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Classical Device Modeling

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Nano-Electronic Devices

Abstract

In this chapter an overview of classical device modeling will be given. The first section is dedicated to the derivation of the Drift–Diffusion Transport model guided by physical reasoning. How to incorporate Fourier’s law to add a dependence on temperature gradients into the description, is presented. Quantum mechanical effects relevant for small devices are approximately covered by quantum correction models. After a discussion of the Boltzmann Transport equation and the systematic derivation of the Drift–Diffusion Transport model, the Hydrodynamic Transport model, the Energy Transport model, and the Six-Moments Transport model via a moments based method out of the Boltzmann Transport Equation, which is the essential topic of classical transport modeling, are highlighted. The parameters required for the different transport models are addressed by an own section in conjunction with a comparison between the Six-Moments Transport model and the more rigorous Spherical Harmonics Expansion model, benchmarking the accuracy of the moments based approach. Some applications of classical transport models are presented, namely, analyses of solar cells, biologically sensitive field-effect transistors, and thermovoltaic elements. Each example is addressed with an introduction to the application and a description of its peculiarities.

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Notes

  1. 1.

    The intrinsic energy \({\mathcal{E}}_{i,0}\) is globally constant.

  2. 2.

    For non-degenerate semiconductors the Fermi–Dirac distribution can be approximated by the Maxwell-Boltzmann distribution (\({\mathcal{E}}_{c} -{\mathcal{E}}_{F} \gg k\mathrm{B}T\mathrm{L}\)).

  3. 3.

    Close to the band edges, the relation between the wave vector \(\vec{k}\) and the energy, also known as dispersion relation, can be approximated by an isotropic and parabolic relation \(\mathcal{E}(\vec{k}) = \frac{{\hslash }^{2}{k}^{2}} {2{m}^{{_\ast}}}\), which corresponds to a free electron without any potential.

  4. 4.

    Stern was the first to recognize, that the finite dimensions of dissolved ions cause a layer depleted from charges at interfaces (q.v. Sect. 8.2).

  5. 5.

    A counter ion is the ion that accompanies an ionic species in order to gain charge neutrality. For instance, in sodium chloride, the sodium cation is the counter ion of the chlorine anion and vice versa.

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Acknowledgements

Special thanks go to Prof. Tibor Grasser, Prof. Hans Kosina, and Neophytos Neophytou for their support in questions related to higher order transport models and modeling transport in thermovoltaic elements. Also the various discussions about higher order transport models and nice pictures regarding higher order transport models and SHE from Martin Vasicek, and the examples related to thermovoltaic elements from Martin Wagner are highly appreciated. This work was partly funded by the Austrian Science Fund project P18316-N13 and partly by the “Klima- und Energiefonds” Austria, project No. 825467.

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    Thomas Windbacher

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  1. Dept. Electrical Engineering &, Center for Solid State, Arizona State University, Tempe, 85287-5706, Arizona, USA

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    Stephen M. Goodnick

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Windbacher, T., Sverdlov, V., Selberherr, S. (2011). Classical Device Modeling. In: Vasileska, D., Goodnick, S. (eds) Nano-Electronic Devices. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-8840-9_1

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