Observation of the 4 π -periodic Josephson effect in indium arsenide nanowires
Dominique Laroche, Daniël Bouman, David J. van Woerkom, Alex Proutski, Chaitanya Murthy, Dmitry I. Pikulin, Chetan Nayak, Ruben J. J. van Gulik, Jesper Nygård, Peter Krogstrup, Leo P. Kouwenhoven, Attila Geresdi
OObservation of the 4 π -periodic Josephson effect in InAs nanowires Dominique Laroche, ∗ Daniël Bouman, ∗ David J. van Woerkom, Alex Proutski, Chaitanya Murthy, Dmitry I. Pikulin, Chetan Nayak,
2, 3
Ruben J. J. van Gulik, Jesper Nygård, Peter Krogstrup, Leo P. Kouwenhoven,
1, 5 and Attila Geresdi † QuTech and Kavli Institute of Nanoscience, Delft University of Technology, 2600 GA Delft, The Netherlands Department of Physics, University of California, Santa Barbara, CA 93106, USA Station Q, Microsoft Research, Santa Barbara, California 93106-6105, USA Center for Quantum Devices and Station Q Copenhagen, Niels Bohr Institute,University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark Microsoft Station Q Delft, 2600 GA Delft, The Netherlands (Dated: December 25, 2017)
Quantum computation by non-Abelian Majo-rana zero modes (MZMs) [1, 2] offers an approachto achieve fault tolerance by encoding quantuminformation in the non-local charge parity statesof semiconductor nanowire networks in the topo-logical superconductor regime [3–5]. Thus far, ex-perimental studies of MZMs chiefly relied on sin-gle electron tunneling measurements [6–11] whichleads to decoherence of the quantum informationstored in the MZM [4, 12]. As a next step towardstopological quantum computation, charge parityconserving experiments based on the Josephsoneffect [13] are required, which can also help ex-clude suggested non-topological origins [14–18] ofthe zero bias conductance anomaly. Here we re-port the direct measurement of the Josephson ra-diation frequency [19] in InAs nanowires with epi-taxial aluminium shells [20]. For the first time, weobserve the π -periodic Josephson effect above amagnetic field of ≈ mT, consistent with theestimated [21, 22] and measured [23] topologicalphase transition of similar devices. The universal relation between the frequency f J of theoscillating current and an applied DC voltage bias V across a superconducting weak link [13] is determinedsolely by natural constants: f J V = 2 eh = Φ − = 483 . / µ V , (1)where e is the single electron charge, h is the Planckconstant and Φ is the superconducting flux quantum.This relation, describing the conventional, π -periodicJosephson effect, can be understood as the tunneling ofCooper pairs with a net charge e (cid:63) = 2 e coupled to pho-tons of energy hf [24]. This coupling, referred to as theAC Josephson effect, has first been measured in super-conducting tunnel junctions [25] and has been shown topersist in metallic weak links [26], carbon nanotubes [27]and semiconductor channels [28, 29], as well as in highcritical temperature superconductors [30].In topological Josephson junctions, the effective tun-neling charge is the single electron charge, e (cid:63) = e , which leads to a factor of two increase in the flux periodicity,giving rise to the so-called π -periodic Josephson effect[21, 22, 31]. Therefore, in this MZM regime, the fre-quency at a given voltage bias V drops by a factor oftwo, f MZM ( V ) = f J ( V ) / , providing a robust signatureof the topological phase transition in the superconductingleads. In real devices however, the finite size of the topo-logical regions [32], poisoning events [22, 31] and Landau-Zener tunneling to the quasiparticle continuum [33] caneffectively restore the π -periodic, trivial state. The lat-ter two parity-mixing effects cause the system to relaxto its ground state, effectively constraining the system inthe lowest topological energy branch (red solid lines inFig. 1a). Nevertheless, out-of-equilibrium measurementsperformed at rates faster than these equilibration pro-cesses can still capture the π -periodic nature of topo-logical junctions [32–34]. In contrast, finite-size effectscan be avoided by biasing the junction at voltages largeenough to overcome the Majorana hybridization gap ε M [33].Here, we report the direct observation of a mag-netic field-induced halving of the Josephson radiationfrequency in InAs nanowire (NW) junctions partiallycovered with an epitaxially grown aluminium shell [20](Fig. 1d). In this system, previous direct transport ex-periments suggest parity lifetimes above . µs [35] andhybridization energies ε M (cid:46) µeV for leads longer than . µm [36]. Thus, a frequency-sensitive measurementin the microwave domain is expected to reveal the π -periodic Josephson effect [37, 38].As a frequency-sensitive microwave detector, we utilizea superconducting tunnel junction with a quasiparticlegap of ∆ DET , wherein the photon-assisted electron tun-neling (PAT) current contributes to the DC current abovea voltage bias threshold eV DET > DET − hf [19, 39](Fig. 1c). This on-chip detector [40], coupled via capaci-tors C C to the NW junction (see Fig. 1b for the schemat-ics and Fig. 1e for optical image of the device) is engi-neered to result in an overdamped microwave environ-ment characterized by a single f c = (2 πRC ) − ≈ GHzcutoff frequency with R = 538 Ω and C = 10 . fF, seeSec. 3 in the Supplementary Information (SI). The re- a r X i v : . [ c ond - m a t . m e s - h a ll ] D ec sulting broadband coupling to the detector [29] inhibitshigher order photon emission, which could mimic the π -periodic Josephson effect [41].The nanowire is deterministically deposited on a setof three gates covered by nm thick SiN x dielectric asshown in Fig. 1d. The Josephson weak link, where theAl shell is removed by wet chemical etching, is locatedabove the central gate (see inset of Fig. 1d). We investi-gated devices with junction lengths ranging from nmto nm. The high quality of the nanowire junctionis apparent from the presence of distinct multiple An-dreev reflection steps in its I NW ( V NW ) characteristics [42](Fig. 1g).The microwave detector, presented in Fig. 1f, is fabri-cated using two angle-evaporated [43] Al/AlO x /Al tunneljunctions, forming a superconducting quantum interfer-ence device (SQUID). This geometry allows us to min-imize the Josephson energy of the detector by applying Φ = Φ / flux through the loop (see Fig. 1h) and thusto limit its backaction to the nanowire. The respectively and nm thick Al layers set an in-plane critical mag-netic field of the detector in excess of T, well above themeasured topological transition in similar devices [23].The circuit parameters and fabrication details are givenin Sec. 1 of the SI.In the presence of a voltage spectral density S V ( f ) , theDC current contribution of the PAT process is as follows[19, 39] in the subgap regime, where eV DET < DET : I PAT ( V DET ) = (cid:90) ∞ d f (cid:18) ehf (cid:19) S V ( f ) I QP , (cid:18) V DET + hfe (cid:19) . (2)Here, I QP, ( eV DET ) is the tunnel junction current in theabsence of absorbed radiation, S V ( f ) = 0 (see Fig. 1h).Note that the quasiparticle gap edge at eV DET = 2∆
DET results in a sharp increase of I QP, ( eV DET ) . In the pres-ence of monochromatic radiation with a frequency f , S V ( f ) ∼ δ ( f − f ) , I PAT thus develops a step-like featureat hf = 2∆ DET − eV DET . With a phenomenological ef-fective charge e (cid:63) of the AC Josephson effect, we write thiscondition in terms of the voltage drop on the nanowire, V NW : e (cid:63) V NW = hf = 2∆ DET − eV DET , (3)where e (cid:63) = 2 e for conventional junctions (see Eq. (1))and e (cid:63) = e in the π -periodic regime. To extract e (cid:63) and thus determine the periodicity of the Joseph-son radiation, we track the transconductance peak d I PAT / d V NW ( V NW , V DET ) measured by standard lock-intechniques at a frequency of . Hz. The experimentswere performed at the base temperature of a dilution re-frigerator ( ∼ mK).Typical experimental datasets are shown in Fig. 2 fortwo nanowire junctions (NW1 and NW2, respectively),and one Al/AlO x /Al tunnel junction (T) as the sourceof Josephson radiation. We limit the detector voltage V DET (mV) I D E T ( n A ) -85085 1-1630-3-6-500 -250 0 250 500 G N / G = 1.2 B = 0 T Φ = Φ /2 V DET ( μ V) I D E T ( n A ) -2 Δ NW - Δ NW =-225 μ eV Δ NW Δ NW Δ NW Δ NW -1 1-50050 V NW (mV) I N W ( n A ) -500 -250 0 250 500-20-1001020 V NW ( μ V) I N W ( n A ) G N / G = 0.6 B = 0 T V DET = 0 V μ m Φ μ m V gmiddle V glower V gupper B μ m Δ DET V DET hfr
DET r NW R NW R CC C I NW I DET C C V NWS V DETS V DET V NW π π Δ NW Δ NW E ϕε M hg fed cba FIG. 1.
Principles of the experiment. (a)
Energy disper-sion of topologically trivial (dashed green line) and nontrivial(solid red line) Andreev levels inside a NW Josephson junc-tion as a function of the phase difference across the junction.The gap ε M arises from the finite MZMs wavefunction over-lap. (b) Equivalent circuit diagram of the device. The NWjunction (in blue box) is capacitively coupled to the supercon-ducting tunnel junction (red box) via the capacitors C C . Themicrowave losses and stray capacitance are modeled by theRC element enclosed by the dashed black box, see text. Theapplied DC bias voltages are V SNW and V SDET with an effectiveinternal resistance r NW and r DET , respectively. (c)
Principleof the frequency sensitive detection based on photon-assistedtunneling: an absorbed photon with an energy hf gives riseto quasiparticle current if hf > DET − eV DET . (d) Scan-ning electron micrograph of the NW junction placed on threeelectrostatic gates. A false color micrograph of the junction isshown in the inset, with the epitaxial Al shell highlighted incyan. (e)
Bright field optical image of the coupling circuitrybetween the NW junction (blue box) and the detector junc-tion (red box). (f )
Micrograph of the split tunnel junction de-tector. The junctions are encircled. (g)
Measured I NW ( V NW ) characteristics of the NW junction at zero in-plane magneticfield exhibiting a supercurrent branch and multiple Andreevreflections. (h) Measured I DET ( V DET ) trace of the detectorsplit junction at zero in-plane magnetic field with a minimizedswitching current. The insets in panels (g) and (h) show thelarge scale I ( V ) trace of each junction. The normal state con-ductance, G N is given in the units of G = 2 e /h . All imagesand data were taken on device NW1. range by the condition d I DET / d V DET < µS where thesubgap quasiparticle current is still negligible, typically I DET (cid:46) nA. A lower limit of the emitter junction volt-age is defined by the phase diffusion regime [44], char-acterized by periodic switching and retrapping events, Δ D E T ( μ e V ) B (mT) Δ = 257 μ eV B C = 1030 mT e * / e d I PA T / d V E M I ( μ S ) V DET ( μ V) V E M I ( μ V ) d I PA T / d V N W ( a . u . ) B = 500 mT e * / e = 1.80 ± 0.41 d I PA T / d V E M I ( μ S ) V E M I ( μ V ) T G N = 0.26 G
420 440 460 480 500 V DET ( μ V) B = 0 mT e * / e = 2.00 ± 0.16 d I PA T / d V N W ( a . u . ) Δ = 265 μ eV B C = 670 mT B (mT) Δ D E T ( μ e V ) e * / e d I PA T / d V N W ( a . u . )
400 420 440 V DET ( μ V) B = 275 mT e * / e = 1.19 ± 0.15 V N W ( μ V ) d I PA T / d V N W ( n S ) NW2 G N = 0.52 G V DET ( μ V) V N W ( μ V ) d I PA T / d V N W ( μ S )
380 400 480 d I PA T / d V N W ( a . u . )
420 440 460 B = 0 mT e * / e = 1.94 ± 0.15 Δ D E T ( μ e V ) B (mT) Δ = 258 μ eV B C = 1180 mT e * / e B = 650 mT e * / e = 1.06 ± 0.58 V N W ( μ V ) d I PA T / d V N W ( a . u . )
380 400 420 V DET ( μ V) d I PA T / d V N W ( n S ) V DET ( μ V)
440 460 480 d I PA T / d V N W ( a . u . ) B = 0 mT e * / e = 1.77 ± 0.30NW1 G N = 0.37 G V N W ( μ V ) d I PA T / d V N W ( n S ) ihgfedcba FIG. 2.
Magnetic field-induced π -periodic Josephson radiation. Differential transconductance d I PAT / d V NW as afunction of V NW and V DET for device NW1 (panels (a) and (b) ), NW2 (panels (d) and (e) ) and T, an Al/AlO x /Al tunneljunction (panels (g) and (h) ) at zero and finite magnetic fields, respectively. The position of the transconductance peak mapsthe frequency of the monochromatic Josephson radiation. A linear fit e (cid:63) V NW = 2∆ DET − eV DET through these peaks is shown asan orange line. Dashed green and red lines show linear fits with a fixed slope corresponding to e (cid:63) = 2 e and e (cid:63) = e , respectively.The shaded regions show the regimes where the fit of the transconductance peak is not reliable, see text. Two normalized andsmoothed horizontal linecuts are plotted, where arrows point at the position of the extracted peaks. The orange, green and reddots denote the position of the best fit, the e (cid:63) = 2 e fit and the e (cid:63) = e fit, respectively. The evolution of e (cid:63) ( B ) and DET ( B ) are presented in panels (c) , (f ) and (i) . The transition from to π - to π -periodic Josephson radiation is observed between and
300 mT for the NW devices as e (cid:63) evolves from values near e (green circles) to values close to e (red triangles). No suchtransition is observed for device T. For all devices, DET ( B ) drops monotonically (black dashed line, see text), independentlyof the change in e (cid:63) . which breaks the validity of Eq. (1) (see Fig. S16 in theSI). We therefore do not consider the low V NW regime,within the supercurrent peak. We show this range, ex-cluded from the linear fits, shaded in grey in Fig. 2 andFig. 3 (see Fig. S5 in the SI on the characterization ofthese limits). We fit the peak positions using Eq. (3) inorder to extract e (cid:63) and ∆ DET as a function of the ap-plied in-plane magnetic field. The standard deviation ofthe fitted parameters is determined using the bootstrap-ping method [45]. For details, see Sec. 5 in the SI.At zero magnetic field (Fig. 2a, d and g), the emit-ted Josephson radiation is always π -periodic with anextracted effective charge close to e (cid:63) = 2 e , as shownby the good agreement between the orange line and thedashed green line (best fit with fixed e (cid:63) = 2 e ). In con-trast, NW1 and NW2 exhibit the π -periodic Josephsoneffect above a threshold magnetic field (Fig. 2b and e),where e (cid:63) ≈ e . The full evolution is shown in Fig. 2cand 2f, respectively, where a sharp transition is visi-ble from e (cid:63) ≈ e (green circles) to e (cid:63) ≈ e (red trian-gles). However, the tunnel junction, where no topologicalphase transition is expected, exhibits e (cid:63) ≈ e in the samemagnetic field range (Fig. 2h). Finally, the fitted ∆ DET (Fig. 2c, f and i) shows a monotonic decrease describedby DET ( B ) = 2∆ (cid:112) − B /B c for all devices (dashedlines), with no additional feature at the transition field.Fig. 3 shows the magnetic field evolution of deviceNW3 at two distinct gate settings. By tuning the chemi-cal potential in the nanowire via changing the gate volt-ages, it is possible to displace the position of the onsetof the π -periodic Josephson radiation from ≈
175 mT (Fig. 3c) to values larger than
375 mT (Fig. 3f). Notethat the additional local maximum at high V NW values,also observed in earlier experiments [29], is attributed tothe shot noise of the nanowire junction.The possibility to tune the nanowire devices into the π -periodic Josephson radiation regime with both mag-netic field and chemical potential is consistent with thepredicted phase diagram of this system [21, 22, 31]. Weobserve the same behaviour in four distinct nanowire de-vices (see Fig. S7 in the SI for NW4), which we can inter-pret within the single subband model of the topologicalphase transition that takes place at a magnetic field B (cid:63) ,where E z = gµ B B (cid:63) / (cid:112) ∆ NW + µ NW . Here g and µ B are the Landé g-factor and the Bohr magnetron, respec-tively. From our device parameters (see table S2 in theSI), lower bounds on the g-factors ranging from g ≈ ( B (cid:63) = 175 mT ) in device NW3 to g ≈ ( B (cid:63) = 190 mT )in device NW4 are obtained, in agreement with values re-ported in similar devices [10, 23, 46]. In contrast, an acci-dental crossing of a trivial Andreev bound state would beinconsistent with the observed field range of ∆ B ∼ . Tof the π -periodic radiation, since within this range, aspinful Andreev level [46] would evolve over the scale ofthe superconducting gap, ∆ NW ∼ gµ B ∆ B suppressingthe π periodicity. a NW3A G N = 0.57 G NW3B G N = 0.73 G fedcb B (mT) Δ D E T ( μ e V ) Δ = 252 μ eV B C = 1205 mT e * / e d I PA T / d V N W ( a . u . )
420 440 460 V DET ( μ V) B = 200 mT e * / e = 1.72 ± 0.19 V N W ( μ V ) d I PA T / d V N W ( n S ) V N W ( μ V ) B = 0 mT e * / e = 1.94 ± 0.15
420 440 480 d I PA T / d V N W ( a . u . ) V DET ( μ V) d I PA T / d V N W ( μ S ) B (mT) Δ D E T ( μ e V ) Δ = 254 μ eV B C = 847 mT e * / e V DET ( μ V) V N W ( μ V ) d I PA T / d V N W ( n S )
430 440 450 d I PA T / d V N W ( a . u . ) B = 375 mT e * / e = 0.9 ± 0.4 V DET ( μ V) V N W ( μ V ) d I PA T / d V N W ( n S )
440 460 480 d I PA T / d V N W ( a . u . ) B = 0 mT e * / e = 1.85 ± 0.37 FIG. 3.
Gate tuning of the π -periodic radiationregime. Differential transconductance d I PAT / d V NW as afunction of V NW and V DET for device NW3 at gate settingA (panels (a) , (b) ) and setting B (panels (d) , (e) ) at zeroand finite magnetic field, respectively. A linear fit and fitswith a fixed slope e (cid:63) = 2 e and e (cid:63) = e are shown as an or-ange line, a dashed green line and a dashed red line, respec-tively. Two normalized and smoothed horizontal linecuts arealso presented, where arrows point at the position of the ex-tracted peaks. The evolution of e (cid:63) ( B ) and ∆ DET ( B ) is shownin panels (c) and (f ) . A transition from to π - to π -periodicJosephson radiation is observed for gate setting A, but theradiation remains π -periodic for setting B. The gate voltagevalues are shown in table S2 of the SI. We observe a single Josephson radiation frequency inthe π -periodic regime, which is consistent with the su-percurrent being predominantly carried by a single trans-mitting mode. While we were not able to reliably ex-tract the transparency and the number of modes in our ab cd S V ( μ V / G H z ) S V ( μ V / G H z ) V N W ( μ V ) ABS ( τ τ =0.9, V =15 μ V) MZM ( V =10 μ V, V =25 μ V)MZM ( V =10 μ V, V =100 μ V) f (GHz) f (GHz) f (GHz) f (GHz) S V ( μ V / G H z ) S V ( μ V / G H z ) V N W ( μ V ) S V ( μ V / G H z ) V N W ( μ V ) S V ( μ V / G H z ) V N W ( μ V ) S V ( μ V / G H z ) S V ( μ V / G H z ) FIG. 4.
The calculated radiation spectrum.
The volt-age spectral density S V ( f ) incident on the detector junction,computed by numerically solving the system of stochastic dif-ferential equations shown in Sec. 7 of the SI. Panels (a) and (b) show results for a junction in the trivial regime (smalltransmission and large transmission, respectively), while pan-els (c) and (d) show the emission spectrum in the topologicalregime. V and V are voltage scales for Landau-Zener tunnel-ing between branches of the junction bound state and for tun-neling to the quasiparticle continuum, respectively, see text.Circuit parameters are set as r NW = 2 . , R NW = 50 kΩ , R = 0 . , C = 10 fF , C C = 400 fF , and I NW = 8 nA . Thenoise temperature is T = 150 mK and the quasiparticle poi-soning rate is Γ q = 100 MHz . As in Fig. 2, the dashed greenand red lines show the frequency of the Josephson radiationcorresponding to e (cid:63) = 2 e and e (cid:63) = e , respectively. The esti-mated phase diffusion region is shaded in gray. devices, the single mode regime was observed earlier insimilar InAs nanowires [46–48]. We also note that an up-per bound on the channel transmission of τ = G N /G can be determined from the normal state conductance G N < G which is shown in Fig. 2 and Fig. 3 for eachdevice.Next, we numerically evaluate the expected voltagespectral density seen by the detector junction in vari-ous regimes. We use the quasiclassical resistively and ca-pacitively shunted junction (RCSJ) model coupled to astochastic differential equation describing the occupationof the single pair of Andreev levels in the NW junction.The equivalent circuit of the device in the microwave do- main is shown in Fig. 1b, where each element is exper-imentally characterized [29] (see Sec. 7 in the SI). Notethat we neglect the load of the detector on the circuit,which is justified by its negligible subgap conductancecompared to that of all other elements in the circuit.Our model of the nanowire junction considers Landau-Zener (LZ) tunneling between branches of the energy-phase dispersion shown in Fig. 1a, as well as tunnel-ing to the continuum, and stochastic quasiparticle poi-soning events [33]. The probability of LZ tunneling isdetermined by the voltage drop V NW according to the P LZ = exp( − V /V NW ) , where eV = 4 πε M / (∆ NW √ τ ) is the characteristic voltage above which P LZ ∼ . Inthis limit, π -periodicity is observed despite the gap ε M caused by finite-size effects [23]. Similarly, LZ tunnelingto the continuum close to ϕ = 2 π defines a voltage scale eV = 2 π ∆ NW (1 − √ τ ) / √ τ , above which π -periodicityis restored [33]. We note that a trivial Andreev boundstate in the short junction limit can be modeled similarlywith eV = π ∆ NW (1 − τ ) and eV = 0 .Fig. 4 shows representative plots obtained by numer-ically evaluating S V ( f, V NW ) (see Sec. 7.5 of the SI),which determines the photon-assisted tunneling currentby Eq. (2). We observe that the numerical results agreewell with the characteristic features of the experimen-tal data. We find that the circuit equations allow fora phase diffusion regime at low V NW values [44], where e (cid:63) V NW < hf , because the junction spends part of thetime in the steady supercurrent state where the voltagedrop is zero. The calculations also reproduce the absenceof higher harmonics in the radiation spectrum, attributedto the low transmission of the junction and overdampednature of the microwave environment [41]. This confirmsour expectation of the suppression of multiphoton pro-cesses due to a low quality factor, justifying the usage ofthe semiclassical junction model.A key result of these simulations is that, with the cir-cuit parameters taking values representative of those inthe experiment, the radiation frequency always reflectsthe internal dynamics of the nanowire Josephson junc-tion both in the π -periodic (Fig. 4a and b) and in the π -periodic emission regime (Fig. 4c and d). Finally, wenote that our results are consistent with V (cid:46) µeVtranslating to an avoided crossing ε M (cid:46) µV. Usingthe exponential cutoff in Ref. [23], this suggests that ourdevices have a continuous topological region longer than . µm on each side of the nanowire junction.In conclusion, we observed the π -periodic Josephsoneffect in multiple InAs nanowires above a threshold mag-netic field in a range of − mT. This effect, whichcan be suppressed by tuning the gate voltages, is consis-tent with the expected signatures of a topological phasetransition. By observing the periodicity of the Joseph-son effect using an on-chip microwave detector, we inves-tigated this system whilst preserving its charge parity,in line with the requirements for prospective topologicalquantum computers. This experimental technique mayalso prove instrumental in identifying more exotic non-Abelian anyon states [49, 50], due to its proven sensi-tivity to the periodicity of the Josephson effect, directlymeasuring the charge fractionalization of the anyon state[51, 52]. DATA AVAILABILITY
The datasets generated and analysed during this studyare available at the 4TU.ResearchData repository, DOI:10.4121/uuid:1f936840-5bc2-40ca-8c32-1797c12cacb1(Ref. [53]).
ACKNOWLEDGEMENTS
The authors acknowledge O. Benningshof,J. Mensingh, M. Quintero-Pèrez and R. Schoutenfor technical assistance. This work has been supportedby the Dutch Organization for Fundamental Researchon Matter (FOM), Microsoft Corporation Station Q, theDanish National Research Foundation and a SynergyGrant of the European Research Council. A. G. ac-knowledges the support of the Netherlands Organizationfor Scientific Research (NWO) by a Veni grant.
AUTHOR CONTRIBUTIONS
D. L., D. B., D. J. v. W. and A. P. fabricated the sam-ples and performed the experiments. P. K. and J. N. con-tributed to the nanowire growth. L. P. K. and A. G. de-signed and supervised the experiments. C. M., D. P. andC. N. developed the theoretical model of the devices.D. L., D. B., D. J. v. W., R. J. J. v. G., L. P. K. andA. G. analyzed the data. The manuscript has been pre-pared with contributions from all the authors. ∗ These authors contributed equally to this work. † To whom correspondence should be addressed; E-mail:[email protected][1] Kitaev, A. Y. Unpaired Majorana fermions in quantumwires.
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