# Schrödinger Poisson theory

1. Feb 7, 2016

### aaaa202

I am reading a lot about how to calculate band bending from solving the Schrödinger equation and Poisson equation self-consistently. To recap some of the central ideas are:
We look at the conduction band of some semiconductor. If we assume that the electrons are free electrons with some effective mass m*, we can solve the Schrödinger equation and calculate the electrondensity. This can then be used to determine the electrostatic potential, which can be plugged back in to the Schrödinger equation and this process can then be carried on until a self-consistent solution is found.
Now some things bother me about this approach. It is assumed that electrons in the conduction band only interact with other electrons in the conduction band and not electrons in any of the filled bands. Why are the electrons in these bands inert? Does this follow from solving the full many-body problem? I don't exactly remmeber how band structure comes about, but I think one uses the periodicity of the coulomb potential from the static ions to show that this creates bands.

2. Feb 8, 2016

### Daz

It sounds like you’re describing the Hartree-Fock method.

Ashcroft & Mermin present an argument based on Liouville’s theorem to show why filled bands are inert in their book Solid State Physics. (Sorry, I don’t have a copy on-hand to hand to give a full reference.)

3. Feb 8, 2016

### aaaa202

Why exactly is it the Hartree Fock method? My supervisor also made that point but I don't see how. Maybe I should work from the Hamiltonian for an electron gas in second quantization?

4. Feb 8, 2016

### Daz

Well, the way you described iterating to a self-consistent many-body solution is pretty much the textbook definition of Hartree Fock. That’s what made me think of it.

I must admit I was at a loss to see how you were going to use that to calculate band-structure given that you are starting with an effective mass, which is itself a consequence of the band structure.

Did you mean instead that you plan to do some sort of perturbation theory? (Like k.p but with a self-consistency loop?)