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Lorenz Bauriedl, Christian Bäuml, Lorenz Fuchs, Christian Baumgartner, Nicolas Paulik, Jonas M. Bauer, Kai-Qiang Lin, John M. Lupton, [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), Christoph Strunk, Nicola Paradiso

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[Supercurrent diode effect and magnetochiral anisotropy in few-layer NbSe2](https://mdr.nims.go.jp/datasets/b26d15df-0587-448d-bb87-c1697313005e)

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Supercurrent diode effect and magnetochiral anisotropy in few-layer NbSe2ARTICLESupercurrent diode effect and magnetochiralanisotropy in few-layer NbSe2Lorenz Bauriedl1, Christian Bäuml1, Lorenz Fuchs1, Christian Baumgartner1, Nicolas Paulik1, Jonas M. Bauer1,Kai-Qiang Lin1, John M. Lupton1, Takashi Taniguchi 2, Kenji Watanabe 2, Christoph Strunk 1 &Nicola Paradiso 1✉Nonreciprocal transport refers to charge transfer processes that are sensitive to the biaspolarity. Until recently, nonreciprocal transport was studied only in dissipative systems,where the nonreciprocal quantity is the resistance. Recent experiments have, however,demonstrated nonreciprocal supercurrent leading to the observation of a supercurrent diodeeffect in Rashba superconductors. Here we report on a supercurrent diode effect in NbSe2constrictions obtained by patterning NbSe2 flakes with both even and odd layer number. Theobserved rectification is a consequence of the valley-Zeeman spin-orbit interaction. Wedemonstrate a rectification efficiency as large as 60%, considerably larger than the efficiencyof devices based on Rashba superconductors. In agreement with recent theory for super-conducting transition metal dichalcogenides, we show that the effect is driven by the out-of-plane component of the magnetic field. Remarkably, we find that the effect becomesfield-asymmetric in the presence of an additional in-plane field component transverse to thecurrent direction. Supercurrent diodes offer a further degree of freedom in designingsuperconducting quantum electronics with the high degree of integrability offered by van derWaals materials.https://doi.org/10.1038/s41467-022-31954-5 OPEN1 Institut für Experimentelle und Angewandte Physik, University of Regensburg, Regensburg, Germany. 2 International Center for MaterialsNanoarchitectonics, National Institute for Materials Science, Tsukuba, Japan. ✉email: nicola.paradiso@physik.uni-regensburg.deNATURE COMMUNICATIONS |         (2022) 13:4266 | https://doi.org/10.1038/s41467-022-31954-5 |www.nature.com/naturecommunications 11234567890():,;http://crossmark.crossref.org/dialog/?doi=10.1038/s41467-022-31954-5&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-022-31954-5&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-022-31954-5&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-022-31954-5&domain=pdfhttp://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0001-9982-0022http://orcid.org/0000-0001-9982-0022http://orcid.org/0000-0001-9982-0022http://orcid.org/0000-0001-9982-0022http://orcid.org/0000-0001-9982-0022http://orcid.org/0000-0002-1797-2901http://orcid.org/0000-0002-1797-2901http://orcid.org/0000-0002-1797-2901http://orcid.org/0000-0002-1797-2901http://orcid.org/0000-0002-1797-2901mailto:nicola.paradiso@physik.uni-regensburg.dewww.nature.com/naturecommunicationswww.nature.com/naturecommunicationsThe archetypal example of a nonreciprocal electronic deviceis the diode. The term nonreciprocity in this context isused to imply a large difference in resistance betweenopposite bias polarities. For a conventional semiconductor diode,nonreciprocity follows from the inequivalence of the two crystalsforming the pn junction, which have different types of doping. Inhomogeneous devices, polarity-dependent resistance is observedwhen both inversion and time-reversal symmetry are brokensimultaneously. As shown by Rikken et al.1,2, in noncentrosym-metric conductors nonreciprocal resistance can be phenomen-ologically described byR ¼ R0½1þ αB2 þ γBI�; ð1Þwhere the coefficient α refers to the usual magnetoresistance andγ is the magnetochiral anisotropy (MCA) coefficient. In normalconductors, γ is typically very small. Its strength is determined bythe ratio between spin–orbit perturbation and the Fermi energy.Nonreciprocal transport therefore becomes discernible in semi-conductors with low Fermi level and large spin–orbit interaction(SOI)3. The MCA effect can be greatly amplified in non-centrosymmetric superconductors4–6,6–8. Here, the energy scalegoverning the fluctuation regime close to the critical temperatureis not the Fermi energy, but the superconducting gap. Anotherway to boost MCA for the resistance is to engineer nonreciprocalvortex motion in a superconductor by asymmetric patterning ofartificial pinning centers9.In the past years, superconductivity-enhanced MCA for theresistance has been studied extensively. There is a satisfactoryunderstanding of the mechanisms producing nonreciprocalresistance and theory predictions successfully describe theexperiments10,11. On this basis, some first applications, such assuperconducting antenna rectifiers8 or spin filtering diodes12,have already been proposed.So far, studies on nonreciprocal transport, even whenexploiting superconductivity to enhance MCA, have mainlyfocused on resistance. The seminal demonstration of a dis-sipationless nonreciprocal supercurrent13 was only recently pro-vided by experiments on synthetic Rashba superconductors basedon Nb/V/Ta multilayers14 and on Josephson junctions15 withstrong Rashba SOI. Such a nonreciprocal supercurrent gives riseto the so-called superconducting diode effect, where the super-current can flow only in one direction that can be switched by amagnetic field.The MCA for the electrical resistance and that for the super-current are two clearly distinct effects. The latter is characterizedby the kinetic inductance, which is uneven in the current, whilethe resistance is zero. In the experiments reported so far, samplesthat showed nonreciprocal supercurrent far below Tc also showednonreciprocal resistance in the fluctuation regime near Tc14,15.This correlation between the two phenomena arises due to thefact that both effects require the same symmetry-breakingmechanisms (i.e., time and inversion symmetry). While there isa satisfactory understanding of MCA for the resistance, the the-oretical study of magnetochiral effects in supercurrents is just inits infancy16–25. Nonreciprocal supercurrent is better understoodin Josephson junctions, where the diode effect can be engineeredby Andreev bound states in the normal weak-link. The observa-tion of non-reciprocal supercurrent in Rashba superconductorsraises the question about its existence in materials with othertypes of SOI. Promising candidates are transition metal dichal-cogenides (TMDs) that feature valley-Zeeman SOI, where mag-netochiral resistance was already measured, while non-reciprocalsupercurrent was predicted26 but not yet observed.Here, we report on the observation of a pronounced super-current diode effect in constrictions of few-layer NbSe2-crystals.Owing to dominant valley-Zeeman SOI, the supercurrent diodebehavior is driven by the out-of-plane component Bz of themagnetic field. Unexpectedly, also the in-plane component affectsthe non-reciprocal supercurrent: we find that it breaks the anti-symmetry of the critical current difference with respect to Bz,boosting the rectification for one Bz direction, and suppressing itfor the opposite one.ResultsOur samples are fabricated using standard exfoliation methodsfor TMDs27–30. A scheme of the typical device is depicted inFig. 1a. The exfoliated NbSe2 crystals considered here are between2 and 5 layers thick29. The flake thickness can be estimated withreasonable accuracy from the optical contrast of the crystal whenstamped on standard SiO2/Si substrates. The parity of the layernumber N and the lattice orientation is determined, a posteriori,by second harmonic generation (SHG) measurements for allsamples. NbSe2 crystals are fully encapsulated in hBN30 and edgecontacts are fabricated by electron beam lithography31. A 250nm-long channel of 250 nm width is patterned by electron beamlithography and reactive ion etching. The etched parts appear asdark purple triangles in Fig. 1b. The purpose of the narrowchannel is to have a well-defined direction of the current densityin the constriction, whose direction is indicated by I in Fig. 1a. Inwhat follows, the x-direction is assumed to be that of the con-striction axis (and thus that of the supercurrent) while the z-direction is perpendicular to the sample plane.The main measurements of this work (sample F, see below)were performed in a dilution refrigerator equipped with a 9T-superconducting coil controlling the in-plane component ofthe magnetic field. Additional coils provide a field perpendicularto the sample plane. A piezo-rotator allows for rotation of thesample around an axis normal to the sample plane. Additionalmeasurements (samples B–E and G, see Supplementary Infor-mation) were performed in a 4He cryostat with a base tempera-ture of 1.3 K, equipped with a single superconducting coil.Figure 1c–e introduces the supercurrent diode effect as measuredin a NbSe2 constriction patterned on sample G. The graphs showthree pairs of current–voltage characteristics (IVs) for appliedout-of-plane magnetic fields of Bz= 0 mT (c), Bz= 32.5 mT (d)and Bz=− 32.5 mT (e). Importantly, all the IVs reported herealways refer to the zero-to-finite (either positive or negative) biassweep direction, in order to rule out heating effects. In the threegraphs, the black (red) symbols refer to current density in thenarrow channel oriented towards the positive (negative) x̂direction. A strong supercurrent diode effect14,15 is evident fromthe comparison of the IVs: there is a marked difference ΔIc �Iþc � jI�c j in the critical current for the two supercurrent orien-tations, the sign of which changes when the magnetic field Bz isinverted. The complete field dependence of both Iþc (black) andjI�c j (red) is plotted in Fig. 1f. The current range between Iþc andjI�c j corresponds to the supercurrent diode regime, where currentflows without dissipation only in one direction, which can beselected by changing the sign of the magnetic field14. As a figureof merit of the supercurrent diode effect, one can take thesupercurrent rectification efficiency Q � 2ΔIc=ðIþc þ jI�c jÞ18, i.e.the difference between Iþc and I�c normalized by their average,which is plotted in Fig. 1g versus Bz. For moderate fields, Qincreases almost linearly with the field. The maximum magnitudeof Q is above 60%, much larger than that (≈5%) observed inref. 14. Beyond a certain breakdown field Bmax;Q � 35 mT thediode effect is gradually suppressed. This behavior is reminiscentof that observed in Rashba superconductors, see, e.g. Fig. 2 inref. 14 and Fig. 3 in ref. 15. Theory models for Rashbasuperconductors16,19 predict a similar suppression, but thethreshold field is expected to be of the order of the paramagneticlimit, i.e. much larger than the value observed in our and in otherARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-022-31954-52 NATURE COMMUNICATIONS |         (2022) 13:4266 | https://doi.org/10.1038/s41467-022-31954-5 | www.nature.com/naturecommunicationswww.nature.com/naturecommunicationsexperiments14,15. Finally, we stress that the observations in Fig. 1f,g do not depend on the field sweep direction, as discussed inthe Supplementary Information.We notice that, for either of the two bias polarities, the criticalcurrent increases by increasing the magnetic field, reaching amaximum at a nonzero field Bmax;Ic. This can be seen, e.g., inFig. 1f, where Bmax;Ic� 10 mT. The remarkable increase of thecritical current (the origin of which certainly deserves furtherstudy) is important since it eliminates the possibility that thenonreciprocal supercurrent originates from Joule heating due to,e.g., nonreciprocal resistance, which in NbSe2 is known to beimportant near Tc owing to magnetochiral effects8. A very recentwork32 on V or Nb superconducting films reported a biaspolarity-dependent critical current increase, which gives rise to aI ±c ðBÞ dependence similar to that in Fig. 1f (i.e., an inverted-W-shaped graph). The authors of ref. 32 attributed the phenomenonentirely to the interplay between Meissner currents and barrierfor vortex entry33. In our work, however, the diode effect is notnecessarily bound to the critical current increase: for instance, insample G and D the rectification efficiency Q keeps increasingwell beyond Bmax;Ic(up to Bmax;Q, which is more than three timesBmax;Ic, cf. Fig. 1f, g), while in sample F, discussed below, thecritical current increase is simply absent (Bmax;Ic¼ 0). As dis-cussed in the Supplementary Information, the experimental evi-dence seems to exclude a contribution of vortices or screeningcurrents (the constriction width is comparable to the penetrationdepth λ and much smaller than the Pearl length λ2/d). However,the remarkable results reported in ref. 32 urge the use of cautionin interpreting the origin of supercurrent nonreciprocity in thepresence of perpendicular fields.We observed supercurrent rectification in other devices withthe same nominal geometry. The supercurrent diode effect wasclearly visible in all the samples where supercurrent could bemeasured, i.e., samples B, D, E, F, and G. Their layer number Nwas determined by combining information from white-lightoptical and SHG microscopy34–37, as discussed in the Supple-mentary Information. We found a layer number N= 3 for sam-ples B, D and F, N= 5 for sample E and N= 2 for sample G. Asfor the Ising superconductivity, the supercurrent diode effect isnot theoretically expected to occur in NbSe2 when N is even,owing to the restored inversion symmetry. However, both Isingsuperconductivity (see Fig. 4 in ref. 34) and the supercurrentdiode effect (this work) are experimentally observed for both evenand odd N. This fact is likely caused by the relatively weakelectronic coupling between the layers, which effectively rendersNbSe2 a collection of monolayers34.The experimental results shown in Fig. 1 clearly demonstratethat an out-of-plane magnetic field is required for the super-current diode effect in materials with valley-Zeeman SOI. How-ever, it is interesting to study whether the in-plane field alsoaffects the rectification. For this reason, we measured one device,sample F, in a setup equipped with a piezo-rotator. This setupallows us to rotate the sample with respect to the main magneticfield Bip, and therefore to control the angle θ between Bip andsupercurrent I, as indicated in Fig. 1a. Additional coils, perpen-dicular to the main one, provide an independent control of theV - RcI (mV)critical current (μA)I (μA)Bz (mT)xyzNb SeB NIadceQgfbipBBz (mT)Ic+|Ic  - | -100 -50 0 50 100-0.6-0.30.00.30.6-100 -50 0 50 10046810120 2 4 6 8 10 1202468 + bias- bias0 2 4 6 8 10 1202468 + bias- bias0 2 4 6 8 10 1202468 + bias- biasV - RcI (mV)V - RcI (mV)Bz = -32.5 mTBz = +32.5 mTBz = 0 mTFig. 1 Supercurrent diode effect in a van der Waals superconductor. a Scheme of the typical device. The central constriction is 250 nm wide and 250 nmlong. The x-direction is chosen to be that of the supercurrent, i.e., the constriction axis. The z-direction is perpendicular to the crystal plane. The device isfabricated starting from a stack of hBN (≈10 nm, tens of layers), NbSe2 (2, 3 or 5 layers) and again hBN (tens of layers). b Optical micrograph of sample B.The dashed white contour highlights the NbSe2 crystal. Electrodes are fabricated by edge contact techniques27,31,47, while the constrictions are made byreactive ion etching. The yellow arrow indicates the supercurrent pathway. The scale bar corresponds to 5 μm. c Current–voltage characteristics (IVs) foropposite bias polarities (i.e., opposite current directions) measured on sample G in a 3-terminal configuration for zero out-of-plane field Bz. The sweepdirection is always from zero to finite bias. A contact resistance Rc= 1 kΩ has been subtracted. d Similar measurements, but for Bz= 32.5 mT. Notice thedifference between the two critical currents. e Same as in panel (d), but with opposite field orientation. Notice that the role of the two bias polarities now isswapped. f Absolute critical current for positive (black) and negative (red) bias as a function of Bz. Each value is the average of 10 consecutivemeasurements. The critical current is maximal for a nonzero Bz, namely for jBzj ¼ Bmax;Ic� 10 mT (black and red arrow). The red and black solid line areguides to the eye, mutually symmetric upon reflection around Bz= 0. g Supercurrent rectification efficiency Q � 2ðIþc � jI�c jÞ=ðIþc þ jI�c jÞ, plotted versus Bz.Q is maximal for Bz ¼ Bmax;Q � 35 mT (arrow). Measurements in (c–g) were performed at 1.3 K. As discussed in the Supplementary Information, an offsetof −2.5 mT and 0.17 μA has been removed from Bz and Ic, respectively.NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-022-31954-5 ARTICLENATURE COMMUNICATIONS |         (2022) 13:4266 | https://doi.org/10.1038/s41467-022-31954-5 |www.nature.com/naturecommunications 3www.nature.com/naturecommunicationswww.nature.com/naturecommunicationsout-of-plane field Bz. Figure 2a–c show the supercurrent rectifi-cation efficiency Q in sample F, plotted as a function of both thein-plane (Bip) and the out-of-plane (Bz) magnetic field, for θ= 90∘(panel (a), Bip transverse to the supercurrent), for θ= 0∘ (panel(b), Bip parallel to the supercurrent), and for θ=− 90∘ (panel (c),Bip antiparallel to that in panel (a). Owing to sample misalign-ment, a large in-plane field produces a significant out-of-planecomponent. This misalignment can be quantified (and accountedfor, as in Fig. 2a–c) by looking at the maximum of Iþc ðBzÞ [orI�c ðBzÞ], as discussed in the Supplementary Material. As a result ofthis offset adjustment, the accessible range in the (Bip, Bz) planeappears as a rhomboid.The most prominent feature in these measurements is thevertical gradient in the color plot, indicating that the rectificationefficiency increases monotonically with Bz and is zero whenBz= 0, at least within the experimental data scatter. Hence, wehave now established experimentally that, as predicted bytheory26, the supercurrent diode effect in TMDs requires an out-of-plane field. In contrast, it is the in-plane field that drives thediode effect in Rashba superconductors. Moreover, unlike what ispredicted for superconductors with pure valley-Zeeman SOI26,we observe that the in-plane field does affect the supercurrentdiode effect. More precisely, the in-plane field componentperpendicular to the current breaks the (anti)symmetry of Q as afunction of Bz. If Bip;y � Bip � ŷ ¼ 0 then Q(Bz)=−Q(− Bz),with no dependence on Bip;x � Bip � x̂, see Fig. 2b. Instead, Fig. 2a,c prove that, for finite Bip,y, Q(Bz) ≠−Q(− Bz), i.e., the diodeeffect becomes asymmetric in Bz. This effect is apparent, e.g., inFig. 2a: for the given field and current orientation, the diode effectis enhanced when Bz and Bip are both positive or negative (firstand third quadrant of the graph, deep red and deep blue region,indicated by the arrows). On the other hand, it is suppressedwhen Bz and Bip have opposite signs (second and fourth quadrant,light red and light blue region). At fixed Bz (dashed lines),inverting the sign of Bip,y corresponds to a transition fromenhanced to suppressed diode effect, i.e. from deep blue to lightblue for the upper dashed line. Clearly, a 180∘ sample rotation isequivalent to a sign change in Bip, as it is evident by comparingFig. 2a, c. It is important to remark that data in Fig. 2a–c are still(anti)symmetric upon simultaneous inversion of both Bz andBip,y, i.e., Q(Bz, Bip,y)=−Q(− Bz,− Bip,y).To better visualize the impact of the in-plane field, it isinstructive to look separately at Iþc and I�c as a function of Bz,both with and without in-plane field. The latter case is shown inFig. 2d. Both Iþc ðBzÞ and I�c ðBzÞ appear as asymmetric, Λ-shapedfunctions. Their difference in slope for positive and negative Bz-2 -1 0 1 2-0.10.00.1Bz (T)Bip (T)-0.10 -0.05 0.00 0.05 0.10-0.10-0.050.000.050.10Q Bz (T) Bip= 2 T Bip= 0 T Bip= -2 T-0.10 -0.05 0.00 0.05 0.101.501.752.00 Bip=+2 T, Ic+ Bip=+2 T, Ic-critical current | Ic| (μA)Bz (T)θ = +90°a b cd e fθ = 0°Q- 0.140.14zxy IzxyBipIzxy IBip-2 -1 0 1 2-0.10.00.1Bz (T)Bip (T)-2 -1 0 1 2-0.10.00.1Bz (T)Bip (T)-0.10 -0.05 0.00 0.05 0.101.501.752.00 Bip=0 T, Ic+ Bip=0 T, Ic-critical current | Ic| (μA)Bz (T)Bipθ = -90°Fig. 2 Supercurrent diode effect versus in- and out-of-plane field. a The color plot shows Q � 2ðIþc � jI�c jÞ=ðIþc þ jI�c jÞ as a function of out-of-plane (Bz)and in-plane (Bip) field, measured in sample F for θ= 90∘ (i.e., for Bip⊥I). Bz here includes both the field produced by the orthogonal coils and the finite z-component of Bip arising due to misalignment. Red and blue arrows indicate the areas where the diode effect is enhanced. b Similar measurements, but forθ= 0∘. c As in (b), but for θ=−90∘. Notice that this graph can be mapped onto that in (a), provided that Bip→−Bip. The color contrast is the same in (a),(b) and (c). d Absolute value of Iþc (black) and I�c (red) plotted versus Bz, for Bip= 0 and for the sample orientation θ=−90∘. e As in (d), but for Bip= 2 T.Note that data in (d, e) were measured in a different session (with higher resolution in Bz) compared to data in c. f Supercurrent rectification efficiency Q asa function of Bz at θ=−90∘ for Bip=−2 T (green), 0 T (gray), and 2 T (magenta). Here, we used the same data as in panel c, for Bip values indicated in thelegend. For Bip= 0 we have substituted three outliers (for Bz=−48,−64 and−80 mT) with the corresponding values for the adjacent in-plane fieldBip=−0.25 T. For Bip=−2 T we have substituted one outlier (for Bz=−43 mT) with the corresponding value for the adjacent in-plane field Bip=−1.75 T,see Supplementary Information. Outliers are also visible in panels (a–c). All measurements reported in this figure were performed at 500 mK.ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-022-31954-54 NATURE COMMUNICATIONS |         (2022) 13:4266 | https://doi.org/10.1038/s41467-022-31954-5 | www.nature.com/naturecommunicationswww.nature.com/naturecommunicationsproduces a supercurrent diode effect increasing monotonicallywith Bz. We note the similarity of Fig. 2d and Fig. 1f with thecorresponding measurements on Rashba Josephson junctions, seeFig. 3e in ref. 15. As for Fig. 1g, the whole graph in Fig. 2d issymmetric if both the sign of Bz and the direction of the super-current are changed simultaneously, i.e. for Bz↔− Bz andIþc $ I�c . This symmetry is broken when an in-plane field isapplied along the direction perpendicular to the supercurrent. Inthe case of Bip,y > 0, shown in Fig. 2e, the difference between theslope of Iþc ðBzÞ and I�c ðBzÞ is reduced for Bz > 0 and enhanced forBz < 0, and vice versa for Bip,y < 0.Figure 2f summarizes our main findings: the supercurrentrectification efficiency Q is plotted as a function of Bz forθ=−90∘ and for Bip= 2 T, 0 T and −2 T. The graph shows thatthe perturbation due to the in-plane field is evident only forsufficiently high out-of-plane field, i.e., ∣Bz∣ > 30 mT. Above thatthreshold, for Bz > 0, the slope of Q versus Bz increases stronglyfor Bip=− 2 T and decreases for Bip= 2 T. The opposite is truefor Bz < 0. We also note that at sufficiently large ∣Bz∣, aboveapproximately 50 mT, the diode effect starts to be suppressed, sothat at around ∣Bz∣ ≳ 100 mT the different curves in Fig. 2fmerge again.The key result of our observations is that the supercurrentdiode effect in NbSe2 is controlled by the out-of-plane field, aspredicted by theory for superconducting TMDs26. For Bz= 0there is no diode effect, independent of Bip. On the other hand, ifBip has a component Bip,y perpendicular to the current, then thesupercurrent diode effect becomes asymmetric in Bz, i.e., it isenhanced for one out-of-plane field polarity and suppressed forthe other. The role of the two polarities is swapped if both thesign of the supercurrent and the sign of Bip,y are inverted. To thebest of our knowledge, there are no predictions to date regardinga possible role of the in-plane field on the supercurrent diodeeffect in TMDs. In contrast, for Rashba superconductors it isprecisely the in-plane field that controls MCA and the diodeeffect14,15. It is clear that further studies are needed to elucidatethe role of the in-plane field in superconducting TMDs, whichmight be related to a possible Rashba-like component of theSOI38,39. In NbSe2 such Rashba components can arise, e.g., due toripples in the crystal40 formed when stamping NbSe2 on hBN30,or due to the substrate41. The existence of a weak Rashba SOIcomponent in NbSe2 has, for example, been invoked to explainthe apparent two-fold anisotropy of the magnetoresistance as afunction of in-plane magnetic field41. The model proposed inref. 41 relies on the presence of p-wave components in the pairingfunction, which might as well play a role in the anisotropy of thediode effect we observe here.Next, we turn to the temperature dependence of Q. Theorypredictions regarding the temperature dependence of the rectifi-cation efficiency are quite diverse. While some models predict asquare-root-like dependence near the critical temperature18,others suggest a more complex functionality19. Figure 3 shows thetemperature dependence of the diode effect in samples D, F andG. In the graph we plot S≡Q/Bz, obtained from Q in the linearregime of low field, where Q∝ Bz. We choose to display S ratherthan Q because it represents a linear interpolation of Q(Bz) andthus it averages over several data points, resulting in a reducedscatter compared to Q for a given Bz.First of all, we observe that the supercurrent diode effectsaturates at low temperature. This saturation is compatible withall theory models and similar to the results on Josephson junc-tions in Rashba superconductors15. This result is not obvious,since in ref. 14 the diode effect is visible only near Tc and it issuppressed for both higher and lower temperatures. We note alsothat the temperature dependence of the diode effect in samples Dand F is nonmonotonic, with a maximum reached nearT/Tc= 0.5. In contrast, a monotonic variation is observed insample G. A nonmonotonic temperature dependence of therectification at a finite field was predicted by recent theory19. It ispossible that the different behavior displayed by samples D, F,and G is due to the fact that, in order to obtain a significantrectification, we need to apply a sufficiently strong field. Note thattheory in ref. 19 predicts that the temperature dependence of therectification is critically affected by a finite magnetic field.DiscussionA comment is in order about features which appear to be sampledependent. While the supercurrent diode effect was observed inevery sample we measured (see Supplementary Information forfurther details), the maximum rectification efficiency stronglyvaries among the samples, ranging from 6 to 60%. Moreover, thesupercurrent increase with Bz is sample dependent, both in termsof relative increase IcðBmax;IcÞ=Icð0Þ and maximum supercurrentfield Bmax;Ic. The supercurrent enhancement seems to be inde-pendent from the efficiency of the supercurrent rectification. Thesupercurrent increase is negligible in sample F, while in sample Gno particular feature is observed in Q(Bz) at Bz ¼ Bmax;Ic. Thenominal differences between the samples are the layer numberand the orientation of I with respect to the lattice. Both featurescan be determined by SHG microscopy performed after thetransport measurements, as described in the SupplementaryInformation. We found that neither the layer number nor thesupercurrent-to-lattice orientation are correlated with the mag-nitude of the rectification efficiency. On the other hand, litho-graphy of narrow constrictions by reactive ion etching producesrandom disorder, in particular at the edges. This randomness,together with the still limited number of studied samples, doesnot allow us to conclusively assess the role of the lattice orien-tation. Further investigation is needed in order to elucidate howthe supercurrent diode effect is influenced by the direction ofcurrent flow with respect to the crystal.In conclusion, we have demonstrated a supercurrent diodeeffect in few-layer NbSe2. We show that the effect is controlled bythe out-of-plane magnetic field, in contrast to what has been0.0 0.2 0.4 0.6 0.8 1.00.00.51.01.52.0sample Dsample Fsample GS/SLTT/TcFig. 3 Temperature dependence of the supercurrent diode effect. Thegraph shows the temperature dependence of S≡Q/Bz for sample D, F, andG. S is evaluated from linear fits of Q(Bz) near Bz= 0, i.e. in the linearregime of the supercurrent rectification efficiency, where Q∝ Bz. The errorbars correspond to the standard error of the fits. The S values arenormalized to the lowest temperature value SLT (e.g. 17.2 T−1 for sampleG), while the temperatures are normalized to the critical temperature Tc ofthe corresponding sample. Orange, blue and purple symbols refer to sampleD, F and G, where the critical temperature of the constriction region is4.3 K, 2.25 K and 4.0 K, respectively.NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-022-31954-5 ARTICLENATURE COMMUNICATIONS |         (2022) 13:4266 | https://doi.org/10.1038/s41467-022-31954-5 |www.nature.com/naturecommunications 5www.nature.com/naturecommunicationswww.nature.com/naturecommunicationsobserved in Rashba superconductors, where the effect is driven bythe in-plane field component directed perpendicular to thesupercurrent. This field component nevertheless plays a role inNbSe2 devices as well, since it suppresses the diode effect for oneout-of-plane field polarity and enhances it for the opposite one.Finally, the temperature dependence of the effect shows satura-tion at low temperature, a maximum or a kink at around T= Tc/2and a suppression near Tc. TMD-based diodes may becomecrucial components in fully superconducting electronics. Theirdissipation-free directional transport makes them suited for logicelements, ultrasensitive detectors, or signal demodulators, whichcan operate at low temperature with no energy loss. Being just afew atoms thick, their performance can conceivably be controlledelectrically by gates, and modified by integrating them intocomplex van der Waals stacks in combination with other 2Dmaterials.During the review process we became aware of relatedexperimental work on supercurrent rectification in NbSe2/CrPS4heterostructures42, in NbSe2-based Josephson junctions43, and onsupercurrent rectification in magic-angle twisted bi-44 andtrilayer45 graphene, and Nb-proximitized NiTe246.MethodsSample preparation. NbSe2 crystals were purchased from HQ Graphene. hBN andNbSe2 crystals were exfoliated following two common techniques. One is thetechnique introduced in ref. 28, where flakes are exfoliated on a poly-dimethylsiloxane (PDMS) film placed on a glass slide. Suitable crystals are thenstamped onto the sample using a micromanipulator placed under a zoom lens. Theother technique is commonly used for the production of fully hBN-encapsulatedgraphene devices27. In this case, flakes are sequentially picked up by a thick PDMSfilm coated with polycarbonate (PC). The flake pick-up takes place at 120∘C, whilethe final release is triggered by melting the PC at 180∘C and by dissolving it inchloroform.Both the fabrication of the contacts and the design of the constriction require anelectron beam lithography step followed by reactive ion etching (RIE). For the RIE,a mixture of 6 sccm O2 and 40 sccm CHF3 is ignited into a plasma with 35 W r.f.forward power at a pressure of 47 mbar. The RIE step etches through hBN andNbSe2 with an approximate etching rate of 0.45 nm/s. When electrodes have to beproduced, immediately after the RIE step a 10 nm-thick layer of Ti and a 100-nm-thick film of Au are deposited. This procedure reflects the well-known recipe foredge contact fabrication in graphene. For sample G the RIE process was substitutedby an Argon plasma etching process with 2 kV acceleration voltage and 20 mAplasma current at about 3 × 10−3 mbar. The etching rate is in this caseapproximately 1 nm per minute.Transport measurements. Measurements on sample F were performed in adilution refrigerator with a base temperature of 30 mK. For electrical measure-ments, we used DC lines with Cu-powder filters. Since one of the four electrodesstopped working after cool-down, we measured the device in a three-terminalconfiguration. The IV characteristics were thus obtained by subtracting the voltageRcI, where Rc= 413Ω is the contact resistance. In the IV curves, the resistivetransition of the constriction at the critical current was clearly visible as a sharpstep, except very close to the critical temperature and field.The critical currents for data in Figs. 1 and 3 were determined from the IVs asthe extrapolation on the current axis (abscissas) of the steep voltage increase at theresistive transition. In Fig. 2, owing to the larger amount of data, we used thealternative criterion V(Ic)= Vthres≡ 100 μV, which is better suited for automaticroutines. Nevertheless, the results shown here depend very weakly on the criterionfor the critical current. We verified in all samples that the I ±c and Q data obtainedwith either criterion were nearly the same (see Supplementary Information forfurther details).Measurements on the other samples (A–E,G) were performed in a 4He cryostatwith only room temperature filtering (π-filters). Except for sample G, the sampleholder is positioned in such a way that the magnetic field produced by thesuperconducting coil is approximately in the plane of the sample surface. Themisalignment (typically of the order of a few degrees) produces an out-of-plane fieldof tens of milliteslas per tesla of the applied field. In sample G, instead, the sample isperpendicular to the main field, thus the field is applied exclusively out-of-plane.In our three- and four-terminal measurements we applied a voltage bias directlyto the source and drain contacts, without using a preresistor (a resistance in seriesis provided by the Au-NbSe2 contact resistance, which is of the order of 1 kΩ). Thevoltage drop across the constriction and the current were then measuredsimultaneously.Second harmonic generation measurements. Optical measurements of secondharmonic generation (SHG) were always performed after transport measurements,in order to minimize the risk of photo-oxidation30. The co-polarized SHG intensityis measured as a function of the relative angle between laser polarization and crystalorientation34–37. The light source used was a pulsed Ti:sapphire laser (80 fs pulseduration, 80 MHz repetition rate, 1 mW power) at 800 nm. Using a microscopeobjective (×40, numerical aperture of 0.6), the light was focused onto the NbSe2samples placed in the vacuum chamber. The reflected SHG signal at 400 nm wascollected with the same objective, filtered by a 680 nm short-pass filter, dispersed ina spectrometer (150 grooves/mm grating) and detected by a CCD camera. A linearpolarizer was placed in front of the spectrometer to ensure acquisition of the signalpolarized parallel to the laser polarization. A 50:50 non-polarizing beam splitterwas used to separate the incident pathway from the signal detection pathway. Inbetween the beam splitter and the objective, an achromatic half-wave plate wasplaced to change the relative angle between the crystal orientation and the laserpolarization. The half-wave plate was rotated using a stepper motor. A 1200grooves/mm grating was used as a reference sample. In the experiments, we usedan exposure time of 1 s per data point.Reporting summary. Further information on research design is available in the NatureResearch Reporting Summary linked to this article.Data availabilityThe data that support the findings of this study are available at the online depositoryEPUB of the University of Regensburg, with the identifier https://doi.org/10.5283/epub.52406.Received: 11 November 2021; Accepted: 7 July 2022;References1. Rikken, G. L. J. A., Fölling, J. & Wyder, P. Electrical magnetochiral anisotropy.Phys. Rev. Lett. 87, 236602 (2001).2. Rikken, G. L. J. A. & Wyder, P. Magnetoelectric anisotropy in diffusivetransport. Phys. Rev. Lett. 94, 016601 (2005).3. Ideue, T. et al. Bulk rectification effect in a polar semiconductor. Nat. Phys. 13,578 (2017).4. Wakatsuki, R. et al. Nonreciprocal charge transport in noncentrosymmetricsuperconductors. Sci. Adv. 3, e1602390 (2017).5. Yasuda, K. et al. Nonreciprocal charge transport at topological insulator/superconductor interface. Nat. Commun. 10, 2734 (2019).6. Itahashi, Y. M. et al. 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Nano Lett.15, 6836 (2015).AcknowledgementsThe authors thank Denis Kochan, Marco Aprili, and Magdalena Marganska for usefuldiscussions. The work was funded by the Deutsche Forschungsgemeinschaft (DFG,German Research Foundation)—Project-ID 314695032—SFB 1277 (subprojects B03,B04, B08, and B11), and by the European Union’s Horizon 2020 research and innovationprogramme under grant agreements No 862660 QUANTUM E-LEAPS.Author contributionsN. Paradiso and C.S. conceived and designed the experiments. L.B., C. Bäuml, and N.Paulik fabricated the samples. L. B., L. F., C. Baumgartner and N. Paradiso, performedthe transport experiments; L.B., J. M. L., K.-Q. L., and J. M. B. conceived and performedSHG measurements. K. W. and T. T. grew hBN crystals. All authors contributed to thepreparation of the manuscript.FundingOpen Access funding enabled and organized by Projekt DEAL.Competing interestsThe authors declare no competing interests.Additional informationSupplementary information The online version contains supplementary materialavailable at https://doi.org/10.1038/s41467-022-31954-5.Correspondence and requests for materials should be addressed to Nicola Paradiso.Peer review information Nature Communications thanks Yong-Lei Wang, and theother, anonymous, reviewer(s) for their contribution to the peer review of this work. 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To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.© The Author(s) 2022NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-022-31954-5 ARTICLENATURE COMMUNICATIONS |         (2022) 13:4266 | https://doi.org/10.1038/s41467-022-31954-5 |www.nature.com/naturecommunications 7https://arxiv.org/abs/2201.00831https://arxiv.org/abs/2201.01775https://arxiv.org/abs/2112.08901https://arxiv.org/abs/2111.05340https://arxiv.org/abs/2111.05340https://arxiv.org/abs/2106.03575v1https://arxiv.org/abs/2205.09276https://arxiv.org/abs/2111.05627https://arxiv.org/abs/2110.01067https://arxiv.org/abs/2110.01067https://arxiv.org/abs/2112.07841https://arxiv.org/abs/2112.11285https://doi.org/10.1038/s41467-022-31954-5http://www.nature.com/reprintshttp://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/www.nature.com/naturecommunicationswww.nature.com/naturecommunications Supercurrent diode effect and magnetochiral anisotropy in few-layer NbSe2 Results Discussion Methods Sample preparation Transport measurements Second harmonic generation measurements Reporting summary Data availability References References Acknowledgements Author contributions Funding Competing interests Additional information