Two-dimensional membrane as elastic shell with proof on the folds revealed by three-dimensional atomic mapping

The great application potential for two-dimensional (2D) membranes (MoS2, WSe2, graphene and so on) aroused much effort to understand their fundamental mechanical properties. The out-of-plane bending rigidity is the key factor that controls the membrane morphology under external fields. Herein we provide an easy method to reconstruct the 3D structures of the folded edges of these 2D membranes on the atomic scale, using high-resolution (S)TEM images. After quantitative comparison with continuum mechanics shell model, it is verified that the bending behaviour of the studied 2D materials can be well explained by the linear elastic shell model. And the bending rigidities can thus be derived by fitting with our experimental results. Recall almost only theoretical approaches can access the bending properties of these 2D membranes before, now a new experimental method to measure the bending rigidity of such flexible and atomic thick 2D membranes is proposed.


Synthesis of 2D Materials and TEM sample preparations
The high quality single layer transition metal dichalcogenide (TMD) membranes (WSe 2 , MoS 2 ) were grown on sapphire substrates using chemical vapor deposition (CVD) in a tube furnace. Details of the synthesis procedure can be found elsewhere 1,2 . A description for the CVD synthesis of WS 2 membranes will be published elsewhere. The monolayer graphene samples were also fabricated using CVD growth over a Ni/Mo substrate 3 . The 2D membranes are transferred from the growth substrate to standard Cu TEM grid using a PMMA transfer route as described 2,3 .

Specifications of TEMs
The HRTEM for graphene was performed on a JEOL 2010F transmission electron microscope equipped with a CEOS spherical aberration corrector 4 . The TEM was operated using an 80 kV accelerating voltage, with an energy spread of 0.3eV. The chromatic aberration Cc was around 1mm, and the spherical aberration, Cs, adjusted to ~1µm. A defocus between 4~5nm was used, with a defocus spread of 3nm.
For the TMD membranes, annular dark field (ADF)-STEM imaging was implemented using a probe aberration corrected JEM ARM200F, operated at 80 kV. High angle annular dark field (HAADF) images were acquired using a 20 mrad convergence angle. The beam size was about 0.15nm. The images acquired with a medium-angle annular dark field detector used a convergence angle between 50 mrad to 180 mrad with acquisition times of 32 µs per pixel.

Procedures to obtain the upper/lower part of the folds and error analysis
By continuum elasticity theory, a fold structure has pure bending strain without any shear strain, which is similar to the tube like structure. The simulation of diffraction of tube structure is shown in Supplementary Figure 1b For a chiral tube, the Fourier components will split into two sets which correspond to the upper and lower half of the tube. The folded structure can be divided into several arc sections of tube-like-structures just with different diameters(curvatures), but the main feature is the same, that both contain several equatorial streaks parallel to the folding direction. However, if the fold is purely zigzag or armchair, this separation cannot work. Therefore, in our analysis, we used the fold structures very close to (2.5~3 o chiral angle) zigzag or armchair to measure the bending properties of these two directions (see supplementary Figure 2). Supplementary In our experiments, along these zigzag or armchair folds, we usually can find some places with very slight chiral angles which makes it possible to separate the two halves and then do 3D reconstructions. The actual chiral angles are shown besides final bending modulus results ( Table 1 in main text) for the zigzag and armchair fold cases. The Supplementary Figure 3 presents the separation of one chiral fold image of WSe 2 .
The purpose for us to do the FFT filtering here is (1) to filter the high frequency noise in the STEM images, (2) to reduce some possible noisy background contrast induced by some amorphous absorption on the sample surface during scanning inside the TEM column, thus to increase the peak finding accuracy in the following steps. Therefore, the images are Fourier filtered, using previously reported methods 5 . In the following we carefully investigate this filtering effects. The masking areas are mainly related to the Fourier components (elongated streaks due to the continuous strain effect on this fold) of these monolayer 2D materials, see In the first case, we just used an opaque mask (unfiltered-opaque mask) only subtracting the Fourier components from the other half of fold but include all the high frequency and low frequency noises. In the inverse FFT image we can see much higher noise level than the filtered one. In the second case, we apply mask(mask_1) on the Fourier components related to the under investigation half fold and remove the high frequency part. The unfiltered image with opaque mask and mask_1 images are both applied with a peak finding procedure based on maximum searching, peak area merging and cropping, and 2D Gaussian fitting using Matlab script. For the unfiltered case (first case), some of the peaks cannot be fitted by the good Gaussian shape, and we just used the local maximum as the atom position in 2D. The identified 2D atomic positions for these two images are presented in Supplementary Figure 5.
We can see that for the filtered and unfiltered cases most of the atomic positions are quite close, especially for the positions near the edge position very good correspondence is still achieved, and we didn't see obvious effects on filtering, but only can see a few atomic positions are a bit deviated.
Then we use our algorithm to do the 3D reconstruction using these two above sets of atomic positions(Supplementary Figure 6). From this cross section view of the fold for these two cases, the determined atomic positions for the unfiltered case is more scattered than the filtered case due to the randomness caused by noise. However, the main shape of the fold can be both well fitted with the continuous mechanics model (γ=0.85). And for the real case, the more smoothly changing structure(filtered case) should be closer to the real situation, and that's the main reason we use filtering before doing analysis. On the other hand, because our modelling depends on the relative position between the atomic positions, and the total dimension of these fold areas(total number of atoms also fixed) is fixed, the randomness in peak finding caused by the noise or amorphous will be naturally compensated in the reconstruction (for details can see Appendix at the end of this section), this is also one advantage of our method. And we checked the filtering effect for the other samples of WSe 2 , MoS 2 , WS 2 and graphene, all of them are fitted well by the similar continuum mechanics model and remain stable in the final fitting parameters.

The error caused by noise on 3D reconstruction
If atom position(X i ) is deviated by δ (randomness) from original position, terms contain higher order of δ can be omitted, and assume the strain distribution is continuous and smooth, where X i -X i-1 =X i+1 -X i +Δx and Y i -Y i-1 =Y i+1 -Y i +Δy, here Δx and Δy are because Δ x is small value compared to (d 2 -(X i+1 -X i ) 2 -(Y i+1 -Y i ) 2 ) 1/2 , it's concluded that total error on the total dimension in the Z axis after reconstruction caused by noise(δ) is small which has a higher order than δ. There is a self-compensation mechanism for the random errors induced by noise in the image in this 3D reconstruction.

3D reconstruction from 2D atomic TEM images
The TEM images are 2D-projection-views (x,y directions) and a few methods can be used to reconstruct the 3D structures. The x and y coordinates of each atom can be extracted directly from HR-ADF images or HR-TEM images. Our technique here is different from the classical tomographic multi-image approach by tilting the sample 6  For all of our bending modulus measurements, we have selected the samples which both sides of the folds can be fitted well with the same continuum mechanics model.

DFT calculations for the adhesive energy between layers of 2D membranes
Spin-polarized density functional theory (DFT) calculations are performed using a plane wave basis set with the projector augmented plane wave (PAW) as implemented in the Vienna ab initio simulation package (VASP) 11 .The Perdew-Burke-Ernzerhof (PBE) functional 12

Discussions on defective and tilted 2D membranes
The lower case in Supplementary Figure 13 has more atomic defects which are highlighted by the red dashed circles. And the 3D reconstruction for the lower case cannot yield a good 3D atomic structure due to the discontinuity in the atomic rows (our method requires continuity in the neighbours of every atom) and can lead to a much lower bending rigidity with a large error. Therefore in graphene, because of the stronger beam irradiation effect which can create more defects, we can see the range in the experimental bending rigidity data (1.8~2.8eV) is relatively larger than the TMD materials.
There could be local tilt or strain in some samples, we can determine the local tilt by measuring the FFT or lattice plane distances in the high resolution images of the flat areas and thus determine the tilt angle of the sample, and then the 3D reconstruction can still be executed, like the example of MoS 2 , similar to the vertical fold, 3D reconstruction of the tilted area (Supplementary Figure 14) is the same as the reconstruction in the horizontal folds, however, when determining the z height of O point (yellow atoms in above figure) during fitting, the tilted sample faces more difficulty and will lead to larger uncertainty due to lacking of symmetry. Therefore we suggest to use the exact horizontal or vertical folded areas to measure the bending rigidity.

Additional explanations on the continuum mechanics models
Because of the folding geometry, there is shear strain whose direction is vertical at point O.
The origin of shear force(V O ) is to balance the bending moment in the folded part. For this continuous mechanics modelling, there is one assumption here that adhesion energy exists on the left side of point O and becomes zero for the right side of O (Fig.1a). Near the point O, the energy caused by the detach of the two layers should be just the same as the increase in the bending energy. In other words, if the two layers of the fold detach more toward the left side of point O (Fig.1a), the adhesion energy consumed is larger than the bending energy while if the layers attach more toward the right side of O (Fig.1a) the adhesion energy accumulated is insufficient to compensate the bending strain energy.