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Tom Anderson

What is the relationship between the potential obtained through the Poisson equation and the vacuum energy level in semiconductor band diagrams?
The Poisson equation describes the potential energy due to a collection of charges, i.e. v'' = - n, where we are in 1S, using units where the permittivity is 1, the potential is v and n is the charge density. The associated electric field is then E= - v'.
 
Assume we have a PN homojunction. At thermal equilibrium, the Fermi-level is flat while the conduction, valence and vacuum (V) energy levels bend.
 
How do I incorporate the potential v into this diagram or, if that is a band thing to do, how are they related? I always assumed v = V, but it seems that the vacuum energy level is the negative of the potential v. This would mean V'' = n. Is this correct, and if so, why?

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