Non-Ohmic hopping transport ina−YSi: From isotropic to directed percolation

Abstract

Electrical transport has been investigated in amorphous ${\mathrm{Y}}_{0.19}{\mathrm{Si}}_{0.81},$ from 30 mK to room temperature. Below 2 K, the conductance G exhibits Shklovskii-Efros behavior $G\ensuremath{\sim}\mathrm{exp}[\ensuremath{-}{(T}_{0}{/T)}^{1/2}]$ at zero electric field, where conduction is expected to occur along a very sinuous path (isotropic percolation). The nonlinear $I\ensuremath{-}V$ characteristics are systematically studied up to very high fields, for which the conductance no longer depends on T and for which the current paths are expected to be almost straight (directed percolation). We show that the contributions of electronic and sample heating to those nonlinearities are negligible. Then, we show that the conductance dependence as a function of low electric fields $(F/T<5000\mathrm{V}{\mathrm{m}}^{\ensuremath{-}1}{\mathrm{K}}^{\ensuremath{-}1})$ is given by $G(F,T)=G(0,T)\mathrm{exp}(\ensuremath{-}{eFL/k}_{B}T).$ The order of magnitude (5--10 nm) and the T dependence $(\ensuremath{\sim}{T}^{\ensuremath{-}1/2})$ of L agree with theoretical predictions. From the ${T}_{0}$ value and the length characterizing the intermediate field regime, we extract an estimate of the dielectric constant of our system. The very high electric field data do not agree with the prediction $I(F)\ensuremath{\sim}\mathrm{exp}[\ensuremath{-}{(F}_{0}{/F)}^{{\ensuremath{\gamma}}^{\ensuremath{'}}}]$ with ${\ensuremath{\gamma}}^{\ensuremath{'}}=1/2:$ we find a F dependence of ${\ensuremath{\gamma}}^{\ensuremath{'}}$ that could be partly due to tunneling across the mobility edge. In the intermediate electric field domain, we claim that our data show both the enhancement of the hopping probability with the field and the influence of the straightening of the paths. The latter effect is due to the gradual transition from isotropic to directed percolation and depends essentially on the statistical properties of the ``returns,'' i.e., of the segments of the paths where the current flows against the electrical force. The critical exponent of this returns contribution, which up to now was unknown both theoretically and experimentally, is found to be $\ensuremath{\beta}=1.15\ifmmode\pm\else\textpm\fi{}0.10.$ An estimation of the length of the returns is also given.