# Twistronics 2023 International Workshop

on twisted bilayer graphene and beyond

**University of Seoul, Seoul, Korea, January 11 ^{th} ~ 13^{th}, 2023**

**• Philip Kim**(Harvard University, USA)

**Title: Experiments in Twisted van der Waals Interface of 2D Materials**[Video] [Slides]

## Abstract

Engineering moire superlattices by twisting and stacking two layers of Van der Waals materials has proved to be an effective way to promote interaction effects and induce exotic phases of matter. In this presentation, I will discuss several research topics related experimental investigations, including: (i) Correlated Electron States in Twisted Multilayer Graphene; (ii) Topological Domain Anti-ferroelectricity in Twisted Bilayer Transition Metal Dichalcogenides and (iii) Novel interfacial superconductivity based on twisted van der Waals junction in high temperature superconductors.

**Philip Kim** is Professor of Physics and Professor Applied Physics at Harvard University. Professor Kim is a world leading scientist in the area of materials research. His research area is experimental condensed matter physics with an emphasis on physical properties and applications of nanoscale low-dimensional materials. The focus of Prof. Kim’s group research is the mesoscopic investigation of transport phenomena, particularly, electric, thermal and thermoelectrical properties of low dimensional nanoscale materials.

**• Francisco (Paco) Guinea**(IMDEA, Spain)

**Title: Electrons, strains, and phonons in twisted and untwisted graphene**[Video] [Slides]

## Abstract

The role on the phase diagram of twisted bilayer graphene of electron-electron interactions and charge fluctuations is discussed. The different ways in which these interactions can lead to polarized phases, and to superconductivity is highlighted. The changes in the electronic structure induced by interactions, substrate effects, strains, and stacking arrangements will also be reviewed

** Francisco Guinea López **obtained his BSc (1975) in Physics from the Universidad Complutense de Madrid, and the Phd at the Universidad Autónoma de Madrid (1980) . He obtained a Fullbright Fellowship and worked at the University of California, Santa Barbara, during the years 1982-1984. He became Assistant Professor at the Universidad Autónoma de Madrid in 1985, and Senior Researcher at the Consejo Superior de Investigaciones Científicas in 1987. He has been visiting Professor at the University of Michigan, 1991-1992, and visiting Researcher at the University of California San Diego, 1997, and Boston University, 2004-2005. He has stayed for shorter periods at a number of institutions worldwide, like IBM Rüschlikon, Kernforschunganlage Jülich, DIPC, San Sebastián, ICTP, Trieste, ENS, Par ́s, and many more. He joined Imdea Nanoscience in January 2005.

F. G. has published over 400 scientific papers, with an h-index of 75 and more than 50 papers with over 100 citations. He has received a number of awards, including the biannual National Prize for Physics (Spain), and the Gold Medal of the Spanish Physical Society.

**• Guy Trambly de Laissardier**(Laboratoire de physique théorique et modélisation, France)

**Title: Electronic, quantum transport and magnetic properties in twisted bilayer graphene**[Video]

## Abstract

The Moiré pattern of the magic-angle twisted bilayers graphene and twisted bilayer MoS_{2} 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).

**Guy Trambly de Laissardière **is an associate professor in condensed matter physics theory at Laboratoire de Physique Théorique et Modélisation (LPTM), CY Cergy Paris Université / CNRS, Cergy-Pontoise, France. He received his PhD in 1996 at the University J. Fourier (Grenoble) in the group of Didier Mayou with a thesis on the electronic properties of quasiperiodic materials. His research activity mainly deals with the electronic confinement, magnetism and quantum transport in graphene, related 2D materials, twisted bilayer and quasiperiodic tilings.

**• Jose Lado**(Aalto University, Finland)

**Title: Atomic defects in topological and correlated twisted van der Waals heterostructures**[Video]

## Abstract

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 NbSe_{2}/CrBr_{3}.
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)

**Jose Lado** is an assistant professor in theoretical physics at Aalto University, in Finland, since 2019. He was an ETH Fellow at the Institute for Theoretical Physics at ETH Zurich, with Prof. Manfred Sigrist and Prof. Oded ZIlberberg from 2017-2019. He got his Ph.D. between 2013-2016 working in the Theory of Nanostructures group at INL, Portugal, led by Prof. Joaquin Fernandez Rossier. His research focuses on the theory of emergent phenomena in topological and correlated quantum materials. In particular, he focus on engineering systems where electronic correlations and topology yield exotic physics such as symmetry broken states, topological excitations and ultimately emerging fractionalized particles. Apart from those purely theoretical research lines, He often work in collaboration with experimental groups studying quantum materials in general, and two-dimensional materials in particular.

**• Oded Hod**(Tel Aviv University, Israel)

**Title: Layered Ferroelectricity: from Geometric Measures to First-Principles Calculations**[Video]

## Abstract

The ability to locally switch a confined electric polarization is vital in modern technologies, aiming to complement or replace traditional magnetic components toward storing, retrieving, and processing large volumes of information. Non-centrosymmetrically stacked layered materials have recently emerged as promising candidates for vertical polarization switching via lateral interlayer shifts – a mechanism known as Slidetronics. In this presentation, I will elucidate the microscopic origins of polarization in layered materials; demonstrate that it is dictated by interlayer registry; explain its cumulative nature and its saturation behavior; show that it can emerge also in non-polar systems, such as graphitic interfaces; and extend the notion of slidetronics to the quasi-one-dimensional case of facetted nanotubes.

**Oded Hod** received his B.Sc. from the Hebrew University, Israel, in 1994 and his Ph.D. from Tel-Aviv University, Israel, in 2005. After completing a postdoctoral term at Rice University, USA, he joined Tel Aviv University in 2008. His research involves computational nanomaterials science including electronic structure, mechanical, electromechanical, and tribological properties, density functional theory, molecular electronics, and electron dynamics and thermodynamics in open quantum systems. Prof. Hod published over 90 papers in leading scientific journals including *Science, Nature, Nature Materials, Nature Nanotechnology, Advanced Materials, Nano Letters, Physical Review Letters, Accounts of Chemical Research, Journal of the American Chemical Society,* and *Proceedings of the National Academy of Sciences.* Furthermore, he delivered over 80 invited talks in prominent national and international conferences and co-organized 11 symposia, conferences, and scientific sessions. Prof. Hod holds the Heineman Chair of Physical Chemistry and is an alumnus of the following organizations: the Global Young Academy, where he co-chaired a working group promoting the importance of fundamental science; the Israeli Young Academy, where he chaired a working group for translating a science game for Hebrew and Arabic speaking high school students; and the Lise Meitner–Minerva Center for Computational Quantum Chemistry. Currently, he is a member of the Raymond and Beverly Sackler Center for Computational Molecular and Materials Science and the Ratner Center for Single Molecule Science. He has established the Israeli CECAM node at Tel-Aviv University and served as its director until 2014. He also served as a co-moderator of the chemical physics section of the arXiv. He received Tel-Aviv University’s 2012, 2015, 2017, and 2019 Rector’s Award for Excellence in Teaching and is the recipient of the 2017 Kadar Family Award for Outstanding Research. As a service to the community, Prof. Hod volunteers in the “Bashaar” organization providing annual popular science lectures in Israeli high-schools located at the periphery, and in the “Different Lesson” organization, where he served as a math teaching assistant in a high school science class on a weekly basis.

**• George Jeil Jung**(University of Seoul, Republic of Korea)

**Title: Commensuration torques and lubricity in double moire systems**[Video]

## Abstract

We study the commensuration torques and layer sliding energetics of alternating twist trilayer
graphene (t3G) and twisted bilayer graphene on hexagonal boron nitride (t2G/BN) that have two
superposed moire interfaces. Lattice relaxations for typical graphene twist angles of~ 1° in t3G
or t2G/BN are found to break the out-of-plane layer mirror symmetry, give rise to layer rotation
energy local minima dips of the order of ~ 10^{-1} meV/atom at double moire alignment angles, and have sliding energy landscape minima between top-bottom layers of comparable magnitude. Moire superlubricity is restored for twist angles as small as ~ 0.03° away from alignment resulting in suppression of sliding energies by several orders of magnitude of typically ~10^{-4} meV/atom, hence indicating the precedence of rotation over sliding in the double moire commensuration process. We discuss the potential implications of our results in the preparation of experimental devices with angle aligned double moire patterns with specific top-bottom layer sliding atomic stacking geometries and how this can impact the electronic structure of the commensurate double moire systems considered.

**George J. Jung** has been a professor at the Physics Department of the University of Seoul in Korea since 2015. Prior to this post he has worked as a senior research fellow at the National University of Singapore (2013~2014) and as a research fellow and lecturer at the University of Texas at Austin (2006~2012) after earning his PhD from UNED in Madrid in year 2005. His current research interests are focused on understanding the physical properties of 2D materials heterostructures in search of new materials that can be of potential interest both from a fundamental viewpoint and for exploring application avenues in future technologies.

Google scholar: https://scholar.google.com/citations?user=0-h3qCYAAAAJ&hl=en

**• Hyoung Joon Choi**(Yonsei University, Republic of Korea)

**Title: Electron-phonon coupling and superconductivity in twisted graphene layers**[Video]

## Abstract

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]

**Hyoung Joon Choi’s** research area is computational studies of solids and nanostructures, including electronic structures, superconductivity, magnetism, and nanometer-scale electronic transport. He received Ph. D. in Physics from Seoul National University in 2000, and received the Miller Research Fellowship (2000-2003) of University of California, Berkeley. He joined the faculty of School of Computational Sciences, Korea Institute for Advanced Study in 2003, and joined the faculty of Department of Physics, Yonsei University in 2005.

**• Lucian Covaci**(University of Antwerp, Belgium)

**Title: Large scale tight-binding methods for 2D moiré heterostructures: hBN encapsulated multilayer graphene**[Video] [Slides]

## Abstract

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 10^{10} 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)

**• Valerio Vitale**(University of Trieste, Italy)

**Title: Multiscale modelling of moiré systems: the case of transition metal dichalcogenides**[Video] [Slides]

## Abstract

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 MoSe_{2} on a 2H-WSe_{2} 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)

**Dr Valerio Vitale** is a Marie Skłowdoska-Curie Fellow working in the Department of Physics at the University of Trieste, Italy. He obtained his PhD in Theoretical and Computational Chemistry in 2017 from the University of Southampton and the Institute for Complex Systems Simulation, UK. He then moved as a postdoctoral research associate to the University of Cambridge, in the Theory of Condensed Matter group at the Cavendish Laboratory and later to Imperial College London, in the Materials department. His research interests focus mainly on developing algorithms to study the electronic and optical properties of large-scale moiré systems from first principles.

**• Mitchell Luskin**(University of Minnesota, USA)

**Title: Exact real space formulation and efficient computation of the electronic properties of 2D heterostructures**[Video] [Slides]

## Abstract

We will present a novel real space approach to the formulation and computation of the mechanical and electronic properties of twisted bilayer graphene and other 2D moiré materials that gives an exact representation of the electronic density of states, the Kubo formula for optical conductivity, and other electronic properties for the true aperiodic structure of 2D moiré materials without a supercell or continuum approximation. We will present efficient computational methods to approximate this formulation that exploits the locality of the hopping functions and the kernel polynomial method.

**• Zhen Zhan**(Wuhan University, China)

**Title: TBPLaS: a Tight-Binding Package for Large-scale Simulation**[Video]

## Abstract

TBPLaS is an open-source software package for the accurate simulation of physical systems with arbitrary geometry and dimensionality utilizing the tight-binding (TB) theory. It has an intuitive object-oriented Python application interface (API) and Cython/Fortran extensions for the performance-critical parts, ensuring both flexibility and efficiency. Under the hood, numerical calculations are mainly performed by both exact diagonalization and the tight-binding propagation method (TBPM) without diagonalization. Especially, the TBPM is based on the numerical solution of the time-dependent Schrödinger equation, achieving linear scaling with system size in both memory and CPU costs. Consequently, TBPLaS provides a numerically cheap approach to calculate the electronic, optical, plasmon and transport properties of large tight-binding models with billions of atomic orbitals. Current capabilities of TBPLaS include the calculations of band structure, density of states, local density of states, quasi-eigenstates, optical conductivity, electrical conductivity, Hall conductivity, polarization function, dielectric function, plasmon dispersion, carrier mobility and velocity, localization length and free path, ℤ_{2}topological invariant, wave-packet propagation, etc. All the properties can be obtained with only a few lines of code. TBPLaS is a powerful tool to tackle complex systems, for example, graphene with vacancies, twisted multilayer graphene, twisted multilayer transition metal dichalcogenides, graphene-boron nitride heterostructures, dodecagonal bilayer graphene quasicrystals and fractals.

*Ref:

**Zhen Zhan** has been involved in the calculations of transport properties of two-dimensional materials. During her PhD studies, she was involved in the development and application of simulators to study nanodevices. During her postdoc period, she focusses on the simulation of quantum system up to billion atoms. She participates in the development of two simulators and has published 21 scientific papers and co-written a book chapter. Her current interest is the novel features of two-dimensional materials with emphasis on correlated properties. The recently developed simulator, named TBPLaS (http://www.tbplas.net) is an open-source package for building and solving tight-binding models, with emphasis on handling large systems. Via the tight-binding propagation method (TBPM), sparse matrices, C/FORTRAN extensions, and hybrid OpenMP+MPI parallelization, TBPLaS is capable of solving models with billions of orbitals on computers with moderate hardware. TBPLaS can be applied to (1) models with arbitrary shape and boundary conditions, (2) defects, impurities and disorders, (3) hetero-structures, quasicrystal, fractals, (4) 1D, 2D and 3D structures. The electronic, transport, optical and plasmonic properties of complex quantum systems can be easily obtained with a few lines of code.

**• Nicolas Leconte**(University of Seoul, Republic of Korea)

**Title: Electronic structure of lattice relaxed alternating twist tG-multilayer graphene and primary and secondary gaps in G/hBN using a reparametrized two-center approximation TB model**[Video] [Slides]

## Abstract

Alternating twist (AT) multilayer graphene systems are at the heart of recent research efforts on flat band superconductivity and therefore precise descriptions of their atomic and electronic structures are desirable. After introducing the methodology behind our reparametrized two-center (TC) tight-binding (TB) model, we firstly present the electronic structure of AA’AA’… stacked AT N-layer (tNG) graphene for $N = $3-10, 20 layers and bulk AT graphite systems where the atomic structure is relaxed using a molecular dynamics simulation code. The low energy bands depend sensitively on the relative sliding between the layers but we show explicitly up to N = 6 that the highly symmetric AA’AA’. . . stacking is energetically preferred among all interlayer sliding geometries of each added layer, justifying why experimental devices consistently show results compatible with this geometry. It is found that lattice relaxation enhances electron-hole asymmetry, and leads to small reductions of the magic angle values with respect to analytical or continuum model calculations with fixed tunneling strengths that we quantify from few layers to bulk AT-graphite. The twist angle error tolerance near the magic angles obtained by maximizing the density of states of the nearly flat bands expand progressively from 0.05° for twisted bilayer graphene to up to 0.2° for AT-graphite, hence allowing a greater twist angle flexibility in multilayers. We further comment on the role of perpendicular electric and magnetic fields in modifying the electronic structure of the system. We secondly illustrate the reparametrized TC-model approach to assess the impact of the substrate relaxation on the primary gap at charge neutrality and secondary valence band gap of graphene on hexagonal boron nitride (G/BN) as a function of twist angle where we observe a substrate-induced reduction of the primary gap and the closing of the primary and secondary gaps for finite angles as well as a slight initial increase for the primary gap at ~ 0.5 degree that coincides with a shallow energetic stabilization of the atomic structure away from perfect alignment.

**Nicolas Leconte** is a postdoc in the group of Prof. J. Jung at the Physics Department of the University of Seoul in Korea since 2016. He obtained his PhD in 2014 at the Universite catholique de Louvain in Belgium with Prof. J.C. Charlier after which he did a Postdoc in Barcelona (2014-2015) in the group of S. Roche and spent 6 months at the University of Texas with Prof. A. MacDonald (2015-2016). He is interested in tight-binding real-space calculations where he maintains his own code to calculate electronic properties including band structures and Kubo transport properties of graphene and other related 2D materials.

Google scholar: https://scholar.google.com/citations?user=_0gSN6wAAAAJ&hl=en