GoKawiilCryogenic QTM
All measurements in this work were performed in a cryogenic QTM system operating at a temperature of 4 K, as described previously45. For conductance measurement, voltage biases are applied using a custom-built digital-to-analogue converter array, capable of supplying both d.c. and a.c. signals with 1 μV resolution. Currents were measured using a FEMTO current amplifier, followed by a National Instruments sampler.
Fabrication and characterization of the QTM tip and van der Waals device
The fabrication of both QTM tips and van der Waals devices on flat substrates were described previously45,46. Briefly, we use the standard dry transfer and polymer membrane transfer techniques to fabricate the TBG sample and the QTM tip, respectively. Extended Data Fig. 1a shows an optical image of the TBG heterostructure, comprising bilayer-WSe 2 /TBG/hexagonal boron nitride (h-BN)/Pt on an approximately 100 μm width and 80 μm tall Si/SiO 2 pillar. Extended Data Fig. 1b shows an optical image of the MLG backed by h-BN and graphite placed on a QTM tip.
In Extended Data Fig. 1c, we characterize the TBG sample by conductive AFM scan of the region of magic twist angle carried out at room temperature (Bruker Icon). To compensate for the thermal drift during this AFM scan, we performed a small window scan at rate 10.9 Hz, such that the time for an entire scan is around 20 s. We apply an approximately 1 V d.c. voltage bias between the tip and sample and measure the tunnelling current. The scan image in Extended Data Fig. 1c shows both the moiré lattice and atomic defects in WSe 2 (bright spots). To accurately correct the thermal drift, we trace the position of three defects inside the scan window. We then obtain the drift velocity by following the shift of defects positions between consecutive scans and assuming that, within a single 2D scan, the drift velocity is constant. Extended Data Figure 1c,d shows the image and corresponding fast Fourier transform (FFT) after thermal drift correction. The three moiré lattice vectors obtained from this image are: 12.72 nm, 12.73 nm and 13.14 nm. This gives a twist angle of θ TBG = 1.1° ± 0.02° and strain of ϵ = 0.03% ± 0.02%.
In Extended Data Fig. 1e, we characterize the QTM tip by in situ imaging of the tip contact area by an atomic defect in the WSe 2 barrier as a localized tunnelling channel. At large bias (V b ≈ −900 mV, well outside the range used in the actual spectroscopy measurements), the defect level becomes energetically accessible and provides an extra, spatially localized current path. When the QTM tip passes over the defect, we observe a small but measurable increase in current. By scanning the tip across such defects and mapping this current enhancement, we obtain a real-space image of the tip’s contact footprint.
Extended Data Fig. 1e presents such a high-bias scan, showing several images of the tip footprint, each produced by a different defect, revealing a contact size of approximately 200 × 50 nm. These real-space dimensions set the momentum-space resolution through the Heisenberg uncertainty relation. The measured footprint corresponds to angular momentum resolutions (in angular units) of roughly δθ QTM ≈ 0.05° and ≈ 0.2° along the two principal directions. In Extended Data Fig. 2, we show that the sharpest momentum features—obtained in the remote bands—exhibit a full width at half maximum (FWHM) of δθ QTM ≈ 0.1°, consistent with the resolution expected from the measured tip dimensions.
Energy and momentum resolution
We determine the energy resolution of our QTM measurements by analysing a sharp spectral feature at ν = −0.7 in Fig. 4a (reproduced in Extended Data Fig. 2a). A linecut through this feature yields a FWHM of δE ≈ 10 meV by fitting a Lorentzian function. This is an upper bound to the energy resolution of the measurements and it may well be coming from the intrinsic lifetime of the energy bands of MATBG at this temperature. To determine the momentum resolution, we analyse a sharp spectral feature of the remote energy bands of MATBG measured under the same conditions as in Fig. 2a. Extended Data Fig. 2b plots a linecut of dI/dV along the dashed white line, cutting through the dispersive bands, yielding an angular FWHM of δθ = 0.1° by fitting a Lorentzian function. This translates into a δk ≈ 0.03 nm−1 momentum resolution. For the flat bands, we trace the dI/dV peak width versus θ QTM at zero bias around the Γ point and identify the peak width δθ ≈ 0.5°. This peak width is composed of two peaks overlapping each other, such that the single peak broadening is δθ ≈ 0.25°.
Electrostatics of the QTM junction
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