First-Principles Calculations of the Mechanical and Elastic Properties of 2H_c- and 2H_a-WS₂/CrS₂ Under Pressure

1 Introduction

There is a growing interest in studding the structures and properties of two-dimensional transition metal dichalcogenides (TMDs) owing to their novel electronic and catalytic properties that differ from their bulk counterparts [1–3]. Because TMDs have in-plane covalent bonding and weak interlayer van der Waals interactions, they could be easily exfoliated down to a monolayer which shows very exotic properties. For example, the band-gap structure of MoS₂ shows an indirect-to-direct semiconductor transition from bulk to single-layer due to a lack of interlayer interaction [4, 5]. Two-dimensional dilute magnetic semiconductors are proposed for substitution of Mo by other transition metal atoms, such as W and Cr. Furthermore, TMDs also have shown exciting prospects for various applications, such as lubricants and catalysts in the petroleum industry [6], possible applications in optoelectronics and nanoelectronics [7], and energy-storage applications [8]. As an important member of TMDs, WS₂ has received a lot of attention. For example, WS₂ has been prepared into nanoparticles, nanotubes, nanoribbons, films, graphene-like WS₂, and WS₂ sheets [9–16]. Some features and properties of WS₂ such as tribological performance [10, 17, 18], structural, electronic [19], optical, and mechanical properties [4, 9, 20–24] have been studied. CrS₂ also attracted some researchers’ interest, and the electronic properties [25, 26], optical, piezoelectric properties, and stability of this single-layer material were investigated and characterized [27].

However, most of the previous theoretical works focused on WS₂ and CrS₂ were carried out at 0 GPa. Very recently, high pressure in situ angle dispersive powder X-ray diffraction using synchrotron radiation together with high-pressure Raman analysis was performed on powder WS₂ samples to investigate the effect of pressure on crystal WS₂. However, it was reported that there was no phase transformations observed in WS₂ under compression for pressures up to 52 GPa [28]. As we have known, for most transition typical TMDs, there is a phase transition from 2H_c to 2H_a under pressure [29, 30]. As we know, due to some reasons, such as the limitations of the experimental conditions, temperature effect, and higher phase transition barrier, some phase transition under pressure cannot be detected experimentally. So it is necessary to investigate structural evolution of WS₂ theoretically. Since chromium belongs to the same group as tungsten, we also investigate the possible pressure-induced transition from 2H_c to 2H_a for CrS₂.

In this paper, we first investigate the possible structural transitions of WS₂/CrS₂ under pressure, then study the anisotropy in elasticity and thermodynamic properties of them under pressure. We found that both CrS₂ and WS₂ will transform from 2H_c to 2H_a phase under pressure, and the anisotropy in elasticity and thermodynamic properties of 2H_c and 2H_a phases for both WS₂/CrS₂ are calculated and compared.

The paper is structured as follows. Section 2 contains the computational details. The results and discussion are provided in Section 3. Finally, we present our conclusions in Section 4.

2 Computational Methods

The calculations reported in this work were performed using the CASTEP code in the Materials Studio software [31, 32]. The density functional theory of the plane-wave pseudopotential method [33] was used to perform geometry optimisation and calculate the elastic parameters. The exchange and correlation functional were treated by the local density approximation [34, 35]. The plane-wave cutoff energy of 800 eV was used for the all structures studied in this work. The k integration over the Brillouin zone was performed up to a 14×14×4 Monkhorst–Pack [36] mesh. Coulomb potential energy caused by electron–ion interaction is described using Norm-Conserving Pseudopotential [37]. In order to make our calculations standard, energy-difference threshold for the convergence is taken as 5*10^–6 eV/atom. These values ensure a satisfactory convergence, and thus provide reliable results.

3 Results and Discussion

The two typical structures of TMDs (2H_c and 2H_a) [29] are shown in Figure 1, and the pressure-induced sliding of layers was reported to be responsible for the transition from 2H_c to 2H_a structure [29]. The calculated lattice constants of 2H_c structures for both CrS₂ and WS₂ at zero pressure together with other theoretical values are listed in Table 1, and our results agree well with other reference’s data. These guarantee the reliability of our calculation.

Figure 1:

(a) Structures of 2H_c-XS₂ (X=W, Cr). (b) Structures of 2H_a-XS₂ (X=W, Cr).

To investigate the possible pressure-induced structural transition of CrS₂ and WS₂, we applied hydrostatic pressure to 2H_c and 2H_a phases, and then fully optimised the lattice parameters and atomic positions. Figure 2a and b plot the relative enthalpy versus pressure for the 2H_c and 2H_a structures of CrS₂ and WS₂, respectively. For CrS₂, 2H_a structure is with lower enthalpy than 2H_c structure for pressures ≥17.5 GPa. For WS₂, enthalpy of 2H_a phase is lower than that of 2H_c phase at 25 GPa. So, for CrS₂ and WS₂, the phase-transition pressures for 2H_c to 2H_a phase are 17.5 and 25 GPa, respectively.

Figure 2:

(a) Enthalpy versus pressure for the 2H_c and 2H_a structures of WS₂. (b) Enthalpy versus pressure for the 2H_c and 2H_a structures of CrS₂.

The elastic properties of a material are very important for understanding and predicting the mechanical stability, and for indications of material-related properties such as brittleness, ductility, based on the analysis of elastic constants C_ij, bulk modulus B, and shear modulus G. The elastic constants can be obtained by calculating the stress response when a small strain was forced to the optimised unit cell [42]. According to the strain energy theory, for a mechanically stable structural phase, the strain energy should be positive, and the matrix of elastic constants should be positive, definite, and symmetrical [43]. For hexagonal structure, the mechanical stability criterion can be expressed as [44]:

According to the calculated elastic constants of the hexagonal 2H_c and 2H_c phase of CrS₂ and WS₂, it is found that they all satisfy their stability conditions up to 80 GPa. That means, for CrS₂ and WS₂, both the 2H_a and 2H_c hexagonal phases are mechanically stable up to 80 GPa. In the view of mechanical stability criteria, 2H_c phase may exist as metastable structure at higher pressure.

The changes of the elastic constants with pressure for both CrS₂ and WS₂ are presented in Figures 3 and 4, respectively. It can be found that the elastic constants C₁₁ and C₃₃ increase quickly and monotonically with pressure, while the elastic constants C₄₄, C₁₂, and C₁₃ increase comparatively slowly. For these two materials, C₁₁ and C₃₃ are larger than other elastic constants, which indicate that they are very incompressible under uniaxial stress along x (ε11) or z (ε33) axis.

Figure 3:

(a) Elastic constants of 2H_a phase of CrS₂ as a function of pressure. (b) Elastic constants of 2H_c-phase of CrS₂ as a function of pressure.

Figure 4:

(a) Elastic constants of 2H_c phase of WS₂ as a function of pressure. (b) The elastic constants of 2H_a phase of WS₂ as a function of pressure.

Bulk modulus reflects the compressibility of the solid under hydrostatic pressure. The bulk modulus (B) and shear modulus (G) can be estimated according to the Voigt–Reuss–Hill approximation [45–48]. For the hexagonal systems, they can be calculated with the computed data using the following relations:

The elastic anisotropy of crystals has a great impact on the properties of physical mechanism, such as anisotropic plastic deformation, elastic instability, and crack behavior. Hence, it is important to calculate the elastic anisotropy to improve its mechanical durability. The universal anisotropic index (A^U) is represented as

The percentage elastic anisotropy in compressibility and shear can be used as follows:

where B and G denote the bulk and shear modulus, and the subscripts V and R represent the Voigt and Reuss bounds. For these two formulas, a value of zero is associated with isotropic elastic constants, while a value of 1 (100 %) is the largest possible anisotropy. The calculated anisotropic index and percent anisotropy factors are shown in Tables 2 and 3, respectively. It can be seen that the two phases of WS₂ and CrS₂ are anisotropic under the ordinary condition. With increasing pressure, the calculated universal anisotropic index (A^U) decreases. So, both the anisotropies of 2H_a- and 2H_c-WS₂ and CrS₂ decrease with pressure. Furthermore, the results also indicate that the 2H_a phase of WS₂ and CrS₂ has stronger mechanical anisotropy than the 2H_c phase. The value of bulk modulus anisotropy and shear modulus anisotropy decreases first, and they will be close to zero under certain pressure (such as the value of A_B of 2H_a-WS₂ under 70 GPa). So, as the pressure increases, the bulk modules and shear modulus of two structural of WS₂ and CrS₂ become slightly anisotropic and, finally, turn to isotropic under certain pressures.

Debye temperature is closely related to many physical properties of solids, such as specific heat, stability of lattices, and melting temperature. At low temperature, it can be calculated using the average sound velocity (v_m) according to the following [49]:

where h is the Plank’s constant, k is the Boltzmann’s constant, and V is the atomic volume. The average sound velocity (v_m) in the polycrystalline material can be obtained using the following [49]:

where ν_s and ν_p are the transverse and longitudinal sound velocity, and are given by Navier’s equation [49]:

Depending upon the above equations, the average sound velocity and Debye temperature of 2H_c- and 2H_a-WS₂/CrS₂ are calculated and listed in Tables 2 and 3, respectively. It can be seen that v_m and Θ of 2H_c- and 2H_a-WS₂/CrS₂ increase with increasing pressure. Additionally, the calculated results predict that Θ of 2H_c phase is higher than that of 2H_a phase for both WS₂ and CrS₂.

Knowing G and B, the Young’s modulus Y and Poisson’s ratio δ, which are frequently measured for polycrystalline materials when investigating their hardness, can be calculated from the isotropic relations:

The pressure dependencies of Young’s modulus Y and shear modulus G of CrS₂ and WS₂ at various pressures are plotted in Figure 5a and b, respectively. It can be seen that both 2H_c and 2H_a phases of CrS₂ and WS₂ show similar trends under pressure. The values of G and Y increase with increasing pressure, and both of them were affected significantly by the pressure. Furthermore, it is noted from Figure 5 that G and Y change approximately linearly with the pressure, and the values of G and Y of 2H_c phase are larger than that of 2H_a phase for both CrS₂ and WS₂.

Figure 5:

(a) Pressure dependencies of the shear modulus G and Young’s modulus Y of CrS₂. (b) Pressure dependencies of the shear modulus G and Young’s modulus Y of WS₂.

Poisson’s ratio δ is related to the volume change during elastic deformation [50]. It can be used to assess the stability of a crystal against shear. The range of values for Poisson’s ratio is –1 to 0.5, with –1 indicating the material does not change its shape and 0.5 indicating the volume remains unchanged during elastic deformation. The obtained Poisson’s ratio for 2H_c- and 2H_a-CrS₂ at pressure 0–80 GPa are 0.194–0.227 and 0.23–0.29, respectively, which indicate that both 2H_c- and 2H_a-CrS₂ have central interatomic forces and are relatively stable against shear. As shown in Figure 6a, the Poisson’s ratio δ of 2H_c- and 2H_a-CrS₂ increases with increasing pressure, and 2H_a phase has larger δ than 2H_c phase. The changes of δ with pressure for 2H_c- and 2H_a-WS₂ are shown in Figure 6b, and it can be seen that the change trends of δ are similar with that of CrS₂.

Figure 6:

(a) Pressure dependencies of the Poisson’s ratio and the ratio of bulk to shear modulus (B/G) of CrS₂. (b) Pressure dependencies of the Poisson’s ratio and the ratio of bulk to shear modulus (B/G) of WS₂.

Considering that the bulk modulus B represents the resistance to fracture, while the shear modulus G represents the resistance to plastic deformation, the ratio of bulk to shear modulus (B/G) was introduced to measure the empirical malleability of polycrystalline materials [51]. The value at the critical point which separates ductile and brittle materials is about 1.75. If B/G>1.75, a material behaves in a ductile manner, and if not, a material will demonstrate brittleness. Higher (lower) values of B/G mean higher ductile (brittle) of a material. The B/G ratios of the two phases of CrS₂ at pressure from 0 to 80 GPa are shown in Figure 6a, and it is found that with increasing pressure, the B/G value of 2H_c phase increases from 1.30 to 1.49, while the B/G value of 2H_a phase increases from 1.53 to 2.07. Therefore, the 2H_c phase of WS₂ is brittle over the pressure range of 0–80 GPa, while 2H_a phase changes from brittle to ductile with increasing pressure. Similar features are also found in the B/G ratios of WS₂. The B/G ratios of WS₂ of the two phases at pressure from 0 to 80 GPa are shown in Figure 6b; with increasing pressure, the B/G value of 2H_c phase increases from 1.24 to 1.58, while the B/G value of 2H_a phase increases from 1.45 to 1.91. Therefore, the 2H_c phase of WS₂ is brittle up to 80 GPa, while the 2H_a phase changes from brittle to ductile with pressure increasing.

4 Conclusions

In this work, the pressure-induced phase transition, mechanical stability, and elastic properties of 2H_c- and 2H_a-WS₂/CrS₂ are investigated using the first-principles method. The calculated equilibrium lattice parameters of 2H_c-WS₂/CrS₂ at 0 GPa are well in agreement with the experimental and other theoretical values. Although the transition for 2H_c to 2H_a phase was not observed experimentally for WS₂, we found that CrS₂ and WS₂ will transform from 2H_c to 2H_a phase at 17.5 and 25 GPa, respectively. According to the mechanical stability criterion, both 2H_c- and 2H_a-WS₂/CrS₂ exhibit mechanical stability in the pressure range of 0–80 GPa. The shear modulus, Young’s modulus, Debye temperature, and average sound velocity of 2H_c- and 2H_a-WS₂/CrS₂ increase with the increasing pressure. The changes of Poisson’s ratio and ratio of bulk to shear modulus (B/G) are also studied, and our results show that the 2H_c phase of WS₂/CrS₂ is brittle on the whole pressure range we investigated, while the 2H_a phase changes from brittle to ductile with pressure increasing. We also found that the two phases of WS₂ and CrS₂ are anisotropic under the ordinary condition, while the mechanical anisotropy of 2H_a- and 2H_c-WS₂ and CrS₂ decreases as pressure increases.