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Therefore, the effect of the substrate supporting or embedding TBG could play a key role in the electronic properties of the system.įrom a theoretical point of view, TBG has been studied using full or effective tight-binding (TB) models as well as continuum models due to the large number of atoms found in the moiré supercell 16, 17. Moreover, a ferroelectric phase has been found in Bernal-stacked bilayer graphene sandwiched between two hBN layers 15. In the latter state, quantum anomalous Hall effect (QAH) has been measured and the material has been thought to be a Chern insulator 13, 14. Interestingly, the appearance of superconducting, insulating, and ferromagnetic phases in TBG supported on a nearly aligned hBN substrate has made this system even more appealing to study 11, 12, 13, 14. Nevertheless, although the interaction between graphene and hBN is small due to the large band gap of hBN 7, in the case of TBG on top of hBN, it has been shown that this small interaction can affect the electronic properties of TBG. Furthermore, the large band gap of hBN has the advantage of having little interaction between the states of hBN and graphene that is opposite to the case with a metallic substrate where a large hybridization between the graphene and metal states can occur 10. In fact, it has been shown that graphene mobility, when deposited on hBN 8, is dramatically improved as compared to those on SiO 2 substrates 9.

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Among all of these, due to its atomically smooth surface that is relatively free of dangling bonds and charge traps 8, hexagonal boron nitride (hBN) stands out due to the cleaner structures that can be fabricated. Interestingly, depending on the experimental setup, it can also be clamped between two layers of material. TBG is usually supported on top of a variety of substrates in the experiment.

Although the properties of twisted bilayer graphene (TBG) have been studied in the past decades 4, 5, recently it has become of great interest due to the finding of highly correlated phases such as superconductivity and Mott insulating phases when the twist angle is in the so-called magic-angle regime 6, 7. Graphene has brought a lot of excitement to the scientific community since its isolation in 2004 1, 2, 3. Our results show that a real-space tight-binding model in combination with semi-classical molecular dynamics is a powerful tool to study the electronic properties of moiré heterostructures, and to explain experimental results in which the effect of the substrate plays an important role. Finally, we develop a continuum model that describes the tight-binding band structure. Interestingly, such degeneracy can be recovered with a second hBN layer. Specifically, the substrate induces a large mass gap and strong pseudo-magnetic fields that break the layer degeneracy.

The hBN substrate has significant effects on the band structure of TBG even in the case where TBG and hBN are not aligned. In this work, we study heterostructures of TBG and hexagonal boron nitride (hBN) using an atomistic tight-binding model together with semi-classical molecular dynamics to consider relaxation effects. Twisted bilayer graphene (TBG) has taken the spotlight in the condensed matter community since the discovery of correlated phases.