Advertisement

Finite length effect on supercurrents between trivial and topological superconductors

  • Jorge Cayao
  • Annica M. Black-Schaffer
Open Access
Regular Article
  • 13 Downloads
Part of the following topical collections:
  1. Topological States of Matter: Theory and Applications

Abstract

We numerically analyze the effect of finite length of the superconducting regions on the low-energy spectrum, current-phase curves, and critical currents in junctions between trivial and topological superconductors. Such junctions are assumed to arise in nanowires with strong spin-orbit coupling under external magnetic fields and proximity-induced superconductivity. We show that all these quantities exhibit a strong dependence on the length of the topological sector in the topological phase and serve as indicators of the topological phase and thus the emergence of Majorana bound states at the end of the topological superconductor.

References

  1. 1.
    A.Y. Kitaev, Phys. Usp. 44, 131 (2001) ADSCrossRefGoogle Scholar
  2. 2.
    C. Nayak, S.H. Simon, A. Stern, M. Freedman, S. Das Sarma, Rev. Mod. Phys. 80, 1083 (2008) ADSCrossRefGoogle Scholar
  3. 3.
    S.D. Sarma, M. Freedman, C. Nayak, Npj Quantum Information 1, 15001 (2015) ADSCrossRefGoogle Scholar
  4. 4.
    C. Schrade, L. Fu, https://doi.org/arXiv:1803.01002 (2018)
  5. 5.
    R.M. Lutchyn, J.D. Sau, S. Das Sarma, Phys. Rev. Lett. 105, 077001 (2010) ADSCrossRefGoogle Scholar
  6. 6.
    Y. Oreg, G. Refael, F. von Oppen, Phys. Rev. Lett. 105, 177002 (2010) ADSCrossRefGoogle Scholar
  7. 7.
    J. Alicea, Phys. Rev. B 81, 125318 (2010) ADSCrossRefGoogle Scholar
  8. 8.
    K.T. Law, P.A. Lee, T.K. Ng, Phys. Rev. Lett. 103, 237001 (2009) ADSCrossRefGoogle Scholar
  9. 9.
    V. Mourik, K. Zuo, S. Frolov, S. Plissard, E. Bakkers, L. Kouwenhoven, Science 336, 1003 (2012) ADSCrossRefGoogle Scholar
  10. 10.
    M.T. Deng, C.L. Yu, G.Y. Huang, M. Larsson, P. Caroff, H.Q. Xu, Nano Lett. 12, 6414 (2012) ADSCrossRefGoogle Scholar
  11. 11.
    A. Das, Y. Ronen, Y. Most, Y. Oreg, M. Heiblum, H. Shtrikman, Nat. Phys. 8, 887 (2012) CrossRefGoogle Scholar
  12. 12.
    A.D.K. Finck, D.J. Van Harlingen, P.K. Mohseni, K. Jung, X. Li, Phys. Rev. Lett. 110, 126406 (2013) ADSCrossRefGoogle Scholar
  13. 13.
    H.O.H. Churchill, V. Fatemi, K. Grove-Rasmussen, M.T. Deng, P. Caroff, H.Q. Xu, C.M. Marcus, Phys. Rev. B 87, 241401 (2013) ADSCrossRefGoogle Scholar
  14. 14.
    E.J.H. Lee, X. Jiang, M. Houzet, R. Aguado, C.M. Lieber, S.D. Franceschi, Nat. Nanotech. 9, 79 (2014) ADSCrossRefGoogle Scholar
  15. 15.
    E.J.H. Lee, X. Jiang, R. Aguado, G. Katsaros, C.M. Lieber, S. De Franceschi, Phys. Rev. Lett. 109, 186802 (2012) ADSCrossRefGoogle Scholar
  16. 16.
    F. Pientka, G. Kells, A. Romito, P.W. Brouwer, F. von Oppen, Phys. Rev. Lett. 109, 227006 (2012) ADSCrossRefGoogle Scholar
  17. 17.
    D. Bagrets, A. Altland, Phys. Rev. Lett. 109, 227005 (2012) ADSCrossRefGoogle Scholar
  18. 18.
    J. Liu, A.C. Potter, K.T. Law, P.A. Lee, Phys. Rev. Lett. 109, 267002 (2012) ADSCrossRefGoogle Scholar
  19. 19.
    D. Rainis, L. Trifunovic, J. Klinovaja, D. Loss, Phys. Rev. B 87, 024515 (2013) ADSCrossRefGoogle Scholar
  20. 20.
    J.D. Sau, S. Das Sarma, Phys. Rev. B 88, 064506 (2013) ADSCrossRefGoogle Scholar
  21. 21.
    R. Žitko, J.S. Lim, R. López, R. Aguado, Phys. Rev. B 91, 045441 (2015) ADSCrossRefGoogle Scholar
  22. 22.
    W. Chang, S.M. Albrecht, T.S. Jespersen, F. Kuemmeth, P. Krogstrup, J. Nygård, C.M. Marcus, Nat. Nanotech. 10, 232 (2015) ADSCrossRefGoogle Scholar
  23. 23.
    A.P. Higginbotham, S.M. Albrecht, G. Kirsanskas, W. Chang, F. Kuemmeth, P. Krogstrup, T.S.J.J. Nygård, K. Flensberg, C.M. Marcus, Nat. Phys. 11, 1017 (2015) CrossRefGoogle Scholar
  24. 24.
    P. Krogstrup, N.L.B. Ziino, W. Chang, S.M. Albrecht, M.H. Madsen, E. Johnson, J. Nygård, C.M. Marcus, T.S. Jespersen, Nat. Mat. 14, 400 (2015) CrossRefGoogle Scholar
  25. 25.
    H. Zhang, Önder Gül, S. Conesa-Boj, K. Zuo, V. Mourik, F.K. de Vries, J. van Veen, D.J. van Woerkom, M.P. Nowak, M. Wimmer et al., Nat. Commun. 8, 16025 (2017) ADSCrossRefGoogle Scholar
  26. 26.
    S.M. Albrecht, A.P. Higginbotham, M. Madsen, F. Kuemmeth, T.S. Jespersen, J. Nygård, , P. Krogstrup, C.M. Marcus, Nature 531, 206 (2016) ADSCrossRefGoogle Scholar
  27. 27.
    M.T. Deng, S. Vaitiekėnas, E.B. Hansen, J. Danon, M. Leijnse, K. Flensberg, J. Nygård, P. Krogstrup, C.M. Marcus, Science 354, 1557 (2016) ADSCrossRefGoogle Scholar
  28. 28.
    F. Nichele, A.C.C. Drachmann, A.M. Whiticar, E.C.T. O’Farrell, H.J. Suominen, A. Fornieri, T. Wang, G.C. Gardner, C. Thomas, A.T. Hatke et al., Phys. Rev. Lett. 119, 136803 (2017) ADSCrossRefGoogle Scholar
  29. 29.
    H.J. Suominen, M. Kjaergaard, A.R. Hamilton, J. Shabani, C.J. Palmstrøm, C.M. Marcus, F. Nichele, Phys. Rev. Lett. 119, 176805 (2017) ADSCrossRefGoogle Scholar
  30. 30.
    M.T. Deng, S. Vaitiekénas, E. Prada, P. San-Jose, J. Nygård, P. Krogstrup, R. Aguado, C.M. Marcus, https://doi.org/arXiv:1712.03536 (2017)
  31. 31.
    H. Zhang, C.X. Liu, S. Gazibegovic, D. Xu, J.A. Logan, G. Wang, N. van Loo, J.D. Bommer, M.W. de Moor, D. Car et al., Nature 556, 74 (2018) ADSCrossRefGoogle Scholar
  32. 32.
    J. Cayao, E. Prada, P. San-José, R. Aguado, Phys. Rev. B 91, 024514 (2015) ADSCrossRefGoogle Scholar
  33. 33.
    C.X. Liu, J.D. Sau, T.D. Stanescu, S. Das Sarma, Phys. Rev. B 96, 075161 (2017) ADSCrossRefGoogle Scholar
  34. 34.
    C. Fleckenstein, F. Domínguez, N. Traverso Ziani, B. Trauzettel, Phys. Rev. B 97, 155425 (2018) ADSCrossRefGoogle Scholar
  35. 35.
    E. Prada, R. Aguado, P. San-Jose, Phys. Rev. B 96, 085418 (2017) ADSCrossRefGoogle Scholar
  36. 36.
    R. Aguado, La Rivista del Nuovo Cimento 40, 523 (2017) Google Scholar
  37. 37.
    R.M. Lutchyn, E.P.A.M. Bakkers, L.P. Kouwenhoven, P. Krogstrup, C.M. Marcus, Y. Oreg, Nat. Rev. Mater. 3, 52 (2018) ADSCrossRefGoogle Scholar
  38. 38.
    L. Fu, C.L. Kane, Phys. Rev. B 79, 161408 (2009) ADSCrossRefGoogle Scholar
  39. 39.
    H. Kwon, K. Sengupta, V. Yakovenko, Eur. Phys. J. B 37, 349 (2003) ADSCrossRefGoogle Scholar
  40. 40.
    Y. Tanaka, T. Yokoyama, N. Nagaosa, Phys. Rev. Lett. 103, 107002 (2009) ADSCrossRefGoogle Scholar
  41. 41.
    D.M. Badiane, M. Houzet, J.S. Meyer, Phys. Rev. Lett. 107, 177002 (2011) ADSCrossRefGoogle Scholar
  42. 42.
    P. San-Jose, E. Prada, R. Aguado, Phys. Rev. Lett. 108, 257001 (2012) ADSCrossRefGoogle Scholar
  43. 43.
    A. Zazunov, R. Egger, Phys. Rev. B 85, 104514 (2012) ADSCrossRefGoogle Scholar
  44. 44.
    D.I. Pikulin, Y.V. Nazarov, Phys. Rev. B 86, 140504 (2012) ADSCrossRefGoogle Scholar
  45. 45.
    F. Domínguez, F. Hassler, G. Platero, Phys. Rev. B 86, (2012) 140503 ADSCrossRefGoogle Scholar
  46. 46.
    Y. Asano, Y. Tanaka, Phys. Rev. B 87, 104513 (2013) ADSCrossRefGoogle Scholar
  47. 47.
    Y. Peng, F. Pientka, E. Berg, Y. Oreg, F. von Oppen, Phys. Rev. B 94, 085409 (2016) ADSCrossRefGoogle Scholar
  48. 48.
    J.I. Väyrynen, G. Rastelli, W. Belzig, L.I. Glazman, Phys. Rev. B 92, 134508 (2015) ADSCrossRefGoogle Scholar
  49. 49.
    E.B. Hansen, J. Danon, K. Flensberg, Phys. Rev. B 93, 094501 (2016) ADSCrossRefGoogle Scholar
  50. 50.
    P. Marra, R. Citro, A. Braggio, Phys. Rev. B 93, 220507 (2016) ADSCrossRefGoogle Scholar
  51. 51.
    A. Nava, R. Giuliano, G. Campagnano, D. Giuliano, Phys. Rev. B 94, 205125 (2016) ADSCrossRefGoogle Scholar
  52. 52.
    J. Cayao, P. San-José, A.M. Black-Schaffer, R. Aguado, E. Prada, Phys. Rev. B 96, 205425 (2017) ADSCrossRefGoogle Scholar
  53. 53.
    F. Domínguez, O. Kashuba, E. Bocquillon, J. Wiedenmann, R.S. Deacon, T.M. Klapwijk, G. Platero, L.W. Molenkamp, B. Trauzettel, E.M. Hankiewicz, Phys. Rev. B 95, 195430 (2017) ADSCrossRefGoogle Scholar
  54. 54.
    A. Zazunov, R. Egger, M. Alvarado, A.L. Yeyati, Phys. Rev. B 96, 024516 (2017) ADSCrossRefGoogle Scholar
  55. 55.
    J. Picó-Cortés, F. Domínguez, G. Platero, Phys. Rev. B 96, 125438 (2017) ADSCrossRefGoogle Scholar
  56. 56.
    A. Nava, R. Giuliano, G. Campagnano, D. Giuliano, Phys. Rev. B 95, 155449 (2017) ADSCrossRefGoogle Scholar
  57. 57.
    M. Trif, O. Dmytruk, H. Bouchiat, R. Aguado, P. Simon, Phys. Rev. B 97, 041415 (2018) ADSCrossRefGoogle Scholar
  58. 58.
    J. Cayao, A.M. Black-Schaffer, E. Prada, R. Aguado, Beilstein J. Nanotechnol. 9, 1339 (2018) CrossRefGoogle Scholar
  59. 59.
    A. Zazunov, A. Iks, M. Alvarado, A.L. Yeyati, R. Egger, https://doi.org/arXiv:1801.10343 (2018)
  60. 60.
    Z.T. Zhang, J. Phys.: Condens. Matter 30, 145402 (2018) ADSGoogle Scholar
  61. 61.
    C. Schrade, L. Fu, Phys. Rev. Lett. 120, 267002 (2018) ADSCrossRefGoogle Scholar
  62. 62.
    L.P. Rokhinson, X. Liu, J.K. Furdyna, Nat. Phys. 8, 795 (2012) CrossRefGoogle Scholar
  63. 63.
    J. Wiedenmann, E. Bocquillon, R.S. Deacon, S. Hartinger, O. Herrmann, T.M. Klapwijk, L. Maier, C. Ames, C. Brüne, C. Gould et al., Nat. Commun. 7, 10303 (2016) Google Scholar
  64. 64.
    R.S. Deacon, J. Wiedenmann, E. Bocquillon, F. Domínguez, T.M. Klapwijk, P. Leubner, C. Brüne, E.M. Hankiewicz, S. Tarucha, K. Ishibashi et al., Phys. Rev. X 7, 021011 (2017) Google Scholar
  65. 65.
    E. Bocquillon, R.S. Deacon, J. Wiedenmann, P. Leubner, T.M. Klapwijk, C. Brüne, K. Ishibashi, , H. Buhmann, L.W. Molenkamp, Nat. Nanotech. 12, 137 (2017) ADSCrossRefGoogle Scholar
  66. 66.
    K. Zuo, V. Mourik, D.B. Szombati, B. Nijholt, D.J. van Woerkom, A. Geresdi, J. Chen, V.P. Ostroukh, A.R. Akhmerov, S.R. Plissard et al., Phys. Rev. Lett. 119, 187704 (2017) ADSCrossRefGoogle Scholar
  67. 67.
    D. Laroche, D. Bouman, D.J. van Woerkom, A. Proutski, C. Murthy, D.I. Pikulin, C. Nayak, R.J.J. van Gulik, J. Nygård, P. Krogstrup et al., https://doi.org/arXiv:1712.08459 (2017)
  68. 68.
    P. San-José, J. Cayao, E. Prada, R. Aguado, New J. Phys. 15, 075019 (2013) ADSCrossRefGoogle Scholar
  69. 69.
    M. Kjaergaard, H.J. Suominen, M.P. Nowak, A.R. Akhmerov, J. Shabani, C.J. Palmstrøm, F. Nichele, C.M. Marcus, Phys. Rev. Appl. 7, 034029 (2017) ADSCrossRefGoogle Scholar
  70. 70.
    M.F. Goffman, C. Urbina, H. Pothier, J. Nygård, C.M. Marcus, P. Krogstrup, New J. Phys. 19, 092002 (2017) ADSCrossRefGoogle Scholar
  71. 71.
    Y.A. Bychkov, E.I. Rashba, Sov. Phys. JETP 39, 78 (1984) Google Scholar
  72. 72.
    Y.A. Bychkov, E.I. Rashba, J. Phys. C: Solid State Phys. 17, 6039 (1984) ADSGoogle Scholar
  73. 73.
    M. Governale, U. Zülicke, Phys. Rev. B 66, 073311 (2002) ADSCrossRefGoogle Scholar
  74. 74.
    P. Středa, P. Šeba, Phys. Rev. Lett. 90, 256601 (2003) ADSCrossRefGoogle Scholar
  75. 75.
    C.H.L. Quay, T.L. Hughes, J.A. Sulpizio, L.N. Pfeiffer, K.W. Baldwin, K.W. West, D. Goldhaber-Gordon, R. de Picciotto, Nat. Phys. 6, 336 (2010) CrossRefGoogle Scholar
  76. 76.
    S. Heedt, N.T. Ziani, F. Crépin, W. Prost, S. Trellenkamp, J. Schubert, D. Grützmacher, B. Trauzettel, T. Schäpers, Nat. Phys. 13, 563 (2017) CrossRefGoogle Scholar
  77. 77.
    S. Porta, F.M. Gambetta, N. Traverso Ziani, D.M. Kennes, M. Sassetti, F. Cavaliere, Phys. Rev. B 97, 035433 (2018) ADSCrossRefGoogle Scholar
  78. 78.
    I. van Weperen, B. Tarasinski, D. Eeltink, V.S. Pribiag, S.R. Plissard, E.P.A.M. Bakkers, L.P. Kouwenhoven, M. Wimmer, Phys. Rev. B 91, 201413 (2015) ADSCrossRefGoogle Scholar
  79. 79.
    J. Kammhuber, M.C. Cassidy, F. Pei, M.P. Nowak, A. Vuik, Önder Gül, D. Car, S.R. Plissard, E.P.A.M. Bakkers, M. Wimmer et al., Nat. Commun. 8, 478 (2017) ADSCrossRefGoogle Scholar
  80. 80.
    Önder Gül, H. Zhang, F.K. de Vries, J. van Veen, K. Zuo, V. Mourik, S. Conesa-Boj, M.P. Nowak, D.J. van Woerkom, M. Quintero-Pérez et al., Nano Lett. 17, 2690 (2017) ADSCrossRefGoogle Scholar
  81. 81.
    M. Eschrig, T. Löfwander, T. Champel, J.C. Cuevas, J. Kopu, G. Schön, J. Low Temp. Phys. 147, 457 (2007) Google Scholar
  82. 82.
    M. Eschrig, Phys. Today 64, 43 (2011) Google Scholar
  83. 83.
    J. Linder, J.W.A. Robinson, Nat. Phys. 11, 307 (2015) CrossRefGoogle Scholar
  84. 84.
    E. Prada, P. San-José, R. Aguado, Phys. Rev. B 86, 180503 (2012) ADSCrossRefGoogle Scholar
  85. 85.
    S. Das Sarma, J.D. Sau, T.D. Stanescu, Phys. Rev. B 86, 220506 (2012) ADSCrossRefGoogle Scholar
  86. 86.
    J. Klinovaja, D. Loss, Phys. Rev. B 86, 085408 (2012) ADSCrossRefGoogle Scholar
  87. 87.
    B. Béri, J.H. Bardarson, C.W.J. Beenakker, Phys. Rev. B 77, 045311 (2008) ADSCrossRefGoogle Scholar
  88. 88.
    O.V. Dimitrova, M.V. Feigel’man, JETP 102, 652 (2006) ADSCrossRefGoogle Scholar
  89. 89.
    B. van Heck, J.I. Väyrynen, L.I. Glazman, Phys. Rev. B 96, 075404 (2017) ADSCrossRefGoogle Scholar
  90. 90.
    E. Nakhmedov, O. Alekperov, F. Tatardar, Y.M. Shukrinov, I. Rahmonov, K. Sengupta, Phys. Rev. B 96, 014519 (2017) ADSCrossRefGoogle Scholar
  91. 91.
    C. Beenakker, Three “Universal” Mesoscopic Josephson Effects, in Transport phenomena in mesoscopic systems: Proceedings of the 14th Taniguchi symposium, Shima, Japan, November 10–14, 1991 (Springer-Verlag, Berlin, Heidelberg, 1992), Vol. 109, p. 235 Google Scholar
  92. 92.
    P.A. Ioselevich, P.M. Ostrovsky, M.V. Feigel’man, Phys. Rev. B 93, 125435 (2016) ADSCrossRefGoogle Scholar
  93. 93.
    C.R. Reeg, D.L. Maslov, Phys. Rev. B 92, 134512 (2015) ADSCrossRefGoogle Scholar
  94. 94.
    J. Cayao, A.M. Black-Schaffer, https://doi.org/arXiv:1805.07948 (2018)

Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.Department of Physics and AstronomyUppsala UniversityUppsalaSweden

Personalised recommendations