Ab initio investigation of phonon-mediated superconductivity in the ternary borides ${\mathrm{Mo}}_{5}X{B}_2$ ($X=P$, Si, Ge): Comparison with $W_5{\mathrm{SiB}}_2$

In this work, we have investigated the elastic and mechanical properties, electronic band structure, lattice dynamics, and electron-phonon interaction in four ternary transition metal borides ${\mathrm{Mo}}_5{\mathrm{PB}}_2$, ${\mathrm{Mo}}_5{\mathrm{SiB}}_2$, $W_5{\mathrm{SiB}}_2$, and ${\mathrm{Mo}}_5{\mathrm{GeB}}_2$ by executing systematic ab initio calculations based on density functional theory within the generalized gradient approximation. The calculated elastic constants and elastic moduli for ${\mathrm{Mo}}_5{\mathrm{SiB}}_2$ agree well with available experimental data. The calculated elastic constants and phonon dispersion relations show that all the considered borides are mechanically and dynamically stable. The electronic density of states near the Fermi level of these compounds is dominated by the transition metal $d$ orbitals, which leads to low-frequency phonon modes arising from the vibrations of transition metal atoms being more strongly involved in the process of scattering of electrons rather than high-frequency phonon modes arising from the vibrations of lighter other two atoms. Our electron-phonon interaction calculations reveal that the electron-phonon coupling strength of ${\mathrm{Mo}}_5{\mathrm{PB}}_2$ is the strongest with a value of 0.959 among all the studied compounds, which in turn yields a superconducting transition temperature ($T_c$) value of 9.527 K, being higher than the $T_c$ values of ${\mathrm{Mo}}_5{\mathrm{SiB}}_2$ (5.845 K), $W_5{\mathrm{SiB}}_2$ (5.931 K), and ${\mathrm{Mo}}_5{\mathrm{GeB}}_2$ (6.193 K). All these $T_c$ values harmonize with their respective experimental values of 9.2, 5.8, 5.8, and 5.8 K, indicating the reliability of our ab initio calculations.