Magnetic Hysteresis

Hysteresis is at the heart of the behavior of magnetic materials. All applications, from electric motors to transformers and permanent magnets, from various types of electronic devices to magnetic recording, rely heavily on particular aspects of hysteresis. The variety of working conditions involved brings to light the fascinating richness of phenomena that may arise and drive the behavior of different materials. One soon realizes that naïve approaches, based on some empirical classification of material properties and on the use of limited phenomenological models developed on purpose, are largely inadequate for gaining convincing insight into the origin of the phenomena observed, or some significant power to predict them under various conditions. On the other hand, strong interest in hysteresis is not just the result of technological pressure. The comprehension of the physical mechanisms responsible for hysteresis and the development of adequate mathematical tools to describe it have attracted the attention of theoretical physicists and mathematicians for years. It is a beautiful example of a physical and mathematical problem of intriguing elegance and challenging complexity that is at the same time the source of pervading technological progress. Nobody can remain indifferent when considering the long but firm interdisciplinary chain that ties spin models of ferromagnetism to engineering applications of magnetic components.

The purpose of this chapter is to give an introductory overview of some general aspects of hysteresis, as they are observed in magnetic materials, and to make the reader acquainted with units, terms, and concepts frequently used in subsequent chapters. We illustrate some properties of hysteresis loops and their fundamental connection with magnetic domains. We discuss rate-independent hysteresis, rate-dependent effects (like eddy-current damping in metals), and thermal relaxation as the three fundamental classes of phenomena that may affect the behavior of a particular system. The presentation emphasizes qualitative aspects of general relevance, with little or no quantitative analysis. Frequent reference is made to subsequent chapters, where the same subjects will be discussed in greater detail.

1.1 HYSTERESIS LOOPS

If we were asked to mention the experimental fact giving the most distinctive fingerprint of ferromagnetism, then, depending on our scientific background, we would probably propose either the existence of the Curie point or the observation of hysteresis loops, examples of which are shown in Fig. 1.1. Loops like these are obtained by applying to the specimen a cyclic magnetic field H and by recording the ensuing change of the magnetization M or of the magnetic induction B along the field direction. M measures the average magnetic moment per unit volume in the material and characterizes its magnetic state. In this book, we shall employ the SI system of units,1 in which the magnetization, M, and the magnetic field, H, are measured in ampères per meter [Am-1], whereas the magnetic induction, B, and the magnetic polarization, I = µ0M, are measured in teslas [T]. µ0 = 4p 10- 7 H m- 1 is the permeability of the vacuum.

1.1.1 Hysteresis loops and magnetic domains

Hysteresis loops may take a variety of different forms and one would like to comprehend what are the basic physical mechanisms governing the observed phenomenology. We shall see that many interesting and highly nontrivial aspects contribute to the picture, and we shall have to go through several chapters of this book before attempting some convincing interpretation. Let us summarize here, as a simplified, introductory approach, the generally accepted view of the problem. A magnetic material can be imagined as an assembly of permanent magnetic moments mi, of quantum-mechanical origin. As an example, iron carries magnetic moments of 2.2µB per atom, where the Bohr magneton µB = hqe/4pme ? 9.27 10-24 A m2 (h is Planck's constant, qe and me are the electron charge and mass) is the natural atomic unit of magnetic moments. The simplest situation one may imagine is the one realized in an ideal paramagnet. This is a system where the individual mi vectors do not interact with each other, but are independently shaken by thermal agitation. As a consequence, they take random orientations in space, which gives zero net magnetization in any macroscopic piece of material. However, a nonzero magnetization can be induced by an external field Ha. The potential energy of a single moment in the field is equal to -µ0mi·Ha. This energy term favors the alignment of the moments along the field direction. Conversely, the tendency of thermal agitation is just to destroy any order possibly present. An appreciable net magnetization component along the field is obtained when the field energy is at least comparable in order of magnitude to the thermal energy, which means that µ0µBHa ~ kBT. At room temperature, this would take place for fields Ha ~ kBT/µ0µB ~ 3 108 Am-1, which is about 106 times the order of magnitude of the fields involved in the upper loop of Fig. 1.1. It is evident that fields of the order of 10-102 Am -1 should be absolutely insufficient to overcome thermal agitation and produce any appreciable magnetization.

In 1907, Weiss suggested that ferromagnetic materials could exhibit a large spontaneous magnetization even at low fields because the elementary magnetic moments were by no means independent, as assumed in the previous picture of a paramagnet, but were strongly coupled by an internal field proportional to the magnetization itself, HW ? M, which he termed molecular field. The molecular field introduces a positive feedback mechanism, because the presence of a net magnetization produces a nonzero molecular field, which in turn acts on all moments trying to further align them. The result is that, below a critical temperature, the Curie temperature Tc, the magnetic moments spontaneously attain long-range order and the material acquires a substantial spontaneous magnetization Ms. The knowledge of the experimental value of Tc permits one to estimate the order of magnitude of HW, because at the Curie point one expects µ0µBHW ~ kBTc. In the case of iron, where Tc ~ 103 K, one finds HW ~ kBTc/µ0µB ~ 109 Am-1. Well below Tc, the moments are nearly perfectly aligned. In iron, at room temperature, this gives a spontaneous magnetization of the order of 2 T, which is the order of magnitude recorded in Fig. 1.1.

The molecular field hypothesis explains the main aspects of the temperature dependence of the spontaneous magnetization, but it raises other conceptual difficulties. If there exists an internal field as enormous as the estimate of 109 Am-1 seems to indicate, then one would expect to find ferromagnetic materials spontaneously magnetized to saturation under all circumstances. Thus how can a hysteresis loop exist at all? It would seem impossible that fields of the order of 10 Am-1 can reduce to zero the magnetization of a ferromagnet, if they are confronted with internal fields of the order of 109 Am-1. Weiss proposed to resolve this difficulty by postulating that a ferromagnetic material is subdivided into regions, called magnetic domains. In each domain, the degree of magnetic moment alignment is dictated by the molecular field, but the orientation of the spontaneous magnetization can vary from domain to domain. As a consequence, when the magnetization is averaged over volumes large enough to contain many domains, one obtains values mainly determined by the relative orientation and volume of domains. The result can be very different from the spontaneous magnetization, and even close to zero.

In a sense, the history of the comprehension of magnetic materials in the twentieth century has been, above all, the history of the elaboration of Weiss' ideas. It took many years before quantum mechanics could give some hint to the microscopic interpretation of the molecular field. It took even longer before the importance of the concept of magnetic domain could be fully appreciated and exploited. In fact, we shall see that magnetic domains are the result of the complex and delicate balance of several competing energy terms, a balance by no means trivial to treat theoretically. However, these difficulties did not prevent domains from acquiring the status of real objects of the physical world, when direct...