Twisted graphene bilayer around the first magic angle engineered by heterostrain

Very recently, twisted graphene bilayers (TGBs) around the first magic angle $\theta \approx 1.1^{\circ }$ have attracted much attention for the realization of exotic quantum states, such as correlated insulator behavior and unconventional superconductivity. Here we elaborately study a series of TGBs around the first magic angle engineered by heterostrain, where each layer is strained independently. Our experiment indicates that a moderate heterostrain enables the structural evolution from the small-angle TGB (θ ∼ 1.5°) to the strained magic-angle TGB (θ ∼ 1.1°), exhibiting the characteristic low-energy flat bands. The heterostrain can even drive the system into highly strained tiny-angle TGBs (θ$\ll 1.1$°) with large deformed tetragonal superlattices, where a unique network of topological helical edge states emerges. Furthermore, the predicted domain wall modes, which are strongly localized and result in a hexagon-triangle-mixed frustrated lattice derived from the Kagome lattice, are observed in the strained tiny-angle TGBs.

Figure 1

(a) Structural transition of bilayer graphene from the AB-stacked mode to rotationlike stacking faults induced by a nonuniform biaxial strain along the zigzag and armchair directions. (b) Schematic of one-step-grown bilayer graphene/$\alpha \text{−}\mathrm{M}{\mathrm{o}}_2\mathrm{C}$ heterostructures. (c) STM topography $\left(V_S=300\mathrm{mV},I=0.2\mathrm{nA}\right)$ of the TGB with different moiré superlattices induced by the heterostrain. I: Almost unstrained triangular moiré superlattice (marked by red circles). II: Deformed triangular moiré superlattice (blue circles). III: Deformed tetragonal moiré superlattice (green circles). Scale bars, 50 nm. (d–f) FFT images of three typical moiré superlattices in I, II, and III, respectively. Scale bars, $0.05\mathrm{n}{\mathrm{m}}^{-1}$. (g) Simulation of the structural transition in panel (c) based on the reduced model where the underlying graphene sheet encounters a gradually increasing (nonuniform) uniaxial strain along the armchair direction and a progressively decreasing twist angle relative to the unstrained topmost layer.

Figure 2

Simulation of deformed moiré supercells by superposition of nonuniform uniaxial strain and twist (a) and that of uniform uniaxial strain and twist (b). Different moiré superlattices are marked by yellow spots, and three basic axes of a moiré unit cell are indicated by red arrows in panel (b). (c) STM topography $\left(V_S=300\mathrm{mV},I=0.2\mathrm{nA}\right)$ of a typical deformed triangular moiré unit cell with three different moiré vectors, ${\mathit{L}}_i^s\left(i=0,\pm \right)$. Scale bar, 10 nm. (d) FFT image of panel (c) exhibiting the corresponding momentum vectors, ${\mathit{k}}_i^s\left(i=0,\pm \right)$. Scale bars, $0.05\mathrm{n}{\mathrm{m}}^{-1}$. (e) Summarizing the deduced twist angles and strain parameters in distinct moiré unit cells with different average wavelengths.

Figure 3

(a) STM morphology $\left(V_S=200\mathrm{mV},I=0.2\mathrm{nA}\right)$ of deformed triangular moiré superlattices with effective twist angles near the first “magic angle.” Areas for the magic-angle TGB and the other TGB are roughly divided by a yellow dashed curve. Scale bar, 20 nm. A transition area is labeled by the white dashed frame. The two moiré patterns with different periods are marked by deformed green and red dotted hexagons, respectively. (b) Theoretical calculated electronic structures of the TGB with twist angle $\theta \sim 1.{43}^{\circ }$ and ∼1.1°, respectively. (c) $d^{2}I/d{V}^2$ map measured along the black arrow in panel (a). The positions of the VHSs $(\mathrm{VH}{\mathrm{S}}^{+}$ and $\mathrm{VH}{\mathrm{S}}^{-})$ and the FB are marked with black dashed lines. (d–f) dI/dV maps obtained over the whole area in panel (a) at the energy of the FB (–9 meV), $\mathrm{VH}{\mathrm{S}}^{-}$ (–50 meV), and away from the FB (–300 meV), respectively. Scale bars, 20 nm.

Figure 4

(a) STM topography $\left(V_S=200\mathrm{mV},I=0.2\mathrm{nA}\right)$ of deformed tetragonal moiré supercells. Scale bar, 20 nm. (b, c) Atomic-resolution STM images $\left(V_S=700\mathrm{mV},I=0.2\mathrm{nA}\right)$ of the AB/BA-stacked and domain-wall regions labeled by the blue and red dots in panel (a), displaying triangular and hexagonal lattices, respectively. Scale bars, 0.5 nm. (d) Upper panel: Schematic of the spectra of the tiny-angle TGB under nonzero external electrical bias $U$. Lower panel: Representative STS spectra taken at the AB/BA, AA, and domain-wall regions. The FB resonance, helical network, and DW modes are marked. (e) The energy of pseudo-Landau-level-like DW modes $E_n–E_D$ deduced from the spectra acquired on three different tetragonal moiré regions under different electric fields $U_i\left(i=1,2,3\right)$ as a function of the corresponding gaps $E_g$ induced by the fields.

Figure 5

(a, b) dI/dV maps obtained over the whole region in Fig. 3 with the fixed sample bias of −46 mV (a) and 200 mV (b). Scale bars, 20 nm. Topological networks (denoted by rose lines) are formed on the DW regions in the map within the gap, while an exotic hexagon-triangle-mixed lattice formed by the DW modes is observed in the map obtained at a resonant level. (c–f) Unstrained and strained networks (c and e, marked by rose lines) and DW modes (d and f, marked by rose spots).