Machine learning the functional form of the superconducting critical temperature

Predicting the critical temperature Tc of superconductors is a difficult task, even for electron-phonon systems. We build on earlier efforts by McMillan [1] and Allen and Dynes [2] to model Tc from various measures of the phonon spectrum and the electron-phonon interaction by using machine learning algorithms. Specifically, we use symbolic regression implemented in the Sure Independence and Sparsifying Operator (SISSO) method [3] to identify new, physically interpretable equations for Tc as a function of a small number of physical quantities. We show that our first model [4], trained using the relatively small Tc < 10K data tested by Allen and Dynes, improves upon the Allen-Dynes fit and can reasonably generalize to superconducting materials with higher Tc such as H₃S. To address the limitations of the Allen-Dynes equation arising from their selection of spectral function α²F(ω) shapes, we generate a new dataset using Eliashberg Theory and more α²F(ω) examples, ranging from bimodal Einstein models to calculated spectra of polyhydrides. Furthermore, we explore the use of variational autoencoders to augment the data. By incorporating physical insights and constraints into a data-driven approach, we demonstrate that machine-learning methods can identify the relevant physical quantities and obtain predictive equations using small but high-quality datasets.

[1] W. L. McMillan, Phys. Rev. 167, 331 (1968).
[2] P. B. Allen and R. C. Dynes, Phys. Rev. B12, 905 (1975).
[3] R. Ouyang, S. Curtarolo, E. Ahmetcik, M. Scheffler, and L. M. Ghiringhelli, Phys. Rev. Materials 2, 083802(2018).
[4] S. R. Xie, G. R. Stewart, J. J. Hamlin, P. J. Hirschfeld, and R. G. Hennig, Phys. Rev. B100, 174513 (2019).