Tensor damping in metallic magnetic multilayers

TL;DR

This paper analyzes spin-pumping in metallic magnetic multilayers, revealing that damping is nonlocal and tensorial, which impacts the understanding of magnetic dynamics and device performance.

Contribution

It introduces a method to calculate nonlocal damping tensors in multilayers, extending previous scalar damping models to tensor form for more accurate descriptions.

Findings

Damping becomes a nonlocal tensor in multilayers.

Explicit analytical results for a 5-layer spin-valve.

Tensor damping influences thermal noise and spin-torque critical currents.

Abstract

The mechanism of spin-pumping, described by Tserkovnyak et al., 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 scalar Gilbert damping constant with a nonlocal matrix of Cartesian damping tensors. The latter are shown to be methodically calculable from a matrix algebra solution of the Valet-Fert transport equations. As an example, explicit analytical results are obtained for a 5-layer (spin-valve) of form NM/FM/NM'/FM/NM. Comparisons with earlier well known results of Tserkovnyak et al. for the related 3-layer FM/NM/FM indicate that the latter inadvertently hid the tensor character of the damping, and instead 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 the tensor components of the damping on thermal noise or spin-torque critical currents are strongly weighted by the relative magnitude of the elements of the nonlocal, anisotropic stiffness-field tensor-matrix, and for in-plane magnetized spin-valves are generally more sensitive to the in-plane element of the damping tensor.