First-principles simulations of Li boracites ${\mathrm{Li}}_4{\mathrm{B}}_7{\mathrm{O}}_{12}\mathrm{Cl}$ and ${\mathrm{Li}}_5{\mathrm{B}}_7{\mathrm{O}}_{12.5}\mathrm{Cl}$

Lithium boracite crystals have been identified as promising ion conductors for possible use in all-solid-state batteries. With the help of first-principles modeling techniques, we are able to show that these materials have structures with natural interstitial sites which play important roles in Li ion migration processes. The arrangements of these natural interstitial sites follow from group theory analyses of the computed and experimentally analyzed structures. Specifically, the low-temperature $\alpha$ phase of ${\mathrm{Li}}_4{\mathrm{B}}_7{\mathrm{O}}_{12}\mathrm{Cl}$ is computed to have the face-centered rhombohedral $R3c$ structure which is closely related to an ideal face-centered cubic $F\overline{4}3c$ structure with 24 natural interstitial sites per conventional unit cell. Li ion diffusion in this material is found to proceed largely by a concerted motion involving these interstitial sites and two neighboring host lattice sites, consistent with the large measured ionic conductivity of this material. Adding one addition ${\mathrm{Li}}_2\mathrm{O}$ cluster per formula unit to ${\mathrm{Li}}_4{\mathrm{B}}_7{\mathrm{O}}_{12}\mathrm{Cl}$ forms ${\mathrm{Li}}_5{\mathrm{B}}_7{\mathrm{O}}_{12.5}\mathrm{Cl}$, which crystallizes in the face-centered cubic $F23$ structure, a subgroup of the $F\overline{4}3c$ structure. While these $F23$ crystals have 16 natural interstitial sites per conventional unit cell, their distribution is such that they do not participate in Li ion diffusion mechanisms, as is consistent with the negligible ionic conductivity measured for this material. Phonon spectral analyses of ${\mathrm{Li}}_4{\mathrm{B}}_7{\mathrm{O}}_{12}\mathrm{Cl}$ in its $R3c$ structure and ${\mathrm{Li}}_5{\mathrm{B}}_7{\mathrm{O}}_{12.5}\mathrm{Cl}$ in its $F23$ structure show that both crystals are dynamically stable. Chemical stability of these materials is indicated by convex hull analysis of ${\mathrm{Li}}_4{\mathrm{B}}_7{\mathrm{O}}_{12}\mathrm{Cl}$ and ${\mathrm{Li}}_5{\mathrm{B}}_7{\mathrm{O}}_{12.5}\mathrm{Cl}$ with respect to their building blocks of LiCl, ${\mathrm{Li}}_2\mathrm{O}$, and ${\mathrm{B}}_2{\mathrm{O}}_3$.

Figure 1

Ball and stick diagram of conventional cell of ${\mathrm{Li}}_4{\mathrm{B}}_7{\mathrm{O}}_{12}\mathrm{Cl}$ in the $F\overline{4}3c$ structure as analyzed from single-crystal samples at room temperature [8]. Ball colors of blue, black, red, and green indicate Li, B, O, and Cl ions, respectively. The polyhedral constructions illustrate the ${\mathrm{BO}}_4$ tetrahedra and ${\mathrm{BO}}_3$ planar triangular structures. The dark blue balls indicate Li ions at the fully occupied $24c$ sites, while the partially filled light blue balls indicate Li ions at the 25% occupied $32e$ sites.

Figure 2

The relative static lattice energy per formula unit for the 28 optimized configurations of ${\mathrm{Li}}_4{\mathrm{B}}_7{\mathrm{O}}_{12}\mathrm{Cl}$ in the primitive lattice setting.

Figure 3

X-ray patterns determined from structural analysis of ${\mathrm{Li}}_4{\mathrm{B}}_7{\mathrm{O}}_{12}\mathrm{Cl}$ for wavelength $\lambda =1.54056$ Å as calculated using mercury software [35] from the computed idealized $R3c$ structure (top) and the analysis of single-crystal data at 298 K based on the $F\overline{4}3c$ space group reported by Jeitschko and coworkers [8].

Figure 4

Ball and stick diagram of ${\mathrm{Li}}_4{\mathrm{B}}_7{\mathrm{O}}_{12}\mathrm{Cl}$ in the (a) primitive and (b) conventional cell settings with the B-O framework represented by thin sticks and Cl indicated by green balls. The host Li ions with labels $b$ and $a$ are represented by dark blue and light blue balls, respectively. The unoccupied Li sites which serve as natural interstitials with Wyckoff label $b$ are represented by brown balls.

Figure 5

(a) Ball and stick diagram of conventional cell of ${\mathrm{Li}}_5{\mathrm{B}}_7{\mathrm{O}}_{12.5}\mathrm{Cl}$ in the $F23$ structure. Ball colors of blue, black, red, and green indicate Li, B, O, and Cl ions, respectively. (b) Thin stick representation of B-O framework to fully display the positions of host Li sites (dark blue for $24g$ and light blue for $16e$) and the lowest energy interstitial Li sites (brown for $16e$).

Figure 6

Calculated phonon spectrum along high-symmetric crystallographic directions and their corresponding projected densities of states for each atom type (PJDOS) for (a) ${\mathrm{Li}}_4{\mathrm{B}}_7{\mathrm{O}}_{12}\mathrm{Cl}$ and (b) ${\mathrm{Li}}_5{\mathrm{B}}_7{\mathrm{O}}_{12.5}\mathrm{Cl}$. The range of the PJDOS plots was fixed at the maximum PJDOS value for ${\mathrm{Li}}_5{\mathrm{B}}_7{\mathrm{O}}_{12.5}\mathrm{Cl}$. The high-symmetry $\mathbf{q}$ lines were selected according to the recommendation from Ref. [38] for each particular space group. For the $R3c$ structure of panel (a), the labels refer to the equivalent hexagonal structure, while for the $F23$ structure of panel (b), the labels refer to the standard face-centered cubic structure.

Figure 7

Plot of Helmholtz free energy and the vibrational free energy (inset) for the left- and right-hand side components of Eq. (12).

Figure 8

Convex hull of the ${\mathrm{Li}}_2\mathrm{O}\text{−}{\mathrm{B}}_2{\mathrm{O}}_3$-LiCl system with filled colored circles indicating values of the energy differences $\mathrm{\Delta }{U}_{\mathrm{SL}}$ as defined in Eq. (13) for each of the stable phases. The color bar indicates the values of $\mathrm{\Delta }{U}_{\mathrm{SL}}$ in units of eV/formula unit. Additionally, the circle radii are drawn in proportion to the magnitudes of $|\mathrm{\Delta }{U}_{\mathrm{SL}}|$.

Figure 9

(a) Ball and stick diagram of Li ion sites of ${\mathrm{Li}}_4{\mathrm{B}}_7{\mathrm{O}}_{12}\mathrm{Cl}$ with the migration path indicated with arrows and by labels ($ha$, host $a$ site; $hb$, host $b$ site; $mb$ and $m{b}^{\prime }$, metastable $b$ sites). (b) NEB energy path diagram for the direct (black) and concerted (red) Li ion migration processes.

Figure 10

(a) Ball and stick diagram of ${\mathrm{Li}}_5{\mathrm{B}}_7{\mathrm{O}}_{12.5}\mathrm{Cl}$ with the Li ions migration pathway indicated by labels and arrows. (b) NEB energy path diagram for this vacancy mechanism.

Figure 11

Diagrams of MD simulations over $t=50$ ps at intervals of $50\mathrm{\Delta }t=0.12$ ps for (a) ${\mathrm{Li}}_4{\mathrm{B}}_7{\mathrm{O}}_{12}\mathrm{Cl}$ at $\langle T\rangle =1162\mathrm{K}$ and (b) ${\mathrm{Li}}_5{\mathrm{B}}_7{\mathrm{O}}_{12.5}\mathrm{Cl}$ at $\langle T\rangle =1171\mathrm{K}$. The little blue balls indicate the time-dependent positions of Li ions. The atomic positions of the initial configurations and natural vacancies are also shown, using the same color conventions as in Figs. 4 and 5, respectively.

Figure 12

Plots of ${\overline{f}}_s\left(t\right)$ as defined in Eq. (15) for (a) ${\mathrm{Li}}_4{\mathrm{B}}_7{\mathrm{O}}_{12}\mathrm{Cl}$ and (b) ${\mathrm{Li}}_5{\mathrm{B}}_7{\mathrm{O}}_{12.5}\mathrm{Cl}$. The subscript $s$ of $f_s\left(t\right)$ denotes the reference site type as described in the text of each figure, using the Wyckoff labels of the $R3c$ and $F23$ structures as appropriate. The average temperature of each plot is listed in the legend.

Figure 13

Plots of simulated ionic conductivities of ${\mathrm{Li}}_4{\mathrm{B}}_7{\mathrm{O}}_{12}\mathrm{Cl}$ and ${\mathrm{Li}}_5{\mathrm{B}}_7{\mathrm{O}}_{12.5}\mathrm{Cl}$, together with the available experimental values obtained by digitizing the published graphs from Refs. [6] (a), [8] (b), and [12] (c). All straight lines represent least squares fits of the $\log \left(T\sigma \right)$ vs $1000/T$ data.