High-spin and low-spin mixed state in ${\text{LaSrCoO}}_4$: An ab initio study

Spin state is an important issue for many cobaltates, and an intermediate spin (IS) state having a half-filled $e_g$ orbital may well be expected for a ${\text{Co}}^{3+}$ ion in a ${\text{CoO}}_6$ octahedron with a remarkable tetragonal distortion. Here the single-layered perovskite cobaltate ${\text{LaSrCoO}}_4$, which has a notable tetragonal elongation, is investigated for its spin state and electronic structure, through a set of local-spin-density approximation plus Hubbard $U$ $\left(\text{LSDA}+U\right)$ calculations including also the multiplet effect and spin-orbit coupling. Counterintuitively, our calculations evidence that the IS state is not the ground state and it would, even if being so, give rise to a wrong ferromagnetic half-metallic solution. We find that a strong band hybridization significantly suppresses a multiplet energy splitting of the IS state. Instead, a high-spin (HS) and low-spin (LS) mixed state turns out to have the lowest total energy among all possibly combined spin states. Moreover, the mixed $\text{HS}+\text{LS}$ ground state well accounts for the experimental paramagnetic insulating behavior, the effective magnetic moment, and the observed optical spectral features. We also predict that ${\text{LaSrCoO}}_4$ in the mixed $\text{HS}+\text{LS}$ ground state has a sizeable out-of-plane orbital moment and a local lattice distortion, which would motivate experimental studies.

Figure 1

(Color online) Crystal structure of ${\text{LaSrCoO}}_4$ in the ${\text{K}}_2{\text{NiF}}_4$ type. The Co-O bond lengths of the tetragonally distorted ${\text{CoO}}_6$ octahedra are $1.902\text{Å}\times 4$ (in plane) and $2.035\text{Å}\times 2$ (out of plane) after Ref. 20.

Figure 2

(Color online) Orbitally resolved $\text{Co}3d$ density of states (DOS), the planar-O (p-O) and apical-O (a-O) $2p$ DOS of ${\text{LaSrCoO}}_4$ in the IS FM state with the formal configuration $t_{2g\uparrow }^3{\left(3z^2-r^2\right)}_{\uparrow }^{1}x{y}_{\downarrow }^1{\left(xz+\text{i}yz\right)}_{\downarrow }^1$ calculated by $\text{LSDA}+U+\text{SOC}$. Blue (bold) lines refer to the up spin, and red (thin) lines the down spin. Fermi level is set at the zero energy. It is a half metal with a broad $x^2-y^2$ conduction band. The $\left(xz,yz\right)$ orbitals are 3/4 occupied with a hole in the $xz-\text{i}yz$ (i.e., $d_{-1}$) complex orbital.

Figure 3

(Color online) The $\text{Co}3d$, the planar-O and apical-$\text{O}2p$ DOS of ${\text{LaSrCoO}}_4$ in the IS FM state with the formal configuration $t_{2g\uparrow }^3{\left(3z^2-r^2\right)}_{\uparrow }^{1}x{z}_{\downarrow }^{1}y{z}_{\downarrow }^1$. See the main text and Table for a comparison with those results shown in Fig. 2.

Figure 4

(Color online) The IS ${\text{Co}}^{3+}$ $3d$ DOS, $2p$ DOS of the apical-O of the IS ${\text{Co}}^{3+}$ ion (IS a-O), planar-$\text{O}2p$ DOS, $2p$ DOS of the apical-O of the HS ${\text{Co}}^{3+}$ ion (HS a-O), and the HS ${\text{Co}}^{3+}$ $3d$ DOS of ${\text{LaSrCoO}}_4$ in the $\text{IS}+\text{HS}$ AF state.

Figure 5

(Color online) The LS ${\text{Co}}^{3+}$ $3d$ DOS, $2p$ DOS of the apical-O of the LS ${\text{Co}}^{3+}$ ion (LS a-O), planar-$\text{O}2p$ DOS, $2p$ DOS of the apical-O of the HS ${\text{Co}}^{3+}$ ion (HS a-O), and the HS ${\text{Co}}^{3+}$ $3d$ DOS of ${\text{LaSrCoO}}_4$ in the $\text{HS}+\text{LS}$ FM state. This is the ground-state solution and it well explains several experiments. See the main text for discussion.

Figure 6

(Color online) The ideal $\text{HS}+\text{LS}$ checkerboard order of ${\text{LaSrCoO}}_4$ in the $ab$ basal plane, marked with the theoretically optimized in-plane (out-of-plane) Co-O bond lengths, and the corresponding energy-level diagrams of the HS and LS ${\text{Co}}^{3+}$ ions as well.

Figure 7

DOS of ${\text{LaSrCoO}}_4$ in the mixed $\text{HS}+\text{LS}$ ground state without/with (solid/dashed lines) a structural optimization. There are no significant changes in the electronic structure. See Fig. 5 for other notations.