  Electric transport of a zigzag graphene nanoribbon through a step-like
potential or a barrier potential is investigated by using the recursive Green's
function method. In the case of the step-like potential, we demonstrate
numerically that scattering processes obey a selection rule for the band
indices when the number of zigzag chains is even; the electrons belonging to
the ``even'' (``odd'') bands are scattered only into the even (odd) bands so
that the parity of wavefunctions is preserved. The so-called valley-valve
effect can be explained by this selection rule. In the case of the barrier
potential, by tuning the barrier height to be an appropriate value, we show
that it can work as the ``band-selective filter'', which transmits electrons
selectively with respect to the indices of the bands to which the incident
electrons belong. Finally, we suggest that this selection rule can be observed
in the conductance by applying two barrier potentials.

Mesoscale and Nanoscale Physics