PHYSICAL REVIEW B VOLUME 53, NUMBER 18 1 MAY 1996-II Eddy currents and spin excitations in conducting ferromagnetic films N. S. Almeida* and D. L. Mills Department of Physics and Astronomy, University of California, Irvine, California 92717-4575 Received 13 October 1995 We explore the influence of the finite conductivity on spin waves in metallic ferromagnetic films. We consider propagation perpendicular to the magnetization, which is parallel to the surface, and wavelengths sufficiently long that the influence of exchange may be ignored. Precession of the magnetization induces eddy currents which damp the spin waves, and also renormalize the dispersion relation of the Damon-Eshbach mode encountered in this geometry. We provide analytic formulas which describe these effects, in various limits. Studies through use of a Green's-function method explore the influence of the conductivity on the spectrum of spin fluctuations, in various wavelength regimes. I. INTRODUCTION are considered. We show below that in certain regimes of wavelength and film thickness, eddy current damping be- The theory of spin-wave excitations in ferromagnetic comes very severe indeed. It is also the case, however, that films is a classic topic in magnetism, treated theoretically in for film thicknesses and wavelengths examined in numerous various limits many years ago.1 Experimentally, these modes recent experiments, its influence is modest. may be probed by ferromagnetic resonance, or by the Bril- We confine our attention to a geometry encountered com- louin scattering of light.2 In such experiments, the modes monly. We consider a ferromagnetic film with magnetization excited have wavelengths very long compared to a lattice parallel to the surfaces, and we consider spin waves which constant. In this regime, exchange interactions contribute propagate perpendicular to the magnetization. This is the negligibly to the excitation energy. Such spin waves are then case studied in most current experiments.2 In this geometry, described accurately by a theory based on magnetostatics, one encounters the Damon-Eshbach wave, a mode which in not only for films, but for samples of diverse shapes.3 the limit of wavelength short compared to the sample thick- Interest in this area has revived in recent years, as a con- ness becomes a surface spin wave,1 bound to either the upper sequence of modern sample preparation techniques, which or lower film surface, depending on its direction of propaga- allow the preparation of very thin ferromagnetic films and tion. The methods used here are readily extended to other multilayers of extraordinary quality, on diverse substrates.2,4 magnetization orientations, or propagation directions. The early studies of spin waves in thin films, spheres or We begin our discussion in Sec. II with a derivation of the ellipsoids, were directed largely toward insulating ferrites, dispersion relation of the Damon-Eshbach wave in the pres- such as YIG.3 The new materials described in the previous ence of finite conductivity. One may extract from this infor- paragraph incorporate films of ferromagnetic metals, such as mation on the linewidth of the mode by extracting the imagi- Fe. Theoretical descriptions of spin excitations directed to- nary part of the frequency, for a given wave vector k parallel ward these materials include features such as the anisotro- to the surface. We are led to simple, useful analytic formulas pies, dipolar couplings and interfilm exchange couplings here, in special limits. found in these samples,5­8 but do not explicitly acknowledge It is difficult to extract useful information from the im- the fact that the constituent films are metallic in nature. It is plicit dispersion relation, in regimes where eddy current the purpose of this paper to present and explore the influence damping and renormalization effects are severe. Thus, in of finite conductivity on the spin-wave excitations of a fer- Sec. III, we derive a set of Green's functions which may be romagnetic film. used to describe the response of the metallic ferromagnet to The issue of concern is the following. When a spin wave an arbitrary external microwave field, applied in the plane is excited, of course the magnetization precesses at each perpendicular to the magnetization. These response functions point in space, generally on a trajectory of elliptical charac- can be used for diverse purposes. By invoking the ter. The precessing magnetization generates a time- fluctuation-dissipation theorem, for example, we can use dependent internal magnetic induction b x,t everywhere. By them to explore the frequency spectrum of thermal spin fluc- Faraday's Law, this time-dependent magnetic induction must tuations, and also to describe the Brillouin spectrum of the induce an electric field e x,t . If the conductivity is finite, film.5,6 By such a study, we extract information on the nature eddy currents are generated by this electric field. The ohmic of the spin excitations, in the frequency and wavelength re- dissipation associated with the eddy currents is a source of gime where the influence of eddy currents is severe. linewidth for the spin-wave mode. We show below that in In the analysis presented here, we ignore the influence of addition, the eddy currents can renormalize the dispersion exchange. The precessing moments, in the present picture, relation of the modes. generate dipolar fields which influence the dispersion rela- It is of interest to inquire if the films and multilayers can tion of the spin waves we consider. Under the conditions be utilized for device applications. The lifetime of the spin- explored here, exchange effects on the Damon-Eshbach wave modes is a critical parameter when such applications waves are quite modest, and may be set aside with little 0163-1829/96/53 18 /12232 10 /$10.00 53 12 232 © 1996 The American Physical Society 53 EDDY CURRENTS AND SPIN EXCITATIONS IN CONDUCTING . . . 12 233 4 M H 1 1 2H i/ 2 2.2a and 4 M i/ 2 2H i/ 2 . 2.2b Here H H0 and M Ms , while is a phenomeno- logical relaxation time. Within the film, we have 4 h FIG. 1. The geometry considered in the present paper. We have c e, 2.3a a ferromagnetic film of thickness D, with magnetization Ms parallel to the surface. An external magnetic field H0 is applied parallel to *b 0, 2.3b the surface. We consider spin waves, with wave vector k that propagate parallel to the x axis. and also e i consequence. One may appreciate this from earlier discus- c b, 2.3c sions, in which exchange is included fully.5 In the particular where e is the electric field generated by the precessing mag- case of Brillouin spectra, exchange influences the spectrum netization. The conductivity of the medium is . For the of standing spin waves importantly, by introducing splittings geometry in Fig. 1, the electric field e is parallel to the z between the various standing wave modes. This feature is direction. We have ignored the displacement current term on absent from the calculations presented below. It is straight- the right-hand side of Eq. 2.3a , an approximation valid so forward, in principle, to extend earlier discussions5 to in- long as we are concerned with lengths smaller c/ . The re- clude both exchange and also the eddy current effects ex- tardation effects introduced by this term will be negligible plored here. The cost in complexity is substantial. We leave for the examples explored here. this for future work, when we wish quantitative contact be- All fields in the above equations are proportional to tween theory and experiment, in the standing spin-wave exp(ik spectra of conducting films. x ) , with a y dependence to be determined. It is a short exercise to find a pair of equations obeyed by hx and hy : II. THE INFLUENCE OF FINITE CONDUCTIVITY ON THE DISPERSION RELATION 2 1 2 2 OF DAMON-ESHBACH WAVES ik2 h h 2 y k 2 x 0 2.4a 0 y i 0 The geometry we consider is illustrated in Fig. 1. We have a ferromagnetic film of thickness D, with magnetization and Ms parallel to the surface. An external magnetic field H0 is applied parallel to M s . The coordinate system is aligned so M i 1k 2 hx 1 hy 0. 2.4b s is along the z axis, the film lies between y 0 and y D, y y 2k and the waves we consider propagate in the x direction. Their wavelength will be sufficiently long that we ignore In these equations, we have introduced the classical skin exchange effects. depth 0, in a medium with conductivity : The spin waves have frequency . As remarked in Sec. I, the precessing magnetization generates magnetic field h and c a magnetic induction b, both of which oscillate in time with 0 2 1/2. 2.5 the frequency . Both h and b lie in the xy plane, for the geometry in Fig. 1. For the ferromagnet, we have the consti- We seek solutions in the medium with the spatial varia- tutive relations9 tion exp Qy . One finds, after a brief calculation, b 2 V x 1hx i 2hy 2.1a Q k2 1/2, 2.6 i 20 and where here and elsewhere in the paper we choose Re(Q) 0. The quantity by 1hy i 2hx , 2.1b 2 2 2 i/ 2 1 2 B where V 1 H B i/ 2 . 2.7 12 234 N. S. ALMEIDA AND D. L. MILLS 53 In Eq. 2.7 , B H 4 M (H0 4 MS). conditions conservation of hx and by . For y D, all fields Some comments on the physical content of Eq. 2.6 are vary as exp k (y D)... and for y 0, they scale as in order. First of all, in a conducting material, the expres- exp( k y). sion in Eq. 2.5 is the well-known classical skin depth. In a It should be remarked that here, and throughout the re- material with a nonzero magnetic permeability, the skin mainder of the paper, we suppose k 0. The frequency depth is modified by the magnetic response. The combina- can then be either positive or negative.13 Disturbances which tion propagate from left to right are described by choosing 0, and those which propagate from right to left by choosing 0. 0 When the analysis is completed, we find the implicit dis- FM 2.8 V persion relation may be written which appears in Eq. 2.6 is the effective skin depth in the ferromagnet. Clearly, this is strongly frequency dependent, Q 2 Vk 2 2 / 1 2k in the vicinity of the resonances in the structure. Recall that Q 2 exp 2QD . 2.11 Vk 2 2 / 1 2k ( H B)1/2 is the ferromagnetic resonance frequency of a thin film with magnetization parallel to the surface.10 Near This may be rearranged to read resonance, V becomes very large, and the skin depth is reduced dramatically from the classical value 0. If H is 8 2 2 k2tanh QD the linewidth, defined as the full width at half maximum of i/ 2 M H B 2 2 , the absorption line then in our phenomenology H 2/ , k Q k i/ tanh 0 QD on resonance the skin depth is well approximated by 2.12 which as 0 , yields the standard dispersion relation1 ap- propriate to the insulating film for : H 1/2 H 1/4 FM H B 1/2... R 0 4 . M B 2.9 2 k 2 2 H 0 2 M s 2 4 2 2M s exp 2k D . In Fe, 4 2.13 Ms 2.1 104 G. If the linewidth H 100 G, and the resonance frequency is in the 10 GHz range, R is smaller From Eq. 2.12 , which remains an implicit dispersion than 0 by a factor of roughly 25. The influence of the spin relation by virtue of the presence of the frequency-dependent system on the penetration depth of electromagnetic radiation quantity Q on the right-hand side, we can extract simple is thus dramatic, near resonance. The skin depth FM be- formulas in special limits. To treat the regime k comes very large near 0 1, for B . The film ``opens up,'' and its example, we replace Q by Q k transmissivity increases dramatically. This is the phenom- [1 i V/( k 0) 2 *** ] and expand the right-hand side in powers of k enon of ``antiresonance,'' discovered some years ago.11 0 2, retaining only the leading term. After some algebra, we obtain we let We recover the theory appropriate to insulating media in for the moment the limit 0 . Then Q k . Suppose we consider an in- finitely thick film, D , and a Damon-Eshbach wave 2 propagating down its surface. The spatial variation of the 2 2 H0 2 Ms 2 4 2 2Msexp 2QD disturbances associated with the wave is controlled by 2 i 2M exp( Qy), which becomes exp( k s y ), as 0 . As the wave vector k k 0 2 H0 2 Ms H0 6 Ms e 2QD 0, the fields penetrate ever more deeply into the material, in a manner identical to Rayleigh surface acous- *** . 2.14 tic waves.12 This behavior is modified dramatically by the presence of the finite conductivity, where now The result in Eq. 2.13 remains an implicit dispersion rela- tion, by virtue of the presence of Q on the right-hand side. There are two limits where we may generate simple results: lim Q 2 1/2. 2.10 the very thin film limit D 0, and the thick film limit D . k i FM 0 We have, restoring the influence of the relaxation time , The fields can penetrate no deeper than the skin depth FM associated with the frequency of the wave. When D i i M FM , in s the long-wavelength limit, the fields of the Damon-Eshbach lim k H0 2 Ms k 0 2 *** , wave will be confined to a channel of depth D FM , near the 2.15a surface. Clearly, in this regime, the eddy currents not only limit the lifetime of the wave, but will modify its effective dispersion relation dramatically as well. i 2i Ms B It is a straightforward matter, following procedures now lim k H0B 1/2 *** . k 0 2 H standard, to determine the implicit dispersion relation of the D 0 2.15b wave. Inside the film, the various fields are supposed a su- perposition of exp( Qy), and exp( Qy). These are Clearly, the last terms in Eqs. 2.14 describe damping pro- matched to fields in the vacuum through standard boundary duced by the eddy currents induced by the spin motion. 53 EDDY CURRENTS AND SPIN EXCITATIONS IN CONDUCTING . . . 12 235 We may also extract the behavior of the effective disper- recalls 4 Ms 21 kG for Fe, one sees the eddy current con- sion relation for small k , from Eq. 2.11 . Quite clearly, as tribution to the linewidth is in the range of a few tens of k 0, tends to ( H B)1/2 i/ , the damped ferromag- Gauss. This is below current experimental resolution, but is netic resonance frequency of the film. The task of extracting by no means negligible. the first correction to this term in tricky, in the small k limit. The limit k 0 1 applies to ferromagnetic resonance ex- For k 2 2 0 1, clearly we may ignore k compared to i / 0 . periments. One sees from Eqs. 2.19 that k must be finite Also, in Eq. 2.11 , we replace Q by simply for eddy current damping to influence the linewidth. This Q (2 2 V/i 0)1/2. Note that as k 0, and ( i/ ) suggests that in most such experiments, eddy current damp- ( H B)1/2, V and thus Q become very large. We shall ing should be very small. We expect k /W, with W the assume QD as k 0, so tanh(QD) 1. The criterion for width of the sample, if one has spin pinning at the edges. the validity of this assumption will be stated when the analy- The approximate formulas obtained above suggest that sis is complete. We then have when k 0 1, the eddy current damping should be very sub- stantial. It is difficult to extract meaningful information from i 8 2 2 k the effective dispersion relation, in this regime. Thus, in Sec. 2 M 0 2 III we turn to a discussion of the Green's-function method H B . 2.16 k 0 2 V /i 1/2 i used in numerous earlier papers. This method can provide We seek a solution of Eq. 2.15 with information on the excitations in the system, even in the limit of strong damping. i/ 2 H B iA k 0 2, 2.17 III. GREEN'S-FUNCTION DESCRIPTION OF THE FILM where A is to be determined. Now RESPONSE IN THE PRESENCE OF EDDY CURRENTS 2 i/ 2 4 B M B In Sec. II, we examined the influence of eddy currents on V B H i/ 2 iA k 0 2 , 2.18 the dispersion relation of the magnetostatic spin waves of the as k ferromagnetic film, for the geometry illustrated in Fig. 1. We 0. When Eq. 2.16 and Eq. 2.17 are inserted into Eq. 2.15 , we find two solutions for A: saw that the eddy currents introduce damping, which could be very severe in the wave-length regime where k 0 1, if the asymptotic formulae of Sec. II are extrapolated into this A 1/2 1/2 2 M B H . 2.19 regime. In principal, at least, we could explore this issue We then have, for each choice of wave vector, two propa- further through examination of the dispersion relation found gating modes, each characterized by a different lifetime and in Sec. II, through numerical studies which trace out the propagation length: complex frequency as a function of wave vector. In the re- gime where damping is strong, such studies are of limited usefulness, since it is often unclear how the dispersion rela- i B 1/2 1/2 tionordampingrateratewhichemergesarerelatedtovari- k H0B 1/2 i MS H H0 0 B ous experimental probes of the system. We recall earlier dis- cussions which led to unphysical conclusions in other k 0 2 *** . 2.20 contexts,14 and as a consequence we turn to the development The origin of the two-mode behavior is the resonance in the of a Green's-function technique within which strong dissipa- Voigt susceptibility tive effects can be incorporated, and related to diverse ex- V which controls the effective skin depth. A discussion of the excitation of these modes requires periments in an unambiguous manner. an analysis of any particular experiment of interest, to assess One proceeds by supposing the film is driven by a weak their relative amplitude. The Green's functions discussed in external magnetic field in the xy plane, given by Sec. III will allow one to perform such analyses, for any desired excitation scheme. h e x,t x h e y y h e y exp ik We remark on the requirement for our assumption x y x i t , 3.1 QD 1 to be valid. From Eq. 2.17 , as k 0, one sees where the profile of the external field in the y direction is D arbitrary. In response to this field, in linear-response theory, QD 8 M B 1/2, 2.21 the magnetization of the film is k 2 A 0 so we must have D k 2 0 , or k 0 ( D / 0). M x,t z Ms m x,t , 3.2a The results in Eqs. 2.14 and Eq. 2.19 allow one to estimate the influence of eddy current damping, for various where experimental situations of interest. For instance, in typical Brillouin-scattering experiments, the modes excited have m x,t x m k x y y my y exp ik x i t . 3.2b 105 cm 1, while for metallic Fe, in the microwave fre- quency range 10 GHz, for example , one finds (e) (e) 0 10 4 cm. The elements of the external field h x (y),h y (y)... are re- We thus have k 0 10 in such studies, and we may use Eqs. lated to the system response mx(y),my(y)... by a matrix of 2.14 to estimate the eddy current damping effects. If one Green's functions introduced below. 12 236 N. S. ALMEIDA AND D. L. MILLS 53 We consider the system driven by an external perturbation d y e y , characterized by only a single frequency , and wave vector Hmx y i my y Mhx Mhx 3.4b k parallel to the surface. Within linear-response theory, straightforward Fourier synthesis may be used to describe while after elimination of the electric field, we may generate the response of the film to a perturbation of arbitrary form in the dipolar field from space and time. By this means, for example, with the use of our Green's functions, one can develop the theory of spin- h d y y wave generation by a current bearing strip or meander line 4 i 2m d x y 2 y2 i 2 hx y ik y 0 modeled as a periodic structure deposited on the surface. In 3.4c this paper, we do confine our attention to propagation per- pendicular to the magnetization, as illustrated in Fig. 1. and The fluctuation-dissipation theorem also may be em- ployed to relate our Green's functions to the amplitude of h d y thermal spin fluctuations in the film, of frequency and 4 i 2m x 2 d y y ik i 2 hy y 0. wave vector k y k . The spatial variation of the thermal spin fluctuations may be explored by this means. Brillouin- 3.4d scattering experiments provide us with a probe of such ther- In these last two equations, 2 2/ 20, where 0 is the classi- mal spin fluctuations, within the optical skin depth. Thus, our cal skin depth defined in Eq. 2.5 . The processing magneti- earlier theories of the Brillouin spectrum of ferromagnetic zation generates fields in the vacuum, which must be films and superlattices utilized similar Green's functions for matched to those in the film through appropriate boundary this purpose.5,6 One may view the present paper as an exten- conditions. These are conservation of tangential h continu- sion of an earlier description5 of spin waves in thin films to ity of h (d) x (y ) at the film surfaces , and normal b continuity include the influence of eddy currents induced by the fluctu- of h (d) y (y ) 4 my(y ) . The fields in the vacuum are de- ating magnetization. The earlier study incorporated exchange scribed by Eqs. 3.4c and 3.4d with 2 set to zero. effects ignored here; as noted in Sec. I the present discussion We rewrite Eqs. 3.4 by introducing two four component is readily extended to include exchange, at a cost in technical vectors. complexity. Our interest will center on the Damon-Eshbach portion of the response presently; exchange effects on this mode are quite modest, in the experiments which motivate mx y m our analysis. u y y d 3.5 We now turn to the formalism. As noted above, our task hx y is to generate a description of the response of the film when h d y y it is driven by the external magnetic field in Eq. 3.1 . The and basic equation we solve describes the precession of the mag- netization of the film in an external field: e y e dm Mhy y f Mhx , 3.6 dt i m M h , 3.3 0 0 where in the spirit of spin-wave theory, we linearize the right-hand side of Eq. 3.3 with respect to the fluctuating so they acquire the form portion of the magnetization defined in Eqs. 3.2 . The quan- tity is the gyromagnetic ratio. The magnetic field h in Eq. 4 3.3 is the externally applied field h(e) x,t described in Eq. Lijuj y fi y 3.7 3.1 , and to this is added the fluctuating dipolar field j 1 h(d) x,t generated by m x,t . The dipolar field is linear in m, with Lij a 4 4 matrix of differential operators. We obtain and is calculated by solving Maxwell's equation. We pose the solution of Eqs. 3.7 by introducing a matrix of Green's the problem of generating formulas for mx(y),my(y) in Eq. functions Gij(y,y ) which satisfy 3.2b when h (e) (e) x (y ),h y (y ) in Eq. 3.1 are arbitrary, un- specified functions of y. All quantities exhibit the time de- 4 pendence exp i t , and vary with x as exp(ik x). The L dipolar field h(d) also lies in the xy plane for the geometry ikGk j y ,y i j y y . 3.8 j 1 considered, and the eddy currents are parallel to the z direc- tion. Thus, the electric field is parallel to z also. We then have We define H H0 , and M Ms . The magnetization components then are found from 4 D u i y Gij y,y f j y dy . 3.9 j 1 0 i m d e x y Hmy y Mhy y Mhy y 3.4a By tracing through the definitions, we see that MG11(y,y ) provides the x component of magnetization mx(y) at point y, and in response to an external driving field applied parallel to the 53 EDDY CURRENTS AND SPIN EXCITATIONS IN CONDUCTING . . . 12 237 y axis, and with the spatial variation (y y ). Similarly MG21(y,y ) gives my(y) for such a driving field, while MG12(y,y ) gives mx(y), in response to a field with this spatial variation, but applied parallel to x. Finally, MG22(y,y ) is my(y) in response to a field applied par- allel to x, localized at y y . Our task is to solve Eqs. 3.7 subject to the boundary conditions that for any choice of y ,G3j(y,y ), considered a function of y, is continuous at the film surface, as is the combination G4j(y,y ) 4 G2j(y,y ). These ensure con- servation of tangential h and normal b, respectively. It is a tedious exercise to carry through the solution of the above set of equations, but the procedure is straightforward in principle. We thus omit details, and present a tabulation of the results in the Appendix. The Green's functions Gij(y,y ) can be used to explore the amplitude of thermal spin fluctuations in the film, as remarked earlier. Let mx x,t and my x,t denote the x and y components of the fluctuating magnetization. We can always Fourier transform these variables: m x,t d2k dt 2 3 eik *x e i tm k ;y , 3.10 where m k ;y is the amplitude, within the plane y const, of the thermal fluctuation of frequency , and wave vector k , in the plane parallel to the film surfaces. If FIG. 2. The spectral density function syy(k , ) defined in Eq. kBT, the limit of interest for the long-wavelength 3.11b , calculated for 0 105 cm 1, 0 10 5 cm, D 10 4 cm, dipolar-dominated spin waves of interest here, then the fluc- H0 0.5 kG, H0 4 MS 4 kG, and 1/ 0.01 , where is the tuation dissipation theorem tells us that Im G12(y,y ) is frequency. proportional to (kBT/ ) mx*(k ;y)mx(k ;y ) T , where the angular brackets denote a statistical average over equal to 105 cm 1. The calculations are for a film whose an ensemble at the temperature T. Similarly Im G21(y,y ) thickness is D 10 4 cm. Finally, H is proportional to (k 0 0.5 kG, and we set BT/ ) my*(k ;y)my(k ;y ) T . A B0 4 kG, while we use a frequency-dependent relaxation precise statement of these connections is found elsewhere.15 rate 1/ 0.01 , with the frequency. Again, the present paper confines its attention to wave vec- These numbers do not describe any real material, but are tors k perpendicular to the magnetization. chosen for convenience in display. Notice that one may scale We shall explore the thermal fluctuations sensed by a the above results to apply to real materials, since the fre- probe that extends into the sample a depth d 1 0 , with an quency dependence is controlled by the two dimensionless exponential profile. The fluctuations of wave vector k and ratios / H and H/ B , and we have two parameters frequency parallel to the surface sample by such a probe 0/D and k 0 which control the wave vector dependence. are described by The prominent peak in Fig. 2 is the structure associated with the Damon-Eshbach wave of the film. In the absence of eddy current effects, from Eq. 2.12 we see that for k D D D , s its frequency is (H xx k 0 2 M s) 2.25 kG, while as k D 0, , dy dy e 0 y y Im G12 y,y 0 0 we have (H0[H0 4 Ms])1/2 1.41 kG. For the largest 3.11a and smallest values of k in Fig. 2, the peak indeed coincides while those normal to the surface are given by with these limiting values. One sees clearly from Fig. 2 the very strong eddy current damping effects when k 0 1. There is some eddy current D D damping in the feature shown for k 4 105 cm 1, the mode syy k , dy dy e 0 y y Im G21 y,y . becomes very broad indeed for k 1.0 105 cm 1 and also 0 0 for k 3.11b 0.5 105 cm 1, and then narrows down when k drops to 0.1 105 cm 1. This behavior is compatible with the be- A description of the Brillouin light-scattering spectrum is havior provided by the asymptotic formulas in Sec. II. obtained by suitably synthesizing these and other closely re- There are two length scales in the problem. One is the lated spectral density functions.5,6,15 film thickness D, and the other by the microwave skin depth In Fig. 2, we show the spectral density syy(k , ) , calcu- 0 . When 0 D, the near surface spectral densities are con- lated for the following parameters, chosen to display clearly trolled by the parameter k 0 , and are insensitive to k D . We the influence of eddy current damping. We have 0 10 5 may see this by calculating spectral density functions for cm for the microwave skin depth, and 0 has been chosen D 10 3 cm, a value ten times larger than that in Fig. 2. The 12 238 N. S. ALMEIDA AND D. L. MILLS 53 FIG. 3. From plots such as that in Fig. 2, we extract the line- width H of the Damon-Eshbach mode dotted line , and its effec- FIG. 4. The same as Fig. 3, but now the film thickness D 10 5 tive dispersion relation, shown as the solid line. The prediction of cm. Eq. 2.12 , appropriate for a film with no eddy current damping, is shown as the dashed line. the eddy current damping will affect the spectra severely only at values of k results are identical to those in Fig. 2, to graphical accuracy. smaller than those accessed in typical experiments, which explore the regime k In essence, the microwave skin depth 105 cm 1. We 0 acts as the active illustrate this in Fig. 5, with a series of spectra calculated for channel, to which the fields of the Damon-Eshbach wave are confined. 0 10 4 cm, appropriate to Fe. Severe eddy current broad- ening is evident in the spectrum for which k We may extract the variation of the linewidth with fre- 0.1 105 cm, but its influence is rather modest at the large wave vectors. quency, and also an effective dispersion relation, from spec- Access to the regime where k tral density plots such as that displayed in Fig. 2. We show 104 cm 1 would require de- tection of scattered light reflected off the sample very close this information in Fig. 3, for the model film used to generate Fig. 2. The dotted line is the frequency variation of the line- width; the peak occurs when k 0 0.85. The effective dis- persion relation is given by the solid line, and this differs qualitatively from that appropriate to the case where eddy current damping is absent Eq. 2.12 . For this film, we show the prediction of Eq. 2.12 as a dotted line. Qualita- tively, the solid line bears resemblance to the prediction of Eq. 2.12 , but with D replaced by 0. In Fig. 4 we show the variation with wavelength of the linewidth dotted line , the effective dispersion relation solid line and the dispersion relation given by Eq. 2.12 , for a film with thickness D 10 5 cm. All other parameters are identical to those used in Figs. 2 and 3. We thus have a case where D 0. The peak in the linewidth occurs very close to k 0 0.85, the same value where we have the peak in Fig. 2. We see in this figure the dramatic fall off in the eddy current damping, as k 0 drops below unity, and also as k 0 increases above 0.85. The dispersion relation now is quite close to that applicable in the absence of eddy current damping. When D 0, it is the film thickness and not the skin depth which controls the effective dispersion relation. As noted earlier, the calculations presented above use pa- rameters chosen to illustrate eddy current damping effects, but the microwave skin depth 0 has been chosen equal to 10 5 cm, an order of magnitude smaller than the value ap- propriate to a transition-metal film such as Fe. The results may be applied to various materials by the appropriate scal- FIG. 5. The spectral density function syy(k , ) defined in Eq. ing procedure, when one realizes the characteristic quantities 3.11b calculated for 0 105 cm 1, 0 10 4 cm, D 5 10 5 cm, enter only in the products k D and k 0 , as remarked above. H0 0.5 kG, and H0 4 MS 4 kG, with also 1/ 0.01 , where If we have actual light-scattering studies of Fe in mind, is the frequency. 53 EDDY CURRENTS AND SPIN EXCITATIONS IN CONDUCTING . . . 12 239 in angle to the specular beam. This would present an experi- for Fe. The additional damping introduced may obscure fine mental challenge, since surface roughness would most likely structure expected in superlattices with a finite number of lead to an elastic component in the same angular regime, and layers. Such finite superlattices have a rich spectrum of spin- the signals detected in such experiments are very weak. wave modes affected sensitively by an external magnetic field.8 The mode structure of finite Fe/Cr 211 superlattices has been studied by BLS recently,16 and compared with theory. The theoretical spectra are considerably richer in de- IV. RESULTS AND DISCUSSION tail than the experimental counterparts. The difference may We have presented the theory of eddy current damping be due, at least in part, to broadening with origin in eddy and frequency renormalization of spin waves in conducting current damping. ferromagnetic films. We focus our attention on the Damon- It is the case that Damon-Eshbach waves may be Eshbach wave, when it propagates perpendicular to the mag- launched in ferromagnetic films via a small scale structures netization. The methods used here are readily extended to laid over the film. Meander lines provide an example of such other propagation geometries. an excitation source. Numerous microwave and magneto- In Sec. II, we obtain an implicit dispersion relation for the optic devices excite modes of finite wave vector by this waves. If k means.17 If a structure used for excitation has a linear dimen- is their wave vector, we see that eddy currents influence the modes modestly in the two regimes k sion w in the direction perpendicular to the direction of 0 1, and k propagation, then the waves excited most efficiently will 0 1, where 0 is the microwave skin depth. We give analytic approximations which apply to these limits in Eqs. have k /w. If w is in the range of 1 m or so, a dimen- 2.13 , 2.14 , and 2.19 . sion quite appropriate to small scale devices, then the waves The approximate formulas, when extrapolated to the re- launched in an Fe film will have k 0 1. Eddy current damp- gime k ing may then have a strong effect on the pulse shape gener- 0 1, suggest eddy current effects can be large in this regime of wave vector. In Sec. III, we present a Green's ated, and its propagation length. The Green's functions in the function analysis which yields forms that may be applied to Appendix will allow the quantitative study of this issue. We analyze the response of the film to a diverse array of probes. plan to address this question in the near future. The fluctuation dissipation theorem allows us to use the same functions to simulate light-scattering spectra. Calculations we present indeed show eddy current damping to be strong when k ACKNOWLEDGMENTS 0 1, and we have also a dramatic renormalization of the dispersion relation, when the film thickness D 0. In This research has been supported by the U.S. Army Re- essence, the skin depth acts as a channel within which the search Office, through Contract No. CS001208. N.S.A. also wave is trapped, and the dispersion relation becomes quali- acknowledges the support of the Brazilian agency CPNq. tatively similar to that of a wave confined to a film of thick- ness 0, rather than a film of thickness D. We conclude with a discussion of the implication of these APPENDIX: THE EXPLICIT FORM OF THE GREEN'S results for various experimental probes of metallic ferromag- FUNCTIONS netic films. We have in mind the case of Fe, for which the microwave skin depth The Green's functions Gij(y,y ) introduced in Sec. III 0 10 4 cm. Most of the transition metals have skin depths very close to this value. are functions not only of y and y , but also k and . In the There are two types of experiments where eddy current interest of brevity, we have omitted reference to k and in effects may play a role: ferromagnetic resonance studies, and the main paper, and in this section as well. In what follows, Brillouin light scattering BLS . we provide the form of the Green's functions only for posi- In an idealized ferromagnetic resonance experiment, one tive values of k . The frequency may be either positive or has k negative.13 One may generate forms valid for k 0, by not- 0, if microwaves strike the film at normal incidence. We see from Eq. 2.11 that in this limit, the film responds at ing that the functions Gij(y,y ) are invariant under simulta- the ferromagnetic resonance neous reversal of the signs k and . H B , unaffected by eddy cur- rent effects. In fact, one expects a mode with nonzero k In the expressions that follow, we have H H0, M to be excited in such experiments. Suppose the sample is finite MS, and B (H0 4 MS), while Q is defined in Eq. in size, possibly in the form of a square or disc with linear 2.5 . We take the root with Re Q 0. Note also the defini- dimension W. Then one may expect the edges to act as pin- tion of 1 and 2 Eqs. 2.2 , and also that of V Eq. ning centers, so one will excite a mode with k 2.7 . Finally, i/ , and we suppose the film lies / W . For typical samples, we expect k between y 0 and y D. 0 to be very small compared to unity under these circumstances, and eddy current effects Each of the Green's functions may be written in the form should be quite negligible. In the BLS studies, as noted above, k 105 cm 1, so for Fe, k Gij y.y G y y Gij y,y , A1 0 10 or so. Eddy current effects are again small; we i j estimated in Sec. II that the linewidth of the modes may contain a contribution in the range of a few tens of Gauss. where G( ) i j (y y ) describes the response of the infinitely This is significant, and may affect light-scattering spectra. extended ferromagnetic medium, and Gij(y y ) correc- For example, we may expect eddy current effects in metallic tions which arise from the presence of the two film surfaces. magnetic superlattices to be comparable to those estimated We have 12 240 N. S. ALMEIDA AND D. L. MILLS 53 i i G 11 y y 2 k G y y 2 k M sgn y y 22 M sgn y y H B 2 H B 2 2 2 i B B Q e Q y y , A2a i e Q y y . A2d Q H B 2 H B 2 1 2 A quantity which enters the expressions for the functions G M 2 12 k B y y Q G H B 2 i j(y ,y ) is 2 2 i B e Q y y , A2b d ,k 2 2 2 e QD Q k H B 2 Q k V 2 k V 2 1 1 2 2 G k e QD. A3 21 Q H y y 2 M 1 H B 2 2 2 i B One sees from the discussion in Sec. II, and Eq. 2.10 Q e Q y y , A2c that this quantity has a pole, for fixed k H B 2 , at the complex frequency of the Damon-Eshbach wave of the film. and Then we have the unfortunately lengthy formulas i k2 G 2k Q 1 V 1 11 y ,y e QD 2 2d 1Q k 1 V 2 B Q H k H B 2 1 ,k 2 k2 2 i i Q eQ y y 1Q k 1 V 2 B Q H k i Q eQ y y H B 2 k2 2 2k Q 1 V 1 e QD e Q y y 2 2 1Q k 1 V 2 B 1d ,k Q H k i Q k2 2 eQD 1Q k 1 V 2 B Q H k i Q e Q y y , A4a 1 k2 G 2k Q 1 V 1 12 y ,y e QD k 2 2d 1 V 2 1Q H Q B k H B 2 1 ,k 2 k2 2 i B B Q eQ y y k 1 V 2 1Q H Q B k i Q eQ y y 1 k2 2k Q 1 V 1 e Qd k 2 2d 1 V 2 1Q H Q B k H B 2) 1 ,k 2 k2 2 i B B Q e Q y y eQd k 1 V 2 1Q H Q B k i Q e Q y y , A4b 1 k2 G 2Q k 1 V 1 21 y ,y e QD 2 2d 1Q k 1 V 2 B Q H k H B 2 1 ,k 2 k2 2 1 i Q eQ y y 1Q k 1 V 2 B Q H k i Q eQ y y H B 2 k2 2 2Q k 1 V 1 e QD e Q y y 2 2 1Q k 1 V 2 B 1d ,k Q H k i Q k2 2 eQD 1Q k 1 V 2 B Q H k i Q e Q y y , A4c and finally 53 EDDY CURRENTS AND SPIN EXCITATIONS IN CONDUCTING . . . 12 241 i k2 G 2Q k 1 V 1 22 y ,y e QD 2 2d 1Q k 1 V 2 H Q B k H B 2 1 ,k 2 k2 2 i B B Q eQ y y 1Q k 1 V 2 ( H Q B)k i Q eQ y y i k2 2Q k 1 V 1 e QD 2 1Q k 1 V 2 H H B 2 2 1d ,k Q B k 2 k2 2 i B B Q e Q y y eQD 1Q k 1 V 2 H Q B k i Q e Q y y . A4d *Permanent address: Departamento de Fisica Teorica e Experimen- 10 C. Kittel, Phys. Rev. 71, 270 1947 ; 73, 155 1948 . tal, Univ. Federal do Rio Grande do Norte, 59072-970, Natal- 11 B. Heinrich and V. F. Mescharyakov, Sov. Phys. JETP 32, 232 RN-Brazil. 1971 . 1 D. L. Mills, Surface Excitations, Modern Problems in Condensed 12 L. D. Landau and E. M. Lifshitz, Theory of Elasticity Pergamon, Matter Science Vol. 9, edited by V. M. Agranovich and R. Lou- Oxford, 1959 , p. 105. don Elsevier, Amsterdam, 1984 , p. 379. 13 In numerical calculations, it must be kept in mind that if one 2 See the chapter by B. Heinrich, in Ultrathin Magnetic Structures chooses 0, one must also let the relaxation time be nega- II, edited by B. Heinrich and J. A. C. Bland Springer-Verlag, tive. This phenomonogical relation time is frequency dependent Heidelberg, 1994 , p. 195. in general, and the Kramers-Kronig relations require it to be an 3 L. R. Walker, in Magnetism, edited by G. Rado and H. Suhl odd function of frequency. Academic, New York, 1963 , Vol. 1, p. 299. 14 For a discussion of difficulties that have resulted from such ap- 4 G. A. Prinz, Surface Excitations Ref. 1 , p. 1. proaches, and their resolution within the Green's-function 5 R. E. Camley, T. S. Rahman, and D. L. Mills, Phys. Rev. B 23, method, see H. Benson and D. L. Mills, Phys. Rev. B 1, 4835 1226 1981 . 1970 . 6 R. E. Camley, T. S. Rahman, and D. L. Mills, Phys. Rev. B 27, 15 R. E. Camley and D. L. Mills, Phys. Rev. B 18, 4821 1978 . 261 1983 . 16 R. W. Wang, D. L. Mills, Eric E. Fullerton, S. Kumar, and M. 7 R. P. Erickson and D. L. Mills, Phys. Rev. B 43, 10 715 1991 . Grimsditch unpublished . 8 R. W. Wang and D. L. Mills, Phys. Rev. B 50, 3931 1994 . 17 See, for example, the device described by Chen S. Tsai and David 9 D. L. Mills and E. Burstein, Rep. Prog. Phys. 37, 817 1974 . Young, IEEE Trans. Microwave Theory Tech. 38, 560 1990 .