Relaxation of the Boltzmann approximation in Döhler's conductivity-thermopower analysis

The theory underlying Döhler's analysis easily generalizes to Fermi-Dirac statistics for independent electronic carriers. The resulting functional equation for Döhler's energy $V$ (= log conductivity prefactor) suggests a simple consistency test for the Boltzmann approximation. This is illustrated by numerical verification for the 1977 $a-\mathrm{S}\mathrm{i}$ data of Beye, Mell, and Overhof considered by Döhler. The less restrictive assumptions suggest interpretation in terms of an intuitively appealing "generalized average mobility." This leads, in turn, to broader perspectives, both theoretical and experimental, upon others among the quasithermodynamic assumptions used by Döhler. This is exemplified for photoexcitation. Expressions are given for the "product of mobility, lifetime, and quantum efficiency," one generalizing the familiar expression in terms of small-signal, steady-state photoconductivity and another unfamiliar expression, in terms of photothermopower and the generalized Döhler energy$V$. Prospects, therefore, remain favorable for applications of generalizations of Döhler's analysis to broader bodies of $a-\mathrm{S}\mathrm{i}$ data, and also to other materials in which combinations of hopping and propagating electronic transport are to be expected.