Abstract

We present results for the mathematical modeling of nanotechnological field-effect sensors. First, we show how a homogenization problem for the electrostatics in these structures can be solved to find an effective equation with two jump conditions. Based on this homogenized equation, an existence and uniqueness (around equilibrium) result and estimates for the solution of a 3d self-consistent charge-transport model are given. This model has been implemented as a parallelized 3d simulator that can simulate realistic structures.

In order to investigate noise in nanostructures, we present a homogenization result for a stochastic elliptic PDE that yields an effective equation for the (co-)variance and a scaling law.

Finally, we present the derivation of a transport equation for 3d structures that are confined in one or two dimensions from the Boltzmann transport equation. Examples for these structures are nanowires, ion channels, etc. In the case of confinement in two directions (which corresponds to tubes) we find a diffusive transport equation. A crucial feature of this equation is that we can give explicit expressions for the transport coefficients as functions of the parabolic confinement potential. Computationally, this means that we have reduced the six-dimensional problem to a 2d diffusion-type equation. Entropy estimates are also given.

Numerical simulation results are presented for all the models as well, as are comparisons to experimental data (field-effect biosensors, ion channels).