At T high compared to freeze out, you can ignore the dopant orbitals -- in essence if n_intrinsic >> D, you can ignore the dopant orbitals since the conduction band orbitals would be more important. In this case, the original law of mass action works. However, when the temperature drops enough that n_intrinsic ~ D, then you have to start worrying about the dopant orbitals. At lower temperature still, n_intrinisc << D, then you get the regime you mentioned -- where mu lies between the dopant band and the conduction band.
If you imagine setting up the "full" problem with all the orbitals -- even without solving the full messy problem -- you should be able to see how these limits will work out.
For very low T, the
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