Carrier concentrations in intrinsic semiconductors

To determine more accurate values for the number of holes and electrons present in an intrinsic semiconductor it is appropriate to use Fermi-Dirac statistics and the density of states at the bottom of the conduction band (see Section S4.8). To a good approximation, it is found that the number of electrons in the conduction band per unit volume, 

Equation (13.2) shows that, at a given temperature np = constant 

Assuming that the effective mass of electrons and holes is independent of temperature, we obtain 

This equation applies to doped semiconductors as well as to intrinsic semiconductors, a finding of considerable practical importance. To a good approximation, the Fermi energy hes at the centre of the band gap (Section S4.8)  

Although the mobility of electrons and holes decreases with temperature, exactly as in a metal, it is found that the conductivity of an intrinsic semiconductor increases with temperature, as the exponential term in Equations (13.2) and (13.3) dominates the other terms. 

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We have shown that the carrier concentration in intrinsic semiconductors is a strong function of temperature. This would allow us to make useful devices such as thermistors or other temperature-sensitive devices, but the property that makes semiconductors so indispensable for modem electronic applications is the ability to drastically alter the electronic properties of the host material by the addition of trace quantities of electrically active impurities called dopants. For example, the addition of a pentavalent (group V) element such as Sb, P, or As to Si results in an additional electron that is loosely bound to the impurity atom. You can think of the orbital for this impiu-ity atom as being similar to that of a hydrogen atom. Recall that the first ionization energy of a hydrogen atom is given by fi. If we replace the mass of a free electron with the effective mass... 

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