The growth of W grains in sputtered W-${\mathrm{Al}}_2$${\mathrm{O}}_3$ films as a consequence of annealing results in a narrowing of the transition region from metallic to nonmetallic conductivity. The conductivity, $\sigma$, as a function of the volume fraction of the metal, $x$, exhibits a critical behavior given by $\sigma \sim {(x-x_c)}^p$, where the critical volume fraction of the metal $x_c=0.47\mathrm{}\pm 0.05$ and the critical exponent $p=1.9\mathrm{}\pm 0.2$. For $x<\mathrm{}0.47\mathrm{}$ the temperature dependence of $\sigma$ has the form $\ln \sigma \sim -\frac{1}{\sqrt{T}}$ which is characteristic of tunneling between isolated grains.

• Received 12 May 1975

DOI:https://doi.org/10.1103/PhysRevLett.35.247