Configurational Electronic Entropy and the Phase Diagram of Mixed-Valence Oxides: The Case of ${\mathrm{Li}}_x{\mathrm{FePO}}_4$

We demonstrate that configurational electronic entropy, previously neglected, in ab initio thermodynamics of materials can qualitatively modify the finite-temperature phase stability of mixed-valence oxides. While transformations from low-$T$ ordered or immiscible states are almost always driven by configurational disorder (i.e., random occupation of lattice sites by multiple species), in ${\mathrm{FePO}}_4\mathrm{\text{−}}{\mathrm{LiFePO}}_4$ the formation of a solid solution is almost entirely driven by electronic rather than ionic configurational entropy. We argue that such an electronic entropic mechanism may be relevant to most other mixed-valence systems.

Figure 1

The ${\mathrm{LiFePO}}_4$ structure shown with (a) ${\mathrm{PO}}_4$ (purple) and ${\mathrm{FeO}}_6$ (brown) polyhedra as well as Li atoms (green), and (b) adjacent layers on Li and Fe sublattices, projected along axis $a$, with nearest-neighbor inter- and intralattice pairs highlighted.

Figure 2

Pair ECI vs site distance (measured from the sites’ ideal coordinates in ${\mathrm{LiFePO}}_4$). The circled points correspond to NN Li-Li, $e\mathrm{\text{−}}e$, and $\mathrm{Li}\mathrm{\text{−}}e$ pairs in Fig. 1b.

Figure 3

${\mathrm{Li}}_x{\mathrm{FePO}}_4$ phase diagram. (a) Experimental phase boundary data taken from Delacourt et al. [15] and from Dodd et al. [16]; (b) calculated with both Li and electron degrees of freedom and (c) with explicit Li only.

Figure 4

Configurational entropy per formula unit: (a) total entropy and the sum $S_{\mathrm{Li}}^{\prime }+S_e^{\prime }$; (b) separate conditional entropy $S_{\mathrm{Li}}^{\prime }$ and $S_e^{\prime }$.