6c}\text{-}\Gamma_{8v})$ band gap. As such, we will now examine both the $E_{g} (\text{X}_{6c}\text{-}\Gamma_{8v})$ and $E_{g} (\text{L}_{6c}\text{-}\Gamma_{8v})$ indirect band gaps. Firstly, analysing the $E_{g} (\text{X}_{6c}\text{-}\Gamma_{8v})$ band gap, we note excellent quantitative agreement with the value 2.384 eV recommended by Vurgaftman et al.~\cite{Vurgaftman_JAP_2001}, with our HSEsol calculated value being within $\approx$ 5 meV. We note that early empirical calculations \cite{Huang_JPCS_1985,Chelikowsky_PRB_1976} suggested values 200 - 300 meV larger than our calculations, and the recommendation of Vurgaftman et al. Secondly, examining $E_{g} (\text{L}_{6c}\text{-}\Gamma_{8v})$, our HSEsol calculations suggest an $\approx$ 200 meV increase vs.~the value recommended by Vurgaftman et al. This is consistent with other HSE-DFT calculations which employ a similar exact exchange mixing $\alpha$ for InP \cite{Bastos_JAP_2018}, and is also consistent with an early empirical pseudopotential calculation that shows good quantitative agreement with our calculated $E_{g} (\text{L}_{6c}\text{-}\Gamma_{8v})$ \cite{Chelikowsky_PRB_1976}.

% SO splitting and CB valley splittings

We will now briefly discuss the spin-orbit splitting $\Delta_{\scalebox{0.6}{\text{SO}}}$ and the CB valley splittings in (Al,Ga,In)P. The spin-orbit splitting $\Delta_{\text{SO}}$ can be difficult to determine directly in experiment, due to the weak spin-orbit coupling in III-P compounds, making $\Delta_{\scalebox{0.6}{\text{SO}}}$ challenging to distinguish in spectroscopic measurements. Our calculations are in excellent agreement with the recommendations of Ref.~\cite{Vurgaftman_JAP_2001}, and those from other HSE-DFT calculations \cite{Bastos_JAP_2018}. Secondly, we examine the CB valley minima splittings in InP. It was noted by Vurgaftman et al.~\cite{Vurgaftman_JAP_2001} that there were uncertainties in the energies of the X- and L-valley CB minima. Our HSEsol calculations predict that the X-valley CB minima lie 0.906 eV above the $\Gamma$-valley, within 6\% of the recommended values of 0.96 eV of Vurgaftman et al.~\cite{Vurgaftman_JAP_2001}. Our HSEsol calculations predict that the L-valley CB minima lie 0.794 eV above the $\Gamma$-valley. This result suggests an increase of 32\% to the recommendation of Vurgaftman et al.~\cite{Vurgaftman_JAP_2001}. We note, that Ref.~\cite{Vurgaftman_JAP_2001} also provided a range for the $\Gamma_{6c}$-L$_{6c}$ CB valley splitting of 0.4 - 0.7 eV. Our revised value suggests an increase to the upper bound of this range by 13\%. Our suggested minor revisions to these aspects of the description of the InP electronic structure are consistent with other HSE-DFT calculations \cite{Bastos_JAP_2018}.

\subsection{Effective masses and Luttinger parameters}
\label{sec:ch_6_unstrained_elec_band_parameters_eff_mass_luttinger}

%\textbf{GaP:}
%\begin{itemize}
%    \item $\Gamma$-valley CB from theory [11,96], some higher values reported [101,106]. Experiments with GaAsP alloys [190] suggest 0.13$m_{0}$, good agreement with TB calculations [191]
%    \item X-valley - number of experimental measurements [192] $\longrightarrow$ camel back structure complicates measurements, X-valley minimum predicted at $k = ( 0.95, 0, 0 )$ [5]. This has given somewhat of a spread in the reported values obtained from experiments and theory
%    \item L-valley masses taken from [11]
%    \item Luttinger first calculated in [106], followed by some cyclotron resonance experiments along [111] and [001] [196,197,198], also determined from acceptor binding energies [199], more detail but required?
%    \item SO mass calculated [106,101] - adjusted based on $E_{P}$ (to increase it)
%\end{itemize}

%\textbf{InP:}
%\begin{itemize}
%    \item $\Gamma$-valley effective mass investigated via numerous cyclotron resonance, magnetophonon resonance, and magnetospectroscopy of donor transitions - giving a spread of values 0.068 - 0.084$m_{0}$ [97,98,108,173,207,215-225] - "sensible exclusions and adjustments" give 0.077 - 0.081$m_{0}$ [99,218] - then averaging gives finalised value
%    \item L- and X-valley masses taken from [213], some debate regarding this and alternatives given from [11]
%    \item Luttinger measured using cyclotron resonance [198], magnetoreflectance [109,226], piezo modulated photovoltac effect [207], magnetoabsorption [227]
%    \item For some(?) of the above - composite parameters constructed?
%    \item SO mass measured [207]
%\end{itemize}

% AlP
Our calculated (relative) effective masses and Luttinger parameters are shown in Table~\ref{tab:III_P_effective_masses}. In Ref.~\cite{Vurgaftman_JAP_2001} Vurgaftman et al.~noted a lack of experimental effective mass data for AlP. Their recommended $m_{\scalebox{0.7}{\text{CB}}}^{\ast} (\Gamma_{6c}) = 0.22$ and $m_{\scalebox{0.7}{\text{SO}}}^{\ast} (\Gamma_{7v}) = 0.30$ for the zone-center CB and SO VB edge (relative) effective masses were calculated ab initio by Krijn \cite{Krijn_SST_1991}. Our HSEsol-calculated $\Gamma_{6c}$ CB ($\Gamma_{7v}$ SO) effective mass is slightly lower (higher) than that predicted by Krijn. Comparing to the HSE calculations of Bastos et al.~\cite{Bastos_JAP_2018}, our HSEsol-calculated $\Gamma_{6c}$ CB effective mass is in close agreement, while our HSEsol-calculated $\Gamma_{7v}$ SO effective mass is $\approx 10$\% lower. The Luttinger parameters $\gamma_{1}$, $\gamma_{2}$ and $\gamma_{3}$ can be computed analytically from a choice of three heavy-hole (HH) and light-hole (LH) VB edge effective masses \cite{Vurgaftman_JAP_2001}. We employ the HH and LH effective masses along [001] to fix $\gamma_{1}$ and $\gamma_{2}$, and the HH effective mass along [111] to fix $\gamma_{3}$. The Luttinger parameters can then be computed as \cite{Vurgaftman_JAP_2001}

\begin{eqnarray}
%
    \gamma_{1} &=& \frac{1}{2} \left[ \left( \frac{1}{m_{\scalebox{0.7}{\text{LH}}}^{\ast}} \right)^{[001]} + \left( \frac{1}{m_{\scalebox{0.7}{\text{HH}}}^{\ast}} \right)^{[001]} \right] \, , \label{eq:ch_6_lutt_1} \\
%    
    \gamma_{2} &=& \frac{1}{4} \left[ \left( \frac{1}{m_{\scalebox{0.7}{\text{LH}}}^{\ast}} \right)^{[001]} - \left( \frac{1}{m_{\scalebox{0.7}{\text{HH}}}^{\ast}} \right)^{[001]} \right] \, , \label{eq:ch_6_lutt_2} \\
%
    \gamma_{3} &=& \frac{1}{4} \left[ \left( \frac{1}{m_{\scalebox{0.7}{\text{LH}}}^{\ast}} \right)^{[001]} + \left( \frac{1}{m_{\scalebox{0.7}{\text{HH}}}^{\ast}} \right)^{[001]} - 2 \left( \frac{1}{m_{\scalebox{0.7}{\text{HH}}}^{\ast}} \right)^{[111]} \right] \, . \label{eq:ch_6_lutt_3}
%
\end{eqnarray}

We note that our Luttinger parameters, computed analytically in this manner, are close to those obtained from numerical fitting to the HSE-calculated VB structure by Bastos et al.~\cite{Bastos_JAP_2018}.

Few data are available for the effective masses of the zone-edge X- and L-point CB valleys. Our HSEsol-calculated longitudinal X-point effective mass $m_{\scalebox{0.6}{\text{CB}},\parallel}^{\ast} (\text{X}_{6c})$ is significantly lower than that calculated ab initio by Huang and Ching \cite{Huang_JPCS_1985}, and than that inferred via fitting to photoluminescence data by Issiki et al.~\cite{Issiki_APL_1995}. However, we note that the X-point CB DOS effective mass per valley, $m_{\scalebox{0.7}{DOS}}^{\ast} = [ m_{\parallel}^{\ast} ( m_{\perp}^{\ast} )^{2} ]^{1/3}$, obtained from our HSEsol-calculated effective masses is within 20\% of the best-fit value of Issiki et al.~\cite{Issiki_APL_1995}. To our knowledge, our calculated L-point CB edge effective masses, $m_{\scalebox{0.6}{\text{CB}},\parallel}^{\ast} (\text{L}_{6c})$ and $m_{\scalebox{0.6}{\text{CB}},\perp}    ^{\ast} (\text{L}_{6c})$, are the first for AlP. Our predicted L-point DOS effective mass per valley is close to that of the X valley, with the L-point DOS effective mass per valley exceeding that at the X point by 8\%.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% III-P Table 3: unstrained electronic band parameters %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{table}[t!]
\centering
	\caption{DFT-calculated zone-centre CB edge effective mass $m_{\scalebox{0.7}{\text{CB}}}^{\ast} (\Gamma_{6c})$, zone-centre VB Luttinger parameters $\gamma_{i}$, SO VB edge effective mass $m_{\scalebox{0.7}{\text{SO}}}^{\ast} (\Gamma_{7v})$, and longitudinal ($\parallel$) and transverse ($\perp$) X- and L-point CB edge effective masses for AlP, GaP and InP. HSEsol- and TB-mBJ-calculated results are respectively denoted by a dagger ($\dagger$) and an asterisk ($\ast$). The exact exchange mixing $\alpha$ (Becke-Roussel mixing $c$) is $\alpha = 0.363$ ($c = 1.218$) for AlP, $\alpha = 0.314$ ($c = 1.193$) for GaP, and $\alpha = 0.274$ ($c = 1.165$) for InP.}
    \begin{tabular}{l|cc|cc|cc} \hline
                                                                                                    & \multicolumn{2}{c}{AlP} \vline          & \multicolumn{2}{c}{GaP} \vline      & \multicolumn{2}{c}{InP}      \\
        Parameter                                                                                   & This work & Reference                   & This work & Reference               & This work & Reference        \\
        \hline                                                                       
        $m_{\scalebox{0.6}{\text{CB}}}^{\ast} (\Gamma_{6c})$ ($m_{0}$)                               & 0.192$^{\dagger}$       & 0.22$^{a}$      & 0.137$^{\dagger}$       & 0.13$^{a}$                   & 0.085$^{\dagger}$       & 0.068$^{a}$    \\
                                                                                                 & 0.242$^{\ast}$    & 0.197$^{b}$                             & 0.175$^{\ast}$    & 0.150$^{b}$                             & 0.106$^{\ast}$    & 0.084$^{a}$                             \\
                                                                                                 &     &                              &     &                              &     & 0.092$^{b}$                             \\                                                                                                 
    \hline
    $\gamma_{1}$                                                                                 & 2.97$^{\dagger}$        & 3.35$^{a}$      & 4.43$^{\dagger}$        & 4.05$^{a}$                   & 5.61$^{\dagger}$        &                5.08$^{a}$    \\
                                                                                                 & 2.31$^{\ast}$     &  2.85$^{b}$                            & 3.51$^{\ast}$     & 4.20$^{b}$                             & 4.96$^{\ast}$     &  5.33$^{b}$                            \\
    \hline
    $\gamma_{2}$                                                                                 & 0.48$^{\dagger}$        & 0.71$^{a}$       & 0.78$^{\dagger}$        & 0.49$^{a}$                   & 1.55$^{\dagger}$        & 1.60$^{a}$    \\
                                                                                                 & 0.36$^{\ast}$     & 0.54$^{b}$                             & 0.60$^{\ast}$     &  0.87$^{b}$                            & 1.40$^{\ast}$     & 1.58$^{b}$                             \\
    \hline
    $\gamma_{3}$                                                                                 & 1.09$^{\dagger}$        & 1.23$^{a}$       & 1.62$^{\dagger}$        & 2.93$^{a}$                   & 2.27$^{\dagger}$        & 2.10$^{a}$    \\
                                                                                                 & 0.84$^{\ast}$     &  1.11$^{b}$                            & 1.27$^{\ast}$     &  1.58$^{b}$                            & 2.02$^{\ast}$     & 2.20$^{b}$                             \\
    \hline
    $m_{\scalebox{0.6}{\text{SO}}}^{\ast}           (\Gamma_{7v})$ ($m_{0}$)                     & 0.318$^{\dagger}$       & 0.30$^{a}$                   & 0.219$^{\dagger}$       & 0.25$^{a}$                   & 0.175$^{\dagger}$       & 0.21$^{a}$    \\
                                                                                                 & 0.412$^{\ast}$    & 0.355$^{b}$                             & 0.275$^{\ast}$    & 0.243$^{b}$                             & 0.212$^{\ast}$    & 0.199$^{b}$                             \\
    \hline
    $m_{\scalebox{0.6}{\text{CB}},\parallel}^{\ast} (\text{X}_{6c})$ ($m_{0}$)                   & 0.704$^{\dagger}$       & 2.680$^{a}$     & 1.316$^{\dagger}$       & 2.0$^{a}$         & 1.625$^{\dagger}$       &                                    1.35$^{d}$                             \\
                                                                                                 & 0.853$^{\ast}$    & 3.670$^{c}$                             & 1.070$^{\ast}$    & 7.0$^{a}$     & 2.988$^{\ast}$    &                              \\
                                                                                                 &     &                              &     & 0.672$^{c}$      &     &                              \\
                                                                                                 &     &                              &     & 1.2$^{e}$      &     &                              \\
    \hline
    $m_{\scalebox{0.6}{\text{CB}},\perp}    ^{\ast} (\text{X}_{6c})$ ($m_{0}$)                   & 0.234$^{\dagger}$       & 0.155$^{a}$     & 0.223$^{\dagger}$       & 0.19$^{a}$      & 0.243$^{\dagger}$       &                             0.275$^{d}$ \\
                                                                                                 & 0.281$^{\ast}$    & 0.212$^{c}$        & 0.257$^{\ast}$    & 0.275$^{a}$         & 0.276$^{\ast}$    &                              \\
                                                                                                 &     &                              &     & 0.193$^{c}$      &     &                              \\
                                                                                                 &     &                              &     & 0.25$^{e}$      &     &                              \\
    \hline
    $m_{\scalebox{0.6}{\text{CB}},\parallel}^{\ast} (\text{L}_{6c})$ ($m_{0}$)                   & 1.571$^{\dagger}$       &                              & 1.655$^{\dagger}$       & 1.2$^{a}$           & 1.967$^{\dagger}$       &                             1.82$^{d}$ \\
                                                                                                 & 1.714$^{\ast}$    &                              & 1.748$^{\ast}$    & 2.0$^{e}$                             & 2.062$^{\ast}$    &                              \\
    \hline
    $m_{\scalebox{0.6}{\text{CB}},\perp}    ^{\ast} (\text{L}_{6c})$ ($m_{0}$)                   & 0.176$^{\dagger}$       &                              & 0.137$^{\dagger}$       & 0.15$^{a}$           & 0.141$^{\dagger}$       &                             0.132$^{d}$ \\
                                                                                                 & 0.216$^{\ast}$    &                              & 0.164$^{\ast}$    & 0.253$^{e}$                             & 0.163$^{\ast}$    &                                                                                                               
	\\ \hline
    \end{tabular}
    %\hrule
	\label{tab:III_P_effective_masses}
         \begin{flushleft}
            $^{a}$Recommended by Vurgaftman et al., see Ref.~\cite{Vurgaftman_JAP_2001} for details \\
            Theory: $^{b}$Ref.~\cite{Bastos_JAP_2018}, $^{c}$Ref.~\cite{Huang_JPCS_1985}, $^{d}$Ref.~\cite{Saidi_JAP_2010}, $^{e}$Ref.~\cite{Fraj_JAP_2007}
        \end{flushleft}
\end{table}

% GaP and InP
For GaP and InP, there exist a larger number of experimental and theoretical investigations of the effective masses \cite{Vurgaftman_JAP_2001}. First, we compare our calculated $m_{\scalebox{0.7}{\text{CB}}}^{\ast} (\Gamma_{6c})$ and $m_{\scalebox{0.7}{\text{SO}}}^{\ast} (\Gamma_{7v})$ effective masses to HSE-DFT calculations \cite{Bastos_JAP_2018} and the recommended values of Vurgaftman et al.~\cite{Vurgaftman_JAP_2001}. Our calculations are in close agreement with these literature results. Our analytically calculated $\gamma_{1}$, $\gamma_{2}$ and $\gamma_{3}$, as in AlP, are in very good agreement with those obtained from fitting to HSE-DFT electronic structure calculations \cite{Bastos_JAP_2018}.

In Ref.~\cite{Vurgaftman_JAP_2001} Vurgaftman et al.~noted factors complicating the measurement and evaluation of the X-valley CB effective masses in GaP. Vurgaftman et al.~noted a resulting large spread in reported values, due to the presence of a ``camel back'' dispersion of the lowest CB near the X-point \cite{Vurgaftman_JAP_2001}. This is consistent with our HSEsol- and TB-mBJ-calculated GaP band structures, where the minimum in this CB valley does not lie precisely at the X-point. To compute our X-point effective masses we perform additional DFT calculations slightly away from the X-point, extracting the longitudinal and transverse effective masses at $\textbf{k} = \frac{ 2 \pi }{a} ( 0.884, 0, 0 )$ in the HSEsol calculation, and $\textbf{k} = \frac{ 2 \pi }{a} ( 0.855, 0, 0 )$ in the TB-mBJ calculation. Both our HSEsol- and TB-mBJ-calculated GaP $m_{\scalebox{0.6}{\text{CB}},\parallel}^{\ast} (\text{X}_{6c})$ effective masses are in good agreement with the 30-band $\textbf{k}\cdot\textbf{p}$ calculations of Fraj et al.~\cite{Fraj_JAP_2007}. Our calculated $m_{\scalebox{0.6}{\text{CB}},\parallel}^{\ast} (\text{X}_{6c})$ effective masses for GaP provide much greater certainty than the range of values originally recommended by Vurgaftman et al.~\cite{Vurgaftman_JAP_2001}. We note that these effective masses do not follow the typical trend of TB-mBJ-calculated effective masses being larger than those calculated using HSEsol. Our calculated $m_{\scalebox{0.6}{\text{CB}},\perp}^{\ast} (\text{X}_{6c})$ effective masses for GaP are within the range recommended by Vurgaftman et al.~\cite{Vurgaftman_JAP_2001}, and find good agreement with the calculations of Fraj et al.~\cite{Fraj_JAP_2007}. In Ref.~\cite{Vurgaftman_JAP_2001} Vurgaftman et al.~originally recommended X-point InP effective masses on the basis of only an X-valley DOS effective mass. This recommendation was later revised in Ref.~\cite{Vurgaftman_book_2020}, citing 40-band $\textbf{k}\cdot\textbf{p}$ results by Saidi et al.~\cite{Saidi_JAP_2010}, with a lack of further available literature data. Our calculated InP $m_{\scalebox{0.6}{\text{CB}},\parallel}^{\ast} (\text{X}_{6c})$ effective masses see a large discrepancy between those calculated using HSEsol and TB-mBJ. Our HSEsol-calculated InP $m_{\scalebox{0.6}{\text{CB}},\parallel}^{\ast} (\text{X}_{6c})$ effective mass is in better agreement with that of Saidi et al.~\cite{Saidi_JAP_2010}. As we will discuss below, HSEsol-calculated effective masses typically provide the most accurate description of semiconductor effective masses. We therefore place greater confidence in our HSEsol-calculated value in this case, while acknowledging further literature data would provide greater clarity regarding this material parameter. Our InP $m_{\scalebox{0.6}{\text{CB}},\perp}^{\ast} (\text{X}_{6c})$ effective masses are in good agreement with those calculated by Saidi et al.~\cite{Saidi_JAP_2010}. In GaP and InP, while there are some studies of the L-point effective masses, there are fewer results available than for those at the X-point. The original recommendations of Vurgaftman et al.~in Ref.~\cite{Vurgaftman_JAP_2001} cited experimental studies, while this was later revised in Ref.~\cite{Vurgaftman_book_2020} again citing the $\textbf{k}\cdot\textbf{p}$ results of Saidi et al.~\cite{Saidi_JAP_2010} for InP, and Fraj et al.~\cite{Fraj_JAP_2007} for GaP. Our calculated GaP $m_{\scalebox{0.6}{\text{CB}},\parallel}^{\ast} (\text{L}_{6c})$ effective masses lie within between the original recommendation of Vurfgaftman et al.~and that of Fraj et al. Our GaP $m_{\scalebox{0.6}{\text{CB}},\perp}^{\ast} (\text{L}_{6c})$ effective masses find better agreement with the recommendation of Vurgaftman et al. than that of Fraj et al., with our calculations straddling the recommendation of Vurgaftman et al. For InP we note that both our $m_{\scalebox{0.6}{\text{CB}},\parallel}^{\ast} (\text{L}_{6c})$ and $m_{\scalebox{0.6}{\text{CB}},\perp}^{\ast} (\text{L}_{6c})$ effective masses are in good agreement with the calculations of Saidi et al.

% The X-valley minima have been reported to occur at $\textbf{k} = \frac{ 2 \pi }{a} \left( 0.95, 0, 0 \right)$ and equivalent points \cite{Vurgaftman_JAP_2001}.
% In Ref.~\cite{Vurgaftman_book_2020} Vurgaftman et al.~recommended revised effective masses for GaP, citing theoretical results from Ref.~\cite{Saidi_JAP_2010}.

% TB-mBJ vs. HSEsol
In line with the known tendency of the TB-mBJ exchange potential to overestimate effective masses \cite{Kim_PRB_2010}, we note that our TB-mBJ-calculated Luttinger parameters -- which depend on the inverse of the zone-centre VB edge effective masses (cf. Eqs.~\eqref{eq:ch_6_lutt_1} - \eqref{eq:ch_6_lutt_3}) -- are lower than those computed using HSEsol-calculated effective masses, noting the exception in our calculated GaP $m_{\scalebox{0.6}{\text{CB}},\parallel}^{\ast} (\text{X}_{6c})$ effective masses. Our results emphasise that careful benchmarking should be undertaken when analysing TB-mBJ band structures, since differences in valley splittings and band edge effective masses can create significant changes in the band edge DOS, particularly in materials possessing multiple valleys lying close in energy. We note that the work of Laurien et al.~\cite{Laurien_PRB_2022} involved rigorous benchmarking of XC functionals for effective mass calculations, resulting in the recommendation that HSEsol XC functionals produce the most accurate prediction of semiconductor effective masses. 


\subsection{Kane parameter and ellipticity}
\label{sec:ch_6_unstrained_elec_band_parameters_kane_ellipticity}

The semi-local nature of the HSEsol XC functional and TB-mBJ exchange potential modifies the commutation relation between the Kohn-Sham Hamiltonian and the position operator \cite{Starace_PRA_1971}. This, and the improper normalisation of the PAW pseudo-wave-functions, preclude direct application of the gradient operator, $\widehat{\textbf{p}} = - i \hbar \nabla$, to compute momentum matrix elements \cite{Gajdos_PRB_2006}. Our Kane parameters $E_{P}$ are obtained via explicit evaluation of the inter-band momentum matrix elements in the PAW formalism \cite{Gajdos_PRB_2006}, and are presented in Table~\ref{tab:III_P_kane_parameter}. Comparing our HSEsol-calculated $E_{P}$ for AlP to that obtained by Bastos et al.~\cite{Bastos_JAP_2018} via numerical fitting of an 8-band $\textbf{k} \cdot \textbf{p}$ Hamiltonian to their HSE band structure, we note good quantitative agreement. Comparing our explicitly calculated $E_{P}$ for GaP and InP those obtained via numerical fitting of an 8-band $\textbf{k} \cdot \textbf{p}$ Hamiltonian to HSE-calculated bands by Bastos et al.~\cite{Bastos_JAP_2018}, we do not see the same close agreement as in AlP. We note that this is a consequence of the choice of Bastos et al.~to fit the exact exchange mixing in their calculations to the direct band gap recommended by Vurgaftman et al.~\cite{Vurgaftman_JAP_2001} which, as discussed in Sec.~\ref{sec:ch_6_elec_structure_validation}, underestimates the true direct band gap by $\approx 1$ eV. Comparing to the range of values recommended in Ref.~\cite{Vurgaftman_JAP_2001} by Vurgaftman et al., our calculations lie near, for GaP, or within, for InP, these ranges. We obtain a significantly lower value of $E_{P}$ in our TB-mBJ calculation. This underestimation of momentum matrix elements by the TB-mBJ exchange potential underlies the method's overestimation of effective masses (cf.~Sec.~\ref{sec:ch_6_unstrained_elec_band_parameters_eff_mass_luttinger}) \cite{Laurien_PRB_2022}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% III-P Table 4: Kane parameter, renormalised inverse zone-centre CB effective mass %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{table}[h!]
\centering
	\caption{DFT-calculated Kane parameter $E_{P}$ and renormalised inverse zone-centre CB effective mass $s_{c}$ for AlP, GaP and InP. HSEsol- and TB-mBJ-calculated results are respectively denoted by a dagger ($\dagger$) and an asterisk ($\ast$). The exact exchange mixing $\alpha$ (Becke-Roussel mixing $c$) is $\alpha = 0.363$ ($c = 1.218$) for AlP, $\alpha = 0.314$ ($c = 1.193$) for GaP, and $\alpha = 0.274$ ($c = 1.165$) for InP.}
    \begin{tabular}{l|cc|cc|cc} \hline
                                                                                                    & \multicolumn{2}{c}{AlP} \vline          & \multicolumn{2}{c}{GaP} \vline      & \multicolumn{2}{c}{InP}      \\
        Parameter                                                                                   & This work & Reference                   & This work & Reference               & This work & Reference        \\
        \hline   
    $E_{P}$ (eV)                                                                                 & 21.23$^{\dagger}$       & 17.7$^{a}$       & 21.53$^{\dagger}$       & 22.2$^{a}$      & 16.86$^{\dagger}$       & 16.6$^{a}$   \\
                                                                                                 & 15.68$^{\ast}$    & 22.1$^{b}$                             & 16.29$^{\ast}$    & 31.4$^{a}$                   & 13.12$^{\ast}$    & 20.7$^{a}$             \\
                                                                                                  &     &                              &     & 25.2$^{b}$                   &     & 18.3$^{b}$              \\
    \hline
    $s_{c}$                                                                                      & 0.65$^{\dagger}$        & $-$0.30$^{a}$  & 0.37$^{\dagger}$  & $-$3.09$^{a}$ & 0.25$^{\dagger}$ & $-$1.62$^{a}$                \\
                                                                                                 & 0.72$^{\ast}$     &            & 0.35$^{\ast}$     &                              & 0.42$^{\ast}$     &
	\\ \hline
    \end{tabular}
    %\hrule
	\label{tab:III_P_kane_parameter}
         \begin{flushleft}
            $^{a}$Recommended by Vurgaftman et al., see Ref.~\cite{Vurgaftman_JAP_2001} for details \\
            Theory: $^{b}$Ref.~\cite{Bastos_JAP_2018}
        \end{flushleft}
\end{table}

In Table~\ref{tab:III_P_kane_parameter} we also present the renormalised inverse zone-center CB effective mass $s_{c}$, calculated using Eq.~\eqref{eq:ch_4_sc}, and related to Kane's parameter $F$ via $s_{c} = 1 + 2F$ \cite{Vurgaftman_JAP_2001}. As described in Chapter~\ref{ch:chapter4}, having $s_{c} < 0$ can, at large wave vector $\vert \textbf{k} \vert$ in the widely-employed 8-band \textbf{k}$\cdot$\textbf{p} Hamiltonian \cite{Bahder_PRB_1990}, produce unphysical states lying energetically within the band gap due to downward bending of the lowest-energy CB \cite{Foreman_PRB_1997}. As demonstrated in Chapter~\ref{ch:chapter4} and Appendix~\ref{ch:appendix1}, even in the absence of downward bending of the CB, a negative value of $s_{c}$ can additionally give rise to spurious solutions in heterostructure calculations via bulk states possessing complex-valued wave vector \cite{Murphy_JPDAP_2023}. This highlights the importance of $s_{c}$ for heterostructure calculations. From our DFT calculations we obtain $s_{c} > 0$, ensuring no downward bending of the CB in AlP, GaP or InP. This also guarantees $s_{c} + 1 > 0$, satisfying one of the conditions that ensure ellipticity of the 8-band Hamiltonian \cite{Veprek_PRB_2007,Sytnyk_arXiv_2018}. In addition to this condition, the 8-band Hamiltonian is elliptic provided \cite{Veprek_PRB_2007,Sytnyk_arXiv_2018}

\begin{equation}
    \gamma_{1} + 4 \gamma_{2} + 6 \gamma_{3} > \frac{ 2 E_{P} }{ E_{g} } \, ,
\end{equation}

\begin{equation}
    \gamma_{1} + 4 \gamma_{2} + 3 \gamma_{3} > \frac{ 3 E_{P}}{ 2 E_{g} } \, ,
\end{equation}

\begin{equation}
    \gamma_{1} - 2 \gamma_{2} + 3 \gamma_{3} > \frac{ E_{P} }{ 2 E_{g} } \, ,
\end{equation}

\begin{equation}
    \gamma_{1} - 2 \gamma_{2} - 3 \gamma_{3} > - \frac{ E_{P} }{ 2 E_{g} } \, ,
\end{equation}

%\begin{eqnarray}
%
%    \gamma_{1} + 4 \gamma_{2} + 6 \gamma_{3} &>& \frac{ 2 E_{P} }{ E_{g} } \, , \\
%
%    \gamma_{1} + 4 \gamma_{2} + 3 \gamma_{3} &>& \frac{ 3 E_{P}}{ 2 E_{g} } \, , \\
%
%    \gamma_{1} - 2 \gamma_{2} + 3 \gamma_{3} &>& \frac{ E_{P} }{ 2 E_{g} } \, , \\
%
%    \gamma_{1} - 2 \gamma_{2} - 3 \gamma_{3} &>& - \frac{ E_{P} }{ 2 E_{g} } \, ,
%
%\label{eq:ch_6_ellipticity_eqns}
%\end{eqnarray}

\noindent
all of which are satisfied by our HSEsol- and TB-mBJ-calculated band parameters. Tables~\ref{tab:III_P_energies}, \ref{tab:III_P_effective_masses} and \ref{tab:III_P_kane_parameter} thus provide a robust set of 8-band $\textbf{k} \cdot \textbf{p}$ parameters for (Al,Ga,In)P, with the ellipticity of the Hamiltonian expected to mitigate the appearance of spurious solutions in heterostructure calculations \cite{Veprek_PRB_2007}.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% Section 5: Strained electronic band parameters %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Strained electronic band parameters}
\label{sec:ch_6_strained_elec_band_parameters}

% Voigt notation $\epsilon_{1} = \epsilon_{xx}$, $\epsilon_{2} = \epsilon_{yy}$, $\epsilon_{3} = \epsilon_{zz}$, $\epsilon_{4} = 2 \epsilon_{yz}$, $\epsilon_{5} = 2 \epsilon_{xz}$ and $\epsilon_{6} = 2 \epsilon_{xy}$

In this Section, having extracted a robust set of parameters for unstrained III-P semiconductors, we investigate the strain dependence of the band structure. This strain dependence is encapsulated by the band edge deformation potential or, equivalently for hydrostatic strain, the band gap pressure coefficients. We now predict the band edge deformation potentials and hydrostatic band gap pressure coefficients for (Al,Ga,In)P.

\subsection{Band edge deformation potentials}
\label{sec:ch_6_strained_elec_band_parameters_def_pot}

Shifts in the band energies of semiconductors under applied strain can be described using deformation potential theory \cite{Bir_book_1974}. These changes in the electronic structure are a result of the change in volume (hydrostatic) and shape (axial or shear) of the unit cell under applied strain, where symmetry-reducing axial or shear strains can lift the degeneracy of the zone-centred VB edge HH and LH states \cite{Bir_book_1974,ODonnell_thesis_2021}. We compute the band edge deformation potentials via strained HSEsol and TB-mBJ band structures. As in Sec.~\ref{sec:ch_6_elastic_structural}, we apply strain branches, denoted in Voigt notation, to the unit cell. In the derivation of the bulk 8-band $\textbf{k} \cdot \textbf{p}$ Hamiltonian described in Eq.~\eqref{eq:ch_2_bulk_8_band_hamiltonian}, the expansion of terms depending on strain can be included \cite{Bogusevschi_thesis_2018}. For an [001]-pseudomorphic strain in the absence of spin-orbit coupling, such that $\epsilon_{1} = \epsilon_{2}$ and $\epsilon_{3} = - 2 C_{12} \epsilon_{1} / C_{11}$, the strain dependent CB and HH and LH VB edge energies can be rewritten in terms of strain as \cite{Bogusevschi_thesis_2018,Bir_book_1974,Krijn_SST_1991,Chuang_book_2009}

\begin{eqnarray}
%
    E_{\scalebox{0.7}{\text{CB}}}^{\epsilon} &=& E_{\scalebox{0.7}{\text{CB}}} + a_{c} ( \epsilon_{1} + \epsilon_{2} + \epsilon_{3} ) \, , \label{eq:ch_6_strain_cb_edge_energy} \\
%
    E_{\scalebox{0.7}{\text{HH}}}^{\epsilon} &=& E_{\scalebox{0.7}{\text{HH}}} + a_{v} ( \epsilon_{1} + \epsilon_{2} + \epsilon_{3} ) - \frac{ b }{ 2 } ( \epsilon_{1} + \epsilon_{2} - 2 \epsilon_{3} ) \, , \label{eq:ch_6_strain_hh_edge_energy} \\
%
    E_{\scalebox{0.7}{\text{LH}}}^{\epsilon} &=& E_{\scalebox{0.7}{\text{LH}}} + a_{v} ( \epsilon_{1} + \epsilon_{2} + \epsilon_{3} ) + b ( \epsilon_{1} + \epsilon_{2} - 2 \epsilon_{3} ) \, , \label{eq:ch_6_strain_lh_edge_energy}
%
\end{eqnarray}

\noindent
where $E_{\scalebox{0.7}{\text{CB}}}$, $E_{\scalebox{0.7}{\text{HH}}}$ and $E_{\scalebox{0.7}{\text{LH}}}$ are, respectively, the CB and HH, LH and SO VB edge energies in the absence of strain (cf.~Eq.~\eqref{eq:ch_2_bulk_8_band_hamiltonian}), $a_{c}$ and $a_{v}$ are the CB and VB edge hydrostatic deformation potentials, and $b$ is the VB edge axial deformation potential. With application of the $( \epsilon, \epsilon, \epsilon, 0, 0, 0 )$ strain branch Eqs.~\eqref{eq:ch_6_strain_cb_edge_energy} and \eqref{eq:ch_6_strain_hh_edge_energy} reduce to

\begin{eqnarray}
%
    E_{\scalebox{0.7}{\text{CB}}}^{\epsilon} &=& E_{\scalebox{0.7}{\text{CB}}} + 3 a_{c} \epsilon \, , \label{eq:ch_6_strain_cb_edge_energy_sb_appl} \\
%
    E_{\scalebox{0.7}{\text{HH}}}^{\epsilon} &=& E_{\scalebox{0.7}{\text{HH}}} + 3 a_{v} \epsilon \, . \label{eq:ch_6_strain_hh_edge_energy_sb_appl}
%
\end{eqnarray}

Using Eqs.~\eqref{eq:ch_6_strain_cb_edge_energy_sb_appl} and \eqref{eq:ch_6_strain_hh_edge_energy_sb_appl} we can compute the change in band gap as

\begin{equation}
    \Delta E_{g} ( \Gamma_{6c}\text{-}\Gamma_{8v} ) = - 3 a_{g} \epsilon = - a_{g} \text{Tr} ( \epsilon ) \, , \label{eq:ch_6_gap_def_pot}
\end{equation}

\noindent
where $a_{g}$ is the hydrostatic band gap deformation potential and $\text{Tr} ( \epsilon ) = \epsilon_{1} + \epsilon_{2} + \epsilon_{3}$ is the trace of the strain tensor.

To obtain the VB edge axial and shear deformation potentials, $b$ and $d$, the application of appropriate strain branches can reduce the 8-band $\textbf{k} \cdot \textbf{p}$ strain Hamiltonian to a block-diagonalised form, from which expressions for $b$ and $d$ can be obtained in terms of the strained band edge energies. To compute $b$ we apply the trace-free biaxial strain $( \epsilon, \epsilon, -2 \epsilon, 0, 0, 0 )$. Taking the zero of energy at the unstrained VB maximum, and in the absence of SO coupling, we obtain eigenvalues $E_{g}$, $-6b\epsilon$ and two of $3 b \epsilon$. Using these band edge energies we compute the LH-HH splitting as

\begin{equation}
    \Delta E_{\scalebox{0.7}{\text{HH-LH}}} = 9 b \epsilon \, . \label{eq:ch_6_b_def_pot}
\end{equation}

We can then compute the VB axial deformation potential $b$ by fitting to the zone-centre HH-LH splitting vs.~$\epsilon$ using Eq.~\eqref{eq:ch_6_b_def_pot}. To obtain $d$ we apply the shear strain $( 0, 0, 0, 0, 0, 2 \epsilon )$. The application of this strain branch, again taking the zero of energy at the unstrained VB maximum and in the absence of SO coupling, provides eigenvalues $E_{g}$, $\pm \sqrt{3} d \epsilon$ and 0. Using these band edge energies we compute

\begin{equation}
    \Delta E_{\scalebox{0.7}{\text{HH-LH}}} = 2 \sqrt{3} d \epsilon \, . \label{eq:ch_6_d_def_pot}
\end{equation}

The VB edge shear deformation potential $d$ can then be computed by fitting to the zone-centre HH-LH splitting vs.~$\epsilon$ using Eq.~\eqref{eq:ch_6_d_def_pot}. Shearing the unit cell produces an internal strain which, akin to calculating $C_{44}$, mandates internal relaxation of the ionic coordinates (cf.~Sec.~\ref{sec:ch_6_elastic_structural}). Calculations are performed for $\vert \epsilon \vert \leq 2$\%, for which these relationships behave linearly. For the HSEsol (TB-mBJ) deformation potential calculations, for consistency with our calculation of the elastic properties, internal relaxation of the ionic positions in response to an applied macroscopic strain was performed using the HSEsol (LDA) XC functional.

In general, the deformation potentials of III-P materials have undergone limited investigation, resulting in a lack of reference values in the literature \cite{Bogusevschi_thesis_2018}. The zone-centre hydrostatic band gap deformation potential $a_{g}$ values reported in literature for InP display large uncertainty \cite{Muller_PRB_1980,Menoni_PRB_1987,Nolte_PRL_1987,Camassel_PRB_1980,Kobayashi_PBC_1986}. In AlP, there have been no experimental investigations of $a_{g}$, with significant uncertainty in the limited set of published theoretical results \cite{Wei_PRB_1999,VandeWalle_PRB_1989}. Additionally, previous recommendations of $d$ for AlP have been based on the values available for GaP \cite{Vurgaftman_JAP_2001}. Given this lack of published information and certainty in the band edge deformation potentials of III-P compounds, we now undertake a rigorous calculation of these parameters.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% III-P Table 5: strained electronic band parameters %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{table}[b!]
\centering
	\caption{DFT-calculated hydrostatic band gap, uniaxial and shear deformation potentials $a_{g}$, $b$ and $d$ and hydrostatic pressure coefficients $\text{d} E_{g} / \text{d} P$ for AlP, GaP and InP. HSEsol- and TB-mBJ-calculated results are respectively denoted by a dagger ($\dagger$) and an asterisk ($\ast$). The exact exchange mixing $\alpha$ (Becke-Roussel mixing $c$) is $\alpha = 0.363$ ($c = 1.218$) for AlP, $\alpha = 0.314$ ($c = 1.193$) for GaP, and $\alpha = 0.274$ ($c = 1.165$) for InP.}
    \begin{tabular}{l|cc|cc|cc} \hline
                                                                                                 & \multicolumn{2}{c}{AlP} \vline                   & \multicolumn{2}{c}{GaP} \vline                   & \multicolumn{2}{c}{InP}                          \\
    Parameter                                                                                    & This work & Reference                            & This work & Reference                            & This work & Reference                            \\
    \hline
    $a_{g} = a_{c} - a_{v}$ (eV)                                                                 & $-$9.43$^{\dagger}$     & $-$8.7$^{a}$   & $-$9.50$^{\dagger}$     & $-$9.9$^{a}$                 & $-$6.39$^{\dagger}$     & $-$6.6$^{a}$                 \\
                                                                                                 & $-$8.56$^{\ast}$  & $-$9.52$^{b}$                & $-$8.72$^{\ast}$  & $-$8.83$^{b}$                             & $-$5.92$^{\ast}$  & $-$5.93$^{b}$                             \\
                                                                                                 &   & $-$8.5$^{b}$ &   & $-$7.99$^{b}$            &   & $-$5.30$^{b}$                             \\
                                                                                                 &   & $-$8.70$^{c}$ &   & $-$9.35$^{c}$            &   & $-$5.95$^{c}$                             \\
    \hline
    $b$ (eV)                                                                                     & $-$1.72$^{\dagger}$     & $-$1.5$^{a}$   & $-$2.06$^{\dagger}$     & $-$1.6$^{a}$                & $-$1.70$^{\dagger}$     & $-$2.0$^{a}$                 \\
                                                                                                 & $-$1.82$^{\ast}$  & $-$1.0$^{c}$                             & $-$2.01$^{\ast}$  & $-$1.5$^{c}$                            & $-$1.61$^{\ast}$  & $-$1.6$^{c}$                             \\
                                                                                                 &   & $-$2.3$^{d}$  &   & $-$2.3$^{d}$            &   &  $-$1.8$^{d}$                            \\
    \hline
    $d$ (eV)                                                                                     & $-$5.30$^{\dagger}$     & $-$4.6$^{a}$                 & $-$5.27$^{\dagger}$     & $-$4.6$^{a}$                 & $-$4.26$^{\dagger}$     & $-$5.0$^{a}$                 \\
                                                                                                 & $-$5.23$^{\ast}$  &                              & $-$4.93$^{\ast}$  &                              & $-$3.95$^{\ast}$  &                              \\
    \hline
    $\frac{ \text{d} E_{g} }{ \text{d} P } (\Gamma_{6c}\text{-}\Gamma_{8v})$                     & 10.04$^{\dagger}$       & 9.40$^{b}$       & 9.72$^{\dagger}$        & 8.90$^{b}$       & 8.27$^{\dagger}$        & 7.40$^{b}$        \\
    (meV kbar$^{-1}$)                                                                            & 9.48$^{\ast}$     & 11.1$^{b}$                             & 9.46$^{\ast}$     &  10.0$^{b}$                            & 8.14$^{\ast}$     & 8.4$^{b}$                             \\
    \hline
    $\frac{ \text{d} E_{g} }{ \text{d} P } (\text{X}_{6c}\text{-}\Gamma_{8v})$                   & $-$1.96$^{\dagger}$     & $-$2.06$^{b}$ & $-$2.05$^{\dagger}$     & $-$2.20$^{b}$ & $-$1.93$^{\dagger}$     & $-$1.40$^{b}$ \\
    (meV kbar$^{-1}$)                                                                            & $-$2.43$^{\ast}$  &  $-$1.60$^{b}$                            & $-$2.53$^{\ast}$  &  $-$1.40$^{b}$                            & $-$2.42$^{\ast}$  & $-$2.26$^{b}$                             \\
    \hline
    $\frac{ \text{d} E_{g} }{ \text{d} P } (\text{L}_{6c}\text{-}\Gamma_{8v})$                   & 4.62$^{\dagger}$        & 4.17$^{b}$       & 3.88$^{\dagger}$        & 3.30$^{b}$       & 4.01$^{\dagger}$        & 3.14$^{b}$       \\
    (meV kbar$^{-1}$)                                                                            & 3.81$^{\ast}$     &  5.10$^{b}$                            & 3.41$^{\ast}$     &  4.30$^{b}$                            & 3.71$^{\ast}$     & 4.20$^{b}$                                     
	\\ \hline
    \end{tabular}
    %\hrule
	\label{tab:III_P_strained_elec}
         \begin{flushleft}
            $^{a}$Recommended by Vurgaftman et al., see Ref.~\cite{Vurgaftman_JAP_2001} for details \\
            Theory: $^{b}$Ref.~\cite{Wei_PRB_1999}, $^{d}$Ref.~\cite{Blacha_PSSB_1984} \\
            Experiment: $^{c}$Ref.~\cite{Ritter_pss_1998}
        \end{flushleft}
\end{table}

We present our calculated band edge deformation potentials in Table~\ref{tab:III_P_strained_elec}. Previous theoretical analysis has produced a spread of predicted values for the band edge deformation potentials of AlP. Our TB-mBJ-calculated $a_{g} = -8.56$ eV is in quantitative agreement with the LDA calculation of Wei and Zunger \cite{Wei_PRB_1999}, and close to that predicted by Van de Walle \cite{VandeWalle_PRB_1989}. Noting that the LDA tends to underestimate the magnitude of $a_{g}$, Wei and Zunger employed an external potential to correct the band gap underestimation in their LDA calculation, resulting in $a_{g} = -9.52$ eV \cite{Wei_PRB_1999}. This is in close quantitative agreement with our HSEsol prediction. In GaP and InP, we note that our TB-mBJ-calculated $a_{g}$ is also in close quantitative agreement with the corrected-LDA predictions of Wei and Zunger \cite{Wei_PRB_1999}. Our HSEsol-calculated $a_{g}$ for GaP and InP demonstrate good quantitative agreement with the original recommendations of Vurgaftman et al. The uniaxial deformation potential $b$ is less studied \cite{Bogusevschi_thesis_2018}. For AlP, Vurgaftman et al.~recommended $b = -1.50$ eV as the average of the values $-1.40$ and $-1.60$ eV respectively proposed by O'Reilly \cite{Reilly_SST_1986} and by Krijn \cite{Krijn_SST_1991}. Our calculated values of $b$ for AlP are slightly larger in magnitude, lying between the tight-binding estimations of O'Reilly \cite{Reilly_SST_1986} and Blacha et al.~\cite{Blacha_PSSB_1984}. Our calculated $b$ for GaP and InP are larger in magnitude than the recommendation of Vurgaftman et al., lying between the values proposed by Ritter et al.~\cite{Ritter_pss_1998} and Blacha et al.~\cite{Blacha_PSSB_1984}. To our knowledge, there have been no first-principles calculations of the shear deformation potential $d$ for AlP. Citing a lack of data, Vurgaftman et al.~\cite{Vurgaftman_JAP_2001} recommended $d = -4.6$ eV for AlP, equal to their recommended value for GaP. In both GaP and InP there also exists a lack of theoretical or experimental data for $d$ \cite{Vurgaftman_JAP_2001}. Our HSEsol- and TB-mBJ-calculated values of $d$ are in close agreement for each of AlP, GaP and InP, providing certainty regarding this little-studied parameter.


\subsection{Pressure coefficients}
\label{sec:ch_6_strained_elec_band_parameters_pressure_coeff}

% Pressure coefficients

Using the calculated direct $\Gamma_{6c}$-$\Gamma_{8v}$, and indirect X$_{6c}$-$\Gamma_{8v}$ and L$_{6c}$-$\Gamma_{8v}$, hydrostatic band gap deformation potentials $a_{g}$ we compute the pressure coefficients as \cite{Wei_PRB_1999}

\begin{equation}
    \frac{ \text{d} E_{g} }{ \text{d} P } = - \frac{ a_{g} }{ B_{0} } \, ,
    \label{eq:ch_6_pressure_coeff}
\end{equation}

where $B_{0}$ is the bulk modulus (cf.~Sec.~\ref{sec:ch_6_elastic_structural}). We compare our calculated pressure coefficients for (Al,Ga,In)P to the LDA and corrected-LDA calculations of Wei and Zunger \cite{Wei_PRB_1999} in Table~\ref{tab:III_P_strained_elec}. The calculated pressure coefficients follow trends common to other III-V semiconductors. First, the $\Gamma_{6c}$-$\Gamma_{8v}$ and L$_{6c}$-$\Gamma_{8v}$ band gaps admit $\text{d} E_{g} / \text{d} P > 0$, with the pressure coefficient associated with the $\Gamma_{6c}$-$\Gamma_{8v}$ band gap being 2-3 times larger in magnitude than that associated with the L$_{6c}$-$\Gamma_{8v}$ band gap. Second, the X$_{6c}$-$\Gamma_{8v}$ band gap admits $\text{d} E_{g} / \text{d} P < 0$, with magnitude approximately one-half of that associated with the L$_{6c}$-$\Gamma_{8v}$ band gap. For the direct $\Gamma_{6c}$-$\Gamma_{8v}$ hydrostatic deformation potential, the band gap correction scheme applied by Wei and Zunger also acts to increase $\text{d} E_{g} / \text{d} P$ for the X$_{6c}$-$\Gamma_{8v}$ and L$_{6c}$-$\Gamma_{8v}$ band gaps. We note a similar trend between our HSEsol and TB-mBJ calculations, with the former yielding increased $\text{d} E_{g} / \text{d} P$. We note that in all cases our HSEsol-calculated pressure coefficients lie within the range suggested by the LDA and corrected-LDA calculations of Wei and Zunger.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% Section 6: valence band offsets %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Valence band offsets}
\label{sec:ch_6_valence_band_offsets}

% In a real heterostructure, the band offsets are critical to determine the nature of carrier confinement, present DFT-calculated unstrained (natural) band offsets, which can be combined with the unstrained bulk band parameters and band edge deformation potentials to provided a complete and consistent set of parameters upon which to base analysis of III-P heterostructures. That overarching messages is what should bookend this chapter. The only think missing are bowing parameters for alloys, so that's something you can mention in your final conclusions/outlook chapter as potential next steps.

% The VBO is not a bulk calculation, and it is explicitly unstrained in this work. Given that you will be including a figure here, I think this merits it own little subsection, with the VBO part of the last table being broken off into a final mini-table to go with it. I also suggest that you give a little theoretical background here for the VBO calculations. The paper by Weston et al. is pretty easy to get through: basically, the task is to calculate the average electrostatic potential in a slab, which then gives you the difference in energy that references energy and the vacuum. Those equations are quick and easy to describe and include. Next, a normal bulk band structure calculation gives you the band energies relative to the average electrostatic potential in the material (which is generally set as the zero of energy in a DFT calculation). In this manner, you can compute the band energies relative to the vacuum via simple subtraction. Repeat this for another material, and you have the band energies in each material relative to the vacuum - i.e. you have their natural band offsets. I suggest trying to work in a little background for this, as it will flesh out a little subsection that you will otherwise likely draw questions regarding - i.e. how were the VBO calculations performed, details of the calculations, etc. Give the Weston paper a read, and we can have a quick meeting for me to explain the idea - there are something like 2 - 3 simple equations that would fully flesh out the theory here.

In a heterostructure, the band offsets are critical to determine the nature of carrier confinement \cite{Vurgaftman_JAP_2001,Weston_CMS_2018}. In this Section we present our DFT-calculated unstrained VBOs. These parameters can be combined with the natural (unstrained) bulk band parameters and band edge deformation potentials to provide a complete and consistent set of parameters upon which to base analysis of III-P heterostructures.

To compute the unstrained VBO slab calculations were performed, which are used to determine the alignment of the average electrostatic (Hartree) potential to the vacuum. An unstrained VBO can be extracted reliably from a SL calculation if the two materials are approximately lattice-matched (e.g.~AlP/GaP), but otherwise introduces strain if there exists a lattice mismatch between the materials (e.g.~AlP/InP or GaP/InP). We therefore use slab calculations to obtain unstrained VBOs. Using the average electrostatic potential energy obtained from a bulk band structure calculation then allows to align the bulk band edge energies to the vacuum, and hence to determine the natural VBO between different materials. We begin by using a slab calculation to determine the difference in energy between the average electrostatic (Hartree) potential and the vacuum. Considering a supercell in which we define a Cartesian coordinate system whose $z$ axis is perpendicular to the slab surface, the planar-average electrostatic potential along the slab is computed as \cite{Weston_CMS_2018}

\begin{equation}
    \overline{V} (z) = \frac{1}{S} \int_{S} V ( \textbf{r} ) \, \text{d}x \, \text{d}y \, \label{eq:ch_6_v_bar}
\end{equation}

\noindent
where $S$ is the area of the slab in the $(x, y)$ plane, and $V ( \textbf{r} )$ is the DFT-calculated Hartree potential. To determine the average electrostatic potential energy within the slab, Eq.~\eqref{eq:ch_6_v_bar} is then macroscopically averaged as \cite{Weston_CMS_2018}

\begin{equation}
    \overline{\overline{V}} = \frac{1}{L} \int_{-L/2}^{L/2} \overline{V} (z) \, \text{d}z \, , \label{eq:ch_6_v_bar_bar}
\end{equation}

\noindent
where $L$ is the unit cell length along $z$. This calculation therefore provides an energy difference in reference to the vacuum. Using bulk HSE band structure calculations, Weston et al.~\cite{Weston_CMS_2018} demonstrated that performing this averaging procedure in the GGA produces minor quantitative difference in VBO compared to far more computationally expensive calculations in which the potential averaging is performed via HSE. We therefore employed the Perdew-Burke-Ernzerhof XC functional for solids (PBEsol), in 32-atom [110]-oriented slab supercells having $> 3$ nm vacuum regions. The TB-mBJ exchange potential is not appropriate for slab vacuum calculations. In the presence of an extended vacuum region the average of $\vert \nabla n \vert / n$, where $n$ is the electron charge density, becomes singular and has no meaning (cf.~Sec.~\ref{sec:ch_3_impl_LDA_TB-mBJ}). The TB-mBJ potential is dependent on this quantity, meaning TB-mBJ slab vacuum calculations are not possible. The TB-mBJ VBOs are therefore, as in the case of the HSEsol-calculated VBOs, computed by aligning the TB-mBJ bulk band structure to the PBE-calculated electrostatic average in vacuum. The \textbf{k}-point grid for BZ integration was downsampled to $1 \times 11 \times 7$. Electrostatic potential averaging via Eqs.~\eqref{eq:ch_6_v_bar} and \eqref{eq:ch_6_v_bar_bar} was implemented using \textsc{VASPKIT} \cite{Wang_CPC_2021}.

We then perform bulk band structure calculations (cf.~Sec.~\ref{sec:ch_6_elec_structure}) to obtain band energies, and a reference between the VB maximum and the average electrostatic potential in bulk. With the VB maximum energy then known relative to the vacuum in the two materials ``A'' and ``B'', their bulk bands can be aligned on a single energy scale and the VBO between two materials, denoted by $A$ and $B$, is computed as

\begin{equation}
    \text{VBO} = \Delta E_{v}^{B} - \Delta E_{v}^{A} + \Delta \overline{\overline{V}} \, , \label{eq:ch_6_vbo}
\end{equation}

\noindent
where $\Delta E_{v}^{A}$ ($\Delta E_{v}^{B}$) is the energy of the VB maximum relative to the average electrostatic potential in material $A$ ($B$), and $\Delta \overline{\overline{V}}$ is the difference in the average electrostatic potential energy between materials $A$ and $B$. The determination of $\Delta \overline{\overline{V}}$ from the DFT slab calculations is illustrated in Fig.~\ref{fig:ch_6_VBO_potential_averaging}.

We employed both HSEsol- and TB-mBJ-calculated bulk band structures to compute the VBO (cf.~Table~\ref{tab:III_P_vbo}). Vurgaftman et al.~\cite{Vurgaftman_JAP_2001} provide natural VBOs, on an energy scale with the zero of energy set to lie at the InSb VB maximum -- i.e.~they provide unstrained VBOs with respect to InSb. We adopt this convention for comparative purposes, where the large lattice mismatch between (Al,Ga,In)P and InSb precludes the use of SL calculations.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% III-P Table 6: VBO %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{table}[t!]
\centering
	\caption{DFT-calculated valence band offsets (VBOs) for AlP, GaP and InP. HSEsol- and TB-mBJ-calculated results are respectively denoted by a dagger ($\dagger$) and an asterisk ($\ast$). The exact exchange mixing $\alpha$ (Becke-Roussel mixing $c$) is $\alpha = 0.363$ ($c = 1.218$) for AlP, $\alpha = 0.314$ ($c = 1.193$) for GaP, and $\alpha = 0.274$ ($c = 1.165$) for InP. Following Ref.~\cite{Vurgaftman_JAP_2001} the VBO is the natural (unstrained) VBO given relative to the VB maximum of InSb.}
    \begin{tabular}{l|cc|cc|cc} \hline
                                                                                                 & \multicolumn{2}{c}{AlP} \vline                   & \multicolumn{2}{c}{GaP} \vline                   & \multicolumn{2}{c}{InP}                          \\
    Parameter                                                                                    & This work & Reference                            & This work & Reference                            & This work & Reference                            \\
    \hline
    VBO (eV)                                                                                     & $-$1.925$^{\dagger}$    & $-$1.74$^{a}$ & $-$1.270$^{\dagger}$    & $-$1.27$^{a}$ & $-$1.137$^{\dagger}$    & $-$0.94$^{a}$ \\
                                                                                                 & $-$1.694$^{\ast}$ & $-$2.07$^{b}$ & $-$1.152$^{\ast}$ & $-$1.33$^{b}$ & $-$1.052$^{\ast}$ & $-$0.88$^{b}$ \\
                                                                                                 &                   & $-$1.72$^{c}$                &                   & $-$1.18$^{c}$                &                   & $-$1.05$^{c}$      \\
                                                                                                 &   & $-$2.00$^{d}$ &   & $-$1.31$^{d}$            &   & $-$0.95$^{d}$                             \\
                                                                                                 &   & $-$2.12$^{e}$ &   & $-$1.59$^{e}$            &   & $-$1.12$^{e}$                             
	\\ \hline
    \end{tabular}
    %\hrule
	\label{tab:III_P_vbo}
         \begin{flushleft}
        $^{a}$Recommended by Vurgaftman et al., see Ref.~\cite{Vurgaftman_JAP_2001} for details \\
        Theory: $^{b}$Ref.~\cite{Qteish_PRB_1992}, $^{c}$Ref.~\cite{Hinuma_PRB_2014}, $^{d}$Ref.~\cite{VandeWalle_PRB_1989}, $^{e}$Ref.~\cite{Li_APL_2009}
        \end{flushleft}
\end{table}

The calculated VBOs for AlP and GaP straddle the recommended value of Vurgaftman et al.~\cite{Vurgaftman_JAP_2001}, with the TB-mBJ calculation deviating by $< 50$ and $< 120$ meV respectively. Our HSEsol-calculated AlP/InSb VBO places the AlP VB maximum 185 meV lower in energy than recommended by Vurgaftman et al.~\cite{Vurgaftman_JAP_2001}. Our HSEsol-calculated GaP VBO is in exact agreement with the recommendation of Vurgaftman et al.~\cite{Vurgaftman_JAP_2001}. The HSEsol- and TB-mBJ-calculated InP VBO place the VB maximum $\approx$ 200 and 110 meV lower in energy than the recommendation of Vurgaftman et al.~\cite{Vurgaftman_JAP_2001}, respectively. The larger HSEsol-calculated AlP and GaP VBOs are in close agreement with other predicted VBOs from the literature, including the model solid theory calculations by Van de Walle \cite{VandeWalle_PRB_1989} and by Qteish and Needs \cite{Qteish_PRB_1992}, and the heterojunction DFT calculations of Li et al.~\cite{Li_APL_2009}. Our TB-mBJ-calculated VBOs are in good agreement with those computed by Hinuma et al.~\cite{Hinuma_PRB_2014}, who combined HSE slab calculations with GW-corrected bulk band edge energies. We note that our HSEsol-calculated AlP and GaP VBOs lie within the range of literature values provided in Table~\ref{tab:III_P_vbo}, while the HSEsol-calculated InP VBO lies outside the range suggested in literature. The DFT-calculated and literature VBOs in Table~\ref{tab:III_P_vbo} can be used to compute the GaP/AlP, InP/AlP and InP/GaP VBOs. Our calculations predict, on average, a decrease in the magnitude of these VBOs. We note that both our HSEsol- and TB-mBJ-calculated InP/GaP VBOs suggest a decrease in the magnitude of the VBO by 260 - 370 meV, where a notable exception is the prediction of Hinuma et al.~\cite{Hinuma_PRB_2014} which is strong agreement with our DFT-calculated results.

\begin{figure}[t!]
	\centering
   \includegraphics[width=0.90\textwidth]{./figures/chapter6/figure_5.pdf}
	\caption{Schematic illustration of the alignment of the average electrostatic (Hartree) potential $\overline{\overline{V}}$ between [110]-oriented AlP (left) and InSb (right) slabs. Solid blue and red lines respectively show the planar and macroscopic averages $\overline{V} (z)$ and $\overline{\overline{V}}$ calculated using Eqs.~\eqref{eq:ch_6_v_bar} and \eqref{eq:ch_6_v_bar_bar}. The vacuum energy is aligned in each slab, shown here by setting the zero of energy at the vacuum level, allowing extraction of $\Delta \overline{\overline{V}}$ between the two materials.}
	\label{fig:ch_6_VBO_potential_averaging}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% Section 7: Conclusions %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Conclusions}
\label{sec:ch_6_conclusions}

% Effective masses: I think you need an overall statement somewhere that the HSEsol parameters are the ones we recommend to use in calculations, and that the TB-mBJ parameters are essentially being provided for benchmarking purposes since that method is useful for performing high-throughput supercell DFT calculations (e.g. for CuPt-ordered GaInP alloys). It would read poorly to do this at the end of each subsection, so I simply suggest that you mention it in the conclusions to the chapter: multiple benchmark studies of structural, elastic and electronic properties suggest that HSEsol delivers - in the context of DFT calculations - state-of-the-art accuracy, and so it is recommended that the HSEsol parameters provided above are employed in heterostructure calculations. Then follow this up with a sentence about DFT alloy supercell calculations, which are much more efficiently performed using TB-mBJ but incurring a slight reduction in accuracy of the description of some band parameters (but that others, such as band gaps, generally remain quite reliable), but that the band parameters you provide constitute an appropriate starting point for such calculations.

% Make this "pop" out a bit more. As described above, there are two key outcomes to this chapter: (i) you have reconciled a decades-old discrepancy regarding the direct band gap of AlP, and (ii) you have provided a complete and consistent set of parameters that are suitable to parametrise empirical continuum (k.p) or atomistic (tight-binding or empirical pseudopotential) models for the analysis of III-P heterostructures. An important feature of (ii) is that all of the parameters are calculated consistently based on an underlying DFT band structure, rather than the disparate combinations of theoretical and experimental parameters that have been assembled ad hoc to support previous theoretical analysis. At the moment, this conclusions section is a bit of a dispassionate summary of what was done, which is fine and need to be there, but a little bit more along the above lines would help set the context/tone (and also tie the work more closely to the other work in the thesis on heterostructures: for III-Sb we were fortunate enough to have a good set of parameters to hand, for III-P further work was found to be necessary to compile those parameters, and you have done that work - try to sell (i) and (ii) as key outcomes of the chapter).

In summary, we presented a theoretical re-evaluation of the electronic band structure of (Al,Ga,In)P based on first-principles calculations. Literature analysis revealed significant uncertainties in key band parameters, notably a decades-old discrepancy regarding the direct band gap of AlP. Other key uncertainties include the ordering of the X-, $\Gamma$- and L-point CB valley minima in AlP, in addition to the values of the band edge effective masses, and band edge deformation potentials. By treating the exact exchange (Becke-Roussel) mixing as an empirical parameter in HSEsol (TB-mBJ) DFT calculations, we demonstrated that fitting to the known fundamental band gap, which can be direct or indirect, produces a reliable description of the band edge electronic structure. This approach produces a direct band gap in AlP that is $\approx 1$ eV larger than has generally been assumed in the literature, reconciling this long-standing discrepancy which has proliferated in the literature as a result of a lack of detailed investigation. To validate this significant revision to the direct band gap of AlP, we extracted critical point energies from the HSEsol-calculated band structure and demonstrated that employing these energies in a constrained fit to experimental SE data accounts accurately for the observed optical transitions. Having validated our HSEsol-calculated band structure, we computed an updated set of elastic and band parameters for (Al,Ga,In)P. These parameters have been calculated consistently based on a single underlying lattice parameter and band structure for each III-P compound rather than the disparate combinations of theoretical and experimental parameters that have been assembled ad hoc to construct previous III-P parameter sets. Our calculated band parameters are in good quantitative agreement with those from beyond-DFT (many-body) electronic structure calculations. Multiple benchmark studies of structural, elastic and electronic properties suggest that HSEsol delivers -- in the context of DFT calculations -- state-of-the-art accuracy, and so it is recommended that the HSEsol parameters provided in this Chapter are employed in heterostructure calculations. However, supercell DFT calculations are much more efficiently performed using the TB-mBJ exchange potential, but incur a reduction in accuracy of the description of some band parameters. The LDA and TB-mBJ parameters constitute an appropriate starting point for many of these types of calculations. The HSE-calculated parameters presented here -- several of which have been computed for the first time -- constitute an improved basis from which to develop predictive analysis of III-P heterostructures, a platform that continues to attract interest for classical and quantum photonics applications.