\chapter{Summary, conclusions and outlook} \label{ch:chapter7}
\lhead{\emph{Summary, conclusions and outlook}}

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%%%% Section 1: Summary and conclusions %%%%
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\section{Summary and conclusions}
\label{sec:ch_7_summary_conclusions}

% Overview and introduction

This Thesis has presented the results of a theoretical investigation of the electronic and optical properties of III-V superlattices (SLs) for mid-infrared (mid-IR) emission and detection, and the determination of robust electronic band parameters for III-P semiconductors from first-principles. For the former, this involved extending plane-wave based multi-band $\textbf{k} \cdot \textbf{p}$ heterostructure calculations to include the miniband dispersion along the growth direction, and detailed benchmarking to ensure accurate and reliable predictive simulations. Analysis of the electronic structure allowed to elucidate the origins of trends in radiative recombination in unstrained broken-gap SLs, and optical absorption in strain-balanced type-II SLs. High-throughput calculations enabled key trends in optoelectronic properties to be identified and quantified, allowing to develop guidelines for epitaxial growth of optimised III-V SLs for applications as high-performance mid-IR emitters and detectors. Following this, literature investigation highlighted the lack of reliable and consistent values for several key electronic band parameters in III-P materials, which are crucial to enable predictive simulations for III-P heterostructures. To address this issue, first principles calculations based on state-of-the-art DFT approaches were performed to produce a set of reliable parameters for AlP, GaP and InP.

% Ch. 4 - InAs/GaSb SLs

Chapter~\ref{ch:chapter4} presented an analysis of radiative recombination in InAs/GaSb SLs for mid-IR light-emitting diode (LED) and laser applications. We established a robust computational framework based on multi-band $\textbf{k} \cdot \textbf{p}$ calculations to compute the full $\textbf{k}$-dependent SL band dispersion and eigenstates. In doing this, we elucidated the origin of so-called spurious solutions in the calculated SL electronic properties, via analysis of the bulk complex band structure. Employing a simple renormalisation of the bulk inter-band momentum matrix element -- encapsulated by the Kane parameter $E_{P}$ -- we demonstrated the ability to reliably mitigate the impact of these spurious solutions in heterostructure calculations. Our analysis, validated via comparison to experimental EL measurements, showed that the occupation by electrons and holes of miniband states possessing non-zero wave vector along the growth direction acts to decrease the temperature dependence of the SE peak energy compared to that of the fundamental zone-centre SL band gap. Considering InAs/GaSb SLs having equal InAs and GaSb layer thicknesses, we quantified the impact of miniband formation on the SE rate and radiative recombination coefficient $B$. Our calculations revealed the important roles played by (i) electron and hole localisation within individual layers of the SL, (ii) carrier occupation of miniband states, and (iii) the strong dependence of the inter-band optical matrix elements on the miniband wave vector $q$. We demonstrated that the interplay between these factors allow for high optical matrix elements, despite the presence of type-II-like (spatially indirect) carrier confinement, and hence for $B$ values that are large relative to those typically associated with type-II structures. Analysis of the SL electronic structure identified that the electron (hole) minibands being strongly (weakly) dispersive resulted in a significant mismatch in the ranges of miniband wave vector $q$ across which electron and hole states are occupied. This mismatch in electron and hole miniband occupancy -- whereby electrons occupy states over a significantly reduced range of $q$ vs.~holes -- combined with the requirement to conserve crystal momentum during electron-hole recombination, prevents holes close to the SL Brillouin zone edge from undergoing radiative recombination. Thus, miniband formation decreases both the magnitude of $B$ and its dependence on temperature. Having undertaken this detailed interrogation of the properties of InAs/GaSb SLs having equal layer thicknesses, we then performed a systematic analysis of SLs having unequal layer thicknesses. We demonstrated that for shorter wavelengths $B$ is maximised in InAs/GaSb SLs in which the InAs layer thickness is close to equal to the GaSb layer thickness. For longer wavelengths $B$ is maximised for InAs layer thickness $\approx 1.5$ -- $2 \; \times$ the GaSb layer thickness. We also identified that $B$ decreases rapidly if the InAs layer thickness is reduced below that of GaSb. To summarise, 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 layer, with the relative thickness increasing from 1 at 3 $\mu$m to 2 at 7 $\mu$m to maximise the radiative recombination rate at fixed carrier injection.

% Ch. 5 - InAs/InAsSb SLs

Chapter~\ref{ch:chapter5} presented an analysis of optical absorption in InAs/InAs$_{1-x}$Sb$_{x}$ strain-balanced SLs (SBSLs) for mid-IR photodetectors operating in the 3 - 5 $\mu$m wavelength range. We extended the computational framework employed 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 identified SBSLs having Sb composition $x = 18$ - 40\%. The upper end of this Sb composition range poses challenges for epitaxial growth, due to the presence of large in-plane compressive strain and the requirement for low layer thicknesses in order to  maintain the required cut-off wavelength. Using the calculated SL electronic structure and eigenstates we computed the band-to-band optical absorption spectrum for each identified SBSL, revealing the ability to engineer an increase in optical absorption in the 3 - 5 $\mu$m range with increasing Sb composition $x$. To provide an appropriate figure of merit to enable SBSL optimisation for photodetection, the calculated SBSL absorption spectra were employed in a model calculation of the internal quantum efficiency (IQE) associated with photodetection of 3 - 5 $\mu$m radiation from an ambient blackbody source. At fixed Sb composition and SBSL absorber thickness, we demonstrated the potential to increase the IQE in the 3 - 5 $\mu$m spectral range by designing structures to have longer $\lambda_{\text{cut}}$. With increasing Sb composition $x$, we predict a significant enhancement of the IQE up to $x = 30$\%, independent of $\lambda_{\text{cut}}$. Our calculations reveal minimal additional enhancement in IQE for $x > 30$\%. Interrogation of the electronic structure evolution vs.~Sb composition $x$ allowed to identify that this trend derives from a competition between the joint density of states and optical matrix elements, which respectively decrease and increase with increasing $x$. The increase in the latter was found to be sufficient to drive a net increase of the IQE. Our calculations predict minimal enhancement for cut-off wavelengths longer than 5.5 $\mu$m.

% Ch. 6 - III-P electronic band parameters

Chapter~\ref{ch:chapter6} presented an analysis of the electronic band parameters of the III-P compounds AlP, GaP and InP. Interrogation of theoretical and experimental data from the literature revealed significant uncertainties in key electronic band parameters for III-P semiconductors. Key uncertainties in these parameters included the direct band gap in AlP, the energy ordering of the CB valley minima in AlP and InP, the band edge effective masses, and the 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 (lowest energy) band gap, reliably accesible via experimental measurement, produces a reliable description of the band edge electronic structure. This fundamental band gap is indirect in AlP and GaP, and direct in InP. Based on a detailed reevaluation of literature data, including a constrained fit to ellipsometry data based on our DFT calculations, we revise the direct band gap of AlP to a value $\approx$ 1 eV larger than generally assumed. Our reevaluation of the AlP band structure reconciles theory and experiment, and revises a decades old underestimation of the direct band gap that has proliferated in the literature due to a lack of additional analysis. Based on a systematic suite of DFT calculations, we extracted a robust and consistent set of electronic band parameters for AlP, GaP and InP, many of which have now been computed explicitly for the first time. Furthermore, our calculated band parameters are in good quantitative agreement with those from beyond-DFT (many-body) electronic structure calculations and, where available, experimental measurements.


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%%%% Section 2: Outlook %%%%
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\section{Outlook}
\label{sec:ch_7_outlook}

% InAs/GaSb SLs

The calculational framework that has been established to treat radiative recombination and optical absorption in SL heterostructures is generally applicable to systems whose constituent material bulk band structures can be well described via a multi-band $\textbf{k} \cdot \textbf{p}$ Hamiltonian. To further develop quantitative in silico design of SL emitters, like the InAs/GaSb SLs investigated in this work, further analysis is required to enable quantitative prediction of the IQE. In particular, this will require accurate computation of the Auger-Meitner recombination coefficient $C$, which quantifies the rate of three-carrier non-radiative recombination processes. This is due to the importance of this non-radiative recombination process as a critical efficiency-limiting loss mechanism in long-wavelength emitters \cite{Alshahrani_ADPR_2022,Sifferman_IEEE_2015,Eales_IEEE_2017}. Our SL simulation framework allows to reliably compute \textbf{k}-dependent SL eigenstates, thereby providing the basis for accurate evaluation of the matrix elements associated with Coulomb scattering, and hence to evaluate the Auger-Meitner recombination rate.

% InAs/InAsSb SBSLs

In the case of absorbers, like the InAs/InAs$_{1-x}$Sb$_{x}$ SBSLs investigated in this work, our systematic analysis focused on the active region of photodetector devices. There exist several strategies to improve the performance of mid-IR photodetectors based on these structures. One strategy is the inclusion of thin Al$_{x}$Ga$_{1-x}$Sb electron-blocking layers which can, for example, reduce dark currents \cite{Delli_ACSP_2019}. Our established calculational framework can be readily extended to analyse and optimise the electronic and optical properties of such structures. Additionally, there is further scope for work using the calculated SBSL electronic structure to develop quantitative calculations of device-specific figures of merit \cite{Datskos_EOPE_2016}. While our analysis provided insight into underlying trends in the optical absorption of SBSLs, the optimisation of specific device figures of merit may be of interest for ongoing experimental research. This type of work is likely to be of significant value in supporting the design of SL-based detectors in industry for real-world applications.

% III-P parameters feeding predictive calculations

Having established a robust parameter set for III-P semiconductors, a clear next step would be to apply these parameters to $\textbf{k} \cdot \textbf{p}$-based calculations to predict accurate electronic and optical properties for visible-wavelength III-P heterostructures. This could include analysis and optimisation of pseudomorphic and metamorphic III-P quantum wells (QWs) and quantum dots (QDs) for application in LEDs and lasers, particularly in the challenging orange/amber (580 - 620 nm) spectral range, in addition to exploration of the electronic structure of coupled GaInP QDs as hosts of Wigner crystallisation for quantum computing applications \cite{Mintairov_PRB_2018,Mintairov_APL_2019,Jauregui_EL_1993}. The plane-wave multi-band $\textbf{k} \cdot \textbf{p}$ framework established in this work can be applied directly to analyse III-P QWs and can also, once implemented in three dimensions, be applied to perform predictive simulations of III-P QDs \cite{Murphy_SPIE_2023}. The bowing parameters in AlP alloys are usually assumed to be close to zero \cite{Vurgaftman_JAP_2001,Bogusevschi_thesis_2018}. This was based on the historical value for the AlP direct band gap and measurements of the experimental band gap of alloys Al$_{x}$Ga$_{y}$In$_{1-x-y}$P alloys grown on GaAs with close to 50\% In composition. First instance reevaluation of the AlP alloy bowing parameters can be done using the alloy experimental measurements with new end-point values. Explicit calculation of the bowing parameters in III-P heterostructures using DFT-based methods would provide clarity and further enhance the accuracy of bowing parameters, enhancing predictive capabilities of future III-P heterostructure simulations. GaInP alloys, and hence GaInP QDs, are subject to CuPt-ordering \cite{Pescaglini_ACSP_2018}. Empirical atomistic analysis is required in order to accurately describe the impacts of this short-range ordering on electronic structure. Our DFT calculations can underpin this type of analysis in III-P heterostructures. Firstly, our DFT-calculated structural and elastic properties allow to parametrise empirical interatomic potentials \cite{Tanner_PRB_2019}, which can underpin atomistic relaxation and strain field calculations for heterostructures. Secondly, our DFT-calculated band structures and band edge deformation potentials provide the information required to parametrise empirical pseudopotential or tight-binding models, which can then underpin atomistic electronic structure calculations for heterostructures \cite{Broderick_JAP_2024}.