Figure 15. Results of a micromagnetic simulations of the coercive field as a function of the system texture (the apex angle, in degrees, of the cone in which the easy axes are distributed). (Courtesy of R. Smirnov-Rueda).

nanostructured materials. This is due to the lack of precise experimental knowledge of the exact exchange value at the grain boundary. The magnetic grain boundary is influenced by the break of atomic bonds, which leads to different values of exchange integrals at different points of space. The higher the degree of perfection of the grain boundary structure, the higher is the exchange interaction across the boundary and thus the stronger is the coupling between neighboring grains. The exchange energy contribution may be found to be a function of the position inside grain [125, 134], since there may exist migration of nonmagnetic atoms toward the grain boundary (such as Cr in Co grains in typical thin films used for magnetic recording). In these films, there is evidence that the exchange value also depends on the angle between easy axes of two neighboring grains [125]. The atomistic models, in which exchange integrals are evaluated at each point, are necessary to correctly take into account the grain boundary. The latter becomes an extremely technically complicated problem for modern state of the art.

As a consequence, different phenomenological models [113, 114, 119, 124, 135] have been suggested to describe the exchange strength in nanostructured materials. Provided the average strength of their "exchange field" is the same, they should give the same influence of the "exchange" contribution, on average, in granular material. Detailed calculations are still necessarily to reveal the differences between different models. Generally speaking, different values for intergrain and intragrain exchange parameters also should be supposed. For granular materials, the intergrain exchange phenomenological model, which is based on the surfaces exchange [124] (strength of the exchange proportional to the surface of the contact grains), is considered to be a rather realistic description. Some preliminarily atomistic calculations in a model of a chain, which takes into account the difference of the concentration of Cr in CoCr films typically used for magnetic recording, has revealed that for Cr rich grain boundaries (more than 30% of Cr), the intragrain exchange parameter could be taken as almost constant [125].

In general, remanence enhancement is attributed to intergrain exchange interactions. Since exchange coupling of neighboring grains favors the nucleation of reversed domains, remanence enhancement generally is achieved at the expense of coercivity [136, 137]. Figure 16 presents simulations of different hysteresis cycles for various exchange parameters C* = Hex/HK (strength of the exchange field versus the anisotropy field) in a Co-based granular media. These effects are confirmed by multiple micromagnetic simulations [111, 138-143], although quantitatively stronger decrease of coercivity or stronger increase of the rema-nence has been predicted. The latter may be attributed to different evaluations of the magnetostatic energy present in the system or to different exchange models. The consequent decrease of remanence for very large exchange values is attributed to cooperative effects, for example, formation of vortices). Figure 14 supposes to simulate a standard Kerr-microscopy image. As expected, the average scale of the observed ripple structure increases strongly with the exchange coupling at grain boundaries.

Calculations show that the magnetostatic interactions in a granular material also influence the coercivity and rema-nence due to formation of magnetic charges at the grain

Applied Field

Figure 16. Simulated hysteresis cycles in a Co-based granular material with different strength of the exchange interactions modeled by an effective parameter C*, introduced by the group of R. Chantrell (see text) (the effective exchange decreases from the top remanence to bottom). The decrease of remanence and coercivity is clearly observed. Reprinted with permission from J. D. Hannay et al., J. Magn. Magn. Mater. 193, 245 (1999). © 1999, Elsevier Science.

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