# Ferromagnetic Resonance

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## ferromagnetic resonance

[¦fe·rō·mag¦ned·ik ′rez·ən·əns]## Ferromagnetic Resonance

a type of electron magnetic resonance. Ferromagnetic resonance is exhibited in the selective absorption of electromagnetic field energy by a ferromagnet at frequencies that coincide with the natural precessional frequencies ω_{0} of the magnetic moments of the electronic system of the ferromagnetic specimen in an internal effective magnetic field *H*_{eff}. In a narrower sense, it is the excitation of uniform precessional oscillations (uniform throughout the entire volume of the specimen) of the magnetization vector **J**, that is, the excitation of spin waves with a wave vector **k = 0**; the oscillations are excited by a superhigh-frequency (SHF) magnetic field **H**_{⊥} that is perpendicular to a static applied magnetic field **H**_{0}.

Uniform ferromagnetic resonance, like electron paramagnetic resonance (EPR), can be detected by the methods of radio-frequency spectroscopy. The magnetic susceptibility (and consequently the absorption) relative to the SHF field is proportional to the magnetic susceptibility relative to the static field: χ_{0}= *J _{s}/H*

_{0}, where

*J*is the saturation magnetization of the ferromagnet. Therefore, the absorption is several orders of magnitude greater in ferromagnetic resonance than in EPR. Because of the spontaneous magnetization of a ferromagnet, the field

_{s}*H*

_{eff}may differ substantially from the external field

*H*

_{0}; the difference is due to magnetic anisotropy and the demagnetizing effects of the specimen’s surface (

*see*DEMAGNETIZING FACTOR). Usually,

*H*

_{eff}≠ 0 even when

*H*

_{0}= 0 (natural ferromagnetic resonance). The main characteristics of ferromagnetic resonance—the resonance frequencies, relaxation, the shape and width of absorption lines, and nonlinear effects—are determined by the collective multi-electron nature of ferromagnetism. The quantum-mechanical theory of ferromagnetic resonance yields the same expression for the ferromagnetic resonance frequency ω

_{0}as does the classical treatment: ω

_{0}= γ

*H*

_{eff}, where γ =

*g*μ

_{B}/ℏ is the gyromagnetic ratio,

*g*is the spectroscopic splitting factor (the Landé

*g*factor), μ

_{B}is the Bohr magneton, and ℏ =

*h*/2π is Planck’s constant. In terms of

*H*

_{eff}, the frequency ω

_{0}depends on the shape of the specimen, the orientation of

*H*

_{0}relative to the symmetry axes of the crystal, and the temperature. The presence of domain structure in a ferromagnet complicates ferromagnetic resonance, leading to the possible occurrence of several resonance peaks.

We usually deal with nonuniform ferromagnetic resonance, that is, the excitation by an SHF magnetic field of nonuniform modes of collective oscillations **J**_{s}, or spin waves with wave vector **k** ≠ 0, that are specific to ferromagnets. The existence of several modes of resonance oscillations and several branches of ferromagnetic resonance (spin waves with **k** ≠ 0), in addition to uniform precessional oscillations (with **k** = 0), completely changes the nature of the magnetic relaxation and the broadening of absorption lines in ferromagnetic resonance as compared with EPR. From the quantum-mechanical viewpoint, relaxation processes are described as the scattering of spin waves by one another, by thermal vibrations, or phonons, and by conduction electrons in metals. For example, in uniform ferromagnetic resonance, relaxation is exhibited in the broadening of an absorption line by a quantity Δω_{0} = (∂ω_{0}/∂*H*) × Δ*H* = 2/τ_{0}, where τ_{0} is the relaxation time, that is, the mean lifetime of a spin wave with **k** = 0. The line width Δ*H* for different ferromagnets ranges from 0.1 to 10^{3} oersteds (Oe). Static inhomogeneities, such as impurity atoms, pores, dislocations, and very fine surface roughness of the specimen, play the main role in line broadening. The narrowest line, with Δ*H* = 0.53 Oe, was observed in a single crystal of the compound Y_{3}Fe_{5}O_{12} (yttrium iron garnet). In ferromagnetic metals, one of the principal line-broadening mechanisms in ferromagnetic resonance is associated with the skin effect; because of eddy currents, the SHF field becomes nonuniform and therefore gives rise to a broad spectrum of spin waves. The interaction of spin waves with conduction electrons also has a substantial role in the scattering of such waves in ferromagnetic metals. The width of the narrowest ferromagnetic resonance line in ferromagnetic metals is of the order of 10 Oe.

The nonlinear effects of ferromagnetic resonance are determined by the relationship between the uniform precession of magnetic moments and the nonuniform modes that are absent in EPR. Because of this relationship, as the amplitude of the strength of the magnetic field **H**_{⊥} increases to some critical value **H**_{⊥, cr}, oscillations with certain wave numbers begin to grow very rapidly (exponentially), that is, they are unstably excited. The threshold nature of the unstable excitation is due to the fact that when **H**_{⊥, cr} is reached, some spin waves with **k** ≠ 0 do not transfer the energy they acquired from waves with **k** = 0 to other spin waves or phonons.

Magnetoelastic interactions in ferromagnets (*see*MAGNETOSTRICTION) may lead to the parametric excitation of unstable vibrations of the crystal lattice (phonons) by an SHF magnetic field and to the opposite effect—the excitation of spin waves by an SHF elastic-stress field (hypersound). The study of ferromagnetic resonance has led to the development of many microwave devices that are based on the phenomenon, such as microwave tubes, circulators, oscillators, amplifiers, parametric frequency converters, and limiters.

The resonance nature of the absorption of centimeter electromagnetic waves by ferromagnets was first pointed out between 1911 and 1913 by V. K. Arkad’ev.

### REFERENCES

*Ferromagnitnyi rezonans i povedenie ferromagnetikov v peremennykh magnitnykh poliakh: Sb*. Moscow, 1952. (Translated from English.)

*Ferromagnitnyi rezonans*. Moscow, 1961.

Gurevich, A. G.

*Ferrity na sverkhvysokikh chastotakh*. Moscow, 1960.

Gurevich, A. G.

*Magnitnyi rezonans v ferritakh i antiferromagnetikakh*. Moscow, 1973.

Monosov, Ia. A.

*Nelineinyi ferromagnitnyi rezonans*. Moscow, 1971.

*Magnetism: A Treatise on Modern Theory and Materials*, vol. 1. New York–London, 1963.

S. V. VONSOVSKII