Dissipation due to pure spin-current generated by spin pumping

Based on spin-dependent transport theory and thermodynamics, we develop a generalized theory of the Joule heating in the presence of a spin current. Along with the conventional Joule heating consisting of an electric current and electrochemical potential, it is found that the spin current and spin accumulation give an additional dissipation because the spin-dependent scatterings inside bulk and ferromagnetic/nonmagnetic interface lead to a change of entropy. The theory is applied to investigate the dissipation due to pure spin-current generated by spin pumping across a ferromagnetic/nonmagnetic/ferromagnetic multilayer. The dissipation arises from an interface because the spin pumping is a transfer of both the spin angular momentum and the energy from the ferromagnet to conduction electrons near the interface. It is found that the dissipation is proportional to the enhancement of the Gilbert damping constant by spin pumping.

Figure 1

Schematic view of the ${\mathrm{F}}_1$/N/${\mathrm{F}}_2$ ferromagnetic multilayer system. The directions of ${\dot{\mathbf{m}}}_1$ and ${\mathbf{m}}_1\times {\dot{\mathbf{m}}}_1$ are indicated by arrows.

Figure 2

Schematic views of the flows of (a) angular momentum and (b) energy from the microwave to the ferromagnetic multilayer, in which L and T define the longitudinal and transverse components with respect to ${\mathbf{m}}_1$.

Figure 3

Dependencies of (a) the dissipation due to pure spin-current, Eq. (19), and (b) the damping ${\alpha }^{\prime }$, Eq. (35), on the cone angle $\theta$.

Figure 4

Examples of the distributions of (a) longitudinal (solid) and transverse (dotted and dashed) spin current densities, (b) longitudinal (solid) and transverse (dotted and dashed) spin accumulations, and (c) bulk dissipations for $\theta ={45}^{\circ }$.