General trends of superconducting pairing and magnetic correlations in the Ruddlesden-Popper nickelate 𝑚-layered superconductors La 𝑚+1 Ni 𝑚 O 3𝑚+1
Here, we report a comprehensive theoretical analysis of the Ruddlesden-Popper layered nickelates La 𝑚+1 Ni 𝑚 O 3𝑚+1 (𝑚 = 1 to 6) under pressure. These materials have recently received significant attention due to the discovery of superconductivity in some nickelates under pressure. Our results suggest that, while these Ruddlesden-Popper layered nickelates display many similarities, they also show noticeable differences. One of the common features of La 𝑚+1 Ni 𝑚 O 3𝑚+1 is that the electronic states near the Fermi level are mainly contributed by Ni 3𝑑 orbitals, slightly hybridized with O 2𝑝 orbitals. The Ni 𝑑 3𝑧 2 −𝑟 2 orbitals display bonding-antibonding, or bonding-antibonding-nonbonding, characteristic splittings, depending on the even or odd number of stacking layers 𝑚. In addition, the ratio of the in-plane interorbital hopping between 𝑑 3𝑧 2 −𝑟 2 and 𝑑 𝑥 2 −𝑦 2 orbitals and in-plane intraorbital hopping between 𝑑 𝑥 2 −𝑦 2 orbitals was found to be large in La 𝑚+1 Ni 𝑚 O 3𝑚+1 (𝑚 = 1 to 6), and this ratio increases from 𝑚 = 1 to 𝑚 = 6, suggesting that the in-plane hybridization will increase as the layer number 𝑚 increases. In contrast to the dominant 𝑠 ± -wave state driven by spin fluctuations in the bilayer La 3 Ni 2 O 7 and trilayer La 4 Ni 3 O 10 , two nearly degenerate 𝑑 𝑥 2 −𝑦 2 -wave and 𝑠 ± -wave leading states were obtained in the four-layer stacking La 5 Ni 4 O 13 and five-layer stacking La 6 Ni 5 O 16 . The leading 𝑠 ± -wave state was recovered in the six-layer material La 7 Ni 6 O 19 with slightly higher calculated pairing strength 𝜆 than that of the 𝑑 𝑥 2 −𝑦 2 -wave state. All this evidence suggests that both 𝑠 ± -wave and 𝑑 𝑥 2 −𝑦 2 -wave channels are strongly competing in the high-order niceklates based on our random-phase approximation calculations. In general, at the level of the random-phase approximation treatment, the superconducting transition temperature 𝑇 𝑐 decreases in stoichiometric bulk systems from the bilayer La 3 Ni 2 O 7 to the six-layer La 7 Ni 6 O 19 , despite the 𝑚-dependent dominant pairing. Both in-plane and out-of-plane magnetic correlations are found to be quite complex. Within the in-plane direction, we obtained the peak of the magnetic susceptibility at 𝐪 = (0.6𝜋, 0.6𝜋) for La 5 Ni 4 O 13 (𝑚 = 4) and La 7 Ni 6 O 19 (𝑚 = 6) and at 𝐪 = (0.7𝜋, 0.7𝜋) for La 6 Ni 5 O 16 (𝑚 = 5). Along the out-of-plane direction, four layers are coupled as ↓−↑−↑−↓ in La 5 Ni 4 O 13 , five layers are coupled as ↑−↑−↓−↑−↑ in La 6 Ni 5 O 16 , and six layers are coupled as ↑−↓−↓−↑−↑−↓ in La 7 Ni 6 O 19 .