Strain-driven spin-state transition and superexchange interaction in LaCoO${}_3$: $Ab$ initio study

Using spin density functional theory with the Hubbard correction, we investigate the magnetic structure of strained LaCoO${}_3$. We show that beyond biaxial tensile strain of 2.5$\%$, local magnetic moments originating from the high spin state of Co${}^{3+}$ emerge in a low spin Co${}^{3+}$ matrix. In contrast, we find that compressive strain is not able to stabilize a magnetic state due to geometric constraints. LaCoO${}_3$ accommodates tensile strain via spin-state disproportionation, resulting in an unusual sublattice structure. In tensile-strained LaCoO${}_3$, the first nearest-neighbor (n.n.) exchange coupling is ferromagnetic (FM), while the second n.n. interaction is stronger and antiferromagnetic (AFM). This unusual feature of the exchange parameters is qualitatively verified with a model superexchange calculation. Due to the competition between the FM and the AFM couplings in the system, we find that the most probable magnetic structure of tensile-strained LaCoO${}_3$ is a canted-spin structure, which may explain the relatively small observed magnetic moment of 0.7${\mu }_B$/Co${}^{3+}$.

Figure 1

(a) Rhombohedral unit cell of LCO with the large sphere representing La, the medium-sized sphere representing Co, and the small sphere representing O. (b) $\sqrt{2}\times \sqrt{2}\times 2$ tetragonal supercell of LCO. $b_{\mathrm{in}}$ and $b_{\mathrm{out}}$ stand, respectively, for the in-plane and the out-of-plane Co-O bond length. ${\theta }_{\mathrm{in}}$ and ${\theta }_{\mathrm{out}}$ stand, respectively, for the in-plane and the out-of-plane Co-O-Co bond angle.

Figure 2

(a) Optimized $c$ lattice constant and (b) cell volume of LCO with different magnetic states. NM means NM LCO. HS/LS (1:3) or (1:1) mean FM LCO with a HS/LS mixed spin state with the ratio of 1:3 or 1:1, respectively.

Figure 3

(a) Energy of LCO with HS/LS mixed spin states per 2 × 2 × 2 cell (8 formula units) under a tensile strain of 3.5$\%$. Energy of NM LCO is set to 0 eV. Insets are schematic pictures of the geometrical arrangement of HS Co${}^{3+}$ ions for each HS/LS configuration (c1–c5). Dark spheres indicate Co${}^{3+}$ sites, with La and O sites omitted for clarity. White arrows in the large spheres represent spins at the HS Co${}^{3+}$ sites. (b) Projected density of states for 3$d$ orbitals in the 1:1 HS/LS mixed state [c5 in Fig. 3a] at the LS Co${}^{3+}$ site (upper panel) and the out-of-plane HS Co${}^{3+}$ site (lower panel). The Fermi energy (dashed vertical line) is set to 0 eV. The positive and negative densities of states are for spin up and spin down, respectively. (c) Cross-sectional valence charge distribution of the 1:1 HS/LS mixed state at the CoO${}_2$ plane with contours overlaid. The contours are drawn for the charge density from 0.5 to 1.0 eÅ${}^{-3}$ using a step interval of 0.04 eÅ${}^{-3}$ to contrast the charge accumulations at the HS Co-O bonds (dotted arrows) with the LS Co-O bonds (solid arrows).

Figure 4

Energy of LCO per $\sqrt{2}\times \sqrt{2}\times 2$ cell (4 formula units) as a function of strain for NM (filled squares), homogeneous (Homo.) IS (up triangles), 1:3 HS/LS (down triangles), and 1:1 HS/LS (right triangles) states.

Figure 5

(a) Energy gap between the $t_{2g}$${}^{*}$ and the $e_g$${}^{*}$ bands of NM LCO as a function of strain. (b) Energy level splitting of LS Co${}^{3+}$ under compressive or tensile strain.

Figure 6

(a) Local tetragonality (${\Delta }_{TD}$ = 2×($b_{\mathrm{in}}$ − $b_{\mathrm{out}}$)/$|b_{\mathrm{in}}$ + $b_{\mathrm{out}}|$) as a function of strain for CoO${}_6$ octahedra in NM LCO (squares), for HS Co${}^{3+}$ sites (down triangles), and LS Co${}^{3+}$ sites (up triangles) in 1:1 HS/LS FM LCO. (b) Schematic of octahedral distortion in 1:1 HS/LS FM LCO above 1.5$\%$ strain. Lateral arrows stand for the epitaxial constraint in the $ab$ plane imposed by biaxial tensile strain, while vertical arrows stand for the contraction of LCO in the $c$ direction due to the Poisson effect. (c) In-plane and (d) out-of-plane Co-O-Co angles as a function of strain for NM LCO (squares) and 1:1 HS/LS FM LCO (triangles). ${\theta }_0$ is the theoretical Co-O-Co angle in the LCO bulk.

Figure 7

Schematic pictures of collinear magnetic structures of HS/LS mixed states. (a) To calculate the first n.n. coupling parameters ($J_{1,\mathrm{out}}$ and $J_{1,\mathrm{in}}$), the 2 × 2 × 2 supercell is used. (b) The 2 × 2 × 4 (left) or 4 × 2 × 2 (right) supercell is used for the second n.n. coupling parameters ($J_{2,\mathrm{out}}$ and $J_{2,\mathrm{in}}$). Subscripts out and in for the coupling parameters stand for out-of-plane and in-plane, respectively. Dark spheres indicate Co${}^{3+}$ sites, with La and O sites omitted for clarity. White and black arrows in the large spheres represent spins at the HS Co${}^{3+}$ sites.

Figure 8

(a) Schematic of the superexchange paths for the first ($J_1$) and second ($J_2$) n.n. couplings of the 1:1: HS/LS mixed state in the $ac$ plane. Only the $e_g$ levels ($|d_{3z^2-r^2}\rangle ,|d_{x^2-y^2}\rangle$) are shown. Diagrams illustrating the (b) FM and (c) AFM superexchange interactions for the first and the second n.n. couplings in the 1:1 HS/LS state, respectively. See the main text for the model description.

Figure 9

(a) Energy of tensile-strained LCO with the 1:1 HS/LS configuration as a function of canting angle θ. The inset shows a canted spin configuration with angle θ in the $ac$ plane. Black arrows represent the local moments at the HS Co${}^{3+}$ sites. The $\sqrt{2}\times \sqrt{2}\times 4$ calculation cell used has 8 formula units. (b) Schematic of a local moment described by polar angle θ and azimuthal angle φ (left). This arrangement is one of the lowest-energy-canted spin structures of tensile-strained LCO (right). La atoms and other small magnetic moments at LS Co${}^{3+}$ and O sites are omitted for clarity.