Higher-order electrical magnetochiral anisotropy effect in high-temperature cuprate superconductors

Abstract

While non-trivial spin-orbit coupling (SOC) was predicted early in high-temperature cuprate superconductors (HTCS), its precise role and microscopic consequences remain unresolved. This challenge is primarily hindered by the lack of suitable detection methods. Here we report a significant electrical magnetochiral anisotropy (EMCA) in an archetypal high-temperature cuprate superconductor NdBa₂Cu₃O_7-δ, which is ascribed to the SOC effect under inversion and time-reversal symmetry violations in HTCS. Notably, a peculiar angular dependence of EMCA on the magnetic field orientation is observed, where an unprecedented higher-order EMCA effect is discovered. Based on Ginzburg-Landau theory analysis, the higher-order EMCA effect is intrinsically linked to magnetic-field-induced finite-momentum pairing states in HTCS. Our findings provide crucial insights into the exotic physics associated with SOC in HTCS, and pave the way for exploring high-temperature superconducting quantum electronics.

Spin-orbit coupling in cuprate superconductors remains poorly understood owing to limited experimental probes. Here, the authors observe electrical magnetochiral anisotropy in NdBa₂Cu₃O₇-δ,including a higher-order response linked to field-induced finite-momentum pairing.

INTRODUCTION

Despite decades of intensive investigation, high-temperature cuprate superconductors (HTCS) persist in unveiling novel phenomena and presenting profound challenges, such as the electron pairing mechanism, nature of the pseudogap state, and physics of various density-wave-ordered states^1–8. In particular, the effect of spin-orbit coupling (SOC) may play a crucial role in HTCS but remains largely underexplored. The interplay between SOC, strong electron correlations, and superconductivity is theorized to give rise to rich quantum phenomena, such as topological superconducting states^9,10 and parity mixing in Cooper pairs¹¹. In HTCS, it was predicted very early that the SOC can manifest itself as the layer-staggered Edelstein effect^12,13 and the anisotropic Dzyaloshinskii-Moriya interaction^14,15. Notably, recent spin- and angle-resolved photoemission spectroscopy experiments^16–18 have suggested the presence of nontrivial SOC features in HTCS. However, a direct and reliable electrical transport probe for quantitatively assessing SOC in HTCS is still lacking, impeding a unified understanding of its role in this superconducting material.

Nonreciprocal charge transport (NCT) occurs in an inversion-symmetry-broken materials structure, exhibiting a distinct current-voltage characteristic upon reversal of the electrical bias polarity^19,20. This phenomenon is exemplified in artificial semiconductor PN junctions, which serve as an important building block of semiconductor electronics. In bulk crystals, the realization of NCT generally requires the absence of time-reversal symmetry (achieved through external magnetic field or magnetic orders²¹), along with the broken inversion symmetry. The resulting unidirectional magnetoresistance effect is referred to as electrical magnetochiral anisotropy (EMCA)^22,23, which has been observed in the Si-based field effect transistor (FET)²³, the oxide heterostructure LaAlO₃/SrTiO₃²⁴, and the polar semiconductor BiTeBr²⁵. In these materials, the broken inversion symmetry will lead to spin splitting of electronic bands when the relativistic SOC exists. The energy scale of spin splitting, as well as that of magnetic energy induced by magnetic fields, is far smaller compared to the Fermi energy of the electronic bands; the EMCA effect is normally small. Nevertheless, EMCA can be significantly enhanced in the superconducting state, where its strength is governed by the ratio between the SOC and the superconducting energy gap, rather than the Fermi energy²⁶. This enhanced sensitivity has made EMCA a valuable tool for investigating novel electronic states in non-centrosymmetric superconducting systems and superconductivity-proximitized heterostructures in recent years^26–29. In this context, the EMCA effect could serve as a sensitive electrical transport probe for assessing the SOC effect in HTCS³⁰.

In this work, we present a comprehensive investigation of EMCA in the HTCS NdBa₂Cu₃O_7-δ (NBCO). Notably, we observe a significant second harmonic signal response under an in-plane magnetic field, which is attributed to the intrinsic NCT within superconducting layers. Additionally, we identify an unprecedented dependence of EMCA on the angle between the magnetic field and current direction. Based on Ginzburg-Landau theory analysis, the higher-order EMCA effect is intrinsically linked to magnetic-field-induced finite-momentum pairing states in HTCS. These findings pave the way for understanding symmetry-broken induced exotic physics in HTCS, providing crucial insights into the fundamental mechanisms governing non-reciprocal superconductivity in low-dimensional systems.

Results

Electrical magnetochiral anisotropy of NBCO films near the BKT transition temperature

As depicted in Fig. 1a, NBCO possesses an orthorhombic crystal structure, with CuO₂ superconducting planes situated between the Nd and Ba layers³¹. Pulsed laser deposition (PLD) is utilized to grow NBCO films on SrTiO₃ (STO) (001) substrates. The high quality of NBCO films is verified by the X-ray diffraction (XRD) (Supplementary Fig. 1), atomic-resolution high-angle annular dark field-aberration-corrected scanning transmission electron microscopy (HAADF-STEM) (Fig. 1b), and energy-dispersive X-ray spectroscopy (EDS) (Supplementary Fig. 2d). Next, Hall bar devices (Fig. 1c) for transport measurements are fabricated by a selective growth technique (see Methods). Figure 1d shows the temperature dependence of the resistance around the superconducting transition of the grown NBCO film. One can see that it exhibits a strange metal state (green area), an intermediate state (yellow area), and a superconducting state (blue area) as the temperature decreases, similar to other reports³¹. The superconducting transition temperature of the device is approximately 70 K, where the resistance starts to drop abruptly in the Fig. 1d. For clarity, we define the channel direction of the Hall bar as X axis, the in-plane direction transverse to the current as Y axis, and the direction perpendicular to the film plane as Z axis.

Fig. 1. NBCO structure and basic electrical characteristics.

a Schematic crystal structure of NBCO. b Atomic-resolution HAADF-STEM image of NBCO. Elemental color map in the upper right corner: Barium (Ba) is represented in red, Neodymium (Nd) in green, and Copper (Cu) in blue, with a scale bar of 2 nm. c Schematic of device structure and electric harmonic measurement configuration. d Temperature dependence of resistance measured under $\mathbf{I}$=10 μA. e Magnetic field dependence of the first (blue) and second harmonic resistance (red) at 40 K.

To assess EMCA phenomenon in NBCO films, we measured the first and second harmonic voltage with an alternating current $I$ applied along the X-axis. If the inversion symmetry of an electronic system is broken, the electrical voltage $V$ under the in-plane magnetic field is phenomenologically described up to a linear order in the magnetic field $\mathbf{B}=B\left(\cos \phi ,\sin \phi ,0\right)$ and second order of current $I$ as²²

$$V=R_{0}I\left(1+\gamma \left(\mathbf{B}\times \mathbf{z}\right)\cdot \mathbf{I}\right)$$ 1

Here, $R_0$ represents current-independent resistance, $\gamma$ is a coefficient representing the strength of the nonreciprocal transport, and $\mathbf{z}$ is a unit vector aligned with the polar axis, i.e. the direction of inversion symmetry breaking. The two terms in the right of Eq. (1) can be characterized via the first ($V^{\omega }$) and second harmonic voltages ($V^{2\omega }$), respectively, when an AC lock-in technique is used (Supplementary Note I). Figure 1e shows the relationship between the first harmonic resistances $R^{\omega }=V^{\omega }/I$ (blue) and second harmonic resistances$R^{2\omega }=V^{2\omega }/I$ (red) and in-plane magnetic field at a temperature of 40 K. When an in-plane magnetic field is applied, the NBCO film transitions from a superconducting state to an intermediate state, as indicated by the magnetic field-dependent first harmonic resistance in Fig. 1e. The second harmonic resistance arises tremendously along with the transition, reaching a maximum value at an intermediate field, and then decreases as the magnetic field is further increased, which indicates a significant EMCA effect in NBCO films and is similar to the behavior of EMCA in non-centrosymmetric superconducting systems and superconductivity-proximitized heterostructures^28,29,32,33. At the low magnetic field regime, the second harmonic resistance demonstrates a linear correlation with the magnetic field, consistent with Eq. (1), while it experiences attenuation due to the considerable suppression of superconducting properties within the elevated magnetic field regime. Specifically, the EMCA effect can only be detected in the intermediate state and eventually disappears in the normal state, which excludes thermoelectric effects as the source of the non-reciprocal signal (see Supplementary Note II). In this work, we have performed harmonics measurements on quite a few devices for EMCA effect in NBCO, which can always be observed once they are in the intermediate state, regardless of having proper device temperature, applied current, or magnetic field magnitude.

Subsequently, we examined the influence of current magnitude on the EMCA effect. Figure 2 (a-b) presents the first and second harmonic resistances in response to the in-plane magnetic field at different currents. One can see that the critical magnetic field for finite first harmonic resistance decreases with the current increasing (Fig. 2a), as the device is more prone to transitioning from a superconducting state to an intermediate state at elevated currents³³. Correspondingly, the position of maximum second harmonic resistance shifts to lower magnetic field with increasing current (Fig. 2b). To further analyze the current effect, we extract the maximum value of the second harmonic resistance and the corresponding current (Fig. 2c). In the small current regime ($I$≤1.5 mA), the maximum value of $R^{2\omega }$ increases linearly with the increase of $I$, consistent with the description of Eq. (1)³⁴. However, as the current continues to rise, the second harmonic resistance appears to decay, which is related to the suppression of superconductivity in NBCO at higher current levels³³, similar to the trend observed under strong magnetic fields.

Fig. 2. Current dependent nonreciprocal charge transport.

a, b The magnetic field dependence of $R^{\omega }$ (a) and $R^{2\omega }$ (b) at $T$=40 K measured under $\mathbf{I}$= 0.5, 0.8, 1.0, 1.2, 1.5, 2.0, and 2.5 mA. c The current-magnitude dependence of second harmonic resistance deduced from the maximum value in (b). d Schematic model of vortex dynamics for the BKT transition under opposite magnetic fields.

Recent studies have reported extrinsic mechanisms, such as field-induced vortices³⁵ and thermoelectric effect³⁶, for the non-reciprocal charge transport, however, which exhibit significant dependence on the out-of-plane magnetic field^37–39. Therefore, we have performed a series of experimental measurements to distinguish the influence from that under the out-of-plane magnetic field (Supplementary Note II). We found that the non-reciprocal response by out-of-plane magnetic fields is far smaller than that by the planar magnetic field, which thus excludes the extrinsic factors for the EMCA effect described above. Another key issue is the origin of inversion symmetry breaking in the centrosymmetric material NBCO. Although lattice distortions along the NBCO growth direction are observed in the grown films (Supplementary Note III), its precise role remains to be determined by future experiments, which is beyond the scope of this work. Notably, the EMCA is detected too in YBa₂Cu₃O_7₋δ film, suggesting that the non-reciprocal transport might be a common feature in high-temperature cuprate superconductors (Supplementary Note IV).

The micro-mechanism of the EMCA in NBCO is ascribed to current-induced motion of vortex and antivortex in the intermediate state (Fig. 2d)⁴⁰, which is above the Berezinskii Kosterlitz Thouless (BKT) phase transition temperature therein. The BKT phase transition, and thereby the two-dimensional superconducting nature of NBCO, is confirmed by the temperature dependence of the exponent $\alpha \left(T\right)$ in the current-voltage power law $V\propto \mid I{\mid }^{\alpha \left(T\right)}$^41,42 (Supplementary Fig. 3). Besides, we find that the non-reciprocal transport signal is significantly enhanced as the temperature decreases, as revealed by the extracted temperature dependent $\gamma$-value (Supplementary Fig. 4). This behavior of $\gamma$-value can be well fitted by the formula ${\left(T-T_{\mathrm{BKT}}\right)}^{-3/2}$(Supplementary Fig. 4), further verifying the BKT transition in NBCO and confirming that the EMCA phenomenon therein is an intrinsic signal as well^29,32,40. We also note that the BKT phase transition in HTCS has been well reported in other studies^43,44. According to the BKT model, vortex and anti-vortex pairs are coupled below the BKT phase transition temperature (left part of Fig. 2d), resulting in a zero-resistance superconducting state. Upon reaching the BKT phase transition temperature, the vortex-antivortex pairs become decoupled, leading to the formation of multiple free vortices. Furthermore, when magnetic fields are applied in different directions, the renormalized BKT transition temperature leads to renormalization of the vortex density (right part of Fig. 2d), thereby exhibiting non-reciprocal charge transport behavior. Consequently, the EMCA effect is introduced by renormalized vortex density due to violations of inversion and time-reversal symmetries in NBCO films. A more complete theoretical model of the BKT free vortex dynamics will be described later.

In-plane angular dependence of the nonreciprocal transport

To verify the symmetry of non-reciprocal transport and the reliability of the free vortex model, we perform an in-plane magnetic-field angular dependence of $R^{2\omega }$. Figure 3a illustrates the in-plane angular dependence of $R^{2\omega }$ under varying magnetic fields. In XOY plane, $R^{2\omega }$ exhibits a sinusoidal dependence at 0.2 T (with minor deviations), that is only the orthogonal y-component of the magnetic field contribute to the EMCA effect, consistent with conventional EMCA effect. However, with the application of higher magnetic fields, the angular dependent $R^{2\omega }$ deviates remarkably from the sinusoidal behavior, hinting other factors beyond conventional EMCA effect might be involved. The main feature is that while the peaks and valleys of the conventional sinusoidal behavior of EMCA are suppressed, there are other extrema emerging with the elevated magnetic field. Such observations are further evidenced by measuring the magnetic field dependence of $R^{\omega }$ and $R^{2\omega }$ when the angle between the magnetic field and the current is fixed at a specific angle. Particularly, when $\phi$ = 50°, the $R^{2\omega }$ transitions from a positive to a negative value as the magnetic field is increased to 8 T (Supplementary Fig. 5), which is consistent with the feature that a reversed extreme value around the angle is emerged at the behavior of the in-plane angular dependence of $R^{2\omega }$ when $\mathbf{B}$ = 4 T and $\mathbf{B}$ = 8 T (Fig. 3a). Though the extrema of the conventional sinusoidal behavior of EMCA at high magnetic fields are suppressed, they remain the same sign. This is distinctly different from the observation at the topological superconducting interface, where the value at $\phi$ = 90° is reversed under a large magnetic field²⁹.

Fig. 3. In-plane angular dependence of second harmonic resistance.

a Angle-dependent second harmonic resistance in different magnetic fields with applied current fixed ($\mathbf{I}$=2 mA). b Angle-dependent second harmonic resistance at different currents with 2 T of in-plane magnetic field. c Angle-dependent second harmonic resistance at different currents with 8 T of in-plane magnetic field. d Plots of the measured second harmonic resistance versus the fitted function ($\mathbf{B}$=0.2 T, $\mathbf{I}$=2 mA). e Plots of the measured second harmonic resistance versus the fitted function ($\mathbf{B}$=4 T, $\mathbf{I}$=2 mA), where the black hollow circle represents the raw data, the red curve represents the fitted data, the blue and purple curves respectively represent the $\sin \phi$ and $\sin 3\phi$ components of the fitted function. f, Magnetic field dependent amplitudes of $\sin \phi$ and $\sin 3\phi$ components ($\mathbf{I}$ = 2 mA), where the lines are drawn for guiding. The red and blue shadow areas represent weak and strong magnetic field regimes, respectively.

In order to investigate the origin of the anomalous EMCA effect, we conducted a series of experiments to evaluate the roles of current and the magnetic field. Initially, we maintained a magnetic field of 2 T and examined the angular dependence of $R^{2\omega }$ with different currents. One can see from Fig. 3b that though there are minor deviations from sinusoidal behaviors, the conventional EMCA effect dominates under a small magnetic field upon increasing the current. It is worth noting that $R^{2\omega }$ does not show finite value and deviates strongly from the sinusoidal behavior at a small current when the magnetic field is applied in the directions far away from the perpendicular ones (Fig. 3b), which is because that the superconducting state of NBCO hasn’t been destroyed in such a status (Supplementary Fig. 6). Next, the magnetic field of 8 T was set and the angular dependence of $R^{2\omega }$ under various currents were measured. In the presence of strong magnetic fields, the angular dependence of $R^{2\omega }$ displays significant anomalous EMCA effect with current magnitude of above 0.5 mA, as shown in Fig. 3c. Supplementary Fig. 7 demonstrates that $R^{2\omega }$ exhibits mirror symmetry when a strong magnetic field is applied in the opposite direction, indicating the similar properties of the anomalous EMCA effect with the conventional one.

Origin of higher-order electrical magnetochiral anisotropy

We now present a comprehensive theoretical analysis of the non-reciprocal transport mechanism in this system. Above the BKT transition temperature $T_{\mathrm{BKT}}$, the proliferation of thermally activated free vortex becomes energetically favorable^45,46. These topological excitations, when subjected to a uniform current density ${\mathbf{j}}_{\mathrm{unif}}$, experience a transverse Magnus force, causing them to move in a direction perpendicular to the applied current (Fig. 2d)⁴⁰. Through the Josephson relation, the motion of vortices results in a voltage drop (and therefore an electric field) along the direction of current, from which the resistivity can be calculated. The magnitude of the electric field is related to the density of these thermally unpaired vortices, which obeys the characteristic scaling law^40,45,46

$$n_v\propto \exp \left(-2\sqrt{b\frac{T_{\mathrm{c}}-T_{\mathrm{BKT}}}{T-T_{\mathrm{BKT}}}}\right)$$ 2

where $b$ is a system-dependent constant and $T_{\mathrm{c}}$ is the mean field transition temperature. The BKT transition temperature $T_{\mathrm{BKT}}$ is directly proportional to the superfluid density $n_s$, which can be tuned by external parameters. When an in-plane magnetic field is applied to the system, interestingly, the superfluid density becomes renormalized, ${\tilde{n}}_s=n_s\left[1+\mathrm{\Lambda }\left(B,\phi \right)v_{s0}\right]$, where $\mathrm{\Lambda }\left(B,\phi \right)$ denotes the magnetic-field-dependent renormalization coefficient (see Eq. (C7) in Supplementary Note VII) and $v_{s0}$ represents the superfluid velocity associated with the current density ${\mathbf{j}}_{\mathrm{unif}}$. The renormalized superfluid density ${\tilde{n}}_s$ modifies the BKT transition temperature to ${\tilde{T}}_{\mathrm{BKT}}$, whose value becomes dependent on the current direction. This directional anisotropy in ${\tilde{T}}_{\mathrm{BKT}}$ subsequently governs the vortex density distribution, which ultimately determines the non-reciprocal resistivity, as schematically illustrated in Fig. 4b, c. Such a current-direction-dependent vortex dynamics fundamentally establishes the EMCA mechanism in our system.

Fig. 4. Microscopic origin of the non-reciprocal charge transport.

a Schematic illustration of finite-momentum Cooper pairing induced by Fermi surface shift. The solid green circle depicts the shifted Fermi surface, whereas the dashed circle corresponds to the equilibrium Fermi surface at zero current ($\mathbf{I}=0$) and zero magnetic field ($\mathbf{B}=0$). The blue arrow indicates the shift direction of the Fermi surface. b, c Schematic illustration of the unpaired vortex density and its dependence on current direction. d Cooper-pair momentum dependence of the GL free energy density $f\left(\mathbf{q}\right)$ for several selected magnetic field strengths, where the pentagram symbols denote the free energy minima, corresponding to the Cooper-pair momentum in equilibrium. e, f The coefficients (e)${\alpha }_3^{xx}\left(\phi \right)$ and (f)${\alpha }_3^{xy}\left(\phi \right)$ of the q-cubic terms, plotted as a function of the in-plane field angle $\phi$ under a strong magnetic field. The first- ($\sin \phi$) and third-order ($\sin 3\phi$) Fourier components are shown, with their amplitudes normalized to the respective peak values of ${\alpha }_3^{xx}\left(\phi \right)$ and ${\alpha }_3^{xy}\left(\phi \right)$.

The origin of superfluid density renormalization can be traced to the synergistic effect between the Zeeman splitting and spin-momentum locking, which generates finite-momentum Cooper pairs (Fig. 4a)^47–49. To quantitatively elucidate the non-reciprocal transport caused by the renormalization of superfluid density, we employ a generalized Ginzburg-Landau (GL) theoretical framework that incorporates Cooper-pair momentum $\mathbf{q}$⁴⁰. The free energy density takes the form$f\left(\mathbf{q},{\mathrm{\Delta }}_{\mathit{q}}\right)={\alpha }_{\mathit{q}}{\mid {\mathrm{\Delta }}_{\mathit{q}}\mid }^2+\frac{1}{2}\beta {\mid {\mathrm{\Delta }}_{\mathit{q}}\mid }^4$, where ${\mathrm{\Delta }}_{\mathit{q}}$ is the superconducting order parameter and quadratic coefficient ${\alpha }_{\mathit{q}}$ can be expanded as a power series in $\mathbf{q}=\left(q_x,q_y\right)$:

$${\alpha }_{\mathit{q}}={\alpha }_0+{\alpha }_1^x{q}_x+{\alpha }_1^y{q}_y+{\alpha }_2{q}^2+{\alpha }_3^{xx}{q}_x^3+{\alpha }_3^{xy}{q}_x{q}_y^2+{\alpha }_3^{yx}{q}_y{q}_x^2+{\alpha }_3^{yy}{q}_y^3+\cdots$$ 3

Notably, the coefficients of the linear and cubic terms in momentum $\mathbf{q}$, known as Lifshitz invariants, arise from the interplay between SOC and the Zeeman effect^47–55. These terms require the breaking of both inversion and time-reversal symmetries. The linear $\mathbf{q}$ term is associated with the Fermi surface shift induced by $\mathbf{B}$, as illustrated in Fig. 4a. It should be realized that this linear term does not play any role in homogeneous systems, since it can be gauged away from the GL free energy. The microscopic origin of nonreciprocal charge transport stems from cubic Lifshitz invariants in the GL expansion, which causes an asymmetry in the free energy with respect to $\mathbf{q}$, as demonstrated in Fig. 4d. Since the nonreciprocal charge transport is detected along a specific current direction (e.g., the x-direction) in experiments, the cubic terms ${\alpha }_3^{yy}{q}_y^3$ and ${\alpha }_3^{yx}{q}_y{q}_x^2$ do not contribute to the EMCA^40,49, leaving ${\alpha }_3^{xx}{q}_x^3$ and ${\alpha }_3^{xy}{q}_x{q}_y^2$ terms as the primary contributors.

At weak magnetic fields, the angle dependence of the cubic Lifshitz invariant ${\alpha }_3^{xx}\left(\phi \right)$ will give rise to the conventional EMCA²², following a sinusoidal $B\sin \phi$ trend. However, in stronger magnetic fields, particularly when the field strength becomes comparable to the critical paramagnetic field (Pauli limit) $B_{c0}$, contribution from $B_y{B}_x^2\left(=B^3\sin \phi {\cos }^2\phi \right)$ and $B_y^3\left(=B^3{\sin }^3\phi \right)$ terms must be taken into account, resulting in a highly non-sinusoidal dependence of ${\alpha }_3^{xx}$ on the field orientation $\phi$, as illustrated in Fig. 4e. It is evident that ${\alpha }_3^{xx}\left(\phi \right)$ contains both the fundamental harmonic term $\sin \phi$ and the third-order harmonic term $\sin 3\phi$. Similarly, the ${\alpha }_3^{xy}\left(\phi \right)$ Lifshitz invariant comprises both $\sin \phi$ and $\sin 3\phi$ (Fig. 4f). As a result, the coefficient of renormalized superfluid density $\mathrm{\Lambda }\left(B,\phi \right)$ will include both first- and third-order harmonic terms. This leads to the following comprehensive expression for the resistance (see Supplementary Note VII.D):

$$R=R_0\left\{1+\left[\left({\gamma }_{1}B-{\gamma }_3{B}^3\right)\sin \phi -{\tilde{\gamma }}_3{B}^3\sin 3\phi \right]I\right\}$$ 4

Here, ${\gamma }_1$ corresponds to the conventional linear non-reciprocal coefficient, which dominates in the weak-field regime. In contrast, the cubic term ${\gamma }_3{B}^3$ (or ${\tilde{\gamma }}_3{B}^3$) captures the essential higher-order EMCA contribution—a central finding of this work.

Thus, guided by the theoretical analysis, we obtain the exact expression for the $R^{2\omega }$ fitting function to the angular dependent EMCA data as follows:

$$y=y_0+A_1\sin \left(\phi -{\theta }_1\right)+A_3\sin \left[3\left(\phi -{\theta }_2\right)\right]$$ 5

In this equation, $A_1$ and $A_3$ denote the amplitudes of different factors, while ${\theta }_1$ and ${\theta }_2$ represent their respective phases. Figure 3d, e illustrates the fitting curves of Eq. (5), as well as the contributions of the sinusoidal components $\sin \phi$ and $\sin 3\phi$ for angular dependent $R^{2\omega }$ measured in $\mathbf{B}$ = 0.2 T and $\mathbf{B}$ = 4 T, respectively (Please refer to Supplementary Fig. 8 for fittings on the data measured at other magnetic fields.). One can see that the fitting function (Eq. (5)) perfectly describes the angular dependent EMCA effect under various magnetic fields, indicating that the higher order EMCA effect must be considered for comprehensive understandings on the angular dependent second harmonic resistance, especially when a strong magnetic field is applied.

Next, the amplitudes of $\sin \phi$ and $\sin 3\phi$ were extracted for further analysis, as shown in Fig. 3f. Under weak magnetic fields, the amplitude of $\sin \phi$ is proportional to the magnetic field (Red shaded area in Fig. 3f), further supported by the magnetic-field-dependent second harmonic resistance (the blue box in Supplementary Fig. 5(b)); while it decreases gradually with increasing magnetic fields in strong magnetic fields (Blue shaded area in Fig. 3f). Because the magnetic field suppresses the superconducting transition temperature^56,57, the amplitude of $\sin \phi$ in the strong magnetic fields regime decreases in accordance with our EMCA theoretical derivation (see Supplementary Note VII.D). However, the suppression effect is not significant for the $\sin 3\phi$ component, as its amplitude is proportional to the cube of the magnetic field. Figure 3f demonstrates that the amplitude of $\sin 3\phi$ remains relatively constant, aligning with the theoretical analysis. Therefore, the ratio of $\sin 3\phi$ to $\sin \phi$ amplitudes increases with the magnetic field, which is consistent with the observation that the higher-order oscillation in EMCA appears when the applied magnetic field is strong enough.

Discussion

It is noteworthy that the higher-order MCA observed in our NBCO films arises primarily from two interrelated factors: the wide temperature window between $T_c$ and $T_{\mathrm{BKT}}$ (typically $~10$ K) allows the cubic-in-$B$ terms to compete with the linear-in-$B$ contribution, and simultaneously the high upper critical field of high-$T_c$ cuprates enables the superconducting state to survive under strong magnetic fields where such higher-order effects can be resolved. In contrast, low-$T_c$ conventional superconductors possess a much narrower temperature window and rapidly enter the normal state under extreme magnetic conditions^29,32, rendering the higher-order angular harmonics unobservable. The absence of even-order harmonics (e.g., $\sin 2\phi$, $\sin 4\phi$) in the measured second-harmonic resistance $R^{2\omega }\left(\phi \right)$ further confirms that the observed $\sin 3\phi$ harmonic originates intrinsically from the $\mathcal{O}\left(B^3\right)$ corrections to the free energy in the presence of Rashba-like SOC^16,30. The excellent agreement with theoretical analysis provides unambiguous evidence for the proposed higher-order EMCA mechanism and demonstrates the intrinsic nature of the nonreciprocal charge transport in HTCS systems.

In conclusion, we have demonstrated the presence of EMCA in the NBCO, indicating that spin-orbit coupling cannot be ignored in HTCS. Additionally, we identify an unprecedented dependence of EMCA on the angle of the applied magnetic field, referred to as the higher-order EMCA effect. Based on Ginzburg-Landau theory analysis up to $\mathcal{O}\left(B^3\right)$ corrections, the higher-order EMCA effect is intrinsically linked to magnetic-field-induced finite-momentum pairing states in HTCS. Our findings not only improve the understanding of the exotic EMCA effect, but also open new avenues for investigating intricate physics in HTCS, as well as exploring their potential applications in quantum electronics^58–60.

Methods

Thin-films growth and devices fabrication

NdBa₂Cu₃O_7-δ (NBCO) thin films were grown by pulsed laser deposition (PLD) using a ceramic NBCO target on SrTiO₃ (STO) (001) substrates, which were treated by buffer-HF solution and annealed at 950 °C for 3 h for attaining high-quality STO surface. The optimized substrate-target distance for NBCO was 76.6 mm in our PLD chamber. The NBCO films were deposited at 850°C with oxygen pressure of 200 mTorr and laser energy of 1 J/cm². A KrF excimer pulsed laser (λ = 248 nm) was used and fired at frequency of 5 Hz. After deposition, the sample was cooled down to room temperature at 10°C/min under oxygen pressure of 150 Torr.

For harmonic transport measurement, Hall-bars with a channel width of 20 μm were fabricated. A selective-growth technique was used for the Hall-bar devices fabrication. First, the photolithography and PLD were utilized to imprint the designed Hall pattern on STO (001) substrates, where the area of the Hall patterns was exposed after lift-off for the following growth while the other part was grown with amorphous AlN with the thickness ≈100 nm by PLD. Then, the NBCO films were grown with the conditions as described above. As the films deposited above the AlN were amorphous and insulating, the epitaxial-grown NBCO films formed the Hall device on the exposed STO area.

Transport measurements

The electrical transport measurements were carried out in a physical properties measurement system using the AC lock-in technique. During the measurement, an AC current (frequency: f = 317.3 Hz) is applied between the source and drain using a Keithley 6221 source meter. The first and second harmonic voltages ($V^{\omega }$ and $V^{2\omega }$) were simultaneously measured by two lock-in amplifiers (SR830). For angular-dependent measurements, the devices were rotated via a rotation sample holder under a constant magnetic field.

XRD measurement

The crystal structure and lattice constants of the films under investigation were analyzed via high-resolution X-ray diffraction (XRD) θ − 2θ scans, employing a Cu Kα radiation source (λ = 0.15406 nm).

STEM measurement and analyses

The cross-sectional samples were prepared using a focused ion beam (FIB; Helios NanoLab 600i, FEI), with operation at 30 kV and final cleaning at 5 kV. High-angle annular dark-field scanning transmission electron microscopy (HAADF-STEM) imaging was conducted using a double Cs-corrected STEM (Titan Themis G2, FEI), operated at 300 kV with collection semi-angles of 48–200 mrad. HAADF-STEM-coupled EDS elemental mapping was performed with 10 minutes of signal collection time.

Model and Hamiltonian

The electronic transport in our cuprate sample is described by an effective continuum model with Rashba SOC and in-plane magnetic field³⁰. The normal-state Hamiltonian is given by

$$H_N\left(\mathbf{k}\right)={\epsilon }_{\mathit{k}}{\sigma }_0+{\alpha }_R\left(k_y{\sigma }_x-k_x{\sigma }_y\right)-{\mu }_B\mathbf{B}\cdot \mathit{\sigma }$$ 6

where ${\epsilon }_{\mathit{k}}={\hslash }^2{\mathbf{k}}^2/2m-\mu$ with $\mu$ being the chemical potential, and $\mathbf{B}$ is the magnetic field. The effective mass $m$ and SOC strength ${\alpha }_R$ can be derived by applying the U(1) slave-boson approach to the extended t-J model with Dzyaloshinskii-Moriya (DM) interaction (Supplementary Note V)³⁰. The superconducting pairing originates from an effective attractive interaction, which takes the form

$$H_{int}=-\frac{1}{\mathrm{V}}{\sum }_{\mathbf{k},{\mathbf{k}}^{\prime },\mathbf{q}}V\left(\mathbf{k},{\mathbf{k}}^{\prime }\right)c_{\mathbf{k}+\mathbf{q}/2\uparrow }^{†}{c}_{-\mathbf{k}+\mathbf{q}/2\downarrow }^{†}{c}_{-{\mathbf{k}}^{\prime }+\mathbf{q}/2\downarrow }{c}_{{\mathbf{k}}^{\prime }+\mathbf{q}/2\uparrow }$$ 7

where V represents the system volume used for normalization, and the momentum-dependent interaction potential is given by $V\left(\mathbf{k},{\mathbf{k}}^{\prime }\right)=V_d\left(k_x^2-k_y^2\right)\left(k_x^{\prime 2}-k_y^{\prime 2}\right)/k^2{k}^{\prime 2}$. Through BCS mean-field decoupling, this leads to the characteristic $d_{x^2-y^2}$-wave pairing potential $\Delta \left(\mathbf{k},\mathbf{q}\right)={\Delta }_{\mathit{q}}\eta \left(\mathbf{k}\right)$ with $\eta \left(\mathbf{k}\right)=\sqrt{2}\left(k_x^2-k_y^2\right)/k^2$. This allows us to construct the Bogoliubov-de Gennes (BdG) Hamiltonian in Nambu space (Supplementary Note V), which serves as the foundation for subsequent GL analysis.

Ginzburg-landau theory

The expansion of GL free energy reveals the crucial nonreciprocal effects^47,49,59. Notably, the quadratic term in GL expansion can be derived using the Matsubara Green’s function formalism⁶¹:

$${\alpha }_{\mathbf{q}}{\mid {\mathrm{\Delta }}_{\mathbf{q}}\mid }^2=\frac{\mid {\mathrm{\Delta }}_{\mathbf{q}}{\mid }^2}{V_d}-k_{B}T{\sum }_n\int \frac{d^2\mathbf{k}}{{\left(2\pi \right)}^2}\mathrm{Tr}\left[G\left(i{\omega }_n,\mathbf{q}/2+\mathbf{k}\right)\widehat{\mathrm{\Delta }}{G}^{\mathrm{T}}\left(-i{\omega }_n,\mathbf{q}/2-\mathbf{k}\right){\widehat{\mathrm{\Delta }}}^{†}\right]$$ 8

where $\widehat{\mathrm{\Delta }}={\mathrm{\Delta }}_{\mathbf{q}}\eta \left(\mathbf{k}\right)i{\sigma }_y$ is the matrix of pairing potential and $G\left(i{\omega }_n,\mathbf{q}/2+\mathbf{k}\right)$ is the Green function for normal state. After performing the integral over momentum $\mathbf{k}$ and the summation over Matsubara frequencies ${\omega }_n$ in Eq. (8), we obtain the cubic Lifshitz invariants ${\alpha }_3^{xx}$ and ${\alpha }_3^{xy}$ up to terms cubic in the magnetic field (see Supplementary Note VI.B):

$${\alpha }_3^{xx}\left(\phi \right)=\frac{217\zeta \left(5\right)m{v}_F^2{\alpha }_R}{512{\pi }^5{T}_c^4}{\mu }_{B}B\sin \phi -\frac{635\zeta \left(7\right)m{v}_F^2{\alpha }_R}{8192{\pi }^7{T}_c^6}{\left({\mu }_{B}B\right)}^3\left(21\sin \phi -5\sin 3\phi \right)$$ 9

$${\alpha }_3^{xy}\left(\phi \right)=\frac{93\zeta \left(5\right)m{v}_F^2{\alpha }_R}{512{\pi }^5{T}_c^4}{\mu }_{B}B\sin \phi -\frac{635\zeta \left(7\right)m{v}_F^2{\alpha }_R}{8192{\pi }^7{T}_c^6}{\left({\mu }_{B}B\right)}^3\left(13\sin \phi +7\sin 3\phi \right)$$ 10

where the in-plane magnetic field is expressed as $\mathbf{B}=\left(B_x,B_y\right)=B\left(\cos \phi ,\sin \phi \right)$ with $\phi$ being the orientation angle. These higher-order terms ($~B^3$) give rise to a distinct third harmonic angular dependence, as shown in Fig. 4(c, d). The numerical computations employ the following parameter set: $k_B{T}_c=0.02$, $T=0.75T_c$, $m=1$, $\mid {\mu }_B\mathbf{B}\mid =1.2T_c$, $v_F=1$, and ${\alpha }_R/v_F=0.1$. Our theoretical framework is built on two key assumptions that define its range of validity: (i)The system must exhibit pronounced two-dimensional character with a BKT transition; the analysis applies in the temperature range $T_{\mathrm{BKT}}<T<T_c$. (ii) We assume a single-$\mathbf{q}$ helical superconducting state, i.e., $\Delta \left(\mathbf{r}\right)\approx {\Delta }_0{e}^{i\mathbf{q}\cdot \mathbf{r}}$, where the order-parameter amplitude is spatially uniform and only the phase varies^47–50.

Vortex dynamics and nonlinear resistance

The $\mathbf{q}-$ cubic terms in Eq. (3) lead to a current-dependent renormalization of superfluid density, which subsequently modifies the BKT transition temperature (see Supplementary Note VII):

$${\tilde{T}}_{\mathrm{BKT}}=T_{\mathrm{BKT}}^{\left(0\right)}\left[1+\left(\frac{24}{7}{\Gamma }_1{B}_y{v}_{s0}-\frac{42}{13}{\Gamma }_3^1{B}_y^3{v}_{s0}-\frac{26}{3}{\Gamma }_3^2{B}_x^2{B}_y{v}_{s0}\right)\right]$$ 11

where $T_{\mathrm{BKT}}^{\left(0\right)}$ is the BKT transition temperature at zero magnetic field and $v_{s0}$ is the magnitude of the superfluid velocity. The coefficients ${\Gamma }_1$, ${\Gamma }_3^1$, and ${\Gamma }_3^2$ originate from the cubic Lifshitz invariants (${\alpha }_3^{xx}$ and ${\alpha }_3^{xy}$) in the GL free energy expansion. The modified ${\tilde{T}}_{\mathrm{BKT}}$ affects the density of unpaired vortices, which eventually leads to the non-reciprocal resistivity. Explicitly, the resistance, exhibiting a characteristic higher-order magnetic field modulation, is given by (see Supplementary Note VII.D):

$$R=R_0\left[1-\sqrt{\frac{b\left(T_c-T_{\mathrm{BKT}}^{\left(0\right)}\right)}{{\left(T-T_{\mathrm{BKT}}^{\left(0\right)}\right)}^3}}\left(\frac{12\pi {\Gamma }_1}{7e^{*}{m}^{*}W}{B}_y-\frac{21\pi {\Gamma }_3^1}{13e^{*}{m}^{*}W}{B}_y^3-\frac{13\pi {\Gamma }_3^2}{3e^{*}{m}^{*}W}{B}_x^2{B}_y\right)I\right]$$ 12

Here, $R_0={\rho }_1{L}_x/W$ represents the linear resistance for a sample of length $L_x$ and width $W$, where ${\rho }_1$ is the linear resistivity. The symbols $e^{*}=2e$ and $m^{*}$ denote the charge and effective mass of Cooper pairs respectively. We emphasize that the pre-factors of the $B^3$ terms are determined by a combination of microscopic parameters: the effective mass $m^{*}$, Fermi velocity $v_F$, Rashba coupling ${\alpha }_R$, critical temperature $T_c$, and the density of states at the Fermi level.

The nonlinear response can be expressed more compactly in terms of nonreciprocal coefficients:

$$R=R_0\left[1+\left({\gamma }_{1}B\sin \phi -{\gamma }_3{B}^3\sin \phi -{\tilde{\gamma }}_3{B}^3\sin 3\phi \right)I\right]$$ 13

where γ₁ (units: T⁻¹A⁻¹) and γ₃ (or ${\tilde{\gamma }}_3$, units: T⁻³A⁻¹) contain the full temperature and SOC dependence. The cubic field terms (~B³) dominate at strong fields and produce the characteristic higher-order EMCA.

Author contributions

J.Guo, W.L., S.Z., Z.W., J.Z., M.F., and H.S. contributed to film growth and device fabrication; J.Guo, W.L., and L.L. contributed to transport data collection; J.Ge and C.L. contributed to STEM characterizations and analysis; J.Guo, W.L., C.H., X.L., Y.L. and Z.P. contributed to data analysis; C.H., and X.L. contributed to theoretical derivation and numerical simulations; J.Guo, C.H., W.L., X.L., J.Ge, C.L. and J.C. contributed to the writing of the manuscript; W.L., X.L., J.C. and C.L. contributed to formal analysis, resources, supervision, and validation. All authors have read and agreed to the published version of the manuscript.

Peer review

Peer review information

Nature Communications thanks the anonymous reviewers for their contribution to the peer review of this work. A peer review file is available.

Funding

W.L. and X.L. discloses support for the research of this work from the National Key R&D Program of China [grant number 2022YFA1602700], the National Natural Science Foundation of China [grant number 52272266 and grant number 12574162], the Fundamental Research Funds for the Central Universities [grant number 20720220035 and grant number 20720250067], the Natural Science Foundation of Jiangxi Province of China [grant number 20224BAB201024] and the Natural Science Foundation of Xiamen [grant number 3502Z202473008].

Data availability

The data generated in this study have been deposited in the Figshare database under accession code [https://doi.org/10.6084/m9.figshare.30336559].

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at https://doi.org/10.1038/s41467-026-77044-8.