Anharmonic Decay of Optical Phonons

An expression is derived from perturbation theory for the lifetime of an optical phonon decaying into two acoustical phonons, and its temperature dependence. The strength of the interaction is expressed in terms of the Gr\"uneisen constant and the effect of a strain, equivalent to the instantaneous strain of an optical mode, on the frequency spectrum of a linear chain with alternating force constants. This interaction depends on the ratio of the acoustical and optical mode frequencies at the zone boundary. Fitting the theory to silicon, the optical mode at k = 0 has a calculated half-width at 300\ifmmode^\circ\else\textdegree\fi{}K of about 1.2% of its frequency. This result is compared with experimental data.