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Super-lattice HAMILTONIAN (Barletti, Bonilla,Escobedo)

Super-lattice HAMILTONIAN (Barletti, Bonilla,Escobedo). Consider the two -band periodic Hamiltonian. written in terms of Pauli matrices, , as. The spectrum of the “free Hamiltonian” is. Wigner spinorial functions.

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Super-lattice HAMILTONIAN (Barletti, Bonilla,Escobedo)

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  1. Super-lattice HAMILTONIAN (Barletti, Bonilla,Escobedo) Consider the two-band periodic Hamiltonian written in terms of Pauli matrices, , as The spectrum of the “free Hamiltonian” is

  2. Wigner spinorial functions • We decompose the Wigner matrix in terms of Pauli matrices where • The expected value of the observable when system is at the state is given by integration over the phase-space of the following function • Electron densities in the band at time t

  3. The evolution equations for the Wigner functions where is the pseudo-differential operator are the Wigner functions at local equilibrium, written in terms of Fermi-Dirac distributions for the two bands, as function of the two band densities

  4. CHAPMAN-ENSKOG expansion SCALING • Hyperbolic scaling: potential and collisions are dominant • Weak coupling: is small with respect to We define the moment operators: band densities Chapman-Enskog ansatz:

  5. Then We obtain a hierarchy of equations for Expanding up to the order m=1, we can write explicitly

  6. Drift-diffusion equations QDD where is the generation-recombination operator etc., etc.

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