A unified theory of direct and indirect interband tunneling under a nonuniform electric field

Abstract

A theory of interband tunneling under a nonuniform electric field has been formulated for both direct and indirect transitions. The formulation starts from the Wannier equation, where the driving force of the transition is a non-diagonal interband matrix element. The Wannier equation is solved in real coordinate space to obtain unperturbed wave functions in the absence of the driving force. The unperturbed wave functions in the tunnel region are obtained by applying the WKB approximation, resulting in an integral form of the function of the electrostatic potential. The integral always exists independently of the complexity of the electrostatic potential. Thus, we always have unified tunneling probabilities for direct and indirect interband transitions by time dependent perturbation. General formulations for tunneling currents were obtained by integrating the probabilities over energy.