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do this when lying along a magnetic easy axis that is in the same direction at every point in the material. However, if the easy axis orientation fluctuates from site to site, a conflict between ferromagnetic coupling and anisotropy arises. As long as we imagine lattice periodicity, a ferromagnetic structure is a consequence of ferromagnetic exchange interactions, the strength of the anisotropy being irrelevant. In this situation, we are assuming a major simplification, namely, the direction of the easy axis is uniform throughout the sample. With this simple picture, we present crucial questions related to the influence of an amorphous structure on magnetic order.

Regarding the magnetic order in amorphous and nanocrystalline materials, we know that it stems from two contributions: exchange and local anisotropy. The exchange arises from the electron-electron correlations. The mechanism of the electrostatic interactions between electrons has no relation to structural order and is sensitive only to overlapping of the electron wave functions. With respect to magnetic anisotropy, it also originated by the interaction of the local electrical field with spin orientation, through the spin-orbit coupling. Therefore, magnetic anisotropy also is a local concept. Nevertheless, the structural configuration of magnetic solids exerts an important influence on the macroscopic manifestation of the local anisotropy. As a consequence, when the local axes fluctuate in orientation owing to the structural fluctuation (amorphous and nanocrystalline materials as examples), calculations of the resultant macroscopic anisotropy become quite difficult.

In the case of amorphous ferromagnetic alloys, the usual approach to the atomic structure of a magnetic order connected to a lattice periodicity is not applicable. These materials can be defined as solids in which the orientation of local symmetry axes fluctuate with a typical correlation length l = 10 A. The local structure can be characterized by a few local configurations with icosahedral, octahedral, and trigonal symmetry. These structural units have randomly distributed orientation. The local magnetic anisotropy would be larger in the units with lower symmetry. In general, these units are characterized by fluctuations of the orientation local axis. It is remarkable that with these types of structures, the correlation length, l, of such a fluctuation is typically the correlation length of the structure and ranges from 10 A (amorphous) to 10 nm (nanocrystals) and 1 mm (polycrystals).

Fluctuations in the interatomic distances associated with the amorphous structure also should contribute to some degree of randomness in the magnetic interactions of the magnetic moments. Nevertheless, such randomness is expected not to affect the magnetic behavior qualitatively [11, 43]. Moreover, random distribution of the orientation of the easy axis drastically affects the magnetic properties. The random anisotropy model developed by Alben et al. [44] provides a successful explanation of how the correlation length, l, exerts a relevant influence on magnetic structure. The important question is What is the range of orientational correlation of the spins? Let L be the correlation length of the magnetic structure. If we assume L > l, the number of oriented easy axes in a volume L3 should be N = (L/l)3.

The effective anisotropy can be written as:

where K is the local anisotropy where strength is assumed to be uniform everywhere. By minimizing the total energy with respect to L, the following expression can be deduced:

where A is the exchange stiffness parameter. If we consider A = 10-11 J/m and l = 10-9 m, which are typical values of ferromagnetic metallic glasses [45], L in equation (2) becomes 105/K2. For 3D-based alloys, we can take the value of K corresponding to crystalline samples (~104 J/m3) leading to L around 10-9 m, which is equal to the structural correlation length of an amorphous material.

In addition, the random anisotropy model provides the following expression for the average macroscopic anisotropy:

Equation (3) points out that the macroscopic structural anisotropy is negligible in 3D amorphous alloys (<K> ~ 10-9 K); this is a consequence of the averaging of several local easy axes, which produces the reduction in magnitude.

Special attention has been paid, in the last decade, to the study of nanocrystalline phases obtained by suitable annealing of amorphous metallic ribbons owing to their attractive properties as soft magnetic materials [1, 15, 19-21, 23, 46-48]. Such soft magnetic character is thought to have originated because the magnetocrystalline anisotropy vanishes and there is a very small magnetostriction value when the grain size approaches 10 nm [1, 12, 46]. As was theoretically estimated by Herzer [12, 46], average anisotropy for randomly oriented a-Fe(Si) grains is negligibly small when grain diameter does not exceed about 10 nm. Thus, the resulting magnetic behavior can be well described with the random anisotropy model [12, 19, 23, 46-48]. According to this model, the very low values of coercivity in the nanocrystalline state are ascribed to small effective magnetic anisotropy (Kef around 10 J/m3). However, previous results [19, 21, 49] as well as recently published results by Varga et al. [50] on the reduction of the magnetic anisotropy from the values in the amorphous precursors (~1000 J/m3) down to that obtained in the nanocrystalline alloys (around 300-500 J/m3), is not sufficient to account for the reduction of the coercive field accompanying the nanocrystallization. The enhancement of the soft magnetic properties should, therefore, be linked to the decrease of the microstructure-magnetization interactions. These interactions, originating in large units of coupled magnetic moments, suggest a relevant role of the magnetostatic interactions, as well a role in the formation of these coupled units [19, 49]. In addition to the suppressed magnetocrystalline anisotropy, low magnetostriction values provide the basis for the superior soft magnetic properties observed in particular compositions. Low values of the saturation magnetostriction are essential to avoid magnetoelastic anisotropies arising from internal or external mechanical stresses. The increase of initial permeability with the formation of the nanocrystalline state is closely related to a simultaneous decrease of the saturation magnetostriction. Partial crystallization of amorphous alloys leads to an increase of the frequency range, where the permeability presents high values [51]. These high values in the highest possible frequency range are desirable in many technological applications involving the use of ac fields.

It is remarkable that a number of workers have investigated the effects on the magnetic properties of the substitution of additional alloying elements for Fe in the Fe73-5Cu1Nb3Si13-5B9 alloy composition, Finemet, to further improve the properties, e.g., Co [52-56], Al [20, 57, 58], varying the degree of success. Moreover, it was shown in [20] that substitution of Fe by Al in the classical Finemet composition led to a significant decrease in the minimum of coercivity, H™ ~ 0-5 A/m, achieved after partial devitrification, although the effective magnetic anisotropy field was quite large (around 7 Oe) [59]. The coercivity behavior was correlated with the mean grain size, and a theoretical low effective magnetic anisotropy field of the nanocrystalline samples was assumed in contradiction with those experimentally found in [49, 50, 58].

Although amorphous Fe-, Co-, and Ni-based ribbons are slightly more expensive compared with conventional soft magnetic materials, such as sendust, ferrites, and supermalloys (mostly due to the significant content of Co and Ni), they found considerable applications in transformers (400 Hz), ac powder distributors (50 Hz), magnetic recording as a magnetic heads, and magnetic sensors. The main reason for using amorphous alloys such as soft magnetic materials is a saving of the electric energy wasted by magnetic cores. Besides, the combination of high magnetic permeability and good mechanical properties of amorphous alloys may be used successfully in magnetic shielding and in magnetic heads [51]. Production of about 3 millions heads per year in Japan in the mid-1980s has been reported [51].

The internal stresses, as the main source of magnetic anisotropy in amorphous and nanocrystalline materials, are due to the magnetoelastic coupling between magnetization and internal stresses through magnetostriction. Consequently, these materials are interesting for field- and stress-sensing elements because the Fe-rich amorphous alloys exhibit high magnetostriction values (As ~ 10-5) and, therefore, many of magnetic parameters (i.e., magnetic susceptibility, coercive field, etc.) are extremely sensitive to the applied stresses.

The discovery of Fe-rich nanocrystalline alloys carried out by Yoshizawa et al. [1] was important owing to the outstanding soft magnetic character of such materials. Typical compositions of the precursor amorphous alloys, which, after partial devitrification, reach the nanostructure character with optimal properties, are FeSi and FeZr, with small amounts of B to allow the amorphization process, and smaller amounts of Cu, which act as nucleation centers for crystallites, and Nb, which prevents grain growth. This effect is provided by Zr in FeZr alloys. After the first step of crystallization, FeSi or Fe crystallites are finely dispersed in the residual amorphous matrix. In a wide range of crystallized volume fraction, the exchange correlation length of the matrix is larger than the average intergranular distance, d, and the exchange correlation length of the grains is larger than the grain size, D. Magnetic softness of Fe-rich nanocrystalline alloys is due to a second complementary reason: the opposite sign of the magnetostriction constant of crystallites and residual amorphous matrix, which allows reduction and compensation of the average magnetostriction.

Figure 3 shows the thermal variation of the coercive field (Hc) in a Finemet-type (Ta-containing) amorphous alloy. This behavior is quite similar to that shown in the case of Nb-containing ones and particularly, evidences the occurrence of a maximum in the coercivity linked to the onset of the nanocrystallization process [60, 61].

Considering the grain size, D, to be smaller than the exchange length, Lex, and the nanocrystals are fully coupled between them, the random anisotropy model implies a dependence of the effective magnetic anisotropy (K), with the sixth power of average grain size, D. The coercivity is understood as a coherent rotation of the magnetic moments of each grain toward the effective axis leading to the same dependence of the coercivity with the grain size [16]:

KD JA3

K4Df A3

where K1 = 8 kJ/m3 is the magnetocrystalline anisotropy of the grains, A = 10-11 J/m is the exchange ferromagnetic constant, Js = 1.2 T, is the saturation magnetic polarization and pc is a dimensionless prefactor close to unity. The predicted D6 dependence of the coercive field has been widely accepted to be followed in a D range below Lex (around 30 to 40 nm) for nanocrystalline Fe-Si-B-M-Cu (m = Iva to Via metal) alloys [21, 46, 62-65]. A clear deviation from the predicted D6 law in the range below Hc = 1 A/m was reported by Hernando et al. [19]. Such deviation was ascribed to effects of induced anisotropy (e.g., magne-toelastic and field induced anisotropies) on the coercivity could be significant with respect those of the random mag-netocrystalline anisotropy. As a consequence, the data of Hc(D) were fitted by assuming a contribution from (i) the spatial fluctuations of induced anisotropies and (ii) (Ku) to Hc (i.e., a dependence Hc = ^(a2 + (bD)6) was found with a = 1 A/m representing the contribution originating from the induced anisotropies).

To investigate the effect of the grain size on coercivity, this dependence of Hc(D) in alloys treated by Joule heating was obtained. Experimental results on this dependence are shown in Figure 4 [21]. The fitting of this dependence appears to follow, surprising, a rough dependence of the Hc a D3-4 type (the best regression was found fitting the D3 law). It must be noted that our data of Hc(D) correspond to a grain size variation between about 10 to 150 nm. As it is well known, an analysis of this Hc(D) data in terms of the random anisotropy model is only justified if the grain size is smaller than Lex and, hence, could not be applicable (in the framework of the random anisotropy model) to the range grain size above Lex, which results in being only two points of our data in Figure 4 [21]. These points should correspond to a magnetic hardening due to the precipitation of the iron borides. In this case, the random anisotropy model should be applied by taking into account the volume fraction and the different anisotropy of the iron borides. This indicates that (K1) should vary as D3, contrary to the theoretical D6 law. This indicates that Hc is mainly governed by (K1), which varies as D3, contrary to the theoretical D6 law. This contradiction of the Hc(D) law between the theory and the experimental has recently been explained by Suzuki et al. [62] considering the presence of long-range uniaxial anisotropy, Ku, which influences the exchange correlation value and length, and yields an anisotropy average given by:

The second part of (5) corresponds to (Kl) and if Ku is coherent in space or if its spatial fluctuations are negligible to (Kl), this second part ultimately determines the grain size influence on the coercivity. Such influence changes from the D6 law to a D3-4 one when the coherent uniaxial anisotropies dominate over the random magnetocrystalline anisotropy. An additional point in order to justify the Eq. (5)

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