Electron dynamics in semiconductors, Markovian and beyond

In earlier calculations of the dynamics of a single electron interacting with lattice vibrations an attempt was made to find a first principles' quantum mechanical description matching the original Haynes-Shockley model for an electron in a semiconductor, a model taken from Einstein's theory of Brownian motion. The equation of motion for the electron's propagator was derived from the exact double path-integral representation of a polaron by Feynman et al. (FHIP). The equation was then converted to the Wigner form of quantum mechanics, and solved in Wigner variables. The purpose of the present calculation is to remove a singularity that arose earlier from scattering by phonons with wave lengths greater than the Einstein length. We find here that the singularity can only be removed by including non-Markovian terms, terms left out in the earlier calculations. The complementary addition of the new terms enhances the plausibility of the modified solution, now no longer Markovian. In this calculation, as a first step, we omit the effect of the electron's recoil energy originating with Markovian scattering by the troublesome long phonon waves. The effect of the neglected recoil energy would be helpful but insufficient in removing the singularities. Of particular interest is the non-Markovian correction to the Einstein diffusion relation, a relationship at the heart of the Haynes-Shockley model. The present calculation, then, may be viewed as yielding an upper limit to the correction, at about 10%.