We derive augmented quasiclassical equations of superconductivity with the Lorentz force in the Matsubara formalism so that the charge redistribution due to supercurrent can be calculated quantitatively. Using it, we obtain an analytic expression for the vortex-core charge of an isolated vortex in extreme type-II materials given in terms of the London penetration depth and the equilibrium Hall coefficient. It depends strongly on the Fermi surface curvature and gap anisotropy, and may change sign even as a function of temperature due to the variation in the excitation curvature under the growing energy gap. This is also confirmed in our numerical study of high-T c 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 $H(H_{{\rm{c}}2}-H)$ with a peak near $H_{{\rm{c}}2}/2$ 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 $\eta\Delta_{0}/(k_{\rm F}\xi_0)\times|e|$ C per $\Delta z=1$ nm along the flux line at low temperatures, where $\eta\equiv\pi\epsilon_0\Delta z/|e|^2=1.09\times10^{18}$ ${\rm{J^{-1}}}$ with $e<0$ the charge of an electron, $\Delta_0$ the energy gap at $T=0$, $k_{\rm F}$ the Fermi wave number, and $\xi_0$ the coherence length at $T=0$.Comment: 5 pages, 4 figure

Besides the magnetic Lorentz force familiar from the Hall effect in metals and semiconductors, there exists a mechanism for charging peculiar to superconductors that is caused by the pair-potential gradient (PPG). We incorporate it in the augmented quasiclassical equations of superconductivity with the Lorentz force to study charging of an isolated vortex in an equilibrium s-wave type-II superconductor. It is found that the PPG mechanism gives rise to charging concentrated within the core whose magnitude at the core center can be 10 to 10 2 times larger than that caused by the Lorentz force. Our detailed calculations on the spatial, temperature, and magnetic-penetration-depth dependences of the vortex-core charge reveal that the PPG mechanism contributes dominantly to the core charging of the isolated vortex over a wide parameter range. The two mechanisms are also found to work additively at the core center for the present model with an isotropic Fermi surface.

We derive augmented Eilenberger equations that incorporate the following missing force terms: (i) the Lorentz force, (ii) the pair-potential gradient (PPG) force, and (iii) the pressure difference arising from the slope in the density of states (DOS). Recently, augmented Eilenberger equations with the Lorentz and PPG forces have been derived microscopically by studying the Hall and charging effects in superconductors, but the pressure due to the slope in the DOS has not yet been considered in augmented Eilenberger equations, despite phenomenological indications that it is a charging mechanism in a vortex of type-II superconductors. This newly added pressure is called "the SDOS pressure". We calculate the charging in an isolated vortex of an s-wave superconductor with a spherical Fermi surface using the augmented Eilenberger equations incorporating the Lorentz force, PPG force, and SDOS pressure. When we compare the charge densities due to the three force terms in the augmented Eilenberger equations, the vortex-core charging due to the SDOS pressure is larger than that due to the other forces near the superconducting transition temperature. Thus, when we calculate the charging in an isolated vortex of a superconductor with a finite slope in the DOS, we should consider not only the Lorentz and PPG forces but also the SDOS pressure.

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