The recent discovery of superconductivity in the infinite-layer nickelates (ILN) [1–4] represents the culmination of a three-decade-long search to successfully dope the 3d ⁹ (Ni¹⁺) configuration in a square planar geometry as a means of replicating the structural and orbital motif found in high-T _c cuprates. Unlike the cuprates, whose parent ground state is a Mott insulator with long-range antiferromagnetic (AFM) order, the undoped ILN were found to be metallic at elevated temperatures with a crossover to a weakly insulating state below approximately 100 K, at which the resistivity starts to develop a moderate upturn. While static AFM order has thus far remained undetected in the nickelates, recent resonant x-ray scattering [5] and nuclear magnetic resonance (NMR) experiments [6, 7] reported signatures consistent with fluctuating AFM paramagnon excitations. Other NMR studies, however, claimed an absence of magnetic order in the nickelates [8]. The occurrence of weakly insulating behavior at a high hole doping level, beyond the range within which superconductivity is realized, further contrasts with the correlated but nonetheless metallic ground state found in highly overdoped cuprates [9]. Numerous theoretical calculations [10–22] have indeed pointed out that the 5d (and possibly 4f) band of the rare-earth (RE) elements contributes a finite density of states at the Fermi level, highlighting a fundamental difference between the two 3d ⁹ oxides. The sizeable negative Hall coefficient [1, 3] and the finite spectral weight at the Fermi level [23] experimentally found in undoped nickelates appear to corroborate this picture.

Despite the recent progress in understanding the low-energy electronic structure of superconducting nickelates, an understanding of the anomalous insulating behavior that is ubiquitously found in ILN is lacking. Here, we present a systematic study of the normal-state transport of two doped families of ILN—the Nd- and La-based systems—by employing high magnetic fields up to 35 T to fully suppress the superconductivity. The effect of varying the rare-earth (RE) element on the functional form of the insulating resistivity, as well as the impact of (hole) doping and disorder level on the transport characteristics are also investigated. By taking into account the evolution of the Hall coefficient in both systems, we arrive at a number of salient points with regards to the origin of the insulating behavior: 1) The resistive upturns at low doping are likely to be due to a partial gapping of the states derived from the RE ions. 2) Hole doping x is much more effective in suppressing the resistivity upturn than a decrease in disorder (as inferred from the residual resistivity ratio). 3) The upturns, though notably weaker in the superconducting samples, nevertheless persist into the superconducting regime, and show a different functional form depending on the choice of RE. 4) In this region of the phase diagram, the insulating behavior is more likely to be associated with the correlated 3d states on the Ni. 5) The field dependence of the magnetoresistance in superconducting samples appears to rule out localization or the Kondo effect as the origin of the resistive upturns. 6) The RE dependence on the functional form of the low-T resistivity, as well as its overall magnitude, are more reminiscent of that seen in electron-doped cuprates than in hole-doped cuprates. 7) Finally, we find that T _c in the nickelates is sensitive to the level of disorder, suggesting that superconductivity in the ILN is unconventional in nature.

La_1−x Sr _x NiO₂ and PrNiO₂ thin films were grown by pulsed laser technique described in [3, 24], respectively. Electrical resistivity was measured with a four-point configuration using the ac lock-in technique, with an alternating current I = 10 μA applied within the ab-plane at a frequency between 13 and 30 Hz. Static magnetic fields up to 35 T, applied parallel to the crystalline c-axis, were generated using a Bitter magnet at the High Field Magnet Laboratory in Nijmegen, the Netherlands.

Figure 1 shows the T-dependent in-plane resistivity ρ _ab (T) of a set of undoped RENiO₂ films (RE = La, Pr, Nd). Several key features of its normal-state resistivity are revealed in these plots. Firstly, for all films, ρ _ab (T) undergoes a resistivity minimum (ρ _min) at T = T _min that delineates the metallic regime from the insulating-like regime at lower temperatures. Secondly, the absolute values of ρ _ab show significant variation between samples, with the newer generation exhibiting lower absolute resistivities as well as a reduced level of disorder, as inferred from the higher ρ _300K/ρ _min ratios. As ρ _300K/ρ _min increases, T _min shifts to lower values, suggesting that disorder plays some role in the insulating behavior, at least in the parent compound(s). (The resistivity of LaNiO₂ from an early report [25] was found to be an exceptionally low yet its ρ _300K/ρ _min ratio is the lowest among all samples investigated, the origin of which is yet unclear). Thirdly, in the high-T metallic regime for PrNiO₂ and NdNiO₂, the slope dρ _ab /dT is found to be very similar despite a large variation in their absolute values. This suggests that the excess disorder, while increasing the impurity scattering rate (and the magnitude of ρ _ab ), does not significantly affect the intrinsic metallic resistivity. Fourthly, the functional form of the resistive upturn over the accessible temperature range depends on the choice of RE. In LaNiO₂, for example, ρ _ab (T) initially follows a log(1/T) behavior for T < T _min but then tends towards a constant value below 10 K. In contrast, ρ _ab (T) in PrNiO₂ and NdNiO₂ ρ _ab (T) ∝ log(1/T) down to the lowest measured temperatures. Whether or not ρ _ab (T) in (Pr, Nd)NiO₂ saturates below ≈2 K, however, remains to be seen.

An emerging picture for the electronic structure of undoped ILN, based on recent spectroscopic studies and realistic theoretical calculations with electron interaction taken into account [15, 16, 23, 26, 27], indicates that its Fermi surface comprises a small electron pocket with a dominant character of the RE 5d band (which hybridizes with the Ni 3 d z 2 band). The Ni 3 d x 2 − y 2 band, on the other hand, is split into the upper and lower Hubbard bands and thus does not directly contribute to the Fermi level. The Hall coefficient R _H(T) in both LaNiO₂ and NdNiO₂ is found to be negative [3, 28], consistent with the notion that the 3d states on the Ni sites are Mott localized and that the longitudinal and Hall conductivities are dominated by the electron pocket derived from the RE 5d states. Hence, it is these states that must be responsible for the resistive upturns in the parent compounds. Secondly, in both systems, T _min is found to mark the onset of a marked increase in R _H(T), possibly indicating some form of gap opening below T _min. Thirdly, the fact that ρ _ab (T) appears to saturate eventually, at least in LaNiO₂ (and possibly in PrNiO₂ too), implies that this gapping is only partial and that a finite density of states remains on the electron pocket(s) whose low-T ground state is ultimately metallic.

According to the conventional Drude transport model, the electrical conductivity σ is given by σ = ∑ i n i e μ i = ∑ i n i e 2 τ i m i * , (1) where i denotes the distinct channel of conducting carriers, e is the elementary charge, and (μ, τ, m*) denote the associated mobility, relaxation time, and effective mass, respectively. Consequently, ρ = 1/(neμ) = R _H/μ for a single-band metal, where R H = 1 n e is the Hall coefficient. From Eq. 1, it can be seen that a reduction of conductivity (i.e. a metal-insulator transition) can be caused by a reduction of n (loss of carrier) or τ (increased scattering rate), an increase in m* (effective mass enhancement), or a combination of these factors. For simplicity, here we estimate the carrier mobility using the measured R _H, known as the Hall mobility μ _H = R _H/ρ for undoped LaNiO₂ and NdNiO₂, for which the single-band picture is most likely to apply. As shown in Figure 2, both LaNiO₂ and NdNiO₂ show a relatively unchanged μ _H above T _min, below which μ _H increases moderately. Crucially, the increase in μ _H below T _min indicates that the increase in ρ below T _min is not related to a reduction of mobility (i.e. a change in τ/m*), but is most likely caused by a reduction in n. The minimization of R _H at low T as x approaches 0.20, at which T _min and ρ ₀ − ρ _min are most suppressed (see Figure 4) further supports this scenario. Possible candidates responsible for the loss of carriers below T _min include the emergence of a secondary order parameter (e.g. magnetic, charge, or stripe order), the opening of a pseudogap that partially depletes the density of states at the Fermi level [29], or a transfer of spectral weight to higher energy [30, 31].

With hole-doping, the situation evolves in a systematic fashion. In Nd_1−x Sr _x NiO₂, we revealed previously by destroying superconductivity with a large magnetic field, that the resistivity upturn, though persisting throughout the doping range of superconductivity, is progressively suppressed, essentially vanishing as x approaches the edge of the superconducting dome x _c ≈ 0.225 [32]. Here, we examine the evolution of the low-T resistivity in La_1−x Sr _x NiO₂ in the field-induced normal state for 0.15 ≤ x ≤ 0.20, i.e. across much of the superconducting doping range [3]. A contrasting behavior manifests in the functional form of ρ _ab (T) below T _c, as shown in Figure 3. Similar to the undoped compound, ρ _ab (T) in the field-induced normal state of superconducting La_1−x Sr _x NiO₂ exhibits an initial log(1/T)-behavior followed by a leveling off as T → 0. The magnitude of the resistive upturn decreases as x increases from 0.15 to 0.18, after which it again increases at x = 0.20.

The overall magnitude of the resistive upturn, on the order of 10% between 0.5 K and T _min, is considerably smaller than that observed in the underdoped hole-doped cuprates (≳ 100%) [33, 34, 35] but is comparable with that reported in the electron-doped cuprates RE _2−x Ce _x CuO₄ below optimal doping [36–38]. A direct comparison of the low-T resistivities in the ILN and RE _2−x Ce _x CuO₄ is shown in Figure 3C. Intriguingly, the functional form of the low-T resistivity in RE _2−x Ce _x CuO₄ also depends on the RE elements in a similar manner to what is seen in the ILN. For La_2−x Ce _x CuO₄ (LCCO, x = 0.08) ρ _ab (T) appears to saturate below 4 K, while for Nd_2−x Ce _x CuO₄ (NCCO, x = 0.14), ρ _ab (T) ∝ log(1/T) down to the lowest measured temperature. This close alignment to the experimental situation in the n-doped cuprates is curious, but may simply be a consequence of the way in which carriers are doped into each system. In the cuprates, doped holes sit preferentially on the O sites while doped electrons reside on the Cu sites [39]. In the ILN, it is thought that the carriers are also introduced directly into the 3 d x 2 − y 2 orbital on the Ni sites [40]. At the same time, the similarities found in the low-T ρ _ab (T) behavior of the ILN (for which no long-range AFM order exists at half-filling) and the n-doped cuprates suggests that the resistive upturns in the latter are not necessarily caused by short-range spin correlations, as is believed to be the case for the p-doped cuprates.

The evolution of R _H(T) with doping in both ILN families is qualitatively the same, with a gradual reduction in the overall magnitude of R _H culminating in a crossover from negative to positive R _H(0) (the Hall coefficient in the low-T limit) around optimal doping [1, 3]. In any two-band metallic system, the sign of R _H(0) reflects the sign of the most mobile carriers. Hence, the observed sign change signals a delocalization of the 3d hole states on the Ni sites with hole-doping until eventually, they become the most mobile carriers in each system. Nevertheless, the fact that ρ _ab (T) continues to exhibit a logarithmic divergence (at least in Nd_1−x Sr _x NiO₂) implies that these carriers are also prone to some form of localization, however weak. (Note that R _H(T) exhibits no upturns within the superconducting doping range, and so it is unlikely that the resistive upturns here are due to partial gapping).

It was noted early on that the insulator-to-metal crossover in the cuprates occurs at a threshold value of k _F ℓ > 10 for both the hole- [35] and electron-doped [36] compounds, far higher than the usual criterion k _F ℓ ≈ 1. Assuming that the suppression of the resistive upturn in Nd_0.775Sr_0.225NiO₂ reflects a metallic ground state and using the two-dimensional free electron model [35]: ρ a b / d = h / e 2 k F ℓ , (2) where d is the c-axis lattice spacing, k _F is the Fermi wavevector, and l is the electronic mean free path, we find a threshold k _F l ≈ 2 − 10 for low-T metallicity in the ILN. We note, however, that the assumption of a single-band, 2D Fermi surface is likely not valid for the entire series of hole-doped nickelates (due to the expected presence of a 3D electron pocket derived from the 5d band of RE elements); therefore the estimates of k _F l here should be interpreted with caution.

Several proposals have been put forward to explain the anomalous upturn in the normal-state resistivity in the nickelates [31, 41, 42]. Two well-known mechanisms to produce a logarithmically diverging resistivity at low T are weak localization due to disorder [43, 44] and Kondo scatterings due to magnetic impurities [14, 45]. In both circumstances, however, a strong negative magnetic-field dependence of the insulating resistivity is expected, which is not observed in the nickelates. Moreover, a monotonic suppression of the insulating behavior with decreasing residual resistivity, expected for a localization-driven origin, is not seen (Figure 4) while the re-entrant insulating behavior found at high dopings also cannot be naturally explained by a Kondo-like mechanism.

In order to gain further insights into the origin of the resistive upturns, we have examined the impact of disorder and doping on the insulating characteristics in the ILN, namely the onset temperature (T _min) and the size of the resistivity upturn (ρ(T → 0) − ρ(T _min), denoted as ρ ₀ − ρ _min), as shown in Figure 4. We find no clear correlations between T _min and ρ _min (Figure 4A) nor between ρ ₀ − ρ _min and ρ _min (Figure 4B), for both Nd_1−x Sr _x NiO₂ and La_1−x Sr _x NiO₂. Meanwhile, T _min and ρ ₀ − ρ _min both appear to collapse near x = 0.20 (Figures 4C,D), though deviations from the overall trends are visible both at zero doping and at the highest dopings. Notably, while T _min and ρ ₀ − ρ _min are gradually suppressed with improved sample quality (Figure 1B), varying x is seen as much more effective in suppressing the insulating behavior, suggesting that it is sensitive to carrier screenings and primarily driven by electron correlation effects. A number of non-Fermi-liquid models have been proposed to explain the anomalous insulating behavior seen in underdoped cuprates, including those based on the marginal Fermi liquid [46], the 2D Luttinger liquid [47] and the polaronic Bose liquid [48] model. The relevance of these more exotic models to the ILN, whose distinction from the (hole-doped) cuprates has become increasingly established, remains to be examined.

Lastly, we examine the impact of disorder on the critical temperature of superconducting nickelates. Figure 5 shows a compilation of Nd_0.8Sr_0.2NiO₂ resistivity data reported to date [1, 28, 32, 49–53]. A large difference in the absolute values of ρ _ab is found, with ρ _ab (300 K) ranging from ∼ 0.1–6.75 mΩ cm for nominally the same samples. Meanwhile, the agreement in the normalized resistivity ρ/ρ _300 K is much better across different reports, as shown in Figure 5B, with a good overlap found in 6 out of 9 traces. This suggests the discrepancy in the absolute resistivities arises from geometric uncertainties. Importantly, we find that T _c depends strongly on the residual resistivity ratio, defined as ρ _300 K/ρ _{20 K}, with T _c increasing from ≈5 K to over 12.5 K as ρ _300 K/ρ _20 K increases. Such a strong dependence of T _c with respect to the level of disorder points to an unconventional nature of the superconductivity in the nickelates, and hints at a possible further increase in T _c with improved sample quality.

In summary, by suppressing superconductivity with high magnetic fields, we find the unusual resistivity upturn in the undoped infinite-layer nickelates persists into the superconducting regime and appears to be maximally suppressed near x = 0.20 in both La- and Nd-based systems. The resilience of the resistivity upturn against magnetic fields rules out localization and Kondo effect as its origin, and points to a partial gapping of the states with dominant RE 5d character as its cause at low doping as supported by Hall mobility analysis. In the superconducting doping range, the resistive upturn is found to be highly reminiscent in both its functional form and its overall magnitude to that found in electron-doped cuprates, which suggests the insulating behaviour is associated with the correlated Ni 3d states. While disorder has only a minor impact on the insulating behavior, the robustness of superconductivity is strongly affected by the level of disorder, pointing towards the unconventional nature of nickelate superconductivity.