Twisted Bilayer Graphene

Twisted bilayer systems of 2D materials have generated great recent interest due to their ability to create new, tunable electronic behavior. Twisted bilayers, for example, have been shown to exhibit novel Mott insulator-like states as well as unconventional superconductivity.1-6 The Crommie group is actively pursuing this area of research.

The most famous twisted bilayer system is obtained by stacking one layer of graphene on top of another and rotating the two layers by a small twist angle, θ. This causes a moiré pattern to arise between the lattices of the two graphene layers(Fig. 1a). One can estimate the resulting twisted bilayer electronic structure by considering the separate Brillouin zones of the two layers (Fig. 1b). If we zoom in on the K points of the two layers (marked K1 and K2 in (Fig. 1b) then we see that the Dirac cones of the two layers shift with respect to one another in k-space as they are rotated by θ, and they intersect at points above and below the Dirac point. This leads to hybridization and avoided crossings in the graphene bands (Fig. 1d), thus inducing the formation of new, flat bands and van Hove singularities (Fig. 1e).

Fig. 1: (a)-(b) Twisted bilayer graphene (TBLG) exhibits a moiré pattern which corresponds to the Brillouin zones of the two graphene layers being rotated by θ. (c)-(d) Rotation of the Brillouin zones causes the Dirac cones of the two layers to rotate and intersect with each other. (e) Band hybridization and avoided crossings cause formation of flat bands to emerge near the Dirac point in TBLG.

The Crommie group has explored how this process creates new electronic structure in twisted bilayers of graphene on BN in a gated, field effect transistor (FET) device configuration.7 The effects of both the graphene-graphene moiré pattern and the graphene-BN moiré pattern were studied. Fig. 2a shows a schematic of the type of device utilized in this investigation. Graphene bilayers were positioned on top of BN flakes one layer at a time, and then contacted and placed inside a cryogenic UHV STM. Figs. 2b-d show our STM images of twisted bilayer graphene (TBLG) oriented at different angles with respect to the underlying BN lattice. In Fig. 2b, the graphene-BN moiré pattern has a longer wavelength, λg-BN, than the graphene-graphene moiré wavelength, λg-g. In Fig. 2c, λg-BN becomes smaller but is still larger than λg-g, while in Fig. 2d it reduces even further until λg-BN ≈ λg-g. The combined graphene-graphene and graphene-BN moiré patterns cause the graphene wavefunctions to change dramatically as the moiré wavelengths are varied.

Fig. 2: (a) Sketch of twisted bilayer graphene (TBLG) device fabricated using BN substrate in field-effect transistor (FET) device configuration, and integrated with cryogenic STM. (b)-(d) STM images of TBLG device shows coexisting graph.-graph. moiré pattern and graph.-BN moiré pattern. Wavelength of graph.-BN moiré pattern decreases from (b) to (c) until it is roughly the same size as the graph.-graph. moiré pattern in (d). (STM images: Crommie group).

In Fig. 3 λg-BN is small and constant, and the evolution of the graphene-graphene moiré pattern can be seen as the graphene-graphene twist angle becomes smaller and λg-g increases.

Fig. 3: Wavelength of graphene-graphene moiré pattern increases as TBLG twist angle decreases from (a) to (c). Expression for moiré pattern wavelength as a function of twist angle shown at bottom. (STM images: Crommie group).

The effect of the graphene-graphene twist angle on TBLG electronic structure can be seen in the dI/dV spectra shown in Fig. 4. The most prominent features here are the two van Hove singularities (vHs) that straddle V = 0. The vHs’s are seen to shift toward V = 0 as the twist angle decreases and λg-g increases. Two “dip” features are also seen (marked by red and green arrows), which also move toward V = 0 as λg-g increases. These mark the locations of bandgaps in the new, moiré-induced TBLG bandstructure.

Fig. 4: dI/dV spectroscopy measured on TBLG for different graphene-graphene moiré pattern wavelengths. Experimentally observed van Hove singularities and “dip” features are marked by arrows. (STM spectroscopy: Crommie group).

The dependence of the three energy-symmetric experimental features (the vHs and the two dips) are compared to theoretical TBLG bandstructure predictions in Fig. 5. The dependence of these features on moiré wavelength follows the theoretical predictions quite closely (Fig. 5a). The TBLG bandstructure and density of states are shown in (Figs. 5c, d) for λg-g = 7.6nm. Our theory/experiment comparison shows that the two graphene layers in TBLG are strongly coupled and cannot be treated via perturbative techniques in the small twist angle regime.7

Fig. 5: (a) Experimental STM spectroscopy features for TBLG (from Fig. 4) compared to theoretical band structure features calculated using hybrid tight-binding technique. (b)-(c) TBLG Brillouin zone and calculated bandstructure. (d) Calculated TBLG density of states shows peaks and dips which correspond to van Hove singularities and bandgaps, respectively, as shown in color-coded bandstructure of (c). (STM spectroscopy: Crommie group; theory: Jung, MacDonald groups).


1. Yuan Cao, Valla Fatemi, Ahmet Demir, Shiang Fang, Spencer L. Tomarken, Jason Y. Luo, J. D. Sanchez-Yamagishi, K. Watanabe, T. Taniguchi, E. Kaxiras, R. C. Ashoori, and P. Jarillo-Herrero. “Correlated insulator behaviour at half-filling in magic-angle graphene superlattices.” Nature 556, 80 (2018).

2. Yuan Cao, Valla Fatemi, Shiang Fang, Kenji Watanabe, Takashi Taniguchi, Efthimios Kaxiras, Raymond C. Ashoori, and P. Jarillo-Herrero. “Unconventional superconductivity in magic-angle graphene superlattices.” Nature 556, 43 (2018).

3. Aaron L. Sharpe, Eli J. Fox, Arthur W. Barnard, Joe Finney, Kenji Watanabe, Takashi Taniguchi, M. A. Kastner, and David Goldhaber-Gordon. “Emergent ferromagnetism near three-quarters filling in twisted bilayer graphene.” arXiv:1901.03520.

4. Xiaomeng Liu, Zeyu Hao, Eslam Khalaf, Jong Yeon Lee, Kenji Watanabe, Takashi Taniguchi, Ashvin Vishwanath, and Philip Kim. “Spin-polarized correlated insulator and superconductor in twisted double bilayer graphene.” arXiv:1903.08130v2.

5. Yuan Cao, Daniel Rodan-Legrain, Oriol Rubies-Bigorda, Jeong Min Park, Kenji Watanabe, Takashi Taniguchi, and Pablo Jarillo-Herrero. “Electric field tunable correlated states and magnetic phase transitions in twisted bilayer-bilayer graphene.” arXiv:1903.08596v2.

6. Xiaobo Lu, Petr Stepanov, Wei Yang, Ming Xie, Mohammed Ali Aamir, Ipsita Das, Carles Urgell, Kenji Watanabe, Takashi Taniguchi, Guangyu Zhang, Adrian Bachtold, Allan H. MacDonald, and Dmitri K. Efetov. “Superconductors, orbital magnets, and correlated states in magic angle bilayer graphene.” arXiv:1903.06513v2.

7. Dillon Wong, Yang Wang, Jeil Jung, Sergio Pezzini, Ashley M. DaSilva, Hsin-Zon Tsai, Han Sae Jung, Ramin Khajeh, Youngkyou Kim, Juwon Lee, Salman Kahn, Sajjad Tollabimazraehno, Haider Rasool, Kenji Watanabe, Takashi Taniguchi, Alex Zettl, Shaffique Adam, Allan H. MacDonald, and Michael F. Crommie. “Local spectroscopy of moiré-induced electronic structure in gate-tunable twisted bilayer graphene.” Phys. Rev. B 92, 155409 (2015).