Magnetic Proximity-Induced Superconducting Diode Effect and Infinite Magnetoresistance in van der Waals Heterostructure

We report unidirectional charge transport in a $\mathrm{NbSe_2}$ noncentrosymmetric superconductor, which is exchange-coupled with a $\mathrm{CrPS_4}$ van der Waals layered antiferromagnetic insulator. The $\mathrm{NbSe_2/CrPS_4}$ bilayer device exhibits bias-dependent superconducting critical-current variations of up to $16\%$, with the magnetochiral anisotropy reaching $\sim 10^5\mathrm{\ T^{-1}A^{-1}}$. Furthermore, the $\mathrm{CrPS_4/NbSe_2/CrPS_4}$ spin-valve structure exhibits the superconducting diode effect with critical-current variations of up to $40\%$. We also utilize the magnetic proximity effect to induce switching in the superconducting state of the spin-valve structure. It exhibits an infinite magnetoresistance ratio depending on the field sweep direction and magnetization configuration. Our result demonstrates a novel route for enhancing the nonreciprocal response in the weak external field regime ($<50\mathrm{\ mT}$) by exploiting the magnetic proximity effect.


I. INTRODUCTION
Nonreciprocal charge transport, where the electrical resistivity varies depending on the current and magnetic-field direction, has attracted considerable attention over the past decade.
In a system with both broken inversion and time-reversal symmetries, the nonlinear resistance R, in general, quantify the nonreciprocal transport, which is phenomenologically expressed as [1][2][3] where B represents the external magnetic field, and I denotes the electric current.The second term represents nonreciprocal transport, which depends on the directions of both I and B. The coupling coefficient  , which is known as magnetochiral anisotropy (MCA), defines the strength of the aforementioned dependence.Typical metals exhibit small MCA values ranging from 10 -3 to 10 0 T -1 A -1 because the spin-orbit interaction and magnetic energies are nominally far lower than the kinetic energy of the electrons, typically within a few electronvolts of the Fermi energy EF [1,2,7,10].However, measuring the MCA in non-centrosymmetric superconducting materials such as transition-metal dichalcogenides (TMD) is a promising route for enhancing the MCA by orders of magnitude (~10 4 T -1 A -1 [4][5][6]) compared with the normal state, by replacing the role of EF by superconducting gap [4,7].Therefore, nonreciprocal charge transport is potentially suitable for developing phase-coherent and direction-selective electronic devices, and a proof-of-principle demonstration has been performed for a broadband antenna [5].
The nonreciprocity quantified by the MCA generally arises from finite inequivalent resistances under different transport directions.In contrast, exotic nonreciprocity can be achieved if the critical current becomes nonreciprocal.In such a case, the resistance is quenched for one direction while remaining finite for the other.If successful, this exotic nonreciprocity can be used to develop dissipation-less electric circuits [18].The recent discovery of such a superconducting diode effect (SDE) in a Nb/V/Ta epitaxial film has prompted research on the nonreciprocal transport in superconductors [8].In subsequent experiments, the SDE was observed in non-centrosymmetric superconductors [19], artificially nanofabricated systems such as a Josephson junction [20], and a non-centrosymmetric superconducting film with conformal-mapped nanoholes [21].
In general, nonreciprocal transport is observed using an external magnetic field for breaking time-reversal symmetry.Several studies have been performed on enhancing the MCA near the superconducting-normal state transition [22].However, the commonly adopted approach typically requires a large magnetic field, on the order of a few teslas [4][5][6], which poses a challenge for practical application.This technical bottleneck provides a solid impetus to developing an alternative method for facilitating time-reversal symmetry breaking.
The proximity effect in superconductor and ferromagnet (S/F) heterostructures offers a possible solution to a large magnetic field requirement.In the case of a superconductor in close contact with a ferromagnet, the magnetic exchange field of the ferromagnet is expected to penetrate into the superconducting layer [23,24].For example, the exchange field in the Sferromagnet insulator (FI) heterostructure has been adequately understood based on this inverse proximity effect [25,26].Exchange fields of a few teslas have often been observed in the spin splitting of the quasiparticle density of states in EuS/Al bilayers [27][28][29] and EuO/Al bilayers [30].This large exchange field paves the way for developing novel superconducting spintronic devices.For instance, superconducting spin-valve structures exhibiting infinite magnetoresistance (MR) have been recently fabricated [31,32].
Here, we show that when a magnetic proximity-coupled van der Waals (vdW) heterostructure is used, the exchange field effectively reduces the otherwise large external magnetic field requirement while preserving or even enhancing the MCA in a noncentrosymmetric superconductor.Using a recently developed polymer-based strong adhesive transfer technique [33], we manufactured a NbSe2/CrPS4 bilayer heterostructure and a CrPS4/NbSe2/CrPS4 trilayer spin-valve structure to leverage the magnetic proximity effect of the adjacent CrPS4 layer-an A-type layered antiferromagnetic insulator (AFI) below 0.7 T B  [34,35].The key observation was that a nonreciprocal SDE with a critical-current variation of up to 16% under a small external magnetic field of <50 mT in the bilayer device and up to 40% in the trilayer device.Moreover, an infinite MR was observed in the trilayer device under specific probe currents, which had never been demonstrated in a vdW heterostructure or S/AFI heterostructure.

A. Device fabrication
The NbSe2/CrPS4 bilayer and CrPS4/NbSe2/CrPS4 trilayer devices are fabricated using the polycaprolactone (PCL) dry-transfer method [33].Specifically, the exfoliated layers of NbSe2 were picked up by the PCL stamp and dropped onto the layers of CrPS4 to minimize the polymer residue at the interface.Then, Ti (5 nm)/Au (70 nm) electrodes were evaporated after the conventional e-beam lithography.

B. Transport measurements
The transport measurement was performed using a commercial variable temperature cryostat (Teslatron PT, Oxford Instruments) with a base temperature of 1.5 K T = . We performed two different types of four-probe transport measurements in this study.The first was total resistance (R in Eqn. ( 1)) measurement using the direct-current voltage V with a Keithley 2182A Nanovoltmeter.The second measurement method involved using the first and second harmonic alternating-current resistances (R0 in Eq. ( 1) and nonlinear correction to R0 due to nonreciprocity, respectively) with a standard lock-in amplification technique (Signal Recovery, 7265).Finally, the device was field-cooled at 8 T B = to increase the domain size of the CrPS4 layers [27,29,[36][37][38].As shown in Fig. 1a, the lattice structure of monolayer 2H-NbSe2 possesses intrinsically broken in-plane inversion symmetry originating from inequivalent Nb and Se sites.

A. Nonreciprocal response in the bilayer device
Consequently, the itinerant electron of NbSe2 is subjected to an effective out-of-plane magnetic field through Ising-type spin-orbit coupling.This effective magnetic field causes the spindependent valley splitting of NbSe2 in Fig. 1b, as theoretically predicted [39] and experimentally confirmed using photoemission spectroscopy [40,41].With locally broken inversion symmetry and spin-valley locking, the nonreciprocal behavior is expected to occur when the time-reversal symmetry is broken by a magnetic field perpendicular to the plane.MCA can then be quantified using R2ω, as follows [4]: where the external field B is used to estimate  solely from experimentally given conditions.
The observed R2ω is different from that observed in the few-layer NbSe2 of the previous study [5].Previously, nonzero R2ω in NbSe2 , which was attributed to the resistive vortex flow regime, started to appear at > 1 T B for T < 4 K [5].However, in our case, nonzero R2ω appears below < 0.06 T B for T < 4 K.We ascribe the unusual behavior of 2 R  to the magnetic ordering of CrPS4.As mentioned, A-type AFI CrPS4 layers consist of antiferromagnetically coupled ferromagnetic monolayers for < 0.7 T B at 1.5 K T = [34,35].Furthermore, because the proximity effect in the S/FI insulator is confined to the interface by the exponential decay of the electronic wave function in the insulator, the exchange coupling between the ferromagnetic CrPS4 monolayer and NbSe2 is expected to decay rapidly within the atomic distance [23,25,31].Therefore, the antiferromagnetically coupled upper layers do not effectively offset the exchange field provided by the nearest ferromagnetic CrPS4 monolayer, resulting in a significant exchange field dominated by the magnetization of the nearest monolayer.This exchange field alone does not affect the orbital dynamics because the exchange field couples only with the quasiparticle spin and does not induce a Lorentz force.
Additionally, the antiferromagnetic nature of CrPS4 results in a negligible stray field [44].Thus, the stray field of CrPS4 and the applied magnetic field do not provide the net magnetic field required to achieve the vortex flow regime, which is > 2 T B for a few-layer NbSe2 at a temperature below 2 K [5].From this perspective, our observation of 2 R  differs significantly from previously reported vortex dynamics-based 2 R  on NbSe2.We next explore the response of the similar proximity-coupled structure with the increased exchange field by CrPS4/NbSe2/CrPS4 trilayer spin valve structure shown in Fig. 3a.
The device is fabricated using the previously mentioned PCL stamp technique, where the picked-up material stack was dropped directly onto pre-patterned Pt(18 nm)/Ti(2 nm) electrodes to avoid contamination. in Fig. 3e, with the noisy stochastic fluctuating behavior depicted in the background of the averaged signal.This stochastic fluctuation may be related to the phase slip line (PSL) and kinematic vortices [45][46][47], the existence of the metastable state near the superconductor-normal state transition [48], or the domain structure of the proximity-coupled CrPS4 and their dynamics to be discussed below.Nonetheless, the overall signal consistently showed the field-dependent switching behavior, as shown in the averaged profile, demonstrating an infinite MR.
We note that the observed c I change near 0 T B = is related to the relative alignment of the adjacent CrPS4 layer.As mentioned previously, CrPS4 provides an effective exchange field with the dominant effect of the nearest ferromagnetic layer.Thus, the system can be effectively identified with a FI/S/FI trilayer structure, and the average exchange field on the quasiparticles is defined as follows: ( ) ( ) where S  represents the ferromagnetic spin multiplied by the coupling constant,  represents the angle between the magnetizations of the two ferromagnetic layers, and a and s d represent the lattice constant and the thickness of the superconductor, respectively [25].In our device, the coercive field of the nearest CrPS4 layers is tuned according to their thicknesses.This condition allows the switching of the device between parallel (P) and antiparallel (AP) configurations through a gradual reduction in the magnetic field.Thus, in the P configuration ( 0  = ), the net exchange field between two CrPS4 layers results in a significant exchange field on NbSe2.
 = ), the net exchange field on NbSe2 is canceled out [31,32].This magnetization reversal of the nearest CrPS4 layers considerably changes the exchange field on NbSe2 near

IV. DISCUSSION
The bilayer device shows no significant difference from the pure NbSe2 device except for the five-fold decrease of the magnetic field for the maximal %Ic and the unusual behavior , which is close to the phase transition of CrPS4 between the A-type antiferromagnetic phase and the canted antiferromagnetic phase [35,49].This vanish of the fluctuation also reveals the connection between the Ic and the magnetization of CrPS4.
However, the exact reason for the magnetization fluctuation remains unclear.We suspect the fluctuation is possibly related to the layer-by-layer switching behavior, as observed in certain layered antiferromagnets [36,37] and ferromagnets [50].Further research on the magnetic structure of CrPS4 is required to elucidate the underlying principle for the fluctuation of Ic, and we leave it for future work.
We next discuss the SDE observed in both devices.As previously noted, the effect of CrPS4 on the Ic of the trilayer structure is revealed by the Ic fluctuation.Furthermore, we also observed that the SDE in NbSe2 disappears when the nearest CrPS4 layers are not present in another bilayer structure [49].These observations suggest that the observed SDE in the based on the FMCP scenario [49,51,52].Additionally, the theory predicts the %Ic of a magnetic proximity-coupled Ising superconductor to be proportional to the magnitude of the exchange field [49].Considering that the effective exchange field of the trilayer structure in the P configuration is expected to be twice that of the bilayer structure, the observed increase of %Ic in the trilayer device in this work compared to the bilayer device is seemingly consistent with the FMCP-based theory.However, it should be noted that the %Ic observed in another device falls short of half of that of the trilayer structure, and that the parameters, including layer numbers and transport direction, are not controlled in the experiments.Moreover, a recent study that observed SDE in pure NbSe2 reports that %Ic shows uncorrelated random behavior regardless of the layer number and supercurrent-to-lattice orientation [19].Thus, further experimental and theoretical investigations are required to fully comprehend the underlying microscopic origin behind the observed behavior.
In conclusion, we demonstrated that the magnetic proximity effect in the NbSe2/CrPS4

Figure 1 .
Figure 1.(a) Structure of non-centrosymmetric NbSe2.Inset: Side view of the structure of

Figure 2 . 2 R 2 R
Figure 2. (a) Voltage drop across the NbSe2 layer with respect to the current and magnetic field of the reduced B field compared with that observed in previous studies[4][5][6].A monotonic decrease in  as a function of T is observed before  becomes negligible at approximately

B. Nonreciprocal response in the trilayer deviceFigure 3 .
Figure 3. (a) Optical microscope image of the trilayer spin-valve device.The boundaries of

Fig. 3b showsI
Fig.3bshows the current-voltage (I-V) characteristics of the device as a function of

I
because it disrupts the Cooper pairs, the analogous switching behavior of coupling between the NbSe2 and CrPS4 layers, along with the hysteresis appearing in the colormap of Fig.3d.This hysteresis becomes more apparent when the field is swept with the fixed current as in Fig.3e, which could be distinguished from the stochastic fluctuation.

of 2 R
 [19].However, the trilayer device shows several exotic features, including infinite MR near 0 T B = that we regard as a fingerprint of the magnetic proximity effect.Another novel feature of the trilayer device is the fluctuation of Ic in the colormap of Fig. 3b.If we interpret this fluctuation as a Fraunhofer-like aperiodic pattern of a Josephson junction, the weak link must have a width of ~10 nm for the ~4-µm-long device.Since there are no reasonable candidates for such a weak link, the fluctuation of Ic most plausibly originates from the fluctuation of the magnetization of CrPS4.In addition, we observe that the Ic fluctuation vanishes 0.7 T B 

NbSe2/
CrPS4 heterostructure is most likely induced by the magnetic proximity effect of the adjacent CrPS4 layer.Although theoretical analyses on the SDE observed in Junction-free superconductors are still in their infancy, recent theories propose finite-momentum Cooper pairing (FMCP) as the microscopic origin of intrinsic SDE.Therefore, it is worth noting that the %Ic of the trilayer device exhibits an abrupt increase near Ic and saturates far below Tc, in line with the theoretically expected formula1 heterostructure provides a large exchange field to induce clear nonreciprocal transport.The NbSe2/CrPS4 bilayer device exhibited the SDE, where the magnetochiral c I We also observed an infinite MR and SDE in the CrPS4/NbSe2/CrPS4 spin-valve structure, with % c I reaching ~40 %.Finally, we expect the time-reversal symmetry breaking caused by the magnetic proximity effect to provide a novel route for investigating the potential application of nonreciprocal transport under small (or even zero) external magnetic fields.