Spin and orbital states in ${\text{La}}_{1.5}{\text{Sr}}_{0.5}{\text{CoO}}_4$ studied by electronic structure calculations

Electronic structure of the layered perovskite ${\text{La}}_{1.5}{\text{Sr}}_{0.5}{\text{CoO}}_4$ with a checkerboard ${\text{Co}}^{2+}/{\text{Co}}^{3+}$ charge order is studied, using the local-spin-density approximation plus Hubbard $U$ calculations including also the spin-orbit coupling and multiplet effect. Our results show that the ${\text{Co}}^{2+}$ ion is in a high spin state (HS, $t_{2g}^5{e}_g^2$) and ${\text{Co}}^{3+}$ low spin state (LS, $t_{2g}^6$). Due to a small ${\text{Co}}^{2+}{t}_{2g}$ crystal field splitting, the spin-orbit interaction produces an orbital moment of $0.26{\mu }_B$ and accounts for the observed easy in-plane magnetism. Moreover, we find that the ${\text{Co}}^{3+}$ intermediate spin state (IS, $t_{2g}^5{e}_g^1$) has a multiplet splitting of several tenths of eV and the lowest-lying one is still higher than the LS ground state by 120 meV, and that the ${\text{Co}}^{3+}$ HS state $\left(t_{2g}^4{e}_g^2\right)$ is more unstable by 310 meV. Either the IS or HS ${\text{Co}}^{3+}$ ions would give rise to a wrong magnetic order and anisotropy.

Figure 1

(Color online) Density of states (DOS) of the ${\text{HS-Co}}^{2+}/{\text{LS-Co}}^{3+}$ ground state of ${\text{La}}_{1.5}{\text{Sr}}_{0.5}{\text{CoO}}_4$ calculated by $\text{LSDA}+U+\text{SOC}$. The HS ${\text{Co}}^{2+}$ has $t_{2g}^5{e}_g^2$ with a $t_{2g}$ hole on the $xy$ orbital, and the LS ${\text{Co}}^{3+}$ closed $t_{2g}^6$ shell. The $2p$ DOS of the planar O1, ${\text{Co}}^{3+}$-apical O2a, and ${\text{Co}}^{2+}$-apical O2b are also shown. Fermi level is set at zero energy.

Figure 2

(Color online) Density of states of the IS ${\text{Co}}^{3+}$ ion in different configurations: (a) $t_{2g\downarrow }^3{\left(3z^2-r^2\right)}_{\downarrow }^{1}x{z}_{\uparrow }^{1}y{z}_{\uparrow }^1$, (b) $t_{2g\downarrow }^3{\left(3z^2-r^2\right)}_{\downarrow }^{1}x{y}_{\uparrow }^1{\left(xz-iyz\right)}_{\uparrow }^1$, and (c) $t_{2g\uparrow }^3{\left(3z^2-r^2\right)}_{\uparrow }^{1}x{y}_{\downarrow }^1{\left(xz+iyz\right)}_{\downarrow }^1$. The corresponding multiplet splitting is calculated to be several tenths of eV (see Table ). Note that the most favorable IS ${\text{Co}}^{3+}$ state [configuration (c)], being ferromagnetically coupled with the robust HS ${\text{Co}}^{2+}$ (not shown here but refer to Fig. 1), is still higher in energy than the LS ${\text{Co}}^{3+}$ ground state by 122 meV and has a wrong easy out-of-plane magnetism (see Table and main text).