Charging in the vortex lattice of type-II superconductors

We study the magnetic-field dependence of the vortex-core charge in the Abrikosov lattice of an $s$-wave superconductor based on the augmented quasiclassical equations, where we incorporate the pair-potential gradient (PPG) terms characteristic of charging in superconductors besides the well-known Lorentz force. Our numerical results at $T=0.2 T_{\rm c}$ and $0.5 T_{\rm c}$ reveal that periodic charge redistribution is superimposed on the magnetic flux-line lattice with different spatial patterns at different fields. The PPG terms are dominant at weak fields over the Lorentz force for accumulating charge in the vortex cores, whereas the Lorentz force prevails at higher fields to give rise to a peak structure in the core charge around $H\sim \frac{1}{2}H_{{\rm c}2}$. We estimate the peak value of the core charge at $T=0.2 T_{\rm c}$ using parameters appropriate for cuprates to obtain a large value of $Q \sim 10^{-2} |e|$ in the core region of radius $0.2 \xi_0$ in the $ab$ plane and length $1 \ {\rm nm}$ along the $c$ axis.