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).

We have studied in earlier chapters that spin-based techniques like neutron scattering, muon spin resonance spectroscopy, and nuclear magnetic resonance can give detailed information on the magnetic structure of a material down to the atomic scale. These techniques, however, cannot provide a real-space image of the magnetic structure and are not sensitive to samples having nanometre-scale volumes. On the other hand, techniques like magnetic force microscopy, scanning hall bars, and superconducting quantum interference devices (SQUIDs) enable real-space imaging of the magnetic fields in nanometre-scale samples. But they have a constraint of finite size, and also they act as perturbative probes working in a rather narrow temperature range. A relatively new technique of magnetometry based on the electron spin associated with the nitrogen-vacancy (NV) defect in diamond combines the powerful aspects of both these classes of experiments. A very impressive combination of capabilities has been demonstrated with NV magnetometry, which sets it apart from other magnetic sensing techniques. That includes room-temperature single-electron and nuclear spin sensitivity, spatial resolution on the nanometre scale, operation under a broad range of temperatures from ∽1 K to above room temperature, and magnetic fields ranging from zero to a few tesla, and most importantly it involves a non-perturbative operation [1]. Here we present a concise introduction to NV magnetometry. There are a few excellent review articles [2, 3] and tutorial article [4] on NV magnetometry, and readers are referred to those for a more detailed exposure to the subject.

Physics of the Nitrogen-Vacancy (NV) Centre in Diamond

Figure 12.1 presents an NV centre in the crystal lattice of a diamond. It is a point defect consisting of a substitutional nitrogen atom and a missing carbon atom in the neighborhood. The NV centres can have negative (NV−), positive (NV+), and neutral (NV0) charge states of which NV− is used for magnetometry [2]. An NV− centre has six electrons, five of which come from the dangling bonds of the three neighbouring carbon atoms and the nitrogen atom. The negative charge state arises from one extra electron captured from an electron donor. The NV axis is defined by the line connecting the nitrogen atom and the vacancy. There can be four NV alignments depending on the four possible positions of the nitrogen atom with respect to the vacancy.

Magnetic imaging techniques enable one to have a direct view of magnetic properties on a microscopic scale. One of the most well-known magnetic microstructures is the magnetic domain. The other example of magnetic microstructures is the nucleation and growth of a magnetic phase across a first-order magnetic phase transition. Such structures can be observed in real space, and their distribution as a function of material and geometric properties can be investigated in a straightforward manner. In this chapter, we will discuss three different classes of magnetic imaging techniques, namely (i) electron-optical methods, (ii) imaging with scanning probes, and (iii) imaging with X-rays from synchrotron radiation sources. There are numerous scientific papers and review articles on these subjects. Instead of going into detail about the individual techniques, this chapter will provide a general overview of the working principles of various magnetic imaging techniques. There are not many specialist books, monographs, or review articles covering all these magnetic imaging techniques under the same cover, but the present author has found the book Modern Techniques for Characterizing Magnetic Materials [1] and the article “Magnetic Imaging” [2] to be quite useful while writing this chapter.

Electron-Optical Methods

Electron-optical methods and electron microscopy encompass a large body of techniques for magnetic imaging. The advanced electron microscopy techniques today can provide images with very impressive resolutions of the order of 1 nm, and show high contrast and sensitivity to detect small changes in magnetization in a material. The particle-like classical picture of the Lorentz force acting on an electron in a magnetic field form the basis of magnetic images of materials observed in various modes of electron microscopy. Electrons are charged particles, and hence electromagnetic fields are utilized as lenses for electrons. A magnetic lens consists of copper wire coils with an iron bore. The magnetic field generated by this assembly acts as a convex lens, which can bring the off-axis electron beam back to focus. Change in the trajectory of an electron in the magnetic field of a magnetic sample results in magnetic contrast and, in turn, provides information on the local magnetization in the material. However, the correct interpretations of the results in many cases involve a wave-like quantum mechanical picture of electrons.

Magnetic Moment and Magnetization

When we examine a magnetic material, it is first essential to identify parameters that characterize the response of the magnetic material to an applied magnetic field. We will see that these parameters are magnetic moment and magnetization.

Magnetic Moment

All of us at some point in our lives have come across magnets and experienced the strange forces of attraction and repulsion between them. These magnetic forces appear to originate in regions called poles, which are located near the ends of, say, a bar magnet. In magnets, poles always occur in pairs, but it is impossible to separate them. A magnetic field is created by a magnetic pole, which pervades the region around the pole [2]. This magnetic field causes a force on a second pole nearby. This magnetic force is directly proportional to the product of the pole strength p and field strength or field intensity H, which can be verified experimentally:

This equation defines H if the proportionality constant k is put equal to 1. A magnetic field of unit strength causes a force of 1 dyne on a unit pole [2]. In CGS units, a field of unit strength has an intensity of 1 oersted (Oe).

Let us now consider a bar magnet with poles of strength p located near each end and separated by a distance l, which is placed at an angle θ to a uniform field B = μ0H(Fig. 3.1). The magnet will experience a torque, which will tend to turn the magnet parallel to the magnetic field. The moment μof this torque is expressed as [2]:

When H= 1 Oe and θ = 900 , the moment is given by μ = pl. The magnetic moment of the magnet is defined as the moment of the torque experienced by the magnet when it is at right angles to a uniform field of 1 Oe. In a non-uniform magnetic field, the magnet will also feel a translational force acting on it. We will see in the subsequent sections that magnetic moment is a fundamental quantity for magnetism in materials.

Electromagnetic (EM) radiation consists of coupled electric and magnetic fields oscillating in directions perpendicular to each other and the direction of propagation of radiation. EM radiation can be an interesting probe to study materials’ properties. It is the electric field component of EM radiation that interacts with molecules and solids in most cases. Two conditions need to be fulfilled for the absorption of EM radiation during such interaction: (i) the energy of a quantum of EM radiation must be equal to the separation between energy levels in the atom/molecule, (ii) the oscillating electric field component must be able to stimulate an oscillating electric dipole in the atom/molecule. EM radiation in the microwave region of the EM spectrum can interact with molecules having a permanent electric dipole moment created by molecular rotation. On the other hand, infrared radiation would interact with molecules in vibrational modes giving rise to a change in the electric dipole moment.

Similarly, a solid or molecule containing magnetic dipoles is expected to interact with the magnetic component of EM radiation. EM irradiation of a molecule over a wide range of spectral frequencies does not normally result in absorption attributable to magnetic interaction. The absorption of EM radiation attributable to magnetic dipole transitions may, however, occur at one or more characteristic frequencies if the material of interest is additionally subjected to a static magnetic field. The application of a magnetic field B can cause precession of a magnetic moment at an angular frequency of |γB|, where γ is the gyromagnetic ratio. A material with magnetic moments placed in a magnetic field can absorb energy at this frequency. It is thus possible to observe a resonant absorption of energy from an EM wave tuned to an appropriate frequency. This phenomenon is known as “magnetic resonance”, and can be studied with a number of different experimental techniques, depending upon the type of magnetic moment involved in the resonance.

The presence of a static magnetic field is a crucial requirement for magnetic dipolar transitions. If there is no static magnetic field, the energy levels will be coincident. The permanent magnetic moments in a material are associated either with electrons or with nuclei. The magnetic dipoles arise from net electronic or nuclear angular momentum.

We have seen in the previous chapter that thermal neutrons with their unique combination of wavelength (≈˚A), magnetic moment (≈ μB), and penetration power in materials can be a very sensitive probe to investigate microscopic magnetic properties. Elastic neutron scattering experiments provide information on the magnitudes and direction of the magnetic moment for complex spin arrangements. Inelastic neutron scattering experiments provide information on spin dynamics such as spin waves or magnons, spin fluctuations, crystal-field excitations, etc.

It had been known for quite some time that X-rays being part of the electromagnetic spectrum can have specific interactions with ordered arrangements of spins in magnetic materials. The absorption cross section and the scattering amplitudes of electromagnetic waves are related through the optical theorem [1], hence magnetic effects are also expected to be observed in X-ray scattering. However, that magnetic interaction is very weak, leading to only very small effects, and as a result for many years, X-rays were not considered as a useful probe of magnetism. In the 1970s–80s in a series of elegant pioneering experiments de Bergevin and Brunel [2, 3] and Gibbs and his collaborators [4, 5] demonstrated that X-ray diffraction effects from magnetic materials could be detected experimentally. These important developments along with the relatively easy access to modern synchrotron radiation sources with the capability of probing small samples with a highly collimated and intense X-ray beam, high degree of photon beam polarization, and energy tunability have made magnetic X-ray diffraction studies a complementary and in some cases competitive technique to neutron scattering studies in the study of magnetic materials [6, 7, 8, 9]. These magnetic X-ray diffraction techniques will be discussed below, and the narrative will closely follow the articles by C. Vettier and L. Paolisini [6, 7].

Magnetic and Resonant X-ray Diffraction

In a purely nonrelativistic limit, X-rays interact only with the electronic charges in solids, hence no information is obtained on the spin densities. The first calculation of the amplitude of X-rays elastically scattered by a magnetically ordered solid was carried out by Platzman and Tzoar [10] in 1970 within the framework of the relativistic quantum theory. That calculation predicted that magnetic X-ray scattering could be observed. Within two years De Bergevin and Brunel [2] reported the first experimental observation of such effects in a single-crystal sample of antiferromagnetic NiO.

A magnetic field is called a steady field if the timescale of the increasing (or decreasing) magnetic field during an experiment is a few minutes or a few tens of minutes. On the other hand, if the magnetic field changes in a very short time, like milliseconds, it is called a pulsed-field [1]. Pulsed-fields with a duration of several hundred milliseconds up to several seconds are sometimes called long pulse fields [1].

Steady Field

Electromagnets are used if the magnetic field requirement is below 2 T. This is because of the saturation of iron cores used in conventional electromagnets. In 1933 the American physicist Francis Bitter introduced a design of an electromagnet to produce a higher magnetic field. This is known as the Bitter magnet where the current flows in a helical path through circular conducting metal plates stacked in a helical configuration with insulating spacers [2]. The schematic of a Bitter magnet is shown in Fig. A.1. The stacked metal plate design helps to withstand the Lorentz force due to the magnetic field acting on the moving electric charges in the plate. The Lorentz force causes an enormous outward mechanical pressure. The other important point in the design of Bitter magnets is about how to remove the huge amount of heat generated in this resistive magnet. This is achieved by flowing water along the axial direction through the holes incorporated in the stacked plates [1]. Bitter magnets are still used today to produce fields of up to 33 T at the National High Magnetic Field Laboratory (NHMFL) at Florida State University, Tallahassee, USA. Fig. A.2 shows a Florida Bitter disk from Tallahassee with highly elongated, staggered cooling holes. In more recent times a magnetic field of 37.5 T is produced at room temperature by a Bitter electromagnet at the High Field Magnet Laboratory in Nijmegen, Netherlands.

Superconducting magnets are more popular nowadays in laboratories worldwide to produce fields up to 20 T. A magnetic field of 7 T using Nb3Sn-based superconducting magnet was first generated in 1961 by Kunzler et al. [3], and the superconducting magnets that produced over 10 T became commercially available in the 1970s [1].