Introduction

The recent discovery of superconductivity at 80 K in (La,Pr)3Ni2O7 under high pressure makes the bilayer Ruddlesden-Popper (RP) nickelates the latest material system exhibiting high-temperature superconductivity1,2. Despite intense interests in the nature of superconductivity in RP bilayer nickelates3,4,5,6,7,8,9,10,11,12, key parameters characterising the normal and superconducting state are still missing. This is in large part due to the high-pressure condition required to induce superconductivity, which limits the type of experiments that can be performed to primarily transport under a static magnetic field and x-ray diffraction. The subsequent realisation of superconductivity in epitaxially strained (La,Pr)3Ni2O7 thin films with a critical temperature Tc ≈ 40 K (refs. 13,14,15) creates a new material platform to which a broader selection of experimental probes can be employed at ambient pressure, including photoemission16,17, x-ray spectroscopy18, and electron energy-loss spectroscopy19. Moreover, the availability of superconducting (La,Pr)3Ni2O7 thin films enables a straightforward implementation of pulsed high-field magnetotransport experiments, which can reveal key information on its superconducting and normal state, including upper critical field (Hc2), scaling laws in magnetotransport, and functional form of low-temperature normal-state resistivity ρn(T).

In superconducting (La,Pr)3Ni2O7 above Tc, ρxx(T) in bulk crystal and thin film shows an intriguing dichotomy: a putative strange-metal T-linear resistivity is found in bulk crystals1,2,20 whereas an apparent Fermi-liquid T-quadratic resistivity is found in thin films14,15. As the identification of the non-superconducting electronic ground state is critical to understanding the mechanism of electron pairing and the formation of superconducting condensate, it is essential to establish the form of ρn(T) below Tc in the bilayer nickelates. It has been recently found that a partial substitution of La by Pr in (La,Pr)3Ni2O7 favours the formation of bilayer structure2,14,15, leading to a higher crystallinity as reflected by a residual resistivity ratio as high as 8 (ref. 15), therefore we choose La2PrNi2O7 (LPNO) thin film as the model platform to study the magnetotransport behaviour of bilayer superconducting nickelates.

Prior magnetotransport experiments on LPNO thin films estimate that μ0Hc2 in the T = 0 limit is well above 50 T13,14,15, necessitating the use of pulsed field conditions to access the low-T normal state. Here, we report detailed magnetotransport experiments on superconducting LPNO thin films under pulsed magnetic fields up to 64 T. We reveal salient characteristics of the normal-state transport in LPNO, including (1) a T2 resistivity in the field-induced low-T normal state, (2) a T2 temperature dependence of the Hall angle, and (3) an observation of Kohler scaling in normal-state magnetoresistance. These findings provide strong evidence that the underlying electronic ground state in compressively strained LPNO thin film is a Fermi liquid. Our measurements also reveal that Hc2(T = 0) in LPNO thin film exhibits an anisotropy with γ ≈ 2.5, measured by the ratio between Hc2 with field applied parallel (Hc2,∥) and perpendicular to the a-b plane (Hc2,⊥), respectively. By estimating the magnitude of key normal-state and superconducting parameters using established approaches, we find that the ratio between Tc and effective Fermi temperature TF in bilayer RP nickelates exhibits a scaling of Tc/TF ~ 0.05, as found in a wide variety of strongly-correlated superconductors21,22,23,24, thereby hinting at a universal underlying principle governing the Tc magnitude in unconventional superconductors.

Results

Form of normal-state resistivity below T c

Figure 1a shows the in-plane resistivity ρxx(H, T) of an LPNO thin film under a pulsed magnetic field up to 53 T. The field is applied parallel to the crystalline c-axis, i.e., perpendicular to the film surface. A clear transition from a superconducting to a resistive state can be seen below Tc = 41 K (defined as the temperature at which the measured ρxx(T) falls below 0.9ρn(T) with ρn(T) obtained by the parallel-resistor fit to the normal-state resistivity; Fig. 1b). The temperature dependence of resistivity measured in the field-induced normal state at 53 T and extrapolated to 0 T are plotted in Fig. 1b. A prominent feature of ρxx(T) is the positive curvature as T → 0 and negative curvature as T → 300 K. As demonstrated in previous work15, ρxx(T) above Tc can be well-described using a parallel resistor formula (PRF)25:

$$1/\rho (T)=1/({\rho }_{0}+{A}_{2}{T}^{2})+1/{\rho }_{\max },$$
(1)

where ρ0, \({\rho }_{\max }\), and A2 are fitting constants (for discussion of the physical significance of \({\rho }_{\max }\), refer to Supplementary Information Section E). By suppressing superconductivity with a large magnetic field, we find that the field-induced normal-state resistivity below Tc closely tracks the PRF fit and shows a clear saturation towards an apparent T2 behaviour as T → 0 (Fig. 1b inset). The same behaviour is reproduced in a second sample (Supplementary Information Section I). Furthermore, the extrapolated zero-field resistivity exhibits essentially the same ρ(T) form as ρxx(53T) (see details for zero-field resistivity extrapolation in Methods and Supplementary Information Section D), demonstrating that the normal-state ρxx(T) of LPNO exhibits a T2 functional form in the T = 0 limit, characteristic of a Fermi-liquid ground state.

Fig. 1: In-plane resistivity of a superconducting La2PrNi2O7 thin film.
Fig. 1: In-plane resistivity of a superconducting La2PrNi2O7 thin film.The alternative text for this image may have been generated using AI.
Full size image

a Magnetoresistivity isotherms ρxx(H) up to 53.4 T measured at the following temperatures: 100, 80, 60, 50, 45, 40, 35, 30, 25, 20, 15, 10, 7.0, 4.1, and 1.5 K. b Temperature-dependent resistivity ρxx(T) measured at 0 T (solid line) and 53 T (filled circles), and the extrapolated 0-T values (open circles). Grey dash line is a fit to the measured zero-field resistivity above between 50 and 300 K using a parallel resistor model: \(1/\rho (T)=1/({\rho }_{0}+{A}_{2}{T}^{2})+1/{\rho }_{\max }\), which finds ρ0 = 102 μΩ cm, \({\rho }_{\max }=780\,{{\mathrm{\mu \varOmega }}}\,{{\mathrm{cm}}}\), and A2 = 8.1 nΩ cm K−2, respectively. Inset: ρxx versus T2 below 60 K, showing a Δρ(T) ∝ T2 behaviour in the measured 53-T resistivity and extrapolated zero-field resistivity. Error bars correspond to the uncertainty associated with achieving magnetoresistance scaling collapse by inspection (see “Methods” for details). For a discussion of the possible origin and impact of the two-step transition below Tc, refer to Supplementary Information Section C.

Hall effect and magnetoresistance

Figure 2a shows the Hall resistivity isotherms ρyx(H, T) measured between 10 and 60 K on the same sample as shown in Fig. 1. H-linear behaviour in ρyx(H) (within the noise level) is found in the (field-induced) normal state, consistent with the low-field measurements above Tc in La3Ni2O7 and LPNO thin films13,15. The magnitude of Hall coefficient RH(T) between 10 and 240 K, plotted in Fig. 2a inset, shows a monotonic increase as T decreases and RH approaches −0.3 mm3/C as T → 10 K. Since the Fermi surface of thin-film LPNO consists of a small electron-like α-pocket and a large hole-like β-pocket16,26,27, the negative sign of RH likely reflects the multiband nature of its electrical transport28, thereby precludes the inference of normal-state carrier density using the measured RH, as similarly found in the infinite-layer nickelates29.

Fig. 2: Hall resistivity, Kohler scaling, and Hall angle in La2PrNi2O7 thin film.
Fig. 2: Hall resistivity, Kohler scaling, and Hall angle in La2PrNi2O7 thin film.The alternative text for this image may have been generated using AI.
Full size image

a Hall resistivity isotherms ρyx(H) measured at the indicated temperatures. Traces at different temperatures are shifted successively by 1 μΩ cm for clarity. Dash lines are fits made to the normal-state ρyx at high fields using RH = ρyx/(μ0H). Inset: Hall coefficient versus temperature RH(T). RH approaches −0.3 mm3/C as T → 0. Vertical dashed line marks Tc. b Normalised magnetoresistance versus magnetic field scaled by zero-field resistivity, i.e., ρxx/ρxx(0) versus μ0H/ρxx(0), known as the Kohler plot. Normal-state data measured at indicated temperatures collapses into a single curve following \({\rho }_{{{{\rm{xx}}}}}/{\rho }_{{{{\rm{xx}}}}}(0)\,\propto \,{({\mu }_{0}H/{\rho }_{{{{\rm{xx}}}}}(0))}^{2}\) shown by the dashed line. Note that traces of T = 10, 20, 30, and 40 K deviate from the Kohler scaling function at low μ0H/ρxx(0) until normal-state behaviour is recovered at sufficiently high fields. ρxx (ρyx) traces measured below Tc are (anti-)symmetrised, whereas the 50- and 60-K data are measured using the positive-polarity trace only, which nonetheless show good agreement with measurements performed at 14 T. c Inverse Hall angle \(\cot {\theta }_{{{{\rm{H}}}}}=| {\rho }_{{{{\rm{xx}}}}}/{\rho }_{{{{\rm{yx}}}}}|\) versus T2. Filled points correspond to measurements at 53 T, and open points correspond to extrapolated 53 T values using Hall resistivity measured at 8 T (i.e., ρyx(53T) = RH(8T) × (53/8)). Dashed line is a fit using \(\cot {\theta }_{{{{\rm{H}}}}}=C+B{T}^{2}\).

The form of normal-state magnetoresistance (MR = [ρxx(H) − ρxx(0)]/ρxx(0)) is examined in Fig. 2b. The MR magnitude is small ( ≲ 4% up to 53 T) and follows a H2-behaviour, i.e., Δρ(H)/ρ(0) ∝ H2. We find that ρxx(H)/ρxx(0) collapses onto a single curve when plotted against μ0H/ρxx(0), known as the Kohler scaling, implying that the MR magnitude is dictated by the carrier mean free path ℓ ~ ωcτ (i.e., the product of cyclotron frequency ωc and carrier lifetime τ), as expected for a Fermi-liquid state characterised by a single quasiparticle lifetime30. The fact that the normal-state MR ∝ H2 obeys Kohler scaling allows us to extract the zero-field normal-state resistivity below Tc as shown in Fig. 1b (see Methods and Supplementary Section D for extraction details). The Fermi-liquid nature of magnetotransport in thin-film LPNO is further supported by the observation of a negligible MR when the magnetic field is applied parallel to the sample surface i.e. H∥ab (Supplementary Information Fig. S5) and a T2 dependence of the Hall angle: \(\cot {\theta }_{{{{\rm{H}}}}}=| {\rho }_{{{{\rm{xx}}}}}/{\rho }_{{{{\rm{yx}}}}}| \,\propto \,{({\omega }_{{{{\rm{c}}}}}\tau )}^{-1}\) (Fig. 2c). These features are in stark contrast to the high-Tc cuprates within the strange-metal regime, in which a sizeable MR is found with H∥ab (ref. 31) and the longitudinal resistivity and Hall angle exhibit distinct T-dependence32, suggesting that the longitudinal and Hall transport are governed by two distinct carrier lifetimes. Overall, we find that the normal-state transport characteristics of superconducting LPNO thin film, including ρxx(T), MR, and Hall angle, can all be well-described using the standard transport theory for a conventional Fermi-liquid, the key finding of this work.

Upper critical field anisotropy

Next, we extract the upper critical field Hc2 in both H∥c and H∥ab configurations (i.e., Hc2,⊥ and Hc2,∥, respectively). Due to the considerable transition width in the magnetic field, the choice of different extraction criteria leads to different forms of Hc2(T) (Supplementary Information Fig. S7). We adopt the 90% ρn criterion, in accordance with previous reports13,15, and extract Hc2 using the condition ρxx(μ0Hc2)/ρxx(53T) = 0.9. The μ0Hc2(T) data, shown in Fig. 3, are fitted using the linearised Ginzburg-Landau form33:

$${H}_{{{{\rm{c2,\perp }}}}}(T)=\frac{{\phi }_{0}}{2\pi {\xi }_{ab}^{2}}\left(1-\frac{T}{{T}_{{{{\rm{c}}}}}}\right),$$
(2)
$${H}_{{{{\rm{c2,\parallel }}}}}(T)=\frac{\sqrt{12}{\phi }_{0}}{2\pi {\xi }_{ab}(0){d}_{{{{\rm{sc}}}}}}{\left(1-\frac{T}{{T}_{{{{\rm{c}}}}}}\right)}^{\frac{1}{2}},$$
(3)

where ϕ0 is the flux quantum, ξab(0) is the in-plane coherence length at zero temperature, and dsc is the effective superconducting thickness. An anisotropic Hc2 is found with μ0Hc2,⊥(0) = (42.8 ± 0.4) T and μ0Hc2,∥(0) = (106 ± 1) T, corresponding to an anisotropic factor γ = Hc2,∥/Hc2,⊥ = 2.48. We further find that the μ0Hc2(θ ≃ 90∘) data measured at 35 K can be well fitted using the two-dimensional (2D) Tinkham model34 (Fig. 3 inset), pointing to a 2D nature of superconductivity (see Supplementary Information Section F for details of Hc2 extraction and model descriptions). This γ value is comparable to optimally doped YBa2Cu3O7−δ (γ ≈ 2.2; ref. 35) and larger than infinite-layer nickelate La0.8Sr0.2NiO2 (γ ≈ 1.7; ref. 36) and iron-based superconductor Ba(Fe,Co)2As2 (γ ≈ 1.1; ref. 37), implying that the anisotropy in electronic structure of thin-film LPNO is similar to the high-Tc cuprates. Furthermore, we find the in-plane coherence length ξab(0) = (2.76 ± 0.01) nm and a superconducting thickness dsc = (3.88 ± 0.02) nm, comparable to the film thickness of 5 nm and suggesting a bulk nature of superconductivity.

Fig. 3: Upper critical fields of La2PrNi2O7 thin film.
Fig. 3: Upper critical fields of La2PrNi2O7 thin film.The alternative text for this image may have been generated using AI.
Full size image

μ0Hc2 are extracted using the criterion of ρxx(μ0Hc2)/ρxx(53 T) = 0.9. Filled points correspond to the configuration that the magnetic field is applied along the film surface normal (i.e., H∥c) and open points the configuration that field applied parallel to the film surface (i.e., H∥ab). Dash and dotted lines are fits made using the linearised Ginzburg-Landau formulae (see main text). The inset shows μ0Hc2 measured at 35 K with the field oriented close to the in-plane configuration (i.e., θ ≃ 90∘). Solid line is a fit made using the 2D Tinkham model34: \({\left(\frac{{H}_{{{{\rm{c2}}}}}\sin \theta }{{H}_{{{{\rm{c2,\parallel }}}}}}\right)}^{2}+\left|\frac{{H}_{{{{\rm{c2}}}}}\cos \theta }{{H}_{{{{\rm{c2,\perp }}}}}}\right|=1\).

Comparing to the previously reported values for La3Ni2O7 and LPNO thin films13,15, our extracted μ0Hc2,⊥(0) is significantly lower. We note that in previous reports, a constant resistivity value at a temperature slightly above Tc (e.g., 50 K) is used to define the 90% ρn criterion for ρxx(H, T) measured at H ≪ Hc2. In LPNO thin films, since the normal-state ρxx(T) follows a T2 behaviour and the Tc ≃ 40 K is relatively high, the 50-K resistivity value considerably overestimates the true normal-state resistivity at T ≲ Tc, thus leading to an overestimation of Hc2(T ≲ Tc) and Hc2(0). It is also likely that there exists a sample variation in Hc2 (see Supplementary Information Fig. S9), whose underlying cause requires further investigation. Our current experiment on a LPNO film with an Hc2,⊥ directly accessible within 53 T provides an accurate quantification of its μ0Hc2,⊥(0), and allows a crude estimate of the superconducting gap magnitude Δ0 ≃ 6.3 meV via ξab(0) = ℏvF/(πΔ0). We caution here that this Δ0 value should be interpreted as an order-of-magnitude estimate, as vF used here is obtained using estimates of EF and \({\bar{m}}^{*}\), both of which are associated with a sizeable uncertainty, and the level of disorder is assumed to play only a minor role in the extraction of Tc and Hc2 via resistivity measurements. Nonetheless, we note that recent tunnelling experiments38 have reported Δ0 = (6 − 20) meV for thin-film LPNO, broadly in agreement with our estimate.

Estimate of effective mass

It is known that the prefactor of T2-resistivity and electronic specific heat γ0 for a variety of correlated electron systems exhibits an empirical relationship39,40: \({A}_{2}/{\gamma }_{0}^{2} \simeq 10\,{\mu \Omega }\,{{{\rm{cm}}}}\,{{{\rm{mol}}}}^{2}\,{{{{\rm{K}}}}}^{2}\,{{{\rm{J}}}}^{-2}\), known as the Kadowaki-Woods ratio. The high-pressure or epitaxial strain condition required for superconductivity in the bilayer nickelates currently precludes the extraction of γ0 via direct calorimetry measurements. Alternatively, by assuming the empirical Kadowaki-Woods ratio of 10 μΩ cm mol2 K2 J−2 and using the measured A2 = (8.1 ± 1.2) nΩ cm K−2, we infer a γ0 = (28 ± 4) mJ mol−1 K−2 for the LPNO film studied in this work. We further estimate the quasiparticle effective mass using41:

$${\gamma }_{0}=\gamma {\prime} {\sum}_{i}{m}_{i}^{*}/{m}_{{{{\rm{e}}}}},$$
(4)

where \(\gamma {\prime}=\frac{\pi {N}_{{{{\rm{A}}}}}{k}_{{{{\rm{B}}}}}^{2}{m}_{{{{\rm{e}}}}}}{3{\hslash }^{2}}{a}^{2}=1.39\) mJ mol−1 K−2 (a = 3.756 Å is the in-plane lattice constant) and \({m}_{i}^{*}\) is the effective mass of the ith distinct Fermi pocket in the first Brillouin zone (BZ). Given that there are two Ni-O layers in the unit cell of LPNO, two sheets of the Fermi surface are expected in the BZ (i.e., i = 2) and verified by photoemission experiment27. This yields an average effective mass \({\bar{m}}^{*}=(10\pm 3)\,{m}_{e}\) for thin-film LPNO, comparable to that found in overdoped cuprates42,43,44,45. We further estimate an effective Fermi temperature TF = (2330 ± 590) K using the carrier density and Fermi energy inferred from photoemission data27 on thin-film LPNO (see details of the uncertainty estimations in \({\bar{m}}^{*}\) and TF in Supplementary Information Sections J and G). Table 1 summarises the key normal-state and superconducting parameters of thin-film LPNO extracted from our transport experiment.

Table 1 Normal-state and superconducting parameters for La2PrNi2O7 thin film discussed in the main text
Full size table

Discussion

It has been noted that, for a wide variety of unconventional superconductors, the ratio between Tc and TF shares a very similar magnitude, i.e., Tc/TF ~ 0.05 (refs. 21,22,23,24). A survey of Tc versus TF for notable superconductors, including high-Tc cuprates21, iron-based superconductors22, heavy fermion materials23, and carbon-based superconductors24, is shown in Fig. 4. We find that for LPNO, the empirical relationship Tc/TF ~ 0.05 also holds, suggesting a strongly correlated nature of superconductivity in the bilayer RP nickelates and the magnitude of Tc is governed by a universal underlying principle independent of material-specific details.

Fig. 4: Superconducting critical temperature Tc versus effective Fermi temperature TF for strongly correlated superconductors.
Fig. 4: Superconducting critical temperature Tc versus effective Fermi temperature TF for strongly correlated superconductors.The alternative text for this image may have been generated using AI.
Full size image

The dotted, dotted-dash, and dashed lines indicate Tc/TF = 0.01, 0.05, and 0.1, respectively. For the referenced materials, TF values are extracted assuming a quadratic energy dispersion \({E}_{{{{\rm{F}}}}}={\hslash }^{2}{k}_{{{{\rm{F}}}}}^{2}/(2{m}^{*})\) and using experimental data of effective mass and carrier density inferred from specific heat, Hall effect, and quantum oscillation measurements (refs. 21,22,23,24 and references therein) or from penetration depth measurements using \({E}_{{{{\rm{F}}}}}=({\hslash }^{2}/2){(3{\pi }^{2})}^{2/3}{n}_{{{{\rm{s}}}}}^{2/3}/{m}^{*}\) (ref. 57). The series of points in gradient blue illustrates the effect of hole doping on Tc and TF of La2−xSrxCuO4 (LSCO) with the darker colour indicating a higher doping level up to x = 0.21 (Tc = 27 K). Error bars for the cuprate materials indicate the variations in Tc and TF with carrier dopings, and for LPNO, the variations in TF inferred from the upper and lower limits of \({\bar{m}}^{*}\) estimates. MATBG magic-angle twisted bilayer graphene, TMTSF tetramethyltetraselenafulvalene, BEDT-TTF bisethylenedithiol-tetrathiafulvalene; Ba122: BaFe2(As1−xPx)2; STO: SrTiO3; YBCO: YBa2Cu3O6+x; Tl2201: Tl2Ba2CuO6+δ; Tl2223: Tl2Ba2Ca2Cu3O10+δ; Bi2223: Bi2Sr2Ca2Cu3O10+δ; Hg2223: Hg2Ba2Ca2Cu3O10+δ.

Our high-field experiment demonstrates that the low-T normal state of compressively strained LPNO thin film exhibits the defining characteristics of Fermi-liquid transport. The dichotomy of normal-state resistivity between pressurised single crystal and strained thin film of superconducting bilayer nickelates thus raises a fundamental question of the appropriate electronic ground state description therein. One possible cause for the contrasting normal-state transport is the impact of effective carrier doping16. It is well established that in unconventional superconductors with a superconducting dome, the ρ(T) functional form depends strongly on the effective doping level; a predominantly T-linear resistivity is often found near the optimal doping level, contrasting a T2 resistivity in the overdoped region on its phase diagram25,29,46,47,48. In such a scenario, our current finding then suggests a possible mapping of epitaxially strained LPNO to the overdoped regime in its putative superconducting dome. Recent transport experiment on La2PrNi2O7−δ with a controlled oxygen deficiency δ (ref. 18), however, shows an absence of T-linear resistivity with 0.65≤δ≤0, implying that an alternative doping strategy is required to explore the putative strange-metal transport in LPNO thin films.

Another possibility to account for the contrasting ρ(T) is the qualitative different effect between hydrostatic pressure and epitaxial strain, leading to an opposite change in c-axis length and possibly critical distinction in the underlying electronic structure. It has been demonstrated that in epitaxial thin film of La2−xSrxCuO4 grown on LaSrAlO4 near optimal doping (p = 0.15), ρ(T) evolves from being T-linear above Tc to being T-superlinear (i.e. ρ(T) ~ Tn with n > 1) as the film thickness reduces from 90 to 3 nm49, accompanied by a reduction in residual resistivity ratio (RRR = ρ(300K)/ρ0) from 6 to 2. We note that in the case of La3Ni2O7 and LPNO, the bulk crystals showing a T-linear resistivity have RRR values of 2.0–2.5 (refs. 1,2,20), whereas our thin films show RRR  ≈ 4 (Fig. 1 and Supplementary Information Fig. S9). It is thus unlikely that the deviation from a T-linear resistivity in LPNO thin films is caused by an enhanced level of disorder, and that the distinction in ρ(T) functional form between that found in bulk crystals and thin films reflects a difference in effective carrier doping and/or strain anisotropy (see further discussion in Supplementary Information Section H).

Lastly, we comment on the surprising fact that high-temperature superconductivity emerges out of a Fermi-liquid ground state in LPNO thin films. While a T2-resistivity is expected for a conventional metallic state as T → 0, it is generally not expected that such resistivity functional form will hold at high temperatures; instead, ρ(T) in conventional metals typically exhibits a T-linear behaviour at intermediate temperature range between TD/3 ≲ T ≲ TD due to dominant electron-phonon coupling (for bulk La3Ni2O7, a Debye temperature TD = 383 K is reported for polycrystals50). In the current case of thin-film LPNO, in no temperature range up to 300 K is a T-linear resistivity found, implying that the electron-electron interaction therein is so dominant that the electron-phonon coupling becomes essentially negligible. This dominant electron-electron coupling is likely to be relevant for understanding the formation of density wave orders in bilayer nickelates51,52,53. To the best of our knowledge, the only other material with an apparent T2 resistivity up to room temperature is electron-doped cuprates R2−xCe2CuO4 (R = La, Nd, Pr) in the overdoped regime, whose origin remains to be understood54; compared to thin-film LPNO, however, the normal-state resistivity of electron-doped cuprates shows no sign of saturation up to 700 K (ref. 55), thus the mechanisms behind the high-temperature T2 resistivity in electron-doped cuprates and thin-film LPNO are possibly distinct. If the compressively strained LPNO thin films are to be mapped to the overdoped regime of its putative superconducting dome, it would then be possible to further optimise the Tc of RP bilayer nickelates, provided a suitable tuning strategy.

Methods

Thin film growth and device fabrication

LPNO thin films of approximately 5-nm thickness with a single-unit-cell capping layer of SrTiO3 were grown on SrLaAlO4(001) substrates using pulsed-layer deposition, as detailed in previous work15, with the crystal structure of thin films confirmed by x-ray diffraction (Supplementary Information Fig. S1). Gold electrodes of 40-nm thickness with Hall-bar (or van der Pauw) geometry were deposited prior to ozone annealing in a tube furnace (Supplementary Information Fig. S2). Following transport characterisation measurements conducted using 4He cryostats, the samples were kept cold during storage at liquid nitrogen temperature and transported to high-field facilities at dry-ice temperature.

Magnetotransport measurements

Ultrasonically bonded aluminium wires were connected to the measurement probe using 25 μm-diameter gold wires and room-temperature silver paint (DuPont 4929N). Samples were exposed to ambient conditions for less than 2 h between taken out of cryogenic storage and loaded into the measurement cryostat. Electrical transport measurements using four-probe method with an AC excitation current of 10 μA applied along the crystalline a-axis were performed in static magnetic fields up to 13 T using a Physical Properties Measurement System by Quantum Design Inc. at a low frequency between 13 − 30 Hz, and in pulsed magnetic fields up to 53.4 T and 64.2 T at the International MegaGauss Science Laboratory (IMGSL) in Institute for Solid State Physics, University of Tokyo and Dresden High Magnetic Field Laboratory (HLD-EMFL), Helmholtz-Zentrum Dresden-Rossendorf, respectively. Measurements at the IMGSL are recorded using a National Instruments PXIe-6124 multifunction DAQ device, operating at a sampling rate of 4 MS/s, which applies an AC excitation current of 50 kHz and simultaneously records the resultant AC voltage response of the sample, yielding approximately 2000 data points over the duration of the field pulse. Measurements at the HLD were conducted using a Stanford Research Systems DS360 function generator (excitation frequency ~3 kHz) in combination with a Yokogawa DL850 ScopeCorder. During the field pulse, voltage drops across the sample and a 10 kΩ series resistor were recorded at a sampling rate of 1 MS/s over 500 ms and subsequently analysed using a digital lock-in procedure. The excitation current was set to  ≈ 10 μA. Magnetic field profiles of the pulse magnets used in this work are shown in Supplementary Material Fig. S3.

Extrapolation of zero-field resistivity below T c

The magnetoresistivity data at magnetic fields above which ρxx(H) shows a positive curvature is fitted to: ρ(H, T) = ρ(0, T) + βH2 for extracting the zero-field resistivity ρxx(0, T) below Tc. We further adjust the extracted ρxx(0, T) values to achieve a scaling collapse of the Kohler plot for ρxx(H, T) measured over the entire temperature range. Details of the normal-state resistivity extractions are described in Supplementary Information Section D. The adjustment of ρxx(0, T) required for Kohler scaling is typically less than 1.0 μΩ cm.

Extraction of normal-state parameters

Electronic specific heat (γ0) and average effective mass (\({\bar{m}}^{*}\)) are calculated using the following equations:

$$\frac{{A}_{2}}{{\gamma }_{0}^{2}}=10\,{\mu \Omega }\,{{{\rm{cm}}}}\,{{{\rm{mol}}}}^{2}\,{{{\rm{K}}}}^{2}\,{{{\rm{J}}}}^{-2},$$
(5)
$${\gamma }_{0}=2\left(\frac{\pi {N}_{{{{\rm{A}}}}}{k}_{{{{\rm{B}}}}}^{2}}{3{\hslash }^{2}}{a}^{2}\right){\bar{m}}^{*},$$
(6)

where NA is the Avogadro number, kB the Boltzmann constant, ℏ the reduced Planck constant, and a the in-plane lattice constant15. Effective Fermi temperature (TF) is estimated using photoemission data via two approaches, as detailed in Supplementary Information Section G. In short, the first approach fits the energy dispersion E(k) of the α-band near the Fermi level using a parabolic function and estimates the Fermi energy (EF) using the difference between the band bottom and the Fermi level. TF and Fermi velocity (vF) are then calculated using:

$${T}_{{{{\rm{F}}}}}={E}_{{{{\rm{F}}}}}/{k}_{{{{\rm{B}}}}},$$
(7)
$${v}_{{{{\rm{F}}}}}=\sqrt{2{E}_{{{{\rm{F}}}}}/{\bar{m}}^{*}}.$$
(8)

The second approach calculates the average Fermi wavevector (\({\bar{k}}_{{{{\rm{F}}}}}\)) using a weighted sum of measured kF for the α- and β-pocket by their carrier density:

$${\bar{k}}_{{{{\rm{F}}}}}=\frac{{n}_{\alpha }{k}_{{{{\rm{F,\alpha }}}}}+{n}_{\beta }{k}_{{{{\rm{F,\beta }}}}}}{{n}_{\alpha }+{n}_{\beta }},$$
(9)
$${E}_{{{{\rm{F}}}}}=\frac{{\hslash }^{2}{\bar{k}}_{{{{\rm{F}}}}}^{2}}{2{\bar{m}}^{*}}.$$
(10)

A good agreement is found between the TF values estimated by these two approaches.