  Two-dimensional Floquet systems consisting of irradiated valley-polarized
metal are investigated. For the corresponding static systems, we consider two
graphene models of valley-polarized metal with either a staggered sublattice or
uniform intrinsic spin-orbital coupling, whose Dirac point energies are
different from the intrinsic Fermi level. If the frequency of irradiation is
appropriately designed, the largest dynamical gap (first-order dynamical gap)
opens around the intrinsic Fermi level. In the presence of the irradiation, two
types of edge state appear at the zigzag edge of semi-infinite sheet with
energy within the first-order dynamical gap: the Floquet edge states and the
strongly localized edge states. In narrow zigzag nanoribbons, the Floquet edge
states are gapped out by the finite size effect, and the strongly localized
edge states remain gapless. As a result, the conducting channels of the
nanoribbons consist of the strongly localized edge states. Under the first and
second model, the strongly localized edge states carry one-way spin polarized
and one-way charge current around the intrinsic Fermi level, respectively.
Thus, the narrow zigzag nanoribbons of the first and second model have
asymmetric spin and charge transmission rates, respectively. Quantum-transport
calculations predict sizable pumped currents of charge and spin, which could be
controlled by the Fermi level.
