Impure semiconductors
So far we have talked about two ways that extra electrons can be put into an otherwise ideally perfect crystal lattice. One way was to inject the electron from an outside source; the other way, was to knock a bound electron off a neutral atom creating simultaneously an electron and a hole. It is possible to put electrons into the conduction band of a crystal in still another way. Suppose we imagine a crystal of germanium in which one of the germanium atoms is replaced by an arsenic atom. The germanium atoms have a valence of $4$ and the crystal structure is controlled by the four valence electrons. Arsenic, on the other hand, has a valence of $5$. It turns out that a single arsenic atom can sit in the germanium lattice (because it has approximately the correct size), but in doing so it must act as a valence $4$ atom—using four of its valence electrons to form the crystal bonds and having one electron left over. This extra electron is very loosely attached—the binding energy is only about $1/100$ of an electron volt. At room temperature the electron easily picks up that much energy from the thermal energy of the crystal, and then takes off on its own—moving about in the lattice as a free electron. An impurity atom such as the arsenic is called a donor site because it can give up a negative carrier to the crystal. If a crystal of germanium is grown from a melt to which a very small amount of arsenic has been added, the arsenic donor sites will be distributed throughout the crystal and the crystal will have a certain density of negative carriers built in. You might think that these carriers would get swept away as soon as any small electric field was put across the crystal. This will not happen, however, because the arsenic atoms in the body of the crystal each have a positive charge. If the body of the crystal is to remain neutral, the average density of negative carrier electrons must be equal to the density of donor sites. If you put two electrodes on the edges of such a crystal and connect them to a battery, a current will flow; but as the carrier electrons are swept out at one end, new conduction electrons must be introduced from the electrode on the other end so that the average density of conduction electrons is left very nearly equal to the density of donor sites.  Since the donor sites are positively charged, there will be some tendency for them to capture some of the conduction electrons as they diffuse around inside the crystal. A donor site can, therefore, act as a trap such as those we discussed in the last section. But if the trapping energy is sufficiently small—as it is for arsenic—the number of carriers which are trapped at any one time is a small fraction of the total. For a complete understanding of the behavior of semiconductors one must take into account this trapping. For the rest of our discussion, however, we will assume that the trapping energy is sufficiently low and the temperature is sufficiently high, that all of the donor sites have given up their electrons. This is, of course, just an approximation. It is also possible to build into a germanium crystal some impurity atom whose valence is $3$, such as aluminum. The aluminum atom tries to act as a valence $4$ object by stealing an extra electron. It can steal an electron from some nearby germanium atom and end up as a negatively charged atom with an effective valence of $4$. Of course, when it steals the electron from a germanium atom, it leaves a hole there; and this hole can wander around in the crystal as a positive carrier. An impurity atom which can produce a hole in this way is called an acceptor because it “accepts” an electron. If a germanium or a silicon crystal is grown from a melt to which a small amount of aluminum impurity has been added, the crystal will have built-in a certain density of holes which can act as positive carriers. When a donor or an acceptor impurity is added to a semiconductor, we say that the material has been “doped.” When a germanium crystal with some built-in donor impurities is at room temperature, some conduction electrons are contributed by the thermally induced electron-hole pair creation as well as by the donor sites. The electrons from both sources are, naturally, equivalent, and it is the total number $N_n$ which comes into play in the statistical processes that lead to equilibrium. If the temperature is not too low, the number of negative carriers contributed by the donor impurity atoms is roughly equal to the number of impurity atoms present. In equilibrium Eq. (14.4) must still be valid; at a given temperature the product $N_nN_p$ is determined. This means that if we add some donor impurity which increases $N_n$, the number $N_p$ of positive carriers will have to decrease by such an amount that $N_nN_p$ is unchanged. If the impurity concentration is high enough, the number $N_n$ of negative carriers is determined by the number of donor sites and is nearly independent of temperature—all of the variation in the exponential factor is supplied by $N_p$, even though it is much less than $N_n$. An otherwise pure crystal with a small concentration of donor impurity will have a majority of negative carriers; such a material is called an “$n$-type” semiconductor.  If an acceptor-type impurity is added to the crystal lattice, some of the new holes will drift around and annihilate some of the free electrons produced by thermal fluctuation. This process will go on until Eq. (14.4) is satisfied. Under equilibrium conditions the number of positive carriers will be increased and the number of negative carriers will be decreased, leaving the product a constant. A material with an excess of positive carriers is called a “$p$-type” semiconductor.  If we put two electrodes on a piece of semiconductor crystal and connect them to a source of potential difference, there will be an electric field inside the crystal. The electric field will cause the positive and the negative carriers to move, and an electric current will flow. Let’s consider first what will happen in an $n$-type material in which there is a large majority of negative carriers. For such material we can disregard the holes; they will contribute very little to the current because there are so few of them. In an ideal crystal the carriers would move across without any impediment. In a real crystal at a finite temperature, however,—especially in a crystal with some impurities—the electrons do not move completely freely. They are continually making collisions which knock them out of their original trajectories, that is, changing their momentum. These collisions are just exactly the scatterings we talked about in the last chapter and occur at any irregularity in the crystal lattice. In an $n$-type material the main causes of scattering are the very donor sites that are producing the carriers. Since the conduction electrons have a very slightly different energy at the donor sites, the probability waves are scattered from that point. Even in a perfectly pure crystal, however, there are (at any finite temperature) irregularities in the lattice due to thermal vibrations. From the classical point of view we can say that the atoms aren’t lined up exactly on a regular lattice, but are, at any instant, slightly out of place due to their thermal vibrations. The energy $E_0$ associated with each lattice point in the theory we described in Chapter 13 varies a little bit from place to place so that the waves of probability amplitude are not transmitted perfectly but are scattered in an irregular fashion. At very high temperatures or for very pure materials this scattering may become important, but in most doped materials used in practical devices the impurity atoms contribute most of the scattering. We would like now to make an estimate of the electrical conductivity of such a material. When an electric field is applied to an $n$-type semiconductor, each negative carrier will be accelerated in this field, picking up velocity until it is scattered from one of the donor sites. This means that the carriers which are ordinarily moving about in a random fashion with their thermal energies will pick up an average drift velocity along the lines of the electric field and give rise to a current through the crystal. The drift velocity is in general rather small compared with the typical thermal velocities so that we can estimate the current by assuming that the average time that the carrier travels between scatterings is a constant. Let’s say that the negative carrier has an effective electric charge $q_n$. In an electric field $\Efieldvec$, the force on the carrier will be $q_n\Efieldvec$. In Section 43–3 of Volume I we calculated the average drift velocity under such circumstances and found that it is given by $F\tau/m$, where $F$ is the force on the charge, $\tau$ is the mean free time between collisions, and $m$ is the mass. We should use the effective mass we calculated in the last chapter but since we want to make a rough calculation we will suppose that this effective mass is the same in all directions. Here we will call it $m_n$. With this approximation the average drift velocity will be \begin{equation} \label{Eq:III:14:5} \FLPv_{\text{drift}}=\frac{q_n\Efieldvec\tau_n}{m_n}. \end{equation}  Knowing the drift velocity we can find the current. Electric current density $\FLPj$ is just the number of carriers per unit volume, $N_n$, multiplied by the average drift velocity, and by the charge on each carrier. The current density is therefore \begin{equation} \label{Eq:III:14:6} \FLPj=N_n\FLPv_{\text{drift}}q_n= \frac{N_nq_n^2\tau_n}{m_n}\,\Efieldvec. \end{equation}  We see that the current density is proportional to the electric field; such a semiconductor material obeys Ohm’s law. The coefficient of proportionality between $\FLPj$ and $\Efieldvec$, the conductivity $\sigma$, is \begin{equation} \label{Eq:III:14:7} \sigma=\frac{N_nq_n^2\tau_n}{m_n}. \end{equation}  For an $n$-type material the conductivity is relatively independent of temperature. First, the number of majority carriers $N_n$ is determined primarily by the density of donors in the crystal (so long as the temperature is not so low that too many of the carriers are trapped). Second, the mean time between collisions $\tau_n$ is mainly controlled by the density of impurity atoms, which is, of course, independent of the temperature. We can apply all the same arguments to a $p$-type material, changing only the values of the parameters which appear in Eq. (14.7). If there are comparable numbers of both negative and positive carriers present at the same time, we must add the contributions from each kind of carrier. The total conductivity will be given by \begin{equation} \label{Eq:III:14:8} \sigma=\frac{N_nq_n^2\tau_n}{m_n}+\frac{N_pq_p^2\tau_p}{m_p}. \end{equation}  For very pure materials, $N_p$ and $N_n$ will be nearly equal. They will be smaller than in a doped material, so the conductivity will be less. Also they will vary rapidly with temperature (like $e^{-E_{\text{gap}}/2\kappa T}$, as we have seen), so the conductivity may change extremely fast with temperature.