Twistronics 2023: International Workshop on twisted bilayer graphene and beyond

  The Moiré pattern of the magic-angle twisted bilayers graphene and twisted bilayer MoS₂ leads to localization of the low energy electrons in the AA-stacking regions, reflected by very flat bands at low energy.^[1],[2] This reduction of the kinetic energy enhances the relative importance of interactions and thus renders the bilayer systems much more susceptible to correlation effects, as show experimentally by the discovery of correlated insulators and superconductivity.^[3] Despite numerous theoretical and experimental studies, the understanding of this new electronic localization is still incomplete. The rotation angle is of course a key parameter, but we have also shown that a small expansion or contraction of one layer with respect to the other (“heterostrain”) can strongly modify the electronic structure of flat bands.^[4]

  Here we present theoretical study of the electronic structure and quantum scattering properties of charge carriers with electronic flat bands, taking into account as well as possible the structural parameters that condition them (rotation angle, bias voltage,^[5] heterostrain and/or local defects^[6]). We also investigate the magnetic instabilities using a combination of real-space Hartree-Fock and dynamical mean-field theories, starting from a tight-binding description of the non-interacting bilayer systems to which we add a local Hubbard interaction U in order to model the Coulomb repulsion between electron.^[7] We find that localized magnetic states emerge for values of the Coulomb interaction U that is significantly smaller than what would be required to render an isolated layer antiferromagnetic. We also show how heterostrain strongly modifies the magnetization and the local magnetic order for realistic values of U.

*Ref
 ^[1] R. Bistritzer, A.H. MacDonald, Proc. Natl. Acad. Sci. 108, 12233 (2011).
 ^[2] G. Trambly de Laissardière, D. Mayou, L. Magaud, Nano Lett. 10, 804 (2010). S. Venkateswarlu, A. Honecker, G. Trambly de Laissardière, Phys. Rev. B 102, 081103(R) (2020).
 ^[3] Y. Cao et al., Nature 556, 43 (2018); Nature 556, 80 (2018).
 ^[4] L. Huder et al., Phys. Rev. Lett. 120, 156405 (2018).
      F. Mesple et al., Phys. Rev. Lett. 127, 126405 (2021).
 ^[5] G. Trambly de Laissardière et al., Phys. Rev. B 93, 235135 (2016).
 ^[6] O. F. Namarvar et al., Phys. Rev. B 101, 245407 (2020).
 ^[7] V. Vahedi et al., SciPost Phys. 11, 083 (2021).

  Twisted van der Waals materials have risen as a powerful platform to engineer artificial quantum matter. Artificial moire heterostructures, in general, display two length scales, the original lattice constant and the emergent moire length. In particular, atomic defects in two-dimensional materials, including substitutional elements and vacancies, have a relevant length scale stemming from the microscopic lattice constant and, therefore, can give rise to a rich interplay with the moire length. Here we will address the impact of atomic defects in twisted graphene bilayers^[1,2], twisted graphene trilayers^[3], and artificial moire topological superconductors^[4]. First^[1,2], we show that local defects in twisted bilayer graphene allow realizing triple point fermions and that, in the presence of interactions, give rise to in-gap excitations in a superconducting moire state. Second^[3], by combining first-principles calculations and low-energy models, we show the impact of local impurities in the flat bands of twisted graphene trilayers and, in particular, the disruption of bulk valley currents. Third^[4], we demonstrate the critical dependence of the topological state on the interplay between atomic defects and the moire pattern in the moire topological superconductor NbSe₂/CrBr₃. These results highlight the key role of impurities in twisted van der Waals materials, revealing the key interplay between length and energy scales in artificial moire systems.

*Ref
 ^[1] Aline Ramires and Jose L. Lado, Phys. Rev. B 99, 245118 (2019)
 ^[2] Alejandro Lopez-Bezanilla and J. L. Lado, Phys. Rev. Materials 3, 084003 (2019)
 ^[3] Alejandro Lopez-Bezanilla and J. L. Lado, Phys. Rev. Research 2, 033357 (2020)
 ^[4] Maryam Khosravian and Jose L. Lado, Phys. Rev. Materials 6, 094010 (2022)

  We investigate electronic structures and electron-phonon interaction in twisted graphene layers based on atomistic calculations^[1-3]. We show electron-phonon coupling strength λ is dramatically different among twisted graphene layers. The total strength λ is very large for magic-angle twisted bilayer graphene and magic-angle twisted trilayer graphene, both of which display robust superconductivity in experiments. However, λ is an order of magnitude smaller in twisted double bilayer graphene (TDBG) and twisted monolayer-bilayer graphene (TMBG) where superconductivity is reportedly rather weak or absent. We find the Bernal-stacked layers in TDBG and TMBG induce sublattice polarization of electronic states, suppressing electron-phonon interaction. Our results suggest that the electron-phonon coupling may play an important role in the superconductivity of twisted graphene layers. This work was supported by NRF of Korea (Grants No. 2020R1A2C3013673 and No. 2017R1A5A1014862).

*Ref:
 ^[1] Y. W. Choi and H. J. Choi, Phys. Rev. B 98, 241412 (2018). [arXiv:1809.08407]
 ^[1] Y. W. Choi and H. J. Choi, Phys. Rev. B 100, 201402 (2019). [arXiv:1903.00852]
 ^[3] Y. W. Choi and H. J. Choi, Phys. Rev. Lett. 127, 167001 (2021). [arXiv:2103.16132]

  The advent of 2D moiré heterostructures, starting with the discovery of flat bands and strong correlations in magically twisted bilayer graphene (TBLG), has shown that there is still a lot of room for uncovering exciting new physics when the layer, twist and strain degrees of freedom are explored. From a theoretical and computational perspective, the 2D moiré heterostructures have in common an increasingly large supercell, with as many as 104 atoms per supercell, leading to increasing difficulties in accurately modeling them. In this case, ab-initio methods are beyond the reach of current computing resources, while continuum effective models might not be valid anymore. In the first part of the presentation I will introduce two tight-binding open-source softwares, Pybinding ^[1] and Quantum-Kite ^[2], that can be deployed in the description of moiré systems. These are based on spectral expansions of the Green’s functions and allow for the description of opto-electronic properties of moiré systems in the presence of disorder, vacancies, magnetic field, with possibilities to model systems with as many as 10¹⁰ degrees of freedom.

  In the second part of the presentation I will exemplify the use of the methods and softwares for the description of electronic properties of hBN encapsulated multilayer graphene. Recent experimental and theoretical works [3] have shown that the combination of marginally aligned graphene/hBN heterostructures will lead to secondary Dirac points, mini-bands and gaps in the spectrum ^[4]. In this presentation we will instead explore the electronic properties of encapsulated graphene multilayer configurations, where the hBN and graphene layers are aligned, paying special attention to the role of precise stacking of the encapsulating hBN layers ^[5]. We find that for selected moiré-stacking configurations an robust gap at the secondary Dirac point, in combination with an applied perpendicular electric field gives rise in encapsulated bernal stacked bilayer graphene to flat bands with bandwidth below 10meV without the need of interlayer twisting. Similar to TBLG, attainable bandwidths are below energy scales related to the Coulomb interaction and thus strongly-correlated electronic states are expected.

*Ref:
 ^[1] Pybinding website: https://docs.pybinding.site/en/stable/
 ^[1] Quantum-Kite website: https://quantum-kite.com/
 ^[3] Z. Wang, Y. B. Wang, J. Yin, E. Tóvári, Y. Yang, L. Lin, M. Holwill, J. Birkbeck, D. J. Perello, Shuigang Xu, J. Zultak, R. V. Gorbachev, A. V. Kretinin, T. Taniguchi, K. Watanabe, S. V. Morozov, M. Anđelković, S. P. Milovanović, L. Covaci, F.M. Peeters, A. Mishchenko, A. K. Geim, K. S. Novoselov, Vladimir I. Fal’ko, Angelika Knothe, C. R. Woods, Science Advances 12, eaay8897 (2019)
 ^[4] M. Anđelković, S. P. Milovanović, L. Covaci, F. M. Peeters, Nano Lett. 20, 979 (2020)
 ^[5] R. Smeyers, L. Covaci, M. V. Milošević, arXiv:2211.16351 (2022)

  Recently, moiré superlattices (MS) from twisted bilayers of transition metal dichalcogenides (TB-TMDs) have gained great interest as a novel and robust platform for simulating quantum phases of matter on emergent 2D lattices^[1-8]. Unlike in twisted bilayer graphene, in TB-TMDs different sets of flat bands can be probed, depending on carrier type and density, chemical composition, twist angle and external fields. These flat bands can derive from states at the Γ point or at Κ/Κ' points in the Brillouin zone of the constituent monolayers, and the corresponding localised Wannier functions sit on sites of different lattices in real space, e.g., honeycomb (for Γ‐derived) or triangular (Κ/Κ'‐derived)^[9]. At small angles the moiré patterns exhibit periods of the order of a few nanometres, and the corresponding moiré unit cells can contain more than 10,000 atoms. From a computational perspective, accessing electronic structure properties of these small-angle systems is challenging because of the unfavourable scaling of standard first-principles techniques with system size. Here, I will present a novel ab initio tight binding method for twisted multilayers TMDs which enables to accurately describe the electronic structure of these systems taking into account atomic relaxation, chemical composition, spin-orbit coupling and external fields at a significantly reduced computational cost^[9]. Finally, I will present first-principles and classical Montecarlo calculations for a trilayer system, namely twisted MoSe₂ on a 2H-WSe₂ bilayer, in which both Γ‐derived and Κ/Κ'‐derived flat bands can be accessed by applying an external electric field. In this set-up it is possible to engineer correlated insulating states on different emerging lattices and study their interactions.

*Ref:
 ^[1] Wang L. et al., Nat. Mater. 19, 861-866 (2020)
 ^[2] Ghiotto A. et al., Nature 597, 345-349 (2021)
 ^[3] Regan E. C. et al, Nature 579, 359-363 (2020)
 ^[4] Xu Y. et al., Nature 587, 214-218 (2020)
 ^[5] Huang X. et al., Nat. Phys.17, 715-719 (2021)
 ^[6] Tang Y. et al., Nature 579, 535-538 (2020)
 ^[7] Kennes D. et al., Nat. Phys. 17, 155-163 (2021)
 ^[8] Li, T., Jiang, S., Shen, B. et al., Nature 600, 641-646 (2021)
 ^[9] Vitale et al., 2D mater. 8, 045010 (2021)