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

TL;DR

This paper studies how edge states in proximitized graphene ribbons and flakes behave under a perpendicular magnetic field, revealing the emergence of pure spin-current states and the conditions for their existence.

Contribution

It demonstrates that a magnetic field above a certain threshold causes only pseudohelical edge states to persist, leading to pure spin-current states in graphene nanostructures.

Findings

Pseudohelical states dominate at high magnetic fields.

Standing waves of pseudohelical states carry pure spin current.

Bulk gap closing relates to the disappearance of intravalley states.

Abstract

We investigate the formation of edge states in graphene ribbons and flakes with proximity induced valley-Zeeman and Rashba spin-orbit couplings in the presence of a perpendicular magnetic field $B$. Two types of edges states appear in the spin-orbit gap at the Fermi level at zero field: strongly localized pseudohelical (intervalley) states and weakly localized intravalley states. 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$_2$, only the pseudohelical edge states remain in zigzag graphene ribbons; the intravalley states disappear. The crossover is directly related to the closing and reopening of the bulk gap formed between nonzero Landau levels. Remarkably, in finite flakes the pseudohelical states undergo perfect reflection at the armchair edges if $B > B_c$, forming standing waves at the zigzag edges. These standing waves comprise two counterpropagating pseudohelical states, so while they carry no charge current, they do carry (pure) spin current.