Introduction

Among the perovskites, the rare-earth nickelates (RENiO3) are interesting because of their tunable metal-insulator transition (MIT)1,2,3,4,5,6,7,8. Recent seminal works directly imaging the evolution of metal and insulating regions, have greatly contributed to a better understanding of the phase transition in these materials9,10,11,12,13 but important questions remain, such as why are the metallic regions still present at low temperatures9 or what is the origin of the critical behavior observed at the (first-order) phase transition12. RENiO3 are also attracting renewed attention due to the discovery of a superconducting phase in the family of the infinite-layer nickelates14,15, which are an oxygen-deficient version of the perovskite nickelates, obtained by removal of a full layer of oxygen atoms. However, most often, oxygen vacancies (Vo) in nickelates are randomly distributed through the lattice, with a different effect on the transport properties16,17,18,19,20. In particular, these randomly distributed Vo are known to give rise to electron localization in defect states16,17,19,21

In addition, NNO films have been reported to show bad metal behavior22, characterized by exceedingly short electron relaxation times; while in other strongly correlated materials, such as cuprate superconductors or heavy fermions, strange metal behavior and Planckian dissipation is observed23,24. This is a recent concept put forward to explain low temperature resistivity that is independent of the strength of the electron interactions and scales linearly with temperature25,26, defying expected quadratic Fermi liquid behavior. However, in NNO, linear behavior of the resistivity has only been rarely claimed27,28. In NNO thin films, other scaling exponent (1 ≤ n < 2) have also been reported29,30,31,32,33,34, which are proposed to arise from spin fluctuations, due to the proximity to a quantum critical point, and to vary with strain29,30. In addition, it has also been shown, in nickelates and in other electron correlated materials, that the apparent exponents experimentally obtained, strongly depend on the quenched disorder induced, for example, by the presence of Vo35,36,37.

These findings emphasize the importance of controlling Vo in RENiO3. In this work, we present a study on the effect of the oxygen content in a NdNiO3 (NNO) film grown on an YAlO3 (YAO) substrate under highly-compressive strain conditions, which allows us to decouple the effects of strain and disorder. Appropriate analysis of the evolution of the resistivity (ρ) versus temperature (T) curves with oxygen content offers a clear picture of the microscopic mechanisms at play, supporting the recent idea that Planckian metal behavior can arise from a two-fluid scenario in a disorder Hubbard model38.

Results

Hidden metal-insulator transition

Epitaxial NdNiO3 (NNO) films are grown on YAlO3 (YAO) substrates by pulsed laser deposition (see Methods). A 20 nm thick NNO film is subjected to high vacuum conditions right after growth, in order to produce a large amount of Vo (see a sketch in Supplementary Fig. 1a). Supplementary Figures 1b, c show, respectively, the atomic force microscopy (AFM) image of the film topography and the X-ray diffraction patterns, attesting the atomically flat, single-crystalline, epitaxial, (001)-oriented nature of the film. Then, the oxygen content is gradually increased by subjecting the same film to increasingly higher annealing temperatures (Ta) in an oxygen atmosphere. As shown in Supplementary Fig. 1c, the (002)-diffraction peak shows a shift towards increasing angles, corresponding to a reduction in the out-of-plane lattice parameter and manifesting the decrease of Vo content (see an estimation in the Supplementary Information) in the film16,17.

Figure 1 a shows the ρ-T curves of the NNO/YAO film measured after different annealing stages, giving rise to various oxygen content levels. The ρ-T dependence of the film annealed either in vacuum or in oxygen at a low Ta, shows semiconducting behavior, as expected for a film with a large amount of defects19. In order to observe a clear change in the sign of the resistivity derivative, a Ta of 800 °C in an oxygen-enriched environment is needed. Under these conditions the film shows a clear metal-to-insulator transition temperature (TMI) at 170 K (see Supplementary Fig. 2). For intermediate states, a progressive decrease of ρ is observed under increasingly Ta, in agreement with a systematic reduction in the amount of Vo in the film. However, the temperature dependence of ρ remains insulator-like for Ta up to 700 °C.

Fig. 1: Modulation of conduction by disorder.
figure 1

a Temperature (T) dependence of resistivity (ρ) (ρ-T) during cooling and heating for the same 20 nm thick NNO thin film grown on an YAO substrate, after annealing at increasingly larger temperatures. b ρ-T measurements during heating (ρh) and cooling (ρc), after annealing at Ta = 250 °C. The inset shows the difference between the heating and cooling curves in a. c Ln(ρ)-1/T measured in the NNO/YAO film annealed in vacuum, together with the fit to a NNH model for T ≥ 170 K. After annealing at Ta = 250 °C d and 650 °C e, the conductivity (σ) versus T fits best to the two-fluid model of Eq. (2) (n = 1). The insets in d and e plotting ln ρ-1/T, highlight the inadequacy of a NNH fitting model. f The same NNO/YAO film after annealing at 800 °C for 1 h displays a metal-like ρ-T (inset) with a scaling exponent n = 2.08.

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In spite of the absence of a change in the temperature derivative of the resistivity, the existence of thermal hysteresis below 170 K, as shown in Fig. 1b and Supplementary Fig. 2a, for Ta above 250 °C, is consistent with the coexistence of metallic and insulating regions in an extended temperature range down to 20 K and it is a clear manifestation of the existence of the metal-insulator transition, as expected in pristine NNO10. In the following, we describe how the metallic regions contribute to the resistivity.

Change of conduction mechanism by modulating disorder

As shown in Fig. 1c and Supplementary Figs. 3 and 4, the ρ-T curves of the sample cooled down from the growth in vacuum, as well as after annealing in O2 at 150 °C (with the largest and second largest amount of Vo), follow a Nearest Neighbors Hopping (NNH) model for T ≥ 170 K. The NNH conduction model is commonly employed in semiconducting systems39,40 and is described by a simple activation process:

$$\rho (T)={\rho }_{0}\exp ({E}_{{{{\rm{a}}}}}/{k}_{{{{\rm{B}}}}}T)$$
(1)

where the ρ0 is a prefactor, Ea is the thermal activation energy of hopping electrons, and kB is the Boltzmann constant. This model, however, does not apply to the same sample with higher oxygen contents (250 °C ≤ Ta ≤ 650 °C), as seen from the worsening of the NNH fit in Fig. 1d, e and Supplementary Fig. 3b. Under these conditions, the film shows hysteresis in the ρ-T curve, as a clear indication of the metal-insulator transition and, thus, of the presence of metallic patches in the samples. The metallic regions are expected to change the resistivity behavior, which deviates from the NNH model. Indeed, the data are shown to follow a two-fluid NNH+M model35,41,42 that describes the combined contribution of semiconducting regions, with localized electrons (NNH) and metallic regions, with itinerant electrons (M). The total ρ is then trivially calculated by the parallel combination of these two components as:

$$\frac{1}{\rho (T)}=\frac{1}{{\rho }_{0}^{* }\exp ({E}_{{{{\rm{a}}}}}/{k}_{{{{\rm{B}}}}}T)}+\frac{1}{{\rho }^{* }(0)+{A}^{* }{T}^{{{{\rm{n}}}}}}$$
(2)

where \({\rho }_{0}^{* }\) is the prefactor of the NNH contribution, normalized by geometrical factors describing the regions of defective insulating phase, while A* measures the electron interactions in the metallic regions, normalized by a geometrical factors associated to the pristine metallic phase and ρ*(0) is the residual resistivity. For convenience, conductivity (σ) rather than ρ was used in the fitting (see Fig. 1d, e and Supplementary Fig. 4). This parallel arrangement implies that the O–Ni–O bonds in the metallic patches form a connected network in the volume between the electrodes, forming a metallic path through the sample (reaching the percolation threshold).

It can also be observed (see inset of Fig. 1b) that both the local TMI and the thermal hysteresis (ΔT = 170–20 K) remain unchanged during oxygenation, thus, displaying no size dependence as the metallic percolated regions grow in size through the material. This shows that the material of the initial percolating metallic path has the same composition (oxygen content) as the subsequently formed paths, strongly indicating that the metallic regions are made of pristine NdNiO3. Altogether, the data shows that metallic conduction at T ≥ TMI takes place as the stoichiometric regions percolate through the sample, even if the material does not show metallic-like overall behavior due to the presence of a too large number of Vo. The Ea values extracted from the fits are plotted as a function of Ta in Supplementary Fig. 5. We will discuss these values in the next section.

After annealing at 800 °C in a oxygen-rich atmosphere, a robust metallic behavior of ρ is obtained in the formerly oxygen-deficient NNO film (see Fig. 1f). However, the film is not fully oxygenated yet, the Vo content can decrease further by increasing the annealing time, which is demonstrated by the progressive decrease of resistivity shown in Fig. 2a.

Fig. 2: Tunable resistivity scaling by regulating disorder.
figure 2

a Resistivity of NNO/YAO film as a function of temperature after annealing in an oxygen-enriched environment at 800 °C with various annealing times. b Evolution of exponents n with the annealing process in 20 nm NNO films grown on three different substrates. Data of NNO/LAO and NNO/STO were extracted from previous work37. In b, error bars are obtained from the fitted model described in the main text.

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As usual, ρ-T in the metallic NNO state can be well fit with a power law added to the residual resistivity (ρ - ρ(0) ∝ ATn). The evolution of the extracted values of n (see Supplementary Fig. 6 for the method of extraction) is plotted in Fig. 2b and shows a gradual decrease with decreasing disorder, reaching n = 1 for sufficiently low vacancy content. The results of similar experiments performed in NNO films under tensile strain (on SrTiO3) and in quasi-strain-free films (on LaAlO3)37 are also plotted in the same figure for comparison.

A direct characterization of the conduction inhomogeneity in oxygen-deficient NNO/YAO films was obtained by using conductive atomic force microscopy (c-AFM). During the scanning, a direct-current sample bias was applied between the tip and the sample (see a sketch in the inset of Fig. 3a and a detailed explanation in the Methods). The sample after annealing at 800 °C for 1 h, displays two distinctly different types of regions, as shown in Fig. 3a. Comparison between Fig. 3a and the height map of the same area (see Supplementary Fig. 7) reveals that the observed contrast is not due to topographic features. I-V measurements were performed in two representative points (1 and 2) of Fig. 3a. Notably, the current of point 1 shows a clear linear dependence with the applied voltage, indicating a robust metallic behavior in this local area (Fig. 3b). On the contrary, the current of point 2 only displays a slight voltage dependence and with much smaller current values than those of point 1 (Fig. 3c).

Fig. 3: Conductive Atomic force microscopy (c-AFM).
figure 3

a c-AFM image of the NNO/YAO film after annealing at 800° for 1 h with a DC sample bias of 500 mV. The inset sketch illustrates the configuration of the c-AFM measurement. b, c show the I-V curves obtained from representative points 1 and 2 in a, respectively. d c-AFM image of the NNO/YAO film after vacuum annealing with a DC sample bias of 1.0 V. e, f show the I-V curves obtained from representative point 3 and 4 in d, respectively.

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A comparable measurement was also performed in the heavily oxygen-deficient NNO/YAO sample (after vacuum annealing). As shown in Fig. 3d, the heavily oxygen-deficient NNO/YAO sample also presents inhomogeneous conduction. However, a voltage of 1.0 V, significantly larger than that applied to the 800 °C annealed sample (500 mV), is needed, indicating a less conducting behavior due to a higher oxygen-deficiency. A more pronounced difference of these two states was revealed by the I-V measurements. While the current of the NNO/YAO sample after annealing at 800 °C shows a robust linear-dependence with voltage, the NNO/YAO sample after vacuum annealing displays a clear semiconductor-like performance (see Fig. 3e, f), manifesting a distinct conduction mechanism in these two states. All of these results are well consistent with our hypothesis discussed above.

Discussion

Our results are consistent with a description of oxygenation giving rise to a percolation process (see a sketch in Fig. 4). In the initial state, the NNO film after vacuum cooling possesses a large amount of Ni2+, acting as a semiconducting matrix (gray regions in Fig. 4). The ρ-T then follows a NNH model for T > 170 K (which is the TMI of the nearly-stoichiometric film), while a Variable Range Hopping (VRH) mechanism is found to dominate in the lower temperature range (see Supplementary Fig. 8), indicating a change in hopping behavior as a precursor of the metal-insulator transition, even if no metallic behavior is yet observed. For Ta = 250 °C, the first percolation paths of stoichiometric regions form across the film. Above TMI, these paths are metallic but, as they are surrounded by the dominant insulating (defective) matrix, the film still shows semiconductor character (dρ(T)/dT < 0). Indeed, an excellent fit to the data is presented by an NNH+M two-fluid model, which includes itinerant electron metallic resistance in parallel with the resistance contributed by regions that host localized hopping electrons, for all intermediate Ta = 250–650 °C (see Fig. 1d, e and Supplementary Fig. 4). For each stage of annealing, and upon decreasing the environment temperature, the stoichiometric percolation paths undergo a metal-insulator transition, which is visible as a thermal hysteresis below the local TMI (marked by green solid lines in Fig. 4). For sufficiently long annealing at 800 °C, the metallic regions cover large enough volume ratio to show a sign change in the ρ-T slope ((dρ(T)/dT > 0)). This is in agreement with previous reports of NNO films grown on YAO substrates showing a robust metallic ρ-T dependence29,30,43.

Fig. 4: Illustration of the double percolation model. The sketch shows the evolution of NNO films as a function of oxygen content (vertical) and temperature (horizontal).
figure 4

Gray, black and white denote electron-localized (defective) NdNiO3−δ, metallic NdNiO3 and insulating NdNiO3, respectively. The evolution of conduction models as a function of oxygen content (from NNH to NNH+M to M) are denoted. The blue horizontal line signifies the metallic percolation threshold. The green horizontal lines indicate the extent of the hysteresis (coexistence of M and I), while the red vertical line points out the measurable TMI.

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The progressive oxygenation of the non-stoichiometric matrix is accompanied by local transformations of different types of polyhedra. In nickelates, oxygen-deficiency usually leads to a square pyramidal NiO5 coordination. However, in heavily deficient cases, NiO4 with either tetrahedral or square-planar configurations can also be formed if two oxygen atoms of the same octahedron are removed19. With increase of oxygen content, the transformation from NiO4 to NiO5 coordination could be expected. This change in the type of polyhedra could be the reason for the observed change in Ea with Ta in the first two states (see Supplementary Fig. 5), different from the constant value of Ea in the intermediate annealing states. Ea ≃ 20 meV is consistent with those reported in the normal semiconducting phases of nickelates44,45,46,47. However, a clear evolution of the A*-coefficients with decreasing Vo is observed in the intermediate states, indicating a progressive increase of the volume fraction of metallic regions, as expected.

While the metal-insulator transition is clearly of first-order, the order of the percolation transition under oxygenation still needs investigation. A second-order percolation transition, as in VO248, could offer an alternative explanation for the fractal antiferromagnetic structures observed in ref. 12. This could perhaps also explain the coexistence of first-order and second-order features also reported in these materials11.

Finally, we discuss the temperature scaling of ρ in the recovered metallic state of annealed NNO films on YAO (under compressive strain) in Fig. 2b. In the same figure, we add the NNO films on STO substrates (NNO/STO), under tensile strain, and on LAO substrates (NNO/LAO), nearly strain-free. In the latter, the scaling of ρ(T) is independent of the Ta and gives rise to n = 137, as in the bulk material27,28 and in agreement with ref. 30. We believe this is attributed to a low Vo content compared with the other two systems, which is supported by values of ρ that are one order of magnitude lower in the metallic state of NNO/LAO film compared with that of the NNO/STO and NNO/YAO films under strain. In the case of tensile strained films (NNO/STO), strain and defect formation are correlated since defects are created during growth as a mechanism to enlarge the lattice and achieve epitaxy37. In the current case of compressively strained NNO/YAO, there is no natural tendency for the film to favor vacancies, so these can be created and removed purposely, and the effects of strain and oxygen content can be separated, as shown in Fig. 2b.

The validity of the n = 1 exponent from above the TMI = 170 K (and even from TMI = 100 K in NNO/LAO), significantly smaller than the Debye temperature of the material (420 K27,30), casts questions about its origin being on electron-phonon interactions and points to the strange metal behavior of NdNiO3. Strange metal behavior is characterized by Planckian dissipation, whose origin is still under debate25, that supports a resistivity that scales linearly with temperature and is independent of the nature of the electron interactions. Recent theoretical calculations have shown that Planckian metal behavior can also originate from a disordered Hubbard model (with electron interactions that do not conserve momentum), from which a two-fluid model emerges that consists of localized Sachdev–Ye–Kitaev (SYK) islands interacting with itinerant fermions12,38,49. Our results seem consistent with this scenario.

To explicate the underlying physics behind this strange linear-T-resistivity, we use a NNO film grown under low strain conditions on LAO substrate to reduce the defect content to the minimum. Cross-sectional high-angle annular dark-field scanning transmission electron microscopy (HAAFAF-STEM) characterizations reported in our previous work37 indicate perfectly epitaxial growth with atomically flat interfaces and a pure phase of this film. Commonly reported lattice imperfections in nickelates, such as dislocations and Ruddlesden–Popper faults, are not observed. The low concentration of the lattice defects present is also supported by the ultra-low residual resistivity (e.g., 8.6 ± 0.22 μΩ cm in the 5 nm film), as shown in Supplementary Fig. 9. Moreover, the ρ values of the NNO/LAO films are well below the Mott–Ioffe–Regel limit22,29 and display a robust linear-T dependence (from 120 to 500 K). All these features make it an ideal candidate to study the intrinsic origin of the linear-T-resistivity in this material.

Hall measurements shown in Fig. 5a display a nearly constant value (within error bars) of the carrier density (N) in the metallic state, with the expected sharp decrease at and below TMI. We note that the measured N values are larger than those expected from 1 e per Ni atom. This discrepancy could be due to the existence of both electron-like and hole-like states at the EF22 or to strong polaronic effects in the metallic state50, which make reliable measurements of carrier density challenging in nickelates2.

Fig. 5: Strange metal-like T-linear resistivity.
figure 5

a Log of carrier density and log of resistivity of a 5 nm NNO/LAO film as a function of temperature. The inset shows the Hall bar used, with channel dimensions 620 × 100 μm2. The dashed line indicates the Mott–Ioffe–Regel limit in nickelates22,29. b 1/τ as a function of temperature is plotted together with the resistivity and their linear fits. c The T-linear resistivities of reported Planckian metals23,24,51,53,54 together with that of the NNO/LAO film. In a and b, error bars indicate the statistical error.

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1/τ values obtained from these measurements, using Drude’s formula ρ = m*/Ne2τ and the reported m* = 7mo22, show linear behavior (see Fig. 5b) and nearly follows the universal dissipation law, 1/τ = αkBT/ℏ, with a constant α = 9.3 ± 4.0, larger than that expected in Planckian metals23,26,51,52. However, if a more physically meaningful value of the carrier density is used (N = 1.0 × 1022cm−322,50), a value of α = 1.8 ± 0.4, in the proximity of the Planckian dissipation bound (α = 1) is obtained. Indeed, the behavior of NdNiO3 is comparable to that of other reported Planckian metals, in the same temperature range, as shown in Fig. 5c, as well as in Supplementary Table 123,24,51,53,54. Our observations, therefore, suggest that further work is required to fully understand the metallicity of this materials family.

To conclude, in oxygen-deficient NdNiO3, metallic regions coexisting with semiconducting defective regions, undergo local metal-insulator transitions, visible above the percolation threshold as thermal hysteresis in the resistivity. Only by further increasing the volume fraction of the pristine state, the metal-insulator transition is unveiled as a change in the slope sign of the resistivity. The n = 1 exponent of ρ-T is recovered upon sufficient oxygenation. This establishes the intrinsic T-linear resistivity in metallic NdNiO3 and postulates its strange metal behavior. Although reliable quantification of the carrier density and the value of α in nickelates remains challenging, these results provide experimental support for the disorder-induced two-fluid origin of Planckian dissipation.

Methods

Synthesis of oxygen-deficient NdNiO3−δ film

An epitaxial NNO film with thickness of 20 nm was deposited on a single-crystal YAlO3 substrate by pulsed laser ablation of a single-phase target (Toshima Manufacturing Co., Ltd.). Before deposition, the YAO substrates were thermally annealed at 1050 °C in a flow of O2 and etched with DI water to obtain an atomically flat surface with single terminated terraces. The growth was performed at 700 °C with an oxygen pressure of 0.2 mbar. The laser fluence on the target was 2 J/cm2. After deposition, the samples were cooled down to room temperature under high vacuum (≤10−7 mbar) to produce an initial NdNiO3−δ state with a large oxygen vacancy content. The thicknesses, crystal orientation and phase purity of the films were assessed using X-ray diffraction by means of 2θ -ω scans on a Panalytical, Xpert MRD Pro diffractometer.

Control of V o content in film

The oxygen content in the NdNiO3−δ film was tuned by annealing in an oxygen-enriched environment (400 cc/min) with a step-by-step increased temperature. In this annealing process, the Vo in the lattice were gradually refilled with the oxygen atoms from the atmosphere. The content of Vo in the lattice can then decreased gradually by increasing the annealing temperature or annealing time.

Electrical properties measurement

The electrical transport properties of the film at each stage of annealing were measured between 5 K and 400 K by the van der Pauw method in a Quantum Design Physical Property Measurement System (PPMS), using a Keithley 237 current source and a Agilent 3458A multimeter. For Hall measurement, the films were patterned into Hall bar (channel dimensions 620 × 100 μm2) using photolithography and ion etching. The Pt electrodes with a thickness of 80 nm were fabricated using e-beam evaporation to provide an ohmic contact with the film.

Conduction maps

For the conductive-AFM measurements, the NdNiO3 film (on top of an insulating substrate) was connected to a bottom metallic plate on the sides of the substrate by silver paste. The bottom plate was connected to the ground of the AFM microscope and a DC bias voltage was applied between the tip and sample. The measurement was performed using a TUNATM amplifier (Bruker Corp.) with Pt/Ir coated silcon tip.