Soon after Enrico Fermi became a professor of physics at Italy’s University of Rome in 1927, Ettore Majorana joined his research group. Majorana’s colleagues described him as humble because he considered some of his work unexceptional. For example, Majorana correctly predicted in 1932 the existence of the neutron, which he dubbed a neutral proton, based on an atomic-structure experiment by Irène Joliot-Curie and Frédéric Joliot-Curie. Despite Fermi’s urging, Majorana didn’t write a paper. Later that year James Chadwick experimentally confirmed the neutron’s existence and was awarded the 1935 Nobel Prize in Physics for the discovery.
Nevertheless, Fermi thought highly of Majorana, as is captured in the following quote: “There are various categories of scientists, people of a secondary or tertiary standing, who do their best but do not go very far. There are also those of high standing, who come to discoveries of great importance, fundamental for the development of science. But then there are geniuses like Galileo and Newton. Well, Ettore was one of them.” Majorana only wrote nine papers, and the last one, about the now-eponymous fermions, was published in 1937 at Fermi’s insistence. A few months later, Majorana took a night boat to Palermo and was never seen again.1
In that final article, Majorana presented an alternative representation of the relativistic Dirac equation in terms of real wavefunctions. The representation has profound consequences because a real wavefunction describes particles that are their own antiparticles, unlike electrons and positrons. Since particles and antiparticles have opposite charges, fermions in his new representation must have zero charge. Majorana postulated that the neutrino could be one of those exotic fermions.
Although physicists have observed neutrinos for more than 60 years, whether Majorana’s hypothesis is true remains unclear. For example, the discovery of neutrino oscillations, which earned Takaaki Kajita and Arthur McDonald the 2015 Nobel Prize in Physics, demonstrates that neutrinos have mass. But the standard model requires that neutrinos be massless, so various possibilities have been hypothesized to explain the discrepancy. One answer could come from massive neutrinos that do not interact through the weak nuclear force. Such sterile neutrinos could be the particles that Majorana predicted. Whereas conclusive evidence for the existence of Majorana neutrinos remains elusive, researchers are now using Majorana’s idea for other applications, including exotic excitations in superconductors.
Majorana quasiparticles in superconductors
From the condensed-matter viewpoint, Majoranas are not elementary particles but rather emergent quasiparticles. Interestingly, the equation that describes quasiparticle excitations in superconductors has the same mathematical structure as the Majorana equation. The reason for the similarity arises from the underlying particle–hole symmetry in superconductors: Unlike quasiparticles in a metal, which have a well-defined charge, quasiparticles in a superconductor comprise coherent superpositions of electrons and holes. For the special zero-energy eigenmode, the electron and the hole, which each contribute half probability, form a quasiparticle. The operators describing the zero-energy particle–hole superpositions are invariant under charge conjugation, and zero-energy modes are therefore condensed-matter Majorana particles.
Particle–hole symmetry dictates that excitations in superconductors should occur in pairs at energies ±E. Therefore, zero-energy excitations are seemingly unreachable because they cannot emerge by any smooth deformation of the Hamiltonian, which would require that one of the solutions disappear. Rather, the only way to generate zero-energy excitations in superconductors is through a topological transition, a process that separates the phase of Majorana zero modes from the phase without them by closing and then reopening the superconducting gap (see the article by Nick Read, Physics Today, July 2012, page 38).
Majorana zero modes are located at topological defects, such as vortices, boundaries, and domain walls in topological superconductors. Remarkably, Majorana zero modes bound to defects do not obey fermion statistics. Unlike the original particles predicted by Majorana, the zero modes possess non-abelian exchange statistics, also known as non-abelian braiding, which makes them promising for applications in topological quantum computing, as detailed in box 1. Quasiparticles with non-abelian exchange statistics were first predicted in 1991 to occur in the filling factor ν = 5⁄2 of the fractional quantum Hall state. In 2000, researchers demonstrated that similar physics occur in superconductors with intrinsic p-wave pairing, an exotic form of superconductivity in which Cooper pairs bind through rare triplet-like pairing instead of the more standard singlet-like pairing in s-wave superconductors.2 Conventional s-wave pairing can be converted to p-wave pairing by combining the superconducting proximity effect in materials with strong spin–orbit interactions and an external magnetic field that breaks time-reversal symmetry.
Quantum mechanics dictates that particles obey either Fermi–Dirac or Bose–Einstein statistics in three dimensions, which means that the wavefunction Ψ of a system of indistinguishable particles is necessarily bosonic or fermionic upon particle exchange. From that point of view, fermions and bosons are not exotic because exchanging them leaves the ground state invariant, up to a sign: Ψ → ±Ψ.
Two dimensions are richer. Now, the possibilities go beyond the fermionic or bosonic cases. A system can exhibit anyon statistics in which the wavefunction picks up an arbitrary phase under an exchange: Ψ → eiθ Ψ. Such behavior generalizes the boson and fermion cases, where the phases can only be θ = 0 or θ = π. Because the phase factors are ordinary commuting numbers, the order of successive exchanges doesn’t matter, and the anyon statistics are called abelian.
The weirdness starts in systems with a degenerate many-body ground state containing several quasiparticles. When quasiparticles are exchanged, the system goes from one ground state, Ψa, to another, MabΨb. Because the unitary transformations Mab that operate in the subspace of degenerate ground states are generally noncommuting, the anyonic statistics take a non-abelian form. The final state of the system, therefore, depends on the order of the exchange operations, similar to braiding cords in a necklace.
Using Majorana zero modes to store and manipulate quantum information is one case where non-abelian braiding statistics form the basis of topological quantum computation (see the article by Sankar Das Sarma, Michael Freedman, and Chetan Nayak, Physics Today, July 2006, page 32). Quantum computation in such a system also benefits from protection against environmental decoherence because of the nonlocal character of Majorana-based qubits.
The nanowire proposal
In 2010 two research groups made an elegant theoretical proposal, shown schematically in figure 1. If a semiconducting nanowire with strong spin–orbit coupling, such as indium arsenide or indium antimonide, is coupled to a standard s-wave superconductor, Majorana zero modes will emerge at both ends of the nanowire, provided that a magnetic field is applied parallel to it.3 The proposal realistically implements the paradigmatic one-dimensional model for p-wave superconductivity that was discussed in 2001 for the first time by Alexei Kitaev.4
(a) The nanowire proposal3 takes a nanowire of a semiconductor, such as indium arsenide or indium antimonide, that has strong spin–orbit coupling and places it in contact with an s-wave superconductor, such as aluminum, in the presence of an external magnetic field B. As in the original model for one-dimensional p-wave superconductors,4 the nanowire device experiences a topological nontrivial phase with exponentially decaying Majorana bound states, denoted γL, at both ends of the nanowire. (b) An actual device from Delft University of Technology includes various metallic gates for tuning it to the topological phase by adjusting the nanowire’s chemical potential. (Panel a adapted from ref. 3, R. M. Lutchyn, J. D. Sau, S. Das Sarma; panel b adapted from H. Zhang et al., Nature 556 74, 2018.)
(a) The nanowire proposal3 takes a nanowire of a semiconductor, such as indium arsenide or indium antimonide, that has strong spin–orbit coupling and places it in contact with an s-wave superconductor, such as aluminum, in the presence of an external magnetic field B. As in the original model for one-dimensional p-wave superconductors,4 the nanowire device experiences a topological nontrivial phase with exponentially decaying Majorana bound states, denoted γL, at both ends of the nanowire. (b) An actual device from Delft University of Technology includes various metallic gates for tuning it to the topological phase by adjusting the nanowire’s chemical potential. (Panel a adapted from ref. 3, R. M. Lutchyn, J. D. Sau, S. Das Sarma; panel b adapted from H. Zhang et al., Nature 556 74, 2018.)
The Majorana zero modes are localized at opposite ends of the wire and decay with position x as e−x/ξ, where ξ is the localization length. But together they form a highly delocalized fermion, which can be seen mathematically as a fermion operator that decomposes to two real, self-adjoint operators. The nonlocal fermion defines two parity states—the empty state and the full fermion one—that are degenerate at zero energy except for exponentially small corrections e−L/ξ, where L is the length of the wire. Those two states can be used to define a qubit. Because the states are stored nonlocally, the qubit is resilient against local perturbations from the environment.
To induce a closing and reopening of an energy gap in the nanowire platform, researchers exploit the competition among three effects. The first, the s-wave superconducting proximity effect, pairs electrons of opposite spin and opens a superconducting gap Δ at the Fermi level. In the second effect, an external magnetic field B generates a Zeeman energy EZ = gμBB/2—with g the nanowire’s Landé factor and μB the Bohr magneton—which tends to break Cooper pairs by aligning their electron spins and closing the gap. The third effect, spin–orbit coupling, negates the external magnetic field by preventing the spins from reaching full alignment.
The competition between the second and third effects creates regions in parameter space where the gap closes and reopens again. At low electron densities, the transition occurs when the Zeeman energy is of the same magnitude as the induced superconducting gap, and it can be reached either by increasing the magnetic field, as shown in figure 2, or by tuning the wire’s chemical potential. Apart from choosing semiconductors with a large spin–orbit coupling and good proximity effect with conventional superconductors, researchers need large g factors to induce a large Zeeman effect with moderate magnetic fields below the critical field of the superconductor. Materials such as the heavy-element semiconductors InAs and InSb have proven to be excellent choices.
Andreev reflections of electrons and holes to form Cooper pairs at the semiconducting– superconducting interface induce superconductivity in a nanowire. As a result, Majorana zero modes (flat red line) emerge in the energy spectrum as the external magnetic field increases. The Majoranas appear beyond some critical value of the external field (black dotted line) where the superconducting gap closes and reopens again, which signals a topological phase transition. Theory predicts that the emergent Majorana zero modes can be detected as a zero-bias anomaly in electrical conductance dI/dV. (Image by R. Aguado and L. P. Kouwenhoven.)
Andreev reflections of electrons and holes to form Cooper pairs at the semiconducting– superconducting interface induce superconductivity in a nanowire. As a result, Majorana zero modes (flat red line) emerge in the energy spectrum as the external magnetic field increases. The Majoranas appear beyond some critical value of the external field (black dotted line) where the superconducting gap closes and reopens again, which signals a topological phase transition. Theory predicts that the emergent Majorana zero modes can be detected as a zero-bias anomaly in electrical conductance dI/dV. (Image by R. Aguado and L. P. Kouwenhoven.)
Topological superconductivity can also be engineered using similar ideas in alternative platforms. Some examples include chains of magnetic impurities above superconductors; proximitized 2D materials; and vortices in proximitized topological insulators such as quantum spin-Hall insulators, quantum anomalous-Hall insulators, and iron-based topological surface states.
Measuring Majoranas
At energies below the superconducting gap, an electron incident on a superconductor (S) from a normal conductor (N) can be reflected either as an electron or as a hole. Whereas the electron process is a standard, normal reflection, the hole process, known as Andreev reflection, is subtler because electrons are reflected as holes in the normal side while creating a Cooper pair in the superconducting side. In a standard NS junction, such Andreev processes are rare in the tunneling limit, and the conductance is small. But in a topological NS junction containing Majorana bound states, an incident electron is always reflected as a hole with unitary probability.
As a result of that resonant Andreev process, the electrical conductance G at zero voltage is expected to be perfectly quantized: G = 2e2/h, where e is the electron charge and h, Planck’s constant. The Andreev process underscores the particle–antiparticle duality of Majorana bound states: Because the electron and hole contribute equally to form a Majorana quasiparticle, the tunneling rates for electrons and holes should be equal. Therefore, researchers can use tunneling spectroscopy to directly detect a Majorana bound state as a zero-bias anomaly (ZBA). The differential conductance dI/dV, with I the current across the junction, is a function of the applied bias voltage V, and the ZBA should emerge as an increasing magnetic field induces a topological transition in the nanowire.
In 2012, researchers showed that the nanowire proposal could indeed be realized.5 A typical measurement from that experiment is illustrated in figure 3a, which shows conductance versus applied bias voltage and magnetic field. For intermediate values of the magnetic field, a clear ZBA emerges in the middle of the superconducting gap and is consistent with the existence of zero-energy Majorana bound states in the nanowire. Subsequent experiments showed similar results.6
(a) A contour plot of dI/dV versus voltage V and external magnetic field B along the axis of an indium antimonide nanowire in contact with niobium titanate nitrate shows that for fields between 100 mT and 400 mT, a clear zero-bias anomaly (ZBA, green dotted oval) emerges in the middle of the superconducting gap, denoted by the dashed green lines. (b) An indium arsenide nanowire coupled to aluminum shows a robust ZBA. (Panel a adapted from ref. 5; panel b adapted from ref. 8.)
(a) A contour plot of dI/dV versus voltage V and external magnetic field B along the axis of an indium antimonide nanowire in contact with niobium titanate nitrate shows that for fields between 100 mT and 400 mT, a clear zero-bias anomaly (ZBA, green dotted oval) emerges in the middle of the superconducting gap, denoted by the dashed green lines. (b) An indium arsenide nanowire coupled to aluminum shows a robust ZBA. (Panel a adapted from ref. 5; panel b adapted from ref. 8.)
The debate
Members of the research community greeted the nanowire experiments with excitement (see Physics Today,June 2012, page 14), and they also challenged the Majorana interpretation. Many features of the experiments, notably the absence of a closing and reopening of the gap and a conductance well below the quantized G = 2e2/h limit, disagreed with model predictions. Importantly, a similar ZBA unrelated to Majoranas may also appear because of various physical mechanisms, such as the Kondo effect and disorder. Those are related to the sizeable subgap conductance that arises from an imperfect superconducting proximity effect.
Fortunately, many of the false-positive scenarios can now be ruled out because of advances in materials and fabrication. Some examples include the epitaxial growth of crystalline superconductor shells directly on the surface of the nanowires and the careful engineering of high-quality semiconductor–superconductor interfaces.7 That progress has generated improved devices with much better induced superconductivity, including negligible subgap conductance. The new generation of devices has produced cleaner data with robust ZBAs,8 as shown in figure 3b, and values close to the expected G = 2e2/h ideal limit.9
Can we now claim that Majoranas have been observed? We cannot, because topological protection has not yet been demonstrated. Variations in the electrostatic potential, including from disorder and inhomogeneous gating, can produce regions where Andreev levels—the superconducting counterparts of particle-in-a-box confined states in quantum mechanics—appear at zero energy without a concomitant topological transition. Theory predicts that those zero modes are ubiquitous.10 Physically, they correspond to single-fermionic Andreev levels that can be decomposed into two Majoranas. Because they partially overlap in space, those Majoranas lack the full topological protection offered by spatial separation. The tunneling coupling to the normal conductor can be distinct, which results in robust nontopological ZBAs. Even without a topological phase in the nanowire, those Majoranas still obey non-abelian statistics.
Now the challenge is to demonstrate that Majorana zero modes can be generated with an exponentially small overlap such that deviations from perfect ground-state degeneracy are exponentially small. In that regime, Majoranas become topologically protected and can be used to define parity qubits. Recent efforts to extract the degree of Majorana nonlocality have started to appear in the literature.11 Researchers have also made experimental advances with the superconductor–semiconductor interface, among them a thin aluminum layer epitaxially covering a high-mobility InAs 2D electron gas and an Al shell wrapping an InAs nanowire core. Both schemes represent a paradigm shift because they allow topological superconductivity to be tuned by controlling the phase of the superconducting order parameter rather than by the Zeeman effect. Researchers can also use those schemes to control Josephson junctions fabricated with a 2D electron-gas hybrid material12 and even full phase windings, akin to vortices, in the full-shell nanowire geometry.13
Majorana islands
Another nanowire option called the floating geometry electrically isolates a nanowire with small capacitors. That geometry produces Majorana islands, which show deviations from ground-state degeneracy. The scheme exploits even–odd effects in small superconductors. Recall that the classical energy to charge a capacitor is inversely proportional to its capacitance; for a sufficiently small island, the charging energy for adding a single extra electron can be significant. Transport through such an island is blocked at low voltages and temperatures, a phenomenon called Coulomb blockade. Current flow—so-called Coulomb blockade peaks—is possible only at special degeneracy points. They occur periodically at gate voltages for which the energy of having N or N + 1 electrons on the island is equal. In the presence of superconductivity, Coulomb blockade still applies, though the energy of N electrons depends also on fermionic parity. If N is even, all quasiparticles couple as Cooper pairs in the ground state. Adding an extra electron costs both electrostatic charging energy and a finite energy that corresponds to the lowest quasiparticle excitation. For a standard superconductor, the energy of the odd-N configuration is the superconducting gap, as shown in figure 4a.
Majorana islands are based on finite-sized nanowire segments in a floating geometry and placed in contact with superconductors. (a) The energy cost for adding an extra electron in a standard superconductor is given by the superconducting gap Δ. When the gap is larger than the Coulomb energy to charge the island, the system can only accommodate electrons in pairs. (b) In a topological superconductor, single electrons can be accommodated at no energy cost by filling the zero-energy fermionic state formed by two nonlocal Majoranas. (c) Observed linear conductance data are graphed as a function of gate voltage for increasing magnetic field. The series of 2e-periodic Coulomb blockade peaks at low magnetic fields become 1e-periodic for larger magnetic fields. (Adapted from ref. 14.)
Majorana islands are based on finite-sized nanowire segments in a floating geometry and placed in contact with superconductors. (a) The energy cost for adding an extra electron in a standard superconductor is given by the superconducting gap Δ. When the gap is larger than the Coulomb energy to charge the island, the system can only accommodate electrons in pairs. (b) In a topological superconductor, single electrons can be accommodated at no energy cost by filling the zero-energy fermionic state formed by two nonlocal Majoranas. (c) Observed linear conductance data are graphed as a function of gate voltage for increasing magnetic field. The series of 2e-periodic Coulomb blockade peaks at low magnetic fields become 1e-periodic for larger magnetic fields. (Adapted from ref. 14.)
If the gap is larger than the charging energy, however, electrons enter the island only as Cooper pairs. As the magnetic field increases, the quasiparticle gap reduces until the superconductor becomes topological. At that point, it can accommodate one extra electron in the nonlocal Majorana zero-mode state with zero energy, regardless of whether it is empty or full, as figure 4b illustrates. Experiments with short proximitized nanowires show a change in periodicity, evident in figure 4c, that is consistent with Majorana theory: The flow of Cooper pairs transitions to single electrons as the magnetic field increases.14
Toward Majorana-based qubits
The Coulomb blockade peaks of Majorana islands have a maximum conductance of G = e2/h, half that of noninteracting wires, because two-charge transfers are strongly suppressed. Only single-electron tunneling is possible at charge-degeneracy points. The transfer occurs through the fermion state formed by two distant Majoranas.15 Researchers can use that nonlocal resonant process for parity readout. And as detailed in box 2, the process enables the non-abelian braiding of Majoranas and the building of Majorana-based qubits.
Researchers have explored various schemes that use two-path electron interferometry for parity readout. One path involves a Majorana island; the other serves as a reference. Part a of the figure shows an interferometer with a Majorana island (green) in the Coulomb blockade regime. In that setup, the amplitudes of the Coulomb blockade conductance peaks display Aharonov–Bohm oscillations as a function of the external magnetic flux Φ piercing the interferometer; the oscillations reflect coherent single-electron transport—electron teleportation—across the island. The plot of the interferometer conductance G (Φ) against Φ shows the oscillations of two successive Coulomb peaks. The π phase shift denotes a change in the fermion parity of the island.16
Proposals for performing parametric braiding, rather than spatial braiding, rely on the interferometer. That reliance avoids the need to move Majoranas around each other in complicated geometries, such as T-junctions.17 In a recent paper on parametric braiding, researchers have proposed the possibility of performing measurement-based topological quantum computation with quantum gates based on interferometers.18 In the schematic shown in part b of the figure, two topological superconductor nanowires (green) are shunted by a superconducting bridge S (orange) to form a floating island hosting four Majorana modes. A reference arm R (yellow) is shorter than the coherence length and closes the interference loop. The measured flux-dependent conductance depends on the fermion parity of Majoranas 2 and 3. Such box qubits allow for measurement-only protocols, including qubit readout for the three Pauli operators in the Majorana basis and full one-qubit control using tunable couplings between Majorana states and quantum dots. (Adapted from D. Aasen et al., Phys. Rev. X 6, 031016 (2016).
Two key steps need to be completed, however, before Majorana qubits can be used in topological quantum computing. Researchers first must establish unambiguously that in the lab they can make fully nonlocal Majoranas with the requisite topological protection, demonstrate ground-state degeneracy, and test for simple measurements, such as fusion rules. The second important step is to demonstrate parity-dependent interferometry, which is at the heart of measurement-only Majorana box qubits. And on the way to topological quantum computing, researchers can explore more exotic physics, including the topological Kondo effect. That phenomenon exploits the analogy between the two degenerate parity ground states formed by two highly nonlocal Majoranas and the spin-½ system of the standard Kondo effect. Each of those milestones would help lay the foundation for topological quantum computing. By themselves, the milestones would represent an unprecedented advancement for fundamental physics.
References
Ramón Aguado is a senior researcher at the Spanish National Research Council (CSIC) in Madrid. Leo Kouwenhoven is a researcher at the Microsoft Quantum Lab Delft and a professor of applied physics at Delft University of Technology in the Netherlands.
