Trapping by a lattice imperfection
There is another interesting situation that can arise if $F$ is a negative number. If the energy of the electron is lower at the impurity atom (at $n=0$) than it is anywhere else, then the electron can get caught on this atom. That is, if $(E_0+F)$ is below the bottom of the band at $(E_0-2A)$, then the electron can get “trapped” in a state with $E<E_0-2A$. Such a solution cannot come out of what we have done so far. We can get this solution, however, if we permit the trial solution we took in Eq. (13.10) to have an imaginary number for $k$. Let’s set $k=\pm i\kappa$. Again, we can have different solutions for $n<0$ and for $n>0$. A possible solution for $n<0$ might be \begin{equation} \label{Eq:III:13:39} a_n\:(\text{for $n<0$})=ce^{+\kappa x_n}. \end{equation}  We have to take a plus sign in the exponent; otherwise the amplitude would get indefinitely large for large negative values of $n$. Similarly, a possible solution for $n>0$ would be \begin{equation} \label{Eq:III:13:40} a_n\:(\text{for $n>0$})=c'e^{-\kappa x_n}. \end{equation}  If we put these trial solutions into Eq. (13.28) all but the middle three are satisfied provided that \begin{equation} \label{Eq:III:13:41} E=E_0-A(e^{\kappa b}+e^{-\kappa b}). \end{equation}  Since the sum of the two exponential terms is always greater than $2$, this energy is below the regular band, and is what we are looking for. The remaining three equations in Eq. (13.28) are satisfied if $a_0=c=c'$ and if $\kappa$ is chosen so that \begin{equation} \label{Eq:III:13:42} A(e^{\kappa b}-e^{-\kappa b})=-F. \end{equation}  Combining this equation with Eq. (13.41) we can find the energy of the trapped electron; we get \begin{equation} \label{Eq:III:13:43} E=E_0-\sqrt{4A^2+F^2}. \end{equation}  The trapped electron has a unique energy—located somewhat below the conduction band. Notice that the amplitudes we have in Eq. (13.39) and (13.40) do not say that the trapped electron sits right on the impurity atom. The probability of finding the electron at nearby atoms is given by the square of these amplitudes. For one particular choice of the parameters it might vary as shown in the bar graph of Fig. 13–7. The probability is greatest for finding the electron on the impurity atom. For nearby atoms the probability drops off exponentially with the distance from the impurity atom. This is another example of “barrier penetration.” From the point-of-view of classical physics the electron doesn’t have enough energy to get away from the energy “hole” at the trapping center. But quantum mechanically it can leak out a little way.