Enhanced dc spin pumping into a fluctuating ferromagnet near $T_C$

A linear-response formulation of the dc spin pumping, i.e., a spin injection from a precessing ferromagnet into an adjacent spin sink, is developed in view of describing many-body effects caused by spin fluctuations in the spin sink. It is shown that when an itinerant ferromagnet near $T_{\mathrm{C}}$ is used as the spin sink, the spin pumping is largely increased owing to the fluctuation enhancement of the spin conductance across the precessing ferromagnet/spin sink interface. As an example, the enhanced spin pumping from yttrium iron garnet into nickel palladium alloy ($T_C\simeq 20$ K) is analyzed by means of a self-consistent renormalization scheme, and it is predicted that the enhancement can be as large as tenfold.

Figure 1

Schematic view of a system considered in the present paper for the FMR-driven spin pumping. A bilayer of a spin injector (SI) and a spin sink (SS) is placed in an external static magnetic field ${\mathit{H}}_0$. The wavy line (green) represents an external ac magnetic field ${\mathit{h}}_{\mathrm{rf}}$ which induces precessional motion of localized spins in the SI.

Figure 2

Feynman diagram representing the process of the dc spin pumping. The red and black solid lines represent conduction-electron Green's function and uniform-mode magnon Green's function, respectively. The transverse susceptibility of itinerant spin density, ${\chi }_{\mathit{k}}\left(\omega \right)$, is defined as a propagator of particle-hole pairs. The green wavy line represents external ac magnetic field ${\mathit{h}}_{\mathrm{rf}}$.

Figure 3

Diagrammatic representation of renormalized Green's function of uniform-mode magnon, $G_0^R\left(\omega \right)$, and the corresponding self-energy, ${\Sigma }_0^R\left(\omega \right)$. The black line with arrow means the bare magnon propagator $G_0^{\left(0\right)R}\left(\omega \right)$.

Figure 4

Magnetization $m$ as a function of reduced temperature $(T-T_C)/T_C$ for $H_0=0$ G (solid line), $H_0=1000$ G (dashed line), and $H_0=10000$ G (dash-dotted line). The data is normalized by its value at $T=0$.

Figure 5

Pumped spin current $I_s^{\mathrm{pump}}$ [Eq. (16)] or additional Gilbert damping $\delta \alpha$ [Eq. (21)] at the resonance condition ${\Omega }_{\mathrm{rf}}=\gamma H_0$, calculated for a fluctuating SS (NiPd alloy) as a function of reduced temperature $(T-T_C)/T_C$ with ${\Omega }_{\mathrm{rf}}{\tau }_{\mathrm{sf}}=0.1$ (solid line), $0.2$ (dashed line), and $0.3$ (dash-dotted line). All the data are normalized by their values at $T/T_C=0.5$. Inset: Inverse spin Hall voltage used to electrically detect the enhanced spin pumping as a function of temperature, calculated using data from Ref. [54]. The dashed curve is given by the enhanced spin pumping predicted in this work, and the deviation from the dashed curve comes from the anomaly in the inverse spin Hall effect reported in Ref. [54]. For more details, see the main text.

Figure 6

(a) Effective longitudinal damping coefficient ${\Gamma }_{\mathrm{eff}}^z$ as a function of reduced temperature. Inset: Temperature dependence of equilibrium spin ${\tilde{S}}_{\mathrm{eq}}$. (b) Effective transverse damping coefficient ${\Gamma }_{\mathrm{eff}}^{+-}$ as a function of reduced temperature. In both figures, $b_{\mathrm{GL}}=1.0$ is used.

Figure 7

Temperature dependence of pumped spin current near $T_C$. The data is normalized by its value at $T=T_C$, and $b_{\mathrm{GL}}=1.0$ is used.