Abstract
Quantum computation by nonAbelian Majorana zero modes (MZMs) offers an approach to achieve fault tolerance by encoding quantum information in the nonlocal charge parity states of semiconductor nanowire networks in the topological superconductor regime. Thus far, experimental studies of MZMs chiefly relied on single electron tunneling measurements, which lead to the decoherence of the quantum information stored in the MZM. As a next step towards topological quantum computation, charge parity conserving experiments based on the Josephson effect are required, which can also help exclude suggested nontopological origins of the zero bias conductance anomaly. Here we report the direct measurement of the Josephson radiation frequency in indium arsenide nanowires with epitaxial aluminium shells. We observe the 4πperiodic Josephson effect above a magnetic field of ≈200 mT, consistent with the estimated and measured topological phase transition of similar devices.
Introduction
The universal relation between the frequency f_{J} of the oscillating current and an applied DC voltage bias V across a superconducting weak link^{1} is determined solely by natural constants.
where e is the single electron charge, h is the Planck constant and Φ_{0} is the superconducting flux quantum. This relation, describing the conventional, 2πperiodic Josephson effect, can be understood as the tunneling of Cooper pairs with a net charge e^{*} = 2e coupled to photons of energy hf^{2}. This coupling, referred to as the AC Josephson effect, has first been measured in superconducting tunnel junctions^{3} and has been shown to persist in metallic weak links^{4}, carbon nanotubes^{5} and semiconductor channels^{6,7}, as well as in high critical temperature superconductors^{8}.
In proximitized semiconductor nanowires, an effective superconducting gap with a pwave symmetry arises due to the breaking of the timereversal symmetry above a threshold magnetic field^{9,10,11,12,13,14,15,16}. When a weak link is formed between two leads, the pwave component leads to a factor of two increase in the flux periodicity, giving rise to the socalled 4πperiodic Josephson effect^{17,18}. Phenomenologically, this phase periodicity is equivalent to an effective tunneling charge e^{*} = e instead of 2e in Eq. (1). Therefore, in this Majorana zero mode (MZM) regime, the frequency at a given voltage bias V drops by a factor of two, f_{MZM}(V) = f_{J}(V)/2, providing a robust signature of the topological phase transition in the superconducting leads. In real devices, however, the finite size of the topological regions^{19}, poisoning events^{9,18} and Landau–Zener (LZ) tunneling to the quasiparticle continuum^{20} can effectively restore the 2πperiodic trivial state. The latter two paritymixing effects cause the system to relax to its ground state, effectively constraining the system in the lowest topological energy branch (red solid lines in Fig. 1a). Nevertheless, outofequilibrium measurements performed at rates faster than these equilibration processes can still capture the 4πperiodic nature of topological junctions^{19,20,21}. In contrast, finitesize effects can be avoided by biasing the junction at voltages large enough to overcome the Majorana hybridization gap ε_{M}^{20}.
Here, we report the direct observation of a magnetic fieldinduced halving of the Josephson radiation frequency^{22} in indium arsenide nanowire (InAs NW) junctions partially covered with an epitaxially grown aluminium shell (Fig. 1d). In this system, possessing a hard induced superconducting gap^{23}, previous direct transport experiments suggest parity lifetimes above 0.1 μs^{24} and hybridization energies ε_{M} ≲ 1 μeV for leads longer than 1.5 μm^{25}. Thus, a frequencysensitive measurement in the microwave domain is expected to reveal the 4πperiodic Josephson effect^{26,27}.
Results
Frequencysensitive detection of Josephson radiation
As a frequencysensitive microwave detector, we utilized a superconducting tunnel junction with a quasiparticle gap of Δ_{DET}, wherein the photonassisted electron tunneling (PAT) current contributed to the DC current above a voltage bias threshold eV_{DET} > 2Δ_{DET}  hf^{22,28} (Fig. 1c). This onchip detector^{29}, coupled via capacitors C_{C} to the NW junction (see Fig. 1b for the schematics and Fig. 1e for an optical image of the device), was engineered to result in an overdamped microwave environment characterized by a single f_{c} = (2πRC)^{1} ≈ 28 GHz cutoff frequency with R = 538 Ω and C = 10.4 fF (see Supplementary Fig. 2). The resulting broadband coupling to the detector^{7} inhibited higher order photon emission, which could mimic the 4πperiodic Josephson effect^{30}.
The nanowire was deterministically deposited on a set of three gates covered by 30nm thick SiN_{x} dielectric as shown in Fig. 1d. The Josephson weak link, where the Al shell was removed by wet chemical etching, was located above the central gate (see inset of Fig. 1d). We investigated devices with junction lengths ranging from 86 to 271 nm. The high quality of the nanowire junction is apparent from the presence of distinct multiple Andreev reflection steps in its I_{NW}(V_{NW}) characteristics^{31} (Fig. 1g), which is a signature of the hard superconducting gap in the nanowire^{23}. The observed curves and linear conductance also establish that no conductive mode with a transmission close to unity exists in the channel, which could contribute to the 4πperiodic signal even in the absence of topological ordering^{20}.
The microwave detector, presented in Fig. 1f, was fabricated using two angleevaporated^{32} Al/AlO_{x}/Al tunnel junctions, forming a superconducting quantum interference device. This geometry allowed us to minimize the Josephson energy of the detector by applying Φ = Φ_{0}/2 flux through the loop (see Fig. 1h) and thus to limit its back action to the nanowire. The respectively 8 and 11nm thick Al layers set an inplane critical magnetic field of the detector in excess of 1 T, well above the measured topological transition in similar devices^{25}. Nevertheless, increasing subgap currents limited our experimental field range to 325–650 mT for different devices. The circuit parameters and fabrication details are given in the Supplementary Tables and in the Methods section, respectively.
In the presence of a voltage spectral density S_{V}(f), the DC current contribution of the PAT process is as follows^{22,28} in the subgap regime, where eV_{DET} < 2Δ_{DET}:
Here, I_{QP,0}(V_{DET}) is the tunnel junction current in the absence of absorbed radiation, S_{V}(f) = 0 (see Fig. 1h). Note that the quasiparticle gap edge at eV_{DET} = 2Δ_{DET} resulted in a sharp increase of I_{QP,0}(V_{DET}). In the presence of monochromatic radiation with a frequency f_{0}, S_{V}(f) ~ δ(f  f_{0}), I_{PAT}(V_{DET}) thus developed a steplike feature at hf_{0} = 2Δ_{DET}  eV_{DET}. With a phenomenological effective charge e^{*} of the AC Josephson effect, we write this condition in terms of the voltage drop on the nanowire, V_{NW}.
where e^{*} = 2e for conventional junctions (see Eq. (1)) and e^{*} = e in the 4πperiodic regime. To extract e^{*} and thus determine the periodicity of Josephson radiation, we tracked the transconductance peak dI_{PAT}/dV_{NW}(V_{NW}, V_{DET}) measured by standard lockin techniques at a frequency of 17.7 Hz (see Supplementary Fig. 1). The experiments were performed at the base temperature of a dilution refrigerator (~20 mK).
Josephson radiation as a function of the magnetic field
Typical experimental datasets are shown in Fig. 2 for two nanowire junctions (NW1 and NW2, respectively) as the source of Josephson radiation. We limited the detector voltage range by the condition dI_{DET}/dV_{DET} < 10 μS, where the subgap quasiparticle current was still negligible, typically I_{DET} ≲ 1 nA. A lower limit of the emitter junction voltage was defined by the phase diffusion regime^{33} characterized by periodic switching and retrapping events, which breaks the validity of Eq. (1) (see Supplementary Note 3). We therefore did not consider the low V_{NW} regime, within the supercurrent peak. We show this range, excluded from the linear fits, shaded in gray in Figs. 2 and 3 (see Supplementary Note 2 on the characterization of these limits). We fit the peak positions using Eq. (3) in order to extract e^{*} and Δ_{DET} as a function of the applied inplane magnetic field. The typical standard deviation was 3.6 GHz for each frequency data point (see Supplementary Note 2). The error bars of the fitted parameters were determined using the bootstrapping method^{34} (see Supplementary Note 2), and they show the full width at half maximum yielding a confidence level of 75% for a Gaussian lineshape.
At zero magnetic field (Fig. 2a, c), the emitted Josephson radiation is always 2πperiodic with an extracted effective charge close to e^{*} = 2e, as shown by the good agreement between the orange line and the dashed green line (best fit with fixed e^{*} = 2e). In contrast, NW1 and NW2 exhibit the 4πperiodic Josephson effect above a threshold magnetic field (Fig. 2b, d), where e^{*} ≈ e. The full evolution is shown in Fig. 2e, g, where a sharp transition is visible from e^{*} ≈ 2e (green circles) to e^{*} ≈ e (red triangles). Finally, the fitted Δ_{DET} (Fig. 2c, f) shows a monotonic decrease described by \(2{\mathrm{\Delta }}_{{\mathrm{DET}}}(B) = 2{\mathrm{\Delta }}_0\sqrt {1  B^2{\mathrm{/}}B_{\mathrm{c}}^2}\) for all devices (dashed lines), with no additional feature at the transition field. In contrast with the nanowire junctions, our control device, an Al/AlO_{x}/Al tunnel junction, exhibits no transition in e^{*} over the entire magnetic field range (see Supplementary Fig. 5).
Josephson radiation at different gate voltages
Figure 3 shows the magnetic field evolution of device NW3 at two distinct gate settings with similar G_{N} and dI_{PAT}/dV_{NW} corresponding to similar Josephson couplings. By tuning the chemical potential in the nanowire via changing the gate voltages, it is possible to displace the position of the onset of the 4πperiodic Josephson radiation from ≈175 mT (Fig. 3b) to values larger than 375 mT (Fig. 3d). Note that the additional local maximum at high V_{NW} values, also observed in earlier experiments^{7}, is attributed to the shot noise of the nanowire junction.
The possibility to tune the nanowire devices into the 4πperiodic Josephson radiation regime with both magnetic field and chemical potential is consistent with the predicted phase diagram of this system^{9,10,18}. We observed the same behavior in four distinct nanowire devices (see Supplementary Fig. 4 for device NW4), which we could interpret within the single subband model of the topological phase transition that takes place at a magnetic field B^{*}, where \(E_{\mathrm{z}} = g\mu _{\mathrm{B}}B^ \ast {\mathrm{/}}2 = \sqrt {{\mathrm{\Delta }}_{{\mathrm{NW}}}^2 + \mu _{{\mathrm{NW}}}^2}\). Here g and μ_{B} are the Landé g factor and the Bohr magnetron, respectively. From our device parameters (see Supplementary Table 2), lower bounds on the g factors ranging from g ≈ 11 (B^{*} = 175 mT) in device NW3 to g ≈ 35 (B^{*} = 190 mT) in device NW4 were obtained, in agreement with the values reported in similar devices^{15,25,35}. In contrast, an accidental crossing of a trivial Andreev bound state would be inconsistent with the observed field range of ΔB ~ 0.3 T of the 4πperiodic radiation, since within this range, a spinful Andreev level^{35} would evolve over the scale of the superconducting gap, Δ_{NW} ~ gμ_{B}ΔB, suppressing the 4π periodicity. We, however, did not observe a continuous variation of the onset magnetic field B^{*} as a function of the applied gate voltages. This behavior is consistent with the calculations of the topological phase diagram based on realistic device simulations including orbital effects of the magnetic field^{36} and multiple spatial dimensions^{37,38} of the device.
We observed a single Josephson radiation frequency in the 4πperiodic regime, which was consistent with the supercurrent being predominantly carried by a single transmitting mode. While we were not able to reliably extract the transparency and the number of modes in our devices, the singlemode regime was observed earlier in similar InAs nanowires^{35,39,40}. We also note that an upper bound on the channel transmission of τ = G_{N}/G_{0} could be determined from the normal state conductance G_{N} < G_{0}, which was measured in the linear regime well above the superconducting gap. This value is shown in Figs. 2 and 3 for each device.
Numerical simulations of the Josephson radiation frequency
Next, we numerically evaluated the expected voltage spectral density seen by the detector junction in various regimes. We used the quasiclassical resistively and capacitively shunted junction model coupled to a stochastic differential equation describing the occupation of the single pair of Andreev levels in the NW junction. The equivalent circuit of the device in the microwave domain is shown in Fig. 1b, in which each element is experimentally characterized^{7} (see Supplementary Fig. 2 and Supplementary Tables). Note that we neglected the load of the detector on the circuit, which is justified by its negligible subgap conductance compared to that of all other elements in the circuit.
Our model of the nanowire junction considers LZ tunneling between branches of the energyphase dispersion shown in Fig. 1a, as well as tunneling to the continuum and stochastic quasiparticle poisoning events^{20}. The probability of LZ tunneling is determined by the voltage drop V_{NW} according to P_{LZ} = exp(V_{0}/V_{NW}), where \(eV_0 = 4\uppi\varepsilon _{\mathrm{M}}^2{\mathrm{/}}\left( {{\mathrm{\Delta }}_{{\mathrm{NW}}}\sqrt \tau } \right)\) is the characteristic voltage above which P_{LZ} ~ 1. In this limit, 4π periodicity is observed despite the gap ε_{M} caused by finitesize effects^{25}. Similarly, LZ tunneling to the continuum close to φ = 2π defines a voltage scale \(eV_1 = 2{\mathrm{\pi\Delta }}_{{\mathrm{NW}}}\left( {1  \sqrt \tau } \right)^2{\mathrm{/}}\sqrt \tau\), above which 2π periodicity is restored^{20}. We note that a trivial Andreev bound state in the short junction limit can be modeled similarly with eV_{0} =π Δ_{NW}(1  τ) and eV_{1} = 0.
Figure 4 shows representative plots obtained by numerically evaluating S_{V}(f, V_{NW}) (see Supplementary Note 3), which determines the photonassisted tunneling current by Eq. (2). We observed that the numerical results agreed well with the characteristic features of the experimental data. We found that the circuit equations allowed for a phasediffusion regime at low V_{NW} values^{33}, where e^{*}V_{NW} < hf, because the junction spent part of the time in the steady supercurrent state in which the voltage drop was zero. The calculations also reproduced the absence of higher harmonics in the radiation spectrum, attributed to the low transmission of the junction and the overdamped nature of the microwave environment^{30}. This confirms our expectation of the suppression of multiphoton processes due to a lowquality factor, justifying the usage of the semiclassical junction model.
A key result of these simulations in a wide range of junction parameters is that with the circuit elements taking values representative of those in the experiment, the radiation frequency always reflects the internal dynamics of the nanowire Josephson junction both in the 2πperiodic (Fig. 4a, b) and in the 4πperiodic emission regimes (Fig. 4c, d). Finally, we note that our results are consistent with V_{0} ≲ 15 μeV translating to an avoided crossing ε_{M} ≲ 10 μeV. Using the exponential cutoff in ref. ^{25}, this suggests that our devices have a continuous topological region of several hundreds of nanometers on each side of the nanowire junction, which is consistent with the scanning electron microscopic images of the devices.
Discussion
In conclusion, we observed the 4πperiodic Josephson effect in multiple InAs nanowires above a threshold magnetic field in a range of 175–300 mT. This effect, which can be suppressed by tuning the gate voltages, is consistent with the expected signatures of a topological phase transition. By observing the periodicity of Josephson effect using an onchip microwave detector, we investigated this system while preserving its charge parity, in line with the requirements for prospective topological quantum computers. This experimental technique may also prove instrumental in identifying more exotic nonAbelian anyon states^{41,42}, due to its proven sensitivity to the periodicity of Josephson effect, directly measuring the charge fractionalization of the anyon state^{43,44}.
Methods
Device fabrication
The devices were fabricated on commercially available undoped Si substrates with a 285nm thick insulating SiO_{x} layer in a similar fashion as in refs. ^{7,35}. All etching and metaldeposition steps were realized using standard positivetone electronbeam lithography techniques. At first, three Ti/Au (5 nm/15 nm) electrostatic gates and the coupling capacitor bottom plates were deposited (see Fig. 1 for design details). These were subsequently covered by an ~30nm thick SiN_{x} dielectric layer deposited by sputtering. Eleven 100nm wide Cr/Pt (5 nm/25 nm) tracks were then defined. These ~100 Ω μm^{1} resistive lines connected the gates, the (yet to be defined) Al/AlO_{x}/Al detector, and the nanowire to the instrumentation setup. Next, the Al/AlO_{x}/Al Josephson junctions were fabricated by evaporating 8 and 11nm thick Al layers with an intermediate insitu oxidation step at 0.5 mbar for 4 min using the Dolan bridge technique^{32}. The nanowires were then deterministically deposited onto the electrostatic gates with a micromanipulator setup equipped with an optical microscope. A gap in the nanowire Al shell was then created by wet etching using Transene D at a temperature of 48.2 °C for 12 s. Next, both the nanowire and the detector junctions were connected to the resistive lines with an 80nm thick sputtered NbTiN film after an insitu Ar plasma milling step. Finally, a Ti/Au (15/100 nm) layer was evaporated to define quasiparticle traps, the upper capacitor plates, and the contact pads. We note that no NbTiN film was used in device NW3. Instead, a Ti/Au (15/100 nm) layer was used to contact the nanowire and the detector. The dimensions and properties of each device are presented in Supplementary Table 1, and the experimental setup is described in Supplementary Fig. 1. We note that the detector was made of narrow and thin aluminum sections (see Fig. 1f) to limit the presence of vortices near the Al/AlO_{x}/Al junctions, and thus to decrease the subgap current in finite magnetic field.
The InAs nanowires used in this work were grown via a twostep process by molecular beam epitaxy. The InAs nanowires were grown at 420 °C using the vapor–liquid–solid method with Au droplets as the catalyst. After cooling the system to 30 °C, Al was epitaxially grown on two of the six nanowire facets^{23}.
The microwave environment of the InAs Josephson junction
We modeled the effective microwave environment of the nanowire Josephson junction with a parallel lumped resistor (R) and capacitor (C) element, which accounted for the lowpass nature of the coupling circuit (see inset of Supplementary Fig. 2a). We determined the effective RC values by measuring a sample in which the nanowire junction was replaced by an Al/AlO_{x}/Al tunnel junction. The supercurrent peak was fitted against the Ivanchenko–Zil’berman model to find the RC values and the noise temperature of the circuit^{33} at zero magnetic field (see Supplementary Fig. 2a). The critical current as a function of the magnetic field was then found using the same model, with the RC and the noise temperature fixed at their zero field values (Supplementary Fig. 2b). We note that the same coupling circuit was used in ref. ^{7}, leading to RC and noise temperature values in good agreement with the current ones. Thus, we conclude that the reproducibility is good for all the samples featured in the current study. These parameters are used to theoretically study the dynamics of Josephson radiation.
Reproducibility of the transition for nanowire devices
Supplementary Figure 3 shows every differential transconductance color plot from which the effective charge e^{*} has been extracted in Fig. 2e. The color plots nominally follow the same trend as the ones presented in Fig. 2. Supplementary Figure 4 shows the magnetic field evolution of e^{*} in device NW4. Device NW4 also exhibits a transition from 2π to 4πperiodic Josephson radiation at B ~ 175 mT. As such, the observation of a magnetic fieldinduced transition in the periodicity of Josephson radiation has been observed in four distinct devices, showcasing the reproducibility of the observation.
Josephson radiation of an Al/AlO_{x}/Al tunnel junction
Supplementary Figure 5 shows our measured data with a conventional Al/AlO_{x}/Al superconducting tunnel junction as the source of Josephson radiation. Evaluating e^{*} as a function of magnetic field in the same range as for Fig. 2 and Supplementary Fig. 4, we observed no transition in the periodicity of Josephson effect, confirming that the 4πperiodic Josephson radiation occurs only in nanowire junctions. We note that, in order to keep the circuit behavior similar, the normal state conductance of the tunnel junction was set to G_{N,T} = 0.26G_{0}.
Data availability
The datasets analyzed during this study are available at the 4TU.ResearchData repository https://doi.org/10.4121/uuid:1f9368405bc240ca8c321797c12cacb1 (ref. ^{45}).
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Acknowledgements
The authors acknowledge O. Benningshof, J. Mensingh, M. QuinteroPèrez, and R. Schouten for technical assistance. This work has been supported by the Dutch Organization for Fundamental Research on Matter (FOM), Microsoft Corporation Station Q, the Danish National Research Foundation, and a Synergy Grant of the European Research Council. A.G. acknowledges the support of the Netherlands Organization for Scientific Research (NWO) by a Veni grant.
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D.L., D.B., D.J.v.W., and A.P. fabricated the samples and performed the experiments. P.K. and J.N. contributed to the nanowire growth. L.P.K. and A.G. designed and supervised the experiments. C.M., D.P., and C.N. developed the theoretical model of the devices. D.L., D.B., D.J.v.W., R.J.J.v.G., L.P.K., and A. G. analyzed the data. The manuscript has been prepared with contributions from all the authors.
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Laroche, D., Bouman, D., van Woerkom, D.J. et al. Observation of the 4πperiodic Josephson effect in indium arsenide nanowires. Nat Commun 10, 245 (2019). https://doi.org/10.1038/s41467018081612
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