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.

弹性是二维（2D）材料的基本机械性能，对于它们的应用以及应变工程至关重要。然而，精确测量2D材料的弹性模量仍然是一个挑战，并且常规的悬挂方法具有许多缺点。在这项工作中，我们演示了一种以高空间分辨率在衬底上映射单层和双层MoS ₂的面内杨氏模量的方法。双峰原子力显微镜用于精确映射显微镜尖端和样品之间的有效弹簧常数，并且开发了一种有限元方法来定量考虑基体刚度对变形的影响。使用这些方法，单层MoS _2的面内杨氏模量可以从衬底上解耦出来并确定为265±13 GPa，尽管不确定性小得多，但与以前的报告大致一致。还发现单层和双层MoS ₂的弹性无法区分，这已通过第一原理计算得到证实。此方法提供了一种方便，可靠且准确的方法，可将2D材料的面内杨氏模量映射到基板上。