Tailoring the Band Structure of Twisted Double Bilayer Graphene with Pressure

Twisted two-dimensional structures open new possibilities in band structure engineering. At magic twist angles, flat bands emerge, which gave a new drive to the field of strongly correlated physics. In twisted double bilayer graphene dual gating allows changing of the Fermi level and hence the electron density and also allows tuning of the interlayer potential, giving further control over band gaps. Here, we demonstrate that by application of hydrostatic pressure, an additional control of the band structure becomes possible due to the change of tunnel couplings between the layers. We find that the flat bands and the gaps separating them can be drastically changed by pressures up to 2 GPa, in good agreement with our theoretical simulations. Furthermore, our measurements suggest that in finite magnetic field due to pressure a topologically nontrivial band gap opens at the charge neutrality point at zero displacement field.


Transport measurements
Transport measurements were carried out in a four-terminal and two-terminal geometry with typical AC voltage excitation of 0.1 mV using a standard lock-in technique at 177.13 Hz. The device was cooled down six times: twice at zero pressure, three times at p = 2 GPa and once at p = 1 GPa. Twice at p = 2 GPa we measured in a two-terminal geometry while for the rest of the cooldowns we measured in a four-terminal geometry as depicted in Fig. S1b. In   Figure S2: Comparison between the measured gap energies at two different cooldowns at ambient pressure. Cooldown 1 is measured before applying the pressure and cooldown 2 is measured after releasing the pressure.

Gate voltage n-D conversion
The charge density (n) and electric displacement field (D) is related to the top and bottom gate voltage by where α TG and α BG are the lever arm for the top and bottom gate respectively, 0 is the vacuum permittivity, V TG and V BG are the top and bottom gate voltage, respectively, is the carrier density when the gate voltages are set to zero, is a built-in offset electric field. V TG0 and V BG0 are the values of the top and bottom gate at the zero density and zero displacement field point.
Vm 2 ) 3.38(5) 3.75(5) 3.84(5) α BG ( 10 15 Vm 2 ) 4.54(6) 4.94(7) 4.91(8) n 0 (10 15 /m) 2.3(2) 2.5(2) 2.4(2) D 0 (V/nm) 0.016(5) 0.018(5) 0.017 (5) The lever arms were obtained from the gate-gate maps (Fig. S3a) and from the quantum oscillations in magnetoconductance measurements (Fig. S4). The extracted lever arms are shown in Table S1. The pressure dependence of the lever arms is originating from the compression of the dielectrics and pressure-dependent dielectric constant of the hBN as already reported in Refs. S2,S3. The extracted lever arms within the margin of error were the same at the same pressures at different cooldowns.  At certain magnetic fields, the magnetic length is commensurable with the lattice periodicity which results in the recovery of the translation symmetry thus the electron feels effectively zero magnetic field. This results an oscillation in the resistance called Brown-Zak (BZ) oscillation. S4,S5 A common method to determine the twist angle (ϑ) in superlattice structures such as the TDBG via transport measurements is to calculate it from the BZ oscillations. It can be calculated using that the BZ oscillations have maxima in the conduc- is the flux quantum, q is an integer number and A s is the area of the superlattice unit cell which is given by where a is the lattice constant of the graphene. We determined the twist angle (ϑ = 1.067°±0.003°) from our data at different pressures and found that they were identical within the uncertainty of our measurements.
We also estimated the twist angle inhomogeneity from the width of the resistance peaks at n = ±n s in Fig. S7 panel a d and f at different pressures which varied in a similar range: at p = 0 GPa ϑ is between 1.06°and 1.09°, at p = 1 GPa ϑ is between 1.06°and 1.1°and at p = 2 GPa ϑ is between 1.05°and 1.08°. In regions containing substantial twist angle inhomogeneity, this inhomogeneity could result in smaller measured gaps values, but it doesn't change our results qualitatively. At the CNP we also estimated the gaps with bias measurements. We measured the twoterminal resistance of the device with the lock-in technique at low frequency and applied a DC voltage bias (V B ) to the sample. If we apply a higher or equal bias voltage to the gap, transport processes become available. We performed bias measurements over a small range of n at fix D (see Fig. S6a). First the maximum resistance (R max ) was determined (occurring in Fig. S6a at n = 0 and V B = 0). We defined the gap value as the point where the resistance drops to 15% of the maximum resistance value, also taking into account a background value. To determine the gap size, the following procedure was followed. At the gap (n = 0, V B = 0) we took the peak value (R max ), while at a slightly different density value, where the resistance R is virtually independent of V B , we took the background value R gapless . We approximated the gap energy with e|V B | where V B = (V B+ + V B-)/2 with R(±V B± ) = R gapless + 0.15(R max − R gapless ) as it is shown in Fig. S6b.

Band structure calculation
The band structure of TDBG is calculated using the non-interacting Bistritzer-Macdonald model S6 and applied to the TDBG using the parameters from Ref. S7. The matrix elements in the model can be written as In our calculations we used a configuration space with cutoff in momentum space with a radius up to 4|b s | = 32 sin(ϑ/2)/( √ 3a) using Hamiltonian matrices with a size of 648×648. In the model for the low-energy Hamiltonian of the BLG (δ l,l H ls,l s (k)) we included the remote hoppings and used the following parameters: γ 0 = 3.1 eV for the intralayer nearest-neighbor hopping, γ 1 = 3ω(p) eV for the interlayer coupling between orbitals on the dimer sites, γ 3 = 0.283 eV for the interlayer coupling between orbitals on the non-dimer sites, γ 4 = 0.138 eV for the interlayer coupling between dimer and non-dimer orbitals and δ = 0.015 eV onsite energy difference between dimer and non-dimer sites. S7,S8 For the matrix elements between the BLGs ([1 − δ l,l ]H ls,l s (K)) which are given by the pressure dependent interlayer couplings ω(p) and ω (p) were taken from Table I. in Ref. S7 as where A = 0.0546 B = 0.0044 C = 0.0031 for ω and A = 0.0561 B = 0.0018 C = 0.0018 for ω .
The external electric field modifies the layer potentials (u i ), where i = {1, 2, 3, 4} marks the 4 layers of graphene from top to bottom. The effect of the external electric field is modelled with u 1 = −u 4 , u 2 = −u 3 , and we introduce u as the potential difference within each BLG and 2u as the potential difference between the BLGs as u 1 = (u + u )/2 and u 2 = (−u + u )/2. In the calculations, we modeled the external electric field with equal interlayer potential drops (u = 2u) and neglected the quantum capacitance corrections and the electron-electron interactions, which could modify the low-energy flat bands.

Magnetic field dependence
In Fig. S7 we present n-D resistance maps at B z = 0 T, 1 T and 2 T at 1.5 K. At ambient pressure a gap opens at the charge density of n = n s /2 which corresponds to a correlated insulating phase. S9-S11 In panel d,e and f,g the measurements are shown for 0 and 2 T at 1 GPa and 2 GPa, respectively. Under pressure this gap disappears for all applied B z fields.
In Fig. S8 we show resistance maps at various in-plane magnetic fields (B x ) at p = 2 GPa.
Comparing B x = 0 (panel a) and B x = 3 T (panel b) it is visible that the effect of the in-plane magnetic field is negligible. At a finite perpendicular magnetic field, the effect of applying an in-plane field is also negligible. In thermal activation measurements, B x also had a negligible and are discussed in the main text. We note that our model does not include the effect of magnetic fields and more advanced model capable of handling that goes beyond the scope of this manuscript. Figure S8: Four-probe resistance of the TDBG as a function of n and D measured at 2 GPa in different in-and out-plane magnetic fields. (a) presents the data for zero magnetic field, (b) for B x = 3 T in-plane magnetic field, (c) for B z = 2 T vertical magnetic field whereas (d) shows the measurement in B z = 2 T and B x = 2 T. Temperature dependence at half and three-quarter fillings Fig. S10 show the temperature dependence of the correlated states at half and near threequarter fillings which are similar what was observed in Ref. S10,S11,S13,S14. At finite pressure the correlated states disappear as shown in Fig. S7.

Comparison between the measurement and the model
To compare experimental findings with our calculations we used the relation of D 0 = eu d , where d = 0.33 nm is the interlayer distance of bilayer graphene and is the relative dielectric constant of bilayer graphene. We also used the same conversion between the top layer of the bottom BLG and the bottom layer of the twisted, top BLG. A qualitatively good agreement at the CNP is achieved using = 5 which is depicted in Fig. S11 where the experiments and the theory are shown in the same figure.