Rotational symmetry breaking in superconducting nickelate Nd_0.8Sr_0.2NiO₂ films

Abstract

The infinite-layer nickelates, isostructural to the high-T_c cuprate superconductors, have emerged as a promising platform to host unconventional superconductivity and stimulated growing interest in the condensed matter community. Despite considerable attention, the superconducting pairing symmetry of the nickelate superconductors, the fundamental characteristic of a superconducting state, is still under debate. Moreover, the strong electronic correlation in the nickelates may give rise to a rich phase diagram, where the underlying interplay between the superconductivity and other emerging quantum states with broken symmetry is awaiting exploration. Here, we study the angular dependence of the transport properties of the infinite-layer nickelate Nd_0.8Sr_0.2NiO₂ superconducting films with Corbino-disk configuration. The azimuthal angular dependence of the magnetoresistance (R(φ)) manifests the rotational symmetry breaking from isotropy to four-fold (C₄) anisotropy with increasing magnetic field, revealing a symmetry-breaking phase transition. Approaching the low-temperature and large-magnetic-field regime, an additional two-fold (C₂) symmetric component in the R(φ) curves and an anomalous upturn of the temperature-dependent critical field are observed simultaneously, suggesting the emergence of an exotic electronic phase. Our work uncovers the evolution of the quantum states with different rotational symmetries in nickelate superconductors and provides deep insight into their global phase diagram.

The authors study transport in the superconducting state of infinite-layer nickelate Nd_0.8Sr_0.2NiO₂ films using a Corbino-disk configuration, finding that the magnetoresistance changes from isotropic to four-fold anisotropic with increasing magnetic field. At even higher field, an additional two-fold component emerges, which coincides with an anomalous upturn of the critical field.

Introduction

The conventional superconductivity with transition temperature (T_c) lower than 40 K was successfully explained by the Bardeen-Cooper-Schrieffer (BCS) theory, in which the electrons with anti-parallel spins and time-reversed momenta form Cooper pairs, and the superconducting order parameter is of isotropic s-wave symmetry^1,2. However, the discovery of high-temperature superconductivity (T_c > 40 K) in cuprates is beyond the expectation of the BCS theory, and the superconducting order parameters of cuprates are believed to be of nodal d-wave symmetry^3,4. Thereafter, the mechanism of unconventional high-T_c superconductivity has become one of the most important puzzles in physical sciences. Recently, the observation of superconductivity in infinite-layer nickelates with a maximal T_c of 15 K in Nd_1-xSr_xNiO₂ has motivated extensive researches in this emerging new superconducting family^5–9. Mimicking the d⁹ electronic configuration and the layered structure including CuO₂ planes of the cuprates, the isostructural infinite-layer nickelates are promising candidates for high-T_c unconventional superconductivity^5–9. Discerning the similarities and the differences between the nickelates and the cuprates, especially in the symmetry of the superconducting order parameters, should be of great significance for understanding the mechanism of unconventional high-T_c superconductivity.

Theoretical calculations have suggested that the nickelates are likely to give rise to a d-wave superconducting pairing, analogous to the cuprate superconductors. However, the consensus has not been reached and there are several proposals, including dominant $d_{x^2-y^2}$-wave^10,11, multi-band d-wave^12,13, and even a transition from s-wave to (d+is)-wave and then to d-wave depending on the doping level and the electrons hoping amplitude¹⁴. Experimentally, through the single-particle tunneling spectroscopy, different spectroscopic features showing s-wave, d-wave, and even a mixture of them are observed on different locations of the nickelate film surface, which complicates the determination of the pairing symmetry in the nickelates¹⁵. The London penetration depths of the nickelates family are also measured, and the results on La-based and Pr-based nickelate compounds support the existence of a d-wave component^16,17. However, the Nd-based nickelate, Nd_0.8Sr_0.2NiO₂, exhibits more complex behaviors that may be captured by a predominantly isotropic nodeless pairing^16,17. The pairing symmetry of the superconducting order parameter in the nickelate superconductors, the fundamental characteristic of the superconducting state, is still an open question, thus further explorations with diverse experimental techniques are highly desired.

In addition to the mystery of the superconducting pairing symmetry, the strong electronic correlation in nickelates is another element that makes the nickelate systems intriguing. The strong correlation is theoretically believed to play an important role in the nickelates systems^8,9,18,19 and the strong antiferromagnetic (AFM) exchange interaction between Ni spins has been experimentally detected²⁰. Generally, the strongly correlated electronic systems are anticipated to host rich phase diagrams and multiple competing states including superconductivity, magnetic order, charge order, pair density wave (PDW), etc^21,22. In the nickelate thin films, the charge order, a spatially periodic modulation of the electronic structure that breaks the translational symmetry, has been experimentally observed by the resonant inelastic X-ray scattering (RIXS)^23–25. However, the charge order is only observable in the lower doping regime where the nickelates are non-superconducting. The interplay between the superconductivity and the charge order as well as other underlying symmetry-broken states is still awaiting explorations.

In this work, we report the polar and azimuthal angular-dependent transport behaviors of the superconducting state in infinite-layer nickelate Nd_0.8Sr_0.2NiO₂ films with the Corbino-disk configuration. The polar angular dependent magnetoresistance shows evident anisotropy, indicating the quasi-two-dimensional nature of the superconductivity. The azimuthal angular dependent magnetoresistance manifests a rotational symmetry breaking from isotropic to four-fold (C₄) rotational symmetric with increasing magnetic field. Strikingly, when approaching lower temperature and larger magnetic field regime, an additional two-fold (C₂) symmetric component emerges in the magnetoresistance, concomitant with an anomalous upturn of the temperature-dependent critical field. The observed successive rotational symmetry breakings in the magnetoresistance may uncover the subtle balance and the intriguing interplay between different competing orders in the Nd_0.8Sr_0.2NiO₂ thin films.

Results

Characterization of quasi-two-dimensional superconductivity

The perovskite precursor Nd_0.8Sr_0.2NiO₃ thin films are firstly deposited on the SrTiO₃ (001) substrates by pulsed laser deposition (PLD). The apical oxygen is then removed by the soft-chemistry topotactic reduction method using CaH₂ power. Through this procedure, the nickelate thin films undergo a topotactic transition from the perovskite phase to the infinite-layer phase, and thus the superconducting Nd_0.8Sr_0.2NiO₂ thin films are obtained⁵. Figure 1a presents the schematic crystal structure of Nd_0.8Sr_0.2NiO₂. In agreement with the previous reports⁵, the temperature-dependence of the resistance R(T) exhibits metallic behavior from room temperature to low temperature followed by a superconducting transition beginning at $T_{\text{c}}^{\text{onset}}$ of 14.8 K (Fig. 1b). Here, $T_{\text{c}}^{\text{onset}}$ is determined at the point where R(T) deviates from the extrapolation of the normal state resistance (R_N). Note that the R(T) curve shows a considerably broad superconducting transition with a smooth tail, which can be described by the Berezinskii-Kosterlitz-Thouless (BKT) transition in two-dimensional (2D) superconductors^26–29. As shown in the inset of Fig. 1b, the R(T) curve under 0 T can be reproduced by the BKT transition using the Halperin-Nelson equation³⁰, $R=R_0\exp \left[-2b{\left(\frac{T_{\mathrm{c}}^{\prime }-T_{\mathrm{BKT}}}{T-T_{\mathrm{BKT}}}\right)}^{1/2}\right]$ (R₀ and b are material-dependent parameters, and $T_{\text{c}}^{\prime }$ is the superconducting critical temperature), yielding the BKT transition temperature T_BKT of 8.5 K. An apparent difference is also noted between the R(T) curves under in-plane and out-of-plane magnetic fields (inset of Fig. 1b), implying the anisotropy of the superconductivity.

Fig. 1. Structure and the quasi-two-dimensional superconductivity in Nd_0.8Sr_0.2NiO₂.

a Crystal structure of the infinite-layer nickelate Nd_0.8Sr_0.2NiO₂. b Temperature dependence of the resistance R(T) at zero magnetic field from 2 K to 300 K. The inset shows the R(T) curves below 20 K at 0 T (black circles), B_∥ = 16 T (red circles), and B_⊥ = 16 T (purple circles). Here, the B_∥ is applied along the a/b-axis and B_⊥ is along the c-axis. The blue solid line represents the BKT transition fitting using the Halperin-Nelson equation. c Schematic image and optical photo (inset) of the Corbino-disk configuration for polar (θ) angular dependent magnetoresistance R(θ) measurements on the Nd_0.8Sr_0.2NiO₂ thin film. Here, θ represents the angle between the magnetic field and the c-axis of the Nd_0.8Sr_0.2NiO₂. d Temperature dependence of the critical magnetic field B_c(T) for the magnetic fields along the c-axis (denoted as ⊥), the a/b-axis (∥, 0°), and the ab diagonal direction (∥, 45°). Here, the B_c is defined as the magnetic field corresponding to 50% normal state resistance. The blue and the orange solid lines are the 2D G-L fittings of the B_c(T) data near T_c. e Polar angular dependence of the critical magnetic field B_c(θ) at T = 6 K. The inset shows a close-up of the B_c(θ) around θ = 90°. The red solid line and the blue solid line are the fittings with the 2D Tinkham model (B_c(θ)sin(θ)/B_c||)² + |B_c(θ)cos(θ)/B_c⊥| =1 and the 3D anisotropic mass model B_c(θ) = B_c||/(sin²(θ) + γ²cos²(θ))^1/2 with γ = B_c||/B_c⊥, respectively. f, g Representative polar angular dependence of the magnetoresistance R(θ) at temperature T = 5 K under B = 12 T (f) and B = 8 T (g).

To obtain more insight into the anisotropic superconductivity in Nd_0.8Sr_0.2NiO₂ thin films, the critical magnetic field and magnetoresistance under different magnetic field orientations are measured. Here, the Corbino-disk configuration is used to eliminate the influence of the current flow in angular-dependent magnetoresistance measurements³¹, which cannot be completely avoided in standard four-probe measurements^32,33. The schematic image and the optical photo of a Corbino-disk device are shown in Fig. 1c. To start with, the temperature-dependence of the critical field B_c is measured under the magnetic field applied along the c-axis (denoted as ⊥), the a/b-axis (∥, 0°), and the ab diagonal direction (∥, 45°). Here, B_c is defined as the magnetic field required to reach 50% of the normal state resistance (R_N = 98.8 Ω), and the B_c(T) curves are collected in Fig. 1d. The T-linear dependence of B_c⊥(T), and the (T_c-T)^1/2-dependence of B_c||,0° and B_c||,45° near T_c show agreement with the phenomenological 2D Ginzburg-Landau (G-L) formula³⁴:

$$B_{\mathrm{c}\perp }\left(T\right)=\frac{{\varphi }_0}{{2\mathrm{\pi }\mathrm{\xi }}_{\mathrm{G}-\mathrm{L}}^2\left(0\right)}\left(1-\frac{T}{T_{\mathrm{c}}}\right)$$ 1

$$B_{\mathrm{c}\mid \mid }\left(T\right)=\frac{\sqrt{12}{\varphi }_0}{{2\mathrm{\pi }\mathrm{\xi }}_{\mathrm{G}-\mathrm{L}}\left(0\right)d_{\mathrm{sc}}}{\left(1-\frac{T}{T_{\mathrm{c}}}\right)}^{\frac{1}{2}}$$ 2

where ϕ₀ is the flux quantum, ξ_G-L(0) is the zero-temperature G-L coherence length and d_sc is the superconducting thickness. The consistency with the 2D G-L formula near T_c indicates the 2D nature of the superconductivity in the Nd_0.8Sr_0.2NiO₂ thin films. To further study the dimensionality of the superconductivity, we measure the polar angular dependence of the critical magnetic field B_c(θ) for Nd_0.8Sr_0.2NiO₂ thin film at T = 6 K. Here, θ represents the angle between the magnetic field and the c-axis of the Nd_0.8Sr_0.2NiO₂. As shown in Fig. 1e, the B_c(θ) curve exhibits a prominent angular dependence of the external magnetic field, and a cusp-like peak is clearly resolved around θ = 90° (B⊥c-axis). The peak in the B_c(θ) around 90° can be well reproduced by the 2D Tinkham model and cannot be captured by the 3D anisotropic mass model, which qualitatively demonstrates the behavior of 2D superconductivity³⁵ (inset of Fig. 1e).

Polar angular dependence of the magnetoresistance R(θ)

To obtain a more comprehensive depiction of the anisotropy, the polar angular dependence of the magnetoresistance R(θ) at various temperatures and magnetic fields are measured, and two representative R(θ) curves at 5 K under 12 T and 8 T are shown in Fig. 1f and g, respectively. The most notable features are two sharp dips at 90° and 270°, corresponding to B⊥c-axis. The two sharp dips correspond to the cusp-like peak in the B_c(θ) curve, resulting from the quasi-2D anisotropy. With varying temperatures and magnetic fields, the two sharp dips are observed in all R(θ) curves measured in the superconducting region (Supplementary Fig. 11), further confirming the quasi-2D nature of the superconducting Nd_0.8Sr_0.2NiO₂ thin films and suggesting that the layered superconducting NiO₂ planes should mainly account for the superconducting properties in our transport measurements. Additionally, small humps at approximately 90° ± 45° and 270° ± 45° are observed under 8 T and 12 T (marked by black arrows in Fig. 1f and g, respectively), while four more subtle kinks at 90° ± 20° and 270° ± 20° can be seen under 12 T (marked by black dashed line Fig. 1g). Considering the crystal structure of the Nd_0.8Sr_0.2NiO₂, the humps and kinks with relatively small variations may originate from the spin-dependent electron scattering with the magnetic moment of the rare-earth Nd³⁺. Further investigations are still desired to prove this scenario.

Azimuthal angular dependence of the magnetoresistance R(φ)

It is noteworthy that in Fig. 1d, the B_c||,0°(T) and B_c||,45°(T) curves overlap with each other below ~4 T (above 9 K), but split with increasing magnetic field (decreasing temperature), suggesting the existence of in-plane anisotropy of the superconductivity of the Nd_0.8Sr_0.2NiO₂ thin films. Thus, we performed the azimuthal angular dependence of the magnetoresistance R(φ) to reveal the in-plane anisotropy of the Nd_0.8Sr_0.2NiO₂ thin films using the Corbino-disk configuration, as schematically shown in Fig. 2a. Here, φ represents the angle between the magnetic field and the a/b-axis of the Nd_0.8Sr_0.2NiO₂. The Corbino-disk configuration guarantees that the electric current flows radially from the center to the outermost electrode, which well excludes the undesired influence of the current flow and ensures that the measured anisotropy is the intrinsic properties of the Nd_0.8Sr_0.2NiO₂ thin films. The representative set of R(φ) at different temperatures under 8 T in polar and rectangular plots are shown in Fig. 2b, c, respectively (more sets of R(φ) at different temperatures and magnetic fields can be found in the Supplementary Information). Remarkably, the R(φ) curves exhibit obvious four-fold (C₄) rotational symmetry in both polar and rectangular plots. The C₄ symmetry of the R(φ) curves shows minima at 0°, 90°, 180°, and 270° (along the a/b-axis) and maxima at 45°, 135°, 225°, and 315° (45° to the a/b-axis) from 8 K to 11 K, and becomes indistinguishable when the temperature is increased to 14 K (the top panel in Fig. 2c).

Fig. 2. Azimuthal (φ) angular dependence of the magnetoresistance in Nd_0.8Sr_0.2NiO₂.

a Schematic of the Corbino-disk device for azimuthal (φ) angular dependent magnetoresistance measurements. Here, φ represents the angle between the magnetic field and the a/b-axis of the Nd_0.8Sr_0.2NiO₂. b, c Azimuthal angle dependence of the magnetoresistance R(φ) at different temperatures under B = 8 T in the polar plot (b) and rectangular plot (c). d, e Azimuthal angle dependence of the magnetoresistance R(φ) at T = 2 K under B = 16 T in the polar plot (d) and rectangular plot (e). Here, the logarithmic scale is used on the resistance-axis to specifically demonstrate the C₂ symmetric feature. The blue solid lines are fits with the trigonometric function: R = R_avg + ΔR_C4 × sin(4φ) + ΔR_C2 × sin(2φ), where R_avg is the averaged magnetoresistance and ΔR_C4 and ΔR_C2 are the C₄ and C₂ components, respectively. The light blue area in (d) is a guide to the eye, representing the C₂ anisotropy. f, g Four-fold components ΔR_C4 (f) and two-fold components ΔR_C2 (g) versus the ratio between the averaged magnetoresistance and the normal state resistance (R_avg/R_N) under different magnetic fields. Here, the values of the C₂ and C₄ components are extracted by the trigonometric function fitting.

To confirm the correspondence between the minima of the C₄ R(φ) and the a/b-axis of the Nd_0.8Sr_0.2NiO₂ thin films, control experiments have been carefully conducted. Specifically, the Nd_0.8Sr_0.2NiO₂ Corbino-disk device is remounted and remeasured after rotating a finite in-plane degree Δφ. Through the comparison between the initial results R(φ) and the remeasured results after rotation R(φ + Δφ), the minima (maxima) of the C₄ symmetry are fixed with the a/b-axis (45° to the a/b-axis), verifying the C₄ symmetry of the R(φ) is an intrinsic property of the Nd_0.8Sr_0.2NiO₂ thin films (Supplementary Fig. 12). Note that the R(φ) curve at 14 K is already larger than the normal state resistance R_N of 98.8 Ω, and is almost isotropic within the experimental resolution. Thus, the observed C₄ symmetry below 11 K should be related to the superconducting characteristics of the Nd_0.8Sr_0.2NiO₂ thin films, evidenced by that the C₄ anisotropy disappears as the superconductivity is suppressed at higher temperatures. Moreover, considering that the quasi-2D nature of the superconductivity in the Nd_0.8Sr_0.2NiO₂ and the large magnetoresistance amplitudes of the C₄ anisotropy (ΔR_C4/R) (approximately 10% under 8 K and 8 T in Fig. 2c, around 20% under 5.5 K and 12 T in the Supplementary Fig. 14), the C₄ symmetry is not likely owing to the magnetic moment of the Nd³⁺ between the NiO₂ planes, but should be ascribed to the superconductivity in the NiO₂ planes. The possible role of the Nd³⁺ magnetic moment in our R(φ) measurements is further discussed in the Supplementary Information. The C₄ anisotropy has ever been observed in the normal state of the cuprates with a magnitude of merely 0.05%, which is attributed to the magnetic order³⁶ and different from our observations. The in-plane critical field of a d-wave superconductor is theoretically predicted to exhibit a C₄ symmetric anisotropy owing to the d-wave pairing symmetry³⁷. The C₄ anisotropic critical field as well as the C₄ anisotropic magnetoresistance has been experimentally used to determine the d-wave superconductivity in cuprate superconductors³⁸ and heavy fermion superconductors³⁹, etc. Therefore, the C₄ symmetry of our R(φ) curves is supposed to imply the C₄ symmetric critical field of the predominant d-wave pairing in the Nd_0.8Sr_0.2NiO₂ thin films. The deduced d-wave pairing in the Nd_0.8Sr_0.2NiO₂ thin films cannot be definitely determined by our transport measurement results solely and requires further experimental investigations (e.g., phases-sensitive measurements).

Remarkably, with further increasing magnetic field, additional two-fold (C₂) symmetric signals are observed as small modulations superimposed on the primary C₄ symmetry in the R(φ) curves. The representative R(φ) curve under 2 K and 16 T in the polar and rectangular plots are shown in Fig. 2d and e, respectively. In addition to the predominant C₄ symmetric R(φ), an additional C₂ signal can be clearly discerned by R(φ = 0° and 180°) being smaller than R(φ = 90° and 270°), indicating the rotational symmetry breaking between a-axis and b-axis. In the following, elaborated experiments and analysis are discussed to exclude the possible extrinsic origin of the C₂ anisotropy. First, a nearly perfect C₂ symmetric feature is confirmed through a two-axis stage rotator, which can basically eliminate the misalignment of the applied magnetic fields (Supplementary Fig. 24). Second, through the aforementioned remounted measurements after an in-plane rotation of Δφ, the C₂ symmetry is confirmed to be invariant with respect to the sample mounting, since R(φ + Δφ) can nicely overlap with R(φ) after slight shifts (Supplementary Fig. 12). Third, the Corbino-disk configuration excludes the anisotropic vortex motion due to the unidirectional current flow, which has been reported in the previous works using the standard four-probe or Hall-bar electrode^32,33. Fourth, the C₂ features superimposed on the C₄ symmetric R(φ) can be consistently observed in many other samples in the large magnetic field, demonstrating the strong reproducibility of the C₂ and C₄ anisotropy (Supplementary Fig. 25). Fifth, the amplitudes of the C₂ anisotropy in our samples (>1 Ω) are orders of magnitude larger than the resistance resolution of our measurement equipment (20 mΩ with the excitation current of 1 μA).

To quantitatively study the evolution of the C₂ and C₄ anisotropy of the R(φ), the C₂ components (ΔR_C2) and C₄ components (ΔR_C4) of each R(φ) curve at different temperatures and magnetic fields are extracted through trigonometric function fitting (Supplementary Fig. 15). Among them, the fitting curve of the R(φ) under 2 K and 16 T is shown in Fig. 2d, e. Here, the ratio between the average resistance of the R(φ) and the normal state resistance (R_avg/R_N) is used as an independent variable for an intuitive comparison. Figure 2f shows ΔR_C4 as a function of R_avg/R_N under different magnetic fields, which exhibit similar parabolic behaviors with maxima approximately around 50% R_N. Differently, the ΔR_C2 are monotonically decreasing with increasing R_avg/R_N as shown in Fig. 2g. With increasing magnetic field, the ΔR_C4 show subtle decreasing tendency, while the ΔR_C2 monotonically increase, exhibiting a magnetic field-mediated competition between the ΔR_C4 and the ΔR_C2. The parabolic R_avg/R_N-dependence of the ΔR_C4 can be understood as the resistance anisotropy due to the superconductivity anisotropy becomes smaller when approaching the superconducting zero-resistance state or the normal state. As the sample is approaching the normal state and losing the superconductivity (R getting close to R_N as well as T getting close to T_c^onset), the amplitudes of the C₄ anisotropy are diminishing, which unambiguously shows that the C₄ anisotropy is associated with the superconducting state. This coincidence between the C₄ anisotropy and superconducting transition is verified for a range of T_c, which varies with applied magnetic fields and among different samples (Supplementary Figs. 22 and 23). However, the monotonic R_avg/R_N-dependence of the ΔR_C2 cannot be explained by such a scenario, suggesting a different origin of the C₂ symmetry. Generally, the observation of spontaneous rotational symmetry breaking in R(φ) curves would indicate the existence of nematicity^31,40. However, in our measurements, the C₂ component has a relatively small weight in the anisotropy of the R(φ) compared with the C₄ symmetry (<12%), inconsistent with previous results of nematic superconductivity with a primary C₂ feature³¹. To understand the origin of the C₂ anisotropy, we recall the RIXS-detected charge order in the nickelates that is along the Ni-O bond direction and exhibits a competitive relationship with the superconductivity^23–25. Considering the Ni-O bond direction of our C₂ feature and the magnetic field-mediated competition between the ΔR_C4 and the ΔR_C2, our observation of the C₂ anisotropy might result from the charge order in Nd_0.8Sr_0.2NiO₂ thin films. The magnetic field suppresses the superconductivity and may alter the competition between the anisotropic superconductivity with C₄ symmetry and the charge order fluctuations with C₂ symmetry in our Nd_0.8Sr_0.2NiO₂, leading to the monotonically magnetic field-dependent decrease of the ΔR_C4 (Fig. 2f) and increase of the ΔR_C2 in our observations (Fig. 2g). As previously reported, the Sr doping dramatically lowers the onset temperature of the charge order in the nickelates²³, which may explain the occurrence of the charge order in our Nd_0.8Sr_0.2NiO₂ only at low temperatures. Although the previous RIXS results show that under zero magnetic field the charge order is absent at a doping level of 0.2^23–25, the strong magnetic fields used in our measurements may suppress the superconductivity and raise the charge order fluctuations. In addition, since the C₂ feature breaks the rotational symmetry, our observation favors a stripe-like charge order in the nickelates, which is supported by theoretical proposals^18,19, while further investigations are still needed.

Magnetic field versus temperature phase diagram

Based on the B_c(T) curves and the anisotropic behaviors of the R(φ), we construct the global phase diagram (Fig. 3a) to develop a comprehensive understanding of the superconductivity in the infinite-layer Nd_0.8Sr_0.2NiO₂ thin films. The B_c||,0°^onset(T) is determined at the point where the R(T) curve deviates from the extrapolation of the normal state resistance under applied magnetic field. Above the B_c||,0°^onset(T), the Nd_0.8Sr_0.2NiO₂ thin films are in the normal state. The B_c||,0°^1%(T) corresponds to the magnetic fields and temperatures below which the resistance R is smaller than 1% of the R_N, enclosing the region labeled as zero-resistance state in the phase diagram. The region between the B_c||,0°^onset(T) and the B_c||,0°^1%(T) represents the superconducting transition, and is further separated into three regions labeled as ~isotropy, C₄, and C₄ + C₂, depending on whether the corresponding R(φ) curves exhibit nearly isotropy, C₄, or C₄ + C₂ rotational symmetry. More R(φ) curves can be found in Supplementary Fig. 14. The representative R(φ) curves are shown in Fig. 3b–e, and the arrows approximately mark the temperatures and magnetic fields where the R(φ) curves are measured. The B_c||,0°^50%(T) and B_c||,45°^50%(T) curves determined by the 50% R_N criterion are also plotted in the phase diagram.

Fig. 3. B versus T phase diagram for the Nd_0.8Sr_0.2NiO₂.

a B versus T phase diagram for in-plane magnetic fields. The white region above B_c||,0°^onset(T) (open circles) represents the normal state, and the dark blue region below B_c||,0°^1%(T) (open triangles) denotes the zero-resistance state (defined by R < 1% of R_N). Between B_c||,0°^onset and B_c||,0°^1%(T) is the superconducting transition region, which is separated into ~isotropy (gray area), C₄ (blue area), and C₄ + C₂ (pink area) regions from small-magnetic-field to large-magnetic-field regime, depending on whether the measured R(φ) curves exhibit nearly isotropy, C₄ or C₄ + C₂ rotational symmetry. B_c||,0°^50%(T) (black dots) and B_c||,45°^50%(T) (red dots) represent the critical field along the a/b-axis (0°) and the ab diagonal direction (45°) determined by the 50% R_N criterion, respectively. The blue solid line is the 2D G-L fit of the B_c||,0°^50%(T) data near T_c. The B_c||,0°^50%(T) and B_c||,45°^50%(T) data are from sample S2, which show a more prominent upturn in the low-temperature regime. b–e The representative R(φ) curves at 3 K and 15 T in the polar plot (b) and rectangular plot (c), 8.5 K and 8 T in the polar plot (d) and 10.8 K and 2 T in the polar plot (e), manifesting different rotational anisotropies. φ represents the azimuthal angle between the magnetic field and the a/b-axis of the Nd_0.8Sr_0.2NiO₂. The light blue area in b is a guide to the eye, representing the C₂ anisotropy. The black arrows in this figure approximately mark the temperature and magnetic field positions in the phase diagram where the R(φ) curves are measured.

The phase diagram demonstrates an evolution of the superconducting states manifesting different rotational symmetries, depending on the external magnetic field. Specifically, in the gray region labeled as ~isotropy (B < 4 T), the B_c||,0°^50%(T) and B_c||,45°^50%(T) curves overlap with each other and, consistently, the R(φ) curves are nearly isotropic within our measurement resolution, indicating the isotropic superconductivity (Fig. 3e). Under the magnetic field from 4 T to 12 T (the blue region labeled as C₄), the R(φ) curves exhibit C₄ rotational symmetric anisotropy (Fig. 3d), which should be ascribed to the superconductivity of the Nd_0.8Sr_0.2NiO₂ films since it disappears in the normal state (Fig. 2c). Simultaneously, B_c||,0°^50%(T) and B_c||,45°^50%(T) curves split in this region, according with the emergence of C₄ anisotropy. When the magnetic field is increased above 12 T (the pink region labeled as C₄ + C₂), an additional C₂ anisotropy is observed in the R(φ) curves as a superimposed modulation on the predominant C₄ anisotropy, which breaks the C₄ symmetry (Fig. 3b, c). At the same time, the B_c||,0°^50%(T) curve shows an anomalous upturn at the low temperatures above 12 T, deviating from the saturating B_c at the low temperatures expected for a conventional superconducting state. The simultaneous occurrences of the rotational symmetry breaking in the R(φ) curves and the enhanced superconducting critical field behavior strongly indicate the emergence of an exotic state.

Discussion

The superconducting phase diagram may reveal two phase transitions characterized by spontaneous rotational symmetry breakings. The first transition occurs at approximately 4 T indicated by the change from isotropic superconductivity to C₄ anisotropy. Considering that the R(φ) curves show the symmetry of the in-plane critical field and could reflect the superconducting pairing³⁷, the first rotational symmetry breaking may suggest a transition from s-wave to d-wave superconductivity. The transition from s-wave to d-wave superconductivity is reminiscent of the theoretical phase diagram hosting s-, d- and (d+is)-wave superconductivity for nickelates with varying parameters¹⁴. Experimentally, the s- and d-wave mixture has been reported by the previous STM study¹⁵. The second transition with the rotational symmetry turning from C₄ to C₄ + C₂ takes place around 12 T. Also, the second transition is accompanied with an anomalous upturn of the in-plane critical field, unveiling the emergence of an electronic state unexplored before. As discussed above, we speculatively ascribe the second transition to the emergence of the charge order in the Nd_0.8Sr_0.2NiO₂ films. It is normally believed that the long-range charge order disfavors the superconductivity^18,22. In our Nd_0.8Sr_0.2NiO₂ films, with increasing magnetic field, the superconductivity gets suppressed and the stripe charge order fluctuation with short-range correlation emerges, accounting for the relatively small C₂ symmetric anisotropy in R(φ) curves. The coexistence between the ordered phases with different broken symmetries is relatively rare, and the intertwined orders would give rise to more unexpected quantum phenomena and more complex phase diagram. In our Nd_0.8Sr_0.2NiO₂ thin films, the short-range stripe order coupled to the superconducting condensation may induce a secondary PDW state, in which the superconducting order parameter is oscillatory in space^22,41,42. Through pairing in the presence of the periodic potential of the charge order, the Cooper pairs gain finite center-of-mass momenta⁴², which is also a signature of the Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) phase that features an upturn of the critical magnetic field at the low temperatures^43,44, resembling our observations. These phases were not expected previously in the Nd_0.8Sr_0.2NiO₂ systems. Our findings suggest that the nickelates would be a potential option to explore these exotic states.

Our experimental observations have important implications on current theoretical debates. The isotropic superconductivity requires a primary pairing mechanism that is not expected in optimally hole-doped superconducting cuprates. The successive phase transitions reveal a subtle balance between several competing interactions that are unique for infinite-layer nickelates and also cannot be explained as in cuprates. In the Mott-Kondo scenario, the phase transition may be attributed to the competition between the Kondo coupling and the AFM spin superexchange coupling¹⁴. The Kondo coupling can produce local spin fluctuations that support isotropic s-wave pairing⁴⁵, while the AFM coupling favors d-wave pairing. In Nd_1-xSr_xNiO₂, the superconductivity emerges around the border of the Kondo regime⁴⁶. Therefore, applying magnetic field may suppress the local spin fluctuations, tilt the balance towards AFM correlation, and thus induce a secondary d-wave component, which explains the emergent C₄ symmetry by either d- or (d+is)-wave pairing¹⁴. The charge order also competes with the AFM correlation¹⁹ and the superconductivity. Previous experiments have shown that the charge order phase boundary may even penetrate into the superconducting dome. Further increasing magnetic field may reduce both the superconductivity and AFM correlation, thus promote the charge order, and lead to the observed C₂ symmetry. The interplay of the Kondo effect, the AFM correlation, the superconductivity, and the charge order provides a potential playground for novel correlated phenomena, which is well beyond a simple scenario. Any candidate theory should be made in conformity with all these experimental observations.

In summary, we systematically investigated the angular dependence of the transport properties of the infinite-layer nickelate Nd_0.8Sr_0.2NiO₂ thin films with Corbino-disk configuration. The evident polar angular dependent anisotropy strongly indicates the quasi-2D nature of the superconductivity in Nd_0.8Sr_0.2NiO₂ thin films. The azimuthal angular dependence of the magnetoresistance R(φ) curves manifests different rotational symmetric structures depending on the external magnetic fields, which are summarized in the B versus T phase diagram. With increasing magnetic field, the R(φ) curves exhibit isotropic, C₄ symmetric, and then C₄ + C₂ symmetric behaviors successively, demonstrating the emergence of the quantum states characterized by spontaneous rotational symmetry breaking. Since the symmetry of the R(φ) curves could reflect the symmetry of the superconducting order parameter, the observed R(φ) curves evolving from isotropy to C₄ anisotropy may suggest an extraordinary superconducting pairing of the Nd_0.8Sr_0.2NiO₂ beyond the familiar cuprate scenario. In the low-temperature and large-magnetic-field regime, we find a striking concurrence of an additional C₂ symmetric modulations in the R(φ) curves and an anomalous upturn of the temperature-dependent critical magnetic field. We speculate that the C₂ feature in the R(φ) curves may result from the stripe charge order fluctuations in the Nd_0.8Sr_0.2NiO₂, which can coexist with the superconductivity, and might potentially give rise to a charge order-driven PDW state. Our findings shed new light on the dimensionality and the pairing symmetry of the superconductivity in the infinite-layer nickelate Nd_0.8Sr_0.2NiO₂ thin films, and reveal that the superconducting Nd_0.8Sr_0.2NiO₂ is a promising material platform to study the unconventional superconductivity and the interplay between the various exotic states.

Methods

Thin-film synthesis

The Nd_0.8Sr_0.2NiO₂ films with infinite-layer structure are prepared by topochemical reduction of perovskite Nd_0.8Sr_0.2NiO₃ films (thickness about 14.5–16 nm) without capping layer⁴⁷. The precursor Nd_0.8Sr_0.2NiO₃ films are deposited on the TiO₂-terminated SrTiO₃ (001) substrates by pulsed laser deposition using 248 nm KrF laser. The substrate temperature is controlled at 620 °C with an oxygen pressure of 200 mTorr during the deposition. A laser fluence of 1 J/cm² was used to ablate the target and the size of laser spot is about 3 mm². After deposition, the samples were cooled down in the same oxygen pressure at the rate of 10 °C/min. To obtain the infinite-layer phase, the samples were sealed in the quartz tube together with 0.1 g CaH₂. The pressure of the tube is about 0.3 mTorr. Then, the tube was heated up to 300 °C in tube furnace, hold for 2 h, and naturally cooled down with the ramp rate of 10 °C/min. The characterizations of the high quality of our samples can be found in the Supplementary Information.

Devices fabrication

The Corbino-disk electrode is fabricated using the standard photolithography technique. After spin coating of PMMA on the Nd_0.8Sr_0.2NiO₂ thin film samples, the annular electrodes were patterned through electron beam lithography in a FEI Helios NanoLab 600i Dual Beam System. Then, the metal electrodes (Ti/Au, 6.5/100 nm) were deposited in a LJUHVE-400 L E-Beam Evaporator. After that, the PMMA layers were removed by the standard lift-off process. Finally, the electrodes are contacted using wire-bonded Al wires for transport measurements.

Transport measurements

The angle-dependent transport measurements were carried out on a rotator with an accuracy of 0.01° in a 16 T-Physical Property Measurement System (PPMS-EverCool-16, Quantum Design).

Author contributions

J.W. conceived and instructed the research. H.R.J., Y.L., Y.N.L., Z.Y.X., C.C.J. and S.C.Q. fabricated the Corbino-disk devices, performed the transport measurements, and analyzed the data under the guidance of J.W. X.D. and L.Q. synthesized the thin film samples. X.D., M.H.X., L.Q., X.Y.G. and P.G. characterized the thin film samples. Y.F.Y. and G.M.Z. contributed to the theoretical explanations. H.R.J., Y.L., Y.N.L. and J.W. wrote the manuscript with input from all authors.

Peer review

Peer review information

Nature Communications thanks the anonymous reviewer(s) for their contribution to the peer review of this work. A peer review file is available.

Data availability

All data needed to evaluate the conclusions in the study are present in the paper and/or the Supplementary Information. The data that support the findings of this study are available from the corresponding author upon request.

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at 10.1038/s41467-023-42988-8.