Spin-orbit coupling and broken spin degeneracy in multilayer graphene

Since the lattices of $ABA$-stacked graphene multilayers with an even number of layers, as well as that of monolayer graphene, satisfy spatial-inversion symmetry, their electronic bands must be spin degenerate in the presence of time-inversion symmetry. In intrinsic monolayer and bilayer graphene, when symmetry is not broken by external fields, the only spin-orbit coupling present at low energy near the corner of the Brillouin zone is the Kane-Mele term, that opens a bulk energy gap but does not break the spin degeneracy of the energy bands [C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801 (2005)]. However, spin splitting is allowed in multilayers with an odd number of layers $\left(\ge 3\right)$ because their lattices do not satisfy spatial-inversion symmetry. We show that, in trilayer graphene, in addition to the Kane-Mele term, there is a second type of intrinsic spin-orbit coupling present at low energy near the corner of the Brillouin zone. It introduces a Zeeman-type spin splitting of the energy bands at each valley with an opposite sign of the effective magnetic field in the two valleys. We estimate the magnitude of the effective field to be $\sim 2\text{T}$.

Figure 1

(a) Plan view of the lattice of a single layer of graphene showing inequivalent $A$ and $B$ sites. (b) Schematic side view of the unit cell of bilayer graphene and (c) of Bernal-stacked trilayer graphene, showing the inversion center for bilayer and the mirror plane for trilayer. (d) Sketch of the trilayer energy bands in the vicinity of the valley center, using Eqs. (7, 8), where we use $a_m=a_b=b_m=b_b=0$, ${\alpha }_m={\alpha }_b={\beta }_m=-{\beta }_b\equiv \alpha$ with the value $\alpha ={\gamma }_1/20$ chosen for illustrative purposes.