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From Andreev to Majorana bound states in hybrid superconductor–semiconductor nanowires

Nature Reviews Physics volume 2, pages 575–594 (2020)Cite this article

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Abstract

Inhomogeneous superconductors can host electronic excitations, known as Andreev bound states (ABSs), below the superconducting energy gap. With the advent of topological superconductivity, a new kind of zero-energy ABS with exotic qualities, known as a Majorana bound state (MBS), has been discovered. A special property of MBS wavefunctions is their non-locality, which, together with non-Abelian braiding, is the key to their promise in topological quantum computation. We focus on hybrid superconductor–semiconductor nanowires as a flexible and promising experimental platform to realize one-dimensional topological superconductivity and MBSs. We review the main properties of ABSs and MBSs, state-of-the-art techniques for their detection and theoretical progress beyond minimal models, including different types of robust zero modes that may emerge without a band-topological transition.

Key points

  • Non-uniform superconductors and hybrid normal metal–superconducting systems can develop discrete bound states inside the superconducting gap, known as Andreev bound states (ABSs), through Andreev reflection processes.

  • Andreev bound states are a superposition of electrons and holes, and can also develop a non-trivial spin structure if the system exhibits spin–orbit coupling and low densities, as is the case in proximitized semiconducting nanowires. Precise experimental characterization of ABSs in proximitized nanowires is now possible through a range of spectroscopic techniques, including tunnelling transport, quantum dot spectroscopy and microwave cavity spectroscopy.

  • The interplay of spin–orbit coupling, Zeeman fields and low densities in proximitized semiconducting nanowires can induce a quantum phase transition into a topological superconducting phase, whereupon ABSs transform into zero-energy, topologically protected Majorana bound states (MBSs) with exotic properties, promising for quantum computation purposes.

  • The experimental search for MBSs in proximitized nanowires has encountered a rich and complex phenomenology that has required standard theoretical models to be extended substantially, revealing in the process a host of mechanisms for the emergence of zero-energy bound states beyond the original band-topological framework. These states are known in the field as quasi-MBSs, partially separated MBSs or non-topological MBSs.

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Fig. 1: Andreev bound states in semiconducting nanowires.
Fig. 2: Andreev bound states in hybrid quantum dots.
Fig. 3: Theory of Andreev reflection and bound-state formation in low-density nanowire junctions.
Fig. 4: Theory of Majorana bound-state formation and length dependence in finite nanowires and Josephson junctions.
Fig. 5: Experimental signatures in the search for Majorana bound states.
Fig. 6: Examples of theoretical results beyond the minimal model.

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References

  1. Kamerlingh Onnes, H. The superconductivity of mercury. Comm. Phys. Lab. Univ. Leiden 122, 122–124 (1911).

    Google Scholar 

  2. van Delft, D. & Kes, P. The discovery of superconductivity. Phys. Today 63, 38–42 (2010).

    Google Scholar 

  3. De Gennes, P.-G. Superconductivity of Metals and Alloys (CRC, 2018).

  4. Tinkham, M. Introduction to Superconductivity (Courier Corporation, 2004).

  5. Martin, J. D. When condensed-matter physics became king. Phys. Today 72, 30 (2019).

    ADS  Google Scholar 

  6. Bardeen, J., Cooper, L. N. & Schrieffer, J. R. Microscopic theory of superconductivity. Phys. Rev. 106, 162–164 (1957).

    ADS  MathSciNet  MATH  Google Scholar 

  7. Cooper, L. N. Bound electron pairs in a degenerate Fermi gas. Phys. Rev. 104, 1189–1190 (1956).

    ADS  MATH  Google Scholar 

  8. Ginzburg, V. L. & Landau, L. D. On the theory of superconductivity. Zh. Eksp. Teor. Fiz. 20, 1064–1082 (1950).

    Google Scholar 

  9. Cyrot, M. Ginzburg–Landau theory for superconductors. Rep. Prog. Phys. 36, 103–158 (1973).

    ADS  Google Scholar 

  10. Caroli, C., de Gennes, P. G. & Matricon, J. Bound fermion states on a vortex line in a type II superconductor. Phys. Lett. 9, 307–309 (1964).

    ADS  MATH  Google Scholar 

  11. Yu, L. Bound state in superconductors with paramagnetic impurities. Acta Phys. Sin. 21, 75–91 (1965).

    Google Scholar 

  12. Shiba, H. Classical spins in superconductors. Prog. Theor. Phys. 40, 435–451 (1968).

    ADS  Google Scholar 

  13. Rusinov, A. Superconductivity near a paramagnetic impurity. Sov. Phys. JETP 9, 85 (1969).

    Google Scholar 

  14. Blonder, G. E., Tinkham, M. & Klapwijk, T. M. Transition from metallic to tunneling regimes in superconducting microconstrictions: excess current, charge imbalance, and supercurrent conversion. Phys. Rev. B 25, 4515–4532 (1982).

    ADS  Google Scholar 

  15. Andreev, A. F. Thermal conductivity of the intermediate state in superconductors. Sov. Phys. JETP 19, 1228–1234 (1964).

    Google Scholar 

  16. Andreev, A. F. Electron spectrum of the intermediate state of superconductors. Sov. Phys. JETP 22, 18–23 (1966).

    Google Scholar 

  17. de Gennes, P. G. & Saint-James, D. Elementary excitations in the vicinity of a normal metal–superconducting metal contact. Phys. Lett. 4, 151–152 (1963).

    ADS  MATH  Google Scholar 

  18. Kulik, I. O. Macroscopic quantization and the proximity effect in SNS junctions. Sov. Phys. JETP 30, 944–950 (1970).

    ADS  Google Scholar 

  19. Sauls, J. A. Andreev bound states and their signatures. Phil. Trans. R. Soc. A 376, 20180140 (2018).

  20. Hasan, M. Z. & Kane, C. L. Colloquium: Topological insulators. Rev. Mod. Phys. 82, 3045–3067 (2010).

    ADS  Google Scholar 

  21. Qi, X.-L. & Zhang, S.-C. Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057–1110 (2011).

    ADS  Google Scholar 

  22. Thouless, D. J., Kohmoto, M., Nightingale, M. P. & den Nijs, M. Quantized Hall conductance in a two-dimensional periodic potential. Phys. Rev. Lett. 49, 405–408 (1982).

    ADS  Google Scholar 

  23. Leijnse, M. & Flensberg, K. Introduction to topological superconductivity and Majorana fermions. Semicond. Sci. Technol. 27, 124003 (2012).

    ADS  Google Scholar 

  24. Alicea, J. New directions in the pursuit of Majorana fermions in solid state systems. Rep. Prog. Phys. 75, 076501 (2012).

    ADS  Google Scholar 

  25. Beenakker, C. Search for Majorana fermions in superconductors. Annu. Rev. Cond. Mat. Phys. 4, 113–136 (2013).

    ADS  Google Scholar 

  26. Sato, M. & Fujimoto, S. Majorana fermions and topology in superconductors. J. Phys. Soc. Jpn 85, 072001 (2016).

    ADS  Google Scholar 

  27. Aguado, R. Majorana quasiparticles in condensed matter. Riv. Nuovo Cimento 40, 523–593 (2017).

    Google Scholar 

  28. Sato, M. & Ando, Y. Topological superconductors: a review. Rep. Prog. Phys. 80, 076501 (2017).

    ADS  MathSciNet  Google Scholar 

  29. Salomaa, M. & Volovik, G. Cosmiclike domain walls in superfluid 3B: instantons and diabolical points in (k,r) space. Phys. Rev. B 37, 9298 (1988).

    ADS  MathSciNet  Google Scholar 

  30. Volovik, G. E. & Volovik, G. The Universe in a Helium Droplet (Oxford Univ. Press, 2009).

  31. Read, N. & Green, D. Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries and the fractional quantum Hall effect. Phys. Rev. B 61, 10267–10297 (2000).

    ADS  Google Scholar 

  32. Kitaev, A. Y. Unpaired Majorana fermions in quantum wires. Phys. Usp. 44, 131 (2001).

    ADS  Google Scholar 

  33. Sato, M. & Fujimoto, S. Topological phases of noncentrosymmetric superconductors: edge states, Majorana fermions, and non-Abelian statistics. Phys. Rev. B 79, 094504 (2009).

    ADS  Google Scholar 

  34. Majorana, E. Teoria simmetrica dell elettrone e del positrone. Il Nuovo Cimento 14, 171–184 (1937).

    ADS  MATH  Google Scholar 

  35. Nishida, Y., Santos, L. & Chamon, C. Topological superconductors as nonrelativistic limits of Jackiw–Rossi and Jackiw–Rebbi models. Phys. Rev. B 82, 144513 (2010).

    ADS  Google Scholar 

  36. Jackiw, R. & Rossi, P. Zero modes of the vortex-fermion system. Nucl. Phys. B 190, 681–691 (1981).

    ADS  Google Scholar 

  37. Fukui, T., Shiozaki, K., Fujiwara, T. & Fujimoto, S. Bulk-edge correspondence for Chern topological phases: a viewpoint from a generalized index theorem. J. Phys. Soc. Jpn 81, 114602 (2012).

    ADS  Google Scholar 

  38. Kitaev, A. Fault-tolerant quantum computation by anyons. Ann. Phys. 303, 2–30 (2003).

    ADS  MathSciNet  MATH  Google Scholar 

  39. Nayak, C., Simon, S. H., Stern, A., Freedman, M. & Das Sarma, S. Non-Abelian anyons and topological quantum computation. Rev. Mod. Phys. 80, 1083–1159 (2008).

    ADS  MathSciNet  MATH  Google Scholar 

  40. Fu, L. & Kane, C. L. Superconducting proximity effect and Majorana fermions at the surface of a topological insulator. Phys. Rev. Lett. 100, 096407 (2008).

    ADS  Google Scholar 

  41. Lutchyn, R. M., Sau, J. D. & Das Sarma, S. Majorana fermions and a topological phase transition in semiconductor–superconductor heterostructures. Phys. Rev. Lett. 105, 077001 (2010).

    ADS  Google Scholar 

  42. Oreg, Y., Refael, G. & von Oppen, F. Helical liquids and majorana bound states in quantum wires. Phys. Rev. Lett. 105, 177002 (2010).

    ADS  Google Scholar 

  43. Středa, P. & Šeba, P. Antisymmetric spin filtering in one-dimensional electron systems with uniform spin–orbit coupling. Phys. Rev. Lett. 90, 256601 (2003).

  44. Stanescu, T. D. & Tewari, S. Majorana fermions in semiconductor nanowires: fundamentals, modeling, and experiment. J. Phys. Condens. Matter 25, 233201 (2013).

    ADS  Google Scholar 

  45. Lutchyn, R. M. et al. Majorana zero modes in superconductor–semiconductor heterostructures. Nat. Rev. Mater. 3, 52–68 (2018).

    ADS  Google Scholar 

  46. Nadj-Perge, S., Drozdov, I. K., Bernevig, B. A. & Yazdani, A. Proposal for realizing Majorana fermions in chains of magnetic atoms on a superconductor. Phys. Rev. B 88, 020407 (2013).

    ADS  Google Scholar 

  47. Nadj-Perge, S. et al. Observation of Majorana fermions in ferromagnetic atomic chains on a superconductor. Science 346, 602–607 (2014).

    ADS  Google Scholar 

  48. Ménard, G. C. et al. Two-dimensional topological superconductivity in Pb/Co/Si(111). Nat. Commun. 8, 2040 (2017).

    ADS  Google Scholar 

  49. Ménard, G. C. et al. Isolated pairs of Majorana zero modes in a disordered superconducting lead monolayer. Nat. Commun. 10, 2587 (2019).

    ADS  Google Scholar 

  50. Palacio-Morales, A. et al. Atomic-scale interface engineering of Majorana edge modes in a 2D magnet–superconductor hybrid system. Sci. Adv. 5 https://doi.org/10.1126/sciadv.aav6600 (2019).

  51. Fu, L. & Kane, C. L. Josephson current and noise at a superconductor/quantum-spin-Hall-insulator/superconductor junction. Phys. Rev. B 79, 161408 (2009).

    ADS  Google Scholar 

  52. Wiedenmann, J. et al. 4π-periodic Josephson supercurrent in HgTe-based topological Josephson junctions. Nat. Commun. 7, 10303 (2016).

    ADS  Google Scholar 

  53. Suominen, H. J. et al. Zero-energy modes from coalescing Andreev states in a two-dimensional semiconductor–superconductor hybrid platform. Phys. Rev. Lett. 119, 176805 (2017).

    ADS  Google Scholar 

  54. Nichele, F. et al. Scaling of Majorana zero-bias conductance peaks. Phys. Rev. Lett. 119, 136803 (2017).

    ADS  Google Scholar 

  55. Bretheau, L., Girit, C. O., Pothier, H., Esteve, D. & Urbina, C. Exciting Andreev pairs in a superconducting atomic contact. Nature 499, 312–315 (2013).

    ADS  Google Scholar 

  56. Janvier, C. et al. Coherent manipulation of Andreev states in superconducting atomic contacts. Science 349, 1199–1202 (2015).

    ADS  Google Scholar 

  57. Pillet, J.-D. et al. Andreev bound states in supercurrent-carrying carbon nanotubes revealed. Nat. Phys. 6, 965–969 (2010).

    Google Scholar 

  58. Eichler, A. et al. Even–odd effect in Andreev transport through a carbon nanotube quantum dot. Phys. Rev. Lett. 99, https://doi.org/10.1103/PhysRevLett.99.126602 (2007).

  59. Dirks, T. et al. Transport through Andreev bound states in a graphene quantum dot. Nat. Phys. 7, 386–390 (2011).

    Google Scholar 

  60. Deacon, R. S. et al. Tunneling spectroscopy of Andreev energy levels in a quantum dot coupled to a superconductor. Phys. Rev. Lett. 104, 076805 (2010).

    ADS  Google Scholar 

  61. Kümmel, R. Dynamics of current flow through the phase-boundary between a normal and a superconducting region. Z. Phys. A 218, 472–494 (1969).

    Google Scholar 

  62. Zhang, H. et al. Ballistic superconductivity in semiconductor nanowires. Nat. Commun. 8, 16025 (2017).

    ADS  Google Scholar 

  63. Xiang, J., Vidan, A., Tinkham, M., Westervelt, R. M. & Lieber, C. M. Ge/Si nanowire mesoscopic Josephson junctions. Nat. Nanotechnol. 1, 208–213 (2006).

    ADS  Google Scholar 

  64. Ridderbos, J. et al. Multiple Andreev reflections and Shapiro steps in a Ge–Si nanowire Josephson junction. Phys. Rev. Mater. 3, 084803 (2019).

    Google Scholar 

  65. Jespersen, T. S., Polianski, M. L., Sørensen, C. B., Flensberg, K. & Nygård, J. Mesoscopic conductance fluctuations in InAs nanowire-based SNS junctions. New J. Phys. 11, 113025 (2009).

    ADS  Google Scholar 

  66. Doh, Y.-J. et al. Tunable supercurrent through semiconductor nanowires. Science 309, 272–275 (2005).

    ADS  Google Scholar 

  67. Günel, H. Y. et al. Supercurrent in Nb/InAs-nanowire/Nb Josephson junctions. J. Appl. Phys. 112, 034316 (2012).

    ADS  Google Scholar 

  68. Goffman, M. F. et al. Conduction channels of an InAs–Al nanowire Josephson weak link. New J. Phys. 19, 092002 (2017).

    ADS  Google Scholar 

  69. Nilsson, H. A., Samuelsson, P., Caroff, P. & Xu, H. Q. Supercurrent and multiple Andreev reflections in an InSb nanowire Josephson junction. Nano Lett. 12, 228–233 (2012).

    ADS  Google Scholar 

  70. Deng, M. T. et al. Anomalous zero-bias conductance peak in a Nb–InSb nanowire–Nb hybrid device. Nano Lett. 12, 6414–6419 (2012).

    ADS  Google Scholar 

  71. Beenakker, C. Three ‘universal’ mesoscopic Josephson effects. In Transport Phenomena in Mesoscopic Systems: Proc. 14th Taniguchi Symposium, 235–253 (Springer, 1992).

  72. Beenakker, C. W. J. Quantum transport in semiconductor–superconductor microjunctions. Phys. Rev. B 46, 12841–12844 (1992).

    ADS  Google Scholar 

  73. Likharev, K. K. Superconducting weak links. Rev. Mod. Phys. 51, 101–159 (1979).

    ADS  Google Scholar 

  74. Furusaki, A. & Tsukada, M. Current-carrying states in Josephson junctions. Phys. Rev. B 43, 10164–10169 (1991).

    ADS  Google Scholar 

  75. Beenakker, C. W. J. & van Houten, H. Josephson current through a superconducting quantum point contact shorter than the coherence length. Phys. Rev. Lett. 66, 3056–3059 (1991).

    ADS  Google Scholar 

  76. Bagwell, P. F. Suppression of the Josephson current through a narrow, mesoscopic, semiconductor channel by a single impurity. Phys. Rev. B 46, 12573–12586 (1992).

    ADS  Google Scholar 

  77. Furusaki, A. Josephson current carried by Andreev levels in superconducting quantum point contacts. Superlattices Microstruct. 25, 809–818 (1999).

    ADS  Google Scholar 

  78. Landauer, R. Can a length of perfect conductor have a resistance? Phys. Lett. A 85, 91–93 (1981).

    ADS  Google Scholar 

  79. Josephson, B. D. Possible new effects in superconductive tunnelling. Phys. Lett. 1, 251–253 (1962).

    ADS  MATH  Google Scholar 

  80. Josephson, B. D. Supercurrents through barriers. Adv. Phys. 14, 419–451 (1965).

    ADS  Google Scholar 

  81. Kos, F., Nigg, S. E. & Glazman, L. I. Frequency-dependent admittance of a short superconducting weak link. Phys. Rev. B 87, 174521 (2013).

    ADS  Google Scholar 

  82. Hofheinz, M. et al. Bright side of the Coulomb blockade. Phys. Rev. Lett. 106, 217005 (2011).

    ADS  Google Scholar 

  83. Holst, T., Esteve, D., Urbina, C. & Devoret, M. H. Effect of a transmission line resonator on a small capacitance tunnel junction. Phys. Rev. Lett. 73, 3455–3458 (1994).

    ADS  Google Scholar 

  84. van Woerkom, D. J. et al. Microwave spectroscopy of spinful Andreev bound states in ballistic semiconductor Josephson junctions. Nat. Phys. 13, 876 EP – (2017).

    Google Scholar 

  85. Blais, A., Huang, R.-S., Wallraff, A., Girvin, S. M. & Schoelkopf, R. J. Cavity quantum electrodynamics for superconducting electrical circuits: an architecture for quantum computation. Phys. Rev. A 69, 062320 (2004).

    ADS  Google Scholar 

  86. Hays, M. et al. Direct microwave measurement of Andreev-bound-state dynamics in a semiconductor–nanowire Josephson junction. Phys. Rev. Lett. 121, 047001 (2018).

    ADS  Google Scholar 

  87. Mourik, V. et al. Signatures of Majorana fermions in hybrid superconductor–semiconductor nanowire devices. Science 336, 1003–1007 (2012).

    ADS  Google Scholar 

  88. Car, D. et al. InSb nanowires with built-in GaxIn1−xSb tunnel barriers for Majorana devices. Nano Lett. 17, 721–727 (2017).

    ADS  Google Scholar 

  89. Jünger, C. et al. Spectroscopy of the superconducting proximity effect in nanowires using integrated quantum dots. Commun. Phys. 2, 76– (2019).

    Google Scholar 

  90. Chang, W. et al. Hard gap in epitaxial semiconductor–superconductor nanowires. Nat. Nanotechnol. 10, 232–236 (2015).

    ADS  Google Scholar 

  91. Lee, E. J. H. et al. Spin-resolved Andreev levels and parity crossings in hybrid superconductor–semiconductor nanostructures. Nat. Nanotechnol. 9, 79–84 (2014).

    ADS  Google Scholar 

  92. Grove-Rasmussen, K. et al. Yu–Shiba–Rusinov screening of spins in double quantum dots. Nat. Commun. 9, 2376 (2018).

    ADS  Google Scholar 

  93. Anselmetti, G. L. R. et al. End-to-end correlated subgap states in hybrid nanowires. Phys. Rev. B 100, 205412 (2019).

    ADS  Google Scholar 

  94. Su, Z. et al. Mirage Andreev spectra generated by mesoscopic leads in nanowire quantum dots. Phys. Rev. Lett. 121, 127705 (2018).

    ADS  Google Scholar 

  95. Spanton, E. M. et al. Current–phase relations of few-mode InAs nanowire Josephson junctions. Nat. Phys. 13, 1177– (2017).

    Google Scholar 

  96. Hart, S. et al. Current–phase relations of inas nanowire Josephson junctions: from interacting to multimode regimes. Phys. Rev. B 100, 064523 (2019).

    ADS  Google Scholar 

  97. Nichele, F. et al. Relating Andreev bound states and supercurrents in hybrid Josephson junctions. Phys. Rev. Lett. 124, 226801 (2020).

    ADS  Google Scholar 

  98. Rifkin, R. & Deaver, B. S. Current–phase relation and phase-dependent conductance of superconducting point contacts from rf impedance measurements. Phys. Rev. B 13, 3894–3901 (1976).

    ADS  Google Scholar 

  99. Chang, W., Manucharyan, V. E., Jespersen, T. S., Nygård, J. & Marcus, C. M. Tunneling spectroscopy of quasiparticle bound states in a spinful Josephson junction. Phys. Rev. Lett. 110, 217005 (2013).

    ADS  Google Scholar 

  100. Deng, M. T. et al. Majorana bound state in a coupled quantum-dot hybrid-nanowire system. Science 354, 1557–1562 (2016).

    ADS  Google Scholar 

  101. Lee, E. J. H. et al. Scaling of subgap excitations in a superconductor–semiconductor nanowire quantum dot. Phys. Rev. B 95, 180502 (2017).

    ADS  Google Scholar 

  102. De Franceschi, S., Kouwenhoven, L., Schönenberger, C. & Wernsdorfer, W. Hybrid superconductor–quantum dot devices. Nat. Nanotechnol. 5, 703– (2010).

    ADS  Google Scholar 

  103. Hewson, A. C. The Kondo Problem to Heavy Fermions (Cambridge Univ. Press, 1993).

  104. Buitelaar, M. R., Nussbaumer, T. & Schönenberger, C. Quantum dot in the Kondo regime coupled to superconductors. Phys. Rev. Lett. 89, 256801 (2002).

    ADS  Google Scholar 

  105. Sand-Jespersen, T. et al. Kondo-enhanced Andreev tunneling in InAs nanowire quantum dots. Phys. Rev. Lett. 99, 126603 (2007).

    ADS  Google Scholar 

  106. Zitko, R., Lim, J. S., López, R. & Aguado, R. Shiba states and zero-bias anomalies in the hybrid normal–superconductor Anderson model. Phys. Rev. B 91, 045441 (2015).

    ADS  Google Scholar 

  107. Grove-Rasmussen, K. et al. Superconductivity-enhanced bias spectroscopy in carbon nanotube quantum dots. Phys. Rev. B 79, 134518 (2009).

    ADS  Google Scholar 

  108. Kumar, A. et al. Temperature dependence of Andreev spectra in a superconducting carbon nanotube quantum dot. Phys. Rev. B 89, 075428 (2014).

    ADS  Google Scholar 

  109. Jellinggaard, A., Grove-Rasmussen, K., Madsen, M. H. & Nygård, J. Tuning Yu–Shiba–Rusinov states in a quantum dot. Phys. Rev. B 94, 064520 (2016).

    ADS  Google Scholar 

  110. Li, S., Kang, N., Caroff, P. & Xu, H. Q. 0−π phase transition in hybrid superconductor–InSb nanowire quantum dot devices. Phys. Rev. B 95, 014515 (2017).

    ADS  Google Scholar 

  111. Island, J. O. et al. Proximity-induced Shiba states in a molecular junction. Phys. Rev. Lett. 118, 117001 (2017).

    ADS  Google Scholar 

  112. Andersen, B. M., Flensberg, K., Koerting, V. & Paaske, J. Nonequilibrium transport through a spinful quantum dot with superconducting leads. Phys. Rev. Lett. 107, 256802 (2011).

    ADS  Google Scholar 

  113. Lee, E. J. H. et al. Zero-bias anomaly in a nanowire quantum dot coupled to superconductors. Phys. Rev. Lett. 109, 186802 (2012).

    ADS  Google Scholar 

  114. Su, Z. et al. Andreev molecules in semiconductor nanowire double quantum dots. Nat. Commun. 8, 585 (2017).

    ADS  Google Scholar 

  115. Saldaña, J. C. E. et al. Two-impurity Yu–Shiba–Rusinov states in coupled quantum dots. Preprint at https://arxiv.org/abs/1812.09303 (2018).

  116. Heinrich, B. W., Pascual, J. I. & Franke, K. J. Single magnetic adsorbates on s-wave superconductors. Prog. Surf. Sci. 93, 1–19 (2018).

    ADS  Google Scholar 

  117. Chen, J. et al. Ubiquitous non-Majorana zero-bias conductance peaks in nanowire devices. Phys. Rev. Lett. 123, 107703 (2019).

    ADS  Google Scholar 

  118. van Dam, J. A., Nazarov, Y. V., Bakkers, E. P. A. M., De Franceschi, S. & Kouwenhoven, L. P. Supercurrent reversal in quantum dots. Nature 442, 667–670 (2006).

    ADS  Google Scholar 

  119. Delagrange, R. et al. Manipulating the magnetic state of a carbon nanotube Josephson junction using the superconducting phase. Phys. Rev. B 91, 241401 (2015).

    ADS  Google Scholar 

  120. Maurand, R. et al. First-order 0−π quantum phase transition in the Kondo regime of a superconducting carbon-nanotube quantum dot. Phys. Rev. X 2, 011009 (2012).

    Google Scholar 

  121. Estrada Saldaña, J. C. et al. Charge localization and reentrant superconductivity in a quasi-ballistic InAs nanowire coupled to superconductors. Sci. Adv. 5, https://doi.org/10.1126/sciadv.aav1235 (2019).

  122. Deng, M. T. et al. Parity independence of the zero-bias conductance peak in a nanowire based topological superconductor–quantum dot hybrid device. Sci. Rep. 4, 7261 (2014).

    Google Scholar 

  123. Deng, M.-T. et al. Nonlocality of Majorana modes in hybrid nanowires. Phys. Rev. B 98, 085125 (2018).

    ADS  Google Scholar 

  124. Cheng, M. & Lutchyn, R. M. Josephson current through a superconductor/semiconductor-nanowire/superconductor junction: effects of strong spin–orbit coupling and Zeeman splitting. Phys. Rev. B 86, 134522 (2012).

    ADS  Google Scholar 

  125. Park, S. & Levy Yeyati, A. Andreev spin qubits in multichannel Rashba nanowires. Phys. Rev. B 96, 125416 (2017).

    ADS  Google Scholar 

  126. van Heck, B., Väyrynen, J. I. & Glazman, L. I. Zeeman and spin–orbit effects in the Andreev spectra of nanowire junctions. Phys. Rev. B 96, 075404 (2017).

    ADS  Google Scholar 

  127. Dmytruk, O., Chevallier, D., Loss, D. & Klinovaja, J. Renormalization of the quantum dot g-factor in superconducting Rashba nanowires. Phys. Rev. B 98, 165403 (2018).

    ADS  Google Scholar 

  128. Tosi, L. et al. Spin–orbit splitting of Andreev states revealed by microwave spectroscopy. Phys. Rev. X 9, 011010 (2019).

    Google Scholar 

  129. Hays, M. et al. Continuous monitoring of a trapped superconducting spin. Nat. Phys. https://doi.org/10.1038/s41567-020-0952-3 (2020).

  130. Kwon, H.-J., Yakovenko, V. M. & Sengupta, K. Fractional AC Josephson effect in unconventional superconductors. Low Temp. Phys. 30, 613–619 (2004).

    ADS  Google Scholar 

  131. Pikulin, D. I. & Nazarov, Y. V. Phenomenology and dynamics of a Majorana Josephson junction. Phys. Rev. B 86, 140504 (2012).

    ADS  Google Scholar 

  132. San-Jose, P., Prada, E. & Aguado, R. AC Josephson effect in finite-length nanowire junctions with Majorana modes. Phys. Rev. Lett. 108, 257001 (2012).

    ADS  Google Scholar 

  133. Klinovaja, J. & Loss, D. Composite Majorana fermion wave functions in nanowires. Phys. Rev. B 86, 085408 (2012).

    ADS  Google Scholar 

  134. Mishmash, R. V., Aasen, D., Higginbotham, A. P. & Alicea, J. Approaching a topological phase transition in Majorana nanowires. Phys. Rev. B 93, 245404 (2016).

    ADS  Google Scholar 

  135. Das, A. et al. Zero-bias peaks and splitting in an Al–InAs nanowire topological superconductor as a signature of Majorana fermions. Nat. Phys. 8, 887 (2012).

    Google Scholar 

  136. Gül, Ö. et al. Ballistic Majorana nanowire devices. Nat. Nanotechnol. 13, 192–197 (2018).

    ADS  Google Scholar 

  137. Grivnin, A., Bor, E., Heiblum, M., Oreg, Y. & Shtrikman, H. Concomitant opening of a bulk-gap with an emerging possible Majorana zero mode. Nat. Commun. 10, 1940 (2019).

    ADS  Google Scholar 

  138. Law, K. T., Lee, P. A. & Ng, T. K. Majorana fermion induced resonant Andreev reflection. Phys. Rev. Lett. 103, 237001 (2009).

    ADS  Google Scholar 

  139. Flensberg, K. Tunneling characteristics of a chain of Majorana bound states. Phys. Rev. B 82, 180516 (2010).

    ADS  Google Scholar 

  140. Wimmer, M., Akhmerov, A. R., Dahlhaus, J. P. & Beenakker, C. W. J. Quantum point contact as a probe of a topological superconductor. New J. Phys. 13, 053016 (2011).

    ADS  Google Scholar 

  141. Prada, E., San-Jose, P. & Aguado, R. Transport spectroscopy of NS nanowire junctions with Majorana fermions. Phys. Rev. B 86, 180503(R) (2012).

    ADS  Google Scholar 

  142. Setiawan, F., Liu, C.-X., Sau, J. D. & Das Sarma, S. Electron temperature and tunnel coupling dependence of zero-bias and almost-zero-bias conductance peaks in Majorana nanowires. Phys. Rev. B 96, 184520 (2017).

    ADS  Google Scholar 

  143. Chen, J. et al. Experimental phase diagram of zero-bias conductance peaks in superconductor/semiconductor nanowire devices. Sci. Adv. 3, https://advances.sciencemag.org/content/3/9/e1701476(2017).

  144. Albrecht, S. M. et al. Exponential protection of zero modes in Majorana islands. Nature 531, 206–209 (2016).

    ADS  Google Scholar 

  145. Shen, J. et al. Parity transitions in the superconducting ground state of hybrid InSb–Al Coulomb islands. Nat. Commun. 9, 4801 (2018).

    ADS  Google Scholar 

  146. Vaitiekenas, S. et al. Flux-induced topological superconductivity in full-shell nanowires. Science 367, https://science.sciencemag.org/content/367/6485/eaav3392 (2020).

  147. van Heck, B., Hassler, F., Akhmerov, A. R. & Beenakker, C. W. J. Coulomb stability of the 4π-periodic Josephson effect of Majorana fermions. Phys. Rev. B 84, 180502 (2011).

    Google Scholar 

  148. Houzet, M., Meyer, J. S., Badiane, D. M. & Glazman, L. I. Dynamics of Majorana states in a topological Josephson junction. Phys. Rev. Lett. 111, 046401 (2013).

    ADS  Google Scholar 

  149. Parker, W. H., Taylor, B. N. & Langenberg, D. N. Measurement of \(\frac{2e}{h}\) using the ac Josephson effect and its implications for quantum electrodynamics. Phys. Rev. Lett. 18, 287–291 (1967).

    ADS  Google Scholar 

  150. Shapiro, S. Josephson currents in superconducting tunneling: the effect of microwaves and other observations. Phys. Rev. Lett. 11, 80–82 (1963).

    ADS  Google Scholar 

  151. Domínguez, F., Hassler, F. & Platero, G. Dynamical detection of Majorana fermions in current-biased nanowires. Phys. Rev. B 86, 140503 (2012).

    ADS  Google Scholar 

  152. Sau, J. D. & Setiawan, F. Detecting topological superconductivity using low-frequency doubled Shapiro steps. Phys. Rev. B 95, 060501 (2017).

    ADS  Google Scholar 

  153. Rokhinson, L. P., Liu, X. & Furdyna, J. K. The fractional a.c. Josephson effect in a semiconductor–superconductor nanowire as a signature of Majorana particles. Nat. Phys. 8, 795–799 (2012).

    Google Scholar 

  154. Kamata, H. et al. Anomalous modulation of Josephson radiation in nanowire-based Josephson junctions. Phys. Rev. B 98, 041302 (2018).

    ADS  Google Scholar 

  155. De Cecco, A., Le Calvez, K., Sacépé, B., Winkelmann, C. B. & Courtois, H. Interplay between electron overheating and ac Josephson effect. Phys. Rev. B 93, 180505 (2016).

    Google Scholar 

  156. Le Calvez, K. et al. Joule overheating poisons the fractional ac Josephson effect in topological Josephson junctions. Commun. Phys. 2, 4 (2019).

    Google Scholar 

  157. Picó-Cortés, J., Domínguez, F. & Platero, G. Signatures of a 4π-periodic supercurrent in the voltage response of capacitively shunted topological Josephson junctions. Phys. Rev. B 96, 125438 (2017).

    ADS  Google Scholar 

  158. Virtanen, P. & Recher, P. Microwave spectroscopy of Josephson junctions in topological superconductors. Phys. Rev. B 88, 144507 (2013).

    ADS  Google Scholar 

  159. Domínguez, F. et al. Josephson junction dynamics in the presence of 2π- and 4π-periodic supercurrents. Phys. Rev. B 95, 195430 (2017).

    ADS  Google Scholar 

  160. Laroche, D. et al. Observation of the 4π-periodic Josephson effect in indium arsenide nanowires. Nat. Commun. 10, 245– (2019).

    ADS  Google Scholar 

  161. Väyrynen, J. I., Rastelli, G., Belzig, W. & Glazman, L. I. Microwave signatures of Majorana states in a topological Josephson junction. Phys. Rev. B 92, 134508 (2015).

    ADS  Google Scholar 

  162. San-Jose, P., Prada, E. & Aguado, R. Mapping the topological phase diagram of multiband semiconductors with supercurrents. Phys. Rev. Lett. 112, 137001 (2014).

    ADS  Google Scholar 

  163. Tiira, J. et al. Magnetically-driven colossal supercurrent enhancement in InAs nanowire Josephson junctions. Nat. Commun. 8, 14984 (2017).

    ADS  Google Scholar 

  164. Cayao, J., San-Jose, P., Black-Schaffer, A. M., Aguado, R. & Prada, E. Majorana splitting from critical currents in Josephson junctions. Phys. Rev. B 96, 205425 (2017).

    ADS  Google Scholar 

  165. Peng, Y., Pientka, F., Berg, E., Oreg, Y. & von Oppen, F. Signatures of topological Josephson junctions. Phys. Rev. B 94, 085409 (2016).

    ADS  Google Scholar 

  166. Potter, A. C. & Lee, P. A. Multichannel generalization of Kitaev’s Majorana end states and a practical route to realize them in thin films. Phys. Rev. Lett. 105, 227003 (2010).

    ADS  Google Scholar 

  167. Potter, A. C. & Lee, P. A. Majorana end states in multiband microstructures with Rashba spin–orbit coupling. Phys. Rev. B 83, 094525 (2011).

    ADS  Google Scholar 

  168. Lutchyn, R. M., Stanescu, T. D. & Das Sarma, S. Search for Majorana fermions in multiband semiconducting nanowires. Phys. Rev. Lett. 106, 127001 (2011).

    ADS  Google Scholar 

  169. Lutchyn, R. M. & Fisher, M. P. A. Interacting topological phases in multiband nanowires. Phys. Rev. B 84, 214528 (2011).

    ADS  Google Scholar 

  170. Nijholt, B. & Akhmerov, A. R. Orbital effect of magnetic field on the Majorana phase diagram. Phys. Rev. B 93, 235434 (2016).

    ADS  Google Scholar 

  171. Winkler, G. W. et al. Unified numerical approach to topological semiconductor–superconductor heterostructures. Phys. Rev. B 99, 245408 (2019).

    ADS  Google Scholar 

  172. Nilsson, H. A. et al. Giant, level-dependent g factors in InSb nanowire quantum dots. Nano Lett. 9, 3151–3156 (2009).

    ADS  Google Scholar 

  173. Winkler, G. W. et al. Orbital contributions to the electron g factor in semiconductor nanowires. Phys. Rev. Lett. 119, 037701 (2017).

    ADS  Google Scholar 

  174. Takei, S., Fregoso, B. M., Hui, H.-Y., Lobos, A. M. & Das Sarma, S. Soft superconducting gap in semiconductor Majorana nanowires. Phys. Rev. Lett. 110, 186803 (2013).

    ADS  Google Scholar 

  175. Krogstrup, P. et al. Epitaxy of semiconductor–superconductor nanowires. Nat. Mater. 14, 400–406 (2015).

    ADS  Google Scholar 

  176. Gazibegovic, S. et al. Epitaxy of advanced nanowire quantum devices. Nature 548, 434–438 (2017).

    ADS  Google Scholar 

  177. Stanescu, T. D. & Tewari, S. Disentangling Majorana fermions from topologically trivial low-energy states in semiconductor Majorana wires. Phys. Rev. B 87, 140504(R) (2013).

    ADS  Google Scholar 

  178. Cole, W. S., Das Sarma, S. & Stanescu, T. D. Effects of large induced superconducting gap on semiconductor Majorana nanowires. Phys. Rev. B 92, 174511 (2015).

    ADS  Google Scholar 

  179. Reeg, C., Loss, D. & Klinovaja, J. Finite-size effects in a nanowire strongly coupled to a thin superconducting shell. Phys. Rev. B 96, 125426 (2017).

    ADS  Google Scholar 

  180. Reeg, C., Loss, D. & Klinovaja, J. Metallization of a Rashba wire by a superconducting layer in the strong-proximity regime. Phys. Rev. B 97, 165425 (2018).

    ADS  Google Scholar 

  181. Awoga, O. A., Cayao, J. & Black-Schaffer, A. M. Supercurrent detection of topologically trivial zero-energy states in nanowire junctions. Phys. Rev. Lett. 123, 117001 (2019).

    ADS  Google Scholar 

  182. Antipov, A. E. et al. Effects of gate-induced electric fields on semiconductor Majorana nanowires. Phys. Rev. X 8, 031041 (2018).

    Google Scholar 

  183. Mikkelsen, A. E. G., Kotetes, P., Krogstrup, P. & Flensberg, K. Hybridization at superconductor–semiconductor interfaces. Phys. Rev. X 8, 031040 (2018).

    Google Scholar 

  184. Vaitiekenas, S., Deng, M.-T., Nygård, J., Krogstrup, P. & Marcus, C. M. Effective g factor of subgap states in hybrid nanowires. Phys. Rev. Lett. 121, 037703 (2018).

    ADS  Google Scholar 

  185. Pan, H., Sau, J. D., Stanescu, T. D. & Das Sarma, S. Curvature of gap closing features and the extraction of Majorana nanowire parameters. Phys. Rev. B 99, 054507 (2019).

    ADS  Google Scholar 

  186. Escribano, S. D., Yeyati, A. L. & Prada, E. Improved effective equation for the Rashba spin–orbit coupling in semiconductor nanowires. Preprint at https://arxiv.org/abs/2001.04375 (2020).

  187. de Moor, M. W. A. et al. Electric field tunable superconductor–semiconductor coupling in Majorana nanowires. New J. Phys. 20, 103049 (2018).

    Google Scholar 

  188. Lim, J. S., Serra, Lmc, López, R. & Aguado, R. Magnetic-field instability of Majorana modes in multiband semiconductor wires. Phys. Rev. B 86, 121103 (2012).

    ADS  Google Scholar 

  189. Das Sarma, S., Sau, J. D. & Stanescu, T. D. Splitting of the zero-bias conductance peak as smoking gun evidence for the existence of the Majorana mode in a superconductor–semiconductor nanowire. Phys. Rev. B 86, 220506 (2012).

    Google Scholar 

  190. Rainis, D., Trifunovic, L., Klinovaja, J. & Loss, D. Towards a realistic transport modeling in a superconducting nanowire with Majorana fermions. Phys. Rev. B 87, 024515 (2013).

    ADS  Google Scholar 

  191. Sharma, G., Zeng, C., Stanescu, T. D. & Tewari, S. Majorana versus Andreev bound state energy oscillations in a 1D semiconductor–superconductor heterostructure. Preprint at https://arxiv.org/abs/2001.10551 (2020).

  192. Domínguez, F. et al. Zero-energy pinning from interactions in Majorana nanowires. NPJ Quant. Mater. 2, 13 (2017).

    ADS  Google Scholar 

  193. Escribano, S. D., Levy Yeyati, A. & Prada, E. Interaction-induced zero-energy pinning and quantum dot formation in Majorana nanowires. Beilstein J. Nanotechnol. 9, 2171–2180 (2018).

    Google Scholar 

  194. Dmytruk, O. & Klinovaja, J. Suppression of the overlap between Majorana fermions by orbital magnetic effects in semiconducting–superconducting nanowires. Phys. Rev. B 97, 155409 (2018).

    ADS  Google Scholar 

  195. Liu, C.-X., Sau, J. D. & Das Sarma, S. Role of dissipation in realistic Majorana nanowires. Phys. Rev. B 95, 054502 (2017).

    ADS  Google Scholar 

  196. Danon, J., Hansen, E. B. & Flensberg, K. Conductance spectroscopy on Majorana wires and the inverse proximity effect. Phys. Rev. B 96, 125420 (2017).

    ADS  Google Scholar 

  197. Avila, J., Peñaranda, F., Prada, E., San-Jose, P. & Aguado, R. Non-hermitian topology as a unifying framework for the Andreev versus Majorana states controversy. Commun. Phys. 2, 133 (2019).

    Google Scholar 

  198. Peñaranda, F., Aguado, R., San-Jose, P. & Prada, E. Quantifying wave-function overlaps in inhomogeneous Majorana nanowires. Phys. Rev. B 98, 235406 (2018).

    ADS  Google Scholar 

  199. Fleckenstein, C., Domínguez, F., Traverso Ziani, N. & Trauzettel, B. Decaying spectral oscillations in a Majorana wire with finite coherence length. Phys. Rev. B 97, 155425 (2018).

    ADS  Google Scholar 

  200. Cao, Z. et al. Decays of Majorana or Andreev oscillations induced by steplike spin–orbit coupling. Phys. Rev. Lett. 122, 147701 (2019).

    ADS  Google Scholar 

  201. Stanescu, T. D., Tewari, S., Sau, J. D. & Das Sarma, S. To close or not to close: the fate of the superconducting gap across the topological quantum phase transition in Majorana-carrying semiconductor nanowires. Phys. Rev. Lett. 109, 266402 (2012).

    ADS  Google Scholar 

  202. Huang, Y. et al. Metamorphosis of Andreev bound states into Majorana bound states in pristine nanowires. Phys. Rev. B 98, 144511 (2018).

    ADS  Google Scholar 

  203. Vuik, A., Eeltink, D., Akhmerov, A. R. & Wimmer, M. Effects of the electrostatic environment on the Majorana nanowire devices. New J. Phys. 18, 033013 (2016).

    ADS  Google Scholar 

  204. Woods, B. D., Stanescu, T. D. & Das Sarma, S. Effective theory approach to the Schrödinger–Poisson problem in semiconductor Majorana devices. Phys. Rev. B 98, 035428 (2018).

    ADS  Google Scholar 

  205. Bagrets, D. & Altland, A. Class D spectral peak in Majorana quantum wires. Phys. Rev. Lett. 109, 227005 (2012).

    ADS  Google Scholar 

  206. Beenakker, C. W. J. Random-matrix theory of Majorana fermions and topological superconductors. Rev. Mod. Phys. 87, 1037–1066 (2015).

    ADS  MathSciNet  Google Scholar 

  207. Altland, A. & Zirnbauer, M. R. Nonstandard symmetry classes in mesoscopic normal–superconducting hybrid structures. Phys. Rev. B 55, 1142–1161 (1997).

    ADS  Google Scholar 

  208. Schnyder, A. P., Ryu, S., Furusaki, A. & Ludwig, A. W. W. Classification of topological insulators and superconductors in three spatial dimensions. Phys. Rev. B 78, 195125 (2008).

    ADS  Google Scholar 

  209. Woods, B. D., Chen, J., Frolov, S. M. & Stanescu, T. D. Zero-energy pinning of topologically trivial bound states in multiband semiconductor–superconductor nanowires. Phys. Rev. B 100, 125407 (2019).

    ADS  Google Scholar 

  210. Chevallier, D., Sticlet, D., Simon, P. & Bena, C. Mutation of Andreev into Majorana bound states in long superconductor–normal and superconductor–normal–superconductor junctions. Phys. Rev. B 85, 235307 (2012).

    ADS  Google Scholar 

  211. Cayao, J., Prada, E., San-Jose, P. & Aguado, R. Sns junctions in nanowires with spin–orbit coupling: role of confinement and helicity on the subgap spectrum. Phys. Rev. B 91, 024514 (2015).

    ADS  Google Scholar 

  212. Liu, C.-X., Sau, J. D., Stanescu, T. D. & Das Sarma, S. Andreev bound states versus Majorana bound states in quantum dot–nanowire–superconductor hybrid structures: trivial versus topological zero-bias conductance peaks. Phys. Rev. B 96, 075161 (2017).

    ADS  Google Scholar 

  213. Ptok, A., Kobiałka, A. & Domański, T. Controlling the bound states in a quantum-dot hybrid nanowire. Phys. Rev. B 96, 195430 (2017).

    ADS  Google Scholar 

  214. Moore, C., Zeng, C., Stanescu, T. D. & Tewari, S. Quantized zero-bias conductance plateau in semiconductor–superconductor heterostructures without topological Majorana zero modes. Phys. Rev. B 98, 155314 (2018).

    ADS  Google Scholar 

  215. Reeg, C., Dmytruk, O., Chevallier, D., Loss, D. & Klinovaja, J. Zero-energy Andreev bound states from quantum dots in proximitized Rashba nanowires. Phys. Rev. B 98, 245407 (2018).

    ADS  Google Scholar 

  216. Vuik, A., Nijholt, B., Akhmerov, A. R. & Wimmer, M. Reproducing topological properties with quasi-Majorana states. SciPost Phys. 7, 61 (2019).

    ADS  Google Scholar 

  217. Stanescu, T. D. & Tewari, S. Robust low-energy Andreev bound states in semiconductor–superconductor structures: importance of partial separation of component Majorana bound states. Phys. Rev. B 100, 155429 (2019).

    ADS  Google Scholar 

  218. Kells, G., Meidan, D. & Brouwer, P. W. Near-zero-energy end states in topologically trivial spin–orbit coupled superconducting nanowires with a smooth confinement. Phys. Rev. B 86, 100503 (2012).

    ADS  Google Scholar 

  219. Moore, C., Stanescu, T. D. & Tewari, S. Two-terminal charge tunneling: disentangling Majorana zero modes from partially separated Andreev bound states in semiconductor–superconductor heterostructures. Phys. Rev. B 97, 165302 (2018).

    ADS  Google Scholar 

  220. Liu, C.-X., Sau, J. D. & Das Sarma, S. Distinguishing topological Majorana bound states from trivial Andreev bound states: proposed tests through differential tunneling conductance spectroscopy. Phys. Rev. B 97, 214502 (2018).

    ADS  Google Scholar 

  221. Pikulin, D. I. & Nazarov, Y. V. Two types of topological transitions in finite Majorana wires. Phys. Rev. B 87, 235421 (2013).

    ADS  Google Scholar 

  222. San-Jose, P., Cayao, J., Prada, E. & Aguado, R. Majorana bound states from exceptional points in non-topological superconductors. Sci. Rep. 6, 21427 (2016).

    ADS  Google Scholar 

  223. Leykam, D., Bliokh, K. Y., Huang, C., Chong, Y. D. & Nori, F. Edge modes, degeneracies, and topological numbers in non-Hermitian systems. Phys. Rev. Lett. 118, 040401 (2017).

    ADS  MathSciNet  Google Scholar 

  224. Shen, H., Zhen, B. & Fu, L. Topological band theory for non-Hermitian Hamiltonians. Phys. Rev. Lett. 120, 146402 (2018).

    ADS  MathSciNet  Google Scholar 

  225. Gong, Z. et al. Topological phases of non-Hermitian systems. Phys. Rev. X 8, 031079 (2018).

    Google Scholar 

  226. McGinley, M. & Cooper, N. R. Classification of topological insulators and superconductors out of equilibrium. Phys. Rev. B 99, 075148 (2019).

    ADS  Google Scholar 

  227. Yu, P. et al. Non-Majorana states yield nearly quantized conductance in superconductor–semiconductor nanowire devices. Preprint at https://arxiv.org/abs/2004.08583 (2020).

  228. Roy, D., Bondyopadhaya, N. & Tewari, S. Topologically trivial zero-bias conductance peak in semiconductor Majorana wires from boundary effects. Phys. Rev. B 88, 020502 (2013).

    ADS  Google Scholar 

  229. Stanescu, T. D. & Tewari, S. Nonlocality of zero-bias anomalies in the topologically trivial phase of Majorana wires. Phys. Rev. B 89, 220507 (2014).

    ADS  Google Scholar 

  230. Szumniak, P., Chevallier, D., Loss, D. & Klinovaja, J. Spin and charge signatures of topological superconductivity in Rashba nanowires. Phys. Rev. B 96, 041401 (2017).

    ADS  Google Scholar 

  231. Chiu, C.-K., Sau, J. D. & Das Sarma, S. Conductance of a superconducting Coulomb-blockaded Majorana nanowire. Phys. Rev. B 96, 054504 (2017).

    ADS  Google Scholar 

  232. Budich, J. C., Walter, S. & Trauzettel, B. Failure of protection of Majorana based qubits against decoherence. Phys. Rev. B 85, 121405 (2012).

    ADS  Google Scholar 

  233. Goldstein, G. & Chamon, C. Decay rates for topological memories encoded with Majorana fermions. Phys. Rev. B 84, 205109 (2011).

    ADS  Google Scholar 

  234. Rainis, D. & Loss, D. Majorana qubit decoherence by quasiparticle poisoning. Phys. Rev. B 85, 174533 (2012).

    ADS  Google Scholar 

  235. Pedrocchi, F. L. & DiVincenzo, D. P. Majorana braiding with thermal noise. Phys. Rev. Lett. 115, 120402 (2015).

    ADS  Google Scholar 

  236. Knapp, C., Karzig, T., Lutchyn, R. M. & Nayak, C. Dephasing of Majorana-based qubits. Phys. Rev. B 97, 125404 (2018).

    ADS  Google Scholar 

  237. Aseev, P. P., Marra, P., Stano, P., Klinovaja, J. & Loss, D. Degeneracy lifting of Majorana bound states due to electron–phonon interactions. Phys. Rev. B 99, 205435 (2019).

    ADS  Google Scholar 

  238. Trif, M. & Tserkovnyak, Y. Resonantly tunable Majorana polariton in a microwave cavity. Phys. Rev. Lett. 109, 257002 (2012).

    ADS  Google Scholar 

  239. Schmidt, T. L., Nunnenkamp, A. & Bruder, C. Majorana qubit rotations in microwave cavities. Phys. Rev. Lett. 110, 107006 (2013).

    ADS  Google Scholar 

  240. Dmytruk, O., Trif, M. & Simon, P. Cavity quantum electrodynamics with mesoscopic topological superconductors. Phys. Rev. B 92, 245432 (2015).

    ADS  Google Scholar 

  241. Aseev, P. P., Klinovaja, J. & Loss, D. Lifetime of Majorana qubits in Rashba nanowires with nonuniform chemical potential. Phys. Rev. B 98, 155414 (2018).

    ADS  Google Scholar 

  242. Schmidt, M. J., Rainis, D. & Loss, D. Decoherence of Majorana qubits by noisy gates. Phys. Rev. B 86, 085414 (2012).

    ADS  Google Scholar 

  243. Lai, H.-L., Yang, P.-Y., Huang, Y.-W. & Zhang, W.-M. Exact master equation and non-Markovian decoherence dynamics of Majorana zero modes under gate-induced charge fluctuations. Phys. Rev. B 97, 054508 (2018).

    ADS  Google Scholar 

  244. Scheurer, M. S. & Shnirman, A. Nonadiabatic processes in Majorana qubit systems. Phys. Rev. B 88, 064515 (2013).

    ADS  Google Scholar 

  245. Sekania, M., Plugge, S., Greiter, M., Thomale, R. & Schmitteckert, P. Braiding errors in interacting Majorana quantum wires. Phys. Rev. B 96, 094307 (2017).

    ADS  Google Scholar 

  246. Hoffman, S., Schrade, C., Klinovaja, J. & Loss, D. Universal quantum computation with hybrid spin–Majorana qubits. Phys. Rev. B 94, 045316 (2016).

    ADS  Google Scholar 

  247. Wakatsuki, R., Ezawa, M. & Nagaosa, N. Majorana fermions and multiple topological phase transition in Kitaev ladder topological superconductors. Phys. Rev. B 89, 174514 (2014).

    ADS  Google Scholar 

  248. Karzig, T. et al. Scalable designs for quasiparticle-poisoning-protected topological quantum computation with Majorana zero modes. Phys. Rev. B 95, 235305 (2017).

    ADS  Google Scholar 

  249. Plugge, S., Rasmussen, A., Egger, R. & Flensberg, K. Majorana box qubits. New J. Phys. 19, 012001 (2017).

    ADS  Google Scholar 

  250. Prada, E., Aguado, R. & San-Jose, P. Measuring Majorana nonlocality and spin structure with a quantum dot. Phys. Rev. B 96, 085418 (2017).

    ADS  Google Scholar 

  251. Clarke, D. J. Experimentally accessible topological quality factor for wires with zero energy modes. Phys. Rev. B 96, 201109 (2017).

    Google Scholar 

  252. Schuray, A., Weithofer, L. & Recher, P. Fano resonances in Majorana bound states–quantum dot hybrid systems. Phys. Rev. B 96, 085417 (2017).

    ADS  Google Scholar 

  253. Ménard, G. C. et al. Conductance-matrix symmetries of a three-terminal hybrid device. Phys. Rev. Lett. 124, 036802 (2020).

    ADS  Google Scholar 

  254. Puglia, D. et al. Closing of the induced gap in a hybrid superconductor-semiconductor nanowire. Preprint at https://arxiv.org/abs/2006.01275 (2020).

  255. Zhang, H., Liu, D. E., Wimmer, M. & Kouwenhoven, L. P. Next steps of quantum transport in Majorana nanowire devices. Nat. Commun. 10, 5128 (2019).

    ADS  Google Scholar 

  256. Frolov, S. M., Manfra, M. J. & Sau, J. D. Topological superconductivity in hybrid devices. Nat. Phys. 16, 718–724 (2020).

    Google Scholar 

  257. Aguado, R. & Kouwenhoven, L. P. Majorana qubits for topological quantum computing. Phys. Today 73, 44–50 (2020).

    Google Scholar 

  258. Peñaranda, F., Aguado, R., San-Jose, P. & Prada, E. Even–odd effect and Majorana states in full-shell nanowires. Phys. Rev. Res. 2, 023171 (2020).

    Google Scholar 

  259. Vaitiekėnas, S., Liu, Y., Krogstrup, P. & Marcus, C. M. Zero-field topological superconductivity in ferromagnetic hybrid nanowires. Preprint at https://arxiv.org/abs/2004.02226 (2020).

  260. Larsen, T. W. et al. Semiconductor-nanowire-based superconducting qubit. Phys. Rev. Lett. 115, 127001 (2015).

    ADS  Google Scholar 

  261. de Lange, G. et al. Realization of microwave quantum circuits using hybrid superconducting–semiconducting nanowire Josephson elements. Phys. Rev. Lett. 115, 127002 (2015).

    ADS  Google Scholar 

  262. Sabonis, D. et al. Destructive Little–Parks effect in a full-shell nanowire-based transmon. Preprint at https://arxiv.org/abs/2005.01748 (2020).

  263. Bargerbos, A. et al. Observation of vanishing charge dispersion of a nearly open superconducting island. Phys. Rev. Lett. 124, 246802 (2020).

    ADS  Google Scholar 

  264. Kringhøj, A. et al. Suppressed charge dispersion via resonant tunneling in a single-channel transmon. Phys. Rev. Lett. 124, 246803 (2020).

    ADS  Google Scholar 

  265. Ginossar, E. & Grosfeld, E. Microwave transitions as a signature of coherent parity mixing effects in the Majorana-transmon qubit. Nat. Commun. 5, 4772 (2014).

    ADS  Google Scholar 

  266. Trif, M., Dmytruk, O., Bouchiat, H., Aguado, R. & Simon, P. Dynamic current susceptibility as a probe of Majorana bound states in nanowire-based Josephson junctions. Phys. Rev. B 97, 041415 (2018).

    ADS  Google Scholar 

  267. Keselman, A., Murthy, C., van Heck, B. & Bauer, B. Spectral response of Josephson junctions with low-energy quasiparticles. SciPost Phys. 7, 50 (2019).

    ADS  Google Scholar 

  268. Avila, J., Prada, E., San-Jose, P. & Aguado, R. Superconducting islands with semiconductor-nanowire-based topological Josephson junctions. Preprint at https://arxiv.org/abs/2003.02852 (2020).

  269. Avila, J., Prada, E., San-Jose, P. & Aguado, R. Majorana oscillations and parity crossings in semiconductor-nanowire-based transmon qubits. Preprint at https://arxiv.org/abs/2003.02858 (2020).

  270. Finocchiaro, F., Guinea, F. & San-Jose, P. Topological π junctions from crossed Andreev reflection in the quantum Hall regime. Phys. Rev. Lett. 120, 116801 (2018).

    ADS  Google Scholar 

  271. Thakurathi, M., Simon, P., Mandal, I., Klinovaja, J. & Loss, D. Majorana Kramers pairs in Rashba double nanowires with interactions and disorder. Phys. Rev. B 97, 045415 (2018).

    ADS  Google Scholar 

  272. Young, A. F. et al. Tunable symmetry breaking and helical edge transport in a graphene quantum spin Hall state. Nature 505, 528–532 (2014).

    ADS  Google Scholar 

  273. Lee, G.-H. et al. Inducing superconducting correlation in quantum Hall edge states. Nat. Phys. 13, 693–698 (2017).

    Google Scholar 

  274. San-Jose, P., Lado, J. L., Aguado, R., Guinea, F. & Fernández-Rossier, J. Majorana zero modes in graphene. Phys. Rev. X 5, 041042 (2015).

    Google Scholar 

  275. Estrada Saldaña, J. C. et al. Supercurrent in a double quantum dot. Phys. Rev. Lett. 121, 257701 (2018).

    ADS  Google Scholar 

  276. Fulga, I. C., Haim, A., Akhmerov, A. R. & Oreg, Y. Adaptive tuning of Majorana fermions in a quantum dot chain. New J. Phys. 15, 045020 (2013).

    ADS  Google Scholar 

Download references

Acknowledgements

Research supported by the Spanish Ministry of Science, Innovation and Universities through grants FIS2015-65706-P, FIS2015-64654-P, FIS2016-80434-P, FIS2017-84860-R, PCI2018-093026 and PGC2018-097018-B-I00 (AEI/FEDER, EU), the Ramón y Cajal programme grant RYC-2011-09345 and RYC-2015-17973, the María de Maeztu Programme for Units of Excellence in R&D (CEX2018-000805-M), the European Union’s Horizon 2020 research and innovation programme under grant agreements 828948 (FETOPEN AndQC), 127900 (Quantera SuperTOP), the European Research Council (ERC) Starting Grant agreements 716559 (TOPOQDot), 757725 (ETOPEX) and 804988 (SiMS), the Netherlands Organization for Scientific Research (NWO), Microsoft, the Danish National Research Foundation, the Carlsberg Foundation, and the Swiss National Science Foundation and NCCR QSIT. We also acknowledge support from CSIC Research Platform on Quantum Technologies PTI-001.

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Authors and Affiliations

  1. Departamento de Física de la Materia Condensada and Condensed Matter Physics Center (IFIMAC), Universidad Autónoma de Madrid, Madrid, Spain

    Elsa Prada & Eduardo J. H. Lee

  2. Instituto de Ciencia de Materiales de Madrid (ICMM), Consejo Superior de Investigaciones Científicas (CSIC), Madrid, Spain

    Pablo San-Jose & Ramón Aguado

  3. QuTech and Kavli Institute of Nanoscience, Delft University of Technology, Delft, Netherlands

    Michiel W. A. de Moor, Attila Geresdi & Leo P. Kouwenhoven

  4. Department of Physics, University of Basel, Basel, Switzerland

    Jelena Klinovaja & Daniel Loss

  5. Center for Quantum Devices, Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark

    Jesper Nygård

  6. Microsoft Station Q at Delft University of Technology, Delft, Netherlands

    Leo P. Kouwenhoven

Authors
  1. Elsa Prada
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  2. Pablo San-Jose
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  3. Michiel W. A. de Moor
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  4. Attila Geresdi
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  5. Eduardo J. H. Lee
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  6. Jelena Klinovaja
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  7. Daniel Loss
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Contributions

L.P.K. initiated this Review. E.P. coordinated the project. All authors discussed the general structure of the manuscript. M.W.A.M. and A.G. wrote ‘ABS spectroscopy’ and ‘MBS spectroscopy’, and contributed to ‘Extensions of the minimal model’. E.J.H.L., J.N. and R.A. wrote ‘ABSs in QDs’. J.K. and D.L. contributed to ‘Zero-energy pinning with a topologically trivial bulk’ and ‘Protection against errors and MBS overlaps’. E.P., P.S.-J. and R.A. wrote everything else. All authors reviewed and polished the manuscript.

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Correspondence to Elsa Prada.

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Nature Reviews Physics thanks Hongqi Xu and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Glossary

Yu–Shiba–Rusinov states

(Also known as Shiba states.) Subgap excitation bound to a magnetic impurity in a superconductor. The bound excitation is formed because the coupling to the impurity reduces the minimal energy for exciting quasiparticles. The magnetic exchange mechanism that creates these excitations is akin to the Kondo effect in metals.

Spatial braiding

Quantum process whereby a pair of non-overlapping Majorana bound states are spatially interchanged (‘braided’). Braiding of several Majorana pairs leads to a non-Abelian transformation of the ground state: that is, the ground-state transformation depends non-commutatively on the order in which the various pairs are braided.

Majorana nanowire

Semiconducting low-density nanowire with strong spin–orbit coupling proximitized by an s-wave superconductor, which is expected to transition to a topological superconducting phase when subject to a strong enough Zeeman field perpendicular to the spin–orbit vector.

Wavefunction non-locality

Property of a fermion formed as a quantum superposition of two Majorana bound states, whereby its wavefunction is split into two spatially separated halves with exponentially suppressed overlap.

Proximitized semiconductor

Semiconductor that acquires superconducting correlations by virtue of its coupling to a superconductor.

Quasi-MBS

Zero-energy modes that emerge in pairs in a topologically trivial proximitized nanowire, typically due to spatially smooth potentials, and which exhibit a partial wavefunction overlap with their partner.

Parametric braiding

Equivalent of spatial braiding that does not require moving Majoranas in space, resorting instead to cyclic paths in parameter space (such as gate voltages and/or magnetic fields).

Topological superconductor

Superconductor that has a bulk characterized by a non-zero topological invariant and that, by virtue of the bulk–boundary correspondence, develops Majorana states confined to its boundaries that are protected against perturbations by the symmetries of the gapped bulk.

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Prada, E., San-Jose, P., de Moor, M.W.A. et al. From Andreev to Majorana bound states in hybrid superconductor–semiconductor nanowires. Nat Rev Phys 2, 575–594 (2020). https://doi.org/10.1038/s42254-020-0228-y

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