In situ manipulation of van der Waals heterostructures for twistronics

Rotation of 2D materials by polymer patch and gel stamp manipulator opens up a new strategy for twistronics.


Identification of the initial contact between polymer gel and epitaxial polymer
In our manipulation technique, the critical step to precisely control the movement of 2D flakes is the identification of the initial contact between the polymer gel (PDMS) and the epitaxial polymer patch (PMMA).
The signature of the contact is a color change in the PMMA patch and an appearance of a sharpened edge of the contact area between the PDMS gel and PMMA patch, which is easily distinguished under optical microscope, Fig. S1.

The motion of incommensurately stacked 2D materials
In our manipulation technique, the motion of 2D materials in the van der Waals heterostructure depends on the balance between external driving force and the adhesion between PMMA patch and the top 2D layer, as well as the kinetic friction originating from the van der Waals force between the adjacent 2D layers. The kinetic friction between 2D layers is dramatically reduced in the incommensurate state due to the cancellation of lateral corrugation forces between two crystals in the sliding direction, the so called superlubricity where the 2D flakes can move smoothly. Inversely, in commensurate state, the two adjacent 2D layers are locked together due to pinning of the boundaries that separate local regions of the commensurate phase (25,29,30), thus the friction is significantly higher. Such boundaries, either topological defects or misfit dislocations, are caused by the lattice mismatch and twist angle between adjacent layers.
The sliding of crystalline interfaces is governed by the friction between the layers. The kinetic friction between graphene and hBN shows rotational anisotropy and six-fold symmetry with respect to the rotation angle between graphene and hBN (48). In incommensurate state where kinetic friction is extremely small, it still varies slightly with the rotation angle (48). Therefore, the friction forces at graphene/top hBN and graphene/bottom hBN interfaces are comparable but slightly different from each other depending on the twist angles t and b, which could lead to simultaneous rotation of both top hBN and graphene. Most likely, due to the superlubricity, as the top hBN rotates under the control of PMMA patch, graphene remains static until top hBN clicks to it, and then with further rotation, the top hBN together with graphene clicks to bottom hBN. More realistically, the interface conditions vary among the whole stack. There could be polymer residues or other contaminants adsorbed on the surface of bottom hBN, causing difference in frictions at the two interfaces, thereby graphene can rotate with top hBN simultaneously.
For graphene/hBN and hBN/hBN interfaces, the commensurate state occurs at a small twist angle between the layers, since these two interfaces already satisfy the condition of a small lattice mismatch (1.65% (34,39) for graphene/hBN interface and 0% for hBN/hBN interface). At commensurate state, the locking of the graphene/hBN and hBN/hBN interfaces limits further motion of the flakes (33), thereby when the external driving force is large enough, the PMMA patch/hBN/graphene/hBN stack will break at the weaker coupled PMMA patch/hBN interface, in agreement with the delamination of PMMA patch when graphene is aligned to both hBN layers after rotation (Fig. 2e in the main text and Fig. S4c). By slightly etching the top hBN and then depositing the polymer patch, the adhesion between PMMA and the top hBN can be dramaticaly improved and PMMA will not delaminate from the top hBN. Then, if one imposes a larger external driving force in order to further rotate or move the crystals after locking, the 2D crystals will be destroyed, as shown in Fig. S2, where we used hBN/graphite flake/hBN stack as an example. Rotation of the top two layers of hBN/graphene/hBN heterostructure with different initial settings of t and b should result in the final stack with commensurate or incommensurate states at the two interfaces, as shown in Fig. S3 and Table S1. The scenarios are based on the fact that graphene can rotate with top hBN simultaneously. Here we consider both the 0⁰ and 60⁰ alignment of graphene and hBN layer, since these two types of alignment have distinct symmetry and affect electronic structure of graphene differently (34,41).
We fabricated hBN/graphene/hBN heterostructure (sample 2) with the manipulation process matching the cases in Fig. S3b (initial alignment) and Fig. S3j (final alignment). To make sure that the crystal orientation of the flakes is known, we first identified hBN fakes which have fractured into two pieces during the mechanical exfoliation procedure (Fig. S4a). Note that for hBN crystals with odd numbers of layers, the top and bottom layers have the same crystal orientation, whereas for hBN crystals with even numbers of layers, the top and bottom layers are of mirror symmetry (Fig. 2c in the main text). Therefore, the common edges of the fractured hBN pieces indicate the same crystal orientation of the atomic layer belonging to the same surface, either top or bottom, of the original crystal. Then we used one of the fractured hBN piece to pick up graphene and flipped over the other fractured piece as the bottom hBN layer, so that the hBN atomic layers adjacent to graphene come from the same crystal surface. See Fig. S5 for the details of this procedure. Next, the initial t and b were set according to Fig. S3b and the following rotation direction led to the final stack with t=0⁰, b=tb=60⁰ (Fig. S4c), which is the case of Fig. S3j.
We carried out Raman spectroscopy to prove the alignment of graphene and hBN in sample 2, as shown in Fig. S4d and e. The behavior of Raman spectra in monolayer and bilayer graphene regions are similar to those observed in sample 1. Whereas the full width at half maximum of 2D peak (FWHM2D) increased from 17 cm -1 to 48.5 cm -1 after rotation, smaller than that in the perfect alignment case, which means that the twist angles t and b are larger than those in sample 1. Figure S4e shows the distribution of FWHM2D among the graphene flake after rotation, indicating a spatially uniform twist angle in both monolayer and bilayer regions. The AFM topography (Fig. S4f) here shows that the rotation process did not damage or crease the graphene layer.
The assembly process of sample 2 is illustrated in Fig. S5. The details of the process are as follows.
Step 1, we used PMMA coated PDMS block mounted on a glass slide to pick-up one of the fractured hBN pieces as the top hBN (hBN piece 1).
Step 2, we used polypropylene carbonate (PPC) coated PDMS block mounted on another glass slide to pick-up the other fractured hBN piece (hBN piece 2), which serves as the bottom hBN.      For other interfaces where the adjacent layers have larger lattice mismatch, such as MoS2/hBN interface, a small or even zero twist angle will not result in the commensurate state. The kinetic friction between the 2D layers always remains at an extremely low value, thereby the 2D flake can move and rotate smoothly even when passing through the point where twist angle is zero without the delamination of PMMA patch, as shown in Fig. S6.

Raman spectra of graphene in the monolayer and bilayer regions before and after rotation
Raman spectra of graphene are strongly modified when graphene is subjected to double moiré superlattices (Fig. 2g). In our sample, for monolayer graphene region, G peak changes its shape from a narrow peak centered at 1582 cm -1 with FWHMG ≈14.5 cm -1 to a broadened asymmetric peak after rotation. The presence of a low energy shoulder at ≈1558 cm -1 of G peak after rotation is attributed to a TO phonon, similar to what was observed in graphene aligned to only one hBN crystal (35). For 2D peak, the broadening effect is strongly enhanced in double-moiré superlattice compared to a simple aligned graphene/hBN heterostructure: ≈20 cm -1 increase in FWHM for graphene in a single moiré superlattice (27,35) and ≈40 cm -1 increase in FWHM for our doubly-aligned sample, indicating that the double moiré superlattices induce a much stronger periodic inhomogeneity originated from charge accumulation, strain, etc.
For bilayer graphene region, G peak splits into two components in the presence of double moiré superlattices. For 2D peak, the four components broadened and their position differences reduced, which is similar to what was reported in bilayer graphene aligned to only one hBN crystal (49).
We found an overall downshift of around 2cm -1 for both G peak and 2D peak among the whole flake after rotation (Fig. S7 a-d), which contradicts the results of previous studies (27,34). Similar behavior is found in Sample 2 (Fig. S8). Near the corner of the folded region (as indicated by the arrows), before rotation, the peak positions of G and 2D peaks are slightly lower than those of the nearby region. We attribute this phonon softening to the strain caused by folding (50). Whereas after rotation, peak position near the corner of the folded region shows an upshift behavior, and is higher than that of the nearby region. Note that during the rotation, the folded graphene rotated as well, therefore we expect an enhanced strain effect near this region.
However the abnormal behavior of the peak position implies a mechanism beyond strain effect.
To further look into the line shape of G peak after rotation, we fit G peak with two Lorentzian peaks for both bilayer and monolayer graphene regions and plot the maps of the peak position difference and intensity ratio of the two components, as shown in Fig. S7e and f. The maps show that the G peak splitting is homogeneous for both bilayer and monolayer graphene regions, which is around 13 cm -1 and 22 cm -1 , respectively. The intensity ratio of the two components is around 1 for the bilayer region, which means that they have comparable weight, and is around 0.6 for the monolayer region.
The overall downshift and broadening of the peak positions for G and 2D peaks and splitting of G peak remind one of the effects of strain. Under uniaxial strain, the doubly degenerate E2g mode splits into two components, one along the strain direction and the other perpendicular, which leads to the splitting of G peak (51). Tensile strain usually gives phonon softening, while compressive strain usually leads to phonon stiffening. In the presence of moiré superlattice, the spatial strain distribution is periodically modulated, and  (53,54). Thus, most likely, the main factor that causes the change in Raman spectra is the periodic strain field induced by moiré patterns.  Based on the Raman spectra of the hBN/graphene/hBN heterostructure, we estimate that the twist angles t and b should be both close to 0⁰, thereby in the transport properties, we expect to see two sets of SDPs close to each other apart from the PDP. For both bilayer and monolayer graphene devices, in longitudinal resistivity (ρxx) vs carrier density (n) dependence, we observed broadened (and with a complex structure) satellite peaks located around n = 210 12 cm −2 (Fig. 3a in the main text and Fig. S11a). At small B = 0.03 T where the Landau quantization is not yet developed, the transversal resistivity ρxy changes sign in the carrier density regions of these satellite peaks, indicating that they are moiré superlattice induced SDPs. These SDPs are more prominent in the hole side compared to the electron side, which is consistent with previous studies (5,6).
In the Landau fan diagram at high magnetic fields, by tracing the linear trajectories we observed two sets of SDPs induced by the two independent moiré patterns for both bilayer and monolayer device. For bilayer device, the SDPs are at ns1 = 2.1510 12 cm −2 , and ns2 = 2.3410 12 cm −2 (Fig. 3a In the fan diagram ∂/∂B(n,B), we observed Brown-Zak (BZ) oscillations originating from the first order magnetic Bloch states (/0=1/q) for the two sets of SDPs in both bilayer and monolayer devices (Fig. S9b, e, f and Fig. S12b, e, f). The first order magnetic Bloch states formed by the two moiré superlattices are also prominent at high temperature (T = 70 K, where the Landau quantization is suppressed), as indicated by the arrows in Figs. S10 and S13. We also observed high order magnetic Bloch states (/0 = 3/q) belonging to the two moiré superlattices, as shown in Fig. S9e, f and Fig. S12e, f.
In principle, the super-moiré pattern generated by the two original moiré patterns should have six possible reciprocal lattice vectors, as described in the previous study (39), whereas the method in Ref (38) only considers one possible reciprocal lattice vector with the largest super-moiré pattern wavelength (sm = 102.3 nm). If we take into account other possible reciprocal lattice vectors, we will get six possible second- We observed satellite peaks in ρxx near most of these carrier densities nsm (Fig. 3a in the main text and Fig. S11a). Note that near nsm corresponding to these resistivity peaks, most of the low-B ρxy regions have sign reversal. In addition, similar to bilayer device, in the plot of ∂xx/∂B(n,B) of monolayer device, we observe prominent horizontal streaks at B = 1.14 T and 0.57 T, which perfectly matches the magnetic field of the first and second order magnetic Bloch state originating from the super-moiré pattern when BA = 0 (where q = 1) and BA = 0/2 (where q = 2), respectively. These features allow us to attribute the origin of the observed carrier densities nsm to the supermoiré pattern with different wavelengths.