Longitudinal Permeability in Thin Permalloy Films

Longitudinal permeability is defined as ${\mu }_L{\mathrm{=(\partial B}}_L{\mathrm{/\partial H}}_L{)}_{H_T=\mathrm{const}}$ , where B_L is the component of applied field parallel to the easy axis, and H_T is a bias field applied perpendicular to the easy axis of a magnetic film with uniaxial anisotropy. The longitudinal permeability is also the hard axis resonance amplitude in the low frequency limit. The longitudinal permeability is calculated using three different models. The first, that of an ideal single domain film, yields an infinite value when the bias equals H_K. Considering the effect of inhomogeneity in easy axes, resulting in many noninteracting domains, gives a maximum ${\mu }_L{\mathrm{/\mu }}_T{\text{=22α}}_{90}^{-\frac{2}{3}}$ , where ${\mu }_T{\text{=4πM/H}}_K$ and α₉₀ is so defined that 90% of the film has a local easy axis within α₉₀ degrees of the average easy axis. Including variation of H_K within a film lowers μ_L/μ_T at $H_T{\mathrm{=H}}_K\hspace{0.17em}\mathrm{to}{\text{\hspace{0.17em}22α}}_{90}^{-\frac{2}{3}}{\text{−34α}}_{90}^{\text{−1}}{\Delta }_{90}^{\frac{1}{2}}$ , where Δ₉₀ is defined so that 90% of the film has an H_K within Δ₉₀ H_K of the average H_K of the film. Comparison with experimental curves for films with measured values of α₉₀ and Δ₉₀ shows that the last model yields good qualitative agreement and correctly predicts the behavior of μ_L/μ_T with finite drive fields, but quantitatively the experimental maxima are 20% to 60% lower than the theoretical curves. This difference is attributed to wall energy terms which are not included as part of the regular calculation.