Theory of the Temperature Derivative of the Refractive Index in Transparent Crystals

A simplified model is introduced to study the temperature dependence of the index of refraction $n$ in the transparent regime of crystals. The dielectric susceptibility is written as a sum of an electronic contribution due to band-to-band transitions and a lattice contribution, viewed as due to a single mode ${\omega }_0$ Each of the latter undergoes a temperature variation consisting of contributions due to thermal expansion, as well as contributions due to the explicit temperature dependence at constant volume. The temperature derivative $\frac{\mathrm{dn}}{\mathrm{dT}}$ is investigated for various materials. It is found that for zinc-blende- and diamond-type semiconductors, electronic effects, in particular the temperature variation of the band gap at constant volume, yield the dominant contribution. Theoretical calculations of the latter are carried out employing a temperature-dependent-pseudopotential band-structure model; the resulting values for $\frac{\mathrm{dn}}{\mathrm{dT}}$ are in good agreement with experiment. For ionic materials, it is found that both lattice and electronic contributions may be important, as are both explicit temperature variation and thermal-expansion effects. Experimental data on the temperature dependence of ${\omega }_0$ and the band gap are employed to obtain good agreement with the frequency variation of $\frac{\mathrm{dn}}{\mathrm{dT}}$ for a variety of ionic crystals. The results demonstrate that the major physical mechanisms responsible for $\frac{\mathrm{dn}}{\mathrm{dT}}$ can be understood within the present simplified model, and that the model is useful in predicting the magnitude and frequency dependence of $\frac{\mathrm{dn}}{\mathrm{dT}}$ for a wide variety of crystals of interest.