\chapter{Introduction and overview} \label{ch:chapter1}
\lhead{\emph{Introduction and overview}}

In this introductory chapter we begin by presenting the background and motivation for the research contained in the subsequent chapters. We then provide an outline of the structure of the Thesis, and an overview of the primary results.


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%%%% Section 1: Background and motivation %%%%
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\section{Background and motivation}
\label{sec:ch_1_background_motivation}

% \textcolor{red}{Include some of the motivation and such which is outlined at beginning of results chapters but no need to overdo what is there already -- think of this section heading and title}

%\begin{itemize}
%    \item Don't go into low level crystal structure details etc.
%    \item Want high-level motivation: importance of mid-IR and visible-wavelength photonics for various applications, the relevant material systems, etc.
%    \item Motivate the specific problems which the thesis looks at
%    \item Introduce: "direct vs.~indirect" - in real space via type-I vs.~type-II/III band offsets (relevant to SLs), in reciprocal space via direct vs.~indirect band gap (relevant to III-P materials, could show the band           structure of GaP and InP from L to $\Gamma$ to X side-by-side to highlight the difference)
%    \item Basically want to set the scene in a way that tells a fairly unified story about various aspects of semiconductor electronic and optical properties
%\end{itemize}

% High-level motivation: mid-IR photonics

Semiconductor photonic devices, such as light-emitting diodes (LEDs), laser diodes, and photodetectors, have long been established for a variety of applications. Many important organic and inorganic molecules possess characteristic absorption lines in their vibrational-rotational spectra at mid-infrared (mid-IR) wavelengths \cite{Popa_Sensors_2019}. The absorption features associated with a number of biologically, environmentally and industrially important molecules are shown in Fig.~\ref{fig:ch_1_mid_ir_abs_spectra}. These absorption spectra admit little overlap in many key absorption features for different molecules, allowing for what is known as ``molecular fingerprinting'' in mid-IR gas sensors \cite{Popa_Sensors_2019}. As a result, mid-IR gas sensing is in demand for a broad range of applications relevant to the environmental, industrial, medical, agricultural and defence sectors \cite{Krier_Springer_mid-IR,Ilev_Springer_2006,Tournie_Elsevier_2012,Wang_OSA_2016,Alshahrani_ADPR_2022,Wang_Small_2019}. The 3 - 5 $\mu$m wavelength range is of particular interest for a number of reasons. Firstly, it is the first IR atmospheric window, within which there is minimal absorption of thermal radiation, thereby allowing for robust detection of signals. Secondly, the 3 - 5 $\mu$m wavelength range, in particular 3 - 4 $\mu$m, has traditionally been a challenging range within which to develop LEDs and diode lasers \cite{Broderick_SST_2018}. For both emitters and detectors, a number of existing technologies operating at these wavelengths pose significant issues for realising the desired efficacy. Mid-IR emission has been achieved with quantum cascade lasers (QCLs). While this technology is well established \cite{Tournie_Elsevier_2012,Vitiello_OSA_2015,Razeghi_OSA_2015}, QCL performance degrades for wavelengths $\lesssim$ 4 $\mu$m and their deployment presents the dual challenges of complex fabrication and high cost. Mid-IR photodetectors are another established technology, with existing devices based on mercury cadmium telluride, Hg$_{1-x}$Cd$_{x}$Te, materials \cite{Alshahrani_ADPR_2022,Delli_ACSP_2019}. Their deployment sees limitations including high fabrication cost, required cryogenic cooling, high non-radiative recombination rates, and large dark current due to band-to-band tunneling at high temperature \cite{Alshahrani_ADPR_2022,Delli_ACSP_2019}. Additionally, Hg$_{1-x}$Cd$_{x}$Te materials contain dangerous components which pose risks to the environment and human health \cite{Alshahrani_ADPR_2022}. 

\begin{figure}[t!]
	\centering
   \includegraphics[width=1.0\textwidth]{./figures/chapter1/figure_1.png}
	\caption{Mid-infrared absorption spectra of selected molecules with their relative intensities. Water (H$_{2}$O), carbon dioxide (CO$_{2}$), carbon monoxide (CO), nitric oxide (NO), nitrogen dioxide (NO$_{2}$), methane (CH$_{4}$), ozone (O$_{3}$), ammonia (NH$_{3}$). Figure reprinted from Popa et al., \textit{Sensors} \textbf{19}, 2076 (2019). Copyright 2019 MDPI.}
	\label{fig:ch_1_mid_ir_abs_spectra}
\end{figure}

% Candidate solutions for mid-IR sensing demands

An alternative platform for these types of mid-IR applications are III-V superlattices (SLs) \cite{Alshahrani_ADPR_2022}. SLs are periodic heterostructures consisting of large numbers of repeats of two, or more, alternating thin layers of III-V semiconductor materials \cite{Sakaki_SSC_1978}. SLs can be considered as an engineered bulk-like material for which quantum confinement along the growth direction in a QW breaks symmetry, such that the component of the carrier wave vector along the growth direction is no longer a ``good'' quantum number -- i.e.~the crystal momentum along the growth direction is not a conserved quantity. Forming a SL restores periodicity along the growth direction, with the quantised QW states acquiring so-called ``miniband'' energy dispersion, a result of the coupling between QWs. III-V SLs offer flexible engineering of the electronic band structure, including tuning the emission/absorption wavelength via the band gap and offer the potential for high temperature device operation \cite{Alshahrani_ADPR_2022}. A number of III-V SL systems have therefore attracted significant and sustained interest. Firstly, research effort on InAs/GaSb SLs and quantum wells (QWs) has been sustained since their initial fabrication in the late-1970s \cite{Sakaki_SSC_1978}. In the calculation of heterostructure electronic properties, the definition of an absolute reference energy level allows for the alignment of the conduction band (CB) and valence band (VB) offsets between constituent semiconductors \cite{Chuang_book_2009,Roul_JPDAP_2015}. An illustration of the three types of heterojunction band alignment is shown in Fig.~\ref{fig:ch_1_band_offsets}. From left to right we see type-I (``straddling-gap''), type-II (``staggered-gap'') and type-III (``broken-gap'') band alignment, in the absence of lattice mismatch. Owing to a low lattice mismatch and a large VB offset \cite{Kroemer_PE_2004}, InAs/GaSb SLs possess type-III (``broken-gap'') band offsets. This means that the VB maximum in GaSb lies higher in energy than the CB minimum in InAs. These broken-gap band offsets produce spatially indirect carrier confinement, akin to that in type-II heterostructures, meaning the electron and hole electronic states are primarily confined in \textit{adjacent} layers of the heterostructure. Prototype inter-band cascade LEDs based on InAs/GaSb SLs have demonstrated promising performance \cite{Zhou_APL_2019}. While there exists previous theoretical studies of this material system, a number of key factors have not been considered. Previous calculations did not examine band filling effects vs.~carrier density and temperature via Fermi-Dirac statistics. As such, these factors were not incorporated into rigorous calculations of the spontaneous emission rate and radiative recombination coefficient $B$. Additionally, previous calculations have not included high-throughput calculations for the in silico optimisation of InAs/GaSb structures to maximise the radiative recombination coefficient $B$ at room temperature for a desired emission wavelength. Our calculations consider these factors in detailed analysis, explicitly accounting for the underlying SL electronic structure.

\begin{figure}[t!]
	\centering
   \includegraphics[width=1.0\textwidth]{./figures/chapter1/figure_2.png}
	\caption{Illustration of the three types of heterojunction band offsets, in the absence of lattice mismatch. Moving left to right: type-I (``straddling-gap''), type-II (``staggered-gap'') and type-III (``broken-gap'') band alignment.}
	\label{fig:ch_1_band_offsets}
\end{figure}

Secondly, III-V SLs have emerged as an alternative to Hg$_{1-x}$Cd$_{x}$Te-based detectors \cite{Alshahrani_ADPR_2022}. In particular, InAs/GaSb and InAs/InAs$_{1-x}$Sb$_{x}$ based detectors have attracted sustained research effort \cite{Mohseni_APL_2001,Wei_APL_2005,Zhang_IEEE_2011,Alshahrani_ADPR_2022,Delli_APL_2020,Delli_ACSP_2019}. Over the past 10 - 15 years the Ga-free InAs/InAs$_{1-x}$Sb$_{x}$ system, possessing type-II band alignment, has garnered a large amount of interest \cite{Alshahrani_ADPR_2022}. This owes, in large part, to the Ga-free design which simplifies growth \cite{Alshahrani_ADPR_2022} and mitigates the deleterious impact of chemically inhomogenous interfaces, which can drive performance degradation via the formation of interfacial defects \cite{Steenbergen_APL_2011,Steenbergen_JVSTB_2012,Olson_APL_2012,Bader_APL_2022}. Previous analysis and optimisation has primarily focused on the zone-centre electronic structure \cite{Lackner_JAP_2012,Webster_APL_2015,Manyk_RP_2018,Ting_JEM_2020,Petticrew_SST_2023}. While such analysis has provided valuable guidance, the associated consideration of the role of the SL electronic structure in determining the optical properties pertinent to device applications is limited. This research gap emphasises the requirement for further research to support the development of mid-IR photodetectors based on InAs/InAs$_{1-x}$Sb$_{x}$ SLs. Our analysis defines an appropriate figure of merit for photodetector operation, which is then used as the basis of high-throughput calculations to identify optimised structures for device applications.

% High-level motivation: visible wavelength photonics and III-P semiconductors

Visible-wavelength ($\sim 380$ - 780 nm) photonics is another area of significant interest for a number of practical applications. Technologies including energy-efficient solid-state lighting, display technologies for virtual and augmented reality applications, and visible-wavelength optical communications require efficient and reliable device platforms, such as LEDs and semiconductor lasers. An area of interest is that of heterostructures based on III-P semiconductor materials. These systems are well established as the basis of red LEDs \cite{Kish_SS_1997} and diode lasers \cite{Kobayashi_EL_1985,Geels_EL_1992,Walter_APL_2001,Ryou_APL_2001,Huang_IEEEJQE_2019}. There remains significant interest in these heterostructures for the development of visible-wavelength emitting devices for applications in solid-state lighting and in display technologies \cite{Oh_OE_2018,Han_OM_2021,Jung_SR_2021}. Additionally, III-P semiconductor heterostructures continue to attract interest for applications in integrated Si photonics \cite{Dhingra_Optica_2021,Dhingra_APL_2020,Chang_ACSAMI_2022} and for quantum computing as robust, high-temperature single- and entangled-photon sources \cite{Mintairov_PRB_2018,Mintairov_APL_2019}.

% Additionally, emerging photonic quantum computing platforms necessitate the development of robust, high-temperature single- and entangled-photon sources.

% Material parameters, direct vs. indirect

\begin{sloppypar}\tolerance 900
Theoretical work, in support of experimental realisation of these technologies, requires accurately parametrised models of the underlying III-P bulk electronic band structure and its dependence on strain. Accurate material parameters, which are required to parametrise such models, describing the individual constituent compounds of semiconductor alloys are crucial to enable predictive alloy heterostructure calculations. This is key for ensuring accurate predictions, e.g.~the indirect to direct band gap transition in III-P alloys. The ``fundamental'' band gap is the lowest energy band gap that exists between the filled VB and empty CB states in a semiconductor. This fundamental gap can be indirect or direct in nature. If the band gap is direct, the VB maximum and CB minimum reside at the same wave vector \textbf{k} in the Brillouin zone. The momentum $p = \hbar \omega / c $ carried by an emitted or absorbed photon of energy $\hbar \omega$ is $\sim 10^{-4}$ times the crystal momentum $\hbar \textbf{k} \sim \hbar \pi / a$ associated with the extent of the Brillouin zone of a semiconductor having lattice parameter $a$. From the perspective of electron-hole pair generation (recombination) via optical absorption (emission), the change $\hbar \, \Delta \textbf{k}$ in crystal momentum is negligible with respect to the extent of the Brillouin zone. Optical transitions are thus characterised by $\Delta \textbf{k} = 0$. This means that radiative recombination involves an electron-hole pair in which both carriers have equal wave vector \textbf{k}. If indirect, the VB maximum and CB minimum reside at different wave vectors \textbf{k} and $\textbf{k}^{\prime}$, rendering electron-hole pair generation optically forbidden. In an indirect-gap semiconductor, optical emission or absorption thus requires simultaneous emission or absorption of a phonon of wave vector $\textbf{k} - \textbf{k}^{\prime}$ to conserve net-zero crystal momentum. The rate of indirect electron-hole pair generation or recombination thus depends on the \textit{product} of (i) an inter-band optical matrix element, and (ii) an electron-phonon matrix element. This most often suppresses the overall generation or recombination rate. The radiative recombination rate in an indirect-gap semiconductor is thus often several orders of magnitude lower than that associated with a direct-gap semiconductor. In Figs.~\ref{fig:ch_1_band_structure}(a) and \ref{fig:ch_1_band_structure}(b) we show the band structure of GaP and InP, respectively. These band structures are calculated via density functional theory (DFT) (cf.~Chapters~\ref{ch:chapter3} and \ref{ch:chapter6}). GaP possesses an indirect fundamental band gap, with the VB maximum residing at the $\Gamma$-point -- i.e.~the Brillouin zone centre at $\textbf{k} = 0$, and the CB minimum at the X-point -- i.e.~at the Brillouin zone edge, close to $\textbf{k} = ( \tfrac{2 \pi}{a}, 0, 0 )$ and equivalent \textbf{k} points. InP possesses a direct fundamental band gap, with both the VB maximum and CB minimum residing at the $\Gamma$-point. In a semiconductor alloy, e.g.~In$_{x}$Ga$_{1-x}$P, the determination of the composition at which the fundamental band gap becomes direct is crucial for the design and realisation of photonic devices.
\end{sloppypar}

\begin{figure}[t!]
	\centering
   \includegraphics[width=1.0\textwidth]{./figures/chapter1/figure_3.pdf}
	\caption{DFT-calculated band structure for (a) GaP and (b) InP. Calculated using HSEsol exchange-correlation functionals. The exact exchange mixing was adjusted to fit to the experimental zero-temperature fundamental indirect (direct) band gap in GaP (InP), taken from Ref.~\cite{Vurgaftman_JAP_2001}. The zero of energy is set at the valence band maximum in both (a) and (b). Further details regarding these calculations follow in Chapter~\ref{ch:chapter6}.}
	\label{fig:ch_1_band_structure}
\end{figure}

% III-P parameters issues

The III-P semiconductor compounds suffer from large uncertainties in key electronic band parameters \cite{Vurgaftman_JAP_2001,Bogusevschi_thesis_2018}, which limits the ability to perform predictive heterostructure calculations, and hence reliable in silico device design. Notable examples for AlP are the direct band gap, and relative energy and ordering of the CB valley minima. Large uncertainties exist for other III-P electronic band parameters, e.g.~the effective masses, and hydrostatic deformation potentials. Additionally, a number of parameters have never been explicitly computed for AlP, GaP and InP, having been inferred based on, e.g., extrapolation of experimental measurements on III-P alloys \cite{Vurgaftman_JAP_2001}. Interrogation of the existing literature revealed the necessity for rigorous examination of the electronic structure of AlP, GaP and InP, with the extraction of key electronic band parameters \cite{Bogusevschi_thesis_2018}. This will underpin accurate predictive simulations of the application rich III-P alloy heterostructures, enabling the guidance and support of experimental device realisations. We perform a comprehensive set of DFT calculations to produce a complete and acccurate set of band parameters for III-P compounds.


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%%%% Section 2: Structure of the thesis %%%%
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\section{Structure of the thesis and overview of primary results}
\label{sec:ch_1_structure_of_thesis}

This Thesis has two main areas of focus. Firstly, the simulation and in silico optimisation of III-V SLs for applications as emitters and detectors in the 3 - 5 $\mu$m mid-IR wavelength range. Secondly, the calculation from first principles of a comprehensive and consistent set of band parameters for III-P semiconductors. We now outline the remaining chapters of this Thesis, and provide an overview of the key results presented therein.

In Chapter~\ref{ch:chapter2} we present the multi-band $\textbf{k} \cdot \textbf{p}$-based calculational framework that is employed for the analysis of III-V SLs. This framework is used in our analysis of radiative recombination in InAs/GaSb SLs in Chapter~\ref{ch:chapter4}, and of optical absorption in InAs/InAs$_{1-x}$Sb$_{x}$ strain-balanced SLs (SBSLs) in Chapter~\ref{ch:chapter5}. We outline the 8-band $\textbf{k} \cdot \textbf{p}$ Hamiltonian, and describe the semi-analytical plane wave expansion method (PWEM) we employ to solve the coupled multi-band $\textbf{k} \cdot \textbf{p}$ envelope function Schr\"{o}dinger equations and compute the heterostructure electronic and optical properties. To accurately analyse the electronic structure of SLs we must consider the miniband dispersion along the growth direction. This dispersion is encapsulated via the miniband wave vector $q$, which is neglected in typical plane wave $\textbf{k} \cdot \textbf{p}$-based heterostructure calculations. Having outlined the inclusion of this miniband dispersion, we then provide a demonstration of the application of the semi-analytical plane wave expansion method for SLs by computing the miniband dispersion of the textbook Kronig-Penney model. We conclude by describing the computation of the inter-band (optical) momentum matrix elements between pairs of electron and hole eigenstates. The spontaneous emission and optical absorption spectra are determined in large part by these momentum matrix elements, and therefore require rigorous computation to analyse III-V SLs for applications as emitters and photodetectors in the mid-IR spectral range.

In Chapter~\ref{ch:chapter3} we present an outline of DFT and considerations for its implementation. This is the basis for our determination of band parameters for III-P semiconductors in Chapter \ref{ch:chapter6}. Here we present an overview of the key aspects and considerations relevant to this work. We begin by outlining Kohn-Sham DFT, and how it enables ab initio quantitative predictions of material properties. We then introduce practical elements related to the implementation of DFT calculations. This includes a description of pseudopotentials, the projector-augmented wave method employed in this work, and the plane wave formalism. We finish with a brief description of the DFT code employed in this work, the Vienna Ab-initio Simulation Package (\textsc{VASP}).

In Chapter~\ref{ch:chapter4} we present an analysis of radiative recombination in InAs/GaSb SLs, quantifying their optimisation in the context of mid-IR LED applications. We implement and apply the multi-band $\textbf{k} \cdot \textbf{p}$-based plane wave calculations described in Chapter~\ref{ch:chapter2} to compute the full \textbf{k}-dependent SL band dispersion and eigenstates. Employing a renormalisation of the bulk inter-band optical matrix element we demonstrate the ability to reliably mitigate the impact of so-called spurious solutions in heterostructure calculations (cf.~Appendix~\ref{ch:appendix1}), thereby ensuring the ability to preform reliable SL electronic structure calculations. To our knowledge, previous work has not computed the full SL spontaneous emission (SE) spectra, performed detailed analysis of the contributing factors to the radiative recombination rate, or performed high-throughput calculations to identify optimised structures. We begin by examining the occupation by electrons and holes of miniband states, possessing non-zero wave vector $q$ along the growth direction, to quantify the temperature dependence of the SE peak energy compared to that of the fundamental zone-centre SL band gap. Our analysis, validated via comparison to experimental electroluminescence measurements, shows that this occupation acts to decrease this temperature dependence. We then consider in detail the nature of radiative recombination in exemplar InAs/GaSb SLs having equal InAs and GaSb layer thicknesses. We investigate and quantify the impact of miniband formation on the SE rate and radiative recombination coefficient $B$. Our calculations predict a high $B$ coefficient, approaching that calculated previously for two novel classes of type-I mid-IR QWs \cite{Broderick_SST_2018,Repiso_JPDAP_2019} despite the presence of type-II-like (spatially indirect) carrier confinement. Detailed analysis reveals the important roles played by carrier localisation, occupation of miniband states, and wave vector dependence of the inter-band optical matrix elements. Our analysis of the SL electronic structure identifies a significant mismatch in the ranges in $q$ over which the electron and hole states occupy the electron and hole minibands, respectively. This mismatch in miniband occupancy, combined with the requirement to conserve crystal momentum during radiative electron-hole recombination, can -- at low temperature -- significantly reduce the number of electron-hole pairs that can undergo optical (direct) recombination. Thus, miniband formation decreases both the magnitude of $B$ and its dependence on temperature. This decrease in the magnitude of $B$ is less pronounced at higher temperatures as states at a greater range of $q$ become occupied. We then performed a systematic analysis of SLs having unequal layer thicknesses, for which we performed optimisation of $B$ as a function of emission wavelength at room temperature. Our calculations predict that, for emission between 3 and 7 $\mu$m, InAs/GaSb SLs should have InAs layer thickness greater than or equal to that of the GaSb layers, with the ratio of the thickness in the InAs layer to that of the GaSb layer increasing from 1 at 3 $\mu$m to 2 at 7 $\mu$m to maximise the room temperature radiative recombination coefficient $B$.

In Chapter~\ref{ch:chapter5} we present an analysis of optical absorption in InAs/InAs$_{1-x}$Sb$_{x}$ SBSLs for mid-IR photodetectors operating in the 3 - 5 $\mu$m wavelength range. This is achieved by extending the computational framework described in Chapter~\ref{ch:chapter2}, and applied to investigate radiative recombination in Chapter~\ref{ch:chapter4}, to compute the optical absorption. By enforcing fixed cut-off wavelength $\lambda_{\text{cut}}$ (= 5, 5.5 and 6 $\mu$m), in addition to strain-balancing with respect to a GaSb substrate, we systematically identify SBSLs at fixed Sb compositions $x = 18$ - 40\%. In targeting structures compatible with established epitaxial growth, our calculations highlight that for large Sb composition $x$ the identified SBSLs are challenging, but achievable, to actualise with existing growth techniques. For each identified SBSL we compute the band-to-band optical absorption spectrum, revealing the ability to engineer an increase in optical absorption in the 3 - 5 $\mu$m range with increasing Sb composition $x$. In order to understand the potential for optimised photodetection and provide an appropriate figure of merit, the calculated SBSL absorption spectra were employed in a model calculation of the internal quantum efficiency (IQE). Our calculations identify target Sb compositions for optimised photodetection in the 3 - 5 $\mu$m spectral range, for all of the targeted cut-off wavelengths. Our calculations reveal the competing factors in the electronic structure which drive the behaviour of the IQE. Additionally, our calculations reveal the extent to which the IQE can be enhanced by designing structures to have longer cut-off wavelength, where we see sizable increase in the IQE at fixed Sb composition $x$ moving from $\lambda_{\text{cut}} = 5$ $\mu$m to $\lambda_{\text{cut}} = 5.5$ $\mu$m. We predict minimal additional enhancement for photodetection in the 3 - 5 $\mu$m window beyond $\lambda_{\text{cut}} = 5.5$ $\mu$m.

In Chapter~\ref{ch:chapter6} we present an analysis of the electronic band parameters of the III-P semiconductors AlP, GaP and InP. Interrogation of theoretical and experimental data from literature revealed significant uncertainties in key electronic band parameters for III-P semiconductors. To address this uncertainty, we carry out a comprehensive set of DFT calculations of the electronic structure of AlP, GaP and InP. We demonstrate that by carrying out appropriate empirical fitting of DFT input parameters we can produce a reliable description of the band edge electronic structure. Where available, our DFT calculations are in good quantitative agreement with literature experimental and theoretical data. For AlP, we identify large uncertainties in several key band parameters, notably the value of the direct band gap, for which a value with large uncertainty has proliferated in the literature over the past several decades. For AlP, using our DFT-calculated electronic structure we undertake a critical assessment of literature spectroscopic ellipsometry data. Our investigation revises a number of key band parameters, most notably the direct band gap in AlP. Additionally, a number of the parameters we present are computed explicitly here for the first time. In order to underpin accurate predictive heterostructure calculations, we extract a robust and consistent set of electronic band parameters for AlP, GaP and InP.

In Chapter~\ref{ch:chapter7} we conclude and summarise our results. This is followed by an outlook, in which we discuss potential next steps to build upon the results developed in this Thesis.