Microscopic equivalence of the vortex-entry current and the depairing current in a superconducting thin-film strip

Technical Summary: Microscopic Equivalence of Vortex-Entry and Depairing Currents

Problem Statement
In narrow superconducting thin-film strips carrying a DC current, the dissipationless state is lost when the current exceeds a threshold. Within the macroscopic Pearl–London theory, this threshold is identified as the "vortex-entry current" ($J_v$), defined by the disappearance of an energy barrier that prevents vortices from entering the strip edge. This barrier arises because, at low currents, moving a vortex inward initially increases the system's energy. The Pearl–London model predicts $J_v \sim B_c/\mu_0\lambda$, which is of the same order as the "depairing current" ($J_{dp}$)—the maximum supercurrent the condensate can sustain before the superconducting order parameter is destroyed.

However, the Pearl–London approach relies on a prescribed vortex coordinate and a short-distance cutoff ($\xi_{cut}$) to define the vortex core, making the precise relationship between $J_v$ and $J_{dp}$ ambiguous. A cutoff-free treatment is required to determine if these two currents are fundamentally distinct or identical. While Ginzburg–Landau (GL) theory offers a cutoff-free description via saddle-point analysis, it is limited to temperatures near the critical temperature ($T_c$). The central question addressed is whether the vortex-entry current and the depairing current are equivalent in a microscopic theory valid at arbitrary temperatures ($0 < T < T_c$).

Methodology
The authors reformulate the problem using the fixed-current Gibbs functional within the dirty-limit Usadel theory, a microscopic framework valid for arbitrary temperatures in the range $0 < T < T_c$. The study focuses on an ideal, homogeneous, narrow thin-film strip ($W \ll \Lambda$, where $\Lambda$ is the Pearl length) at zero applied magnetic field, neglecting self-field effects, edge roughness, and current crowding.

The methodology involves a stability analysis of the vortex-free, current-carrying state:

1. Functional Formulation: The system is described by the Usadel equation, the gap equation, and supercurrent conservation, derived from the stationarity of a Gibbs functional $G$ with respect to the spectral angles $\theta_n$, the gap $\Delta$, and the gauge-invariant momentum $q$.
2. Stability Criterion: The vortex-entry barrier is interpreted as the energy difference between the vortex-free state (a local minimum of $G$) and a saddle point. The barrier disappears when the local minimum and the saddle merge (a saddle-node bifurcation), causing the local stability of the vortex-free state to be lost. This occurs when the second variation of the Gibbs energy, $\delta^2 G$, ceases to be positive definite.
3. Perturbation Analysis: The authors examine the stability of the uniform vortex-free state against two types of perturbations:
   • Spatially Uniform Perturbations: Homogeneous changes in momentum, gap, and spectral angles.
   • Spatially Nonuniform Perturbations: Expansions in normal modes (cosine/sine modes across the strip width and Fourier modes along the length) to account for spatial variations.

Key Results

1. Uniform Mode Dominance: The analysis of spatially nonuniform perturbations reveals that they possess an additional positive energy cost (stiffness) compared to the uniform mode. Specifically, the minimized curvature of the energy functional for nonuniform modes is strictly larger than that of the uniform mode. Consequently, in an ideal zero-field strip, nonuniform modes cannot become unstable before the uniform mode.
2. Equivalence of Currents: The stability limit for the uniform mode is determined by the condition $KL - M^2 = 0$, where $K$, $L$, and $M$ are coefficients derived from the linearized Usadel and gap equations. The authors demonstrate that this condition is mathematically equivalent to the condition where the derivative of the self-consistent current-momentum curve vanishes ($dJ/dQ = 0$).
3. Final Conclusion: The current at which the vortex-free state loses local stability (the barrier-disappearance current) is exactly the depairing current. For an ideal homogeneous narrow strip, the paper establishes the equality:
   $$J_v(T) = J_{dp}(T)$$
   This result holds for all temperatures $0 < T < T_c$.

Significance and Claims
The paper claims to provide a "cutoff-free microscopic counterpart" to the Pearl–London barrier-disappearance current. By utilizing the Usadel theory, the work resolves the ambiguity inherent in the Pearl–London model regarding the dependence on short-distance cutoffs.

The primary significance is the theoretical unification of two distinct instability concepts:

• Vortex Entry: Traditionally viewed as a topological event where a vortex penetrates the edge.
• Depairing: Traditionally viewed as a bulk thermodynamic limit where the superconducting condensate collapses.

The authors conclude that for an ideal system, these are not separate instabilities; the loss of stability against vortex entry coincides exactly with the loss of stability of the superconducting condensate itself.

Caveats and Scope
The authors explicitly state that this equality ($J_v = J_{dp}$) applies strictly to an ideal, homogeneous strip at zero applied field with neglected self-field effects. They caution that this should not be interpreted as a universal statement for all experimental switching currents. Real devices may switch at lower currents due to:

• Local imperfections, constrictions, or bends that enhance local current density or lower the vortex-entry barrier.
• Thermal or fluctuation-assisted vortex crossing, which can trigger switching below the deterministic limit.

Conversely, the vortex-entry current can be engineered to be higher than the depairing current in non-ideal geometries (e.g., via nonuniform Pearl lengths). The paper identifies extending this microscopic stability framework to include such nonidealities and finite-barrier fluctuation processes as a direction for future work.