The semiclassical theory of hopping conductivity

This work develops the semiclassical theory of electrical conduction due to electrons in localized states, and compares the resultant formulae with a variety of experimental data. We begin by using the equivalent electrical network, derived from the phenomenological rate equations, to deduce the dc conductivity of a number of model systems. Percolation arguments are used to derive both the exponent and the prefactor when the dc conductivity is written in the form a = a exp [-sp]. In particular, we derive formulae for the cases when the energies of the electron states are distributed over a very narrow range, a very wide range and an intermediate range. In the first two cases the formulae are in excellent agreement with computer generated data obtained by numerical solutions of Kirchhoff's equations for the equivalent network. Experimental data obtained from studies of a number of systems are analysed, namely, impurity conduction in crystalline germanium and amorphous silicon, the conductivity of evaporated films of amorphous germanium and finally the conductivity due to electrons in an inversion layer formed in a metal-oxide-silicon-field-effecttransistor. In all cases very good agreement is found between experiment and theory. Formulae are also derived for the ac conductivity. Comparison of these formulae with computer data again shows good agreement between theory and experiment. We show how detailed considerations indicate that the ac data obtained from evaporated films of amorphous germanium cannot be due to hopping at the Fermi level, as is normally assumed. In conclusion, this work develops the simple hopping theory which adequately describes experimental data obtained from a variety of systems. Various problems are isolated; which relate to the model adopted rather than any approximation inherent in the deduction of the analytical formulae