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Strained graphene with lattice deformations has been demonstrated to giverise to large pseudomagnetic fields and host many exotic properties. Here, wepropose a non-magnetic approach to realize a momentum-dependent out-of-planespin splitting in strained graphene nanoribbons with a moderate spin-orbitcoupling. This unique spin splitting distincts from the well-known Zeeman-typespin splitting and the Rashba-type spin splitting. Our analysis indicates thatthe competition between quantum spin Hall edge states and valley edge states inthe nanoribbon leads to the unique spin splitting. The quantum spin Hall statesat one edge of the nanoribbon are suppressed by the counterpropagating edgemodes induced by the pseudomagnetic field. At the opposite edge, the quantumspin Hall states are not affected at all. Therefore, the degenerate quantumspin Hall states of opposite spin orientation, which propagate at the twoopposite edges of the nanoribbon, are lifted. This result reveals a new methodto manipulate the spin degrees of freedom of electrons.

Spin Splitting Induced by a Competition between Quantum Spin Hall Edge States and Valley Edge States