Theory of two magnon scattering microwave relaxation and ... · Theory of two magnon scattering...

JOURNAL OF APPLIED PHYSICS VOLUME 83, NUMBER 8 15 APRIL 1998

Theory of two magnon scattering microwave relaxation and ferromagneticresonance linewidth in magnetic thin films

M. J. Hurben and C. E. Pattona)

Department of Physics, Colorado State University, Fort Collins, Colorado 80523

~Received 22 July 1997; accepted for publication 5 January 1998!

A detailed analysis of the two magnon scattering contribution to the microwave relaxation andferromagnetic resonance linewidth in isotropic and anisotropic films and disks has been made. Theanalysis is based on the Sparks, Loudon, and Kittel~SLK! theory for the scattering of uniform modemagnons into degenerate spin wave states for isotropic spherical samples in the presence ofmagnetic inhomogeneities in the form of spherical voids or pores. The SLK theory has beenextended to include:~i! thin film and thick film samples magnetized in an oblique out-of-planedirection;~ii ! uniaxially anisotropic materials with either easy-axis or easy-plane anisotropy and ananisotropy axis perpendicular to the disk plane;~iii ! a modified density of degenerate states toaccount for the nonzero relaxation rate of the scattered spin waves; and~iv! two limiting cases of thescattering interaction:~a! the original SLK case where the inhomogeneities are modeled as sphericalvoids and the coupling to the degenerate spin waves varies with the spin wave propagation directionand~b! an isotropic scattering model where the coupling is independent of the propagation direction.The formulation is valid for thick films for which the discrete nature of the spin wave modes maybe neglected. The two magnon linewidth as a function of field orientation is calculated for threeclasses of material parameters corresponding to yttrium iron garnet and bariumM -type and zincY-type hexagonal ferrites. The linewidth versus static field angle profiles show characteristicprofiles which depend on the crystalline anisotropy, the sample dimensions, the nature of thescattering interaction, the inhomogeneity size, and the inhomogeneity volume fraction. Theseparameters, as well as the shape and evolution of the spin wave band as a function of the field angleunder ferromagnetic resonance conditions, play critical roles in determining the linewidth versusangle profiles. ©1998 American Institute of Physics.@S0021-8979~98!01208-0#

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I. INTRODUCTION

The discovery of yttrium iron garnet~YIG!1 has led tomany advances in the fundamental understanding of mnetic materials. From a technological point of view, the mimportant of these properties concerns the losses which lthe high speed switching of the magnetization for memelements and control the insertion loss for microwavevices. The basic loss mechanisms for such high frequeprocesses in YIG and related ferrite materials are best exined by microwave techniques at ferromagnetic resona~FMR! as well as off resonance.2 The main microwave lossmechanisms which are usually operative in ferrite materhave been reviewed in detail by Sparks.3

In addition to so-called ‘‘intrinsic’’ mechanisms in verpure, high quality single crystals of YIG materials, whicyield FMR half power field swept linewidths in the 0.0Oe/GHz range at room temperature and in the mOe ranglow temperature, there are two other generic processes wtend to increase the microwave losses above these ‘‘insic’’ levels. The first such ‘‘nonintrinsic’’ process is oftetermed ‘‘two magnon scattering.’’ The second nonintrinsprocess is related to impurities. The present article is ccerned with two magnon scattering.

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A. Two magnon scattering losses andinhomogeneities

In ferromagnetic resonance, the two magnon scatteprocess involves coupling between the uniform mode aspin waves over a range of wave vectors which are degeate with the microwave pump and the FMR response. Tcoupling is typically strongest for spin wave wavelengtwhich are on the order of whatever inhomogeneities maypresent in the material. This process is termed ‘‘two magnscattering’’ because the mechanism can be expressedretically by a second quantization formalism in which a uform precession or FMR magnon is destroyed and a swave magnon at the same frequency is created. In the etheories,4–7 the uniform mode source magnons and the swaves were taken to be degenerate. Since the participamagnons~i! are degenerate and~ii ! have different wave vec-tors, momentum is not conserved for the two magnon stering process. From a formal point-of-view, the pseudmomentum from the spatial variation in the internal fieldsone sort or another which derives from the inhomogeneiserves to conserve the momentum in the scattering calction. Two magnon scattering requires, therefore, the preseof inhomogeneities. Later theories included the possiblefects of secondary scattering.8,9

The two magnon scattering contribution to the FMlinewidth has been investigated theoretically and experim

4 © 1998 American Institute of Physics

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4345J. Appl. Phys., Vol. 83, No. 8, 15 April 1998 M. J. Hurben and C. E. Patton

tally for three types of inhomogeneities in bulk YIG, namesurface pits, volume pores or voids, and randomly distribusingle crystal grains in a polycrystal.4,7 The most detailedpublished theory is for scattering due to an isolated spherpore imbedded in a large sample.3,4 Theoretical and experimental results have been reviewed in Ref. 2.

Critical evidence for the importance of two magnon sctering is contained in the data of Buffler10 on linewidth ver-sus frequency for YIG spheres which were polished to pduce pits of different sizes. Various studies of FMlinewidths in ferrite materials have also demonstrated tdetailed information on the two magnon scattering contrition to the loss may be obtained if the number of spin wawhich are degenerate with the FMR frequency can bejusted experimentally through the use of different samshapes, different field orientations, etc.11–14

B. Thin films and degenerate spin waves

A considerable amount of related work has been donethin films. Ferromagnetic resonance measurements inpermalloy films,15,16 for example, demonstrated the potentimportance of two magnon scattering processes throughuse of the thin film geometry and field orientation to chanthe degeneracy condition. For FMR with an in-plane stamagnetic field and an in-plane magnetized thin film, thare a large number of spin wave states which are degenwith the uniform mode and a significant contribution to ttwo magnon scattering linewidth is possible. If the stamagnetic field is perpendicular to the film and sufficientsaturate the magnetization perpendicular to the film, thereessentially no spin wave states degenerate with the Ffrequency and there should be almost no two magnon stering contribution to the linewidth. For an obliquely magntized film, the FMR frequency lies between these two limThe number of degenerate spin waves to which the unifomode can couple via the two magnon process, and hencrelaxation rate and the linewidth, are strongly dependenthe external field orientation.

The angle dependence of the two magnon scatteringtribution to the linewidth for films has been treated preously by Sparks.17 This treatment was done for two limitecases, and only for isotropic materials. The first partSparks’ treatment developed an approximate expressionthe linewidth due to scattering from such spherical vowithin the film, but did not explicitly consider the role of thvoid size. The second part considered scattering from laetch pits which extended through the film thickness. Twork shows the wealth of two magnon effects which aimportant for thin films.

C. Objective of this work

The objective of this work is to provide a specific, anlytical, and operational theoretical formalism for two manon scattering relaxation and the FMR linewidth in thfilms. A summary of the theory is given in Ref. 18.

The analysis is based on the transition probability callation of Sparks, Loudon, and Kittel4 ~SLK! for isotropicspherical samples. In this approach, the microstructur

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modeled in terms of spherical voids or pores. The calculatresults in a relaxation rate expression which reflects thesumption of spherically symmetric inhomogeneities. Whileis possible to extend the analysis to include nonspherinhomogeneities,17,19,20no attempt has been made to inclusuch effects here. For a wide distribution of pore or voshapes, it is reasonable to adopt an ‘‘isotropic scatterinlimit in which the angular coupling term which derives frothe dipole field distribution around a spherical void is rplaced by some average value. The ‘‘spherical void scating’’ and ‘‘isotropic scattering’’ limits give very differenttwo magnon linewidth versus field angle profiles. These dferences will provide a way to separate very different kinof scattering processes.

Previously two magnon analyses have been limitedmaterials which were either isotropic or with relatively smlevels of magnetocrystalline anisotropy such that the behior of the spin wave band was essentially the same asisotropic materials. Schlo¨mann et al.,21 among others, hasshown that anisotropy can have a significant effect onspin wave dispersion and the corresponding spin wave bExplicit examples of such effects and further references mbe found in Refs. 21–24.

The modifications in the spin wave band due to anisropy have a large effect on the two magnon linewidth verangle profiles. From a technological point of view, thesefects are extremely important. This is due, in part, to tcontinuing interest in hexagonal ferrite materials fmillimeter-wave device applications,25 and the recent development of pulse laser deposited~PLD! single crystal bariumferrite ~Ba–M! films.26 These new Ba–M PLD films, as weas other PLD ferrite films,27,28 have rather large FMR linewidths. As these materials are developed and refinedmillimeter-wave device applications, it will be important tconsider the possibility of a two magnon scattering contribtion to the high frequency losses, the types of microstructresponsible for these losses, and ways to modify the fimorphology to eliminate these effects and reduce the los

The objective of this work, then, has been:~i! to developa clear, practical, and operational formalism for the two mnon scattering relaxation rate in anisotropic ferrite films;~ii !to make explicit the role of different types and sizes ofhomogeneities on the two magnon losses;~iii ! to make ex-plicit the role of anisotropy modifications to the spin waband on these processes; and~iv! to provide example calculations of profiles of linewidth versus external field angwhich demonstrate these effects.

Section II defines basic parameters and establishes oating equations for static equilibrium and the uniform moFMR response when the static magnetic field is appliedsome angleu relative to the film normal. Section II alsoestablishes operational equations for the FMR frequenvarious effective field parameters, and a phenomenologrelaxation rate which will be used to characterize intrinlosses. Section III provides a brief review of spin wave proerties for isotropic and uniaxially anisotropic materials.Sec. IV, the relaxation rate or inverse relaxation timescattering between the uniform mode and the degeneratewaves is developed along the same lines as used in the o

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4346 J. Appl. Phys., Vol. 83, No. 8, 15 April 1998 M. J. Hurben and C. E. Patton

nal SLK theory. Here, however, specific working equatioare obtained for scattering into a band of spin waves of nzero width. In Sec. V, these working equations are usedcalculate the two magnon linewidth as a function of theternal field orientation for representative cases of isotroeasy-axis, and easy-plane ferrite films and disks, and a vety of microstructure options which correspond more-or-lto those expected in real materials.

II. UNIFORM MODE ANALYSIS

This section considers various aspects of the unifomode response in thin film ferromagnetic resonance. Thetopics are~i! static equilibrium,~ii ! dynamic response aneffective fields,~iii ! FMR field as a function of static fieldangle relative to the film normal at fixed frequency, and~iv!FMR linewidth versus angle for fixed frequency and fixrelaxation rate. The dynamic response analysis is basethe Kittel29 formulation of ferromagnetic resonance andlaxation according to the Bloch–Bloembergen formulation30

The effect of anisotropy is included through a free eneapproach.

Example results are given for YIG, Ba–M, and Zn–materials. Because it has a relatively small cubic anisotroYIG is treated as an isotropic material. The hexagonalrites, on the other hand, exhibit large uniaxial anisotropwith the easy direction along thec axis for Ba–M and in thec plane for Zn–Y. For these materials, thec axis is taken tobe oriented normal to the disk. This uniaxial anisotropy mbe characterized in terms of a anisotropy energy densitthe formEK52KU cos2 f, whereKU is the uniaxial anisot-ropy energy density in erg/cm3 and f is the angle betweenthe magnetization vectorM and thec axis. PositiveKU cor-responds to an easy axis material with the easyM directionalong the disk normal. NegativeKU corresponds to an easplane material with the easyM direction in the disk plane. Itwill prove convenient to define an effective anisotropy fieHA52KU /Ms , whereMs is the saturation magnetization othe material in emu/cm3. Positive values of the anisotropfield HA correspond to easy-axis anisotropy and negavalues to easy-plane anisotropy. AnHA value of zero corre-sponds to an isotropic material. Typical room temperatvalues ofHA for hexagonal ferrite materials are 16.3 kOe fBa–M and 29.0 kOe for Zn–Y.31 Typical values of theroom temperature saturation induction 4pMs are 1.75 kG forYIG, 4.7 kG for Ba–M, and 2.1 kG for Zn–Y.31 Gaussianunits will be used throughout this work.

A. Sample geometry and static equilibrium

Consider a thin, uniaxially anisotropic disk or film manetized by the application of an external static magnefield. Figure 1 shows the sample oriented relative to soright-handedX-Y-Z frame such that the sample normalparallel to theZ axis. This direction also corresponds to tuniaxial crystallinec axis. The disk is assumed to have symmetry about thec axis and to be characterized by the iplane and out-of-plane demagnetization factorsNXY andNZ

which satisfy the conditionNZ12NXY51. A static externalmagnetic fieldHext is applied at an angleu relative to thesample normal and is taken to lie in theY-Z plane. At static

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equilibrium, the static magnetization vectorM s also lies inthe Y-Z plane and is directed at an anglef relative to thesample normal. The specific situation in Fig. 1 would appto an isotropic material, an easy-plane material withHA

,0, or an easy-axis material withHA,4pMs . For a Ba–Mdisk with easy-axis anisotropy andHA.4pMs , the magne-tization anglef would be less than the field angleu.

The condition for static equilibrium is found if the netorque onM s is set equal to zero. The net torque is a resultthe external field, the demagnetization field, and the aniropy which acts to pull the magnetization into an easy dirtion. This condition yields an expression which relatesfield and magnetization anglesu andf

4Hext sin~u2f!5@4pMs~123NZ!12HA#sin~2f!.~1!

Note that forHext.uHAu, the internal field and magnetizatiovectors will be parallel withHext applied either perpendiculato the disk, withf5u50°, or in-plane, withf5u590°.

B. Dynamic response, relaxation, and effective fields

In the small signal limit, the conditionum(t)u!Ms issatisfied and the total time dependent magnetization veM (t) may be resolved into a static component oriented pallel to the saturation magnetization vectorM s and a smalldynamic componentm(t) perpendicular toM s

M ~ t !'M s1m~ t !. ~2!

In the uniform mode analysis,m(t) is assumed to be independent of position throughout the sample.

The precessional motion of the total magnetization vtor about the equilibrium direction can be driven by the aplication of a microwave fieldh(t) perpendicular to the staticmagnetization direction. The microwave field is taken touniform throughout the sample, to vary sinusoidally withfrequencyv, and to be directed along theX axis in Fig. 1.This insures thath(t) will be perpendicular toM s regardlessof the anglesu andf. Ferromagnetic resonance results whthe microwave frequency corresponds to the naturalquency of the precession.

There are two standard ways to perform the FMR eperiment. In the first approach, the static fieldHext is appliedat some angleu and held at a fixed strength while the pum

FIG. 1. Disk, field, and static magnetization geometry for the static equirium and FMR analysis. The external fieldHext is applied at an angleurelative to the sample normal and crystallinec axis and lies in theY-Zplane. The static magnetizationM s lies in the Y-Z plane at an anglefrelative to theZ axis.

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frequencyv is varied. The response is then characterizedthe FMR frequencyvFMR at which the maximum power isabsorbed and a corresponding frequency linewidthDv. Thislinewidth is the interval, in frequency units, between thalf-power points of the microwave absorption profile.

In the second approach, and the one which is favoexperimentally, the microwave frequency is fixed atv whilethe strength of the external fieldHext is varied. The two basicparameters of interest are then the FMR fieldHFMR and fieldlinewidth DH. The fieldHFMR is defined to be the value oHext at which the maximum microwave power is absorbeThe linewidthDH is the interval, in field units, between thhalf power points.

In order to determine the FMR parametersvFMR,HFMR, Dv, andDH, a modified version of the torque eqution is often used. The modification involves the additionphenomenological damping terms which account for thelaxation of the magnetization. The analysis leads to relativsimple working equations for the FMR frequencyvFMR andlinewidth Dv. Expressions for the corresponding field prametersHFMR and DH are much more involved. Determnation of these parameters is discussed shortly.

The two most commonly used phenomenological mofications to the torque equation which account for lossthe Bloch–Bloembergen~BB!30 and Landau–Lifshitz~LL !32

approaches. In this work, the BB formalism will be usexclusively, because it is physically consistent with the tmagnon process. A thorough discussion of the various reation mechanisms and the phenomenological dampingmalisms is given by Sparks.3 Various phenomenologicadamping approaches to ferromagnetic resonance are alsocussed by Lax and Button33 and Patton.2

The modified torque equation of motion for the tranverse dynamic magnetizationm(t) with BB loss includedmay be written as

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HereH(t) represents the total effective magnetic field in tsample,g is the absolute value of the electron gyromagneratio, and 1/T is the transverse and longitudinal BB relaation rates. A typical value forg is 1.763107 rad/s Oe or 2.8GHz/kOe in practical units. For a thin disk or film magntized out-of-plane, a 1/T value of 1.763107 rad/s would cor-respond to a field swept linewidth of 1 Oe. Note that 1/T is inunits of rad/s. The calculations presented below will yieexpressions for 1/T. When numerical results are presentethese will be given in terms of 1/gT, in Oe, corresponding tofield linewidths. The BB formulation also includes a longtudinal relaxation term for thez component of the total dynamic magnetization. This term is not considered in a lintheory.

The net magnetic fieldH(t) consists of the applied fieldHext andh(t) along with the static and dynamic fields whicresult from sample demagnetization and non-Maxwellianfective fields related to magnetocrystalline anisotropy. Tnet effective fieldH(t) can be expressed in terms of a modfied demagnetizing tensorA according to

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H~ t !5Hext1h~ t !1A–M ~ t !. ~4!

In the X-Y-Z frame, the tensorA is given by

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For purposes of the usual uniform mode analysis,components of the dynamic magnetization are assumehave aneivt dependence. The analysis leads to an expresfor the FMR frequencyvFMR which is valid to first order inthe loss term 1/T

vFMR5g~HxHy!1/2, ~6!

Hx5Hextcos~u2f!1 12@2HA24pMs~3NZ21!#

3cos2 f, ~7!

Hy5Hext cos~u2f!1 12@2HA24pMs~3NZ21!#

3cos~2f!. ~8!

TheHx andHy parameters may be viewed as effective stiness fields which characterize the instantaneous torqueerted onM (t) when it is tipped parallel toward theh(t)direction or perpendicular to theh(t) direction, respectively.

Although the above results givevFMR as a function ofHext, the explicit dependence of the value of the extermagnetic field,HFMR, on the field angleu for resonance at agiven frequencyv, can also be determined. Once the samparametersg, 4pMs , HA , and NZ are specified, Eqs.~1!,~6!, ~7!, and~8!, with Hext andvFMR replaced byHFMR andv, respectively, can be solved forHFMR as a function ofu.

C. FMR field versus angle at fixed frequency

Figure 2 shows the calculated FMR fieldHFMR as afunction of the external field angleu for the three representative materials described above and in the infinite thin filimit where NZ51 is satisfied. Figure 2~a! is for YIG atv/2p510 GHz, Figure 2~b! is for Ba–M at 50 GHz, andFigure 2~c! is for Zn–Y at 10 GHz. The frequencies werchosen to be representative of typical operating frequenfor these materials. These same material—frequency comnations will be used throughout this work for purposesnumerical evaluations.

Consider the YIG results in~a! first. For an external fieldapplied perpendicular to the film plane atu5f50°, theFMR field is a maximum, while the minimum occurs for thin-plane configuration atu5f590°. The variation inHFMR

with the field angle is due to the change in the samplemagnetization and stiffness fields. For the Ba–M case in~b!,the situation is reversed because of the very strong easyanisotropy perpendicular to the film plane. Here, the FMfield is a minimum foru5f50°, which corresponds to theeasy direction, and a maximum when the field is in-plaalong a hard direction. For the Zn–Y result in~c!, the anisot-ropy acts together with the demagnetization fields to pullmagnetization vector in-plane. For the out-of-plane oriention, the field required for ferromagnetic resonance is v

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FIG. 2. Calculated static external field for ferromagnetic resonance,HFMR , as a function of the external field angleu for the three representative materialsthe infinite thin film limit. ~a! is for YIG at 10 GHz.~b! is for Ba–M at 50 GHz.~c! is for Zn-Y at 10 GHz.

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large, but this field then rapidly decreases as the fieldtipped away. The rapid change inHFMR reflects the severemisalignment between the static field and magnetization vtors. For field angles above about 30° or so, the static mnetization is essentially in-plane.

The results in Fig. 2 demonstrate an important effectthe external field orientation on the FMR response for a cstant microwave frequency, namely, that the strength ofapplied static field and hence the net field within the matevary strongly with field angle. This effect will play a cruciarole in the two magnon scattering and linewidth versus antheory developed shortly.

D. Linewidth considerations and linewidth versusangle

The FMR analysis also yields an expression for thefrequency swept linewidthDv. This linewidth is conve-niently expressed in magnetic field units asD(v/g). The BBlinewidth in field units is

D~v/g!51

gT. ~9!

In contrast with the FMR frequency result of Eq.~6!, forwhich vFMR varied as the geometric mean of two stiffnefields, the BB linewidthD(v/g) is constant and proportionato the transverse relaxation rate.

Unlike the simpleD(v/g)51/gT result of Eq. ~9!, acorresponding expression for the field swept linewidthDH inthe field swept FMR experiment at constant frequencymuch more complicated. This can be understood fromexamination of the field and magnetization vectors durthe course of an FMR experiment. For the frequency swcase, the orientation and strength ofHext are held fixed, sothat the equilibrium direction of static magnetizationM s alsoremains constant. For the field swept case, the directioHext is fixed while its strength is varied. As can be seen froEq. ~1!, the orientation anglef for M s will then vary asHext

is swept, unlessu is either 0° or 90°. As a result,DH can bea complicated function ofu even for constant 1/T. It will beseen from the two magnon scattering theory that the reation rate 1/T itself shows strong angle dependences as w

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It is important from the outset, therefore, to separate aunderstand the geometric linewidth angle dependenwhich result in field swept experiments.

For a given 1/T value and a particular field angleu, thefield linewidth DH can be determined directly from the explicit microwave susceptibility expressions obtained frothe uniform mode analysis. Although it is not possibleobtain simple closed form expressions forDH as a functionof the field angleu, it is possible to evaluate the susceptibity response numerically and determineDH as the separationin field values which correspond to the half-power pointsthe absorption. A simpler approach, and one which canused to demonstrate the intuitive connections betweenDHand D(v/g), is to invoke a simple connection between ttwo linewidths which is strictly valid only in the limit ofsmall linewidths. This connection may be written as

DH'g]HFMR

]vFMRD~v/g!. ~10!

The derivative can be determined from Eqs.~1!, ~6!, ~7!, and~8!. In the limit thatDv!vFMR is satisfied, the linewidthspredicted by the two approaches agree to within severalAs will be discussed below, the linewidth conversion facg]HFMR/]vFMR is equal to one foru50°, greater than onewhen magnetization rotation effects dominate the field swline broadening at angles betweenu50° andu590°, and isless than one for parallel resonance atu590°.

The effect of the field orientation on the FMR linewidtfor a constant relaxation rate is illustrated in Fig. 3. Tfigure follows the form of Fig. 2, with~a! for YIG, ~b! forBa–M, and~c! for Zn–Y. The calculations are based on thinfinite thin film limit and the same material parameters afrequencies used in Fig. 2, along with 1/gT values of 0.5, 20,and 15 Oe, for the YIG, Ba–M, and Zn–Y cases, resptively. These values correspond to reasonable out-of-plresonance linewidths due to intrinsic losses in these mateat the chosen frequencies.

Consider the results for the YIG film in Fig. 3~a!. Here,the field swept linewidthDH is equal to 1/gT for the out-of-plane orientation and shows only a small increase at inmediate angles. There is a small but indiscernible dropDH from 1/gT at u590°. In the case of YIG at 10 GHz, th

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FIG. 3. Calculated ferromagnetic resonance field swept linewidthDH as a function of the external field angleu for the three representative materials in thinfinite thin film limit and for constant values of the BB relaxation rate 1/T. ~a! is for YIG at 10 GHz with 1/gT50.5 Oe.~b! is for Ba–M at 50 GHz with1/gT520 Oe. ~c! is for Zn–Y at 10 GHz with 1/gT515 Oe.

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conversion factor between the field linewidth and the relation rate given by Eq.~10! is very close to one and essetially independent of angle. The small peak aroundu535° isdue to the small difference between the field angleu and themagnetization anglef as the field is rotated at intermediaangles. For in-plane ferromagnetic resonance atu5f590°, the YIG linewidth for BB damping is slightly smallethan the perpendicular linewidth at 0.47 Oe. This relatinsensitivity of the YIG field swept linewidth to rotation idue to the near alignment ofM s with Hext as the field isrotated andHext is maintained at the value needed for ferrmagnetic resonance at 10 GHz.

The situation is quite different for the Ba–M linewidtresults in~b! of Fig. 3. Here, the field swept linewidth showa very significant variation with angle. Note that the laruniaxial anisotropy for the Ba–M film, with the easy axperpendicular to the film, causes the FMR field to be smau5f50° and large atu5f590°. For intermediateu val-ues, theM s vector tends to be somewhat closer to the finormal than doesHext and the magnetization anglef lagsbehind the field angleu. For ferromagnetic resonance at 5GHz and the combination of Ba–M parameters used hthe field swept linewidth for a constant 1/gT values of 20 Oeincreases to a maximum value of about 33 Oe near an enal field angle of 60° and then drops to a value slighbelow the perpendicular field linewidth atu5f590°.

As shown in~c!, this effect is even more pronounced fZn–Y film parameters and 10 GHz ferromagnetic resonanIn this case, the anisotropy is in-plane and serves to enhthe demagnetizing effects. Here,DH increases very rapidlyas the external FMR field is rotated away from the perpdicular FMR orientation atu50°. The peak inDH occurs atu'8° and is more than a factor of 5 greater than the liwidth atu50°. It is important to keep in mind that the 1/gTvalues in each of these calculations were taken to be cstant, independent ofu.

It is instructive to compare the angle dependences oflinewidths in Fig. 3 with the angle dependences of the FMfields shown in Fig. 2. It can be seen that the variation in

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linewidth with the field angle roughly tracks the changethe slope of theHFMR vs u curve in each scenario.

III. SPIN WAVE DISPERSION AND DEGENERATE SPINWAVES

This section presents a brief review of spin wave cosiderations which will be important for the two magnon prcess. Dispersion relations of spin wave frequencyvk versuswave vectork are obtained. This section also providesexplicit examination of the change and frequency shift inspin wave band for YIG, Ba–M, and Zn–Y films as onmoves from the perpendicular FMR configuration with tfield at u50° to the parallel FMR case with the field atu590°. These considerations lead to an intuitive understaing of the change in the degenerate magnon situationhence, the states available for two magnon scattering, asmoves from the perpendicular to the parallel FMR configrations. The presentation is limited to so-called bulk spwaves for which the wave numberk is much greater than2p/S, where S is the film thickness. The magnetostatmode limit in whichk may be on the order of or smaller tha2p/S, but still above the pure electromagnetic wave limvk /c, wherec is the speed of light, will not be considerehere. Possible effects due to magnetostatic modes andthickness effects other than demagnetizing factors willbriefly considered at the end of the article.

A. Classical bulk spin waves

For a bulk material, the sample boundaries can be tato lie at infinity and do not influence the dispersive propertof the spin wave modes. For thin film materials, however,boundary conditions of the magnetic field vectors can snificantly modify these dispersion relations. Calculationthe so-called dipole-exchange modes, which are the normodes for thin film materials, is in general extremely coplicated and is beyond the scope of this work. Referencesdipole-exchange modes include Wolfram and De Wame34

De Wames and Wolfram,35 and Kalinikoset al.36

In this work, the approach of Sparks18 is followed. Thebulk spin wave dispersion relations are used to approximthe normal modes of the film. This results in a considera

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simplification of the two magnon scattering analysis. As loas the microstructure responsible for the two magnon inaction is much smaller than the film thickness, the bulk swave dispersion relations provide a reasonable modeltwo magnon calculations.

In order to determine the bulk spin wave dispersionlations, Eq.~3! can be re-written in the lossless form, withe magnetization and field vectors taken to be functionsspace as well as time

dm~r ,t !

dt5gM ~r ,t !3H~r ,t !. ~11!

Analogous to Eq.~2!, the total magnetization is now of thform M (r ,t)'M s1m(r ,t). The spatial dependences in thH(r ,t) enter through the spatial dependence ofm(r ,t) andthe associated dipole–dipole interactions, exchange intetions, and anisotropy considerations. These effects aresidered shortly.

The inclusion of spin waves in the model requiresextension of the basicX-Y-Z coordinate system defined iFig. 1. The two important extensions are depicted bydiagrams in Fig. 4. First, it is useful to introduce an adtional reference frame as defined in Fig. 4~a!. Here, in addi-tion to the sample or laboratoryX-Y-Z frame, one considersanx-y-z frame of reference in which the static componentthe magnetizationM s is directed along thez axis. For thepresent purposes, thisz axis is taken to lie in theY-Z planeof the sample reference frame and is at an anglef relative totheZ axis. Recall thatf is the angle between theM s and thefilm normal. WithM s along thez axis, the precession of thfull magnetization vectorM yields a dynamic magnetizatiom which, to first order, only hasx andy components. Hencethe x-y-z frame may be termed the ‘‘precessional’’ framThis frame, as depicted in Fig. 4~a!, has thex axis alongXand they axis in theY-Z plane and rotated away from theYaxis by the anglef. Note that for a static magnetizatioangle f50°, the x-y-z frame is equivalent to theX-Y-Zframe. For other orientations, thex-y-z frame is obtained bya clockwise rotation of theX-Y-Z frame about theX axis byan anglef.

One may now establish parameters to define the swave wave vectork. A suitable definition for these parameters may be based on~b! of Fig. 4. The wave vector is

FIG. 4. ~a! Relative orientations of the sample (X-Y-Z) and precessiona(x-y-z) frames. Thez axis is taken to be parallel to the static magnetizatvectorM s and thex axis is taken to be parallel to theX axis.~b! Orientationof the propagation wave vectork relative to thex, y, and z axes in theprecessional frame. The polar angleuk is measured relative to thez axis.The azimuthal anglefk is measured relative to thex axis.

gr-nor

-

f

c-n-

e-

f

in

specified in terms of a magnitudek, a polar angleuk , whichis measured relative to thez axis, and an azimuthal anglfk , which corresponds to the angle between the projecof k on thex-y plane and thex axis.

In order to determine the spin wave dispersion relatiofrom Eq. ~11!, the dynamic magnetization associated withgiven spin wave is written in the form of a plane wave wia wave vectork and frequencyvk

m~r ,t !5mke2 i ~k–r2vkt !. ~12!

The two components of the vectormk are the complexx andy amplitudes of the spin wave andr denotes a general spaccoordinate. The magnetic fieldH(r ,t) consists of a numbeof terms which may be written as

H~r ,t !5Hiz1han~r ,t !1hdip~r ,t !1hexch~r ,t !. ~13!

Here,Hi is an effective internal static magnetic field and tthree additional fields denoted by lower caseh are dynamicfields associated with the spin wave dynamic magnetizaof Eq. ~12!. The first dynamic field,han(r ,t), is associatedwith the crystalline anisotropy, the fieldhdip(r ,t) is the spinwave dipole field, andhexch(r ,t) is an effective exchangefield for the spin wave.

The various terms in the effective field expression of E~13! have been developed in the literature in various formaThe basic approach is given in Ref. 23, among others. Iassumed that the static equilibrium condition of Eq.~1! issatisfied. This means that the static effective field is parato the static magnetization vector, and hence, along thzaxis. The static field termHi may be written as

Hi5Hext cos~u2f!22pMs@sin2 f1NZ~3 cos2 f21!#

1HA cos2 f. ~14!

The first term on the right side of Eq.~14! is simply thecomponent of the static external field along the directionthe static magnetization vector. The second term is the ctribution from the static demagnetization field. The thiterm in Eq.~14! is the contribution to the static effective fieldue to the uniaxial anisotropy. It is important to emphasthat this term does not represent a real magnetic field inMaxwellian sense. It is an effective field based on the scific form taken for the anisotropy energy. If a different forfor EK(f) was used,EK51KU sin2 f, for example, this lastterm in Eq. ~14! would change, as would Eq.~15! for thedynamic anisotropy fieldhan.

The next term in Eq.~13! is the effective dynamic anisotropy fieldhan(r ,t). This term is needed to account for thdynamic effects of the uniaxial anisotropy associated wthe dynamic magnetization. For anisotropy as defined abthe dynamic effective anisotropy field is given by

han~r ,t !5HAmk–y

MS~y sin2 f2z sin f cosf!

3e2 i ~k–r2vkt !. ~15!

This effective anisotropy field is specified in thex-y-zframe. Procedures to obtain such effective fields arecussed in Ref. 23.

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The next field,hdip(r ,t), accounts for the dipole–dipolinteraction due to the misalignment of nearby spins assated with the plane wave excitation of Eq.~12!. For the caseof the uniform precession, wherek50, there is no such di-pole field because all of the spins are parallel. The dipfield can be conveniently expressed in terms of the wvector and the dynamic magnetization for the spin wavecording to

hdip~r ,t !524p

k2 k@k–mk#e2 i ~k–r2vkt !. ~16!

The dipole field given in Eq.~16! is valid in the so-calledmagnetostatic approximation in which the wave numbekfor a given frequency is much greater than the correspondpure electromagnetic wave wave number. These consiations are discussed in Ref. 37.

The last field term in Eq.~13! is the effective exchangefield hexch(r ,t). This field, like the static and dynamic anisoropy fields, is not a true Maxwellian field, but is neededorder to account for the quantum mechanical exchange inaction. For a ferromagnetic material, this field can be writas

hexch~r ,t !52D

Msk2mke

2 i ~k–r2vkt !. ~17!

The form of this exchange field is discussed in Ref. 3, amothers. TheD parameter characterizes the strength ofexchange. For ferrite materials,D has a nominal value in the531029 Oe cm2/rad2 range. This value will be used for thexplicit numerical results presented below. It is assumedD is independent ofk. Note that for the hexagonal ferritmaterials,D may actually show a strong dependencepropagation direction.11,21 Such effects have not been included in the present analysis.

In order to obtain dispersion relations of frequencyvk asa function ofk, the torque equation of Eq.~11!, together withthe above effective field equations, are first linearized inx and y components of the dynamic magnetizationmk ,taken asmkx andmky , respectively. One then obtains a settwo homogeneous equations inmkx and mky . The seculardeterminant from these equations leads to an expressiothe spin wave frequencyvk as a function of the wave vectok. The general spin wave dispersion relation for a uniaxiaplanar anisotropy material is obtained as

vk25g2~Hi1Dk2!~Hi1Dk214pMs sin2 uk2HA sin2 f!

2g24pMSHA sin2 f sin2 uk cos2 fk . ~18!

The validity of this dispersion relation is contingent upon tsatisfaction of the stability condition of Eq.~1!, as well as theother conditions specified above.

Equation~18! gives the general form of the bulk spiwave dispersion relation for a uniaxially anisotropic matermagnetized at any anglef relative to the crystallinec axis inthe magnetostatic approximation. If the anisotropy fieldHA

is set to zero, this equation reduces to the well known dpersion relation for an isotropic ferrite material

vk25g2~Hi1Dk2!~Hi1Dk214pMs sin2 uk!. ~19!

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For the case of a uniaxial material with the external fieldthec plane, such thatf590° is satisfied, Eq.~18! reduces tothe hexagonal ferrite dispersion relation of Joseph, Sch¨-mann, and Bady.21

Two other special cases of the spin wave dispersionlation will be useful for the two magnon calculations. Thfirst is the dispersion relation corresponding to propagatalong the internal field and static magnetization directiosuch thatuk50 is satisfied. This represents the lowest posible spin wave frequency. This frequency is denoted asvmin

and is given by

vmin2 5g2~Hi1Dk2!~Hi1Dk22HA sin2 f!. ~20!

The second special dispersion relation is for propagationpendicular to the static magnetization direction atuk590°.This frequency, denoted asvmax, corresponds to the top othe spin wave band

vmax2 5g2~Hi1Dk2!~Hi1Dk22HA sin2 f!

1g24pMs~Hi1Dk22HA sin2 f cos2 fk!.

~21!

Note thatvmax depends on the azimuthal spin wave propgation anglefk , while vmin does not depend on this angle

It is clear from Eqs.~14! and ~18! that the spin wavedispersion will depend strongly on the orientation of tstatic field and magnetization vectors. As shown in Fig.this field value can vary over a wide range as the orientaangleu of the external field is varied over the entire rangevalues from 0° to 90°. In the case of an isotropic materthe effect of this field variation is to shift the spin wavdispersion frequency band, relative to the FMR frequencyu is changed. For anisotropic materials, the effect is mumore involved. The spin wave dispersion curves are not oshifted, but change in shape as well.

B. Spin wave dispersion and two magnon scattering

It is instructive to consider representative spin wave dpersion band diagrams for the three material cases introduabove. Such diagrams for YIG, Ba–M, and Zn–Y materiare shown in Figs. 5–7, respectively. In each case, bdiagrams are shown for a range of external field orientatifrom out-of-plane atu50° to in-plane atu590°. For eachdiagram in a given figure, the value of the internal fieldHi

was adjusted to produce the FMR peak response at a parlar fixed FMR frequency and a range of illustrative externfield orientations. Figure 5 for YIG shows the band diagrain the usual two dimensional format of spin wave frequenvk versus wave numberk. Figures 6 and 7 for Ba–M andZn–Y show both two dimensional and three dimensioplots.

Consider the diagrams in Fig. 5 for an isotropic thin YIfilm and an FMR frequency of 10 GHz. The values ofHext

needed to obtain the diagrams in Fig. 5 are the same asHFMR values in Fig. 2~a!. Figure 2~a! is for the field appliedout of the plane atu50°, 2~b! is for u545°, and 2~c! is foru590°. In each case, two specific dispersion branchesspin wave frequencyvk versus wave numberk are shown.

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FIG. 5. Diagrams showing bulk spin wave dispersion curves of spin wfrequencyvk vs wave numberk for a thin YIG film and three orientations othe external static field, with the sample biased at the static field needeFMR at 10 GHz. The lower curve in each diagram designates spin wpropagating parallel to the static internal field direction and the upper cis for spin waves propagating perpendicular to the internal field directThe region between the two curves represents the range of spin wbetween these limiting case propagation directions. For~a! the static field isapplied perpendicular to the film plane atu50°. For ~b! the static field isapplied atu545°. For ~c! the static field is applied in the film plane atu590°. The FMR frequency is indicated by the large solid circles. In~b! and~c!, the narrow band of spin wave modes, which is degenerate with the Ffrequency, is indicated by the shaded region.

FIG. 6. Bulk spin wave dispersion diagrams for a thin Ba–M film and foorientations of the external static field, with the sample biased at the sfield needed for FMR at 50 GHz. The top part of each set of diagrams shcalculated dispersion curves in the usual spin wave frequencyvk vs wavenumberk format used in Fig. 5. For the upper diagrams, the top curves in~a!and the top pair of curves in~b!, ~c!, and~d! correspond to a polar spin wavangleuk of 90°, and the bottom curves in~a!–~d! correspond touk50°. Thehorizontal dashed line in~a! and the hatched line in~b!–~d! indicates theFMR frequency. The bottom part of each set shows schematic illustratof the spin wave bands in three dimensions, with the azimuthal spin wpropagation anglefk out of the page. For these lower diagrams, the tshaded surface corresponds touk590° and the bottom shaded surface is fuk50°. The middle horizontal plane, which crosses all the diagrams, icates the FMR frequency. For diagram set~a! the static field is appliedperpendicular to the film plane atu50°. For set~b! the static field isapplied atu560°. For set~c! the static field is applied atu575°. For set~d!the static field is applied in the film plane atu590°.

The lower curve corresponds to spin waves atuk50°, orpropagation parallel to the static magnetization direction,specified by Eq.~19! or by the frequencyvmin of Eq. ~20! forHA50. The upper curve corresponds to spin waves atuk

590°, or propagation perpendicular to the static magnettion direction, as specified by Eq.~19! or by the frequencyvmax of Eq. ~21! for HA50. The band of available spinwaves consists of the regions between these lower and ucurves. The FMR frequency is indicated in each diagramthe solid circle. The horizontal shaded strips in~b! and ~c!indicate the range of spin waves which are degeneratethe FMR frequency. These states will be important for ttwo magnon scattering interaction.

The important result in Fig. 5 for two magnon scatteriis the shift of the band as the external field orientationrotated from perpendicular to parallel. In the perpendicuconfiguration of~a!, the bottom of the spin wave band atk50 is coincident with the FMR frequency and there arenonzerok spin wave states at this frequency. In the paraconfiguration of~c!, the band has dropped down so that ttop of the spin wave band atk50 is coincident with theFMR frequency. There is now an extended range of swave states degenerate with the FMR frequency. Th

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FIG. 7. Bulk spin wave dispersion diagrams for a thin Zn–Y film and throrientations of the external static field, with the sample biased at the sfield needed for FMR at 10 GHz. The top part of each set of diagrams shcalculated dispersion curves in the usual spin wave frequencyvk vs wavenumberk format used in Fig. 5. For the upper diagrams, the top curves in~a!and the top pair of curves in~b! and ~c! correspond to a polar spin wavangleuk of 90°, and the bottom curves in~a!–~c! correspond touk50°. Thehorizontal dashed line in~a! and the hybrid dashed-hatched line in~b!–~c!indicate the FMR frequency. The bottom part of each set shows schemillustrations of the spin wave band in three dimensions with the azimuspin wave propagation angle out of the page. Note that the orientation ofk axis is reversed from thefk axis in Fig. 6. For these lower diagrams, thtop shaded surface corresponds touk590° and the bottom shaded surfacefor uk50°. The middle horizontal plane which crosses all the diagraindicates the FMR frequency. For diagram set~a!, the static field is appliedperpendicular to the film plane atu50°. For set~b!, the static field isapplied atu57.5°. For set~c!, the static field is applied atu590°.

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states range from spin waves atk50 for uk590° to ratherlargek values atuk50°. Note that the maximum value ofkfor degenerate states is in the range 3 – 43105 rad/cm, orwavelengths in the 0.2mm range.

The shift in the frequency of the spin wave band relatto the YIG film FMR frequency asHext is rotated fromu50° to u590° is due to two effects. First, it is evident fromFig. 2~a! that the external field required for ferromagneresonance at 10 GHz,HFMR, is much larger atu50° than atu590°. From the results in Fig. 2~a!, HFMR drops from over5 kOe foru50° to less than 1 kOe foru590°. The corre-sponding internal fieldHi also drops as the static fieldrotated from perpendicular to parallel. Second, the changrole of the static and dynamic demagnetizing fields worksposition the thin film FMR frequency precisely at the bottoof the k50 spin wave band limit atu50° and precisely atthe top of thek50 band limit foru590°.

The net effect of these FMR and spin wave band shifta variation in the availability of spin wave states at the FMfrequency with angle. Since these are the spin wave stwhich can contribute to two magnon scattering relaxatione is able to vary this contribution to the overall relaxatisimply by rotating the direction of the static field usedproduce the FMR response. If two magnon processesimportant, the profile of FMR linewidthDH versus the fieldangleu at fixed frequency will be different from the consta1/T profiles shown in Fig. 3. These differences can providsignature profile for the two magnon scattering process.

Turn now to the dispersion relations for anisotropBa–M and Zn–Y films. As made explicit in Eqs.~18!, ~20!,and ~21!, the spin wave frequencies in these materialspend, in general, on the azimuthal spin wave propagaanglefk as well as the polar angleuk and the wave numbek. It is now necessary to includefk as a parameter whedispersion curves are computed. Examples of the anisotrspin wave band for the case of a Ba–M film and an FMfrequency of 50 GHz are shown in Fig. 6. The upper dgrams show calculated curves in the two dimensionalvk vsk format of Fig. 5. The lower diagrams show three dimesional plots ofvk as a function ofk andfk . Figure 5~a! isfor u50°. Figures 5~b! and 5~c! are for u560° and u575°, respectively. Figure 5~d! is for u590°. Note thefk

axis scale in the lower set of diagrams, which was choseprovide the best perspective view of the spin wave band

The curves in the upper diagrams and the two shacurved surfaces in the lower diagrams correspond to diffe(uk ,fk) combinations. The bottom curved surfaces in tlower diagrams are foruk50° and the upper curved surfaceare for uk590°. Corresponding projections of the edcurves forfk50° andfk590° from the bottom diagramyield the curves shown in the upper diagrams. The bottcurves in the upper diagrams, for example, are all foruk

590° and do not depend onfk . The other projections areclear from the diagrams. The main effect of the anisotropto produce spin wave frequency surfaces as a functionkandfk for fixed values ofuk . When the applied field is sucthat the rotational symmetry is broken, e.g., whenu is anyangle except zero anduk is also any angle except zero, theuk

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surface is warped. The warp is most pronounced atu590°anduk590°.

Turn now to the FMR response. The dashed line inupper diagrams and the horizontal plane in the lower dgrams, both labeled as FMR, indicate the FMR frequency50 GHz which is common to all the diagrams. This FMfrequency is the same as for Fig. 2~b!. The shaded strip in theupper diagrams and the shaded part of the FMR plane inlower diagrams indicate the range of spin wave modes wvk the same as the FMR frequency.

The basic effect conveyed by the diagrams in Fig. 6the c plane Ba–M film is the same as for the isotropic YIfilm. As one moves from the perpendicular FMR configurtion at u50° to the parallel FMR case atu590°, with theexternal field adjusted to keep the FMR frequency constthe spin wave band of frequencies shifts relative to the FMfrequency and there is a substantial change in the spin wstates which are degenerate at this frequency. The situatiou50° shown in the left-most diagrams in Fig. 6, is the saas for the YIG case: no degenerate spin wave states excek50. Now, asu is increased from zero, the entire banmoves down in frequency and various spin wave states minto the FMR frequency strip in the upper diagrams or FMfrequency cut in the lower diagrams of Fig. 6.

The situation here, however, is more complicated thfor YIG. In order to understand the evolution of the degeerate spin wave states with angleu, it is necessary to followin detail the change in the shaded part of the FMR frequecut in the lower diagrams in Fig. 6. Consider the lower pof ~b!, for example. As long asu is not increased too muchthe FMR frequency cut extends across the spin wave bansuch a way that there are degenerate states for all valueall fk from 0° to 90° andk values from zero out to sommzximum cut-off defined by theuk50° edge of the bandThis range of degenerate spin wave states is indicated byshaded part of the FMR frequency cut in the lower partFig. 6~b! for u560° and the shaded strip in the upper paThe situation changes as one moves tou575° and Fig. 6~c!.Now, as shown by the shaded part of the FMR frequencyin the lower diagram of the set, the available degeneratewaves become more limited. Here, for small values ofk, theallowed fk values for degenerate spin waves range frsome lowerfk limit up to fk590°. This lowerfk limitincreases and approaches 90° as one moves up to an exfield angleu590° and a parallel FMR configuration. Thidegenerate spin wave situation is indicated by the very smshaded section of the FMR frequency cut for the lower pin Fig. 6~d!. The details of these changes in the availadegenerate spin wave states asu is varied will have impor-tant consequences for two magnon scattering.

Figure 7 shows the evolution in the spin wave band wthe external field angleu for Zn–Y and an FMR frequencyof 10 GHz. The format is the same as for the Ba–M diagrin Fig. 6 and will not be described in detail. There are seveimportant differences between the Zn–Y and Ba–M dgrams. The Zn–Y film represents an easy-plane rather tan easy-axis situation. The first effect of this change inisotropy is that the warped spin wave band surfaces areversed. Although the warped surfaces look the same in Fi

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as in Fig. 6, note that thefk axes are reversed. The secoeffect is in the movement in the band relative to the FMfrequency as the field angleu is changed. For the perpendicular configuration andu50°, the FMR frequency point islocated at the very bottom of the band, as before. Forparallel configuration andu590°, however, the FMR fre-quency point is at the bottom of the warpeduk590° spinwave surface atk50, rather than at the top of this surfacThere are never any excludedfk modes from the two magnon scattering process. As the field angleu is rotated from 0°to 90°, the spin wave band moves only part way down athere is no drastic change in the degenerate state situatiofor Ba–M. One would expect that the two magnon linewidprofile for Zn–Y would be much simpler than for eitheBa–M or for YIG films.

IV. TWO MAGNON SCATTERING THEORY

The two magnon relaxation rate corresponds to theat which energy is transferred from the uniform precessiondegenerate spin waves due to the presence of magneticmogeneities. It is assumed that these excited spin wavesrelax to the lattice extremely rapidly and do not affect ttwo magnon decay. The basic formalism has been takenfrom the analysis developed by SLK.4 This section is dividedinto three parts. The first part takes the SLK scattering reation rate as a starting point, and introduces several mocations which make the analysis more practical and apcable. The second part provides a detailed and spedevelopment of the scattering integrals and limits whmust be invoked to perform practical two magnon scattercalculations. The third part presents a qualitative and dgrammatic discussion of these spin wave regimes which ctribute to the scattering.

A. A modified SLK two magnon scattering model

For a scattering inhomogeneity of a given size, the Smodel predicts the rate at which energy is coupled fromuniform precession of the magnetization vector over thetire sample, which is excited through the FMR response,particular spin wave due to the dipolar interaction betwethe scatterer and the spin wave. The net relaxation ratthen found from adding up the rates for all degenerate swave states for all of the scatterers in the sample. Inoriginal SLK treatment, the scattering inhomogeneity wmodeled as a spherical void in an infinite sample. Thesumption of spherical symmetry leads to ak-dependent cou-pling between the uniform precession and the degenespin waves. That is, spin waves which propagate in cerdirections and at certaink values couple more strongly to thuniform mode than others. Thisk dependence is due to thgeometry of the dipole fields associated with the void. TSLK treatment then applied this result for a single sphervoid to a sample with many identical scatterers by assumthat these scatterers operate independently.

One problem in the SLK formalism is thatk-dependentscattering is associated directly with the spherical naturethe void used for the analysis. Modifications were also intduced to account for the expected occurrence of many nspherical scatterers. Nonspherical scatterers were inclu

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through the addition of anad hoc constant term to thek-dependent scattering factor in the coupling function.19 Fur-ther comments on this modification will be given below. TSLK calculation has also been applied to the case of anliquely magnetized thin film,18 but the analysis was restricteto isotropic materials and spherical void scattering.

It is important to establish a clear connection betwethe relaxation rate obtained from the SLK approach andlinewidths which are obtained in the experiment. For a givtwo magnon relaxation rate 1/TTM , the FMR frequency line-width is simplyD(v/g)51/gTTM . Once the frequency linewidth is determined, the field linewidthDH for two magnonscattering can be found from Eq.~10!. It should be noted thamany treatments specify the relaxation timetm associatedwith the decay of the dynamic magnetization ampliturather than the decay of the energy associated with thenamic magnetization. In that case, the frequency linewidtgiven byD(v/g)52/gtm .38 The presentation here will usenergy decay as the basic decay rate.

Several modifications have been made to the SLKproach. First, in order to deal with anisotropic films, the SLtreatment has been extended to account for the effectanisotropy on both the FMR response and the spin wdispersion properties. Second, the scattering calculationbeen modified to include scattering into a band of neadegenerate spin waves with a frequency width which cosponds to an average frequency linewidth for the scattespin waves. This allows the subsequent relaxation of thscattered spin waves to be included in the analysis isimple and intuitive manner. Third, specific calculationsangle dependent linewidths have been made for two lim~i! a ‘‘spherical void scattering’’ limit in which thek-dependent scattering factor from the SLK model is assumto apply and~ii ! an ‘‘isotropic scattering’’ limit in which thisscattering factor is assumed to bek-independent and equal tits average value over all angles. It is clear that purely sphcal voids represent a poor approximation to the inhomogeities in real materials. One would expect that these two limwould bracket the real physical situation for ferrite films wivarious microstructure properties.

The main result of the SLK theory is an expressionthe relaxation rate due to a single scatterer. This rate istermined from a Fermi golden rule calculation and canwritten as

1

TSLK5

2p

\ (kÞ0

@F~k!#2d~\vk2\v!. ~22!

Here,\ is Planck’s constant,d(\vk2\v) is the Dirac deltafunction, and the@F(k)#2 factor represents the coupling between the spherical void and the degenerate spin waves.sum spans the variousk values in the spin wave band, buwith the uniform mode excluded.

The F(k) coupling factor is given by

F~k!516p2MsR3

m

V~3 cos2 uk21!

j i~kR!

kR. ~23!

Here,R is the scatterer radius,m is the Bohr magneton,V isthe sample volume, andj 1(kR) is the first spherical Besse

K

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ite

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function. In addition to the development in the original SLpaper, a detailed derivation of Eqs.~22! and~23! is given byHaas and Callen.39 It is useful to note thatg, the magnitudeof the electron gyromagnetic ratio, is given byg52m/\.The matter of units will be considered shortly.

Equations~22! and ~23! are the key working equationfor two magnon scattering calculations. The net relaxatrate is in the form of a sum over allk states. The sum isweighted by the Dirac delta function. This insures that odegenerate spin waves, that is, spin waves withvk5v, areincluded in the summation. Each relaxation rate sum termproportional to@F(k)#2, which is related to the interactioenergy between the uniform mode and the spin wave at wvectork.

In practice, the Dirac delta function in Eq.~22! should bereplaced by a function of finite width in order to account fthe subsequent relaxation of the degenerate spin waves.simple approximation to the delta function which has finwidth and unit area is

G~v,vk!5H 1

\Dv i, v2

Dv i

2,vk,v1

Dv i

20, otherwise.

~24!

Here,Dv i is the intrinsic frequency linewidth which charaterizes the magnetic loss for the material in the absenctwo magnon scattering. This linewidth is taken to be an orof magnitude estimate to describe the intrinsic relaxationthe degenerate spin waves.

Turn now to the@F(k)#2 term. Note that@F(k)#2 variesasMs

2. This dependence on the square of the magnetizareflects the dipole–dipole origin of the two magnon scatting interaction. The specific form of the (3 cos2 uk21)2 termis due to the assumption of spherically shaped scatteBecause of the spatial dependence of the dipole field assated with the spherical void, spin waves which propagatecertain directions couple more strongly to the scatterer tothers. For a spin wave propagating parallel to the stmagnetization direction, atuk50, (3 cos2 uk21)2 is equal to4. For a spin wave propagating perpendicular to thez axis atuk590°, (3 cos2 uk21)2 is equal to 1. For cos2 uk51/3, oruk554.7°, this angular coupling function is equal to zeSpin waves at this propagation angle do not participate intwo magnon process at all. The use of the (3 cos2 uk21)2

term will be identified as ‘‘spherical void scattering.’’The vanishing of the two magnon coupling atuk

554.7° for spherical void scattering is related to propertof the dipole field around the void. If one considers a conisurface defined by all rays which extend away from the vat an angle of 54.7° relative to the saturation magnetizadirection, the static dipole field along this surface is alwaperpendicular to the static magnetization. The dynamicpole field associated with the uniform precession is, thefore, parallel to the direction of the static magnetization. TSLK analysis shows that this parallel dynamic field doescouple to spin waves propagating in the coincident directat uk554.7°.

In addition to the assumption of spherical inhomogeities and the corresponding (3 cos2 uk21)2 coupling term, itwill be useful to consider possible angle dependences for

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case of nonspherical scatterers. In order to model the efof nonspherical inhomogeneities, it is reasonable to repl(3 cos2 uk21)2 with its average value over all solid angles0.8. This approach is designated ‘‘isotropic scattering.’’

It is important to note that Sparks,18 Seiden andSparks,19 and Sage20 have all proposed various modificationto the SLK model to account for nonspherical scattererseach case, the resultant angular coupling function incluboth spherical void and isotropic scattering terms. Tpresent approach appears to be intuitively plausible anderationally convenient.

Consider now the role of the@ j 1(kR)/kr#2 term in therelaxation rate expression. This function has a value offor kR50 and is essentially zero forkR.p. This factorinsures that the degenerate spin waves which participatthe two magnon process have wavelengths on the ordethe scatterer size or larger. This means that for large inmogeneities, only spin waves with low wave numbers ccontribute. For smaller inhomogeneities, higher wave nuber spin waves can participate. For very smallR values,degenerate spin waves with all availablek values can con-tribute appreciably to the scattering. It will be seen thatsize of the scattering inhomogeneities will have a significeffect on the shape of the linewidth versus field angle profi

For simplicity, and following the SLK approach, it iassumed that the material containsN independent, identicascatterers of radiusR. The net linewidth is then given by Eq~22! multiplied by a factor ofN.

B. Two magnon scattering integrals and limits

In order to evaluate the two magnon relaxation rates,useful to convert the sum overk states to an integration ovek space. The sum in Eq.~22! is replaced by an integral.

(k→

V

8p3 E E E d3k. ~25!

The factorV/8p3 represents the number of states per uvolume in k space. The two magnon relaxation rate 1/TTM

for N scatterers is given by

1

TTM5

NV

4p2\ E0

`E0

2pE0

p

@F~k!#2G~v,vk!

3k2 sin ukdukdfkdk. ~26!

It is more useful, however, to cast Eq.~26! in a form forwhich the integration limits directly reflect the spin wavband and range of degenerate states. To do so, it is neceto eliminate theG(v,vk) function in the integrand. SinceG(v,vk) is equal to 1/\Dv i for degenerate spin waves buis zero otherwise, this function can be replaced by 1/\Dv i ifthe integration limits are modified appropriately. The firintegration, over the cosuk variable, reflects the range ofuk

values for the degenerate spin waves for a givenk andfk .The second integration is overfk . For an isotropic materialthis integration is trivial because the spin wave dispersrelations show no dependence on thefk . For a uniaxialmaterial, thefk dependence can play a significant role a

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may result in a modification to the integration limits as weThe third integration is over the wave numberk.

In terms of the new integration limits, the two magnorelaxation rate may be written as

1

TTM5

6pg2~4pMs!2R3

pDv i

3E0

`Ef ~k!

p/2Ea~k,fk!

b~k,fk!S ~3 cos2 uk21!j i~kR!

kR D 2

3d~cosuk!dfkk2 dk. ~27!

Here, p is the ratio of total scatterer volume to the sampvolume, 4pR3/3V. The conditiong52m/\ has been in-voked to eliminatem and\ from the relaxation rate expression. In the format of Eq.~27!, the various units are cleaBoth g4pMs andDv i have units of rad/s, so that 1/TTM alsohas units of rad/s.

The a(k,fk) andb(k,fk) limit functions for the cosuk

integral are derived from the anisotropic spin wave dispsion relation of Eq.~18!. Thea(k,fk) anda(k,fk) functionsgive the value of cosuk when the spin wave frequencyvk isat the top or bottom of the finite width band of nearly degeerate states, respectively, or atvk5v6Dv i /2. These func-tions may be written as

a~k,fk!

b~k,fk!

551, v6

Dvi

2,vmin

Avmax2 2S v6

Dv i

2 D 2

vmax2 2vmin

2 ,vmin,v6

Dv i

2,vmax

0, v6Dv i

2.vmax

.

~28!

The 6 signs apply toa(k,fk) and a(k,fk), respectively.The frequenciesvmin andvmax are the same as given by Eq~20! and ~21!. The limit values of zero and one fora(k,fk)andb(k,fk) apply when the wave numberk is greater thanthe largest value ofk allowed by the dispersion or whenk isless than the smallest allowed value, respectively. When blimits are zero or one, of course, the integral will vanish.

The function f (k) represents the minimum value offk

at vk5v for a given field orientation. Thisf (k) may bewritten in implicit form through the condition

cos2@ f ~k!#

5g2~Hi1Dk2!~Hi1Dk22HA sin2 f14pMs!2v2

4pMsg2HA sin2 f

.

~29!

Equation~29! is valid provided that the right hand sidepositive and less than or equal to 1. Otherwise, the funcf (k) is simply zero. By this definition,f (k) is zero for anisotropic material whereHA50.

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C. Discussion of two magnon scattering regimes

The a(k,fk), b(k,fk), and f (k) limits can best be un-derstood from graphical illustrations of the cosuk and fk

integrations. The cosuk integration is placed in perspectivin Fig. 8. The figure shows various spin wave disperscurves of frequencyvk as a function of wave numberk, allfor a given value and orientation of the external magnefield and some specific azimuthal spin wave propagatanglefk . The format is the same as with the previous twdimensional spin wave band diagrams. The top and botdispersion branches are shown as solid curved lines labvmin andvmax, respectively. These curves correspond todispersion curves foruk590° anduk50°, as in previousdiagrams. The point labeledvFMR denotes the FMR fre-quency and the operating frequency of interest. Givenorientation anglef for the static external field,Hext, thevalue of the field has been set to yield this FMR frequen

The solid horizontal lines in Fig. 8 delineate the rangedegenerate spin wave states which are within 1/2Dv i of theoperating frequency. This width has been exaggerated forsake of illustration. For FMR frequencies in the GHz rangthis degenerate band of states typically will have a widththe range of several tenths of a MHz to tens of MHz, dpending on the intrinsic loss level of the ferrite film undconsideration. The intersection of this horizontal band wthe overall spin wave band fromvmin to vmax gives the re-gion of states ink anduk , for a givenfk , which contributeto the two magnon scattering. In order to illustrate the cosuk

integration limits, the diagram shows a particulark value asindicated by the vertical dashed line. The intersections of

FIG. 8. Schematic illustration of the spin wave dispersion curves of swave frequencyvk vs wave numberk at some particular azimuthal spinwave anglefk , shown by the curved solid lines, and the band of allowspin wave frequencies for two magnon scattering, shown by the horizosolid lines. The bottom solid curve labeledvmin corresponds touk50°. Thetop solid curve labeledvmax corresponds touk590°. The scattering banddefined by the solid horizontal lines is centered on the FMR frequeindicated asvFMR and has a widthDv i . For a particular wave numberk, asindicated by the vertical dashed line, and the selected azimuthal anglefk ,the curves and the lines intersect at the solid circles as shown. The usolid circle identifies the integration limit functiona(k,fk) given in Eq.~28!and the value of cosuk which matches the upper dashed dispersion culabeled asa in the figure. The lower solid circle identifies theb(k,fk)integration limit function given in Eq.~29! and the value of cosuk whichmatches the lower dashed dispersion curve labeled asb in the figure.

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line with the horizontal scattering band limits are indicatby the solid circles. The dashed curves through thesesolid circles further identify spin wave dispersion curvesvalues of cosuk equal toa andb, as indicated. Thesea andb values correspond to thea(k,fk) andb(k,fk) integrationlimit functions introduced above.

It is straightforward to understand the cosuk integral inEq. ~27! from the diagram. This integral simply scans thostates inuk along the vertical dashed line at constantk, fromthea intersection point specified by cosuk5a(k,fk), to thebintersection point specified by cosuk5b(k,fk). If the chosenk or fk values are such that either or both intersection pofall below the solidvmin curve, one simply sets the corresponding cosuk limit to zero. If the chosenk or fk valuesare such that either or both intersection points fall abovesolid vmax curve, one simply sets the corresponding cosuk

limit to one. These assignments fora(k,fk) andb(k,fk) areexpressed analytically through Eqs.~28! and ~29!.

It is clear from Fig. 8 that thea(k,fk) and b(k,fk)integration limits are related to the widthDv i , while a1/Dv i dependence is present in the prefactor of Eq.~27!. Aslong asDv i!v is satisfied, the two magnon linewidthessentially independent ofDv i . It should also be noted thabecause there is another spin wave state atp2uk which isalso degenerate, the introduction of thea(k,fk) andb(k,fk)limits is accompanied by an overall multiplication by a factof 2.

Turn now to the azimuthal anglefk integration. Notethat the integration in Eq.~26! was from 0 to 2p. By sym-metry this integral can be evaluated over 0 top/2 and thenmultiplied by a factor of 4. This factor of 4 is taken intaccount in Eq.~27!. Equation~27! shows that, in additionthe lower limit on thefk integration isf (k) rather than zero.As discussed above, thef (k) function will be zero for iso-tropic materials withHA50. For anisotropic materials, however, the distortion of the spin wave band and the additiodependences of the spin wave frequency onfk can lead tothe exclusion of lowfk states from the scattering integraThis effect is evident from the three dimensional dispersdiagram in~c! of Fig. 6 for the Ba–M film andu575°.

This same diagram is shown in Fig. 9~a!. Figure 9~b!shows the same constant frequency cut as a graph offk vs k.These graphs show that, under appropriate circumstaand for values of the wave numberk below some upper limitvalue, the range of degenerate spin wave states is from snonzero value offk up to p/2. Under such circumstancethe f (k) function of Eq.~30! gives this lower limitfk value.For the example of Fig. 9, thef (k) function describes thecurved lower left boundary for the shaded region of tgraph in~b!. Note that this lowerfk limit is a function of thewave numberk for low k but becomes zero at larger wavnumber. The point where this function becomes zero cosponds to the wave number for which the right hand sideEq. ~30! becomes zero. Note that the topfk limit is always90°, independent ofk.

It should also be noted that thef (k) is nonzero only forthe case of an easy-axis material magnetized near the ouplane direction. Figures 6~a! and 6~b! show no suchfk cuts

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es

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of the sort evident for~c! and ~d!. Moreover, for an easy-plane material such as Zn–Y, thef (k) function is alwayszero, regardless of the field orientation. This result is evidfrom the three dimensional diagrams in Fig. 7. In these dgrams, the constant frequency cuts all include the full ranof fk values.

Turn finally to the integration over wave numberk.While the limits on thek integral are shown as zero to infinity, there is always an upper limit onk for which the line ofdegenerate states extends outside the spin wave band. Tkcut-off is evident from all of the diagrams in Figs. 5–9. Thexplicit cut-off value ofk, kcut, is obtained from Eq.~20! forvmin . One simply setsvmin in Eq. ~20! equal tovFMR andsolves fork5kcut. Note that the two control parametersEq. ~20!, the internal static effective fieldHi and the equilib-rium static magnetization anglef, are specified throughHext

and u. The FMR frequencyvFMR is also specified throughHext andu. In addition to thisk state cut-off because of thspin wave band limits, there is also a practical cut-through the j 1(kR) function. The value of this sphericaBessel function in the integrand of Eqs.~26! and ~27! issmall for large values ofkR. For practical purposes,j 1(kR)can be set to zero fork values greater thanp/R.

In closing this section, it is useful to enumerate thequence of operations to calculate two magnon relaxarates.~1! One first chooses the film material, the basic marials parametersg, 4pMs , KU and HA , and an intrinsicfrequency linewidthDv i . ~2! One then chooses an operatinfrequencyv and solves~i! the static equilibrium problem and~ii ! the FMR problem to obtain the FMR field,HFMR, theinternal fieldHi , the static magnetization anglef, and thevalue ofkcut, all as a function of the external field angleu forall values of interest between 0° and 90°.~3! One thenchooses inhomogeneity parametersR and p. ~4! One thenchooses either the ‘‘spherical void scattering’’ limit and icorporates the full (3 cos2 uk21)2 angle term in Eq.~27!, or

FIG. 9. Schematic illustration of the regime of degenerate spin wave stfor a Ba–M sample magnetized in an oblique direction such that the Fcut in Fig. 6 intersects the topuk590° surface.~a! shows the spin waveband in three dimensions, in the same format as in the lower diagram o~c! in Fig. 6. As in Fig. 6, the thin band of degenerate spin wave states aFMR frequency is indicated by the cross-hatched region.~b! shows thishatched region in a two dimensional plot of azimuthal spin wave anglefk

vs wave numberk. The functionf (k) for the lower limit of thefk integra-tion in Eq.~27!, defined mathematically through Eq.~30!, is indicated by thecurve labeledfk5 f (k) in ~b!.

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selects the ‘‘isotropic scattering’’ limit in which this term ireplaced by its average value of 0.8 over all solid angles

One is now in a position to evaluate a two magnonlaxation rate 1/TTM at these different values off, and hence,to obtain information on the two magnon frequency linwidth 1/gTTM or the corresponding field linewidthDH as afunction of f or u. For eachf, this 1/TTM is obtained fromEq. ~27!. For a givenf value, the first step is to determinthe effective internal fieldHi as given by Eq.~14!. One canthen write explicit expressions fora(k,fk) and b(k,fk)based on Eqs.~28!, ~20!, and ~21!. In order to evaluate Eq~27!, the integral over cosuk should be performed first. Thiintegration can be done analytically. The result is a comcated double integral overk andfk which can be handled bya number of numerical methods. One approach would buse a quadrature algorithm for thefk integration nestedwithin a second quadrature algorithm for thek integration.

V. EXPLICIT TWO MAGNON LINEWIDTH VERSUSANGLE RESULTS

In this section, some representative two magnon resare presented. These are shown in terms of the frequeswept linewidthD~v/g! and field swept linewidthDH versusthe angleu between the static magnetic field and the finormal. Various linewidth angle profiles for the YIG, Ba–Mand Zn–Y materials, different size voids, and different sctering limits are shown. These results will demonstratedifferent types of linewidth angle profiles which can resfrom two magnon scattering.

Recall from Eq.~9! that the frequency linewidthD~v/g!is simply proportional to the relaxation rate. This means tthe angle dependences forD~v/g! are identical to those fo1/T. The only difference is thatD~v/g! conveniently ex-presses the calculated losses in terms of magnetic field uwhich correspond to linewidths. Recall as well that the fieswept resonance linewidthDH can have significant angldependences even for a constant 1/T or, therefore, a constanD~v/g!. For YIG films and frequencies in the 10 GHz ranand above, theD~v/g! andDH profiles are almost the sameThis is due to the relatively small saturation induction athe nearly parallel alignment ofHext and M s as the field isrotated from out-of-plane to in-plane andHext is adjusted tosatisfy the requisite FMR condition at a given angle. Funiaxial and planar anisotropy materials, however, the lar4pMs values and large anisotropy fields cause the anbetweenHext andM s to vary considerably as one sweeps tfield through the FMR response curve. This leads to anificant difference in the angle profiles forD~v/g! andDH inthese materials.

Theg, 4pMs , andHA values used for the results showbelow are the same as those used to calculate the field veangle and linewidth versus angle results shown in Figs. 23. The exchange constantD in the spin wave dispersion relations is taken to be independent of the propagation anand to have a nominal value of 531029 Oe cm2/rad2, thesame value used for the spin wave dispersion relationFigs. 5–7.

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A. Linewidth versus angle profiles: Basicdependences

It will be useful to develop first a qualitative picture fothe expected angle dependences and the effect of scasize and type on the linewidth versus angle profiles. Thissues are best addressed for the case of an isotropic mawhereHA50 because of the considerable simplificationsthe FMR field and spin wave dispersion expressions whresult in this limit.

As a further simplification, the angle dependence ofrelaxation rate is easiest to understand for the case of isopic scattering. The effect of scatterer size on the linewiversusu profile will be first examined for this limit. Forisotropic scattering, the coupling between the uniform moand the degenerate spin waves does not depend on thewave propagation direction. In this case, the relaxation ratsimply related to the number of degenerate spin waves whlie below the cut-off wave numberk5p/R and the corre-sponding weighting of these states according@ j 1(kR)/kr#2 coupling term in the scattering integral. For thcase of very small scatterers, such thatp/R is much greaterthan the largest availablek value atu590°, all of the avail-able degenerate spin waves at a givenu participate equally inthe scattering. In this situation, the only consideration whcontributes to the angle dependence of the FMR linewidthconstant frequency is the variation in the number of degerate states withu. All states out to the largest availablekvalue for the specifiedu value, taken askcut, contribute.These contributions are weighted, of course, according@ j 1(kR)/kr#2.

This effect was shown through the dispersion diagrain Fig. 5. These same dispersion diagrams are repeateFig. 10, along with an additional diagram which shows tresulting two magnon frequency linewidth 1/gTTM as a func-tion of u. Figures 10~a!, 10~b!, and 10~c! show the spin wavedispersion diagrams for three different field orientations asFig. 5. These diagrams include, however, additional vertdashed lines which indicate the position of thep/R cut-offwave number for two magnon scattering from thej 1(kR)term in theF(k) coupling function of Eq.~23!. For verysmall scatterer sizes,R is also very small and thesep/R linesare always outside the spin wave band at the FMR frequepoint. This means that the entire band of degenerate stindicated by the shaded horizontal strip of modes in~b! and~c! contributes to the two magnon scattering losses.

Figure 10~d! shows the frequency linewidth as a functioof the field angleu which results. For small angles, the spwave band is high in frequency, few states are degeneand the resulting relaxation rate is small. In the limitu→0,the number of degenerate states goes to zero as well fthin film of infinite extent, and the two magnon linewidtalso goes to zero. As the angle is increased, however,spin wave band shifts down in frequency relative to the FMfrequency position, more and more states become degenwith the FMR frequency, and the linewidth increases. As oapproaches the parallel field FMR condition atu590°, onereaches the configuration with the maximum number ofgenerate spin waves and the two magnon linewidth reachmaximum. It should be emphasized that the linewidth ver

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angle profile result in Fig. 10~d! is for: ~i! isotropic scatteringand~ii ! very small scatterer sizes. The profile which resultsa reflection of the variation in the number of degenerate swaves with angle, and no other effects.

For YIG parameters and an FMR frequency in the Gregime, the results in Fig. 10 would correspond to smscatterers with sizes on the order of 0.05mm or smaller. Ifone were to increase the size of the scatterers, one cmove thep/R cut-off wave number inside the spin wavband foru values above some lower limit. This effect, thresulting change in the scattering, and the change in thewidth versus field profile for a large scatterer size are shoin Fig. 11. The format of Fig. 11 is the same as Fig. 10, wspin wave band diagrams in~a!, ~b!, and~c!, and a linewidthversusu diagram in~d!. Note that for~b! and~c!, the verticaldashed lines atk5p/R are now well inside the spin wavband at the FMR frequency. The solid line in~d! shows thecalculated two magnon linewidth versus angle profile anddashed line repeats the result from Fig. 10~d!.

As shown by the kink in the solid curve in Fig. 11~d!nearu515° or so, and departure of the solid line from tdashed line at largeru values, the change from a small scaterer size to a large scatterer size has a drastic effect on

FIG. 10. ~a!–~c! show schematic spin wave dispersion curves of spin wfrequencyvk vs wave numberk. These diagrams are for a thin isotropfilm and three orientations of the external static field with the sample biaat the static field needed for FMR at some constant frequency, as indicThese diagrams follow the same format as Fig. 5. For~a! the static field isapplied perpendicular to the film plane atu50°. For ~b! the static field isapplied at some intermediate angle. For~c! the static field is applied in thefilm plane atu590°. The shaded horizontal regions in~b! and~c! show theband of available degenerate spin wave states for two magnon scattereach case. The vertical dashed line atk5p/R in each diagram shows theposition of the effective wave number associated with an inhomogenradiusR. For these diagrams, a small size inhomogeneity is assumed, sothe k5p/R point is always outside the spin wave band at the FMR fquency. The solid line in~d! shows the variation in the two magnon scatering frequency swept linewidth 1/gTTM with the external field angle.

sin

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e

the

linewidth versus angle profile. At small values ofu, as longas kcut is less thanp/R, the two magnon linewidth is thesame as in Fig. 10. This result is shown by the part ofsolid curve in Fig. 11~d! out to 10° or so. The kink in~d!occurs at theu value for whichp/R5kcut is satisfied. Forlarger u values, kcut is now greater thanp/R. Now, themodes which can contribute to the two magnon scattertruncate atk5p/R rather than atkcut and the two magnonlinewidth is reduced from the dashed line result in~d! to thesolid line result. The space between the solid line anddashed line in~d! represents those modes betweenk5p/Randk5kcut which are now excluded from the scattering prcess. As a result, the linewidth continues to increase wangle, but at a lower rate than for the case of small scatteThe increase in linewidth withu is now due to the increase ithe number of degenerate states for the lowk modes only,that is, those modes which satisfy the conditionk,p/R.

The results in Figs. 10 and 11 show how the detailsthe microstructure affect the two magnon linewidth versfield angle profile for thin films. The truncation effect whicis evident from the solid and dashed curves in Fig. 11~d!shows that the size of the scatterer plays a crucial roleshaping these profiles. For the example shown, one canthat the position of the kink in the profile, if such a kink cabe observed,~i! indicates a relatively large scatterer size a~ii ! provides a direct indication of the size of the scatterer.the case of YIG parameters, a kink should be observable

e

ded.

in

tyhat-

FIG. 11. ~a!–~c! show schematic spin wave dispersion curves of spin wafrequencyvk vs wave numberk in the same format and under the samconditions as for Fig. 10. Also as in Fig. 10, the vertical dashed line ak5p/R in each diagram shows the position of the effective wave numassociated with an inhomogeneity radiusR. For these diagrams, however,large size inhomogeneity is assumed, so that thek5p/R point is alwaysinside the spin wave band at the FMR frequency point. The solid line in~d!shows the variation in the two magnon scattering frequency linewi1/gTTM with the external field angle. The dashed line repeats the smallinhomogeneity result from Fig. 10.

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scatterer sizes greater than 0.1mm or so. Specific numericaresults which demonstrate these effects will be shoshortly.

For the isotropic scattering option used for Figs. 10 a11, all of the degenerate spin waves withk,p/R and all thecorresponding availableuk values are taken to scatteequally. If one considers the case of spherical void scating, one must insert theuk dependent scattering factor bacinto the analysis. Recall that as one moves across the hzontal band of degenerate spin waves indicated byshaded stripes in~b! and ~c! of Figs. 10 and 11, the spinwave angleuk is changing. For the diagrams as shown,uk isgenerally decreasing as one moves from lower to highekvalues. The presence of the (3 cos2 uk21)2 term in the scat-tering integral results in certain spin waves being weighmore heavily than others. This weighting produces additioangle dependences in the linewidth versus field angleu pro-files. Generally, these additional effects are more pronounfor large scatterer sizes than for small scatterers.

For small scatterers, as indicated above, all of the degerate modes across the band must be considered. One heffect, an average of the additional (3 cos2 uk21)2 angle fac-tor over all the availablek values from zero tokcut. This willtend to smear out the explicituk dependences which arintroduced through the (3 cos2 uk21)2 factor. When the scatterer radiusR is large, on the other hand, the scattering cotributions become limited to relatively low wave numbewhich satisfyk,p/R!kcut over all but the smallest valueof the field angleu. This limitation translates to the selectioof a small range ofuk values for eachu value which gener-ally ranges fromuk'0° at u50° to uk'90° atu590°. Inthis limit, the (3 cos2 uk21)2 coupling term for sphericavoid scattering can dominate the linewidth versusu profilecharacter.

B. Linewidth angle profiles for thin and thick YIGfilms

Figure 12 shows numerical results for the effects jdiscussed. The graphs show the calculated two magnontering frequency linewidthD(v/g)51/gTTM as a functionof external field angleu for a YIG film of infinite extent. Ateachu value, the external field was adjusted for an FMfrequency of 10 GHz. The YIG material and microwave prameters were taken to be the same as those used fonumerical results shown earlier. Figure 12~a! is for scattererswith a radiusR50.04mm. Figure 12~b! is for scatterers witha radiusR50.1mm. Figure 12~c! is for scatterers with aradiusR50.2mm. These particularR values were chosen tdemonstrate the role of the scatterer size on the linewprofiles. The isotropic scattering results are shown bysolid lines and the results for spherical void scatteringshown by the dashed lines. The ratio of the total scattevolume to the sample volumep was always taken to be 0.01for a nonmagnetic volume of 1%.

Consider the results for the isotropic scattering mofirst. For the smallest inhomogeneity size,R50.04mm, thesolid curve in ~a! shows that the relaxation rate increassteadily with field angle. In this case, the small scatterer splacesp/R well above the highest value ofk for degenerate

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modes atkcut;3.83105 rad/cm for u590°. As in Fig. 10,the increase in linewidth with angle shown by the solid liin Fig. 12~a! simply reflects the increase in the numberdegenerate states with angle and thek-dependent@ j 1(kR)/kr#2 weighting factor in the scattering integraTurn now to the smallest inhomogeneity size atR50.2mmand the solid curve in~c!. Here, the maximum spin wavwave number for scattering is atp/R51.63105 rad/cm.This k value is well below the maximumkcut value citedabove. In accord with the Fig. 11 discussion, one nowserves a distinct kink in the linewidth versus angle profiThis kink is evident for the solid curve in Fig. 12~c! at a uvalue near 15°. This kink point corresponds to the anglewhich the conditionkcut;p/R is satisfied. The solid curve in~b! for R50.1mm represents an intermediate situatioWhile the shape of the solid curve profile is different frothat in ~a!, the change in character at thekcut;p/R is not aspronounced.

It is important to take not of the different vertical axscales for the graphs in Fig. 12. From~a! to ~b! to ~c!, thisscale increases from 0–8 Oe to 0–50 Oe to 0–100 Oe.growth in the two magnon linewidth asR is increased is duemainly to theR3 factor in Eq.~27!. The increase does noscale strictly with the cube of the scatterer size becausthe decrease in thep/R cut-off point in wave number asR isincreased.

FIG. 12. Calculated two magnon frequency linewidth 1/gTTM as a functionof the external field angleu for an isotropic YIG film at 10 GHz in theinfinite thin film limit. ~a! is for scatterers with a radiusR50.04mm, ~b! isfor R50.1mm, and~c! is for R50.2mm. The porosity parameterp was setat 0.01 for all computations. The solid lines show results for the isotroscattering approximation and the dashed lines are for sphericalscattering.

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Turn now to the results for spherical void scatterinshown by the dashed lines. The linewidth versus angle pfiles now show more complicated variations than for isotpic scattering. Note, in particular, the appearance of a pnounced shoulder in dashed line profile in~a! and the broadpeaks in the 15°–30° range for~b! and ~c!. The ~b! and ~c!results show that these peaks become somewhat sharpeshift to smalleru values as the scatterer size is made largFor the largestR value and~c!, the dashed curve providesreasonable representation of the (3 cos2 uk21)2 coupling, inwhich the spin waveuk propagation angle at lowk mapsmore-or-less into the variation in the field angleu. The cal-culated linewidth in~c! for the larger scatterers is similar tthe angle dependence shown in Fig. 3 of Ref. 18.

The results in Fig. 12, as well as most of the considations in previous sections, have been for the case oinfinite thin film where the sample demagnetizing factorsgiven byNZ51 andNXY50. One result, as shown in Figs10–12, is that the two magnon linewidth vanishes in the limof a film magnetized perpendicular to the film plane. Thistrue because the FMR point then lies at the bottom ofband and there are no degenerate spin wave states to coute to the scattering. This is not the situation for a film fwhich NZ is no longer unity andNXY is no longer zero.

It is instructive to consider modifications to the twmagnon linewidth versus angle profiles for the case of anite size film or disk, whereNZ,1 andNXY.0 are satisfied.The results are shown in Fig. 13. The calculations were mwith the same YIG material and FMR parameters as for F12, a scatterer radiusR of 0.04mm, spherical void scatteringand two different disk diameterd to disk thicknessS aspectratios,d/S5100 andd/S510. Numerical values for the demagnetizing factors can be found by the Osborn metho40

Figures 13~a! and 13~b! show spin wave bands for the twcases, as indicated, and the effect of disk size on themagnon frequency linewidth 1/gTTM versus field angleuprofile is shown in~c!. The solid and dashed curves in Fi13 show the two magnon linewidth versus angle for thin fid/S5100 case and the thick filmd/S510 case, respectively

The format for the spin wave band diagrams in~a! and~b! is the same as for previous graphs. The top and botsolid curves show the bulk spin wave frequencyvk versuswave numberk for uk590° anduk50°, respectively. Forpurposes of comparison, these diagrams are shown forcommon value of the static internal field. The positionsthe FMR frequencies when the corresponding external fiis oriented atu50° andu590° are shown by the horizontadashed lines labeled with the small disk diagrams with eita perpendicular arrow or an in-plane arrow, respectively.the thin film d/S5100 case and Fig. 13~a!, the two FMRfrequencies are at the bottom and the top of thek50 bandlimits, as expected. For the thick filmd/S510 case and Fig13~b!, the two FMR frequencies are no longer at thek50band limits. As already indicated, this change affects themagnon linewidth versus angle profile significantly.

The effects of finite size on the two magnon linewidare clear from the curves in Fig. 13~c!. The solid curve isessentially the same as shown in Fig. 12~a!. The dashedcurve for the thick film case shows a modified profile. Th

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curve no longer goes to a zero linewidth in the perpendicuu50° limit and has a slightly reduced value in the paralu590° limit. These changes are due to the relative shiftsthe FMR frequency positions relative to the spin wave bafor the thicker film. Note that the basic profile has the sacharacter. This is because the basic evolution in the denof states with the external field angleu has not changed.

C. Linewidth versus angle profiles for Ba–M andZn–Y

The effects of inhomogeneity size, type of scatteringteraction, and film thickness on the angle dependence oftwo magnon linewidth for isotropic films hold for thuniaxial materials as well. Here, however, there are adtional complications for both the uniform mode FMR rsponse and the spin wave dispersion relations.

For an easy-axis material such as Ba–M, there aremain effects. First, the anisotropic distortion of the top of tspin wave band shown in Figs. 6 and 9 has a significeffect on the number of degenerate modes available for stering as one moves from the perpendicular FMR conditat u50° to the parallel FMR condition atu590°. Note, inparticular, that the FMR position moves from the very botom of the spin wave band foru50°, through the bottom ofthe top spin wave band surface atuk590° andfk50° for

FIG. 13. ~a! and ~b! show schematic spin wave dispersion curves of swave frequencyvk vs wave numberk which illustrate the relative FMRfrequencies and spin wave band position at a fixed value of the staticfor two different film diameterd to thicknessS ratios.~a! is for a d/S ratioof 100. ~b! is for d/S510. The upper and lower horizontal dashed linesthese diagrams show the FMR frequency for the in-plane static fieldperpendicular-to-field configurations, as indicated.~c! shows calculated fre-quency linewidths for a YIG film or disk at an FMR frequency of 10 GHza function of the external field angleu for 0.04mm scatterers and sphericavoid scattering. The solid curve is for a relatively thin slab withd/S5100. The dashed curve is for a relatively thick slab withd/S510. Theporosity parameterp was set at 0.01.

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some intermediateu value, and then to the very top of theband atuk590° andfk590° for u590°. Second, the ef-fects which derive from this movement of the FMR pointfrom the bottom of the band to the top of the band becomcompressed in angle. Subject to the constraint of a constaFMR frequency, the large uniaxial anisotropy tends to holthe magnetization anglef at values close to 0° untilu be-comes relatively large. Most of the rotation of the static magnetization and the corresponding change inf takes place for45°,u,90°.

The above effects are demonstrated by the numericresults on the frequency swept two magnon linewidth versufield angle in Fig. 14. The calculations were based on thsame Ba–M parameters cited above and for an FMR frequency of 50 GHz. Figure 14~a! is for scatterers withR50.01mm, 14~b! is for R50.04mm, and 14~c! is for R50.08mm. The porosity parameterp was taken as 0.01 forall graphs. Curves for isotropic scattering and spherical voscattering are shown by the solid and dashed lines, respetively, in each graph.

Consider the isotropic scattering results first. For versmall scatterers andR50.01mm, as in ~a!, the linewidthversus angle profile is similar to that for the isotropic YIGmaterial in Fig. 12~a!. The only real difference is that themain field angle effects have been pushed to largeu values.This is due to the rotation effects noted above. If the hor

FIG. 14. Calculated two magnon frequency linewidth 1/gTTM as a functionof the external field angleu for an easy axis Ba–M film at 50 GHz in theinfinite thin film limit. The easy axis is perpendicular to the film plane.~a! isfor scatterers with a radiusR50.01mm, ~b! is for R50.04mm, and~c! isfor R50.08mm. The porosity parameterp was set at 0.01 for all compu-tations. The solid lines show results for the isotropic scattering approximtion and the dashed lines are for spherical void scattering.

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zontal axis displayed the magnetization anglef rather thanthe field angleu, the profile would be very close to thashown in Fig. 12~a!. As the size of the scatterer is increasehowever, there are some significant new dependences.solid curves in Fig. 14~b! and Fig. 14~c! show that the in-crease inR is accompanied by the development of a prnounced high angle peak. This high angle peak has nothto do with the spherical void scatteringuk-dependent scattering which produced the low angle peak shown by the dascurves in Fig. 12. This high angle isotropic scattering pefor the Ba–M films is due to the distortion of the top of thspin wave band as illustrated in Figs. 6 and 9, coupled wdrop in p/R to values well belowkcut. The f (k) functionwhich gives the minimumfk for scattering becomes nonzerwhen the FMR frequency cut is across the warpeduk590°top surface of the spin wave band, as in Figs. 6~c!, 6~d!, and9. When this occurs, there is a loss of degenerate statelow wave numbers. If only lowk states can be coupled, asthe case whenp/R is small, this results in a decreaselinewidth. This is the origin of the peak in linewidth au;70° shown by the solid curve in Fig. 14~c!.

The spherical void scattering results are shown bydashed lines in the graphs of Fig. 14. All of the previodiscussion for isotropic scattering in the Ba–M film apphere as well. The profiles are pushed over to higheru valuesand there is a highu peak associated with the warped tosurface of the spin wave band and the minimumfk effect.Now, however, this peak is somewhat obscured by the pat loweru values which comes in for spherical void scatteing whenR is large. This loweru peak is directly related tothe (3 cos2 uk21)2 contribution to the scattering in Eq.~27!.Here too, the previous discussion for the YIG film applieThe peak in linewidth nearu555°, evident from the dashecurve in Fig. 14~c!, derives from this term. As with the YIGexample, the small value ofp/R serves to select out a narow range ofuk values which allow the linewidth versus thfield angleu to reflect this coupling in an almost one-to-onmanner.

As a final example, consider the case of an easy-planisotropic material. Figure 15 shows calculated two mnon scattering linewidths as a function of external field anfor a Zn–Y film in the infinite thin film limit and for ferro-magnetic resonance at 10 GHz. Figure 15~a! is for R50.04mm and 15~b! is for R50.1mm. As with the twoprevious figures, the results for isotropic scattering aspherical void scattering are shown by the solid lines anddashed lines, respectively. Thep value was 0.01. The Zn–Yparameters were the same as cited previously.

Consider the isotropic scattering results first. The solines in the graphs of Fig. 15 appear at first glance to be qdifferent from either of the two previous examples. For boof the R values examined, the two magnon linewidth icreases very rapidly from zero tou50° to a relatively con-stant value foru values above 10°–15° or so. This initiarapid increase withu is somewhat misleading, however. Fthis easy plane material, the anisotropy serves to enhanctendency for the static magnetization vector to lie in-plaAs the external field angle is moved away from 0°, the manetization anglef moves from 0° to angles close to 90

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rather quickly. This is a result of the large easy-plane anisropy together with the demagnetization fields which actpull the magnetization vector in the plane of the film evenrelatively low values of the external field angle. For thesults in Fig. 15, the range ofu values from 0° to 15° corresponds to a change inf from 0° to nearly 90°. If theselinewidths were displayed as a function off, the profiles inFig. 15 would have a very similar appearance to the profiin Fig. 12 for YIG films. Most of the YIG discussion appliehere as well. The small shoulder atu;3° on the solid curveZn–Y result in Fig. 15~b! for R50.1mm, for example, isrelated to the samekcut5p/R condition which caused theshoulder to appear on the solid curve in Fig. 12~c!. Here, theshoulder occurs at a much smalleru value because of theplanar anisotropy.

For spherical void scattering, the effects also trackresults for the YIG films in terms of thef rotation. Thesingle peak which is evident for both of the dashed curveFig. 15 is due to the (3 cos2 uk21)2 term in Eq. ~27!. Thepeak is more distinct for the largerR value. This is becausethe reduced coupling to modes withk values abovep/Rtends to select out a more limited range ofuk values for thescattering, and this allows the functional form of th(3 cos2 uk21)2 coupling term to be reflected directly in thlinewidth versusu response. The arguments here are exathe same as for the YIG case.

It is important to note that for Zn–Y there is no addtional peak at higheru values associated with the warped tuk590° surface of the spin wave band, as was the casethe Ba–M analysis. For ferromagnetic resonance in ac-plane

FIG. 15. Calculated two magnon frequency linewidth 1/gTTM as a functionof the external field angleu for an easy plane Zn–Y film at 10 GHz in thinfinite thin film limit. The easy plane is the film plane.~a! is for scattererswith a radiusR50.1mm and~b! is for R50.4mm. The porosity parametep was set at 0.01 for all computations. The solid lines show results forisotropic scattering approximation and the dashed lines are for sphevoid scattering.

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film of Zn–Y, the resonance frequency lies at the bottomthe spin wave band foru50°, but then moves only to thebottom of the warped surface of the band atu590°. Thisresult was emphasized in connection with Fig. 7. For Ba–films, the FMR frequency cut moves into the warped partthe band for largeu values. This is not the case for Zn–Y

The above YIG, Ba–M, and Zn–Y examples demostrate explicitly the various processes which can affecttwo magnon linewidth. There are density of states effeThere are anisotropy effects. There are microstructurefects. There is one final effect to be considered, the diffence between the frequency swept linewidthD~v/g! and thefield swept linewidthDH. Recall that there is, in general,variation in the field swept linewidthDH with the field angleu even for a constantD~v/g!. This point was made explicitlyin connection with Fig. 3. The effect was relatively small fYIG materials, about a 50% effect for Ba–M materials, aan over 600% effect for Zn–Y films.

It is only necessary to consider the differences betwD~v/g! andDH for the Ba–M and Zn–Y cases. Figure 3~a!shows that the effect is small for YIG films. Figure 16 showa comparison of frequency swept and the corresponding fiswept two magnon linewidth versus field angle profilesBa–M and Zn–Y thin films. Figure 16~a! is for Ba–M at 50GHz. Figure 16~b! is for Zn–Y at 10 GHz. In both cases, thcalculations were based on a scatterer size of 0.1mm, theinfinite thin film limit, spherical void scattering, and a poro

eal

FIG. 16. Diagrams of calculated two magnon frequency linewidth 1/gTTM

and the corresponding field swept two magnon linewidthDHTM as a func-tion of the external field angleu. ~a! is for an easy axis Ba–M film at 50GHz in the infinite thin film limit.~b! is for an easy plane Zn–Y film at 10GHz in the infinite thin film limit. Both diagrams are for scatterers withradiusR50.1mm, a porosity parameterp of 0.01, and spherical void scattering. The dashed lines show the results for the frequency swept linew1/gTTM . The solid lines show the results for the field swept linewidDHTM .

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ity p parameter of 0.01. The dashed lines show the two mnon frequency swept linewidthD(v/g)51/gTTM angle pro-files. The solid lines show the two magnon field swelinewidth DHTM profiles. The field swept linewidths werobtained from the frequency swept two magnon linewidand Eq. ~10!, with the FMR derivative factorg]HFMR/]vFMR evaluated numerically at eachu. From theresults in Fig. 3, it is evident that this factor is equal to oneu50°, greater than one whenM s rotation effects are important at intermediate angles, and less than one for resonwhen the static field is in plane atu590°.

The results in Fig. 16 are self explanatory. In the Ba–case, the roughly 50% effect atu;60° from Fig. 3 servesmainly to magnify the angle profile forDHTM . The basicprofile structure found for 1/gTTM is retained. Although it isnot readily evident from the curves shown, the field swlinewidth at u590° is slightly smaller than the frequencswept linewidth. This is a direct consequence of the fact tthe rotation effects foru values different from 0° and 90cause g]HFMR/]vFMR to be greater than unity and tbroaden the field swept FMR line, while dynamic demagtizing effects which affect the FMR frequency field equatiat u590° give ag]HFMR/]vFMR multiplier which is lessthan unity.

The situation is quite different for Zn–Y. Here, th600% effect for the 1/gTTM to DHTM conversion, combinedwith relatively sharp peaks for both 1/gTTM andg]HFMR/]vFMR over roughly the same range inu values,leads to a very significant change. One finds that~i! the peakin DHTM is accentuated to the extreme and shifted inu, and~ii ! the DHTM for large values ofu actually falls below1/gTTM . For Zn–Y, this decrease inDHTM below 1/gTTM atlargeu is much more distinct than for Ba–M, as was also tcase in Fig. 3.

VI. ADDITIONAL CONSIDERATIONS

As discussed in Sec. I, an important consideration whhas been neglected in this work involves the effect of fithickness on the normal spin wave modes. For thin films,bulk spin wave dispersion relations are not the true normodes of the system, due to the film surfaces and the cosponding boundary conditions on the dynamic magnettion. Instead of the continuous spectra of spin wave dispsion curves within the band, the spin wave spectrumdiscretized. One would then expect the angle dependencthe linewidth to reflect the discrete nature of the normmodes. The dipole-exchange mode dispersion relationsveloped by Kalinikoset al.36 for obliquely magnetized anisotropic thin films, for example, might be used in place of tbulk dispersion relations in order to determine the effectfilm thickness.

There are a number of additional considerations whmay be important in the two magnon calculation but habeen neglected in this analysis. First, the exchange conD has been taken to be independent of propagation anglementioned above, this may not be the case for highly antropic Ba–M hexagonal ferrite materials. Second, the effof the ellipticity of the uniform mode and the spin wave h

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also been neglected. It is reasonable to expect that the eticity would influence the coupling between the inhomogneities and the degenerate spin waves. Third, a scatteHamiltonian based on fluctuations in the effective anisotrofield HA rather than the saturation induction 4pMs mightprovide a better model of the microstructure in highly anistropic ferrites.

VII. SUMMARY

The two magnon scattering contribution to the ferromanetic resonance relaxation rate and linewidth has been evated as a function of field angle for isotropic and anisotrodisks and films. The analysis was based on the original Streatment but has been extended to include anisotropicterials, nonspherical scatterers, sample demagnetizationtors, and a nonzero linewidth for the scattered spin wamodes. The analysis yields characteristic angle dependein the two magnon scattering linewidth which reflect tscatterer size, sample shape, scattering interaction, andtalline anisotropy. Specific numerical evaluations were dofor three types of materials: YIG, Ba–M, and Zn–Y. Theexplicit calculations serve to demonstrate the various effeof the spin wave band shift relative to the FMR frequencanisotropy, and microstructure on the two magnon relaxaprocess. The results demonstrate that measurement oFMR linewidth as a function of angle for film and disk materials can provide useful information on the microstructuand two magnon losses in the sample.

ACKNOWLEDGMENTS

The authors acknowledge Dr. P. Kabos, Dr. B.Kalinikos, Dr. M. A. Wittenauer, and Dr. A. K. Srivastavfor helpful discussions during the course of this study. Dr.K. Srivastava provided a critical reading of the final manscript. This work was supported, in part, by the United StaOffice of Naval Research Grant Nos. N00014-94-1-0096 aN00014-91-J-1324, the United States Army Research OffiGrant No. DAAH04-95-1-0325, and the National ScienFoundation, Grant No. DMR-9400276.

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itz

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