Hopping conductivity with distributed energy-barrier heights

An alternative expression for the temperature dependence of hopping conductivity is proposed. A conduction model is proposed, based on a collection of many independent Arrhenius-type processes. The density in the material considered has a Λ-shaped distribution as a function of activation energy, for example Gaussian, isosceles, and scalene distributions. The validity of the model has been checked with the electrical-conductivity data of disordered carbon fibers which show a metal-insulator transition. The result is that the conductivity data between 4 and 250 K fit well to the form ${\mathit{T}}^2$[1-exp(-E/kT)${]}^2$, where T is temperature and E activation energy, and is related to the degree of disorder in the system. This form is the simplest form derived from the isosceles distribution; however, a better fit is obtained from the scalene distribution with more complex form. By using the proposed conduction model, the activation energy E is found to decrease systematically as the insulator-metal transition is approached by heat treatment. Also this model can yield the widely observed fractional temperature dependences of hopping conduction: x=1, 1/2, 1/3, and 1/4 in the conventional form exp-(${\mathit{T}}_0$/T${)}^{\mathit{x}}$.