Dispersive Stiffness of Dzyaloshinskii Domain Walls

It is well documented that subjecting perpendicular magnetic films that exhibit the interfacial Dzyaloshinskii-Moriya interaction to an in-plane magnetic field results in a domain wall (DW) energy $\sigma$, which is highly anisotropic with respect to the orientation of the DW in the film plane $\mathrm{\Theta }$. We demonstrate that this anisotropy has a profound impact on the elastic response of the DW as characterized by the surface stiffness $\tilde{\sigma }(\mathrm{\Theta })=\sigma (\mathrm{\Theta })+{\sigma }^{\prime \prime }(\mathrm{\Theta })$ and evaluate its dependence on the length scale of deformation. The influence of stiffness on DW mobility in the creep regime is assessed, with analytic and numerical calculations showing trends in $\tilde{\sigma }$ that better represent experimental measurements of domain wall velocity in magnetic thin films compared to $\sigma$ alone. Our treatment provides experimental support for theoretical models of the mobility of anisotropic elastic manifolds and makes progress toward a more complete understanding of magnetic domain wall creep.

Figure 1

(a) ${\sigma }^{\text{eq}}$ and long wavelength $\tilde{\sigma }$ vs ${\mu }_0{H}_x$ for varying ${\mu }_0{H}_{\mathrm{DMI}}$ with ${\mu }_0{H}_k=1\mathrm{T}$, $M_s=600\mathrm{kA}/\mathrm{m}$, and $t_f=1.8\mathrm{nm}$. (b) Wulff construction (gray), ${\sigma }^{\text{eq}}(\mathrm{\Theta })$ (red), and $\mathrm{\Gamma }(\mathrm{\Theta })$ (blue) for a Dzyaloshinski DW with ${\mu }_0{H}_x=300\mathrm{mT}$ and ${\mu }_0{H}_{DMI}=360\mathrm{mT}$. The red point of the inset indicates the origin. (c) Example calculation of wall faceting in mumax³ for the same conditions as (b). (d) Calculated driving force for faceting $\mathrm{\Delta }\sigma$. (e) Calculated facet orientation $\mathrm{\Delta }\mathrm{\Theta }$ with superimposed numerical results.

Figure 2

(a)–(b) Normalized $\tilde{\sigma }$ vs ${\mu }_0{H}_x$ for (a) $D=0.25\mathrm{mJ}/{\mathrm{m}}^2$ and (b) $D=0.5\mathrm{mJ}/{\mathrm{m}}^2$ from analytic and 2D numerical calculations. Dashed lines indicate DW energy. (c)–(d) Corresponding $\uparrow \downarrow$ and $\downarrow \uparrow \mathrm{DW}$ velocity behavior calculated from (a),(b). (e)–(f) Antisymmetric component of the velocity, $A_{\text{creep}}=\ln (v_{\uparrow \downarrow }/v_{\downarrow \uparrow })$ calculated from (c),(d).

Figure 3

(a) Subtractive Kerr images of $\mathrm{Co}/\mathrm{Ni}$ where bounds of the light gray region represent the displacement of an $\uparrow \downarrow \mathrm{DW}$. (b) Experimental velocity vs ${\mu }_0{H}_x$ for ${\mu }_0{H}_z=4.25\mathrm{mT}$ with fit from the dispersive stiffness model. The fitting parameters use $v_0$ and $\alpha$ taken from experimental results at $H_x=0$. (c) DW velocity vs $({\mu }_0{H}_z{)}^{-1/4}$ for a series of ${\mu }_0{H}_x$. Dashed lines correspond to creep parameters used in our stiffness model where $v_0$ is fixed.