Intrinsic supercurrent diode effect in NbSe2 nanobridge

The significance of the superconducting diode effect lies in its potential application as a fundamental component in the development of next-generation superconducting circuit technology. The stringent operating conditions at low temperatures have posed challenges for the conventional semiconductor diode, primarily due to its exceptionally high resistivity. In response to this limitation, various approaches have emerged to achieve the superconducting diode effect, primarily involving the disruption of inversion symmetry in a two-dimensional superconductor through heterostructure fabrication. In this study, we present a direct observation of the supercurrent diode effect in a NbSe2 nanobridge with a length of approximately 15 nm, created using focused helium ion beam fabrication. Nonreciprocal supercurrents were identified, reaching a peak value of approximately 380 $\mu$A for each bias polarity at $B_{z}^{max} =\pm 0.2$ mT. Notably, the nonreciprocal supercurrent can be toggled by altering the bias polarity. This discovery of the superconducting diode effect introduces a novel avenue and mechanism through nanofabrication on a superconducting flake, offering fresh perspectives for the development of superconducting devices and potential circuits.


INTRODUCTION
Nonreciprocal charge transport is a phenomenon commonly observed in semiconductors [1,2] , characterized by an electron-hole asymmetric junction that produces an asymmetric current in response to positive and negative voltages.This behavior finds extensive applications in electronic devices, including diodes, a.c./d.c.converters, optical isolators, circulators, and microwave diodes across a wide frequency spectrum [3][4][5] .Among these, the p-n junction, a well-established device in logic and computation, is structured by the hetero-interface of a p-type and an n-type semiconductor, resulting in an asymmetric current-voltage characteristic (IVC).However, the applicability of semiconductor junctions in quantum circuits is limited by the requirement to operate at extremely low temperatures to avoid thermal excitation.To address this challenge, a practical solution is found in the nonreciprocal supercurrent, known as the superconducting diode effect (SDE) [6] .The SDE exhibits nonreciprocity in non-dissipative superconducting current, allowing it to flow exclusively in one direction.Serving as the superconducting counterpart to a semiconducting diode, the SDE has the potential to emerge as a novel non-dissipative circuit element, akin to traditional diodes.This characteristic of SDE opens up exciting possibilities in the realms of superconducting electronics [7] , superconducting spintronics [8,9] , and quantum information and communication technology [10,11] .
In recent years, various methods have been proposed to realize the SDE.The initial discovery of SDE was reported by F. Ando et al., who investigated a junction-free superconducting [Nb/V/Ta]n superlattice, breaking both spatial-inversion and time-reversal symmetries [12] .Subsequent research by the same team introduced another implementation of zero-field SDE using noncentrosymmetric [Nb/V/Co/V/Ta] superconducting films with 20 multilayers, demonstrating the achievability of fieldfree SDE through noncentrosymmetric superconductor/ferromagnet multilayers [13] .A notable advancement stems from the study of a NbSe2/Nb3Br8/NbSe2 Josephson junction, functioning as a field-free SDE due to the asymmetric Josephson tunneling induced by rotational symmetry breaking from Nb3Br8 on NbSe2/Nb3Br8 interfaces [14] .Moreover, a supercurrent diode effect was observed in few-layer NbSe2 sandwiched between BN.This observation results from the breaking of inversion symmetry caused by the presence of a few layers of NbSe2 [15] .The Ising superconductivity nature of the few-layer NbSe2 or rotational symmetry breaking on the NbSe2/BN interfaces may dominate the mechanism.It is crucial to note that prior instances of SDE were based on rotational or time-reversal symmetry breaking on the interface of a two-dimensional superconductor through other quantum material [16][17][18] .This approach requires a complex technique for interface control, especially concerning electron or magnetic correlation effects on the interface.Therefore, there is a need to explore intrinsic SDE in noncentrosymmetric superconductors.Remarkably, a few-layer NbSe2 has been identified as a noncentrosymmetric superconductor, possessing unique intrinsic Ising-type spin-orbit coupling with a locked electron spin along the out-of-plane axis.Consequently, the pairing symmetry can also be disrupted to generate a nonreciprocal supercurrent [19] .Thus, the observation of intrinsic SDE in lowdimensional patterned NbSe2 appears highly promising.
This study presents the observation of intrinsic Superconducting Diode Effect (SDE) in a NbSe2 nanobridge, created using a focused helium ion microscope.Nonreciprocal critical currents are evident without the need for artificially breaking inversion symmetry when a nonzero magnetic field is applied.
I-V mapping illustrates the asymmetry of the critical current (Ic) concerning the magnetic field under bias polarities at Bz= ± 0.2 mT.A demonstrated example of a bias-polarity-controlled superconductivity diode is provided.

MATERIALS AND METHODS
Thin NbSe2 flakes are mechanically exfoliated onto a polydimethylsiloxane (PDMS) film positioned on a glass slide.Subsequently, the PDMS is stamped onto the pre-prepared electrode using a micromanipulator located beneath a microscope.The sample transfer process is conducted within a glove box filled with argon gas.
The microbridge sample was fabricated using a Laser Direct-Write Lithography System (Durham Magneto Optics Ltd) and a reactive ion beam etching system (Advance Vacuum Scandinavia AB).The ẑ axis is parallel to the stacking direction of NbSe2.Subsequently, the sample was introduced into a Zeiss Orion NanoFab helium ion microscope, and a 30 kV helium beam was scanned across the microbridges to create nanobridges.A lower linear fluence may slightly influence the superconductivity of NbSe2, whereas a higher linear fluence induces insulating behavior in the barrier.Within these extremes, linear fluences (600 ions/nm in this sample) were determined to selectively diminish superconductivity to a certain depth, effectively reducing the dimension of NbSe2.The regions bombarded by helium ions can be identified by SEM (shown in Figure 1c, d).
The as-patterned nanobridge measures approximately 15 nm in length and 1 μm in width, depicted as a red and blue gradient area in Figure 1a.The red area signifies the region where superconductivity is disrupted, while the blue area represents NbSe2 reaching the two-dimensional (2D) limit.A schematic diagram of the helium-ion-beam milling process of NbSe2 is presented in Figure 1d.During the process, He + implantation into the NbSe2 crystal can lead to preferential sputtering of selenium atoms within the two-dimensional transition-metal dichalcogenides (TMDs) [20], resulting in the degradation of superconductivity in the irradiated region (red area).Notably, vacancy defects caused by recoil ions serve as the primary source of superconductivity disruption.Consequently, the surface, in contrast to the underlying NbSe2, can be considered nearly undamaged.As the radiation linear fluence increases, the influence depth of He ions extends, subsequently reducing the thickness of the superconducting NbSe2.This process may facilitate the formation of Ising pairing in few-layer NbSe2 (blue area).
Electrical transport measurements were conducted utilizing a Physical Property Measurement System (PPMS-Dynacool, Quantum Design) with the external electric meter comprising a Keithley 2400 as the current source and a Keithley 2182 as the voltage meter.The investigation of transport properties employed conventional four-terminal methods.

RESULTS AND DISCUSSION
In monolayer or few-layer NbSe2, the in-plane inversion symmetry is disrupted.Consequently, the concurrent impact of the Zeeman effect and substantial intrinsic spin-orbit interactions gives rise to an electron-spin-locking phenomenon along the out-of-plane direction [19,[21][22][23] (Figure 1e).Ising-type superconducting pairing symmetry emerges in NbSe2 due to the reverse spin splitting within the valence bands near valleys K and K'.This phenomenon gives rise to intervalley spin-momentumlocked spin-singlet Cooper pairing between two electrons, characterized by opposing momenta and antiparallel out-of-plane spins.
The directions of the inversion symmetry break () and the current ( ̂) have been illustrated in Figure 1e.Consequently, the supercurrent diode behavior of this nanobridge can be influenced by the out-ofplane component (Bz) of the magnetic field.When the applied magnetic field (B), electric current (), and polar axis () are mutually orthogonal, the critical current (Ic) magnitude depends on both B and , leading to a magnetic chiral effect [24] .Therefore, the supercurrent diode behavior is determined by the out-of-plane component Bz.Coupling (SOC) is likely ineffective, and it is reasonable to assume the preservation of inversion symmetry [22] .The minimal irradiation of 300 ions/nm appears insufficient to impact the superconductivity of the nanobridge region, as evidenced by the absence of nonreciprocal transport.
As the irradiation linear fluence increases to 600 ions/nm, a robust SDE becomes evident, as depicted in the I-V curves in The I-V loop further excludes the influence of thermal effects (shown in Figure S2).It is essential to note that the nanobridge appears to be slightly damaged under helium ion irradiation, as evidenced by optical microscope and scanning electron microscope images (Figure 1c, d).A zoom-in image can clearly show the area of helium ion radiation, framed by a dashed rectangle.We hypothesize that helium ions might influence chemical bonds or even induce doping [25,26] , rather than causing a direct etching effect.Similar to the outcomes of electron beam irradiation [27] , helium ion irradiation can  The substantial rise in the critical current is vital as it dismisses the possibility of the nonreciprocal supercurrent originating from Joule heating [15] .Although the origin of the supercurrent diode effect in NbSe2 requires further investigation, several potential explanations can be considered.These include the interplay between Meissner currents and barriers for vortex entry [28,29] , vortex flow in asymmetric pinning potentials [19] , and the influence of valley-Zeeman spin-orbit interaction [15] .It is noteworthy that the observed behavior closely resembles the characteristics of NbSe2 in the 2D limit [19,21,22,30,31] , despite our sample having a thickness of approximately 15.4 nm.However, upon dimension reduction in the nanobridge, phenomena akin to those observed in Josephson junctions emerge (refer to Figure S4).It is noteworthy that in non-uniform Josephson junctions, the dominance of Josephson vortex motion leads to an asymmetric Fraunhofer pattern [32,33] .A recent experimental study has successfully achieved SDE by introducing non-uniform current and a single Abrikosov vortice in the junction electrodes [34] .
Actually, the non-uniform penetration of helium ions can induce non-uniform supercurrent, even when the current is uniform based on the device's geometry, as shown in

Figure 1a
Figure 1a illustrates the schematic representation of the typical device.The chosen NbSe2 crystal for this study has a thickness of approximately 15.4 nm (shown in Figure S1).Electrical contacts and microbridge strips (1 μm wide and 4 μm long) were defined through conventional photolithography, as depicted in Figure 1b.The narrow channel serves the purpose of establishing a well-defined current direction in the constriction, indicated by the orange arrow in Figure 1a and defined as the x̂ direction.

Figure 1 .
Figure 1.a: Schematic illustration of the measurement configuration.The orange arrow indicates the supercurrent pathway, and the red laser indicates the focused helium ion beam creating a weakly coupled junction in the multilayer NbSe2.The central constriction is about 1 μm wide and 15 nm long.The direction of the supercurrent is defined along the x-axis, while the z-axis is perpendicular to the crystal plane.b: Artistic representation of the focused helium ion beam etching the multilayer NbSe2.The crystal structure is that of NbSe2.c: SEM (upper) and optical (lower) image of the device.The nanobridge is made by ion beam etching, and an Ohmic contact is made by the pick-up and transfer method; scale bar, 20 μm.d: A zoom-in image of the dashed rectangular area shown in b.The color change reflects the change in electrical conductivity, which can be observed.e: Illustration depicting type-I Ising superconductivity: the pairing of electrons in valleys with opposite spin splitting.̂ ( ̂) refers to the direction of space inversion symmetry breaking,  ̂ () refers to the direction of time inversion symmetry breaking, and  ̂ ( ̂) refers to the current direction.

Figure 2
Figure 2 illustrates the nonreciprocal transport properties of the NbSe2 nanobridge under varying irradiation linear fluences.Figures 2 a-c depict three pairs of current-voltage (I-V) characteristics corresponding to the magnetic fields applied along -axis.I-V curves are recorded under Bz = 0 (top), Bz = 0.7 mT (middle), Bz = -0.7 mT (bottom) for three linear fluences.All curves represent zero-tofinite (either positive or negative) bias sweep directions, mitigating potential heating effects.The orange (blue) curve denotes the current density in the nanobridge oriented towards the positive (negative)  ̂ direction.Two additional devices, fabricated with NbSe2 of identical thickness, were irradiated at different linear fluences.Figure 2a and 2b display the I-V curves for the devices under irradiation linear fluences of 0 and 300 ions/nm.Notably, nonreciprocal charge transport is absent in these samples.Given the intrinsic thickness of the NbSe2 flake at 15.4 nm, the Ising Spin-Orbit Figure 2c.Notably, a significant disparity exists in linear fluences between  c + and | c − | in the critical current for the two supercurrent orientations.The sign of ∆ =  c + − | c − | changes with the reversal of the magnetic field Bz, confirming that ∆ is intrinsically determined by the magnetic field.
introduce disorders into thin crystals, potentially suppressing superconductivity.Temperature dependence of resistance (Figures 2 d-f) exhibits a broadened superconducting transition region with increasing linear fluences, which indicates the degradation of superconductivity in nanobridge.Consequently, our focus shifts to the study of the sample under 600 ions/nm irradiation.The results of another two devices (Device#4 and Device#5) have been shown in Figure S3-5.

Figure 2 .
Figure 2. a-c: Current-voltage characteristics were measured at 2 K, considering opposite bias polarities (current directions) in a 4-terminal configuration, under zero magnetic field Bz = 0 T, which is always applied parallel to z direction.The bias sweep direction consistently proceeds from 0 to a finite bias.But for Bz = -0.7 mT. a and b: Measurement for 0 ions/nm and 300 ions/nm sample.c: 600 ions/nm sample.There is a difference between two critical currents.With the orientation of the magnetic field reversed, the roles of the two bias polarities are also exchanged.d-f: The temperaturedependent resistance under different linear fluences ranging from 0 ions/nm to 600 ions/nm.

Figure S6 .
This inhomogeneity can amplify the SDE, particularly in the presence of a large critical current density, 26.7  •  −2 in this instance.

Figure 3 .
Figure 3. a: A color map of critical current in the plane of magnetic field at 2 K, and a color bar shows the value of dV/dI.An apparent current asymmetry with respect to the magnetic field can be observed.The critical current can be extracted as shown in c. b: A magnification of a.The red dotted line marks the zero-field position.c: Nonreciprocal critical current (Ic) as a function of Bz, with positive bias (orange) and negative bias (red), exhibiting symmetry across reflection about Bz= 0. The critical current peaks at a nonzero |Bzmax| = 0.2 mT, indicated by labeled orange and blue arrows.

Figure 4 .
Figure 4. Controllable superconducting diode.The top panel displays the square-wave excitation applied at 2 K with an amplitude of 340 μA (between  c + and | c − |) under a magnetic field of 0.2 mT.The coincidentally measured junction voltage is presented in the bottom panel.In this depiction, the blue shaded region represents the superconducting state, where the voltage remains zero under negative current bias.Conversely, the white region signifies the normal state, with a high voltage observed during positive current bias.The red dotted line denotes the zero line.