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In the frame of the Harrison bonded-orbital method, the variations of the force constant, the Young modulus, the torsional modulus and the phonon frequency with temperature are given through the relevant theory or method of the solid state physics with considering the non-harmonic effect and the short-range interaction of atoms. Results show that the force constant, the Young modulus, the torsional modulus, the phonon frequency and the Poissons coefficient all vary with temperature. The results show that the first three quantities increase with temperature but not very much; the phonon frequency increases with temperature rapidly; the Poissons coefficient decreases fast with the increase of temperature. There are transverse vibrations along the direction perpendicular to the bond-length direction and the longitudinal vibrations along the bond-length direction, in which the longitudinal vibrations are dominant. The nonharmonic effect of the longitudinal vibration is much larger than that of the transverse vibration. The first and the second non-harmonic coefficient of the transverse vibration are both much less than those of the longitudinal vibration, where 0/0 8.477 and 2/2 156. The above five physical quantities are constant at different temperatures if the first and second nonhamonic effects are omitted, which does not conform to the experimental results. After the first and second nonhamonic effects are considered, they all increase with temperature and results are in good agreement with experimental data. The anharmonic effect increases with temperature.
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Keywords:
- graphene /
- non-harmonic effect /
- Young modulus /
- phonon frequency
[1] Novoselov K S, Ceim A K, Morozov S V, Jiang D, Zhang Y, Dubonos S, Grigorieva I V 2004 Science 306 666
[2] Katsnelson M I 2007 Materials Today 10 20
[3] Bolotin K I, Sikes K J, Jiang Z, Klima M, Eudenberg G, Hone J, Stormer H L 2008 Sol. Sta. Com. 146 351
[4] Wang L, Feng W, Yang L Q, Zhang J H 2014 Acta Phys. Sin. 63 176801 (in Chinese) [王浪, 冯伟, 杨连乔, 张建华 2014 63 176801]
[5] Yang Y L, Lu Y 2014 Chin. Phys. B 23 106501
[6] Tian W, Yuan P F, Yu Z L, Tao B B, Hou S Y, Ye C, Zhang Z H 2015 Acta Phys. Sin. 64 046102 (in Chinese) [田文, 袁鹏飞, 禹卓良, 陶斌斌, 侯森耀, 叶聪, 张振华 2015 64 046102]
[7] Zhou P, He D W 2016 Chin. Phys. B 25 017302
[8] Chang X 2014 Acta Phys. Sin. 63 086102 (in Chinese) [常 旭 2014 63 086102]
[9] Mi C G, Liu G P, Wang J J, Guo X L, Wu S X, Yu J 2014 Acta Phys. Chem. Sin. 30 1230 (in Chinese) [米传国, 刘国平, 王家佳, 郭新立, 吴三械, 于 金 2014 物理化学学报 30 1230]
[10] Davydov S Y, Subinova G Y 2011 Phys. Stat. Sol. 53 608 (in Russian)
[11] Davydov S Y, Subinova G Y 2015 Phys. Stat. Sol. 57 1017 (in Russian)
[12] Mounet N, Marzari N 2005 Phys. Rev. B 71 205214
[13] Zakharchenko K V, Katsnelson M I, Fasolino A 2009 Phys. Rev. Lett. 102 046808
[14] Jiang J W, Wang J S, Li B 2009 Phys. Rev. B 80 205429
[15] Pozzo M, Alfe D, Lacovig P, Hofmann P, Lizzit S, Baraldi A 2011 Phys. Rev. Lett. 106 135501
[16] Bao W, Miao F, Chen Z, Zhang H, Jang W, Dames C, Lau N 2009 Nat. Nanotechol. 4 562
[17] Davydov S Y 2011 Tech. Phys. Lett. 37 42 (in Russian)
[18] Davydov S Y 2012 Phys. Solid State 54 875
[19] Davydov S Y 2013 Phys. Stat. Sol. 55 813 (in Russian)
[20] Zheng R L, Hu X Q 1994 College Physics 13 15 (in Chinese) [郑瑞伦, 胡先权 1994 大学物理 13 15]
[21] Davydov S Yu, Bosledlik O W 2015 Phys. Stat. Sol. 57 819 (in Russian)
[22] Davidov S Yu. 2009 Phys. Stat. Sol. 51 2041 (in Russian)
[23] Solin S A, Ramdas A K 1970 Phys. Rev. B 1 1687
[24] Zheng R L, Hu X Q 1996 Solid Theory and Application (Chongqing: southwest normal university press) pp267-271 (in Chinese) [郑瑞伦, 胡先权 1996 固体理论及其应用 (重庆: 西南师范大学出版社) 第 267-271 页]
[25] Blaksly O L, Proctor D G, Seldin E J, Spence G B, Weng T 1970 J. Appl. Phys. 41 3373
[26] Kudin K N, Scuseria G E, Yakobson B I 2001 Phys. Rev. B 64 235406
[27] Lu Q, Arroyo M, Huang R 2009 J. Appl. Phys. 42 102002
[28] Cadelano E C, Palla P L, Giordano S, Colombo L 2009 Phys. Rev. Lett. 102 235502
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[1] Novoselov K S, Ceim A K, Morozov S V, Jiang D, Zhang Y, Dubonos S, Grigorieva I V 2004 Science 306 666
[2] Katsnelson M I 2007 Materials Today 10 20
[3] Bolotin K I, Sikes K J, Jiang Z, Klima M, Eudenberg G, Hone J, Stormer H L 2008 Sol. Sta. Com. 146 351
[4] Wang L, Feng W, Yang L Q, Zhang J H 2014 Acta Phys. Sin. 63 176801 (in Chinese) [王浪, 冯伟, 杨连乔, 张建华 2014 63 176801]
[5] Yang Y L, Lu Y 2014 Chin. Phys. B 23 106501
[6] Tian W, Yuan P F, Yu Z L, Tao B B, Hou S Y, Ye C, Zhang Z H 2015 Acta Phys. Sin. 64 046102 (in Chinese) [田文, 袁鹏飞, 禹卓良, 陶斌斌, 侯森耀, 叶聪, 张振华 2015 64 046102]
[7] Zhou P, He D W 2016 Chin. Phys. B 25 017302
[8] Chang X 2014 Acta Phys. Sin. 63 086102 (in Chinese) [常 旭 2014 63 086102]
[9] Mi C G, Liu G P, Wang J J, Guo X L, Wu S X, Yu J 2014 Acta Phys. Chem. Sin. 30 1230 (in Chinese) [米传国, 刘国平, 王家佳, 郭新立, 吴三械, 于 金 2014 物理化学学报 30 1230]
[10] Davydov S Y, Subinova G Y 2011 Phys. Stat. Sol. 53 608 (in Russian)
[11] Davydov S Y, Subinova G Y 2015 Phys. Stat. Sol. 57 1017 (in Russian)
[12] Mounet N, Marzari N 2005 Phys. Rev. B 71 205214
[13] Zakharchenko K V, Katsnelson M I, Fasolino A 2009 Phys. Rev. Lett. 102 046808
[14] Jiang J W, Wang J S, Li B 2009 Phys. Rev. B 80 205429
[15] Pozzo M, Alfe D, Lacovig P, Hofmann P, Lizzit S, Baraldi A 2011 Phys. Rev. Lett. 106 135501
[16] Bao W, Miao F, Chen Z, Zhang H, Jang W, Dames C, Lau N 2009 Nat. Nanotechol. 4 562
[17] Davydov S Y 2011 Tech. Phys. Lett. 37 42 (in Russian)
[18] Davydov S Y 2012 Phys. Solid State 54 875
[19] Davydov S Y 2013 Phys. Stat. Sol. 55 813 (in Russian)
[20] Zheng R L, Hu X Q 1994 College Physics 13 15 (in Chinese) [郑瑞伦, 胡先权 1994 大学物理 13 15]
[21] Davydov S Yu, Bosledlik O W 2015 Phys. Stat. Sol. 57 819 (in Russian)
[22] Davidov S Yu. 2009 Phys. Stat. Sol. 51 2041 (in Russian)
[23] Solin S A, Ramdas A K 1970 Phys. Rev. B 1 1687
[24] Zheng R L, Hu X Q 1996 Solid Theory and Application (Chongqing: southwest normal university press) pp267-271 (in Chinese) [郑瑞伦, 胡先权 1996 固体理论及其应用 (重庆: 西南师范大学出版社) 第 267-271 页]
[25] Blaksly O L, Proctor D G, Seldin E J, Spence G B, Weng T 1970 J. Appl. Phys. 41 3373
[26] Kudin K N, Scuseria G E, Yakobson B I 2001 Phys. Rev. B 64 235406
[27] Lu Q, Arroyo M, Huang R 2009 J. Appl. Phys. 42 102002
[28] Cadelano E C, Palla P L, Giordano S, Colombo L 2009 Phys. Rev. Lett. 102 235502
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