Spectral Gaps of Dirac Operators Describing Graphene Quantum Dots

  • Rafael D. Benguria
  • Søren Fournais
  • Edgardo Stockmeyer
  • Hanne Van Den BoschEmail author


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.


Dirac operator Spectral gap Graphene flakes Infinite mass boundary conditions Armchair boundary conditions 

Mathematics Subject Classification (2010)

35 P xx 81 Q 05 81 Q 10 



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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Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.Instituto de FísicaPontificia Universidad Católica de ChileSantiagoChile
  2. 2.Department of MathematicsAarhus UniversityAarhusDenmark

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