PHYSICAL REVIEW B VOLUME 62, NUMBER 13 1 OCTOBER 2000-I Local modes of thin magnetic films A. L. Dantas,1 A. S. Carric¸o,2,* and R. L. Stamps3 1Departamento de Fi´sica, Universidade do Estado do Rio Grande do Norte, 59.610-210 - Mossoro´, RN, Brazil 2Departamento de Fi´sica Teo´rica e Experimental, Universidade Federal do Rio Grande do Norte, 59.072-970 - Natal, RN, Brazil 3Department of Physics, The University of Western Australia, Nedlands WA 6907, Australia Received 27 May 2000 We calculate the frequency of rigid displacement domain wall excitations of a Ne el wall in a thin uniaxial ferromagnetic film. The domain wall is pinned by a line defect running along the uniaxial axis. We study the effect of an external field applied along the magnetization of one of the domains. The restoring force originates from energy fluctuations resulting from spin motion within the domain wall width and the excitation frequency turns zero when the external field approaches the threshold value for depinning the domain wall from the defect. The results are applied to the study of excitations of a Ne el wall in a thin uniaxial ferromagnetic film exchange coupled to a uniaxial two-sublattice antiferromagnetic substrate. There is a wide recognition of the central role played by the magnetic structure in a local manner. We show that, con- domain walls in the leading features of phenomena of current trary to the long wave-length domain excitations, measured interest in a large class of magnetic artificial structures of by FMR, the frequency of RDWDM is a decreasing function nanometer size, made out of transition metal thin films. Do- of the external field and turns zero at the value of the external main walls participate in key processes, such as magnetiza- field which depins the domain wall from the local pinning tion reversal and affect the transport of charge. center. Most techniques currently used to characterize artificial We obtain the field dependence of the frequency of magnetic systems, such as magnetization measurements, are RDWDM for a general model of a Ne el wall. We keep the based on methods that sample large areas and thus average energy density of the wall in general form and obtain the out the microscopic details. These methods do not inform, frequency of excitations by examining energy fluctuations for instance, on the possible modifications in the domain around the equilibrium state. We allow the field to displace wall profile and the nature of domain wall pinning forces. the wall from the pinning center and calculate the restoring These features may have a relevant impact in key aspects of force constant and the Do¨ring mass in terms of the equilib- phenomena of current interest. We cite only a few examples: rium profile functions. 1 the reduction in remanence of thin films on compensated We consider a wall of a uniaxial ferromagnet, pinned antiferromagnetic AF interfaces,1,2 2 the effective inter- face exchange leading to short period oscillations in Fe/Cr by a line defect running along the z axis at y 0. The mag- wedges,3 3 spin selective domain wall scattering in chemi- netization is in the yz plane and its orientation with respect cally homogeneous materials,4 4 interface roughness in- to the uniaxial axis, in the plane, is given by the function duced giant magnetoresistance,5 5 domain wall resistivity (y). In the domain wall center /2 and the domains of submicrometer wires,6 6 macroscopic quantum tunnel- have 0 and , as shown in Fig. 1, for the particular ling in domain wall junctions,7 7 domain wall jumps and case of an interface step defect. the resonant frequency in magnetic force microscopy We start from an equilibrium profile 0(y) which mini- measurements,8 8 domain wall mobility and the mizes the magnetic energy Barkhausen effect,9 and 9 the compression of domain walls during the magnetization reversal in domain wall junctions.10 In this paper we show that the excitations of domain walls, pinned by local defects, are controlled by the magnetic structure in regions of microscopic dimensions. We study rigid domain wall displacement modes RDWDM and we show that, provided the pinning energy is of the same order of magnitude as the anisotropy energy of the ferromagnet F , these domain wall excitations can be accessed by reso- nance experiments in experimental setups designed for fer- romagnetic resonance FMR . This is the case, as we show later, of domain walls pinned by interface defects in F/AF bilayers.2,3 Although the domain walls might be of microscopic size and constitute a minor fraction of the whole sample, the mea- surement of the field effects on the frequency of the domain FIG. 1. Schematic representation of a Ne el wall pinned at a step wall excitations provides a promising means for accessing defect on an antiferromagnetic substrate. 0163-1829/2000/62 13 /8650 4 /$15.00 PRB 62 8650 ©2000 The American Physical Society PRB 62 BRIEF REPORTS 8651 L The factor ( 0)2 in the integrand of Eq. 6 restricts the E y eq dy f , y , 1 contribution to the excitation energy E to the region of the L domain wall. Notice also, from Eq. 3 , that the main contri- where L is the width of the domains at each side of the bution to the magnetostatic energy comes from the domain domain wall and f ( , wall region, since the function sin2 is zero in the domains. y) is the magnetic energy density, including intrinsic exchange and anisotropy energies of the The out-of-plane fluctuation is assumed to be small and we ferromagnet as well as Zeeman energy and the domain wall use the equilibrium function 0(y) in Eq. 3 . pinning energy. The leading terms for small amplitude rigid displacement 0(y) is the function that corresponds to the equilibrium oscillations (q/ 0 1 and 0) are given by Eqs. 3 and profile. Thus, it satisfies the Euler-Lagrange equation 6 . The total energy, E Eeq EM is of the form f f 1 1 E E0 y 0. 2 2 k q2 2 b 2, 7 y y 0 is the position of the domain wall center in the absence where E0 is the equilibrium value of the energy, as given by of external magnetic field. For a given value of the external Eq. 1 , using the profile 0(y). The Landau-Lifshitz's field strength H the equilibrium profile, represented by torque equations are integrated throughout the domain wall11 0(y), includes the field-induced displacement of the domain leading to wall center. dq E Rigid displacement domain wall excitations are character- ized by a rigid displacement of the angular profile of the dt 2M , 8a domain wall. We consider the variations induced in the en- ergy by small amplitude displacements around the equilib- d E rium pattern, using the function (y q), with q q dt 2M q , 8b 0ei t. We also introduce an extra term in the energy corresponding where is the gyromagnetic factor. to a small out of plane angle 0ei t. The out of plane From Eqs. 8a and 8b we obtain the frequency of do- oscillations induce surface charges and the demagnetizing main wall oscillations as energy is approximated by L k b. 9 E 2M M 2 M2 sin2 sin2 dy. 3 L The restoring force constant k is a decreasing function of the The total energy is the sum of Eqs. 1 and 3 . We cal- external field strength. When the external field approaches culate the variations in E the threshold value H*, which makes the domain wall free eq , when (y ) 0(y q) is used in Eq. 1 in the place of 0(y) and add to it the demagnetizing from the defect, the center of the domain wall is far from the energy, given by Eq. 3 . The variations in and defect line at y 0. Assuming the defect contribution to the y are given by q 0 0 magnetic energy to be of finite range, centered at y 0, when y(y ) and y q yy(y ). In order to calculate the leading term of the excitation energy we expand H H* the function 2f / 2 is practically zero, since in the the function f ( , defect range the magnetization is uniform. Thus, the fluctua- y) up to second order of the displacement tions in the domain wall position produce no extra energy variable q. Considering that the function f ( 0, 0y) is a solu- and k 0. tion of the Euler-Lagrange equations, we find that Notice that the results, so far, are valid for any kind of magnetic domain wall structure, provided that the equilib- E 0 eq , y Eeq 0, y E, 4 rium structure corresponds to having the magnetization in a plane. This covers Ne el walls as well as Bloch walls. Fur- where thermore the domain wall pinning mechanism, as well as the internal structure of the ferromagnet have not been specified. q2 L 2f 2f E 0 0 0 0 Thus the results apply equally well for a variety of 2 dy 2f y 2 yy 2 y yy . L 2 2 systems.2,3,13,14 y y 5 The nucleation and pinning of domain walls has been re- cently studied for an uncompensated F/AF interface.12 It has For a good number of magnetic systems of current inter- been shown that ferromagnetic narrow domain walls are est there is no cross derivative of the energy density nucleated at interface step defects. ( 2f / We calculate the excitation of a Ne el wall pinned at a step y 0). Furthermore, for a rigid displacement the intrinsic exchange energy does not change, thus we do not defect in a F/AF interface. The system consists of a thin have a term involving the ferromagnetic film, with in-plane magnetization, on a two- y derivative of the energy density. We then find sublattice uniaxial antiferromagnetic substrate as shown in Fig. 1. The anisotropy axis of the antiferromagnet is parallel q2 to the easy direction of the ferromagnet the z axis . The L 2f E 0 substrate step edge runs along the z axis and divides the 2 dy y 2. 6 L 2 interface in two regions, each one containing spins from a 8652 BRIEF REPORTS PRB 62 sublattice of the antiferromagnet. In our model no relaxation is allowed for the substrate spins, which are held fixed along the anisotropy direction. We do not consider any variation of the magnetization along the z- or x-axis directions. The nucleation of a Ne el wall in the ferromagnetic film follows from the discontinu- ous change of direction of the interface exchange field at the step edge. The magnetic energy density is given by f , y A y 2 HM J y cos K cos2 , 10 where J y J, y 0, 11 J, y 0. The first term in the Eq. 10 is the intrinsic exchange energy FIG. 2. Frequency of rigid displacement domain wall oscilla- density, the second term is the Zeeman energy density for an tions. The numbers by the curves indicate the values of HJ /HA . external field of strength H applied along the direction z , the third term is the interface coupling energy density and the last term is the uniaxial anisotropy energy. J E q, E q 2. The intrinsic exchange and the anisotropy energies make H,0 4 M 2 2 cosh2 qH / no contribution to the restoring force and k 15 (1/q)( EJ,H / q) where EJ,H is the sum of the Zeeman energy and the interface coupling energy. As the wall moves In Eq. 15 E(q rigidly out of the equilibrium position by a small displace- H,0) is the equilibrium value of the energy and the field effects are contained in q ment, it induces a change in the Zeeman energy due to the H and . From Eq. 9 we obtain the frequency of the domain wall modification in the sizes of the domains. The interface en- oscillations ergy is also changed since the displacement of the wall in- duces changes in the orientation of the magnetization with respect to the interface field. 4 MHJ In order to study rigid domain wall displacement oscilla- 2 , 16 tions around the equilibrium position we take 0 HA HA 4 M cosh2 tanh 1 H/HJ tan y,t where 0 HA(HA 4 M) is the frequency of the uni- 2 exp y qH t 12a form mode of the domains in the absence of interface effects and external field. In Fig. 2 we show (H)/ and 0. We selected a few values of the interface exchange field for an anisotropy t , 12b field of HA 0.55 kOe. (H) is a monotonically decreasing function of H with an upper limit of the order of 0. where (t) is the dynamical variable which describe the os- The upper limit of the excitation frequency (H) is for cillations of the domain wall center around the equilibrium H 0. As seen in Eq. 16 (0)/ 0 is proportional to the position qH , and (t) is the angle between the projection of square root of HJ /HA . Thus a large increase in HJ /HA does the magnetization in the yx plane and the y axis, describes not lead to a correspondingly large increase in (0). the out-of-plane component of the magnetization. The restoring force constant k is a decreasing function of qH and are the equilibrium values of the position of the H and turns zero for H HJ . For H 0 the energy fluctua- domain wall center and the domain wall width. They are tions include in full the oscillations of the domain wall obtained from the minimization of the energy and are given around the step edge. The equilibrium position of the wall by center moves away from the step defect when H increases. For H HJ the step defect is at the tail of the domain wall. q Thus, there is no variation in the angular profile, near the H tanh 1 H H 13 J step edge, for small displacement oscillations (y) 0 and and y(y) 0 , and there is no variation of the interface energy due to small oscillations of the domain wall position. H The shift of the hysteresis in F/AF bilayers, attributed to J H HJ H H HJ 1 2 ln 2 ln 1 HJ , is commonly found to be of the order of the anisotropy 0 HA HA HJ HA field of the F film.15 However, HJ may be larger than HA by H two to three orders of magnitude.16 Our results for 1 ln 1 1/2 H , 14 H J J /HA 103, not shown here for brevity, indicate that (0) is of the same order of magnitude of 0. Thus it where 0 A/K, HJ J/M and HA 2K/M. should be possible to observe interface pinned domain wall Using the magnetic profile defined by Eqs. 12 we obtain modes in experimental setups designed for FMR. 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