Exchange coupled domain walls: Resonance in multilayers R. L. Stamps and P. E. Wigen Department of Physics, Ohio State University, Columbus, Ohio 43210 A. S. Carric¸o Departamento de Fisica, Centro de Ciencias Exatas, Universidade Federal do Rio Grande do Norte, 59072-970 Natal/RN, Brazil Exchange coupling between magnetic films can lead to an attractive force between domain walls in the separate films. The coupling between the films allows for small amplitude oscillations around the equilibrium configuration of the walls, analogous to optic and acoustic type spin wave resonance modes. Since the restoring force acts only over the length of a domain wall, this suggests the possibility of sensitive measurements of the local coupling by studying domain wall resonances. The effects of applied fields are also examined. With antiferromagnetic coupling, small static fields push the walls apart and result in different behaviors of the acoustic and optic domain wall resonances. © 1997 American Institute of Physics. S0021-8979 97 50508-1 Studies of domain configurations and domain walls in structure and result in torques on the spins in the walls of magnetic multilayers have provided valuable insights into each film. Harmonic oscillations are then possible with a the effects and character of interactions within and between natural frequency that depends on the magnitude and sign of magnetic layers.1 An interesting feature of multilayers are the interfilm coupling. correlations between domain walls in separated films.2 Oscil- Domain wall resonance differs from spin wave reso- lations of domain wall pairs in single thin films have been nance in that the precession of spins in the domain walls predicted for certain wall configurations.3,4 In this article we results in a translation of the domain wall along the film. consider a new kind of restoring force responsible for wall Wall resonance frequencies are therefore determined by in- oscillations-interlayer exchange coupling between mag- ertia involved in the translation and the nature of restoring netic films. ``forces'' incurred by the relative motion of the walls. We To date, domain walls in multilayers have only been describe below a theory for the frequencies of oscillations studied in terms of their static properties. Exchange coupling when the restoring force is due to interfilm coupling. between domain walls on adjacent films can, however, lead We consider two exchange coupled ferromagnetic films. to interesting dynamic effects. Consider two antiparallel The films have identical magnetic properties and are as- coupled thin ferromagnetic films. The coupling energy will sumed to be very thin so that Nee´l type walls are preferred. be smallest if walls in each film are positioned directly be- The wall profile is assumed to depend on position in only neath one another as shown in Fig. 1 a for two Nee´l walls. one direction. At equilibrium the spins lie in the film planes. In this figure the arrows represent the local orientation of the Angles and are used to specify the orientation of the magnetization in each film. Small deviations from this con- magnetization as a function of position x in each film. The figuration, depicted in Fig. 1 b , increase the energy of the magnetization in the separate films is labeled m1 and m2. The energy for the uncoupled wall system is assumed to have a usual form for one-dimensional walls5 and includes a uniaxial anisotropy, intrafilm exchange, applied static field hs , and a demagnetizing factor for the thin film geometry. Our problem differs by the inclusion of an interfilm coupling J, Eex J sin 1 cos 1 sin 2 cos 2 sin 1 sin 1 sin 2 sin 2 cos 1 cos 2 dx. 1 The subscripts 1 and 2 identify the film. Interfilm and intrafilm magnetostatic are assumed small in comparison to other energies in the problem and are ne- glected for simplicity. Approximate solutions for the profiles specified by 1, 2, 1, and 2 are found by using a varia- tional method involving trial solutions for the coupled equa- FIG. 1. Schematic illustration of relative motion and orientation of the mag- tions. The trial solutions describe independent walls in un- netizations for Nee´l walls in two antiparallel coupled films. In a the equi- coupled films. For antiparallel coupling these are given by librium configuration is shown and in b the walls are displaced a small amount with a corresponding increase in energy due to the coupling. 1 cos 1 tanh x / and 1 0, 2 5370 J. Appl. Phys. 81 (8), 15 April 1997 0021-8979/97/81(8)/5370/3/$10.00 © 1997 American Institute of Physics 2 cos 1 tanh x / and 2 . 3 The wall width is used as a variational parameter and the energy of the static coupled wall structure is found by substituting the solutions from Eqs. 2 and 3 into the wall energy and minimizing with respect to . Corresponding ex- pressions for parallel walls are used in the case of parallel coupling. Wall motion involves translations of 1 and 2. This per- turbs the wall profile by an amount x(d /dx) to first order where x is a small translation of the wall. Translation of the walls also involve fluctuations out of the film plane. These are given by 1 and 2 representing deviations of 1 and 2 from their equilibrium values. We expand the wall energy to second order in the translation variables x1 ,x2 and the out- of-plane fluctuations 1, and 2, and evaluate the resulting FIG. 2. Frequencies of the spin wave and domain wall resonances for the integrals using the trial wall profile solutions. antiparallel coupled configurations as functions of coupling strength He . A key feature is the form of the interfilm coupling term. The solid lines are acoustic a and optic o domain wall resonances and the dotted lines are the acoustic and optic spin wave resonances. After expanding the energy to second order in x1 and x2, the coupling has the form J(x1 x2)2, which leads to the inter- film exchange restoring force on the walls. We note that this The frequencies for domain wall resonance with antipar- is intrinsic to the interfilm coupling and does not involve any allel coupling are shown in Fig. 2 as functions of H defects or pinning centers. e . The results are given in unitless frequency and field variables The validity of this approach depends on the magnitude defined by / 4 M and H of x and the strength of J. When x 0, the trial solutions e/4 M . In all cases H are identically the uncoupled wall solutions regardless of the K/4 M 0.05 which allows for a nonzero acoustic mode. Spin wave resonance frequencies are also shown for strength of J. Interfilm exchange energy is only involved comparison.7,8 The difference in spin wave resonance and when relative motions of the walls lead to deviations from wall resonance frequencies is due mostly to the in-plane perfect antiparallel alignment of the spins in the separate uniaxial anisotropy K. Note that a large interfilm exchange films. When the energy of these deviations from antiparallel can cause the optic wall resonance mode to have a frequency alignment are large, then the walls can be strongly deformed greater than the acoustic spin wave branch. and the variational procedure will fail. This is discussed in The motion of the walls can be thought of as similar to relation to a numerical check on the validity of the approach two masses connected by a spring. Acoustic- and optic-type in Ref. 6. In the following we restrict our calculations to oscillations are possible with an effective mass for each wall cases where the x are much smaller than a domain wall determined by interfilm coupling and magnetostatic energies. width. The oscillation frequencies follow directly as a ratio of the Equations of motion can be constructed using the meth- restoring force to the effective mass. Because magnetic an- ods of Ref. 5. These have the form (2M/ )dxi/dt d /d i isotropy determines the wall profile, and walls are able to and (2M/ )d i/dt d /dxi , where is the gyromag- move without changing shape, a result is that the frequencies netic ratio. Resonance frequencies can then be determined for wall resonance do not contain the anisotropy gap of spin by assuming time varying solutions of the form exp i t wave resonance and are usually much smaller than spin wave for x1, x2, 1, and 2. frequencies. We note that the effective masses are different One nonzero mode exists for antiparallel coupling with for the two modes. In the present case, the acoustic mode the frequency effective mass contains contributions from interlayer mag- netic coupling whereas the optic mode effective mass de- o / 2 4 MHe , 4 pends only on M. where H We now consider the case of a small static applied field. e 2J/ M . The subscript o denotes optic meaning that the separate walls oscillate out of phase with one an- The presence of a static applied field creates pressure on the other. An in-phase acoustic mode can exist only if there are walls and causes them to move apart in the case of antipar- other restoring forces present. For example, the inclusion of allel interfilm coupling. The equilibrium positions for the a phenomenological effective restoring field H walls are found by minimizing the wall energy with respect K a simplified representation of pinning by a defect, for example gives an to the position variables x1, x2, 1, and 2 with a nonzero acoustic mode frequency for the antiparallel case of applied field. Calculation of the resonance frequencies then procede along the lines describe above. Wall resonance frequencies in the antiparallel configura- a / 2 He 4 M HK . 5 tion are shown in Fig. 3 as a function of field hs . The field is The optic mode is also modified by the presence of HK .6 small in order that the walls are not pushed too far apart and When the interfilm coupling is zero, both frequencies reduce so overlap. The dotted lines are the acoustic modes a and to the single uncoupled wall resonance frequency. the solid lines are optic modes o . Coupling parameters J. Appl. Phys., Vol. 81, No. 8, 15 April 1997 Stamps, Wigen, and Carric¸o 5371 mode. The field dependence of the optic mode, however, appears because of higher-order deformations of the domain wall width.6 In conclusion, we note that since the domain wall reso- nance is due to restoring forces localized to the region of the domain wall, the frequencies of the resonances are deter- mined by coupling across areas with dimensions determined by the domain wall widths. These lengths are on the order of 100 Å in high-anisotropy ferromagnetic metals. Observation of domain wall resonances would therefore allow for inves- tigations of coupling mechanisms on a much smaller length scale than possible with ferromagnetic resonance or Brillouin light scattering. These measurement techniques provide val- ues for the interfilm coupling averaged over lengths deter- mined by the wavelength of the probing microwave or opti- cal field, which puts the length scales at 1000 Å lengths and FIG. 3. Frequencies for antiparallel coupling as a function of applied field. more. The static applied field hs pushes the walls apart and the frequencies repre- Work by R.L.S. and A.S.C. was supported in part by the sent small oscillations about the equilibrium position. The acoustic modes dotted lines increase linearly with field in contrast to the optic modes. CNPq. R.L.S. also thanks the UFRN for support. He/4 M between 0 and 0.1 are used. In this example HK 0 so the only restoring force is due to He . 1 The striking feature is the existence of the acoustic mode M. Ru¨hrig, R. Scha¨fer, A. Hubert, R. Mosler, J. A. Wolf, S. Demokritov, and P. Gru¨nberg, Phys. Status Solidi A 124, 635 1991 . for a small applied field. In the absence of an external ap- 2 L. J. Heyderman, H. Niedoba, H. O. Gupta, and I. B. Puchalska, J. Magn. plied field, the acoustic mode has zero frequency since it Magn. Mater. 96, 125 1991 . takes no energy for the walls to translate equal amounts in 3 J. C. Slonczewski, J. Appl. Phys. 55, 2536 1984 . 4 the same direction. In the antiparallel configuration with a H. Braun and O. Brodbeck, Phys. Rev. Lett. 70, 3335 1993 . 5 A. P. Malozemoff and J. C. Slonczewski, in Applied Solid State Science, nonzero applied field, acoustic mode motion always costs edited by R. Wolfe Academic, London, 1979 . Zeeman energy for translation of one of the walls. Transla- 6 R. L. Stamps, P. E. Wigen, and A. S. Carrico, Phys. Rev. B to be pub- tion of the other wall in the acoustic mode gains Zeeman lished . 7 energy. This means that the two walls experience unequal B. Heinrich, Z. Celinski, J. F. Cochran, A. S. Arrott, K. Myrtle, and S. T. Purcell, Phys. Rev. B 47, 5077 1993 . forces with the result of a nonzero frequency for the acoustic 8 R. L. Stamps, Phys. Rev. B 49, 339 1994 . 5372 J. Appl. Phys., Vol. 81, No. 8, 15 April 1997 Stamps, Wigen, and Carric¸o