Vbe = 0.7v, Ib = 1 mA, and Ic = 10 mA.
In practice, Vce might be 0.2v or so, depending on the transistor.
How can Vce < Vbe, when C-B-E essentially form a series circuit?
I liked this video, but I don't think this model can explain this observed behavior.
That is, the built-in potentials of the two diode junctions are in opposite directions, so in series combination those voltages mostly cancel out.
Since the base-emitter current heats up the junction, but the collector-emitter current doesn't (or rather it heats it up to the tune of 0.2 milliwatts per milliamp rather than 0.7) it stands to reason that this collector-base transition must be sucking heat out of the junction when the electrons cross it. And I think this is actually how thermoelectric generators (TEGs) and Peltier coolers made out of semiconductors like bismuth telluride work.
One problem with this account is that I think the band-bending diagram of a p–n junction has the electrons in the P material at a higher energy level, although I guess the relationship of the Fermi levels is opposite?
In particular, factors that have to be considered are the built-in potentials of the space charge layers (aka depletion zones), and the different doping profile of the collector versus emitter. The doping is very important, and is often neglected. But without it, transistors would basically operate equally well in forward mode as in reverse, and they definitely don't do that. These two things have a big effect on the Fermi/band-bend diagram and probably help clear up your last point.
I barely feel like I actually understand these things on a good day, and I pay the bills with them....
Just like QM itself, I guess. So, “shut up an calculate” all over again?
QM isn't too bad. The basic ideas are alien but straightforward enough. It's hard to extend small examples to larger systems, though, and that plus the innate weirdness really messes with your intuition.
BJTs are complicated devices built from multiple simple-enough effects all happening at the same time. Each piece isn't too bad, but you've got to track them all and combine them into an overall function.
QFT (quantum field theory, such as QED) is basically all the bad parts of both of those: the basics aren't too bad, but it's difficult to scale, there's a lot going on, and it's hard to develop intuition. QCD then makes the basic calculations intractable, which is why it's still poorly understood to this day (in my opinion).
So, yeah, in conclusion, complexity is complex? Not so profound, I guess.
But it's way more complicated than that.