Phase transition into the superconducting mixed state and the de Haas-van Alphen effect

The Landau expansion for the free energy of the superconducting mixed state near the upper critical field in powers of the square modulus of the order parameter averaged over an Abrikosov lattice is derived. The analytical ca lculations have been carried out in the framework of the Gor'kov formalism for a 3-dimensional isotropic BCS model beyond the limits of the quasiclass ical approximation, that is, with Landau quantization of the quasiparticle energy levels taken into consideration. The derivation is performed at low temperature for finite but high enough crystal purity. The effect of Pauli paramagnetic terms is taken into account. The quantum oscillations of the c ritical temperature, the order parameter's amplitude and the magnetization (de Haas-van Alphen effect) in the mixed state are found. The limitation fo r the validity of a mean field approach due to critical fluctuations (Ginzb urg criterion) for the phase transition under consideration is established.