Temperature effect on the magnetic oscillations in 2D materials

We study the magnetic oscillations (MO) in 2D materials with a buckled honeycomb lattice, considering a perpendicular electric and magnetic field. At zero temperature the MO consist of the sum of four sawtooth oscillations, with two unique frequencies and phases. The values of these frequencies depend on the Fermi energy and electric field, which in turn determine the condition for a beating phenomenon in the MO. We analyse the temperature effect in the MO by considering its local corrections over each magnetization peak, given by Fermi-Dirac like functions. We show that the width of these functions is related to the minimum temperature necessary to observe the spin and valley properties in the MO. In particular, we find that in order to observe the spin splitting, the width must be lower than the MO phase difference. Likewise, in order to observe valley mixing effects, the width must be lower than the MO period. We also show that at high temperatures, all the maxima and minima in the MO shift to a constant value, in which case we obtain a simple expression for the MO and its envelope. The results obtained show unique features in the MO in 2D materials, given by the interplay between the valley and spin.