Abstract
The recent discovery of Srdoped infinitelayer nickelate NdNiO_{2} offers a new platform for investigating unconventional superconductivity in nickelatebased compounds. Most intriguingly, the resistivity minimum and Hall coefficient drop were identified simultaneously in the experiment, reflecting a novel electronic structure and transport property of NdNiO_{2}. Driven by this pioneering work, we present a firstprinciples calculation for the electronic and magnetic structure of undoped parent NdNiO_{2}. By taking into account experimentally relevant interaction strength, we found that (π, π, π) antiferromagnetic NdNiO_{2} is a compensated bad metal with small Fermi pockets. However, due to the small exchange coupling between 3delectrons of Ni and strong hybridization with 5delectrons of Nd, the discovered antiferromagnetic ordering is very weak. Crucially, with the decreasing of temperature, there exists a phase transition between good paramagnetic metal and bad AFM metal. The estimated transition temperature is ~70–90 K, which is consistent with that for observing the resistivity minimum and Hall coefficient drop. In this regarding, our results provide a plausible physical interpretation for these significant experimental observations.
Introduction
Since the discovery of hightemperature (highT_{c}) superconductivity in cuprates^{1}, extensive effort has been devoted to investigate unconventional superconductors, ranging from nonoxide compounds^{2,3} to ironbased materials^{4,5}. Exploring highT_{c} materials could provide a new platform to understand the fundamental physics behind highT_{c} phenomenon, thus is quite valuable. Very recently, the discovery of superconductivity in Srdoped NdNiO_{2}^{6} potentially raises the possibility to realize highT_{c} in nickelate family^{7,8}.
One key experimental observation for the infinitelayer NdNiO_{2} is that its resistivity exhibits a minimum ~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, signaling the loss of charge carriers^{6}. Interestingly, no longrange 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 firstprinciples calculations combined with classical Monte Carlo (MC) calculations. For the paramagnetic (PM) phase, its Fermi surface includes one large sheet and two electron pockets at Γ and A point, respectively. This can be approximately described by a threeband lowenergy effective model that captures the main physics of exchange coupling mechanism, which is in line with previous works^{13,14,15,16,17,18}. Moreover, by including the HubbardU, (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, our results provide a physical understanding of two experimental observations. First, 3delectrons of Ni tend to form AFM ordering ~70–90 K, coinciding with the minimum in resistivity and drop in Hall coefficient. Second, the (π, π, π) AFM ordering could be weak (compared with cuprates), because of small effective exchange coupling and strong hybridization with itinerant 5delectrons of Nd. This could be the reason why AFM ordering is missing in previous studies, which calls for more careful neutron scattering measurements on NdNiO_{2}.
Results
Electronic structure in PM phase
Firstly, we present the band structure of PM phase without Hubbard U. The orbital resolved band structure of PM phase is shown in Fig. 1b. Comparing with typical cuprates CaCuO_{2}^{13}, 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^{19}, this indicates that the physics of NdNiO_{2} is close to MottHubbard rather than chargetransfer^{13,20,21}; (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 pureband) and the other one has a complicated orbital compositions (called mixedband). 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, and the pocket at Γ point is called heavy electron pocket (HEP). In k_{z} = 0.5 plane, the mixedband 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, and the pocket at A point is called light electron pocket (LEP). As a comparison, one notices that there is only one pureband crossing the Fermi level in CaCuO_{2}^{13}. The Fermi surface of PM phase is shown in Fig. 1c. There is a large sheet contributed by the pureband, as the case in CaCuO_{2}^{13}. This Fermi surface is obviously twodimensional (2D), because of the weak dispersion along ΓZ. In addition, there are two electron pockets residing at Γ and A point, respectively, showing a feature of threedimensional (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 “oxideintermetalic” compound^{20,22}.
The existence of mixedband also reflects the inherent interactions between Nd 5d and Ni 3d electrons. To explore the lowenergy 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. 1d, one can see the good agreement between firstprinciples and Wannierfitting bands near the Fermi level. The little deviation between DFT and Wannierfitting bands above the Fermi level between Z and R is due to the hybridization between Nd 5d and Ni 3d orbitals. Such a hybridization causes the gap open which is missing in Wannierfitting bands. The corresponding three maximally localized WFs are shown in Fig. 1e, 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 deviations from standard atomic orbitals, that is, WF1 and WF2 are mixed 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^{23,24,25}, these deviations (or hybridizations) will give clues for the magnetic properties.
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. 2a. Within all Hubbard U ranges, we found that AFM1 configuration always has the lowest energy, as shown in Fig. 2b, indicating a stable (π, π, π) AFM phase with respect to electron–electron interactions and is in accordance with random phase approximation treatment^{16}. 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 noncollinear magnetic states are further checked by including the spinorbit 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 and the collinear magnetic states are used 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. 3a, b. Physically, the origin of these four pockets can be easily understood through the comparison of orbital resolved band structures between PM phase (Fig. 1b) and (π, π, π) AFM phase (Fig. 4). Because of the Zeeman field on Ni, its spinup and down bands are split away from each other. The original pureband (\({d}_{{x}^{2}{y}^{2}}\) orbital of Ni) in PM phase becomes partially occupied in spinup channel (forming two hole pockets) and totally unoccupied in spindown channel. Hence, the two hole pockets in AFM phase are inherited from large sheet in PM phase, showing a 2D character with negligible 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. 1a]. 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 spindown (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 does not disappear until U = 6 eV. In addition, the orbital components of lighter electron pockets are purified by electron–electron interaction and it is 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. 3d. 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. 3e. Further increasing the value of Hubbard U, the system enters an AFM insulator, just like cuprates. Therefore, the metaltoinsulator phase transition point is near U ~ 6 eV. If Hubbard U is <6 eV, NdNiO_{2} is an AFM metal with relatively small amount of holes that are selfdoped^{16,17,18,20,26} into d orbitals of Ni. Interestingly, there is an orbital switching from \(3{d}_{{x}^{2}{y}^{2}}\) orbital of Ni at U = 0 eV to \(3{d}_{{z}^{2}}\) orbital of Ni at U = 6 eV in the NiO_{2} plane, as depicted in Fig. 3f. We speculate that this orbital switching may change the paradigm after doping^{8,27,28,29,30}. 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 MottHubbard metal to chargetransfer insulator.
For the magnetic ground state, so far, no consensus has been reached. For example, most of studies assumed the (π, π, 0) AFM order as the ground state, without a detailed energetic investigation, e.g. Gu et al.^{22}, Botana et al.^{13}, and Lee et al.^{8}. Addtionally, Zhang et al.^{31} and Hepting et al.^{20} calculated band structures based on (π, π, π) AFM order. Most importantly, the detailed analysis of magnetic ground state is missing in all previous works. In this work, we compare a large number of magnetic configurations and found (π, π, π) (rather than (π, π, 0)) to be the magnetic ground state.
PMAFM transition
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. 5a. 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 infinitelayer 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 labeled in Fig. 2a), 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. Only (π, π, π) and (π, π, 0) are stable at U = 0 eV, and we ignore results at U = 0 eV in the following discussions. The calculated exchange coupling parameters as a function of Hubbard U are shown in Fig. 5b. 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 infinitelayer SrFeO_{2}^{32,33,34}. 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 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. 5c. Then, T is extracted from the peak position in the curve of C(T), as shown in Fig. 5c. 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. 5e, the C(T) vs C(T) plot shows a broaden peak at a lower temperature. Since MerminWagner theorem prohibit magnetic order in 2D isotropic Heisenberg model at any nonzero temperatures^{35}, the broad peak in C(T) vs T plot implies the presence of shortrange order. Regarding the small drop of T_{N} (~30 K in Fig. 5f), our MC simulations indicate that the weak interactions between NiO_{2} planes.
Experimental relevance
To address the relevance of the current calculations and experimental observations, we first estimate the value of Hubbard interaction strength U. Although the exact value of Hubbard U cannot be directly extracted from the firstprinciples calculations, its value range can still be estimated based on similar compounds. The infinitelayer nickelate is believed to be a worse metal compared to elemental nickel with U ~ 3 eV^{36}, which can be considered as a lower bound of Hubbard U. The Coulomb interaction in infinitelayer nickelates should be smaller than that in the chargetransfer insulator NiO with U ~ 8 eV^{37}, which can be considered as a upper bound of Hubbard U. Therefore, a reasonable value of Hubbard U in NdNiO_{2} will be between 3 eV and 8 eV. In the following discussion, we would like to use the median value, U ~ 5–6 eV.
Second, if the interaction strength is around U ~ 5–6 eV, the estimated magnetic exchange interaction is around J_{1} ~ 10 meV, J_{2} ~ 10 meV, J_{3} ~ −13 meV (see Fig. 5b). This indicates that, effective exchange interactions in NdNiO_{2} are about oneorder smaller than those of cuprates ~112 meV^{21,38,39,40,41,42,43,44}. Also it results in a relative weaker magnetic ordering and lower Neel temperature T_{N} in NdNiO_{2}, compared with cuprates (see below).
Third, to further discuss the magnetic transition temperature, we show the Neel temperature dependence on interaction strength in Fig. 5d. Within the estimated interaction range U ~ 5–6 eV, we find the Neel temperature, which separates the PM phase from the (π, π, π) AFM phase, is near 70–90 K. Interestingly, this coincides with the minimum in resistivity and the drop in Hall coefficient^{6}. It provides a novel understanding of minimum in resistivity, i.e. partial electrons tend to localize and form magnetic order, therefore the conducting carriers decreases. Thus, our calculation indicates that, the resistivity minimum and the drop in Hall coefficient relates to the magnetic phase transition at the Neel temperature.
Fourth, previous power neutron scattering measurement on NdNiO_{2} did not find the signal of AFM ordering^{45,46}. This could be attributed to the following reason. The estimation of Neel temperature (in Fig. 5d) does not include the influence of conduction electrons. We notice that, there are small Nd 5delectron pockets near the Fermi surface at the critical interaction strength (Fig. 3e). By concluding the hybridization effect between Nd 5delectrons and Ni 3delectrons, there are at least two influences. On one hand, it is expected that the local magnetic AFM ordering below the Neel temperature is further weakened. On the other hand, the magnetic phase transition is replaced by a crossover from normal metal to bad AFM metal around T_{N} ~ 70–90 K, which is more likely to occur in the experiment^{6}. Both of the above effects lead to further weaken the AFM ordering^{26}, which could be hard to detect in neutron scattering measurement.
Lastly, some recent experiments fail to find bulk superconductivity in NdNiO_{2} systems, and the parent samples show strong insulating behaviors^{47}. The insulating behavior could be attributed to strong inhomogenious disorder or improper introduction of H during the reaction with CaH_{2}^{48}. Especially, it is worth noting, these 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^{45,46}.) 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.
Taken as a whole, we conclude that, our calculations provide a selfconsistent understanding on the experimental observation^{6}: the resistivity minimum and drop in Hall coefficient is related to the crossover from normal metal to bad AFM metal around T_{N} ~ 70–90 K. The AFM ordering at lower temperature is weak due to the hybridization or selfdoping effect, which is hard to detect in the previous measurements. In this regarding, the upcoming inelastic neutron scattering on highquality samples is highly desired.
Discussion
In this work, we present a systematic study of the electronic and magnetic properties of parent compound NdNiO_{2} by firstprinciples calculations combined with classical Monte Carlo calculations. The Fermi surface of PM phase is quite large with two 3Dliked electron pockets, which is consistent with the previous work^{13,14,15,16,18}. By tuning the interaction strength, before it enters correlated insulator, NdNiO_{2} 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. We identify the interaction selects a (π, π, π) AFM order as the correct magnetic ground state. In the reasonable estimation U = 5–6 eV^{20,37}, we find several conclusions related to the experiment^{6}: (1) the exchangecoupling parameters are ~10 meV, which is one order smaller than cuprates^{21,38,39,40,41,42,43,44} and results in a low T_{N} compared with cuprates; (2) 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 longrange AFM order unstable^{6,45,49,46}; (3) with the decreasing of temperature, there is a phase transition from PM phase to (π, π, π) AFM phase near 70–90 K. Therefore, there could exist a transition from normal metal to bad AFM metal around T_{N} ~ 70–90 K, which provides a plausible understanding of minimum of resistivity and Hall coefficient drop in infinitelayer NdNiO_{2}^{6}. We envision that our calculations will intrigue intensive interests for studying the magnetic properties of highquality infinitelayer NdNiO_{2} samples.
Methods
The firstprinciples calculations are carried out with the plane wave projector augmented wave method as implemented in the Vienna ab initio simulations package (VASP)^{50,51,52}. The Perdew–Burke–Ernzerhof (PBE) functionals of generalized gradient approximation (GGA) is used for PM phase^{53}. To incorporate the electronelectron interactions, DFT + U is used for AFM phase, which can reproduce correctly the gross features of correlatedelectrons in transition metal oxides^{54,55,56}. There are two kinds of strongly localized orbitals in NdNiO_{2}: 4f of Nd and 3d of Ni. The 4f electrons of Nd are expected to display the local magnetic moment as Nd^{3+} in Nd_{2}CuO_{4}^{57}. The hybridization between f electrons and superconducting plane is very complicated. In EuRbFe_{4}As_{4}^{58}, superconductivity in FeAs layers and ferromagnetism in Eu layers can coexist. In PrBa_{2}Cu_{3}O_{7}, where the local environment of the 4f ion is the same as in NdNiO_{2}, the antibonding coupling of the Pr \({f}_{z({x}^{2}{y}^{2})}\) orbital to the O 2p orbitals makes PrBa_{2}Cu_{3}O_{7} not superconducting^{59}. Recently, it has been proposed that the intraatomic 4f–5d exchange coupling of Nd may change the understanding of the superconducting properties in NdNiO_{2}^{60}. Since the local magnetic moment of Nd^{3+} is weakly coupled in NdNiO_{2}^{60}, here we ignore the possibility of local moment formed by 4f electron of Nd^{3+} in our calculation by treating them as the corelevel electrons and focus only on the magnetic order fromed by Ni. The Hubbard U (0–8 eV) term is added to 3d electrons of Ni. The energy cutoff of 600 eV, and MonkhorstPack 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^{61,62}. The structure of infinitelayer NdNiO_{2} is shown in Fig. 1a, 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^{45,46}. Due to apical O vacancies, the lattice constant in c direction shrinks (smaller than a direction) and the space group becomes P4/mmm. The experimental lattice constant a = b = 3.92 Å and c = 3.28 Å are used in our calculations.
Data availability
The numerical datasets used in the analysis in this study, and in the figures of this work, are available from the corresponding author on reasonable request.
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Acknowledgements
W.Z. thanks Chao Cao for sharing their unpublished DMFT results, and thanks Filip Ronning, H. H. Wen, G. M. Zhang for helpful discussion. This work was supported by NSFC (No. 11774325, 21603210, 21603205, and 21688102), National Key Research and Development Program of China (No. 2017YFA0204904 and 2016YFA0200604), Anhui Initiative in Quantum Information Technologies (No. AHY090400), Fundamental Research Funds for the Central Universities and the Startup Funding from Westlake University. We thank Supercomputing Center at USTC for providing the computing resources.
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W.Z. and J.Y. proposed the idea. Z.L. carried out the calculations and drafted the article. All authors contribute to the analysis of date, writing, and revision of the manuscript.
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Liu, Z., Ren, Z., Zhu, W. et al. Electronic and magnetic structure of infinitelayer NdNiO_{2}: trace of antiferromagnetic metal. npj Quantum Mater. 5, 31 (2020). https://doi.org/10.1038/s4153502002291
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