Phonon-Assisted Jump Rate in Noncrystalline Solids

An exact calculation of the acoustic-phonon-assisted jump rate has been carried out. It is found that while the Miller-Abrahams (single-phonon) approximation is adequate for the study of shallow-impurity conduction at helium temperatures, it is typically inappropriate for the study of hopping between deep states in noncrystalline solids. In particular, the multiphonon jump rate will display a nonactivated temperature dependence below the Debye temperature which is similar to the often-seen $\exp [-{\left(\frac{T_0}{T}\right)}^{\frac{1}{4}}]$ behavior.