石墨烯（graphene）是由碳原子組成六角呈蜂巢狀晶格的單層二維材料，具有良好的機械性質、電性與熱性質，此外可藉由堆疊方式調變其能帶結構(band structure)，而成為奈米電子元件發展的潛力材料。然當其應用於奈米電子元件時，高功率密度、製程或是環境測試所產生的高溫，常會改變材料之物理性質或行為而使元件性能下降，甚至帶來高熱應力/應變進而造成元件結構材料的破壞。因此，石墨烯的熱力與熱機械性質之探討，已成為一重要研究課題。
過往文獻常結合分子動力學(molecular dynamics)及熱容法(thermostat)模擬分析探討各式固態材料，如石墨烯，於奈米尺度以及正則叢集下(canonical ensemble)溫度相依的材料特性。熱容法包括速度縮放法(velocity-rescaling)、Berendsen熱容法、標準型Nosé-Hoover(NH)熱容法及大規模鏈狀Nosé-Hoover(massive NH chain)熱容法等。但上述各熱容法係以單原子氣體模型為基礎，並未考慮原子間之交互作用力，即原子的勢能(potential energy)，此意謂忽略聲子(phonon)或是量子效應，終將造成無法準確預估極低溫下如石墨烯等固體材料的熱力與熱機械性質以及高溫時系統能量的高度振盪而造成系統穩定度變差及原子鍵結提早斷裂等問題。
本論文為解決上述之缺失，採用一結合修正型Nosé-Hoover熱容法的分子動力學模式深入探討石墨烯於恆溫(constant temperature)或正則叢集下之熱力與熱機械性質，包含楊氏模數、蒲松比、剪切模數、比熱、線性熱膨脹係數與熱傳導係數等以及其與溫度、尺寸、掌性(chirality)與層數間的關係。基本上，此法係藉由Debye理論的晶格振動能(lattice vibration energy)及零點能(zero-point energy) 納入聲子或是量子效應，用以描述固體之能量與溫度間關係以精準掌握固體材料於低溫下之總能量以及維持高溫下固體材料能量及結構的穩定性。計算所得結果與文獻中之理論與實驗結果深入比對，以驗證本論文分析方法之正確及有效性。本論文有助於未來以石墨烯為本之奈米電子元件的設計與開發。

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 Lett 8(3): 902-907.

Bao, W.; Miao, F.; Chen, Z.; Zhang, H.; Jang, W.; Dames, C.; Lau, C. N. (2009): Controlled ripple texturing of suspended graphene and ultrathin graphite membranes. Nat Nanotechnol 4(9): 562-566.

Bao, W.; Jing, L.; Velasco, J.; Lee, Y.; Liu, G.; Tran, D.; Standley, B.; Aykol, M.; Cronin, S. B.; Smirnov, D.; Koshino, M.; McCann, E.; Bockrath, M.; Lau, C. N. (2011): Stacking-dependent band gap and quantum transport in trilayer graphene. Nature Physics 7 (12): 948-952.

Bartolucci, S. F.; Paras, J.; Rafiee, M. A.; Rafiee, J.; Lee, S.; Kapoor, D.; Koratkar, N. (2011): Graphene–aluminum nanocomposites. Materials Science and Engineering A 528(27): 7933-7937.

Battezzatti, L.; Pisani, C.; Ricca, F. (1975): Equilibrium conformation and surface motion of hydrocarbon molecules physisorbed on graphite. Journal of the Chemical Society, Faraday Transactions 2: Molecular and Chemical Physics 71: 1629-1639.

Beeler, J. (1966): Displacement Spikes in Cubic Metals. I. Alpha-Iron, Copper, and Tungsten. Physical Review 150: 470-487.

Berendsen, H. J. C.; Postma, J. P. M.; van Gunsteren, W. F.; Dinola, A; Haak, J. R. (1984): Molecular dynamics with coupling to an external bath. Journal of Chemical Physics 81: 3684.

Brenner, D. W. (1990): Empirical Potential for Hydrocarbons for Use in Simulating the Chemical Vapor Deposition of Diamond Films. Physical Review B 42: 9458-9471.

Castro Neto, A. H.; Peres, N. M. R.; Novoselov, K. S.; Geim, A. K. (2009): The electronic properties of graphene. Reviews of Modern Physics 81(1): 109-162.

Castro, E. V.; Peres, N. M. R.; Santos, J. M. B. L. d.; Guinea, F.; Neto, A. H. C. (2008): Bilayer graphene: gap tunability and edge properties. Journal of Physics: Conference Series 129: 012002.

Chen, W. H.; Cheng, H. C.; Hsu, Y. C. (2007): Mechanical Properties of Carbon Nanotubes Using Molecular Dynamics Simulations with the Inlayer van der Waals Interactions. Computer Modeling in Engineering & Sciences 20: 123-145.

Chen, W. H.; Wu, C. H.; Cheng, H. C. (2011a): Modified Nosé–Hoover Thermostat for Solid State for Constant Temperature Molecular Dynamics Simulation. Journal of Computational Physics 230: 6354–6366.

Chen, W. H.; Wu, C. H.; Cheng, H. C. (2011b): Temperature-dependent Thermodynamic Behaviors of Carbon Fullerene Molecules at Atmospheric Pressure. Computers, Materials, & Continua 25: 195–214.

Chen, W. H.; Liu, Y. L.; Wu, C. H.; Cheng, H. C. (2012): A Theoretical Investigation of Thermal Effects on Vibrational Behaviors of Single-Walled Carbon Nanotubes. Computational Materials Science: 226-233.

Cheng, H. C.; Hsu, Y. C.; Chen, W. H. (2009a): The Influence of Structural Defect on Mechanical Properties and Fracture Behaviors of Carbon Nanotubes. Computers, Materials, & Continua 11: 127-146.

Cheng, H. C.; Liu, Y.-L; Chen, W. H. (2009b): Atomistic-Continuum Modeling for Mechanical Properties of Single-Walled Carbon Nanotubes. International Journal of Solids and Structures 46: 1695-1704.

Cheng, H. C.; Chen, W. H.; Wu, C. H. (2012): Low Temperature Thermal Conductivity of Short Single-Wall Carbon Nanotubes Using a Modified Nosé-Hoover Thermostat. Nanoscale and Microscale Thermophysical Engineering 16: 242-259.
Chuang, Y. C.; Wu, J. Y.; Lin, M. F. (2013): Electric field dependence of excitation spectra in AB-stacked bilayer graphene. Sci Rep 3: 1368.

Cranford, S.; Buehler, M. J. (2011): Twisted and coiled ultralong multilayer graphene ribbons. Modelling and Simulation in Materials Science and Engineering 19(5): 054003.

Frank, I. W.; Tanenbaum, D. M.; van der Zande, A. M.; McEuen, P. L. (2007): Mechanical properties of suspended graphene sheets. Journal of Vacuum Science & Technology B: Microelectronics and Nanometer Structures 25(6): 2558.

Fukushima, H.; Drzal, L. T. (2002): Graphite nanoplatelets as reinforcements for polymers: structural and electrical properties. In: Proceedings of the American society for composites, the 17th technical conference: 21–23.

Gao, W.; Huang, R. (2014): Thermomechanics of monolayer graphene: Rippling, thermal expansion and elasticity. Journal of the Mechanics and Physics of Solids 66: 42-58.

Geim, A. K.; Novoselov, K. S. (2007): The rise of graphene. Nat Mater 6(3): 183-191.

Ghosh, S.; Calizo, I.; Teweldebrhan, D.; Pokatilov, E. P.; Nika, D. L.; Balandin, A. A.; Bao, W.; Miao, F.; Lau, C. N. (2008): Extremely high thermal conductivity of graphene: Prospects for thermal management applications in nanoelectronic circuits. Applied Physics Letters 92 (15): 151911.

Ghosh, S.; Bao, W.; Nika, D. L.; Subrina, S.; Pokatilov, E. P.; Lau, C. N.; Balandin, A. A. (2010): Dimensional crossover of thermal transport in few-layer graphene. Nat Mater 9(7): 555-558.

Güneş, F.; Han, G. H.; Kim, K. K.; Kim, E. S.; Chae, S. J.; Park, M. H.; Jeong, H. K.; Lim, S. C.; Lee, Y. H. (2009): Large-Area Graphene-Based Flexible Transparent Conducting Films. Nano 04 (02): 83-90.

Hajgató, B.; Güryel, S.; Dauphin, Y.; Blairon, J.-M.; Miltner, H. E.; Van Lier, G.; De Proft, F.; Geerlings, P. (2012): Theoretical Investigation of the Intrinsic Mechanical Properties of Single- and Double-Layer Graphene. The Journal of Physical Chemistry C 116 (42): 22608-22618.

Hajgató, B.; Güryel, S.; Dauphin, Y.; Blairon, J.-M.; Miltner, H. E.; Van Lier, G.; De Proft, F.; Geerlings, P. (2013): Out-of-plane shear and out-of plane Young’s modulus of double-layer graphene. Chemical Physics Letters 564: 37-40.

Han, T. W.; He, P. F.; Wang, J.; Wu, A. H. (2009): Numerical Simulation of Temperature Dependence of Tensile Mechanical Properties for Single Graphene Sheet. Journal of Tongji University (Natural Science) 37(12).

Hibbeler, R. C. (2004): Mechanics of materials (Singapore; London: Pearson Prentice Hall.)

Hoover, W. G. (1985): Canonical dynamics: Equilibrium phase-space distributions. Physical Review A 31: 1695.

Hosseini Kordkheili, S. A.; Moshrefzadeh-Sani, H. (2013): Mechanical properties of double-layered graphene sheets. Computational Materials Science 69: 335-343.

Hu, J.; Ruan, X.; Chen, Y. P. (2009): Thermal Conductivity and Thermal Rectification in Graphene Nanoribbons: A Molecular Dynamics Study. Nano Lett 9(7): 2730-2735.

Huntington, H. B. (1958): The elastic constants of crystals. Academic Press. (New YorK)

Incropera, F. P. and DeWitt, D. P. (2006): Fundamentals of Heat and Mass Transfer 6th Edition (John Wiley & Sons, New York.)

Jiang, H.; Huang, Y.; Hwang, K. C. (2005): A Finite-Temperature Continuum Theory Based on Interatomic Potentials. Transactions of the ASME 127: 408–416.

Jung, N.; Kim, N.; Jockusch, S.; Turro, N. J.; Kim, P.; Brus, L. (2009): Charge Transfer Chemical Doping of Few Layer Graphenes: Charge Distribution and Band Gap Formation. Nano Lett 9(12): 4133-4137.

Kelly, B.T. (1981): Physics of Graphite (Applied Science, London)

Kittel, C. (1996): Introduction to Solid State Physics 7th edition (John Wiley & Sons, New York.)

Lee, C.; Wei, X.; Kysar, J. W.; Hone, J. (2008): Measurement of the Elastic Properties and Intrinsic Strength of Monolayer Graphene. Science 321: 385–388.

Lee, C.; Wei, X.; Li, Q.; Carpick, R.; Kysar, J. W.; Hone, J. (2009): Elastic and frictional properties of graphene. Physica Status Solidi B 246(11-12): 2562-2567.

Lee, Y. H.; Biswas, R.; Soukoulis, C. M.; Wang, C. Z.; Chan, C. T.; Ho, K. M. (1991): Molecular-dynamics simulation of thermal conductivity in amorphous silicon. Physical Review B 43: 6573-6580.

Lennard-Jones, J. E. (1924): The determination of molecular fields. I. From the variation of the viscosity of a gas with temperature. (Proceedings of Royal Society, London)

Lin, M. F.; Chuang, Y. C.; Wu, J. Y. (2012): Electrically tunable plasma excitations in AA-stacked multilayer graphene. Physical Review B 86(12).

Liu, C.; Yu, Z.; Neff, D.; Zhamu, A.; Jang, B. Z. (2010): Graphene-based supercapacitor with an ultrahigh energy density. Nano Lett 10(12): 4863-4868.

Liu, X.; Metcalf, T. H.; Robinson, J. T.; Houston, B. H.; Scarpa, F. (2012): Shear Modulus of Monolayer Graphene Prepared by Chemical Vapor Deposition. Nano Letters 12: 1013–1017.

Lukers, J. R.; Zhong, H. (2007): Thermal conductivity of individual single-wall Carbon nanotubes. Journal of Heat Transfer 129: 705-716.

Maiti, A.; Mahan, G. D.; Pantelides, S. T. (1997): Dynamical simulations of nonequilibrium processes-heat flow and the Kapitza resistance across grain boundaries. Solid State Communications 102: 51.

Marion, J. B.; Thornton, S. T. (2004): Classical dynamics of particles and systems 5th edition. (Belmont, CA)

Martyna, G. J.; Tuckerman, M. E.; Klein, M. L. (1992): Nosé-Hoover chains: The canonical ensemble via continuous dynamics. Journal of Chemical Physics. 97: 2635-2643.

Maruyama, S. (2000): Molecular Dynamics Method for Microscale Heat Transfer. Advances in Numerical Heat Transfer 2: 189-226.

Min, K.; Aluru, N. R. (2011): Mechanical properties of graphene under shear deformation. Applied Physics Letters 98(1): 013113.

Mohanty, N.; Berry, V. (2008): Graphene-based single-bacterium resolution biodevice and DNA transistor: interfacing graphene derivatives with nanoscale and microscale biocomponents. Nano Letters 8: 4469–4476.

Moran, M. J. and Shapiro, H. N. (2007): Fundamentals of Engineering Thermodynamics 6th Edition (John Wiley & Sons, New York.)

Morgan, W. C. (1972): Thermal expansion coefficients of graphite crystals. Carbon 10(1): 73-79.

Mounet, N.; Marzari, N. (2005): First-principles determination of the structural, vibrational and thermodynamic properties of diamond, graphite, and derivatives. Physical Review B 71(20).

Nosé, S. (1984): A unified formulation of the constant temperature molecular dynamics methods. Journal of Chemical Physics 81: 511.

Novoselov, K. S.; Geim, A. K.; Morozov, S. V.; Jiang, D.; Zhang, Y.; Dubonos, S. V.; Grigorieva, I. V.; Firsov, A. A. (2004): Electric field effect in atomically thin carbon films. Science 306(5696): 666-669.

Novoselov, K. S.; Geim, A. K.; Morozov, S. V.; Jiang, D.; Katsnelson, M. I.; Grigorieva, I. V.; Dubonos, S. V.; Firsov, A. A. (2005a): Two-dimensional gas of massless Dirac fermions in graphene. Nature 438(7065): 197-200.

Novoselov, K. S.; Jiang, D.; Schedin, F.; Booth, T. J.; Khotkevich, V. V.; Morozov, S. V. (2005b): Two-dimensional atomic crystals. Proceedings of the National Academy of Sciences 102: 0451–0453.

Pan, W.; Xiao, J.; Zhu, J.; Yu, C.; Zhang, G.; Ni, Z.; Watanabe, K.; Taniguchi, T.; Shi, Y.; Wang, X. (2012): Biaxial compressive strain engineering in graphene/boron nitride heterostructures. Scientific reports 2: 893.

Pop, E.; Varshney, V.; Roy, A. K. (2012): Thermal properties of graphene: Fundamentals and applications. MRS Bulletin 37(12): 1273-1281.

Rapaport D. C. (2004): The art of molecular dynamics simulation. Cambridge University Press, U.K.

Rickayzen, G.; Powles, J. G. (2000): Temperature in the classical microcanonical ensemble. Journal of Chemical Physics 114: 4333.

Riley, D. P. (1945): The thermal expansion of graphite: part II. Theoretical. Proceedings of the Physical Society 57(6): 486.

Scarpa, F.; Adhikari, S.; Srikantha Phani, A. (2009): Effective elastic mechanical properties of single layer graphene sheets. Nanotechnology 20(6): 065709.

Singh, V.; Sengupta, S.; Solanki, H. S.; Dhall, R.; Allain, A.; Dhara, S.; Pant, P.; Deshmukh, M. M. (2010): Probing thermal expansion of graphene and modal dispersion at low-temperature using graphene nanoelectromechanical systems resonators. Nanotechnology 21 (16): 165204.

Stankovich, S.; Dikin, D. A.; Dommett, G. H. B.; Kohlhaas, K. M.; Zimney, E. J.; Stach, E. A.; Piner, R. D.; Nguyen, S. T.; Ruoff, R. S. (2006): Graphene-based composite materials. Nature 442: 282–286.

Swope, W. C.; Andersen, H. C.; Berens, P. H.; Wilson, K. R. (1982): A computer simulation method for the calculation of equilibrium constants for the formation of physical clusters of molecules: Application to small water clusters. The Journal of chemical physics 76 (1): 637-649.

Tohei, T.; Kuwabara, A.; Oba, F.; Tanaka, I. (2006): Debye temperature and stiffness of carbon and boron nitride polymorphs from first principles calculations. Physical Review B 73(6): 064304.

Tserpes, K. I.; Papanikos, P. (2014): Finite Element Modeling of the Tensile Behavior of Carbon Nanotubes, Graphene and Their Composites 188: 303-329.

Tuckerman, M. E.; Parrinello, M. (1994): Integrating the Car-Parrinello equations. I. Basic integration techniques. Journal of Chemical Physics 101: 1302-1315.

Verlet, L. (1967): Computer Experiments on Classical Fluids. I. Thermodynamical Properties of Lennard-Jones Molecules. Physical Review 159: 98-103.

Wagner, P.; Ivanovskaya, V. V.; Rayson, M. J.; Briddon, P. R.; Ewels, C. P. (2013): Mechanical properties of nanosheets and nanotubes investigated using a new geometry independent volume definition. Journal of Physics: Condensed Matter 25 (15): 155302.

Wang, M.; Yan, C.; Ma, L. (2012): Graphene nanocomposites. In N. Hu (Ed.), Composites and their Properties , InTech, Rijeka, Croatia: 17-36.

WenXing, B.; ChangChun, Z.; WanZhao, C. (2004): Simulation of Young's modulus of single-walled carbon nanotubes by molecular dynamics. Physica B: Condensed Matter 352(1-4): 156-163.

Yi, L.; Chang, T. (2012): Loading direction dependent mechanical behavior of graphene under shear strain. Science China Physics, Mechanics and Astronomy 55(6): 1083-1087.

Yoon, D.; Son, Y. W.; Cheong, H. (2011): Negative thermal expansion coefficient of graphene measured by Raman spectroscopy. Nano Lett 11(8): 3227-3231.

Zahabul Islam, M.; Mahboob, M.; Robert Lowe, L.; Stephen Bechtel, E. (2013): Characterization of the thermal expansion properties of graphene using molecular dynamics simulations. Journal of Physics D: Applied Physics 46(43): 435302.

Zakharchenko, K.; Katsnelson, M.; Fasolino, A. (2009): Finite Temperature Lattice Properties of Graphene beyond the Quasiharmonic Approximation. Physical Review Letters 102(4): 046808.

Zakharchenko, K. V.; Los, J. H.; Katsnelson, M. I.; Fasolino, A. (2010): Atomistic simulations of structural and thermodynamic properties of bilayer graphene. Physical Review B 81(23): 235439.

Zhang, Y.; Tan, Y. W.; Stormer, H. L.; Kim, P. (2005): Experimental observation of the quantum Hall effect and Berry's phase in graphene. Nature 438(7065): 201-204.

Zhang, Y. Y.; Gu, Y. T. (2013): Mechanical properties of graphene: Effects of layer number, temperature and isotope. Computational Materials Science 71: 197-200.

Zhong, W. R.; Zhang, M. P., Ai, B. Q.; Zheng, D. Q. (2011): Chirality and thickness-dependent thermal conductivity of few-layer graphene: A molecular dynamics study. Applied Physics Letters 98(11).

Zhou, L.; Wang, Y.; Cao, G. (2013): Elastic properties of monolayer graphene with different chiralities. J Phys Condens Matter 25(12): 125302.

Zhu, S. E.; Krishna Ghatkesar, M.; Zhang, C.; Janssen, G. C. A. M. (2013): Graphene based piezoresistive pressure sensor. Applied Physics Letters 102(16): 161904.