Article pubs.acs.org/cm
Role of Dynamically Frustrated Bond Disorder in a Li+ Superionic Solid Electrolyte Nicole Adelstein*,† and Brandon C. Wood* Materials Science Division, Lawrence Livermore National Laboratory, Livermore, California 94550, United States S Supporting Information *
ABSTRACT: Inorganic lithium solid electrolytes are critical components in next-generation solid-state batteries, yet the fundamental nature of the cation−anion interactions and their relevance for ionic conductivity in these materials remain enigmatic. Here, we employ first-principles molecular dynamics simulations to explore the interplay among chemistry, structure, and functionality of a highly conductive Li+ solid electrolyte, Li3InBr6. Using local-orbital projections to dynamically track the evolution of the electronic charge density, the simulations reveal rapid, correlated fluctuations between cation−anion interactions with different degrees of directional covalent character. These chemical bond dynamics are shown to correlate with Li+ mobility and are enabled thermally by intrinsic frustration between the preferred geometries of chemical bonding and lattice symmetry. We suggest that the fluctuating chemical environment from the polarizable anions functions like a solvent, contributing to the superionic behavior of Li3InBr6 by temporarily stabilizing configurations favorable for migrating Li+. The generality of these conclusions for understanding solid electrolytes and key factors governing the superionic phase transition is discussed.
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surface.21−24 In this case, the dynamic polarizability of the large I− anion was thought to be important for enhancing the covalent interaction. Subsequent investigations of prototypical Ag+ and Cu+ superionic materials provided additional support for the relevance of covalent bond character for diffusion dynamics25 by suggesting the presence of highly correlated, polymer-like chains of interacting cation−anion pairs with welldefined local structural motifs.23,26−30 By contrast, despite the intense interest in them as solid electrolytes, the dynamical bonding properties of Li + conductors have received comparatively little attention. Nevertheless, there is reason to believe that interactions beyond simple Coulomb forces may also be relevant for understanding Li+ conductors, because conductivity in many Li+-conducting halides and oxides has been shown to improve with the introduction of more polarizable, lower-electronegativity anions that are likely to reduce the level of ionic bond character.31 Understanding the specific nature and role of the cation−anion interaction in Li+ conductors could therefore provide new engineering strategies for solid-state electrolytes and aid in the development of meaningful descriptors for rapid screening. Here, we combine first-principles molecular dynamics simulations with local chemical bond analysis to isolate details of the cation−anion interactions within a Li+ solid electrolyte,
INTRODUCTION Novel and highly conductive inorganic solid electrolytes offer the promise of safer and higher-energy lithium-ion batteries with a broader range of thermodynamic, chemical, mechanical, and electrochemical stability.1 The discovery and design of new solid electrolytes rely on a better understanding of the underlying principles that govern superionic behavior, such as high Li+ concentrations or low-vacancy formation energies, depending on the conductivity mechanisms. However, many details of what drives superionic conductivity remain elusive, impeding the prediction of new electrolytes. Focusing on the connection between atomic structure and diffusive properties, several recent studies have investigated diffusion mechanisms and proposed design rules for certain classes of superionic Li conductors.2−18 Although illustrative, many of these principles lack universality, suggesting additional fundamental motivations have yet to be discovered. One factor that has attracted limited attention is the specific nature of the cation−anion interaction. Oftentimes, this interaction is thought to be dictated by simple ionic bonding. However, motivated by the initial work of Phillips19,20 and later refined by Aniya and collaborators,21−23 the idea that covalent character may play an equally important role has also been suggested, as many superionic conductors have bonding that lies between covalent and ionic. On the basis of studies of α-AgI and its various relatives, it was proposed that thermally driven fluctuations in bond character may aid conduction by generating multiple local minima in the potential energy © XXXX American Chemical Society
Received: February 24, 2016 Revised: September 15, 2016
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DOI: 10.1021/acs.chemmater.6b00790 Chem. Mater. XXXX, XXX, XXX−XXX
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Li3InBr6. Short-range interactions with covalent bond character are shown to be present, strengthening the connection to other superionic systems for which this property has been shown to be important. Correlations between Li+ mobility and covalent bond character suggest that superionic conductivity in Li3InBr6 is aided by thermally activated, low-barrier fluctuations in Li− Br bond chemistry, much as Aniya has proposed for other ionic conductors.22 These fluctuations combine with cooperative factors that introduce frustration, mitigate local electrostatic penalties, and prevent the system from fully ordering.32 We suggest that the superionic phase transition in Li3InBr6 can be interpreted as the product of an order−disorder transition in bond character. Li3InBr6 demonstrates superionic conductivity of 1 × 10−3 S cm−1 at 298 K and was first reported in 1998 as part of a broader class of lithium solid electrolytes based on halides with substituted group III metals.33,34 Its structure is shown in Figure 1a and involves a monoclinic (C2/m) lattice of Br−
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COMPUTATIONAL DETAILS
First-principles Car−Parrinello molecular dynamics37 simulations were run on 2 × 2 × 1 supercells of Li3InBr6 using the Quantum ESPRESSO plane-wave density functional theory code.38 A time step of 6 au, a fictitious mass of 500 au, and kinetic energy and wave function cutoffs of 30 and 300 Ry, respectively, were used. Ultrasoft pseudopotentials39 for Li, In, and Br were employed with the Perdew− Burke−Ernzerhof (PBE) exchange-correlation functional.40 Simulations were run in 100 K increments from 500 to 900 K, with temperatures maintained using Nosé−Hoover chains.41 Except where indicated, results presented in the text are derived from the 700 K simulations, which retain the same dominant mechanisms as at lower temperatures while ensuring adequate statistics. To account for the intrinsic configurational variability, we prepared three different configurations with disordered In3+ sublattice occupancies, keeping the In3+ within the same a−c plane, and one ordered configuration with In-occupied sites aligned along the b crystallographic axis. Each simulation was run for 25 ps following an initial equilibration period, and results from these four simulations were averaged to generate 100 ps of total dynamics at each temperature. At 500, 700, and 900 K, additional electronic bond and polarization analyses were performed by computing the maximally localized Wannier functions (MLWFs), sampling at intervals of 1.5 fs over a period covering 1000 frames. This fine sampling permitted full resolution of the dynamics of the electronic degrees of freedom alongside the ionic degrees of freedom, which provided a novel way to define bond breaking and forming events. Details of the MLWF interpretation and construction are provided in the Results and Supporting Information. For the analysis, Li+ cations at each simulation frame were further separated according to “jumping” Li+ ions that are actively involved in an octahedral-to-octahedral site jump (typically via an intermediate tetrahedral site) and “ordinary” Li+ cations that encompass all others. Because site occupancy is difficult to uniquely define in a highly diffusive superionic system, Li+ site jumps were instead determined using an algorithm based on changes in coordinating neighbors. The initial frame of the Li+ jump was designated when three of the initial Br− neighbors have changed, and the final frame is designated when four of the initial Br− neighbors have changed. This algorithm ensures that our analysis of jumping Li+ includes passage through the tetrahedral site in addition to the two saddle points at the intersection between the octahedral and tetrahedral sites. An additional restriction requiring the jumps to be completed within 400 fs was introduced to ensure these events are representative of the rapid diffusion process. Further details can be found in the Supporting Information.
Figure 1. Structure of Li3InBr6. (a) Structure of Li3InBr6, with Br− anion locations colored yellow. Green squares indicate locations of octahedral sites, which are occupied by Li+, vacancies, and In3+ in a 3:2:1 ratio. (b) Top view of the structure, showing atomic density isosurfaces for ordinary Li+ occupying octahedral sites (blue), as well as for Li+ jumping through tetrahedral sites (red). The most common pathway for the conduction of Li+ between octahedral sites is indicated with a green arrow. (c) Schematic of the Li+ jump mechanism between octahedral sites through a tetrahedral site (indicated with orange Br−).
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RESULTS Conduction Pathway. Figure 1 shows a schematic of the Li+ conduction pathway, which nominally involves transitions between octahedral interstitial sites via tetrahedral sites. The octahedral sites form vacancy-rich channels along the [100], [010], and [001] crystallographic directions (for instance, into the page along the green squares in Figure 1b). Surprisingly, the dynamics reveal that Li+ does not typically travel along these channels. Rather, the dominant pathway involves Li+ skirting around Br− as it jumps between neighboring sites along the [110] direction. This pathway, which maximizes interaction with the Br− anion, is indicated by an arrow in panels a and b of Figure 1. Additionally, we find that jumping Li+ cations do not travel through the symmetric center of the tetrahedral site but rather remain closer to a Br− anion throughout the jump. This phenomenon is visible in the shape of the Li+ atomic density isosurface for the region nearest the tetrahedral site in Figure 1b (red isosurface). It is also reflected in the proximity between jumping Li and Br in the Li−Br pair distribution function in Figure 2a (2.45 Å, compared to a mean value of 2.57 Å for the
anions with two-thirds of its octahedral interstitial sites occupied by Li+ or In3+ in a 1:3 Li:In ratio.34,35 The lattice is a distortion of the rock salt LiBr, with In3+ substitution serving to introduce intrinsic vacancies (one-third of octahedral sites) that enhance superionic behavior. It can be classified as a type I superionic conductor with configurational variability in the octahedral lattice site occupations.36 Beyond its relevance as a Li+ solid electrolyte, Li3InBr6 has two features that make it an interesting case study for investigating alternative physicochemical motivations for superionic behavior. First, it maintains exceptional Li+ conductivity despite its relatively low-symmetry monoclinic lattice structure, in contrast to many materials for which the superionic phase possesses high symmetry. Second, its structure is based on Li+ occupation of octahedral sites, which places it outside of the recent analysis by Wang et al. based on percolation of tetrahedral sites within structures with bcc-like symmetry.14 B
DOI: 10.1021/acs.chemmater.6b00790 Chem. Mater. XXXX, XXX, XXX−XXX
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Figure 2. Structure and bonding in Li3InBr6. (a) Li−Br radial pair distribution functions g(r) (solid lines) and corresponding integrated coordination numbers (dashed lines) for Li3InBr6, separated according to “jumping” and “ordinary” Li+. Distances to the centroids of the octahedral and tetrahedral sites in the atomically relaxed structure are shown as vertical dotted lines, and thin solid vertical lines indicate the cutoffs for the first coordination shell. (b) Histogram of Br−WC distances (gray) resolved according to the associated target of the bond (colored lines; see the text for details).
centroid of a tetrahedral site). The behavior implies that a Br− anion “hosts” the jumping Li+ cation, maintaining a stronger local anion−cation interaction as the latter migrates from an octahedral site to a tetrahedral site and onward to a new octahedral site. Note that this explanation transcends classical point charge descriptions and suggests a more specific cation− anion interaction. A trajectory/movie provided in the Supporting Information further illustrates the conduction pathway. Local Bond Character. To elucidate the nature of cation− anion bonding, we employ MLWF analysis as a novel way to track the dynamics of the electronic degrees of freedom. MLWFs transform the otherwise complex ground-state charge density into a local-orbital basis. This provides a key advantage over conventional charge density descriptions, because one can define Wannier centers (WCs) as virtual dynamical “coordinates” that can be tracked to collectively describe the time evolution of the electronic orbitals.42 These WC dynamics can be correlated to the nuclear motion to understand how bonding correlates with diffusion. In addition, various physically meaningful quantities can be extracted from the WCs. For instance, the vector sum of WCs around an ion center rigorously defines its polarization, and individual WCs typically orient along chemical bonds when they are present. Accordingly, the WC analysis gives an intuitive picture of how polarization and bonding change during the course of the dynamics. Four WCs are associated with each Br−, consistent with the full eight-electron valence shell expected for the monovalent anion in an ionic crystal. Accordingly, we expect that cation− anion bonding is predominantly ionic. Note that an isolated Br− ion would have a spherically symmetric charge distribution, which translates to perfect tetrahedral coordination of WCs at a fixed distance from the Br nucleus. Therefore, any deviations from this rigid ideal reflect distortions of the ionic charge density due to the effects of the ion environment within the crystal, e.g., dipole/multipole responses, crystal field effects, or covalent interactions. The Br-associated WCs in Li3InBr6 exhibit a variety of distinct bond characters consistent with different chemical environments. This can be seen in Figure 2b, which shows clear structure in the overall distribution (gray) of Br−WC distances
(rWC) in the vicinity of Br− ions. By correlating them with the positions of nearby cations, we can further decompose the peaks and assign each to a different type of local interaction (colored lines). These assignments were made by finding the WC closest to each cation within the first Li−Br coordination shell in Figure 2a (described more fully in the Supporting Information). Within our analysis, WCs were found to fall into categories corresponding to Br−In bonds (purple), Br−Li bonds (blue and orange), and “free” WCs that were not assigned to any cation (green). We find that the most distant peak (rWC > 0.6 Å) in Figure 2b corresponds exclusively to WCs that point along the direction of the Br−In bond (purple); the significant distortion of the electronic distribution in this case is a consequence of the trivalent cation. “Free” WCs (green) lie exclusively within the shortest-distance peak (rWC < 0.5 Å), as expected for a weak interaction. On the other hand, Br−Li bonds (blue and orange) are split between the shortest-distance peak (rWC < 0.5 Å) and the center peak (0.6 Å > rWC > 0.5 Å). Interestingly, this indicates that at least two distinct electronic states are present in the interactions of Li+ with Br−. Those comprising the shortest-distance peak in Figure 2a share a bond character with free, nonbonded WCs. In other words, the character of the corresponding WCs is largely unaffected by the presence of nearby Li+, meaning these Br−Li interactions are predominantly governed by classical Coulombic behavior. In Figure 2b, these are labeled as “unbound” Li+ (blue). The remainder of the WCs associated with Br−Li bonds sit at larger Br−WC distances, indicating more appreciable distortion of the Br− electronic orbitals by nearby Li+. These are labeled as “bound” Li+ (orange). The rationale and specific criteria for defining Br−Li bonds in Figure 2b as “bound” or “unbound” are illustrated in the probability density maps in panels a and b of Figure 3. Specifically, we define a “bond angle” θ according to the Li− Br−WC angle for the Li+ nearest each WC and assess how it relates to the Br−Li “bond distance” R for ordinary versus jumping Li+ (see Figure 3b for schematic definitions of R and θ). This allows us to relate the orientations of the electronic WCs to the surrounding Li+ environment. The histograms of R versus θ are normalized by the subtended solid angle (2πR sin θ)−1 to give an unbiased probability map. Note that the data in C
DOI: 10.1021/acs.chemmater.6b00790 Chem. Mater. XXXX, XXX, XXX−XXX
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Figure 3. Polar−covalent Li−Br bonding. Normalized probability density map of Li−Br distance R vs Li−Br−WC angle θ for (a) all Li+ and (b) jumping Li+. The criteria for polar−covalent Li−Br bond formation (R < 3.15 Å and θ < 23°) are indicated by the orange box. Orange arrows indicate the “capture region” defined by small θ and larger R values. Histograms show the distributions of R and θ summed over the probability map. Average values of R as a function of θ and average values of θ as a function of R are indicated by red and white dashed lines, respectively. (c) Schematic of polar−covalent capture and release of Li+ by Br−. As Li+ (blue) enters within a narrow solid angle defined by the orientation of the WC (purple) about Br− (yellow), a polar−covalent bond is formed and the Br−Li distance contracts. At the same time, the Br−WC distance increases to signal formation of the bond. Li+ release is described by the reverse process.
panels a and b of Figure 3 offer an additional analysis of the Li+associated WCs in the blue and orange curves of Figure 2b. It is evident from Figure 3a that there is an extremely high probability of configurations with short bond distances (R < 3.0 Å) and small bond angles (θ < 5°). Such interactions are highly directional and short-range. This matches the expected behavior of covalent bond character (where the term is employed generally to describe perturbations of the charge density away from the classically Coulombic description). Effectively, these WCs trace the dynamics of the cations. Further evidence of covalent-like bonding can be seen upon averaging the values of R and θ over the range of the figure (dashed lines). With Figure 3a as the focus, if we express the average bond angle (white dashed line) as a function of distance, it decreases markedly to θ = 5° in the region of highest probability. At the same time, the average bond distance (red dashed line) in this region is a near-constant value but increases quickly for bond angles beyond ∼20°. This behavior
is illustrated in the picture shown in Figure 3c, in which bond distance R decreases as Li+ enters a narrow solid angle θ of influence. We note that covalent character in the Li−Br bond is predicted from Fajan’s rules in inorganic chemistry. Although its precise origin and nature are difficult to determine from the MLWF analysis alone, covalent character can generally be ascribed to two effects, both of which turn out to be relevant in Li3InBr6: orbital hybridization and multipole perturbations of the charge density. The former aspect is evident in the representative projected electronic density of states (see Figure S1), whereas the latter aspect is discussed further in the sections below. Note that the range of bond angles is significantly broader for jumping Li+ (Figure 3b) than for ordinary Li+ (Figure 3a). This can be seen easily in the corresponding bond angle histograms. The broader angular distribution suggests that the Br−Li bonds for jumping Li+ are not as ideally aligned; we will return to this D
DOI: 10.1021/acs.chemmater.6b00790 Chem. Mater. XXXX, XXX, XXX−XXX
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Chemistry of Materials Table 1. Analysis of the Polar−Covalent Li−Br Bond Coordination Environment no. of Li−Br polar−covalent bonds
overall fraction of Li (%)
fraction among jumping Li (%)
0 1 2 3 4 5 6
1.6 1.6 12.5 30.0 38.0 14.7 1.5
2.3 9.7 16.5 47.5 24.1 0 0
mean no. of polar−covalent bonds a
overall Li
jumping Li
3.5
2.8
overall fraction of Br (%)a 19.2 (0.5, 9.3, 9.4) 40.2 (3.2, 31.0, 6.0) 30.8 (8.2, 22.0, 0.6) 7.8 (6.8, 1.0, 0) 2.0 (2.0, 0, 0) overall Br 1.3
fraction among Br with jumping Li (%)a 14.2 (0.2, 3.3, 10.7) 25.4 (2.5, 22.0, 0.9) 37.3 (11.1, 25.7, 0.5) 17.5 (12.8, 3.5, 1.2) 6.8 (6.8, 0, 0) Br with jumping Li 1.8
Numbers in parentheses represent further decomposition by the number of In neighbors (0, 1, 2).
formation of a bond as the Li+ cation approaches a nearby host Br− and is captured, as illustrated schematically in Figure 3c. Note that this region is more pronounced for the jumping Li+ (Figure 3b), which supports the notion that the capture and release processes play a role in the overall ionic mobility mechanism. Using the bond classification criteria in Figure 3 to distinguish polar−covalent Li−Br bonds, we can determine how many such bonds on average are active around each Li+ or Br−. The results are listed in Table 1. The average coordination of polar−covalent bonds to Li+ about Br− is 1.3 and to Br− about Li+ is 3.5. Any remaining Li−Br interactions within the first coordination shell of Li+ or Br− correspond to unbound Li+ configurations. For jumping Li+, the average number of polar−covalent bonds to Br− decreases to 2.8. However, this decrease is far more modest than one would expect from the change in the overall coordination of Br− neighbors as tetrahedral sites become occupied (Figure 2a). If we instead normalize by the number of coordinating Br− neighbors assuming six for ordinary Li+ and four for jumping Li+, then we find that 58% of bonds formed by ordinary Li+ are polar−covalent bonds, compared with 70% of bonds formed by jumping Li+. Note that this implies that the tetrahedral sites provide more covalent character on average than octahedral sites do, a point to which we will later return. Moreover, whereas ordinary Li+ cations show a fairly broad distribution of different chemical environments with multiple polar−covalent bonds to Br−, jumping Li+ cations show an unusually strong preference for having exactly three polar−covalent bonds (comprising nearly half of all configurations). By comparison, configurations with two or four polar−covalent bonds are far less likely. Accordingly, jumping Li+ cations prefer to maintain a 3-fold coordination of polar− covalent Br− anions throughout conduction. This suggests a local conduction mechanism following the schematic in Figure 7, in which three directional polar−covalent Br−Li bonds anchor the Li+ at any given time as it travels between octahedral sites via a tetrahedral interstitial site. Bonds with former Br− hosts are broken, giving way to new ones formed with destination Br− anions in the tetrahedral site, such that the total number of polar−covalent bonds is largely conserved. Viewing conduction from the point of view of Br− makes comparisons more straightforward. Because Br− anions participate in both octahedral and tetrahedral sites simultaneously, we can fully isolate bonding changes from the geometric differences between these sites that complicate interpretation of the Li+ data. Table 1 shows that the average coordination of polar−covalent bonds for each Br− to Li+ is
important point later, as it is associated with bond switching induced during jumping. The orange box in panels a and b of Figure 3 defines the distance and angle criteria used to select “bound” Li+ in Figure 2b (R < 3.15 Å and θ < 23°). Note that these criteria capture polar−covalent character in both the ordinary (Figure 3a) and jumping Li+ (Figure 3b) cases. Equally important, however, are the significant numbers of Br−Li bonds that lie well outside the orange box, possessing larger angles or distances that make the “bound” classification problematic. Collectively, these give rise to the blue “unbound” Li+ distribution in Figure 2b, for which the shorter WC−Br distances closely match those for free WCs. The probability density associated with the unbound Li+ distributions is more diffuse, is generally less directional, and falls off more slowly with angle and distance. This matches the expected behavior for an ionic-type bond that lacks preferred directionality and derives from long-range Coulomb interactions. Significantly, the coexistence of bound and unbound Li+ in the neighborhood of the anions suggests that polar−covalent bonds may be formed and broken in the course of the finitetemperature dynamics. This echoes previous findings for superionic α-AgI,26 implying the coexistence of bound and unbound cations (representing various degrees of ionic and covalent bond character) may be a more universal indicator of ionic conductivity, as has been proposed.21,22,25 As discussed below, the anion−cation bond dynamics end up making an important contribution to the conduction mechanism in Li3InBr6. Two regions in Figure 3 corresponding to unbound Li+ merit further discussion. First, there are significant numbers of unbound Li+ cations with short distances (R < 3.0 Å) but with much larger angles (θ > 30°). These indicate imperfect alignment between Br− orbital symmetry and Li+ coordination, which gives rise to a lower level of covalent bond character despite the relatively short Br−WC distance. Note that this behavior is analogous to molecular coordination chemistry, where specific bond alignment with d-orbital symmetry often determines the bond strength of a metal cation−ligand complex with mixed ionic−covalent character. Here, it instead manifests for a halide anion within a well-defined crystalline environment.22 An additional interesting region in Figure 3 is the concentration of configurations with small angles (θ < 10°) but with larger distances (R > 3.5 Å). This region, marked with an orange arrow in panels a and b of Figure 3, features bonds that are highly directional like covalent bonds yet longer-range like ionic bonds. These configurations may signal the initial E
DOI: 10.1021/acs.chemmater.6b00790 Chem. Mater. XXXX, XXX, XXX−XXX
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Figure 4. Distribution of Br− polarization magnitudes. Gray regions show the overall distributions, whereas colored lines show the distributions decomposed into constituents according to the local bond environment. Panels a and b correspond to ordinary Li+ and panels c and d to jumping Li+. Panels a and c are decomposed into the number of polar−covalent Br−Li bonds and panels b and d into the number of Br−In bonds.
and d in Figure 4), we can also determine which Br− coordination environments promote Li+ mobility. Panels b and d of Figure 4 show that the higher- and lower-polarization peaks reflect Br− anions with and without In3+ neighbors, respectively. Because a lower level of polarization is more preferred for jumping Li+, having no In3+ neighbors is clearly beneficial. More specifically, Br− anions that are highly coordinated with polar−covalent Li−Br bonds but lack In3+ neighbors provide the most favorable environment for Li+ mobility. This generally agrees with the results of Table 1, in which further decomposition of the Br− coordination environment in terms of In3+ neighbors demonstrates the increased relevance of highly coordinated anions without In3+ neighbors. The results in Figure 4 imply that the strong polarizing influence of In3+ cations on Br− limits the charge density redistribution necessary for Li+ migration. This justifies the smaller role in Li+ conduction played by Br− configurations with Br−In bonds. The lower probability of Li+ diffusion about a host Br− ion with high In3+ coordination in Figure 4 is generally consistent with expectations from electrostatic cation repulsion. However, this consideration does not explain the Li+ conduction pathway along the [110] direction, nor does it lead to a lower statistical occupancy of octahedral sites with a high level of In3+ coordination as one might expect. If all octahedral sites were filled with equal probability according to a random cation distribution, then the average number of neighbor In3+ cations around each Li+ would be 2.5. If Li+ prefers to occupy sites with fewer In3+ neighbors, then the average number of In3+ neighbors should be lower than this value, remembering that there are also available sites in which Li+ has two instead of three coordinating In3+ ions. Instead, for configurations with disordered In3+, this average of 2.5 neighboring In3+ cations around Li+ is unchanged. Accordingly, the primary effect of the local In3+ configuration on the conduction pathway is best described through a detrimental increase in the level of Br− polarization. Additionally, we emphasize that a lower average level of Br− polarization with many Li+ bonds does not translate to weaker or less polarized individual Li−Br bonds. In fact, the average of all four Br−WC distances for hosts of jumping Li+ (0.57 Å) is essentially equivalent to the analogous quantity for the ordinary nonjumping case. Rather, a lower average level of Br− polarization is best explained as a more centrosymmetric
enhanced from 1.3 to 1.8 if the bond is associated with a jumping Li+. Closer analysis shows that this difference chiefly reflects a low jump likelihood of Br− with a single polar− covalent Li+ neighbor. Note that the number of polar−covalent bonds to the jumping Li+ decreases, indicating that the increase in the number of polar−covalent bonds to the Br− is due to bonds with neighboring, nonjumping Li+. The relevance of the Br− environment for Li+ conduction is particularly significant, because it confirms that Li+ jumps are not merely a product of local, single-particle cation−anion interactions but rather are highly sensitive to the broader chemical environment of the ions. In particular, the results suggest a degree of Li−Li correlation, mediated through the Br− anion. Many superionic materials exhibit hallmarks of highly correlated motion,17,18,25 and the subsequent sections introduce an atomic-scale origin for this phenomenon in Li3InBr6. Cooperative Phenomena. In light of the apparent link between covalent character and diffusion, we next turn to a collective analysis of the electronic redistribution about Br− due to all cation neighbors. To do so, we replace our description based on individual Br−Li bonds with one based instead on the total polarization of Br−, computed by the vector sum of the WCs about the Br− nucleus. Analyzing conduction from the reference frame of the anions rather than the mobile Li+ highlights the role of the anion crystalline sublattice in diffusion. Figure 4 shows histograms of Br− polarization magnitudes, separated into Br− surrounded by ordinary Li+ (Figure 4, top row) and those hosting at least one jumping Li+ (Figure 4, bottom row). The overall distributions (gray regions) show two significant peaks. Comparing the distributions for ordinary and jumping Li+, we find that for jumping Li+ the lower polarization peak gains prominence and the higher polarization peak shifts to slightly lower values. This change indicates that Br− anions hosting jumping Li+ cations are on average less polarized than their ordinary counterparts, and that a lowered level of anion polarization apparently enhances local Li+ mobility on average. Note that this result contrasts with the current understanding of many other superionic materials, in which increases in the level of polarization have been invoked to explain diffusion.27 Further insight into the origin of the polarization distributions can be obtained by decomposing them according to the quantity and identity of bonds formed by each Br−. By comparing ordinary versus jumping configurations (a and c vs b F
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Chemistry of Materials distribution of the electron density about Br−, whose distortion is poorly described by dipole interactions alone and instead involves higher-order multipoles. Notably, the relevance of higher-order multipoles and the preferred [110] conduction pathway offer a possible interpretation of nuclear magnetic resonance (NMR) results on Li3InBr6 showing enhancement of 7 Li nuclear quadrupole effects above the superionic transition temperature, for which Li+ mobility is activated within an axially symmetric environment.36 In Figure 4, one of the passive roles of collective ordering of Li+ ions around Br− becomes clear, in that they can cooperatively prevent excessive polarization of their Br− host. This maintains a low barrier for the electronic redistribution that accompanies the breaking and forming of polar−covalent Br−Li bonds, suggesting that the local interaction of cations through Br− plays a role beyond conventional cation repulsion effects. Bond Dynamics. In the previous sections, we focused on the relevance to the Li+ conduction mechanism of local Br−Li bond formation, as well as collective ordering of cations about a shared Br− host. However, Figures 2−4 are based on a static analysis that does not fully capture the dynamical nature of these processes or the intrinsic frustration that ultimately aids broader Li+ mobility. Here, the power of the MLWF analysis becomes evident, because we can extract such dynamical information directly by treating the WCs as a virtual electronic trajectory. We use our formal definition for polar−covalent Br−Li bond formation based on the contour map of Figure 3 to compute the time scale of bond breaking and forming. Figure 5a shows
5a) and the “intermittent” definition in which bond re-forming is allowed (dashed lines in Figure 5a). Note that the latter has a shallower decay because it accounts for Li+ vibration within an octahedral cage as Li−Br bonds are broken and re-formed after a full vibrational period. The short-time decay rate of the bond autocorrelation function can be used to extract a characteristic time constant for the exponential decay of the polar−covalent bond. For a Li−Br polar−covalent bond, this characteristic time is very short (∼220 fs for the continuous definition) and for a Li−Br−Li polar−covalently bound complex is shorter still (∼110 fs). Although these values depend on the chosen cutoff for polar− covalent bond formation, it is clear that the fluctuations are rapid, similar to or even faster than the hydrogen bond fluctuations in liquid water that typically drive aqueous proton conduction.43 The results indicate that the polar−covalent bonds are in a continual state of flux that changes faster than the mean period between Li+ jumps (calculated as 2.95 ps per Li+ at 700 K). Actual Li−Br polar−covalent bond lifetimes can be much shorter when bond breaking or forming is correlated, such as during a jump event. Evidence of such correlation can be seen in Figure 5b, which shows the bond-breaking frequency distribution obtained via the Fourier transform of bondbreaking events for polar−covalent Br−Li bonds. The collective results for all Li+ cations are provided alongside results obtained by averaging all individual Li+ bond-breaking frequencies (details are provided in the Supporting Information). The latter reflects the correlations of a single Li+ atom as it forms and breaks bonds with nearby Br−; differences between the two curves can therefore be uniquely attributed to correlations between bond fluctuations associated with nearby Li+ atoms. The spectrum for all Li+ cations (black curve, Figure 5b) contains structure indicating both correlated and anticorrelated behavior, particularly at ∼42 and
