Pressure dependence of dynamically screened Coulomb interactions in NiO: Effective Hubbard, Hund, intershell, and intersite components

In this work, we report the pressure dependence of the effective Coulomb interaction parameters (Hubbard $U$) in paramagnetic NiO within the constrained random phase approximation (cRPA). We consider five different low-energy models starting from the most expensive one that treats both Ni-$d$ and O-$p$ states as correlated orbitals ($dp-dp$ model) to the smallest possible two-orbital model comprising the $e_g$ states only ($e_g-e_g$ model). We find that in all the considered models, the bare interactions are not very sensitive to the compression. However, the partially screened interaction parameters show an almost linear increment as a function of compression, resulting from the substantial weakening of screening effects upon compression. This counterintuitive trend is explained from the specific characteristic changes of the basic electronic structure of this system. We further calculate the nearest-neighbor intersite $d-d$ interaction terms, which also show substantial enhancement due to compression. The computed interaction parameters for antiferromagnetic NiO are almost identical to their paramagnetic counterparts. Results for other prototypical $3d$ transition metal monoxides, FeO, CoO, and CuO, further demonstrate that the monotonic increase of the partially screened interaction parameters under the application of pressure is a generic characteristic of transition metal monoxides. Finally, the effective parameters of $3d$ states grow as a function of the atomic number of the transition metal ion.

Figure 1

(a) The band dispersion and (b) the partial density of states (PDOS) of fcc NiO. Calculations are carried out with experimental lattice parameters.

Figure 2

(a) The band dispersion and (b) the partial density of states (PDOS) of fcc NiO. Calculations are carried out with the reduced lattice parameters, corresponding to the volume $V=59\%V_0, V_0$ being the experimental volume.

Figure 3

Local Coulomb interactions (diagonal element of $U_{mm}$) of the $t_{2g}$ (black line) and $e_g$ (red line) states as a function of the percentage of unit cell volume contraction for the five relevant low-energy models. Orbital resolved diagonal elements of the bare interactions $v$ for (a) $dp-dp$, (b) $d-dp$, (c) $d-d$, (d) $e_g-dp$, and (e) $e_g-e_g$ models. Partially screened interaction $U$ for (f) $dp-dp$, (g) $d-dp$, (h) $d-d$, (i) $e_g-dp$, (j) $e_g-e_g$ models and fully screened interaction $W$ for (k) $dp-dp$, (l) $d-dp$, (m) $d-d$, (n) $e_g-dp$, (o) $e_g-e_g$ models at zero frequency limit.

Figure 4

Effective correlation strength as a function of compression.

Figure 5

Frequency dependence of the (a) real and (b) imaginary parts of the partially screened interaction $U\left(\omega \right)=\frac{1}{5}{\sum }_m{U}_{mm}\left(\omega \right)$ for the considered five models with the experimental structure. (c) Experimental electron energy loss spectrum is shown to compare the prominent features in $\mathrm{Im}U\left(\omega \right)$. Experimental spectrum is taken from Ref. [80].

Figure 6

Frequency dependence of the partially screened $U_d,U_p$, and $U_{dp}$ in the $dp-dp$ model for the (a) experimental structure ($V=V_0$), (b) moderately compressed structure ($V=75.7\%V_0$), and (c) the highest compressed structure in our study ($V=59.1\%V_0$).

Figure 7

Frequency dependence of the partially screened interaction $U\left(\omega \right)$ for three different volumes of the unit cell, computed within (a) $d-dp$, (b) $d-d$, (c) $e_g-dp$, and (d) $e_g-e_g$ models.

Figure 8

Total density of states and site projected partial density of states of Ni-$d$ and O-$p$ orbitals in antiferromagnetic NiO for the (a) experimental structure ($V=V_0$), and (b) the highest compressed structure ($V=59.1\%V_0$). In the insets, we have compared the total antiferromagnetic DOS with its nonmagnetic counterpart.

Figure 9

The band dispersion of (a) FeO, (b) CoO, (c) NiO, and (d) CuO. Calculations are carried out for experimental volume of the unit cell. For CuO, we have considered an artificial fcc structure like other transition metal oxides.