Homogenization of Boundary Layers in the Boltzmann--Poisson System




Heitzinger, Clemens
Morales Escalante, José A.

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Society for Industrial and Applied Mathematics


We homogenize the Boltzmann--Poisson system where the background medium is given by a periodic permittivity and a periodic charge concentration. The domain is the half-space with a periodic charge concentration on the boundary. Hence the domain consists of cells in R3 that are periodically repeated in two dimensions and unbounded in the third dimension. We obtain formal results for this homogenization problem. We prove the existence and uniqueness of the solution of the Laplace and Poisson problems in the fast variables with periodic and surface charge boundary conditions generating an electric field at infinity, obtaining formal solutions for the potential in terms of Magnus expansions for the case where the diagonal permittivity matrix depends on the vertical fast variable. Further on, splitting the potential into a stationary part and a self-consistent part, performing the two-scale homogenization expansions for the Poisson and the Boltzmann equations, and applying a solvability condition, we arrive at the drift-diffusion equations for the boundary-layer problem.


Copyright is held by the Society for Industrial and Applied Mathematics.


homogenization, Boltzmann--Poisson system, nanowire sensors


Heitzinger, C., & Morales Escalante, J. A. (2021). Homogenization of Boundary Layers in the Boltzmann--Poisson System. Multiscale Modeling & Simulation, 19(1), 506-532. doi:10.1137/18m1193888


Physics and Astronomy