Negative exchange J < 0 leads to magnetic order that depends on lattice topology. Structures with more than one magnetic sublattice include antiferromagnets and ferrimagnets. An antiferromagnet has two equal but oppositely directed sublattices, where the sublattice magnetization disappears above the Néel point TN. Two unequal oppositely directed magnetic lattices constitute a ferrimagnet. The molecular field theory is extended to cover these cases. A wealth of more complex noncollinear magnetic structures exist. The subtle effects of a noncrystalline structure are manifest in amorphous magnets, where spins sometimes freeze in random orientations. Magnetic model systems highlight the influence of some particular feature on collective magnetic order, such as reduced space or spin dimensionality, a particular distribution of exchange interactions, special topology or lack of crystal structure. Examples include the two-dimensional Ising model, frustrated antiferromagnets and canonical spin glasses.

Antiferromagnetism is an occult magnetic order. A crystal lattice is subdivided into two or more atomic sublattices which order in such a way that their net magnetization is zero. Louis Néel, who was a student of Pierre Weiss, first discussed this possibility in 1936 for two equal and oppositely aligned sublattices. The antiferromagnetic ordering transition, known as the Néel point, is marked by a small peak in the magnetic susceptibility, and a substantial specific heat anomaly, similar to that found at the Curie point of a ferromagnet (Fig. 5.36).

After a short historical summary, the central concepts of magnetic order and hysteresis are presented. Magnet applications are summarized, and magnetism is situated in relation to physics, materials science and industrial technology.

A brief history of magnetism

The history of magnetism is coeval with the history of science. The magnet's ability to attract ferrous objects by remote control, acting at a distance, has captivated countless curious spirits over two millenia (not least the young Albert Einstein). To demonstrate a force field that can be manipulated at will, you need only two chunks of permanent magnet or one chunk of permanent magnet and a piece of temporary magnet such as iron. Feeble permanent magnets are quite widespread in nature in the form of lodestones – rocks rich in magnetite, the iron oxide Fe3O4 – which were magnetized by huge electric currents in lightning strikes. Priests and people in Sumer, ancient Greece, China and pre-Colomban America were familiar with the natural magic of these magnets.

A lodestone carved in the shape of a Chinese spoon was the centrepiece of an early magnetic device, the ‘South pointer’. Used for geomancy in China at the beginning of our era (Fig. 1.1), the spoon turns on the base to align its handle with the Earth's magnetic field. Evidence of the South pointer's application can be seen in the grid-like street plans of certain Chinese towns, where the axes of quarters built at different times are misaligned because of the secular variation of the direction of the horizontal component of the Earth's magnetic field.

The quantum mechanics of a single electron in a central potential leads to classification of the one-electron states in terms of four quantum numbers. The individual electrons' spin and orbital angular momenta are coupled in an isolated manyelectron ion to give total spin and orbital quantum numbers S and L. Spin-orbit coupling then operates to split the energy levels into a series of J -multiplets, the lowest of which is specified by Hund's rules. Curie-law susceptibility, χ = C /T, is calculated for a general value of J. When placed in a solid, the ion experiences a crystal field due to the charge environment which disturbs the spin-orbit coupling, leaving either S or J as the appropriate quantum number. The crystal field modifies the structure of the lowest MS or MJ magnetic sublevels which are split by the Zeeman interaction and it introduces single-ion anisotropy.

Atomic physics is concerned with the energy levels of an atom or ion and the possible transitions between them, which are usually in the optical or ultraviolet energy range (1–10 eV). Magnetism is concerned with the energy levels that are occupied at ambient temperature, which usually means only the ground state, and its sublevels resulting from interactions with magnetic or electric fields, which produce a splitting of less than 0.1 eV. At ambient temperature kBT is about 25 meV.

Back to basics

The dipole moment m is the elementary magnetic quantity, and magnetization M(r) is its mesoscopic volume average. The primary magnetic field B is related to the auxiliary magnetic field H and the magnetization by B = µ0(H + M). Sources of magnetic field are electric currents and magnetized material. The field produced by a given distribution of magnetization can be calculated by integrating the dipole field due to each volume element M(r)dV, or using the equivalent distributions of electric currents or magnetic charge. Magnetic scalar and vector potentials ϕm and A are defined for H and B, respectively. The internal, external and demagnetizing fields are distinguished. Internal field may be defined on a mesoscopic or a macroscopic scale, the latter in terms of the demagnetizing factor N. Magnetic forces and energies are related to magnetization and external field.

We begin with magnetostatics, the classical physics of the magnetic fields, forces and energies associated with distributions of magnetic material and steady electric currents. The concepts presented here underpin the magnetism of solids. Magnetostatics refers to situations where there is no time dependence.

The magnetic dipole moment

The elementary quantity in solid-state magnetism is the magnetic moment m. On an atomic scale, intrinsic magnetic moments are associated with the spin of each electron and a further contribution is associated with its orbital motion around the nucleus.

Order, order, order!

Ferromagnetism and the Curie temperature were explained by Weiss in terms of a huge internal ‘molecular field’ proportional to the magnetization. The theory is applicable both to localized and delocalized electrons. No such magnetic field really exists, but it is a useful way of approximating the effect of the interatomic Coulomb interaction in quantum mechanics, which Heisenberg described by the Hamiltonian ℋ = −2JS1 · S2, where S1 and S2 are operators describing the localized spins on two adjacent atoms. When J > 0, ferromagnetic exchange leads to ferromagnetic order in three dimensions. Spin waves are the low-energy excitations of the exchange-coupled magnetic lattice. In the delocalized electron picture, a ferromagnet has spontaneously spin-split energy bands. The density of ↑ and ↓ states is calculated using spin-dependent density functional theory. Important physical phenomena associated with ferromagnetism are discussed in this chapter, including magnetic anisotropy and, magnetoelastic, magneto-optic and magnetotransport effects.

The characteristic feature of a ferromagnet is its spontaneous magnetization Ms, which is due to alignment of the magnetic moments located on an atomic lattice. The magnetization tends to lie along easy directions determined by crystal structure, atomic-scale texture or sample shape. Heating above a critical temperature known as the Curie point, which ranges from less than 1 K for magnetically dilute salts to almost 1400 K for cobalt, leads to a reversible collapse of the spontaneous magnetization.

The keenest desire of matter is form

Almost all magnetically ordered materials involve 3d or 4 f elements. The late 3d metals and their alloys, including interstitial alloys and intermetallic compounds, are frequently ferromagnetic. Large magnetocrystalline anisotropy in 3d–4 f intermetallics, due mainly to the rare-earth, gives useful hard magnets. Conversely, the anisotropy can be reduced practically to zero in certain 3d alloys and, when magnetostriction is also vanishingly small, perfect soft ferromagnetism results. Oxides and other ionic compounds are usually insulators with localized electrons. There, antiferromagnetic superexchange coupling leads to antiferromagnetic or ferrimagnetic order. Some oxides, however, are metals, with the d-electrons forming a conduction band. Occasionally the 3d band is half-metallic. A few examples are included of materials showing magnetic order which involves neither 3d nor 4 f electrons.

Introduction

This chapter is a catalogue of representative magnetically ordered materials. The selection is biased towards materials that are practically useful, or illustrate some interesting aspect of magnetic order. Included are the common iron-group metals and alloys, the rare-earths, intermetallic and interstitial compounds, as well as a range of oxides with ferromagnetic or antiferromagnetic interactions. The catalogue covers insulators, semiconductors, semimetals and metals. Ferromagnetic, antiferromagnetic, ferrimagnetic and noncollinear spin structures are encountred. Examples of noncrystalline metals and insulators are also included. Table 11.1 collects information on the 38 representative materials. Each is described more fully on a data sheet, where its properties and significance are indicated, and related materials are presented.

Disciplines grow at their boundaries. The interdisciplinary topics considered here fall into three groups. One is mainly concerned with liquids: paramagnetic liquids, ferrofluids, magnetic levitation and confinement, and magnetoelectrochemistry. The second relates to life sciences: magnetism in biology and medicine, magnetic imaging and magnetically aided diagnostics. Finally there is planetary and stellar magnetism, covering the magnetism of rocks and the Earth's magnetic field, as well as and those of other planets, the Sun and stars.

Magnetism has been a spur to human curiosity for centuries. The force field with its attractive and repulsive interactions led to dreams of levitation and perpetual motion, and hopes for cures of illness, as well as a striving for understanding. These hopes and dreams have been realized in unexpected ways. Magnetism is a mature discipline with a secure physical foundation, which allows it to engage in interdisciplinary joint ventures with other branches of science.

If perpetual motion has proved to be a pipe dream – periodically revived to peddle to gullible investors – it nevertheless finds an echo in the stationary states of quantum mechanics where the electrons occupy quantized orbits. There, they enjoy undiminished motion, at least until they exchange a quantum of energy with their environment. But they can do no work in their stationary states. Energy conservation is inviolate.

Levitation is a more practical proposition, but again not as people imagined long ago – for example, Jonathan Swift's island of Laputa, the ‘coffin of the Prophet’ in Medina or the golden idol in the temple of Somnath.

Conventional electronics has ignored the spin on the electron

Spin electronics exploits the angular momentum and magnetic moment of the electron to add new functionality to electronic devices. A first generation of devices comprised magnetoresistive sensors and magnetic memory. The sensors have numerous applications, especially in digital recording. Magnetic recording uses semihard magnetic thin films as the recording media. Write heads are miniature thin-film electromagnets, while read heads are usually spin-valves exhibiting giant magnetoresistance (GMR) or tunnelling magnetoresistance (TMR). Magnetic randomaccess memory (MRAM) is based on switchable spin valve cells, similar in structure to the read head. New generations of spin electronic devices are under development in which the angular momentum of a spin-polarized current is used to exert spin transfer torque, or the flow of spin-polarized electrons is controlled via a third electrode in a transistor-like structure.

A hugely successful electronics technology has been built around the manipulation of electronic charge in semiconductor microcircuits. The operations needed for computation are conducted using complementary metal-oxide semiconductor (CMOS) logic. The semiconductors can be doped n- or p-type so that the charge carriers may be electrons or holes. Binary data are stored as charge on the gates of field-effect transistors (FETs). An important feature of CMOS logic, Fig. 14.1 is that it only consumes power when the transistors are switching between the on and off states. It is scalable technology, which has been repeatedly miniaturized since its introduction in 1982.

Permanent magnets deliver magnetic flux into a region of space known as the air gap, with no expenditure of energy. Hard ferrite and rare-earth magnets are ideally suited to generate flux densities comparable in magnitude to their spontaneous polarization Js. Applications are classified by the nature of the flux distribution, which may be static or time-dependent, as well as spatially uniform or nonuniform. Applications are also discussed in terms of the physical effect exploited (force, torque, induced emf, Zeeman splitting, magnetoresistance). The most important uses of permanent magnets are in electric motors, generators and actuators. Their power ranges from microwatts for wristwatch motors to hundreds of kilowatts for industrial drives. Annual production for some consumer applications runs to tens or even hundreds of millions of motors.

The flux density Bg in the airgap (equal to µ0Hg) is the natural field to consider in permanent magnet devices because flux is conserved in a magnetic circuit, and magnetic forces on electric charges and magnetic moments all depend on B.

A static uniform field may be used to generate torque or align pre-existing magnetic moments since г = m × Bg. Charged particles moving freely through the uniform field with velocity v are deflected by the Lorentz force f = qv × Bg, which causes them to move in a helix, turning with the cyclotron frequency (3.26) fc = qB/2πm.

In this chapter we temporarily leave the arena of many-body physics and second quantization and, at least superficially, return to single-particle quantum mechanics. By establishing the path integral approach for ordinary quantum mechanics, we will set the stage for the introduction of field integral methods for many-body theories explored in the next chapter. We will see that the path integral not only represents a gateway to higher-dimensional functional integral methods but, when viewed from an appropriate perspective, already represents a field theoretical approach in its own right. Exploiting this connection, various concepts of field theory, namely stationary phase analysis, the Euclidean formulation of field theory, instanton techniques, and the role of topology in field theory, are introduced in this chapter.

The path integral: general formalism

Broadly speaking, there are two approaches to the formulation of quantum mechanics: the “operator approach” based on the canonical quantization of physical observables and the associated operator algebra, and the Feynman path integral.

In this chapter, the concept of path integration is extended from quantum mechanics to quantum field theory. Our starting point is a situation very much analogous to that outlined at the beginning of the previous chapter. Just as there are two different approaches to quantum mechanics, quantum field theory can also be formulated in two different ways: the formalism of canonically quantized field operators, and functional integration. As to the former, although much of the technology needed to efficiently implement this framework – essentially Feynman diagrams – originated in high-energy physics, it was with the development of condensed matter physics through the 1950s, 1960s, and 1970s that this approach was driven to unprecedented sophistication. The reason is that, almost as a rule, problems in condensed matter investigated at that time necessitated perturbative summations to infinite order in the non-trivial content of the theory (typically interactions).