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We present and compare three distinct atomistic models -- based on firstprinciples and semi-empirical approaches -- of the structural and electronicproperties of Ge$_{1-x}$Sn$_{x}$ alloys. Density functional theory calculationsincorporating Heyd-Scuseria-Ernzerhof (HSE) and modified Becke-Johnson (mBJ)exchange-correlation functionals are used to perform structural relaxation andelectronic structure calculations for a series of Ge$_{1-x}$Sn$_{x}$ alloysupercells. Based on HSE calculations, a semi-empirical valence force field(VFF) potential and $sp^{3}s^{\ast}$ tight-binding (TB) Hamiltonian areparametrised. Comparing the HSE, mBJ and TB models, and using the HSE resultsas a benchmark, we demonstrate that: (i) mBJ calculations provide an accuratefirst principles description of the electronic structure at reducedcomputational cost, (ii) the VFF potential is sufficiently accurate tocircumvent the requirement to perform first principles structural relaxation,and (iii) TB calculations provide a good quantitative description of the alloyelectronic structure in the vicinity of the band edges. Our results alsoemphasise the importance of Sn-induced band mixing in determining the nature ofthe conduction band structure of Ge$_{1-x}$Sn$_{x}$ alloys. The theoreticalmodels and benchmark calculations we present inform and enable predictive,computationally efficient and scalable atomistic calculations for disorderedalloys and nanostructures. This provides a suitable platform to underpinfurther theoretical investigations of the properties of this emergingsemiconductor alloy.

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