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

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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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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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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