Optical Phonons and Dynamic Charge in Trigonal Se and Te

Lattice vibrations in elemental crystals can possess a first-order electric moment, and thus exhibit one-phonon infrared absorption (reststrahlen), by the mechanism of displacement-induced charge redistribution (dynamic charge). Trigonal Se and Te, with three atoms per unit cell, belong to the simplest class of reststrahlen-displaying elemental crystals. We have performed a group-theoretical, shell-model analysis of the zone-center optical phonons in these monatomic semiconductors using the most general dynamical matrix consistent with crystal symmetry and translational invariance. The dynamic charges associated with the infrared-active modes have been derived, and the effective-charge tensor ($\frac{\partial \mathbf{p}}{\partial \mathbf{u}}$) constructed. In this model the macroscopic polarization is associated entirely with the shell motion, and the three independent entries appearing in the effective-charge tensor correspond to coupling coefficients connecting acoustical shell displacements with optical core displacements. For the $A_2$ mode, the chain-twisting core motion induces an axial shell motion responsible for the electric moment so that a vibration with atomic displacements $\perp c$ interacts with radiation polarized $\parallel c$. For this mode both dynamic charge and restoring force depend upon next-nearest-neighbor interactions between chains, and vanish in the limit of isolated chains. The expressions obtained for the two $E$-mode effective charges provide a means for understanding the small oscillator strength observed for one $E$-mode pair in both Se and Te. The lattice sums entering into the long-range electrostatic interaction matrix have been evaluated numerically for the detailed geometry of both crystals. In addition, a simple central-force model for the short-range interactions is discussed to illustrate some of the more generally derived results.