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.