Hopping transport on a fractal: ac conductivity of porous silicon

Abstract

We have measured the frequency dependence of the conductivity and the dielectric constant of various samples of porous Si in the regime 1 Hz to 100 kHz at different temperatures. The conductivity data exhibit a strong frequency dependence. When normalized to the dc conductivity, our data obey a universal scaling law, with a well-defined crossover in which the real part of the conductivity σ’ changes from an ${\mathrm{\omega }}^{1/2}$ dependence to being proportional to ω. We explain this in terms of activated hopping in a fractal network. The low-frequency regime is governed by the fractal properties of porous Si, whereas the high-frequency dispersion comes from a broad distribution of activation energies. Calculations using the effective-medium approximation for activated hopping on a percolating lattice give fair agreement with the data.