Magnetoresistance of a multilayered film in a current-perpendicular-to-plane geometry

The method to calculate the electrical resistance of a multilayered film consisting of any finite number of ferromagnetic and nonmagnetic layers for a given collinear magnetic configuration is proposed. The problem is solved for the geometry when the current flows perpendicular to the film plane. The method is based on the description of the resistance of a multilayered film by the differential equation for the current polarization which satisfies two boundary conditions at the interface between adjacent ferromagnetic layers. It is demonstrated that the description of the stack resistance depending on the mutual collinear orientation of ferromagnetic layers can be reduced to the routine mathematical calculations. Also it is shown that in the case of tunneling barriers between ferromagnetic layers one has to determine the barrier resistance asymmetry coefficient in order to calculate magnetoresistance of a multilayer film as a function of ferromagnetic layer collinear magnetic configuration. It is essential that these asymmetry parameters cannot be obtained directly from experiment. In the present work, the expressions for these parameters are derived within Slonczewski model from similar parameters in a ferromagnetic-tunneling barrier-nonmagnetic stack which can be measured experimentally. The obtained results allow one to evaluate the magnetoresistance values in a multilayered film consisting of any finite number of ferromagnetic layers.