Domain wall dynamics and Barkhausen jumps in thin-film permalloy microstructures

Domain wall jump-amplitude and velocity distributions associated with Barkhausen jumps in a 300-\AA{}-thick thin-film permalloy and a $200\ifmmode\times\else\texttimes\fi{}200\phantom{\rule{0.3em}{0ex}}{(\ensuremath{\mu}\mathrm{m})}^{2}\ifmmode\times\else\texttimes\fi{}220\text{\ensuremath{-}}\mathrm{\AA{}}$-thick permalloy sample microstructrue are measured using a high-speed magneto-optic Kerr effect polarimeter. The jump-amplitude and velocity distributions are obtained for applied-field sweep rates from $0.9\phantom{\rule{0.3em}{0ex}}\mathrm{Oe}∕\mathrm{s}$ to $6.3\ifmmode\times\else\texttimes\fi{}{10}^{4}\phantom{\rule{0.3em}{0ex}}\mathrm{Oe}∕\mathrm{s}$. The velocity distributions exhibit statistical properties consistent with the stochastic description of field-driven domain wall motion developed by ABBM [B. Alessandro, C. Beatrice, G. Bertotti, and A. Montorsi, J. Appl. Phys. 68, 2901 (1990)]. Averaged velocity distributions exhibit the expected increase of dynamic coercivity as the sweep rate is increased, and the maximum domain wall velocity measured as a function of the applied field at the stochastic depinning threshold is shown to be governed by the mobility limit imposed by local spin damping. The averaged velocity $⟨\ensuremath{\nu}(H\ensuremath{-}{H}_{0})⟩$ obtained from the distributions is observed to depart from the commonly accepted linear-dependence model, especially at higher drive-field sweep rates. This departure is interpreted as an indication of a sweep-rate-dependent mobility. Sweep-rate-dependent jump-amplitude distributions $P(\ensuremath{\Delta}M)$ versus $\ensuremath{\Delta}M$ are obtained from the Barkhausen effect data. These distributions exhibit power-law behavior with a sharp cutoff at large values of $\ensuremath{\Delta}M$. Attempts to reconcile the measured jump-amplitude distributions and sweep-rate-dependent exponents with various models of universal scaling are described. Power-law fits to $P(\ensuremath{\Delta}M)$ distributions measured to optimize temporal resolution (required for the velocity-distribution studies) yield a sweep-rate-dependent exponent that varies from $\ensuremath{\beta}=1.45\ifmmode\pm\else\textpm\fi{}0.05$ to $\ensuremath{\beta}=1.0$ as $dH∕dt$ is varied from $25\phantom{\rule{0.3em}{0ex}}\mathrm{Oe}∕\mathrm{s}$ to $6.3\ifmmode\times\else\texttimes\fi{}{10}^{4}\phantom{\rule{0.3em}{0ex}}\mathrm{Oe}∕\mathrm{s}$. This range of sweep-rate-dependent $\ensuremath{\beta}$ agrees with the ABBM model and is consistent with the sweep-rate dependence exponent rule that predicts linear sweep-rate scaling for the adiabatic value $\ensuremath{\beta}=3∕2$ [R. A. White and K. A. Dahmen, Phys. Rev. Lett. 91, 085702 (2003)]. Additional experiments on both 300-\AA{}-thick continuous films and the same microstructures optimized for $\ensuremath{\Delta}M$ sensitivity yield a more accurate value of $\ensuremath{\beta}\phantom{\rule{0.3em}{0ex}}(\ensuremath{\beta}=1.33\ifmmode\pm\else\textpm\fi{}0.01)$, which corresponds to a CZDS model $(\ensuremath{\beta}=4∕3)$ [C. Cizeau, S. Zapperi, G. Durin, and H. E. Stanley, Phys. Rev. Lett. 79, 4669 (1997); Phys. Rev. B 58, 6353 (1998)]. Adiabatic limit values of $\ensuremath{\beta}$ determined from all of the experiments are definitively outside the range $\ensuremath{\beta}\ensuremath{\sim}1.0$ for self-organized criticality in a two-dimensional system [P. Bak, C. Tang, and K. Wiesenfeld, Phys. Rev. Lett. 59, 381 (1987)], and also appear to be inconsistent with experimental results for 900 \AA{} Fe films on MgO in which $\ensuremath{\beta}=1.1$ was obtained from rescaled $P(\ensuremath{\Delta}M)$ distributions [E. Puppin, Phys. Rev. Lett. 84, 5415 (2000)]. The sweep-rate dependence of $\ensuremath{\beta}$ determined in a system that manifests an adiabatic limit of $\ensuremath{\beta}$ that is not equal to $3∕2$ also violates the model of White and Dahman that prohibits sweep-rate-dependent scaling for $\ensuremath{\beta}=4∕3$.