Mobility of Polarons

The dc mobility of electrons is considered for scattering by acoustical and optical phonons. It is shown that the vertex terms (ladder diagrams) in the mobility integral can be partially summed to all orders of perturbation theory. These vertex terms are shown to be important for acoustic-phonon scattering. The sum of the ladder diagrams yields an effective energy width (inverse lifetime) with the $1-cos\theta$ term, in agreement with transport theory. For polar coupling to optical phonons, the ladder diagrams can still be summed but are shown to be unimportant at low temperatures.