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Phase Diagram of Infinite-layer Nickelate Compounds from First- and Second-principles Calculations

The fundamental properties of infinite-layer rare-earth nickelates (RNiO2) are carefully revisited and compared with those of CaCuO2 and RNiO3 perovskites. Combining first-principles and finite-temperature second-principles calculations, we highlight that bulk NdNiO2 compound are far from equivalent to CaCuO2, together at the structural, electronic, and magnetic levels. Structurally, it is shown to be prone to spin-phonon coupling induced oxygen square rotation motion, which might be responsible for the intriguing upturn of the resistivity. At the electronic and magnetic levels, we point out orbital-selective Mott localization with strong out-of-plane band dispersion, which should result in the isotropic upper critical fields and weakly three-dimensional magnetic interactions with in-plane local moment and out-of-plane itinerant moment. We further demonstrate that as in RNiO3 perovskites, oxygen rotation motion and rare-earth ion controlled electronic and magnetic properties can give rise in RNiO2 compounds to a rich phase diagram and high tunability of various appealing properties. In line with that, we reveal that key ingredients of high-Tc superconductor such as orbital polarization, Fermi surface, and antiferromagnetic interactions can be deliberately controlled in NdNiO2 through epitaxial strain. Exploiting strain-orbital engineering, a crossover from three- to two-dimensional magnetic transition can be established, making then NdNiO2 thin film a true analog of high-Tc cuprates.

2 engineering, a crossover from three-to two-dimensional magnetic transition can be established, making then NdNiO2 thin film a true analog of high-Tc cuprates.

I. INTRODUCTION
Infinite-layer nickelates RNiO2 are currently attracting intense research interest owing to the discovery of superconductivity in NdNiO2 [1] and PrNiO2 [2]. These intriguing observations have reinvigorated theoretical and experimental investiagtions  to rationalize the puzzling mechanism and explore effective approach to improve the superconducting temperature Tc.
Comparing with the RNiO3 perovskites, RNiO2 can be viewed as a defective NdNiO3 perovskite with missing apical oxygen atoms. In RNiO3, the motion and amplitude of antiferrodistortive (AFD) rotations, electronic state, Ńel temperature, and metal-insulator transition temperature vary strongly dependent on the R-site ionic radius rR [27,28]. Interestingly, in the RNiO2 family, depending on the R-site ion, there is also a large diversity of their properties: hole doped LaNiO2 remains a metallic state, while hole doped NdNiO2 and PrNiO2 exhibit a superconducting phase with a higher Tc in NdNiO2 [1,2]; another intriguing feature of NdNiO2 is the unusual upturn of its resistivity at low temperature [1], which does not appear in LaNiO2 [29]. These results indicate that rare-earth ion determined property should be an essential factor to differentiate the behaviors of different compounds within the RNiO2 family like in

RNiO3.
Magnetic interactions in high-Tc superconductors are commonly believed to play essential roles for the emergence of superconductivity. Moreover, the rare-earth element is a key factor affecting the AFM coupling and superconducting temperature [30,31]. Recently, the magnetic property in NdNiO2 has been verified by experiments [5]. Consequently, rationalizing the origin of the magnetism and magnetic interactions as well as the evolution of magnetic interactions in RNiO2 series is at the heart of understanding the superconductivity mechanism and controlling Tc. Investigations on the exchange interactions of RNiO2 series primarily focused on the high-symmetry P4/mmm phase [21,22], while structural effect received much less attention. Identifying the fundamental role of R-site cation and achieving a more global description of the interplay between lattice, electron, orbital, and spin degrees of freedom in RNiO2 compounds is therefore not only of academic interest but might also reveal of practical relevance for the optimization of their superconducting properties.
More broadly, despite significant progresses have been realized so far regarding the fundamental properties of RNiO2 [21,22], there are still two key questions that need to be addressed. On the one hand, the temperature dependent phase diagram is still lacking, it is not only fundamental interesting but also critical for practical control of electronic and magnetic properties by strain engineering, rotation engineering, and dimensionality engineering which play important roles in RNiO3 perovskites. On the other hand, despite it is gradually recognized that the electronic and magnetic properties of RNiO2 are at odds with that of cuprates, there is still lack of effective approach to make RNiO2 analogous to cuprate superconductors to try to improve the Tc in infinitelayer nickelates.
In this work, the structural, electronic, and magnetic properties of RNiO2 compounds are investigated theoretically combining first-principles calculations at zero Kelvin and second-principles calculations at finite-temperature. Our calculations reveal the presence of strong spin-phonon coupling triggered out-of-phase rotation motion of NiO4 square in NdNiO2, reminiscent of the rotations of BO6 octahedra in ABO3 perovskites. From this, the microscopic origin of the abrupt upturn of resistivity is further rationalized. Moreover, as in RNiO3 perovskites, rare-earth cation, oxygen rotation, and strain are demonstrated to have a profound impact on the magnetic coupling strength and electronic structure and a completely temperature-structuralmagnetic phase diagram is achieved. Based on our knowledge, the spin splitting, orbital polarization and magnetic corrections are further controlled by strain engineering, yielding strained NdNiO2 thin films that show marked electronic and magnetic similarities with CaCuO2. So, although infinite layer nickelates and cuprates are shown to exhibit striking differences at the bulk level, our work reveal the possibility to make them similar in thin films from strain engineering, providing so a potential pathway to achieve rational control of superconductivity. 4

II. STRUCTURAL PROPERTIES
A. Spin-phonon coupling triggered rotation motion in NdNiO2.
We start our study by a careful re-investigation of the structural properties of RNiO2 compounds (R = La, Pr, Nd, Sm, Eu, Gd, Tb, Dy, Er, Tm, and Lu), relaxing first their P4/mmm phase. First-principles density functional theory (DFT) results reported in this Section were obtained using the generalized gradient approximation (GGA) [32] but are similar to those obtained using more advanced GGA + U [33] or strongly constrained and appropriately normed (SCAN) [34] as discussed in the following. The calculations have been performed with a G-type antiferromagnetic order (G-AFM) which was checked to be the GGA magnetic ground state for all compounds, consistently with previous works [16]. We will discuss further in Section III and Appendix C, Fig. 13, that the magnetic ground state should be better a C-type AFM order for GGA + U and SCAN functionals but this is not affecting the present discussions on structural property.
To probe the dynamical stability of RNiO2 in the high-symmetry P4/mmm phase, we then calculated the phonon dispersion curves of LaNiO2, NdNiO2, TbNiO2, and LuNiO2 in both the magnetic and non-magnetic phase. Surprisingly, although LaNiO2 appears as dynamically stable (see Fig. 1(a)), NdNiO2, TbNiO2 and LuNiO2 in their magnetic phases all show sizable phonon instabilities (imaginary frequencies in Figs. 1(b)-1(d)). Moreover, decreasing the size of the R-site cation radii when going from LaNiO2 to LuNiO2, the number and amplitude of the instabilities progressively increase.
The dominant instability is always an A mode associated to AFD out-of-phase rotation of the NiO4 squares around c axis. As the cation size decreases, additional instabilities appear that are further characterized in Appendix A, Fig. 9. The progressive destabilization of oxygen square rotation motion as the size of the R cation decreases is reminiscent of what is observed for AFD motions of the oxygen octahedra in RNiO3 perovskites [35,36] (see also Appendix A, Fig. 10). Note that the AFM magnetic interactions have recently been experimentally confirmed in NdNiO2, the dynamically stable of NdNiO2 in the non-magnetic phase (see Fig. 1(a) and Ref. [24,25] indicates that there is strong spin-phonon coupling that triggers the appearance of rotation motion.

B. The ground state structure
In order to identify the ground-state structure of RNiO2 compounds, we fully relaxed various possible structures condensing individual and combined phonon instabilities, taking NdNiO2 and LuNiO2 as prototypical examples. In both cases, the identified ground state is an I4/mcm phase obtained from the condensation of the A unstable mode of the P4/mmm phase. 6 For NdNiO2, this ground state is natural since the A mode is the only phonon instability. For LuNiO2, the situation is more complicated since there are more unstable modes. However the A instability remains dominant. As illustrated in Fig. 1(e), the double well associated the A mode is significantly deeper than that related to other instabilities. Then, it is further clarified in Fig. 1(f) that, when condensing the A mode with its natural amplitude, other weaker instabilities disappear (i.e. all curves switch from double-to single-well shape). This reveals an inherent competition between outof-phase rotation and other unstable modes: the appearance of the stronger out-of-phase rotation completely suppresses the other instabilities, stabilizing the I4/mcm structure as the ground state.
Further comparison of the energy difference between I4/mcm and P4/mmm phases for all investigated RNiO2 compounds indicates that, except for LaNiO2 which remains P4/mmm, all other compounds stabilize in an I4/mcm distorted ground state. This makes RNiO2 compounds (R = Pr-Lu) distinct from CaCuO2 which, like LaNiO2, does not exhibit any phonon instability in its P4/mmm phase. In these latter compounds, out-ofphase oxygen rotation can however be easily induced via strain engineering (see  Although previously overlooked, the appearance of AFD oxygen rotation motion giving rise to an I4/mcm ground state is a robust result that seems to be also confirmed independently by other authors [25,26]. It appears as an intrinsic feature of the RNiO2 infinite layer compounds that is perfectly in line with the physics of related RNiO3 compounds, and more generally of ABO3 perovskites. First, as illustrated in Figs. 2(a), the tendency to develop oxygen rotation decreases as the size of the A cation increases (Sm, Nd, Pr, La) similarly to the trend observed in the family of RNiO3 perovskites [35].
Second, the AFD rotation motion competes together (e.g. in Fig. 1(f), the out-of-phase A rotation suppresses the instability associated to in-phase X rotation) as typically reported in perovskites [37] and compete also with polar distortions [38][39][40] (Fig. 2(b)), as also observed in YNiO3 [35]. Third, oxygen rotation shows strong strain coupling like in ABO3 compounds [41,42], but with an opposite trend ( Fig. 2(c), AFD distortion increases under tensile biaxial strain). Finally, we see in Fig. 2 (d) that the rotation instability is strongly related to the covalency of the metal-oxygen bond which can be controlled by Hubbard U [43]: reducing the covalency through increasing the U value continuously enhances the rotation instability and the amplitude in line with perovskite oxide [43]. All these confirm that RNiO2 infinite layer compounds show marked resemblance to that of ABX3 perovskites from both structural and chemical viewpoints, providing further evidence that rotation instability should be a natural feature of ABX2 compounds.

C. Finite temperature behavior
Having established that RNiO2 infinite layer compounds are prone to AFD distortions, the natural question that arises concerns the temperature at which those distortions appear. In ABX3 perovskites, AFD distortions survive up to extremely high temperatures [35] and it might be questioned if the same is true in infinite layers. Results of the Monte Carlo simulations for NdNiO2 are reported in Fig. 3(a) that 9 summarizes the temperature evolution of the average displacement associated to AFD oxygen rotation. The Figure clearly highlights a structural phase transition from the high-symmetry P4/mmm phase to the low-symmetry I4/mcm phase with a phase transition temperature (TC) that is consistently predicted by SCAN, GGA, and GGA + U (U = 1-3 eV) at a relatively low temperature in the range from 39 to 63 K, which is not far from 70 K at which the resistance begins to increase [1].
In order to better assess the the mechanism of the phase transition and its rather displacive [46,47] or order-disorder [48] character, we have explored the probability distribution of oxygen displacements in NdNiO2 with SCAN functional below (30 K) and above (60 K) the transition temperature. As displayed in Fig. 3(b), the distribution evolves from a single peak centred at zero above TC to a double peak below TC, in line with a displacive character.

A. Orbital-selective Mott transition
We now turn our attention to the electronic structure of NdNiO2 to reveal the physical origin behind the isotropic Hc2. For that purpose, we use the SCAN functional that is expected to be more accurate and reliable than GGA (+U) for superconductors like La2CuO4 [49]. Note that all the following conclusions obtained from SCAN can be reproduced by GGA + U (see Appendix C). Consistent with previous works on NdNiO2 [22], we found that C-AFM order is always the most stable for R = La-Lu .   [17][18][19][20][21], and is reminiscent of the orbital-selective Mott physics that found in alkaline iron selenides [50,51] and (Ca,Sr)2RuO4 [52,53].
Importantly, the PDOS of NdNiO2 and Fermi surface (Fig. 12) reveals a clear signature of dispersion along the kz direction, similarly to ferropnictides [54,55]. The comparable out-of-plane hopping between Ni orbitals (0.46 meV) and inplane hopping between Ni and O p orbitals (1.21 meV) provides another direct confirmation of the strong inter-layer coherence. It has been revealed that kz dispersion naturally facilitates the circulating currents at all field orientations and eventually results in the isotropic Hc2 as found in ferropnictides [55]. Therefore, our results support the fact that the kz dispersion in NdNiO2 are likely correlated with the isotropic Hc2 as in Iron superconductors.

B. Unusual upturn of resistivity
Having characterized and discussed the electronic structure of the ground state structure, we explore further the electronic properties of structures at finite temperatures in order to investigate whether the predicted structural phase transition and progressive increase of rotation amplitude highlighted in Fig. 3 is the key factor that leads to the unusual upturn of resistivity of NdNiO2 observed at low temperature [1].
To shed light on the actual connection between temperature and electronic structure, the PDOS of NdNiO2 at different temperatures are compared in Figs. 4(c) and 4(d). Since the atomic structure appears to be spatially homogeneous at all temperatures (Figs. 3(c)-(f)), this is simply achieved by computing the DOS of homogeneous structures with the amplitude of rotation related to each temperature ( Fig. 3(a)).
Clearly, the PDOS in Figs. 4(c) and 4(d) reflect the fact that the electronic structure at the Fermi level is very sensitive to the amplitude of rotation, thus confirming the significant influence of temperature. In details, we found that although Nd orbital is insensitive to temperature and rotation amplitude, the bandwidth of Nd orbital decreases as the temperature decreases and the band edge shifts to an energy higher than the Fermi level, which significantly reduces the self-doping effect. The dramatic impact of temperature can also be immediately recognized by the observation that the elliptical Fermi surface of the ground state structure at the  point induced by the self-doping effect of Nd is strongly reduced (see Appendix B, Fig. 12) at low temperature. Additionally, the decrease of temperature is accompanied with a decrease of Ni states. All these effects seem to favor a less metallic phase at low temperature, suggesting that the low-temperature resistance upturn in NdNiO2 is likely to be associated with the appearance of rotation.
To clarify in practise whether rotation could be a reliable experimental fingerprint for the upturn of resistivity, a directly comparison of resistivity obtained from the Boltzmann transport equation [56] in the high-temperature P4/mmm phase without rotation and low-temperature I4/mcm phase with rotation is shown in Fig. 4(e), it is obvious that the resistivity around the Fermi level increases significantly in the low-temperature phase, confirming the important role of rotation.

IV. MAGNETIC PROPERTIES
A. Unified theory for weakly 3D magnetic correlations Having elucidating the electronic structure, we further build a unified theory for the magnetic correlations in nickelate infinite-layer superconductors by a step by step analysis from the macroscopic spin-charge density to the microscopic orbitalcontributed magnetic interactions. responsible for the magnetization of CaCuO2 consistent with its 2D magnetic dimensionality [57]. In contrast, the spin-density of NdNiO2 is characterized by the mixture of Ni and electrons. The results are in full agreement with the spin splitting of the majority and minority spin orbitals shown in Fig. 4(b).
Therefore, we highlight that the substantial differences between NdNiO2 and CaCuO2 not only lie in the electronic properties but also the magnetic order (C-AFM vs G-AFM) and the source of the magnetization. Interestingly, the in-plane local moment The variation of dominant exchange constants across the RNiO2 series is shown in Fig. 6 (b), it is obvious that both the in-plane and out-of-plane nearest-neighbor magnetic couplings are progressively increased when decreasing rR in the P4/mmm phase similarly with recent work [23]. In contrast, in-plane AFM coupling in the I4/mcm phase is in contrast Ref. [23] as it gradually decreases as rR decreases. Therefore, the decreased in-plane AFM coupling for smaller rR in the ground state I4/mcm phase should be attributed to the increase of rotation whose effect is partly compensated by the lattice constant effect. The rR dependent AFM coupling is completely in line with rare-earth perovskites like RCrO3 [65], RFeO3 [66] and RNiO3 [67], hence, the same superexchange mechanism and superexchange coupling J developed for perovskites should also be applicable to RNiO2. Thus, to make an accurate description of the variation of electronic and magnetic properties across the RNiO2 series, it is crucial to consider the rotation effect like in RNiO3.

VI. ELECTRONIC AND MAGNETIC TRANSITIONS
From the above analysis, we notice that with its orbital and 3D magnetic order closely similar to iron superconductors, NdNiO2 is far from being an analog of CaCuO2 which shows pure single band and 2D AFM order. These distinct electronic and magnetic properties may be the microscopic origin of much lower Tc in NdNiO2. Nevertheless, if the cuprate-like characters could be engineered in NdNiO2, it may become more favorable for superconductivity. As emphasized before, the key difference between NdNiO2 and CaCuO2 arises from the itinerant electrons, which are spin-polarized and have a higher energy level than the electrons.
Therefore, it seems plausible that pushing down the band edge of orbital 19 below the Fermi level would make NdNiO2 more analogous CaCuO2. Doing so, the orbital would be nearly fully occupied similarly to CaCuO2, the 3D magnetic interactions would correspondingly be reduced to 2D AFM interactions and the states around the Fermi lever would be mainly dominated by the single Ni band. Strain engineering has long been demonstrated to be an effective approach to modulate the on-site energy of orbitals and eventually tailor the related properties [69,70]. In ABO3 perovskites, compressive epitaxial biaxial strain directly leads to the 20 elongation (contraction) of the out-of-plane (in-plane) lattice constant, which typically weakens (strengthens) the hybridization of ( ) orbital with the surrounding O p orbitals, thus shifting the on-site energy of ( ) orbital to a lower (higher) level. To verify whether this scenario could be applied to infinitelayer NdNiO2, biaxial strain dependence of energy difference E between C-AFM and G-AFM phases is firstly explored and represented in Fig. 8(a). We see that E gradually decreases upon increasing the compressive strain and there is a crossover from C-AFM to G-AFM order at around 4.5% compressive strain. To verify the above qualitative analysis, we calculate the exchange constant to provide a quantitative picture. We found that the out-of-plane exchange constant is reduced to about -0.87 meV and the nearestneighbor in-plane exchange constant is about 37 times larger than the nearest-neighbor out-of-plane exchange constant, highlighting a strong quasi-2D magnetic coupling.
As we mentioned above, the appearance of out-of-plane exchange interaction is connected to the spin-polarized itinerant electron. It is obvious from the PDOS, displayed in Fig. 8(b), that the spin polarization of orbital almost vanishes when compressive strain reaches 5% and thereby yields the quasi-2D AFM order. Moreover, we see that the band edge of orbital goes across the Fermi level and governs the states around the Fermi level. The design principle of strain engineering is summarized in Fig. 8(c), a crossover from original 3D to cuprate-like 2D magnetic dimensionality dominated by orbital is triggered. The strained NdNiO2 thin film then obviously combine the key ingredients of CaCuO2 (see Figs. 4(a) and 5(a)): strong quasi-2D AFM order, almost pure single band near the Fermi level and large orbital polarization between the and orbitals.

VII. OUTLOOK
Our first-and second-principles results highlight that, despite sharing the same reference atomic structure, RNiO2 infinite-layer compounds differ significantly from high-Tc cuprate superconductors such as CaCuO2. Our calculations unveil that, 21 contrarily to CaCuO2, RNiO2 (with R=Pr-Lu) compounds are prone to develop AFD oxygen-rotation distortions similarly to RNiO3 perovskites and exhibit a displacive phase transition from P4/mmm to I4/mcm at low temperature.
These AFD oxygen-rotation motion appear to be the key to provide a consistent description of the structural, electronic and magnetic properties of infinite layers. At first, AFD oxygen rotation motion simultaneously localize the electron and suppress the self-doping effect from Nd atom, which can explain the unusual upturn of resistivity observed at low temperature [1]. The rare-earth ion rotation evolution, the rotation controlled bandwidth and the magnetic coupling strength are completely in line with RNiO3 perovskites. Additionally, unlike CaCuO2, we emphasize that the Ni orbital dominates the states at the Fermi level. The strong kz dispersion of itinerant electron is not only critical in determining the strong interlayer magnetic coupling and 3D features of exchange interactions (that contrast with 2D features in CaCuO2), but is also relevant to explain that the upper critical field is surprisingly isotropic at low temperature [4,5]. At the electronic and magnetic levels, RNiO2 compounds are therefore closely similar to ferropnictides [50,51,[58][59][60]. As a summary of our calculations, we then successfully built a complete phase diagram, summarizing the temperature evolution of structural and magnetic phases in the whole family of RNiO2 compounds (Fig. 7).
Having revealed the similarities between RNiO3 perovskites and RNiO2 infinite layers, it naturally appears that the strategies currently used to control the properties of perovskites (rotation engineering [35,71], strain engineering [72], R-site element substitution [73], and interfacial coupling [74]) could be exploited in infinite layers to achieve the continuous control of their superconducting properties. Relying then on strain-orbital engineering, we highlight that applying epitaxial compressive strains, as practically achievable in thin films, it is possible to shift the energy of Ni In conclusion, our work not only provides a deeper fundamental understanding of RNiO2 compounds that explicitly combines the interplay between their structural, electronic and magnetic degrees of freedom but also exploits this knowledge to provide new pathways to improve their properties. As such, we hope that our work will motivate experimentalists to further exploit the coupling between rotation, charge, orbital, spin, and strain in infinite layer RNiO2 and other ABX2 system to realize the practical control of their properties.

VIII. METHODS
We perform first-principles calculations using density functional theory (DFT) for the ground state properties calculations, finite-temperature second-principles model to calculate the structural phase transition, TB2J code [61] to extract orbital-dependent exchange constant from maximally localized Wannier functions (MLWF) and vampire code to calculate the Néel temperature [75].
A. First-principles calculations.
Our first-principles calculations are performed using the projector augmented wave (PAW) method implemented in VASP code [76,77]. Recent work shows that the electronic and magnetic properties of superconducting La2CuO4 is significantly improved [49] with SCAN [34] functional. Thus, widely used GGA functional [32] is employed for lattice dynamical analysis and we employ the SCAN for calculating of electronic and magnetic properties. Results from GGA and GGA + U [33] are also compared and discussed (see Appendix C). For SCAN functional, an energy cutoff of 900 eV is used, the force and energy convergence criterions are set to be 10 -8 eV and 10 -4 eV/Å for the structural relaxation. Lower kinetic energy of 700 eV and force and energy convergence criterions of 10 -7 eV and 10 -3 eV/Å are used for GGA and GGA + U. The kpoint meshes [78] are set to be 9  9  7 and 6  6  7 for √2  √2  2 and 2  2  2 supercells.

B. Second-principles calculations
First-principles based second-principles method is applied to investigate the evolution of rotation motion at different temperature. We build a second-principles model with displacements of oxygen atom $\mu$ as the degree of freedom. The total energy can be expressed as: Where ( ) in the first term = ∑ ( ) represents the energy of an isolated oxygen atom at i-th location with amplitude . It's truncated at fourth order since we found that's enough to describe the PESs. Due to the symmetry consideration, ( ) can be written as: Where and refers to the parameters to be determined from first-principles calculations. In the second term = ∑ , , and are the amplitude of oxygen atom displacements at i-th and j-th location and is the coupling parameter between them. This term is the energy contribution from the short-range interactions between neighboring oxygen atoms. The length of short-range interactions is truncated at one unit cell, due to the symmetry consideration, there are only four independent interaction parameters (j1, j2, j3 and j4) for the short-range interactions as sketched in Appendix D, Fig. 19(a). All these parameters are obtained from firstprinciples calculations.
Based on the second-principles model, the Monte Carlo simulations are performed to investigate the rotation motion at different temperature. The total energies obtained from first-principles calculations and second-principles model are compared in Appendix D, Fig. 19(b). The close match of energy guarantees the accuracy and validity of our model.

C. Exchange constant calculations
The exchange constants and the orbital contributions are obtained by the magnetic force theorem [79] as implemented in the TB2J code [61]   The results are obtained from GGA functional.   AFM orders, I4/mcm phase always has lower energy than the P4/mmm phase for GGA and GGA + U methods, the larger the U values, the larger the energy difference between 27 two phases. Note that G-AFM is the magnetic ground state for GGA functional, while C-AFM becomes the ground state when Hubbard U equals to 1-3 eV.