This study resolves a previously reported mechanical instability of the experimentally observed superconducting P4/ncc phase of ${\mathrm{Mg}}_4{\mathrm{AlSi}}_3$ using first-principles calculations. Previous theoretical work reported a negative ${\mathrm{C}}_{66}$ elastic constant for the P4/ncc structure and consequently proposed a lower symmetry Pccn structure as the mechanically stable phase. Here, we reinvestigate the elastic properties of P4/ncc ${\mathrm{Mg}}_4{\mathrm{AlSi}}_3$ using both energy-strain and stress-strain approaches. We find that all independent elastic constants are positive and satisfy the mechanical stability criteria, with the two approaches yielding consistent results. The absence of imaginary modes in the calculated phonon dispersion further establishes the dynamical stability of the phase, with the negative formation enthalpy indicating its thermodynamic stability. We further investigate for the first time the microscopic origin of its superconductivity through the electron-phonon coupling calculations based on Migdal-Eliashberg theory. The calculated electron-phonon coupling constant $\lambda$ of 0.618 and a logarithmic average frequency of 205.174 K, lead to a superconducting transition temperature $T_c$ of 5.123 K, in close agreement with the experimental value of 5.2 K. Therefore our results resolve the previous discrepancy concerning the mechanical stability of the experimentally realized P4/ncc phase and provide a consistent microscopic description of its lattice stability and phonon-mediated superconductivity.

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• Density of states
• Elasticity
• Electronic structure
• Fermi surface
• Hardness
• Phonons
• Superconductivity
• First-principles calculations