Gate-tunable semiconductor nanowires with superconducting leads have great potential for quantum computation1,2,3 and as model systems for mesoscopic Josephson junctions4,5. The supercurrent, I, versus the phase, ϕ, across the junction is called the current–phase relation (CPR). It can reveal not only the amplitude of the critical current, but also the number of modes and their transmission. We measured the CPR of many individual InAs nanowire Josephson junctions, one junction at a time. Both the amplitude and shape of the CPR varied between junctions, with small critical currents and skewed CPRs indicating few-mode junctions with high transmissions. In a gate-tunable junction, we found that the CPR varied with gate voltage: near the onset of supercurrent, we observed behaviour consistent with resonant tunnelling through a single, highly transmitting mode. The gate dependence is consistent with modelled subband structure that includes an effective tunnelling barrier due to an abrupt change in the Fermi level at the boundary of the gate-tuned region. These measurements of skewed, tunable, few-mode CPRs are promising both for applications that require anharmonic junctions6,7 and for Majorana readout proposals8.
In superconductor–normal–superconductor (SNS) Josephson junctions, the critical current is predicted to be quantized as IC = NΔ0/ℏ︀ (refs 4,5), where N is the number of occupied subbands and Δ0 is the superconducting gap, as long as N is sufficiently small and the junctions are short, ballistic and adiabatically smooth. This quantization shows the number of Andreev bound states or modes, each coming from a single occupied subband, that carry the supercurrent. Single-mode junctions are desirable for the observation of Majoranas. Quantized supercurrent has been difficult to realize experimentally, and it is difficult from transport measurements alone to determine which conditions for quantization are unmet. In transport measurements of proximitized InAs and InSb nanowires, critical currents are far below the expected quantized values for perfectly transmitting modes, and fluctuate with gate voltage9,10,11,12,13. Nanowire Josephson junction qubits also display similar fluctuations in the measured resonant frequency6,7. Here, we make direct, non-contact measurements (Fig. 1a) of Al/InAs nanowire/Al junctions (Fig. 1b, c) using an inductively coupled14 scanning superconducting quantum interference device15,16 (SQUID) to measure the CPR (Methods). The contact between Al and InAs nanowires is epitaxial17, greatly reducing the presence of in-gap states2. In the low-temperature, few-mode limit, the shape of the CPR and its dependence on gate voltage, junction length and temperature yield unique insights into the number and transmission of Andreev bound states.
The vast majority of CPRs we measured were forward-skewed; their first maximum after zero was forward of a sine wave (Fig. 1d). Fourier transforms of the CPRs revealed that they were well described by a Fourier series with up to seven harmonics (for example, Fig. 1e). In the absence of time-reversal symmetry breaking, the theoretical CPR can be decomposed into a sine Fourier series18; fits to the experimental data include appropriate instrumental phase shifts (Methods, Supplementary Section 3).
The Fourier amplitudes (An) obtained by fitting the CPR characterize its shape. A1 is the amplitude of the 2π-periodic component and is approximately IC. The ‘shape parameters’ an ≡ (−1)n+1An/A1 characterize the shape of the CPR independent of its amplitude. Positive a2 indicates a forward-skewed CPR; negative a2 indicates a backward-skewed CPR. Higher-order terms yield more detailed information about the shape of the CPR and can be used to extract information about multiple modes, to compare with CPR theories, and to differentiate between the effects of elevated temperature and lowered transmission. Higher harmonics of the CPR can potentially reveal information about high-transmission modes even in junctions with many modes, similar to higher-order multiple Andreev reflections observed in superconducting point contacts19,20 (Supplementary Section 7).
The amplitude and shape of the CPR of an L = 150 nm nanowire junction fluctuated as a function of applied bottom gate voltage, VBG. The fluctuations were most dramatic close to depletion of the nanowire as seen in Fig. 2. The most forward-skewed CPR observed in this study and a backward-skewed CPR both occurred in the same gate-tuned junction (Fig. 2a). Fluctuations in A1 remained similar in amplitude for all VBG (Fig. 2b), while fluctuations in an decreased with increased VBG (Fig. 2c) and with elevated temperature (Supplementary Section 1). The phase of the CPR remained constant for all gate voltages, indicating that the junction never underwent a 0 to π-junction transition (Fig. 2a).
An and an both displayed peaked behaviour near depletion, where we expect a single subband to be occupied (Fig. 3a, b). The peaks in both a2 and An were asymmetric, with the side at more positive VBG appearing more broad (Fig. 3a, b). The most forward-skewed CPR occurred in this peaked regime (Fig. 3c).
We observed a backward-skewed CPR only for a narrow gate voltage range in the gated junction, and never in ungated junctions (Fig. 4a). a2 was negative in the backward-skewed region (Fig. 4c) and was nearly coincident with a minimum in A1 (Fig. 4b). a3 switched sign when a2 was at its most negative. Our observation of a backward-skewed CPR cannot be explained by noise rounding, as in ref. 21, as the junction studied here was not near the hysteretic regime (Supplementary Section 4).
Junctions with different lengths (Fig. 5) exhibited a range of IC and forward-skewness (Fig. 5a). A1 did not depend strongly on the length of the junction (L) (Fig. 5b) and a2 trended downward with increasing L, albeit with significant scatter (Fig. 5c). We detected no clear correlation between A1 and a2 (Supplementary Fig. 1), which suggests that variations in the Fermi level from junction to junction were responsible for fluctuations in the amplitude and shape of the CPR.
For SNS junctions in the short-junction limit (when L is much smaller than the superconductor’s coherence length), each occupied subband (mode) leads to a single Andreev bound state whose properties depend only on its normal-state transmission (τ) (ref. 4). The short-junction limit holds for the junctions studied here except very close to the onset of a new subband, where a low Fermi velocity makes the junction in effect very long (Supplementary Section 6). The CPR is predicted to be sinusoidal when T ∼ TC or when τ ≪ 1, but forward-skewed at low temperatures and high transmissions, as evident in the short-junction expression for the CPR: where T is the temperature, kB is the Boltzmann constant, ℏ︀ is the reduced Planck constant, Δ0(T) is the superconducting gap, and e is the electron charge.
We can estimate the typical transmission (τ) from the skewness, a2, and the number of modes (N) from the amplitude, A1, assuming a value of T based on our mixing chamber temperature. For VBG > 4 V in Fig. 2, a2 ≈ 0.2 and A1 ≈ 100 nA; we estimate that τ ≈ 0.8 and N ≈ 4–5. For the CPRs measured on many junctions in Fig. 5, N varies from 0–10 and τ varies from 0.5–0.9 for CPRs with A1 > 10 nA. Therefore, the junctions are in the few-mode regime and often have modes with very high transmission. Realistically, a junction would have an integer number of modes with different transmissions. Multi-mode fits to equation (1) could potentially resolve multiple values of τp and would be particularly sensitive to the presence of high-transmission modes (Supplementary Section 7).
Peaks in An versus gate voltage at low densities indicate resonant tunnelling behaviour (Figs 3 and 4). Equation (1) remains valid for a junction with tunnel barriers, with τ given by the Breit–Wigner transmission4: where ΓL, R are tunnel rates into the left and right barriers, respectively, and E0 is the energy of the resonance with respect to the Fermi energy, EF. Equation (2) is valid for SINIS junctions only when ΓL + ΓR ≫ Δ0 and shows that perfect transmission (τ = 1) requires the left and right tunnel barriers to have identical tunnel rates. The most skewed CPR we observed, shown in Fig. 3c, fits well to equation (1) with four parameters: T, τ, a scaling parameter (ϵ), and a phase shift (ϕ0). Fixing T to its measured value, 0.03 K, gives a good fit with ϵ and ϕ0 consistent with our experiment, with a best-fit τ = 0.98. Allowing T to vary gives a best-fit value of T = 0.13 K and τ = 1.00 (Supplementary Section 2). A fit to equation (1) with two free τ gave best-fit values of τ1 = 1.00 and τ2 = 0.0 (Supplementary Section 2). These results indicate a single mode with perfect or nearly perfect transmission.
We attribute the high-transmission mode (Fig. 3c) to supercurrent through a resonant mode formed by symmetric, weak barriers. Perfectly symmetric tunnel barriers are unlikely to accidentally occur due to disorder-induced quantum dot behaviour, so their origin is more likely an intrinsic barrier. Symmetric barriers can exist at the epitaxial InAs/Al interface and at the border between etched and Al-coated regions of the nanowire (Fig. 1c). Different band bending at the etched and Al-coated surfaces can lead to different radial wavefunctions and subband occupation, resulting in an effective tunnel barrier (wavefunction mismatch).
Simulations that include wavefunction mismatch reproduce the essential features of Figs 2 and 3 (Supplementary Section 5). First, we simulated a two-dimensional InAs junction with delta-function barriers that depend on the difference in Fermi velocity between the superconductor and normal parts of the junction5. At low Fermi energy, the model shows peaked behaviour in An and an, indicating a single resonant mode with a highly forward-skewed CPR. When many subbands are occupied at higher Fermi energies, the CPR’s shape and amplitude fluctuate with Fermi energy. Second, we also performed tight-binding simulations, which use a more accurate cylindrical geometry, to capture the electron wavefunctions and the effects of wavefunction mismatch more directly. The tight-binding model also shows resonant single-mode CPRs at low Fermi energy and fluctuations when multiple subbands are occupied, showing that wavefunction mismatch is a likely candidate to describe the behaviour of forward-skewed CPRs in these nanowire junctions22 (Supplementary Section 5). Our models do not include spin–orbit coupling, which is essential in describing the high-magnetic-field behaviour of InAs nanowire Josephson junctions. In the absence of a magnetic field, spin–orbit coupling can shift the energies of resonant Andreev levels and change the shapes of the resonant peaks, but each individual CPR is still described by equation (1) in the absence of interactions (ref. 23 and references therein, Supplementary Section 8).
The backward-skewed CPR (Fig. 4), however, does not arise in theories of short-junction SNS or resonant tunnelling behaviour (equations (1) and (2)). Interaction effects in the nanowire can lead to qualitatively similar backward-skewed CPRs and gate voltage dependence when the charging energy of the wire is finite, but not large enough to fully induce a 0–π junction transition24,25 (Supplementary Section 8). The presence of both a fully transmitting single mode (which requires negligible charging energy) and a backward-skewed CPR implies that the charging energy of the junction is tuned by the gate voltage.
The shape of the CPR has important consequences for nanowire Josephson junction devices. Our observation of a single τ ∼ 1 mode that occurs at a peak in Ic, rather than a step, suggests that weak, symmetric tunnel barriers can form due to wavefunction mismatch in the nanowire. The resonance condition (equation (2)) leads to the maximum possible tunability of the skewness, and therefore the anharmonicity of the Josephson potential, with small gate voltages. Control of anharmonicity can tune nanowire-based qubits to a flux-qubit-like regime near ϕ = π/2 (ref. 7). Our data also indicate that the likelihood of a non-sinusoidal CPR must be considered when attempting to extract the spatial distribution of current in Josephson junctions26,27.
Inductive measurements of nanowire Josephson junctions with spin–orbit coupling28, such as the Al/InAs nanowire/Al junctions studied here, or quantum spin Hall junctions29 have been proposed as a readout mechanism for the parity of Majoranas. We have measured the CPR with the requisite sensitivity for non-contact readout of Majorana parity and observed the CPR of a single occupied subband that has close-to-perfect transmission. Challenges remain in realizing Majorana readout. First, the in-plane fields required to create Majoranas in InAs nanowires are a major technical challenge for scanning SQUID microscopy. Second, the timescale of the measurements presented here (up to tens of minutes) may not allow a measurement of a full CPR before environmental factors cause the parity to switch. Nevertheless, our sensitivity to such parity switching would allow us to use noise measurements to observe such environmental switching as long as the parity lifetime is longer than ≈10 μs.
InAs nanowires were grown along the B direction and Al was deposited epitaxially to fully coat the nanowires17. A length, L, of the coated nanowire was etched to form a Josephson junction. We evaporated Ti/Al (5 nm/120 nm) ex situ to form the ring geometry required for CPR measurements. We measured the dimensions of the nanowires using scanning electron microscopy after the scanning SQUID measurements (Supplementary Fig. 2 and Supplementary Table 1).
We measured three samples, each with many single-junction rings, in a dilution refrigerator. In the gated junction, a gold bottom gate made of Ti/Au (5 nm/20 nm) with a AlOx dielectric (40 nm) was used to tune the density of the nanowire. A number of devices were hysteretic; scanning electron microscopy revealed that they were under-etched (inset of Supplementary Section 4). We used a SQUID microscope in a dilution refrigerator with a nominal base temperature of ∼30 mK (refs 30,31). Temperatures were measured at the mixing chamber plate using a Rox thermometer. Data in the main text were all taken at T < 50 mK. We centred the scanning SQUID’s pickup loop and field coil over the ring using the diamagnetic response of the evaporated Al to navigate, measured by applying an a.c. current through the field coil and standard lockin techniques (Fig. 1b). With the SQUID centred over the ring, we swept the current through our local field coil (IFC) sinusoidally at ≈200 Hz and recorded the flux through the SQUID’s pickup loop (ΦPU) (Fig. 1a). We set the amplitude of IFC to thread multiple flux quanta through the ring. We subtracted linear and experimental background from the measured ΦPU versus IFC to account for the diamagnetic response of the ring and imperfect geometric cancellation of the applied local field.
In single-junction rings (Fig. 1a), the CPR can be directly measured inductively as long as the self-inductance of the ring is small, Lself < Φ0/2πIc, where Φ0 is the superconducting flux quantum14. We converted measured ΦPU versus IFC to I versus ϕ (the CPR) using the periodicity of the signal and calculations of the mutual and self-inductances32. Although we included corrections for self-inductance effects in the conversion, they were not important in determining the shape of the CPR because the (non-shorted) junctions were always in the limit of β = 2πLsIC/Φ0 ≪ 1. We confirmed our calculation of the self-inductance by measuring the height of steps in the response of hysteretic rings (Supplementary Section 4). Errors in centring the SQUID’s pickup loop over the ring of a few micrometres would result in a systematic error of ≈10% in the extracted current and Fourier components An. The shape parameters an, however, are insensitive to this form of systematic error. We fitted both forward and backward sweeps of the CPR to extract their Fourier components. We fit the CPRs presented in Figs 2–4 using: which fully accounted for the shape of the CPR. The phase shifts between forward and backward sweeps (ϕbw, fw) were subtracted from ϕ in the figures presented in the main text. In Fig. 5, some CPRs exhibited phase shifts between harmonics. The harmonics were shifted in opposite directions for forward and backward sweeps, indicating that they were instrumental rather than intrinsic to the junction (see Supplementary Methods). Phase shifts between harmonics were not present in the CPRs in Figs 2–4, as is evident in the lack of an out-of-phase component in the fast Fourier transform (Fig. 1e). The errors bars in Figs 2–5 are 90% confidence intervals obtained from bootstrapping.
The data that support the findings of this study are available from the corresponding author on reasonable request.
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We thank S. Hart, J. Kirtley and C. Beenakker for useful discussions and C. Watson, Z. Cui and I. Sochnikov for useful discussions and experimental assistance. The scanning SQUID measurements were supported by the Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering, under Contract No. DE-AC02-76SF00515. Nanowire growth and device fabrication was supported by Microsoft Project Q, the Danish National Research Foundation, the Lundbeck Foundation, the Carlsberg Foundation, and the European Commission. C.M.M. acknowledges support from the Villum Foundation.