Background

Molybdenum disulphide (MoS2) is an interesting material for applications in nanoelectronic applications due to its unique mechanical, electronic, and optical properties [1, 2]. It is a typical layered inorganic material, which is similar to graphite. MoS2 triple layers are held together by weak van der Waals (vdW) interactions. It is a typical example of layered transition-metal dichalcogenides family, which has been studied in recent years. The MoS2 attracts investigation due its distinctive industrial applications from use as a lubricant [3] and a catalyst [4] as well as in photovoltaics. By chemical bath deposition method and the mechanochemical route, MoS2 films have been obtained in the experiments [5]. Due to the interlayer vdW interaction, the bulk MoS2 tends to form a bilayer which is known to be an indirect semiconductor. It has indirect energy band gap of 1.23 eV [6]. The bulk MoS2 has been used in conventional industries as an intercalation agent and a dry lubricant for many years. In addition, a two-dimensional MoS2 is expected for applications in nanoelectronic devices [7].

In recent years, properties of MoS2 and its related structures have been theoretically studied [8–13] such as stability of structure, band gap, functionalization through adatom adsorption, and vacancy defects. By means of density functional theory computations, Chen et al. have been systematically investigated the stability and magnetic and electronic properties of MoS2 nanoribbons [14, 15]. The defect structure of MoS2 has also been studied [16]. Besides, the creation of magnetic and metallic characteristics in low-width MoS2 nanoribbons has been studied by the first principles calculations [17]. The electronic structure of MoS2 has been also studied[3]. Up to date, many works about MoS2 have been done, but some questions are still worth studying. For example, the electronic properties of bulk MoS2 under strain have not been enough investigated. Current studies have confirmed that the properties of low-dimensional materials can be modified by strain, therefore, the response of electronic properties of bulk MoS2 to the strain would be an interesting issue for discussion.

In the present work, we investigate the strain-dependent structural and electronic properties of low-dimensional MoS2 materials using first principles calculations. We apply uniaxial strain onto the bulk MoS2. The structural and electronic band structures of MoS2 with relaxed unit cells are optimized and calculated by the dispersion-corrected density functional theory (DFT-D2). The effects of strain on the band gap and the dependence of formation energy on strain of MoS2 are also studied and discussed.

Methods

The MoS2 bulk belongs to the space group P 63/mmc. It is a layered material and a single layer consists of an S-Mo-S sandwich. In each such layer, the Mo atoms are arranged in a hexagonal lattice and are positioned in a trigonal prismatic coordination with respect to the two S layers. This implies that each Mo atom is coordinated by six S atoms. The hexagonal MoS2 was selected, and the lattice parameters, a = 3.16 Å, c/a ratio of 0.89 are taken as a starting point for the geometry optimization [6, 18]. Model of bulk MoS2 includes two layers of S-Mo-S sandwich, which consist of one Mo atom and two S atoms in each layer, as seen in Fig. 1.

Fig. 1
figure 1

Atomic structure of the bulk MoS2. Relaxed atomic structure of the bulk MoS2 from DFT-D2 calculations: a top view and b side view

Full size image

The present calculations are performed within density functional theory using accurate frozen-core full-potential projector augmented-wave (PAW) pseudopotentials [19, 20], as implemented in the Quantum Espresso code [21]. We use the generalized gradient approximation (GGA) with the parametrization of Perdew-Burke-Ernzerhof (PBE) with added van der Waals (vdW) corrections. They are important for describing the interaction between Mo S 2 layers. Beside DFT-D2 method, calculations within the local density approximation (LDA) and GGA using PAW potentials were also performed for specific cases. This combination is for the comparison with DFT-D2 calculations when the vdW interactions were introduced. These vdW interactions were included using the method of Grimme (DFT (PBE)-D2) [22]. This approach has been successful in describing graphene-based structures [23]. For the plane waves used in the expansion of the pseudowave functions, the cutoff energy varies in the range from 450 to 545 eV. The results of the calculations for convergence in the surface energy and interplanar distances confirmed that the cutoff energy of higher than 400 eV and the planar grid with dimensions of 6×6×1 are quite sufficient. For the different layers of the MoS2, the supercells are constructed with a vacuum space of 20 Å along the z direction. The Brillouin zones are sampled with the Γ-centered K point grid of 18×18×1. The strain is simulated by setting the lattice parameter to a fixed larger value and relaxing the atomic positions. The magnitude of strain is defined as: ε=(a−a 0)/a 0, where a 0 and a are the lattice parameters of the unstrained and strained systems, respectively.

The traditional density functionals are unable to give a correct description of the vdW interactions because of the dynamical correlations between fluctuating charge distributions. In the present study, we considered the vdW interaction within the DFT framework using a semi-empirical potential via the total energy functional (DFT-D2) as defined by Grimme et al. [22]. This method has been successfully applied for calculations of graphene nanoribbon/h-BN [24–26] and graphene nanoribbon/AlN [27, 28] interfaces. The total energy E tot can be expressed as follows [22]:

$$ E_{\text{tot}} = E_{\text{KS--DFT}} + E_{\text{disp}} = E_{\text{KS--DFT}} + E_{\text{vdW}}, $$
((1))

where E KS-DFT is the usual self-consistent Kohn–Sham energy as obtained from the chosen DFT and E disp is an empirical dispersion correction (vdW)

$$E_{\text{disp}} = - \frac{1}{2}\sum_{ij} C_{6ij} \sum_{\mathbf{R}}\left|\mathbf{r}_{ij} + \mathbf{R}_{ij}\right|^{-6} f_{\text{damp}}\left(| \mathbf{r}_{ij}+ \mathbf{R}_{ij}|\right) $$

and

$$f_{\text{damp}}\left(| \mathbf{r}_{ij}+ \mathbf{R}_{ij}|\right) \!= s_{6}\! \left\{1 + \exp\! \left[ -d\!\left(\frac{| \mathbf{r}_{ij} + \mathbf{R}_{ij}|}{r_{0}} - 1\! \right) \!\right] \right\}^{-1}, $$

where r ij =(r j −r i ) is the atom–atom distance vector, R=l a+m b+n c is the lattice vector, s 6 is the functional-dependent scaling parameter, and d is a parameter that tunes the steepness of the damping function (d=20, s 6=0.75 for PBE). The C 6i j coefficients are computed for each atom pair by the geometric mean of atomic terms \({C_{6ij}} = \sqrt {{C_{6i}}.{C_{6j}}} \), and the r 0 term is computed by the simple sum of vdW radii of the atom pairs r 0=r 0i +r 0j .

Results and Discussion

Structural and Electronic Properties of the Bulk MoS2

Our calculations for the geometrical parameters and band gap of the bulk MoS2 using different methods are listed in Table 1. By using three different methods, our calculations show that the lattice parameter a for the bulk MoS2 is 3.116, 3.172, and 3.176 Å corresponding to the LDA, GGA, and DFT-D2 methods, respectively. This result is in good agreement with other theoretical [29, 30] and experimental [6, 18] studies, as shown in Table 1.

Table 1 Calculated structural parameters and band gap of the bulk MoS2 using LDA, GGA, and DFT-D2 methods
Full size table

We calculate the electronic band gap of the bulk MoS2 by using different methods (LDA, GGA, and DFT-D2). We see that the band gap value calculated by the LDA and GGA methods is smaller than that of the experimental study. Our result for the band gap of the bulk MoS2 form is 0.72 eV (0.96 eV) using LDA (GGA) functionals, which is in good agreement with the available theoretical data of 0.72 eV (0.85 ev) using the same LDA-PAW (GGA) funtionals [30, 31]. These values are smaller than that of the experimental study (1.23 eV) [6]. This difference is due to the inherent drawback of standard LDA/GGA functionals. However, the DFT-D2 calculations for the band gap give results (1.20 eV) that are in good agreement with the experimental data (1.23 eV) [6]. Besides, our DFT-D2 calculations give the bond length d M o−S being 1.41 Å. That is the same value as in the experimental study. The match between the DFT-D2 method and the experimental study can be explained by the existence of the vdW interaction in MoS2. Our DFT-D2 calculations is including the vdW interaction. We believe that the DFT-D2 method is a suitable method for the structural and electronic properties of the bulk MoS2.

Figure 2 shows the projected density of state (PDOS) of the ideal bulk MoS2 form. It shows that the valence band consists of two main bands, in which the lower band (from −14.0 eV to −13.0 eV) below the Fermi level E F (E F =0) is mainly due to S3s states. Upper valence bands (from −7.0 eV to −0.5 eV) involve the noticeable contributions of the Mo4d and S3p states. This results in hybridized Mo4d-S3p interactions and provides the covalent component of the Mo-S bonds in bulk MoS2. Besides, the lower edge of the conduction band of the bulk MoS2 involves mainly the antibonding Mo4d states. The bands around the energy band gap are relatively flat, as expected from the d character of the electron states at these energies. Our DFT-D2 band structure calculations show that the bulk MoS2 has an indirect band gap of 1.22 eV opening between the lowest energy of the conduction band (located at between the Γ and K points) and the highest energy of the valence band (located at the Γ point) (see Fig. 2).

Fig. 2
figure 2

Electronic band structure of the bulk MoS2. Band structure and projected density of state (PDOS) of the bulk MoS2 by using LDA (a) and DFT-D2 (b) methods. The Fermi energy was set to zero

Full size image

Effect of Uniaxial Strain on the Structural and Electronic Properties of MoS2

In this part, we consider the influence of uniaxial strain on the structural and electronic properties of the bulk MoS2. Uniaxial strain along both x and z directions is considered in our study (see, Fig. 1). The components of strain along the x and z directions are noted as ε x and ε z , respectively. The strains are evaluated as the lattice stretching percentage. We defined ε x =(a−a 0)/a 0 and ε z =(c−c 0)/c 0, where a 0 and c 0 are the lattice constants at the equilibrium state, and a and c are strained lattice constants. A wide range of strain along (up to 10 %) both directions with step Δ ε=2 % has been employed in the present study.

Figure 3 shows the dependence of the total energy, the Fermi energy E F , and bond length d M o−S on the strain. At the equilibrium state (unstrained), the total energy of the bulk MoS2 is minimum. The dependence of total energy on uniaxial strain can be described by a hyperbolic shape as shown in Fig. 3 a. Figure 3 a also shows that the total energy of bulk MoS2 along the x direction is higher than that along the z direction. When the strain along the x direction is applied, the bulk MoS2 turns out to be less stable than that of the strain along the z direction.

Fig. 3
figure 3

MoS2 under uniaxial strain. Dependence of the total energy (a), Fermi energy (b), and bond length d M o−S (c) on the strain along both x and z directions

Full size image

Figure 3 b, c shows the dependence of Fermi level E F and bond length d M o−S in the bulk MoS2 on the strain along x and z directions. Under uniaxial strain, the Fermi energy and the Mo-S bond length change linearly with strain. As the uniaxial strain increases, the Fermi energy is decreased and the bond length d M o−S increases. We can see that the total energy, Fermi energy, and bond length depend not only on strain strength but also depend strongly on the direction of the applied strain.

The effect of the uniaxial strain on the electronic band structure and energy band gap of bulk MoS2 is shown in Figs. 4 and 5, respectively. Both x and z directions of strain are taken into account. We can see that the electronic properties of bulk MoS2 are sensitive to the uniaxial strain. The band gap strongly depends not only on the elongation but also on the strain direction (see, Fig. 5). As shown in Fig. 4, we can see that in the bulk MoS2, the orbitals will be shifted towards a higher/lower energy level at the K point when strain is applied along the z/x direction, respectively. The states at top of the valence band and bottom of the conduction band near the Γ point, which originate mainly from d orbitals on Mo atoms and contributions of p z orbitals on S atoms, are accordingly independent on the uniaxial strain. When ε x strain is applied, we observe a shift of orbitals in the conduction bands towards the lower energy region, while all orbitals of valence bands shift toward the Fermi level, which is in result of a reduction of the band gap energy. Similarly, when ε z strain is applied, the shift of orbitals are also observed but in the valence band. It shifts towards the higher energy region.

Fig. 4
figure 4

Band structure of the bulk MoS2 under uniaxial strain. The applied strain is along the x axis (a) and along z axis (b). The blue and red lines stand for the lowest conduction (CBM) and highest valence (VBM) subbands, respectively. For comparison, the energies are aligned with respect to the valence band top at the Γ point

Full size image
Fig. 5
figure 5

Dependence of the MoS2 band gap on uniaxial strain

Full size image

At equilibrium, the lowest energy of the conduction band \(\left (E_{K}^{\text {CBM}}\right)\) and the highest energy of the valence band \(\left (E_{K}^{\text {VBM}}\right)\) are 0.903 and −0.736 eV, respectively. These values of energy are decreased due to increasing the strain strength. Especially, when the strain increases from 0 to 10 %, \(E_{K}^{\text {CBM}}\) decreases from 0.903 to 0.08 eV.

Figure 5 depicts the energy band gap as a function of the applied ε xz strain. Under ε x strain, the band gap decreases monotonically with strain. We can see that at ε x =2 %, the band gap is equal to 0.78 eV. This band gap decreases to 0.13 eV when ε x =10 %. The effect of the ε z on the band gap of the MoS2 is negligible (in comparison to the case of strain applied along the x direction). We see that when the ε z <6 %, the point in k-space corresponding to the highest energy of the valence band is located at the Γ point and it will be shifted to the K point in the first Brillouin zone when the ε z >6 %. The band gap of the bulk MoS2 is strongly dependent on the applied strain along the x direction and we expect that a phase transition will occur in the case of larger deformation.

In addition, we also calculate band gap of bulk MoS2 under uniaxial strain along the armchair direction (y direction). Similar to the case of strain along the zigzag direction, band gap of MoS2 reduces concurrently with strain. Our calculations show that the change in band gap of MoS2 under uniaxial strain along the zigzag and armchair directions is almost the same. This result is in good agreement with the previous works [32, 33].

The strain dependence of the projected density of states (PDOS) of the Mo4d and S3p states is shown in Fig. 6. It shows that, under the uniaxial strain, an energy splitting of Mo4d states in the range from 0 to 2 eV is observed. The effect may be associated with crystal field theory, in which the loss of degeneracy of Mo4d orbitals in transition metal complexes is described. The energy splitting of the Mo4d orbitals is observed in both cases of strain, as shown in Fig. 6. Figure 7 shows the isosurface of lowest unoccupied crystal orbitals at the K point in the first Brillouin zone of the bulk MoS2 in the case of with and without strain. Based on the partial charge density distribution, we see that the change in electronic properties of the bulk MoS2 is determined by the strength of the Mo-S bond. Figure 7 also shows that the lowest energy of the conduction band at K point is mainly contributed by the coupling between the Mo4d and S3p orbitals.

Fig. 6
figure 6

The PDOS of the Mo4d and S3p states under strain. The PDOS of the Mo4d and S3p states of the bulk MoS2 form under applied ε x (a) and ε z (b) strain, respectively

Full size image
Fig. 7
figure 7

Isosurfaces of lowest unoccupied crystal orbitals (CBM) at the K point in the first Brillouin zone of the bulk MoS2 under the applied strain. a ε xz =0, b ε x =8 %, c ε z =8 %

Full size image

Conclusions

In this paper, we studied the effect of uniaxial strain on the structural and electronic properties of the bulk MoS2 using first principles calculations. Methodologically, we pointed out that the DFT-D2 calculations are a suitable method for calculations of structural and electronic properties of the bulk MoS2. Our calculations showed that the electronic properties of the bulk MoS2 are very sensitive to the uniaxial strain, especially when the strain is applied along the x direction. The band gap of the bulk MoS2 decreases linearly with an increase of the strain strength and we can control the energy splitting and band gap of the bulk MoS2 by the strain. This makes MoS2 becoming a promising material for application in nanoelectronic device such as nanosensors.