Novel $\pi$-type vortex in a nanoscale extreme type-II superconductor: Induced by quantum-size effect

By numerically solving the Bogoliubov-de Gennes equations, we report a novel $\pi$-type vortex state whose order parameter near the core undergoes an extraordinary $\pi$-phase change for a quantum-confined extreme type-II $s$-wave superconductor. Its supercurrent behaves as the cube of the radial coordinate near the core, and its local density of states spectrum exhibits a significant negative-shifted zero-bias peak. Such $\pi$-type vortex state is induced by quantum-size effect, and can survive thermal smearing at temperatures up to a critical value $T_\tau$. The Anderson's approximation indicates that the $\pi$-type vortex may remain stable under sufficiently week magnetic field in the case less deep in the type-II limit. Moreover, we find that its appearance is governed by the sample size and $k_F\xi_0$ with $k_F$ the Fermi wave number and $\xi_0$ the zero-temperature coherence length. Similar effects may be expected in quantum-confined ultracold superfluid Fermi gasses, or even high-$T_c$ superconductors with proper $k_F\xi_0$ value.