Activity coefficient and Einstein relation for different densities of states in semiconductors

A recently established general relation between the activity coefficient of a charge carrier and the Einstein diffusion-mobility ratio for that charge carrier is elucidated by considering various densities of states Nₑ. A reduced carrier concentration {ν} is the independent variable. Simple results are found if Nₑ(E) is a constant or if it is a {δ} function in energy (localized states). In other cases it is convenient to define a new transform of Nₑ, and use the Joyce-Dixon expansions to obtain useful results. A study is made of the case when Nₑ(E) is a power law and also when Nₑ(E) incorporates weak departures from band parabolicity.