Theory of Impurity-Induced Infrared Absorption

The problem of impurity-induced absorption of light due to a localized mode is studied on the basis of the Kubo linear-response theory. The equation of motion for the dipole-moment correlation function with respect to the localized mode is derived using Zwanzig's projection-operator method, and solved with the assumption that the anharmonic coupling between the localized mode and the remaining modes (the bath) is of the form $Q\Gamma$ and weak enough to allow a perturbation treatment. The variables $Q$ and $\Gamma$ represent the localized-mode normal coordinate and an arbitrary function of the bath coordinates, respectively. The first nonvanishing contribution is shown to be of second order in the coupling and to give an absorption band of Lorentzian shape whose width and shift are given by the real and imaginary parts of the half-interval Fourier transform of the correlation function ${〈[\Gamma ,\Gamma \mathrm{}(t\left)\right]〉}^0$, where ${〈\cdots 〉}^0$ denotes an average over the canonical ensemble which represents the uncoupled bath. It follows directly from this result that the width and shift functions satisfy the Kramers-Kronig relations. In an Appendix, the width and shift expressions for the special case of an anharmonic crystal having up to quartic terms in the potential are derived and shown to be equivalent to Maradudin's calculation based on many-body diagram techniques.