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Why do hysteresis loops stabilise in a few runs?
Volume 121, number 3
PHYSICS LETTERS A
13 April 1987
WHY DO HYSTERESIS LOOPS STABILISE IN A FEW RUNS? J.L. PORTESEIL Laboratoire Louis Née!, CNRS—USTMG, 38042 Grenoble Cedex, France Received 4 December 1986; revised manuscript received 15 January 1987; accepted for publication 9 February 1987
Hysteresis loops of ferromagnets are usually stabilised after four or five field cycles. A tentative explanation is given in the spirit of learning models of spin glasses. It is suggested that loops approach their limit like 2—”, due to the binary nature of magnetic elements (Preisach grains).
1. Stabilisation of hysteresis loops: experiment Successive hysteresis loops of a cyclically magnetised ferromagnet are not exactly reproducible. It was observed that, after demagnetising a steel sample, the first four or five loops recorded between upper and lower fields HA and HB do not feature the same magnetisation amplitude AM= MA MB [1]. The changes in Ls.M can amount to a few percent if the field cycles are symmetrical (HB = HA) and if HA lies in the vicinity of the coercive field. This phenomenon, often called “bascule” (tilt), was further studied from a macroscopic standpoint in polycrystalline FeSi and single-crystal FeA1 [2,3]. It was demonstrated more recently that the virgin domain structure obtained by cooling a ferromagnet from above 7’. (thermal demagnetisation) exhibits a similar behaviour [4]. However, the evolution of loops is much more conspicuous, as the amplitude tIM’ ofthe first loop can be as large as three times its assymptotic value. Loops of rank n approach their limit like exp( —fin), with fi~0.7l.In other words, the ioops are again practically stabilised after four or five runs, whichever the amplitude of field cycles, Moreover, the sequences ofirreversible (Barkhausen) magnetisation jumps change significantly during the first few loops following a demagnetisation showingthat the domain structure undergoes noticeable reorganisations [5—71. This was confirmed by direct domain observations on a FeSi polycrystal [8]. A mechanical counterpart for this phenomenon was observed by studying the acoustic emission of —
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0375-960 1/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
cyclically strained superconductors [9,10]. Such a behaviour is technically relevant, as it entails a “training” process of superconducting magnets which is practically completed after a small number ofstrain cycles. Let us make the following remark: when recording an audio signal, the magnetic tape is subjected both to the audiofrequency field H which carries the information to be recorded and to an alternating, high-frequency field H’ which helps in stabilising the corresponding magnetic configuration of fine partides in the tape. The frequency of this field is generally adjusted so that a magnetic grain experiences about five decreasing field cycles during the time it travels past the recording head [11]. The aim of this paper is to suggest a tentative explanation of why the number of runs needed to stabilise hysteresis ioops seems to be four or five as a rule, and not 1 or iO~.
2. Preisach model, interacting grains and learning in spin glass models In the classical Preisach model, a ferromagnet is pictured by an assembly offictitious, non-interacting “grains”, each of them featuring a rectangular hysteresis loop with two stable (up and down) states [12]. Indeed any microscopic hysteresis loop can be mimicked by means of a suitable distribution of grains, but in the absence of interactions Preisach’s 145
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13 April 1987
very rapidly untractable as the number of grains is increased, and statistical results on large assemblies ofgrains are difficult to obtain. The approach used in this paper is somewhat different and was entirely inspired by recent studies on learning phenomena in spin-glass models [16,17]. Consider a set of N Ising variables S, = ±1 interacting via the hamiltonian H= >JiJJIJSISJ~in which the are defined as follows: M patterns S~’~= ~ S~’~’ S~P),~ ~ —
...,
1.. U
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~
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t
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2
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Fig. 1 (a) Detail of a hysteresis loop obtained with four negatively coupled fictitious Preisach grains (analog simulation [151). (b) Sketch of a system of four real single-domain particles with dipolar couplings which would feature the same behaviour,
model cannot exhibit any loop evolution. Néel discussed the extremely simple situation in which two grains interact and showed that negative interactions could give rise to “bascule” phenomena [13]. However, loop stabilisation is completed after two runs in this oversimplified model. The analysis was further extended by Wohlfarth [141. More recently, an attempt was made to simulate an assembly of Preisach grains [15]. With 30 interacting bistable electronic circuits, it was possible to create “bascule” effects which affected the first four loops. Fig. 1 a is reproduced from ref. [15] and shows a detail of a loop obtained with four interacting electronic “grains” subjected to asymmetric cycles: HB=O—~.HA>O--~HB=O—~..... Starting from the mitial demagnetised configuration, the magnetisation jumps during the first cycle to a positive value M1, then settles down to M2
among the 2N possible patterns are “stored” by choosing the interactions J,~such that for each pattern S~all spins (or most of the spins) point parallel to the internal field. This can be achieved by choosing J~=> IS~S~ (Cooper prescription [161). We limit ourselves here t,p the simplest possible problem of storing only one pattern S, so that the S are simply given by J• = S,S. In other words, the .
.
. ~
network of interactions is constructed ad hoc so that two spins which are parallel (anti-parallel) in the pattern to be stored are connected by a positive (negative) interaction. S is obviously the ground state of the hamiltonian. In the next step, an initial set of S,= ±1 is selected at random, then the system is allowed to relax toward the stored pattern. This is achieved by the simple usual algorithm which consists of testing all spins sequentially. Each spin S, remains unchanged if S~1,~,J~S>0 (which means S, parallel to the local field), and is reversed otherwise. In the special case ~ ~f~ = 0, the sign ofS, is selected at random with equal probabilities. The algorithm is then repeated until retrieval of the stored pattern. The relaxation toward a given pattern is conveneniently represented by defining the overlap of two states k and/as q~~=N’ S~k) 5~1).The (Hamming) distance between the two states is dk,= (1 —q,~)/2; it vanishes when the two states are identical. ~,
3. Assumptions of the model and their physical meanings The above formal description is based of interacting binary elements which evolve as an algorithm is applied. In this paper, we attempt to apply such concepts to the problem of ferromagnetic hysteresis loops.
Volume 121, number 3
PHYSICS LETTERS A
The first question to be answered pertains to the actual nature of possible binary elements, or Preisach grains, in a ferromagnet. The answer is rather straightforward in the case of single-domain particles, as each of them features a real rectangular hysteresis loop. It is widely accepted that Preisach’s model, implemented with thermal activation con cepts and computer studies of partical interactions, affords a realistic description of magnetisation processes in particulate media. The situation is far less clear in bulk materials where magnetisation processes take place by displacements of domain walls. Irreversibility arises from pinning of walls by obstades of various types, and the magnetisation flips irreversibly in a finite volume as a wall jumps past an energy barrier. It was demonstrated [181 that a double energy wall is the physical counterpart for a fictitious Preisach grain (this concept was also used in the study of spin glasses [19]). Unfortunately, the decomposition of the random potential in which a wall moves into double wells is not unique, so that at first sight the grain—well equivalence is only statistically valid, and no real existence can be attributed to Preisach grains. However, careful examination of hysteresis loops in samples with small numbers ofbig domains suggest that reproducible events can be identified in sequences ofjumps (see for instance ref. [20]). This is rather surprising at first sight as the irreversible motion of a domain wall involves great numbers of Preisach grains. Let us assume for instance that two neighbouring wells of the potential energy of a wall are typically separated by the wall thickness, say 100 nm. In the experiments, reported in ref. [20], the free path of a wall can be as big as 1 mm, so that the movements of a wall would have to be pictured by 1 0~Preisach grains. In fact, a wide majority of these grains can be disregarded when applying given field cycles (HA, BB). On the other hand, the strong obstacles which would be overcome by a field H such that I HI> sup (I HA I~IHB I) are never crossed during these cycles, and the corresponding Preisach grains remain in their initial state. On the other hand, the very numerous weak obstades are always overcome, and the corresponding small jumps cannot be separated because they are embedded in bigger jumps. This behaviour is illustrated by fig. 2. Finally, one is left with a small number ofobservable grains. It must be stressed that this
13 April 1987
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Fig. 2. Decomposing the potential energy curve V(x) in which a wall moves into Preisach grains (schematical). (a) The slope of obstacle 8 (1) is assumed to be too strong for the upper field HA (lower field HB) to overcome it. On the other hand, obstacles 4 and 6 (3 and 4) are weak and embedded in the irreversible ascending jump 2—.8 (descending jump 7=1). (b) In the cornplex corresponding hysteresis loop, thedotted paths can be ignored and the phenomena can be pictured by a single Preisach grain (solid curve).
decomposition critically depends on the field span (HA, HB). Let us also remark that simple relationships for the number and size of observable grains can be derived if some scaling property of the potential energy curve V(x) is assumed [21]. Let L be the total possible displacement ofa wall, and assume that the number of obstacles which are overcome by a field H decays like H with a> 1. Then the number of unsuperable obstacles is proportional to W and the typical jump size scales like LH”’’. The next question is about the physical origin of interactions between Preisach grains. Here again, the situation is rather clear for single-domain particles which are coupled by dipolar fields. In this case, the coupling network merely depends on the geometrical features of the array of particles (this is known to be technically important, as particle aggregation for instance may severely affect the performance of magnetic tape). In bulk materials, an essential role is ~,
~,
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13 April 1987
can be assimilated to five negatively magnetised Preisach grains.
inside a FeSicycles grain0~H~0. at the upper of successive asymmetric The end volumes of down domains increase by small amounts between the first and sixth cyles, thus depressing the net magnetisation under field H. The sense of evolution is consistent with the macroscopic negative “bascule” of asymmetric loops and the results of analog simulations (fig. 1). The dashed areas of fig. 3 can be regarded as Preisach grains which flip from their up to their down states. The decomposition of the remaining parts of the domain structure into Preisach grains could only be precised by knowing all the irreversible events which take place during the first few cycles, but it can be asserted that the pattern reached after six cycles contains the sequence Moreover, the coupling terms J1~(1 ~ I ~ 5) are obviously negative as a volume element
played by the boundary conditions (outer edges for a single crystal, grain boundaries for a polycrystal). Fig. 3 is reproduced from ref. [8] and shows the domain structure inside a metallurgical grain of polycrystalline FeSi. The movements of walls are obviously coupled since each of them modifies the pole distribution on the grain boundaries, and hence the local field acting upon the other walls. Using the above definitions of Preisach grains and their interactions, one may wonder about the meaning of “pattern” and its evolution. A simple example is given by the analog simulation of fig. 1 a. This loop was obtained by coupling 4 electronic elements featuring rectangular loops with the following interaction scheme: J1~=J~ <0, J23=J3~<0,J34=J43-’zO, and I I> I I and I I. The other matrix elements can be made zero or moderately strong without qualitatively changing the results. Any microscopic state of the four-element array can be pictured by a symbol ±±±±;starting from the mitial state + + +, the system flips to + + + as the field is applied for the first time, then to + + for the second, and all subsequent, applications. The latter state can be regarded as the stable pattern for asymmetric field cycles 0—H—~0.Fig. lb sketches a real physical situation in which singledomain particles would be coupled by the above interaction network and would exhibit the same phenomenon of loop stabilisation. Fig. 3 shows the evolution of the wall positions
magnetised in a given sense tends to reduce the other domain with the same orientation, due to demagnetizing effects. The next (and difficult) step would consist of clarifying to what extent completing a field cycle could be assimilated to interacting an algorithm. Possibly this would be done in the following way: a magnetic element of volume v, experiences a total field H1 = H+ ~ (H= applied field, h~internal coupling fields). It will jump irreversibly (Barkhausen jump) when H1 becomes equal to some critical field H’~.The corresponding critical applied field H” = H~ ~1h~is not a constant as it depends on the microscopic configuration ofthe system via the coupling fields. Given a cycle of applied field between upper and lower bounds HA and H8, the material can be divided into two types of regions. In part 1, the distribution of critical fields H’~of any element of volume is such that no microscopic combination of surrounding domains can change the interaction term ~ by an amount sufficient to shift one ofthe 11* inside or outside the span (HA, HB) of the applied field. In such regions, the sequences of jumps which take place during successive cycles are globally unchanged, except for fluctuations in the criticial values of the applied field. As a consequence, the macroscopic changes of magnetisation exhibit no stabilisation effect. On the other hand, the sequences of jumps in regions of type 2 are modified by inversion, suppression or addition ofjumps. We expect from ref. [13]
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a Fig. 3. Solid lines: domain structure in a FeSi grain when applying a field H for the first time [8]. At the sixth cycle, the walls have moved to the new positions indicated by dotted lines (not to scale). The dashed areas have flipped opposite to the field and
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Volume 121, number 3
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that this can occur if two conditions are fulfilled: (a) some of the coupling coefficients are negative, and (b) the corresponding coupling fields have at least the same order of magnitude as the typical interval
d 0.5 0.4 0.3
between two critical fields H’~.The weakness of stabilisation phenomena (at most a few percent near the coercive field) tends to show that regions of type 2 represent only a small fraction of an AC demagnetised material. Probably this fraction is more important in a thermally demagnetised sample, which is a more metastable initial situation. Performing a field cycle could be viewed as “shaking” all domains before returning to the same external conditions, so that the state of any domain at a given recurrent point of the cycles would have to be tested at every cycle, and eventually changed depending on the coupling fields exerted by neighbouring domains.
4. Relaxation of an array of binary elements to a stable pattern The relevant parameters in the model are a priori the dimensionality D, size N and connectance of the network of binary elements. We tested the following cases:
D = 2, N= 16 and 36 (4 x 4 and 6 x 6 arrays), p = ~
nearest neighbours; 8 D=2, N=25 and 49, P= D=3,N=64 (4x4x4array),p=6. Cyclic conditions were used to eliminate boundary effects. Fig. 4 is a logarithmic plot ofthe distance to the stored pattern versus the number of iteration steps. Although the points exhibit noticeable scatter, due to the small size of arrays, they cluster around a straight line. A least-squares fit yields: d~=0.47 exp( —0.75n) with 0.81 correlation. The initial average distance 0.47 is close to the value 0.5 expected for two completely non-correlated patterns ofsufficient size. The uncertainty on the logarithmic decrement 0.75 is estimated to ±0.1. The points corresponding to various values for D, N and p seem to be fairly well distributed in the cluster of points. Hence the dynamics of relaxation appears to be little dependent on these parameters. On the other hand, we tested the same algorithm on similar arrays of three-level spins: S~=0, ±1. Relax-
13 April 1987
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Fig. 4. Decay ofthe distance dto the stored pattern (log—log plot) for arrays of various types versus the number of iteration steps n. Dots: 4 x 4; open circles: 5 x 5; closed triangles: 6 x 6; open triangles: 7 x 7; squares: 4 x 4 x 4. The broken line represents the decay d= 0.5 x 2”.
ation toward the stored pattern was much slower, and due to the scatter of results it was impossible to determine whether the average decay of d~was again exponential, d~= d0 exp ( —fin) with /3<0.1, or obeyed some other law, e.g. a power law d~= d0n So it seems that only two-state interacting elements can result that in fairly fast relaxation. It can bedecrealso remarked the numerically determined ment, /3=0.75±0.1,is not so far from the experimental rate of loop stabilisation, /3 = 0.71, nor from ln 2 = 0.693.... Hence the following conjecture: hysteresis loops approach their asymptotic limit like e in 2 = 2”, owing to the binary nature of the physical element involved. Accordingly, stabilisation is practically completed in a few runs as 2—~ 0.03 for instance. ~.
—
5. Conclusion We would conclude by coming back to the physics of ferromagnetic hysteresis. Indeed the present attempt of reproducing 1oop stabilisation phenomena can by no means be regarded as a realistic description of intricated magnetisation phenomena. 149
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The concept of “hidden pattern” is intended to be a schematic picture of a stable domain configuration which settles down after a few cycles. It is widely accepted, as a consequence of the hysteretic nature of magnetisation processes, that this configuration cannot be unique for a given field, as it depends on the previous history of the sample. Only for periodic field cycles (HA, HB) can one hope to observe reproducible domain structures. Moreover, thermal activation effects can trigger at random irreversible events which would not have occured at zero ternperature, thus destroying the deterministic character ofjump sequences. In the two physical examples discussed above (single-domain particles and polycrystal), the stable patterns, if they exist, are entirely determined by geometrical features of the particle distribution or the grain morphology via the dipolar coupling scheme. It can be said that they were built-in during preparation of the material.. Another physical situation was entirely disregarded here and deserves special attention: in practically defect-free single crystals such as garnet layers (the so-called bubble materials), the energy barriers to be overcome do not arise from domain wall pinning but from topological constrains. In such a case the domain structure features an enormous number ofquasi-continuous degrees of freedom. As a consequence, a very wide manifold of states is available for the domain structure, and any of these states can be selected at random as a result ofdomain nucleation processes. Then the system selftraps and gives rise to an intrinsically metastable domain configuration in which finite regions can jump irreversibly. The very first optical observations tend to show that local stable patterns emerge after a small number of field cycles. If so, this situation would be a new and interesting one since the “stored patterns” could be renewed at will (and at random) by merely saturating the sample. Finally, we would stress that, as suggested by Ned, loop stabilisation effects originate from negative
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short-range couplings between domains. These result in many possible states with neighbouring energies. We hope that the extremely rough approach presented here could foster theoretical studies, maybe inspired by spin-glass concepts, on metastability and frustration in the much older and more macroscopic situation of ferromagnetic domains. Acknowledgement The author is deeply indebted to the unknown referee for his expert criticism and suggestions.
References [1] N. Van Dang, J. Phys. Radium 20 (1959) 222. [2] V. Hajko, J. Daniel-Szabo and V. Kavecansky, Czech. J. Phys. B. 12 (1962) 867. [3] J. Daniel.Szabo and H. Gengnagel, Phys. Stat. Sol. 1 (1961) 512. [4] P. Molho and J.L. Porteseil, J. Phys. (Paris) 46 (1985). [5] A. Zentko, V. Hajko and L. Potocky, Czech. J. Phys. B 16 (1966) 939. [6] V. Hajko, A. Zentko and L. Hucko, Acta Phys. Slov. 23 (1973) 53. [7] V. Hajdo, J. Daniel-Szabo, L. Potocky Trans. Magn. MAG-lO (1974) 128. and A. Zentko, IEEE [8] V. Kavecansky, V.C. Hajko and J. Daniel-Szabo, Mat. Fyz. Cas. (Bratislava) 13 (1963) 64. [9] G. Pasztorand C. Schmidt, J. Appl. Phys. 49 (1987) 886. [10] M. Pappe, IEEE Trans. Magn. MAG-l7 (1981) 2082. [111 [12] P. F. Mollard, Preisach, private Z. Phys.communication. 94 (1935) 277. [13] L. Néel, C.R. Acad. Sci. 246 (1985) 2313. [14] [15] [16] [17] [18] [19] [20]
E.P. Wohlfarth, J. AppI. Phys. 35(1964) 783. J. Vuillod, Phys. Stat. Sol. (a) 31(1975) 235. J.J. Hopfield, Proc. Natl. Acad. Sci. USA 79 (1982) 2554. W. Kinzel, Z. Phys. B 60(1982) 205. L. Ned, Cah. Phys. 12 (1942) 1. J. Souletie, J. Phys. (Paris) 44 (1983) 1095. J.L. Porteseil and R. Vergne, J. Phys. (Paris) 37 (1976) 1211. [21] J.L. Porteseil and R. Vergne, CR. Acad. Sci. B 288 (1979)
PHYSICS LETTERS A
13 April 1987
WHY DO HYSTERESIS LOOPS STABILISE IN A FEW RUNS? J.L. PORTESEIL Laboratoire Louis Née!, CNRS—USTMG, 38042 Grenoble Cedex, France Received 4 December 1986; revised manuscript received 15 January 1987; accepted for publication 9 February 1987
Hysteresis loops of ferromagnets are usually stabilised after four or five field cycles. A tentative explanation is given in the spirit of learning models of spin glasses. It is suggested that loops approach their limit like 2—”, due to the binary nature of magnetic elements (Preisach grains).
1. Stabilisation of hysteresis loops: experiment Successive hysteresis loops of a cyclically magnetised ferromagnet are not exactly reproducible. It was observed that, after demagnetising a steel sample, the first four or five loops recorded between upper and lower fields HA and HB do not feature the same magnetisation amplitude AM= MA MB [1]. The changes in Ls.M can amount to a few percent if the field cycles are symmetrical (HB = HA) and if HA lies in the vicinity of the coercive field. This phenomenon, often called “bascule” (tilt), was further studied from a macroscopic standpoint in polycrystalline FeSi and single-crystal FeA1 [2,3]. It was demonstrated more recently that the virgin domain structure obtained by cooling a ferromagnet from above 7’. (thermal demagnetisation) exhibits a similar behaviour [4]. However, the evolution of loops is much more conspicuous, as the amplitude tIM’ ofthe first loop can be as large as three times its assymptotic value. Loops of rank n approach their limit like exp( —fin), with fi~0.7l.In other words, the ioops are again practically stabilised after four or five runs, whichever the amplitude of field cycles, Moreover, the sequences ofirreversible (Barkhausen) magnetisation jumps change significantly during the first few loops following a demagnetisation showingthat the domain structure undergoes noticeable reorganisations [5—71. This was confirmed by direct domain observations on a FeSi polycrystal [8]. A mechanical counterpart for this phenomenon was observed by studying the acoustic emission of —
—
0375-960 1/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
cyclically strained superconductors [9,10]. Such a behaviour is technically relevant, as it entails a “training” process of superconducting magnets which is practically completed after a small number ofstrain cycles. Let us make the following remark: when recording an audio signal, the magnetic tape is subjected both to the audiofrequency field H which carries the information to be recorded and to an alternating, high-frequency field H’ which helps in stabilising the corresponding magnetic configuration of fine partides in the tape. The frequency of this field is generally adjusted so that a magnetic grain experiences about five decreasing field cycles during the time it travels past the recording head [11]. The aim of this paper is to suggest a tentative explanation of why the number of runs needed to stabilise hysteresis ioops seems to be four or five as a rule, and not 1 or iO~.
2. Preisach model, interacting grains and learning in spin glass models In the classical Preisach model, a ferromagnet is pictured by an assembly offictitious, non-interacting “grains”, each of them featuring a rectangular hysteresis loop with two stable (up and down) states [12]. Indeed any microscopic hysteresis loop can be mimicked by means of a suitable distribution of grains, but in the absence of interactions Preisach’s 145
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very rapidly untractable as the number of grains is increased, and statistical results on large assemblies ofgrains are difficult to obtain. The approach used in this paper is somewhat different and was entirely inspired by recent studies on learning phenomena in spin-glass models [16,17]. Consider a set of N Ising variables S, = ±1 interacting via the hamiltonian H= >JiJJIJSISJ~in which the are defined as follows: M patterns S~’~= ~ S~’~’ S~P),~ ~ —
...,
1.. U
~
j~
~
~‘
t
t
1
2
j
31,
~
-
t
3
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4
Fig. 1 (a) Detail of a hysteresis loop obtained with four negatively coupled fictitious Preisach grains (analog simulation [151). (b) Sketch of a system of four real single-domain particles with dipolar couplings which would feature the same behaviour,
model cannot exhibit any loop evolution. Néel discussed the extremely simple situation in which two grains interact and showed that negative interactions could give rise to “bascule” phenomena [13]. However, loop stabilisation is completed after two runs in this oversimplified model. The analysis was further extended by Wohlfarth [141. More recently, an attempt was made to simulate an assembly of Preisach grains [15]. With 30 interacting bistable electronic circuits, it was possible to create “bascule” effects which affected the first four loops. Fig. 1 a is reproduced from ref. [15] and shows a detail of a loop obtained with four interacting electronic “grains” subjected to asymmetric cycles: HB=O—~.HA>O--~HB=O—~..... Starting from the mitial demagnetised configuration, the magnetisation jumps during the first cycle to a positive value M1, then settles down to M2
among the 2N possible patterns are “stored” by choosing the interactions J,~such that for each pattern S~all spins (or most of the spins) point parallel to the internal field. This can be achieved by choosing J~=> IS~S~ (Cooper prescription [161). We limit ourselves here t,p the simplest possible problem of storing only one pattern S, so that the S are simply given by J• = S,S. In other words, the .
.
. ~
network of interactions is constructed ad hoc so that two spins which are parallel (anti-parallel) in the pattern to be stored are connected by a positive (negative) interaction. S is obviously the ground state of the hamiltonian. In the next step, an initial set of S,= ±1 is selected at random, then the system is allowed to relax toward the stored pattern. This is achieved by the simple usual algorithm which consists of testing all spins sequentially. Each spin S, remains unchanged if S~1,~,J~S>0 (which means S, parallel to the local field), and is reversed otherwise. In the special case ~ ~f~ = 0, the sign ofS, is selected at random with equal probabilities. The algorithm is then repeated until retrieval of the stored pattern. The relaxation toward a given pattern is conveneniently represented by defining the overlap of two states k and/as q~~=N’ S~k) 5~1).The (Hamming) distance between the two states is dk,= (1 —q,~)/2; it vanishes when the two states are identical. ~,
3. Assumptions of the model and their physical meanings The above formal description is based of interacting binary elements which evolve as an algorithm is applied. In this paper, we attempt to apply such concepts to the problem of ferromagnetic hysteresis loops.
Volume 121, number 3
PHYSICS LETTERS A
The first question to be answered pertains to the actual nature of possible binary elements, or Preisach grains, in a ferromagnet. The answer is rather straightforward in the case of single-domain particles, as each of them features a real rectangular hysteresis loop. It is widely accepted that Preisach’s model, implemented with thermal activation con cepts and computer studies of partical interactions, affords a realistic description of magnetisation processes in particulate media. The situation is far less clear in bulk materials where magnetisation processes take place by displacements of domain walls. Irreversibility arises from pinning of walls by obstades of various types, and the magnetisation flips irreversibly in a finite volume as a wall jumps past an energy barrier. It was demonstrated [181 that a double energy wall is the physical counterpart for a fictitious Preisach grain (this concept was also used in the study of spin glasses [19]). Unfortunately, the decomposition of the random potential in which a wall moves into double wells is not unique, so that at first sight the grain—well equivalence is only statistically valid, and no real existence can be attributed to Preisach grains. However, careful examination of hysteresis loops in samples with small numbers ofbig domains suggest that reproducible events can be identified in sequences ofjumps (see for instance ref. [20]). This is rather surprising at first sight as the irreversible motion of a domain wall involves great numbers of Preisach grains. Let us assume for instance that two neighbouring wells of the potential energy of a wall are typically separated by the wall thickness, say 100 nm. In the experiments, reported in ref. [20], the free path of a wall can be as big as 1 mm, so that the movements of a wall would have to be pictured by 1 0~Preisach grains. In fact, a wide majority of these grains can be disregarded when applying given field cycles (HA, BB). On the other hand, the strong obstacles which would be overcome by a field H such that I HI> sup (I HA I~IHB I) are never crossed during these cycles, and the corresponding Preisach grains remain in their initial state. On the other hand, the very numerous weak obstades are always overcome, and the corresponding small jumps cannot be separated because they are embedded in bigger jumps. This behaviour is illustrated by fig. 2. Finally, one is left with a small number ofobservable grains. It must be stressed that this
13 April 1987
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Fig. 2. Decomposing the potential energy curve V(x) in which a wall moves into Preisach grains (schematical). (a) The slope of obstacle 8 (1) is assumed to be too strong for the upper field HA (lower field HB) to overcome it. On the other hand, obstacles 4 and 6 (3 and 4) are weak and embedded in the irreversible ascending jump 2—.8 (descending jump 7=1). (b) In the cornplex corresponding hysteresis loop, thedotted paths can be ignored and the phenomena can be pictured by a single Preisach grain (solid curve).
decomposition critically depends on the field span (HA, HB). Let us also remark that simple relationships for the number and size of observable grains can be derived if some scaling property of the potential energy curve V(x) is assumed [21]. Let L be the total possible displacement ofa wall, and assume that the number of obstacles which are overcome by a field H decays like H with a> 1. Then the number of unsuperable obstacles is proportional to W and the typical jump size scales like LH”’’. The next question is about the physical origin of interactions between Preisach grains. Here again, the situation is rather clear for single-domain particles which are coupled by dipolar fields. In this case, the coupling network merely depends on the geometrical features of the array of particles (this is known to be technically important, as particle aggregation for instance may severely affect the performance of magnetic tape). In bulk materials, an essential role is ~,
~,
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can be assimilated to five negatively magnetised Preisach grains.
inside a FeSicycles grain0~H~0. at the upper of successive asymmetric The end volumes of down domains increase by small amounts between the first and sixth cyles, thus depressing the net magnetisation under field H. The sense of evolution is consistent with the macroscopic negative “bascule” of asymmetric loops and the results of analog simulations (fig. 1). The dashed areas of fig. 3 can be regarded as Preisach grains which flip from their up to their down states. The decomposition of the remaining parts of the domain structure into Preisach grains could only be precised by knowing all the irreversible events which take place during the first few cycles, but it can be asserted that the pattern reached after six cycles contains the sequence Moreover, the coupling terms J1~(1 ~ I ~ 5) are obviously negative as a volume element
played by the boundary conditions (outer edges for a single crystal, grain boundaries for a polycrystal). Fig. 3 is reproduced from ref. [8] and shows the domain structure inside a metallurgical grain of polycrystalline FeSi. The movements of walls are obviously coupled since each of them modifies the pole distribution on the grain boundaries, and hence the local field acting upon the other walls. Using the above definitions of Preisach grains and their interactions, one may wonder about the meaning of “pattern” and its evolution. A simple example is given by the analog simulation of fig. 1 a. This loop was obtained by coupling 4 electronic elements featuring rectangular loops with the following interaction scheme: J1~=J~ <0, J23=J3~<0,J34=J43-’zO, and I I> I I and I I. The other matrix elements can be made zero or moderately strong without qualitatively changing the results. Any microscopic state of the four-element array can be pictured by a symbol ±±±±;starting from the mitial state + + +, the system flips to + + + as the field is applied for the first time, then to + + for the second, and all subsequent, applications. The latter state can be regarded as the stable pattern for asymmetric field cycles 0—H—~0.Fig. lb sketches a real physical situation in which singledomain particles would be coupled by the above interaction network and would exhibit the same phenomenon of loop stabilisation. Fig. 3 shows the evolution of the wall positions
magnetised in a given sense tends to reduce the other domain with the same orientation, due to demagnetizing effects. The next (and difficult) step would consist of clarifying to what extent completing a field cycle could be assimilated to interacting an algorithm. Possibly this would be done in the following way: a magnetic element of volume v, experiences a total field H1 = H+ ~ (H= applied field, h~internal coupling fields). It will jump irreversibly (Barkhausen jump) when H1 becomes equal to some critical field H’~.The corresponding critical applied field H” = H~ ~1h~is not a constant as it depends on the microscopic configuration ofthe system via the coupling fields. Given a cycle of applied field between upper and lower bounds HA and H8, the material can be divided into two types of regions. In part 1, the distribution of critical fields H’~of any element of volume is such that no microscopic combination of surrounding domains can change the interaction term ~ by an amount sufficient to shift one ofthe 11* inside or outside the span (HA, HB) of the applied field. In such regions, the sequences of jumps which take place during successive cycles are globally unchanged, except for fluctuations in the criticial values of the applied field. As a consequence, the macroscopic changes of magnetisation exhibit no stabilisation effect. On the other hand, the sequences of jumps in regions of type 2 are modified by inversion, suppression or addition ofjumps. We expect from ref. [13]
3~ ‘Ni.
N
‘~
.~
“\
“
a Fig. 3. Solid lines: domain structure in a FeSi grain when applying a field H for the first time [8]. At the sixth cycle, the walls have moved to the new positions indicated by dotted lines (not to scale). The dashed areas have flipped opposite to the field and
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that this can occur if two conditions are fulfilled: (a) some of the coupling coefficients are negative, and (b) the corresponding coupling fields have at least the same order of magnitude as the typical interval
d 0.5 0.4 0.3
between two critical fields H’~.The weakness of stabilisation phenomena (at most a few percent near the coercive field) tends to show that regions of type 2 represent only a small fraction of an AC demagnetised material. Probably this fraction is more important in a thermally demagnetised sample, which is a more metastable initial situation. Performing a field cycle could be viewed as “shaking” all domains before returning to the same external conditions, so that the state of any domain at a given recurrent point of the cycles would have to be tested at every cycle, and eventually changed depending on the coupling fields exerted by neighbouring domains.
4. Relaxation of an array of binary elements to a stable pattern The relevant parameters in the model are a priori the dimensionality D, size N and connectance of the network of binary elements. We tested the following cases:
D = 2, N= 16 and 36 (4 x 4 and 6 x 6 arrays), p = ~
nearest neighbours; 8 D=2, N=25 and 49, P= D=3,N=64 (4x4x4array),p=6. Cyclic conditions were used to eliminate boundary effects. Fig. 4 is a logarithmic plot ofthe distance to the stored pattern versus the number of iteration steps. Although the points exhibit noticeable scatter, due to the small size of arrays, they cluster around a straight line. A least-squares fit yields: d~=0.47 exp( —0.75n) with 0.81 correlation. The initial average distance 0.47 is close to the value 0.5 expected for two completely non-correlated patterns ofsufficient size. The uncertainty on the logarithmic decrement 0.75 is estimated to ±0.1. The points corresponding to various values for D, N and p seem to be fairly well distributed in the cluster of points. Hence the dynamics of relaxation appears to be little dependent on these parameters. On the other hand, we tested the same algorithm on similar arrays of three-level spins: S~=0, ±1. Relax-
13 April 1987
t
0.2
•
.
.
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I
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0.1 I
0 ~
•
A
I
.
\o .~
A
0.05
~
0
• 0.02
A
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n 1
I
2
I
3
4
5
Fig. 4. Decay ofthe distance dto the stored pattern (log—log plot) for arrays of various types versus the number of iteration steps n. Dots: 4 x 4; open circles: 5 x 5; closed triangles: 6 x 6; open triangles: 7 x 7; squares: 4 x 4 x 4. The broken line represents the decay d= 0.5 x 2”.
ation toward the stored pattern was much slower, and due to the scatter of results it was impossible to determine whether the average decay of d~was again exponential, d~= d0 exp ( —fin) with /3<0.1, or obeyed some other law, e.g. a power law d~= d0n So it seems that only two-state interacting elements can result that in fairly fast relaxation. It can bedecrealso remarked the numerically determined ment, /3=0.75±0.1,is not so far from the experimental rate of loop stabilisation, /3 = 0.71, nor from ln 2 = 0.693.... Hence the following conjecture: hysteresis loops approach their asymptotic limit like e in 2 = 2”, owing to the binary nature of the physical element involved. Accordingly, stabilisation is practically completed in a few runs as 2—~ 0.03 for instance. ~.
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5. Conclusion We would conclude by coming back to the physics of ferromagnetic hysteresis. Indeed the present attempt of reproducing 1oop stabilisation phenomena can by no means be regarded as a realistic description of intricated magnetisation phenomena. 149
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The concept of “hidden pattern” is intended to be a schematic picture of a stable domain configuration which settles down after a few cycles. It is widely accepted, as a consequence of the hysteretic nature of magnetisation processes, that this configuration cannot be unique for a given field, as it depends on the previous history of the sample. Only for periodic field cycles (HA, HB) can one hope to observe reproducible domain structures. Moreover, thermal activation effects can trigger at random irreversible events which would not have occured at zero ternperature, thus destroying the deterministic character ofjump sequences. In the two physical examples discussed above (single-domain particles and polycrystal), the stable patterns, if they exist, are entirely determined by geometrical features of the particle distribution or the grain morphology via the dipolar coupling scheme. It can be said that they were built-in during preparation of the material.. Another physical situation was entirely disregarded here and deserves special attention: in practically defect-free single crystals such as garnet layers (the so-called bubble materials), the energy barriers to be overcome do not arise from domain wall pinning but from topological constrains. In such a case the domain structure features an enormous number ofquasi-continuous degrees of freedom. As a consequence, a very wide manifold of states is available for the domain structure, and any of these states can be selected at random as a result ofdomain nucleation processes. Then the system selftraps and gives rise to an intrinsically metastable domain configuration in which finite regions can jump irreversibly. The very first optical observations tend to show that local stable patterns emerge after a small number of field cycles. If so, this situation would be a new and interesting one since the “stored patterns” could be renewed at will (and at random) by merely saturating the sample. Finally, we would stress that, as suggested by Ned, loop stabilisation effects originate from negative
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short-range couplings between domains. These result in many possible states with neighbouring energies. We hope that the extremely rough approach presented here could foster theoretical studies, maybe inspired by spin-glass concepts, on metastability and frustration in the much older and more macroscopic situation of ferromagnetic domains. Acknowledgement The author is deeply indebted to the unknown referee for his expert criticism and suggestions.
References [1] N. Van Dang, J. Phys. Radium 20 (1959) 222. [2] V. Hajko, J. Daniel-Szabo and V. Kavecansky, Czech. J. Phys. B. 12 (1962) 867. [3] J. Daniel.Szabo and H. Gengnagel, Phys. Stat. Sol. 1 (1961) 512. [4] P. Molho and J.L. Porteseil, J. Phys. (Paris) 46 (1985). [5] A. Zentko, V. Hajko and L. Potocky, Czech. J. Phys. B 16 (1966) 939. [6] V. Hajko, A. Zentko and L. Hucko, Acta Phys. Slov. 23 (1973) 53. [7] V. Hajdo, J. Daniel-Szabo, L. Potocky Trans. Magn. MAG-lO (1974) 128. and A. Zentko, IEEE [8] V. Kavecansky, V.C. Hajko and J. Daniel-Szabo, Mat. Fyz. Cas. (Bratislava) 13 (1963) 64. [9] G. Pasztorand C. Schmidt, J. Appl. Phys. 49 (1987) 886. [10] M. Pappe, IEEE Trans. Magn. MAG-l7 (1981) 2082. [111 [12] P. F. Mollard, Preisach, private Z. Phys.communication. 94 (1935) 277. [13] L. Néel, C.R. Acad. Sci. 246 (1985) 2313. [14] [15] [16] [17] [18] [19] [20]
E.P. Wohlfarth, J. AppI. Phys. 35(1964) 783. J. Vuillod, Phys. Stat. Sol. (a) 31(1975) 235. J.J. Hopfield, Proc. Natl. Acad. Sci. USA 79 (1982) 2554. W. Kinzel, Z. Phys. B 60(1982) 205. L. Ned, Cah. Phys. 12 (1942) 1. J. Souletie, J. Phys. (Paris) 44 (1983) 1095. J.L. Porteseil and R. Vergne, J. Phys. (Paris) 37 (1976) 1211. [21] J.L. Porteseil and R. Vergne, CR. Acad. Sci. B 288 (1979)