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Ge$_{1-x}$Sn$_{x}$ alloys are a promising candidate material to realise direct-gap group-IV semiconductors for applications in Si-compatible electronic and photonic devices. Here, we present a combined theoretical and experimental analysis of Raman spectroscopy in Ge$_{1-x}$Sn$_{x}$ alloys. We describe liquid-vapour-solid growth and structural characterisation of Ge$_{1-x}$Sn$_{x}$ ($x \leq 8$%) nanowires displaying high crystalline quality, and investigate the structural and vibrational properties of the nanowires using Raman spectroscopy. Our theoretical analysis is based on a fully analytic anharmonic valence force field (VFF) potential, which describes exactly - i.e. without recourse to numerical fitting - the second-order elastic constants, third-order bulk modulus, selected second- and third-order inner elastic constants and, as a consequence, the zone-centre transverse optical phonon mode frequency and its hydrostatic and axial strain dependence. We compute bulk elastic properties via density functional theory to parametrise the VFF potential for Ge$_{1-x}$Sn$_{x}$ alloys, and apply the VFF potential to explicitly compute the Raman spectra of realistic, disordered Ge$_{1-x}$Sn$_{x}$ alloy supercells. Our atomistic theoretical calculations quantitatively capture: (i) the evolution of the measured Raman spectra with Sn composition $x$, (ii) demonstrate explicitly that the presence of short-range alloy disorder can significantly impact the shift coefficients $a$ and $b$ that respectively describe the dependence of the Raman shift on Sn composition and pseudomorphic strain, (iii) elucidate the origin of the so-called "disorder activated" mode identified in previous experimental investigations, and (iv) allow for detailed atomic-scale interpretation of measured Raman spectra. Overall, our analysis provides insight relevant to the characterisation of this emerging material system.

We present a detailed theoretical analysis of the electronic and optical properties of c-plane InGaN/GaN quantum-well structures with In contents ranging from 5% to 25%. Special attention is paid to the relevance of alloy-induced carrier-localization effects to the “green gap” problem. Studying the localization length and electron-hole overlaps at low and elevated temperatures, we find alloy-induced localization effects are crucial for the accurate description of (In,Ga)N quantum wells across the range of In content studied. However, our calculations show very little change in the localization effects when moving from the blue to the green spectral regime; that is, when the internal quantum efficiency and wall-plug efficiencies reduce sharply, for instance, the in-plane carrier separation due to alloy-induced localization effects changes weakly. We conclude that other effects, such as increased defect densities, are more likely to be the main reason for the green-gap problem. This conclusion is further supported by our finding that the electron localization length is large, when compared with that of holes, and changes little in the In composition range of interest for the green-gap problem. Thus, electrons may become increasingly susceptible to an increased (point) defect density in green emitters and as a consequence the nonradiative-recombination rate may increase.

We present and compare three distinct atomistic models -- based on first principles and semi-empirical approaches -- of the structural and electronic properties of Ge$_{1-x}$Sn$_{x}$ alloys. Density functional theory calculations incorporating Heyd-Scuseria-Ernzerhof (HSE) and modified Becke-Johnson (mBJ) exchange-correlation functionals are used to perform structural relaxation and electronic structure calculations for a series of Ge$_{1-x}$Sn$_{x}$ alloy supercells. Based on HSE calculations, a semi-empirical valence force field (VFF) potential and $sp^{3}s^{\ast}$ tight-binding (TB) Hamiltonian are parametrised. Comparing the HSE, mBJ and TB models, and using the HSE results as a benchmark, we demonstrate that: (i) mBJ calculations provide an accurate first principles description of the electronic structure at reduced computational cost, (ii) the VFF potential is sufficiently accurate to circumvent the requirement to perform first principles structural relaxation, and (iii) TB calculations provide a good quantitative description of the alloy electronic structure in the vicinity of the band edges. Our results also emphasise the importance of Sn-induced band mixing in determining the nature of the conduction band structure of Ge$_{1-x}$Sn$_{x}$ alloys. The theoretical models and benchmark calculations we present inform and enable predictive, computationally efficient and scalable atomistic calculations for disordered alloys and nanostructures. This provides a suitable platform to underpin further theoretical investigations of the properties of this emerging semiconductor alloy.

We present a detailed theoretical analysis of the electronic structure of $c$-plane InGaN/GaN quantum wells with indium contents varying between 10\% and 25\%. The electronic structure of the quantum wells is treated by means of an atomistic tight-binding model, accounting for variations in strain and built-in field due to random alloy fluctuations. Our analysis reveals strong localisation effects in the hole states. These effects are found not only in the ground states, but also the excited states. We conclude that localisation effects persist to of order 100~meV into the valence band, for as little as 10\% indium in the quantum well, giving rise to a significant density of localised states. We find, from an examination of the modulus overlap of the wave functions, that the hole states can be divided into three regimes of localisation. Our results also show that localisation effects due to random alloy fluctuations are far less pronounced for electron states. However, the combination of electrostatic built-in field, alloy fluctuations and structural inhomogeneities, such as well-width fluctuations, can nevertheless lead to significant localisation effects in the electron states, especially at higher indium contents. Overall, our results are indicative of individually localised electron and hole states, consistent with the experimentally proposed explanation of time-dependent photoluminescence results in $c$-plane InGaN/GaN QWs.