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Mio Hashimoto, [Takako Konoike](https://orcid.org/0000-0002-6037-5782), Tomoki Kobayashi, Shintaro Hoshino, Takuya Kawada, Tomoyuki Yokouchi, [Shinya Uji](https://orcid.org/0000-0001-9351-6388), Atsutaka Maeda, Yuki Shiomi

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[Nonlinear Planar Hall Effect from Superconducting Vortex Motion](https://mdr.nims.go.jp/datasets/c92131d6-5d57-40a7-9df6-f56dd62c4e2e)

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Nonlinear planar Hall effect from superconducting vortex motionNonlinear planar Hall effect from superconducting vortex motionMio Hashimoto,1 Takako Konoike,2 Tomoki Kobayashi,1 Shintaro Hoshino,3 TakuyaKawada,1 Tomoyuki Yokouchi,1, 4 Shinya Uji,2 Atsutaka Maeda,1 and Yuki Shiomi11Department of Basic Science, The University of Tokyo, Meguro, Tokyo 153-8902, Japan2Research Center for Materials Nanoarchitectonics (MANA),National Institute for Materials Science, 3-13 Sakura, Tsukuba, Ibaraki 305-0003, Japan3Department of Physics, Saitama University, Shimo-Okubo, Saitama 338-8570, Japan4RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan(*takuyakawada@g.ecc.u-tokyo.ac.jp)(Dated: September 9, 2025)We report the nonreciprocal charge transport along the longitudinal and transverse directions in the vortexflow regime of FeSe superconducting films. Clear nonreciprocal signals under an inplane magnetic field revealssymmetry breaking at the film surfaces since the crystal structure of FeSe is centrosymmetric. Although thesymmetry in such polar superconductors allows the nonreciprocal transverse response under a magnetic fieldparallel to the electric current, its observation is physically counterintuitive because vortex motion is not ex-pected in this configuration. We propose that thermally excited (anti)vortices due to the two-dimensional natureof FeSe give rise to the nonreciprocal transverse signals when the mirror symmetry is broken by the inplanemagnetic field.Introduction- Research on rectification effects has a verylong history in condensed matter physics since their discoveryat metal-semiconductor interfaces [1, 2], which establishes thefundamentals of modern electronics, such as p-n junctions.The emergence of rectification effects requires a broken spa-tial inversion symmetry at the interfaces. The asymmetry inenergy levels at the interfaces generates a unidirectional elec-tron flow. However, broken inversion symmetry alone is insuf-ficient to observe the rectification effect in single materials. Inaddition to breaking the spatial inversion symmetry, breakingthe time-reversal symmetry is also necessary. This type of rec-tification effect is commonly referred to as the nonreciprocaltransport effect or magnetochiral anisotropy effect [3], whichhas recently been explored in various materials [4, 5].In nonreciprocal transport effects, the longitudinal andtransverse resistivities are different for electric current I flow-ing to the right (+I) and to the left (−I). The symmetry argu-ment predicts that for polar systems such as material surfacesand hetero-interfaces, the longitudinal and transverse nonre-ciprocal signals are expected when the magnetic field (H)is applied in the inplane direction perpendicular and parallelto the I direction, respectively. Beyond the symmetry argu-ment, several microscopic mechanisms have been proposedand found to be material-dependent, such as an asymmetricelectron scattering [6–9], the deformation of the Fermi surfacecaused by the Zeeman term [10, 11], and the Berry curvatureeffect [12].Recently, the study of nonlinear transport effects has alsogained increasing attention in superconductors, exemplifiedby the superconducting diode effects [13–17]. To date, nonre-ciprocal response along the longitudinal direction (so-callednonreciprocal magnetoresistance) has been reported in var-ious noncentrosymmetric superconductors, for example, inlow-symmetry bulk superconductors [14, 18, 19], and in po-lar two-dimensional (2D) superconductors [20–23]. In mostcases, the nonreciprocal magnetoresistance is enhanced in su-perconductors compared to normal metals [20, 21, 23, 24].An origin of the enhanced rectification is the contribution ofthe ratchet motion of vortices in superconductors. Asymmet-ric pinning potentials for vortices induce nonreciprocal vortextransport in the vortex flow regime, resulting in giant nonre-ciprocal signals. In addition, a recent theoretical work pointedout that thermally activated vortices and antivotices also con-tribute to nonreciprocal magnetoresistance in 2D supercon-ductors [25].Although the nonreciprocal longitudinal response has beencomprehensively studied in superconductors experimentallyand theoretically, the nonreciprocal transport along the Halldirection has been less studied and limited in noncentrosym-metric superconductors [18, 26, 27]. In fact, previous the-oretical works predicted the nonreciprocal transverse signalgeneration even in 2D centrosymmetric superconductors byapplying inplane magnetic field to break the mirror symmetrywith respect to the magnetic field direction [25, 28]. To thebest of our knowledge, however, no experiments have explic-itly demonstrated this effect yet.In this letter, we study the nonreciprocal transport effectsin superconducting FeSe films in detail by measuring current-nonlinear contributions of longitudinal and transverse resis-tances under magnetic fields. Although the crystal structureof FeSe is centrosymmetric, we observed nonreciprocal trans-port signals in the vortex flow regime, suggesting that theirorigin is vortex ratchet motions. The inversion symmetry isonly broken at the film interfaces, but clear nonreciprocal sig-nals are observed owing to the topological nature of vortices[29]. Remarkably, we found that the nonreciprocal signals areobserved in the transverse (Hall) direction when H is appliedparallel to I. This is highly counterintuitive, since it is be-lieved that the driving force of magnetic-field induced vortexmotion is generated in the direction of I⃗ × H⃗ and thus shouldbe zero in the configuration of H||I. We tentatively attributethe origin of the nonreciprocal transverse signals to thermallyexcited (anti)vortices arising from the 2D nature of FeSe.Methods- 23-nm-thick FeSe epitaxial films were grown on(001) LaAlO3 (LAO) substrates and capped with amorphousSi by the pulsed laser deposition method [30, 31]. The amor-arXiv:2509.06313v1  [cond-mat.supr-con]  8 Sep 2025https://arxiv.org/abs/2509.06313v12FIG. 1. (a) Sample and measurement setup. (b) Temperature (T )dependence of resistance (Rxx). The inset shows a magnified viewof the low-T range. (c),(d) Electric-current (I) dependence of (c) Rxxand (d) nonreciprocal longitudinal voltage ∆Vx at normal state (4.2 K)and superconducting state (2.1 K). The dotted line in (d) representsthe position of the depinning current.phous Si layer is insulating at low temperatures. As shown inFig. 1(a), the films were deposited in a Hall-bar shape usinga metal mask. Transport measurements were conducted usingthe standard five-terminal method under magnetic fields ap-plied with an Oxford 17 T superconducting magnet. The Hdirections were rotated with respect to the electric-current di-rection (||x) using a homemade two-axis rotator system. Themeasurement was mainly performed at 2.1 K below the super-conducting transition temperature (Tc); the samples were im-mersed in superfluid helium with high thermal conductivity,and effects of Joule heating were negligible even when elec-tric current of 20 mA was applied to the films as discussedlater.We used Keithley 2401 Sourcemeter as dc current sourceand Keithley 2182A Nanovoltmeters for detecting the longitu-dinal and transverse voltages; whereas lock-in detection meth-ods have been used in most studies of nonreciprocal transporteffects, we directly measured the nonreciprocal componentsof the longitudinal (||x) and transverse (||y) voltages by takingthe difference between the voltages for +I and −I currents ateach H: ∆Vi = {Vi(+I)+Vi(−I)}/2 (i = x, y). Here, to elim-inate background voltage on the transverse voltage Vy, we fur-ther estimated ∆Ṽy(θ) = {∆Vy(θ)−∆Vy(θ +π)}/2 in each Hdirection θ(= α,β , and γ; see Fig. 3 for the definition). Bydividing ∆Vx and ∆Ṽy by I, the nonreciprocal resistances alongthe longitudinal and transverse directions, ∆Rxx and ∆Ryx, areobtained, respectively.Results and Discussion- Figure 1(b) presents the tempera-ture (T ) dependence of the electrical resistance (Rxx) of theFeSe film. The sample is metallic and exhibits superconduc-tivity below Tc= 2.8 K, where Tc is defined as the temperaturecorresponding to the midpoint of the superconducting transi-tion. Figure 1(c) shows the current (I) dependence of electricresistance Rxx measured at 2.1 K (< Tc) and 4.2 K (> Tc),FIG. 2. Nonreciprocal longitudinal resistance ∆Rxx measured (a) forH ∥ I and (b) for H ⊥ I and nonreciprocal transverse resistance ∆Ryx(c) for H ∥ I and (d) for H ⊥ I as a function of electric current (I).both measured under an applied H of 0.6 T along the +y di-rection. At 4.2 K, Rxx is constant with respect to the amplitudeof I, as expected from the Ohm’s law. In contrast, Rxx at 2.1K is almost zero in the small I regime below ∼ 2 mA, butsuddenly increases with a further increase in I. Notably, Rxxvalues for +I (shown in red) and −I (blue) are slightly differ-ent in the middle I region. Such deviation is not significantat 4.2 K above Tc; Rxx for +I (shown in pink) and −I (lightblue) is almost the same over the entire I range, showing thatnonreciprocal transport of normal-state electrons is negligiblebecause of the centrosymmetric crystal structure.By taking the difference between the longitudinal voltagesmeasured in +I and −I conditions, ∆Vx is estimated at 2.1 Kand 4.2 K, as shown in Fig. 1(d). Whereas ∆Vx is negligiblysmall in the normal state, a nonzero ∆Vx signal is observed inthe superconducting state at 2.1 K. |∆Vx| exhibits a peak at ap-proximately 8 mA and decreases with higher I values. This Idependence is attributed to the dynamics of superconductingvortices driven by the applied current. Vortices are pinned byimpurities and/or defects inside the superconductor at a lowI range, while they are depinned and begin to move as I in-creases. When the amplitude of I further increases above 8mA, the superconducting state is gradually destroyed by largeelectric current (∼ 108 A/m2) and ∆Vx approaches zero.The I dependence of ∆Rxx(= ∆Vx/I) measured under var-ious magnetic fields applied along the x and y axes is shownin Figs. 2(a) and 2(b). ∆Rxx is only observed for H||y [Fig.2(b)], and hardly observed for H|| − x [Fig. 2(a)]. For H||y,the magnitude of ∆Rxx is negligibly small without H becauseof the preserved time-reversal symmetry, and then increaseswith H owing to the enhanced driving force on the vorticesand/or the increased vortex density. At higher magnetic fields,superconductivity begins to diminish, and the nonreciprocalresponse becomes difficult to detect. Note that, in the low-Iregion, a sign change is observed in the nonreciprocal signalat 1.2 T and 3 T. Similar sign changes have been frequentlyobserved in the nonreciprocal magnetoresistance due to vor-3tex ratchet motion [14, 18, 19, 22–24, 32] and also in formerstudies on vortex ratchet effects in nanopatterned supercon-ductors with asymmetric pinning potentials [33, 34]. At rel-atively high magnetic fields, the strong vortex-vortex interac-tion due to the high vortex density may cause the sign reversal[34, 35].The nonzero ∆Vx signal under H||y indicates that spatial in-version symmetry is broken along the z direction. Lustikova etal. reported the nonreciprocal magnetoresistance in supercon-ductor/magnet bilayer films originating from the asymmetricsurface barriers due to different magnetic environment at in-terfaces [29]. In our case, crystal structure of FeSe is cen-trosymmetric [36] and thus the symmetry breaking is ascribedto the FeSe/LaAlO3 and FeSe/Si interfaces. The different sur-face barriers at the FeSe/LaAlO3 and FeSe/Si interfaces breakthe inversion symmetry along the z axis, satisfying the sym-metry requirement for nonreciprocal vortex transport. Notethat the breaking of time-reversal symmetry is also required,as the nonreciprocal signal was negligibly small under zeromagnetic field [31].We also measured the nonreciprocal transport responsealong the transverse (Hall) direction. Figures 2(c) and 2(d)show the I dependence of ∆Ryx under H||− x and H||y. Non-reciprocal signals are observed for H|| − x but negligible forH||y. This is in stark contrast to the case of nonreciprocalmagnetoresistance, where the signals appear for H||y but notfor H|| − x, as shown in Figs. 2(a) and 2(b). The overall Idependence of ∆Ryx for H|| − x is similar to that of ∆Rxx forH||y; ∆Ryx shows a broad dip at a middle I value. The simi-lar I dependences of ∆Rxx and ∆Ryx suggest a vortex origin ofthe nonreciprocal transverse signal. In terms of the symme-try condition, nonreciprocal transverse signals are expectedwhen H||I for polar systems. Hence, our observation is con-sistent with the prediction of the symmetry argument. How-ever, we stress that H-induced superconducting vortices can-not move when H||I because of the absence of driving forces.In contrast to electronic nonreciprocal transport, where elec-tron flow is always driven by the application of I, transport ofH-induced vortex strongly depends on the mutual directionsof I and H. Within the framework of the conventional theoryof vortex transport, the emergence of the nonreciprocal trans-verse signal is nontrivial.In Fig. 3, we measured ∆Rxx and ∆Ryx as a function ofmagnetic-field angles in three different scan geometries at 2.1K: α-scan in Fig. 3(a), β -scan in Fig. 3(b), and γ-scan inFig. 3(c). The definition of α , β , and γ is shown in the upperpanels of Fig. 3. The results suggest that both ∆Rxx and ∆Ryxsatisfy the symmetry condition. In the α-scan in Fig. 3(a),∆Rxx reaches its maximum magnitude at H||y while ∆Ryx isthe largest in H||x, both of which show an almost sinusoidalangular dependence. Consistently, ∆Rxx and ∆Ryx are barelyobserved in the entire γ [Fig. 3(c)] or β values [Fig. 3(b)], re-specetively. As for ∆Rxx, similar sinusoidal in-plane angulardependence has been observed in previous reports on nonre-ciprocal magnetoresistance [14, 23]. When the magnetic fieldis rotated from being parallel to the z axis toward the y axis,∆Rxx remains small up to β ∼ 50◦ but rapidly increases whenthe H direction approaches the y direction, exhibiting a peak(a)(b)(c)(d)FIG. 3. Definitions of magnetic field angles α , β and γ . Greenarrows represent the direction of the magnetic field (H). (b-d)Magnetic-field angle dependence of ∆Rxx and ∆Ryx in (b) α scan (ro-tation within the xy plane), (c) β scan (rotation within the yz plane),and (d) γ scan (rotation within the xz plane). The amplitude of I is8 mA, and the H strength is 0.6 T for α and β scans and 1 T for γscan.at β =−90◦, 90◦, and 270◦ in Fig. 3(b). This nonmonotonicangular dependence is likely related to anisotropic pinning ef-fects owing to the layered structure of FeSe [37]. As shownin Fig. 3(c), the angular dependence of ∆Ryx in the γ-scan issimilar to that observed in the β -scan of ∆Rxx.The results shown in Fig. 3 suggest that the emergence of∆Rxx and ∆Ryx is not explained by the vortex Nernst effect[31, 38–40]. The absence of ∆Rxx and ∆Ryx in H||z indicatesthat the vortex Nernst voltage due to the inplane temperaturegradient is negligibly small. In the α-scan in Fig. 3(a), peakvalues of ∆Rxx is larger than those of ∆Ryx by a factor of three,which denies that the nonreciprocal signals solely originatefrom the vortex Nernst effect caused by the out-of-plane tem-perature gradient: if so, the ratio of them should be equal tothat of the sample size along x (∼ 1.7 mm) and y (∼ 1.2 mm).Since H-induced vortex motion is not expected when H||I,the origin of the nonreciprocal transverse response is notstraightforward. A clue to approach the mechanism is a 2Dcharacter in superconductivity in FeSe films [41–43]. We con-firmed the 2D nature of our FeSe films by measuring the uppercritical field Hc2 at different β angles at 2.1 K. Hc2 is evaluatedfrom the H dependence of Rxx plotted in Fig. S1(a) in Supple-4(a)3D model2D modelMirror symmetry breakingvortexantivortexFeSe film(b)FIG. 4. (a) (b) β dependence of the upper critical field Hc2 for severalRxx/RN values. Solid and dotted curves stand for the fits to 2D and3D models, respectively. See SM about the model curves [31]. (b)Schematic illustration of asymmetric transport of thermally excited(anti)vortices when H||I. Orange (blue) arrows represent the velocityof vortex (antivortex).mental Material (SM) [31], which is defined as the position atRxx = 0.5RN [41]. Here RN is the normal state resistance at2.1 K and estimated to be 34 Ω. The β dependence of Hc2 isplotted in Fig. 4(a). We found that the β dependence of Hc2 isbetter fitted with a 2D model than a three-dimensional model,suggesting that the superconductivity in our FeSe films is 2Din nature. Note that this tendency is robust regardless of thevalue of Rxx/RN to determine Hc2: see Fig. S1(b) in SM forthe analysis of Hc2 under different Rxx/RN values [31].Based on the 2D nature of our FeSe thin film, a plau-sible origin of the nonreciprocal transverse response is theratchet motion of (anti)vortices excited without H [Fig. 4(b)].In 2D superconductors, vortex and anti-vortex pairs are ex-cited thermally or by electric current, in addition to the H-induced vortices. The magnetic flux of these (anti)vorticespoints in the direction perpendicular to the 2D plane, and these(anti)vortices contribute to the nonreciprocal magnetoresis-tance, as demonstrated theoretically [25] and experimentallyin several superconductors [20–23, 27, 44]. Since the direc-tion of the magnetic flux (||z) is perpendicular to the I direc-tion (||x), the motion of (anti)vortices is allowed along the yaxis. If the vortex Hall effect is incorporated in the vortex dy-namics, it becomes asymmetric for superconductors lackingmirror symmetry with respect to the xz plane [27, 31]. Theasymmetric (anti)vortex transport along the x axis gives rise tononreciprocal transverse voltage (||y), as shown in Fig. 4(b).It is notable that a large vortex Hall angle of ∼ 0.5 was in-deed reported for FeSe single crystals [45, 46]. Although ourFeSe film exhibits inversion symmetry breaking only along z,the application of H along the x axis further breaks the mirrorsymmetry with respect to the xz plane, and thus nonrecipro-cal transverse signal is allowed in terms of symmetry. Wealso investigated temperature dependence of the nonrecipro-cal signals and we find that it is consistent with the scenarioof thermally-excited (anti)vortices (see SM about the detail[31]).We also perform a model calculation for a 2D superconduc-tor under the application of a magnetic field by incorporatingthe Rashba spin-orbit coupling in the Ginzburg-Landau freeenergy [25]. As detailed in SM [31], the vortex velocitiescalculated by the derivative of the free energy with respectto wave vectors include nonlinear force terms. The struc-ture of the force-velocity relation is similar to that reportedfor a ratchet potential model in the noncentrosymmetric trig-onal superconductor PbTaSe2 [27]. The nonlinear force termscause asymmetric transport of (anti)vortices along the x di-rection in the presence of the vortex Hall effect, leading tothe nonreciprocal transverse voltage that is proportional to theHall angle under H||x, as illustrated in Fig. 4(b). This calcu-lation explicitly describes the nonreciprocal transverse signalowing to the mirror symmetry breaking by the magnetic field.We note that our model calculation also derives the nonrecip-rocal magnetoresistance in H||y under the same mechanism.In addition, the nonreciprocal magnetoresistance is predictedto be three times larger than the nonreciprocal transverse sig-nal when the width and length of the system are nearly equal,which is satisfied in this study. Using the peak values of ∆Rxxand ∆Ryx shown in Fig. 2(b) and (c), the ratios of ∆Rxx to∆Ryx can be estimated as 2 ∼ 4, implying that the asymmetrictransport of (anti)vortices mainly contributes to the nonrecip-rocal signals. We remark that the nonreciprocal motion ofH-induced vortices can give rise to ∆Rxx, but not ∆Ryx, whichmight deviate the value of ∆Rxx/∆Ryx from the theoretical ex-pectation. See SM about the detail of this scenario [31].Conclusion- We successfully observed nonreciprocalcharge transport due to vortex ratchet motion in FeSe films notonly along the longitudinal direction but also the transversedirection. The nonreciprocal transverse response under I||His allowed in terms of symmetry, but the microscopic mecha-nism is nontrivial since vortex motion is not expected in thisconfiguration. The vortex-antivortex pairs thermally excitedby the 2D character of FeSe may contribute to the nonrecip-rocal transverse signal. Since this mechanism is not unique toFeSe, we anticipate that similar phenomena will be observedin other 2D superconductors. The present results highlightrich nonlinear transport phenomena of superconducting vor-tices.ACKNOWLEDGMENTSThe authors are grateful to Dr. T. Kawamura, Dr. S.Sumita, and Prof. Y. Kato for fruitful discussions. 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