Hessian characterization of the pinning landscape in a type-II superconductor

Recent advances in vortex imaging allow for tracing the position of individual vortices with high resolution. Pushing an isolated vortex through a superconducting sample with the help of a controlled dc transport current and measuring its local ac response, the pinning energy landscape could be reconstructed along the vortex trajectory [L. Embon et al., Sci. Rep. 5, 7598 (2015)], providing unprecedented access to the mesoscopic, i.e., sample dependent, properties of a superconducting device. The controlled motion of objects through such tilted energy landscapes is fundamentally limited to those areas of the landscape developing local minima under appropriate tilt. We introduce the Hessian stability map and the Hessian character of a pinning landscape as quantities that help characterizing a pinning landscape, or short, pinscape. We determine the Hessian character, the area fraction admitting stable vortex positions, for various types of pinning potentials: Assemblies of cut parabolas, Lorentzian- and Gaussian-shaped traps, as well as a Gaussian random disordered energy landscape, with the latter providing a universal result of (3 - root 3)/6 approximate to 21% of stable area. Our Hessian analysis sheds light on various aspects of mesoscopic vortex physics in the single vortex limit and opens interesting avenues in strong pinning theory.