Semiclassical theory of spin transport in magnetic multilayers

Abstract

A semiclassical model of the spin momentum transfer in ferromagnetic film ͑FM͒/normal metal ͑NM͒ structures is presented. It is based on the Landau-Lifshitz equation of motion and the exchange interaction in FM, and on the spin diffusion equation in NM. The internal magnetic field is treated by employing Maxwell's equations. A precessing magnetization in FM creates a spin current which is described by spin pumping proposed by Tserkovnyak et al. In our recent ferromagnetic resonance ͑FMR͒ studies [1][2][3][4] it was shown that the transfer of the spin momentum across ferromagnetic ͑FM͒/normal metal ͑NM͒ interfaces can result in nonlocal interface Gilbert damping GЈ. The generation of spin momentum in magnetic ultrathin films was theoretically described by Tserkovnyak et al. 5 and the effect was called ''spin pumping.'' The presence of a second magnetic layer creates a spin sink. 3,4,6,7 The combination of spin pump and spin sink in the ballistic limit leads to an additional interface Gilbert damping. In this article we extend the spin pump and spin sink mechanisms to the nonballistic electron transport which includes a full treatment of the Landau-Lifshitz ͑LL͒ equation of motion in FM and diffusion equation in NM and Maxwell's equations accounting for a finite penetration of the rf fields.The coordinate system was chosen in such a way that the sample normal is parallel to the z axis. The external dc field, H, lies in the sample plane and is parallel to the y axis, and the internal electromagnetic rf fields are hϭ(h,0,0), e ϭ(0,e,0). The LL equations of motion in FM and NM layers can be written aswhere ␥ is the absolute value of the electron gyromagnetic ratio, M s is the saturation magnetization of FM, G 0 is the intrinsic Gilbert damping, D is the diffusion constant in NM (Dϭv F 2 el /6, v F is the Fermi velocity and el is the electron momentum relaxation time͒, s f is the spin-flip relaxation time, and ␦M N ϭM N Ϫ P h is the excess magnetization in NM, where P is the Pauli susceptibility. The effective field H eff F is derived from the total Gibbs free energy which contains the external fields, magnetocrystaline anisotropies, and exchange interaction. 8 The effective field H eff N in NM contains only the external dc, internal field H, and the demagnetizing field perpendicular to the sample plane. Equations ͑1͒ and ͑2͒ were solved in a small angle approximation using

Scattering of spin current injected in Pd(001)

“…A similar behavior was found for the Au thin films grown on Fe/ GaAs͑001͒. 15 Since the momentum mean free path in our Pd overlayers is larger than the film thicknesses the spin-diffusion theory 16 is not applicable in the interpretation of our results.…”

Spin-pumping-enhanced magnetic damping in ultrathin Cu(001)/Co/Cu and Cu(001)/Ni/Cu films

“…In this work UHV FMR studies have been performed on single crystalline ultrathin Co and Ni films on Cu(0 0 1) with Cu cap layers of variable thickness, which are much thinner than the spin diffusion lengths of these systems [11]. In this thickness regime the interface-related relaxation mechanisms like spinpump effect are dominant [12], especially at first contact between the vacuum side of the FM film and the NM cap layer.…”

Magnetization dynamics in the presence of pure spin currents in magnetic single and double layers in spin ballistic and diffusive regimes

“…The conservation of total angular momentum leads to an additional damping in the F1 layer, which follows the phenomenology of Gilbert damping; i.e., the damping effective field is proportional to M ជ ϫ ‫ץ‬M ជ / ‫ץ‬t. In addition the strength scales inversely with the ferromagnetic film thickness 8,9 as a consequence of the interface nature of spin pumping,…”