Vortex states and quantum magnetic oscillations in conventional type-II superconductors

The theory of pure type-II superconductors at high magnetic fields and low temperatures has recently attracted much attention due to the discovery of de Haas–van Alphen oscillations deep in the vortex state. In this article the authors review the state of the art in this rapidly growing new field of research. The very existence of quantum magnetic oscillations deep in the vortex state poses challenging questions to the theorists working in this field. For a conventional extreme type-II superconductor in magnetic fields just below the upper critical field $H_{c2\mathrm{}},$ the quasiparticle spectrum is gapless and the de Haas–van Alphen effect is suppressed with respect to the corresponding normal-state signal due to superconducting induced currents near the vortex cores, which are of paramagnetic nature. Numerical simulations of the quasiparticle band structure in the Abrikosov vortex lattice show the existence of well-separated Landau bands below $H_{c2\mathrm{}}.$ An analytical perturbative approach, which emphasizes the importance of phase coherence in quasiparticle scattering by the pair potential in the Abrikosov lattice, predicts a relatively weak magnetic breakdown of the corresponding cyclotron orbits. In contrast to the situation in the Abrikosov lattice state, a theory based on a random vortex lattice model yields large exponential decay of the de Haas–van Alphen oscillations with the superconducting order parameter below $H_{c2\mathrm{}}.$ The disordered nature of the vortex state near $H_{c2\mathrm{}}$ in real superconductors, where long-range phase coherence in the superconducting order parameter is destroyed, could explain the success of this model in interpreting experimental data below $H_{c2\mathrm{}}.$ In the Abrikosov vortex lattice state, which usually stabilizes well below $H_{c2\mathrm{}},$ the residual damping of the de Haas–van Alphen amplitude is significantly reduced. In quasi-two-dimensional superconductors, phase fluctuations associated with sliding Bragg chains along principal axes in the vortex lattice lead to a weak first-order melting transition far below the mean-field $H_{c2\mathrm{}}.$ Superconducting fluctuations dominate the additional damping of the de Haas–van Alphen oscillations in this vortex liquid state. Below the first-order freezing point, this damping is predicted to weaken signifiFantly.