Ground state magnetization of conduction electrons in graphene with Zeeman effect

Abstract

In this work we address the ground state magnetization in graphene, considering the Zeeman effect and taking into account the conduction electrons in the long wavelength approximation. We obtain analytical expressions for the magnetization at T = 0, where the oscillations given by the de Haas van Alphen (dHvA) effect are obtained. We find that the Zeeman effect modifies the magnetization by introducing new peaks associated with the spin splitting of the Landau levels. These peaks are very small for typical carrier densities in graphene, but become prominent for higher densities. The results obtained provide insight of the way in which the Zeeman effect modifies the magnetization, which can be useful to control and manipulate the spin degrees of freedom.

Magnetic oscillations in silicene

“…It shows that the MO peaks are produced whenever ξ q , M ′ or M P changes discontinuously, U being continuous always. Thus the magnetization in pristine silicene at T = 0 K oscillates in a sawtooth pattern, as in graphene [23,25] and in general 2DEG with a Dirac-like spectrum [22]. This is also in agreement with the results found in [29], where the MO at T = 0 K in a pristine buckled honeycomb lattice are expressed as an infinite sum of harmonics k of the form sin(k)/k, which gives a sawtooth oscillation.…”

“…Hence n q still changes only q changes by four, so ξ L q gives the frequency ω 1 = π n e /2e = 10.34 T. On the other hand, now s q changes whenever q changes, so for ξ S q we have ∆q = 1 in Eq. (25). This gives a new frequency ω 3 = 2π n e /e which implies ω 3 = 41.36 T for n e = 0.01 nm −2 .…”

“…1(a), and taking into account the valley degeneracy at E z = 0, we get that ξ L q changes periodically when q changes by four, so ∆q = 4 in Eq. (25), giving the frequency ω 1 = π n e /2e. On the other hand, ξ S q changes periodically when q changes by two, so ∆q = 2 in Eq.…”

“…On the other hand, ξ S q changes periodically when q changes by two, so ∆q = 2 in Eq. (25), giving the frequency ω 2 = π n e /e. Thus the magnetization in Eq.…”

“…Moreover, considering the Zeeman effect, the LL for each spin split, loosing then the spin degeneracy. This loss of valley and spin degeneracy gives discontinuous changes in the last energy level, which in turn is expected to produce new magnetization peaks, as occur in graphene [25]. Motivated by this we have studied the magnetic oscillations (MO) at T = 0 K in a general silicene-like system with a conduction electron density n e , which could be due to an applied gate voltage.…”