Electric transport of a zigzag graphene nanoribbon through a step-like potential or a potential barrier is studied by using the recursive Green's function method. The results for a step-like potential show that scattering processes in a zigzag graphene nanoribbon obey a following selection rule: when the number of zigzag chains N is even, electrons in the band m are only scattered into the bands m+2n, where n is an integer. According to this selection rule, a step-like potential blocks the current when the potential height exceeds the incident energy as long as only the low-energy region is treated. Then, replacing a step-like potential with a potential barrier, we also show that it can play the role of a "band-selective filter".
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