Dielectric Constant and Charging Energy in Array of Touching Nanocrystals

TL;DR

This paper derives analytical expressions for the dielectric constant of nanocrystal arrays near touching points, revealing limitations of traditional models and linking dielectric properties to charging energy in these systems.

Contribution

The authors develop new asymptotic and interpolated formulas for the dielectric constant of nanocrystal arrays near contact, improving upon mean-field models and connecting dielectric behavior to charging energy.

Findings

Derived asymptotic expressions for dielectric constant near touching points.

Interpolated a new approximate formula matching numerical data.

Established a relationship between charging energy and dielectric constant.

Abstract

We calculate the effective macroscopic dielectric constant $\varepsilon_a$ of a periodic array of spherical nanocrystals (NCs) with dielectric constant $\varepsilon$ immersed in the medium with dielectric constant $\varepsilon_m \ll \varepsilon$. For an array of NCs with the diameter $d$ and the distance $D$ between their centers, which are separated by the small distance $s=D-d \ll d$ or touch each other by small facets with radius $\rho\ll d$ what is equivalent to $s < 0$, $|s| \ll d$ we derive two analytical asymptotics of the function $\varepsilon_a(s)$ in the limit $\varepsilon/\varepsilon_m \gg 1$. Using the scaling hypothesis we interpolate between them near $s=0$ to obtain new approximated function $\varepsilon_a(s)$ for $\varepsilon/\varepsilon_m \gg 1$. It agrees with existing numerical calculations for $\varepsilon/\varepsilon_m =30$, while the standard mean-field Maxwell-Garnett and Bruggeman approximations fail to describe percolation-like behavior of $\varepsilon_a(s)$ 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 $\varepsilon_a $, namely $E_c = \alpha e^2/\varepsilon_a d$, where $\alpha$ varies from 1.59 to 1.95 depending on the studied three-dimensional lattices. Our approximation for $\varepsilon_a(s)$ 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.