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A finite-size effect leads to more electrons to occupy high-energy states in a graphene nanoribbon. Therefore, the thermionic-emission (TE) current density of a metal/graphene nanoribbon changes with its width when the graphene nanoribbon is a semiconductor. This letter demonstrates that the TE current density increases rapidly with the decreasing width of the graphene nanoribbon. It is also found that the finite-size effect on TE current density is more significant at low temperatures.