Electronic and Magnetic Structure of Infinite-layer $\textrm{NdNiO}_2$: Trace of Antiferromagnetic Metal

The recent discovery of Sr-doped infinite-layer nickelate $\textrm{NdNiO}_2$ [D. Li et al. Nature 572, 624 (2019)] offers an exciting platform for investigating unconventional superconductivity in nickelatebased compounds. In this work, we present a first-principles calculations for the electronic and magnetic properties of undoped parent $\textrm{NdNiO}_2$. Intriguingly, we found that: 1) the paramagnetic phase has complex Fermi pockets with 3D characters near the Fermi level; 2) by including electronelectron interactions, 3d-electrons of Ni tend to form $(\pi, \pi, \pi)$ antiferromagnetic ordering at low temperatures; 3) with moderate interaction strength, 5d-electrons of Nd contribute small Fermi pockets that could weaken the magnetic order akin to the self-doping effect. Our results provide a plausible interpretation for the experimentally observed resistivity minimum and Hall coefficient drop. Moreover, we elucidate that antiferromagnetic ordering in $\textrm{NdNiO}_2$ is relatively weak, arising from the small exchange coupling between 3d-electrons of Niand also hybridization with 5d-electrons of Nd.

Since the discovery of high-temperature (high-T c ) superconductivity in cuprates 1 , extensive effort has been devoted to investigate unconventional superconductors, ranging from non-oxide compounds 2,3 to ironbased materials 4,5 . Exploring high-T c materials could provide a new platform to understand the fundamental physics behind high-T c phenomenon, thus is quite valuable. Very recently, the discovery of superconductivity in Sr-doped NdNiO 2 6 potentially raises the possibility to realize high-T c in nickelate family 7,8 .
One key experimental observation for the infinitelayer NdNiO 2 is that its resistivity exhibits a minimum around 70 K and an upturn at a lower temperature 6 . At the same time that the resistivity reaches minimum, the Hall coefficient drops towards a large value, signalling the loss of charge carriers 6 . Interestingly, no long-range magnetic order has been observed in powder neutron diffraction on NdNiO 2 when temperature is down to 1.7 K 6 . This greatly challenges the existing theories, since it is generally believed that magnetism holds the key to understand unconventional superconductivity [9][10][11][12] . Therefore, it is highly desirable to study the magnetic properties of undoped parent NdNiO 2 and elucidate its experimental indications.
In this work, the electronic and magnetic properties of NdNiO 2 are systemically studied by first-principles calculations combined with classical Monte Carlo calculations. Firstly, the paramagnetic (PM) phase is studied. Its Fermi surface includes one large sheet and two electron pockets at Γ and A point, respectively. This can be described by a three-band low-energy effective model that captures the main physics of exchange coupling mechanism. Then, the magnetic properties are studied by including Hubbard U and (π, π, π) antiferromagnetic (AFM) ordering is confirmed to be the magnetic ground state. Most significantly, the Fermi surface of AFM phase is simpler than that of PM phase, demonstrating an interaction induced elimination of Fermi pockets. Before NdNiO 2 enters correlated insulator, it is a compensated metal with one small electron pocket formed by d xy orbital of Nd and four small hole pockets formed by d z 2 orbital of Ni. The estimated phase transition temperature (T N ) from PM phase to (π, π, π) AFM phase is 70 ∼ 90 K for moderate interaction strength of U = 5 ∼ 6 eV.
Through these studies, we identify two key messages that are distinguishable from the cuprates: 1) NdNiO 2 is dominated by the physics of Mott-Hubbard instead of charge-transfer; 2) effective exchange coupling parameters are about one-order smaller than those of cuprates. In this regarding, supposed that the ground state is magnetic, our calculations demonstrate (π, π, π) AFM ordering is energetically favorable. Moreover, our results provide a natural understanding of two experimental observations. First, 3delectrons of Ni tend to form AFM ordering around 70 ∼ 90 K, coinciding with the minimum in resistivity and the drop in Hall coefficient. Second, the (π, π, π) AFM ordering could be weak (compared with cuprates), because of the small effective exchange coupling and the hybridization with itinerant 5d-electrons of Nd. This could be the reason why AFM ordering is missing in previous study, which calls for more careful neutron scattering measurements on NdNiO 2 .
The first-principle calculations are carried out with the plane wave projector augmented wave method as implemented in the Vienna ab initio simulations package (VASP) [13][14][15] . The Perdew-Burke-Ernzerhof (PBE) functionals of generalized gradient approximation (GGA) is used for PM phase 16 . To incorporate the electron-electron interactions, DFT + U is used for AFM phase, which can reproduce correctly the gross features of correlated-electrons in transition metal oxides [17][18][19] . The 4f electrons of Nd 3+ are expected to display the local magnetic moment as Nd 3+ in Nd 2 CuO 4 20 and are treated as the corelevel electrons. The Hubbard U (0 ∼ 8 eV) term is added to 3d electrons of Ni. The energy cutoff of 600 eV, and Monkhorst-Pack k point mesh of 11 × 11 × 11 and 18 × 18 × 30 is used for PM and AFM phase, respectively. The maximally localized Wannier functions (WFs) are constructed by using Wannier90 package 23,24 . The structure of infinite-layer NdNiO 2 is shown in Fig. 1(a), including NiO 2 layers sandwiched by Nd, which can be obtained from the perovskite NdNiO 3 with reduction of apical O atoms in c direction 21,22 . Due to apical O vacancies, the lattice constant in c direction shrinks (smaller than a direction) and the space group becomes P 4/mmm. The experimental lattice constant a = b = 3.92 Å and c = 3.28 Å are used in our calculations.
Firstly, We present the band structure of PM phase without Hubbard U. The orbital resolved band structure of PM phase is shown in Fig. 1(b). Comparing with typical cuprates CaCuO 2 25 , two significant differences are noted: 1) there is a gap ∼ 2.5 eV between 2p orbitals of O and 3d orbitals of Ni. According to Zaanen-Sawatzky-Allen classification scheme 26 , this indicates that the physics of NdNiO 2 is close to Mott-Hubbard rather than charge-transfer; 2) there are two bands crossing the Fermi level, in which one is mainly contributed by d x 2 −y 2 orbital of Ni (called pure-band) and the other one has a complicated orbital compositions (called mixed-band). In k z = 0 plane, the mixedband is mainly contributed by d z 2 orbital of Nd and Ni. The dispersion around Γ point is relatively small, called heavy electron pocket (HEP). In k z = 0.5 plane, the mixed-band is mainly contributed by d xy (d xz , d yz and d z 2 ) orbital of Nd (Ni). The dispersion around A point is relatively large, called light electron pocket (LEP). As a comparison, one notices that there is only one pure-band crossing the Fermi level in CaCuO 2 25 . The Fermi surface of PM phase is shown in Fig. 1 There is a large sheet contributed by the pure-band, as the case in CaCuO 2 25 . This Fermi surface is obviously two-dimensional (2D), because of the weak dispersion along Γ-Z. In addition, there are two electron pockets residing at Γ and A point, respectively, showing a feature of three-dimensional (3D) rather than 2D (see labels HEP and LEP in Fig. 1(c)). Therefore, the 3D metallic state will be hybridized with the 2D correlated state in NiO 2 plane, suggesting NdNiO 2 to be an "oxide-intermetalic" compound 27,28 .
The existence of mixed-band also reflects the inherent interactions between Nd 5d and Ni 3d electrons. To explore the low energy physics of NdNiO 2 , a threeband model consisting of Ni d x 2 −y 2 , Nd d z 2 and Nd d xy orbitals is constructed by Wannier90 package. As shown Fig. 1(d), one can see the good agreement between first-principles and Wannier-fitting bands near the Fermi level. The corresponding three maximally localized WFs are shown in Fig. 1(e), demonstrating the main feature of d z 2 (WF1) and d xy (WF2) orbital of Nd, and d x 2 −y 2 (WF3) orbital of Ni. However, these WFs still have some derivations from standard atomic orbitals, that is, WF1 and WF2 are mixed Energy of (π, π, π) AFM is set to zero.
with d z 2 orbital of Ni, and WF3 is mixed with p x/y orbital of O in the NiO 2 plane. According to the classical Goodenough-Kanamori-Anderson rules 29-31 , these derivations (or hybridizations) will give clues for the magnetic properties.
To determine the magnetic ground state of NdNiO 2 , six collinear spin configurations are taken into account in a 2 × 2 × 2 supercell, that is, AFM1 with q = (π, π, π), AFM2 with q = (π, π, 0), AFM3 with q = (0, 0, π), AFM4 with q = (π, 0, 0), AFM5 with q = (π, 0, π) and FM with q = (0, 0, 0), as shown in Fig. 2(a). Within all Hubbard U ranges, we found that AFM1 configuration always has the lowest energy, as shown in Fig. 2(b), indicating a stable (π, π, π) AFM phase with respect to electron-electron interactions and is in accordance with random phase approximation treatment 32 . This can be attributed to the special orbital distributions around the Fermi level. The intralayer NN exchange coupling is the typical 180 • typed Ni-O-Ni superexchange coupling, that is, the coupling between d x 2 −y 2 orbital of Ni is mediated by p x/y orbital of O (see WF3), preferring a (π, π) AFM phase in NiO 2 plane. The interlayer NN exchange coupling is due to the superexchange between the Ni d z 2 orbitals mediated by Nd d z 2 orbital as shown in WF1, preferring a (π, π) AFM phase between NiO 2 planes. Therefore, the superexchange coupling results in a stable (π, π, π) AFM phase in NdNiO 2 . Moreover, the magnetic anisotropy is further checked by including the spin-orbit coupling (SOC). We found that the spin moment prefers along c direction with the magnetic anisotropic energy of ∼0.5 meV/Ni. Thus, the tiny SOC effect can be safely neglected in the following phase transition temperature calculations.
In cuprates, the Fermi surface is unstable with electron-electron interactions, making its parent phase to be an AFM insulator. However, this is apparently not the case in NdNiO 2 , because of the extra electron pockets and the inherent interaction between Nd 5d and Ni 3d electrons. At U=0 eV, there are two electron pockets at Γ point and two hole pockets along X-R direction as shown in Fig. 3(a)-(b). Physically, the origin of these four pockets can be easily understood through the comparison of orbital resolved band structures between PM phase ( Fig. 1(b)) and (π, π, π) AFM phase (Fig. 4). Because of the Zeeman field on Ni, its spin-up and -down bands are split away from each other. The original pure-band (d x 2 −y 2 orbital of Ni) in PM phase becomes partially occupied in spinup channel (forming two hole pockets) and totally unoccupied in spin-down channel. Hence, the two hole pockets in AFM phase are inherited from large sheet in PM phase, showing a 2D character with neglectable dispersion along Γ-Z direction. For the electron pockets at Γ point, the heavier one is mainly contributed by d z 2 orbital of Nd and Ni, so it comes from the HEP at Γ point of PM phase. While for the lighter one, it comes from the LEP at A point of PM phase which is folded into the Γ point of (π, π, π) AFM phase [see Fig. 1(a)]. The orbital composition can also be used to check this folded band, which is contributed by d xy orbital of Nd, d xz/yz (d z 2 ) orbital of spin-down (-up) Ni.
These pockets have a different evolution with the increasing value of Hubbard U. For electron pockets, the heavier one is very sensitive to Hubbard U and disappears at U = 1 eV. Meanwhile the lighter one doesn't appear until U = 6 eV. In addition, the orbital components of lighter electron pockets are purified by electron-electron interaction and it mainly contributed by d xy of Nd in the large U limit as shown in Fig.  4. The case for hole pockets is rather complicated. Firstly, the bands of hole pockets become flat with the increasing value of Hubbard U. Secondly, the original hole pockets formed by d x 2 −y 2 orbital of Ni gradually disappear, meanwhile, a new hole pocket formed by d z 2 orbital of Ni appears along Γ-M as shown in Fig.  3(d). At U = 6 eV, NdNiO 2 is a compensated metal with a small electron pocket at Γ point and four hole pockets along Γ-M as displayed in Fig. 3(e). Further increasing the value of Hubbard U, the system enters an AFM insulator, just like cuprates. Therefore, the metal-to-insulator phase transition point is near U ∼ 6 eV. If Hubbard U is less than 6 eV, NdNiO 2 is an AFM metal with relatively small amount of holes that are self-doped 27,32-35 into d orbitals of Ni. Interestingly, there is an orbital shift from 3d x 2 −y 2 orbital of Ni at U = 0 eV to 3d z 2 orbital of Ni at U = 6 eV in the NiO 2 plane, as depicted in Fig. 3(f). We speculate that this orbital shift may change the paradigm after doping 8,36 . Moreover, without the Hubbard U, the 2p orbital of O is far away from the Fermi level, just like the case of PM phase. However, with the increasing value of Hubbard U, the gap between 3d orbital of Ni and 2p orbital of O gradually decreases (Fig. 4), demonstrating an evolution from Mott-Hubbard metal to charge-transfer insulator.
In order to quantitatively describe such a phenomenon, the phase transition temperature is further calculated. For (π, π, π) AFM phase, the magnetic momentum of Ni increases from 0.58 µ B (U = 0 eV) to 1.04 µ B (U = 8 eV) and becomes gradually saturated, as shown in Fig. 5(a). This is also consistent with the fact that d x 2 −y 2 orbital of Ni is closer to single occupation with the increasing value of Hubbard U. Therefore, Ni is spin one half (S = 1/2) in infinite-layer NdNiO 2 , just like the case in cuprates. To extract the exchange coupling parameters of J 1 , J 2 , J 3 and J 4 (as labelled in Fig. 2(a)), the total energy of five AFM configurations obtained from DFT + U calculations are mapped onto the Heisenberg spin Hamiltonian. In the 2 × 2 × 2 supercell, there are 8 Ni atoms and the total energy of different AFM configurations are: where E 0 is the reference energy without magnetic order. The calculated exchange coupling parameters as a function of Hubbard U are shown in Fig. 5(b). We would like to make several remarks here: 1) the NN intralayer exchange coupling (J 1 ), mediated by d x 2 −y 2 orbital of Ni, demonstrates a 1/U law; 2) the NN interlayer exchange coupling (J 2 ) is ∼ 10 meV with little variation. The positive value of J 2 indicates an AFM coupling between NiO 2 planes; 3) the next NN intralayer exchange coupling (J 3 ), mediated by d xy orbital of Nd, is comparable to J 1 at large value of Hubbard U, which is dramatically different to that in infinite-layer SrFeO 2 37-39 . This large value could be attributed to the relative robustness of lighter electron pocket and orbital purification; 4) the next NN interlayer exchange coupling (J 4 ) is ∼ 0 meV, indicating the validity of our Hamiltonian up to the third NN; 5) J 1 , J 2 and J 3 have the same strength at large U, suggesting NdNiO 2 is a 3D magnet rather than 2D magnet.
Based on the above exchange coupling parameters, the phase transition temperature (T N ) is calculated by Figure 4: Orbital resolved band structures of (π, π, π) AFM phase with different values of Hubbard-U. The first, second, third and forth row represents Nd, spin-up Ni, spin-down Ni and O, respectively. The filled circles with different colors have the same meaning as those in Fig. 1. The name of d orbitals in the AFM supercell has been aligned to that of unit cell.
classical Monte Carlo method in a 12 × 12 × 12 supercell based on the classical spin Hamiltonian: where the spin exchange parameters J ij have been defined above. First, we calculate the specific heat (C ) after the system reaches equilibrium at a each given temperature (T ), as shown in Fig. 5(c). Then, T N is extracted from the peak position in the curve of C(T ), as shown in Fig. 5(c). For U = 1 eV, T N is as high as 220 K, which can be ascribed to the large value of J 1 .
With the increasing value of Hubbard U, T N gradually decreases and becomes ∼ 70 K at U = 6 eV. To further check the effect of interlayer exchange coupling on 3D magnet, an additional Monte Carlo calculation is performed without J 2 and J 4 . As shown in Fig. 5(e), the C(T ) vs T plot shows a broaden peak at a lower temperature. Since Mermin-Wagner theorem prohibit magnetic order in 2D isotropic Heisenberg model at any nonzero temperatures 40 , the broad peak in C(T ) vs T plot implies the presence of short-range order. Regarding the small drop of T N (about 30 K in Fig.  5(f)), our MC simulations indicate that the weak interactions between NiO 2 planes. Although the exact value of Hubbard U cannot be directly extracted from the first principles calculations, its value range can still be estimated based on similar compounds. The infinite-layer nickelates are undoubtedly worse metals compared to elemental nickel with U ∼ 3 eV 41 , which can be considered as a lower bound of Hubbard U. The Coulomb interaction in infinite-layer nickelates should be smaller than that in the chargetransfer insulator NiO with U ∼ 8 eV 17 , which can be considered as a upper bound of Hubbard U. Therefore, a reasonable value of Hubbard U in NdNiO 2 will between 3 eV and 8 eV. In the following discussions, we use U = 5 ∼ 6 eV 27,42 to draw our conclusions: 1) with the decreasing of temperature, there is a phase transition from PM phase to (π, π, π) AFM phase near 70 ∼ 90 K; 2) the exchange-coupling parameters are ∼ 10 meV, which is one order smaller than cuprates [43][44][45][46][47] and results in a low T N compared with cuprates; 3) the selfdoing effect from 5d orbital of Nd and 3d orbital of Ni may screen the local magnetic momentum in d x 2 −y 2 orbital of Ni, which gives a small magnetic momentum less than 1 µ B and makes the long-range AFM order unstable 6,21,22,48 ; 4) the Fermi surface of PM phase is quite large with two 3D-liked electron pockets, while the Fermi surface of (π, π, π) AFM phase is quite small with one 3D-liked electron pocket and four 2D-liked hole pocket. Therefore, there could exist a crossover from normal metal to bad AFM metal around T N ∼ 70 − 90 K, which provides a plausible understanding of minimum of resistivity and Hall co-efficient drop in infinite-layer NdNiO 2 6 . We envision that our calculations will intrigue intensive interests for studying the magnetic properties of high quality infinite-layer NdNiO 2 samples.
Lastly, we would like to make some remarks on the existing experiments. Some recent experiments fail to find bulk superconductivity in NdNiO 2 systems, and the parent samples show strong insulating behaviors 49 . The insulating behavior could be attributed to strong inhomogenious disorder or improper introduction of H during the reaction with CaH 2 50 . Especially, it is worth noting, the experiments cannot rule out the possibility of weak AFM ordering, due to the presence of Ni impurities in their samples. (Actually this problem has been pointed out before 21,22 .) The strong ferromagnetic order from elemental Ni would dominate over and wash out the weak signal of AFM ordering from NdNiO 2 as we suggested in this work. In this regarding, the upcoming inelastic neutron scattering on high-quality samples is highly desired.