A silicon sample is doped with atoms having a donor state 0.11 eV below the bottom of the conduction band. If each of these states is occupied with probability at 290 K, we have to find the Fermi level relative to the top of the valence band. (b) We have to calculate the probability that a state at the bottom of the conduction band is occupied. The energy gap in silicon is 1.1 eV.

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(a)

We have been given that the donor state lies 0.11 eV below the bottom of the conduction band. That is

Let

We have

It is given that the probability that a donor state to be occupied at 290 K is .

Therefore,

We note that

We thus have the equation

Therefore, relative to the top of the valence band the Fermi level will be at

The Fermi level is at above the top of the valence band.

(b)

We calculate next the probability that a state at the bottom of the conduction band will be occupied. It is given by