Abstract
Topological superconductors that support Majorana fermions have been predicted when onedimensional semiconducting wires are coupled to a superconductor^{1,2,3}. Such excitations are expected to exhibit nonAbelian statistics and can be used to realize quantum gates that are topologically protected from local sources of decoherence^{4,5}. Here we report the observation of the fractional a.c. Josephson effect in a hybrid semiconductor–superconductor InSb/Nb nanowire junction, a hallmark of topological matter. When the junction is irradiated with a radiofrequency f_{0} in the absence of an external magnetic field, quantized voltage steps (Shapiro steps) with a height Δ V = h f_{0}/2e are observed, as is expected for conventional superconductor junctions, where the supercurrent is carried by charge 2e Cooper pairs. At high magnetic fields the height of the first Shapiro step is doubled to h f_{0}/e, suggesting that the supercurrent is carried by charge e quasiparticles. This is a unique signature of the Majorana fermions, predicted almost 80 years ago^{6}.
Main
In 1928 Dirac reconciled quantum mechanics and special relativity in a set of coupled equations that became the cornerstone of quantum mechanics^{7}. Its main prediction that every elementary particle has a complex conjugate counterpart—an antiparticle—has been confirmed by numerous experiments. A decade later Majorana showed that Dirac’s equation for spin1/2 particles can be modified to permit real wavefunctions^{6,8}. The complex conjugate of a real number is the number itself, which means that such particles are their own antiparticles. Although the search for Majorana fermions among elementary particles is continuing^{9}, excitations with similar properties may emerge in electronic systems^{4}, and are predicted to be present in some unconventional states of matter^{10,11,12,13,14,15}.
Ordinary spin1/2 particles or excitations carry a charge, and thus cannot be their own antiparticles. In a superconductor, however, free charges are screened, and chargeless spin1/2 excitations become possible. The Bardeen–Cooper–Schrieffer (BCS) theory of superconductivity allows fermionic excitations that are a mixture of electron and hole creation operators, γ_{i} = c_{i}^{†}+c_{i}. This creation operator is invariant with respect to charge conjugation, . If the energy of an excitation created in this way is zero, the excitation will be a Majorana particle. However, such zeroenergy modes are not permitted in ordinary swave superconductors.
The present work is inspired by ref. 15, where the authors show that Majorana fermions can be formed in a coupled semiconductor/superconductor system. Superconductivity can be induced in a semiconductor material by the proximity effect. At zero magnetic field electronic states are doubly degenerate and Majorana modes are not supported. In semiconductors with strong spin–orbit interactions the two spin branches are separated in momentum (k) space, but spin–orbit interactions do not lift the Kramer’s degeneracy. However, in a magnetic field there is a range of energies where double degeneracy is lifted^{16}, see the schematic in Fig. 1c. If the Fermi energy E_{F} is tuned to be within this singlemode range of energies, , (where Δ is the proximity gap, E_{Z} = g μ_{B}B/2 is the Zeeman energy, μ_{B} is the Bohr magneton, and g is the Landé gfactor), the proximity effect from a conventional swave superconductor induces pwave pairing in the semiconductor material and drives the system into a topological superconducting state which supports Majorana particles. Theoretically, it has been predicted that proper conditions for this to occur can be realized in twodimensional^{15,17} and, most relevant to the current work, in onedimensional systems^{1,2}. Moreover, multiband nanowires are also predicted to support topological superconductivity^{18,19,20}.
What are the experimental signatures of Majorana particles? Majorana particles should have zero energy, and zeroenergy Andreev endmodes localized at the ends of a wire can be probed in tunnelling experiments^{11,21,22}. Indeed, there are reports of a zero bias anomaly observed in topological insulator–superconductor^{23} and semiconductor–superconductor^{24,25} structures. However, conductivity enhancement near zero bias can also be a signature of diverse phenomena in mesoscopic physics, such as the Kondo effect in quantum dots^{26,27} or the ‘0.7 anomaly’ in nanowires^{28,29}. Fusion of two Majorana modes produces an ordinary fermion and, uniquely to Majorana particles, modifies the periodicity of the Josephson relation from 2π (Cooper pairs) to 4π (Majorana particles)^{1,4,30,31,32}. In the d.c. Josephson effect, fluctuations between filled and empty Majorana modes will mask the 4π periodicity and, indeed, we observe only 2π periodicity in a d.c. SQUID (superconducting quantum interference device) configuration. In the a.c. Josephson effect, however, the 4π periodicity due to Majorana modes should be fully revealed.
In our experiments Nb/InSb/Nb Josephson junctions are fabricated lithographically from a shallow InSb quantum well. Superconductivity in InSb is induced by the proximity effect from a Nb film placed on top of the InSb nanowire. The selfaligning fabrication process which we use is described in the Supplementary Information. A pattern of multiple Josephson junctions is defined by ebeam lithography, and a 45 nm layer of Nb is deposited by d.c. sputtering on top of the InSb quantum well. An image of a Josephson junction region is shown in Fig. 1a. A weak link is formed between two 120 nmwide and 0.6 μmlong Nb wires, with gaps ranging from 20 to 120 nm in different devices. During deposition of Nb, a thin (2–4 nm) layer of Nb is extended 70–80 nm outside the pattern, including the space inside the gaps, which can be seen as a brown halo around the Nb wire on the atomic force microscope (AFM) image. This layer is used as an etch mask to define a nanowire in the underlying semiconductor selfaligned to the Nb wire. After the etching, a continuous w≲290 nmwide InSb wire is formed under the Nb, as shown schematically in Fig. 1b. The thin Nb layer is not conducting at low temperatures, therefore the supercurrent is carried by the proximityinduced superconductivity in InSb.
The InSb wires have rectangular cross section w≫d and spin–orbit interactions are dominated by the Dresselhaus term H_{D} = γ_{D}〈k_{z}^{2}〉(k_{x}σ_{x}−k_{y}σ_{y}), where 〈k_{z}^{2}〉 = (π/d)^{2}, d = 20 nmis the quantum well thickness, γ_{D} = 760 eV Å^{3} for InSb, σ_{i} are Pauli matrices, and and are the principal crystallographic axes. We estimate E_{so}≈1 meV (refer to the Supplementary Informartion). The wires are oriented along the [110] crystallographic direction, and expected direction of the effective spin–orbit magnetic field B_{so} is perpendicular to the current, as indicated by the green arrow on the AFM image. From the lithographical dimensions we estimate that only a few (1–3) onedimensional subbands should be populated in InSb nanowires, however, we expect the density of states in the nanowires to be modified by the strong coupling to Nb and the actual number of filled subbands may be larger.
As devices are cooled down, a series of superconducting transitions T_{c1}−T_{c3} and T_{c} are observed (Fig. 2). The first transition, T_{c1}∼6.4 K, is for wide areas, T_{c2} = 5.8 K and T_{c3} = 1.9 K are for the 1 and 0.12 μmwide Nb wires, and T_{c} is for the Josephson junctions. From T_{c} = 1.17 K for Josephson junction JJ8 we estimate a proximity gap Δ≈180 μ eV and a semiconductorsuperconducting coupling λ≈2.6Δ (ref. 33). Lithographically our devices consist of two Josephson junctions in parallel, and we can measure the ratio of the critical currents in the two arms r = I_{c1}/I_{c2} by measuring the magnitude of current modulation in a d.c. SQUID configuration. The ratio r = 7.3 for JJ7 and r>10 for JJ8, indicating that conduction is dominated by a single junction. In the following analysis we will treat our devices as containing a single Josephson junction.
As seen in Fig. 2, the V (I) characteristic for JJ8 measured at the base temperature of 20 mK exhibits a clear supercurrent region (V = 0), with an abrupt appearance of a finite voltage. A small hysteresis is observed for the return critical current I_{R}∼I_{c}, characteristic of a resistively shunted Josephson junction in an intermediate damping regime^{34}. Indeed, we measure high leakage to the substrate, and estimate the shunting resistance R_{s}≲1 kΩ. The measured resistance in the normal state is R^{′}_{N} = 650 Ω, and the actual normal resistance of the JJ8 R_{N}≈2–6 kΩ, consistent with the number of onedimensional channels estimated from the size of the wire. The product R_{N}I_{c}≈1 mV>2Δ indicates that JJ8 is in a clean limit (weak link length L_{eff}<ξ,l, where ξ is the coherence length and l is the mean free path), a proper condition for the formation of Majorana particles.
Normalized differential conductance, plotted in the right panel of Fig. 2, shows enhancement at low voltages. This excess current is a signature of Andreev reflection^{35,36}. Most important for our measurements is that the excess current, and thus coherent electron transport, is observed at high inplane magnetic fields up to 4 T, as shown in the the inset.
In the presence of microwave excitation, phase locking between the rf field and the Josephson supercurrent gives rise to constantvoltage Shapiro steps in the V (I) characteristics at V _{n} = n h f_{0}/q, where h is Planck’s constant, q is the charge of quasiparticles, f_{0} is the microwave frequency, and n = 0,±1,±2… (ref. 37). Shapiro steps for f_{0} = 3 GHz are shown in Fig. 3. At B = 0we observe steps with a height ΔV = 6 μV, consistent with the Cooper pair tunnelling (q = 2e). ΔV scales linearly with f_{0} (refer to the Supplementary Information). The evolution of the steps with V _{rf} is best visualized in the dV/dI plots, where steps with 0<n<10 are seen at high rf powers. A transition from a low to high rf power regime is clearly seen in the dV/dI plot near V _{rf}≈4 mV and is marked by a green horizontal line. At high rf powers V _{rf}>4 mV the evolution of the width of the Shapiro steps ΔI_{n} follows a Bessel function pattern as a function of power, ΔI_{n} = AJ_{n}(2e v_{rf}/h f_{0}), where v_{rf} is the rf amplitude at the junction. We can find the rf power attenuation from the fit to the ΔI_{0}, v_{rf} = 5×10^{−3}V _{rf} for f_{0} = 3 GHz, Fig. 4. Here V _{rf} is the rf amplitude at the top of the fridge. Thus, V _{rf} = 4 mV corresponds to , where V _{R}≈60 μV (see Fig. 2).
For V _{rf}<4 mV the junction is in the small microwave signal regime^{38}. The linear response of a Josephson junction has a singularity at ω = ±ω_{J}, where Ω_{J} = 2e/ℏV is the Josephson frequency, and the Josephson junction performs a parametric conversion of the external frequency. Josephson junction JJ8 is in the intermediate damping regime and the V (I) characteristic is expected to become nonhysteretic in the vicinity of the first step. Indeed, we observe no hysteresis for V _{rf}>1.8 mV. Although nonlinear effects can be present at high I and V _{rf}, we want to stress that the first step at the onset of the normal state is due to phase locking between the external and the Josephson frequencies, ω = ±ω_{J}.
When an inplane magnetic field is applied, Shapiro steps at V = 6,12 and 18 μV are clearly visible at low fields, B<2 T. Steps at 12 and 18 μV remain visible up to B≈3 T, whereas the step at 6 μV disappears above B≈2 T. The disappearance of all steps above 3 T is consistent with suppression of the excess current and Andreev reflection at high fields.
Quantitative comparison of the width of the Shapiro steps ΔI_{n}(V _{rf}) for different B extracted from dV/dI data is plotted in Fig. 4. For B<2 T steps at 0, 6, 12, 18 and 24 μV evolve according to Bessel functions with an amplitude A≈150 nA. For B>2 T the step at 6 μV vanishes at low V _{rf}, and reappears at high V _{rf}>10 mV for B = 2.2 T. We also observe that the lowfield rf attenuation does not fit the evolution of the ΔI_{0} plateau. Moreover, a minimum of the 12 μV plateau at V _{rf}≈7 meV coincides with the minimum of the J_{1} Bessel function, suggesting that indeed the 12 μV plateau became the n = 1 Shapiro step. Evolution of plateaux at B>2 T is poorly described by Bessel functions, suggesting that oscillations with different frequencies may contribute to the width of the plateaux. At high fields Majorana particles are expected to form inside the InSb wire close to the ends of the Nb wires. At these fields the supercurrent is dominated by the fusion of two Majorana particles across the gap, which amounts to the 1e charge transfer and leads to the doubling of the Shapiro steps. We emphasize that the doubling of the Shapiro step height is a unique signature of a topological quantum phase transition.
Theoretically it has been argued that Josephson currents with both 2π (I_{c}sin(ϕ)) and 4π (I_{M}sin(ϕ/2)) periodicity should be present in the topological state^{3,39,40,41}, especially in multichannel wires. However, in currentbiased junctions odd steps are expected to vanish even in the presence of large supercurrents carried by the charge2e quasiparticles I_{c}≫I_{M} (ref. 42). In this case the width of even steps is expected to be defined primarily by I_{c}, and the coefficient A remains almost unchanged across the transition, as is observed experimentally, especially if a large number of subbands is occupied in the nanowire owing to the strong coupling to the superconductor. At high voltages across the junction we expect enhanced mixing between gapless and gapped modes, which may explain the prominence of the 18 μV and higher plateaux at B>2 T.
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Acknowledgements
The work was partially supported by ARO grant W911NF0910498 (L.P.R.) and by NSF grant DMR1005851 (J.K.F., X.L.). L.P.R. benefited from discussions with Roman Lutchyn.
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L.P.R. conceived and performed the experiments; J.K.F. and X.L. designed and grew the heterostructures; all authors contributed to the writing of the manuscript.
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Rokhinson, L., Liu, X. & Furdyna, J. The fractional a.c. Josephson effect in a semiconductor–superconductor nanowire as a signature of Majorana particles. Nature Phys 8, 795–799 (2012). https://doi.org/10.1038/nphys2429
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DOI: https://doi.org/10.1038/nphys2429
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