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Spectral Gaps of Dirac Operators Describing Graphene Quantum Dots

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Abstract

The two-dimensional Dirac operator describes low-energy excitations in graphene. Different choices for the boundary conditions give rise to qualitative differences in the spectrum of the resulting operator. For a family of boundary conditions, we find a lower bound to the spectral gap around zero, proportional to |Ω|−1/2, where \({\Omega } \subset \mathbb {R}^{2}\) is the bounded region where the Dirac operator acts. This family contains the so-called infinite mass and armchair cases used in the physics literature for the description of graphene quantum dots.

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Acknowledgments

This work has been supported by the Iniciativa Científica Milenio (Chile) through the Millenium Nucleus RC–120002 “Física Matemática”. R.B. has been supported by Fondecyt (Chile) Projects # 112–0836, # 114–1155 and # 116–0856. S.F. acknowledges partial support from a Sapere Aude grant from the Danish Councils for Independent Research, Grant number DFF–4181-00221. E.S has been partially funded by Fondecyt (Chile) project # 114–1008. H. VDB. acknowledges support from Conicyt (Chile) through CONICYT–PCHA/Doctorado Nacional/2014. This work was carried out while S.F. was invited professor at Pontificia Universidad Católica de Chile.

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Authors and Affiliations

  1. Instituto de Física, Pontificia Universidad Católica de Chile, Vicuña Mackenna 4860, Santiago, 7820436, Chile

    Rafael D. Benguria, Edgardo Stockmeyer & Hanne Van Den Bosch

  2. Department of Mathematics, Aarhus University, Ny Munkegade 118, DK-8000, Aarhus, Denmark

    Søren Fournais

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  1. Rafael D. Benguria
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  2. Søren Fournais
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  3. Edgardo Stockmeyer
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  4. Hanne Van Den Bosch
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Correspondence to Hanne Van Den Bosch.

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Benguria, R.D., Fournais, S., Stockmeyer, E. et al. Spectral Gaps of Dirac Operators Describing Graphene Quantum Dots. Math Phys Anal Geom 20, 11 (2017). https://doi.org/10.1007/s11040-017-9242-4

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