Dielectric Constant and Charging Energy in Array of Touching Nanocrystals

Abstract

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 ${\epsilon }_m\ll \epsilon$. 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 ${\epsilon }_a(s)$ in the limit $\epsilon /{\epsilon }_m\gg 1$. Using the scaling hypothesis we interpolate between them near $s=0$ to obtain new approximated function ${\epsilon }_a(s)$ for $\epsilon /{\epsilon }_m\gg 1$. It agrees with existing numerical calculations for $\epsilon /{\epsilon }_m=30$, 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 $E_c=\alpha e^2/{\epsilon }_{a}d$, 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.

References in corpus (4)

• Metal-insulator transition in films of doped semiconductor nanocrystals
• Theory of hopping conduction in arrays of doped semiconductor nanocrystals
• Hopping conductivity and insulator-metal transition in films of touching semiconductor nanocrystals
• Carrier Transport in Films of Alkyl-Ligand-Terminated Silicon Nanocrystals

Cited by in corpus (3)

• Tuning Nanocrystal Surface Depletion by Controlling Dopant Distribution as a Route Toward Enhanced Film Conductivity
• Hopping conductivity and insulator-metal transition in films of touching semiconductor nanocrystals
• Exciton Transfer in Array of Epitaxially Connected Nanocrystals