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.