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

ARTICLE Received 10 Jul 2015 | Accepted 15 Oct 2015 | Published 19 Nov 2015 DOI: 10.1038/ncomms9935 OPEN Two-dimensional membrane as elastic shell with proof on the folds revealed by three-dimensional atomic mapping Jiong Zhao1,2, Qingming Deng3, Thuc Hue Ly1,2, Gang Hee Han1,2, Gorantla Sandeep3 & Mark H. Rümmeli1,2 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. 1 Center for Integrated Nanostructure Physics, Institute for Basic Science, Sungkyunkwan University, Room 86175, Suwon 440-746, Republic of Korea. 2 Department of Energy Science, Sungkyunkwan University, Suwon 440-746, Republic of Korea. 3 IFW Dresden, Institute of Solid State Research, PO Box 270116, Dresden D-01171, Germany. Correspondence and requests for materials should be addressed to J.Z. (email: ). NATURE COMMUNICATIONS | 6:8935 | DOI: 10.1038/ncomms9935 | www.nature.com/naturecommunications & 2015 Macmillan Publishers Limited. All rights reserved. 1 ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms9935 O ut-of-plane bending properties are of great significance for graphene as well as two-dimensional (2D) transition metal dichalcogenide (TMD) membranes such as 2H-MoS2, WS2 and WSe2 (ref. 1), because it can explain the morphology of such materials under external fields and is thus important with regards to their use in applications2. Here we propose a simple approach to three dimensionally map the folded edges of these 2D membranes at the atomic scale, based on the high-resolution transmission electron microscopy (HRTEM). A quantitative comparison with the continuum mechanics shell model confirms that all the bending behaviour of the investigated 2D materials can be well explained by the linear elastic shell model. Excellent elasticity for these 2D materials has been demonstrated in which they approach their theoretical in-plane tensile strength and exhibit ultrahigh in-plane modulus3. Nano-indentation tests on suspended membranes using atomic force microscope tips4 or direct in-plane tensile testing involving microelectromechanical systems gave rather scattered results for the Young’s modulus and strength, which suggest many unknown factors in sample preparation, loading system and, in particular, identifying the correct thickness for these membranes5. In most of these measurements, a linear and isotropic elasticity were assumed. However, some theoretical studies using density functional theory (DFT), molecular dynamics or microscopic analytical derivation from the empirical interatomic potential suggest a nonlinear and anisotropic elasticity in large strain regimes3. In contrast to the above in-plane tests, the bending test (for flexural rigidity and flexural strength), which is also an important elastic measurement for thin plates or shells6, is still lacking for most of the 2D membranes. The applicability of elastic shell theory on 2D membranes, which are only 1 to 3 atoms thick, has not been verified experimentally, although various theoretical studies have looked at the problem7,8. The primary experimental difficulty is that such 2D membranes are too flexible, and their bending rigidities are too small as compared with their in-plane rigidities. One experimental work on the nanometer scale rippling structure of graphene shows violation to the continuum mechanics shell model9. Another recent study reported the bending stiffness measured in monolayer graphene by the thermal method and laser method10. Moreover, the direct observation of the cross-sections of these 2D membranes after bending is either inaccurate by atomic force microscope11 or interfered by supporting layers in transmission electron microscopy (TEM)12, thus no intrinsic mechanical properties can be derived. The naturally folded or buckled structures of 2D membranes have been observed in many previous reports13,14, however, their focus was mainly on the stacking order after folding. In addition, one of the studies also conducted in an investigation of the 2D strain mapping of the folded graphene using Geometric Phase Analysis15. A few studies have correlated the van der Waals (vdW) interaction and critical diameter for collapsed carbon nanotubes16,17. However, in this study, based on a 3D atomic mapping by using high-resolution HRTEM and scanning TEM (STEM), we confirmed the continuum mechanics model and derived the bending rigidity of 2D materials. and a JEOL 2010F TEM equipped with CEOS image aberration corrector was employed for HRTEM imaging, both with an acceleration voltage of 80 kV (see Methods and Supplementary Discussion for details). The schemes of the folded and buckled (vertically folded) structures of 2D membranes are presented in Fig. 1a,b, respectively. It is not hard to find regions of sample with folded areas for all of our observed 2D specimens. All the examined areas are monolayers, with some parts folded during transfer process or due to crack (Fig. 2a). Figure 2b shows one WSe2 fold, which has no inclusions inside, leaving the folded area an intrinsic structure. A typical high-resolution ADF image of the folded edge of WSe2 is presented in Fig. 2c. It shows a Moiré Pattern from one folded chiral edge (whereas zigzag and armchair folds are achiral). The fast Fourier transform (FFT) of this image shows similar streaking of the certain reflexes in a particular direction (west to east in this case; Fig. 2d). Such streaking of reflexes can also be found with carbon nanotubes. The FFT consisting of two sets of spots corresponding to the upper and lower half of the fold (see Supplementary Figs 1 and 2 for details). To view a single side of the fold, we carefully select the reflexes corresponding to the chosen side (forming a selective mask) and then apply inverse FFT. This process provides a micrograph comprised solely of one side chosen from the fold (Fig. 2e). In this reconstructed ADF image, the position of the brightest spots reflect the real atomic positions of Tungsten (W) atoms. The Se atoms (which have a reduced Z contrast) are not well resolved in part because of the relative tilt of the two Se atoms in one col (...truncated)