Machine learning of superconducting critical temperature from Eliashberg theory

The Eliashberg theory of superconductivity accounts for the fundamental physics of conventional electron-phonon superconductors, including the retardation of the interaction and the effect of the Coulomb pseudopotential, to predict the critical temperature Tc and other properties. McMillan, Allen, and Dynes derived approximate closed-form expressions for the critical temperature predicted by this theory, which depends essentially on the electron-phonon spectral function α²F(ω), using α²F for low-Tc superconductors. Here we show that modern machine learning techniques can substantially improve these formulae, accounting for more general shapes of the α²F function. Using symbolic regression and the sure independence screening and sparsifying operator (SISSO) framework, together with a database of artificially generated α²F functions, ranging from multimodal Einstein-like models to calculated spectra of polyhydrides, as well as numerical solutions of the Eliashberg equations, we derive a formula for Tc that performs as well as Allen-Dynes for low-Tc superconductors and substantially better for higher-Tc ones. The expression identified through our data-driven approach corrects the systematic underestimation of Tc while reproducing the physical constraints originally outlined by Allen and Dynes. This equation should replace the Allen-Dynes formula for the prediction of higher-temperature superconductors and the estimation of λ from experimental data. This repository contains CSV tables of model inputs, prefactors, and critical temperature predictions used for machine learning and validation in this work. Additionally, the inputs and outputs of the SISSO framework for both rounds of symbolic regression are included. Please see https://github.com/henniggroup/symbolic-regression-utilities for tools to create and analyze SISSO inputs and outputs.