Dynamic electronic correlation effects in ${\mathrm{NbO}}_2$ as compared to ${\mathrm{VO}}_2$

In this paper we present a comparative investigation of the electronic structures of ${\mathrm{NbO}}_2$ and ${\mathrm{VO}}_2$ obtained within a combination of density functional theory and cluster-dynamical mean-field theory calculations. We investigate the role of dynamic electronic correlations on the electronic structure of the metallic and insulating phases of ${\mathrm{NbO}}_2$ and ${\mathrm{VO}}_2$, with a focus on the mechanism responsible for the gap opening in the insulating phases. For the rutile metallic phases of both oxides, we obtain that electronic correlations lead to a strong renormalization of the $t_{2g}$ subbands, as well as the emergence of incoherent Hubbard subbands, signaling that electronic correlations are also important in the metallic phase of ${\mathrm{NbO}}_2$. Interestingly, we find that nonlocal dynamic correlations do play a role in the gap formation of the [body-centered-tetragonal (bct)] insulating phase of ${\mathrm{NbO}}_2$, by a similar physical mechanism as that recently proposed by us in the case of the monoclinic ($M_1$) dimerized phase of ${\mathrm{VO}}_2$ [Phys. Rev. Lett. 117, 056402 (2016)]. Although the effect of nonlocal dynamic correlations in the gap opening of bct phase is less important than in the ($M_1$ and $M_2$) monoclinic phases of ${\mathrm{VO}}_2$, their presence indicates that the former is not a purely Peierls-type insulator, as it was recently proposed.

Figure 1

Crystal structures of (a) rutile ($R$) (space group $P{4}_2/mnm$), (b) monoclinic $M_1$ (space group $P{2}_1/c$), and (c) monoclinic $M_2$ (space group $C2/m$) phases of ${\mathrm{VO}}_2$. (d) Crystal structure of the body-centered-tetragonal (bct) phase of ${\mathrm{NbO}}_2$ (space group $I{4}_1/a$). Vanadium and niobium atoms are represented by black spheres while the oxygens by the orange ones. The local axis system used throughout this paper [10] is shown schematically in (a).

Figure 2

DFT+DMFT-based total (black dashed line) and projected density of states of $R$ phases of (a) ${\mathrm{VO}}_2$ and (b) ${\mathrm{NbO}}_2$, at $T=390$ K and $T=1132$ K, respectively. The projections to $a_{1g}$, $e_g^{\pi }\left(1\right)$, $e_g^{\pi }\left(2\right)$, and $e_g^{\sigma }$ states are shown in blue, red, green, and brown lines, respectively. Shaded regions indicate the DFT obtained total density of states.

Figure 3

Spectral function and projected density of states of (a) $M_1$ and (b) bct phases at 332 and 1000 K, respectively. The projections to $a_{1g}$, $e_g^{\pi }\left(1\right)$, and $e_g^{\pi }\left(2\right)$ dimer states are shown in blue, red, and green lines, respectively. The solid (dashed) blue line corresponds to the projection on the bonding (antibonding) $a_{1g}$ dimer state.

Figure 4

(a) Imaginary part, on the real frequency axis, of bonding (dashed lines) and antibonding (solid lines) self-energies associated with the $a_{1g}$ dimer electronic states of the $M_1$ (black) and bct (red) phases. In (b) and (c), we show the valence histograms of vanadium and niobium dimers in the $M_1$ and bct phases, respectively. In these figures, $N$ denotes the number of electrons in distinct dimer states. Singlet states ($S_z=0$) are indicated by blue arrows.

Figure 5

DFT+DMFT spectral function and projected density of states of the $M_2$ phase at 332 K, considering (a) antiferromagnetic and (b) paramagnetic states. In (a) and (b), the projections to $t_{2g}$ states associated with dimerized and zigzaglike chains of V atoms are shown in the central and right panels, respectively. With respect to the dimerized atoms, projections to $a_{1g}$, $e_g^{\pi }\left(1\right)$, and $e_g^{\pi }\left(2\right)$ “molecular” states are shown in blue, red, and green, respectively. For the zigzag V atoms, we use the same colors for each $t_{2g}$ state, with the contributions of the up and down (dn) spins shown as solid and dashed lines, respectively. In (c) and (d), we show the valence histograms of the V dimer and V states of dimerized and zigzag atoms, respectively.

Figure 6

(a) Imaginary part, on the real frequency axis, of bonding and antibonding self-energies of the $a_{1g}$ and $e_g^{\pi }\left(1\right)$ states of V dimers (on the left) and of the $a_{1g}$ and $e_g^{\pi }$ states of zigzag vanadium atoms (on the right). (b) Real part of $a_{1g}$ and $e_g^{\pi }$ self-energies of zigzag (unpaired) vanadium atoms in the antiferromagnetic phase.

Figure 7

Real part of intersite self-energies, on the imaginary frequency axis, of the $a_{1g}\text{−}{a}_{1g}$ (solid lines) and $e_g^{\pi }\left(1\right)\text{−}{e}_g^{\pi }\left(1\right)$ (dashed lines) states of $M_1$ (indigo), $M_2$ (AFM) (red), and bct (green) phases. We considered $T=332$ K for the monoclinic phases and $T=1000$ K in the case of the bct phase.