De Haas-van Alphen effect in two- and quasi two-dimensional metals and   superconductors
  An analytical form of the quantum magnetization oscillations (de Haas-van Alphen effect) is derived for two- and quasi two-dimensional metals in normal and superconducting mixed states. The theory is developed under condition that the chemical potential is much greater than the cyclotron frequency, which is proved to be valid for using grand canonical ensemble in the systems of low dimensionality. Effects of impurity, temperature, spin-splitting and vortex lattice - in the case of superconductors of type II -, are taken into account. Contrary to the three dimensional case, the oscillations in sufficiently pure systems of low dimensionality and at sufficiently low temperatures are characterized by a saw-tooth wave form, which smoothened with temperature and concentration of impurities growth. In the normal quasi two-dimensional systems, the expression for the magnetization oscillations includes an extra factor expressed through the transfer integral between the layers. The additional damping effect due to the vortex lattice is found. The criterion of proximity to the upper critical field for the observation of de Haas-van Alphen effect in the superconducting mixed state is established. 
T. Champel, V. P. Mineev