Frequency-dependent optical properties

Ever since the early days of atomistic simulation within the shell model context it has been routine to calculate the static and high frequency dielectric constant tensors. Indeed this data has often been used during the fitting process as well. However, the values obtained from experiment will always be measured at a particular frequency and this will never exactly correspond to the limiting values determined by the direct means of calculation. As described in the background theory section, it is possible to evaluate the dielectric properties and refractive indices as a function of radiation frequency for the gamma point.

Here we present results for the frequency variation of the dielectric constant tensor of quartz, shown in Figure 2.2, as calculated using the previously mentioned shell model potential of Sanders et al [120]. Note that the limiting values are the same as the ones obtained without reference to the phonon frequencies.

In accord with experiment, the dielectric constant decreases slowly with the frequency of measurement until the highest phonon mode of quartz - the Si-O stretch - is approached. At frequencies below this the curve contains considerable variation caused by the discontinuities when a lattice phonon mode is reached. For simplicity, the curve shown is for the dielectric constant in the ab plane only. The corresponding curve for the principal tensor component parallel to the 001 direction is almost identical, except that the limiting values are slightly different and the positions of the discontinuities due to coincidence with phonons are displaced to higher frequency.

While the qualitative agreement with experimental data is good, there is a quantitative discrepancy in the dielectric constant variation. This is a result of the imperfection of the original fit to the dielectric data for quartz, though there are also issues concerning the variation with temperature since the calculations are formally performed at absolute zero, while the experiment data was measured at 293 K. However, the use of empirical fitting implies that the interatomic potentials do partially account for the temperature difference already. At the lowest measured frequency, the calculated values are an almost perfect match due to the faster rate of decrease of the dielectric constant in the experiment data.

Figure 2.2: The on-diagonal component of the dielectric constant tensor for $\alpha $-quartz in the ab plane as a function of frequency of measurement. The solid line represents the calculated shell model values, while the crosses represent values estimated from experimental measurements of the refractive index as a function of frequency.