Spin-signal propagation in time-dependent noncollinear spin transport

Using a macroscopic analysis, we investigate the signal propagation in time-dependent noncollinear spin transport through magnetic multilayers. As in the collinear case, we find a wavelike character in addition to its diffusive character, which allows one to extract a finite spin-signal-propagation velocity. We numerically study the dynamics of a pure spin current pumped into a nonmagnetic layer by a precessing magnetization in an adjacent ferromagnetic layer for precession frequencies ranging from GHz to THz. The wave character of spin transport produces deviations from pure spin-diffusion dynamics for modulation frequencies of several 10 GHz. In this frequency range, the polarization of the pumped spin current is still approximately parallel to the injected spin at the interface at all times. Above 100 GHz, the wave character of the spin current becomes obvious because the polarization is no longer parallel to the injected spin.

Figure 1

Snapshots of the spin-current density ${\vec{ȷ}}_s^{\perp }$ at $t=5/4$ $T_a$, $5/4T_b$, and $5/4T_c$, for the frequencies, ${\nu }_a$, ${\nu }_b$, and ${\nu }_c$, respectively (see text). ${\vec{ȷ}}_s^{\perp }$ is plotted as vector starting from its $x$ coordinate.

Figure 2

Snapshots of the spin density ${\vec{n}}_s^{\perp }$ for the same parameters as in Fig. 1.

Figure 3

Snapshots of ${\vec{ȷ}}_s^{\perp }$ at $t=T_d$, $2T_d$, and $3T_d$, where $T_d=0.12\text{ps}$. The solid (dashed) curve is for $j_s^x$ $\left(j_s^y\right)$.

Figure 4

Snapshot of ${\vec{ȷ}}_s^{\perp }$ (plotted as vector) at $t=T_d$.