Runaway “Fingerlike” Instability of Magnetic Walls in Ultrathin Layers

We show that smooth domain walls in ultrathin ferromagnetic films can develop jaggedness even in the absence of random defects when confronted with a sufficiently large tilt between the uniaxial anisotropy direction and the external field. From the Kerr imaging of 0.7 nm thin Co films and from numerical simulations we report a previously unseen runaway fingerlike instability in a magnetic wall that begins on nanoscales but grows to macroscopic lengths for sufficiently large tilt angles. A threshold for the instability is controlled by the ferromagnet's parameters and the applied field.

Figure 1

Kerr images of two up-spin (in lighter shade) domains approaching the crest of a nanoscale ridge in a 0.7 nm thin Co film are “snapshots” at $t=400\mathrm{s}$ shown for $H^{\mathrm{e}\mathrm{x}\mathrm{t}}=980\mathrm{O}\mathrm{e}$ in (a), $1190\mathrm{O}\mathrm{e}$ in (b), and $1232\mathrm{O}\mathrm{e}$ in (c). The time sequences of images were recorded either with the field on or briefly off [19]. Domain walls are subject to an effective anisotropy field $H_{\mathrm{e}\mathrm{f}\mathrm{f}}^{\mathrm{a}\mathrm{n}\mathrm{i}}$ (main panel) induced by strain from the nanoscale ridge sketched on top. The ridge's position is indicated by a dotted red line. At each $x$, $H_{\mathrm{e}\mathrm{f}\mathrm{f}}^{\mathrm{a}\mathrm{n}\mathrm{i}}$ was obtained from its balancing the applied field. A continuous viewing in the Kerr microscope showed an instability suddenly appearing on the steeper side of $H_{\mathrm{e}\mathrm{f}\mathrm{f}}^{\mathrm{a}\mathrm{n}\mathrm{i}}(x)$ when $H^{\mathrm{e}\mathrm{x}\mathrm{t}}\gtrsim 1.2\mathrm{k}\mathrm{O}\mathrm{e}$. (d) Unlike the square hysteresis loops $\sim$mm away (or without) the ridge [7], the gradual saturation of $\mathbf{m}$ near the ridge indicates an anisotropy tilt.

Figure 2

Simulation of a magnetic wall driven up a ramp by an external field. The scale of the active grid is $48\times 96$ dipoles, which corresponds to $33.6\times 67.2\mathrm{n}\mathrm{m}$. The initial perturbation is a sine wave with an amplitude of $1$ dipole spacing. Dipoles are shown as arrows (pointing up at right) footed in the layer for the time sequence 0.425, 0.65, to 0.8 ns ($c\to a$). Here $H^{\mathrm{e}\mathrm{x}\mathrm{t}}=2\times {10}^4\mathrm{O}\mathrm{e}$, $M_s=2\times {10}^4/(4\pi )\mathrm{O}\mathrm{e}{\mathrm{c}\mathrm{m}}^{-3}$, $S_x=1.3\times {10}^{-6}\mathrm{e}\mathrm{r}\mathrm{g}{\mathrm{c}\mathrm{m}}^{-1}$, and $K_u=6.7\times {10}^7\mathrm{e}\mathrm{r}\mathrm{g}{\mathrm{c}\mathrm{m}}^{-3}$. We used a fourth-order Runge-Kutta method [26] with a variable step, typically $2.5\times {10}^{-5}\mathrm{n}\mathrm{s}$. The height $z$ of the 0.7 nm thick layer is given by $dz/dx=\tan 27°\times \exp [-(x-48{)}^2/{20}^2]$ in units of the layer thickness. A periodic boundary condition in the $y$ direction is enforced by wrapping the exchange field partners on these edges and by reproducing on each $y$ side all the active dipoles shown here for the evaluation of $H^{\mathrm{d}\mathrm{i}\mathrm{p}}$. See Ref. 29 for the simulation movie.

Figure 3

Five frames are shown for a simulation with $192\times 192$ active dipoles corresponding to times of $2.5t\mathrm{p}\mathrm{s}$ for indicated values of $t$. The grid scale is $196\times 196$ dipoles, or $137.2\times 137.2\mathrm{n}\mathrm{m}$. In (a)–(d), the dipoles are $+1$ out of the page and $-1$ into the page. The parameters are as in Fig. 2 but now the anisotropy is tilted with respect to the layer by a constant ${\theta }_0=27°$ while the layer is flat, parallel to the page, and perpendicular to $H^{\mathrm{e}\mathrm{x}\mathrm{t}}$, which is out of page; see diagram in (e). The initial perturbation is shown in (a); it is a quadruple sine wave with an amplitude of 2.5 dipole spacings. The dipoles just ahead of the wall tend to be pointed more antiparallel to $H^{\mathrm{e}\mathrm{x}\mathrm{t}}$ than their equilibrium direction, and the dipoles just behind the wall tend to be pointed more parallel to $H^{\mathrm{e}\mathrm{x}\mathrm{t}}$. Left inset: a blowup of a fingerlike protrusion in (c). Right inset: intermediate bubble formation resulting from the merger of two fingers. (f) Large gradient of $\mathbf{m}$ inside the wall produces an “explosion” in $H^{\mathrm{e}\mathrm{x}\mathrm{c}}$, indicated by the arrow length and shade; a circular pattern of spin waves is emitted. See Ref. 29 for the simulation movie.

Figure 4

Calculated wall speed $v$ for misaligned layers relative to the aligned (${\theta }_0=0$) wall speed $v_0$ versus the external field and the driving parameter $\xi$ for several anisotropy tilts ${\theta }_0$. The wall speed increases at high $H^{\mathrm{e}\mathrm{x}\mathrm{t}}$ as the external field approaches $H^{\mathrm{a}\mathrm{n}\mathrm{i}}$. The instability operates when $\sin {\theta }_0\gtrsim 0.1\xi$ for small $\xi$, defining a threshold external field $H_{\mathrm{t}\mathrm{h}}$. For steep tilts, $H_{\mathrm{t}\mathrm{h}}$ in the simulations (arrows) shifts to lower fields. Consistent with experiment, at a lower $H^{\mathrm{e}\mathrm{x}\mathrm{t}}$ the wall speed decreases with increasing tilt. Inset: $H_{\mathrm{t}\mathrm{h}}$ (in tesla) vs tilt angle. Within the error bars, the experimental threshold in Fig. 1 (half-shaded triangle) corresponds to $50°\lesssim {\theta }_0\lesssim 70°$.