Theory of the current-field relation in silicon-rich silicon dioxide

The dependence of the current on the electric field in silicon-rich silicon dioxide (Si-rich ${\mathrm{SiO}}_2$) is studied with the use of a theoretical model based on quantum-mechanical tunneling between a random array of small semiconducting Si islands in a large-band-gap ${\mathrm{SiO}}_2$ insulator matrix. The current $J$ is calculated in the presence of an electric field $F$ by a simple percolation method for various regimes of external voltage. In the high-field limit, the current is found to obey a Fowler-Nordheim law, 1n $J\sim \frac{-\overline{F}}{F}$, but with $\overline{F}$ weakly dependent on the field $F$.