Conductivity and tunneling density of states in granular Cr films

Abstract

We have measured the tunneling differential conductances, $G(V)$, of four $\text{Al}/{\text{AlO}}_{x}/\text{Cr}$ planar tunnel junctions as well as the conductivities, $\ensuremath{\sigma}(T)$, of the Cr electrodes at liquid-helium temperatures. The Cr electrodes were made to be granular with dimensionless intergrain tunneling conductance spanning from $g\ensuremath{\simeq}1$ to $g⪢1$, and the dimensionality of the granular array $d=3$. For the samples with $g⪢1$, we found that the measured $G(V)$ curves display large zero-bias singularities which obey a $\text{ln}\text{ }V$ law at low bias voltages ($\ensuremath{\lesssim}$ a few millielectron volt) while crossing over to a $\sqrt{V}$ law at high bias voltages. Simultaneously, the conductivities of the Cr electrodes reveal $\text{ln}\text{ }T$ dependence below a characteristic temperature. These results are explained in terms of the recent theory of granular metals. In a sample with $g\ensuremath{\simeq}1$, in addition to the conductivity dependence $\ensuremath{\sigma}\ensuremath{\propto}\text{ln}\text{ }T$, we observed a universal scaling behavior of the normalized differential conductance $[G(V,T)\ensuremath{-}G(0,T)]/\sqrt{T}$ with the combined parameter $\sqrt{e|V|/{k}_{B}T}$ in a wide temperature interval of 2.5--32 K. This result is not yet understood.