Charles Kittel notes, in his book Introduction to Solid State Physics: “The difference between a good conductor and a good insulator is striking. The electrical resistivity of a pure metal may be as low as 10^–10 ohm-cm…. The resistivity of a good insulator may be as high as 10²² ohm-cm. This range of 10³² may be the widest of any common physical property of solids.”1 This comment on such a remarkable property range applies to what we call normal materials. The range is expanded even further when we consider superconductors in Chapter 19, just ahead. But to appreciate superconductors, we first need to understand metals, insulators, and semiconductors.

METALS

The elements listed toward the left and top of Table IV in Chapter 13 or Table B.2 in Appendix B (those elements in the region without shading) are all classified as metals. Most of these elements in solid or liquid form (mercury, as an example of the latter) can carry an electrical current.

As also noted before, the atoms of metals in either solid or liquid form are held together by a collective sea of all of their outer electrons. Thinking classically, these electrons are essentially free to move around within the solid or liquid, and it is these electrons that can carry electrical current. They are spread more or less evenly throughout the metal, and their electrical charges attract them to locally balance the positive charges of the ions that they left behind when they broke loose from their atoms to roam free.

Because some of these metals are malleable and ductile, they can be pressed into shapes or drawn into wire. When used electrically, the wires are usually connected with other electrical components, which include some sort of power source that forces electrons to flow through a complete circuit. For example, in a flashlight they are repelled from the end of the negative terminal of a set of batteries to pass through a wire and a light bulb (circuit element) and back through another wire to the attracting positive terminal of the battery set.

Though it is handy to use this classical view descriptively, there is a problem with it: individual electrons, as particles, would be expected to collide with each other and the lattice array of ions, with the result that the metal would exhibit a much higher resistance to current flow than is actually found.

THE QUANTUM VIEW OF ELECTRICAL CONDUCTION: METALS, INSULATORS, AND SEMICONDUCTORS DEFINED

In the quantum view, all of the electrons are indistinguishable one from another and it is really impossible to know which electron is where, except that we do know that integral multiples of one electron charge, e, move out of the negative terminal of the battery, and the same amount of charge simultaneously moves into the positive terminal of the battery at the other end of the circuit. When Schrödinger's equation is solved for the entire collection of all of the electrons in all of the atoms, what results is a “band” of states at a very large number of very closely spaced energy levels.2 Just as for the individual atom, the indistinguishability of electrons and their ½ spin requires that no two electrons may be in the same state. When all of the electrons in the metal settle into the lowest of the energy states, one electron per state, the band is filled to what has been called a Fermi level.³ States having electrons moving in different directions are equally occupied, so there is no net flow of electrons.

When an electric potential, a voltage, is applied to the metal, some of the electrons are shifted preferentially to states that lie above the Fermi level, having motion in one particular direction, and so there is a net flow of electrons in that direction and the conduction of an electrical current.

But the lattice of ions causes gaps in the band of available states, band gaps, energy regions for which there are no solutions to Schrödinger's equation.4 Electrons simply cannot exist at these energies.

If the electrons occupy the available states to a Fermi level that is comfortably below or above one of these band gaps, the shift in states for net motion can easily occur. The electrons near this Fermi level are said to be in a conduction band, and they are impervious to the lattice of ions. The electrons also tend not to interfere with each other, because they are all subject to exclusion and in separate states.⁵ Conduction can easily take place, and one has a metal.

If the electrons occupy the available states to a Fermi level that just reaches one of these band gaps, and if the gap is sufficiently large (larger than the applied voltage, so that the voltage cannot lift them in energy to states above the gap), their part of the band is essentially capped, there are no states that the electrons can shift into, there is no electrical conduction, and one has an insulator. That part of the overall set of bands that they fill is called a valence band.

If the electrons occupy the available states to a Fermi level that is just below or above one of the band gaps, that is at the top of a valence band or at the bottom of a conduction band, or if the gap isn't too large so that the electrons can be excited thermally or by the application of voltage across the gap to the conduction band above, one has a limited amount of conduction and what is called a semimetal or semiconductor, with characteristics that will be described in Chapter 23.

DIRECT CURRENT (DC), ALTERNATING CURRENT (AC), RESISTANCE, AND TEMPERATURE

Thinking classically again, we visualize a flow of billions of electrons through the circuit from the negative terminal of the battery to the positive terminal. But the electrons have a negative charge. If negative charges flow in one direction, it is the same electrically as what would happen if positive charges were to flow in the other direction, and so we say that a positive direct current (DC) effectively flows from the positive terminal of the battery through the circuit to the negative terminal (under the electrical pressure, i.e., the electrical potential described in Chapter 11, provided chemically as voltage by the battery). When electrical pressure is made to drive the current first in one direction and then the other direction, as happens when we use the voltage provided at wall outlets from far distant generators (using transformers and transmission lines in between), we say that we have an alternating current (AC).

Because electron flow through normal metals involves collisions with impurities and other defects in the metal (which unlike the ions of the metal are typically not of uniform size and charge, or not on a regular lattice), the impurities and defects are jostled, and they in turn jostle the ions of the metal and cause vibrations throughout the entire crystalline lattice. We call the collective effect of these collisions that impede the flow of electrons resistance.

The introduction of this jostling of the lattice is called heating; and the degree to which the atoms are jostled around in the general location of their lattice positions is a measure of their temperature. (We could of course cause or increase the jostling, i.e., heat the metal, by letting it come into contact with more rapidly moving, hotter, molecules in the air of an oven, or by touching the metal with other hotter materials whose atoms are already jostling with even higher energy [i.e., at a higher temperature]. When the atoms are jostled sufficiently that the crystal lattice breaks down and the atoms find themselves continuously bounced into new locations, we say that the solid melts and becomes a liquid.)

Although we think of copper wire as being a good conductor, the electrical resistance of copper and other normal metal wires is high enough so that for the miles and miles of wire in electrical generators, transmission lines, and transformers, a great deal of electrical energy is lost as heat. Even if all of the defects could be eliminated, heating would still occur in metals because the normal jostling of the atoms at ambient (typical outdoor or indoor) temperatures distorts the crystal lattice of atoms enough that it causes collisions that interfere with electron conduction, in a similar manner to the interference caused by defects. Cooling would reduce the jostling and reduce this loss, but the cost of refrigeration to cool away the remaining losses would be prohibitive.

So it would seem that we are stuck with resistance and its associated energy losses, even with the best of metals. But nature has supplied a way to avoid the jostling and this heating: the quantum phenomenon of superconductivity, soon to be described in Chapter 19.