Conductors

Type of physical science: Condensed matter physics

Field of study: Solids

The special atomic structure of metals enables them to conduct electricity well.

Overview

A classical atomic model of electrical conductivity was first proposed by Paul Drude in 1900 and further developed by Hendrick Antoon Lorentz around 1909. The classical model attempted to show that conductors should obey Ohm's law, and it also sought to relate resistance to atomic properties.

First, it is necessary to understand Ohm's law. Suppose that an ammeter, various conductors, and a 12-volt automobile battery were used in an experiment. An ammeter is a device that measures current, which is the amount of charge that flows by a given point in one second. Current is measured in amperes (named for Andre-Marie Ampere). An ammeter has two terminals, one for the current to enter and the other for the current to leave. Suppose that the "in" terminal of the ammeter were connected to the plus terminal of the battery and that one of the conductors, such as a long copper wire, connected the "out" ammeter terminal to the negative side of the battery. The result is a complete circuit, and current will flow. Ohm's law says that the product of the current in the wire (measured by the ammeter), multiplied by the resistance of the wire, is equal to the voltage across the wire (the 12 volts of the battery).

Consistent with Ohm's law, the resistance of the wire may be defined as the voltage across the wire divided by the current through the wire. Resistance is measured in ohms, named for Georg Simon Ohm, who originated Ohm's law. Resistance is thus a measure of the hindrance that the wire offers to the flow of current for a given voltage. For example, if wire A has ten times as much resistance as wire B, then the current in wire A will be only one-tenth of that in wire B when the same voltage is applied to the wires.

If these measurements were continued using wires of various lengths and different thicknesses, then it would be discovered that longer wires have more resistance and thicker wires have less resistance. The resistivity of a substance is defined to be the measured resistance of the substance multiplied by its cross-sectional area and divided by its length. With this definition, resistivity depends only on the substance, since the geometric factors of length and cross-sectional area have been canceled out. Resistivity is measured in ohm meters. When examining atomic properties, it is more convenient to compare the resistivities of substances than their resistance levels. The resistivity of copper, a good conductor, is 1.7 x 10^-8 ohm meters, while that of sulfur, a good insulator, is 1 x 10¹⁵ ohm meters.

If one could see the atoms in a copper wire, then one would observe that they are packed so closely together that they touch one another, the centers of adjacent atoms being 0.26 nanometer apart. The atoms form a more or less regular, three-dimensional array called a crystal lattice.

Each copper atom has a tiny nucleus composed of twenty-nine protons and either thirty-four or thirty-six neutrons. The nucleus is surrounded by a cloud of twenty-nine electrons, which takes up most of the space in an atom. Most of these electrons remain bound to the nucleus (opposite charges attract); however, the outermost electrons are not very tightly bound and are free to wander from atom to atom throughout the metal. These wandering electrons are called conduction electrons, since they can constitute an electric current. The atoms left behind by the conduction electrons are the ions of the crystal lattice. These atoms are positively charged because they have one or two more protons than electrons. Copper and silver both contribute an average of 1.3 conduction electrons per atom, while aluminum averages 3.5.

Most of these conduction electrons race about at fantastic speeds, generally more than 1 million meters per second. Since their motion is random, however, on the average, there is no net flow of electrons in any particular direction. Therefore, these racing electrons do not constitute a net electric current. A typical conduction electron in copper will pass about three hundred ions of the crystal lattice before it strikes one and bounces off in a different direction.

When an electric current flows in a wire, the conduction electrons have a small drift velocity down the wire in addition to their random high-speed velocities. A typical drift velocity is only 0.0001 meter per second. While this figure may seem insignificant when compared to the million-meter-per-second velocities of conduction electrons, higher velocities are random and will therefore average to zero.

The conduction electrons behave very much like air molecules, which are also in constant motion. Air molecules sail about in random directions faster than the speed of sound, frequently colliding with one another and with any objects present. Since their motion is in random directions, there is no net flow of air. Nevertheless, if a light breeze arises, there is a net flow of air as the molecules now move in some particular direction at the "drift velocity," even if this is only a tiny fraction of their random velocities.

In the classical model of electrical conduction, the conduction electrons are pictured as being accelerated by an electric field to produce the drift velocity. For simple situations, the electric field is numerically equal to the voltage across the conductor divided by the length of the conductor. Drift velocities are small because the electrons travel for very short times before they collide with lattice ions and lose their acquired drift velocities. The average distance traveled between collisions is called the mean free path, which is usually less than 100 nanometers. The mean time between collisions is only 10^-13 to 10^-14 seconds.

The classical model then gives the electric current as the product of four factors: the cross-sectional area of the conductor, the number of conduction electrons per cubic meter, the charge of the electron, and the drift velocity. The model yields Ohm's law, which states that the voltage across a conductor is directly proportional to the current. Finally, the model predicts that resistivity equals the mass of the electron divided by a product of three factors: the density of charge carriers (number of electrons per cubic meter), the square of the charge on a carrier, and the mean time between the collisions.

Quantum mechanics is the branch of physics that applies specifically to atoms.

Quantum mechanics states that the conduction electrons must obey a strange rule: At most, only two electrons can have the same energy, and if they do, they must have different spins. These two spin states are often represented as spinning clockwise and spinning counterclockwise.

Among the conduction electrons, one pair will have almost no kinetic energy (energy of motion), another pair will have slightly more energy, and another will have even more energy.

All the conduction electrons are ranked in this manner. If the temperature of the sample is absolute zero (the coldest that it can be), then the energy of the most energetic electron is called the fermi energy or fermi level of energy, named for Enrico Fermi. Since there are many electrons in any macroscopic sample, the fermi energy is high. Electrons with energies near the fermi energy travel about 1 million meters per second, and most of the electrons that can conduct electricity will have approximately this speed.

Quantum mechanics dictates that these electrons play a rather stringent version of "musical chairs": If there is no empty chair available, then they not only cannot sit down but also cannot stop bouncing around. If other electrons have a little more or a little less energy than a given electron, then that electron cannot slow down or speed up because other electrons already have energies corresponding to those speeds. It is likely that only electrons near the fermi energy can change their energy, and they can do so only if there are empty energy states above the fermi energy. Therefore, two characteristics of a good conductor are that it must have a large supply of electrons near the fermi energy and that it must have a large supply of empty energy states near the fermi energy.

The fermi energy of a good conductor lies in the conduction band, a series of very closely spaced energy levels. The conduction band has an abundance of empty energy states, which allows the conduction electrons to change energy easily. If a conductor is warmed above absolute zero, then electrons from just beneath the fermi level will gain energy from the heat and rise to energy levels just above the fermi level. If the sample is warmed further, then the process will continue. The fermi level can be redefined as the energy roughly midway between the most energetic electron above the fermi level and the energy of the least energetic empty space below the fermi level. Therefore, the fermi energy is approximately the average energy of those electrons that can change energy. This conclusion explains why conduction electrons travel at 1 million meters per second and why only those near the fermi level can respond to an electric field by accelerating or decelerating (changing kinetic energy little by little).

Applications

A good conductor, according to the classical model, should have a high density of charge carriers and a long mean time between collisions. Both of these requirements make sense.

The classical model itself takes one no further, but it does provide hints. The mean time will be shorter if collisions are more frequent, and the electrons collide most often when there are irregularities in the crystal lattice. These irregularities are of three kinds: a gap or misalignment in the lattice, an ion of a different size (an impurity ion), and thermal oscillations by the ions of the lattice. Thermal oscillation is the vibration of an ion around its home position caused by heat energy. An electron continues on its course until a lattice ion thrusts itself into the electron's path and causes a collision. The higher the temperature, the more violent and frequent become the movements of the ions. Therefore, one can expect that the resistance of metals increases with increasing temperature.

The resistance of the white-hot tungsten filament of an incandescent light is many times that of its room temperature value. The resistivity of copper falls off by a factor of fifty to one hundred (depending on the purity of the copper) when the temperature is reduced from 300 Kelvins to 4 Kelvins. In a test, the resistivity of a very pure single crystal of copper drops by more than twenty thousand as the temperature is lowered to 4 Kelvins. Yet, the resistivity of brass, an alloy of zinc and copper, falls only by a factor of four over the same temperature range.

(Neither metal is a superconductor at these temperatures.) Evidently, much of the resistivity of brass is attributable to the irregularity of the copper-zinc crystal lattice. The resistivity of a semiconductor decreases with increasing temperature, causing the density of charge carriers to increase rapidly with increasing temperature.

How fast does electricity travel down a wire? The drift velocity is very slow. For example, if someone wanted to call his grandmother who lives 2,000 kilometers away, it could take a thousand years for electrons traveling at drift velocity to get there. In fact, an electric field can travel in a copper cable at nearly the speed of light. The field causes the electrons throughout the wire to move at the drift velocity. Electrons moving with the drift velocity constitute a current in the small speaker in the grandmother's telephone, which reconstructs the sound of the person calling, but the message traveled through the telephone cable as an electric field at nearly the speed of light.

The concept of the fermi level can explain some very interesting phenomena. While electrons near the fermi level have much energy, they do not have enough energy to escape from the metal. The work function is defined to be the amount of work necessary to lift an electron with the fermi energy up out of the metal--that is, give the electron enough energy to escape the metal. If light is shone onto a clean metal surface, then electrons may absorb it. If the photons that comprise the light have greater energy than the value of the work function, then electrons can absorb light and leave the metal. This is the essence of the photoelectric effect, which has many uses. For example, a light meter is simply a properly prepared piece of metal along with an ammeter to measure the current of the "photoejected" electrons.

Suppose that wires of two different metals, such as silver and zinc, were connected together at one end. Work functions are measured in electronvolts, which is the energy that an electron gains when accelerated through 1 volt. The work function of silver is about 4.2 electronvolts and that of zinc is about 3.2 electronvolts, which means that the fermi level of zinc is about 1 electronvolt higher than that of silver. With the two metals in contact, electrons will flow from the higher energy levels of zinc to those of silver, which are lower. The flow will continue until the fermi levels are equal. There will now be a voltage difference between the two metals of about 1 volt. This voltage difference is called the contact potential. Contact potentials occur anytime two dissimilar metals are connected. If the other ends of the wires are also connected, nothing further will happen because the fermi energies are already the same. If the two junctions are kept at different temperatures, however, then the fermi levels will not be the same. A small current will flow in the circuit as long as the junctions are maintained at different temperatures, which is called the Seebeck effect. This effect, named for Thomas Johann Seebeck, is the basis for the thermocouple thermometer. The net contact potential is proportional to the temperature difference of the two junctions.

The opposite of the Seebeck effect is the Peltier effect, named for Jean Charles Athanase Peltier. If a current is applied to a junction of dissimilar metals, then the junction will heat with the current flowing in one direction and cool with the current flowing in the opposite direction. Maintaining this current requires an external power source. Because the Peltier refrigerator has no moving parts, it does not vibrate. This makes it ideal for astronomers who wish to keep photographic film cold while taking a long exposure, which improves the quality of the photographs. Astronomers also use this vibrationless refrigerator to chill photocubes and other light sensors in order to reduce their background noise, or static.

Context

The accumulated knowledge of conductors has made possible the development of fabulous technology, as well as deep explorations of the structure of nature. Classical physicists created Ohm's law and discovered other such insights on which the "electrical revolution" was based. Electric lights, motors, generators, and power lines were developed, and these inventions enabled factories to break away from water and steam power. This independence changed where factories could be built, which in turn changed where people lived. Being more widely dispersed and more efficient, electric power plants helped reduce the deadly black fogs of the industrial centers of England and Europe. Telephones, radios, and televisions helped to keep people more in touch over longer distances than ever before. Then, another revolution occurred, this time an electronic revolution. The first practical digital electronic computer, ENIAC (Electronic Numerical Integrator and Computer) was completed in 1946. The marvel of its day, it weighed 30 tons and consumed 174 kilowatts of power. As the knowledge of quantum mechanics and the ability to supply it grew, the transistor and then the integrated circuit were developed, leading to the "computer revolution." It is difficult to overestimate the effect of these revolutions on Western civilization.

One application of the Seebeck effect powers the Voyager spacecraft. The Seebeck effect converts heat directly into electricity and involves no moving parts. While the process is not efficient, it is dependable. The Voyager spacecraft were built to operate far from the sun, where solar cells are unusable. Instead, they used radioisotope thermoelectric generators (RTGs) to supply their power. For a heat source, they used radioactive plutonium-238 oxide. The device was designed to produce 450 watts at launch, an amount that would slowly decrease over the eighty-six-year half-life of the radioisotope.

Understanding the behavior of conductors on an atomic scale has added profoundly to an understanding of nature. As can be seen by a discussion of the fermi level, the rather bizarre predictions of quantum mechanics are borne out. The study of electrons in conductors has also guided scientists to a more unified view of nature. Hans Christian rsted was the first to comment on the fact that an electric current in a wire produced a magnetic field around the wire. It was soon learned that the reverse also occurred, that changing magnetic fields produces currents in wires. Scientists have since learned that electric and magnetic fields are merely two aspects of the same force.

Principal terms

BAND GAP: the energy difference between the top of the valence band and the bottom of the conduction band

CONDUCTION BAND: a series of very closely spaced energy levels; electrons with these energies can move freely throughout a conductor

FERMI LEVEL OR FERMI ENERGY: the energy of the highest energy-free electron in a substance at absolute zero temperature

MEAN FREE PATH: the average distance that a conduction electron travels before it collides with a lattice ion

MEAN TIME: the average time between collisions for a conduction electron

RESISTANCE: the hindrance to the flow of electricity through a material, which depends on the material, its size, and its shape

RESISTIVITY: the intrinsic resistance of a substance independent of the size and shape of the sample

VALENCE BAND: a series of very closely spaced energy levels occupied by the outermost electrons of each atom in a solid

Bibliography

Banigan, Sharon, ed. ELECTRONS ON THE MOVE. New York: Walker, 1964. This book by the "scientists of the Westinghouse Research Laboratories" is an effort to communicate clearly with the layperson. It is fairly easy to read, but it does contain some mathematics. Included are treatments of conductors, semiconductors, and band theory.

Epstein, Arthur J., and Joel S. Miller. "Linear-Chain Conductors." SCIENTIFIC AMERICAN 241 (October, 1979): 52-61. This article examines molecules, which are actually long chains that conduct well only along the axis of the chain. Explains how their conductivity is measured and reviews simple band theory, including the band structures of metals, semimetals, semiconductors, and insulators.

Faughn, Jerry S., and Karl F. Kuhn. PHYSICS FOR PEOPLE WHO THINK THEY DON'T LIKE PHYSICS. Philadelphia: W. B. Saunders, 1976. Those with little knowledge of physics would do well to consult this simplified text. Included in the several chapters on electricity and magnetism are treatments of the simpler properties of conductors. The book features topics with popular appeal, humorous cartoons, and simple experiments to try at home.

Holden, Alan. THE NATURE OF SOLIDS. New York: Columbia University Press, 1965. This book is nonmathematical, but it requires a thoughtful reader who has had at least some high school chemistry or physics. The presentation is clear and straight-forward. Treats, among other topics, insulators and conductors in terms of the conduction band and the band gap.

Millikan, Robert Andrews. THE ELECTRON. Chicago: University of Chicago Press, 1963. This is a facsimile of the original 1917 edition. Millikan is credited with the first accurate measurement of the charge of the electron. Here, he traces views of electricity from ancient times up to the time of his own work. Mathematical proofs are contained in the appendices.

Charges and Currents

Conductors and Resistors

Forces on Charges and Currents

Insulators and Dielectrics

Electrical Properties of Solids