Behavior of the Electronic Dielectric Constant in Covalent and Ionic Materials

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 $n^2-1=\frac{E_d{E}_0}{(E_0^2-{\hslash }^2{\omega }^2)}$, where $\hslash \omega$ 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 $E_d=\beta N_c{Z}_a{N}_e$, 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 $N_e=8\mathrm{}$), and $\beta$ is essentially two-valued, taking on the "ionic" value ${\beta }_i=0.26\mathrm{}\pm 0.04$ eV for halides and most oxides, and the "covalent" value ${\beta }_c=0.37\mathrm{}\pm 0.05$ eV for the tetrahedrally bonded $A^N{B}^{8-N}$ 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 ${\epsilon }_2$ spectrum via appropriately defined moment integrals. It is found, using relationships between moment integrals, that for a particularly simple choice of a model ${\epsilon }_2$ 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 ($E_g,\hslash {\omega }_p$), i.e., ${\left(\hslash {\omega }_p\right)}^2=E_a{E}_d \mathrm{and} E_g^2=E_a{E}_0$. 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.