# Functional Form of the Superconducting Critical Temperature from Machine Learning

## TL;DR

This paper uses machine learning to improve the analytical prediction of superconducting critical temperatures, revealing limitations of existing models and suggesting new descriptors for better accuracy in superconductor discovery.

## Contribution

It introduces a machine learning approach that refines analytical formulas for $T_c$, highlighting the need for new descriptors beyond traditional electron-phonon parameters.

## Key findings

- Improved analytical expression for $T_c$ over Allen-Dynes fit.
- Reasonable prediction for high-pressure H$_3$S superconductor.
- Identification of the need for Fermi surface descriptors.

## Abstract

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 by McMillan and Allen and Dynes to improve on the weak-coupling BCS formula 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 $T_c < 10$ K data tested 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 H$_3$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' 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 new approach to superconductor materials discovery which may eventually replace the serendipitous discovery paradigm begun by Kamerlingh Onnes.