Trigonal warping and anisotropic band splitting in monolayer graphene due to Rashba spin-orbit coupling

We study the electronic band structure of monolayer graphene when Rashba spin-orbit coupling is present. We show that if the Rashba spin-orbit coupling is stronger than the intrinsic spin-orbit coupling, the low-energy bands undergo trigonal-warping deformation and that for energies smaller than the Lifshitz energy, the Fermi circle breaks up into separate parts. The effect is very similar to what happens in bilayer graphene at low energies. We discuss the possible experimental implications, such as threefold increase in the minimal conductivity for low electron densities, anisotropic, wave-number-dependent spin splitting of the bands, and the spin-polarization structure.

Figure 1

(Color online) The $\mathbf{k}$ dependence and spin polarization of the energy bands: (a) $E_1^{+}\left(\mathbf{k}\right)$ and (b) $E_2^{+}\left(\mathbf{k}\right)$ around the $\mathbf{K}$ point for $\beta =0.034$. Solid lines show the constant-energy contours, the arrows indicate the direction of the spin at certain points in the BZ. The wave-vector components $k_x$ and $k_y$ are in units of $3{\lambda }_R/\left(2\hslash v_F\right)$. The inset (a) shows a closeup of $E_1^{+}\left(\mathbf{k}\right)$ in the vicinity of the $\mathbf{K}$ $\left({\mathbf{K}}^{\prime }\right)$ point. The pocket structure has a $2\pi /3$ rotational symmetry and the distance of the pocket minima from the $\mathbf{K}$ $\left({\mathbf{K}}^{\prime }\right)$ point is $k_{\text{SO}}$ (see the text). The dashed line in (b) is a circle, it indicates that the isoenergy lines of $E_2^{+}\left(\mathbf{k}\right)$ are also slightly distorted. [(c)–(f)] A cut through the energy bands for $k_x=0$ and (c) ${\Delta }_{\text{SO}}=0$, (d) ${\Delta }_{\text{SO}}⪡{\lambda }_R$, (e) $2{\Delta }_{\text{SO}}\lesssim 3{\lambda }_R$, and (f) $2{\Delta }_{\text{SO}}>3{\lambda }_R$. Solid (blue) lines denote electron bands while dashed (red) lines correspond to valance bands.

Figure 2

(a) The spin splitting along the $\overline{\Gamma \text{K}}$ line of the BZ using the Hamiltonian of Eq. (3). The inset shows the comparison of the experimental data from Ref. 6 and the calculated band splitting with finite TW (solid line) and with no TW (constant dashed line). (b) The band splitting along $\overline{\Gamma \text{M}}$ for ${\lambda }_{R_1}=80\text{meV}$, ${\lambda }_{R_2}=60\text{meV}$, ${\lambda }_{R_3}=40\text{meV}$, and ${\lambda }_{R_4}=10\text{meV}$ obtained from Hamiltonian (3). Vertical dashed line indicates the minimum of the curves. The TB hopping parameter was ${\gamma }_0=3\text{eV}$.