Chapter 1
Dielectric Properties of Materials

1.1 Energy Band in Crystals

In crystallography, a crystal structure is a unique arrangement of atoms, ions, or molecules in a crystalline solid. It describes a highly ordered structure, occurring due to the intrinsic nature of its constituents to form symmetric patterns. The crystal lattice can be thought of as an array of "small boxes" infinitely repeating in all three spatial directions. Such a unit cell is the smallest unit of volume that contains all of the structural and symmetry information to build up the macroscopic structure of the lattice by translation. The crystal structure and symmetry play a role in determining many of its physical properties, such as electronic band structure and optical transparency.

To discuss the behavior of electrons in a crystal, we consider an isolated atom of the crystal. If Z is the atomic number, the atomic nucleus has a positive charge Ze. At a distance r from the nucleus, the electrostatic potential due to the nuclear charge is (in SI units)

where ?0 is the permittivity of free space. Since an electron carries a negative charge, the potential energy of an electron at a distance r from the nucleus is

V(r) is positive, while Ep(r) is negative. Both V(r) and Ep(r) are zero at an infinite distance from the nucleus. Figure 1.1a,b shows the variation of V(r) and Ep(r), respectively, with r.

Figure 1.1 Variation of (a) potential in the field of a nucleus with distance and (b) potential energy of an electron with its distance from the nucleus.

We now consider two identical atoms placed close together. The net potential energy of an electron is obtained as the sum of the potential energies due to the two individual nuclei. In the region between the two nuclei, the net potential energy is clearly smaller than the potential energy for an isolated nucleus (Figure 1.2).

Figure 1.2 Potential energy variation of an electron with distance between two identical nuclei.

The potential energy along a line through a row of equispaced atomic nuclei, as in a crystal, is diagrammatically shown in Figure 1.3. The potential energy between the nuclei is found to consist of a series of humps. At the boundary AB of the solid, the potential energy increases and approaches zero at infinity, there being no atoms on the other side of the boundary to bring the curve down.

Figure 1.3 Potential energy of an electron along a row of atoms in a crystal.

The total energy of an electron in an atom, kinetic plus potential, is negative and has discrete values. These discrete energy levels in an isolated atom are shown by horizontal lines in Figure 1.4a. When a number of atoms are brought close together to form a crystal, each atom will exert an electric force on its neighbors. As a result of this interatomic coupling, the crystal forms a single electronic system obeying Pauli's exclusion principle. Therefore, each energy level of the isolated atom splits into as many energy levels as there are atoms in the crystal, so that Pauli's exclusion principle is satisfied. The separation between the split-off energy levels is very small. A large number of discrete and closely spaced energy levels form an energy band. Energy bands are represented schematically by the shaded regions in Figure 1.4b.

Figure 1.4 Splitting of energy levels of isolated atoms into energy bands as these atoms are brought close together to produce a crystal.

The width of an energy band is determined by the parent energy level of the isolated atom and the atomic spacing in the crystal. The lower energy levels are not greatly affected by the interaction among the neighboring atoms and hence form narrow bands. The higher energy levels are greatly affected by the interatomic interactions and produce wide bands. The interatomic spacing, although fixed for a given crystal, is different for different crystals. The width of an energy band thus depends on the type of the crystal and is larger for a crystal with a small interatomic spacing. The width of a band is independent of the number of atoms in the crystal, but the number of energy levels in a band is equal to the number of atoms in the solid. Consequently, as the number of atoms in the crystal increases, the separation between the energy levels in a band decreases. As the crystal contains a large number of atoms (~1029 m-3), the spacing between the discrete levels in a band is so small that the band can be treated as continuous.

The lower energy bands are normally completely filled by the electrons since the electrons always tend to occupy the lowest available energy states. The higher energy bands may be completely empty or may be partly filled by the electrons. Pauli's exclusion principle restricts the number of electrons that a band can accommodate. A partly filled band appears when a partly filled energy level produces an energy band or when a totally filled band and a totally empty band overlap.

As the allowed energy levels of a single atom expand into energy bands in a crystal, the electrons in a crystal cannot have energies in the region between two successive bands. In other words, the energy bands are separated by gaps of forbidden energy.

The average energy of the electrons in the highest occupied band is usually much less than the zero level marked in Figure 1.4b. The rise of the potential energy near the surface of the crystal, as shown in Figure 1.4b, serves as a barrier, preventing the electrons from escaping from the crystal. If sufficient energy is imparted to the electrons by external means, they can overcome the surface potential energy barrier and come out of the crystal surface.

1.2 Conductor, Insulator, and Semiconductor

On the basis of the band structure, crystals can be classified into conductors, insulators, and semiconductors.

1.2.1 Conductors

A crystalline solid is called a metal if the uppermost energy band is partly filled or the uppermost filled band and the next unoccupied band overlap in energy as shown in Figure 1.5a. Here, the electrons in the uppermost band find neighboring vacant states to move in and thus behave as free particles. In the presence of an applied electric field, these electrons gain energy from the field and produce an electric current, so that a metal is a good conductor of electricity. The partly filled band is called the conduction band. The electrons in the conduction band are known as free electrons or conduction electrons.

Figure 1.5 Energy band structure of (a) a conductor, (b) an insulator, and (c) a semiconductor.

1.2.2 Insulators

In some crystalline solids, the forbidden energy gap between the uppermost filled band, called the valence band, and the lowermost empty band, called the conduction band, is very large. In such solids, at ordinary temperatures, only a few electrons can acquire enough thermal energy to move from the valence band into the conduction band. Such solids are known as insulators. Since only a few free electrons are available in the conduction band, an insulator is a bad conductor of electricity. Diamond having a forbidden gap of 6 eV is a good example of an insulator. The energy band structure of an insulator is schematically shown in Figure 1.5b.

1.2.3 Semiconductors

A material for which the width of the forbidden energy gap between the valence and the conduction band is relatively small (~1 eV) is referred to as a semiconductor. Germanium and silicon having forbidden gaps of 0.78 and 1.2 eV, respectively, at 0 K are typical semiconductors. As the forbidden gap is not very wide, some of the valence electrons acquire enough thermal energy to go into the conduction band. These electrons then become free and can move about under the action of an applied electric field. The absence of an electron in the valence band is referred to as a hole. The holes also serve as carriers of electricity. The electrical conductivity of a semiconductor is less than that of a metal but greater than that of an insulator. The band diagram of a semiconductor is given in Figure 1.5c.

1.3 Fermi-Dirac Distribution Function

The free electrons are assumed to move in a field-free or equipotential space. Due to their thermal energy, the free electrons move about at random just like gas particles. Hence these electrons are said to form an electron gas. Owing to the large number of free electrons (~1023 cm-3) in a metal, principles of statistical mechanics are employed to determine their average behavior. A useful concept is the distribution function that gives the probability of occupancy of a given state by the electrons. The Fermi-Dirac (FD) distribution function can be used to determine the energy distribution of free electrons in a metal. From statistical mechanics, the FD distribution function is found to be

where f(E) is the occupation probability of a state with energy E, EF is a characteristic energy (chemical potential) for a particular solid and is referred to as the Fermi level,...