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We study the dynamics of a ferromagnetic domain wall driven by an externalmagnetic field through a disordered medium. The avalanche-like motion of thedomain walls between pinned configurations produces a noise known as theBarkhausen effect. We discuss experimental results on soft ferromagneticmaterials, with reference to the domain structure and the sample geometry, andreport Barkhausen noise measurements on Fe$_{21}$Co$_{64}$B$_{15}$ amorphousalloy. We construct an equation of motion for a flexible domain wall, whichdisplays a depinning transition as the field is increased. The long-rangedipolar interactions are shown to set the upper critical dimension to $d_c=3$,which implies that mean-field exponents (with possible logarithmic correction)are expected to describe the Barkhausen effect. We introduce a mean-fieldinfinite-range model and show that it is equivalent to a previously introducedsingle-degree-of-freedom model, known to reproduce several experimentalresults. We numerically simulate the equation in $d=3$, confirming thetheoretical predictions. We compute the avalanche distributions as a functionof the field driving rate and the intensity of the demagnetizing field. Thescaling exponents change linearly with the driving rate, while the cutoff ofthe distribution is determined by the demagnetizing field, in remarkableagreement with experiments.Comment: 17 RevTeX pages, 19 embedded ps figures + 1 extra figure, submitted  to Phys. Rev.

Dynamics of a ferromagnetic domain wall: Avalanches, depinning transition, and the Barkhausen effect