Main

Experimentally, the parity modulation of a superconducting island can be observed via the ground state charge2, the even–odd modulation of the charge stability diagram11,12, or the parity dependence of the switching current, Isw (refs 13, 14). The interplay of the charging energy Ec = e2/2C and the Josephson coupling energy EJ = Icℏ/2e where Ic is the critical current makes the Cooper-pair transistor (CPT) a single, gate-modulated Josephson junction13,15 with a 2e charge periodicity in the absence of parity switches—meaning, infinitely long parity lifetime, τp.

Recent, direct measurements of τp (refs 4, 16) yielded values up to the millisecond regime for aluminium devices. Despite considerable efforts, however, no 2e periodicity has been reported for non-aluminium superconductors17,18, such as niobium or vanadium. Comparative studies of aluminium and niobium CPTs suggested that the elusiveness of parity effects is related to the material properties18, in particular the formation of a surface oxide layer under ambient conditions. Whereas aluminium tends to form a few-nanometre-thick insulating oxide layer19, niobium is prone to oxidize through the bulk material20, mostly forming metallic NbOx compounds. This process then leads to localized metallic states in the island, which prevents parity control.

In contrast, nitridized niobium compounds, such as niobium titanite nitride (NbTiN), have been shown to be less prone to form oxides20 and hence are good candidates for parity-conserving superconducting circuits. Furthermore, NbTiN forms transparent contacts with spin–orbit coupled semiconductor nanowires9, and has become a preferred superconductor to investigate Majorana bound states.

Our device features a NbTiN island sputtered onto Al leads (Fig. 1a). The tunnel barriers between the island and the leads are created by means of controlled in situ surface oxidation of Al, resulting in amorphous AlOx barriers19.

Figure 1: Device layout and parity-dependent switching current.
figure 1

a, Scanning electron microscope image and schematic cross-section of a typical Al/NbTiN/Al hybrid Cooper-pair transistor (CPT). The NbTiN island dimensions are 250 nm × 450 nm. Scale bar denotes 200 nm. The AlOx oxide barriers are indicated by thick red lines. b, Energy level diagram as a function of the gate charge, ng, in the presence of low-energy subgap states restoring 1e periodicity. The maximum even–odd energy difference is denoted by δE. Grey lines represent energy levels in the absence of Josephson coupling, EJ = 0. Red and blue lines show energy levels for even and odd charge parity, respectively, both for EJ = Ec. Parity switches occur on a timescale of τp owing to quasiparticle tunnelling. c,d, Measured switching current histogram (c) and calculated Ic(ng) (d) in the low-temperature limit. Note that in d the two possible Ic(ng) values corresponding to the even and odd charge states are denoted by blue and red lines, respectively. In the measured data (c) the two branches are superimposed, see text.

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We extract a charging energy Ec ≍ 50 μeV and EJ ≍ 30–50 μeV for different devices. A detailed list of parameters and characterization methods are presented in the Supplementary Methods. Our devices are in the intermediate coupling regime with EJ ∼ Ec, where the energy diagram (Fig. 1b) and the critical current (Fig. 1d) are sensitive to the charge parity. It is important to note that our CPTs are in the optimal regime to establish flux-controlled braiding of Majorana bound states with EJ ∼ Ec ≫ kBT (ref. 10), and hence a useful platform to establish the parity lifetime for Majorana circuits21.

We model the CPT as a two-level system which can exist in either parity state (red and blue bands in Fig. 1, respectively), and switches state on the timescale of τp (ref. 22) owing to quasiparticle tunnelling. We collect the switching current histograms by repetitively sweeping the bias current from zero (non-dissipative state) to beyond the switching current. Here, in the resistive state, quasiparticle tunnelling causes a random reinitialization of the parity state of the CPT for the next measurement. This results in the apparent 1e periodicity in Fig. 1c. Nevertheless, as long as the parity remains constant during each sweep, we expect to find the two branches as a bimodal histogram, as we indeed observe in Fig. 2a. In these measurements, the current ramp time is much shorter than the parity lifetime, τp (fast measurement limit).

Figure 2: Characterization and temperature dependence of the parity lifetime.
figure 2

a, Linecut histogram at integer ng of Fig. 1c showing a bimodal distribution. We attribute the two peaks to the two parity states of the CPT (coloured red and blue, respectively). b, Schematic representation of our model. For a fast current ramp (upper panel), the histograms of the two parity states are independently probed, showing the characteristics of data in a. In the slow limit (lower panel), parity switches occur during the current ramp, leading to an exponential tail of the distribution (shown in green), quantifying τp. The Ibias(t) current ramp is represented by a dark grey line. c, Experimental data set in the slow limit. Note the change in the current- and timescale compared with a. We show the exponential cutoff in green, and extract τp = 49 s (solid black line). The dashed black lines are guide to the eye envelopes for the peaks in a and c. d, Experimental data at different temperatures show the temperature dependence of τp. e, τp as a function of temperature for non-shielded device N, and shielded devices S1, S2. For detailed comparison, see the main text and the Supplementary Methods. All CPTs exhibit an activated behaviour with Δ ≍ 170–210 μV in the high-temperature limit, corresponding to the gap of the aluminium leads. Saturation of device N without shielding and no quasiparticle traps is observed below T ≍ 100 mK. Shielded devices S1 and S2 exhibit a minigap-activated behaviour Δ⋆ ≍ 20 μV in the low-temperature limit. The fitted τp(T) function is defined in the main text. Star symbol shows τp = 15 s at T = 12 mK extracted from parity-distilled data for device S2 (see Fig. 3d).

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We quantify τp in the slow measurement limit. In this regime parity switches occur during the current ramp (Fig. 2b lower panels) such that reaching the upper branch (depicted as blue in Fig. 2a, b) becomes exponentially suppressed (Fig. 2c). The exponential tail represents parity switches during the current bias ramp, resulting in an observable decay of the upper branch (depicted as green in Fig. 2b, c), pu(t) = pu(0)exp(−t/τp). Thus, from the decay of the histogram (black solid line in Fig. 2c), we can directly obtain τp.

The observed τp is a result of single-electron tunnelling events through the junctions of the CPT. In the zero-voltage state, we can link τp to ρsubgap, the subgap density of states on the island, and nqp, the quasiparticle density in the leads (for details see the Supplementary Methods):

where RN is the normal state resistance of the CPT. It is instructive to note that equation (1) is similar to the tunnelling rate derived in ref. 23 for a device with a normal metallic island, which stems from our assumption of a finite subgap density of states in the island. In our case, the parity lifetime is determined by the quasiparticle density in the leads21 and the phenomenological Dynes parameter24, ρsubgap/ρNbTiN, of the island material. Assuming a thermal quasiparticle density

in the leads, we find ΔAl ≍ 170–210 μeV (Fig. 2e) for temperatures exceeding 120 mK, in good agreement with the superconducting gap of the aluminium leads extracted from the charge stability diagram. Furthermore, by inserting equation (2) into equation (1), we get an estimate of the Dynes parameter to be ∼10−2. This is in good agreement with the value of ∼10−3 that we get based on our measurements of a single NbTiN/AlOx/Al junction shown in the Supplementary Methods. We can therefore attribute the observed parity lifetime for T > 120 mK to the thermal quasiparticle population in the leads.

For device N, however, we find a saturated τp = 9.5 ms in the low-temperature limit, a common observation in superconducting qubits4 and hybrid single-electron transistors23 signifying the presence of non-thermal quasiparticle excitations.

A common strategy to drain non-thermal quasiparticles is to introduce traps—normal metal junctions attached to the leads. We note that the trapping efficiency of a quasiparticle trap is influenced by its distance from the CPT with respect to the quasiparticle diffusion length on the timescale of the recombination time25. Our choice of a nominal 5 μm separation is much shorter than the typical diffusion length of ≳100 μm, even at the highest temperature of T = 150 mK. However, as this distance is much longer than the estimated coherence length of ξAl ≍ 100 nm, the inverse proximity effect does not influence the transport at the junction of the CPT (ref. 26).

By introducing quasiparticle traps and microwave-tight shielding coated with infrared absorber painting for devices S1 and S2, we find a non-saturated behaviour of τp (Fig. 2e). We observe a minigap-activated temperature dependence with Δ⋆ ≍ 20 μeV for both devices, which we attribute to a minimum excitation energy on the island. This activation energy is consistent with the maximum even–odd energy difference of δE ≍ 20–30 μeV (Fig. 1b), which promotes τp ∼ exp(δE/kBT) (ref. 27).

It is to be stressed that the lack of saturation signals that the effective quasiparticle temperature of the CPT follows the bath temperature down to the 10 mK regime, which is in clear contrast to the saturated behaviour of device N. We find τp = 49 s at T = 12 mK for device S2. To put this number into context, we note that the Josephson frequency fJ = EJ/h ≍ 10 GHz and thus a single-quasiparticle event occurs only once for every τpfJ ∼ 1011 Cooper pairs tunnelling through the junctions. This signifies the low probability of parity switches for an open device with EJ ≍ Ec required for flux-tunable Majorana braiding schemes.

Thus far, we started each switching current measurement from an unknown parity state because of the random reinitialization in the dissipative state of the CPT for Ibias > Isw. To reproducibly select the same parity state, we employ a parity-distillation protocol (Fig. 3) where, by selecting a single parity state without switching to the resistive state, we ensure that the parity remains well defined for the subsequent measurement. This protocol indeed results in a 2e periodic switching current pattern (Fig. 3c), which is observed for the first time for a non-aluminium CPT.

Figure 3: Parity-state distillation.
figure 3

a, Schematic current bias and gate voltage waveforms applied to the device. First, a current bias pulse with an amplitude between the two branches is applied (1). At the same time the measured voltage Vsd(t) is recorded to initialize the parity state. Then the gate voltage is ramped (2) and, finally, the switching current is recorded (3). b, Schematic representation of the applied waveforms overlayed on the parity-dependent critical current. c, Parity-distilled switching current histogram exhibiting 2e periodicity with the wait time twait = 100 ms. d, Linecut histograms for an even (blue) and odd (red) ng at the positions denoted by arrows in c. e, Raw histogram without parity distillation exhibiting 1e periodicity. Slight shifts in c and e are caused by gate charge switches that occur on a timescale of several 10 min. f, Parity-distillation fidelity as a function of twait, see main text. Solid red line denotes the fit ∼exp(−twait/τp) with τp = 15 s. All data was recorded on device S2.

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We quantify the effectiveness of the parity distillation by defining the fidelity as:

where pu, f is the conditional probability of the upper branch in the final step (3), and pu, i is the initial probability. We note that the above expression is valid for an arbitrary 0 < pu, i < 1 value set by the average quasiparticle occupation of the CPT. For device S2 we find for twait = 100 ms, demonstrating the high degree of parity distillation (Fig. 3d). By changing twait between the parity initialization (1) and measurement (3), we acquire an independent measurement τp = 15 s for device S2 at T = 12 mK (Fig. 3f).

Finally, we investigate the evolution of τp(B) in different magnetic field directions. In parallel magnetic field, we observe a gradual decrease of τp. The onset of the steep decay at B‖ = 110 mT (yellow arrow in Fig. 4a) is in agreement with the condition for vortex penetration through a mesoscopic superconducting island28 with Φ ≍ 1.1Φ0 ≳ Φ0, where Φ0 = h/2e is the magnetic flux quantum. We note that the measurement geometry shown in the inset of Fig. 4a may give rise to a Fraunhofer-like interference on the same magnetic field scale; however, the upper critical magnetic field of 320 mT of the Al leads did not allow the evaluation of τp over a larger magnetic field range—that is, several flux quanta.

Figure 4: Influence of the magnetic field on the CPT.
figure 4

a,b, The parity lifetime as a function of in-plane field, B‖ (a) and perpendicular field, B⊥ (b), respectively. In a, the shaded region and the purple arrows depict the condition for inducing Majorana bound states (MBS) in spin–orbit-coupled semiconductor wires7,8,9 and the yellow arrow shows the onset of the steeper decay at B‖ ≍ 110 mT. The inset depicts the relative orientation of the CPT with respect to the magnetic field. In b, the inset shows the false-colour electron microscope image of the device, with the white scale bar denoting 5 μm. In both panels, the dashed lines serve as a guide to the eye.

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These results underline the significance of the sample geometry of the CPTs in maintaining parity control in a finite magnetic field. However, our device exhibits τp > 10 ms in B‖ > 100 mT, required to induce Majorana bound states7,8,9.

In contrast, applying a perpendicular field first results in an increase of τp, reaching a maximum value τp = 94 s at B⊥ ≍ 10 mT, before dropping at higher magnetic fields (Fig. 4b). Notably, the typical switching current values, shown in the Supplementary Methods, do not follow the same trend. Making use of the relation between the lower critical field and the stripe width, Bc1, ⊥(w) ∼ Φ0/w2 (ref. 29), we attribute the initial increase to vortex formation and hence more effective quasiparticle trapping in the wide lead sections (w ≍ 2 μm) of the device (see the yellow sections in the inset of Fig. 4b). On reaching B⊥ ≍ 10 mT, the vortex phase becomes stable in the close vicinity of the CPT (w ≍ 250 nm), causing a gradual decrease of τp. We note that owing to ξAl ≍ 100 nm ∼ w for our devices, we cannot make a quantitative comparison between our measurement data and theoretical expressions for the critical magnetic field of thin stripes. However, we conclude that vortices induced by a perpendicular magnetic field can increase the efficiency of quasiparticle traps, but the formation of a vortex phase in the near vicinity of the CPT enhances quasiparticle transport in agreement with earlier observations29,30.

Methods

Device fabrication.

The Cooper-pair transistors (CPTs) were fabricated using electron beam lithography and thin film deposition, starting with a p++ doped silicon wafer with a 285-nm-thick thermally grown SiO2 surface layer. First, aluminium leads were defined and evaporated in a high-vacuum chamber (pbase ∼ 10−7 torr) at a rate of 0.2 nm s−1 with a thickness of 30–35 nm. Subsequently, the mask for the NbTiN island was defined in a second lithography step. The sample was loaded into an ultrahigh vacuum AJA International ATC 1800 sputtering system (pbase ∼ 10−9 torr), where first a ∼5 nm Al layer was removed by means of argon plasma etching at p = 3 mtorr. This step was followed by in situ oxidation to create AlOx tunnel barriers. Without breaking vacuum, the NbTiN island was then sputtered with a layer thickness of 70–100 nm.

We used a Nb0.7Ti0.3 target with a diameter of 3 inches. Reactive sputtering resulting in nitridized NbTiN films was performed in an Ar/N process gas with a typical 10 at% nitrogen content at a pressure of 3 mtorr using a d.c. magnetron source. During deposition, a −45 V bias was applied to the sample with respect to the target. The critical temperature of the superconducting transition of thin films with a layer thickness of 100 nm was measured to be 14.1 K in zero magnetic field.

For the shielded samples S1 and S2, quasiparticle traps were fabricated by first cleaning the Al surface by means of argon plasma milling at p = 0.2 mtorr and in situ evaporation of 25 nm Ti and 100 nm Au films at a base pressure of ∼10−7 torr.

Care was taken to remove resist mask residues after each electron beam writing step using a remote oxygen plasma etch at a pressure of 1 mbar.

We show the scanning electron microscope images of the devices investigated in Supplementary Fig. 1 and define the fabrication parameters in Supplementary Table 1.

Measurement set-up.

The measurements were performed in a Leiden Cryogenics CF-1200 dry dilution refrigerator with a base temperature of 12 mK equipped with Cu/Ni shielded twisted pair cables thermally anchored at all stages of the refrigerator.

The current bias and gate voltage were applied through battery operated and optically isolated sources to reduce line interference. Similarly, the first stage of the Vsd amplifier was isolated from the commercial readout electronics. Filtering of the measurement lines was achieved by room temperature LC π filters with a cutoff frequency of ∼100 MHz followed by a sequence of a high-frequency copper powder filter31 and a two-pole RC filter with a nominal f−3 dB = 50 kHz, both thermally anchored to the 12 mK stage.

Special care was taken to avoid stray microwave radiation by using outer and inner copper shields enclosing the device. The inner surface of both shields was treated with commercially available Aeroglaze Z306 paint to absorb far-infrared stray radiation32. We note that the inner shield was not present for device N.

The switching current histograms were acquired using a Rigol DG4062 arbitrary waveform generator controlling the isolated current bias source with a triangular wave signal, resulting in a dI/dt current ramp. A finite voltage response above the preset Vref ∼ 10 μV triggers the recording of Isw (Supplementary Fig. 2c). We note that the delay of the low-pass filters was accounted for on the basis of circuit simulations. Subsequent Isw measurements were taken by setting zero Ibias for approximately 100 ms in between to avoid overheating effects. We did not observe a difference in the switching current histograms taken with waiting times in the range between 20 ms and 10 s.

The schematics of the measurement and typical waveforms are shown in Supplementary Fig. 2.

Basic characterization of the CPT.

We estimate the charging energy, Ec = e2/2C, based on the periodicity of characteristic resonances visible for eVsd = ΔAl…2ΔAl. We attribute these lines to resonant Andreev tunnelling through single levels of the CPT (ref. 33). The resonances occur with a vertical periodicity of eVsd = 2Ec and the onset defines eVsd = ΔAl as well.

The onset of quasiparticle transport is defined by eVsd = 2(ΔNbTiN + ΔAl), which then defines ΔNbTiN. Typical I(V) and dI/dV (V) traces are shown in Supplementary Fig. 3 and the recovered parameters are tabulated in Supplementary Table 2.

We evaluate the Josephson coupling for a single tunnel junction34:

with K(x) being the complete elliptic integral of the first kind. This expression is valid in the zero-temperature limit, assuming symmetric tunnel junctions with a resistance of RN/2.

Alternatively, we can estimate the EJ/Ec ratio based on the modulation of Isw as a function of the gate charge, ng (last column of Supplementary Table 2)35. We find a good agreement between the EJ/Ec values acquired by the two independent methods. Finally, we provide δE(Ec, EJ) in the main text based on ref. 36.