Insulating state and the importance of the spin-orbit coupling in ${\mathrm{Ca}}_3\mathrm{Co}\mathrm{Rh}{\mathrm{O}}_6$

We have carried out a comparative theoretical study of the electronic structure of the one-dimensional ${\mathrm{Ca}}_3\mathrm{Co}\mathrm{Rh}{\mathrm{O}}_6$ and ${\mathrm{Ca}}_3\mathrm{Fe}\mathrm{Rh}{\mathrm{O}}_6$ systems. The insulating antiferromagnetic state for ${\mathrm{Ca}}_3\mathrm{Fe}\mathrm{Rh}{\mathrm{O}}_6$ can be well explained by band structure calculations with the closed-shell high-spin $d^5$ $\left({\mathrm{Fe}}^{3+}\right)$ and low-spin $t_{2g}^6$ $\left({\mathrm{Rh}}^{3+}\right)$ configurations. We found for ${\mathrm{Ca}}_3\mathrm{Co}\mathrm{Rh}{\mathrm{O}}_6$ that Co has a strong tendency to be $d^7$ $\left({\mathrm{Co}}^{2+}\right)$ rather than $d^6$ $\left({\mathrm{Co}}^{3+}\right)$ and that there is an orbital degeneracy in the local Co electronic structure. We argue that it is the spin-orbit coupling which will lift this degeneracy thereby enabling local spin density $\text{approximation}+\text{Hubbard}$ U band structure calculations to generate the band gap. We predict that the orbital contribution to the magnetic moment in ${\mathrm{Ca}}_3\mathrm{Co}\mathrm{Rh}{\mathrm{O}}_6$ is substantial, i.e., significantly larger than $1{\mu }_B∕\mathrm{f.u.}$. Moreover, we propose a model to explain the contrasting intrachain magnetism in both materials.

Figure 1

Density of states (DOS) of ${\mathrm{Ca}}_3\mathrm{Fe}\mathrm{Rh}{\mathrm{O}}_6$ [panel (a)] and ${\mathrm{Ca}}_3\mathrm{Co}\mathrm{Rh}{\mathrm{O}}_6$ [panel (b)] in the ferromagnetic (FM) state from LSDA. The Fermi level is set at zero energy. Both panels show, from top to bottom, the total DOS, the $\mathrm{Rh}4d$, $\mathrm{Fe}∕\mathrm{Co}3d$, and $\mathrm{O}2p$ partial DOS. Solid (dashed) lines depict the spin-up (-down) states.

Figure 2

Density of states (DOS) of ${\mathrm{Ca}}_3\mathrm{Fe}\mathrm{Rh}{\mathrm{O}}_6$ [panel (a)] and ${\mathrm{Ca}}_3\mathrm{Co}\mathrm{Rh}{\mathrm{O}}_6$ [panel (b)] in the ferromagnetic (FM) state from $\mathrm{LSDA}+U$. Inset: Close-up of the down-spin $\mathrm{Co}{d}_0$, $d_2$, and $d_{-2}$ DOS. See Fig. 1 caption for other notations.

Figure 3

Density of states (DOS) of ${\mathrm{Ca}}_3\mathrm{Co}\mathrm{Rh}{\mathrm{O}}_6$ in the ferromagnetic (FM) state from $\mathrm{LSDA}+\mathrm{U}+\mathrm{SOC}$. Inset: Close-up of the down-spin $\mathrm{Co}{d}_0$, $d_2$, and $d_{-2}$ DOS. See Fig. 1 caption for other notations.

Figure 4

Model of the intrachain AF in ${\mathrm{Ca}}_3\mathrm{Fe}\mathrm{Rh}{\mathrm{O}}_6$ (a) and of the intrachain FM in ${\mathrm{Ca}}_3\mathrm{Co}\mathrm{Rh}{\mathrm{O}}_6$ (b).