$\mathrm{DFT}+\mathrm{DMFT}$ calculations of the complex band and tunneling behavior for the transition metal monoxides MnO, FeO, CoO, and NiO

We report complex band structure (CBS) calculations for the four late transition metal monoxides MnO, FeO, CoO, and NiO in their paramagnetic phase. The CBS is obtained from density functional theory plus dynamical mean field theory (DMFT) calculations to take into account correlation effects. The so-called $\beta$ parameters, governing the exponential decay of the transmission probability in the nonresonant tunneling regime of these oxides, are extracted from the CBS. Different model constructions are examined in the DMFT part of the calculation. The calculated $\beta$ parameters provide theoretical estimation for the decay length in the evanescent channel, which would be useful for tunnel junction applications of these materials.

Figure 1

Nonmagnetic NiO ground-state band structure (solid lines), and band characters (open circles) calculated by projecting Bloch states onto atomic orbital states. The radius of the open circles is proportional to the weight of the atomic states. Fermi level is at zero.

Figure 2

Nonmagnetic ground-state band structures (solid curves) of NiO, CoO, FeO, and MnO. And the TM $4s$ orbital character only (open circles). Horizontal solid lines are placed at the maximum of $d$-like bands, to make the separation or overlap of $s$-like and $d$-like bands clearly seen. Fermi level is always at zero.

Figure 3

DFT band structure of NiO (solid line), and reconstructed bands (dots) in symmetry-preserving Wannier orbital basis, which are identical to DFT bands by construction. Fermi level is at zero.

Figure 4

Complex plane of ${\mathcal{C}}_1$ for searching poles of $G({\mathcal{C}}_1,\omega )$, for the direction ${\widehat{k}}_{\perp }={\vec{b}}_3/|{\vec{b}}_3|$. Shaded area is the searching area, which includes the real axis and boundaries of the first Brillouin zone.

Figure 5

Total (black), $t_{2g}$ (blue), $e_g$ (green), and $p$ (red) spectral functions $A\left(\omega \right)$ of the two models of NiO, CoO, FeO, and MnO.

Figure 6

$\vec{k}$-resolved spectral functions of the $d-dp$ model, of (a) NiO, (b) CoO, (c) FeO, and (d) MnO.

Figure 7

Complex band structures of the $d-dp$ model. The decay direction being studied is ${\widehat{k}}_{\perp }={\vec{b}}_1/|{\vec{b}}_1|$, with $C_2=C_3=0$ [see Eq. (5) for definition]. The path for the band plot is $(\text{Re}\left[{\mathcal{C}}_1\right],\text{Im}\left[{\mathcal{C}}_1\right])=(0,-0.5)\to (0,0)\to (0.5,0)$. The $\mathrm{DFT}+\mathrm{DMFT}$ Fermi level $E_F^{\text{DMFT}}$ is at zero.

Figure 8

The CBS of the $d-dp$ model of NiO. The $x$ axis is the path of $(\text{Re}\left[{\mathcal{C}}_1\right],\text{Im}\left[{\mathcal{C}}_1\right])$. $E_F^{\text{DMFT}}$ is at zero (eV). $E_B^{\mathrm{Mott}}$ is located at the crossing of the imaginary extension of conduction band (small slope arrow) and the imaginary extension of valence band (large slope arrow). $E_F^{\text{DMFT}}$ is at zero. $E_B^{d-p}$ is found within the imaginary band connecting $d$- and $p$-valence bands where $dE/dk\to \infty$. The conduction-band minimum $E_{\min }^{\mathrm{cond}.}$ (short horizontal dashed line) is found at $\mathrm{\Gamma }$ at a slightly higher energy than $E_B^{\mathrm{Mott}}$. The Schottky barrier height is measured to be ${\mathrm{\Phi }}_B=E_{\min }^{\mathrm{cond}.}-E_B^{\mathrm{Mott}}\approx 0.35\mathrm{eV}.$

Figure 9

The distribution of $\beta (C_2,C_3)$ at $E_B^{\mathrm{Mott}}$ for the decay direction (1,0,0). (a) NiO, (b) CoO, (c) FeO, and (d) MnO.

Figure 10

The distribution of $\beta (C_2,C_3)$ at $E_B^{\mathrm{Mott}}$ for the decay direction (1,1,0). (a) NiO, (b) CoO, (c) FeO, and (d) MnO.

Figure 11

The distribution of $\beta (C_2,C_3)$ at $E_B^{\mathrm{Mott}}$ for the decay direction (1,1,1). (a) NiO, (b) CoO, (c) FeO, and (d) MnO.

Figure 12

$\beta$-resolved density of state $n\left(\beta \right)$ for the three directions: (0,0,1) upper, (0,1,1) middle, (1,1,1) down. The $\beta$ is measured at $E_B^{\mathrm{Mott}}$. The $y$-axis scale is the same for the three plots.

Figure 13

Relative transmission probability on linear and log scale for the directions: (0,0,1), upper two plots (a) and (b) (log scale); (0,1,1), middle two plots (c) and (d) (log scale); (1,1,1), lower two plots (e) and (f) (log scale).