Theory of band-to-band tunneling under nonuniform electric fields for subbreakdown leakage currents

A new expression for the band-to-band tunneling in the presence of a nonuniform electric field is derived, with emphasis on the importance of the exact evaluation of integrals over wave vectors of a tunneling electron. Nonuniformity of the electric field is taken into account in calculating the interband matrix element, although it is neglected in the electronic wave functions. Nonuniformity of the electric field tends to decrease the tunneling rate. The new expression shows a different electric field dependence from E.O. Kane's (1959, 1961) tunneling rate especially in high electric fields. The appropriate evaluation of wave-vector integrations and the consideration of the spatial dependence of the electric field are indispensable to obtaining a correct electric-field dependence of the tunneling rate