  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.
