  We treat a tunneling electron coupled to acoustical phonons through a
realistic electron phonon interaction: deformation potential and piezoelectric,
in two or three-dimensional tunneling configurations. Making use of slowness of
the phonon system compared to electron tunneling, and using a Green function
method for imaginary time, we are able to calculate the change in the
transition probability due to the coupling to phonons. It is shown using
standard renormalization procedure that, contrary to the one-dimensional case,
second order perturbation theory is sufficient in order to treat the
deformation potential coupling, which leads to a small correction to the
transmission coefficient prefactor. In the case of piezoelectric coupling,
which is found to be closely related to the piezoelectric polaron problem,
vertex corrections need to be considered. Summing leading logarithmic terms, we
show that the piezoelectric coupling leads to a significant change of the
transmission coefficient.
