Abstract
The Eliashberg theory of superconductivity accounts for the fundamental physics of conventional superconductors, including the retardation of the interaction and the Coulomb pseudopotential, to predict the critical temperature Tc. McMillan, Allen, and Dynes derived approximate closed-form expressions for the critical temperature within this theory, which depends on the electron–phonon spectral function α2F(ω). 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 SISSO framework, together with a database of artificially generated α2F functions and 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. This 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.
Highlights
The theory of electron–phonon superconductivity due to Bardeen–Cooper–Schrieffer, Gor’kov, Eliashberg, Migdal, and others is well-established, it has not historically aided in the discovery of new superconductors
The total λ is equal to the sum of the λi, which simplifies sampling of the space of spectral functions
The present work demonstrates the application of symbolic regression to a curated dataset of α2F(ω) spectral functions, yielding an improved analytical correction to the McMillan equation for the critical temperature of a superconductor
Summary
The theory of electron–phonon superconductivity due to Bardeen–Cooper–Schrieffer, Gor’kov, Eliashberg, Migdal, and others is well-established, it has not historically aided in the discovery of new superconductors. Recent computational developments may allow a new approach to superconducting materials discovery based on ab-initio and materials-genome type methods[1,2,3] One approach to this problem, pioneered by McMillan[4] and Allen and Dynes[5], is to search for a formula for Tc based on materials-specific parameters derived from the Eliashberg equations of superconductivity. These parameters, mostly moments of the electron–phonon spectral function α2F(ω), can be determined by experiment or, more recently, calculated within ab initio approaches. It is based on 217 Eliashberg solutions of three types of α2F(ω) shapes (those obtained from tunneling data on Hg and Pb, and those obtained for a single Einstein mode)
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