Numerical study of magnetoresistance for currents perpendicular to planes in spring ferromagnets

Magnetoresistance (MR) for currents perpendicular to planes is calculated within the ballistic limit for spring ferromagnets in which an artificial domain wall is formed by the external magnetic fields. We consider two contributions to the MR: one is caused by a twisting of the magnetization and the other is due to a mismatch of the electronic structure between the two ferromagnets comprising the spring ferromagnets. We show that the resulting MR may show a nonmonotonic dependence on the width of the domain walls and can be either positive or negative according to the magnitude of these two contributions. We further show that oscillatory behavior appears in the MR when the soft ferromagnet is sandwiched between hard ferromagnets.