Magnetic domain wall creep and depinning: A scalar field model approach

Magnetic domain wall motion is at the heart of new magnetoelectronic technologies and hence the need for a deeper understanding of domain wall dynamics in magnetic systems. In this context, numerical simulations using simple models can capture the main ingredients responsible for the complex observed domain wall behavior. We present a scalar field model for the magnetization dynamics of quasi-two-dimensional systems with a perpendicular easy axis of magnetization which allows a direct comparison with typical experimental protocols, used in polar magneto-optical Kerr effect microscopy experiments. We show that the thermally activated creep and depinning regimes of domain wall motion can be reached and the effect of different quenched disorder implementations can be assessed with the model. In particular, we show that the depinning field increases with the mean grain size of a Voronoi tessellation model for the disorder.

Figure 1

Evolution of the effective domain radius (circles), when a field square pulse of $h=0.07$ is applied (dashed lines) in a system at zero temperature and with uniform disorder. The straight black line is a linear fit of the data during the application of the field pulse, whose slope is indistinguishable from the domain wall velocity obtained at this field, as $\mathrm{\Delta }R/\mathrm{\Delta }t$ (see the text). In the inset, the spatial distribution of $\varphi$ for a system with $L=4096$ cells is shown. Black indicates the value $\varphi =-1$, while gray and white correspond to $\varphi =+1$. The gray circle corresponds to the initial domain (before the field pulse) and the white part is the growth of the initial domain after the field pulse.

Figure 2

(a) Velocity as a function of the radius of the initial domain $R_0$, for field square pulses of duration $\mathrm{\Delta }t$, at two values of the applied field $h$, as indicated. The initial domain radius should be large enough in order to ensure that the obtained velocities are not dependent on $R_0$. (b) Velocity as a function of the field pulse duration $\mathrm{\Delta }t$ for two values of the applied field and two sizes for the initial domain. Velocities may be overestimated if the pulse duration is not large enough, especially for small values of $h$.

Figure 3

Velocities and domain wall profiles calculated in a system at zero temperature. (a) Domain wall velocities as a function of magnetic field in a system without disorder, $\varepsilon =0$ in Eq. (9) (squares) and in a system with uniform disorder $\varepsilon =1$ (circles). The dashed black line is the linear velocity obtained from Eq. (14) with $\delta =1.4$, the domain wall width obtained from fitting domain wall profiles with Eq. (15). (b) Close-up view of the curve corresponding to the disordered system. (c) Domain wall profiles as a function of distance for three simulation snapshots, separated by $t=10$ in a nondisordered system for $h=1$. Fits of these curves with the function $\varphi \left(x\right)=tanh[(x-x_0)/\delta ]$ are also shown.

Figure 4

Velocity as a function of magnetic field at three temperatures in a system with uniform disorder. Dashed line indicates the flow regime, where velocities grow linearly with slope $\delta =1.4$. The pointed vertical line indicates the depinning field $h_d=0.0598$. The inset shows $\beta$ values as a function of $h_d^{\mathrm{test}}$. The horizontal dashed line indicates the expected value $\beta =0.245$ from which the depinning field $h_d$ is estimated.

Figure 5

Creep plot for a system with uniform disorder at two different temperatures. A linear behavior, highlighted by the black dashed lines, is observed for small field values, indicating that the system is in a regime compatible with the creep regime. The inset shows the data corresponding to $T=0.01$ again with open squares. These stationary velocities were computed from the simulation of systems where the field was applied during time lapses $\mathrm{\Delta }t$, varying from $2.6\times {10}^6$ to ${10}^5$. Closed squares in the inset correspond to velocities obtained at $T=0.01$, but with fixed $\mathrm{\Delta }t={10}^5$, and are shown in order to emphasize that some care should be taken in order to avoid the overestimation of velocities.

Figure 6

Velocity field curve at $T=0.01$, analyzed with the method proposed by Diaz Pardo et al. [10] for experimental curves. The dashed black line is a fit of data below the depinning field $h_d$, denoted by a vertical black line, following the creep law (1) and (2). The dashed gray line corresponds to the prediction for the depinning transition [see Eq. (16)] for the limit $v(h\to h_d,T=0)$. It is obtained with the same adjusted parameters as the black line. The closed black diamond indicates the point $(h_d,v_d=v\left(h_d\right))$, the upper boundary of the creep regime, while the open diamond corresponds to $v_T=v_d{(T_d/T)}^{\psi }$ (see the discussion in the text).

Figure 7

Differences between the disorder types used illustrated with a single realization. Each pair of images shows in top the value of the disorder parameter $\varepsilon$ along the first line of cells of each grid showed in the bottom: (a) uniform disorder, (b) Voronoi disorder, and (c) filtered disorder. Bottom images correspond to a portion of $50\times 50$ cells.

Figure 8

(a) Velocity field curves for three different disorders types at $T=0.01$. (b) Creep plot for the three velocity field curves. (c)–(e) Images showing the final configuration of the domains for different fields values corresponding to similar velocities in the creep regime, as indicated by the large closed symbols in (b). The shown domains correspond to systems with (c) filtered, (d) uniform, and (e) Voronoi disorders.

Figure 9

(a) Velocity field curves for four different numbers of Voronoi grains $N_V$ in the implementation of the disorder in a system at $T=0.01$. When the number of Voronoi cells is equal to the system size ($N_V=2^{24}\approx 1.7\times {10}^7$), the uniform disorder is recovered. The shown domains correspond to systems with (b) $N_V=2^{24}$, (c) $N_V={10}^7$, (d) $N_V=5\times {10}^6$, and (e) $N_V=1.5\times {10}^6$, for simulations with different field values corresponding to similar velocities, indicated by the dotted horizontal line.

Figure 10

Velocity field curves of Figs. 8 and 9 rescaled with the mobility $\delta$ of the flow regime and the depinning field of each curve. The curves correspond to a system at $T=0.01$ for the five different disorder implementations considered in this work (i.e., filtered, uniform, and Voronoi with three different mean grain sizes $\sigma$). The inset shows depinning fields plotted as a function of $\sigma$ (see the discussion in the text).