Hysteresis and avalanches in disordered systems

qThis work has received "nancial support from the CICyT(Spain), Project MAT98-0315 and from the CIRIT (Catalonia),Project 1998SGR48.

*Corresponding author. Fax: #39-34-934-021-174.E-mail address: [email protected] (E. Vives).

Journal of Magnetism and Magnetic Materials 221 (2000) 164}171

Hysteresis and avalanches in disordered systemsq

Eduard Vives*, Antoni Planes

Departament d+Estructura i Constituents de la Mate% ria, Facultat de Fn&sica, Universitat de Barcelona, Diagonal 647, 08028 Barcelona,Catalonia, Spain

Abstract

Rate-independent hysteresis is studied in magnetic systems driven by an external "eld for which the in#uence ofthermal #uctuations is negligible. In such systems, the hysteresis cycles are not continuous, but rather are composed ofa sequence of magnetisation jumps or avalanches between metastable states; the so-called Barkhausen noise. The studyof the statistical distribution of such avalanches provides an alternative description to the more common procedureof measuring properties of the loop shape. We focus on four di!erent zero-temperature 3d lattice models: the random"eld Ising model, the random bond Ising model, the site-diluted Ising model and the random anisotropy Ising model.By de"ning appropriate local dynamics, we have studied the metastable evolution by numerical simulations. We analysethe avalanche size distribution as a function of the degree of quenched disorder in these systems. For speci"c amountsof disorder, the distributions exhibit critical behaviour that can be characterised by universal exponents. ( 2000Published by Elsevier Science B.V. All rights reserved.

Keywords: Lattice models; Hysteresis; Avalanches

1. Introduction

Hysteresis is an out-of-equilibrium phenomenonoccurring in "eld-driven systems which exhibitmulti-valued responses [1]. Here we are interestedin the so-called rate-independent hysteresis, whichis observed at very slow driving rates, that is, closeto the quasistatic limit. In such circumstances, thesystem exhibits a dependence on the history but notan explicit time dependence. Strictly speaking in thequasistatic limit, at "nite temperature, hysteresis

will not occur. However, if thermal #uctuations areweak and the "eld driving rates are not in"nitelysmall, hysteresis does, in fact, occur. This is fa-voured by the existence of disorder in the system[2].

Consider the case of solid magnetic materials ofinterest here. Disorder may take various di!erentforms; among others, structural defects such asvacancies or dislocations, polycrystallinity andnon-stoichiometry are important since they can actas possible pinning sources of the magnetic do-mains. Actually, this is because they induce local#uctuations of the exchange and anisotropy con-stants or e!ective local magnetic "elds. Suchcomplex systems can be viewed as described bya multidimensional energy landscape containingmany local minima which correspond to meta-stable states. These states are separated by energy

0304-8853/00/$ - see front matter ( 2000 Published by Elsevier Science B.V. All rights reserved.PII: S 0 3 0 4 - 8 8 5 3 ( 0 0 ) 0 0 3 8 0 - 2

barriers arising from pinning e!ects. Magnetisationchanges can thus take place only if such barrierscan be overcome. In many cases of practical inter-est, these barriers are high compared with the ther-mal energy (k

B¹), and the only way to force the

system to evolve is by modifying the external "eld,which in the landscape picture means to tilt theenergy pro"le. The evolution of the system pro-ceeds as a sequence of jumps or avalanchesconnecting metastable states. The characteristicBarkhausen noise is the manifestation of theseavalanche events [3]. In the strict rate-independentlimit, the sequence of energy minima that the sys-tem passes through is the only important featureand time does not play any role. In practice, thisrequires that the "eld remains constant during theoccurrence of the avalanches.

Preisach modelling is considered a commonmethod for the study of hysteresis in magnetism[4]. It is based on the decomposition of the systeminto independent elementary hysteresis units. Itallows reproduction of many features of hysteresisloops. Nevertheless, such an approach lacks a phys-ical basis and is not adequate for the study ofcollective behaviour. A more physical description isprovided by micromagnetics. This method con-siders the system free-energy as the starting pointfor hysteresis studies [5]. This approach rests ona classical and continuum description of matterand includes the long-range magnetostatic e!ect.Nevertheless, this framework is not so suitable forthe inclusion of disorder and, for instance, the studyof Barkhausen signals.

Recently, a di!erent approach has been proposedfor the analysis of hysteresis. It is based on spin-lattice models with di!erent kinds of disorder suchas random "elds [6,7], random-bonds [8,9], va-cancies [10}12], random anisotropy, and a fewothers. Usually, two-state spins (Ising spins) areconsidered, but also more complex situations havebeen studied [13]. Such a discrete description of thematerials allows a simpler numerical analysis of thee!ect of the inhomogeneous character of the modelparameters such as the exchange and anisotropyconstants. The inclusion of the long-range mag-netostatic forces is also possible within this frame-work [11,14,15]. Simulation of rate-independenthysteresis in such lattice models is performed at

zero temperature (absence of thermal #uctuations)by means of algorithms corresponding to local en-ergy relaxation.

It might seem that such models are thought todescribe the bulk of the materials at a microscopiclevel, with the characteristic lattice spacing beingrelated to the interatomic distance. However, theycan also be used, for the description of the systemsat a mesoscopic scale. Actually, this point of view isimplicitly adopted in many studies of hysteresisbased on lattice models. From this point ofview, the lattice spacing should be related to thecharacteristic size of the uniformly magnetised do-mains. This interpretation is especially adequatenear critical points where physical properties arescale free. Thus such models are suitable for thedescription of hysteresis in di!erent magnetic ma-terials ranging from pure bulk crystals to granularmaterials.

In this paper we compare the properties of hys-teresis in four di!erent lattice models. All of theminclude a (basically ferromagnetic) exchange energyterm, interaction with an external "eld and di!erenttypes of disorder: random "elds, random bonds,dilution and random anisotropy, which are ana-lysed separately. We focus our attention on thestatistical properties of Barkhausen noise and theexistence of scale-free critical phenomena which arefound for speci"c amounts of disorder.

2. Models

Modelling of hysteresis phenomena in extendedsystems requires both a description of the systemenergy (Hamiltonian) and the speci"cation of therules determining metastable local dynamics.

2.1. Hamiltonians

The studied models are all based on the 3d Isingmodel with di!erent types of quenched disorder.This corresponds to the case of strong anisotropymodels for magnets. Spin variables S

itaking

values $1 are de"ned on the N"¸]¸]¸ sitesof a cubic lattice. The di!erent Hamiltoniansstudied are

E. Vives, A. Planes / Journal of Magnetism and Magnetic Materials 221 (2000) 164}171 165

2.1.1. Random xeld Ising model (RFIM)

H"!

/./.+i,j

SiSj!B

N+i

Si!

N+i

Sihi, (1)

where the "rst term, which extends to all nearest-neighbour (n.n.) pairs on the lattice stands for thebasic Ising ferromagnetic exchange interaction, thesecond term takes into account the interaction withthe driving external "eld, and the last term repres-ents the interaction with local quenched random"elds. We assume that such random "elds h

iare

Gaussian distributed with zero mean and standarddeviation p. Note that in this Hamiltonian we haveconsidered the exchange energy J as the unit ofenergy and that the external "eld and the random"elds are measured in units of J/k where k is themagnetic moment at each site (which is supposed tobe the same for all the atoms). A similar procedurefor obtaining dimensionless Hamiltonians will beperformed for the models below.

2.1.2. Random bond Ising model (RBIM)

H"!

/./.+i,j

SiSj!B

N+i

Si!

/./.+i,j

JijSiSj, (2)

where the "rst two terms are the same as in theprevious model, but the disorder is now introducedby adding noise to the nearest-neigbour exchangeinteractions. The values of J

ijare also Gaussian

distributed with zero mean and standard deviationp. Note that for large values of p some bonds maybe antiferromagnetic and that the Hamiltonian issymmetric under reversal of all the spins.

2.1.3. Site diluted Ising model (SDIM)

H"!

w+k/1

k~/.+i,j

JkeiSiejSj!B

N+i

eiSi. (3)

In this case, the model contains exchange interac-tions J

kup to the wth nearest neighbour and the

interaction with the external "eld. The variablesei

de"ne the presence or absence (ei"1, 0) of an

atom with a magnetic moment at site i. TheJkconstants are typically oscillating functions with

the neighbouring distance (for instance of theRKKY type). In this work the model parameters

have been chosen in order to reproduce data forCu}Al}Mn [12]. The exchange interactions havebeen cut at the 7th neighbour (w"7). The actualvalues of J

kused are J

1"!2.822, J

2"1 (unit of

energy), J3"0.5376, J

4"0.3978, J

5"!0.315,

J6"0.477 and J

7"!0.593. Notice that the near-

est-neighbour interaction is strongly antiferromag-netic. Dilution is obtained by randomly settingsome e

ivariables to zero. This is not done uniform-

ly but the cubic lattice is subdivided into two equalalternating FCC sublattices with N/2 sites each.The total number N

4of occupied sites (x"N

4/N is

the concentration) is randomly distributed as fol-lows: a fraction pN

4of magnetic atoms are placed

in one sublattice and the rest (1!p)N4

are placedin the other sublattice. Thus, disorder is controlledby two parameters p and x. By this method, theparameter p controls the average amount of n.n.antiferromagnetic interactions. The results present-ed in this paper correspond to a "xed value p"0.1.

2.1.4. Random anisotropy Ising model (RAIM)In this case, the local strong anisotropy axes are

randomly distributed. Besides the variables Si, we

de"ne a unit vector n(i(or equivalently two angles

0(hi(p/2 and 0(/

i(2p) at each lattice site

which account for the anisotropy axis orientation.The vectors n(

iare uniformly distributed within

a cone with maximum angle h0. The case h

0"p/2

corresponds to a completely random distributionof anisotropy axis. In the in"nite anisotropy limit,the magnetic moments should sit either parallel(S

i"1) or antiparallel (S

i"!1) to n(

i. In this case,

the Hamiltonian reads:

H"!

/./.+ij

SiSjn(in(j!

N+i

Sin(iB, (4)

where the "rst term stands for the ferromagneticexchange interaction and the second term for theexternal "eld interaction. Note that n(

iB is nothing

more than B cos(hi).

Hysteresis cycles in the above models are studiedby measuring the magnetisation of the system mas a function of B, which is de"ned as m"+S

i/N

for the RFIM, RBIM and SDIM and m"

+Sicos h

i/N for the RAIM.

166 E. Vives, A. Planes / Journal of Magnetism and Magnetic Materials 221 (2000) 164}171

Fig. 1. Hysteresis cycles corresponding to the RFIM, RBIM,SDIM and RAIM for di!erent amounts of disorder.

2.2. Relaxation dynamics

The four models presented above are studied byusing the same local relaxation dynamics. We "xa certain con"guration of the disorder and letB"#R. This guarantees that the con"gurationwith maximum magnetisation is stable. We thendecrease B and compute the change of energy *H

iassociated with the independent reversal of anyspin S

i. This energy change can always be expressed

as,

*Hi"2S

iBi, (5)

where Bi"!RH/RS

iis the total local "eld acting

on spin Si. When for a certain value of B, one of the

spins gets unstable (*Hi(0), we keep B constant

and #ip that spin. This may destabilise some neigh-bouring spins, and triggers what is called an ava-lanche. Then, in a second avalanche step, all theunstable spins are simultaneously #ipped. The pro-cedure continues with B constant until all the spinsbecome stable again. This is the end of the ava-lanche. We then proceed by decreasing B. Note thatthe "eld remains constant during the avalanchewhich, as mentioned above, is the condition forrate-independent hysteresis. By this method, we canmeasure both the size (number of spins reversed)and duration (number of steps) of each avalanche.When antiferromagnetic interactions are present inthe system (RBIM, SDIM), inverse avalanches mayoccur (increase of m with decreasing B). Moreover,for the RBIM and the SDIM, when the amount ofantiferromagnetic bonds is large or for certain con-"gurations of the disorder with very low probabil-ity, it is possible to reach never ending situations inwhich the system alternates between two states. Ifthis occurs, we stop the avalanche by an ad hocmechanism and continue decreasing the "eld.Nevertheless, such situations occur far from thevalues of disorder of interest in this work.

3. Results

Fig. 1 shows examples of the hysteresis cyclescorresponding to the four studied models. Notethat the amount of disorder increases by increasingthe parameter p in the RFIM and RBIM, by de-

creasing the concentration x in the SDIM and byincreasing the angle h

0in the RAIM. Besides other

interesting features, the common e!ect is that thehysteresis cycles become smoother when the dis-order increases. For low values of disorder, thecycle displays large jumps (avalanches), in which animportant fraction of the magnetisation is reversed,while for large amounts of disorder, the hysteresiscycle is much more continuous and displays onlytiny jumps involving few spins. Fig. 2 displays thedetails of the hysteresis cycles showing the sequenceof avalanches that, in some cases, can even beinverse.

Other interesting features of the hysteresis loopsof the above models can be described. Loops al-ways look symmetrical, although strictly speakingfor the RFIM the detailed sequence of avalanches isdi!erent in the forward and backward processes.Regarding remanence, as expected, in all cases itdecreases with increasing disorder. The magnetisa-tion saturation is independent of disorder for theRFIM and RBIM while it decreases with increasingdisorder for the SDIM (the total magnetic momentdecreases with dilution) and for the RAIM (thein"nite anisotropy condition leads to a decreaseof the maximum magnetic moment in the "elddirection when anisotropy axes become morespread). Finally, concerning coercivity, one wouldexpect that it decreases with increasing disorder.


Fig. 2. Details of the hysteresis cycles.

Nevertheless, the actual behaviour will depend onspeci"c details (symmetry, disorder characteristics,etc.) of each model. For the RBIM it is clear fromFig. 1, that the coercive "eld increases with increas-ing disorder. A systematic study of these featureswill be published in Ref. [16].

We have also analysed the existence of the returnpoint memory property. This is an interesting fea-ture that can be reproduced by Preisach models,which allow a simple characterisation of the mem-ory properties of the system. Return point memoryappears when two conditions are satis"ed: (i) "rstly,the quasistatic character of the evolution (which isful"lled for the four presented models with the localrelaxation dynamics), and (ii) secondly, and moreimportant, the fact that the dynamics preserves thepartial ordering of the metastable states. This hasbeen demonstrated for the RFIM but it is clearlyviolated by the RBIM and SDIM, as revealed bythe numerical simulations.

Fig. 3 shows the derivative of the upwardsbranch in the hysteresis loops of Fig. 1 correspond-ing to the RBIM. This corresponds to the Bar-khausen signal. One can see that for small amountsof disorder the noise exhibits one single burst, re-lated to the reversal of almost all the systems and

a few small signals. For large amounts of disorder,all the Barkhausen peaks are small. For the inter-mediate situation peaks of all sizes are present. Bystudying the full sequence of avalanches in a for-ward or backward branch of the hysteresis cyclesand by averaging over many di!erent con"gura-tions of the disorder one can determine the prob-ability p(s) of having an avalanche of size (numberof reversed spins) s for a certain amount of disorderfor the di!erent models. Fig. 4 shows such prob-abilities in a log}log plot. Note that for low valuesof the disorder, the distribution of the avalanchesexhibits a bump in the large s region correspondingto the existence of large avalanches [17]. On theother hand, for large amounts of disorder, onlysmall avalanches appear. For a certain intermedi-ate (&critical') amount of disorder the distributionbecomes power law p(s)&s~q{ indicating the ab-sence of any characteristic size. Actually, accordingto Sethna and collaborators [17], criticality occursat a certain value H

#of the driving "eld, where

p(s)&s~q. The exponent q@ corresponding to thedistribution of avalanches over the entire hysteresisloop is related to q through a simple relation in-volving other critical exponents. We will concen-trate on the determination of q@. This has two clear


Fig. 3. Barkhausen noise of the RBIM for di!erent amounts of disorder. The magnetisation jumps are measured in arbitrary units. Notethat the two largest avalanches that have been cut in the plot, reach a size *m"24330 and *m"17122 for p"0.9 and p"1.2,respectively.

Fig. 4. Log}log plot of the distribution of avalanches p(s). The thin continuous line indicates the behaviour of a power law with q"2.0.The di!erent histograms correspond to the same cases as in Fig. 1. Note that disorder increases from left to right.


1This analysis does not pretend to be an exhaustive com-parison.

advantages: (i) there is no need to "t the actualvalue of H

#and (ii) one has better statistics of

avalanches.A numerical analysis of such probability distri-

butions p(s) can be performed by "tting the func-tion:

p(s)"As~q{e~js, (6)

where j is the parameter of the exponential correc-tion and A is a normalisation constant. Sucha two-parameter "t is carried out using a maximumlikelihood method, which is independent of anybinning process or logarithmic representation. Theamount of disorder for which j"0 determines thecritical point (p

#for the RFIM and RBIM, x

#for

the SDIM and h#

for the RAIM). The obtainedvalues of these quantities are: p

#"2.4$0.1 for the

RFIM (¸"30), p#"1.21$0.05 for the RBIM

(¸"24), x#"0.35$0.05 for the SDIM (¸"32)

and h#"1.4$0.1 for the RAIM (¸"30). It

should be remarked that such values exhibit a cer-tain dependence on the system size ¸. This can becorrected by standard "nite-size scaling analysis, ashas been presented elsewhere for the RFIM andRBIM [8,13]. Nevertheless, the value of q@ at thecritical point exhibits weak dependence on ¸. Wehave found the following values: for the RFIM(obtained from scalings of systems with ¸ up to 40)q@"1.8$0.1 [13]; for the RBIM (¸ up to 40)q@"2.0$0.2 [13]; for the SDIM (¸ up to 32)q@"1.9$0.2 [18]; and for the RAIM (¸ up to 40)q@"2.05$0.05. It should be mentioned that forthe RFIM, a value of q@"2.03$0.03 has beenfound from simulations of systems with ¸ up to 320[19]. In the four cases, the exponents lay withinq@"2.0$0.1. This power-law behaviour is plottedin Fig. 4.

4. Discussion and conclusions

All the above-studied models exhibit the follow-ing common properties: (i) a basic instability whichis a "rst-order phase transition that appears whendriven by an external "eld, (ii) the existence ofquenched disorder in#uencing the characteristics ofthis phase transition and (iii) the absence of thermal#uctuations which force the systems to evolve fol-

lowing a metastable path, thus exhibiting hyster-esis. This allows the classi"cation of these modelswithin the framework of #uctuation-less "rst-orderphase transitions (FLFOPT) [13].

Despite the fact that the shape of the hysteresisloops may be di!erent in the four studied models,the critical behaviour that appears when theamount of disorder increases exhibits universalproperties as occur in standard equilibrium criticalpoints. Experimental evidence for such a criticalpoint has been found in di!erent systems:

(i) On the one hand, Barkhausen signals havebeen studied in many di!erent systems. The distri-bution of sizes and duration of the signals typicallyexhibit a power-law behaviour exponentially cor-rected. Two kinds of measurements have been per-formed: (i) distributions over the entire hysteresisloop [20}22] and (ii) distributions around a smallportion of the loop usually close to the coercive"eld [23,25]. Here we only compare with exponentsq@ obtained over the whole cycle.1 The second classof experiments, aimed to obtain q, are more sensi-tive to the exact location of H

#(which is not neces-

sarily coincident with the coercive "eld), and thusa larger spreading of the exponents is found. Valuesof q@"1.77 and 2.1 (depending on heat treatments)were reported for Ni}Fe alloys [20]. The [email protected] and 2.0 (for di!erent treatments) can beestimated using scaling relations [26] from the ex-perimental measurements of signal durations andjoint probability distributions [21,22]. More re-cently, another value q@"1.77$0.09 has beenmeasured directly from the distributions of areas ofthe Barkhausen signals [24] in nonsaturatingloops. Surprisingly, these exponents are in reason-ably good agreement with the universal criticalexponent obtained from the simulations in 3d sys-tems. According to the above models such &critical'behaviour must appear only for a certain amountof disorder. Experimentally, this does not seem tobe the case, and the amount of disorder is usuallyan uncontrolled property of the studied samples.Why then is criticality found? In the light of thepresent studies, two answers are possible. Firstly, it


has been shown [19] that critical regions in themodels extend far from the critical point, and thusit would not be so unprobable to "nd samplesbehaving close to criticality (one should take intoaccount the fact that the measured amplitudesare limited to a range, which extends to no morethan 3 decades). A second explanation wouldbe related to the existence of an internalmechanism for relaxation (or rearrangement) of thedisorder in the system. Such a mechanism wouldtake place in a much larger time scale but would,after cycling or aging, drive the experimentalsystems towards the critical point. This mechanismwould correspond to a self-organisation of thesystem disorder.

(ii) On the other hand, for Cu}Al}Mn alloys, theSDIM presented above has been used for the studyof the hysteresis cycles [12]. The model predictsa transition from a smooth to sharp hysteresis loopwith decreasing Mn concentration. This providesan explanation for the transition from the spin-glass to the glassy ferromagnetic behaviour foundat very low temperatures [12] when thermal #uctu-ations are irrelevant. The predicted critical concen-tration of Mn agrees reasonably well with thatfound experimentally.

Finally, we want to point out that such FLFOPTlattice models also allow the study of systems withlow dimensionality and the spatial morphology ofthe magnetic domains during the magnetisationprocess. Some results have already been publishedfor the diluted RFIM [27]. Moreover, in order toreproduce the observed microstructure in magneticmaterials, more realistic models must be for-mulated. Such models must be de"ned on latticeswith the actual symmetry of the systems and in-clude di!erent kinds of disorder and long-rangemagnetostatic forces, which play a very relevantrole. The long-range interactions could modify thecritical behaviour of these systems [28]. More stud-ies along these lines are needed in order to clarify

the advantages of lattice models for the study ofhysteresis.

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Hysteresis and avalanches in disordered systems