Hall Effect in the Abrikosov Lattice of Type-II Superconductors

Abstract

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

Hall Effect in the Vortex Lattice ofd-Wave Superconductors with Anisotropic Fermi Surfaces

“…3) As the magnetic field is increased further, the vortex lattice symmetry generally starts to affect the charge distribution owing to the overlapping of vortices. 1) We observe the effect of overlapping clearly in Fig. 4 as compared with in Fig.…”

“…It was shown previously 1) on the basis of a microscopic theory 2,3) that the vortex-core charge accumulated by the Lorentz force on supercurrents exhibits a characteristic magneticfield dependence in two-dimensional s-wave superconductors. Specifically, the charge density at the core center ρ 0 approximately obeys ρ 0 (H) ∝ H(H c2 − H) owing to the competition between the increasing magnetic field H and decreasing pair potential (H c2 : upper critical field).…”

“…Our calculations are based on the augmented quasiclassical equations with the Lorentz force in the Matsubara formalism. 3) We can divide them into the standard Eilenberger equations [6][7][8][9][10] and an electric-field equation 1,3,5) through an expansion in terms of the dimensionless quasiclassical parameter δ ≡ (k F ξ 0 ) −1 ≪ 1, where k F is the Fermi wave number and ξ 0 is the coherence length at T = 0. These equations can be written as follows.…”

Charging in a Superconducting Vortex Due to the Three Force Terms in Augmented Eilenberger Equations

“…On the other hand, our previous works have shown that the charge accumulation by the Lorentz force in the core is strongly enhanced compared with that of an isolated vortex as the external magnetic field increases from a lower critical field. 37,38) Thus, the contribution of the Lorentz force to the core charge may become dominant over the contributions of other forces in finite magnetic fields. Hence, we should study the magnetic-field dependence of the vortex-core charging due to the three forces by solving these augmented Eilenberger equations with three forces for the vortex lattice system.…”

Charging due to Pair-Potential Gradient in Vortex of Type-II Superconductors

“…In this context, it is worth noting that ρ(0) has been shown to have a strong magnetic-field dependence with a peak structure and can be enhanced significantly over the value of an isolated vortex as the magnetic field is increased. 7,8) Hence, it is not appropriate to conclude that the Lorentz-force contribution is negligible in all cases.…”