Edge states and enhanced spin-orbit interaction at graphene/graphane interfaces

We study the electronic properties of interfaces between graphene and graphane, a hydrogenated version of graphene. It is shown that these interfaces are useful for creating an effective edge for the $\pi$-electrons of graphene. If the interface is oriented along a zigzag direction, edge states are found. We consider two different interface types, corresponding to usual zigzag and bearded graphene edges. It is shown that, because of a broken symmetry, the spin-orbit interaction is strongly amplified by the graphene/graphane interface and that the edge states are particularly susceptible to this amplification. As an application, we propose a device which is capable of converting spin polarizations to valley polarizations and vice versa. Exploiting the amplification of the spin-orbit interaction, this conversion may be performed at temperatures near one Kelvin. We show that these edge states give rise to quantum spin and/or valley Hall effects.

Figure 1

(Color online) $\alpha$- and $\beta$-interfaces in a graphene/graphane heterostructure. The large (red) spheres represent the carbon atoms and the small (blue) spheres the hydrogen atoms. The brighter spheres correspond to the graphene region while the darker spheres correspond to graphane.

Figure 2

(Color online) The band structure of a graphane-terminated $\alpha \beta$-ribbon with $W=10\text{nm}$. The dashed horizontal line is a typical Fermi energy ${ϵ}_F$ for the QVHE regime chosen such that ${\xi }_{k_F}⪡W/2$. The gray bands correspond to exponentially localized states at the outer graphane edge, which are not important here. The full curves are results of numerical calculations within the extended tight-binding model. The dashed (blue) curves show the effective model dispersion [Eq. (5)]. The inset shows the conductance $G$ of the graphene region as a function of ${ϵ}_F$.

Figure 3

(Color online) SOI-induced spin-splitting of the edge states. The solid black (blue) curve shows the absolute value of the spin-splitting $\Delta {ϵ}_{SO}\left(k\right)$ of the edge states calculated from the extended tight-binding model with the full SOI. This calculation has been done for a 10 nm wide $\alpha \beta$-ribbon. The dashed (red) line shows the spin-splitting calculated from the effective Hamiltonian (9). The spin-splitting at $K,K^{\prime }$ which is slightly larger than the spin-orbit splitting in bulk graphene is due to finite-size effects. The gray (green) curve shows the spin-orbit splitting of the edge states without graphene termination, scaled up by a factor of 100. This calculation has also been done within the extended tight-binding model (without the graphane regions), as described in Appendix .

Figure 4

(Color online) Energy dispersion of the edge states of a 2 nm wide $\alpha \beta$-ribbon near the $K$-valley. The two bands have opposite spin direction. The dashed line indicates a typical ${ϵ}_F$. The inset shows a larger region of the band structure. The gray lines are bulk graphane states and states at the outer graphane edge which are not of interest here.

Figure 5

(Color online) Definitions in the simplified interface model. The gray (green) dots represent the carbon atoms of the -C-H groups and the small black (blue) dots represent the hydrogen atoms. The number labels indicate the $n_1$ coordinate of the carbon atoms in the graphene region.

Figure 6

(Color online) Relative orientation of the carbon $s$- and $p$-orbitals. ${\mathbf{r}}_b$ is the bond vector, connecting the positions of the two atoms.

Figure 7

(Color online) Bulk band structure of graphane calculated from our tight-binding model [Eqs. (C12, C15)].

Figure 8

(Color online) Different band structures for $z_J=0$ (red), $z_J=z_0/2$ (green) and $z_J=z_0$ (blue). The graphene part of the $\alpha \beta$- ribbon is 21 nm wide in this calculation and the graphane terminations are 2.1 nm wide. The bulk states and the states at the outer graphane edges are not significantly affected by $z_J$. The corresponding energy bands lie on top of each other for the three cases.

Figure 9

(Color online) Spin-orbit strength ${\Gamma }_{\theta }\left(k\right),\theta =i,R$ from the effective model (dashed lines) and from our extended tight-binding model (solid lines). The gray (green) line represents the Rashba term of the spin-orbit Hamiltonian and the nearby dashed lines represent the corresponding terms from the effective models. The black (blue) line represents the intrinsic term of the spin-orbit Hamiltonian. The effective Hamiltonian of the $\alpha$-edge state is Eq. (F2). The effective Hamiltonian of the $\beta$-edge state on the left-hand side (the region labeled by linear fit) is given by the linear version Eq. (F3) while on the right-hand side (the region labeled by 5th order fit), Eq. (F4) is shown. Part (a) shows a calculation for a flat interface $\left(z_J=0\right)$ and part (b) shows a calculation for a maximally tilted interface $\left(z_J=z_0\right)$.