Comparison of first principles and semi-empirical models of the structural and electronic properties of Ge$_{1-x}$Sn$_{x}$ alloys

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.