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.
This is a preview of subscription content, access via your institution
Access options
Access Nature and 54 other Nature Portfolio journals
Get Nature+, our best-value online-access subscription
£14.99 / 30 days
cancel any time
Subscribe to this journal
Receive 12 digital issues and online access to articles
£79.00 per year
only £6.58 per issue
Buy this article
- Purchase on SpringerLink
- Instant access to full article PDF
Prices may be subject to local taxes which are calculated during checkout
Similar content being viewed by others
References
Kamerlingh Onnes, H. The superconductivity of mercury. Comm. Phys. Lab. Univ. Leiden 122, 122–124 (1911).
van Delft, D. & Kes, P. The discovery of superconductivity. Phys. Today 63, 38–42 (2010).
De Gennes, P.-G. Superconductivity of Metals and Alloys (CRC, 2018).
Tinkham, M. Introduction to Superconductivity (Courier Corporation, 2004).
Martin, J. D. When condensed-matter physics became king. Phys. Today 72, 30 (2019).
Bardeen, J., Cooper, L. N. & Schrieffer, J. R. Microscopic theory of superconductivity. Phys. Rev. 106, 162–164 (1957).
Cooper, L. N. Bound electron pairs in a degenerate Fermi gas. Phys. Rev. 104, 1189–1190 (1956).
Ginzburg, V. L. & Landau, L. D. On the theory of superconductivity. Zh. Eksp. Teor. Fiz. 20, 1064–1082 (1950).
Cyrot, M. Ginzburg–Landau theory for superconductors. Rep. Prog. Phys. 36, 103–158 (1973).
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).
Yu, L. Bound state in superconductors with paramagnetic impurities. Acta Phys. Sin. 21, 75–91 (1965).
Shiba, H. Classical spins in superconductors. Prog. Theor. Phys. 40, 435–451 (1968).
Rusinov, A. Superconductivity near a paramagnetic impurity. Sov. Phys. JETP 9, 85 (1969).
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).
Andreev, A. F. Thermal conductivity of the intermediate state in superconductors. Sov. Phys. JETP 19, 1228–1234 (1964).
Andreev, A. F. Electron spectrum of the intermediate state of superconductors. Sov. Phys. JETP 22, 18–23 (1966).
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).
Kulik, I. O. Macroscopic quantization and the proximity effect in SNS junctions. Sov. Phys. JETP 30, 944–950 (1970).
Sauls, J. A. Andreev bound states and their signatures. Phil. Trans. R. Soc. A 376, 20180140 (2018).
Hasan, M. Z. & Kane, C. L. Colloquium: Topological insulators. Rev. Mod. Phys. 82, 3045–3067 (2010).
Qi, X.-L. & Zhang, S.-C. Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057–1110 (2011).
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).
Leijnse, M. & Flensberg, K. Introduction to topological superconductivity and Majorana fermions. Semicond. Sci. Technol. 27, 124003 (2012).
Alicea, J. New directions in the pursuit of Majorana fermions in solid state systems. Rep. Prog. Phys. 75, 076501 (2012).
Beenakker, C. Search for Majorana fermions in superconductors. Annu. Rev. Cond. Mat. Phys. 4, 113–136 (2013).
Sato, M. & Fujimoto, S. Majorana fermions and topology in superconductors. J. Phys. Soc. Jpn 85, 072001 (2016).
Aguado, R. Majorana quasiparticles in condensed matter. Riv. Nuovo Cimento 40, 523–593 (2017).
Sato, M. & Ando, Y. Topological superconductors: a review. Rep. Prog. Phys. 80, 076501 (2017).
Salomaa, M. & Volovik, G. Cosmiclike domain walls in superfluid 3B: instantons and diabolical points in (k,r) space. Phys. Rev. B 37, 9298 (1988).
Volovik, G. E. & Volovik, G. The Universe in a Helium Droplet (Oxford Univ. Press, 2009).
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).
Kitaev, A. Y. Unpaired Majorana fermions in quantum wires. Phys. Usp. 44, 131 (2001).
Sato, M. & Fujimoto, S. Topological phases of noncentrosymmetric superconductors: edge states, Majorana fermions, and non-Abelian statistics. Phys. Rev. B 79, 094504 (2009).
Majorana, E. Teoria simmetrica dell elettrone e del positrone. Il Nuovo Cimento 14, 171–184 (1937).
Nishida, Y., Santos, L. & Chamon, C. Topological superconductors as nonrelativistic limits of Jackiw–Rossi and Jackiw–Rebbi models. Phys. Rev. B 82, 144513 (2010).
Jackiw, R. & Rossi, P. Zero modes of the vortex-fermion system. Nucl. Phys. B 190, 681–691 (1981).
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).
Kitaev, A. Fault-tolerant quantum computation by anyons. Ann. Phys. 303, 2–30 (2003).
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).
Fu, L. & Kane, C. L. Superconducting proximity effect and Majorana fermions at the surface of a topological insulator. Phys. Rev. Lett. 100, 096407 (2008).
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).
Oreg, Y., Refael, G. & von Oppen, F. Helical liquids and majorana bound states in quantum wires. Phys. Rev. Lett. 105, 177002 (2010).
Středa, P. & Šeba, P. Antisymmetric spin filtering in one-dimensional electron systems with uniform spin–orbit coupling. Phys. Rev. Lett. 90, 256601 (2003).
Stanescu, T. D. & Tewari, S. Majorana fermions in semiconductor nanowires: fundamentals, modeling, and experiment. J. Phys. Condens. Matter 25, 233201 (2013).
Lutchyn, R. M. et al. Majorana zero modes in superconductor–semiconductor heterostructures. Nat. Rev. Mater. 3, 52–68 (2018).
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).
Nadj-Perge, S. et al. Observation of Majorana fermions in ferromagnetic atomic chains on a superconductor. Science 346, 602–607 (2014).
Ménard, G. C. et al. Two-dimensional topological superconductivity in Pb/Co/Si(111). Nat. Commun. 8, 2040 (2017).
Ménard, G. C. et al. Isolated pairs of Majorana zero modes in a disordered superconducting lead monolayer. Nat. Commun. 10, 2587 (2019).
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).
Fu, L. & Kane, C. L. Josephson current and noise at a superconductor/quantum-spin-Hall-insulator/superconductor junction. Phys. Rev. B 79, 161408 (2009).
Wiedenmann, J. et al. 4π-periodic Josephson supercurrent in HgTe-based topological Josephson junctions. Nat. Commun. 7, 10303 (2016).
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).
Nichele, F. et al. Scaling of Majorana zero-bias conductance peaks. Phys. Rev. Lett. 119, 136803 (2017).
Bretheau, L., Girit, C. O., Pothier, H., Esteve, D. & Urbina, C. Exciting Andreev pairs in a superconducting atomic contact. Nature 499, 312–315 (2013).
Janvier, C. et al. Coherent manipulation of Andreev states in superconducting atomic contacts. Science 349, 1199–1202 (2015).
Pillet, J.-D. et al. Andreev bound states in supercurrent-carrying carbon nanotubes revealed. Nat. Phys. 6, 965–969 (2010).
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).
Dirks, T. et al. Transport through Andreev bound states in a graphene quantum dot. Nat. Phys. 7, 386–390 (2011).
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).
Kümmel, R. Dynamics of current flow through the phase-boundary between a normal and a superconducting region. Z. Phys. A 218, 472–494 (1969).
Zhang, H. et al. Ballistic superconductivity in semiconductor nanowires. Nat. Commun. 8, 16025 (2017).
Xiang, J., Vidan, A., Tinkham, M., Westervelt, R. M. & Lieber, C. M. Ge/Si nanowire mesoscopic Josephson junctions. Nat. Nanotechnol. 1, 208–213 (2006).
Ridderbos, J. et al. Multiple Andreev reflections and Shapiro steps in a Ge–Si nanowire Josephson junction. Phys. Rev. Mater. 3, 084803 (2019).
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).
Doh, Y.-J. et al. Tunable supercurrent through semiconductor nanowires. Science 309, 272–275 (2005).
Günel, H. Y. et al. Supercurrent in Nb/InAs-nanowire/Nb Josephson junctions. J. Appl. Phys. 112, 034316 (2012).
Goffman, M. F. et al. Conduction channels of an InAs–Al nanowire Josephson weak link. New J. Phys. 19, 092002 (2017).
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).
Deng, M. T. et al. Anomalous zero-bias conductance peak in a Nb–InSb nanowire–Nb hybrid device. Nano Lett. 12, 6414–6419 (2012).
Beenakker, C. Three ‘universal’ mesoscopic Josephson effects. In Transport Phenomena in Mesoscopic Systems: Proc. 14th Taniguchi Symposium, 235–253 (Springer, 1992).
Beenakker, C. W. J. Quantum transport in semiconductor–superconductor microjunctions. Phys. Rev. B 46, 12841–12844 (1992).
Likharev, K. K. Superconducting weak links. Rev. Mod. Phys. 51, 101–159 (1979).
Furusaki, A. & Tsukada, M. Current-carrying states in Josephson junctions. Phys. Rev. B 43, 10164–10169 (1991).
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).
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).
Furusaki, A. Josephson current carried by Andreev levels in superconducting quantum point contacts. Superlattices Microstruct. 25, 809–818 (1999).
Landauer, R. Can a length of perfect conductor have a resistance? Phys. Lett. A 85, 91–93 (1981).
Josephson, B. D. Possible new effects in superconductive tunnelling. Phys. Lett. 1, 251–253 (1962).
Josephson, B. D. Supercurrents through barriers. Adv. Phys. 14, 419–451 (1965).
Kos, F., Nigg, S. E. & Glazman, L. I. Frequency-dependent admittance of a short superconducting weak link. Phys. Rev. B 87, 174521 (2013).
Hofheinz, M. et al. Bright side of the Coulomb blockade. Phys. Rev. Lett. 106, 217005 (2011).
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).
van Woerkom, D. J. et al. Microwave spectroscopy of spinful Andreev bound states in ballistic semiconductor Josephson junctions. Nat. Phys. 13, 876 EP – (2017).
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).
Hays, M. et al. Direct microwave measurement of Andreev-bound-state dynamics in a semiconductor–nanowire Josephson junction. Phys. Rev. Lett. 121, 047001 (2018).
Mourik, V. et al. Signatures of Majorana fermions in hybrid superconductor–semiconductor nanowire devices. Science 336, 1003–1007 (2012).
Car, D. et al. InSb nanowires with built-in GaxIn1−xSb tunnel barriers for Majorana devices. Nano Lett. 17, 721–727 (2017).
Jünger, C. et al. Spectroscopy of the superconducting proximity effect in nanowires using integrated quantum dots. Commun. Phys. 2, 76– (2019).
Chang, W. et al. Hard gap in epitaxial semiconductor–superconductor nanowires. Nat. Nanotechnol. 10, 232–236 (2015).
Lee, E. J. H. et al. Spin-resolved Andreev levels and parity crossings in hybrid superconductor–semiconductor nanostructures. Nat. Nanotechnol. 9, 79–84 (2014).
Grove-Rasmussen, K. et al. Yu–Shiba–Rusinov screening of spins in double quantum dots. Nat. Commun. 9, 2376 (2018).
Anselmetti, G. L. R. et al. End-to-end correlated subgap states in hybrid nanowires. Phys. Rev. B 100, 205412 (2019).
Su, Z. et al. Mirage Andreev spectra generated by mesoscopic leads in nanowire quantum dots. Phys. Rev. Lett. 121, 127705 (2018).
Spanton, E. M. et al. Current–phase relations of few-mode InAs nanowire Josephson junctions. Nat. Phys. 13, 1177– (2017).
Hart, S. et al. Current–phase relations of inas nanowire Josephson junctions: from interacting to multimode regimes. Phys. Rev. B 100, 064523 (2019).
Nichele, F. et al. Relating Andreev bound states and supercurrents in hybrid Josephson junctions. Phys. Rev. Lett. 124, 226801 (2020).
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).
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).
Deng, M. T. et al. Majorana bound state in a coupled quantum-dot hybrid-nanowire system. Science 354, 1557–1562 (2016).
Lee, E. J. H. et al. Scaling of subgap excitations in a superconductor–semiconductor nanowire quantum dot. Phys. Rev. B 95, 180502 (2017).
De Franceschi, S., Kouwenhoven, L., Schönenberger, C. & Wernsdorfer, W. Hybrid superconductor–quantum dot devices. Nat. Nanotechnol. 5, 703– (2010).
Hewson, A. C. The Kondo Problem to Heavy Fermions (Cambridge Univ. Press, 1993).
Buitelaar, M. R., Nussbaumer, T. & Schönenberger, C. Quantum dot in the Kondo regime coupled to superconductors. Phys. Rev. Lett. 89, 256801 (2002).
Sand-Jespersen, T. et al. Kondo-enhanced Andreev tunneling in InAs nanowire quantum dots. Phys. Rev. Lett. 99, 126603 (2007).
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).
Grove-Rasmussen, K. et al. Superconductivity-enhanced bias spectroscopy in carbon nanotube quantum dots. Phys. Rev. B 79, 134518 (2009).
Kumar, A. et al. Temperature dependence of Andreev spectra in a superconducting carbon nanotube quantum dot. Phys. Rev. B 89, 075428 (2014).
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).
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).
Island, J. O. et al. Proximity-induced Shiba states in a molecular junction. Phys. Rev. Lett. 118, 117001 (2017).
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).
Lee, E. J. H. et al. Zero-bias anomaly in a nanowire quantum dot coupled to superconductors. Phys. Rev. Lett. 109, 186802 (2012).
Su, Z. et al. Andreev molecules in semiconductor nanowire double quantum dots. Nat. Commun. 8, 585 (2017).
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).
Heinrich, B. W., Pascual, J. I. & Franke, K. J. Single magnetic adsorbates on s-wave superconductors. Prog. Surf. Sci. 93, 1–19 (2018).
Chen, J. et al. Ubiquitous non-Majorana zero-bias conductance peaks in nanowire devices. Phys. Rev. Lett. 123, 107703 (2019).
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).
Delagrange, R. et al. Manipulating the magnetic state of a carbon nanotube Josephson junction using the superconducting phase. Phys. Rev. B 91, 241401 (2015).
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).
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).
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).
Deng, M.-T. et al. Nonlocality of Majorana modes in hybrid nanowires. Phys. Rev. B 98, 085125 (2018).
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).
Park, S. & Levy Yeyati, A. Andreev spin qubits in multichannel Rashba nanowires. Phys. Rev. B 96, 125416 (2017).
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).
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).
Tosi, L. et al. Spin–orbit splitting of Andreev states revealed by microwave spectroscopy. Phys. Rev. X 9, 011010 (2019).
Hays, M. et al. Continuous monitoring of a trapped superconducting spin. Nat. Phys. https://doi.org/10.1038/s41567-020-0952-3 (2020).
Kwon, H.-J., Yakovenko, V. M. & Sengupta, K. Fractional AC Josephson effect in unconventional superconductors. Low Temp. Phys. 30, 613–619 (2004).
Pikulin, D. I. & Nazarov, Y. V. Phenomenology and dynamics of a Majorana Josephson junction. Phys. Rev. B 86, 140504 (2012).
San-Jose, P., Prada, E. & Aguado, R. AC Josephson effect in finite-length nanowire junctions with Majorana modes. Phys. Rev. Lett. 108, 257001 (2012).
Klinovaja, J. & Loss, D. Composite Majorana fermion wave functions in nanowires. Phys. Rev. B 86, 085408 (2012).
Mishmash, R. V., Aasen, D., Higginbotham, A. P. & Alicea, J. Approaching a topological phase transition in Majorana nanowires. Phys. Rev. B 93, 245404 (2016).
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).
Gül, Ö. et al. Ballistic Majorana nanowire devices. Nat. Nanotechnol. 13, 192–197 (2018).
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).
Law, K. T., Lee, P. A. & Ng, T. K. Majorana fermion induced resonant Andreev reflection. Phys. Rev. Lett. 103, 237001 (2009).
Flensberg, K. Tunneling characteristics of a chain of Majorana bound states. Phys. Rev. B 82, 180516 (2010).
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).
Prada, E., San-Jose, P. & Aguado, R. Transport spectroscopy of NS nanowire junctions with Majorana fermions. Phys. Rev. B 86, 180503(R) (2012).
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).
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).
Albrecht, S. M. et al. Exponential protection of zero modes in Majorana islands. Nature 531, 206–209 (2016).
Shen, J. et al. Parity transitions in the superconducting ground state of hybrid InSb–Al Coulomb islands. Nat. Commun. 9, 4801 (2018).
Vaitiekenas, S. et al. Flux-induced topological superconductivity in full-shell nanowires. Science 367, https://science.sciencemag.org/content/367/6485/eaav3392 (2020).
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).
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).
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).
Shapiro, S. Josephson currents in superconducting tunneling: the effect of microwaves and other observations. Phys. Rev. Lett. 11, 80–82 (1963).
Domínguez, F., Hassler, F. & Platero, G. Dynamical detection of Majorana fermions in current-biased nanowires. Phys. Rev. B 86, 140503 (2012).
Sau, J. D. & Setiawan, F. Detecting topological superconductivity using low-frequency doubled Shapiro steps. Phys. Rev. B 95, 060501 (2017).
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).
Kamata, H. et al. Anomalous modulation of Josephson radiation in nanowire-based Josephson junctions. Phys. Rev. B 98, 041302 (2018).
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).
Le Calvez, K. et al. Joule overheating poisons the fractional ac Josephson effect in topological Josephson junctions. Commun. Phys. 2, 4 (2019).
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).
Virtanen, P. & Recher, P. Microwave spectroscopy of Josephson junctions in topological superconductors. Phys. Rev. B 88, 144507 (2013).
Domínguez, F. et al. Josephson junction dynamics in the presence of 2π- and 4π-periodic supercurrents. Phys. Rev. B 95, 195430 (2017).
Laroche, D. et al. Observation of the 4π-periodic Josephson effect in indium arsenide nanowires. Nat. Commun. 10, 245– (2019).
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).
San-Jose, P., Prada, E. & Aguado, R. Mapping the topological phase diagram of multiband semiconductors with supercurrents. Phys. Rev. Lett. 112, 137001 (2014).
Tiira, J. et al. Magnetically-driven colossal supercurrent enhancement in InAs nanowire Josephson junctions. Nat. Commun. 8, 14984 (2017).
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).
Peng, Y., Pientka, F., Berg, E., Oreg, Y. & von Oppen, F. Signatures of topological Josephson junctions. Phys. Rev. B 94, 085409 (2016).
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).
Potter, A. C. & Lee, P. A. Majorana end states in multiband microstructures with Rashba spin–orbit coupling. Phys. Rev. B 83, 094525 (2011).
Lutchyn, R. M., Stanescu, T. D. & Das Sarma, S. Search for Majorana fermions in multiband semiconducting nanowires. Phys. Rev. Lett. 106, 127001 (2011).
Lutchyn, R. M. & Fisher, M. P. A. Interacting topological phases in multiband nanowires. Phys. Rev. B 84, 214528 (2011).
Nijholt, B. & Akhmerov, A. R. Orbital effect of magnetic field on the Majorana phase diagram. Phys. Rev. B 93, 235434 (2016).
Winkler, G. W. et al. Unified numerical approach to topological semiconductor–superconductor heterostructures. Phys. Rev. B 99, 245408 (2019).
Nilsson, H. A. et al. Giant, level-dependent g factors in InSb nanowire quantum dots. Nano Lett. 9, 3151–3156 (2009).
Winkler, G. W. et al. Orbital contributions to the electron g factor in semiconductor nanowires. Phys. Rev. Lett. 119, 037701 (2017).
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).
Krogstrup, P. et al. Epitaxy of semiconductor–superconductor nanowires. Nat. Mater. 14, 400–406 (2015).
Gazibegovic, S. et al. Epitaxy of advanced nanowire quantum devices. Nature 548, 434–438 (2017).
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).
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).
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).
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).
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).
Antipov, A. E. et al. Effects of gate-induced electric fields on semiconductor Majorana nanowires. Phys. Rev. X 8, 031041 (2018).
Mikkelsen, A. E. G., Kotetes, P., Krogstrup, P. & Flensberg, K. Hybridization at superconductor–semiconductor interfaces. Phys. Rev. X 8, 031040 (2018).
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).
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).
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).
de Moor, M. W. A. et al. Electric field tunable superconductor–semiconductor coupling in Majorana nanowires. New J. Phys. 20, 103049 (2018).
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).
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).
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).
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).
Domínguez, F. et al. Zero-energy pinning from interactions in Majorana nanowires. NPJ Quant. Mater. 2, 13 (2017).
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).
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).
Liu, C.-X., Sau, J. D. & Das Sarma, S. Role of dissipation in realistic Majorana nanowires. Phys. Rev. B 95, 054502 (2017).
Danon, J., Hansen, E. B. & Flensberg, K. Conductance spectroscopy on Majorana wires and the inverse proximity effect. Phys. Rev. B 96, 125420 (2017).
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).
Peñaranda, F., Aguado, R., San-Jose, P. & Prada, E. Quantifying wave-function overlaps in inhomogeneous Majorana nanowires. Phys. Rev. B 98, 235406 (2018).
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).
Cao, Z. et al. Decays of Majorana or Andreev oscillations induced by steplike spin–orbit coupling. Phys. Rev. Lett. 122, 147701 (2019).
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).
Huang, Y. et al. Metamorphosis of Andreev bound states into Majorana bound states in pristine nanowires. Phys. Rev. B 98, 144511 (2018).
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).
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).
Bagrets, D. & Altland, A. Class D spectral peak in Majorana quantum wires. Phys. Rev. Lett. 109, 227005 (2012).
Beenakker, C. W. J. Random-matrix theory of Majorana fermions and topological superconductors. Rev. Mod. Phys. 87, 1037–1066 (2015).
Altland, A. & Zirnbauer, M. R. Nonstandard symmetry classes in mesoscopic normal–superconducting hybrid structures. Phys. Rev. B 55, 1142–1161 (1997).
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).
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).
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).
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).
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).
Ptok, A., Kobiałka, A. & Domański, T. Controlling the bound states in a quantum-dot hybrid nanowire. Phys. Rev. B 96, 195430 (2017).
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).
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).
Vuik, A., Nijholt, B., Akhmerov, A. R. & Wimmer, M. Reproducing topological properties with quasi-Majorana states. SciPost Phys. 7, 61 (2019).
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).
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).
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).
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).
Pikulin, D. I. & Nazarov, Y. V. Two types of topological transitions in finite Majorana wires. Phys. Rev. B 87, 235421 (2013).
San-Jose, P., Cayao, J., Prada, E. & Aguado, R. Majorana bound states from exceptional points in non-topological superconductors. Sci. Rep. 6, 21427 (2016).
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).
Shen, H., Zhen, B. & Fu, L. Topological band theory for non-Hermitian Hamiltonians. Phys. Rev. Lett. 120, 146402 (2018).
Gong, Z. et al. Topological phases of non-Hermitian systems. Phys. Rev. X 8, 031079 (2018).
McGinley, M. & Cooper, N. R. Classification of topological insulators and superconductors out of equilibrium. Phys. Rev. B 99, 075148 (2019).
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).
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).
Stanescu, T. D. & Tewari, S. Nonlocality of zero-bias anomalies in the topologically trivial phase of Majorana wires. Phys. Rev. B 89, 220507 (2014).
Szumniak, P., Chevallier, D., Loss, D. & Klinovaja, J. Spin and charge signatures of topological superconductivity in Rashba nanowires. Phys. Rev. B 96, 041401 (2017).
Chiu, C.-K., Sau, J. D. & Das Sarma, S. Conductance of a superconducting Coulomb-blockaded Majorana nanowire. Phys. Rev. B 96, 054504 (2017).
Budich, J. C., Walter, S. & Trauzettel, B. Failure of protection of Majorana based qubits against decoherence. Phys. Rev. B 85, 121405 (2012).
Goldstein, G. & Chamon, C. Decay rates for topological memories encoded with Majorana fermions. Phys. Rev. B 84, 205109 (2011).
Rainis, D. & Loss, D. Majorana qubit decoherence by quasiparticle poisoning. Phys. Rev. B 85, 174533 (2012).
Pedrocchi, F. L. & DiVincenzo, D. P. Majorana braiding with thermal noise. Phys. Rev. Lett. 115, 120402 (2015).
Knapp, C., Karzig, T., Lutchyn, R. M. & Nayak, C. Dephasing of Majorana-based qubits. Phys. Rev. B 97, 125404 (2018).
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).
Trif, M. & Tserkovnyak, Y. Resonantly tunable Majorana polariton in a microwave cavity. Phys. Rev. Lett. 109, 257002 (2012).
Schmidt, T. L., Nunnenkamp, A. & Bruder, C. Majorana qubit rotations in microwave cavities. Phys. Rev. Lett. 110, 107006 (2013).
Dmytruk, O., Trif, M. & Simon, P. Cavity quantum electrodynamics with mesoscopic topological superconductors. Phys. Rev. B 92, 245432 (2015).
Aseev, P. P., Klinovaja, J. & Loss, D. Lifetime of Majorana qubits in Rashba nanowires with nonuniform chemical potential. Phys. Rev. B 98, 155414 (2018).
Schmidt, M. J., Rainis, D. & Loss, D. Decoherence of Majorana qubits by noisy gates. Phys. Rev. B 86, 085414 (2012).
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).
Scheurer, M. S. & Shnirman, A. Nonadiabatic processes in Majorana qubit systems. Phys. Rev. B 88, 064515 (2013).
Sekania, M., Plugge, S., Greiter, M., Thomale, R. & Schmitteckert, P. Braiding errors in interacting Majorana quantum wires. Phys. Rev. B 96, 094307 (2017).
Hoffman, S., Schrade, C., Klinovaja, J. & Loss, D. Universal quantum computation with hybrid spin–Majorana qubits. Phys. Rev. B 94, 045316 (2016).
Wakatsuki, R., Ezawa, M. & Nagaosa, N. Majorana fermions and multiple topological phase transition in Kitaev ladder topological superconductors. Phys. Rev. B 89, 174514 (2014).
Karzig, T. et al. Scalable designs for quasiparticle-poisoning-protected topological quantum computation with Majorana zero modes. Phys. Rev. B 95, 235305 (2017).
Plugge, S., Rasmussen, A., Egger, R. & Flensberg, K. Majorana box qubits. New J. Phys. 19, 012001 (2017).
Prada, E., Aguado, R. & San-Jose, P. Measuring Majorana nonlocality and spin structure with a quantum dot. Phys. Rev. B 96, 085418 (2017).
Clarke, D. J. Experimentally accessible topological quality factor for wires with zero energy modes. Phys. Rev. B 96, 201109 (2017).
Schuray, A., Weithofer, L. & Recher, P. Fano resonances in Majorana bound states–quantum dot hybrid systems. Phys. Rev. B 96, 085417 (2017).
Ménard, G. C. et al. Conductance-matrix symmetries of a three-terminal hybrid device. Phys. Rev. Lett. 124, 036802 (2020).
Puglia, D. et al. Closing of the induced gap in a hybrid superconductor-semiconductor nanowire. Preprint at https://arxiv.org/abs/2006.01275 (2020).
Zhang, H., Liu, D. E., Wimmer, M. & Kouwenhoven, L. P. Next steps of quantum transport in Majorana nanowire devices. Nat. Commun. 10, 5128 (2019).
Frolov, S. M., Manfra, M. J. & Sau, J. D. Topological superconductivity in hybrid devices. Nat. Phys. 16, 718–724 (2020).
Aguado, R. & Kouwenhoven, L. P. Majorana qubits for topological quantum computing. Phys. Today 73, 44–50 (2020).
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).
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).
Larsen, T. W. et al. Semiconductor-nanowire-based superconducting qubit. Phys. Rev. Lett. 115, 127001 (2015).
de Lange, G. et al. Realization of microwave quantum circuits using hybrid superconducting–semiconducting nanowire Josephson elements. Phys. Rev. Lett. 115, 127002 (2015).
Sabonis, D. et al. Destructive Little–Parks effect in a full-shell nanowire-based transmon. Preprint at https://arxiv.org/abs/2005.01748 (2020).
Bargerbos, A. et al. Observation of vanishing charge dispersion of a nearly open superconducting island. Phys. Rev. Lett. 124, 246802 (2020).
Kringhøj, A. et al. Suppressed charge dispersion via resonant tunneling in a single-channel transmon. Phys. Rev. Lett. 124, 246803 (2020).
Ginossar, E. & Grosfeld, E. Microwave transitions as a signature of coherent parity mixing effects in the Majorana-transmon qubit. Nat. Commun. 5, 4772 (2014).
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).
Keselman, A., Murthy, C., van Heck, B. & Bauer, B. Spectral response of Josephson junctions with low-energy quasiparticles. SciPost Phys. 7, 50 (2019).
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).
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).
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).
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).
Young, A. F. et al. Tunable symmetry breaking and helical edge transport in a graphene quantum spin Hall state. Nature 505, 528–532 (2014).
Lee, G.-H. et al. Inducing superconducting correlation in quantum Hall edge states. Nat. Phys. 13, 693–698 (2017).
San-Jose, P., Lado, J. L., Aguado, R., Guinea, F. & Fernández-Rossier, J. Majorana zero modes in graphene. Phys. Rev. X 5, 041042 (2015).
Estrada Saldaña, J. C. et al. Supercurrent in a double quantum dot. Phys. Rev. Lett. 121, 257701 (2018).
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).
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.
Author information
Authors and Affiliations
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.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Peer review information
Nature Reviews Physics thanks Hongqi Xu and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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.
Rights and permissions
About this article
Cite this article
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
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1038/s42254-020-0228-y
This article is cited by
-
Room-temperature anisotropic in-plane spin dynamics in graphene induced by PdSe2 proximity
Nature Materials (2025)
-
Supercurrent mediated by helical edge modes in bilayer graphene
Nature Communications (2024)
-
Atomically precise engineering of spin–orbit polarons in a kagome magnetic Weyl semimetal
Nature Communications (2024)
-
Nonlocal correlations transmitted between quantum dots via short topological superconductor
Scientific Reports (2024)
-
Correlation between two distant quasiparticles in separate superconducting islands mediated by a single spin
Nature Communications (2024)