Accurate calculation of the transverse anisotropy of a magnetic domain wall in perpendicularly magnetized multilayers

Abstract

The transverse anisotropy constant and the related Döring mass density are key parameters of the one-dimensional model to describe the motion of magnetic domain walls. So far, no general framework is available to determine these quantities from static characterizations such as magnetometry measurements. Here, we derive a universal analytical expression to calculate the transverse anisotropy constant for the important class of perpendicular magnetic multilayers. All the required input parameters of the model, such as the number of repeats, the thickness of a single magnetic layer, and the layer periodicity, as well as the effective perpendicular anisotropy, the saturation magnetization, and the static domain wall width are accessible by static sample characterizations. We apply our model to a widely used multilayer system and find that the effective transverse anisotropy constant is a factor 7 different from the when using the conventional approximations, showing the importance of using our analysis scheme.

Accurate model of the stripe domain phase of perpendicularly magnetized multilayers

“…To describe the domain wall profile theoretically we can use the expression for the stray field energy associated with the volume charges inside of the isolated domain wall obtained by Büttner et al 17 : with…”

“…17, the integration along y and z can be performed analytically. In the limit L → ∞, the integration kernel reads…”

“…as a single integral in Fourier space 17 . Here, the hat above the functions denotes a Fourier transform…”

“…By following the approach of Büttner et al 17 we find in the limit L → ∞ (compare with Eq. 24 from 17):…”

“…1b) can be described as a periodic repetition of the single domain wall profile (1)-(3), as long as the distance between the domain walls is at least 8∆, as shown later in this paper. The magnetostatic energy associatied with interactions between volume charges (i.e., the volume stray field energy) formed in a single wall state with a given profile has been calculated in single layer films and multilayers 17 . The result is a thickness-dependent transverse anisotropy, i.e., a local hard-axis anisotropy that describes the energetic costs to tilt the spins from Bloch to Néel orientation.…”