Electrical Properties of Solids

Type of physical science: Condensed matter physics

Field of study: Solids

The molecular structure of solids determines which will be conductors, semiconductors, or insulators as well as which will be transparent or opaque to ultraviolet, visible, or infrared radiation.

Overview

In 1819, Hans Christian Orsted performed a classroom demonstration in which he brought a magnetic compass near a current-carrying wire. To his surprise, the compass needle turned about. Orsted reversed the direction of the current, and the compass needle reversed its direction. Thus, he proved that an electric current in a conductor produces a magnetic field. By 1831, Michael Faraday not only had demonstrated the complementary effect that changing magnetic fields produce electric currents in conductors but also had constructed the first electric motors and generators. While these inventions eventually led to widespread practical use of electricity, of equal importance to science was the demonstration that electricity and magnetism are closely linked. In order to discover how tight the linkage is, science had to learn which effects depended upon the properties of the materials used and which effects were more fundamental.

It was eventually established that an electric current in a wire is a stream of electrons and that such moving charged particles produce magnetic fields. Conversely, a magnetic field exerts a force on moving charged particles. Yet, why is it that under the same conditions (such as bringing a magnet up to a wire at the same speed each time), a large current will be briefly engendered in a copper loop, a smaller current in an iron loop, and no detectable current in a silk loop? Why are some materials good conductors and others are not? It cannot simply depend upon the number of electrons that a material has, since lead is a worse conductor than copper, and yet each lead atom has eighty-two electrons and each copper atom has only twenty-nine electrons.

Consider a simple experiment using only a single substance: copper. If various lengths and thicknesses of copper wire are taken and connected one at a time across the terminals of a battery, then current will flow in the wire. It will be evident that less current flows if the wire is longer, while more current flows if the wire is thicker.

The voltage across the wire (between its ends) divided by the amount of current that flows through the wire is defined to be the resistance of the wire. Resistance is measured in ohms, named for Georg Simon Ohm.

Resistivity 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 upon the substance, since the geometric factors of length and cross-sectional area have been canceled out. Resistivity is measured in ohm meters. A good insulator such as porcelain has a resistivity of 3 x 10¹² ohm meters. Quartz is even better as an insulator, with a resistivity of 5 x 10¹⁶ ohm meters. A typical resistivity for the semiconductor germanium is 0.45 ohm meter, while that of copper is 1.7 x 10^-8 ohm meters. The resistivity of a very good insulator is more than a million billion billion times that of a good conductor.

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

Each copper atom is composed of a tiny positively charged nucleus surrounded by a cloud of electrons. The outermost regions are not very tightly bound, and they 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. They are positively charged since they have 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 a million meters per second. Their motion is random, however, and they frequently collide with lattice ions. The average distance traveled between collisions is called the mean free path; it is usually less than 100 nanometers. The mean time between collisions is only 10^-13 to 10^-14 seconds.

If one assumes that an electric field accelerates electrons to give them a common small drift velocity between collisions with the crystal lattice, then there will be a current, a net flow of charge. It can then be shown that the resistivity should equal 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 collisions. This means that a good conductor should have a high density of charge carriers that go a long time between collisions.

Anytime atoms are brought close together as they are in a solid, their outermost electrons will interact. That is, the outermost electrons of each atom behave differently when other atoms are nearby. In such circumstances, these electrons obey a rather curious rule: At most, only two electrons can have the same energy, and if they have the same energy, they must be in different spin states. A useful model for these spin states is to picture one electron spinning clockwise and the other spinning counterclockwise.

If the conduction electrons are examined carefully, one will find one pair with almost no kinetic energy (energy of motion). Another pair will have slightly more energy, another will have even more than that, and so on, until the most energetic electron is found. If the temperature of the sample is absolute zero (the coldest it can be), 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. For example, in silver, an electron with the fermi energy speeds along at 1.4 million meters per second.

Applications

According to the model of resistivity, a good conductor should have a high density of charge carriers that go a long time between collisions. The fact that the density of charge carriers in a conductor is millions of times greater than that of a semiconductor is largely responsible for their different resistivities. Furthermore, the increase in the density of charge carriers with temperature in a semiconductor correctly predicts its decrease in resistivity with temperatures.

The resistivity of metals increases with temperature. This can be related to the mean time between collisions. The mean time will be shorter if collisions are more frequent. The electrons collide most often with 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 in the ions of the lattice. Thermal oscillations are the vibrations of an ion around its home position caused by heat energy. A conduction electron may be pictured as racing through the crystal lattice when suddenly a lattice ion lunges into the electron's path and causes a collision. The higher the temperature, the more violent and frequent the lunges of the ions will become. Therefore, one expects (correctly) that the resistance of metals should increase with increasing temperature. On the other hand, the resistivity of brass, an alloy of zinc and copper, is high because its crystal lattice cannot be as regular as that of a single metal.

Atoms in solids have various well-spaced energy levels and also a series of closely spaced levels called bands. Electrons may then occupy these energy levels. The outermost electrons of each atom are called valence electrons. In a solid, these electrons occupy the valence energy band. At absolute zero, the fermi level will be at the top of the valence band.

Metals have an energy band called the conduction band in which a large portion of the energy states are unoccupied. In a metal, the valence band overlaps the conduction band. This arrangement provides numerous conduction electrons along with numerous nearby empty energy states. These electrons can respond to a voltage applied to a conductor because they can be accelerated, they can go a little faster (or slower). Electrons that have no empty energy states close to their own energies cannot change speeds gradually.

As the temperature rises above absolute zero, the fermi level will rise beyond the top of the valence band and further into the conduction band. The fermi level is now roughly the average energy of those electrons that have enough empty states available so that they can gain and lose energy.

Insulators also have conduction bands, but there is a band gap between the top of the valence band and the bottom of the conduction band. The valence band is filled and has no empty energy states. The fermi level will lie at the top of the valence band even at fairly high temperatures, since extremely few electrons can gain enough energy to jump the gap and enter the conduction band. Better insulators have larger band gaps.

Take diamond as an example. Free electrons cannot travel very far in this insulator before they collide with an atom of the crystal lattice and change directions. It would take 100 million volts to give an electron enough energy to cross the band gap if that electron had to gain this energy during the short time between collisions. Since the conduction band is almost empty of electrons, only the most minuscule current can flow in an insulator.

As might be expected, semiconductors represent the intermediate case in which the valence band lies just beneath the conduction band so that the band gap is small. If electrons are introduced into the conduction band by heating the sample or by the presence of impurity atoms, the fermi level will rise into the band gap. The semiconductor can now carry a modest current. A key point is that the number of conduction electrons, and hence the current, can be controlled by applying a voltage to the semiconductor.

Using the band gap concept, further insight can be gained into the optical properties of dielectrics and conductors. All really good insulators must have a large band gap, and therefore they should be transparent to visible light. A visible light photon does not have enough energy to excite an electron near the fermi level up to the conduction band. This in turn means that light cannot be absorbed, and hence good insulators such as diamond, glass, quartz, and amber are transparent. Insulators that are not transparent contain impurities or irregularities that scatter (reflect) light in all directions. Most insulators are opaque to ultraviolet light. Ultraviolet light can be absorbed because it does have enough energy to excite electrons across the band gap into the conduction band.

Since semiconductors have narrow band gaps, it follows that they are opaque to ultraviolet and visible light. The semiconductor germanium is transparent to infrared waves, and infrared lenses are made of this material. Infrared photons have even less energy than the narrow germanium band gap. Finally, since metals have no band gap, they should be able to absorb photons of any energy. Metals are opaque to ultraviolet, visible, infrared, microwaves, and radio waves. The luster characteristic of metals results from their nearly complete absorption and reemission of visible light.

Consider a semiconductor with a band gap such that visible light can easily excite electrons from the valence band up into the conduction band. In the dark, such a material has high resistance; however, when exposed to light, it becomes a relatively good conductor. Such material is called a photoconductor; it is the basis for the dry photocopy process.

The photoconductor in xerography (dry writing) is often a thin coating on the outside of a metal cylinder. The metal is referred to as a metal substrate for the photoconductor. As the cylinder revolves, several processes take place in rapid succession. First, with the photoconductor in the dark, the cylinder rotates past a wire carrying several thousand volts. This places a positive charge on the surface of the photoconductor. With the metal substrate grounded, the positive charges attract negative charges from the ground and through the metal to the underside of the photoconductor. These negative charges help hold the positive surface charges in place.

The second step is to reflect onto the cylinder the image of the document to be copied.

Wherever light strikes the photoconductor, electrons are promoted from the valence band up into the conduction band. The conduction electrons neutralize the positive surface charge, while electrons from the metal substrate fill the vacancies in the valence band. All the surface charge is neutralized wherever the full light intensity falls on the photoconductor. Some charge remains where the light level is lower, and all the charge remains where no light strikes the photoconductor. Thus, the surface charge on the photoconductor is patterned into an image of the document.

Fine, negatively charged toner particles are now brought to the cylinder. The toner may be either black or colored. Toner particles will cling to the photoconductor in the pattern of the positive charge. Next, the toner is transferred from the photoconductor to a positively charged sheet of paper and is briefly heated to fuse it to the paper. Finally, any residual toner is scraped from the photoconductor, and it is flooded with light to clear it of any remaining charge patterns.

The system is now ready to make another copy.

As a final application, consider the effect of placing a conductor in a static (unchanging) electric field. Suppose the conductor is a solid cube, and that the field enters the right side of the cube and exits from the left side. Conduction electrons will flow from one side of the cube to the other because of the force exerted by the field. They will continue to flow until the repulsive force from the electrons already there (like charge repels) is equal to the force exerted by the original field. At this point, no more charge flows inside the cube, because the net field there is zero. The electrons on the side of the cube and the positive ions they left behind on the opposite side have established an electric field that canceled the external field. This also works if the conductor is hollow but completely enclosed. Such a device is called a Faraday cage, named for Michael Faraday, and is used to shield sensitive electrical components from electrical fields.

Context

The fact that the electric field is zero inside a conductor in static equilibrium is an important property of a conductor and is closely related to the form of Coulomb's law.

Charles-Augustin de Coulomb proposed, on the basis of experiments he performed, that the force between two charges should become weaker in proportion to the square of the distance between charges.

Benjamin Franklin was the first to check the accuracy of Coulomb's law by the following method. He charged a small conducting sphere and lowered it by an insulating thread into a deep metal can. As the sphere touched the bottom of the can, charge flowed from the sphere to the can. Now, if all the charge flows from the sphere to the can and then rearranges itself on the surface of the can so that the field within the conductor is zero, then Coulomb's law holds. To see if all the charge had left the sphere, Franklin withdrew the sphere and checked it for electric charge. Finding none, he deduced that Coulomb's law is correct. The experiment would be more accurate if one used a hollow conducting sphere instead of a deep can and if one had a more accurate instrument with which to detect the presence of charge. This more accurate experiment has been done with the result that the power of the distance in Coulomb's law can differ from the square by no more than one part in 10¹⁶.

The fact that a charge conductor behaves in this fashion ultimately depends upon the coulomb force falling off exactly as the square of the distance and that the surface area of a sphere grows larger exactly in proportion to the square of the sphere's radius. Thus, the fact that the electric field inside a conductor in static equilibrium is zero depends on the geometry followed in the universe.

If the universe were much different, humans could not exist. Suppose the force between charges fell off as the cube of the distance. Then, forces between charges at a given distance would be weaker. Solids and liquids would exist only near absolute zero, all elements would easily ionize, and chemical reactions would be rare. At the opposite extreme, if the charge force decreased with the first power of the distance, forces between charges would be stronger than they are now. Electrons would be more tightly bound to atoms, again making chemical reactions less likely. Forces between atoms would be stronger so liquids and solids would be less volatile and gases would exist only at higher temperatures. Life would be difficult if not impossible in either case.

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 of these energies can move freely throughout a conductor

CRYSTAL LATTICE: the regular array of points marking the equilibrium positions of the atoms in a crystalline solid

DIELECTRIC: an insulator

FERMI LEVEL or FERMI ENERGY: the energy of the highest-energy free electron in a substance at absolute zero temperature; at higher temperatures, the fermi energy may increase

INSULATOR: a substance that is an extremely poor electrical conductor

MEAN FREE PATH: the average distance 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; it depends upon 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

Asimov, Isaac. THE HISTORY OF PHYSICS. New York: Walker, 1984. Asimov is well known as a writer of both science fiction and popular-level science books. True to form, Asimov has produced a book that can be easily read by someone with little background in science. Contains several chapters on electricity and the electrical properties of matter. While the main treatment is classical, some quantum mechanical effects can be located through the unusually complete index.

Burland, Donald M., and Lawrence B. Schein. "Physics of Electrophotography." PHYSICS TODAY 39 (May, 1986): 46-53. The photocopy machine is a nice application of electrostatic charging and of the photoconductivity of insulators. This article contains relatively little mathematics but still explains the processes involved in detail.

Dohler, Gottfried H. "Solid-State Superlattices." SCIENTIFIC AMERICAN 249 (November, 1983): 144-151. The article is about semiconductors that have been formed from layers of thin films and can be made to have unusual properties. Contains a nice review of simple band theory in which semiconductors and conductors are compared.

Ehrenreich, Henry. "The Electrical Properties of Materials." SCIENTIFIC AMERICAN 217 (September, 1967): 195-204. A nonmathematical overview of what makes different materials conductors, semiconductors, or insulators. Discusses classical model of conductivity, fermi level, band gap, and superconductivity.

Slater, John C. "Energy Bands in Solids." PHYSICS TODAY 21 (April, 1968): 61-71. Slater shows how band theory arises from the wavelike behavior of electrons. While the article itself is nonmathematical, Slater describes the results of various calculations and compares them with experiments. To gain real benefit, the reader should at least be familiar with the energy levels of the simple hydrogen atom.

Charges and Currents

Conductors and Resistors

Forces on Charges and Currents

Insulators and Dielectrics