Investigation of the structural properties of Na$_{\mathrm{4}}$P$_{\mathrm{2}}$S$_{\mathrm{6}}$ and Li$_{\mathrm{4}}$P$_{\mathrm{2}}$S$_{\mathrm{4}}$

Li-ion batteries were proposed in the 1970's and are currently commercially produced globally for many technological purposes; however, recent studies have suggested that Na-ion conducting materials might be a feasible alternative that could have some advantages over Li-ion batteries. First principles simulations are used in this investigation to examine the structural and physical properties of Na$_{\mathrm{4}}$P$_{\mathrm{2}}$S$_{\mathrm{6}}$, a$_{\mathrm{\thinspace }}$Na-ion conductor, in comparison with its Li-ion conducting analog, Li$_{\mathrm{4}}$P$_{\mathrm{2}}$S$_{\mathrm{4}}$. Four model structures are considered including the $C$2\textit{/m\textasciitilde }structure recently reported by Kuhn and co-workers from their analysis of single crystals of Na$_{\mathrm{4}}$P$_{\mathrm{2}}$S$_{\mathrm{6}}$ (ZAAC, 640(5):689-692, April 2014), and three structures related to the $P$6$_{\mathrm{3}}$\textit{/mcm }structure with P site disorder found in 1982 by Mercier and co-workers from their analysis of single crystals of Li$_{\mathrm{4}}$P$_{\mathrm{2}}$S$_{\mathrm{4}}$ (JSSC, 43(2):151-162, July 1982). The computational results indicate that both Na$_{\mathrm{4}}$P$_{\mathrm{2}}$S$_{\mathrm{6}}$ and Li$_{\mathrm{4}}$P$_{\mathrm{2}}$S$_{\mathrm{4}}$ have the same disordered ground state structures consistent with the P6$_{\mathrm{3}}$/mcm space group, while the optimized $C$2$/m $structures are meta-stable by 0.1 eV and 0.4 eV per formula unit for Na$_{\mathrm{4}}$P$_{\mathrm{2}}$S$_{\mathrm{6\thinspace }}$and Li$_{\mathrm{4}}$P$_{\mathrm{2}}$S$_{\mathrm{4}}$, respectively.