Fermi-statistics revised for Degenerately Doping with impurities forming Band-tail

Abstract

Fermi-Dirac (FD) distribution function is generally known as Fermi-Statistics (FS) applicable for non-degenerately doped semiconductors, because in this case, the band structure prevails. As the dopings are increased strongly, the semiconductor is degenerate and its band-structure is perturbed with the formation of band-tail. It is convinced that the FD distribution function must be revisited for modification under this present condition. In the present attempt, modified Fermi-statistics has been derived for the degenerate semiconductor having band-tail due to Gaussian distribution of the impurity potential. Unlike FS for non-degenerate semi conductor, which is an exponential function, our present result is general, involving exponential and error functions with series summation solution. In the classical limits, the normal FS and the Maxwell-Boltzmann’s (MB) distribution functions can be retrived for non-degenerate semiconductor from our results. Also, MB distribution function for degenerately dopings are obtained from our approximate results of the general conclusions.