Parity transitions in the superconducting ground state of hybrid InSb-Al Coulomb islands

The number of electrons in small metallic or semiconducting islands is quantized. When tunnelling is enabled via opaque barriers this number can change by an integer. In superconductors the addition is in units of two electron charges (2e), reflecting that the Cooper pair condensate must have an even parity. This ground state (GS) is foundational for all superconducting qubit devices. Here, we study a hybrid superconducting-semiconducting island and find three typical GS evolutions in a parallel magnetic field: a robust 2e-periodic even-parity GS, a transition to a 2e-periodic odd-parity GS,and a transition from a 2e- to a 1e-periodic GS. The 2e-periodic odd-parity GS persistent in gate-voltage occurs when a spin-resolved subgap state crosses zero energy. For our 1e-periodic GSs we explicitly show the origin being a single zero-energy state gapped from the continuum, i.e. compatible with an Andreev bound states stabilized at zero energy or the presence of Majorana zero modes.

A superconductor can proximitize a semiconductor and open a gap in its energy spectrum. If the two materials are strongly coupled, the induced gap can be as large as the original gap in the superconductor. The two gaps respond differently to an applied magnetic field, e.g. when the Landé gfactors differ in the two materials. A large g-factor in the semiconductor can cause the induced gap to close long before the closing of the original gap. If, in addition, the semiconductor has strong spin-orbit interaction, the induced gap can re-open, signalling a transition to a topological superconducting phase 1,2 . This phase contains pairs of Majorana zero modes (MZMs) that can accommodate either zero or one fermion, and thus allows for both even-parity and odd-parity ground states (GSs) 3,4 .
When a conductor has a finite size, it forms an island restricting the charge to an integer times the elementary charge, e 5 . The resulting Coulomb blockade effects have been widely studied in metallic and superconducting islands, the latter often referred to as Cooper pair boxes 6,7 . A major breakthrough was the demonstration of charge quantisation in units of 2e in aluminium (Al) islands [8][9][10][11][12][13] , indicating that the even-parity superconducting GS was not poisoned by quasiparticles on the time scale of the measurement. The 2e quantisation could be destroyed by subjecting the Al to an external magnetic field, B, which causes a transition to the metallic state with 1e charge quantisation 10,11 .
Hybrid superconducting-semiconducting islands have also shown a 2e charge quantisation at low B-fields [14][15][16] . These observations imply that the low-energy spectrum in the semiconductor is completely proximitized with no Andreev bound states (ABSs) at low energies. Also, for these hybrid islands a Bfield can cause a 2e to 1e transition 15,16 . A recent breakthrough demonstrated that under particular circumstances the 1e quantisation is not due to the transition to the metallic state-but rather due to a topological superconducting phase 16 . These pioneering experiments used InAs as the semiconductor.
Here, we harness the large g-factor (g ≈ 50) and the ballistic transport properties of InSb nanowires 17,18 , and find additional B-field-induced transitions, including a recurrence of a 2e quantisation at higher B-fields.

Results
Different B-field GS evolutions at controllable gate configurations. Our device (Fig. 1a) consists of a hexagonal InSb nanowire with two of its facets covered by a thin epitaxial layer of Al (see ref. 19 for materials details). Two top gates (TG) can induce adjustable tunnel barriers separating the InSb-Al island from the two normal leads. The voltage, V PG , applied to the top plunger gate (PG), can be used to tune the charge on the island, as well as the spatial charge density profile in the semiconductor. A bias voltage, V b , is applied between source (S) and drain (D), yielding a current, I, that is measured in a dilution refrigerator at a base temperature of~15 mK. Figure 1b shows the differential conductance, dI/dV b , vs. V b and the voltage applied to the tunnel gate at B = 0. The charge is fixed in the current-blockaded Coulomb diamonds (with mostly blue colour) with a periodicity in gate voltage corresponding to a charge increment of 2e. For V b > 120 µV the periodicity is halved to 1e, indicating the onset of single electron transport. Linecuts in the right panel show that dI/dV b can be enhanced as well as suppressed, even down to negative values (black colour). These are known features for hybrid islands and can be used to extract values for the charging energy, E c = e 2 /2C ≈ 25 µeV and the lowest-energy subgap state, E 0 = 50-90 µeV ( Supplementary  Fig. 2). The Al superconducting gap, Δ = 220 µeV, is extracted from tunnelling spectroscopy measurements ( Supplementary  Fig. 3).
To understand parity transitions induced by a B-field, we illustrate four different scenarios in Fig. 1c-f. We sketch a slow reduction of Δ for a B-field, B ∥ , along the nanowire axis ( Fig. 1c-e) and a rapid decrease of Δ for a perpendicular field, B ⊥ (Fig. 1f). We consider a single lowest-energy subgap state (i.e. an ABS) with energy E 0 , which is two-fold spin-degenerate at B = 0 and becomes spin-split in a B-field. Fig. 1c sketches the case where E 0 decreases very slowly with B ∥ , remaining above E c such that the GS parity remains even. This translates to 2e-periodic conductance oscillations for all B ∥ -fields (Fig. 1k). Note that conductance peaks occur when the lowest-energy parabolas cross. In Fig. 1g the lowest crossings are always between even-charge parabolas. These crossings, for instance at N g = −1 and +1, are 2e-periodic. The odd-charge parabolas remain above these lowest-energy crossings and thus do not participate in the lowenergy transport. Figure 1d sketches a second case where E 0 varies more rapidly with B ∥ . When E 0 crosses E c the odd-charge parabolas pass the lowest-energy crossings of the even parabolas, thereby adding degeneracy points between even-charge and odd-charge parabolas (Fig. 1h). This results in alternating smaller and larger peak spacings, where the smaller valleys have odd-parity for E 0 > 0 and even-parity for E 0 < 0. Note that an equal spacing with 1eperiodicity occurs when E 0 = 0. At negative energies, when E 0 crosses −E c the 2e-periodicity is restored, however, now with an odd-parity GS. In terms of the charge parabolas this corresponds to odd-charge parabolas being always lower in energy than the even ones (red parabolas in Fig. 1h). This case of 2e-periodicity for an odd-parity GS has not been reported before.
In the third case we illustrate the possible consequence of strong spin-orbit interaction. The zero-energy crossing of the subgap state can now be followed by a transition to a topological phase containing MZMs rigidly fixed at E 0 = 0 (Fig. 1e). (Note that the re-opening of the gap is not shown in Fig. 1e since only the lowest-energy subgap state is sketched.) As E 0 decreases below E c , the peak spacing gradually evolves from 2e to 1e with alternating even-parity and odd-parity GSs (Figs. 1i and m). It is important to note that the even/odd degeneracy of the topological phase in bulk materials is lifted here by the charging energy 3,20 . This fundamental degeneracy is however visible by comparing the lowest energies for the even and odd parity GSs, which are both zero, albeit at different gate voltages. We note that ABSs confined by a smooth potential may also give rise to a similar phenomenon as sketched in Fig. 1e 21 .
Finally, in the fourth case ( Fig. 1f), the superconducting gap in Al closes at its critical perpendicular magnetic field. This transition to the normal state also causes equidistant 1e-periodic oscillations (Figs. 1j and n), which in the peak evolution is similar to the topological case. However, we show below that finite-bias spectroscopy is significantly different in these two cases.
In Fig. 2, we present exemplary data for the four cases illustrated in Fig. 1k-n. Fig. 2a-d show four panels of dI/dV b measured at zero bias as a function of V PG and B-field (B ∥ or B ⊥ ).
The bottom row of panels shows representative linecuts at high Bfields. Fig. 2e-h show the corresponding peak spacings for even (S e ) and odd (S o ) GSs. These spacings are converted from gate voltage to energy via the gate lever arm and reflect the energy difference between even-parity and odd-parity states. In Fig. 2a the peaks are 2e-periodic with even-parity GS up to a field of~0.9 T. This observation reflects that up to this B-field our Al thin film remains superconducting without any low-energy subgap state. Above~0.9 T the gap is significantly suppressed such that 1etransport sets in 11 .
In Fig. 2b, the conductance peaks split into pairs around 0.11 T with alternating small and large spacings. These split peaks merge with neighbouring split peaks, leading to the recurrence of 2e-periodic oscillations, but strikingly with an odd-parity in the valleys (cf. Fig. 1l). The parity transition is also illustrated in Fig. 2f by the single crossing between S e and S o . Similar to Fig. 2a, above 0.9 T the oscillation becomes 1e-periodic (see linecuts).
For the case shown in Fig. 2c (cf. Fig. 1m), the 2e-periodicity gradually changes to uniform 1e-periodicity above~0.35 T. S e and S o exhibit slight but visible parity-changing oscillations up to 0.9 T, whose amplitude decreases with field. This case resembles the experiment of ref. 15 , where the 1e oscillations are associated with MZMs. Additionally, we found that the peaks are alternating in height. To quantify this effect, we extract from the data an , which amounts to 0.5 for peaks with equal heights 22 . Here, G e→o (G o→e ) is the peak height at an even-to-odd (odd-to-even) transition occurring upon increasing V PG . Fig. 2g shows that Λ undergoes drastic oscillations around 0.5 as B ∥ is varied, and levels off at 0.5 above 0.9 T.
The data in Fig. 2d are taken for the same gate configuration as Fig. 2c but in a perpendicular B-field, which turns the Al into a normal state around B ⊥ = 0.18 T (see also Supplementary Fig. 3). In the normal state the oscillations are 1e-periodic and both Λ and S e /S o are constant, in agreement with established expectations 6 .
The four columns in Figs. 1 and 2 represent distinct phases at high B: an even-parity GS, an odd-parity GS, a superconducting phase of alternating even and odd parities due to a single state at zero energy, and a gapless normal phase of alternating even and odd parities. These distinct phases can be reached by varying V PG and thereby the spatial profile of the wave functions, which determines the coupling strength to Al and to the external magnetic field 23-25 . In Fig. 2a, a very negative V PG pushes the wave functions against and partly into the Al, leading to a robust induced gap with weak sensitivity to the B-field, which indeed never induces a parity change. Fig. 2b, at more positive V PG , reflects the presence of a subgap state with larger weight in the InSb (as indicated by its estimated g-factor, g ≈ 7-15; see another example in Fig. 3f). In Fig. 2c, at even more positive V PG , the involved wave function has an even larger weight in the semiconductor, yielding a large g-factor of~10, possibly augmented by orbital effects 26 . This leads to the appearance of robust zero-energy modes at B-fields much lower than the critical field of the thin Al shell. This V PG -dependent 2e-periodic to 1eperiodic transition has been repeated for another device, which is not depicted here.
2e-periodic odd-parity GS. The 2e-periodicity with even-parity valleys persists up to 0.9 T, above which quasiparticle poisoning occurs. b The 2e-periodic peaks split at~0.11 T and merge again at~0.23 T. For B ∥ > 0.23 T, the oscillations are again 2e-periodic, but here the GS parity is odd, consistent with Fig. 1h. c 2e-periodicity transitioning to uniform 1e-periodicity at B ∥ ≈ 0.35 T, accompanied by peak spacing and peak height ratio oscillations up to 0.9 T (see also panel g). d 2eperiodicity transitioning to 1e-periodicity at B ⊥ = 0.18 T (the vertical dashed line), coinciding with the critical B-field of the Al layer (see Supplementary  Fig. 3). Above the critical field, peak heights are constant (see linecuts) and the even/odd peak spacings are equal (h). A few common offsets in V PG are introduced to compensate the shifts in gate voltage, and the raw data are listed in Supplementary Fig. 4 Fig . 1 Hybrid semiconducting-superconducting island and its energy spectrum. a False-colour scanning electron microscope image of the device consisting of an InSb nanowire (green) with an 800-900 nm long Al-shell (light-blue) covering the top facet and one side facet. Inset: schematic cross-section at the centre of the plunger gate (PG) indicated by the yellow line. The Si/SiO x substrate contains a global back gate that we keep at zero voltage. The InSb wire is contacted by Cr/Au leads (yellow) and then covered by a 30 nm-thick dielectric layer of SiN x (light-grey). Ti/Au top gates (blue) that wrap around the wire allow for local electrostatic control of the electron density. Two gates are used to induce tunnel barriers (TG) and one plunger gate (PG) controls the electron number on the island. The scale bar indicates 500 nm. b dI/dV b vs. tunnel gate voltage and V b showing 2e-periodic Coulomb diamonds (one diamond is outlined by yellow dashed lines). The lower panel shows horizontal linecuts with 2e-periodic Coulomb oscillations at V b = 0 (black trace) and 1e-periodic oscillations at V b = 150 μV (red trace). The panel on the right shows a vertical linecut through the Coulomb peak at the degeneracy point (blue trace) and through the centre of the Coulomb diamond (purple trace). Below are four scenarios for the B-dependence of a single Andreev level (c-f), the resulting energies as a function of the induced charge, N g (g-j), and the Coulomb oscillations (k-n). In panels c-f, the grey regions represent the continuum of states above Δ. The coloured traces represent the energy, E 0 , of the lowest-energy subgap state. Panels g-j show the energies of the island with N excess electrons, E(N g ) = E c (N g − N) 2 + p N E 0 , where N g is the gate-induced charge, N is the electron occupancy number, and p N = 0 (1) for N = even (odd). Parabolas for N = even are shown in black, while parabolas for N = odd are shown in colours in correspondence to the colours in the other rows. Crossings in the lowest-energy parabolas correspond to Coulomb peaks as sketched in panels k-n, again with the same colour coding. Labels in the Coulomb valleys between the peaks indicate the GS parity being either even (e) or odd (o) with an odd electron number can also have odd parity 27 . In our case all states are strongly hybridised with the superconductor and one bound state drops below −E c , which causes the evenparity state to become an excited state above the odd-parity GS. (We describe different types of bound states in the Supplementary Note 5.) Fig. 3a-c show Coulomb diamonds for the gate settings of Fig. 2b at three values of B ∥ . Fig. 3a at B ∥ = 0 shows 2e-periodic diamonds with 1e and 2e linecuts shown at the bottom (similar data was presented in Fig. 1b). Fig. 3b at B ∥ = 0.17 T corresponds to the even-odd regime. Note that the conductance near V b = 0 is suppressed, indicating that the subgap state causing the even-toodd transition is weakly coupled to at least one of the normal leads (see also linecuts). Fig. 3c at B ∥ = 0.35 T shows again 2eperiodic diamonds but now the GS inside the diamonds has an odd parity. The diamond structure, including the presence of regions with negative dI/dV b , is very similar to the even-parity GS diamonds, except for the shift in gate charge by 1e. Figure 3f shows another example of the transition from the 2eperiodic even GS, via a region of even-odd spacings, to a 2eperiodic odd GS. Note again that the even-odd peak heights are significantly suppressed. These peaks correspond to a crossing of an even-parity parabola with an odd-parity parabola in Fig. 1h, where transport occurs via single electron tunnelling. In contrast, transport at the 2e-periodic peaks, both for even and odd GSs, occurs via Andreev reflection. The two cartoons (Fig. 3d, e) illustrate these different transport mechanisms.

Isolated zero-energy modes and Coulomb valley oscillations.
Richer sequences of GS transitions as a function of magnetic field are also possible. For instance, the sketch in Fig. 4a illustrates the occurrence of multiple zero-energy crossings at low B ∥ followed by the appearance of a stable zero-energy state at higher B ∥ . This type of behaviour is observed in Fig. 4b, showing large oscillations of S e and S o for B ∥ < 0.6 T, and a stable 1e-periodicity for B ∥ > 0.7 T (see Fig. 4c). Similarly, Fig. 4d also shows large oscillations of S e and S o below 0.6 T, followed by a region of almost equally spaced peaks above 0.6 T (see Fig. 4e).
The features at low B ∥ (such as the position where the 2eperiodic peaks first split) can depend on the precise value of V PG . In contrast, the features at high B ∥ are strikingly regular, with only a weak dependence on V PG . This suggests that the 1eperiodicity at high fields originates from a state which is remarkably robust against gate variations. Furthermore, we note Λ are similar in number and period to the corresponding oscillations in S e and S o , indeed suggesting a possible connection between the two (another example is presented in Supplementary  Figs. 8c-f). Fig. 4f shows finite-bias spectroscopy in the 1eperiodic regime of Fig. 4d, at a high parallel field B ∥ = 0.7 T. This spectroscopy reveals that the marked asymmetry of the peak heights originates from a discrete state that is gapped from a continuum of states at higher bias. As a comparison, Fig. 4g shows that for Al in the normal state no discrete features are observed. These are important verifications that substantiate our conclusion that the scenario in Fig. 1e is the proper description of the 1e oscillations in the experimental figures (Figs. 2c and 4d) at high B ∥ .

Discussion
In summary, we have revealed distinct types of fermion parity transitions occurring as a function of magnetic field and gate voltages in a Coulomb-blockaded InSb-Al island. These transitions provide a complete picture of all the parity phases in mesoscopic Cooper pair boxes 28 . Among these, in a finite field, we find a novel odd-parity phase with 2e periodicity in gate voltage. Additionally, we find 1e-periodic oscillations at a high field, which we show to originate from isolated zero-energy modes. The thorough understanding of the involved physics is important, since such islands form the building blocks of future Majorana qubits [29][30][31] .

Methods
Device fabrication. An isolated Al segment is formed by selectively shadowing the nanowire during Al evaporation. InSb-Al nanowires with double shadows (Supplementary Fig. 1) were transferred from the InP growth chip to a doped-Si/SiO x substrate using a mechanical nanomanipulator installed inside an SEM. Au is used as leads and the top gates. A 30 nm dielectric of SiN x separates the nanowire from the top gates.
Transport measurements. The device is cooled down to~15 mK in an Oxford dry dilution refrigerator. The effective electron temperature is estimated to be 20-50 mK. Conductance across the devices was measured using a standard low-frequency lock-in technique (amplitude is 5 μV). The voltage bias V b = V S − V D is applied symmetrically between the two leads (V S = −V D = V b /2). A magnetic field is applied using a 6-1-1 T vector magnet. The direction of the magnetic field is aligned carefully with respect to the nanowire axis ( Supplementary Fig. 3). The sweeping rate of the datapoints is very slow and one V PG -dependent linetrace at fixed B in Fig. 2 takes a few minutes. We measure from 0 to 1 T with 0.01 T steps, bringing the measurement time for each panel to 3-4 h. Because of this, we suffer from an ultra-slow drift that effectively changes the island potential. It is important to stress, however, that the slow drift affects the peak positions but not the peak spacings (since each trace only takes a few minutes to acquire). We extract subgap states and parity transitions from peak spacings and thus our conclusions are not affected by the ultra-slow drift.

Data availability
The figures are created from the raw data. All data and the code used for peak fitting are available at https://doi.org/10.4121/uuid:e0ecafef-7f45-4475-b2f8-351c2af4a2b0.