Using the techniques described in the previous chapter, we can in a unified way discuss properties of the electron gas with Coulomb interaction and consider such effects as optical response, screening, plasmons, etc. In many textbooks these properties are obtained using a variety of methods. The virtue of the Green function method is its universality and, I would say, not much simpler, but standardized form. This method permits one to obtain all the properties mentioned above in the form of one general expression, and it also gives the possibility to generalize the results quite easily to the cases of low-dimensional (1d, 2d) systems, or to take into account the details of the band structure of the material, etc. But more important is the fact that it leads naturally to a number of special interesting consequences which would be rather difficult to obtain with the usual classical methods. In this and in the next two chapters I will demonstrate how to reproduce, using this method, the familiar results such as Debye screening or the plasmon energy, but I will mostly concentrate on less frequently discussed effects which are quite naturally obtained using this technique.

Dielectric function, screening: random phase approximation

We start by studying the form of the effective electron–electron interaction in metals. The ordinary Coulomb interaction V(q) = 4πe2/q2 is modified by the reaction of the electronic system. The first, well-known effect is just the screening of the electric charge. But there are other interesting effects as well.

Introduction

Fast and affordable computing power, especially in the form of personal computers and workstations, has enabled the expansion of the study of soft condensed matter physics over the last two decades. The use of computing power to not only analyze image data but acquire it from high-resolution and high-speed digital sources has also made many significant investigations in soft condensed matter experiments accessible to students while still pursuing their undergraduate degrees. Unlike students just a generation before, contemporary undergraduates are well versed in the use of computing power, even operating systems such as Linux, when they first arrive on campus. Because of outreach initiatives such as REU programs, intentions to recruit future graduate students, and an increasingly competitive trend in the careers of undergraduate majors, the opportunities to engage undergraduates in research has flourished over nearly the same period of time.

Granular systems, with simple hard sphere interactions and inter-particle friction, tend to be investigated in experiments that are tabletop in scale. As a subset of soft condensed matter systems, the macroscopic nature of granular physics makes the systems conceptually accessible to students as early as the sophomore or junior year of their baccalaureate careers. This no way trivializes the investigations or minimizes the advances in knowledge that a properly trained undergraduate can contribute to the larger scientific community when mentored well.

In the previous section we have already seen that in the case of strong electron–electron interactions, when the average interaction energy becomes larger than the corresponding kinetic energy, one can expect drastic changes of the properties of the system. Notably, the electrons will have a tendency to localize, so as to minimize their repulsion at the expense of a certain increase in kinetic energy. Materials and phenomena for which this factor plays an important role are now at the centre of activity of both experimentalists and theoreticians; this interest was especially stimulated by the discovery of high-Tc superconductivity in which electron correlations play a very important role. But even irrespective of the high-Tc problem, there are a lot of other interesting phenomena which are connected with strong electron–electron interactions. These phenomena include electron localization, orbital ordering and certain structural phase transitions, insulator–metal transitions, mixed valence and heavy fermion behaviour. The very existence of localized magnetic moments in solids, both in insulators and in metals, is actually determined by these correlations. That is why this is one of the most actively studied classes of phenomena at present.

Real materials to which one applies the models and the treatment presented in this chapter are mostly transition metal and rare earth compounds, although general ideas developed in this context are now applied to many other systems, including organic materials, nanoparticles or supercooled atoms. The typical situation in transition metal compounds is the one with partially filled d-shells.

Introduction

Though the movement of a single, isolated bacterium is reasonably well understood, when a large number of interacting bacteria are put together they produce beautiful and often complex phenomena. “Large” here typically means from 106 to 1012 individual cells: a small population by thermodynamic standards, but certainly unwieldy for any except statistical descriptions. Even restricting ourselves to the simple system of bacteria moving on or in solidified agar plates, the colony structures produced are surprisingly rich. In dilute solutions, swimming cells move independently, interacting with each other through their common consumption of a reservoir of nutrient. A point source of swimming cells expands in concentric rings as successive waves of bacteria chase gradients of nutrients, sometimes condensing into regular geometric patterns by chasing self-generated gradients. More concentrated solutions of swimming cells interact hydrodynamically through the fluid, producing large-scale swirls reminiscent of turbulence. This swirling occurs in two-dimensional surface motility as well, where uncorrelated motion turns into large-scale swirling as surface density increases. At extremely high density, bacteria jam and stop moving, as occurs in colloids. These high densities occur naturally on hard surfaces, where colony expansion is driven by cell growth rather than by motion; as the surface property becomes softer and wetter and bacteria begin to move, the resulting colonies change from fractal-like to radially symmetric and finally to a form dominated by a fast-growing, single-cell-thick outer layer.

Introduction

Langmuir monolayers have proven themselves to be a powerful experimental system for the study of a range of issues in soft condensed matter [1–4]. Essentially, Langmuir monolayers are a single layer of insoluble molecules at the air–water interface. As such, they form an almost ideal two-dimensional system. This offers the opportunity to study fundamental questions in the phase behavior and material properties of two-dimensional systems. Secondly, they are readily transferred from the surface to a solid substrate. This has been the motivation for studying these systems for a range of technological applications. Finally, they are a natural system for the study of biological questions given that much of biology relies on processes at interfaces, such as cellular membranes and membranes of cellular components.

The experimental techniques associated with Langmuir monolayers can loosely be divided into two classes: formation of the monolayer and characterization of the monolayer. In this chapter, we will briefly review important issues in the formation of Langmuir monolayers. The focus will be mostly on the general concepts and issues, with some specific recipes given for illustrative purposes. However, it should be recognized that many groups specializing in Langmuir monolayers have refined and developed specific methods that are optimized for their system. So, some care needs to be taken in applying any given specific recipe. In the area of characterization, we will briefly review standard surface pressure characterization and provide a survey of commercial tools, including x-ray and neutron scattering techniques.

Introduction

This chapter will review aspects of soft random solids, including particulate gels, compressed emulsions, and other materials having similar basic features. Soft random solids are appealing for a number of reasons – and not just because of their taste (as in foods) or appearance (as in cosmetics). They offer the potential for insight into much broader and quite elegant problems in nonequilibrium thermodynamics such as the dynamics of phase transitions, the origin of the glass transition, and stresses and flows in granular media. They are also central players in a host of industries in the form of paint, ink, concrete, asphalt, dairy foods, and cosmetics. From the range of examples and the techniques involved, it should be apparent that investigation of these materials is a multi-disciplinary process, combining contributions from physicists, chemists, and engineers.

Since the 1980s, studies of soft random solids have benefited enormously from advances in microscopy, computer-aided image analysis, computer simulations, and new methods to synthesize colloidal particles with controlled shape, surface chemistry, and interactions. The chapter's purpose is to summarize areas of recent investigations and point out experimental breakthroughs, remaining questions, and relevant experimental methods. Despite the recent progress, a number of fundamentally interesting and practically relevant questions remain, and it is hoped that this chapter will stimulate further work in these areas.

In the previous chapter we studied two contributions to the magnetic moment of atoms – the electron spin and orbital angular momenta. Next we are going to investigate the third (and final) contribution to the magnetic moment of a free atom. This is the change in orbital motion of the electrons when an external magnetic field is applied.

The change in orbital motion due to an applied field is known as the diamagnetic effect, and it occurs in all atoms, even those in which all the electron shells are filled. In fact diamagnetism is such a weak phenomenon that only those atoms which have no net magnetic moment as a result of their shells being filled are classified as diamagnetic. In other materials the diamagnetism is overshadowed by much stronger interactions such as ferromagnetism or paramagnetism.

Observing the diamagnetic effect

The diamagnetic effect can be observed by suspending a container of diamagnetic material, such as bismuth, in a magnetic field gradient, as shown in Fig. 4.1. Since diamagnetic materials exclude magnetic flux, their energy is increased by the presence of a field, and so the cylinder swings away from the high-field region, towards the region of lower field (the north pole in the figure).

The purpose of this chapter is to understand the origin of the magnetic dipole moment of free atoms. We will make the link between Ampère's ideas about circulating currents, and the electronic structure of atoms. We'll see that it is the angular momenta of the electrons in atoms which correspond to Ampère's circulating currents and give rise to the magnetic dipole moment.

In fact we will see that the magnetic moment of a free atom in the absence of a magnetic field consists of two contributions. First is the orbital angular momenta of the electrons circulating the nucleus. In addition each electron has an extra contribution to its magnetic moment arising from its “spin.” The spin and orbital angular momenta combine to produce the observed magnetic moment.

By the end of this chapter we will understand some of the quantum mechanics which explains why some isolated atoms have a permanent magnetic dipole moment and others do not. We will develop some rules for determining the magnitudes of these dipole moments. Later in the book we will look at what happens to these dipole moments when we combine the atoms into molecules and solids.

Introduction

The data storage industry is huge. Its revenue was tens of billions of U.S. dollars per year at the end of the 20th century, with hundreds of millions of disk, tape, optical, and floppy drives shipped annually. It is currently growing at an annual rate of about 25%, and the growth rate can only increase as the storing and sending of digital images and video becomes commonplace, with the phenomenal expansion of the world wide web and in ownership of personal computers and mobile computing platforms.

Magnetic data storage has been widely used over the last decades in such applications as audio tapes, video cassette recorders, computer hard disks, floppy disks, and credit cards, to name a few. Of all the magnetic storage technologies, magnetic hard-disk recording is currently the most widely used. In this chapter, our main focus will be on the technology and materials used in writing, storing, and retrieving data on magnetic hard disks. Along the way we will see how some of the phenomena that we discussed in Part II, such as magnetoresistance and single-domain magnetism in small particles, play an important role in storage technologies.

RAMAC, the first computer containing a hard-disk drive, was made by International Business Machines Corporation (IBM) in 1956.

The term “magnetic anisotropy” refers to the dependence of the magnetic properties on the direction in which they are measured. The magnitude and type of magnetic anisotropy affect properties such as magnetization and hysteresis curves in magnetic materials. As a result the nature of the magnetic anisotropy is an important factor in determining the suitability of a magnetic material for a particular application. The anisotropy can be intrinsic to the material, as a result of its crystal chemistry or its shape, or it can be induced by careful choice of processing method. In this chapter we will discuss both intrinsic and induced anisotropies in some detail.

Magnetocrystalline anisotropy

In Chapter 7 we introduced the concept of magnetocrystalline anisotropy, which is the tendency of the magnetization to align itself along a preferred crystallographic direction. We also defined the magnetocrystalline anisotropy energy to be the energy difference per unit volume between samples magnetized along easy and hard directions. The magnetocrystalline anisotropy energy can be observed by cutting a {110} disk from a single crystal of material as shown in Fig. 11.1, and measuring the M–H curves along the three high-symmetry crystallographic directions and contained within the disk.

Schematic results for single-crystal samples of ferromagnetic metals such as iron and nickel were shown in Fig. 7.4. Body-centered cubic Fe has the 〈100〉 direction as its easy axis.

In Chapter 8 we described the original 1956 experiment on Co/CoO nanoparticles in which the shift in hysteresis loop known as exchange bias or exchange anisotropy was first observed. The goal of this chapter is to describe the exchangebias phenomenon in more detail and to point out open questions in the field, which remains an active area of research. Significantly, a simple theoretical model that accounts for all experimental observations is still lacking.

Remember that exchange bias appears when a ferromagnetic/antiferromagnetic interface is cooled in the presence of a magnetic field through the Néel temperature of the antiferromagnet (Fig. 14.1). The Curie temperature of the ferromagnet should be above the Néel temperature of the antiferromagnet so that its moments are already aligned in the field direction; this is usually the case for typical FM/AFM combinations. In a simple model, the neighboring moments of the antiferromagnet then align parallel to their ferromagnetic neighbors when their Néel temperature is reached during the field cooling process. An exchange-biased system shows two characteristic features: first, a shift in the magnetic hysteresis loop of the ferromagnet below the TN of the AFM, as though an additional biasing magnetic field were present, resulting in a unidirectional magnetic anisotropy; and second, an increase in coercivity and a wider hysteresis loop, which can even occur independently of the field cooling process.

In Chapter 2 we introduced the concept of ferromagnetism, and looked at the hysteresis loop which characterizes the response of a ferromagnetic material to an applied magnetic field. This response is really quite remarkable! Look at Figs. 2.3 and 2.4 again – we see that it is possible to change the magnetization of a ferromagnetic material from an initial value of zero to a saturation value of around 1000 emu/cm3 by the application of a rather small magnetic field – around tens of oersteds.

The fact that the initial magnetization of a ferromagnet is zero is explained by the domain theory of ferromagnetism. The domain theory was postulated in 1907 by Weiss and has been very successful. We will discuss the details of the domain theory, and the experimental evidence for the existence of domains, in the next chapter.

The subject of this chapter is: How can such a small external field cause such a large magnetization? In Exercise 6.2(b), you'll see that a field of 50 Oe has almost no effect on a system of weakly interacting elementary magnetic moments. Thermal agitations act to oppose the ordering influence of the applied field, and, when the atomic magnetic moments are independent, the thermal agitation wins.