Microscopic considerations of long wavelength acoustic phonons in symmetry-induced zero gap semiconductors

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Abstract
The effective dynamical matrix for long wavelength acoustic phonons in a symmetry-induced zero-gap semiconductor is derived from microscopic considerations of the electron response to lattice displacement. A nonadiabatic theory is shown to be necessary in the very long wavelength limit for a pure system at T=0. Macroscopic correspondence is shown to require an effective charge-neutrality sum rule. This sum rule is used to examine acoustic phonons at somewhat shorter, but still long, wavelengths, and leads to the prediction of two regions of linear dispersion with different sound velocities separated in alpha -Sn at about a wavelength 103 cm. The degenerately doped system is also considered at low temperatures where it is shown that for wavenumbers much less than the Fermi wavenumber the acoustic phonons have a linear dispersion determined primarily by intraband polarization but with a change of character above kF.

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