Approximations for the ac Impurity Hopping Conduction

Abstract

A previous approximation of the ac impurity hopping conduction in the high-temperature, low-concentration limit is extended to low temperatures and to highly compensated material. Only the real part of the conductivity is considered, and random distribution is assumed. The lengths $r_D\equiv r_{\mathrm{maj}}={\left(\frac{4\pi N_D}{3}\right)}^{\frac{-1\mathrm{}}{3}}$, $r_T=\frac{e^2}{4\pi \kappa \mathrm{kT}}$ ($\kappa$ is the dielectric constant) and $r_{\omega }$ are defined for the sake of simple expressions. The latter is a distance characteristic of the frequency, proportional to the radius $a$ of the localized impurity state and only weakly dependent on other parameters. All the expressions for $\sigma$, written as functions of these variables, are explicitly proportional to $a$, to the imaginary conductivity $\kappa \omega$, and for low compensations, to $N_A=N_{min}$. In addition to the distribution of spacings between impurities already considered in the high-temperature limit, the distribution in energies is now taken into account. The low-temperature treatment holds in the region where $\sigma$ can be expanded in $\frac{r_D}{r_T}$ and $\frac{r_{\omega }}{r_T}$. Due to the existence of a zero-order term, $\sigma$ is almost independent of temperature at very low temperatures. At extremely low temperatures, however, where $\mathrm{kT}$ is much smaller than the resonance energy for a separation $r_{\omega }$, $\sigma$ is proportional to the temperature. The low-compensation and high-compensation results are basically identical at high temperatures. At very low temperatures, they differ mainly in the concentration dependence. At intermediate temperatures, the high-compensation case is expected to interpolate smoothly between the two temperature extremes; the low-compensation case is not. For both cases, the frequency dependence at very low temperatures is slightly more pronounced than at high temperatures, as is borne out by experiments. The following additional results are of interest for low compensation. At very low temperatures, the previously reported experimental result, that $\sigma$ is practically independent of $N_D$, is accounted for. The magnitudes of the calculated and measured conductivities are in very satisfactory agreement, particularly when compensated for the experimentally found $N_A^{}{}_{}{}^{0.85}$ dependence. It is shown that the results are valid up to much higher concentrations than the previous high-temperature treatment. At higher temperatures, the situation is less satisfactory. A tendency for pairing and an alteration of the radii $a$ from those calculated by Miller and Abraham is necessary to get reasonable agreement. The radii have to be altered so as to make $\frac{a_P}{a_{A_s}}=1.14\mathrm{}$ instead of 1.05. To test the validity of the previously described model at intermediate temperatures, similarity relations based on statistical equivalence are developed. Comparison with data again necessitates the assumption $\frac{a_P}{a_{A_s}}=1.14\mathrm{}$. The results on the heavy compensation cannot be evaluated because of lack of experimental data.