Ferromagnetic domain wall pinning by a random array of inhomogeneities

The pinning site inhomogeneities interacting with a domain wall are characterized by the maximum restoring force, f, acting on an area A of domain wall. The critical field, H 0, required to release this area from the pin is given by f/2IA where I is the magnetic moment per unit volume. The area A is not independent of the applied field H and a statistical treatment is required to evaluate it. The Friedel ‘steady state’ model assumes that the volume swept out when a wall breaks away from a pin contains, on average, one replacement pin. This leads to the relation A ∝ 1/H 1/2 and hence to the critical field H 0 = 3ρf 2/(4πγI) where ρ is the pin density and γ is the wall energy per unit area. If the parameter β0 = 3f/(8πγb) > 1, where 4b is the interaction range of the pin, then the Friedel model holds and the pinning is ‘strong'. If β < 1 the Friedel model breaks down and the pinning is ‘weak'. The unpinning process is then no longer connected with unit steps related to breakaway from single pins bu...