Hopping conductivity in lightly doped semiconductors. II. Three dimensions

In the preceding paper, we presented a theory for hopping conductivity in a two-dimensional lightly doped semiconductor. In this paper, we extend our theory to three dimensions. As before, we choose a flat density of states with width Δɛ and model the semiconductor as a Miller-Abrahams-type resistor network; we again use the full form of the resistance and do not take the low-temperature asymptotic form because we are interested in temperatures where the reduced temperature t≡kT/Δɛ is of order unity. One of the major results of this paper is that we establish that the small experimentally observed $T_0$ values in the relation & (s∼(1/4) for the conductivity σ may not be accounted for using a flat density of states in conjunction with the full resistor-network model. We also demonstrate that, in the high-temperature regime, for a flat density of states, our theory predicts a conductivity of the form σ=${\sigma }_0$ $e^{\mathrm{-}{\varepsilon }_3/\mathrm{kT}}$ where the activation energy ${\varepsilon }_3$=0.20 Δɛ for three dimensions. For two dimensions, we found ${\varepsilon }_3$=0.28 Δɛ. Our values for ${\varepsilon }_3$ differ considerably from those reported by Skal, Shklovskii, and Efros and by Hayden and Butcher.