⇦Chinese Physics Letters, 2023, Vol. 40, No. 6, Article code 067301 Gate-Tunable Negative Differential Conductance in Hybrid Semiconductor–Superconductor Devices Ming-Li Liu (刘明黎)1,2†, Dong Pan (潘东)3†, Tian Le (乐天)1, Jiang-Bo He (贺江波)1, Zhong-Mou Jia (贾仲谋)1,2, Shang Zhu (朱尚)1,2, Guang Yang (杨光)1, Zhao-Zheng Lyu (吕昭征)1, Guang-Tong Liu (刘广同)1,4, Jie Shen (沈洁)1,4, Jian-Hua Zhao (赵建华)3*, Li Lu (吕力)1,2,4*, and Fan-Ming Qu (屈凡明)1,2,4* Affiliations 1Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China 2School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China 3State Key Laboratory of Superlattices and Microstructures, Institute of Semiconductors, Chinese Academy of Sciences, Beijing 100083, China 4Songshan Lake Materials Laboratory, Dongguan 523808, China Received 4 April 2023; accepted manuscript online 4 May 2023; published online 19 May 2023 †These authors contributed equally to this work.
*Corresponding authors. Email: jhzhao@semi.ac.cn; lilu@iphy.ac.cn; fanmingqu@iphy.ac.cn Citation Text: Liu M L, Pan D, Le T et al. 2023 Chin. Phys. Lett. 40 067301    Abstract Negative differential conductance (NDC) serves as a crucial characteristic that reveals various underlying physics and transport process in hybrid superconducting devices. We report the observation of gate-tunable NDC outside the superconducting energy gap on two types of hybrid semiconductor–superconductor devices, i.e., normal metal–superconducting nanowire–normal metal and normal metal–superconducting nanowire–superconductor devices. Specifically, we study the dependence of the NDCs on back-gate voltage and magnetic field. When the back-gate voltage decreases, these NDCs weaken and evolve into positive differential conductance dips; and meanwhile they move away from the superconducting gap towards high bias voltage, and disappear eventually. In addition, with the increase of magnetic field, the NDCs/dips follow the evolution of the superconducting gap, and disappear when the gap closes. We interpret these observations and reach a good agreement by combining the Blonder–Tinkham–Klapwijk (BTK) model and the critical supercurrent effect in the nanowire, which we call the BTK-supercurrent model. Our results provide an in-depth understanding of the tunneling transport in hybrid semiconductor–superconductor devices.
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 DOI:10.1088/0256-307X/40/6/067301 © 2023 Chinese Physics Society Article Text The research on hybrid semiconductor–superconductor devices based on nanowires with strong spin-orbit coupling (SOC) has been particularly attractive in recent years, due to the underlying exotic physics and potential applications.[1-20] When the system is configured to present topological superconductivity, theories predict the existence of Majorana bound states (MBSs) which could be the building blocks of topological quantum computation.[21-24] Experimentally, measurement of the differential conductance using a tunneling probe or a point contact provides plentiful physical information of the system, such as the superconducting energy gap, Andreev bound states, the strength of the barrier, and the zero-bias conductance peak that may be characteristics of MBSs.[25,26] In addition to the ordinary positive differential conductance, there also exists negative differential conductance (NDC) (or a positive differential conductance dip) in the tunneling spectroscopy or point-contact spectroscopy, reflecting rich physical properties of the device including the superconducting gap, the sub-gap states, the coupling strength, the relaxation of quasiparticles, and the contact conditions, etc. In the exploration of hybrid superconducting devices, NDC in the tunneling spectroscopy has been investigated in various device geometries, which can be roughly classified into the following three categories: (1) Superconductor-quantum dot-superconductor (S-QD-S) devices.[2,9,27-34] NDC appears in the coulomb diamond with an odd number of electrons, which is caused by the extremely asymmetric coupling strength of the two superconducting electrodes. (2) Majorana island devices.[12,35-40] When the quasiparticle takes a long time to relax to the bound state, the differential conductance $dI/dV$ will be negative (where $I$ is the current, and $V$ is the voltage), i.e., NDC appears. (3) Superconducting tunnel junction with superconductors of different energy gaps (S$_{1}$–I/N–S$_{2}$, where I/N is the insulator/normal metal that functions as the tunneling barrier).[41-43] When the bias voltage increases to be $|\varDelta_{1}-\varDelta_{2}|/e < V < |\varDelta_{1}+\varDelta_{2}|/e$ (where $\varDelta_{1}$ and $\varDelta_{2}$ are the superconducting gaps of S$_{1}$ and S$_{2}$, respectively, and $e$ is the elementary charge), the NDC could be observed due to a sharp decrease in the density of states. On the other hand, for the measurement of point-contact spectroscopy, the positive differential conductance dip has been extensively reported[44-50] and can be generally attributed to the critical supercurrent effect due to the Maxwell resistance in the non-ballistic regime, as reported by Sheet et al.[45] Moreover, Shan et al.[44] explained the dip using the combination of superconducting proximity in the normal metal and Josephson tunnelling in the polycrystalline superconductor. However, NDCs outside the superconducting gap and a gate-tunable evolution from NDC to a differential conductance dip have been elusive. In this Letter, we report the observation of such NDCs and their gate tuning in tunneling spectroscopy of two types of hybrid devices, i.e., the normal metal–superconducting nanowire–normal metal (N-SNW-N) and normal metal–superconducting nanowire–superconductor (N-SNW-S) devices. These NDCs have several distinct characteristics. First, the amplitude of the NDCs is tunable by gate. The NDC is deep at the high-conductance regime (large gate voltage), and weakens to be a dip (positive value) at the low-conductance regime (small gate voltage). Second, the NDC/dip appears at a bias voltage outside the superconducting gap, and goes away from the gap towards large bias voltage with the decrease of conductance (gate voltage). Third, with the increase of magnetic field, the NDC/dip gradually approaches to zero bias voltage and disappears when the superconducting gap closes. We provide a phenomenological interpretation and achieve a good agreement with the observations, by considering the Blonder–Tinkham–Klapwijk (BTK) model[51] combined with the critical supercurrent in the superconducting nanowire, which is hereinafter referred to as the BTK-supercurrent model. The superconducting nanowires (SNWs) in our hybrid N-SNW-N and N-SNW-S devices are InAsSb nanowires with an epitaxial superconductor Al layer ($\sim$ $12$ nm), grown in situ on two or three facets of the hexagonal InAsSb core by molecular beam epitaxy method.[33,52] The ternary compound InAsSb is predicted to possess a stronger SOC[53] than its binary compounds InAs and InSb, so that it could potentially provide a more promising platform for research of MBSs, which is our original intention in selecting this material. The fabrication details of the superconducting (S) and normal (N) electrodes can be found in the Supplementary Material. The junction segment of device A [Fig. 1(a)] is $\sim$ $10$ nm on the left and $\sim$ $14$ nm on the right, and that of device B is $\sim$ $30$ nm (only one junction). The doped silicon substrate with 300 nm-thick SiO$_{2}$ is used as the global back gate, and the Ti/Au near the junction segment could be used as the side gate but are not functional in this work. The differential conductance $dI/dV$ is measured in a dilution refrigerator with a base temperature of $\sim$ $10$ mK by using a standard lock-in-amplifier technique. We first show the results of device A, which is of N-SNW-N-type as shown in Fig. 1(a). Three normal metal electrodes (N) are marked as I, II, and III, respectively. Figure 1(b) shows the differential conductance $dI/dV$ as a function of bias voltage $V$ and back-gate voltage $V_{\rm bg}$ measured by applying a voltage (voltage driven) between electrodes III$\to$I at zero magnetic field. The two horizontal peaks around $V=0$ represent the superconducting gap, which is harder at larger barrier strength (lower conductance; see the Supplementary Material). Next, we focus on another obvious feature in Fig. 1(b), that is, the two blue stripes outside the superconducting gap, which changes its color from dark blue to light blue and goes to higher bias voltage $\vert V\vert$ with decreasing the back-gate voltage from $V_{\rm bg}=2$ V to $V_{\rm bg}=-2$ V. After that, these stripes move faster towards higher $\vert V\vert$ as marked by the yellow arrows. Three typical linecuts as indicated by the black, red and blue bars are plotted in Fig. 1(c). At $V_{\rm bg}=2$ V (black curve), NDCs outside the superconducting gap can be clearly recognized. For $V_{\rm bg}=1$ V (red curve) and $V_{\rm bg}=-1$ V (blue curve), the NDCs weaken to be positive differential conductance dips and move towards higher $\vert V\vert$.
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Fig. 1. NDCs in device A between electrodes III$\to$I. (a) Schematic diagram of device A with three normal metal electrodes (green) marked as I, II, III, respectively. The nanowire is InAsSb (blue) with epitaxial superconductor Al (yellow). (b) The differential conductance $dI/dV$ as a function of bias voltage $V$ and back-gate voltage $V_{\rm bg}$ for electrodes III$\to$I. Electrode III is the source, electrode I is the drain, and electrode II is floating. (c) Linecuts taken from (b). (d) The differential resistance $dV/dI$ as a function of $I$ and $V_{\rm bg}$ for electrodes III$\to$I. The two yellow arrows indicate the differential resistance peaks. (e) Integrated linecuts from (d). The black, red and blue curves in (c) and (d) correspond to $V_{\rm bg} = 2$, $1$, $-$1 V, respectively. A line resistance 3 k$\Omega$ was subtracted from the experimental data to obtain the real resistance of the device.
In order to compare with the differential conductance spectroscopy [Fig. 1(b)] and understand the NDCs/dips, we measured the differential resistance $dV/dI$ spectroscopy for the same electrodes III$\to$I in the current-driven mode, as shown in Fig. 1(d). The NDCs/dips in Fig. 1(b) manifest as a pair of differential resistance peaks in Fig. 1(d), as marked by the two yellow arrows, whose occurrence in $I$ could be attributed to the critical supercurrent of the proximity-induced SNW (as shown later in our BTK-supercurrent model). The three typical linecuts as indicated by the black, red and blue bars are integrated to obtain the $V$–$I$ curves, as displayed in Fig. 1(e). The critical supercurrent is roughly estimated to be 120 nA (as indicated by the black arrow) at $V_{\rm bg}=2$ V. Note that the NDCs/dips in Fig. 1(b) and the differential resistance peaks in Fig. 1(d) are both tunable by back-gate voltage [the transform from $dI/dV$ to $dV/dI$ of Fig. 1(b) is almost the same as Fig. 1(d); see the Supplementary Material], and we initially speculate that the occurrence of NDCs/dips is related to the supercurrent (as discussed later). It is interesting to note that the differential conductance $dI/dV$ spectroscopy for electrodes III$\to$II has no obvious NDCs, but only small positive dips (not reaching negative values), as shown in Figs. 2(a) and 2(b). However, the evolution of the dips when sweeping $V_{\rm bg}$ is almost the same as that for electrodes III$\to$I [Fig. 1(b)]. In particular, when a perpendicular magnetic field $B_{Z}$ is applied, the differential conductance dips and correspondingly the differential resistance peaks disappear as the superconducting gap closes, which indicates once again that these features are related to superconductivity, as shown in Figs. 2(c) and 2(d). See the Supplementary Material for the results of electrodes I$\to$II and additional details of device A.
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Fig. 2. NDCs/dips in device A for electrodes III$\to$II. (a) $dI/dV$ as a function of $V$ and $V_{\rm bg}$ for electrodes III$\to$II (electrode I is floating). (b) Linecuts taken from (a). [(c), (d)] Two-dimensional differential conductance $dI/dV$ and differential resistance $dV/dI$ maps, respectively, showing the evolution of the dips and peaks in magnetic field at $V_{\rm bg}=2$ V.
We next present the results on device B, which is of the N-SNW-S type. Figure 3(a) shows a false-colored scanning electron microscope image of device B, and Fig. 3(b) is the zoom-in of the red box area in Fig. 3(a), showing that the junction segment is $\sim$ $30$ nm. Figure 3(c) shows the differential conductance at $V_{\rm bg}=20$ V and 0 V, and NDC appears at $V\approx \pm 0.36$ mV and $\pm 0.6$ mV, respectively. As the same as in Figs. 1(b) and 2(a), the NDCs in Fig. 3(e) weaken to be positive dips and go away from the superconducting gap with decreasing $V_{\rm bg}$, as shown by the blue stripes (marked by the two yellow arrows). In the pinch-off regime ($V_{\rm bg} < -8$ V), the dips disappear (go towards very large $\vert V\vert$), and a hard superconducting gap appears, whose size $\varDelta \sim 0.3$ meV. See the Supplementary Material for additional details of device B. Gate-tunable NDCs/dips outside the superconducting gap can be observed in both two-terminal and three-terminal devices, indicating that a fundamental mechanism may originate from the N-SNW junction. We model the N-SNW junction in both devices as a series of the tunneling resistance $R_{\scriptscriptstyle{\rm BTK}}$ and the SNW resistance $R_{\scriptscriptstyle{\rm SC}}$, as shown in Fig. 4(a). By combining the BTK model and the critical supercurrent effect (i.e., the BTK-supercurrent model), the behavior of NDCs/dips can be well understood.
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Fig. 3. NDCs in device B. (a) False-colored scanning electron microscope image of device B, where the normal metal (green) electrode is marked as N, and the Al electrode (purple) is marked as S. (b) Red box in (a). The nanowire is InAsSb (blue) with epitaxial superconductor Al (yellow). (c) $dI/dV$ as a function of $V$ at $V_{\rm bg}=20$ V and 0 V for electrodes N$\to$S. (d) $dI/dV$ shows the superconducting gap at $V_{\rm bg}=-10$ V. (e) Two-dimensional $dI/dV$ map as a function of $V$ and $V_{\rm bg}$. Two yellow arrows point to the blue stripes that indicate NDCs/dips.
The tunneling resistance $R_{\scriptscriptstyle{\rm BTK}}$ is described by the BTK model, and the current could be calculated as \begin{align} I=\,&2N(0)e\nu_{\scriptscriptstyle{\rm F}}S\int\limits_{-\infty }^{+\infty } {[f_{0}(E-eV)-f_{0}(E)]}\notag\\ &\cdot [1+A(E)-B(E)]dE,\notag \end{align} where $N(0)$ is the density of states at the Fermi level, $S$ is the effective area of the NS interface, $f_{0}$ is Fermi–Dirac distribution function, $A(E)$ is the probability of Andreev reflection, and $B(E)$ is the probability of normal electron reflection. A one-dimensional potential $V_{\rm NS}=V_{0}\delta (x)$ is used to model the NS interface. The barrier strength $Z$ of the NS interface is described by $Z=V_{0}/\hslash \nu_{\scriptscriptstyle{\rm F}}$. We introduce $\varGamma$ to account for the strength of inelastic scattering at the interface, $\varGamma =\hslash /\tau$, where $\tau$ is the lifetime of the quasiparticles. The SNW resistance $R_{\scriptscriptstyle{\rm SC}}$ has typical characteristics of the induced superconducting nanowire, i.e., when the SNW is superconducting, $R_{\scriptscriptstyle{\rm SC}}=0$; when the current $I$ is greater than the critical supercurrent $I_{\rm c}$, $R_{\scriptscriptstyle{\rm SC}}=R_{\scriptscriptstyle{\rm SC}}^{\scriptscriptstyle{\rm N}}$ (the normal-state resistance of the SNW).
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Fig. 4. Model and the simulation results of NDCs/dips. (a) Model of an N-SNW junction, which consists of the tunneling resistance $R_{\scriptscriptstyle{\rm BTK}}$ between N and SNW, and the SNW resistance $R_{\scriptscriptstyle{\rm SC}}$. (b) $dI/dV$ as a function of $V$ at $V_{\rm bg}=2$ V for device A (black), the same as the black curve in Fig. 1(c). The red curve is the simulation result using the BTK theory. The green curve is the simulation result using the BTK-supercurrent model. [(c), (d)] $I$–$V$ and $V$–$I$ curves (black) of device A for electrodes III$\to$I, obtained by integrating the experimental $dI/dV$ vs $V$ and $dV/dI$ vs $I$ curve at $V_{\rm bg}=2$ V, respectively. The red lines are the simulation result using the BTK theory. The blue lines are the result of subtracting the red line by the black line using the current $I$ as the independent variable, respectively. (e) Simulation results for different $R_{\scriptscriptstyle{\rm BTK}}^{\scriptscriptstyle{\rm N}}$. Parameters: $\varGamma =0.03$ meV, $\varDelta =0.46$ meV, $T=0.03$ K, $Z=0.4$, $I_{\rm c}=120$ nA, $R_{\scriptscriptstyle{\rm SC}}^{\scriptscriptstyle{\rm N}}=0.2\,h/e^{2}$, $\gamma_{\rm c}=0.01$ nA. (f) Simulation results for different $R_{\scriptscriptstyle{\rm SC}}^{\scriptscriptstyle{\rm N}}$. Parameters: $\varGamma =0.03$ meV, $\varDelta =0.46$ meV, $T=0.03$ K, $Z=0.4$, $R_{\scriptscriptstyle{\rm BTK}}^{\scriptscriptstyle{\rm N}}=1.2\,h/e^{2}$, $I_{\rm c}=120$ nA, $\gamma_{\rm c}=0.01$ nA. (g) Simulation result obtained by assuming that parameters $Z$, $R_{\scriptscriptstyle{\rm BTK}}^{\scriptscriptstyle{\rm N}}$, $I_{\rm c}$, and $\gamma_{\rm c}$ evolve with $V_{\rm bg}$ in certain function forms (see the Supplementary Material).
The black curve in Fig. 4(b) shows the $dI/dV$ curve of device A for electrodes III$\to$I at $V_{\rm bg}=2$ V, which is the same as the black curve in Fig. 1(c). First of all, we use the BTK model to simulate the central segment (the part between the two NDC structures) of the black curve, and plot out the red line using the simulated parameters, as shown in Fig. 4(b). The parameters are the barrier strength $Z=0.4$, superconducting gap $\varDelta =0.46$ meV, temperature $T=0.03$ K, the normal-state resistance $R_{\scriptscriptstyle{\rm BTK}}^{\scriptscriptstyle{\rm N}}=0.96h/e^{2}$ ($R_{\scriptscriptstyle{\rm BTK}}$ at high-bias voltage), and $\varGamma =0.03$ meV. Although the simulation could capture the structure inside the energy gap well, the NDCs cannot be simulated. Therefore, it is necessary to consider the critical supercurrent effect and the role of $R_{\scriptscriptstyle{\rm SC}}$ based on BTK theory, i.e., the BTK-supercurrent model (see the Supplementary Material). The $I$–$V$ curve incorporating $R_{\scriptscriptstyle{\rm SC}}$ can be expressed as $V_{\scriptscriptstyle{\rm SC}}=R_{\scriptscriptstyle{\rm SC}}^{\scriptscriptstyle{\rm N}} [(I+i\gamma_{\rm c})^{2}-I_{\rm c}^{2}]^{1/2}$, where $\gamma_{\rm c}$ is used to adjust the broadening near the critical supercurrent resulting from finite temperature and disorder. In order to simulate the NDCs, we consider the voltage-driven measurement (sweeping voltage $V$). The current can be expressed as $I=V/(R_{\scriptscriptstyle{\rm BTK}}+R_{\scriptscriptstyle{\rm SC}})$. Here, $dI/dV$ can be extracted after differentiation, and the contribution of $R_{\scriptscriptstyle{\rm SC}}$ is a key to the NDCs. Then, we simulate the measured black curve in Fig. 4(b) using the BTK-supercurrent model and obtain the green curve, which shows a good agreement with the experimental result (see Table 2 in the Supplementary Material for parameters). To further verify the applicability of our BTK-supercurrent model, we extract the critical supercurrent from both the voltage-driven (sweeping voltage) and current-driven (sweeping current) measurements, as shown in Figs. 4(c) and 4(d), respectively. The black $I$–$V$ and $V$–$I$ curves in Figs. 4(c) and 4(d) are the integral of the measured $dI/dV$ vs $V$ and $dV/dI$ vs $I$ curve at $V_{\rm bg}=2$ V, and the red curves are the integral of the simulated red curve in Fig. 4(b). The $I$–$V$ and $V$–$I$ curves of the induced SNW can be obtained by subtracting the red curve from the black curve using $I$ as the independent variable, as shown by the blue curves in Figs. 4(c) and 4(d). As indicated by the blue arrows, the critical supercurrent is $I_{\rm c} \sim 120$ nA. As expected, the $dI/dV$ spectroscopy and $dV/dI$ spectroscopy give the same $I_{\rm c}$. Remarkably, a sudden current drop as marked by the black arrow in Fig. 4(c) can be clearly recognized. Specifically, in the voltage-driven measurement, when the current is larger than $I_{\rm c}$, $R_{\scriptscriptstyle{\rm SC}}$ is not zero but jumps to $R_{\scriptscriptstyle{\rm SC}}^{\scriptscriptstyle{\rm N}}$, so the current decreases, resulting in NDC. In the contrary, for the current-driven measurement, i.e., the $dV/dI$ spectroscopy, even if $I$ increases to be larger than $I_{\rm c}$, the increase of total resistance only enhances the voltage without NDCs, as can be clearly seen from the black $V$–$I$ curve in Fig. 4(d). The three experimental characteristics can be well explained by the BTK-supercurrent model. (1) Why do the NDCs weaken with the decrease of $V_{\rm bg}$? This is because the amplitude (depth) of the NDC depends on $R_{\scriptscriptstyle{\rm BTK}}^{\scriptscriptstyle{\rm N}}$ and $R_{\scriptscriptstyle{\rm SC}}^{\scriptscriptstyle{\rm N}}$, which are the main parameters that are modified by $V_{\rm bg}$. By fixing other parameters and adjusting $R_{\scriptscriptstyle{\rm BTK}}^{\scriptscriptstyle{\rm N}}$ or $R_{\scriptscriptstyle{\rm SC}}^{\scriptscriptstyle{\rm N}}$, respectively, we can simulate the $dI/dV$ spectroscopy, as shown in Figs. 4(e) and 4(f). When $R_{\scriptscriptstyle{\rm BTK}}^{\scriptscriptstyle{\rm N}}$ decreases or $R_{\scriptscriptstyle{\rm SC}}^{\scriptscriptstyle{\rm N}}$ increases, the NDCs become deeper. When $V_{\rm bg}$ decreases, $R_{\scriptscriptstyle{\rm SC}}^{\scriptscriptstyle{\rm N}}$ hardly changes due to the coverage of Al, but $R_{\scriptscriptstyle{\rm BTK}}^{\scriptscriptstyle{\rm N}}$ increases. Thus, the contribution of $R_{\scriptscriptstyle{\rm SC}}$ to the current reduction is less obvious and the NDC weakens. (2) Why do the NDCs/dips gradually move to the high bias voltage with the decrease of $V_{\rm bg}$? When $R_{\scriptscriptstyle{\rm BTK}}^{\scriptscriptstyle{\rm N}}$ increases, a higher voltage is required to reach the current that surpasses $I_{\rm c}$. Figure 4(e) illustrates such behavior clearly. (3) Why do the NDCs/dips follow the evolution of the superconducting gap in a magnetic field? The NDCs/dips originate from the critical supercurrent effect. When the magnetic field increases, the gap and $I_{\rm c}$ decrease, and eventually NDCs disappear as $I_{\rm c}$ reaches zero. Regarding the gate tuning, by assuming that parameters $Z$, $R_{\scriptscriptstyle{\rm BTK}}^{\scriptscriptstyle{\rm N}}$, $I_{\rm c}$, and $\gamma_{\rm c}$ evolve with $V_{\rm bg}$ in certain function forms (see the Supplementary Material), we obtain the $V_{\rm bg}$-dependent $dI/dV$ spectroscopy, as shown in Fig. 4(g), which is in good agreement with the characteristics of NDCs/dips in our experimental data [Figs. 1(b) and 3(e)]. One more thing to note is that the NDCs/dips for electrodes III$\to$I are deeper than those for III$\to$II in the same range of $V_{\rm bg}$ [see Figs. 1(b) and 2(a)]. By comparing $R_{\rm N}$ (the measured resistance at high bias voltage) at the same $V_{\rm bg}$, for example, $R_{\rm N}\approx 1.16\,h/e^{2}$ (III$\to$I) and $R_{\rm N}\approx 0.62\,h/e^{2}$ (III$\to$II) at $V_{\rm bg}=2$ V, and according to the correspondence between a larger $R_{\scriptscriptstyle{\rm SC}}^{\scriptscriptstyle{\rm N}}$ and a deeper NDC/dip as shown in Fig. 4(f), the reason could be that $R_{\scriptscriptstyle{\rm SC}}^{\scriptscriptstyle{\rm N}}$ is larger for electrodes III$\to$I due to the longer segment of the nanowire. Note that SOC is not considered in our model, indicating that it does not play a decisive role in the evolution of the NDC. In conclusion, we have studied NDCs/dips outside the superconducting gap in N-SNW-N and N-SNW-S devices. The observed distinct characteristics as a function of gate voltage and magnetic field could be interpreted by combining the BTK model and the critical supercurrent effect, i.e., the BTK-supercurrent model. Our results provide an in-depth understanding and serve as a foundation for further research on the tunnelling spectroscopy of hybrid semiconductor–superconductor devices. Such a stable and repeatable NDC may offer some practical applications, such as low-temperature oscillator[54] and amplifier.[55] Acknowledgments. Supported by the National Key Research and Development Program of China (Grant Nos. 2022YFA1403400 and 2017YFA0304700), the National Natural Science Foundation of China (Grant Nos. 12074417, 92065203, 92065106, 61974138, 11774405, 11527806, and 12104489), the Strategic Priority Research Program B of Chinese Academy of Sciences (Grant Nos. XDB28000000 and XDB33000000), the Synergetic Extreme Condition User Facility sponsored by the National Development and Reform Commission, and the Innovation Program for Quantum Science and Technology (2021ZD0302600). D.P. also acknowledges the support from Youth Innovation Promotion Association, Chinese Academy of Sciences (Grant Nos. 2017156 and Y2021043). References Tunable Supercurrent Through Semiconductor NanowiresHybrid superconductor–quantum dot devicesAnomalous Zero-Bias Conductance Peak in a Nb–InSb Nanowire–Nb Hybrid DeviceSignatures of Majorana Fermions in Hybrid Superconductor-Semiconductor Nanowire DevicesAnomalous Modulation of a Zero-Bias Peak in a Hybrid Nanowire-Superconductor DeviceParity independence of the zero-bias conductance peak in a nanowire based topological superconductor-quantum dot hybrid deviceSpin-resolved Andreev levels and parity crossings in hybrid superconductor–semiconductor nanostructuresMajorana bound state in a coupled quantum-dot hybrid-nanowire system 0 − π phase transition in hybrid superconductor–InSb nanowire quantum dot devicesAndreev bound states versus Majorana bound states in quantum dot-nanowire-superconductor hybrid structures: Trivial versus topological zero-bias conductance peaksNonlocality of Majorana modes in hybrid nanowiresEngineering hybrid epitaxial InAsSb/Al nanowires for stronger topological protectionMagnetic-Field-Independent Subgap States in Hybrid Rashba NanowiresRelating Andreev Bound States and Supercurrents in Hybrid Josephson JunctionsFrom Andreev to Majorana bound states in hybrid superconductor–semiconductor nanowiresNontopological zero-bias peaks in full-shell nanowires induced by flux-tunable Andreev statesKondo effect in a hybrid superconductor–quantum-dot–superconductor junction with proximity-induced p -wave pairing statesClosing of the induced gap in a hybrid superconductor-semiconductor nanowireInAs-Al Hybrid Devices Passing the Topological Gap ProtocolMicrowave spectroscopy of Andreev states in InAs nanowire-based hybrid junctions using a flip-chip layoutUnpaired Majorana fermions in quantum wiresMajorana Fermions and a Topological Phase Transition in Semiconductor-Superconductor HeterostructuresHelical Liquids and Majorana Bound States in Quantum WiresGeneric New Platform for Topological Quantum Computation Using Semiconductor HeterostructuresNon-Abelian quantum order in spin-orbit-coupled semiconductors: Search for topological Majorana particles in solid-state systemsMajorana zero modes in superconductor–semiconductor heterostructuresEven-Odd Effect in Andreev Transport through a Carbon Nanotube Quantum DotSuperconductivity-enhanced bias spectroscopy in carbon nanotube quantum dotsAndreev bound states in supercurrent-carrying carbon nanotubes revealedNonequilibrium Transport through a Spinful Quantum Dot with Superconducting LeadsZero-Bias Anomaly in a Nanowire Quantum Dot Coupled to SuperconductorsTransport Measurement of Andreev Bound States in a Kondo-Correlated Quantum DotNonequilibrium interplay between Andreev bound states and Kondo effectMirage Andreev Spectra Generated by Mesoscopic Leads in Nanowire Quantum DotsParity lifetime of bound states in a proximitized semiconductor nanowireExponential protection of zero modes in Majorana islandsParity transitions in the superconducting ground state of hybrid InSb–Al Coulomb islandsCoherent transport through a Majorana island in an Aharonov–Bohm interferometerIn Situ Epitaxy of Pure Phase Ultra-Thin InAs-Al Nanowires for Quantum DevicesMajorana-like Coulomb spectroscopy in the absence of zero-bias peaksDirect Measurement of the Superconducting Energy GapElectron Tunneling Between Two SuperconductorsDevice Applications of Superconductive Tunneling s -wave pairing in MgCNi 3 revealed by point contact tunnelingRole of critical current on the point-contact Andreev reflection spectra between a normal metal and a superconductorPoint contact spectroscopy of superconductors via nanometer scale point contacts formed by resistive switchingDistinction between critical current effects and intrinsic anomalies in the point-contact Andreev reflection spectra of unconventional superconductorsTransport and Point Contact Measurements on Pr1−xCexPt4Ge12 Superconducting PolycrystalsTip-induced superconductivity on the topological semimetals TaAs 2 and NbAs 2 Nonballistic transport characteristics of superconducting point contactsTransition from metallic to tunneling regimes in superconducting microconstrictions: Excess current, charge imbalance, and supercurrent conversionQuantifying quantum coherence of multiple-charge states in tunable Josephson junctionsTopological Phases in InAs 1 − x Sb x : From Novel Topological Semimetal to Majorana Wire
[1] Doh Y J, van Dam J A, Roest A L, Bakkers E P, Kouwenhoven L P, and De Franceschi S 2005 Science 309 272
[2] De Franceschi S, Kouwenhoven L, Schonenberger C, and Wernsdorfer W 2010 Nat. Nanotechnol. 5 703
[3] Deng M T, Yu C L, Huang G Y, Larsson M, Caroff P, and Xu H Q 2012 Nano Lett. 12 6414
[4] Mourik V, Zuo K, Frolov S M, Plissard S R, Bakkers E P, and Kouwenhoven L P 2012 Science 336 1003
[5] Finck A D K, Van Harlingen D J, Mohseni P K, Jung K, and Li X 2013 Phys. Rev. Lett. 110 126406
[6] Deng M T, Yu C L, Huang G Y, Larsson M, Caroff P, and Xu H Q 2014 Sci. Rep. 4 7261
[7] Lee E J H, Jiang X C, Houzet M, Aguado R, Lieber C M, and De Franceschi S 2014 Nat. Nanotechnol. 9 79
[8] Deng M T, Vaitiekenas S, Hansen E B, Danon J, Leijnse M, Flensberg K, Nygard J, Krogstrup P, and Marcus C M 2016 Science 354 1557
[9] Li S, Kang N, Caroff P, and Xu H Q 2017 Phys. Rev. B 95 014515
[10] Liu C X, Sau J D, Stanescu T D, and Das S S 2017 Phys. Rev. B 96 075161
[11] Deng M T, Vaitiekėnas S, Prada E, San-Jose P, Nygård J, Krogstrup P, Aguado R, and Marcus C M 2018 Phys. Rev. B 98 085125
[12] Sestoft J E, Kanne T, Gejl A N, von Soosten M, Yodh J S, Sherman D, Tarasinski B, Wimmer M, Johnson E, Deng M, Nygård J, Jespersen T S, Marcus C M, and Krogstrup P 2018 Phys. Rev. Mater. 2 044202
[13] Jünger C, Delagrange R, Chevallier D, Lehmann S, Dick K A, Thelander C, Klinovaja J, Loss D, Baumgartner A, and Schönenberger C 2020 Phys. Rev. Lett. 125 017701
[14] Nichele F, Portoles E, Fornieri A, Whiticar A M, Drachmann A C C, Gronin S, Wang T, Gardner G C, Thomas C, Hatke A T, Manfra M J, and Marcus C M 2020 Phys. Rev. Lett. 124 226801
[15] Prada E, San-Jose P, de Moor M W A, Geresdi A, Lee E J H, Klinovaja J, Loss D, Nygård J, Aguado R, and Kouwenhoven L P 2020 Nat. Rev. Phys. 2 575
[16] Valentini M, Penaranda F, Hofmann A, Brauns M, Hauschild R, Krogstrup P, San-Jose P, Prada E, Aguado R, and Katsaros G 2021 Science 373 82
[17] Li L, Sun J H, Su W, Wang Z H, Xu D H, Luo H G, and Chen W Q 2021 Phys. Rev. B 103 125144
[18] Puglia D, Martinez E A, Ménard G C, Pöschl A, Gronin S, Gardner G C, Kallaher R, Manfra M J, Marcus C M, Higginbotham A P, and Casparis L 2021 Phys. Rev. B 103 235201
[19] Aghaee M, Akkala A, Alam Z et al. 2022 arXiv:2207.02472 [cond-mat.mes-hall]
[20] Zellekens P, Deacon R, Perla P, Grutzmacher D, Lepsa M I, Schapers T, and Ishibashi K 2022 Commun. Phys. 5 267
[21] Kitaev A Y 2001 Phys.-Usp. 44 131
[22] Lutchyn R M, Sau J D, and Das S S 2010 Phys. Rev. Lett. 105 077001
[23] Oreg Y, Refael G, and von Oppen F 2010 Phys. Rev. Lett. 105 177002
[24] Sau J D, Lutchyn R M, Tewari S, and Das S S 2010 Phys. Rev. Lett. 104 040502
[25] Sau J D, Tewari S, Lutchyn R M, Stanescu T D, and Das S S 2010 Phys. Rev. B 82 214509
[26] Lutchyn R M, Bakkers E P A M, Kouwenhoven L P, Krogstrup P, Marcus C M, and Oreg Y 2018 Nat. Rev. Mater. 3 52
[27] Eichler A, Weiss M, Oberholzer S, Schonenberger C, Levy Y A, Cuevas J C, and Martin-Rodero A 2007 Phys. Rev. Lett. 99 126602
[28] Grove-Rasmussen K, Jorgensen H I, Andersen B M, Paaske J, Jespersen T S, Nygard J, Flensberg K, and Lindelof P E 2009 Phys. Rev. B 79 134518
[29] Pillet J D, Quay C H L, Morfin P, Bena C, Yeyati A L, and Joyez P 2010 Nat. Phys. 6 965
[30] Andersen B M, Flensberg K, Koerting V, and Paaske J 2011 Phys. Rev. Lett. 107 256802
[31] Lee E J, Jiang X, Aguado R, Katsaros G, Lieber C M, and De Franceschi S 2012 Phys. Rev. Lett. 109 186802
[32] Kim B K, Ahn Y H, Kim J J, Choi M S, Bae M H, Kang K, Lim J S, Lopez R, and Kim N 2013 Phys. Rev. Lett. 110 076803
[33] He J B, Pan D, Yang G, Liu M L, Ying J H, Lyu Z, Fan J, Jing X N, Liu G T, Lu B, Liu D E, Zhao J H, Lu L, and Qu F M 2020 Phys. Rev. B 102 075121
[34] Su Z, Zarassi A, Hsu J F, San-Jose P, Prada E, Aguado R, Lee E J H, Gazibegovic S, Op H V R L M, Car D, Plissard S R, Hocevar M, Pendharkar M, Lee J S, Logan J A, Palmstrom C J, Bakkers E, and Frolov S M 2018 Phys. Rev. Lett. 121 127705
[35] Higginbotham A P, Albrecht S M, Kiršanskas G, Chang W, Kuemmeth F, Krogstrup P, Jespersen T S, Nygård J, Flensberg K, and Marcus C M 2015 Nat. Phys. 11 1017
[36] Albrecht S M, Higginbotham A P, Madsen M, Kuemmeth F, Jespersen T S, Nygard J, Krogstrup P, and Marcus C M 2016 Nature 531 206
[37] Shen J, Heedt S, Borsoi F, van Heck B, Gazibegovic S, Op H V R L M, Car D, Logan J A, Pendharkar M, Ramakers S J J, Wang G, Xu D, Bouman D, Geresdi A, Palmstrom C J, Bakkers E, and Kouwenhoven L P 2018 Nat. Commun. 9 4801
[38] Whiticar A M, Fornieri A, O'Farrell E C T, Drachmann A C C, Wang T, Thomas C, Gronin S, Kallaher R, Gardner G C, Manfra M J, Marcus C M, and Nichele F 2020 Nat. Commun. 11 3212
[39] Pan D, Song H D, Zhang S, Liu L, Wen L J, Liao D Y, Zhuo R, Wang Z C, Zhang Z T, Yang S, Ying J H, Miao W T, Shang R N, Zhang H, and Zhao J H 2022 Chin. Phys. Lett. 39 058101
[40] Valentini M, Borovkov M, Prada E, Marti-Sanchez S, Botifoll M, Hofmann A, Arbiol J, Aguado R, San-Jose P, and Katsaros G 2022 Nature 612 442
[41] Nicol J, Shapiro S, and Smith P H 1960 Phys. Rev. Lett. 5 461
[42] Giaever I 1960 Phys. Rev. Lett. 5 464
[43] Taylor B N 1968 J. Appl. Phys. 39 2490
[44] Shan L, Tao H J, Gao H, Li Z Z, Ren Z A, Che G C, and Wen H H 2003 Phys. Rev. B 68 144510
[45] Sheet G, Mukhopadhyay S, and Raychaudhuri P 2004 Phys. Rev. B 69 134507
[46] Dvoranová M, Plecenik T, Moško M, Vidiš M, Gregor M, Roch T, Grančič B, Satrapinskyy L, Kúš P, and Plecenik A 2018 AIP Adv. 8 125217
[47] He G, Wei Z X, Brisbois J, Jia Y L, Huang Y L, Zhou H X, Ni S L, Silhanek A V, Shan L, Zhu B Y, Yuan J, Dong X L, Zhou F, Zhao Z X, and Jin K 2018 Chin. Phys. B 27 047403
[48] Romano P, Avitabile F, Nigro A, Grimaldi G, Leo A, Shu L, Zhang J, Di Bartolomeo A, and Giubileo F 2020 Nanomaterials 10 1810
[49] Zhang M D, Hou X Y, Wang Q, Wang Y Y, Zhao L X, Wang Z, Gu Y D, Zhang F, Xia T L, Ren Z A, Chen G F, Hao N, and Shan L 2020 Phys. Rev. B 102 085139
[50] Kumar R and Sheet G 2021 Phys. Rev. B 104 094525
[51] Blonder G E, Tinkham M, and Klapwijk T M 1982 Phys. Rev. B 25 4515
[52] He J, Pan D, Liu M, Lyu Z, Jia Z, Yang G, Zhu S, Liu G, Shen J, Shevchenko S N, Nori F, Zhao J, Lu L, and Qu F 2023 arXiv:2303.02845 [cond-mat.mes-hall]
[53] Winkler G W, Wu Q, Troyer M, Krogstrup P, and Soluyanov A A 2016 Phys. Rev. Lett. 117 076403
[54]Whitaker J C 2018 The Electronics Handbook (New York: CRC Press)
[55]Iniewski K 2007 Wireless Technologies: Circuits, Systems, and Devices (New York: CRC Press)