Theory of sodium ordering in ${\mathrm{Na}}_x\mathrm{Co}{\mathrm{O}}_2$

The ordering of Na ions in ${\mathrm{Na}}_x\mathrm{Co}{\mathrm{O}}_2$ is investigated systematically by combining detailed density functional theory (DFT) studies with model calculations. Various ground state ordering patterns are identified, and they are in excellent agreement with available experimental results. Our results suggest that the primary driving force for the Na ordering is the screened Coulomb interaction among Na ions. Possible effects of the Na ordering on the electronic structure of the $\mathrm{Co}{\mathrm{O}}_2$ layer are discussed. We propose that the nonexistence of a charge ordered insulating state at $x=2∕3$ is due to the lack of a commensurate Na ordering pattern, whereas an extremely stable Na ordering at $x=0.5$ enhances the charge ordering tendency, resulting in an insulating state as observed experimentally.

Figure 1

DFT formation energies for 82 ordered structures of ${\mathrm{Na}}_x\mathrm{Co}{\mathrm{O}}_2$. The red dots represent the ground-state structures, joined together by the convex hull (red line). The arrows indicate the experimentally observed ground states.[4]

Figure 2

Relevant ordered structures of ${\mathrm{Na}}_x\mathrm{Co}{\mathrm{O}}_2$. The dashed lines represent the projected triangular lattice of Co atoms. Large red open circles and small red dots represent projected Na positions in different planes, at $z=0$ and $z=0.5c$, respectively. The Na(1) sites fall on top of the triangular lattice sites and Na(2) sites fall in the center of the triangles. Blue lines indicate the minimum unit cell in each case. A detailed discussion of each structure is presented in the text.

Figure 3

DFT versus electrostatic model formation energies.