Time-resolved buildup of twisted indirect exchange interaction in two-dimensional systems

We study theoretically the time-domain dynamics of the spin-dependent Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction between driven magnetic impurities localized in a spin-orbit-coupled two-dimensional system. Particular attention is given to the influence of the spin-orbit coupling (SOC) on the system's dynamical response to a time-dependent precessional motion of the localized magnetic moment. We show that, via the RKKY mechanism, a flip of the spin $z$ component of one localized moment affects all $x,y$, and $z$ spin components of the other localized moment. The Friedel oscillations and the transient spin current triggered by the time-varying localized spin depend strongly on the orientation of the spin with respect to the two-dimensional structure. Our results demonstrate how the dynamic interplay between SOC and the RKKY interaction can be used to manipulate and fine-tune the characteristics of impurity spin reversal and to control the generated spin current and the local magnetization.

Figure 1

A schematics of a possible realization of pumping the RKKY interaction via time variations of the local spin moment. A scanning tip acts locally on the magnetic moment leading to a transversal local spin dynamics. The time resolution may be achieved for instance by voltage bias pulses or THz pulses. We study then the buildup of the spin-resolved indirect exchange interaction.

Figure 2

Spin-density oscillations for the dimensionless time $t_0=1$, spin-flip time $T_0=20$, and relaxation time ${\tau }_0=$ 100. Panel (a) shows the dependence on the dimensionless Rashba constant ${\kappa }_R$, as indicated in the legend. Panel (b) shows the dependence on the anisotropy angle $\alpha$ as listed in the legend. The curves in panel (b) correspond to ${\kappa }_R=0.3$.

Figure 3

Same as in Fig. 2 for different values of $t_0$ [see the legend in (a)]. Parts (a)–(c) correspond to different values of $T_0$ as shown in each panel. The static case (11) is shown by the broken line in (a)–(c). Dimensionless parameters: ${\tau }_0=100$, ${\kappa }_R=0.3$. (a)–(c) $\alpha ={90}^{\circ }$ [shown in (a)]; (d) the $\alpha$-dependence for the parameters indicated in the panel.

Figure 4

Spin-density Friedel oscillations at different spatial points $z$ (legends in the panels). The left panel reports the time dependence at $z=1$, where the corresponding values of spin density are around 100 times larger than those for $z=6$ and 12 (right panel). Dimensionless parameters: ${\kappa }_R=0.3,T_0=20$, and ${\tau }_0=$ 100.

Figure 5

Spatial oscillations of the spin-current component $J_x^y$ at different values of $t_0$ [legend in (b)]. Panels (a)–(c) correspond to different $T_0$ as shown in each panel. Dimensionless parameters: ${\tau }_0=$ 100, ${\kappa }_R=0.3$. (a)–(c) $\alpha ={30}^{\circ }$ [shown in (a)]. Panel (d) shows the $\alpha$-dependence for the parameters as indicated.

Figure 6

Temporal evolution of the spin current $J_x^y$ taken at different spatial points (indicated near the curves). The left panel shows the time dependence for $k_{F0r}=1$ and 6, while the right panel (due to different scale) shows the case $z=12$. The curves are plotted for ${\kappa }_R=0.3,\alpha ={30}^{\circ },T_0=20$, and ${\tau }_0=100$.