Solid materials can be classified by their ability to conduct electrical current as metals, semiconductors, and insulators. The ability to conduct is directly related to the availability of and mobility of charge carriers (e.g., electrons). As you may recall from your basic chemistry course, electrons can only occupy discrete energy levels in atoms. Hence, you will remember filling orbitals with electrons in unique states as you learned about the periodic table. For atoms in a crystalline solid, discrete atomic levels with similar energy blend together to form energy bands as illustrated in Figure 15.1 for a material with an energy gap between the two bands. The valence band is the highest energy band that contains electrons, and the conduction band is the lowest energy band that is unoccupied. Bands that are completely filled with electrons or are empty do not conduct. The energy band situation for solid materials is illustrated in Figure 15.2. Electrons in a metal reside in a band that is only partially filled. Hence, a large number of electrons can easily become excited from the occupied states to the unoccupied states where they are free to move between atoms and provide electrical conduction. In a semiconductor, the valence band is completely full (0 K), and there is an energy gap between the valence band and the conduction band. This difference in energy between the valence band maximum (EV) and conductivity band minimum (EC) is known as the band gap (Eg). In order for conduction to occur, electrons must be excited from the valence band to the conduction band. Fortunately, the energy gap is sufficiently small that thermal energy is able to excite a certain number of electrons into the conduction band, leading to conductivity. Importantly, the number of charge carriers is critical for semiconductor conductivity. With insulators, the energy gap is too large for a significant number of charge carriers to be excited into the conduction band, leading to very low electrical conductivity.
Figure 15.1 Illustration of energy bands formed in a crystalline solid.
Figure 15.2 Energy bands and energy (band) gaps for different types of solid materials.
Figure 15.3 illustrates thermal excitation of an electron from the valence band to the conduction band, leaving in its place an electron vacancy that is known as a hole. This results in the ability to move charge in both bands and gives rise to conductivity. In a pure semiconductor devoid of impurities or dopants, the number of free electrons in the conduction band and holes in the valence band are equal, and the resulting conductivity is known as the intrinsic conductivity. The charge carriers in the conduction band are electrons, while the holes (missing electrons) are the charge carriers in the valence band. These holes move like positive charges, and are a convenient way to represent charge movement in the valence band. In reality, it is electrons that move by breaking and reforming bonds in the valence band. Because of this, electrons in the conduction band have a higher mobility, meaning that they move more easily in response to a potential gradient than do holes since their movement does not require the making and breaking of chemical bonds. In intrinsic (pure) semiconductors, the number of charge carriers, and thus the intrinsic conductivity, is strongly influenced by the crystallographic quality of the semiconductor because defects in the crystal structure increase the recombination rate of the electron–hole pairs. The rate of recombination is closely tied to a parameter called carrier lifetime, which is the average amount of time that a charge carrier can exist in an excited state before recombination occurs. Carrier lifetimes are a few microseconds for high-quality silicon, but can be much shorter for defect-riddled oxide semiconductors. The number of carriers depends on temperature, as you would expect, with more carriers and higher conductivity at higher temperatures. For intrinsic silicon, the number of carriers at room temperature is 9.65 × 109 carriers per cubic centimeter, and the room temperature conductivity is 3.1 × 10−6 S·cm−1.
Figure 15.3 Excitation of an electron from the valence band to the conduction band of a semiconductor.
If intrinsic conductivity were the end of the story, semiconductors would not be particularly interesting. One of the key advantages of semiconductors is that their conductivity can be changed by doping with impurity atoms. For silicon, which is the most commonly used semiconductor material in photovoltaic cells, each Si atom in the pure material shares an electron with the four adjacent Si atoms in order to provide a stable outer shell with eight electrons as illustrated in Figure 15.4a. Dopants for silicon typically come from the elements in the columns just to the left and right of silicon in the periodic table. Elements to the right of silicon, such as phosphorous or arsenic, have an additional valence electron compared to silicon. Therefore, when one of these elements displaces a silicon atom in the crystal matrix, there is an extra electron that is easily excited or donated to the conduction band (see Figure 15.4b). These donated electrons in the conduction band serve as charge carriers and lead to extrinsic conductivity, or conductivity caused by doping. Donors such as phosphorous are called n-type dopants (n is for negative, since the resulting charge carriers are negative electrons).
Figure 15.4 (a) Undoped Si. (b) An illustration of an n-type extrinsically doped Si with electrons as the majority carriers.
In contrast, elements in the column to the left of silicon, such as boron, contain one less valence electron. When integrated into the silicon matrix, these elements tie up or “accept” an electron, giving the dopant atoms a net negative charge, as shown in Figure 15.5. Since that electron comes from the valence band, it generates a hole in the valence band, resulting in extrinsic conductivity or conductivity that is due to the presence of a dopant. Acceptors such as boron are called p-type dopants (p is for positive, since the resulting charge carriers are positive holes). As noted previously, electrical conduction can occur by the holes or the electrons. The more abundant carriers are called majority carriers, and the less abundant carriers are called minority carriers. In a p-type semiconductor, holes are the majority carriers while electrons are the minority carriers.
Figure 15.5 Illustration of a p-type extrinsically doped semiconductor with holes as the majority carriers.
Figure 15.6 qualitatively illustrates the energy levels associated with the dopant atoms and their relation to the energy band of the semiconductor. Because of the proximity of the energy levels to the respective band, the thermal energy at room temperature is sufficient to excite essentially all of the electrons from the dopant atoms to the conduction band (n-type), or to permit all of the dopant atoms to capture an electron from the valence band (p-type). This condition is referred to as full ionization, and it is a good assumption for all temperatures at or above room temperature. The net result is a large concentration of the majority carriers, a smaller concentration of the minority carriers, and fixed charges associated with ionized dopant atoms that are incorporated into the crystal structure of the semiconductor. This situation is illustrated in Figure 15.7, where the charges that are circled represent the fixed charges. The net charge is zero as one would expect. The fixed charges associated with the dopant atoms play an important role, especially at the interface, in the behavior of semiconductors in electrochemical systems as will be seen later in this chapter.
Figure 15.6 Illustration of energy levels and ionization for (a) n-type and (b) p-type dopants.
Figure 15.7 Carriers and immobile ions in extrinsic semiconductors.
Some of the important quantities associated with semiconductors are defined below. Note that the “number of” portion of the units if frequently left off and is implied. For example, n is frequently expressed in cm−3 rather than number of electrons cm−3.
n = number of free electrons cm−3
ni = number of free electrons cm−3 in the intrinsic (undoped) semiconductor.
p = number of holes cm−3
ND = number of donor atoms cm−3
NA = number of acceptor atoms cm−3
Assuming full ionization,
n∼ND for n-type semiconductors,
p∼NA for p-type semiconductors.
If the concentration of the majority carriers is known (e.g., from the dopant concentration), the following relation, known as the law of mass action, can be used to calculate the concentration of the minority carriers:
(15.1)
Illustration 15.1 demonstrates how to determine the number of carriers from the dopant level. The conductivity of intrinsic semiconductors is limited by the number of charge carriers available, and even low dopant concentrations can dramatically influence the number of charge carriers and the electrical conductivity.
ILLUSTRATION 15.1
Extrinsic Doping of Silicon
Silicon is doped with 1 ppb (by weight) of indium. What is the dopant concentration (cm−3) at room temperature? Is the doping level sufficient to alter the properties of the silicon? Which is the majority carrier, electrons or holes? The density of silicon is 2.33 g·cm−3; the molecular weight of indium is 114.82 g·mol−1.
SOLUTION:
Assume 1 × 10−9 g indium and 1 g silicon. First calculate the number of In atoms in 1 × 10−9 g.
Now calculate the volume from the silicon density:
The dopant concentration is
Note that “atoms” in the numerator is implied when you write cm−3.
Since In is a p-type dopant (three electrons in outer shell), the majority carriers are holes. The intrinsic carrier concentration for silicon is about 1010 cm−3. Therefore, even 1 ppb is sufficient to significantly change the carrier concentration and hence the electronic properties of the semiconductor.
Typical doping levels for silicon are listed below for reference, where “−” refers to light levels of doping and “+” to heavy levels of doping.
ni = pi = 1010 electrons or holes per cm3 (intrinsic concentration for silicon)
n−−, p−− < 1014 electrons or holes per cm3 n−, p− = 1015 electrons or holes per cm3 n, p = 1017 electrons or holes per cm3 n+, p+ = 1019 electrons or holes per cm3 n++, p++ > 1020 electrons or holes per cm3
The resistivity of silicon as a function of the dopant concentration is shown in Figure 15.8 for both n-type and p-type dopants. Note that the resistivity of the p-type semiconductors is higher, owing to the lower mobility of the holes relative to the mobility of electrons. Perhaps more important, all of the values of resistivity shown in Figure 15.8 are significantly higher than those typically associated with metals. Why, then, are semiconductors useful? The answer to this question lies, at least in part, in the fact that we can locally modify semiconductor properties to create very small devices called transistors that form the basis for the microelectronics industry. Such devices are beyond the scope of this text, but we encourage students to explore this topic on their own! In the rest of this chapter, we will examine the behavior of semiconductors as part of electrochemical systems.
Figure 15.8 Resistivity of silicon at room temperature as function of dopant concentration. For n-type, the dopant is phosphorous; for p-type, it is boron.
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