Classification of solids

Condensed matter physics and solid state physics usually refer to the same area of physics, but in principle the former title is broader. Condensed matter is meant to include solids, liquids, liquid crystals, and some plasmas in or near solids. This is the largest branch of physics at this time, and it covers a broad scope of physical phenomena. Topics range from studies of the most fundamental aspects of physics to applied problems related to technology

The focus of this book will be primarily on the quantum theory of solids. To begin, it is useful to start with the concept of a solid and then describe the two commonly used models that form the basis for modern research in this area. The word “solid” evokes a familiar visual picture well described by the definition in the Oxford Dictionary: “Of stable shape, not liquid or fluid, having some rigidity.” It is the property of rigidity that is basic to the early studies of solids. These studies focused on the mechanical properties of solids. As a result, until the nineteenth century the most common classification of solids involved their rigidity or mechanical properties. The Mohs hardness scale (talc – 1; calcite – 3; quartz – 7; diamond – 10) is a typical example. This is a useful but limited approach for classifying solids.

The advent of atomic theory brought more microscopic concepts about solids. Solids were viewed as collections of more or less strongly interacting atoms. From the point of view of atomic theory, a gas is described in terms of a collection of almost independent atoms, while a liquid is formed by atoms that are weakly interacting. This picture leads to a description of the formation of solids, under pressure or by freezing, in which the distances between atoms are reduced and, in turn, this causes them to interact more strongly. Molecular solids are formed by condensing molecular gases.

Hence, the development of atomic physics and chemical analysis led to a more detailed classification of solids according to chemical composition. Although for most studies of solids it is necessary to establish the identity of the constituent atoms, such a scheme provides limited insight into many of the basic concepts of condensed matter physics.

The field of condensed matter physics is the largest branch of physics worldwide and probably the most diverse. Undergraduate courses in this area are ubiquitous and most research universities offer graduate courses. Over the past 50 years, the undergraduate course has been open to physicists, chemists, materials scientists, engineers, and, to a smaller extent, biologists. The graduate course slowly evolved in many institutions from a course for theorists to one that welcomed students interested in a career in experimental condensed matter physics and materials research. In recent years, the proportion of chemists, materials scientists, engineers, and researchers in nanoscience has increase significantly in graduate courses in condensed matter physics.

There are numerous undergraduate texts. The prime example is Introduction to Solid State Physics (ISSP) authored by C. Kittel. At the graduate level, no single text has emerged as the canonical choice. N. Ashcroft and N. D. Mermin's book Solid State Physics is sometimes chosen since it contains advanced topics going beyond Kittel's ISSP, although Ashcroft & Mermin is often used as an undergraduate text. J. Ziman's Principles of the Theory of Solids is at roughly the same level as Ashcroft & Mermin, with excellent physical examples and discussions of concepts. C. Kittel's Quantum Theory of Solids, which was written for the graduate course at the University of California, Berkeley, at a time when students taking the course were predominantly theorists, is somewhat limited in scope and generally considered difficult by graduate students not intent on a career in theoretical condensed matter physics. Other texts, such as those by J. Callaway, O. Madelung, M. Marder, and J. Patterson and B. Bailey, are considered to be at the right level and suitable. Many others are in the recommended, but not required, category and are useful when specific subjects are considered. Examples include those authored by E. Kaxiras, R. Martin, M. Balkanski and R. Wallis, P. Yu and M. Cardona, and M. Cohen and J. Chelikowsky for electronic and optical properties of solids; M. Tinkham, P. de Gennes, and R. Schrieffer for superconductivity; G. Mahan for many-particle physics; and P. Chailken and T. Lubensky for phase transitions and soft matter systems. There are also many other excellent texts on specific subjects in this field.

The electronic, transport, optical, and other properties of a material often undergo drastic changes, and new phenomena emerge, when the system is reduced in size such that quantum confinement of electrons (i.e. the wavelength of the electron is comparable to the confining structure) occurs in one or more dimensions. These reduced-dimensional systems and nanostructures have physical properties and phenomena that are of basic interest and are also potentially useful in applications. Because many of their properties are fundamentally derived from restricted geometry, the behaviors of these reduced dimensional systems and nanostructures are tunable by changing their size, and they are strongly influenced by considerations such as quantum confinement, enhanced many-electron interactions, reduced number of degrees of freedom, and symmetry effects. In this chapter, we shall discuss some basic elements of the electronic, transport, and optical properties of reduced-dimensional systems. We consider several systems, including semiconductor two-dimensional electron gas (2DEG) systems, quantum dots, graphene, carbon nanotubes, atomically thin quasi two-dimensional (2D) crystals, and molecular junctions, illustrating that small is different.

Density of states and optical properties

Electrons confined in a semiconductor quantum well or at a metal-oxide–1h are fragments of semiconductor crystals consisting of hundreds to many thousands of atoms with surface states eliminated by passivating adsorbates, or enclosure in a material that has a larger bandgap, is another class of quantum dots. Semiconductor wires of several nanometers in diameter can also be grown. Examples of other fascinating nanostructures that have been synthesized and behave like reduced-dimensional systems even at room temperature include graphene, carbon nanotubes, atomically thin quasi-2D materials consisting of mono- and few-layer Van der Waals crystals (such as hexagonal BN, transition metal dichalcogenides, etc.), and structures derived from these systems, such as nanoribbons.

The electron density of states in a reduced-dimensional system has distinct characteristics, i.e. van Hove singularities of different nature than for three dimensional systems, leading to many of its characteristic properties. Let us idealize the carriers in a semiconductor quantum well, e.g. a thin layer of GaAs of thickness L sandwiched between two AlAs crystals, as free electrons with an effective mass m*. The larger bandgap of AlAs effectively forms a potential confining the lower-energy carriers (both electrons and holes) to move within the GaAs layer, as illustrated in Fig. 16.1 for the states in the conduction band of GaAs.

In Chapter 6, the many-electron problem was discussed and electron–electron interactions were considered using the standard approaches of many-body quantum theory. The quasi-particle and collective excitations of the interacting system are expected to emerge as elementary excitations of the interacting system, and their properties should dominate the spectra associated with various response functions. Since the elementary excitations are emergent properties, which can be viewed in terms of excitations or particles that can be created or destroyed, it is convenient to use techniques associated with the fields of quantum electrodynamics (QED) and statistical physics. The essential point is that in QED, because relativistic effects are included, particles can be created and destroyed, and the formalism accounts for the changes associated with these events.

For condensed matter systems, we are not usually probing the systems in energy ranges where relativistic effects are important. However, the techniques of QED are useful if they are applied to cases where elementary excitations are created or destroyed. In this chapter, we focus on recipes for calculations of this kind and refer the reader to books in this area that provide the proofs and more complete discussions.

General formalism

The first step in setting up the formalism for a many-electron- or many-phonon-type study is to rely on second quantization techniques. The approach, which is described in many texts on quantum mechanics, consists in replacing operators in the usual quantum theory, such as the kinetic energy, that act on the coordinate variables in a wavefunction by other operators that operate on a wavefunction describing the number of particles in a given state. These new operators change the number of particles in specific states, and the result is the same as would be obtained when using ordinary quantum theory techniques. So second quantization is a convenient method, which although in principle is not required for the study of solids, allows efficient calculations and provides new insights. Analogies like the one comparing electron–hole excitations by photons with Dirac's electron–positron theory, and viewing a collection of phonons as a Bose system, are very useful. Because particles can be created and destroyed at will, the grand canonical ensemble approach of statistical physics is employed. Another very convenient aspect of second quantization is that the task of symmetrizing wavefunctions for many-body systems is achieved through commutation relations.

Brief discussion of the experimental background

Superconductivity is one of the most studied areas of condensed matter physics. The discovery by H. Kamerlingh-Onnes in 1911 demonstrated that, at 4.3 K, Hg had zero resistance and remained in this state at temperatures below the transition temperature Tc (Fig. 14.1). It appeared that the solid had made a transition to a new state, and this visualization of the superconducting transition as a change of state remained. The transition was found to be very sharp with ΔT ~ 10−2 K, and no crystallographic changes were noted.

Further tests of various materials yielded new superconductors, and it became clear that non-metals were not superconducting, while metals with higher room-temperature resistivities tended to be more likely candidates for superconductivity. Another feature of superconductors was that at finite frequencies ω, the resistance ρ(ω) remained zero up to a critical value ω = ωc, where ħωc became known as the optical superconducting energy gap (Fig. 14.2). The relation ħω c ≈ 3.5kBTc was established experimentally and later derived theoretically for a large class of superconductors.

The maximum Tc for materials increased at an average rate of 1 K every three years until the mid 1980s when it was around 25 K. In 1986, Bednorz and Müller found superconductivity in a copper oxide material in the 50 K range, and, soon afterward, Chu and collaborators achieved Tc's above liquid nitrogen temperature and, in 1996, achieved Tc's around 160 K. At this time, this is the highest Tcachieved in this class of materials.

An important early discovery was the existence of persistent currents, where supercurrents, once generated, persist in a superconducting ring for as long as the ring is kept below Tc. A magnetic field is induced in the hole of the ring and the system is very stable. Theoretical estimates of the time for the decay of the currents at T ≪ Tc are enormously long, perhaps of the order of the age of the universe. Another important magnetic field effect is the destruction of superconductivity in the presence of a magnetic field above a value called the critical field Hc (Fig. 14.3). The critical field is temperature dependent, rising from Hc (Tc) = 0 to a maximum Hc(T = 0), which for superconductors like Hg is only around several hundred G.