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 $T_c<10$ 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.

Figure 1

Beginning with feature space ${\mathrm{\Phi }}_0$, consisting of ${\omega }_{log}, \lambda$, and ${\mu }^{*}$, each additional tier ${\mathrm{\Phi }}_i$ is constructed by applying four binary operators ($+, -, \times$, and /) and seven unary operators (exp, log, $\sqrt{}, \sqrt[3]{}, {}^{-1}, {}^2$, and ${}^3$) to features from preceding tiers. This procedure is applied up to tier ${\mathrm{\Phi }}_3$, after which sure-independence screening is applied to eliminate features with correlation factors (inner product) below 0.5 with respect to $T_c$. Physical constraints as listed are then applied to further reduce the feature space. We fit coefficients to the 6021 features and obtain the 100 models with lowest root-mean-square error in predictions on the testing set.

Figure 2

Machine learning of optimal analytical expression for $T_c$ as a function of three parameters (${\omega }_{log}, \lambda$, and ${\mu }^{*}$) trained on the $T_c<10$ K dataset of Allen and Dynes [19] using the SISSO algorithm [22]. (a) The three-parameter machine-learned equation results in a smaller RMSE than the four-parameter Allen-Dynes or the three-parameter McMillan equation. (b) The testing of the machine-learned equation using nine different superconductors assumes that ${\mu }^{*}=0.1$ and takes ${\omega }_{log}$ and $\lambda$ from tunneling measurements [23, 24, 25, 26, 27, 28, 29, 30, 31]. This extrapolation shows larger deviations with a testing RMSE = 3.4 K or 17%. To compare, we also show four materials $[{\mathrm{Nb}}_3\mathrm{Sn}, {\mathrm{MgB}}_2$(1), ${\mathrm{La}}_3{\mathrm{Ni}}_2{\mathrm{B}}_2{\mathrm{N}}_3$, and ${\mathrm{LuNi}}_2{\mathrm{B}}_2\mathrm{C}]$ for which ${\omega }_{log}$ is obtained from low-temperature specific heat measurements and $\lambda$ from high-temperature resistivity [32] and ${\mathrm{MgB}}_2$(2), for which $\lambda$ is from density-functional calculations [33]. The extrapolation reveals two outliers, ${\mathrm{NbS}}_2$ at low temperatures and ${\mathrm{MgB}}_2$ at high temperatures.

Figure 3

$\lambda$ dependence of $T_c$ in the top 100 models, ranked by testing error assuming ${\mu }^{*}=0.1$. Two red curves correspond to the Allen-Dynes equation with the minimum and maximum values of ${\overline{\omega }}_2/{\omega }_{log}$ in the training set. The modified McMillan equation systematically predicts smaller $T_c$'s, and over the range of available $\lambda$ values the simple machine-learned model closely matches the more complex Allen-Dynes equation.