Low-Frequency Conductivity Due to Hopping Processes in Silicon

The complex conductivity has been measured in $n$-type silicon with various kinds of impurities at frequencies between ${10}^2$ and ${10}^5$ cps and temperatures between 1 and 20°K. In most cases it is orders of magnitude larger than the measured dc conductivity and is attributed to polarization caused by hopping processes. The observed frequency dependence in the measured range can be expressed as $A{\omega }^{0.8}$, where $A$ is a complex constant. At the low-temperature end the conductivity is roughly proportional to minority impurity concentration and is almost independent of the majority impurity concentration and At higher temperatures the conductivity becomes approximately proportional to the product of both concentrations. A simple theory, based on the currently accepted model of impurity conduction, is given for the higher temperature range. It accounts well for the observed frequency and concentration dependences. However, only order-of-magnitude absolute agreement is obtained.