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Research Papers: Micro/Nanoscale Heat Transfer

Thermal Conductivity of Graphene Nanoribbons: Effect of the Edges and Ribbon Width

[+] Author and Article Information
Paul Plachinda

Graduate Research Assistant
Department of Physics,
Portland State University,
Portland, OR 97201
e-mail: plachind@pdx.edu

David Evans

Program Manager
Sharp Laboratories of America, Inc.,
Camas, WA 98607
e-mail: devans@sharplabs.com

Raj Solanki

Professor
Department of Physics,
Portland State University,
Portland, OR 97201
e-mail: solanki@pdx.edu

We use the “crystallographic” definition of the Fourier transform, i.e., F[f(x)]=-+f(x)exp(-2πikx)dx.

Original definition of the Fejér kernel Fn-1(x)=sin2(nx/2)/sin2(x/2), the Fejér kernel has a property: 12π-ππFn(x)dx=1.

1Corresponding author.

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received August 9, 2011; final manuscript received February 21, 2012; published online October 5, 2012. Assoc. Editor: Pamela M. Norris.

J. Heat Transfer 134(12), 122401 (Oct 05, 2012) (7 pages) doi:10.1115/1.4006297 History: Received August 09, 2011; Revised February 21, 2012

We have calculated thermal conductance of graphene nanoribbons (GNRs) and their dependence on the type of ribbon edge termination (zigzag or armchair) and the width of the ribbon, which ranges from 50 Å to 50 μm. Our model incorporates the effect of edge roughness and includes edge roughness correlation functions for both types of termination. The dependence of thermal conductance on the width of the ribbons and relative contribution of different scattering mechanisms are also analyzed by means of the Green’s function approach to the edge scattering. High temperature thermal conductance of the nanoribbons was found to be 0.15 nW/K and 0.18 nW/K (corresponding to thermal conductivity, 4641 and 5266 W/mK, respectively, for 10 μm long GNRs) which is in a good agreement with the experimental results.

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References

Geim, A. K., and Novoselov, K. S., 2007, “The Rise of Graphene,” Nature Mater., 6(3), pp. 183–191. [CrossRef]
Rao, C. N. R., Sood, A., Subrahmanyam, K., and Govindaraj, A., 2009, “Graphene: The New Two-Dimensional Nanomaterial,” Angew. Chem., Int. Ed., 48(42), pp. 7752–7777. [CrossRef]
Castro Neto, A., Guinea, F., Peres, N., Novoselov, K., and Geim, A., 2009, “The Electronic Properties of Graphene,” Rev. Mod. Phys., 81, pp. 109–162. [CrossRef]
Hartnagel, H., Katilius, R., and Matulionis, A., 2001, Microwave Noise in Semiconductor Devices, Wiley, New York.
Nakada, K., Fujita, M., Dresselhaus, G., and Dresselhaus, M., 1996, “Edge State in Graphene Ribbons: Nanometer Size Effect and Edge Shape Dependence,” Phys. Rev. B, 54(24), pp. 17954–17961. [CrossRef]
Wang, Z. F., Shi, Q. W., Li, Q., Wang, X., Hou, J. G., Zheng, H., Yao, Y., and Chen, J., 2007, “Z-Shaped Graphene Nanoribbon Quantum Dot Device,” Appl. Phys. Lett., 91(5), p. 053109. [CrossRef]
Balandin, A. A., Ghosh, S., Bao, W., Calizo, I., Teweldebrhan, D., Miao, F., and Lau, C. N., 2008, “Superior Thermal Conductivity of Single-Layer Graphene,” Nano Lett., 8(3), pp. 902–907. [CrossRef] [PubMed]
Ghosh, S., Calizo, I., Teweldebrhan, D., Pokatilov, E. P., Nika, D. L., Balandin, A. A., Bao, W., Miao, F., and Lau, C. N., 2008, “Extremely High Thermal Conductivity of Graphene: Prospects for Thermal Management Applications in Nanoelectronic Circuits,” Appl. Phys. Lett., 92(15), p. 151911. [CrossRef]
Ghosh, S., Bao, W., Nika, D. L., Subrina, S., Pokatilov, E. P., Lau, C. N., and Balandin, A. A., 2010, “Dimensional Crossover of Thermal Transport in Few-Layer Graphene,” Nature Mater., 9(7), pp. 555–558. [CrossRef]
Ghosh, S., Nika, D. L., Pokatilov, E. P., and Balandin, A. A., 2009, “Heat Conduction in Graphene: Experimental Study and Theoretical Interpretation,” New J. Phys., 11(9), p. 095012.
Balandin, A. A., 2011, “Thermal Properties of Graphene and Nanostructured Carbon Materials,” Nature Mater., 10(8), pp. 569–581. [CrossRef]
Munoz, E., Lu, J., and Yakobson, B. I., 2010, “Ballistic Thermal Conductance of Graphene Ribbons,” Nano Lett., 10(5), pp. 1652–1656. [CrossRef] [PubMed]
Wang, Z., Xie, R., Bui, C. T., Liu, D., Ni, X., Li, B., and Thong, J. T. L., 2011, “Thermal Transport in Suspended and Supported Few-Layer Graphene,” Nano Lett., 11(1), pp. 113–118. [CrossRef] [PubMed]
Ghosh, S., Subrina, S., Goyal, V., Nika, D., Pokatilov, E., and Balandin, A. A., 2010, “Extraordinary Thermal Conductivity of Graphene: Possibility of Thermal Management Applications,” 12th IEEE Intersociety Conference on Thermal and Thermomechanical Phenomena in Electronic Systems, IEEE, pp. 1–5.
Zhong, W.-R., Zhang, M.-P., Ai, B.-Q., and Zheng, D.-Q., 2011, “Chirality and Thickness-Dependent Thermal Conductivity of Few-Layer Graphene: A Molecular Dynamics Study,” Appl. Phys. Lett., 98(11), p. 113107. [CrossRef]
Aksamija, Z., and Knezevic, I., 2011, “Lattice Thermal Conductivity of Graphene Nanoribbons: Anisotropy and Edge Roughness Scattering,” Appl. Phys. Lett., 98(14), p. 141919. [CrossRef]
Nika, D. L., Ghosh, S., Pokatilov, E. P., and Balandin, A. A., 2009, “Lattice Thermal Conductivity of Graphene Flakes: Comparison With Bulk Graphite,” Appl. Phys. Lett., 94(20), p. 203103. [CrossRef]
Nika, D., Pokatilov, E., Askerov, A., and Balandin, A. A., 2009, “Phonon Thermal Conduction in Graphene: Role of Umklapp and Edge Roughness Scattering,” Phys. Rev. B, 79(15), p. 155413. [CrossRef]
Mingo, N., 2003, “Calculation of Si Nanowire Thermal Conductivity Using Complete Phonon Dispersion Relations,” Phys. Rev. B, 68(11), p. 113308. [CrossRef]
Tan, Z. W., Wang, J.-S., and Gan, C. K., 2011, “First-Principles Study of Heat Transport Properties of Graphene Nanoribbons,” Nano Lett., 11(1), pp. 214–219. [CrossRef] [PubMed]
Santamore, D. H., and Cross, M. C., 2001, “Effect of Surface Roughness on the Universal Thermal Conductance,” Phys. Rev. B, 63(18), p. 184306. [CrossRef]
Huang, Y., Wu, J., and Hwang, K., 2006, “Thickness of Graphene and Single-Wall Carbon Nanotubes,” Phys. Rev. B, 74(24), pp. 1–9. [CrossRef]
Wang, L., Zheng, Q., Liu, J., and Jiang, Q., 2005, “Size Dependence of the Thin-Shell Model for Carbon Nanotubes,” Phys. Rev. Lett., 95(10), pp. 2–5. [CrossRef]
Brenner, D. W., Shenderova, O. A., Harrison, J. A., Stuart, S. J., Ni, B., and Sinnott, S. B., 2002, “A Second-Generation Reactive Empirical Bond Order (REBO) Potential Energy Expression for Hydrocarbons,” J. Phys.: Condens. Matter, 14(4), pp. 783–802. [CrossRef]
Evans, W. J., Hu, L., and Keblinski, P., 2010, “Thermal Conductivity of Graphene Ribbons From Equilibrium Molecular Dynamics: Effect of Ribbon Width, Edge Roughness, and Hydrogen Termination,” Appl. Phys. Lett., 96(20), p. 203112. [CrossRef]
Clark, S. J., Segall, M. D., Pickard, C. J., Hasnip, P. J., Probert, M. I. J., Refson, K., and Payne, M. C., 2005, “First Principles Methods Using CASTEP,” Z. Kristallogr., 220(5–6), pp. 567–570. [CrossRef]
Becke, A. D., 1993, “Density-Functional Thermochemistry. III. The Role of Exact Exchange,” J. Chem. Phys., 98(7), pp. 5648–5652. [CrossRef]
Stephens, P. J., Devlin, F. J., Chabalowski, C. F., and Frisch, M. J., 1994, “Ab Initio Calculation of Vibrational Absorption and Circular Dichroism Spectra Using Density Functional Force Fields,” J. Phys. Chem., 98, pp. 11623–11627. [CrossRef]
Gale, J. D., and Rohl, A. L., 2003, “The General Utility Lattice Program (GULP),” Mol. Simul., 29(5), pp. 291–341. [CrossRef]
Maultzsch, J., Reich, S., Thomsen, C., Requardt, H., and Ordejón, P., 2004, “Phonon Dispersion in Graphite,” Phys. Rev. Lett., 92(7), p. 075501. [CrossRef] [PubMed]
Gan, C., and Srolovitz, D., 2010, “First-Principles Study of Graphene Edge Properties and Flake Shapes,” Phys. Rev. B, 81(12), p. 125445.
Saito, R., Dresselhaus, M. S., and Dresselhaus, G., 1998, Physical Properties of Carbon Nanotubes, Vol. 3, Imperial College Press, London.
Ziman, J., 1960, Electrons and Phonons: The Theory of Transport Phenomena in Solids, Oxford University Press, New York.
Yamamoto, T., Konabe, S., Shiomi, J., and Maruyama, S., 2009, “Crossover From Ballistic to Diffusive Thermal Transport in Carbon Nanotubes,” Appl. Phys. Express, 2(9), p. 095003. [CrossRef]
Srivastava, G., 1990, The Physics of Phonons, A. Hilger, London.

Figures

Grahic Jump Location
Fig. 1

Cuts along two directions a graphene sheet, to produce zigzag (horizontal gray, or red in the online version) and armchair (vertical and oblique dark grey, or green and blue in the online version) terminations of the GNRs

Grahic Jump Location
Fig. 2

(a) Phonon dispersion relations, (b) ballistic transmission function, and (c) phonon density of states for the Γ–M (zigzag) (left) and Γ–K (armchair) (right) directions in graphene

Grahic Jump Location
Fig. 3

Shape functions and their Fourier transforms. (a) Shape function for the armchair boundary, (b) shape function for the zigzag boundary, (c) Fourier transform of the shape function for the armchair boundary, and (d) Fourier transform of the shape function for the zigzag boundary.

Grahic Jump Location
Fig. 4

Scaled attenuation coefficient (mW4/a3L)t-n,m for armchair (a) and zigzag (b) types of boundary as a function of k-vector: solid—from mode m = 0 to mode −n, n = 0, …, 2; dashed—from mode m = 1 to mode −n, n = 0, …, 2; and dashed-dotted—from mode m = 2 to mode −n, n = 0, …, 2

Grahic Jump Location
Fig. 5

Thermal conductance of armchair (a) and zigzag (b) types of boundary as function of temperature

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