搜索

x

留言板

尊敬的读者、作者、审稿人, 关于本刊的投稿、审稿、编辑和出版的任何问题, 您可以本页添加留言。我们将尽快给您答复。谢谢您的支持!

姓名
邮箱
手机号码
标题
留言内容
验证码

人工带隙材料的拓扑性质

孙晓晨 何程 卢明辉 陈延峰

引用本文:
Citation:

人工带隙材料的拓扑性质

孙晓晨, 何程, 卢明辉, 陈延峰

Topological properties of artificial bandgap materials

Sun Xiao-Chen, He Cheng, Lu Ming-Hui, Chen Yan-Feng
PDF
导出引用
  • 近年来,人工带隙材料(如声子晶体和光子晶体)由于其优异的性能,已成为新一代智能材料的研究焦点.另一方面,材料拓扑学由凝聚态物理领域逐渐延伸到其他粒子或准粒子系统,而研究人工带隙材料的拓扑性质更是受到人们的广泛关注,其特有的鲁棒边界态,具有缺陷免疫、背散射抑制和自旋轨道锁定的传输等特性,潜在应用前景巨大.本文简要介绍拓扑材料特有的鲁棒边界态的物理图像及其物理意义,并列举诸如光/声量子霍尔效应、量子自旋霍尔效应、Floquet拓扑绝缘体等相关工作;利用Dirac方程,从原理上分析光/声拓扑性质的由来;最后对相关领域的发展方向和应用前景进行了相应的讨论.
    Recently, artificial bandgap materials (such as photonic crystals and phononic crystals) have been becoming the research hotspot of the next generation intelligent materials, because of its extremely designable, tunable and controllable capacity of classical waves. On the other hand, topological material phase, originally proposed and first demonstrated in Fermionic electronic systems, has been proposed in more and more Bosonic systems. In this review paper, we first focus on some of the representative photonic/phononic topological models, and four common types of topological photonic system are discussed:1) photonic/phononic quantum Hall effect with broken time-reversal symmetry; 2) photonic topological insulator and the associated pseudo-time-reversal symmetry protected mechanism; 3) time/space periodically modulated photonic Floquet topological insulator; 4) a summary and outlook including a brief introduction of Zak phase in one-dimensional systems and Weyl point in three-dimensional systems. Finally, the underlying Dirac model is analyzed.
      通信作者: 卢明辉, luminghui@nju.edu.cn
    • 基金项目: 国家重点研发计划(批准号:2017YFA0303702)、国家自然科学基金(批准号:11134006,11474158,11404164)、国家自然科学基金杰出青年基金(批准号:11625418)、江苏省自然科学基金(批准号:BK20140019)和江苏高校优势学科建设工程项目(批准号:PAPD)资助的课题.
      Corresponding author: Lu Ming-Hui, luminghui@nju.edu.cn
    • Funds: Project supported by the National Key RD Program of China (Grant No. 2017YFA0303702), the National Natural Science Foundation of China (Grant Nos. 11134006, 11474158, 11404164, 11625418), the Natural Science Foundation of Jiangsu Province, China (Grant No. BK20140019), and the support from Academic Program Development of Jiangsu Higher Education (Grant No. PAPD).
    [1]

    Pancharatnam S 1956 Proc. Indian Acad. Sci. Sect. A 44 398

    [2]

    Berry M V 1984 Proc. Royal Soci. London A:Math. Phys. Engineer. Sci. 392 45

    [3]

    Tomita A, Chiao R Y 1986 Phys. Rev. Lett. 57 937

    [4]

    Asorey M 2016 Nat. Phys. 12 616

    [5]

    Hall E H 1879 Am. J. Math. 2 287

    [6]

    Klitzing K V, Dorda G, Pepper M 1980 Phys. Rev. Lett. 45 494

    [7]

    Thouless D J, Kohmoto M, Nightingale M P, den Nijs M 1982 Phys. Rev. Lett. 49 405

    [8]

    Haldane F D M 1988 Phys. Rev. Lett. 61 2015

    [9]

    Shen S Q 2012 Topological Insulator. (Springer, Berlin)

    [10]

    Bernevig B A, Hughes T L, Zhang S C 2006 Science 314 1757

    [11]

    Knig M, Wiedmann S, Brne C, Roth A, Buhmann H, Molenkamp L W, Qi X L, Zhang S C 2007 Science 318 766

    [12]

    Kane C L, Mele E J 2005 Phys. Rev. Lett. 95 146802

    [13]

    Sheng L, Sheng D N, Ting C S, Haldane F D M 2005 Phys. Rev. Lett. 95 136602

    [14]

    Zhang H J, Liu C X, Qi X L, Dai X, Fang Z, Zhang S C 2009 Nat. Phys. 5 438

    [15]

    Chen Y L, Analytis J G, Chu J H, Liu Z K, Mo S K, Qi X L, Zhang H J, Lu D H, Dai X, Fang Z, Zhang S C, Fisher I R, Hussain Z, Shen Z X 2009 Science 325 178

    [16]

    Fu L 2011 Phys. Rev. Lett. 106 106802

    [17]

    Hsieh T H, Lin H, Liu J, Duan W, Bansil A, Fu L 2012 Nat. Commun. 3 982

    [18]

    Wan X, Turner A M, Vishwanath A, Savrasov S Y 2011 Phys. Rev. B 83 205101

    [19]

    Cayssol J, Dora B, Simon F, Moessner R 2013 Phys. Status Solidi Rapid Res. Lett. 7 101

    [20]

    Chang C Z, Zhang J, Feng X, Shen J, Zhang Z, Guo M, Li K, Ou Y, Wei P, Wang L L, Ji Z Q, Feng Y, Ji S, Chen X, Jia J, Dai X, Fang Z, Zhang S C, He K, Wang Y, Lu L, Ma X C, Xue Q K 2013 Science 340 167

    [21]

    Lu L, Joannopoulos J D, Soljacic M 2016 Nat. Phys. 12 626

    [22]

    Haldane F D M, Raghu S 2008 Phys. Rev. Lett. 100 013904

    [23]

    Wang Z, Chong Y D, Joannopoulos J D, Soljacic M 2008 Phys. Rev. Lett. 100 013905

    [24]

    Wang Z, Chong Y, Joannopoulos J D, Soljacic M 2009 Nature 461 772

    [25]

    Poo Y, Wu R X, Lin Z, Yang Y, Chan C T 2011 Phys. Rev. Lett. 106 093903

    [26]

    Yannopapas V 2011 Phys. Rev. B 84 195126

    [27]

    Liu K, Shen L, He S 2012 Opt. Lett. 37 4110

    [28]

    Asatryan A A, Botten L C, Fang K, Fan S, McPhedran R C 2013 Phys. Rev. B 88 035127

    [29]

    Skirlo S A, Lu L, Igarashi Y, Yan Q, Joannopoulos J, Soljačić M 2015 Phys. Rev. Lett. 115 253901

    [30]

    Fleury R, Sounas D L, Sieck C F, Haberman M R, Al A 2014 Science 343 516

    [31]

    Yang Z, Gao F, Shi X, Lin X, Gao Z, Chong Y, Zhang B 2015 Phys. Rev. Lett. 114 114301

    [32]

    Ni X, He C, Sun X C, Liu X P, Lu M H, Feng L, Chen Y F 2015 New J. Phys. 17 053016

    [33]

    Khanikaev A B, Mousavi S H, Tse W K, Kargarian M, MacDonald A H, Shvets G 2013 Nat. Mater. 12 233

    [34]

    He C, Sun X C, Liu X P, Lu M H, Chen Y, Feng L, Chen Y F 2016 Proc. Natl. Acad. Sci. USA 113 4924

    [35]

    Chen W J, Jiang S J, Chen X D, Zhu B, Zhou L, Dong J W, Chan C T 2014 Nat. Commun. 5 5782

    [36]

    Wu L H, Hu X 2015 Phys. Rev. Lett. 114 223901

    [37]

    He C, Ni X, Ge H, Sun X C, Chen Y B, Lu M H, Liu X P, Chen Y F 2016 Nat. Phys. 12 1124

    [38]

    Susstrunk R, Huber S D 2015 Science 349 47

    [39]

    Fang K, Yu Z, Fan S 2012 Phys. Rev. Lett. 108 153901

    [40]

    Fang K, Yu Z, Fan S 2012 Nat. Photon. 6 782

    [41]

    Rechtsman M C, Zeuner J M, Plotnik Y, Lumer Y, Podolsky D, Dreisow F, Nolte S, Segev M, Szameit A 2013 Nature 496 196

    [42]

    Hafezi M, Demler E A, Lukin M D, Taylor J M 2011 Nat. Phys. 7 907

    [43]

    Hafezi M, Mittal S, Fan J, Migdall A, Taylor J M 2013 Nat. Photon. 7 1001

    [44]

    Liang G Q, Chong Y D 2013 Phys. Rev. Lett. 110 203904

    [45]

    Petrescu A, Houck A A, Le Hur K 2012 Phys. Rev. A 86 053804

    [46]

    Rechtsman M C, Zeuner J M, Tunnermann A, Nolte S, Segev M, Szameit A 2013 Nat. Photon. 7 153

    [47]

    Wang Y H, Steinberg H, Jarillo-Herrero P, Gedik N 2013 Science 342 453

    [48]

    Guzmn-Silva D, Meja-Corts C, Bandres M A, Rechtsman M C, Weimann S, Nolte S, Segev M, Szameit A, Vicencio R A 2014 New J. Phys. 16 063061

    [49]

    Lin Q, Fan S 2014 Phys. Rev. X 4 031031

    [50]

    Mittal S, Fan J, Faez S, Migdall A, Taylor J M, Hafezi M 2014 Phys. Rev. Lett. 113 087403

    [51]

    Ozawa T, Carusotto I 2014 Phys. Rev. Lett. 112 133902

    [52]

    Teras H, Flayac H, Solnyshkov D D, Malpuech G 2014 Phys. Rev. Lett. 112 066402

    [53]

    Tzuang L D, Fang K, Nussenzveig P, Fan S, Lipson M 2014 Nat. Photon. 8 701

    [54]

    Hu W, Pillay J C, Wu K, Pasek M, Shum P P, Chong Y D 2015 Phys. Rev. X 5 011012

    [55]

    Schmidt M, Kessler S, Peano V, Painter O, Marquardt F 2015 Optica 2 635

    [56]

    Gao F, Gao Z, Shi X, Yang Z, Lin X, Xu H, Joannopoulos J D, Soljacic M, Chen H, Lu L, Chong Y, Zhang B 2016 Nat.Commun. 7 11619

    [57]

    Fleury R, Khanikaev A B, Al A 2016 Nat.Commun. 7 11744

    [58]

    He C, Li Z, Ni X, Sun X C, Yu S Y, Lu M H, Liu X P, Chen Y F 2016 Appl. Phys. Lett. 108 031904

    [59]

    Peng Y G, Qin C Z, Zhao D G, Shen Y X, Xu X Y, Bao M, Jia H, Zhu X F 2016 Nat. Commun. 7 13368

    [60]

    Harper P G 1955 Proc. Phys. Soci. Sec. A 68 874

    [61]

    Aubry S, Andr G 1980 Ann. Israel Phys. Soc. 3 18

    [62]

    Kohmoto M, Kadanoff L P, Tang C 1983 Phys. Rev.Lett. 50 1870

    [63]

    Kraus Y E, Lahini Y, Ringel Z, Verbin M, Zilberberg O 2012 Phys. Rev. Lett. 109 106402

    [64]

    Verbin M, Zilberberg O, Kraus Y E, Lahini Y, Silberberg Y 2013 Phys. Rev. Lett. 110 076403

    [65]

    Zak J 1989 Phys. Rev. Lett. 62 2747

    [66]

    Su W P, Schrieffer J R, Heeger A J 1979 Phys. Rev. Lett. 42 1698

    [67]

    Xiao M, Zhang Z Q, Chan C T 2014 Phys. Rev. X 4 021017

    [68]

    Xiao M, Ma G, Yang Z, Sheng P, Zhang Z Q, Chan C T 2015 Nat. Phys. 11 240

    [69]

    Kitaev A Y 2001 Physics-Uspekhi 44 131

    [70]

    Poddubny A, Miroshnichenko A, Slobozhanyuk A, Kivshar Y 2014 ACS Photon. 1 101

    [71]

    Lu L, Fang C, Fu L, Johnson S G, Joannopoulos J D, Soljacic M 2016 Nat. Phys. 12 337

    [72]

    Wang H, Xu L, Chen H, Jiang J H 2016 Phys. Rev. B 93 235155

    [73]

    Lu L, Fu L, Joannopoulos J D, Soljacic M 2013 Nat. Photon. 7 294

    [74]

    Lu L, Wang Z, Ye D, Ran L, Fu L, Joannopoulos J D, Soljačić M 2015 Science 349 622

    [75]

    Xiao M, Chen WJ, He W Y, Chan C T 2015 Nat. Phys. 11 920

    [76]

    Yang Z, Zhang B 2016 Phys. Rev. Lett. 117 224301

    [77]

    Noh J, Huang S, Leykam D, Chong Y D, Chen K P, Rechtsman M C 2017 Nat. Phys. 13 611

  • [1]

    Pancharatnam S 1956 Proc. Indian Acad. Sci. Sect. A 44 398

    [2]

    Berry M V 1984 Proc. Royal Soci. London A:Math. Phys. Engineer. Sci. 392 45

    [3]

    Tomita A, Chiao R Y 1986 Phys. Rev. Lett. 57 937

    [4]

    Asorey M 2016 Nat. Phys. 12 616

    [5]

    Hall E H 1879 Am. J. Math. 2 287

    [6]

    Klitzing K V, Dorda G, Pepper M 1980 Phys. Rev. Lett. 45 494

    [7]

    Thouless D J, Kohmoto M, Nightingale M P, den Nijs M 1982 Phys. Rev. Lett. 49 405

    [8]

    Haldane F D M 1988 Phys. Rev. Lett. 61 2015

    [9]

    Shen S Q 2012 Topological Insulator. (Springer, Berlin)

    [10]

    Bernevig B A, Hughes T L, Zhang S C 2006 Science 314 1757

    [11]

    Knig M, Wiedmann S, Brne C, Roth A, Buhmann H, Molenkamp L W, Qi X L, Zhang S C 2007 Science 318 766

    [12]

    Kane C L, Mele E J 2005 Phys. Rev. Lett. 95 146802

    [13]

    Sheng L, Sheng D N, Ting C S, Haldane F D M 2005 Phys. Rev. Lett. 95 136602

    [14]

    Zhang H J, Liu C X, Qi X L, Dai X, Fang Z, Zhang S C 2009 Nat. Phys. 5 438

    [15]

    Chen Y L, Analytis J G, Chu J H, Liu Z K, Mo S K, Qi X L, Zhang H J, Lu D H, Dai X, Fang Z, Zhang S C, Fisher I R, Hussain Z, Shen Z X 2009 Science 325 178

    [16]

    Fu L 2011 Phys. Rev. Lett. 106 106802

    [17]

    Hsieh T H, Lin H, Liu J, Duan W, Bansil A, Fu L 2012 Nat. Commun. 3 982

    [18]

    Wan X, Turner A M, Vishwanath A, Savrasov S Y 2011 Phys. Rev. B 83 205101

    [19]

    Cayssol J, Dora B, Simon F, Moessner R 2013 Phys. Status Solidi Rapid Res. Lett. 7 101

    [20]

    Chang C Z, Zhang J, Feng X, Shen J, Zhang Z, Guo M, Li K, Ou Y, Wei P, Wang L L, Ji Z Q, Feng Y, Ji S, Chen X, Jia J, Dai X, Fang Z, Zhang S C, He K, Wang Y, Lu L, Ma X C, Xue Q K 2013 Science 340 167

    [21]

    Lu L, Joannopoulos J D, Soljacic M 2016 Nat. Phys. 12 626

    [22]

    Haldane F D M, Raghu S 2008 Phys. Rev. Lett. 100 013904

    [23]

    Wang Z, Chong Y D, Joannopoulos J D, Soljacic M 2008 Phys. Rev. Lett. 100 013905

    [24]

    Wang Z, Chong Y, Joannopoulos J D, Soljacic M 2009 Nature 461 772

    [25]

    Poo Y, Wu R X, Lin Z, Yang Y, Chan C T 2011 Phys. Rev. Lett. 106 093903

    [26]

    Yannopapas V 2011 Phys. Rev. B 84 195126

    [27]

    Liu K, Shen L, He S 2012 Opt. Lett. 37 4110

    [28]

    Asatryan A A, Botten L C, Fang K, Fan S, McPhedran R C 2013 Phys. Rev. B 88 035127

    [29]

    Skirlo S A, Lu L, Igarashi Y, Yan Q, Joannopoulos J, Soljačić M 2015 Phys. Rev. Lett. 115 253901

    [30]

    Fleury R, Sounas D L, Sieck C F, Haberman M R, Al A 2014 Science 343 516

    [31]

    Yang Z, Gao F, Shi X, Lin X, Gao Z, Chong Y, Zhang B 2015 Phys. Rev. Lett. 114 114301

    [32]

    Ni X, He C, Sun X C, Liu X P, Lu M H, Feng L, Chen Y F 2015 New J. Phys. 17 053016

    [33]

    Khanikaev A B, Mousavi S H, Tse W K, Kargarian M, MacDonald A H, Shvets G 2013 Nat. Mater. 12 233

    [34]

    He C, Sun X C, Liu X P, Lu M H, Chen Y, Feng L, Chen Y F 2016 Proc. Natl. Acad. Sci. USA 113 4924

    [35]

    Chen W J, Jiang S J, Chen X D, Zhu B, Zhou L, Dong J W, Chan C T 2014 Nat. Commun. 5 5782

    [36]

    Wu L H, Hu X 2015 Phys. Rev. Lett. 114 223901

    [37]

    He C, Ni X, Ge H, Sun X C, Chen Y B, Lu M H, Liu X P, Chen Y F 2016 Nat. Phys. 12 1124

    [38]

    Susstrunk R, Huber S D 2015 Science 349 47

    [39]

    Fang K, Yu Z, Fan S 2012 Phys. Rev. Lett. 108 153901

    [40]

    Fang K, Yu Z, Fan S 2012 Nat. Photon. 6 782

    [41]

    Rechtsman M C, Zeuner J M, Plotnik Y, Lumer Y, Podolsky D, Dreisow F, Nolte S, Segev M, Szameit A 2013 Nature 496 196

    [42]

    Hafezi M, Demler E A, Lukin M D, Taylor J M 2011 Nat. Phys. 7 907

    [43]

    Hafezi M, Mittal S, Fan J, Migdall A, Taylor J M 2013 Nat. Photon. 7 1001

    [44]

    Liang G Q, Chong Y D 2013 Phys. Rev. Lett. 110 203904

    [45]

    Petrescu A, Houck A A, Le Hur K 2012 Phys. Rev. A 86 053804

    [46]

    Rechtsman M C, Zeuner J M, Tunnermann A, Nolte S, Segev M, Szameit A 2013 Nat. Photon. 7 153

    [47]

    Wang Y H, Steinberg H, Jarillo-Herrero P, Gedik N 2013 Science 342 453

    [48]

    Guzmn-Silva D, Meja-Corts C, Bandres M A, Rechtsman M C, Weimann S, Nolte S, Segev M, Szameit A, Vicencio R A 2014 New J. Phys. 16 063061

    [49]

    Lin Q, Fan S 2014 Phys. Rev. X 4 031031

    [50]

    Mittal S, Fan J, Faez S, Migdall A, Taylor J M, Hafezi M 2014 Phys. Rev. Lett. 113 087403

    [51]

    Ozawa T, Carusotto I 2014 Phys. Rev. Lett. 112 133902

    [52]

    Teras H, Flayac H, Solnyshkov D D, Malpuech G 2014 Phys. Rev. Lett. 112 066402

    [53]

    Tzuang L D, Fang K, Nussenzveig P, Fan S, Lipson M 2014 Nat. Photon. 8 701

    [54]

    Hu W, Pillay J C, Wu K, Pasek M, Shum P P, Chong Y D 2015 Phys. Rev. X 5 011012

    [55]

    Schmidt M, Kessler S, Peano V, Painter O, Marquardt F 2015 Optica 2 635

    [56]

    Gao F, Gao Z, Shi X, Yang Z, Lin X, Xu H, Joannopoulos J D, Soljacic M, Chen H, Lu L, Chong Y, Zhang B 2016 Nat.Commun. 7 11619

    [57]

    Fleury R, Khanikaev A B, Al A 2016 Nat.Commun. 7 11744

    [58]

    He C, Li Z, Ni X, Sun X C, Yu S Y, Lu M H, Liu X P, Chen Y F 2016 Appl. Phys. Lett. 108 031904

    [59]

    Peng Y G, Qin C Z, Zhao D G, Shen Y X, Xu X Y, Bao M, Jia H, Zhu X F 2016 Nat. Commun. 7 13368

    [60]

    Harper P G 1955 Proc. Phys. Soci. Sec. A 68 874

    [61]

    Aubry S, Andr G 1980 Ann. Israel Phys. Soc. 3 18

    [62]

    Kohmoto M, Kadanoff L P, Tang C 1983 Phys. Rev.Lett. 50 1870

    [63]

    Kraus Y E, Lahini Y, Ringel Z, Verbin M, Zilberberg O 2012 Phys. Rev. Lett. 109 106402

    [64]

    Verbin M, Zilberberg O, Kraus Y E, Lahini Y, Silberberg Y 2013 Phys. Rev. Lett. 110 076403

    [65]

    Zak J 1989 Phys. Rev. Lett. 62 2747

    [66]

    Su W P, Schrieffer J R, Heeger A J 1979 Phys. Rev. Lett. 42 1698

    [67]

    Xiao M, Zhang Z Q, Chan C T 2014 Phys. Rev. X 4 021017

    [68]

    Xiao M, Ma G, Yang Z, Sheng P, Zhang Z Q, Chan C T 2015 Nat. Phys. 11 240

    [69]

    Kitaev A Y 2001 Physics-Uspekhi 44 131

    [70]

    Poddubny A, Miroshnichenko A, Slobozhanyuk A, Kivshar Y 2014 ACS Photon. 1 101

    [71]

    Lu L, Fang C, Fu L, Johnson S G, Joannopoulos J D, Soljacic M 2016 Nat. Phys. 12 337

    [72]

    Wang H, Xu L, Chen H, Jiang J H 2016 Phys. Rev. B 93 235155

    [73]

    Lu L, Fu L, Joannopoulos J D, Soljacic M 2013 Nat. Photon. 7 294

    [74]

    Lu L, Wang Z, Ye D, Ran L, Fu L, Joannopoulos J D, Soljačić M 2015 Science 349 622

    [75]

    Xiao M, Chen WJ, He W Y, Chan C T 2015 Nat. Phys. 11 920

    [76]

    Yang Z, Zhang B 2016 Phys. Rev. Lett. 117 224301

    [77]

    Noh J, Huang S, Leykam D, Chong Y D, Chen K P, Rechtsman M C 2017 Nat. Phys. 13 611

  • [1] 方静云, 孙庆丰. 石墨烯p-n结在磁场中的电输运热耗散. 物理学报, 2022, 71(12): 127203. doi: 10.7498/aps.71.20220029
    [2] 隋文杰, 张玉, 张紫瑞, 王小龙, 张洪方, 史强, 杨冰. 拓扑自旋光子晶体中螺旋边界态单向传输调控研究. 物理学报, 2022, 71(19): 194101. doi: 10.7498/aps.71.20220353
    [3] 吕新宇, 李志强. 石墨烯莫尔超晶格体系的拓扑性质及光学研究进展. 物理学报, 2019, 68(22): 220303. doi: 10.7498/aps.68.20191317
    [4] 王子, 张丹妹, 任捷. 声子系统中弹性波与热输运的拓扑与非互易现象. 物理学报, 2019, 68(22): 220302. doi: 10.7498/aps.68.20191463
    [5] 孔令尧. 磁斯格明子拓扑特性及其动力学微磁学模拟研究进展. 物理学报, 2018, 67(13): 137506. doi: 10.7498/aps.67.20180235
    [6] 贾子源, 杨玉婷, 季立宇, 杭志宏. 类石墨烯复杂晶胞光子晶体中的确定性界面态. 物理学报, 2017, 66(22): 227802. doi: 10.7498/aps.66.227802
    [7] 陈泽国, 吴莹. 声子晶体中的多重拓扑相. 物理学报, 2017, 66(22): 227804. doi: 10.7498/aps.66.227804
    [8] 杨 波. 直线加速Kinnersley黑洞中Dirac粒子的热辐射. 物理学报, 2008, 57(2): 1278-1284. doi: 10.7498/aps.57.1278
    [9] 张民仓, 王振邦. 一类环状非球谐振子势场中相对论粒子的束缚态解. 物理学报, 2007, 56(7): 3688-3692. doi: 10.7498/aps.56.3688
    [10] 张民仓, 王振邦. 第二类P?schl-Teller势场中相对论粒子的束缚态. 物理学报, 2006, 55(2): 525-528. doi: 10.7498/aps.55.525
    [11] 曹江陵. 任意加速带电动态黑洞中Dirac粒子的Hawking辐射. 物理学报, 2006, 55(6): 2682-2686. doi: 10.7498/aps.55.2682
    [12] 张民仓, 王振邦. Manning-Rosen标量势与矢量势的Klein-Gordon方程和Dirac方程的束缚态. 物理学报, 2006, 55(2): 521-524. doi: 10.7498/aps.55.521
    [13] 张民仓, 王振邦. Makarov势的Dirac方程的束缚态解. 物理学报, 2006, 55(12): 6229-6233. doi: 10.7498/aps.55.6229
    [14] 李 宁, 鞠国兴, 任中洲. 一类相对论性非球谐振子系统的束缚态. 物理学报, 2005, 54(6): 2520-2523. doi: 10.7498/aps.54.2520
    [15] 陈 刚. 具有Wood-Saxon势的Dirac方程的束缚态. 物理学报, 2004, 53(3): 680-683. doi: 10.7498/aps.53.680
    [16] 张靖仪. 一般球对称带电蒸发黑洞Dirac场的熵. 物理学报, 2003, 52(9): 2354-2358. doi: 10.7498/aps.52.2354
    [17] 张靖仪, 赵 峥. 电磁直线加速动态黑洞时空中Dirac粒子的Hawking辐射. 物理学报, 2003, 52(8): 2096-2101. doi: 10.7498/aps.52.2096
    [18] 张靖仪, 赵峥. 直线加速动态黑洞Dirac场的熵. 物理学报, 2002, 51(10): 2399-2406. doi: 10.7498/aps.51.2399
    [19] 冉扬强, 薛立徽, 胡嗣柱. 具有一维Coulomb型对称势Dirac方程的精确解. 物理学报, 2002, 51(11): 2435-2439. doi: 10.7498/aps.51.2435
    [20] 陈刚. 具有P?schl-Teller型标量势与矢量势的Klein-Gordon方程和Dirac方程的束缚态. 物理学报, 2001, 50(9): 1651-1653. doi: 10.7498/aps.50.1651
计量
  • 文章访问数:  10326
  • PDF下载量:  1456
  • 被引次数: 0
出版历程
  • 收稿日期:  2017-09-20
  • 修回日期:  2017-11-06
  • 刊出日期:  2017-11-05

/

返回文章
返回