2Dmechanics.com


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2	Review: Tailoring the mechanical properties of 2D materials and heterostructures
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5		Charalampos Androulidakis(1), Kaihao Zhang(1), Matthew Robertson(1), and Sameh Tawfick(1,2)
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8		1. Department of Mechanical Science and Engineering, University of Illinois Urbana-Champaign, 1206 W. Green St., Urbana, IL 61801
9		2. The Beckman Institute for Advanced Science and Technology, University of Illinois Urbana-Champaign, 1206 W. Green St., Urbana, IL 61801.
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1	Table 4 Mechanical properties of various 2D crystals. The derived method in the last column (experiment or theory) refers to the modulus and strength.
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3	2D Material (nL=No layers)		E2D (N/m)	E (GPa)	Strength (GPa)	Fracture Strain (%)	Interlayer shear modulus (GPa)	Bending rigidity (eV)	Poisson’s ratio	Method
4	Graphene	1L[1]	340	1000	130	30	-	1-2.4[2, 3]	0.13-0.20[4-7]	Exp.
5		2L[8]	698	1040	126	-	0.40[9]	3.35[9]	«	Exp.
6		3L[8]	986	980	101	-	0.49	6.92	«	Exp.
7		4L[10]	-	1020	-	-	0.47	12.5	«	Theory
8		5L[10]	-	1020	-	-	0.4	18.1	«	Theory
9		6L	-	-	-	-	0.36	28.29	«	Exp.
10		CVD (1L)[11]	324-328	1000	90-99	-	-	-	«	Exp.
11	MoS2[12]	1L	180	270	22	6-11	-	9.61-10.2[2, 13]	0.27[14]	Exp.
12	MoS2[12]	2L	260	200	21	6-11	19[15]	74[16]	«	Exp.
13	h-BN[17]	1-9L	-	865	70.5	17	7.7[18]	0.86-1.54[19-22]	0.21[21]	Exp.
14	WS2[23]	1L	177	272	-	-	-	13.4[2]	0.21[24]	Exp.
15	WSe2[25]	ML	596-1615**	167.3	12.4	7.3	-	11.9[2]	0.19[24]	Exp.
16	MoSe2[26]	1L	-	177.2	4.8	-	18.7[15]	6.39-10.14[27]	0.23	Exp.
17	MoTe2	1L	79.9-94.07[24]	-	-	-	21.7[15]	-	0.24[24]	Theory
18	WTe2[28]	1L	71.29/106.54**	-	-	19	-	-	0.26/0.38	Theory
19	Black Phosphorus	1L[29]	23/92.3	41.3/106.4	-	48/11	-	-	0.40/0.93	Theory
20	Black Phosphorus	FL[30]	-	46-276	2.1-25	8-17	-	-	«	Exp.
21	Borophene[31]	1L(α=0)⁑	212/212	-	~22/13♂	~9/15‖	-	0.79/0.79	0.14/0.14	Theory
22		Triangular	399/163	-	20.9/12.2♂	8.7/14.3‖	-	4.76/1.39	−0.23/0	Theory
23		v1/5	196/208	-	-	-	-	0.52/0.54	0.11/0.12	Theory
24		v1/6	189/210	-	16.4/15.4♂	12.5/10.6‖	-	0.56/0.39	0.15/0.17	Theory
25		v1/8	216/222	-	-	-	-	0.74/0.59	0.17/0.18	Theory
26		v1/12	208/161	-	-	-	-	1.33/0.92	0.08/0.09	Theory
27	Silicene	1L	61.7/59[32-35]	-	5.85-4.78♂	18/9	-	38.63[35]	0.29/0.33[36]	Theory
28	Stanene	1L	25.2/23.5[34, 37]	-	2.6/2.2♂[36]	17/18*	-	-	0.36/0.42[36]	Theory
29	Germanene	1L	44/43.4[34, 36]	-	4.7/4.1♂[36]	20/20.5*	-	-	0.29/0.35[36]	Theory
30	Be5C2[38]	1L	32.68/130.03	32.9/130.89	-	-	-	-	−0.041/−0.16	Theory
31	Bi2Se3[39]	7-12L	-	17.86-25.45	-	4-8.3	-	-	0.27[40]	Exp.
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34	♂ These values correspond to 2D stress in units Nm−1. ‖ These strains correspond to the peak stress where phase transition occurs.
35	⁑ All borophenes are monolayers but with different HH concentration as discussed in the main text. * The tensile strain at the peak stress (not at fracture).
36	** With the ‘-‘ we denote range of values between minimum and maximum while with ‘/’ we separate the armchair and zigzag directions.
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38	References: activated links redirecting to the journal website
39	[1] Lee C, Wei X, Kysar J W and Hone J 2008 Measurement of the elastic properties and intrinsic strength of monolayer graphene science 321 385-8
40	[2] Zhao J, Deng Q, Ly T H, Han G H, Sandeep G and Rümmeli M H 2015 Two-dimensional membrane as elastic shell with proof on the folds revealed by three-dimensional atomic mapping Nat Commun 6 8935
41	[3] Lu Q, Arroyo M and Huang R 2009 Elastic bending modulus of monolayer graphene Journal of Physics D: Applied Physics 42 102002
42	[4] Liu F, Ming P and Li J 2007 Ab initio calculation of ideal strength and phonon instability of graphene under tension Phys Rev B 76 064120
43	[5] Blakslee O, Proctor D, Seldin E, Spence G and Weng T 1970 Elastic constants of compression‐annealed pyrolytic graphite Journal of Applied Physics 41 3373-82
44	[6] Bosak A, Krisch M, Mohr M, Maultzsch J and Thomsen C 2007 Elasticity of single-crystalline graphite: Inelastic x-ray scattering study Phys Rev B 75 153408
45	[7] Kalosakas G, Lathiotakis N, Galiotis C and Papagelis K 2013 In-plane force fields and elastic properties of graphene Journal of Applied Physics 113 134307
46	[8] Lee C, Wei X, Li Q, Carpick R, Kysar J W and Hone J 2009 Elastic and frictional properties of graphene physica status solidi (b) 246 2562-7
47	[9] Chen X, Yi C and Ke C 2015 Bending stiffness and interlayer shear modulus of few-layer graphene Applied Physics Letters 106 101907
48	[10] Wei X D, Meng Z X, Ruiz L, Xia W J, Lee C, Kysar J W, Hone J C, Keten S and Espinosa H D 2016 Recoverable Slippage Mechanism in Multilayer Graphene Leads to Repeatable Energy Dissipation Acs Nano 10 1820-8
49	[11] Lee G-H, Cooper R C, An S J, Lee S, van der Zande A, Petrone N, Hammerberg A G, Lee C, Crawford B and Oliver W 2013 High-strength chemical-vapor–deposited graphene and grain boundaries Science 340 1073-6
50	[12] Bertolazzi S, Brivio J and Kis A 2011 Stretching and breaking of ultrathin MoS2 ACS nano 5 9703-9
51	[13] Jiang J-W, Qi Z, Park H S and Rabczuk T 2013 Elastic bending modulus of single-layer molybdenum disulfide (MoS2): finite thickness effect Nanotechnology 24 435705
52	[14] Feldman J 1976 Elastic constants of 2H-MoS2 and 2H-NbSe2 extracted from measured dispersion curves and linear compressibilities Journal of Physics and Chemistry of Solids 37 1141-4
53	[15] Kuzuba T and Ishii M 1989 Elastic Constant c44 of Uniaxial Layered Crystals physica status solidi (b) 155
54	[16] Elder R M, Neupane M R and Chantawansri T L 2015 Stacking order dependent mechanical properties of graphene/MoS2 bilayer and trilayer heterostructures Applied Physics Letters 107 073101
55	[17] Falin A, Cai Q, Santos E J, Scullion D, Qian D, Zhang R, Yang Z, Huang S, Watanabe K and Taniguchi T 2017 Mechanical properties of atomically thin boron nitride and the role of interlayer interactions Nat Commun 8 15815
56	[18] Bosak A, Serrano J, Krisch M, Watanabe K, Taniguchi T and Kanda H 2006 Elasticity of hexagonal boron nitride: Inelastic x-ray scattering measurements Phys Rev B 73 041402
57	[19] Singh S K, Neek-Amal M, Costamagna S and Peeters F 2013 Thermomechanical properties of a single hexagonal boron nitride sheet Phys Rev B 87 184106
58	[20] Thomas S, Ajith K, Chandra S and Valsakumar M 2015 Temperature dependent structural properties and bending rigidity of pristine and defective hexagonal boron nitride Journal of Physics: Condensed Matter 27 315302
59	[21] Kudin K N, Scuseria G E and Yakobson B I 2001 C 2 F, BN, and C nanoshell elasticity from ab initio computations Phys Rev B 64 235406
60	[22] Wu J, Wang B, Wei Y, Yang R and Dresselhaus M 2013 Mechanics and mechanically tunable band gap in single-layer hexagonal boron-nitride Materials Research Letters 1 200-6
61	[23] Liu K, Yan Q, Chen M, Fan W, Sun Y, Suh J, Fu D, Lee S, Zhou J and Tongay S 2014 Elastic properties of chemical-vapor-deposited monolayer MoS2, WS2, and their bilayer heterostructures Nano Lett 14 5097-103
62	[24] Fan Z, Wei-Bing Z and Bi-Yu T 2015 Electronic structures and elastic properties of monolayer and bilayer transition metal dichalcogenides MX2 (M= Mo, W; X= O, S, Se, Te): a comparative first-principles study Chinese Physics B 24 097103
63	[25] Zhang R, Koutsos V and Cheung R 2016 Elastic properties of suspended multilayer WSe2 Applied Physics Letters 108 042104
64	[26] Yang Y C, Li X, Wen M R, Hacopian E, Chen W B, Gong Y J, Zhang J, Li B, Zhou W, Ajayan P M, Chen Q, Zhu T and Lou J 2017 Brittle Fracture of 2D MoSe2 Adv Mater 29
65	[27] Lai K, Zhang W-B, Zhou F, Zeng F and Tang B-Y 2016 Bending rigidity of transition metal dichalcogenide monolayers from first-principles Journal of Physics D: Applied Physics 49 185301
66	[28] Torun E, Sahin H, Cahangirov S, Rubio A and Peeters F 2016 Anisotropic electronic, mechanical, and optical properties of monolayer WTe2 Journal of Applied Physics 119 074307
67	[29] Jiang J-W and Park H S 2014 Mechanical properties of single-layer black phosphorus Journal of Physics D: Applied Physics 47 385304
68	[30] Wang J-Y, Li Y, Zhan Z-Y, Li T, Zhen L and Xu C-Y 2016 Elastic properties of suspended black phosphorus nanosheets Applied Physics Letters 108 013104
69	[31] Zhang Z H, Yang Y, Penev E S and Yakobson B I 2017 Elasticity, Flexibility, and Ideal Strength of Borophenes Advanced Functional Materials 27
70	[32] Zhao H J 2012 Strain and chirality effects on the mechanical and electronic properties of silicene and silicane under uniaxial tension Phys Lett A 376 3546-50
71	[33] Qin R, Wang C H, Zhu W J and Zhang Y L 2012 First-principles calculations of mechanical and electronic properties of silicene under strain Aip Adv 2
72	[34] John R and Merlin B 2016 Theoretical Investigation of Structural, Electronic, and Mechanical Properties of Two Dimensional C, Si, Ge, Sn Crystal Structure Theory and Applications 5 43
73	[35] Roman R E and Cranford S W 2014 Mechanical properties of silicene Comp Mater Sci 82 50-5
74	[36] Mortazavi B, Rahaman O, Makaremi M, Dianat A, Cuniberti G and Rabczuk T 2017 First-principles investigation of mechanical properties of silicene, germanene and stanene Physica E 87 228-32
75	[37] Tao L, Yang C, Wu L, Han L, Song Y, Wang S and Lu P 2016 Tension-induced mechanical properties of stanene Modern Physics Letters B 30 1650146
76	[38] Wang Y, Li F, Li Y and Chen Z 2016 Semi-metallic Be5C2 monolayer global minimum with quasi-planar pentacoordinate carbons and negative Poisson's ratio Nat Commun 7
77	[39] Yan H, Vajner C, Kuhlman M, Guo L, Li L, Araujo P T and Wang H-T 2016 Elastic behavior of Bi2Se3 2D nanosheets grown by van der Waals epitaxy Applied Physics Letters 109 032103
78	[40] Gao X, Zhou M, Cheng Y and Ji G 2016 First-principles study of structural, elastic, electronic and thermodynamic properties of topological insulator Bi2Se3 under pressure Philosophical Magazine 96 208-22


1		2D Material	Toughness	Stress intensity factor
2			(energy release rate)	KIC
3			(Jm^−2)	(MPa√m)
4		Graphene (1L)	15.9	4
5		Graphene (2L)	-	8.5
6		Graphene (3L)	-	9.17-16.8
7		MoS2	33.48-39.73	1.3-1.8
8		h-BN	0.038-0.072	5.56/5.35
9		Black Phosphorus	5.66 to 16.66	-
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13	References: activated links redirecting to the journal website
14	[1] Zhang P, Ma L L, Fan F F, Zeng Z, Peng C, Loya P E, Liu Z, Gong Y J, Zhang J N, Zhang X X, Ajayan P M, Zhu T and Lou J 2014 Fracture toughness of graphene Nat Commun 5
15	[2] Jang B, Kim B, Kim J-H, Lee H-J, Sumigawa T and Kitamura T 2017 Asynchronous cracking with dissimilar paths in multilayer graphene Nanoscale
16	[3] Wang S, Qin Z, Jung G S, Martin-Martinez F J, Zhang K, Buehler M J and Warner J H 2016 Atomically Sharp Crack Tips in Monolayer MoS2 and Their Enhanced Toughness by Vacancy Defects ACS nano 10 9831-9
17	[4] Wang X, Tabarraei A and Spearot D E 2015 Fracture mechanics of monolayer molybdenum disulfide Nanotechnology 26 175703
18	[5] Becton M and Wang X 2015 Grain-size dependence of mechanical properties in polycrystalline boron-nitride: a computational study Phys Chem Chem Phys 17 21894-901
19	[6] Tabarraei A and Wang X 2015 A molecular dynamics study of nanofracture in monolayer boron nitride Materials Science and Engineering: A 641 225-30
20	[7] Liu N, Hong J, Pidaparti R and Wang X 2016 Fracture patterns and the energy release rate of phosphorene Nanoscale 8 5728-36


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3	2D material	Graphene	MoS2	Boron Nitride	WS2	Black Phosphorus	Stanene
4	Graphene		M[1], M[2], M[3]	M[5], M[6], M[7], M[8]			M[4]
5	MoS2
6	Boron Nitride	S[2]
7	WS2		S[1]
8	Black Phosphorus
9	Stanene
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11	M ->Mechanical Properties		E -> Electrical	O -> Optical	S-> Synthesis
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14	References: activated links redirecting to the journal website
15	1. Liu, Kai, et al. "Elastic properties of chemical-vapor-deposited monolayer MoS2, WS2, and their bilayer heterostructures." Nano letters 14.9 (2014): 5097-5103.
16	2. Jiang, Jin-Wu, and Harold S. Park. "Mechanical properties of MoS2/graphene heterostructures." Applied Physics Letters 105.3 (2014): 033108
17	3. Elder, Robert M., Mahesh R. Neupane, and Tanya L. Chantawansri. "Stacking order dependent mechanical properties of graphene/MoS2 bilayer and trilayer heterostructures." Applied Physics Letters 107.7 (2015): 073101
18	4. Chung, J. Y., Sorkin, V., Pei, Q. X., Chiu, C. H., & Zhang, Y. W. (2017). Mechanical properties and failure behaviour of graphene/silicene/graphene heterostructures. Journal of Physics D: Applied Physics, 50(34), 345302.
19	5. Zhao, Shijun, and Jianming Xue. "Mechanical properties of hybrid graphene and hexagonal boron nitride sheets as revealed by molecular dynamic simulations." Journal of Physics D: Applied Physics 46.13 (2013): 135303.
20	6.Ding, Ning, Xiangfeng Chen, and Chi-Man Lawrence Wu. "Mechanical properties and failure behaviors of the interface of hybrid graphene/hexagonal boron nitride sheets." Scientific reports 6 (2016): 31499.
21	7. Li, Yinfeng, et al. "Mechanical and thermal properties of grain boundary in a planar heterostructure of graphene and hexagonal boron nitride." Nanoscale 10.7 (2018): 3497-3508.
22	8. Wei, Anran, et al. "Mechanical properties of graphene grain boundary and hexagonal boron nitride lateral heterostructure with controlled domain size." Computational Materials Science 126 (2017): 474-478.
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24	References for Synthesis
25	1. Kang, Kibum, et al. "Layer-by-layer assembly of two-dimensional materials into wafer-scale heterostructures." Nature 550.7675 (2017): 229.
26	2. Liu, Zheng, et al. "In-plane heterostructures of graphene and hexagonal boron nitride with controlled domain sizes." Nature nanotechnology 8.2 (2013): 119.


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3	2D material	Graphene	MoS2	Boron Nitride	WS2	Black Phosphorus	Stanene
4	Graphene		M[1], M[2], M[3]	M[5], M[6], M[7], M[8]			M[4]
5	MoS2
6	Boron Nitride
7	WS2
8	Black Phosphorus
9	Stanene
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11	M ->Mechanical Properties		E -> Electrical	O -> Optical	S-> Synthesis
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14	References: activated links redirecting to the journal website
15	1. Liu, Kai, et al. "Elastic properties of chemical-vapor-deposited monolayer MoS2, WS2, and their bilayer heterostructures." Nano letters 14.9 (2014): 5097-5103.
16	2. Jiang, Jin-Wu, and Harold S. Park. "Mechanical properties of MoS2/graphene heterostructures." Applied Physics Letters 105.3 (2014): 033108
17	3. Elder, Robert M., Mahesh R. Neupane, and Tanya L. Chantawansri. "Stacking order dependent mechanical properties of graphene/MoS2 bilayer and trilayer heterostructures." Applied Physics Letters 107.7 (2015): 073101
18	4. Chung, J. Y., Sorkin, V., Pei, Q. X., Chiu, C. H., & Zhang, Y. W. (2017). Mechanical properties and failure behaviour of graphene/silicene/graphene heterostructures. Journal of Physics D: Applied Physics, 50(34), 345302.
19	5. Zhao, Shijun, and Jianming Xue. "Mechanical properties of hybrid graphene and hexagonal boron nitride sheets as revealed by molecular dynamic simulations." Journal of Physics D: Applied Physics 46.13 (2013): 135303.
20	6.Ding, Ning, Xiangfeng Chen, and Chi-Man Lawrence Wu. "Mechanical properties and failure behaviors of the interface of hybrid graphene/hexagonal boron nitride sheets." Scientific reports 6 (2016): 31499.
21	7. Li, Yinfeng, et al. "Mechanical and thermal properties of grain boundary in a planar heterostructure of graphene and hexagonal boron nitride." Nanoscale 10.7 (2018): 3497-3508.
22	8. Wei, Anran, et al. "Mechanical properties of graphene grain boundary and hexagonal boron nitride lateral heterostructure with controlled domain size." Computational Materials Science 126 (2017): 474-478.


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3	2D Materials	[1] [3] [4] [9] [10]	[6] [10]	[8]
4	2D Heterostructures	[1] [10]	[10]
5	Graphene	[2] [10]	[5] [10]	[7]
6	MoS2	[10]	[10]
7	BN	[10]
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15	1. Review: Tailoring the mechanical properties of 2D materials and heterostructures
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