Random-Phase-Approximation Dielectric Function ofa-Sn in the Far Infrared

The frequency-dependent random-phase-approximation (RPA) dielectric function of a symmetry-induced zero-gap semiconductor is examined in the far infrared. Expressions are derived for the temperature, wavelength, and impurity-concentration dependence in the degenerate and nondegenerate limits. It is shown that, at absolute zero, the real part of the dielectric function has a logarithmic singularity at a frequency corresponding to excitations to the Fermi surface. This singularity is removed by temperature and lifetime broadening. Numerical calculations are presented for cases of intermediate degeneracy in $\alpha$-Sn at a number of impurity concentrations, temperatures, and lifetimes. It is shown that strong temperature dependences exist at liquid-helium temperatures and below. A recent experiment on the temperature dependence of the reflectivity minimum is analyzed and shown to be, with the exception of one low-temperature point, in excellent agreement with the theory. A value of 19 is determined for the background dielectric constant ${\epsilon }_0$. Although the anomalous low-temperature datum can be quantitatively accounted for by small errors introduced by sample inhomogeneity into the Hall measurement of the carrier concentration, it is pointed out that the value of ${\epsilon }_0$ inferred from this point yields electron mobilities in excellent agreement with experiment in the one-band region.