Multi-band Wigner Function Formulation of Quantum Transport
  A Wigner function representation of multi-band quantum transport theory is developed in this paper. The equations are derived using non-equilibrium Green's function formulation with the generalized Kadanoff-Baym ansatz and the multi-band $\bf{k.p}$ Hamiltonian including spin. The results are applied to a two-band resonant inter-band tunneling structure. 
Mehmet Burcin Unlu, Bernard Rosen, Hong-Liang Cui, Peiji Zhao