Subgap spectroscopy along hybrid nanowires by nm-thick tunnel barriers

Hybrid nanowires that utilize nm-thick tunnel barriers are introduced in Fig. 1. A false-colored scanning-electron microscopy (SEM) image of a representative device is shown in Fig. 1A and a schematical longitudinal cross-section along the device is displayed in Fig. 1B. A superconducting Al (red) film is grown by the shadow-wall lithography technique  (?, ?) on a semiconducting InSb (light blue) nanowire  (?). By a subsequent in-situ oxidation, the Al film is partially oxidized to form a dielectric Al oxide - $Al_{2}O_{3}$ (pink) layer that covers the hybrid. The shadow-wall lithography technique is used to define three normal Ag (navy) leads along the nanowire on top of the Al oxide layer. Two Au (yellow) leads contact the bare semiconducting nanowire part on the left and the hybrid nanowire part on the right. Two Pd (dark grey) gates are coupled to the nanowire via a dielectric $HfO_{2}$ (light grey) layer. The gate under the nanowire section with the superconducting shell (super gate) controls the electro-chemical potential in the hybrid. The gate under the bare nanowire section (tunnel gate) tunes a tunnel barrier at the semiconducting junction between the left Au lead and the hybrid. Four normal leads are tunnel-coupled to the hybrid and denoted as tunnel probes P0, P1, P2 and P3 in Fig. 1B. The fifth lead forms a contact to the hybrid and is denoted as a drain lead. The tunnel probe P0 utilizes the semiconducting tunnel barrier controlled by the tunnel gate, while in the tunnel probes P1, P2 and P3 the Al oxide layer serves as a nm-thick tunnel barrier. The widths of probes P1, P2 and P3 are designed to be $200\,\mathrm{nm}$ and the lateral edge-to-edge distances between the neighboring probes are designed to be $200\,\mathrm{nm}$. Schematical transversal cross-sections of the device are displayed in Fig. 1C-1E. The cross-section through the probes P1, P2 and P3 in Fig. 1D indicates that the nanowire has the superconducting Al shell on one of its facets and that the Al oxide layer extends over the entire contact area between the hybrid and the Ag leads. The Al oxide layer in the drain area is removed by Ar ion milling prior to the deposition of the gold contacts - as shown in the cross-section through the drain lead in Fig. 1E. Details of the steps in device fabrication can be found in the Device fabrication section and Fig. S1 in the Supplementary Materials.

A critical step in the fabrication of our hybrid nanowire devices is the formation of the Al oxide layer by an in-situ oxidation of the superconducting Al film. In order to test this fabrication step, we fabricate a planar tunnel junction which perpendicular cross-section is schematically shown in the top panel of Fig. 1F. The junction leads are a superconducting Al (red) and a normal Ag (navy) film that partially overlap and that are separated by a thin dielectric Al oxide (pink) layer. The Al oxide is formed by an in-situ oxidation of the Al film, prior to the deposition of the Ag film. The tunnel junction is characterized in the bottom panel of Fig. 1F by measuring the junction conductance as a function of a bias voltage $V_{b}$ and an in-plane magnetic field $B$. The result represents a typical tunneling spectroscopy of superconducting Al. As shown in the panel, coherence peaks spin-split with magnetic field due to the Zeeman effect. This demonstrates that the in-situ oxidation of Al can yield an Al oxide layer as a nm-thick tunnel barrier for tunneling spectroscopy. Next, we perform such in-situ oxidation on hybrid nanowires and characterize these hybrid nanowire devices in electrical transport measurements..

We have studied three hybrid nanowire devices (Device 1, 2 and 3) that use nm-thick tunnel barriers. Devices 1 and 2 are nominally identical and are fully described in Fig. 1. For Device 3, Al is deposited instead of Ag, so that three superconducting leads are defined on top of the Al oxide layer (see the Device fabrication section in the Supplementary Materials). Therefore, the probes P1, P2 and P3 of Device 3 form three Josephson junctions with the hybrid nanowire. The replacement of Ag by Al in Device 3 is motivated by proposals that studying supercurrent in hybrid devices is an alternative way of detecting MZMs  (?, ?) and realizing MZM-based qubits  (?, ?).

As an initial step, the superconducting properties of Device 1 are characterized in conductance measurements by different probes in a voltage-bias setup. Each of the four probes P0, P1, P2 and P3 is in turn connected in Setup V1 to measure the tunneling conductance (see the Measurement setups section in the Supplementary Materials) and the results are shown in Fig. 2. A voltage $V_{TG}=2.1\,\mathrm{V}$ is applied to the tunnel gate to define a tunnel barrier in the semiconducting junction of probe P0. The super gate is set to $V_{SG}=0\,\mathrm{V}$. In the top row of Fig. 2, the tunneling conductance $G$ is measured by each probe as a function of a bias voltage $V_{b}$ and a parallel magnetic field $B$ along the nanowire of Device 1. In the parallel $B$-field, superconducting gaps detected by all four probes close at $3-3.5\,\mathrm{T}$. The variations in critical fields can be explained by small misalignments of the applied fields in different probes - as the nanowire is not perfectly straight. Furthermore, in comparison with the tunneling spectroscopy of the Al film in Fig. 1F, generally there is no splitting of the coherence peaks measured by probes P1, P2 and P3. This is most likely due to spin-mixing mechanisms induced by the spin-orbit interaction from the semicoducting nanowire  (?). The phenomenon of non-splitting coherence peaks is widely seen in tunneling spectroscopies with semiconducting junctions - as shown here in Fig. 2A. From each 2D-map in Fig. 2, a linecut at zero field is taken and shown in logarithmic scale as a red trace in the bottom row of Fig. 2. In some traces, large negative values appear at the coherence peaks and their origin is explained in the Measurement setups section in the Supplementary Materials. A superconducting gap $\Delta\sim 300\,\mathrm{\mu eV}$ of the hybrid can be obtained by extracting the positions of the coherence peaks. Noticeably, in all four probes the tunneling conductance at $V_{b}<\Delta$ (in-gap conductance) is roughly two orders of magnitude lower than the tunneling conductance at $V_{b}>\Delta$ (out-of-gap conductance). However, the out-of-gap conductance in these probes is two orders of magnitude larger than in probe P0 – due to the nm-thick tunnel barriers allowing for tunnel coupling of a large number of modes in the metallic leads of P1, P2 and P3. In order to measure the high out-of-gap conductance by P1, P2 and P3, the measurement sensitivity is adjusted, and consequently the modulations of the in-gap conductance cannot be precisely detected in Fig. 2B-2D. The subgap spectra in the probes P1, P2 and P3 is in detail studied in Fig. 4 and Fig. 5 of this work. A characterization measurement like the one of Device 1 in Fig. 2 has been performed for Device 2 and similar results are shown in Fig. S4 in the Supplementary Materials.

In order to test the Al oxide layer as a weak link for supercurrent measurements, current-bias measurements are performed on Device 3 and the results are shown in Fig. 3. Fig. 3B is a schematical cross-section through the probe P1 (or P2, or P3). It is shown that the probe P1, as well as P2 and P3, uses a superconducting lead made of thick Al. Together with the underlying Al oxide layer and the superconducting Al shell on the nanowire, the three superconducting leads of P1, P2 and P3 form three asymmetric Josephson junctions – JJ1, JJ2 and JJ3. In order to characterize JJ1, probes P1 and P2 are connected in Setup I2 and probe P1 is current-biased (see the Measurement setups section in the Supplementary Materials). A voltage drop $V$ over JJ1 is measured as a function of a bias current $I_{b}$ and a parallel magnetic field $B$ and the dependence is shown in Fig. 3A. The linecuts taken at $B=0\,\mathrm{T}$ and $B=0.4\,\mathrm{T}$ are displayed in Fig. 3C. The linecut taken at $B=0\,\mathrm{T}$ (bottom panel of Fig. 3C) shows a zero voltage plateau due to the non-dissipative Josephson supercurrent with a switching current of $\sim 200\,\mathrm{nA}$. This demonstrates that at low fields the probe P1 is in the SIS transport regime (S-thin superconducting Al shell, I-thin dielectric Al oxide, S-thick superconducting Al lead). As the field increases in Fig. 3A, the zero voltage plateau shrinks and disappears at $B\sim 0.2\,\mathrm{T}$ due to the suppressed superconductivity in the thick Al lead. Consequently, the SIS transport regime is altered by the SIN transport regime as the thick Al lead changes from being superconducting (S) to being normal (N). The linecut taken at $B=0.4\,\mathrm{T}$ (top panel of Fig. 3C) confirms this as it resembles an $I-V$ characteristic of the tunneling transport between a superconductor and a normal metal. This suggests that a parallel field of $0.4\,\mathrm{T}$ is sufficient to turn the thick Al lead fully normal and that at high fields the probes P1, P2 and P3 of Device 3 can be used as normal metal leads for tunneling spectroscopy.

In Fig. 2 and Fig. 3 we demonstrate that the probes with nm-thick tunnel barrier can serve to characterize superconductivity in hybrid nanowires both in tunneling conductance measurements and supercurrent measurements. In the rest of this work, we focus on measuring in-gap conductance by different probes with a goal to study subgap states in hybrid nanowire devices.

The capability of probes with nm-thick tunnel barriers to detect subgap states is examined for Device 1 in Fig. 4. In-gap conductance is measured by two tunnel probes - probe P0 that utilizes the semiconducting tunnel barrier and probe P1 as the nearest probe that utilizes the nm-thick tunnel barrier. Probes P0 and P1 are connected in Setup V2 (see the Measurement setups section in the Supplementary Materials) and the super gate voltage is set at $V_{SG}=0\,\mathrm{V}$. In-gap conductance is measured by both probes as a function of a bias voltage $V_{b}$ and a parallel magnetic field $B$ (Fig. 4A) or a tunnel gate voltage $V_{TG}$ (Fig. 4B). Upon setting a value of $B$ or $V_{TG}$, $V_{b}$ is first swept on probe P0 with probe P1 at zero bias voltage and then $V_{b}$ is swept on probe P1 with probe P0 at zero bias voltage. The conductance dependences in the top panels of Fig. 4A and 4B show that a single subgap state is detected by probe P0 for the given ranges of $B$ and $V_{TG}$. The strong modulation by $V_{TG}$ (Fig. 4B top) suggests that the subgap state is localized close to the semiconducting junction of probe P0. Such subgap states are commonly detected in tunneling spectroscopy measurements with semiconducting tunnel barriers in two-terminal  (?) and three-terminal  (?, ?, ?, ?, ?) hybrid nanowire devices. Interestingly, the conductance dependences in the bottom panels of Fig. 4A and 4B show that the same subgap state is also detected by probe P1. This is additionally demonstrated by the linecuts taken from Fig. 4A (Fig. 4B) and displayed in Fig. 4C (Fig. 4D) in which aligned conductance peaks correspond to the same subgap state detected by the two probes. In addition, there are conductance peaks measured by probe P1 that are not measured by probe P0 – indicating that these subgap states most likely reside near P1 and are decoupled from P0. An additional tunnel gate sweep at a finite $B$-field and positive super gate shows that the subgap states detectable by both P0 and P1 remain detectable by P1 even when the semiconducting junction is pinched-off (see Fig. S5 in the Supplementary Materials). This means that the probe P1 can substitute the probe P0 when probing the hybrid nanowire end and even allow for tunneling spectrosopy in broader parameter ranges that are inaccessible to P0. An analogous measurement to the one in Fig. 4 has been carried out on Device 2 (see Fig. S6 in the Supplmentary Materials) and the capability of Al oxide tunnel probes to detect hybrid states is validated there as well. As we demonstrate that tunnel probes utilizing nm-thick tunnel barriers can detect subgap states in hybrid nanowires, in the rest of this work we use only these probes to study the subgap spectra in our hybrids.

An appealing advantage of the tunnel probes with nm-thick Al oxide barriers is the opportunity to use multiple probes along a single hybrid nanowire for exploring the spatial distribution of subgap states. In Fig. 5 tunneling spectroscopy is performed by the probes P1, P2 and P3 of Device 1 and Device 3 in order to study the subgap spectra at different positions along the hybrid nanowires. The three probes of Device 1 are in pairs consecutively connected in Setup V2 (see the Measurement setups section in the Supplementary Materials) and the tunneling spectroscopy results are shown in Fig. 5A. For Device 3, the three probes are in turn connected in Setup V1 (see the Measurement setups section in the Supplementary Materials) and the tunneling spectroscopy results are shown in Fig. 5B. A high magnetic field is applied for the measurements of Device 3 in order to fully suppress the superconductivity in thick Al leads of the probes P1, P2 and P3.

The measurement in Fig. 5A is performed with the super gate of Device 1 at $V_{SG}=0.6\,\mathrm{V}$ and the tunnel gate floated. Tunneling conductance is measured by probes P1, P2 and P3 as a function of a bias voltage $V_{b}$ and a parallel magnetic field $B$. The subgap spectra obtained by the three probes show different evolution with $B$ field. By probe P1, two kinds of subgap states are detected – subgap states insensitive to the $B$ field and subgap states with high $g$ factor that cross zero energy as the $B$ field is increased (Fig. 5A left). The measurement in Fig. S5 in the Supplementary Materials demonstrates that the subgap states detected by the P1 reside at the hybrid end even when the semiconducting junction is pinched-off. This indicates that the states detected by the probe P1 are not junction states but subgap states localized at the end of the hybrid. These states appear to be strongly localized at the hybrid end, as no subgap states are detected by probe P2 (Fig. 5A middle). Another subgap state with low $g$ factor is measured to be localized in the hybrid bulk - as it is detected by probe P3 (Fig. 5A right), but is not detected by probe P2. The correlation between the states detected by different probes has been examined while varying the super gate of Device 1 (see Fig. S7 and Fig. S8 in the Supplementary Materials). The absence of correlations indicates that the subgap states detected in Device 1 are localized over less than $\sim 200\,\mathrm{nm}$ at the end or inside the bulk of the hybrid. A similar qualitative picture is observed for Device 2 and the corresponding measurements are shown in Fig. S9 in the Supplementary Materials. Besides confirming the strong localization of the subgap states in Device 1, the measurements of Fig. S7 and Fig. S8 show some additional features of the subgap states that can be used to better understand their nature. This is elaborated in the Disucssion section.

For the measurements of Device 3, a parallel magnetic field $B=1\,\mathrm{T}$ is applied along the nanowire and the tunnel gate is floated. The superconductivity in the thick Al leads of probes P1, P2 and P3 is fully suppressed due to the high field, and they are used as tunnel probes. In Fig. 5B, each of the three probes is in turn connected in Setup V1 (see the Measurement setups section in the Supplementary Materials) and the tunneling conductance is measured as a function of a bias voltage $V_{b}$ and a super gate voltage $V_{SG}$. The order in which the 2D maps are measured is P2, P1, P3 (middle, left, right in Fig. 5B). Even the three 2D maps are obtained in three consecutive $V_{SG}$ sweeps, a striking similarity between the two subgap features detected by probes P2 and P3 indicate the presence of a subgap state coupled to both probes. However, the absence of any similar feature in the tunneling spectroscopy by probe P1 (taken in between the measurements by P2 and P3) suggests that the same state is not detectable at the hybrid nanowire end. This implies that the subgap state extends over more than $400\,\mathrm{nm}$ in the hybrid bulk, but does not reach the hybrid end. Importantly, detecting such state shows the capability of probes with nm-thick tunnel barriers to detect extended subgap states. Another extended subgap state is detected in the same device in other $V_{SG}$ range (see Fig. S10 in the Supplementary Materials). Additional tunneling spectroscopy in a broad super gate range ($-10\,\mathrm{V}<V_{SG}<10\,\mathrm{V}$) in all three probes shows that applying positive $V_{SG}$ weakly reduces the superconducting gap (see Fig. S11A in the Supplementary Materials). At $V_{SG}=10\,\mathrm{V}$ (and $B=1\,\mathrm{T}$) the gap remains open along the hybrid of Device 3. Additional supercurrent measurement at zero field shows that sweeping $V_{SG}$ from $-2\,\mathrm{V}$ to $2\,\mathrm{V}$ has no effect on the supercurrent measured by probe P1 (see Fig. S11B in the Supplementary Materials). Together with the large switching current value, such insensitivity to the super gate indicates that the hybrid states have a negligible contribution to the supercurrent that is dominantly carried by the condensate in the Al shell.