First-principles calculation of coherence length and penetration depth based on density functional theory for superconductors

We develop a first-principles framework for evaluating the fundamental length scales of superconductivity, namely the coherence length $\xi_0$ and the magnetic penetration depth $\lambda_\mathrm{L}$, within superconducting density functional theory (SCDFT). By incorporating finite-momentum Cooper pairs, we formulate a microscopic scheme that enables a consistent and parameter-free determination of $\xi_0$, $\lambda_\mathrm{L}$, and the superconducting transition temperature $T_\mathrm{c}$ on the same theoretical footing. Applying the method to representative elemental superconductors, the A15 compound V$_3$Si, and H$_3$S under high pressure, we obtain results in good agreement with available experimental data. Furthermore, the unified access to $\xi_0$ and $\lambda_\mathrm{L}$ allows us to construct the Uemura plot entirely from first principles, demonstrating that conventional elemental superconductors systematically exhibit small $T_\mathrm{c}$/$T_\mathrm{F}$, while higher-$T_\mathrm{c}$ systems are characterized by the simultaneous realization of strong pairing and large phase stiffness. Our results establish a predictive first-principles route to superconducting length scales and provide a microscopic interpretation of empirical correlations in superconductivity.