Machine learning the relationship between Debye temperature and superconducting transition temperature

Recently a relationship between the Debye temperature $\Theta_D$ and the superconducting transition temperature $T_c$ of conventional superconductors has been proposed [npj Quantum Materials $\mathbf{3}$, 59 (2018)]. The relationship indicates that $T_c \le A \Theta_D$ for phonon-mediated BCS superconductors, with $A$ being a pre-factor of order $\sim 0.1$. In order to verify this bound, we train machine learning (ML) models with 10,330 samples in the Materials Project database to predict $\Theta_D$. By applying our ML models to 9,860 known superconductors in the NIMS SuperCon database, we find that the conventional superconductors in the database indeed follow the proposed bound. We also perform first-principles phonon calculations for H$_{3}$S and LaH$_{10}$ at 200 GPa. The calculation results indicate that these high-pressure hydrides essentially saturate the bound of $T_c$ versus $\Theta_D$.