Symmetry-breaking polymorphous descriptions for correlated materials without interelectronic U

Correlated materials with open-shell $d$ and $f$ ions having degenerate band-edge states show a rich variety of interesting properties ranging from metal-insulator transition to unconventional superconductivity. The textbook view for the electronic structure of these materials is that mean-field approaches are inappropriate, as the interelectronic interaction U is required to open a band gap between the occupied and unoccupied degenerate states while retaining symmetry. We show that the latter scenario often defining what Mott insulators are is in fact not needed for the $3d$ binary oxides MnO, FeO, CoO, and NiO. The mean-field-like band theory can indeed lift such degeneracies in the binaries when nontrivial unit-cell representations (polymorphous networks) are allowed to break symmetries, in conjunction with a recently developed nonempirical exchange and correlation density functional without an on-site interelectronic interaction U. We explain how density-functional theory in the polymorphous representation achieves band-gap opening in correlated materials through a separate mechanism from the Mott-Hubbard approach. We show the method predicts magnetic moments and gaps for the four binary monoxides in both the antiferromagnetic and paramagnetic phases, offering an effective alternative to symmetry-conserving approaches for studying a range of functionalities in open $d$- and $f$-shell complex materials.

Figure 1

SCAN band structures of the four G-type AFM transition-metal monoxides calculated with the experimental NaCl crystal structures. Orbital characters are indicated where red circles ($○$) are the $3d$ states in the majority spin channel, green squares ($\square$) the $3d$ states in the minority spin channel, and blue crosses ($\times$) the transition-metal $4s$ sates. Interaction patterns of the $3d$ states, i.e., bonding $\left({}^b\right)$, antibonding $\left({}^a\right)$, and nonbonding $\left({}^n\right)$, are also labeled. For FeO, the inset shows the charge density distribution of the highest occupied band $a_{1g}$ band, which is a linear combination of the $d_{xy}$, $d_{yz}$, and $d_{xz}$ orbitals [51, 56]. The valence-band maximum (VBM) is located at the “M” point, which is not included in the standard K path of this figure. The regions between occupied and unoccupied states are shaded with yellow. Calculation details are given at end of the main text.

Figure 2

Density of states of four transition-metal monoxides with the G-type AFM and SQS-PM spin configurations calculated by SCAN. All crystal structures are fixed to the experimental data (“Unrelaxed” in Table 1). The SQS model contains 216 atoms. The band gaps are indicated by the vertical dashed lines. Calculation details are shown at the end of the main text.

Figure 3

Spin and orbital disorder patterns in the paramagnetic phase simulated in the 64-atom-supercell from the SCAN SQS-PM model. (a) MnO, spin disorder visualized from the charge density distributions of the two $e_g^a$ bands. (b) FeO, the $a_{1g}$ band. The two $e_g^a$ bands in MnO and the $a_{1g}$ band in FeO are selected because they are just below the valence-band maximum and are well separated from the other transition-metal $3d$ and O-$2p$ states. Similar information for NiO and CoO is difficult to extract because of the significant orbital mixing. The blue and red isosurfaces denote the two spin channels on each transition-metal ion. Oxygen atoms are omitted for simplicity. The 216-atom supercell contains similar information, but the smaller 64-atom supercell is selected here for clarity.

Figure 4

Self-interaction error reduction and electron localization with SCAN. (a) The dissociation energy curves of the ${{\mathrm{H}}_2}^{+}$ molecule from Hartree-Fock theory (HF), PBE, and SCAN. HF theory is exact for the chosen basis set for single electron ${{\mathrm{H}}_2}^{+}$. (b) Difference of electron density calculated by SCAN and PBE, i.e., $n_{\mathrm{SCAN}}-n_{\mathrm{PBE}}$. AFM MnO is selected because $\mathrm{M}{\mathrm{n}}^{2+}$ is fully spin polarized with a simple completely filled $3d^5$ shell in one spin channel. (c) Total energy of isolated Mn ion as a function of fractional orbital occupation, normalized such that the $3d^5$ ($\mathrm{M}{\mathrm{n}}^{2+}$) state is at zero. (d) Deviation from ideal linear behavior between integer electron numbers for the total energies of (c).

Figure 5

Performance of unrestricted SCAN approach with spin-symmetry breaking for various hydrogen structures. Binding energies are plotted as a function of interatomic distances for (a) ${\mathrm{H}}_2$, (b) ${\mathrm{H}}_{10}$, and (c) the thermodynamic limit of the hydrogen chain $\left({\mathrm{H}}_{\infty }\right)$. In (a), the coupled-cluster-singles-doubles (CCSD) (red $•$) is exact and used as the reference. In (b) and (c), the HF data $<1.9\AA$ (blue $+$) and reference auxiliary-field quantum Monte Carlo data (red $•$) are from Ref. [30]. ${\mathrm{H}}_{\infty }$ has an antiferromagnetic order when the magnetic moment persists. Atomic spacing is uniform in ${\mathrm{H}}_{10}$ and ${\mathrm{H}}_{\infty }$ systems, the same value as used in Ref. [30]. The insets show the energy difference between SCAN and the references.

Figure 6

Gap opening mechanism in rocksalt FeO with the G-AFM magnetic ordering. (a) PBE predicted band structure by keeping the $t_{2g}$ orbital symmetry. (b) PBE band structure but with the $t_{2g}$ orbital symmetry removed. Spin-orbit coupling effect is used to guide the calculation to find the electronic ground state. (c) SCAN band structure also with orbital symmetry removed. In (a)–(c), the $t_{2g}$ wave functions are superimposed onto the band structures. The insets of subplots (a) and (b) show the band degeneracies of the zoomed-in areas. (d) Brillouin zone and K path of the G-AFM phase for the band-structure calculations, together with the energy surfaces that are 0.1 eV below the valence-band maximum in (c). (e)–(g) The $d_{xy}$, $d_{yz}$, $d_{xz}$ orbitals extracted from the Wannier function construction, from SCAN calculation. The blue and pink colors denote the signs of Wannier function. (h) The $a_{1g}$ orbital. (i) The difference of charge density distribution between calculations with and without orbital symmetry, both from SCAN calculations. The blue and pink colors denote charge accumulation and depletion, respectively. (j) The difference of charge density distribution between SCAN and PBE calculations, both without orbital symmetry.

Figure 7

Charge density of the $a_{1g}$ orbital of FeO in the 64-atom supercell. The G-type spin configuration is shown as FM coupling within (111) plane and AFM coupling between the planes. The $a_{1g}$ band shows a disordered pattern in charge density. Oxygen atoms are omitted for simplicity.