Mapping the elastic properties of two-dimensional MoS2 via bimodal atomic force microscopy and finite element simulation

Elasticity is a fundamental mechanical property of two-dimensional (2D) materials, and is critical for their application as well as for strain engineering. However, accurate measurement of the elastic modulus of 2D materials remains a challenge, and the conventional suspension method suffers from a number of drawbacks. In this work, we demonstrate a method to map the in-plane Young's modulus of mono- and bi-layer MoS₂ on a substrate with high spatial resolution. 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. Using these methods, the in-plane Young's modulus of monolayer MoS₂ can be decoupled from the substrate and determined as 265 ± 13 GPa, broadly consistent with previous reports though with substantially smaller uncertainty. It is also found that the elasticity of mono- and bi-layer MoS₂ cannot be differentiated, which is confirmed by the first principles calculations. This method provides a convenient, robust and accurate means to map the in-plane Young's modulus of 2D materials on a substrate.