Small coherence length limit for a two-dimensional quantum transport model

In a previous work [N. Ben Abdallah, C. Negulescu, A one-dimensional transport model with small coherence lengths, Transp. Theory and Stat. Phys. 31, no. 4-6 (2002), 559-578] a one dimensional transport model accounting for both, quantum phenomena and smallness of coherence lengths, has been analyzed. In the limit of infinitely small coherence lengths, this model leads to the classical Vlasov equation. In this paper the two dimensional situation is considered by adding to the one dimensional transport direction a transversal confined one. The transport direction is splitted into several regions of a size comparable to the coherence length. A quantum model, based on the 2D Schrödinger equation, discribes the electron transport within each cell, while on larger distances, statistics defined on the interfaces of the cells, determine the electron motion. Reflection and transmission coefficients, deduced from the wavefunctions, solutions of the Schrödinger equation, relate neighbouring statistics with one another. In the limit of small coherence lengths, we obtain a collisionless subband model, which is classical in the transport direction and keeps quantum features in the confined one.