Behavior of the Electronic Dielectric Constant in Covalent and Ionic Materials

TLDR

Wurtzite-structure crystals occupy a transitional position between ionic and covalent crystal classes. The study proposes that refractive-index behavior can be understood through a localized molecular theory of optical transitions, based on the dependence of \(E_d\) on coordination number and valency. The authors analyze refractive-index dispersion below the interband edge using a single-effective-oscillator model and relate the results to the \(\varepsilon_2\) spectrum through moment integrals. They find that the dispersion energy \(E_d\) follows a simple empirical law dependent on coordination number, anion valency, and valence electrons, is largely independent of bandgap and electron density, and links single-oscillator parameters to static dielectric constants via moment-integral relations.

Abstract

Refractive-index dispersion data below the interband absorption edge in more than 100 widely different solids and liquids are analyzed using a single-effective-oscillator fit of the form ${n}^{2}\ensuremath{-}1=\frac{{E}_{d}{E}_{0}}{({E}_{0}^{2}\ensuremath{-}{\ensuremath{\hbar}}^{2}{\ensuremath{\omega}}^{2})}$, where $\ensuremath{\hbar}\ensuremath{\omega}$ is the photon energy, ${E}_{0}$ is the single oscillator energy, and ${E}_{d}$ is the dispersion energy. The parameter ${E}_{d}$, which is a measure of the strength of interband optical transitions, is found to obey the simple empirical relationship ${E}_{d}=\ensuremath{\beta}{N}_{c}{Z}_{a}{N}_{e}$, where ${N}_{c}$ is the coordination number of the cation nearest neighbor to the anion, ${Z}_{a}$ is the formal chemical valency of the anion, ${N}_{e}$ is the effective number of valence electrons per anion (usually ${N}_{e}=8$), and $\ensuremath{\beta}$ is essentially two-valued, taking on the "ionic" value ${\ensuremath{\beta}}_{i}=0.26\ifmmode\pm\else\textpm\fi{}0.04$ eV for halides and most oxides, and the "covalent" value ${\ensuremath{\beta}}_{c}=0.37\ifmmode\pm\else\textpm\fi{}0.05$ eV for the tetrahedrally bonded ${A}^{N}{B}^{8\ensuremath{-}N}$ zinc-blende- and diamond-type structures, as well as for scheelite-structure oxides and some iodates and carbonates. Wurtzite-structure crystals form a transitional group between ionic and covalent crystal classes. Experimentally, it is also found that ${E}_{d}$ does not depend significantly on either the bandgap or the volume density of valence electrons. The experimental results are related to the fundamental ${\ensuremath{\epsilon}}_{2}$ spectrum via appropriately defined moment integrals. It is found, using relationships between moment integrals, that for a particularly simple choice of a model ${\ensuremath{\epsilon}}_{2}$ spectrum, viz., constant optical-frequency conductivity with high- and low-frequency cutoffs, the bandgap parameter ${E}_{a}$ in the high-frequency sum rule introduced by Hopfield provides the connection between the single-oscillator parameters (${E}_{0},{E}_{d}$) and the Phillips static-dielectric-constant parameters (${E}_{g},\ensuremath{\hbar}{\ensuremath{\omega}}_{p}$), i.e., ${(\ensuremath{\hbar}{\ensuremath{\omega}}_{p})}^{2}={E}_{a}{E}_{d} \mathrm{and} {E}_{g}^{2}={E}_{a}{E}_{0}$. Finally, it is suggested that the observed dependence of ${E}_{d}$ on coordination number and valency implies that an understanding of refractive-index behavior may lie in a localized molecular theory of optical transitions.