Polymorphous band structure model of gapping in the antiferromagnetic and paramagnetic phases of the Mott insulators MnO, FeO, CoO, and NiO

The existence of band gaps in both the antiferromagnetic (AFM) and paramagnetic (PM) phases of the classic NaCl-structure Mott insulators MnO, FeO, CoO, and NiO is traditionally viewed and taught as a manifestation of strong correlation whereby insulation results from electrons moving across the lattice forming states with doubly occupied $d$ orbitals on certain atomic sites and empty $d$ orbitals on other sites. Within such theories, the gap of the AFM and PM phases of these oxides emerges even in the absence of spatial symmetry breaking. The need for such a correlated picture is partially based on the known failure of the commonly used band models for the PM phase that assume for such a spin disordered state the macroscopically averaged NaCl structure, where all transition metal (TM) sites are symmetry-equivalent (a monomorphous description), producing a gapless PM state with zero magnetic moments, in sharp conflict with experiment. Here, we seek to understand the minimum theoretical description needed to capture the leading descriptors of ground state Mott insulation in the classic, $3d$ monoxide Mott systems—gapping and moment formation in the AFM and PM phase. As noted by previous authors, the spin-ordered AFM phase in these materials already shows in band theory a significant band gap when one doubles the NaCl unit cell by permitting different potentials for transition-metal atoms with different spins. For the spin-disordered PM phase, we allow analogously larger NaCl-type supercells where each TM site can have different spin direction and local bonding environments (i.e., disordered), yet the total spin is zero. Such a polymorphous description has the flexibility to acquire symmetry-breaking energy-lowering patterns that can lift the degeneracy of the $d$ orbitals and develop large on-site magnetic moments without violating the global, averaged NaCl symmetry. Electrons are exchanged between spin-up and spin-down bands to create closed-shell insulating configurations that lend themselves to a single determinental description. It turns out that such a polymorphous description of the structure within the single-determinant, mean-field, Bloch periodic band structure approach (based on $\mathrm{DFT}+U$) allows large on-site magnetic moments to develop spontaneously, leading to significant (1–3 eV) band gaps and large local moments in the AFM and PM phases of the classic NaCl-structure Mott insulators MnO, FeO, CoO, and NiO in agreement with experiment. We adapt to the spin disordered polymorphous configurations the “special quasirandom structure” (SQS) construct known from the theory of disordered substitutional alloys whereby supercell approximants which represent the best random configuration average (not individual snapshots) for finite (64, 216 atoms, or larger) supercells of a given lattice symmetry are constructed. We conclude that the basic features of these paradigmatic Mott insulators can be approximated by the physics included in energy-lowering symmetry broken DFT.

Figure 1

Percent fraction $F\left(n_{NN}^{\uparrow }\right)$ of cation sites as a function of the number $n_{NN}^{\uparrow }$ of spin-up nearest-neighbor (NN) metal site of a central cation site in (a) the AFM-II phase and (b) the PM phase modeled here by the 216-atom rock-salt SQS shown in the insert (in this model only the metal sublattice of the underlying rock-salt structure is shown). In the AFM phase, $n_{NN}^{\uparrow }=6$ for all cations, while in the PM phase, $n_{NN}^{\uparrow }$ varies between 0 and 12 and $F\left(n_{NN}^{\uparrow }\right)$ approximates a binomial distribution. The frequency of local environment types $F\left(n_{NN}^{\uparrow }\right)$ was calculated averaging over a 216-atom SQS with $A_{0.5}{B}_{0.5}$ composition and its complementary $B_{0.5}{A}_{0.5}$ spin configuration. The band gap and the TM magnetic moment are reported for CoO in the AFM phase and the PM phase, which was calculated using the displayed 216-atom $3\times 3\times 3$ SQS.

Figure 2

Schematic of the sequence of level splittings and combinations for the $d$ orbitals in MnO, NiO, FeO, and CoO as the exchange coupling and the crystal field of the symmetry appropriate to each phase are progressively imposed: (a) splitting of the $d$ orbitals into the transition-metal atoms subjected to exchange coupling. (b) Splitting of the spin-up and spin-down $d$ levels subjected to a cubic $O_h$ crystal field: this is the case of a rock-salt structure with hypothetical $Fm\overline{3}m$ magnetic ordering. (c) Splitting of the $d$ level in the $D_{3d}$ crystal field in the distorted rock-salt lattice with the rhombohedral $R\overline{3}m$ AFM-II magnetic ordering. (d) Splitting of the $d$ orbitals in a tetragonal crystal field as in the monoclinic $C2/m$ phases of FeO and CoO. Note that in this schematic we emphasize the effect of the relevant interaction in progressively removing the degeneracy of the $d$ orbitals, while we do not intend to reproduce to scale the position of the spin-up and spin-down energy levels. The reader should inspect the projected DOSs of Figs. 3 and 4 to obtain the actual calculated energy position of the $d$ bands.

Figure 3

Projected density of states (PDOS) on the transition metal $s$ and $d$ orbitals ($t_{2g}$ and $e_g$ components) calculated by $\mathrm{DFT}+U$ ($U=5\mathrm{eV}$) for MnO and NiO in (a) and (b) the AFM phase with fully relaxed $R\overline{3}m$ structures, and [(c) and (d)] the PM phase modeled by a cubic 64-atom $2\times 2\times 2$ SQS. The lattice parameters of the SQSs are set so that the volume per formula unit is equal to the calculated volume per formula unit of the $\mathrm{DFT}+U$ relaxed $R\overline{3}m$ structures of MnO and NiO.

Figure 4

Projected density of states (PDOS) on the transition metal $s$ and $d$ orbitals ($t_{2g}$ and $e_g$ components) calculated by $\mathrm{DFT}+U$ ($U=5\mathrm{eV}$) for FeO and CoO, in (a) and (b) the AFM phases with the fully relaxed monoclinic $C2/m$ structures, and in (c) and (d) the paramagnetic phases modeled by a cubic 64-atom $2\times 2\times 2$ SQS. The lattice parameters of the SQSs are set so that the volume per formula unit is equal to the calculated volume per formula unit of the $\mathrm{DFT}+U$ relaxed $C2/m$ structures of FeO and CoO.

Figure 5

(a) and (b) depict the minority-spin electron density ${\rho }^{\downarrow }\left(r\right)$ at the Fe and Co site in the AFM monoclinic phases of, respectively, FeO and CoO calculated by $\mathrm{DFT}+U$. (c) and (d) depict the minority-spin electron density ${\rho }^{\downarrow }\left(r\right)$ at the Fe and Co site in the PM phase of, respectively, FeO and CoO modeled by the magnetic 64-atom cubic SQS used for the calculation of the PDOS shown in Figs. 5 and 5. The full ${\rho }^{\downarrow }\left(r\right)$ of PM FeO and CoO in the SQS is shown in Figs. 6 and 6.

Figure 6

Minority-spin electron density ${\rho }^{\downarrow }\left(r\right)$ in the regions with positive magnetization $m\left(r\right)={\rho }^{\uparrow }\left(r\right)-{\rho }^{\downarrow }\left(r\right)>0$ within the magnetic 64-atom SQS cell used in the $\mathrm{DFT}+U$ ($U=5\mathrm{eV}$) calculations of the PM phases of (a) FeO and (b) CoO. To avoid visual clutter, we masked out the regions of space within spheres of of radius centered at the oxygen sites.

Figure 7

Analysis of the effects of different symmetry breaking modes on the electronic structure, total energy, band gap, and local moment of paramagnetic CoO. We show the projected density of states (PDOS) on the transition-metal $s$ and $d$ orbitals ($t_{2g}$ and $e_g$ components) and oxygen $p$ orbitals from levels (a) IV, (b) III, (c) II, and (d) I (see Sec. VII A for definition of these symmetry breaking levels) of rock-salt CoO. (a)–(c) are calculated in a 216-atom cubic rock-salt supercell with SQS, while (d) is calculated in a two-atom rock-salt primitive cell. All lattice parameters are the same as the one used in Fig. 4.