Spectral gaps of Dirac operators describing graphene quantum dots

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 $|\Omega {|}^{-1/2}$, 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.

final version, improved introduction on boundary conditions in physics literature, self-adjointness-result deferred to other paper, references added

References in corpus (8)

• The electronic properties of graphene
• Substrate-induced band gap opening in epitaxial graphene
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• Electronic States of Graphene Nanoribbons
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Cited by in corpus (13)

• On the MIT bag model: self-adjointness and non-relativistic limit
• Self-adjoint Dirac operators on domains in ${\mathbb{R}}^3$
• Resolvent convergence to Dirac operators on planar domains
• A sharp upper bound on the spectral gap for graphene quantum dots
• A variational formulation for Dirac operators in bounded domains. Applications to spectral geometric inequalities
• Multiple solutions for a self-consistent Dirac equation in two dimensions
• Spectral optimisation of Dirac rectangles
• Eigenvalue curves for generalized MIT bag models
• Boundary problems for three-dimensional Dirac operators and generalized MIT bag models for unbounded domains
• Spectral inequality for Dirac right triangles
• Convergence of generalized MIT bag models to Dirac operators with zigzag boundary conditions
• Faber-Krahn inequalities for Schrödinger operators with point and with Coulomb interactions
• A connection between quantum dot Dirac operators and $\overline{\partial }$-Robin Laplacians in the context of shape optimization problems