PHYSICAL REVIEW B VOLUME 57, NUMBER 1 1 JANUARY 1998-I Low-frequency dynamic response and hysteresis in magnetic superlattices S. Rakhmanova and D. L. Mills Department of Physics and Astronomy, University of California, Irvine, California 92697 Eric E. Fullerton Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439 Received 28 April 1997; revised manuscript received 5 August 1997 We study theoretically the low-frequency dynamic response of magnetic superlattices. We have the Fe/ Cr 211 structure in mind, which has been demonstrated to have a surface or bulk spin-flop phase, depending on the number of magnetic layers. We proceed by integrating the equations of motion of the coupled magnetic films in time, for an extended period. We include Landau-Lifshitz damping in the equations of motion, and drive the structure with an appropriate low-frequency field. The externally applied nominal dc magnetic field is increased slowly. We can follow the structure through the sequence of magnetic-field-induced phase tran- sitions. By this means, we obtain the magnetic phase diagram, 1 and 2 , along with hysteresis curves in a single calculation. We also provide data on the magnetic-field dependence of the low-field susceptibility, which is in good accord with theory. S0163-1829 97 07746-1 I. INTRODUCTION exchange couplings are very weak, such a phase may be studied with very modest magnetic fields. In recent years, magnetic multilayers of diverse character Some years ago, it was noted that if the external field is have been synthesized, and their properties studied exten- applied antiparallel to the surface layer moments in an MnF2 sively. In structures that incorporate films of ferromagnetic structure with a 100 surface, then at fields well below the materials such as Fe, exchange couplings between adjacent bulk spin-flop field, the surface region ``flops'' first.3 One Fe films are transmitted through the spacer layers between thus has a surface spin-flop phase for magnetic fields below the Fe films. These are very weak compared to the strong those that induce the bulk spin-flop transition. Recent experi- effective exchange couplings between spins in a given Fe mental studies of the Fe/Cr 211 superlattices provide clear film. Thus, we may model the structure by representing each evidence for the presence of the surface spin-flop transition.4 Fe film as a very large and hence classical spin S, formed In a finite structure with an even number of layers, in the from the spins within the film tightly linked by intra film low-field antiferromagnetic state, necessarily one of the two exchange. The various classical spins then interact by the surface films has its moment antiparallel to the applied field, inter film exchange, and experience anisotropy or dipolar a condition required for the surface spin-flop transition to coupling relevant to any structure of interest. occur.3 If odd number of films is present, one realizes a Thus, magnetic multilayers such as those just described ``bulk'' spin-flop transition, modified in character near the are a physical realization of one dimensional lines of classi- surface. cal spins. The inter film coupling is commonly antiferromag- In an earlier paper,4 theoretical studies were presented netic in character, so in fact such systems are isomorphic to that trace the evolution of the superlattice from the low-field one-dimensional classical antiferromagnets. When placed in antiferromagnetic state, to the high-field ferromagnetic state, an external field, they then may exhibit a spin-flop phase, where the Fe film moments are all parallel to the field. For very much as found in crystalline antiferromagnets.1 the case where the surface flop transition occurs, just above Of particular interest as a model system are Fe/Cr 211 the critical field, the surface moment initially antiparallel to superlattices.2 The Fe magnetizations lie within the plane the external field rotates nearly 180°, to become almost par- parallel to the surface of the structure, and there is an easy allel to it. In effect, a 180° twist has been applied to the axis within this plane, by virtue of the fact that the underly- antiferromagnet. A domain wall forms in the structure, ini- ing unit cell is rectangular in shape. When the inter film tially located off center, in the direction of the ``flopped'' exchange coupling is antiferromagnetic, the energy func- surface moment. Further increases in field cause the domain tional that describes the orientation of the moments is iden- wall to move to the center of the superlattice, in a sequence tical to that which applies to the 100 sheets of spins in the of discrete hops. In effect, there is a magnetic analogue of classical antiferromagnets FeF2 and MnF2. Thus, in zero ex- the Peierls-Nabarro barrier experienced by a dislocation in a ternal magnetic field, the ground state of the superlattice is crystal lattice.5 Each hop of the domain wall introduces a antiferromagnetic, with sublattice magnetizations aligned spike into dM/dH, as the domain wall moves to the center.4 along the easy axis in the plane. Application of a magnetic Further increases in field cause its width to increase, and a field parallel to the easy axis will induce a spin-flop phase, bulk spin-flop-like configuration is realized when the width just as it does in MnF2 and FeF2 However, since the interfilm of the wall becomes comparable to the size of the structure. 0163-1829/98/57 1 /476 9 /$15.00 57 476 © 1998 The American Physical Society 57 LOW-FREQUENCY DYNAMIC RESPONSE AND . . . 477 Very interesting subsequent work by Griffiths and his method, in a relatively straightforward manner, within a collaborators6 shows a surface phase not discussed in earlier framework of a single calculation. While we could use spin- work4 appears in a narrow field interval just above the sur- wave theory to generate expressions for 1( ) and 2( ), face spin-flop field. as remarked above, the algebraic analysis required would be The present paper is motivated by the appearance of the quite involved, for the complex magnetic phases of the finite domain wall mentioned, which hops through the lattice very structure. The equation of motion method is both conceptu- easily. The dynamic response of the structure should prove ally simple and elegant and, as we demonstrate here, works of great interest, in this field regime. Thus, we present theo- remarkably well. retical studies of the low-frequency response of the superlat- Our decision to approach this problem by this method was tice structures, for the cases where an even or odd number of influenced by an interesting paper on domain walls in anti- Fe layers is present and compare these calculations to ac ferromagnets published by Papanicolaou.9 Rather than gen- susceptibility results of a 22-layer Fe/Cr 211 superlattice. erate a description of domain walls by minimizing the energy We confine our attention to the regime where the response of of an antiferromagnet to which a 180° twist is applied at one the structure, as measured by the total ac moment induced by end, Papanicolaou began at time t 0 with the spins ar- the external field, is linear. The theory accounts nicely for ranged to mimic a domain wall in an approximate and crude the principal features observed, though our model is not suf- manner. He then numerically integrated the equations of mo- ficient for us to obtain a full and complete account of the tion of the damped spin system forward in time, to find the data. spins relaxed into their lowest energy state at long times. In the linear response regime, we could describe the re- With the constraint that one end spin is twisted 180°, he sponse of the structure within the framework of spin-wave obtained impressively accurate descriptions of domain walls, theory, where the externally applied ac field couples to the noting in the process that these walls posses a ferromagnetic collective spin-wave modes of the superlattice. The theory of moment parallel to the easy axis. Here we show that by the collective spin-wave modes has been developed and de- applying an ac field, and varying the dc field slowly in time scribed earlier, with attention to the surface spin-flop as described above, the equation of motion method may be regime.7 These modes have been studied experimentally by used to obtain the magnetic phase diagram, hysteresis loops, Brillouin light scattering, and the key features of the Bril- along with 1( ) and 2( ) in a single calculation. louin spectrum are reproduced in theoretical calculations.8 Instead, we introduce here an approach that, in a single II. THE MODEL AND THE METHOD OF CALCULATION numerical study, provides us with a remarkably complete description of the response of the structure both to the exter- We consider a superlattice which consists of N ferromag- nally applied static field H0 , and the ac field h sin t. netic films, and Mi is the magnetization of the ith film. The We proceed as follows. We begin by placing the system y axis will be chosen perpendicular to the interfaces between in a weak external field H0 , so it resides in its antiferromag- films, and the z axis, in the plane parallel to the interfaces, is netic ground state. The ac field is turned on, and we integrate the easy axis. An external dc field H0 is applied parallel to the equations of motion of the spin system forward in time. the easy axis. We then describe the system by an effective We add damping for each spin, of the Landau-Lifshitz form Hamiltonian (Si S i). In the presence of this damping, the transients die down, 2 and the spins settle into steady state motion. We may calcu- H A M i *M i 1 H0 Mz i K Mz i late the total transverse moment as a function of time, and Fourier transform this to obtain the real and imaginary part 2 M2 i h sin t M of the low-frequency ac susceptibility, y x i . 1 1( ) and 2( ). We then increase the dc field H0 very slowly; in this manner we obtain Here K is an anisotropy constant which renders the z axis an 1 and 2 as functions of H0 . When we 2 cross the spin-flop fields, the structure relaxes into its lowest easy axis, and the term in My(i) is the shape anisotropy with energy state, by virtue of the damping present. Thus, the origin in the dipolar field generated by tipping the magneti- system spontaneously ``flops,'' and by monitoring the total zation out of the xz plane. While this term plays no role in transverse moment we obtain the energetics of the various magnetic ground states realized 1 and 2 in the spin-flop phase. At the same time this is done, we may calculate the in the external field H0 , it enters the description of the dy- magnetic moment parallel to the nominally dc field H namical response of the structure, since the spins precess in 0 . We thus obtain the dc magnetization, as a function of H an elliptical manner, tipping out of the plane as they do. We 0 at the same time. The magnetization curves we obtain in this assume A 0, so we have antiferromagnetic coupling be- manner are in excellent agreement with those calculated tween adjacent films. The last term is the weak externally earlier,4 by minimizing the energy of a static spin array for applied ac field discussed in Sec. I. each value of H The equations of motion we study can be written in the 0 . The surface and bulk spin-flop transitions are first order, form and thus display hysteresis. By first increasing H0 until we reach the high-field saturated ferromagnetic state, then de- M i creasing this field until it changes sign, we may also generate t M i Heff i M i M i , 2 hysteresis curves for the structure. We thus obtain a large amount of information with this where the effective field Heff acting on the ith moment is 478 S. RAKHMANOVA, D. L. MILLS, AND ERIC E. FULLERTON 57 Heff i H0 2KMz i z 2A M i 1 M i 1 tion by HE/2, and rescales the time in this manner. These numbers are somewhat different than those employed in 4 My i y x h sin t. 3 Refs. 4 and 7. We have reduced the strength of the anisot- ropy field, to bring the value of the surface spin-flop field We have added damping of the Landau-Lifshitz form to closer to those in the sample studied here. the right hand of Eq. 2 . From the point of view of the To proceed, we exploit the fact that the ground state of the questions we wish to explore here, the virtue of damping of system for zero external dc field H this form is that M(i) relaxes without changing its length. 0 is known exactly. This is the simple antiferromagnetic state of NeŽel character, which We rewrite Eq. 2 in terms of the unit vector corresponds to choosing all i /2, and i to be 0 for odd M i i, and for even i. We use this condition as an initial con- n i figuration. We then increase the external field linearly with M , 4 s time, integrating the equations of motion continuously as we where M do so. The slope d(H s is the saturation magnetization of the film. Then 0 /HE)/d is chosen to be 10 4. We we introduce the effective anisotropy field select (g/HE) 0.1, so that in a dimensionless time interval 10, the system relaxes in response to any change. In the HA 2KMs 5 time interval 10, (H0 /HE) changes by only one part in and the exchange field 103. The system thus adiabatically follows changes in the dc field. The ac field discussed further below weakly excites H the system, so as soon as the energy of a spin-flop state drops E 2AM s , 6 below that of the low-field antiferromagnetic state, as H0 to write increases, the system is stimulated to make a transition into the new low-energy spin-flop state. The Landau-Lifshitz n i damping allows the spins to lose energy through dissipation t n i Heff i g n i n i , 7 into the reservoir responsible for the damping, so to speak. where now To integrate the equations of motion, we use the DDEBDF code from the package of differential equation solvers DEPAC H that was developed at the Lawrence Livermore Laboratory. H E eff i H0 HAn z i z 2 n i 1 n i 1 The code uses the backward differentiation formulas to solve first-order stiff differential equations. It advances the solu- 4 M tion using step sizes that are automatically selected so as to sn y i x h sin t 8 achieve the desired accuracy. Ordinarily, a time step is of and g Ms . Of course, it is essential to realize the films at order 1, except in the vicinity of a phase-transition point each end of the superlattice are exchange coupled to only one where it becomes the order of 10 1. To sweep out a typical neighbor in the film interior. magnetization curve such as those shown below requires in- We next turn to a discussion of the procedure we have tegrating the equations of motion for a dimensionless time used to integrate the equations of motion. First of all, we interval in the range of 5 104. This requires about five min- write n (i) in spherical coordinates: utes on a DEC Alpha workstation. If in addition, one obtains n i sin 1( ) and 2( ), perhaps a half hour of computer time is i sin i ,cos i ,sin i cos i , 9 required. with the angle It is useful, for the purpose of obtaining a physical feeling i measured from the y axis. We apply free end boundary conditions, wherein each end film is exchange for the time scales discussed above, to examine the collective coupled to only one interior neighbor. Upon substitution of spin-wave frequencies of the superlattice structure. For the Eq. 9 into Eq. 6 , we may reduce the problem to the solu- finite superlattice modeled as we do here, one finds detailed tion of 2N equations, where N is the number of Fe films in calculations in Ref. 7. For a structure of infinite length, one the superlattice. To facilitate comparison with calculations may work out the dispersion relation from the equations of reported in Refs. 4 and 7, most of the results reported here motion given above. We have done this to find a two branch are for N 15 and N 16. We have performed calculations dispersion relation in the low-field antiferromagnetic state. If for N in the range of a few hundred, it should be remarked, we consider a spin wave that propagates in the y direction to obtain accurate results very quickly. with wave vector q, we find We scale the various quantities that enter by measuring them in units of kilogauss. We shall choose 4 Ms /HE 2 2 2 2 2 21, and H q q H 4 M q A /HE 0.125. The ratio of HA /HE just given is 0 0 s HA HE 2 H0 0 appropriate for the Fe/Cr 211 structure, and we choose HE 4 H2M 2 kG. We take 4 M 0 s HA HE M s s to be 21 kG, appropriate to bulk Fe. We remark that is not the purpose of this paper to provide a 4 2M2H2 cos2 q/2 1/2, 10 full quantitative account of the data on the samples discussed s E here and in Ref. 4. To do so would require elaboration of the basic model, with inclusion of biquadratic exchange that is where surely present. The parameters just stated provide us with transition fields rather close to those observed. The unit of 2 time will be (H cos2 q/2 . 11 E/2)t; one divides the equations of mo- 0 q HA HE 2 HE 57 LOW-FREQUENCY DYNAMIC RESPONSE AND . . . 479 cussed below. For the case N 15, we realize a ``bulk'' spin flop, of course modified by the presence of the two surfaces. The discontinuity in Mz just above H0 1.0 is the signature of the spin-flop transition. So far as we can tell, the transition occurs right at the field where the static energy of the flopped state drops below that of the low-field antiferromagnetic state. For the case N 16, we realize the surface spin-flop tran- sition, which occurs for a value of H0 reduced from that for the case N 15 by roughly a factor of &, as expected3 when HA HE . We turn to one difference between the present results, and those reported earlier4 for the case N 16. The static calcu- lations showed that as H0 is increased above the surface spin-flop field, the surface moment rotates by nearly 180°, to FIG. 1. The component of magnetization, parallel to the dc field become nearly parallel, rather than antiparallel to the exter- H0 , as a function of H0 for a sample with 15 Fe layers, and a nal field. It is as if the antiferromagnet has one end spin sample with 16 Fe layers. H0 is dimensionless and is measured in twisted by nearly 180°. There is then a domain wall in the units of HE . structure, between two nearly antiferromagnetic regions. As H0 is increased, the domain wall executes discontinuous In an antiferromagnetic resonance experiment, one excites jumps, as it migrates to the center of the structure, to ulti- the q 0 modes. In zero external magnetic field we have for mately widen and evolve into a bulk spin-flop-like state. It the two modes should be remarked that the energy differences between the states with the domain wall in different locations were very 2 0 HA 4 Ms HA 2HE , 12a small indeed. While, as we discuss below, we see the domain wall form in the present calculations and migrate toward the 2 0 HA HA 4 Ms 2HAHE . 12b center of the structure with increasing field, it does so The parameters above give smoothly so far as we can discern by eye. It may be difficult (0) 9.5, and (0) 2.51, in units of kilogauss. to perceive the jumps, or possibly in the presence of the ac In dimensionless form, the ac field is written field, the wall moves smoothly through the structure. (h/H In Fig. 2, we illustrate how the system evolves from the E)sin( ), where for most of the calculations below, we choose 0.1. In this paper, we are thus exploring the re- low-field antiferromagnetic state, to the high-field saturated sponse to frequencies well below the spin-wave frequencies ferromagnet, for N 15 and for N 16. We can see clearly just discussed. We have chosen (h/H that when N 15, the entire structure ``flops'' at once. The E) 10 4 in all results shown below. The period of the driving field is 20 spins at the two ends of the structure are pulled closer to the 62.8 time units. This is very long compared to the time dc field than those at the center, because they are exchange 10 for transients to die out in the system, but still so coupled to only one neighbor, rather than two, as is the case short that dc field changes very little over one cycle of os- for the interior spins. cillation. When we calculate For N 16, we illustrate the surface spin flop, with for- 1( ) and 2( ), we need to have the dc field constant over many cycles of oscillation of mation of the domain wall and its subsequent migration to the ac field. For this purpose, we thus increase it in a step- the center. Once it is centered, its width increases continu- wise fashion, rather than the linear manner discussed above. ously, and the system evolves into a bulk spin-flop-like state. Each step in the field is taken to be 150 periods of the ac There is one interesting aspect to the sequence of events field. We use data from the last 10 periods in this sequence to illustrated in Fig. 2, for the case N 16. If one looks at the fit the total transverse moment to the form pattern of arrows in the low-field antiferromagnetic state, the 1 sin( ) picture is odd under reflection through its midpoint. The final 2 cos( ) by a -squared procedure. We have also Fou- rier transformed the transverse moment, to confirm only a ferromagnetic state is even. There is a field at which the single frequency is present in the output, for the range of domain wall is centered precisely in the film center, and at (h/H this point the ``pattern of arrows'' is even under reflection E) employed. We now turn to our results. for all higher fields. We thus have a mechanism for evolution from the odd- to the even-parity state. For N 15, the pattern III. RESULTS AND DISCUSSION of arrows is even under reflection through the midpoint of the structure at all fields. In Fig. 1, for the case where the superlattice has 15 Fe In Figs. 3 and 4, we show the evolution of the surface films, and also 16 Fe films, we show the z component of spin-flop state with increasing magnetic field, in a very nar- magnetization Mz as a function of the dc magnetic field H0 . row interval of field just above the surface spin-flop field. If these are compared to the magnetization curves calculated We have a sequence of ``snapshots''; the dc field changes by by minimizing the static energy of the Hamiltonian in Eq. 1 about 0.2% in magnitude from the beginning to the end of with h 0 , one sees excellent agreement, save for the fact the sequence. In essence, the domain wall continuously that the spin-flop fields are somewhat lower, because of our creeps into the structure from right to left, as the dc field use of a smaller anisotropy field. One further point is dis- increases. In the first illustration in Fig. 3, there is a very 480 S. RAKHMANOVA, D. L. MILLS, AND ERIC E. FULLERTON 57 FIG. 3. Selected ``snapshots'' of the spin lattice, for a narrow interval of time in the near vicinity of the surface spin-flop transi- tion. Again we have N 16. From the beginning to the end of the sequence, the dc magnetic field changes by 0.2%. The view is side view, but with the axis canted out of the plane a bit, to assist in viewing the spin array. Each spin configuration shown is a stable, relaxed configuration. FIG. 2. Moment configurations as the strength of the dc field is increased, for a the case where there are 15 Fe films, and b the case where there are 16 Fe films. These are not strict side views of the structure, but the backbone of the structure is canted out of the page a bit, to provide perspective. slight tipping of the unstable spin; the tail of the domain wall has crept in a bit at this point. The system evolves very quickly, over a narrow field interval, to a state where the unstable moment is twisted nearly 180°. We believe that in each of the panels in Figs. 3 and 4, we have a fully relaxed moment configuration. We see no clear evidence of the ``true surface spin-flop state'' discussed in Ref. 6, unfortunately. The surface spin does seem to ``hang up'' a bit when it makes an angle of roughly 60° with the external field, as expected for their state, but we cannot identify a clear signal of this phase. By sweeping the field first upward, until we reach the saturated ferromagnetic state, and then decreasing the field back downward past the spin-flop transition, we can generate hysteresis curves. The magnetization exhibits irreversible be- havior only in the near vicinity of the spin-flop transition. In Fig. 5 a , for N 15, we show hysteresis curves calcu- lated by ramping the field up linearly in time, and then ramp- ing it back downward linearly in time, until the field changes sign and we reach the saturated ferromagnetic state with all moments pointing downward. Fig. 1, as we have discussed, gives the total magnetization as a function of field, when one starts at zero field and ramps it up to the saturated ferromag- FIG. 4. An end on view of the spin array for the same fields netic state. The dashed line in Fig. 5 a give the magnetiza- used in Fig. 3. 57 LOW-FREQUENCY DYNAMIC RESPONSE AND . . . 481 FIG. 6. Calculations of 1 and 2 as a function of field, as one begins in zero external field, and increases it to reach the saturated ferromagnetic state. Again we show results for a the case N 15, and b the case N 16. FIG. 5. Hysteresis curves, for a the case N 15, and b the case N 16, for the case where the dc magnetic field is ramped up then testing for stability. The results are remarkably similar or down linearly with time. In c , we show a curve when the field to those in Figs. 5 a and 5 b , except the equivalent of the is increased in small steps. Strictly speaking, the horizontal axis dashed line in Fig. 5 b extended down to zero dc field. We should be labeled with time rather than field, but the steps in field believe the small difference has its origin in the presence of are small. In Fig. 1, we show the magnetization when one begins in the ac field in our simulation, which can stimulate the tran- zero field, and increase the field to achieve the saturated ferromag- sition. Also, if the time profile of the dc field is decreased in netic state. In these figures, dashed curves describe the magnetiza- a steplike manner, as one comes down in field, the transition tion as one decreases the field from a large positive value, to a large to the asymmetric state occurs much sooner for the case negative value, while the solid curve describes the magnetization where the system receives a sequence of impulsive ``blows'' when one starts from large negative field, and increases the field. through the sudden change in the dc field. We illustrate this tion as a function of field, as one comes down in field. Then in Fig. 5 c . once the saturated state is reached for strong negative fields, Now we turn our attention to calculations of the dynamic the solid line is the magnetization as one ramps the field back susceptibilities 1( ) and 2( ). We change the dc field in up. It is striking that the coercive field for N 15, defined as steps, as mentioned earlier. After a given step, when the the field where the net magnetization rotates to align with the transients die down, we fit the total transverse moment to the field after its sign reverses, agrees very accurately with the expression value of the surface spin-flop field for the case N 16. m In Fig. 5 b , we show the hysteresis curves calculated for T t h 1 sin t 2 cos t , 13 the case N 16. Notice that the amount of hysteresis is very where in the convention of Eq. 13 , in fact 2( ) is nega- much larger than for N 15. We believe this is because, at tive. high fields, the system is in the symmetric state described In Fig. 6, we show 1 and 2 as a function of field, with above. It is difficult for the spins to make the transition back 1 given as a solid line and 2 as a dashed line. The calcu- to the low-field asymmetric state. They remain ``locked'' in lations assume one begins in zero field, and increases the the symmetric state until very low fields. field until the saturated ferromagnetic state is reached. For Hysteresis curves were also calculated earlier in the paper N 15, we see a clear signature at the spin-flop transition, by Wang and Mills,7 by searching for the limits of stability while for N 16, we see a feature at the surface spin-flop of various states, through minimizing their static energy, transition and a second bump located near the bulk spin-flop 482 S. RAKHMANOVA, D. L. MILLS, AND ERIC E. FULLERTON 57 FIG. 7. The same as Fig. 6, but now we begin in the saturated ferromagnetic state, and decrease the field to reach large negative fields. FIG. 8. For N 16, we show a series of snapshots of the field excursions that interchange the two degenerate antiferromagnetic ground states. field. As the number of spins N increases, the feature at the surface spin-flop transition becomes smaller, and the bump at with the BA structure in the final state. This operation has the bulk spin-flop grows and sharpens. interchanged the two degenerate antiferromagnetic ground In Fig. 7, we show calculations in which the system is states. initially saturated in the high-field state, and the field is then As remarked briefly in Sec. I, we have measured the ac decreased until we reach the saturated state at large negative susceptibility, to obtain values of 1 and 2 as a function of fields. For the case N 15, we see a two peaked structure at field, on the Fe/Cr 211 superlattice described in Ref. 4, negative fields. That at lowest negative field is associated which exhibited the surface spin-flop transition. The ac sus- with the coercive field of the structure where its total mo- ceptibility and dc magnetization were measured in a Quan- ment reverses, and the second with the subsequent bulk spin- tum Design PPMS 6000 with a 14 G, 1000 Hz ac field was flop transition. For N 16, we see a very large asymmetry, parallel to the dc applied field. Thus, when comparing theory between the structures at positive and negative fields. and experiment, one must recognize that in the experiment, We have found very striking behavior of the structure, the ac field is parallel to the dc field, while the calculations upon traversing the hysteresis loop in Fig. 5 b appropriately. generate the transverse response. This difference clearly de- We begin with the antiferromagnetic state in zero field which serves comment. In our theoretical studies of the phase dia- we might call the AB state. By this we mean that the leftmost gram and hysteresis curves, application of a time varying moment, designated by A, points upward, while the right- field would not stimulate spin reorientation transition, when most moment, designated by B, is directed downward. Note one approached the transition field from either the low-field this state is degenerate in energy with configuration BA. antiferromagnetic state, or the high-field ferromagnetic state, Once again, we begin in zero field, and increase the dc field since such a field exerts no torque on the spins. Thus, we past the surface spin-flop field to, say, the knee in the revers- have used a transverse field in our theoretical studies. In the ible part of the M-H diagram in Fig. 5 b . Then we decrease data on 1 and 2 , the signal is only large in the spin-flop the field back to zero. region. We expect a rather small difference between the dy- The behavior of the structure is illustrated in the series of namic susceptibilities calculated for longitudinal or trans- panels displayed in Fig. 8. We see the initial AB state, and verse fields in this state, so we believe it appropriate to com- the domain wall that enters the structure from the right, and pare the theory and the experimental data, for progresses to the center of the structure. As the field is de- semiquantitative purposes. In Fig. 9 a we show the dc mag- creased in this state, the wall keeps moving from right to left, netization as a function of dc field. The surface spin-flop to exit the structure from its left side, so to speak. We are left transition occurs at 1000 G in this sample. There is clear 57 LOW-FREQUENCY DYNAMIC RESPONSE AND . . . 483 FIG. 9. Experimental measurements for a total magnetization of the sample described in the text as a function of dc field, b 1 and c 2 the real and imaginary parts of susceptibility, respec- tively. hysteresis in the near vicinity of the surface spin-flop transi- tion, though one sees less hysteresis than is evident in our simulations. The experimental results more closely resemble the simulation in Fig. 5 c where the applied field is changed in steps. FIG. 10. Calculations for a 2 , and c 1 as a function of Figure 9 b shows field for 22-layer structure. In b , we show the corresponding hys- 1 , the real part of the susceptibility, taken on a field sweep that begins in the saturated ferromag- teresis curve. Here H0 is given in kilogauss. netic state (H 3 kG), with H along z to saturation along z . We see features near 1000 G, the bulk spin-flop state frequencies as low as those used in the experiments, since of an infinitely long structure in agreement with previous the integration times would be prohibitively long. In the static susceptibility (dM/dH) results of Ref. 4. We only see samples used, the spin-wave frequencies are in the range of a very modest signature of the surface spin-flop transition that few tens of GHz.8 Thus, if SW is a typical spin-wave fre- was prominent in the static susceptibility. We believe that quency, in the experimental measurement, / SW 3 the reason for the absence of structure in 1 at the surface 10 8. In our simulations, from the spin-wave frequencies spin flop results from the hysteresis near this transition. A quoted in Sec. II, / SW 5 10 2. However, so long as minor loop about the surface spin-flop transition determined the ac frequency is very small compared to the spin-wave a coercive field of 30 G that is large compared to the ac frequencies, we are in the regime where the spins follow the driving field. Therefore, in these measurements, the ac field ac field nearly adiabatically, with a phase lag provided by the is not sufficient to sample reversibly the surface spin-flop finite value of . In this low-frequency regime, where the transition. ac field has simple relaxational form, the field dependence Shown in Fig. 9 c is the imaginary part 2 of the sweep realized is not sensitive to frequency. In the simulations re- in Fig. 9 b . We again see features at the bulk spin flop that ported here, the ac driving field is also smaller than that used have roughly equal strength at positive and negative fields in the experiments, by a bit more than a factor of 10. We and are enhanced to the low-field side of the transition. have carried out calculations for ac driving fields in the range Above the bulk spin flop the value of 2 is below the sensi- used in the experiments indeed we have explored even tivity of our magnetometer. Surprisingly, there is also struc- larger fields , to find results for 1 and 2 identical to those ture at the surface spin-flop transition that is highly asym- reported. metric. We find that the dramatic peak in 2 at the spin-flop Nonetheless the simulations reproduce the key features of transition when the applied field is swept from H 0 toward the data very nicely. We show the field dependence of 2 in saturation is missing. Fig. 10 a , as the field is decreased from high positive values, We have carried out simulations for the 22-layer film, to large negative values. We see the two symmetric struc- employing values of HA and HE used in the calculations tures at the bulk spin-flop field, and the very large, dramatic discussed earlier. It should be noted that we cannot employ spike at HSSF , with HSSF the surface spin-flop field. We do 484 S. RAKHMANOVA, D. L. MILLS, AND ERIC E. FULLERTON 57 not see the small structure evident in the data at H IV. CONCLUDING REMARKS SSF . The reason for this is the very different hysteresis curve found in The method of calculation employed here allows one to the simulations. As we lower the field, the 22-layer model generate magnetic phase diagrams, hysteresis curves, and ac- structure remains locked in the symmetric state until rather counts of the dynamic response of one-dimensional classical large negative fields are reached. We show our calculated arrays of spins, in one single computation. While the results hysteresis curve in Fig. 10 b , for this model structure. The reported here explore a relatively small number of spins, coercive field at which the system returns to the antiferro- since we wish to compare our new results with data and the magnetic ground state is negative; note the small jump in Mz earlier theoretical literature, we emphasize that we have ap- just before one reaches HSSF . One sees a small structure in plied the technique to several hundred spins with no diffi- 2 at this field, in Fig. 10 a . culty. Thus, the direct integration of the equations of motion Save for the difference between the theoretical and ex- should prove a most useful technique for studies of a wide perimental coercive fields, we thus obtain an excellent ac- range of magnetic nanostructures. count of the 2 data in our simulations. In Fig. 10 c , we show ACKNOWLEDGMENTS 1 as a function of field. We again see symmetric structures at the bulk spin-flop field very similar to those in The research at Irvine was completed with support from the data , but now we see a major feature at HSSF as well. the Army Research Office, under Contract No. CS0001028, We expect this structure should be present in the data as and that at Argonne National Laboratory by the U.S. Depart- well, if the experiments could be carried out in a transverse ment of Energy, BES-Materials Science, under Contract No. ac field. W-31-109-ENG-38. 1 S. Foner, in Magnetism, edited by G. Rado and H. Suhl Aca- 5 J. Friedel, Dislocations Pergamon, Oxford, 1964 , p. 54. demic, New York, 1993 , Vol. 1, Chap. 9, p. 338. 6 C. Micheletti, R. B. Griffiths, and J. M. Yeomans, J. Phys. A 30, 2 Eric E. Fullerton, M. J. Conover, J. E. Mattson, C. H. Sowers, and L233 1997 . S. D. Bader, Phys. Rev. B 48, 15 755 1993 . 7 R. W. Wang and D. L. Mills, Phys. Rev. B 50, 3931 1994 . 3 D. L. Mills, Phys. Rev. Lett. 20, 18 1968 ; F. 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