Theory of the de Haas–van Alphen effect in type-II superconductors

Abstract

Theories of quasiparticle spectra and the de Haas--van Alphen (dHvA) oscillation in type-II superconductors are developed based on the Bogoliubov--de Gennes equations for vortex-lattice states. As the pair potential grows through the superconducting transition, each degenerate Landau level in the normal state splits into quasiparticle bands in the magnetic Brillouin zone. This brings Landau-level broadening, which in turn leads to the extra dHvA oscillation damping in the vortex state. We perform extensive numerical calculations for three-dimensional systems with various gap structures. It is thereby shown that (i) this Landau-level broadening is directly connected with the average gap at $H=0$ along each Fermi-surface orbit perpendicular to the field $\mathbf{H},$ (ii) the extra dHvA oscillation attenuation is caused by the broadening around each extremal orbit. These results imply that the dHvA experiment can be a unique probe to detect band- and/or angle-dependent gap amplitudes. We derive an analytic expression for the extra damping based on the second-order perturbation with respect to the pair potential for the Luttinger-Ward thermodynamic potential. This formula reproduces all our numerical results excellently, and is used to estimate band-specific gap amplitudes from available data on ${\mathrm{NbSe}}_{2},$ ${\mathrm{Nb}}_{3}\mathrm{Sn},$ and ${\mathrm{YNi}}_{2}{\mathrm{B}}_{2}\mathrm{C}.$ The obtained value for ${\mathrm{YNi}}_{2}{\mathrm{B}}_{2}\mathrm{C}$ is fairly different from the one through a specific-heat measurement, indicating presence of gap anisotropy in this material.