For a system you can think of something like a Si crystal. In regards to energy configuration, I was wondering if T=0K would imply that the electrons will all have zero thermal energy, lead to them occupying the same energy states, which would be against pauli's exclusion principle because they can't occupy the same energy state, and I know that they don't. But I don't know why, i.e. why do the electrons have any thermal energy at all at T=0K

when we plot the energy band diagram, the diagram represents all the electrons within the crystal. Now the electrons corresponding to the dopants are part of the same entire system, i.e. the crystal. So, it only makes sense that the electrons corresponding to the dopant atoms, each have an unique and quantized energy state. Thus if I plot the band diagram at 0K, I should get almost continuous energy levels present somewhere around that discrete level of dopant electrons. So Pauli's principle, according to me, is not being followed. I'll be grateful if you can clear this doubt.

The E you are writing in the equation is the net electric field or the applied electric field? @Urgje

I did not just write thrown around... It actually is thrown around a lot amongst my peers, and I'm the only one who never agrees... Sorry if it seemed a little unpleasant...