Journal
PHYSICAL REVIEW B
Volume 80, Issue 6, Pages -Publisher
AMER PHYSICAL SOC
DOI: 10.1103/PhysRevB.80.064412
Keywords
damping; ferromagnetism; magnetic multilayers; magnetic thin films; magnetisation; spin dynamics; spin valves
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The mechanism of spin pumping, described by Tserkovnyak [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 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.
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