Volume 34, Issue 4, 01 April 1963
 SPIN‐WAVE INSTABILITIES AND MAGNETOELASTIC COUPLING


Extension of Coincidence Limiting Frequency Range in Ferrimagnets
View Description Hide DescriptionPower limiting using the first‐order spin‐wave instability has been observed in a ferrimagnetically coupled filter‐limiter at higher frequencies than had previously been believed possible. For example, a filter‐limiter, using orthogonal strip‐line resonators coupled by the dipolar fields of a YIG sphere, was constructed at 4000 Mc. Output power was limited with the sharp break at the threshold and flat limiting curve characteristic of coincidence limiters. This result is in contrast with the operation of similar limiters at this frequency which use the second‐order spin‐wave instability in which the limiting characteristic has a much more gradual slope transition above the threshold. Coincidence limiters have previously been believed to have a maximum operating frequency ω_{ c }=2ω _{M}N_{T} (3.27 kMc for a YIG sphere) since, with an isolated sample, coincidence of the uniform precession resonance and the subsidiary absorption occurs only for ω<ω_{ c }.
These results are observed for a coupled mode resonance in these ferrimagnetically coupled circuits which occurs for dc magnetic field values in the subsidiary absorption region below the uncoupled uniform precession value (ω/γ). It is found that a minimum value of the circuit‐sample coupling is required to observe this extension of coincidence type limiting. Limiting thresholds as low as −10 dBm have been observed with the strip‐line circuit and lower levels are believed to be possible. Wide low‐level bandwidths (>10%) are also obtained with these circuits, but with an increase in the limiting threshold.
The above results are explained by computing the normal modes of the microwave‐ferrimagnetic resonator circuit. It is found for tight enough coupling that one of the branches of the normal modespectrum is in coincidence with the subsidiary absorption at a higher frequency than the uniform precession resonance. The usual coincidence limiter operation (ω<ω_{ c }) is thus the small coupling case of the more general type of operation.

Even and Odd Instabilities in Coupled Ferrite Spheroids
View Description Hide DescriptionA theoretical analysis supported by experimental evidence shows that coupling between two ferrimagnetic ellipsoids can lead to a type of instability analogous to the second‐order, Suhl spin‐wave instability in a single sample. Two ferrite samples coupled through their dipolar fields constitute a pair of coupled nonlinear resonators. If the samples are identical, the uniform precession of the system is describable in terms of an even mode in which the transverse components of magnetization in the two samples are parallel, and an odd mode in which the transverse components are antiparallel. The frequency separation of these modes is determined by the strength of the coupling. At sufficiently low power levels the even and odd modes are independent, and the response of the system to a driving field exhibits the usual double‐peaked shape characteristic of coupled linear resonators. At high power levels, however, nonlinear terms, arising primarily from shape anisotropy, produce significant mode interaction. If, for example, the even mode is excited to a large amplitude at ω_{ p } by applying identical driving fields to the two samples, there results a torque at 2ω_{ p } that can lead to exponential growth of the odd mode. Conversely, excitation of the odd mode to a large amplitude can lead to parametric excitation of the even mode. The theoretical analysis predicts the conditions under which growth is attained. Experiments conducted on narrow linewidth disks of single‐crystal YIG are in substantial agreement with the theory. The instability is observed as saturation of the directly driven mode and as radiation from the unstable mode.

Absorption Phenomena in Parallel‐Pump Experiments
View Description Hide DescriptionExperimental results using longitudinally pumped YIG at X band are described. It is shown that absorption as a function of applied field depends on the manner in which the YIG sample is supported. When the sample is loosely mounted the absorption characteristics show evidence of magnetoelastic coupling. A value for the exchange constant determined under this condition is in good agreement with that obtained by Turner. When the sample is more rigidly mounted the absorption characteristic changes markedly and it appears that the magnetoelastic coupling is suppressed. A discontinuity in the absorption characteristic is also observed at a field strength well below the critical threshold value. Differences in absorption between up and down cycles of the sweep field are also reported.

Study of Magnetoelastic Interactions by Parallel Pumping
View Description Hide DescriptionMagnetoelastic interactions, the coupling of spin waves with phonons, have been studied in single crystals of yttriumirongarnet using the parallel pumping technique. Observations of the coupling have been made for various temperatures and crystalline orientations at 4.75 and 11.6 kMc.
An increase in the threshold field for spin‐wave excitation occurs when there is magnetoelastic coupling, and the form of the increase provides a line shape for the spin‐wave‐phonon interaction. From the magnitude, width, and location of the interaction line are found the acoustic Q, the magnetoelastic constant b _{2}, and the spin‐wave exchange constant. The quantities obtained (i.e., Q=2×10^{4} and b _{2}=4.7×10^{6} ergs/cm^{3} at room temperature) are in reasonable agreement with those from lower frequency measurements, assuming a f ^{−1} variation for Q. The magnetoelastic constant b _{2} is found to increase with decreasing temperature. Correlation of the temperature variation of b _{2} with theoretical predictions is good and, if the comparison is valid, provides an evaluation of the higher‐order magnetoelastic constant b _{5}.

Magnetostriction Constants from Ferromagnetic Resonance
View Description Hide DescriptionThe saturationmagnetostrictive constants of a magnetic material may be determined by means of ordinary ferromagnetic resonance. For small strains, the magnetic field required for resonance at a given frequency is a linear function of the applied uniaxial stress. The magnitude of this shift may be directly related to the magnetostrictive constants of the substance under investigation. The constants for YIG, YIG doped with Er^{3+}, YIG doped with Yb^{3+}, nickelferrite, and nickelferritedoped with cobalt have been measured by this new technique in a temperature range from 77° to 350°K. Where comparisons are possible, the results agree quite well with previous measurements. In particular, the measured magnetostrictive constants of YIG are in good agreement with values that have been obtained from magnetoacoustic and strain gauge experiments.

Parallel Pumping in Planar Zn_{2}Y
View Description Hide DescriptionParallel pump absorption measurements have been made on a single‐crystal sphere and a single‐crystal thin flake of Zn_{2}Y. The imaginary part of the susceptibility in each sample reaches a maximum value of about 2 for an rf magnetic field in the vicinity of 25 Oe. The dependence of the critical rf magnetic field upon dc magnetic field is compared to the theory of Bady and Schlömann and the agreement is found to be reasonably good. From the minimum rf critical field, spin‐wave linewidths of 8.0 Oe for the sphere and 9.24 Oe for the thin flake have been calculated.

Unstable Growth of Phonons at High Microwave Power Levels in Magnetically Ordered Single Crystals
View Description Hide DescriptionConditions are given under which transverse and longitudinal microwavephonon standing waves present in a ferrimagneticsingle crystal can be parametrically excited by the uniform precession of the magnetization. The instability thresholds for both first‐ and second‐order processes are derived for crystals having cubic symmetry and are found to depend on higher‐order magnetoelastic energy terms. The phonons need not be degenerate with any portion of the spin‐wave band; in which case, the usual magnetoelastic coupling is absent. Competition from spin‐wave processes is considered, and the conditions necessary for phonon domination are presented. The possibility of either maximizing or forbidding a particular phonon process is also discussed.

Longitudinal Pumping of Magnetoelastic Waves in Ferrimagnetic Ellipsoids
View Description Hide DescriptionThe instability threshold of parametrically excited magnetoelasticwaves 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 ferrimagneticsingle 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 magnetoelasticwaves 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 magnetoelasticwave 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 ∼b ^{2}(2ΔH_{k} )Q (where b is the magnetoelastic energy coefficient, 2ΔH_{k} is the appropriate spin‐wave linewidth, and Q is the appropriate phononQ) 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 ∼λω^{2} Q ^{−1}, where ω is the frequency of the magnetoelasticwave 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 ∼bM ^{2}ωλ^{½}, where M is the value of the saturation magnetization.

Spin‐Wave Growth Under Parallel Pumping
View Description Hide DescriptionThe transient growth of the spin‐wave population immediately after the application of a strong microwave pulse has been calculated, assuming that the spin waves are in thermal equilibrium before the pulse is applied. The theory is compared with experiments performed at room temperature on a spherical single crystal of yttriumirongarnet at X band. Good agreement is found for a time interval of a few μsec starting several μsec after the pulse has been applied. Before this time interval, the absorption coefficient is too small to be measured. After this interval, the transient behavior is influenced by saturation effects which are not taken into account in the theory.

Magnetoelastic Coupling Constants of Terbium and Europium Iron Garnets
View Description Hide DescriptionThe purpose of this paper is twofold: (1) to present data on the magnetoelastic constants of rare‐earth irongarnets, a subject which for various reasons has been little studied thus far, and (2) to describe a new method for obtaining these constants which is applicable to many ferromagneticinsulators and does not require the use of strain gauges. The new method was developed from a study of the coupling introduced between long wavelength acoustic modes and long wavelength spin‐wave modes by ions in which the orbital moment is not quenched, such as rare‐earth ions, Co^{2+}, and Fe^{2+}. The effect of the coupling caused by such ions is observed as a magnetic field dependence of the resonant frequency of an acoustic resonator made from the given material, even above dc saturation. The acoustic Q above dc saturation is also field‐dependent. A theory has been obtained which explains the observed field dependences of both v(H) and Q(H) as well as other important features of the magnetoacoustic interaction. It will appear elsewhere.^{1}
One of the results of this theory is a relation between the cubic magnetoelastic constants, B _{1} and B _{2}, and the acoustic resonant frequency v of a thin disk vibrating in a thickness shear mode. For a (110) disk with the external field H _{0} in the plane along [100] and the rf driving field perpendicular to the disk, we obtain,where H _{eff}=H _{0}+H _{anis}+4πM, and c _{44} is the usual shear elastic modulus, which may be obtained directly from v _{∞} and the thickness of the disk. Equation (1) applies to the mode with displacement along [100]. B _{1} is obtained similarly, only now H _{0} is along [110] in the plane, and the shear displacement is along [110]. Here we have.All data thus far have given excellent fits to these equations as H _{eff} is varied, making it easy to obtain the unknown constants. It should be noted that Eqs. (1) and (2) are high‐temperature approximations and at low temperatures will not in general be as simple.
For TbIG and EuIG at room temperature we find the following (all in ergs cm^{−3}): For TbIG, B _{1}≤3.5×10^{6}, B _{2}=30×10^{6}; and for EuIG, B _{1}=45×10^{6} and B _{2}≤4.2×10^{6}. The signs must still be determined from strain gauge measurements. The observed reversal of the magnetoelastic anisotropy (ratio of B _{1} to B _{2}) is being investigated, as well as the temperature dependences of B _{1} and B _{2}.

Effects of Electrical Conductivity on High‐Power Linearity in Lithium‐Titanium Ferrite
View Description Hide DescriptionNonlinear effects at high rf power levels were investigated in substituted polycrystallinelithiumferrite with respect to changes in the dc conductivity of the material. Selected samples of a lithium‐titanium ferrite system were thermally annealed to produce material with dcresistivities in the range 10^{10} to 100 Ω‐cm. The threshold field, spin‐wave linewidth, and field‐dependent characteristics of the subsidiary absorption curve were investigated by the perpendicular pump technique in a 5600‐Mc cavity. The rf threshold field required for the onset of nonlinearity exhibited a definite dependence upon the resistivity, increasing by a factor of three as the resistivity was decreased to 390 Ω‐cm. The initial permeability spectrum maintained its characteristic form and no significant change was observed in the intrinsic magnetic properties over the range of resistivity values. Data are presented showing the observed relationships between the magnetic and dielectric properties and the dcresistivity. The application of this technique to improved peak power performance in a C‐band wye‐junction circulator is discussed.

Saturation Magnetostriction of Single‐Crystal YIG
View Description Hide DescriptionThe magnetostriction of single‐crystal yttriumirongarnet (YIG) has been measured from liquid nitrogen temperature to 450°K. The theory of magnetostriction of cubic crystals has also been extended to Néel ferrimagnets. The theoretical predictions of the temperature‐dependent magnetostriction coefficients h _{1} and h _{2} are in excellent agreement with the measured values of −h _{1}, which has a maximum (2.1×10^{−6}) near room temperature, and of −h _{2}, which falls monotonically with rising temperature (7.7×10^{−6} at 100°K and 4.2×10^{−6} at 300°K).

Spin‐Wave Propagation and the Magnetoelastic Interaction in Yttrium Iron Garnet
View Description Hide DescriptionExcitation of spin waves by the method of nonuniform internal dc magnetic fields in disk‐shaped samples is demonstrated in two types of experiments. The first is a cw microwave absorption experiment in YIG disks. At high gain an absorption continuum is observed on the low field side of the magnetostatic mode spectrum of a disk magnetized perpendicular to its plane. This is shown to correspond to the expected absorption due to certain spin‐wave modes in terms of the field values at which it occurs, its intensity and general shape. The second type of experiment permits observation of the propagation of microwave pulses via short wavelength spin waves and coupled magnetoelasticwaves. The propagation time of the pulses is analyzed in terms of the group velocity of these waves and found to be in good agreement using known values of the exchange constant and velocities of elastic waves in YIG. The magnetoelastic coupling constant b _{2} in YIG is estimated from these experiments to be of the order of 10^{7} ergs/cm^{3} or less.
