Effect of Structural Phases on Mechanical Properties of Molybdenum Disulfide.

Molybdenum disulfide (MoS2) is a promising layer-structured material for use in many applications due to its tunable structural and electronic properties in terms of its structural phases. Its performance including efficiency and durability is often dependent on its mechanical properties. To understand the effects of the structural phase on its mechanical properties, a comparative study on the mechanical properties of bulk 2H, 3R, 1T, and 1T' MoS2 was conducted using the first-principles density functional theory. Since considerable applications of MoS2 are developed through strain engineering, the impact of the external pressure on its mechanical properties was also considered. Our results suggest a strong relationship between the mechanical properties of MoS2 and the structural symmetry of its crystal. Accordingly, the impacts of the external pressure on the mechanical properties of MoS2 also greatly vary with respect to the structural phases. Among all of the considered phases, the 2H and 3R MoS2 have a larger bulk modulus, Young's modulus, shear modulus, and microhardness due to their higher stability. Conversely, 1T and 1T' MoS2 are less strong. As such, 1T and 1T' MoS2 can be a better candidate for strain engineering.

Introduction

Layer-structured materials, such as transition metal dichalcogenides (TMDs), have attracted intensive attention recently due to their specific physical and chemical properties and potential applications.[1−3] One of the most important and widely studied layer-structured materials is molybdenum disulfide (MoS2), which has the demonstrated high capacity for use in electronic devices, hydrogen storage, catalysts, solar cells, hydrogen production, and solid lubricants.[4] Many large-scale applications of MoS2 are directly or indirectly determined by its mechanical characteristics, such as elastic constants, bulk modulus (B), shear modulus (G), Young’s modulus (Y), Poisson’s ratio (σ), and microhardness (H).[5−7] The bulk modulus indicates the resistance to fracture deformation, which is closely related to the cohesive energy of materials or bonding energy between atoms in the crystal structure. The shear modulus represents the plastic deformation of the material. Young’s modulus is a parameter representing the stiffness of a solid, which is also a function of atomic bond strength. Poisson’s ratio is a measure of the Poisson effect, the phenomenon in which a material tends to expand in directions perpendicular to the direction of compression, and microhardness indicates the resistance of the physical object against compression of the contacting parts.

Most previous studies have particularly focused on the mechanical properties of two-dimensional (2D) MoS2 monolayers.[7−11] However, few-layered MoS2 is also widely used in practical applications,[12,13] whose properties are more similar to the bulk MoS2.[14] This is because only the in-plane mechanical properties need to be taken into consideration for the monolayers due to the absence of interlayer interactions. For bulk MoS2, however, the interlayer breathing and shear modes become important to holistically understand their mechanical properties,[15] and the in-plane mechanical properties of MoS2 have also been demonstrated to be directly related to their thickness.[15,16] It was also suggested that the mechanical properties of MoS2, particularly Young’s modulus, is strongly dependent on the interlayer sliding in the multilayers or bulk.[17] As such, due to the many different properties of bulk TMDs with respect to their monolayers, the mechanical properties of bulk MoS2 with different structural phases are needed to be investigated. Previous studies have evidenced that bulk MoS2 has at least five phases, e.g., 2H, 3R, 1T, 1T′, and 1T″, with significantly different structural and electronic properties.[4,8,18,19] As such, their performance in specific applications is strongly related to their structural phases. For example, Zhao et al. found that 3R MoS2 has distinct electronic and optical properties arising from its crystal structure.[20] 2H MoS2 is a thermodynamically stable semiconductor. Its edge has been found to be the active site for electrocatalytic hydrogen evolution reactions (HERs).[21] 2H-MoS2 has also been widely used as a support for catalysis due to the specific 2D structure and high thermodynamic and chemical stability. The 1T phase displays a comparatively higher rate of charge transfer, which provides potential advantages in electrochemical applications,[22,23] and the 1T MoS2 may be an auxetic material with an intrinsic in-plane negative Poisson’s ratio.[24] Also, the S atoms in the basal plane of the 1T′ phase are active to HERs.[25−28] Previously, the mechanical properties of bulk 2H MoS2 have been reported.[29−31] Since the other phases of MoS2 also have great potential applications, it is crucial to investigate the mechanical properties of MoS2 with other structural phases. Additionally, the introduction of external pressure has a remarkable influence on the mechanical and electronic properties of MoS2.[32−37] For example, Feng et al. proposed that the strain-engineered MoS2 may capture a broad range of the solar spectrum.[35] Liu et al. demonstrated that thermally induced nonuniform tensile strain of monolayer MoS2 played a significant role in grain boundary optical properties.[36] Yan et al. revealed that properties of materials could alter under external pressure.[37] The pressure effects on the properties of MoS2 are essential to its application as a lubricant.[38] To this end, the understanding of the impact of external pressure is also of paramount importance. However, accurate experimental measurements of the mechanical properties of bulk MoS2 materials with various phases are impeded by several issues, such as the availability of their high-quality single crystals.[26,39] The alternative route is to compute the mechanical properties of MoS2 since the high-level computations have the demonstrated capacity to provide accurate mechanical properties of single-crystal materials. The significant pressure-induced effects on the mechanical properties of 2H-MoS2 have already been theoretically confirmed by Wei and Peelaers et al.[30,40] However, the pressure impact on the mechanical properties of MoS2 with other structural phases is largely unknown.

In this study, the effects of both the structural phase and external pressure on the mechanical properties of 2H, 3R, 1T, and 1T′ MoS2 have been systematically investigated using the first-principles density functional theory (DFT) method. The 1T″ phase was not considered here because it is least thermodynamically stable and no layered bulk 1T″ MoS2 has been reported.[8,19] Our results reveal the strong dependence of mechanical properties of MoS2 on the symmetry of its prismatic unit and thermodynamic stability.

Results and Discussion

Figure shows the atomic configurations of MoS2 in different phases (2H, 1T, 3R, and 1T′). 2H and 3R MoS2 have the trigonal prismatic unit with the symmetry of D3. The hexagonal 2H-polytype has two layers per unit cell along the c-axis, and the rhombohedral 3R-polytype has three layers per unit cell. 1T MoS2 has molybdenum atoms octahedrally coordinated by S atoms to form a unit cell. 1T′ is a lightly distorted atomic structure of 1T. After structural optimization, 2H, 3R, 1T, and 1T′ possess hexagonal (crystal class: 6/mmm), trigonal (crystal class: 3m), trigonal (crystal class: 3®m), and monoclinic lattice systems, respectively. The optimized structures are provided in the Supporting Information.

Figure 1

Optimized atomic structures of 2H, 3R, 1T, and 1T′ MoS2. Enclosed by red parallelograms are the unit cells used in this study.

The calculated lattice constants, monolayer thicknesses, and Mo–S bond lengths are listed in Table in comparison with available experimental and theoretical data. The initial structures of 1T, 2H, and 3R MoS2 are from the experiments.[41−43] The 1T′ structure is built based on a similar structure of 1T′ ReS2 due to the lack of direct experimental data.[44] Our results of 2H, 1T, and 3R MoS2 using the DFT-D3 approach are in good agreement with the experimental results and the theoretical results with dispersion corrections. From the optimized results, the 2H and 3R phases have similar structures. Additionally, the 1T′ phase has the shortest interlayer distance, as evidenced by the smallest average layer thickness, and 1T′ MoS2 has the lowest monoclinic structural symmetry. Consequently, two sets of Mo–S bond lengths and S–Mo–S bond angles are provided in Table .

Table 1

Calculated Lattice Constants a (Å) and c (Å), Average Monolayer Thickness t (Å), Bond Length d (Å), and S–Mo–S Bond Angles θ (deg) of Bulk MoS2 in Comparison with Reported Data from the Literature

phase		a (Å)	c (Å)	t (Å)	d (Å)	θ (deg)
2H	present	3.16	12.31	6.16	2.40	81.2
	exp.[42]	3.16	12.29	6.15
	exp.[41]	3.15	12.30	6.15
	LDA[45]	3.13	12.06	6.03
	PBE[45]	3.09	14.01	7.01
	PBE-D2[45]	3.20	12.42	6.21
1T	present	3.18	6.04	6.04	2.42	82.0
	exp[46]	3.26	6.14	6.14
3R	present	3.16	18.40	6.13	2.40	81.0
	exp.[41]	3.15	18.30	6.10
	LDA[45]	3.14	17.93	5.98
	PBE[45]	3.19	22.00	7.33
	PBE-D2[45]	3.20	18.56	6.19
1T′	present	6.39	5.85	5.85	2.37, 2.40	80.8, 83.5

The electrical conductivity of different phases of MoS2 was also investigated through the analyses of their total density of states (TDOS), as shown in Figure . It can be found that there is a band gap for the 2H and 3R structures, indicating the fact that the material in both phases behaves as a semiconductor. On the other hand, 1T phases have metallic characteristics. Our results agree with the reported conclusion.[4,18] Dobson and his co-workers reported that the interactions between layers of 2D materials are significantly determined by their electrical conductivity.[47] As such, 1T MoS2 should possess stronger interlayer interactions, which has been confirmed by the calculated interlayer interaction energies of different phases (see Figure S2). The stronger interlayer interaction in 1T MoS2, therefore, leads to the smallest average layer thickness. 1T′ has the weakest interlayer interaction energies, which may be due to its zigzag atomic structures (see Figure S2). Our calculation suggests that the MoS2 1T′ phase is metallic while the evolution at Fermi energy level is small, which is different from recent results, which show that the 1T′ monolayer has a very narrow band gap.[48,49] The discrepancy can be caused by 2-fold reasons. First, the narrow band gap can only be identified using a high-level and time-consuming method. Second, studies on 2H-MoS2 demonstrate that the 2H monolayer has a much larger band gap value when compared to the few layers and bulk system. The same trend is also expected here. Since the mechanical properties of MoS2 are focused on and it has been demonstrated later that reasonable mechanical properties can be obtained at the PBE-D3 level, the PBE-D3 method has been used throughout this study.

Figure 2

Total DOS of 2H, 3R, 1T, and 1T′ MoS2.

To validate our calculation method for the investigation of mechanical properties, we then compared our calculated mechanical properties of 2H MoS2 with the reported data, which are listed in Table . The similarities of the experimental data with ours can be ascribed to the consideration of vdW interactions.[40,50] One exception is C12, which is positive from our calculations, while the experimental value is negative.[51] However, our result matches the reported theoretical data.[30,40] Our elastic constants are also in good agreement with the experimental data and previous studies using the high-level hybrid DFT method with consideration of dispersion corrections. As a comparison, previous data obtained using the method without the vdW correction show the relatively large deviation of C13, C33, and C44 values from the experiments.[40] It demonstrates the importance of the inclusion of vdW corrections when the mechanical properties of layer-structured materials are investigated.

Table 2

Elastic Constants (C, GPa) of 2H-MoS2

elastic constants	present	exp.[51]	exp.[15]	GGA[40]	HSE06-D2[30]
C11	223	238		211	238
C12	53	–54		49	64
C13	10	23		3	12
C33	49	52	52	37	57
C44	15	19	16	30	18

Our calculated elastic constants of 2H, 3R, 1T, and 1T′ MoS2 are shown in Figure . According to the Born–Huang criteria,[52] a mechanically stable 2D sheet would satisfy C66 = (C11 – C12)/2 > 0 and C11C22 – C122 > 0. Based on our results, it can be found that all four phases of MoS2 are mechanically stable. The elastic constants of 2H and 3R MoS2 are similar. This can be ascribed to the subtle structural difference because they have similar prismatic units. As a comparison, Mo atoms are octahedrally coordinated by S atoms in the 1T and 1T′ phases (see Figure ). As such, the elastic constants of 1T and 1T′ are different from those of 2H and 3R MoS2. Additionally, the elastic constants of 1T are also considerably different from those of 1T′. This could be attributed to the different structural symmetry (trigonal and monoclinic crystal systems for the 1T and 1T′, respectively), while their prismatic units are similar.

Figure 3

Calculated elastic constants of 2H, 1T, 3R, and 1T′ MoS2.

Figure shows the different mechanical properties of the four structural phases. Due to the similarities of the prismatic unit structure, the 2H and 3R phases exhibit a comparatively similar bulk modulus of 55 GPa. 1T and 1T′ possess smaller bulk moduli, 43 and 49 GPa, respectively. Bulk modulus is employed to describe the resistance of a material under compressive loads, which reveals the response of a material to stress and strain. As such, all phases hold comparatively similar resistances against external loads, and 2H and 3R phases show a higher withstanding ability compared to the other two structural phases. Moreover, 1T and 1T′ MoS2 also have a lower shear modulus (32 and 34 GPa for 1T and 1T′, respectively) compared to the other two phases (39 and 38 GPa for 2H and 3R, respectively). The lower G value suggests the material to be less rigid and easy for its deformation. Figure also shows that Y and H have similar trends as B and G values and 1T and 1T′ have lower values in comparison with the 2H and 3R phases. These characteristics are directly related to the structural stability of the material. Figure S2 shows that 2H and 3R have stronger cohesive interactions, which matches the experiments since they are more thermodynamically stable. As such, the positive relationship between B, G, Y, and H values with cohesive energies is confirmed based on our results. The lowest B, G, Y, and H values of 1T suggest that the 1T phase exhibits the least stiff and the most flexible configuration among all other phases. Additionally, similar Poisson’s ratio of the four phases suggests that their resistance capacities to the shape deformation are close. Pugh once suggested using the ratio of B/G to distinguish ductility and brittleness of a material.[53] The critical value that separates ductile and brittle materials is around 1.75. The material with a B/G ratio lower than 1.75 behaves in a brittle manner. From Figure , it can be found that all four phases have a similar B/G ratio of ∼1.4, which is lower than the critical value of 1.75. It indicates that all MoS2 tend to behave more in a brittle manner.

Figure 4

Bulk modulus, shear modulus, Young’s modulus, Poisson’s ratio, microhardness, and bulk modulus/shear modulus ratio of 2H, 1T, 3R, and 1T′ MoS2.

While the average mechanical properties have been provided and discussed, it should be kept in mind that all mechanical properties of MoS2 are anisotropic. To provide a comprehensive understanding of the impact of structural phases, the three-dimensional (3D) plots of Y, G, linear compressibility (the material volume change along a specific axis of the structure), and σ are also analyzed and shown in Figures and S3–S6.[54,55]

Figure 5

Three-dimensional plots of mechanical properties of MoS2 in its (a) 2H, (b) 1T, (c) 3R, and (d) 1T′ structural phases.

To understand the influence of the strain caused by the external pressure on the mechanical properties of MoS2, the mechanical features have been investigated under external pressure from 0 to 60 GPa. The pressure-induced structural strains are represented in Figure . Our computational results show that the average monolayer thickness of the structure with all possible phases decreases with pressure. However, the largest change of the relative interlayer distance can be found in the 1T phase since it has a small monolayer thickness caused by the strongest interlayer interaction strength (see Figure S2). The Mo–S bond lengths also decrease under the external pressure, as shown in Figure . The changes of the bond length of the four phases are similar, which suggests that the reduction of bond length is mainly affected by external pressure.

Figure 6

Variation of the interlayer distances and the bond lengths of 2H, 1T, 3R, and 1T′ MoS2 under external pressure, which are referenced to their equilibrium values (corresponding to 100%).

Accordingly, the mechanical properties of MoS2 are also affected by external pressure. Almost all elastic constants of MoS2 in four different phases increase significantly over the pressure range of 0–60 GPa, as shown in Figure . The most significant change is C33 values, followed by C11 by increasing the pressure. The pressure-induced change of mechanical properties of 2H MoS2 was previously investigated by Wei and Peelaers et al. using GGA and HSE06-D2 methods.[30,40] In their study, the increase of C11 and C33 over pressure was also significant. The C33 value of layered materials is related to the interlayer interactions. The significant change of C33 suggests that the external pressure has a greater impact on weak van der Waals interlayer interactions. Interestingly, the C11 and C44 values of the 1T′ MoS2 decrease when the pressure is higher than 54 GPa. This is because the 1T′ phase undergoes an inverse Peierls phase transition. As evidenced by the change of its bond angles shown in Figure , the increase in the S–Mo–S bond angle over the pressure indicates increased symmetry, which leads to the structural transition similar to 1T.

Figure 7

Elastic constants of 2H, 1T, 3R, and 1T′ MoS2 at different external pressures. Symbols indicate the calculated values.

Figure 8

Structural change of 2H, 1T, 3R, and 1T′ MoS2 over pressure.

The changes of bulk modulus, shear modulus, Young’s modulus, Poisson’s ratio, microhardness, and bulk modulus/shear modulus ratio under external pressure are shown in Figure . It can be found that the values of B, G, and Y increase significantly with the external pressure, which suggests that exerting external pressure can enhance the hardness of the MoS2 materials. All of the phases have a similar trend of changes of B, G, Y, and H. It demonstrates that the impact of the pressure on these mechanical properties of bulk MoS2 is insensitive to the structural phase. There are relatively large fluctuations of σ and B/G values for 1T at 20 GPa. Again, it may be related to the significant change of their monolayer thickness and C33 values under this pressure, which suggests the importance of the accurate description of interlayer interactions during calculations. Similar to the change of elastic constants, the G and H values of 1T′ MoS2 decrease when the external pressure is higher than 54 GPa, which can also be ascribed to the inverse Peierls phase transition.

Figure 9

Bulk modulus, shear modulus, Young’s modulus, and microhardness parameter of 2H, 1T, 3R, and 1T′ MoS2 at different external pressures. Symbols indicate the calculated values.

Conclusions

In summary, first-principles DFT calculations were carried out to investigate the mechanical characteristics of 2H, 3R, 1T, and 1T′ MoS2 with the consideration of the influence of external pressure. Based on the change of the total energies with the strain, the elastic constants, B, G, Y, σ, H, and B/G values were calculated. The comparative results reveal that the structural phase transition can greatly change the corresponding mechanical properties. Among the four phases, the 2H and 3R phases have a stronger resistance to structural deformation as evidenced by the larger bulk modulus, shear modulus, Young’s modulus, and microhardness. This can be ascribed to the higher cohesive energy and thermodynamic stability of these phases. Additionally, 1T MoS2 possesses the smallest B, G, Y, and H values, which suggests that 1T MoS2 has the smallest stiffness value and is easier to deform. Therefore, 1T MoS2 is more promising for strain engineering. Furthermore, 1T′ MoS2 will undergo an inverse Peierls phase transition under high external pressure and become 1T with higher symmetry. The findings of this comparative study have offered a theoretical knowledge base for engineering mechanical behaviors of the layer-structured MoS2 materials for their specific applications.

Computational Methods

All DFT computations were performed using the Vienna ab initio simulation package (VASP).[56−58] The generalized gradient approximation (GGA) with the format of Perdew–Burke–Ernzerhof (PBE) was applied for the exchange–correlation functional.[59] Since traditional DFT calculations at the PBE level cannot correctly include the nonlocal van der Waals (vdW) interactions,[60] the calculations without dispersion corrections may underestimate the interaction strengths between layers of 2D materials. In this regard, the DFT-D3 method was used for dispersion corrections here.[61] A plane-wave basis set with a cut-off kinetic energy of 520 eV was employed to expand the smooth part of the wave function (see Figure S1), and γ point centered (18 × 18 × 4), (18 × 18 × 3), (18 × 18 × 9), and (9 × 9 × 9) k-point grids for 2H, 3R, 1T, and 1T′ phases of MoS2 were employed, respectively. Before the calculation of mechanical properties, both the lattice constants and the atomic coordinates were optimized. All of the atoms were allowed to relax until the Hellmann–Feynman forces were smaller than 0.02 eV/Å and the convergence criterion for the self-consistent electronic optimization loop was set to 1 × 10–5 eV. Pulay stress was added to the stress tensor during the optimization of lattice constants to consider the impact of the external pressure.

To calculate the elastic constants of MoS2 according to generalized Hooke’s law, the energies as a function of train (ε) in the strain range −2.0% ≤ ε ≤ 2.0% with an increment of 0.5% were calculated. The elastic constants C were obtained by fitting a second-order polynomial to the change of the total energy versus applied strain by postprocessing the VASP calculated data using the VASPKIT code.[62] The G and B of MoS2 are calculated by using the universal Voigt–Reuss–Hill average methodHere, Voigt bulk modulus (BV), the Reuss bulk modulus (BR), the Voigt shear modulus (GV), and the Reuss shear modulus (GR) were calculated aswhere s =C–1, which is the compliance tensor.

In addition, Young’s modulus Y, Poisson’s ratio σ, and the microhardness H can be obtained as