Visualizing delocalized correlated electronic states in twisted double bilayer graphene

The discovery of interaction-driven insulating and superconducting phases in moiré van der Waals heterostructures has sparked considerable interest in understanding the novel correlated physics of these systems. While a significant number of studies have focused on twisted bilayer graphene, correlated insulating states and a superconductivity-like transition up to 12 K have been reported in recent transport measurements of twisted double bilayer graphene. Here we present a scanning tunneling microscopy and spectroscopy study of gate-tunable twisted double bilayer graphene devices. We observe splitting of the van Hove singularity peak by ~20 meV at half-filling of the conduction flat band, with a corresponding reduction of the local density of states at the Fermi level. By mapping the tunneling differential conductance we show that this correlated system exhibits energetically split states that are spatially delocalized throughout the different regions in the moiré unit cell, inconsistent with order originating solely from onsite Coulomb repulsion within strongly-localized orbitals. We have performed self-consistent Hartree-Fock calculations that suggest exchange-driven spontaneous symmetry breaking in the degenerate conduction flat band is the origin of the observed correlated state. Our results provide new insight into the nature of electron-electron interactions in twisted double bilayer graphene and related moiré systems.


Supplementary Note 1: Determination of three stacking orders
Our STM image of tDBLG in Fig. 1c shows three distinct regions within each moiré unit cell, which can be called "bright", "intermediate" and "dark" based on their apparent heights for 100 mV < |VBias| < 500 mV. We determined their stacking orders by analyzing structural as well as electronic contributions to the apparent height. In tDBLG, ABBC stacking is energetically unfavorable (due to the strong repulsion between inner layers of carbon atoms) and has been predicted to exhibit an out-of-plane structural displacement of ~0.1 Å. 1 This allows us to identify the "bright" region in Fig. 1c   6

Supplementary Note 2: Additional data on different devices and/or spots
The LDOS suppression around  = 2 was observed on different devices and/or spots, as shown in the gate-dependent dI/dV spectra of three stacking sites in Supplementary Fig. 3.
Here, the first set of data was taken on the same device as the data shown in the main text, but at a different spot on the surface (micrometers away) with a different local twist angle (a-c), while the second set was taken on a different device. The STM tips were freshly prepared and calibrated on Cu(111) before each set of measurements (Methods). initial VBias = 100 mV, I0 = 0.25 nA).

Supplementary Note 3: Comparison of STM/STS and transport measurements
Transport measurements on tDBLG devices are usually performed with both a top gate and a bottom gate so that the carrier density n and the vertical E-field can be independently tuned. Results from several groups 2-6 are summarized in the schematic shown in Supplementary Figure 4. A correlated high-resistivity phase surrounded by a "halo" feature can be seen at  = 2 (labeled as "ns/2"). Signals of incipient insulating states also show up at In our STM/STS study, on the other hand, only a single back-gate is present (Fig. 1b) and n and E are always proportionally changed when we adjust VG (Methods). The parameter space in STM/STS therefore corresponds to a "diagonal" line-cut in the n-E plane. This line cut touches the half-filling correlated phase but misses many of the other correlation features.
Consequently, we only observe splitting of the CFB peak around  = 2. We note that this limitation of STM/STS can be potentially overcome by using tips with different work functions, thus "moving" the line-cut in the n-E plane to access other interesting regimes. This is technically extremely challenging, however, and controllable "work function-tunable" tips are currently beyond STM state-of-the-art.

Supplementary Note 4: Extracting the splitting magnitude from dI/dV spectra
The magnitude of the energy-splitting was extracted by fitting the dI/dV spectra with a sum of two Lorentzian peak functions and a linear background as shown in Supplementary Fig. 5a. Here y0 and bx are background terms, xi (i = 1, 2) represents the center position of peak #i, Ai is the area under peak #i, and w is the peak FWHM. The peak separation  = |x1 -x2| represents the splitting magnitude. The error bars shown in Fig. 5b were estimated by combining fitting uncertainty, finite temperature broadening, and an instrumental broadening of ~1 mV.
The same fitting procedure was applied to the dI/dV spectra measured on different devices and/or spots. Supplementary Figure 5b shows the maximum splitting magnitude (squares with error bars) as well as the filling range over which the splitting can be extracted (blue bars) as a function of twist angle . More systematic study is required before we can reach a definite conclusion regarding the -dependence of the correlation-induced splitting. 13

Supplementary Note 5: Spatial distribution of layer-summed CFB and RCB wavefunctions
The theoretical LDOS shown in Fig. 3 and Supplementary Fig. 2 Fig. 7c) and the corresponding histogram ( Supplementary Fig. 7e) at this energy. The CFB peak centered at ℰ = -13 meV has significant weight in all three regions, although the intensity in the ABBC region is higher than in the ABCA and ABAB regions.
The layer-summed LDOS map at the CFB peak energy shows more spatial variation ( Supplementary Fig. 7b,d) compared to the top-layer-projected LDOS map at the same energy (Fig. 3g,i). Nevertheless, the LDOS max/min ratio of 2.5 is still significantly smaller than that of 4.4 in the RCB state, and the CFB remains reasonably delocalized when multiple layers are accounted for.

Supplementary Note 6: Layer polarization of flat band wavefunctions in an E-field
The flat band wavefunctions in tDBLG exhibit layer polarization in a gate-induced Efield. To illustrate this effect within the single-particle continuum model, we plot the theoretical total DOS as well as LDOS projected onto the topmost graphene layer for both VG = 27 V,  = +0.15 V/nm (Supplementary Fig. 9a) and VG = -27 V,  = -0.15 V/nm ( Supplementary Fig. 9b). The conduction and valence flat bands appear as two peaks in the total DOS (gray curves). At VG = 27 V, however, only the CFB peak is prominent in the toplayer LDOS for all three regions. The VFB peak shows up in the ABAB region with much smaller intensity compared to the CFB peak and is absent in the ABBC and ABCA regions.
This explains why the VFB signal is experimentally observed only in the ABAB region for positive gate voltage (Fig. 4f). For opposite gate-voltage (VG = -27 V) the VFB peak shows up clearly in the top layer while the CFB peak is barely present. This explains why experimental dI/dV spectra measured in the negative gate range do not exhibit the CFB signal for all three stacking regions (Fig. 4d-f).

Supplementary Note 7: Screened Coulomb interaction in Hartree-Fock calculations
The single-plane-screened Coulomb potential ( ) Ref. 6 . The tip apex contributes another empirical dielectric factor of 2 to 4 (if we assume a conical shape and solve the Laplace equation to the lowest order).
Considering the contributions above, the overall effective dielectric constant lies in the range 3 < eff < 30.

Supplementary Note 8: Competition between different symmetry-breaking correlated states
The simplest explanation for reduction in DOS at integer filling is spontaneous breaking of the spin and valley symmetry. Within Hartree-Fock theory, symmetry breaking is found to be of two distinct types: isospin-polarized (ISP), or inter-valley coherent (IVC). 11 In  Fig. 12a). Nevertheless, due to the large gap away from Γ, the DOS at the Fermi level is reduced compared to the bare single-  (Fig. 3g) and exhibit a small max/min ratio, corroborating our finding of a delocalized correlated state.
 Phase diagram at  = 2 and comparison with experiment. Supplementary Figure   10a shows the energy difference between the ISP and IVC states as a function of eff.
A crossover from the ISP ground state to the IVC ground state occurs slightly above eff = 14. Supplementary Figure 10b shows the effective splitting magnitude of the CFB peak in these two phases. In the main text we utilize eff = 14 since this yields the splitting that is closest to the experimental value of ~19 meV.

Supplementary Note 9: Tip-bias-induced gating
The presence of a bias voltage across the tip-sample junction can modify both the carrier density n and the E-field in the graphene layers. Here we estimate the tip gating effect by applying the simplest approximation and treating the tip-sample junction as a parallel plate capacitor. We have 0 Bias where VBias is the sample bias relative to the tip and dT ≈ 0.7 nm is the tip-sample distance.
Supplementary Figure 13 shows the theoretical LDOS in an ABBC region when these tipinduced corrections are taken into consideration. Since tip-gating is seen to only cause a slight shift in the LDOS curves for the bias range of our measurements, we do not include this effect in the main text. We also note that a more accurate determination of tip-biasinduced corrections would require detailed knowledge of the shape of the tip apex, which is beyond our capability at present.