de Haas–van Alphen effect in the superconducting state of a two-dimensional metal

We present a theory of the de Haas–van Alphen (dHvA) effect in the superconducting mixed state within the framework of Gorkov’s scheme near ${\mathit{H}}_{\mathit{c}2}$ in a two-dimensional (2D), extremely type-II superconductor. A semiclassical approximation is developed whose results are compared to the exact quantum-mechanical result. A remarkable agreement between the two methods is found even for extremely strong fields and low temperatures, where strong quantum oscillations exist. The method is used to calculate variationally the superconducting free energy to the fourth order in the Gorkov expansion. The resulting variational equation for the order parameter is solved exactly, and the mean-square order parameter over the entire vortex lattice, as well as the superconducting magnetization, is calculated as functions of the magnetic field and the temperature. It is found that, despite significant smearing effects of the collective pairing process which involves many Landau levels, strong quantum oscillations exist in the superconducting order parameter. The calculated superconducting magnetization oscillations are of the same order of magnitude as the normal-electron ones. The envelope of these oscillations is found to decay exponentially with increasing magnetic field. At sufficiently low temperatures the fine structure of the oscillations may reflect the repulsive nature of the interaction between vortex lines. The transition to superconductivity is investigated along, and above, the classical Ginzburg-Landau phase boundary: Reentrance of the superconducting state above ${\mathit{H}}_{\mathit{c}2}$(T), a dramatic manifestation of the 2D dHvA effect, is found to occur within an experimentally accessible range of fields and temperatures if the cyclotron-effective mass is an integral multiple of the free-electron mass.