% Cónal Murphy, Photonics Theory Group, Tyndall National Institute, University College Cork
%
% murphyconal7@gmail.com
% 116301431@umail.ucc.ie
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\title  {Predictive simulation and design of III-V heterostructures for light emission and detection}
\authors{C\'{o}nal Murphy}
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%%%% Declaration %%%%
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\Declaration{

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\vspace{2em}

I, C{\'o}nal Murphy (student no.~116301431), hereby declare that, unless otherwise stated, this work is my own, and that it has not been submitted for another degree, either at University College Cork or elsewhere.

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Signed: \hspace{3.0em} \includegraphics[width=0.32\textwidth]{./figures/signature.pdf}
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\dedicatory{\scalebox{1.6}{\textit{To my Mother}}}


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%%%% Acknowledgements %%%%
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\acknowledgements{

% PTG and Stefan

Firstly, I would like to thank all of those, both former and current, in the Photonics Theory Group at Tyndall National Institute, University College Cork. Thank you all for your helpful discussions, valuable advice and the welcoming atmosphere that has been present in the office and in our group meetings. I would like to extend my sincere thanks to Dr.~Stefan Schulz for all of his help and advice over the last couple of years.

% IRC, Research-Ireland

I would like to thank Taighde {\'E}ireann - Research Ireland, formerly the Irish Research Council, for the funding that supported my work via a Government of Ireland Postgraduate Scholarship (GOIPG; Grant No. GOIPG/2020/1252).

% Friends and family

I would like to thank my family for their support and encouragement throughout both my PhD and undergraduate studies. In particular, I would like to thank my mother, Margaux, for her kindness and ever present motivation. Your determination and resilience will always serve as such a tremendous inspiration to me, and I would not have been able to reach this point if it wasn't for you. I thank all of my friends, your kindness and understanding over the last number of years is something I cannot appropriately thank you for. A special thanks also goes to JJ and James, who have been tremendously understanding and supportive while living with me in the final months of completing my PhD studies.

% Eoin and Chris

Finally, I would like to express my sincere gratitude to my supervisors, Prof.~Eoin P.~O'Reilly and Dr.~Christopher A.~Broderick, for their patience, advice and support. Thank you both for allowing me the opportunity to carry out research as an undergraduate intern, for supporting my application to undertake my PhD studies, and for the guidance provided throughout and thereafter. I would like to thank Eoin for his guidance and the opportunity to learn from his immense experience and knowledge. I wish you all the best in your retirement, which is most certainly deserved. I would like to thank Chris for his support and guidance. Without his technical and scientific expertise I would not have been able to reach this point. I wish him every success in his future research. I have been extremely fortunate to have such patient, understanding and present supervisors throughout my PhD. I sincerely appreciate all of your advice, both in science and in life.

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\abstract{

% General intro

%Semiconductor heterostructures have been long established for a variety of applications, particularly in photonic devices such as light-emitting diodes (LEDs) and lasers, in addition to photodetectors and optical amplifiers. Predictive simulations are key to support the development of these technologies. Such simulations can largely be separated into two categories: (i) atomistic first-principles calculations, which offer significant predictive power but at extremely high computational cost, and (ii) continuum empirical calculations, which strive to incorporate first-principles accuracy for targeted material properties in parametrised models that are sufficiently computationally inexpensive to enable predictive analysis on length scales beyond those practically accessible to first-principles calculations. In this Thesis we employ both first-principles and continuum empirical -- based respectively on density functional theory (DFT) and the $\textbf{k} \cdot \textbf{p}$ method -- to analyse and optimise III-V semiconductor heterostructures for light emission and detection.

Semiconductor heterostructures have been long established for a variety of applications, particularly in photonic devices such as light-emitting diodes (LEDs) and lasers, in addition to photodetectors and optical amplifiers. Predictive simulations are key to support the development of these technologies. In this Thesis we employ both first-principles and continuum empirical -- based respectively on density functional theory (DFT) and the $\textbf{k} \cdot \textbf{p}$ method -- calculations to analyse and optimise III-V semiconductor heterostructures for light emission and detection.

% III-V SLs

III-V superlattices (SLs) have attracted significant interest for emission and detection at mid-infrared (mid-IR) wavelengths, for applications relevant to the environmental, industrial, medical, agricultural and defence sectors. We develop a robust framework to rigorously analyse the electronic and optical properties of III-V SLs. We employ a semi-analytical plane wave expansion method in conjunction with an 8-band $\textbf{k}\cdot\textbf{p}$ Hamiltonian to compute SL electronic structure, paying careful attention to the identification and mitigation of spurious solutions. The calculated SL eigenstates are then used directly to compute optical emission and absorption spectra, which are in turn used to define figures of merit that allow for in silico optimisation of SL structures for applications as mid-IR emitters and photodetectors.

Firstly, we present a theoretical analysis of mid-IR radiative recombination in InAs/GaSb SLs. The calculated SL eigenstates are used directly to compute spontaneous emission spectra and the radiative recombination coefficient $B$. We elucidate the origin of the relatively large $B$ coefficients in InAs/GaSb SLs which, despite the presence of spatially indirect (type-II-like) carrier confinement, are close to that of bulk InAs and compare favourably to those calculated for mid-IR type-I pseudomorphic and metamorphic quantum well structures having comparable emission wavelengths. Our analysis explicitly quantifies the roles played by carrier localisation (specifically, partial delocalisation of bound electron states) and miniband formation (specifically, miniband occupation and optical selection rules) in determining the magnitude of $B$ and its temperature dependence. Across the 3.5 - 7 $\mu$m wavelength range we use the room temperature $B$ coefficient as a figure of merit. We identify that $B$ is governed by the relative thickness of the electron-confining InAs and hole-confining GaSb layers, with our analysis of $B$ vs.~relative thickness providing guidance for the growth of optimised SLs for mid-IR light emitters. These results, combined with the expected low non-radiative Auger recombination rates in structures having spatially indirect electron and hole confinement, corroborate recently observed high output power in prototype InAs/GaSb SL inter-band cascade LEDs.

\begin{sloppypar}\tolerance 900
Secondly, we perform a systematic optimisation of 3 \text{-} 5 $\mu$m photodetection in strain-balanced InAs/InAs$_{1-x}$Sb$_{x}$ SLs, which constitute an emerging platform for the development of efficient mid-IR photodetectors. Growing strain-balanced SLs allows to realise high material quality in structures containing up to several hundred SL periods, by mitigating defect formation associated with lattice relaxation. For a given target detector cut-off wavelength we employ continuum elasticity theory to identify a strain-balanced SL at each Sb composition $x$. We then utilise the calculated SL electronic structure to compute the optical absorption spectrum. The SL absorption spectra are used in a model calculation of the internal quantum efficiency (IQE) for photocurrent generation which is employed as a figure of merit to quantify photodetector performance. We demonstrate that increasing the Sb composition to $x \approx 30$\% can increase the (relative) IQE at fixed cut-off wavelength by $\approx 15$\%, with minimal additional improvement in performance expected at higher Sb compositions.
\end{sloppypar}

% III-P material parameters

% Both (i) and (ii) have significant implications for the design of III-P heterostructures, particularly at orange/amber wavelengths which are close to a direct- to indirect-gap crossover.

Thirdly, we then develop an improved set of band parameters for the III-P semiconductors, which we extract from a comprehensive suite of DFT calculations. Despite their relevance to the development of quantum-confined heterostructures for classical and quantum applications, there remains significant uncertainty regarding several key band parameters for the III-P semiconductors AlP, GaP and InP. In literature reports, there is significant uncertainty regarding: (i) the direct band gap in AlP, the magnitude of which varies by approximately 1 eV, and (ii) the reported band edge deformation potentials for III-P compounds. Using hybrid-functional first-principles calculations we demonstrate that the $\Gamma_{6c}$-$\Gamma_{8v}$ band gap of AlP is approximately 1 eV larger than the widely assumed value of 3.63 eV. This prediction is validated by using the predicted inter-band transition energies to reevaluate literature spectroscopic ellipsometry data. Having validated our calculated band structures vs.~experimental and theoretical literature data, we compute a comprehensive and consistent set of band parameters for III-P semiconductors, including: direct and indirect band gaps, band edge effective masses, inter-band momentum (Kane) matrix elements, band edge deformation potentials, direct and indirect band gap pressure coefficients, and valence band offsets. Our revised parameter sets reconcile the large spread in reported values for the direct band gap of AlP, and the energy ordering of the AlP conduction band valleys, in addition to providing a consistent set of band edge deformation potentials for GaP and InP. Overall, our DFT calculations enable us to extract a complete and consistent set of 8-band $\textbf{k} \cdot \textbf{p}$ parameters for III-P semiconductors, providing an improved basis upon which to develop predictive calculations of the electronic and optical properties of pseudomorphic and metamorphic III-P heterostructures.

% Band parameters are also calculated using the Tran-Blaha modified Becke-Johnson exchange potential.

} \clearpage % Abstract finished, start new page


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%%%% List of Publications %%%%
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\publications{

\addtocontents{toc}{\vspace{2em}}  % Add a gap in the Contents, for aesthetics

The following is a list of published work in which aspects of the research presented in this Thesis have featured.

% The following is a list of published work and presentations in which aspects of the research presented in this thesis have featured.

% Refereed journal articles

\paragraph*{\scalebox{1.25}{1.~Refereed journal articles}}

\begin{itemize}
%
	\item ``Theory and optimisation of radiative recombination in broken-gap InAs/GaSb superlattices'',
    		 \underline{C{\'o}nal Murphy}, Eoin P. O'Reilly, and Christopher A. Broderick;
    		 \emph{J.~Phys.~D:~Appl.~Phys.} \textbf{57}, 035103 (2024).
%
	\item ``Revised direct band gap and band parameters for AlP: hybrid-functional first-principles calculations vs. experiment'',
    		 \underline{C{\'o}nal Murphy}, Eoin P. O'Reilly, and Christopher A. Broderick;
    		 \emph{APL Mater.} \textbf{13}, 011110 (2025).
%
	\item ``In silico optimization of strain-balanced InAs/InAs$_{1-x}$Sb$_{x}$ type-II superlattices for mid-infrared photodetectors'',
    		 \underline{C{\'o}nal Murphy}, George F. Seager, Eoin P. O'Reilly, Andrew R. J. Marshall, and Christopher A. Broderick;
    		 \emph{in preparation}.
%
\end{itemize}


% Conference proceedings

\paragraph*{\scalebox{1.25}{2.~Conference proceedings}}

\begin{itemize}
%
	\item ``Efficient multi-band \textbf{k}$\cdot$\textbf{p} calculations of superlattice electronic and optical properties using plane waves'',
    		 \underline{C{\'o}nal Murphy}, Eoin P. O'Reilly, and Christopher A. Broderick;
    		 in \emph{Proceedings of 2021 International Conference on Numerical Simulation of Optoelectronic Devices} (NUSOD), Online (2021).
%
\end{itemize}

\newpage
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% Conference talks

\paragraph*{\scalebox{1.25}{3.~Conference talks}}

\noindent
\newline \newline During the course of my doctoral research I have personally presented the following conference talks:

\begin{itemize}
%
	\item ``Theory of radiative recombination in broken-gap InAs/GaSb superlattices'',
    		 \underline{C{\'o}nal Murphy}, Eoin P. O'Reilly, and Christopher A. Broderick;
    		 \emph{Semiconductor Integrated Optoelectronics} (SIOE), Online (2021).
%
	\item ``Theory of high radiative recombination in broken-gap InAs/GaSb superlattices for mid-infrared emitters'',
    		 \underline{C{\'o}nal Murphy}, Eoin P. O'Reilly, and Christopher A. Broderick;
    		 \emph{Photonics Ireland}, Online (2021).
%
	\item ``Efficient multi-band \textbf{k}$\cdot$\textbf{p} calculations of superlattice electronic and optical properties using plane waves'',
    		 \underline{C{\'o}nal Murphy}, Eoin P. O'Reilly, and Christopher A. Broderick;
    		 \emph{Numerical Simulation of Optoelectronic Devices} (NUSOD), Online (2021).
%
	\item ``In silico optimisation of radiative recombination in InAs/GaSb superlattices'',
    		 \underline{C{\'o}nal Murphy}, Eoin P. O'Reilly, and Christopher A. Broderick;
    		 \emph{UK Semiconductors}, Sheffield, U.K., (2023).
%
    \item ``Quantitative in silico design of strain-balanced InAs/InAs$_{1-x}$Sb$_{x}$ type-II superlattices for mid-infrared photodetectors'',
    		 \underline{C{\'o}nal Murphy}, George F. Seager, Eoin P. O'Reilly, Andrew R. J. Marshall and Christopher A. Broderick;
    		 \emph{UK Semiconductors}, Sheffield, U.K., (2024).
%
\end{itemize}

\paragraph*{\scalebox{1.25}{4.~Conference posters}}

\noindent
\newline \newline During the course of my doctoral research I have personally presented the following conference posters:

\begin{itemize}
%
     \item ``Computationally efficient \textbf{k}$\cdot$\textbf{p} calculations for semiconductor superlattices: plane waves and multi-band Hamiltonians'',
     \underline{C{\'o}nal Murphy}, Eoin P. O'Reilly, and Christopher A. Broderick;
     \emph{International Workshop on Quantum Antimonides Research and Upscaling} (IQARUS), Donostia-San Sebasti{\'a}n, Spain (2024).
%
\end{itemize}


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%%%% Abbreviations %%%%
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\makenomenclature

\renewcommand{\nomname}{Abbreviations}

\renewcommand{\nompreamble}{This list defines some of the commonly used abbreviations which appear throughout this Thesis. For the readers ease these abbreviations are also defined in their instance in each chapter.}

\setlength{\nomlabelwidth}{1.5cm}

\nomenclature{QCL}{Quantum cascade laser -- A semiconductor laser that operates through inter-subband transitions within the conduction band, achieving lasing through a cascade of electrons through neighbouring quantum wells.}
\nomenclature{LED}{Light-emitting diode -- A semiconductor device that converts electrical current into light through radiative recombination of electrons and holes.}
\nomenclature{Mid-IR}{Mid-infrared -- Electromagnetic radiation having wavelengths 3 - 30 $\mu$m, an important spectral range for biological, environmental and industrial applications.}
\nomenclature{QW}{Quantum well --  A quantum-confined heterostructure that confines particles (electrons or holes) in one dimension while allowing free motion in the other two dimensions.}
\nomenclature{SL}{Superlattice -- A periodic structure of alternating thin semiconductor layers with different band gaps, creating a one-dimensional artificial crystal with modified electronic properties.}
\nomenclature{SBSL}{Strain-balanced superlattice -- A superlattice design where tensile and compressive strains from different layers admit net zero stress, preventing the accumulation of strain that can drive formation of structural defects.}
\nomenclature{MIA}{Microscopic interface asymmetry -- The atomic-scale structural differences between nominally equivalent interfaces that break the assumed mirror symmetry in continuum electronic structure calculations.}
\nomenclature{PWEM}{Plane wave expansion method -- A computational technique that expresses electronic envelope functions as linear combinations of plane wave basis functions, exploiting the periodicity of crystalline materials.}
\nomenclature{EFA}{Envelope function approximation -- A quantum mechanical approximation that factorises a wave function into the product of a rapidly oscillating atomic-scale Bloch function, and a slowly varying envelope functions.}
\nomenclature{SE}{Spontaneous emission -- Radiative recombination of an electron-hole pair, emitting a photon without external stimulation.}
\nomenclature{SB}{Spurious band -- A non-physical solution of the coupled envelope function (Schr\"{o}dinger) equations for a heterostructure, that ``pollutes'' numerical calculations, with potentially deleterious effects on the predicted properties.}
\nomenclature{DFT}{Density functional theory -- A quantum mechanical method allows to calculate the structural and electronic properties of materials by expressing the total energy as a functional of the electron density.}
\nomenclature{HSEsol}{Heyd-Scuseria-Ernzerhof hybrid functional for solids -- A hybrid density functional that includes a fraction of exact (Hartree-Fock) exchange, optimised for solid-state calculations with improved band gap predictions.}
\nomenclature{LDA}{Local density approximation -- A simple approximation in DFT that treats the exchange-correlation energy as if the electron density were locally uniform.}
\nomenclature{TB-mBJ}{Tran-Blaha modified Becke-Johnson -- An exchange potential that, when combined with the LDA to the electronic correlation, serves as a meta-generalised gradient approximation functional that provides accurate band gaps for semiconductors at low computational cost.}

\printnomenclature

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