Theory of multiphonon absorption due to nonlinear electric moments in crystals

We formulate a correlation-function theory of the multiphonon absorption due to nonlinear electric moments in anharmonic crystals, at frequencies $\omega$ far above the reststrahl ${\omega }_0$ employing the method of cumulants. In contrast to previous treatments, direct expansion of the moment and anharmonicity potential in powers of displacements is avoided; we thus obtain expressions containing various classes of phonon processes summed to infinite order. The frequency and temperature dependence of the absorption coefficient $\alpha$ is calculated for various approximations and simplified limits, including: the harmonic limit, for which computations with Debye and Einstein models are carried out; the quadratic anharmonic approximation to the cumulant, for which computations are carried out in the noninteracting-cell picture; and the single-particle model, within which "exact" computations are pursued. The results indicate exponentiallike frequency dependences for $\alpha$ for all cases, with enhancement and broadening of $\alpha$ for large anharmonicity, behavior similar to that predicted for anharmonicity absorption stemming from linear moments alone. Application of the single-particle model to cubic diatomic crystals demonstrates significant effects of anharmonicity on $\alpha$, especially for $\frac{\omega }{{\omega }_0}\gg 1$. Moreover, for covalent crystals nonlinear moment contributions to $\alpha$ dominate, while for highly ionic crystals the linear term can exceed or compete with the nonlinear ones for small to intermediate values of $\frac{\omega }{{\omega }_0}$.