No Access Submitted: 20 March 2020 Accepted: 12 May 2020 Published Online: 27 May 2020
Journal of Applied Physics 127, 204301 (2020); https://doi.org/10.1063/5.0008271
This paper reveals the existence of a critical separation distance ( d c) beyond which the elastic interactions between a pair of monovacancies in graphene or hexagonal boron nitride become inconsequential for the strength and toughness of the defective lattice. This distance is independent of the chirality of the lattice. For any inter-defect distance higher than d c, the lattice behaves mechanically as if there is a single defect. For a distance less than d c, the defect–defect elastic interactions produce distinctive mechanical behavior depending on the orientation ( θ) of the defect pair relative to the loading direction. Both strength and toughness of the lattice containing a pair of “interacting monovacancies (iMVs)” are either higher or smaller than that of the lattice containing a pair of “non-interacting monovacancies (nMVs),” suggesting the existence of a critical orientation angle θ c. For θ < θ c, the smaller the distance between the iMVs, the higher the toughness and strength compared to the lattice containing nMVs, whereas, for θ θ c, the smaller the separation distance between the iMVs, the smaller the toughness and strength compared to the lattice containing nMVs. The transitional behavior has a negligible dependence on the chirality of the lattice, which indicates that the crystallographic anisotropy has a much weaker influence on toughness and strength compared to the anisotropy induced by the orientation angle itself. These observations underline an important point that the elastic fields emanating from vacancy defects are highly localized and fully contained within a small region of around 1.5 nm radius.
  1. 1. N. Y. Lin, M. Bierbaum, P. Schall, J. P. Sethna, and I. Cohen, Nat. Mater. 15, 1172 (2016). https://doi.org/10.1038/nmat4715, Google ScholarCrossref
  2. 2. Z. Dai, L. Liu, and Z. Zhang, Adv. Mater. 31, 1805417 (2019). https://doi.org/10.1002/adma.201805417, Google ScholarCrossref
  3. 3. J. Dong, L. Zhang, and F. Ding, Adv. Mater. 31, 1801583 (2019). https://doi.org/10.1002/adma.201801583, Google ScholarCrossref
  4. 4. F. Banhart, J. Kotakoski, and A. V. Krasheninnikov, ACS Nano 5, 26 (2010). https://doi.org/10.1021/nn102598m, Google ScholarCrossref
  5. 5. M. E. Turiansky, A. Alkauskas, L. C. Bassett, and C. G. Van de Walle, Phys. Rev. Lett. 123, 127401 (2019). https://doi.org/10.1103/PhysRevLett.123.127401, Google ScholarCrossref
  6. 6. J. D. Caldwell, I. Aharonovich, G. Cassabois, J. H. Edgar, B. Gil, and D. Basov, Nat. Rev. Mater. 4, 552 (2019). https://doi.org/10.1038/s41578-019-0124-1, Google ScholarCrossref, ISI
  7. 7. M. Toth and I. Aharonovich, Annu. Rev. Phys. Chem. 70, 123–142 (2019). https://doi.org/10.1146/annurev-physchem-042018-052628, Google ScholarCrossref
  8. 8. X. Liu and M. C. Hersam, Nat. Rev. Mater. 4, 669 (2019). https://doi.org/10.1038/s41578-019-0136-x, Google ScholarCrossref
  9. 9. N. R. Jungwirth, B. Calderon, Y. Ji, M. G. Spencer, M. E. Flatté, and G. D. Fuchs, Nano Lett. 16, 6052 (2016). https://doi.org/10.1021/acs.nanolett.6b01987, Google ScholarCrossref
  10. 10. J. Zhang, Y. Yu, P. Wang, C. Luo, X. Wu, Z. Sun, J. Wang, W. D. Hu, and G. Shen, InfoMat 1, 85 (2019). https://doi.org/10.1002/inf2.12002, Google ScholarCrossref
  11. 11. A. L. Gibb, N. Alem, J.-H. Chen, K. J. Erickson, J. Ciston, A. Gautam, M. Linck, and A. Zettl, J. Am. Chem. Soc. 135, 6758 (2013). https://doi.org/10.1021/ja400637n, Google ScholarCrossref
  12. 12. Z. Lin, B. R. Carvalho, E. Kahn, R. Lv, R. Rao, H. Terrones, M. A. Pimenta, and M. Terrones, 2D Mater. 3, 022002 (2016). https://doi.org/10.1088/2053-1583/3/2/022002, Google ScholarCrossref
  13. 13. W. Zhou, X. Zou, S. Najmaei, Z. Liu, Y. Shi, J. Kong, J. Lou, P. M. Ajayan, B. I. Yakobson, and J.-C. Idrobo, Nano Lett. 13, 2615 (2013). https://doi.org/10.1021/nl4007479, Google ScholarCrossref, ISI
  14. 14. J. C. Meyer, C. Kisielowski, R. Erni, M. D. Rossell, M. Crommie, and A. Zettl, Nano Lett. 8, 3582 (2008). https://doi.org/10.1021/nl801386m, Google ScholarCrossref
  15. 15. S. Wang, A. Robertson, and J. H. Warner, Chem. Soc. Rev. 47, 6764 (2018). https://doi.org/10.1039/C8CS00236C, Google ScholarCrossref
  16. 16. M. M. Ugeda, I. Brihuega, F. Hiebel, P. Mallet, J.-Y. Veuillen, J. M. Gómez-Rodríguez, and F. Ynduráin, Phys. Rev. B 85, 121402 (2012). https://doi.org/10.1103/PhysRevB.85.121402, Google ScholarCrossref
  17. 17. S. Wang, Z. Qin, G. S. Jung, F. J. Martin-Martinez, K. Zhang, M. J. Buehler, and J. H. Warner, ACS Nano 10, 9831 (2016). https://doi.org/10.1021/acsnano.6b05435, Google ScholarCrossref
  18. 18. B. Liu and K. Zhou, Prog. Mater. Sci. 100, 99 (2019). https://doi.org/10.1016/j.pmatsci.2018.09.004, Google ScholarCrossref
  19. 19. P. Y. Huang, C. S. Ruiz-Vargas, A. M. Van Der Zande, W. S. Whitney, M. P. Levendorf, J. W. Kevek, S. Garg, J. S. Alden, C. J. Hustedt, Y. Zhu et al., Nature 469, 389 (2011). https://doi.org/10.1038/nature09718, Google ScholarCrossref, ISI
  20. 20. Y. L. Huang, Z. Ding, W. Zhang, Y.-H. Chang, Y. Shi, L.-J. Li, Z. Song, Y. J. Zheng, D. Chi, S. Y. Quek et al., Nano Lett. 16, 3682 (2016). https://doi.org/10.1021/acs.nanolett.6b00888, Google ScholarCrossref
  21. 21. Z. Zhang, X. Zou, V. H. Crespi, and B. I. Yakobson, ACS Nano 7, 10475 (2013). https://doi.org/10.1021/nn4052887, Google ScholarCrossref
  22. 22. T. H. Ly, D. J. Perello, J. Zhao, Q. Deng, H. Kim, G. H. Han, S. H. Chae, H. Y. Jeong, and Y. H. Lee, Nat. Commun. 7, 10426 (2016). https://doi.org/10.1038/ncomms10426, Google ScholarCrossref
  23. 23. M. M. Ugeda, I. Brihuega, F. Guinea, and J. M. Gomez-Rodriguez, Phys. Rev. Lett. 104, 096804 (2010). https://doi.org/10.1103/PhysRevLett.104.096804, Google ScholarCrossref, ISI
  24. 24. S. Mishra, D. Beyer, K. Eimre, S. Kezilebieke, R. Berger, O. Groning, C. A. Pignedoli, K. Mullen, P. Liljeroth, P. Ruffieux et al., Nat. Nanotechnol. 15(1), 1 (2019). Google Scholar
  25. 25. A. Bommer and C. Becher, Nanophotonics 8, 2041–2048 (2019). https://doi.org/10.1515/nanoph-2019-0123, Google ScholarCrossref
  26. 26. P. A. Denis and F. Iribarne, J. Phys. Chem. C 117, 19048 (2013). https://doi.org/10.1021/jp4061945, Google ScholarCrossref
  27. 27. Z. Zhang, F. Li, G. Malpuech, Y. Zhang, O. Bleu, S. Koniakhin, C. Li, Y. Zhang, M. Xiao, and D. Solnyshkov, Phys. Rev. Lett. 122, 233905 (2019). https://doi.org/10.1103/PhysRevLett.122.233905, Google ScholarCrossref
  28. 28. S. M. Hus and A.-P. Li, Prog. Surf. Sci. 92, 176 (2017). https://doi.org/10.1016/j.progsurf.2017.07.001, Google ScholarCrossref, ISI
  29. 29. L. Qiu, N. Zhu, Y. Feng, E. E. Michaelides, G. Żyła, D. Jing, X. Zhang, P. M. Norris, C. N. Markides, and O. Mahian, Phys. Rep. 843, 1–81 (2019). https://doi.org/10.1016/j.physrep.2019.12.001, Google ScholarCrossref
  30. 30. D. L. Nika and A. A. Balandin, Rep. Progress Phys. 80, 036502 (2017). https://doi.org/10.1088/1361-6633/80/3/036502, Google ScholarCrossref
  31. 31. P. Venezuela, M. Lazzeri, and F. Mauri, Phys. Rev. B 84, 035433 (2011). https://doi.org/10.1103/PhysRevB.84.035433, Google ScholarCrossref, ISI
  32. 32. D. L. Nika and A. A. Balandin, J. Phys. Condens. Matter 24, 233203 (2012). https://doi.org/10.1088/0953-8984/24/23/233203, Google ScholarCrossref, ISI
  33. 33. Z.-X. Xie, K.-Q. Chen, and W. Duan, J. Phys. Condens. Matter 23, 315302 (2011). https://doi.org/10.1088/0953-8984/23/31/315302, Google ScholarCrossref, ISI
  34. 34. T. Feng, X. Ruan, Z. Ye, and B. Cao, Phys. Rev. B 91, 224301 (2015). https://doi.org/10.1103/PhysRevB.91.224301, Google ScholarCrossref
  35. 35. J. F. Kong, L. Levitov, D. Halbertal, and E. Zeldov, Phys. Rev. B 97, 245416 (2018). https://doi.org/10.1103/PhysRevB.97.245416, Google ScholarCrossref
  36. 36. L. Wang, M. Yin, B. Zhong, J. Jaroszynski, G. Mbamalu, and T. Datta, J. Appl. Phys. 126, 084305 (2019). https://doi.org/10.1063/1.5100813, Google ScholarScitation, ISI
  37. 37. L. Li and F. Peeters, Phys. Rev. B 97, 075414 (2018). https://doi.org/10.1103/PhysRevB.97.075414, Google ScholarCrossref
  38. 38. K. Balasubramanian, T. Biswas, P. Ghosh, S. Suran, A. Mishra, R. Mishra, R. Sachan, M. Jain, M. Varma, R. Pratap et al., Nat. Commun. 10, 1090 (2019). https://doi.org/10.1038/s41467-019-09000-8, Google ScholarCrossref
  39. 39. D. Akinwande, C. J. Brennan, J. S. Bunch, P. Egberts, J. R. Felts, H. Gao, R. Huang, J.-S. Kim, T. Li, Y. Li et al., Extreme Mech. Lett. 13, 42 (2017). https://doi.org/10.1016/j.eml.2017.01.008, Google ScholarCrossref, ISI
  40. 40. W. Zhang, M. Kim, R. Cheng, W.-C. Lu, H.-X. Zhang, K.-M. Ho, and C.-Z. Wang, J. Phys. Chem. C 152, 144304 (2020). https://doi.org/10.1063/5.0002505, Google ScholarScitation
  41. 41. H. Terrones, R. Lv, M. Terrones, and M. S. Dresselhaus, Rep. Progress Phys. 75, 062501 (2012). https://doi.org/10.1088/0034-4885/75/6/062501, Google ScholarCrossref
  42. 42. H. Ding, Z. Zhen, H. Imtiaz, W. Guo, H. Zhu, and B. Liu, Extreme Mech. Lett. 32, 100507 (2019). https://doi.org/10.1016/j.eml.2019.100507, Google ScholarCrossref
  43. 43. T. Ahmed, Z. Zhang, C. McDermitt, and Z. M. Hossain, J. Appl. Phys. 124, 185108 (2018). https://doi.org/10.1063/1.5052500, Google ScholarScitation, ISI
  44. 44. T. Ahmed, A. Procak, T. Hao, and Z. M. Hossain, Phys. Rev. B 99, 134105 (2019). https://doi.org/10.1103/PhysRevB.99.134105, Google ScholarCrossref
  45. 45. H. I. Rasool, C. Ophus, and A. Zettl, Adv. Mater. 27, 5771 (2015). https://doi.org/10.1002/adma.201500231, Google ScholarCrossref
  46. 46. A. C. Ferrari and D. M. Basko, Nat. Nanotechnol. 8, 235 (2013). https://doi.org/10.1038/nnano.2013.46, Google ScholarCrossref, ISI
  47. 47. L. G. Cançado, M. G. Da Silva, E. H. M. Ferreira, F. Hof, K. Kampioti, K. Huang, A. Pénicaud, C. A. Achete, R. B. Capaz, and A. Jorio, 2D Mater. 4, 025039 (2017). https://doi.org/10.1088/2053-1583/aa5e77, Google ScholarCrossref
  48. 48. L. G. Cançado, A. Jorio, E. M. Ferreira, F. Stavale, C. A. Achete, R. B. Capaz, M. V. D. O Moutinho, A. Lombardo, T. Kulmala, and A. C. Ferrari, Nano Lett. 11, 3190 (2011). https://doi.org/10.1021/nl201432g, Google ScholarCrossref, ISI
  49. 49. M. M. Lucchese, F. Stavale, E. M. Ferreira, C. Vilani, M. V. d. O. Moutinho, R. B. Capaz, C. A. Achete, and A. Jorio, Carbon 48, 1592 (2010). https://doi.org/10.1016/j.carbon.2009.12.057, Google ScholarCrossref, ISI
  50. 50. R. Beams, L. G. Cançado, and L. Novotny, J. Phys. Condens. Matter 27, 083002 (2015). https://doi.org/10.1088/0953-8984/27/8/083002, Google ScholarCrossref, ISI
  51. 51. A. Zandiatashbar, G.-H. Lee, S. J. An, S. Lee, N. Mathew, M. Terrones, T. Hayashi, C. R. Picu, J. Hone, and N. Koratkar, Nat. Commun. 5, 1 (2014). https://doi.org/10.1038/ncomms4186, Google ScholarCrossref
  52. 52. G. Rajasekaran, P. Narayanan, and A. Parashar, Critical Rev. Solid State Mater. Sci. 41, 47 (2016). https://doi.org/10.1080/10408436.2015.1068160, Google ScholarCrossref
  53. 53. C. Sevik, H. Sevinçli, G. Cuniberti, and T. Cagın, Nano Lett. 11, 4971 (2011). https://doi.org/10.1021/nl2029333, Google ScholarCrossref, ISI
  54. 54. P. Gehring, B. F. Gao, M. Burghard, and K. Kern, Nano Lett. 12, 5137 (2012). https://doi.org/10.1021/nl3019802, Google ScholarCrossref, ISI
  55. 55. I. M. Felix and L. F. C. Pereira, Sci. Rep. 8, 2737 (2018). https://doi.org/10.1038/s41598-018-20997-8, Google ScholarCrossref
  56. 56. F. Zhang, Y. Wang, C. Erb, K. Wang, P. Moradifar, V. H. Crespi, and N. Alem, Phys. Rev. B 99, 155430 (2019). https://doi.org/10.1103/PhysRevB.99.155430, Google ScholarCrossref
  57. 57. D. Edelberg, D. Rhodes, A. Kerelsky, B. Kim, J. Wang, A. Zangiabadi, C. Kim, A. Antony, J. Ardelean, M. Scully et al., Nano Lett. 19, 4371-–4379 (2019). https://doi.org/10.1021/acs.nanolett.9b00985, Google ScholarCrossref
  58. 58. U. Chandni, K. Watanabe, T. Taniguchi, and J. Eisenstein, Nano Lett. 15, 7329 (2015). https://doi.org/10.1021/acs.nanolett.5b02625, Google ScholarCrossref, ISI
  59. 59. G. S. Jung, J. Yeo, Z. Tian, Z. Qin, and M. J. Buehler, Nanoscale 9, 13477 (2017). https://doi.org/10.1039/C7NR04455K, Google ScholarCrossref
  60. 60. J. Yang, X. Shen, C. Wang, Y. Chai, and H. Yao, Extreme Mech. Lett. 29, 100473 (2019). https://doi.org/10.1016/j.eml.2019.100473, Google ScholarCrossref
  61. 61. Z. Dai, G. Wang, Z. Zheng, Y. Wang, S. Zhang, X. Qi, P. Tan, L. Liu, Z. Xu, Q. Li et al., Carbon 147, 594 (2019). https://doi.org/10.1016/j.carbon.2019.03.014, Google ScholarCrossref
  62. 62. G. López-Polín, C. Gómez-Navarro, V. Parente, F. Guinea, M. I. Katsnelson, F. Perez-Murano, and J. Gómez-Herrero, Nat. Phys. 11, 26 (2015). https://doi.org/10.1038/nphys3183, Google ScholarCrossref
  63. 63. F. H. Stillinger and T. A. Weber, Phys. Rev. B 31, 5262 (1985). https://doi.org/10.1103/PhysRevB.31.5262, Google ScholarCrossref, ISI
  64. 64. M. Z. Hossain, T. Ahmed, B. Silverman, M. S. Khawaja, J. Calderon, A. Rutten, and S. Tse, J. Mech. Phys. Solids 110, 118 (2018). https://doi.org/10.1016/j.jmps.2017.09.012, Google ScholarCrossref
  65. 65. S. Plimpton, P. Crozier, and A. Thompson, Sandia Natl. Lab. 18, 43 (2007). Google Scholar
  66. 66. A. Stukowski, Model. Simul. Mater. Sci. Eng. 18, 015012 (2009). https://doi.org/10.1088/0965-0393/18/1/015012, Google ScholarCrossref, ISI
  67. 67. M. Hossain, C.-J. Hsueh, B. Bourdin, and K. Bhattacharya, J. Mech. Phys. Solids 71, 15 (2014). https://doi.org/10.1016/j.jmps.2014.06.002, Google ScholarCrossref
  68. 68. M. A. Meyers, J. McKittrick, and P.-Y. Chen, Science 339, 773 (2013). https://doi.org/10.1126/science.1220854, Google ScholarCrossref
  69. 69. S. Keten, Z. Xu, B. Ihle, and M. J. Buehler, Nat. Mater. 9, 359 (2010). https://doi.org/10.1038/nmat2704, Google ScholarCrossref
  70. 70. T. Giesa, M. Arslan, N. M. Pugno, and M. J. Buehler, Nano Lett. 11, 5038 (2011). https://doi.org/10.1021/nl203108t, Google ScholarCrossref
  71. 71. F. Barthelat and R. Rabiei, J. Mech. Phys. Solids 59, 829 (2011). https://doi.org/10.1016/j.jmps.2011.01.001, Google ScholarCrossref
  72. 72. G. Marc and W. McMillan, Adv. Chem. Phys. 58, 209 (1985). Google ScholarISI
  73. 73. A. K. Subramaniyan and C. Sun, Int. J. Solids Struct. 45, 4340 (2008). https://doi.org/10.1016/j.ijsolstr.2008.03.016, Google ScholarCrossref, ISI
  74. 74. S. Timoshenko, J. Goodier Inc., New York, 1951. Google Scholar
  75. 75. A. W. Robertson, B. Montanari, K. He, C. S. Allen, Y. A. Wu, N. M. Harrison, A. I. Kirkland, and J. H. Warner, ACS Nano 7, 4495 (2013). https://doi.org/10.1021/nn401113r, Google ScholarCrossref
  76. 76. A. V. Krasheninnikov and R. Nieminen, Theor. Chem. Acc. 129, 625 (2011). https://doi.org/10.1007/s00214-011-0910-3, Google ScholarCrossref, ISI
  77. 77. T. Trevethan, C. D. Latham, M. I. Heggie, P. R. Briddon, and M. J. Rayson, Nanoscale 6, 2978 (2014). https://doi.org/10.1039/C3NR06222H, Google ScholarCrossref
  78. 78. J. Kotakoski, A. Krasheninnikov, and K. Nordlund, Phys. Rev. B 74, 245420 (2006). https://doi.org/10.1103/PhysRevB.74.245420, Google ScholarCrossref
  79. 79. R. Dettori, E. Cadelano, and L. Colombo, J. Phys. Condens. Matter 24, 104020 (2012). https://doi.org/10.1088/0953-8984/24/10/104020, Google ScholarCrossref
  80. 80. J. T. Robinson, M. K. Zalalutdinov, C. D. Cress, J. C. Culbertson, A. L. Friedman, A. Merrill, and B. J. Landi, ACS Nano 11, 4745 (2017). https://doi.org/10.1021/acsnano.7b00923, Google ScholarCrossref
  81. 81. J. Los, J. Kroes, K. Albe, R. Gordillo, M. Katsnelson, and A. Fasolino, Phys. Rev. B 96, 184108 (2017). https://doi.org/10.1103/PhysRevB.96.184108, Google ScholarCrossref
  82. 82. J. Tersoff, Phys. Rev. B 38, 9902 (1988). https://doi.org/10.1103/PhysRevB.38.9902, Google ScholarCrossref, ISI
  83. 83. F. Banhart, J. Kotakoski, and A. V. Krasheninnikov, ACS. Nano 5, 26 (2011). https://doi.org/10.1021/nn102598m, Google ScholarCrossref, ISI
  1. © 2020 Author(s). Published under license by AIP Publishing.