Probing dynamical properties at equilibrium

By definition, the macroscopic properties of a system at equilibrium do not change with time. This does not mean, however, that the system is dynamically inert. On the contrary, the equilibrium state is associated with permanent motion at the molecular level. Although this motion is sometimes described as random ‘thermal noise’, it is in fact quite well organized, and reflects the microscopic processes that govern the dynamics of the system. As these same processes will also determine the system response to an external perturbation, understanding their organization and time evolution is of primary importance for the determination of the material response, and of its relationship to the microscopic structure.

As a simple illustration of how seemingly random dynamical processes are organized in a coherent fashion, one may consider the case of vibrations in a crystalline solid. Each atom oscillates around its equilibrium position, in a way which is apparently very random. The well known harmonic analysis, however, shows that this motion is really caused by the superposition of well defined sound waves, the phonons, with different phases and directions of propagation. This organization of the atomic motions into excitations of well defined spatial and temporal structure determines many thermodynamic and transport properties of the crystal.

The harmonic crystal is of course particularly simple, since it is possible in that case to deduce analytically from the interaction potential the structure of the coherent excitations. In more complex, disordered systems, an analytical treatment is usually out of reach. Nevertheless, the information on the way atomic motions are organized is encoded in the correlation functions of atomic positions.

While the solid state of matter is mostly associated with the mineral world and much of modern technology, liquids are more closely related to living matter and biological processes. In fact life is generally believed to have emerged in the primordial ocean which was formed when the right temperature conditions came to prevail on the young Earth, providing a striking illustration of the ‘marginal’ character of the liquid state, compared to the solid and gaseous phases of the same substances, which exist over much wider ranges of temperature and pressure. The liquid state arises from a delicate balance between ‘packing’ of molecules and cohesive forces or, more formally, between entropy and energy, which renders a statistical description very difficult, due to the absence of any obvious ‘small parameter’. This may explain that, while the gaseous and crystalline states of matter were well understood by the 1950s, significant theoretical progress on the liquid state only just started around that time, and was then speeded up by early neutron scattering and computer simulation data.

From the start, the exploration of the liquid state was interdisciplinary par excellence, thanks to the combined efforts of physicists, physical chemists and chemical engineers. However, early theoretical work evolved along two lines of thought. ‘Simple’ liquids were studied on a molecular scale, using statistical mechanics and computer simulations as basic tools, while ‘complex’ fluids (sometimes referred to as ‘soft matter’ following P.G. de Gennes) were mostly examined on a more coarse-grained level, epitomized in the scaling approach to the theory of polymer solutions.

Lattice models and mean field treatment

Changes in temperature, pressure or chemical composition can lead to continuous or discontinuous phase changes, as illustrated schematically in figure 1.1. Phase transitions are ubiquitous in simple and complex fluids, and the thermodynamic conditions for phase equilibria have been spelled out in section 2.3. The most common phase transitions, like vapour condensation, freezing or the isotropic to nematic transition in liquid crystals, are first-order transitions, characterized by discontinuities of first derivatives of the free energy, like the entropy or the molar volume. Two (or more phases) can coexist over a range of temperatures or pressures and metastable thermodynamic states of one phase can exist for appreciable times under conditions where another phase has a lower free energy. Most first-order transitions can be reasonably well described within mean field theory, which is the leitmotiv of the present chapter. A key quantity in the description of first-order transitions is the order parameter, which generally characterizes some broken symmetry, taking a finite value in the phase of lower symmetry, and vanishing discontinuously at the transition towards the phase of higher symmetry. In many cases the order parameter can be associated with some microscopic variable, like the orientation vector of mesogenic molecules, but in the case of a transition between two isotropic fluids, with full rotational and translational invariance, like a liquid and its coexisting vapour, or two liquid mixtures of different compositions, the order parameter is a macroscopic characteristic, conveniently chosen to be the difference in density or concentration between the two phases.

Thermodynamics is the branch of physical sciences developed in the 19th century to provide a systematic theoretical framework for the characterization of physical and chemical transformations of substances, involving exchange of heat, work and matter. It is an essentially macroscopic and phenomenological theory, which ignores the molecular nature of matter; in fact at the time the laws of thermodynamics were formulated, atomistic theories were far from being universally accepted. The link between macroscopic thermodynamics and the modern microscopic description of matter was provided later by statistical mechanics, which is the central theoretical tool for the study of complex fluids; this link will be briefly recalled in section 2.2.

A brief summary of macroscopic thermodynamics is provided in section 2.1. This reminder, which emphasizes key concepts based on the two fundamental laws of thermodynamics, more than specific thermodynamic relations, will serve as a constant reference throughout the book. In particular the summary will provide the basis for the phenomenology of phase transitions and interfacial phenomena (section 2.3), as well as for the macroscopic theory of fluctuations in chapter 3, which in turn can be generalized to mesoscopic and microscopic scales (chapter 3, sections 3.4 to 3.5).

State variables and thermodynamic equilibrium

Numerous measurements carried out on a broad variety of gaseous, liquid or solid substances, clearly show that the macroscopic equilibrium states of any substance can be fully characterized by specifying the values of a small number of state variables. In other words all measurable, macroscopic properties of a given substance, say its specific heat, its thermal expansivity or its viscosity, are well defined functions of these state variables.

We begin with a brief overview of magnetic behavior in different types of solids. We first define the terms used to describe the various types of magnetic behavior. A system is called paramagnetic if it has no inherent magnetization, but when subjected to an external field it develops magnetization which is aligned with the field; this corresponds to situations where the microscopic magnetic moments (like the spins) tend to be oriented in the same direction as the external magnetic field. A system is called diamagnetic if it has no inherent magnetization, but when subjected to an external field it develops magnetization that is opposite to the field; this corresponds to situations where the induced microscopic magnetic moments tend to shield the external magnetic field. Finally, a system may exhibit magnetic order even in the absence of an external field. If the microscopic magnetic moments tend to be oriented in the same direction, the system is described as ferromagnetic. A variation on this theme is the situation in which microscopic magnetic moments tend to have parallel orientation but they are not necessarily equal at neighboring sites, which is described as ferrimagnetic behavior. If magnetic moments at neighboring sites tend to point in opposite directions, the system is described as antiferromagnetic. In the latter case there is inherent magnetic order due to the orientation of the microscopic magnetic moments, but the net macroscopic magnetization is zero.

• 1954 Nobel prize for Chemistry: Linus Carl Pauling, for his research into the nature of the chemical bond and its application to the elucidation of the structure of complex substances.

• 1956 Nobel prize for Physics: William Shockley, John Bardeen and Walter Houser Brattain, for their research in semiconductors and their discovery of the transistor effect.

• 1962 Nobel prize for Medicine: Francis Harry Compton Crick, James Dewey Watson and Maurice Hugh Frederick Wilkins, for their discoveries concerning the molecular structure of nucleic acids and its significance for information transfer in living material.

• 1968 Nobel prize for Medicine: Robert W. Holley, Hav Gobind Khorana and Marshall W. Nirenberg, for their interpretation of the genetic code and its function in protein synthesis.

• 1972 Nobel prize for Physics: John Bardeen, Leon N. Cooper and J. Robert Schrieffer, for their jointly developed theory of superconductivity, usually called the BCS theory.

• 1977 Nobel prize for Physics: Philip W. Anderson, Sir Nevill F. Mott and John H. van Vleck, for their fundamental theoretical investigations of the electronic structure of magnetic and disordered systems.

• 1985 Nobel prize for Chemistry: Herbert A. Hauptman and Jerome Karle, for their outstanding achievements in the development of direct methods for the determination of crystal structures.

• 1985 Nobel prize for Physics: Klaus von Klitzing, for the discovery of the quantized Hall effect.

• 1986 Nobel prize for Physics: Gerd Binning and Heinrich Rohrer, for their design of the scanning tunneling microscope; Ernst Ruska, for his fundamental work in electron optics, and for the design of the first electron microscope.

• […]

Solids are the physical objects with which we come into contact continuously in our everyday life. Solids are composed of atoms. This was first postulated by the ancient Greek philosopher Demokritos, but was established scientifically in the 20th century. The atoms (ατομα = indivisible units) that Demokritos conceived bear no resemblance to what we know today to be the basic units from which all solids and molecules are built. Nevertheless, this postulate is one of the greatest feats of the human intellect, especially since it was not motivated by direct experimental evidence but was the result of pure logical deduction.

There is an amazing degree of regularity in the structure of solids. Many solids are crystalline in nature, that is, the atoms are arranged in a regular three-dimensional periodic pattern. There is a wide variety of crystal structures formed by different elements and by different combinations of elements.

In the previous two chapters we examined in detail the effects of crystal periodicity and crystal symmetry on the eigenvalues and wavefunctions of the single-particle equations. The models we used to illustrate these effects were artificial free-electron models, where the only effect of the presence of the lattice is to impose the symmetry restrictions on the eigenvalues and eigenfunctions. We also saw how a weak periodic potential can split the degeneracies of certain eigenvalues at the Bragg planes (the BZ edges). In realistic situations the potential is certainly not zero, as in the free-electron model, nor is it necessarily weak. Our task here is to develop methods for determining the solutions to the single-particle equations for realistic systems. We will do this by discussing first the so called tight-binding approximation, which takes us in the most natural way from electronic states that are characteristic of atoms (atomic orbitals) to states that correspond to crystalline solids. We will then discuss briefly more general methods for obtaining the band structure of solids, whose application typically involves a large computational effort. Finally, we will conclude the chapter by discussing the electronic structure of several representative crystals, as obtained by elaborate computational methods; we will also attempt to interpret these results in the context of the tight-binding approximation.

The tight-binding approximation

The simplest method for calculating band structures, both conceptually and computationally, is the so called Tight-Binding Approximation (TBA), also referred to as Linear Combination of Atomic Orbitals (LCAO). The latter term is actually used in a wider sense, as we will explain below.

Superconductivity was discovered in 1911 by Kamerling Onnes and remains one of the most actively studied aspects of the behavior of solids. It is a truly remarkable phenomenon of purely quantum mechanical nature; it is also an essentially many-body phenomenon which cannot be described within the single-particle picture. Because of its fascinating nature and of its many applications, superconductivity has been the focus of intense theoretical and experimental investigations ever since its discovery. Studies of superconductivity have gained new vigor since the discovery of high-temperature superconductors in 1987.

Overview of superconducting behavior

Superconductivity is mostly characterized by a vanishing electrical resistance below a certain temperature Tc, called the critical temperature. Below Tc there is no measurable DC resistance in a superconductor and, if a current is set up in it, it will flow without dissipation practically forever: experiments trying to detect changes in the magnetic field associated with current in a superconductor give estimates that it is constant for 106 – 109 years! Thus, the superconducting state is not a state of merely very low resistance, but one with a truly zero resistance. This is different than the case of very good conductors. In fact, materials which are very good conductors in their normal state typically do not exhibit superconductivity. The reason is that in very good conductors there is little coupling between phonons and electrons, since it is scattering by phonons which gives rise to resistance in a conductor, whereas electron-phonon coupling is crucial for superconductivity.

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.