Tunneling density of states of granular metals

We investigate the effect of Coulomb interactions on the tunneling density of states (DOS) of granular metallic systems at the onset of Coulomb blockade regime in two and three dimensions. Using the renormalization group technique we derive the analytical expressions for the DOS as a function of temperature $T$ and energy $ϵ$. We show that samples with the bare intergranular tunneling conductance $g_T^0$ less than the critical value $g_{\scriptscriptstyle T}^{\scriptscriptstyle C}=(1/2\pi d) \ln(E_{\scriptscriptstyle C}/\delta)$,where$E_{\scriptscriptstyle C}$and$\delta$ are the charging energy and the mean energy level spacing in a single grain respectively, are insulators with a {\it hard gap} in the DOS at temperatures $T\to 0$. In 3d systems the critical conductance $g_T^C$ separates insulating and metallic phases at zero temperature, whereas in the granular films $g_T^C$ separates insulating states with the hard (at $g^0_{\scriptscriptstyle T}<g_{\scriptscriptstyle T}^{\scriptscriptstyle C}$)andsoft(at$g^0_{\scriptscriptstyle T}>g_{\scriptscriptstyle T}^{\scriptscriptstyle C}$) gaps. The gap in the DOS begins to develop at temperatures $ T^* \sim E_{\scriptscriptstyle C} g_{\scriptscriptstyle T}^{\scriptscriptstyle 0} \exp (-2\pi d g_{\scriptscriptstyle T}^{\scriptscriptstyle 0})$andreachesthevalue$\Delta \sim T^*$at$T\to 0$.