Elastic Deformations in 2D van der waals Heterostructures and their Impact on Optoelectronic Properties: Predictions from a Multiscale Computational Approach

Recent technological advances in the isolation and transfer of different 2-dimensional (2D) materials have led to renewed interest in stacked Van der Waals (vdW) heterostructures. Interlayer interactions and lattice mismatch between two different monolayers cause elastic strains, which significantly affects their electronic properties. Using a multiscale computational method, we demonstrate that significant in-plane strains and the out-of-plane displacements are introduced in three different bilayer structures, namely graphene-hBN, MoS2-WS2 and MoSe2-WSe2, due to interlayer interactions which can cause bandgap change of up to ~300 meV. Furthermore, the magnitude of the elastic deformations can be controlled by changing the relative rotation angle between two layers. Magnitude of the out-of-plane displacements in graphene agrees well with those observed in experiments and can explain the experimentally observed bandgap opening in graphene. Upon increasing the relative rotation angle between the two lattices from 0° to 10°, the magnitude of the out-of-plane displacements decrease while in-plane strains peaks when the angle is ~6°. For large misorientation angles (>10°), the out-of-plane displacements become negligible. We further predict the deformation fields for MoS2-WS2 and MoSe2-WSe2 heterostructures that have been recently synthesized experimentally and estimate the effect of these deformation fields on near-gap states.


Moiré periodicity
The interaction energy of a unit cell in layer A, as a function of its position relative to the unit cell in layer B, can be written as: 1. Mean interlayer separation z 0 is determined as the separation with minimum average energy: 2. Spatial variations of the interlayer interaction energy within Moiré unit cell are tabulated by assigning the energy to each primitive unit cell according to its local stacking configuration, for a mean interlayer separation z 0 .
3. Tabulated values of the interlayer interaction energy are used to determine the unknown Fourier coefficients g mn (z 0 ) in Eq. (S.2) by taking the inverse Fourier transform: where N uc is the number of primitive unit cells in the Moiré unit cell and sum extends over the all unit cells within Moiré unit cell.
4. Once, we have determined the unknown Fourier coefficients g kl , in-plane spatial variation of the interaction energy can be obtained from Eq. (S.2) for mean interlayer separation.
Using this information, we determine the forces due to interlayer interactions. For small deformations, energy given by Eq. (S.2) can be expanded around mean spacing z 0 : This spatial variation of the energy leads to the following functional form of the different force components.
If the forces are known, this can be used in the von-karman plate model to predict the deformation of the sheet. While coefficients g mn (z 0 ) and v 0 (z 0 ) are known, their derivatives can be estimated numerically by repeating the step (2,3,4) for two other separations around the mean separation.

Validity of the analytical solutions
To check the validity of the the analytical solutions, we have compared these solutions with the numerical solutions obtained using finite element simulations for the perfectly aligned graphene-hBN bilayer. Magnitude of the out-of-plane displacement estimated from the finite element simulations is 0.23Å while analytical solution gives a magnitude of 0.21Å.
Similarly the magnitude of the in-plane displacement in the X direction computed from finite element simulations is 0.0026Å and the analytical solution results in 0.0023Å. We have also checked the validity of these solutions against finite elements results for different interaction parameters. In Figure S1, the magnitude of the out-of-plane displacements computed from both the approaches have been plotted for different magnitudes of energy modulation g m . It is clear that the closed form solutions remain valid for the interactions that are 20 times stronger as compared to those between graphene/hBN. Only for stronger modulation magnitudes, do the analytical solutions start deviating from the finite element solutions. Figure S1: Magnitude of the out-of-plane displacements computed using analytical solutions compared with finite element simulations for different magnitudes of the energy modulation (g m )

Deformation in Freestanding Bilayer
In case of freestanding bilayer, equal and opposite forces will act on the substrate layer. Due to these forces, deformation can be computed same method Where different displacement coefficients u m β are given by: Where subscript s denotes elastic constant corresponding to substrate layer. E s is Young's modulus, σ s is the Poission's ration and h s is the thickness of the substrate layer.