Li-ion diffusion mechanisms in crystalline Li$_3$PO$_4$ electrolytes

Using first principles electronic structure methods and ``nudged elastic band'' optimization techniques,\footnote{Using the Quantum ESPRESSO package http://www.pwscf.org/.} we examine ideal Li-ion diffusion in crystalline Li$_3$PO$_4$ electrolytes, considering both vacancy and interstitial mechanisms. The simulations are performed in supercells containing 16 Li$_3$PO$_4$ units. We determine the activation barriers for several plausible diffusion paths, considering the effects of the exchange-correlation functional forms, of the crystalline form in the $\beta$- and $\gamma$- structures, and also the effects of substitutional N. Using the generalized gradient approximation, results for $\gamma$-Li$_3$PO$_4$ show diffusion barriers of 0.6-0.7~eV for the vacancy mechanism with a small dependence on the crystallographic direction. For the interstitial mechanism, the diffusion barriers are 0.8~eV and 1.3~eV along the ${\bf{b}}$- and ${\bf{c}}$-axes, respectively. The larger activation barriers of the interstitial mechanism are closer to experimental measurements on polycrystalline and single crystal samples\footnote{J. Solid St. Chem. {\bf{115}}, 313 (1995), Cryst. Rep. {\bf{46}}, 864 (2001) } which find $E_a \approx 1.1-1.3$~eV.