Stacking order, interaction, and weak surface magnetism in layered graphene sheets

Recent transport experiments have demonstrated that rhombohedral-stacking trilayer graphene is an insulator with an intrinsic gap of 6 meV and Bernal-stacking trilayer graphene is a metal. We propose a Hubbard model with a moderate $U$ for layered graphene sheets, and show that the model well explains the experiments on the stacking-dependent energy gap. The moderate on-site Coulomb repulsion drives the metallic phase of the noninteracting system to a weak surface antiferromagnetic insulator for the rhombohedral stacking layers, while the interaction-opened energy gaps for the Bernal stacking layers are much smaller.

Figure 1

Schematic diagrams of trilayer graphene sheets. (a) Bernal (ABA) stacking and (b) rhombohedral (ABC) stacking. (c) and (d) are their side views. Blue and pink colors represent carbon atoms on sublattices $A$ and $B$, respectively.

Figure 2

(a) Low-energy bands of ABC-stacking trilayer. Solid red line, $U=0$; dashed black line, $U=6.2$ eV. (b) Low-energy bands of ABA trilayer for $U=0$. (c) Mean-field energy gaps as functions of $U$ for ABC and ABA trilayer graphene. (d) Logarithmic plots of energy gaps as functions of $t/U$. The parameters are $t=3.16\mathrm{eV}$ and $t_{\perp }=0.39\mathrm{eV}$.

Figure 3

Schematic illustration of the spin ordering and spin polarization per site with $U=6.2\mathrm{eV}$: (a) and (b) for ABC-stacking trilayer graphene, (c) and (d) for the ABA-stacking case. The spin polarization as a function of $U$: (e) for ABC stacking and (f) for ABA stacking.

Figure 4

ABC-stacking $N$-layer graphene: (a) energy band and (b) spin polarization for $N=24$; (c) the gaps ${\epsilon }_{g1}$ and ${\epsilon }_{g2}$; and (d) surface spin polarization as functions of $N$. ABA-stacking $N$-layer graphene: (e) energy gap as a function of $N$, and (f) spin polarization for $N=9$. $U=6.2$ eV and the hopping parameters are the same as in Fig. 2.