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Cross validation of discontinuous Galerkin method and Monte Carlo simulations of charge transport in graphene on substrate

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Abstract

The aim of this work is to simulate the charge transport in a monolayer graphene on a substrate. This requires the inclusion of the scatterings of the charge carriers with the impurities and the phonons of the substrate, besides the interaction mechanisms already present in the graphene layer. As physical model, the semiclassical Boltzmann equation will be assumed. Two approaches will be used for the simulations: a numerical scheme based on the Discontinuous Galerkin method for finding deterministic (non stochastic) solutions and a new Direct Monte Carlo Simulation formulated in Romano et al. (J Comput Phys 302:267–284, 2015) in order to deal in the appropriate way with the Pauli exclusion principle for degenerate Fermi gases. A cross validation of the deterministic and stochastic solutions shows the robustness and accuracy of both the approaches.

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Notes

  1. We expect an exponential decay of the distribution function as \(|\mathbf {k}| \mapsto +\infty \). This is proved, under suitable conditions, for the classical Boltzmann equation of rarefied monatomic gases. In our simulations, we check if, after each time step, the values of f at the boundary of the domain \(\Omega \) are sufficiently low; otherwise, we enlarge the domain \(\Omega \) and repeat the integration starting from the initial time.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, University of Catania, viale A. Doria 6, 95125, Catania, Italy

    Marco Coco, Armando Majorana & Vittorio Romano

Authors
  1. Marco Coco
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  2. Armando Majorana
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  3. Vittorio Romano
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Correspondence to Vittorio Romano.

Additional information

This work has been partially supported by the University of Catania, project F. I. R. Charge transport in graphene and low dimensional systems, and by INDAM.

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Coco, M., Majorana, A. & Romano, V. Cross validation of discontinuous Galerkin method and Monte Carlo simulations of charge transport in graphene on substrate. Ricerche mat 66, 201–220 (2017). https://doi.org/10.1007/s11587-016-0298-4

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  • DOI: https://doi.org/10.1007/s11587-016-0298-4

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