Wigner Approach to the Two-Band Kane Model for a Tunneling Diode

In this article we study a simple model for describing the electron behavior in a heterogeneous material characterized by two energy bands. The model, the so-called Kane system, is composed of two Schrödinger-like equations coupled by a k · P term which describes the interband tunneling. We give a physical justification of the Kane system and reformulate it in terms of Wigner functions. The well-posedness of the corresponding Wigner system is investigated in a suitable Hilbert space.