Current-driven dynamics of coupled domain walls in a synthetic antiferromagnet

We develop the theory of magnetic domain-wall motion in coupled double-layer systems where electrons can hop between the layers, giving rise to an antiferromagnetic coupling. We demonstrate that the force from the interlayer coupling drives the walls and the effect of the extrinsic pinning is greatly reduced if the domain walls are initially separated. The threshold current density for metastable spin-aligned configurations is also much lower. We conclude that the interlayer coupling has a significant effect on domain-wall mobility in double-layer systems.

Figure 1

(a) Ground-state geometry of Néel-type domain walls in a synthetic double-layer antiferromagnet. (b) Coupling between the layers gives rise to an attractive force between the walls at finite separation $Z_1-Z_2$. The figure shows a spin configuration which has out-of-plane angles ${\varphi }_{1,2}$. Spin orientations at the center of the walls are shown in the inset.

Figure 2

Potentials $u\left(Z\right)$ and $w\left(Z\right)$ and their derivatives, $u^{\prime }\left(Z\right)$ and $w^{\prime }\left(Z\right)$.

Figure 3

Motion of coupled domain walls in double-layer systems under extrinsic pinning. The current is switched on at time $t=0$ and positions $Z$ of the two domain walls in the system are calculated for pinning potentials which are located at $Z_1=5$ and at $Z_2=0$. When the current is insufficient to unpin the walls they oscillate in the pinning potentials with dampening amplitude (left). When the current is sufficiently large, it unpins the walls aided by the force from the interlayer coupling and the walls start moving (right). Eventually the walls move together with a vanishing phase difference. The terminal velocity is calculated from this limiting motion.

Figure 4

Averaged terminal domain-wall velocity (given by the color bar) in double-layer systems under weak nonadiabatic force ($\beta /\alpha =0.5$). The velocity is calculated as a function of antiferromagnetic in-plane interlayer coupling ${\Delta }_{\left|\right|}$ and velocity of driving electrons ${\nu }_e$. The thicknesses of the layers are ${\mu }_1=1$ and ${\mu }_2=1/2$. The velocities are calculated for different distances between the pinning potentials $\ell$. The interlayer coupling helps unpin the domain walls and lowers the threshold current at $\ell >0$. Note that the range for ${\Delta }_{\left|\right|}$ is larger at $\ell =5$.

Figure 5

Averaged terminal domain-wall velocity (given by the color bar) in double-layer systems in the regime of strong nonadiabatic driving ($\beta /\alpha =6$) and different distances between the pinning potential sites, $\ell$. In this regime the unpinned domain-wall velocity is proportional to $\beta /\alpha$ until the point of Walker breakdown at ${\nu }_e\simeq 0.2$. In-plane interlayer coupling is antiferromagnetic ($\Delta ={\Delta }_{\left|\right|}>0$) and improves domain-wall mobility at finite $\ell$. The layers have unequal thickness ${\mu }_1=1$, ${\mu }_2=1/2$, and $\alpha =0.01$.

Figure 6

Averaged terminal domain-wall velocity in double-layer systems in the nonadiabatic driving regime with fixed in-plane coupling strength ${\Delta }_{\parallel }=+0.5$ and ferromagnetic (${\Delta }_{\perp }=-0.5$), vanishing (${\Delta }_{\perp }=0$), and antiferromagnetic (${\Delta }_{\perp }=+0.5$) out-of-plane component, respectively. Layer thickness ${\mu }_1=1$ and ${\mu }_2=0.5$. Distance between the pinning potential sites is $\ell =2$.

Figure 7

(a) A metastable initial state with parallel spin orientations at the center of the walls (${\varphi }_1={\varphi }_2=0$). (b) Averaged terminal domain-wall velocity for the initial metastable states. The in-plane interlayer coupling is antiferromagnetic (${\Delta }_{\left|\right|}>0$) and the out-of-plane component is ${\Delta }_{\perp }=0$. Figure shows the regime of weak nonadiabatic force ($\beta /\alpha =0.5$) and the regime of strong nonadiabatic driving ($\beta /\alpha =6$). The threshold current decreases with increasing interlayer coupling in both cases. The layers have unequal thickness ${\mu }_1=1$, ${\mu }_2=1/2$, the pinning potential strength is $k_1=k_2=0.1$, and the pinning sites are at the same position, $\ell =0$.

Figure 8

Terminal domain-wall velocity under intrinsic pinning in the absence of extrinsic pinning ($k_1=k_2=0$). Interlayer coupling is assumed to be antiferromagnetic and isotropic, ${\Delta }_{\left|\right|}={\Delta }_{\perp }=\Delta$. The domain-wall velocity as a function of the velocity of driving electrons, ${\nu }_e$, at $\Delta =0$, $0.25$, and $0.5$ at equal layer thickness ${\mu }_1={\mu }_2=1$ shown at left. The threshold ${\nu }_e$ at different layer thicknesses ${\mu }_2$ fixing ${\mu }_1=1$ shown at right.

Figure 9

Averaged terminal domain-wall velocity under intrinsic pinning. Layers have unequal thickness ${\mu }_1=1$, ${\mu }_2=1/2$. In the absence of nonadiabatic torque the interlayer coupling moderately increases the threshold current needed to move the domain walls.