We perform first-principles electron-phonon calculations to evaluate the superconducting transition temperature $T_c$ for ternary superhard metals ${\mathrm{B}}_2{\mathrm{C}}_3\mathrm{N}$ and ${\mathrm{B}}_4{\mathrm{C}}_5{\mathrm{N}}_3$. An ambient-pressure $T_c$ of $\sim 40\mathrm{K}$ and $\sim 20\mathrm{K}$ is obtained respectively for ${\mathrm{B}}_2{\mathrm{C}}_3\mathrm{N}$ and ${\mathrm{B}}_4{\mathrm{C}}_5{\mathrm{N}}_3$ from the Eliashberg gap equations. The relatively high $T_c$ of these compounds is due in part to their high Debye temperatures associated with superhardness. The materials under study are potentially synthesizable, as their formation energies are comparable to those of other recently synthesized superhard B-C-N compounds. Therefore, studying superhard metals could hold the promise of realizing new higher-$T_c$ superconductors at ambient pressure.

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• Superconductivity
• Density functional theory
• Eliashberg theory