Other states in a lattice
According to Eq. (13.24) the electron states we have been talking about can have energies only in a certain “band” of energies which covers the energy range from the minimum energy \begin{equation*} E_0-2(A_x+A_y+A_z)\hphantom{.} \end{equation*}  to the maximum energy \begin{equation*} E_0+2(A_x+A_y+A_z). \end{equation*} Other energies are possible, but they belong to a different class of electron states. For the states we have described, we imagined base states in which an electron is placed on an atom of the crystal in some particular state, say the lowest energy state. If you have an atom in empty space, and add an electron to make an ion, the ion can be formed in many ways. The electron can go on in such a way as to make the state of lowest energy, or it can go on to make one or another of many possible “excited states” of the ion each with a definite energy above the lowest energy. The same thing can happen in a crystal. Let’s suppose that the energy $E_0$ we picked above corresponds to base states which are ions of the lowest possible energy. We could also imagine a new set of base states in which the electron sits near the $n$th atom in a different way—in one of the excited states of the ion—so that the energy $E_0$ is now quite a bit higher. As before there is some amplitude $A$ (different from before) that the electron will jump from its excited state at one atom to the same excited state at a neighboring atom. The whole analysis goes as before; we find a band of possible energies centered at a higher energy. There can, in general, be many such bands each corresponding to a different level of excitation.  There are also other possibilities. There may be some amplitude that the electron jumps from an excited condition at one atom to an unexcited condition at the next atom. (This is called an interaction between bands.) The mathematical theory gets more and more complicated as you take into account more and more bands and add more and more coefficients for leakage between the possible states. No new ideas are involved, however; the equations are set up much as we have done in our simple example. We should remark also that there is not much more to be said about the various coefficients, such as the amplitude $A$, which appear in the theory. Generally they are very hard to calculate, so in practical cases very little is known theoretically about these parameters and for any particular real situation we can only take values determined experimentally. There are other situations where the physics and mathematics are almost exactly like what we have found for an electron moving in a crystal, but in which the “object” that moves is quite different. For instance, suppose that our original crystal—or rather linear lattice—was a line of neutral atoms, each with a loosely bound outer electron. Then imagine that we were to remove one electron. Which atom has lost its electron? Let $C_n$ now represent the amplitude that the electron is missing from the atom at $x_n$. There will, in general, be some amplitude $iA/\hbar$ that the electron at a neighboring atom—say the $(n-1)$st atom—will jump to the $n$th leaving the $(n-1)$st atom without its electron. This is the same as saying that there is an amplitude $iA/\hbar$ for the “missing electron” to jump from the $n$th atom to the $(n-1)$st atom. You can see that the equations will be exactly the same—of course, the value of $A$ need not be the same as we had before. Again we will get the same formulas for the energy levels, for the “waves” of probability which move through the crystal with the group velocity of Eq. (13.18), for the effective mass, and so on. Only now the waves describe the behavior of the missing electron—or “hole” as it is called. So a “hole” acts just like a particle with a certain mass $m_{\text{eff}}$. You can see that this particle will appear to have a positive charge. We’ll have some more to say about such holes in the next chapter. As another example, we can think of a line of identical neutral atoms one of which has been put into an excited state—that is, with more than its normal ground state energy. Let $C_n$ be the amplitude that the $n$th atom has the excitation. It can interact with a neighboring atom by handing over to it the extra energy and returning to the ground state. Call the amplitude for this process $iA/\hbar$. You can see that it’s the same mathematics all over again. Now the object which moves is called an exciton. It behaves like a neutral “particle” moving through the crystal, carrying the excitation energy. Such motion may be involved in certain biological processes such as vision, or photosynthesis. It has been guessed that the absorption of light in the retina produces an “exciton” which moves through some periodic structure (such as the layers in the rods we described in Chapter 36, Vol. I; see Fig. 36–5) to be accumulated at some special station where the energy is used to induce a chemical reaction. 