Dielectric constant and charging energy in array of touching nanocrystals

We calculate the effective macroscopic dielectric constant ε_a of a periodic array of spherical nanocrystals (NCs) with dielectric constant ε immersed in the medium with dielectric constant ε_m 蠐 ε. For an array of NCs with the diameter d and the distance D between their centers, which are separated by the small distance s = D - d 蠐 d or touch each other by small facets with radius ρ 蠐 d what is equivalent to s < 0, | s | 蠐 d we derive two analytical asymptotics of the function ε_a(s) in the limit ε/ε_m 蠑 1. Using the scaling hypothesis, we interpolate between them near s = 0 to obtain new approximated function ε_a(s) for ε/ε_m 蠑 1. It agrees with existing numerical calculations for ε/ε_m = 30, while the standard mean-field Maxwell-Garnett and Bruggeman approximations fail to describe percolation-like behavior of ε_a(s) near s = 0. We also show that in this case the charging energy E_c of a single NC in an array of touching NCs has a non-trivial relationship to ε_a, namely, E_c = αe²/ε_ad, where α varies from 1.59 to 1.95 depending on the studied three-dimensional lattices. Our approximation for ε_a(s) can be used instead of mean field Maxwell-Garnett and Bruggeman approximations to describe percolation like transitions near s = 0 for other material characteristics of NC arrays, such as conductivity.