Probability density function of the potential energy due to random impurities in semiconductors

We have calculated the probability density function of the potential energy for the case of heavily doped uncompensated semiconductors due to randomly located ionized impurities. The approach combines both analytical and numerical techniques to form a piecewise approximation of P(E) and can be used in conjunction with all types of screening potentials. We obtained a measure of the quality of our results by comparing the moments of our approximated function for P(E) with those moments obtained directly from the known generating function. The agreement between the two sets of moments is extremely good for lower and higher order moments which demonstrates the high accuracy of our piecewise approximation for P(E). Our result turns out to be non-Gaussian with a peak that is higher than and shifted in comparison to, the results of previous Gaussian approximations based on lower order moments only.