Polaron-ion correlations in ${\mathrm{Li}}_x{\mathrm{FePO}}_4$ studied by x-ray nuclear resonant forward scattering at elevated pressure and temperature

Valence fluctuations of ${\mathrm{Fe}}^{2+}$ and ${\mathrm{Fe}}^{3+}$ were studied in a solid solution of ${\mathrm{Li}}_x{\mathrm{FePO}}_4$ by nuclear resonant forward scattering of synchrotron x rays while the sample was heated in a diamond-anvil pressure cell. The spectra acquired at different temperatures and pressures were analyzed for the frequencies of valence changes using the Blume-Tjon model of a system with a fluctuating Hamiltonian. These frequencies were analyzed to obtain activation enthalpies and an activation volume for polaron hopping. There was a large suppression of hopping frequency with pressure, giving an activation volume for polaron hopping of $5.8\pm 0.7 {\AA }^3$. This large, positive value is typical of ion diffusion, which indicates correlated motions of polarons and ${\mathrm{Li}}^{+}$ ions that alter the dynamics of both. Monte Carlo simulations were used to estimate the strength of the polaron-ion interaction energy.

Figure 1

Olivine-type structure of ${\mathrm{LiFePO}}_4$ with chains of ${\mathrm{Li}}^{+}$ ions (green), planes of ${\mathrm{FeO}}_6$ octahedra (red), and phosphate tetrahedra (yellow) [20].

Figure 2

Schematic of randomly populated 1D coupled ${\mathrm{Li}}^{+}$ ion and electron chains.

Figure 3

Six subprocesses describing ion and electron jumps on coupled 1D chains, where the energy barrier for each subprocess is listed below the schematic. $E_{\mathrm{p}}$ and $E_{\mathrm{i}}$ are the free polaron and ion activation barriers, respectively, $E_{\mathrm{pi}}$ is the polaron-ion interaction energy, and $V_{\mathrm{i}}$ is the activation volume for ion hopping. For ${\mathrm{Li}}^{+}$ ion jumps, depicted in the lower frames, the energy barrier depends only on the initial 1NN electron site; the final 1NN site on the electron chain is not depicted.

Figure 4

Temperature series of NFS spectra taken at 0, 3.5, 7.1, and 17 GPa. The fits (black curves) overlay experimental data (red points). Temperatures are listed to the left of spectra in Kelvin. The $x$ axis is the delay in nanoseconds after the arrival of the synchrotron pulse. The spectra have been scaled by their maximum value and offset for comparison.

Figure 5

(a) Polaron hopping frequencies, $\Gamma (T,P)$, at 0, 3.5 and 7.1 GPa, as determined from the solid curves in Fig. 4. Solid curves are Arrhenius-type fits using a pressure-independent prefactor. (b) Activation enthalpies, $\Delta H=E_a+P{V}_a$, versus pressure, where $E_a=470$ meV. Black triangles are results using a fixed prefactor and red circles are for a pressure dependent prefactor.

Figure 6

Electron MSD versus time for six series of MC simulations for a pair of coupled 1D ion and electron chains. Units for MSD are site index squared. Time is dimensionless. Each subplot shows the results for a different $E_{\mathrm{pi}}$. Subplots are labeled with $-E_{\mathrm{pi}}$ (0, 50, 150, 250, 350, and 450 meV). In each series, the MSD is shown for three different pressures: 0 (black), 3 (red), and 7 GPa (green).