Mechanisms of Li${}^{+}$ transport in garnet-type cubic Li${}_{3+x}$La${}_3{M}_2$O${}_{12}$ ($M$ = Te, Nb, Zr)

We have studied a promising lithium-ion conductor, garnet-type cubic Li oxides, at various Li concentrations. The ab initio calculations performed on these materials revealed two distinct mechanisms of Li-ion transport, with very different energy barriers and a strong dependence on Li distribution. Our findings explain the origin of the vastly varying ionic conductivities at different Li concentrations and suggest possible principles to improve such materials for solid-state electrolyte applications.

Figure 1

The atomic structure of garnet-type $c$-LLM. The green (dark gray) polyhedrons are tetrahedral Li(1) sites and the yellow (light gray) ones are octahedral Li(2) sites. Each Li(1) site is enclosed by four neighboring Li(2) sites and each Li(2) site connects two neighboring Li(1) sites. The occupancy of Li(1) and Li(2) sites depends on the Li concentration. The snapshot below is the Wyckoff positions that the Li ions could possibly be located. The centers of Li(1) and Li(2) sites are noted as 24$d$ and 48$g$ sites, respectively, and the 96$h$ sites are slightly displaced off the 48$g$ sites but they are still inside the Li(2) octahedra.

Figure 2

(a)–(c) The coordination numbers (CN) of Li ions in LLT, LLN, and LLZ. CN = 4, 5, 6 represent Li ions that are inside tetrahedral sites [Li(1) sites], near the tetrahedral-octahedral borders [Li(1)-Li(2) borders], and inside octahedral sites [Li(2) sites], respectively. The unrelaxed structures mean all Li(1) sites are occupied and the remaining Li ions randomly occupy the Li(2) sites, while the stable structures are obtained by relaxing the proposed ones. (d)–(f) 2D schematics of the Li positions in three corresponding compositions. Triangular and square sites denote Li(1) and Li(2) sites, and the open and solid circles are proposed and stable positions of Li ions, respectively (the arrows indicate the direction of displacement).

Figure 3

(a) The energy cost to move one Li${}^{+}$ ion from the Li(1) site to the neighboring Li(2) site in LLT. (b), (c) Two predominant mechanisms of Li migration, (a) Route A in LLN and (b) Route B in LLZ. The energy barriers are calculated to be 0.8 and 0.26 eV, respectively. The dots are the calculated points and the lines are the energy of the extrapolated migration paths. The arrows in the insets (b) and (c) indicate these two distinct migration paths.

Figure 4

The bulk ionic conductivities of various garnet-type Li oxides measured at room temperature (Refs. 5, 6, 7 and 18, 19, 20, 21, 22, 23, 24, 25, 26), together with the calculated ones based on our simulations.