Spin wave excitations of a magnetic pillar with dipolar coupling between the layers

It is demonstrated analytically that the spectrum of small-amplitude spatially uniform magnetization excitations in an in-plane magnetized magnetic pillar with two ferromagnetic layers coupled by dipole-dipole interaction can be approximately described by the traditional Kittel formula with reduced saturation magnetization and effective anisotropy field. The spectrum consists of a quasi-symmetric and a quasi-antisymmetric mode, and the apparent reduction of saturation magnetization for the quasi-symmetric mode (≤50%) is much larger than that for the quasi-antisymmetric mode (≤10%). The effect of dynamic dipolar coupling between the nano-pillar layers could be partly responsible for the apparent reduction of static magnetization seen in many spin-torque experiments performed on magnetic nano-pillars.