Adiabatic Theory of Nearly Small Polarons

Abstract

An adiabatic theory of polarons is developed under conditions when the polarons are "nearly small," that is, when their binding energy is greater than or of the order of half the rigid-lattice bandwidth, but is not so great that small-polaron theory is applicable. In such an adiabatic theory there are a set of minimum-energy nuclear configurations, each of which has associated electronic wave-functions concentrated mainly on one positive ion, but spreading slightly to its neighbors. At and near these configurations trial electronic wave functions are taken in the form of linear combinations of single-ion functions for a particular ion and its nearest neighbors, with coefficients determined by minimizing the energy. Conditions for localized normal modes to be associated with any minimum are examined. Properties of wave functions describing the nuclear motion are studied within the framework of a generalized tight-binding approach. It is shown that, if localized modes are not formed, then bandwidths will decrease with increasing temperature as in small-polaron theory, but that when localized modes are present, then at nonzero temperatures, a discrete distribution of bandwidths will occur, and the thermal average of these widths may increase with increasing temperature. Optical absorption due to transitions between a wide valence band for which electron-phonon coupling is neglected and an adiabatic nearly-small-polaron conduction band with localized modes is considered. Results for absorption at absolute zero are similar to those obtained previously for a small-polaron conduction band without localized modes, but the temperature dependence of the absorption obtained here shows some new features. Parameters occurring in the theory are estimated using a continuum-polarization model for electron-phonon interactions, and numerical values are found for a simplified model of a possible conduction band in SrTi${\mathrm{O}}_3$. The calculations indicate that electronic states may exist from which both adiabatic nearly small polarons and weak-coupling large polarons can be formed, and that the lowest energy polaron state may suddenly change from one type to the other as electronic overlap integrals or electron-phonon interactions are altered in magnitude.