Electronic structure, self-doping, and superconducting instability in the alternating single-layer trilayer stacking nickelates La$_3$Ni$_2$O$_7$
Yang Zhang Ling-Fang Lin Adriana Moreo Thomas A. Maier Elbio Dagotto





Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996, USA

Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996, USA
Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA

Computational Sciences and Engineering Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA

Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996, USA
Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA




Motivated by the recently proposed alternating single-layer trilayer stacking structure for the nickelate
La_3Ni_2O_7, we comprehensively study this system using ab initio and random-phase approximation techniques. Our analysis unveils similarities between this novel La_3Ni_2O_7 structure and other Ruddlesden-Popper nickelate superconductors, such as a similar charge-transfer gap value and orbital-selective behavior of the e_g orbitals. However, different from other Ruddlesden-Popper nickelate superconductors, we do not observe any obvious reconstruction of the Fermi surface from ambient conditions (Cmmm phase) to high pressures (P4/mmm phase). Pressure primarily increases the bandwidths of the Ni e_g bands, suggesting an enhancement of the itinerant properties of those e_g states. Furthermore, the d_3z^2-r^2 orbital also has a layer-selective behavior because the antibonding-bonding-nonbonding splitting can only be obtained in the trilayer. In addition, we observe a “self-doping” effect from the trilayer to the single-layer sublattices and this effect will be enhanced by overall electron doping. Moreover, we find a leading d_x^2-y^2-wave pairing state that is restricted to the single-layer. Because the effective coupling between the single layers is very weak – due to the non-superconducting trilayer in between – this suggests that the superconducting transition temperature T_c in this structure should be much lower than in the bilayer structure.


Electronic structure, self-doping, and superconducting instability in the alternating single-layer trilayer stacking nickelates La_3Ni_2O_7
    Elbio Dagotto
    April 29, 2024
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Introduction.– The Ruddlesden-Popper (RP) perovskite nickelate systems have recently attracted considerable attention due to the discovery of superconductivity under pressure in La_3Ni_2O_7 (327-LNO) with a d^7.5 configuration <cit.>, opening a remarkable and challenging avenue for the study of nickelate-based unconventional high-T_ c superconductivity <cit.>.


For the 327-LNO system, the original experiments reported a bilayer (BL) stacking structure <cit.>, where superconductivity was found in a very broad pressure range, with a high transition temperature up to 80 K <cit.>. This occurs after a first-order structural transition from the ambient-pressure Amam phase with distorted NiO_6 octahedra to the high-pressure Fmmm phase without tilting of those oxygen octahedra (see Fig. <ref>(a)). Based on the BL structure, many follow-up experimental <cit.> and theoretical <cit.> efforts were presented to understand the interesting physical properties and possible superconducting pairing mechanism in this compound. Very recently, J. Li et. al. <cit.> observed the Meissner effect of the superconducting state using the ac magnetic susceptibility, with the superconducting volume fraction being around 50 %.




Furthermore, an interesting alternating single-layer (SL) – trilayer (TL) stacking structure was proposed experimentally (with the same chemical formula 327-LNO) as an alternative to the BL stacking structure (see Fig. <ref>(a)), by several groups <cit.>. Interestingly, signatures of superconductivity under pressure were also reported in another RP  nickelate La_4Ni_3O_10 with only a trilayer stacking structure <cit.>. However, superconductivity was absent in SL La_2NiO_4 under pressure <cit.>.


Similarly to 327-LNO (BL), the corner-sharing NiO_6 octahedra were also found to be strongly distorted in TL La_4Ni_3O_10 at ambient conditions <cit.>, leading to a monoclinic P2_1/c structure (No. 14). Increasing pressure suppresses the distortion of the NiO_6 octahedra, yielding a high-symmetry I4/mmm phase without the tilting of oxygen octahedra around 15 GPa, where the signature of superconductivity was found with a T_c of about 20-30 K <cit.>. However, in 327-LNO (SL-TL), the NiO_6 oxygen octahedra do not display any substantial tilting in neither the ambient-pressure Cmmm nor the high-pressure P4/mmm phases <cit.>.


Considering these developments, several interesting questions naturally arise: What are the similarities and differences between the 327-LNO (SL-TL) and other RP nickelate superconductors under pressure? What interesting physics can be obtained in this alternating SL and TL stacking 327-LNO, as compared to the first proposed bilayer 327-LNO structure? Could we obtain superconductivity in 327-LNO (SL-TL) and is the pairing mechanism the same as in other RP nickelate superconductors?


Electronic structure of 327-LNO (SL-TL) – In the 327-LNO (BL) system, due to the “dimer” physics induced by its bilayer geometry <cit.>, the d_3z^2-r^2 orbital splits into antibonding and bonding states, while the d_x^2-y^2 orbital remains decoupled among planes, as shown in Fig. <ref>(b), resulting in an orbital-selective behavior <cit.>. Considering the nominal valence  Ni^2.5+, close to two electrons would occupy the bonding states of the d_3z^2-r^2 orbital and approximately one electron stays in the d_x^2-y^2 band <cit.> (see Fig. <ref>(b)).


Because 327-LNO (SL-TL) has an alternating SL and TL stacking structure, the d_3z^2-r^2 orbital from the TL sublattice also splits into antibonding, nonbonding, and bonding states due to the “trimer” physics induced by the TL structure <cit.>, as displayed in Fig. <ref>(b). Meanwhile, the d_x^2-y^2 orbital does not participate in the formation of antibonding and bonding states, as well as the e_g states from the SL sublattice. Considering an average of 1.5 electrons in e_g states per Ni in the 327-LNO,  two electrons would occupy the bonding states of the d_3z^2-r^2 orbital and one electron enters the nonbonding state in the TL sublattice, while an extra 1.5 electrons would remain in the d_x^2-y^2 state [Fig. <ref>(b)]. For the SL sublattice, 1.0 and 0.5 electrons would occupy the states of the d_3z^2-r^2 and d_x^2-y^2 orbitals, respectively. Being a SL, these are, obviously, only nonbonding states. Then, the e_g orbitals of 327-LNO (SL-TL) have both orbital-selective and layer-selective characteristics.





To better understand the physical properties of 327-LNO (SL-TL), we have calculated the electronic structure based on the experimental crystal structure under pressure <cit.> using first-principles density functional theory (DFT) <cit.>. The states near the Fermi level  mainly arise from Ni 3d orbitals hybridized with O p-states, which are located at lower energies than the Ni 3d states, indicating a charge-transfer picture similar to other nickelates <cit.>. As expected, the d_3z^2-r^2 orbital from the TL sublattice splits into three antibonding, nonbonding, and bonding related bands in the 327-LNO (SL-TL) band structure, as shown in Fig. <ref>(c). In addition, the d_x^2-y^2 orbital from the TL sublattice and the e_g orbitals from the SL sublattice all remain nearly decoupled between different planes.


Similarly as it occurs in 327-LNO (BL), due to the strong in-plane hybridization of the e_g states <cit.> electrons transfer between the d_3z^2-r^2 and d_x^2-y^2 orbitals in 327-LNO (SL-TL) as well, leading to a noninteger electronic population in both orbitals. A crucial observation is that different from the TL La_4Ni_3O_10, the bonding band of the d_3z^2-r^2 states from the TL sublattice does not touch the Fermi surface (FS). Therefore, the TL states do not create a Fermi surface γ pocket in 327-LNO (SL-TL) (see Fig. <ref>(d)). Instead, a γ pocket does exist in the TL La_4Ni_3O_10 context <cit.>), basically due a different crystal field. Note that the γ_1 sheet shown in Fig. <ref>(d) arises entirely from the SL sublattice, and should not be confused with the γ pocket of the BL and TL systems. The TL sublattice states give rise to two β sheets and an α_1 sheet of mixed e_g orbital character, while the σ pocket is made up of the d_3z^2-r^2 orbital in the 327-LNO (SL-TL), similar to La_4Ni_3O_10. Moreover, the SL sublattice also gives rise to the α_2 sheet displayed in Fig. <ref>(d).


Pressure effect – To understand better the different phases of 327-LNO (SL-TL) and 327-LNO (BL), we also studied the energies when changing the ratio V/V_0, where V_0 is the conventional-cell volume of the P4/mmm phase of 327-LNO (SL-TL) at 16 GPa. As shown in Fig. <ref>(a), 327-LNO (BL) always has lower energy than 327-LNO (SL-TL) when changing the ratio V/V_0 from 0.9 to 1.10. In addition, as the volume increases, corresponding to a decrease in pressure, the Amam phase has lower energy than the Fmmm phase in the 327-LNO (BL) structure, in agreement with a previous experiment <cit.>. For 327-LNO (SL-TL), the P4/mmm and Cmmm phases have nearly degenerate energies in the entire V/V_0 region that we studied because the distortions from the high symmetry P4/mmm phase are quite small.


In addition, we calculated the electronic structure of the Cmmm phase of the 327-LNO (SL-TL) crystal structure based on the primitive unit cell, by using the experimental results without pressure. As displayed in Fig. <ref>(b), the band structure of the Cmmm phase looks quite similar to the P4/mmm phase under pressure. By comparing the band structures of the ambient and high-pressure cases, the bandwidths of the Ni e_g bands increase upon pressure by ∼ 16% at 16 GPa, suggesting an enhancement of itinerant properties of the e_g states. Furthermore, we also do not observe any obvious reconstruction of the FS between the P4/mmm and Cmmm phases, as displayed in Fig. <ref>(c). Taken together, all these results suggest that the structural phase transition Cmmm-P4/mmm should not induce the stabilization of superconductivity in 327-LNO (SL-TL).


Tight-binding model and self-doping effect – To understand the low-energy physics under pressure, we constructed an eight-band e_g-orbital tight-binding model for the SL and TL sublattice states for the P4/mmmm phase of 327-LNO (SL-TL) at 16 GPa, with overall filling n = 6, including the longer-range hoppings between the SL and TL sublattices. The entire hopping file used in the present work can be found in Supplementary Material <cit.>, where additional details are provided for the tight-binding model.





As shown in Figs. <ref>(a) and (b), the tight-binding band structure and FS fit very well with the DFT results discussed in the previous section. By using the maximally localized Wannier functions (MLWFs) method based on the WANNIER90 package <cit.>, we obtained the crystal-field splitting of the e_g orbitals for different Ni sites in the P4/mmm phase of 327-LNO (SL-TL), as summarized in Fig. <ref>(c). The Ni1 site from the SL sublattice has a larger crystal-field splitting (Δ∼ 0.666 eV) than the Ni2, Ni3, and Ni4 sites from the TL sublattice. In the TL sublattice, the outer layers (Ni2 and Ni4 sites) have lower on-site energies than the middle layer (Ni3 site), similarly to the case of the TL for La_4Ni_3O_10. Furthermore, we find that the ratios t^12/t^22  are 0.425, 0.511, and 0.548 for the Ni1, Ni2/Ni4 and Ni3 sites, respectively (where the in-plane inter-orbital hopping between the d_3z^2-r^2 and d_x^2-y^2 orbitals is t^12, and the intra-orbital  d_x^2-y^2 hopping is t^22). This  indicates that the in-plane hybridization of the e_g orbitals is stronger in the TL sublattice.


Considering the average electronic density of the e_g states (1.5 per Ni) in 327-LNO (SL-TL), the calculated electronic densities are approximately 1.770, 1.436, and 1.360 for Ni1, Ni2/Ni4 and Ni3 sites, respectively. This indicates a “self-doping” effect, where electrons are transferred from the TL to the SL sublattice. Note that here the sum of the electronic populations of Ni2, Ni3, and Ni4 (∼ 4.23), locates the system in the superconducting region of La_4Ni_3O_10, where d_3z^2-r^2 γ pockets are obtained according to Ref. <cit.>. The primary intuitive reason why TL La_4Ni_3O_10 has a γ pocket, and thus superconductivity, but TL from 327-LNO (SL-TL) does not, is because of the crystal-field value: it is larger in TL from 327-LNO (SL-TL) than in TL La_4Ni_3O_10.


Moreover, the outer Ni layers (Ni2 and Ni4 sites) of the TL sublattice have more electrons than the middle layer (Ni3 site). In addition, we also calculated the electron occupations for different Ni sites by changing the overall filling n from 5.0 to 7.0, as displayed in Figs. <ref>(d) and (e). For electronic doping, extra electrons move to the SL Ni1 site with an increased difference in electron densities between the SL and TL sublattices, suggesting an enhanced “self-doping” effect. Specifically, in the hole-doping region, the SL Ni1 site loses more electrons than other sites in the TL sublattice. As shown in Fig. <ref>(e), for the doping region we studied, the electronic occupation of the d_3z^2-r^2 orbitals is larger than that in the d_x^2-y^2 orbital.







RPA pairing tendencies – To analyze the superconducting pairing tendencies, we used multi-orbital random phase approximation (RPA) calculations to assess the tight-binding model including the SL and TL sublattices. The RPA calculations of the pairing vertex are based on a perturbative weak-coupling expansion in the local Coulomb interaction matrix, which contains intra-orbital (U), inter-orbital (U'), Hund's rule coupling (J), and pair hopping (J') terms  <cit.>. The pairing strength λ_α for pairing channel α and its corresponding pairing structure g_α( k) are obtained by solving an eigenvalue problem of the form

    ∫_FS d k' Γ( k -  k') g_α( k') = λ_α g_α( k),

where the momenta k and k' are restricted to the FS, and Γ( k - k') is the irreducible particle-particle vertex. In the RPA approximation, the dominant term entering Γ( k-k') is the RPA spin susceptibility χ_s( k-k').



By solving the eigenvalue problem in Eq. <ref> for the RPA calculated pairing vertex Γ( k-k') of 327-LNO (SL-TL) at n = 6.0, we find that the d_x^2-y^2 channel is the leading pairing instability caused by spin fluctuations, followed by subleading d_xy- and g-wave solutions, as displayed in Figs. <ref>(a-c). Here we used U=0.5 eV, U'=U/2, J=J'=U/4. We find that larger U causes a spin-density wave instability in the SL sublattice with q∼ (π, π, 0) arising from the near perfect nesting of the SL α_2 and γ_1 sheet Fermi surface states (magenta regions in Fig. <ref>(d)). Despite the small value of U, the pairing strength λ=0.152 of the leading d_x^2-y^2 state is already significant. As a comparison, for the bilayer 327 structure, we obtained λ∼ 0.2 at 15 GPa
for U=0.8 eV <cit.>. Moreover, we note that for all three leading pairing states, the pairing gap is only finite on the regions of the Fermi surface made up of states from the SL sublattice (magenta regions in panel (d) in Fig. <ref>), and negligible in the TL dominated regions. This suggests that pairing, while strong, is restricted to the SL sublattice, and caused by strong intra-SL-sublattice spin fluctuations between the α_2 and γ_1 Fermi surface sheets. We believe that it is the absence of a TL γ-pocket that pairing in the TL sublattice is weak. Based on these results, we speculate that a possible three-dimensional superconducting transition involving the SL system would be suppressed to very low temperatures, because the effective coupling between different SL sublattices is negligible due to the presence of the TL in between.



Conclusions.– In summary, we presented a systematic study of the newly reported 327-LNO (SL-TL), with an alternating single-layer and trilayer stacking structure, by using DFT and RPA calculations. We found a large charge-transfer energy, a common character with other nickelates. In addition, more importantly, we did not observe any obvious reconstruction of the Fermi surface from the ambient-pressure Cmmm to the high-pressure P4/mmm phases. By changing the volume ratio V/V_0 from 0.9 to 1.10, corresponding to the effect of pressure, we found that 327-LNO (BL) always has lower energy than 327-LNO (SL-TL). Furthermore, the e_g orbitals of 327-LNO (SL-TL) have both orbital-selective and layer-selective behaviors. Moreover, we observed a “self-doping” effect from the TL to the SL sublattice, which is enhanced by electron doping. Based on RPA calculations, we obtained a leading d_x^2-y^2-wave pairing state with sizable pairing strength λ. However, this pairing state is restricted to the SL sublattice and, therefore, a three-dimensional superconducting state will be suppressed to very low temperatures due to the weak coupling between the SL sublattices, which will reduce the coherence between those layers. These results suggest that a possible superconducting T_c in 327-LNO (SL-TL) may not be sufficiently high to match the experimentally observed T_c in 327-LNO. Consequently, we believe that a bilayer stacking structure remains the most likely in 327-LNO with T_c ∼ 80 K.

This work was supported by the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES), Materials Sciences and Engineering Division.


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