DOMAIN WALLS AND MICROMAGNETICS

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DOMAIN WALLS AND MICROMAGNETICS

A. Aharoni

To cite this version:

A. Aharoni. DOMAIN WALLS AND MICROMAGNETICS. Journal de Physique Colloques, 1971, 32 (C1), pp.C1-966-C1-971. �10.1051/jphyscol:19711344�. �jpa-00214379�

DOMAIN WALLS AND MICROMAGNETICS

A. AHARONI

Department of Electronics, Weizmann Institute of Science, Rehovoth, Israel

R&mC. - Une classification et un compte rendu des theories recentes pour la structure et l'energie des parois des domaines a 180° de differents types, sont faits, en mettant un accent spkcial sur la possibilite d'approcher la rigueur des calculs Micromagnetiques. La verification de la self-consistance des modtles des parois est gCneralis5e. On discute des methodes experimentales pour mesurer 1'Cnergie et la largeur des parois, et on fait voir que les resultats theoriques r h n t s sont en accord raisonnable avec les experiences.

Abstract. - Recent theories for the structure and energy of the different types of 180° domain walls are classified and reviewed, with a particular emphasis on the possibility of approaching the rigorousness of Micromagnetic calculations.

The test for self-consistency of wall models is generalized. Experimental methods for measuring the wall energy and width are discussed, and it is shown that the recent theoretical results are in reasonable agreement with experiment.

I. Introduction. - Micromagnetics and domain theory both started with the classical p a p q of Landau and Lifshitz [I], but they developed along different directions. In domain theory [2] one assumes a cer- tain spatial configuration for the magnetization, and compares the energy to that of some other configura- tions, or at most minimizes the energy with respect to some adjustable parameters. Such an approach ignores the possibility that the lower energy state is inaccessible because of an energy barrier. Besides, domain theory always leaves open the question of existence of a completely differeni configuration whose energy is still lower. Yet, such limitations can be offset to a large extent if the configurations are taken from some experiments, or if the calculation is done with a lot of ingenuity and physical insight, and indeed domain theory has reached many successful achieve- ments. A well-known example is the prediction by NCel [3] of spike-shaped domains attached to non- magnetic inclusions, which were found experimentally [4] only 3 years later. A calculation of the detailed structure of such a spike [5] agrees very well with experiment.

Micromagnetics [6], on the other hand, has tried to avoid arbitrary assumptions by minimizi~g the energy with respect to all configurations, and to take the hysteresis into account. However, it involves compli- cate mathematics, and was, therefore, successful only in treating relatively simple and somewhat ideali- sed geometries [7]. For other cases it is not possible, with presently known techniques, to obtain rigorous, micromagnetic solutions, and one has to use certain approximations, or impose certain constraints, or both.

In the particular study of 1800 domain wall, Micro- magnetics and domain theory have approached each other considerably in recent years. There are numerical computations that minimize the energy with respect to all the configurations, although these are still done under constraints, and do not contain hysteresis.

On the other hand, the models used in the Ritz- method calculations have become more sophisticated, they answer at least some self-consistency require- ments, and their results are approaching those of the numerical computations. It is the purpose of this paper to review these recent developments, and discuss

steps which should be taken in order to have data which are both reliable and useful.

In particular, all calculations done so far yield in principle only an upper bound to the wall energy.

For a reliable estimate, one needs also a lower bound.

One could probably use a method developed by Brown [8, 61, and used successfully [9-11, 71 in nuclea- tion problems, but this has not been done yet. Another possibility is to obtain the wall energy from an expe- riment, but the accuracy of experiments reported so far rules this out, as will be discussed in section IV 1.

However, even when one obtains lower and upper bounds to the wall energy, which come very close to each other, one does not necessarily have a good appro- ximation to the actual physical structure of the wall, because quite different structures might yield, in principle, approximately the same energy. For the structure to be reliable, the model should either fit the details of some experiment, or fulfill a certain theoretical test which is sufficient for the model to be close to the minimum energy structure. No such sufficient test has been found yet, but at least there is a necessary test, which allows the elimination of models that are certainly wrong. The following chapter I1 will discuss, and somewhat generalize, this test.

In the review which follows, the surface anisotropy will be ignored for lack of adequate data, although it might [12] be not negligible. We shall also ignore magnetostriction [13, 141, and anisotropy perpendi- cular to the film [15-181. We shall assume that the temperature is low compared to the Curie point, where special wall structure [19] take place, and we shall consider only antiparalled domains with the wall parallel to them, in particular ignoring a wall perpen- dicular to the domains [20] which occurs e. g. in magne- tic recording [21]. We shall use for the exchange constant in permalloy the usual value

which is the average of the experimental values [22]

for Ni and Fe, but an experimental evaluation for this material would have been more desirable.

11. Self-consistency tests. - Consider a homoge- neous ferromagnetic slab, 1 y 1 < b, which is infinite in the x and z directions. Let the z-axis be an easy

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19711344

DOMAIN WALLS AND MICROMAGNETICS C 1 - 967

axis for a uniaxial anisotropy. Let the direction of the magnetization vector M ⁼ M,(ai

+

+

+

y = 5 I , for x = & a . (1) Along the z-axis it is assumed that the wall is periodic, with the periodicity 2 c, namely that a, P, y and their derivatives at z = c have the same values as at z = - c.

Minimizing the total energy under the constraint

leads to the Brown equations [23]

and to the boundary conditions [besides (1) and the periodicity]

Here C and K are the exchange and anisotropy cons- tants respectively, and V is the potential of surface and volume charges. Using the notation

multiplying (3a) by a, (3b) by P, adding, integrating, and using (2),

Here y, and y, are respectively the anisotropy and magnetostatic energies, per unit wall area per cycle, according to the conventional definitions. These, and the exchange term, make up the total wall energy, per cycle per unit wall area, which is, in zero magnetic field,

Y?JI

+

+

+

+

+

+

YcJl

I

it is seen by substituting (8) in ( 6 ) and comparing to (7), that

The relation (10) has thus been proved for functions a, /I, y which fulfill (3), but it does not necessarily hold for these only. Therefore, this leads to anecessary, but not sufficient, condition that Ritz modelsof walls, or energy minimization under constraint, yield results which are close to the energy minimum, which is the solution of (3). In these one should calculate both energy expressions (7) and (9), then define the self- consistency parameter [23]

According to the foregoing, a necessary condition for a configuration to be near an energy minimum is that this S is approximately 1. No scale can be established, but one would expect the inaccuracy in the structure of any wall model to be at least as large as the devia- tion of S from 1. It should be noted that (9) has been given here for the particular case of 1800 wall in a uniaxial film, at zero applied field. Hubert [24] has derived a more general expression.

Two other such necessary tests can be obtained from (3). We define

Dividing (3a) by a, integrating and using (2), (8) and (9), and similarly in (3b), it is seen that for func- tions which fulfill (3),

Near an energy minimum one should, therefore, approximately fulfill the two equalities in (14), as well as (10). Although these are necessary conditions only, a configuration should have a higher probability of approximating an energy minimum if it passes several independent tests. Unfortunately, however, (12) and (13) are useful only if a is finite, whereas most of the wall models use an infinite a.

Another qualitative conclusion [25] from (3b), is that p ⁼ 0 is not a solution, if aV/ay # 0. Hence, NCel wall models which assume P ^{= 0} can be a good approximation only if they yield negligibly small magnetostatic self-energy, and similar argument holds for Bloch walls with a = 0.

111. Theoretical Studies. - 1 . NUMERICAL COMPU- TATIONS. - Minimization of wall energy for all possible configurations was first done by Brown and LaBonte [26], assuming that a, P, and y depend only on x of chapter 11. Some later arguments [27,28], and then detailed computations [29], demonstrated that a factor of 2 reduction in the energy is obtained in moderately thick permalloy films by allowing varia-

C 1 - 968 A. AHARONI tion in the y-direction (see Fig. I), Still, this is in prin-

ciple only an upper bound to the wall energy because the magnetization is not allowed to vary along the z-direction. In the computations of La Bonte [29], S of (1 1) differed from 1 by only a few percent, but this is only a necessary condition, for the structure to approximate the tree-dimensional minimum. In this case a is finite, and one could check if (14) is also approximately fulfilled, but this is not possible to find out from the published data.

Experimentally, walls are known to have a definite structure along the z-axis, not only in the cross-tie region (i. e. roughly 200 to 1 000 A in permalloy [30]), but even for thicker films, where Bloch walls consist of segments, separated by NCel lines [31], which play an important part in the motion of these walls [32].

Moreover, this structure was observed in bulk mate- rial (e. g. iron whiskers [33]), even though there are some claims that the subdivided wall is a metastable configuration, and its energy is higher than that of a wall in which the magnetization does not depend on z.

However, this is based on very crude calculations [34], and the observations [35], which qualitatively support this view, are [36] not conclusive. Calculations [37,38]

which use consistently the Ritz method show that subdividing the Bloch wall reduces its energy, but it should be noted that these ignore the y-dependence, which is not a very good approximation according to figure 1.

O . . J L I I ~ . I L ~ . . J

!GOO 2000 3cm 4 m ZccO cl'a

Film t:ric:;n~ss, 2b .in A

FIG. 1 . - Two-dimensional calculations of the Bloch wall ener- gy per unit wall area, as a function of film thickness, with C = 2 x 10-berg/ cm., Ms = 800 e. m. u., and K = 103 erg/cm3.

Numerical minimization of LaBonte [29] are compared to Ritz-method model of Aharoni [28] and of Hubert [52,54].

The dashed curve is the lowest energy possible in one dimension [26].

The two-dimensional computations can thus be at most an approximation for those parts of the wall which are relatively pure Bloch wall (if such regions exist), between NCel lines, and only for the thicker films. For these sections the approximation should be about as good as all the study of wall structure without considering the domains. The change in the magnetostatic energy of the domains, when pushed apart by the width of the wall, was found by a crude estimate [39] to be negligible compared to

energy of wall models used at that time. This contribu- tion does not seem negligible any more, with the lower energy and larger width of the 2-dimensional wall, and unless a better estimation is found, it will have to be included in the minimization process. This was done in one case [40] of one-dimensional wall compu- tations. The relative importance of this energy term has not been reported.

For the thinner films, only one-dimensional com- putations were done. The first of these [26] did not converge. Kirchner and Doring [41] said their method converged, but gave results only for a l 000 A per- malloy film, which is out of the NCel wall region.

Oredson and Torok [40] gave more detailed results, and also looked into the interesting possibility that the angle between the magnetization in adjacent domains is not 1800. This turned out to be energeti- cally favorable under certain conditions, but in view of the foregoing, it seems rather risky to draw conclu- sions from such one-dimensional computations. These authors [25, 401 define as ⁽⁽ Intermediate Wall ⁾⁾ any configuration in which ap # 0, but the present author does not find any justification for this nomen- clature.

2. Rrrz METHOD CALCULATIONS. - Numerical computations are difficult to use in problems other than those which have been specifically included in the program. In particular, domain wall calculations are needed as a starting point for studies of domain structure, or wall mobility, etc., and the numerical results cannot be used for these. One thus needs relatively simple analytic approximations, which can be used in a variety of problems. If these are found to be sufficiently good approximations for the results of the numerical minimization, one can rather safely and conveniently use them for parameters not covered by the numerical computations. The following review of these models will be restricted to the parameters of typical permalloy films, but it should be noted that the same models can be used for walls [42, 431, or domains with walls [44, 451, in thin Co (or Fe) films, or even for the similar wall [46] in ferroelectrics.

The first two-dimensional wall calculation was a perturbation scheme [47], but in this only first order perturbation was done in detail, and this is not suffi- cient for any practical case. Only for coupled walls in double films were calculations actually carried out in 2 dimensions [48-511. However, it was soon realized that the flux-closure arrangement of a double film can be used in a single film, which led [28] to a model of a wall without surface charge, in which the volume charge exists only in a small kernel region, and vanishes in the wide flanks of the wall. The upper bound thus calculated for the wall energy is plotted in figure 1.

Using almost the same x-dependence, Hubert [52]

chose the y-dependence so that the wall had no sur- face or volume charges. Under the constraint (2), this could be achieved only by defining numerically the dependence on y, which is a severe limitation for using the model in other problems unless the use of some analytic approximation for this function (e. g. as in [53]) can be justified. The wall energy com- puted from this model [54] is also plotted in figure 1, and is only about 10 % above the minimum energy in

DOMAIN WALLS AND MICROMAGNETICS C 1 - 9 6 9 2 dimensions for the thinner films, and the difference

decreases for the thicker films. The self-consistency parameters, S, was also within 10 % of 1 1551, and the whole structure seems [54] a good approximation to the structure computed by LaBonte [29].

The model of Hubert for Bloch walls [52, 541 is thus a relatively useful approximation for many problems. It also has the particularly attractive feature that the calculations do not involve the film thickness directly. However, if one is specifically interested in the stray field outside the wall (e. g. in order to study visibility by Bitter powder [56, 651, or in an electron microscope [57], or wall interaction with disloca- tions 1581 and other defects [59], stresses, or magnetic fields [60, 611, especially for special geometries [62, 631 which are particularly handy for experimental obser- vations), a configuration for which the stray field vanishes is obviously a very poor approximation. But some methods for modifying the structure to include a stray field, and approach the boundary conditions (4), have been proposed [52], and were demonstrated [54]

to reduce the energy somewhat.

Hubert [52] also studied what he calls a two-dimen- sional N&el wall, which he defines [54] as a wall whose energy varies linearly with a field applied perpendi- cular to the wall. We prefer to reserve the name NCel wall to a configuration whose energy (in zero applied field) increases with an increasing film thickness, because this fits better the basic idea of Nkel. Accord- ing to this definition, almost all NCel wall models are still one-dimensional, and all the recent ones are some generalization of the one-parameter model of Dietze and Thomas [64], as extended by Feldtkeller and Fuchs 1311 to 3 parameters, c, p,, p,

Aharoni [66] introduced a 4th adjustable parameter in a power form,

a = c

[

]

⁺

[A]

PI + ^{x 2} P z + x

while Holz and Hubert 1531 added together similar terms in the form

to obtain 2 n - 1 adjustable parameters, ci and ^pi.

The magnetostatic self-energy of (17) is obtained by adding known [64] contributions from each term, and is very simple. But the calculation of the exchange energy becomes very complicated when extra terms are added, and this sets the practical limit to n in (17).

Actual computations where done for n = 4.

For typical permalloy films, the self-consistency parameter, S, of the model (16) varied between 0.94 and 1.23 [66]. The first computations [53] of the model (17) yielded S = 1.75 for a 250 fi thick permal- loy film, but later, refined data [55] for the same model gave a peak value of S ⁼ 1.12. The wall energy in

these [52, 551 computations was about 2 % lower than that computed 1661 from the model (16). The model (17) is also the only one for which some attempt was made to add some form of y-dependent P, instead of j? = 0 taken in all other N&el wall calculations. One form [53] was abandoned, because i t reduced the energy only by some 1 to 2 %. Another trial [54], to add stray-field-free structure, did not make much diffe- rence either, for a NCel wall as defined here, in zero field. For the energy of a cross-tie wall, only one upper bound is known [67]. I t must be too high, or the wall could not be so abundant experimentally, although certain structures (e. g. 360° walls [18]) are sometimes observed without being energetically favorable [68], because of an energy barrier 1691. The estimation of Middelhoek [70] was criticized by Holz and Hubert [53]

and approximations were given by Prutton [71] and in [54].

IV. Experimental Methods. - 1. ENERGY. -The first scheme for measuring wall energy by twisting it 1721 involved a crude analysis. More detailed study [73] of the equilibrium shape of a wall twisted by a magnetic field Hz, led to the following differential equation, for y, z defined as in 11,

Here y is the wall energy per unit wall area, and y, is the angle between the wall and the y-axis. Equa- tion (18) was integrated for particular cases [73, 741, but actually its general solution is

2 M,

1

⁺

⁺

as can be verified by substitution. The integration of Hz should be very simple in a given geometry, but one needs a detailed theory of the wall structure in order to know y as a function of p, in (19). Experiments on Fe-Si were analysed [74, 751 using an old theory of NCel[76], in which magnetostatic energy was neglected, but this can no longer be taken as a good approxi- mation for the wall structure. Similar difficulties must also be encountered when the wall is streched to a sinusoidal shape by a periodic magnetic field, obtained [77, 781 by passing a current through grid of wires, placed on the sample, which was done in orthoferrites.

But when the wall energy is obtained from the size of the domains in a narrow strip [79, 801, thean alysis is independent of the wall structure, and should be more reliable, although there are other approximations.

In permalloy, this method yielded [SO] energies which were smaller than the theoretical values available a that time, but they are larger than more recent theoretical values. Detailed comparison is not possible, since the values of the anisotropy were [not reported.

Much larger wall energy was deduced from domain tip propagation [81], but this involves an oversimpli- fied description of the creation and motion of walls.

Formation of walls is a basic problem in Microma- gnetics, for which only approximate solutions for specific cases are known [61, 82-85], and must be connected to imperfections [7].

C 1 - 970 A. AHARONI 2. STRUCTURE. - The use of Lorentz electron microscopy to study walls was reviewed by Cohen [86].

The main difficulty is in the analysis of the observed intensity pattern. One way [86, 871 is to compute the wall image for some wall model, and find the values of the parameters which give the best fit to the observed image. The wall structure can also be computed directly from the observed pattern [88, 891, but appa- rently the accuracy of the numerical integration is rather poor. Therefore, results are mainly reported as the value of the wall << width ^)), W, defined by :

These data of Cohen [89] are reporduced in figure 2, correcting an error in reproducing the results of Col- lette [90], and adding the more recent [66] model (16).

The wall widths (20) for the model (17) are somewhat below the corresponding curves for (16). It is seen that the scattering in the experimental results might be partly, but not wholly, due to different film anisotropies.

For thicker permalloy [91] and Co [92] films, wall width is rather large, and increases with film thickness.

The theoretical value for two-dimensional walls [52,54]

has approached considerably the experimental results [91] for permalloy. Better agreement can be obtained by extrapolating widths measured for various defocu- sing distances [93-961.

Wall width has also been measured using the Kerr effect [97], but the accuracy was not high. Besides, in this technique one has to use the rather doubtful assumption that the wall does not change its structure when it moves.

F I L M THICKNESS (i)

FIG. 2. -Wall << width ^D, defined by eq. (20), in permalloy films. The experimental data and various theoretical models are reproduced from 1891. The values marked on the curves for the N6el wall model (16) of Aharoni 1661 are the anisotropy

constant, in erg/cm3.

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