Paul F. Weller
State University College Fredonia, New York 14063
An Analogy for Elementary Band Theory C0nceph in Solids
The concepts of solid state physics and chemistry are foreign to most chemists and particularly to undergraduate students. The author has found the following analogies to be valuable in the study of e l e mentary solid state principles. Insulators and Metals
One of the major distinctions between metals and insulators is the ability to conduct an electric current. This is generally explained on the basis of the band theory of solids (a description of this theory as well as all of the other principles described below can be found in the excellent monograph edited by N. B. Hannay1). The band theory assumes that isolated free atom energy levels are broadened into a baud of allowed energy levels, separated by forbidden energy regions or gaps, when the atom exists in a solid substance. A typical picture of the band structure of a metal and an insulator is given in Figure 1. Under the influence of an electric field, the electrons that ~artiallvfill the conduction band in the metal can move'. In the insulator, however, all of the electrons CONDUCTION BAND
Semiconductors
VALENCE BAND METAL
INSULATOR
Figure 1. A representation of energy bands in solids The diagonal lines indicate electron populations. The electrons in the valence bonds ore flxod while those in the metal condudion bond are mobile.
are used for bonding purposes, i.e., all of the energy levels of the valence band are occupied, and there are essentially no electrons in the conduction band. In order to move an electron through an insulator, sufficient energy must be provided to promote the electron from the valence band to the empty energy levels of the conduction band. Energy gaps for typical insula tors are five or more electron volts (1 ev = 23 kcal/ mole). Consequently, very few electrons are promoted under ordinary conditions. A rather good analogy can be drawn between the energy bands in metals and insulators and a sectioned water bottle as shown in Figure 2. The large sections of the bottle, connected by capillary tubing, correspond
' HANNAY,N.
B., (Editor), "Semiconductors," Reinhold
Publishing Cop., New York, 1959.
to the conduction and valence bands of the band theory, shown in Figure 1. In the bottle correspondimg to the metal, water fills thelower section, "valence band," and part of the upper section, "conduction band." The application of an electric field is analogous to tipping the bottle. When the bottle is tipped, the water in the "conduction band" is free to flow to the low side of the bottle, corresponding to the flow of electrons in a metal. The water contained in the L'valenceband" is not free to move and remains essentially in place, just as the valence band electrons are thought to do in real solids. The bottle corresponding to the insulator has only the "valence band" filled with water, i.e., there are no electrons in the conduction band. Now when the bottle is tipped, there is no water flow, corresponding to the lack of electrical conductivity in an insulator. The forbidden energy gap corresponds to the distance between the upper and lower sections of the bottle, i.e., to the length of the capillary tubmg. The larger the energy gap, the longer the tube. In order to transport water to the "conduction band," the bottle must be tipped or jostled vigorously, corresponding to the expenditure of high energy. As the capillary tubing increases in length, it becomes more diicult to get water into the "conduction band," i.e., the energy gap has increased with a subsequent increase in the electrical resistance of the insulator. Semiconductors can be divided into two broad categories, intrinsic and extrinsic. Extrinsic semiconductivity is considered further in the following section. For the case of an intrinsic semiconductor the band picture and correspondingwater bottle analogy for an insulator in Figures 1 and 2 can be used. The only difference between an intrinsic semiconductor and an insulator is in the size of the forbidden energy gap. Insulators have much larger band gaps, i.e., the capillary tube is much longer than in the semiconductor
WATER
\
METAL
INSULATOR
Figure 2. A water bottle onology to the energy bands depicted in Figure 1. The lower and upper sections correspond to the valence ond conduction bands, rerpoctively. The water is ondogour to electrons and the capillary tube to the forbidden energy gap.
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case. With the relatively short capillary tube, water (electrons) can be "excited" into the "conduction hand" with much less vigorous tipping. This allows for a greater electron concentration in the conduction hand of the semiconductor and, hence, for a material with lower electrical resistance. It is interesting to note what happens when a drop of water is transferred from the lower bottle section to the upper section-a hubhle is simultaneously formed in the lower section. This bubble exhibits two interesting properties. First, it floats at the top of the "valence band." It has a tendency to escape up (the capillary tube) into the upper bottle section. This is exactly the opposite behavior displayed by the water droplet. The lower section of the bottle is the preferred level, and energy must be expended to transfer the water to the upper section. Second, note what happens to the water droplet in the "conduction band" and the hubhle in the "valence hand" when the bottle is tipped. The water runs to the low side of the bottle while the bubble floats to the high side, still in the "valence band.'' This is an excellent analogy to what happens in a semiconductor. When an electron (the water droplet) is excited from the valence hand into the conduction hand (lower to upper bottle section), an electron d o ficiency (the bubble) is produced in the valence hand. This deficiency is called a hole. Electron and hole behavior in solids is similar to that of the water droplet and bubble. Electrons prefer low-energy levels while holes tend to "float," i.e., prefer high-energy levels. Electrons move in one direction in an applied electric field while holes move in exactly the opposite direction. If it is imagined that the water droplet carries a negative charge and the bubble a positive charge, then the electrical charge transport properties of electrons and holes are illustrated. If the electrons dominate in the charge transport process, the semiconductor is n-type (negative charge carriers). If the holes are dominant, the semiconductor is p-type (positive charge carriers). The dominance of one charge carrier type in intrinsic semiconductors is caused by the relative ease of movement of one of the carriers over the other. Generally, electrons are more mobile than holes. This is not shown by the water bottle analogy. Extrinsic Semiconductors
Electrical charge carriers, electrons or holes, can be produced in semiconducting materials in ways that require much less energy than in the case of an intrinsic semiconductor. Defects in the semiconductor crystal lattice can serve as electron or hole sources. These lattice defects are generally lattice vacancies, i.e., the absence of a host cation or anion, or impurity ions dissolved in t,he semiconductor host. These defects give rise to energy levels within the forbidden energy gap of the semiconductor. Generally, levels introduced near the conduction band edge are called donor levels, i.e., an electron can be easily excited from the level into the conduction baud, and levels occurring near the valence band edge are called acceptor levels, i.e., they will readily accept an electron from the valence hand, there by producing a hole in the valence baud. Donor and acceptor levels are shown in a typical band picture in 392
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CONDUCTION
r
r
r r
A -
,ACCEPTORS VALENCE BAND
Figure 3. Donor ond acceptor lsvelr, occupied b y electrons and holes, respectively, of on extrinsic tamiconductor reprarontod b y a typical bond picture and a water bottle analogy.
Figure 3. In this same figure is the correspondingwater bottle analogy. Donor and acceptor levels are simulated by drilling small "wells" in the "conduction band" section and erecting small "water storage tanks" on the "valence baud" section. The "wells" are partly filled with water, i.e., they are occupied by electrons. The "storage tanks" are empty, i.e., they are occupied by holes. I t is now relatively easy to transport water droplets from the "wells" into the "conduction band," producing electron charge carriers, or water into the "storage tanks" from the "valence band," producing bubbles (holes). Tipping the bottle only slightly, corresponding to a small applied voltage, can produce charge carriers (contrasted with the vigorous tipping necessary to transport water from the "valence" to the "conduction hand"). In semiconducting materials there is a variation in donor or acceptor level position, depending on the type of defect present. This is easily shown with the water bottle by having "wells" of various depths or "storage tanks" with various heights. I t takes more energy to transport a water droplet from a deep "well'' than from a shallow "well." A deep donor level in a semiconductor is one with a high energy of activation, i.e., it lies farther in energy below the conduction baud edge than does a shallow donor. A corresponding analogy exists for the "storage tanks" and hole activation energies. The production of charge carriers, say electrons, in semiconductors by heat energy can also be shown by the water bottle analogy. The most easily visualized case is with an n-type, extrinsic semiconductor with a shallow donor level. When the bottle is heated slightly, water contained in the donor level expands, flows up the short connecting tube and into the "conduction band." This is analogous to the production of free conduction electrons in the semiconductor. No water (electron) is transferred from the "valence" to the "conduction" bauds since the capillary tube (energy gap) is too long. The p-n Junction
The technological importance of p-n junctions is very great. One of the most useful characteristics of such a junction is rectification, i.e., allowing a current to flow in one direction through the junction but not in the other direction. This property can he visualized by using the water bottle analogy as shown in Figure 4. One section of the bottle has been made p-type by transferring water droplets (electrons) into the "storage tanks" (acceptor levels), leaving bubbles (holes) in the "valence hand." The second section of the bottle has been made n-type by transferring water from the "wells" (donor levels) into the "conduction band."
These two sections of the bottle me separated from one another by an energy barrier, the p-n junction which is simulated by the jog between the water bottles, high enough to keep the water and bubbles in their respective sections. This jog between the bottles corresponds to the real energy barriers found in solids when junctions are formed.
p -TYPE
n-TYPE
Figure 4. A water b o n k analogy for o p-n junction. The junction occurs at the iog between the n- and p-type regions. Electron and hole charge carriers, onologour to the woter in the upper section and bubbler in the lower 3ection. orire from the simulated donor and occeptor Isvelr
Now note what happens when the bottle is tipped, i.e., an electric field is applied to the p-n junction. If the n-type side is elevated (corresponding to the application of a negative charge to the n-type region, often called a positive bias), water will run over the barrier into the empty "conduction hand" of the ptype region while bubbles will flow from the p-type region, over their barrier (actually down) and up into the n-type "valence band." Again, imagining that the water droplets are negatively charged and the bubbles positively charged shows that there is charge transfer or current flow across the p-n junction. If the bottle is tipped so that the p-type side is higher than the n-type side (applying a positive charge to the n-type region, often called negative bias), no water flows from the n-type region and no buhbles from the p-type region. Hence, no charge transfer and no current flow across the junction. Alternate application of negative and positive charge to the n-type region will result in current flow only during the negative charge (forward bias) part of the cycle. Hence, the change of alternating to direct current.
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