Energy levels and Aharonov-Bohm oscillations in twisted bilayer graphene quantum dots and rings

We present a systematic study of the energy levels of twisted bilayer graphene (tBLG) quantum dots (QDs) and rings (QRs) under an external perpendicular magnetic field. The confinement structures are modeled by a circular dot-like- and ring-like-shaped site-dependent staggered potential, which prevents edge effects and leads to an energy gap between the electron and hole states. Results are obtained within the tight-binding model with interlayer hopping parameters defined by the Slater-Koster form for different interlayer twist angles $\ensuremath{\theta}$. Our findings show that, for $\ensuremath{\theta}$ around ${0}^{\ensuremath{\circ}}$ or ${60}^{\ensuremath{\circ}}$, the energy spectra exhibit features resulting from the interplay between characteristics of the AA and AB/BA stacking orders that compose the moir\'e pattern of such tBLG, while the low-energy levels are shown to be nearly independent on the rotation angle for ${10}^{\ensuremath{\circ}}\ensuremath{\lesssim}\ensuremath{\theta}\ensuremath{\lesssim}{50}^{\ensuremath{\circ}}$. In the absence of a magnetic field, the energy levels of the QR scale with its width $W$ according to a power law ${W}^{\ensuremath{-}\ensuremath{\alpha}}$, whose exponent $1⪅\ensuremath{\alpha}⪅2$ depends on the twist angle. Most interestingly, the lowest-energy states of tBLG QRs oscillate as a function of its average radius, with the oscillation period matching half of the moir\'e period. In the presence of an intense magnetic field, the lowest-energy levels for the tBLG QDs and QRs match almost perfectly, regardless of whether the external radius of the quantum confinement structure is smaller or on the order of the moir\'e period, which is due to the interplay of the trigonal warping effect and the preferential localization of the eigenstates. Our results reveal relevant information about the moir\'e pattern in tBLG and its role in charge particle confinement.