Probing the wavefunctions of correlated states in magic angle graphene

Using scanning probe microscopy and spectroscopy, we explore the spatial symmetry of the electronic wavefunctions of twisted bilayer graphene at the"magic angle"of 1.1 degrees. This small twist angle leads to a long wavelength moir\'e unit cell on the order of 13 nm and the appearance of two flat bands. As the twist angle is decreased, correlation effects increase until they are maximized at the magic angle. At this angle, the wavefunctions at the charge neutrality point show only C2 symmetry due to the emergence of a charge ordered state. As the system is doped, the symmetry of the wavefunctions change at each commensurate filling of the moir\'e unit cell pointing to the correlated nature of the spin and valley degeneracy broken states.

When two layers of graphene are twisted away from each other at the "magic angle", around 1.1 degree, the long wavelength moiré pattern leads to the folding of the band structure into a mini-Brillouin zone and the formation of ultra-flat bands [1][2][3][4][5][6][7].As the twist angle approaches the magic angle, the bands become flatter and the ratio between the Coulomb interaction and their bandwidth increases leading to increased correlation effects [8][9][10][11][12][13][14][15][16][17][18][19][20] and broken symmetry states [21][22][23].Superconductivity and correlated insulating states [24][25][26][27][28] have been discovered in magic angle twisted bilayer graphene (MATBG) when the flat bands are tuned to commensurate filling factors ν = 0, ±1, ±2, ±3 electron per moiré unit cell.For ν = ±1(± 3), there is only one electron(hole) in one of the flat bands, hence both the spin and valley degeneracy must be broken while for even filling factors one of these degeneracies must be restored.Due to the correlated nature of these states, small changes in the filling factors can lead to dramatic changes in the wavefunctions.Transport studies [15,[26][27][28] have revealed that the phenomenology of the commensurate states at odd filling levels are different from those at even filling levels.Theory calculations [15,21,29,30] suggest that odd and even filling levels will lead to different symmetry breaking phases.Although there is still considerable theoretical uncertainty in the exact nature of the wavefunctions at each of the commensurate fillings.Our STM/STS study focuses on the difference in the wavefunctions between even and odd filling factors, which has not been explored by previous STM studies on MATBG [22,23,31,32].We find that adding electrons to the flat bands leads to change in the symmetry and localization properties of the wavefunctions.levels, while the gate voltage (Vg) applied between the Si substrate and the sample is used for tuning the doping level of the MATBG.Figure 1(b) shows an STM topography image of a 1.10° MATBG moiré pattern, the twist angle is determined by measuring the moiré wavelengths in 3 different directions (L1, L2, L3) and using a uniaxial heterostrain model [23].The bright spots in the topography image correspond to the AA sites of the moiré superlattice, other high symmetry sites are labeled as AB, BA.The atomic arrangements of these high symmetry points are illustrated in the schematic for a 5° moiré pattern in Fig. 1(c).Figure 1(d) shows the scanning tunneling spectroscopy (STS) on the AA and AB sites when the flat bands are fully occupied.The two sharp peaks between -0.05V and 0V in the spectroscopy of the AA site correspond to the strong local density of states (LDOS) of the flat bands, while the LDOS in this energy range is much weaker on the AB site, indicating that the flat bands are localized on the AA site [3,5].
To study the electronic properties of the flat bands at different doping levels, we performed STS measurements on the AA site as a function of both Vb and Vg, as shown in Fig. 2(a) for a 1.07° twist angle moiré.For each Vg, the tip height was stabilized at a bias voltage of 0.1V and tunnel current of 50 pA.Then the feedback circuit was switched off, a small ac voltage (1 mV) was applied to the bias voltage and the differential conductance dI/dV was measured as a function of bias voltage using lock-in detection.The tip gating effect was corrected from the data [32], see supplementary materials [34] for details.When the two flat bands are fully filled (Vg >14 V) or fully depleted (Vg <-50 V), they are close together with a peak to peak separation of ~14 mV and their energy shifts linearly with respect to Vg with a slope of ~3 mV per volt in Vg.In contrast, in the region where the flat bands are overlapping with the Fermi level (-50 V< Vg<14 V), the energy of the flat bands changes much slower with respect to Vg.On top of this, various new features of the flat bands emerge: (1) The total band width of the flat bands, defined as the full width at the half maximum on either side of the flat bands, indicated in Fig. 2(b) and Fig. 2(c), is broadened in this region, shown by the black curve in Fig. 2(g).The overall broadening indicates stronger electron-electron interactions and the breakdown of a single particle picture [31,35].Such broadening is not present in the spectroscopy of non-magic angle devices, as shown in supplementary materials [34].
(2) The separation between the two flat bands is enhanced near the charge neutrality point (CNP, defined as where the conduction band and valence band are equidistant from the Fermi level and have the same peak LDOS) around Vg = -26 V, which is a result of stronger interlayer exchange interactions [31,32,35].To illustrate that this effect is strongest near the magic angle of 1.1°, we measured the total band width at CNP (WCNP) and when both bands are fully depleted (WFD) from dI/dV measurements of different twist angle devices.The total band width as a function of angle is plotted in Fig. 2(e), while the band width decreases with the twist angle, the difference between WCNP and WFD increases at small angle.As shown in Fig. 2(f), the ratio between WCNP and WFD shows a peak around 1.07°, confirming that the exchange interaction is indeed strongest around the magic angle.Other angle dependent quantities such as separation between the two flat bands, individual band widths and the separation between the flat bands and remote bands are presented in supplementary materials [34].
(3) Near the CNP, the width of the upper conduction band is wider than the lower valence band in the pdoped region (Vg < -26 V) and vice versa in the n-doped region (Vg > -26 V), see the red and blue curves in Fig. 2(g).The spectral weight redistribution between the two flat bands as a function of doping is consistent with previous experiments suggesting a correlated chargeordered phase [22], Mott-insulating states as well as superconductivity near CNP have also been observed in transport measurements [27].
(4) In the range of -15 V< Vg<13 V when the lower band is fully filled and the upper band is partially filled, both bands show various distortions.Dips appear in the LDOS in the upper band at certain doping levels and the lower band is broadened when the upper band is partially filled.To better identify the location of the dips and compare our STS results with transport experiments [24][25][26][27][28], we plot the LDOS at the Fermi level as a function of Vg in Fig. 3(a), which is a horizontal line cut from Fig. 2(a) at zero bias voltage.Consistent with transport measurements [24][25][26][27][28], strong insulating states appear when the bands are fully filled (Vg >14 V), fully depleted (Vg <-50V) and around the CNP (-46 V<Vg < -20 V).Additional dips in the LDOS show up around Vg = -12 V, -1 V and 9 V, similar to the insulating states with filling factors of ν = 1, 2 and 3 observed in transport measurements [24][25][26][27][28]. From the individual dI/dV curves at these gate voltages in Fig. 3(b), a soft gap like structure appeared at the Fermi level for Vg = -12V (ν = 1) and Vg = -1V (ν = 2) but not for Vg = 9V (ν = 3).
The asymmetry of the dI/dV as a function of Vg as well the missing a gap at ν = 3 is because of tip-induced band bending [31].With different tip conditions, we observed symmetric gate dependent spectroscopy with no tip-induced bending as well as asymmetric gate dependent spectroscopy with tip-induced bending in a different direction, as shown in supplementary materials [34].The dI/dV signal in Fig. 2(a) is much stronger when the flat bands are fully filled, this is because we are setting the tip at a positive voltage (0.1V) when starting to take the spectroscopy.When the bands are fully filled, there is almost no density of states present from the Fermi level up to the positive tip voltage, thus the tip height will be stabilized closer to the sample in order to reach the set current, which in turn increases the dI/dV signal.By setting the starting voltage at a negative voltage, the dI/dV signal is stronger when the flat bands are fully depleted, see supplementary materials [34].Although gap opening and insulating behavior has been observed at all commensurate fillings: ν = 0 (CNP), ±1, ±2 and ±3.The intrinsic gap driving mechanism for these states can be very different [15,26,30,35].Parallel field dependent transport measurements [26] have found that the ν = ±1 and ν = ±3 states are spin-polarized while the ν = ±2 states are not.Ferromagnetic state [28] and intrinsic quantized anomalous Hall effect [19] have been found in the ν = 3 state, Chern insulating states have been discovered for the ν = ±1 states [27].To highlight the difference between the different commensurate filling states, which have not been addressed in the previous STM studies on MATBG [22,23,31,32], we measure the spatial profile of the wave functions at various energy and doping levels by mapping the LDOS with different Vb and Vg.The LDOS maps at the energies of the flat bands for different commensurate fillings are shown in Fig. 4, each one of them are averaged over several unit cells and plotted in the Wigner-Seitz cell of the moiré lattice centered at the AA site.The dI/dV value is normalized so that the summation of dI/dV over the unit cell is one then the average value is subtracted, thus the average value will always appear as white color in the color plot, independent of the setting of the color scale.In order to compare these LDOS maps between different fillings, all images are plotted under the same color scale.The strain directions that are extracted from the uniaxial heterostrain model [23] are plotted in Fig. 4(a) with respect to the unit cell, where the strain percentage was found to be ε = 0.17% using the Poisson ratio δ = 0.16 for graphene.Figure 4(b) shows the unit cell topography of the same area, which has almost the full C6 symmetry, only slightly altered by the strain.In contrast, the LDOS maps at CNP, figure 4(c), show roughly only C2 symmetry with the symmetry axes perpendicular to each other between the lower band (Vb = -18 mV) and the upper band (Vb = 14 mV), which is a result of a stripe charge ordered state [22].When both bands are fully filled, figure 4(d), similar broken symmetry still exists but the shape of the wavefunctions for the lower (Vb =-106 mV) and upper (Vb =-90 mV) bands are not as perpendicular to each other as they are in Fig. 4(b), indicating weaker interactions.LDOS maps at other energies and doping levels, as well as a qualitative study of the anisotropy and localization are present in supplementary materials [34].
Interestingly, the nature of the wavefunctions changes dramatically when the bands are partially filled, as shown in Fig. 4(e) and Fig. 4(f) for ν = 1 and ν = 2.The wavefunctions of the fully filled lower band (Vb = -32 mV for ν = 1, Vb = -23 mV for ν = 2) are much less localized compared to Fig. 4(b) and Fig. 4(e).This delocalization could be related to the broadening of the lower band when the upper band is partially filled as shown previously in Fig. 2.
For the partially filled upper band, the wavefunctions are no longer localized at AA sites when ν = 1, the occupied state (Vb = -7 mV) and the unoccupied state (Vb = 5 mV) are localized on opposite locations somewhere between AA (center of the unit cell) and the boundary of the unit cell.On the other hand, the wavefunctions of the upper band for ν = 2 are still localized on the AA sites.The shape of these wave functions also show a certain degree of rotational symmetry breaking but they are not perpendicular to each other, which provides further evidence that the perpendicularity of wavefunctions at the CNP originates from a charge ordered state [22], since both the occupied and unoccupied wavefunctions here at ν = 2 are from the upper band.The stark difference of the conduction band wave functions between ν = 1 and ν = 2 fillings indicates that they are intrinsically different correlated states, consistent with the fact that the ν = 1 state is both spin and valley polarized while the ν = 2 state can only either be spin-polarized or valley polarized [15,26,27,30,35].
In summary, we have shown that the wavefunctions of the insulating state at CNP is consistent with stripe charge order [22].The insulating states at ν = 1 and ν = 2 have distinct wavefunctions, their lower valence band wavefunctions are delocalized while their conduction band wavefunctions are localized on different sites.
Our findings are consistent with transport measurements [24][25][26][27][28] and previous STM studies [22,23,31,32] but highlight the intrinsic difference between the correlated insulating states at different commensurate fillings, which was not discovered previously.By direct measurement of the LDOS maps and the localization of these wavefunctions, we visualized the spatial dependence of flat band wavefunctions as a function of doping level, providing a deeper understanding towards the nature of correlated states in MATBG at various commensurate fillings.

Tip gating correction for dI/dV measurements
In the moiré region close to the magic angle, quantum dots can be induced by the formation of insulating states [1], they appear as constant density resonance lines in Fig. S1, indicated by the red arrows.In the uncorrected image, Fig. S1(a), the constant density lines are slopped due to tip gating effects.In Fig. S1(b) we correct the tip gating effect by shifting the dI/dV curves to make the constant density lines vertical, thus the bias voltage and carrier density are decoupled in the corrected image.The tip gating correction does not change the appearance of the correlated effects as we discussd in the main text, however, it enables more accuate measurement of band widths.

Non-magic angle spectroscopy
To show that the correlation effects we discussed in the main text are only present in the moiré flat bands close to the magic angle, Fig. S2 shows the gate dependent STS spectroscopies for 1.30° and 2.31° twist angles.Consistent with pervious observations [2], the van Hove singularity (VHS) peaks are still present for both areas, however, in contrast to the magic angle area, there is no broadening of the peaks when they are partially filled.The separation between the peaks is almost constant for all gate voltages, and no significant distortion of the peaks occurs near commensurate fillings.In the main text, we have shown the total band width of both flat bands decreases with angle and the difference between the total band width at the CNP and FD is maximum around the magic angle.Figure S3(a, b) shows that the band widths of the individual bands also decrease with angle, but the difference between the CNP and FD do not show clear indication of enhancement around magic angle, suggesting that the enhanced separation at CNP is originated from inter-band interactions.Figure S3(c) shows the VHS separation decreases with angle, the ratio of VHS separation at the CNP compared to FD also increases at small angle as shown in Fig. S3(d).Figures S3(e,f) show the gap between the valence flat band and lower dispersive band and the gap between conduction flat band and the higher dispersive both decrease with angle, consistent with theoretical predictions [3] and previous STS measurements [1].The gate dependent STS measurements often show asymmetric shapes in dI/dV as a function of gate voltages, this tip band bending effect can be explained by the difference in the work function between the STM tip and the sample [4].With different tip conditions, we observed symmetric spectroscopy in Fig.

FIG. 1 .
FIG. 1.(a) Schematic of the experimental setup showing the STM tip and an optical microscope image of the measured MATBG sample.(b) Atomic resolution STM topography of the 1.10° moiré superlattice.(c) Schematic of different atomic stacking arrangement arising from the twist of the two layers of graphene.(d) STS spectroscopy of the MATBG on AA and AB sites of 1.10° moiré when the flat bands are fully filled, where Vg = 8V.

Figure 1 (FIG. 2 .
Figure1(a) shows a schematic of the experimental setup.All measurements were performed in ultra-high vacuum at a temperature of 4.6 K. Samples were fabricated by a dry transfer technique with controlled rotational alignment between the two layers of graphene[33].A bias voltage (Vb) applied between the tip and the sample is used for probing different energy

FIG. 3 .
FIG. 3. (a) Gate dependent LDOS at the Fermi level on the 1.07° moiré superlattice, colored dashed line corresponding to different commensurate fillings.(b) dI/dV curves at different commensurate fillings, vertical dashed line marks the fermi level.

Figure
Figure S1: (a) Gate dependent STS spectroscopy on the AA site of the 1.07° moiré superlattice before the tip gating correction.(b) Gate dependent STS spectroscopy after the bias voltage is decoupled from the carrier density.

Figure
Figure S2: (a) Gate dependent STS spectroscopy on the AA site of the 1.30° twist angle device.(b) Gate dependent STS spectroscopy on the AA site of the 2.31° twist angle device.

Figure S3 :
Figure S3: Angle dependence of (a) conduction band width, (b) valence band width, (c) VHS separation, (d) ratio of VHS separation between CNP and HD, (e) gap between the valence flat band and the lower dispersive band, (f) gap between the conduction flat band and the higher dispersive band.

Figure
Figure S4: (a) Gate dependent STS spectroscopy on the AA site of the 1.11° moiré superlattice without strong tip band bending effect.(b) Gate dependent STS spectroscopy on the AA site of the 1. 09° moiré superlattice with strong tip band bending effect.