Quantum Percolation in Granular Metals

Theory of quantum corrections to conductivity of granular metal films is developed for the realistic case of large randomly distributed tunnel conductances. Quantum fluctuations of intergrain voltages (at energies $E$ much below the bare charging energy scale $E_C$) suppress the mean conductance $\overline{g}(E)$ much more strongly than its standard deviation $\sigma (E)$. At sufficiently low energies $E_{*}$ any distribution becomes broad, with $\sigma (E_{*})\sim \overline{g}(E_{*})$, leading to strong local fluctuations of the tunneling density of states. The percolative nature of the metal-insulator transition is established by a combination of analytic and numerical analysis of the matrix renormalization group equations.

Figure 1

Conducting bonds (with $g_{ij}>0$) at different values of the RG time $t$. Results of numerical simulation of Eq. (2) on the lattice $20\times 20$ with $P_0(g)=[2g_0\theta (g)/\pi ]/(g^2+g_0^2)$. (a) $t=0.64g_0$ with the fraction of conducting bonds $N_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}}=0.95$; (b) $t=1.08g_0$ with $N_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}}=0.55$.

Figure 2

Histograms for the distribution of the ZBA factors $Z(E)$ near the percolation threshold $t_c=0.99{\overline{g}}_0$, obtained numerically for ${\sigma }_0/{\overline{g}}_0=0.32$: $t=0.8t_c$ (solid line); $t=0.9t_c$ (dashed line); $t=0.95t_c$ (dotted line).

Figure 3

Results of the numerical simulation (solid lines) and EMA (dashed lines) for the global conductivity $g_{\mathrm{e}\mathrm{f}\mathrm{f}}(t)$ and the fraction of conducting bonds $N_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}}(t)$ (see text for details).