Extended Data Fig. 1: High-resolution momentum-resolved elastic tunnelling near the commensurate angle of 21.8°.

a, Measured conductance (G) as a function of twist angle (θ) and bias voltage (V_b), with a finite carrier density in both graphene layers induced by a back-gate voltage of V_bg = 4 V. b, The second derivative, \(\frac{{{\rm{d}}}^{2}I}{{\rm{d}}{V}_{{\rm{b}}}^{2}}\), obtained numerically from panel a. c, The derivative of conductance with respect to twist angle, \(\frac{{\rm{d}}G}{{\rm{d}}\theta }\), obtained numerically from panel a. The measured conductance clearly reveals a distinct ‘double X’ structure, within which the conductance is enhanced. This structure exhibits a mirror symmetry around θ = 21.8° and an approximate mirror symmetry with respect to bias. The boundaries of this structure are even more pronounced in the \(\frac{{\rm{d}}G}{{\rm{d}}\theta }\) plot, in which they appear as narrow red and blue lines. Superimposed on these features are nearly horizontal conductance steps, which are more prominent in the second derivative \(\frac{{{\rm{d}}}^{2}I}{{\rm{d}}{V}_{{\rm{b}}}^{2}}\) plot, in which they manifest as horizontal lines. These lines correspond to inelastic, momentum-conserving phonon emission processes, as described in detail in the main text. The extra ‘double X’ structure around θ = 21.8° arises from elastic momentum-conserving tunnelling between overlapping Fermi surfaces at the corners of the third BZ (see illustration in Extended Data Fig. 2). These Fermi surfaces are associated with high-momentum components of the in-plane wavefunctions, which decay more rapidly in the z direction, explaining why the experiment operates in the tunnelling regime despite the two layers being in contact. The dashed black lines in all panels represent theoretically calculated conditions for which the Fermi surface of one layer touches the unoccupied energy bands of the other layer, marking the onset of momentum-conserving tunnelling. The various conditions are illustrated in panel d and their calculated trajectories in the θ–V_b plane show excellent agreement with the experimental data. Notably, this agreement is achieved with no free parameters (the geometric capacitance used in these calculations was determined by fitting the phonon spectrum measurements in the main text). As explained in the corresponding Methods section, this correspondence puts a tight upper bound on the strains of the graphene layers in the tip and on the flat substrate, to be smaller than 0.1%.