Intrinsic and substrate induced spin-orbit interaction in chirally stacked trilayer graphene

We present a combined group-theoretical and tight-binding approach to calculate the intrinsic spin-orbit coupling (SOC) in ABC stacked trilayer graphene. We find that compared to monolayer graphene (S. Konschuh, M. Gmitra, and J. Fabian [Phys. Rev. B 82, 245412 (2010)]), a larger set of $d$ orbitals (in particular the $d_{z^2}$ orbital) needs to be taken into account. We also consider the intrinsic SOC in bilayer graphene, because the comparison between our tight-binding bilayer results and the density functional computations of S. Konschuh, M. Gmitra, D. Kochan, and J. Fabian [Phys. Rev. B 85, 115423 (2012)] allows us to estimate the values of the trilayer SOC parameters as well. We also discuss the situation when a substrate or adatoms induce strong SOC in only one of the layers of bilayer or ABC trilayer graphene. Both for the case of intrinsic and externally induced SOC we derive effective Hamiltonians which describe the low-energy spin-orbit physics. We find that at the $K$ point of the Brillouin zone the effect of Bychkov-Rashba type SOC is suppressed in bilayer and ABC trilayer graphene compared to monolayer graphene.

Figure 1

Lattice and band structure of ABC trilayer graphene. (a) Lattice structure, where atoms on different layers are indicated with different symbols. ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$ are the two lattice vectors. (b) Schematic side view of the unit cell with the most important hopping amplitudes. (c) Schematic of the Brillouin zone with reciprocal lattice vectors ${\mathbf{b}}_1$, ${\mathbf{b}}_2$, the high-symmetry points $\Gamma$, $K$, and $K^{\prime }$, and the momentum $\mathbf{p}=(p_x,p_y)$ measured from $K$. (d) Schematic of the band structure at the $K$ point of the Brillouin zone. The velocity $v_0$ is given by $v_0=\sqrt{3}/2a{\gamma }_0/\hslash$, where $a$ is the lattice constant: $a=|{\mathbf{a}}_1|$.

Figure 2

Low-energy bands of ABC trilayer graphene at the $K$ point as a function of $p_y$ for $p_x=0$: using the full $\mathbf{k}·\mathbf{p}$ Hamiltonian with zero spin-orbit coupling (solid black), with finite spin-orbit coupling (red dashed), and using the effective Hamiltonian ${\widehat{H}}_{ABC}^{\mathrm{eff}}$ (dashed-dotted blue). The parameters $v_i$ were taken from Ref. 11 and we have chosen ${\lambda }_{1/2}={\lambda }_1=0.2{\gamma }_2$.