Electronic Structure and Hole Concentration of Ce1-xSrxNiO2 and Nd1-xSrxNiO2 by Density Functional Theory

Recently, superconductivity was discovered in nickelate Nd0.8Sr0.2NiO2. It is predicted to be a cuprate-like superconductor. Cerium, with its natural oxidation states 3+ and 4+, has a stronger effect on the modification of the Ni electronic structure near the Fermi level than neodymium. This work investigates the electronic states and the charge distribution of CeNiO2 and NdNiO2 compounds. Hole concentration and holes doped relative to the parent compound of Ce1-xSrxNiO2 and Nd1-xSrxNiO2 with x = 0, 0.2, 0.33 are calculated. The results shown that the Ni ion in NdNiO2 have half-filled 3dx2-y2 orbital with ionic charge of Ni+, while Ni ion in CeNiO2 is in mixed charges of Ni+ and Ni2+. The undoped CeNiO2 has almost the same hole concentration as that of 20% Sr-doped NdNiO2.


Introduction
The superconductivity in non-stochiometric compound Nd0.8Sr0.2NiO2with Tc of 15 K was discovered in 2019 by Harold Hwang, his postdoctoral student Danfeng Li, and their colleagues at Stanford University [1].A perovskite phase Nd0.8Sr0.2NiO3thin film was first synthesized on a SrTiO3 (001) substrate by pulsed laser deposition, followed by an oxygen reduction step using CaH2 to obtain an infinite-layer Nd0.8Sr0.2NiO2thin film.Another group of S. Zeng and colleagues at the National University of Singapore confirmed the superconductivity in nickelates in paper published in 2020 [2].In their research work, infinite-layer Nd1-xSrxNiO2 thin films with doping levels x ranging from 0.08 to 0.3 were synthesized and investigated.By studying the phase diagram of Nd1-xSrxNiO2, a superconducting dome with a maximum Tc of about 15 K was found for x between 0.12 and 0.235.The Stanford University research group also further studied the phase diagrams of Nd1-xSrxNiO2 and Pr1-xSrxNiO2 and observed superconducting domes [3], [4].The discovery of nickelate superconductors brought excitement.Ni is next to Cu in the element table, and ions Ni and Cu have incompletely filled 3d orbitals, so they have similar electronic structures near the Fermi level.Therefore, nickelate superconductors are expected to be cuprate-like superconductors and can help to better understand superconductivity.
The first discovered cuprate superconductor was the Ba-La-Cu-O system, discovered in 1986 by Bednorz and Müller of the IBM Zurich Research Laboratory [5].The following year, YBa2Cu3O7-x (in which lanthanum was replaced by yttrium) was found to have a Tc of 93 K [6].Since then, cuprate superconductors have been extensively studied, and as of 2015, more than 200 cuprate superconductors have been discovered, including doped or self-doping of their parent compounds (i.e., YBCO, BSCCO, TBCCO, and HBCCO) [7]- [10].The research on cuprate superconductor is the most mature superconductivity research at present, but it still lacks the final comprehensive theoretical support.Based on what we have learned so far from cuprate superconductors, the key ingredients for cuprate superconductivity to occur are: 1. quasi-2D layered crystal structure, 2. half-filled single band near the Fermi level, 3. antiferromagnetic correlations in the parent compound, and 4. hybridization with ligands orbitals [11].
This work is to investigate whether nickelate superconductors are cuprate-like superconductors by studying the behaviour of nickelates to see if key ingredients of cuprate superconductors are observed.Inspired by Ce has higher natural oxidation state than Nd, this work also investigated Ce substitution on Nd to understand how Ce substitution affects the electronic structure of the compound.The work calculated the electronic structures and hole concentration in Ce1-xSrxNiO2 and Nd1-xSrxNiO2 for x = 0, 0.2, and 0.33 using density functional theory.The hole concentrations of the compounds are also calculated.The relationship between nickelate and cuprate superconductors is discussed in Section Results and Discussion.

Method
Nitrate compounds with two general formulae Ce1-xSrxNiO2 and Nd1-xSrxNiO2 with x = 0, 0.2, 0.33 were investigated.Their unit cells are shown in Figure 1, where Figure 1(a) is the parent compound (undoped) nickelate superconductor with chemical formula CeNiO2 or NdNiO2, while Figure 1(b) and Figure 1(c) are Sr doping of x = 0.2 and 0.33, respectively.They are the models used for computations by the Quantum Espresso code based on density functional theory (Giannozzi et al., 2009).A Marzari-Vanderbilt smearing method [12] with degauss value of 0.01 Ry was used to expand the occupation function around the Fermi level.The exchange-correlation energy are calculated by generalized gradient approximation (GGA) of Perwel et al. [13].The Brillouin zone is sampled using a uniform (unbiased) k-point Monkhorst-Pack grid, with 4 × 4 × 4 k-points for self-consistent and 8 × 8 × 8 k-points for band calculation.The convergence threshold and kinetic cutoff energy were set to 1 × 10 -8 Ry and 500 Ry, respectively.The Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm [14], [15] was used for structural relaxation.Structural optimization is achieved when all forces acting on atoms in the unit cell are less than 1 × 10 -3 Ry/Bohr.

Results and Discussion
The structure of the nickelate superconductors are shown in Figure 1, which is a layered structure, with a Ni-O2 layer as the main layer for electrical conductivity and other layers as electronically inert layer.These structures are called quasi-2D structures because the dimensionality, in an electronic sense, is reduced from 3 to 2. Dimensionality reduction often leads to quantum metal behaviour that cannot be explained by conventional band theory.The parent compounds of YBCO is an insulator [16].It insulating behaviour can be explained based on the Mott-Hubbard model [17].Cu ion in undoped compounds have half-filled 3d orbitals with an odd number of electrons per Cu-O2 unit cell.The large repulsive energy between electrons (Hubbard U) prevents electrons from moving easily from one site to another.The resulting electrons are localized and highly correlated, that is, one electron hopping to another site and the electron on that side has to shift out.Also due to the Pauli exclusion principle, two electrons occupying the same state require anti-spin, so that the electron in the equilibrium state has anti-spin with the adjacent site, thereby forming an antiferromagnetic ordered phase.Figure 2 shown the calculated band structures of (a) CeNiO2 and (b) NdNiO2.Similar to the case where the energy band of the Cu-O2 layer of YBCO crossed the Fermi level [18], [19], the energy band of the Ni-O2 layer also crossed the Fermi level (along the momentum paths of X-M and M-G), forming a maximum at the point M. Since the states are occupied only upto the Fermi level, it is suggested that the orbital of Ni ion near the Fermi level have incompletely filled bands, just like the Cu case.However, the band structure of nickelates does not have a profound gap as in the case of YBCO, as can be observed that there are bands crossing the Fermi level from the conduction to the valence band.The origin of these energy bands can be determined by Density of State (DOS) calculations, for the CeNiO2 case, they originate from the 4f orbital of Ce, and for the NdNiO2 case, they originate from the 5d orbital of Nd. Figure 3 shown the density of state of CeNiO2 and NdNiO2.From Figure 3(a) and Figure 3(c), it can be seen that the oxygen 2p state is located fairly below the Fermi level, while the nickel 3d state spans around the Fermi level.Therefore, the results indicate that the oxygen 2p orbitals are not hybridized with the nickel 3d orbitals, which is consistent with the work of Hepting et al. [20].In Figure 3(a), the 4f orbital of Ce has a kurtosis near the Fermi level, and the states above the Fermi level are coupled with the states of the Ni 3d orbital.Whereas in Figure 3(c), the 5d orbital of Nd contributes a considerable number of states near and above the Fermi level.Indeed, the energy bands from the conduction band cross the Fermi level to the valence band in Figure 2(a) and Figure 2(b) come from 4f orbital of Ce and 5d orbital of Nd, respectively.These results, however, indicates that the Ni-O2 layer is not completely insulated by the rare earth layers, hence, they may are conductive and have almost isotropic resistivity.This result is consistent with the metallic prediction for LaNiO2 in the work of Lee et al. [21].The results differ from cuprate superconductors because the anisotropic transport property is a key characteristic of cuprates.The experimental work of Wang et al. also observed isotropic magneto-transport properties in nickelate superconductors [22] which against the anisotropic character of cuprate superconductors.The Ni 3d orbital states of CeNiO2 and NdNiO2 have almost the same pattern, but only the CeNiO2 3d orbital has a wider bandwidth.The wider bandwidth is due to the larger ionic charge of Ce 3+ attracts nearby oxygen ions and leads to smaller layer separation.Figure 3(b) and Figure 3(d) shows the DOS of 5 sublevel orbitals of 3d for CeNiO2, and NdNiO2, respectively.The results show that the eg orbital has John-Teller distortion, where spitting of sub-orbitals 3dz 2 and 3dx 2 -y 2 occurred, and 3dx 2 -y 2 is the main orbital near the Fermi level for both compounds.From the partial density of states results in Figure 3  Figure 4 shown the spatial charge distribution plots of CeNiO2 and NdNiO2 near the Fermi level, with the Ni atom at the centre for all 4 plots.The shape of 3dx 2 -y 2 orbital is clearly observed for both compounds.This result is consistent with the DOS results, both showing that 3dx 2 -y 2 is the dominant orbital near the Fermi level.Deformation of 3dx 2 -y 2 orbital have been observed for both compounds.With the help of DOS data, the origin of the orbital deformation was determined.In Figure 4(b), a dumbbell-shaped charge distribution that parallel to the z direction can be observed.The round head of the dumbbell-shaped charge distribution is contributed by the Ni 3dx 2 -y 2 orbital being coupled with the 4fz 3 orbital of Ce in the adjacent layer, while the Ni 3dz 2 contributes to the small sphere in the centre.For NdNiO2, the Ni 3dx 2 -y 2 is coupled to the 5dx 2 -y 2 orbital of Nd.The wavefunction nodes of NdNiO2 cannot be observed in Figure 4(c) because of the other 3d orbitals of Ni provide the charges and form a sphere in the centre.Overall, the orbital size of CeNiO2 is larger than that of NdNiO2, which may be due to the fact that Ce atoms with larger oxidation states provide stronger charge interactions with Ni.This results in stronger electronic connections between atoms/layers.Cuprate superconductivity is caused by charge carrier doping into its parent compound.The same is true for nickelate superconductivity, which only appear when they are doped with a certain amount of impurities.Table 1 presents the calculated hole concentrations and holes doped relative to the parent compound of Ce1-xSrxNiO2 and Nd1-xSrxNiO2.The h is the hole concentration, it is calculated by summing all empty states of Ni 3d orbitals.P is the holes doped with relative to its parent compound.
The calculated hole concentration of NdNiO2 is 1.16.From the inherent uncertainty of quantum electronic states, the wavefunction of Ni 3d orbitals is in mixed state of α|3d 8 ⟩ + β|3d 9 ⟩, where α and β are the coefficient of the states and with property of α 2 + β 2 = 1.Therefore, from the calculated hole concentration of NdNiO2, it suggested that, α is much smaller than β, so Ni is in a strong 3d 9 state and therefore in ionic charge Ni + .While for CeNiO2 with hole concentration of 1.48, the Ni 3d orbital is in mixed state of 3d 8 and 3d 9 , therefore Ni ions are in mixed charges of Ni + and Ni 2+ .The hole concentration of Sr-doped NdNiO2 increases to 1.50 and 1.23 with Sr doping amounts of 20% and 33%, respectively.This gives improved hole counts of 0.34 and 0.07 for 20% and 33% Sr doping levels, respectively.For the CeNiO2 case, Sr doping has little effect on hole concentration, which is 1.48 and 1.50 when Sr doping is 20% and 33%, respectively.Although doping Sr on CeNiO2 does not affect much on the hole concentration, however, the hole concentration of undoped CeNiO2 is almost the same as that of 20% Sr-doped NdNiO2.
Superconducting dome is frequently observed in doping phase diagrams for cuprate superconductors.The maximum Tc is achieved at optimal hole doping, Popt.The region with P < Popt is called under-doped, and the region with P > Popt is called over-doped.Below the superconducting-under-doped region is the antiferromagnetic insulating phase, and above the superconducting-over-doped region is the Fermi liquid metallic phase.While similar superconducting dome is observed in nickelate superconductors, however, the phase above the superconducting-Sr-over-doped region of Nd1-xSrxNiO2 is a weakly insulating phase [3].For cuprate superconductors, the improvement in the number of carriers is usually proportional to the number of impurities doped.For example, holes number doped are proportional to the number of oxygen atom doped in YBCO (exceptions for Ca-doped YBCO [19], [23]).Sr-doped NdNiO2 is behaved differently, where 20% Sr doping corresponds to 0.34 hole doping, but 33% Sr doping yields only 0.07 hole doping.This low number of hole doped of higher Sr doping percentages is consistent with the observation of weak insulating phase above the superconducting-Sr-over-doped regions.If mimicking the cuprate superconductor phase diagram with the doping number of holes as the horizontal axis, this Sr-over-doped region can be considered as a hole-under-doped region.

Conclusion
The discovery of the nickelate superconductor is exciting because researchers believe it may be a cuprate-like superconductor, helping to improve the understanding of superconductors.For YBCO, other rare earth elements can replace yttrium and still exhibit superconductivity with roughly the same Tc, however, exception for Ce, Pr, and Tb [24], [25].That being said the discovery of Pr1-xSrxNiO2 superconductor brings a contradiction to the study of the similarity between Ni superconductor and Cu superconductor.
From the results of this work, the following conclusions can be drawn: 1.For both CeNiO2 and NdNiO2 compounds, the oxygen 2p orbitals are not hybridized with the nickel 3d orbitals.2. Ni ion in NdNiO2 have half-filled 3dx 2 -y 2 orbital with ionic charge of Ni + , while Ni ion in CeNiO2 are in mixed charges of Ni + and Ni 2+ .3. Sr-doped NdNiO2 improved hole counts of 0.34 and 0.07 for 20% and 33% Sr doping levels, respectively.4. Sr-doped on CeNiO2 does not affect much on the hole concentration, however, the hole concentration of undoped CeNiO2 is almost the same as that of 20% Sr-doped NdNiO2.According to the results, nickelate and cuprate superconductors have similarities and differences.Not only does this help determine whether a Ni superconductor is a Cu superconductor, but it also helps narrow down what is most relevant to superconductivity.The substitution of Ce to Nd has a significant effect on the electronic properties of the compounds.Ce with strong oxidation states leads to larger energy level splitting and leads to larger orbital bandwidth.This results in stronger electronic connections for CeNiO2 than for NdNiO2.
Nickelate superconductors may have potential use in the development of high-temperature superconducting wires and other electronic devices.However, more research is needed to fully understand and optimize their properties for practical applications.Future investigation could focus on investigating the effect of doping with different elements on the electronic properties and superconductivity of nickelate compounds.

Figure 2 .
Figure 2. Calculated band structures of (a) CeNiO2 and (b) NdNiO2, the vertical axis is the total energy level of the calculated system with reference to the Fermi level, and the horizontal axis is the wavefunction along the highly symmetrical points.(c) shows the highly symmetry points in tetragonal Brillouin zone.
(b) and Figure 3(d), the John-Teller gaps of CeNiO2 and NdNiO2 are estimated to be about 2 eV and 1 eV, respectively.

Figure 3 .
Figure 3. Calculated density of states.Partial density of states of (a) CeNiO2, and (c) NdNiO2 of 2P of O, 3d of Ni, and 4f and 5d of Ce and Nd, respectively.The DOS of 5 sublevel orbitals of 3d for (b) CeNiO2, and (d) NdNiO2.The horizontal axis is the total energy level with reference to the Fermi level.

Figure 4 .
Figure 4. Spatial charge distribution surface plots of CeNiO2 on (a) (2,0,0) plane, and (b) (0,0,2) plane, with the Ni atom at the centre.Spatial charge distribution surface plots of NdNiO2 on (c) (2,0,0) plane, and (d) (0,0,2) plane, with the Ni atom at the centre.The charge distribution was calculated by integrating the total number of states within ±0.5 eV of the Fermi level.The colour bar legend shows the total number of states.

Table 1 .
The calculated hole concentration per unit Ni and holes doped relative to the parent compound of Ce1-xSrxNiO2 and Nd1-xSrxNiO2 with x = 0, 0.2, 0.33.h is the hole concentration; it is calculated by summing all empty states of 3d orbitals of Ni.P is the holes doped with relative to its parent compound.