Hopping conductivity and insulator-metal transition in films of touching semiconductor nanocrystals

This paper is focused on the the variable-range hopping of electrons in semiconductor nanocrystal (NC) films below the critical doping concentration $n_c$ at which it becomes metallic. The hopping conductivity is described by the Efros-Shklovskii law which depends on the localization length of electrons. We study how the localization length grows with the doping concentration $n$ in the film of touching NCs. For that we calculate the electron transfer matrix element $t(n)$ between neighboring NCs for two models when NCs touch by small facets or just one point. We study two sources of disorder: variations of NC diameters and random Coulomb potentials originating from random numbers of donors in NCs. We use the ratio of $t(n)$ to the disorder-induced NC level dispersion to find the localization length of electrons due to the multi-step elastic co-tunneling process. We found three different phases at $n<n_c$ depending on the strength of disorder, the material, sizes of NCs and their facets: 1) "insulator" where the localization length of electrons increases monotonically with $n$ and 2) "oscillating insulator" when the localization length (and the conductivity) oscillates with $n$ from the insulator base and 3) "blinking metal" where the localization length periodically diverges. The first two phases were seen experimentally and we discuss how one can see the more exotic third one. In all three the localization length diverges at $n=n_c$. This allows us to find $n_c$.