Quantitative Calculation of the Magnetic Ripple of Uniaxial Thin Permalloy Films

Abstract

The magnetic ripple of thin films with uniaxial anisotropy is calculated for the field and the magnetization lying along the easy and the hard direction of the uniaxial anisotropy. For small dispersion of the direction of magnetization, the variational principle, applied to all energy terms including the exchange, crystalline anisotropy, uniaxial anisotropy, magnetostriction, magnetostatic, and stray field energies, yields a Bessel's differential equation. Its solutions (cylindrical functions) describe the two‐dimensional magnetic ripple of the film. Along the mean direction of the magnetization the fluctuations of the magnetization vector are much greater than perpendicular to it. In the first case these fluctuations are coupled by exchange forces, and in the second case by the stray field. The dispersion of the magnetization depends on the dispersion of the anisotropy, on the magnetic constants of the film, on the thickness of the film, and on the applied field. The theory is able to explain the image of the ripple observed by the electron microscope and to explain the variation of the ripple quantitatively, if the applied field is varied. The estimation of the mean magnetization dispersion agrees with the experimental data of Fuller and Hale. The statements of the theory concerning the mean wavelength λ of the ripple for the mean direction of magnetization along the easy axis $${\lambda }_L{\text{=2πA}}^{\frac{1}{2}}{K}_u^{-\frac{1}{2}}{\mathrm{(h}}_L{\text{+1)}}^{-\frac{1}{2}}$$ , and along the hard axis $${\lambda }_s{\text{=2πA}}^{\frac{1}{2}}{K}_u^{-\frac{1}{2}}{\mathrm{(h}}_s{\text{−1)}}^{\frac{1}{2}}$$ , are verified by experimental observations. (A is the exchange constant, K_u the uniaxial anisotropy constant, h_L = H_L/H_k is the reduced field along the easy axis, h_s = H_s/H_k is the reduced field along the hard axis.) The calculated domain width, occuring at magnetization reversal along the hard axis, is in good agreement with the experiments of Middelhoek, Feldtkeller, and Smith. The domain splitting is fine at a great dispersion of the anisotropy axis and is less fine at smaller dispersions. The smallest domain width of 80–20 Permalloy films is of the order of 2 μ. Furthermore the theory gives a new method for the determination of the anisotropy field strength H_k.