Nondilute diffusion from first principles: Li diffusion in ${\text{Li}}_x{\text{TiS}}_2$

We investigate Li diffusion in the favorable intercalation compound ${\text{Li}}_x{\text{TiS}}_2$ as a function of Li concentration, $x$, from first principles. We find that Li ions hop between neighboring octahedral interstitial sites of the ${\text{TiS}}_2$ host by passing through an adjacent tetrahedral site. The migration barriers for these hops are significantly reduced when the end points belong to a divacancy. We use a cluster expansion within kinetic Monte Carlo simulations to describe the configuration dependence of the migration barriers and predict a diffusion coefficient that varies by several orders of magnitude with Li concentration, exhibiting a maximum close to $x=0.5$. The kinetic Monte Carlo simulations predict that diffusion is mediated predominantly by divacancies. We also find that the migration barriers depend strongly on the $c$-lattice parameter, which decreases as Li ions are removed from ${\text{Li}}_x{\text{TiS}}_2$.

Figure 1

(Color online) (a) Crystal structure of ${\text{Li}}_x{\text{TiS}}_2$ consisting of a periodic stacking of ${\text{TiS}}_2$ slabs between which Li ions can intercalate. The sulfur ions (circles) form a hexagonal close-packed sublattice, while the Ti and Li ions occupy alternating layers of octahedrally coordinated interstitial sites. Li migration between neighboring octahedral sites passes through adjacent tetrahedral sites [redrawn from (Ref. 43)]. (b) Li hop into an isolated vacancy (circles are Li, squares are vacancies, and triangles are tetrahedral sites within the Li layer). (c) Li hop into a divacancy.

Figure 2

(Color online) Calculated formation energies of 195 Li-vacancy configurations within the Li layers of ${\text{Li}}_x{\text{TiS}}_2$. The formation energies were calculated with LDA using the PAW method as implemented in VASP.

Figure 3

(Color online) Calculated voltage curves at (a) 300 K and at (b) 450 K. The voltage curves were calculated with grand canonical Monte Carlo simulations applied to a cluster expansion for the configurational energy of Li-vacancy disorder over the octahedral sites of ${\text{Li}}_x{\text{TiS}}_2$.

Figure 4

Calculated temperature composition phase diagram of ${\text{Li}}_x{\text{TiS}}_2$.

Figure 5

(Color online) Calculated thermodynamic factor as a function of Li composition in ${\text{Li}}_x{\text{TiS}}_2$ at 300 K.

Figure 6

(Color online) (a) Energy along minimal energy migration path for a Li hop into an isolated vacancy [see Fig. 1b] as calculated with the nudged-elastic band method using VASP. (b) Energy along paths I and II of Fig. 1c for a Li hop into a divacancy.

Figure 7

(Color online) Energy along the migration path into a tetrahedral site belonging to a divacancy at four different Li concentrations. The increase in energy of the tetrahedral site with decreasing Li is primarily due to a contraction of the $c$-lattice parameter as Li is removed from ${\text{Li}}_x{\text{TiS}}_2$.

Figure 8

(Color online) The tracer (black squares), self (filled circles), and chemical (empty circles) diffusion coefficients calculated with kinetic Monte Carlo simulations.

Figure 9

(Color online) Fraction of Li hops into divacancies as a function of Li concentration. Only at concentrations close to $x=1$ is there any appreciable hopping into isolated Li vacancies.

Figure 10

(Color online) Equilibrium number of divacancies surrounding each Li ion as a function of Li concentration calculated with grand canonical Monte Carlo simulations.

Figure 11

(Color online) Variation of the energy of tetrahedral occupancy relative to octahedral occupancy next to a divacancy as a function of $c$-lattice parameter [$\Delta E$ is the difference in energy between the tetrahedral and octahedral sites of Fig. 1c].