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Quantization of plasmon in two-dimensional square quantum dot system

Wu Reng-Lai Xiao Shi-Fa Xue Hong-Jie Quan Jun

Quantization of plasmon in two-dimensional square quantum dot system

Wu Reng-Lai, Xiao Shi-Fa, Xue Hong-Jie, Quan Jun
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  • Plasmon in quantum dot system is one of the most notable research topics in the field of optoelectronics. With the development of nanotechnology, plasmon in nano-structure has received considerable attention due to its potential applications in future natural science areas. To better understand the quantum effect and the properties of plasmon, in this paper we use the linear response theory and the tight-binding approximation to investigate the collective response of charge in a twodimensional square quantum dot system. The results show that when the frequency of the external field equals the frequency of the plasmon, there are strong charge collective oscillations in the quantum dot system, accompanied by great energy absorption and near-field enhancement. Owing to the quantization of plasmon, the collective charge oscillations in a two-dimensional square quantum dot system are found at different frequencies. The number of quantum modes of plasmon increases with the size and electron number of square quantum dots increasing, this behaviour of quantum mode of plasmon is similar to the one of phonon. The reasons for this behaviour are as follows. First, with the increase of quantum dot size, there are more energy levels around the fermi energy, and the electrons can jump from more energy levels to the outside of fermi circle, so there are more collective excitation frequencies (i.e., more quantum modes of plasmon) in a larger size system. Second, with the increase of electron number in quantum dots, there are more energy levels occupied by electrons, so there are more quantum modes of plasmon too. Furthermore, the size dependence of plasmon shows that with the increase of quantum dot size, the frequency interval between two neighbouring modes of plasmon is smaller, and the discrete modes of plasmon will gradually display quasi-continuous characteristic and transform gradually into the classical continuous modes of plasmon, and the frequency spectrum of plasmon turns into the classical dispersion relation. Such a characteristic is in accord with Bohr's correspondence principle, implying that the quantum plasmon and classical plasmon are gradually unified in a macroscopic size. The dependence of plasmon on the size and electron number of quantum dots also show that with the increase of the quantum dot size, the frequencies of the plasmon is red-shifted and the excitation intensity of the plasmon increases; with the increase of the electron number in quantum dot, the frequency of the plasmon is blue-shifted and the excitation intensity of the plasmon increases.
      Corresponding author: Quan Jun, quanj@lingnan.edu.cn
    • Funds: Project supported by the National Natural Science Foundation of China (Grant No. 11647156) and Natural Science Foundation of Guangdong Province, China (Grant No. 2014A030307035).
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    Menegazzo N, Kegel L L, Kim Y C, Allen D L, Booksh K S 2012 Rev. Sci. Instrum. 83 095113

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    Cassidy A, Grigorenko I, Haas S 2008 Phys. Rev. B 77 245404

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    Zuloaga J, Prodan E, Nordlander P 2009 Nano Lett. 9 887

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    Li X, Xiao D, Zhang Z 2013 New J. Phys. 15 23011

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    Moaied M, Yajadda M M A, Ostrikov K 2015 Plasmonics 10 1615

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    Gu S J, Deng S S, Li Y Q, Lin H Q 2004 Phys. Rev. Lett. 93 086402

    [16]

    Niehaus T A, Suhai S, Della Sala F, Lugli P, Elstner M, Seifert G, Frauenheim T 2001 Phys. Rev. B 63 247

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    Xu Z, Chiesa S, Yang S, Su S Q, Sheehy D E, Moreno J 2011 Phys. Rev. A 84 9325

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    Yu Y Q, Yu Y B, Xue H J, Wang Y X, Chen J 2016 Physica B 496 26

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    Muniz R A, Haas S, Levi A F J, Grigorenko I 2009 Phys. Rev. B 80 1132

    [20]

    Xin W, Wu R L, Xue H J, Yu Y B 2013 Acta Phys. Sin. 62 177301 (in Chinese) [辛旺, 吴仍来, 薛红杰, 余亚斌 2013 物理学报 62 177301]

    [21]

    Wu R, Xue H, Yu Y, Hu H, Liu Q 2014 Europhys. Lett. 108 27001

    [22]

    Wu R, Xue H, Yu Y, Hu H 2014 Phys. Lett. A 378 2295

    [23]

    Li Z Z 2002 Solid State Theory (2nd Ed.) (Beijing: Higher Education Press) p108 (in Chinese) [李正中 2002 固体理论 (第二版)(北京: 高等教育出版社) 第 108 页]

    [24]

    Liu D D, Zhang H 2011 Chin. Phys. B 20 097105

    [25]

    Liu D D, Zhang H, Cheng X L 2012 J. Appl. Phys. 112 053707

  • [1]

    Huang S Y, Chew W C, Liu Y G, Wu B I, Choi H W 2012 J. Appl. Phys. 111 034308

    [2]

    Haran G 2010 AIP Conf. Proc. 1267 59

    [3]

    Menegazzo N, Kegel L L, Kim Y C, Allen D L, Booksh K S 2012 Rev. Sci. Instrum. 83 095113

    [4]

    Koller D M, Hohenau A, Ditlbacher H, Galler N, Reil F, Aussenegg F R, Leitner A, List E J W, Krenn J R 2008 Nat. Photon. 2 684

    [5]

    Walters R J, van Loon R V A, Brunets I, Schmitz J, Polman A 2009 Nat. Mater. 9 21

    [6]

    Liu N, Tang M L, Hentschel M, Giessen H, Alivisatos A P 2011 Nat. Mater. 10 631

    [7]

    Lal S, Clare S E, Halas N J 2008 Acc. Chem. Res. 41 1842

    [8]

    de Abajo F J 2012 Nature 483 417

    [9]

    Yuan Z, Gao S 2008 Phys. Rev. B 78 235413

    [10]

    Yan J, Yuan Z, Gao S 2007 Phys. Rev. Lett. 98 216602

    [11]

    Cassidy A, Grigorenko I, Haas S 2008 Phys. Rev. B 77 245404

    [12]

    Zuloaga J, Prodan E, Nordlander P 2009 Nano Lett. 9 887

    [13]

    Li X, Xiao D, Zhang Z 2013 New J. Phys. 15 23011

    [14]

    Moaied M, Yajadda M M A, Ostrikov K 2015 Plasmonics 10 1615

    [15]

    Gu S J, Deng S S, Li Y Q, Lin H Q 2004 Phys. Rev. Lett. 93 086402

    [16]

    Niehaus T A, Suhai S, Della Sala F, Lugli P, Elstner M, Seifert G, Frauenheim T 2001 Phys. Rev. B 63 247

    [17]

    Xu Z, Chiesa S, Yang S, Su S Q, Sheehy D E, Moreno J 2011 Phys. Rev. A 84 9325

    [18]

    Yu Y Q, Yu Y B, Xue H J, Wang Y X, Chen J 2016 Physica B 496 26

    [19]

    Muniz R A, Haas S, Levi A F J, Grigorenko I 2009 Phys. Rev. B 80 1132

    [20]

    Xin W, Wu R L, Xue H J, Yu Y B 2013 Acta Phys. Sin. 62 177301 (in Chinese) [辛旺, 吴仍来, 薛红杰, 余亚斌 2013 物理学报 62 177301]

    [21]

    Wu R, Xue H, Yu Y, Hu H, Liu Q 2014 Europhys. Lett. 108 27001

    [22]

    Wu R, Xue H, Yu Y, Hu H 2014 Phys. Lett. A 378 2295

    [23]

    Li Z Z 2002 Solid State Theory (2nd Ed.) (Beijing: Higher Education Press) p108 (in Chinese) [李正中 2002 固体理论 (第二版)(北京: 高等教育出版社) 第 108 页]

    [24]

    Liu D D, Zhang H 2011 Chin. Phys. B 20 097105

    [25]

    Liu D D, Zhang H, Cheng X L 2012 J. Appl. Phys. 112 053707

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  • Received Date:  19 May 2017
  • Accepted Date:  19 August 2017
  • Published Online:  05 November 2017

Quantization of plasmon in two-dimensional square quantum dot system

    Corresponding author: Quan Jun, quanj@lingnan.edu.cn
  • 1. College of Physics Science and Technology, Lingnan Normal University, Zhanjiang 524048, China;
  • 2. School of Electronic Engineering, Xi'an Aviation University, Xi'an 710077, China
Fund Project:  Project supported by the National Natural Science Foundation of China (Grant No. 11647156) and Natural Science Foundation of Guangdong Province, China (Grant No. 2014A030307035).

Abstract: Plasmon in quantum dot system is one of the most notable research topics in the field of optoelectronics. With the development of nanotechnology, plasmon in nano-structure has received considerable attention due to its potential applications in future natural science areas. To better understand the quantum effect and the properties of plasmon, in this paper we use the linear response theory and the tight-binding approximation to investigate the collective response of charge in a twodimensional square quantum dot system. The results show that when the frequency of the external field equals the frequency of the plasmon, there are strong charge collective oscillations in the quantum dot system, accompanied by great energy absorption and near-field enhancement. Owing to the quantization of plasmon, the collective charge oscillations in a two-dimensional square quantum dot system are found at different frequencies. The number of quantum modes of plasmon increases with the size and electron number of square quantum dots increasing, this behaviour of quantum mode of plasmon is similar to the one of phonon. The reasons for this behaviour are as follows. First, with the increase of quantum dot size, there are more energy levels around the fermi energy, and the electrons can jump from more energy levels to the outside of fermi circle, so there are more collective excitation frequencies (i.e., more quantum modes of plasmon) in a larger size system. Second, with the increase of electron number in quantum dots, there are more energy levels occupied by electrons, so there are more quantum modes of plasmon too. Furthermore, the size dependence of plasmon shows that with the increase of quantum dot size, the frequency interval between two neighbouring modes of plasmon is smaller, and the discrete modes of plasmon will gradually display quasi-continuous characteristic and transform gradually into the classical continuous modes of plasmon, and the frequency spectrum of plasmon turns into the classical dispersion relation. Such a characteristic is in accord with Bohr's correspondence principle, implying that the quantum plasmon and classical plasmon are gradually unified in a macroscopic size. The dependence of plasmon on the size and electron number of quantum dots also show that with the increase of the quantum dot size, the frequencies of the plasmon is red-shifted and the excitation intensity of the plasmon increases; with the increase of the electron number in quantum dot, the frequency of the plasmon is blue-shifted and the excitation intensity of the plasmon increases.

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