Self-consistent electronic structure of a vortex line in a type-II superconductor

The electronic structure of a vortex line in a type-II superconductor is calculated self-consistently in the framework of the Bogoliubov–de Gennes theory. The Debye frequency, the Fermi velocity, and the coupling constant of the electron-electron attractive interaction are used as microscopic input parameters. The resulting quasiparticle-excitation spectrum, the pair potential, and the current distribution are studied as a function of temperature, and can be used to define a coherence length and to determine the magnetic penetration depth. The local density of one-particle excitations, calculated from the quasiparticle amplitudes, explains the results of scanning-tunneling-microscopy (STM) experiments by Hess et al. [Phys. Rev. Lett. 62, 214 (1989)] on ${\mathrm{NbSe}}_2$. The main spectroscopic features in the experimental results are caused by bound states in the vortex cores. Spatial distortions of the bound-state wave functions caused by neighboring vortices and by the crystalline lattice are discussed in terms of a simplified two-band model. In the case of ${\mathrm{NbSe}}_2$, the resulting local density of states has a characteristic star shape in real space, whose orientation is energy dependent, in agreement with recent STM experiments [Phys. Rev. Lett. 64, 2711 (1990)].