The magnetic properties in solids originate mainly from the magnetic moments associated with electrons. The nuclei in solids also carry a magnetic moment. That, however, varies from isotope to isotope of an element. The nuclear magnetic moment is zero for a nucleus with even numbers of protons and neutrons in its ground state. The nuclei can have a non-zero magnetic moment if there are odd numbers of either or both neutrons and protons. However, the magnetic moment of a nucleus is three orders of magnitude less than that of the electron.

The microscopic theory of magnetism is based on the quantum mechanics of electronic angular momentum, which has two distinct sources: orbital motion and the intrinsic property of electron spin [1]. The spin and orbital motion of electrons are coupled by the spin–orbit interaction. The magnetism observed in various materials can be fundamentally different depending on whether the electrons are free to move within the material (such as conduction electrons in metals) or are localized on the ion cores. In a magnetic field, bound electrons undergo Larmor precession, whereas free electrons follow cyclotron orbits. The free-electron model is usually a starting point for the discussion of magnetism in metals. This leads to temperature-independent Pauli paramagnetism and Landau diamagnetism. This is the case with noble metals and alkali metals. On the other hand, localized non-interacting electrons in 3d-transition metals, 4f-rare earth elements, 5f-actinide elements, and their alloys and intermetallic compounds with incompletely filled inner shells exhibit Curie paramagnetism. Many transition metal-based insulating oxide and sulfide compounds also show Curie paramagnetism. In the presence of magnetic interactions, many such systems eventually develop long-range magnetic order if the magnetic interaction can overcome thermal fluctuations in some temperature regimes.

Against the above backdrop, in the next three chapters, we will introduce the readers to the basic phenomenology of magnetism, concentrating mainly on solid materials with some electrons localized on the ion cores. There are some excellent textbooks available on the subject, including those by J. M. D. Coey [1], B. D. Cullity and C. D. Graham [2], D. Jiles [3], S. J. Blundell [4], and N. W. Ashcroft and N. D. Mermin [5].

In this chapter, we shall study different types of ordered magnetic states that can arise as a result of various kinds of magnetic interactions as discussed in the previous section. In Fig. 5.1 we present some of these possible ground states: ferromagnet, antiferromagnet, spiral and helical structures, and spin-glass. There are other more complicated ground states possible, the discussion of which is beyond the scope of the present book. For detailed information on the various magnetically ordered states in solids, the reader should refer to the excellent textbooks by J. M. D. Coey [1] and S. J. Blundell [4].

Ferromagnetism

In a ferromagnet, there exists a spontaneous magnetization even in the absence of an external or applied magnetic field, and all the magnetic moments tend to point towards a single direction. The latter phenomenon, however, is not necessarily valid strictly in all ferromagnets throughout the sample. This is because of the formation of domains in the ferromagnetic samples. Within the individual domains, the magnetic moments are aligned in the same direction, but the magnetization of each domain may point towards a different direction than its neighbour. We will discuss more on the magnetic domains later on.

The Hamiltoninan for a ferromagnet in an applied magnetic field can be expressed as:

The exchange interaction Jij involving the nearest neighbours is positive, which ensures ferromagnetic alignment. The first term on the right-hand side of Eqn. 5.1 is the Heisenberg exchange energy, and the second term is the Zeeman energy. In the discussion below it is assumed that one is dealing with a system with no orbital angular momentum, so that L = 0 and J= S.

In order to solve the equation it is necessary to make an assumption by defining an effective molecular field at the ithsite by:

Now the total energy associated with ith spin consists of a Zeeman part gμB_Si._B and an exchange part. The total exchange interaction between the ith spin and its neighbours can be expressed as:

The factor 2 in Eqn. 5.3 arises due to double counting. The exchange interaction is essentially replaced by an effective molecular field Bmf produced by the neighbouring spins.

A neutron is a nuclear particle, and it does not exist naturally in free form. Outside the nucleus, it decays into a proton, an electron, and an anti-neutrino. The scattering of low energy neutrons in solids forms the basis of a very powerful experimental technique for studying material properties. A neutron has a mass mn= 1.675 × 10−27 kg, which is close to that of the proton and a lifetime τ = 881.5 ±1.5 s. This lifetime is considerably longer than the time involved in a typical scattering experiment, which is expected to be hardly a fraction of a second.

A neutron has several special characteristics, which makes it an interesting tool for studying magnetic materials as well as engineering materials and biological systems. It is an electrically neutral, spin-1/2 particle that carries a magnetic dipole moment of μ = -1.913 μN, where nuclear magneton μN = eh/mp = 5.051 ×10−27 J/T. The zero charge of neutron implies that its interactions with matter are restricted to the short-ranged nuclear and magnetic interactions. This leads to the following important consequences:

• 1. The interaction probability is small, and hence the neutron can usually penetrate the bulk of a solid material.

• 2. Additionally, a neutron interacts through its magnetic moment with the electronic moments present in a magnetic material strong enough to get scattered measurably but without disturbing the magnetic system drastically. This magnetic neutron scattering has its origin in the interaction of the neutron spin with the unpaired electrons in the sample either through the spin of the electron or through the orbital motion of the electron. Thus, the magnetic scattering of neutrons in a solid can provide the most direct information on the arrangement of magnetic moments in a magnetic solid.

• 3. Energy and wavelength of a neutron matches with electronic, magnetic, and phonon excitations in materials and hence provide direct information on these excitations.

Neutrons behave predominantly as particles in neutron scattering experiments before the scattering events, and as waves when they are scattered. They return to their particle nature when they reach the detectors after the scattering events.

Magneto-optical Effects

The magneto-optical effect arises in general as a result of an interaction of electromagnetic radiation with a material having either spontaneous magnetization or magnetization induced by the presence of an external magnetic field. Michael Faraday in 1846 demonstrated that in the presence of a magnetic field the linear polarization of the light with angular frequency w was rotated after passing through a glass rod. This rotation is now termed as Faraday rotation, and it is proportional to the applied magnetic field B. The angle of rotation θ(w) can be expressed as [1].

Here V(w) is a constant called the Verdet constant, which depends on the material and also on the frequency w of the incident light; |B| is the magnitude of the applied magnetic field, and l thickness of the sample. The Faraday effect is observed in non-magnetic as well as magnetic samples. For example, the Verdet constant of SiO2 crystal is 3.25 × 10−4 (deg/cm Oe) at the frequency w = 18300cm−1 [1]. This implies that a Faraday rotation of only a few degrees can be observed in a sample of thickness 1 cm in a magnetic field of 10 kOe. A much larger Faraday rotation can, however, be observed in the ferromagnetic materials in the visible wavelength region under a magnetic field less than 10 kOe.

In 1877 John Kerr showed that the polarization state of light could be modified by a magnetized metallic iron mirror. This magneto-optical effect in the reflection of light is now known as the magneto-optical Kerr effect (MOKE), and it is proportional to the magnetization M of the light reflecting sample. Today MOKE is a popular and widely used technique to study the magnetic state in ferromagnetic and ferrimagnetic samples. With MOKE it is possible to probe samples to a depth, which is the penetration depth of light. This penetration depth can be about 20 nm in the case of metallic multilayer structures. In comparison to the conventional magnetometers like vibrating sample magnetometer and SQUID magnetometer which measure the bulk magnetization of a sample, MOKE is rather a surface-sensitive technique.

Phenomenology

Maxwell equations in free space are expressed as:

The electric field E(r,t) and magnetic field B(r,t) are averaged over elementary volume ΔVcentred around the position r. Similarly ρ and j represent electric charge density and current density, respectively. Equation C.1 indicates the absence of magnetic charge and Eqn. C.2 represents Faraday's law of indication in differential form. These two equations do not depend on the sources of an electric field or magnetic field, and they represent the intrinsic properties of the electromagnetic field. Eqns. C.3 and C.4 contain ρ and j, and they describe the coupling between the electromagnetic field and its sources.

Let us now consider a sample of ferromagnetic material through which no macroscopic conduction currents are flowing. A ferromagnet is characterized by the presence of spontaneous magnetization that can produce a magnetic field outside the sample. The microscopic current density jmicro producing such a magnetic field can be associated with the electronic motion inside the atoms and electron spins, or elementary magnetic moments of the ferromagnetic materials. Such microscopic currents present in an elementary volume ΔVcentred about a position r gives rise to an average current [1]:

j M is termed as magnetization current and represents the current density in Maxwell Eqn. C.4 for a ferromagnetic material. This magnetization current jM does not represent any macroscopic flow of charges across the sample. It can rather be crudely associated with current loops confined to atomic distances. This, in turn, implies that the surface integral jM over any generic cross section Sof this ferromagnetic sample must be zero:

This, in turn, tells that jM(r) can be expressed as the curl of another vector M(r):

Now inserting Eqn. C.6 into Eqn. C.7 and with the help of Stoke's theorem, one can convert Eqn. C.6 into a line itegral along some contour completely outside the ferromagnetic sample:

The Eqn. C.8 will be satisfied under all circumstances provided M (r) = 0 outside the sample. This latter condition is true if we take M as the magnetization or magnetic moment density of the ferromagnetic sample. It can be seen from Eqn. C.7 that the magnetic field created by the ferromagnetic sample is identical to the field that would be created by a current distribution jM(r) = ∇×M(r).