Fundamental limits on the electron-phonon coupling and superconducting $T_c$

Fundamental upper bounds on the electron-phonon interaction strength and superconducting transition temperature $T_c$ in metals are established based on the intrinsic instability of the equilibrium between electrons and the crystal lattice under strong interaction. This instability explains why observed electron-phonon coupling constants are limited to $\lambda \lesssim 4$. The theory also accounts for the mechanism of metastable superconductivity with enhanced $T_c$, which emerges near the instability threshold. Based on theoretical analysis and comparison with experimental data, room-temperature phonon-mediated superconductivity is found to be feasible exclusively in hydrogen compounds.