Layered transition-metal di-chalcogenides (LTMDCs) have been extensively reviewed in the recent past [1] . MoS₂ is a typical example of LTMDC family of materials which attracts investigation because of its distinctive industrial applications, ranging from use as a lubricant [2] and a catalyst [3] as well as in photo-voltaics [4] and energy storage [5] . In its bulk form MoS₂ is a semiconductor with an indirect band gap of about 1.23 eV while its monolayer has a direct energy gap of 1.8 eV [6] . A special attention has been paid on single layer MoS₂ in the recent years. Upon thinning from the bulk [7] -[9] the electronic structure of MoS₂ undergoes an interesting transition. Recently, a monolayer MoS₂-based field effect transistor (FET) with HfO₂ as gate insulator has been successfully implemented [10] . These ideal properties make monolayer MoS₂ a very promising candidate for next generation FET and as optoelectronic devices [11] . This has raised enormous interest in exploring the extraordinary properties of mono layers of MoS₂.

Theoretically, there are various possibilities of energy gap manipulation in MoS₂. By reducing the layer thickness from bulk to monolayer, the indirect band gap energies in the bulk are shifted relative to the direct band gap in the monolayer limit [12] . It undergoes a transition from an indirect to direct gap exhibiting strong photoluminescence when confined in a 2D monolayer [13] . It has been suggested a way to engineer 3D semiconducting MoS₂ nanoparticles with direct band gaps as well as metallic di-chalcogenides nanowires with promising catalytic and thermoelectric properties [14] . Eellis et al. carried out a study using HSE screened hybrid functional and offered improvement over semi local density functional [13] . All electron calculations including spin orbit coupling were performed [15] and confirmed indirect to direct band gap transition. Electronic structure of transition metal di-chalcogenides has been studied using ab initio theory using Troullier-Martin norm conserving, relativistic pseudopotentials in fully separable Kleinman and Bylander form [16] . They used exchange and correlation energies within LDA.

The properties of transition metal di-chalcogenides (TMDs) not only can be tuned by varying number of layers, but also can be modified by application of external field or strain engineering. Studies [17] [18] have confirmed that applying strain is one of the best possible strategies to tune the band gap, since it neither attenuates the properties nor is inefficacious for single layers. It has been predicted that straining MoS₂ modifies the band gap energy and the carrier effective mass. Moreover, at strains larger than 1% the lowest lying band gap changes from direct to indirect [19] -[22] . It has been suggested that strain engineering of the band structure of MoS₂ could be used to increase carrier mobility of MoS₂, to create tunable photonic devices and solar cells [23] and even to control the magnetic properties of MoS₂ [19] [20] . While strain perturbs the band structure of all materials, two-dimensional materials such as MoS₂ can sustain strains greater than 11% [24] allowing exceptional control of material properties by strain engineering. In a recent study [25] the conduction band valley structure of a few layer MX₂ by close examination of temperature dependent indirect excitation emission peaks has been explored. A study on elastic constants and electronic structures of two-dimensional monolayer MoS₂ under elastic strain using first principle calculations has been made [26] . It is shown that the band gap of monolayer MoS₂ undergoes a descent trend with the increase in strain. They observed a direct to indirect transition at strain of 0.01 and semiconductor to metal transition at strain of 0.10.

With the goal of understanding the electronic properties of bulk and monolayer MoS₂ and strain engineering, we carried out ab initio calculations of bulk and monolayer MoS₂ using gradient corrected exchange-correlation functional in DFT framework and observed a transition from indirect to direct band gap. However if this band gap can be tuned such that a semiconductor with a lower band gap or a semiconductor to metal transition can be achieved with the application of strain, then a wide range of tunable nano device can be fabricated. In the present work, therefore we study the effects of mechanical strains on the electronic properties of monolayer of MoS₂. Our results suggest a way of band gap engineering in MoS₂.

The calculations were performed using self consistent plane wave pseudopotential total energy method based on density functional technique as implemented in Quantum Espresso code [27] . This method has been previously used to study the electronic properties of undoped and doped graphene [28] [29] . The exchange correlation potential was approximated by generalized gradient approximation using Perdew-Wang 91 functional (GGA- PW91) [30] . The atomic positions and cell parameters were fully relaxed until an energy convergence of 10⁻⁹ eV reached. We used wave function- and charge-density cut-offs of 70 Ryd and 300 Ryd, respectively. First, we obtained lattice constants a and c by the process of total energy minimization. Optimized structure (coordinates) was used to perform self consistent calculations with a Monkhorst-Pack [31] k-mesh followed by the non-self consistent calculations for band structures, density of states and partial density of states of bulk MoS₂. We used k-points mesh along the path Γ-K-M-Γ in the irreducible Brillouin zone to obtain the band structure with a very fine mesh points. However, for monolayer we used and Monkhorst-Pack of k-points respectively for sampling Brillouin zone for calculations of structural properties and electronic structure. In case of monolayer MoS₂, we created 15 Å vacuum along Z axis to isolate it and to prevent any interaction between the layers. The cohesive energy per atom of bulk MoS₂ was calculated as E_coh = E(MoS₂) − E(Mo) − 2E(S), where E(MoS₂) is the total energy of the unit cell of Molybdenum disulphide, E(Mo) is the energy of Mo atom and E(S) is the energy of S atom. The cohesive energies per atom of monolayer MoS₂ was also calculated accordingly. A uniform tensile strain ranging from 0 to 10% were applied on monolayer MoS₂ to study the change in behavior of its band gap.

In summary, we studied the structural and electronic properties of MoS₂ using plane wave pseudopotential method under GGA scheme based DFT calculations. Electronic band structure and density of states calculation show many similarities between monolayer-MoS₂ and bulk-MoS₂ except the nature of the band gap which is found direct for monolayer-MoS₂ as compared to indirect for bulk-MoS₂. This observation is consistent with the theoretical prediction of indirect to direct band gap transition in going from bulk to monolayer. Such behavior, arising from d-orbital related interaction in MoS₂, may also arise in other layered transition metal di-chalcoge- nides. A further variation in band gap has been observed in MoS₂ monolayer on applying strain. It points out a new direction of band engineering, hence such capability can lead to engineering novel behaviors and holds promise for new applications.

Part of the calculations was performed on High-Performance cluster computer of SNBNCBS, Kolkata for which authors are thankful. SA acknowledges valuable discussion with T.P. Kaloni. This work was initiated at S.N. Bose National Centre, when one of us (SA) was a visiting scientist under the Extended Visitors Linkage Programme (EVLP), which is gratefully acknowledged.