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Considering the anharmonic vibrations and the interactions between electron and phonon of atoms, in this article we study the temperature dependence of Grneisen parameter, thermal expansion coefficient at low temperature and phonon relaxation time by using the theory and method of solid state physics. The influences of the anharmonic vibration of the atom on the above parameters are further discussed. The obtained results are as follows. 1) The thermal expansion coefficient of graphene is a negative value when the temperature drops below room temperature. The absolute value of the thermal expansion coefficient of graphene increases monotonically with the increase of temperature. The thermal expansion coefficient of graphene is-3.64×10-6 K-1 at room temperature. 2) The value of Grneisen parameter is zero in the harmonic approximation. If the anharmonic vibration is considered, the Grneisen parameter will increase slowly with the increase of temperature. Its value is between 1.40 and1.42 and the change is almost linear. And we find that the influence of the second anharmonic term is less than that of the first anharmonic term on Grneisen parameter. 3) The phonon relaxation time decreases with the increase of temperature. The rate changes rapidly at low temperature (T<10 K), then it changes very slowly. The phonon relaxation time is almost inversely proportional to temperature when the temperature is higher than 300 K.
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Keywords:
- graphene /
- thermal expansion coefficient at low temperature /
- Grüneisen parameter /
- relaxation time
[1] Novoselov K S, Ceim A K, Morozov S V, et al. 2004 Science 306 666
[2] Katsnelson M I 2007 Mater. 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] Tian W, Yuan P F, Yu Z L, Tao B B, Hou S Y, 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, 张振华 2015 物理学报 64 046102]
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[6] Mounet N, Marzari N 2005 Phys. Rev. B 71 205214
[7] Zakharchenko K V, Katsnelson M I, Fasolino A 2009 Phys. Rev. Lett. 102 046808
[8] Jiang J W, Wang J S, Li B 2009 Phys. Rev. B 80 205429
[9] Pozzo M, Alfe D, Lacovig P, Hofmann P, Lizzit S, Baraldi A 2011 Phys. Rev. Lett. 106 135501
[10] Bao W, Miao F, Chen Z, Zhang H, Jang W, Dames C, Lau N 2009 Nat. Nanotechol. 4 562
[11] Yu D S (in Russian)
[12] Cheng Z F, Zheng R L 2016 Chin. Phys. Lett. 33 046501
[13] Cheng Z F, Zheng R L 2016 Acta Phys. Sin. 65 104701 (in Chinese) [程正富, 郑瑞伦 2016 物理学报 65 104701]
[14] Yoon D, Son Y W, Cheong H 2011 Nano Leet. 11 3227
[15] Zha X H, Zhang R Q, Lin Z 2014 J. Chem. Phys. 141 064705
[16] Ge X J, Xao K L, Lil J T 2016 Phys. Rev. B 94 165433
[17] Kang K, Abdula D, Cahill D G, Shim M 2010 Phys. Rev. B 81 165405
[18] Lindsay L, Broido D A, Mingo N 2011 Phys. Rev. B 83 235428
[19] Bonini N, Lazzeri M, Marzari N, Mauri F 2007 Phys. Rev. Lett. 99 176802
[20] Ye Z Q, Cao B Y, Guo Z Y 2014 Acta Phys. Sin. 63 154704 (in Chinese) [叶振强, 曹炳阳, 过增元 2014 物理学报 63 154704]
[21] Davydov S Yu 2012 Phys. Solid Stat. 54 875
[22] Jiang J W, Wang B S, Wang J S, Park S A 2015 J.Phys.: Condens. Matter 27 083011
[23] Zheng R L, Hu X Q 1996 Solid Theory and Application (Chongqing: Southwest Normal University Press)pp316–325 (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 316—325 页]
[24] yu D S, Jihonov S K 1996 Phys. Semicond. Technol. 30 968
[25] Ren X X, Kang W, Cheng Z F, Zheng R L 2016 Chin.Phys. Lett. 33 126501
[26] Nika D L, Pokatilov E P, Askerov A S, Balandin A A 2009 Phys. Rev. B 79 155413
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[1] Novoselov K S, Ceim A K, Morozov S V, et al. 2004 Science 306 666
[2] Katsnelson M I 2007 Mater. 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] Tian W, Yuan P F, Yu Z L, Tao B B, Hou S Y, 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, 张振华 2015 物理学报 64 046102]
[5] Yu D S 2013 Phys. Stat. Sol. 55 813 (in Russian)
[6] Mounet N, Marzari N 2005 Phys. Rev. B 71 205214
[7] Zakharchenko K V, Katsnelson M I, Fasolino A 2009 Phys. Rev. Lett. 102 046808
[8] Jiang J W, Wang J S, Li B 2009 Phys. Rev. B 80 205429
[9] Pozzo M, Alfe D, Lacovig P, Hofmann P, Lizzit S, Baraldi A 2011 Phys. Rev. Lett. 106 135501
[10] Bao W, Miao F, Chen Z, Zhang H, Jang W, Dames C, Lau N 2009 Nat. Nanotechol. 4 562
[11] Yu D S (in Russian)
[12] Cheng Z F, Zheng R L 2016 Chin. Phys. Lett. 33 046501
[13] Cheng Z F, Zheng R L 2016 Acta Phys. Sin. 65 104701 (in Chinese) [程正富, 郑瑞伦 2016 物理学报 65 104701]
[14] Yoon D, Son Y W, Cheong H 2011 Nano Leet. 11 3227
[15] Zha X H, Zhang R Q, Lin Z 2014 J. Chem. Phys. 141 064705
[16] Ge X J, Xao K L, Lil J T 2016 Phys. Rev. B 94 165433
[17] Kang K, Abdula D, Cahill D G, Shim M 2010 Phys. Rev. B 81 165405
[18] Lindsay L, Broido D A, Mingo N 2011 Phys. Rev. B 83 235428
[19] Bonini N, Lazzeri M, Marzari N, Mauri F 2007 Phys. Rev. Lett. 99 176802
[20] Ye Z Q, Cao B Y, Guo Z Y 2014 Acta Phys. Sin. 63 154704 (in Chinese) [叶振强, 曹炳阳, 过增元 2014 物理学报 63 154704]
[21] Davydov S Yu 2012 Phys. Solid Stat. 54 875
[22] Jiang J W, Wang B S, Wang J S, Park S A 2015 J.Phys.: Condens. Matter 27 083011
[23] Zheng R L, Hu X Q 1996 Solid Theory and Application (Chongqing: Southwest Normal University Press)pp316–325 (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 316—325 页]
[24] yu D S, Jihonov S K 1996 Phys. Semicond. Technol. 30 968
[25] Ren X X, Kang W, Cheng Z F, Zheng R L 2016 Chin.Phys. Lett. 33 126501
[26] Nika D L, Pokatilov E P, Askerov A S, Balandin A A 2009 Phys. Rev. B 79 155413
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