Towards the discovery of high critical magnetic field superconductors

TL;DR

The paper tackles the underexplored role of critical magnetic fields in superconductors and develops a high-throughput, first-principles workflow to predict $H_{c}$, $H_{c1}$, and $H_{c2}$ for ~7300 electron-phonon superconductors. It combines density functional theory with clean-limit Eliashberg theory, using $α^2F(ω)$ and accurate Fermi surfaces to compute $ξ$, $λ_{L}$, and the Ginzburg-Landau parameter $κ=λ_{L}/ξ$, from which the three critical fields are derived. Key findings include an unexpectedly large population of Type-I superconductors and clear trends that larger unit cells yield higher fields and promote Type-II behavior; strong-coupling corrections and electron-phonon mass renormalization are essential for accurate predictions. The resulting database enables AI-driven inverse materials design for high-$T_c$ and high-critical-field superconductors and has practical implications for high-field magnets and superconducting technologies.

Abstract

Superconducting materials are of significant technological relevance for a broad range of applications, and intense research efforts aim at enhancing the critical temperature $T_{c}$. Intriguingly, while numerous studies have explored different computational and machine-learning routes to predict $T_{c}$, the fundamental role of the critical magnetic field has so far been overlooked. Here we open a new frontier in superconductor discovery by presenting a consistent computational database of critical fields $H_{c}$, $H_{c1}$, and $H_{c2}$ for over 7300 electron-phonon-paired superconductors covering distinct materials classes. A theoretical framework is developed that combines $α^2F(ω)$ spectral functions and highly accurate Fermi surfaces from density functional theory with clean-limit Eliashberg theory to obtain the coherence lengths, London penetration depths, and Ginzburg-Landau parameters. We discover an unexpectedly large number of Type-I superconductors and show that larger unit cells generically support higher critical fields and Type-II behavior. We identify the importance of going beyond BCS theory by including strong-coupling corrections to the superconducting gap and electron-phonon renormalizations of the effective mass for predictions of critical fields across materials. These results provide a framework for foundational AI models that realize the concept of inverse materials design for high-$T_{c}$ and high-critical-field superconductors.

Figures (4)

• Figure 1: First-principles database of superconducting critical magnetic fields.a The workflow developed here to obtain a consistent database of critical fields and related superconducting properties entirely from first principles. b The fundamental ingredient is a new database of Fermi surfaces, colored here by the Fermi velocities; a selection has been compiled in the form of a mosaic where each pixel is a Fermi surface. c Explanation of the different critical fields. d Upper critical fields at zero temperature for all Type-II superconductors in our database, systematically ranked in a $(T_{c}, 1/v_\text{F})$ scatterplot. The blue-yellow background shows $H_{c2}^\text{BCS}$ to structure the data, as described in the text. The point colors reflect the actual upper critical fields $H_{c2} > H_{c2}^\text{BCS}$ determined by using the calculated Eliashberg gaps, which include strong-coupling corrections, and considering effective-mass renormalizations of the Fermi velocities due to electron-phonon interactions $\lambda_\text{el-ph}$.
• Figure 2: Critical fields as a function of the Ginzburg-Landau parameter $\kappa$. Type-I critical fields (left) are represented by a single point per material. For Type-II superconductors (right), upper and lower critical fields are represented by a square-diamond pair, respectively, connected by a vertical line. Red colors encode the electron-phonon renormalized density of states at the Fermi level $n(E_\text{F})$, with is particularly relevant for $H_{c}$ and $H_{c1}$. Green-purple colors represent the electron-phonon renormalized Fermi velocity $v_\text{F}$. Selected materials are labeled for reference. The histogram visualizes the distribution of $\kappa$ and thus the number of Type-I versus Type-II superconductors. The figure focuses on materials with $T_{c}>1$ K. All properties have been predicted consistently from DFT.
• Figure 3: Statistical analysis of the superconducting properties.a Distribution of $T_{c}$ for Type-I versus Type-II superconductors. b The distribution of $\Delta(0) / k_\text{B} T_{c}$ shows a clear maximum near the BCS reference, yet concomitantly a substantially extended tail reaching up to twice that value. The inset displays the Eliashberg superconducting gaps (diamonds) versus $T_{c}$, colored by $\Delta(0) / k_\text{B} T_{c}$. Comparison with the BCS expression (dashed line) highlights the strong-coupling corrections to the gaps. c Correlation of the Ginzburg-Landau parameter $\kappa$ with the unit-cell size. d Correlation of the combined $H_c$ (orange; Type I) and $H_{c2}$ (blue; Type II) data with the unit-cell size. The linear regression lines are shown in red. Panels a, c, and d focus on materials with $T_{c}>1$ K.
• Figure 4: Comparison of predicted and experimental critical field properties.a Analysis of the Type-I (orange) versus Type-II (blue) superconductor classification, based on the Ginzburg-Landau parameter $\kappa$ predicted entirely from first principles. The vertical axis corresponds to the experimental $T_{c}$, which shows that the classification cannot be trivially deduced from the critical temperature. b Parity plot of predicted versus experimental critical fields ($H_{c}$: circles; $H_{c2}$: squares). A table of the data, compiled from Refs. Roberts-SC:1976Eisenstein:1954MatthiasGeballeCompton:1963Schwochau-Tc:2000MgB2-Hc2:2002Cr3Ru:2022, is provided in the Supplemental Information. For those materials where the predicted $T_{c}$ is in significant disagreement with the experimental value, we show in addition a rescaled critical field value using the experimental $T_{c}$ ($H_{c}^{*}$: crosses; $H_{c2}^{*}$: pluses), which generally enhances the quantitative agreement as evidenced by the log RMSE and R$^2$ metrics.