![]() It is also nontrivial to fabricate the pre-patterned substrate and transfer the sample onto the designated holes, and mechanical properties measured from such suspended membrane may not be directly applicable to device structures, since for most applications 2D materials are usually directly deposited onto the substrate without suspension. 23 This is not unexpected since the method is based on the precise alignment of AFM probe with the center of the suspended membrane, which is very difficult to accomplish, and a slight misalignment will result in large errors. 21 Note that there is substantial difference in Young’s modulus between mono- and bi-layer MoS 2 measured, inconsistent with previous DFT calculations. 19, 20, 21, 22 While having provided considerable insight into the mechanical properties of 2D materials, Young’s modulus measured as such scatters over a large range and suffers from large uncertainty, for example 1000 ± 100 GPa for graphene, 19 270 ± 100 and 200 ± 60 GPa for mono- and bi-layer MoS 2, 20 and 330 ± 70 GPa for five-layer MoS 2. 19 on graphene using suspension method, wherein 2D samples suspended over pre-patterned holes are indented by atomic force microscope (AFM) probe, and the corresponding force–displacement curves obtained are analyzed within the frameworks of continuum mechanics. To date, majorities of the mechanical testing of 2D materials follow the pioneering work of Hone et al. 18 However, to date even the basic mechanical properties of most 2D materials, critical not only for device reliability but also for the emerging applications such as flexible electronics, remain largely uncharacterized and poorly understood, and it is still very challenging to accurately measure mechanical properties of 2D materials, especially when they are deposited on a substrate as in most devices instead of being suspended as in a testing structure. 14, 15, 16, 17 Understanding the mechanical behavior of 2D materials also plays a central role in tuning their electronic and optoelectronic properties, wherein strain engineering can be highly effective. 13 Of particular interest is the mechanical properties of 2D materials and structures, which are extremely stiff when stretched, yet exceptionally flexible in membrane form, and such extraordinary combination opens exciting opportunities for their engineering applications in microelectromechanical systems (MEMS) as well as flexible and stretchable electronics and photonics. Since the first successful exfoliation of graphene in 2004, 1 atomically thin two-dimensional (2D) materials such as graphene, 2, 3 hexagonal boron nitride (h-BN), 4 transition metal dichalcogenides (TMDs) 5, 6, 7, 8, 9, 10 and black phosphorus (BP) 11, 12 have generated great excitement because of their unique and exotic functionalities. This method provides a convenient, robust and accurate means to map the in-plane Young’s modulus of 2D materials on a substrate. It is also found that the elasticity of mono- and bi-layer MoS 2 cannot be differentiated, which is confirmed by the first principles calculations. Using these methods, the in-plane Young’s modulus of monolayer MoS 2 can be decoupled from the substrate and determined as 265 ± 13 GPa, broadly consistent with previous reports though with substantially smaller uncertainty. Bimodal atomic force microscopy is used to accurately map the effective spring constant between the microscope tip and sample, and a finite element method is developed to quantitatively account for the effect of substrate stiffness on deformation. In this work, we demonstrate a method to map the in-plane Young’s modulus of mono- and bi-layer MoS 2 on a substrate with high spatial resolution. However, accurate measurement of the elastic modulus of 2D materials remains a challenge, and the conventional suspension method suffers from a number of drawbacks. Elasticity is a fundamental mechanical property of two-dimensional (2D) materials, and is critical for their application as well as for strain engineering.
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