First Principles Investigation of the Geometrical and Electrochemical Properties of Na$_4$P$_2$S$_6$ and Li$_4$P$_2$S$_6$.

First principles simulations are used to examine the structural and physical properties of Na$_4$P$_2$S$_6$ in comparison with its Li$_4$P$_2$S$_6$ analog. Four model structures are considered including the $C2/m$ structure recently reported by Kuhn and co-workers\footnote{ZAAC {\bf{640}}(5):689-692 (2014)} from their analysis of single crystals of Na$_4$P$_2$S$_6$, and three structures related to the $P6_3/mcm$ structure with P site disorder found in 1982 by Mercier and co-workers\footnote{JSSC {\bf{43}}:151-162 (1982)} from their analysis of single crystals of Li$_4$P$_2$S$_6$. The computational results indicate that both Na$_4$P$_2$S$_6$ and Li$_4$P$_2$S$_6$ have the same disordered ground state structures consistent with the $P6_3/mcm$ space group, while the optimized $C2/m$ structures have higher energies by 0.1~eV and 0.4~eV per formula unit for Na$_4$P$_2$S$_6$ and Li$_4$P$_2$S$_6$, respectively. %Activation energies for Na-ion %vacancy migration were computed to be % smaller than the Li analogs in %all of the structural models. Simulations of ion migration suggest that Na$_4$P$_2$S$_6$ may have more favorable ionic conductivity compared to Li$_4$P$_2$S$_6$.