Tensor damping in metallic magnetic multilayers

The mechanism of spin pumping, described by Tserkovnyak et al. [Phys. Rev. B 67, 140404 (2003)], is formally analyzed in the general case of a magnetic multilayer consisting of two or more metallic ferromagnetic (FM) films separated by normal-metal (NM) layers. It is shown that the spin-pumping-induced dynamic coupling between FM layers modifies the linearized Gilbert equations in a way that replaces the usual local, scalar Gilbert damping constant with a nonlocal matrix of Cartesian damping tensors. As an example, explicit analytical results are obtained for a five-layer (spin valve) of form NM/FM/NM^'/FM/NM. These are compared with earlier well-known results of Tserkovnyak et al. for the related three-layer FM/NM/FM, which are shown to have singled out the diagonal element of the local damping tensor along the axis normal to the plane of the two magnetization vectors. For spin-valve devices of technological interest, the influence of tensor damping on thermal noise fluctuations and/or spin-torque critical currents is shown to necessarily be coupled to the nonlocal tensor properties of the magnetostatic interaction as well.