Thermodynamic theory of domain structures

Abstract

A detailed description is given of certain results of domain structure theory which have been obtained recently and which have not, so far, been treated in monographs or reviews. A large part of the article consists of an analysis of the fundamentals of the theory; the properties of domain structures which are independent of those of the model are then considered in detail. Domain structures in uniaxial and cubic ferromagnets serve as practical examples ; experimental results are given simply by way of illustration. Certain generally accepted ideas are shown to be mistaken and are criticized here. This criticism appears to be necessary because the ideas in question are widespread and are given in many text books.

Bounds on the micromagnetic energy of a uniaxial ferromagnet

“…For example, it is well-known that the optimal one-dimensional pattern has energy of order ε 1/2 L 1/2 and domain width proportional to L 1/2 ; we shall review this calculation below (see Figures 1.2 and 1.3). Optimizing in a more complicated class of "branched" domain structures achieves an energy of order ε 2/3 L 1/3 and a basic domain width proportional to L 2/3 : this was shown in slightly different settings by Lifshitz [15], Privorotskiȋ [17], and Hubert [5]. This calculation has been used to explain why magnetic domains branch and their widths scale as L 2/3 when ε/L is sufficiently small.…”

“…They require rather different arguments, both for the upper bound (obtained in each case by a suitable branched construction) and for the lower bound (which must consider an arbitrary divergence-free magnetization in one case, and an arbitrary anisotropyfree magnetization in the other). The branched construction we use for (P2) is essentially due to Privorotskiȋ [17].…”

“…This idea goes back at least to Landau, Lifshitz, and Kittel, see, e.g., [14], [15], [9], and [10]. More recent work related to the uniaxial case includes that of Privorotskiȋ [17], Hubert [5], Kaczer [8], Marchenko [16], and Gabay and Garel [4].…”

Energy-driven pattern formation

“…One must simply give an example of an m with the desired scaling. This is done in [12], [52] for a slightly different model in which interfaces are sharp rather than diffuse. (See see also [21] for a concise summary.)…”

“…Therefore it should not be surprising that many of our results were guessed long ago. For example, the scaling of the minimum energy for a uniaxial ferromagnet (Section 2.2) has been "known" for decades [33], [52]. The cross-tie wall, however, is an exception to this rule.…”