Twistronics in graphene-based van der Waals structures

doi:10.1088/1674-1056/abbbe2

Abstract

The electronic properties of van der Waals (vdW) structures can be substantially modified by the moiré superlattice potential, which strongly depends on the twist angle among the compounds. In twisted bilayer graphene (TBG), two low-energy Van Hove singularities (VHSs) move closer with decreasing twist angles and finally become highly non-dispersive flat bands at the magic angle (∼ 1.1°). When the Fermi level lies within the flat bands of the TBG near the magic angle, Coulomb interaction is supposed to exceed the kinetic energy of the electrons, which can drive the system into various strongly correlated phases. Moreover, the strongly correlated states of flat bands are also realized in other graphene-based vdW structures with an interlayer twist. In this article, we mainly review the recent scanning tunneling microscopy (STM) advances on the strongly correlated physics of the magic-angle TBG (MATBG) and the small-angle twisted multilayer graphene. Lastly we will give out a perspective of this field.

Keywords: twisted bilayer graphene correlated states superconductivity

Received: 21 August 2020
Revised: 02 September 2020
Accepted manuscript online: 28 September 2020

Corresponding Authors: ^†These authors contributed equally to this work. ^‡Corresponding author. E-mail: helin@bnu.edu.cn

Cite this article:

Ya-Ning Ren(任雅宁), Yu Zhang(张钰), Yi-Wen Liu(刘亦文), and Lin He(何林) Twistronics in graphene-based van der Waals structures 2020 Chin. Phys. B 29 117303

Fig. 1.

Abundant physical phenomena in TBG. On the logarithmic axis of twist angle, the blue regions correspond to the commensurate angle, and the red regions correspond to the incommensurate angle. The arrow points out several typical physical phenomena, the angles from small to large correspond to: Aharonov–Bohm oscillations along the triangular network of AB/BA domain walls (θ ∼ 0.1°). Reproduced with permission from Ref. [113]. Copyright 2019, Nature Publishing Group. Topologically protected helical edge states on the domain wall network (θ ∼ 0.245°). Reproduced with permission from Ref. [111]. Copyright 2018, American Physical Society. Strongly correlated phases for the partially filled moiré flat minibands on MATBG systems (θ ∼ 1.1°). Reproduced with permission from Ref. [79]. Copyright 2020, American Physical Society. Reproduced with permission from Ref. [154]. Copyright 2018, National Academy of Sciences. The higher-order topological insulator with topological corner states (θ ∼ 21.78°). Reproduced with permission from Ref. [125]. Copyright 2019, American Physical Society. The emergence of mirrored Dirac cones in graphene quasicrystal (θ ∼ 30°). Reproduced with permission from Ref. [99]. Copyright 2018, American Association for the Advancement of Science. Reproduced with permission from Ref. [100]. Copyright 2018, National Academy of Sciences. Flat bands caused by the electrons confined in a geometrically frustrated network of topologically protected modes (θ ∼ 38.21°). Reproduced with permission from Ref. [148]. Copyright 2019, American Physical Society.

Fig. 2.

VHSs separation as a function of twist angle θ. Solid colorful circles are the experimental data measured in different TBG from different references. The data are taken from Refs. [32,35–40,67–72,97]. Error bars in energy represent the peak position change caused by slight doping change. The inset shows the electronic band structure of TBG. Left: The first Brillouin zone of TBG with twist angle θ. K₁ and K₂ are the Dirac points of top and bottom layers. Middle: Energy dispersion relation of the overlap of the two Dirac cones, giving rise to two VHSs which generate peaks in the DOS. Right: Diagram of the energy dependence of DOS near the Fermi level.

Fig. 3.

The sample differences of TBG in different references. (a) A schematic diagram of the MATBG band structure and the corresponding DOS showing VHS peaks. The colored arrows point out three typical parameters: the band gap between the flat and the upper dispersive band (Δ_gap marked by blue), the full-width at half-maximum of the VHSs (FWHM marked by red), the energy separations of two VHSs (ΔE_VHS marked by green). (b) VHSs separation ΔE_VHS as a function of twist angle θ near MA-TBG. Solid colorful circles are the experimental data measured in different TBG, and the chemical potential is in between the two VHSs. The data are taken from Refs. [35,36,65,67–72]. The open stars symbols represent data for the band fully filled or fully unfilled. The data are taken from Refs. [32,37,65,67,68,70,74]. (c) The FWHM of the VHSs as a function of twist angle θ, each of VHS is fully occupied or unoccupied. The data are taken from Refs. [35–37,67–72]. Solid colorful circles represent the FWHM of conduction VHS. Open colorful circles represent FWHM of valence VHS. (d) and (e) The energy gap Δ_gap between the flat bands and the higher-energy dispersive bands as a function of twist angle θ. (d) Data from STM experiments. The data are taken from Refs. [67,68]. (e) Data from transport experiments. The data are taken from Refs. [73,75–77,81,126,204]. Solid colorful circles represent the gap between the electron flat band and the upper dispersive bands. Open colorful circles represent the gap between the hole flat band and the lower dispersive bands. Error bars in (b)–(e) originate from the slight doping change.

Fig. 4.

The STS spectra of TBG when one of the flat bands is partially filled. (a) The schematic diagram of the partially filled DOS peak in MATBG. (b) The partially filled flat band splits into four DOS peaks in a non-magic-angle TBG with θ ∼ 1.49°. Reproduced with permission from Ref. [71]. Copyright 2020, American Chemical Society.

Fig. 5.

Correlated insulator and superconductivity in the MATBG. Top panel: Conductance at the Fermi level as a function of filling factors ν in STM experiments. Grey areas correspond to fully occupied or unoccupied flat bands. Green areas correspond to the CNP (ν = 0). Twist angles include 1.05°,^[66] 1.07°,^[70] 0.99°,^[68] and 1.01°.^[67] Bottom panel: Superconducting phase diagrams acquired from the transport measurements. Solid superconducting domes indicate the coexistence of superconductivity and insulating states at ν = ± 2. Twist angles include 1.14°,^[90] 1.10°,^[84] 1.05°,^[80] 1.16°,^[80] and 1.27°.^[90] Dotted superconducting domes indicate that the superconductivity persists in the absence of the correlated insulators at ν = ± 2. Twist angles include 1.04°,^[83] 1.15°,^[83] and 1.18°.^[82] The data are taken from Refs. [66–68,70,80,82–84,90].

Fig. 6.

Spatially resolved conductance maps at the energies of the flat bands (VHSs) in MATBG. (a) Schematic atomic structure of the TBG with a twist angle θ. The AA stacking configurations are marked in the panel, exhibiting the C₆ rotational symmetry. (b) The conductance maps at the energies around the VHS peaks show the same features as topographies and maintain the C₆ symmetry when the two VHSs are fully filled or empty. (c) The conductance maps at the energies around the VHS peaks show a pronounced anisotropy in each moiré when the Fermi level lies in one of the VHSs, reducing the symmetry from the initial C₆ to C₂.

Fig. 7.

Quantized anomalous Hall effect in TBG (θ ∼ 1.15°) at θ = 3. (a) Longitudinal resistance R_xx and Hall resistance R_xy measured at B = 150 mT and T = 1.6 K as a function of carrier density over the entire flat bands. Near the ν = 3, R_xy approaches h / e² and R_xx reaches a deep minimum, illustrating the quantized anomalous Hall state. (b) Magnetic-field-dependent R_xx and R_xy at ν = 3. The sweep directions are indicated by the arrows, showing an obvious hysteresis. Reproduced with permission from Ref. [76]. Copyright 2020, American Association for the Advancement of Science.

Fig. 8.

Schematic of the electrical-displacement-field-dependent low energy moiré bands and DOS in the TDBG. The two parabolic dispersive bands are originated from the Bernal-stacked BLG. The two low-energy flat bands are separated from the high-energy dispersing bands by the moiré gaps. The bandwidth of the flat bands W and the band gap Δ at the charge neutral point can be easily tuned via the perpendicular electrical displacement field D. In a specific range of D, the correlated insulating states emerge when the isolated flat band is half filled, exhibiting the spin-polarized ordering.

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