Strong pinning transition with arbitrary defect potentials

• Published in: Physical review research 2023-08, Vol.5 (3), p.033098, Article 033098
• Main Authors: Gaggioli, Filippo, Blatter, Gianni, Buchacek, Martin, Geshkenbein, Vadim B.
• Format: Article
• Language: English
• ISSN: 2643-1564; 2643-1564
• DOI: 10.1103/PhysRevResearch.5.033098

Dissipation-free current transport in type II superconductors requires vortices, the topological defects of the superfluid, to be pinned by defects in the underlying material. The pinning capacity of a defect is quantified by the Labusch parameter κ∼f_{p}/ξC[over ¯], measuring the pinning force f_{p} relative to the elasticity C[over ¯] of the vortex lattice, with ξ denoting the coherence length (or vortex core size) of the superconductor. The critical value κ=1 separates weak from strong pinning, with a strong defect at κ>1 able to pin a vortex on its own. So far, this weak-to-strong pinning transition has been studied for isotropic defect potentials, resulting in a critical exponent μ=2 for the onset of the strong pinning force density F_{pin}∼n_{p}f_{p}(ξ/a_{0})^{2}(κ−1)^{μ}, with n_{p} denoting the density of defects and a_{0} the intervortex distance. This result is owed to the special rotational symmetry of the defect producing a finite two-dimensional trapping area S_{trap}∼ξ^{2} at the strong pinning onset. The behavior changes dramatically when studying anisotropic defects with no special symmetries: the strong pinning then originates out of isolated points with length scales growing as ξ(κ−1)^{1/2}, resulting in a different force exponent μ=5/2. The strong pinning onset is characterized by the appearance of unstable areas U_{R[over ̃]} of elliptical shape whose boundaries mark the locations where vortices jump. The associated locations of asymptotic vortex positions define areas B_{R[over ¯]} of bistable vortex states and assume the shape of a crescent. The geometries of unstable and bistable regions are associated with the local differential properties of the Hessian determinant D(R) of the pinning potential e_{p}(R), specifically, its minima, maxima, and saddle points. Extending our analysis to the case of a random two-dimensional pinning landscape, a situation describing strong pinning in a thin superconducting film, we discuss the topological properties of unstable and bistable regions as expressed through the Euler characteristic, with the latter related to the local differential properties of D(R) through Morse theory.