Van der Waals stacking of twisted two-dimensional (2D) atomic sheets provides a versatile platform for engineering exotic electronic states through rotational misalignment that folds dispersive electronic bands into flat mini-bands within a moiré Brillouin zone1,2. The resulting suppression of kinetic energy relative to electron–electron interactions can lead to correlated insulating states as well as unconventional superconductivity3,4. Moiré flat bands also inherit the large Berry curvature of the individual atomic layers which can result in topologically non-trivial phases5,6,7. Electron-doped twisted monolayer–bilayer graphene (tMBLG)—a graphene monolayer rotationally misaligned with a Bernal-stacked bilayer—stands out among these since it exhibits the quantum anomalous Hall (QAH) effect (i.e., quantized Hall conductance in the absence of external magnetic field) accompanied by doping-controlled switching of its Chern number, an effect not observed in other moiré QAH systems8. Such behaviour is expected to be sensitive to local structural parameters such as twist angle and hetero-strain (i.e., the relative strain between adjacent layers). For example, twist angle directly affects the moiré mini-band structure while even small hetero-strains (<0.5%) can be magnified by the moiré superlattice to induce large moiré distortions, thus altering the energetics of mini-bands and the behaviour of emergent correlated and topological phases9,10. Understanding the rich physics of moiré systems requires understanding the relationship between exotic electronic phases and local structure, something difficult to achieve using macroscopic probes that only explore spatially-averaged behaviour.

Here we show how scanning tunnelling microscopy and spectroscopy (STM/STS) enables determination of how changes in local structure alter correlated and topological electronic behaviour in tMBLG field-effect transistor devices. We find that tuning the electron doping concentration of tMBLG results in the emergence of charge gaps observable to STS at filling levels ν = 2 and ν = 3 (i.e., two and three electrons per moiré unit cell), indicating the formation of correlated insulating states. STS performed in an out-of-plane magnetic field allows us to detect non-trivial topology in the ν = 3 QAH insulating state which has total Chern number Ctot = ±2, and to demonstrate its dependence on local twist angle and hetero-strain. In addition to observing strong variation of correlation and topological properties at different twist angles, we find that regions having nearly identical twist angle but different hetero-strain values exhibit very different behaviour. In the small-strain regime, the correlation gap evolves into two separate gaps at different gate voltages that correspond to Ctot = +2 and Ctot = –2, indicating doping-controlled switching of valley polarization consistent with previous electrical transport results8. Such behaviour is absent, however, when large hetero-strain is present, in which case only a single correlation gap with Ctot = +2 is observed. This behaviour can be understood using a continuum model for tMBLG that reveals how Chern number switching results from a competition between the bulk and edge contributions to orbital magnetization that is highly sensitive to local hetero-strain. These results demonstrate the crucial role that local structural parameters play in shaping correlation and topological effects in twisted moiré systems.


Correlated insulating behaviour at integer fillings

Figure 1a shows a schematic of our experiment, which incorporates a gate-tunable graphene device into an STM measurement geometry. A Bernal-stacked bilayer graphene is placed on top of a monolayer graphene with a twist angle θ between them, and the stack is supported by a hexagonal boron nitride (hBN) substrate placed on a Si/SiO2 wafer (Methods, Supplementary Fig. 1). The carrier density n of the graphene stack can be tuned continuously via voltage VG applied to the Si back-gate. Our devices were annealed in ultra-high vacuum before being loaded into the STM system at T = 4.7 K for measurement (Methods). Figure 1b shows a representative topographic image of the monolayer–bilayer moiré pattern which exhibits an average wavelength of lM = 11.2 nm, from which we extracted a local twist angle of θ = 1.25° (Methods). Within each moiré unit cell (dashed box) we observe three representative regions with different apparent heights that correspond to the three local tMBLG stacking orders: BAB, ABC, and AAB (Supplementary Note 1).

Fig. 1: Correlated insulating states in gate-tunable tMBLG.
figure 1

a Schematic of the gate-tunable tMBLG device used in our STM/STS measurements. MLG monolayer graphene, BLG Bernal-stacked bilayer graphene. VBias is the sample bias voltage and VG is the gate voltage referenced to the sample. b Representative STM topographic image of tMBLG (VBias = –1 V, tunnelling current I0 = 0.02 nA). The dashed box outlines the moiré unit cell. The local stacking orders BAB, ABC, and AAB are shown in the side view. c Gate-dependent dI/dV density plot for the BAB stacking region over the gate range –70 V ≤ VG ≤ 70 V. The vertical dashed line denotes the Fermi energy. dh dI/dV spectra measured at d VG = 0 V (ν = 0), e VG = 31.5 V (ν = 2), f VG = 39 V (ν = 2.5), g VG = 47 V (ν = 3), and h VG = 62.5 V (ν = 4). Spectroscopy parameters: modulation voltage VRMS = 1 mV; setpoint VBias = 100 mV, I0 = 1.15 nA for –70 V ≤ VG ≤ –2 V in (c); setpoint VBias = –100 mV, I0 = 0.8 nA for 0 V ≤ VG ≤ 70 V in (c) and (d); setpoint VBias = –60 mV, I0 = 0.5 nA for (eh). VFB valence flat band, CFB conduction flat band, CFB– lower branch of CFB, CFB + upper branch of CFB.

We access correlated electronic states of tMBLG by tuning the carrier concentration via VG and performing dI/dV spectroscopy. Figure 1c shows a density plot of gate-dependent dI/dV spectra obtained in the BAB region. Estimation of the device capacitance allows us to convert VG to the filling factor ν, defined as the average number of electrons/holes per moiré unit cell referenced to charge neutrality (Methods). At ν = 0 (VG = 0 V) we observe two narrow peaks in the dI/dV spectrum that are centred at VBias = 6 mV and VBias = –‍14 mV (Fig. 1d) that we identify as originating from van Hove singularities of the fourfold degenerate conduction flat band (CFB) and valence flat band (VFB). Increasing VG leads to partial occupation of the CFB and shifts both peaks toward lower energy. As the filling level approaches ν = 2 the CFB peak gradually splits into two branches, CFB– and CFB+, that are located below and above the Fermi energy EF (VBias = 0 mV). At ν = 2 (VG = 31.5 V) these two branches have roughly the same spectral weight and a clear charge gap can be observed across EF (Fig. 1e). As the doping level is further increased from ν = 2 to ν = 2.5 (VG = 39 V) the energy splitting between CFB– and CFB+ becomes smaller and the gap feature evolves into a shallow dip (Fig. 1f). At ν = 3 (VG = 47 V) an insulating gap reappears at EF with CFB– having significantly greater weight compared to CFB+ (Fig. 1g). Finally, at ν = 4 (VG = 62.5 V, full filling of the CFB) the CFB– and CFB+ branches merge into a single peak that lies completely below EF (Fig. 1h). The presence of charge gaps at ν = 2 and ν = 3 demonstrates the formation of correlated insulating states at these filling factors, corroborating results from previous electrical transport studies8,11,12,13 (Supplementary Note 2, Supplementary Figs. 2, 3).

Gate-switchable QAH insulating state

To discern the nature of the ν = 2, 3 correlated insulating states in tMBLG, we applied an out-of-plane magnetic field B = (0, 0, B) to our sample and performed gate-dependent dI/dV spectroscopy. Figure 2a–c shows density plots of gate-dependent dI/dV spectra measured near ν = 2 for B = 0, 1, and 2 T, respectively. The insulating gap feature, marked by vanishing dI/dV at EF and maximum CFB peak splitting, always appears at the same filling level (white arrows) regardless of the B value. dI/dV spectra measured near ν = 3 (Fig. 2d–i), however, exhibit very different field-dependent behaviour. The charge gap (white arrows) is seen to remain constant in energy splitting (Supplementary Note 3) but to evolve into two separate gaps for B > 0 T. These two gaps bracket ν = 3 and split away from it as B increases.

Fig. 2: Topological behaviour of correlated insulating states in an out-of-plane magnetic field.
figure 2

ac Gate-dependent dI/dV density plot for the BAB region near ν = 2 at a B = 0.0 T, b B = 1.0 T, and c B = 2.0 T (modulation voltage VRMS = 1 mV; setpoint VBias = –60 mV, I0 = 0.5 nA). Arrows indicate correlation gaps. di Gate-dependent dI/dV density plot for the BAB region near ν = 3 at d B = 0.0 T, e B = 0.4 T, f B = 0.8 T, g B = 1.2 T, h B = 1.6 T, and i B = 2.0 T (modulation voltage VRMS = 1 mV; setpoint VBias = –60 mV, I0 = 0.1 nA). Arrows indicate correlation gaps. j Normalized dI/dV at VBias = 0 mV (EF) as a function of gate voltage and magnetic field. Dashed lines are guides to the eye following the Středa formula with total Chern number Ctot = ±2.

We can better visualize the magnetic field evolution of the ν = 3 correlated insulating state by plotting normalized dI/dV at VBias = 0 mV (EF) as a function of both ν (VG) and B (Fig. 2j). The dark region in the plot indicates vanishing dI/dV due to the emergence of a charge gap, which forms a V-shape (white dashed lines) that is roughly symmetric about the ν = 3 horizontal line. This linear scaling of the correlation gap position with magnetic field is reminiscent of correlated Chern insulating states reported previously in magic-angle twisted bilayer graphene (MA-tBLG)14,15,16 where the change in carrier concentration n of a Chern insulating state is related to the out-of-plane field B through the Středa formula \(\frac{\Delta n}{\Delta B}=\frac{{C}_{{{{{{\rm{tot}}}}}}}}{{\varPhi }_{0}}\) (Ctot is the total Chern number and Φ0 = h/e is the magnetic flux quantum)17. Our observations thus imply that the ν = 3 insulating state in tMBLG has Chern number Ctot = ±2 as derived from the slope of the lines in Fig. 2j. Two significant differences, however, distinguish our results from those reported in MA-tBLG. First, ΔnB linear scaling is observed in MA-tBLG only under high external fields (B > 3 T) that break time-reversal symmetry and stabilize the Chern insulating states, whereas such behaviour in tMBLG can be resolved in fields as low as B = 0.2 T (Supplementary Fig. 4) and can be traced back to B = 0 T. Combined with the robust charge gap at ν = 3, this indicates that the zero-field ground state of tMBLG is a topologically non-trivial QAH insulator with spontaneous time-reversal symmetry breaking (Supplementary Note 4). Second, each non-zero integer filling of MA-tBLG features only a single correlation gap that shifts monotonically with increasing B, while in tMBLG we observe two separate gaps corresponding to Ctot = +2 and Ctot = –2. This indicates that the total Chern number for the tMBLG QAH state is switchable between +2 and –2 by simply tuning the carrier concentration across ν = 38.

Tuning Chern number switching with twist angle and strain

Simultaneous structural measurement via STM and local electronic characterization via STS provide a unique opportunity to investigate how correlation and topological effects in tMBLG are affected by local structural variations at the moiré scale. Figure 3a summarizes our experimental results as a function of local twist angle and local hetero-strain obtained through analysis of moiré anisotropy in our STM topographs (Methods). While the emergence of an insulating gap at ν = 2 is robust in all of our data, the behaviour at ν = 3 depends strongly on both local twist angle and local hetero-strain. Gate tunability of the Chern number is only observed when the twist angle is between 1.25° and 1.28°, as indicated by the green data points in Fig. 3a. When the twist angle increases slightly above this range (orange data points) the charge gap at ν = 3 persists but Chern number switching is suppressed. Figure 3b–d shows representative data from this regime in which the gap feature (white arrows) evolves monotonically toward higher filling factors as the magnetic field is increased instead of developing into two separate gaps. When the twist angle deviates even further the correlation gap at ν = 3 disappears (red data points in Fig. 3a; see Supplementary Fig. 5).

Fig. 3: Local structural effects on correlation and topology.
figure 3

a Electronic phase diagram of tMBLG at ν = 3 in the parameter space of local twist angle and local hetero-strain. Ctot is the total Chern number. bd Gate-dependent dI/dV density plot for the BAB region at b B = 0.0 T, c B = 0.8 T, and d B = 1.6 T where only the Ctot = +2 gap is observed (modulation voltage VRMS = 1 mV; setpoint VBias = –‍75 mV, I0 = 0.2 nA). Arrows indicate correlation gaps. e STM topographic image of a region with θ = 1.25° and small hetero-strain of 0.10% (VBias = –1 V, I0 = 0.02 nA). Standard deviations for angle and strain are calculated from uncertainties in the moiré wavelength. f Evolution of the correlation gap position near ν = 3 in an out-of-plane magnetic field shows two branches for small strain. The data points were extracted from dI/dV spectra taken in the BAB region marked in (e) with the error bars determined via linear fitting (Supplementary Note 5). Dashed lines are guides to the eye following the Středa formula with Ctot = ±2. g STM topographic image of a region with θ = 1.26° and large hetero-strain of 0.24% (VBias = –1 V, I0 = 0.02 nA). h Evolution of the correlation gap position near ν = 3 in an out-of-plane magnetic field shows only one branch for large strain. The data points were extracted from dI/dV spectra taken in the BAB region marked in (g). The dashed line is a guide to the eye following the Středa formula with Ctot = +2.

To reveal the effect of hetero-strain, we directly compare two regions with almost identical twist angle (~1.26°) but different hetero-strain values (0.10% versus 0.24%) for the same device, thus allowing other variables such as carrier concentration, electric field, and correlation strength to be kept mostly constant. In the region with a smaller hetero-strain (Fig. 3e), the ν = 3 insulating gap develops into two branches under application of an out-of-plane magnetic field (Fig. 3f), indicating gate-induced switching between Ctot = +2 and Ctot = –2 QAH insulating states. In contrast, the region with a larger hetero-strain (Fig. 3g) exhibits only one branch of the ν = 3 insulating gap (Fig. 3h; see Supplementary Fig. 6), corresponding to Ctot = +2 with no gate-controlled switching.


The emergence of correlated QAH insulating states in electron-doped tMBLG can be understood as resulting from spontaneous symmetry breaking driven by electron–electron Coulomb interactions. The CFB of tMBLG is fourfold degenerate due to spin and valley degrees of freedom. Each CFB sub-band in the unfolded graphene K+ (K–) valley hosts a non-zero Chern number of C = +2 (–2) due to the large Berry curvature inherited from constituent graphene layers18,19. At integer fillings (e.g., ν = 2 and ν = 3) strong correlation can drive spontaneous polarization along a certain axis in the spin–valley space, splitting the CFB into occupied lower sub-bands (CFB–) and unoccupied upper sub-bands (CFB+) separated by a charge gap as observed in the experimental dI/dV (Fig. 1e, g). At ν = 3 the spin and valley polarization leads to breaking of time-reversal symmetry and topologically non-trivial states with Ctot ≠ 0. Figure 4a shows one possible filling configuration with double occupancy of CFB sub-bands in the K+ valley, resulting in a QAH insulating state with Ctot = +2. Similarly, double occupancy of K– valley sub-bands can result in a Ctot = –2 state that is energetically equivalent to the Ctot = +2 state in the absence of an external magnetic field.

Fig. 4: Bulk–edge magnetization competition and gate-controlled Chern number switching.
figure 4

a Energy configuration of the K+ valley-polarized state and the K– valley-polarized state at ν = 3 in tMBLG. Arrows represent electron spin. \({{{{{{\mathscr{E}}}}}}}_{{{{{{\rm{g}}}}}}}\) is the size of the correlation gap. The total Chern number Ctot is the sum of Chern number C for all occupied sub-bands. b Schematic of orbital magnetization for both valley-polarized states when EF is at the top of the correlation gap and the chiral edge state band is occupied (\(\nu\, \gtrsim\, 3\)). Grey arrows represent the direction of current flow in edge states. Mbulk and Medge are bulk and edge orbital magnetic moments. Under an applied out-of-plane magnetic field B (green arrow on right) the Ctot = +2 state is the energetically favourable ground state. c Same as b, but EF is at the bottom of the correlation gap and the edge state band is depleted (\(\nu \, \lesssim \, 3\)). Dashed grey lines represent empty edge states. Here the Ctot = –2 state is the ground state. d Bulk and edge orbital magnetization for both valley-polarized states (Ctot = ±2) calculated based on a continuum model of 1.26° tMBLG. The vertical dashed line divides the parameter space into a small-strain regime where Mbulk and Medge have opposite sign and a large-strain regime where they have the same sign. e Top sketch shows the depletion/occupation of the edge states (represented by dashed/solid grey lines) for different filling levels. Orbital Zeeman energy is plotted for both valley-polarized states (Ctot = ±2) as a function of filling factor when EF is inside the correlation gap. Solid (dashed) lines represent the energetically favourable (unfavourable) states. CFB– lower branch of conduction flat band, CFB+ upper branch of conduction flat band.

The large Berry curvature in the tMBLG moiré flat bands produces out-of-plane orbital magnetic moments that respond to an external magnetic field. The competition between bulk orbital magnetization (Mbulk) due to self-rotation of electron wave packets and edge orbital magnetization (Medge) due to circulation of electrons in the topologically-protected in-gap chiral edge states is responsible for the gate-controlled switching behaviour at ν = 3 (spin magnetism plays no explicit role in the switching due to negligible spin–orbit coupling in tMBLG)8. When the chemical potential resides in the bulk correlation gap the sign of Medge is the same as the sign of Ctot and its magnitude is determined by how much the edge state band is filled20. The sign and magnitude of Mbulk, on the other hand, are sensitive to the detailed band structure and are not simply related to Ctot other than the fact that a reversal in Ctot is accompanied by a reversal in Mbulk (Supplementary Note 6). The total orbital magnetization is the sum of Mbulk and Medge and can be altered electrostatically by changing the filling of the edge states.

Our data is consistent with the special case illustrated in Fig. 4b, c which shows magnetic states for both Ctot = +2 and Ctot = –2 near ν = 3 under the condition that Mbulk and Medge are antiparallel and |Medge| > |Mbulk|. When EF is placed at the top of the correlation gap (\(\nu \,\gtrsim \,3\), Fig. 4b), the chiral edge states are fully filled, and the total orbital magnetization is in the same direction as Medge. When EF is moved to the bottom of the correlation gap (\(\nu \,\lesssim \,3\), Fig. 4c), however, the edge states are empty and the total magnetization is in the direction of Mbulk. In an external magnetic field the system will be driven into a magnetic ground state that aligns the total orbital magnetic moment with the applied field to minimize the orbital Zeeman energy. For the case shown here the energetically favourable states are Ctot = +2 for \(\nu \,\gtrsim \,3\) and Ctot = –2 for \(\nu \,\lesssim \,3\), thus illustrating how electrostatic gating can cause Chern number switching in tMBLG consistent with our experimental observations.

We have performed theoretical simulations that support the assumptions made in the switching picture described above (i.e., that Mbulk and Medge are antiparallel and that |Medge| > |Mbulk|) and that also explain why some tMBLG regions do not exhibit switching. For the experimental regimes shown in Fig. 3e–h (where the tMBLG twist angle lies close to 1.26°) we find that the key physical parameter that controls magnetic switching functionality is hetero-strain. To see this we calculated the strain-induced behaviour of Mbulk and Medge for a continuum model of 1.26° tMBLG in the ν = 3 QAH insulating state with EF set to the top of the correlation gap (a single phenomenological parameter characterizing the electron–electron interaction strength is included similar to ref. 8; see Supplementary Note 6 for details). Figure 4d shows a plot of the resulting Medge and Mbulk for Ctot = +2 (top) and Ctot = –2 (bottom) as a function of hetero-strain with all other parameters kept constant. For small strain (<0.15%) |Medge| is significantly greater than |Mbulk| while Medge and Mbulk are anti-aligned. For large hetero-strain (>0.15%), however, the situation changes significantly. The sign of Medge remains the same as Ctot, but Mbulk flips its sign due to a strain-induced redistribution of Berry curvature and band dispersion throughout the mini-Brillouin zone. This divides the strain parameter space into two regimes separated by the vertical dashed line in Fig. 4d: the small-strain regime where Mbulk and Medge have opposite sign (and switching can occur), and the large-strain regime where they have the same sign (and switching does not occur). To better illustrate this behaviour, Fig. 4e shows a schematic of the resulting orbital Zeeman energy for B > 0 as a function of filling factor near ν = 3. Here the solid lines show the energetically favourable ground state and the dashed lines show the unfavourable state having opposite Ctot. For small hetero-strain Mbulk dominates for \(\nu \,\lesssim \,3\) (when the edge state band is empty) and results in Ctot = –2 whereas Medge dominates for \(\nu \,\gtrsim \,3\) (as the edge states are filled), resulting in a switching to Ctot = +2. For large hetero-strain, Mbulk and Medge both line up parallel to the applied field at all fillings and the system stays at Ctot = +2 without switching, consistent with our experimental observations.

In conclusion, we have observed local spectroscopic signatures of strong correlation and non-trivial topology at the moiré scale in tMBLG, and have demonstrated local control of the QAH Chern number via electrostatic gating. Combining STM and STS allows us to characterize the topological and magnetic switching phase diagram of tMBLG in the parameter space of local twist angle and hetero-strain. We observe magnetic switching only at low strain, revealing the sensitive interplay between correlation, topology, and local structural parameters that determines electronic and magnetic ground states in twisted moiré systems. This provides insight into the many phase diagrams observed in related moiré systems via measurements that average over regions having different structural parameters, and creates new opportunities for future manipulation of QAH insulating domains and the chiral edge states that lie between them.


Sample preparation

Samples were prepared using the “flip-chip” method21 followed by a forming-gas anneal22,23. Electrical contacts were made by evaporating Cr/Au (5 nm/70 nm) through a silicon nitride shadow-mask onto the heterostructure. The sample surface cleanliness was confirmed using contact-AFM prior to STM measurements. Samples were annealed at 300 °C overnight in ultra-high vacuum before insertion into the low-temperature STM stage.

STM/STS measurements

STM/STS measurements were performed in a commercial CreaTec LT-STM held at T = 4.7 K using tungsten (W) tips. STM tips were prepared on a Cu(111) surface and calibrated against the Cu(111) Shockley surface state before measurements to avoid tip artifacts. A voltage VG was applied to the Si back-gate to change the carrier density n in the tMBLG stack (\(n=\frac{{\varepsilon }_{{{{{{\rm{D}}}}}}}{\varepsilon }_{0}{V}_{{{{{{\rm{G}}}}}}}}{e{d}_{{{{{{\rm{D}}}}}}}}\) where εD ≈ 3.6 is the average out-of-plane dielectric constant of hBN and SiO2, ε0 is the vacuum permittivity, e is the elementary charge, and dD = 335 nm is the thickness of the dielectric layers). dI/dV spectra were recorded using standard lock-in techniques with a small bias modulation VRMS = 1 mV at 613 Hz. All STM images were edited using WSxM software24.

Determination of local twist angle and hetero-strain

The local twist angle and hetero-strain were determined by analysing the monolayer–bilayer moiré pattern. The moiré wavelength was obtained for three directions by measuring the spatial separation between peaks in STM topographs and averaging over several moiré unit cells. The reciprocal primitive vectors Ki (i = 0, 1, 2) were derived through Fourier transform analysis. In the presence of hetero-strain, Ki can be approximately written as \({K}_{i}=k\left(\theta+\left(1+{\nu }_{{{{{{\rm{P}}}}}}}\right)\epsilon {{\cos }}\left(\alpha+i\frac{2\pi }{3}\right){{\sin }}\left(\alpha+i\frac{2\pi }{3}\right)\right)\) where k = 4.694 nm–1 is the length of the graphene reciprocal primitive vectors, θ is the twist angle, \(\epsilon\) is the hetero-strain amplitude, νP = 0.16 is Poisson’s ratio for graphene, and α is the angle between the principal axis of the strain tensor and one of the graphene reciprocal primitive vectors25. This allows us to solve both θ and \(\epsilon\) from the extracted Ki.