Ab initio materials design of superconductivity in d⁹ nickelates

Unconventional superconductivity in strongly correlated electron systems such as the cuprates,¹ iron-based superconductors,² Sr₂RuO₄,³ and Na_xCoO₂·yH₂O⁴ has been one of the central issues in condensed matter physics. While we still lack a generally accepted explanation for their pairing mechanism, there is one characteristic common feature in these compounds: They have a crystal structure with stacked two-dimensional layers. For such layered materials, various families have been synthesized by inserting different layers or changing the stacking patterns.

It is of great interest to note that the recently discovered d⁹ nickelate superconductors^5–11, RNiO₂ (R = La, Pr, Nd) also share this common feature (for recent reviews, see, e.g., Refs. 12–20). Namely, they consist of the NiO₂ layer and block layer, forming the infinite-layer structure. The maximum transition temperature T_c is about 15 K.^5–11 It is further enhanced to 20–30 K by changing the substrate or applying pressure.^21–23 A theoretical phonon calculation has shown that the electron–phonon coupling is too weak to explain the experimental T_c.²⁴ Thus, the conventional mechanism is unlikely, and various unconventional mechanisms have been discussed theoretically.^25–38 On the experimental side, all the reports so far are consistent in that the pairing symmetry is not a simple s-wave, although there exists a discrepancy in the proposed symmetry.^39–41 

More recently, a quintuple-layer nickel superconductor Nd₆Ni₅O₁₂ has been synthesized.⁴² This has proven that superconductivity can be realized in nickelates other than infinite-layer compounds. Therefore, it is interesting to consider different layered structures and think of manipulating the electronic structure of nickelate superconductors. Indeed, a variety of theoretical materials design has been performed.^43–45 In particular, it has been shown that we can control the charge transfer from the NiO₂ layer to the block layer.⁴³ 

In Fig. 1, we show the crystal structure, phonon dispersion, and electronic band dispersion of NdNiO₂.^17,24 We see that the electronic structure of NdNiO₂ is similar to that of the cuprates in which only the 3$dx2−y2$ orbital among the five 3d orbitals contribute to the formation of the Fermi surface. The dispersion can be represented by a simple tight-binding model on the two-dimensional square lattice. However, there are also several distinct differences between NdNiO₂ and the cuprates: In NdNiO₂, the oxygen 2p level is far below the Fermi level, and the system belongs to the Mott–Hubbard regime^46–48 in the Zaanen–Sawatzky–Allen phase diagram.⁴⁹ Thus, the hybridization between the oxygen 2p and Ni 3d orbitals is weaker than that in the cuprates that reside in the charge-transfer regime. In addition, there is substantial charge transfer from the NiO₂ layer to the block layer,^50,51 and the Nd 5d states and interstitial s state form Fermi pockets around the Γ and A points.

In Ref. 43, three of the present authors have shown that there are dynamically stable layered d⁹ nickelates for which the charge transfer from the NiO₂ layer to the block layer is suppressed. Especially, the charge transfer is completely suppressed for RbCa₂NiO₃ and A₂NiO₂Br₂. There, only the Ni 3$dx2−y2$ orbital forms a two-dimensional Fermi surface, whose shape is very similar to that of the cuprates. Analyzing these nickelate compounds will provide new insights into the similarity/difference between cuprate and nickelate superconductivity.

For these nickelates, effective low-energy models in terms of the Wannier functions⁵² for the Ni 3$dx2−y2$ orbital and O 2p orbitals have been derived using the constrained random phase approximation (cRPA).^53,54 In Ref. 55, the magnetic exchange coupling J in the NiO₂ plane has also been estimated. While it has been known that the cuprate superconductors have a large exchange coupling of ∼140 meV,⁵⁶ these nickelates also have a significant exchange coupling as large as 80–100 meV.⁵⁷ 

In this paper, we extend these studies and calculate the phase diagram of RbCa₂NiO₃ and A₂NiO₂Br₂ (A is a cation with a valence of 2.5+) by means of the dynamical vertex approximation (DΓA).^58–60 The DΓA was also applied to study the phase diagram of the infinite-layer nickelate NdNiO₂³¹ and quintuple-layer nickelate Nd₆Ni₅O₁₂,⁶¹ which allows us to compare the phase diagram among nickelates. In RbCa₂NiO₃ and A₂NiO₂Br₂, differently from the infinite-layer nickelate NdNiO₂, superconductivity emerges by a carrier doping into an antiferromagnetic insulator. The superconducting transition temperature (T_c) shows a dome-like shape as a function of doping. Therefore, RbCa₂NiO₃ and A₂NiO₂Br₂ may provide us with a unique opportunity to study the electron–hole asymmetry of the Mott insulating states in the nickelate superconductors.

The structure of the paper is as follows: In Sec. II, we discuss the phonon dispersion and electronic band dispersion of RbCa₂NiO₃ and A₂NiO₂Br₂. We show that these nickelates do not have imaginary phonon modes, indicating that they are dynamically stable. Thus, although the crystal structures of these nickelates may not be the most stable structure, they are one of the meta-stable structures. Regarding the electronic structure, in contrast with the case of RNiO₂, the charge transfer from the NiO₂ layer to the block layer in RbCa₂NiO₃ and A₂NiO₂Br₂ is absent. In Sec. III, we show the results of ab initio derivation of the effective low-energy model (i.e., the single-orbital Hubbard model) for these nickelates. We see that RbCa₂NiO₃ and A₂NiO₂Br₂ are more strongly correlated than NdNiO₂. In Sec. IV, we solve the single-orbital Hubbard model derived in Sec. III using the DΓA. We obtain the phase diagram for RbCa₂NiO₃ and A₂NiO₂Br₂ and compare it with that of NdNiO₂ and Nd₆Ni₅O₁₂. In Sec. V, we summarize the results obtained in this study.

Let us move on to the materials design of new nickelate superconductors.⁴³ Following the idea for the cuprate superconductors,⁶² four types of block layers and six types of crystal structures for the nickelate superconductors have been proposed. The strategy to suppress the charge transfer from the NiO₂ layer to the block layer is the following: If we make the energy level of the block layer sufficiently higher than that of the Ni 3$dx2−y2$ orbital, the charge transfer will not occur. Here, it should be noted that the level of Ni 3$dx2−y2$ orbital is considerably higher than that of the Cu 3$dx2−y2$ orbital in the cuprates. Thus, we should properly choose elements in the 1–3 groups (such as Sr and La) that strongly favor closed-shell electronic configuration. Meanwhile, such materials keep having NiO₂-plane structures and similar (Ni¹⁺) valence states. Then, the hopping structure and the charge-transfer energy on NiO₂-planes would not change much. Following this strategy, the dynamical stability of 57 materials was examined in Ref. 43. While 16 compounds do not have imaginary phonon modes, we hereafter focus on RbCa₂NiO₃ and A₂NiO₂Br₂ as representative compounds.

In Figs. 2 and 3, we show the crystal structure, phonon dispersion, and electronic structure of RbCa₂NiO₃ and A₂NiO₂Br₂, respectively. For the calculation of the phonon dispersion, the frozen-phonon method with a 2 × 2 × 2 or 4 × 4 × 2 supercell was employed. We see that calculation for the 2 × 2 × 2 supercell is enough to examine the presence/absence of imaginary modes. Since the calculation of the convex hull of ternary compounds is extremely expensive, we will not discuss whether these compounds are thermodynamically stable. However, from the results in Figs. 2(b) and 3(b), we can safely conclude that they are at least dynamically stable. Here, it is interesting to note that there is an experimental report of the synthesis of the d⁸ nickelate Sr₂NiO₂Cl₂,⁶³ which has the same crystal structure as A₂NiO₂Br₂. The phonon band width is about 60–70 meV, which is similar to that of NdNiO₂ (see Fig. 1).

In Figs. 2(c) and 3(c), we show the electronic structure of RbCa₂NiO₃ and A₂NiO₂Br₂, respectively. Starting from these results, one can derive a single-orbital tight-binding model for the Ni 3$dx2−y2$ orbital.⁴³ We highlight the band dispersion of the effective model with green dotted curves. As we discuss in Sec. III, the dispersion can be represented by a simple tight-binding model on the 2D square lattice. We also show the band minimum of the block-layer band with open circles. We see that the band minimum is higher than the Fermi level so that the charge transfer from the NiO₂ layer to the block layer does not occur. Therefore, the effective Coulomb interaction between the Ni 3$dx2−y2$ electrons tends to be stronger than that in RNiO₂ since the screening from the block layer becomes weaker.

In Table I, we list the values of the transfer integrals between the nearest neighbor sites (t), next nearest neighbor sites (t′), third nearest neighbor sites (t″), and Hubbard U.^24,43,55 We see that the ratio between U and t in RbCa₂NiO₃ and A₂NiO₂Br₂ is substantially larger that in NdNiO₂.

Next, let us analyze the correlation effect of these effective low energy models for studying the superconductivity property. Based on Table I, we studied the two-dimensional Hubbard model with hoppings: t′/t = −0.28, t″/t = 0.13, U = 9.5t for RbCa₂NiO₃ and t′/t = −0.26, t″/t = 0.12, U = 10.5t for A₂NiO₂Br₂. Here, we employ the DΓA,^58–60 which is one of the diagrammatic extensions of the dynamical mean field theory (DMFT).^64–66 The DΓA can capture the strongly correlated effect and the long-range fluctuation effect beyond DMFT, both of which are essential for describing the layered unconventional superconductivity with strong correlations. The method has succeeded in describing the phase diagram of the unconventional superconductivity,^31,67 e.g., cuprates and nickelates. In calculation details, we follow the previous paper about the infinite layer nickelates.³¹ Please see that paper or review articles^60,68,69 for more information about the method.

In Fig. 4, we show the DΓA result of leading eigenvalues of the linearized gap function in the d-wave (singlet, even-frequency) channel against the hole doping level. Here, we also show NdNiO₂ results taken from Ref. 31, where the charge transfer effect is also taken into account. The eigenvalue is often used as the measure of T_c since it usually monotonically increases as decreasing the temperature, and the phase transition occurs when the eigenvalue reaches the unity. We can see that both RbCa₂NiO₃ and A₂NiO₂Br₂ show the dome structure of the superconductivity instability peaked at around 15%–20% doping like cuprates and the infinite-layer nickelates. On the other hand, eigenvalues are lower than the infinite-layer system³¹ or the quintuple-layer system.⁶¹ Some theoretical studies suggest that the infinite-layer nickelate material resides in the strong-coupling regime of superconductivity.^25,31 Indeed, the T_c change in recent pressure and strain experiments^21–23 can be reasonably understood by the change of the effective interaction strength. Both materials considered here have further large U/t values due to the weaker screening effect from the relatively distanced surrounding bands, and then, we obtained weaker superconductivity instability. The relation between the effective interaction U/t and T_c is further analyzed recently,⁷⁰ which suggests palladate compounds for a higher transition temperature.

We also show the temperature dependence of eigenvalues for RbCa₂NiO₃ in Fig. 5. At the optimal doping regime at n = 0.85, we obtained the transition temperature T_c ∼ 0.0045t ∼ 18 K by extrapolating the DΓA result with the fit function λ(T) = a − b ln(T).^31,71 Here, we just used the U/t value obtained from cRPA calculations. If we employ a somewhat larger U/t value like that in Ref. 31, T_c will further decrease.

Furthermore, we show the filling dependence of the spectrum and spin fluctuation in Fig. 6. As a proxy of the spectrum, we here show the momentum dependence of the imaginary part of the Green function at the lowest Matsubara frequency $−1πIG(k,ωn=π/β)$ and the momentum averaged Green function at τ = β/2. Without doping, we can see the similarity with cuprates: the spectral weight at the Fermi level is strongly suppressed by the self-energy, and spin fluctuation becomes strongly enhanced so that Néel ordered antiferromagnetism will stabilize once we consider the weak three dimensionality.

In conclusion, we performed a calculation based on the dynamical vertex approximation and obtain the phase diagram of RbCa₂NiO₃ and A₂NiO₂Br₂. In these materials, the charge transfer from the NiO₂ layer to the block layer is absent, and the effective Coulomb interaction between Ni 3$dx2−y2$ electrons is stronger than that in RNiO₂ or Nd₆Ni₅O₁₂. We obtained the dome shaped superconductivity instability like infinite layer nickelates. While the estimated T_c is lower than RNiO₂,³¹ we here demonstrate that we can control the charge transfer and correlation effect among nickelate compounds by changing block layers. Furthermore, RbCa₂NiO₃ and A₂NiO₂Br₂ may provide us with a unique opportunity to study the electron–hole asymmetry of the Mott insulating states in the nickelate superconductors.

We are grateful for fruitful discussions with Karsten Held, Liang Si, Paul Worm, Terumasa Tadano, Takuya Nomoto, and Kazuma Nakamura. We acknowledge the financial support by Grant-in-Aids for Scientific Research (JSPS KAKENHI) [Grant Nos. JP20K22342 (M.K.), JP21K13887 (M.K.), JP20K14423 (Y.N.), JP21H01041 (Y.N.), and JP19H05825 (R.A.)] and MEXT as “Program for Promoting Researches on the Supercomputer Fugaku” (Basic Science for Emergence and Functionality in Quantum Matter—Innovative Strongly-Correlated Electron Science by Integration of “Fugaku” and Frontier Experiments—) (Grant No. JPMXP1020200104). A part of the calculation has been done on MASAMUNE-IMR of the Center for Computational Materials Science, Tohoku University.

The authors have no conflicts to disclose.

Motoharu Kitatani: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Writing – original draft (lead). Yusuke Nomura: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Writing – original draft (supporting). Motoaki Hirayama: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Writing – original draft (supporting). Ryotaro Arita: Conceptualization (equal); Funding acquisition (lead); Project administration (lead); Resources (lead); Supervision (equal); Writing – original draft (supporting); Writing – review & editing (lead).

The data that support the findings of this study are available within the article.