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The magnetic impurity effects and the existence of bound states (i.e. Yu-Shiba-Rusinov states) in superconductors have been a topic of great interest. Recently, the existence of Yu-Shiba-Rusinov states in graphene-based superconducting materials has been successfully observed in the laboratory. In this work, an effective Hamiltonian in real space is established to describe the superconducting state of graphene materials by considering a single magnetic impurity. Thus the Bogoliubov-de Gennes (BdG) equation is constructed and the self-consistency calculations of the superconducting order parameter are conducted. On this basis, the effects of magnetic impurities on graphene-like superconductors are investigated theoretically. The numerical results show that the Yu-Shiba-Rusinov state can only appear in the symmetry of the superconducting pair of the traditional s-wave coupling. The position and strength of the bound state are related to the magnetic moment of the impurity, showing a notable electron-hole asymmetry. There are no bound states in the energy gap for other pairing symmetries. This theoretical calculation not only provides a reasonable explanation for experimental phenomena, but also demonstrates that the heterojunction system composed of graphene and traditional superconductors has an s-wave superconducting pairing induced by the proximity effect in the graphene layer.
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Keywords:
- graphene /
- superconductor /
- magnetic impurity /
- bound state
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图 2 (a) 平均粒子数
$ n_i $ 的空间分布; (b) 局域磁矩$ m_i $ 的空间分布Figure 2. (a) Spatial distribution of the on-site particle number
$ n_i $ ; (b) spatial distribution of the local magnetic moment$ m_i $ .
图 3 s波配对情况下磁性杂质附近的局域电子态密度的计算结果 (a) 杂质点位置的局域电子态密度; (b) 杂质最近邻格点上的局域电子态密度
Figure 3. Numerical results of the local density of states near the magnetic impurity site for the s-wave graphene based superconductor: (a) The local density of states at the impurity site; (b) the local density of states at the nearest neighbor site of the impurity.
图 4 d+id波和p+ip波配对情况下磁性杂质附近的局域电子态密度的计算结果 (a) d+id配对杂质点位置的局域电子态密度; (b) d+id 配对杂质最近邻格点上的局域电子态密度; (c) p+ip配对杂质点位置的局域电子态密度; (d) p+ip配对杂质最近邻格点上的局域电子态密度
Figure 4. Numerical results of the local density of states near the magnetic impurity site for the d+id-wave and p+ip-wave graphene based superconductor: (a) The local density of states at the impurity site for the d+id pairing symmetry; (b) the local density of states at the nearest neighbor site of the impurity for the d+id pairing symmetry; (c) the local density of states at the impurity site for the p+ip pairing symmetry; (d) the local density of states at the nearest neighbor site of the impurity for the p+ip pairing symmetry.
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[1] Tonnoir C, Kimouche A, Coraux J, Magaud L, Delsol B, Gilles B, Chapelier C 2013 Phys. Rev. Lett. 111 246805
Google Scholar
[2] Ichinokura S, Sugawara K, Takayama A, Takahashi T, Hasegawa S 2016 Acs Nano 10 2761
Google Scholar
[3] Ichinokura S, Sugawara K, Takayama A, Takahashi T, Hasegawa S 2016 Sci. Rep. 6 23254
Google Scholar
[4] Zhou H X, Xie T, Taniguchi T, Watanabe K, Young A F 2021 Nature 598 434
Google Scholar
[5] Zhou H, Holleis L, Saito Y, Cohen L, Huynh W, Patterson C L, Yang F, Taniguchi T, Watanabe K, Young A F 2022 Science 375 774
Google Scholar
[6] Cao Y, Fatemi V, Fang S, Watanabe K, Taniguchi T, Kaxiras E, Jarillo-Herrero P 2018 Nature 556 43
Google Scholar
[7] Cao Y, Park J M, Watanabe K, Taniguchi T, Jarillo-Herrero P 2021 Nature 595 526
Google Scholar
[8] Lee G H, Lee H J 2018 Rep. Prog. Phys. 81 056502
Google Scholar
[9] Bernardo A D, Millo O, Barbone M, et al. 2017 Nat. Commun. 8 14024
Google Scholar
[10] Ma T, Yang F, Yao H, Lin H Q 2014 Phys. Rev. B 90 245114
Google Scholar
[11] Faye J P L, Sahebsara P, Senechal D 2015 Phys. Rev. B 92 085121
Google Scholar
[12] Nandkishore R, Levitov L S, Chubukov A V 2012 Nat. Phys. 8 158
Google Scholar
[13] Kiesel M L, Platt C, Hanke W, Abanin D A, Thomale R 2012 Phys. Rev. B 86 020507(R
Google Scholar
[14] Nandkishore R, Thomale R, Chubukov A V 2014 Phys. Rev. B 89 144501
Google Scholar
[15] Xiao L Y, Yu S L, Wang W, Yao Z J, Li J X 2016 Europhys. Lett. 115 27008
Google Scholar
[16] Balatsky A V, Vekhter I, Zhu J X 2006 Rev. Mod. Phys. 78 373
Google Scholar
[17] Li Y Q, Zhou T 2021 Front. Phys. 16 43502
Google Scholar
[18] Awoga O A, Black-Schaffer A M 2018 Phys. Rev. B 97 214515
Google Scholar
[19] Wehling T O, Dahal H P, Lichtenstein A I, Balatsky A V 2008 Phys. Rev. B 78 035414
Google Scholar
[20] 于渌 1965 物理学报 21 75
Google Scholar
Yu L 1965 Acta Phys. Sin. 21 75
Google Scholar
[21] Shiba H 1968 Progress of theoretical Physics 40 435
Google Scholar
[22] Rusinov A I 1969 Sov. Phys. JETP 29 1101
[23] Yazdani A, Jones B A, Lutz C P, Crommie M F, Eigler D M 1997 Science 275 1767
Google Scholar
[24] Lado J L, Fernandez-Rossier J 2016 2D Mater. 3 025001
Google Scholar
[25] Río E C, Lado J L, Cherkez V, et al. 2021 Adv. Mater. 33 2008113
Google Scholar
[26] Cervenka J, Katsnelson M I, Flipse C F J 2009 Nat. Phys. 5 840
Google Scholar
[27] Akhukov M A, Fasolino A, Gornostyrev Y N, Katsnelson M I 2012 Phys. Rev. B 85 115407
Google Scholar
[28] Dutta S, Wakabayashi K 2015 Sci. Rep. 5 11744
Google Scholar
[29] Zuo X J, An J, Gong C D 2008 Phys. Rev.B 77 144512
Google Scholar
[30] Zuo X J, An J, Gong C D 2008 Phys. Rev.B 77 212508
Google Scholar
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