Electronic band structure of a superconducting nickelate
probed by the Seebeck coefficient in the disordered limit

Unconventional superconductivity remains one of the most active and challenging subfields of strongly correlated electron research, with cuprates posing some of the toughest experimental and theoretical challenges over the past three decades [1]. The origin of high-$T_{\rm c}$ superconductivity in the cuprates remains a mystery in part due to the complex interplay of several competing states and relatively strong disorder. One approach to understanding the physics of high-$T_{\rm c}$ is to replace copper entirely, for example with ruthenium or nickel, while maintaining the same square-lattice, transition metal oxide motif. Sr₂RuO₄ is a success of this approach [2], but it does not share the complex phase diagram of the cuprates.

The recent discovery of superconductivity in strontium-doped NdNiO₂ [3, 4, 5] and stoichiometric Nd₆Ni₅O₁₂ [6] presents an opportunity to explore the key ingredients for unconventional superconductivity by contrasting the physical properties of the nickelates with the cuprates. The nickelates contain cuprate-like NiO₂ planes, and the family we study here is Nd_n+1Ni_nO_2n+2, where $n$ indicates the number of NiO₂ planes per unit cell [7, 8, 9, 10, 11]. While nickel in the $n=\infty$ member of the series—NdNiO₂—has the same nominal $3d^{9}$ electronic configuration as copper does in the cuprates, the finite-$n$ members have the nominal configuration of $3d^{9-\delta}$, where $\delta=1/n$. This offers a mechanism for exploring the hole-doped phase diagram without introducing cation disorder.

Superconducting nickelates exhibit many similarities with the cuprates. These include a phase diagram with a superconducting dome maximized around similar $3d^{8.8}$ electron concentrations, evidence for a nodal superconducting gap [12], magnetism[13, 14], charge density waves [15, 16], and even a strange metal phase [17] (Fig. 1a). Conspicuously absent from this list are experimental comparisons of the electronic structure. To understand which aspects of the electronic dispersion are favorable for unconventional superconductivity, one must first understand how electrons interact in the normal metallic state.

The central difficulty is that most of the experimental techniques used to study electronic structures are incompatible with current superconducting nickelate samples. There have been attempts to measure the angle-integrated density of states [18], and there are recent angle-resolved photoemission spectroscopy (ARPES) measurements on non-superconducting, single crystal nickelates [19], but ARPES remains out of reach for superconducting nickelate films due to surface quality issues. Similarly, quantum oscillations require metals with a defect density lower than what is currently available in even the cleanest films. This calls for the use of other techniques that are sensitive to the electronic structure and that are compatible with higher levels of elastic scattering from defects and with thin films.

Thermoelectricity—as measured by the Seebeck coefficient $S$—provides an alternative to probe the electronic band structure of a material. Unlike electrical transport, which is only sensitive to the electronic states in the immediate vicinity of the Fermi energy ($E_{F}$) (Fig. 2a), the Seebeck effect is sensitive to details of the electronic dispersion away from $E_{F}$. Specifically, the Seebeck coefficient reflects the asymmetry of the dispersion above and below $E_{F}$—it probes the asymmetry between occupied and unoccupied states (Fig. 2b), also called particle-hole asymmetry or energy asymmetry [20, 21]. In general, the Seebeck coefficient is defined by both the band structure and the energy dependence of the scattering rate. However, we will demonstrate that this coefficient is only determined by the band structure in the disordered limit, which is analogous to how the Hall coefficient becomes independent of scattering rate in the high-field limit. As the high-field limit is usually inaccessible in most metals, this makes the Seebeck effect a new powerful probe of the electronic dispersion of relatively disordered materials.

To investigate the electronic structure of the nickelates, we measured the Seebeck coefficient of a superconducting 5-layer nickelate Nd₆Ni₅O₁₂ ($n=5$ nickelate) with a transition onsetting at $T_{\rm c}\approx 10$ K (Fig. 1b)—as well as a more-overdoped, non-superconducting, 3-layer nickelate Nd₄Ni₃O₈ ($n=3$ nickelate) for comparison (Fig. 1b). We find that both the $n=5$ and $n=3$ nickelates share a similar temperature independent, negative $S/T$. We show the electronic dispersion obtained from density functional theory (DFT) accounts for both the magnitude and sign of the temperature-independent Seebeck coefficient for the two compounds when calculated in the disordered limit.

To justify the disordered limit, we compare the nickelate data to previous measurements of the Seebeck coefficient in hole doped cuprates with a similar electron count to the $n=5$ nickelate. First, we compare with measurements performed on a single crystal of La_1.36Nd_0.4Sr_0.24CuO₄ (Nd-LSCO $p=0.24$) [20] with a Seebeck coefficient that is positive and qualitatively different from that of the nickelates. Second, we compare with measurements performed on a single crystal of (Bi,Pb)₂(Sr,La)₂CuO_6+δ (Bi2201 $p=0.23$) [21, 23] with an almost identical Seebeck coefficient to the nickelates. Despite their disparities, we show that the differences in Seebeck coefficients between nickelates and cuprates come from strong dissimilarities in impurity concentrations, and not necessarily from fundamental differences in the nature of the metallic state. Despite the presence of strong electronic correlations, the success of DFT and semi-classical transport calculations in our study provides evidence of well-defined quasiparticles responsible for charge and heat transport in both nickelates and cuprates.

Samples. The perovskite-like parent Nd_n+1Ni_nO_3n+1 films ($n=5$ and $n=3$) were synthesized by molecular beam epitaxy on (110)-orientated NdGaO₃. The growth process used distilled ozone, substrate temperatures of $\sim$650-690 ^∘C, and the NdNiO₃ calibration procedure described in Ref. [24]. This synthesis was followed by a reduction process contained in a sealed glass ampoule, optimized with a process at $\sim 290^{\circ}$C lasting three hours in order to reach the square-planar Nd_n+1Ni_nO_2n+2 phases (this process is similar to the procedure in Ref. [6]). Using an electron-beam evaporator, contacts consisting of a 10 nm chromium sticking layer and 150 nm of gold were deposited in a Hall bar geometry such that the applied thermal gradient and measured Seebeck voltage were along the [001]-direction of the substrate.

The substrate material NdGaO₃ has a high thermal conductivity that increases 30-fold between room temperature and $\sim$30 K [25], weakening the applied thermal gradient along the nickelate film. To mitigate this effect, we polished the NdGaO₃ substrate to reduce its thickness from $500$ microns down to $\sim$100 - 150 microns using diamond lapping film. This served to increase the thermal gradient that generates the Seebeck voltage, which allowed us to measure the Seebeck effect down to $\sim$60 K, below which the thermal gradient becomes too small and the experiment cannot be performed reliably. This process necessarily involves a brief heat exposure during sample mounting. We minimized the degradation risk to the sample [26] by using low temperature crystal wax and mounting in an argon glove box; resistivity measurements taken before and after polishing showed no substantial changes.

Measurements. We measured the Seebeck coefficient using an AC technique used previously for cuprates [20]. An AC thermal excitation is generated by passing an electric current at frequency $\omega\sim 0.1$ Hz through a 5 k$\Omega$ strain gauge used as a heater to generate a thermal gradient in the sample. While the heat is carried primarily by the substrate, this also generates a thermal gradient $\Delta T_{\rm AC}$ along the film. We detect this AC thermal gradient at frequency $2\omega$, as well as the absolute temperature shift, using two type E thermocouples. An AC Seebeck voltage, $\Delta V_{\rm AC}$, is also generated at a frequency $2\omega$ in response to the thermal gradient. We measure this voltage with phosphor-bronze wires attached to the same contacts where the thermocouples measure $\Delta T_{\rm AC}$: this eliminates uncertainties associated with the geometric factor.

The thermocouple and Seebeck voltages were amplified using EM Electronics A10 preamplifiers and detected using a MCL1-540 Synktek lock-in amplifier at the thermal excitation frequency $2\omega$. The Seebeck coefficient is then given by $S=-\Delta V_{\rm AC}/\Delta T_{\rm AC}$. The frequency $\omega$ was adjusted so that the thermoelectric voltage and the thermal gradient remained in phase.

Band structure calculations. The paramagnetic electronic structure of the $n=5$ and $n=3$ layered nickelates was calculated using density functional theory (DFT) combined with the projector augmented wave method, as implemented in the Vienna ab-initio simulation package [27]. We used a pseudopotential that treats the Nd $4f$ electrons as core electrons. The in-plane lattice parameters were set to match the NdGaO₃ substrate, and we optimized the out-of-plane lattice parameter. See Appendix A for more details on the band structure calculations.

Boltzmann transport. We fit a tight-binding model (Tables 1 and 2) to the DFT band structure calculated for the nickelates (Fig. 6). We combined the tight-binding model and Boltzmann transport theory to calculate the Seebeck coefficient. We applied the same algorithm that was used successfully in the cuprates [28, 29, 20, 30] to numerically evaluate the Seebeck coefficient for the nickelates.

Seebeck coefficient. Fig. 3b shows the in-plane Seebeck coefficient of both the $n=5$ and $n=3$ samples. Both samples show an $S/T$ that is similar in magnitude, negative, and independent of temperature. We reproduced the Seebeck coefficient of the $n=5$ layer nickelate on a second sample (Appendix C), and the measured $S/T$ of the $n=3$ sample is similar to what was measured previously on the 3-layer nickelate La₄Ni₃O₈ above its metal-to-insulator transition at 105 K [31].

The Seebeck coefficients of both nickelate samples are also comparable in magnitude and sign to that of the overdoped cuprate Bi2201 $p=0.23$ [21]. All of these measurements contrast with the optimally-doped cuprate Nd-LSCO $p=0.24$ [20], whose Seebeck coefficient is strongly temperature dependent and changes sign near room temperature (Fig. 3b). Both cuprates have a similar electron count to the $n=5$ nickelate.

The Seebeck coefficient in overdoped cuprates has been a puzzle for decades, with most cuprates showing a positive Seebeck coefficient similar to Nd-LSCO at low temperature, but Bi2201 showing a negative Seebeck coefficient. Our analysis is able to account for the differences in sign between Bi2201 and Nd-LSCO and explain the temperature dependence of Bi2201 for the first time, presenting a unified picture of the Seebeck coefficient across nickelates and overdoped cuprates.

Boltzmann calculations. We performed Boltzmann transport calculations to interpret the temperature dependence and the negative sign of $S/T$ in both the $n=5$ and $n=3$ nickelates (see Appendix B for more details). For a free-electron model (i.e. a circular Fermi surface), the sign of the Seebeck coefficient reflects the sign of the charge carriers—hole (positive) or electron (negative)—which is similar to the Hall coefficient. For a real material, the Seebeck coefficient is sensitive to the particle-hole asymmetry of the electronic dispersion (Fig. 2b), as well as to the particle-hole asymmetry of the scattering rate, and the resulting Seebeck coefficient can be of either sign.

To perform Boltzmann transport calculations of the Seebeck coefficient, we require the electronic band dispersions for each material. For $n=5$ and $n=3$ nickelates, we fit a tight-binding model $E(\bf{k})$ to the calculated DFT band structure [10, 11, 6]. For both materials, a single $d_{x^{2}-y^{2}}$ band per NiO₂ layer crosses $E_{\rm F}$ (Fig. 4a and 4a). For the $n=5$ compound, one additional band of Nd character crosses $E_{\rm F}$ while for the $n=3$ material the Nd bands are well above the Fermi level (see Appendix A for more details). For the cuprates, we used the tight-binding models obtained from fitting angle dependent magnetoresistance and ARPES for Nd-LSCO [28, 32], and ARPES for Bi2201 [21] (Fig. 4g and 4g). The tight-binding model $E(\bf{k})$ provides the velocities $\bf{v}=\frac{1}{\hbar}\nabla_{\rm\bf{k}}E(\bf{k})$ that serves to calculate the Seebeck coefficient.

We obtain excellent agreement between the calculated and measured $S/T$ for the nickelates by using the DFT band dispersions and a constant (energy and temperature independent) scattering rate, $1/\tau_{\rm 0}$ (Fig. 4c, 4f). We will justify this choice of scattering rate below.

For the cuprates, the calculations with a constant scattering rate predicts also exactly the right magnitude and sign for Bi2201; while the constant scattering rate calculation for Nd-LSCO gets both the sign and the temperature dependence incorrect. To obtain agreement between the Nd-LSCO data and Boltzmann transport calculations, an inelastic, particle-hole asymmetric scattering rate must be invoked (Fig. 4l from Gourgout et al. [20]). In this case, the scattering rate is not only energy dependent in Nd-LSCO $p=0.24$ (denoted $1/\tau(\epsilon)$), but it is also linear-in-energy, with a different slope above and below the Fermi energy (see Appendix B and Gourgout et al. [20] for more details).

Effect of impurity scattering. The stark difference in $S/T$ between the nickelates and cuprates is somewhat surprising given the similarity of their electronic structures. These compounds have predominantly 3$d^{9}$ bands crossing the Fermi energy, and the curvatures of the Fermi surfaces are not all that different—the single Fermi surface in Nd-LSCO essentially interpolates between the hole and electron-like Fermi surfaces found in the multi-layer nickelates (Fig. 4a, b, c). Given that the band structure is largely temperature-independent, the disparities in $S/T$ between the two families must originate in a difference in the scattering rate.

To understand this difference, we examine the relative amounts of disorder in the cuprate and nickelate samples by comparing the residual resistivities, $\rho_{0}$. For the $n=5$ and $n=3$ nickelates, $\rho_{0}=1450$ $\mu\Omega$ cm and $920$ $\mu\Omega$ respectively as measured by Pan et al. [6], which is significantly larger than the $\rho_{0}=23$ $\mu\Omega$ cm of Nd-LSCO $p=0.24$ [33] and $\rho_{0}\approx 120$ $\mu\Omega$ cm of Bi2201 $p=0.23$ [34, 35, 23]. To quantify the disorder, we use the same Boltzmann transport framework we use to calculate $S/T$ to fit the elastic scattering rate $1/\tau_{0}$ to $\rho_{0}$ for each material. We find that $1/\tau_{0}$ is approximately $350$ times higher for the $n=5$ nickelate, and $180$ times for $n=3$, compared to Nd-LSCO $p=0.24$ ($1/\tau_{\rm 0}=10$ ps^-1).

In the disordered limit, the scattering rate is predominately energy-independent (elastic), $1/\tau_{0}\gg 1/\tau(\epsilon)$, the Seebeck coefficient becomes independent of scattering because it is the ratio of two quantities that are proportional to the scattering time: $S=\alpha/\sigma$, with the Peltier coefficient $\alpha\propto\tau_{\rm 0}$ and the electrical conductivity $\sigma\propto\tau_{\rm 0}$ (more details are given in Appendix B). The high $\rho_{0}$ and the measured temperature-independent $S/T$ suggest that elastic scattering is indeed dominant in the nickelates, whereas we know the inelastic scattering plays a dominant role in the physics of Nd-LSCO [20]. The cuprate Bi2201, with a significantly higher level of disorder than Nd-LSCO, lies also in the disordered limit like the nickelates, which is confirmed by its temperature independent $S/T$.

To further illustrate this argument, we recalculate the Seebeck coefficient for Nd-LSCO $p=0.24$ with the total scattering rate $1/\tau_{\rm 0}+1/\tau(\epsilon)$ from Gourgout et al. [20] where we increase the relative amount of elastic scattering $1/\tau_{\rm 0}$ compared to the amount of inelastic scattering $1/\tau(\epsilon)$, going from the clean limit of Nd-LSCO to the disordered limit of the nickelates. Increasing $1/\tau_{\rm 0}$ from $10$ ps^-1 to $3500$ ps^-1 while holding $1/\tau(\epsilon)$ fixed, the calculated $S/T$ drops to a temperature-independent, negative value—very similar to what we measured in the nickelates (Fig. 5a) and what was measured for Bi2201. This confirms that the nickelate films and Bi2201 are dominated by elastic scattering and, in this limit, $S/T$ directly reflects the properties of the electronic bands rather than the energy dependence of the scattering rate. This proves the effectiveness of our approach to probe the electronic band dispersion of the nickelates using the Seebeck coefficient in the disordered limit. Note that the elastic scattering rate in the infinite layer nickelates is about 10 times smaller [17] than in the $n=5$ nickelate. However, Fig. 5b shows that the infinite layer nickelates are still in the limit where the elastic scattering rate dominates over the energy-asymmetric scattering and thus should also exhibit a negative and temperature-independent Seebeck coefficient.

Retrospectively, we can understand that the differences in $S/T$ between cleaner cuprates like Nd-LSCO [20] and LSCO [36], and dirtier cuprates like Bi2201 [21] comes only from the differences in impurity concentrations. The residual resistivity in Bi2201 is typically 5 to 20 larger [34, 35] than in LSCO and Nd-LSCO and the Seebeck coefficient in overdoped Bi2201 is therefore similar to the ones measured in the nickelates (Fig. 10 and Appendix G).

Fortuitously, the larger level of elastic disorder in the nickelates makes the Seebeck coefficient entirely insensitive to the scattering rate; similar to the high-field limit of the Hall coefficient, the high-elastic-scattering limit of $S/T$ reflects only the underlying electronic band structure.

The idea that the Seebeck coefficient is solely determined by the electronic band structure in the disordered limit is further supported by two additional examples from the literature: the infinite-layer superconducting nickelate Nd_0.8Sr_0.2NiO₂ [37] and the delafossite PdCoO₂ [38]. Quirk et al. [37] measured the Seebeck effect in Nd_0.8Sr_0.2NiO₂. This film had a residual resistivity of $\rho_{\rm 0}\sim 800$ $\mu\Omega$.cm—similar to our 3-layer nickelate sample and also in the disordered limit. We used Boltzmann transport and a tight-binding model based on the ARPES experiments of Sun et al. [39] to calculate the Seebeck coefficient. We find excellent agreement between the Seebeck data of Quirk et al. [37] and calculations in the disordered limit (see Fig. 9 and Appendix F).

Yordanov et al. [38] measured the Seebeck effect in thin films of the delafossite PdCoO₂. These films were also in the disordered limit, with a residual resistivity 1000 larger than in single-crystal samples. Yordanov et al. [38] followed a procedure similar to the one we took for the 5-layer and 3-layer nickelates: they used the band structure from DFT and a standard Boltzmann transport package (that uses elastic scattering by default) to evaluate the Seebeck coefficient. They find perfect agreement between the measurements and the calculations. While Yordanov et al. [38] did not connect the success of their calculations to the disordered limit and the constant scattering rate hypothesis, this is another example of the broader validity of our conclusions.

Quasiparticles. $T$-linear and $T+T^{2}$ resistivity was recently reported down to the lowest temperature in infinite layer nickelates [17] and Nd₃Ni₂O₇ [40] under pressure. This suggests a strange-metal phase is present in the nickleates. A $T+T^{2}$ fit of its resistivity (Fig. 8)—as standardized in cuprates [22]—suggests that the $n=5$ nickelate is in proximity to this strange metal regime. While some theories propose that the strange metal regime is a phase without quasiparticles [41], the equally good determination of the Seebeck coefficient for both the $n=5$ or $n=3$ nickelates based on a semi-classical Boltzmann transport suggests that this is not the case here. This is in line with the success of several recent studies in cuprates that have demonstrated the validity of the semi-classical approach to describe transport in strange metals [28, 29, 20, 30] and Fermi liquids [36]. In addition, our study suggests that the band structure of the nickelates as calculated by DFT is a reliable description of the electronic structure of these materials, despite the absence of ARPES measurements to date.

We report the first Seebeck effect study of superconducting nickelates. We used the Seebeck coefficient in the disordered limit to probe the electronic band structure of both a superconducting 5-layer nickelate, as well as metallic 3-layer nickelate. We find the measured Seebeck coefficient is well-described by the band dispersion calculated with DFT, combined with semi-classical transport calculations. The calculated $S/T$ reproduces the amplitude, sign, and temperature dependence of the measured Seebeck coefficient, a rare achievement in predicting transport coefficients in quantum materials, and demonstrates that we have been able to probe the nature of the electronic states in superconducting nickelates—the first report of its kind.

Because of the similar electronic band structures between the nickelates and cuprates, we compare the Seebeck effect in the nickelates with Nd-LSCO – a cleaner cuprate – and Bi2201 – a more disordered cuprate. We find that the Seebeck coefficient for Bi2201 is in perfect agreement with experimental and theoritical data for nickelates. In the case of Nd-LSCO, however, we find a qualitative disagreement despite similarities in the electronic structure of the families. We show that the higher level of disorder present in nickelate thin films and in Bi2201, compared to Nd-LSCO, explains this difference.

As a corollary to our main result, our study highlights that the disordered limit of the Seebeck effect is a powerful and scattering-rate-independent probe of the electronic structure. This is opposite of other transport coefficients like the Hall effect, whose interpretation is opaque in the high scattering rate limit, especially for materials with anisotropic Fermi surfaces like the nickelates. This opens a new avenue of applications for the Seebeck effect in quantum materials by intentionally disordering otherwise-clean materials—for example with electron irradiation—to access intrinsic information about their electronic structure.

The work of B.J.R. and G.G. was supported as part of the Institute for Quantum Matter, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Basic Energy Sciences under Award No. DE-SC0019331. G.A.P. and D.F.S. are primarily supported by U.S. Department of Energy (DOE), Office of Basic Energy Sciences, Division of Materials Sciences and Engineering, under Award No. DE-SC0021925; and by NSF Graduate Research Fellowship Grant No. DGE-1745303. G.A.P. acknowledges additional support from the Paul & Daisy Soros Fellowship for New Americans. Q.S. was supported by the Science and Technology Center for Integrated Quantum Materials, NSF Grant No. DMR-1231319. J.A.M. acknowledges support from the Packard Foundation and the Gordon and Betty Moore Foundation’s EPiQS Initiative, Grant No. GBMF6760. Materials growth was supported by PARADIM under National Science Foundation (NSF) Cooperative Agreement No. DMR-2039380. We acknowledge the Cornell LASSP Professional Machine Shop for their contributions to designing and fabricating equipment used in this study. H.L and A.S.B acknowledge the support from NSF Grant No. DMR 2045826, the ASU Research Computing Center and the Extreme Science and Engineering Discovery Environment (XSEDE) through research allocation TG-PHY220006, which is supported by NSF grant number ACI-1548562 for HPC resources.