Effects of nanosized constriction on thermal transport properties of graphene
 WenJun Yao^{1, 2},
 BingYang Cao^{2}Email author,
 HeMing Yun^{1} and
 BaoMing Chen^{1}
DOI: 10.1186/1556276X9408
© Yao et al.; licensee Springer. 2014
Received: 28 May 2014
Accepted: 9 August 2014
Published: 21 August 2014
Abstract
Thermal transport properties of graphene with nanosized constrictions are investigated using nonequilibrium molecular dynamics simulations. The results show that the nanosized constrictions have a significant influence on the thermal transport properties of graphene. The thermal resistance of the nanosized constrictions is on the order of 10^{7} to 10^{9} K/W at 150 K, which reduces the thermal conductivity by 7.7% to 90.4%. It is also found that the constriction resistance is inversely proportional to the width of the constriction and independent of the heat current. Moreover, we developed an analytical model for the ballistic thermal resistance of the nanosized constrictions in twodimensional nanosystems. The theoretical prediction agrees well with the simulation results in this paper, which suggests that the thermal transport across the nanosized constrictions in twodimensional nanosystems is ballistic in nature.
PACS
65.80.CK; 61.48.Gh; 63.20.kp; 31.15.xv
Keywords
Graphene Ballistic resistance Nanosized constriction Molecular dynamics simulationBackground
Graphene is a twodimensional (2D) material formed of the honeycomb lattice of sp^{2}bonded carbon atoms. The strong bonding and perfect lattice structure give its unique thermal properties [1–3]. As Balandin et al. [1, 2] demonstrated, the thermal conductivity of graphene is up to 5,400 W/(m · K), which makes it one of the most promising base materials for nextgeneration electronics and thermal management [2–6]. Additionally, compared with other highconductivity materials, such as carbon nanotubes [7–9], graphene is much easier to be fashioned into a broad range of shapes. Such flexibility makes possible the utilization of graphene.
Usually, limited by the synthesis and fabrication procedure, graphene inevitably has a variety of defects, such as vacancies, StoneWales defects, and impurities [10, 11]. Many scholars have demonstrated that these defects are obstacles to heat transfer and create additional sources of phonon scattering in graphene [12–16], especially when the characteristic dimension is less than the phonon mean free path (approximately 775 nm at room temperature) [2]. Hao et al. [13] performed molecular dynamics (MD) simulations on defected graphene sheets. They observed that the increasing defect concentration dramatically reduces the thermal conductivity of graphene. Chien et al. [14] investigated the effect of impurity atoms in graphene and found a rapid drop in thermal conductivity, where hydrogen coverage down to as little as 2.5% of the carbon atoms reduces the thermal conductivity by about 40%. So we can conclude that the thermal transport properties of graphene are very sensitive to its own structures. Besides these defects, the structural configuration is another important but less studied factor impacting the thermal properties, and thus, it can affect the lifetime and reliability of the graphenebased nanodevices further because these devices have more complex shapes in engineering situations. Therefore, from a practical point of view, the investigation on how to predict or tune the thermal transport properties of graphene with a variety of shapes is especially useful for thermal management.
Recently, Xu et al. [17] investigated the transport properties of various graphene junctions and quantum dots using nonequilibrium Green's function method and found that the thermal conductance is insensitive to the detailed structure of the contact region but substantially limited by the narrowest part of the system. Huang et al. [18] constructed a sandwich structure with atomistic Green's function method, where two semiinfinite graphene sheets are bridged by a graphene nanoribbon (GNR). They mainly focused on the phonon transport behavior in GNR and observed that the thermal conductance increases with the width of GNR at fixed length and decreases with GNR length at fixed width.
This paper presents the effect of the nanosized constrictions on the thermal transport properties of graphene studied by the nonequilibrium molecular dynamics (NEMD) simulations. We calculate the thermal transport properties of graphene with those constrictions, and the effects of the heat current and the width of the constriction were explored in detail. Further, based on the phonon dynamics theory, we develop an analytical model for the ballistic resistance of the nanosized constrictions in twodimensional nanosystems, which agrees well with the simulation results in this paper.
Methods
where N is the number of atoms per slab, k_{B} is the Boltzmann constant, and P_{ i } is the momentum of the i th atom.
Results and discussion
Nine graphene sheets with differentsized constrictions are simulated in this paper, and the corresponding pristine one is also designed for comparison. The constriction widths of nine cases are 0.216, 0.648, 1.08, 1.512, 1.944, 2.376, 2.808, 3.24, and 3.672 nm, respectively. And four heat currents (i.e., J = 0.2097, 0.3146, 0.4195, and 0.5243 μW) are performed for all the cases.
where ω is the frequency of phonons, ω_{m} is the maximum frequency, ℏ is the reduced Planck constant, $\frac{1}{exp\left(\frac{\mathit{\hslash \omega}}{{k}_{\mathrm{B}}T}\right)1}$ is the occupation of phonons given by the BoseEinstein distribution, D(ω) is the phonon density of states, v_{g}(ω) is the phonon group velocity, τ(ω,θ) is the transmissivity of phonons, θ is the polar angle, and φ is the azimuthal angle. What is more, in the ballistic limit, two limiting cases of phonon transmission behavior are further discussed, which is differentiated depending on the characteristic size of the constriction (a) relative to the dominant phonon wavelength λ_{d}. If a is much larger than λ_{d}, which is the geometric scattering limit, the transmissivity of phonons is described as τ(ω,θ) = cosθ. If a is near or smaller than λ_{d}, which is the Rayleigh scattering limit, the effect of the wave diffraction should be considered and the calculation of the transmissivity is more complex [33]. It can be seen that the theoretical modeling of the constriction resistance is based on the threedimensional (3D) system so far. But for graphene, a 2D material, it is invalid.
From Equation 9, the ballistic constriction resistance is inversely proportional to the cross section area (A), i.e., the width of the constriction (w), which is consistent with the conclusion of MD. And the predicted results, obtained by substituting c_{v} = 6.81 × 10^{5} J/(m^{3} · K) [34] and v_{g} = 17.45 km/s into Equation 9, are compared quantitatively with MD results in Figure 4. It can be seen that the present model predicts well the thermal resistance of the constriction in graphene, which suggests that thermal transport across the nanosized constrictions in 2D nanosystems is ballistic in nature.
Conclusions
Graphene has shown great potential for the applications in highefficiency thermal management and nanoelectronics due to its exceptional thermal properties in the past few years. Understanding the underlying mechanism of controlling the thermal properties of various structures is of considerable interest. In this paper, systems of rectangular graphene sheets with various nanosized constrictions are constructed by embedding linear vacancy defects and the thermal transport properties are investigated by using nonequilibrium molecular dynamics method. The results show that the nanosized constriction has a significant influence on the thermal transport properties of graphene. And the constriction resistance is on the order of 10^{7} to 10^{9} K/W at 150 K, which reduces the thermal conductivity by 7.7% to 90.4%. Besides, the constriction resistance is inversely proportional to the constriction width and independent of the heat current. These findings indicate that the desired thermal conduction can be achieved via nanosized constrictions. Moreover, we develop a ballistic constriction resistance model for 2D nanosystems, which corresponds to the case when the mean free path of phonon is much larger than the characteristic dimension of the constriction. The predicted values of this model agree well with the simulation results in this paper, which suggests that the thermal transport across nanosized constrictions in 2D nanosystems is ballistic in nature.
Abbreviations
 2D:

twodimensional
 3D:

threedimensional
 GNR:

graphene nanoribbon
 MD:

molecular dynamics
 NEMD:

nonequilibrium molecular dynamics.
Declarations
Acknowledgements
This work was supported by the National Natural Science Foundation of China (Grant Nos. 51322603, 51136001, and 51356001), Science Fund for Creative Research Groups (No. 51321002), the Program for New Century Excellent Talents in University, Tsinghua University Initiative Scientific Research Program, the Tsinghua National Laboratory for Information Science and Technology of China, and the Foundation of Key Laboratory of Renewable Energy Utilization Technologies in Buildings of the National Education Ministry in Shandong Jianzhu University (No. KF201301).
Authors’ Affiliations
References
 Balandin AA, Ghosh S, Bao W, Calizo I, Teweldebrhan D, Miao F, Lau CN: Superior thermal conductivity of singlelayer graphene. Nano Lett 2008, 8: 902–907. 10.1021/nl0731872View ArticleGoogle Scholar
 Ghosh S, Calizo I, Teweldebrhan D, Pokatilov EP, Nika DL, Balandin AA, Bao W, Miao F, Lau CN: Extremely high thermal conductivity of graphene: prospects for thermal management applications in nanoelectronic circuits. Appl Phys Lett 2008, 92: 151911–13. 10.1063/1.2907977View ArticleGoogle Scholar
 Pop E, Varshney V, Roy AK: Thermal properties of graphene: fundamentals and applications. MRS Bull 2012, 37: 1273–1281. 10.1557/mrs.2012.203View ArticleGoogle Scholar
 Novoselov KS, Geim AK, Morozov SV, Jiang D, Zhang Y, Dubonos SV, Grigorieva IV, Firsov AA: Electric field effect in atomically thin carbon films. Science 2004, 306: 666–669. 10.1126/science.1102896View ArticleGoogle Scholar
 Geim AK, Kim P: Carbon wonderland. Sci Am 2008, 298: 90–97.View ArticleGoogle Scholar
 Soldano C, Mahmood A, Dujardin E: Production, properties and potential of graphene. Carbon 2010, 48: 2127–2150. 10.1016/j.carbon.2010.01.058View ArticleGoogle Scholar
 Fujii M, Zhang X, Xie H, Ago H, Takahashi K, Ikuta T, Abe H, Shimizu T: Measuring the thermal conductivity of a single carbon nanotube. Phys Rev Lett 2005, 95: 065502–14.View ArticleGoogle Scholar
 Pop E, Mann D, Wang Q, Goodson K, Dai H: Thermal conductance of an individual singlewall carbon nanotube above room temperature. Nano Lett 2006, 6: 96–100. 10.1021/nl052145fView ArticleGoogle Scholar
 Choi TY, Poulikakos D, Tharian J, Sennhauser U: Measurement of the thermal conductivity of individual carbon nanotubes by the fourpoint threeω method. Nano Lett 2006, 6: 1589–1593. 10.1021/nl060331vView ArticleGoogle Scholar
 Hashimoto A, Suenaga K, Gloter A, Urita K, Iijima S: Direct evidence for atomic defects in graphene layers. Nature 2004, 430: 870–873. 10.1038/nature02817View ArticleGoogle Scholar
 Lee GD, Wang CZ, Yoon E, Hwang NM, Kim DY, Ho KM: Diffusion, coalescence, and reconstruction of vacancy defects in graphene layers. Phys Rev Lett 2005, 95: 205501–14.View ArticleGoogle Scholar
 Nika DL, Pokatilov EP, Askerov AS, Balandin AA: Phonon thermal conduction in graphene: role of Umklapp and edge roughness scattering. Phys Rev B 2009, 79: 155413–112.View ArticleGoogle Scholar
 Hao F, Fang D, Xu Z: Mechanical and thermal transport properties of graphene with defects. Appl Phys Lett 2011, 99: 041901–13. 10.1063/1.3615290View ArticleGoogle Scholar
 Chien S, Yang Y, Chen C: Influence of hydrogen functionalization on thermal conductivity of graphene: nonequilibrium molecular dynamics simulations. Appl Phys Lett 2011, 98: 033107–13. 10.1063/1.3543622View ArticleGoogle Scholar
 Yang P, Wang XL, Li P, Wang H, Zhang LQ, Xie FW: The effect of doped nitrogen and vacancy on thermal conductivity of graphenenanoribbon from nonequilibrium molecular dynamics. Acta Phys Sin 2012, 61: 076501–18. in Chinese in ChineseGoogle Scholar
 Yao HF, Xie YE, Tao O, Chen YP: Thermal transport of graphene nanoribbons embedding linear defects. Acta Phys Sin 2013, 62: 068102–17. in Chinese in ChineseGoogle Scholar
 Xu Y, Chen X, Wang JS, Gu BL, Duan W: Thermal transport in graphene junctions and quantum dots. Phys Rev B 2012, 81: 195425–17.View ArticleGoogle Scholar
 Huang Z, Fisher TS, Murthy JY: Simulation of thermal conductance across dimensionally mismatched graphene interfaces. J Appl Phys 2010, 108: 114310–17. 10.1063/1.3514119View ArticleGoogle Scholar
 Ye ZQ, Cao BY, Guo ZY: High and anisotropic thermal conductivity of bodycentered tetragonal C_{4} calculated using molecular dynamics. Carbon 2014, 66: 567–575.View ArticleGoogle Scholar
 Hu GJ, Cao BY: Thermal resistance between crossed carbon nanotubes: molecular dynamics simulations and analytical modeling. J Appl Phys 2013, 114: 224308–18. 10.1063/1.4842896View ArticleGoogle Scholar
 Li YW, Cao BY: A uniform sourceandsink scheme for calculating thermal conductivity by nonequilibrium molecular dynamics. J Chem Phys 2010, 133: 024106–15. 10.1063/1.3463699View ArticleGoogle Scholar
 Hu GJ, Cao BY: Molecular dynamics simulations of heat conduction in multiwalled carbon nanotubes. Mol Simulat 2012, 38: 823–829. 10.1080/08927022.2012.655731View ArticleGoogle Scholar
 Cao BY, Kong J, Xu Y, Yung K, Cai A: Polymer nanowire arrays with high thermal conductivity and superhydrophobicity fabricated by a nanomolding technique. Heat Transfer Eng 2013, 34: 131–139. 10.1080/01457632.2013.703097View ArticleGoogle Scholar
 Yao WJ, Cao BY: Thermal wave propagation in graphene studied by molecular dynamics simulations. Chin Sci Bull 2014, 27: 3495–3503.View ArticleGoogle Scholar
 Hu J, Ruan X, Chen YP: Thermal conductivity and thermal rectification in graphene nanoribbons: a molecular dynamics study. Nano Lett 2009, 9: 2730–2735. 10.1021/nl901231sView ArticleGoogle Scholar
 Brenner DW: Empirical potential for hydrocarbons for use in simulating the chemical vapor deposition of diamond films. Phys Rev B 1990, 42: 9458–9471. 10.1103/PhysRevB.42.9458View ArticleGoogle Scholar
 Hoover WG: Canonical dynamics: equilibrium phasespace distributions. Phys Rev A 1985, 31: 1695–1697. 10.1103/PhysRevA.31.1695View ArticleGoogle Scholar
 MüllerPlathe F: A simple nonequilibrium molecular dynamics method for calculating the thermal conductivity. J Chem Phys 1997, 106: 6082–6085. 10.1063/1.473271View ArticleGoogle Scholar
 Jiang JW, Chen J, Wang JS, Li BW: Edge states induce boundary temperature jump in molecular dynamics simulation of heat conduction. Phys Rev B 2009, 80: 052301–14.View ArticleGoogle Scholar
 Cooper MG, Mikic BB, Yovanovish MM: Thermal contact conductance. Int J Heat Mass Transfer 1969, 12: 279–300. 10.1016/00179310(69)900118View ArticleGoogle Scholar
 Prasher R: Predicting the thermal resistance of nanosized constrictions. Nano Lett 2005, 5: 2155–2159. 10.1021/nl051710bView ArticleGoogle Scholar
 Prasher R: Ultralow thermal conductivity of a packed bed of crystalline nanoparticles: a theoretical study. Phys Rev B 2006, 74: 165413–15.View ArticleGoogle Scholar
 Prasher R, Tong T, Majumdar A: Diffractionlimited phonon thermal conductance of nanoconstrictions. Appl Phys Lett 2007, 91: 143119–13. 10.1063/1.2794428View ArticleGoogle Scholar
 Mounet N, Marzari N: Firstprinciples determination of the structural, vibrational and thermodynamic properties of diamond, graphite, and derivatives. Phys Rev B 2005, 71: 205214–114.View ArticleGoogle Scholar
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