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

Mitsuaki Kawamura; Takuya Nomoto; Niklas Witt; Ryotaro Arita

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

Mitsuaki Kawamura, Takuya Nomoto, Niklas Witt, Ryotaro Arita

Abstract

We develop a first-principles framework for evaluating the fundamental length scales of superconductivity, namely the coherence length $ξ_0$ and the magnetic penetration depth $λ_\mathrm{L}$, 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 $ξ_0$, $λ_\mathrm{L}$, 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 $ξ_0$ and $λ_\mathrm{L}$ 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.

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

Abstract

We develop a first-principles framework for evaluating the fundamental length scales of superconductivity, namely the coherence length

and the magnetic penetration depth

, 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

, and the superconducting transition temperature

on the same theoretical footing. Applying the method to representative elemental superconductors, the A15 compound V

Si, and H

S under high pressure, we obtain results in good agreement with available experimental data. Furthermore, the unified access to

and

allows us to construct the Uemura plot entirely from first principles, demonstrating that conventional elemental superconductors systematically exhibit small

, while higher-

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.

Paper Structure (15 equations, 2 figures, 1 table)

This paper contains 15 equations, 2 figures, 1 table.

Figures (2)

Figure 1: (Top) Fermi-surface-averaged superconducting gap function $\langle\Delta_{n\mathbf{k}}^{(\mathbf{Q})}\rangle$ plotted as a function of the momentum $|\mathbf{Q}|$ of the Cooper pair. (Bottom) Supercurrent density $\bar{\mathbf{j}}_{\mathrm{sc}}^{(Q)}$ as a function of $|\mathbf{Q}|$.
Figure 2: Uemura plot: Log-log plot of the superconducting critical temperature $T_{\mathrm{c}}$ versus the Fermi temperature $T_{\mathrm{F}}$. Ab initio SCDFT results are highlighted by saturated symbols, while experimental values for comparison across a wide range of superconducting materials are shown in lighter tones WittPhDThesis2024Nakagawa2021Uemura2019. The solid line corresponds to $T_{\mathrm{c}} = T_{\mathrm{F}}$, the dashed line indicates the critical temperature of a non-interacting BEC in 3D with $T_{\mathrm{BEC}} = T_{\mathrm{F}}/4.16$, and the dotted lines serve as visual guides for successive decades of the $T_{\mathrm{c}}/T_{\mathrm{F}}$ ratio. The gray shaded area highlights the region where most unconventional superconductors are located, spanning $T_{\mathrm{c}}/T_{\mathrm{F}}$ ratios between 0.1 and 0.01. Horizontal bars on the theoretical points represent the range of Fermi temperatures obtained from SCDFT.