Quantum mechanical effects in continuum charge flow models

This paper concerns mathematical modelling of charge transport across a thin poorly conducting layer between two electrodes. We describe and analyse two alternative approaches to model quantum effects within a continuum theory: the Density Gradient Confinement (DGC) and Density Gradient Tunnelling (DGT) theories. In either case, quantum effects are characterised by a small parameter, which we exploit to analyse the problems asymptotically. We thus find simplified approximate solutions which show excellent agreement with numerical solutions of the full models, and demonstrate previously undocumented oscillatory behaviour both in the charge density profile and in the variation of current with applied potential difference.