Semiconductor junctions
We would like to discuss now what happens if we take two pieces of germanium or silicon with different internal characteristics—say different kinds or amounts of doping—and put them together to make a “junction.” Let’s start out with what is called a $p$-$n$ junction in which we have $p$-type germanium on one side of the boundary and $n$-type germanium on the other side of the boundary—as sketched in Fig. 14–8. Actually, it is not practical to put together two separate pieces of crystal and have them in uniform contact on an atomic scale. Instead, junctions are made out of a single crystal which has been modified in the two separate regions. One way is to add some suitable doping impurity to the “melt” after only half of the crystal has grown. Another way is to paint a little of the impurity element on the surface and then heat the crystal causing some impurity atoms to diffuse into the body of the crystal. Junctions made in these ways do not have a sharp boundary, although the boundaries can be made as thin as $10^{-4}$ centimeters or so. For our discussions we will imagine an ideal situation in which these two regions of the crystal with different properties meet at a sharp boundary. On the $n$-type side of the $p$-$n$ junction there are free electrons which can move about, as well as the fixed donor sites which balance the overall electric charge. On the $p$-type side there are free holes moving about and an equal number of negative acceptor sites keeping the charge balanced. Actually, that describes the situation before we put the two materials in contact. Once they are connected together the situation will change near the boundary. When the electrons in the $n$-type material arrive at the boundary they will not be reflected back as they would at a free surface, but are able to go right on into the $p$-type material. Some of the electrons of the $n$-type material will, therefore, tend to diffuse over into the $p$-type material where there are fewer electrons. This cannot go on forever because as we lose electrons from the $n$-side the net positive charge there increases until finally an electric voltage is built up which retards the diffusion of electrons into the $p$-side. In a similar way, the positive carriers of the $p$-type material can diffuse across the junction into the $n$-type material. When they do this they leave behind an excess of negative charge. Under equilibrium conditions the net diffusion current must be zero. This is brought about by the electric fields, which are established in such a way as to draw the positive carriers back toward the $p$-type material. The two diffusion processes we have been describing go on simultaneously and, you will notice, both act in the direction which will charge up the $n$-type material in a positive sense and the $p$-type material in a negative sense. Because of the finite conductivity of the semiconductor material, the change in potential from the $p$-side to the $n$-side will occur in a relatively narrow region near the boundary; the main body of each block of material will have a uniform potential. Let’s imagine an $x$-axis in a direction perpendicular to the boundary surface. Then the electric potential will vary with $x$, as shown in Fig. 14–9(b). We have also shown in part (c) of the figure the expected variation of the density $N_n$ of $n$-carriers and the density $N_p$ of $p$-carriers. Far away from the junction the carrier densities $N_p$ and $N_n$ should be just the equilibrium density we would expect for individual blocks of materials at the same temperature. (We have drawn the figure for a junction in which the $p$-type material is more heavily doped than the $n$-type material.) Because of the potential gradient at the junction, the positive carriers have to climb up a potential hill to get to the $n$-type side. This means that under equilibrium conditions there can be fewer positive carriers in the $n$-type material than there are in the $p$-type material. Remembering the laws of statistical mechanics, we expect that the ratio of $p$-type carriers on the two sides to be given by the following equation: \begin{equation} \label{Eq:III:14:10} \frac{N_p(\text{$n$-side})}{N_p(\text{$p$-side})}= e^{-q_pV/\kappa T}. \end{equation}  The product $q_pV$ in the numerator of the exponential is just the energy required to carry a charge of $q_p$ through a potential difference $V$. We have a precisely similar equation for the densities of the $n$-type carriers: \begin{equation} \label{Eq:III:14:11} \frac{N_n(\text{$n$-side})}{N_n(\text{$p$-side})}= e^{-q_nV/\kappa T}. \end{equation}  If we know the equilibrium densities in each of the two materials, we can use either of the two equations above to determine the potential difference across the junction. Notice that if Eqs. (14.10) and (14.11) are to give the same value for the potential difference $V$, the product $N_pN_n$ must be the same for the $p$-side as for the $n$-side. (Remember that $q_n=-q_p$.) We have seen earlier, however, that this product depends only on the temperature and the gap energy of the crystal. Provided both sides of the crystal are at the same temperature, the two equations are consistent with the same value of the potential difference.  Since there is a potential difference from one side of the junction to the other, it looks something like a battery. Perhaps if we connect a wire from the $n$-type side to the $p$-type side we will get an electrical current. That would be nice because then the current would flow forever without using up any material and we would have an infinite source of energy in violation of the second law of thermodynamics!  There is, however, no current if you connect a wire from the $p$-side to the $n$-side. And the reason is easy to see. Suppose we imagine first a wire made out of a piece of undoped material. When we connect this wire to the $n$-type side, we have a junction. There will be a potential difference across this junction. Let’s say that it is just one-half the potential difference from the $p$-type material to the $n$-type material. When we connect our undoped wire to the $p$-type side of the junction, there is also a potential difference at this junction—again, one-half the potential drop across the $p$-$n$ junction. At all the junctions the potential differences adjust themselves so that there is no net current flow in the circuit. Whatever kind of wire you use to connect together the two sides of the $p$-$n$ junction, you are producing two new junctions, and so long as all the junctions are at the same temperature, the potential jumps at the junctions all compensate each other and no current will flow in the circuit. It does turn out, however—if you work out the details—that if some of the junctions are at a different temperature than the other junctions, currents will flow. Some of the junctions will be heated and others will be cooled by this current and thermal energy will be converted into electrical energy. This effect is responsible for the operation of thermocouples which are used for measuring temperatures, and of thermoelectric generators. The same effect is also used to make small refrigerators. If we cannot measure the potential difference between the two sides of an $p$-$n$ junction, how can we really be sure that the potential gradient shown in Fig. 14–9 really exists? One way is to shine light on the junction. When the light photons are absorbed they can produce an electron-hole pair. In the strong electric field that exists at the junction (equal to the slope of the potential curve of Fig. 14–9) the hole will be driven into the $p$-type region and the electron will be driven into the $n$-type region. If the two sides of the junction are now connected to an external circuit, these extra charges will provide a current. The energy of the light will be converted into electrical energy in the junction. The solar cells which generate electrical power for the operation of some of our satellites operate on this principle. In our discussion of the operation of a semiconductor junction we have been assuming that the holes and the electrons act more-or-less independently—except that they somehow get into proper statistical equilibrium. When we were describing the current produced by light shining on the junction, we were assuming that an electron or a hole produced in the junction region would get into the main body of the crystal before being annihilated by a carrier of the opposite polarity. In the immediate vicinity of the junction, where the density of carriers of both signs is approximately equal, the effect of electron-hole annihilation (or as it is often called, “recombination”) is an important effect, and in a detailed analysis of a semiconductor junction must be properly taken into account. We have been assuming that a hole or an electron produced in a junction region has a good chance of getting into the main body of the crystal before recombining. The typical time for an electron or a hole to find an opposite partner and annihilate it is for typical semiconductor materials in the range between $10^{-3}$ and $10^{-7}$ seconds. This time is, incidentally, much longer than the mean free time $\tau$ between collisions with scattering sites in the crystal which we used in the analysis of conductivity. In a typical $p$-$n$ junction, the time for an electron or hole formed in the junction region to be swept away into the body of the crystal is generally much shorter than the recombination time. Most of the pairs will, therefore, contribute to an external current.