A $d^{8}$ anti-Hund’s Singlet Insulator in an Infinite-layer Nickelate

The discovery of superconductivity, after 30 years of attempts, in hole-doped ‘infinite layer’ nickelates ${\cal R}$NiO₂ (${\cal R}$=lanthanide)H.Hwang2019 , has broadened the scope of high temperature superconductivity (HTS). Such nickelates have identical crystal structures and underlying $d^{9}$ metal atom configurations, yet nickelate superconductivity has been remarkably elusive. Although the achieved temperatures so farH.Hwang2020 ; Ariando2020 , T_c up to 15 K, are not yet HTS, the connection is clear. The nickelates have the same square planar transition metal-oxygen layers, and the same underlying formal oxidation state with a single hole in the $3d$ shell: Ni¹⁺ rather than Cu²⁺. In both systems, doping the $d^{9}$ configuration toward $d^{8}$ leads to superconductivity.H.Hwang2020 ; Ariando2020 Yet with these and other similarities, essential differences have been uncovered lee2004 ; choi_prr2020 ; arita2021 ; wen2021 ; review_our2021 .

An early study by some of the authors of the ‘infinite layer’ compound LaNiO₂ (Ni¹⁺, $d^{9}$) revealed substantial differences which emphasized that, while relations between the two ions are evident, Ni¹⁺ does not behave like Cu²⁺ in these structures lee2004 . When strong correlation effects are included in the band calculation, there was a strong tendency toward the formation of peculiar on-site ‘singlet’ spin character in a metallic background arising from spin compensation between the two $e_{g}$ orbitals. This behavior arises in spite of the formal spin-half (not spin $S$=0 or $S$=1) criterion of the formal oxidation state. The carriers at the Fermi level (and below – the hole states) consisted of roughly equal $d_{x^{2}-y^{2}}$ spin up, $d_{z^{2}}$ spin down. This incipient singlet leaves oxygen ions straying from the O^2- configuration, forming the metallic background. In the same sort of calculation for Cu²⁺, a spin-half antiferromagnetic (AFM) insulating state is obtained, consistent with experiment.

Another distinction from cuprates is that a $d^{8}$ nickelate Ba₂NiO₂(AgSe)₂ (BNOAS) has been reported bnoas_exp , and it contains an infinite-layer NiO₂ layer. The formally $d^{8}$ cuprate LaCuO₃ is a conventional although metastable metal, with samples being oxygen deficient karpinen and typically prepared under pressure. Other formally $d^{8}$ cuprates are rare. The magnetic character of BNOAS is enigmatic. Conventionally, the $d^{8}$ configuration should be a non-magnetic singlet, with two holes in the $d_{x^{2}-y^{2}}$ orbital, or a triplet, with spin parallel holes in each of the $e_{g}$ orbitals. The former has zero (spin) susceptibility, and the latter would have a Curie-Weiss susceptibility with AFM ordering at some temperature, supposing conventional antialigned neighbors.

What was observed by Matsumoto et al. bnoas_exp is (when zero field cooled) a constant value, on top of which is a rounded peak centered at T^∗=130 K with half width $\sim$10 K – apparently some magnetic reconstruction in an otherwise magnetically inert state. When field cooled, the susceptibility is higher below T^∗, and a weak peak is seen at ($\frac{1}{2},\frac{1}{2}$,0) in polarized neutron scattering at 5 K, suggesting antialigned neighboring moments. Matsumoto et al. suggested a canted AFM state of $S$=1 spins to account for the field cooling dependence below T^∗, but other aspects of the susceptibility remain to be explained. These are some of the questions we address.

The report of BNOAS has reinvigorated study of the two dimensional (2D) Kondo necklace model bnoas_ours , the 1D version of which was proposed by Doniach Doniach . The 2D model consists of a square lattice of two-orbital sites, with in-plane coupling $J$ between neighboring orbital₁ states and on-site Kondo coupling $K$ between orbital₁ and orbital₂. Wu et al. have applied the iPEPS (infinite projected entangled pair state) formalism to obtain the phase diagram of the anisotropic XXZ version tu_bnoas , finding spin-singlet, supersolid, and Bose-Einstein condensed phases for various ratios $K/J$. Finite temperature studies by Singh RRPSingh included the local $S=1$ excited states that are exchange coupled to neighboring $S=1$ states. This spin model leads to a ground state first-order transition between singlet and antiferromagnetic phases, with finite temperature behavior being amenable to cluster expansion techniques. Separately, Kitamine et al. have pursued design of $d^{8}$ nickelate materials kitamine2020 , focusing on $A_{2}$NiO${}_{2}X_{2}$ compounds with $A$ being a divalent alkaline earth cation and anions $X$=H, F, Cl, Br, I, the aim being to isolate the $d_{x^{2}-y^{2}}$ orbital and leave other $d$ orbitals just below the Fermi energy rather than to engineer magnetic behavior.

The organization of the paper is as follows. Section II provides the input to the theoretical work: crystal structure and computational methods. The non-magnetic electronic structure, including a comparison of $d^{8}$ and $d^{9}$ ions in infinite layer environments, is provided in Sec. III, and the magnetic calculations are introduced. In Sec. IV effects of correlations are introduced, and the peculiar band singlet that arises is discussed and its origin analyzed. Then some analysis of this anti-Hund’s singlet and a corresponding correlated Hartree-Fock wavefunction is given in Secs. V and VI, with the intent that it might shed light on the peculiar observed magnetic behavior of BNOAS. A Discussion and Summary is provided in Sec. VII.

BNOAS crystal structure. BNOAS is usefully considered as [Ba(AgSe)₂Ba]NiO₂ – an active $\infty$-layer Ni²⁺O₂ plane separated by thick Ba(AgSe)₂Ba blocking layers bnoas_exp . The space group is body-centered tetragonal $I4/mmm$ (#139) bnoas_exp , and the Ni ion has $4/mmm$ symmetry. As displayed in Fig. 1(a), the (AgSe)₂ blocking layer has the FeSe structure: Ag in the central layer is coordinated tetrahedrally by Se atoms, which in the next layers are neighbored by a plane of Ba²⁺ ions. Ni has no apical oxygen neighbors.

The compound is apparently metastable: it was synthesized under 7 GPa pressure at 850 ^∘C. This is reminiscent of the superconducting ‘infinite layer’ nickelates, which are only superconducting when grown in strained perovskite form using layer-by-layer synthesis techniques H.Hwang2019 , with O later removed topotactically. In BNOAS, strain supplied by the (AgSe)₂ layer leads to an anomalously large Ni-O separation of 2.105 Å, which Matsumoto et al.bnoas_exp argued should put BNOAS beyond an $S=0$ to $S=1$ transition at 2.05 Å. Our earlier results also indicated a change in electronic behavior bnoas_ours around 2.05 Å.

Theoretical methods. Our calculations are based at the density functional theory (DFT) level using the generalized gradient approximation (GGA) gga implemented in wien2k wien2k . In wien2k, we used the same input parameters in our previous studies of BNOAS bnoas_ours . The in-plane lattice constant is $a$=3.96 (4.21) Å  for LaNiO₂  (BNOAS). The latter anomalously large value is believed to be key to the unusual behavior of the Ni ion bnoas_exp ; bnoas_ours .

Correlation effects were included as conventionally done using the DFT+U approach ylvisaker2010 as in Ref. bnoas_ours , using GGA for the semi-local exchange-correlation effects. In DFT+U, Hund’s coupling was fixed at $J=0.7$ eV. For the magnetic calculations on BNOAS, the on-site Coulomb repulsion $U$=$7$ eV was chosen for most of the studies. We did check that results are fairly insensitive to somewhat smaller values of $U$. For example, at $U=5$ eV, the hopping parameters differed by at most 10 meV. The energy gap (which is not the Mott-Hubbard gap, see following sections) is small in this range of $U$, closing around $U_{c}\approx$5.5 eV (specifically, $U_{c}$= 5.2 eV and 5.8 eV in fplo (Full Potential Local Orbital)fplo and wien2k, respectively). We note that in wien2k $U$ and $J$ are applied within an atomic sphere. The sphere does not include quite all of the atomic orbital (whose tails are not well defined anyway), also the atomic function is distinct from the Wannier function (see below) that might be argued to be a more realistic ‘local orbital’. Both of these aspects would require, and perhaps justify, a somewhat smaller value of ‘atomic’ $U$ to achieve the same impact of on-site repulsion.

To construct the maximally localized Wannier functions, we used the wannier90 wan90 and wien2wannier wien2wan codes, projecting onto symmetry adapted Wannier functions. Two frozen energy windows were investigated: –5.7 eV to 1.3 eV (–7 eV to 3.5 eV) for 27 (37) starting orbitals, given below, for nonmagnetic (respectively magnetic, textured singlet) BNOAS. The parameters for BNOAS, included in Table I, are obtained for Ni $d$, O $p$, Se $(s),p$, and Ag $(s,p),d$ orbitals (the orbitals inside parentheses are the additional orbitals used for the singlet state).

The excellent quality of the Wannier bands for NM BNOAS are shown in Fig. 2; the corresponding result for magnetic BNOAS is displayed in Appendix A. For comparison, we also calculated Wannier parameters for NM LaNiO₂. For a fit with more flexibility compared with the previous results of Botana and Norman botana2020 where Ni $d$, O $p$, and La only $d_{z^{2}}$ orbitals were used, we used 23 starting orbitals of La $d,f$, Ni $d$, and O $p$ orbitals within an energy widow of –8.5 eV to 1.8 eV. The excellent representation of the bands is shown in Appendix A.

The character of doped holes and their relationship to superconductivity in HTS cuprates have been a central concern. The CT, versus MH, characterization of the energy gap leads to a large amount of Zhang-Rice (ZR) singlet character zhangrice in doped holes and the electron removal spectrum. The ZR singlet is a coupling of spin-half Cu spin with an oppositely directed linear combination of planar O $p_{\sigma}$ orbitals of $d_{x^{2}-y^{2}}$ symmetry. Holes in a Mott insulator would have simple $d_{x^{2}-y^{2}}$ character, with small (negligible) O character. The $d^{8}$ Ni ion in BNOAS presents a different reference state, yet singlet versus triplet character becomes a central feature, albeit on-site rather than involving oxygen.

Including $U$ in the electronic structure calculation to account for on-site repulsion in a mean field picture, and thereby obtaining magnetic character, causes important changes in the electronic structure and the derived band parameters, as shown in Table 1 under the two rightmost BNOAS columns. First, it should be expected that all Ni $d$ orbitals will be shifted: strongly downward if occupied, somewhat upward if unoccupied. These differences will appear most apparently in the on-site parameters, but a secondary but still significant effect can be seen in Wannier hopping amplitudes.

BNOAS presents in the calculated ground state not the trivial singlet of a completely unoccupied $d_{x^{2}-y^{2}}$ orbital, but a singlet with internal character as described in the previous section: there is exchange coupling within the NiO₂ layer and weaker coupling between layers, leading to the Kondo sieve model bnoas_ours for the insulating spin system. The results that we present for BNOAS are for this singlet phase, which persists in our calculations for $U>4$ eV, using two different all-electron DFT codes wien2kwien2k and fplofplo and with various starting spin configurations. In BNOAS, for each spin direction all five majority orbitals are occupied while one minority orbital is unoccupied. This peculiar singlet that we find to be persistent differs in which $e_{g}$ spin-orbital is unoccupied. Because the spin splitting between majority and minority is so large, the mean $d$ site energy in the magnetic state becomes less relevant.

For the Ni $t_{2g}$ orbitals, the in-plane $d_{xy}$ orbital is more strongly bound than the other two by 0.3-0.4 eV, an effect of environment and bonding-type hybridization in-plane. The $e_{g}$ orbitals are of primary interest. The mean $e_{g}$ on-site energies (majority, minority; see Table I) are nearly degenerate. Individually, the exchange (up minus down) site energies are around 8 eV, equal to $U$=7 eV plus some contribution from Hund’s $J$ and hybridization effects. The mean $d$ sites energies are 1.0-1.3 eV lower for the singlet case, because there are more occupied majority orbitals (moved down) than minority orbitals (moved up). Regarding oxygen, only the $p_{\sigma}$ orbitals show a significant spin dependence, with majority being 0.2 eV lower than minority.

For the hopping amplitudes of BNOAS, only one shows a significant difference due to singlet formation relative to the NM state. This one is the $d_{z^{2}}$-$p_{\sigma}$ hopping, normally considered to be insignificant in square-planar geometry. While it is -0.28 eV in NM BNOAS, in the singlet case it becomes -0.7 eV for both spins. The effect is to spread $d_{z^{2}}$ character into band positions (bonding and antibonding) where it would otherwise be negligible.

Origin of the Singlet. The underlying feature of significance of the GGA band parameters for anti-Hund’s singlet formation is the effective degeneracy of the two $e_{g}$ orbital energies. One could consider a unitary transformation among the $e_{g}$ orbitals but reducing the symmetry without any evident usefulness does not appear attractive. The anti-Hund’s orientation for spin-half moments costs $J_{H}/2$ = 0.35 eV in exchange energy, for the value of $J_{H}\approx J$=0.70 eV we have used in our DFT+$U$ calculations (see Sec. II). The value of pair spin $S$=0 or $S$=1 (i.e. the relative spin density orientation) affects the band structure and hence the kinetic energy, the charge and spin densities, and the correlation energy, with some contribution from the purely Coulombic (charge density) terms. The energy sources favoring the anti-Hund’s orientation are not clear. It is worth noting that in LaNiO₂, where the $e_{g}$ subsplitting is of the usual sign with a value of 0.7 eV, even then in the large $U$ regime a similar singlet character (opposite orientation of $e_{g}$ spins) became dominant lee2004 . Studies varying the lattice constant, hence the interatomic and interionic separations, could be helpful in understanding the mechanism of singlet formation.

Spin singlet physics in 2D systems has played a large role in cuprate HTS, and similarities of nickelates to cuprates have encouraged extension of this concept to superconducting nickelates. The dominant concept has been that of ZR singlet zhangrice , building on the characterization of undoped cuprates as CT insulators rather than explicit MH insulators. The latter would involve only $d$ orbital reoccupation by doping or low energy excitations, while in CT insulators doped holes will lie primarily on neighboring O $p$ orbitals. A linear combination on four O sites will form a molecular orbital of ($d_{x^{2}-y^{2}}$) symmetry, and a hole in this molecular orbital will pair with the central spin-half Cu²⁺ to form a spin singlet, the ZR singlet. While most studies have assigned nickelates as having more Mott than CT character, the ZR singlet concept is sprinkled through the recent nickelate literature. The infinite layer nickelates are more often considered from a Hund-Hubbard viewpoint chang2019 . We find that electronic structure parameters suggest that BNOAS lies in the vicinity of a charge transfer to Mott-Hubbard (CT-MH) crossover, making the basic ‘physics’ harder to pinpoint.

Zhang, Yang, and Zhang (ZYZ) have proposed ZYZ2020 a generalized singlet construct. Noting that even undoped NdNiO₂ has a small concentration of Nd $5d$ carriers mixing somewhat with Ni $d$ states, they have proposed that ZR-like combinations of these carriers form a local molecular orbital and form a singlet. Each occupied Nd singlet removes a spin from the Ni square lattice, and it is proposed that the few percent provided by Nd inhibits AFM ordering of the Ni spins. These ZYZ singlets might also account for observed low-T transport behavior. The model was built on the Mott insulator viewpoint before the CT insulator picture of nickelates became more prevalent. Photoemission spectroscopies are now indicating charge or spin ordering Tam_CDW_2021 ; Krieger_CDW-SDW_2021 ; Rossi_CO_2021 in ${\cal R}$NiO₂ (${\cal R}$ a lanthanide atom), complicating any connection between such singlets and inhibition of geometrical symmetry breaking.

Singlet phases arise in the 2D Shastry-Sutherland spin model in certain parameter regimes, but they have a different character and origin than those discussed above. On a square lattice of $S$=$\frac{1}{2}$ spins, a specific regime of exchange parameters can result in the formation and ordering of singlets from spins on neighboring sites or across the diagonal of the square Haravifard2014 . Such exchange processes enrich the phase diagram.

In this paper it has been shown by comparison in what ways electronic structure parameters differ between $d^{8}$ BNOAS and $d^{9}$ LaNiO₂. Both share the infinite-layer local environment of the Ni ion, but the further environment (Ba²⁺ versus La³⁺ etc.) affects several parameters. Separately, the strongly increased in-plane a lattice constant, with related perpendicular changes, has a strong influence on the physics.

We acknowledge communications with A. S. Botana, M. R. Norman, and V. Pardo, and conversations with R. R. P. Singh, and feedback from Wei-Lin Tu and Hyun-Yong Lee. H.S.J. and K.W.L. were supported by National Research Foundation of Korea Grant No. NRF-2019R1A2C1009588. W.E.P. was supported by NSF Grant No. DMR 1607139.