CONDUCTORS AND INSULATORS

In a gas the average distance between molecules (or atoms) is large compared to mole- cular dimensions. In liquids and solids, however, the intermolecular distance is compar- able to a molecular diameter (Problem 2.3). Consequently the intermolecular forces are roughly comparable in strength to the interatomic bonding forces in the molecules.

The molecules in liquids and solids are thus influenced very strongly by their neighbors.

What is generally called “solid-state physics” is mostly the study of crystalline solids, that is, solids in which the molecules are arranged in a regular pattern called a crystal lattice. The central fact of the theory of crystalline solids is that the discrete energy levels of the individual atoms are split intoenergy bands, each containing many closely spaced levels (Fig. 2.11). Between these allowed energy bands are gaps with no allowed energies. The way this happens is easy to explain with a simple example.

Imagine a sodium atom with its 11 electrons distributed according to the Pauli exclu- sion principle over its 1slevel (2 electrons), 2slevel (2 electrons), 2plevel (6 electrons) and 3s level (1 electron). A second sodium atom has exactly the same energy levels occupied by 11 electrons in the same way. If the two sodium atoms are brought close together their two equal-energy 1s levels turn into two levels of “di-sodium,” and these two levels of di-sodium have slightly different energies from their Na values and slightly different energies from each other. Similarly, the two 2s levels of Na become two slightly different levels of di-sodium, and so on for the higher levels.

2349cm^–1 (001)

1388cm^–1 (100)

Symmetric stretch mode

Asymmetric stretch mode Bending mode

667cm^–1 (010)

(000) (020) (030)

Figure 2.10 The first few vibrational energy levels of the CO2molecule.

2.6 CONDUCTORS AND INSULATORS 35

The same process occurs for three sodium atoms, in which case the 1slabel applies to three slightly separated levels, the 2slabel applies to three slightly separated levels, and so on. When the number of atomsNis as large as is appropriate to a macroscopic piece of sodium metal theNslightly separated levels are so closely bunched that they constitute an effectively continuous band of energies. The relatively large gap between the 1sand 2s levels in sodium atoms becomes the “forbidden gap” between 1sand 2s bands in sodium metal, where there are no longer any distinguishable levels. This is sketched in Fig. 2.12 for 1, 2, 6, andN10²³atoms.

A one-dimensional model showing how such band structure arises in quantum theory is discussed in the Appendix to this chapter.

We can reach a crude understanding of the formation of energy bands by beginning with the case of two identical atoms. When the atoms are far apart and effectively non- interacting, their electron configurations and energy levels are identical. As they are brought closer together, however, the electrons of each atom begin to feel the presence

First excited

state

1 atom 2 atoms 6 atoms N atoms

Ground state

Figure 2.12 Sketch of the change in the energy values originally assigned to the ground and first excited states of an atom as more and more atoms are combined to form a solid. Note that the band gap energy can be identified with the original atomic level spacing, but is generally different in size.

Allowed energy band

Forbidden energy gap

Allowed energy band

Allowed energy band Forbidden energy gap

Figure 2.11 In a crystalline solid the allowed electron energy levels occur inbandsof closely spaced levels. Between these allowed energy bands are forbidden gaps.

of the other atom, and the energy levels become those of the two atoms as a whole.⁵The difference between these new energy levels depends upon the interatomic spacing (Fig. 2.12).

The difference between the highest and the lowest of these N levels depends on the interatomic distances, amounting typically to several electron volts for atomic spacings of a few angstroms, typical of solids. Now if we increase N, keeping the interatomic spacing fixed as in a crystalline solid, the total energy spread of the N levels stays about the same, but the levels become more densely spaced. For the large values of N typical of a solid (say, something like 10²⁹ atoms/m³), each set of N levels thus becomes in effect a continuous energy band (as in Fig. 2.13), which in some solids can be even wider then the original atomic level spacing.

The chemical and optical properties of atoms are determined primarily by their outer electrons. In solids, similarly, many important properties are determined by the electrons in the highest energy bands, the bands evolving out of the higher occupied states of the individual atoms. Consider, for instance, a solid in which the highest occupied energy band is only partially filled, as illustrated in Fig. 2.14a. In an applied electric field the electrons in this band can readily take up energy and move up within the band. A solid whose highest occupied band is only partially filled is a good conductor of electricity.

Now consider a solid whose highest occupied band is completely filled with electrons, as illustrated in Fig. 2.14b. In this case it is quite difficult for an electron to move because all the energetically allowed higher states in the band already have their full measure of electrons permitted by the Pauli principle. Therefore, a solid whose highest occupied energy band is filled will be an electricalinsulator; in other words, its electrons will not flow freely when an electric field is applied. Implicit in

E_n+1

E_n

E_n–1

Figure 2.13 Crystalline solid energy bands formed from energy levels of isolated atoms.

5These two energy levels correspond to symmetric and antisymmetric spatial wave functions for the electrons, with correspondingly antisymmetric and symmetric spin eigenfunctions. The twofold exchange degeneracy in the case of widely separated atoms is broken when their wave functions begin to overlap.

2.6 CONDUCTORS AND INSULATORS 37

this definition of an insulator is the assumption that the forbidden energy gap between the highest filled band and the next allowed energy band, denotedEgin Fig. 2.14, is large compared to the amount of energy an electron can pick up in the applied field.

Solids in which this band gap is not so large are called semiconductors. Their band structure is indicated in Fig. 2.14c. At the absolute zero of temperature thevalence band of a semiconductor is completely filled, whereas theconductionband, the next allowed energy band, is empty. At room temperature, however, electrons in the valence band may have enough thermal energy to cross the narrow energy gap and go into the conduction band. Thus, diamond, which has a band gap of about 7 eV, is an insulator, whereas silicon, with a band gap of only about 1 eV, is a semiconductor. In a metallic conductor, by contrast, there is no band gap at all; the valence and conduction bands are effectively overlapped.

This characterization of solids as insulators, conductors, and semiconductors is obviously more descriptive than explanatory. To understandwhy a given solid is an insulator, conductor, or semiconductor, we must consider the nature of the forces binding the atoms (or molecules) together in the solid.

In covalent solids the atoms are bound by the sharing of outer electrons in partially filled configurations. In a true covalent solid, there are no free electrons, and so such solids do not conduct electricity very well. The covalent bonds tend to hold the atoms tightly together, thus causing covalent solids to be rather hard and have high melting points. A good example is diamond, in which carbon atoms are arranged in a lattice such that each atom is at the center of a tetrahedron formed by its four nearest neighbors.

Inionic solidssuch as NaCl, the binding is produced by electrostatic forces between oppositely charged ions. The reason for the binding is the same as in ionic molecules. In NaCl, for instance, the energy required to remove the 3selectron from Na and transfer it to Cl, to form Na^þand Cl², is less than the electrostatic energy of attraction between the ions. Here again there are no free electrons available to conduct heat or electricity, and so ionic solids are not good conductors.

The so-called molecular solids, which include many organic compounds (e.g., teflon), are also poor conductors. In such solids the binding is due to the very weak van der Waals forces, which were originally postulated by J. D. van der Waals (1873) to explain deviations from the ideal gas law. The van der Waals energy of attraction between two molecules ordinarily varies as the inverse sixth power of the distance

Conduction band

Conduction band

Valence band Valence

band Valence

band

(a) (b) (c)

E_g E_g

Figure 2.14 In a good conductor of electricity (a), the highest occupied band is only partially filled with electrons, whereas in a good insulator (b) it is filled. In (b) the energy gapE_gbetween the valence band and the conduction band is large. In the case (c) of a semiconductor, however, this gap is small, and electrons in the valence band can easily be promoted to the conduction band.

between the molecules. Because of the weakness of the van der Waals interaction, molecular solids are much easier to deform or compress than covalent or ionic solids.

Of course, electrical technology as we know it would be impossible withoutmetallic solids, which are good conductors of electricity (copper, silver, etc.). In a metallic solid the electrons are not all tightly bound at crystal lattice sites. Some of the electrons are free to move over large distances in the metal, much as atoms move freely in a gas.

This occurs because metals are formed from atoms in which there are one, two, or occasionally three outer electrons in unfilled configurations. The binding is associated with these weakly held electrons leaving their parent ions and being shared by all the ions, and so we can regard metallic binding as a kind of covalent binding. We can also think of the positive ions as being held in place because their attraction to the “electron gas” exceeds their mutual repulsion.

It is sometimes a useful approximation to regard the conduction electrons of a metal as completely free to move about. Of course, conduction electrons are not really completely free, as evidenced by the fact that even the very best conductors—copper, silver, and gold—have a finite resistance to the flow of electricity.

It should be emphasized that many solids do not fit so neatly into the covalent, ionic, molecular, or metallic categories. Furthermore many important properties of various solids are determined by imperfections such as impurities and dislocations in the crystal lattice. Steel, for instance, is much harder than pure iron because of the small amount of carbon that was mixed into the iron melt. Impurities can also determine the color of a crystal, as in the case of ruby (Section 3.1). We will shortly discuss how the addition of certain impurities in semiconductors is responsible for modern electronic technology.