Spectral gaps of Dirac operators describing graphene quantum dots

TL;DR

This paper analyzes the spectral gaps of Dirac operators with various boundary conditions in graphene quantum dots, establishing a lower bound proportional to the inverse square root of the domain size, relevant for physical models.

Contribution

It provides a rigorous lower bound on the spectral gap for a family of boundary conditions in the Dirac operator modeling graphene quantum dots.

Findings

Lower bound on spectral gap proportional to |7|^{-1/2}

Includes infinite mass and armchair boundary conditions

Results applicable to physical models of 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.