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Pengyuan Shi, [Xiaoyu Wang](https://orcid.org/0009-0002-8450-4937), Lihao Zhang, Wenqin Song, Kunlin Yang, [Shuxi Wang](https://orcid.org/0000-0002-1789-4376), Ruisheng Zhang, Liangliang Zhang, [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), Sen Yang, Lei Zhang, [Lei Wang](https://orcid.org/0000-0003-3509-4011), Wu Shi, [Jie Pan](https://orcid.org/0000-0002-5024-9192), [Zhe Wang](https://orcid.org/0000-0001-5664-2932)

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[Magnetoresistance Oscillations in Vertical Junctions of 2D Antiferromagnetic Semiconductor <math display="inline">  <mrow>    <msub>      <mrow>        <mi>CrPS</mi>      </mrow>      <mrow>        <mn>4</mn>      </mrow>    </msub>  </mrow></math>](https://mdr.nims.go.jp/datasets/6ba6bf2c-e702-4fbe-b92e-027019847e46)

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Magnetoresistance Oscillations in Vertical Junctions of 2D Antiferromagnetic Semiconductor CrPS4Magnetoresistance Oscillations in Vertical Junctionsof 2D Antiferromagnetic Semiconductor CrPS4Pengyuan Shi,1,* Xiaoyu Wang ,1,* Lihao Zhang,1 Wenqin Song,2 Kunlin Yang,2 Shuxi Wang ,1Ruisheng Zhang,1 Liangliang Zhang,3 Takashi Taniguchi ,4 Kenji Watanabe ,5 Sen Yang,1 Lei Zhang,1Lei Wang ,6 Wu Shi,2,7 Jie Pan ,1,† and Zhe Wang 1,‡1MOE Key Laboratory for Nonequilibrium Synthesis and Modulation of Condensed Matter,Shaanxi Province Key Laboratory of Advanced Materials and Mesoscopic Physics, School of Physics,Xi’an Jiaotong University, Xi’an 710049, China2State Key Laboratory of Surface Physics and Institute for Nanoelectronic Devicesand Quantum Computing, Fudan University, Shanghai 200433, China3State Key Laboratory for Manufacturing Systems Engineering,Xi’an Jiaotong University, Xi’an 710049, China4Research Center for Materials Nanoarchitectonics,National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan5Research Center for Electronic and Optical Materials,National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan6Key Laboratory of Quantum Materials and Devices of Ministry of Education, School of Physics,Southeast University, 211189, Nanjing, China7Zhangjiang Fudan International Innovation Center, Fudan University, Shanghai 201210, China(Received 14 April 2024; revised 26 October 2024; accepted 8 November 2024; published 13 December 2024)Magnetoresistance (MR) oscillations serve as a hallmark of intrinsic quantum behavior, traditionallyobserved only in conducting systems. Here we report the discovery of MR oscillations in an insulatingsystem, the vertical junctions of CrPS4 which is a two-dimensional A-type antiferromagnetic semi-conductor. Systematic investigations of MR peaks under varying conditions, including electrode materials,magnetic field direction, temperature, voltage bias, and layer number, elucidate a correlation between MRoscillations and spin-canted states in CrPS4. Experimental data and analysis point out the important role ofthe in-gap electronic states in generating MR oscillations, and we propose that spin selected interlayerhopping of localized defect states may be responsible for it. Our findings not only illuminate the unusualelectronic transport in CrPS4 but also underscore the potential of van der Waals magnets for exploringinteresting phenomena.DOI: 10.1103/PhysRevX.14.041065 Subject Areas: Electronics, SpintronicsI. INTRODUCTIONMagnetoresistance (MR) is a phenomenon widely stud-ied for its demonstration of intriguing physics and potentialapplications in industrial technologies. Magnetoresistanceoscillation, characterized by nonmonotonic changes multi-times in resistance with respect to an external magneticfield, often manifests the inherent quantum mechanicalnature of the studied system. The well-known examplesinclude Shubnikov–de Haas oscillations, where the elec-tron density of states on the Fermi surface varies withmagnetic field, and Aharonov-Bohm effect, where thephase of conducting electrons shift by an external magneticfield [1]. However, these phenomena usually present inconducting systems, and observations of MR oscillations ininsulating systems are rare.MR of insulating magnetic systems actually has been afocus of research over the past decades, forming thefoundation of spintronics, which has had a significantimpact on current information technologies [2–4]. Oneclassic model is the magnetic tunneling junction, consistingof two ferromagnetic metals separated by a thin insulator.Electron transport in such systems occurs via tunneling,where resistance depends on the alignment of electronmagnetizations [2,4,5]. Another notable model is the spinfilter of magnetic semiconductor, characterized by its spin-split band structure, thus leading to different tunneling*These authors contributed equally to this work.†Contact author: jiepan@xjtu.edu.cn‡Contact author: zhe.wang@xjtu.edu.cnPublished by the American Physical Society under the terms ofthe Creative Commons Attribution 4.0 International license.Further distribution of this work must maintain attribution tothe author(s) and the published article’s title, journal citation,and DOI.PHYSICAL REVIEW X 14, 041065 (2024)2160-3308=24=14(4)=041065(11) 041065-1 Published by the American Physical Societyhttps://orcid.org/0009-0002-8450-4937https://orcid.org/0000-0002-1789-4376https://orcid.org/0000-0002-1467-3105https://orcid.org/0000-0003-3701-8119https://orcid.org/0000-0003-3509-4011https://orcid.org/0000-0002-5024-9192https://orcid.org/0000-0001-5664-2932https://ror.org/017zhmm22https://ror.org/013q1eq08https://ror.org/017zhmm22https://ror.org/026v1ze26https://ror.org/026v1ze26https://ror.org/04ct4d772https://ror.org/013q1eq08https://crossmark.crossref.org/dialog/?doi=10.1103/PhysRevX.14.041065&domain=pdf&date_stamp=2024-12-13https://doi.org/10.1103/PhysRevX.14.041065https://doi.org/10.1103/PhysRevX.14.041065https://doi.org/10.1103/PhysRevX.14.041065https://doi.org/10.1103/PhysRevX.14.041065https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/barrier heights for different spins. After the pioneeringworks on Eu-based systems [6,7], recent advancements in2D magnetic semiconductors [8–13], particularly in sys-tems like CrI3 [14–17], have demonstrated substantialprogress in achieving large magnetoresistance. In bothmagnetic tunneling junctions and spin filters with magneticsemiconductor, MR oscillation is not expected as resistancetypically correlates monotonically with the total magneti-zation of the system.Here we report observations of MR oscillations invertical junctions of 2D antiferromagnetic semiconductorCrPS4. While the total magnetization of few-layer CrPS4 ismonotonically changed by magnetic field, several well-defined resistance peaks appear in the MR measurements.The MR oscillation is robust against different directions ofapplied magnetic field, and smoothly vanishes as thetemperature goes above Néel temperature of CrPS4. NoMR oscillation is observed when the junction barrier isthinned down to monolayer, which is a ferromagneticsemiconductor. These findings demonstrate that the MRoscillations are correlated to the spin-canted states in thesystem. We further discuss the possible transport mecha-nism in CrPS4 and point out that the in-gap states wouldplay an important role.II. EXPERIMENTAL RESULTSCrPS4 is a van der Waals layered semiconductor withband gap of around 1.3 eV [18–23], the structure ofwhich is depicted in Fig. 1(a). Below the Néel temperatureof 38 K [refer to Fig. 1(b)], Cr atoms within the ab plane ofthe crystal exhibit ferromagnetic coupling, while theinterlayer coupling is antiferromagnetic [24–26], beingA-type antiferromagnet. Upon the application of a magneticfield along the c axis, the spins undergo a spin-floptransition at approximately 0.7 T, followed by a smoothalignment along the direction of the magnetic field untilreaching the saturation field of approximately 8.25 T.FIG. 1. Basic characteristics of bulk CrPS4 and vertical junctions of few-layer CrPS4. (a) Crystal and magnetic structure of CrPS4,depicting its A-type antiferromagnetic behavior below the Néel temperature with the easy axis along the c axis. Above the spin-flip field,the magnetic moment aligns with the external magnetic field. (b) Temperature dependence of magnetization under zero-field-cooling(ZFC) and field-cooling (FC) conditions, indicating a Néel temperature of around 38 K. (c) Magnetic field dependence of magnetizationwith the magnetic field parallel and perpendicular to the c axis. (d) Schematic of vertical junctions, with graphene serving as electrodesand h-BN used to encapsulate the entire device. (e) IV curves of different vertical junctions at 2 K, all devices exhibiting very largeresistivity. Here 7Lðh-BNÞ denotes device with h-BN inserted between graphene and CrPS4, 7LðNbSe2Þ and 8LðNbSe2Þ representdevices with NbSe2 as electrodes. All other devices use graphene as electrodes. (f) Temperature dependence of resistance of a ten-layerdevice, with a kink clearly identified at the Néel temperature. The upper inset shows the Arrhenius plot, while the lower inset displaysthe optical image of the device.PENGYUAN SHI et al. PHYS. REV. X 14, 041065 (2024)041065-2Conversely, when the magnetic field is applied perpen-dicular to the c axis, no spin-flop transition occurs, and themagnetization increases smoothly with the magnetic fielduntil reaching a saturation field of approximately 8.45 T, asshown in Fig. 1(c). These observations confirm the easyaxis of CrPS4 to be along the c axis, consistent withprevious bulk measurements [25–27]. Based on thespin-flop transition field and the saturation field in twodifferent directions, we estimate the interlayer couplingenergy to be 0.16 meVand the on-site anisotropy energy tobe 0.0045 meV, which are found to be very close to theneutron scattering measurement results [28]. This weakanisotropy energy results in similar spin-canted states formagnetic fields applied along both directions, as evidencedby nearly identical magnetization in both cases.To investigate the electronic transport propertiesof few-layer CrPS4, we fabricated vertical junctions ofgraphene=CrPS4=graphene as illustrated in Fig. 1(d) (seethe Appendix for details of fabrications). Figure 1(e)presents the IV curves of vertical junctions comprisingmultilayer CrPS4 at 2 K. Some thicker devices exhibitclearly nonlinear IV behavior with ultrahigh resistivity atzero voltage bias. Devices exhibiting relatively linear IVbehavior also demonstrate high resistivity, typically over105 Ω cm. These findings indicate the insulating behaviorof our devices at low temperatures, consistent with previousreports on both bulk and thin flakes of CrPS4 [23,29–34].We also fabricated a field-effect transistor to measure thelongitudinal transport (see Supplemental Material Note 1[35]); the resistance at zero gate voltage is very large andbeyond our measurement capability. The insulating state isfurther corroborated by temperature-dependent resistancemeasurements, as depicted in Fig. 1(f). The resistanceincreases by approximately 2 orders of magnitude as thedevice is cooled from room temperature to 50 K. Analysisof the Arrhenius plot [inset in Fig. 1(f)] reveals a thermalactivation energy of 17.8 meV. At lower temperatures, thedeviation from thermal activation transport behavior sug-gests electron transport occurs via tunneling or hoppingprocesses, which will be discussed in detail later. Notably, adistinct feature is observed at the Néel temperature in the Rvs T curve of all devices, confirming the consistent qualityof our atomically thin devices and bulk samples.Having established that our devices exhibit insulatingbehavior at low temperatures, we proceeded to investigatetheir electronic transport behavior under the influenceof a magnetic field. Figure 2(a) illustrates the MR of aten-layer device, defined as MR ¼ ðRH − R0Þ=R0, when amagnetic field is applied parallel to the c axis at 2 K. In thelow magnetic field region, resistance undergoes a suddenchange at approximately �0.6 T, closely matching thespin-flop transition field of bulk CrPS4. In the highmagnetic field region, resistance saturates at around 8 T,aligning with the magnetic field at which the magnetizationsaturates, known as the spin-flip transition field. Theseobservations underscore the correlation between the elec-tronic transport behavior of few-layer vertical junctions andthe magnetic states of CrPS4.MR oscillations are prominently observed in the inter-mediate magnetic field region between the spin-floptransition and the spin-flip transition. The MR exhibitsthree distinct peaks at around 2, 4.7, and 6.4 T, labeled asFIG. 2. MR oscillations in the vertical junction of ten-layer CrPS4. (a) Magnetic field dependence of resistance measured with the fieldparallel to the c axis, obtained with a constant ac bias voltage of 2 mV at 2 K. Clear MR peaks are observed, along with the spin-floptransition at approximately �0.6 T. (b) MR measured with the magnetic field perpendicular to the c axis. The persistence of MRoscillations indicates their origin from CrPS4 rather than the graphene electrodes. (c) MR measured with different dc bias voltages. TheMR peaks gradually shift to lower fields with increasing voltage, as summarized in (d). (e) MR at different temperatures measured withan ac bias voltage of 2 mV. The peak positions shift to lower fields as the temperature increases, as summarized in (f).MAGNETORESISTANCE OSCILLATIONS IN VERTICAL … PHYS. REV. X 14, 041065 (2024)041065-3P1, P2, and P3, respectively. These peaks are evident inboth magnetic field sweep directions and exhibit symmetrywith respect to zero magnetic field. It is noteworthy that themagnetization of bulk CrPS4 monotonically changes withthe magnetic field, as previously demonstrated in Fig. 1(c).Similarly, the magnetization of few-layer flakes would alsoexhibit monotonic changes when considering the totalmagnetic energy of the system (see SupplementalMaterial Note 4 and Fig. 15(b) [35]), a behavior exper-imentally verified in various few layer A-type antiferro-magnets [36,37]. In this context, the observation of MRoscillations in CrPS4 vertical junctions is unexpected andstimulates further investigation.One plausible explanation for these MR oscillations is achange in the density of states of the graphene electrodesinduced by the perpendicular magnetic field. To explorethis possibility, we conducted MR measurements with amagnetic field applied perpendicular to the c axis, and thecorresponding data are presented in Fig. 2(b). In the lowmagnetic field region, the MR exhibited a smooth variation,as no spin-flop transition was anticipated when the mag-netic field was perpendicular to the easy axis. In the highmagnetic field region, the MR saturated at around 8.2 T,close to the bulk spin-flip transition field. Interestingly, inthe intermediate magnetic field region, the MR oscillationsremained largely unchanged in terms of both peak ampli-tudes and positions.An alternative explanation is that the MR oscillationsmight originate from new states in the graphene electrodesinduced by the proximity effect with magnet CrPS4,persisting even when the magnetic field is perpendicularto the c axis. To test this hypothesis, We conducted twocontrol experiments. First, we fabricated a device with thestructure graphene=h-BN=CrPS4=h-BN=graphene. Theinclusion of thin h-BN layers between the grapheneelectrodes and CrPS4 is intended to largely reduce thewave function overlap between them, thus eliminating anypotential new states in graphene induced by proximityeffect. Second, we constructed another type of device withthe structure NbSe2=CrPS4=NbSe2, where 2D metal NbSe2served as the electrodes, eliminating graphene entirely. Asdetailed in the Appendix, MR oscillations were clearlyobserved in both types of control devices.These experiments provide robust evidence that the MRoscillations originated from CrPS4 itself, rather than fromthe graphene electrodes. In the following we focus on thedevices of graphene=CrPS4=graphene. Figure 2(c) illus-trates the MR measurements of ten-layer device conductedunder different voltage biases at 2 K. The MR oscillationspersisted across all measured biases, albeit with decreasingamplitudes as the bias increased. At a bias of 60 mV, theidentification of the first peaks became less straightforward.Interestingly, the positions of the MR peaks shifted to lowermagnetic fields with increasing bias, as summarized inFig. 2(d).We further investigated the temperature dependence ofthe MR behavior. As shown in Fig. 2(e), the amplitudes ofthe MR oscillations decreased with increasing temperature,ultimately vanishing at around 40 K, close to the Néeltemperature of CrPS4. In addition to affecting the amplitudeof the MR oscillations, increasing temperature also shiftedall three peaks to lower magnetic field, as summarized inFig. 2(f). This temperature dependence contrasts with thatof typical quantumMR oscillations observed in conductingsystems, such as Shubnikov–de Haas oscillations and theAharonov-Bohm effect, where MR peak positions remainindependent of temperature. The similar trend in temper-ature dependence between the MR oscillations and thespin-flip transition suggests a connection to the cantedmagnetic states of CrPS4.Before delving deeper into the origin of MR oscillations,we investigated this behavior in devices with varyingthicknesses. Figure 3(a) presents the MR of 12-layerdevice, characterized by a clearly nonlinear IV curve,indicative of even more insulating compared to ten-layerdevice. Three MR peaks are distinctly observed in bothperpendicular and parallel magnetic field configurations,with the middle peak (P2) dominating in comparison to ten-layer device. Turning to the results of the six-layer device,as shown in Fig. 3(b), only two MR peaks are observed inmost measurements, with an additional small MR peakemerging when the voltage is less than 5 mV (seeSupplemental Material Fig. 9 [35]). This specific flakehas clear crystal crossed with angle of 67.5°, meaning thecrystal edge is a crystal axis [25,38], as indicated in theinset of Fig. 3(b). We applied the magnetic field along thisaxis, and the measured MR is almost the same as the resultof randomly applied in-plane field.We have measured total 11 multilayer devices andFigs. 3(c) and 3(d) summarize the featured magnetic fieldsof all devices (see Supplemental Material for additionalelectronic transport data of devices [35]). Figure 3(c)illustrates MR peak positions measured under variousvoltages at 2 K, where each color represents data fromone device and each symbol denotes the same peak in thatdevice. It is evident that all MR peaks shift to lowermagnetic fields with increasing voltage, and they can beroughly divided into four groups. Figure 3(d) depicts thethickness dependence of the spin-flip field. The dotsrepresent data obtained from the saturation of magneto-conductance in the parallel field configuration, particularlythe bilayer and three-layer device reveals a clear decrease.This reduction aligns with expectations, as the effectiveinterlayer coupling energy per layer decreases. This phe-nomenon can be fully elucidated using the antiferromag-netic linear chain model, as discussed in the SupplementalMaterial [35].Expanding our investigation to the atomically thin limit,we studied monolayer CrPS4, which has been identified asa ferromagnet [25]. Its zero-field temperature dependencePENGYUAN SHI et al. PHYS. REV. X 14, 041065 (2024)041065-4of resistance monotonic increase deviated at around 38 K[the arrow in Fig. 4(a)], which is around the same as theNéel temperature of bulk CrPS4 [25,34,38]. Figure 4(b)presents measurement results with a magnetic fieldperpendicular to the c axis, where no MR oscillationsare observed in this monolayer device. The MR is aroundzero at 2 K and becomes negative as temperature rises.The MR amplitude increases as temperature rises andreaches maximum at around Curie temperature, exhibitinga clear triangle shape when plotted as a function of tem-perature in a 2D format (see Supplemental Material [35]).This behavior closely resembles that observed in verticaljunctions of other 2D ferromagnetic semiconductors,such as CrBr3 [39] and monolayer CrSBr [40]. In contrastto antiferromagnets, no spin-canted states are expectedin ferromagnets. The absence of MR oscillations in themonolayer device thus further confirms that the oscillatorybehavior is related to spin-canted states.III. DISCUSSIONIn summary of the aforementioned experimental inves-tigations across various devices, no MR oscillations areobserved in monolayer device due to its ferromagneticnature. Conversely, for antiferromagnetic few-layer devi-ces, MR oscillations occur within the region betweenspin-flop and spin-flip transitions, with peak positionsshifting in tandem with the spin-flip transition. Thesefindings strongly suggest that MR oscillations are intri-cately linked to the magnetic canted states induced by theapplied magnetic field.To gain deeper insights into the specifics of these cantedstates in multilayer CrPS4, we employ the antiferromag-netic linear-chain model [36,37]. Within this framework,the magnetization of each CrPS4 layer is depicted as amacrospin with an on-site anisotropy energy K, coupledto its nearest neighbors through the interlayer coupling J.The magnetic state under a certain magnetic field can beelucidated by minimizing the total magnetic energy of theFIG. 4. Absence of MR oscillations in ferromagnetic mono-layer CrPS4 device. (a) Temperature dependence of resistance atzero field. The inset shows the IV curve of this vertical junction.(b) MR at different temperatures, measured with a magnetic fieldperpendicular to the c axis; no MR oscillation is observed.FIG. 3. Statistics of MR oscillations in few-layer CrPS4 devices. (a) MR of a 12-layer device measured with a dc bias voltage of0.15 V. Despite being more resistive than the ten-layer device, MR peaks persist in both magnetic field orientations. (b) MR of a six-layer device measured with an ac bias voltage of 10 mV. The inset shows the optical image of the flake, with the vertical junction locatedat the black point. The flake exhibits two sharp edges intersecting at an angle of 67.5°. Magnetic fields are applied parallel to the c axis,ab plane, and a� axis. (c) Summary of MR peak positions for all few-layer devices. Here the seven- and eight-layer devices use NbSe2 aselectrodes. (d) Spin-flip transition fields determined from in-plane MR measurements at 2 K, with the solid line representing calculatedresults based on parameters derived from bulk magnetization data. The layer number is determined using a combination of atomic forcemicroscopy and optical contrast, with approximate �1 layer error for thick CrPS4 flakes.MAGNETORESISTANCE OSCILLATIONS IN VERTICAL … PHYS. REV. X 14, 041065 (2024)041065-5system (detailed in Supplemental Material Notes 3 and 4[35]). This model can nicely describe the layer numberdependence of spin-flip field shown in Fig. 3(d), where thesolid orange line represents the calculated results withparameters derived from bulk magnetization measure-ments. Our analysis reveals that while the total magneti-zation monotonically increases [Supplemental MaterialFig. 15(b)], the canting angle undergoes nonmonotonicchanges as a function of the magnetic field, for example,the canting angle of layer 2 and 5 in the six-layer CrPS4[Supplemental Material Fig. 15(a)].Despite our insights into the magnetic states through theantiferromagnetic linear-chain model, establishing the linkbetween the magnetic states and electronic transport proveselusive. Our calculations of MR based on the tunnelingmodel used in CrCl3 (see details in Supplemental MaterialNote 4 [35]) reveal a monotonic behavior, with nooscillations observed. This points out the importance ofevaluating the predominant transport mechanism in verticaljunctions of CrPS4. Our analysis of the saturation of low-temperature resistance in the Arrhenius plot [see the inset ofFig. 1(f)] suggests that thermally activated electrons are notthe predominant carriers within the examined temperaturerange. These imply the presence of states inside the gap ofCrPS4. This conclusion is also consistent with previousreport on field-effect transistors of CrPS4 [30], where onlymonotonic MR is observed when the Fermi level is tunedinto the bands of CrPS4, as the transport is not mediated bythe in-gap states.The nature of these in-gap states remains elusive.Previous observations of quantum MR oscillations inKondo insulator YbB12 and topological insulator WTe2suggest an unconventional Fermi surface inside the insu-lating gap [41,42], yet their origins remain inconclusive.These oscillations, resembling Shubnikov–de Haas oscil-lations in metals, differ from those in our device, and theband gap opening mechanism in these systems also differsfrom CrPS4. Besides the possibility of these exotic states,defects offer a trivial explanation for in-gap states in ourdevices. At high temperatures, hopping between defectstates leads to thermally active behavior, with the activationbarrier determined by the energy spacing between thesestates [43,44]. This scenario elucidates the ultrasmallthermal activation energy (17.8 meV) obtained from theArrhenius plot, notably lower than the band gap (1.3 eV) ofCrPS4. With these defect states, variable-range hoppingtypically dominates at low temperatures in standard longi-tudinal devices [43,45]. However, given the small distancebetween the two graphene electrodes of our verticaljunctions (few nanometers), direct application of classicvariable-range hopping theory is untenable. Nevertheless,we can conceptualize the transport process in a similarmanner, where electrons tunnel from one defect state toanother suitable defect state, either within the same layeror in neighboring layers, until they reach the grapheneelectrode on the opposite side. The average hopping lengthis determined not only by the sample thickness but also bythe defect density and the energy difference between thesedefects, potentially explaining why some thicker devicesexhibit smaller resistance than thinner devices.To further investigate the correlation between the elec-tronic transport properties and the magnetic states in CrPS4,we calculated the band structure of CrPS4 containing sulfur(S) vacancies, a common type of defect in 2D transitionmetal chalcogenides. As detailed in the SupplementalMaterial [35], our calculations reveal localized states withinthe band gap of CrPS4. Notably, the S vacancy is spinpolarized, aligning with the spin direction of the Cr atomsin the layer. In the antiferromagnetic ground state, thesubstantial energy difference between spin-polarized defectstates in adjacent layers (when considering spin conservedhopping) necessitates electron hopping to more distantlayers, which increases the hopping length and, conse-quently, the resistance. When an external magnetic field isapplied, spin canting reduces the energy difference betweenneighboring layers, facilitating hopping and thereby reduc-ing the hopping length and the resistance. Concurrently,within conventional theory of hopping conduction in amagnetic field [45], the magnetic field compresses the wavefunction of the localized states, decreasing the probabilityof hopping and thus increasing the hopping length andresistance [46]. The interplay between these competingeffects—spin canting and wave function contraction—accounts for the MR peaks observed in our measurements.Additionally, the application of an electric field caneffectively reduce the energy difference between defectstates, leading to a decrease in the hopping length [47–49].This dependence on electric field strength modulates thecompetition between the spin-canting effect and the mag-netic field effect, resulting in shifts in the positions of theMR peaks.The interlayer hopping of spin-polarized defect states inCrPS4 offers a plausible framework to quantitativelyexplain our key findings. We employed principles ofconventional variable-range hopping theory to develop atoy model, the spin selected interlayer hopping model asdetailed in the Supplemental Material [35], to calculate theMR. This model successfully captures the MR peaks andtheir shifts under an electric field. However, it does notaccurately predict the peak positions observed in theexperimental results. The exact MR oscillations’ behavior,such as the MR magnitudes and peak positions, areinfluenced by the intricate behavior of the localized wavefunction in response to the magnetic field, as well asdetailed calculations of hopping resistance. It is importantto note that a comprehensive theory of interlayer hoppinginvolving spin-polarized states in 2þ 1 dimensions—con-sidering the strong anisotropy along the c axis in oursystem—has yet to be developed. Further efforts are neededto clarify the exact nature of the in-gap states and toPENGYUAN SHI et al. PHYS. REV. X 14, 041065 (2024)041065-6establish a complete hopping conduction framework forsuch systems if spin-polarized defect states are indeedfound to be responsible for the observed MR oscillations.At the same time, we note that the position of MR peakscan be more precisely described when we consider theinterference between spin Berry phases. In the simplifiedscenario, the spin can follow two distinct evolutionarypaths in a single hopping process due to the coexistence ofspin-up and spin-down channels. Different spin Berryphases emerge for each of these cyclic quantum evolutionpaths [50]. The interference between these two paths,which is strongly dependent on the spin-canting configu-ration, further affects the overall hopping probability asdetailed in the Supplemental Material. However, moreefforts are needed to further support the existence of spinBerry phase coherence and investigate its precise ori-gin [51].IV. SUMMARYIn conclusion, our study reveals the presence of robustMR oscillations in vertical junctions of few-layer CrPS4.These oscillations persist regardless of the direction of theapplied magnetic field, whether parallel or perpendicular tothe c axis. They are observed across a spectrum of samples,ranging from highly insulating (exhibiting clearly nonlinearIV curves with unmeasurable resistance at zero voltagebias) to those with relatively lower resistance (still in theorder of megohm), as long as the device comprises multi-layer CrPS4. Moreover, the MR peaks exhibit a gradualshift to lower magnetic fields as the temperature increases,ultimately disappearing above the Néel temperature, mir-roring the behavior of the spin-flip transition. This corre-lation underscores the connection between these MRoscillations and the canted magnetic state in CrPS4.While further investigations are necessary to preciselydetermine the proposed spin-polarized defect states andpossible coherence between spin Berry phase, our findingshighlight CrPS4 as a unique example of MR oscillations, aphenomenon rarely observed in insulating systems. Thiswork underscores the potential of 2D van der Waalsmagnets not only for spintronics applications but also asintriguing platforms for exploring novel physics.Note added in proof. Recently, we became aware of similarworks [51,52].ACKNOWLEDGMENTSThis work is financially supported by National NaturalScience Foundation of China (Grants No. 12374121,No. 12304232, No. 12274090, and No. 92265103),Shaanxi Fundamental Science Research Project forMathematics and Physics (22JSY026, 23JSQ011),Natural Science Basic Research Program of Shaanxi(2022JC-DW5-02), the Fundamental Research Funds forthe Central Universities, and the Natural ScienceFoundation of Shanghai (Grant No. 22ZR1406300). S. Y.acknowledge support from the National Key Research andDevelopment Program of China (2022YFE0109500).K.W. and T. T. acknowledge support from the JSPSKAKENHI (Grants No. 20H00354 and No. 23H02052)and World Premier International Research Center Initiative(WPI), MEXT, Japan.APPENDIX1. Bulk crystal growth and characterizationsCrPS4 crystals were grown using the chemical vaportransport method, following established protocols [53].Chromium, red phosphorus, and sulfur powders weremeasured stoichiometrically (Cr∶P∶S ¼ 1∶1∶4) with 5%additional sulfur added as transport agent. The precursorswere mixed and sealed into quartz ampoules under vacuum(10−2 Pa), followed by loading into a two-zone furnace.The hot and cold ends of the ampoules were kept at 680 °Cand 600 °C, respectively, for 8 days. At the end of thegrowth process, the furnace was turned off for naturalcooling and the CrPS4 crystals could be collected from thecold zone of the ampoule. X-ray diffraction was performedto determine the structure of the grown crystals. One crystalweighing 1.13 mg was utilized for magnetization measure-ments performed in a physical property measurementsystem (PPMS) VSM magnetometer (Quantum Design),with the magnetic field oriented parallel or perpendicular tothe crystallographic c-axis.2. Vertical junction fabricationand transport measurementsAtomically thin CrPS4 flakes were obtained throughmechanical exfoliation from bulk crystals. The verticaljunctions of graphene=CrPS4/graphene were assembledusing a standard pickup technique with stamps of poly-dimethylsiloxane(PDMS)/polycarbonate(PC). To ensurethe high quality of the vertical junctions, exfoliation andassembly were conducted within a glove box filled withnitrogen gas, and the junctions were encapsulated with h-BN. Standard nanofabrication processes, including electronbeam lithography, reactive ion etching, and electron beamevaporation (10 nm=50 nm Cr=Au), were employed tomake contacts to the graphene electrodes. Electronic trans-port measurements were conducted in a cryostat fromCryogenic or PPMS from Quantum Design, using aKeithley 2400 and SR830 lock-in amplifier. MR of multi-layer devices were measured with constant bias voltage,and the monolayer device was measured with constantcurrent of 40 nA. Frequency of 17.773 Hz was used in acmeasurements.MAGNETORESISTANCE OSCILLATIONS IN VERTICAL … PHYS. REV. X 14, 041065 (2024)041065-73. Electronic transport data of vertical junctionswith different electrodesAs discussed in the main text, one possible explanationof MR oscillations is that they might originate from newstates in the graphene electrodes induced by the proximityeffect with magnet CrPS4. To address this, we conductedtwo control experiments. First, we fabricated a device withthe structure graphene=h-BN=CrPS4=h-BN=graphene. Theproximity effect typically involves the expansion of thewave function from one material into another, resulting innew properties in the latter. This wave function expansiondecreases exponentially with increasing distance betweenthe materials. For instance, ferromagnetism in grapheneinduced by proximity to a yttrium iron garnet (YIG)substrate disappears when a very thin Al2O3 layer isinserted between them [54]. In our device, the thinh-BN layer inserted between CrPS4 and graphene wouldeliminate any potential new states in graphene induced byproximity to CrPS4. As shown in Fig. 5, MR oscillationsFIG. 5. MR oscillations in vertical junction of graphene=h-BN=CrPS4=h-BN=graphene. (a) IV curve of the device at 2 K and zerofield, the difference thickness of inserted h-BN results in the asymmetry of the device. The inset shows schematic of vertical junctiongraphene=h-BN=CrPS4=h-BN=graphene, where thin h-BN was inserted between graphene electrodes and CrPS4. (b) MR oscillations at2 K with magnetic field applied parallel and perpendicular to the c axis. The measurement is done with constant dc voltage of 0.6 V. Theinset shows the optical image of the device. (c) MR at different temperatures; the magnetic field is applied perpendicular to c axis.FIG. 6. MR oscillations in vertical junction of NbSe2=CrPS4=NbSe2. (a) Schematic of vertical junction NbSe2=CrPS4=NbSe2.(b) Optical image of device NS1. (c) IV curve measured at 2 K at zero magnetic field. (d) MR oscillation at 2 K with magnetic fieldparallel and perpendicular to the c axis. Applied dc bias voltage is −0.2 V. (e) MR oscillation at 2 K with dc bias voltage −0.185 V.(f) MR oscillation at 2 K with different dc bias voltage; magnetic field is perpendicular to c axis.PENGYUAN SHI et al. PHYS. REV. X 14, 041065 (2024)041065-8were still clearly observed when magnetic field was appliedparallel or perpendicular to the c axis of the device,providing evidence that these oscillations originate fromCrPS4 itself.Second, we constructed another type of device with thestructure NbSe2=CrPS4=NbSe2 as illustrated in Fig. 6(a).We use 2D metal NbSe2 as the electrodes and omitgraphene in this type of device. The fabrication processfor this device was similar to that described in the main textfor the graphene=CrPS4=graphene device. To avoid theoxidation of NbSe2, the device was immediately loadedinto the e-beam evaporator’s vacuum chamber after etchingthe h-BN, with NbSe2 exposure to air minimized to lessthan one minute. Figure 6 shows the results for one suchdevice, NS1. The nonlinear IV curve at 2 K and 0 T ispresented in Fig. 6(c). We observed features around�0.2 V in the IV curve, which were also seen in anotherNbSe2=CrPS4=NbSe2 device NS2, though their origin isunclear. 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X 14, 041065 (2024)041065-11https://doi.org/10.1103/PhysRevLett.96.246403https://doi.org/10.1103/PhysRevLett.96.246403https://doi.org/10.1103/PhysRevLett.58.1593https://doi.org/10.1038/s41567-024-02675-xhttps://arXiv.org/abs/2410.17930https://doi.org/10.1002/adma.202001200https://doi.org/10.1103/PhysRevLett.114.016603 Magnetoresistance Oscillations in Vertical Junctions of 2D Antiferromagnetic Semiconductor CrPS4 I. INTRODUCTION II. EXPERIMENTAL RESULTS III. DISCUSSION IV. SUMMARY Note added in proof.  ACKNOWLEDGMENTS APPENDIX  1. Bulk crystal growth and characterizations 2. Vertical junction fabrication and transport measurements 3. Electronic transport data of vertical junctions with different electrodes References