Electronic structure of perovskite-type transition metal oxides $\text{La}M{\text{O}}_3$ $\left(M=\text{Ti}\sim \text{Cu}\right)$ by $\text{U}+\text{GW}$ approximation

We investigate electronic structures of $\text{La}M{\text{O}}_3$ $\left(M=\text{Ti}\sim \text{Cu}\right)$ systematically by means of $\text{U}+\text{GW}$ approximation. In these strongly correlated systems, it is important to treat large on-site Coulomb interactions and their dynamical screening effects. Transition-metal ions in perovskite-type lanthanum oxides are trivalent and their physics is qualitatively different from that of divalent transition-metal ions in transition-metal mono-oxides. The localization of wave functions of $\text{La}4f$ and $3d$ orbitals of Ti, V, and Co is crucial. On the other hand, the screening effect for other transition-metal $3d$ orbitals is strong enough so as to reduce the on-site static-screened Coulomb interaction in trivalent oxides. The band gaps, the magnetic moments, and energy spectra are discussed in comparison with the experimentally observed results. Calculated energy spectra of $\text{La}M{\text{O}}_3$ $\left(M=\text{V}\sim \text{Cu}\right)$ are in good agreement with experimental results.

Figure 1

Convergence behavior of the spectrum (the imaginary part of the total Green’s function $\frac{1}{\pi }\left|\text{Im}G\left(\omega \right)\right|$) of ${\text{LaFeO}}_3$ by the $e$-only self-consistent calculation. The calculation is based on $\text{U}+\text{GWA}$ and $U$ and $J$ values for $\text{La}4f$ orbital are 7.5 and 0.5 eV, respectively. $\text{LSDA}+\text{U}$: thin solid line, $\text{U}+\text{GWA}$ (first iteration): dotted line, $\text{U}+\text{GWA}$ (second iteration step): thin broken line, $\text{U}+\text{GWA}$ (third iteration step): bold solid line. The energy zeroth is fixed at the Fermi energy $E_F$ of respective iteration step.

Figure 2

Feynman diagrams of the basic contributions to GWA; (a) the polarization and (b) the exchange self-energy. The products of wave functions, ${\psi }_{k^{\prime }}^{\ast }{\psi }_{k^{\prime }+q}$ or ${\psi }_k^{\ast }{\psi }_{k-q}$, appear at each vertex.

Figure 3

Quasiparticle density of states of ${\text{LaMnO}}_3$. (a) LSDA, (b) GWA quasiparticle (QP) (Ref. 5), (c) $\text{LSDA}+\text{U}$, and (d) $\text{U}+\text{GWA}$ QP. (c) and (d) are the results of the present work with the parameters of $U=7.5\text{eV}$ and $J=0.5\text{eV}$ for $\text{La}4f$.

Figure 4

(Color online) Imaginary part of Green’s functions $\left(1/\pi \right)\left|\text{Im}G\left(\omega \right)\right|$ (per unit cell) of $\text{La}M{\text{O}}_3$ $\left(M=\text{V}\sim \text{Cu}\right)$ where the energy zeroth is set at the Fermi energy $E_F$. (a) those by $\text{LSDA}+\text{U}$ (starting spectra), and (b) those by $\text{U}+\text{GWA}$ at third iteration. In all cases, $\text{La}4f$ orbitals are treated by $\text{LSDA}+\text{U}/\text{U}+\text{GWA}$ with $U=7.5\text{eV}$ and $J=0.5\text{eV}$. For transition-metal ions, $U$ and $J$ are used for Ti ($U=2.5\text{eV}$, $J=1.0\text{eV}$), V ($U=3.0\text{eV}$, $J=1.0\text{eV}$) and Co ($U=2.7\text{eV}$, $J=1.3\text{eV}$). It should be noted that the unit cells of $\text{La}M{\text{O}}_3$ $\left(M=\text{Ti},\text{V},\text{Cr},\text{Mn},\text{Fe}\right)$ contain four molecular units and those of $M=\text{Co},\text{Ni},\text{Cu}$ contain one unit. The dotted lines in (a) and (b) are the experimental spectra and are shown in arbitrary intensity scale. The original experimental data can be found in the following references: ${\text{LaTiO}}_3$ (Ref. 48), ${\text{LaVO}}_3$ (Ref. 49), ${\text{LaCrO}}_3$ (Ref. 50), ${\text{LaMnO}}_3$ (Ref. 35), ${\text{LaFeO}}_3$ (Ref. 38), ${\text{LaCoO}}_3$ (Ref. 39), ${\text{LaNiO}}_3$ (Ref. 40), and ${\text{LaCuO}}_3$ (Ref. 41).

Figure 5

(Color online) Imaginary part of the projected Green’s functions per atom $\left(1/\pi \right)\left|\text{Im}G\left(\omega \right)\right|$ of $\text{La}M{\text{O}}_3$ $\left(M=\text{V}\sim \text{Cu}\right)$ by $\text{U}+\text{GWA}$ at third iteration. The energy zeroth is set at the Fermi energy $E_F$. The spectra of oxygen atom are the ones averaged over inequivalent sites. As for the transition-metal atoms, the spectra are those per atom of the spin-up site. The projected $\text{Im}G$ is depicted by using the local coordinate system where the $z$ axis is along the direction of $M\text{-O}$ of the longest interatomic distance and other $x$ and $y$ axes direct to the other $M\text{-O}$. For all cases, $\text{La}4f$ orbitals are treated by $\text{U}+\text{GWA}$ ($U=7.5\text{eV}$, $J=0.5\text{eV}$. Transition metal $U$ is used for Ti ($U=2.5\text{eV}$, $J=1.0\text{eV}$), V ($U=3.0\text{eV}$, $J=1.0\text{eV}$) and Co ($U=2.7\text{eV}$, $J=1.3\text{eV}$). Unit cells of $\text{La}M{\text{O}}_3$ $\left(M=\text{Ti},\text{V},\text{Cr},\text{Mn},\text{Fe}\right)$, contain four molecular units and those of $M=\text{Co}$, Ni and Cu contain one unit.

Figure 6

(Color online) Real and imaginary parts of $W_{C_{n^{\prime }n}}^{\mathbf{k}\mathbf{q}}\left(\omega \right)$ of ${\text{LaMnO}}_3$, where $n=n^{\prime }=\text{valence}$ band bottom, $\mathbf{k}=\left(0,0,0\right)$ and $\mathbf{q}=\frac{\pi }{c}\left(0,0,\frac{1}{2}\right)$, where $c$ is the lattice constant along the $c$ axis. Marks represent the calculated points, 128 points and 32 points data, and the solid and dotted lines are the results fitted by the rational function Eq. (A1).

Figure 7

(Color online) The mixing ratios of wave functions ${\left(\sqrt{x^2+1}-x\right)}^2$ are shown for ${\text{LaFeO}}_3$ and ${\text{LaVO}}_3$ with the parameters of $U=7.5\text{eV}$ and $J=0.5\text{eV}$ for $\text{La}4f$. (a) $\text{La}4f$ and $5d$ region $\left(3\text{eV}<E<13\text{eV}\right)$ in ${\text{LaFeO}}_3$ where $U=0$ and $J=0$ for Fe, (b) near Fermi level $E_F$ $\left(-7\text{eV}<E<3\text{eV}\right)$ in ${\text{LaFeO}}_3$ where $U=0$ and $J=0$ for Fe, (c) $U=0$ and $J=0$ for $\text{V}3d$ in ${\text{LaVO}}_3$, (d) $U=3.0\text{eV}$ and $J=1.0\text{eV}$ for $\text{V}3d$ in ${\text{LaVO}}_3$. The vertical and horizontal axis are the $\text{LSDA}+\text{U}$ eigenenergies ${ϵ}_{\mathbf{k}n}$ and ${ϵ}_{\mathbf{k}{n}^{\prime }}$ for $\text{U}+\text{GWA}$ and the energy zeroth is set to the Fermi energy of $\text{LSDA}+\text{U}$ calculation.