Longitudinal Pumping of Magnetoelastic Waves in Ferrimagnetic Ellipsoids

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Abstract
The instability threshold of parametrically excited magnetoelastic waves is derived, as a function of the dc magnetic field strength, for that region of field where the phonon and magnon spectra intersect. The case considered involves the longitudinal pumping (with an rf magnetic field of constant frequency 2ω), of a small ferrimagnetic single crystal having cubic symmetry. The crystal is ellipsoidal in shape and is magnetized along a [100] direction. (The analysis is also applicable to a uniaxial crystal magnetized along its principal axis.) In this instance, the unstable magnetoelastic waves are a mixture of transverse phonons (polarized in the longitudinal direction) and magnons (spin waves) which propagate at (or very near) right angles to the dc field. If, as is usual, the two branches of the magnetoelastic wave spectrum are split by a frequency large compared to the phonon relaxation frequency, an approximate, and greatly simplified expression may be deduced from the general solution. In this case, it is found that the line shape of the phonon contribution to the threshold is asymmetric, and that the peak of the curve occurs for a smaller value of dc field than is generally believed. This shift in field is ∼b2(2ΔHk)Q (where b is the magnetoelastic energy coefficient, 2ΔHk is the appropriate spin‐wave linewidth, and Q is the appropriate phonon Q) and must be taken into account for an accurate determination of the exchange constant (particularly at the lower microwave frequencies). The peak of the phonon contribution to the threshold curve is independent of this shift and also of b; it is ∼λω2Q−1, where ω is the frequency of the magnetoelastic wave and λ is the exchange constant. The width of the peak at half‐height is essentially independent of Q (except at very low frequencies) and is ∼|b|M2ωλ½, where M is the value of the saturation magnetization.

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