Solution of a new nonlinear equation for the distribution of charge carriers in a semiconductor

The solution of a recently obtained nonlinear differential equation for the distribution function of charge carriers in a semiconductor in an electric field is derived. It is given by ${\mathit{f}}_{\mathrm{SL}}$(x)={1+B[s/(x+s${\left)\right]}^{\mathrm{s}}$ ${\mathrm{e}}^{\mathrm{x}}$ ${\mathit{\}}}^{\mathrm{-}1}$. This solution represents the symmetric part of the total distribution function. The nondimensional energy and applied electric field are x and √s , respectively, and B is a constant determined by normalization. The total distribution is given by the above and its derivative and is found to be rotationally symmetric about the electric field. This distribution reduces to the shifted Fermi-Dirac distribution for small s and to the Druyvesteyn distribution in the classical limit. An analytic expression for electrical conductivity is derived together with an approximate expression for the chemical potential in the small-electric-field limit. A generalized criterion for the classical versus quantum domains is discussed relevant to the present study. It is found that otherwise quantum domains become classical for sufficiently large applied electric fields.