The Kronig-Penney (KP) model is a classic model that is used to show that a periodic lattice of finite well potential sites will give rise to a band gap. The typical process in solving the KP seems to be:
- Solve the Schrodinger Equation for a single period of the lattice to get the wave functions in the "well" part of the period, and the "free" part of the period.
- Put the wave functions in the form of Bloch functions $\psi(x) = e^{ikx}u(x)$.
- Apply periodic boundary conditions to the Bloch functions and perodicity conditions to the $u(x)$ periodic parts of them.
- Do some algebraic/matrix solving to get the constraints on the wave vector $k$.
- Take the limit as the well becomes infinitely deep and narrow, so it is a dirac delta function, which simplifies the constraint equation even further to: $cos( Ka) = cos(ka) - \frac{mg} {\hbar^2 k} sin( ka ) $
In 1D, because the LHS of the equation must be between -1 and 1, that limits the values $k$ can take and satisfy this equation, and thus constrains $E = \hbar^2 k^2/(2m)$ as well.
Expanding this to 3D, my naive approach would be to say that the potential is now basically $\sum_{l,m,n}\delta(x-la)\delta(y-mb)\delta(z-nc)$ (where $l,m,n$ are integers and $a,b,c$ are the periodicities), which can be separated into $(\sum_l\delta(x-la))(\sum_m\delta(y-mb))(\sum_n\delta(z-nc))$. Then it seems like you can do the common practice of separating the Schrodinger Equation, which would then just leave you with three copies of the previous process.
But I'm pretty sure this can't be right, because it would lead to no band gaps. Each equation would have its own constraints on its respective $k_i$ (and thus $E_i$) still, but I think any value of the overall energy $E = E_x + E_y + E_z$ could be achieved by carefully selecting different valid values of $k_i$.
But I'm pretty sure I'm wrong, but I don't see where. Could anyone tell me what's wrong with my approach?
