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Figure 1.  Moiré pattern of TBG.^[9,10] (a) Schematic of general twisted bilayer systems with the twist angle θ = 4.4°, where the largest black hexagon denotes moiré superlattices including AA and AB/BA stacked lattices. Blue (red) dots represent the sites in L1 (L2). (b) Schematic of BZs. Sky blue and red hexagons represent first BZs for L1 and L2 while black hexagons represent moiré BZs (mBZ). (c) Scanning tunneling microscope (STM) images of TBG with θ = 9.5°. The scale bar is 5.0 nm.

Figure 2.  Superlattice of hBN/graphene heterojunction.^[11,13] (a) Schematic diagram of graphene/hBN superlattice formed with twist angle of 0°, with hBN represented in blue, and graphene in red. Black hexagons mark the moiré plaquette and red arrows mark the basis vectors, with carbon atoms in gray. Different stacking regions are present within the superlattice. (b) The upper part shows the morphology image of graphene/boron nitride characterized by AFM, while the lower part represents the height variations along the green line. (c) Spatial distribution of the interlayer interaction potential in the graphene/boron nitride superlattice with twist angle of 0°.

Figure 3.  Atomic reconstruction of moiré superlattice.^[13–16] (a) Schematic diagram depicting the stacking types of unreconstructed twisted bilayer graphene (left) and reconstructed twisted bilayer graphene (right). (b) STM images of the graphene/hBN heterojunction sample with twist angle of 0° and their local magnifications (as shown in the upper-right inset), with a scale bar of 10 nm. The size of the square region is 3 nm × 3 nm. Inset: A magnified view of the black square region, showing clear atomic structures with a scale bar of 1 nm. (c) Relative lattice constants of different color regions within the images in panel (b), corresponding to the colors in panel (b). (d) Dark-field transmission electron microscope images obtained by selecting graphene diffraction peaks in TBG, with twist angles of 0.1°, 0.4°, 0.8°, and 1.2°, respectively. (e) Due to the spatially modulated vdW adhesion potential, the graphene lattice periodically expands and contracts in-plane. An out-of-plane corrugation profile also develops, both matching the moiré.

Figure 4.  Van Hove singularities.^[10,21–26] (a) Schematic diagram illustrating the formation of Van Hove singularities (VHS). VHS forms between the two Dirac cones, resulting in a higher electron density at the singularity point. (b) Scanning tunneling spectroscopy (STS) curves for graphene. G represents the STS curve for a single layer graphene, while M1 and M2 represent STS curves for twisted bilayer graphene with θ = 1.79°, where VHS is clearly observed on both sides of the Dirac point. (c) When the angle is between 2° and 10°, the E_vhs value decreases as the angle θ decreases, but the energy center of the two VHS points remains almost unchanged with varying angles. (d) STS spectra of twisted bilayer graphene with θ = 1.79°, where 1 corresponds to the AA-stacked region, and 2 and 3 correspond to other stacking regions. While all spectra display two peaks, the electron density at position 1 is significantly higher than that at other positions. (e) Relationship between the reorganized Fermi velocity and the interlayer twist angle. The red curve represents the fitted curve obtained from formula (10), while the blue triangles correspond to experimentally measured data. The localization phenomenon at the AA-stacked position occurs around a twist angle of approximately 1.16° as mentioned in the text.

Figure 5.  Secondary Dirac points.^[3,39,40] (a) The electronic band structure of graphene near the κ point on boron nitride with a small misalignment angle, θ≪1. In the left image, there is an isolated saddle point (SDP) in the valence band at the κ point. In the right image, there is an SDP at the μ point. (b) Transport properties of Dirac fermions in moiré superlattices of hBN/graphene heterojunction, θ = 0±1°. Upper image: Longitudinal resistivity as a function of carrier concentration, exhibiting a peak in resistivity at the SDP with a strong temperature dependency. Lower image: Hall resistivity as a function of carrier concentration, showing a change in sign of the Hall resistivity at the SDP. (c) and (d) ARPES observations of gapped main Dirac point (c) and SDP (d), for 0° aligned graphene/hBN.

Figure 6.  Flat bands.^[43,44] (a) The three equivalent Dirac points in the first Brillouin zone result in three distinct hopping processes. Interference between hopping processes with different wave vectors captures the spatial variation of interlayer coordination that defines the moiré pattern. (b) Energy dispersion for the 14 bands closest to the Dirac point plotted along the k-space trajectory A → B → C → D → A for w = 110 meV, and θ = 5° (left), 1.05° (middle), and 0.5° (right). (c) At certain angles corresponding to α, the reorganized Fermi velocity becomes zero, leading to flat bands. α = w/(vk_θ); w is the hopping energy; k_θ = 2k_D sin(θ/2), where k_D is the magnitude of the Brillouin-zone corner wave vector for a single layer; v is the Dirac velocity. (d) Moiré bands of WSe₂ on MoSe₂ at twist angle θ = 2.0°. The red dashed line is a tight-binding-model fit to the highest valence band that includes hopping up to the third nearest neighbor and has a narrow bandwidth (about 11 meV).

Figure 7.  Correlated insulating states and superconducting states.^{[6,7,51–55]} (a) The conductivity in TBG with θ = 1.08° and T = 0.3 K as a function of carrier concentration, exhibiting an insulating state at half-filling ν = ±2. (b) Phase diagram near ν = −2 in TBG with θ = 1.16°, showing superconducting phases slightly away from half-filling on both sides of the carrier concentration. (c) Adjustment of the electric displacement field and carrier concentration in TDBG with θ = 1.09° by tuning the top and bottom gate voltages. Under electric field tuning, the emergence of correlated insulating states is observed at the electron half-filling. (d) Phase diagram of high-quality MATTG devices, concerning carrier concentration and electric displacement field, with the presence of superconducting states near integer fillings. (e) Longitudinal resistivity in MATTG at ν = −2.28 as a function of temperature and parallel magnetic field, showing that TTG’s superconductivity is not limited by the Pauli limit and can exist under magnetic fields as high as 10 T. (f) Longitudinal resistivity in MATTG as a function of filling and parallel magnetic field. As the magnetic field increases, the appearance of a second superconducting phase can be observed, possibly connected by a first-order phase transition, indicating that superconducting electrons in TTG may be of spin-triplet nature. (g) Phase diagram of tMBG device with θ = 1.08° concerning carrier concentration and electric displacement field. The tMBG exhibits significant tunability with strong differences in phase diagrams for D > 0 and D < 0. (h) Evidence of superconductivity observed from the temperature (T) dependence of resistance (R_xx) in TDBG in proximity to WSe₂ for two different samples with twist angles of 1.37° and 1.24°, referred to hereafter as D1 and D2, where ν is the moiré filling factor and D is the displacement field. (i) I–V characteristics of superconductivity as a function of temperature, obtained by integrating dV/dI (I_dc) at the red marker in (a) (lower inset). V(I_dc) was used to determine the value of the Berezinskii–Kosterlitz–Thouless transition temperature, T_BKT = 64 mK (upper inset).

Figure 8.  Chern insulator and time-reversal symmetry in MATBG.^{[55,91,93,96–98]} (a) The Chern numbers associated with each flat band depend on the specific symmetry-breaking mechanism. This panel illustrates how the band Chern numbers in MATBG change when C_2z symmetry and time-reversal symmetry are broken. (b) The upper part of the panel depicts the cascade filling sequence in MATBG, while the lower part shows the corresponding relationship between the chemical potential and the filling factor in MATBG. A cascade phase transition occurs before integer fillings, driven by exchange interactions that redistribute carriers into one band while elevating the other three bands to higher energies. (c) A schematic diagram illustrating the evolution of the system’s Chern numbers during a cascade phase transition under time-reversal symmetry breaking in MATBG. (d) Color plot of R_xx versus ν and B_⊥, measured at T = 1.5 K, showing experimentally observed Chern insulator states at various filling factors of MATBG. At fillings ν = ±1, ±2, ±3, 0, the corresponding Chern numbers C = ±1, ±2, ±3, ±4 are consistent with the Chern number changes induced by breaking time-reversal symmetry. (e) The variation in longitudinal resistance and Hall conductance with filling of MATBG, showing flat band reconstruction induced by a magnetic field. The upper part of the panel shows the occupied reconstructed bands. (f) Around ν = 0.8, MATBG undergoes a transition from a Chern number with an absolute value of 1 to 3 at a magnetic field of approximately 0.5 T.

Figure 9.  Chen insulators and spatial symmetry in MATBG.^[101–104] (a) Schematic illustration of the Brillouin zone folding due to the breaking of translational symmetry, leading to the emergence of a new band with a Chern number of zero and a doubling of the total number of bands. (b) Local inverse compressibility coefficients between B = 0 T and 3 T for fillings ν = 2.5 to 4, showing the presence of a SBCI state with (t,s) = (1,8/3) at ν = 8/3. (c) Schematic representation of the formation of the SBCI state with (t,s) = (1,8/3) observed in (b), with a tripling of the Brillouin zone. (d) Local inverse compressibility coefficients between B = 3 T and 11 T for fillings ν = 3 and 4, illustrating the presence of different states. (e) Wannier diagrams obtained from measurements in (d), showing the transition from a charge-density wave (CDW) state to a fractional Chern insulator (FCI) state with increasing magnetic field. (f) Schematic depiction of the formation of the FCI state with (t,s) = (2/3,10/3), which can be considered as a superposition of the previous (1,3) Chern insulator state and (−1/3,1/3) FCI state.

Figure 10.  Quantum anomalous Hall effects.^[110–112] (a) and (b) QAH effect in TBG (θ ≈ 1.15° ±0.01°). (a) R_xy approaching h/e² in a narrow range of density near ν = 3 (n = 2.37 × 10¹² cm⁻²), concomitant with a deep minimum in R_xx, measured at a magnetic field B = 150 mT and temperature T = 1.6 K. (b) R_xy and R_xx as a function of B measured at various temperatures for n = 2.37 × 10¹² cm⁻². (c) and (d) Topological states in AB-stacked WSe₂/MoTe₂ moiré superlattices. (c) Schematic illustrations of electric field induced topological phase transitions. A band insulator to a quantum spin Hall insulator transition is possible when the first moiré mini band is full-filled, and a Mott insulator to a quantum anomalous Hall insulator transition could occur when the first moiré mini band is half-filled. (d) At the quantum anomalous Hall region, the measured R_xx and R_xy versus B field at various temperatures are shown respectively. (e) FQAH effect in pentalayer graphene-hBN moiré superlattice, with the twist angle of 0.744°. Magnetic hysteresis scans of R_xy and R_xx at ν = 2/5, 3/7, 4/9, 4/7, 3/5 and 2/3, showing quantized values of Rxy=hνe2 and much smaller R_xx.