Impacts of f-d Kondo cloud on superconductivity of nickelates

Kondo effect in NdNiO₂. We found the Kondo effect driven by the strong correlation of Nd-4$f$ and Ni-3$d$ electrons. Figure 1(a) shows crystal structure of tetragonal phase (P4/mmm) NdNiO₂, and the high symmetry lines in the first Brillouin Zone. Rare-earth Nd atoms make bridges between the NiO₂-layers. Figure 1(b) shows spectral functions and density of states (DOS) of NdNiO₂ at $T=$ 100 and 300 K. There are two prominent differences between electronic structures at these two temperatures. First, the energies of Nd-$f$-driven states increase significantly as temperature decreases. The DOS of Nd-4$f$ (red line) became sharp and the flat Nd-4$f$ bands appeared at the Fermi level ($E_{\textrm{F}}$) in the spectra function at 100 K. The flat Nd-4$f$ bands are hybridized with the mixed conduction band of Ni-3$d$ and Nd-5$d$ and give rise to a kink-like band structure at the Fermi level along $\Gamma$-Z and R-A-Z high symmetry lines. That is a hallmark of the Kondo screening[29]. The table in Figure 1(c) shows significant occupancy of Nd-4$f$ in $j$=5$/$2 states. The calculated total occupation in Nd-4$f$ orbitals is 1.9 for both temperatures, indicating a large local magnetic moment of Nd-4$f$ is screened by conduction electrons at low temperatures. This result is reminiscent of heavy fermion superconductor UTe₂, where flat U-5$f$ bands with occupancy of 2.27 lead to orbital selective Kondo effect[29]. For convenience in this work, $d$ orbitals are labelled, and grouped based on the main orbital character at the vicinity of the Fermi level (-0.5 $<$ $E$-$E_{\textrm{F}}$ $<$ 0.5 eV) as shown in Fig. 1(c). As shown in Fig. 1(d), Ni-$d_{\alpha}$ is hybridized with Nd-$d_{\alpha}$ along the R-A-Z high symmetry line. Ni-$d_{\beta}$ is hybridized with Nd-$d_{\beta}$ along the $\Gamma$-X and M-R-Z high symmetry lines. We also group Nd-4$f$ orbitals into $f_{\alpha}$ and $f_{\beta}$ based on occupation. At $T=$ 300 K, Nd-$f_{\alpha}$ is hybridized with Nd-$d_{\alpha}$, Ni-$d_{\alpha}$, and Ni-$d_{\beta}$, whereas Nd-$f_{\beta}$ is hybridized with Nd-$d_{\beta}$ and Ni-$d_{\beta}$ in the vicinity of the Fermi level. At $T=$ 100 K, three Kondo scatterings appeared with momentum-dependent (see Fig. 1(b) and Fig. 2(b) ): i) Nd-$f_{\alpha}$ with Nd-$d_{\alpha}$, Ni-$d_{\alpha}$ and Ni-$d_{\beta}$ along R-A-Z high symmetry line, ii) Nd-$f_{\beta}$ with Nd-$d_{\beta}$ and Ni-$d_{\beta}$ at the X-point and along the M-$\Gamma$-Z high symmetry line, iii) Nd-$f_{\alpha}$ with Ni-$d_{\alpha}$ at the X point and near the $\Gamma$ point. This indicates that not only Nd-5$d$ but also Ni-3$d$ in NiO₂ layer are involved to conduction electrons, which screen local spin momentum of Nd-4$f$. Therefore, the inter-atomic Kondo effect changes the topology of the Fermi surface and leads to the 3D nature of electronic structure at the Fermi surface of NdNiO₂ at low temperatures. Another important aspect is that the DOS of Ni-3$d$-driven states (green line) around -1 eV becomes slightly higher with lowering temperature. This describes the hybridization between Ni-3$d$ and Nd-5$d$. The details will be demonstrated in next section.

We examined the effect of the lattice modulation on the Nd-4$f$-driven Kondo screening in NdNiO₂. The Kondo effect, which manifests an inter-atomic interaction between Nd-4$f$ and the itinerant Ni-3$d$ electrons, should be sensitive to the distance between Ni and Nd, i.e., the lattice modulation, owing to localized characters of strongly correlated Nd-4$f$ and Ni-3$d$ electrons. We use two artificial lattices, namely L_large and L_small, in addition to the experimental lattice constants L_exp ($a$ = 3.92 Å, $c$ = 3.28 Å) of bulk NdNiO₂[23]. For larger lattice L_large ($a$ = 3.96 Å, $c$ = 3.38 Å), we adopted the experimental lattice constants of bulk LaNiO₂[30]. The difference between L_large and L_exp yields $\Delta a$ $\approx$0.04Å and $\Delta c$ $\approx$0.1Å for in-plane and out-of-plane, respectively. We define the smaller lattice L_small ($a$ = 3.88 Å, $c$ = 3.18 Å) to be [L${}_{\mathrm{exp}}-(\Delta a,\Delta c)$]. We, therefore, examine the effect of lattice modulation on the Kondo screening mainly due to the expansion along the $c$-axis. All lattice constants are shown in Fig. 2(a). Note that the geometry optimization using many-body methods is currently not feasible. Although our test with those lattice constants does not provide quantitative results comparable to the experimental data, we expect to uncover some distinct features induced by the variation of interlayer distance. The examination of the lattice modulation effect should help the understanding of the physics in the Sr-doped film with larger lattice. The measured data in Nd_1-xSr_xNiO₂/SrTiO₃[31] indicate that the $c$-axis lattice constant increased from 3.28 Å to 3.42 Å by Sr-doping $x$ $=$ 0.25. For Nd_0.8Sr_0.2NiO₂/SrTiO₃, the $c$-axis lattice constant becomes smaller, as the film is thicker: from $\sim$3.42 Å for the 4.6-nm film to $\sim$3.36 Å for the 15.2-nm film[32].

Figure 2(b) shows that the spectral weight of Nd-4$f$ and the band kinks at the Fermi level are found to be significantly weaker in L_large, which indicates the Kondo effect diminished by the increase of inter-atomic distance. This tendency is also described by the local total angular momentum susceptibility $\chi_{\mathrm{loc}}^{J_{Z}}$ of Nd-4$f$ behavior in Fig. 2(a). $\chi_{\mathrm{loc}}^{J_{Z}}$ is given as

The $\chi_{\mathrm{loc}}^{J_{Z}}$ of Nd-4$f$ in L_small and L_exp deviate from the Curie-Weiss behavior at $\sim$400 K, which indicates the onset temperature of the Kondo scattering process[29]. On the contrary, in L_large, $\chi_{\mathrm{loc}}^{J_{Z}}$ of Nd-4$f$ keeps dropping with lowering temperature, which indicates that Kondo screening effect disappears[29].

The lattice modulation effect on Kondo screening can be also understood through the calculation of self-energy ($\Sigma$), as shown in Fig. 2(c). At 900 K, in all three lattices, the imaginary part (Im) of $\Sigma$ of Nd-$f_{\alpha}$ exhibit a singularity on the imaginary frequency axis, which indicates that the electronic structure at high temperature is governed by Mott-like physics. It also leads to the peak of Im $\Sigma$ at Fermi-level on the real frequency axis and the appearance of a gap in the DOS. On the other hand, at a lower temperature, the self-energy exhibits Fermi-liquid-like behavior giving rise to a formation of quasi-particle peak at the vicinity of the Fermi level. The high-$T$ Mott-like feature becomes pronounced in L_large owing to the suppressed inter-orbital hopping, which is manifested by the reduced hybridization function, as shown in Supplementary Fig. S2. The prominent Mott-like characteristic of Nd-4$f$ in L_large hinders the formation of quasi-particle peak at the Fermi level, weakening the Kondo screening. Our results are reminiscent of Fermi-liquid behavior of 5$f_{5/2}$ states with larger Kondo scale than the 5$f_{7/2}$ states which are at the edge of a Mott transition in PuCoGa₅[33]. Interestingly, the two features are competing in the same states of NdNiO₂.

Our results propose that the Kondo temperature driven from Nd-4$f$ decrease with the increase of inter-atomic distance, which gives an explanation to the experimental data reported for the electrical resistivity upturn at low temperature. The upturn was measured around 70 K for NdNiO₂ film ($c$-axis lattice constant: 3.31 Å)[5], whereas insulating resistivity behavior arises below 300 K in NdNiO₂ bulk with smaller $c$-axis lattice constant of 3.24 Å[34]. While the Kondo effect is not the only possibility for the resistivity upturn, these observations could indicate higher onset temperature of Kondo scattering in NdNiO₂ bulk than film, and be tied to our result of weaker Kondo effect in larger lattice.

$d$-$d$ hybridization in LaNiO₂ and NdNiO₂. LaNiO₂ has the same crystal structure as NdNiO₂, however La-4$f$ states are far away from the Fermi level. To investigate aforementioned hybridization effect, we first calculated the electronic structure of LaNiO₂. The two left panels of Fig. 3(b) show DOS of Ni-$d_{\alpha}$ and La-$d_{\beta}$. In this work, we focus on the $d$-$d$ hybridization which manifests the evolution of Ni-$d_{\alpha}$ and La-$d_{\beta}$ peaks around -1.0 eV with decreasing the temperature. The peaks are absent at 2000 K, and developed along the X-M-R symmetry line with lowering temperature, as shown in Fig. 3(a). The apparent coincidence of peak positions of Ni-3$d$ and La-5$d$ is presented by one orbital projected DOS in Supplementary Fig. S3. Due to formation of the hybridized peaks, Ni-$d_{\alpha}$ peak at the Fermi level shifted to lower energy with lowering temperature, as shown in the right top panel of Fig. 3(b). The shift reduces the DOS of Ni-3$d$ (Ni-$d_{\alpha}$ $+$ Ni-$d_{\beta}$) at the Fermi level (see circle markers, red line in the left panel of Fig. 3(c)), resulting in decreased local total angular momentum susceptibility $\chi_{\mathrm{loc}}^{J_{Z}}$ of Ni-3$d$ (see square markers, dotted red line in the left panel of Fig. 3(c)) with lowering temperature. Here, the DOS at the Fermi level was calculated by

where $\omega_{0}$ is the first Matsubara frequency and $G$ is the calculated local Green’s function.

We tested the effect of lattice modulation on the $d$-$d$ hybridization. We took three lattices L_small, L_exp, and L_large, similarly to NdNiO₂. Here, L_small uses the experimental lattice parameters of bulk NdNiO₂, L_exp uses the experimental lattice constants of bulk LaNiO₂, and L_large uses the L_exp added with the difference between L_exp and L_small (see Fig. 3(c)). As shown in Fig. 3(c), the DOS at the Fermi level $D(\mathrm{i}\omega_{0})$ of Ni-$d_{\beta}$ are almost the same in magnitude, together with similar temperature dependence in all three lattices. This indicates a negligible effect of lattice modulation on Ni-$d_{\beta}$. All $D(\mathrm{i}\omega_{0})$ of Ni-$d_{\beta}$ gradually increase upon cooling (see also the right bottom panel of Fig. 3(b)). This behavior is reminiscent of evolution of quasi-particle peaks in strongly correlated materials, where the spectral weight at the Fermi level is transferred from the upper and lower Hubbard bands [29, 35, 36].

On the contrary, as shown in Fig. 3(c), all $D(\mathrm{i}\omega_{0})$ of Ni-$d_{\alpha}$ exhibit overall enhancement as the lattice is larger, indicating the significant lattice modulation effect on the $d$-$d$ hybridization, unlike Ni-$d_{\beta}$. They increase upon cooling to a certain temperature, below which they start decreasing (Data calculated above 1000 K are not shown here. But this is clearly shown in NdNiO₂). The temperature of maximum $D(\mathrm{i}\omega_{0})$ is proportional to the strength of the $d$-$d$ hybridization. As the lattice is larger, L_small $\rightarrow$ L_large, in LaNiO₂, the temperature of maximum $D(\mathrm{i}\omega_{0})$ of Ni-$d_{\alpha}$ decreases from above 1000 K to around 700 K, and $D(\mathrm{i}\omega_{0})$ of Ni-$d_{\alpha}$ and $\chi_{\mathrm{loc}}^{J_{Z}}$ of Ni-3$d$ become higher. The hybridized peaks of Ni-$d_{\alpha}$ and La-$d_{\beta}$ are together weakened and shifted to higher energy (see Supplementary Fig. S3 and S4). These indicate that as the lattice is larger, the $d$-$d$ hybridization between Ni-$d_{\alpha}$ and La-$d_{\beta}$ is weakened, making the DOS of Ni-$d_{\alpha}$ at the Fermi level higher.

The $d$-$d$ hybridization in NdNiO₂ is manifested in the same fashion as in LaNiO₂. As lattice sizes are larger L_small $\rightarrow$ L_large, the $d$-$d$ hybridization peaks of Ni-$d_{\alpha}$ and Nd-$d_{\beta}$ become weaker. As shown in Fig.3(c), $D(\mathrm{i}\omega_{0})$ of Ni-$d_{\alpha}$, and $\chi_{\mathrm{loc}}^{J_{Z}}$ of Ni-3$d$ for both LaNiO₂ and NdNiO₂ show the similar trend in response to the lattice modulations down to about 200 K, below which the $D(\mathrm{i}\omega_{0})$ of Ni-3$d$ and Nd-4$f$ increases due to the emerging Kondo effect (see Fig. 3 (c)). However, the temperatures of maximum $D(\mathrm{i}\omega_{0})$ of Ni-$d_{\alpha}$ and $\chi_{\mathrm{loc}}^{J_{Z}}$ of Ni-3$d$ in NdNiO₂ are much lower, about half, than those in LaNiO₂. As L_small $\rightarrow$ L_large in NdNiO₂, the temperature of maximum $D(\mathrm{i}\omega_{0})$ drops significantly from about 700 K to about 300 K. Thus the $d$-$d$ hybridization in NdNiO₂ is suggested to be much weaker than that in LaNiO₂. This is also described by the weaker hybridization function in NdNiO₂, as shown in Supplementary Fig. S7.

Discussion

We have shown that the electronic structure at Fermi level in bulk nickelates has 3D nature through the Kondo effect driven by Nd-4$f$ and Ni-3$d$ at low temperature, i.e., the Fermi surface of bulk NdNiO₂ consists of 3D multi-bands: i) Kondo band by Nd-4$f$, ii) Ni-$d_{\alpha}$, and iii) mixed bands of Nd-$d_{\beta}$, Ni-$d_{\beta}$ and Nd-$d_{\alpha}$. The third one can be lifted from the Fermi level by hole doping[37, 8]. According to these results and the experimental observation of insulating behaviors of non-superconducting Nd_0.8Sr_0.2NiO₂ bulk[34], we suggest that the hole doped NdNiO₂ still has 3D multi-band by the Kondo bands of the Ni-$d_{\alpha}$ and Nd-4$f$ at the Fermi level. As schematically described by Fig. 4, as the distance between Nd and Ni is larger, (i) the Kondo effect is suppressed or eliminated, thus the 3-dimensional Kondo cloud around Fermi-liquid-like Nd-4$f$ can be vanished, by which the quasi-2D-like Fermi surface is formed, (ii) the $d$-$d$ hybridization between Ni and rare-earth is weakened, by which the DOS of Ni-3$d$ at the Fermi level is increased.

The superconductivity of nickelates occurs only in films with the larger interlayer distance than that of bulk. In addition, the recent measure of significant angle-dependent $T_{\textrm{c}}$, also, emphasizes a more important role of quasi-2D bands comprised of Ni-3$d_{x^{2}-y^{2}}$ for the pair formations in nickelates[15]. A recent theoretical work also shows the presence of dominant pairing instability on the Ni-3$d_{x^{2}-y^{2}}$ 2D single band channel[38]. It is suggested that quasi-2D-like Fermi surface is essential for unconventional superconductivity in cuprates[3]. These observations, reflecting our finding about lattice modulation effect, suggest that the 2D framework, applied to the cuprates, may also be valid for the nickelates, and the superconductivity in the nickelates can be promoted by the 3D-to-2D transition.

Therefore, the concerted effect of the 3D-to-2D transition and the increased DOS achieved by lattice modulation can boost superconductivity. These can give explanations to the following three experimental observations of appearance (absence) of superconductivity for the Nd_0.8Sr_0.2NiO₂ film (bulk) with the large (small) $c$-axis lattice constant. We assume that these films and bulk have the same doping concentration, and only difference is in lattice size. First, the Sr doping arise the lattice modulation effect of the monotonic increment of interlayer distance from 3.28 Å to 3.42 Å upon zero to 25$\%$ Sr doping in Nd_1-xSr_xNiO₂/SrTiO₃ thin films, where superconducting dome appear for 0.125 $<$ $x$ $<$ 0.25[31]. The interlayer distance is $\sim$3.38 Å for Nd_0.8Sr_0.2NiO₂/SrTiO₃. Second, the superconductivity is observed in films with thickness ($c$-axis lattice constant) ranging from 4.6-nm ($\sim$3.42 Å) to 15.2-nm ($\sim$3.36 Å) in Nd_0.8Sr_0.2NiO₂/SrTiO₃[32]. Third, no superconductivity is observed in Nd_0.8Sr_0.2NiO₂ bulk[34]. Its interlayer distance is 3.33 Å, which is smaller than that of the films.

Our work sheds light on the superconductivity mechanism in nickelates and suggest that the superconductivity can be readily enhanced by engineering the 3D-to-2D transition through lattice modulation. It provides with an immediate route to the manipulation of the superconductivity.