Single-Layer MoS₂: A Two-Dimensional Material with Negative Poisson’s Ratio

Abstract

Negative Poisson’s ratio (NPR) materials have broad applications such as heat dissipation, vibration damping, and energy absorption because of their designability, lightweight quality, and high strength ratio. Here, we use first-principles calculations to find a two-dimensional (2D) auxetic material (space group R$\overline{3}$m), which exhibits a maximum in-plane NPR of −0.0846 and a relatively low Young’s modulus in the planar directions. Calculations show that the NPR is mainly related to its unique zigzag structure and the strong interaction between the 4d orbital of Mo and the 3p orbital of S. In addition, molecular dynamics (MD) simulations show that the structure of this material is thermodynamically stable. Our study reveals that this layered MoS₂ can be a promising 2D NPR material for nanodevice applications.

1. Introduction

Young’s modulus is a physical quantity that characterizes the resistance of a material to tensile or compressive forces within the limits of elasticity [1,2]. Poisson’s ratio (PR) is defined as the ratio of the transversal strain to the strain along the applied stress direction when the material is deformed elastically [3]. Generally, PR, by definition, is positive for most commonly available engineering materials when the applied strain and the resultant transverse strain have opposite signs [4]. NPR materials refer to the phenomenon of lateral expansion when subjected to uniaxial tensile stress [5]. NPR materials possess certain advantageous properties including strong designability [6], light weight [7], high strength ratio [8], and high damping [9], and moreover have good performance in heat dissipation [10], vibration reduction [11], and energy absorption [12]. These excellent mechanical properties greatly expand the scope of the application of NPR materials. At present, there are broad application prospects in the fields of aerospace, equipment armor, intelligent manufacturing, coating materials, and so on.

Quantum confinement caused by the low-dimensional effect of two-dimensional (2D) materials makes them more prone to NPR [13,14,15,16]. Therefore, it is possible to find new 2D NPR materials for novel nanomaterial applications. Until now, some NPR materials have been identified, such as black phosphorus (BP) [17], Be₅C₂ [18], SnSe [19], TiN [20], α-phosphorene [21], δ-phosphorene [22], Cd₂C [23], Zn₂C [24], and Ag₂S [25]. However, the number is still small considering the numerous materials [26,27] that have been discovered and investigated [28]. MoS₂ is one of the well-known materials possessing several different configurations, which have different structures. MoS₂ [29] is a material composed of two chemical elements, molybdenum and sulfur, with excellent optical, electrical, magnetic, force, and thermal properties, and is widely used in energy storage, catalysis, semiconductors, lubrication, and other fields. It is notable that in the mechanical properties, both positive and negative PR are present in the MoS₂ configuration, which is rare for the same material of other compounds. It has been reported that MoS₂ has both positive and negative PR, and small changes in the structure will greatly affect the PR. Yu et al. [30], in 2017, reported 42 1T-type crystals in which the presence of NPR for MoS₂ was suggested, while Hung et al. [31], in 2018, reported three different structures of MoS₂ with positive and negative PR. These results coincide with our calculations. However, for the same material (MoS₂), there is no exact explanation of what causes its many configurations to have both positive and negative PR. The main reason is that there is a general lack of understanding of the mechanism of NPR materials. Therefore, in-depth analysis from the perspective of electron interaction is needed to understand the mechanisms behind the formation of NPR materials. Such understanding is important for the search and design of new materials with widened application prospects.

There are three different phases of monolayer MoS₂: 1T-phase, 1H-phase, and 1T’-phase MoS₂, respectively [32]. To investigate in-depth the formation mechanism behind NPR, 1T-phase MoS₂ with NPR and three other materials were selected for comparison, and their respective Young’s modulus, PR, and projected density of states (PDOS) were calculated. Their geometries and electronic structures were compared to analyze their effects on the NPR generation of the materials, and the mechanism of NPR formation was demonstrated in terms of both geometric deformation and electronic structure. Finally, the formation mechanism of NPR materials was further investigated from the perspective of layer spacing and intralayer forces.

2. Computational Methods

All calculations were performed using the CASTEP (version 8.0.) [33,34] module of the Materials Studio package. During the computational process, self-consistent periodic density functional theory (DFT) [35,36] was adopted by the Generalized Gradient Approximation (GGA) method using the Perdew–Burke–Ernzerhof (PBE) exchange–correlation functional [37]. The energy cutoff was 340 eV and the Self-Consistent Field (SCF) tolerance was 5.0 × 10⁻⁷ eV atom⁻¹. The convergence criteria for max force, max displacement, max stress, and energy were set to lower than 0.01 eV Å⁻¹, 5.0 × 10⁻⁴ Å, 0.02 GPa, and 5.0 × 10⁻⁶ eV atom⁻¹, separately [38]. The k-point meshes were set as 5 × 5 × 5 for the bulk. The Young’s modulus $Y\left(\theta \right)$ and PR $\mathsf{\upsilon }\left(\theta \right)$ could be indicated as follows [39]:

$$\begin{gathered} Y\left(\theta \right)=\frac{C_{11}{C}_{22}-C_{12}^2}{C_{11}si{n}^4\theta +Asi{n}^2\theta co{s}^2\theta +C_{22}co{s}^4\theta } \\ \mathsf{\upsilon }\left(\theta \right)=\frac{C_{12}si{n}^4\theta -Bsi{n}^2\theta co{s}^2\theta +C_{12}co{s}^4\theta }{C_{11}si{n}^4\theta +Asi{n}^2\theta co{s}^2\theta +C_{22}co{s}^4\theta } \end{gathered}$$

where $C_{ij}$ are the elastic constants, $A=\left(C_{11}{C}_{22}-C_{12}^2\right)/C_{66}-2C_{12}$ and $B=C_{11}+C_{22}-\left(C_{11}{C}_{22}-C_{12}^2\right)/C_{66}$. In order to verify the calculation, we calculated the elastic constant of MoTe₂, which was in agreement with the available theoretical values [40]. The calculated elastic constants of MoS₂ and MoTe₂ are included in Table S1. The molecular dynamics (MD) simulation in the NVT (Number of particles, Volume, and Temperature) ensemble was performed at 300 K with a 3 × 3 × 1 supercell for 6ps with a time step of 1fs. MD was performed with periodic boundary conditions. In order to ensure the accuracy of the simulation, a vacuum layer of 15 Å was added to the four models to eliminate a certain degree periodic interaction and the bottom layer of MoS₂ was fixed in the calculation process. The force field used in the MD simulation was a universal force field (UFF). UFF was applied to optimize the model and assign charges to ensure the accuracy of the calculation results. The designation of the algorithm used in the geometry optimization calculations was a quasi-Newton method. The initial model was minimized by the quasi-Newton method until the gradient was less than 0.1 kcal/mol. All materials involved were from the crystallographic database: materials project.

3. Results and Discussion

1T-phase MoS₂ is a member of transition metal dichalcogenides (TMD) [41]. The unique zigzag structure of MoS₂ brings many unique mechanical properties. In the current work, we explore the mechanical properties by computing the planar Young’s modulus $Y\left(\theta \right)$ and PR $\mathsf{\upsilon }\left(\theta \right)$. Results are drawn in Figure 1a. Young’s modulus $Y\left(\theta \right)$ is a physical quantity that describes a material’s ability to resist elastic deformation. More rigid materials have a higher Young’s modulus. As Figure 1a shows, Young’s modulus of MoS₂ does not show anisotropy, which means that the mechanical response to the same strain varies little along different in-plane directions. Young’s modulus attains a maximum value of 183.1 GPa at $\theta =62°$, $118°$, $224°,$ and $298°$, and a minimum value of 178.9 GPa at $\theta =0°$ and $180°$. This indicates that MoS₂ exhibits moderate deformation-resistant stiffness in all directions. For comparison, Young’s modulus of graphene [42] and TiS₂ [43] is added in the bar chart of Figure 1a. Compared to these two well-known materials, Young’s modulus of MoS₂ is comparatively low. This means that MoS₂ exhibits considerable elastic compliance, which can be traced to its special atomic structure (to be discussed later). The PR $\mathsf{\upsilon }\left(\theta \right)$ of MoS₂ is shown in Figure 1b. It exhibits an NPR in the whole region and reaches the highest value of −0.0846 at 46°, 134°, 226°, and 314°, indicating that MoS₂ is an auxetic material. The planar NPR value is bigger than that of penta-graphene ($\mathsf{\upsilon }=-0.068)$ [44], PN ($\mathsf{\upsilon }=-0.078$) [45], and borophene ($\mathsf{\upsilon }=-0.053$) [46].

The crystal structure of bulk MoS₂ is displayed in Figure 1c. The band structure as well as the density of states (DOS) of MoS₂ is plotted in Figure S1. It could be regarded as a laminar structure in which the zigzag layers are linked by the Mo-S bonds. Its primitive cell consists of six sulfur atoms and three molybdenum atoms in a trigonal crystal system. The lattice constants are a = b = 3.190 Å, and the space group is R$\overline{3}$m [47]. From the crystal structure diagram, the structure of this material can be considered as a network of herringbone Mo-S atomic chains in two directions, one along the monoclinic lattice and the other perpendicular to the lattice direction. The angles of S-Mo-S along these two directions are 81.567° and 98.433°, respectively (See Figure 1e).

To investigate how the atomic geometry affects PR, three additional materials were selected for comparison: the MoS₂ with zigzag structure (space group P6₃/mmc), the MoS₂ without the special undulations (space group F$\overline{4}3$m), and the WS₂ with the same R$\overline{3}$m space group but different elements. The calculated results are plotted in Figure 2 and Figure S2. The lattice constants and the atomic coordinates in every cell after optimization are included in Table S2. Clearly, among the three compared materials, MoS₂ with space group F$\overline{4}3$m has no obvious zigzag geometry, and it shows a positive Poisson’s ratio (PPR). This is in agreement with the previous conclusion that the unique sawtooth-like geometry is necessary for an NPR to occur [48]. However, the PR of the other two materials with similar sawtooth geometries (MoS₂, space group P6₃/mmc; WS₂, space group R$\overline{3}$m) is also positive. This suggests the NPR is not only related to the material geometry. Other factors may play a role too.

To gain a more in-depth insight into the deformation mechanism behind the NPR, we further analyzed the electronic interactions. We calculated the PDOS of the four materials to observe their electronic structures, as shown in Figure 3. The PDOS peak shape and position of the 4d orbital of the Mo atom and the 3p orbit of the S atom are similar in energy, indicating the strong orbital interaction of the Mo atom and S atom (the yellow highlighted portion of Figure 3a). It can be seen that both sets of orbitals overlap above the Fermi energy level, indicating that the atoms are antibonding. It is well-known that the bonding state leads to attractive interactions, while the possession of the antibonding state causes repulsive interactions. This is most likely a key factor in the generation of NPR for MoS₂. Moreover, the MoS₂ that shows an NPR has an overlap area of 5.854 above the Fermi level, while the other three materials showing PPR have overlaps of 1.807, 0, and 2.508 above the Fermi level, respectively, which all indicate that their coupling is not as strong as that of the NPR materials at the p-d orbitals.

Figure 3d indicates that the interaction of the p orbital and d orbital of WS₂ is significantly weaker than MoS₂ in the energy range of 0 to 6 eV. Although both MoS₂ (space group R$\overline{3}$m) and WS₂ (space group R$\overline{3}$m, and the value of PR is 0.25 in Figure 2h) share the same space group, there are differences in their electronic structures that lead to very different results. Therefore, the electronic structure is one of the characteristics of the NPR of MoS₂. In summary, an NPR material should have a special geometry in terms of structure (i.e., undulating structure), while in microscopic terms, it should have a strong enough coupling between different orbitals (i.e., electron interaction), and the atoms are anti-bonding (the atoms will tend to interact with each other in a repulsive manner). Materials that have all these characteristics will have a much higher probability of NPR.

Electronic interactions can be divided into interlayer and intralayer based on the geometric configuration of MoS₂. Furthermore, we investigated how the electronic structure affects the interatomic forces that lead to the NPR in the material. Figure 4a displays the structure of the initial MoS₂, and Figure 4b displays the structure with the new alignment rebuilt by cleaving the (0 0 1) surface of MoS₂. The difference between them is that the initial structure is staggered and aligned, while the modeled structure is overlapping and aligned. By varying their layer spacing, it is possible to investigate how much the interlayer forces of these two very different alignments contribute to the generation of NPR. As illustrated in Figure 4c, the PR of the material varies in value but remains negative overall. The PR varies from −0.67 to −0.82 and from −0.10 to −0.13, with a moderate variation. It indicates that the value of the PR does not change greatly with an increase in the layer spacing, indicating that the layer spacing has almost no impact on the NPR of the material. Therefore, the interatomic forces caused by the intralayer electronic structure of monolayer MoS₂ may be the dominant factor.

Based on the above analyses, the force generated by the electronic structure between the layers has little effect on the deformation of the whole structure. In the next step, we applied a strain to the single-layer MoS₂ (space group R$\overline{3}$m) and observed how the interaction between the electrons in the structure within the layer affected the material’s PR. Figure 5a shows a sketch of the original structure with a monolayer of MoS₂. This is a material with a folded structure with upper and lower pyramidal shapes. Figure 5b shows a schematic of the structure after it has been stretched. When two Mo atoms, Mo_A and Mo_B, are pulled in the Y-axis, the structure produces a force of compression in the Z-axis because of the intense interaction of the p-d orbitals, resulting in the structure being compressed in that direction and pulled in the X-axis. The bond angles of $\angle$CAB and $\angle$CBA are reduced behind the deformation. In principle, the change of $\angle$CAB is not impacted by the stretching of the Mo_A and Mo_B atoms, which is only related to the reciprocal action of the p-d orbitals. That is, the displacements occurring in the Mo_A and Mo_B during the deformation are only relevant to the strength inside the cell. Therefore, the angular variation of $\angle$CAB could be used to describe the mechanical behavior generated within the cell. When a 5% stretch strain is applied in the Y-axis, the bond angle of $\angle$CAB decreases from 49.216° to 45.376° (a decrease of 3.84°), and the interplanar spacings d_AC and d_CE decrease by 0.052 Å and 0.039 Å, respectively. The strong coupling of the p-d orbitals causes a large amount of strain energy generated during deformation to be stored in the reduced $\angle$CAB and $\angle$CBA, and this strain energy is released by increasing the distance between Mo atoms D and Mo atoms A, leading to an NPR. This confirms the above inference that the p-d orbital interactions lead to a planar NPR in monolayered MoS₂. Further observing the PDOS of MoS₂ in Figure 3a, the p-orbital overlaps with the d-orbital at positions above the Fermi level, which indicates that the metal atoms exhibit a strong antibonding property, which leads to mutual repulsion between the atoms. As a result, the NPR phenomenon appears.

For comparison, we applied tensile strain on both WS₂ (space group R$\overline{3}$m) and MoS₂ (space group P6₃/mmc), which exhibited PPR. Figure 5 shows that after applying a tensile strain, the whole structure shrunk to a certain extent. Combined with the previous analysis, these two materials used as comparisons have a weaker coupling in their p-d orbitals, resulting in a reduction in the strain energy that can be stored in the bond angles during tensile strain. As a result, the changing bond angles release less energy and have a reduced effect on the surrounding atoms when the structure shrinks. The PDOS of the two compared materials did not exhibit strong antibonding properties and the atoms did not tend to repel each other, meaning they ended up exhibiting PPR. To better verify our speculation, we also calculated PDOS for four other NPR materials, and the outcomes are listed in Figure S4. It is clear from the figure that the four NPR materials have similar degrees of overlap in the positions of the peaks and energies of the PDOS, indicating that they all have strong interactions in their p-d orbitals. Such a comparison shows that the change in interatomic forces due to the interaction between p-d orbitals is a major factor in the NPR behavior of the 2D materials, which can be used as a feature for screening NPR materials. More specifically, the NPR of single-layer MoS₂ is related to the strong coupling between the 4d orbital of the Mo atom and the 3p orbital of the S atom, and the antibonding orbits of the atoms present. This eventually causes the interaction forces between the Mo and S atoms to tend to repel each other, leading to their extension and triggering NPR.

Finally, we performed molecular dynamics (MD) simulations at 500 K for 6 ps with NVT (Number of particles, Volume, and Temperature) ensemble to evaluate the structural stability of MoS₂ [49]. In the process of structural evolution, temperature fluctuations and the mean potential energy of atoms are displayed in Figure 6. Throughout the simulation process, the changes in potential energy remained near the average. Figure 6 and Figure S5 give three different configurations of MoS₂ and WS₂ structural snapshots at the end of 6 ps. The diagram shows that the geometry is well-saved and no significant structured fractures are observed, which indicates that the structure of MoS₂ is stable. The calculated elastic constants of the 2D molybdenum disulfide meet the Born−Huang criteria [50], in which $C_{11}$, $C_{22}$, $C_{66}>0$ and $C_{11}+C_{22}-2C_{12}>0$. Judging from the data in Table S1, it is confirmed that the MoS₂ structure is stable.

4. Conclusions

To sum up, we used first-principles calculations for the elastic constants of 2D molybdenum disulfides and have succeeded in finding a molybdenum disulfide with NPR (space group R$\overline{3}$m). This serrated structure with unique layer overlap exhibits a separate NPR of υ_x = −0.0736 and υ_y = −0.0750 in the X- and Y-axis. In addition, we explored the effect of layer spacing on its NPR behavior. Finally, we found that the unusual NPR behavior of single-layer MoS₂ is linked to its unique geometry and the strong interaction between the 4d orbitals of Mo and the 3p orbitals of S. Moreover, we also found that the strong coupling of p-d orbitals and the zigzag structure of the antibonding state are notable features for screening 2D NPR materials. These discoveries provide insight into the influence of the geometry and electronic structure of 2D molybdenum disulfide on the mechanical performance of the material. These results could propel progress in the screening and design of other 2D materials in the future.