I am an EE undergraduate with a lot of interest in semiconductors. I took an introductory semiconductors course which was helpful but left me with some holes in my understanding, as the course (rightfully) assumed that the EE students had not yet had quantum or solid state physics. As I dig deeper into the topic, I find many papers that will give a complex layer structure (not exactly complex, just a "stack" of parameters) and then show the conduction band profile of the device, which is probably calculated using some simulator.
I have looked at many, many books to find the steps involved in calculating such a thing given a layer structure, but most explain the pn junction, metal metal junction, and then a heterojunction, and stop there without showing any concrete examples. What I have learned is that there is both a classical (I'm assuming extending the junction problem and electrostatics) and quantum way ("introducing new long-wavelength boundary conditions")
I am simply interested in the steps a simulator would go through, classically or otherwise, to take a stack of semiconductor layers and spit out a conduction band profile. Answers like "it iteratively solves such and such equation" is fine/what I'm looking for, or a direct reference that will explain to me what I am looking for is also great, as I have been searching hard for a clear answer.
As a bonus, I am mostly interested in III-V materials, how all of this relates to quantum wells, wires, and dots. I also understand the complexity behind this, things like strain and lattice matching and the electrostatics involved, and would also appreciate knowing under what assumptions your answer applies to.