Robust electronic states due to inhomogeneous spin-orbit couplings in graphene heterostructure

Recent experimental progress in designing electronic states in layered heterostructures raises questions about inducing topological states in inhomogeneous systems. In particular, we investigate numerically and analytically several types of inhomogeneous spin-orbit coupling terms in graphene. We begin by studying domain walls in Kane-Mele spin-orbit coupling, addressing the robustness of gapless helical electronic modes and the emergence of gapped Landau-level-like modes bound to the domain wall. Our numerical analysis shows that the domain-wall modes are gapped by Rashba spin-orbit coupling of any strength, even while the Kane-Mele topological bulkgap remains open. The domain-wall states are also gapped by randomness in the Kane-Mele term even without breaking the conserved spin component.

On the other hand, the combination of homogeneous Valley-Zeeman and Rashba types of spin-orbit coupling also opens a bulkgap, which is not topological as in the case of Kane-Mele coupling. Nevertheless, we find that should one of these two spin-terms have a sign-changing domain wall, it binds electronic modes which are robust to inclusion of the competing Kane-Mele term, as long as the bulkgap remains open.