Strain engineering of the mechanical properties of two-dimensional WS₂

Abstract

Tuning the physical properties of two-dimensional (2D) materials is crucial for their successful integration into advanced applications. While strain engineering demonstrated an efficient means to modulate the electrical and optical properties of 2D materials, tuning their mechanical properties has not been carried out. Here we applied compressive strain through the buckling metrology to 2D tungsten disulfide (WS₂), which demonstrated mechanical softening manifested by the reduction of its effective Young's modulus. Raman modes analysis of the strained WS₂ also showed strain-dependent vibrational modes softening and revealed its Grüneisen parameter (γ_E2g = 0.29) and its shear deformation potential (β_E2g = 0.56) – both are similar to the values of other 2D materials. In parallel, we conducted a molecular dynamic simulation that confirmed the validity of continuum mechanics modeling in the nanoscale and revealed that due to sequential atomic-scale buckling events in compressed WS₂, it shows a mechanical softening. Therefore, by tuning the mechanical properties of WS₂ we shed light on its fundamental physics, thus making it an attractive candidate material for high-end applications, such as tunable sensors and flexible optoelectronic devices.

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Introduction

Two-dimensional (2D) tungsten disulfide (WS₂), a layered material that comprises tungsten atoms sandwiched between two layers of sulfide atoms, is a member of transition metal dichalcogenides (TMDs) 2D materials and has attracted immense scientific attention due to its extraordinary physical properties. WS₂ is among the strongest members in the TMDs family with Young's modulus exceeding 300 GPa for a single-layer,¹ it has high electrical conductivity and high excitation band energy.² These properties made WS₂ attractive for a wide range of applications, such as flexible photodetectors,³ field effect transistors (FET),⁴ resonators,⁵ and a variety of sensors.^6,7

The mechanical properties of TMDs have been under intense focus as they have a crucial influence on the operation of functional devices. While the indentation method was widely used to investigate the mechanical properties of suspended TMDs,^8–12 other methods were suggested as well, including resonance characterization,^13,14 bulge test (pressurized cavity test),¹⁵ and the micro-Brillouin light scattering method.¹⁶ The buckling method, which we use in the current study, was recently employed on 2D materials and allowed the mechanical analysis of thin layers (e.g., both TMDs¹⁷ and other 2D materials¹⁸). In this method, a controlled uniaxial strain is transferred from a compliant substrate to a thin film, and the deformed (wrinkled) shape of the latter is used to extract its elastic properties.

The studies carried out so far demonstrated the resilience and high stretchability of WS₂ and other TMDs,¹⁹ that makes them suitable materials for strain engineering analyses. In strain engineering experiments the atomic structure of the investigated material is modified by means of controlled application of mechanical deformation, which allows the tuning of their properties. Specifically, since energy states are tightly bound in TMDs, they show strong coupling between the applied strain and their energy structure. As a result, many studies used strain engineering for bandgap modulation. For example, WSe₂ demonstrated optical band-gap reduction under uniaxial strain,²⁰ while bandgap tuning of WS₂ was demonstrated through local buckling deformation.²¹ An enhanced photoluminescence signal of WS₂ was observed under tensile strain due to direct excitonic recombination.²² Furthermore, the bandgap modulation of MoS₂ was reported under bi-axial compression and tension,²³ uniaxial tension,²⁴ and pure shear.²⁵ Other works demonstrated the use of strain engineering to enhance the piezoelectric properties of TMDs,²⁶ their magnetic properties,²⁷ and phonon vibration.²⁸ First principle calculations often accompanied the experimental studies and theoretically confirmed the impact of strain engineering on the band structure of TMDs.²⁴ Despite the intense research on strain engineering of TMDs, its impact on their mechanical behavior is yet to be understood.

In this study, we bridge this knowledge gap. We first applied compression to thin layered WS₂. Prior to investigating it using a continuum mechanics model, we verified its validity by comparing it to an atomistic MD model. Then we extract the effective Young's modulus using Bernoulli–Euler continuum mechanics model and use a modified molecular dynamics (MD) simulation to further clarify our results. We also conducted a Raman analysis of compressed WS₂ to improve our understanding and validate our results. We showed that compressed WS₂ demonstrates mechanical softening, manifested by reduction of its effective Young's modulus and redshift of its Raman vibrational modes, due to atomic-scale buckling events.

Methods

The buckling metrology

We studied the behavior of few-layered WS₂ flakes under compressive strains ranging between 0.3% and 4%. We clamped polydimethylsiloxane (PDMS) substrate to movable stages (Fig. 1a(I)) and stretched the PDMS by bringing the stages apart (Fig. 1a(II)). Then, we mechanically transferred a thin WS₂ flake via scotch tape exfoliation from a solid crystal to the substrate of the stretched PDMS (Fig. 1a(III)). Next, by bringing the stages closer we transferred the strain from the PDMS to the WS₂, thus compressing it and creating wrinkles due to buckling (Fig. 1a(IV)). A high-resolution digital optical microscope (Keyence VHX-5000, magnification ×2000) was used to capture the wrinkles obtained under different strains, Fig. 1b. A scanning electron microscope (SEM) image of a compressed flake is shown in Fig. 1c. Optical images of the wrinkled WS₂ were post-processed using a fast Fourier transform (FFT) analysis implemented on MATLAB software that revealed the spatial frequency of the wrinkles (inset of Fig. 1b).

Fig. 1 (a) Illustration of the buckling metrology. PDMS is shown as the blue layer while the WS₂ flake is represented as the green layer. (b) Optical image of strained WS₂. Scale bar: 5 μm. Inset: FFT of the image. (c) SEM image of strained WS₂. Scale bar: 1 μm. The arrows denote the direction of the applied strain.

Knowing the thickness of the WS₂ flakes is essential for their analysis (see details below). However, since measuring the flakes thicknesses on the relatively soft PDMS substrate is challenging, after completing the buckling experiments we transferred the flakes to Si/SiO₂ wafer and measured their thicknesses using an atomic force microscope (AFM).

Molecular dynamics simulations

We performed molecular dynamics (MD) simulations to improve our understanding of the behavior of the WS₂. We employed the open-source code LAMMPS²⁹ for the MD simulation. We adopted a modified version of the Stillinger–Weber (SW) potential for TMDs, as was done in similar studies.³⁰ This interatomic potential was found to properly describe the interlayer interaction between W and S atoms. We initiated our theoretical analysis with an MD simulation of a monolayer WS₂ and then performed an analysis of a few layered WS₂ structure.

In the monolayer MD simulation, a rectangular computational cell was constructed with a unit cell composed of six atoms (two W atoms and four S atoms, Fig. 2a). We considered periodic boundary conditions along the sheet axes in the x- and y-directions. An example of a sheet with five unit cells in each direction is shown in Fig. 2b. First, the atom positions were relaxed, in combination with the relaxation of the unit cell size in the x- and y-directions to nullify the internal stresses, using the conjugate gradient algorithm. Then, initial velocities are assigned to the atoms that correspond to a velocity of 300 K, followed by MD timesteps using the microcanonical ensemble. During the MD simulations, a constant strain rate of 10⁷ s⁻¹ is applied along the x-direction.

Fig. 2 MD simulation. W and S atoms are denoted as purple and red colored spheres. (a) MD unit cell. (b) MD sheet.

Experimental results and discussion

Buckling experiment

We initiated our experiments by applying 4% strain to the PDMS, transferring the WS₂ flake, and gradually releasing the stretch of the PDMS, thereby compressing the flake. Importantly, prior to the application of the buckling, we did not observe wrinkles in the WS₂. During the transfer of the strain to the WS₂ we captured optical images of the wrinkles (e.g., Fig. 1b), from which we extracted their wavelength, λ, that is inversely proportional to the spatial frequency. The thickness was obtained after the buckling experiment, as we transferred the WS₂ flakes to a Si/SiO₂ wafer and measured their thickness using AFM (Cypher-ES, Oxford Instruments), Fig. 3a. The relationship between the wavelength and the compressive strain demonstrated a reduction of the wavelength as we increased the strain, see Fig. 3b obtained for a flake subjected to compressive strain.

Fig. 3 (a) Thickness measurement of the WS₂ flake using AFM after transfer to Si/SiO₂ wafer. Scale bar: 10 μm. (b) The relationship between the wavelength and the compressive strain.

Validation of continuum model

Several studies analyzed the results obtained from buckling metrology through continuum models. Contrary to first principal calculations, such continuum models are beneficial since they are computationally cheap and they take into account large-scale phenomena, such as the micron-scale wrinkles we observed in our experiments. In addition, previous studies showed that continuum mechanics models reliably describe mechanical behavior at the nanoscale. For example, continuum model previously described the buckling of carbon nanotubes,³¹ while the Bernoulli–Euler beam theory was employed to describe the vibrations of a nano foam.³²

We checked the validity of the continuum model through a comparison of its results to those obtained from the MD atomistic model. We examined a WS₂ free-standing monolayer under compression. When the WS₂ monolayer is subjected to compression in the MD simulation, it first linearly and uniaxially compresses, followed by its buckling, insets of Fig. 4a. An example of the stress–strain curve and the buckled shape is shown in Fig. 4a. We extracted pre-buckled Young's modulus of WS₂ monolayer from the slope of the linear part of the stress–strain curve, such that E_monolayer = 404 ± 17 GPa. This value is in reasonable agreement with theoretical estimations reported elsewhere.¹ Importantly, since we do not control the crystal orientation in the experiments, a similar analysis was conducted by applying the strain in a perpendicular direction to the one shown in Fig. 4a. For brevity, we do not show the stress–strain response in this direction, since a similar behavior was obtained, with Young's moduli in the range 418 ± 6 GPa. In addition, we found that the physical size of the MD model (i.e., the number of atoms per unit cells included in the model) weakly influenced the calculated Young's modulus.

Fig. 4 MD analysis results of WS₂ monolayer. (a) Stress–strain response. The solid lines are the averaged linear elastic curve (blue line) and the buckled curve (gray line). Insets: the pre-buckled (lower inset) and buckled (upper inset) shapes. (b) The critical buckling strain as a function of (n/L)². A linear fit is drawn as a dashed line.

The linear pre-buckling deformation is terminated by the buckling of the film, accompanied by stress fluctuations. At this stage, we limit ourselves to a small strain range after buckling, and we consider the average post-buckling stress to be constant at this strain range (gray line in Fig. 4a). This average stress represents the bending rigidity of the buckled monolayer. The fluctuations are a result of the high-frequency waves generated after buckling. Since we are employing periodic boundary conditions, the length of the box affects the buckling strain. To examine this dependency, we compare the MD simulation results to those obtained from a continuum model. We model the WS₂ monolayer as a sheet having bending rigidity and subjected to an axial force, such that it obeys the following governing equation

where E_f and ν_f are Young's modulus and Poisson's ratio of the WS₂ flake, respectively, I = wh³/12 is the second moment of area of the thin layer cross-section with respect to the natural axis (h and w are the thickness and width of the flake, respectively). We denote u as the deflection of the flake and F is the axial force. Importantly, the two terms on the left-hand side of eqn (1) represent the unit length forces due to the bending of the flake and the axial load applied to the flake that generates compression. Namely, these terms represent the bending and axial stiffnesses of the flake. When considering a periodic cell of size L, the wavelength of the buckled sheet is λ_n = L/n, where n is the buckling mode. Note that the first buckling mode corresponds to a sine shape with a period L (we provide a detailed analysis in the ESI†). The corresponding uniaxial force, at which the sheet buckles into the n-th mode, is

We assume that the atomic sheet behaves like a continuum sheet of width h and ν_f = 0, such that the corresponding buckling strain is

We plot in Fig. 4b the buckling strains extracted from the MD simulations (red markers), which correspond to the first buckling mode, and the continuum model prediction obtained from eqn (3) (dashed line) as a function of (n/L)². Fitting the continuum model to the MD results yields the effective thickness of h = 4.3 Å. When applying the load in the perpendicular direction, a similar relation between the critical buckling strain and the cell size was found, with an effective thickness of h = 4.2 Å. The similar values in both directions, in addition to the outcome of eqn (3) which is indifferent to the compression direction, demonstrate the accuracy of treating this atomically-thin system on the continuum level, regardless of the compression direction. Additionally, this effective thickness is comparable to the reported distance between the atoms along the thickness of the layer film (about 3.1 Å). Therefore, we conclude that although some reasonable quantitative difference arises from the continuum model, this model gives a good description of the mechanical behavior and a rough quantitative estimation of the buckling stress and strain.

Buckling experiment analysis

We modify our continuum model to include the unit force applied by the elastic substrate (the PDMS), such that the WS₂ monolayer obeys the following governing equation³³Here, E_s and ν_s are Young's modulus and Poisson's ratio of the substrate (PDMS, see ESI†), respectively. The third term on the left-hand side of eqn (4) is the unit-length force applied by the elastic substrate due to the deflection of the flake.³⁴ We assume that owing to the hexagonal structure of the WS₂ lattice, many directions show symmetry in their mechanical behavior. In addition, our thickness measurement demonstrated that samples consist of several layers, each has different direction. As a result, we consider our samples to be isotropic, namely they show similar properties in different in-plane directions. A similar assumption was considered in previous studies.³⁵

Eqn (4) describes the behavior of a slender mechanical element with bending rigidity and constrained to an elastic support. Our observations indicate that due to the micron-scale deflection of the buckled WS₂ compared to the millimeter-scale thickness of the PDMS, both materials maintained contact after buckling took place. Importantly, we excluded from our analysis experiments, in which the WS₂ detached from the PDMS and slipped. A solution of eqn (4) is a sinusoidal deflection of the flake, namely

where A is the amplitude of the deflected flake. Substituting eqn (5) into eqn (4) gives the compressive force, as follows

The axial force has a minimum at a certain wavelength, λ (Fig. 5a). If the load is lower than that minimum, the flake is stable (i.e., it does not wrinkle/buckle). On the other hand, if the load is higher than the minimum value, the flake is unstable and wrinkles according to eqn (5). Thus, the minimum force value, obtained at dF/dλ = 0, represents the force at which buckling occurs, which yields the following term

Fig. 5 (a) Axial force versus wavelength derived from eqn (6). (b) Experimental effective Young's modulus of WS₂ as the function of the applied strain. (c) Post-buckling stress–strain curve extracted from the experiment.

The critical buckling strain, namely the strain at which wrinkles first appear, is evaluated as follows

Eqn (8) implies that when Young's modulus of the thin flake is significantly higher than that of the substrate, as occurs in our case, the buckling appears under small compressive strain.

Eqn (7) gives the relationship between the Young's modulus and the measured wavelength, thus we use it to extract the effective Young's modulus of the wrinkled WS₂. The measured effective Young's modulus is shown in Fig. 5b for a representative compressed flake, and demonstrates a decrease in its value when increasing the compressive strain, namely softening behavior. Notably, previously reported density functional theory (DFT) analysis predicted the opposite trend, namely stiffening under compression (we clarify this discrepancy in the discussion below).³⁶

We substitute the extracted Young's modulus, the wavelength, and material constants into the force term written in eqn (6), and we calculated the stress (σ = F/Â, where  is the cross-section area) to get an experimental evaluation of the stress–strain curve of the WS₂, see Fig. 5c (the flake dimensions were estimated as w = 10 μm and h = 6 nm). This curve demonstrates a decrease in the stress when increasing the compressive strain. This behavior also describes softening, as it reflects lower elastic resistance of the flake.

Raman analysis under compression

We performed in situ Raman spectra experiments, in which we acquired the Raman spectra of strained WS₂ samples (compressive strain ranged from 0% to 2%). We used a LabRam HR Evolution Horiba Raman system with a 532 nm laser, ×100 objective, and an intensity of 0.6 mW to avoid damage to the samples. The Raman spectra of WS₂ (ref. 37 and 38) and other strain-engineered TMDs were thoroughly investigated in the literature.^39–41 TMDs presented Raman vibrational modes that are highly sensitive to the applied strain due to phonon coupling to the strain. We show the Raman spectra of a representative strained WS₂ in Fig. 6a. Unstrained WS₂ usually shows the doubly degenerated in-plane vibrational mode that appears at 350 cm⁻¹ and the out-of-plane vibrational mode at 410 cm⁻¹. Previous studies showed that when increasing the strain, the degenerated peak splits into two subpeaks, and due to the symmetry breaking of the WS₂ crystal.⁴² In some of our samples we observed the splitting of the in-plane vibrational mode peak already in the unstrained spectra, which we attribute to some preliminary stresses that exist due to the exfoliation process (an example for a sample with no initial splitting is shown in the ESI†). When we applied compressive strain all peak positions ( and modes) redshifted with the increasing strain (Fig. 6b–d), indicating vibrational modes softening in the WS₂. Importantly, this observation agrees with the mechanical softening observed in our buckling experiment. Also, similar Raman mode softening observed in other strained TMDs was reported in the literature.^10,39,42,43

Fig. 6 (a) Raman spectra of strained WS₂. (b–d) Raman peak's location shifts for A_1g, and

The strain-dependent Raman peaks allow us to calculate the Grüneisen parameter, γ_E2g, which is proportional to the shift rate of phonons and determines the thermodynamic properties of the crystal. In addition, the symmetry breaking of the lattice allows us to estimate the shear deformation potential, β_E2g, which represents the required energy to induce shear deformation, such that⁴⁴

where is the frequency of the modes, and are the frequency shifts of the and peaks, respectively. In addition, ν_f is the Poisson's ratio of the WS₂ flake and ε is the applied compressive strain. We obtained that the Grüneisen parameter is γ_E2g = 0.29, which is similar to the values obtained for other TMDs (e.g., 1.1 for MoS₂ and 0.38 for WSe₂) and to the value reported for WS₂ (0.5),⁴⁵ and smaller than Grüneisen parameter values obtained for graphene (1.99)⁴⁶ and hexagonal boron-nitride (hBN, 1.88).⁴⁵ We calculated the shear deformation potential of β_E2g = 0.56, which is again similar to other TMDs (e.g., 0.78 for MoS₂ and 0.10 for WSe₂) and to the values reported for WS₂ (0.14), while graphene (0.99) and hBN (0.91) showed larger values.

Bilayer MD simulation

We modified our MD model to gain more insights regarding the softening observed in our experiments. We assumed a bilayer WS₂ and employed a Lennard-Jones (LJ) potential as a simple model for the adhesion between the layers. The LJ potential is defined between the S atoms, with a LJ distance of 5.8 Å. Different LJ energies were examined to represent different interlayer adhesions. In Fig. S1 in the ESI,† we show the buckling strain when considering two layers for different LJ energies. The main difference was found to be in short periodicity, where interlayer shear stress is more influential, and thus, the approximation of the Kirchhoff–Love plate theory is not valid. However, for long films, as taken in our simulations, the effect of the adhesion between the layers on the buckling strain is smaller and can be included in the value of the effective thickness, h. When considering the LJ energy of 0.1, 0.2, and 0.5 eV, the effective widths were found to be 13.6 Å, 14.9 Å, and 15.1 Å, respectively. These values are comparable to the reported distance between the atoms in the bilayer film (about 12.1 Å).

To date, we are not familiar with a proper interatomic potential for the interlayer interaction, however, we do not aim in this work to make a quantitative comparison with the experimental results. Nonetheless, the simulations above demonstrate the validity of the continuum analysis for these thin 2D structures, which ratifies the use done in the analysis of the experimental study, and we employ the atomistic model to qualitatively explain the strain-dependent Young's modulus in the experiments. Thus, in the following, we consider a bilayer WS₂ with LJ interlayer energy of 0.1 eV to extract insights regarding the post-buckling behavior.

Similar to the case of monolayer simulation, the MD simulation of the WS₂ bilayer shows that first, the WS₂ demonstrates a linear pre-buckling relationship between the stress and strain. Again, since the buckling is abrupt, stress waves are generated and to follow the overall stress response, we reduced the simulated post-buckling strain rate by an order of magnitude to capture these waves. Additionally, the stress response is averaged over a strain range of 0.035% to clarify the mechanical response of the WS₂ to compression.

In Fig. 7 we plot the normalized stress (the stress divided by the buckling stress)–strain curve of a 540 Å long film with periodic boundary conditions. The film first deforms elastically up to a buckling stress, σ_cr, after which it buckles into the first mode (Fig. 7b). As explained, after buckling the stress of the WS₂ largely fluctuates, but on average the buckled film still resists compression. Right after buckling, the average compressive stress reduces by ∼25%. During compression, the thin film retains its buckling mode shape up to about a certain strain (1.5% in the chosen example). Beyond this value, we observed an additional stress drop which corresponds to a shape change, in which the buckled shape developed straight segments connected with rounded parts with a small radius of curvature (we further demonstrate this buckled shape in ESI†). This shape change, which is beyond the scope of the present study, can be related to faceting that was found to be related to the interatomic interaction between layers in 2D materials.⁴⁷

Fig. 7 (a) MD simulation for the normalized stress–strain response of a bilayer WS₂ including the post-buckling response. To reduce the effect of the fluctuations, a moving average is applied to the post-buckling response (red dots). (b) The buckled shape.

Discussion

Previous theoretical studies showed that the interatomic distance decreases under compression, which results in the dominance of repelling forces in the interatomic interactions.³⁶ As a result, the WS₂ is more resistant to an external load, namely, it shows stiffening. However, our buckling experiments demonstrated different behavior, as we observed a decrease in the stress under compression, which indicates a softening of the WS₂. Our MD simulations show that atomic scale buckling events occur under compressive strain, which results in a significant shape change of the WS₂ to the buckled shape, and thus, also a reduction of its capability to resist external loads. Consequently, we witnessed a mechanical softening. This was further backed by Raman measurements that demonstrated the softening of vibrational modes.

Some differences exist between the MD simulations and the experiments. For example, due to limited computation power, only a few layers and a small number of atoms can be included in our simulations. In addition, the nature of the interlayer interaction still requires further study to fully understand its behavior. Nevertheless, and although we did not aim to make a quantitative comparison with the experiments, the MD analysis provided important qualitative insights that clarify our experiments and revealed the reason for the observed softening behavior.

Previous studies demonstrated the ability of various strain engineering methods to modulate the energy structure of the investigated materials, and therefore, tune their Raman spectra and photoluminescence emission. However, the ability to tune the mechanical properties has not been reported. Therefore, we shed light on the fundamental coupling between the atomic structure of WS₂ and its mechanical behavior. This understanding will impact the operation of WS₂ in applicative systems where they are subjected to mechanical strain, such as flexible electronics and reinforcement of composite materials. This, in turn, will also open the path for new applications that will benefit from the tunability of the mechanical properties of 2D materials, such as smart sensors and adjustable materials.

Summary

We investigated how strain engineering of thin-layered WS₂ influences its mechanical behavior. We experimentally applied buckling load to thin layered WS₂ and witnessed the softening of its effective Young's modulus. Raman analysis of compressed WS₂ backed this behavior as we observed redshift and softening of its vibrational modes. In parallel, we applied MD simulations that confirmed the validity of continuum mechanics models and revealed that under compressive load the thin layer presented atomic-scale buckling events. As a result, the WS₂ is subjected to a significant shape change that alters its ability to resist external load and is manifested by mechanical softening. Therefore, we uncover the mechanics underlying strained WS₂ and demonstrate the ability to tune its mechanical properties. This ability will be useful for future smart applications.

Data availability

The data supporting the article “Strain Engineering of the Mechanical Properties of Two-Dimensional WS₂” by Y. M. Jahn, G. Alboteanu, D. Mordehai, and A. Ya'akobovitz are included within the figures in the article and its ESI† files.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

We would like to thank Juergen Jopp for his assistant and devotion regarding the AFM measurements. We also want to thank Roxana Golan for her assistance with the SEM measurements.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d3na00990d

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