Quasi-Bound States for Circular Bilayer-Graphene Quantum Dots: Subtle Detail of Real and Complex-Valued Analysis

Graphene quantum dots (GQDs) are assumed to be a perspective systems for quantum computing. The simplest one can be considered as a circular dot described by two-dimensional Dirac-like equation within the known graphene model of massless pseudo-Dirac fermions. The system admits an analytical treatment with a stair-case confining potential. The existence of quasi-bound states for such systems has been declared in a series of papers, an opposite point of view has been proposed by us in [H. Grushevskaya & G. Krylov J NPCS,25 21(2022)]. Detailed analysis of eigenfunctions and possibility of quasi-bound states for monolayer and bilayer graphene circular quantum dots with a step like potential is the goal of the current research. It has been demonstrate that anomalous density of states emerge in bilayer graphene cases for the real energies in the vicinity of the point of vanishing determinant of the linear algebraic system for matching eigenfunctions and their derivative at the boundary of the quantum dot. The effect is stipulated by the significant contribution of the Bessel Km function at the distance up to several dot's radii.