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We investigate the formation of edge states in graphene ribbons and flakeswith proximity induced valley-Zeeman and Rashba spin-orbit couplings in thepresence of a perpendicular magnetic field $B$. Two types of edges statesappear in the spin-orbit gap at the Fermi level at zero field: stronglylocalized pseudohelical (intervalley) states and weakly localized intravalleystates. We show that if the magnetic field is stronger than a crossover field$B_c$, which is a few mT for realistic systems such as graphene/WSe$_2$, onlythe pseudohelical edge states remain in zigzag graphene ribbons; theintravalley states disappear. The crossover is directly related to the closingand reopening of the bulk gap formed between nonzero Landau levels. Remarkably,in finite flakes the pseudohelical states undergo perfect reflection at thearmchair edges if $B > B_c$, forming standing waves at the zigzag edges. Thesestanding waves comprise two counterpropagating pseudohelical states, so whilethey carry no charge current, they do carry (pure) spin current.

physical review. b./physical review. b

Edge states in proximitized graphene ribbons and flakes in a  perpendicular magnetic field: emergence of lone pseudohelical pairs and pure  spin-current states