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Anomalous heat conduction and anomalous diffusion in low dimensional nanoscale systems

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Abstract

Heat conduction is an important energy transport process in nature. Phonon is the major energy carrier for heat in semiconductors and dielectric materials. In analogy to Ohm’s law of electrical conduction, Fourier’s law is the fundamental law of heat conduction in solids. Although Fourier’s law has received great success in describing macroscopic heat conduction in the past two hundred years, its validity in low dimensional systems is still an open question. Here we give a brief review of the recent developments in experimental, theoretical and numerical studies of heat conduction in low dimensional systems, including lattice models and low dimensional nanostructures such as nanowires, nanotubes and graphene. We will demonstrate that phonons transport in low dimensional systems superdiffusively, which leads to a size dependent thermal conductivity. In other words, Fourier’s law is not applicable in low dimensional structures.

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References

  1. N. Li, J. Ren, L. Wang, G. Zhang, P. Hänggi, B. Li, Rev. Mod. Phys. 84, 1045 (2012)

    Article  ADS  Google Scholar 

  2. G. Zhang, B. Li, Nanoscale 2, 1058 (2010)

    Article  ADS  Google Scholar 

  3. E. Pop, Nano Research 3, 147 (2010)

    Article  Google Scholar 

  4. S. Lepri, R. Livi, A. Politi, Phys. Rep. 377, 1 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  5. A. Dhar, Adv. Phys. 57, 457 (2008)

    Article  ADS  Google Scholar 

  6. R. Yang, G. Chen, M. Laroche, Y. Taur, J. Heat Transfer. 127, 298 (2005)

    Article  Google Scholar 

  7. C.W. Chang, D. Okawa, H. Garcia, A. Majumdar, A. Zettl, Phys. Rev. Lett. 101, 075903 (2008)

    Article  ADS  Google Scholar 

  8. Z. Rieder, J.L. Lebowitz, E. Lieb, J. Math. Phys. 8, 1073 (1967)

    Article  ADS  Google Scholar 

  9. H. Nakazawa, Prog. Theor. Phys. Suppl. 39, 236 (1968)

    Google Scholar 

  10. S. Lepri, R. Livi, A. Politi, Phys. Rev. Lett. 78, 1896 (1997)

    Article  ADS  Google Scholar 

  11. S. Lepri, R. Livi, A. Politi, Europhys. Lett. 43, 271 (1998)

    Article  ADS  Google Scholar 

  12. T. Hatano, Phys. Rev. E 59, R1 (1999)

    Article  ADS  MathSciNet  Google Scholar 

  13. A. Dhar, Phys. Rev. Lett. 86, 3554 (2001)

    Article  ADS  Google Scholar 

  14. P. Grassberger, W. Nadler, L. Yang, Phys. Rev. Lett. 89, 180601 (2002)

    Article  ADS  Google Scholar 

  15. G. Zhang, B. Li, J. Chem. Phys. 123, 014705 (2005)

    Article  ADS  Google Scholar 

  16. N. Yang, G. Zhang, B. Li, Nano Today 5, 85 (2010)

    Article  Google Scholar 

  17. P. Kim, L. Shi, A. Majumdar, P.L. Mceuen, Phys. Rev. Lett. 87, 215502 (2001)

    Article  ADS  Google Scholar 

  18. S. Shen, A. Henry, J. Tong, R. Zheng, G. Chen, Nat. Nanotechnol. 5, 251 (2010)

    Article  ADS  Google Scholar 

  19. D. Li, Y. Wu, P. Kim, L. Shi, P. Yang, A. Majumdar, Appl. Phys. Lett. 83, 2934 (2003)

    Article  ADS  Google Scholar 

  20. R. Chen, A.I. Hochbaum, P. Murphy, J. Moore, P. Yang, A. Majumdar, Phys. Rev. Lett. 101, 105501 (2008)

    Article  ADS  Google Scholar 

  21. A.I. Hochbaum, R. Chen, R.D. Delgado, W. Liang, E.C. Garnett, M. Najarian, A. Majumdar, P. Yang, Nature 451, 163 (2008)

    Article  ADS  Google Scholar 

  22. D. Li, Y. Wu, R. Fan, P. Yang, A. Majumdar, Appl. Phys. Lett. 83, 3186 (2003)

    Article  ADS  Google Scholar 

  23. C.T. Bui, R. Xie, M. Zheng, Q. Zhang, C.H. Sow, B. Li, J.T.L. Thong, Small 8, 738 (2012)

    Article  Google Scholar 

  24. A.L. Moore, M.T. Pettes, F. Zhou, L. Shi, J. Appl. Phys. 106, 034310 (2009)

    Article  ADS  Google Scholar 

  25. J. Chen, G. Zhang, B. Li, J. Chem. Phys. 135, 104508 (2011)

    Article  ADS  Google Scholar 

  26. M.C. Wingert, Z.C.Y. Chen, E. Dechaumphai, J. Moon, J.H. Kim, J. Xiang, R. Chen, Nano Lett. 11, 5507 (2011)

    Article  ADS  Google Scholar 

  27. L. Shi, D. Li, C. Yu, W. Jang, D. Kim, Z. Yao, P. Kim, A. Majumdar, J. Heat Transfer. 125, 881 (2003)

    Article  Google Scholar 

  28. A. Henry, G. Chen, Phys. Rev. Lett. 101, 235502 (2008)

    Article  ADS  Google Scholar 

  29. X. Huang, G. Liu, X. Wang, Adv. Mater. 24, 1482 (2012)

    Article  MathSciNet  Google Scholar 

  30. K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, Y. Zhang, S.V. Dubonos, I.V. Grigorieva, A.A. Firsov, Science 306, 666 (2004)

    Article  ADS  Google Scholar 

  31. S. Ghosh, I. Calizo, D. Teweldebrhan, E.P. Pokatilov, D.L. Nika, A.A. Balandin, W. Bao, F. Miao, C.N. Lau, Appl. Phys. Lett. 92, 151911 (2008)

    Article  ADS  Google Scholar 

  32. A.A. Balandin, S. Ghosh, W. Bao, I. Calizo, D. Teweldebrhan, F. Miao, C.N. Lau, Nano Lett. 8, 902 (2008)

    Article  ADS  Google Scholar 

  33. S. Ghosh, W. Bao, D.L. Nika, S. Subrina, E.P. Pokatilov, C.N. Lau, A.A. Balandin, Nat. Mater. 9, 555 (2010)

    Article  ADS  Google Scholar 

  34. J.H. Seol et al., Science 328, 213 (2010)

    Article  ADS  Google Scholar 

  35. W. Cai, A.L. Moore, Y. Zhu, X. Li, S. Chen, L. Shi, R.S. Ruoff, Nano Lett. 10, 1645 (2010)

    Article  ADS  Google Scholar 

  36. S. Chen et al., ACS Nano 5, 321 (2010)

    Article  Google Scholar 

  37. J.U. Lee, D. Yoon, H. Kim, S.W. Lee, H. Cheong, Phys. Rev. B 83, 081419 (2011)

    Article  ADS  Google Scholar 

  38. C. Faugeras, B. Faugeras, M. Orlita, M. Potemski, R.R. Nair, A.K. Geim, ACS Nano 4, 1889 (2010)

    Article  Google Scholar 

  39. A.A. Balandin, Nat. Mater. 10, 569 (2011)

    Article  ADS  Google Scholar 

  40. Z. Wang, R. Xie, C.T. Bui, D. Liu, X. Ni, B. Li, J.T.L. Thong, Nano Lett. 11, 113 (2011)

    Article  ADS  Google Scholar 

  41. X. Xu et al. (2010), arXiv:1012.2937

  42. N. Mingo, D.A. Broido, Phys. Rev. Lett. 95, 096105 (2005)

    Article  ADS  Google Scholar 

  43. L. Lindsay, D.A. Broido, N. Mingo, Phys. Rev. B 82, 115427 (2010)

    Article  ADS  Google Scholar 

  44. X. Xu et al. (2012), unpublished

  45. M.T. Pettes, I. Jo, Z. Yao, L. Shi, Nano Lett. 11, 1195 (2011)

    Article  ADS  Google Scholar 

  46. D.L. Nika, S. Ghosh, E.P. Pokatilov, A.A. Balandin, Appl. Phys. Lett. 94, 203103 (2009)

    Article  ADS  Google Scholar 

  47. O. Narayan, S. Ramaswamy, Phys. Rev. Lett. 89, 200601 (2002)

    Article  ADS  Google Scholar 

  48. L. Yang, P. Grassberger, B. Hu, Phys. Rev. E 74, 062101 (2006)

    Article  ADS  Google Scholar 

  49. S. Iijima, Nature 354, 56 (1991)

    Article  ADS  Google Scholar 

  50. J. Yang, S. Waltermire, Y. Chen, A.A. Zinn, T.T. Xu, D. Li, Appl. Phys. Lett. 96, 023109 (2010)

    Article  ADS  Google Scholar 

  51. T. Yamamoto, S. Watanabe, K. Watanabe, Phys. Rev. Lett. 92, 075502 (2004)

    Article  ADS  Google Scholar 

  52. G. Zhang, B. Li, J. Chem. Phys. 123, 114714 (2005)

    Article  ADS  Google Scholar 

  53. S. Maruyama, Physica B 323, 193 (2002)

    Article  ADS  Google Scholar 

  54. B. Li, J. Wang, Phys. Rev. Lett. 91, 044301 (2003)

    Article  ADS  Google Scholar 

  55. Y. Cui, Q. Wei, H. Park, C.M. Lieber, Science 293, 1289 (2001)

    Article  ADS  Google Scholar 

  56. G.J. Zhang, G. Zhang, H.J. Chua, R.E. Chee, E.H. Wong, A. Agarwal, K.D. Buddharaju, N. Singh, Z. Gao, N. Balasubramanian, Nano Lett. 8, 1066 (2008)

    Article  ADS  Google Scholar 

  57. J. Xiang, W. Lu, Y. Hu, Y. Wu, H. Yan, C.M. Lieber, Nature 441, 489 (2006)

    Article  ADS  Google Scholar 

  58. A.I. Boukai, Y. Bunimovich, J. Tahir-Kheli, J.K. Yu, W.A. Goddard III, J.R. Heath, Nature 451, 168 (2008)

    Article  ADS  Google Scholar 

  59. G. Zhang, Q. Zhang, C.T. Bui, G.Q. Lo, B. Li, Appl. Phys. Lett. 94, 213108 (2009)

    Article  ADS  Google Scholar 

  60. G. Zhang, Q. Zhang, D. Kavitha, G.Q. Lo, Appl. Phys. Lett. 95, 243104 (2009)

    Article  ADS  Google Scholar 

  61. D. Donadio, G. Galli, Nano Lett. 10, 847 (2010)

    Article  ADS  Google Scholar 

  62. T. Markussen, A.P. Jauho, M. Brandbyge, Phys. Rev. Lett. 103, 055502 (2009)

    Article  ADS  Google Scholar 

  63. F. Sansoz, Nano Lett. 11, 5378 (2011)

    Article  ADS  Google Scholar 

  64. D. Donadio, G. Galli, Phys. Rev. Lett. 102, 195901 (2009)

    Article  ADS  Google Scholar 

  65. L. Shi, D. Yao, G. Zhang, B. Li, Appl. Phys. Lett. 95, 063102 (2009)

    Article  ADS  Google Scholar 

  66. D. Yao, G. Zhang, G.Q. Lo, B. Li, Appl. Phys. Lett. 94, 113113 (2009)

    Article  ADS  Google Scholar 

  67. Y. Chen, D. Li, J.R. Lukes, A. Majumdar, J. Heat Transfer. 127, 1129 (2005)

    Article  Google Scholar 

  68. J. Chen, G. Zhang, B. Li, J. Chem. Phys. 135, 204705 (2011)

    Article  ADS  Google Scholar 

  69. R. Yang, G. Chen, M.S. Dresselhaus, Nano Lett. 5, 1111 (2005)

    Article  ADS  Google Scholar 

  70. J. Chen, G. Zhang, B. Li, Nano Lett. 10, 3978 (2010)

    Article  ADS  Google Scholar 

  71. M. Hu, K.P. Giapis, J.V. Goicochea, X. Zhang, D. Poulikakos, Nano Lett. 11, 618 (2011)

    Article  ADS  Google Scholar 

  72. J. Chen, G. Zhang, B. Li, Nano Lett. 12, 2826 (2012)

    Article  Google Scholar 

  73. N. Yang, G. Zhang, B. Li, Nano Lett. 8, 276 (2008)

    Article  ADS  Google Scholar 

  74. J. Chen, G. Zhang, B. Li, Appl. Phys. Lett. 95, 073117 (2009)

    Article  ADS  Google Scholar 

  75. L. Shi, D. Yao, G. Zhang, B. Li, Appl. Phys. Lett. 96, 173108 (2010)

    Article  ADS  Google Scholar 

  76. J. Chen, G. Zhang, B. Li, J. Phys. Soc. Jpn 79, 074604 (2010)

    Article  ADS  Google Scholar 

  77. S. Denisov, J. Klafter, M. Urbakh, Phys. Rev. Lett. 91, 194301 (2003)

    Article  ADS  Google Scholar 

  78. Y.B. Zhang, Y.W. Tan, H.L. Stormer, P. Kim, Nature 438, 201 (2005)

    Article  ADS  Google Scholar 

  79. C. Berger et al., Science 312, 1191 (2006)

    Article  ADS  Google Scholar 

  80. X. Wang, H. Dai, Nature Chem. 2, 661 (2010)

    Article  ADS  Google Scholar 

  81. Y. Xu, X. Chen, B.L. Gu, W. Duan, Appl. Phys. Lett. 95, 233116 (2009)

    Article  ADS  Google Scholar 

  82. Y. Xu, X. Chen, J. Wang, B.L. Gu, W. Duan, Phys. Rev. B 81, 195425 (2010)

    Article  ADS  Google Scholar 

  83. W.J. Evans, L. Hu, P. Keblinski, Appl. Phys. Lett. 96, 203112 (2010)

    Article  ADS  Google Scholar 

  84. X. Ni, G. Zhang, B. Li, J. Phys.: Condens. Matter 23, 215301 (2011)

    Article  ADS  Google Scholar 

  85. S.K. Chien, Y.T. Yang, C.K. Chen, Appl. Phys. Lett. 98, 033107 (2011)

    Article  ADS  Google Scholar 

  86. Q.X. Pei, Z.D. Sha, Y.W. Zhang, Carbon 49, 4752 (2011)

    Article  Google Scholar 

  87. J. Hu, S. Schiffli, A. Vallabhaneni, X. Ruan, Y.P. Chen, Appl. Phys. Lett. 97, 133107 (2010)

    Article  ADS  Google Scholar 

  88. Z. Aksamija, I. Knezevic, Appl. Phys. Lett. 98, 141919 (2011)

    Article  ADS  Google Scholar 

  89. T. Ouyang, Y.P. Chen, K.K. Yang, J.X. Zhong, Europhys. Lett. 88, 28002 (2009)

    Article  ADS  Google Scholar 

  90. J.W. Jiang, B.S. Wang, J.S. Wang, Appl. Phys. Lett. 98, 113114 (2011)

    Article  ADS  Google Scholar 

  91. Z.X. Xie, K.Q. Chen, W. Duan, J. Phys.: Condens. Matter 23, 315302 (2011)

    Article  ADS  Google Scholar 

  92. X. Ni, G. Liang, J.S. Wang, B. Li, Appl. Phys. Lett. 95, 192114 (2009)

    Article  ADS  Google Scholar 

  93. X. Zhai, G. Jin, Europhys. Lett. 96, 16002 (2011)

    Article  ADS  Google Scholar 

  94. N. Wei, L. Xu, H.Q. Wang, J.C. Zheng, Nat. Nanotechnol. 22, 105705 (2011)

    ADS  Google Scholar 

  95. Z.X. Guo, D. Zhang, X.G. Gong, Phys. Rev. B 84, 075470 (2011)

    Article  ADS  Google Scholar 

  96. Z.Y. Ong, E. Pop, Phys. Rev. B 84, 075471 (2011)

    Article  ADS  Google Scholar 

  97. Z.X. Guo, J.W. Ding, X.G. Gong, Phys. Rev. B 85, 235429 (2012)

    Article  ADS  Google Scholar 

  98. G. Zhang, H. Zhang, Nanoscale 3, 4604 (2011)

    Article  ADS  Google Scholar 

  99. N. Yang, X. Ni, J.W. Jiang, B. Li, Appl. Phys. Lett. 100, 093107 (2012)

    Article  ADS  Google Scholar 

  100. W.R. Zhong, M.P. Zhang, B.Q. Ai, D.Q. Zheng, Appl. Phys. Lett. 98, 113107 (2011)

    Article  ADS  Google Scholar 

  101. H. Cao, Z. Guo, H. Xiang, X. Gong, Phys. Lett. A 376, 525 (2012)

    Article  ADS  Google Scholar 

  102. N. Yang, G. Zhang, B. Li, Appl. Phys. Lett. 93, 243111 (2008)

    Article  ADS  Google Scholar 

  103. N. Yang, G. Zhang, B. Li, Appl. Phys. Lett. 95, 033107 (2009)

    Article  ADS  Google Scholar 

  104. J. Hu, X. Ruan, Y.P. Chen, Nano Lett. 9, 2730 (2009)

    Article  ADS  Google Scholar 

  105. H. Tian, D. Xie, Y. Yang, T.L. Ren, G. Zhang, Y.F. Wang, C.J. Zhou, P.G. Peng, L.G. Wang, L.T. Liu, NPG Sci. Rep. 2, 523 (2012)

    Google Scholar 

  106. W.R. Zhong, W.H. Huang, X.R. Deng, B.Q. Ai, Appl. Phys. Lett. 99, 193104 (2011)

    Article  ADS  Google Scholar 

  107. T. Ouyang, Y. Chen, Y. Xie, G.M. Stocks, J. Zhong, Appl. Phys. Lett. 99, 233101 (2011)

    Article  ADS  Google Scholar 

  108. Z. Guo, D. Zhang, X.G. Gong, Appl. Phys. Lett. 95, 163103 (2009)

    Article  ADS  Google Scholar 

  109. A. Eletskii, I. Inskandarova, A. Knizhnik, D. Krasikov, Molecular Dynamic Simulation of the Thermal Conductivity of Graphene and Graphene Oxide, in Graphene 2012 International Conference (Brussels, 2012)

  110. K.R. Allen, J. Ford, Phys. Rev. 176, 1046 (1968)

    Article  ADS  Google Scholar 

  111. H. Matsuda, K. Ishii, Prog. Theor. Phys. Suppl. 45, 56 (1970)

    Article  ADS  MathSciNet  Google Scholar 

  112. A. Casher, J.L. Lebowitz, J. Math. Phys. 12, 1701 (1971)

    Article  ADS  Google Scholar 

  113. R.J. Rubin, W.L. Greer, J. Math. Phys. 12, 1686 (1971)

    Article  ADS  Google Scholar 

  114. T. Verheggen, Commun. Math. Phys. 68, 69 (1979)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  115. A. Dhar, Phys. Rev. Lett. 86, 5882 (2001)

    Article  ADS  Google Scholar 

  116. D. Roy, A. Dhar, Phys. Rev. E 78, 051112 (2008)

    Article  ADS  Google Scholar 

  117. A. Dhar, J.L. Lebowitz, Phys. Rev. Lett. 100, 134301 (2008)

    Article  ADS  Google Scholar 

  118. S. John, H. Sompolinsky, M.J. Stephen, Phys. Rev. B 27, 5592 (1983)

    Article  ADS  Google Scholar 

  119. L.W. Lee, A. Dhar, Phys. Rev. Lett. 95, 094302 (2005)

    Article  ADS  Google Scholar 

  120. L. Yang, Phys. Rev. Lett. 88, 094301 (2002)

    Article  ADS  Google Scholar 

  121. S. Lepri, Phys. Rev. E 58, 7165 (1998)

    Article  ADS  Google Scholar 

  122. A. Pereverzev, Phys. Rev. E 68, 056124 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  123. M.S. Green, J. Chem. Phys. 22, 398 (1954)

    Article  ADS  MathSciNet  Google Scholar 

  124. R. Kubo, M. Yokota, S. Nakajima, J. Phys. Soc. Jpn 12, 1203 (1957)

    Article  ADS  MathSciNet  Google Scholar 

  125. L. Delfini, S. Lepri, R. Livi, A. Politi, J. Stat. Mech. 2007, P02007 (2007)

    Article  Google Scholar 

  126. J.S. Wang, B. Li, Phys. Rev. Lett. 92, 074302 (2004)

    Article  ADS  Google Scholar 

  127. J.S. Wang, B. Li, Phys. Rev. E 70, 021204 (2004)

    Article  ADS  Google Scholar 

  128. T. Mai, O. Narayan, Phys. Rev. E 73, 061202 (2006)

    Article  ADS  Google Scholar 

  129. P.I. Hurtado, Phys. Rev. Lett. 96, 010601 (2006)

    Article  ADS  Google Scholar 

  130. J. Lukkarinen, H. Spohn, Commum. Pure Appl. Math. 61, 1753 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  131. K. Aoki, J. Lukkarinen, H. Spohn, J. Stat. Phys. 124, 1105 (2006)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  132. H. Kaburaki, M. Machida, Phys. Lett. A 181, 85 (1993)

    Article  ADS  Google Scholar 

  133. T. Mai, A. Dhar, O. Narayan, Phys. Rev. Lett. 98, 184301 (2007)

    Article  ADS  Google Scholar 

  134. L. Wang, T. Wang, Europhys. Lett. 93, 54002 (2011)

    Article  ADS  Google Scholar 

  135. D. Xiong, J. Wang, Y. Zhang, H. Zhao, Phys. Rev. E 85, 020102 (2012)

    Article  ADS  Google Scholar 

  136. F. Muller-Plathe, J. Chem. Phys. 106, 6082 (1997)

    Article  ADS  Google Scholar 

  137. B. Hu, B. Li, H. Zhao, Phys. Rev. E 57, 2992 (1998)

    Article  ADS  Google Scholar 

  138. K. Aoki, D. Kusnezov, Phys. Lett. A 265, 250 (2000)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  139. B. Hu, B. Li, H. Zhao, Phys. Rev. E 61, 3828 (2000)

    Article  ADS  Google Scholar 

  140. D.N. Payton, W.M. Visscher, Phys. Rev. 156, 1032 (1967)

    Article  ADS  Google Scholar 

  141. B. Li, H. Zhao, B. Hu, Phys. Rev. Lett. 86, 63 (2001)

    Article  ADS  Google Scholar 

  142. A. Dhar, K. Saito, Phys. Rev. E 78, 061136 (2008)

    Article  ADS  Google Scholar 

  143. A. Lippi, R. Livi, J. Stat. Phys. 100, 1147 (2000)

    Article  MATH  Google Scholar 

  144. P. Grassberger, L. Yang (2002), arXiv:cond-mat/0204247v1

  145. H. Shiba, N. Ito, J. Phys. Soc. Jpn 77, 054006 (2008)

    Article  ADS  Google Scholar 

  146. D. Alonso, R. Artuso, G. Casati, I. Guarneri, Phys. Rev. Lett. 82, 1859 (1999)

    Article  ADS  Google Scholar 

  147. B. Li, L. Wang, B. Hu, Phys. Rev. Lett. 88, 223901 (2002)

    Article  ADS  Google Scholar 

  148. D. Alonso, A. Ruiz, I. de Vega, Phys. Rev. E 66, 066131 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  149. B. Li, G. Casati, J. Wang, Phys. Rev. E 67, 021204 (2003)

    Article  ADS  Google Scholar 

  150. A. Blumen, G. Zumofen, J. Klafter, Phys. Rev. A 40, 3964 (1989)

    Article  ADS  Google Scholar 

  151. A. Dhar, D. Dhar, Phys. Rev. Lett. 82, 480 (1999)

    Article  ADS  Google Scholar 

  152. P. Cipriani, S. Denisov, A. Politi, Phys. Rev. Lett. 94, 244301 (2005)

    Article  ADS  Google Scholar 

  153. J. Klafter, G. Zumofen, Physica A 196, 102 (1993)

    Article  ADS  Google Scholar 

  154. H. Zhao, Phys. Rev. Lett. 96, 140602 (2006)

    Article  ADS  Google Scholar 

  155. V. Zaburdaev, S. Denisov, P. Hänggi, Phys. Rev. Lett. 106, 180601 (2011)

    Article  ADS  Google Scholar 

  156. S. Liu, N. Li, J. Ren, B. Li (2012), arXiv:1103.2835

  157. A. Henry, G. Chen, Phys. Rev. B 79, 144305 (2009)

    Article  ADS  Google Scholar 

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Authors and Affiliations

  1. NUS Graduate School for Integrative Sciences and Engineering, National University of Singapore, 117456, Singapore, Republic of Singapore

    S. Liu & B. W. Li

  2. Centre for Computational Science and Engineering, Graphene Research Centre, Department of Physics, National University of Singapore, 117546, Singapore, Republic of Singapore

    S. Liu, X. F. Xu, R. G. Xie & B. W. Li

  3. Key Laboratory for the Physics and Chemistry of Nanodevices and Department of Electronics, Peking University, Beijing, 100871, P.R. China

    G. Zhang

  4. NUS-Tongji Center for Phononics and Thermal Energy Science and Department of Physics, Tongji University, Shanghai, 200092, P.R. China

    B. W. Li

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Liu, S., Xu, X.F., Xie, R.G. et al. Anomalous heat conduction and anomalous diffusion in low dimensional nanoscale systems. Eur. Phys. J. B 85, 337 (2012). https://doi.org/10.1140/epjb/e2012-30383-8

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