Hall Effect in the Abrikosov Lattice of Type-II Superconductors

We study vortex charging caused by the Lorentz force on supercurrent based on the augmented quasiclassical equa- tions of superconductivity. Our numerical study on an s-wave vortex lattice in the range $H_{{\rm{c}}1} < H < H_{{\rm{c}}2}$ reveals that each vortex core with a single flux quantum also accumulates charge due to the circulating supercurrent and has a Hall voltage across the core. The field dependence of the charge density at the core center is well described by $H(H_{\mathrm{c}2}-H)$ with a peak near $H_{\mathrm{c}2}\mathrm{/}2$ originating from competition between the increasing magnetic field and the decreasing pair potential. The peak value of the accumulated charge in a core region of radius $0.5{\xi }_0$ is estimated to be about $\eta\Delta_{0}/(k_{\rm F}\xi_0)\times|e|$Cper$\Delta z=1$ nm along the flux line at low temperatures, where $\eta \equiv \pi {ϵ}_0\mathrm{\Delta }z\mathrm{/}\mathrm{\mid }e{\mathrm{\mid }}^2=1.09\times {10}^{18}$ ${\mathrm{J}}^{-1}$ with $e<0$ the charge of an electron, ${\mathrm{\Delta }}_0$ the energy gap at $T=0$, $k_{\mathrm{F}}$ the Fermi wave number, and ${\xi }_0$ the coherence length at $T=0$.