Phase Shifts and Local Charge Neutrality in Semiconductors

We imagine solving the Schrödinger equation in a piece of semiconductor with $N$ atoms containing a dislocation, vacancy, or other defect. There will be some probability density ${\left|\phi \right|}^2$ inside the vacancy and the normalization of each state will be affected by it. The mean charge density of each state far from the vacancy will be $\frac{1}{N}$ electrons per atom minus a term of order $\frac{1}{N^2}$, as occurs in metals and leads to the Friedel phase-shift theorem. Thus when we sum over the whole band, we might expect the vacancy and immediate surroundings to carry a nonintegral net charge $\gamma$, which is compensated by a charge-density deficit $\frac{\gamma }{N}$ over the whole crystal. There can be no screening (at 0°K) and the effect would lead to strong electric fields of the type found in fact around dislocations in semiconductors. However, we find in a one-dimensional calculation that $\gamma$ is identically zero and the effect does not exist. This comes about because the Bloch states at the band edges are standing waves, and the phase shift tends to a multiple of $\frac{\pi }{2}$. The result may plausibly be extended to three dimensions, so that local charge neutrality appears to be maintained, apart from the usual donor and acceptor states, of course.