Superconducting Geometric Potential and Curvature-Enhanced Superconductivity in Curved Thin Films

Abstract

We derive the linearized Ginzburg-Landau (GL) equation for a curved ultra-thin superconducting film in the presence of a magnetic field. By introducing a novel transverse order parameter that varies slowly along the film with the superconducting/vacuum boundary condition, we decouple the linearized GL equation into a transverse component and a surface component in the thin-layer quantization scheme. A superconducting geometric potential (GP) is present in the surface equation, which can substantially affect the nucleation of the superconducting state in the curved ultra-thin superconducting film. From the perspective of the GL free energy, the superconducting GP reduces the coefficient of the quadratic term of the order parameter, which enables the curved film to stay in the superconducting state even when the superconducting parameter $α$ becomes positive. Based on our equivalent equation, for a superconducting thin film with uniform curvature, the relative increase of the critical temperature is proportional to the magnitude of the superconducting GP. As an example, we numerically investigate the phase transition of a rectangular superconducting film bent around a cylindrical surface, with the numerical results in good agreement with the theoretical expectations. We further propose a strain-free experimental validation using ultracold atomic condensates, where nested superfluid shells enforce Neumann boundary conditions to isolate the superconducting GP.

Figures (3)

• Figure 1: (Color online) Schematic representation of a curved superconducting film $\Omega$ with uniform thickness $d$, bounded by surfaces $S_1, S_2$ and sidewall $S_0$. A point on the central surface $S$ is parameterized by $\bm{r}(q^1, q^2)$, where $q^1, q^2$ are curvilinear coordinates. Vectors $\bm{e}_1$ and $\bm{e}_2$ form the tangent basis on $S$, while $\bm{e}_3$ is the unit normal vector.
• Figure 2: (Color online) A rectangular thin superconducting film of width $U$, height $V$, and thickness $d$, bent along its width to conform to a cylindrical surface of radius $R$. The stress-free central surface $S$ is parameterized by orthogonal coordinates $(u,v)$ aligned with the azimuthal and axial directions, respectively. Coordinate $q^3$ denotes the normal distance from $S$. An external magnetic field $\bm{B}$ (dark green arrows) is applied perpendicular to the film surface with uniform intensity.
• Figure 3: (Color online) (a) Phase diagram of the critical temperature $\tilde{T}^\ast$ (in units of $T_\mathrm{c}$) for the cylindrical film model ($U = 8 \xi_0, V = 6 \xi_0$) as a function of mean curvature $M$ ($\xi_0^{-1}$) and magnetic field $B$ ($\hbar c \xi_0^{-2} Q^{-1}$). Solid black curves are contour lines of $\tilde{T}^\ast$. Dashed lines marked with circles delineate boundaries between regions with different vorticity $L$. (b) $\tilde{T}^\ast$ versus curvature $M$ for selected magnetic fields $B=0.0, 0.30, 0.60, 0.90$. The quadratic increase confirms the $M^2$ dependence. (c) $\tilde{T}^\ast$ versus magnetic field $B$ for curvatures $M=0, 0.05, 0.08, 0.12$. This illustrates the competition between curvature-induced enhancement and magnetic field suppression.