In a simplified model of an undoped semiconductor, the actual distribution of energy states may be replaced by one in which there are states in the valence band, all 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.
We are given the condition that the number of electrons in the conduction band and the number of holes in the valence band are equal. Thus, using the probability equation in the density of occupied states equation, we can get the individual equations for both cases. This will imply the given condition. Similarly, for the second case, we can consider the condition that the Fermi level in the energy state is relatively large to the value kT.
The density of occupied states, (i)
The probability of the condition that a particle will have energy E according to Fermi-Dirac statistics, (ii)
The number of electrons occupying the valance band is given by substituting equation (ii) in equation (i) as follows:
Since there are a total of states in the valence band; the number of holes in the valence band is given by:
Now, the number of electrons in the conduction band is given by substituting equation (ii) in equation (i) as follows:
As we are given that
Thus, using equations (a) and (b), we get that
where and .
In this case, and
Thus, using this condition in the above equation (c), we get that
Hence, the above condition is proved.
A silicon sample is doped with atoms having donor states 0.110eV below the bottom of the conduction band. (The energy gap in silicon is 1.11eV ) If each of these donor states is occupied with a probability of at , (a) is the Fermi level above or below the top of the silicon valence band and (b) how far above or below? (c) What then is the probability that a state at the bottom of the silicon conduction band is occupied?
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