  On the basis of tight-binding lattice model, the edge states of monolayer and
bilayer graphene nanoribbons (GNRs) with different edge terminations are
studied. The effects of edge-hopping modulation, spin-orbital coupling (SOC),
and bias voltage on bilayer GNRs are discussed. We observe the following: (i)
Some new extra edge states can be created by edge-hopping modulation for
monolayer GNRs. (ii) Intralayer Rashba SOC plays a role in depressing the band
energy gap $E_g$ opened by intrinsic SOC for both monolayer and bilayer GNRs.
An almost linear dependent relation, i.e., $E_g\sim \lambda_R$, is found. (iii)
Although the bias voltage favors a bulk energy gap for bilayer graphene without
intrinsic SOC, it tends to reduce the gap induced by intrinsic SOC. (iv) The
topological phase of the quantum spin Hall effect can be destroyed completely
by interlayer Rashba SOC for bilayer GNRs.
