Mechanisms of ${\mathrm{Li}}^{+}$ diffusion in crystalline $\gamma$- and $\beta \text{−}{\mathrm{Li}}_3\mathrm{P}{\mathrm{O}}_4$ electrolytes from first principles

Recently, there has been significant interest in developing solid-state lithium ion electrolytes for use in batteries and related technologies. We have used first-principles modeling techniques based on density-functional theory and the nudged elastic band method to examine possible Li ion diffusion mechanisms in idealized crystals of the electrolyte material ${\mathrm{Li}}_3\mathrm{P}{\mathrm{O}}_4$ in both the $\gamma$ and $\beta$ crystalline forms, considering both vacancy and interstitial processes to find the migration energies $E_m$. We find that interstitial diffusion via an interstitialcy mechanism involving the concerted motion of an interstitial Li ion and a neighboring lattice Li ion may provide the most efficient ion transport in ${\mathrm{Li}}_3\mathrm{P}{\mathrm{O}}_4$. Ion transport in undoped crystals depends on the formation of vacancy-interstitial pairs requiring an additional energy $E_f$, which results in a thermal activation energy $E_A=E_m+E_f∕2$. The calculated values of $E_A$ are in excellent agreement with single crystal measurements on $\gamma \text{−}{\mathrm{Li}}_3\mathrm{P}{\mathrm{O}}_4$. Our results examine the similarities and differences between the diffusion processes in the $\gamma$ and $\beta$ crystal structures. In addition, we analyze the zone center phonon modes in both crystals in order to further validate our calculations with experimental measurements and to determine the range of vibrational frequencies associated with Li ion motions which might contribute to the diffusion processes.

Figure 1

(Color online) Ball and stick drawing of the equilibrium structures of the (a) $\gamma \text{−}{\mathrm{Li}}_3\mathrm{P}{\mathrm{O}}_4$ and (b) $\beta \text{−}{\mathrm{Li}}_3\mathrm{P}{\mathrm{O}}_4$ supercells used in the simulations. The $\mathrm{P}{\mathrm{O}}_4$ groups are indicated with bonded yellow and blue spheres. Li ions are indicated by light and dark gray spheres, representing the crystallographically distinct sites. The number labels on some of the Li sites are used in Sec. 4 to describe vacancy diffusion.

Figure 2

(Color online) Partial densities of states $⟨N^a\left(E\right)⟩$ for ${\mathrm{Li}}_3\mathrm{P}{\mathrm{O}}_4$ comparing results for the $\gamma$ and $\beta$ structures, calculated using Eq. (1) and averaging over all spheres within the unit cell for each atomic species. The zero of energy is taken as the top of the highest occupied valence state.

Figure 3

(Color online) Spectrum of Raman active modes of $\gamma \text{−}{\mathrm{Li}}_3\mathrm{P}{\mathrm{O}}_4$, comparing experimental results (A–D) corresponding to Refs. 45, 43 (at room temperature) and Refs. 43, 36 (at liquid nitrogen temperature), respectively, to the results of the LDA and GGA simulations.

Figure 4

(Color online) Spectrum of Raman active modes of $\beta \text{−}{\mathrm{Li}}_3\mathrm{P}{\mathrm{O}}_4$, comparing experimental results corresponding to Ref. 36 (at liquid nitrogen temperature) to the results of the LDA and GGA simulations.

Figure 5

(Color online) Diagram of LDA and GGA energy paths for $\mathbf{a}$, $\mathbf{b}$, and two alternative $\mathbf{c}$-axis vacancy diffusions for $\gamma \text{−}{\mathrm{Li}}_3\mathrm{P}{\mathrm{O}}_4$ plotted along the configuration coordinate defined in Eq. (5). The zero of energy is taken as the energy of a $c$-type Li ion vacancy as calculated from our supercell. (Note: In order to visualize the comparison of the LDA and GGA results, the value of $\Delta X_{if}$ was taken from the experimental data of Ref. 35.)

Figure 6

(Color online) Diagram of LDA energy paths for $\mathbf{b}$, $\mathbf{a}$, and two alternative $\mathbf{c}$-axis vacancy diffusions for $\beta \text{−}{\mathrm{Li}}_3\mathrm{P}{\mathrm{O}}_4$, plotted along the configuration coordinate defined in Eq. (5). The zero of energy is taken as the energy of an $a$-type Li ion vacancy as calculated from our supercell $\Delta X_{\mathit{\text{if}}}$ was taken from our LDA calculations as reported in Table .

Figure 7

(Color online) Ball and stick drawing of metastable interstitial Li ion configurations in $\beta \text{−}{\mathrm{Li}}_3\mathrm{P}{\mathrm{O}}_4$ viwed along the $\mathbf{c}$ axis using the same conventions as Fig. 1, with the interstitial sites labeled I (top) and II (bottom) (green). These correspond to the ${\mathit{I}}_0$ and ${\mathit{II}}_0$ sites, respectively, as listed in Table .

Figure 8

(Color online) Diagram of LDA and GGA energy paths plotted along the configuration coordinate defined in Eq. (5) for interstitial diffusion in $\gamma \text{−}{\mathrm{Li}}_3\mathrm{P}{\mathrm{O}}_4$ (top) along the $\mathbf{b}$ and $\mathbf{c}$ axes between adjacent $I_0$ configurations via an interstitialcy mechanism and (bottom) along the $\mathbf{a}$ axis between adjacent $I$ and $II$ channel sites using a combination of interstitialcy and direct-hop steps. The energy of the $I_0$ configuration was taken as the zero of energy. (Note: In order to visualize the comparison of the LDA and GGA results, the value of $\Delta X_{if}$ was taken from the experimental data of Ref. 35.)

Figure 9

(Color online) Diagram of LDA energy path plotted along the configuration coordinate defined in Eq. (5) for interstitial diffusion in $\beta \text{−}{\mathrm{Li}}_3\mathrm{P}{\mathrm{O}}_4$ (top) along the $\mathbf{a}$ and $\mathbf{c}$ axes between adjacent $I_0$ configurations via an interstitialcy mechanism and (bottom) along the $\mathbf{b}$ axis between adjacent $I$ and $II$ channel sites using a combination of interstitialcy and direct-hop steps. The energy of the $I_0$ configuration was taken as the zero of energy. $\Delta X_{if}$ was taken from our LDA calculations as reported in Table .