non-oscillatory contribution since the valence band is always full if $\mu>0$. Therefore we omit for the moment $\Omega_{V}$ and consider only the conduction grand potential $\Omega_{C}=\sum_{m=0}^{f}D\left(\varepsilon_{m}-\mu\right)$. Separating $\varepsilon_{m}=\varepsilon_{m}^{0}-s_{m}\mu_{\mathrm{B}}B$, with $\varepsilon_{m}^{0}=\alpha\sqrt{n_{m}B}$, we get

$$\Omega_{C}=\Omega_{0}-D\mu_{\mathrm{B}}B\sum_{m=0}^{f}s_{m},\label{GP2}$$

where $\Omega_{0}=\sum_{m=0}^{f}D\left(\varepsilon_{m}^{0}-\mu\right)$. The last term in equation (\[GP2\]) is related to the Pauli paramagnetism associated with the spin population. Indeed, we have $D\sum_{m=0}^{f}s_{m}=N_{+}-N_{-}$, where $N_{+}$($N_{-}$) is the total number of spin up(down) states. Thus $\mu_{\mathrm{B}}D\sum_{m=0}^{f}s_{m}=\mu_{\mathrm{B}}D\left[1+\left(-1\right)^{f}\right]/2=\mu_{\mathrm{B}}\left(N_{+}-N_{-}\right)=M_{p}$, so equation (\[GP2\]) becomes

$$\Omega_{C}=\Omega_{0}-BM_{P}.\label{GP3}$$

The conduction magnetization is given by $M_{C}=-\mathcal{A}^{-1}\left(\partial\Omega_{C}/\partial B\right)_{\mu}$, where $\mathcal{A}$ is the graphene area. From equation (\[GP3\]) we get

$$M_{C}=M_{0}+\frac{1}{\mathcal{A}}\left(M_{P}+B\frac{\partial M_{P}}{\partial B}\right),\label{Mc-1}$$

where $M_{0}=-\mathcal{A}^{-1}\left(\partial\Omega_{0}/\partial B\right)_{\mu}$. Given that $\partial D/\partial B=D/B$, while $\partial\varepsilon_{m}^{0}/\partial B=\varepsilon_{m}^{0}/2B$, we can write

$$M_{C}=-\frac{1}{2B}\left(\frac{3\Omega_{C}}{\mathcal{A}}+n\mu\right)+\frac{1}{2}m_{P},\label{Mc}$$

where $n=N/\mathcal{A}=\sum_{m=0}^{f}D/\mathcal{A}=D\left(f-1\right)/\mathcal{A}$ is the density of conduction electrons and $m_{p}=M_{P}/\mathcal{A}$. Equation (\[Mc\]) shows that the MO have a sawtooth oscillation (SO) produced whenever $n$ or $m_{P}$ changes discontinuously, $\Omega_{C}$ being continuous always. The sawtooth peaks amplitude $\Delta M$ are given by

$$\Delta M=-\frac{\mu}{2B}\Delta n+\frac{1}{2}\Delta m_{P}.\label{DeltaM}$$

Each contribution $\Delta n$ and $\Delta m_{P}$ is determined by the discontinuous change in the parameters $n_{f}$ and $s_{f}$, which define the last energy level occupied. Considering the possible changes of LL and spin, we can write the MO as a sum of two sawtooth. 