### Abstract

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.

Original language | English (US) |
---|---|

Pages (from-to) | 693-716 |

Number of pages | 24 |

Journal | Communications in Mathematical Sciences |

Volume | 10 |

Issue number | 2 |

State | Published - Jun 2012 |

Externally published | Yes |

### Fingerprint

### Keywords

- Biosensor
- DNA sensor
- Existence
- Field-effect sensor
- Gas sensor
- Homogenization
- Interface conditions
- Local uniqueness
- Nanowire
- Self-consistent model
- System of elliptic pdes

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Communications in Mathematical Sciences*,

*10*(2), 693-716.

**Existence and local uniqueness for 3d self-consistent multiscale models of field-effect sensors.** / Baumgartner, Tefan; Heitzinger, Clemens.

Research output: Contribution to journal › Article

*Communications in Mathematical Sciences*, vol. 10, no. 2, pp. 693-716.

}

TY - JOUR

T1 - Existence and local uniqueness for 3d self-consistent multiscale models of field-effect sensors

AU - Baumgartner, Tefan

AU - Heitzinger, Clemens

PY - 2012/6

Y1 - 2012/6

N2 - 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.

AB - 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.

KW - Biosensor

KW - DNA sensor

KW - Existence

KW - Field-effect sensor

KW - Gas sensor

KW - Homogenization

KW - Interface conditions

KW - Local uniqueness

KW - Nanowire

KW - Self-consistent model

KW - System of elliptic pdes

UR - http://www.scopus.com/inward/record.url?scp=84855438993&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84855438993&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:84855438993

VL - 10

SP - 693

EP - 716

JO - Communications in Mathematical Sciences

JF - Communications in Mathematical Sciences

SN - 1539-6746

IS - 2

ER -