Simple crystal lattices

Atoms in molecules and solids are bound together by the valence electrons which have been supplied by them and are shared among them. In this chapter we will consider the motion of atoms under given interatomic forces without asking about their nature – the electronic origin of the bonds. We will confine ourselves to solids and study some simple characteristics of the atomic motions within them – lattice vibrations – brought about by the periodic structure of the crystal lattice. The lattice vibrations will turn out to be a prototype of elementary excitations in a solid, other examples of which will be given in later sections.

Consider an assemblage of a great number (N) of identical atoms with interatomic potential ν(r) which typically is attractive except for the hard-core repulsion, and let them be arrayed in one-dimensional space with positions x1, x2, … in increasing order. In the approximation that only the nearest neighbor potentials are considered, as is allowed when ν(r) is short ranged, the total potential energy is given by U ≃ ν(x1 – x2) + ν(x2 – x3) + …, and the most stable arrangement of the atoms is a periodic array with equal distance a at which ν(r) is a minimum.

Statistical mechanics is the theory that describes the behavior of macroscopic systems in terms of thermodynamic variables (such as the entropy, volume, average number of particles, etc.), using as a starting point the microscopic structure of the physical system of interest. The difference between thermodynamics and statistical mechanics is that the first theory is based on empirical observations, whereas the second theory is based on knowledge (true or assumed) of the microscopic constituents and their interactions. The similarity between the two theories is that they both address the macroscopic behavior of the system: thermodynamics does it by dealing exclusively with macroscopic quantities and using empirical laws, and statistical mechanics does it by constructing averages over all states consistent with the external conditions imposed on the system (such as temperature, pressure, chemical potential, etc.). Thus, the central theme in statistical mechanics is to identify all the possible states of the system in terms of their microscopic structure, and take an average of the physical quantities of interest over those states that are consistent with the external conditions. The average must involve the proper weight for each state, which is related to the likelihood of this state to occur, given the external conditions.

As in thermodynamics, statistical mechanics assumes that we are dealing with systems composed of a very large number of microscopic particles (typically atoms or molecules). The variables that determine the state of the system are the position, momentum, electric charge and magnetic moment of the particles. All these are microscopic variables, and their values determine the state of individual particles.

While the crystalline state is convenient for describing many properties of real solids, there are a number of important cases where this picture cannot be used, even as a starting point for more elaborate descriptions, as was done for defects in chapters 9–11. As an example, we look at solids in which certain symmetries such as rotations, reflections, or an underlying regular pattern are present, but these symmetries are not compatible with three-dimensional periodicity. Such solids are called quasicrystals. Another example involves solids where the local arrangement of atoms, embodied in the number of neighbors and the preferred bonding distances, has a certain degree of regularity, but there is no long-range order of any type. Such solids are called amorphous. Amorphous solids are very common in nature; glass, based on SiO2, is a familiar example.

In a different class of non-crystalline solids, the presence of local order in bonding leads to large units which underlie the overall structure and determine its properties. In such cases, the local structure is determined by strong covalent interactions, while the variations in large-scale structure are due to other types of interactions (ionic, hydrogen bonding, van der Waals) among the larger units. These types of structures are usually based on carbon, hydrogen and a few other elements, mostly from the first and second rows of the Periodic Table (such as N, O, P, S). This is no accident: carbon is the most versatile element in terms of forming bonds to other elements, including itself.

In this final chapter we deal with certain structures which, while not of macroscopic size in 3D, have certain common characteristics with solids. One such example is what has become known as “clusters”. These are relatively small structures, consisting of a few tens to a few thousands of atoms. The common feature between clusters and solids is that in both cases the change in size by addition or subtraction of a few atoms does not change the basic character of the structure. Obviously, such a change in size is negligible for the properties of a macroscopic solid, but affects the properties of a cluster significantly. Nevertheless, the change in the properties of the cluster is quantitative, not qualitative. In this sense clusters are distinct from molecules, where a change by even one atom can drastically alter all the physical and chemical properties. A good way to view clusters is as embryonic solids, in which the evolution from the atom to the macroscopic solid was arrested at a very early stage.

Clusters composed of either metallic or covalent elements have been studied extensively since the 1980s and are even being considered as possible building blocks for new types of solids (see the collection of articles edited by Sattler, mentioned in the Further reading section). In certain cases, crystals made of these units have already been synthesized and they exhibit intriguing properties. One example is crystals of C60 clusters, which when doped with certain metallic elements become high-temperature superconductors. There are also interesting examples of elongated structures of carbon, called carbon nanotubes, which can reach a size of several micrometers in one dimension.

Overview

In this chapter, we will be concerned with one of the principal subjects of this book, the spectroscopic study of microscopic dynamical processes in matter, with the exciton in the phonon field as a model system. The linear response of matter to an electromagnetic wave of definite frequency reveals a component, with the same frequency, of the motion of the charged particles (electrons and nuclei) in the matter. The linearity of this response is usually assured under not-too-intense light due to the weak radiation–matter interaction. Therefore, the frequency dependence of the linear response such as the lineshape of absorption spectra (imaginary part of susceptibility apart from an unimportant factor), tells us what is going on in the microscopic world.

Both light and matter have the dual nature of wave and particle and obey the uncertainty principle of quantum mechanics, facts which govern all aspects of the spectroscopic study of matter. An exciton, a typical elementary excitation in an insulating solid, behaves as a quasi-particle with definite dispersion (energy–momentum relation), and can be created by annihilation of an incoming photon. Due to the energy–momentum conservation rule, this elementary process can take place only with a photon with definite energy equal to that of the exciton with the same wave vector; namely, the absorption spectrum consists of an infinitely sharp line. Monochromatic light with infinite duration time corresponds to an exciton with definite energy and hence infinite lifetime.

This book is addressed to first-year graduate students in physics, chemistry, materials science and engineering. It discusses the atomic and electronic structure of solids. Traditional textbooks on solid state physics contain a large amount of useful information about the properties of solids, as well as extensive discussions of the relevant physics, but tend to be overwhelming as introductory texts. This book is an attempt to introduce the single-particle picture of solids in an accessible and self-contained manner. The theoretical derivations start at a basic level and go through the necessary steps for obtaining key results, while some details of the derivations are relegated to problems, with proper guiding hints. The exposition of the theory is accompanied by worked-out examples and additional problems at the end of chapters.

The book addresses mostly theoretical concepts and tools relevant to the physics of solids; there is no attempt to provide a thorough account of related experimental facts. This choice was made in order to keep the book within a limit that allows its contents to be covered in a reasonably short period (one or two semesters; see more detailed instructions below). There are many sources covering the experimental side of the field, which the student is strongly encouraged to explore if not already familiar with it. The suggestions for further reading at the end of chapters can serve as a starting point for exploring the experimental literature. There are also selected references to original research articles that laid the foundations of the topics discussed, as well as to more recent work, in the hope of exciting the student's interest for further exploration.

Elasticity theory considers solids from a macroscopic point of view, and deals with them as an elastic continuum. The basic assumption is that a solid can be divided into small elements, each of which is considered to be of macroscopic size, that is, very large on the atomic scale. Moreover, it is also assumed that we are dealing with small changes in the state of the solid with respect to a reference configuration, so that the response of the solid is well within the elastic regime; in other words, the amount of deformation is proportional to the applied force, just as in a spring. Although these assumptions may seem very restrictive, limiting the applicability of elasticity theory to very large scales, in most cases this theory is essentially correct all the way down to scales of order a few atomic distances. This is due to the fact that it takes a lot of energy to distort solids far from their equilibrium reference state, and for small deviations from that state solids behave in a similar way to elastic springs.

The strain tensor

We begin with the definitions of the strain and stress tensors in a solid. The reference configuration is usually taken to be the equilibrium structure of the solid, on which there are no external forces. We define strain as the amount by which a small element of the solid is distorted with respect to the reference configuration.

Two-dimensional defects in crystals consist of planes of atomic sites where the solid terminates or meets a plane of another crystal. We refer to the first type of defects as surfaces, to the second as interfaces. Interfaces can occur between two entirely different solids or between two grains of the same crystal, in which case they are called grain boundaries.

Surfaces and interfaces of solids are extremely important from a fundamental as well as from a practical point of view. At the fundamental level, surfaces and interfaces are the primary systems where physics in two dimensions can be realized and investigated, opening a different view of the physical world. We have already seen that the confinement of electrons at the interface between a metal and a semiconductor or two semiconductors creates the conditions for the quantum Hall effects (see chapters 7 and 9). There exist several other phenomena particular to 2D: one interesting example is the nature of the melting transition, which in 2D is mediated by the unbinding of defects. Point defects in 2D are the equivalent of dislocations in 3D, and consequently have all the characteristics of dislocations discussed in chapter 10. In particular, dislocations in two dimensions are mobile and have long-range strain fields which lead to their binding in pairs of opposite Burgers vectors. Above a certain temperature (the melting temperature), the entropy term in the free energy wins and it becomes favorable to generate isolated dislocations; this produces enough disorder to cause melting of the 2D crystal. At a more practical level, there are several aspects of surfaces and interfaces that are extremely important for applications.

The crystalline structure of solids forms the basis for our understanding of their properties. Most real systems, however, are not perfect crystals. To begin with, even in those real solids which come close to the definition of a perfect crystal there are a large number of defects. For instance, in the best quality crystals produced with great care for the exacting requirements of high-technology applications, such as Si wafers used to manufacture electronic devices, the concentration of defects is rarely below one per billion; this corresponds to roughly one defect per cube of only 1000 atomic distances on each side! These imperfections play an essential role in determining the electronic and mechanical properties of the real crystal. Moreover, there are many solids whose structure has none of the characteristic symmetries of crystals (amorphous solids or glasses) as well as many interesting finite systems whose structure has some resemblance to crystals but they are not of infinite extent. This second part of the book is concerned with all these types of solids and structures.

A natural way to classify defects in crystalline structure is according to their dimensionality:

1. (i) Zero-dimensional or “point” defects consist of single atomic sites, or complexes of very few atomic sites, which are not in the proper crystalline positions; examples of point defects are missing atoms called vacancies, or extra atoms called interstitials. Point defects may also consist of crystalline sites occupied by atoms foreign to the host crystal; these are called substitutional impurities.

2. (ii) One-dimensional defects consist of lines of atomic sites perturbed from their ideal positions; these can extend for distances comparable to the linear dimension of the solid. The linear defects are called dislocations.

3. […]

Line defects in crystals are called dislocations. Dislocations had been considered in the context of the elastic continuum theory of solids, beginning with the work of Volterra, as a one-dimensional mathematical cut in a solid. Although initially viewed as useful but abstract constructs, dislocations became indispensable in understanding the mechanical properties of solids and in particular the nature of plastic deformation. In 1934, Orowan, Polanyi and Taylor, each independently, made the connection between the atomistic structure of crystalline solids and the nature of dislocations; this concerned what is now called an “edge dislocation”. A few years later, Burgers introduced the concept of a different type of dislocation, the “screw dislocation”. The existence of dislocations in crystalline solids is confirmed experimentally by a variety of methods. The most direct observation of dislocations comes from transmission electron microscopy, in which electrons pass through a thin slice of the material and their scattering from atomic centers produces an image of the crystalline lattice and its defects (see, for example, Refs.). A striking manifestation of the presence of dislocations is the spiral growth pattern on a surface produced by a screw dislocation. The field of dislocation properties and their relation to the mechanical behavior of solids is enormous. Suggestions for comprehensive reviews of this field, as well as some classic treatments, are given in the Further reading section.

Solids exhibit an extremely wide range of properties, which is what makes them so useful and indispensable to mankind. While our familiarity with many different types of solids makes this fact seem unimpressive, it is indeed extraordinary when we consider its origin. The origin of all the properties of solids is nothing more than the interaction between electrons in the outer shells of the atoms, the so called valence electrons. These electrons interact among themselves and with the nuclei of the constituent atoms. In this first chapter we will give a general description of these interactions and their relation to the structure and the properties of solids.

The extremely wide range of the properties of solids is surprising because most of them are made up from a relatively small subset of the elements in the Periodic Table: about 20 or 30 elements, out of more than 100 total, are encountered in most common solids. Moreover, most solids contain only very few of these elements, from one to half a dozen or so. Despite this relative simplicity in composition, solids exhibit a huge variety of properties over ranges that differ by many orders of magnitude. It is quite extraordinary that even among solids which are composed of single elements, physical properties can differ by many orders of magnitude.

One example is the ability of solids to conduct electricity, which is measured by their electrical resistivity. Some typical single-element metallic solids (such as Ag, Cu, Al), have room-temperature resistivities of 1–5μΩ·cm, while some metallic alloys (like nichrome) have resistivities of 102μΩ·cm. All these solids are considered good conductors of electrical current.

In the previous chapter we examined in detail methods for solving the single-particle equations for electrons in solids. The resulting energy eigenvalues (band structure) and corresponding eigenfunctions provide insight into how electrons are arranged, both from an energetic and from a spatial perspective, to produce the cohesion between atoms in the solid. The results of such calculations can be useful in several other ways. The band structure of the solid can elucidate the way in which the electrons will respond to external perturbations, such as absorption or emission of light. This response is directly related to the optical and electrical properties of the solid. For example, using the band structure one can determine the possible optical excitations which in turn determine the color, reflectivity, and dielectric response of the solid. A related effect is the creation of excitons, that is, bound pairs of electrons and holes, which are also important in determining optical and electrical properties. Finally, the band structure can be used to calculate the total energy of the solid, from which one can determine a wide variety of thermodynamic and mechanical properties. In the present chapter we examine the theoretical tools for calculating all these aspects of a solid's behavior.

Density of states

A useful concept in analyzing the band structure of solids is the density of states as a function of the energy. To illustrate this concept we consider first the free-electron model. The density of states g(∈)d∈ for energies in the range [∈, ∈ + d∈] is given by a sum over all states with energy in that range.