Absolutely Small: How Quantum Theory Explains Our Everyday World - Michael D. Fayer (2010)

Chapter 19. Metals, Insulators, and Semiconductors

FIGURE 19.1 IS A SCHEMATIC DIAGRAM of a battery connected to a metal rod. We will discuss sodium metal below as an example, but the rod could be any metal. The positive end of the battery pulls electrons from the metal rod. Electrons flow out of the rod into the battery.


FIGURE 19.1. A metal rod of sodium, for example, is connected by wires to a battery. Negatively charged electrons are pulled from the metal rod to the positive side of the battery. Electrons flow from the negative side of the battery back into the metal rod.

However, to keep the rod from developing a positive charge that would pull back on the electrons and stop the flow, the rod must be connected to the negative side of the battery.

Electrons flow from the negative side of the battery into the rod, keeping the rod neutral, that is, the rod does not develop an electrical charge. Instead of a rod, the electrons could flow through the filament of a lightbulb in a flashlight. The flow of electrons through the filament causes it to get very hot and produce black body radiation in the visible part of the spectrum.


Delocalize Molecular Orbital for a Metal

How can electrons move through a piece of metal? What is the difference between a metal and an insulator? What is a semiconductor? Why does a metal get hot when electrons flow through it? What is superconductivity? To answer the first three questions, we will extend our discussion of the types of delocalized molecular orbitals found in aromatic molecules such as benzene and naphthalene (see Chapter 18) to the MOs of a macroscopic piece of metal and other materials. To answer the last two questions, we will expand the discussion to the effect of the thermal vibrations of the atoms that make up a piece of metal on electron motion in a metal.

In Chapter 10 on the hydrogen molecule, we saw two hydrogen atomic orbitals combine to form two molecular orbitals, one bonding and one antibonding. In benzene, we saw that six pz atomic orbitals, one from each carbon, combined to form six MOs, three bonding MOs and three antibonding MOs. For naphthalene, 10 pz atomic orbitals combined to form 10 MOs, five bonding and five antibonding. In each case, the MOs span the entire molecule. In Chapter 11 on the Periodic Table of Elements, we said that sodium, Na, is a metal because it has one electron, the 3s, past the neon filled shell configuration. Na can readily give up an electron to form salts, such as table salt, NaCl. In water NaCl dissolves to become Na+ and Cl-. We said that Na as a solid was a metal and conducted electricity. Now we are in a position to see why.

First consider the 3s orbitals of two sodium atoms that are next to each other and interacting. For sodium, the 3s electron is the valence electron, which will participate in bonding. The top portion of Figure 19.2 shows the energy levels of the two 3s atomic orbitals combining to form MOs. One of the MOs has lower energy than the atomic orbitals. It is the bonding MO. The other MO has higher energy; it is the antibonding MO. The middle shows that three atomic orbitals will form three MOs. The bottom illustrates the situation for six interacting sodium atoms. The six 3s atomic orbitals combine to form six MOs, three bonding and three antibonding.

Each Na has one 3s electron, which it will contribute to fill the MOs. For the system with six sodium atoms, there will be six electrons to fill the MOs. Each MO can be occupied by two electrons with opposite spins (one up arrow and one down arrow). Therefore, the three lowest energy MOs, which are the bonding MOs, will be filled by the six electrons. The three higher energy MOs are empty.


FIGURE 19.2. Top: Two sodium 3s atomic orbitals interact to produce two molecular orbitals, one lower in energy (bonding) and one higher in energy (antibonding). Middle: Three 3s atomic orbitals interact to form three MOs. Bottom: Six 3s atomic orbitals interact to form six MOs.

Now we must consider what happens when we have a very large number of interacting sodium atoms. Consider a metal rod composed of sodium atoms that is 10 cm long and 1 mm in diameter, such as that in Figure 19.1. For the dimensions given, the rod contains N = 2 × 1021 Na atoms, where N is the number of atoms. This number is two billion trillion atoms. The two billion trillion 3s atomic orbitals combine to form two billion trillion molecular orbitals. Like the MOs of benzene or naphthalene, the MOs of the sodium rod should be thought of as spanning the entire system, that is, the entire piece of metal.

A Piece of Metal Has a Vast Number of MO Energy Levels Called a Band

Figure 19.3 illustrates the energy levels of this system. Each of the N sodium atoms has an electron in a 3s atomic orbital. In the absence of interactions between the atoms, all of these atomic orbitals have the same energy. In the figure, this is represented by the collection of closely spaced lines on the left-hand side. To show that there are many atomic levels, the lines have been spread out, but they all have the same energy. When the atoms interact, the N atomic orbitals form N MOs. As we have seen previously for molecules, the MOs have different energies. Some of the MOs have lower energies than the atomic orbital energy and some have higher energy. This is represented on the right side of the figure by the spread-out but closely spaced lines. The MO energy levels in Figure 19.3 are equivalent to those shown in Figures 18.8, 18.9, and 19.2, except that there are vastly more energy levels, which are much more closely spaced. These are referred to as a band of states.


FIGURE 19.3. In a piece of sodium metal, there are N atoms. Each has an electron in a 3s orbital, represented by the closely spaced lines on the left. These all have the same energy. The N 3s atomic orbitals interact to form N molecular orbitals with energy levels shown on the right. The MO energy levels are so close together that the energy is effectively a continuous band of states. The Fermi level marks the highest occupied molecular orbital.

Quantum theory shows that the width of the band of states, that is, the difference in energy between the highest energy MO and the lowest energy MO, is only a few times the energy splitting of the MOs that arise from a pair of interacting sodium atoms (see Figure 19.2, top). Then in our example of two billion trillion Na atoms, there are this many energy levels in a relatively narrow range of energy. The result is that the energy levels are so closely spaced that the energy is effectively continuous within the band.

Putting in the Electrons

There are N sodium atoms, each with a single 3s electron. We need to take these N electrons and put them in the appropriate MOs, as we did with small molecules in Chapters 12 and 13 and as shown in Figures 18.8 and 18.9. These sodium metal delocalized MOs are orbitals like any others, so we must obey the three rules for putting in the electrons discussed in Chapter 11. They are lowest energy first, no more than two electrons in an orbital that must have paired spins (Pauli Exclusion Principle), and don’t pair spins unless necessary (Hund’s Rule). Figure 19.3 illustrates putting in the electrons. The first electron goes in the lowest energy level. The next electron goes into the same level with the opposite spin, that is, one up arrow and one down arrow. The third electron can’t go into the lowest energy level because that would violate the Pauli Principle. So, it goes into the level one up from the lowest. The fourth electron goes into this same level with paired spins. This will continue until all N electrons are in MOs.

The Fermi Level

There are N MO energy levels and N electrons. But two electrons can go in each level. Therefore, only the bottom half of the energy band of levels will be filled. This is like benzene (Figure 18.8) and naphthalene (Figure 18.9), which also only have the bottom half of their MOs filled. The energy of the highest filled orbital is called the Fermi level, for Enrico Fermi (1901-1954). Fermi was a physicist who worked in many areas of science including the theory of solids, such as metals, and the theory of nuclear reactions. His work contributed to the development of nuclear energy. He won the Nobel Prize in Physics in 1938 “for his demonstrations of the existence of new radioactive elements produced by neutron irradiation, and for his related discovery of nuclear reactions brought about by slow neutrons.” As we will see, the Fermi level is very important.

The Fermi level is the level of the highest filled MO at the absolute zero of temperature. This temperature is 0° K, where K is the Kelvin unit of temperature. One degree K is the same as one degree C (centigrade) except the scale begins at the absolute zero of temperature. 0° K is -273° C or -459° F. We have briefly discussed how heat in systems of molecules, such as water, causes the molecules to jiggle around. In Chapter 15, it was pointed out that the thermal motions of water were responsible for the breaking of hydrogen bonds between water molecules. As temperature is decreased, there is less and less heat (thermal energy) and the motions of atoms and molecules decrease. The absolute zero of temperature, 0° K, is the temperature at which there is no heat to cause atoms and molecules to move. The Fermi level is actually defined to be the energy of the highest filled MO at 0° K.

How Electrons Move Through Metal

As shown in Figure 19.1, electrons enter one side of the metal rod and leave the other. This is possible because the electrons are in delocalized MOs that span the entire piece of metal. However, quantum theory shows that if all of the electrons occupy only the MOs below the Fermi level, the electrons will not move in a particular direction. Real metals are three dimensional, but for this discussion let’s consider only one dimension at a time. In our metal rod, when it is not connected to the battery, the electrons in the MOs are, nonetheless, constantly moving. Although the electrons are described in terms of quantum mechanical wavefunctions, the electrons have kinetic energy. Therefore, it is possible to calculate an electron velocity. The electrons in some MOs can be thought of as moving to the right. There are corresponding MOs of identical energy with electrons moving to the left. With all of the MOs filled, as shown in Figure 19.3, there will be no net electron current because there are an equal number of electrons moving to the right and to the left. In three dimensions, for any direction you pick, there will be equal probability of electrons moving in that direction and in the exact opposite direction.

However, when the metal rod in Figure 19.1 is connected to a battery, things change. One end of the rod is connected to the positive side of the battery and the other end is connected to the negative end of the battery. The connection to the battery changes what the electrons feel. Without the battery, the electrons feel the positive charges of the sodium nuclei and the negative charges of other electrons. Any one electron in the middle of the rod sees no difference between right and left. But with the battery there is an additional influence, the electric field produced across the metal rod by the battery. The electrons are attracted to the positive end and repelled from the negative end. The effect is to modify the system with the result that some electrons are in levels above the Fermi level that existed without the battery (see Figure 19.4). The electron states of the system are changed such that there are more electrons moving in the direction toward the positive end of the metal rod than toward the negative end.


FIGURE 19.4. Schematic of the sodium metal 3s band of levels as shown in Figure 19.3, but now with the influence of being connected to the battery. The effect is to put some electrons above the no battery Fermi level, taking them from filled MOs to empty MOs. These electrons are represented by the arrows above the Fermi level.

Quantum theory shows that it is necessary to have electrons above the Fermi level for electron conduction to occur. Because there are only infinitesimal differences in energy between the levels, even a very low voltage applied to the rod, which produces a tiny electric field, is sufficient to put some electrons above the Fermi level. The result is an electrical current flowing throw the metal rod. Electrons leave the positive end of the rod and are replaced by electrons entering from the negative end. For a bigger electric field (higher voltage), more electrons will be above the zero field Fermi level, and the electrical current is bigger. The detailed quantum theory of electrical conductivity in metals says that current will flow when an electric field is applied even at absolute zero temperature. Heat is not required for a metal to conduct. We will see below that this is not the case for semiconductors, and also, that heat, which is present at all temperatures above 0° K, actually interferes with electrical conductivity in metals.


Insulators Do Not Conduct Electricity Because the Band Is Filled

Metals conduct electricity easily even at 0° K because the electrons only fill part of the band of states, as shown in Figure 19.3 and 19.4. A very small electric field (voltage) will put electrons above the Fermi level. An insulator is a material, such as glass or plastic, that does not conduct electricity at any temperature. A schematic illustration of the band structure of an insulator is shown in Figure 19.5. In sodium metal, the 3s electrons are the valence electrons. The valence band is only half filled. In an insulator, like quartz (SiO2, silicon dioxide), which is very similar to glass, sharing of the electrons completes the shell of electrons. The interactions in a quartz crystal produce a band of states, with delocalized MOs, like in a metal. However, the valence band is completely filled. There are two electrons in each MO because there are N MOs but 2N electrons. So, all of the MOs have two electrons, from the lowest energy level to the highest energy level in the band. The filled band is shown in Figure 19.5 by the presence of arrows from the bottom of the band to the top of the band. This filled band should be compared to the half-filled band of Na metal shown in Figure 19.3.


FIGURE 19.5. Schematic of the band structure of an insulator. There is a filled band, with two electrons in each MO. At much higher energy, there is an empty band.

The Band Gap Is Large in Insulators

There are empty atomic orbitals at much higher energy, and these form MOs. However, there are no electrons to put in these MOs. Therefore, the next higher energy band is completely empty. The difference in energy between the top of the filled band and the bottom of the unfilled band is called the band gap. The Fermi level is at the top of the filled band.

As discussed qualitatively above and shown in detail by quantum theory, conductivity requires electrons in MOs with energies above the Fermi level. When an electric field is applied across the material (connected to a battery or other source of voltage), the nature of the delocalized states shifts. In a metal, because the band is only half filled and the energy levels are only separated by an infinitesimal amount, an applied electric field will result in a change such that some electrons are above the zero field Fermi level, and electrons flow through the metal. In an insulator, the next level above the Fermi level is in the empty band. The band gap is large, and the application of an electric field cannot change the system enough to put electrons into the empty band. Therefore, application of an electric field to an insulator is insufficient to produce conductivity, in contrast to a metal.

Another possibility is that thermal energy could excite electrons from the filled band to the empty band. An insulator has the property that the band gap energy is much greater than the thermal energy. As the temperature is increased, the amount of thermal energy increases. But an insulator has a band gap that is so large that the insulating material will be destroyed at temperatures that are still insufficient to thermally excite electrons from the filled band to the empty band. The net result is that application of an electric field cannot modify the states in a way to produce conductivity and thermal excitation of electrons cannot occur. Therefore, insulators do not conduct electricity.


In a Semiconductor the Band Gap Is Small

A semiconductor is like an insulator, but with a small band gap. A schematic of the band structure of a semiconductor is shown in Figure 19.6. In a semiconductor, such as silicon (Si), there are enough electrons to fill the valence band completely. At 0° K, where there is no thermal energy to excite electrons, all of the electrons are paired in the valence band. The Fermi level is at the top of the filled valence band. Therefore, silicon is an insulator at 0° K. However, in silicon and other semiconductors, the band gap is small. At room temperature, there is sufficient thermal energy to excite some electrons above the Fermi level into the next band. The thermal energy is contained in the motions of the atoms in a piece of semiconductor. The thermal excitation of electrons above the Fermi level into the next band is illustrated in Figure 19.6. The electrons that have been excited from filled MOs of the valence band to empty MOs of the conduction band are represented in the figure by arrows above the Fermi level. Because there are electrons above the Fermi level, a piece of semiconductor like silicon can conduct electricity. The electrons in the conduction band are called the conduction electrons.

Semiconductors do not conduct electricity as well as metals because they have far fewer conduction electrons. In a metal, there is no band gap. Large numbers of electrons are easily promoted above the Fermi level. In a semiconductor, there is a band gap, but it is small enough that thermal energy can excite some electrons above the Fermi level into the conduction band. As the temperature of a semiconductor is reduced, there are fewer and fewer conduction electrons to carry electrical current. At sufficiently low temperature, semiconductors become insulators. The only difference between a semiconductor and an insulator is the size of the band gap. Your computer chips, which are composed mainly of silicon semiconductors, will not work if they are sufficiently cold. The computers and electronics on a satellite must be kept warm or they will cease to function.


FIGURE 19.6. Schematic of the band structure of a semiconductor. The valence band is essentially filled. The gap in energy to the next band is relatively small. Some electrons are thermally excited above the Fermi level into the conduction band.

Thermal Energy Affects Electrical Conduction in Metals

Thermal energy is necessary in semiconductors to generate conduction electrons. Thermal energy also strongly influences electrical conduction in metals, although thermal energy is not necessary to generate the conduction electrons. In a piece of metal wire connected to a battery, there are electrons moving toward the positive end. As electrons leave the wire, they are replenished by electrons entering from the negative side of the battery. Current (electrons) flowing through a piece of wire will cause its temperature to rise. The heating elements in an electric stove or an electric space heater get very hot from a large current flowing through them. They get so hot that they glow red. The red color is black body radiation from the hot metal. We have said that electrons can readily flow through a piece of metal because the electrons are in delocalized MOs that span the metal. It only requires an electric field (connection to a battery or other voltage source) to get them moving in a particular direction. So the question is why does the flow of electrons cause the metal to heat up?

The electrons in a metal should be thought of as wave packets that are more or less localized. We discussed wave packets in Chapter 6 in connection with the Heisenberg Uncertainty Principle. The electron wave packets in a metal are formed from superpositions of the delocalized electronic MO wavefunctions in a manner analogous to photon wave packets or electron wave packets in a vacuum that are superpositions of the delocalized momentum states. Electrons are negatively charged so an electron wave packet carries a negative charge. The electron is accelerated toward the positive end. The acceleration gives the electron increased kinetic energy.

Vibrations of a Solid Are Phonons

In Chapter 17, we briefly discussed the quantized vibrations of molecules in connection with the greenhouse gas, carbon dioxide. A piece of metal, which is made up of atoms, also has quantized vibrations. The atoms in a metal crystal lattice can jiggle around in their positions. Although they jiggle, an atom remains in the same spot on average. The motion of each atom is connected to the motions of the other atoms in the same manner that the motions of the atoms of a CO2 molecule are connected to each other (see Figure 17.2). CO2 has several distinct vibrations, symmetric and antisymmetric stretches, and two bending modes. These three different types of modes have vibrational energies (frequencies) that are very different from each other.

In a metal crystal lattice, each atom can move in all three dimensions. For N atoms, there are 3N lattice vibrations, where again, N is the number of atoms in the piece of metal. For any finite size piece of metal, this huge number of vibrations results in a band of vibrations rather than several discreet frequencies. At low temperature, only the lowest energy part of the band of vibrations is thermally excited. At higher temperature more lattice vibrations are excited and higher energy vibrations are excited. The excited vibrations have kinetic energy. The energy of the excited vibrations is what we think of as heat.

The quantized vibrations of a lattice are called phonons. This name came about because phonons in certain fundamental quantum theory properties bear some resemblance to photons. Each phonon is a delocalized vibrational wave that spans the entire crystal lattice. The lattice waves can form more or less localized wave packets through the superposition of a range of wavelengths of lattice waves. The more or less localized phonon wave packet is completely analogous to the photon and electron wave packets mentioned just above and discussed in Chapter 6. The phonons are moving wave packets of mechanical thermal energy. The phonon wave packet can be thought of as a moving region of relatively localized jiggling of the atoms.

Electron Wave Packets and Phonon Wave Packets Scatter

An electron wave packet that is being accelerated toward the positive direction can interact with a phonon. The phonon causes the positively charged atomic nuclei to move. The negatively charged electron is influenced by the moving positive charges. The interaction of the electron and phonon is called a scattering event and is shown schematically in Figure 19.7. The electron and phonon wave packets are propagating in certain directions. The electron is being accelerated by the electric field when it “collides” with a phonon. Following the scattering event, in general, both wave packets will move in new directions. The electron will again be accelerated by the electric field toward positive. After some time, it will again encounter a phonon, and scatter. Each time the electron scatters, it gives up some of its kinetic energy to a phonon that it got from being accelerated by the electric field (voltage source).


FIGURE 19.7. Cartoon of an electron-phonon scattering event. The interaction of the electron and phonon causes the directions of the wave packet motions to change.

The scattering events do two things. First, they prevent the electrons from moving directly to the positive battery connection. Second, the scattering events add kinetic energy to the phonons. The electron loses energy, and the phonon gains it. The electron-phonon scattering reduces the electrical conductivity of metals because the electrons keep getting bumped, which causes them to move in different directions as they are trying to move to the electrically positive end of the wire. This is called electrical resistance. At very low temperature, there are few phonons, so the electrons can move a long way between scattering events. This makes it easier for the electrons to reach the positive connection. As the temperature is increased, there are more and more phonons because phonons are heat. As the temperature goes up, the electrons propagate less distance before their direction is changed, reducing their ability to reach the positive electrode. The result is that electrical conductivity decreases (the resistance increases) as the temperature increases.

Electron-Phonon Scattering Heats the Metal

Because the scattering events add kinetic energy to the phonons, they raise the metal’s temperature. Temperature is a measure of the heat content of a piece of material. Heat is the kinetic energy of the atomic motions. If there are a lot of electrons moving through the metal undergoing scattering events, then a lot of heat is added to the wire, and the temperature goes up. However, when the temperature goes up, there are even more phonons, more scattering events, and so the temperature goes up more.

This process is what you see when you turn an electric stove to high, and it takes some time for the heating element to glow red. When you first turn on the stove, the heating wire is at room temperature. Current (electrons) is flowing and electron-phonon scattering is occurring, causing the temperature to go up. The increased temperature means there are more phonons and more scattering events, and even more heat is added to the wire. The temperature goes up even more. However, as the temperature goes up, the current goes down because the additional scattering slows the progress of the electrons through the wire. The wire will come to a constant high temperature determined by the amount of current (the setting of the knob on the stove) that was initially applied at room temperature when the stove was first turned on.

Electrons in a normal metal will undergo electron-phonon scattering at any finite temperature. Therefore, a piece of wire has electrical resistance at any temperature other than absolute zero, 0° K. At absolute zero, there is no heat, so there are no phonons. However, it is impossible to reach absolute zero. To cool something down, you need something colder to take heat away. It is possible to achieve very low temperatures, for example, one millionth of a degree above absolute zero, using very specialized experimental methods, but even at this unbelievably low temperature, there are still some phonons and some electron-phonon scattering. In addition, if a piece of normal wire is at very low temperature, and you flow a significant amount of current through it, it will heat up. As mentioned in Chapter 17, electrical transmission lines from power plants to cities lose a lot of electricity. We now see why that is. It is caused by the electrical resistance of the wire, that is, electron-phonon scattering.


Materials that have no electrical resistance at finite temperature are called superconductors, and the flow of electrons through a superconducting piece of wire is called superconductivity. In metals, superconductivity only occurs at very low temperature. The Dutch physicist Heike Kamerlingh Onnes (1853-1926) discovered superconductivity in 1911, when he cooled mercury metal to 4° K (-269° C, -452° F). He observed that the resistance went to zero. Some other metals and the highest temperature at which they are superconducting are niobium: 9.26° K, lead: 7.19° K, vanadium: 5.3° K, aluminum: 1.2° K, and zinc: 0.88° K. Superconductivity was not explained until decades later. In 1972, three American physicists, John Bardeen (1908-1991), Leon Cooper (1930-), and John Schrieffer (1931-) won the Nobel Prize in Physics “for their jointly developed theory of uperconductivity, usually called the BCS- theory.” The BCS Theory, developed in 1957, is a detailed quantum mechanics description of electron-phonon interactions at low temperature. In 1956, Leon Cooper showed that electron-phonon interactions can cause electrons to pair. Two electrons will in some sense be joined together even though they are physically far apart. BCS used this idea to show that these Cooper pairs do not undergo electron-phonon scattering of the type discussed above that leads to electrical resistance. When electron-phonon scattering is absent, the electrons move through the metal with no resistance even though the temperature is not absolute zero. Since there is no resistance, there is no loss of electrical energy in spite of the fact that a large current may be flowing.

Superconductors are used today in a variety of applications, and there is promise for very important widespread applications in the future. Magnetic resonance imaging (MRI) requires a very large magnet. The large cylinder that an MRI subject is placed in is a superconducting electromagnet. A magnetic field is produced when electrical current flows through a coil of wire. To get a large magnetic field, it is necessary to have a great deal of current flow through a lot of wire that comprises the coil. Before superconducting magnets existed, magnetic fields were limited. The wire would get very hot, and cooling was a great problem. Now the wire is made from a superconducting metal, such as niobium. Once the flow of electrons around the coil is started, the two ends of the coil are joined. The electrons keep whizzing around coil. Because there is no resistance, there is no dissipation of energy, and no additional electricity needs to be added to the coil. Without superconductivity, there would be no MRI.

One great hope is to make electrical transmission lines that are superconducting. Such transmission lines would eliminate electrical loss in power. It would be possible to move electricity over much greater distances than is possible today. The problem is that the metallic superconductors need to be so cold that they are not practical for transmission lines. There are new high temperature superconducting materials. They were discovered in 1986 by Karl Müller (1927-) and Johannes Bednorz (1950-). They were awarded the Nobel Prize in Physics in 1987 “for their important break-through in the discovery of superconductivity in ceramic materials.” To date, superconductivity in these ceramic materials is not understood theoretically. Such materials can be superconducting at temperatures as high as 138° K. This temperature is high enough to make many applications practical. Because the high temperature superconductors are ceramic, the materials cannot be formed into wire as metals can. However, future advances may produce usable high temperature superconductivity, which will revolutionize power transmission and other fields of electronics.