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

Mitsuaki Kawamura, Takuya Nomoto+2 more|arXiv (Cornell University)|Mar 5, 2026

Iron-based superconductors research0 citations

TL;DR

The paper develops a first-principles SCDFT framework that incorporates finite-momentum Cooper pairs to compute superconducting coherence length, penetration depth, and Tc on the same footing, and validates against experiments across several materials including H3S under pressure.

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.

Motivation & Objective

• Motivate a predictive, parameter-free description of fundamental superconducting length scales beyond Tc.
• Develop a SCDFT-based method that includes finite-momentum Cooper pairs to extract xi0 and lambdaL alongside Tc.
• Demonstrate numerical stability and applicability to a range of materials, including extreme-pressure hydrides.
• Provide a microscopic interpretation of empirical correlations like the Uemura plot from first principles.

Proposed method

• Formulate a gap equation for finite-Q Cooper pairs within superconducting DFT.
• Use a decoupling/Generalized Bloch theorem to reduce the problem to a set of 2x2 secular equations for each n,k and Q.
• Introduce an auxiliary energy-dependent gap function to stabilize k-point integrations at small Q.
• Compute the supercurrent density from the Q-dependent superconducting state and extract lambdaL from its slope near Q=0.
• Obtain xi0 from the Q-dependence of the averaged gap via the Q2 criterion, xi0 = 1/(sqrt(2) Q2).
• Validate Tc, xi0, and lambdaL across materials (Al, Nb, Pb, V3Si, H3S) and compare with experimental data.

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 $|\mathbf{Q}|$ .

Experimental results

Research questions

• RQ1 Can SCDFT, extended to finite-momentum Cooper pairs, predict coherence length xi0 and magnetic penetration depth lambdaL on the same footing as Tc?
• RQ2 Do first-principles results for xi0 and lambdaL reproduce the type-I/type-II classification and experimental trends across elemental metals and high-pressure hydrides?
• RQ3 How does the Uemura relation between Tc and TF emerge from a fully first-principles treatment of pairing and phase stiffness?
• RQ4 What is the impact of strong electron-phonon coupling and extreme conditions (e.g., H3S at ~200 GPa) on xi0 and lambdaL?

Key findings

• The framework yields xi0 and lambdaL in good agreement with experimental data for Nb and other materials.
• For Nb, xi0 = 34 nm and lambdaL = 34–40 nm depending on the evaluation method, aligning with experimental ranges.
• H3S under 200 GPa yields xi0 ≈ 3.0 nm and lambdaL ≈ 19–22 nm, consistent with upper critical-field inferences and low-temperature magnetometry estimates.
• H3S exhibits a very large depairing current Jdp ≈ 697 × 10^7 A/cm^2, indicating potential for exceptionally large supercurrents (theoretically predicted).
• The unified approach allows constructing the Uemura plot from first principles, showing conventional elemental superconductors have small Tc/TF ratios while higher-Tc systems show strong pairing and large phase stiffness.
• Across materials, the method correctly reproduces the type-I/type-II classification via the xi0/lambdaL ratio.

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