Absence of diode effect in chiral type-I superconductor NbGe2
Introduction
Stemming from the nonreciprocal phenomena in quantum materials1,2,3, novel functionalities such as the superconducting diode effect (SDE) have been achieved in prototype devices through symmetry engineering4. Analogous to the semiconductor p-n junctions, superconducting diodes carry dissipationless supercurrent in one direction and show finite resistance in the opposite direction. An asymmetry parameter, called the diode efficiency (Qequiv frac{|{{{rm{I}}}}_{{{rm{c}}}}^{+}-{{{rm{I}}}}_{{{rm{c}}}}^{-}|}{({{{rm{I}}}}_{{{rm{c}}}}^{+}+{{{rm{I}}}}_{{{rm{c}}}}^{-})/2}), was introduced to quantify the difference between the opposite critical currents Ic+ and Ic− in SDE5. The SDE is crucial in realizing energy-efficient and quantum-based superconducting circuits, attracting extensive attention from the superconducting electronics community6. However, intense controversy has recently emerged regarding the mechanism of SDE7. Specifically, mainstream theories suggest that the SDE arises from the intrinsic inequality of Cooper-depairing currents along opposite directions8,9,10,11. In contrast, the experimentally measured Ic is considered by other groups to be the vortex-depinning current from a practical viewpoint12 and the SDE is hence thought to originate from the asymmetric vortex pinning potential7,13,14,15.
The central question is how to avoid or even regulate the vortex effect in the study of SDE. Vortices are inevitably generated by the negative interface energy in type-II superconductors but are absent in type-I superconductors with positive interface energy16. On the other hand, as one kind of nonreciprocal response, the superconducting diodes require intrinsic space-inversion symmetry breaking. Therefore, investigating SDE in chiral type-I superconductors is an elegant method to circumvent the complicated vortex effect. However, both type-I superconductivity and chiral structures are unusual, rendering the overlap extremely rare. The only identified candidates for chiral type-I superconductors are polycrystalline BeAu (space group P213)17 and single crystalline NbGe2 (space group P6222/P6422)18 so far. NbGe2 belongs to the Kramers-Weyl semimetals19 with a noticeable antisymmetric spin-orbit coupling (SOC) effect20. Meanwhile, the chiral symmetry facilitates the formation of a coupled electron-phonon liquid in NbGe2 below T′ ~ 50 K21. The electron fluid property leads to an ultralow resistivity in NbGe2 crystals, posing a large obstacle to comprehensively investigating the subtle nonreciprocal charge transport22,23.
This challenge can be overcome by preparing micro-sized devices using the well-established focused ion beam (FIB) technique24. FIB offers an excellent method for investigating quantum materials in the mesoscopic regime, particularly for materials like NbGe2 that are incompatible with mechanical exfoliation. Another advantage of FIB devices is their flexibility in tailoring geometry, enabling diverse experimental configurations for transport measurements. Here, we successfully fabricated the NbGe2 devices using FIB and systematically investigated their nonreciprocal transport. An unexpected outcome of FIB milling is the superconducting amorphous NbGe surface, which coincidentally exhibits type-II superconductivity with a superconducting critical temperature Tc higher than that of NbGe2 bulk. This artificial structure allows for the incorporation of vortex dynamics into NbGe2 devices by suppressing the bulk superconductivity. While most nonreciprocal transport results align well with theoretical predictions, the field dependence of diode efficiency Q is in strong contrast to anticipations. The main message of this paper is that the SDE is negligible (Q < 2%) at low field regimes but emerges (Q exceeding 50%) at high field regimes, corresponding to the strongly suppressed or activated vortex dynamics. Our study reveals that, in addition to the well-known symmetry rules, the ubiquitous vortex effect plays a nonnegligible and even critical role in realizing the SDE.
Results and discussion
Type-I superconductivity in chiral NbGe2
NbGe2 belongs to the hexagonal C40 structural group, with right- and left-handed helix structures along the c-axis corresponding to space groups P6222 (No. 180) and P6422 (No. 181), respectively (Fig. 1a). The left-handedness of these two enantiomers in the studied NbGe2 crystal was confirmed using single-crystal diffraction (Supplementary Table 1). Superconductivity in NbGe2 was first found in 197825 and has recently been revisited due to its chiral crystal structure18,20,21,26,27. Figure 1b shows the temperature-dependent resistivity of a NbGe2 crystal with a residual resistivity ratio RRR = 96. The Tc is consistent across electrical transport (Tc = 1.96 K) and magnetic susceptibility measurements (Tc = 2.01 K). It is worth noting that the magnetic susceptibility is identical in both the zero-field-cooled (ZFC) and field-cooled (FC) measurements (Fig. 1c). In type-II superconductors, the diamagnetism measured in the FC mode is smaller than that of the ZFC mode due to the trapped vortices. However, it has been confirmed that the superconductivity in NbGe2 exhibits rare type-I behavior18,20, which is responsible for the absence of the vortex pinning effect in our FC measurement. As the temperature decreases from 1.8 K to 400 mK (Fig. 1d), we observe a crossover in NbGe2 from type-I superconductivity at low fields to type-II/1 superconductivity at higher fields28. The so-called type-II/1 superconductivity significantly differs from the conventional type-II superconductivity, although magnetic flux will penetrate both superconductors. The interaction between flux quanta is attractive, rather than repulsive, in type-II/1 superconductors29. This feature strongly suppresses the vortex dynamics in type-II/1 superconductivity, rendering it more like type-I than the conventional type-II superconductivity.

a The chiral crystal structure of NbGe2. The right-handed and left-handed Ge-Ge bond enantiomers are highlighted by blue and red lines, respectively. b The temperature dependent resistivity of a NbGe2 crystal up to 300 K. The residual resistivity ratio RRR = R300 K/R5 K = 96. The inset shows a two-terminal resistance under different magnetic fields, in which the superconducting critical temperature Tc is determined as 1.96 K from the onset of the resistance drop. c The temperature dependent dc magnetic susceptibility χ of NbGe2 crystal under the zero-field-cooled (ZFC) and field-cooled (FC) modes with an applied magnetic field of μ0H = 0.1 mT. Tc is determined as 2.01 K from the onset of diamagnetism. The inset shows the expulsion of the magnetic flux from type-I superconductors. d The isothermal magnetization of NbGe2 crystal from 1.8 K to 400 mK. The data is corrected by the demagnetization factor. The magnetization is expressed as μ0M to align with the external magnetic field μ0H.
To enhance the charge transport signals, we fabricated micro-sized NbGe2 devices using FIB (“Methods”). As shown in Fig. 2a, we have deliberately designed the NbGe2 device D3 to simultaneously measure the c-axis and a-axis transport properties. The a-axis resistivity is higher than the c-axis resistivity in NbGe2, consistent with previous report21. As the temperature-dependent resistivity remains consistent between the NbGe2 bulk and NbGe2 devices in their normal states (Fig. 2b), we believe that the transport properties measured in these devices reliably reflect the bulk characteristics of NbGe2 in the normal state. For instance, one of the chirality-induced anomalies in NbGe2 is the coupled electron-phonon liquid21, leading to a kink feature in the Kohler scaling30. We also measured the magnetoresistance and indeed observed similar results in NbGe2 device D1 (Supplementary Fig. 1), demonstrating the high quality of our NbGe2 devices. According to the crystal symmetry, the chirality-induced nonreciprocal transport response (R={R}_{0}left[1+gamma left({{boldsymbol{B}}}cdot {{boldsymbol{I}}}right)right]) appears when I and B are applied along the helix axis1,31. And the nonreciprocal coefficient (gamma =frac{2{R}^{2{{rm{omega }}}}}{{R}^{{{rm{omega }}}}{BI}}) is obtained by the associated first-harmonic resistance Rω and second-harmonic resistance R2ω in lock-in measurements32. Figures 2c and d show the raw data of field dependent R2ω/Rω measured under red I//B//c and black I//a⊥B//c configurations, respectively. The R2ω/Rω data consists of extrinsic symmetric contribution and intrinsic antisymmetric signals. We then extract the intrinsic R2ω/Rω response from the asymmetric part. The c-axis nonreciprocal response (I//B//c) is significantly stronger than the a-axis one (I//a, B//c) as anticipated (Fig. 2e). With decreasing temperature, the chirality-induced nonreciprocal coefficient γ eventually saturates at approximately 7.0 × 10−2 T−1 A−1 at low temperatures (Fig. 2f), which is of the same order as that observed in the chiral molecular conductor33. In principle, the two-fold rotational symmetry (2001) prohibits the generation of nonreciprocal response in any direction other than along the c-axis. The observation of a tiny R2ω response along the a-axis (I//a⊥B//c) suggests the presence of additional symmetry breaking in the NbGe2 device beyond its intrinsic crystal symmetry, potentially arising from the polar symmetry along the b-axis of the artificial surface, as discussed below.

a Scanning electron microscope image of NbGe2 device D3. The NbGe2 microstructures and electrodes are highlighted in purple and yellow colors, respectively. b The temperature dependent resistivity of NbGe2 devices D1-D3. The purple c-axis transport was obtained from D3. The residual resistivity ratio RRR is calculated from R300 K/R5 K. c, d Raw data of field dependence of second-harmonic resistance R2ω/ first-harmonic resistance R1ω measured under configurations of current I// magnetic field B//c-axis and I ⊥ B at T = 10 K, respectively. e Asymmetric analysis of the corresponding raw data shown in (a, b). f Temperature dependence of chirality-induced nonreciprocal coefficient γ in the normal state. To amplify the subtle R2ω signal, the applied current was as large as 5 mA. All R2ω data were obtained on device D3.
Type-II superconductivity in artificial NbGe surface
It is well known that the FIB damages the material surface, leaving an artificial amorphous layer due to ion irradiation24. The FIB-induced surface damage is gentle and confined to a typical thickness of around 10 nm, ensuring that the transport properties remain unaffected in FIB-fabricated microdevices, as confirmed by our measurement in the normal state (Fig. 2b). However, the situation became complicated when we investigated the superconducting state. As shown in Fig. 3, we found that both Tc and upper critical fields Hc2 are significantly enhanced from the NbGe2 bulk (Tc = 2.01 K, Hc2 = 40 mT) to NbGe2 devices (Tc = 2.95 K, Hc2 = 4 T in D1). The enhanced superconductivity originates from the amorphous NbGe surfaces induced by FIB irradiation, as demonstrated by the following observations. Firstly, the scanning transmission electron microscope (STEM) image provides direct evidence of an amorphous surface, with a thickness ranging from 10 to 15 nm (Fig. 3b). The six-fold symmetry is observed in the crystalline region 2 and absent in the amorphous region 1 as expected. The composition of the amorphous surface is identified as NbGe using spatial-resolved electron energy loss spectroscopy (EELS) (Supplementary Fig. 2). Secondly, the broad superconducting resistive transition is fitted well by the Halperin-Nelson formula34, showing a Berezinskii-Kosterlitz-Thouless (BKT) transition with a higher Tc = 2.95 K in device D1 (Fig. 3c). The fitted BKT transition temperature TBKT in the resistivity is 2.50 K, which is almost identical to TBKT = 2.49 K derived from the fitted exponent V∝I α at α (TBKT) = 3 (Fig. 3d). Note that the BKT transitions were reproducibly observed in another NbGe2 device D2 with a different Tc = 2.50 K (Supplementary Figs. 3a–d). Lastly, the upper critical fields Hc2 of NbGe2 device are two orders of magnitude higher than that of NbGe2 bulk (Fig. 3e, f), but are consistent with the Hc2 of NbGe35 (Supplementary Note 3). It is anomalous to observe such a considerable enhancement of Hc2 if the superconductivity comes from NbGe2 bulk.

a Schematic illustration of NbGe2 device in the transport measurements. The crystalline NbGe2 core is covered by a thin amorphous NbGe shield, which induces the excitations of vortex-antivortex pairs. b High-angle annular dark field (HAADF) image of focused-ion-beam irradiated NbGe2. The corresponding fast Fourier transform (FFT) images of the amorphous region 1 and crystalline region 2 are shown in the top panel. c Detailed scan of resistivity in device D1 with temperature step of 5 mK. The resistive curve is fitted well by the Berezinskii-Kosterlitz-Thouless (BKT) transition using the Halperin-Nelson formula (red line). The superconducting critical temperature Tc is determined at 90% normal state resistivity ρn and the fitted TBKT is 2.50 K. The inset is resistivity replotted on a [d(ln ρ)/dT]-2/3 scale. d Temperature dependence of the power-law fitted exponent α from zero-field voltage-current (V–I) curves V α Iα. The corresponding BKT transition temperature TBKT ∼ 2.49 K is obtained at α (TBKT) = 3. e The resistivity of NbGe2 device D1 as a function of temperature under different magnetic fields. The upper critical field Hc2 was obtained at 90% ρn. f Superconducting phase diagram of NbGe2 devices. The solid and open symbols represent the corresponding Hc2 of NbGe2 bulk and devices D1–D3, respectively. The regime of type-I (type-II/1) superconductivity of the NbGe2 core, type-II superconductivity of the NbGe surface, and the normal state are shown in purple, orange, and blue colors, respectively.
Based on these results, the amorphous NbGe shell as depicted in Fig. 3a results in enhanced superconducting properties, such as Tc and Hc2, in transport measurements. Similar amorphous niobium surfaces were created and responsible for the induced superconductivity in non-superconducting NbAs devices via FIB milling36. Since twice the superconducting coherence length 2ξ = 32 nm is larger than the thickness of NbGe surface (see Supplementary Note 4), our results strongly suggest that the NbGe surface hosts 2D superconductivity in NbGe2 devices37,38. Additionally, it is worth noting that the vortex dynamics is always absent (strongly suppressed) below the critical field Hc (Hc2) of NbGe2 due to its type-I (-II/1) superconductivity and its significantly larger cross-section compared to the NbGe surface. However, as confirmed by the observations of BKT transitions, the excitations of vortices/antivortices could penetrate the devices due to the type-II nature of the amorphous NbGe surfaces at higher temperature. This result indicates that we can easily switch between type-I and type-II superconductivity in different regimes, as delineated in the H–T phase diagram of Fig. 3f.
Nonreciprocal transport near T
c in NbGe2 devices
The amorphous NbGe surface is distinctly separated from the crystalline NbGe2 core, resulting in a well-defined NbGe2/NbGe interface. This sharp boundary is attributed to the self-annealing process occurring at room temperature24. Akin to other artificial interfaces, this interface will induce the Rashba-like SOC, which gives rise to polarization in the perpendicular direction. To investigate this additional space-inversion symmetry breaking, we systematically investigated the nonreciprocal electrical transport in NbGe2 devices near Tc. The polarization-induced second harmonic resistance R2ω is expressed as ({R}^{2omega }propto left({{boldsymbol{P}}}times {{boldsymbol{B}}}right)cdot {{boldsymbol{I}}}=-{BI}sin theta), where θ represents the angle between the current I and magnetic field B when they are perpendicular to the polarization P22,39. We indeed observed significant R2ω responses in NbGe2 device D2 under the polar configuration of I⊥B⊥P (Fig. 4a) and its typical angular dependence (Fig. 4b). Figure 4c shows the temperature dependence of γ, which features a peak at TBKT with a value of 3.2 × 105 T−1 A−1. Figure 4d illustrates that R2ω progressively deviates from the anticipated linear behavior in the low-current regime and exhibits a decline in the high-current regime, consistent with previous observations in gated SrTiO340. Figure 4e, f show the magnetic field dependence of first-harmonic resistance Rω and R2ω at various T down to 0.3 K respectively. It shows that the R2ω responses are noticeable in the superconducting transition regions but undetectable in the normal state. All the current, temperature, and field dependencies of R2ω are well reproducible in another NbGe2 device D1 (Supplementary Fig. 4).

a The field dependence of second-harmonic resistance R2ω at T = 1.9 K under polar configuration. Insets show the interfaces between purple NbGe2 bulk and orange NbGe surface. The direction of current I (red arrow), external magnetic field B (blue arrow), and polarization P (green arrow) are perpendicular to each other. b Angular dependence of R2ω measured under 60 mT, 1.9 K. The evolution is fitted well by a gray sine wave. c Temperature dependence of the nonreciprocal coefficient γ and first-harmonic resistance Rω near the superconducting transition. The red and orange fitting line is Berezinskii-Kosterlitz-Thouless (BKT) transition induced nonreciprocal response (gamma (T),propto {(T-{T}_{{{rm{BKT}}}})}^{-3/2}) near BKT transition temperature TBKT and (gamma (T),propto ({T}_{{{rm{c}}}}-T)) near superconducting critical temperature Tc. d R2ω as a function of current I at 1.9 K, 0.2 T. The R2ω is linear with I in the low current region but then gradually deviates the linearity with an increasing I. e The field dependence of Rω and R2ω at various temperatures. The curves of R2ω are separated vertically by 10 milliohms for clarity.
These observations confirm the presence of a polar NbGe2/NbGe interface, which induces a significant nonreciprocal transport in NbGe2 devices near Tc. There are two major factors that contribute to the substantial increase of γ from around 10−1 T−1 A−1 in the normal state to approximately 105 T−1 A−1 in the superconducting state. First, due to the remarkable reduction of the electronic energy scale from Fermi energy to superconducting gap, the γ value is gigantically increased when electrons condense into Cooper pairs41. The second factor is the BKT transition, which further enhances the γ value in the 2D NbGe2/NbGe interfaces. This involves a two-step evolution of γ(T)41, characterized by ({{rm{gamma }}}left(Tright) sim ({T}_{{{rm{c}}}},-T)) and ({{{rm{gamma }}}left(Tright) sim (T-{T}_{{{rm{BKT}}}})}^{-3/2}), as the temperature decreases from Tc to TBKT. With a further decreasing T, γ falls to zero associated with the bound vortex/antivortex pairs in the superconducting ground state. This evolution explains why we observe a peak feature in Fig. 4c. It is worth noting that our observations are highly similar to previous reports in various platforms including gated MoS2 (γ ~ 8 × 102 T−1 A−1)32, gated SrTiO3 (γ ~ 3.2 × 106 T−1 A−1)40, Bi2Te3/FeTe interfaces (2D γ’ ~ 6.5 × 10−3 T−1 A−1 m)42, NbSe2 antennas (γ ~ 3.4 × 104 T−1 A−1)43, etc. Albeit these materials possess various origins of broken space-inversion symmetry, the theoretical analysis indicated that the nonreciprocal responses are similar in these 2D superconductors due to the BKT transition41. Our results show that the motion of unpaired vortices boosts γ significantly in NbGe2 devices, emphasizing the importance of vortex dynamics in nonreciprocal transport41,44.
Absence of SDE in type-I NbGe2 regime
Albeit resistance, whether nonreciprocal or not, vanishes in the superconducting ground state, the supercurrents retain the nonreciprocal characteristics, as manifested itself by the SDE3. Owing to a small penetration depth, the theoretical depairing critical current Ic of NbGe2 is significantly larger than that of the NbGe surface (see Supplementary Note 6). This feature suggests that the superconductivity of NbGe2 can be detected in terms of Ic (Fig. 5a), in contrast to the H–T phase diagram (Fig. 3f), where the superconductivity of NbGe2 is obscured by the NbGe surface. Hence, the I–V characteristics can be divided into two different regimes based on the Hc2 of NbGe2: the type-I (or type-II/1) NbGe2 regime at low magnetic fields and the type-II NbGe regime at high magnetic fields. We characterized the I–V curves in both positive and negative directions under different magnetic fields at 0.3 K, and summarized the field dependence of Ic in Fig. 5b. The difference between Ic+ and Ic− becomes noticeable once B exceeds 40 mT, aligning with the Hc2 of NbGe2 at 0.3 K. The associated I–V curves exhibit clear distinctions between the pale purple NbGe2 regime and the pale orange NbGe regime. In the NbGe2 regime like B = 20 mT, the Ic is equal in opposite directions Ic+ = Ic−. A tiny nonreciprocity appears in the middle dissipative regime, which is difficult to define as the SDE phenomenon (Fig. 5c). In contrast, when the applied magnetic field increases to 100 mT, Ic becomes distinct along opposite directions, Ic+ ≠ Ic−, showing the emergence of SDE in the NbGe regime (Fig. 5d). One vital functionality of SDE is the perfect rectification effect without dissipation, which is also verified in Supplementary Fig. 5. We notice that its polarity is effectively switched by reversing the magnetic fields, as marked by the contrasting red and gray colors. The switchable polarity of SDE in NbGe2 device is similar to that in polar artificial multilayers4,45, patterned thin-films13,15, few-layer NbSe246 and others, but in contrast to the strained trigonal superconductor PbTaSe247, where the polarity of SDE is fixed and the underlying diode mechanism should be distinct from others.

a Field dependence of theoretical critical current Ic in NbGe2 devices. Ic maximum is calculated from the self-field Ic, and the empirical evolution is capped at the measured upper critical field Hc2. b The experimental field dependence of Ic+ (positive direction) and Ic− (negative direction). The low-field NbGe2 and high-field NbGe regimes are marked by pale purple and orange backgrounds. The opposite polarity of SDE is noted by contrasting red and black colors. c, d The typical current-voltage I–V curves in low-field NbGe2 and high-field NbGe superconducting regime, respectively. Black and red lines mark the positive and negative current directions. We extract Ic values using the same voltage criterion of 1 μV. The inset of (c) shows an equal critical current Ic along opposite directions. The SDE was only observed in high-field regime, as marked the gray region in (d). e The vortex creep rate S as a function of fields. The slope of S changes abruptly as superconductivity transitions from the NbGe2 to NbGe superconducting regime. The solid and open symbols present the data obtained from positive and negative magnetic fields, respectively. f The field dependence of Q in NbGe2 devices D1-D3. We measured I–V curves under both polar and chiral configurations: current I ⊥ magnetic field B ⊥ polarization P (devices D1 and D2) and I//B//c-axis (device D3). The Q values are anomalously absent in the NbGe2 superconducting regime.
To characterize the difference in vortex dynamics between NbGe2 and NbGe regimes, we extract the vortex creep rate S using the relation S ~ 1/(N-1), where the flux creep exponent N is derived from I–V curves V(I) ∝ IN in the vicinity of Ic48. Figure 5e displays the corresponding field dependence of S, in which the slope of S abruptly increases at the same threshold of 40 mT. This result confirms that the vortex dynamics are strongly suppressed in the NbGe2 regime but effectively activated in the NbGe regime. Additionally, the structural disorder within the amorphous NbGe layer could broaden the I–V transition in the type-II regimes (Fig. 5d) compared to the type-I regimes (Fig. 5c). Given that two sources of space-inversion symmetry breaking are observed in NbGe2 devices, we measured I–V curves under both configurations: I⊥B⊥P (devices D1 and D2) and I//B//c (device D3). The I–V characteristics are highly consistent across different NbGe2 devices, and the field dependence of their diode efficiency Q are shown in Fig. 5f. Distinct from the linear enhancement of Q (B) in previous reports of SDE4,13, the Q value fluctuates within 2% below 40 mT, indicating the absence of SDE even under conditions of simultaneous space-inversion and time-reversal symmetry breaking. The similar evolution of Q(B) and S(B) suggests an underlying relationship between them, highlighting the significance of vortex dynamics in triggering SDE. We also observed small Q values ~ 0.2 when the external magnetic fields were parallel to the currents in device D3, a configuration in which the unidirectional Lorentz force should be absent, and thus the SDE is expected to be negligible. However, the applied currents inevitably generate vortex and antivortex pairs, known as the self-field effect12. The excitation of vortex/antivortex pairs could induce giant nonreciprocality even when B = 044, providing a possible explanation for our observations in the B//I configuration.
Conclusions
In summary, we systematically performed nonreciprocal transport measurements on FIB fabricated NbGe2 devices and confirmed the critical role of vortex dynamics in realizing a significant SDE. By inducing artificial NbGe surfaces through FIB irradiation, we observed enhanced superconductivity in these devices. Both types of space-inversion symmetry breaking, chirality in the NbGe2 bulk and polarization at the NbGe2/NbGe interface, were detected in the nonreciprocal transport measurements. Additionally, two-dimensional vortex excitations emerged on the NbGe surfaces as the type-I superconductivity of NbGe2 bulk was suppressed, leading to the BKT transition and a dramatic increase in the nonreciprocal coefficient. The SDE was negligible in the NbGe2 bulk at low magnetic fields but became pronounced in the NbGe surface under high magnetic fields, aligning with the abrupt increase in vortex creep rates.
Although we did not observe the SDE in chiral NbGe2, likely due to the weak antisymmetric SOC, we cannot dismiss the possibility of exotic superconductivity induced SDE, such as the finite momentum Cooper pairing49. Due to the inevitable excitation of vortices in type-II superconductors, it is challenging to distinguish between contributions to SDE from intrinsic crystal symmetry and those from extrinsic vortex effects. Therefore, it is crucial to exercise caution when attributing the widely observed SDE to the inherent symmetry breaking in superconductivity, rather than to the artificial asymmetry. Additional phase-sensitive experiments, such as those measuring the interference patterns50 or Little–Parks oscillations51, are necessary and promising to support the existence of exotic superconductivity in these platforms.
From the perspective of electronic applications, we have confirmed that artificial superconducting surfaces can host the SDE. This functionality can be readily replicated in various quantum materials containing superconducting elements, such as niobium, using FIB technology. Consequently, we envision a significant expansion of the SDE application into a broader range of electronic devices, including potential integration with non-superconducting quantum materials, such as the Weyl semimetal NbAs. Our findings suggest that the unique properties of these materials could be harnessed in conjunction with SDE to develop advanced electronic components with enhanced performance and functionality.
Methods
Crystal synthesis and characterizations
NbGe2 single crystals were synthesized by vapor transport using iodine as the transport agent. At first, the polycrystalline NbGe2 samples, prepared via solid-state reaction at a heating temperature of 850 °C, were ground into a powder and mixed with iodine in a glove box. The reactants were then sealed under vacuum in a quartz tube and placed in a two-zone furnace, with the hot end maintained at 850 °C and the cold end at 800 °C. After two weeks, hexagonal crystals with typical sizes of 0.5 mm × 0.5 mm × 0.3 mm were obtained after ultrasonic cleaning in ethanol. Magnetization measurements were performed in a Quantum Design MPMS-XL system. The isothermal magnetization down to 400 mK was collected in an MPMS-3 system equipped with the He3 option. The NbGe2 crystal structures are drawn by VESTA52.
Devices fabrication
All NbGe2 devices were fabricated by a FEI Helios-600i FIB under an acceleration voltage of up to 30 kV with Ga ion sources. Similar to the methods developed by Moll et al.24, the whole procedure of FIB device preparation can be summarized as four steps.
First step: preparing the lamella from bulk samples
The crystallographic orientation of the NbGe2 bulk was checked by the Laue diffraction and mounted in the SEM sample holder. We then cut the lamella with a typical size of 150 × 40 × 2 μm3 from bulk like the routine sample preparation for the transmission electron microscope. A high 65 nA was used to dig the grooves, and a smaller current of 2.5 nA was then used to polish the sidewalls into a flat shape under grazing incidence.
Second step: transferring the lamella to substrate
Firstly, we place a hundred-micrometer-sized droplet of Araldite epoxy glue on the sapphire substrate with lithographically prepatterned gold contact pads. We vertically picked up the lamellas from crystals and then horizontally, flat sides down, put them onto the droplet using an ex-situ micromanipulator. The continuous and smooth contacting profile between lamella and droplet was built by the capillary forces of epoxy liquid.
Third step: evaporating gold electrical contacts
The droplet was first heated at 150 °C for one hour to release the bubble from the epoxy droplet. A brief argon etching was used to clean the surface of the lamellas before the following gold evaporation. By using a homemade shadow mask, typical Ti/Au (5/300 nm) layers were deposited on the region covering lamellas and contact pads. The deposited gold layers serve as the electrical contacts with a typical contact resistance of a few Ohms.
Final step: shaping the measured region
We put the substrate with covered lamellas back into the FIB chamber and proceeded with the final cutting. The covered gold layers were carefully removed by FIB milling at 1.6 nA, 5 kV. Then, the overall shape of the measured region and the position of the corresponding contact was cut by FIB at 0.79 nA, 30 kV. Finally, the outer gold layer is selectively cut through to ensure the applied current flows only as designed for transport measurements.
STEM on amorphous surfaces
To characterize the FIB induced amorphous damage, STEM specimens were prepared using the same FIB machine with the same milling parameters as those used for NbGe2 devices. Atomically resolved high-angle annular dark-field scanning transmission electron microscopy (HAADF-STEM) and electron energy loss spectroscopy (EELS) were conducted on a JEOL ARM200F transmission electron microscopy (TEM) equipped with a probe-forming spherical-aberration corrector.
Electrical transport measurements
The temperature-dependent resistance and magnetoresistance were conducted in a Quantum Design PPMS-9 system. The dc I–V measurements were collected by two Keithley meters 6221 and 2182. The ac Rω and R2ω were obtained by the Stanford SR830 lock-in amplifiers with a frequency of 137.77 Hz, and a phase offset as −90° was set to collect R2ω in the nonreciprocal transport measurements. To obtain a higher cooling power during the current-voltage measurements at 300 mK, similar transport measurements were performed in a top-loading He-3 refrigerator with a 15 T superconducting magnet.
Note added: We noticed a similar study on FIB fabricated NbGe2 device during the peer review process53.
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