Introduction

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.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.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.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.

To date, majorities of the mechanical testing of 2D materials follow the pioneering work of Hone et al.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.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 MoS2,20 and 330 ± 70 GPa for five-layer MoS2.21 Note that there is substantial difference in Young’s modulus between mono- and bi-layer MoS2 measured, inconsistent with previous DFT calculations.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. 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. In such case, spatial mappings of mechanical properties are highly desirable, for which the point-wise suspension method is inefficient, while the method to accurately map the mechanical properties of 2D materials directly on the substrate with high spatial resolution and sensitivity has yet to be developed.

In this work, we demonstrate a method to map in-plane Young’s modulus of 2D materials directly on the substrate with high spatial resolution, which can be easily applied to any 2D materials as processed, eliminating the need to transfer 2D samples onto pre-patterned substrate with holes. The method is also insensitive to the exact loading configuration and thus overcomes one of the major difficulties of suspension method, making the data robust with much less uncertainty. We demonstrate this method using mono- and bi-layer MoS2, one of the most investigated 2D TMDs due to its excellent electronic properties.5,8,9,24 We apply bimodal AFM technique to accurately map the effective spring constant between the AFM tip and sample under both amplitude modulation (AM) and frequency modulation (FM),25,26 and then develop finite element method (FEM) to quantitatively account for the effect of substrate stiffness, which has substantial influence on the tip-sample interactions. As a result, the in-plane Young’s modulus of MoS2 can be decoupled from the substrate, and is determined to be 265 ± 13 GPa for monolayer, broadly consistent with previous reports yet with much smaller uncertainty. It is also found that the elasticity of mono- and bi-layer MoS2 cannot be differentiated, which is supported by our DFT calculations showing virtually no interaction between the charges of S atoms on the top and bottom layers. The method thus provides a convenient, robust and accurate means to map the in-plane Young’s modulus of 2D materials on a substrate, which can be easily applied to a wide range of materials and systems.

Results

Mono- and bi-layer MoS2

Mono- and bi-layer MoS2 were micromechanically exfoliated from bulk MoS2 crystal onto Si/SiO2 substrate with 300 nm thick thermal oxide using Scotch tape, as detailed in Materials and Methods section,27,28 and the process is illustrated in Fig. S1 in the Supplementary Information (SI). The optical image of exfoliated MoS2 is shown in Fig. 1a overlaid with a schematic MoS2 lattice, wherein mono- and bi-layer as well as thicker MoS2 can be seen judged from their distinct optic contrasts.29 To confirm the layer counts, AFM scans in tapping mode were carried out over cyan box marked in Fig. 1a, with the resulting topography mappings in Fig. 1b showing a clear interface between MoS2 and Si/SiO2. The corresponding line scan of AFM topography reveals a step of approximately 0.8 nm high, and thus confirm monolayer MoS2 in the corresponding region.30 The red box marked in Fig. 1a exhibits an interface between mono- and bi-layer MoS2, which will be examined later. These layer counts are further verified by Raman measurements using 532 nm laser for excitation as detailed in Materials and Methods section, revealing peaks centered at 385.9 cm−1 and 404.5 cm−1 (cyan area) and 383.9 cm−1 and 405.4 cm−1 (red area) as shown in Fig. 1c, corresponding to out-of-plane (A1g) and in-plane \(\left( {E_{{\mathrm{2g}}}^1} \right)\) phonon modes of MoS2, respectively. The differences of Raman shift between two peaks in cyan and red areas are 18.6 cm−1 and 21.5 cm−1, respectively, confirming that there are monolayer and bilayer MoS2.31,32,33 The Raman spectra excited by 633 nm laser were also obtained as shown in Fig. S2, revealing consistent data.

Fig. 1
figure 1

Exfoliated MoS2 on Si/SiO2 substrate. a Optical image; b AFM topography mapping of cyan area marked in a with an interface between Si/SiO2 and a monolayer MoS2; c Raman spectra of mono- and bi-layer MoS2

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Bimodal atomic force microscopy (AM–FM)

We seek to map the elastic property of monolayer MoS2 on Si/SiO2 substrate quantitatively using bimodal AFM, which utilizes simultaneously the first and second flexural modes of the cantilever having spring constants k1 and k2, with corresponding resonant frequencies f1 and f2, amplitude \(A_1^0\) and \(A_2^0\), and quality factors Q1 and Q2 under free vibration in air. When interacting with the sample in tapping mode during scanning, the cantilever has its first and second mode vibration amplitudes as well as first mode frequency preset at A1, A2, and f1, with the second mode kept at resonance by maintaining a constant phase ϕ2 of π/2, as schematically shown in Fig. 2a.26 Under such configuration, the first mode cantilever deflection is used to track the topography of the sample based on amplitude modulation (AM) with a large amplitude, and its amplitude A1 and phase ϕ1 are used to measure the tip–sample interaction stiffness. In addition, measuring the frequency shift Δf2 of a higher eigenmode with a small amplitude A2 based on frequency modulation (FM) provides complementary information about the tip–sample interaction. The method thus is referred to as AM–FM,25,34 which allows distinguishing between changes in indentation depth and sample modulus, and both of these independent measurements can be used to solve for the effective spring constant of the tip–sample system,34

$$k_ \ast = \pi \sqrt {\frac{2}{3}} \left( {\frac{{k_1}}{{Q_1}}\frac{{A_1^0}}{{A_1}}{\mathrm{cos}}\varphi _1} \right)^{ - 1/2}\left( {\frac{{2k_2{\mathrm{\Delta }}f_2}}{{f_2 + {\mathrm{\Delta }}f_2}}} \right)^{3/2},$$
(1)

and the corresponding indentation depth δ during scanning is determined as25,34

$$\delta = \frac{3}{4}\left( {\frac{{k_1}}{{Q_1}}A_1^0{\mathrm{cos}}\,\phi _1} \right)\left( {\frac{{2k_2{\mathrm{\Delta }}f_2}}{{f_2 + {\mathrm{\Delta }}f_2}}} \right)^{ - 1}.$$
(2)
Fig. 2
figure 2

AM-FM probing of monolayer MoS2 on Si/SiO2 substrate. a Schematic of AM–FM imaging; b–d mappings of b phase ϕ1 of the first flexural mode, c frequency shift Δf2 of the second flexural mode, and d maximum indentation depth δ, all overlaid on topography of monolayer MoS2 (right) and Si/SiO2 (left), with A1 increased from 70 nm to 120 nm in 5 steps and the scan size of 5 μm; and e corresponding effective spring constant k* overlaid on topography

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Note that the loss tangent of either first or second eigenmodes can be used to extract information about the dissipation at either of the frequencies f1 and f2. As an example, mappings of phase ϕ1 of the first flexural mode and frequency shift Δf2 of the second flexural mode obtained from AM-FM scanning of monolayer MoS2 on Si/SiO2 substrate are shown in Fig. 2b, c, along with the corresponding maximum indentation depth δ in Fig. 2d. During the scan, A1 was increased incrementally from 70 nm to 120 nm in 5 steps, resulting in different phases, frequency shifts and indentation depths evident from these mappings, as expected from the theory.34,35 Nevertheless, the effective spring constant of the sample k* calculated from Eq. (1) is stable and does not change under these different experimental conditions for both SiO2 and MoS2, as seen in Fig. 2e. While the intentional adjustment of set point results in horizontal bands in the mappings of phase ϕ1, frequency shift Δf2, and indentation depth δ, the mapping of the effective spring constant only reveals a contrast between MoS2 and SiO2 without any bands, demonstrating that the method is robust and the effective spring constant measured is insensitive to the applied experimental parameters. Of particular interest is the indentation depth shown in Fig. 2d, in the range of a few nanometer. This is much smaller than 10% of the 300 nm thickness of SiO2 as required for indentation test, and thus effective spring constant for SiO2 is accurate. The depth, however, is thicker than the thickness of MoS2, and thus the effective spring constant mapped on MoS2 reflects the influences of both MoS2 and the underlying SiO2. We thus resort to FEM based on contact mechanics to account for the effect of substrate and help us determine the intrinsic elastic property of monolayer MoS2 from the mapping of effective spring constant.

Contact mechanics and finite element analysis

The effective spring constant measured from AM-FM makes it possible to determine Young’s modulus of the sample using contact mechanics theory. For a flat punch of radius R in contact with a homogeneous elastic material in half-space, the effective spring constant k* is given by25,34,36

$$k_ \ast = 2RE_ \ast ,$$
(3)

such that the effective Young’s modulus E* can be deduced from Eq. (1) as

$$E_ \ast = \frac{\pi }{R}\sqrt {\frac{1}{6}} \left( {\frac{{k_1}}{{Q_1}}\frac{{A_1^0}}{{A_1}}{\mathrm{cos}}\phi _1} \right)^{ - 1/2}\left( {\frac{{2k_2{\mathrm{\Delta }}f_2}}{{f_2 + {\mathrm{\Delta }}f_2}}} \right)^{3/2},$$
(4)

with

$$\frac{1}{{E_ \ast }} = \frac{{1 - \nu _{\rm t}^2}}{{E_{\rm t}}} + \frac{{1 - \nu _{\rm s}^2}}{{E_{\rm s}}}$$
(5)

where in Et, Es, νt and νs are Youn'g modulus and Poisson’s ratio of tip and sample, respectively, from which Young’s modulus Es of the sample can be determined. Note that explicitly characterizing geometrical shape of the tip is not necessary, since reference material with known Young’s modulus and Poisson’s ratio can be used for calibration instead to obtain the normalized effective spring constant \(\bar k\) of the sample with respect to that of reference material. In our case, Young’s modulus and Poisson’s ratio of SiO2 are known, providing us a natural choice of reference for calibration. The effective spring constant of MoS2 measured as such, however, is affected by the underlying SiO2, and we thus resort to FEM to account for this effect. Such continuum analysis has proven to be appropriate for in-plane deformation of monolayer.20,21

MoS2 is a transversely isotropic layered material with five independent elastic constants,37 namely in-plane and out-of-plane Young’s modulus E11 and E33, in-plane and out-of-plane Poisson’s ratio υ12 and υ13, and out-of-plane shear modulus G44, as detailed in Materials and Methods section. Based on first-principles calculations and molecular dynamics simulations, Poisson’s ratios of MoS2 can be assumed as 0.25.38,39,40,41 Hence, there are only three unknown material constants E11, E33 and G44 to be determined. Note that we have ignored viscoelasticity in our current model, since the sample is stiff. The FEM model configuration is shown in Fig. 3a, wherein a transversely isotropic layered material with thickness of 0.8 nm and radius of 150 nm on top of bulk SiO2 substrate with thickness of 99.2 nm is indented by a flat punch made of silicon with radius of 5 nm, and the radial stress distribution in the probe and sample is shown when the tip press onto sample surface. What we are really interested in, however, is the slope of simulated force–displacement curve for direct comparison with AM-FM experiment, and it turns out that such curve is rather insensitive to the out-of-plane Young’s modulus E33 (Fig. 3b) and shear modulus G44 (Fig. S3a), wherein large changes in E33 and G44 results in no appreciable difference in force–displacement curve. This can be understood from the distribution of resulted in-plane and out-of-plane displacement versus radial distance away from the probe (Fig. S4), which shows that the in-plane displacement is much larger than out-of-plane displacement and its variation spans much larger regions, suggesting that the deformation mode is largely in-plane that justifies the continuum treatment. As a result, the normalized effective spring constant \(\bar k\) simulated shows little variation over large range of Young’s modulus E33 and shear modulus G44 (Fig. 3c), making it possible for us to determine E11, regardless of E33 and G44. We note that under flat punch approximation, the effective spring constant \(\bar k\) is insensitive to the tip radius for sufficient large computation domain and sufficient thick SiO2, as shown in Fig. S5 and Fig. S6, and the effect of Si underneath of SiO2 is negligible.

Fig. 3
figure 3

FEM simulation of transversely isotropic layered material of different out-of-plane modulus on Si/SiO2 substrate subjected to indentation by a silicon tip, with G44 and E11 of 50 and 300 GPa, respectively. a Radial stress distribution with E33 of 300 GPa; b force–displacement curves evaluated for different out-of-plane Young’s modulus; and c normalized effective spring constant \(\bar k\) evaluated under different out-of-plane Young’s modulus E33 and shear modulus G44

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Because of insensitivity of force–displacement curve to out-of-plane modulus of 2D materials, it becomes possible to determine their in-plane Young’s modulus E11 from the AM-FM experiment. To appreciate this, we evaluated from FEM the force–displacement curves under different E11 with out-of-plane Young’s modulus E33 and shear modulus G44 set as 100 GPa and 50 GPa based on first-principles calculations of MoS2,37,42 which shows distinct slope and thus k* (Fig. 4a). The in-plane and out-of-plane displacement under different E11 confirms again that the dominant deformation mode is in-plane (Fig. 4b), and the normalized effective spring constant \(\bar k\) simulated for monolayer MoS2 shows strong dependence on in-plane Young’s modulus E11 (Fig. 4c). Importantly, we can directly compare this curve of normalized effective spring constant \(\bar k\) with the experimentally measured data from different areas of sample and under different conditions, which falls into the reddish strip with \(\bar k\) measured between 1.0669 and 1.0795, which can be converted into in-plane Young’s modulus for monolayer MoS2 using this curve.

Fig. 4
figure 4

FEM simulation of transversely isotropic layered material of different in-plane modulus on Si/SiO2 substrate subjected to indentation by silicon tip, with E33 and G44 of 100 and 50 GPa, respectively. a Force–displacement curves evaluated for different in-plane Young’s modulus; b in-plane and out-of-plane displacement on the surface of layered material; and c normalized effective spring constant \(\bar k_{}^{}\) simulated under different in-plane Young’s modulus versus the experimental data

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Young’s modulus of mono- and bi-layer MoS2

The relationship between the normalized effective spring constant \(\bar k\) and in-plane Young’s modulus E11 shown in Fig. 4c enables us to convert AM-FM mapping of the effective spring constant (Fig. 2e) into a mapping of Young’s modulus of MoS2. To this end, the mappings of topography and in-plane Young’s modulus obtained under the optimal experimental conditions are shown in Fig. 5a, b for monolayer MoS2 and SiO2, revealing distinct in-plane Young’s modulus of monolayer MoS2 and SiO2. The corresponding histogram in Fig. 5c reveals that the distribution of in-plane Young’s modulus is quite sharp, being 70 ± 1 GPa for SiO2 and 265 ± 13 GPa for monolayer MoS2, and it is consistent with earlier report yet with much smaller scattering.20 As discussed, we used SiO2 as a natural reference material for calibration, enabling us to carry out the analysis in just one mapping involving both monolayer MoS2 and Si/SiO2 substrate. While they have substantially different Young’s modulus, the difference in indentation depth is rather small, as shown in Fig. S7, ensure the accuracy of AM-FM using reference method. Unlike the previous suspension method that derives Young’s modulus at just a single point on micro-patterned substrate, our method allows spatial mapping of Young’s modulus directly on an ordinary substrate, and thus it is particularly convenient for studying mechanical properties of 2D materials in a device configuration, wherein both substrate and/or interface, structural heterogeneity, and defects are important.

Fig. 5
figure 5

In-plane Young’s modulus of monolayer MoS2 on Si/SiO2, with mappings of a topography, b in-plane Young’s modulus, and c histogram distribution of in-plane Young’s modulus; the scan size is 5 μm

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Of particular interest is Young’s modulus of bilayer MoS2, which is not expected to be different from that of monolayer due to weak van der Waals interactions between layers, yet previous study reported substantial difference, 270 ± 100 GPa for monolayer and 200 ± 60 GPa for bilayer.20 We thus examined the red box in Fig. 1a that contains both mono- and bi-layer MoS2, with its AFM topography shown in Fig. 6a revealing a clear 0.8 nm step between mono- and bi-layers. The AM-FM mapping of the effective spring constant in Fig. 6b, however, exhibits no appreciable difference between mono- and bi-layers, which is evident from the histogram distribution in Fig. 6c as well. To help understand this result, we carry out the first-principles DFT calculation for bi-layer MoS2, for which interlayer spacing h is a critical parameter. As seen from the charge density profile in Fig. 6d, there is a strong interaction between S atoms on the top and bottom layers at a small interlayer space of 1.50 Å, which becomes weaker and weaker with the increased h (Fig. S8). When h approaches 3.00 Å as shown in Fig. 6d, there is barely any interaction between the S atoms on the top and bottom layers, corresponding to weak van der Waals instead of covalent interaction between layers. Indeed, energetic analysis shows that the energy of bilayer MoS2 per primitive cell decreases rapidly with increased h and converges to that of monolayer MoS2 at around 3.00 Å, upon which the out-of-plane strain becomes zero and the energetic behavior is expected to be identical to that of monolayer MoS2. As a result, the stress–strain curve and the corresponding Young’s modulus of bilayer MoS2 shown in Fig. 6e evaluated under the uniaxial loading converges with that of monolayer MoS2, consistent with previous report.23 In other words, the in-plane Young’s modulus of perfect mono- and bi-layer MoS2 are confirmed to be identical by DFT calculations, as observed in our AM-FM experiments.

Fig. 6
figure 6

In-plane Young’s modulus of bi-layer MoS2 interfacing with monolayer ones, with mappings of a topography, b effective spring constant k* and (c) histogram distribution of effective spring constant k* for mono- and bi-layer MoS2, respectively; the scan size is 5 μm. d Charge density of bi-layer MoS2 with respect to the interlayer space h; e energetics of bi-layer MoS2 with respect to the interlayer space; f stress–strain relation in Zigzag (ZZ) direction of bi-layer MoS2 with difference interlayer spacing h

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Discussion

Mechanical testing of 2D materials is highly challenging, and the main difficulty lies in the effect of substrate that imposes substantial influence on the mechanical response of 2D materials. Previous experimental efforts were primarily based on suspension method, requiring transferring of the sample onto substrates with pre-patterned holes, which is a nontrivial task. Even more importantly, such pointwise loading for force–displacement curve requires precise alignment of AFM at the symmetric center of the suspended membrane, or large error will be induced. This explains large scattering of experimentally reported data under suspension approach, up to 10% for graphene and 37% for monolayer MoS2, since precise alignment is very difficult. Our method largely overcomes such difficulties. By mapping elastic response of 2D materials directly on the substrate using AM-FM, we eliminate the need of transferring the sample to pre-patterned substrate, and the precise alignment is no longer required as in a pointwise measurement, making the method widely applicable for other 2D materials and systems on any substrate, especially in a device configuration. The effect of substrate constraint is accurately accounted for by FEM, enabling us to decouple the deformation of 2D materials from the substrate. As a result, the in-plane Young’s modulus of monolayer MoS2 is measured to be 265 ± 13 GPa with less than 5% uncertainty, and that of bilayer MoS2 is found to be indistinguishable from monolayer, which is rationalized by our DFT calculations. Importantly, it is the elasticity of 2D materials on the substrate instead of suspended one that is relevant for most applications, and the method provides a powerful technique to spatially map elasticity of 2D materials on the substrate with high resolution and sensitivity.

Methods

Sample preparation

Mono- and few-layer of MoS2 flakes were mechanically exfoliated from commercially available MoS2 crystals (Graphene Supermarket) onto a Si substrate with a 300 nm SiO2 layer using Scotch tape as the transfer medium, followed by annealing in argon at standard atmospheric pressure and 350 °C for 1 h in order to enhance the contact between sample and substrate.27 Before exfoliating, the substrates were ultrasonically cleaned in acetone, ethanol and deionized water, respectively, and then subjected to oxygen plasma for 10 min to remove ambient adsorbates from surface. The mechanical exfoliation process consists of following steps. The tape is pressed into contact with MoS2 crystals by thumb first, and the thick MoS2 flakes are stuck on the tape. The process is then repeated 4 or 5 times to thin the flakes, after which the substrate is pressed onto the tape with thin MoS2 flakes for transferring. The obtained sample containing tape and substrates is packed by silver paper and put onto a hot plate to enhance contact between SiO2 and MoS2 for 3 min at 100 °C. Finally, the tape is removed from the substrates at approximately constant speed when the sample temperature is dropped to ambient temperature, which completes the exfoliation process, as illustrated in Fig. S1(a)–(f). While some tape residues may remain on the sample, as seen in the top right corner of Fig. 2b, clean areas can be found after careful search for further study.

Raman measurement

Raman scattering measurement was performed on a Horiba Jobin-Yvon LabRam HR-VIS high-resolution confocal Raman microscope equipped with a 532 nm laser and a 633 nm laser as the excitation source at room temperature to identify layer thickness of MoS2 based on the difference between the out-of-plane vibrational (A1g) and the in-plane vibration \(\left( {E_{{\mathrm{2g}}}^1} \right)\) peaks, as reported by Lee et al.43 A XYZ motorized sample stage controlled by LabSpec software was used to move sample accurately and the output power was controlled by neutral density filters to protect the MoS2 samples.

Bimodal atomic force microscopy (AM-FM)

Bimodal atomic force microscopy (AM-FM) uses two driving forces with frequencies matching the first and the second flexural resonance of the cantilever to excite the oscillation of probe. The tip deflection z(t) can be written as follows,35,44 and the first two modes can be measured via lock-in.

$$\begin{array}{*{20}{l}} {z(t)} \hfill & = \hfill & {z_0 + \mathop {\sum}\limits_n {A_n\cos (\omega _nt - \phi _n)} } \hfill \cr {} \hfill & {} \hfill & {\mathop{\longrightarrow}\limits^{{Lock - in}}z_0 + A_1{\mathrm{cos}}(\omega _1t - \phi _n) + A_2{\mathrm{cos}}\left( {\omega _2t - \frac{\pi }{2}} \right).} \hfill \end{array}$$
(6)

In Eq. (6) An, ϕn, and ωn = 2πfn are amplitude, phase and angular frequency of nth mode. During the experiment, the cantilever parameters k1, k2, f1, f2, Q1, and Q2 are determined by fitting the thermal signals of the cantilever away from the surface before experiment, and the sensitivity of the optical beam deflection is deduced from the force–displacement curve obtained when approaching a hard sample.45 Before tip–sample interaction, the probe is driven close to its first resonant frequency f1 with a large amplitude \(A_1^0\) (~100 nm) and its second resonant frequency f2 with a small amplitude \(A_2^0\) (~1 nm) in air. Upon approaching the sample, the responses of the two resonance frequencies are tracked by two independent lock-in amplifiers. For the first flexural resonance mode, the frequency f1, amplitude A1 and the phase ϕ1 are extracted from the deflection signal of the probe and the feedback makes adjustment to keep a predetermined amplitude set point A1. Meanwhile, the feedback for the second flexural mode adjusts the frequency by a small shift Δf2 to maintain the phase ϕ2 of the second flexural mode at π/2 with a constant amplitude set point A2. This makes sure the amplitude A2 of the second flexural mode is considerable compared to the noise while relatively small compared to the indentation depth δ,25 as shown in Fig. S9.

Our AM-FM experiments were performed on Asylum Research MFP-3D atomic force microscope in ambient using OMCL-AC160TS-R3 probe with k1 = 31.19 N/m, f1 = 283.58 kHz, k2 = 996.30 N/m, and f2 = 1.60 MHz in air. For monolayer MoS2, the free amplitudes were \(A_1^0 = {\mathrm{260}}\,{\mathrm{nm}}\) and \(A_2^0{\mathrm{ = 5}}\,{\mathrm{nm}}\), and the image was taken at A1 = 100 nm. For bilayer MoS2, the experimental values for the probe were k1 = 42.20 N/m, f1 = 301.41 kHz, k2 = 1358.3 N/m, and f2 = 1.70 MHz in air. The free amplitudes were \(A_1^0 = {\mathrm{186}}\,{\mathrm{nm}}\) and \(A_2^0 = {\mathrm{5}}\,{\mathrm{nm}}\), and the image was taken at A1 = 56 nm. Note that assumed Euler-Bernoulli dynamic stiffness scaling for AC160 cantilever, where k2 = k1(f2/f1)2. As shown in Ref. 45, due to the presence of probe tip and/or not ideal thin diving boards, the scaling exponent for an AC160 was actually found to be ~1.67. Nevertheless, our analysis indicating that different scaling exponents have negligible influence on the results, as shown in Fig. S10, which is also consistent with earlier report by Kocun et al.25

FEM simulation

Transversely isotropy layered material is considered, with elastic matrix given by46,47

$$C^{ - 1} = \left[ {\begin{array}{*{20}{c}} {1/E_{11}} & { - \nu _{12}/E_{11}} & { - \nu _{13}/E_{33}} & 0 & 0 & 0 \cr { - \nu _{12}/E_{11}} & {1/E_{11}} & { - \nu _{13}/E_{33}} & 0 & 0 & 0 \cr { - \nu _{13}/E_{33}} & { - \nu _{13}/E_{33}} & {1/E_{33}} & 0 & 0 & 0 \cr 0 & 0 & 0 & {1/G_{44}} & 0 & 0 \cr 0 & 0 & 0 & 0 & {1/G_{44}} & 0 \cr 0 & 0 & 0 & 0 & 0 & {2(1 + \nu _{12})/E_{11}} \end{array}} \right]$$
(7)

COMSOL Multiphysics software was chosen to simulate this problem because of its parametric modeling. In order to save computing time, a central symmetry model was developed, consisting of cantilever tip (punch), layered material and substrate as shown in Fig. 3a. The height and radius are 7.5 nm and 5 nm for tip, 0.8 nm and 150 nm for layered material, and 99.2 nm and 150 nm for substrate, respectively. The material of cantilever tip and substrate are silicon and silicon dioxide with respective Poisson’s ratio and Young’s modulus of 0.28 and 170 GPa for silicon and 0.17 and 70 GPa for silicon dioxide. For convergence and simplification, the displacement instead of force of tip was selected as ‘Loads’, which is defined by pressing the tip into sample surface for 1 nm under 100 steps for each trace and retrace. An eight-node rectangular unit was used as basic element of mesh, and convergence test were carried out by changing the number of elements, as shown in Fig. S11.

DFT calculations

First-principles DFT calculations were carried out using the Vienna Ab initio Simulation Package (VASP).48,49 We adopt the PAW pseudopotential50 for electron–ionic core interaction and the generalized gradient approximation (GGA) for the electron exchange and correlation with the Perdew-Burke-Ernzerhof (PBE)51 formulation. We also use the conjugate-gradient algorithm for ion relaxation with the cutoff energy set at 300 eV. As shown in Fig. S12, we consider monolayer MoS2 and bi-layer MoS2 with AA stacking and AB stacking, respectively. Periodic boundary condition is applied with a vacuum region of 40 Å along the out-of-plane direction to avoid the inter-layer interactions. A primitive cell in all the cases is adopted for the calculation. A K-point mesh of 16 × 16 × 1 is used with the Monkhorst–Pack sampling scheme.