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We calculate the effective macroscopic dielectric constant εa of a periodic array of spherical nanocrystals (NCs) with dielectric constant ε immersed in the medium with dielectric constant εm 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 εa(s) in the limit ε/εm 蠑 1. Using the scaling hypothesis, we interpolate between them near s = 0 to obtain new approximated function εa(s) for ε/εm 蠑 1. It agrees with existing numerical calculations for ε/εm = 30, while the standard mean-field Maxwell-Garnett and Bruggeman approximations fail to describe percolation-like behavior of εa(s) near s = 0. We also show that in this case the charging energy Ec of a single NC in an array of touching NCs has a non-trivial relationship to εa, namely, Ec = αe2/εad, where α varies from 1.59 to 1.95 depending on the studied three-dimensional lattices. Our approximation for εa(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.
Bibliographical noteFunding Information:
The authors would like to thank Han Fu and B. Skinner for helpful discussions. This work was supported primarily by the National Science Foundation through the University of Minnesota MRSEC under Award No. DMR-1420013.
© 2016 AIP Publishing LLC.
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