Heterostrain-enabled dynamically tunable moiré superlattice in twisted bilayer graphene

The ability to precisely control moiré patterns in two-dimensional materials has enabled the realization of unprecedented physical phenomena including Mott insulators, unconventional superconductivity, and quantum emission. Along with the twist angle, the application of independent strain in each layer of stacked two-dimensional materials—termed heterostrain—has become a powerful means to manipulate the moiré potential landscapes. Recent experimental studies have demonstrated the possibility of continuously tuning the twist angle and the resulting physical properties. However, the dynamic control of heterostrain that allows the on-demand manipulation of moiré superlattices has yet to be experimentally realized. Here, by harnessing the weak interlayer van der Waals bonding in twisted bilayer graphene devices, we demonstrate the realization of dynamically tunable heterostrain of up to 1.3%. Polarization-resolved Raman spectroscopy confirmed the existence of substantial heterostrain by presenting triple G peaks arising from the independently strained graphene layers. Theoretical calculations revealed that the distorted moiré patterns via heterostrain can significantly alter the electronic structure of twisted bilayer graphene, allowing the emergence of multiple absorption peaks ranging from near-infrared to visible spectral ranges. Our experimental demonstration presents a new degree of freedom towards the dynamic modulation of moiré superlattices, holding the promise to unveil unprecedented physics and applications of stacked two-dimensional materials.


Supplementary Note 2: Mechanical stacking of twisted 2D materials
A 135-m thick PET substrate was cut into 1 cm × 5 cm strips. PET substrates were cleaned by acetone, isopropanol alcohol (IPA) and deionized (DI) water, and then dried at 65 ºC for 5 min.
About 400 nm polymethyl methacrylate (PMMA, 950 PMMA A6, MICROCHEM) was spincoated onto PET at 4500 rpm for 90 sec. Twisted bilayer graphene was mechanically stacked on PMMA-coated PET with a home-made transfer stage, following the steps shown in Fig. S2.  layer that has been successfully transferred onto a PMMA-coated PET substrate with long side aligned to bending direction as highlighted by the black arrow. Scale bar is 20 m.

Supplementary Note 3: Heterostrain measurements by Raman spectroscopy
After stacking TBG on PMMA-coated PET, the flexible PET substrate with TBG on top was bended and fixed on a glass slide. Strain applied on graphene was tuned by changing the bending radius ( Fig. S4). Graphene G peaks were measured by Raman spectroscopy (Alpha300 M+, WITec) with a 100× objective and 1800 g/mm grating. Laser wavelength was 532 nm. Before measurement, laser power was adjusted to low level to avoid heating effect. Lorentz fitting was applied to determine peak position and FWHM.
At each bending radius which was measured from the PET arch, strain in graphene was extracted from Raman shift of G (Gpeak when G peak is split), using the strain coefficient of 31.7 cm -1 /%. Table 1 summaries G peak position, red shift and derived strain.

Figure S4 | Bending apparatus.
Bending radius is tuned by changing the distance between two ends of PET strip. Table S1. Raman shift and strain on stretched graphene.
Bending radius

Supplementary Note 4: Simulation on band structure of heterostrained TBG
To calculate the energy spectrum of TBG with the heterostrain, the single-orbital tight-binding model was used, which can be expressed as: 6 = ∑ ( ⃗ ) † + ℎ. .
where the hopping integral ( ⃗ ) between any of two carbon atoms with distance ⃗ = ( , , ) where = 0,1,2 … we choose = 2, and get the rotation angle 2 = 13.174 °, which matches the rotation angle of TBG in our experiment. Thus, we have the superlattice constant vectors: where ⃗ 1 and ⃗ 2 are the lattice constant vectors of pristine graphene unit cell. | ⃗ 1 | = | ⃗ 2 | = 1.072 . With the heterostrain, the superlattice structure will be distorted, the lattice constants | ⃗ 1 | = 1.081 , | ⃗ 2 | = 1.072 , and the angle between ⃗ 1 and ⃗ 2 is 60.4 °. Lattice relaxation in TBG may alter the electronic band structure and related optical and electrical properties. Nam et al. 9 theoretically demonstrated that lattice relaxation only causes a significant difference when rotation angle is smaller than 2°, and the energy change is up to 20 meV, which is a very small energy modification compared to our large band width of 1.34 eV. Experimental analysis of lattice relaxation was performed by using the selected-area electron diffraction (SAED) 10 , which showed that no appreciable lattice reconstruction occurs when rotation angle is larger than 4°. Therefore, atomic-scale reconstruction in our TBG with twist angle of 13.2° is very small and negligible. Our model and calculations were performed without considering lattice relaxation.

Supplementary Note 5: Simulation on DOS of heterostrained TBG
The density of the states can be calculated by the expression: where , ⃗⃗ is the eigenvalue of Bloch state based on the tight binding model, and = 0.03 is the broadening factor due to the impurities and disorder.

Supplementary Note 6: Simulation on dynamic conductivity of heterostrained TBG
The frequency-dependent optical conductivity tensor, expressed by the Kubo-Greenwood formula in the bases of Bloch states can be formulated as: 11,12 ( ) = where index labels the atoms in the primitive cell, labels the orbitals on the given atom , R are the lattice vectors for the different unit cells, and | , ⃗⃗ ⟩ is the tight-binding orbital. Usually, we use the atomic orbital in graphene system. Since we only consider the single-orbital tightbinding model, thus the index can be neglected. By solving the tight-binding eigen equations: we have the eigenenergy ⃗⃗ and eigenvector ⃗⃗ . ⃗⃗ is the vector of the coefficient ⃗⃗ .
With the coefficient ⃗⃗ , we have the constructed Bloch states, and the momentum matrix element ⟨ ⃗⃗ | | ⃗⃗ ⟩ can be calculated. 13

Supplementary Note 7: Heterostrain-enabled optical transition between saddle points
The forbidden transition from S2 to S1 at point between highest valence band and lowest conduction band in pristine TBG is due to the symmetry protection: Σ −1 Σ = − * , where Σ is the 4 × 4 matrix defined as: However, it is not the case in the heterostrained TBG. Basically, the stain-induced gauge potential breaks the certain symmetry in the pristine TBG, and gives out a non-zero optical matrix elements at the saddle points. We start it from the effective low energy Hamiltonian 14 with heterostrain in one layer: where Δ 1 and Δ 2 are the wave vector shifts of Dirac cones by rotation in the first layer and second layer, respectively. is the interlayer hopping matrix, ⃗ is the gauge potential induced by heterostrain: 15 ⃗ = 2 0 ( 11 − 22 , −2 12 ) where ( 11 , 22 , 12 ) are the elements of the strain tensor. One can easily check that the symmetry transformation is broken with strain induced gauge potential: