#
Magnetic and Electronic Properties of Sr Doped Infinite-Layer NdNiO_{2} Supercell: A Screened Hybrid Density Functional Study

^{*}

## Abstract

**:**

_{2}, we carried out the screened hybrid density functional study on the Nd

_{9-n}Sr

_{n}Ni

_{9}O

_{18}(n = 0–2) unit cells. Geometries, substitution energies, magnetic moments, spin densities, atom- and lm-projected partial density of states (PDOS), spin-polarized band structures, and the average Bader charges were studied. It showed that the total magnetic moments of the Nd

_{9}Ni

_{9}O

_{18}and Nd

_{8}SrNi

_{9}O

_{18}unit cells are 37.4 and 24.9 emu g

^{−1}, respectively. They are decreased to 12.6 and 4.2 emu g

^{−1}for the Nd

_{7}Sr

_{2}Ni

_{9}O

_{18}-Dia and Nd

_{7}Sr

_{2}Ni

_{9}O

_{18}-Par unit cells. The spin density distributions demonstrated that magnetic disordering of the Ni atoms results in the magnetism decrease. The spin-polarized band structures indicated that the symmetry of the spin-up and spin-down energy bands around the Fermi levels also influence the total magnetic moments. Atom- and lm-projected PDOS as well as the band structures revealed that $\mathrm{Ni}({d}_{{x}^{2}-{y}^{2}})$ is the main orbital intersecting the Fermi level. As a whole, electrons of Sr atoms tend to locate locally and hybridize weakly with the O atoms. They primarily help to build the infinite-layer structures, and influence the electronic structure near the Fermi level indirectly.

## 1. Introduction

_{1.85}Ba

_{0.15}CuO

_{4}by Bednorz and Müller in 1986 [2], the superconducting mechanism has not achieved consensus in spite of considerable quantities of experimental and theoretical studies [3]. In recent years, the discovery of a new class of superconductivity in infinite-layer nickelate has attracted tremendous attention in the superconductivity area [4,5,6,7,8,9,10,11], which has sparked researchers’ expectations of addressing this issue. That is because the novel Sr-doped NdNiO

_{2}superconductors were assumed as a structural and electronic analog of the well-known cuprate superconductor [12,13,14]. Specifically, despite certain differences [15,16], both Ni

^{+}and Cu

^{2+}ions are formally d

^{9}in the respective parent compounds. Near the Fermi level, Ni and Cu also have similar electronic structures. Thereby, a study on the nickelate superconductors would have a great impact on not only the nickelates but also the other layered transition-metal oxides [17,18,19]. It may provide a new opportunity for further understanding the unconventional superconductivity.

_{0.8}Sr

_{0.2}NiO

_{2}, but arise from the interface or the stress effect [23]. However, based on the comparison between NdNiO

_{2}and Nd

_{0.8}Sr

_{0.2}NiO

_{2}, D. Li’s opinion was that the interface effect alone does not lead to superconductivity [4]. Theoretical studies by Si et al. indicated that the unexpected topotactic hydrogen intercalation has dramatic consequences for the superconductivity of the infinite-layer Sr-doped NdNiO

_{2}[12]. Malyi et al. even emphasized that stoichiometric NdNiO

_{2}is significantly unstable, and the incorporation of hydrogen can reduce its instability [24].

_{0.8}Sr

_{0.2}NiO

_{2}has also been set in [21,23,32,33,34]. Regarding the magnetism being different from the NdNiO

_{3}and cuprate, antiferromagnetic coupling in nickelates is substantially weaker [4,5,15,16,32]. That was attributed to the competition between ferromagnetic (FM) and antiferromagnetic (AFM) exchanges, which significantly suppressed the magnetic tendency [3]. LDA + DMFT studies by Ryee et al. indicated that hole-doping induced magnetic two-dimensionality is the key to superconducting Nd

_{1−x}Sr

_{x}NiO

_{2}[35]. Zhang et al. supported that the self-doping effect, namely strong Kondo coupling-induced holon-doublon excitations, suppresses the antiferromagnetic (AF) long-range order and produces the paramagnetic metallic ground state even in the parent compound NdNiO

_{2}. Sr doping introduces extra holes on the Ni sites, which further drives the system away from the AF Mott insulating phase [32]. A multiorbital description within the spin-freezing theory was employed by Werner et al. They concluded that local moment fluctuations rather than antiferromagnetic fluctuations induce the pairing mechanism of nickelate superconductors [15,36].

_{2}hosts an almost isolated $\mathrm{Ni}({d}_{{x}^{2}-{y}^{2}})$ orbital system, namely its hybridization with the states in the Nd layer is tiny [20].

_{2}. Compared with the molecular dynamics simulations [41,42], the static hybrid density functional method was employed to calculate the geometrical structures, stabilities, magnetism, electronic structures, and charges of the Nd

_{9−n}Sr

_{n}Ni

_{9}O

_{18}(n = 0–2) unit cells. The results were discussed and analyzed in Section 2. Brief conclusions were given in the last Section.

## 2. Results and Discussions

#### 2.1. Structures and Stabilities

_{9-n}Sr

_{n}Ni

_{9}O

_{18}(n = 0–2) are shown in Figure 1. Figure 1 displays the top views (upper diagram) and side views (lower diagram) of Nd

_{9}Ni

_{9}O

_{18}, Nd

_{8}SrNi

_{9}O

_{18}, Nd

_{7}Sr

_{2}Ni

_{9}O

_{18}-Par, and Nd

_{7}Sr

_{2}Ni

_{9}O

_{18}-Dia unit cells, respectively. Among them, Nd

_{9}Ni

_{9}O

_{18}is the pure nickelate with no doping Sr ions. Nd

_{8}SrNi

_{9}O

_{18}is endowed with one Sr ion. Nd

_{7}Sr

_{2}Ni

_{9}O

_{18}-Par and Nd

_{7}Sr

_{2}Ni

_{9}O

_{18}-Dia denote two inequivalent Nd

_{7}Sr

_{2}Ni

_{9}O

_{18}models. One possesses parallel aligned Sr ions, the other has diagonally aligned Sr ions with the lattice vector. Calculated lattice constants of the Nd

_{9}Ni

_{9}O

_{18}supercell are a = b = 11.73 Å (3.91 Å for single cell) and c = 3.31 Å, which consist well with the experimental results of a = b = 3.92 Å and c = 3.31 Å [34]. With respect to the Nd

_{8}SrNi

_{9}O

_{18}and Nd

_{7}Sr

_{2}Ni

_{9}O

_{18,}the a and b stay almost invariable, whereas the c axes rise to 3.32 and 3.34 Å consecutively. The value 3.34 Å also matches well with the experimental value 3.37 Å (3.34~3.38 Å) of Nd

_{0.8}Sr

_{0.2}NiO

_{2}film [4]. That is to say, the more Nd atoms were replaced, the longer the c axis is.

_{2}. The substitution energy of the second Sr atom is 1.03 eV and 0.69 eV for Nd

_{7}Sr

_{2}Ni

_{9}O

_{18}-Par and Nd

_{7}Sr

_{2}Ni

_{9}O

_{18}-Dia, respectively. In comparison with the first Sr atom, the value of the second Sr atom for the Nd

_{7}Sr

_{2}Ni

_{9}O

_{18}-Par is almost invariable, while for the Nd

_{7}Sr

_{2}Ni

_{9}O

_{18}-Dia structure, the substitution energy decreases apparently. That is, the alignment of the Sr atoms influenced the stabilities. The two Sr atoms are more prone to aligning along the diagonal line instead of the parallel line of the lattice.

#### 2.2. Magnetism

_{2}supercell was delegated to test the spin-orbital coupling effect first. The <0 0 1> and <0 1 0> spin quantization axes were taken into account. Only small energy differences of 0.01 eV were found for these two directions. Except for that, the obtained magnetic moments for both cases were (0, 0, 9) ${\mu}_{b}$, which resembled the collinear calculation. Therefore, the spin-orbital coupling effect was omitted, and collinear magnetic structures were calculated in this study.

_{2}planes and mainly localized on the Ni atoms.

_{9}Ni

_{9}O

_{18}displays mostly a paramagnetic ground state. Obtained total magnetic moment of the Nd

_{9}Ni

_{9}O

_{18}unit cell is 37.4 emu g

^{−1}(~1 ${\mu}_{b}$ per Ni ion). It is also comparable with the experimental data of 26.7 emu g

^{−1}at ~2 K and 30 kOe [23]. With the Sr substitution for Nd atoms, magnetic disordering namely, a slow fluctuation of the magnetic moment appears [15]. Via a two-orbital model, Hu et al. interpreted this as the competition between ferromagnetism and anti-ferromagnetism exchanges, thus the magnetic tendency is significantly suppressed [3]. The magnetic moment of the Nd

_{8}SrNi

_{9}O

_{18}shrinks to 24.9 emu g

^{−1}. For the Nd

_{7}Sr

_{2}Ni

_{9}O

_{18}, spins of certain Ni atoms turn negative. Total magnetic moments of the Nd

_{7}Sr

_{2}Ni

_{9}O

_{18}-Dia and Nd

_{7}Sr

_{2}Ni

_{9}O

_{18}-Par models decrease furtherly to 12.6 and 4.2 emu g

^{−1}, respectively. The calculated value of 12.6 emu g

^{−1}for Nd

_{7}Sr

_{2}Ni

_{9}O

_{18}-Dia is in concordance with the experimental value of 19.1 emu g

^{−1}for Nd

_{0.8}Sr

_{0.2}NiO

_{2}[23]. The unfounded value of 4.2 emu g

^{−1}may be due to the higher energy of Nd

_{7}Sr

_{2}Ni

_{9}O

_{18}-Par than the Nd

_{7}Sr

_{2}Ni

_{9}O

_{18}-dia. In other words, the fabrication of Nd

_{7}Sr

_{2}Ni

_{9}O

_{18}-Dia is easier than the Nd

_{7}Sr

_{2}Ni

_{9}O

_{18}-Par during the experimental assembly. The decreasing tendency of magnetic moments demonstrated that doping of Sr elements obviously suppress the magnetic moment of NdNiO

_{2}[23].

#### 2.3. Electronic Structure

_{9−n}Sr

_{n}Ni

_{9}O

_{18}(n = 0–2) models are listed in Figure 3. The Fermi levels read as 0 eV. On one hand, for the Sr substituted models, electrons of Sr atoms tend to locate locally at about −18 eV and hybridize weakly with the O atoms. Except for that, no conspicuous hybridization was found among Sr atoms and the other atoms. Apart from the O–Sr hybridization, the primary hybridization can be classified approximately into two energy ranges, namely, −24~−19 eV for O–Nd atoms and −9~0 eV for the O–Ni atoms. Further analysis based on our lm-projected DOS illuminated them as the $\mathrm{O}\left(s\right)\text{-}\mathrm{Sr}\left({p}_{x},{p}_{y},{p}_{z}\right)$,$\mathrm{O}\left(s\right)\text{-}\mathrm{Nd}\left({p}_{x},{p}_{y},{p}_{z}\right)$, and the $\mathrm{O}\left({p}_{x},{p}_{y},{\mathrm{p}}_{z}\right)\text{-}\mathrm{Ni}\left({\mathrm{d}}_{xy},{\mathrm{d}}_{yz},{\mathrm{d}}_{xz},{\mathrm{d}}_{{z}^{2}},{\mathrm{d}}_{{x}^{2}-{y}^{2}}\right)$ orbital overlaps. Hybridization between the Ni electronic states and the states of the Nd layer is tiny [20].

_{9-n}Sr

_{n}Ni

_{9}O

_{18}(n = 0–2) models are listed in Figure 5. The bands intersecting the Fermi levels were marked with orange, red, and blue. Combing with the lm-decomposed calculations for each ion at each K-points and based on the analysis at G (0, 0, 0) point, it found that these three colored bands corresponded to the $\mathrm{Ni}({d}_{xz},{d}_{yz})$, $\mathrm{Ni}({d}_{{x}^{2}-{y}^{2}})$ and $\{\mathrm{Ni},\mathrm{Nd}\}\text{-}({d}_{{z}^{2}})$ orbitals, respectively. That is, the band couple primarily acts as the hybridization among these three kinds of orbitals near the Fermi level. The $\{\mathrm{Ni},\mathrm{Nd}\}\text{-}({d}_{{z}^{2}})$ orbitals close to the Fermi level were also discovered by Tam et al. using NiL

_{3}resonant X-ray scattering [49]. Moreover, $\mathrm{Ni}({d}_{{x}^{2}-{y}^{2}})$ is the main orbital intersecting the Fermi level.

_{9-n}Sr

_{n}Ni

_{9}O

_{18}(n = 0–2) models, it was also found that the symmetry of the bands has an impact on the magnetic moments. The highest symmetry of the spin-up and spin-down energy bands for the Nd

_{7}Sr

_{2}Ni

_{9}O

_{18}-Par lead to the smallest magnetic moment. While the lowest symmetry of the spin-up and spin-down energy bands for the Nd

_{9}Ni

_{9}O

_{18}induces the biggest magnetic moment. Considering the DOS and Energy bands comprehensively, Sr atoms influence the electronic structure indirectly.

#### 2.4. Charge

_{0.8}Sr

_{0.2}NiO

_{2}was reported as the hole-doped superconductor [4]. But what doped holes are introduced upon the chemical substitution of the Sr atom, the Ni or O sites? Hirsch et al. assumed that, like the cuprates, the added holes go into the oxygen pπ orbitals [50]. Whereas, Zhang and Chang et al. suggested that the doped holes enter the Ni orbitals and form a conducting band [15,32,51].

_{9-n}Sr

_{n}Ni

_{9}O

_{18}(n = 0–2) models. Average Bader charges for each kind of atom were listed in Table 1. The involved numbers of valence electrons for the isolated Nd, Sr, Ni, and O atoms are 11, 10, 10, and 6, respectively. As can be seen, the average charge of the Ni atom is approximately +1, which agrees with the widely accepted opinion [3,52]. Both the Ni and O atoms lose electrons with the increasing substitution of Sr atoms. Specifically, the average numbers of electrons decrease from 7.39 to 7.33 and 7.36 for the O atoms. The average numbers of electrons decrease from 9.29 to 9.16 and 9.18 for Ni atoms. However, for the two Nd

_{7}Sr

_{2}Ni

_{9}O

_{18}models, the Ni atoms nearest both the two Sr atom favor getting the minimums of 9.08 (Dia) and 9.13 (Par).

## 3. Models and Computational Details

_{2}possesses a space group crystalline symmetry of P4/mmm. The Ni-site surrounded by four O anions establishes a planar square NiO

_{2}structure [3,53]. Each Ni ion is surrounded by four O ions in the basal plane and is absent of apical oxygens above and below it. These NiO

_{2}planes are simply separated by Nd layers. In this work, the supercells of size 3 × 3 relative to the primitive P4/mmm cell were employed to explore the effect of Sr ions. Four different models were built with the increasing Sr substitution for Nd ions.

^{2}6s

^{2}5p

^{6}5d

^{1}), Sr (4s

^{2}5s

^{2}4p

^{6}), Ni (3d

^{9}4s

^{1}), and O (2s

^{2}2p

^{4}) electrons were treated as valence electrons. Fully relaxed configurations namely, both lattice parameters and atomic positions were optimized using the Quasi-Newton optimization scheme. Architectures were relaxed until the Feynman-Hellman force on each atom was less than 0.01 eV Å

^{−1}. A fairly robust mixture of the blocked Davidson and RMM-DIIS iteration schemes was used for the electronic minimization algorithm during the geometry optimization [60]. Moreover, the screened exchange hybrid density functional based on the Heyd−Scuseria−Ernzerhof (HSE06) method was adopted to get the exact magnetic and electronic properties [61]. Meanwhile, the iterative matrix diagonalization algorithm which updates all orbitals simultaneously was employed for the static calculations [62,63]. Convergence threshold for the self-consistent field (SCF) total energy was 10

^{−5}eV. For both the geometry relaxation and the static calculations, a Monkhorst-Pack grid of 1 × 1 × 4 was used to sample the first Brillouin zone of K space. The kinetic energy cutoff for the plane-wave basis set was 630 eV. Both of them had been carefully checked. Spin polarized calculations with full relaxation of spins were performed. It should be mentioned that non-magnetic calculations as well as improper initial magnetic moments for spin-polarized calculation hardly converge.

## 4. Conclusions

_{9-n}Sr

_{n}Ni

_{9}O

_{18}(n = 0–2) unit cells were studied with the HSE06 method. Geometries and formation energies as well as the spin densities, site- and lm-projected DOS, spin-polarized band structures, and Bader charges were calculated.

_{2}planes and mainly localized on the Ni atoms. The total magnetic moment of the Nd

_{9}Ni

_{9}O

_{18}shrinks apparently with the Sr substitution for Nd atoms. According to the spin density distributions, the magnetic disordering of the Ni atoms is responsible for the magnetism decrease. While, from a perspective of spin-polarized band structures, it showed that the symmetry of the spin-up and spin-down energy bands around the Fermi levels also influence the total magnetic moments.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Sample Availability

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**Figure 1.**Unit cells of the Nd

_{9-n}Sr

_{n}Ni

_{9}O

_{18}(n = 0–2). The red, green, blue, and purple balls denote O, Nd, Ni, and Sr atoms, respectively.

**Figure 2.**Spin densities contour for the Nd

_{9-n}Sr

_{n}Ni

_{9}O

_{18}(n = 0–2) (001) surface. The blue and red fillings represent negative and positive electron spin densities. The unit of the contour is e.

**Figure 3.**Site-projected partial density of states for the Nd

_{9-n}Sr

_{n}Ni

_{9}O

_{18}(n = 0–2) models. The vertical dashed line denotes the Fermi level. The arrows underline the apparent variation near the Fermi level.

**Figure 4.**The lm-projected partial density of states for the Ni (

**left**) and O (

**right**) atoms of the Nd

_{9-n}Sr

_{n}Ni

_{9}O

_{18}(n = 0–2) models.

**Figure 5.**Spin-up (

**left**) and spin-down (

**right**) band structures for the Nd

_{9-n}Sr

_{n}Ni

_{9}O

_{18}(n = 0–2) models. The green horizontal solid line denotes the Fermi level. The black lines represent the bands being away from the Fermi levels. The orange, red, and blue lines correspond to the $\mathrm{Ni}({d}_{xz},{d}_{yz})$, $\mathrm{Ni}({d}_{{x}^{2}-{y}^{2}})$ and $\{\mathrm{Ni},\mathrm{Nd}\}\text{-}({d}_{{z}^{2}})$ orbitals intersecting the Fermi levels, respectively.

**Table 1.**Average Bader charges for the atoms in Nd

_{9-n}Sr

_{n}Ni

_{9}O

_{18}(n = 0–2) models. The two values in the parentheses represent the minimums and maximums.

Nd | Sr | Ni | O | |
---|---|---|---|---|

Nd_{9}Ni_{9}O_{18} | 8.93 (8.93, 8.93) | 9.29 (9.29, 9.29) | 7.39 (7.39, 7.39) | |

Nd_{8}SrNi_{9}O_{18} | 8.89 (8.89, 8.90) | 8.40 (8.40, 8.40) | 9.26 (9.26, 9.29) | 7.39 (7.37, 7.40) |

Nd_{7}Sr_{2}Ni_{9}O_{18}-Par | 9.01 (8.83, 9.14) | 8.43 (8.43, 8.43) | 9.18 (9.13, 9.20) | 7.36 (7.32, 7.39) |

Nd_{7}Sr_{2}Ni_{9}O_{18}-Dia | 9.07 (8.87, 9.35) | 8.55 (8.55, 8.55) | 9.16 (9.08, 9.23) | 7.33 (7.31, 7.35) |

Isolated atom | 11 | 10 | 10 | 6 |

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## Share and Cite

**MDPI and ACS Style**

Hua, Y.; Wu, M.; Qin, Q.; Jiang, S.; Chen, L.; Liu, Y.
Magnetic and Electronic Properties of Sr Doped Infinite-Layer NdNiO_{2} Supercell: A Screened Hybrid Density Functional Study. *Molecules* **2023**, *28*, 3999.
https://doi.org/10.3390/molecules28103999

**AMA Style**

Hua Y, Wu M, Qin Q, Jiang S, Chen L, Liu Y.
Magnetic and Electronic Properties of Sr Doped Infinite-Layer NdNiO_{2} Supercell: A Screened Hybrid Density Functional Study. *Molecules*. 2023; 28(10):3999.
https://doi.org/10.3390/molecules28103999

**Chicago/Turabian Style**

Hua, Yawen, Meidie Wu, Qin Qin, Siqi Jiang, Linlin Chen, and Yiliang Liu.
2023. "Magnetic and Electronic Properties of Sr Doped Infinite-Layer NdNiO_{2} Supercell: A Screened Hybrid Density Functional Study" *Molecules* 28, no. 10: 3999.
https://doi.org/10.3390/molecules28103999