Kronig-Penney model

Question

I am studying the Kronig-Penney model as treated in the book by Kittel: Introduction to Solid State Physics.

In this model one considers a period potential which is zero in the region $[0,a]$ (define as region I), $U_0$ in the region $[a,a+b]$ (define as region II) and again zero in the region $[a+b, 2a+b]$ etc. etc., so the period of repetition is $a+b$.

One can solve the Schrödinger equation seperately in region I and II, yielding complex exponentials in region I. In region II one can have both complex and real exponentials, depending on the sign of $U_0 - \epsilon$. Kittel takes real exponentials without saying a word, but I guess that one should demand explicitly that $\epsilon<U_0$? Why can we do that?

Furthermore, the solutions in region I and II are tied together by demanding $\psi_I(0)=\psi_{II}(0)$ and $\psi_I'(0) = \psi_{II}'(0)$. But then the crucial step comes in: Kittel writes that Bloch's theorem states (in this specific case) that $$ \psi_{II}(a<x<a+b) = \psi_{I}(-b<x<0)e^{ik(a+b)} $$ and uses this to write $\psi_{II}(a) = \psi_I(b)e^{ik(a+b)}$ and likewise for the derivatives. These two conditions, together with the previous two yield two equations which give a consistency equation on $k$, which will have no solutions in certain regions, yielding band gaps etc.

I have some trouble understanding this. I understand that the crystal periodicity is $a+b$, but why does that mean that this combination occurs in the complex exponential? Furthermore, why can't we say $\psi_{II}(a) = \psi_{II}(-b)e^{ik(a+b)}$ (besides the fact that it will not help us solve the problem)? I do not really understand why one should tie the two different wavefunctions together in this specific way.

Answer 1

(1) The real vs imaginary exponential question: As you said, whether the exponent of the exponential is real or imaginary depends on the sign of $U_0 - \epsilon$. Well, $U_0$ is fixed by the system you are considering, but $\epsilon$ is a parameter you get to input. In other words, you are asking for how the system behaves as a function of $\epsilon$. So you can ask questions where $\epsilon$ is greater than, less than, or equal to $U_0$. It turns out to be more useful for the study of conductivity to study bound states, which means $\epsilon < U_0$.This is because you are interested in the behavior of electrons that are bound to the solid, you are not firing electrons from infinity at the solid.

(2) The matching condition question: The underlying requirement is that the wavefunction be continuous and differentiable everywhere. It is guaranteed to be continuous and differentiable within the two regions $0<x<a$ and $a<x<a+b$, so the only places where things can go wrong are (I) at $a$ and (II) at $0$ (which is the same as $a+b$). You may impose the conditions at these locations using any coordinates you like. In particular, you can impose continuity of the wavefunction at the location (I) by demanding $\psi(a)=\psi(-b)$. If you do this correctly you will get the same answer as Kittel, even though he phrased that same condition using slightly different coordinates $\psi_<(a) = \psi_>(a)$. It is a useful exercise to see that this works out. The main reason Kittel doesn't do things that way is because it is a little less efficient (requires more work to get to the same answer).