Intermittent collective dynamics of domain walls in the creep regime

We study the ultraslow domain-wall motion in ferromagnetic thin films driven by a weak magnetic field. Using time-resolved magneto-optical Kerr effect microscopy, we access to the statistics of the intermittent thermally activated domain-wall jumps between deep metastable states. Our observations are consistent with the existence of creep avalanches: roughly independent clusters with broad size and ignition waiting-time distributions, each one composed by a large number of spatiotemporally correlated thermally activated elementary events. Moreover, we evidence that the large-scale geometry of domain walls is better described by depinning rather than equilibrium universal exponents.

Figure 1

(a), (b) Velocity-field characteristics for ultrathin Pt/Co/Pt magnetic thin film in different scales. In (b), the fitted solid line confirms agreement with the creep velocity law: $lnv\sim H^{-1/4}$. The red point corresponds to a single long pulse of duration $t=27000$ s at a small field of $H=46.1\mathrm{Oe}$ and room temperature (RT). The total reversed area over this long pulse is indicated in (c) and corresponds to $v=1.7{10}^{-9}$ m/s. During this long pulse, PMOKE images were taken every $t_0=15$ s, allowing us to identify $N_{\mathtt{WE}}(t_0,t)=1151$ magnetization reversal events or “window events” (WEs), highlighted over the image. The color scale corresponds to the time at which each WE was observed.

Figure 2

Sequences of magnetization reversal areas (WEs) detected for different time windows of duration $\mathrm{\Delta }t$, for $T=\mathrm{RT}$ and $H=46.1\mathrm{Oe}$. The color scale corresponds to the time at which each WE, delimited by contours lines, was detected.

Figure 3

WE area distributions for increasing window times $\mathrm{\Delta }t$ (as indicated) at RT and $H=46.1\mathrm{Oe}$ (a) and at $T=50{}^{\circ }\mathrm{C}$ and $H=24.2\mathrm{Oe}$ (b). In both cases, $v\sim 1\mathrm{nm}{\mathrm{s}}^{-1}$. At small $S$, we compare the initial decay of $P_{\mathtt{WE}}\left(S\right)$ with $S^{-{\tau }_{\mathtt{CE}}}$, with ${\tau }_{\mathtt{CE}}\approx 1.11$, where ${\tau }_{\mathtt{CE}}$ corresponds to depinning avalanches. (c) The collapse scaling shows that the data of (a) and (b) displays a large size cutoff scaling $S_{\mathtt{WE}}\sim {(\mathrm{\Delta }t/t^{*})}^{1/2}$, with $t^{*}$ an $H$- and $T$-dependent characteristic time. (d) Effective power-law exponents ${\tau }_{\mathtt{WE}}$ for $P_{\mathtt{WE}}\left(S\right)$ vs $\mathrm{\Delta }t/t^{*}$.

Figure 4

(a) Aspect ratio scaling of $\mathrm{\Delta }t=15\mathrm{s}$ WEs. The solid (dashed) line shows the expected depinning scaling $S_i\sim L_i^{1+{\zeta }_d}$ for qEW (qKPZ) class. (b) Scaling of the square width $W^2$ of DW segments of size $L$, for two typical configurations at RT. The solid (dashed) line shows the expected qEW (qKPZ) scaling at depinning, $W^2\sim L^{2{\zeta }_d}$, with ${\zeta }_d=1.25\left(0.63\right)$.

Figure 5

(a) Aspect-ratio of WEs obtained experimentally, between the area $S_i$ and major axis length $L_i$ of individual WE. We display the data $S_i$ vs $L_i$ for WEs corresponding to all values of $\mathrm{\Delta }t$ to enhance the crossover behaviour and access large WEs. Small WEs fairly scale as CEs or depinning avalanches in the qEW class, $S_i\sim L_i^{1+{\zeta }_d}$ with ${\zeta }_d\approx 1.25$ and thus $S_i\sim L_i^{2.25}$ (solid line), as compared to the qKPZ class, with ${\zeta }_d=0.63$, predicting $S_i\sim L_i^{1.63}$ (dashed line). The dotted-dashed line indicates a fair $S_i\sim L_i^{1.4}$ scaling, which can be rationalized using a simple model. (b) Probability distribution (nonnormalized) for the major axis length $L_i$ of the events shown in (a). In (a) and (b) a vertical line indicates the approximate location of the aspect-ratio crossover, $L^{*}$.

Figure 6

Normalized mean square distance $\langle {\delta }^{2}x\left(\mathcal{T}\right)\rangle /\mathcal{C}$ as a function of $\mathcal{T}$, measured at $T=\mathrm{RT}$ and $H=46.1\mathrm{Oe}$.

Figure 7

A simple model for understanding the area vs major axis length of large WEs. (a) Simulation results for the model [compare them with experimental results in Fig. 5]. The solid line indicates the qEW depinning scaling, and the dashed dotted line the effective power law observed at large WEs. The color bar indicates number of randomly deposited CEs. Results are reported in arbitrary units. (b) Probability distribution for percolation, as a function of the number of randomly deposited CEs. (c) Average aspect-ratio scaling versus number of CEs. (d) From bottom to top, four actograms showing the growth of compact WEs as we increase the number of deposited CEs (as indicated by the color bar). The actograms display segments centered at uniformly distributed epicenters, whose size equals the lateral extension of the growing WE. When the system percolates, i.e., a single WE spans the system (set to unity), the actogram is reset and a new deposition process starts.

Figure 8

Histograms for the $N_{\mathtt{tot}}(t_0,t)=1151$ WEs shown in Fig. 1, obtained by comparing consecutive images taken every $\mathrm{\Delta }t=t_0=15\mathrm{s}$ at $T=\mathrm{RT}$ and $H=46.1\mathrm{Oe}$. The histogram with uniform binning is presented in the left panel, while in the right panel a logarithmic binning is used for the same WEs. Power-law behavior at small sizes can be described with $P_{\mathtt{WE}}\left(S\right)\sim S^{-{\tau }_{\mathtt{WE}}}$ with the distribution exponent ${\tau }_{\mathtt{WE}}\approx 0.76\pm 0.1$.

Figure 9

WE area histograms obtained from a different region (upper panel) of the same Pt(4.5 nm)/Co(0.7 nm)/Pt(3.5 nm) sample used to report most our results in the main text, and obtained for a different $\mathrm{Pt}\left(6\mathrm{nm}\right)/{[\mathrm{Co}\left(0.2\mathrm{nm}\right)/\mathrm{Ni}\left(0.6\mathrm{nm}\right)]}_3/\mathrm{Al}\left(5\mathrm{nm}\right)$ sample (lower panel).

Figure 10

Numerical test for the practical implementation used to compute $W^2$ for a DW. We compute $W^2$ for numerically generated self-affine Gaussian signals of size $L_0=1024$ for several precise values of $\zeta$. We average over ten samples for each $\zeta$. The solid lines show agreement with the expected $W^2\sim L^{2\zeta }$. The method allows us to measure super-rough cases $\zeta >1$.