One Dimensional Phonon Coupled Electron Tunneling: A Realistic Model
  The transition probability for a one dimensional tunneling electron coupled to acoustical phonons is calculated, with the Feynman path-integral method for zero temperature. We considered a realistic electron phonon interaction (deformation potential, piezoelectric), making use of slowness of the phonon system compared to electron tunneling. We show that the problem of the complex non-linear coupling of a tunneling electron to the zero point fluctuations of a phonon field is equivalent to that of an electron tunneling through a slow fluctuating spatially uniform barrier, thus resulting in an increase of the tunneling probability due to electron coupling with zero-point phonon oscillations. We calculated also the the energy change of the tunneling electron due to phonon emission. 
E.Pazy, B.Laikhtman