In a simplified model of an intrinsic semiconductor (no doping), the actual distribution in energy of states is replaced by one in which there are states in the valence band, all these states having the same energy , and states in the conduction band, all these states having the same energy . The number of electrons in the conduction band equals the number of holes in the valence band. (a) We have to show that this last condition implies that

(b) If the Fermi level is in the gap between the two bands and is far from both ends compared to , then the exponentials dominate in the denominators. Under these conditions, we have to show that

,

and therefore that, if , the Fermi level is close to the centre of the gap.

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(a)

In the given model of an intrinsic semiconductor it is assumed that the actual distribution in energy of states is replaced by one in which there are states in the valence band, all these states having the same energy , and states in the conduction band, all these states having the same energy .

The probability of occupation of a state with energy is therefore

Therefore, the number of electrons in the conduction band will be given by

The probability of finding a hole at energy is , which is

Therefore, the number of holes in the valence band will be given by

, which is

We require that the number of electrons in the conduction band equals the number of holes in the valence band. This condition implies that

(b)

We assume that the Fermi level is in the gap between the two bands and is far from both ends compared to , and thus the exponentials dominate in the denominators. Under this assumption, we have

Taking ln of both sides of the above equation, we get

Therefore, if ,

and

that is the Fermi level is close to the centre of the gap between the conduction band and the valence band.