Density Matrix Formulation of Small-Polaron Motion

A density-matrix treatment of small-polaron motion is presented for the case in which the electronic overlap term of the total Hamiltonian is a small perturbation. The principal result of the density matrix formalism is that total small-polaron mobility can be expressed as the sum of a band part $〈{v_o}^2\tau 〉$, characteristic of the low-temperature regime ($T<\mathrm{}{T}_t$), plus a part ($W_T{a}^2$) describing the hopping motion dominant at high temperatures ($T>\mathrm{}{T}_t$). This verifies the separation of the above two types of motion made on the basis of physical arguments. In addition, the present treatment avoids certain formal divergences in the integrals for the jump probabilities. Furthermore, the hopping contribution does not require localization of the polaron at a particular site, but follows from a translationally invariant formulation. These results are obtained, in part, from lowest order Boltzmann equations which are derived both in the local-site and polaron-band representations. The principal contributions to the scattering terms of the Boltzmann equations are determined by interference effects between the matrix elements, which are examined in some detail.