Structural transitions, octahedral rotations, and electronic properties of A 3 Ni 2 O 7 rare-earth nickelates under high pressure

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Very recently, signatures of superconductivity with a high T c ∼ 80 K were reported in the bilayer nickelate La 3 Ni 2 O 7 subject to ∼ 14-43.5 GPa external pressure in a diamond anvil cell [1].This structure is a member of the Ruddlesden-Popper family, A n+1 Ni n O 3n+1 (n = 2), and contributes another exciting perspective to this field, specifically due to the formal valence of Ni 2.5+ .Independent experimental works have already claimed confirmation of high-temperature superconductivity in this system in a similar pressure range [36,37].
This raises the fundamental questions (i) how the structural transition and the accompanying lack of octahedral rotations are related to the emergence of superconductivity, and (ii) if we can reduce the critical pressure by chemical precompression (a key concept in superconducting hydrides [58,59]) via isoelectronic A-site variation.In oxides, octahedral rotations are known to be sensitively related to the electronic structure and the degree of electronic correlation: In the isomorphic bilayer compound Sr 3 Ru 2 O 7 , for example, similar octahedral rotations of around ∼6 • [60,61] are suppressed by Mn doping, which drives the emergence of antiferromagnetic order [60,62,63] and a low-temperature Mott insulating phase [64,65].Thus, the avoidance of octahedral relaxations may be coupled to unconventional superconductivity, although the exact mechanism remains elusive.
Here we provide a comprehensive and consistent exploration of the structural and electronic properties in A 3 Ni 2 O 7 bilayer nickelates (A = La-Lu, Y, Sc) as a function of hydrostatic pressure (0-150 GPa) from first principles including a Coulomb repulsion term.We compile a structural phase diagram with particular emphasis on the orthorhombic distortion (b/a ratio), octahedral anisotropy, and octahedral rotations.Surprisingly, the results establish chemical and external pressure as two distinct and counteracting control parameters, which limits the perspectives of chemical precompression in this system.We trace this phenomenon back to the enhancement of octahedral rotations with reducing A-site ionic radius.The response of the lattice parameters to exter-nal pressure is found to be highly anisotropic.In La 3 Ni 2 O 7 at ∼ 20 GPa, we observe an orthorhombic-to-tetragonal transition to an I4/mmm phase at variance with recent x-ray diffraction data, which points to yet unresolved complexities near the onset of superconductivity, e.g., electron-or holedoped samples due to variations in the oxygen stoichiometry, and suggests a careful reassessment of the so-far proposed superconductivity mechanisms.The critical pressure associated with this transition, which coincides universally with vanishing octahedral tilts, is found to increase quadratically over the rare-earth series.For A = Nd-Lu, Y, Sc, two novel structural phases are uncovered at ambient conditions that are characterized by the emergence of significant in-plane c + octahedral rotations as well as in-plane bond disproportionations and exhibit a surprising pressure-driven electronic reconstruction involving the rotation of the Ni d z 2 orbital.The successive quenching of the distinct rotational degrees of freedom by pressure demonstrates that the potential energy landscape of the in-plane octahedral rotations is significantly shallower than for the octahedral tilts.Moreover, we highlight unexpected correlations between T c and the in-plane Ni-O-Ni bond angles for La 3 Ni 2 O 7 and discuss their possible relation to superconductivity.Finally, by disentangling the involvement of basal versus apical oxygen ions in the Fermi surface, we identify Tb 3 Ni 2 O 7 as interesting candidate for superconductivity at ambient pressure.

II. METHODOLOGY
We performed first-principles simulations in the framework of density functional theory (DFT [66]) as implemented in the Vienna Ab initio Simulation Package (VASP) [67,68], employing a wave-function cutoff of 520 eV.Exchange and correlations were described by using the generalized gradient approximation as parameterized by Perdew, Burke, and Ernzerhof [69].The rare-earth 4f electrons were frozen in the core.We focus on the nonmagnetic phase here, but discuss additional results on the magnetic interactions in the Appendix.Static correlation effects were considered within the DFT+U formalism [70,71], employing an effective U = 3 eV at the Ni sites, in line with previous nickelate work [9,10,19,27,[72][73][74][75][76][77] and a recent analysis of the optical spectrum of La 3 Ni 2 O 7 [78].
To account for octahedral rotations and in-plane bond disproportionations, the A 3 Ni 2 O 7 bilayer nickelates (A = La-Lu, Y, Sc) were modeled by using 24-atom unit cells.In this geometry, the Brillouin zone was sampled employing 8 × 8 × 8 Monkhorst-Pack ⃗ k-point grids [79] in conjunction with a Gaussian smearing of 5 mRy.We confirmed that these parameters provide converged energies and lattice parameters.Accurate densities of states were obtained on 12 × 12 × 12 ⃗ kpoint grids.The compounds can equivalently be described by orthorhombic 48-atom unit cells [Fig.1(a)]; for these, we used 8 × 8 × 2 ⃗ k-point grids.
The lattice parameters a, b and c (given with respect to the more convenient orthorhombic representation in the following) and the internal ionic positions were accurately optimized in each case in DFT+U under zero and finite external pressure, reducing ionic forces below 1 mRy/a.u.

III. STRUCTURAL PROPERTIES OF A3Ni2O7 RARE-EARTH NICKELATES UNDER HIGH PRESSURE
Over the rare-earth series and the considered pressure range (p = 0-150 GPa), we observe an unexpected richness of the structural properties of A 3 Ni 2 O 7 (Fig. 1).Our comprehensive study puts us in position to compile an ab initio structural phase diagram [Fig.1(d)], which underlines the possible potential of high-pressure experiments to realize exotic new quantum states in the bilayer nickelates in particular and in correlated transition metal oxides in general.We will explore the phase diagram and the underlying data step by step in the following and discuss their implications for superconductivity.

A. Pressure-induced orthorhombic-to-tetragonal transition in La3Ni2O7
We begin by analyzing the pressure dependence of the lattice parameters of La 3 Ni 2 O 7 [Fig.1(a,b)].The fully ab initio relaxed results for a, b and c agree nicely with the experimental lattice parameters refined from x-ray diffraction (XRD) [1] between 0-10 GPa.In particular, the difference between a and b, i.e., the degree of orthorhombic distortion, is accurately captured.This further corroborates that the Hubbard U = 3 eV employed here is appropriate.Consistent with previous experimental [80,81] and theoretical assessments [1,52], we identify a structure of Cmcm space group (or Amam, No. 63) at low pressure, referred to as phase II in the following [Fig.1(d)].It is characterized by finite octahedral tilts θ and b/a > 1, where we define b as the axis along which the tilts are expressed [Fig. 1(e)].Simultaneously, octahedral rotations in the basal plane are absent (ϕ = 0, c 0 ).Notably, without octahedral rotations of any kind (F mmm, I4/mmm), the assignment of a and b is arbitrary.
Between 10-20 GPa, the experimental lattice parameters a and b display the unambiguous trend to converge, accompanied by a plateau in the c curve.However, a sudden increase can be observed at ∼ 20 GPa, after which they continue to decrease monotonically with a finite separation of around 0.075 Å.It has been suggested earlier that the octahedral tilts are quenched in this pressure range (θ = 0), corresponding to inter-layer Ni-O-Ni bond angles of 180 • and resulting in a geometry with orthorhombic F mmm space group (No. 69) [1].This geometry was explored in detail in subsequent work [38, 40-42, 44, 45, 47, 48, 50, 52].
Surprisingly, we observe a clear orthorhombic-totetragonal transition at p ∼ 20 GPa, resulting in a structure of I4/mmm symmetry (phase I, space group No. 139) rather than the so-far reported F mmm geometry.The enthalpy difference between the F mmm (lattice parameters from XRD [1]) and the I4/mmm geometry obtained here, exemplarily for La 3 Ni 2 O 7 at 30 GPa, amounts to ∆H = 90 meV/Ni, which additionally corroborates our findings.Moreover, we confirmed the orthorhombic-to-tetragonal transition by variation of U at the Ni site, additional application of U to the La 4f and 5d states, and different patterns of the octahedral rotations (c 0 , c + , c − ).We recently became aware of higher-resolution x-ray experiments that report the observation of the I4/mmm phase above p ∼ 19 GPa [82].
We demonstrate below for La 3 Ni 2 O 7 that the key features of the electronic structure are preserved between the I4/mmm and F mmm phases.Nevertheless, these observations suggest so-far unresolved complexities at the onset of superconductivity which may be key in identifying the underlying mechanism: On the one hand, the application of pressure may not be uniform, as indicated by the role of different pressure-transmitting media employed in the diamond anvil cells [1,36,37].On the other hand, these results may indicate that the experimental samples are electron or hole doped, most probably due to variations in the oxygen stoichiometry.There is considerable evidence that the electronic properties are extremely sensitive to small changes in the oxygen vacancy concentration [80,83].Due to the strong coupling of the charge and lattice degrees of freedom in correlated transition metal oxides [84][85][86][87][88], this would alter their structural response to pressure and potentially lead to the experimentally observed finite orthorhombic distortion at higher pressure.In that case, the so-far proposed s ± superconductivity mechanism needs a careful reassessment.

B. Can we substitute external by chemical pressure?
Next, we explore the role of chemical pressure by isoelectronic variation of the A-site element.The ionic radii of the considered elements range from 1.172 (La 3+ ) to 1.001 (Lu 3+ ), as well as 1.04 (Y 3+ ) and 0.885 (Sc 3+ ).As a measure of this chemical pressure, we employ the Goldschmidt tolerance factor, a traditional descriptor for perovskite oxides: where t = 1 corresponds to size-balanced A and Ni sites.It is reasonable to order the A-site elements with decreasing t [Fig.1(d)].Here we find consistently t < 1. Can we exploit this strategy to lower the critical external pressure driving the structural phase transition and thus facilitate the emergence of superconductivity in the bilayer nickelates?
We can directly conclude from the phase diagram in Fig. 1(d), which is not symmetric, that this hypothesis does not hold.The transition from the Cmcm to the I4/mmm phase occurs at critical pressures that increase monotonically from p ∼ 20 (A = La, Ce), 30 (Pr), and 40 GPa (Nd) to p ∼ 125-150 GPa (Tb), as shown in Fig. 1(d), with a parabolashaped phase boundary as a function of t [see also the b/a ratios in Fig. 2(a) and the octahedral tilts θ in Fig. 3(a)].While finalizing this manuscript, we became aware of another very recent work that suggests a similar shape of this phase boundary, albeit classifying the high-pressure phase as F mmm [89].If superconductivity is linked to this structural phase transition, our results imply that continuously increasing external pressures are required as A progresses through the rare-earth series.In the following, we will identify enhanced octahedral rotations as the impeding physical mechanism.
These results unambiguously show that chemical and external pressure generally constitute two distinct control parameters that independently and non-interchangeably allow the designing of the quantum state in bilayer rare-earth nickelates.

C. Anisotropic pressure response of the lattice parameters
Figure 1(c) shows that the cell height c varies overall from ∼ 15 (A = Sc at 150 GPa) to 20.6 Å (La at 0 GPa) and generally reduces with decreasing ionic radius of the A-site element.Some pronounced discontinuities can be observed at the phase boundaries, e.g., between phases II and III [Fig.1(c)].Surprisingly, a counter-intuitive increase of c can be identified with increasing pressure for ∼ 0-50 GPa in Dy For the in-plane lattice parameters a and b, which range from 4.41 (A = Sc at 150 GPa) to 5.95 Å (Tm at 0 GPa), a simple decreasing A-site dependence can only be identified for the early rare-earth metals up to Nd [Fig.1(c)].Between Pm and Tm, even a sharp upturn in the average of a and b can be observed, which also translates to the basal Ni-Ni distances (not shown).
This coincides with the stabilization of the novel structural phases III and IV [Fig.1(d)].Both have Cmc2 1 symmetry (space group No. 36) and are characterized by finite ferrodistortive (c + ) octahedral rotations arising as an additional degree of freedom (ϕ > 0; Fig. 3).These are accompanied by substantial A-site displacements in the basal plane [Fig.1(e)], which are typical for this octahedral rotation pattern [87,88] and a useful fingerprint in transmission electron microscopy.The transition to the Cmcm phase II is associated with pronounced jumps of a and b [Fig.1(c)], resulting generally in a reversal of the b/a ratio [Fig.2(a)].
We see that the response of the lattice parameters to uniform external pressure is highly anisotropic.The cell height c is generally compressed more strongly than a and b.This can be traced back to a partial accommodation of the pressure by the more elastic bilayer separation.For example, we find that the latter reduces from 6.4 to 5.34 Å for 0-150 GPa in La 3 Ni 2 O 7 , which clearly exceeds the concomitant compression of the bilayer height from 3.91 to 3.54 Å (each measured between the Ni planes).
Inspection of the phase diagram in Fig. 1(d) reveals that the boundaries between phases II-III and III-IV are roughly parabola-shaped as a function of t, similar to the boundary between phases I-II.This further corroborates the opposing impact of external pressure versus A-site variation.Beyond the phases I-IV [top-right corner of Fig. 1(d)], we found that these counteracting forces lead to bizarre deformations of the NiO 6 octahedra (not shown), which indicates that the compounds become unstable under such extreme conditions.
The richness of the structural phase diagram and the unexpected transitions identified in the later rare-earth nickelates provide additional opportunities for e.g.superconducting phases.We will shed more light on this aspect below.

D. Octahedral anisotropy and bond disproportionation
The anisotropic pressure response of the lattice parameters is directly related to shape modifications of the NiO 6 octahedra, which in turn determine the spatial orientation of the two Ni e g orbitals and their relative occupation, i.e., the orbital polarization [76,77,90].We measure the deviations from an ideal, regular octahedron by defining The octahedral anisotropy R is a measure of the apical versus basal extension of the octahedra [Fig.2(b)].It ranges overall from 5.3% (A = La at 0 GPa) to −3.5% (Tm at 0 GPa).We find R > 0 for most rare-earth nickelates, implying an elongation of the Ni octahedra in the z ∼ c direction and thus a Ni d z 2 orbital that points in the apical direction, whereas the Ni d x 2 −y 2 orbital is oriented in the basal plane.
Notably, the broken octahedral connectivity between the bilayer slabs results in an internal asymmetry of each NiO 6 octahedron, visible in the structural side views in Fig. 1(e).Specifically, for La 3 Ni 2 O 7 , the Ni-O bond lengths involving the 'outer' apical oxygen ions (pointing into the structural gap) are generally enhanced [1,41].Simultaneously, we find  that these Ni-O bonds are reduced more strongly due to pressure than their 'inner' analogs.For instance, we observe 2.31, 1.97 Å (0 GPa), 2.08, 1.90 Å (30 GPa), and 1.91, 1.81 Å (100 GPa) for the outer and inner apical Ni-O bonds, respectively.This mechanism facilitates the vertical elongation of the octahedra and thus the occupation of the Ni d z 2 orbital.
The early rare-earth elements exhibit a monotonically decreasing trend of R with increasing pressure, i.e., the anisotropy is reduced [Fig.2(b)].This is in line with the more rapid decrease of c with respect to a and b.Specifically, for La 3 Ni 2 O 7 , we find that R decreases from 5.3% at 0 GPa to 3.3% at 150 GPa.The central rare-earth elements exhibit a more complex pressure dependence [Fig.2(b)].Sharp discontinuities at the phase transition II-III can be observed, and correlations with the orthorhombic distortion (b/a ratio) are clearly visible, particularly in phase II [Fig.2(a)].
At zero pressure, R decreases rapidly over the rare-earth series [Fig.2(b)].Intriguingly, it becomes negative for A = Ho-Lu, Y, Sc, which corresponds to vertically compressed octahedra (phase IV).The respective compounds show a pressureinduced transition from R < 0 to R > 0 at critical pressures ranging from ∼ 10-75 GPa.This behavior is highly distinct from all other bilayer rare-earth nickelates.Fig. 2(c) quantifies the in-plane disproportionation of the O-Ni-O distances (Q).We observe that it vanishes in phases I and II.In phase III, where R > 0, typical values are finite but small, e.g., −0.06% (A = Nd) and 0.79% (Tb) at zero pressure.Surprisingly, Q becomes very pronounced in phase IV, which coincides with R < 0, and reaches 9% at A = Tm at zero pressure.This corresponds to cigar-shaped octahedra that alternate in the plane in conjunction with c + octahedral rotations (Fig. 3), a pattern that closely resembles LaMnO 3 [91,92].Most importantly, this coincides with a considerable electronic reconstruction involving the rotation of the Ni d z 2 orbital into the basal plane.Intriguingly, we see that this phase can be lifted by pressure of around 10-40 GPa [Fig.2(c)].Such extreme tunability of the geometry and the electronic structure is highly promising and deserves further exploration.In the Cmcm phase II, where ϕ = 0, we find that θ ranges from ∼ 2-8 • .In the Cmc2 1 phases III and IV, ϕ and θ are both enhanced to ∼ 4-15 • .With increasing pressure, the in-plane rotations ϕ, if present, are quenched first (for A = Nd-Tm at 10-75 GPa), which corresponds to the boundary between phases II and III.For Lu and Sc, the octahedral rotations are so pronounced that we observe finite values up to 150 GPa.Subsequently, the octahedral tilts θ are quenched at yet higher pressure values (for A = La-Tb at 20-150 GPa).Intriguingly, we find that these vanishing tilts coincide universally with the orthorhomic-to-tetragonal transition from phase II to phase I.

E. The pivotal role of octahedral rotations
This demonstrates that external pressure generally reduces or even quenches the octahedral rotations, whereas chemical pressure rather enhances them.The subsequent suppression of the distinct rotational degrees of freedom indicates that different energy scales are involved, with a significantly shallower potential energy landscape related to ϕ than to θ.
The superconducting transition has been so far associated with a straightening of the inter-layer (apical) Ni-O-Ni bond angles towards 180 • [1].The respective top-right panel in Fig. 3 1(d)].Surprisingly, we observe an overall maximum in the in-plane Ni-O-Ni bond angles for A = La at 20 GPa, i.e., near the experimental onset of superconductivity (marked by the arrow).For higher pressure, the in-plane bond angles are enhanced again, which correlates with the experimentally observed reduction of Tc.(b) Structural impact of the in-plane octahedral rotations ϕ in A3Ni2O7 at p = 0 GPa.While absent for A = La-Pr, ferrodistortive c + rotations stabilize for Nd-Lu, Y, Sc and induce a reversal from b/a > 1 to b/a < 1.Moreover, they couple strongly to the octahedral tilts θ and enhance them considerably for the later rare-earth compounds (t < 0.82).
we observe a value of 168 • for La 3 Ni 2 O 7 , in perfect agreement with previous work [1].The late rare-earth nickelates exhibit considerably smaller values (i.e., more pronounced bond angles), reaching even below 140 • for Lu and Sc.
Motivated by the superconducting infinite-layer nickelates, which exhibit in-plane Ni-O-Ni bond angles of 180 • [2, 10, 19, 21], we additionally explore these quantities for the bilayer nickelates.They directly impact the basal Ni 3d x 2 −y 2 -O 2p x,y hybridization and thus the electronic structure at the Fermi energy [see also Fig. 4(a) below] including the Ni e g orbital polarization as well as possible superexchange mechanisms.Intriguingly, the comprehensive perspective provided in Fig. 3(a), top-left panel, unveils an overall maximum for A = La at 20 GPa, i.e., close to the experimental onset of superconductivity.For higher pressure, the in-plane bond angles are enhanced again, which we find to surprisingly correlate with the experimentally observed reduction of T c [1,36,37].It is important to note here that the in-plane Ni-O-Ni bond angles can deviate from 180 • despite θ = ϕ = 0, since the basal oxygen ions show an increasing tendency towards a uniform inwards relaxation under high pressure that does not correspond to finite octahedral rotations.These observations suggest that the in-plane Ni-O-Ni bond angles are an important aspect in understanding the superconducting phase in bilayer rare-earth nickelates.
Figure 3(b) analyzes the structural impact of the in-plane octahedral rotations ϕ in A 3 Ni 2 O 7 at zero pressure, i.e., highlighting the chemical effect of A-site variation.While absent for A = La-Pr, ferrodistortive c + rotations stabilize for A = Nd-Lu, including Y and Sc, with a considerable energy difference ∆E ∼ −0.6 eV/Ni relative to the metastable c 0 case.We see that they are strongly coupled to the b/a ratio and induce a reversal from b/a > 1 to b/a < 1, which for antifer-rodistortive c − rotations occurs only for Lu and Sc and without octahedral rotations (c 0 ) only for Sc.Moreover, they couple strongly to the octahedral tilts θ and enhance them considerably for the later rare-earth nickelates (phase IV, t < 0.82), an effect that is clearly absent for c − and c 0 .This demonstrates that the correct description of the octahedral rotations is key for obtaining accurate structural properties for A = Nd-Lu, Y, Sc.

IV. ELECTRONIC STRUCTURE OF A3Ni2O7 RARE-EARTH NICKELATES UNDER HIGH PRESSURE
Finally, we explore how the structural observations we discussed above relate to the electronic properties of A 3 Ni 2 O 7 rare-earth nickelates.Figure 4(a) displays the total density of states (DOS) at the Fermi energy as a function of the A-site element and the external pressure, which is an important indicator for enhanced superconducting properties.Furthermore, Fig. 4(a) disentangles the relative contributions from the Ni, basal oxygen, and apical oxygen ions.This allows us to track the composition of the Fermi surface, particularly variations in the basal versus apical oxygen involvement, which also reflects the Ni 3d x 2 −y 2 versus 3d z 2 contributions, respectively.
Substantial correlations of all four panels with the distinct structural phases established above can be identified.Along the transition from phase III to II, as well as for A = La-Pr at ∼ 10 GPa, we observe a sudden enhancement in the total DOS from roughly 3.5 to 6.5 States/eV.This is accompanied by a reduction of the basal oxygen contributions from ∼ 15% to 10%, corresponding to the Ni 3d x 2 −y 2 -O 2p x,y system, and a concomitant increase of the apical oxygen contributions from ∼ 5% to 11%, related to the Ni 3d z 2 -O 2p z hybrid states.Si-   1(d)].Noteworthy are the high contributions from the apical oxygen ions in phases I and II.Intriguingly, Tb3Ni2O7 at zero pressure presents a considerably higher DOS at the Fermi level (∼ 8.5 States/eV) than La3Ni2O7 reaches at around 100 GPa (∼ 7.2 States/eV) in addition to a sizable involvement of the apical oxygen ions.(b) Evolution of the total DOS as a function of the external pressure for selected compounds.An overall pressure-induced broadening of the O 2p-derived valence band can be observed, as well as of the Ni eg manifold located at the Fermi energy.Characteristic peaks clearly shift to lower energies, e.g., the Ni t2g peak visible between −2 and −1 eV at zero pressure, or the oxygen peak around −5.5 eV (A = La-Tb).The feature directly below the Fermi energy (marked by the arrow) corresponds to the lower set of Ni 3d z 2 -O 2pz hybrid states [Fig.5(a)].
multaneously, the absolute Ni contributions increase as well, even though we see a slight reduction of the relative Ni contributions, which implies that the overall relevance of oxygen increases.
It is surprising that the enhanced involvement of apical oxygen clearly correlates with phase II (ϕ ∼ 0) rather than with phase I (θ ∼ 0).Nevertheless, within phase I, the apical oxygen contribution is generally further boosted and reaches its maximum > 12% for Nd at 150 GPa.In sharp contrast, in phase III and particularly IV the basal ions dominate the oxygen contributions to the Fermi surface.An exception are the unexpectedly high apical oxygen contributions arising along the transition from phase III to IV, i.e., at the verge to the strongly bond-disproportionated state which is accompanied by in-plane Ni 3d z 2 orbital order and vertically compressed octahedra discussed above [Fig.2(b,c)].
The total DOS in La 3 Ni 2 O 7 ranges from ∼ 3.5 States/eV at 0 GPa to ∼ 7.2 States/eV at 100 GPa.By screening the A-site elements at p = 0 GPa, we identify even higher values particularly for Tb 3 Ni 2 O 7 (∼ 8.5 States/eV).This compound also exhibits the characteristic reversal from basal to apical oxygen involvement that usually appears exclusively under finite pressure.The maximum of the total DOS is observed for Tm 3 Ni 2 O 7 at 0 GPa.However, the contributions from the Ni and the apical oxygen ions are relatively low for this compound, and its band structure (not shown) differs considerably from that of e.g.La 3 Ni 2 O 7 due to the Ni e g electronic reconstruction.
Figure 4(b) tracks the evolution of the total DOS as a function of the external pressure for a selection of representative compounds.We observe an overall pressure-induced broaden-ing of the O 2p-derived valence band.Characteristic peaks are clearly shifted to lower energies with increasing pressure, e.g., the Ni t 2g peak visible between −2 and −1 eV at 0 GPa, and, even more strongly, the oxygen peak visible between −4.5 and −5.5 eV at 0 GPa.
The Ni e g manifold is located around the Fermi energy.We see that its band width is considerably reduced across the rare-earth series, whereas finite external pressure broadens it.This observation reflects once again the counteracting nature of these two control parameters.The feature directly below the Fermi level has Ni 3d z 2 -O 2p z character [see also Fig. 5(a)].Figure 4(b) demonstrates the pressure-induced occupation of these states for A = La, Pr, and Nd.Surprisingly, for Tb 3 Ni 2 O 7 , we observe that they are already partly occupied at 0 GPa.
Finally, the DOS of Y 3 Ni 2 O 7 and Lu 3 Ni 2 O 7 has a distinct shape around the Fermi energy since the Ni 3d z 2 orbital is oriented in the plane at 0 GPa (phase IV), as we discussed above in the context of Fig. 2(c).It is interesting to follow the evolution of these curves as pressure drives the structural transition to phase III, which uncovers a concomitant electronic reconstruction in the e g manifold, i.e., a re-alignment of the Ni 3d z 2 orbital with the vertical axis.The high apical oxygen values at the phase boundary render this transition even more compelling, and A = Ho at ∼ 10 GPa as well as A = Y-Tm at ∼ 20 GPa emerge as further candidates for future in-depth exploration.
Figure 5(a) shows for the prototypical La 3 Ni 2 O 7 system that the Ni 3d z 2 bands are split into an occupied lower and an empty upper set, separated by a band gap of ∼ 0.25 eV at 0 GPa.Simultaneously, the system is metallic due to highly dispersed Ni 3d x 2 −y 2 states.Both Ni e g orbitals hybridize substantially with each other around the M point [38] and, in addition, present an overall hybridization with the O 2p states.Application of external pressure enhances the energy difference between the two Ni 3d z 2 sets, but simultaneously also the band width of the 3d x 2 −y 2 and 3d z 2 states, which results in a reduction of the band gap.Moreover, the lower 3d z 2 set crosses the Fermi energy, and a fraction of its charge is transferred to the 3d x 2 −y 2 bands.Thus, a 'self-doping' hole pocket around the Γ point appears [Fig.5(b); folded back from the M point of the primitive cell] [1, 38-41, 50, 52], which plays a key role in the suggested s ± superconductivity mechanism [39,40,[50][51][52].We furthermore see an increase in Ni 3d z 2 orbital weight specifically in the α Fermi sheet.Notably, the Fermi surface of pressurized I4/mmm La 3 Ni 2 O 7 strongly resembles previous reports based on the F mmm geometry.Thus, moderate orthorhombic distortions rather impact details of the electronic structure.
The reduction of the octahedral anisotropy R with pres- This results in enhanced (reduced) DFT+U occupation numbers of the Ni 3d x 2 −y 2 (3d z 2 ) orbitals.These findings are consistent with recent reports based on a model study [52] and demonstrate explicitly that the orbital polarization can be tuned by external pressure.While the formal Ni 2.5+ (d 7.5 ) configuration implies that 1.5 electrons per Ni ion occupy the e g manifold, the consistently higher values observed here highlight the involvement of the oxygen system, reminiscent of the d 8 L configuration characteristic of rare-earth nickelates [32].Finally, we find a Fermi surface with a quasi identical topology as in pressurized La 3 Ni 2 O 7 in Tb 3 Ni 2 O 7 at zero pressure [Fig.5(b)], intriguingly despite the presence of pronounced octahedral rotations (space group Cmc2 1 ).We highlighted this compound already above due to its considerably enhanced DOS at the Fermi energy and a concomitant strong involvement of the apical oxygen ions [Fig.4(a)].Therefore, we suggest this bilayer nickelate as a candidate for superconductivity at ambient pressure.Investigations of possible superconducting pairing in this system would be highly interesting.

V. SUMMARY
The structural and electronic properties of A 3 Ni 2 O 7 bilayer nickelates (A = La-Lu, Y, Sc) and their pressure dependence (0-150 GPa) were investigated by performing first-principles simulations including a Coulomb repulsion term, with particular emphasis on the role of orthorhombic distortion (b/a ratio), octahedral anisotropy, and octahedral rotations.
The lattice parameters were found to exhibit a highly anisotropic response to external pressure.Surprisingly, in La 3 Ni 2 O 7 at ∼ 20 GPa, we observed an orthorhombicto-tetragonal transition to an I4/mmm phase at variance with recent x-ray diffraction data, which points to yet unresolved complexities near the onset of superconductivity, e.g., electron-or hole-doped samples owing to variations in the oxygen stoichiometry.Due to the sensitivity of particularly the Ni 3d z 2 -derived states and their energy relative to the Fermi level, this calls for a careful reassessment of the superconductivity mechanisms so-far proposed for the undoped compound.We showed that this transition coincides universally with vanishing octahedral tilts.The associated critical pressure increases quadratically over the rare-earth series, which we traced back to the enhancement of octahedral rotations with reducing A-site ionic radius.This establishes chemical and external pressure as two distinct control parameters in these complex oxides that independently and noninterchangeably allow the designing of their quantum state.
We compiled an ab initio structural phase diagram, which unveils two novel phases for A = Nd-Lu, Y, Sc at ambient conditions that are characterized by the emergence of significant in-plane c + octahedral rotations as well as in-plane bond disproportionations and exhibit a surprising pressure-driven electronic reconstruction involving the rotation of the Ni 3d z 2 orbital.The consecutive quenching of the distinct rotational degrees of freedom by pressure demonstrates that the potential energy landscape of the in-plane octahedral rotations is significantly shallower than for the octahedral tilts.
Moreover, we found unexpected correlations between T c and the in-plane Ni-O-Ni bond angles for La 3 Ni 2 O 7 , which determine the basal Ni 3d x 2 −y 2 -O 2p x,y hybridization.An overall maximum near the onset of superconductivity indicates enhanced superexchange interactions and promotes the in-plane bond angles as an important aspect in understanding the superconductivity mechanism in bilayer rare-earth nickelates.
Finally, by disentangling the contributions of basal versus apical oxygen states at the Fermi level, we identified Tb 3 Ni 2 O 7 as an interesting candidate for superconductivity at ambient pressure, with a considerably higher density of states at the Fermi energy and a Fermi surface similar to pressurized La 3 Ni 2 O 7 .This is even more astonishing since our results show that the perspectives of conventional chemical precompression are limited in this system.We suggest further exploration of the superconducting properties of Tb 3 Ni 2 O 7 at zero pressure.
This comprehensive study uncovers a profound tunability of the structural and electronic phases in this novel materials class.The richness of the structural phase diagram and the unexpected transitions identified specifically for the later rare-earth nickelates provide additional opportunities for the discovery of e.g.superconducting phases.It also emphasizes that future work needs to carefully assess the impact of defects, especially oxygen excess (hole doping) or oxygen vacancies (electron doping).
BG and PJH conceived of the project.JJH, GRS, RGH, and PJH supervised the research.BG performed the theoretical simulations and corresponding analysis.BG and PJH wrote the manuscript.All authors discussed the results and revised the manuscript.

DATA AVAILABILITY
The data is available upon reasonable request to the authors.Appendix A: Magnetic interactions in La3Ni2O7 under pressure: Ferromagnetic in-plane coupling and proximity of an insulating site-disproportionate state The experimental evidence on the magnetic properties of La 3 Ni 2 O 7 at ambient pressure is so far ambivalent.While recent measurements suggested the presence of a spin density wave under appropriate sample conditions [93,94], earlier neutron diffraction and nuclear magnetic resonance studies found no long-range magnetic order [81,95].
To shed some light into this issue, we explore the magnetic properties of La 3 Ni 2 O 7 as a function of the on-site Coulomb repulsion parameter U and the external pressure p (Fig. 6).At U = 0 eV, the energy differences between the considered magnetic orders are negligible (< ±5 meV/Ni), reflecting the possibility of substantial magnetic fluctuations [Fig.6(b)].With increasing U , ferromagnetic (FM) Ni-Ni interactions stabilize in the NiO 2 planes (orange and red curve).This observation is in sharp contrast to the cupratelike in-plane antiferromagnetic (AFM) interactions (blue and green curve) that have been reported for the infinite-layer nickelates [14,26].Simultaneously, the Ni magnetic moments are strongly enhanced with U [Fig. 6(d)].Moreover, we observe that the magnetic interactions in the bilayer nickelate are highly susceptible to external pressure, which counteracts these trends [Fig.6(c,e)].
Interestingly, we identify an A-type AFM ground state for U ≥ 1 eV, characterized by FM in-plane interactions and AFM-coupled NiO 2 layers [Fig.6(a,b)].Moreover, we find this state to be superimposed by a checkerboard site disproportionation, expressed in the emergence of two distinct Ni magnetic moments [Fig.6(d,e); denoted as Site 1 and Site 2].This phenomenon is accompanied by breathing-mode structural distortions, i.e., larger and smaller NiO 6 octahedra, as reported for (LaNiO 3 ) 1 /(LaAlO 3 ) 1 (001) superlattices [75,77,96].A band gap opens around the Fermi energy, rendering an insulating state [Fig.6 -G) − E(AFM-A) ∼ 145 meV/Ni reflect the in-plane magnetic coupling, which we find to unambiguously exceed the inter-layer coupling.Hence, the magnetic interactions are highly anisotropic.At the same time, these values show that the in-plane magnetic coupling is strongly affected by the inter-layer coupling.This interplay is also mirrored in the Ni magnetic moments, which are highly distinct in all four considered magnetic phases [Fig.6(d,e)].

Figure 1 .
Figure 1.(a) Structure of the bilayer Ruddlesden-Popper nickelate A3Ni2O7.(b) Lattice parameter analysis for La3Ni2O7 as a function of the external pressure p.While our DFT+U predictions agree closely with experimental observations [1] for low pressure, we observe an unexpected transition to a tetragonal I4/mmm (a = b) instead of an orthorhombic F mmm phase (a ̸ = b) around p ∼ 20 GPa.(c) Lattice parameters for further selected A3Ni2O7 nickelates.The vertical dashed lines mark structural phase boundaries [see panel (d)].(d) Structural phase diagram, contrasting the impact of chemical versus external pressure in A3Ni2O7.It has been compiled from the DFT+U data discussed in Figs. 2 and 3 (marked by small circles).(e) Corresponding top and side views of representative bilayer slabs.The arrows mark characteristic differences between the distinct structural phases, involving octahedral rotations ϕ and tilts θ, distinct orthorhombic distortions b/a, typical displacements of the A-site ions, and different bond disproportionations of the NiO6 octahedra.
d y from the in-plane O-Ni-O distances d x and d y (which may alternate among the Ni sites in a checkerboard pattern, resulting in Q ̸ = 0), their average d x,y , and the out-of-plane O-Ni-O distances d z .

Figure 2 .
Figure 2. Structural trends of A3Ni2O7 across the rare-earth series (including Y and Sc) for varying external pressure p from 0-150 GPa.The A-site elements are ordered according to their ionic radius.The dashed lines separate the distinct structural phases [Fig.1(d)].(a) The b/a ratio clearly identifies the tetragonal I4/mmm phase I and separates it from the Cmcm phase II with b/a > 1.The remaining phases show largely b/a < 1.(b) Apical versus basal extension of the octahedra (R).In phases I-III, the octahedra are elongated in vertical z ∼ c direction, particularly for La3Ni2O7 at ambient conditions.In contrast, phase IV exhibits vertically compressed octahedra.(c) The in-plane disproportionation of the O-Ni-O distances (Q) is very pronounced in phase IV, which is characterized by cigar-shaped octahedra alternating in the plane, reminiscent of LaMnO3, and a considerable electronic reconstruction of the Ni 3d z 2 orbital.

Figure 3 (
Figure 3(a) analyzes the basal and apical Ni-O-Ni bond angles as well as the octahedral rotations ϕ and tilts θ [defined in Fig. 1(e)] in A 3 Ni 2 O 7 , which provide insightful and distinct perspectives on the rotational degrees of freedom.In the Cmcm phase II, where ϕ = 0, we find that θ ranges from ∼ 2-8 • .In the Cmc2 1 phases III and IV, ϕ and θ are both enhanced to ∼ 4-15• .With increasing pressure, the in-plane rotations ϕ, if present, are quenched first (for A = Nd-Tm at 10-75 GPa), which corresponds to the boundary between phases II and III.For Lu and Sc, the octahedral rotations are so pronounced that we observe finite values up to 150 GPa.Subsequently, the octahedral tilts θ are quenched at yet higher pressure values (for A = La-Tb at 20-150 GPa).Intriguingly, we find that these vanishing tilts coincide universally with the orthorhomic-to-tetragonal transition from phase II to phase I.This demonstrates that external pressure generally reduces or even quenches the octahedral rotations, whereas chemical pressure rather enhances them.The subsequent suppression of the distinct rotational degrees of freedom indicates that different energy scales are involved, with a significantly shallower potential energy landscape related to ϕ than to θ.The superconducting transition has been so far associated with a straightening of the inter-layer (apical) Ni-O-Ni bond angles towards 180 •[1].The respective top-right panel in Fig.3(a) closely resembles the bottom-right panel displaying the octahedral tilts θ and shows that pressures up to 150 GPa successfully engineer this state for A = La-Tb.In particular, ∼ 20 GPa for La 3 Ni 2 O 7 are sufficient to obtain inter-layer Ni-O-Ni bond angles close to 180 • .For Ce 3 Ni 2 O 7 , already slightly lower pressures result in this state, whereas Pr 3 Ni 2 O 7 and Nd 3 Ni 2 O 7 require rather ∼ 30-40 GPa.At zero pressure,

Figure 3 .
Figure 3(a) analyzes the basal and apical Ni-O-Ni bond angles as well as the octahedral rotations ϕ and tilts θ [defined in Fig. 1(e)] in A 3 Ni 2 O 7 , which provide insightful and distinct perspectives on the rotational degrees of freedom.In the Cmcm phase II, where ϕ = 0, we find that θ ranges from ∼ 2-8 • .In the Cmc2 1 phases III and IV, ϕ and θ are both enhanced to ∼ 4-15• .With increasing pressure, the in-plane rotations ϕ, if present, are quenched first (for A = Nd-Tm at 10-75 GPa), which corresponds to the boundary between phases II and III.For Lu and Sc, the octahedral rotations are so pronounced that we observe finite values up to 150 GPa.Subsequently, the octahedral tilts θ are quenched at yet higher pressure values (for A = La-Tb at 20-150 GPa).Intriguingly, we find that these vanishing tilts coincide universally with the orthorhomic-to-tetragonal transition from phase II to phase I.This demonstrates that external pressure generally reduces or even quenches the octahedral rotations, whereas chemical pressure rather enhances them.The subsequent suppression of the distinct rotational degrees of freedom indicates that different energy scales are involved, with a significantly shallower potential energy landscape related to ϕ than to θ.The superconducting transition has been so far associated with a straightening of the inter-layer (apical) Ni-O-Ni bond angles towards 180 •[1].The respective top-right panel in Fig.3(a) closely resembles the bottom-right panel displaying the octahedral tilts θ and shows that pressures up to 150 GPa successfully engineer this state for A = La-Tb.In particular, ∼ 20 GPa for La 3 Ni 2 O 7 are sufficient to obtain inter-layer Ni-O-Ni bond angles close to 180 • .For Ce 3 Ni 2 O 7 , already slightly lower pressures result in this state, whereas Pr 3 Ni 2 O 7 and Nd 3 Ni 2 O 7 require rather ∼ 30-40 GPa.At zero pressure,

Figure 4 .
Figure 4. Electronic structure of A3Ni2O7 rare-earth nickelates.(a) Total DOS at the Fermi energy and relative contributions from the Ni, basal oxygen, and apical oxygen ions.The dashed lines separate the distinct structural phases [Fig.1(d)].Noteworthy are the high contributions from the apical oxygen ions in phases I and II.Intriguingly, Tb3Ni2O7 at zero pressure presents a considerably higher DOS at the Fermi level (∼ 8.5 States/eV) than La3Ni2O7 reaches at around 100 GPa (∼ 7.2 States/eV) in addition to a sizable involvement of the apical oxygen ions.(b) Evolution of the total DOS as a function of the external pressure for selected compounds.An overall pressure-induced broadening of the O 2p-derived valence band can be observed, as well as of the Ni eg manifold located at the Fermi energy.Characteristic peaks clearly shift to lower energies, e.g., the Ni t2g peak visible between −2 and −1 eV at zero pressure, or the oxygen peak around −5.5 eV (A = La-Tb).The feature directly below the Fermi energy (marked by the arrow) corresponds to the lower set of Ni 3d z 2 -O 2pz hybrid states [Fig.5(a)].

Figure 5 .
Figure 5. (a) Band structures of La3Ni2O7 at p = 0 (Cmcm) and 30 GPa (I4/mmm) as well as Tb3Ni2O7 at 0 GPa (Cmc21) from DFT+U .The band character is represented by the different colors, and the Ni eg occupation numbers n are explicitly provided.(b,c) The corresponding Fermi surfaces unravel an increase in Ni 3d z 2 orbital weight (color bar) in pressurized La3Ni2O7, specifically for sheet α, and an emerging hole pocket (γ) around the Γ point.Tb3Ni2O7 presents a similar Fermi surface topology at zero pressure, despite finite octahedral rotations, in conjunction with a considerably enhanced DOS at the Fermi energy [Fig.4(a)].
sure discussed above [Fig.2(b)] lowers the energy of the Ni 3d x 2 −y 2 orbital relative to the Ni 3d z 2 states [Fig.5(a)].

Figure 6 .
Figure 6.(a) Illustration of the four considered magnetic states at the bilayer Ni sites in La3Ni2O7.(b,c) Energies of the magnetic phases as a function of the on-site Coulomb repulsion parameter U and the external pressure p, using G-type AFM as reference.(d,e) Corresponding Ni magnetic moments, together with a visualization of the site disproportionation (SD) and the concomitant breathing-mode structural distortions identified in the A-type AFM state.The DOS inset in panel (b) uncovers this phase to be insulating (black: total DOS; red: oxygen; gray: Ni).

Figure 7 .
Figure 7. (a) Band structure of Tb3Ni2O7 at ambient pressure, keeping the Tb 4f electrons frozen in the core [see Fig. 5(a)].(b) The explicit treatment of the Tb 4f electrons unveils an identical electronic structure near the Fermi energy.