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Majorana corner modes in mesoscopic superconducting square systems with mixed pairing in the presence of spin–orbit interaction

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Abstract

In the framework of the Bogoliubov-de Gennes theory, we investigate the Majorana zero-energy states in mesoscopic superconducting square systems with spin–orbit (SO) interaction. The mixed d-wave and extended s-wave condensates as well as the favored \(s+id\) state can be obtained by suitable choice of model parameters. We find that the energy gap is highly sensitive to the introduced SO interaction and the Majorana zero modes can emerge at four outer corners of a perfect square with only the Rashba SO interaction or with combined Rashba and Dresselhaus SO couplings. Furthermore, for a square sample with a centered hole, energy levels can cross the Fermi energy at appropriate Rashba SO-coupling strengths, accompanied with the occurrence of additional four Majorana corner states localized around the inner corners of the loop. The effect of Dresselhaus SO interaction on the number and location of Majorana corner states is also examined, and novel zero-energy modes mainly located at four opposite inner and outer corners along the (anti-)diagonal direction of the square loop can be realized.

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Data availability statement

This manuscript has no associated data or the data will not be deposited. [Authors’ comment: The results and data presented in this work can be replicated using the numerical procedures described in the text.]

References

  1. E.I. Rashba, Sov. Phys. Solid State 2, 1109 (1960)

    Google Scholar 

  2. Y.A. Bychkov, E.I. Rashba, JETP Lett. 39, 78 (1984)

    ADS  Google Scholar 

  3. G. Dresselhaus, A.F. Kip, C. Kittel, Phys. Rev. 95, 568 (1954)

    Article  ADS  Google Scholar 

  4. G. Dresselhaus, Phys. Rev. 100, 580 (1955)

    Article  ADS  Google Scholar 

  5. L.P. Gorkov, E.I. Rashba, Phys. Rev. Lett. 87, 037004 (2001)

    Article  ADS  Google Scholar 

  6. J.K. Xin, J. Liu, C.-X. Liu, Phys. Rev. Lett. 113, 227002 (2014)

    Article  ADS  Google Scholar 

  7. C.R. Reeg, D.L. Maslov, Phys. Rev. B 92, 134512 (2015)

    Article  ADS  Google Scholar 

  8. I.V. Bobkova, A.M. Bobkov, Phys. Rev. B 95, 184518 (2017)

    Article  ADS  Google Scholar 

  9. Y. Tanaka, Y. Mizuno, T. Yokoyama, K. Yada, M. Sato, Phys. Rev. Lett. 105, 097002 (2010)

    Article  ADS  Google Scholar 

  10. K. Yada, M. Sato, Y. Tanaka, T. Yokoyama, Phys. Rev. B 83, 064505 (2011)

    Article  ADS  Google Scholar 

  11. A.P. Schnyder, S. Ryu, Phys. Rev. B 84, 060504(R) (2011)

    Article  ADS  Google Scholar 

  12. P.M.R. Brydon, A.P. Schnyder, C. Timm, Phys. Rev. B 84, 020501(R) (2011)

    Article  ADS  Google Scholar 

  13. A.P. Schnyder, P.M.R. Brydon, C. Timm, Phys. Rev. B 85, 024522 (2012)

    Article  ADS  Google Scholar 

  14. Y. Tanaka, M. Sato, N. Nagaosa, J. Phys. Soc. Jpn. 81, 011013 (2012)

    Article  ADS  Google Scholar 

  15. C.L.M. Wong, J. Liu, K.T. Law, P.A. Lee, Phys. Rev. B 88, 060504(R) (2013)

    Article  ADS  Google Scholar 

  16. M. Sato, Y. Takahashi, S. Fujimoto, Phys. Rev. Lett. 103, 020401 (2009)

    Article  ADS  Google Scholar 

  17. M. Sato, Y. Takahashi, S. Fujimoto, Phys. Rev. B 82, 134521 (2010)

    Article  ADS  Google Scholar 

  18. J.D. Sau, R.M. Lutchyn, S. Tewari, S. Das Sarma, Phys. Rev. Lett. 104, 040502 (2010)

    Article  ADS  Google Scholar 

  19. M. Sato, S. Fujimoto, Phys. Rev. Lett. 105, 217001 (2010)

    Article  ADS  Google Scholar 

  20. G.-Q. Zha, L. Covaci, F.M. Peeters, S.-P. Zhou, Phys. Rev. B 92, 094516 (2015)

    Article  ADS  Google Scholar 

  21. Chao Li, Feng Zhai, J. Appl. Phys. 109, 093306 (2011)

    Article  ADS  Google Scholar 

  22. L. DellAnna, G. Mazzarella, L. Salasnich, Phys. Rev. A 86, 053632 (2012)

    Article  ADS  Google Scholar 

  23. Z. Li, L. Covaci, F. Marsiglio, Phys. Rev. B 85, 205112 (2012)

    Article  ADS  Google Scholar 

  24. Xu. Yan, Gu. Qiang, Solid State Commun. 187, 68–71 (2014)

    Article  ADS  Google Scholar 

  25. C.O. Dias, H.O. Frota, A. Ghosh, Phys. Status Solidi B 253, 1824–9 (2016)

    Article  ADS  Google Scholar 

  26. M. Biderang, H. Yavari, M.-H. Zare, P. Thalmeier, A. Akbari, Phys. Rev. B 98, 014524 (2018)

    Article  ADS  Google Scholar 

  27. B. Scharf, F. Pientka, H. Ren, A. Yacoby, E.M. Hankiewicz, Phys. Rev. B 99, 214503 (2019)

    Article  ADS  Google Scholar 

  28. Lu. Xiancong, Hongxu Liu, J. Phys. Condens. Matter 32, 455601 (2020)

    Article  Google Scholar 

  29. Yanick Volpez, Daniel Loss, Jelena Klinovaja, Phys. Rev. Lett. 122, 126402 (2019)

    Article  ADS  Google Scholar 

  30. Xiaoyu Zhu, Phys. Rev. Lett. 122, 236401 (2019)

    Article  ADS  Google Scholar 

  31. Majid Kheirkhah, Zhongbo Yan, Yuki Nagai, Frank Marsiglio, Phys. Rev. Lett. 125, 017001 (2020)

    Article  ADS  MathSciNet  Google Scholar 

  32. S. Ikegaya, W.B. Rui, D. Manske, A.P. Schnyder, Phys. Rev. Res. 3, 023007 (2021)

    Article  Google Scholar 

  33. Xiaoyu Zhu, Phys. Rev. B 97, 205134 (2018)

    Article  ADS  Google Scholar 

  34. Max Geier, Luka Trifunovic, Max Hoskam, Piet W. Brouwer, Phys. Rev. B 97, 205135 (2018)

    Article  ADS  Google Scholar 

  35. Eslam Khalaf, Phys. Rev. B 97, 205136 (2018)

    Article  ADS  Google Scholar 

  36. Zhongbo Yan, Phys. Rev. Lett. 123, 177001 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  37. Zhongbo Yan, Phys. Rev. B 100, 205406 (2019)

    Article  ADS  Google Scholar 

  38. R. Micnas, J. Ranninger, S. Robaszkiewicz, Rev. Mod. Phys. 62, 113 (1990)

    Article  ADS  Google Scholar 

  39. K. Kuboki, J. Phys. Soc. Jpn. 70, 2698 (2001)

    Article  ADS  Google Scholar 

  40. Guo-Qiao. Zha, EPL 130, 67005 (2020)

    Article  ADS  Google Scholar 

  41. Rui-Feng. Chai, Guo-Qiao. Zha, Eur. Phys. J. B 94, 193 (2021)

    Article  ADS  Google Scholar 

  42. R.-F. Chai, G-Q. Zha, Eur. Phys. J. B (2022) (accepted)

  43. P.G. de Gennes, Superconductivity of Metals and Alloys (Addison-Wesley, New York, 1994)

    MATH  Google Scholar 

  44. L.-F. Zhang, V.F. Becerra, L. Covaci, M.V. Milosevic, Phys. Rev. B 94, 024520 (2016)

    Article  ADS  Google Scholar 

  45. Guo-Qiao. Zha, Phys. Rev. B 95, 014510 (2017)

    Article  ADS  Google Scholar 

  46. K.M. Shen, J.C.S. Davis, Mater. Today 11, 14 (2008)

    Article  Google Scholar 

  47. Y. Zhong, Y. Wang, S. Han, Y.-F. Lv, W.-L. Wang, D. Zhang, H. Ding, Y.-M. Zhang, L. Wang, K. He, R. Zhong, J.A. Schneeloch, G.-D. Gu, C.-L. Song, X.-C. Ma, Q.-K. Xue, Sci. Bull. 61, 1239 (2016)

  48. K. Jiang, X. Wu, J. Hu, Z. Wang, Phys. Rev. Lett. 121, 227002 (2018)

    Article  ADS  Google Scholar 

Download references

Acknowledgements

This work was supported by National Natural Science Foundation of China under Grants no. 62171267 and no. 61771298.

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Authors and Affiliations

  1. Department of Physics and Shanghai Key Laboratory of High Temperature Superconductors, Shanghai University, Shanghai, 200444, China

    Yue Xie & Guo-Qiao Zha

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  1. Yue Xie
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  2. Guo-Qiao Zha
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All the authors were involved in the preparation of the manuscript. All the authors have read and approved the final manuscript.

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Correspondence to Guo-Qiao Zha.

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Xie, Y., Zha, GQ. Majorana corner modes in mesoscopic superconducting square systems with mixed pairing in the presence of spin–orbit interaction. Eur. Phys. J. B 95, 120 (2022). https://doi.org/10.1140/epjb/s10051-022-00383-0

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