Domain-wall roughness in GdFeCo thin films: Crossover length scales and roughness exponents

Domain wall dynamics and spatial fluctuations are closely related to each other and to universal features of disordered systems. Experimentally measured roughness exponents characterizing spatial fluctuations have been reported for magnetic thin films, with values generally different from those predicted by the equilibrium, depinning, and thermal reference states. Here we study the roughness of domain walls in GdFeCo thin films over a large range of magnetic field and temperature. Our analysis is performed in the framework of a model considering length-scale crossovers between the reference states, which is shown to bridge the differences between experimental results and theoretical predictions. We also quantify the size of the depinning avalanches below the depinning field at finite temperatures.

Figure 1

Roughness crossover diagram. Lengths ${\ell }_{\mathrm{opt}}$ and ${\ell }_{\mathrm{av}}$ separate regions with roughness exponents characterized by the equilibrium and depinning states, and by the depinning and thermal states, respectively.

Figure 2

(a) PMOKE image of a domain wall (red line) obtained at a temperature $T=275\mathrm{K}$ after applying a magnetic field with amplitude ${\mu }_{0}H=5.71\mathrm{mT}$ ($H/H_{\mathrm{d}}=0.3$). (b) The corresponding $B\left(r\right)$ (black dots) is fitted with Eq. (4) (red line) in the low-$r$ region to find the best parameters $\zeta$ and $B_0$. Light gray curves show the $B\left(r\right)$ functions for the rest of the ensemble of $\mathcal{N}=63$ domain walls imaged under the same experimental conditions, which can be fitted analogously. All these individual values are then averaged to obtain $\langle \zeta \rangle$ and $\langle B_0\rangle$ for this temperature and field.

Figure 3

Experimentally obtained mean roughness parameters as functions of the field for different temperatures. (a) Mean roughness exponent, which does not coincide with any of the expected values ${\zeta }_{\mathrm{eq}}, {\zeta }_{\mathrm{th}}$ or 1 (dotted lines). (b) Mean roughness amplitude, which decreases as field increases in the studied range.

Figure 4

Proposed model for (a) the structure factor defined in Eq. (6) and (b) the roughness function deduced from Eq. (5) using the data in (a). The parameters used to plot $S\left(q\right)$ are $q_{\mathrm{av}}=0.28\mu {\mathrm{m}}^{-1}, q_{\mathrm{opt}}=9.88\mu {\mathrm{m}}^{-1}$, and $S_0=1.13\times {10}^{-6}\mu {\mathrm{m}}^2$, and $r_{\min }=\delta$ and $r_{\max }=100\mu \mathrm{m}$. The structure factor shows two crossovers at $q_{\mathrm{av}}\equiv 2\pi /{\ell }_{\mathrm{av}}$ and $q_{\mathrm{opt}}\equiv 2\pi /{\ell }_{\mathrm{opt}}$ between the three regimes of interest (thermal, depinning, and equilibrium). The corresponding roughness function (black dots) can be fitted in the low-$r$ range with a power law (blue line) characterized by an exponent which does not coincide with any of the used theoretical roughness exponents (${\zeta }_{\mathrm{th}}, {\zeta }_{\mathrm{eq}}$, and 1). We also present the $B\left(r\right)$ obtained with a different long-range cutoff $r_{\max }=1000\mu \mathrm{m}$ (gray dots), showing that the power-law fitting does not contain finite size effects. The parameters of the functions shown in both panels were chosen to reproduce the same roughness parameters obtained using $T=275\mathrm{K}, {\mu }_{0}H=5.71\mathrm{mT}$ ($H/H_{\mathrm{d}}=0.3$), as shown in the key.

Figure 5

Structure factor parameters for all temperatures and fields. (a) The characteristic length of the depinning avalanches ${\ell }_{\mathrm{av}}=2\pi /q_{\mathrm{av}}$ (empty symbols) compared to that of the creep events ${\ell }_{\mathrm{opt}}=2\pi /q_{\mathrm{opt}}$ (full symbols). The typical fitting region $[r_0,r_1]$ is highlighted in gray. The horizontal dotted lines indicate the short-range cutoff $r_{\min }$ and finite size $r_{\max }$ used to compute the model structure factor and $B\left(r\right)$, see Fig. 4. (b) The structure factor amplitude, on its turn, grows with decreasing field for all temperatures. In most cases, error bars are smaller that the points.

Figure 6

Relation between the roughness and structure factor parameters. Horizontal green lines show the values of the roughness exponent and amplitude found experimentally, with shaded regions representing their uncertainty ranges. (a) Given a fixed $q_{\mathrm{opt}}$ (dashed line), the roughness exponent varies smoothly from $\zeta \approx 1$ to $\zeta ={\zeta }_{\mathrm{th}}$ as $q_{\mathrm{av}}$ increases, although only $q_{\mathrm{av}}<q_{\mathrm{opt}}$ have physical meaning. Dot-dashed lines highlight the limits of the working range. The best value for $q_{\mathrm{av}}$ is that whose corresponding $\zeta$ coincides with $\langle \zeta \rangle$. (b) The best value of $S_0$ is determined analogously by comparing the amplitude $B_0$ of the generated $B\left(r\right)$ with $\langle B_0\rangle$. Both panels correspond to the $T=275\mathrm{K}, {\mu }_{0}H=5.71\mathrm{mT}$ ($H/H_{\mathrm{d}}=0.3$) case.