Impact of junction length on supercurrent resilience against magnetic field in InSb-Al nanowire Josephson junctions

Semiconducting nanowire Josephson junctions represent an attractive platform to investigate the anomalous Josephson effect and detect topological superconductivity by studying Josephson supercurrent. However, an external magnetic field generally suppresses the supercurrent through hybrid nanowire junctions and significantly limits the field range in which the supercurrent phenomena can be studied. In this work, we investigate the impact of the length of InSb-Al nanowire Josephson junctions on the supercurrent resilience against magnetic fields. We find that the critical parallel field of the supercurrent can be considerably enhanced by reducing the junction length. Particularly, in 30 nm-long junctions supercurrent can persist up to 1.3 T parallel field - approaching the critical field of the superconducting film. Furthermore, we embed such short junctions into a superconducting loop and obtain the supercurrent interference at a parallel field of 1 T. Our findings are highly relevant for multiple experiments on hybrid nanowires requiring a magnetic field-resilient supercurrent.

field of the supercurrent can be considerably enhanced by reducing the junction length.
Particularly, in 30 nm-long junctions supercurrent can persist up to 1.3 T parallel fieldapproaching the critical field of the superconducting film. Furthermore, we embed such short junctions into a superconducting loop and obtain the supercurrent interference at a parallel field of 1 T. Our findings are highly relevant for multiple experiments on hybrid nanowires requiring a magnetic field-resilient supercurrent.
Semiconducting nanowire Josephson junctions (JJs) are widely used as a versatile platform for studying various physical phenomena that arise in semiconductor-superconductor hybrid systems. Therein, the III-V semiconductors have attracted a particular interest in exploring the anomalous Josephson effect 1-3 , topological superconductivity 4-8 and the Josephson diode effect 9-11 , due to their strong spin-orbit interaction and large g factor. In such experiments, an indispensible ingredient is the Zeeman energy introduced by an external magnetic field. However, an external magnetic field generally suppresses the supercurrent through a hybrid nanowire JJ -therefore significantly limiting the parameter space for addressing the aforementioned effects in hybrid nanowires. Preserving the supercurrent in hybrid nanowire JJs at high magnetic fields becomes thus critically important. Selecting high critical field superconductors, such as NbTiN 12 , Pb, 13 Sn 14 or Al doped by Pt, 15 seems to be an option for improving the magnetic field compatibility of the supercurrent. However, none of these material platforms have yielded a supercurrent at high magnetic fields.
Moreover, it has been observed that the supercurrent of nanowire JJs generally vanishes at magnetic fields far below the critical field of the superconducting film 16,17 . Searching for an alternative way to improve the supercurrent resilience against magnetic field in nanowire JJs is thus needed.
In this work, we have studied InSb-Al nanowire JJs with the junction length L varying from 27 nm to 160 nm. The junction length has been found to be an essential parameter that determines the supercurrent evolution in a parallel magnetic field, as well as the critical parallel field of the supercurrent. In the long devices (L ∼ 160 nm), the supercurrent is suppressed quickly in a magnetic field and fully vanishes at parallel fields of ∼ 0.7 T. In contrast, the supercurrent in short devices (L ∼ 30 nm) persists up to parallel fields of ∼ 1.3 T, approaching the critical in-plane magnetic field of the Al film (∼ 1. 5 T 15,16,18 ). The evolution of supercurrent in parallel magnetic field is strongly influenced by the electro-chemical potential in all junctions, however, the resilient supercurrent is present only in the short devices (L ∼ 30 nm). We exploit this property to realise a magnetic field-resilient superconducting quantum interference device (SQUID). At the magnetic field of 1 T parallel to the SQUID arms, the supercurrent through the device displays the characteristic oscillatory pattern as a function of the magnetic flux through the loop. We expect that our demonstration of magnetic field resilient supercurrent in remarkably short nanowire JJs offers a new approach to improving the field-compatibility of not only SQUIDs but many other hybrid nanowire devices utilizing the Josephson effect at high magnetic field.
The hybrid nanowire JJs are fabricated by the recently developed shadow-wall deposition techniques 16,18 . In Fig. 1a, a scanning electron microscope (SEM) image of a representative InSb-Al nanowire JJ device is taken at a tilted angle and shown with false colors. Source (S) and drain (D) superconducting leads (blue) are formed via an in-situ angle deposition of Al film after the preparation of a clean and oxide-free InSb nanowire 19 interface (see the Methods section in the Supporting Information). Pre-patterned dielectric shadow-walls (yellow) selectively define the nanowire sections that are exposed to the Al flux during the deposition. The junction length is determined by the width of the shadow-wall in the vicinity of which the nanowire is deposited. In comparison with the etched dielectric shadow-walls used in recent works 16,18 , here we use lithographically defined shadow-walls which dimensions therefore can be as small as 20 nm. This allows us to precisely control the length of nanowire JJs and to achieve surpassingly short junctions, as shown in the inset SEM image in Fig. 1a.
In this work, nine nanowire JJ devices are presented (Device 1-9) with the junction length L in the range of 27 nm − 160 nm. The diameter of the nanowires is ∼ 100 nm. An overview of these devices is shown in Fig. S2 in the Supporting Information. False-colored SEM image depicting a representative JJ device with a semiconducting InSb junction defined between the source (S) and drain (D) superconducting Al leads (blue). The junction length is determined by the hydrogen silsesquioxane (HSQ) (yellow) shadow-wall structure. A zoom-in at the junction is shown in the inset. The back side of the substrate is used as a global back gate. (b) Zero-field dependence of switching current I sw (red) and normal state conductance G n (blue) on the back gate voltage V g , overlapped onto the I b -V g two-dimensional (2D) map taken for Device 1 (with junction length L = 37 nm).
We first characterize the nanowire Josephson junction devices by means of quantum transport at zero magnetic field and ∼ 20 mK base temperature. The back side of the substrate is used as a back gate and the applied voltage V g acts globally on the entire nanowire. In Fig. 1b we show how the switching current I sw (red) and the normal state conductance G n (blue) depend on V g for Device 1. In order to obtain the switching current I sw , a four-terminal measurement is employed, where the voltage drop V over the junction is measured while sweeping the current bias I b . I sw is extracted from the (V, I b ) traces as the bias value at which the junction switches to the resistive quasiparticle regime (see the Data analysis section in the Supporting Information for the I sw extraction algorithm). The normal state conductance G n is obtained in the voltage-bias range 1 mV < |V b | < 2 mV -well above the double value of the induced superconducting gap of the leads (2∆ ∼ 0.5 meV).
The conductance measurements from which G n and ∆ are extracted are shown in Fig. ??
and Fig. ?? and the procedure for obtaining G n is explained in the Data analysis section in the Supporting Information. The details on the measurement setups are given in the Measurement section in the Supporting Information. By increasing V g , both I sw and G n , in spite of the fluctuations, become larger as the carrier states in the junction get populated and more subbands contribute to transport. At V g = 15 V, G n and I sw reach up to ∼ 5 G 0 (G 0 = 2e 2 /h) and ∼ 50 nA, respectively. The remaining nanowire JJs (Device 2-9) show comparable zero-field properties, as shown in Fig. ?? and Fig. ?? in the Supporting Information. The high tunability of G n as well as of I sw enables the systematic investigation of the junctions in different electro-chemical potential regimes.
Hybrid nanowire JJs have been shown to exhibit a supercurrent evolution in a parallel magnetic field-B that is strongly affected by the electro-chemical potential of the semiconducting junction 17 . Therefore, when exploring the resilience of switching current in a parallel B-field, the electro-chemical potential of a junction has to be taken into account. In the following, the switching current dependence on the back gate voltage V g and the parallel B-field is studied for two JJ devices with significantly different lengths. In Fig. 2a and 2b, we show how the switching current I sw evolves with V g and B for Device 2 (L = 31 nm) and Device 7 (L = 157 nm), respectively. I sw is extracted from the corresponding (V, I b ) traces taken at each setting of V g and B. As shown in Fig. 2a, the short device shows a remarkable supercurrent resilience with the supercurrent persisting above a parallel field of 1 T. A linecut at 1 T (red bar) is taken and the corresponding data is shown in Fig. 2c. The switching current I sw (red trace) continuously persists over a ∼ 3.5 V interval of V g . As a comparison, I sw drops more rapidly with magnetic field in the long device, as shown in Fig. 2b. Fig. 2d shows that at 0.6 T the supercurrent is barely detectable. Besides this apparent difference, the switching current behaviours in Fig. 2a  and 2b is that, as the magnetic field is increased, certain intervals in the intermediate gate regime support more resilient supercurrent. In these V g intervals we define the "resilient gate settings V g,res " (blue markers in Fig. 2c and 2d) and the supercurrent at such gate settings is examined for all devices in the following paragraph.
In Fig. 3 we focus on the supercurrent at the resilient gate settings V g,res . For Device 1-7 we determine the V g,res values as shown and described in Fig. S6, while for Device 8-9 we choose V g = 15 V (see the Data selection and reproducibility section in the Supporting Information). Fig. 3a shows the voltage drop V over the junction as a function of the current bias I b and the parallel magnetic field B for Device 1 (L = 37 nm). The red dotted line marks the extracted switching current I sw at different B-fields. Three linecuts (black, red and blue) are shown in Fig. 3c -demonstrating more than 1 nA supercurrent at the parallel field of 1.2 T. Fig. 3b and Fig. 3d show the results for Device 6 (L = 160 nm) obtained at its V g,res setting. From the overlaid red trace it can be seen that the supercurrent vanishes at ∼ 0.75 T, as confirmed by the linecuts shown in Fig. 3d. Analogous measurements of the switching current evolution with parallel field are carried out for all nine devices (see Fig. S7 in the Supporting Information). Finally, these I sw (B) dependences allow for the extraction of the maximal critical parallel magnetic field of switching current B Ic for each Device 1-9.
The details of the B Ic extraction are given in the Data analysis section in the Supporting Information. By plotting B Ic versus the junction length L in Fig. 3e, it can be seen how the junction length influences the measured critical field of the supercurrent. We reproducibly reach the critical fields of ∼ 1.3 T in the sub-40 nm junctions while B Ic drops gradually to ∼ 0.7 T in the longest junctions.
As a next step, we evaluate the supercurrent resilience over a broader gate interval -in a range of the electro-chemical potential. As our nanowire JJs are highly tunable, in Fig.   4 their supercurrent resilience against the parallel magnetic field is studied over the gate ranges in which the junctions are in the few mode regimes. Fig. 4a shows the voltage drop V as a function of the current bias I b and the gate voltage V g at the parallel field of 0.6 T for Device 2 (L = 31 nm), together with the switching current I sw (red trace) and the normal state conductance G n (blue trace). To quantify the supercurrent resilience, the switching current in Fig. 4a is averaged in the V g range corresponding to 0.01G 0 < G n (V g ) < 2G 0 (denoted by the two white dotted vertical lines) and the obtained average switching current is I avg sw (0.6 T) = 2.73 nA. An analogous averaging is done for the I sw (V g ) dependence measured at zero field and the obtained average switching current at zero field is I avg sw (0 T) = 11.29 nA (see Fig. S4 for the zero-field dependence and the average value). By calculating the ratio Dependence of the switching current I sw (red) on the gate voltage V g at the parallel magnetic field B = 0.6 T for Device 2 (L = 31 nm). Two white vertical lines mark the gate interval over which the normal state conductance G n of the device (blue) is tuned from 0.01G 0 to 2G 0 . In this gate range the switching current is averaged and I avg sw (0.6 T) value is obtained. Analogously, from the switching current dependence on V g at zero-field the average value  Fig. 4b. It can be noticed that at finite parallel field the shorter junctions preserve larger fractions of the corresponding zero field supercurrent in the described conductance ranges. Moreover, only negligible fractions of switching current (less than 2 %) systematically remain in the longer junctions -emphasizing their poor performance when the supercurrent resilience in tunable junctions is of interest. In comparison with Fig. 3e, the less strong dependence on the junction length can be seen in Fig. 4b. This can be attributed to particular devicespecific shapes of the switching current and the normal state conductance dependences on the gate voltage. We emphasize that the particular shape of the dependence in Fig. 4b could also vary depending on the choice of the normal conductance tunability range and the subsequently determined gate intervals for averaging. However, the main qualitative features of such dependence would still remain (see Fig. S8 in the Supporting Information).
In Fig. 3  We find that the measured induced gaps at zero field and the parallel field at which the induced gaps close are influenced not only by the back-gate voltage but by the junction length as well. This is in accordance with the known phenomenon that the proximity effect in nanowire hybrids can be controlled by the electric field 20-23 -that is influenced by both the gate setting and the device geometry. The measurements in Fig. S9 show that reducing the junction length of a hybrid nanowire JJ enhances the proximity effect inside the junction.
Consequently, the resilient supercurrent in the short junctions could be attributed to the enhancement of the proximity effect. Possible origins of such enhancement are discussed in more detail in the corresponding section in the Supporting Information.
Measurements on an additional short JJ device (Device 10, L = 40 nm) are shown in Multimode interference is an additional mechanism that could cause the prominent supercurrent dependence on the gate voltage and magnetic field -resulting in the observation of resilient gate settings. Differences between the accumulated phases of different transversal nanowire modes increase with the junction length and the flux applied through the conductive cross-section of a junction. 24 Therefore, destructive supercurrent interference due to large accumulated phase differences could be causing poor supercurrent resilience in long junctions and in open regime in all the junctions -where the conductive section increases due to high positive back gate. Furthermore, in the previous study 17 the scattering on disorder was shown to enhance multimode interference. With assuming comparable linear densities of disorder in the junctions in our study, the scattering on disorder would be more prominent in longer junctions and could therefore additionally diminish their supercurrent resilience.
From the above results, we find that significantly reducing the nanowire JJ length is essential for preserving supercurrents in a high parallel magnetic field. Here, we take a step further and incorporate remarkably short nanowire JJs into the SQUID architecture. In conclusion, we demonstrate that the length of a hybrid nanowire Josephson junction is an essential parameter that determines its supercurrent resilience against magnetic fields.
Nanowire JJs with a length of less than 40 nm can be precisely defined by the shadow-wall angle-deposition technique and are shown to reproducibly preserve supercurrent at parallel

Data analysis
All the codes used for the data analysis in this work are available in the data repository.
The details of the data analysis procedures performed in these codes are described in the following subsections.
Extracting normal state conductance G n Normal state conductance G n is extracted from the data collected in the voltage-bias measurements of the nanowire JJ devices. After correcting for the series resistance R s (as explained in the previous section), the normal state conductance is obtained as G n (V g ) = (G + n (V g )+G − n (V g ))/2 where G + n (V g ) = dI dV (V g , 1 mV < V < 2 mV) and G − n (V g ) = dI dV (V g , −2 mV < V < −1 mV) are averaged conductances at the positive and the negative source-drain volt-ages much larger than the double value of the superconducting gap (2∆ ∼ 500 µV).

Extracting switching current I sw
Switching current is extracted for each (V, I b ) trace measured in the current-bias setup.
Four-examples of (V, I b ) traces are shown in the top parts of Fig. S1a-d switching current is small. Therefore, when extracting I sw , we rather look at the maximum in the differential resistance, as it resembles the sharpness of a switch in a (V, I b ) trace.
For each differential resistance (dV /dI b , I b ) trace, the maximal value (peak) of dV /dI b is found and divided by the third value of the same (dV /dI b , I b ) trace sorted in decreasing order. In this way we quantify how dominant the peak in the differential resistance is. If the obtained value is smaller than the analogous value obtained from the trace in Fig. S1d with clearly no switch in it -the peak in differential resistance is not dominant and the switching current is extracted as a "not a number" (NaN) value. These NaN values correspond to the interruptions in the red I sw traces plotted over 2D maps throughout the study.
The trace in Fig. S1c depicts that the range over which the dominant peak in (dV /dI b , I b ) is searched for can affect the extracted value. For example, there is a dominant peak in Fig. S1c at I b ∼ 1.7 nA, but it does not correspond to the switching current. Therefore, the range in which the switching current is searched for is an important input parameter that is marked by the blue lines in Fig. S1. This parameter is commonly set at sufficiently high values and subsequently adjusted for particular traces where it leads to mistakes as the one described in Fig. S1c. The red lines in Fig. S1 mark the extracted switching current values and nicely match the dominant peaks of the differential resistance in the relevant ranges of the current bias.
The described algorithm successfully identifies the switching current in most of the traces. Extracting critical magnetic field B Ic By applying the above described algorithm to extract the switching current I sw , we extract I sw (B) from the 2D maps shown in Fig. S7 where the voltage drop V is measured as the current bias I b and the parallel magnetic field B are swept. By analyzing the evolution of the (V, I b ) linecuts in B field, it can be noticed that the algorithm may give an isolated NaN value for I sw at some B value even if the switching current is correctly extracted at higher fields. Therefore, defining the critical field of switching current B Ic as the lowest B field for which the algorithm gives NaN value for I sw can lead to underestimations of B Ic .
However, if the algorithm gives NaN values for two consecutive B field values, then even occasionally extracted I sw values different from NaN at higher fields are most often falsepositive extracted values. We therefore determine the critical field B Ic as the lowest field such that two consecutive extracted values for I sw are NaN. In Fig. S7 I sw is plotted up to the determined B Ic while the entire I sw (B) data is available in the data repository.

Effects of junction length and global back gate on induced superconducting gap
In order to measure the induced superconducting gap for Device 1-9 and study its evolution in parallel magnetic field, tunneling spectroscopy is performed in the voltage bias setup.
In Fig. S9  These differences in the induced gap sizes and their evolution in parallel magnetic field for junctions of different lengths are accompanied by differences in the gate settings at which different devices are set into the tunneling regime. Namely, it can be noticed that shorter devices mostly require low or even negative back gate voltages for reaching the tunneling regime, while this value is higher for the longer junctions. A valid question that arises is whether the differences in the tunneling spectroscopy in Fig. S9 are due to the differences in the junction lengths or due to the differences in the electrical fields induced by the different gate voltages.
Despite the differences present among the nine devices in the tunneling regime regarding the back gate settings, the junction lengths and the conductance values, some conclusions can be made by looking at specific subsets of the devices for which some of these parameters are comparable. By comparing the data for Device 4, 5 and 7, it can be seen that with almost the same gate settings of V g ∼ 2.15 V and the comparable tunneling conductance values G n ∼ 0.3 − 0.4 G 0 , the shortest device out of the three (Device 4) exhibits the largest induced gap that closes at the highest field. The data for the other two devices (Device 5 and 7) suggest that gradual increases of the junction length lead to weaker proximity effect with gradually smaller induced gap and gradually lower critical parallel field of the induced gap.
Furthermore, the shortest device in the study (Device 8) requires the largest gate voltage to be tuned into the tunneling regime (V g = 5.7 V) and still exhibits larger induced gap than the longest devices (Device 6-7) measured at the lower gate voltages. Despite the high gate voltage, the induced gap of Device 8 closes at ∼ 1.3 T. However, in comparison to the remaining short junctions measured at significantly lower gate voltages (Device 1,2 and 3), Device 8 has poorer induced superconducting properties, probably due to the the high gate voltage and reduced superconductor-semiconductor coupling.
We can conclude that junction length is an important parameter that influences the induced superconducting gap. This does not exclude an effect that the applied back gate voltage has on induced superconductivity. Moreover, the data in Fig. S9 demonstrates that both the junction length and the back gate voltage determine the semiconductor-superconductor hybridization. This confirms that the electrostatic profile inside a hybrid nanowire JJ device -influenced by both device geometry and gate voltage -can control the strength of the semiconductor-superconductor hybridization 5,6 .
The stronger proximity effect in the short JJs could originate from an electron layers accumulated at the interfaces between the semiconducting nanowire and the superconducting leads. Namely, the band offset at an InSb-Al interface can cause a bending of the InSb conduction band and results in a strongly proximitized electron layer at the interdace with Al. Because of a finite lateral extension of such layers from the two sides of a short JJ, the junction superconducting properties could be enhanced. Note that in some short JJs in our study the normal conductance and supercurrent have been measured to be finite when no back gate voltage is applied (see the data for Device 2 and 3 in Fig. S4). This could suggest that the accumulation layers can fully extend over a ∼ 30 nm junction by extending ∼ 15 nm laterally at each side.

Effects of local gates on supercurrent resilience
As an additional measurement, we perform current bias measurements on a single Josephson junction (Device 10) which is one arm of the SQUID (see Fig. S10a

Data selection and reproducibility
By systematically sweeping the back gate voltage V g when measuring Device 1-7, we could identify the resilient gate settings V g,res , as described in the main text. However, at the initial phase of the study, when measuring the chips from which Device 8-9 originate, the resilience of supercurrent against magnetic field was only examined at V g = 15 V. Therefore, for these devices the identification of the resilient gate setting V g,res (like those shown in Fig.   S6) was not performed. Still, we include Device 8-9 in our study as they manifest resilient supercurrent even at V g = 15 V which is not necessarily their V g,res . Other short junction devices from these chips did not manifest such resilient supercurrent (critical parallel field of ∼ 0.7 T at V g = 15 V) and long junction devices from these chips showed very poor supercurrent resilience (critical parallel field of ∼ 0.4 T at V g = 15 V). We do not include these devices in our study as their critical parallel fields at V g = 15 V may be significantly smaller in comparison to their critical fields at the back gate tuned to their V g,res settings. Importantly, we have never measured any long junction device (with or without back gate tuning) that showed better supercurrent resilience than the long junction devices (Device 6-7) presented in the study.   Device 8 Figure S3: Differential conductance at zero-field: Measured differential conductance G through Device 1-8 as a function of the voltage drop V between the source and drain and the back gate voltage V g . The normal state conductance dependences G n (V g ) for Device 1-8 are obtained from these 2D maps. as described in the Data analysis section. The analogous 2D map was not taken for Device 9 and the G n (V g ) dependence for this device was measured as a single trace at V > 1 mV.   Figure S4: Tunable switching current and normal conductance at zero-field: -For Device 1-9 the extracted switching current I sw (red) and measured normal conductance G n (blue) are plotted over I b − V g 2D maps obtained in the current bias measurements at zerofield. All devices show tunability by the back gate voltage V g from the pinch-off regime with no supercurrent to the open regime with I sw of several tenths of nA and G n of few G 0 (with G 0 = 2e 2 /h). G n (V g ) dependences are obtained from the data shown in Fig. S3. The white dotted vertical lines mark the ranges of V g over which G n increases from 0.01G 0 to 2G 0 . The average switching currents in these intervals are shown as insets.  Figure S6: Identifying the resilient gate settings V g,res : The back gate voltage V g is swept at high parallel magnetic field for Device 1-7. The red markers denote the resilient gate settings V g,res . V g is set to these values for obtaining the magnetic field dependences shown in Fig. 3 and Fig. S7. The analogous measurements were not performed for Device 8-9.  Figure S7: Evolution of switching current in parallel magnetic field: Dependence of the switching current I sw (red) on the parallel magnetic field B for Device 1-9. The back gate is set at the resilient gate setting V g = V g,res for Device 1-7 and at V g = 15 V for Device 8-9 (see the Data selection and reproducibility section). The corresponding extracted critical field B c is shown as an inset. The gate settings for Device 1-7 are marked by the red markers in Fig. S6.  Figure S8: Switching current at B = 0.6 T parallel magnetic field: Dependence of the extracted switching current I sw (red) on the back gate voltage V g at the parallel field B = 0.6 T for Device 1-7. The white dotted vertical lines indicate the ranges of V g over which the normal state conductance G n at zero-field of the corresponding device increases from 0.01G 0 to 2G 0 . The average switching currents over these intervals are shown as insets. The analogous measurement was not performed for Device 8-9. Dependence of the tunneling conductance G on the parallel magnetic field B for Device 1-9. Extracted induced superconducting gap at zero field ∆ and the back gate voltage V g at which the tunneling spectroscopy is measured for each device are shown as insets.