To interpret our observations, we consider various mechanisms that have been theoretically proposed to induce potential. Following ref. 10, the moiré Hamiltonian can be decomposed into three terms: an effective pseudoelectric potential (H 0 ), a pseudomagnetic field (H xy ) and a local mass term (H z ), each associated with a corresponding sublattice Pauli matrix (Supplementary Information section 13). The dominant H 0 term arises from two effects: (1) the spatially varying stacking potential generated by changes in the local G/hBN alignment within the moiré cell and (2) the in-plane relaxation of the graphene lattice, which stretches and compresses the C–C bonds, leading to local variations in the Dirac point energy relative to the vacuum level and captured by the deformation potential9,44.
Figure 4 shows the predicted stacking (Fig. 4a) and deformation (Fig. 4b) pseudopotential terms comprising H 0 , along with the pseudomagnetic field H xy (Fig. 4c), with the three high-symmetry CB (carbon above boron), CN (carbon above nitrogen) and AA (carbon above both boron and nitrogen) stacking sites marked. Generally, a pseudoelectric potential may reflect energy changes that are not electrostatic in nature. However, theory44 and density functional theory calculations45,46 suggest that the pseudopotentials in Fig. 4a,b are directly accounted for by charge polarization perpendicular to the layers and therefore manifest as real electrostatic potentials.
Fig. 4: Theoretical breakdown of the various physical mechanisms contributing to the moiré potential in aligned G/hBN. a, Stacking pseudopotential due to the relative stacking of G and hBN, which varies within the moiré unit cell. The three high-symmetry points corresponding to local CB, CN and AA stacking are marked. b, Deformation pseudopotential due to atomic relaxation within the graphene layer. The terms in a and b both appear in the H 0 part of the Bloch Hamiltonian, corresponding to the identity matrix in the sublattice basis. c, Magnitude of the pseudomagnetic field, which appears in the H xy part of the Hamiltonian, corresponding to the σ x and σ y Pauli matrices in the sublattice basis. d–f, Self-consistent electrostatic potentials obtained after considering the screening by the graphene carriers, calculated by including a self-consistent Hartree potential response using a carrier density corresponding to v = 4. All terms show strong C 3 symmetry around the moiré centre, in contrast to the C 6 symmetry observed in the experiments. g, Self-consistent stacking and deformation potentials are plotted along a linecut through the moiré centre (dashed white line, bottom inset). The CB, CN and AA high-symmetry points are labelled. Visibly, each of the two terms (blue, pink) shows a strong C 3 symmetry. However, owing to cancelling contributions, their sum (purple) exhibits an approximate C 6 symmetry, with only a small difference between the potential minima at the CB and AA stacking sites. h, Total self-consistent potential calculated for v = 0. This potential resembles the experiment in terms of the approximate C 6 symmetry, but its magnitude is half of that measured experimentally. Full size image
Graphene carriers redistribute to screen these pseudopotentials, producing the self-consistent electrostatic potential measured by our detector. We model this using self-consistent Hartree calculations (Supplementary Information section 13). The resulting screened potentials from the components in Fig. 4a–c are shown in Fig. 4d–f. Notably, screening preserves the shape of the pseudoelectric potentials but reduces their magnitude by approximately 2.2, consistent with the predicted random-phase-approximation dielectric constant ϵ = 1 + πα/2 ≈ 2.0, where \(\alpha =\frac{{e}^{2}}{4{\rm{\pi }}\kappa {{\epsilon }}_{0}\hbar {v}_{{\rm{F}}}}\) is the fine-structure constant of graphene, v F is its Fermi velocity and κ = 3.5 is the hBN dielectric constant47. Furthermore, we find that electronic screening converts the pseudomagnetic field H xy into an electrostatic potential (Fig. 4f). This potential is small compared with the other two, scales linearly with ν, and becomes identically zero at v = 0 (same for H z ; Supplementary Information section 13). As our experiments show only a minor v dependence, we omit the H xy and H z terms in further discussions.
Both leading potential terms (Fig. 4d,e) exhibit a clear C 3 symmetry around the central CN site, in contrast to the approximate C 6 symmetry observed experimentally. However, examining the minima at the CB and AA sites shows that these C 3 symmetries are inverted—for the first term ϕ AA > ϕ CB and for the second term ϕ CB > ϕ AA . The two terms compensate each other to form an almost C 6 -symmetric total potential with a pronounced central peak (Fig. 4g). The resulting total self-consistent potential (Fig. 4h) strongly resembles the experimental result, with one notable exception—the experimental potential scale is double the theoretical prediction. One explanation might be that theory underestimates strain in the moiré interface21. However, increasing strain alone would yield a more C 3 -symmetric potential, contrasting our observations. This large discrepancy demonstrates that, despite G/hBN being one of the most relevant and extensively studied moiré interfaces, there are substantial gaps in its theoretical understanding, which has direct consequences for recent experiments that use this interface to design new states of matter (for example, FQAHE in moiré pentalayer graphene).
Finally, we show the dependence of the moiré potential on the distance from the moiré interface. Extended Data Fig. 1 presents potential traces measured by two defects, located at approximately 0.8 nm (D2) and 1.5 nm (D1) from the interface (Supplementary Information section 3). The measured potential decays rapidly, even over these small distances. This significant drop suggests that if the detector were at a moiré distance (h = λ m ) away from the interface, it would have detected only \({{\rm{e}}}^{-\frac{4{\rm{\pi }}}{\sqrt{3}}\left(\frac{h}{{\lambda }_{{\rm{m}}}}\right)\sqrt{\frac{{{\epsilon }}_{\parallel }}{{{\epsilon }}_{\perp }}}}\) ≈ 10–4 of the potential, underscoring the importance of our atomic SET operating at extremely close standoff distances. At the same time, these measurements also demonstrate that in thin flakes, such as pentalayer graphene, electrons can still experience a significant moiré potential (tens of mV) even in the farthest graphene layer.
The atomic SET scanning probe technique demonstrated here has a combination of features that are extremely powerful for studying a wide range of quantum materials. Its QTM geometry allows it to scan within the pristine interfaces of a variety of vdW materials. Similar to existing SETs, this technique will allow quantitative measurements of thermodynamic properties such as the electronic compressibility24,25,48 and entropy49, but now with two orders of magnitude improved spatial resolution—below the Fermi wavelength, magnetic length and moiré scales of many systems. This advance extends this powerful imaging method to a much broader class of physical phenomena occurring on small scales such as Wigner crystals, topological edge states, vortex charges, symmetry-broken phases and fractionally charged quasiparticles.