Probing the wave functions of correlated states in magic angle graphene

Using scanning probe microscopy and spectroscopy, we explore the spatial symmetry of the electronic wave functions of twisted bilayer graphene at the “magic angle” of 1.1°. This small twist angle leads to a long wavelength moiré 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 wave functions at the charge neutrality point show reduced symmetry due to the emergence of a charge ordered state. As the system is doped, the symmetry of the wave functions changes at each commensurate filling of the moiré unit cell pointing to the correlated nature of the spin and valley degeneracy broken states.

Figure 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 arrangements arising from the twist of the two layers of graphene. (d) $dI/dV$ spectroscopy of the MATBG on AA and AB sites of 1.10° moiré when the flat bands are fully filled, where $V_g=8\mathrm{V}$.

Figure 2

(a) Gate-dependent STS spectroscopy on the AA site of the 1.07° moiré superlattice. Dashed lines indicate the position of commensurate fillings: $\nu =-4$ (red), $\nu =0$ (black), $\nu =1$ (purple), $\nu =2$ (green). (b) $dI/dV$ curve at CNP, $V_g=-26\mathrm{V}$. (c) $dI/dV$ curve at FD, $V_g=-54\mathrm{V}$. (d) Flat band width at two different fillings as a function of twist angle. (e) The ratio of flat band width at two different fillings as a function of twist angle. (f) The flat band width extracted from (a) as a function of $V_g$.

Figure 3

(a) Gate-dependent LDOS at the Fermi level on the 1.07° moiré superlattice, colored dashed lines correspond to different commensurate fillings. (b) $dI/dV$ curves at different commensurate fillings, vertical dashed line marks the Fermi level. (c) Illustration of the evolution of the flat bands at different filling levels. Pink and gray correspond to the upper band and lower band, respectively, while solid dots are electrons and circles are holes.

Figure 4

(a) Uniaxial strain directions relative to the unit cell. (b) Topography of the unit cell. (c) LDOS maps at CNP ($\nu =0$), dashed ovals highlight the shape of the wave functions, and dashed lines show the symmetry axes. (d) LDOS maps when the upper band is fully filled ($\nu =4$). (e) LDOS maps when the upper band has one electron ($\nu =1$). (f) LDOS maps when the upper flat band is half filled ($\nu =2$).

Figure 5

(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 6

(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 7

Angle dependence of (a) the conduction-band width, (b) the valence-band width, and (c) the VHS separation. The red circles are when the band is fully depleted, and the black circles are at the CNP. Angle dependence of (d) the ratio of the VHS separation between CNP and FD, (e) the gap between the valence flat band and the lower dispersive band, and (f) the gap between the conduction flat band and the higher dispersive band.

Figure 8

(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.

Figure 9

(a) Gate-dependent STS spectroscopy on the AA site of the 1.08° moiré superlattice with a set point voltage of 0.15 V. (b) Gate-dependent STS spectroscopy on the AA site of the 1.08° moiré superlattice with a set point voltage of −0.15 V.

Figure 10

(a) LDOS maps at different energies for $\nu =1$. (b) Anisotropy extracted from (a) as a function of bias voltage. (c) Radial distribution extracted from (a) as a function of bias voltage. (d) Color plot plotted against left axis: histogram of (a); black curve ploted against right axis: STS spectroscopy on the center of the AA site.

Figure 11

(a) LDOS maps at different energies for $\nu =0$. (b) Anisotropy extracted from (a) as a function of bias voltage. (c) Radial distribution extracted from (a) as a function of bias voltage. (d) Color plot plotted against left axis: histogram of (a); black curve plotted against right axis: STS spectroscopy on the center of the AA site.

Figure 12

(a) LDOS maps at different energies for $\nu =2$. (b) Anisotropy extracted from (a) as a function of bias voltage. (c) Radial distribution extracted from (a) as a function of bias voltage. (d) Color plot plotted against left axis: histogram of (a); black curve plotted against right axis: STS spectroscopy on the center of the AA site.

Figure 13

(a) LDOS maps at different energies for $\nu =3$. (b) Anisotropy extracted from (a) as a function of bias voltage. (c) Radial distribution extracted from (a) as a function of bias voltage. (d) Color plot plotted against left axis: histogram of (a); black curve plotted against right axis: STS spectroscopy on the center of the AA site.

Figure 14

(a) LDOS maps at different energies for $\nu =4$. (b) Anisotropy extracted from (a) as a function of bias voltage. (c) Radial distribution extracted from (a) as a function of bias voltage. (d) Color plot plotted against left axis: histogram of (a); black curve plotted against right axis: STS spectroscopy on the center of the AA site.

Figure 15

(a) LDOS maps at different energies for ν = −4. (b) Anisotropy extracted from (a) as a function of bias voltage. (c) Radial distribution extracted from (a) as a function of bias voltage. (d) Color plot plotted against left axis: histogram of (a); black curve plotted against right axis: STS spectroscopy on the center of the AA site.