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Direct observation of magneto-electric Aharonov-Bohm effect in moir\'e-scale quantum paths of minimally twisted bilayer graphene

Aharonov-Bohm (AB) effect, the well-known archetype of electron-wave interference phenomena, has been explored extensively through transport measurements. However, these techniques lack spatial resolution that would be indispensable for studying the magnetic and electrostatic AB oscillations at the nanometer scale. Here, we demonstrated that scanning tunneling microscopy (STM) can be used as an AB interferometer operating on nanometer length scales and the magneto-electric Aharonov-Bohm effect in minimally twisted bilayer graphene (TBG) was directly measured by using STM. In the minimally TBG, there is a triangular network of chiral one-dimensional states hosted by domain boundaries due to structural reconstruction. Taking advantage of the high spatial resolution of the STM, both the magnetic and electrostatic AB oscillations arising from electron interference along moir\'e-scale triangular quantum paths in the minimally TBG were measured. Our work enables measure and control of the AB effect and other electron-wave interference at the nanoscale.


2
The phase of electron wave packets circling along a closed trajectory can be modulated by electronmagnetic potentials, which is now known as the Aharonov-Bohm (AB) effect [1][2][3][4][5][6][7][8][9][10][11] . As schematically shown in Fig. 1a, one coherent electron beam splits into two moving along two paths and they interfere after recombination. The phase difference between two paths of electron beams is ∆ = 2 ( /ℎ)Φ ( Φ is the magnetic flux encircled by the paths, h is Planck's constant and e is the electron charge), which generates cycles of destructive and constructive interference of the electron wave functions. The phase tuning strongly affects the magnetoresistance in small devices and gives rise to conductance oscillations with the period of one flux quantum Φ 0 = ℎ/ in transport measurements. Consequently, the AB effect has been studied extensively in transport techniques [2][3][4][5][6][7][8][9] . In the meanwhile, the density of states (DOS) along the paths of electrons are also sensitive to the interference of the wave functions. Scanning tunneling microscopy (STM) is known as a powerful tool to measure the local DOS.
Therefore, it was proposed to use STM as an AB interferometer to directly study the AB effect 12,13 . However, such a measurement has so far not been achieved. Here we report on the first measurement of the AB effect by using STM and our experiment demonstrates that the high spatial resolution of the STM enables us to detect both the magnetic and electrostatic AB oscillations in a moiré-scale triangular quantum path in minimally twisted bilayer graphene (TBG).
In the TBG with twist angle θ smaller than a characteristic angle θc (θc ~ 1.1°), there is a strain-accompanied lattice reconstruction that results in large triangular Bernal (AB and BA) stacking domains and a triangular network of domain walls (DWs) [14][15][16][17][18] , as schematically shown in Fig. 1b. When the bandgap is opened in the Bernal-stacking regions, the domain walls can host topological chiral conducting states [19][20][21][22][23][24] , which provide a unique platform to explore the AB effect [25][26][27] . In a recent transport experiment, giant magnetic AB oscillations were observed in the triangular network of domain walls in the minimally TBG and the main period of the oscillations was attributed to interference in paths encircling a AB or BA domain 26 . Very recently, analysis based on a phenomenological scattering theory indicated that the main period of the magnetic 3 AB oscillations should correspond to paths encircling a moiré cell 27 , comprising an AB and BA triangle. The discrepancy between the experiment and theory is attributed to the ambiguity in the determination of the twist angle because the lack of spatial resolution in transport techniques 27 . The high spatial resolution of the STM can naturally overcome this limitation and accurately measure the area encircled by the paths. In this work, not only the magnetic AB oscillations, but also the electrostatic AB effect in a moiré-scale triangular quantum path in the minimally TBG are directly measured by using the STM.
In our experiment, the TBG were obtained by transferring layer by layer of the largearea aligned monolayer graphene onto a Nb-doped SrTiO3 (0.7%) substrate with controlled twist angles, as reported in ref. 16  To further characterize the TBG, we also carried out scanning tunneling spectroscopy (STS) measurements in different stacking regions (see Supplementary Sections 3-5 for details). In the AA stacking regions, a pronounced DOS peak, which arises from the low-energy flat bands in the minimally TBG 28,29 , is obtained. In the AB and BA stacking 4 regions, a finite gap, which is generated by the inequivalent charge densities of the two topmost adjacent layers [22][23][24][30][31][32][33] , is observed in the tunneling spectra. In the meanwhile, There are other possible origins, such as Shubnikov-de Haas (SdH) oscillations 34,35 , that can lead to oscillations of the DOS modulated by magnetic fields. Usually, the period of these oscillations is sensitive to the measured energy, which is quite different from the AB effect with the period that is independent of the energy. To further explore the origin of the observed DOS oscillations in the 0.18° TBG, we measured the DOS oscillations at different energies, as shown in Fig. 3a. Obviously, the periodicities of the DOS oscillations are almost independent of the energy, further confirming that the oscillations arise from the AB effect. In the 0.18° TBG, there is a large heterostrain that strongly modifies the local period of the morié superlattice 17,18,36,37 , as schematically shown in Fig. 3b, which strongly affects the observed AB oscillations. With assuming that the oscillations arise from the AB effect, the expected periods of magnetic field ∆B can be roughly estimated as 2.58 T, 1.97 T, 1.18 T, and 0.66 T for electrons propagating along four different paths around the measured position (Fig. 3b). It is interesting to note that the calculated values of the ∆B are consistent with the observed AB resonances in our experiment (Fig. 3a), further confirming that the DOS oscillations arise from the magnetic AB effect. The advantage of our experiment is the ability to obtain the area encircled by the quantum paths and the periods of magnetic field simultaneously.
According to our experiment, the main period of the oscillations should be attributed to interference in paths encircling a AB or BA domain, rather than a morié pattern.
The high-spatial resolution of the STM enables us to measure the AB effect at the nanoscale. In conclusion, we demonstrated for the first time that STM can be used to study the AB effect at the nanoscale. Thanks to the high spatial resolution of the STM, both the magnetic and electrostatic AB effects are measured in a moiré -scale triangular quantum path in the minimally TBG.