Spin Damping in Ultrathin Magnetic Films-Spin-Damping

7/28/2019 Spin Damping in Ultrathin Magnetic Films-Spin-Damping

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Spin Damping in Ultrathin Magnetic Films

Douglas L. Mills1 and Sergio M. Rezende2

1 Department of Physics and Astronomy, University of CaliforniaIrvine, CA, 92697 [email protected]

2 Departamento de Fisica, Universidade Federal de Pernambuco50670-901 Recife, Brazil

Abstract. This chapter reviews the origin of the damping of spin motions in ultra-thin ferromagnetic films and multilayer structures, with focus on the linear responseregime probed by ferromagnetic resonance or Brillouin light scattering. We beginwith a description of the spin response provided by the LandauLifshitz equation,which ascribes damping to dissipative processes of intrinsic origin. It is noted thatthe form of the damping term should be modified in anisotropic materials, andexplicit expressions are provided for the form of a generalized damping term inbulk matter. We then turn to an extrinsic damping mechanism, the two-magnonprocess, which, recent experiments illustrate, plays a major role in spin dampingin ultrathin films and multilayer structures. The history of this mechanism in fer-romagnetic resonance studies is reviewed, the physical reasons for it to be active inultrathin ferromagnetic films are discussed, and we the review recent experimentalstudies that have verified central predictions of the theory.

1 Introduction

Understanding spin dynamics in nanoscale magnetic structures is a centralissue in current research for several reasons. In very small-scale devices, suchas GMR sensors or spin valves to be incorporated as storage elements inMRAM chips, the speed at which the magnetization of a constituent filmcan be reversed or reoriented is a key consideration in assessing the potentialusefulness of such structures. Clearly, this rate is controlled by the damping

experienced by magnetization as it precesses or rotates about the equilibriumdirection. Also, thin metallic ferromagnetic films deposited on semiconduct-ing or insulating microwave waveguides may serve as the basis for very highfrequency microwave devices [1,2,3]. Here, the spin damping rate, which de-termines the ferromagnetic resonance line width, is a key parameter thatcontrols the performance potential realizable in such devices.

In ferromagnetic media, spin motions are described commonly throughuse of the LandauLifshitz equation that contains a phenomenological pa-rameter G, the Gilbert damping constant. This controls the dissipation rate

associated with either the small amplitude motions probed in ferromagneticresonance (FMR) or Brillouin light scattering (BLS) studies of long wave-length spin excitations in ferromagnetic media and also that of large am-plitude spin motions associated with magnetization reversals. The damping

B. Hillebrands, K. Ounadjela (Eds.): Spin Dynamics in Confined Magnetic Structures II,Topics Appl. Phys. 87, 2759 (2003)c Springer-Verlag Berlin Heidelberg 2003


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28 Douglas L. Mills and Sergio M. Rezende

constant G is generally extracted from data on a particular system of inter-est; in most analyses, it is assumed either explicitly or implicitly that thedamping rate is intrinsic to the material from which the sample is fabricated.

However, for many years, it has been known that ferromagnetic linewidths, and other measures of spin damping in magnetic media can depend,in part, on growth conditions or on sample topology. There are thus extrinsiccontributions to the spin damping rate and other parameters that may controlthe dynamic response of magnetization. Such contributions are of great inter-est because they are subject to control through sample preparation. Carefulsample preparation can thus eliminate extrinsic sources of spin damping toproduce very narrow FMR lines. Conversely, in selected instances, one maywish to see spin motions more heavily damped. As an example of the lattercase, it is desirable to suppress ringing of magnetization after reversal in

certain devices. Thus, identification of specific extrinsic mechanisms of spindamping is of great interest.

This chapter is devoted to a discussion of spin damping and related mat-ters in ultrathin ferromagnetic films and multilayer structures that incorpo-rate such films. It has been known for a considerable time that the FMR linewidths in such films are controlled importantly by the manner in which suchfilms are prepared [4,5]. In the recent literature, it has been argued [6,7] thata particular process, two-magnon scattering, can be active in ultrathin films,and in fact explicit calculations [7] suggest that this mechanism can accountfor extrinsic contributions to line width comparable to those observed. Inaddition, associated with this source of line width is a shift in the resonancefield [7]. Recent experiments confirm key predictions of the theory [8,9,10].It now appears as if two-magnon scattering accounts very nicely for the verylarge FMR and BLS line widths found in exchange-biased structures, alongwith the differences found with the two methods of probing spin waves [10].In this chapter, we review general aspects of the description of spin dampingin ferromagnetic media, the history of two-magnon scattering and its rolein FMR line widths, and we then discuss its applications to the analysisof line widths realized in ultrathin film ferromagnetic films and multilayer

structures. Our attention is directed to the small amplitude spin motions ofinterest in the analysis of linear response characteristics of these systems.

While the focus of this chapter is on damping provided by the two-magnonmechanism, we call the readers attention to recent discussions of other pro-cesses wich influence damping in ultrathin films [11,12,13].

Section 2 is devoted to a review of the LandauLifshitz equation, includ-ing generalizations of the structure of the damping term if one wishes toexplore materials with strong uniaxial anisotropy. We turn our attention tothe theoretical basis of the two-magnon mechanism of spin damping in Sect. 3,

and discuss recent data in Sect. 4. Section 5 is devoted to brief concludingremarks.


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Spin Damping in Ultrathin Magnetic Films 29

2 Ferromagnetic Resonance in Ultrathin

Ferromagnetic Films. Phenomenology

In this section, we describe the phenomenological treatment of ferromagnetic

resonance in ultrathin films, with attention to the expression for the linewidth provided by the standard phenomenological description of such films,based on the LandauLifshitz equation. We divide this discussion into threesections. In the first, we examine the description provided by what we maywish to call the classical LandauLifshitz equation. Then in a second sec-tion, we digress a bit from the main thrust of this chapter to argue that,in fact, for some applications, a generalized form of this standard equationshould be employed. In the third, we comment on data reported for ultrathinferromagnetic films.

2.1 The LandauLifshitz Equation

and the Ferromagnetic Resonance Response

of Ultrathin Ferromagnetic Films Frequency Dependence

of Line Width in FMR and BLS

In a ferromagnetic material, the atomic spins are coupled together by verystrong exchange interactions of microscopic origin. As a consequence, if thesystem is driven by an externally applied microwave field of some frequency or by some other perturbation that drives the magnetization away from its

equilibrium orientation and if we examine the spins within a very small vol-ume d3r, the spins remain locked tightly parallel to each other by virtue ofexchange. A consequence is that the system may be described completelyby its magnetization per unit volume M(r, t), a vector of fixed length as itprecesses. The LandauLifshitz equation describes the motion of this vectorin space and time:

dM(r, t)

dt= [Heff(r, t) M(r, t)] + G

M2S

M(r, t) dM(r, t)

dt

. (1)

Here is the gyromagnetic ratio, the magnetization per unit volume inthe equilibrium state is zMS, and G is the Gilbert damping constant. In thefirst term on the right, Heff(r, t) is an effective magnetic field to which thespins respond. There are several contributions to the effective field. First,we assume that a static Zeeman field is present, zHz = z(H0 NzMS),where H0 is the external dc field and Nz is the demagnetizing field in thez direction. Of course, in a film magnetized in a plane, Hz = H0. A microwavedriving field h(r)exp(it), assumed perpendicular to the dc field, drivesthe precession of the magnetization. This precession generates a dipole fieldHD(r, t) which, in the magnetostatic approximation where

HD = 0,

may be written HD(r, t) = M(r, t) if desired. The magnetic potentialM is found from

2M(r, t) + 4 M(r, t) = 0 , (2)


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with appropriate boundary conditions appended. There are also contributionsfrom crystalline anisotropy, ignored here in the interest of simplicity. Finally,if the driving field is not spatially uniform, the magnetization density willvary with position, and spatial gradients will be resisted by exchange. Onemay describe this by appending an effective exchange operator D2 to theZeeman field, where the parameter D is referred to as the exchange stiffness.The effective field in (1) is the sum of these various fields:

Heff(r, t) = z(Hz D2) +HD(r, t) + h(r) exp(it) . (3)The first term in (1) describes the precession of magnetization in the effectivefield, and as we shall see, the second term is the equivalent of a viscousdamping term.

The film we consider is magnetized in-plane, so that the z-axis is parallel

to the film surfaces. We take the y-direction perpendicular to the film surface,so that the x-axis lies in-plane as well. The film is assumed ultrathin, so themicrowave driving field inside the film, which lies in the xy-plane, is spatiallyuniform. Thus, h(r) = xhx + yhy, with the right-hand side independent ofposition. Of interest are small amplitude motions of the spin system, whereinthe magnetization precesses uniformly in space. Thus, M(r, t) = zMS +(xmx+ymy)exp(it), where if we wish to discuss small amplitude motions,the LandauLifshitz equation is linearized with respect to m and h. Foruniform precession of the magnetization vector in a thin film, magnetized

in-plane, the dipole field is given byH

D(r

, t) = 4myy exp(it). Thesolution of the linearized LandauLifshitz equation then may be writtenmx = ,()hx i,()hy (4a)

and

my = ,()hy + i,()hx . (4b)

Before we display the susceptibilities which appear in (4), we introducenotation we shall use in this chapter. We let H = Hz, B = (Hz +

4MS), M = MS, and FM = [Hz(Hz + 4MS)]]1

2 . We shall see that

a dimensionless measure of damping is provided by the parameter = G/M.The damping parameter is small compared to unity for the materials tobe considered here. For example, for bulk Fe, G is 0.8 108 s1 at roomtemperature, and 4MS is 21 kG. Because the g factor is quite close to 2, 2.7 103. We shall keep this number in mind in the discussion below.

The explicit forms for the response functions in (4) are

,() =M(B i)

(B i)(H i) 2 , (5a)

,() =M(H

i)

(B i)(H i) 2 , (5b)

,() =M

(B i)(H i) 2 . (5c)


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Of interest is the rate at which energy is absorbed by the system. Fora magnetic material placed in a magnetic field H(r, t) with exchange ignored,a limit suitable to the case where the driving field is spatially uniform, therate of energy dissipation per unit volume is given by dU/dt =

H(r, t)

dM(r, t)/dt, where the angular brackets denote a time average. We find, toquote the complete expression for a moment,

dU

dt=

2M |Bhx ihy|2 + |Hhy ihx|2 + 22|hx + ihy|2

[2FM (1 + 2)2] + 22(B + H)2. (6)

For a weakly damped spin system, we have resonant absorption of energyfor frequencies near FM, which is the ferromagnetic resonance frequencyof a parallel magnetized film. As we have seen, our interest is in the limitwhere the dimensionless damping constant is small compared to unity. The

expression in (6) then reduces to the simpler form

dU

dt=

M4

(B + H)(B|hx|2 + H|hy|2) + 4iMFM(hxhy hyhx)

(FM )2 + (2 )2(B + H)2

. (7)

We are now in a position to deduce an expression for the line width

associated with the resonant response of the film, as provided by the LandauLifshitz phenomenology. This requires discussion because the result dependson the means of probing the response function displayed on the right-handside of (7).

In an FMR experiment, one places the sample in a resonant cavity withfixed resonance frequency and sweeps the dc magnetic field H0, so that FMis driven through the resonant frequency of the cavity. Thus, we consider thevariation of dU/dt with H0, at fixed frequency. If Hr is the field at whichmaximum absorption occurs, write H0 = Hr + H, and note that for smallexcursions in field, with r = [Hr(Hr + 4MS)]

1

2 ,

FM = r + [(B + H)/2r] H + . . . . (8)When (8) is inserted in (7), the factor (B + H) common to both terms

may be factored out, and a resonant denominator proportional to H2 +(r/)2 remains. We define the FMR line width as the full width of theLorentzian absorption line at half maximum. Then,

HFMR =2r

. (9)

The result in (9) is a central result of the LandauLifshitz phenomenology:the FMR line width should scale linearly with the FMR resonance frequency,as we see.


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Now, suppose instead of the FMR experiment, we consider Brillouin lightscattering (BLS). In the very thin limit, one may argue that BLS probes theresponse functions given above, with suitable modification of the numerator,to reflect the different means of coupling to the spin system. Thus, (7) de-scribes the Brillouin spectrum. However, one probes the response functiondifferently in BLS from that in FMR. In a BLS measurement, the magneticfield is fixed, and one explores the dependence of film response on frequency.It follows from (7) that the full width at half maximum of the spin-wavefeature in the Brillouin spectrum is given by (in magnetic field units)

HBLS = 2(Hz + 2MS) . (10)

The point of these remarks is that the BLS line width does not scalelinearly with the frequency of the spin-wave feature observed in the spec-

trum, which is roughly [Hz(Hz + 4MS)]1

2 . Note that the result in (10) isapproximate and applies in the regime where HBLS is small compared toFM.

We have traced through this example because it is often stated that thefact that the damping term in the LandauLifshitz equation is proportionalto dM/dt means that the width of a spin-wave feature in a spectrum shouldnecessarily display a linear variation with frequency, within the frameworkof this phenomenology. We see from (10) that this need not be so. One mustanalyze the manner in which the system is probed.

The discussion in this section has, in the interest of simplicity, confined itsattention to an in-plane magnetized ultrathin film, viewed as so thin that theexciting fields may be regarded as spatially uniform. When this is not the case,the magnetization components mx and my will also depend on y, the coordi-nate normal to the film. In this circumstance, one must perform a completeanalysis, with exchange incorporated into the response of the spin system.In microwave experiments such as FMR studies, it would seem that the ap-proximation of uniform exciting field should work for a wide range of metallicfilm thicknesses because the classical skin depth for a typical transition metalis in the range of one micron at 10 GHz. However, for in-plane magnetized

ferromagnetic films, the skin depth is in fact given by 0/

V(), where thequantity V() = [(B i)2 2]/[(FM i)2 2] is referred toas the Voigt permeability. The skin depth is thus reduced, possibly dramati-cally, near the FMR frequency. In high-quality Fe films, for instance, the skindepth can be as small as 300 A on resonance in the 10 GHz-regime. We referthe reader to discussions of this issue in [2].

In a BLS experiment, the relevant length scale for penetration of the ex-citing field is the optical skin depth, which is in the range of 150200 A forthe typical ferromagnetic metals of interest. The early theoretical descrip-

tions of BLS provide the formalism for addressing the response of the spinsystem to spatially inhomogeneous optical fields [14,15]. The treatment ofspin systems given is straightforwardly adapted to the description of FMRunder conditions where the exciting field is spatially inhomogeneous.


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A feature of the BLS probe that will enter importantly in the discussionsbelow is that the exciting fields associated with the laser also have a spatialvariation within the surface plane. The wavelength of visible light is in therange of 5000 A, and this has the consequence that the spin waves excited insuch experiments have wave vectors in the range of 105 cm1. In contrast, inthe microwave excitation process employed in FMR experiments, to excellentapproximation, the exciting fields may be viewed as spatially uniform withinthe surface plane. We shall return to this issue below, when line widths, asobserved in FMR and BLS, are compared for exchange-biased structures.

2.2 Comments on Extensions

of LandauLifshitz Phenomenology

The remarks in this section are stimulated by arguments set forward recentlyby Safonov and Bertram [16], who derive relaxation terms in the equationof motion of an anisotropic ferromagnet by coupling its magnetization toa heat bath. These authors argue that their results suggest that a dampingterm should be used which is more complex in structure than that providedby (1). Here, we examine this question from the viewpoint of symmetry con-siderations for a uniaxial ferromagnet. We find that an expanded form forthe damping term should be employed in such materials, and we argue thatif spin damping has its origin in spin-orbit-induced processes, the new termscan be of appreciable magnitude. For the case examined here, we also find

that for small amplitude spin motions, the extended equation provides re-sults compatible with those obtained from (1), provided that its coefficientsare renormalized. However, for large amplitude motions of magnetization,such as those involved in magnetization reversal, the new terms may assertthemselves.

We consider bulk ferromagnetic matter, of uniaxial character, with theeasy axis parallel to the symmetry axis. As the magnetization precesses insuch a material, it must remain fixed in length, for the reasons discussedabove. If the damping term in the equation of motion is designated as

(dM/dt)d, then for the magnetization to be fixed in length as it precessesabout, we must have M (dM/dt)d = 0, so that (dM/dt)d must necessar-ily lie in the plane perpendicular to M. The most general such vector maybe written (dM/dt)d = (G/M

2S )(M), where we inquire into the form

of. For low-frequency spin motions, we expect the dissipative term in theequation of motion to scale linearly with the frequency of the motion, so thecomponents of clearly should be proportional to those of dM/dt. The mostgeneral linear relation has tensor form:

=

dM

dt

. (11)

Landau and Lifshitz have required the damping term to be form-invariantunder arbitrary spatial rotations. If one requires this, then the only choice


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for is the unit matrix, and the classical LandauLifshitz equation (1),necessarily follows.

However, if we are considering a uniaxial ferromagnet, there is no reasonto require the damping term to be form-invariant under arbitrary spatialrotations. It is only necessary for it to be form-invariant under rotationsabout the easy axis. With this more restricted constraint applied, one canshow that

x =dMx

dt dMy

dt, (12a)

y =dMy

dt+

dMxdt

, (12b)

and

z = (1 + )

dMz

dt . (12c)

We are not quite finished because the forms in (12) are incompatible withreflection symmetry. Recall that M is an axial vector, so that reflection ofit through any plane changes the sign of components of M parallel to theplane but leaves components perpendicular to the reflection plane unchanged.Thus, upon reflection through the xz plane, Mx and Mz change sign, but Mydoes not. Thus, the expressions in (12) lead to an equation of motion thatis not form-invariant under such reflections. If, however, is required to beproportional to either Mz or H0 (the latter is the dc field assumed parallel

to z), then the structure is compatible with reflection symmetry.With the remarks above in mind, if n is a unit vector parallel to the

anisotropic axis, for a uniaxial ferromagnet, we may write

=dM

dt+ n

n dM

dt

+

a

n M

MS

+ b

n H0

MS

n dM

dt

. (13)

Here H0 is an applied magnetic field, not necessarily parallel to the

anisotropic axis. Of course, the new coefficients can be functions of the squareof the projection ofH0 onto n and the square of its projection onto the planeperpendicular to n, as well as of temperature.

One may inquire if one expects the new coefficients to be appreciablein magnitude. This depends on the microscopic origin of the damping pro-cesses. If they have their origin in processes that originate from terms inthe underlying microscopic Hamiltonian which are form-invariant under ar-bitrary spin rotations, then the new coefficients must vanish. An examplewould be multimagnon relaxation in magnetic insulators, whose origin is in

interactions between spin waves derived from isotropic exchange couplings.If the damping results from spin-orbit-mediated processes, where the basiccoupling terms reflect the symmetry of the crystal lattice, then there is noreason to expect that the new terms will be small in magnitude.


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Suppose we apply (13) to the description of small amplitude spin motionsas in our discussion of FMR. Then again we write M = zMS + (xmx + ymy)and linearize the equation of motion. We find

dmxdt

= 1 +

(HeffM)x GMS(1 + )

dmydt

, (14a)

anddmy

dt=

1 + (HeffM)y + G

MS(1 + )

dmxdt

. (14b)

Equations (14) are identical in form to the results provided by the classicalLandauLifshitz equation in (1), provided that one replaces the gyromagneticratio and Gilbert damping constants by = /(1 + ) and G = G/(1 + ).Thus, with these replacements, the results in the previous subsection stand,

as do those of similar analyses of small amplitude motions of magnetizationin uniaxial ferromagnets. Of course, conclusions about the field or frequencydependence of the line width and related issues would be affected if the newcoefficients are appreciable in magnitude. For large amplitude spin motions,the new contributions to the damping term may assert themselves more sub-stantialy.

Finally, note that the fact that the energy dissipation rate must be positivedefinite imposes the constraint , > 1.

2.3 Brief Comments on Experimental Data

In large single crystals of ferromagnetic metals, the phenomenology basedon the classical form of the LandauLifshitz equation accounts for the fre-quency variation of the line width, as one sees from the analysis presentedby Bhagat and Lubitz [17]. To extract the Gilbert damping constant fromexperimental data, one needs to combine the LandauLifshitz equation withMaxwells equations [18] because the finite skin depth creates spatial gradi-ents in transverse magnetization. The exchange terms then assert themselves.

A consequence is that the FMR line is no longer a symmetrical Lorentzian asin the simple treatment above, and a full analysis is required. This is quitea straightforward exercise, but consequently the Gilbert damping constantmay not be read directly from the data in a simple manner.

In ultrathin films of transition metal ferromagnets, the line width is foundas a linear function of frequency, as illustrated in Fig. 1, which we have repro-duced from a paper by Celinski and Heinrich [19]. However, as the frequencyis extrapolated to zero, one finds a residual line width that is sensitive to filmquality; the highest quality films exhibit the smallest residual, or zero-fieldline width. The data is commonly fitted by the empirical expression

HFMR = H(0) +2r

. (15)


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Fig.1. Data reported by Celinsky and Heinrich on the frequency dependence of theline width for a sequence of ultrathin Fe films grown under various circumstances.The figure was reproduced from [19]

Quite clearly, the zero-field line width H(0) is extrinsic in origin.Note also that in ultrathin films, the value of the Gilbert damping constantextracted from such a fit is substantially larger than that for single crystalbulk materials, as we see from Table I in [19]. One may compare the valuestabulated there with the data on single crystals in [17].

3 Extrinsic Contributions to the Line Width:

The Two-Magnon Mechanism

As remarked earlier, Arias and Mills have put forth a description of a spe-cific extrinsic mechanism that can provide contributions to the line widthcomparable to the zero-field line width discussed in the previous section [7].In the theory, there is also a shift in the resonance field associated with themechanism. As we shall see in Sect. 4, data presently in hand provide strongsupport for this picture for a variety of samples. In addition, Rezende andco-workers have shown that the discussion in [7] can be extended to accountfor the systematics of their striking data on FMR and BLS line widths inexchange-biased multilayer structures [10]. This section is devoted to a de-scription of the two-magnon mechanism and its physical basis. In Sect. 4, weshall review the current experimental situation.

In fact, two-magnon scattering was proposed several decades ago, as thesource of extrinsic contributions to the FMR line width of YIG samples [20].Thus, we begin this section with a brief history of this early work. Then we

turn to the reasons why the two-magnon mechanism is active in ultrathinferromagnetic films and the predictions of the theory for this case.


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3.1 Historical Comments

The discovery of the ferromagnetic insulator yttrium iron garnet (YIG) wasa development of major importance in research on the ferromagnetism of mat-

ter. Although the atomic arrangement within the unit cell is complicated, infact the crystal is an insulator of cubic symmetry. Crystalline anisotropy isthus very small as a consequence, and to an excellent approximation, thematerial may be modeled by a cubic array of localized spins coupled by near-est neighbor interactions of the Heisenberg form, supplemented by dipolarinteractions between elementary moments.

In the early generation of ferromagnetic resonance studies of YIG, it wasfound that the ferromagnetic line widths were substantially larger than ex-pected from intrinsic processes, as calculated from the model Hamiltonian

just described. LeCrawet al. [21] made the crucial observation that line widthwas controlled by the size of the grit used to polish the YIG spheres probedin the FMR measurements. It thus became clear that the origin of increasedline widths resided in an extrinsic process, that originates in surface de-fects associated with the sample polishing procedures. In a classic paper,Sparksetal. [20] developed the theory of a particular extrinsic mechanism,the two-magnon contribution to line width. The results provided an excellentaccount of the data.

To understand the manner in which this mechanism operates, we needto discuss the spin-wave spectrum of such spherical samples. First, consider

the spin motions induced by the microwave driving field in the ferromagneticresonance experiment. To an excellent approximation, the exciting field insuch insulating samples (macroscopic in size) is spatially uniform throughoutthe sample, as in our thin film example above. The precessing magnetizationgenerates a macroscopic dipole field, very much like that we encountered inthe description of the ultrathin film given in Sect. 2.1. However, in a sample ofspherical shape, the dipolar field generated by the spin motion of frequency is HD = (4/3)(xmx + ymy)exp(it), where again the equilibrium mag-netization of the sphere is zMS. If one describes the resonant response of the

sphere to the uniform driving field by using the LandauLifshitz equation,the ferromagnetic resonance frequency is given by

FM =

Hz +

4

3MS

= H0 . (16)

The uniform precession mode just described is, formally, a spin wave ormagnon mode in the spherical sample in which all the elementary spins pre-cess perfectly in phase; it is the equivalent of a spin wave of zero wave vector kin an infinitely extended medium, to speak somewhat loosely, though its fre-

quency is controlled by the nature of the dipole field generated by the spinmotion, and the character of this reflects the spherical sample shape. Sucha sample, in fact, possesses a whole spectrum of spin-wave modes. Suppose


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for the moment that we consider modes whose wavelength is very small com-pared to the radius R of the sphere. Such modes have a wave vector thatsatisfies kR >> 1. For such short wavelength modes, the dispersion rela-tion will be indistinguishable from the spin-wave dispersion relation in aninfinitely extended medium. In this limit, it is straightforward to generate anexpression for the dispersion relation of spin waves from (1), with the damp-ing term set to zero. For such a spin wave, the transverse magnetization willhave the plane waveform

m = (xmx + ymy) ex p [ ik r i(k)t ] , (17)where we wish to find the dispersion relation of the spin waves or magnons,as described by (k). The wave-like disturbance in the spin system generatesa dipolar field which may be calculated from (2):

HD(r, t) = 4k2k(kxmx + kymy) ex p [ ik r i(k)t ] . (18)

It is essential now for such short wavelength disturbances in the spinsystem to include the influence of exchange. From the discussion of Sect. 2.1,this may be done by replacing the internal dc field zHz by z(Hz + Dk

2). Onemay then obtain (k) from the homogeneous version of (1),

(k) = [(Hz + Dk2)(Hz + 4MS sin

2 k + Dk2)]

1

2 . (19)

In this expression, k is the angle between the wave vector k and z, thedirection in which the equilibrium magnetization is directed. This dependenceon the direction of propagation originates in the fact that the strength of thedipolar field generated by the spin motions depends on the orientation ofk.

A central observation is that there are short wavelength spin-wave modes,as described by (19), that are degenerate with the ferromagnetic resonancemode. To see this, consider the modes described by (19) whose wavelengthis sufficiently long that the influence of exchange may be ignored. We thenhave a band of modes whose frequency is bounded from below by m = Hz

(here k = 0) and above by M = [Hz(Hz + 4MS)]

1

2 (for which k =/2). Because FM lies above m and below M, necessarily there are shortwavelength spin waves degenerate with the ferromagnetic resonance modein some range of propagative directions. This is ensured by the exchangecontributions Dk2. The use of parameters characteristic of YIG show thatthese modes have wave vectors in the range of 5 105 cm1 [20]. Such modeshave wavelengths much smaller than the diameter of the spheres used in theexperiments, but also it should be noted, very much longer than the latticeconstant.

In this circumstance, any static defect in the system may scatter energyfrom the uniform precessional ferromagnetic resonance mode, here viewedas a k = 0 magnon, to a finite wave vector magnon which is degeneratein frequency. The process is a magnetic analog of the elastic scattering of


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electrons in metals, viewed as quantum mechanical de Broglie waves, fromimpurities and defects. However, the de Broglie wavelength of electrons inmetals lie in the range of two or three angstroms; consequently, these wavesare strongly scattered by defects of atomic size. In contrast, the magnonsinvolved in the two-magnon process have much longer wavelengths, so thecross section associated with scattering from impurities of atomic size is verysmall. Larger defects are responsible for the scattering, such as the polishingpits left on the surface during sample preparation. In [20], it is argued thatsuch surface defects couple the AF resonance mode to the short wavelengthdegenerate waves via a matrix element that originates in perturbation ofthe dipole field associated with the defect. Explicit calculations provide anaccount of the extrinsic contribution to line widths observed in the YIGsamples.

If we wish to construct a dynamical equation which describes the timeevolution of the total magnetization of the sample, under conditions wheretwo-magnon scattering is operative, the LandauLifshitz equation is inappro-priate, for the following reason. First recall that in a ferromagnet with mag-netization zMS creation of a magnon (as in excitation of the FMR mode)lowers the z component of angular momentum by precisely h. In a two-magnon scattering event, the z component of total magnetization remainsunchanged, since the number of magnons excited in the system remains un-changed. However, since the final state magnon has finite wave vector, thetransverse component of total magnetization is decreased. In the languageof resonance physics, two magnon scattering is a dephasing event, whichcontributes to the transverse relaxation time T2 but leaves the longitudinalrelaxation time T1 unaffected. This suggests the total magnetization may bedescribed by the Bloch equations, which may be written

dMx,ydt

= [HeffM]x,y 1

T2(Mx,y Mx,y) , (20a)

and

dMz

dt = [H

effM]z 1

T1 (Mz MS) , (20b)where Mx,y are the expectation values of the total transverse magnetiza-tion, in the presence of the applied microwave field. One may see that ifT2 = 2T1, the length of the magnetization is preserved upon relaxation asin the LandauLifshitz equation. Clearly, this equality is violated by two-magnon scattering. We note that in terms of the quantities which enter theBloch equation, the FMR linewidth (full width at half maximum) is given by2/(T2[dFM/dH]) = 2FM/(

2T2[H + 2MS]).Within volumes with linear dimensions small compared to the wavelength

of the final state magnon (which, we recall from the estimates above is verylong compared to a lattice constant), the length of the local magnetizationis conserved in the relaxation process as envisioned in the LandauLifschitz


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description. Then a complete dynamical equation for the time and spaceevolution of the magnetization density must be non-local in character. Onecan see this from the analysis of small amplitude spin motions in the paperby Arias and Mills [7]. In their general discussion, the response of the spinsystem is described by a self energy matrix with explicit wave vector depen-dence; in real space the dynamical equation thus has a non-local character.The form of such an equation is a most interesting topic for further study.

3.2 The Nature of Spin Waves in Ultrathin

Ferromagnetic Films

In Sect. 2.1, we used the phenomenology provided by the classical LandauLifshitz equation to describe the ferromagnetic resonance response of an ul-

trathin film magnetized in-plane, with anisotropy ignored. The resonance fre-quency associated with a spatially uniform precessional motion of the magne-tization, as shown, is FM = (H0B0)

1

2 , where H0 is the in-plane applied dcmagnetic field and B0 = H0 + 4MS. In this section, we discuss the nature ofspin waves of a finite wave vector in such an ultrathin film, with attention tothe issue of whether finite wave vector modes exist that are degenerate withthe ferromagnetic resonance mode. We shall see that the answer is in theaffirmative, as a consequence of the unusual contributions to the dispersionrelation from dipolar interactions between the spins in such systems.

The coordinate system we are using is illustrated in Fig. 2a. Thus, thexz plane is parallel to the film surfaces. Throughout this section, we assume,the film is so thin that, as spin waves are excited, the variation in magneti-zation in the direction normal to the surfaces may be ignored. We are thusin the limit where the system is regarded as a two-dimensional structure, toan excellent approximation.

Fig.2. (a) The geometry considered in the discussion of the nature of spin wavesin ultrathin ferromagnetic films. We suppose for the initial part of the discussion

that the magnetization is in-plane and the spin-wave vector makes an angle withthe magnetization and the applied dc magnetic field. (b) A sketch of the spin-wavedispersion relation in the ultrathin film for the two propagative regimes indicated.The angle is defined in the text, shortly after (22)


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There is additional physics we require before we proceed. In an ultrathinferromagnetic film, a large fraction of the magnetic moment bearing ions re-sides in surface sites or in sites at the interface with the substrate. Theseions experience very strong local anisotropy of spin-orbit origin as a conse-quence. Because all spins in the sample are coupled strongly by exchangeinteractions, this strong surface or interfacial anisotropy affects the responseof the system. In many circumstances, the influence of this anisotropy maybe accounted for by adding to the Hamiltonian of the system a term of theform

HA =KSM2S

M2y (x, z)dxdz . (21)

The parameter KS in (23) can be either positive or negative. In the former

case, the axis normal to the surface is a hard axis, and the magnetization willlie in-plane. If KS is negative, the axis normal to the surface is an easyaxis. In fact, one may see that in this case, if |KS| > 4MS, the surfaceanisotropy is sufficiently strong to orient the magnetization perpendicularlyto the surface. In practice, in ultrathin ferromagnets, one encounters bothcases, where in some instances the film is magnetized in-plane and in others,it is magnetized perpendicularly to the surfaces. Both the strength of thesurface anisotropy and the film magnetization vary with temperature. So, asone raises the temperature, a perpendicularly magnetized film may reorient

to magnetize in-plane at higher temperatures [22]. We confine our attentionto in-plane magnetized films in this section, save for a few brief remarks.When a spin wave of finite wave vector is excited in an ultrathin film,

the transverse components of magnetization vary as exp

ik r i(k)t

,where k lies in the xz plane, as illustrated in Fig. 2a. We are interested in thelimit kd


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of the spin wave and the magnetization, as illustrated in Fig. 2a, and D isthe spin-wave exchange stiffness.

For our purposes, the significant term in (22) is the term that is linearin the wave vector. Note that if the propagation angle k satisfies either

or |k | < C or | k| < C, where C = sin1 [H0/(B0 + HS)]1

2 , thenthe coefficient of the linear term in the wave vector is negative in sign. Quiteclearly, with the influence of exchange noted, the minimum spin-wave fre-quency is not at k = 0 as one customarily assumes, but rather occurs ata finite wave vector kM(k), as illustrated in Fig. 2b. Hence, by virtue of thelinear term in the dispersion relation, for modes that propagate in the an-gular regions just outlined, finite wave vector modes are degenerate with theferromagnetic resonance mode. Thus, the two-magnon process is operative.

We note one very important point. If we decrease the magnitude of the

external field H0 to zero, the critical angle C vanishes, and in the limit, nomodes are degenerate with the ferromagnetic resonance mode. Hence, as theFMR frequency approaches zero, necessarily the two-magnon damping shutsoff. Thus, when this mechanism is invoked in ferromagnetic ultrathin films,we arrive at a picture of the extrinsic contributions to the line width ratherdifferent from those based on using the empirical form in (15).

The discussion above has supposed that the dc magnetic field H0 liesentirely in-plane. It is interesting to inquire about the nature of the spin-wavedispersion when an out-of-plane component in the dc magnetic field is present

as well. The general discussion given by Erickson and Mills [23] includes thispossibility, though we need to change notation to adapt their discussion toour purposes here. First, their parameter a0 must be replaced by

2d. Let

the in-plane magnetic field component be denoted by H0 , let the out-of-

plane component be denoted by H0 , and define B0 = H

0 + 4MS. Then the

parameter HA of [23] must be replaced by the combination H0 4MSHS,

in terms of the notation used here. The spin-wave dispersion relation has theform [23]

(k

) = [A1

(k

)A2

(k

)]1

2 . (23a)

If H0 < B0 + 4MS, then the magnetization continues to lie in-plane, as in

the discussion above. Then,

A1(k) = H0 + 2MSkd sin

2(k) + Dk2 (23b)

and

A2(k) = (B0 + HS H0 ) 2MSkd + Dk2 . (23c)

If we expand the spin-wave dispersion relation in powers of k,

(k)2 =

H0 (B0 + HS H0 )

2MSkd

H0 (B0 + HS H0 )sin2 k

+(B

0 + HS H0 )Dk2 . (24)


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Quite clearly, as the perpendicular component of the magnetic field isincreased, with the parallel component held fixed in magnitude, the angularrange over which one realizes modes degenerate with the FMR mode in-

creases. Note that if we set H0

= B

0+ HS, the nominal value of the perpen-

dicular field where the magnetization just tips out of plane, the approximatedispersion relation in (24) is isotropic.

As just remarked, the nominal value of the perpendicular field required to

lift the magnetization out of plane is H0 = B0 + HS. However, as discussed

by Erickson and Mills [23], as the perpendicular component of the field isincreased toward this value, one enters a regime where the spatially uniform

ferromagnetic state is rendered unstable. Let H0 = B0 + HS H. When

H < HC1 = (MSd)2/D, there is a regime of wave vectors where

the quantity A2(k) is negative and the spin-wave frequency is negative. The

spatially uniform ferromagnetic state is unstable with respect to a modulatedstate, where the magnetization direction oscillates in the yz-plane with thecharacteristic length scale oflD D/MSd. For a 10 A thick Fe film, HC 100 G, and the characteristic length is in the range of 500 A.

As the perpendicular component of the field is increased, the magneti-

zation tips out of plane. Suppose that H0 > B0 + HS. Let be the angle

between the magnetization and the xz-plane, parallel to the film surfaces(in [23] the discussion is phrased in terms of = /2 , the angle betweenthe magnetization and the normal to the film surfaces). Then,

cos =H0

H0 HS 4MS, (25)

A1(k) = (H0 HS 4MS) + 2MSk sin2(k) + Dk2 , (26a)

and

A2(k) = (H0 HS 4MS)sin2

+2MSkd sin2 cos2(k) cos2

+ Dk2 . (26b)

For a regime of values of H0 above B0 + HS, we encounter againa regime where the uniform ferromagnetic state is unstable. If we write H0 =

B0 + HS + H

, then ferromagnetism is unstable when H < HC2 =(MSd)

2/2D, because in this field regime, there is again a regime of wavevectors for which A2(k) is negative.

In the regime where the magnetization is tipped out of plane and ferro-magnetism is stable, the dispersion relation for spin waves, through termsquadratic in the wave vector, is

(k

)2 = H0

HS

4MS

(H0 HS 4MS)sin2 2MSkd cos(2)

+(1 + sin2 )Dk2

. (27)


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There are two interesting features of the dispersion relation in (27). First,surprisingly, to the order considered the dispersion relation is independentof the direction of propagation, despite the fact that the magnetization iscanted with respect to the film normal. Second, when the magnetization istipped out of the plane, finite wave vector modes are degenerate with theFMR mode only when < /4. Thus, the two-magnon mechanism shouldbe operative in this regime of the tipping angle, but it should shut off forlarger values. In Sect. 4, we shall see that this conclusion is in excellent accordwith data reported in [6].

3.3 Two-Magnon Scattering and the FMR Response

of Ultrathin Ferromagnetic Films

As noted above, Arias and Mills [7] have presented a detailed theory of therole of the two-magnon processes in the ferromagnetic resonance responseof ultrathin films, when the magnetization and the applied dc field are in-plane. We summarize the ingredients of the theory in this section, along witha description of the results. In Sect. 4, we shall examine the experimentalevidence which suggests that the two-magnon mechanism is operative in typ-ical samples, along with a most important extension of the theory providedby the discussion presented by Rezendeet al. [10].

Surface defects in the form of islands and/or pits in the shape of platelets,it was assumed, activate two-magnon scattering. To proceed with a quantita-tive theory, one requires an explicit description of the coupling between thek = 0 spin-wave mode or magnon excited in the FMR measurement and theshort wavelength magnons degenerate with it in frequency. In [7], a detailedstudy is presented of three contributions to the matrix element. First, suchsurface or interfacial defects may perturb the Zeeman energy of the system.Second, as pointed out in the classical paper by Sparksetal. [20], the defectswill perturb the dipolar fields generated by the spin motions, and the nonuni-formity of the dipolar field is a source of mixing. Finally, in an ultrathin film,the strong surface or interfacial anisotropy discussed in Sect. 3.1 also provides

a source of mixing. One may describe this by applying the form in ( 21) toa film whose surface is not planar: Then

HA =KSM2S

[n M]2dS , (28)

where now the integration is over the actual physical surface perturbed by theislands or pits and n is the local normal to the surface. When the analysis ofthe various contributions to the matrix elements is completed, if one assumesthat H0, HS, and 4MS are roughly comparable in magnitude, the analysis

in [7] suggests that the dominant contribution to the matrix element comesfrom surface and interfacial anisotropy. Thus, in what follows, we shall assumethat this is the case. It should be noted that if one wishes to explore theinfluence of dipolar and Zeeman perturbations on the results quoted below,


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one may do this readily from the matrix elements provided in (81) and (82)of [7].

With a description of the coupling between the uniform FMR mode andshort wavelength spin waves in hand, one requires a description of the influ-ence of this mixing on the response of the film. This is done by generatinga description of equivalent response functions such as those displayed in (5)above, where the influence of the defect induced mixing between the FMRmode and short wavelength magnons is treated to second order. The generalresults are complex in structure, but the principal conclusions are readilysummarized.

First, of course, there is an extrinsic contribution to the line width.Azevedo and co-workers arrange the result in a simple form [8]. One replacesthe empirical form in (15) by

HFMR =32

3

sH2SD

sin1

H

1

2

0

(B0 + HS)1

2

+

2

FM . (29)

In this expression, the parameter s = pb2 [ a/c 1], where p is the frac-tion of the surface area of the film covered by defects, b is their average heightabove the surface or depth, and a and c are their lateral dimensions, with cthe length of the side parallel to the magnetization. The angular brackets inthe expression describe an average of the ratio across the ensemble of defects.

It is clear that the description of the extrinsic contribution to the linewidth provided by (29) is very different from that contained in the empiricalform described in (15). There is, in fact, clear dependence of the extrinsiccontribution on the applied field absent in (15). The extrinsic line widthvanishes in the limit H0 0, so the term zero-field line width used oftenin the literature is inappropriate. The physical origin of the field dependenceof the extrinsic contribution in the line width is clearly in the fact that asthe external field is decreased, the critical angle C discussed in the previoussection does as well, so the two-magnon mechanism shuts off as the fieldis decreased.

In Fig. 3, we reproduce a plot of the magnitude and frequency depen-dence of the two-magnon contribution to the line width, taken from [7]. Theparameters chosen are typical of those encountered in ultrathin Fe films. Thecalculation shows that with a reasonable choice of parameters, the mecha-nism can provide contributions to the line width comparable to those foundin samples. Notice that in the frequency range often explored in FMR exper-iments, 1036 GHz, the curve is rather well approximated by a straight lineof the form A + BFMR. Thus, if one has data taken in this regime, the linewidth can be fitted rather well by the empirical form in (15), with an appro-

priate choice of H(0) and G. However, it is quite incorrect to view H(0)as a low field line width. It is also inappropriate to interpret G so obtained asthe Gilbert damping constant operative in the bulk of the film. The appar-ent slope is in fact controlled both by Gilbert damping, surely present, and


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Fig.3. The frequency variation and magnitude of the two-magnon contribution to

the line width for an in-plane magnetized film, as calculated by Arias and Mills fora particular model of the surface defects. The curve is reproduced from [7]

a contribution from the two-magnon process which can be viewed as roughlylinear in frequency across a limited frequency domain. As remarked earlier,from Table 1 in [19], we see that the apparent Gilbert damping parameterextracted for ultrathin Fe films by using (15) is substantially larger than thatin bulk Fe, suggesting that the two-magnon mechanism is also contributing tothe apparent slope in the samples examined. Of course, the electronic struc-

ture in ultrathin films may be very different from that in the bulk material, soin fact there is no reason to expect to find the same Gilbert damping constantfor the two cases. The point of our remarks, however, is that it is difficultto separate the two-magnon contribution to the apparent slope from thatprovided by Gilbert damping by using data taken across a limited frequencyrange. It is the case as well that other mechanisms have been put forwardwhich also lead to enhanced Gilbert damping constants in the ultrathin filmlimit [10,11], and their presence is supported by recent data [11,12].

There is a further consequence of the two-magnon mixing terms in theHamiltonian not noted in the early discussion [20]. The analysis in [7] showsthat in addition to providing an extrinsic contribution to the line width,there is a two-magnon induced shift in the resonance field. If we view thetwo-magnon process as a coupling of the FMR mode to a reservoir of shortwavelength spin waves, then there is a reactive as well as dissipative compo-nent as a consequence of the coupling. In fact, within the framework of theclassical LandauLifschitz analysis, there is a reactive component as well, asone sees from (6). In the presence of damping, the FMR frequency of the film

is not simply FM = (H0B0)1

2 , but rather is given by FM/(1 + 2)

1

2 . Thus,the introduction of damping results in a shift in resonance frequency as well,

as we know from discussions of elementary harmonic oscillators.However, within the framework of the LandauLifshitz analysis, the shift

in resonance frequency is very small in all practical situations. This is not at


21/33


all the case with the two-magnon mechanism. In the language of quantumtheory, one can think that the two-magnon shift in resonance field describedin [7] originates in a shift in frequency of the FMR magnon, as given bysecond-order perturbation theory. The FMR mode is thus coupled to thewhole manifold of short wavelength spin waves, not only those degenerate infrequency with it. Calculations in [7], based on the same picture that providesthe extrinsic contribution to the line width displayed in Fig. 3, show that theshift in resonance field is roughly 100 G and is rather insensitive to the dcmagnetic field H0. Azevedo and colleagues [8] argue that this shift is wellrepresented by a simplified version of the expression derived in [7]. Theywrite for the shift HR in resonance field

HR = rH2S , (30)

r =16SD

ln

qmq0

12

+

1 +

qmq0

12

, (31)

where qm 1/a is a cutoff wave vector controlled by the lateral length scaleof the surface and interfacial defects, and q0 = 2MSd/D. We shall see in thenext section that the two-magnon extrinsic contribution to the line width isindeed accompanied by a shift in the resonance field, as given in (30).

The discussion so far has assumed that the defects that initiate two-magnon-scattering are randomly distributed islands, pits, or related struc-tures. Of interest is the case where the perturbations in surface topologyhave a unidirectional character. For example, the surface may have perfectlyparallel steps. If all of the steps have identical height and perfectly uniformterrace width, then the problem of describing spin-wave propagation is thatof spin waves in a perfectly periodic structure. Minigaps will be opened inthe dispersion relation for wave vectors whose component perpendicular tothe steps is equal to (

w+n 2

w). However, if the terrace widths are randomly

distributed, with the width of terrace i given by wi = w + wi, where w isthe mean terrace width and wi is randomly distributed, then the array of

steps will act as a scattering potential.Suppose then that the k = 0 FMR mode is excited, and we inquire

about the nature of the finite wave vector modes to which it may decay.Clearly, if the steps are perfectly straight and parallel, wave vector conserva-tion considerations require that the wave vector of the final state magnon beperpendicular to the step edges.

The statements in the previous paragraph then have a most importantimplication for the contribution to the line width from such scatterings: therewill be large in-plane anisotropy of the two-magnon contribution to the line

width. Imagine that we rotate the magnetization in-plane, and let be theangle between the magnetization and the normal to the step edges. Then ifC < < C, we see from Fig. 2b that there are no final state spin wavesdegenerate with the FMR mode and the two-magnon process is suppressed.


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If, however, 0 < < C, or C < < , the final state spin-wavepropagation direction is such that the coefficient of the linear term in thewave vector in the dispersion relation is negative, and we do have a finitewave vector mode degenerate with the FMR mode.

Thus, for a surface with parallel steps or with disorder that is unidirec-tional in nature, we expect that the two-magnon process will lead to in-plane anisotropy of the line width. This situation was discussed by Arias andMills [24], who provide an explicit expression for the angular variation of theline width. The result is striking. If nS is the number of steps per unit lengthand each has height h, then, for the step-induced two-magnon contributionto the line width,

HFMR =nSh

2H2S

4MSd

f()g()

sin2

(C) sin2

()

. (32)

In (32), f() =

cos(2)(B0 + HS) + cos2()H02

/ [(B0 + HS + H0)(B0 + HS)], whereas g() is controlled by the degree of disorder in the ter-race widths. If we assume that the terrace width distribution is Gaussian,with w the mean width and w the width of the distribution,

g() =sinh2

(kM() w/2)

2

2sinh2

(kM() w/2

2)2

+ sin2 [kM()w/2]

. (33)

In (33), kM() is the wave vector of the spin wave whose frequency isdegenerate with that of the FMR mode, when the magnetization is cantedwith respect to the normal to the steps by the angle . One has

kM() = {[2MSd(B0 + HS)] / [D(B0 + HS + H0)]} sin2(C) sin2() . (34)

The interesting feature of the result in (32) is the divergence as the anglebetween the magnetization and the step edges approaches the critical angleC (or C) from inside the regime where the two-magnon process isallowed. This divergence is, of course, an artifact of the model, in the sensethat in any real sample, step edges or linear defects will never be perfectlystraight and strictly parallel, as assumed in the analysis. Nonetheless, theobservation of a peak in the in-plane line width, as the magnetization isrotated through the critical angle C, accompanied by a rapid falloff as onemoves outside the allowed region will be a clear signal that the two-magnonmechanism is operative. As we shall see in the next section, this feature hasbeen observed in recent experiments.

In the next section, we discuss recent experimental studies of the mi-crowave response of ultrathin ferromagnetic films, with the above theoreticalpicture in mind.


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4 Experimental Evidence for the Role

of Two Magnon-Processes in the Microwave Response

of Ultrathin Ferromagnets

In this section, we review recent experimental FMR and BLS studies of themicrowave response of ultrathin ferromagnetic films. These provide evidencethat the two-magnon processes discussed in Sect. 3 influence both the linewidth and the resonance field.

We begin with a summary of the study by Azevedo et al. [8] of the thicknessvariation in FMR line width and resonance field in a series of Ni 50Fe50 films.In all measurements, magnetization lies in-plane. The films were prepared ona Si(100) substrate by a sputtering procedure, and the eight samples rangedin thickness from 20 to 160 A. FMR measurements on each sample were

carried out at various frequencies in the range 8.53 to 10.83 GHz. Both theline width and the resonance field increased dramatically as the film thicknessdecreased. For example, line widths of roughly 20 G were found in thick filmsat the higher frequency, and the line width increased to nearly 40 G in thethinnest sample.

The first step in analyzing the data was to assume that the FMR frequencyis found by combining the FMR frequency contained in (22) with the two-magnon-induced shift, so

FM =

{[H0(B0 + HS)]

1

2

rH2S

}. (35)

It is assumed, as discussed above, that HS varies inversely with film thick-ness. There are then four fitting parameters, , MS, r and the prefactor of 1/din HS. The procedure leads to HS = 82/d kG, with the thickness measuredin angstroms.

A check on the surface anisotropy field determined by this means wasobtained by directly measuring HS of the 27 A thick film by studying theangular variation in the FMR frequency as the magnetization was tipped outof plane. The two results agree. Thus, using data on the thickness variationof the resonance field allows one to deduce the strength of the anisotropy

field, in this instance. For this film, the shift in resonance field produced bytwo-magnon-induced renormalization of the FMR frequency is 77 G, which isappreciable. The size of the shift is comparable to that estimated by Ariasand Mills [7]. The authors of [8] remark that the frequency shift associatedwith the classical LandauLifshitz damping term is of the order of 0.01G.

The thickness dependence of the line width is then described by (29),where now all quantities are known except s and G. The value of s deducedby fitting the line width data can be accounted for by a reasonable model ofthe defects. One may take p = 0.5, b = 8 A and the ratio a/c = 1.5. If theseparameters are then employed to calculate the parameter r which enters theexpression for the shift in resonance field, one estimates r = 7 106 Oe1whereas from the earlier fit to the thickness variation of the resonance field,the value r = 8 106 was obtained.


24/33


Clearly, the analysis in [8] shows that the thickness variation of both theresonance field and the line width may be accounted for very reasonably byinvoking the two-magnon mechanism, where the variation in the directionof the anisotropy field across the nonideal film surfaces provides the cou-pling mechanism. The simple expressions given in (29) and (30) appear quitesufficient for this purpose, and clearly there is internal consistency in theparameters deduced from the data.

We have seen in Sect. 3.2 that if the magnetization is tipped out of theplane, one expects the two-magnon mechanism to be quenched. From (27), weexpect the mechanism to be inoperative when the magnetization tipping an-gle, measured from the plane of the film, is greater than 45. McMichael etal.[6] observed precisely this behavior, for a 10- nm thick Permalloy film de-posited on antiferromagnetic NiO, a structure that exhibits the phenomenon

of exchange bias. These authors, in fact, suggested that this observation iscompatible with two-magnon damping that originates in fluctuations of ei-ther the anisotropy field or the exchange field at the interface. They presenteda numerical calculation of the angular variation expected for the line widthin their discussion. We find it most striking that as one sees from their Fig. 3,there is very little dependence of the line width on tipping angle in the rangefrom 45 to 90. Virtually all of the decrease occurs in the angular rangewhere the tipping angle lies in the range from 0 to 45. The behavior theyobserve is thus quite compatible with the expectations based on the discus-sion in Sect. 3.2.

In [6], two samples were studied. One was the 10- nm thick Permalloy filmdeposited on NiO discussed above, and in addition they also explored similarfilms deposited on a Ta substrate. The line width found for the exchange-biased system was substantially larger than that observed for the Permalloyfilm deposited on the Ta substrate, and for the latter case the variation intipping angle was modest as well, though one can perceive a clear decreasein line width in the range from 0 to 45. Other authors have observed verylarge line widths in exchange-biased samples as well [25,26,27]. A systematicstudy of line widths in exchange-biased films and those in films deposited on

nonmagnetic substrates has been carried out by Rezende and his collabora-tors [10]. Of particular interest is their comparison between the line widthsobserved in FMR and the much larger line widths they find in spin wavesobserved via BLS. We turn to a description of their data and the associatedanalysis.

The samples studied were Fe50Ni50 films, deposited on NiO. The filmsranged in thickness from 140 A down to 37 A. An apparently extrinsic con-tribution to line width was observed, which increased with film thicknessroughly as 1/d2. Although the extrinsic contribution to FMR line width for

a 37A thick film deposited on Si(100) and studied in [8] was 10 G, for the37 A film deposited on NiO, this was 450 G, nearly two orders of magnitude

larger. In Fig. 4 reproduced from [10], we show a comparison between the


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Fig.4. Data reported by Rezende and co-workers on the thickness dependence ofthe FMR line width of ultrathin Fe50Ni50 films. The squares refer to the line widthof films grown on a Si substrate, and the circles show the line width for an exchange-biased structure, where the ferromagnetic film is in contact with NiO. The figureis reproduced from [10]

line widths of films deposited on NiO (circles) and films deposited on Si(100).The difference is dramatic. The inset shows the FMR spectrum for two cases,

d = 37 A, and d = 137 A. Before we turn to a discussion of the origin of thevery large line widths in the exchange-biased samples, we remind the readerof early discussions of the origin of this phenomenon.

Some years ago, Malozemoff argued that the phenomenon of exchangebiasing had its origin in random exchange fields felt by a ferromagnet fromsmall randomly arranged patches of antiferromagnetic grains at the interfacebetween the ferromagnetic film and the antiferromagnet substrate on which itwas grown [28,29]. It has been known since the discovery of exchange biasingmany decades ago [30,31] that the effective exchange fields responsible forthe shifted hysteresis loops were roughly two orders of magnitude smaller

than those associated with direct interfacial exchange between moments inthe two films. Malozemoff argued that in small antiferromagnetic domainslocated at the interface, the number of spins in each domain was sufficientlysmall and each domain contained a net moment due to the fact that ona statistical basis, there would be a slight imbalance in the number of spinson each sublattice within the grain. The ferromagnet could then have a netexchange coupling with the unbalanced spins in each grain, and the effectivefield responsible for exchange biasing had its origin in a nonzero averagealignment of the residual fields associated with the array of grains.

Rezendeet al. argue that the large line widths in exchange-biased filmsoriginate in microscopic roughness or defects such as pits and bumps ran-domly distributed on the interface between the antiferromagnetic NiO and


26/33


the NiFe ferromagnetic film. These are sensed in the ferromagnetic film asperturbations of the exchange coupling to the antiferromagnet. Their anal-ysis produces the following expression for the line width, when the applieddc field H0 is small compared to 4MS:

HFMR =4pAdcos2 H

1

2

03DB

1

2

0

H2I . (36)

Here, p is the fraction of the interface covered by the random defects inarea Ad, is the angle between the magnetization of the ferromagnet andthe easy axis within an antiferromagnetic domain, and HI is the strength ofexchange coupling within a domain. A fit to the magnitude and thicknessdependence of the FMR line width data is achieved by the choice p = 0.3,

Ad = (20 A)2

, < cos2

>= 0.5 and JI = 11.6 ergs/ cm2

, where HI = JI/MSd.The parameters deduced from the fit to FMR line width are surely reason-

able from the physical point of view, but the key point of the analysis in [10]is the ability of the picture just outlined to account also for the BLS linewidth data, by using exactly the same set of parameters. Although the linewidths found in the BLS spectra also increase dramatically as the thicknessof the ferromagnetic film decreases, for a given film they are very much largerthan found in FMR. In Fig. 5, we reproduce the data on BLS line widthsfrom [10]. The inset shows spectra for two different films, and one notes thatfor the 37 A thick sample, the mode excited in BLS is heavily damped.

The authors of [10] argue that the difference in line widths observed inthe two means of probing spin waves resides in the fact that in FMR, the

Fig.5. Thickness dependence and magnitude of the line width observed in BLSexcited spin waves for an exchange-biased Fe50Ni50 film. The figure is reproducedfrom [10]


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microwave field excites a k = 0 spin wave, whereas in the BLS experiment,the spin wave excited has a wave vector in the range of 10 5 cm1. In thelatter case, the magnon can scatter to a much larger number of final statemodes. We illustrate this in Fig. 6, where the spin-wave excitation spectrumis displayed for an Fe50Ni50 film 50 A thick. The mode excited in BLS canscatter to final state spin waves with the full range of propagative directions,whereas the FMR mode can decay to only the small angular range indicatedin the inset. In addition, for the BLS magnon, the density of final magnonstates at larger wave vectors is also very much higher than that accessibleto the FMR mode. For these reasons, the two-magnon mechanism leads tomuch stronger damping of the BLS mode than for the k = 0 excited in FMR.

Rezendeet al. provide an expression for the line width appropriate to theBLS regime. They show that, within the framework of the picture used to

derive (36), the BLS line width (in frequency units) is given by

q =2p Ad cos2 4MS

2DqH2I , (37)

where q is the frequency of the BLS magnon and is a dimensionless numberof order unity, given by the integral

=

xx0

x0x x

12

1 + x x0

x

12

dx , (38)

where x0 = q/k0, where q is the wave vector of the BLS magnon, k0 =(2MSd/D)

1

2 , and xm = km/k0. Here km is a cutoff wave vector, controlled bythe linear size of the defects. As remarked above, (37) provides a remarkably

Fig.6. An illustration of the final state spin waves that can be accessed throughtwo-magnon scattering by the spin wave excited in Brillouin light scattering for anFe50Ni50 film 50 A thick. The inset shows the final states accessible to the FMRmode. The figure was reproduced from [10]


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fine account of the BLS line width data, with parameters virtually identicalto those employed to fit FMR line widths.

Thus, we see that the two-magnon mechanism accounts for quite a numberof systematic features found in the data on extrinsic contributions to linewidths and to FMR resonance field shifts in ultrathin ferromagnetic films,including those incorporated into exchange-biased structures. We concludethis section with an account of data that addresses the prediction [24] of thein-plane anisotropy of the extrinsic line width, induced by linear defects ona surface such as steps.

McMichael and colleagues [9] produced a sample that allowed them toverify the prediction in (32). They grew a 65-nm thick Permalloy film on a Sisubstrate that had been subjected to mechanical abrasion with a 0.25 mdiamond slurry. This produced a sample that had roughly parallel grooves

etched in its surface, with an average spacing of 140 nm. The separationbetween adjacent grooves, along with other features such as their depth,were nonuniform. This is not the stepped surface discussed in [24], but quiteclearly, any random array of linear features on the surface will play preciselythe same role as steps. These authors then measured the azimuthal variationof the line width, as the magnetization was rotated in-plane. Under theirexperimental conditions, the angle C was roughly 15. When the angle was in the region C < < +C, the line width was roughly 40 G largerthan when the magnetization was canted at larger angles with respect to thenormal to the grooves. The data provide very clear evidence for the singularityin line width displayed in (32). We see peaks at the critical angle displayedclearly in their Fig. 5. In the view of the present authors, the data providevery strong evidence that the two-magnon mechanism controls the extrinsiccontribution to line width in this sample. Although the angular variation inthe extrinsic line width displays a clear peak at the critical angles = C,it does not plummet to zero suddenly when angle is increased. The linewidth falls off rapidly, but there are clear wings that extend out to largerangles. Of course, the grooves in the sample are not perfectly parallel, so,the singularity predicted by the simple model should be blurred. It should

be remarked that in the angular regime where the two-magnon mechanismis active, an upward shift in the resonance field is seen as well, as expectedon general grounds from the discussion in [7].

5 Concluding Remarks

From the discussion in Sect. 4, it is clear that recent experiments have verifiedall principal features contained in the theory of two-magnon scattering asa source of extrinsic line widths in ultrathin ferromagnetic films and alsoas an important source of extrinsic shifts in the resonance field. The lattercan be quantitatively significant, as one appreciates from the analysis in [8].Noteworthy is the understanding that has been achieved of the large line


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widths found in exchange-biased structures and of the difference betweenline widths observed in FMR and BLS studies of such samples.

A number of issues remain to be addressed. One can inquire about therole of two-magnon scattering in nanoscale structures other than the filmgeometry. This question has been addressed recently in a discussion of themicrowave response of ferromagnetic nanowires magnetized parallel to theirsurfaces [32], and the remarks there in fact can be viewed as applying toother systems as well. Suppose we consider a ferromagnetic wire of ratherlarge radius, so the ferromagnetic resonance mode is well described by simplemagnetostatic theory. Its frequency is then FM = (H0 + 2MS). The spinwaves whose wavelength is short compared to R may be described by (19),where we suppose the wave vector k = k + zk. For R quite large, we canregard both k and k as continuous variables in the regime of interest. In this

limit, the two-magnon contribution to line width can be explored within theframework offered in [20], though with modifications here and there. Now,as R is decreased, we can always regard k as a continuous variable for thenanowire, but we must recognize that k is quantized in units of/R. Whenthe radius R becomes so small that D(/R)2 becomes larger than the FMRfrequency, there are no longer degenerate short wavelength modes, and wecan expect the two-magnon mechanism to shut off. Thus, we may expectthat there is a critical radius of the order of (D/2MS)

1

2 below which thetwo-magnon mechanism is suppressed. (The criterion follows by comparingD(/R)2 with 2M

S.) If we have Fe in mind, the critical radius is in the

range of 100 A. These comments should apply to any small-scale magneticstructure, not just the nanowire, if one replaces the critical radius by thesmallest linear dimension that characterizes the structure. It would thus beof great interest to see systematic studies of the size variation in line widthwith this criterion in mind. This will be a challenge, of course, because anensemble of particles or small objects would be probed in any experiment.Thus, there will surely be inhomogeneous broadening of the resonance lineby virtue of the distribution in size and shape of the particles or entities inthe sample.

Another intriguing issue is extension of the theory to large amplitudespin motions, such as encountered when magnetization is reversed. The the-oretical discussions of the role of the two-magnon mechanism in the spindynamics of ultrathin film structures have confined their attention entirelyto the linear response regime probed in (low power) FMR and in BLS. Wehave seen by example in Sect. 2.2 that in bulk magnetic matter, the struc-ture of the damping term in the LandauLifshitz equation may have to begeneralized and that the resulting form may influence the description of largeamplitude spin motions, though there have been no explicit studies of this

issue, so far as we know. It would be of great interest to phrase the discus-sion of spin damping by extrinsic mechanisms in terms that lead one to anequation of motion for magnetization as a whole, so that the adequacy of the


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classical LandauLifshitz equation may be accessed. Even for small ampli-tude motions, it may prove possible to obtain an effective equation of motionthrough suitable modification of the formalism presented in [7].

We conclude by reminding the reader of the very interesting discussions ofspin transport based mechanisms which may produce increases in the Gilbertdamping constants of ultrathin metallic ferromagnetic films [11,12,13]. Thedynamic magnetization excited in FMR or BLS induces spin currents in theconduction electrons, which transport angular momentum out of the film.The signature of such processes is a linear dependence of the damping an spinprecession frequency, and a contribution to the apparent damping constantwhich varies inversely with the film thickness, a variation with thicknessdifferent than that associated with the two magnon mechanism.

Acknowledgements

The work at UFPE was supported by CNPq, CAPES, and FINEP, and hadmost important contributions from Prof. A. Azevedo, Prof. F. M. de Aguiar,Dr. M. A. Lucena, and A. B. Oliveira. The effort of DLM was supported bythe Army Research Office Durham, under Contract No. CS0001028.

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Index

Brillouin light scattering, 32, 49

degenerate spin-wave modes, 38, 42, 44dipolar interaction, 41

exchange-bias system, 50

ferromagnetic resonance, 31, 44, 49

Gilbert damping constant, 29, 35, 46

interfacial anisotropy, 41, 44interfacial defects, 44iron garnet, 37

LandauLifshitz equation, 29, 34, 40line width, 31, 36, 42, 49

magnetostatic approximation, 29

parallel steps, 47Permalloy, 49

residual line width, 35

skin depth, 32, 35spin-wave dispersion, 41surface anisotropy, 41, 44, 49surface defects, 44

two-magnon scattering, 36, 42, 44

uniform precession, 30, 37

zero-field line width, 35

Spin Damping in Ultrathin Magnetic Films-Spin-Damping

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