Communications in Mathematical Sciences

Volume 10 (2012)

Number 2

Existence and local uniqueness for 3D self-consistent multiscale models of field-effect sensors

Pages: 693 – 716



Stefan Baumgartner (Department of Mathematics and Wolfgang Pauli Institute, University of Vienna)

Clemens Heitzinger (Department of Applied Mathematics and Theoretical Physics, University of Cambridge)


We present existence and local uniqueness theorems for a system of partial differential equations modeling field-effect nano-sensors. The system consists of the Poisson(-Boltzmann) equation and the drift-diffusion equations coupled with a homogenized boundary layer. The existence proof is based on the Leray-Schauder fixed-point theorem and a maximum principle is used to obtain a-priori estimates for the electric potential, the electron density, and the hole density. Local uniqueness around the equilibrium state is obtained from the implicit-function theorem. Due to the multiscale problem inherent in field-effect biosensors, a homogenized equation for the potential with interface conditions at a surface is used. These interface conditions depend on the surfacecharge density and the dipole-moment density in the boundary layer and still admit existence and local uniqueness of the solution when certain conditions are satisfied. Due to the geometry and the boundary conditions of the physical system, the three-dimensional case must be considered in simulations. Therefore a finite-volume discretization of the 3d self-consistent model was implemented to allow comparison of simulation and measurement. Special considerations regarding the implementation of the interface conditions are discussed so that there is no computational penalty when compared to the problem without interface conditions. Numerical simulation results are presented and very good quantitative agreement with current-voltage characteristics from experimental data of biosensors is found.


existence, local uniqueness, system of elliptic PDEs, self-consistent model, homogenization, interface conditions, nanowire, field-effect sensor, biosensor, DNA sensor, gas sensor

2010 Mathematics Subject Classification

35A01, 35A02, 35J25, 35Q70, 65N06, 82D80

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