Hubbard $U$ and Hund exchange $J$ in transition metal oxides: Screening versus localization trends from constrained random phase approximation

In this work, we address the question of calculating the local effective Coulomb interaction matrix in materials with strong electronic Coulomb interactions from first-principles. To this purpose, we implement the constrained random phase approximation into a density functional code within the linearized augmented plane-wave framework. We apply our approach to the 3$d$ and 4$d$ early transition metal oxides Sr$M$O${}_3$ ($M=$ V, Cr, Mn) and ($M=$ Nb, Mo, Tc) in their paramagnetic phases. For these systems, we explicitly assess the differences between two physically motivated low-energy Hamiltonians: The first is the three-orbital model comprising the $t_{2g}$ states only, which is often used for early transition metal oxides. The second choice is a model where both metal $d$ and oxygen $p$ states are retained in the construction of Wannier functions, but the Hubbard interactions are applied to the $d$ states only (“$d$-$dp$ Hamiltonian”). Interestingly, since (for a given compound) both $U$ and $J$ depend on the choice of the model, so do their trends within a family of these compounds. In the 3$d$ perovskite series Sr$M$O${}_3$, the effective Coulomb interactions in the $t_{2g}$ Hamiltonian decrease along the series due to the more efficient screening. The inverse, generally expected, trend, increasing interactions with increasing atomic number, is however recovered within the more localized “$d$-$dp$ Hamiltonian.” Similar conclusions are established in the layered 4$d$ perovskites series Sr${}_2$$M$O${}_4$ ($M=$ Mo, Tc, Ru, Rh). Compared to their isoelectronic and isostructural 3$d$ analogs, the 4$d$ perovskite oxides Sr$M$O${}_3$ ($M=$ Nb, Mo, Tc) exhibit weaker screening effects. Interestingly, this leads to an effectively larger $U$ on 4$d$ than on 3$d$ shells when a $t_{2g}$ model is constructed.

Figure 1

Electronic band structures (top) and projected density of states (bottom) of Sr$M$O${}_3$ ($M=\mathrm{V}$, Cr, Mn from left to right), obtained from DFT-LDA paramagnetic calculations. The $3d$ $t_{2g}$ states are highlighted in red (dashed line), the $3d$ $e_g$ states in blue (dashed-dotted line), and the oxygen $p$ states in maroon (solid line).

Figure 2

Electronic band structures (top) and projected density of states (bottom) of Sr$M$O${}_3$ ($M=\mathrm{Nb}$, Mo, Tc from left to right), obtained from DFT-LDA paramagnetic calculations. The 4$d$ $t_{2g}$ states are highlighted in red (dashed line), the 4$d$ $e_g$ states in blue (dashed-dotted line), and the oxygen $p$ states in maroon (solid line).

Figure 3

Middle panel: Onsite Hubbard interaction ${\overline{U}}_{mm}$ (left) and exchange interaction ${\overline{J}}_{mm}$ (right) between $t_{2g}$ orbitals within the $d$-$d$$p$ (black curve) low-energy Hamiltonians for 3$d$ Sr$M$O${}_3$, compared to the onsite $\mathcal{U}$ and $\mathcal{J}$ within $t_{2g}$-$t_{2g}$ (red curve). In dashed line with open circles, the Hubbard parameter $U={\text{F}}^0$ and Hund's exchange $J=({\text{F}}^2+{\text{F}}^4)/14$ are shown. Top panel: Same but for the bare interactions between $t_{2g}$ orbitals. Bottom panel: Same but for the fully screened interactions between $t_{2g}$ orbitals.

Figure 4

Middle panel: Onsite Hubbard-Kanamori interaction $\mathcal{U}$ (left) and exchange interaction $\mathcal{J}$ (right) between $t_{2g}$ orbitals within the $t_{2g}$-$t_{2g}$ low-energy Hamiltonian for 3$d$ (red curve) and 4$d$ (indigo curve) Sr$M$O${}_3$ (see also Table ). Top panel: Same but for the bare interaction $\mathcal{V}$ and ${\mathcal{J}}_{\mathrm{bare}}$ between $t_{2g}$ orbitals. Bottom panel: Same but for the fully screened interaction $\mathcal{W}$ and ${\mathcal{J}}_{\mathrm{screened}}$ between $t_{2g}$ orbitals.

Figure 5

DFT-LDA band structure of the four-layer hexagonal unit cell of SrMnO${}_3$ in the paramagnetic phase. Lattice parameters $a=5.461\text{Å},c=9.093\text{Å}$ are taken from Ref. 76.

Figure 6

DFT-LDA paramagnetic band structures for layered perovskites Sr${}_2$$M$O${}_4$. (Top) From left to right: $M=$ Mo, Tc. (Bottom) From left to right: $M=$ Ru, Rh.

Figure 7

Bottom panel: Onsite interaction $U_{mm}$ (left) and exchange interaction $J_m$ (right) between $t_{2g}$ orbitals within the $d$-$d$$p$ (black curve) model for Sr${}_2$$M$O${}_4$ and onsite interactions $\mathcal{U}$ (left) and $\mathcal{J}$ (right) but within $t_{2g}$-$t_{2g}$ (red curve). In dashed line with open circles, the Hubbard parameter $U={\text{F}}^0$ and Hund's exchange $J=({\text{F}}^2+{\text{F}}^4)/14$ are shown. Top panel: Same but for the bare interactions between the $t_{2g}$ orbitals.