PHYSICAL REVIEW B VOLUME 51, NUMBER 17 1 MAY 1995-I

Individual-domain-wall motion in Nio 77Mno 23 observed via resistance Auctuations

C. D. Keener* and M. B. WeissmanDepartment of Physics, Uniuersity of Illinois at Urbana Ch-ampaign, 1110West Green Street, Urbana, Illinois 61801

(Received 18 November 1994)

Both equilibrium magnetic noise and Barkhausen noise in Nio 77Mno 23 films were observed via electri-cal resistance. The duty cycle of one two-state switcher could be adjusted by changing the magneticfield, as is expected for thermally activated single-domain-wall motion. Two switchers showed non-Arrhenius behavior near the Curie point. Temperature and frequency dependence of the noise indicatedthe presence of smaller fiuctuating domains throughout the ferromagnetic regime. At some tempera-tures when a large oscillating magnetic field was applied, individual Barkhausen jumps were seen atreproducible points in each field cycle.

I. INTRODUCTION

Electrical transport noise in ferromagnetic materials isacquiring technical importance as magnetoresistive de-vices are adopted for magnetic memory applications, andespecially as the size of such devices is reduced. ' Thedetection of individual domain fluctuations via transportmeasurements is also becoming of interest as a techniquewhich should be able to test theories of macroscopic mag-netic quantum tunneling when extended to low enoughtemperatures.

There are two broad categories of domain fluctuations.The most familiar is Barkhausen noise, ' which resultswhen domains rotate or domain walls move in discretesteps as the magnetic field is swept. The relative impor-tance of domain-wall pinning by defects and of the col-lective self-organization of domain walls in theBarkhausen effect has become a matter of dispute.

Although spontaneous thermal fluctuations have notbeen studied as extensively as the Barkhausen effect, fluc-tuations of this second category can be observed inquasiequilibrium when the magnetic field is not inducinga change in magnetization on experimental timescales. ' In this paper we show both quasiequilibriumand Barkhausen fluctuations of individual domains inpartially short-range-ordered NiMn thin films. ' An in-teresting experiment, complementary to this one, ' hasshown reproducible magnetization reversals of single-domain ferromagnetic particles, partly due to thermal en-ergy, after large changes in the applied field.

II. BACKGROUND

Resistance fluctuations are often quantified in terms ofa convenient, dimensionless parameter:

X,fSIt(f, T)ct(f, T)=

R

where f is the frequency, R is resistance, T istemperature, Sz (f, T) is the power spectral densityof resistance fluctuations, and X, is the number ofatoms in the sample. We also define spectral slope

y = —t) in'� (f, T)/d 1nf. a can depend not only on f an'd

T but also on magnetic field H with the form of thesedependences determined by the detailed mechanism gen-erating the noise in a particular material. '

In many ferromagnets, the main factor that couplesmagnetization fIuctuations to resistance fluctuations isthe spontaneous resistive anisotropy' (SRA), which isthe fractional difference between the resistivity paralleland perpendicular to the magnetic moment within adomain of constant magnetization. (Other factors, suchas giant magneto-resistive effects at domain walls, can bemore important in some materials. ") When the magneti-zation in some region rotates by any angle other than180' (e.g. , when a domain wall moves), the local resistivi-ty tensor changes. At equilibrium, in samples suKcientlysmall that only a few FM domain walls fluctuate in theexperimental frequency range, the resistance of the sam-ple will have observable discontinuous jumps as a func-tion of time. These are similar to Barkhausen jumps ex-cept that they occur reversibly, under constant condi-tions, such as a constant magnetic field. In the tempera-ture range explored here, thermal energy activates thedomain walls over some energy barrier. Arrhenius fluc-—E /kTtuations obey the relation f~ =foe- ', where f is a' speak frequency, fo is the attempt rate, F., is the activa-tion energy, and k is Boltzmann's constant.

Ferromagnetic Ni& Mn„has a large SRA, ' ' up to10% in bulk disordered material at 4 K and low Mn con-centrations. When a domain switches between twodifferent orientations, the fractional step size in the resis-tance b,R /R due to the SRA is'

«&p VD(cos Oi cos O2),

p V,

where hp/p is the SRA, O, and Oz are the two angles ofthe magnetic moment of the domain with respect to thecurrent density, and VD and V, are the volumes of thedomain and the sample, respectively.

Besides VD, it is helpful to find pD, the magnetic mo-ment of the fluctuating region. p~ can be inferred fromthe effect of H on the energy asymmetry between the twostates. This energy difference appears in the logarithm of

0163-1829/95/51(17)/11463(6)/$06. 00 11 463 1995 The American Physical Society

11 464 C. D. KEENER AND M. B.WEISSMAN 51

the Boltzmann factor between the two states, an equilibri-um property which does not depend on the details of thekinetic pathway between the states, which afFect only E, .The Boltzmann factor is simply the ratio t„/t&, of theaverage times spent in the two states. If we apply Halong the direction defined as z, we then find

6, jn(t„/t~) =(bpD), H/kT,where (b,pD), is the difference between the magneticmoment's z components in the two states, and k isBoltzmann's constant. We define H, to be the field thatcauses t„/t& to change by a factor of e. Then

E2

— 2

0

.e~I I I I l I I I I I I t I I I I I I ~ I I I I I I I I I 0 I I I I

5 0 1 0 0 150 200 250 300 350T (K)

I& ~ II

I I I II

I I I ~I

I I I II

~ I I II

~ I j II

I I I II

.~ --~ '4 - - ~ .4

) kTPD —

2H

For a single FM domain,

PD ) kTp, n 2H, p, n

(4)FIG. 1. M vs T of sample 1 . The onset of strong coercivity

occurred at about 130 K. Below 150 K, the upper curve showsdata obtained while cooling in 20 Oe. The lower curve corre-sponds to cooling in near zero field, although there was a slightnegative offset from zero.

where p, is the average magnetic moment per atomwithin the domain and n is the atomic density. If p, andthe SRA are known at the temperature where the depen-dence of t„/t„on H and hR/R are measured, Eqs. (2)and (5) provide independent estimates of the domainvolume.

XiMn is a material known to have a ferromagnetic re-gime and a reentrant-spin-glass regime, although the ex-act nature of these regimes, including whether or notthey are true phases, is disputed. ' ' Previous electricalnoise measurements in relatively large samples[(6X1280X0.11) pm ] found large, discrete switchingobjects which were identified as clusters of ferromagneticdomains, since the sizes obtained from the resistanceswitches greatly exceeded the net moments consistentwith the relatively weak field dependences. ' In this pa-per we examine the noise in smaller samples, in which in-dividual domains should be detectable.

III. SAMPLES AND TECHNIQUES

A Nlo 77Mno p3 film was deposited onto a sapphire sub-strate by dual electron beam evaporation at 3 X 10 Torrbase pressure, at a rate near 3 A/sec.

The Mn concentration was determined by the induc-tively coupled argon-ion plasma (ICP) technique to be 23at. fo. The two samples were fabricated from the samefilm at difFerent locations on the substrate. A sample fordc magnetization (M) measurements, 1,was grown on aglass substrate. M vs T, shown in Fig. 1, and hysteresisloops were measured. The Curie temperature T& of sam-ple 1 was 285+10 K, in agreement with the T& ob-tained from ac susceptibility. In samples 1 and 2, the T&may have been higher due to annealing at 90 C during fa-brication„which could have increased the amount ofatomic short-range ordering. From ac-susceptibility mea-surements, the crossover into the ferro-spin-glass regimeoccurred at about 65 K, below the temperature range ofthe FM domain noise presented here.

Although the SRA was not measured in these samples,similar filrns' had a low-temperature SRA of 0.5% below

—100 K. Ap/p decreased gradually to near zero as theT& was approached. p„ the average magnetic momentper atom within a FM domain, was estimated from thesaturation magnetization of sample 1 to be 0.2pz at 5K. From earlier results, the temperature dependence ofp, and Ap/p can be estimated, yielding the domainvolume at higher temperatures to within a factor of 3,

Because the T& was higher than it would be in atorni-cally disordered bulk samples,

' we conclude that thesamples were partially short-range ordered (SRO). Itshould be noted that such SRO films are truly ferromag-netic below T& and above the reentrant spin-glass tem-perature. In Ref. 10, as well as in the data of this paper,magnetoresistive jumps corresponded to magnetically or-dered volumes that were too large to be consistent withsuperparamagnetism in this temperature regime. Also,hysteresis loops and noise results of similar samples' sug-gested a long-range antiferromagnetic interaction amongferromagnetic domains at low T, implying a domainstructure similar to that of bulk disordered samples. ' Fi-nally, in a difFerent sample, with T~ well above 300 K,domains were directly observed by scanning electron mi-croscopy. '

Samples were fabricated into five-point resistancebridges by standard optical lithography. The film wasetched by ion milling in an Ar atmosphere of 3X10Torr. To rninirnize contact resistance, Ag was evaporat-ed onto the contact areas, and then 36 gauge Cu wire wasconnected by using pressed In pads. The samples were 32pm long and 55 nm thick. Sample 1 was 1.8 pm wide, andsample 2 was 2.0 pm wide. These had 300 times smallervolume than the samples of Ref. 10.

Most of the noise measurements were made with five-probe balanced-bridge samples using a 2.41-kHz ac probecurrent and a homemade lock-in amplifier, limiting theupper measurement frequency to about 1 kHz. The ap-plied voltages were close to 100-mV rms. The actual volt- .

age fluctuation time traces shown were filtered at about 5Hz. All spectral data used a standard antialias filter be-fore the analog-to-digital conversion. The Barkhausen

INDIVIDUAL-DOMAIN-WALL MOTION IN Nio 77Mno 23. . . 11 465

experiments were made with a four-probe arrangementand dc current giving about 200 mV. The Barkhausenvoltages were sampled at 10 kHz, after standard antialiasfiltering.

IV. RESULTS

0 ~ . - - - ~

0 50 100 150Time (sec)

~ I I I I IIII I I I I I IIII I I I I ~ I k ~I

200 250

A. Equilibrium noise 10

84

CI

4

I

—(b)I

0 50 1O0Time (sec)

3] 0 & i »I

i I »I

i i & sI

& r

I

150 200

102

~ 10

1 p0

10e,

'

1 p ~ »» I » i i I & s & & I & &» I »» I «» I »» I

-1.5 — 1 -0.5 0 0.5 1 1.5 2 2.5

H(oe)T (K)

FIG. 2. (a) Time series of sample 1 at H=1.5 Oe andT=307.8 K. (b) H = —0.7 Oe. (c) Ratio of the times spent inthe "up" and "down" states, t„/td vs H. H was 6rst increasedfrom 0 to 2.4 Oe and then decreased to —1.4 Oe. (a) and (b) areportions of time series from which two points on (c) are calcu-lated.

Several easily resolved two-state switchers were foundin the time-dependent resistance. Typical time traces areshown in Figs. 2(a), 2(b), and 3(a). For one of these, wemeasured the magnetic-field dependence of t„ /td, shownin Fig. 2(c). The strong field dependence clearly illustratesthat the switcher is magnetic in nature. Figure 2(c) showssome weak hysteresis, possibly from the environment ofthe particular magnetic Auctuator, but it is not largeenough to prevent an accurate fit of t„/td to an exponen-tial dependence on H. Equation (3) gives (b,)rI,D ),=(1.0+0. 1)X 10 prI.

Between 300 and 311 K, the magnitude of theswitcher's step size, AR/R, varied by less than about12%. Because the SRA must vanish as T approaches Tc,the lack of a decrease in hR /R means that Tc for thisparticular region was at least about 400 K. Then theSRA would be at least -0.2 of its low-temperature (5 K)value, and p, was at least -0.4XILr, (5 K). Assumingthat this Auctuator is all or part of a FM domain whichswitches between two states, Eq. (2) then givesVD=1.5X10 ' cm, and Eq. (5) gives VD=3X10

cm . Given that our estimates of p, and the SRA are un-

certain to about a factor of 2, the two estimates are in sa-tisfactory agreement. Thus, in all probability this Auc-

tuator consisted of a single domain or a portion of a

10I I I I I I III ~ i i s islil

10 10f(Hz)

10

j I II

I I c II

I I I II

~ I I

10

II

I ~ I I

c) =

100

0

3.5I I I I l I I I I I I I I I I I I I I I I I I~ l 'I t I

36 37 38 39 40 411000 I (K )

4.2

FIG. 3. (a) Time series of sample 1 at 240.5 K. (b) Spectruma vs f at 252 K. (c) Arrhenius plot of the peak frequency vs

1/T.

domain near a domain wall. This result contrasts sharplywith the larger Auctuators seen in bigger samples, whichrequired a picture of clusters of antiferromagneticallyaligned domains.

The temperature dependence of this Auctuator'scharacteristic rate could be tracked only over a narrowrange of T, over which significant history-dependenteffects were found. If the temperature dependence wereto be fit with an Arrhenius form, it would have an activa-tion energy of about ( l. 7+0.3) eV and an attempt rate foof roughly 10 +— Hz, but these numbers may be distortedby the dependence of the rate on thermal history. Clear-ly, since the attempt rate is not physically plausible, theactivation energy is somewhat temperature dependent ifnot also dependent on thermal history.

Another switcher was seen in sample 1, somewhatsmaller than the one described above. This one was ob-served at four temperatures between 240 and 270 K. Aspectrum and a time series are plotted in Figs. 3(a) and3(b). Here, an Arrhenius plot, Fig. 3(c), was obtainedfrom the frequencies of the peak, yielding an E, of0.9+0.2 eV and an fo of 1X 10' Hz, also slightly toohigh to be consistent with a temperature-independentbarrier.

In addition to the easily resolved individual switchers,apparently given by individual domain-wall motion, theequilibrium noise showed many bumps as a function ofboth f and T, as illustrated in Fig. 4(a). The features in

11 466 C. D. KEENER AND M. B.WEISSMAN

-20 I I ~ I I I I Iaj

the T dependence generally matched well with deviationsof the slope from y = 1, as expected for thermally activat-ed processes. The temperatures at which features ap-peared in a given frequency range varied from sample tosample and to some extent from one cooldown to thenext. (There was a similar variation of spectral featuresin 300-times-larger samples. '

) At 75 K the noise was

slightly non-Gaussian, with a variance in successive mea-sures of the power in an octave around 0.5 Hz being1.10+0.02 of the Gaussian value. '

The peak in a vs T at 40 K behaved difFerently fromthe other peaks, in that the temperature and frequencydependence were nearly identical in samples 1 and 2, andthe noise was Gaussian to within experimental uncertain-ty. Figures 4(b) —4(d) show that, as the sample was cooledthrough -70 K, spectra became nearly power law, i.e.,featureless.

1 {)10

-sL

a

~ I I I ~ ~ I II I I I I I I

1010

1-30 I I ~ l I I I I I I I I

- --- ~ --- 89 6 K77.0 K70.1 K

----~—- slope

a a ~ a a I

100T(K)

~ ~ I I I ~ I II

~ I s ~ I ~ ~ lI I I I I I I

Il

~ ~ ~ ~

~ ~ ~ I~, sl lI ~ I~ ~ I~ I Ili~ ~ I~ I ~ ~

~ ~ ~

~ ~ I~ ~

~i

~ ~ ~

c)

1.0

B. Barkhausen noise

Since some domain walls jump via thermal activation,it is reasonable to expect the appearance of Barkhausenjumps when domain walls are forced to move by an ap-plied magnetic field. We observed such jumps when asinusoidal field (at frequency fH and amplitude H„) wasapplied to sample 2. At some other temperatures, spikesin the derivative were absent, and domain-wall motion ei-ther occurred in smaller jumps or in a nearly continuousmotion.

Figure 5(a) shows the time-dependent part of the rmssample voltage (measured with an applied ac current toavoid direct electrical pickup). Most of the response wasat frequency fH or 2fH. After subtracting the fundamen-tal from the top trace, the voltage scale could be expand-ed to reveal jumps [Fig. 5(b)] or higher harmonics of fH.As seen in Figs. 5(c)—5(f), the spikes in the time deriva-tive (from the difference of successive points) clearly illus-trate sharp jumps in resistance. Using Eq. (2), the volumeaff'ected by the largest jumps of magnetization was

0

/0

~ 1 ~ ~ 1 ~ l~ lj I ~ 1 I

p -o-~/

/0--0 -pP 60.0 K

55.0 K49.8 K45.0 K39.8 K

0

42

Qgj„as~Jm& J LJI~ill J sascjaiiklsisa lakalliltllas Isl TJas ~EJ gJ ILhhl IsdaaLllw~ J at Lalsr ls+ J~ata. as sl »l I »l g0 ~ ~Ts ~ ~P/liPr-r««'FP 'Tlr«srsal+(PT~~ ~a [ll~gqPI~) IQP7&T-(tat% I+ftl1- «~a' ~r IT(~~2—

- 0 0 P--0 -v3- -0- - O0 0 IJlal la IIJIJ ll»ssaJ J g JgaJJsla lsaJ Jules~ all l ass»l saI g Jal~llgsllsss as' Jasti~(sla ~Q)~~.~~,k~ai/Tssssl IIIJQQ~~r [r ~pqlprrv~gr7I-1~ ~)~I T «vpmr ra~le~ /l ta Tq~lr~wllr ~"~elm~2

34.5 K.

30.1 K

25.0 K20.0 K.

1 5.0 K6.6 K

(e):0 )~ggg~IILlgJJIJ L~4IJI ~ silfMa~d LlaLII a Jla»~~tigjJ +alsal4a41(hasJlssallul 1LilaatlLs ~L~2

TIIT r-~ rrssqi ~~V-T-T» T»srm - rTIt 'lm IIII r Tr I T r»TIIIIr ITTTarars»S

ialtl 4~lJk I also J cLIr JJ ILlhl~ aa»JassJJI&ssJI M I lUIILI S IJLJI l»»isLllLJU l» ISJ Ijlal hl ~J)su kL adaJul flu j~ JJ»IJ a ILI ll }Llf raassara ls

ItsqmT ~lt q pirl~i laqalm Ir»~l~pn~~&lllrrrr r~~pa'[r~~~p~r y~ T r q

] O I s ~ s l

0. 1

I I Sl

f(Hz)0 O. i 0.2 0.3

Time {sec)0.4 0»5

FICx. 4. (a) a vs T and y vs T of sample 1 in the range0.41—10.6 Hz. (b) —(d) Spectra of sample 2 at several differenttemperatures. Note the large spectral features at 89.6 and 77.0K. Spectra became approximately power law below 70 K. Thetemperatures given do not include corrections for Joule heatingof the sample, which are significant only at 6.6 K.

FICx. 5. Barkhausen noise of sample 2 with T = 88 K,H„=53 Oe, fH =9.5 Hz, average rms sample voltage of 0.234V, and time between successive points of 10 sec. (a) Time-dependent part of sample voltage, V(t). (b) V(t) with the fun-damental subtracted away. (c)—(f) One long time trace of thedifference between successive points, proportional to the timederivative, of data plotted in (b).

INDIVIDUAL-DOMAIN-WALL MOTION IN Nio 77Mno 23. . . 11 467

around 1X10 ' cm, somewhat larger than the thermal-ly activated switcher at 310 K.

These spikes reproduced on every cycle, although withslightly fluctuating sizes. The upward and downwardspikes under particular field and temperature conditionswere not symmetrical, although there is no reason to ex-pect any systematic asymmetry. Just before and aftereach upward spike in Fig.5, the noise increased.

The size of the jumps increased with increasing H„,i.e., with increasing dH/dt As.mall H„(5 Oe) producedno perceptible jumps larger than background noise. Atother combinations of T and H„, there were no obviousspikes in the time derivative of the magnetoresistance.Sometimes there were no discontinuities in R at all, butonly harmonics off~.

V. DISCUSSION

The most obvious explanation of the many features inthe equilibrium noise above 70 K is spontaneous thermalAuctuation of ferromagnetic domains. Thermally activat-ed domain-wall motion was previously proposed' basedon the overall suppression of such noise in larger samplesin fields of about 1 kOe. Here, we have explicitly shownthat this noise includes discrete two-state Auctuators. Byshowing the very strong dependence of its duty cycle onthe applied magnetic field, we found that one such Auc-tuator must come from a small ferromagnetic region,probably the consequence of the motion of a single fer-romagnetic domain wa11. This Auctuating region has avolume of about 2X10 ' cm . From the size of spectralfeatures, following the techniques of Ref. 10, we estimatethat the volume of typical fluctuating domain regions atT & 70 K was somewhat smaller, about 3 X 10 ' cm . Inlarger samples Auctuating domain clusters with volumesup to 4X10 ' cm (Ref. 10) have been found.

Although the kinetics of these domain motions are

thermally activated, the attempt rates inferred by assum-ing fixed barriers were unphysically high. Most likely thebarriers to rotation or domain-wall movement shrankover a considerable range of T as Tc was approached.

The loss of the spectral features in these small samplesbelow 70 K is consistent with FM domain motion freez-ing below the ferro-spin-glass crossover temperature.Such FM domain stiffening or slowing down was alsoseen in neutron scattering, ' Lorentz microscopy, fer-romagnetic resonance, Barkhausen noise, and the on-set of large remanence in dc magnetization data. ' Fur-thermore, it probably was reAected in low-field ac mag-netic susceptibility measurements, where the FMresponse nearly vanished below the reentrant spin-glasstemperature. ' '

In their repeatability as H was cycled, the Barkhausenjumps were similar to other data showing partially repro-ducible jumps in dM/dt of multidomain samples whilemeasuring hysteresis loops with an ac field. Such a re-sult is obviously consistent with standard pictures basedon individual domain walls moving in a fixed environ-ment of defects, e.g. , Ref. 5. Since the domains see thesame potential barriers, or defects, each time the field iscycled, jumps tend to occur repeatedly at the same partof the cycle. However, the enhanced noise which bothprecedes and follows some of the large jumps suggests anavalanchlike process, perhaps having something in com-mon with the phenomenon dubbed self-organization,without any criticality.

ACKNOWLEDGMENTS

This work was supported by the National ScienceFoundation through Grant No. DMR 93-05763 and,through the facilities and staff of the Materials ResearchLaboratory, including R. S. Averback's film depositionapparatus, Grant No. DMR 89-20538.

*Present address: Department of Physics, Ohio State Universi-

ty, Columbus, Ohio 43210.V. S. Speriosu, J. D. A. Herman, I. L. Sanders, and T. Yogi,

IBM J. Res. Develop. 34, 884 (1990).2Anupam Garg, Phys. Rev. Lett. 71, 4249 (1993); 70, 1541

(1993).H. Barkhausen, Physik. Z. 20, 401 (1919).

4D. A. Bell, Noise and the Solid State (Wiley, New York, 1985),pp. 61—82.

5B. Alessandro, C. Beatrice, G. Bertotti, and A. Montorsi, J.Appl. Phys. 68, 2901 (1990);68, 2908 (1990).

P. J. Cote and L. V. Meisel, Phys. Rev. Lett. 67, 1334 (1991).J. Brophy, in Fluctuation Phenomena in Solids, edited by R. E.

Burgess (Academic, New York, 1965), pp. 1 —35.8J. W. Eberhard and P. M. Horn, Phys. Rev. B 18, 6681 (1978).B. M. Lebed, I. I. Marchik, V. A. Noskin, and V. G. Shirko,

Fiz. Tverd. Tela. 24, 1997 (182) [Sov. Phys. Sol. State 24,1140 (1983)];B. M. Lebed, I. I. Marchik, V. A. Noskin, V. Cs.

Shirko, and Y. S. Erov, ibid 27, 757 (1985). [ibid 27, 466.(1985)].C. D. Keener and M. B. Weissman, Phys. Rev. B 49, 3944(1994).

i H. T. Hardner, M. B. Weissman, M. B. Salamon, and S. S. P.Parkin, Phys. Rev. B 48, 16 156 (1993);H. T. Hardner, S. S. P.Parkin, M. B. Weissman, M. B. Salamon, and E. Kita, J.Appl. Phys. 75, 6531 (1994).G. B.Alers, Ph.D. thesis, University of Illinois, 1991.M. Lederman, S. Schultz, and M. Ozaki, Phys. Rev. Lett. 73,1986 (1994).

M. B.Weissman, Rev. Mod. Phys. 60, 537 (1988).~~I. A. Campbell and A. Fert, in Ferromagnetic Materials: 3

Handbook on the Properties of Magnetically Ordered Substances, edited by E. P. Wohlfarth (North-Holland, NewYork, 1980), Vol. 3, pp. 748 —805.

60. Jaoul, I. A. Campbell, and A. Fert, J. Magn. Magn. Mater.5, 23 (1977); I. A. Campbell, A. Fert, and O. Jaoul, J. Phys. C1, S95 (1970).H. Kunkel, R. M. Roshko, W. Ruan, and G. Williams, J.Appl. Phys. 69, 5060 (1991);Philos. Mag. 64, 153 (1991).R. M.Roshko and W. Ruan, J. Phys. I (France) 1, 1809 (1991);J. Magn. Magn. Mater. 104-107, 1613 (1992).

' I. Mirebeau, S. Itoh, S. Mitsuda, T. Watanabe, Y. Endoh, M.Hennion, and P. Calmettes, Phys. Rev. B 44, 5120 (1991).

2 See the SRA vs T, Fig. 4(a) of Ref. 10, for the approximate

11 468 C. D. KEENER AND M. B.%'EISSMAN

form of (bp/p)(T). SRA in I'eZr [H. Ma, Z. Wang, H. P.Kunkel, G. Williams, and D. H. Ryan, J. Appl. Phys. 67,5964 (1990)] and several Ni alloys [Yu. V. Vasilev, Phys.Status Solidi 38, 479 (1970)] are similar. Temperature depen-dence of p, is discussed by Abdul-Razzaq et al. in Ref. 21.To within less than a factor of 2, we can approximate it to beproportional to M(T) except for T very near Tc.J. S. Kouvel, W. Abdul-Razzaq, and K. Ziq, Phys. Rev. B 35,1768 (1987); W. Abdul-Razzaq and J. S. Kouvel, ibid. 35,1764 (1987); J. S. Kouvel and W. Abdul-Razzaq, J. Magn.Magn. Mater. 53, 139 {1985).P. Dutta, P. Dimon, and P. M. Horn, Phys. Rev. Lett. 43, 646(1979); P. Dutta and P. M. Horn, Rev. Mod. Phys. 53, 497

(1981).S. Senoussi, S. Hadjoudj, and R. Fourmeaux, Phys. Rev. Lett.6l, 1013 (1988).B. Aktas and Y. Oner, J. Phys. Condens. Matter 5, 5443(1993); B. Aktas, Y. Oner, and H. Z. Durusoy, J. Magn.Magn. Mater. 119, 339 (1993).

25R. L. Sommer, J. E. Schmidt, and A. A.Gomes, J. Magn.Magn. Mater. 103, 2S (1992);J. Appl. Phys. 73, 5497 (1993).J. Horvat, E. Babic, and Z. Marohnic, J. Magn. Magn. Mater.86, L1 (1990).

27P. Bak, C. Tang, and K. Wiesenfeld, Phys. Rev. Lett. 59, 381(1987);Phys. Rev. A 38, 364 (1988).