二维纳米材料异质结构的原子间相互作用模型

谭雅文; 江进武

doi:10.6052/1000-0992-19-010

二维纳米材料异质结构的原子间相互作用模型

doi: 10.6052/1000-0992-19-010

谭雅文,
江进武^,

上海大学, 上海市应用数学和力学研究所, 力学与工程科学学院, 上海 200072

详细信息

作者简介:
江进武, 上海大学教授、博士生导师.2013年中组部第五批"青年千人"计划入选者,基金委2018年"优秀青年科学基金"获得者.2003年北京师范大学本科毕业; 2008年中科院理论物理研究所博士毕业;此后在新加坡国立大学和德国包豪斯大学做博士后;2013年底到上海大学力学所(力学与工程科学学院)工作至今.研究方向是固体力学, 长期致力于晶格动力学理论的基础研究.擅长使用晶格动力学理论发展原子间相互作用势能模型,并用于分析负泊松比等力学现象; 通过晶格动力学理论,研究纳米材料的超高热导率、以及高温热障涂层超强隔热性能的力学调控和结构设计.

通讯作者:
江进武

中图分类号: TB383.1
计量
- 文章访问数: 2060
- HTML全文浏览量: 411
- PDF下载量: 265
- 被引次数: 0
出版历程
- 收稿日期: 2019-07-04
- 刊出日期: 2020-10-08

The empirical potential of two-dimensional nanomaterials and their heterostructures

Tan Yawen,
Jiang Jinwu^,

Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai Institute of Applied Mathematics and Mechanics, School of Mechanics and Engineering Science, Shanghai University, Shanghai 200072, China

Funds:

The work is supported by the National Natural Science Foundation of China (NSFC) under Grant No. 11822206, and the Innovation Program of Shanghai Municipal Education Commission under Grant No. 2017-01-07-00-09-E00019.

More Information

Corresponding author: Jiang Jinwu

摘要

摘要: 近些年二维纳米材料得到了大量的研究,其中一个热点研究方向是将不同的二维纳米材料堆垛成纳米异质结构,从而实现多功能的纳米器件.这些二维纳米材料可以从面外和面内两个方向上进行堆垛从而形成两类不同的异质结构.在关于这类二维纳米材料及其异质结构的理论研究中,原子间的相互作用起到类似于连续介质力学中本构关系的作用.因此学者提出了多种方案用于描写原子间相互作用,主要包括第一性原理计算和经验势能模型等.本文主要是对比和分析各种描写二维纳米材料及其异质结构的常见经验势能模型,从而为研究人员选择相互作用模型提供一些参考.
- 二维纳米材料 /
- 异质结构 /
- 相互作用
Abstract: The atomic interaction is a fundamental ingredient in the research of two-dimensional (2D) nanomaterials that have been widely investigated for decades. The atomic interaction can be described by empirical models, which are of both high accuracy and efficiency owing to their physics-inspired functional forms. We outline some typical empirical potential models for the 2D material and its heterostructures, including the vertical van der Waals heterostructure and the lateral heterostructure. The present survey shall offer some help in choosing potential models for the simulation of these 2D-material-based systems. We also discuss some prospects and current challenges at the end of the article.
- two-dimensional nanomaterial /
- empirical potential /
- molecular dynamics simulation

HTML全文

参考文献(111)

[1]	Albe K, Moller W, Heinig K H. 1997. Computer simulation and boron nitride. Rad. Eff. Def. Sol., 141:85.
[2]	Allinger N L. 1976. Calculation of molecular structure and energy by force-field methods. Advances in Physical Organic Chemistry, 13:1-82.
[3]	Allinger N L. 1977. Conformational analysis. 130. MM2. A hydrocarbon force field utilizing V1 and V2 torsional terms. Journal of the American Chemical Society, 99:8127-8134.
[4]	Allinger N L, Yuh Y H, Lii J H. 1989. Molecular Mechanics. The MM3 force field for hydrocarbons. 1. Journal of the American Chemical Society, 111:8551.
[5]	Allinger N L, Chen K, Katzenellenbogen J A, Wilson S R, Anstead G M. 1996. Hyperconjugative effects on carbon-carbon bond lengths in molecular mechanics (MM4). Journal of Computational Chemistry, 17:747-755.
[6]	Artrith N, Urban A, Ceder G. 2017. Efficient and accurate machine-learning interpolation of atomic energies in compositions with many species. Physical Review B, 96:014112.
[7]	Balandin A A, Ghosh S, Bao W, Calizo I, Teweldebrhan D, Miao F, Lau C N. 2008. Superior thermal conductivity of single-layer graphene. Nano Letters, 8:902-907.
[8]	Bartok A P, Payne M C, Kondor R, Csanyi G. 2010. Gaussian approximation potentials: The accuracy of quantum mechanics, without the electrons. Physical Review Letters, 104:136403.
[9]	Bartok A P, Kondor R, Csanyi G. 2013. On representing chemical environments. Physical Review B, 87:184115.
[10]	Bartok A P, Csanyi G. 2015. Gaussian approximation potentials: A brief tutorial introduction. International Journal of Quantum Chemistry, 115:1051-1057.
[11]	Bartok A P, De S, Poelking C, Bernstein N, Kermode J R, Csanyi G, Ceriotti M. 2017. Machine learning unifies the modeling of materials and molecules. Science Advances, 3:e1701816.
[12]	Bartok A P, Kermode J, Bernstein N, Csanyi G. 2018. Machine learning a general purpose interatomic potential for silicon. Preprint at http://arxiv.org/abs/1805.01568.
[13]	Becker C A, Tavazza F, Trautt Z T, Macedoc R B. 2013. Considerations for choosing and using force fields and interatomic potentials in materials science and engineering. Current Opinion in Solid State and Materials Science, 17:277-283.
[14]	Behler J, Parrinello M. 2007. Generalized neural-network representation of high-dimensional potential-energy surfaces. Physical Review Letters, 98:146401.
[15]	Behler J. 2011. Atom-centered symmetry functions for constructing high-dimensional neural network potentials. Journal of Chemical Physics, 134:074106.
[16]	Bertolazzi S, Brivio J, Kis A. 2011. Stretching and breaking of ultrathin $MoS_{2}$. ACS Nano, 5:9703-9709.
[17]	Blank T B, Brown S D, Calhoun A W, Doren D J. 1995. Neural network models of potential energy surfaces. Journal of Computational Physics, 103:4129.
[18]	Blakslee O L, Proctor D G, Seldin E J, Spence G B, Weng T. 1970. Elastic constants of compression-annealed pyrolytic graphite. Journal of Applied Physics, 41:3373.
[19]	Brenner D W, Shenderova O A, Harrison J A, Stuart S J, Ni B, Sinnott S B. 2002. A second-generation reactive empirical bond order (REBO) potential energy expression for hydrocarbons. Journal of Physics: Condensed Matter, 14:783-802.
[20]	Bu H, Chen Y, Zou M, Yi H, Bi K, Ni Z. 2009. Atomistic simulations of mechanical properties of graphene nanoribbons. Physics Letters A, 373:3359-3362.
[21]	Buckingham R A. 1938. The classical equation of state of gaseous Helium, Neon and Argon. Proc. R. Soc. Lond. A, 168:264-283.
[22]	Bukkapatnam S, Malshe M, Agrawal P M, Raff L M, Komanduri R. 2006. Parametrization of interatomic potential functions using a genetic algorithm accelerated with a neural network. Physical Review B, 74:224102.
[23]	Cai W, Moore A L, Zhu Y, Li X, Chen S, Shi L, Ruoff R S. 2010. Thermal transport in suspended and supported monolayer graphene grown by chemical vapor deposition. Nano Letters, 10:1645-1651.
[24]	Carre A, Horbach J, Ispas S, Kob W. 2008. New fitting scheme to obtain effective potential from Car-Parrinello molecular-dynamics simulations: Application to silica. Europhysics Letters, 82:17001.
[25]	Casewit C J, Colwell K S, Rappii A K. 1992 a. Application of a universal force field to main group compounds. Journal of the American Chemical Society, 114:10046-10053.
[26]	Casewit C J, Colwell K S, Rappii A K. 1992 b. Application of a universal force field to organic molecules. Journal of the American Chemical Society, 114:10035-10046.
[27]	Chowdhury R, Adhikari S, Wang C Y, Scarpa F. 2010. A molecular mechanics approach for the vibration of single-walled carbon nanotubes. Computational Materials Science, 48:730-735.
[28]	Cooper R C, Lee C, Marianetti C A, Wei X, Hone J, Kysar J W. 2013 a. Erratum: Nonlinear elastic behavior of two-dimensional molybdenum disulfide (Physical Review B—Condensed matter and materials physics (2013) 87 (035423)). Physical Review B, 87:079901.
[29]	Cooper R C, Lee C, Marianetti C A, Wei X, Hone J, Kysar J W. 2013 b. Nonlinear elastic behavior of two-dimensional molybdenum disulfide. Physical Review B, 87:035423.
[30]	Deringer V K, Bernstein N, Bartok A P, Cliffe M J, Kerber R N, Marbella L E, Grey C P, Elliott S R, Csanyi G. 2018. Realistic atomistic structure of amorphous silicon from machine-learning-driven molecular dynamics. Preprint at http://arxiv.org/abs/1803.02802.
[31]	Deringer V L, Csanyi G. 2017. Machine learning based interatomic potential for amorphous carbon. Physical Review B, 95:094203.
[32]	Deringer V L, Pickard C J, Csanyi G. 2018. Data-driven learning of total and local energies in elemental boron. Physical Review Letters, 120:156001.
[33]	Ercolessi F, Adams J B. 1994. Interatomic potentials from first-principles calculations: the force-matching method. Europhysics Letters, 26:583-588.
[34]	Felix C, Mocanu F C, Konstantinou K, Lee T H, Bernstein N, Volker L, Deringer V L, Csányi G, Elliott S R. 2018. Modeling the phase-change memory material ($Ge_{2}$Sb_{2}$Te_{5}$) with a machine-learned interatomic potential. Journal of Physical Chemistry B, 122:8998-9006.
[35]	Fujikake S, Deringer V L, Lee T H, Krynski M, Elliott S R, Csányi G. 2018. Gaussian approximation potential modeling of lithium intercalation in carbon nanostructures. Journal of Computational Physics, 148:241714.
[36]	Gale J D. 1997. GULP: A computer program for the symmetry-adapted simulation of solids. J. Chem. Soc., Faraday Trans., 93:629-637.
[37]	Gao Q, Yao S, Schneider J, Widom M. 2015. Machine learning methods for interatomic potentials: Application to boron carbide. Preprint at http://arxiv.org/abs/1512.09110.
[38]	Guo W, Zhu C Z, Yu T X, Woo C H, Zhang B, Dai Y T. 2004. Formation of sp3 bonding in nanoindented carbon nanotubes and graphite. Physical Review Letters, 93:245502.
[39]	Guo Y, Tokmakov I, Thompson D L, Wagner A F, Minkoff M. 2007. Interpolating moving least-squares methods for fitting potential energy surfaces: Improving efficiency via local approximants. Journal of Computational Physics, 127:214106.
[40]	Himanen L, Jager M O J, Morooka E V, Canova F F, Ranawat Y S, Gao D Z, Rinke P, Foster A S. 2019. Dscribe: Library of descriptors for machine learning in materials science. Preprint at http://arxiv.org/abs/-1904.08875.
[41]	Imbalzano G, Anelli A, Giofre D, Klees S, Behler J, Ceriotti M. 2018. Automatic selection of atomic fingerprints and reference configurations for machine-learning potentials. Journal of Chemical Physics, 148:241730.
[42]	Ischtwan J, Collins M A. 1994. Molecular potential energy surfaces by interpolation. Journal of Computational Physics, 100:8080.
[43]	Izvekov S, Parrinello M, Burnham C J, Voth G A. 2004. Effective force fields for condensed phase systems from ab initio molecular dynamics simulation: A new method for force-matching. Journal of Computational Physics, 120:10896.
[44]	Jacobsen T L, J?rgensen M S, Hammer B. 2018. On-the-fly machine learning of atomic potential in density functional theory structure optimization. Physical Review Letters, 120:026102.
[45]	Jiang J W, Wang J S, Li B. 2009. Thermal conductance of graphene and dimerite. Physical Review B, 79:205418.
[46]	Jiang J W, Wang J S. 2011 a. Manipulation of heat current by the interface between graphene and white graphene. Europhysics Letters, 96:16003.
[47]	Jiang J W, Wang J S. 2011 b. Theoretical study of thermal conductivity in single-walled boron nitride nanotubes. Physical Review B, 84:085439.
[48]	Jiang J W, Qi Z, Park H S, Rabczuk T. 2013. Elastic bending modulus of single-layer molybdenum disulphide ($MoS_{2}$): Finite thickness effect. Nanotechnology, 24:435705.
[49]	Jiang J W, Park H S, Rabczuk T. 2013. Molecular dynamics simulations of single-layer molybdenum disulphide ($MoS_{2}$): stillinger-weber parametrization, mechanical properties, and thermal conductivity. Journal of Applied Physics, 114:064307.
[50]	Jiang J W, Park H S. 2015. A Gaussian treatment for the friction issue of Lennard-Jones potential in layered materials: Application to friction between graphene, MoS2, and black phosphorus. Journal of Applied Physics, 117:124304.
[51]	Jiang J W, Rabczuk T, Park H S. 2015. A stillinger-weber potential for single-layer black phosphorus, and the importance of cross-pucker interactions for negative Poisson's ratio and edge stress-induced bending. Nanoscale, 7:6059-6068.
[52]	Jiang J W. 2015. Parametrization of stillinger-weber potential based on valence force field model: Application to single-layer $MoS_{2}$ and black phosphorus. Nanotechnology, 26:315706.
[53]	Jiang J W, Zhou Y P. 2017. Parameterization of stillinger-weber potential for two-dimensional atomic crystals. Preprint at http://arxiv.org/abs/1704.03147.
[54]	Jiang J W. 2019. Misfit strain induced buckling in MX2 (with $M=Mo$, $W$ and $X=S$, $Se$, $Te)$ lateral heterostructures: A molecular dynamics study. Acta Mechanica Solida Sinica, 32:17-28.
[55]	Jones J E. 1924. On the determination of molecular fields. II. From the equation of state of a gas. Proc. R. Soc. Lond. A, 106:463-477.
[56]	Kandemir A, Yapicioglu H, Kinaci A, Cagin T, Sevik C. 2016. Thermal transport properties of $MoS_{2}$ and $MoSe_{2}$ monolayers. Nanotechnology, 27:055703.
[57]	Keating P N. 1966. Effect of invariance requirements on the elastic strain energy of crystals with application to the diamond structure. Physical Review, 145:637-645.
[58]	Kolmogorov A N, Crespi V H. 2000. Smoothest bearings: Interlayer sliding in multiwalled carbon nanotubes. Physical Review Letters, 85:4727.
[59]	Kolmogorov A N, Crespi V H. 2005. Registry-dependent interlayer potential for graphitic systems. Physical Review B, 71:235415.
[60]	Kong B D, Paul S, Nardelli M B, Kim K W. 2009. First-principles analysis of lattice thermal conductivity in monolayer and bilayer graphene. Physical Review B, 80:033406.
[61]	Kudin K N, Scuseria G E, Yakobson B I. 2001. $C_2$F, BN, and C nanoshell elasticity from ab initio computations. Physical Review B, 64:235406.
[62]	Landa A, Wynblatt P, Siegel D J, Adams J B, Mryasov O N, Liu X Y. 2000. Development of glue-type potentials for the Al-Pb system: Phase diagram calculation. Acta Materialia, 48:1753-1761.
[63]	Lebedeva I V, Knizhnik A A, Popov A M, Lozovik Y E, Potapkin B V. 2011. Interlayer interaction and relative vibrations of bilayer graphene. Phys. Chem. Chem. Phys, 13:5687-5695.
[64]	Lee C, Wei X, Kysar J W, Hone J. 2008. Measurement of the elastic properties and intrinsic strength of monolayer graphene. Science, 321:385.
[65]	Li M, Wan Y, Tu L, Yang Y, Lou J. 2016. The effect of $VMoS_3$ point defect on the elastic properties of monolayer $MoS_2$ with REBO potentials. Nanoscale Res Lett, 11:155.
[66]	Li M Y, Shi Y, Cheng C C, Lu L S, Lin Y C, Tang H L, Tsai M L, Chu C W, Wei K H, He J H, Chang W H, Suenaga K, Li L J. 2015. Epitaxial growth of a monolayer $WSe_{2}$-$MoS_{2}$ lateral $p$-$n$ junction with an atomically sharp interface. Science, 349:524.
[67]	Li Y, Siegel D J, Adams J B, Liu X Y. 2003. Embedded-atom-method tantalum potential developed by the force-matching method. Physical Review B, 67:125101.
[68]	Li Z, Kermode J R, Vita A D. 2015. Molecular dynamics with on-the-fly machine learning of quantum-mechanical forces. Physical Review Letters, 114:096405.
[69]	Liang T, Phillpot S R, Sinnott S B. 2009. Parametrization of a reactive many-body potential for Mo-S systems. Physical Review B, 79:245110.
[70]	Lii J H, Allinger N L. 1989 a. Molecular mechanics. The MM3 force field for hydrocarbons. 2. Vibrational frequencies and thermodynamics. Journal of the American Chemical Society, 111:8566-8575.
[71]	Lii J H, Allinger N L. 1989 b. Molecular mechanics. The MM3 force field for hydrocarbons. 3. The van der Waals' potentials and crystal data for aliphatic and aromatic hydrocarbons. Journal of the American Chemical Society, 111:8576-8582.
[72]	Lindsay L, Broido D A. 2010. Optimized Tersoff and Brenner empirical potential parameters for lattice dynamics and phonon thermal transport in carbon nanotubes and graphene. Physical Review B, 81:205441.
[73]	Liu F, Ming P, Li J. 2007. Ab initio calculation of ideal strength and phonon instability of graphene under tension. Physical Review B, 76:064120.
[74]	Liu X Y, Adams J B, Ercolessi F, Moriarty J A. 1996. EAM potential for magnesium from quantum mechanical forces. Modelling Simul. Mater. Sci. Eng., 4:293-303.
[75]	Liu Y, Huang Y, Duan X. 2019. Van der Waals integration before and beyond two-dimensional materials. Nature, 567:323.
[76]	Lu J P. 1997. Elastic properties of carbon nanotubes and nanoropes. Physical Review Letters, 79:1297-1300.
[77]	Mills M J L, Popelier P L A. 2011. Intramolecular polarisable multipolar electrostatics from the machine learning method Kriging. Computational and Theoretical Chemistry, 975:42-51.
[78]	Mortazavi B, Ahzi S, Toniazzo V, Remond Y. 2012. Nitrogen doping and vacancy effects on the mechanical properties of graphene: A molecular dynamics study. Physics Letters A, 376:1146-1153.
[79]	Narayanan B, Kinaci A, Sen F G, Davis M J, Gray S K, Chan M K Y, Sankaranarayanan S K. 2016. Describing the diverse geometries of gold from nanoclusters to bulk—a first-principles based hybrid bond order potential. Journal of Physical Chemistry C, 120:13787-13800.
[80]	Norouzzadeh P, Singh D J. 2017. Thermal conductivity of single-layer $WSe_2$ by a Stillinger-Weber potential. Nanotechnology, 28:075708.
[81]	Novoselov K S, Mishchenko A, Carvalho A, Neto A H C. 2016. 2D materials and van der Waals heterostructures. Science, 353: aac9439.
[82]	Ostadhossein A, Rahnamoun A, Wang Y, Zhao P, Zhang S, Crespi V H, Duin A C T. 2017. ReaxFF reactive force-field study of molybdenum disulfide $MoS_{2}$. Journal of Physical Chemistry Letters, 8:631-640.
[83]	Pahari P, Chaturvedi S. 2012. Determination of best-fit potential parameters for a reactive force field using a genetic algorithm. J Mol. Model, 18:1049-1061.
[84]	Pierro M D, Elber R, Leimkuhler B. 2015. A stochastic algorithm for the isobaric-isothermal ensemble with Ewald summations for all long rang forces. J. Chem. Theory Comput, 11:5624-5637.
[85]	Popov V N, Van Doren V E, Balkanski M. 2000. Elastic properties of single-walled carbon nanotubes. Physical Review B, 61:3078-3084.
[86]	Rappe A K, Casewit C J, Colwell K S, III W A G, Skiff W M. 1992. UFF, a full periodic table force field for molecular mechanics and molecular dynamics simulations. Journal of the American Chemical Society, 114:10024-10035.
[87]	Rappe A K, Colwell K S, Casewit C J. 1993. Application of a universal force field to metal complexes. Norg. Chem, 32:3438-3450.
[88]	Reddy C D, Rajendran S, Liew K M. 2006. Equilibrium configuration and continuum elastic properties of finite sized graphene. Nanotechnology, 17:864.
[89]	Rowe P, Csanyi G, Alfe D, Michaelides A. 2018. Development of a machine learning potential for graphene. Physical Review B, 97:054303.
[90]	Sahoo P K, Memaran S, Xin Y, Balicas L, Gutierrez H R. 2018. One-pot growth of two-dimensional lateral heterostructures via sequential edge-epitaxy. Nature, 553:63.
[91]	Scandolo S, Bernasconi M, Chiarotti G L, Focher P, Tosatti E. 1995. Pressure-induced transformation path of graphite to diamond. Physical Review Letters, 74:4015-4018.
[92]	Schutt K T, Glawe H, Brockherde F, Sanna A, Muller K R, Gross E K U. 2014. How to represent crystal structures for machine learning: Towards fast prediction of electronic properties. Physical Review B, 89:205118.
[93]	Seko A, Takahashi A, Tanaka I. 2014. Sparse representation for a potential energy surface. Physical Review B, 90:024101.
[94]	Seko A, Takahashi A, Tanaka I. 2015. First-principles interatomic potentials for ten elemental metals via compressed sensing. Physical Review B, 92:054113.
[95]	Stillinger F H, Weber T A. 1985. Computer simulation of local order in condensed phases of silicon. Physical Review B, 31:5262.
[96]	Sun H. 1998. COMPASS: An ab initio force-field optimized for condensed-phase applicationssoverview with details on alkane and benzene compounds. Journal of Physical Chemistry B, 102:7338-7364.
[97]	Tangney P, Scandolo S. 2002. An ab initio parametrized interatomic force field for silica. Journal of Computational Physics, 117:8898.
[98]	Tersoff J. 1986. New empirical model for the structural properties of silicon. Physical Review Letters, 56:632-635.
[99]	Tersoff J. 1988 a. Empirical interatomic potential for carbon, with applications to amorphous carbon. Physical Review B, 61:2879-2882.
[100]	Tersoff J. 1988 b. Empirical interatomic potential for silicon with improved elastic properties. Physical Review B, 38:9902-9905.
[101]	Tersoff J. 1988 c. New empirical approach for the structure and energy of covalent systems. Physical Review B, 37:6991-7000.
[102]	Tersoff J. 1989. Modeling solid-state chemistry: Interatomic potentials for multicomponent systems. Physical Review Letters, 39:5566-5568.
[103]	Thompson A P, Swiler L P, Trott C R, Foiles S M, Tucker G J. 2015. Spectral neighbor analysis method for automated generation of quantum-accurate interatomic potentials. Journal of Computational Physics, 285:316-330.
[104]	Wen M, Carr S, Fang S, Kaxiras E, Tadmor E B. 2018. Dihedral-angle-corrected registry-dependent interlayer potential for multilayer graphene structures. Physical Review B, 98:235404.
[105]	Xie S, Tu L, Han Y, Huang L, Kang K, Lao K U, Poddar P, Park C, Muller D A, Jr R A D, Park J. 2018. Coherent, atomically thin transition-metal dichalcogenide superlattices with engineered strain. Science, 359:1131-1136.
[106]	Xin Chen X, J?rgensen M S, Li J, Hammer B. 2018. Atomic energies from a convolutional neural network. J. Chem. Theory Comput., 14:3933-3942.
[107]	Xu W, Zhu L, Cai Y, Zhang G, Li B. 2015. Direction dependent thermal conductivity of monolayer phosphorene: Parameterization of Stillinger-Weber potential and molecular dynamics study. Journal of Applied Physics, 117:214308.
[108]	Yang L, Dacek S, Ceder G. 2014. Proposed definition of crystal substructure and substructural similarity. Physical Review B, 90:054102.
[109]	Yu P Y. 2010. Fundamentals of Semiconductors. Springer, New York.
[110]	Zhang Z W, Chen P, Duan X D, Zang K T, Luo J, Duan X F. 2017. Robust epitaxial growth of two-dimensional heterostructures, multiheterostructures, and superlattices. Science, 357:788-792.
[111]	Zhao H, Min K, Aluru N R C. 2009. Size and chirality dependent elastic properties of graphene nanoribbons under uniaxial tension. Nano Letters, 9:3012-3015.