Magnetic Oscillations and Quasiparticle Band Structure in the Mixed   State of Type-II Superconductors
  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. 
M.R. Norman, A.H. MacDonald, and H. Akera