First principles study of the structural, electronic, optical and thermodynamic properties of cubic quaternary Al${}_x$In${}_{1-x}$P${}_y$Bi${}_{1-y}$ alloys

The non-relativistic full potential linearized augmented plane wave (FP LAPW) method was applied to investigate the structural, electronic, optical and thermodynamic properties of (ZB)-AlP, AlBi, InP and InBi compounds and their ternary Al${}_x$In${}_{1-x}$P, Al${}_x$In${}_{1-x}$Bi, AlP${}_x$Bi${}_{1-x}$ and InP${}_x$Bi${}_{1-x}$ and the ordered Al${}_x$In${}_{1-x}$P${}_y$Bi${}_{1-y}$ quaternary alloys. For the exchange-correlation potential, the LDA, GGA and WC-GGA have been used to calculate structural parameters. The TB-mBJ approximation was used to compute the band structures. Our results for binary compound agree well with available data found in literature. The lattice constants and the bulk modulus versus compositions x and y deviate from the linearity. All quaternary alloys are semiconductors with direct band gap with the exception for Al${}_{0.25}$In${}_{0.75}$P${}_{0.25}$Bi${}_{0.75}$ and Al${}_{0.25}$In${}_{0.75}$P${}_{0.50}$Bi${}_{0.50}$, which exhibit a half metallic character (the band gap tends to zero). Furthermore, the optical properties such as the dielectric constants, refractive index, absorption, reflectivity and the energy loss have been calculated and analysed in the energy range varying from 0 to 14 eV. At the end, we have investigated some thermodynamic properties, where the lattice constants, the Debye temperature θ${}_D$, the heat capacity C${}_V$ and the entropy S were carried out, plotted and discussed.