Rigorous Electrodynamic Approach to Ferromagnetic Resonance in Cavity‐Coupled Ferrimagnetic Films

Abstract

A rigorous electrodynamic approach to the modeling of ferromagnetic resonance (FMR) in ferrimagnetic films of arbitrary thickness placed in a rectangular microwave cavity is presented in this letter. It is shown that the coupling of the cavity mode with the FMR mode of an in-plane magnetized ferrimagnetic film occurs when the real part of its intrinsic diagonal permeability component m approaches zero. The obtained theoretical prediction of frequency vs. magnetic bias is confirmed by experiments performed on a 3 mm thin thin yttrium iron garnet (YIG) layer grown on a 500 mm thick gadolinium gallium garnet (GGG) substrate for both even and odd cavity modes. It is also shown that the proposed rigorous electrodynamic approach agrees well with perturbation theory in the high damping regime, however, leads to qualitatively different conclusions for low damping. Finally, it is demonstrated that the experimentally observed higher-order FMR modes can be attributed to the longitudinal modes distributed across the thickness of the film at frequencies corresponding to negative values of m approaching zero. This letter paves the way for further study of, among others, the magnonmicrowave photon coupling strength and the development of ferromagnetic linewidth characterization methods.Strong coupling between cavity modes and magnetic samples, also known as the hybrid CMP (cavity-magnon-polariton) mode, currently represents a highly active research area [1] that can lead to applications in quantum information processing. If a microwave cavity is loaded with a magnetic sample, hybrid CMP modes can occur when the cavity mode and ferromagnetic resonance (FMR) mode couple. [2,3] The coupling takes place for samples that are sufficiently large with respect to the cavity and whose ferromagnetic linewidth ΔH is narrow enough. Many existing descriptions of this phenomenon utilize quantum mechanics [3,4] in the macrospin formalism. [5] A few involve electrodynamics, [6,7] however, to the authors' knowledge all of these methods have their limitations as compared to the rigorous but straightforward approach discussed in this letter. The presented method concerns not the more common extrinsic [8] but intrinsic permeability tensor of the ferromagnet [9] related to internal sample fields, with the relationship between the internal and external fields imposed by boundary conditions for electromagnetic fields. The diagonal and offdiagonal components m and κ of the relative tensor permeability of a saturated ferromagnet in the low magnetic loss regime (α ( 1) can be conveniently expressed as follows: [9] where α is the Gilbert damping factor, H r ¼ H= 4πM s ð Þis the relative internal magnetic bias,ŵ ¼f = γ4πM s ð Þis the relative; 4πM s is the saturation magnetization of the sample, γ % 2:8 MHz Oe À1 is the gyromagnetic ratio, and Q 0 is the unloaded quality factor of the resonant system including the sample. In this work, a model of an in-plane (IP)-magnetized [111] yttrium iron garnet (YIG) layer grown on a gadolinium gallium garnet (GGG) su...

Electromagnetic Characterization of Shielded Spherical Gyromagnetic Resonators

“…It has been chosen to associate the factor of 2 with the filling factor p rather than with tan δ m in order to conform with the definition of the dielectric loss tangent. It is crucial to note that according to definition (21), tan δ m > 0, while applying a standard definition for a scalar permeability μ d = μ d − j μ d , [38] tan…”

“…In the limiting case of electrically small samples, for which w = H + (1/3) is a very good approximation, (21) simplifies to (see also [1], [27])…”

“…In the low-loss regime (α 1), the diagonal and off-diagonal components μ and κ in (1) can be expressed as follows [21]:…”

Cavity Spintronics: An Early Review of Recent Progress in the Study of Magnon–Photon Level Repulsion

“…For example, Maksymov et al [102] used numerical solutions of the LLG and Maxwell equations, obtained using a finite-difference time-domain method, to consider a resonator formed by a dielectric-magnetic multilayer, Cao et al [103] and Yao et al [104] used a scattering approach in a simplified 1D configuration to examine the effects of coupling, with the influence of spin waves explicitly accounted for in Ref. [103] and finally, the behaviour of the dispersion was examined by Krupka et al [105] and Pacewicz et al [106]. In the former work the authors revisited the role of perturbation theory for sample characterization in traditional cavity based FMR measurements by numerically computing the cavity quality factor in regimes where magnon-photon coupling may play a dominant role.…”

“…Similarly, in Ref. [106] the authors accounted for the intrinsic permeability of the ferromagnetic material and numerically solved the transcendental equation for the electromagnetic field.…”