Recently, twist bilayer van der Waals heterostructures have been the subject of substantial theoretical and experimental works due to many fascinating electrical, optical, and magnetic properties, such as unconventional superconductivity, ferroelectricity, and correlated insulator behavior for rotation angle between layers of order θ ∼ 1◦. [1, 2, 3] Moreover, quantum dots (QDs) in bilayer graphene (BLG) are a promising quantum information platform because of their long spin decoherence times, high sample quality, and tunability, whereas quantum rings (QRs) are the most natural systems to investigate quantum interference phenomenon in transport properties, Aharonov–Bohm oscillations and persistent currents. Within the context of moiré superlattice and quantum confinement systems, [4] in this talk, we present a systematic study of the energy levels of twisted BLG QDs and QRs, both in the absence and presence of an external perpendicular magnetic field. Results are obtained within the tight-binding model, with
interlayer hopping parameters defined by the Slater-Koster form, which takes into account the distance between the lattice points, which is fundamental for obtaining the Hamiltonian for inter-layer twisted systems. The confinement structures are modeled by a circular dotlike and ringlike-shape site-dependent staggered potential, which prevents edge effects. Due to a non-zero interlayer twist angle, the energy spectra exhibit features resulting from the interplay between characteristics of the AA and AB stacking orders that compose the moiré pattern of such twisted bilayer. Our findings show that, in the absence of a magnetic field, the energy levels of the QR scale with its width W according to a power law W^{−α}, whose exponent 1 ⪅ α ⪅ 2 depends on the twist angle. Moreover, assuming the so-called magic angle (θ = 1.08◦) for the interlayer twist, the lowest energy state oscillates as a function of the average radius of the ring, as a consequence of the different distributions of AA and AB stacking regions for each value of radius. In the presence of a perpendicular magnetic field, two sets of energy levels, which approach the Landau levels of infinite AA-staked and AB-staked BLG sheets, are observed, from which a variety of crossings between energy states emerges. Interestingly, these sets of energy states exhibit periodic (Aharonov-Bohm) oscillations as a function of the magnetic field, even for a QD, which reveals information about the moiré pattern of AA and AB stacked regions covered by the ring area.

[1] Y. Cao, V. Fatemi, S. Fang, K. Watanabe, T. Taniguchi, E. Kaxiras, and P. Jarillo-Herrero,
Nature 556(7699), 43 (2018).
[2] M. Yankowitz, S. Chen, H. Polshyn, Y. Zhang, K. Watanabe, T. Taniguchi, D. Graf, A. F.
Young, and C. R. Dean, Science 363(6431), 1059 (2019).
[3] Y. Cao, V. Fatemi, A. Demir, S. Fang, S. L. Tomarken, J. Y. Luo, J. D. Sanchez-Yamagishi,
K. Watanabe, T. Taniguchi, E. Kaxiras, et al, Nature 556(7699), 80 (2018).
[4] M. Mirzakhani, F. M. Peeters, and M. Zarenia. Physical Review B 101(7), 075413 (2020).