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In a n-type semiconductor electrons from the donor level move up into the conduction band but can they move back down? Also in a p-type semiconductor electrons move to the acceptor level from the valence band but can these return back to the valence band?

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Yes, the electrons from the donor states are thermally excited into the conduction band and the carrier density is an equilbrium between thermal excitation and relaxation back into the donor states. The carrier density is given by:

$$ n = N_C e^{-\frac{E_C - E_F}{kt}} $$

where $N_C$ is the density of states in the conduction band, $E_C$ is the conduction band energy level and $E_F$ is the Fermi energy level (which is basically the energy of the donor states). At low enough temperatures the carrier density falls to effectively zero because all the electrons have relaxed back into the donor states. The temperature range over which this occurs is called the ionization regime, and Googling this phrase will find you many related articles.

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