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Graphene is a single atomic layer of carbon atoms forming a dense honeycomb crystal lattice. Now tremendous results of two dimensional (2D) graphene have been obtained recently in the electronic properties both experimentally and theoretically due to the massless energy dispersion relation of electrons and holes with zero (or close to zero) bandgap. In addition, through the process of stimulated emission in population inverted graphene layers, the coupling of the plasmons to interband electron-hole transitions can lead to plasmon amplification. Recently, research results have also shown that at moderate carrier densities (109-1011/cm2), the frequencies of plasma waves in graphene are in the terahertz range.In this paper, based on the Maxwell's equations and material constitutive equation, the gain characteristics of the surface plasmon in graphene are theoretically studied in the terahertz range. In the simulations process we assume a nonequilibrium situation in graphene, where the densities of the electron and the hole are equal. And the gain characteristics for different carrier concentrations, graphene temperature and the momentum relaxation time are calculated. The calculated results show that the peak gain positions shift towards the higher frequencies with the increase of the quasi Fermi level of electron and hole associated with electron-hole concentrations. The reason may be that the change rate of the electron quasi Fermi level is higher than the hole's and thus the distributions of electrons and holes in energy are broader, resulting in the peak gain frequency shifting towards higher frequencies. However, the results also indicate that the temperature of the graphene has little effect on both the peak gain value and the peak gain position of the plasmon. It is maybe because in the simulation process the temperature is taken to be less than 50 K, which is corresponding to the energy of the 1 THz. However the calculated results show that the frequencies of the gain peak positions are all larger than 1 THz, hence, the effects of the temperature on the peak gain value and peak position both could be neglected. Moreover, it is obviously seen that the peak gain value is a function of momentum relaxation time in graphene. This is because when the momentum relaxation time increases, more electrons will be excited, and this will increase the plasmon gain probability in graphene. However, the momentum relaxation time has no effect on the position of the gain peak. It is maybe because the momentum relaxation time has little effect on radiation frequency in the whole momentum relaxation period.
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Keywords:
- graphene /
- surface plasmon /
- terahertz /
- gain characteristic
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[1] Novoselov K S, Geim A K, Morozov S V, Jiang D, Zhang Y, Dubonos S V, Grigorieva I V, Firsov A A 2004 Science 306 666
[2] Han P Y, Liu W, Xie Y H, Zhang X C 2009 Physics 38 395 (in Chinese) [韩鹏昱, 刘伟, 谢亚红, 张希成 2009 物理 38 395]
[3] Geim A K, Macdonald A H 2007 Phys. Today 60 35
[4] Castro Neto A H, Guinea F, Peres N M R, Novoselov K S, Geim A K 2009 Rev. Mod. Phys. 81 109
[5] Wu H Q, Linghu C Y, L H M, Qian H 2013 Chin. Phys. B 22 098106
[6] Rzhii V, Rzhii M, Otsuji T 2007 J. Appl. Phys. 101 083114
[7] Satou A, Vasko F T, Ryzhii V 2008 Phys. Rev. B 78 115431
[8] Ryzhii V, Ryzhii M, Satou A, Otsuji T, Dubinov A A, Ya V 2009 J. Appl. Phys. 106 084507
[9] Ryzhii V, Ryzhii M, Otsuji T 2008 Phys. Stat. Sol. 5 261
[10] Zhang Y P, Zhang X, Liu L Y, Zhang H Y, Gao Y, Xu S L, Zhang H H 2012 Chin. J. Lasers 39 0111002 (in Chinese) [张玉萍, 张晓, 刘凌玉, 张洪艳, 高营, 徐世林, 张会会 2012 中国激光 39 0111002]
[11] Ryzhii V, Ryzhii M, Mitin V, Otsuji T 2011 J. Appl. Phys. 110 094503
[12] Victor R, Maxim R, Vladimir M 2011 Jpn. J. Appl. Phys. 50 094001
[13] Zhang Y P, Zhang H Y, Yin Y H, Liu L Y, Zhang X, Gao Y, Zhang H H 2012 Acta Phys. Sin. 61 047803 (in Chinese) [张玉萍, 张洪艳, 尹贻恒, 刘凌玉, 张晓, 高营, 张会会 2012 61 047803]
[14] Zhang Y P, Liu L Y, Chen Q, Feng Z H, Zhang X, Zhang H Y, Zhang H H 2013 Acta Phys. Sin. 62 097202 (in Chinese) [张玉萍, 刘凌玉, 陈琦, 冯志红, 张晓, 张洪艳, 张会会 2013 62 097202]
[15] Sun Y F, Sun J D, Zhou Y, Tan R B, Zeng C H, Xue W, Qin H, Zhang B S, Wu D M 2011 Appl. Phys. Lett. 98 252103
[16] Guo N, Hu W D, Chen X S, Wang L, Lu W 2013 Opt. Express 21 1606
[17] Wu S Q, Liu J S, Wang S L, Hu B 2013 Chin. Phys. B 22 104207
[18] Hanson G W 2008 J. Appl. Phys. 103 064302
[19] Vafek O 2006 Phys. Rev. Lett. 97 266406
[20] Falkovsky L A, Varlamov A A 2007 Eur. Phys. J. B 56 281
[21] Jablan M, Buljan H, Soljacic M 2009 Phys. Rev. B 80 245435
[22] Chen P, Al A 2011 ACS Nano 5 5855
[23] Vakil A, Engheta N 2011 Science 332 1291
[24] Dubinov A A, Aleshkin V Y, Mitin V 2011 J. Phys.: Conden. Matter 23 145302
[25] Ju L, Geng B S, Horng J, Girit C, Martin M, Hao Z, Bechtel H A, Liang X G, Zettl A, Shen Y R, Wang F 2011 Nat. Nanotechnol. 6 630
[26] Fei Z, Rodin A S, Andreev G O, Bao W, McLeod A S, Wagner M, Zhang L M, Zhao Z, Thiemens M, Dominguez G, Fogler M M, Castro Neto A H, Lau C N, Keilmann F, Basov D N 2012 Nature 487 82
[27] Chen J, Badioli M, AIonso-Gonzalez P, Thongrattanasiri S, Huth F, Osmond J, Spasenović M, Centeno A, Pesquera A, Godignon P, Elorza A Z, Camara N, Garca de Abajo F G, Hillenbrand R, Koppens F H L 2012 Nature 487 77
[28] Watanabe T, Fukushima T, Yabe Y 2013 New J. Phys. 15 075003
[29] Chen L, Zhang T, Li X, Wang G P 2013 Opt. Express 21 28628
[30] George P A, Strait J, Dawlaty J, Shivaraman S, Chandrashekhar M, Rana F, Spencer M G 2008 Nano Lett. 8 4248
[31] Batke E, Heitmann D, Tu C W 1986 Phy. Rev. B 34 6951
[32] Allen S J, Tsyi D C, Logan R A 1977 Phys. Rev. Lett. 38 980
[33] Yan H, Li X, Chandra B, Tulevski G, Wu Y, Freitag M, Zhu W, Avouris P, Xia F 2012 Nat. Nanotechnol. 7 330
[34] Rana F 2007 Lasers and Electro-Optics Society, LEOS 2007. the 20th Annual Meeting of the IEEE Lake Buena Vista, USA, October 21-25, 2007 p862
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