We give a criterion for a nef divisor D to be semi-ample on a Calabi-Yau threefold X when D3 = 0 = c2 (X)center dot D and c3 (X) 6= 0. As a direct consequence, we show that on such a variety X, if D is strictly nef and.(D) 6= 1, then D is ample; we also show that if there exists a Cariter divisor D 6= 0 in the boundary of the nef cone of X, then X contains a rational curve when its topological Euler characteristic is not 0