Generalization of Snoek’s law to ferromagnetic films and composites

The present paper establishes characteristics of the relative magnetic permeability spectrum $\mu \left(f\right)$ of magnetic materials at microwave frequencies. The integral ${\int }_0^{\infty }{\mu }^{″}\left(f\right)fdf$ of the imaginary part of $\mu \left(f\right)$ multiplied with the frequency $f$ gives remarkable properties. A generalization of Snoek’s law consists in this quantity being bounded by the square of the saturation magnetization multiplied with a constant. While previous results have been obtained in the case of nonconductive materials, this work is a generalization to ferromagnetic materials and ferromagnetic-based composites with significant skin effect. The influence of truncating the summation to finite upper frequencies is investigated, and estimates associated with the finite summation are provided. It is established that, in practice, the integral does not depend on the damping model under consideration. Numerical experiments are performed in the exactly solvable case of ferromagnetic thin films with uniform magnetization, and these numerical experiments are found to confirm our theoretical results. Microwave permeability measurements on soft amorphous films are reported. The relation between ${\int }_0^{6\mathrm{GHz}}{\mu }^{″}\left(f\right)fdf$ and $4\pi M_s$ is verified experimentally, and some practical applications of the theoretical results are introduced. The integral can be used to determine the average magnetization orientation in materials with complex configurations of the magnetization, and furthermore, to demonstrate the accuracy of microwave measurement systems. For certain applications, such as electromagnetic compatibility or radar absorbing materials, the relations established herein provide useful indications for the design of efficient materials, and simple figures of merit to compare the properties measured on various materials.

Figure 1

(Color online) Various inclusion shapes, with the associated functions $A\left(ka\right)$ used for the expression of the permeability in the presence of skin effect according to Ref. 19, 20, 21; $k$ is the wave vector inside the inclusion and $a$ is its radius (or half thickness in the case of a plate). The expression of the second order approximation of $A\left(ka\right)$ is also given.

Figure 2

(Color online) The imaginary part of the permeability ${\mu }^{″}$ computed for films with a thickness $2a$ ranging from $0.1\text{to}2\mu \mathrm{m}$ using the Landau–Lifschitz–Gilbert model and taking into account skin effect.

Figure 3

(Color online) The efficient dynamic magnetization $M_{\mu }\left(F\right)$ associated with the calculated permeabilities represented in Fig. 2, obtained either by numerical integration (symbols) or by analytical estimates (lines). The error bars represent typical experimental errors and the dashed line corresponds to the saturation magnetization.

Figure 4

(Color online) The imaginary part of the permeability ${\mu }^{″}$ measured on CoZr amorphous thin films of varying thicknesses.

Figure 5

(Color online) The efficient dynamic magnetization $M_{\mu }\left(F\right)$ associated with the measured permeabilities represented in Fig. 4, and a comparison with the saturation magnetization values (dashed line with error bars). The extrapolation of the saturation magnetization from $M_{\mu }\left(F\right)$ using analytical estimates is also represented, with appropriate error bars.