Electron-correlation induced phase separation in Li$_x$FePO$_4$

Li$_x$FePO$_4$ is a promising rechargeable Li-ion battery cathode material. The Li$_x$FePO$_4$ system is known experimentally to phase separate into FePO$_4$ end LiFePO$_4$ up to $\sim$ 400 K, and form solid solutions above $\sim$ 600 K. Phase separation in this system is surprising as one would expect Li$^+$ ions to repel each other and form ordered compounds at compositions between those of the end members FePO$_4$ and LiFePO$_4$. This is exactly what is found in the local density or generalized gradient approximations to DFT, though it is in stark disagreement with experiments. The fact that the LDA+U method corrects this qualitative error points at the role of electron correlation in inducing phase separation in this material. We have thoroughly studied the LixFeO4 phase diagram with LDA+U. We found that the charge ordering of Fe$^{2+}$/Fe$^{3+}$ on the iron sub-lattice, induced by $d$ electron localization, couples delicately with Li/vacancy ordering on the lithium sub-lattice. Although the repulsion within each sub-lattice favors compound formation, the Li-e$^-$ attraction favors phase separation. It is the balance of interactions within and between the two sub-lattices that gives rise to the unexpected phase behavior of Li$_x$FePO$_4$. Possible relevance of this novel phase transition mechanism to other alkali metal-intercalated materials will be discussed.