Variable range cotunneling and conductivity of a granular metal

The Efros–Shklovskii (E–S) law for the conductivity of granular metals is interpreted as a result of a variable range cotunneling process. The cotunneling between distant resonant grains is predominantly elastic at low $T \leq T_c$, while it is inelastic (i.e., accompanied by creation of electron-hole pairs on a string of intermediate non-resonant grains) at $T \geq T_c$. The corresponding E–S temperature $T_{ES}$ in the latter case is slightly (logarithmically) $T$-dependent. The magnetoresistance in the two cases is different: it may be relatively strong and negative at $T \ll T_c$, while at $T>T_c$ it is suppressed due to inelastic processes which destroy the interference.