Analytical Charge Density Profile of Vortex Core in Weak-Coupling Superconductor

Abstract

Self-consistent Bogoliubov-de Gennes calculations have long shown that solving the Poisson equation inside a superconducting vortex turns a one-signed charge depletion into a modulation that alternates in sign with period $π/k_F$. We give an elementary account of that result. Taking the Caroli-de Gennes-Matricon bound states in a step-like gap, we show that the normalization of the bound-state spinor is nearly independent of angular momentum, which collapses the mode sum into closed form. Inside the core the vortex winding removes one Bessel channel from a completeness sum, so the density vanishes on the vortex line and carries Friedel-like oscillations of wavevector $2k_F$; outside it the sum gives a $1/r$ envelope decaying over a coherence length, with a residual ripple. The bound-state charge does not integrate to zero, so neutrality obliges the extended states to compensate it exactly. That compensation is complete at long wavelength but fails at the diameter of the Fermi circle, and what survives is a sign-alternating $2k_F$ modulation reduced only by $4k_F^2/(4k_F^2+k_{TF}^2)$, a factor lying between one-half and three-quarters for any metal. The oscillation is therefore not a delicate effect but a consequence of neutrality and the inefficiency of screening at large momentum transfer: in the screened total the smooth terms cancel and only the ripple is left.