Diffusion and Homogenization approximation for semiconductor Boltzmann-Poisson system Nader Masmoudi 1 , Mohamed Lazhar Tayeb 2 1 Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, NY 10012, USA. email : masmoudi@courant.nyu.edu N.M is partially supported by an NSF grant DMS 0403983. 2 Facult´ e des Sciences de Tunis. Campus Universitaire El-Manar, 1060 Tunis Tunisia. Courriel : lazhar.tayeb@fst.rnu.tn Abstract. We are concerned with the study of the diffusion and homogenization ap- proximation of the Boltzmann-Poisson system in presence of a spatially oscillating electrostatic potential. By analyzing the relative entropy, we prove uniform en- ergy estimate for well prepared boundary data. An averaging lemma and two scale convergence techniques are used to prove rigorously the convergence of the scaled Boltzmann equation (coupled to Poisson) to a homogenized Drift-Diffusion-Poisson system. Key words: semiconductors, Boltzmann-Poison system, Drift-Diffusion equation, renor- malized solution, diffusion approximation, homogenization. two-scale convergence, averaging lemma.