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

This paper presents a parameter-free, first-principles framework based on superconducting density functional theory to simultaneously calculate the coherence length, penetration depth, and transition temperature of superconductors, successfully validating the method against experimental data and providing a microscopic explanation for the Uemura plot's empirical correlations.

The Gist

Imagine you are trying to understand how a superconductor works. A superconductor is a special material that conducts electricity with zero resistance, but only when it's very cold. To truly understand these materials and design better ones for things like MRI machines or future quantum computers, scientists need to know two specific "rulers" that measure the material's behavior:

1. The Coherence Length ($\xi_0$): Think of this as the "size of a dance pair." In a superconductor, electrons don't move alone; they pair up (called Cooper pairs) and dance together. This length tells you how far apart the two dancers can be while still holding hands and staying in sync.
2. The Penetration Depth ($\lambda_L$): Think of this as the "shield thickness." Superconductors are famous for kicking out magnetic fields (the Meissner effect). This length measures how deep a magnetic field can poke its nose into the material before the superconductor pushes it back out.

The Problem

For a long time, scientists had a powerful computer program (based on Density Functional Theory) that could predict when a material would become a superconductor (its critical temperature, $T_c$). However, this program couldn't accurately measure those two "rulers" (the dance pair size and the shield thickness) from first principles. It was like having a weather forecast that told you if it would rain, but not how hard it would pour or how long the storm would last.

The New Solution: "The Finite-Momentum Twist"

The authors of this paper developed a new way to use that computer program to measure those rulers. Here is the clever trick they used, explained with an analogy:

The Analogy: The Conga Line
Imagine a line of people (electrons) holding hands and dancing in a perfect circle. In a normal superconductor, everyone is standing still relative to the dance floor, just spinning in place.

The researchers asked: "What happens if we make the whole line of dancers move forward slightly while they spin?"

In physics terms, they gave the electron pairs a tiny bit of momentum (a push). They didn't just look at the stationary pairs; they looked at pairs that were "walking" through the material.

• By seeing how the "dance" (the superconducting state) gets weaker as the "walk" gets faster, they could calculate the Coherence Length (how robust the dance is).
• By seeing how much "current" (the flow of the walking dancers) is generated by that push, they could calculate the Penetration Depth (how strong the magnetic shield is).

Why This Matters

This method is like upgrading from a black-and-white sketch to a high-definition 3D movie. Before, scientists had to guess these values or rely on messy experiments. Now, they can calculate them directly from the atoms themselves.

What they found:

1. It Works: They tested this on common metals like Aluminum and Niobium, and the computer results matched real-world experiments almost perfectly.
2. Extreme Conditions: They even tested a material called H3S (Hydrogen Sulfide) under immense pressure (like the center of a planet). In these conditions, it's nearly impossible to run experiments. The computer said, "This material is a superconductor with a tiny dance-pair size and a very strong shield," and later experiments confirmed they were right.
3. The "Uemura Plot" Mystery: Scientists have long noticed a pattern: materials with higher superconducting temperatures tend to have stiffer "dance floors" (stronger phase stiffness). The authors used their new method to draw this pattern entirely from scratch using only math and physics, proving that the "stiffness" of the electron dance is the key to high-temperature superconductivity.

The Big Picture

This paper gives scientists a predictive toolkit. Instead of mixing chemicals in a lab and hoping for a breakthrough, they can now simulate materials on a computer, measure their "dance pair size" and "shield thickness," and predict if they will be good superconductors before they are even built.

It's like having a GPS for the world of superconductivity, allowing us to navigate directly to the materials that could power the next generation of technology.

Technical Summary

Here is a detailed technical summary of the paper "First-principles calculation of coherence length and penetration depth based on density functional theory for superconductors" by Kawamura et al.

1. Problem Statement

While Superconducting Density Functional Theory (SCDFT) has become a powerful tool for predicting the superconducting transition temperature ($T_c$) of conventional phonon-mediated superconductors, it has historically lacked a first-principles framework for calculating fundamental length scales that characterize the superconducting state itself. Specifically, the coherence length ($\xi_0$) and the magnetic penetration depth ($\lambda_L$) have not been consistently determined within SCDFT without phenomenological inputs.

These parameters are critical because:

• They determine the classification of superconductors (Type-I vs. Type-II).
• They define critical magnetic fields and depairing currents.
• They are essential for materials design and understanding the interplay between pairing strength and phase stiffness.

Existing methods (like Dynamical Mean-Field Theory) have explored finite-momentum Cooper pairs, but a unified, parameter-free SCDFT approach for these quantities was missing.

2. Methodology

The authors developed a first-principles framework within SCDFT that explicitly incorporates finite-momentum Cooper pairs (momentum $Q$).

• Finite-Momentum Formalism:

  • The superconducting order parameter is treated as periodically modulated with a finite center-of-mass momentum $Q$: $\chi(r+R, r'+R) = e^{iQ\cdot R}\chi(r, r')$.
  • This leads to a generalized Bloch theorem where the Kohn-Sham-Bogoliubov-de Gennes (KS-BdG) eigenvectors are shifted in momentum space ($k \pm Q/2$).
  • A gap equation is derived for Cooper pairs with finite $Q$, allowing the calculation of the gap function $\Delta^{(Q)}_{nk}$ and the supercurrent density $j^{(Q)}_{sc}$.

• Numerical Stability Techniques:

  • Since $\xi_0$ is typically orders of magnitude larger than the lattice constant, the required $Q$ values are extremely small relative to the Brillouin zone grid.
  • To handle this, the authors employ Taylor expansions for the Kohn-Sham energies at shifted $k$-points ($k \pm Q/2$) using first and second derivatives.
  • An auxiliary energy-dependent gap function is introduced to stabilize the $k$-point integration near the Fermi surface, ensuring numerical convergence for tiny $Q$.

• Extraction of Physical Quantities:

  • Coherence Length ($\xi_0$): Determined by analyzing the momentum dependence of the Fermi-surface-averaged gap function $\langle \Delta^{(Q)} \rangle$. It follows the Ginzburg-Landau form $\langle \Delta^{(Q)} \rangle \propto \sqrt{1 - \xi_0^2 Q^2}$. $\xi_0$ is extracted from the wave number $Q$ where the gap drops to $1/\sqrt{2}$ of its zero-momentum value.
  • Penetration Depth ($\lambda_L$): Calculated from the linear response of the supercurrent density $j^{(Q)}_{sc}$ to $Q$ at $Q \to 0$. Alternatively, it is derived from the depairing current ($J_{dp}$), the maximum supercurrent the system can sustain, using the relation $\lambda_L \propto (\xi_0 J_{dp})^{-1/2}$.

3. Key Contributions

1. Unified First-Principles Framework: The paper establishes a single SCDFT-based methodology to simultaneously and consistently calculate $T_c$, $\xi_0$, and $\lambda_L$ without empirical parameters.
2. Robust Numerical Implementation: The introduction of Taylor expansion techniques and auxiliary gap functions allows for the stable calculation of supercurrents and gap functions at the extremely small $Q$ values required for realistic coherence lengths.
3. First-Principles Uemura Plot: By calculating the Fermi temperature ($T_F$) and $T_c$ from first principles, the authors construct a theoretical Uemura plot ($T_c$ vs. $T_F$), linking pairing strength and phase stiffness microscopically.

4. Results

The method was applied to a diverse set of materials: elemental superconductors (Al, Nb, Sn, In, Ta, Pb), the A15 compound V$_3$Si, and the high-pressure hydride H$_3$S.

• Quantitative Agreement:

  • Nb: Calculated $\xi_0 = 34$ nm (Exp: $39 \pm 1$nm) and$\lambda_L = 34$nm (Exp:$27-39$ nm).
  • H$_3$S (at 200 GPa): Calculated $\xi_0 = 3.0$ nm and $\lambda_L = 19-22$ nm. This agrees with experimental estimates derived from upper critical fields and magnetization, despite the extreme difficulty of measuring these directly under megabar pressures.
  • Classification: The method correctly reproduces the Type-I vs. Type-II classification for all tested materials based on the ratio $\xi_0/\lambda_L$.

• Depairing Currents ($J_{dp}$):

  • The framework predicts theoretical upper limits for critical currents. For H$_3$S, an exceptionally high $J_{dp} \approx 697 \times 10^7$ A/cm$^2$ was predicted, exceeding typical values in cuprates, suggesting H$_3$S can sustain massive supercurrents.

• Uemura Plot Analysis:

  • The first-principles data confirms that conventional elemental superconductors exhibit small $T_c/T_F$ ratios.
  • Higher-$T_c$ systems (like V$_3$Si and H$_3$S) are characterized by the simultaneous realization of strong pairing (short $\xi_0$) and large phase stiffness (short $\lambda_L$), providing a microscopic basis for empirical correlations previously observed only experimentally.

5. Significance

• Predictive Power for Extreme Conditions: The method provides a vital tool for characterizing superconductors in regimes where experiments are impossible or extremely challenging, such as high-pressure hydrides (H$_3$S) or novel materials yet to be synthesized.
• Microscopic Understanding: It bridges the gap between macroscopic phenomenological parameters (like $\xi_0$ and $\lambda_L$) and microscopic electronic structure, offering a deeper understanding of the mechanisms governing superconductivity.
• Materials Design: By enabling the prediction of critical length scales and currents, this approach aids in the rational design of superconducting materials for technological applications, including quantum technologies and high-field magnets.

In summary, this work extends SCDFT beyond the calculation of $T_c$ to a comprehensive description of the superconducting state's spatial and magnetic properties, establishing a rigorous, parameter-free route to predicting fundamental superconducting length scales.