Symmetry and Reststrahlen in Elemental Crystals

Lattice vibrations in elemental crystals can possess a first-order electric moment, and thus exhibit reststrahlen (symmetry-allowed one-phonon infrared absorption), by the mechanism of displacement-induced charge redistribution (dynamic charge). By a group-theoretical investigation of the relation between symmetry and reststrahlen, we show that a necessary and sufficient condition for the existence of reststrahlen in an elemental crystal is a structure with at least three atoms in the primitive unit cell, $s\ge 3$. Using ${n_{\nu }}^{\mathrm{ir}}$ for the number of infrared-active phonon frequencies (reststrahlen bands), this minimum-complexity condition states: (a) ${n_{\nu }}^{\mathrm{ir}}=0\mathrm{}\leftrightarrow s=1\mathrm{} or 2$; (b) ${n_{\nu }}^{\mathrm{ir}}\ge 1\leftrightarrow s\ge 3$. To derive (a) and (b), group-theoretical arguments are used to determine the number of infrared-active phonons and phonon frequencies, and thereby the form of the effective charge tensor ($\frac{\partial \mathbf{p}}{\partial \mathbf{u}}$), in terms of crystal symmetry (group characters) and unit-cell structural complexity (structure factors specifying the number of sublattices invariant under factor-group symmetry operations). The proof of (a), $s\ge 3$ as a structural requirement for a reststrahlen-displaying elemental crystal, follows from these results and the observation that all $s=2\mathrm{}$ elemental crystals possess an inversion operation which interchanges the two sublattices; (a) is equivalent to a generalization of the Lax-Burstein argument for the vanishing first-order moment in Ge. The demonstration of (b), $s\ge 3$ as a sufficient condition for a first-order moment, is obtained by developing an inequality relating ${n_{\nu }}^{\mathrm{ir}}$, $s$, and $g$ (the order of the factor group), and by considering the highest-symmetry crystal classes in some detail. Other applications of his approach, to compounds as well as elemental crystals, are discussed.