Importance of intersite Hubbard interactions in $\beta \text{−}{\mathrm{MnO}}_2$: A first-principles $\mathrm{DFT}+U+V$ study

We present a first-principles investigation of the structural, electronic, and magnetic properties of pyrolusite ($\beta \text{−}\mathrm{Mn}{\mathrm{O}}_2$) using conventional and extended Hubbard-corrected density-functional theory ($\mathrm{DFT}+U$ and $\mathrm{DFT}+U+V$). The onsite $U$ and intersite $V$ Hubbard parameters are computed using linear-response theory in the framework of density-functional perturbation theory. We show that while the inclusion of the onsite $U$ is crucial to describe the localized nature of the $\mathrm{Mn}\left(3d\right)$ states, the intersite $V$ is key to capture accurately the strong hybridization between neighboring $\mathrm{Mn}\left(3d\right)$ and $\mathrm{O}\left(2p\right)$ states. In this framework, we stabilize the simplified collinear antiferromagnetic (AFM) ordering (suggested by the Goodenough-Kanamori rule) that is commonly used as an approximation to the experimentally-observed noncollinear screw-type spiral magnetic ordering. A detailed investigation of the ferromagnetic and of other three collinear AFM spin configurations is also presented. The findings from Hubbard-corrected DFT are discussed using two kinds of Hubbard manifolds—nonorthogonalized and orthogonalized atomic orbitals—showing that special attention must be given to the choice of the Hubbard projectors, with orthogonalized manifolds providing more accurate results than nonorthogonalized ones within $\mathrm{DFT}+U+V$. This paper paves the way for future studies of complex transition-metal compounds containing strongly localized electrons in the presence of pronounced covalent interactions.

Figure 1

Four collinear AFM configurations of $\beta \text{−}\mathrm{Mn}{\mathrm{O}}_2$: (a) A1-AFM, (b) A2-AFM, (c) C-AFM, and (d) G-AFM. Yellow and blue octahedra contain Mn atoms in the center with up and down spin alignments (shown with black and white arrows), respectively, and oxygen atoms are represented as red balls.

Figure 2

Experimental crystal structure of $\beta \text{−}\mathrm{Mn}{\mathrm{O}}_2$ [85]. Mn atoms are indicated in yellow, while O atoms are indicated in blue and red to highlight longer and shorter Mn–O bonds, respectively. Three types of O–Mn–O bond angles are indicated with ${\phi }_1$, ${\phi }_2$, and ${\phi }_3$, while two types of Mn–O–Mn bond angles are indicated with ${\theta }_1$ and ${\theta }_2$. $a$ and $c$ are the lattice parameters. The experimental values of bond lengths and bond angles are indicated in Fig. 3.

Figure 3

Bond lengths and bond angles in $\beta \text{−}\mathrm{Mn}{\mathrm{O}}_2$ (see Fig. 2) as computed in this work using DFT, $\mathrm{DFT}+U$, and $\mathrm{DFT}+U+V$ (PBEsol functional) with two types of Hubbard projectors (NAO and OAO) and corresponding first-principles parameters $U$ and $V$ (see Table 1), and as measured in experiments (dashed horizontal lines) [85]. Theoretical results are shown for five collinear magnetic orderings (FM, A1-AFM, A2-AFM, C-AFM, G-AFM), while the experimental results correspond to the noncollinear helical structure. (a) Mn-O bond lengths (in Å), (b) Mn-O-Mn bond angles ${\theta }_1$ and ${\theta }_2$ (in degrees), and (c) O-Mn-O bond angles ${\phi }_1$, ${\phi }_2$, and ${\phi }_3$ (in degrees). Solid lines that connect squares are guides to the eye.

Figure 4

Total energy difference per formula unit (in meV) for five collinear magnetic orderings (FM, A1-AFM, A2-AFM, C-AFM, G-AFM) computed at three levels of theory (DFT, $\mathrm{DFT}+U$, and $\mathrm{DFT}+U+V$) using the PBEsol functional. For each case, the Hubbard parameters $U$ and $V$ were computed using DFPT, using two types of Hubbard projectors (NAO and OAO), and they are listed in Table 1.

Figure 5

Magnetic moment (in ${\mu }_{\mathrm{B}}$) on Mn atoms in $\beta \text{−}\mathrm{Mn}{\mathrm{O}}_2$ for five collinear magnetic orderings (FM, A1-AFM, A2-AFM, C-AFM, G-AFM) computed at three levels of theory (DFT, $\mathrm{DFT}+U$, and $\mathrm{DFT}+U+V$) using the PBEsol functional. For each case, the Hubbard parameters $U$ and $V$ were computed using DFPT, using two types of Hubbard projectors (NAO and OAO), and they are listed in Table 1. The experimental magnetic moment at $T=10$ K is 2.35 ${\mu }_{\mathrm{B}}$ [19] and it is indicated with a horizontal dashed line.

Figure 6

Band gap (in eV) in $\beta \text{−}\mathrm{Mn}{\mathrm{O}}_2$ for five collinear magnetic orderings (FM, A1-AFM, A2-AFM, C-AFM, G-AFM) computed at three levels of theory (DFT, $\mathrm{DFT}+U$, and $\mathrm{DFT}+U+V$) using the PBEsol functional. For each case, the Hubbard parameters $U$ and $V$ were computed using DFPT, using two types of Hubbard projectors (NAO and OAO), and they are listed in Table 1. The experimental band gap is 0.26 eV [11, 12], 0.27 eV [13], 0.28 eV [14]; the minimum and maximum values are indicated by the horizontal dashed lines.

Figure 7

Spin-resolved PDOS and total DOS for five collinear magnetic orderings of $\beta \text{−}\mathrm{Mn}{\mathrm{O}}_2$ (FM, A1-AFM, A2-AFM, C-AFM, and G-AFM) computed using standard DFT (PBEsol functional). The zero of energy corresponds to the Fermi energy (in the case of metallic ground states) or top of the valence bands (in the case of insulating ground states). The intensity of PDOS and total DOS was rescaled to the 6-atoms unit cell for A2-AFM, C-AFM, and G-AFM. The spin-up (upper part) and spin-down (lower part) components of the PDOS are shown on each panel (summed over all atoms of the same type). Mn1 and Mn2 correspond to two Mn atoms with the opposite spin polarizations in the AFM cases.

Figure 8

Spin-resolved PDOS and total DOS for five collinear magnetic orderings of $\beta \text{−}\mathrm{Mn}{\mathrm{O}}_2$ (FM, A1-AFM, A2-AFM, C-AFM, and G-AFM) computed at three levels of theory (DFT, $\mathrm{DFT}+U$, and $\mathrm{DFT}+U+V$) using the PBEsol functional. For each case, the Hubbard parameters $U$ and $V$ were computed using DFPT, using two types of Hubbard projectors (NAO and OAO), and they are listed in Table 1. The zero of energy corresponds to the Fermi energy (in the case of metallic ground states) or top of the valence bands (in the case of insulating ground states). The intensity of PDOS and total DOS was rescaled to the 6-atoms unit cell for A2-AFM, C-AFM, and G-AFM. The spin-up (upper part) and spin-down (lower part) components of the PDOS are shown on each panel (summed over all atoms of the same type). Mn1 and Mn2 correspond to two Mn atoms with the opposite spin polarizations in the AFM cases.

Figure 9

Spin-resolved PDOS for the $t_{2g}$ and $e_g$ states of one Mn atom of $\beta \text{−}\mathrm{Mn}{\mathrm{O}}_2$ with the A1-AFM magnetic ordering computed at three levels of theory (DFT, $\mathrm{DFT}+U$, and $\mathrm{DFT}+U+V$) using the PBEsol functional. OAO Hubbard projectors are used. The upper and lower panels on each figure correspond to the spin-up and spin-down channels, respectively. The zero of energy corresponds to the Fermi energy (in the DFT case) or top of the valence bands (in the case of $\mathrm{DFT}+U$ and $\mathrm{DFT}+U+V$).