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Technical Brief

# Characterization of the Mechanical Properties of Monolayer Molybdenum Disulfide Nanosheets Using First PrinciplesOPEN ACCESS

[+] Author and Article Information
R. Ansari

Department of Mechanical Engineering,
University of Guilan,
P.O. Box 3756,
Rasht, Iran
e-mail: r_ansari@guilan.ac.ir

S. Malakpour, S. Ajori

Department of Mechanical Engineering,
University of Guilan,
P.O. Box 3756,
Rasht, Iran

M. Faghihnasiri

Department of Physics,
University of Guilan,
P.O. Box 1914,
Rasht, Iran

1Corresponding author.

Manuscript received July 15, 2013; final manuscript received December 5, 2013; published online January 29, 2014. Assoc. Editor: Abraham Wang.

J. Nanotechnol. Eng. Med 4(3), 034501 (Jan 29, 2014) (4 pages) Paper No: NANO-13-1041; doi: 10.1115/1.4026207 History: Received July 15, 2013; Revised December 05, 2013

## Abstract

Recently, synthesized inorganic two-dimensional monolayer nanostructures are very promising to be applied in electronic devices. This article explores the mechanical properties of a monolayer molybdenum disulfide ($MoS2$) including Young's bulk and shear moduli and Poisson's ratio by applying density functional theory (DFT) calculation based on the generalized gradient approximation (GGA). The results demonstrate that the elastic properties of $MoS2$ nanosheets are less than those of graphene and hexagonal boron-nitride (h-BN) nanosheets. However, their Poisson's ratio is found to be higher than that of graphene and h-BN nanosheet. It is also observed that due to the special structure of $MoS2$, the thickness of nanosheet changes when the axial strain is applied.

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## Introduction

Synthesizing covalent bonded two-dimensional nanostructures such as graphene [1] is responsible for significant progress in designing flexible electronic devices. Comprehensive investigations conducted on graphene have revealed the distinguished mechanical and physical properties of this structure [2-8]. However, it was found out that the lack of band gap in graphene restricts its application in semiconducting devices. There are two ways to overcome this problem. One of them is finding a way to open a band gap in the graphene [9-12] and another one is using covalent bonded two-dimensional structures with intrinsic large band gap [13,14]. $MoS2$ is a member of transition metal dichalcogenides which also involve $MoSe2$, $WS2$, and $WSe2$ monolayer structures. Unlike the graphene, $MoS2$ possesses a large band gap of 1.8 eV [15,16] and high mobility larger than 200 cm2V−1s−1 [17]. These advantages of monolayer $MoS2$ make it one of the most suitable choices in semiconducting applications as an alternate or complement to graphene.

Studying the mechanical properties of such nanomaterials becomes important as they are employed in the flexible devices. Cooper et al. [18], applying density functional calculations based on local density approximation (LDA), estimated Young's modulus of $MoS2$ nanosheet equal to 210 GPa. Employing atomic force microscope (AFM), Castellanos-Gomez et al. [19] calculated the stiffness of molybdenum disulfide nanosheet around 330 GPa. Moreover, Bertolazzi et al. [20] approximately computed Young's modulus of $MoS2$ nanosheet equal to 270 ± 100 GPa.

This paper aims to perform a comprehensive study on the mechanical properties of $MoS2$ nanosheets such as Young's bulk, shear moduli and also Poisson's ratio. To this end, based on the Perdew–Burke–Ernzerhof exchange correlation, the DFT calculations are utilized to compute strain energies for prediction of aforementioned elastic properties.

## Methodology

In the present study, the Quantum-Espresso code [21] is utilized to perform DFT calculations within the framework of generalized gradient approximation and the Perdew–Burke–Ernzerhof exchange correlation [22,23]. A Monkhorst–Pack [24] k-point mesh of 20 × 20 × 1 is used in Brillouin zone integration and 80 Ry is the cut-off energy for plane wave expansion. In these calculations, as the results are insensitive to increase in unit cell dimension, the smallest unit cell is taken into consideration.

## Results and Discussion

Figure 1 is demonstrated to visualize monolayer $MoS2$ nanosheet. In this figure, molybdenum and sulfur atoms are respectively shown by blue and yellow colors. It can be observed that sulfur atoms are out of molybdenum plane. Each molybdenum atom form covalent bonds with six sulfur atoms, three of them are in the top of the molybdenum atom plane (P1) and remaining atoms are under the molybdenum plane (P2). Initially, the system was relaxed enough to reach the equilibrium state and optimized structure with minimal energy. The DFT calculations show that the distance between two sulfur atoms at P1 and P2 planes is 3.163 Å with Mo–S bond length of 2.437 Å. Moreover, the angle between two Mo–S bonds is determined around 80.9 deg. The aforementioned parameters are illustrated in Fig. 2.

In order to calculate Young's modulus and Poisson's ratio, an axial tensile load is imposed to the unit cell of $MoS2$ nanosheet. The strain energy with respect to the applied axial strain in the harmonic elastic deformation ranging between −2% and 2% is obtained and presented in Fig. 3. By polynomial fitting on the data set, Young's modulus is computed as the second derivative of the strain energy with respect to the strain, expressed by Eq. (1) [26,27]

The thickness of $MoS2$ nanosheet is assumed to be the distance between P1 and P2 planes. From the calculations, Young's modulus is computed around 397.87 GPa which is in close agreement with experimental data [19,20]. Comparing the result reported herein with the ones for graphene and h-BN nanosheet [5] reveals that Young's modulus of $MoS2$ nanosheet is approximately 63% and 55% lower than that of graphene and h-BN nanosheet, respectively.

Considering Poisson's ratio as the ratio of the transverse strain ($ɛtrans=Δb/b,b$ denotes the second lattice constant) to $ɛaxial$ uniaxial strain ($ɛaxial=Δa/a,a$ expresses the first lattice constant), the value of 0.39 is calculated for $MoS2$ nanosheet which is respectively, 60% and 45% higher than that of graphene and h-BN nanosheet [5]. Because of the special structure of $MoS2$, during applying axial strain, the angle between two Mo–S bonds decreases in the case of tensile strain, while it increases in the case of compressive strain as tabulated in Table 1. The discrepancy between the distance of P1 and P2 planes is around 0.7% in both tensile and compressive strain with respect to the structure with minimal energy state.

Biaxial strain is applied to the unit cell of $MoS2$ nanosheet in order to compute the bulk modulus and strain energies are determined, Fig. 4. Utilizing the second derivation of strain energy [8], the bulk modulus of $MoS2$ nanosheet is obtained around 221 GPa which is 65% and 54% lower than that of graphene and h-BN nanosheet [8].

Similar behavior of $MoS2$ nanosheet for the angle between two Mo–S bonds and also distance of P1 and P2 planes can be observed which are presented in Table 2. According to the results, the discrepancy of distance between these planes compared to structure with minimal energy state is calculated around 1.4% which is higher than that of uniaxial strain.

As presented in Fig. 5, by imposing shear strain in the harmonic elastic range up to 0.015 and considering that unit cell area should not be changed during imposing strain, the shear modulus of $MoS2$ nanosheet is determined around 184 GPa using data set of energy-shear strain (Fig. 6) and the following equation

where A0 denotes the area of unit cell, E is the total strain energy and $γxy$ expresses the shear strain. It is observed that shear modulus of $MoS2$ nanosheet is considerably lower than that of graphene reported by Min and Aluru [25] based on molecular dynamics simulation. Obtained results from the calculations reveal that the distances between P1 and P2 planes do not considerably change in the presence of shear strain in comparison with axial ones and accordingly it can be neglected (Table 3).

## Conclusion

In this examination, employing DFT based on the GGA-PBE exchange correlation, a comprehensive investigation into the mechanical properties of $MoS2$ nanosheets such as Young's bulk, shear moduli and Poisson's ratio was carried out. From the results generated, it was observed that the aforementioned elastic properties of $MoS2$ nanosheets other than Poisson's ratio are considerably lower than those of graphene and its inorganic analogous of h-BN nanosheet. It was also observed that the thickness of $MoS2$ nanosheets, which is defined as the distance between two sulfur atoms in the both sides of Molybdenum atoms, changes when an axial load is applied. Moreover, the thickness of $MoS2$ nanosheets remains almost unchanged in the presence of shear load.

## References

Novoselov, K. S., Geim, A. K., Morozov, S. V., Jiang, D., Zhang, Y., Dubonos, S. V., Grigorieva, I. V., and Firsov, A. A., 2004, “Electric Field Effect in Atomically Thin Carbon Films,” Science, 306, pp. 666–669. [PubMed]
Zhang, Y., Tan, Y. W., Stormer, H. L., and Kim, Ph., 2005, “Experimental Observation of the Quantum Hall Effect and Berry's Phase in Grapheme,” Nature, 438, pp. 201–204. [PubMed]
Lv, R., and Terrones, M., 2012, “Towards New Graphene Materials: Doped Graphene Sheets and Nanoribbons,” Mater. Lett., 78, pp. 209–218.
Morozov, S., Novoselov, K., Katsnelson, M., Schedin, F., Elias, D. C., Jaszczak, J. A., and Geim, A. K., 2008, “Giant Intrinsic Carrier Mobilities in Graphene and Its Bilayer,” Phys. Rev. Lett., 100, p. 016602. [PubMed]
Topsakal, M., Cahangirov, and S., Ciraci, S., 2010, “The Response of Mechanical and Electronic Properties of Graphane to the Elastic Strain,” Appl. Phys. Lett., 96, pp. 091912–091914.
Ni, Zh., Bu, H., Zou, M., Yi, H., Bi, K., and Chen, Y., 2010, “Anisotropic Mechanical Properties of Graphene Sheets from Molecular Dynamics,” Physica B, 405, pp. 1301–1306.
Ansari, R., Ajori, S., and Motevalli, B., 2012, “Mechanical Properties of Defective Single-Layered Graphene Sheets Via Molecular Dynamics Simulation,” Superlattices Microstruct., 51, pp. 274–289.
Nag, A., Raidongia, K., Hembram, K. P. S. S., Datta, R., Waghmare, U. V., and Rao, C. N. R., 2010, “Graphene Analogue of BN: Novel Synthesis and Properties,” ACS Nano, 4, pp. 1539–1544. [PubMed]
Li, Y., and Chen, Zh., 2013, “XH/π (X = C, Si) Interactions in Graphene and Silicene: Weak in Strength, Strong in Tuning Band Structures,” J. Phys. Chem. Lett., 4, pp. 269–275.
Chen, Z., Lin, Y., Rooks, M. J., and Avouris, P., 2007, “Graphene Nano-Ribbon Electronics,” Physica E, 40, pp. 228–232.
Han, M., Özyilmaz, B., Zhang, Y., and Kim, P., 2007, “Energy Band-Gap Engineering of Graphene Nanoribbons,” Phys. Rev. Lett., 98, p. 206805. [PubMed]
Li, X., Wang, X., Zhang, L., Lee, S., and Dai, H., 2008, “Chemically Derived, Ultra Smooth Graphene Nanoribbon Semiconductors,” Science, 319, pp. 1229–1233. [PubMed]
Ayari, A., Cobas, E., Ogundadegbe, O., and Fuhrer, M. S., 2007, “Realization and Electrical Characterization of Ultrathin Crystals of Layered Transition-Metal Dichalcogenides,” J. Appl. Phys., 101, p. 014507.
Podzorov, V., Gershenson, M. E., Kloc, C., Zeis, R., and Bucher, E., 2004, “High Mobility Field-Effect Transistors Based on Transition Metal Dichalcogenides,” Appl. Phys. Lett., 84, pp. 3301–3303.
Lee, C., Yan, H., Brus, L. E., Heinz, T. F., Hone, J., and Ryu, S., 2010, “Anomalous Lattice Vibrations of Single- and Few-Layer MoS2,” ACS Nano, 4, pp. 2695–2700. [PubMed]
Splendiani, A., Sun, L., Zhang, Y., Li, T., Kim, J., Chim, C.-Y., Galli, G., and Wang, F., 2010, “Emerging Photoluminescence in Monolayer MoS2”Nano Lett., 10, pp. 1271–1275. [PubMed]
Radisavljevic, B., Radenovic, A., Brivio, J., Giacometti, V., and Kis, A., 2011, “Single-Layer MoS2 Transistors,” Nat. Nanotechnol., 6, pp. 147–150. [PubMed]
Cooper, R. C., Lee, Ch., Marianetti, Ch. A., Wei, X., Hone, J., and Kysar, J. W., 2013, “Nonlinear Elastic Behavior of Two-Dimensional Molybdenum Disulfide,” Phys. Rev. B, 87, pp. 035423–035433.
Castellanos-Gomez, A., Poot, M., Steele, G. A., van derZant, H. S. J., Agraït, N., and Rubio-Bollinger, G., 2012, “Elastic Properties of Freely Suspended MoS2 Nanosheets,” Adv. Mater., 24, pp. 772–775. [PubMed]
Bertolazzi, S., Brivio, J., and Kis, A., 2011, “Stretching and Breaking of Ultrathin MoS2”ACS Nano, 5, pp. 9703–9709. [PubMed]
Giannozzi, P., Baroni, S., Bonini, N., Calandra, M., Car, R., Cavazzoni, C., Ceresoli, D., Chiarotti, G. L., Cococcioni, M., Dabo, I., Dal Corso, A., Fabris, S., Fratesi, G., de Gironcoli, S., Gebauer, R., Gerstmann, U., Gougoussis, C., Kokalj, A., Lazzeri, M., Martin-Samos, L., Marzari, N., Mauri, F., Mazzarello, R., Paolini, S., Pasquarello, A., Paulatto, L., Sbraccia, C., Scandolo, S., Sclauzero, G., Seitsonen, A. P., Smogunov, A., Umari, P., and Wentzcovitch, R. M., 2009, “Quantum Espresso,” J. Phys.: Condens. Matter, 21, p. 395502.
Perdew, J. P., Burke, K., and Ernzerhof, M., 1996, “Generalized Gradient Approximation Made Simple,” Phys. Rev. Lett., 77, pp. 3865–3868. [PubMed]
Perdew, J. P., Burke, K., and Wang, Y., 1996, “Generalized Gradient Approximation for the Exchange-Correlation Hole of a Many-Electron System,” Phys. Rev. B., 54, pp. 16533–16539.
Monkhorst, H. J., and Pack, J. D., 1976, “On Special Points for Brillouin Zone Integrations,” Phys. Rev. B, 13, pp. 5188–5192.
Min, K., and Aluru, N. R., 2011, “Mechanical Properties of Graphene Under Shear Deformation,” Appl. Phys. Lett., 98, p. 013113.
Wagner, P., Ivanovskaya, V. V., Rayson, M. J., Briddon, P. R., and Ewels, C. P., 2013, “Mechanical Properties of Nanosheets and Nanotubes Using a New Geometry Independent Volume Definition,” J. Phys.: Condens. Matter, 25, pp. 155302–155323. [PubMed]
Zhukovskii, Y. F., Piskunov, S., Pugno, N., Berzina, B., Trinkler, L., and Bellucci, S., 2009, “Ab Initio Simulations on the Atomic and Electronic Structure of Single-Walled BN Nanotubes and Nanoarches” J. Phys. Chem. Solids, 70, pp. 796–803.
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## References

Novoselov, K. S., Geim, A. K., Morozov, S. V., Jiang, D., Zhang, Y., Dubonos, S. V., Grigorieva, I. V., and Firsov, A. A., 2004, “Electric Field Effect in Atomically Thin Carbon Films,” Science, 306, pp. 666–669. [PubMed]
Zhang, Y., Tan, Y. W., Stormer, H. L., and Kim, Ph., 2005, “Experimental Observation of the Quantum Hall Effect and Berry's Phase in Grapheme,” Nature, 438, pp. 201–204. [PubMed]
Lv, R., and Terrones, M., 2012, “Towards New Graphene Materials: Doped Graphene Sheets and Nanoribbons,” Mater. Lett., 78, pp. 209–218.
Morozov, S., Novoselov, K., Katsnelson, M., Schedin, F., Elias, D. C., Jaszczak, J. A., and Geim, A. K., 2008, “Giant Intrinsic Carrier Mobilities in Graphene and Its Bilayer,” Phys. Rev. Lett., 100, p. 016602. [PubMed]
Topsakal, M., Cahangirov, and S., Ciraci, S., 2010, “The Response of Mechanical and Electronic Properties of Graphane to the Elastic Strain,” Appl. Phys. Lett., 96, pp. 091912–091914.
Ni, Zh., Bu, H., Zou, M., Yi, H., Bi, K., and Chen, Y., 2010, “Anisotropic Mechanical Properties of Graphene Sheets from Molecular Dynamics,” Physica B, 405, pp. 1301–1306.
Ansari, R., Ajori, S., and Motevalli, B., 2012, “Mechanical Properties of Defective Single-Layered Graphene Sheets Via Molecular Dynamics Simulation,” Superlattices Microstruct., 51, pp. 274–289.
Nag, A., Raidongia, K., Hembram, K. P. S. S., Datta, R., Waghmare, U. V., and Rao, C. N. R., 2010, “Graphene Analogue of BN: Novel Synthesis and Properties,” ACS Nano, 4, pp. 1539–1544. [PubMed]
Li, Y., and Chen, Zh., 2013, “XH/π (X = C, Si) Interactions in Graphene and Silicene: Weak in Strength, Strong in Tuning Band Structures,” J. Phys. Chem. Lett., 4, pp. 269–275.
Chen, Z., Lin, Y., Rooks, M. J., and Avouris, P., 2007, “Graphene Nano-Ribbon Electronics,” Physica E, 40, pp. 228–232.
Han, M., Özyilmaz, B., Zhang, Y., and Kim, P., 2007, “Energy Band-Gap Engineering of Graphene Nanoribbons,” Phys. Rev. Lett., 98, p. 206805. [PubMed]
Li, X., Wang, X., Zhang, L., Lee, S., and Dai, H., 2008, “Chemically Derived, Ultra Smooth Graphene Nanoribbon Semiconductors,” Science, 319, pp. 1229–1233. [PubMed]
Ayari, A., Cobas, E., Ogundadegbe, O., and Fuhrer, M. S., 2007, “Realization and Electrical Characterization of Ultrathin Crystals of Layered Transition-Metal Dichalcogenides,” J. Appl. Phys., 101, p. 014507.
Podzorov, V., Gershenson, M. E., Kloc, C., Zeis, R., and Bucher, E., 2004, “High Mobility Field-Effect Transistors Based on Transition Metal Dichalcogenides,” Appl. Phys. Lett., 84, pp. 3301–3303.
Lee, C., Yan, H., Brus, L. E., Heinz, T. F., Hone, J., and Ryu, S., 2010, “Anomalous Lattice Vibrations of Single- and Few-Layer MoS2,” ACS Nano, 4, pp. 2695–2700. [PubMed]
Splendiani, A., Sun, L., Zhang, Y., Li, T., Kim, J., Chim, C.-Y., Galli, G., and Wang, F., 2010, “Emerging Photoluminescence in Monolayer MoS2”Nano Lett., 10, pp. 1271–1275. [PubMed]
Radisavljevic, B., Radenovic, A., Brivio, J., Giacometti, V., and Kis, A., 2011, “Single-Layer MoS2 Transistors,” Nat. Nanotechnol., 6, pp. 147–150. [PubMed]
Cooper, R. C., Lee, Ch., Marianetti, Ch. A., Wei, X., Hone, J., and Kysar, J. W., 2013, “Nonlinear Elastic Behavior of Two-Dimensional Molybdenum Disulfide,” Phys. Rev. B, 87, pp. 035423–035433.
Castellanos-Gomez, A., Poot, M., Steele, G. A., van derZant, H. S. J., Agraït, N., and Rubio-Bollinger, G., 2012, “Elastic Properties of Freely Suspended MoS2 Nanosheets,” Adv. Mater., 24, pp. 772–775. [PubMed]
Bertolazzi, S., Brivio, J., and Kis, A., 2011, “Stretching and Breaking of Ultrathin MoS2”ACS Nano, 5, pp. 9703–9709. [PubMed]
Giannozzi, P., Baroni, S., Bonini, N., Calandra, M., Car, R., Cavazzoni, C., Ceresoli, D., Chiarotti, G. L., Cococcioni, M., Dabo, I., Dal Corso, A., Fabris, S., Fratesi, G., de Gironcoli, S., Gebauer, R., Gerstmann, U., Gougoussis, C., Kokalj, A., Lazzeri, M., Martin-Samos, L., Marzari, N., Mauri, F., Mazzarello, R., Paolini, S., Pasquarello, A., Paulatto, L., Sbraccia, C., Scandolo, S., Sclauzero, G., Seitsonen, A. P., Smogunov, A., Umari, P., and Wentzcovitch, R. M., 2009, “Quantum Espresso,” J. Phys.: Condens. Matter, 21, p. 395502.
Perdew, J. P., Burke, K., and Ernzerhof, M., 1996, “Generalized Gradient Approximation Made Simple,” Phys. Rev. Lett., 77, pp. 3865–3868. [PubMed]
Perdew, J. P., Burke, K., and Wang, Y., 1996, “Generalized Gradient Approximation for the Exchange-Correlation Hole of a Many-Electron System,” Phys. Rev. B., 54, pp. 16533–16539.
Monkhorst, H. J., and Pack, J. D., 1976, “On Special Points for Brillouin Zone Integrations,” Phys. Rev. B, 13, pp. 5188–5192.
Min, K., and Aluru, N. R., 2011, “Mechanical Properties of Graphene Under Shear Deformation,” Appl. Phys. Lett., 98, p. 013113.
Wagner, P., Ivanovskaya, V. V., Rayson, M. J., Briddon, P. R., and Ewels, C. P., 2013, “Mechanical Properties of Nanosheets and Nanotubes Using a New Geometry Independent Volume Definition,” J. Phys.: Condens. Matter, 25, pp. 155302–155323. [PubMed]
Zhukovskii, Y. F., Piskunov, S., Pugno, N., Berzina, B., Trinkler, L., and Bellucci, S., 2009, “Ab Initio Simulations on the Atomic and Electronic Structure of Single-Walled BN Nanotubes and Nanoarches” J. Phys. Chem. Solids, 70, pp. 796–803.

## Figures

Fig. 1

Monolayer MoS2 nanosheet, (a) isometric view and (b) top view

Fig. 2

Mo–S bonds with defined parameters

Fig. 3

Variation of strain energy with uniaxial strain

Fig. 4

Variation of strain energy with biaxial strain

Fig. 5

Schematic representation of applied shear strain

Fig. 6

Variation of strain energy with shear strain

## Tables

Table 1 Values of Mo–S bond length, angles between Mo–S bond and distance of P1 and P2 planes in the presence of axial strain
Table 2 Values of Mo–S bond length, angles between Mo–S bond and distance of P1 and P2 planes in the presence of biaxial strain
Table 3 Values of Mo–S bond length, angles between Mo–S bond and distance of P1 and P2 planes in the presence of shear strain

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