Energetics and cathode voltages of $\mathrm{Li}M{\mathrm{PO}}_4$ olivines ($M=\mathrm{Fe}$, Mn) from extended Hubbard functionals

Transition-metal compounds pose serious challenges to first-principles calculations based on density-functional theory (DFT), due to the inability of most approximate exchange-correlation functionals to capture the localization of valence electrons on their $d$ states, essential for a predictive modeling of their properties. In this work we focus on two representatives of a well known family of cathode materials for Li-ion batteries, namely the orthorhombic $\mathrm{Li}M{\text{PO}}_4$ olivines ($M$ = Fe, Mn). We show that extended Hubbard functionals with on-site ($U$) and intersite ($V$) interactions (so called DFT+$U$+$V$) can predict the electronic structure of the mixed-valence phases, the formation energy of the materials with intermediate Li contents, and the overall average voltage of the battery with remarkable accuracy. We find, in particular, that the inclusion of intersite interactions in the corrective Hamiltonian improves considerably the prediction of thermodynamic quantities when electronic localization occurs in the presence of significant interatomic hybridization (as is the case for the Mn compound), and that the self-consistent evaluation of the effective interaction parameters as material- and ground-state-dependent quantities allows the prediction of energy differences between different phases and concentrations.

Figure 1

The unit cell of ${\mathrm{FePO}}_4$ (a) and ${\mathrm{LiFePO}}_4$ (b). In evidence both the corner-sharing Fe-O octahedra and the P-O tetrahedra acting as linkers between them. The comparison between the two cells highlights the position of Li ions along channels parallel to the $b$ axis of the orthorhombic cell.

Figure 2

The density of states of ${\mathrm{LiFePO}}_4$ obtained with different approximations: (a) GGA (PBEsol), (b) DFT+$U_{\text{ave}}$, and (c) DFT+$U$+$V$. In all the graphs the black dashed line represents the total density of state while solid red, green, and blue ones designate iron $d$ state spin up, iron $d$ state spin down, and oxygen $p$ states total contributions, respectively. All energies are referred to the Fermi level or to the top of the valence band in the presence of a gap.

Figure 3

The four magnetic configurations of ${\mathrm{LiFePO}}_4$ compared in the text: (a) ${\mathrm{AF}}_1$, (b) ${\mathrm{AF}}_2$, (c) ${\mathrm{AF}}_3$, and (d) FM. Note that the Li atoms have been removed from the figures to improve clarity.

Figure 4

The two Li configurations considered for ${\mathrm{Li}}_{0.5}{\mathrm{FePO}}_4$ with Li occupying alternate $yz$ (a) and $xz$ (b) planes. The two supercells present a view of the crystal along the [001] direction.

Figure 5

The density of states of ${\mathrm{MnPO}}_4$ obtained with different approximations: GGA (PBEsol) (a), DFT+$U_{\text{ave}}$ (b), and DFT+$U$+$V$ (c). In all the graphs the black dashed line represents the total density of state while solid red, green, and blue ones designate manganese $d$ state spin up, manganese $d$ state spin down, and oxygen $p$ states total contributions. All energies are referred to the Fermi level or to the top of the valence band in the presence of a gap.

Figure 6

The two Li configurations of ${\mathrm{Li}}_{0.5}{\mathrm{MnPO}}_4$ considered in this work with Li ions occupying alternate $yz$ planes (a) and (c) or $xz$ planes (b) and (d). The figure compares views of a $4\times 4\times 2$ supercell of the crystals along the [001] (a) and (b) and along the [010] (c) and (d) directions.