Phenomenological theory of permeability in films having no in-plane magnetic anisotropy: Application to spin-sprayed ferrite films

In terms of the first-order magnetocrystalline anisotropy energy constant K1, we determined magnetic anisotropy in single-crystalline cubic films having their normal lines parallel to (hkl) axes. We found that the films have biaxial magnetic anisotropy, with which we derived permeability μ(ω)=μ′(ω)−jμ″(ω) (ω: angular frequency) and natural resonance angular frequency ω0, due to magnetization rotation in single-crystalline and polycrystalline films. For films that have preferential orientation of major crystal axes (such as [100] and [110]) perpendicular to the film plane but have random orientation in the film plane (i.e., exhibiting no in-plane anisotropy), the low-frequency permeability μ(ω⪡ω0) and ω0 follow the product rule μ(ω⪡ω0)ω02=γ2Ms2∕2μ02 (Ms: spontaneous magnetization, γ: gyromagnetic ratio, and μ0: permeability of free space). This is similar to that which applies for films having an in-plane uniaxial magnetic anisotropy field. Our phenomenological theory gave a plausible explanation for the permeability of spin-sprayed ferrite films that have no in-plane magnetic anisotropy and yet have high μ′(ω) of a few tens up to the gigahertz region.