Effective Hamiltonian for nickelate oxides ${\mathrm{Nd}}_{1-x}{\mathrm{Sr}}_x{\mathrm{NiO}}_2$

We derive the effective single-band Hamiltonian in the flat ${\mathrm{NiO}}_2$ planes for nickelate compounds ${\mathrm{Nd}}_{1-x}{\mathrm{Sr}}_x{\mathrm{NiO}}_2$. We implement the first-principles calculation to study electronic structures of nickelates using the Heyd-Scuseria-Ernzerhof hybrid density functional and derive a three-band Hubbard model for Ni-O $pd\sigma$ bands of ${\mathrm{Ni}}^{+}3d_{x^2-y^2}$ and ${\mathrm{O}}^{2-}2p_{x/y}$ orbitals in the ${\mathrm{NiO}}_2$ planes. To obtain the effective one-band $t-t^{\prime }-J$ model Hamiltonian, we perform the exact diagonalization of the three-band Hubbard model for the ${\mathrm{Ni}}_5{\mathrm{O}}_{16}$ cluster and map the low-energy spectra onto the effective one-band models. We find that the undoped ${\mathrm{NiO}}_2$ plane is a Hubbard Mott insulator and the doped holes are primarily located on Ni sites. The physics of the ${\mathrm{NiO}}_2$ plane is a doped Mott insulator, described by the one-band $t-t^{\prime }-J$ model with $t=265\mathrm{meV}, t^{\prime }=-21\mathrm{meV}$, and $J=28.6\mathrm{meV}$. We also discuss the electronic structure for the self-doping effect and heavy fermion behavior of electron pockets of ${\mathrm{Nd}}^{3+}5d$ character in ${\mathrm{Nd}}_{1-x}{\mathrm{Sr}}_x{\mathrm{NiO}}_2$.

Figure 1

(a) Crystal structure of ${\mathrm{NdNiO}}_2$ with space group $P4/mmm$ (No. 123). (b) HSE06 band structures of the nonmagnetic state in the parent compound ${\mathrm{NdNiO}}_2$ along the $\mathrm{\Gamma }(0,0,0)-X(0,0.5,0)-M(0.5,0.5,0)-\mathrm{\Gamma }-Z(0,0,0.5)-R(0,0.5,0.5)-A(0.5,0.5,0.5) Z$ directions. The projected band structures of $d$ orbitals (red for $d_{x^2-y^2}$, blue for $d_{z^2}$, brown for $d_{xz/yz}$, and green for $d_{xy}$) in Ni and Nd and $2p$ orbitals (orange for $p_{x/y}$ and purple for $p_z$) in O are also shown. The Fermi level is set at 0 eV. (c) Nd and its eight nearest-neighbor oxygens. (d) Ni and its four nearest-neighbor oxygens. (e) Crystal field splitting of Nd $5d$ and $\mathrm{Ni}3d$ orbitals.

Figure 2

GGA band structures of the nonmagnetic state in the simulated ${\mathrm{SrNiO}}_2$ with the same structure as ${\mathrm{NdNiO}}_2$. The projected band structures of $d$ orbitals in Sr and $4p$ orbitals in Ni are also shown. The color in (a) has the same meanings as in Fig. 1. The color turquoise in (b) indicates the projected Ni $4s$ orbital (mainly located at the $\mathrm{\Gamma }$ point at around 2.4 eV).

Figure 3

HSE06 band structures of $G$-type AFM states for a $\sqrt{2}\times \sqrt{2}\times 2$ supercell in (a) ${\mathrm{NdNiO}}_2$ and (b) ${\mathrm{Nd}}_{0.75}{\mathrm{Sr}}_{0.25}{\mathrm{NiO}}_2$. The notation $\mathrm{\Gamma }, X, M, Z, R$, and $A$ is in the folded magnetic Brillouin zone. The color has the same meanings as in Fig. 1.

Figure 4

Schematic representation of the orbitals ($\mathrm{Ni}3d_{x^2-y^2}$ and O $p_{x/y}$) included in the three-band Hubbard model in the ${\mathrm{Ni}}_5{\mathrm{O}}_{16}$ cluster. The effective one-band model has the degree of freedom only on the five $\mathrm{Ni}3d_{x^2-y^2}$ orbitals connected by the solid ($t$ and $J$) and dashed ($t^{\prime }$ and $J^{\prime }$) lines.

Figure 5

Low-energy spectrum for the ${\mathrm{Ni}}_5{\mathrm{O}}_{16}$ cluster calculated in the three-band Hubbard model in comparison to the mappings onto the Heisenberg Hamiltonian for the five-hole case.

Figure 6

Low-energy spectrum for ${\mathrm{Ni}}_5{\mathrm{O}}_{16}$ cluster mapping onto the effective one-band $t-t^{\prime }-J$ and Hubbard models.

Figure 7

GGA band structures of $G$-type AFM states for a supercell in (a) ${\mathrm{NdNiO}}_2$ and (b) ${\text{Nd}}_{0.75}{\text{Sr}}_{0.25}{\text{NiO}}_2$ calculated with the $4f$ valence electron pseudopotential within the $\mathrm{GGA}+\mathrm{U}$ ($U_{4f}=10$ eV) scheme with SOC. The notation $\mathrm{\Gamma }, X, M, Z, R$, and $A$ is in the folded magnetic Brillouin zone. The projected band structures of $f$ orbitals (red) in Nd are also shown. The Fermi level is set at 0 eV.

Figure 8

GGA band structures of ${\mathrm{Nd}}_{1-x}{\mathrm{Sr}}_x{\mathrm{NiO}}_2$, corresponding to (a) Fig. 1 (the nonmagnetic state for ${\mathrm{NdNiO}}_2$), (b) Fig. 3 (the $G$-type AFM state for ${\mathrm{NdNiO}}_2$), and (c) Fig. 3 (the $G$-type AFM state for ${\mathrm{Nd}}_{0.75}{\mathrm{Sr}}_{0.25}{\mathrm{NiO}}_2$), without SOC and (d), (e), and (f), respectively, with SOC.

Figure 9

Similar to Fig. 8, but for the SCAN band structures of ${\mathrm{Nd}}_{1-x}{\mathrm{Sr}}_x{\mathrm{NiO}}_2$.

Figure 10

HSE06 band structures of ${\mathrm{La}}_{1-x}{\mathrm{Sr}}_x{\mathrm{NiO}}_2$ without SOC: (a) nonmagnetic state for ${\mathrm{LaNiO}}_2$, (b) $G$-type AFM state for ${\mathrm{LaNiO}}_2$, and (c) $G$-type AFM state for ${\mathrm{La}}_{0.75}{\mathrm{Sr}}_{0.25}{\mathrm{NiO}}_2$, corresponding to Figs. 1, 3, and 3, respectively.