Studies of polaron motion

Abstract In this paper is described a model for polaron motion which incorporates, in simplified form, the principal physical features of the problem. The (crystalline) medium, within which a single excess electron (or hole) is contained, is taken to be a one-dimensional molecular crystal, consisting of diatomic molecular sites; each site possesses a single vibrational degree of freedom, represented by the deviation, x n , of its internuclear separation from equilibrium. The motion of the electron in this medium is treated by a tight-binding approach, in which the wave function is represented as a superposition of local “molecular” functions, φ (r − n a, x n ). In line with the x n dependence of the δ's, it is also assumed that the “local” electronic energy, E n , (which, in the conventional tight-binding theory, has one and the same value for all sites) depends (linearly) on x n . This dependence gives rise to electron-lattice interaction; alternatively, it may be regarded as removing the electronic translational degeneracy, characteristic of the undistorted crystal, and thereby providing the possibility for electron trapping. On the basis of the above-described model, the zeroth order adiabatic treatment of the polaron problem is developed. For values of the parameters such that the linear dimension of the polaron is large compared to a lattice spacing (“large” polaron), an exact solution is obtained; the correspondence between it and Pekar's zeroth-order solution is established. The conditions under which the size of the polaron becomes comparable to a lattice spacing (“small” polaron) are discussed. Finally, by way of exhibiting the relationship of the molecular-crystal concept to the real situation, a description is given of an alternate molecular-crystal model which, in the case of the large polaron, is completely equivalent to the continuum-polarization model of conventional polaron theory.