How Can We Calculate Band Bending Using Schrödinger-Poisson Theory?

In summary, the conversation discusses the process of calculating band bending by solving the Schrödinger equation and Poisson equation self-consistently. This involves looking at the conduction band of a semiconductor and assuming that the electrons are free with an effective mass. The conversation also addresses the assumption that electrons in the conduction band only interact with each other and not with electrons in filled bands. The Hartree-Fock method is mentioned as a possible approach and its connection to Liouville's theorem is discussed. There is also mention of using perturbation theory, such as k.p with a self-consistency loop, to calculate band structure.
  • #1
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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 into 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.
 
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  • #2
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
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
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?)
 

1. What is the Schrödinger Poisson theory?

The Schrödinger Poisson theory is a mathematical framework used to describe the behavior of quantum particles in a classical electromagnetic field. It combines the Schrödinger equation, which describes the wave function of a quantum particle, with the Poisson equation, which describes the electric potential in a classical electromagnetic field.

2. How does the Schrödinger Poisson theory differ from the Schrödinger equation?

The Schrödinger Poisson theory takes into account the effects of a classical electromagnetic field on quantum particles, while the Schrödinger equation only describes the behavior of quantum particles in the absence of an external field.

3. What are the applications of the Schrödinger Poisson theory?

The Schrödinger Poisson theory is used in various fields of physics, such as quantum mechanics, solid state physics, and plasma physics. It is also used in engineering applications, such as the design of electronic devices and semiconductor materials.

4. How does the Schrödinger Poisson theory relate to other quantum theories?

The Schrödinger Poisson theory is a non-relativistic theory, meaning it does not take into account the effects of special relativity. It is often used in conjunction with other quantum theories, such as the Dirac equation, which does incorporate relativistic effects.

5. Who developed the Schrödinger Poisson theory?

The Schrödinger Poisson theory was developed by Erwin Schrödinger, a renowned physicist who also developed the Schrödinger equation, and Siméon Denis Poisson, a French mathematician and physicist. Both of their contributions were combined to create the Schrödinger Poisson theory.

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