Generalized phonon-assisted Zener tunneling in indirect semiconductors   with non-uniform electric fields : a rigorous approach
  A general framework to calculate the Zener current in an indirect semiconductor with an externally applied potential is provided. Assuming a parabolic valence and conduction band dispersion, the semiconductor is in equilibrium in the presence of the external field as long as the electronphonon interaction is absent. The linear response to the electron-phonon interaction results in a non-equilibrium system. The Zener tunneling current is calculated from the number of electrons making the transition from valence to conduction band per unit time. A convenient expression based on the single particle spectral functions is provided, enabling the numerical calculation of the Zener current under any three-dimensional potential profile. For a one dimensional potential profile an analytical expression is obtained for the current in a bulk semiconductor, a semiconductor under uniform field and a semiconductor under a non-uniform field using the WKB (Wentzel-Kramers-Brillouin) approximation. The obtained results agree with the Kane result in the low field limit. A numerical example for abrupt p - n diodes with different doping concentrations is given, from which it can be seen that the uniform field model is a better approximation than the WKB model but a direct numerical treatment is required for low bias conditions. 
William G. Vandenberghe and Bart Sor\'ee and Wim Magnus and Massimo V.   Fischetti
cond-mat.mes-hall