The occurrence of superconductivity in proximity to various strongly correlated phases of matter has drawn extensive focus on their normal state properties, to develop an understanding of the state from which superconductivity emerges [1][2][3][4] . The recent finding of superconductivity in layered nickelates raises similar interests [5][6][7][8] . However, transport measurements of doped infinite-layer nickelate thin films have been hampered by materials limitations of these metastable compounds: in particular, a high density of extended defects [9][10][11] . Here, by moving to a substrate (LaAlO 3 ) 0.3 (Sr 2 TaAlO 6 ) 0.7 that better stabilizes the growth and reduction conditions, we can synthesize the doping series of Nd 1-x Sr x NiO 2 essentially free from extended defects. In their absence, the normal state resistivity shows a low-temperature upturn in the underdoped regime, linear behaviour near optimal doping and quadratic temperature dependence for overdoping. This is phenomenologically similar to the copper oxides 2,12 despite key distinctions-namely, the absence of an insulating parent compound 5,6,9,10 , multiband electronic structure 13,14 and a Mott-Hubbard orbital alignment rather than the charge-transfer insulator of the copper oxides 15,16 . We further observe an enhancement of superconductivity, both in terms of transition temperature and range of doping. These results indicate a convergence in the electronic properties of both superconducting families as the scale of disorder in the nickelates is reduced.

The idea that superconductivity can arise from doping a correlated insulator has been a pervasive guiding principle since the discovery of the copper oxide superconductors, with an impact on materials as far-ranging as twisted bilayer graphene [1][2][3] . On doping the insulator, a 'strange metal' with unconventional electrical transport often occurs and nucleates superconductivity, before further doping gives way to more conventional Fermi-liquid-like behaviour. The extent to which this phenomenology requires that the parent compound shows a strongly insulating ground state, and whether or not it should show magnetism has been discussed for decades. A further dichotomy, whether the strange metallic behaviour reflects the proximity to the correlated insulator or follows from a zero-temperature phase transition to a broken symmetry phase, remains actively debated and largely unresolved.

The observation of superconductivity in a family of layered nickelates presents an opportunity to address some of these perplexing issues. The parent compounds of the infinite-layer nickelates show a weak resistive upturn at low temperatures without a strongly insulating ground state or indications of a gap (NdNiO 2 and PrNiO 2 ), and even evidence of superconductivity in LaNiO 2 (refs. 5,6,9,10). Moreover, so far, long-range magnetic order has not been observed in this system [17][18][19] . Nevertheless, upon doping, we find that Nd 1-x Sr x NiO 2 shows strange metal behaviour with resistivity linearly increasing with temperature T for Sr doping x at the peak of the superconducting dome. Further hole doping results in a metallic state with resistivity varying as T 2 , with reduced and ultimately vanishing superconducting scales. Our results indicate, therefore, that much of the emergent behaviour of this class of unconventional superconductors does not strictly require a Mott insulating parent compound with a hard gap to charge excitations. Features in the Hall effect directly correlate with the evolution of the resistivity and may suggest a broken symmetry associated with Fermi surface reconstruction 2,20,21 . Our results can be considered in two contexts: the multiband nature of the electronic structure 13,14,21 and the possibility of a quantum phase transition underlying the strange metallic behaviour 2,4 .

A central issue for the synthesis and study of superconducting infinitelayer nickelates is material control due to the poor thermodynamic stability of this system [9][10][11] , as evident from the orders-of-magnitude variations in the resistivity of infinite-layer nickelates reported across the literature 9,10,[22][23][24][25] . Just as in the development of copper oxides 1,26,27 , minimizing disorder and extrinsic defects is critical for elucidating the nature of the normal state and superconducting phase diagram.

We have achieved substantial advances in the crystallinity of Nd 1-x Sr x NiO 2 (x = 0.05-0.325) by using (LaAlO 3 ) 0.3 (Sr 2 TaAlO 6 ) 0.7 (LSAT; lattice constant a = 3.868 Å) substrates to optimize the epitaxial mismatch for both the perovskite precursor and the infinite-layer phases (Extended Data Table 1; see Supplementary Information for an extended comparison of LSAT and SrTiO 3 ). We note that the use of LSAT substrates and the enhancement of the superconducting onset transition temperature T c was first reported by ref. 24 for Pr 0.8 Sr 0.2 NiO 2 . As shown in the high-angle annular dark-field (HAADF)-scanning transmission electron microscopy (STEM) cross-sectional images (Fig. 1a,b), the Ruddlesden-Popper-type vertical stacking faults 9,11,28 (marked by yellow dashed outlines in Fig. 1a) that densely populate films grown on the widely used substrate SrTiO 3 (a = 3.905 Å) are now essentially eliminated on LSAT, leaving a macroscopically clean thin film with minimal defects (Fig. 1b and Extended Data Figs. 1 and2). This is also reflected in the substantial decrease in resistivity ρ (Fig. 1c). Note that the in-plane lattice constants of the films are locked to the substrate in both cases (Fig. 1a,b and Extended Data Fig. 3d-g). X-ray diffraction θ-2θ symmetric scans show prominent film peaks with out-of-plane lattice constant trends associated with systematic Sr doping (Extended Data Fig. 3a-c). Overall, these data indicate that high crystallinity is uniformly established throughout the probed range of Sr doping (Extended Data Figs. 3 and4), minimizing extended-defect contributions to electrical transport.

This optimized sample series on LSAT provides a robust platform to investigate the phase diagram of the infinite-layer nickelates. We first observe that ρ(20 K) now maintains a similar range of 0.1-0.3 mΩ cm across all x, notably below the scale of a resistance quantum R q per NiO 2 plane (Fig. 2a; see Extended Data Fig. 5 for all individual ρ(T) curves). This includes the underdoped and overdoped regimes, with ρ(20 K) roughly 5-30 times lower than any previously reported (roughly 0.6-9 mΩ cm) 5,[9][10][11] . This suggests that the low-temperature normal state scattering rate is comparable across the phase diagram, although multiband effects should be considered.

It is noteworthy that a superconducting dome is still observed for films on LSAT (Fig. 2b-d and Extended Data Table 2). For previous films on SrTiO 3 , a direct correlation was found between the presence of superconductivity and the magnitude of the normal state resistance with respect to R q (refs. 6,9). The fact that the normal state resistance depends substantially on the substrate (LSAT versus SrTiO 3 ), whereas the existence of the superconducting dome does not, suggests that the dome itself is not a result of disorder in doped nickelates. However, at a quantitative level, the superconducting dome in the lower resistance films on LSAT is notably larger, with T c above 20 K for optimal doping at x ≅ 0.15-0.175 (Fig. 2b), and with a width Δx ≅ 0.2 that is now very comparable to that of the copper oxides (Δx ≅ 0.21 for La 2-x Sr x CuO 4 ) 12 (Extended Data Fig. 6). Further data (field suppression of superconductivity and mutual-inductance measurements) for optimal doping are given in Extended Data Fig. 7. We also note that the experimentally observed range of the superconducting dome here shows good agreement with the theoretical calculations in ref. 29. Both the robustness of the dome and the higher T c indicate that superconductivity here is probably unconventional and cannot be explained purely by electronphonon mechanisms 30 .

Figure 2b shows the variation of the slope of ρ(T) normalized to the room-temperature value across the phase diagram. Three regimes of behaviour are observed, with representative data shown in Fig. 2e-g: a resistive upturn in the underdoped region characterized by T upturn (the temperature at which the resistivity minimum occurs (dρ/dT = 0)), ρ ∝ T 2 in the overdoped region, and a narrow range of ρ ∝ T at the peak of the superconducting dome.

The evolution of the resistivity is accompanied by systematic features in the Hall coefficient R H . At high temperatures and low doping R H is negative, whereas it is positive in the low-temperature limit beyond optimal doping (Fig. 2c; see Extended Data Fig. 8 for all individual R H (T) curves). The boundary defining the sign change in R H extrapolates to T = 0 at optimal doping. The second clear feature in R H can be seen in the underdoped region. Here, R H is negative at all temperatures and shows a pronounced local maximum (Fig. 2d). The temperature at which this maximum occurs decreases as a function of doping and tracks the resistive upturn (dark-blue triangles in Fig. 2b), such that both features extrapolate to vanish under the peak of the superconducting dome. 11,28 , making the regions with stacking faults easily distinguishable from the pristine infinite-layer (Extended Data Fig. 1a). b, HAADF-STEM cross-sectional image of a Nd 0.85 Sr 0.15 NiO 2 thin film grown on LSAT, synthesized using the conditions specified in Extended Data Table 1. The film now displays highly uniform crystallinity largely free from extended defects. c, Resistivity ρ versus temperature T for the films in a (red dotted curve) and b (blue solid curve). A substantial decrease in the normal state resistivity reflects the crystallinity improvement. Scale bars, 5 nm.

In the underdoped region, the low-temperature resistivity has been noted to vary as roughly log(1/T) for films on SrTiO 3 (ref. 31). Comparing data for films on LSAT, we see that although the resistivity itself is a factor of around four smaller in the cleaner samples, T upturn is nearly the same (Fig. 2e). T upturn is also essentially constant as a function of magnetic field, with an overall relatively small magnetoresistance (Fig. 3). These observations suggest that the resistive upturn cannot be directly ascribed to disorder or localization effects (with or without interaction corrections), nor to Kondo physics 32 . Indeed, by extending measurements down to lower temperatures, we observe a saturation in the resistivity below 2 K (Fig. 3). Both the resistive upturn and ultimate saturation are closely tracked by R H (T), with identical functional form (Fig. 3). This behaviour is naturally and most simply explained within a two-band model with both electron and hole carriers. Electronic structure calculations for the nickelates indicate the presence of a large hole pocket with d

character, and electron pockets arising from Nd 5d and Ni 3d hybridization, making this an intrinsically multiband system 13,14,21 . A two-band analysis concludes that the observed magnetotransport behaviour corresponds to a decrease in the hole carrier contribution in the presence of parallel electron conduction (Supplementary Information).

The relative insensitivity of T upturn to disorder strength (Fig. 2e) further suggests that the upturn is caused by strong correlation effects that effectively freeze out the hole contribution to conduction. The recent observations of charge order in infinite-layer nickelates with an incommensurate wave vector of roughly (1/3, 0) reciprocal lattice units [33][34][35] provide a candidate for the correlations driving the upturn. This would also be consistent with the lack of magnetic field dependence of T upturn . Notably, the upturn in resistivity-with T upturn decreasing as a function of x-and the presence of strong correlation effects are

)

) (mΩ cm) highly reminiscent of the underdoped region of the copper oxide phase diagram (Extended Data Fig. 6) 2,12,36,37 .

In contrast to the underdoped regime, the temperature onset of the resistive upturn in the overdoped region is considerably reduced when lowering disorder (Fig. 2f) and under applied magnetic field (Fig. 4a-d). Indeed, T upturn shows a strong linear correlation with the low-temperature normal state resistivity (Fig. 4e). We also note that the suppression of resistive upturn shows minimal dependence on epitaxial strain (Extended Data Fig. 9). These observations indicate that the resistive upturn in the overdoped region is driven by disorder physics, unlike in the underdoped region where, as we have discussed, it appears to arise from correlation effects. Suppressing the upturn (or superconductivity) with a 14 T magnetic field reveals metallic Fermi-liquid-like ρ ≅ AT 2 at low temperatures, with the quadratic coefficient A essentially uncorrelated with T upturn or x (Fig. 4e). In addition, the temperature at which the measured resistivity starts to deviate from the low-temperature T 2 fit increases as a function of x (Fig. 4a-d), in a manner analogous to the crossover in the functional form of ρ from T + T 2 or T n (1 < n < 2) to T 2 in the overdoped region of the copper oxide phase diagram 2,38 . Furthermore, R H (T) rather resembles that in overdoped La 2-x Sr x CuO 4 in terms of magnitude, functional form and a sign change at around 100 K (ref. 39).

A notable feature that emerges for the highly crystalline films on LSAT is the strange metallic ρ ∝ T behaviour in the normal state near optimal doping of x ≅ 0.15-0.175 (Fig. 2b,g). The evolution of the functional form and the emergence of T-linear behaviour with reduced disorder (Fig. 1c) are similarly observed in disorder studies of the copper oxides 40 . The T-linear slope of the infinite-layer nickelates (roughly 11 mΩ cm K -1 ) is also close to that of La 2-x Sr x CuO 4 (roughly 6-11 mΩ cm K -1 ) (Fig. 2g) 36,37,41 . Among the copper oxides, this comparison appears most apt given that both systems share the solid-solution cation disorder associated with chemical hole doping. The T-linear resistivity in copper oxides has been a longstanding puzzle, and it is one of several examples that have raised consideration of fundamental bounds on scattering rates 42 .

For the nickelates, the lack of experimental measures of the multiband effective masses and carrier densities preclude attempts at a quantitative analysis. However, it is surprising that the value of the nickelate T-linear slope itself is so close to that of La 2-x Sr x CuO 4 despite the notable differences in the electronic structure that result in parallel conduction channels. Whereas the ultimate origins of strange metallicity (and the upturn in resistance on the underdoped side) are yet unclear, the observations here directly indicate that a parent compound with a hard insulating gap is not a crucial ingredient for the strange metal physics that ensues near optimal doping for superconductivity.

If strange metallicity is not directly tied to a proximate Mott insulator, a commonly invoked alternative would involve scattering off the soft Temperature dependence of ρ (left, at 0 and 14 T) and R H (right, note the reverse vertical scale) for x = 0.075. The functional form of ρ and R H converges as the resistive upturn onsets at lower temperatures. At T < 5 K, ρ deviates from a roughly log(1/T) upturn and saturates instead, with the saturation also observed in R H .

order parameter fluctuations associated with a continuous quantum phase transition occluded by the superconducting dome 2,4 . Further studies are needed to determine the nature of such fluctuations if present-the charge stripe phase is a plausible candidate [33][34][35] -and their effect on the resistive upturn in the underdoped regime. When all of the magnetotransport features are summarized on a common phase diagram (Extended Data Fig. 10), it is suggestive of such a quantum critical scenario: the monotonic decrease of the resistive T upturn (and associated R H maximum) with doping and its apparent vanishing near optimal doping, and Hall measurements that mark a locus of doping dependent temperatures on the overdoped side in which the effective sign of the carriers changes, which again vanishes approaching optimal doping. These features have been presented in a two-band picture appropriate for the nickelates, and strictly within this framework the evolution of R H could be coincidental, reflecting the details of the underlying fermiology of the material: there are analogous debates on multiband effects in the copper oxides, for both hole and electron doping 20 . Alternatively, the vanishing Hall number could be ascribed to Fermi surface reconstruction associated with a density wave order parameter for both nickelates and copper oxides, and thus of more fundamental significance.

With a resistive upturn in the underdoped region driven by strong electron correlations, a non-Fermi-liquid T-linear resistivity near optimal doping and T 2 resistivity in the overdoped region, the superconducting phase diagram of the infinite-layer nickelates bears a close resemblance to that of La 2-x Sr x CuO 4 (Extended Data Fig. 10). This is surprising, especially considering the key distinctions between the two systems. The undoped parent state of the nickelates is not an antiferromagnetic insulator 6,9,10,[17][18][19] . The hybridization between the Nd 5d and Ni 3d bands introduces electron pockets in the Fermi surface, making the nickelates an intrinsically multiband system 13,14,21 . Spectroscopic measurements suggest that the orbital alignment of the nickelates is closer to the Mott-Hubbard regime, rather than the chargetransfer regime such as the hole-doped copper oxides 15,16 . And yet, the Nd 1-x Sr x NiO 2 phase diagram-in particular the superconducting dome and the electrical transport in the normal state-is similar to that of La 2-x Sr x CuO 4 . The good agreement in the superconducting dome with calculations assuming pairing predominantly in the Ni d 3 x y -2 2 band 29 suggests that the same mechanism could be at play in both systems. Similarities in the phase diagram extend beyond these oxides to materials as disparate as twisted bilayer graphene and chalcogenides 3,4 , hinting at an underlying universality in their electrical transport that remains to be understood.

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Polycrystalline Nd 1-x Sr x Ni 1.15 O 3 targets (x = 0.05-0.325) were prepared by pelletizing mixtures of Nd 2 O 3 , SrCO 3 and NiO powders, decarbonating at 1,200 °C for 12 h, regrinding and repelletizing and then sintering at 1,350 °C for 12 h (Extended Data Fig. 11) 5,11 . Roughly 15 unit cells of Nd 1-x Sr x NiO 3 epitaxial thin films were grown by pulsed-laser deposition with a KrF excimer laser (λ = 248 nm) on 5 × 5 mm 2 LSAT (001) and SrTiO 3 (001) substrates, with the substrate surface prepared ex situ by standard acetone-isopropyl alcohol ultrasonication. The films were synthesized using the conditions specified in Extended Data Table 1. Roughly four unit cells of SrTiO 3 (001) were grown in situ as a capping layer 11 .

After cutting into two 2.5 × 5 mm 2 pieces, the perovskite samples were vacuum-sealed (below 0.1 mTorr) with roughly 0.1 g of CaH 2 powder in a Pyrex glass tube, loosely wrapped with aluminium foil to avoid direct contact with CaH 2 . The glass tube was first heated at 240-260 °C for 2 h, with a temperature ramp rate of 10 °C min -1 . Then, X-ray diffraction (XRD) θ-2θ symmetric scan and ρ(T) measurements were performed ex situ to evaluate the degree of topotactic transition. Thirty-minute reductions at 240-260 °C and ex situ characterizations were incrementally continued until the out-of-plane lattice constant, superconducting transition temperature and residual resistivity ratio saturated, indicating complete reduction. The total reduction time across all samples was roughly 2.5-4 h.

XRD θ-2θ symmetric scans and reciprocal space maps were measured using a monochromated Cu K α1 source (λ = 1.5406 Å). Cross-sectional STEM specimens were prepared by a standard focused ion beam (FIB) lift-out process on a Thermo Scientific Helios G4 UX FIB. HAADF-STEM images of the specimens were acquired on an aberration-corrected Thermo Fisher Scientific Spectra 300 X-CFEG operated at 300 kV with a probe convergence semi-angle of 30 mrad and inner (outer) collection angles of 66 (200) mrad. The measurements of ρ(H, T) and Hall effect were conducted in a six-point Hall bar geometry using aluminium wire-bonded contacts. The Hall effect was measured to be linear up to the highest measured magnetic field of 14 T. For two-coil mutual-inductance measurements, a pickup coil of 400 turns and lateral dimensions of roughly 0.5 × 0.5 mm 2 and a drive coil of 50 turns and lateral dimensions of roughly 0.25 × 0.25 mm 2 were made using 20 μm diameter copper wires. The leakage around the film was calibrated using a 100-nm thick aluminium film with the same lateral dimensions as the nickelate samples (2.5 × 5 mm 2 ) 43 .