Percolation conductivity in hafnium sub-oxides

In this study, we demonstrated experimentally that formation of chains and islands of oxygen vacancies in hafnium sub-oxides (HfO$_x$, $x<2$) leads to percolation charge transport in such dielectrics. Basing on the model of \'{E}fros-Shklovskii percolation theory good quantitative agreement between the experimental and theoretical data of current-voltage characteristics were achieved. Based on the percolation theory suggested model shows that hafnium sub-oxides consist of mixtures of metallic Hf nanoscale clusters of 1-2 nm distributed onto non-stoichiometric HfO$_x$. It was shown that reported approach might describe low resistance state current-voltage characteristics of resistive memory elements based on HfO$_x$.