Spectroscopy of the superconducting proximity effect in nanowires using integrated quantum dots

The superconducting proximity effect has been the focus of significant research efforts over many years and has recently attracted renewed interest as the basis of topologically non-trivial states in materials with a large spin orbit interaction, with protected boundary states useful for quantum information technologies. However, spectroscopy of these states is challenging because of the limited spatial and energetic control of conventional tunnel barriers. Here, we report electronic spectroscopy measurements of the proximity gap in a semiconducting indium arsenide (InAs) nanowire (NW) segment coupled to a superconductor (SC), using a spatially separated quantum dot (QD) formed deterministically during the crystal growth. We extract the characteristic parameters describing the proximity gap which is suppressed for lower electron densities and fully developed for larger ones. This gate-tunable transition of the proximity effect can be understood as a transition from the long to the short junction regime of subgap bound states in the NW segment. Our device architecture opens up the way to systematic, unambiguous spectroscopy studies of subgap bound states, such as Majorana bound states.


Majorana bound states. Introduction
Coupling a superconductor (SC) to a metal or a semiconductor results in the so called superconducting proximity effect in the normal material 1 . If the proximitised material is a low-dimensional semiconductor, this phenomenon can, for example, be used as a source of spin-entangled electrons 2,3 or superconducting magnetometers 4 . By combining an s-wave SC with a one-dimensional semiconducting nanowire (NW) with large spin orbit interaction, one can artificially create a proximity region with superconductivity of p-wave character. This can give rise to exotic quantum states at the ends of the SC, such as Majorana bound states [5][6][7][8] , potentially useful as building blocks for topological quantum information processing 9,10 . However, an unambiguous characterization of superconducting subgap states and the proximity region in NWs remains challenging. Several theoretical proposals suggest to use a quantum dot (QD) as a spectrometer to investigate Majorana bound state lifetime 11,12 , spin texture 13 or parity 14 . A first experiment was reported recently in which the QD was defined by electrical gating 7 . However, a systematic and deterministic spectroscopy, requires a spatially well defined, weakly coupled, QD with sharp tunnel barriers across the complete NW, which does not hybridise with the bound states or the SC under investigation.
Here, we introduce a new material platform that allows us to perform ideal tunnel spectroscopy: we use an in-situ grown axial QD in an InAs NW covering the complete diameter of the NW, to probe the superconducting proximity region of a NW segment close to the superconducting contact. The QD is defined by two potential barriers that form when the NW crystal structure is changed from zincblende (ZB) to wurtzite (WZ), which can be achieved with atomic precision by controlling the growth parameters 15,16 . These QDs are electrically and spatially well defined, which allows us to probe the induced gap at a precise distance from the QD and with predictable coupling parameters. In-situ grown barriers have previously been used to investigate double QD physics 17,18 .
Here, we use such artificial QDs to study the evolution of the proximity induced superconductivity in a controlled and systematic manner. We demonstrate this type of spectroscopy using two different transport regimes of the QD: in the cotunneling regime, where the QD can be seen as a single tunnel barrier, and in the sequential tunneling regime, where the QD acts as an energy filter with Coulomb blockade (CB) resonances. The complementary measurements allow us to draw a clear picture of how the proximity gap forms in a NW segment, for which we present an intuitive explanation supported by numerical simulations.

Results
Device and characterization. We use an InAs NW with an in situ grown axial QD formed by controlling its crystal phase structure in the growth direction 16 : two thin segments (30 nm) of WZ phase are grown in the otherwise ZB NW. These segments act as atomically precise hard-wall tunnel barriers for electrons, because the ZB and WZ bandstructure align with a conduction band offset of ∼100 meV 19,20 . Consequently, in the ZB section (20 nm) between the two barriers a QD is formed 19 . A false color scanning electron micrograph of the actual device is shown in Fig.1a, together with a transmission electron micrograph of the atomically precise interface in a represen-  Figure 1: Crystal phase engineered QD in an S -N -QD device.  Fig.S1a) for which we find a systematic increase of the total tunnel coupling Γ with increasing V BG (see Fig.S1b), consistent with the lowering of the tunnel barrier when the bandstructure is shifted to lower energies with respect to the Fermi energy (see schematic in Fig.S2) 19 .
In the device discussed here, the QD is located about L≈350 nm from the Al electrode, leaving a bare NW segment of this length between the QD and the SC, which we refer to as the "lead segment" (LS). This provides a new experimental situation, as the QD is not directly coupled to the SC (as it might be the case in other experiments [21][22][23] ) and can be used to probe the LS. V BG here directly tunes the chemical potential in both the LS and the QD. We record G as a function of the bias voltage V SD applied between the SC and the normal metal contact. If the electrons would tunnel directly from the QD to the superconducting electrode, we would expect to see a gap similar to the one of the bulk ∆≈210 µeV 24 of the SC, independent of V BG . Since most of the bias drops over the QD, G is proportional to the density of states (DOS) in the LS. This is the case for positive V BG where carriers accumulate in the NW. Therefore we can perform spectroscopy on the LS by tunneling from the QD (see Fig.1b).
An interesting transition in the conductance can already be found in the overview data in Fig.1c,d presenting regular CB diamonds in the superconducting state (charging energy E c ≈ 6 meV, level spacing ≈ 1.5 meV to 2 meV) for two different regimes of V BG . While for positive gate voltages V BG 3 V we observe a superconducting gap around zero bias (c.f. Fig.1d), for low gate voltages (c.f. Fig.1c) we do not find any features related to superconductivity, but rather the standard sequence of diamonds.
Measurements. The QD can be used as a spectrometer in two different regimes. In the first regime, the QD is kept deep in the CB regime where the charge is fixed for bias voltages in the range of the proximity induced gap ∆ * , since ∆ * << E c . In this regime transport is mediated by cotunneling, which is a second order process involving the virtual occupation of a QD state 25 .
Here, the QD can be thought of as a single tunnel barrier. In the second regime, the QD electrochemical potential is tuned to to a CB resonance, with transport occurring as first order sequential tunneling. In the following, we discuss the experiments in the two regimes one after the other.
First, we investigate the cotunneling regime, where we can understand the SC -LS as an S-N junction which is weakly coupled to the QD. The measured spectrum is shown in Fig.S3a. It can be characterized by four quantities: the magnitude of the observed gap ∆ * , the full width at half the maximum (FWHM) of the peaks at the gap edge, the normal state conductance G N measured at a bias |V SD | > ∆ * /e and G S the conductance at V SD = 0. Since the conductance is not fully suppressed at zero bias, we characterise this softness by defining the suppression factor S = G S /G N . suppression of G at V SD = 0 and ∆ * ≈ 150 µeV (position of the peak maxima). The FWHM and S are shown to be roughly constant over the investigated gate range, see Fig.S3b, with F W HM ≈ 65 µeV±10 µeV and S ≈ 0.5 ± 0.1, respectively. In an ideal SC and weak coupling, S should be close to zero for a strong tunnel barrier. Larger values are often observed for proximity induced gaps, referred to as soft gap [26][27][28] . For lower gate voltages V BG <2.6 V (Fig.2b) it is difficult to perform this analysis because the cotunneling signal is very low, compared to the noise floor of the experiment (G ≈ 10 −5 e 2 /h). However, we can still observe broad peaks down to V BG ≈ 1.8 V.
For V BG <1.8 V the differential conductance is too low to observe any signatures of superconductivity with the resolution of our measurement. For even lower gate voltages V BG ≈ 0.2 V (Fig. 2c) no features inside the CB diamonds can be resolved anymore. In summary, in the regime accessible by the cotunneling experiments, the proximity gap characteristics are roughly independent of the gate voltage.
To extract characteristics of the LS for a larger gate range than in the cotunneling regime, we now From the QD characteristics in the normal state (see Fig. S1a), we find that the tunnel coupling Γ increases significantly for V BG > 2.5 V. For large Γ, e.g. at V BG ≈ 6.6 V, a CB pattern is shown in Fig.3a. The CB resonances are broad, but clearly show a suppressed conductance around zero bias. We find a gap of ∆ * ≈ 150 µeV, confirming the value obtained in the cotunneling regime ( Fig. 2).
A CB resonance with a smaller Γ value at V BG ≈ 2.1 V is shown in Fig. 3b, which is similar to the one in Fig. 3a, but with an additional resolved resonance. As expected (see Fig. 3e), the "CB diamond tips" are shifted in energy by ±∆ * /e and ∆V BG in gate voltage, yielding a consistent value of ∆ * ∼ 150 µeV. We observe an additional resonance that crosses through the gap (white arrows).
This line corresponds to the alignment of the Fermi level of the two reservoirs with the QD state.
We attribute this to the tunneling through the non-zero DOS remaining at zero bias (see position I in Fig. 3e).
At V BG ≈ 0.2 V the conductance suppression is significantly reduced, as shown in Fig.3c. The CB diamond tips appear to be only slightly separated and shifted. We note that in this gate range we cannot resolve any signal in the cotunneling spectrum, as discussed in Fig. 2.
At even lower gate voltages, e.g. V BG ≈ 80 mV (Fig. 3d), we do not observe any influence of the superconducting contact, but a regular CB resonance, as found in the normal state.
To extract the characteristic numbers from these data, we use a resonant tunneling model for a S -QD -N junction. The current is then given by 22,30 By adjusting the magnitude of the gap ∆ * , the QD resonance broadening Γ and the Dynes parameter δ, we get the conductance maps represented in the central panels in Fig.3a-d ("model").
We can reproduce the characteristics of the CB resonance in Fig. 3a using Γ = 150 µeV, which is slightly smaller than what we obtained in the normal state. For the size of the gap we find ∆ * ≈165 µeV, and δ = 0.4 · ∆ * (δ =65 µeV), resulting in a suppression of S ≈ 0.5, similar to what is found in the cotunneling regime. To reproduce the data in Fig. 3b we use Γ = 40 µeV (similar to the numbers in the normal state), ∆ * ≈ 145 µeV and δ = 0.4 · ∆ * (δ ≈ 60 µeV), i.e. the parameters are almost identical to the ones we obtained in the cotunneling regime at higher gate voltages.
The corresponding cross sections agree well with the experiment. We note, that the conductance enhancement at the gap edge, as well as the negative differential conductance are reproduced by the model. We point out that in the sequential tunneling regime, we can extract gap characteristics down to back-gate voltages of 0.2V, which is not possible in the cotunneling regime.
However, the size of the superconducting energy gap is found to be ∆ * ≈ 85 µeV, which is significantly smaller than that found at larger gate voltages. Also here, the model reproduces the data well, illustrated in the corresponding cross sections.
Using the resonant tunneling model to simulate the resonance around V BG ≈ 80 mV (Fig. 3d) we extract Γ = 50 µeV and an upper limit for ∆ * of 10 µeV. The model reproduces very well the characteristics of the CB resonance, which essentially corresponds to an N -QD -N device. The resonances in the experiment at higher bias outside the CB are due to excited states, which are not included in the model.
To summarize the measurements in the CB resonance regime (Fig.3 Discussion This evolution of the induced gap ∆ * as a function of V BG is summarised on Fig. 4a.
The curve shows a sharp transition from a clearly resolved energy gap for V BG > 0 to a fully suppressed gap at V BG < 0.
While the observed proximity feature can well be fitted with a broadened BCS DOS, this approach is not an adequate description, since there are only few states in the quasi one-dimensional NW lead. Qualitatively, one can understand the transition by considering only a few modes in the LS.
All electrons at energies within the gap of the SC are Andreev reflected (AR) at the SC, giving rise to Andreev-bound states (ABSs).
We interpret the observed transition in ∆ * qualitatively as a gate-tunable transition of ABSs forming in the LS from the long to the short junction limit. Both limits are defined by comparing the physical length of the junction, L, to the characteristic length-scale L c = v F / ∆, which is the coherence length in the ballistic limit 32 . Here v F is the Fermi velocity in the LS, the Planck In the short junction limit (L << L c ) the energy of the ABSs (E ABS ) in the LS is dominated by the phase change due to AR at the SC/LS interface (dashed line in schematic), where ∆ is assumed to change abruptly. In this limit, the energies of the ABSs are "pushed" to E ABS ∼ = ∆, resulting in the superconducting proximity gap in the LS that is similar to ∆, i.e. ∆ * ∼ ∆ 33 . In the long junction limit (L >> L c ), E ABS is determined by the phase acquired in the LS, which scales with with m * the effective electron mass. In this limit, E ABS can take on smaller values, thus filling the proximity gap 32 . We therefore can tune E ABS by tuning v F and the electron density in the LS using V BG . For very positive gate voltages, E F is relatively far up in the con-duction band with a correspondingly large v F , resulting in a large L c , bringing the LS to the short junction limit, L < L c . In contrast, when we tune E F to the bottom of the conduction band by lowering V BG , v F and L c are strongly reduced, bringing the LS to the long junction limit, L > L c , and E ABS < ∆. As a result, the apparent gap ∆ * is reduced by the ABSs moving into the gap. We note that ∆ in the NW segment located below the SC is screened by the SC and is therefore not gate tunable.
To support this qualitative picture, we employ a numerical model, in which we combine the Green's function method with a tight binding model. The properties of the superconductor are taken into account as a self energy dressing the bare Green's function in a NW section below S, where a superconducting gap ∆ is induced depending only on the coupling to S (details can be found in the supplementary materials). This region is coupled to a bare NW segment (of length L = 350 nm) modelling the LS (see inset of Fig. 4a). We then investigate the local density of states (LDOS) at a distance L =350 nm as a function of the gate voltage in the NW (Fig. 4b). Like for the experimental data, we extract ∆ * as the distance between the maxima in the DOS. Here the number of ABSs as well as the ABS energy is determined by the total length of the LS. The resulting ∆ * of this calculation is plotted as a function of V BG as a red line in Fig. 4a. In excellent agreement with the experiment, we observe a sharp transition in the detected gap, corresponding to the transition from the short to the long junction limit. In the model, ∆ * tends to zero when E F is aligned with the bottom of the conduction band (V BG ≈ − 0.2 V), just before the LS is fully depleted and v F → 0.
For a better understanding we study the dependence of an individual ABS as a function of V BG in the supplementary (see Fig. S4). Evolving from the long to the short junction limit, the states with energy E < ∆ * move towards the gap, resulting in a fully formed smooth proximitised gap ∆ * .
We note that in this model the width of the ABS is a free parameter, which we set to 25 µeV. In seeming contradiction to the depleted LS, the experiment still shows several CB resonances below this gate voltage, which we attribute to evanescent modes from the NW segment below S (which is not depleted) that couples weakly to the QD wavefunctions, yielding the QD couplings highly asymmetric with very low transmission amplitudes, as observed in the experiment.
The gap in our devices is soft, i.e. the conductance suppression at zero bias is significantly lower than in NWs with an epitaxial Al shell 34,35 . The reason for this fact remains unclear and is a priori not expected from the model. The evaporated bulk Al shows a hard gap (S ≈ 0.01) when used in standard, large-area metallic S -I -S tunnel junctions measured in a similar experimental setup (see Fig.S5). Introducing random spatial potential fluctuations at the NW-S interface 36,37 in the presented numerical model does not account for the observed small suppression, either (see the QD, which should exhibit a similar tunability as the QD life time, 2) single particle tunneling to the NW segment below S, for example by an inverse proximity effect due to the gold nanoparticle used for the NW growth, and 3) quasiparticle excitation by microwave radiation absorption that might be different in NW devices compared to metallic ones.
In conclusion, we present a systematic study of the apparent transport gap in a NW segment induced by a proximity coupled superconductor. For this purpose we introduce QD tunnel spectroscopy enabled by in-situ grown axial tunnel barriers. We observe a gate-tunable transition of the gap amplitude from a fully developed, constant proximity gap at large electron densities to smaller values and ultimately a complete suppression of the gap at low densities. The data are consistent with a transition from the short junction limit of an S-N device to the long junction limit with ABSs forming at energies also below the gap energy. This transition occurs when the Fermi energy is close to the bottom of the conduction band and the respective Fermi velocity tends towards zero. Our experiments demonstrate that NWs with in-situ grown barriers, in our example with crystal phase engineered barriers, are very useful to perform unambiguous transport spectroscopy in superconductor-semiconductor hybrid systems. We have thus introduced a novel spectroscopy tool, which is well suited to study superconducting bound states in semiconducting NWs, and open, for example, an unambiguous path for battling fundamental limitations found in recent studies of Majorana bound states 38,39 .

Fabrication
The InAs nanowires were grown by metal-organic-vapor-phase epitaxy (MOVPE) and have an average diameter of 70 nm. The two segments of wurtzite crystal phase forming the tunnel barriers have a thickness of 30 nm. The zinc-blend segment in between, which defines the QD, has a width of 25 nm.
The nanowires were transferred mechanically from the growth substrate to a degenerately p-doped silicon substrate with a SiO 2 capping layer (400 nm). The substrate is used as a global back gate.
For the electron beam lithography we employ pre-defined markers and contact pads, made of Ti/Au (5 nm/45 nm). The normal metal contact to the NW is made of Ti/Au (5 nm/70 nm) and the superconducting contact of Ti/Al (5 nm/80 nm). Before each evaporation step, the native oxide of the NW is removed by an Argon ion sputtering.
All measurements were carried out in a dilution refrigerator at a base temperature of 20 mK. Differential conductance has been measured using standard lock-in techniques (V ac = 10 µV, f ac = 278 Hz).
Research Council (VR). The authors thank C. Reeg and S. Hoffman for fruitful discussions.
Competing Interests The authors declare that they have no competing financial interests.

Author contributions
with σ(τ ) the Pauli matrices in the spin (particle-hole), µ the electrochemical potential and α the strength of the Rashba spin-orbit coupling. To be able to study the gate dependence/length dependence in our system, the Hamiltonian of the NW is described by a lattice tight-binding model where the continuum limit Hamiltonian becomes with c † j is the creation operator of an electron in the nanowire on site j written in the Nambu basis.
The Hamiltonian of the bulk superconductor can be written as with ξ k = k 2 2m − µ and Ψ k,σ the annihilation operator for an electron in the superconductor with spin σ and momentum k. The tunneling Hamiltonian describing the coupling between the bulk SC and the nanowire takes on the form where the operator c † j,σ creates an electron of spin σ on site j. Since the total Hamiltonian is quadratic in the SC degrees of freedom, we can integrate them out, so that all the properties of the bulk superconductor and the tunneling to the nanowire are taken into account in the superconducting self-energy. The total Green's function of the system can be obtained by dressing the Green's function of the nanowire by the superconducting self-energy such that The total retarded superconducting self-energy has the following formΣ S R (ω) = I ⊗Σ S j,R (ω), where I is the unity matrix in the space of sites, andΣ S j,R the on-site retarded self-energỹ Σ S j,R (ω) = t j g R (ω) the Green's function of the superconductor. This Green's function is well known 40,41 and can be incorporated using the superconducting self-energŷ where 1 is the unity matrix in spin and particle-hole space, and Γ j,S = πν(0) t j 2 are the tunneling rates. The retarded bare Green's function of the semiconducting NW electrons is obtained by where δ is an infinitesimal quasiparticle inverse lifetime introduced to avoid diverging terms in the numerical evaluations. This quantity determines the width of the bound states in the system.
We are interested in getting the local density of states evaluated at a given site j in the NW. Such a quantity can be obtained by taking the imaginary part of the total Green's function The hopping amplitude is defined as t = 2 /2m * a 2 with m * = 0.0025m e being the effective mass, a the lattice constant, and the Planck constant. The relation between the experimental and the tight binding parameters are determined by t. We use the maximum value of the gap ∆ * =160 µeV, in the short junction limit when it is fully formed, which corresponds to t ≈ 1.6meV and a lattice constant of 70 nm. With these values, we can fully determine the parameters in the experiment.
The only free parameters in our model is the lever arm which is taken to be 0.1 meV/mV similar as in the experiment (0.06 meV/mV). For the spin-orbit interaction strength we used 70 µeV, similar to what has been reported before.    Figure S6: Local density of states as a function of bias V SD without (black) and with (red) disorder.

Extracted QD and LS characteristics
The disorder LDOS was averaged 30 times.
Gaussian distribution. The plotted cross section was averaged 30 times. As a result, the peaks at the gap edge are smaller in amplitude and also broadened, compared to the clean case. The edges of the gap appear to be slightly smoother.