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A general analytical method for finding the quantum capacitance of graphene

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Abstract

The quantum capacitance of graphene has been modeled in numerous articles using approximate analytical expressions and numerical methods. However, no article shows how to analytically find the quantum capacitance of graphene for the full temperature range and for any Fermi level. This, of course, would require a complete analytical evaluation of Fermi–Dirac-type integrals valid for any temperature. This article will illustrate a method for finding the quantum capacitance of monolayer graphene in the presence of electron–hole puddles for any Fermi level and for any temperature. The method employed is easily generalized to find the quantum capacitance of bilayer graphene as well as any other material when the density of states is a known function of energy.

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Acknowledgements

The author would like to express his thanks to Jerry A. Selvaggi for insightful discussions concerning the Fermi–Dirac integral. This ultimately leads the author to analytically evaluate a wide range of seemingly intractable integrals. The author would also like to express his gratitude to Jessica Diaz for helpful comments on various ways to numerically integrate Fermi–Dirac-type and Bose–Einstein-type integrals. Some of her ideas assisted the author in developing some useful numerical schemes to evaluate the quantum capacitance of MLG and BLG, and these were employed to check all the numerical results in this article.

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Authors and Affiliations

  1. Rensselaer Polytechnic Institute, Troy, NY, USA

    Jerry P. Selvaggi

  2. SUNY New Paltz, New Paltz, NY, USA

    Jerry P. Selvaggi

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  1. Jerry P. Selvaggi
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Correspondence to Jerry P. Selvaggi.

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Selvaggi, J.P. A general analytical method for finding the quantum capacitance of graphene. J Comput Electron 17, 1268–1275 (2018). https://doi.org/10.1007/s10825-018-1202-0

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