Electrons and holes in semiconductors
One of the remarkable and dramatic developments in recent years has been the application of solid state science to technical developments in electrical devices such as transistors. The study of semiconductors led to the discovery of their useful properties and to a large number of practical applications. The field is changing so rapidly that what we tell you today may be incorrect next year. It will certainly be incomplete. And it is perfectly clear that with the continuing study of these materials many new and more wonderful things will be possible as time goes on. You will not need to understand this chapter for what comes later in this volume, but you may find it interesting to see that at least something of what you are learning has some relation to the practical world. There are large numbers of semiconductors known, but we’ll concentrate on those which now have the greatest technical application. They are also the ones that are best understood, and in understanding them we will obtain a degree of understanding of many of the others. The semiconductor substances in most common use today are silicon and germanium. These elements crystallize in the diamond lattice, a kind of cubic structure in which the atoms have tetrahedral bonding with their four nearest neighbors. They are insulators at very low temperatures—near absolute zero—although they do conduct electricity somewhat at room temperature. They are not metals; they are called semiconductors. If we somehow put an extra electron into a crystal of silicon or germanium which is at a low temperature, we will have just the situation we described in the last chapter. The electron will be able to wander around in the crystal jumping from one atomic site to the next. Actually, we have looked only at the behavior of electrons in a rectangular lattice, and the equations would be somewhat different for the real lattice of silicon or germanium. All of the essential points are, however, illustrated by the results for the rectangular lattice. As we saw in Chapter 13, these electrons can have energies only in a certain energy band—called the conduction band. Within this band the energy is related to the wave-number $\FLPk$ of the probability amplitude $C$ (see Eq. (13.24)) by \begin{equation} \label{Eq:III:14:1} E=\!E_0\!-\!2A_x\!\cos k_xa\!-\!2A_y\!\cos k_yb\!-\!2A_z\!\cos k_zc. \end{equation}  The $A$’s are the amplitudes for jumping in the $x$-, $y$-, and $z$-directions, and $a$, $b$, and $c$ are the lattice spacings in these directions. For energies near the bottom of the band, we can approximate Eq. (14.1) by \begin{equation} \label{Eq:III:14:2} E=E_{\text{min}}+A_xa^2k_x^2+A_yb^2k_y^2+A_zc^2k_z^2 \end{equation}  (see Section 13–4). If we think of electron motion in some particular direction, so that the components of $\FLPk$ are always in the same ratio, the energy is a quadratic function of the wave number—and as we have seen of the momentum of the electron. We can write \begin{equation} \label{Eq:III:14:3} E=E_{\text{min}}+\alpha k^2, \end{equation}  where $\alpha$ is some constant, and we can make a graph of $E$ versus $k$ as in Fig. 14–1. We’ll call such a graph an “energy diagram.” An electron in a particular state of energy and momentum can be indicated by a point such as $S$ in the figure.  As we also mentioned in Chapter 13, we can have a similar situation if we remove an electron from a neutral insulator. Then, an electron can jump over from a nearby atom and fill the “hole,” but leaving another “hole” at the atom it started from. We can describe this behavior by writing an amplitude to find the hole at any particular atom, and by saying that the hole can jump from one atom to the next. (Clearly, the amplitudes $A$ that the hole jumps from atom $a$ to atom $b$ is just the same as the amplitude that an electron on atom $b$ jumps into the hole at atom $a$.) The mathematics is just the same for the hole as it was for the extra electron, and we get again that the energy of the hole is related to its wave number by an equation just like Eq. (14.1) or (14.2), except, of course, with different numerical values for the amplitudes $A_x$, $A_y$, and $A_z$. The hole has an energy related to the wave number of its probability amplitudes. Its energy lies in a restricted band, and near the bottom of the band its energy varies quadratically with the wave number—or momentum—just as in Fig. 14–1. Following the arguments of Section 13–3, we would find that the hole also behaves like a classical particle with a certain effective mass—except that in noncubic crystals the mass depends on the direction of motion. So the hole behaves like a positive particle moving through the crystal. The charge of the hole-particle is positive, because it is located at the site of a missing electron; and when it moves in one direction there are actually electrons moving in the opposite direction. If we put several electrons into a neutral crystal, they will move around much like the atoms of a low-pressure gas. If there are not too many, their interactions will not be very important. If we then put an electric field across the crystal, the electrons will start to move and an electric current will flow. Eventually they would all be drawn to one edge of the crystal, and, if there is a metal electrode there, they would be collected, leaving the crystal neutral. Similarly we could put many holes into a crystal. They would roam around at random unless there is an electric field. With a field they would flow toward the negative terminal, and would be “collected”—what actually happens is that they are neutralized by electrons from the metal terminal. One can also have both holes and electrons together. If there are not too many, they will all go their way independently. With an electric field, they will all contribute to the current. For obvious reasons, electrons are called the negative carriers and the holes are called the positive carriers. We have so far considered that electrons are put into the crystal from the outside, or are removed to make a hole. It is also possible to “create” an electron-hole pair by taking a bound electron away from one neutral atom and putting it some distance away in the same crystal. We then have a free electron and a free hole, and the two can move about as we have described. The energy required to put an electron into a state $S$—we say to “create” the state $S$—is the energy $E^-$ shown in Fig. 14–2. It is some energy above $E^-_{\text{min}}$. The energy required to “create” a hole in some state $S'$ is the energy $E^+$ of Fig. 14–3, which is some energy greater than $E^+_{\text{min}}$. Now if we create a pair in the states $S$ and $S'$, the energy required is just $E^-+E^+$. The creation of pairs is a common process (as we will see later), so many people like to put Fig. 14–2 and Fig. 14–3 together on the same graph—with the hole energy plotted downward, although it is, of course a positive energy. We have combined our two graphs in this way in Fig. 14–4. The advantage of such a graph is that the energy $E_{\text{pair}}=E^-+E^+$ required to create a pair with the electron in $S$ and the hole in $S'$ is just the vertical distance between $S$ and $S'$ as shown in Fig. 14–4. The minimum energy required to create a pair is called the “gap” energy and is equal to $E^-_{\text{min}}+E^+_{\text{min}}$. Sometimes you will see a simpler diagram called an energy level diagram which is drawn when people are not interested in the $k$ variable. Such a diagram—shown in Fig. 14–5—just shows the possible energies for the electrons and holes.1 How can electron-hole pairs be created? There are several ways. For example, photons of light (or x-rays) can be absorbed and create a pair if the photon energy is above the energy of the gap. The rate at which pairs are produced is proportional to the light intensity. If two electrodes are plated on a wafer of the crystal and a “bias” voltage is applied, the electrons and holes will be drawn to the electrodes. The circuit current will be proportional to the intensity of the light. This mechanism is responsible for the phenomenon of photoconductivity and the operation of photoconductive cells. Electron hole pairs can also be produced by high-energy particles. When a fast-moving charged particle—for instance, a proton or a pion with an energy of tens or hundreds of MeV—goes through a crystal, its electric field will knock electrons out of their bound states creating electron-hole pairs. Such events occur hundreds of thousands of times per millimeter of track. After the passage of the particle, the carriers can be collected and in doing so will give an electrical pulse. This is the mechanism at play in the semiconductor counters recently put to use for experiments in nuclear physics. Such counters do not require semiconductors, they can also be made with crystalline insulators. In fact, the first of such counters was made using a diamond crystal which is an insulator at room temperature. Very pure crystals are required if the holes and electrons are to be able to move freely to the electrodes without being trapped. The semiconductors silicon and germanium are used because they can be produced with high purity in reasonable large sizes (centimeter dimensions). So far we have been concerned with semiconductor crystals at temperatures near absolute zero. At any finite temperature there is still another mechanism by which electron-hole pairs can be created. The pair energy can be provided from the thermal energy of the crystal. The thermal vibrations of the crystal can transfer their energy to a pair—giving rise to “spontaneous” creation. The probability per unit time that the energy as large as the gap energy $E_{\text{gap}}$ will be concentrated at one atomic site is proportional to $e^{-E_{\text{gap}}/\kappa T}$, where $T$ is the temperature and $\kappa$ is Boltzmann’s constant (see Chapter 40, Vol. I). Near absolute zero there is no appreciable probability, but as the temperature rises there is an increasing probability of producing such pairs. At any finite temperature the production should continue forever at a constant rate giving more and more negative and positive carriers. Of course that does not happen because after a while the electrons and holes accidentally find each other—the electron drops into the hole and the excess energy is given to the lattice. We say that the electron and hole “annihilate.” There is a certain probability per second that a hole meets an electron and the two things annihilate each other.  If the number of electrons per unit volume is $N_n$ ($n$ for negative carriers) and the density of positive carriers is $N_p$, the chance per unit time that an electron and a hole will find each other and annihilate is proportional to the product $N_nN_p$. In equilibrium this rate must equal the rate that pairs are created. You see that in equilibrium the product of $N_n$ and $N_p$ should be given by some constant times the Boltzmann factor: \begin{equation} \label{Eq:III:14:4} N_nN_p=\text{const}\,e^{-E_{\text{gap}}/\kappa T}. \end{equation}  When we say constant, we mean nearly constant. A more complete theory—which includes more details about how holes and electrons “find” each other—shows that the “constant” is slightly dependent upon temperature, but the major dependence on temperature is in the exponential. Let’s consider, as an example, a pure material which is originally neutral. At a finite temperature you would expect the number of positive and negative carriers to be equal, $N_n=N_p$. Then each of them should vary with temperature as $e^{-E_{\text{gap}}/2\kappa T}$. The variation of many of the properties of a semiconductor—the conductivity for example—is mainly determined by the exponential factor because all the other factors vary much more slowly with temperature. The gap energy for germanium is about $0.72$ eV and for silicon $1.1$ eV. At room temperature $\kappa T$ is about $1/40$ of an electron volt. At these temperatures there are enough holes and electrons to give a significant conductivity, while at, say, $30^\circ$K—one-tenth of room temperature—the conductivity is imperceptible. The gap energy of diamond is $6$ or $7$ eV and diamond is a good insulator at room temperature.