The magnetic aftereffect in CoCr films:A modelD. Lottis, R. White, and E. Dan Dahlberg Citation: Journal of Applied Physics 67, 5187 (1990); doi: 10.1063/1.344657 View online: http://dx.doi.org/10.1063/1.344657 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/67/9?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Magnetization reversal in Co/Cr multilayer films J. Appl. Phys. 73, 6353 (1993); 10.1063/1.352646 Magnetic interactions in CoCr thin films J. Appl. Phys. 73, 6671 (1993); 10.1063/1.352551 Magnetization decay in CoCr films J. Appl. Phys. 63, 2923 (1988); 10.1063/1.340959 Magnetic aftereffect in CoCr films J. Appl. Phys. 63, 2920 (1988); 10.1063/1.340958 Influence of deposition conditions on magnetic properties of CoCr thin films J. Appl. Phys. 57, 4016 (1985); 10.1063/1.334656

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The magnetic aftereffect in Coer films: A model D. LoUis, R. White, and E. Dan Dahlberga)

School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455

We have previously reported experimental studies of the magnetic aftereffect in CoCr films. In that work data were compared to phenomenological models, and it was recognized that the decay ofthe magnetization was driven by the demagnetizing field. The difficulty of incorporating such fields into phenomenological models, which rely on a broad distribution of relaxation times or activation energies to account for the quasilogarithmic decays, motivated the present study. Here we present a simple alternate model which does not assume a dis~ributi::m of energies. ~nst~ad, we focus on the time dependence of the local magnetic field h WhlCh dnves the magnetlzatlOn. We approximate h by the average internal field Hi = Ha - 41TM, where H,: is the applied field, Mis the magnetization averaged over regions

large compared to domams, and 417 is the demagnetizing factor appropriate for a planar geometry. The magnetization is found to relax quasiIogarithmically with time in this model. We have also made a more detailed comparison with experimental results. We conclude that this simple model is more transparent to the essential physics of the problem than previously used models, and may be useful in analyzing decays in other materials.

I. INTRODUCTION

Previous researchers [·-3 have measured the decay of the remanent magnetization in CoCr films prepared to have an easy axis perpendicular to the plane of the films. Thev have found that the decay is quasilogarithmic in time which is similar to that found in spin glasses and structural glasses. Although at first it was surprising that the remanent magne­tization of the films decayed, it has been determined that the decay is driven by the demagnetization field of the sam­pIes. ',2 It is usual in modeling the decay of the remanence in magnetic systems and structural glasses to assume that dis­order, which creates a distribution of energy barriers, is re­sponsible for the slow or logarithmic decay in these systems. We have developed a simpie model which decays with the appropriate time scale, i.e., quasilogarithmic, which does not require disorder to provide the slow decay. Using the structure of the films, which we approximate as simple co­lumns, and the fact that the demagnetizing field drives the decay, we have simulated the decay of the magnetization of the CoCr films by modifying the fine-grain model ofNeel4 to include the demagnetization field. It is this simulation which we report here. In the next section, we review Ned's model and show how we have modified it to include the demagnet­ization field (which is the mean-field approximation to the dipole-dipole interactions of the columns or grains). This section is followed by a comparison between the model and the experimental magnetization decays, The last section pro­vides our conclusions.

II. THEORETICAL MODEL

Briefly, the model consists of a collection of magnetic grains or spins lying in a plane. The spins have an anisotropy energy such that their magnetizations, in the absence of any

.) Also at 3M Corporation, St. Paul, MN 55144.

fields, point perpendicular to the plane which is the direction defined as the z axis. The spins possess a magnetization and therefore interact with each other via a dipole-dipole interac­tion which in the mean-field limit is just the demagnetization field. If the spins are first polarized in the + z direction, the dipolar coupling or the demagnetization field is such that it appears as a field in the - z direction, which tends to demag­netize the film. As the magnetization of the sample de­creases, the driving field also decreases, which, in turn, slows down the decay of the magnetization.

The starting point of the details of the model is, as men­tioned previously, Neefs fine-grain model. He developed an expression to describe the flipping of the magnetization in a system of particles, each of which has a saturation magneti­zation M, (initially polarized in the + zdirection), a uniax­ial anisotropy energy K u' a volume for each grain, 11, and, for the purposes of the present work, a magnetic field H which is directed in the - z direction. By assuming the magnetiza­tion process occurs by a rotation of the magnetization in the individual grains, he obtains an expression for the decay of the magnetization in external fields for which the decay rate has a peak at the coercive field of the grains. This peak at the coercive field has been measured in CoCr films previously,!

In modifying his model to applicable to the decay of remanent magnetization in CoCr films, we have replaced the externally applied field with the demagnetization field of 417M(t) , where M(t) is the value of the remanent magneti­zation5 and 41T is the appropriate demagnetization factor for a planar geometry, This is effectively a mean-field approxi­mation in the sense that the demagnetization field arises from the dipolar fields of the grains and therefore the inter­action responsible for the decay ofthe remanent magnetiza­tion should actually be a discrete sum over the grains in the vicinity of a single grain of interest.

Making the above substitution and defining MU) as M,on(t), where 8n = (Nup - Nd )/N and N = N • own up

+ Ndown , one obtams the following expression for the time dependence of the magnetization decay:

5187 J. Appl. Phys. 67 (9), 1 May 1990 0021-8979/90/095187-03$03.00 @ 1990 American Institute of Physics 5187

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don = _ w (1 + on) dt +

xexp[ -/3(vI4K" )(47TM;8n - 2Ku )2]

+ w_ (1- on)

xexp[ -,B(v/4KuH41rM;on+2Ku)2], 0) where f3 is 1/ k B T and N is the total number of grains in the system. The w 1-/ ... are the attempt frequencies for the mag­netization of the individual grains to hop over the energy barrier due to the anisotropy energy. In general, the w's de­pend on the applied magnetic field and therefore in the pres­ent model the magnetization of the sample. However, we have found the decay to be rather insensitive to the magneti­zation dependence of the w's, and for computational ease we equate the w's and replace them with a fixed value which is appropriate to the average value of the remanent magnetiza­tion under investigation. This replacement works wen since the decay is, in fact, very slow and because the dynamics are dominated by the exponentials in Eq. ( 1 ). Equation ( 1 ) pro­vides the basis for a comparison between the fine-grain mod­el with interactions and the experimentally measured values of CoCr films. We have used a third-order Runge-Kutta a1-gorithm6 to numerically sOlve Eq. (1) for this comparison.

III. COMPARISON OF THEORY AND EXPERIMENT

Equation (1) has a number of parameters to be specified prior to determining the relaxation of the magnetization. We have used published values7 for Ms and Ku in fitting to ex­perimentally detennined magnetization decays 1,2 and values of w on the order of 104 Is, which is a typical number calcu­lated from the model of Browns for attempt frequencies. With these assignments, it is only the volume of the grains which is left as an adjustable parameter. To fix both Ms and Ku and only vary the volume u would appear reasonable

:/0.65 .... ::ii

0.60

1000

t (in seconds)

0.104

0.102 "' ::Ii .... ::I!

0.100

FIG. 1. An example of the decay of the remanent magnetization modeled usingEq, (!) (solid line and left-hand scale) and the measured decay of the remanent magnetization of film of CoCr (right-hand scale) at room tem­perature (data from a film different than that used in table). In the modeled decay, the saturation magnetization is taken to be 200 emu/crn', Ku is 5X 105 ergs/em3, the particle volume is 4X 10- '" crn-C<, and w is 2X 104

S . I, The magnetization decay ofthe CoCr film is measured after the sample is first saturated in a magnetic field of 5 T. In both situations the time depen­dence of the magnetization is measured as a fraction of the saturation mag­netization.

5188 J. Appl. Phys., Vol. 67, No.9, 1 May 1990

since the reversal of the magnetization may nucleate in a volume smaller than the grain volume measured microscopi­cally or may involve several exchange-coupled grains. On the other hand, it is difficult to imagine magnetizations or anisotropy energies grossly different than those measured.

In Fig. 1 we show a typical decay of the magnetization as predicted by the model described here (solid line). For a comparison, also displayed in this figure are magnetization­decay data from a CoCr film measured previously. The two things to note are that the volume of a grain in the real film measured microscopically 7 is on the order of 10- 15 cm3

,

whereas the simulated grain has a volume of 4 X 10- 18, and

that the remanent magnetization of the simulation is larger than that of the measured film. The large discrepancy between the model and the real system grain volume most likely indicates that the magnetization-reversal process in an individual grain does not take place by a magnetization rota­tion, but instead by some other process, such as the nuclea­tion of a domain wall within a single grain and its subsequent propagation. In this situation, the volume determined in the simulations may be indicative of the nucleation volume of a reversed domain in a single grain. The second point, the lact that the remanence of the simulation is larger than that of the film, probably indicates a distribution of sizes in the film and that most ofthe smaner volumes have already flipped. In this sense, the decay at a fixed temperature actually is only sensi­tive to a small range of particle volumes. This will be dis­cussed in more detail later.

Another feature of the experimental results is the non­monotonic behavior of the temperature dependence of the slope. It is found 2 that the magnitude of the decay of the magnetization (measured as the fraction of the remanent magnetization which decays per decade of time in a fixed time window) has a maximum at approximately room tem­perature. The model described in Eq. (1) also has this fea­ture. This is due to the temperature dependence of the in­stantaneous value of bn, which appears in the argument of the exponentials. Increasing the temperature for fixed on would speed the decay by reducing the absolute value ofthe arguments of the exponentials (which are always negative). However, in a fixed time window, the decrease in On with growing temperature has the opposite effect.

A more transparent way to show that the model predicts this nonmonotonic behavior can also be shown by the fol­lowing simple arguments. In the zero-temperature limit, the magnetization cannot decay since there is no thermal energy available for the magnetization in a grain to fiuctuate and finally hop over the barrier. In this limit the slope would be zero and the remanent magnetization would be ll1s • In the other extreme limit, that of infinite temperature, where f3 is zero, the exponentials are equal to unity and the relaxation is therefore described by the usual exponential decay where the characteristic time is given by l/w. In the present case, the w's are on the order of 104 and after 1 s the remanent magne­tization will be effectively zero, and therefore the decay over the time window of the measurement (on the order of lOs minimum) will be zero again. Clearly, at temperatures between zero and infinity there will be a nonzero slope in the decay, Since the decay slope is zero at large and small tem~

Lottis, White, and Dahlberg 5188

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TABLE I. A comparison of the experimental decay rates compared to the theoretical decay. The only adjustable parameter is the volume of the re­versed grain or region. The other quantities were fixed at valu~s as deter­mined experimentally, e.g., M, "'" 220 emu/em', Ku ~ 2X 10' ergslcm', andw=2Xl0's- l

.

M 1M, Temperature Measured decay slope Fitted volume (time = 60 s) (K) M 1M, per decade time (10- 18 em')

0.065 400 .- 0,(lO28 7.0 0.137 150 .- 0.0037 3.4 0.179 50 -··0.0028 1.4 0.2 10 - 0.00085 0.32

peratures, there will be a maximum value at an intermediate temperature,

Although the above supports the applicability of the model, the fact that the value of the measured magnetization versus the simulated magnetization is problematic (see Fig, 1) is an indication that disorder is still important. Even a small variation in grain volumes can greatly affect the dy~ namics of a real system. As an example, we have taken the experimentally determined values of the remanent magneti­zation as the initial magnetization and again fixed the values of the saturation magnetization and the anisotropy energy as those determined experimentally, and have replicated the nonmonotonic behavior oftne slope using only the value of the volume as an adjustable parameter. The results are shown in Table I.

It is clear from the sensitivity of the decay slope to the nucleation volume that a small variation in the grain or nu~ cleation volume can generate a large distribution of tempera­ture~dependent decay slopes. Physically, what would occur is that at a fixed temperature all of the grains in the superpar­amagnetic limit would decay rapidly, leaving only those grains with a size which makes them slightly more stable than those in the superparamagnetic regime. It is these parti­cles that would have observable decays in the time limit of a decay experiment. This, of course, implies that one may de~ tennine the particle-size or nucleation-volume distribution

5189 J. Appl. Phys., Vol. 67, No, 9,1 May 1990

by utilizing both the remanent magnetization at a fixed tem­perature (determines the number of particles not in the su­perparamagnetic limit) and the decay slope at the same tem­perature (determines the corresponding volume for that temperature) ,

IV. CONCLUSIONS

We have presented a model based on the fine-grain mod­el of Nee! which provides a model system for studying the slow decay of the remanent magnetization in magnetic sys­tems. The nove! feature of the model is that the remanence decays quasilogarithmically with time even though there is no disorder present in the modeL The reason for the slow decay is that the driving field for the decay, as represented by the demagnetization field, diminishes as the system relaxes to equilibrium or the demagnetized state. In addition, we have shown that the model also exhibits the nonmonotonie temperature dependence of the magnitude of the decay slope, However, concerning this last point, we have also

. shown that a rather small distribution of volumes can also give rise to the nonmonotcnic temperature dependence of the decay slope. In real systems there is certainly a distribu­tion of grain sizes and it may be that both the model de­scribed here and a distribution of volumes are necessary in describing the real systems,

'D. K. Lottis, E. Dan Dahlberg, J, Christner, J. I. Lee, R. Peterson, and R. White, J, Phys. (Paris) Colloq. 49, C8-1989 (1988).

2n. K. Lottis, E. Dan Dahlberg, J. Christner, J. L Lee, R. Peterson, and R. White, App\. Phys. 63, 2920 (1988).

3Bucknell C. Webb and S. Schultz, App!. Phys, 63, 2923 (\988). 4L, Nee!, Rev. Mod. Phys. 25, 293 (1953); S. Chikazumi, PhysicsofMagne­tLfm (Wiley, New York, 1972), Chap. 15.

'For purposes of analyzing experimental data on CoCr films, the demagne­tizing field can be approximated by 41TM(t = 60 s), which differs very little from the instantaneous value 4rrM( t) during the experimental time window of 2-3 time decades.

"w. H. Press, B. p, Flannery, S. A. Teukolsky, and W. T. Vetterling, Nu­merical Recipes (Cambridge University Press, Cambridge, 1986).

7M. R. Khan, J. Lee, D. J. Seagle, and N. C, Fernelius, App!. Phys., 63, 833 (1988).

·William Fuller Brown, Jr., Appl. Phys. 30, 130 (1959),

Lottis, White, and Dahlberg 5189

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