Scaling theory applied to strong and weak pinning in ferromagnets

Abstract

Scaling theory is used to determine the form of the relation between pin site density ρ, maximum pinning force f _o, wall energy γ_o, spontaneous magnetization I _s, range parameter δ_o, and the coercive field H _c (neglecting thermal activation effects). The results depend on whether the pinning is strong or weak, that is, whether B = 3f _o/(8γ_oδ_o) is bigger or less than one. The form assumed for the demagnetizing field acting on the bowed domain walls also influences the results. If the demagnetizing field is approximated as H _D ∝ [L _z ⁻² L _y L _x], where L _z, L _y and L_x are the semi-axes of the bowed ellipsoidal wall, then H _c ∝ ρ^5/4 f _o ^9/4γ^−3/4 I _s ⁻² for strong pinning. A better approximation with H _d ∝ [L _z ^−3/2 L _x ^1/2 L _y] gives H_c ∝ [ρ⁴ f _o ⁷γ_o ⁻² I _s ⁻⁷]^1/3 for strong pinning. If demagnetizing terms are sufficiently small to be neglected, then the coercive field takes the same form for both strong and weak pinning with H _c ∝ ρf _o ²/(γ_o I _s). Since for weak pinning the wall breaks away simultaneously from many pinning sites, H _c does not depend on the demagnetizing energy associated with wall bowing between the sites and is given by H _c ∝ ρf _o ²/(γ_o I _s).