Effect of Tensile Strain on Performance Parameters of Different Structures of MoS2 Monolayer

The present work is based on the computational study of MoS2 monolayer and effect of tensile strain on its atomic level structure. The bandgap for MoS2 monolayer, defected MoS2 monolayer and Silicon-doped monolayer are 1.82 eV (direct bandgap), 0.04 (indirect bandgap) and 1.25 eV (indirect bandgap), respectively. The impact of tensile strain (0-0.7 %) on the bandgap and effective mass of charge carriers of these MoS2 structures has been investigated. The bandgap decrease of 5.76 %, 31.86 % and 6.03 % has been observed in the three structures for biaxial strain while the impact of uniaxial strain is quite low. The impact of higher temperature on the bandgap under biaxial tensile strain has been also analyzed in this paper. These observations are extremely important for 2D material-based research for electronic applications.


Introduction
The nanotechnology has allowed the researchers to work up to a nanoscale and offered extraordinary results which were never possible earlier. The nanotechnologies include highly sensitive sensors, high-performance nanoelectronics, nanocomposites, and innovative medicines and cancer therapy based on nanoparticles [1][2][3][4]. The development of nanotechnology has also proposed the different classes of dimensional systems which are zerodimensional (0D, e.g., nanoparticles), one-dimensional (1D, e.g., nanowires), two-dimensional (2D, e.g., graphene), and three-dimensional (3D, e.g., bulk materials). The low dimensional systems play major role to differentiate the nanomaterials, as they explore the atomic structure of materials and properties of the low dimensional materials [5].
In 2004, the experimental discovery of graphene is one of the most important scientific revolution of the 20th century [6]. The two-dimensional (2D) crystal of graphene is created by separating it from graphite material [7][8][9]. Graphene is a single layer structure of carbon atoms arranged in the honeycomb form. Graphene material has the potential for highspeed nanoelectronics applications because of its unique features like extremely high carrier mobility and half-integer quantum effects. Because of high mechanical and chemical stability, we can use it in different working conditions [10,11]. Looking into the potential of 2D materials, graphenelike inorganic monolayer materials Gallium Nitride (GaN) [12], Silicon Carbide (SiC) [13,14], Boron Nitride (BN) [15], Zinc Oxide (ZnO) [16,17], Aluminum Nitride (AlN) [18][19][20] and Transition-Metal Dichalcogenides (TMDs) [21,22] have been synthesized and analyzed for various parameters. Among these 2D materials, the TMDs have shown a broad range of mechanical, electronics, optical, thermal, and chemical properties [23,24]. The monolayer of TMDs (such as MX 2 where M = Mo, W; and X = S, Se, Te, etc.) having a honeycomb structure where transition metal atoms (M) are switched between chalcogen atoms X. All the atoms in a monolayer of TMD are attached by the covalent force and the different layers are attached by van der Waals forces. For electronics applications, the TMDs like MoS 2 , MoSe 2 , MoTe 2 , WS 2 , and WSe 2 have shown extraordinary features [25,26]. In bulk form, TMDs are indirect bandgap semiconductors. By thinning down to a single layer, the indirect bandgap changes into a direct bandgap. The monolayer direct bandgap of TMD is in the range of (1.1-2.0 eV) at the K-point in the Brillion zone [27][28][29]. In this work, we have focused on bandgap and effective mass of MoS 2 monolayer for without strain and with strain structure. We also have investigated the same for defected monolayer and silicon-doped (Si-doped) monolayer.

Related Work
S. Deng et al., [30] have investigated the response of 2D TMDs on a mechanical strain by using first-principles methods. By studying the mechanical properties of 2D TMDs it has been found that the stable range of strain is determined by the young's modulus. Moreover, it has been analyzed that the strain induces the electronic bandgap properties. X Wang e al., [31] have presented a theoretical study report for the interaction of NO 2 molecule with the Mo-edge of MoS 2 zigzag nanoribbon, and for its simulations, the Density Functional Theory (DFT) method has been used. The effect of uniaxial tensile strain on the physical and structural properties of MoS 2 Nanoribbon and the absorption process has been discussed in detail. It has been observed that there is increase in the magnitude of the adsorption energy and a more stable structure is obtained with strain. S. Yua et al., [32] have presented the computational study of MoS 2 monolayer under tensile strain. The bandgap transition from direct to indirect and semiconductor to metal has been investigated under the tensile strain along with both x-direction and y-direction. The phase transition, carrier mobility, and effective mass of MoS 2 monolayer have been studied under tensile strain. The mobility increases when the biaxial strain Ɛ x =Ɛ y =9.5 % has been applied. Additionally, the mobility parameter with an increase in temperature has been decreased monotonically. J. Ni et al., [33] have demonstrated the modulation of bandgap transition of two heterostructures i.e. Blue P/GeC and Blue P/SiC with strain engineering. According to the authors, the electronic structure and optical properties of the Blue P/GeC and Blue P/SiC have changed under strain. R. Beiranvand [34] has investigated the electrical and optical properties of TMDs MoX 2 (X = S, Se, Te) based on the DFT method. Also, the author has studied the optical absorption coefficient, real and imaginary parts of the dielectric functions, reflectivity, and energy loss functions for external electric fields along with two directions in detail. The high absorption coefficient value is the remarked property of the MoX 2 monolayer which makes these materials best for optoelectronics applications. A. O. M. Almayyali et al., [35] have investigated the optical and electronics of Zinc Iodide (ZnI 2 ) material under the impact of the biaxial strain. For the ZnI 2 materialbased calculation of molecular dynamic simulations, binding energy, and phonon dispersion curve it has been observed that  [36] proposed a new class of 2D materials XBi (where X = Si, Ge, Sn, and Pb) along with the metal monochalogenide structure to provide tunable orbital properties. It has been found that the spin-orbit coupling shifts the SnBi electronics properties from semiconductor to metal, and the applied strain can lead toward a novel Dirac electronic state. Surface chemical decoration has been confirmed to be an effective path to achieve the Bi-pz filtering effect and p-p inversion in the orbital. Y. Solyaev et al., [37] have investigated the electric field, inertia gradient, and strain impact on anti-plane wave propagation of piezoelectric materials. It has been analyzed that the model represents a normal dispersion of short wave in piezoelectric materials. Another model parameter on the phase velocity, attenuation of shear horizontal, and coupled electromechanical factor have been investigated. It has been concluded that the results obtained in this article can be applied for the analysis of small-scale piezoelectric structures and high-frequency MEMS/NEMS. H. Li et al., [38] have explored a novel 2D bonding heterostructure that consists of a hexagonal borophene monolayer attached with two blue phosphorene layers. The proposed heterostructure having good conductivity, high stability, and high in-plane stiffness. Later, the proposed novel 2D heterostructure has been investigated by using the DFT method as an acceptable

Computational Details
For this study, we have used the DFT method in Virtual Nanolab Atomistix ToolKit (ATK) [42] for first-principle calculation. In the basis setting of the ATK-DFT method, we have used Localized Density Approximation (LDA) exchange-correlation along with density mesh cut-off energy value of 75 Hartree and a Double Zeta Polarized (DZP) [43]. LDA successfully predict properties of a wide class of electronics systems. All the simulations with strain and without strain are done at room temperature. The unit cell selected for

Simulation Results
At first, we have discussed the structures of MoS 2 monolayer. Figure 1 show the atomistic lattice structure of MoS 2 monolayer, where the supercell geometry is orthorhombic and the small rectangular structure shown with solid line is used for elastic constant calculation. To study the carrier conduction in the armchair [1,0] and zigzag [0,1] directions, the orthorhombic geometry is created in supercell. The one S atom is removed to create a defected MoS 2 structure and Si-doped MoS2 monolayer is created by replacing one S atom with a Silicon atom. shown in Fig. 2. First parameter that we discuss is band structure of different additions of MoS 2 monolayer. In band structure calculation we have set the 100 points per segment and Г-Y-X-Г Brillouin zone route. The K-points which is the fractional reciprocal coordinates in hexagonal lattice is equal to (1/3,1/3), but in orthorhombic supercell [45] it is equivalent to (1/3,0) that lies as midpoint between Г and Y points. The band structure and density of states (DoS) of unstrained atomic structures of MoS 2 monolayer is shown in Fig. 3 and in Fig. 4. The bandgap value for MoS 2 Monolayer, Defected MoS 2 Monolayer and Si-Doped MoS 2 monolayer are 1.820001 eV (direct b a n d g a p ) , 0 . 0 4 4 1 7 e V ( i n d i r e c t b a n d g a p ) a n d 1.252501 eV (indirect bandgap), respectively. Fig. 7 Temperature variation in bandgap with respect to biaxial tensile strain Next, we have explored the electronic properties of three MoS 2 structures with tensile strain. The vector components of strain along x-direction [1,0] and y-direction [0,1] are denoted as Ɛ x and Ɛ y , respectively. This study calculates the tensile strain as the relative lattice stretching percentage, given by Where, a 0 and b 0 are the lattice constant of unstrained structure. The a and b are the new lattice dimensions due to tensile strain. In this work the uniaxial strain in the range of 0-0.7 % of lattice constant with 0.1 % step size in X and Y direction has been applied. Figure 5 shows the results of uniaxial and biaxial strains on different structures. From Fig. 5(a), it is observed that when a strain is applied in X and Y direction the bandgap percentage decreases for monolayer MoS 2 by 1.99 % and for biaxial direction the percentage decrease by 5.45 %. For defected layer the percentage decrease in bandgap is 14.32 %, 20.17 % and 31.86 % for the strain in X-direction, Y-direction and biaxial, respectively. For Si-doped monolayer the percentage decrease in bandgap for uniaxial strain in Xdirection, Y-direction and biaxial are 4.69 %, 1.36 % and 6.03 %, respectively.
When the strain applied to a structure, it also effects the energy band curvature which is proportional to the carrier effective mass(m): Where, h = Planck Constant, E = Energy and k = Momentum.
For monolayer we investigate the electron effective mass (m e *) at fractional K-points (0,0.335,0), hole effective mass(m h *) at fractional K-points (0,0.335,0) and Г (0,0,0) under tensile strain. The same effective mass calculation done for defected monolayer and Si-doped monolayer. Table 2 shows the values of carrier effective mass of different structures of MoS 2 monolayer without strain. Figure 6 shows the hole and electron effective mass at different tensile value from 0 to 0.7 %. As shown in Fig. 6 the effective mass value under tensile strain change with different rate along the three directions.
We have also analyzed the effect of increase in temperature on the bandgap. Figure 7 shows the change in bandgap with respect to biaxial tensile strain under various temperature conditions. In the case of defected monolayer for temperature 25ºC and 50ºC the bandgap is almost same under the strain, depicted by the almost overlaping lines in Fig. 7(b). The bandgap percentage decrease for various temperature is calculated and listed in Table 3.

Conclusion
In summary, we have investigated the different MoS 2 monolayers such as simple monolayer, defect of S atom in monolayer, and Silicon doped monolayer using first principle calculation method. After that, we have studied these structures under the tensile strain of range 0-0.7 % with steps size of 0.1 %. For the analysis purpose, we explore the two electronics properties i.e., band structure and effective mass for holes and electrons. It is observed that the bandgap is decreased harmoniously and monotonically with the rise of strain. The strain in [1,0], [0,1] and [1,1] directions have same role for bandgap modulation. However, the observed effective mass values are behaving differently for tensile strain along [1,0], [0,1], and [1,1] directions. The bandgap modulation analysis at higher temperature also has been done. In conclusion, the computational study achieved in this work can help the research on further applications of 2D materials and devices.
Funding The authors are thankful to National Institute of Technology Hamirpur (HP) for supporting this research work. However, no funding was received with the preparation of this manuscript.
Data Availability The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.
Code Availability We have used VNL-ATK QuantumWise Software for implementation.

Declarations
Conflicts of Interest/Competing Interest Authors have no conflict of interest.
Ethics Approval The reported work did not involve any human's participation and/or did not harm welfare of animals.