Ferromagnetic domain wall pinning by a random array of inhomogeneities

Abstract

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 but with cooperative breakaway from many pinning sites. The effective pin density is expressed as ρ’ = √(ρ/ν) where v is the volume swept out between minimum and maximum energy positions of the wall. The value predicted for H ₀ due to weak pinning is not however significantly different from the strong pinning result.

The strong and weak pinning models do differ significantly in their predictions about the activation energy required to unpin a wall in a field H < H₀. The strong pinning model predicts that the coercive field, H _c at temperature T, should fall (due to thermal activation effects) as temperature rises so that H _c ^1/2 = H ₀ ^1/2 — const T ^2/3. The weak pinning model predicts a simple linear relationship with H _c = H ₀ — const T. The difference between these two predictions is sufficiently strong for experimental test, and some experimental comparisons are made.

The bowed walls produced by the application of a field give rise to unpaired spins and demagnetizing energies. The demagnetizing effects are considered in detail for both strong and weak pinning. For large demagnetizing energies and strong pinning H ₀ ³ ∝ ρ⁴ f ⁷γ⁻² I ⁻⁷. In the weak pinning model, the maximum pinning force per unit area is a function of field but goes to the zero demagnetizing energy value in the absence of thermal activation effects.