Protected Pseudohelical Edge States in ${\mathbb{Z}}_2$-Trivial Proximitized Graphene

We investigate topological properties of models that describe graphene on realistic substrates which induce proximity spin-orbit coupling in graphene. A ${\mathbb{Z}}_2$ phase diagram is calculated for the parameter space of (generally different) intrinsic spin-orbit coupling on the two graphene sublattices, in the presence of Rashba coupling. The most fascinating case is that of staggered intrinsic spin-orbit coupling which, despite being topologically trivial, ${\mathbb{Z}}_2=0$, does exhibit edge states protected by time-reversal symmetry for zigzag ribbons as wide as micrometers. We call these states pseudohelical as their helicity is locked to the sublattice. The spin character and robustness of the pseudohelical modes is best exhibited on a finite flake, which shows that the edge states have zero $g$ factor, carry a pure spin current in the cross section of the flake, and exhibit spin-flip reflectionless tunneling at the armchair edges.

Figure 1

Schematics of proximity induced properties in graphene. (a) Graphene placed on a symmetry-breaking substrate. (b) Hopping parameters used in our model. Sublattice $A$ is represented as empty dots and sublattice $B$ as filled dots. Symbols colored in red (blue) denote spin-up (spin-down) characteristics. Dashed red lines encode spin-up intrinsic SOC hoppings (signs indicated by arrows), for the uniform case of ${\lambda }_I^A={\lambda }_I^B$, within a hexagon. Helical states and their velocity directions are indicated by long arrows. Panel (c) shows reciprocal $K$ and $K^{\prime }$ directions with respect to the lattice. Intrinsic SOC hoppings are shown for staggered intrinsic SOC, ${\lambda }_I^A=-{\lambda }_I^B$, by red dashed lines. Solid (dashed) gray arrows indicate valley edge states located in the $\kappa =1$ ($-1$) valley. Red and blue arrows show pseudohelical states carrying finite spin current along the ribbon.

Figure 2

Spectra of 100 unit cells wide zigzag ribbons. The color code in (a) and (b) for the spinless case denotes the sublattice expectation value: red for sublattice $A$ and blue for sublattice $B$. The spectrum of the spinful case with additional Rashba SOC in (c) and (d) is color coded with the spin expectation value: red for spin-up and blue for spin-down. Left-hand column shows the uniform case, ${\lambda }_I^A={\lambda }_I^B$, and the right-hand column shows the staggered case with strong spin-valley locking, ${\lambda }_I^B=-{\lambda }_I^A$.

Figure 3

${\mathbb{Z}}_2$ phase space and bulk gap landscape of graphene in the ${\lambda }_I^A-{\lambda }_I^B$ plane. Color denotes the size of the gap in graphene. Solid lines are analytical expressions for a (global) bulk graphene gap closing, which separate trivial ($\mathbb{0}$) and nontrivial ($\mathbb{1}$) phases from each other. Orbital parameters and Rashba SOC are the same as in the text.

Figure 4

Finite-sized zigzag ribbons and flakes of width of ten zigzag unit cells. Left-hand column is for the case of ${\lambda }_I^A={\lambda }_I^B$ and right-hand column for ${\lambda }_I^B=-{\lambda }_I^A$. Panels (a) and (b) show the band structure of an infinite zigzag ribbon over half of the Brillouin zone with spin expectation values as color code (up in red, down in blue). Panels (c) and (d) show finite flakes of length of 100 zigzag cells and properties of a state that lies at energy indicated by dashed lines in (a) and (b), respectively. Empty dots denote the lattice, full dots indicate the site expectation value color coded for spin polarization, and black arrows show probability bond currents. Orbitals in the middle areas of (c) and (d) have been removed, acting as short-range scatterers. Images of flakes have been cut due to size constraints.