Predicting the compositional phase stability of strongly correlated electron materials is an outstanding challenge in condensed matter physics. In this work, we employ the density functional theory plus $U$ $\left(\mathrm{DFT}+U\right)$ formalism to address the effects of local correlations due to transition metal $d$ electrons on compositional phase stability in the prototype phase stable and separating materials ${\mathrm{Li}}_x{\mathrm{CoO}}_2$ and olivine ${\mathrm{Li}}_x{\mathrm{FePO}}_4$, respectively. We introduce a spectral decomposition of the $\mathrm{DFT}+U$ total energy, revealing the distinct roles of the filling and ordering of the $d$ orbital correlated subspace. The on-site interaction $U$ drives both of these very different materials systems towards phase separation, stemming from enhanced ordering of the $d$ orbital occupancies in the $x=0$ and $x=1$ species, whereas changes in the overall filling of the $d$ shell contribute negligibly. We show that $\mathrm{DFT}+U$ formation energies are qualitatively consistent with experiments for phase stable ${\mathrm{Li}}_x{\mathrm{CoO}}_2$, phase separating ${\mathrm{Li}}_x{\mathrm{FePO}}_4$, and phase stable ${\mathrm{Li}}_x{\mathrm{CoPO}}_4$. However, we find that charge ordering plays a critical role in the energetics at intermediate $x$, strongly dampening the tendency for the Hubbard $U$ to drive phase separation. Most relevantly, the phase stability of ${\mathrm{Li}}_{1/2}{\mathrm{CoO}}_2$ within $\mathrm{DFT}+U$ is qualitatively incorrect without allowing charge ordering, which is problematic given that neither charge ordering nor the band gap that it induces are observed in experiment. We demonstrate that charge ordering arises from the correlated subspace interaction energy as opposed to the double counting. Additionally, we predict the Li order-disorder transition temperature for ${\mathrm{Li}}_{1/2}{\mathrm{CoO}}_2$, demonstrating that the unphysical charge ordering within $\mathrm{DFT}+U$ renders the method problematic, often producing unrealistically large results. Our findings motivate the need for other advanced techniques, such as DFT plus dynamical mean-field theory, for total energies in strongly correlated materials.

• Received 13 November 2016

DOI:https://doi.org/10.1103/PhysRevB.95.045141