Three-dimensional optimal multi-level Monte–Carlo approximation of the stochastic drift–diffusion–Poisson system in nanoscale devices
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The three-dimensional stochastic drift–diffusion–Poisson system is used to model charge transport through nanoscale devices in a random environment. Applications include nanoscale transistors and sensors such as nanowire field-effect bio- and gas sensors. Variations between the devices and uncertainty in the response of the devices arise from the random distributions of dopant atoms, from the diffusion of target molecules near the sensor surface, and from the stochastic association and dissociation processes at the sensor surface. Furthermore, we couple the system of stochastic partial differential equations to a random-walk-based model for the association and dissociation of target molecules. In order to make the computational effort tractable, an optimal multi-level Monte–Carlo method is applied to three-dimensional solutions of the deterministic system. The whole algorithm is optimal in the sense that the total computational cost is minimized for prescribed total errors. This comprehensive and efficient model makes it possible to study the effect of design parameters such as applied voltages and the geometry of the devices on the expected value of the current.
KeywordsStochastic drift–diffusion–Poisson system Multi-level Monte–Carlo Optimal method Uncertainty quantification Fluctuations Noise Transistor Nanowire
The authors acknowledge support by FWF (Austrian Science Fund) START Project No. Y660 PDE Models for Nanotechnology. The authors also acknowledge discussions with Gerhard Tulzer.
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