  We theoretically study the magnetoresistance of a CPP-GMR system with current
confined paths (CCP) in the framework of Valet-Fert theory. The continuity
equations for charge and spin currents are numerically solved with the
three-dimensional CCP geometry by use of finite element method. It is confirmed
that the MR ratio is enhanced by the CCP structure, which is consistent with
the experimental results. Moreover, we find that there exists a certain contact
width which maximize the MR ratio. We show that the contact width which
maximize the MR ratio is well described by the effective resistance matching.
