Theory of Large-Angle Ripple in Magnetic Films

Abstract

A theory of planar fluctuations of the direction of magnetization in a thin film which takes into consideration the nonlinear longitudinal magnetostatic force is outlined, and the resultant ripple is compared with that predicted by linear theories. The longitudinal magnetostatic force arises from the change along the direction of mean magnetization m₀ of the component of magnetization M(r) parallel to m₀; it leads to a torque that varies as the cube of the ripple amplitude. As a result of this torque, the linear theory breaks down for dispersion δ≳δ₁, where δ is the rms angular deviation of M from m₀ and δ₁ is typically only 1° or 2° for a 1000 Å Permalloy film in zero field and varies as L^−½, where 2L is the film thickness. With the inclusion of this force the range of validity of the theory has been extended to δ≲20°. A number of interesting effects have been found, including (1) an effective field in the direction of m₀ (and proportional to δ⁴ for δ>δ₁) acting on ripple components with wavelengths greater than 4πL; (2) an effective field 4πm₀δ², acting on shorter wavelength components; (3) as a result of (1), a decrease in the longitudinal and transverse ripple coherence lengths r_l and r_t (or, depending on viewpoint, a decrease in the longitudinal and transverse cutoff wavelengths); (4) a dispersion δ∝ (Kr_c)^2/5, where K=local anisotropy energy and r_c=crystallite size (in the linear theory δ∝ Kr_c); (5) almost no change in the mean ripple wall spacing as observed by Lorentz microscopy, ∼πr_c, provided r_c<r_l; and (6) a spin‐wave reaction torque coefficient in high‐speed switching R∝δ^10/3 (in the linear theory R∝δ²).