Dielectric Constant and Charging Energy in Array of Touching Nanocrystals

We calculate the effective macroscopic dielectric constant ${\epsilon }_a$ of a periodic array of spherical nanocrystals (NCs) with dielectric constant $\epsilon$ immersed in the medium with dielectric constant $\varepsilon_m \ll \varepsilon$\mathrm{.}ForanarrayofNCswiththediameter$d$andthedistance$D$betweentheircenters,whichareseparatedbythesmalldistance$s=D-d \ll d$ortoucheachotherbysmallfacetswithradius$\rho\ll d$ what is equivalent to $s<0$, $\mathrm{\mid }s\mathrm{\mid }\ll d$ we derive two analytical asymptotics of the function ${\epsilon }_a(s)$ in the limit $\epsilon \mathrm{/}{\epsilon }_m\gg 1$. Using the scaling hypothesis we interpolate between them near $s=0$ to obtain new approximated function ${\epsilon }_a(s)$ for $\varepsilon/\varepsilon_m \gg 1$\mathrm{.}Itagreeswithexistingnumericalcalculationsfor$\varepsilon/\varepsilon_m =30$, while the standard mean-field Maxwell-Garnett and Bruggeman approximations fail to describe percolation-like behavior of ${\epsilon }_a(s)$ 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 ${\epsilon }_a$, namely $E_c=\alpha e^2\mathrm{/}{\epsilon }_{a}d$, where $\alpha$ varies from 1.59 to 1.95 depending on the studied three-dimensional lattices. Our approximation for ${\epsilon }_a(s)$ 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.