Computational study of ${\mathrm{Li}}_3{\mathrm{BO}}_3$ and ${\mathrm{Li}}_3{\mathrm{BN}}_2$ II: Stability analysis of pure phases and of model interfaces with Li anodes

Both ${\mathrm{Li}}_3{\mathrm{BO}}_3$ and ${\mathrm{Li}}_3{\mathrm{BN}}_2$ materials have promising properties for use in all solid-state batteries and other technologies dependent on electrolytes with significant ionic conductivity. As the second of a two-part study, the structural properties of ${\mathrm{Li}}_3{\mathrm{BO}}_3$ and three reported phases of ${\mathrm{Li}}_3{\mathrm{BN}}_2$ are investigated using first-principles modeling techniques. For $\alpha \text{−}{\mathrm{Li}}_3{\mathrm{BN}}_2$, the tetragonal $P{4}_2/mnm$ structure reported in the literature is found to be unstable as evidenced by imaginary phonon modes near the M point of its Brillouin zone. Our simulations within the harmonic approximation suggest that the real $\alpha$ phase has the orthorhombic space group symmetry $Pmmn$ formed with twice as many formula units and tiny adjustments of the equivalent lattice parameters and fractional coordinates. Extending the analysis of the $Pmmn\alpha \text{−}{\mathrm{Li}}_3{\mathrm{BN}}_2$ structure to the quasiharmonic approximation improves the agreement between the room-temperature x-ray pattern reported in the literature and the corresponding simulation results. In anticipation of the use of the monoclinic phases of ${\mathrm{Li}}_3{\mathrm{BO}}_3$ and ${\mathrm{Li}}_3{\mathrm{BN}}_2$ in Li ion conducting applications, chemical stability is investigated in terms of free-energy differences of possible decomposition and Li reaction processes, finding encouraging results. As further investigations of ${\mathrm{Li}}_3{\mathrm{BO}}_3$ and $\beta \text{−}{\mathrm{Li}}_3{\mathrm{BN}}_2$ as electrolyte or coating materials, particularly for use with Li metal anodes, idealized electrolyte/Li interfaces were investigated in terms of their geometric, energetic, and electronic properties. The results find the electrolyte/Li interfaces to be quite favorable, perhaps comparable to the pioneering LiPON/Li system.

Figure 1

Ball and stick visualization of ${\mathrm{Li}}_3{\mathrm{BO}}_3$ in the monoclinic $P{2}_1/c$ (No. 14) structure with Li, B, and O represented by blue, black, and red balls, respectively. Three shades of blue are used to indicate the three inequivalent Li sites in this structure.

Figure 2

Phonon dispersion curves and the corresponding projected density states for ${\mathrm{Li}}_3{\mathrm{BO}}_3$ in the $P{2}_1/c$ (No. 14) structure. The Brillouin zone diagram for this structure was reproduced from Hinuma et al. in Ref. [29] with permission from the publisher.

Figure 3

Ball and stick visualization of a conventional unit cell of $\beta \text{−}{\mathrm{Li}}_3{\mathrm{BN}}_2$ having $P{2}_1/c$ (space group No. 14). Li, B, and N ions are represented by three shades of blue, black, and gray balls, respectively.

Figure 4

Phonon dispersion curves and the corresponding projected density states for ${\mathrm{Li}}_3{\mathrm{BN}}_2$ in the $P{2}_1/c$ ($\beta$ form) structure. The Brillouin zone diagram for this structure was reproduced from Hinuma et al. in Ref. [29] with permission from the publisher.

Figure 5

Ball and stick visualization of a conventional unit cell of $\gamma \text{−}{\mathrm{Li}}_3{\mathrm{BN}}_2$ having the $I{4}_1/amd$ space group No. 141). Li, B, and N ions are represented by two shades of blue, black, and gray balls, respectively.

Figure 6

Phonon dispersion curves and the corresponding projected density states for $\gamma \text{−}{\mathrm{Li}}_3{\mathrm{BN}}_2$ in the $I{4}_1/amd$ structure. The Brillouin zone diagram for this structure was reproduced from Hinuma et al. in Ref. [29] with permission from the publisher.

Figure 7

Ball and stick visualization of a conventional unit cell of $\alpha \text{−}{\mathrm{Li}}_3{\mathrm{BN}}_2$ having the $P{4}_2/mnm$ (space group No. 136). Li, B, and N ions are represented by two shades of blue, black, and grey balls, respectively.

Figure 8

Phonon dispersion curves and the corresponding projected density states for ${\mathrm{Li}}_3{\mathrm{BN}}_2$ in the $P{4}_2/mnm$ ($\alpha$ form) structure. The Brillouin zone diagram for this structure was reproduced from Hinuma et al. in Ref. [29] with permission from the publisher.

Figure 9

Ball and stick diagram of the $P{4}_2/mnm$ structure of ${\mathrm{Li}}_3{\mathrm{BN}}_2$, similar to Fig. 7. The up and down arrows indicate the dominating amplitudes of the motions associated with the two eigenvectors of the imaginary phonon modes at the M point of the Brillouin zone involving the Li(2) atoms at $b$ sites. The green and purple arrow colors are used to indicate separate site amplitudes for the group of atoms.

Figure 10

Ball and stick diagrams projected onto the $\mathbf{bc}$ plane for (a) the supercell of the tetragonal $P{4}_2/mnm$ structure as defined by Eq. (5) and (b) the optimized orthorhombic $Pmmn$ structure. The ball conventions are the same as for Fig. 7 except that the three crystallographically distinct Li sites of the $Pmmn$ structure are indicated with three shades of blue balls.

Figure 11

Phonon dispersion curves along high symmetry directions of the Brillouin zone and the corresponding projected density states for ${\mathrm{Li}}_3{\mathrm{BN}}_2$ in the $Pmmn$ structure. The Brillouin zone diagram for this structure was reproduced from Hinuma et al. in Ref. [29] with permission from the publisher.

Figure 12

Plots of $F_{\text{vib}}\left(T\right)$ [Eq. (4)] for ${\mathrm{Li}}_3{\mathrm{BO}}_3$ and ${\mathrm{Li}}_3{\mathrm{BN}}_2$ in the $\alpha$ (as corrected in the $Pmmn$ structure), $\beta$, and $\gamma$ phases. The curves for the three phases of ${\mathrm{Li}}_3{\mathrm{BN}}_2$ are superposed in the figure.

Figure 13

Lattice parameters $a$, $b$, and $c$ in units of Å for $\alpha \text{−}{\mathrm{Li}}_3{\mathrm{BN}}_2$ in the $Pmmn$ structure. Full lines represent the temperature-dependent results of the quasiharmonic approximation. The constrained quasiharmonic results are indicated with the full blue line for a range of temperatures near 300 K. Dashed lines represent the temperature-independent results of the harmonic approximation. Experimental results presumed to be measured at room temperature as reported in Ref. [30] and transformed to the $Pmmn$ structure are indicated with *.

Figure 14

Contour plots of (a) $F^{\mathrm{QH}}(T=300\mathrm{K},a,b,c_{\min })$ and (b) $F^{\mathrm{QH}}(T=300\mathrm{K},a,b_{\min },c)$ of $\alpha \text{−}{\mathrm{Li}}_3{\mathrm{BN}}_2$ in the $Pmmn$ structure. Values of the Helmholtz free energy in units of eV/FU are represented in terms of the color chart shown beside each contour.

Figure 15

X-ray diffraction patterns of ${\mathrm{Li}}_3{\mathrm{BN}}_2$ in the $P{4}_2/mnm$ and the $Pmmn$ structures approximated in the harmonic phonon approximation (top two panels), in the $Pmmn$ structure approximated in the quasiharmonic approximation at $T=300$ K (red and blue curves), and experiment of Ref. [30]. The Mercury software [23] at an x-ray wavelength of $\lambda =1.54056\AA$ was used to generate all the x-ray patterns.

Figure 16

Plot of Helmholtz free energy in eV/formula unit as a function of $T$ in Kelvin for ${\mathrm{Li}}_3{\mathrm{BN}}_2$. The results for the $Pmmn\alpha$ (turquoise), $\beta$ (purple), and the $\gamma$ (blue) phases are obtained from harmonic approximations, and the quasiharmonic analysis of $F_{min}\left(T\right)$ for the $Pmmn\alpha$ phase are displayed in red curve and circles.

Figure 17

Ball and stick diagrams of optimized interface structures between (a) $3\times 1\times 1$ supercell of ${\mathrm{Li}}_3{\mathrm{BO}}_3$ and Li slab in $\left[100\right]$ direction, viewed in a projection onto the $\mathbf{ac}$ plane, and (b) $2\times 1\times 2$ supercell of ${\mathrm{Li}}_3{\mathrm{BO}}_3$ and Li slab in $\left[001\right]$ direction viewed in a projection onto the $\mathbf{bc}$ plane. Both configurations contain 24 Li ions in the electrode. The Li, B, and O sites are represented with blue, black, and red balls, respectively. (c) Plot of interface energy ${\gamma }_{ab}$ as a function of the number of Li ions in Li metal.

Figure 18

(a) Ball and stick diagrams of optimized interface structures between $2\times 1\times 1$ supercell of $\beta \text{−}{\mathrm{Li}}_3{\mathrm{BN}}_2$ and 24 metallic Li ions in $\left[001\right]$ direction, viewed in a projection of the $\mathbf{ac}$ plane. The Li, B, and N sites are represented with blue, black, and gray balls, respectively. (b) Plot of interface energy ${\gamma }_{ab}$ as a function of the number of Li ions in Li metal.

Figure 19

Plots of partial densities of states, in units of states (approximate charge) within the augmentation sphere per eV, for (a) ${\mathrm{Li}}_3{\mathrm{BO}}_3$ corresponding to configuration shown in Figs. 17 and (b) $\beta \text{−}{\mathrm{Li}}_3{\mathrm{BN}}_2$ corresponding to configuration shown in Fig. 18. The zero of energy for each subplot was adjusted to the top of the filled valence band.