Induced magnetization spiral in a nonmagnetic metal sandwiched between two ferromagnets

Calculation of the magnetic moment induced in a non-magnetic metal, sandwiched between two ferromagnets with magnetizations at an arbitrary angle, is reported. It is found that the induced magnetization rotates along a complex three-dimensional spiral and can undergo many complete 360 deg. rotations. A simple free-electron model is used to derive an analytic formula for the twist angle phi inside the spacer. This demonstrates that, contrary to the behavior of magnetization inside a domain wall in a ferromagnet, phi varies non-uniformly inside the spacer and exhibits plateaus of almost constant rotation separated by regions of sharp rotations by large angles. The calculation is extended to the case of a realistic Co/Cu/Co(0 0 1) trilayer described by s, p, d tight-binding bands fitted to an ab initio band structure. An analytic formula for the components of the induced moment (and hence, for phi) is derived using the stationary phase approximation. Its validity is tested against a fully numerical calculation using the same band structure. The formula shows that the components of the induced magnetization each oscillate with a predominant short period determined by the Cu Fermi surface neck extrema. The twist angle again displays the same remarkable behavior as in the free-electron model and depends in an intricate manner on geometrical properties of the spacer Fermi surface as well as on the degree of confinement of carriers in the spacer quantum well