  We consider magnetic oscillations due to Landau quantization in the mixed
state of type-II superconductors. Our work is based on a previously developed
formalism which allows the mean-field gap equations of the Abrikosov state to
be conveniently solved in a Landau level representation. We find that the
quasiparticle band structure changes qualitatively when the pairing self-energy
becomes comparable to the Landau level separation. For small pairing self-
energies, Landau level mixing due to the superconducting order is weak and
magnetic oscillations survive in the superconducting state although they are
damped. We find that the width of the quasiparticle Landau levels in this
regime varies approximately as $\Delta_0 n_{\mu}^{-1/4}$ where $\Delta_0$ is
proportional to the magnitude of the order parameter and $n_{\mu}$ is the
Landau level index at the Fermi energy. For larger pairing self-energies, the
lowest energy quasiparticle bands occur in pairs which are nearly equally
spaced from each other and evolve with weakening magnetic field toward the
bound states of an isolated vortex core. These bands have a weak magnetic field
dependence and magnetic oscillations vanish rapidly in this regime. We discuss
recent observations of the de Haas-van Alphen effect in the mixed state of
several type II superconductors in light of our results.
