Signal propagation in time-dependent spin transport

This paper analyzes theoretically the signal propagation in spin transport by modulating the current passing through magnetic multilayers. Using a macroscopic description of spin transport based on the dynamical Boltzmann equation, we show that time-dependent spin transport possesses a wavelike character that leads to modifications of pure spin-diffusion dynamics. In particular, the wavelike characteristics allow one to extract a finite spin signal-propagation velocity.

Figure 1

Variation of $l_d/{\lambda }_{s,z}$ and $\lambda /{\lambda }_{s,z}$ with $\omega {\tau }_s^{\prime }$ for three different values of $\eta$. The short-dashed, solid, and long-dashed curves correspond to $\eta =0.1$, 0.02, and 0.01, respectively.

Figure 2

Variation of $v_p/c$ with $\omega {\tau }_s^{\prime }$ for three different values of $\eta$. The short-dashed, solid, and long-dashed curves correspond to $\eta =0.1$, 0.02, and 0.01, respectively. The thick vertical dot-dashed line indicates the critical angular frequencies ${\omega }_c{\tau }_s^{\prime }$ for the three different values of $\eta$, which are very close to each other according to Eq. (31).

Figure 3

Spin-current density $J_m\left(z,t\right)$ as a function of $z$. The solid curves in (a) and (b) are $J_m\left(z,t\right)$ at $t=1.75T_a$ and $t=5.75T_a$ [charge current $J\left(t\right)=-J_0$] with AC drive, respectively. The dashed curves in (a) and (b) are the spin-current density $J_m\left(z\right)$ resulting from the dc density, $J=-J_0$.

Figure 4

Spin accumulation ${\mu }_m\left(z,t\right)$ for the same parameters as in Fig. 3b.

Figure 5

Spin-current density $J_m\left(z,t\right)$ as a function of $z$. The solid curve is $J_m\left(z,t\right)$ at $t=1.75T_b$ [charge current $J\left(t\right)=-J_0$] with AC drive. The dashed curve is the spin-current density $J_m\left(z\right)$ for the case of a dc density $J=-J_0$.

Figure 6

Spin accumulation ${\mu }_m\left(z,t\right)$ for the same parameters as in Fig. 5.

Figure 7

Spin-current density $J_m\left(z,t\right)$ as a function of $z$. The solid, dot-dashed, and dashed curves are $J_m\left(z,t\right)$ at $t=T_b/16,T_b/8,T_b/4$, respectively, with ac drive.

Figure 8

Spin accumulation ${\mu }_m\left(z,t\right)$ for the same parameters as in Fig. 7.

Figure 9

Spin-current density $J_m\left(z,t\right)$ as a function of $z$. The solid curves in (a)–(c) are $J_m\left(z,t\right)$ at $t=0,0.5,3.5\text{ps}$, respectively. The dashed curves in (b) and (c) are $J_m\left(z,t\right)$ at $t=0\text{ps}$ plotted again as a reference.

Figure 10

Spin accumulation ${\mu }_m\left(z,t\right)$ for the same parameters as in Fig. 9.