PHYSICAL REVIEW B VOLUME 55, NUMBER 10 1 MARCH 1997-II Domain-wall resonance in exchange-coupled magnetic films R. L. Stamps* and A. S. Carric¸o Departamento de Fisica, Centro de Ciencias Exatas, Universidade Federal do Rio Grande do Norte, 59072-970 Natal/RN, Brazil P. E. Wigen Department of Physics, Ohio State University, Columbus, Ohio 43210 Received 27 November 1995; revised manuscript received 9 July 1996 Exchange coupling between magnetic films in multilayer geometries can strongly influence magnetization behavior and spin-wave energies by correlating the motion of spins in one film with the motion of spins in adjacent films. In a similar fashion correlations can be expected between domain walls existing in neighboring exchange coupled films. In this paper we show that the motion of domain walls in neighboring films can depend strongly on interfilm exchange coupling. A static equilibrium configuration exists for the domain walls, and small amplitude oscillations about equilibrium can result in domain-wall resonances that involve interfilm exchange energies. Frequencies for optic- and acoustic-type domain-wall resonances are calculated and effects of a small static applied magnetic field are examined, revealing distinctly different behaviors for the acoustic and optic domain-wall resonances. The possibility for sensitive measurements of the local coupling by study- ing domain-wall resonance is discussed. Resonances with parallel and antiparallel coupling are considered and the response to a small in-plane driving field is calculated. S0163-1829 97 00206-3 I. INTRODUCTION resonance involving interlayer coupling as a restoring force thus offers interesting possibilities for new studies of inter- Numerous investigations of domain-wall motion and reso- layer exchange coupling. nance have been made in various systems in order to study In this paper we investigate the effect of coupling on dy- phenomena such as magnetization reversal, domain stability, namic correlations between domain walls in separate films and material homogeneity. In a similar vein, studies of do- and examine possibilities of using domain-wall resonance to main configurations and domain walls in magnetic multilay- study interlayer coupling on length scales the order of the ers have provided valuable insights into the effects and char- width of a domain wall. In order to place the results for wall acter of interactions within and between magnetic layers.1 resonance in perspective, it is useful to review known results An interesting feature of multilayers are correlations between for spin-wave excitations in uniformly magnetized coupled domain walls in separated films.2 To date, domain walls in multilayers have only been stud- ied in terms of their static properties. Exchange coupling between domain walls on adjacent films can however lead to interesting dynamic effects. Consider two antiparallel coupled thin ferromagnetic films. The coupling energy will be smallest if walls in each film are positioned directly be- neath one another as shown in Fig. 1 a for two Ne´el walls. In this figure, the arrows represent the local orientation of the magnetization in each film. Small deviations from this con- figuration, depicted in Fig. 1 b , increase the energy of the structure and result in torques on the spins in the walls of each film. We will show in this paper how harmonic oscil- lations are possible with a natural frequency that depends on the magnitude and sign of the interfilm coupling. Breathing oscillations of domain-wall pairs in single thin films have been predicted for certain wall configurations.3,4 In this paper, we consider a very different kind of restoring force responsible for wall oscillations-interlayer exchange coupling between magnetic films. In this regard it is useful to FIG. 1. Schematic illustration of relative motion and orientation note that a continuing experimental problem is the quantita- of the magnetizations for Ne´el walls in two antiparallel coupled tive measurement of interlayer magnetic coupling in layered films. In a the equilibrium configuration is shown and in b walls magnetic film structures.5­7 The sign and average strength of are displaced a small amount with a corresponding increase in en- the coupling are usually found through magnetization mea- ergy due to the coupling. The geometry is also defined with the y surements, ferromagnetic resonance, light scattering, and axis normal to the film planes, the x axis normal to the domain magnetoresistance measurements.8­10 Coupled domain-wall walls, and a static applied field in the z direction. 0163-1829/97/55 10 /6473 12 /$10.00 55 6473 © 1997 The American Physical Society 6474 R. L. STAMPS, A. S. CARRIC¸O, AND P. E. WIGEN 55 layers. For a sufficiently thin ferromagnetic film, only the II. DOMAIN-WALL RESONANCE FOR COUPLED FILMS lowest energy spin-wave branch is easily observed in Bril- The geometry is shown in Fig. 1 with the y direction louin light scattering or ferromagnetic resonance measure- normal to the film planes. The films have identical magnetic ments. The energy of this branch at zero wave vector van- properties and are assumed to be very thin so that Ne´el-type ishes in the limit of zero applied field unless anisotropies are walls are preferred and the wall profile is assumed to depend present. A uniaxial in-plane anisotropy produces a gap typi- only on x. At equilibrium the spins lie in the xz plane and far cally on the order of a few hundred Oe for ferromagnets such from the walls the magnetization of each film is collinear as Fe or Co. Two such films coupled by an effective mag- with the z axis. Angles and are used to specify the netic coupling will support combinations of spin-wave orientation of the magnetization as a function of x in each modes where the spins in each film precess either together in film. For the magnetization m1 in film 1, the components are phase or out of phase. These modes are usually referred to as acoustic and optic. The difference in frequency between the m1x x M sin 1 x ...cos 1 x ..., 1 optic and acoustic modes is determined by the effective ex- change field coupling the films, in addition to any differences m between effective internal fields acting in each film. 1y x M sin 1 x ...sin 1 x ..., 2 Similar behavior can be expected for domain-wall reso- nance in coupled multilayers. Domain-wall resonance differs m1z x M cos 1 x .... 3 from spin-wave resonance, however, in that the precession of spins in the domain walls results in a translation of the do- M is the magnitude of the magnetization vector and assumed main wall along the film. Wall resonance frequencies are to be the same in both films. A magnetization m2(x) for the therefore determined by inertia involved in the translation second film is defined similarly using 2(x) and 2(x). The parameters entering into the wall energy are as fol- and the nature of restoring ``forces'' incurred by the relative lows. A uniaxial anisotropy K with an easy axis along the z motion of the walls. In multilayers, interfilm coupling can act direction is assumed for each film. The magnitude of the as a restoring force on each domain wall. The motion of the intrafilm exchange energy is specified with a constant A and walls can then be thought of as similar to two masses con- the interfilm coupling has strength J here J is averaged over nected by a spring. Acoustic- and optic-type oscillations are the film thickness and is in units energy per volume . A possible with an effective mass for each wall determined by small magnetic field h is also applied in the z direction. The interfilm coupling and magnetostatic energies. The oscilla- energy per wall area of the two-film system in the continuum tion frequencies follow directly as a ratio of the restoring limit is given by force to the effective mass. Because magnetic anisotropy de- termines the wall profile, and walls are able to move without changing shape, a result is that the frequencies for wall reso- E A d 1 2 2 dx d 2 dx K sin2 1 sin2 2 dx nance do not contain the anisotropy gap of spin-wave reso- nance and are usually much smaller than spin-wave frequen- cies. 2 M2 sin2 1sin2 1 sin2 2sin2 2 dx Since the domain-wall resonance is due to restoring forces localized to the region of the domain wall, the frequencies of the resonances are determined by coupling across areas with pJ sin 1 cos 1sin 2cos 2 sin dx 1sin 1sin 2sin 2 cos 1cos 2 dimensions determined by the domain-wall widths. These lengths are on the order of 100 Å in high anisotropy ferro- Mh cos magnetic metals. Observation of domain-wall resonances 1 cos 2 dx. 4 would therefore allow for investigations of coupling mecha- nisms on a much smaller length scale than possible with Our notation is chosen so that J 0 always. We use p to ferromagnetic resonance or Brillouin light scattering. These denote the type of coupling by defining p 1 to mean par- measurement techniques provide values for the interfilm cou- allel coupling and p 1 to mean antiparallel coupling. pling averaged over lengths determined by the wavelength of The first set of terms contain the intrafilm exchange and the probing microwave or optical field, which puts the length uniaxial anisotropy that determine the shape and energy of scales at 1000 Å lengths and more. the uncoupled domain walls in film. We note that magneto- A picture of domain-wall resonance is easily expressed by static energies due to the divergence of the magnetization in formulating the problem of domain-wall resonance in the the film plane can be approximated by including a position- coupled film structure in terms of exchange, dipolar, and dependent demagnetizing energy that varies as sin2 . This magnetic anisotropy contributions to the domain-wall en- has the same functional form as the uniaxial anisotropy and ergy. We do this in the next section where coupled equations is included in the definition of K. The second set of terms in of motion for each wall are derived and solved for the al- Eq. 4 are demagnetizing energies for out-of-plane fluctua- lowed frequencies. In Sec. III, we examine the effects of a tions. The third and fourth integrals are the interfilm cou- small static applied magnetic field on the walls and wall pling energy and the Zeeman energy in an applied field h. resonance frequencies. Finally in Sec. IV, we describe an The magnetostatic energies are assumed small in com- alternate formulation of the problem in order to provide a parison to an uncoupled domain-wall energy per area o : unified description of spin-wave and wall resonance excita- tions and derive response functions to an ac driving field. o 4 AK 1/2. 5 55 DOMAIN-WALL RESONANCE IN EXCHANGE-COUPLED . . . 6475 An approximate solution for the profiles specified by 1, resulting integrals over x are then evaluated using the trial 2, 1, and 2 are found by using a variational method in- wall profile solutions and treated as functions of the varia- volving trial solutions for the coupled equations. The trial tional parameter . solutions describe independent walls in uncoupled films. For We consider first the case of zero applied field h. The parallel coupling these are given by integrals are straightforward and so only the results are pre- sented. The energy per area of the wall pair, quadratic in cos 1 tanh x / and 1 0, 6 the fluctuation variables x and , can be put in the form cos 2 tanh x / and 2 0, 7 4 A/ K 4 M2 1 2 2 2 and for antiparallel coupling, by J 2 1 2 p 1 2 2 2 J x1 x2 2/ . cos 1 tanh x / and 1 0, 8 15 cos 2 tanh x / and 2 . 9 This energy is measured with respect to the total interfilm The wall width is used as the variational parameter. The coupling energy for uniformly magnetized films, Jtot J dx. energy of the static coupled wall structure is then found by The last term of Eq. 15 shows how wall separations substituting the appropriate solutions from Eqs. 6 ­ 9 , as involve interlayer exchange coupling. This means that an determined by the sign of p, into Eq. 4 and minimizing interlayer exchange restoring force can exist so that domain- with respect to the wall width . This procedure assumes wall resonance is possible without considering any other ad- that the wall profile in the coupled film system is described ditional restoring forces. This is in contrast to domain-wall by functions of the form used in Eqs. 6 ­ 9 . This is a good resonance in single films which requires some sort of addi- approximation as long as the interfilm coupling and magne- tional restoring force due, for example, to pinning by defects. tostatic interactions introduce only small corrections to the Note however that an interlayer exchange coupling restoring profile of the walls in the individual films.11 In the special force only makes sense when the walls overlap such that case of no applied field, for example, the interfilm exchange x1 x2 . For the remainder of the paper we consider coupling does not deform the wall profiles at all in the static only cases where the amplitude of the wall oscillation is configuration. In this case, the two walls are simply centered small so that this condition is fulfilled. over one another. The profiles are then given by Eqs. 6 ­ 9 In order to calculate the allowed resonance frequencies, it with is useful to include terms in the energy that describe possible o where additional restoring forces. Pinning effects due, for example, to inhomogeneities in the films can result in effective restor- o A/K 1/2. 10 ing forces. We treat these effects in an approximate Wall motion involves translations of 1 and 2 along the x manner11,12 by including terms in the wall energy propor- direction. Deviations of the walls from their equilibrium po- tional to x2. This approximation is only useful for displace- sitions perturb the wall profiles and change the wall width so ments from a defect smaller than the wall width since the that it is no longer given by Eq. 10 . If we know 1 and 2 quadratic form is not an accurate representation of the pin- at equilibrium, the effect of deviations x1 and x2 away from ning potential far from a point defect. This is acceptable equilibrium can be written to first order in x1 and x2 as because our entire discussion is valid only for small ampli- tude oscillations of the walls. 1 1o x1 1x , 11 Magnetostatic energies also enter the problem as a contri- bution to the effective coupling between the two walls.13,14 2 2o x2 2x . 12 The dipole interaction is long ranged and an accurate de- scription of dipolar effects between coupled film domain The profile at equilibrium is io and ix d i/dx is evaluated walls is quite involved. Qualitatively, antiparallel alignment in the equilibrium configuration of the wall pair. of the magnetization is preferred by stray magnetic fields Fluctuations out of the film plane are given by deviations between the films due the divergence of the magnetization of 1 and 2 from their equilibrium values. With the fluc- within each film. This acts as an additional attractive force tuations denoted by 1 and 2, then in the case of parallel when the walls overlap. Parallel alignment, in contrast, is coupling where the equilibrium values of 1 and 2 are repulsive when the walls overlap. In order to represent these zero we have effects, we restrict our calculations to structures where mag- netostatic energies are weak compared to the interlayer cou- 1 1 and 2 2 . 13 pling and consider only cases where the walls overlap. Be- For antiparallel coupling the equilibrium values of cause the magnetostatic interaction between the walls will 1 and 2 are zero and , so we use instead either add or subtract from the magnitude of the effective coupling correlating the wall motion, we represent the aver- age magnetostatic contribution by an additional small qua- 1 1 and 2 2 . 14 dratic dependence on wall separation (x1 x2)2. The approximations given by Eqs. 11 ­ 14 are substituted Based on the above considerations, approximate effects of into the energy of Eq. 4 and the cos and sin terms expanded magnetostatic interactions and pinning potentials are ex- to second order in the variables x1, x2, 1, and 2. The pressed by postulating an energy ED of the form 6476 R. L. STAMPS, A. S. CARRIC¸O, AND P. E. WIGEN 55 E The energy of Eq. 15 together with Eq. 16 is next D KD / x1 2 x2 2 p KD / x1 x2 2, 16 minimized with respect to the variational wall width. First, where K we set D controls the pinning potential assumed for sim- plicity to be the same in each film and KD represents the strength of the magnetostatic interaction between domain- d ED /d 0 17 wall pair. The sign p on KD describes the relative orienta- tion of induced poles on the walls in the two films. in order to determine . The result is 4A J pK 2 2 2 2 D x1 x2 2 KD x1 x2 / 4K 4 M2 1 2 J 1 p 2 2 1/2. 18 Equation 18 is then substituted into the wall energy per sented by KD reduces the optic mode frequency, as can be area ED and terms to second order in the fluctuations x expected since in the parallel coupling configuration magne- and are kept. The resulting expression for the energy is tostatic energies acting between the films are presumed to decrease the energy in Eq. 16 . 2 o 4 M2 o 1 2 2 2 J 0 1 p 2 2 The frequencies for parallel coupling are shown in Fig. 2 J pK as functions of He for different values of HK D x1 x2 2/ o KD / 0 x1 2 x2 2 . . The results are given in unitless frequency and field variables defined by 19 / 4 M and He/4 M. In all cases HK/4 M 0.01 which For the small displacements assumed here, the walls move allows for a nonzero acoustic mode. Note that the acoustic rigidly so that the wall widths are simply given by mode is independent of H 0 . Also e . note that in the absence of fluctuations, the energy is simply The main effect of magnetostatic interaction between the that of two unperturbed walls. walls is to open a gap between the frequencies of the acous- The equations of motion can now be formed using the tic and optic resonances for He 0. Also note that competi- methods of Ref. 11 from the wall energy per area, Eq. 4 , tion between HK and the parallel He coupling can be seen in and appropriate wall profiles, Eqs. 6 and 7 or 8 and 9 . Fig. 2. The magnetostatic repulsion lowers the frequency of We note that some care must be taken in determining the the optic mode but leaves the acoustic mode unchanged. If proper signs using the solutions in the different films. With He HK HK 0, magnetostatic energies overcome the at- given in Eq. 19 , the results are tractive interlayer coupling and the domain walls repel each other. Thus we observe that the frequencies are nonzero and 2M/ dxi /dt d /d i , 20 the structure is stable for HK /4 M less than 0.01. The be- havior of the domain-wall resonance in antiparallel coupled 2M/ d i /dt d /dxi . 21 films is similar but differs in that the gap introduced by HK is the gyromagnetic ratio and the subscript ``i'' identifies increases the frequency of the optic mode relative to the the film 1 or 2 . In order to solve these equations, time acoustic. varying solutions of the form exp i t are assumed for x It is interesting to interpret the optic- and acoustic-mode 1 , x2 , 1, and 2. Substitution into Eqs. 20 and 21 results in a set of four linear coupled equations. Four allowed reso- nance frequencies can then be determined. For parallel coupling, with p 1, the frequencies are a / 2 4 MHK 22 and o / 2 He 4 M He HK HK . 23 The effective fields appearing in Eqs. 22 and 23 are de- fined as follows: He 2J/M, 24 HK 2KD /M, 25 FIG. 2. Frequencies in zero applied field as a function of inter- H film coupling strength. Frequencies for parallel coupling are shown K 2KD /M. 26 for different values of HK , which approximately represents a mag- The subscripts a and o refer to acoustic and optic and de- netostatic restoring force. HK/4 M 0.01 in all cases. The fre- scribe the relative phase between translations of the two quency of the optic mode solid line is strongly dependent on the walls. Note that the interfilm magnetostatic coupling repre- interlayer coupling. 55 DOMAIN-WALL RESONANCE IN EXCHANGE-COUPLED . . . 6477 frequencies in terms of the harmonic oscillator analogy. The resonance frequency can be interpreted as a ratio between a Mh s cos 1 cos 2 dx Mhs x1 x2 xsin o restoring force and an effective mass. The effective mass in Eqs. 22 and 23 is the inverse of 4 M for the acoustic 1/2 x2 2 2 1 x2 x cos o 1/6 mode and the inverse of He 4 M for the optic mode. The 3 3 3 restoring forces are H x x sin K and HK HK He for the acoustic 1 2 x o dx. 29 and optic modes, respectively. The acoustic-mode frequency is identical to the resonance mode frequency of a single un- The x2 terms vanish when the integrals in Eq. 29 are coupled wall given by [4 MH evaluated. The term linear in x1 and x2 determines the equi- K]1/2, with an effective mass that depends only on M and a restoring force that depends librium position of the walls but does not enter the resonance only on restoring forces from pinning centers. The optic frequencies. Only the x3 terms enter into the frequencies by mode involves relative displacements of the walls with re- disturbing the wall profile and changing the wall width. spect to one another, so that the restoring force includes con- Evaluation of integrals in Eq. 29 give the result tributions to the interlayer magnetic coupling He in addition to effective forces due to K D and KD . Note that when the Mhs cos 1 cos 2 dx 2hs x1 x2 films are uncoupled (HK He 0) all frequencies reduce to the uncoupled single wall resonance frequency. x3 x3 / 3 2 . 30 Two different modes also exist for antiparallel coupling 1 2 with p 1. These have frequencies: Only the third-order term involves the wall width . This, as will be shown below, leads to a field dependence of the wall width and consequently, the resonance frequencies. a / 2 He 4 M HK 27 The equilibrium separation between the walls is again found by setting d /dx1 and d /dx2 to zero. When the in- and terfilm coupling is antiparallel, the equilibrium x values are x1 x2 d with d/ 3J/Mh o / 2 4 M He HK HK . 28 s 1 1 1/3 Mhs /J 2 1/2 . 31 Our discussion assumes that d is less than , and so we are The acoustic-mode vanishes for both the parallel and antipar- only interested in the behavior for small h. The root gives allel coupling if the only restoring force is interfilm coupling. the proper behavior of d for small hs whereas the root When the interfilm coupling is zero and the films are un- gives unphysical answers for vanishing hs . We also note that coupled, both frequencies reduce to the single uncoupled the equilibrium values for 1 and 2, determined by wall resonance frequency. We also note that the effective d /d 1,2 0, are still zero even to third order in the expan- masses are different for the two modes, as in the case of sion. parallel coupling. In the present case, the acoustic-mode ef- The separation is nearly linear in hs for a wide range of fective mass contains contributions from interlayer magnetic fields hs . For J much larger than Mhs , coupling whereas the optic-mode effective mass depends only on M. d/ Mhs/2J. 32 This relationship is exactly what one finds when solving for the equilibrium position of the walls keeping only terms to III. DOMAIN-WALL RESONANCE IN AN APPLIED second order in the energy. STATIC FIELD We next transform the position variables in according to x1 x1 d and x2 x2 d. Minimizing the result with We now consider domain-wall resonance for two antipar- respect to as before, we arrive at an expression for the wall allel interlayer exchange coupled films in the presence of a energy small static applied field, h hs . For simplicity we neglect other energies represented by KD or KD . The static applied 2 h h J 4 M2 1 2 2 2 2J 1 2 field creates pressure on the walls and causes them to move 2 apart. This is countered by antiparallel coupling that instead 1/ h J M2hs/3J x1 2 x2 2 2Jx1x2 , tries to bring the walls together. A static equilibrium exists 33 with the walls positioned at d1 and d2 when the correspond- ing torques balance. The equilibrium positions d where 1 and d2 are found by minimizing the wall energy with respect to the 2 4 position variables x h 2 o A/J Mhs 2 1/2, 34 1 , x2 , 1 , and 2 . The frequencies of the wall resonance modes are un- changed when this is done using the energy per area of Eq. h o / 1 Mhs 2/ 4JK 1/2. 35 4 expanded to second order in the x and variables. Field The field dependence of the wall width and energy are effects in the energies only appear for higher-order correc- clearly shown in Eqs. 34 and 35 and are due entirely to tions to and in the energy. To see this, consider the the third-order term appearing in Eq. 29 . Competition be- expansion of the last term in Eq. 4 using Eqs. 11 and 12 tween the applied field and the antiparallel interfilm coupling with antiparallel coupling: causes a distortion of the wall, leading to the field depen- 6478 R. L. STAMPS, A. S. CARRIC¸O, AND P. E. WIGEN 55 FIG. 4. Frequencies for antiparallel coupling as a function of applied field. The static applied field hs pushes the walls apart and the frequencies represent small oscillations about the equilibrium position. Note that the acoustic modes dotted lines increase lin- early with field unlike the optic modes at small field strengths. FIG. 3. Field dependence of a wall separation and b wall width. The solid lines are calculated using the analytical results of Zeeman energy for translation of one of the walls. Transla- Eqs. 31 and 35 and the dots are from the numerical calculation tion of the other wall in the acoustic mode gains Zeeman described in Appendix A. Field dependence of the wall width enters energy. This means that the two walls experience unequal through higher-order terms in the wall energy expansion. forces with the result of a nonzero frequency for the acoustic mode, thus giving rise to the nonzero frequency of Eq. 36 . dence of the wall width. As long as the walls overlap, the The optic-mode frequency is increased by the applied distortion of the walls increases as the separation increases. field, as can be seen by comparing Eqs. 28 and 37 . The It is interesting to examine the severity of the approxima- quadratic dependence on field is interesting because it means tions made in deriving these equations. In Appendix A, a that field effects on the optic mode are only visible for large description is given of a numerical solution for the static h configuration of two antiparallel coupled walls in an applied s . This is illustrated in Fig. 4 where wall resonance fre- quencies in the antiparallel configuration are shown as a field. The results are shown in Fig. 3 along with the analyti- function of field h cal approximation of Eqs. 31 and 35 . The solid lines are s . The dotted lines are the acoustic modes, the analytic results and the dots are the numerical results. In a , and the solid lines are optic modes, o . The frequencies are given in dimensionless units as before with the field vari- Fig. 3 a the separation d/ o is shown as a function of able h Mh s/4 M . Coupling parameters He/4 M between 0 and s/J. The wall width h/ o is shown as a function of 0.1 are used. Note the linear dependence on h Mh s for the s/J in Fig. 3 b . In both cases, J/K 0.5 and A/K 400. acoustic-mode frequency whereas for small fields the optic- The agreement is very good for small hs , and only deterio- mode frequency is insensitive to h rates for larger h s . Also note that the sen- s . sitivity to changes in h The equations of motion are next constructed from the s becomes less as the interlayer cou- pling H energy in Eq. 33 according to Eqs. 20 and 21 . Solution e is made stronger. of these give the following acoustic- and optic-mode fre- quencies: IV. RELATION TO SPIN WAVES AND DYNAMIC RESPONSE 2 a / 2 Mhs /3J He 4 M 36 The theory presented in the previous sections was ad- and equate for describing domain-wall profiles and motions in exchange-coupled films. In this section we construct a theory 2 based on a discrete spin Hamiltonian capable of describing o / 2 4 M He Mhs /3J . 37 spin waves as well as domain-wall dynamics. This will allow We again emphasize that these results are valid in the anti- a comparison of spin-wave frequencies and domain-wall parallel configuration and only for small wall separations resonances obtained using the same theory, and will also less than a wall width. The striking feature is the existence of provide a general framework for constructing dynamic sus- the acoustic mode for a small applied field. Reference to Eq. ceptibilities. Being a different approach, the theory of the 27 shows that in the absence of an external applied field, present section can also be used as a check on the previous the acoustic mode has zero frequency. This is because it results. takes no energy for the walls to translate equal amounts in We will see that a comparison of the frequencies for the same direction so the acoustic-mode motion does not coupled film spin waves and domain-wall resonances pro- incur a restoring force. In the antiparallel configuration with vides a surprising result. As noted in the Introduction, a nonzero applied field, acoustic-mode motion always costs anisotropies usually cause spin-wave frequencies to be larger 55 DOMAIN-WALL RESONANCE IN EXCHANGE-COUPLED . . . 6479 than domain-wall resonance frequencies. In this section we neighbors is represented by the stiffness constant D. The two show how interfilm exchange coupling can increase the optic films are allowed to have different thicknesses; ta for the top domain-wall resonance frequency into the spin-wave region. film and tb for the bottom film. The theory of this section will also be applied in the calcu- Film thickness effects the strength of the interfilm ex- lation of susceptibilities for small applied time varying mag- change coupling experienced by the spins and can also con- netic fields. trol the strength of anisotropies in thin films. Out-of-plane Our approach is an extension of that used by Winter in anisotropies do not affect the profile of the in-plane walls describing wall waves in single Bloch walls.12 Because the assumed here, but they do influence the energies of spin formulation will begin with discrete variables, slightly differ- wave and wall resonance excitations. The out-of-plane ent notations and geometry are used. To begin, the magnetic anisotropies enter the energy equations in the same manner moments are not described by a continuous variable as be- as dipolar demagnetizing energies. We represent the total fore, but instead are assigned positions on a three- out-of-plane anisotropy with an effective perpendicular an- dimensional periodic lattice with the index i. Spins in the top isotropy, Keff , defined for each film as film 1 have amplitude ai and spins in the bottom film 2 have amplitude b Ka 2 M2 K i . eff p /ta 38 The saturation magnetization in both films is M and the anisotropy and exchange constants in each film are also as- and sumed identical. The anisotropy K is uniaxial as before with Kb 2 M2 K an easy axis in the z direction. The geometry is the same as eff p /tb , 39 in Fig. 1 and the parameters and notation are as follows. where Kp is an out-of-plane uniaxial anisotropy with units Unlike the previous treatment, here a small time-dependent energy per area. Anisotropies of this form often appear in magnetic field h is applied along the z direction. The thin films as a surface or interface anisotropy. strength of the intrafilm exchange coupling between nearest The energy per wall area for the system in this notation is D K Ka Kb E eff eff M2 taai*aj tbbi*bj ta ai,x 2 1 ai,y 2 tb bi,x 2 1 bi,y 2 i,j M2 i K K J h taai,z tbb1,z p ai*bi . 40 i M2 i The sums over i, j indicate sums over nearest neighbors. axis defines the equilibrium orientation of the magnetic mo- The first two sets of terms contain the exchange, anisotropy, ment at site i. For example, ai is rotated into ai via and demagnetizing energies. Note that the uniaxial anisot- ropy K is written here in terms of the x and y components of ai, x ai,x cos i ai,z sin i , 41 the magnetization rather than the z component. This is more convenient for the present representation using variables in ai, y ai,y , 42 Cartesian coordinates. The third term represents the interac- tion energy with a driving field h. The last term represents a a the interfilm coupling which has the form of an effective i,z i,x sin i ai,z sin i . 43 Heisenberg interfilm exchange interaction. Note that the in- In our convention, primed variables are in the local system. terfilm coupling has not been averaged over the film as be- Before continuing, we also include a description of pin- fore and has units energy per area. The parameter ``p'' is ning effects as in previous sections. This is done in the used to specify parallel p 1 or antiparallel p 1 cou- present notation by including into the total energy an addi- pling as before. Note also that the films are assumed to be tional uniaxial anisotropy energy very thin such that the magnetization is uniform across the film thickness. This is a reasonable approximation for strong Eu KD ax 2 bx 2 /M2. 44 ferromagnets such as Fe and Co for film thicknesses less than 50 Å. This can be shown to result in a restoring force in the equa- The orientation of a magnetic moment at site i is specified tions of motion of the same form used in Eq. 16 .12 Magne- by an angle tostatic effects, discussed in Sec. II, are ignored in the i in the top film and the angle i in the bottom film. Both angles are referenced from the z axis as shown in present calculation. Fig. 5. The equilibrium positions of the magnetic moments The next step in the calculation is to expand the z com- are always assumed to be in the xz plane because of demag- ponents of the moments in terms of transverse fluctuations in netizing fields in a thin-film geometry. The energy per area the local x and y directions: Eq. 40 is then written with the magnetic moments at each position i rotated into a local coordinate system. The local z ai, z M ai, 2x ai, 2y / 2M , 45 6480 R. L. STAMPS, A. S. CARRIC¸O, AND P. E. WIGEN 55 bi, z M bi, 2x bi, 2y / 2M . 46 da dt a Ha, 55 The resulting energy describes second-order fluctuations in the spin variables that are discrete functions of position. db Effective fields at a given site are next determined using Ha b dt b Hb. 56 i a i ( E E u) and Hi bi (E Eu). We are prima- rily concerned with long wavelength excitations and materi- The equations of motion given in Eq. 55 and 56 to- als with wide domain walls so that a , b , , and vary little gether with the effective fields given in Eqs. 49 ­ 52 are between neighboring lattice sites. We therefore transform the capable of describing both spin-wave and domain-wall dy- effective fields from a discrete form to a continuum limit namics in the coupled system. We consider zero wave-vector with ai a (x) and bi b (x) in the usual manner. We note excitations for both spin-wave excitations and domain-wall that in order to convert the discrete variables i and i to translations. In each case we first calculate the frequencies continuous functions (x) and (x) we expand quantities for parallel and antiparallel coupling and then discuss the such as x , where is a nearest-neighbor distance, in a results in terms of response to the driving field h. The thick- Taylor's expansion under the assumption that there is a nesses ta and tb are set equal in the following discussion domain-wall structure in the films. In particular, we use the unless otherwise specified. relations Zero wave-vector spin waves d /dx sin / 0 , 47 In order to compare with results for spin waves in uni- d /dx sin / formly magnetized films, we set K 0 . 48 D to zero. A time depen- dence for a , b , and h of the form exp i t is assumed. In the following D is contained in the exchange parameter A Zero wave-vector spin-wave resonance solutions to the which includes lattice constants and coordination numbers. torque equations are The final effective fields Ha and Hb are a 2K 2K J x Axtanh x/ exp i t , 57 Ha D a x 2A M2 2 M2 cos2 sin2 M2 p M2 ax ay Aytanh x/ exp i t , 58 J a b M2 bx h sin , 49 x Bxtanh x/ exp i t , 59 b 2K 2Ka J y Bytanh x/ exp i t . 60 Ha eff a y 2A M2 2 M2 cos2 sin2 M2 p M2 ay Substitution of these solutions into the equations of motion Eqs. 55 and 56 give a set of four coupled equations. J These equations are shown in Appendix B, Eqs. B1 . a M2 by , 50 We set p 1 for parallel coupling. The allowed frequen- cies are found by setting the determinate of the correspond- 2K 2K J Hb D b x 2A M2 2 M2 cos2 sin2 M2 p M2 bx J b M2 ax h sin , 51 2K 2Kb J Ha eff b y 2A M2 2 M2 cos2 sin2 M2 p M2 by J b M2 ay , 52 The different film thicknesses scale the interfilm exchange appearing in each film and is represented by the quantities Ja J/ta , 53 Jb J/tb , 54 in the expressions for the effective fields. We note that these expressions also agree with those of Winter in the case of FIG. 5. In the discrete spin formulation, the coordinates are zero interfilm coupling.12 rotated locally in each film about the y axis in order to place the Equations of motion are found with the usual torque equa- primed x and z axes along the equilibrium directions of the local tions magnetizations. 55 DOMAIN-WALL RESONANCE IN EXCHANGE-COUPLED . . . 6481 ing homogeneous with h 0 set of equations to zero. De- fining an effective exchange field He as in Eq. 24 , and effective anisotropy fields Ha and Heff with Ha 2K/M, 61 Heff 2Keff /M, 62 the allowed frequencies are 2o 2 Ha He Ha He Heff 63 and 2a 2Ha Ha Heff . 64 The subscripts o and a refer to ``optic'' and ``acoustic'' re- spectively, as before. For antiparallel coupling, p 1. The frequencies are 2o 2 Ha He Ha Heff , 65 2a 2Ha Ha He Heff . 66 The results of Eqs. 63 ­ 66 agree with those of previous spin-wave and ferromagnetic resonance calculations on coupled films.8­10 The main points we now wish to emphasize are the dif- ference in the optic-mode frequencies between parallel and antiparallel coupled cases and the existence of an energy ``gap'' which increases the frequencies of the excitations. The difference in frequencies between parallel and antiparal- lel is due to demagnetizing fields and the gap is due to the uniaxial in-plane anisotropy. FIG. 6. Frequencies of the spin-wave and domain-wall reso- nances for the a antiparallel coupled and b parallel coupled con- Domain-wall resonance figurations as functions of coupling strength He . The solid lines are Solutions of the form acoustic a and optic o domain-wall resonances and the dotted lines are the acoustic and optic spin-wave resonances. The films are a identical with anisotropies K/2 M2 1.5 and K x Axsin exp i t , 67 p 0. A restoring force for the wall resonances is included with KD/2 M2 0.05. a Note that the optic resonance mode has a frequency greater than the y Aysin exp i t , 68 acoustic spin-wave mode for large He . bx Bxsin exp i t , 69 The difference in spin-wave resonance and wall reso- b nance frequencies for both the parallel and antiparallel cases y Bysin exp i t , 70 is due mostly to the in-plane uniaxial anisotropy K with a correspond to translation of the domain wall in the x direc- small contribution from KD . The acoustic modes are less tion. The corresponding equations of motion are shown in sensitive to the interfilm exchange than the optic modes for Appendix B in Eqs. B2 . antiparallel coupling and independent of the exchange for We allow KD to be nonzero and define a corresponding parallel coupling. Interestingly, a large interfilm exchange field HK as before. The frequencies found by setting the can cause the optic wall resonance mode to have a frequency determinate of homogeneous h 0 set of equations to zero greater than the acoustic spin-wave branch. This occurs for a agree exactly with the HK 0 form of Eqs. 22 and 23 , for smaller value of the exchange in the parallel case than in the p 1 and with Eqs. 27 and 28 for p 1. Comparison antiparallel case. of the spin-wave and domain-wall resonance frequencies As a final comment, we note that it is a simple matter to show that the wall resonance frequencies are lower because generalize the equations of motion in Eq. B2 to describe of the absence of anisotropy terms in the frequencies. domain-wall resonance in multilayers consisting of several Spin-wave resonance and wall resonance frequencies are coupled films. A band of wall resonances would appear, shown in Figs. 6 a and 6 b as functions of exchange cou- analogous to collective excitations in multilayers. pling. In both cases a small additional restoring force KD is included KD/2 M2 0.05 with frequencies for antiparallel coupling shown in a and frequencies for parallel coupling Dynamic response shown in b . The other parameters are: ta tb , As in ferromagnetic resonance on coupled films, standard K/2 M2 1.5. techniques for observing domain-wall resonances in multi- 6482 R. L. STAMPS, A. S. CARRIC¸O, AND P. E. WIGEN 55 layers require a net fluctuating magnetic moment for the coupled film structure. This is the situation for the optic wall resonance mode in antiparallel coupled films since out-of- phase translations lead to a change in the magnetization in the z direction averaged over both films. This is not the case however for parallel coupled films since optic wall resonance translations lead to zero average change in the total magne- tization. This difficulty can be overcome by allowing Keff to be different for each film. This can be accomplished using dif- ferent film thicknesses, i.e., ta tb , as done by Zhang et al.,7 in measurements of the acoustic and optic spin-wave reso- nances in multilayers with thickness dependent anisotropies. In order to gain a feeling for how the intensities of the wall FIG. 7. Imaginary part resonance modes depend on unequal anisotropies in the pres- x of the in-plane response x as a func- tion of driving field frequency for the parallel coupling configura- ence of small rf driving fields, we calculate the response of tion. The ratio r controls the magnitude of the internal fields acting the coupled wall structure to the time-dependent driving field in film b relative to those in film a and the response is shown for h. The results are presented as averaged susceptibilities zz four different r. The dotted line is the response for r 1. The pa- and yz that measure fluctuations in the z and y directions in rameters are J/2 M2 0.1, K/2 M2 1.5, Kp/2 M2 0.5, and the film geometry due to a driving field in the z direction. KD/2 M2 0. The response becomes larger with increasing r while The susceptibilities are defined as averages over a film of the peak response shifts lower in frequency as explained in the text. length L: Here again the resonance is only visible for the in-plane fluc- 1 tuations of the magnetization. When Ka Kb , then eff eff zz hL Az Bz dx, 71 2 MH eff zz L / 2 , 77 1 and there is no longer a pole at the wall resonance in either yz hL Ay By dx. 72 component. This can be understood by considering the am- plitude of precession for the magnetic moments in each film. When the effective fields acting within each film are the The susceptibilities are found by solving the equations of same, the precession amplitudes are the same. The walls then motion Eq. B2 for the amplitudes A and B. For simplicity move equal amounts in the same direction, which takes no we present results with KD 0. The general expressions are energy if K not particularly illuminating so it is useful instead to examine D 0. There is then also no fluctuation in the net magnetization. When the effective internal fields are differ- the susceptibilities for equal film thicknesses but with differ- ent for the two films, the two precession amplitudes are dif- ent Keff . For antiparallel coupling the susceptibilities are ferent and fluctuations in the net magnetization are possible. The behavior of M Ha Hb zz for parallel coupling is illustrated in eff eff Fig. 7 for ta tb . Here the susceptibility is found by solving zz L H a b e Heff Heff /2 / 2 , 73 the equations of motion for the sum of the in-plane fluctua- tions Az Bz . A small imaginary part was added to the fre- quency to give a width to the peaks in simulation of damp- ing. We note that this should not be understood as properly yz 0. 74 representing actual damping processes. The parameters for Fig. 7 are K In this case the z susceptibility still has a pole at the wall p/2 M 2 0.5, Ja/2 M2 0.1, and ta held constant. The imaginary part resonances when Ka b eff Keff . Note that the total response de- fined by a b is only in the z direction. zz of the response zz is shown for three values of r tb/ta indicated on the plot with the dotted line calculated for r 1. For parallel coupling we find As expected, the response to the optic wall resonance pole becomes larger as r is increased. However the frequency at which the peak occurs decreases with increasing r. This de- M Ha Hb crease is due to the reduction of the internal fields as t eff eff b zz L / 2 H2 a b becomes larger, which weakens the interfilm coupling and e He Heff Heff /2 / 2 anisotropies. Note also the large values of zz for small driv- 2H a Hb ing field frequencies due to the second-order pole at 0. H2 eHeff eff e Ha b / 2 , 75 eff Heff V. CONCLUSIONS 2iM We have examined spin-wave and domain-wall resonance yz / . 76 in two thin ferromagnetic films coupled either parallel or 55 DOMAIN-WALL RESONANCE IN EXCHANGE-COUPLED . . . 6483 antiparallel with an effective Heisenberg exchange interac- the corresponding Euler equations: tion. The coupling acts as a restoring force correlating the wall motion in the separate films and only one wall reso- nance appears in the absence of other restoring forces. This resonance corresponds to an optic-type oscillation involving 2 2A 1 opposing motion of the walls. An acoustic-type oscillation x2 K sin 2 1 Mhssin 1 J sin 1 2 0, appears when other restoring forces within the individual A1 films exist. For small interfilm coupling, the frequencies of the wall resonances lie below the frequencies for spin-wave reso- 2 nance in the domain walls by an amount dependent on the 2A 2 in-plane anisotropy of the films. A feeling for the frequencies x2 K sin 2 2 Mhssin 2 J sin 1 2 0. can be obtained by estimating the resonances for coupled A2 ferromagnetic films such as Fe. With 4 M 21 kG, K 106 erg/cm3, Kp 0, and Hex 0.1 kG and in the absence of other restoring forces, the spin-wave resonance fields are 5.2 and These can be written as two coupled finite difference equa- 5.4 kG for the acoustic and optic modes, respectively, in the tions where the value of at position x is determined by antiparallel configuration. The optic wall resonance mode is values at neighboring positions a distance x away at x x at 1.45 kG. Domain-wall resonances in coupled transition- and x x. These equations are metal magnetic structures should then occur in a range below 5 GHz for weak to moderate coupling. We also find that for larger interfilm coupling it is pos- sible for the optic wall resonance to have a frequency com- 1 2 sin 2 parable or even larger than the acoustic spin-wave resonance. 1 x 2 1 x x 1 x x x 2 1 x 0 This is interesting because it offers the possibility for large interactions between spin-wave and domain-wall excitations Mh J s in strongly coupled films. K sin 1 x K sin 1 x 2 x , A3 The effects of a small static applied magnetic field were examined for antiparallel coupling. The walls on the separate films separate until the pressure on the walls due to the field balances the pressure due to the antiparallel coupling. Both 1 2 the acoustic- and the optic-mode oscillations about the equi- 2 x 2 2 x x 2 x x x 2 sin 2 2 x librium position of the walls are then possible. The acoustic 0 mode increases linearly with field unlike the optic mode at Mhs J small field strengths. K sin 2 x K sin 1 x 2 x . A4 The possibility of observing domain-wall resonances in coupled film structures may allow the study of coupling mechanisms across regions with dimensions on the order of a Numerical solution of these two equations is done by di- domain-wall width. With experiments of this type in mind, viding a long interval in position x into N segments each of we have examined the response of the domain-wall reso- length x. Fixed values for 1 and 2 at the ends determine nances to a small driving field. The optic wall resonance the boundary conditions. In this problem, the boundary con- should be visible by looking at in-plane fluctuations of the ditions are 1 0, 2 in the first segment, and 1 , magnetization if the films are antiparallel coupled. In the 2 2 in the last segment. The values of 1 and 2 at each case of parallel coupling, the optic-mode resonance will only other segment in the interval are then adjusted according to be visible if the effective internal fields acting in one film Eqs. A3 and A4 iteratively. The values in each segment differ from those acting in the other film. Finally, we note relax toward a solution of Eqs. A1 and A2 relatively that the theory presented in this paper applies to any layered quickly. magnetic system capable of supporting hard domain walls as The solutions shown in Fig. 3 where obtained with 1000 long as the interlayer coupling is weak compared to intra- iterations with N 800 and x/ 0 1/20. The angles in this layer coupling. example converged within 10 4 radians and N was chosen so that finite-size effects were negligible. The wall separation ACKNOWLEDGMENTS and width were determined by calculating the position and slope of the wall profiles where We acknowledge stimulating discussions with A. Dantas 1 /2 and 2 3 /2. and T. Dumelow. Work by R.L.S. was supported in part by the CNPq. R.L.S. also thanks the UFRN for support. APPENDIX B: EQUATIONS OF MOTION APPENDIX A: NUMERICAL SOLUTION The torque equations Eqs. 55 and 56 , using the effec- FOR STATIC CONFIGURATION tive fields given in Eqs. 49 ­ 52 , are shown below. When solutions of the form presented in Eqs. 57 ­ 60 are used, Consider the case of antiparallel coupling p 1 . The equations of motion for spin-wave excitations on coupled energy per area given in Eq. 4 can be minimized by solving domain walls result: 6484 R.L.STAMPS,A.S.CARRIC¸O,ANDP.E.WIGEN 55 i / 2 0 J a a 2 M K Keff pJa J M a M K pJa i / M 0 AxAy 0Mh 0 J . B1 b i / 2 B 0 b x J p M 2 p M K Keff pJb B Mh p b y M 0 p M K pJb i / When solutions of the form given in Eqs. 67 ­ 70 are used to construct the torque equations, one obtains equations of motion for domain-wall oscillations: i / 2 0 J a a 2 M Keff pJa J M a M KD pJa i / M 0 AxAy 0Mh 0 J . B2 b i / 2 B 0 b x J p M 2 p M Keff pJb B Mh p b y M 0 p M KD pJb i / *Permanent address: Dept. of Physics, Ohio State University, Co- Lett. 73, 336 1994 . lumbus, OH 43210. 8 P. Kabos, C. E. Patton, M. O. Dima, D. B. Church, R. L. Stamps, 1 M. Ru¨hrig, R. Scha¨fer, A. Hubert, R. Mosler, J. A. Wolf, S. and R. E. Camley, J. Appl. Phys. 75, 3553 1994 . Demokritov, and P. Gru¨nberg, Phys. Status Solidi A 124, 635 9 B. Heinrich, Z. Celinski, J. F. Cochran, A. S. Arrott, K. Myrtle, 1991 . and S. T. Purcell, Phys. Rev. B 47, 5077 1993 . 2 L. J. Heyderman, H. Niedoba, H. O. Gupta, and I. B. Puchalska, 10 R. L. Stamps, Phys. Rev. B 49, 339 1994 . J. Magn. Magn. Mater. 96, 125 1991 . 11 A. P. Malozemoff and J. C. Slonczewski, in Applied Solid State 3 J. C. Slonczewski, J. Appl. Phys. 55, 2536 1984 . Science, edited R. Wolfe Academic, London, 1979 . 4 H. Braun and O. Brodbeck, Phys. Rev. Lett. 70, 3335 1993 . 12 J. M. Winter, Phys. Rev. 124, 452 1961 . 5 P. Gru¨nberg, R. Schreiber, Y. Pang, M. B. Brodsky, and H. Sow- 13 S. Middelhoek, J. Appl. Phys. 37, 1276 1966 . ers, Phys. Rev. Lett. 57, 2442 1986 . 14 6 J. C. Slonczewski and S. Middelhoek, Appl. Phys. Lett. 6, 139 S. S. P. Parkin, Phys. Rev. Lett. 67, 3598 1991 . 7 1965 . Z. Zhang, L. Zhou, P. E. Wigen, and K. Ounadjela, Phys. Rev.