  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.
