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 (d=2,3). 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_T⁰ less than the critical value g_T^C=(1∕2πd)ln(E_C∕δ), where E_C and δ are the charging energy and the mean energy level spacing in a single grain, respectively, are insulators with a hard gap in the DOS at temperatures T→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_T⁰<g_T^C) and soft (at g_T⁰>g_T^C) gaps. The gap in the DOS begins to develop at temperatures T*∼E_Cg_T⁰ exp(−2πdg_T⁰) and reaches the value Δ∼T* at T→0.