expression for the oscillation envelope. Finally, our conclusions follow in section 4.

MO at zero temperature
======================

In order to study the MO, we will consider that the Fermi energy $\mu$ is held constant. Moreover, we shall always take $\mu>0$, so that the valence band is always full. Given that $\mu$ is fixed, while the number of electrons $N$ may change, it is convenient to work with the grand potential $\Omega$. We will consider the long wavelength approximation, with energies close to the Fermi level. This condition is satisfied as long as [@Neto2009] $\mu\ll\left|t\right|\sim3$ eV ($t$ is the nearest neighbor hopping amplitude). In this case the electrons in graphene behave as relativistic massless fermions, whose dynamics is given by the corresponding Dirac Hamiltonian. In the presence of a perpendicular magnetic field $B$ one obtains the discrete Landau level, as well as a spin splitting due to the Zeeman effect [@Zeeman1897]. Then it can be shown [@Escudero2017] that in graphene the energy levels are given by $\varepsilon_{\lambda,n,s}=\lambda\alpha\sqrt{nB}-s\mu_{\mathrm{B}}B$, where $\alpha=\upsilon_{F}\sqrt{2e\hbar}$ and $\upsilon_{F}\sim10^{6}$ m/s is the Fermi velocity, while $\lambda=\pm1$ for the conduction and valence bands, $n=0,\:1,\,2,\ldots$ for the Landau level (LL) and $s=\pm1$ for the spin (throughout this paper we will use 