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 1.25eV (indirect bandgap), respectively. The impact of tensile strain (0-0.7%) on the bandgap and effective mass of charge carriers of these three MoS2 structure 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.
Research Article
Effect of Tensile Strain on Performance Parameters of Different Structures of MoS2 Monolayer
https://doi.org/10.21203/rs.3.rs-503274/v1
This work is licensed under a CC BY 4.0 License
Journal Publication
published 26 Jul, 2021
You are reading this latest preprint version
Two-dimensional materials
MoS2 monolayer
First principles
Strain effect
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 [1-4]. The development of nanotechnology has also proposed the different classes of dimensional systems which are zero-dimensional (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 [7-9]. Graphene is a single layer structure of carbon atoms arranged in the honeycomb form. Graphene material has the potential for high-speed 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, graphene-like inorganic monolayer materials gallium nitride (GaN) [12], silicon carbide (SiC) [13,14], boron nitride (BN) [15], zinc oxide (ZnO) [16,17], aluminum nitride (AlN) [18-20] 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 MX2 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 MoS2, MoSe2, MoTe2, WS2, and WSe2 have shown extraordinary features. 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 (1.1-2.0eV) at the K-point in the Brillion zone [25-27]. In this work, we have focused on bandgap and effective mass of MoS2 monolayer for without strain and with strain structure. We also investigated the same for defected monolayer and silicon-doped (Si-doped) monolayer.
1.1 Related Work
S. Deng et al., [28] 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 also analyzed that the strain also induces the electronic bandgap properties. X Wang e al., [29] have presented a theoretical study report for the interaction of No2 molecule with the Mo-edge of MoS2 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 MoS2 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., [30] have presented the computational study of MoS2 monolayer under tensile strain. The 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 MoS2 monolayer have been studied under tensile strain. The mobility increases when the biaxial strain Ɛx=Ɛy=9.5% has been applied. Additionally, the mobility parameter with an increase in temperature has been decreased monotonically. J. Ni et al., [31] 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 [32] has investigated the electrical and optical properties of TMDs MoX2 (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 MoX2 monolayer which makes these materials best for optoelectronics applications. A. O. M. Almayyali et al., [33] have investigated the optical and electronics of zinc iodide (ZnI2) material under the impact of the biaxial strain. For the ZnI2 material-based calculation of molecular dynamic simulations, binding energy, and phonon dispersion curve it has been observed that the ZnI2 has high stability. According to the results it has been observed that the ZnI2 monolayer behave as a semiconductor having an indirect bandgap under PBF and HSE06 methods and values of bandgaps are 2.018eV and 2.94eV, respectively. M. Y. Liu et al. [34] have 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., [35] 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., [36] 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 material for lithium-sulfur batteries. The results have shown that the lithium-polysulfides based on the proposed structure having proper adsorption energies i.e., 0.60-2.68 eV, and moderate diffusion barrier i.e., 0.09-0.31 eV. D.M. Hoat et al., [37] have investigated the electronic structure and optical properties of Hafnium disulfide (HfS2) under vertical strain using DFT calculations. By calculating the phonon dispersion curves the dynamical stability of the HfS2 monolayer has been examined. The HfS2 monolayer has shown a high absorption coefficient of 49.6000 (104 /cm) and 88.122 (104 /cm) in the visible and ultraviolet regions, respectively which display promising optoelectronic applicability.
For this study, we use the DFT method in Virtual Nanolab Atomistix ToolKit (ATK) [38] for first-principle calculation. In the basis setting of the ATK-DFT method, we use localized density approximation (LDA) exchange-correlation along with density mesh cut-off energy value of 75 Hartree and a double Zeta polarized (DZP) [39]. All the simulations with strain and without strain are done at room temperature. The unit cell selected for simulation containing two Mo and four S atoms with a periodic boundary. The Monkhorst-Pack k-grid mesh value for our simulation is 11x11x1. All the lattice constant, atomic positions, and atomic bonds are optimized by applying the limited-memory Broyden–Fletcher–Goldfarb–Shanno (L-BFGS) [40] which is a quasi-Newton- type minimization technique, and set the value for force and stress tolerance equal to 0.05 eV/Å. The Pulay-mixer algorithm with a 10− 5 tolerance value is applied as iteration control in ATK-DFT. The 100 steps are set for maximum numbers of full self-consistent field (SCF) iteration. The Hamiltonian variable is used in the mixing variable. The Fast Fourier Transform (FFT) Poisson solver under ATK-DFT with a periodic boundary condition. The simple orthorhombic lattice constant and angles of TMDs are described in Table 1.
At first, we discuss the structures of MoS2 monolayer that going to investigated in this work. Fig. 1 show the atomistic lattice structure of MoS2 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 defected MoS2 structure where one S atom is removed from structure and Si-doped MoS2 monolayer where one S atom is replaced with one Silicon atom shown in fig. 2. First parameter that we discuss is band structure of different additions of MoS2 monolayer. In band structure calculation we 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 equal to (1/3,1/3), but in orthorhombic supercell [41] it is equivalent to (1/3,0) lies a midpoint between Г and Y points. The band structure along with density of states (DoS) of unstrained atomic structures of MoS2 monolayer is shown in fig. 3. The bandgap value for MoS2 Monolayer, Defected MoS2 Monolayer and Si-Doped MoS2 monolayer are 1.820001eV (direct bandgap), 0.04417 eV (indirect bandgap) and 1.252501eV (indirect bandgap), respectively.
Table 1 Lattice parameters of TMDs monolayers
|
Material |
a(Å) |
b(Å) |
Bond Length(Å) |
α |
β |
γ |
Bandgap (eV) |
|
MoS2 |
5.47397 |
3.1604 |
2.41763 |
90º |
90º |
90º |
1.820001 |
|
MoSe2 |
5.69498 |
3.288 |
2.53048 |
90º |
90º |
90º |
1.581859 |
|
MoTe2 |
6.0937 |
3.5182 |
2.71074 |
90º |
90º |
90º |
1.17133 |
|
WS2 |
5.4615 |
3.5132 |
2.40475 |
90º |
90º |
90º |
1.887673 |
|
WSe2 |
5.68459 |
3.282 |
2.52168 |
90º |
90º |
90º |
1.623808 |
Next, we explore the electronics properties of three MoS2 structures with tensile strain. The vector components of strain along x-direction [1,0] and y-direction [0,1] are denoted as Ɛx and Ɛy, respectively. This study calculated the tensile strain as the relative lattice stretching percentage, given by
Where, a0 and b0 are the lattice constant of unstrained structure. The a and b are the new lattice dimension 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 along X and Y direction has been applied. Fig. 4 shows the results of uniaxial and biaxial strains on different structures. From figure 4(a), it is observed that the bandgap percentage decrease in monolayer MoS2 is 1.99% for strain along both X and Y directions but in the case of biaxial direction the percentage decrease is 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 X-direction, Y-direction and biaxial are 4.69%, 1.36% and 6.03%.
When the strain applied to a structure, it also effects the energy band curvature which is proportional to the carrier effective mass(m):
Where, ℏ = Planck Constant, E = Energy and k = Momentum.
For monolayer we investigate the electron effective mass (me*) at fractional K-points (0,0.335,0), hole effective mass(mh*) 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 MoS2 monolayer without strain. Fig. 5 shows the hole and electron effective mass at different tensile value from 0-0.7%. As shown in fig. 5 the effective mass value under tensile strain change with different rate along the three directions.
Table 2 Effective mass of different MoS2 structures without strain
|
Structures |
me* at K-points |
mh* at K-points |
mh* at Г-points |
|
MoS2 Monolayer |
0.475 |
0.602 |
2.65 |
|
Defected MoS2 Monolayer |
0.301 |
0.209 |
0.139 |
|
Si-doped MoS2 Monolayer |
0.492 |
0.743 |
1.987 |
We have also analyzed the effect of increase in temperature on the bandgap. Fig. 6 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. 6(b). The bandgap percentage decrease for various temperature is calculated and listed in Table 3.
Table 3 Bandgap Decreasing rate at different higher temperature.
|
Structure |
25˚C |
50ºC |
100ºC |
|
MoS2 Monolayer |
5.49% |
5.55% |
5.55% |
|
Defected MoS2 Monolayer |
31.81% |
33.63% |
36.58% |
|
Silicon Doped MoS2 Monolayer |
6.4% |
6.77% |
8.31% |
In summary, we have investigated the different MoS2 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.
Conflicts of Interest/Competing interest:
Authors have no conflict of interest.
Availability of Data and Material:
All data and materials comply with field standards.
Code availability: We have used VNL-ATK QuantumWise Software for implementation.
Author’s Contributions:
All authors contributed to the study conception, simulation and analysis. All the authors have contributed in writing the manuscript and approved the final manuscript.
Ethics approval
The reported work did not involve any human’s participation and/or did not harm welfare of animals.
Consent to Participate:
Not required as this manuscript does not contain participation of humans/children/animals.
Consent for publication:
Not required as this manuscript does not contain participation of humans/children/animals.
- Davis ME, Zuckerman JE, Choi, CHJ, Seligson D, Tolcher A, Alabi CA, Yen Y, Heidel J, & Ribas A (2010) Evidence of RNAi in humans from systemically administered siRNA via targeted nanoparticles. Nature 464:1067-1070. https://doi.org/10.1038/nature08956
- Joseph JD, Kumaragurubaran B, & Sathish S (2019) Effect of MoS2 on the Wear Behavior of Aluminium (AlMg0. 5Si) Composite. Silicon 12:1-9. https://doi.org/10.1007/s12633-019-00238-x
- Karakoti AS, Tsigkou O, Yue S, Lee PD, Stevens MM, Jones JR, & Seal S (2010) Rare earth oxides as nanoadditives in 3-D nanocomposite scaffolds for bone regeneration. Journal of Materials Chemistry 20:8912-8919. https://doi.org/10.1039/C0JM01072C
- Tiwari JN, Tiwari RN, & Kim KS (2012) Zero-dimensional, one-dimensional, two-dimensional and three-dimensional nanostructured materials for advanced electrochemical energy devices. Progress in Materials Science 57:724-803. https://doi.org/10.1016/j.pmatsci.2011.08.003
- Mas-Balleste R, Gomez-Navarro C, Gomez-Herrero J, & Zamora F (2011) 2D materials: to graphene and beyond. Nanoscale 3:20-30. https://doi.org/10.1039/C0NR00323A
- Boochani A, & Veisi S (2018) The Vanadium Effect on Electronic and Optical Response of MoS2 Graphene-Like: Using DFT. Silicon 10:2855-2863. https://doi.org/10.1007/s12633-018-9825-0
- Novoselov KS, Jiang D, Schedin F, Booth TJ, Khotkevich VV, Morozov SV, & Geim AK (2005) Two-dimensional atomic crystals. Proceedings of the National Academy of Sciences 102:10451-10453. https://doi.org/10.1073/pnas.0502848102
- Novoselov KS, Geim AK, Morozov SV, Jiang D, Katsnelson MI, Grigorieva IV, Dubonos SV & Firsov AA (2005) Two-dimensional gas of massless Dirac fermions in graphene. nature, 438:197-200. https://doi.org/10.1038/nature04233
- Geim AK & Novoselov KS (2010) The rise of graphene.: a review. Nanoscience and technology 11-19. https://doi.org/10.1142/9789814287005_0002
- Choi W, Lahiri I, Seelaboyina R, & Kang YS (2010) Synthesis of graphene and its applications: a review. Solid State and Materials Sciences 35:52-71. https://doi.org/10.1080/10408430903505036
- Allen MJ, Tung VC, & Kaner RB (2010) Honeycomb carbon: a review of graphene. Chemical reviews 110:132-145. https://doi.org/10.1021/cr900070d
- Beiranvand R, & Valedbagi S (2016) Electronic and optical properties of advance semiconductor materials: BN, AlN and GaN nanosheets from first principles. Optik,127:1553-1560. https://doi.org/10.1016/j.ijleo.2015.10.194
- Shi Z, Zhang Z, Kutana A, & Yakobson BI (2015) Predicting two-dimensional silicon carbide monolayers. ACS nano 9:9802-9809. https://doi.org/10.1021/acsnano.5b02753
- Javan MB (2016) Electronic and magnetic properties of monolayer SiC sheet doped with 3d-transition metals. Journal of Magnetism and Magnetic Materials 401:656-661. https://doi.org/10.1016/j.jmmm.2015.10.103
- Beiranvand R & Valedbagi S (2015) Electronic and optical properties of h-BN nanosheet: A first principles calculation. Diamond and Related Materials 58:190-195. https://doi.org/10.1016/j.diamond.2015.07.008
- Chaurasiya R, Dixit A, & Pandey R (2019) Strain-driven thermodynamic stability and electronic transitions in ZnX (X= O, S, Se, and Te) monolayers. Journal of Applied Physics 125: 082540. https://doi.org/10.1063/1.5053680
- Pradhan D, & Kar JP (2021) Role of Process Parameters on Microstructural and Electronic Properties of Rapid Thermally Grown MoS2 Thin Films on Silicon Substrates. Silicon 1-11. https://doi.org/10.1007/s12633-021-00959-y
- Aghili S, Beiranvand R, Elahi SM, & Abolhasani MR (2016) Half-metallic ferromagnetism in Mn-doped zigzag AlN nanoribbon from first-principles. Journal of Magnetism and Magnetic Materials 420:122-128. https://doi.org/10.1016/j.jmmm.2016.06.067
- Beiranvand R (2016) Electronic and magnetic properties of Cd-doped zigzag AlN nanoribbons from first principles. Rare Metals 35:771-778. https://doi.org/10.1007/s12598-015-0471-z
- Chegeni M, Beiranvand R & Valedbagi S (2017) Generating Tunable Magnetism in AlN Nanoribbons Using Anion/Cation Vacancies: a First-Principles Prediction. Brazilian Journal of Physics 47:137-144. https://doi.org/10.1007/s13538-016-0480-x
- Papageorgiou DG, Kinloch IA, & Young RJ (2017) Mechanical properties of graphene and graphene-based nanocomposites. Progress in Materials Science 90:75-127. https://doi.org/10.1016/j.pmatsci.2017.07.004
- Peres NM, Araújo MA, & Bozi D (2004) Phase diagram and magnetic collective excitations of the Hubbard model for graphene sheets and layers. Physical Review B 70:195122. https://doi.org/10.1103/PhysRevB.70.195122
- Wang QH, Kalantar-Zadeh K, Kis A, Coleman J N, & Strano MS (2012) Electronics and optoelectronics of two-dimensional transition metal dichalcogenides. Nature nanotechnology 7:699-712. https://doi.org/10.1038/nnano.2012.193
- Chhowalla M, Liu Z, & Zhang H (2015) Two-dimensional transition metal dichalcogenide (TMD) nanosheets. Chemical Society Reviews 44:2584-2586. https://doi.org/10.1039/C5CS90037A
- Mak KF, Lee C, Hone J, Shan J, & Heinz TF (2010) Atomically thin MoS2: a new direct-gap semiconductor. Physical review letters 105:136805. https://doi.org/10.1103/PhysRevLett.105.136805
- Gutierrez HR, Perea-Lopez N, Elías AL, Berkdemir A, Wang B, Lv R, Lopez-Urias F, Crespi VH, Terrones H & Terrones M (2013) Extraordinary room-temperature photoluminescence in triangular WS2 Nano letters 13:3447-3454. https://doi.org/10.1021/nl3026357
- Chang CH, Fan X, Lin SH, & Kuo JL (2013) Orbital analysis of electronic structure and phonon dispersion in MoS2, MoSe2, WS2, and WSe2 monolayers under strain. Physical Review B 88:195420. https://doi.org/10.1103/PhysRevB.88.195420
- Deng S, Li L, & Li M (2018) Stability of direct band gap under mechanical strains for monolayer MoS2, MoSe2, WS2 and WSe2. Physica E: Low-dimensional Systems and Nanostructures 101:44-49. https://doi.org/10.1016/j.physe.2018.03.016
- Wang X, & Shi J (2017) Strain effects on the interaction between NO2 and the Mo-edge of the MoS2 zigzag nanoribbon. IEEE Transactions on Nanotechnology 16:982-990. https://doi.org/10.1109/TNANO.2017.2737942
- Yu S, Xiong HD, Eshun K, Yuan H, & Li Q (2015) Phase transition, effective mass and carrier mobility of MoS2 monolayer under tensile strain. Applied Surface Science 325:27-32. https://doi.org/10.1016/j.apsusc.2014.11.079
- Ni J, Quintana M, Jia F, & Song S (2021) Tailoring the electronic and optical properties of layered blue phosphorene/XC (X= Ge, Si) vdW heterostructures by strain engineering. Physica E: Low-dimensional Systems and Nanostructures 127:114460. https://doi.org/10.1016/j.physe.2020.114460
- Beiranvand R (2021) Theoretical investigation of electronic and optical properties of 2D transition metal dichalcogenides MoX2 (X= S, Se, Te) from first-principles. Physica E: Low-dimensional Systems and Nanostructures 126:114416. https://doi.org/10.1016/j.physe.2020.114416
- Almayyali AO, Muhsen HO, Merdan M, Obeid MM, & Jappor HR (2021) Two-dimensional ZnI2 monolayer as a photocatalyst for water splitting and improvement its electronic and optical properties by strains. Physica E: Low-dimensional Systems and Nanostructures 126:114487. https://doi.org/10.1016/j.physe.2020.114487
- Liu MY, Gong L, Li WZ, Zhang ML, He Y, & Cao C (2021) Band engineering of XBi (X= Si, Ge, Sn, and Pb) single layers via strain and surface chemical-modulation. Applied Surface Science 540:148268. https://doi.org/10.1016/j.apsusc.2020.148268
- Solyaev Y, & Lurie S (2021) Electric field, strain and inertia gradient effects on anti-plane wave propagation in piezoelectric materials. Journal of Sound and Vibration 494:115898. https://doi.org/10.1016/j.jsv.2020.115898
- Li H, Hou J, Duan Q, & Jiang D (2021) Hexagonal borophene sandwiched between blue phosphorenes: A novel bonding heterostructure as an anchoring material for lithium-sulfur batteries. Applied Surface Science 545:148770. https://doi.org/10.1016/j.apsusc.2020.148770
- Hoat DM, Ponce-Perez R, Vu TV, Rivas-Silva JF, & Cocoletzi GH (2021) Theoretical analysis of the HfS2 monolayer electronic structure and optical properties under vertical strain effects. Optik 225:165718. https://doi.org/10.1016/j.ijleo.2020.165718
- Atomistix ToolKit (ATK), https://quantumwise.com/
- Kohn W, & Sham LJ (1965) Self-consistent equations including exchange and correlation effects. Physical review 140:A1133. https://doi.org/10.1103/PhysRev.140.A1133
- Liu DC & Nocedal J (1989) On the limited memory BFGS method for large scale optimization. Mathematical programming 45:503-528. https://doi.org/10.1007/BF01589116
- Cai Y, Zhang G, & Zhang YW (2014) Polarity-reversed robust carrier mobility in monolayer MoS2 Journal of the American Chemical Society 136:6269-6275. https://doi.org/10.1021/ja4109787
- Yun WS, Han SW, Hong SC, Kim IG, & Lee JD (2012) Thickness and strain effects on electronic structures of transition metal dichalcogenides: 2H-M X 2 semiconductors (M= Mo, W; X= S, Se, Te). Physical Review B 85:033305. https://doi.org/10.1103/PhysRevB.85.033305
- Yue Q, Kang J, Shao Z, Zhang X, Chang S, Wang G, Qin S & Li, J (2012) Mechanical and electronic properties of monolayer MoS2 under elastic strain. Physics Letters A 376:1166-1170. https://doi.org/10.1016/j.physleta.2012.02.029
- Sengupta A, Ghosh RK, & Mahapatra S (2013) Performance Analysis of Strained Monolayer MoS2 IEEE transactions on electron devices, 60:2782-2787. https://doi.org/10.1109/TED.2013.2273456