Magnetic oscillations and frequency mixing in a two-band conductor

Exact analytical results of the de Haas-van Alphen (dHvA) effect in an idealized two-band Fermi liquid with parabolic dispersion are presented. We consider a Fermi surface consisting in two electron bands with different band edges and band masses. Magnetic breakthrough (MB) between the bands is negligible. Analytical expressions of the dHvA Fourier amplitudes are derived in the case where the total number of electron is fixed (Canonical Ensemble, CE). As already reported in the literature, the oscillations of the chemical potential yield frequency mixing and Lifshitz-Kosevich (LK) theory, which is valid in the Grand Canonical Ensemble (GCE), does not apply at very low temperature. We show that the corresponding Fourier amplitudes depend on the commensurability between the two effective masses and also the two fundamental frequencies