Low-Field Limit for a Nonlinear Discrete Drift-Diffusion Model Arising in Semiconductor Superlattices Theory

Charge transport in semiconductor superlattices can be described through a discrete drift-diffusion model. In this model, we identify some small parameter h > 0, by means of physically relevant dimensionless quantities. Precisely, we investigate a regime where the length of the superlattice period is small while the doping profile is high. In the limit h → 0, we are led to a nonlinear drift diffusion model, coupled to the Poisson equation. 1 Discrete Drift-Diffusion Model A semiconductor superlattice is a periodic array of layers of two different semiconductors whose lateral dimension is much larger than the length ` of one period. Since the two semiconductors have different energy gaps, the conduction band of a superlattice (SL) can be viewed as a periodic array of potential wells and barriers, of widths w and d, respectively, with ` = d + w. We assume that scattering times are shorter than escape times from quantum wells, the latter being shorter than dielectric relaxation times. In such weakly coupled semiconductor SL, the dominant mechanism of charge transport is sequential resonant tunneling. In the simplest situation, the center of each quantum well is n-doped. Then electronic transport in these devices can be described by a discrete drift-diffusion model; see Aguado, Platero, Moscoso, Bonilla [1] and Bonilla, Platero, Sánchez [3]. This model has been extended by taking into account stochastic effects by Bonilla, Sánchez, Soler [4], in comparison with the experimental results of [8]. In such a modeling, we consider an array of 2N + 1 consecutive cells, which are well-barrier pairs, labelled by the index i ∈ {−N, . . . ,+N}. The barrier separating the injecting contact from the first well of the SL is considered as the (−N − 1)-th barrier, while the barrier of the N -th SP period separates the N -th well from the collector. The model assumes that the electrons are singularly ∗This research was partially supported by the EU financed network IHP-HPRN-CT-2002-00282 and by MCYT (Spain), Proyecto BFM2002–00831. †Labo. J. A. Dieudonné, Université Nice-Sophia Antipolis, Parc Valrose, 06108 Nice cedex 02, France ([email protected]). ‡Departamento de Matematica Aplicada, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain (ossanche, [email protected]). §Departamento de Matemáticas, Escuela Politécnica Superior, Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911 Leganés, SPAIN Also: Unidad Asociada al Instituto de Ciencia de Materiales (CSIC), 28049 Cantoblanco, SPAIN, ([email protected])