Robustness of hybrid monolayer-bilayer graphene quantum dot energy spectra to realistic edge terminations, dot shape, and atomistic boundary structure

A 2016 continuum-model study (Mirzakhani, Zarenia, Ketabi, da Costa, and Peeters, Phys. Rev. B 93, 165410) predicted the discrete energy spectra of circular hybrid quantum dots in which a single-layer graphene (SLG) region and a bilayer graphene (BLG) region of the same sheet meet at a lateral interface, imposing zigzag boundary conditions at the interface in order to obtain analytical results. The authors noted that a real dot cut from bilayer graphene presents both zigzag and armchair edge segments, so the sensitivity of the predicted bound and interface-localized states to realistic mixed edge terminations, noncircular dot shapes, and atomistic lattice-scale boundary physics was left undetermined. This problem asks for that determination: an atomistic-level characterization of which predicted features (existence of bound states, level energies, valley selection rules, zero-energy edge-state structure, and spatial localization) survive realistic edge and shape conditions, together with a benchmark against the original continuum predictions. Follow-up work has treated ideal zigzag or ideal armchair terminations of hexagonal hybrid dots and one-dimensional junctions, but no identified study addresses mixed edge terminations within one dot, edge disorder or reconstruction, a direct continuum-versus-atomistic comparison for the same geometry, or experimental detection; the core question remains open.

Background

Graphene at low energy is described by a two-dimensional Dirac-Weyl equation rather than an ordinary Schrödinger equation: in single-layer graphene (SLG) the electronic bands form cones that touch at two inequivalent points of the hexagonal Brillouin zone, called valleys and labeled $K$ and $K'$, so carriers behave as massless Dirac fermions. Bilayer graphene (BLG) consists of two graphene sheets in the AB (Bernal) stacking arrangement, in which half of the carbon atoms of one layer sit directly above atoms of the layer below; its low-energy bands are parabolic, and unlike SLG it acquires a tunable band gap when a perpendicular electric field (an interlayer bias potential) is applied. Because massless Dirac fermions can tunnel through smooth electrostatic barriers (Klein tunneling), confining carriers in SLG generally requires real boundaries or magnetic fields. Graphene quantum dots (QDs) — nanoscale islands with discrete carrier spectra — are therefore often made by cutting flakes, and their spectra depend strongly on the dot's shape and on the atomic termination of its boundary. The honeycomb lattice has two canonical edge terminations, zigzag (outermost atoms on one sublattice) and armchair (boundary alternating between sublattices): zigzag edges support zero-energy surface states, whereas armchair edges mix the two valleys and open a gap. In continuum calculations the edge enters as a boundary condition on the Dirac spinor; the zigzag boundary condition sets one spinor component to zero at the boundary, while so-called infinite-mass conditions simulate a large mass outside the dot. In a perpendicular magnetic field $B$, extended graphene instead shows discrete Landau levels (LLs), which differ for SLG (massless) and BLG (massive) carriers.

Exfoliated graphite samples routinely contain islands of different layer number, so SLG and BLG regions meet at lateral monolayer-bilayer junctions whose interface states measurably affect transport; ideal one-dimensional SLG-BLG interfaces had been analyzed before 2016 for both zigzag and armchair interface matching. The source paper asked whether such a junction can itself confine carriers in a QD-like structure, realizable by (accidental) nanostructuring of one layer of a bilayer sheet. It considered two circular hybrids: a circular SLG dot embedded in an infinite BLG sheet, and a circular BLG dot embedded in infinite SLG. Solving the Dirac-Weyl equation in both regions with zigzag boundary matching at the circular interface, it obtained analytical energy levels and wave spinors, treating both valleys separately because the interface breaks inversion symmetry, and studied the effects of a perpendicular magnetic field and an interlayer bias. Its central predictions are: at $B=0$ without bias, neither hybrid geometry supports bound states; under bias, the SLG-dot-in-BLG geometry develops bound states and an electron-hole gap that closes as the dot radius grows; zero-energy states appear with a geometry-dependent valley selection rule (for the SLG-dot-in-BLG geometry, only for angular momentum $m \ge 0$ in the $K$ valley and $m \le 0$ in the $K'$ valley, while for the BLG-dot-in-SLG geometry zero-energy states occur for all angular momenta in the $K'$ valley and only for one sign of $m$ in the $K$ valley); the $K$-valley spectrum obeys the symmetry $E^K(m)=E^K(-m)$, which the $K'$ valley violates; interface states appear near the band edges of the Mexican-hat-shaped low-energy dispersion of biased BLG (the energy range where the biased-BLG dispersion turns around, at energies near $\pm g/2$ for gap $g$), predominantly confined along the zigzag edge of the second layer; and in a magnetic field the spectra show anticrossings between dot-, interface-, and layer-localized states and cross over to SLG or BLG Landau-level sequences as the dot radius or field changes. The authors proposed scanning tunneling microscopy (STM), which measures the local density of states (LDOS), as the experimental probe of these levels.

The open question arises from the idealizations behind these analytical results, which the authors state explicitly: the circular geometry was adopted for convenience to allow analytical results, and the zigzag boundary condition was imposed at the SLG-BLG junction to obtain analytical expressions, even though — as they note — a circular dot actually cut out of BLG would present both armchair and zigzag edge segments. A real hybrid dot is therefore generally noncircular, has an interface running through a mixture of zigzag and armchair segments, and at the lattice scale involves edge disorder, reconstruction, and atomic relaxations that the continuum zigzag matching does not represent. Which of the analytical predictions survive these realistic conditions determines whether the predicted spectra can be used to interpret measurements on real samples, and no systematic treatment of this robustness question had been given. A concurrent companion study by the same groups (da Costa et al., Phys. Rev. B 94, 035415 (2016)) treated ideal zigzag and ideal armchair terminations of hexagonal hybrid dots atomistically, informing — but not settling — parts of the question.

Problem Statement

Determine how the analytical predictions of Mirzakhani et al. (Phys. Rev. B 93, 165410 (2016)) for circular hybrid monolayer-bilayer graphene quantum dots — obtained from the continuum Dirac-Weyl equation with zigzag boundary conditions imposed at the SLG-BLG interface — are modified when the idealizations of that treatment are lifted. Specifically, characterize the bound and interface-localized states of hybrid dots in both geometries treated by the source (an SLG dot surrounded by BLG, and a BLG dot surrounded by SLG) when (i) the SLG-BLG interface termination mixes zigzag and armchair segments, as a real dot cut from a bilayer sheet would present, (ii) the dot shape deviates from circular, and (iii) the boundary is treated at the atomistic lattice scale, including edge disorder, reconstruction, or relaxation beyond the continuum zigzag matching. The quantities to be determined are the existence of bound states (with and without interlayer bias and perpendicular magnetic field), the discrete energy spectra, the zero-energy edge-state structure and its valley selection rules (in the source treatment, zero-energy states occur only for $m \ge 0$ in the $K$ valley and $m \le 0$ in the $K'$ valley for the SLG-dot-in-BLG geometry, while for the BLG-dot-in-SLG geometry they occur for all angular momenta in the $K'$ valley and only for one sign of $m$ in the $K$ valley; and the $K$-valley symmetry $E^K(m)=E^K(-m)$), the bias-induced electron-hole gap and its radius dependence, the interface states near the biased-BLG Mexican-hat band edges, the magnetic-field anticrossings and Landau-level crossovers, and the spatial localization of the eigenstates. The overall scientific objective is a robustness assessment of these predictions — a determination of which features survive, which are destroyed, and which are qualitatively modified under realistic edge terminations, shapes, and atomistic boundary structure — including a direct comparison with the original continuum circular zigzag-condition results for matched geometry and parameters. All requested results jointly support this one objective: establishing the domain of validity of the analytical hybrid-dot theory for real samples. The three lifts (i)–(iii) and all of the quantities listed above, for both geometries, are jointly posed parts of this single robustness assessment.

References

1. M. Mirzakhani, M. Zarenia, S. A. Ketabi, D. R. da Costa, and F. M. Peeters, 'Energy levels of hybrid monolayer-bilayer graphene quantum dots', Phys. Rev. B 93, 165410 (2016). DOI: 10.1103/PhysRevB.93.165410, https://doi.org/10.1103/PhysRevB.93.165410
2. D. R. da Costa, M. Zarenia, Andrey Chaves, J. M. Pereira, Jr., G. A. Farias, and F. M. Peeters, 'Hexagonal-shaped monolayer-bilayer quantum disks in graphene: A tight-binding approach', Phys. Rev. B 94, 035415 (2016). DOI: 10.1103/PhysRevB.94.035415, https://doi.org/10.1103/PhysRevB.94.035415
3. M. Mirzakhani, M. Zarenia, and F. M. Peeters, 'Edge states in gated bilayer-monolayer graphene ribbons and bilayer domain walls', J. Appl. Phys. 123, 204301 (2018). DOI: 10.1063/1.5025937, https://doi.org/10.1063/1.5025937
4. M. Mirzakhani, D. R. da Costa, and F. M. Peeters, 'Isolated and hybrid bilayer graphene quantum rings', Phys. Rev. B 105, 115430 (2022). DOI: 10.1103/PhysRevB.105.115430, https://doi.org/10.1103/PhysRevB.105.115430 (preprint arXiv:2106.00947)
5. Bouchaib Lemaalem, Youness Zahidi, and Ahmed Jellal, 'Band structures of hybrid graphene quantum dots with magnetic flux', Phys. Lett. A 426, 127898 (2022). DOI: 10.1016/j.physleta.2021.127898, https://doi.org/10.1016/j.physleta.2021.127898 (preprint arXiv:2107.14568)
6. Abdelhadi Belouad, Abdellatif Kamal, El Bouâzzaoui Choubabi, Rachid Houça, and Mohammed El Bouziani, 'Energy levels of monolayer–AA-stacked bilayer graphene quantum dots', arXiv:2107.14366 (2021). arXiv ID: 2107.14366, https://arxiv.org/abs/2107.14366
7. N. S. Bandeira, Andrey Chaves, L. V. de Castro, R. N. Costa Filho, M. Mirzakhani, F. M. Peeters, and D. R. da Costa, 'Energy levels and Aharonov-Bohm oscillations in twisted bilayer graphene quantum dots and rings', Phys. Rev. B 111, 125409 (2025). DOI: 10.1103/PhysRevB.111.125409, https://doi.org/10.1103/PhysRevB.111.125409 (preprint arXiv:2410.12014)
8. Włodzimierz Jaskólski, 'Topological Electric Field-Defined Quantum Dots in Bilayer Graphene: An Atomistic Approach', Materials 19, 2790 (2026). DOI: 10.3390/ma19132790, https://doi.org/10.3390/ma19132790 (preprint arXiv:2603.13055)
9. Zhehao Ge, Dillon Wong, Juwon Lee, Frederic Joucken, Eberth A. Quezada-Lopez, Salman Kahn, Hsin-Zon Tsai, Takashi Taniguchi, Kenji Watanabe, Feng Wang, Alex Zettl, Michael F. Crommie, and Jairo Velasco, 'Imaging Quantum Interference in Stadium-Shaped Monolayer and Bilayer Graphene Quantum Dots', Nano Lett. 21, 8993-8998 (2021). DOI: 10.1021/acs.nanolett.1c02271, https://doi.org/10.1021/acs.nanolett.1c02271
10. Xiang Liu, Zheng Tao, Wenchen Luo, and Tapash Chakraborty, 'Correlated quantum phenomena in confined two-dimensional hexagonal crystals', Physica E 182, 116575 (2026). DOI: 10.1016/j.physe.2026.116575, https://doi.org/10.1016/j.physe.2026.116575 (preprint arXiv:2603.16677)