Structural and elastic properties of $\mathrm{Fe}$-$\mathrm{Ge}$ alloys: ab initio studies

In this paper, with the help of the density functional theory, the structural and elastic properties of $\mathrm{A2}$, $\mathrm{B2}$, $\mathrm{D}0_3$, and $\mathrm{L}1_2$ phases of $\mathrm{Fe}_{100-x}\mathrm{Ge}_{x}$ alloys ($12,5 \le x \le 28,125$ at. $\%$) have been studied. The electronic and full ionic relaxations were used for the investigation of crystal structures. The concentration dependencies of the atomic volumes, structural phase transition temperatures, tetragonal and rhombohedral shear moduli have been calculated. We show that the atomic volume curves correlate with the sequence of phase transitions observed experimentally: $\mathrm{A}2\to \mathrm{B}2\to \mathrm{D}0_3$ ($x \le 22$ at. $\%$ of $\mathrm{Ge}$ content). The structural phase transition temperatures increase with the $\mathrm{Ge}$ concentration. The calculated tetragonal moduli for the $\mathrm{D}0_3$, $\mathrm{A2}$, and $\mathrm{L}1_2$ structures decrease with the increasing of the $\mathrm{Ge}$ content, what agrees with the experimental results. The dependence of rhombohedral shear moduli as a function of $\mathrm{Ge}$ concentration does not change significantly with increasing $\mathrm{Ge}$ atoms. The $\mathrm{C}_{44}$ is increased for the $\mathrm{D}0_3$ phase, while for $\mathrm{A2}$, $\mathrm{B2}$, and $\mathrm{L}1_2$, it decreases.