Effect of doping in semiconductors

To appreciate the importance of the results of the preceding discussion, we calculate first the number of conduction electrons or holes that exist in a semiconductor at finite temperature T, in the absence of any dopant impurities. The number of conduction electrons, denoted by ndT), will be given by the sum over occupied states with energy above the Eermi level ep. In a semiconductor, this will be equal 

Similarly, the number of holes, denoted by pdT), will be equal to the integral over all valence states of the density of valence states gv( ) multiplied by one minus the Fermi occupation number  

In the above equations we have used the traditional symbols tic for electrons and pv for holes, which come from the fact that the former represent negatively charged carriers (with the subscript c to indicate they are related to conduction bands) and the latter represent positively charged carriers (with the subscript v to indicate they are related to valence bands). We have mentioned before (see chapter 5) that in a perfect semiconductor the Fermi level lies in the middle of the band gap, a statement that can be easily proved from the arguments that follow. For the moment, we will use this fact to justify an approximation for the number of electrons or holes. Since the band gap of semiconductors is of order 1 eV and the temperatures of operation are of order 300 K, i.e. 1/40 eV, we can use the approximations 

These approximations are very good for the values e = ecbm and e = e bm in the expressions for the number of electrons and holes, Eqs. (9.36) and (9.37), respectively, and become even better for values larger than ecbm or lower than vbm, so that we can use them for the entire range of values in each integral. This gives 

These considerations make it possible to evaluate explicitly the integrals in the above expressions for ndT) and pdT), giving 

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