SuperFlow: Two-Stage Joint Flow Matching for Superconductor Doping

Finding effective doping strategies is a difficult but valuable step in superconductor development: both the substitution choice and its doping rate matter, and an unfavorable change can lower the superconducting critical temperature $T_c$. Locating an optimum experimentally requires synthesizing and characterizing multiple compositions along a substitution trajectory, with substantial time and resource costs. We introduce a benchmark for parent-conditioned optimal-doping prediction. The benchmark combines literature-labeled ICSD structures with ICSD and Materials Project data from 3DSC. We use a large language model to extract experimental $T_c$ values from source papers and correct selected 3DSC labels based on the literature. The benchmark measures doping-rate error and whether a proposed change moves toward an observed optimum. We further propose SuperFlow, a two-stage joint flow-matching framework that predicts doping rate together with crystal structure for a specified substitution axis. A first flow generates a candidate crystal. A goal head reads this result and guides a second joint refinement. SuperFlow achieves the best moving-parent and large-move doping-rate errors and direction accuracies among the evaluated methods on both datasets. Moving-parent doping-rate MAE is 0.11633 on ICSD and 0.08847 on MP. Selected case studies show that predicted doping rate changes are accompanied by local structural changes toward dataset targets. The benchmark and method provide a basis for prioritizing experiments and, with further validation, supporting industrial searches for optimal doping.