Communications in Mathematical Sciences

Volume 15 (2017)

Number 8

Analysis of the drift-diffusion-Poisson–Boltzmann system for nanowire and nanopore sensors in the alternating-current regime

Pages: 2303 – 2325



Clemens Heitzinger (Institute for Analysis and Scientific Computing, Technical University Vienna, Austria)

Leila Taghizadeh (Institute for Analysis and Scientific Computing, Technical University Vienna, Austria)


The basic analytical properties of the drift-diffusion-Poisson–Boltzmann system in the alternating-current (AC) regime are shown. The analysis of the AC case differs from the direct-current (DC) case and is based on extending the transport model to the frequency domain and writing the variables as periodic functions of the frequency in a small-signal approximation. We first present the DC and AC model equations to describe the three types of material in nanowire field-effect sensors: The drift-diffusion-Poisson system holds in the semiconductor, the Poisson–Boltzmann equation holds in the electrolyte, and the Poisson equation provides self-consistency. Then the AC model equations are derived. Finally, existence and local uniqueness of the solution of the AC model equations are shown. Real-world applications include nanowire field-effect bio- and gas sensors operating in the AC regime, which were only demonstrated experimentally recently. Furthermore, nanopore sensors are governed by the system of model equations and the analysis as well.


drift-diffusion-Poisson–Boltzmann system, existence, local uniqueness, charge transport, alternating current, field-effect sensor, nanowire sensor, nanocapacitor, nanopore sensor, nanotechnology

2010 Mathematics Subject Classification

35Q20, 62P30, 76R50, 82D37, 82D80

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The authors acknowledge support by the FWF (Austrian Science Fund) START project no. Y660 PDE Models for Nanotechnology.

Received 24 March 2016

Accepted 23 August 2017

Published 20 December 2017