Spin-polarized and valley helical edge modes in graphene nanoribbons

We investigate the electronic properties of zigzag-terminated graphene nanoribbons in the presence of a staggered sublattice potential. We show that due to the edge ferromagnetism, spin-polarized dispersive edge modes with well-defined valley indices can appear inside the bulk band gap opened by the inversion symmetry breaking. These edge modes are helical with respect to their valley indices, hence are robust against scattering from smooth disorder potentials. We further propose a concrete system with a zigzag graphene nanoribbon grown on top of a hexagonal boron-nitride substrate to realize such edge modes. These edge states could be utilized as perfect spin filters or analyzers in spintronics applications.

Figure 1

Schematic figure of a zigzag-edged graphene nanoribbon along $y$ axis. $A$ ($\circ$) and $B$ ($•$) sublattices are subjected to staggered potentials: $U_A=+\Delta /2$ and $U_B=-\Delta /2$. The edge magnetization $M$ is considered only at the outmost boundary atoms labeled as 1 and $N$.

Figure 2

Evolution of the band structure of the zigzag-edged graphene nanoribbon with a fixed width $N=800$. (a) When the staggered sublattice potential $\Delta =0.4$ is applied, an bulk gap is opened, and the flat-bands are doubly-degenerate; (b)–(d) The edge magnetism is further switched on with $M=0.6,1.0,$ and 1.4, respectively. The flat bands become spin split: spin-up edge band bends upward, while spin-down edge band bends downward. The green solid (red dashed) curves represent the edge states from the left (right) boundary.

Figure 3

Schematic plot of the edge states propagation directions. (a) When the spin polarizations on the two boundaries are parallel, spin-up (spin-down) polarized valley helical edge states propagate along the right (left) boundary; (b) when the spin polarizations on the two boundaries are antiparallel, the edge states at each boundary have the same spin polarization.

Figure 4

Average conductance $⟨G⟩$ in units of $e^2/h$ versus the effective disorder strength $W$ for scalar-type disorders at $\Delta =0.4$ and $M=0.6$ for different Fermi energies $\varepsilon =-0.1$ (square), 0 (circle), 0.1 (triangle), respectively. (a) For long-range disorder ($\xi =4$), the edge states can be exactly quantized for disorder strength up to $W=0.8$; (b) for short-range disorder ($\xi =0$), the conductance quickly decreases as $W$ increases above $0.1$. $20000$ ensembles are collected for each data point in the figure. Note that in this calculation we only consider the ferromagnetism at one boundary to emphasize the disorder effect of only one conducting channel.

Figure 5

(a) Atomic structure of the hydrogen-terminated zigzag-edged graphene nanoribbons on top of a single layer of hexagonal boron-nitride. The red square shows a supercell. Upper: side view; lower: top view. (b) Band structure of spin antiparallel configurations (between two boundaries). In the bulk energy gap (the narrow energy window of the projected bulk band structure shown in light gray, $\Delta$ around $78\text{meV}$), only the spin-up states exist. A very small gap appears because of the weak interaction between the edges states on opposite boundaries, which is a finite-size effect. (c) Band structure for spin parallel case. Spin-up and spin-down states coexist in the gap. (d) When a voltage bias of $0.27\text{V}$ is applied transversely, the upper (lower) edge states are shifted upward (downward), leaving only the spin-down states in the gap. Bands in red and green color represent the spin-up and the spin-down edge bands, respectively.

Figure 6

Spin polarization for the carbon atoms inside a supercell. (a) For the spin-parallel configuration, the spin polarization is symmetric about the ribbon center; (b) for the spin antiparallel configuration, the spin polarization is antisymmetric about the ribbon center. Note that the amplitude of the spin polarization is exponentially decreased accompanying with oscillations between sublattice sites.