Domain wall propagation in ferromagnetic semiconductors: Beyond the one-dimensional model

We have investigated experimentally the field-driven propagation of domain walls (DWs) in perpendicularly magnetized ferromagnetic semiconductor layers. The results were then compared with the historical one-dimensional (1D) DW propagation model widely used in spintronics studies of magnetic nanostructures. Anomalous velocity peaks, not predicted by the 1D model, were observed experimentally. Using micromagnetic simulations we show indeed that, in the particular regime of layer thickness ($h$) of the order of the exchange length, velocity peaks appear in the precessional regime, their shape and position shifting with $h$. Analyses of the simulations show a distinct correlation between the curvature of the DW and the twist of the magnetization vector within it and the velocity peak. Associating a phenomenological description of this twist with a four-coordinate DW propagation model, we show that the velocity peaks result from the torque exerted by the stray field created by the domains on the twisted magnetization. The position of the peaks is well predicted from the DW’s first flexural mode frequency and depends strongly on the layer thickness. Comparison of the proposed model with data obtained on GaMnAs and GaMnAsP shows that the anomalies observed close to Walker breakdown are indeed induced by the flexion resonance of the domain wall.

Figure 1

Experimental velocity versus field data (curves shifted for clarity where needed). The polygons indicate the velocity anomaly regions, labeled 1 (low field) and 2 (high field). The arrows point to the $H=0$ resonance fields of the DW flexural modes determined numerically (see Section 6). (a) Ga${}_{0.93}$Mn${}_{0.08}$As${}_{0.0915}$P${}_{0.085}$, $h=50$ nm. (b) Ga${}_{0.93}$Mn${}_{0.07}$As, $h=50$ nm (reproduced from Ref. 10). (c) Ga${}_{0.93}$Mn${}_{0.07}$As, $h=20$ nm. (T $=40$ K), (d) $h=40$ nm (T $=$ 60 K).

Figure 2

(a) Geometry of the simulation, side view of the layer. The DW propagates toward the left; its mean position along $y$ is given by $q_0$(t) (not to scale). (b) DW propagation evidencing curvature, shown here for $h=80$ nm to highlight the effect. (c) DW free oscillation ($H_z=$ 0), after initializing the simulation from either of the top two positions. Magnetic parameters used for the simulations are indicated in the text.

Figure 3

Simulated velocity versus field (squares): (a) $h=10$ nm, (b) $h=20$ nm, (c) $h=30$ nm, and (d) $h=40$ nm. In (a), (b) and (d), the solid line is a fit to the 1D model, and the stars are the maximum DW elongation ${\eta }_0$. In (c) the open circles are the twist amplitude $S_0$ and its Lorentzian fit (solid line). Extrema values of the twist are 8${}^{ˆ}$ and 108${}^{ˆ}$.

Figure 4

Dynamics over one precession period under ${\mu }_{0}H=70$ mT for the sample $h=20$ nm. (a) Instantaneous velocity (open squares) and mean $\varphi (t)$ (full squares) over one precession period. Arrows indicate maximum (minimum) velocities for the canting angle close to 45${}^{ˆ}$ (modulo 180${}^{ˆ}$) [135${}^{ˆ}$ (modulo 180${}^{ˆ}$)]. (b) DW elongation ($\eta$, solid line) and DW twist ($S$, line with diamonds).

Figure 5

Depth dependence of $\psi (z,t)$ over half a precession period for (a) $h=20$ nm, ${\mu }_{0}H=70$ mT and (b) $h=40$ nm, ${\mu }_{0}H=35$ mT. Weakly localized horizontal Bloch lines appear in the thicker layer.

Figure 6

Velocity computed analytically using the ($q$, $\psi$, $\nabla q$, $\nabla \psi$) model for $h=30$ nm and $H>H_W$, and the corresponding simulated OOMMF curve. Inset close-up of the kink area evidencing the respective contributions to the velocity of the applied field ($A$), the demagnetizing field ($B$), and the domains’ stray field ($C$). See Eq. (10) and text for details.

Figure 7

Velocity versus field curves (closed symbols, shifted vertically for clarity), and precession frequency (solid line): the fields giving the peaks in the v(H) curves coincide with the ones yielding the simulated free-oscillation frequencies ($H=0$). Inset comparison between the thickness dependence of the analytical resonance frequencies ($f_{\mathit{SL}}$) given in Ref. 29 and the numerically simulated ones ($f_{\mathit{FO}}$).