Modeling interfaces between solids: Application to Li battery materials

We present a general scheme to model an energy for analyzing interfaces between crystalline solids, quantitatively including the effects of varying configurations and lattice strain. This scheme is successfully applied to the modeling of likely interface geometries of several solid state battery materials including Li metal, ${\mathrm{Li}}_3{\mathrm{PO}}_4, {\mathrm{Li}}_3{\mathrm{PS}}_4, {\mathrm{Li}}_2\mathrm{O}$, and ${\mathrm{Li}}_2\mathrm{S}$. Our formalism, together with a partial density of states analysis, allows us to characterize the thickness, stability, and transport properties of these interfaces. We find that all of the interfaces in this study are stable with the exception of ${\mathrm{Li}}_3{\mathrm{PS}}_4/\mathrm{Li}$. For this chemically unstable interface, the partial density of states helps to identify mechanisms associated with the interface reactions. Our energetic measure of interfaces and our analysis of the band alignment between interface materials indicate multiple factors, which may be predictors of interface stability, an important property of solid electrolyte systems.

Figure 1

Structural diagrams of supercells for ${\mathrm{Li}}_2\mathrm{O}/\mathrm{Li}$ interfaces with Li and O represented by small gray and larger blue balls respectively. The lattice directions refer to the Miller indices of the cubic ${\mathrm{Li}}_2\mathrm{O}$ lattice. In both structures the ${\mathrm{Li}}_2\mathrm{O}$ was cleaved along the (110) and an interface was formed with one of two different Li configurations. The Li structure in subfigure (a), labeled the ${\mathrm{Li}}_2\mathrm{O}/\mathrm{Li}({\tilde{\mathrm{\Omega }}}_1$) configuration, is an orthorhombic structure derived from the Li positions within ${\mathrm{Li}}_2\mathrm{O}$. The Li structure in subfigure (b), the ${\mathrm{Li}}_2\mathrm{O}/\mathrm{Li}({\tilde{\mathrm{\Omega }}}_2$) configuration, corresponds to a strained fcc Li structure cleaved along the (110) plane.

Figure 2

Plot of ${\tilde{\gamma }}_{ab}$ for the ${\mathrm{Li}}_2\mathrm{O}/\mathrm{Li}({\tilde{\mathrm{\Omega }}}_1$) and the ${\mathrm{Li}}_2\mathrm{O}/\mathrm{Li}({\tilde{\mathrm{\Omega }}}_2$) interfaces showing the linear relationship described in Eq. (6). The $y$ intercept of the graph corresponds to ${\tilde{\gamma }}_{ab}^{\lim }$ and the slope corresponds to $\sigma$. The numerical values are summarized in Table 2.

Figure 3

Structural diagrams of supercells for ${\mathrm{Li}}_2\mathrm{S}/\mathrm{Li}$ interfaces with Li and S represented by small gray and larger yellow balls respectively. The lattice directions reference the Miller indices of cubic ${\mathrm{Li}}_2\mathrm{S}$. Four configurations are presented. (a) The ${\mathrm{Li}}_2\mathrm{S}/\mathrm{Li}({\tilde{\mathrm{\Omega }}}_1$) and (b) the ${\mathrm{Li}}_2\mathrm{S}/\mathrm{Li}({\tilde{\mathrm{\Omega }}}_2$) configurations are both based on a [110] cleave of ${\mathrm{Li}}_2\mathrm{S}$ with the Li slab positions derived from the Li structure in ${\mathrm{Li}}_2\mathrm{S}$, although the interfaces have two different alignments. (c) The ${\mathrm{Li}}_2\mathrm{S}/\mathrm{Li}({\tilde{\mathrm{\Omega }}}_3$) has a configuration with a Li structure based on optimizing randomly generated Li positions. (d) The ${\mathrm{Li}}_2\mathrm{S}/\mathrm{Li}({\tilde{\mathrm{\Omega }}}_4$) configuration shows the interface between the (100) face of ${\mathrm{Li}}_2\mathrm{S}$ and a Li structure based on optimizing randomly generated Li positions.

Figure 4

Plot of ${\tilde{\gamma }}_{ab}$ for the ${\mathrm{Li}}_2\mathrm{S}/\mathrm{Li}({\tilde{\mathrm{\Omega }}}_i$) interfaces for $i=1–4$ showing the linear relationship described in Eq. (6). The numerical values are summarized in Table 2.

Figure 5

Structural diagrams of supercells for ${\mathrm{Li}}_3{\mathrm{PO}}_4/\text{Li}$ interfaces with Li, P, and O atoms represented by small gray, tiny black, and medium blue balls, respectively. (a) ${\mathrm{Li}}_3{\mathrm{PO}}_4/\mathrm{Li}({\tilde{\mathrm{\Omega }}}_1$) interface of [010] $\beta -{\mathrm{Li}}_3{\mathrm{PO}}_4$ and a Li slab. (b) ${\mathrm{Li}}_3{\mathrm{PO}}_4/\mathrm{Li}({\tilde{\mathrm{\Omega }}}_3$) interface of [010] $\gamma -{\mathrm{Li}}_3{\mathrm{PO}}_4$ and a Li slab.

Figure 6

Plot of ${\tilde{\gamma }}_{ab}$ for the ${\mathrm{Li}}_3{\mathrm{PO}}_4/\mathrm{Li}({\tilde{\mathrm{\Omega }}}_i$) interfaces for $i=1–3$ showing the linear relationship described in Eq. (6). The numerical values are summarized in Table 2.

Figure 7

Structural diagram of $\gamma -{\mathrm{Li}}_3{\mathrm{PS}}_4\left[010\right]/\mathrm{Li}$ interface with 24 Li with Li, P, and S represented by small gray, tiny black, and medium yellow balls respectively.

Figure 8

Interface energy for the $\gamma -{\mathrm{Li}}_3{\mathrm{PS}}_4\left[010\right]/\mathrm{Li}$ interface. The large negative value of ${\tilde{\gamma }}_{ab}$ is due to chemical reactions that occur at the interface. The bond breaking and bond formation at the interface dominates lattice strain effects.

Figure 9

Structural diagram of $\gamma -{\mathrm{Li}}_3{\mathrm{PS}}_4\left[010\right]/{\mathrm{Li}}_2\mathrm{S}\left[110\right]$ interface with Li, P, and S represented by small gray, tiny black, and medium yellow balls respectively. The asymmetry of the two interfaces is evident.

Figure 10

Plot of ${\tilde{\gamma }}_{ab}$ for the $\gamma -{\mathrm{Li}}_3{\mathrm{PS}}_4\left[010\right]/{\mathrm{Li}}_2\mathrm{S}\left[110\right]$ interface. Numerical values for ${\tilde{\gamma }}_{ab}^{\lim }$ and $\sigma$ are reported in Table 2.

Figure 11

Partial densities of states for bulk (a) ${\mathrm{Li}}_2\mathrm{O}$, (b) ${\mathrm{Li}}_2\mathrm{S}$, (c) $\beta -{\mathrm{Li}}_3{\mathrm{PO}}_4$, and (d) $\gamma -{\mathrm{Li}}_3{\mathrm{PS}}_4$. In this case, the $N^s\left(E\right)$ sets are defined by the atom types, and $N^{\mathrm{Li}}\left(E\right)$ has been scaled by a factor of 10. The energy scale is adjusted so the $E=0$ corresponds to the top of the valence band of each material.

Figure 12

$N^{\mathrm{Li}}\left(E\right)$ scaled by a factor of 10, comparing results for the hexagonal (hcp), face centered cubic (fcc), and body centered cubic (bcc) structures.

Figure 13

Partial densities of states for the ${\mathrm{Li}}_2\mathrm{O}/\mathrm{Li}$ interfaces in the (a) ${\tilde{\mathrm{\Omega }}}_1$ and (b) ${\tilde{\mathrm{\Omega }}}_2$ configurations.

Figure 14

Partial densities of states for the ${\mathrm{Li}}_2\mathrm{S}/\mathrm{Li}$ interfaces for the (a) ${\tilde{\mathrm{\Omega }}}_1$, (b) ${\tilde{\mathrm{\Omega }}}_2$, (c) ${\tilde{\mathrm{\Omega }}}_3$, and (d) ${\tilde{\mathrm{\Omega }}}_4$ configurations.

Figure 15

Partial density of states for ${\mathrm{Li}}_3{\mathrm{PO}}_4/\text{Li}$ interface in the ${\tilde{\mathrm{\Omega }}}_1$ configuration.

Figure 16

Partial density of states for $\gamma -{\mathrm{Li}}_3{\mathrm{PS}}_4/\mathrm{Li}$ interface (left) and corresponding optimized structure (right). The three panels of the $N^s\left(E\right)$ plot correspond to the Li slab, the interface, and the electrolyte regions, respectively.

Figure 17

Partial density of states for the $\gamma -{\mathrm{Li}}_3{\mathrm{PS}}_4/{\mathrm{Li}}_2\mathrm{S}$ interface. The zero of energy is set to the top of the valence band of $\gamma -{\mathrm{Li}}_3{\mathrm{PS}}_4$

Figure 18

Structural diagram of a supercell of $\beta -{\mathrm{Li}}_3{\mathrm{PO}}_4\left[010\right]/\mathrm{Li}$ with an interstitial defect and 11 Li atoms in the metallic slab. Li, P, and O sites are indicated with small gray, tiny black, and medium blue balls respectively. The vertical direction of the diagram is oriented along the interface normal direction $\left(y\right)$. The red arrows indicate the direction, extent (length of arrow) and magnitude (width of arrow) of the electric fields $E_1$ and $E_2$ within the electrolyte.

Figure 19

Partial density of states for $\beta -{\mathrm{Li}}_3{\mathrm{PO}}_4\left[010\right]/\mathrm{Li}$ interface corresponding to the structure shown in Fig. 18 calculated according Eq. (9). The contributions for the Li interstitial $\left({\mathrm{Li}}^{+}\right)$ and its nearest neighbor oxygen are plotted separately. The zero of energy is set approximately at the top of the electrolyte valence band.

Figure 20

Plot of minimum energy barrier $E_A$ (in eV) as a function of the normalized reaction coordinate for the breaking of a P-O bond at a $\beta -{\mathrm{Li}}_3{\mathrm{PO}}_4/\mathrm{Li}$ interface as determined with the NEB approximation. The three inserts represent structural diagrams of the initial, maximal, and final configurations of the process using small gray, tiny black, and medium blue balls to represent Li, P, and O, respectively.