Phase diagram of the ${\mathrm{Fe}}_{1\mathrm{-}\mathit{x}}$${\mathrm{Co}}_{\mathit{x}}$${\mathrm{Si}}_2$ alloy in the fluorite form

The phase diagram of the ${\mathrm{Fe}}_{1\mathrm{-}\mathit{x}}$ ${\mathrm{Co}}_{\mathit{x}}$ ${\mathrm{Si}}_2$ alloy in the fluorite form has been computed from first principles. The alloy has been represented as an ensemble of tetrahedral clusters in which the corners are occupied by Fe or Co atoms in five possible configurations around a central Si atom. The formation energy of each cluster as a function of the lattice constant has been evaluated by means of self-consistent total-energy calculations of the corresponding ordered structure based on the linear muffin-tin orbitals method within the density-functional theory. Using the cluster-variation method for the evaluation of the entropy, we have obtained the phase diagram, showing that a disordered ${\mathrm{Fe}}_{1\mathrm{-}\mathit{x}}$ ${\mathrm{Co}}_{\mathit{x}}$ ${\mathrm{Si}}_2$ alloy (in fluorite form) could be easily grown above 160 K. We have also fitted the total energies of the end compounds to derive the bond-stretching constants ${\mathrm{\alpha }}_{\mathrm{FeSi}2\mathrm{}}$ and ${\mathrm{\alpha }}_{\mathrm{CoSi}2\mathrm{}}$. Through these quantities and an elastic model of the alloy we have evaluated the displacement of the Si atom from the center of each tetrahedron and the mean Fe-Si and Co-Si distances as a function of the concentration. The energy connected to this relaxation has been subtracted from the previously obtained formation energies, leading to a final value of the critical temperature (${\mathit{T}}_{\mathit{c}}$=115 K) definitely lower than the previous one. This fact confirms the importance of the internal relaxation also in systems with small lattice mismatch.