Lattice Relaxation and Small-Polaron Hopping Motion

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Abstract
The three-dimensional analog of Holstein's molecular crystal model is utilized as the basis for a study of the relaxation of the lattice after a small-polaron hop. In particular, it is shown explicitly that the time-dependent activationlike energy arising in the previously developed theory of correlated small-polaron hopping motion is directly related to the actual relaxation of the lattice from the distorted configuration it must assume to facilitate a small-polaron hop. That is, the time dependence of this "activation energy" and the concomitant relaxation of the hop-related lattice displacements are governed by a single entity denoted as the relaxation function. Furthermore it is demonstrated that this function is directly expressible in terms of the transfer of vibrational energy from the initially distorted sites to successive (initially undistorted) neighbor sites. In fact, for the most part, the relaxation of the lattice after a hop is associated with the transfer of vibrational energy to only nearest-neighbor sites, this being essentially a local phenomenon independent of the periodic nature of the lattice. Finally, although the lattice relaxation for our three-dimensional model is found to proceed much faster than in the previously developed one-dimensional model, its effect on small-polaron hopping motion may not be inconsequential. In particular, the small-polaron drift mobility is shown to be significantly affected by lattice relaxation effects when the mean time between small-polaron hops is less than or comparable to the lattice relaxation time; this time being essentially the reciprocal of the optical-phonon bandwidth parameter.