  It was known that the Sierpi\'{n}ski carpets can appear as the Julia sets in
the families of some rational maps. In this article we present a criterion that
guarantees the existence of the carpet Julia sets in some rational maps having
exactly one fixed (super-) attracting or parabolic basin. We show that this
criterion can be applied to some well known rational maps, such as McMullen
maps and Morosawa-Pilgrim family. Moreover, we give also some special examples
whose Julia sets are Sierpi\'{n}ski carpets.
