Influence of temperature on the magnetic oscillations in graphene with spin splitting: a new approach

Abstract

We analyze the magnetic oscillations (MO) due to the de Haas-van Alphen effect, in pristine graphene under a perpendicular magnetic field, taking into account the Zeeman effect. We consider a constant Fermi energy, such that the valence band is always full and only the conduction band is available. At zero temperature the MO consist of two sawtooth peaks, one for each spin. Both peaks have the same frequency, but different amplitude and phase. We show that, in order to observe the spin splitting in the MO, Fermi energy of about 0.1 eV is required. At low temperatures we obtain that the MO can be expressed as the MO at zero temperature, plus small Fermi-Dirac like functions, each centered around the MO peaks. Using this expression, we show that the spin splitting is observable in the MO only when the thermal energy is smaller than the Zeeman energy. We also analyze the shift of the MO extrema as the temperature increases. We show that it depends on the magnetic field, which implies a broken periodicity at nonzero temperature. Finally, we obtain an analytical expression for the MO envelope. The results obtained could be used to infer temperature changes from the MO extrema shift and vice versa.

A general formulation for the magnetic oscillations in two dimensional systems

“…Equation ( 18) can be conveniently rewritten in different ways, depending on the situation. For instance, if ψ γ ≫ 1 such that M 0 osc,γ is given by equation ( 9), then one can use the properties of the arc tangent function to rewrite [37]…”

“…Damping effects, due to temperature or impurity scattering, tend to broaden and reduce the MO amplitude [35][36][37]. Besides the need for relatively strong magnetic fields, this damping of the MO is the reason why they are difficult to observe [38].…”

“…In 2D materials, this loss can be significant, as many properties of the system, such as the interplay between valley and spin, would no longer be observed in the MO. Another approach to treat the damping of the MO, recently developed in pristine graphene and 2D materials [37,38], considers the temperature effect by its modification to the MO in the ground state. At zero impurities, this modification is given by Fermi-Dirac distribution functions, and thus they represent the damping in the MO due to the change in the occupancy of the LL.…”

Temperature effect on the magnetic oscillations in 2D materials

“…To study the influence of temperature over the observation of the spin splitting (SP) in the MO, we follow the same lines as we did in the graphene case [54], applying it to each of the frequencies now present. Then we consider two MO peaks with frequency ω, at a given LL n, separated due to the SP, located in general at 1/B 1 = n/ω − ∆ and 1/B 2 = n/ω + ∆.…”

“…We shall now obtain an expression for the MO envelope, restricting ourselves to the beating condition, so that (ω 1 − ω 2 ) /ω 1 ≪ 1. In the general case, at a given temperature one should numerically obtain the shift δ as function of B, and from it construct the MO envelope, as was done in graphene [54]. The generalization to 2D materials with broken valley degeneracy is trivially done by taking into account the two frequencies involved and the resulting beating phenomenon.…”