Aharonov–Bohm flux and dual gaps effects on energy levels in graphene magnetic quantum dots

We address the question of how the Aharonov–Bohm flux ${\Phi }_{AB}$ can affect the energy levels of graphene magnetic quantum dots (GMQDs) of radius $R$. To answer this question, we consider GMQDs induced by a magnetic field $B$ and subjected to two different gaps — an internal gap ${\Delta }_1$ and an external gap ${\Delta }_2$. After determining the eigenspinors and ensuring continuity at the boundary of the GMQDs, we formulate an analytical equation describing the corresponding energy levels. Our results show that the energy levels can exhibit either a symmetric or an asymmetric behavior depending on the valleys $K(\tau =1)$ and $K^{\prime }(\tau =-1)$ together with the quantum angular momentum $m$. In addition, we find that ${\Phi }_{AB}$ causes an increase in the band gap width when ${\Delta }_1$ is present inside the GMQDs. This effect is less significant when a gap is present outside, resulting in a longer lifetime of the confined electronic states. Further increases in ${\Phi }_{AB}$ reduce the number of levels between the conduction and valence bands, thereby increasing the band gap. These results demonstrate that the electronic properties of graphene can be tuned by the presence of the AB flux, offering the potential to control the behavior of graphene-based quantum devices.