Machine-learning approach for discovery of conventional superconductors

First-principles computations are the driving force behind numerous discoveries of hydride-based superconductors, mostly at high pressures, during the last decade. Machine-learning (ML) approaches can further accelerate the future discoveries if their reliability can be improved. The main challenge of current ML approaches, typically aiming at predicting the critical temperature $T_{\mathrm{c}}$ of a solid from its chemical composition and target pressure, is that the correlations to be learned are deeply hidden, indirect, and uncertain. In this paper, we show that predicting superconductivity at any pressure from the atomic structure is sustainable and reliable. For a demonstration, we curated a diverse data set of 584 atomic structures for which $\lambda$ and ${\omega }_{log}$, two parameters of the electron-phonon interactions, were computed. We then trained some ML models to predict $\lambda$ and ${\omega }_{log}$, from which $T_{\mathrm{c}}$ can be computed in a postprocessing manner. The models were validated and used to identify two possible superconductors whose $T_{\mathrm{c}}\simeq 10–15$ K at zero pressure. Interestingly, these materials have been synthesized and studied in some other contexts. In summary, the proposed ML approach enables a pathway to directly transfer what can be learned from the high-pressure atomic-level details that correlate with high-$T_{\mathrm{c}}$ superconductivity to zero pressure. Going forward, this strategy will be improved to better contribute to the discovery of new superconductors.

Figure 1

A typical workflow to compute $T_{\mathrm{c}}$. Existing ML efforts are devoted to the following lines: (i) using ML potentials to accelerate the structure prediction step, (ii) deriving new formulas of $T_{\mathrm{c}}$, and (iii) predicting $T_{\mathrm{c}}$ from chemical composition (comp.). This work is in line (iv), predicting $\lambda$ and ${\omega }_{log}$ from atomic structures.

Figure 2

A summary of the computed $\lambda , {\omega }_{log}$, and $T_{\mathrm{c}}$ data set of 584 superconducting materials reported and curated, including (a) the periodic table coverage, (b) the ten most frequent species in the data set, (c) 567 values of $T_{\mathrm{c}}$ computed and arranged at different pressures, and the distribution of (d) 584 computed values of $\lambda$ and (e) 567 values of ${\omega }_{log}$. Among the 53 species found in our data set, 47 of them are shown in (a), and the other 6 species are Ac, Ce, La, Nd, Pm, and Pr. In (c), each solid circle represents a combination of a chemical composition and a pressure, while error bars are for cases where $T_{\mathrm{c}}$ was computed for different atomic structures, using different methods, e.g., using the McMillan formula and solving Éliashberg equations, and/or for different values of ${\mu }^{*}$.

Figure 3

Fitting procedure used to compute (a) $\lambda$ and (b) ${\omega }_{log}$ of mp-24287 and mp-24208, two atomic structures identified from the Materials Project database. Circles and inverted triangles show $\lambda$ and ${\omega }_{log}$ computed with some finite $\mathbf{q}$-point grids, while stars represent the extrapolated values of $\lambda$ and ${\omega }_{log}$ at the limit of an infinite $\mathbf{q}$-point grid, i.e., $1/q=0$.

Figure 4

Procedure to down-select 35 atomic structures for predicting $\lambda$ and ${\omega }_{log}$ from 83 989 atomic structures of the Materials Project database.

Figure 5

Learning curves obtained by learning two data sets of (a) $\lambda$ and (b) ${\omega }_{log}$, a typical model trained on 90% of the data of (c) $\lambda$ and (d) ${\omega }_{log}$ and validated on the remaining unseen 10% data, and two ML models trained on 100% of the data of (e) $\lambda$ and (f) ${\omega }_{log}$. In (a) and (b), each data point is associated with an error bar obtained from 100 models that were independently trained.

Figure 6

Computed superconducting properties of mp-24287 and mp-24208, whose atomic structures are visualized in (a) and (b) and for which the spectral function ${\alpha }^{2}F\left(\omega \right)$ and the accumulative $\lambda \left(\omega \right)$ are shown in (c) and (d). The $\mathbf{k}$-point and $\mathbf{q}$-point grids used for (c) are $24\times 24\times 24$ and $8\times 8\times 8$, respectively, while those used for (d) are $21\times 21\times 21$ and $7\times 7\times 7$, respectively. In (e) and (f), the computed (using the extrapolation procedure described in Sec. 2e) and the predicted values of $\lambda , {\omega }_{log}$, and $T_{\mathrm{c}}$ (computed from $\lambda$ and ${\omega }_{log}$ using the McMillan formula with ${\mu }^{*}=0.1$) at $P=0$, 50, and 100 GPa are shown; solid and dashed lines show the computed and predicted values, respectively.

Figure 7

Distribution of the zero-pressure superconducting gap function ${\mathrm{\Delta }}_0\left(T\right)$ computed by numerically solving the Éliashberg equations for mp-24287 and mp-24208. The dashed curves, joining the middle point of the distributions, serve as a guide to the eye. The critical temperature $T_{\mathrm{c}}$ is estimated to be at the middle point of the downward-sloping segment of the ${\mathrm{\Delta }}_0\left(T\right)$ curves.