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 rl and rt (or, depending on viewpoint, a decrease in the longitudinal and transverse cutoff wavelengths); (4) a dispersion δ∝ (Krc)2/5, where K=local anisotropy energy and rc=crystallite size (in the linear theory δ∝ Krc); (5) almost no change in the mean ripple wall spacing as observed by Lorentz microscopy, ∼πrc, provided rc<rl; and (6) a spin-wave reaction torque coefficient in high-speed switching R∝δ10/3 (in the linear theory R∝δ2).

Out of plane component of the magnetization of sputtered Fe72Ga28 layers

“…The MFM contrast present in our samples is a corrugation known as magnetic ripple that has been previously observed in other polycrystalline magnetic systems, including FeGa thin films [21][22]. This corrugation reflects the competition between different contributions, and its origin is an inhomogeneous magnetization due to fluctuations of the magnetic anisotropy [21,[28][29]. Therefore, a random magnetic anisotropy contribution is able to break the uniform orientation of the magnetization giving raise to the irregular magnetic contrast as that presented in figure 2.…”

Magnetic ripple domain structure in FeGa/MgO thin films

“…Magnetic force microscopy images presented in Fig 2a) shows lines on the film surface, interpreted as magnetic domain walls, separating areas without contrast, indicating that M is confined in the plane, and also some tip induced features; the image displayed in Fig 2b, taken for film 24-20, shows a fine structure which is not observed in the areas separated by the domains walls of the image 2a for film [13][14][15][16][17]. Fig 2c and 2d shows the same kind of magnetic contrast, revealing a non uniform magnetic configuration, in more detail for films with 28-56 and 28-21, respectively.…”

“…The explanation of the fluctuations of M is based on the irregular magnetocrystalline anisotropy of the randomly distributed crystallites. The model description incorporates the statistical treatment of local randomly oriented anisotropy and a uniform magnetic anisotropy [16].…”

Quasistatic remagnetization processes in two-dimensional systems with random on-site anisotropy and dipolar interaction: Numerical simulations

“…35, which can be accessed much more easily. The only nonlinear ripple theory known to us 36 ͑which, according to Ref. 36, should be valid for permalloy films if the magnetization deviation angle does not exceed Ϸ20°) predicts the growth of the ripple correlation length with the exchange constant ͑however, only a linear growth͒ and does not predict any nonmonotonic behavior of the correlation lengths as functions of the film thickness.…”