Functional form of the superconducting critical temperature from machine learning

Predicting the critical temperature ${T}_{c}$ of new superconductors is a notoriously difficult task, even for electron-phonon paired superconductors, for which the theory is relatively well understood. Early attempts to obtain a simple ${T}_{c}$ formula consistent with strong-coupling theory, by McMillan and by Allen and Dynes, led to closed-form approximate relations between ${T}_{c}$ and various measures of the phonon spectrum and the electron-phonon interaction appearing in Eliashberg theory. Here we propose that these approaches can be improved with the use of machine-learning algorithms. As an initial test, we train a model for identifying low-dimensional descriptors using the ${T}_{c}l10$ K dataset by Allen and Dynes, and show that a simple analytical expression thus obtained improves upon the Allen-Dynes fit. Furthermore, the prediction for the recently discovered high-${T}_{c}$ material ${\mathrm{H}}_{3}\mathrm{S}$ at high pressure is quite reasonable. Interestingly, ${T}_{c}$'s for more recently discovered superconducting systems with a more two-dimensional electron-phonon coupling, which do not follow Allen and Dynes's expression, also do not follow our analytic expression. Thus, this machine-learning approach appears to be a powerful method for highlighting the need for a new descriptor beyond those used by Allen and Dynes to describe their set of isotropic electron-phonon coupled superconductors. We argue that this machine-learning method, and its implied need for a descriptor characterizing Fermi-surface properties, represents a promising approach to superconductor materials discovery which may eventually replace the serendipitous discovery paradigm begun by Kamerlingh Onnes.